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Lightweight Classification of IoT Malware based on Image Recognition Jiawei Su arXiv:1802.03714v1 [cs.CR] 11 Feb 2018 School of Information Science and Electrical Engineering, Kyushu University, Japan jiawei.su@inf.kyushu-u.ac.jp Daniele Sgandurra ISG Smart Card and IoT Security Centre, Royal Holloway University of London, UK daniele.sgandurra@rhul.ac.uk Danilo Vasconcellos Vargas Faculty of Information Science and Electrical Engineering, Kyushu University, Japan vargas@inf.kyushu-u.ac.jp Yaokai Feng Faculty of Information Science and Electrical Engineering, Kyushu University, Japan fengyk@ait.kyushu-u.ac.jp ABSTRACT The Internet of Things (IoT) is an extension of the traditional Internet, which allows a very large number of smart devices, such as home appliances, network cameras, sensors and controllers to connect to one another to share information and improve user experiences. Current IoT devices are typically micro-computers for domain-specific computations rather than traditional functionspecific embedded devices. Therefore, many existing attacks, targeted at traditional computers connected to the Internet, may also be directed at IoT devices. For example, DDoS attacks have become very common in IoT environments, as these environments currently lack basic security monitoring and protection mechanisms, as shown by the recent Mirai and Brickerbot IoT botnets. In this paper, we propose a novel light-weight approach for detecting DDos malware in IoT environments. We firstly extract one-channel gray-scale images converted from binaries, and then utilize a lightweight convolutional neural network for classifying IoT malware families. The experimental results show that the proposed system can achieve 94.0% accuracy for the classification of goodware and DDoS malware, and 81.8% accuracy for the classification of goodware and two main malware families. CCS CONCEPTS • Network security → IOT network security; • Machine learning → Convolutional neural network; KEYWORDS Internet of things, Malware image, Convolutional neural network, Light-weight detection Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). Conference’17, July 2017, Washington, DC, USA © 2018 Copyright held by the owner/author(s). ACM ISBN 978-x-xxxx-xxxx-x/YY/MM. https://doi.org/10.1145/nnnnnnn.nnnnnnn Sanjiva Prasad Department of Computer Science and Engineering, Indian Institute of Technology Delhi, India Sanjiva.Prasad@cse.iitd.ac.in Kouichi Sakurai Faculty of Information Science and Electrical Engineering, Kyushu University, Japan sakurai@csce.kyushu-u.ac.jp ACM Reference Format: Jiawei Su, Danilo Vasconcellos Vargas, Sanjiva Prasad, Daniele Sgandurra, Yaokai Feng, and Kouichi Sakurai. 2018. Lightweight Classification of IoT Malware based on Image Recognition. In Proceedings of ACM Conference (Conference’17). ACM, New York, NY, USA, 7 pages. https://doi.org/10.1145/ nnnnnnn.nnnnnnn 1 INTRODUCTION Nowadays the notion of the “Internet” has extended from the connection between personal computers to networks composed of a much larger range of devices. Traditional micro devices, such as many kinds of sensors and controllers, are typically only able to perform function-specific tasks based on pre-defined rules. By substituting these function-specific devices with CPU-controlled ones, and by enabling interconnection among them through the Internet, these “things” become “smart” and can now deal with complex tasks. In addition, by enabling Cloud services on these smart-devices, users can easily receive data reported by them and control them. Despite these advantages, smarter devices imply more vulnerabilities, due to the complexity in hardware and software, with more chances for potential adversaries to threaten them. In addition, IoT systems are generally unsecured due to the difficulty of creating unified standards for the various types of IoT hardware and software platforms. Finally, even if smarter compared with traditional sensors, IoT devices still lack sufficient computational resources to be able to use existing PC-based security solutions. In some cases, Cloud services provide a way for developing security protection for IoT devices, e.g. for malware detection [18, 19]. In this paper, we consider a solution to protect local IoT devices from being abused to perform DDoS attacks by being enslaved in botnets of IoT devices, which is currently a common attack. To accomplish this, we first classify IoT DDoS malware samples recently collected in the wild on two major families, namely Mirai and Linux.Gafgyt. We then propose a lightweight solution for detecting and classifying IoT DDoS malware and benign applications locally on the IoT devices by converting the program binaries to gray-scale images, and by feeding these images to a small size convolutional neural network for classification. In this way, resource-constrained Conference’17, July 2017, Washington, Jiawei Su, DC, Danilo USA Vasconcellos Vargas, Sanjiva Prasad, Daniele Sgandurra, Yaokai Feng, and Kouichi Sakurai IoT devices can afford the computation needed for running the proposed detection system locally. Experimental results show that the proposed system can achieve 94.0% accuracy for classifying goodware and DDoS malware, and 81.8% accuracy for the classification of goodware and two main malware families. The main contributions of this research are the following ones: • this is the first classification system tested on real IoT malware samples: previous works have used regular or mobile malware samples instead, due to the difficulty in obtaining IoT malware samples [2, 11, 13]. Specifically there is currently no publicly available IoT malware dataset and the first IoT honeypot for collecting samples of IoT threats was released relatively recently [1]; • the IoT malware classification system can be deployed on real IoT devices. We show in detail the feasibility of using lightweight image classifier for recognizing IoT malware through malware images. Malware image classification has been proposed for classifying regular malware [4]; however, IoT malware is functionally different. For example, many IoT malware may try to kill other malware to guarantee enough computational resource for themselves; • according to the experimental results, we prove that the proposed system can reliably classify goodware and IoT DDoS malware. • to the best of our knowledge, there is currently no reference to describe the time complexity of CNNs. However, the proposed CNN-based approach is empirically considered to be lightweight since it does not need to maintain any training data for classification, differently from other types of classifiers for malware, such as Support Vector Machine and K-nearest neighbours. The computation of CNN for classification is rather simple, and only involves summation and activation. In addition, the proposed system is based on a two layer shallow network which is much more efficient than common deep learning models. The paper is structured as follows. In Sect. 2 we discuss related work. Sect. 3 explains procedures for extracting IoT DDoS malware images and implementing a small size convolutional neural network for classification. In Sect. 4 the detection results in two different scenarios are listed and in Sect. 5 the achievement of this research is summarized and future work is discussed. 2 RELATED WORKS Even if IoT security is an important topic, few defensive solutions exist in the literature [12]. Only recently, the first honeypot specifically for collecting IoT malware has been established by Pa et al. [1]. Their honeypot systems simulated 8 different CPU architectures and are built for observing attacks coming through the Telnet protocol. Initially they collected 43 distinct malware samples which are mostly DDoS attack malware. Their results show that the DDoS attack is the most common security threat in current IoT network environments. These authors kindly shared their observed data set with us which we have used in this research for evaluating our proposal. To the best of our knowledge, while most other works focus on Android malware detection [24, 25], the“Cloudeye” [2] is in practice the only current work specific for IoT malware detection. The system is a signature matching-based malware detection solution. IoT clients are only responsible for preliminary scanning the software locally, and then sending hashed abstracts of suspicious files to Cloud servers for deep analysis, therefore guaranteeing data privacy and low-cost communications. However, in IoT environments the inherent weakness of signature matching-based detection still exists: for example, Cloudeye is not able to deal with new variants of existing samples. Apart from signature matching, machine learning-based malware detection has been proved as effective in various scenarios [3, 14–16, 22, 23]. In IoT environments, also heavy computation machine learning methods are expected to be suitable too because of the availability of Cloud services. In fact, in a possible scenario, the training can be performed on Cloud server, while resourceconstrained IoT devices can receive the trained classifiers from the servers and run the algorithm locally. Note that several machine learning classifiers are heavy at training but efficient during test phase. Classifying malware images has been proven as an effective way for recognizing common PC malware [9, 26]. It is essentially a method for comparing two malware binaries. Nataraj et al. first utilize malware images for classifying regular Internet malware with knearest neighbors [4]. However, the system requires pre-processing of filtering to extract the image texture as features for classification, which might not fit the resource-constrained IoT environments. Similaly, the artificial neural network (ANN) malware classification proposed by Makandar [28] might not be a good candidate to be run on IoT devices to handle due to the heavy computational cost of multiple fully connected layers in ANN for classification. Yue utilized convolutional neural network for malware family classification [5]. In this research, we use malware images for IoT malware classification and show it is a feasible approach. 3 METHODOLOGY In this Section, we describe the methodology of feeding malware images as features, to a small two-layer convolutional neural network for detection. 3.1 Lightweight IoT DDoS Malware Filter For the scenario of detecting IoT DDoS malware detection locally, as previously pointed out, the main difficulty of deploying malware filters lies in the fact that the computational resources available on current IoT devices is limited. A direct solution under such a condition is relying on the security protection services provided by powerful remote servers, such as in Cloud-enabled IoT environments. Cloud servers are usually better protected against node failures, e.g. due to DDoS attacks. Another advantage of using Cloud servers is that a malware databases can be made more comprehensive and can be updated more rapidly than on IoT devices. For these reasons, we propose a two-tier detection architecture, based on a local IoT detection system and a remote, Cloud-based, classification system. In more detail, firstly a lightweight malware classification system is responsible of recognizing suspicious programs and behaviors locally. Note that, at this stage, the main goal is to provide a score on binary suspiciousness. Lightweight Classification of IoT Malware based on Image Recognition Conference’17, July 2017, Washington, DC, USA detecting zero-day attacks. Finally, converting malware binaries to the corresponding images only requires creating the input vectors to the convolutional neural network, i.e. 8-bit vectors, which is a very fast operation. 3.4 Figure 1: The proposed light-weight malware detection scheme. The local detector is located on the client side and captures potentially suspicious programs by relying on cloud backend for final decision • automatic feature extraction: neural network can automatically extract higher level features from the input raw features. That is, the network can learn deep non-linear features that can be hardly discovered and understood by humanbeings. These are sometimes actually counter-intuitive, but indeed effective. Note that many previous works have focused on extracting effective features for malware detection. However, most of them are only effective under specific scenarios, and this might lead to poor scalability. • test-phase efficiency: the training progress of a convolutional neural network requires heavy computation and, for instance, high-end graphic cards are necessary for accelerating training large networks. However, once trained, the network itself is rather lightweight and can be run with tiny computational resources, since only the trained parameters and information of network structure are kept [29, 30]. In contrast, another supervised lightweight classifier, the oneclass support vector machine (OCSVM), though simpler than normal Two-class SVM, still needs to keep a certain amount of training data when running the classification, while a convolutional neural network does not need to keep any. In practice, the training can be handled by the Cloud servers and only the trained network is sent to IoT nodes. On the local IoT side, the convolutional neural network is run to detect malware. In such a case, the system delivers the files or the corresponding abstracts to a remote Cloud server for deeper analysis. In addition, the cloud side can update and distribute new trained detectors to the clients periodically. In the following, we discuss the local malware filter on the client side. We assume that a set of Cloud servers are able to analyze malware samples and retrain the classifiers using standard machine learning algorithms. The proposal system structure is shown in Fig. 1 3.2 IoT DDoS Malware Families According to recent observations and preliminary analysis [1], even if IoT DDoS malware are functionally similar to existing DDoS malware on PC platforms, they contain specific features that are rarely observed on PCs. For example, some samples try to kill other ones of competitive families to get more system resources for themselves, due to the limited computational capability of IoT devices. In addition, IoT malware often target a wide range of devices, such as Internet cameras, DVR and so on. Finally, IoT malware can be also compatible with different processor architectures, ensuring the maximum possible successful infections. 3.3 Malware Image Classification An interesting and novel way of performing malware classification is to analyze their converted binary images. In particular, a malware binary can be reformatted as an 8-bit sequence and then be converted to a gray-scale image which has one channel and pixel values from 0 to 255 [4]. The resulting image can then be fed into machine learning image classifiers for classification. Note that running machine learning classifiers typically needs fewer local storage than signature-matching systems, which is the most common used malware detection method. This is important for storage-constrained IoT devices. In fact, in a matching signatures system, the signature database is typically large in size as it has to contain information for each malware sample and all of its possible variants. In the case of machine learning, little information has to be kept for classification. For example, k-means clustering needs only the information of centroids and radii for classification once trained. Support vector machine merely keeps a small set of training data (i.e., the support vectors) in the test phase. In addition, machine learning methods overcome signature matching on Neural Network for Malware Detection Convolutional neural networks have been proven to have better performance for image recognition than many other kinds of classifiers. A convolutional neural network has two important characteristics that make it fit the scenario of preliminary filtering malware on local IoT devices: 4 EXPERIMENT AND RESULTS In this section we discuss the experimental setup and the results of the classification of the proposed system. 4.1 Preparing the Dataset For these experiments, we have used an IoT DDoS malware dataset newly collected by IoTPOT [1], the first honeypot for collecting IoT threat samples. The malware samples are labelled using VirusTotal [8] with the majority rule. The dataset originally contains 500 malware samples, where most of them are classified into four big families: Linux.Gafgyt.1, Linux.Gafgyt (other variants of Linux.Gafgyt family) , Mirai [10] and Trojan.Linux.Fgt. The rest of the samples belong to relatively rare families such as Tsunami, Hajime, LightAidra. Then we cluster the samples into two categories: Mirai family, which contains Mirai and Trojan.Linux.Fgt1 , and Linux.Gafgyt family which contains Linux.Gafgyt.1 and the other variants. Instead, the benign binary samples (goodware) are collected from 1 Mirai has been shown to have similar features to Trojan.Linux.Fgt [17]. Conference’17, July 2017, Washington, Jiawei Su, DC, Danilo USA Vasconcellos Vargas, Sanjiva Prasad, Daniele Sgandurra, Yaokai Feng, and Kouichi Sakurai convolution layer(kernel=3, stride=1, depth=32) max pooling layer(kernel=2, stride=2) convolution layer(kernel=3, stride = 1, depth=72) max pooling layer(kernel=2, stride=2) fully connected layer(size=256) softmax classifier Table 1: Structure of the Implemented Convolutional Neural Network Ubuntu 16.04.3 system files. The number of samples are balanced for each family by randomly removing the samples that belong to classes that are too large. After the preprocessing phase, we analyzed 365 samples where each class has the same number of samples. Among them, we utilize 45 sample (each class has 15 samples) for testing, and the rest for training. According to the discussion above, the system proposed is only responsible for preliminary detection. That is, the goal is to identify whether a sample is benign or belongs to one of the big malware families: Mirai and Linux.Gafgyt, but there is no need to understand exactly which kind of variant it is. 4.2 Obtaining the Malware Images We then convert each sample of the dataset into the corresponding malware gray-scale image by following the same procedures implemented in [4]. In particular, a malware binary can be reformatted as an 8-bit string sequence, whose decimal encoding represents the value of a one-channel pixel (in the range [0, 255]). Therefore the entire sequence represents a gray-scale image. We rescale the images to the size of 64X64 to be used as input to a convolutional neural network. Some examples of malware and benign-ware images are shown by Fig. 2, 3 and 4. In these images, the structural difference between malware and goodware images can be easily identified. For example, it can be seen that malware images always are more dense. In particular, the majority of the Mirai malware images have a dense central code payload. On the other hand, the images of goodware tend to have larger header parts than malwares. 4.3 Convolutional Neural Network Configuration To have a lightweight detection system, we have implemented a small, two layer shallow convolutional neural network, compared with common image recognition models, such as ImageNet [20] and VGG [7]. The network structure is shown in Table. 1. The network is trained with 5000 iterations with a training batch size of 32 and learning rate 0.0001. 4.4 Results The classification results are shown in Tables 2 and 3 for the cases of two (benign and malicious) and three-class (benign and two malware families: Mirai and gafgyt) classification. The experiments were conducted five times which each time with a completely different training/test data combination (i.e., there are no shared test samples between any two of five test data sets). Figure 2: Images of Goodware PP PPPredict Benign PP True P Benign Gafgyt Mirai 94.67% 6.67% 0% Gafgyt Mirai 2.67% 72.00% 21.33% 2.67% 21.33% 78.67% Table 2: Confusion Matrix for 3-class Classification PPPredict Benign PP True P Benign 94.67% Malicious 6.67% PP Malicious 5.33% 93.33% Table 3: Confusion Matrix for 2-class Classification According to the results of two-class classification, we find the proposed system can predict the existence of maliciousness with about 94.0% accuracy on the average. The accuracy of three-class Lightweight Classification of IoT Malware based on Image Recognition Conference’17, July 2017, Washington, DC, USA Figure 4: Malware Image Examples of the Mirai Family Figure 3: Malware Image Examples of the Linux.Gafgyt Family Class Time consumption in second Goodware 0.0241 Gafgyt 0.0011 Mirai 0.0003 Table 4: Practical results of time complexity for classifying one image of goodware and two malware families. classification is relatively lower. Specifically, there are 6.67% malicious samples that are mis-classified as benign, all of which belong to Gafgyt family, while there is no misclassification of Mirai family to benign. This indicate the Gafgyt has more similar binaries to goodware. On the other side, the rate of misclassification between Mirai and Gafgyt is exactly the same. Generally, the difference between benign and malicious samples is more recognizable than the difference between two malware families. Comparing with misclassification between benign and malicious samples (i.e., two-class classification), the system is more likely to misclassify the samples of two malware families in the case of three-class classification. This indicates the similarity between these two families. Specifically, samples of two families might be obfuscated in similar ways, Conference’17, July 2017, Washington, Jiawei Su, DC, Danilo USA Vasconcellos Vargas, Sanjiva Prasad, Daniele Sgandurra, Yaokai Feng, and Kouichi Sakurai XXX Systems XXX XXX Metrics Accuracy Classifier Num of layers Num of nodes Fully connect layer Preprocess Input dimension Our method ANN with random projection [5] Weighted loss [3] 94.0% CNN 2 104 256 Re-organizing binary 64X64 scalar matrix 99.5% ANN 2 1536 2048 N-gram binary, Random projection 179 thousand binaries 96.9% CNN (VGG-s) 5 1888 4096X2 Re-organizing binary Unknown Table 5: Comparing proposed system with two previous related works. In particular, the number of hidden layers, number of neurons and the number of nodes in fully connect layers, are shown by “Num of Layers”,“Num of nodes”,“Fully connect layer” respectively. It can be seen that the proposed system is more lightweight than references due to the smaller size of network model and lower dimensions of input, as well as simpler preprocessing. or/and share a part of the malicious functions. In fact, the basic botnet functions of different DDoS malware are similar, and mainly include receiving instructions from the control server and spreading the infection. Also consider that Mirai source code has been available online since its beginning, and several new families of IoT botnets include some portions of Mirai code. In addition, IoT malware has to be lightweight and their functions have to be relatively direct and simple. Our accuracy results compete with similar previous works [3, 5]. In specific, Yue [5] also utilized convolutional neural networks and malware images for classifying several PC malware families. However the results are carried out by using much bigger and complex network structures, namely very deep networks (VGG) which contain more than 10 layers while ours only has two layers. Similarly, a very complex preprocess procedure is needed in [5] which involves initial feature selection and random projection while our proposal directly uses raw features for classification. According to the accuracy results, the proposed system can be utilized as a regular malware detector, or a first-stage malware classifier. That is, it can perform a precise classification to identify benign and maliciousness .The exact classification of the malware family can then be performed on a Cloud backend. A comparison of corresponding experimental accuracy and settings is shown by Table 5. The practical time cost of classifying images is depicted in Table 4. In detail, the experiment is conducted with python and tensorflow on a system running Ubuntu 14.04, with a i7-7700 processor, GTX1080Ti graphic card and 16G memory. The code of this research can be found in: https://github.com/Carina02/IotMalwareImage. 5 CONCLUSION AND FUTURE WORK In this paper we have proposed a lightweight malware image classification scheme for locally detecting IoT DDoS malware, and shown its feasibility. The malware filter proposed in this paper is based on convolutional neural networks and can be tuned to be more efficient by using various techniques of reducing network size. For example, removing the neurons and links that are not critical in the network can reduce the number of parameters needed for classification [21]. Such further optimization can make the proposed system implementable on IoT devices with even less computation resources. In addition, new malware image extraction methods can be proposed to obtain more representative features of malware for classification. For improving the detection rate of IoT malware, more complicated cases in practice can be considered such as malware obfuscation. To the best knowledge, there is currently no systematical evaluation on IoT malware obfuscation and several critical questions are yet to be answered, such as whether IoT malware is obfuscated in a similar way to traditional malware, and how limited resources influence obfuscation methods. ACKNOWLEDGMENTS This research was partially supported by Collaboration Hubs for International Program (CHIRP) of SICORP, Japan Science and Technology Agency (JST), and Project of security in the IoT space funding by Department of Science and Technology (DST), India. REFERENCES [1] Pa, Y.M.P., Suzuki, S., Yoshioka, K., Matsumoto, T., Kasama, T. and Rossow, C., 2015. IoTPOT: analysing the rise of IoT compromises. EMU, 9, p.1. 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Secure Adaptive Group Testing Alejandro Cohen ∗ Department ∗ Asaf Cohen ∗ Sidharth Jaggi † Omer Gurewitz∗ of Communication Systems Engineering, Ben-Gurion University, Israel, {alejandr, coasaf, gurewitzs}@bgu.ac.il of Information Engineering, Chinese University of Hong Kong, Hong Kong, jaggi@ie.cuhk.edu.hk arXiv:1801.04735v1 [cs.IT] 15 Jan 2018 † Department Abstract—Group Testing (GT) addresses the problem of identifying a small subset of defective items from a large population, by grouping items into as few test pools as possible. In Adaptive GT (AGT), outcomes of previous tests can influence the makeup of future tests. This scenario has been studied from an information theoretic point of view. Aldridge 2012 showed that in the regime of a few defectives, adaptivity does not help much, as the number of tests required for identification of the set of defectives is essentially the same as for non-adaptive GT. Secure GT considers a scenario where there is an eavesdropper who may observe a fraction δ of the outcomes, and should not be able to infer the status of the items. In the non-adaptive scenario, the number of tests required is 1/(1 − δ) times the number of tests without the secrecy constraint. In this paper, we consider Secure Adaptive GT. Specifically, when an adaptive algorithm has access to a private feedback link of rate Rf , we prove that the number of tests required for both correct reconstruction at the legitimate user, with high probability, and negligible mutual information at the eavesdropper is 1/min{1, 1 − δ + Rf } times the number of tests required with no secrecy constraint. Thus, unlike non-secure GT, where an adaptive algorithm has only a mild impact, under a security constraint it can significantly boost performance. A key insight is that not only the adaptive link should disregard test results and send keys, these keys should be enhanced through a “secret sharing" scheme before usage. I. I NTRODUCTION Group Testing (GT) was introduced in the seminal study by Dorfman to identify syphilis infected draftees while dramatically reducing the number of required assays [1]. Specifically, the objective of GT is to identify a small subset of K unknown defective items within a much larger set of N items, conducting as few measurements T as possible. This problem has been analyzed in various scenarios [2], one of which, Non-Adaptive Group Testing (NGT), is when the entire pooling strategy is decided on beforehand. This scenario has also been formulated as a channel coding problem, e.g., [3]–[6], where each codeword is associated with the pool tests its associated item participates in. Secure GT protects the items’ privacy such that an eavesdropper who may observe only a fraction of the pool-tests, will not be able to infer the status of any of the items (negligible mutual information between the captured pool-tests and the status of the items). [7] addressed the Secure Nonadaptive Group Testing (SNGT) model. In order to confuse the eavesdropper, instead of each item having a single test vector, determining in which pool-tests it should participate, each item has a random vector, chosen from a known set. [7] proved that when the fraction of tests observed by the eavesdropper (Eve) is 0 ≤ δ < 1, the number of tests required for both correct Figure 1: Analogy between channel scheme with feedback to Secure Adaptive Group Testing. reconstruction at the legitimate user and negligible mutual information at Eve’s side is 1/(1 − δ) times the number of tests required with no secrecy constraint. Thus, the solution in [7] relies on information theoretic security, which offers security at the price of rate, or, in the GT case, trades security and the number of tests. In this paper, we consider Adaptive Group Testing (AGT), in which the outcomes of previous tests can influence the construction of future pools. In general, the adaptivity may benefit from both a reduced number of tests (T ) and efficient decoding techniques [8]. However, it has been shown that in many interesting cases, the decrease in T is negligible. Several studies have analyzed AGT as a channel coding problem with feedback (e.g., [9]–[13]). Utilizing the same techniques as Shannon’s seminal study [14] which proved that feedback does not increase channel capacity, the authors in [9] showed that feedback from the lab does not decrease the number of tests required significantly. That is, if T is the minimal number of tests required for reconstruction with negligible error, when N → ∞ the gain due to the adaptivity is marginal. Even with zero-error, the difference in T is only between O(K 2 log N/ log K) and O(K log N ) [2]. Surprisingly, we show that in secure-adaptive group testing, the adaptive feedback link may decrease the number of tests required significantly, which in some cases coincides with the number of pool tests required with no secrecy constraint at all. While from an information theoretic perspective, previous results indeed show that in communication feedback can increase secrecy capacity [15]–[17], herein, previous techniques do not apply directly, as the test transferred are physical entities which cannot be, e.g., one-time-padded, and, as we explain in the sequel, the “encoder” in this case has no knowledge on which K of the N items it needs to protect. Main Contribution We propose a new Secure Adaptive Group Testing (SAGT) algorithm. This algorithm significantly reduces the number of tests required, yet is sufficient for the legitimate user to identify the defective items, and to keep an eavesdropper ignorant regarding any of the items. The model is depicted in Figure 1. In the suggested solution, the set of indices describing the defective items takes the place of a confidential message; the testing matrix represents the design of the pools, where each row corresponds to a separate item. Each such matrix row is associated with a codeword where 1 denotes the pool tests that the corresponding item participates. The decoding algorithm is analogous to a channel decoding process, yet now the adaptive link from the lab (who examines the pools) to the mixer (who mixes the samples and creates the pooled tests) takes the place of a feedback link. The eavesdropper observation is analogous to the output of an erasure channel, such that only part of the tests sent from the legitimate source (the mixer) to the legitimate receiver are observed by the eavesdropper. We assume a private adaptive link, which is not observable to the eavesdropper. We use the feedback link from the lab to the mixer in order to modify the testing matrix, according to the shared information between them, in a way which can be comprehended by both the lab and the mixer, yet confuses the eavesdropper regarding which item participates in each pool test. However, unlike wiretap channels with feedback, we cannot use the data on the link directly, and must use a coding scheme, to increase the number of keys we can generate. Although the keys generated are dependent, any subset which Eve eventually observes is independent and thus protected. Accordingly, this adaptive algorithm decreases the factor 1/(1 − δ) given in SNGT, and one may use a smaller number of tests. In the case that the information rate on the feedback is equal to the rate of the eavesdropper’s observation, this factor is completely rescinded. Thus, we achieve the same sufficiency bound on T as given for the non-secure GT. to denote their realizations, and calligraphic letters to denote the alphabet. Logarithms are in base 2 and hb (·) denotes the binary entropy function. In general, GT is defined by a testing matrix X = {Xj (t)}1≤j≤N,1≤t≤T ∈ {0, 1}N ×T , where each row corresponds to a separate item j ∈ {1, . . . , N }, and each column corresponds to a separate pool test t ∈ {1, . . . , T }. For the j-th item, entry Xj (t) = 1 if item j participates in the t-th pool test and Xj (t) = 0 if is not. Denoting by Aj ∈ {0, 1} an indicator function indicating whether the j-th item belongs to the defective set, the pool test outcome Y (t) ∈ Y T is Y (t) = N _ j=1 Xj (t)Aj = _ Xd (t), d∈K W where is used to denote the boolean OR operation. In AGT, we assume that the makeup of a testing pool can depend on the outcomes of earlier tests, by adaptive feedback link from the lab to the mixer, such that for any test t > 1 and any item j, Xj (t) = Xj (t\|Y (1), . . . , Y (t − 1)). In the secure model, we assume this link is private, and at a limited rate Rf . That is, symbols Ft , t ∈ {1, . . . , T } are sent over an adaptive private feedback link, secretly from the eavesdropper. The feedback alphabets are denoted by {F1 , . . . , FT }. Their cardinalities must satisfy T 1X log(\|Ft \|) ≤ Rf . T t=1 At each time instant t, Ft−1 is computed by the lab and revealed to the mixer. The symbol Ft at time instant t may depend on Y t−1 , the t−1 prior outcomes of the pool tests, the previous feedback symbols, F t−1 , and some randomness. That is, we assume there exists a distribution p(Ft \|Y t−1 , F t−1 ). Hence, in the secure AGT II. P ROBLEM F ORMULATION Xj (t) = Xj (t\|F t ). In SAGT, a legitimate user wishes to identify a small unknown subset K of defective items form a larger set N , while reducing the number of measurements ,T , as much as possible and keeping an eavesdropper, which is able to observe a subset of the tests results ignorant regarding the status of the N items. The adaptive link allows the outcomes from previous tests to influence the makeup of future tests in order to further reduce the total number of measurements. N = \|N \|, K = \|K\| denote the total number of items, and the number of defective items, respectively. The status of the items, defective or not, should be kept secure from the eavesdropper, but detectable by the legitimate user. We assume that in each round the number K of defective items is known apriori. This is a common assumption in the GT literature [18]. Figure 2 gives a graphical representation of the model. Throughout the paper, we use boldface to denote matrices, capital letters to denote random variables, lower case letters Note that in the classic adaptive case, where simply the previous outcome is revealed to the mixer, we have Ft = Yt−1 , hence \|Ft \| = 2 for all t and therefor Rf = 1. In SAGT, we assume an eavesdropper, which observes a noisy vector Z T = {Z(1), . . . , Z(T )}, generated from the outcome vector Y T . We concentrate on the erasure case, where the probability of erasure is 1 − δ, i.i.d. for each test. That is, on average, T δ outcomes are not erased and are available to the eavesdropper via Z T . N Denote by W ∈ W , {1, . . . , K } the index of the subset of defective items. We assume W is uniformly distributed, that is, there is no apriori bias to any specific subset. Denote by Ŵ (Y T , F T ) the index recovered by the legitimate decoder, after observing Y T . We refer to the adaptive procedure of creating the testing matrix, together with the decoder as a SAGT algorithm. We are interested in the asymptotic behavior of a SAGT algorithm. Figure 2: Noiseless secure adaptive group-testing setup. Definition 1. A sequence of SAGT algorithms with parameters N, K,T and Rf is asymptotically reliable and weakly secure if: (1) At the legitimate receiver, observing Y T , lim P (Ŵ (Y T , F T ) 6= W ) = 0. T →∞ (2) At the eavesdropper, observing Z T , lim T →∞ 1 I(W ; Z T ) = 0. T III. R ELATED W ORK Recently works [3]–[5], [7], [9]–[13] adopted an information theoretic perspective on GT, presenting it as a channel coding problem. We briefly review the most relevant results. In [3], the authors mapped the NGT model to an equivalent channel model, where the defective set takes the place of the message, the testing matrix rows are codewords, and the test outcomes are the received signal. They let Ŝ(X T , Y T ) denote the estimate of the defective subset S, which is random due to the randomness in X and Y . Furthermore, let Pe denote the average probability of error, averaged over all subsets S of cardinality K, variables X T and outcomes Y T , i.e., Pe = P r[Ŝ(X T , Y T ) 6= S]. Then, where (S 1 , S 2 ) denote the partition of defective set S into disjoint sets S 1 and S 2 with cardinalities i, the flowing bounds on the total number of tests required in Bernoulli NGT was given by T ≤ T ≤ T , where for some ε > 0 independent of N and K, log N −K i T = (1 + ε) · max , i=1,...,K I(XS 1 ; XS 2 , Y ) log N −K+i i T = max , i=1,...,K I(XS 1 ; XS 2 , Y ) In [9], the authors considered the AGT model as channel coding with feedback, where future inputs to the channel can depend on past outputs. Shannon proved that feedback does not improve the capacity of a single-user channel [14]. However, due to the non-tightness of the bounds on testing in the non-adaptive case, the authors could not show that adaptive GT requires the same number of tests as non-adaptive testing exactly. Alternatively, they showed that it obeys the same lower bound and requires no more tests than the non-adaptive case. Moreover, in [11], using a similar analogy to channel coding N problem, the authors defined the AGT rate R as log2 K /T and introduced the capacity C, that is, if for any ε > 0, there exists a sequence of algorithms with N log2 K ≤C −ε lim N →∞ Ta and success probability approaching one, yet with N limN →∞ log2 K /T > C it approaches zero. In our earlier work [7], we focused on SNGT. We considered a scenario where there is an eavesdropper which is able to observe a subset of the outcomes. We proposed a SNGT algorithm, which keeps the eavesdropper with leakage probability δ, ignorant regarding the items’ status. Specifically, when the fraction of tests observed by Eve is 0 ≤ δ < 1, [7] proved that the number of tests required for both correct reconstruction at the legitimate user and negligible mutual information at Eve’s 1 times the number of tests required with no secrecy side is 1−δ constraint. Then, where I(XS 1 ; XS 2 , Y ) ≥ i/K, according [7, Claim 1], the flowing bounds on the total number of tests required in Bernoulli SNGT was given by T s ≤ Ts ≤ T s , where for some ε > 0 independent of N and K, K N −K 1+ε Ts = max log , 1 − δ i=1,...,K i i 1 N Ts = log . 1−δ K IV. M AIN R ESULTS Under the model definition given in Section II, our main results are the following sufficiency (direct) and necessity (converse) conditions, characterizing the maximal number of tests required to guarantee both reliability and security. The direct proof and leakage are given in Section V. The remaining proofs are deferred to Section VI. A. Direct (Sufficiency) With a private rate limited feedback link, the sufficiency part is given by the following theorem. Theorem 1. Assume a SAGT model with N items, out of which K = O(1) are defective. For any δ < 1 and private adaptive rate limited feedback 0 < Rf < 1, if K N −K 1+ε Tsa ≥ max log , min{1, 1 − δ + Rf } i=1,...,K i i for some ε > 0 independent of N and K, then there exists a sequence of SAGT algorithms which are reliable and weakly secure. That is, as N → ∞, both the average error probability approaches zero and an eavesdropper with leakage probability δ is kept ignorant. It is important to note that if Rf ≥ δ, as the direct proof will show, the information obtained over the adaptive link between the lab and the mixer is powerful enough to obtain security without increasing T . Hence, in this case, the direct bounds of the non-secure and secure adaptive group testing are equal, that is, T = Tsa . Even when Rf < δ, the information obtained over the feedback between the lab and the mixer reduces the upper bound on the number of the tests required in SNGT, thus, T < Tsa ≤ T s . Using an upper bound, such that, log N −K ≤ i i log (N −K)e , the maximization in Theorem 1 can be solved i easily, leading to a simple bound on T . Corollary 1. For SAGT with parameters K << N and T , reliability and secrecy can be maintained with Tsa ≥ 1+ε K log(N − K)e. min{1, 1 − δ + Rf } Remark 1. In this paper, we consider the asymptotic case in T and negligible error. In this case, without a secrecy constrain, it is well known that feedback does not help [9]. Nonetheless, with a secrecy constraint we show that the link is used only for shared randomness. However, for finite T and zero error [2], the mixer may use the previous outcomes to reduce the number of tests T . Thus, for finite T , zero error and a secrecy constraint, is not trivial what should be shared over the link, either pure randomness in order to cope with the secrecy constraint or single previous outcomes in order to try adaptively reduce the number of tests. B. Converse (Necessity) The necessity part is given by the following theorem. Theorem 2. Let T be the minimum number of tests necessary to identify the defective set Sw of cardinality K among population of size N in a SAGT model with private adaptive rate limited feedback 0 ≤ Rf ≤ 1 while keeping eavesdropper, with pooling outcome test leakage probability δ < 1, ignorant regarding the status of the items. Then, 1 N a log . Ts ≥ K min{1, 1 − δ + Rf } The proof is deferred to Section VI. Note that, compared to the lower bound without a security constraints, there is an increase by a multiplicative factor of 1/ min{1, 1 − δ + Rf }. When Rf ≥ δ, the lower bounds of the non-secure and secure adaptive group testing are equal, i.e., T = Tsa . C. Secrecy capacity in SAGT Returning to the analogy in [11] between channel capacity and group testing, one might define by Cs the (asymptotic) N minimal threshold value for log K /T , above which no reliable and secure scheme is possible. Under this definition, where C is the capacity without the security constraint, the result in this paper show that Cs ≥ (min{1, 1 − δ + Rf })C . Clearly, this can be written as, Cs ≥ min{C, C − C(δ) + C(Rf )}, raising the usual interpretation as the difference between the capacity to the legitimate decoder and that to the eavesdropper, yet, with the capacity of the information obtained over the feedback link between the legitimate decoder and encoder [15]–[17]. Note that as the effective number of tests Eve sees is Te = δT , hence her GT capacity is (δ)C. V. C ODE C ONSTRUCTION WITH A DAPTIVE P RIVATE F EEDBACK AND A P ROOF FOR T HEOREM 1 The goal, in general, is to design a proper testing matrix, or, specifically, an algorithm to adaptively update it. Remember that each row describes the tests an item participates in. We thus construct this matrix in batches (of T tests each), each time selecting an appropriate row for each item. In a batch, for each item we generate a bin, containing several sub-bins, with several rows in each sub-bin (see Figure 3). Internal randomness in the mixer, which is not shared with any other party, is used to select a sub-bin for each item, while data received from the adaptive link is used to select the right row from the sub-bin. While this solution is inspired by codes for wiretap channels with rate limited feedback [16], there are several key differences, which not only change the construction, but also require non-trivial processing of the data received from the feedback. Specifically, first, unlike a wiretap channel, herein there are N items, only K of them, unknown to the mixer, actually participate in the output (“transmit”). Thus, bins and sub-bins sizes should be properly normalized. More importantly, the mixer, which acts as an encoder, does not know which K messages it should protect. Thus, the mixer should artificially blow-up the data it receives from the private feedback: from bits (used as keys) intended to protect the K defective items, it generates a larger number of keys, sufficient N items, satisfying the property that any K out of the N which will eventually participate, will still be protected. In other words, the keys received from the feedback cannot be used as is, and an interesting secret sharing-type scheme must be used. Formally, a (batch-processing) SAGT code consists of an N index set W = {1, 2, . . . K }, its w-th item corresponding to the w-th subset K ⊂ {1, . . . , N }; A discrete memoryless source of randomness at the mixer (RX , pRX ); A discrete memoryless source of randomness at the lab (RY , pRY ); A feedback at rate Rf bits per test, resulting in an index I ∈ {1, . . . , 2T Rf } after T uses; We use a single index due to the batch processing (Section V-A describes a test-by-test adaptive algorithm based on the one herein); The mixer, of course, does not know which items are defective, thus it needs to select a row for each item. However, since only the rows of the defective items affect the output Y T , it is beneficial to define an “encoder" GX : W × RX × I → XSw ∈ {0, 1}K×T , Figure 3: Encoding process for a SAGT code. that is, mapping the message, the randomness and input from the adaptive link to the K codewords which are summed to give the tests output Y T . Note that a stochastic encoder and the causally known feedback message I are similar to encoders ensuring information theoretic security, as randomness is required to confuse the eavesdropper about the actual information [16], [19]. A decoder at the legitimate user is a map Ŵ : Y T × I → W. The probability of error is P (Ŵ 6= W ). The probability that an outcome test leaks to the eavesdropper is δ. We assume a memoryless model, i.e., each outcome Y (t) depends only on the corresponding input XSw (t), and the eavesdropper observes Z(t), generated from Y (t) according to p(Y T , Z T \|XSw ) = T Y p(Y (t)\|XSw (t))p(Z(t)\|Y (t)). t=1 Next we provide the detailed construction and analysis. 1) Codebook Generation: Choose integers F and M such that log2 (F ) = T (Rf /K) and log2 (M ) = T (δ − Rf − )/K. For each item we generate bin of M · F independent and identically distributed codewords. Each codeword of size T is generated randomly, where each Xj (t) is chosen according to P (x) ∼ Bernoulli(ln(2)/K). Consequently, T P (X ) = T Y P (xi ). i=1 Then we split each bin to sub-bins of codewords xT (m, f ), 1 ≤ m ≤ M and 1 ≤ f ≤ F (illustrated in Figure 3). 2) Key Generation: We now describe the generation of the shared keys created from the information sent over the adaptive link. This link is of rate Rf . We do not use it to send information about test results, and simply send random bits. Therefore, in a block of length T we receive S = T Rf secret bits. We divide the secret bits to K secret keys, each constituting T Rf /K = SK bits. Our goal is to take these K keys, and use them to create N new keys, with the property that if the original K keys had a random uniform i.i.d. distribution, then any set of K keys out the N new ones will have the same random uniform i.i.d. distribution. This can be done using a generator matrix of an [N, K] MDS code. Such a generator matrix has the property that any K columns are linearly independent. Thus, taking the K original keys, as an SK × K matrix, and multiplying it by the generator matrix GK×N creates a matrix of size SK × N , where each column is used as the new key. Since any subset of K columns of G is invertible, each set of K new keys is simply a 1 : 1 transformation of the K original keys. The importance of this scheme in our context is as follows: for any subset of K new keys (out of N ), if an eavesdropper has no access to the original K keys, he/she is ignorant regarding the new keys. Moreover, it is important to note that unlike protection using a one-time-pad, these keys cannot be XORed with the rows of the testing matrix, as this operation will change the probability of each item participating in a pool-test, deviating from the optimal distribution of the testing matrix. 3) Testing: The mixer receives the T Rf feedback secret bits, divides them to K keys and uses the MDS code to create N new keys. Each new key is of length T Rf /K. Therefore, at each round and for each item j, the mixer selects a subbin using its internal randomness, and a message within it using the key. The result is a codeword xT (m, f ). Therefore, the SAGT matrix contains N randomly selected codewords of length T , one for each item (defective or not). In the first round of tests, the mixer has no available (feedback) key, hence it operates using a larger number of tests for that round as given for the non-adaptive SGT in [7]. Amortized over multiple rounds, this loss is negligible. 4) Decoding at the Legitimate Receiver: The decoder looks for a collection of K codewords XSTŵ , one from each sub-bin, for which Y T is most likely. Namely, P (Y T \|XSTŵ ) > P (Y T \|XSTw ), ∀w 6= ŵ. Then, the legitimate user declares Ŵ (Y T × F ) as the set of bins in which the rows reside. 5) Reliability: Let I(XS 1 ; XS 2 , Y ) denote the mutual information between XS 1 and (XS 2 , Y ), under the i.i.d. distribution with which the codebook was generated and remembering that Y is the output of a Boolean channel. The following lemma is a key step in proving the reliability of the decoding algorithm suggested herein. This Lemma is a direct consequence of [7, Lemma 1], under the enhancement that the index of each sub-bin is set according to the known key sheared between the legitimate decoder and the mixer at each round of the algorithm. Lemma 1. If the number of tests satisfies i log N −K M i , T ≥ (1 + ε) · max i=1,...,K I(XS 1 ; XS 2 , Y ) then, under the codebook above, as N → ∞ the average error probability approaches zero. Applying [7, Claim 1], which lower bounds the mutual information between XS 1 to (XS 2 , Y ) by i/K, to the expression δ−Rf − in Lemma 1, and substituting M = 2T K , a sufficient condition for reliability is 1+ε N −K i T ≥ max log + T (δ − Rf ) i 1≤i≤K K i K with some small . Rearranging terms results in 1+ε N −K T ≥ max log . 1≤i≤K min{1, 1 − (1 + ε)(δ − Rf )}i/K i This complete the reliability part. Note that the bound holds for large K and N , and ε independent of K and N . 6) Information Leakage at the Eavesdropper: We now prove the security constraint is met. Hence, we wish to show that I(W ; Z T )/T → 0, as T → ∞. Denote by CT the random codebook and by XST the set of codewords corresponding to the true defective items. We have, 1 T I(W ; Z T \|CT ) = 1 T F F I(W, RK RK ; Z T \|CT ) + I(RK RK ; Z T \|W CT ) , (1) where RK is the random variable used by the encoder to choose the sub-bins. That is, the encoder has a random variable for each item, RK denotes the union of K such variables, for F is again a union, this time of K the K defective items. RK keys. These are K “shares", out of the N shares generated F uniquely define XST , by the MDS code. Since W, RK , RK continuing from (1), we have: F = T1 I(XST ; Z T \|CT ) − I(RK RK ; Z T \|W, CT ) F = T1 (I(XST ; Z T \|CT ) − H(RK RK \|W, CT ) F +H(RK RK \|Z T , W, CT )) (a) 1 =T (b) F F \|Z T , W, CT ) ) + H(RK RK (XST ; Z T \|CT ) − H(RK RK δ−R − R ≤δ − T1 K T Kf + T Kf + T1 H(RK \|Z T , W, CT ) (c) ≤T , F where T → 0 as T → ∞. (a) is since both RK and RK are independent of W and the codebook. (b) is since both keys are Rf uniform, the first includes K variables of T ( δ− K − K ) bits R each, and the second K shares of T Kf bits each. (c) follows from [7, Section V.B]. In short, given the true K defectives, the codebook and her output Z T , Eve sees a simple MAC channel, at a rate slightly below her capacity. Therefore, she can identify which codeword was selected for each item, hence F identify both RK and RK . A. Test design depends on the outcome of previous pool-test In this solution, the outcome of each separate pool-test, according to the rate of the private adaptive link, can be available at the mixer to influence on the next test. It is important to note that unlike the first solution, we assume that by the adaptive link, the lab and the mixer can agree\know which pool-test, t ∈ T , depends on the symbols sent over the link. This assumption is not trivial, however, in various Figure 4: Encoding process for a SAGT code where the test design depends on the outcome of previous pool-test. scenarios based on GT [2] this assumption holds, e.g., in the channel problem, when the encoder (mixer) and the decoder (lab) schedule the transmission over the main channel and the feedback. In the code construction phase, we generate code as in Section V, yet, with one codeword in each sub-bin (i.e., the feedback will not used to select the rows in the original testing matrix). Hence for each item we have a row-bin with M rowcodewords. Moreover, for each possible column in the original testing matrix, we generate a column-bin with two columncodewords. Then for each j-th item, the mixer randomly select one row for the original testing matrix randomly as in the first setup from the j-th row-bin, yet, per pool test, one column-codeword is selected from his column-bin to define which items will participate in this pool test. This column selected according to the previous outcome feedback sent after each separate pool-test by the legitimate user to the mixer. If the feedback of the previous outcome is not available to the mixer, since the rate of the feedback is limited, the mixer use the column of the original testing matrix. Figure 4 gives a graphical representation of the code. Specifically, in the Q codebook generation phase, using a T distribution P (X T ) = i=1 P (xi ), for each item we generate a row-bin with M independent and identically distributed row-codewords. QN Then in the same way, using a distribution P (X̃ N ) = i=1 P (x̃i ), for each possible column xN (t) we generate a column-bin with two independent and identically distributed column-codewords. In the testing phase, for each item j, the mixer selects uniformly at random one row-codeword xTj (m) from his rowbin. The lab select randomly one index in I ∈ {1, . . . , 2T Rf } and before which pool-test t ∈ T , one bit of the index will sent to the mixer over the feedback to influence on the next test. Note that the lab is available to send at most T Rf bits. Then, in each pool-test t, using the previous feedback outcome Ft , the N mixer selects one column-codeword x̃N t (Ft ) from the x (t) column bin. In the case that the previous feedback outcome is not available at the mixer, the mixer use the original column. The decoder at the legitimate user (lab) know exactly which pool test was depended on the bits sent over the feedback and what was the indexes of the columns selected from the columnbins by the mixer for each pool-tests. Hence, the decoder procedure after T pool-tests, and the analysis of the reliability and the information leakage at the eavesdropper are almost a direct consequence of Section V. Thus, H(W ) ≤ VI. C ONVERSE (N ECESSITY ) In this section, we derive the necessity bound on the required number of tests, in the setup with rate limited feedback between the legitimate user to the lab. Similar to [16], using Fano’s inequality, for Ŵ = ŵ(Y T , F T ), N H(W \|Ŵ ) ≤ 1 + Pe log = T , K where T → 0 as T → ∞ if Pe → 0. Since Ŵ is a function of Y T , F T , H(W \|Y T , F T ) ≤ H(W \|Ŵ ) ≤ T T . I(W ; Z T ) = T γT , 1 T PT i=1 log(\|Fi \|) ≤ Rf and To conclude, the well-known technique of introducing a time sharing random variable is used. Assume Q is independent of XTS w , Y T , Z T and uniform on {1, . . . , T }, this results in Rf + = Rf + T 1X (I(XSiw ; Yi \|Zi ) + δT T i=1 T 1X (I(XSiw ; Yi \|Zi , Q = i) + δT T i=1 = Rf + I(XSQw ; YQ \|ZQ , Q) + δT H(W \|Z T ) + I(W ; Z T ) = Rf + I(XSw ; Y \|Z, Q) + δT T = H(W \|Z ) + T γT I(W ; Y T , F T \|Z T ) + H(W \|Y T , Z T , F T ) + T γT ≤ I(W ; Y T , F T \|Z T ) + T T + T γT (c) I(W ; F T \|Z T ) + I(W ; Y T \|F T , Z T ) + T δT (d) ≤ H(Fi ) + T δT . i=1 (2) = = T X T 1 1X H(W ) ≤ I(XSiw ; Yi \|Zi ) + Rf + δT . T T i=1 H(W ) (b) ; Yi \|Zi ) + We now use the constraint normalize by T . We have, where γT → 0 as T → ∞. Consequently, (a) I(X Siw i=1 1 H(W ) ≤ T Using the secrecy constraint, = T X T T H(F \|Z ) + I(W, XTS w ; Y T \|F T , Z T ) where XSw := XSQw , Y := YQ , Z := ZQ ,. Letting T → ∞, δT → 0 and T → 0, hence 1 H(W ) ≤ T ≤ = + T δT Rf + I(XSw ; Y \|Z, Q) Rf + I(XSw , Q; Y \|Z) Rf + I(XSw ; Y \|Z). where (a) follows from (2), (b) follows from Fano’s inequality and (c) follows by defining δT = T + γT . The following recursive lemma is now required. Hence, Lemma 2 ( [16]). For each t ∈ {1, . . . T }, From the above bound, and since the eavesdropper can obtain information only from the outcomes of the legitimate user, such that, p(y, z\|x) = p(y\|x)p(z\|y), H(F t \|Z t ) + I(W, XtS w ; Y t \|F t , Z t ) t−1 ≤ H(F t−1 \|Z t−1 ) + I(W, Xt−1 \|F t−1 , Z t−1 ) Sw ; Y t−1 + H(Ft \|W, Xt−1 , Z t−1 ) + I(XStw ; Yt \|Zt ). Sw , F To continue, we use Lemma 2 recursively starting from (d): H(W ) ≤ H(F T \|Z T ) + I(W, XTS w ; Y T \|F T , Z T ) + T δT ≤ H(F T −1 \|Z T −1 ) +I(W, XTS w−1 ; Y T −1 \|F T −1 , Z T −1 ) 1 H(W ) ≤ min{I(XSw ; Y ), I(XSw ; Y \|Z) + Rf }. T 1 H(W ) ≤ min{I(XSw ; Y ), I(XSw ; Y ) − I(XSw ; Z) + Rf }. T N Since H(W ) = log K , we have 1 N log ≤ min{I(XSw ; Y ), I(XSw ; Y )−I(XSw ; Z)+Rf }. T K Now let Z̄ denote the random variable corresponding to the tests which are not available to the eavesdropper. Hence, +I(XSTw ; YT \|ZT ) + H(FT ) + T δT H(Z̄) = I(XSw ; Y ) − I(XSw ; Z). ≤ H(F T −2 \|Z T −2 ) +I(W, XTS w−2 ; Y T −2 \|F T −2 , Z T −2 ) +I(XSTw−1 ; YT −1 \|ZT −1 ) + H(FT −1 ) +I(XSTw ; YT \|ZT ) + H(FT ) + T δT ≤ ... T T X X ≤ I(XSiw ; Yi \|Zi ) + H(Fi ) + δT . i=1 i=1 Denote by Ē the set of tests which are not available to Eve and by Ēγ the event {\|Ē\| ≤ T (1 − δ)(1 + γ)} for some γ > 0. We have H(Z̄) = P (Ēγ )H(Z̄\|Ēγ ) + P (Ēγc )H(Z̄\|Ēγc ) ≤ T (1 − δ)(1 + γ) + T P (Ēγc ) ≤ T (1 − δ)(1 + γ) + T 2−T (1−δ)f (γ) , where the last inequality follows from the Chernoff bound for i.i.d. Bernoulli random variables with parameter (1 − δ) and is true for some f (γ) such that f (γ) > 0 for any γ > 0. Thus, we have N log ≤ T (1 − δ)(1 + γ) + T 2−T (1−δ)f (γ) + Rf . K That is, Tsa N 1 log . ≥ min{1, 1 − δ + Rf } K This completes the converse proof. R EFERENCES [1] R. Dorfman, “The detection of defective members of large populations,” The Annals of Mathematical Statistics, vol. 14, no. 4, pp. 436–440, 1943. [2] D.-Z. Du and F. K. Hwang, “Combinatorial group testing and its applications (applied mathematics),” 2000. [3] G. K. Atia and V. Saligrama, “Boolean compressed sensing and noisy group testing,” Information Theory, IEEE Transactions on, vol. 58, no. 3, pp. 1880–1901, 2012. A minor corection appered in vol. 61, no. 3, pp. 1507–1507, 2015. [4] M. B. Malyutov, “Search for sparse active inputs: A review.” in Information Theory, Combinatorics, and Search Theory, 2013, pp. 609–647. [5] H. Aydinian, F. Cicalese, and C. Deppe, Information Theory, Combinatorics, and Search Theory: In Memory of Rudolf Ahlswede. Springer, 2013, vol. 7777. [6] G. K. Atia, V. Saligrama, and C. 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Arrhythmia Classification from the Abductive Interpretation of Short Single-Lead ECG Records Tomás Teijeiro*, Constantino A. García, Daniel Castro and Paulo Félix arXiv:1711.03892v1 [cs.AI] 10 Nov 2017 Centro Singular de Investigación en Tecnoloxías da Información (CITIUS), University of Santiago de Compostela, Santiago de Compostela, Spain Abstract In this work we propose a new method for the rhythm classification of short single-lead ECG records, using a set of high-level and clinically meaningful features provided by the abductive interpretation of the records. These features include morphological and rhythm-related features that are used to build two classifiers: one that evaluates the record globally, using aggregated values for each feature; and another one that evaluates the record as a sequence, using a Recurrent Neural Network fed with the individual features for each detected heartbeat. The two classifiers are finally combined using the stacking technique, providing an answer by means of four target classes: Normal sinus rhythm (N), Atrial fibrillation (A), Other anomaly (O) and Noisy (~). The approach has been validated against the 2017 Physionet/CinC Challenge dataset, obtaining a final score of 0.83 and ranking first in the competition. 1. Introduction The potential of Artificial Intelligence and machine learning techniques to improve the early detection of cardiac diseases using low-cost ECG tests is still largely untapped. The 2017 Physionet/Computing in Cardiology challenge defies the scientific community to propose solutions to the automatic detection of Atrial Fibrillation from short single lead ECG signals [1]. The challenge is posed as a classical machine learning problem: A labeled training set is provided, and the proposals are evaluated against a hidden test set of records. However, even if the only metric for the final ranking is the accuracy of the proposed models, a number of additional properties should be considered for the final adoption of each proposal in the clinical practice. Here, we emphasize on the interpretability of the automatic detection of Atrial Fibrillation, a major concern to ensure trust by the care staff [2]. In this sense, our proposal is based on a high-level description of the target signal by means of the same features used by cardiologists in ECG analysis. This description is generated with a pure knowledge-based approach, using an abductive framework for time series interpretation [3] that looks for the set of explanatory hypotheses that best account for the observed evidence. Only after this description has been built, machine learning methods were used to make up for the lack of the expert criteria applied in the labeling of the training set, and to alleviate the effect of possible errors in the interpretation process. 2. Methods The global architecture of the proposal is depicted in Figure 1, and the processing stages are explained in the following subsections. 2.1. Preprocessing The preprocessing stage aims at improving the quality of the data to be interpreted in the following stages, and involves two different tasks: 2.1.1. Data relabeling: The labeling of the training set was performed by a single expert in a single pass, and as a consequence some inconsistencies appear in the classification criteria. Thus, a thorough manual relabeling was carried out, but trying to be conservative and guided by preliminary classification results. We focused on records classified as N but showing what we consider clear anomalies. A total number of 197 out of 8528 records were relabeled. 2.1.2. Lead inversion detection: A number of records in the training set were found to be inverted, probably due to electrode misplacement. Inverted records are more likely to be classified as abnormal due to the presence of infrequent QRS and T wave morphologies, as well as to the greater difficulty to identify P waves. The inverted records were first identified manually, and then a simple logistic regression classifier was trained considering 14 features obtained from the raw signal and a tentative delineation of the P wave, QRS complex and T wave of every heartbeat detected by the gqrs application from the Physionet library [4]. This delineation was performed using the Construe algorithm [3], limiting the interpretation to the conduction level, that is, avoiding the rhythm interpretation. ECG Signal Lead Inversion Detection Global Features Global Classiﬁcation Per-Beat Features Sequence Classiﬁcation Abductive Interpretation Classiﬁcation Stacking Final Class Figure 1. Classification algorithm steps. 2.2. Abductive interpretation The abductive interpretation of the ECG signal is the most significant stage in the proposed approach. Its objective is to characterize the physiological processes underlying the signal behavior, building a description of the observed phenomena in multiple abstraction levels. This responsibility lies with the Construe algorithm, which applies a non-monotonic reasoning scheme to find the set of hypotheses that best explain the observed evidence, by means of a domain-specific knowledge base composed of a set of observables and a set of abstraction grammars. The knowledge base is the same used in [3], that allows to explain the ECG at the conduction and rhythm abstraction levels, thus providing the same features used by cardiologists in ECG analysis. The initial evidence is the set of waves identified in the wave delineation step, that are abstracted by a set of rhythm patterns to describe the full signal as a sequence of cardiac rhythms, including normal sinus rhythms, bradycardias, tachycardias, atrial fibrillation episodes, etc. The non-monotonic nature of the interpretation process allows us to modify the initial set of evidence, by discarding heartbeats that cannot be abstracted by any rhythm pattern, or by looking for missed beats that are predicted by the pattern selected as the best explanatory hypothesis for a signal fragment. This ability to correct the initial evidence is the main strength of our proposal, since it discards many false anomalies generated by the presence of noise and artifacts in the signal. Figure 2 shows an example of a noisy signal in which the gqrs application detects many false positive beats, that are removed or modified in the final interpretation that concludes with a single normal rhythm hypothesis that explains the full fragment. As we can also see in the Figure, the result of the interpretation stage is a sequence of P waves, QRS complexes and T waves observations, as well as a sequence of cardiac rhythms abstracting all those waves. 2.3. Global feature extraction Considering that each ECG record has to be classified globally, providing a single label for the entire signal duration, after the interpretation stage a set of features are calculated trying to summarize the information provided by Construe. A total number of 79 features are calculated, that are comprehensively described in the published software documentation. The feature set is divided into three main groups: • Rhythm features: This includes statistical measures on the RR sequence, such as the limits, median or median absolute deviation; heart rate variability features such as the PNN5, PNN10, PNN50 and PNN100 measures [5]; and information about the rhythm interpretation, such as the median duration of each rhythm hypothesis. • Morphological features: This includes information about the duration, amplitude and frequency spectrum of the observations in the conduction abstraction level, including P and T waves, QRS complexes, PR and QT intervals, and the TP segments. • Signal quality features: Their purpose is to assess the importance of the morphological features showing conduction anomalies, such as wide QRS complexes or long PR intervals. They are based on the sum of the absolute differences of the signal, which we refer to as profile. Some of the profiled areas of the signal are the baseline segments and the P wave area before each heartbeat (taking a constant window of 250 ms). 2.4. Global classification If a precise definition of the expert knowledge leading to the labeling of the training set were available, then the final classification could be directly developed with a basic rule-based system operating on the features extracted from the abductive interpretation stage, and the accuracy of the system would depend mainly on the accuracy of the interpretations. However, the challenge does not publish any guidelines for the classification, specially for the O class. Therefore, an automatic classifier was trained with two objectives: 1) To reveal the criteria leading to the training set labeling; and 2) to make the classification more accurate by learning possible mistakes of the abductive interpretation. The classification method selected for this stage was the Tree Gradient Boosting algorithm, and particularly the XGBoost implementation [6], which showed a high performance and a certain level of interpretability through the importance given to the classification features. The optimization of the hyperparameters was performed using exhaustive grid search and 8-fold cross-validation, leading to the following values: Maximum tree depth: 6, Learning rate: 0.2, Gamma: 1.0, Column subsample by tree: 0.9, Min. child weight: 20, Subsample: 0.8, and Number of boosting rounds: 60. Figure 2. How the abductive interpretation can fix errors in the initial evidence. [Source: First 10 seconds of the A02080 record. Grey: Original gqrs annotations. Blue: QRS observations. Yellow: T wave observations. Green: P wave observation and Normal rhythm hypothesis.] With respect to the first objective, we were able to formalize a number of specific anomalies that lead to classify a record as O. This identification helped to optimize the training set by defining more specific features to be calculated from the interpretation results. Some of the identified anomalies sharing this class were: • Tachycardia (Mean heart rate over 100 bpm). • Bradycardia (Mean heart rate under 50 bpm). • Wide QRS complex (Longer than 110 milliseconds). • Presence of ventricular or fusion beats. • Presence of at least one extrasystole. • Long PR interval (Longer than 210 milliseconds). • Ventricular tachycardia. • Atrial flutter. For some of these anomalies the classification in the training set seems a bit inconsistent, since examples can be found in several classes. For example, there are various records labeled as normal with PR interval longer than 210 milliseconds, as long as examples of records labeled as atrial fibrillation showing an atrial flutter pattern. Regarding the second objective, even after discovering some of the expert criteria distinguishing the target classes a rule-based system was not still competitive against automatic learned models. From our point of view this shows that the XGBoost classifier is able to improve the results of the interpretation alone. 2.5. Per-beat feature extraction Some of the conditions leading to a certain classification may not be present for the entire duration of a record, so the global features are not the best option to characterize episodic events of abnormalities. For example, a normal record with a single ectopic ventricular beat that does not break the rhythm is quite difficult to classify as abnormal by the global classifier. For this reason, some of the features calculated from the abductive interpretation are disaggregated to the individual heartbeat scope, such as the morphology, duration and amplitude of the P wave, the QRS complex and the T wave. Also the RR interval and the RR variation before and after each beat is included, as long as the profile of the P wave area. A sequence classification approach is then used to learn characteristic temporal patterns of each target class. 2.6. Sequence classification In the proposed approach, sequence classification relies on Recurrent Neural Networks (RNNs), a family of neural networks specialized for recognizing sequences of values. Among the different RNN implementations, we focused on Long Short Term Memory networks (LSTMs) [7], since they are capable of remembering information for long periods of time through the use of a cell state. Furthermore, they are able to avoid vanishing and exploding gradients when doing backpropagation through time. The architecture of the neural net is shown in Figure 3. The timedistributed Multilayer Perceptron (MLP) preprocesses the features described in Section 2.5 to transform the data into a space with easier temporal dynamics. The number of hidden units of the MLP was 256, and the dimension of the output space 128. A Rectified Linear Unit (ReLU) was used as activation function. The LSTM_0 layer preprocesses the resulting sequence of transformed features and returns a new sequence, which is subsequently used by the other LSTMs. The LSTM_2 layer just returns the final state of the network, whereas LSTM_1 and LSTM_3 return new transformed sequences. The pooling layers after LSTM_1 and LSTM_3 remove the temporal dimension by computing the temporal mean and maximum of each feature of the sequences, respectively. All the LSTMs used 128 units. Another MLP (with the same configuration of the time-distributed one) joins and transforms the outputs of each LSTM before a Softmax layer, which outputs a probability for each of the 4 classes. L2 -regularization was applied to all layers, using 10−4 as regularization strength. Finally, dropout was also used to improve generalization by preventing feature co-adaptation [8]. The neural network was trained using the categorical cross-entropy as loss function, a batch size of 32, and Adam [9] as optimizer. Furthermore, 15% of all the data was used as validation set to monitor the performance of the neural network. This permitted us to decrease the learning rate when the validation loss got stuck in a plateau and to avoid overfitting by using early stopping. The initial √ learning rate was set to 0.002 and it was decreased by 2 when the validation loss did not improve for at least 3 epochs. Training was ended after 15 epochs without improvement. Temporal mean pooling LSTM_1 Padded sequence of beat features Time-distributed MLP LSTM_2 LSTM_0 MLP Temporal max pooling LSTM_3 Softmax Sequence Class Figure 3. The neural network architecture. 2.7. Classification stacking of interpretability by the use of high-level and meaningful features. The XGBoost classifier based on global features and the RNN classifier based on the per-beat features were combined using the stacking technique. Stacking (also referred to as stacked generalization) involves training a new classification algorithm to combine the predictions of several classifiers [10]. Usually, the stacked model achieves better performance than the individual models due to its ability to discern when each base model performs best and when it performs poorly. Prior to the application of stacking, the predictions of 3 RNNs were averaged to decrease the variance of the RNN classifier arising from the random initialization of the RNN weights and the random split between test and validation set. Averaging similar models also helps in reducing overfitting. Note that this averaging can be seen as a simple bagging method. The probabilities predicted by the XGBoost and the averaged RNNs are then combined through a Linear Discriminant Analysis (LDA) classifier, which acts as stacker. To avoid possible collinearity issues, only 3 probabilities from each model are used. 3. Evaluation To evaluate the performance of the algorithm, we followed the challenge guidelines and metrics. The final score is assigned as the mean F1 measure of the N, A, and O classes. Table 1 shows an example of the results that the proposed method is able to achieve using 8-fold crossvalidation. Note that the stacker usually achieves better scores than the base models and, furthermore, it has lower variance (not shown in the Table). Table 1. Example of stratified 8-fold cross-validation. Method 0 1 2 Fold Number 3 4 5 6 7 XGBoost 0.84 0.84 0.85 0.85 0.82 0.80 0.82 0.82 RNN 0.82 0.81 0.84 0.83 0.86 0.83 0.83 0.83 LDA-stacker 0.85 0.84 0.86 0.86 0.85 0.83 0.84 0.85 4. Mean 0.83 0.83 0.85 Conclusions This work proves that the combination of knowledgebased and learning-based approaches is effective to build classification systems that exploit sophisticated machine learning methods while maintaining a remarkable degree Acknowledgements This work was supported by the Spanish Ministry of Economy and Competitiveness under project TIN201455183-R. Constantino A. García is also supported by the FPU Grant program from the Spanish Ministry of Education (MEC) (Ref. FPU14/02489). References [1] Clifford G, Liu C, Moody B, Lehman L, Silva I, Li Q, Johnson A, Mark R. AF classification from a short single lead ECG recording: The Physionet Computing in Cardiology Challenge 2017. In Computing in Cardiology. 2017; . [2] Caruana R, Lou Y, Gehrke J, Koch P, Sturm M, Elhadad N. Intelligible Models for HealthCare. In 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM Press, 2015; . [3] Teijeiro T, Félix P, Presedo J, Castro D. Heartbeat classification using abstract features from the abductive interpretation of the ECG. IEEE Journal of Biomedical and Health Informatics 2016;. [4] Goldberger A, et al. PhysioBank, PhysioToolkit and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals. Circulation 2000;101:215–220. [5] Mietus JE, Peng CK, Henry I, Goldsmith RL, Goldberger AL. The pNNx files: re-examining a widely used heart rate variability measure. Heart British Cardiac Society oct 2002; 88(4):378–80. ISSN 1468-201X. [6] Chen T, Guestrin C. XGBoost: A Scalable Tree Boosting System. In 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. mar 2016; . [7] Hochreiter S, Schmidhuber J. Long short-term memory. Neural computation 1997;9(8):1735–1780. [8] Srivastava N, Hinton GE, Krizhevsky A, Sutskever I, Salakhutdinov R. Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research 2014;15(1):1929–1958. [9] Kingma D, Ba J. Adam: A method for stochastic optimization. arXiv preprint arXiv14126980 2014;. [10] Wolpert DH. Stacked generalization. Neural networks 1992;5(2):241–259. Address for correspondence: Tomás Teijeiro Campo Rúa de Jenaro de la Fuente Domínguez, S/N, CITIUS Building 15782 Santiago de Compostela, SPAIN tomas.teijeiro@usc.es	2
An experimental exploration of Marsaglia’s xorshift generators, scrambled arXiv:1402.6246v5 [cs.DS] 13 Oct 2016 Sebastiano Vigna, Università degli Studi di Milano, Italy Marsaglia proposed xorshift generators as a class of very fast, good-quality pseudorandom number generators. Subsequent analysis by Panneton and L’Ecuyer has lowered the expectations raised by Marsaglia’s paper, showing several weaknesses of such generators. Nonetheless, many of the weaknesses of xorshift generators fade away if their result is scrambled by a non-linear operation (as originally suggested by Marsaglia). In this paper we explore the space of possible generators obtained by multiplying the result of a xorshift generator by a suitable constant. We sample generators at 100 points of their state space and obtain detailed statistics that lead us to choices of parameters that improve on the current ones. We then explore for the first time the space of high-dimensional xorshift generators, following another suggestion in Marsaglia’s paper, finding choices of parameters providing periods of length 21024 − 1 and 24096 − 1. The resulting generators are of extremely high quality, faster than current similar alternatives, and generate long-period sequences passing strong statistical tests using only eight logical operations, one addition and one multiplication by a constant. Categories and Subject Descriptors: G.3 [PROBABILITY AND STATISTICS]: Random number generation; G.3 [PROBABILITY AND STATISTICS]: Experimental design General Terms: Algorithms, Experimentation, Measurement Additional Key Words and Phrases: Pseudorandom number generators 1. INTRODUCTION xorshift generators are a simple class of pseudorandom number generators introduced by Marsaglia [2003]. In Marsaglia’s view, their main feature is speed: in particular, a xorshift generator with a 64-bit state generates a new 64-bit value using just three 64-bit shifts and three 64-bit xors (i.e., exclusive ors), thus making it possible to generate hundreds of millions of values per second. Subsequent analysis by Brent [2004] showed that the bits generated by xorshift generators are equivalent to certain linear feedback shift registers. Panneton and L’Ecuyer [2005] analyzed in detail the theoretical properties of the generators, and found empirical weaknesses using the TestU01 suite [L’Ecuyer and Simard 2007]. They proposed an increase in the number of shifts, or combination with another generator, to improve quality. In the first part of this paper, as warm-up we explore experimentally the space of xorshift generators with 64 bits of state using statistical test suites. We sample generators at 100 points of their state space, to easily identify spurious failures. Marsaglia proposes some choice of parameters, that, as we will see, and as already reported by Panneton and L’Ecuyer [2005], are not particularly good. We report results that are actually worse than those of Panneton and L’Ecuyer as we use the entire 64-bit output of the generators. While we can suggest some good parameter choices, the result remains poor. Thus, we turn to the idea of scrambling the result of a xorshift generator using a multiplication, as it is typical, for instance, in the construction of practical hash functions due to the resulting avalanching behavior (bits of the result depend on several bits of the input). This method is actually suggested in passing in Marsaglia’s paper. The third edition of the classic “Numerical Recipes” [Press et al. 2007], indeed, proposes this construction for This work is supported the EU-FET grant NADINE (GA 288956). This paper is an extended version of the paper with the same title published in the ACM Transactions on Mathematical Software [Vigna 2016]. Author’s addresses: Sebastiano Vigna, Dipartimento di Informatica, Università degli Studi di Milano, via Comelico 39, 20135 Milano MI, Italy. 2 S. Vigna A0 A1 A2 A3 A4 A5 A6 A7 x x x x x x x x ^= ^= ^= ^= ^= ^= ^= ^= x x x x x x x x << >> << >> << >> >> << a; a; c; c; a; a; b; b; x x x x x x x x C code ^= x >> ^= x << ^= x >> ^= x << ^= x << ^= x >> ^= x << ^= x >> b; b; b; b; c; c; a; a; x x x x x x x x ^= ^= ^= ^= ^= ^= ^= ^= x x x x x x x x << >> << >> >> << << >> c; c; a; a; b; b; c; c; X1 X3 X2 X4 X5 X6 X7 X8 Fig. 1. The eight possible xorshift64 algorithms. The list is actually derived from Panneton and L’Ecuyer [2005], as they correctly remarked that two of the eight algorithms proposed by Marsaglia were redundant, whereas two (A6 and A7 ) were missing. On the right side we report the name of the linear transformation associated to the algorithm as denoted by Panneton and L’Ecuyer [2005]. With our numbering, algorithms A2i and A2i+1 are conjugate by reversal. Note that contiguous shifts in the same direction can be exchanged without affecting the resulting algorithm. We normalized such contiguous shifts so that their letters are lexicographically sorted. a basic, all-purpose generator. From the wealth of data so obtained we derive generators with better statistical properties than those suggested in “Numerical Recipes”. In the last part of the paper, we follow the suggestion about high-dimensional generators contained in Marsaglia’s paper, and compute several choices of parameters that provide full-period xorshift generators with a state of 1024 and 4096 bits. Once again, we propose generators that use a multiplication to scramble the result. At the end of the paper, we apply the same methodology to a number of popular noncryptographic generators, and we discover that our high-dimensional generators are actually faster and of higher or equivalent statistical quality, as assessed by statistical test suites, than the alternatives. The software used to perform the experiments described in this paper is distributed by the author under the GNU General Public License. Moreover, all files generated during the experiments are available from the author. They contain a large amount of data that could be further analyzed (e.g., by studying the distribution of p-values over the seeds). We leave this issue open for further work. 2. AN INTRODUCTION TO xorshift GENERATORS The basic idea of xorshift generators is that their state is modified by applying repeatedly a shift and an exclusive-or (xor) operation. In this paper we consider 64-bit shifts and states made of 2n bits, with n ≥ 6. We usually append n to the name of a family of generators when we need to restrict the discussion to a specific state size. For xorshift64 generators Marsaglia suggests a number of possible combination of shifts, shown in Figure 1. Not all choices of parameters give a full (264 − 1) period: there are 275 suitable choices of a, b and c and eight variants, totaling 2200 generators. In linear-algebra terms, if L is the 64 × 64 matrix on Z/2Z that effects a left shift of one position on a binary row vector (i.e., L is all zeroes except for ones on the principal subdiagonal) and if R is the right-shift matrix (the transpose of L), each left/right shift and xor can be described as a linear multiplication by I + Ls or I + Rs , respectively, where s is the amount of shifting.1 For instance, algorithm A0 of Figure 1 is equivalent to the Z/2Z-linear transformation X1 = I + La I + Rb I + Lc . 1A more detailed study of the linear algebra behind xorshift generators can be found in [Marsaglia 2003; Panneton and L’Ecuyer 2005]. An experimental exploration of Marsaglia’s xorshift generators, scrambled 3 It is useful to associate with a linear transformation M its characteristic polynomial P (x) = det(M − xI). The associated generator has maximum-length period if and only if P (x) is primitive over Z/2Z. This happens if P (x) is irreducible and if x has maximum period in the ring of n polynomial over Z/2Z modulo P (x), that is, if the powers x, x2 , . . . , x2 −1 are distinct modulo P (x). Finally, to check the latter condition is sufficient to check that n x(2 −1)/p 6= 1 mod P (x) n for every prime p dividing 2 − 1 [Lidl and Niederreiter 1994]. The weight of P (x) is the number of terms in P (x), that is, the number of nonzero coefficients. It is considered a good property for generators of this kind that the weight is close to n/2, that is, that the polynomial is neither too sparse nor too dense [Compagner 1991]. Note that the family of algorithms of Figure 1 is intended to generate 64-bit values. This means that the entire output of the algorithm should be used when performing tests. We will see that this has not always been the case in previous literature. 3. SETTING UP THE EXPERIMENTS In this paper we want to explore experimentally the space of a number of xorshift-based generators. Our purpose is to identify variants with full period which have particularly good statistical properties, and test whether claims about good parameters made in the previous literature are confirmed. The basic idea is that of sampling the generators by executing a battery of tests starting with 100 different seeds that are equispaced in the state space. More precisely, if the state is made of n bits we use the seeds 1 + ib2n /100c, 0 ≤ i < 100. The tests produce a number of statistics, and we decided to use as score the number of failed tests. A higher score, thus, means lower quality. Running multiple tests makes it easy to rule out spurious failures, as suggested also by Rukhin et al. [2001] in the context of cryptographic applications.2 We use two tools to perform our tests. The first and most important is TestU01, a test suite developed by L’Ecuyer and Simard [2007] that contains several tests oriented towards the generation of uniform real numbers in [0 . . 1).3 We also perform tests using Dieharder, a suite of tests developed by Brown [2013], both as a sanity check and to compare the power of the two suites. Dieharder contains all original tests from Marsaglia’s Diehard, plus many more additional tests. We refer frequently to the specific type of tests failed: the reader can refer to the TestU01 and Dieharder documentation for more information. We consider a test failed if its p-value is outside of the interval [0.001 . . 0.999]. This is the interval outside which TestU01 reports a test by default. Sometimes a much stricter threshold is used (For instance, L’Ecuyer and Simard [2007] use [10−10 . . 1 − 10−10 ] when applying TestU01 to a variety of generators), and weaker p-values are called suspicious values, but since we are going to repeat the test 100 times we can use relatively weak pvalues: spurious failures will appear rarely, and we can catch borderline cases (e.g., tests failing on 50% of the seeds) that give us useful information. We call systematic a failure that happens for all seeds. For all such failures in our tests, p-values are smaller than 10−15 . Thus, all conclusions drawn in this paper based on system2 We remark that, arguably, a more principled choice would be choosing seeds that are equispaced in the sequence of states traversed by the generator. Unfortunately, this is possible only for generators with “jumpahead” primitives, and we want our methodology to be universal. We checked that all sequences of states used in our tests on generators with 64 bits of state do not overlap. The chance that this happens with more than 128 bits of state is negligible. 3 We use the double-dot notation for intervals introduced by C. A. R. Hoare and Lyle Ramshaw [Graham et al. 1994]. 4 S. Vigna atic failures would not change even if we lowered significantly the failure threshold. More generally, 90% of the p-values of failed tests are actually smaller than 10−6 . We remark that our choice (counting the number of failures) is somewhat rough; for example, we consider the same failure a p-value very close to 0 and a p-value just below 0.001. Indeed, other, more sophisticated methods might be used to aggregate the result of our samples: combining p-values, for instance, or computing a p-value of p-values [Rukhin et al. 2001]. However, our choice is very easy to interpret, and multiple samples partially compensate this problem (spurious failures will appear in few samples). Of course, the number of experiments is very large—in fact, our experiments were carried out using hundreds of cores in parallel and, overall, they add up to more than a century of computational time. Our strategy is to apply a very fast test to all generators and seeds, in the hope of isolating a small group of generators that behave significantly better with respect to these tests. Stronger tests can then be applied to this subset. The same strategy has been followed by Panneton [2004] in the experimental study of xorshift generators contained in his Ph.D. thesis. TestU01 offers three different predefined batteries of tests (SmallCrush, Crush and BigCrush) with increasing computational cost and increased difficulty. Unfortunately, Dieharder does not provide such a segmentation. Note that Dieharder has a concept of “weak” success and a concept of “failure”, depending on the p-value of the test, and we used command-line options to align its behavior with that of TestU01: a p-value outside of the range [0.001 . . 0.999] is a failure. Moreover, we disabled the initial timing tests so that exactly the same stream of 64-bit numbers is fed to the two test suites. In both cases we implemented our own xorshift generator. Some care is needed in this phase, as both TestU01 and Dieharder are inherently 32-bit test suites: since we want to test xorshift as a 64-bit generator, it is important that all bits produced are actually fed into the test. For this reason, we implemented the generation of a uniform real value in [0 . . 1) by dividing the output of the generator by 264 , but we implemented the generation of uniform 32-bit integer values by returning first the lower and then the upper 32 bits of each 64-bit generated value.4 A possible downside of this approach is that we might fail to detect some failure in the high bits (of the 64-bit, full output) due to the interleaving process: however, the fact that in our tests xorshift generators generate many more failures than those reported previously [Panneton and L’Ecuyer 2005] suggests that the approach is well founded. An important consequence of this choice is that some of the bits are actually not used at all. When analyzing pseudorandom real numbers in the unit interval, there is an unavoidable bias towards high bits, as they are more significant. The very lowest bits have lesser importance and will in any case be perturbed by numerical errors. For this reason, it is a good practice to run tests both on a generator and on its reverse5 [Press et al. 2007]. In our case, this is even more necessary, as the lowest eleven bits returned by the generator are not used at all due to the fact that the mantissa of a 64-bit floating-point number is formed by 53 bits only. A recent example shows the importance of testing the reverse generator. Saito and Matsumoto [2014] propose a different way to eliminate linear artifacts: instead of multiplying the output of an underlying xorshift generator (with 128 bits of state and based on 32-bit shifts) by a constant, they add it (in Z/232 Z) with the previous output. Since the sum in Z/232 Z is not linear over Z/2Z, the result should be free of linear artifacts. However, while their generator passes BigCrush, its reverse fails systematically the LinearComp, Ma- 4 If a real value is generated when the upper 32 bits of the last value are available, they are simply discarded. is, on the generator obtained by reversing the order of the 64 bits returned. 5 That An experimental exploration of Marsaglia’s xorshift generators, scrambled 5 trixRank, MaxOft and Permutation test of BigCrush, which highlights a significant weakness in its lower bits. We remark that in this paper we do not pursue the search for equidistribution—the property that all tuples of consecutive values, seen as vectors in the unit cube, are evenly distributed, as done, for instance, by Panneton and L’Ecuyer [2005]. Brent [2010] has already argued in detail that for long-period generators equidistribution is not particularly desirable, as it is a property of the whole sequence produced by the generator, and in the case of a long-period generator only a minuscule fraction of the sequence can be actually used. Moreover, equidistribution is currently impossible to evaluate exactly for long-period nonlinear generators, and in the formulation commonly used in the literature it is known to be biased towards the high bits [L’Ecuyer and Panneton 2005]: for instance, the WELL1024a generator has been designed to be maximally equidistributed [Panneton et al. 2006], and indeed it has measure of equidistribution ∆1 = 0, but the generator obtained by reversing its bits has ∆1 = 366: a quite counterintuitive result, as in general we expect all bits to be equally important. Another problem with equidistribution is that it is intrinsically unstable, unless we restrict its usage to the class of linear generators, only. Indeed, if we take a maximally equidistributed sequence, no matter how long, and we flip the most significant bit of a single element of the sequence, the new sequence will have the worst possible ∆1 . For instance, by flipping the most significant bit of a single chosen value out of the output of WELL1024a we can turn its equidistribution measure to ∆1 = 4143. But for any statistical or practical purpose the two sequences are indistinguishable—we are modifying one bit out of 25 (21024 − 1). However, in general this paradoxical behaviour is not a big issue, because the modified sequence can no longer be emitted by a linear generator. We note that since multiplication by an invertible constant induces a permutation of the space of 64-bit values (and thus of t-tuples of such values), it preserves some of the equidistribution properties of the underlying generator (this is true of any bijective scrambling function); more details will be given in the rest of the paper. 4. RESULTS FOR xorshift64 GENERATORS First of all, all generators fail at all seeds the MatrixRank test from TestU01’s SmallCrush suite.6 A score-rank plot7 of the SmallCrush scores for all generators is shown in Figure 2. The plot associates with abscissa x the number of generators with x or more failures. We observe immediately that there is a wide range of quality among the generators examined. The “bumps” in the plot corresponds to new tests failed systematically. A closer inspection would confirm that there is just a weak correlation between scores of algorithms conjugate by reversal, because of the bias of TestU01 towards high bits. We thus report in Table I reports the best four generators by combined scores (i.e., adding the scores of conjugate generators), which are the only ones failing systematically just the MatrixRank test. The table reports also results for the generator A0 (13, 7, 17) suggested by Marsaglia in his original paper, claiming that it “will provide an excellent period 264 − 1 RNG, [. . . ] but any of the above 2200 choices is likely to do as well”. Clearly, this is not the case: A0 (13, 7, 17)/A1 (13, 7, 17) ranks 655 in the combined SmallCrush ranking and fails systematically several tests. 6 Panneton and L’Ecuyer [2005] reports that half of the generators fail this test, but the authors have chosen to use only 32 of the 64 generated bits as output bits, in practice applying a kind of decimation to the output of the generator. 7 Score-rank plots are the numerosity-based discrete analogous of the complementary cumulative distribution function of scores. They give a much clearer picture than frequency dot plots when the data points are scattered and highly variable. 6 S. Vigna 2500 2000 Rank 1500 1000 500 0 100 1000 SmallCrush score 10000 Fig. 2. Score-rank plot of the distribution of SmallCrush scores for the 2200 possible full-period xorshift64 generators. Table I. Best four xorshift64 generators following SmallCrush. Algorithm A2 (11, 31, 18) A2 (8, 29, 19) A0 (8, 29, 19) A0 (11, 31, 18) A0 (13, 7, 17) Failures 111 155 159 130 276 Conjugate A3 (11, 31, 18) A3 (8, 29, 19) A1 (8, 29, 19) A1 (11, 31, 18) A1 (13, 7, 17) Failures 120 115 112 150 802 Overall 231 270 271 280 1078 W 25 35 35 25 25 Table II. The generators of Table I tested with BigCrush. Algorithm A2 (11, 31, 18) A2 (8, 29, 19) A0 (8, 29, 19) A0 (11, 31, 18) A2 (4, 35, 21) A0 (13, 7, 17) Failures 762 747 749 748 961 1049 Conjugate A3 (11, 31, 18) A3 (8, 29, 19) A1 (8, 29, 19) A1 (11, 31, 18) A3 (4, 35, 21) A1 (13, 7, 17) Failures 750 780 884 926 1444 5454 Overall 1512 1527 1633 1674 2405 6503 Sanity check 1. Is the result of our experiments dependent on our seed choice? To answer this question, we repeated our experiments on xorshift64 generators with SmallCrush on a different set of seeds, namely the integers in the interval [1 . . 100]. Kendall’s τ [Kendall 1938; Kendall 1945] between the two rankings is 0.98, which makes it clear that the dependence on the seed is negligible. In particular, the four best conjugate pairs in Table I are the same with both seeds. To gather more information, we ran the full BigCrush suite and Dieharder on our four best generators, on Marsaglia’s choice and on the best choice from “Numerical Recipes”: the results are given in Tables II and III. Even the four best generators fail now systematically the BirthdaySpacings, MatrixRank and LinearComp tests. The first two generators, however, turn out to perform slightly better than other two. We also notice that BigCrush draws a much thicker line between our four best generators and the other ones, which now fail several more tests. Not surprisingly, Dieharder cannot really separate our four best generators from A2 (4, 35, 21)/A3 (4, 35, 21). An experimental exploration of Marsaglia’s xorshift generators, scrambled 7 Table III. The generators of Table I tested with Dieharder. Algorithm A2 (11, 31, 18) A2 (8, 29, 19) A0 (8, 29, 19) A0 (11, 31, 18) A2 (4, 35, 21) A0 (13, 7, 17) Failures 182 179 176 181 189 183 Conjugate A3 (11, 31, 18) A3 (8, 29, 19) A1 (8, 29, 19) A1 (11, 31, 18) A3 (4, 35, 21) A1 (13, 7, 17) Failures 162 181 182 186 187 1352 Overall 344 360 358 367 376 1535 Equidistribution score (combined) 300 250 200 150 100 50 0 0 500 1000 1500 2000 2500 3000 SmallCrush score (combined) Fig. 3. Scatter plot of the combined SmallCrush score of conjugate xorshift64 generators versus the combined equidistribution score. 4.1. Equidistribution It is interesting to compare the ranking provided by equidistribution properties and that provided by statistical tests. Note that a xorshift64 generator is 1-dimensionally equidistributed, that is, every 64-bit value appears exactly once except for zero. We refer to the already quoted paper by Panneton and L’Ecuyer [2005] for a detailed description of the equidistribution statistics ∆1 , the sum of dimension gaps: a lower value is better. A maximally distributed generator has ∆1 = 0, and we will refer to ∆1 as to the equidistribution score. We computed the equidistribution score for all generators using the implementation of Harase’s algorithm [Harase 2011] contained in the MTToolBox package from Saito [2013]. Similarly to SmallCrush scores, ∆1 has high-bits bias, and a quite strong one [L’Ecuyer and Panneton 2005]. For a fair comparison, we to thus combine the ∆1 score of a generator and of its reverse. Figure 3 shows that there is some correlation (τ = 0.58) between combined SmallCrush scores and combined equidistribution scores. Nonetheless, even if equidistribution is able to detect reliably generators with a very bad SmallCrush score, is not so good at detecting the generators with the best score, as is visible from the quite noisy lower left part of the plot. Indeed, when we restrict our attention to the best 30 generators (by combined SmallCrush scores) Kendall’s τ drops to 0.3. The first two generators by combined equidistribution score, A4 (8, 29, 19) and A6 (8, 29, 19), rank 20 (combined score 361) and 170 (score 596) in the combined SmallCrush test. When analyzed with the more powerful lens of BigCrush, they have combined scores 3441 and 4082, respectively, and fail systematically almost twenty additional tests with respect to the top four generators of Table II. Definitely, choosing among xorshift64 generators by equidistribution score alone is not a good idea. 8 S. Vigna Table IV. The three multipliers used in the rest of the paper. The subscripts recalls the t for which they have good figures of merit. M32 = 2685821657736338717 M8 = 1181783497276652981 M2 = 8372773778140471301 5. AN INTRODUCTION TO xorshift64* GENERATORS Since a xorshift64 generator exhibits evident linearity artifacts, the next obvious step is to perturb its output using a nonlinear (in Z/2Z sense) transformation. A natural candidate is multiplication by a constant, also because such operation is very fast in modern processors. Note that the current state of the generator is multiplied by a constant before returning it, but the state itself is not affected by the multiplication: thus, the period is the same. We call such a generator xorshift. By choosing a constant invertible modulo 264 (i.e., odd), we can guarantee that the generator will output a permutation of the sequence output by the underlying xorshift generator. This approach was noted in passing in Marsaglia’s paper, and it is also proposed in a more systematic way in the third edition of “Numerical Recipes” [Press et al. 2007] to create a very fast, good-quality pseudorandom number generator. However, in the latter case the authors first compute allegedly good triples for xorshift using Diehard (with results markedly different from ours, and in strident contrast with TestU01’s results, as discussed in Section 4) and then choose a multiplier. There is no reason why the best triples for a xorshift64 generator (which are computed empirically) should continue to be such in a xorshift64 generator: and indeed, we will see that this is not the case. We thus repeated the experiments of the previous section on xorshift64* generators. To choose scrambling constants, we followed the heuristic considerations of [Press et al. 2007]. We consider primitive (e.g., full-period) elements of the multiplicative group of Z/264 Z: these elements have no fixed point except for zero, which is a very desirable property for a scrambling function. Moreover, we choose from L’Ecuyer [1999] primitive elements that have good qualities as multiplicative congruential linear generators, as we expect that multiplication by such elements will combine bits in a non-trivial way. We use a standard theoretical measure of quality, the figure of merit, which is a normalized best distance between the hyperplanes of families covering tuples of length t given by successive outputs of the generators (see L’Ecuyer [1999] for details). Since t is an additional parameter, to further understand the dependency on the multiplier we used three different multipliers, shown in Table IV, which have good figures of merit for different t’s. The first multiplier, M32 (the one used in [Press et al. 2007]) and the second, M8 , have been taken from L’Ecuyer [1999]. The third, M2 , was kindly provided by Richard Simard. We remark that many other choices for scrambling the output of a generator are possible, like adding or xoring a fixed word, xoring the output with the output of another generator, or using a bijective function with strong avalanching behavior, such as those used in the construction of high-quality hash functions. The three factors we considered in our choice are: speed, good results in statistical test suites, and preservation of some equidistribution properties (similarly to the approach taken in [L’Ecuyer and Granger-Piché 2003]). For instance, xoring with an additive Weyl generator (another suggestion in Marsaglia’s paper) makes it in general impossible to prove any equidistribution property—not even that all 64bit value except for zero are output by the generator. Multiplication by a constant is a very fast operation in modern processor, and mixing linear operations on Z/2Z with operations in the ring Z/264 Z is a standard technique to avoid visible artifacts from either type of algebraic structure. A drawback is that the lowest bit is, in fact, not scrambled, and thus it is identical to the lowest bit of the underlying xorshift generator.8 8 As remarked by one of the referees, since our multipliers are all equal to 1 modulo 4, this is true also of the second-lowest bit. An experimental exploration of Marsaglia’s xorshift generators, scrambled 9 SmallCrush score (reverse) 1000 100 10 1 1 10 100 SmallCrush score (standard) 1000 Fig. 4. Scatter plot of the SmallCrush score of xorshift64* generators and their reverse. 2500 Standard Reverse Combined 2000 Rank 1500 1000 500 0 0 1 10 100 SmallCrush Score 1000 Fig. 5. Score-rank plot of the distribution of SmallCrush scores for the 2200 possible xorshift64* generators with multiplier M32 . 6. RESULTS FOR xorshift64* GENERATORS The scatter plot in Figure 4 shows that there is essentially no correlation between the scores assigned by SmallCrush to a generator and its reverse (τ = 0.15).9 Another interesting observation on Figure 4 is that the lower right half is essentially empty. So bad generators have a bad reverse, but there are good generators with a very bad reverse. This suggests that the quality of a xorshift64* generator can vary wildly from the low to the high bits. A score-rank plot of the SmallCrush scores for all generators shown in Figure 5 provides us with further interesting information: almost all generators have no systematic failure, but only about half of the reverse generators have no systematic failure. Moreover, the distribution of standard generators degrades smoothly, whereas the distribution of reverse generators sports again the “bump” phenomenon we observed in Figure 2. Since we need to reduce the number of candidates to apply stronger tests, in the case of M32 we decided to restrict our choice to generators with 3 overall failed tests or less, which left us with 152 generators. Similar cutoff points were chosen for M8 and M2 . 9 We report plots only for M32 , as the ones for the other multipliers are visually identical. 10 S. Vigna 1000 Dieharder score (reverse) Crush score (reverse) 10000 1000 100 10 1 1 10 100 1000 10000 Crush score (standard) 100 10 1 1 10 100 1000 Dieharder score (standard) Fig. 6. Scatter plots for Crush (left) and Dieharder (right) scores on xorshift64* generators with multiplier M32 and their reverse, for the 152 best generators. Dieharder score 1000 100 10 10 100 1000 Crush score 10000 Fig. 7. A scatter plot of Crush and Diehard combined scores of the 152 SmallCrush-best xorshift64* generators. The plot is in log-log scale to accommodate some very high values returned by Crush on reverse generators. The lower-left “sweet spot” corner contains generators that never fail systematically (not even reversed) in both test suites. These generators were few enough so that we could apply both Crush and Dieharder. Once again, we examine the correlation between the score of a generator and its reverse by means of the scatter plots in Figure 6, which confirm the high-bits bias, albeit less so in the Dieharder case. In Figure 7 we compare instead the two scores (Crush and Dieharder) available. The most remarkable feature is there are no points in the upper left corner: there is no generator that is considered good by Crush but not by Dieharder. On the contrary, Crush heavily penalizes (in particular because of the score on the reverse generator) a large number of generators. The generators we will select in the end all belong to the small cloud in the lower left corner, where the two test suite agree. The score-rank plot in Figure 8 shows that our strategy pays off: we started with 152 generators with less than three failures, but analyzing them with the more powerful lens provided by Crush we get a much more fine-grained analysis: in particular, only 73 of them give no systematic failure, and they all belong to the “sweet spot” of Figure 7, that is, they do not give any systematic failure in Dieharder, too. An experimental exploration of Marsaglia’s xorshift generators, scrambled 11 160 Standard Reverse Combined 140 120 Rank 100 80 60 40 20 0 100 1000 Crush Score Fig. 8. Score-rank plot of the distribution of Crush scores for the 152 SmallCrush-best xorshift64* generators using multiplier M32 . Finally, we selected for each multiplier the eight generators with the best Crush scores, and applied the BigCrush suite: we obtained several generators failing systematically the MatrixRank test only and shown in Table V (which should be compared with Table II). 6.1. Equidistribution Multiplication by an invertible element just permutes the elements of Z/264 Z leaving zero fixed, so a xorshift64* generator, like the underlying xorshift64 generator, is 1dimensionally equidistributed. 7. HIGH DIMENSION Marsaglia [2003] describes a strategy for xorshift generators in high dimension: the idea is to use always three low-dimensional shifts, but locating them in the context of a larger t × t block matrix of the form   0 0 0 · · · 0 (I + La )(I + Rb )  I 0 0 ··· 0  0    0 I 0 ··· 0  0 M =  0  0 0 I ··· 0  · · · · · · · · · · · · · · ·  ··· 0 0 0 ··· I (I + Rc ) Marsaglia notes that even in this restricted form there are matrices of full period (he provides examples for 32-bit shifts up to 160 bits). However, this route has not been explored for high-dimensional (say, more than 1024 bits of state) generators. The only similar approach is that proposed by Brent [2007] with his xorgens generators, which however uses four shifts. The obvious question is thus: is the additional shift really necessary to pass a strong statistical test such as BigCrush? We are thus going to look for good, full-period generators with 1024 or 4096 bits of state using 64-bit basic shifts.10 10 The reason why the number 4096 is relevant here is that we know the factorization of Fermat’s numbers k 22 + 1 only up to k = 11. When more Fermat numbers will be factorized, it will be possible to design xorshift or xorgens generators with larger state space [Brent 2007]. Note that, however, in practice a period of 21024 − 1 is more than sufficient for any purpose. For example, even if 2100 computers were to generate sequences of 2100 numbers starting from random seeds using a generator with period 21024 , the chances that two sequences overlap would be less than 2−724 . 12 S. Vigna Table V. Results of BigCrush on the best eight xorshift64* generators found by SmallCrush and Crush in sequence. The generators fail systematically only MatrixRank. Algorithm A7 (11, 5, 45) A7 (17, 23, 52) A1 (12, 25, 27) A1 (17, 23, 29) A5 (14, 23, 33) A5 (17, 47, 29) A1 (16, 25, 43) A7 (23, 9, 57) A5 (11, 5, 32) A2 (8, 31, 17) A5 (3, 21, 31) A3 (17, 45, 22) A4 (8, 37, 21) A3 (13, 47, 23) A3 (13, 35, 30) A4 (9, 37, 31) A7 (13, 19, 28) A3 (9, 21, 40) A1 (14, 23, 33) A7 (19, 43, 27) A1 (17, 47, 28) A5 (16, 11, 27) A4 (4, 35, 15) A7 (13, 21, 18) Failures S R M32 226 128 232 130 230 133 229 137 238 132 231 141 238 138 242 134 M8 229 122 229 126 230 141 241 133 239 136 232 144 244 136 243 141 M2 228 128 228 132 234 142 239 137 240 137 234 144 230 149 238 144 + W 354 362 363 366 370 372 376 376 23 25 31 21 32 24 31 19 351 355 371 374 375 376 380 384 13 21 33 27 33 27 27 27 356 360 376 376 377 378 379 382 23 35 29 23 25 25 35 31 The output of such generators will be given by the last 64 bits of the state. It is well known [Brent 2004; Niederreiter 1992] that every bit of state satisfies a linear recurrence (defined by the characteristic polynomial) with full period, so a fortiori the last 64 bits have full period, too. Since we already know that some deficiencies of low-dimensional xorshift generators are well corrected by multiplication by a constant, we will follow the same approach, thus looking for good xorshift* generators of high dimension.11 Note that since multiplication by an integer invertible in Z/264 Z is a permutation of Z/264 Z, a high-dimension xorshift* generator has the same period of the underlying xorshift generator. We cannot in principle claim full period if we look at a single bit of the output of a xorshift* generator; but this property can be easily proved by purely combinatorial means: 11 As in the xorshift64 case, different choices for the shifts are possible. We will not pursue them here. An experimental exploration of Marsaglia’s xorshift generators, scrambled 13 Proposition 7.1. Let x0 , x1 , . . . , x2n −2 be a list of 2t -bit values, t ≤ n, such that every value appears 2n−t times, except for 0, which appears 2n−t − 1 times. Then, for every fixed bit k the associated sequence has period 2n − 1. Proof. Suppose that there is a k and a p \| 2n − 1 such that the k-th bit of x0 , x1 , . . . , x2n −2 has period p (that is, the sequence of bits associated with the k-th bit is made by (2n − 1)/p repetitions of the same sequence of p bits). The k-th bit runs through 2n−1 − 1 zeroes and 2n−1 ones (as there is a missing zero in the output sequence). This means that (2n − 1)/p \| 2n−1 , too, as the same number of ones must appear in every repeating subsequence, and since (2n − 1)/p is odd this implies p = 2n − 1. Corollary 7.2. Every bit of the output of a full-period xorshift* generator has full period. 7.1. Finding good shifts The first step is identifying values of a, b and c for which the generator has maximum period using the primitivity check on the characteristic polynomial. We performed these computations using the algebra package Fermat [Lewis 2013], with the restriction that a + b ≤ 64 and that a is coprime with b (see [Brent 2007] for the rationale behind this choices, which significantly reduce the search space). The resulting sets of values are those shown in Table VI and VIII. For a state of 1024 bits, we obtain 20 possible parameter choices, which we examined in combination with our three multipliers both through BigCrush and through Dieharder. The results, reported in Table VI and VII, are excellent: with the exception of two pathological choices, no test is failed systematically. For a state of 4096 bits (Table VIII and IX) there are 10 possible parameter choices, and no generator fails a test systematically. 7.2. Equidistribution Looking at the shape of the matrix defining high-dimensional xorshift generators it is clear that if the state is made of n bits the last n/64 output values, concatenated, are equal to the current state. This implies that such generators are n/64-dimensionally equidistributed (i.e., every n/64-tuple of consecutive 64-bit values appears exactly once, except for a missing tuple of zeroes), so xorshift1024 generators are 16-dimensionally equidistributed and xorshift4096 generators are 64-dimensionally equidistributed. Since multiplication by a constant just permutes the space of tuples, the same is true of the associated xorshift* generators. 8. JUMPING AHEAD The simple form of a xorshift generator makes it trivial to jump ahead quickly by any number of next-state steps. If v is the current state, we want to compute vM j for some j. But M j is always expressible as a polynomial in M of degree lesser than that of the characteristic polynomial. To find such a polynomial it suffices to compute xj mod P (x), where P (x) is the characteristic polynomial of M . Such a computation can be easily carried k out using standard techniques (quadratures to find x2 mod P (x), etc.), leaving us with a polynomial Q(x) such that Q(M ) = M j . Now, if Q(x) = n X αi xi , i=0 we have vM j = vQ(M ) = n X i=0 αi vM i , S. Vigna 14 S 25 28 37 37 23 28 34 32 44 37 45 42 35 38 34 43 45 39 31 50 M32 Failures R + 31 56 32 60 24 61 26 63 40 63 37 65 34 68 36 68 28 72 36 73 29 74 34 76 43 78 40 78 45 79 40 83 39 84 51 90 890 921 902 952 275 79 223 65 167 265 113 155 227 59 89 281 363 77 233 81 255 69 111 99 W 1, 13, 7 3, 26, 35 40, 11, 31 15, 16, 19 22, 7, 48 9, 14, 41 41, 7, 29 31, 11, 30 2, 11, 61 10, 11, 61 7, 16, 55 16, 23, 30 25, 8, 15 27, 13, 46 31, 10, 27 9, 5, 60 31, 33, 37 10, 9, 63 51, 1, 46 47, 1, 41 a, b, c S 28 29 24 30 29 32 25 33 25 42 32 35 25 39 40 40 39 31 60 67 M8 Failures R + 19 47 22 51 33 57 32 62 33 62 30 62 38 63 32 65 41 66 25 67 35 67 34 69 45 70 32 71 32 72 36 76 39 78 49 80 896 956 907 974 113 89 77 255 223 167 265 363 81 155 65 59 281 275 233 227 79 69 111 99 W 3, 26, 35 27, 13, 46 25, 8, 15 31, 10, 27 9, 5, 60 1, 13, 7 15, 16, 19 2, 11, 61 41, 7, 29 9, 14, 41 22, 7, 48 31, 11, 30 7, 16, 55 31, 33, 37 10, 11, 61 16, 23, 30 40, 11, 31 10, 9, 63 51, 1, 46 47, 1, 41 a, b, c S 29 41 38 36 24 28 36 40 36 33 37 45 36 37 41 44 38 48 31 47 M2 Failures R + 24 53 20 61 24 62 31 67 43 67 42 70 34 70 30 70 34 70 37 70 35 72 27 72 39 75 39 76 37 78 37 81 48 86 48 96 799 830 799 846 89 275 281 233 227 113 255 81 265 167 223 363 65 79 155 59 77 69 111 99 W Table VI. Results of BigCrush on the xorshift1024* generators. The last two generators fail systematically CouponCollector, Gap, HammingIndep, MatrixRank, SumCollector and WeightDistrib. a, b, c 27, 13, 46 31, 33, 37 22, 7, 48 7, 16, 55 9, 14, 41 41, 7, 29 1, 13, 7 10, 11, 61 9, 5, 60 16, 23, 30 3, 26, 35 25, 8, 15 31, 11, 30 40, 11, 31 31, 10, 27 2, 11, 61 15, 16, 19 10, 9, 63 51, 1, 46 47, 1, 41 31, 33, 37 31, 11, 30 16, 23, 30 41, 7, 29 9, 14, 41 10, 9, 63 22, 7, 48 51, 1, 46 27, 13, 46 25, 8, 15 3, 26, 35 2, 11, 61 40, 11, 31 31, 10, 27 47, 1, 41 9, 5, 60 10, 11, 61 15, 16, 19 7, 16, 55 1, 13, 7 a, b, c M32 Failures S R + 57 67 124 65 61 126 74 56 130 71 61 132 74 64 138 74 66 140 66 75 141 78 63 141 63 79 142 80 64 144 81 66 147 79 71 150 74 76 150 82 71 153 74 79 153 81 75 156 75 84 159 72 88 160 94 68 162 87 76 163 79 363 59 265 167 69 223 111 275 281 89 81 77 233 99 227 155 255 65 113 W 25, 8, 15 16, 23, 30 7, 16, 55 3, 26, 35 10, 11, 61 31, 10, 27 31, 33, 37 47, 1, 41 27, 13, 46 31, 11, 30 10, 9, 63 41, 7, 29 2, 11, 61 9, 14, 41 40, 11, 31 15, 16, 19 51, 1, 46 22, 7, 48 1, 13, 7 9, 5, 60 a, b, c M8 Failures S R + 67 56 123 77 54 131 66 66 132 60 75 135 63 74 137 74 69 143 86 58 144 82 62 144 78 69 147 85 62 147 65 86 151 84 68 152 88 65 153 77 80 157 82 78 160 85 76 161 92 74 166 90 82 172 79 95 174 97 89 186 281 59 65 89 155 233 79 99 275 363 69 265 81 167 77 255 111 223 113 227 W 22, 7, 48 15, 16, 19 10, 9, 63 51, 1, 46 1, 13, 7 40, 11, 31 2, 11, 61 31, 11, 30 25, 8, 15 10, 11, 61 47, 1, 41 9, 5, 60 16, 23, 30 27, 13, 46 7, 16, 55 9, 14, 41 41, 7, 29 31, 10, 27 3, 26, 35 31, 33, 37 a, b, c M2 Failures S R + 56 76 132 66 67 133 70 71 141 65 78 143 80 64 144 80 67 147 85 65 150 75 75 150 74 77 151 79 76 155 70 86 156 70 86 156 81 76 157 78 80 158 92 70 162 87 80 167 87 81 168 82 87 169 92 79 171 98 88 186 Table VII. Results of Dieharder on xorshift1024* generators. No test is failed systematically. 223 255 69 111 113 77 81 363 281 155 99 227 59 275 65 167 265 233 89 79 W An experimental exploration of Marsaglia’s xorshift generators, scrambled 15 S. Vigna 16 Algorithm 14, 41, 15 5, 22, 27 30, 29, 39 25, 3, 49 7, 12, 59 19, 34, 19 12, 11, 61 5, 27, 21 23, 26, 29 11, 9, 25 241 45 177 441 103 291 195 187 49 567 W 5, 22, 27 5, 27, 21 25, 3, 49 7, 12, 59 11, 9, 25 12, 11, 61 19, 34, 19 14, 41, 15 30, 29, 39 23, 26, 29 Algorithm M8 Failures R + 35 69 35 71 37 72 39 73 34 74 33 74 35 74 34 77 37 79 43 81 S 34 36 35 34 40 41 39 43 42 38 45 187 441 103 567 195 291 241 177 49 W 11, 9, 25 5, 27, 21 25, 3, 49 19, 34, 19 23, 26, 29 30, 29, 39 12, 11, 61 14, 41, 15 7, 12, 59 5, 22, 27 Algorithm Table VIII. Results of BigCrush on xorshift4096* generators. M32 Failures R + 27 60 30 64 32 65 38 68 25 68 36 70 39 71 41 75 42 78 44 79 S 33 34 33 30 43 34 32 34 36 35 M2 Failures R + 33 63 27 64 34 67 36 75 35 75 37 75 37 77 42 78 44 82 50 88 S 30 37 33 39 40 38 40 36 38 38 W 567 187 441 291 49 177 195 241 103 45 25, 3, 49 12, 11, 61 30, 29, 39 5, 22, 27 11, 9, 25 19, 34, 19 14, 41, 15 7, 12, 59 23, 26, 29 5, 27, 21 Algorithm M32 Failures S R + 70 70 140 58 83 141 67 77 144 62 84 146 73 75 148 85 66 151 83 74 157 73 85 158 73 88 161 98 67 165 441 195 177 45 567 291 241 103 49 187 W 25, 3, 49 14, 41, 15 30, 29, 39 11, 9, 25 12, 11, 61 19, 34, 19 5, 22, 27 23, 26, 29 5, 27, 21 7, 12, 59 Algorithm M8 Failures S R + 67 70 137 72 69 141 70 75 145 73 77 150 75 80 155 89 67 156 93 65 158 72 87 159 75 84 159 90 77 167 441 241 177 567 195 291 45 49 187 103 W 19, 34, 19 5, 22, 27 25, 3, 49 5, 27, 21 11, 9, 25 14, 41, 15 23, 26, 29 12, 11, 61 7, 12, 59 30, 29, 39 Algorithm Table IX. Results of Dieharder on xorshift4096* generators. M2 Failures S R + 75 64 139 67 77 144 77 71 148 77 71 148 81 76 157 79 78 157 74 84 158 74 85 159 84 79 163 78 89 167 291 45 441 187 567 241 49 195 103 177 W An experimental exploration of Marsaglia’s xorshift generators, scrambled 17 18 S. Vigna and now vM i is just the i-th state after the current one. If we known in advance the αi ’s, computing vM j requires just computing the next state for n times, accumulating by xor the i-th state iff αi 6= 0.12 In general, one needs to compute the αi ’s for each desired j, but the practical usage of this technique is that of providing subsequences that are guaranteed to be non-overlapping. We can fix a reasonable jump, for example 2512 for a xorshift1024* generator, and store the αi ’s for such a jump as a bit mask. Operating the jump is now entirely trivial, as it requires at most 1024 state changes. In Figure 12 we show the jump function for the generator of Figure 11. By iterating the jump function, one can access 2512 non-overlapping sequences of length 2512 (except for the last one, which will be of length 2512 − 1). 9. COMPARISON How do our best xorshift* generators score with respect to more complex generators in the literature? We decided to perform a comparison with the popular Mersenne Twister MT19937 [Matsumoto and Nishimura 1998],13 with WELL1024a/WELL19937a, two generators introduced by Panneton et al. [2006] as an improvement over the Mersenne Twister, and with xorgens4096, a very recent 4096-bit generator introduced by Brent [2007] we mentioned in Section 7. All these generators are non-cryptographic and aim at fast, high-quality generation. As usual, 100 tests are performed at 100 equispaced points of the state space. We choose generators from the xorshift* family that perform well on both BigCrush and Dieharder, have a good weight score and enough large parameters (which provide faster state change spreading): more precisely, the xorshift64* generator A1 (12, 25, 27) · M32 (Figure 10), xorshift1024* with parameters 31, 11, 30 and multiplier M8 (Figure 11), and xorshift4096* with parameters 25, 3, 49 and multiplier M2 . 9.1. Quality Table X compares the BigCrush scores of the generators we discussed. The results are quite interesting. A simple 64-bit xorshift* generator has less linear artifacts than MT19937, WELL1024a or WELL19937a and, thus, a significantly better score. High-dimension xorgens4096 and xorshift* generators perform significantly better, in spite of being extremely simple, and have no systematic failure. The 64-bit xorshift* generator suggested by “Numerical Recipes” fails systematically the BirthdaySpacings test, contrarily the one we have selected.14 We do not report the results of Dieharder, as at this level of quality the suite is unable to make any significant distinction among the generators. 9.2. Escaping zeroland We show in Figure 9 the speed at which a few of the generators of Table X “escape from zeroland” [Panneton et al. 2006]: purely linearly recurrent generators with a very large state space need a very long time to get from an initial state with a small number of ones to a state in which the ones are approximately half. The figure shows a measure of escape time given by the ratio of ones in a window of 4 consecutive 64-bit values sliding over the first 100 000 generated values, averaged over all possible seeds with exactly one bit set (see [Panneton et al. 2006] for a detailed description). As it is known, MT19937 needs hundreds of thousands of iterations to start behaving correctly. xorshift4096* and xorgens4096 need a few thousand (but xorgens4096 oscillates 12 Brent’s ranut generator [Brent 1992] contains one of the first applications of this technique. precisely, with its 64-bit version. 14 Note that we report the number of failed tests on our 100 seeds. L’Ecuyer and Simard [L’Ecuyer and Simard 2007] report the number of types of failed tests (e.g., failing two distinct RandomWalk tests counts as one) on a single run, so some care must be taken when comparing the results we report and those reported by them. 13 More An experimental exploration of Marsaglia’s xorshift generators, scrambled 19 Table X. A comparison of generators using BigCrush. Algorithm A1 (12, 25, 27) · M32 A3 (4, 35, 21) · M32 xorshift1024* xorshift4096* xorgens4096 MT19937 WELL1024a WELL19937a S 230 240 33 33 42 258 441 235 Failures R 133 223 32 34 40 258 441 233 W/n + 363 463 65 67 82 516 882 468 Systematic 0.48 0.38 0.35 0.11 0.23 0.34 0.40 0.43 MatrixRank MatrixRank, BirthdaySpacings — — — LinearComp MatrixRank, LinearComp LinearComp xorgens4096 MT19937 WELL1024a WELL19937a xorshift64* xorshift1024* xorshift4096* Average number of ones 0.6 0.5 0.4 0.3 0.2 0.1 0 1 10 100 1000 10000 100000 Samples Fig. 9. Convergence to “half of the bits are ones in average” plot. always around 1/2), WELL19937a and xorshift1024* a few hundreds, whereas WELL1024a just a few dozens, and xorshift64* is almost unaffected. Table XI condenses Figure 9 into the mean and standard deviation of the displayed values. Clearly, the multiplication step helps in reducing the correlation between the number of ones in the state and the number of ones in the output values. Also, the slowness in recovering from states with too many zeroes it directly correlated to the size of the state space—a very good argument against linear generators with too large state spaces. 9.3. Speed Finally, we benchmark the generators of Table X. Our tests were run on an Intel R CoreTM i7-4770 CPU @3.40GHz (Haswell), and the results are shown in Table XII (variance is undetectable, as we generate 1010 values in each test). We also report as a strong baseline results about SFMT19937, the SIMD-Oriented Fast Mersenne Twister [Saito and Matsumoto 2008], a 128-bit version of the Mersenne Twister based on the SSE2 extended instruction set of Intel processors (and thus not usable, in principle, on other processors). We used suitable options to keep the compiler from unrolling loops or extracting loop invariants. 20 S. Vigna Table XI. Mean and standard deviation for the data shown in Figure 9. Algorithm xorshift64* xorgens4096 xorshift1024* WELL1024a xorshift4096* WELL19937a MT19937 Mean 0.5000 0.5000 0.5000 0.4999 0.4992 0.4983 0.2823 Standard deviation 0.0039 0.0031 0.0035 0.0036 0.0110 0.0185 0.1705 Table XII. Time to emit a 64-bit integer on an Intel R CoreTM i7-4770 CPU @3.40GHz (Haswell). Algorithm xorshift64* xorshift1024* xorshift4096* xorgens4096 MT19937 (64-bit version) SFMT19937 WELL1024a WELL19937a Speed (ns/64 bits) 1.58 1.36 1.36 2.06 2.84 1.80 10.31 7.45 The highest speed is achieved by the high-dimensional xorshift* generators. SFMT19937 is a major improvement in speed over MT19937, albeit slightly slower than a high-dimensional xorshift* generator; it fails systematically, moreover, the same tests of MT19937. A xorshift64* generator is actually slower than its high-dimensional counterparts. This is not surprising, as the three shift/xors in a xorshift64* generator form a dependency chain and must be executed in sequence, whereas two of the shifts of a higher-dimension generator are independent and can be internally parallelized by the CPU. WELL1024a and WELL19937a are heavily penalized by their 32-bit structure. #include <stdint.h> uint64_t x; uint64_t next(void) { x ^= x >> 12; // a x ^= x << 25; // b x ^= x >> 27; // c return x * UINT64_C(2685821657736338717); } Fig. 10. The suggested xorshift64* generator in C99 code. The variable x should be initialized to a nonzero seed before calling next(). 10. CONCLUSIONS After our careful experimental analysis, we reach the following conclusions: A xorshift1024* generator is an excellent choice for a general-purpose, highspeed generator. The statistical quality of the generator is very high (it has, actually, An experimental exploration of Marsaglia’s xorshift generators, scrambled 21 #include <stdint.h> uint64_t s[16]; int p; uint64_t next(void) { const uint64_t s0 = s[p]; uint64_t s1 = s[p = (p + 1) & 15]; s1 ^= s1 << 31; // a s[p] = s1 ^ s0 ^ (s1 >> 11) ^ (s0 >> 30); // b,c return s[p] * UINT64_C(1181783497276652981); } Fig. 11. The suggested xorshift1024* generator in C99 code. The array s should be initialized to a nonzero seed before calling next(). #include <stdint.h> #include <string.h> void jump(void) { static const uint64_t JUMP[] = { 0x84242f96eca9c41d, 0xa3c65b8776f96855, 0x4489affce4f31a1e, 0x2ffeeb0a48316f40, 0x3659132bb12fea70, 0xaac17d8efa43cab8, 0x5ee975283d71c93b, 0x691548c86c1bd540, 0x0b5fc64563b3e2a8, 0x047f7684e9fc949d, 0x284600e3f30e38c3 }; 0x5b34a39f070b5837, 0xdc2d9891fe68c022, 0xc4cb815590989b13, 0x7910c41d10a1e6a5, 0xb99181f2d8f685ca, uint64_t t[16] = { 0 }; for(int i = 0; i < sizeof JUMP / sizeof JUMP; i++) for(int b = 0; b < 64; b++) { if (JUMP[i] & 1ULL << b) for(int j = 0; j < 16; j++) t[j] ^= s[(j + p) & 15]; next(); } for(int j = 0; j < 16; j++) s[(j + p) & 15] = t[j]; } Fig. 12. The jump function for the xorshift1024 generator of Figure 11 in C99 code. It is equivalent to 2512 calls to next(). the best results in BigCrush), and its period is so large that the probability of overlapping sequences is practically zero, even in the largest parallel simulation (and strictly nonoverlapping sequences can be easily generated using the jump function). Nonetheless, the state space is reasonably small, so that seeding it with high-quality bits is not too expensive, and recovery from states with a large number of zeroes happens quickly. The generator is also blazingly fast (it is actually the fastest generator we tested). The reasonable state space makes it also easier, in case a large number of generators is used at the same time, to fit their state into the cache. In any case, with respect to other generators, the state is accessed 22 S. Vigna in a more localized way, as read and write operations happen at two consecutive locations, and thus will generate at most one cache miss. In case memory is an issue, or array access is expensive, a very good generalpurpose generator is a xorshift64* generator. While the generator A1 (12, 25, 27)·M32 fails systematically the MatrixRank test, it has less linear artifacts than MT19937, WELL1024a or WELL19937a, which fail systematically even more tests. It is a very good choice if memory footprint is an issue and a very large number of generators is necessary. It can also be used, for instance, to generate the initial state of another generator with a larger state space using a 64-bit seed. We remark that a xorshift64* generator can also actually be faster than a xorshift1024* generator if the underlying language incurs significant costs when accessing an array: for instance, in Java a xorshift64* generator emits a value in 1.62 ns, whereas a xorshift1024* generator needs 2.06 ns. Linear generators with an excessively long period have a number of problems that are not compensated by higher statistical quality. WELL19937a is almost four slower than xorshift1024*, and has a worse performance in BigCrush; moreover, recovery from states with many zeroes, albeit enormously improved with respect to MT19937, is still very slow, and seeding properly the generator requires almost twenty thousands random bits. In the end, it is in general difficult to motivate state spaces larger than 21024 . Similar considerations are made by Press et al. [2007] and L’Ecuyer and Panneton [2005]. Surprisingly simple and fast generators can produce sequences that pass strong statistical tests. The code in Figure 11 is extremely shorter and simpler than that of MT19937, WELL1024a or WELL19937a. Yet, it performs significantly better on BigCrush. It is a tribute to Marsaglia’s cleverness that just eight logical operations, one addition and one multiplication by a constant can produce sequences of such high quality. xorgens generators are similar with this respect, but use several more operations due to the additional shift and to combination with a Weyl generator to hide linear artifacts [Brent 2007]. The t for which the multiplier has a good figure of merit has no detectable effect on the quality of the generator. If our tests, we could not find any significant difference between the behavior of generators based on M32 , M8 or M2 . It could be interesting to experiment with multipliers having very bad figures of merit, or more generally with multipliers chosen using different heuristics. Equidistribution is more useful as a design feature than as an evaluation feature. While designing generators around equidistribution might be a good idea, as it leads in general to good generators, evaluation by equidistribution is a more delicate matter because of high-bits bias, instability issues, and failure to detect the generators having the best scores in statistical suites. TestU01 has significantly more resolution than Dieharder as a test suite. In particular in the high-dimension case, TestU01 is able to provide useful information, whereas Dieharder scores flatten down. However, TestU01 (as any other test suite with high-bits bias) must always be applied to the reverse generator, too. REFERENCES Richard P. Brent. 1992. Uniform Random Number Generators for Supercomputers. In Supercomputing, the competitive advantage: proceedings of the Fifth Australian Supercomputing Conference. 5ASC Organising Committee, Melbourne, 95–104. Richard P. Brent. 2004. Note on Marsaglia’s Xorshift Random Number Generators. Journal of Statistical Software 11, 5 (2004), 1–5. 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arXiv:1707.09652v1 [math.GR] 30 Jul 2017 Choosing elements from finite fields Michael Vaughan-Lee November 2012 1 Introduction Graham Higman wrote two immensely important and influential papers on enumerating p-groups in the late 1950s. The papers were entitled Enumerating p-groups I and II, and were published in the Proceedings of the London Mathematical Society in 1960 (see [1] and [2]). In these two papers Higman proved that for any given n, the function f (pn ) enumerating the number of p-groups of order pn is bounded by a polynomial in p, and he formulated his famous PORC conjecture concerning the form of the function f (pn ). He conjectured that for each n there is an integer N (depending on n) such that for p in a fixed residue class modulo N the function f (pn ) is a polynomial in p. For example, for p ≥ 5 the number of groups of order p6 is 3p2 + 39p + 344 + 24 gcd(p − 1, 3) + 11 gcd(p − 1, 4) + 2 gcd(p − 1, 5). (See [3].) So for p ≥ 5, f (p6 ) is one of 8 polynomials in p, with the choice of polynomial depending on the residue class of p modulo 60. The number of groups of order p6 is Polynomial On Residue Classes. As evidence in support of his PORC conjecture Higman proved that, for any given n, the function enumerating the number of p-class 2 groups of order pn is a PORC function of p. He obtained this result as a corollary to a very general theorem about vector spaces acted on by the general linear group. As another corollary to this general theorem, he also proved that for any given n the function enumerating the number of algebras of dimension n over the field of q elements is a PORC function of q. A key step in Higman’s proof of these results is Theorem 2.2.2 from [2]. Theorem 1 (Higman [2]) The number of ways of choosing a finite number of elements from Fqn subject to a finite number of monomial equations and inequalities between them and their conjugates over Fq , considered as a function of q, is PORC. 1 The statement of this theorem probably requires some explanation! Here we are choosing elements x1 , x2 , . . . , xk (say) from the finite field Fqn (where q is a prime power) subject to a finite set of equations and non-equations of the form xn1 1 xn2 2 . . . xnk k = 1 and xn1 1 xn2 2 . . . xnk k 6= 1, where n1 , n2 , . . . , nk are integer polynomials in the Frobenius automorphism x 7→ xq of Fqn . Higman calls these equations and non-equations monomial. For example, suppose we want to choose x1 , x2 ∈ Fqn such that x1 is the root of an irreducible quadratic over Fq and such that x22 is the product of the roots of this quadratic. Then we require that x1 and x2 satisfy 2 −1 x1q −2 = 1, x1q−1 6= 1, xq+1 1 x2 = 1. (1) 2 The equation x1q −1 = 1 guarantees that x1 is the root of a quadratic over Fq , and the non-equation x1q−1 6= 1 guarantees that x1 ∈ / Fq so that the quadratic is irreducible. The other root of the quadratic is then xq1 , so the last equation guarantees that x22 is the product of the roots. To make sure q n −1 q n −1 = 1, x2 = 1. Higman’s that x1 , x2 ∈ Fqn , we also require that x1 theorem is that the function enumerating the number of solutions to (1) in Fqn is a PORC function of q. In this note we give very precise information about the exact form of the PORC functions needed to enumerate the number of solutions to a set of monomial equations. Higman’s proof of his Theorem 2.2.2 involves five pages of homological algebra. A shorter more elementary proof can be found in [4]. The proof in [4] shows that one way to calculate the number of solutions to a set of monomial equations is to write the equations as the rows of a matrix. So we represent the equations 2 −1 x1q n −1 q −2 = 1, xq+1 1 x2 = 1, x1 n −1 = 1, xq2 =1 by the matrix  q2 − 1 0  q+1 −2  .  n  q −1 0  0 qn − 1  For any given value of q the matrix becomes an integer matrix, and it is shown in [4] that the number of solutions is the product of the elementary 2 divisors in the Smith normal form of this integer matrix. To obtain the number of solutions to (1) in Fqn we subtract the number of solutions to the equations n q n −1 −2 x1q−1 = 1, xq+1 = 1, xq2 −1 = 1. 1 x2 = 1, x1 The number of solutions to these equations is just the product of the elementary divisors in the Smith normal form of the matrix   q−1 0  q+1 −2  .  n  q −1 0  0 qn − 1 (In [4] q is assumed to be prime, but the proof is still valid when q is a prime power.) Matrices used in this way to represent a set of monomial equations have entries which are integer polynomials in q. The columns correspond to the unknowns we are solving for, and since there will always be rows (q n − 1, 0, 0, . . . , 0), (0, q n − 1, 0, . . . , 0), . . . , (0, . . . , 0, q n − 1) corresponding to the requirement that the unknowns are elements in Fqn , it follows that the rank of one of these matrices is the number of columns. So the product of the elementary divisors in the Smith normal form of one of these matrices is the greatest common divisor of the k × k minors, where k is the number of columns. These k × k minors are integer polynomials in q, and it is proved in [4] that the greatest common divisor of a set of integer polynomials in q is a PORC function of q. There is some ambiguity about what “the greatest common divisor of a set of polynomials” means here. Suppose we have some integer polynomials f1 (q), f2 (q), . . . , fs (q). For any given value of q these polynomials evaluate to integers, and by “greatest common divisor of the polynomials” we actually mean “greatest common divisor of the values of the polynomials at q”. It is this integer valued function of q which we claim is PORC, and it turns out that we can be quite precise about the form that this PORC function takes. Theorem 2 The greatest common divisor of a set of integer polynomials in q can be expressed in the form df where f is an integer polynomial in q and where r X d=α+ αi gcd(q − ni , mi ) i=1 for some rational numbers α, α1 , α2 , . . . , αr , some integers m1 , m2 , . . . , mr with mi > 1 for all i, and some integers n1 , n2 , . . . , nr with 0 < ni < mi for all i. 3 Corollary 3 The number of ways of choosing a finite number of elements from Fqn subject to a finite number of monomial equations and inequalities between them and their conjugates over Fq can be expressed as a linear combination of terms of the form df , where f and d are as described in Theorem 2. 2 Choosing field elements To make this note self contained, we give a proof here that we can use a matrix to represent a set of monomial equations over a finite field, and that the number of solutions to the equations is the product of the elementary divisors in the Smith normal form of the matrix. So suppose we have a set of monomial equations in unknowns x1 , x2 , . . . , xk , and suppose that we want to find the number of solutions to these equations in the field Fqn . We represent the equations in a matrix A with k columns, with a row (n1 , n2 , . . . , nk ) for each monomial equation xn1 1 xn2 2 . . . xnk k = 1. We also add in rows (q n − 1, 0, 0, . . . , 0), (0, q n − 1, 0, . . . , 0), . . . , (0, 0, . . . , 0, q n − 1) corresponding to the requirement that x1 , x2 , . . . , xk ∈ Fqn . Note that the entries in the matrix A are integer polynomials in q. We now take a particular value for q so that the matrix becomes a matrix with integer entries. Let ω be a primitive element in Fqn , and write xi = ω mi for i = 1, 2, . . . , k , taking the exponents mi as elements in Zqn −1 . Then a row (β1 , β2 , . . . , βk ) in the matrix A corresponds to a relation β1 m1 + β2 m2 + . . . + βk mk = 0 which we require the exponents mi to satisfy. The matrix A can be reduced to Smith normal form over Z by elementary row and column operations. As we apply these operations, the relations encoded in the matrix change. But we show that at each step the number of solutions to the relations stays constant. This is clear for elementary row operations, since an elementary row operation replaces the relations by an equivalent set of relations. So we need to consider the effect of elementary column operations. We can consider the k-tuples (m1 , m2 , . . . , mk ) as elements in the additive group G = Zqn −1 × Zqn −1 × . . . × Zqn −1 . Let A be one of these relation matrices, and let B be the matrix obtained from A after applying an elementary column operation. For each such operation we define an automorphism σ of G with the property that g ∈ G satisfies the relations given by the rows of A if and only if gσ satisfies the relations given by the rows of B. This shows that 4 the number of elements in G satisfying the relations given by A is the same as the number of elements in G satisfying the relations given by B. If the elementary column operation swaps two columns of A then we let σ be the automorphism which swaps the corresponding entries in (m1 , m2 , . . . , mk ), and if the elementary column operation multiplies a column by −1 we let σ be the automorphism which multiplies the corresponding entry in (m1 , m2 , . . . , mk ) by −1. Finally, if the elementary column operation subtracts α times column j from column i, then we let σ be the automorphism which leaves all the entries in (m1 , m2 , . . . , mk ) fixed except for the j-th entry, which it replaces by mj + αmi . The argument above shows that the number of g ∈ G satisfying the original set of relations given by the rows of A is the same as the number of g ∈ G satisfying the relations given by the Smith normal form A. If the elementary divisors in the Smith normal form are d1 , d2 , . . . , dk , then (m1 , m2 , . . . , mk ) is a solution to these equations if and only if d1 m1 = d2 m2 = . . . = dk mk = 0. Provided we can show that di \|q n − 1 for all i, this shows that the number of solutions is d1 d2 . . . dk , as claimed. If A is one of these relation matrices with k columns, then the rows of A are elements in the free Z-module F = Zk . We let R(A) denote the Zsubmodule of F generated by the rows of A. Our claim that di \|q n − 1 for all i amounts to the claim that (q n − 1)F ≤ R(S), where S is the Smith normal form of our initial relation matrix. The Smith normal form is obtained from the initial matrix by a sequence of elementary row and column operations, and we show that (q n − 1)F ≤ R(B) for all the matrices B generated in this sequence. Let A be the starting matrix. Then it contains rows (q n − 1, 0, 0, . . . , 0), (0, q n − 1, 0, . . . , 0), . . . , (0, 0, . . . , 0, q n − 1), so it is clear that (q n −1)F ≤ R(A). Suppose that at some intermediate stage in the reduction of A to Smith normal form we have two matrices B and C, where C is obtained from B by an elementary row operation or an elementary column operation. We assume by induction that (q n − 1)F ≤ R(B), and we show that this implies that (q n − 1)F ≤ R(C). This is clear if C is obtained from B by an elementary row operation, since then R(B) = R(C). So consider the case when C is obtained from B by an elementary column operation. This column operation corresponds to an automorphism σ of F , and if r is a row of B then the corresponding row of C is rσ. So R(C) = 5 R(B)σ, and the fact that (q n −1)F is a characteristic submodule of F implies that (q n − 1)F ≤ R(C). This completes the proof that the number of solutions to the relations given by the rows of the matrix is equal to the product of the elementary divisors in the Smith normal form. As mentioned in the introduction, the product of the elementary divisors in the Smith normal form of an integer matrix with k columns and rank k is the greatest common divisor of the k ×k minors. In the situation we are concerned with, these minors are integer polynomials in q. So the number of solutions to our monomial equations is the greatest common divisor of a set of integer polynomials in q. More precisely, we have a set of integer polynomials in q, and for any given value of q the number of solutions to our monomial equations is the greatest common divisor over Z of the values of these polynomials at q. 3 Proof of Theorem 2 Let f1 (q), f2 (q), . . . , fs (q) be a set of integer polynomials in q. We want to compute the function whose value at q is the greatest common divisor of the integers f1 (q), f2 (q), . . . , fs (q). In this section there is no requirement that q be a prime power, and to make this clear we define a function h : Z → Z by setting h(x) = gcd(f1 (x), f2 (x), . . . , fs (x)) for x ∈ Z. It is the function h we want to compute. As mentioned above, there is some ambiguity about what we mean by “the greatest common divisor of f1 (x), f2 (x), . . . , fs (x)”. We now exploit this ambiguity, and treat x as an indeterminate and treat f1 (x), f2 (x), . . . , fs (x) as elements of the Euclidean domain Q[x]. We can use the Euclidean algorithm to compute the greatest common divisor f (x) of f1 (x), f2 (x), . . . , fs (x) in Q[x] and we can take f (x) to be a primitive polynomial in Z[x]. We then obtain polynomials g1 , g2 , . . . , gs ∈ Q[x] such that f1 g1 + f2 g2 + . . . + fs gs = f. Let m be the least common multiple of the denominators of the coefficients in g1 , g2 , . . . , gs . Then for any given value of x in Z, the greatest common divisor of the integers f1 (x), f2 (x), . . . , fs (x) is df (x) for some d dividing m. Furthermore, as a function of x, the value of d at x depends only on the 6 residue class of x modulo m. We show that we can express d(x) in the form α+ r X αi gcd(x − ni , mi ) i=1 described in the statement of Theorem 2. Furthermore we show that we can take the integers mi to be of the form dmi for some square free divisors di of m with di < m. This shows that h(x) = d(x)f (x) has the form described in Theorem 2. If m = 1 then d = 1 for all x, and we are done. So suppose that m > 1 and let S be the set of prime factors of m. For each subset T ⊂ S let Y dT = p, p∈T and consider the function k : Z → Z defined by X m k(x) = (−1)\|T \| gcd(x, ). dT T ⊂S Clearly the value of k at any given value of x depends only on the residue class of x modulo m. First consider the case when x = m. X Y m 1 k(m) = (−1)\|T \| = m (1 − ) 6= 0. dT p T ⊂S p∈S Next suppose that 1 ≤ x < m. Then there is at least one p ∈ S with the property that the power of p dividing x is less than the power of p dividing m. Pick one such p and let U = S\{p}. Then X m m \|T \| gcd(x, ) − gcd(x, k(x) = (−1) ) = 0, dT pdT T ⊂U since gcd(x, dmT ) = gcd(x, pdmT ) for all T ⊂ U. So if we let c = k(m) then 1 k(x) takes values 0, 0, . . . , 0, 1 as x takes values 1, 2, . . . , m modulo m. It c follows that if 0 < a < m then 1c k(x − a) takes values 0, . . . , 0, 1, 0, . . . , 0 as x takes values 1, 2, . . . , m modulo m (with the 1 in the ath place). So we can express d(x) as a rational linear combination of the functions k(x − a) for 0 ≤ a < m. This implies that we can express d(x) as a rational linear combination of functions of the form gcd(x − ni , mi ) where mi = dmT for some 7 T ⊂ S. If mi = 1 then we can replace gcd(x − ni , mi ) by the constant 1. Also, we can assume 0 ≤ ni < mi . Finally, using the fact that m i −1 X gcd(x − a, mi ) a=0 is a constant function, we can assume that 0 < ni < mi , provided we add a constant term into our expression for d(x). This completes the proof of Theorem 2. Note that the proof shows that we can assume that the denominators of the rational coefficients which appear in the expression for d(x) divide the constant k(m). References [1] G. Higman, Enumerating p-groups. I: Inequalities, Proc. London Math. Soc. (3) 10 (1960), 24–30. [2] G. Higman, Enumerating p-groups. II: Problems whose solution is PORC, Proc. London Math. Soc. (3) 10 (1960), 566–582. [3] M.F. Newman, E.A. O’Brien, and M.R. Vaughan-Lee, Groups and nilpotent Lie rings whose order is the sixth power of a prime, J. Algebra 278 (2004), 383–401. [4] Michael Vaughan-Lee, On Graham Higman’s famous PORC paper, Internat. J. Group Theory 1 (2012), 65–79. 8	4
Deep Layer Aggregation Fisher Yu Dequan Wang Evan Shelhamer Trevor Darrell UC Berkeley arXiv:1707.06484v2 [cs.CV] 4 Jan 2018 Abstract Visual recognition requires rich representations that span levels from low to high, scales from small to large, and resolutions from fine to coarse. Even with the depth of features in a convolutional network, a layer in isolation is not enough: compounding and aggregating these representations improves inference of what and where. Architectural efforts are exploring many dimensions for network backbones, designing deeper or wider architectures, but how to best aggregate layers and blocks across a network deserves further attention. Although skip connections have been incorporated to combine layers, these connections have been “shallow” themselves, and only fuse by simple, one-step operations. We augment standard architectures with deeper aggregation to better fuse information across layers. Our deep layer aggregation structures iteratively and hierarchically merge the feature hierarchy to make networks with better accuracy and fewer parameters. Experiments across architectures and tasks show that deep layer aggregation improves recognition and resolution compared to existing branching and merging schemes. Dense Connections Feature Pyramids + Deep Layer Aggregation Figure 1: Deep layer aggregation unifies semantic and spatial fusion to better capture what and where. Our aggregation architectures encompass and extend densely connected networks and feature pyramid networks with hierarchical and iterative skip connections that deepen the representation and refine resolution. riers, different blocks or modules have been incorporated to balance and temper these quantities, such as bottlenecks for dimensionality reduction [29, 39, 17] or residual, gated, and concatenative connections for feature and gradient propagation [17, 38, 19]. Networks designed according to these schemes have 100+ and even 1000+ layers. Nevertheless, further exploration is needed on how to connect these layers and modules. Layered networks from LeNet [26] through AlexNet [23] to ResNet [17] stack layers and modules in sequence. Layerwise accuracy comparisons [11, 48, 35], transferability analysis [44], and representation visualization [48, 46] show that deeper layers extract more semantic and more global features, but these signs do not prove that the last layer is the ultimate representation for any task. In fact, skip connections have proven effective for classification and regression [19, 4] and more structured tasks [15, 35, 30]. Aggregation, like depth and width, is a critical dimension of architecture. In this work, we investigate how to aggregate layers to better fuse semantic and spatial information for recognition and localization. Extending the “shallow” skip connections of current approaches, our aggregation architectures incor- 1. Introduction Representation learning and transfer learning now permeate computer vision as engines of recognition. The simple fundamentals of compositionality and differentiability give rise to an astonishing variety of deep architectures [23, 39, 37, 16, 47]. The rise of convolutional networks as the backbone of many visual tasks, ready for different purposes with the right task extensions and data [14, 35, 42], has made architecture search a central driver in sustaining progress. The ever-increasing size and scope of networks now directs effort into devising design patterns of modules and connectivity patterns that can be assembled systematically. This has yielded networks that are deeper and wider, but what about more closely connected? More nonlinearity, greater capacity, and larger receptive fields generally improve accuracy but can be problematic for optimization and computation. To overcome these bar1 porate more depth and sharing. We introduce two structures for deep layer aggregation (DLA): iterative deep aggregation (IDA) and hierarchical deep aggregation (HDA). These structures are expressed through an architectural framework, independent of the choice of backbone, for compatibility with current and future networks. IDA focuses on fusing resolutions and scales while HDA focuses on merging features from all modules and channels. IDA follows the base hierarchy to refine resolution and aggregate scale stage-bystage. HDA assembles its own hierarchy of tree-structured connections that cross and merge stages to aggregate different levels of representation. Our schemes can be combined to compound improvements. Our experiments evaluate deep layer aggregation across standard architectures and tasks to extend ResNet [16] and ResNeXt [41] for large-scale image classification, finegrained recognition, semantic segmentation, and boundary detection. Our results show improvements in performance, parameter count, and memory usage over baseline ResNet, ResNeXT, and DenseNet architectures. DLA achieve stateof-the-art results among compact models for classification. Without further architecting, the same networks obtain stateof-the-art results on several fine-grained recognition benchmarks. Recast for structured output by standard techniques, DLA achieves best-in-class accuracy on semantic segmentation of Cityscapes [8] and state-of-the-art boundary detection on PASCAL Boundaries [32]. Deep layer aggregation is a general and effective extension to deep visual architectures. 2. Related Work We review architectures for visual recognition, highlight key architectures for the aggregation of hierarchical features and pyramidal scales, and connect these to our focus on deep aggregation across depths, scales, and resolutions. The accuracy of AlexNet [23] for image classification on ILSVRC [34] signalled the importance of architecture for visual recognition. Deep learning diffused across vision by establishing that networks could serve as backbones, which broadcast improvements not once but with every better architecture, through transfer learning [11, 48] and metaalgorithms for object detection [14] and semantic segmentation [35] that take the base architecture as an argument. In this way GoogLeNet [39] and VGG [39] improved accuracy on a variety of visual problems. Their patterned components prefigured a more systematic approach to architecture. Systematic design has delivered deeper and wider networks such as residual networks (ResNets) [16] and highway networks [38] for depth and ResNeXT [41] and FractalNet [25] for width. While these architectures all contribute their own structural ideas, they incorporated bottlenecks and shortened paths inspired by earlier techniques. Network-innetwork [29] demonstrated channel mixing as a technique to fuse features, control dimensionality, and go deeper. The companion and auxiliary losses of deeply-supervised networks [27] and GoogLeNet [39] showed that it helps to keep learned layers and losses close. For the most part these architectures derive from innovations in connectivity: skipping, gating, branching, and aggregating. Our aggregation architectures are most closely related to leading approaches for fusing feature hierarchies. The key axes of fusion are semantic and spatial. Semantic fusion, or aggregating across channels and depths, improves inference of what. Spatial fusion, or aggregating across resolutions and scales, improves inference of where. Deep layer aggregation can be seen as the union of both forms of fusion. Densely connected networks (DenseNets) [19] are the dominant family of architectures for semantic fusion, designed to better propagate features and losses through skip connections that concatenate all the layers in stages. Our hierarchical deep aggregation shares the same insight on the importance of short paths and re-use, and extends skip connections with trees that cross stages and deeper fusion than concatenation. Densely connected and deeply aggregated networks achieve more accuracy as well as better parameter and memory efficiency. Feature pyramid networks (FPNs) [30] are the dominant family of architectures for spatial fusion, designed to equalize resolution and standardize semantics across the levels of a pyramidal feature hierarchy through top-down and lateral connections. Our iterative deep aggregation likewise raises resolution, but further deepens the representation by nonlinear and progressive fusion. FPN connections are linear and earlier levels are not aggregated more to counter their relative semantic weakness. Pyramidal and deeply aggregated networks are better able to resolve what and where for structured output tasks. 3. Deep Layer Aggregation We define aggregation as the combination of different layers throughout a network. In this work we focus on a family of architectures for the effective aggregation of depths, resolutions, and scales. We call a group of aggregations deep if it is compositional, nonlinear, and the earliest aggregated layer passes through multiple aggregations. As networks can contain many layers and connections, modular design helps counter complexity by grouping and repetition. Layers are grouped into blocks, which are then grouped into stages by their feature resolution. We are concerned with aggregating the blocks and stages. 3.1. Iterative Deep Aggregation Iterative deep aggregation follows the iterated stacking of the backbone architecture. We divide the stacked blocks of the network into stages according to feature resolution. Deeper stages are more semantic but spatially coarser. Skip connections from shallower to deeper stages merge scales Block OUT Existing Proposed Stage Aggregation Node OUT IN OUT IN (a) No aggregation IN (b) Shallow aggregation OUT IN IN (d) Tree-structured aggregation (c) Iterative deep aggregation OUT OUT IN (e) Reentrant aggregation (f) Hierarchical deep aggregation Figure 2: Different approaches to aggregation. (a) composes blocks without aggregation as is the default for classification and regression networks. (b) combines parts of the network with skip connections, as is commonly used for tasks like segmentation and detection, but does so only shallowly by merging earlier parts in a single step each. We propose two deep aggregation architectures: (c) aggregates iteratively by reordering the skip connections of (b) such that the shallowest parts are aggregated the most for further processing and (d) aggregates hierarchically through a tree structure of blocks to better span the feature hierarchy of the network across different depths. (e) and (f) are refinements of (d) that deepen aggregation by routing intermediate aggregations back into the network and improve efficiency by merging successive aggregations at the same depth. Our experiments show the advantages of (c) and (f) for recognition and resolution. and resolutions. However, the skips in existing work, e.g. FCN [35], U-Net [33], and FPN [30], are linear and aggregate the shallowest layers the least, as shown in Figure 2(b). We propose to instead progressively aggregate and deepen the representation with IDA. Aggregation begins at the shallowest, smallest scale and then iteratively merges deeper, larger scales. In this way shallow features are refined as they are propagated through different stages of aggregation. Figure 2(c) shows the structure of IDA. The iterative deep aggregation function I for a series of layers x1 , ..., xn with increasingly deeper and semantic information is formulated as ( x1 if n = 1 I(x1 , ..., xn ) = (1) I(N (x1 , x2 ), ..., xn ) otherwise, where N is the aggregation node. 3.2. Hierarchical Deep Aggregation Hierarchical deep aggregation merges blocks and stages in a tree to preserve and combine feature channels. With HDA shallower and deeper layers are combined to learn richer combinations that span more of the feature hierarchy. While IDA effectively combines stages, it is insufficient for fusing the many blocks of a network, as it is still only sequential. The deep, branching structure of hierarchical aggregation is shown in Figure 2(d). Having established the general structure of HDA we can improve its depth and efficiency. Rather than only routing intermediate aggregations further up the tree, we instead feed the output of an aggregation node back into the backbone as the input to the next sub-tree, as shown in Figure 2(e). This propagates the aggregation of all previous blocks instead of the preceding block alone to better preserve features. For efficiency, we merge aggregation nodes of the same depth (combining the parent and left child), as shown in Figure 2(f). The hierarchical deep aggregation function Tn , with depth n, is formulated as n n Tn (x) = N (Rn−1 (x), Rn−2 (x), ..., R1n (x), Ln1 (x), Ln2 (x)), (2) where N is the aggregation node. R and L are defined as Ln2 (x) = B(Ln1 (x)), Ln1 (x) = B(R1n (x)), ( Tm (x) if m = n − 1 n Rm (x) = n Tm (Rm+1 (x)) otherwise, where B represents a convolutional block. 3.3. Architectural Elements Aggregation Nodes The main function of an aggregation node is to combine and compress their inputs. The nodes learn to select and project important information to maintain Hierarchical Deep Aggregation Iterative Deep Aggregation Aggregation Node Downsample 2x Conv Block OUT IN Figure 3: Deep layer aggregation learns to better extract the full spectrum of semantic and spatial information from a network. Iterative connections join neighboring stages to progressively deepen and spatially refine the representation. Hierarchical connections cross stages with trees that span the spectrum of layers to better propagate features and gradients. the same dimension at their output as a single input. In our architectures IDA nodes are always binary, while HDA nodes have a variable number of arguments depending on the depth of the tree. Although an aggregation node can be based on any block or layer, for simplicity and efficiency we choose a single convolution followed by batch normalization and a nonlinearity. This avoids overhead for aggregation structures. In image classification networks, all the nodes use 1×1 convolution. In semantic segmentation, we add a further level of iterative deep aggregation to interpolate features, and in this case use 3×3 convolution. As residual connections are important for assembling very deep networks, we can also include residual connections in our aggregation nodes. However, it is not immediately clear that they are necessary for aggregation. With HDA, the shortest path from any block to the root is at most the depth of the hierarchy, so diminishing or exploding gradients may not appear along the aggregation paths. In our experiments, we find that residual connection in node can help HDA when the deepest hierarchy has 4 levels or more, while it may hurt for networks with smaller hierarchy. Our base aggregation, i.e. N in Equation 1 and 2, is defined by: different backbones. Our architectures make no requirements of the internal structure of the blocks and stages. The networks we instantiate in our experiments make use of three types of residual blocks [17, 41]. Basic blocks combine stacked convolutions with an identity skip connection. Bottleneck blocks regularize the convolutional stack by reducing dimensionality through a 1×1 convolution. Split blocks diversify features by grouping channels into a number of separate paths (called the cardinality of the split). In this work, we reduce the ratio between the number of output and intermediate channels by half for both bottleneck and split blocks, and the cardinality of our split blocks is 32. Refer to the cited papers for the exact details of these blocks. 4. Applications We now design networks with deep layer aggregation for visual recognition tasks. To study the contribution of the aggregated representation, we focus on linear prediction without further machinery. Our results do without ensembles for recognition and context modeling or dilation for resolution. Aggregation of semantic and spatial information matters for classification and dense prediction alike. 4.1. Classification Networks N (x1 , ..., xn ) = σ(BatchNorm( X Wi xi + b)), (3) i where σ is the non-linear activation, and wi and b are the weights in the convolution. If residual connections are added, the equation becomes N (x1 , ..., xn ) = σ(BatchNorm( X Wi xi + b) + xn ). (4) i Note that the order of arguments for N does matter and should follow Equation 2. Blocks and Stages Deep layer aggregation is a general architecture family in the sense that it is compatible with Our classification networks augment ResNet and ResNeXT with IDA and HDA. These are staged networks, which group blocks by spatial resolution, with residual connections within each block. The end of every stage halves resolution, giving six stages in total, with the first stage maintaining the input resolution while the last stage is 32× downsampled. The final feature maps are collapsed by global average pooling then linearly scored. The classification is predicted as the softmax over the scores. We connect across stages with IDA and within and across stages by HDA. These types of aggregation are easily combined by sharing aggregation nodes. In this case, we only need to change the root node at each hierarchy by combin- Iterative Deep Aggregation 2s OUT Aggregation Node Upsample 2x 2s 4s 4s 8s Stage 2s IN 2s 2s 4s 8s 16s 4s 8s 16s 32s Figure 4: Interpolation by iterative deep aggregation. Stages are fused from shallow to deep to make a progressively deeper and higher resolution decoder. ing Equation 1 and 2. Our stages are downsampled by max pooling with size 2 and stride 2. The earliest stages have their own structure. As in DRN [46], we replace max pooling in stages 1–2 with strided convolution. The stage 1 is composed of a 7×7 convolution followed by a basic block. The stage 2 is only a basic block. For all other stages, we make use of combined IDA and HDA on the backbone blocks and stages. For a direct comparison of layers and parameters in different networks, we build networks with a comparable number of layers as ResNet-34, ResNet-50 and ResNet-101. (The exact depth varies as to keep our novel hierarchical structure intact.) To further illustrate the efficiency of DLA for condensing the representation, we make compact networks with fewer parameters. Table 1 lists our networks and Figure 3 shows a DLA architecture with HDA and IDA. 4.2. Dense Prediction Networks Semantic segmentation, contour detection, and other image-to-image tasks can exploit the aggregation to fuse local and global information. The conversion from classification DLA to fully convolutional DLA is simple and no different than for other architectures. We make use of interpolation and a further augmentation with IDA to reach the necessary output resolution for a task. IDA for interpolation increases both depth and resolution by projection and upsampling as in Figure 4. All the projection and upsampling parameters are learned jointly during the optimization of the network. The upsampling steps are initialized to bilinear interpolation and can then be learned as in [35]. We first project the outputs of stages 3–6 to 32 channels and then interpolate the stages to the same resolution as stage 2. Finally, we iteratively aggregate these stages to learn a deep fusion of low and high level features. While having the same purpose as FCN skip connections [35], hypercolumn features [15], and FPN top-down connections [30], our aggregation differs in approach by going from shallow-todeep to further refine features. Note that we use IDA twice in this case: once to connect stages in the backbone network and again to recover resolution. 5. Results We evaluate our deep layer aggregation networks on a variety of tasks: image classification on ILSVRC, several kinds of fine-grained recognition, and dense prediction for semantic segmentation and contour detection. After establishing our classification architecture, we transfer these networks to the other tasks with little to no modification. DLA improves on or rivals the results of special-purpose networks. 5.1. ImageNet Classification We first train our networks on the ImageNet 2012 training set [34]. Similar to ResNet [16], training is performed by SGD for 120 epochs with momentum 0.9, weight decay 10−4 and batch size 256. We start the training with learning rate 0.1, which is reduced by 10 every 30 epochs. We use scale and aspect ratio augmentation [41], but not color perturbation. For fair comparison, we also train the ResNet models with the same training procedure. This leads to slight improvements over the original results. We evaluate the performance of trained models on the ImageNet 2012 validation set. The images are resized so that the shorter side has 256 pixels. Then central 224×224 crops are extracted from the images and fed into networks to measure prediction accuracy. DLA vs. ResNet compares DLA networks to ResNets with similar numbers of layers and the same convolutional blocks as shown in Figure 5. We find that DLA networks can achieve better performance with fewer parameters. DLA-34 and ResNet-34 both use basic blocks, but DLA-34 has about 30% fewer parameters and ∼ 1 point of improvement in top-1 error rate. We usually expect diminishing returns of performance of deeper networks. However, our results show that compared to ResNet-50 and ResNet-101, DLA networks can still outperform the baselines significantly with fewer parameters. DLA vs. ResNeXt shows that DLA is flexible enough to use different convolutional blocks and still have advantage in accuracy and parameter efficiency as shown in Figure 5. Our models based on the split blocks have much fewer parameters but they still have similar performance with ResNeXt models. For example, DLA-X-102 has nearly the half number of parameters compared to ResNeXt-101, but the error rate difference is only 0.2%. DLA vs. DenseNet compares DLA with the dominant architecture for semantic fusion and feature re-use. DenseNets are composed of dense blocks that aggregate all of their layers by concatenation and transition blocks that reduce dimensionality for tractability. While these networks can Name DLA-34 DLA-48-C DLA-60 DLA-102 DLA-169 DLA-X-48-C DLA-X-60-C DLA-X-60 DLA-X-102 Block Basic Bottleneck Bottleneck Bottleneck Bottleneck Split Split Split Split Stage 1 16 16 16 16 16 16 16 16 16 Stage 2 32 32 32 32 32 32 32 32 32 Stage 3 1-64 1-64 1-128 1-128 2-128 1-64 1-64 1-128 1-128 Stage 4 2-128 2-64 2-256 3-256 3-256 2-64 2-64 2-256 3-256 Stage 5 2-256 2-128 3-512 4-512 5-512 2-128 3-128 3-512 4-512 Stage 6 1-512 1-256 1-1024 1-1024 1-1024 1-256 1-256 1-1024 1-1024 Table 1: Deep layer aggregation networks for classification. Stages 1 and 2 show the number of channels n while further stages show d-n where d is the aggregation depth. Models marked with “-C” are compact and only have ∼1 million parameters. aggressively reduce depth and parameter count by feature reuse, concatenation is a memory-intensive fusion operation. DLA achieves higher accuracy with lower memory usage because the aggregation node fan-in size is log of the total number of convolutional blocks in HDA. Compact models have received a lot of attention due to the limited capabilities of consumer hardware for running convolutional networks. We design parameter constrained DLA networks to study how efficiently DLA can aggregate and re-use features. We compare to SqueezeNet [20], which shares a block design similar to our own. Table 2 shows that DLA is more accurate with the same number of parameters. Furthermore DLA is more computationally efficient by operation count. SqueezNet-A SqueezNet-B DLA-46-C DLA-46-C DLA-X-60-C Top-1 42.5 39.6 36.8 34.0 32.5 Top-5 19.7 17.5 15.0 13.7 12.0 Params 1.2M 1.2M 1.3M 1.1M 1.3M FMAs 1.70B 0.72B 0.58B 0.53B 0.59B Table 2: Comparison with compact models. DLA is more accurate at the same number of parameters while inference takes fewer operations (counted by fused multiply-adds). 5.2. Fine-grained Recognition We use the same training procedure for all of fine-grained experiments. The training is performed by SGD with a minibatch size of 64, while the learning rate starts from 0.01 and is then divided by 10 every 50 epochs, for 110 epochs in total. The other hyperparameters are fixed to their settings for ImageNet classification. In order to mitigate over-fitting, we carry out the following data augmentation: Inception-style Bird Car Plane Food ILSVRC #Class 200 196 102 101 1000 #Train (per class) 5994 (30) 8144 (42) 6667 (67) 75750 (750) 1,281,167 (1281) #Test (per class) 5794 (29) 8041 (41) 3333 (33) 25250 (250) 100,000 (100) Table 3: Statistics for fine-grained recognition datasets. Compared to generic, large-scale classification, these tasks contain more specific classes with fewer training instances. scale and aspect ratio variation [39], AlexNet-style PCA color noise[23], and the photometric distortions of [18]. We evaluate our models on various fine-grained recognition datasets: Bird (CUB) [40], Car [22], Plane [31], and Food [5]. The statistics of these datasets can be found in Table 3, while results are shown in Figure 6. For fair comparison, we follow the experimental setup of [9]: we randomly crop 224×224 in resized 256×256 images for [5] and 448×448 in resized 512×512 for the rest of datasets, while keeping 224×224 input size for original VGGNet. Our results improve or rival the state-of-the-art without further annotations or specific modules for fine-grained recognition. In particular, we establish new state-of-the-arts results on Car, Plane, and Food datasets. Furthermore, our models are competitive while having only several million parameters. However, our results are not better than stateof-the-art on Birds, although note that this dataset has fewer instances per class so further regularization might help. 5.3. Semantic Segmentation We report experiments for urban scene understanding on CamVid [6] and Cityscapes [8]. Cityscapes is a largerscale, more challenging dataset for comparison with other methods while CamVid is more convenient for examining ResNet ResNeXt DenseNet DLA DLA-X 28 Top-1 Error Rate 27 26 25 24 23 22 21 20 20 40 60 # Params (Million) 80 2 4 6 8 10 12 # Multi-Add (Billion) 14 16 Figure 5: Evaluation of DLA on ILSVRC. DLA/DLA-X have ResNet/ResNeXT backbones respectively. DLA achieves the highest accuracies with fewer parameters and fewer computation. Bird Car 90 86 84 .3 85 .2 70 84 .7 81 .6 78 .4 80 75 92 9 91 2.4 .2 95 84 85.085.1 .4 82 .7 80 .5 .7 88 91 Plane 94 93 94 .0 .9 .1 .5 .0 .6 86 .9 84 84 .1 .7 85 .7 88 .7 84 .2 8 81 2.4 .2 81 .6 79 .2 79 .8 85 .5 89 90. 89 .6 0 .7 87 . .7 8 86 8 82 3.2 .1 74 73 .1 .1 VGGNet Food 9 9 91 92.4 2.9 2.6 .6 92 ResNet Baseline DLA VGGNet Compact ResNet Kernel DLA DLA-X-60-C VGGNet ResNet DLA-34 DLA DLA-60 VGGNet DLA-X-60 ResNet DLA DLA-102 Figure 6: Comparison with state-of-the-art methods on fine-grained datasets. Image classification accuracy on Bird [40], Car [22], Plane [31], and Food [5]. Higher is better. P is the number of parameters in each model. For fair comparison, we calculate the number of parameters for 1000-way classification. V- and R- indicate the base model as VGGNet-16 and ResNet-50, respectively. The numbers of Baseline, Compact [13] and Kernel [9] are directly cited from [9]. ablations. We use the standard mean intersection-over-union (IoU) score [12] as the evaluation metric for both datasets. Our networks are trained only on the training set without the usage of validation or other further data. CamVid has 367 training images, 100 validation images, and 233 test images with 11 semantic categories. We start the training with learning rate 0.01 and divide it by 10 after 800 epochs. The results are shown in Table 8. We find that models with downsampling rate 2 consistenly outperforms those downsampling by 8. We also try to augment the data by randomly rotating the images between [-10, 10] degrees and randomly scaling the images between 0.5 and 2. The final results are significantly better than prior methods. Cityscapes has 2, 975 training images, 500 validation images, and 1, 525 test images with 19 semantic categories. Following previous works [49], we adopt the poly learnepoch−1 0.9 ing rate (1 − total with momentum 0.9 and train the epoch ) model for 500 epochs with batch size 16. The starting learning rate is 0.01 and the crop size is chosen to be 864. We also augment the data by randomly rotating within 10 degrees and scaling between 0.5 and 2. The validation results are shown in 9. Surprisingly, DLA-34 performs very well on this dataset and it is as accurate as DLA-102. It should be noted that fine spatial details do not contribute much for this choice of metric. RefineNet [28] is the strongest network in the same class of methods without the computational costs of additional data, dilation, and graphical models. To make a fair comparison, we evaluate in the same multi-scale fashion as that approach with image scales of [0.5, 0.75, 1, 1.25, 1.5] and sum the predictions. DLA improves by 2+ points. 5.4. Boundary Detection Boundary detection is an exacting task of localization. Although as a classification problem it is only a binary task of whether or not a boundary exists, the metrics require precise spatial accuracy. We evaluate on classic BSDS [1] with multiple human boundary annotations and PASCAL boundaries [32] with boundaries defined by instances masks of select semantic classes. The metrics are accuracies at different thresholds, the optimal dataset scale (ODS) and Method DLA-34 8s DLA-34 2s DLA-102 2s Split Val FCN-8s [35] RefineNet-101 [28] DLA-102 DLA-169 Test mIoU 73.4 74.5 74.4 Method SegNet [2] DeepLab-LFOV [7] Dilation8 [45] FSO [24] DLA-34 8s DLA-34 2s DLA-102 2s 65.3 73.6 75.3 75.9 Table 4: Evaluation on Cityscapes to compare strides on validation and to compare against existing methods on test. DLA is the best-in-class among methods in the same setting. 1 Precision Table 5: Evaluation on CamVid. Higher depth and resolution help. DLA is state-of-the-art. Method SE [10] HED [43] DLA-34 2s DLA-102 2s 0.9 0.8 [F=.80] Human [F=.80] DLA-102 [F=.79] DLA-34 [F=.79] HED [F=.77] DLA-34 4s [F=.76] DLA-34 8s 0.7 0.6 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mIoU 46.4 61.6 65.3 66.1 66.7 68.5 71.0 DSBD [32] M-DSBD [32] DLA-34 2s DLA-102 2s Train ODS OIS AP BSDS 0.541 0.553 0.642 0.648 0.570 0.585 0.668 0.674 0.486 0.518 0.624 0.623 PASCAL 0.643 0.652 0.743 0.754 0.663 0.678 0.757 0.766 0.650 0.674 0.763 0.752 Recall Figure 7: Precision-recall evaluation on BSDS. DLA is the closest to human performance. Method ODS OIS AP SE [10] DeepEdge [3] DeepContour [36] HED [42] CEDN [43]† UberNet [21] (1-Task)† 0.746 0.753 0.756 0.788 0.788 0.791 0.767 0.772 0.773 0.808 0.804 0.809 0.803 0.807 0.797 0.840 0.821 0.849 DLA-34 8s DLA-34 4s DLA-34 2s DLA-102 2s 0.760 0.767 0.794 0.803 0.772 0.778 0.808 0.813 0.739 0.751 0.787 0.781 Table 6: Evaluation on BSDS († indicates outside data). ODS and OIS are state-of-the-art, but AP suffers due to recall. See Figure 7. more lenient optimal image scale (OIS), as well as average precision (AP). Results are shown in for BSDS in Table 6 and the precision-recall plot of Figure 7 and for PASCAL boundaries in Table 7. To address this task we follow the training procedure of HED [42]. In line with other deep learning methods, we Table 7: Evaluation on PASCAL Boundaries. DLA is stateof-the-art. take the consensus of human annotations on BSDS and only supervise our network with boundaries that three or more annotators agree on. Following [43], we give the boundary labels 10 times weight of the others. For inference we simply run the net forward, and do not make use of ensembles or multi-scale testing. Assessing the role of resolution by comparing strides of 8, 4, and 2 we find that high output resolution is critical for accurate boundary detection. We also find that deeper networks does not continue improving the prediction performance on BSDS. On both BSDS and PASCAL boundaries we achieve state-of-the-art ODS and OIS scores. In contrast the AP on BSDS is surprisingly low, so to understand why we plot the precision-recall curve in Figure 7. Our approach has lower recall, but this is explained by the consensus ground truth not covering all of the individual, noisy boundaries. At the same time it is the closest to human performance. On the other hand we achieve state-of-the-art AP on PASCAL boundaries since it has a single, consistent notion of boundaries. When training on BSDS and transferring to PASCAL boundaries the improvement is minor, but training on PASCAL boundaries itself with ∼ 10× the data delivers more than 10% relative improvement over competing methods. 6. Conclusion Aggregation is a decisive aspect of architecture, and as the number of modules multiply their connectivity is made all the more important. 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It is only used when there are more than 100 layers in the classification network. Three types of convolutional blocks are studied in this paper, as shown in Figure 9, since they are widely used in deep learning literature. Because the convolutional blocks will be combined with additional linear projection in the aggregation nodes, we reduce the ratio of bottleneck blocks from 4 to 2. We compare DLA and DLA-X to other networks in Figure 5 in the submitted paper in terms of network parameters and classification accuracy. DLA includes the networks using residual blocks in ResNet and DLA-X includes those using the block in ResNeXt. For fair comparison, we design DLA and DLA-X networks with similar depth and channels with their counterparts. The ResNet models are ResNet-34, ResNet-50, ResNet-101 and ResNet-152. The corresponding DLA model depths are 34, 60, 102, 169. The ResNeXt models are ResNeXt-50 (32x4d), ResNeXt-101 (32x4d), and ResNeXt-101 (64x4d). The corresponding DLA-X model depths are 60 and 102, while the third DLA-X model double the number of 3 × 3 bottleneck channels, similar to ResNeXt-101 (64x4d). B. Semantic Segmentation We report experiments for semantic segmentation on CamVid and Cityscapes. Table 8 shows a breakdown of the accuracies for the categories. We also add data agumention in the CamVid training, as shown in the third group of Table 8. It includes random rotating the images between -10 and 10 degrees and random scaling between 0.5 and 2. We find the results can be further improved by the augmentation. Table 9 shows the breakdown of categories in Cityscapes on the validation set. We also test the models on multiple scales of the images. This testing procedure is used in evaluating the models on the testing images in the previous works. ReLU BN (b) Plain 3-node Conv Concat ReLU BN Conv Concat 3-Node Conv Block Conv Block 2-Node Conv Block Conv Block (a) 3-level hierarchical aggregation (c) Residual 3-node Figure 8: Illustration of aggregation node architectures. n, 1 x 1, n/2 n/2, 3 x 3, n/2 n/2, 1 x 1, n n, 3 x 3, n n, 3 x 3, n (a) Basic (b) Bottleneck n/32, 1 x 1, n n/32, 3 x 3, n/32 n, 1 x 1, n/32 32 Paths 32 Paths 32 Paths n/32, 1 x 1, n n/32, 3 x 3, n/32 n, 1 x 1, n/32 (c) Split . Tree Sky Car Sign Road Pedestrian Fence Pole Sidewalk Bicyclist mean IoU DLA-60 DLA-102 DLA-169 Building DLA-34 8s DLA-60 8s DLA-34 DLA-60 DLA-102 Data Aug Figure 9: Convolutional blocks used in this paper. Our aggregation architecture is as general as stacking layers, so we can use the building blocks of existing networks. The layer labels indicate output channels, kernel size and input channels. (a) and (b) are derived from [16] and (c) from [41]. No 83.2 83.0 83.2 84.4 84.9 77.2 77.0 76.4 77.7 78.0 91.2 91.4 92.5 92.6 92.5 83.6 84.1 84.6 87.1 86.4 48.8 46.9 52.1 51.4 50.8 94.3 94.1 94.4 95.3 94.9 58.6 58.3 61.5 62.2 62.8 32.0 32.8 29.4 32.1 45.4 27.8 26.0 35.1 36.2 35.7 81.1 81.3 82.0 84.5 83.7 55.4 56.8 57.8 64.1 65.8 66.7 66.5 68.1 69.8 71.0 Yes 86.6 86.6 86.9 79.3 78.8 78.9 92.5 92.2 92.5 90.9 90.3 89.9 55.3 57.9 58.5 96.2 96.5 96.5 65.5 66.7 66.1 48.6 49.6 55.4 37.4 38.7 39.0 86.9 87.9 87.7 66.5 66.7 67.7 73.2 73.8 74.4 Road Sidewalk Building Wall Fence Pole Light Sign Vegetation Terrain Sky Person Rider Car Truck Bus Train Motorcycle Bicycle mean IoU Table 8: Semantic segmentation results on the CamVid dataset. DLA-34 8s DLA-34 DLA-102 DLA-169 97.9 98.0 98.0 98.2 83.2 83.5 84.3 84.8 91.9 92.1 92.3 92.5 47.7 51.0 43.2 45.9 57.7 56.8 56.9 60.0 62.4 64.9 67.2 68.0 68.6 69.6 71.6 72.3 77.3 78.5 80.9 81.1 92.2 92.4 92.5 92.7 60.4 62.9 61.4 61.9 94.8 95.1 94.6 95.1 81.1 81.5 82.7 83.4 59.8 59.6 61.5 63.3 94.1 94.5 94.5 95.3 57.5 59.0 60.3 70.9 76.6 78.4 77.7 80.8 54.2 57.8 53.8 48.1 59.7 62.9 62.2 65.4 76.6 76.9 78.5 79.1 73.4 74.5 74.4 75.7 DLA-34 MS DLA-102 MS DLA-169 MS 98.2 84.7 92.5 54.3 59.5 65.9 71.1 79.5 92.7 64.1 95.3 82.6 61.8 94.7 63.3 83.7 64.6 64.2 77.6 98.5 85.0 92.5 47.1 56.7 66.9 74.4 78.6 93.6 71.7 95.1 85.8 67.4 95.3 55.8 63.5 57.8 68.1 76.1 98.3 85.9 92.8 48.3 61.2 69.0 73.4 82.2 92.9 63.1 95.4 84.2 65.1 95.7 76.3 82.9 49.6 68.5 80.2 76.3 76.1 77.1 Table 9: Performance of DLA on the Cityscapes validation set. s8 indicates the input image is downsampled by 8 in the model output. It is 2 by default. Lower downsampling rate usually leads to higher accuracy. “MS” indicates the models are tested on on multiple scales of the input images.	1
MNRAS 000, 1–13 (2017) Preprint 15 November 2017 Compiled using MNRAS LATEX style file v3.0 Uncertainty quantification for radio interferometric imaging: II. MAP estimation Xiaohao Cai1? , Marcelo Pereyra2? and Jason D. McEwen1? 1 Mullard Space Science Laboratory, University College London (UCL), Surrey RH5 6NT, United Kingdom Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom 2 Maxwell arXiv:1711.04819v1 [astro-ph.IM] 13 Nov 2017 Accepted —. Received —; in original form — ABSTRACT Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Statistical sampling approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from which uncertainties can then be quantified. However, for massive data sizes, like those anticipated from the Square Kilometre Array (SKA), it will be difficult if not impossible to apply any MCMC technique due to its inherent computational cost. We formulate Bayesian inference problems with sparsity-promoting priors (motivated by compressive sensing), for which we recover maximum a posteriori (MAP) point estimators of radio interferometric images by convex optimisation. Exploiting recent developments in the theory of probability concentration, we quantify uncertainties by post-processing the recovered MAP estimate. Three strategies to quantify uncertainties are developed: (i) highest posterior density credible regions; (ii) local credible intervals (cf. error bars) for individual pixels and superpixels; and (iii) hypothesis testing of image structure. These forms of uncertainty quantification provide rich information for analysing radio interferometric observations in a statistically robust manner. Our MAP-based methods are approximately 105 times faster computationally than state-of-the-art MCMC methods and, in addition, support highly distributed and parallelised algorithmic structures. For the first time, our MAP-based techniques provide a means of quantifying uncertainties for radio interferometric imaging for realistic data volumes and practical use, and scale to the emerging big-data era of radio astronomy. Key words: techniques: image processing – techniques: interferometric – methods: data analysis – methods: numerical – methods: statistical. 1 INTRODUCTION Radio interferometric (RI) telescopes provide observations of the radio emission of the sky with high angular resolution and sensitivity, and provide a wealth of valuable information for astrophysics and cosmology (Ryle & Vonberg 1946; Ryle & Hewish 1960; Thompson et al. 2008). Radio interferometers essentially acquire Fourier measurements of the sky image of interest. Imaging observations made by radio interferometers thus requires solving an ill-posed linear inverse problem (Thompson et al. 2008), which is an important first step in many subsequent scientific analyses. Since the inverse problem is ill-posed (sometimes seriously), uncertainty information (e.g. error estimates) regarding reconstructed images is critical. Nevertheless, uncertainty information is currently lacking in all RI imaging techniques used in practice. In Cai et al. (2017), ? E-mail: x.cai@ucl.ac.uk (XC); m.pereyra@hw.ac.uk (MP); jason.mcewen@ucl.ac.uk (JDM) c 2017 The Authors the first of these companion articles, we propose uncertainty quantification strategies for RI imaging based on state-of-the-art Markov chain Monte Carlo (MCMC) methods that sample the full posterior distribution of the image, with the sparsity-promoting priors that have been shown in practice to be highly effective (e.g. Pratley et al. 2016). Excellent results were achieved and a variety of different uncertainty quantification strategies were presented. However, it is difficult to scale these strategies to big-data due to their high computational overhead. We address this issue in the current article. Over the coming decades radio astronomy will transition into the so-called big-data era. Generally speaking, the new generation of radio telescopes, such as the LOw Frequency ARray (LOFAR1 ), the Extended Very Large Array (EVLA2 ), the Australian 1 2 http://www.lofar.org http://www.aoc.nrao.edu/evla 2 Cai et al. Square Kilometre Array Pathfinder (ASKAP3 ), and the Murchison Widefield Array (MWA4 ), will achieve much higher dynamic range and angular resolution than previous instruments and will acquire very large volumes of data. The Square Kilometer Array (SKA5 ) will provide a considerable step again in dynamic range (six or seven orders of magnitude beyond prior telescopes) and angular resolution, and will acquire massive volumes of data, ushering in the big-data era of radio astronomy. This emerging era of big-data, inevitably, will bring further challenges and so uncertainty quantification will be increasingly important. As discussed in Cai et al. (2017), existing image reconstruction techniques, such as CLEAN-based methods (Högbom 1974; Bhatnagar & Corwnell 2004; Cornwell 2008; Stewart et al. 2011), the maximum entropy method (MEM) (Ables 1974; Gull & Daniell 1978; Cornwell & Evans 1985), and compressed sensing (CS) methods (Wiaux et al. 2009a,b; McEwen & Wiaux 2011; Li et al. 2011a,b; Carrillo et al. 2012, 2014; Wolz et al. 2013; Dabbech et al. 2015; Dabbech et al. 2017; Garsden et al. 2015; Onose et al. 2016, 2017; Pratley et al. 2016; Kartik et al. 2017), do not provide uncertainty information regarding their reconstructed images. The approaches that do provide some form of uncertainty quantification (Sutter et al. 2014; Junklewitz et al. 2016; Greiner et al. 2017) cannot scale to bigdata due to their high computational cost, are typically restricted to Gaussian or log-normal priors, and are not currently used in practice. Please see our first article in this companion series (Cai et al. 2017) for a more thorough review of RI imaging techniques and their properties. The current state of the field thus triggers an urgent need to develop efficient uncertainty quantification methods for RI imaging that scale to big-data. Furthermore, we seek to support the sparsity-promoting priors that have been demonstrated in practice to be highly effective for RI imaging (e.g. Pratley et al. 2016). In Cai et al. (2017) (the first part of this companion series), we proposed uncertainty quantification methods to address the RI imaging problem with sparse priors. In the current article (the second part of this companion series), we present fast uncertainty quantification methods that not only support sparse priors but also scale to big-data. The techniques presented in this article are very different to those presented in Cai et al. (2017) but support the same forms of uncertainty quantification. The uncertainty quantification methods proposed in Cai et al. (2017) are based on two proximal MCMC sampling methods, i.e. the Moreau-Yoshida unadjusted Langevin algorithm (MYULA) (Durmus et al. 2016) and the proximal Metropolis-adjusted Langevin algorithm (Px-MALA) (Pereyra 2015). The main steps of the uncertainty quantification strategies presented in Cai et al. (2017) can be briefly summarised as follows: firstly, the posterior distribution of the image is MCMC sampled; then, uncertainty quantification is performed by using the generated samples to compute local (pixel-wise) credible intervals, highest posterior density (HPD) credible regions, and to perform hypothesis testing of image structure. Two frameworks – analysis and synthesis models – are considered. While excellent results were achieved in Cai et al. (2017), when it comes to big-data, the proposed approach would suffer due to the long computation time required to sample the posterior distribution (as would be the case for any MCMC sampling approach). 3 4 5 http://www.atnf.csiro.au/projects/askap http://www.mwatelescope.org/telescope http://www.skatelescope.org/ In this article we exploit an analytic method to approximate HPD credible regions from maximum a posteriori (MAP) estimators, as derived in Pereyra (2016a), in order to develop very fast methods to perform uncertainty quantification for RI imaging. Our approach supports sparse priors and scales to massive data sizes, i.e. to big-data. We begin by formulating Bayesian MAP estimation for RI imaging as unconstrained convex optimisation problems, for analysis and synthesis forms. These are subsequently solved efficiently by using convex minimisation algorithms (e.g. Combettes & Pesquet 2010). Recent advances in convex optimisation have resulted in techniques that achieve excellent reconstruction fidelity (with convergence guarantees), are flexible, and exhibit relatively low computational costs. They also afford algorithmic structures that can be highly distributed and parallelised (e.g. Carrillo et al. 2014; Onose et al. 2016). Note, specifically, that only one point estimator is computed here for the analysis or synthesis form, in contrast to sampling approaches that seek to explore the full posterior distribution as in Cai et al. (2017), which is very time consuming. MAP estimation is then followed by various strategies to quantify uncertainties. Precisely, first the method of Pereyra (2016a) is used to obtain approximate HPD credible regions for the recovered image. These HPD regions are then used, for the first time, to compute local credible intervals (cf. error bars) that analyse uncertainty spatially and at different scales (pixles or superpixels). Finally, we also use the HPD credible regions to perform hypothesis tests of image structure. We test our proposed approaches on simulated RI observations to demonstrate their effectiveness and compare with the MCMC methods presented in Cai et al. (2017). The remainder of this article is organised as follows. In Section 2 we review the RI imaging inverse problem. In Section 3 we apply convex optimisation algorithms to solve the MAP estimation problem for RI imaging in the context of sparse priors. Uncertainty quantification techniques for RI imaging based on MAP estimation are formulated in Section 4. The performance of the proposed methods is then evaluated numerically in Section 5, where we compare uncertainties quantified by proximal MCMC methods and by MAP estimation. Finally, we conclude in Section 6 with a summary of our main contributions and a discussion of planned extensions. 2 RADIO INTERFEROMETRIC IMAGING In this section the inverse problem related to RI image reconstruction is introduced. We briefly recall the use of proximal MCMC methods to solve this problem (Cai et al. 2017), which we use as a benchmark in the experiments that follow. Finally, an introduction to Bayesian MAP estimation approaches for RI imaging is presented, which may be solved by efficient convex optimisation strategies. 2.1 Radio interferometry Here, we concisely recall the inverse problem of RI imaging (for further details see Cai et al. 2017 and references therein). When the baselines in an array are co-planar and the field of view is narrow, the visibilities, y, can be measured by correlating the signals from pairs of antennas, separated by the baseline components u = (u, v). Let x represent the sky brightness distribution, described in coordinates l = (l, m), and A(l) represent the primary beam of the telescope. The general RI equation for acquiring y can MNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II be represented as (Thompson et al. 2008) Z y(u) = A(l)x(l)e−2πiu·l d2 l. (1) Recovering the sky intensity signal x from the measured visibilities y acquired according to equation (1) then amounts to solving a linear inverse problem (Rau et al. 2009). In the discretised setting, let x ∈ RN represent the sampled intensity signal (the sky brightness distribution). In particular, x can be represented by X x = Ψa = Ψi ai , (2) i where Ψ ∈ CN ×L is a basis or dictionary (e.g., a wavelet basis or an over-complete frame) and vector a = (a1 , · · · , aL )> represents the synthesis coefficients of x under Ψ. In particular, x is said to be sparse if a contains only K non-zero coefficients, K N , or compressible if many coefficients of a are nearly zero. In practice, it is ubiquitous that natural images are sparse or compressible for approriate choices of Ψ. Refer to Cai et al. (2017) for more details about sparse representation. Let y ∈ CM be the M visibilities observed under a linear measurement operator Φ ∈ CM ×N modelling the realistic acquisition of the sky brightness distribution. Then, we have y = Φx + n or y = ΦΨa + n, (3) where n ∈ CM is the instrumental noise. Without loss of generality, we subsequently consider independent and identically distributed (i.i.d.) Gaussian noise. In practice, y is only observed partially or with limited resolution and thus solving (3) for x presents an ill-posed inverse problem. 2.2 Bayesian inference The RI inverse problem (3) can be solved elegantly in the Bayesian statistical framework, which provides tools to estimate x as well as to quantify the uncertainty in the estimated solutions. Let p(y\|x) be the likelihood function of the statistical model associated with (3). In the case of i.i.d. Gaussian noise this reads p(y\|x) ∝ exp(−ky − Φxk22 /2σ 2 ), (4) where σ represents the standard deviation of the noise level. Recovering x directly from y is not possible because the problem is not well posed. Bayesian methods use prior knowledge to address this difficulty. Precisely, they use a prior distribution p(x) to regularise the problem, reduce uncertainty, and improve estimation results. Here we consider both analysis and synthesis formulations because they are both widely used in RI imaging. For analysis models we use Laplace-type priors of the form p(x) ∝ exp(−µkΨ† xk1 ), (5) where Ψ† denotes the adjoint of Ψ, µ > 0 is a regularisation parameter, and k·k1 is the `1 norm. Analogously, for synthesis models we use p(a) ∝ exp(−µkak1 ). (6) Observe that both formulations are equivalent when Ψ is an orthogonal basis. However, for redundant dictionaries the approaches have very different properties. Please see Cai et al. (2017) for more details about this model and other models used in RI imaging. MNRAS 000, 1–13 (2017) 3 The observed and prior information are then combined by using Bayes’ theorem to obtain the posterior distribution, which models our knowledge about x after observing y. For the analysis formulation this is given by p(x\|y) = R RN p(y\|x)p(x) . p(y\|x)p(x)dx (7) Similarly, for the synthesis model the posterior reads p(y\|a)p(a) , p(y\|a)p(a)da p(a\|y) = R (8) RN with p(y\|a) = p(y\|x) for x = Ψa. Refer to Cai et al. (2017) for more detailed discussion about Bayesian inference in the context of RI imaging. 2.3 Proximal MCMC methods To solve the ill-posed inverse problem in (3) with sparsitypromoting priors, which have been shown in practice to be highly effective (Pratley et al. 2016), while also performing uncertainty quantification, two proximal MCMC methods to perform Bayesian inference for RI imaging were developed in the companion article (Cai et al. 2017). These proximal MCMC methods seek to sample the full posterior density p(x\|y) that models our understanding of the image x given data y, in the context of prior information. From the full posterior, summary estimators of x and other quantities of interest can be computed. In particular, in Cai et al. (2017) these methods are used to perform a range of uncertainty quantification analysis for RI images. One of the proximal MCMC methods presented in Cai et al. (2017), MYULA, scales efficiently to high dimensions but suffers from some estimation bias (Durmus et al. 2016). The other, PxMALA, corrects this bias by using a Metropolis-Hastings correction step, at the expense of a higher computational cost and slower convergence (Pereyra 2015). Since Px-MALA can provide results with corrected bias and thus is more accurate, we use it as a benchmark in the subsequent numerical tests presented in this work. Nevertheless, the MCMC methods discussed in Cai et al. (2017) will suffer when scaling to big-data (as will any MCMC method), which motives us to explore alternative faster methods that can scale to big-data. In this article we develop methods for uncertainty quantification based on MAP estimation. We emphasise that while MCMC methods such as Px-MALA are not as efficient as MAP estimation (the main focus in this article), and do not scale to large RI datasets, they are useful for smaller datasets and as a benchmark for the efficient alternative methods that we propose in Section 4. 2.4 Maximum a posteriori (MAP) estimation As discussed in the previous sections, sampling the full posterior p(x\|y) or p(a\|y) by MCMC methods is difficult because of the high dimensionality involved. Instead, Bayesian estimators that summarise p(x\|y) or p(a\|y) are often computed. In particular, one common approach is to compute MAP (maximum-a-posteriori) estimators given by n o xmap = argmin µkΨ† xk1 + ky − Φxk22 /2σ 2 , (9) x for the analysis model, and for the synthesis model by n o xmap = Ψ × argmin µkak1 + ky − ΦΨak22 /2σ 2 . a (10) 4 Cai et al. As we discuss below, a main computational advantage of the MAP estimators (9) and (10) is that they can be computed very efficiently, even in high dimensions, by using convex optimisation algorithms (e.g. Combettes & Pesquet 2010; Green et al. 2015). There is also abundant empirical evidence suggesting that these estimators deliver accurate reconstruction results (see Pereyra 2016b also for a theoretical analysis of MAP estimation). However, since MAP estimation results in a single point estimator, we typical lose uncertainty information that MCMC methods can provide (Cai et al. 2017). On the contrary, however, as we show in this article it is possible to approximately quantify the uncertainties associated with MAP estimators by leveraging recent results in the theory of probability concentration (Pereyra 2016a). Consequently, using the techniques presented later in this article MAP estimation can provide fast methods that scale to big-data and that quantify uncertainties. 2.5 Convex optimisation methods for MAP estimation There are several convex optimisation methods that can be used to solve the MAP estimation problems (9) and (10) efficiently, such as forward-backward splitting, Douglas-Rachford splitting, or alternating direction method of multipliers (ADMM) (see Combettes & Pesquet 2010). In our experiments (9) and (10) are solved by adopting the simple forward-backward algorithm, which we detail below. Forward-backward splitting algorithms solve optimisation problems of the form argmin(f + g)(x), (11) x∈RN by using a splitting of (f + g)(x). We consider the setting where f ∈ / C 1 is proper, convex and lower semi-continuous (l.s.c.) and g ∈ C 1 is l.s.c. convex and βLip -Lipchitz differentiable, i.e., k∇g(ẑ) − ∇g(z̄)k ≤ βLip kẑ − z̄k, ∀(ẑ, z̄) ∈ CN × CN . (12) Precisely, forward-backward algorithms solve (11) by using the iteration x (i+1) = proxλ(i) f (x (i) (i) (i) − λ ∇g(x )), (13) where λ(i) is the step size in a suitable bounded interval (see, e.g., Combettes & Pesquet 2010). The proximity operator of λf is defined as (Moreau 1965) proxλf (z) ≡ argmin f (u) + ku − zk2 /2λ . (14) u∈RN There are several refinements of (13) with better convergence properties. For example, using relaxation leads to the iteration x(i+1) = (1 − β (i) )x(i) + β (i) x̃(i+1) , (15) where x̃(i+1) is computed by (13), β (i) is a sequence of relaxation parameters, λ(i) ∈ (, 2/βLip − ), β (i) ∈ (, 1), and ∈ (0, min{1, 1/βLip }) (Combettes & Wajs 2005); or with λ(i) = 1/βLip , β (i) ∈ (, 3/2 − ), and ∈ (0, 3/4) (Bauschke & Combettes 2011). Furthermore, algorithmic structures that allow computations to be highly distributed and parallelised can also be developed (e.g. Carrillo et al. 2014; Onose et al. 2016) to assist in scaling to big-data. estimation problems for both the analysis setting (9) and synthesis setting (10). For the sake of brevity, henceforth the labels ¯ and ˆ denote symbols related to the analysis and synthesis models, respectively. 3.1 Analysis For the analysis setting (9), set f¯(x) = µkΨ† xk1 and ḡ(x) = ky − Φxk22 /2σ 2 . Then n o argmin f¯(x) + ḡ(x) (16) x can be solved using the forward-backward iteration formula (13), leading to the iterations x(i+1) = proxλ(i) f¯(x(i) − λ(i) ∇ḡ(x(i) )). (17) Assume for now Ψ† Ψ = I, where I is identity matrix (although this assumption is not essential and relaxed later). We have, ∀z̄ ∈ RN , proxλf¯(z̄) = z̄ + Ψ softλµ (Ψ† z̄) − Ψ† z̄ , (18) and ∇ḡ(x) = Φ† (Φx − y)/σ 2 , (19) where softλ (z) = softλ (z1 ), softλ (z2 ), · · · is the pointwise soft-thresholding operator of vector z defined by ( zj (\|zj \| − λ)/\|zj \| if \|zj \| > λ, softλ (zj ) = (20) 0 otherwise, for every component zj . Refer to Cai et al. (2017) for the derivation of (18). Substituting (18) and (19) into (17), the analysis problem (9) can be solved iteratively by v (i+1) = x(i) − λ(i) Φ† (Φx(i) − y)/σ 2 , x(i+1) = v (i+1) +Ψ softλ(i) µ (Ψ† v (i+1) )−Ψ† v (i+1) ) . (21) (22) As initialisation use, e.g., x(0) = Φ† y, i.e. the dirty image. Remark 3.1. In the analysis form (9), if choosing Ψ such that Ψ† Ψ 6= I, i.e. an over-complete frame Ψ, then proxλf¯(z̄) can be computed in an iterative manner: ! 1 1 u(t− 2 ) (t) † (t) u(t+ 2 ) = λite (1 − proxλk·k /λ(t) ) + Ψ u , (23) (t) 1 ite λite 1 u(t+1) = z̄ − Ψu(t+ 2 ) , (24) (t) λite 2 where ∈ (0, 2/βPar ) (βPar is a constant satisfying kΨzk2 ≤ βPar kzk , ∀z ∈ RL ) is a predefined step size and u(t) → proxγ f¯(z̄); refer to Fadili & Starck (2009) and Jacques et al. (2011) and references therein for details. 3.2 Synthesis For the synthesis setting (10), set fˆ(a) = µkak1 and ĝ(a) = ky − ΦΨak22 /2σ 2 . Then n o argmin fˆ(a) + ĝ(a) (25) x 3 SPARSE MAP ESTIMATION FOR RI IMAGING In this section we present the algorithmic details of implementing the forward-backward splitting algorithm to solve the sparse MAP can be solved using the forward-backward iteration formula (13), leading to the iterations a(i+1) = proxλ(i) fˆ(a(i) − λ(i) ∇ĝ(a(i) )). (26) MNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II We have, ∀ẑ = (ẑ1 , · · · , ẑL ) ∈ RL , 5 Observed visibilities in RI imaging: y proxλfˆ(ẑ) = proxλk·k1 (ẑ) = argmin λµkuk1 + ku − ẑk2 /2 u∈RL MAP image estimation: xmap (27) Approximate HPD 0 credible regions: Cα = softλµ (ẑ) and † † 2 ∇ĝ(a) = Ψ Φ (ΦΨa − y)/σ . (28) Finally, substituting (27) and (28) into (26), the synthesis problem (10) can be solved iteratively by a(i+1) = softλ(i) µ a(i) − λ(i) Ψ† Φ† (ΦΨa(i) − y)/σ 2 . (29) Remark 3.2. Note that in both the analysis and synthesis settings various terms can be precomputed. For example, in (21) and (28) the operators Φ† Φ and Ψ† Φ† ΦΨ can be precomputed offline. Similarly, the terms of Φ† y (the so-called dirty map) and Ψ† Φ† y respectively in (21) and (28) can also be precomputed to improve computation efficiency. We summarise the forward-backward splitting algorithms for the analysis and synthesis reconstruction forms in Algorithms 1 and 2. We consider stopping criteria based on a maximum iteration number and when the relative difference between solutions at two consecutive iterations is within some tolerance, i.e., kx(i+1) − x(i) k2 /kx(i) k2 (for Algorithm 1) and kΨa(i+1) − Ψa(i) k2 /kΨa(i) k2 (for Algorithm 2). The iteration is terminated when either of the stopping criteria are reached. Algorithm 1: Forward-backward algorithm for analysis 2 Input: y ∈ RM , x(0) ∈ RN , σ and λ(i) ∈ (0, ∞) Output: x0 3 do 1 8 update v (i+1) = x(i) − λ(i) Φ† (Φx(i) − y)/σ 2 compute u = Ψ† v (i+1) update x(i+1) = v (i+1) + Ψ softλ(i) µ (u) − u) i=i+1 while Stopping criterion is not reached; 9 set x0 = x(i) 4 5 6 7 Algorithm 2: Forward-backward algorithm for synthesis 2 Input: y ∈ RM , a(0) ∈ RL , σ and λ(i) ∈ (0, ∞) Output: a0 3 do 1 7 compute u = a(i) − λ(i) Ψ† Φ† (ΦΨa(i) − y)/σ 2 update a(i+1) = softλ(i) µ (u) i=i+1 while Stopping criterion is not reached; 8 set a0 = a(i) 4 5 6 Approximate local credible intervals: (ξ− , ξ+ ) Hypothesis testing Figure 1. Our proposed uncertainty quantification procedure for RI imaging based on MAP estimation. The light green areas on the right show the types of uncertainty quantification developed. Firstly, an image is reconstructed by MAP estimation using convex optimisation techniques, which scale to big-data. Then, various forms of uncertainty quantification are performed. Global approximate Bayesian credible regions are computed. These are then used to compute local credible intervals (cf. error bars) corresponding to individual pixels and superpixels and to perform hypothesis testing of image structure to test whether a structure is physical or an artefact. 4 BAYESIAN UNCERTAINTY QUANTIFICATION: MAP ESTIMATION The analysis and synthesis reconstruction models address inverse problems which are generally ill-conditioned or ill-posed (especially when the measurements are only observed partially or with limited resolution). Consequently, the corresponding estimators have significant intrinsic uncertainty that is very challenging to analyse and quantify. In Pereyra (2016a) a general methodology was proposed to use MAP estimators to accurately approximate Bayesian credible regions for p(x\|y). These credible regions indicate the regions of the parameter space where most of the posterior probability mass lies. A remarkable property of the approximation is that it only requires knowledge of xmap and therefore it can be computed very efficiently, even in very large-scale problems. The diagram in Figure 1 shows the main components of our proposed uncertainty quantification methodology based on MAP estimation. As is shown, firstly, an image is reconstructed by MAP estimation. MAP estimation can be computed extremely rapidly and is therefore ideal for application to big-data. Then, various forms of uncertainty quantification are performed. Firstly, global approximate Bayesian credible regions are computed. These are then used to compute local credible intervals (cf. error bars) corresponding to individual pixels and superpixels. Finally, again using the global approximate Bayesian credible regions, hypothesis testing of image structure can be performed to test whether a structure is physical or an artefact. For consistency, we adopt the same notation as in the companion article (Cai et al. 2017). 4.1 Approximate highest posterior density (HPD) credible regions The first step in our uncertainty quantification methodology is to compute a credible region for p(x\|y). A posterior credible region with credible level (1 − α)% is a set Cα ∈ RN that satisfies Z p(x ∈ Cα \|y) = p(x\|y)1Cα dx = 1 − α, (30) x∈RN MNRAS 000, 1–13 (2017) 6 Cai et al. where 1Cα is the indicator function for Cα , defined by 1Cα (u) = 1 if u ∈ Cα and 0 otherwise. Many regions satisfy the above property. We focus on the HPD (Highest Posterior Density) region defined by Cα := {x : f (x) + g(x) ≤ γα }, (31) where the threshold γα which defines an isocontour or level-set of the log-posterior is set such that (30) holds, and we recall that p(x\|y) ∝ exp{−f (x) − g(x)}. This region is decisiontheoretically optimal in the sense of minimum volume (Robert 2001). Computing HPD credible regions in (31) is difficult because of the high-dimensional integral in (30). For RI models that are not too high dimensional, Cα can be computed efficiently by using proximal MCMC method as described in Cai et al. 2017. However, this is not possible in big-data settings. Here we use an approximation of Cα proposed recently in Pereyra (2016a) for convex inverse problems solved by MAP estimation. The approximation is given by Cα0 := {x : f (x) + g(x) ≤ γα0 }, (32) where γα0 is an approximation of the HPD threshold γα given by √ γα0 = f (xmap ) + g(xmap ) + τα N + N, (33) p with universal constant τα = 16 log(3/α). Recall that N is the dimension of x and (1 − α)% the credible level considered. After computing xmap by using modern convex optimisation algorithms, γα0 can be calculated straightforwardly using (33), even in very high dimensions. The approximation given in (33) was motivated from recent results in information theory in terms of a probability concentration inequality (refer to Pereyra 2016a for more details). For any α ∈ (4exp(−N/3), 1), the error between γα0 and γα is bounded by the following inequality √ (34) 0 ≤ γα0 − γα ≤ ηα N + N, p p 0 where ηα = 16 log(3/α) + 1/α. Since the error γα − γα grows at most linearly with respect to N when N is large, the credible region Cα0 associated with γα0 is a stable approximation of Cα . Moreover, since γα0 − γα ≥ 0 the approximation is theoretically conservative in the sense that Cα0 overestimates Cα . Precisely, in the analysis formulation, we first compute the reconstructed image xmap by using Algorithm 1, and then obtain an approximate HPD credible region C̄α0,map := {x : f¯(x) + ḡ(x) ≤ γ̄α0 } (35) with √ γ̄α0 = f¯(xmap ) + ḡ(xmap ) + τα N + N. (36) Similarly, in the synthesis setting we compute amap via Algorithm 2, and then construct Ĉα0,map := {Ψa : fˆ(a) + ĝ(a) ≤ γ̂α0 } (37) with √ γ̂α0 = fˆ(amap ) + ĝ(amap ) + τα N + N. γ̄α0 γ̂α0 (38) Note that and define the HPD credible regions implicitly. The HPD credible regions can be used to quantify uncertainties in a variety of manners. In the reminder of this section we describe two such strategies. 4.2 Local credible intervals The first strategy we propose is a novel approach to compute local credible intervals corresponding to pixels and superpixels, as a means for quantifying uncertainty spatially at different scales. This presents a new form of Bayesian uncertainty quantification tailored for image data and is easy to visualise and interpret. The method is based on the HPD credible regions discussed above and is applicable for any method for which HPD credible regions can be computed. Here we promote the MAP-based approach, based on the approximations (36) and (38), and benchmark our results against the MCMC approach Px-MALA, introduced in Cai et al. (2017). Let Ω = ∪i Ωi be a partition of the image domain Ω into subsets or superpixels Ωi such that Ωi ∩ Ωj = ∅, i 6= j. The image domain can be partitioned at different scales, from a single pixel to larger scales involving blocks of several pixels. To index superpixels we define the index operator ζΩi = (ζ1 , · · · , ζN ) ∈ RN on Ωi , which satisfies ( 1, if k ∈ Ωi , (39) ζk = 0, otherwise. To quantify the uncertainty associated with the region Ωi we calculate the points ξ−,Ωi and ξ+,Ωi that saturate the HPD credible region Cα0,map from above and from below at Ωi , given by ξ−,Ωi = min ξ\|f (xi,ξ ) + g(xi,ξ ) ≤ γα0 , ∀ξ ∈ [0, +∞) , (40) ξ ξ+,Ωi = max ξ\|f (xi,ξ ) + g(xi,ξ ) ≤ γα0 , ∀ξ ∈ [0, +∞) , (41) ξ where xi,ξ = x∗ (I − ζΩi ) + ξζΩi represents a point estimator generated by replacing the intensity of x∗ in Ωi by ξ. We recall that γα0 is the threshold or isocontour level defining Cα0,map . We then construct the interval (ξ−,Ωi , ξ+,Ωi ) that represents the range of intensity values ξ of Ωi for which xi,ξ ∈ Cα0,map . Finally, for visualisation, we gather all the lower and upper bounds ξ−,Ωi , ξ+,Ωi , ∀i, into the following two images: X X ξ− = ξ−,Ωi ζΩi , ξ+ = ξ+,Ωi ζΩi . (42) i i We typically consider the difference image (ξ+ − ξ− ) that shows the length of the local credible intervals (cf. error bars). These images can be constructed at different scales to analyse structure of different sizes. In our experiments, as examples, we consider superpixels of sizes 10 × 10, 20 × 20, and 30 × 30 pixels. 4.3 Hypothesis testing of image structure In a manner akin to the companion article Cai et al. (2017), we use knock-out posterior tests to assess specific areas or structures of interest in the reconstructed images. These tests proceed by constructing a surrogate test image x∗,sgt by carefully replacing the structure of interest in an point estimator x∗ (or Ψa∗ ) with background information. If removing the structure has pushed x∗,sgt outside of the HPD credible region (i.e. x∗,sgt ∈ / Cα0,map ), this indicates that the data strongly supports the structure under consideration. Conversely, if x∗,sgt remains inside of the HPD credible region (i.e. x∗,sgt ∈ Cα0,map ), then the likelihood is insensitive to the modification, indicating lack of strong evidence for the scrutinised structure. Algorithmically, a surrogate x∗,sgt for a test area ΩD ⊂ Ω is generated by performing segmentation-inpainting of x∗ , for example by applying a wavelet filter Λ iteratively by using x(m+1),sgt = x∗ 1Ω−ΩD + Λ† softλthd (Λx(m),sgt )1ΩD , (43) MNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II with x(0),sgt = x∗ or x(0),sgt = Ψa∗ for the synthesis formulation (usually 100 iterations are sufficient for convergence). To determine if x∗,sgt ∈ Cα0,map , it suffices to check if f (x∗,sgt ) + g(x∗,sgt ) ≤ γα0 . 5 (44) Table 1. CPU time in minutes for the proximal MCMC method Px-MALA (generating full posterior samples) and MAP-based methods (computing a point estimator), for the analysis and synthesis models and for test images of M31, Cygnus A, W28 and 3C288. MAP estimation is approximately 105 times faster than Px-MALA and can be scaled to big-data. EXPERIMENTAL RESULTS We now investigate the performance of the proposed uncertainty quantification methodology for the three strategies discussed in Section 4. We also report a detailed comparison with the proximal MCMC method Px-MALA, which is one of the MCMC methods introduced in the companion article (Cai et al. 2017) and that can also support sparsity-promoting priors. Px-MALA produces (asymptotically) exact inferences and therefore we use it here as an accurate benchmark for the methods proposed in this article. 5.1 Simulations In a manner akin to Cai et al. (2017), we perform our experiments with the following four RI images: M31 galaxy (size 256 × 256), Cygnus A galaxy (size 256 × 512), W28 supernova remnant (size 256×256), and 3C288 (size 256×256). These images are depicted in Figure 2 (a) and Figure 3 (a). Radio interferometric observations are simulated for these ground truth images in a similar manner as in Cai et al. (2017). The numerical experiments performed in this article for MAP estimation were run on a Macbook laptop with an i7 Intel CPU and memory of 16 GB, running MATLAB R2015b. The Px-MALA algorithm used as a benchmark is significantly more computationally expensive and required a high-performance workstation (see Cai et al. 2017). For further details about the experiment setup and the implementation of Px-MALA please see Cai et al. (2017). Regarding the models used for the experiments, the `1 regularisation parameter µ in the analysis and synthesis models is set to 104 and the dictionary Ψ in the analysis and synthesis models is set to Daubechies 8 wavelets. In Algorithms 1 and 2, we use λ(i) = 0.5, with stopping criteria set by a maximum iteration number of 500 and relative difference between solutions of 10−4 . In formulas (36) and (38), the range of values for α is [0.01, 0.99]. In particular, credible regions and intervals are reported at α = 0.01, corresponding to the 99% credible level. The maximum number of iterations for segmented-inpainting in (43) is set to 200. 5.2 Image reconstruction As the first step in our analysis we perform Bayesian image reconstruction for the four images considered. Precisely, for each image we compute two Bayesian estimators, the MAP estimator computed by convex optimisation and the sample mean estimator computed with Px-MALA. For completeness, we consider both the analysis and the synthesis models (9) and (10). The Bayesian estimators related to the analysis model are shown in Figures 2 and 3. Observe that both estimators produce similar, excellent reconstruction results. For comparison, dirty maps (reconstructed by applying the inverse Fourier transform directly to the visibilities) of the test images are shown in Figure 2 (b) and Figure 3 (b). As expected, the results of the analysis and synthesis models (9) and (10) under an orthogonal basis Ψ are nearly MNRAS 000, 1–13 (2017) 7 CPU time (min) Analysis Synthesis Images Methods M31 (Fig. 2 ) Px-MALA MAP 1307 .03 944 .02 Cygnus A (Fig. 3 ) Px-MALA MAP 2274 .07 1762 .04 W28 (Fig. 3 ) Px-MALA MAP 1122 .06 879 .04 3C288 (Fig. 3 ) Px-MALA MAP 1144 .03 881 .02 undistinguishable6 (see results for M31 in Figure 2; to avoid redundancy the results for the other images are not reported here). For this reason, in the reminder of this article only the results for the analysis model are presented. We emphasise again that MAP estimators computed by convex optimisation are significantly faster to compute than the estimators that require MCMC methods. In particular, in our experiments there is a gain of order 105 in terms of computation time (see Table 1 for the computation time comparisons with Px-MALA). Furthermore, MAP estimation based on convex optimisation supports algorithmic structures that can be highly distributed (e.g. Carrillo et al. 2014; Onose et al. 2016) to further assist in scaling to bigdata. MCMC algorithms cannot typically be distributed to such a high degree. We have not yet considered distributed MAP algorithms here; our MAP-based methods therefore provide additional performance improvements over MCMC beyond the already dramatic improvements shown in Table 1. 5.3 Approximate HPD credible regions We compute the HPD credible regions for the four images considered. Precisely, we use formulas (36) and (38) to approximate the threshold or isocontour value γα0 defining the HPD regions for the analysis and synthesis models (recall that these are highly efficient approximations derived from the MAP estimates xmap and amap ). Figure 4 shows the threshold values obtained for each image and model, for α ∈ [0.01, 0.99]; observe again that the results of the analysis and synthesis models are consistent with each other, as expected. To assess the approximation error involved in using (36) and (38) instead of an MCMC method, we also computed the HPD threshold values using the Px-MALA algorithm which is asymptotically exact (cf. Cai et al. 2017, Figure 6). Recall than Px-MALA is several orders of magnitude more computationally expensive than MAP estimation (see Table 1). This comparison revealed approximation errors of between 1% and 5% over all cases, which is in close agreement with the results reported in Pereyra (2016a). These experiments confirm that the MAP-based approximations (36) and Note that, when Ψ† Ψ = I, as considered here, the analysis and synthesis models are identical. However, when Ψ† Ψ 6= I, they are very different and we expect different reconstructed images. 6 8 Cai et al. (a) ground truth (b) dirty map (c) Px-MALA for analysis model (d) MAP for analysis model (e) Px-MALA for synthesis model (f) MAP for synthesis model Figure 2. Image reconstructions for M31 (size 256 × 256). All images are shown in log10 scale. (a): ground truth; (b): dirty image (reconstructed by inverse Fourier transform); (c) and (d): point estimators for the analysis model (9) computed by Px-MALA and MAP estimation, respectively; (e) and (f): the same as (c) and (d) but for the synthesis model (10). In particular, the point estimators of Px-MALA are the sample mean. Clearly, consistent results between Px-MALA and MAP estimation and between the analysis and synthesis models are obtained. (a) ground truth (b) dirty map (c) Px-MALA for analysis model (d) MAP for analysis model Figure 3. Image reconstructions for Cygnus A (size 256 × 512), W28 (size 256 × 256), and 3C288 (size 256 × 256) radio galaxies (first to third rows). All images are shown in log10 scale. First column: (a) ground truth. Second to forth columns: (b) dirty images; (c) and (d) point estimators for the analysis model (9) computed by Px-MALA and MAP estimation, respectively. Clearly, consistent results between Px-MALA and MAP estimation are obtained. (38) deliver accurate estimates of the HPD credible regions with a dramatically lower computational cost. 5.4 Approximate local credible intervals We use the approximate HPD regions to calculate local credible intervals for image superpixels. Precisely, Figures 5–8 report the length of local credible intervals for the four test images for superpixel grid sizes of 10 × 10, 20 × 20, and 30 × 30 pixels, computed MNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II 1−α (a) M31 1−α (b) Cygnus A 1−α (c) W28 9 1−α (d) 3C288 MAP Px-MALA 0 and γ̂ 0 computed using MAP-based methods, for test images (a) M31, (b) Cygnus A, (c) W28, and (d) Figure 4. HPD credible region isocontour levels γ̄α α 3C288. In particular, MAP-ana (resp. MAP-syn) represents the results by MAP estimation for the analysis (resp. synthesis) model. Note that the red line in plot (d) is overlaid by the blue line and thus may not be visible, due to the high degree of similarity between the two results. In all cases the results of the analysis and synthesis models are in close agreement. (a) point estimators (b) local credible interval length grid size 10 × 10 pixels (c) local credible interval length grid size 20 × 20 pixels (d) local credible interval length grid size 30 × 30 pixels MAP Px-MALA Figure 5. Length of local credible intervals (99% credible level), cf. error bars, computed for M31 for the analysis model (9). First column: (a) point estimators. Second to fourth columns: (b)–(d) local credible intervals at grid sizes of 10 × 10, 20 × 20, and 30 × 30 pixels, respectively. First row gives exact inferences computed with the MCMC method Px-MALA (Cai et al. 2017). Second row gives MAP-based approximate inferences computed by convex optimisation. Clearly, MAP-based approximations provide estimates of the length of local credible intervals (cf. error bars) that are extremely consistent with the ones obtained by Px-MALA, while the MAP estimates can be computed several orders of magnitude more rapidly (Table 1). Moreover, the length of the approximate credible intervals computed by the MAP-based approach are theoretically conservative and can be seen to slightly overestimate the lengths computed by MCMC sampling). (a) point estimators (b) local credible interval length grid size 10 × 10 pixels (c) local credible interval length grid size 20 × 20 pixels Figure 6. Same as Figure 5 but for Cygnus A. MNRAS 000, 1–13 (2017) (d) local credible interval length grid size 30 × 30 pixels Cai et al. MAP Px-MALA 10 (a) point estimators (b) local credible interval length grid size 10 × 10 pixels (c) local credible interval length grid size 20 × 20 pixels (d) local credible interval length grid size 30 × 30 pixels MAP Px-MALA Figure 7. Same as Figure 5 but for W28. (a) point estimators (b) local credible interval length grid size 10 × 10 pixels (c) local credible interval length grid size 20 × 20 pixels (d) local credible interval length grid size 30 × 30 pixels Figure 8. Same as Figure 5 but for 3C288. w.r.t. the analysis model (the results for the synthesis model are very similar). For comparison, Figures 5–8 also show the exact estimates obtained by Px-MALA based on its posterior sample mean. We conclude the main observations as follows. Firstly, the results obtained with both approaches are extremely consistent with each other, indicating that the approximate credible intervals derived from the MAP estimation are very accurate. Secondly, the length of the approximate local credible intervals computed by MAP estimation are theoretically conservative and can be seen to slightly overestimate the lengths computed by MCMC sampling, and so are trustworthy. Thirdly, note that (i) coarser scales have shorter credible intervals than narrower scales, and (ii) superpixels at object boundaries generally have longer credible intervals than superpixels in homogenous regions. These two observations are direct consequences of the fact that there is more uncertainty about high-frequency image components because of the sampling profile associated with the measurement operator Φ, which mainly covers low frequencies (see Cai et al. 2017, Figure 2). 5.5 Hypothesis testing of image structure We conclude our experimental results by demonstrating our methodology for testing structure in reconstructed images. We consider the same images and structures of interest as in Cai et al. (2017), shown in the yellow rectangular areas in the first column of Figure 9. All of these structures are physical (i.e. present in the ground truth images), except for structure 2 in 3C288 which is a reconstruction artefact. Recall that the methodology proceeds as follows. First, we construct a surrogate image x∗,sgt by modifying the MAP estiMNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II 11 Table 2. Hypothesis test results for test structures shown in Figure 9 for M31, Cygnus A, W28, and 3C288. Note that γα represents the isocontour defining the HPD credible region at credible level (1 − α), where here α = 0.01, x∗,sgt represents the surrogate generated from point estimator x∗ (in particular, for Px-MALA x∗ is the sample mean of the MCMC samples), and (f + g)(·) represents the objective function; symbols with labels ¯ and ˆ are related to the analysis model (9) and the synthesis model (10), respectively. Symbol 7 indicates that the test area is artificial (and no strong statistical statement can be made as to the area), while 3 indicates that the test area is physical. All values are in units 106 . Clearly, both Px-MALA and MAP estimation give convincing and consistent hypothesis test results. Note that MAP estimation is dramatically more computationally efficient that Px-MALA (Table 1). Images Test areas Ground truth Method (f¯ + ḡ)(x̄∗,sgt ) Isocontour γ̄0.01 (fˆ + ĝ)(Ψ† x̂∗,sgt ) Isocontour γ̂0.01 Hypothesis test M31 (Fig. 9 ) 1 3 Px-MALA MAP 2.44 2.29 2.34 2.25 2.43 2.29 2.34 2.25 3 3 Cygnus A (Fig. 9 ) 1 3 Px-MALA MAP 1.17 1.02 1.26 1.14 1.18 1.02 1.27 1.14 7 7 W28 (Fig. 9 ) 1 3 Px-MALA MAP 3.38 3.47 1.84 1.89 3.37 3.47 1.85 1.89 3 3 1 3 2 7 Px-MALA MAP Px-MALA MAP 3.27 3.11 1.971 1.844 2.02 1.91 2.027 1.912 3.25 3.11 1.954 1.844 2.01 1.91 2.010 1.912 3 3 7 7 3C288 (Fig. 9 ) mator xmap to remove the structure of interest via segmentationinpaiting, computed using formula (43). Each structure is assessed individually. Second, we check if x∗,sgt ∈ / Cα0,map (i.e. if f (x∗,sgt ) + g(x∗,sgt ) > γα0 ) to determine whether there exists strong evidence in favour of the structure considered. Conclusions are generally not highly sensitive to the exact value of α; here we report results for α = 0.01 related to a 99% credible level. The resulting surrogate images are displayed in the second column of Figure 9. The results of these tests are shown in Table 2. For comparison, we also include the results obtained with the reference method Px-MALA (Cai et al. 2017). Again, the two methods produce excellent results that are consistent with each other. From Table 2, we observe that the methods have correctly classified the three main physical structures of M31, W28, and 3C288, and correctly identified the minor structure of 3C288 as a potential reconstruction artefact. Moreover, the methods have found that it is not possible to make a strong statistical statement about the small physical structure in image Cygnus A, which is difficult because it is only a few pixels in size, isolated, and significantly weaker in intensity than the other structures in the image. Before closing this section, we emphasise again that the methods presented in this article deliver a variety of forms of uncertainty quantification with a very low computational cost. While these new forms of uncertainty quantification can also be achieved by using state-of-the-art proximal MCMC methods, such as Px-MALA and MYULA, as presented in the companion article Cai et al. (2017), MCMC techniques cannot scale to massive data sizes. Nevertheless, they are useful for medium-scale problems and provide accurate benchmarks for the highly efficient methods presented herein, which will scale very well to the emerging big-data era of radio astronomy. 6 CONCLUSIONS Uncertainty quantification is an important missing component in RI imaging that will only become increasingly important as the bigdata era of radio interferometry emerges. No existing RI imaging techniques that are used in practice (e.g. CLEAN, MEM or CS approaches) provide uncertainty quantification. In this article, as an MNRAS 000, 1–13 (2017) alternative to MCMC methods, such as Px-MALA and MYULA that were presented in Cai et al. (2017), we present new uncertainty quantification methods based on MAP estimation by convex optimisation. The proposed uncertainty quantification methods exhibit extremely fast computation speeds and allow uncertainty quantification to be performed practically and in a manner that will scale to the emerging big-data era of RI imaging. Our proposed methods, which inherit the advantages of convex optimisation methods, are much more efficient than proximal MCMC methods that explore the entire posterior distribution of the image. Note, however, that the methods proposed here give an approximation of HPD credible regions and, consequently, the additional forms of uncertainty quantification that are built on the approximate HPD credible regions are also approximate. Nevertheless, we show these approximations are very accurate. Moreover, the approximations are conservative so that uncertainties are not underestimated. In contrast, proximal MCMC methods can theoretically provide HPD credible regions and other forms of uncertainty quantification that are more accurate. Therefore, the proposed fast MAP-based methods and the proximal MCMC methods complement each other, rather than being mutually exclusive. We anticipate that when it comes to the big-data era, we will use predominantly fast uncertainty quantification methods such as those based on MAP estimation, and reserve MCMC methods for benchmarking and detailed comparison. A variety of forms of uncertainty quantification for MAP estimation were constructed, including HPD credible regions, local credible intervals (cf. error bars) for individual pixels and superpixels, and tests for image structure. Our methods were evaluated on four test images that are representative in RI imaging. These experiments demonstrated that our MAP-based methods exhibit excellent performance and can reconstruct images with sharp detail. Moreover, they simultaneously underpin highly accurate approximate techniques to quantify uncertainties. In terms of computation time, MAP techniques were found to be approximately 105 times faster than state-of-the-art proximal MCMC methods, even when MAP estimation is run on a standard laptop and proximal MCMC methods on a high-performance workstation. Moreover, they lead to algorithmic structures that can be highly distributed and parallelised. In the near future we will consider alternative approaches to 12 Cai et al. M31 tations and data. It is our hope that uncertainty quantification, e.g. in the form of recovering error bars (Bayesian credible intervals) and hypothesis testing of image structure, will become an important standard component in RI imaging for statistically principled and robust scientific inquiry. For the first time, we propose techniques for the practical quantification of uncertainties in RI imaging. These techniques can be applied not only to observations made by existing telescopes but also to the emerging big-data era of radio astronomy. W28 Cygnus A 1 1 ACKNOWLEDGEMENTS This work is supported by the UK Engineering and Physical Sciences Research Council (EPSRC) by grant EP/M011089/1, and Science and Technology Facilities Council (STFC) ST/M00113X/1. 1 REFERENCES 3C288 2 1 (a) MAP point estimators (b) inpainted surrogate Figure 9. Hypothesis testing for M31, Cygnus A, W28, and 3C288. The five structures depicted in yellow are considered, all of which are physical (i.e. present in the ground truth images), except for structure 2 in 3C288, which is a reconstruction artefact. First column (a): point estimators obtained by MAP estimation for the analysis model (9) (shown in log10 scale). Second column (b): segmented-inpainted surrogate test images with information in the yellow rectangular areas removed and replaced by inpainted background (shown in log10 scale). Hypothesis testing is then performed to test whether the structure considered is physical by checking whether the surrogate test images shown in (b) fall outside of the HPD credible regions. Results of these hypothesis tests are specified in Table 2. Note that for the case shown in the last row the structures within areas 1 and 2 are tested independently. perform hypothesis testing of image structure. For example, substructure can be tested by creating surrogate test images with the substructure in question removed, for example by smoothing the corresponding region. Such an approach would be valuable for improving our understanding of the astrophysical processes involved in radio galaxies, for example in the processes governing radio jets. Most importantly, we plan to apply the uncertainty quantification techniques presented in this article to RI observations acquired by a variety of different telescopes and to make the methods publicly available. 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Soft + Hardwired Attention: An LSTM Framework for Human Trajectory Prediction and Abnormal Event Detection arXiv:1702.05552v1 [cs.CV] 18 Feb 2017 Tharindu Fernandoa,∗, Simon Denmana , Sridha Sridharana , Clinton Fookesa a Image and Video Research Laboratory, SAIVT, Queensland University of Technology, Australia. Abstract As humans we possess an intuitive ability for navigation which we master through years of practice; however existing approaches to model this trait for diverse tasks including monitoring pedestrian flow and detecting abnormal events have been limited by using a variety of hand-crafted features. Recent research in the area of deeplearning has demonstrated the power of learning features directly from the data; and related research in recurrent neural networks has shown exemplary results in sequenceto-sequence problems such as neural machine translation and neural image caption generation. Motivated by these approaches, we propose a novel method to predict the future motion of a pedestrian given a short history of their, and their neighbours, past behaviour. The novelty of the proposed method is the combined attention model which utilises both “soft attention” as well as “hard-wired” attention in order to map the trajectory information from the local neighbourhood to the future positions of the pedestrian of interest. We illustrate how a simple approximation of attention weights (i.e hard-wired) can be merged together with soft attention weights in order to make our model applicable for challenging real world scenarios with hundreds of neighbours. The navigational capability of the proposed method is tested on two challenging publicly available surveillance databases where our model outperforms the currentstate-of-the-art methods. Additionally, we illustrate how the proposed architecture can ∗ Corresponding author Email address: t.warnakulasuriya@qut.edu.au. (Tharindu Fernando ) Preprint submitted to Neural Networks February 21, 2017 be directly applied for the task of abnormal event detection without handcrafting the features. Keywords: human trajectory prediction, social navigation, deep feature learning, attention models. 1. Introduction Understanding and predicting crowd behaviour in complex real world scenarios has a vast number of applications, from designing intelligent security systems to deploying socially-aware robots. Despite significant interest from researchers in domains such as 5 abnormal event detection, traffic flow estimation and behaviour prediction; accurately modelling and predicting crowd behaviour has remained a challenging problem due to its complex nature. As humans we possess an intuitive ability for navigation which we master through years of practice; and as such these complex dynamics cannot be captured with only a 10 handful of hand-crafted features. We believe that directly learning from the trajectories of pedestrians of interest (i.e. pedestrian who’s trajectory we seek to predict) along with their neighbours holds the key to modelling the natural ability for navigation we posses. The approach we present in this paper can be viewed as a data driven approach 15 which learns the relationship between neighbouring trajectories in an unsupervised manner. Our approach is motivated by the recent success of deep learning approaches (Goroshin et al. (2015); Madry et al. (2014); Lai et al. (2014)) in unsupervised feature learning for classification and regression tasks. 1.1. Problem Definition The problem we have addressed can be defined as follows: Assume that each frame in our dataset is first preprocessed such that we have obtained the spatial coordinates of each pedestrian at every time frame. Therefore the trajectory of the ith pedestrian for the time period of 1 to Tobs can be defined as, xi = [x1 , y1 , . . . , xTobs , yTobs ]. 2 (1) Figure 1: A sample surveillance scene (on the left): The trajectory of the pedestrian of interest is shown in green, and has two neighbours (shown in purple) to the left, one in front and none on right. Neighbourhood encoding scheme (on the right): Trajectory information is encoded with LSTM encoders. A soft attention context vector Cts is used to embed the trajectory information from the pedestrian of interest, and a hardwired attention context vector C h is used for neighbouring trajectories. In order to generate Cts we use a soft attention function denoted at in the above figure, and the hardwired weights are denoted by w. The merged context vector is then used to predict the future trajectory for the pedestrian of interest (shown in red). 20 The task we are interested in is predicting the trajectory of the ith pedestrian for the period of Tobs+1 to Tpred , having observed the trajectory of the ith pedestrian from time 1 to Tobs as well as the trajectories all the other pedestrians in the local neighbourhood during that period. This can be considered a sequence to sequence prediction problem where the input sequence captures contextual information corresponding to the spatial 25 location of the pedestrian of interest and their neighbours, and the output sequence contains the predicted future path of the pedestrian of interest. 1.2. Proposed Solution To solve this problem we propose a novel architecture as illustrated in Fig. 1. For encoding and decoding purposes we utilise Long-Short Term Memory networks 30 (LSTM) due to their recent success in sequence to sequence prediction (Bahdanau et al., 2014; Xu et al., 2015; Yoo et al., 2015). We demonstrate the social navigational capability of the proposed method on two challenging publicly available surveillance 3 databases. We demonstrate that our approach is capable of learning the common patterns in human navigation behaviour, and achieves improved predictions for pedestri35 ans paths over the current state-of-the-art methodologies. Furthermore, an application of the proposed method for abnormal human behaviour detection is shown in Section 5. 2. Related Work 2.1. Trajectory Clustering 40 When considering approaches for learning motion patterns through clustering, Giannotti et al. (2007) have proposed the concept of “trajectory patterns”, which represents the descriptions of frequent behaviours in terms of space and time. They have analysed GPS traces of a fleet of 273 trucks comprising a total of 112,203 points. Deviating from discovering common trajectories, Lee et al. (2007) proposed to discover 45 common sub-trajectories using a partition-and-group framework. The framework partitions each trajectory into a set of line segments, and forms clusters by grouping similar line segments based on density. Morris and Trivedi (2009) evaluated different similarity measures and clustering methodologies to uncover their strengths and weaknesses for trajectory clustering. With reference to their findings, the clustering method had 50 little effect on the quality of the results achieved; however selecting the appropriate distance measures with respect to the properties of the trajectories in the dataset had great influence on final performance. 2.2. Human Behaviour Prediction When predicting human behaviour the most common motion models are social 55 force models (Helbing and Molnár, 1995; Koppula and Saxena, 2013; Pellegrini et al., 2010; Yamaguchi et al., 2011; Xu et al., 2012) which generate attractive and repulsive forces between pedestrians. Several variants of such approaches exist. Alahi et al. (2014) represents it as a social affinity feature by learning the pedestrian trajectories with relative positions where as Yi et al. (2015a) observed the behaviour of stationary 60 crowd groups in order to understand crowd behaviour. With the aid of topic models the 4 authors in Wang et al. (2008) were able to learn motion patterns in crowd behaviour without tracking objects. This approach was extended to incorporate spatio-temporal dependencies in Hospedales et al. (2009) and Emonet et al. (2011). Deviating from the above approaches, a mixture model of dynamic pedestrian 65 agents is presented by Zhou et al. (2015), who also consider the temporal ordering of the observations. Yet, this model ignores the interactions among agents, a key factor when predicting behaviour in real world scenarios. The main drawback in all of the above methods is that they utilise hand-crafted features to model human behaviour and interactions. Hand-crafted features may only 70 capture abstract level semantics of the environment and they are heavily dependent on the domain knowledge that we posses. In Alahi et al. (2016) an unsupervised feature learning approach was proposed. The authors have generated multiple LSTMs for each pedestrian in the scene at that particular time frame. They have observed the position of all the pedestrians from time 75 1 to Tobs and predicted all of their positions for the period Tobs+1 to Tpred . They have pooled the hidden states of the immediately preceding time step for the neighbouring pedestrians when generating their positions in the current time step. A more detailed comparison of this model with our proposed model is presented in Sec. 3.3 80 2.3. Attention Models Attention-based mechanisms are motivated by the notion that, instead of decoding based on the encoding of a single element or fixed-length part of the input sequence, one can attend a specific area (or important areas) of the whole input sequence to generate the next output. Importantly, we let the model learn what to attend to based on 85 the input sequence and what it has produced so far. In Bahdanau et al. (2014) have shown that attention-based RNN models are useful for aligning input and output word sequences for neural machine translation. This was followed by the works by Xu et al. (2015) and Yao et al. (2015) for image and video captioning respectively. According to Sharma et al. (2015), attention based models 90 can be broadly categorised into soft attention and hard attention models, based on the 5 method that it uses to learn the attention weights. Soft attention models (Bahdanau et al., 2014; Xu et al., 2015; Yao et al., 2015; Sharma et al., 2015) can be viewed as “supervised” guiding mechanisms which learn the alignment between input and output sequences through backpropagation. Hard attention (Mnih et al., 2014; Williams, 95 1992) is used by Reinforcement Learning to predict an approximate location to focus on. With reference to Sharma et al. (2015), learning hard attention models can become computationally expensive as it requires sampling. Still, soft aligning multiple feature sequences is computationally inefficient as we need to calculate an attention value for each combination of input and output elements. 100 This is feasible in cases such as neural machine translation where we have a 50-word input sequence and generate a 50-word output sequence, but prohibitively expensive in a surveillance setting when a target has hundreds of neighbours, and we have to learn the attention weight values for all possible value combinations for each of the neighbouring trajectories. We tackle this problem via the merging of soft attention and 105 hardwired attention in our framework. 3. Proposed Approach 3.1. LSTM Encoder-Decoder Framework In contrast to past attention models (Bahdanau et al., 2014; Xu et al., 2015) which align a single input sequence with an output sequence, we need to consider multiple 110 feature sequences in the form of trajectory information from the pedestrian of interest and their neighbours when predicting the output sequence. Aligning all features together is not optimal as they have different degrees of influence (i.e. a person walking directly next to the target has greater influence than a person several meters away). Soft attention models are deterministic models which are trained using back-propagation. 115 Therefore, aligning each input trajectory sequence separately via a separate soft attention model is computationally expensive. We show that we can overcome this problem with a set of hardwired weights which we calculate based on the distance between each neighbour and the pedestrian of interest. When considering navigation, as distance is the key factor which determines the 6 yi,t-1 yi,t St-1 St Decoder Cx*t Cst at,1 Encoder h1 h2 at,2 at,3 at,T h3 xi,1 xi,2 xi,3 Pedestrian of interest hT xi,t Ch w1,1 w1,2 w1,T h’1,1 h’1,2 x1,1 x1,2 wj,1 h’1,T x1,t Neighbour 1 wj,2 … wj,T h’j,1 h’j,2 xj,1 xj,2 h’j,T xj,t Neighbour j Figure 2: The proposed Soft + Hardwired Attention model. We utilise the trajectory information from both the pedestrian of interest and the neighbouring trajectories. We embed the trajectory information from the pedestrian of interest with the soft attention context vector Cts , while neighbouring trajectories are embedded with the aid of a hardwired attention context vector C h . In order to generate Cts we use a soft attention function denoted at in the above figure, and the hardwired weights are denoted by w. Then the merged context vector, Ct∗ , is used to predict the future state yi(t) 120 neighbour’s influence, it acts as a good generalisation. The proposed LSTM Encoder-Decoder framework is shown in Fig. 2. Due to its computational complexities, soft attention (denoted at in Fig. 2) is used only when embedding the trajectory information from the pedestrian of interest. We show that by approximating the required attention for the neighbours through hardwired atten- 125 tion weights (wj ), which we calculate based on the distance between the neighbouring pedestrian and the pedestrian of interest, we can generate a good approximation of their influence. The methodology of generating soft and hardwired attentions is outlined in the following subsections. 3.1.1. LSTM Encoder In a general Encoder-Decoder framework, an encoder recieves an input sequence x from which it generates an encoded sequence h. In the context of this paper, the input 7 sequence for the pedestrian i is given in Equation 1 and the encoded sequence is given by, hi = [h1 , . . . , hT ]. (2) The encoding function is an LSTM, which can be denoted by, ht = LSTM(xt , ht−1 ). 130 (3) With the aid of above equation we encode the trajectory information from the pedestrian of interest as well as each trajectory in the local neighbourhood. 3.1.2. LSTM Decoder Before considering how the combine context vector Ct∗ is formulated, the concept of time dependent context vector can be illustrated as follows. For a general case, let st−1 be the decoder hidden state at time t − 1, yt−1 be the decoder output at time t − 1, Ct be the context vector at time t and f be the decoding function. The decoder output at time t is given by, yt = f (st−1 , yt−1 , Ct ), (4) as defined by Bahdanau et al. (2014) such that distinct context vectors are given for each time instant. The context vector depends on the encoded input sequence h = 135 [h1 , . . . , ht ]. In the proposed approach, the given trajectory (i.e x = [x1 , y1 , . . . , xTobs , yTobs ]) for the pedestrian of interest is encoded and used to generate a soft attention context vector, Cts . With the aid of distinct context vectors we are able to focus different degrees of attention towards different parts of the input sequence, when predicting the output sequence. The soft attention context vector Cts can be computed as a weighted sum of hidden states, Cts = T obs X αtj hj . (5) j=1 In Bahdanau et al. (2014), the authors have shown that the weight αtj can be computed by, exp(etj ) αtj = PT , k=1 exp(etk ) 8 (6) etj = a(st−1 , hj ), (7) and the function a is a feed forward neural network for joint training with other components of the system. As an extension to the decoder model proposed in Bahdanau et al. (2014), we have added a set of hardwired weights which we use to generate hardwired attention (C h ). 140 Utilising the hardwired attention model we combine the encoded hidden states of the neighbouring trajectories in the local neighbourhood. Hardwired attention weights are designed to incorporate the notion of distance between the pedestrian of interest and his or her neighbours into the trajectory prediction model. The closer a neighbouring pedestrian, the higher their associated weight, be- 145 cause that pedestrian has a greater influence on the trajectory that we are trying to predict. The simplest representation scheme can be given by, 1 w(n,j) = , dist(n, j) (8) where dist(n, j) is the distance between the nth neighbour and the pedestrian of interest at the j th time instance, and w(n,j) is the generated hardwired attention weight. This idea can be extended to generate the context vector for the hardwired attention 150 model. 0 Let there be N neighbouring trajectories in the local neighbourhood and h(n,j) be the encoded hidden state of the nth neighbour at the j th time instance, then the context vector for the hardwired attention model is defined as, Ch = N T obs X X 0 w(n,j) h(n,j) . (9) n=1 j=1 We then employ a simple concatenation layer to combine the information from individual attentions. Hence the combined context vector can be denoted as, Ct∗ = tanh(Wc [Cts ; C h ]), (10) where Wc is referred to as the set of weights for concatenation. We learn this weight value also through back-propagation. 9 The final prediction can now be computed as, yt = LSTM(st−1 , yt−1 , Ct∗ ), (11) where the decoding function f in Eq. 4 is replaced with a LSTM decoder as we are employing LSTMs for encoding and decoding purposes. 155 3.2. Model Learning The given input trajectories in the training set are clustered based on source and sink positions and we run an outlier detection algorithm for each cluster considering the entire trajectory. For clustering we used DBSCAN (Ester et al. (1996)) as it enables us to cluster the data on the fly without specifying the number of clusters. Hyper 160 parameters of the DBSCAN algorithm were chosen experimentally. In the training phase trajectories are clustered based on the entire trajectory and after clustering we learnt a separate trajectory prediction model for each generated cluster. When modelling the local neighbourhood of the pedestrians of interest, we have encoded the trajectories of those closest 10 neighbours in each direction, namely 165 front, left and right. If there exist more than 10 neighbours in any direction, we have taken the first (closest) 9 trajectories and the mean trajectory of the rest of the neighbours. If a trajectory has less than 10 neighbours, we create dummy trajectories such that we have 10 neighbours, and set the weight of these dummy neighbours to 0. When testing the model, we are concerned with predicting the pedestrian trajectory 170 given the first Tobs locations. To select the appropriate prediction model to use, the mean trajectory for each cluster for the period of 1 to Tobs is generated and in the testing phase, the given trajectories are assigned to the closest cluster centre while considering those mean trajectories as the cluster centroids. 3.3. Comparison to the Social-LSTM model of Alahi et al. (2016) In this section we draw comparisons between the current state-of-the-art technique and the proposed approach. In Alahi et al. (2016), for each neighbouring pedestrian, the hidden state at time t−1 is extracted out and fed as an input to the prediction model of the pedestrian of interest. Let there be N neighbours in the local neighbourhood and 10 0 h(n,t−1) be the hidden state of the nth neighbour at the time instance t − 1. Then the process can be written as, 0 Ht = N X 0 h(n,t−1) , (12) n=1 and the hidden state of the pedestrian of interest at the tth time instance is given by, 0 ht = LSTM(ht−1 , xt−1 , Ht ), 175 (13) where ht−1 refers to the hidden state and xt−1 refers to the position of the pedestrian 0 of interest at the t − 1 time instance. The authors are passing Ht and xt−1 through embedding functions before feeding it to the LSTM model, but in order to draw direct comparisons we are using the above notation. In Alahi et al. (2016), the hidden state of the pedestrian of interest at the tth time instance depends only on his or her previous 180 hidden state, the position in the previous time instance and the pooled hidden state of the immediately preceding time step for the neighbouring pedestrians (see Eq. 13). Comparing to our model, we are considering the entire set of hidden states for the pedestrian of interest as well as the neighbouring pedestrians when predicting the tth output element (see Eq. 5-11). 185 As humans we tend to vary our intentions time to time. For an example consider the problem of navigating in a train station. A person may start walking towards the desired platform and then realise that he hasn’t got a ticket and then make a sudden change and move towards the ticket counter. When applying the LSTM model proposed by Alahi et al. (2016) to such real world scenarios, by observing the immediate preceding 190 hidden state one can generate reactive behaviour to avoid collisions but when doing long term path planning, even though the LSTM is capable of handling long term relationships, the prediction process may go almost “blindly” towards the end of the sequence (Jia et al. (2015)) as we are neglecting vital information about pedestrian’s behaviour under varying contexts. In contrast, the proposed combined attention model 195 considers the entire sequence of hidden states for both the pedestrian of interest and his or her neighbours and then we utilise time dependent weights which enables us to vary their influence in a timely manner. Additionally, we observed that even in unstructured scenes such as train stations, 11 (a) First 2 clusters (b) Next 5 clusters (c) First 5 clusters (d) Next 5 clusters Figure 3: Clustering results for Grand Central (a, b) and Edinburgh Informatics Forum (c, d) Datasets airport terminals and shopping malls where multiple source and sink positions are 200 present, still there exists dominant motion patterns describing the navigation preference of the pedestrians. For instance taking the same train station example, although the main problem that we are trying to solve here is to navigating while avoiding collisions, humans demonstrate different preferences in doing so. One pedestrian may be there to meet passengers and hence is wandering in a free area, while another pedestrian 205 may aim to get in or out of the train station as quickly as possible. Therefore one single LSTM model is not sufficient to capture such ambiguities in navigational patterns. We observed that such distinct preferences in navigation generate unique trajectory patterns which can be easily segmented via the proposed clustering process. Therefore in contrast to Alahi et al. (2016), we are learning a different trajectory prediction model 210 for each trajectory cluster. 4. Experiments We present the experimental results on two publicly available human trajectory datasets: New York Grand Central (GC) (Yi et al. (2015b)) and Edinburgh Informat- 12 ics Forum (EIF) database (Majecka (2009)). The Grand Central dataset consist of 215 around 12,600 trajectories where as the Edinburgh Informatics Forum database contains around 90,000 trajectories. We have conducted 2 experiments. For the first experiment on the Grand Central dataset, after filtering out short and fragmented trajectories 1 , we are left with 8,000 trajectories, and train our model on 5,000 trajectories and evaluate the prediction accuracy on 3,000 trajectories. In the next experiment we con220 sidered 3 days worth trajectories from Edinburgh Informatics Forum database, trained our model on 10,000 trajectories and tested on 5,000 trajectories. Prior to learning trajectory models, we employ clustering to separate the different modes of human motion. This allows us to learn separate models for different behaviours, such as one model for a pedestrian who is buying tickets and another for 225 those who are directly entering or leaving the train station. We believe that these different motion patterns generate unique pedestrian behavioural styles and that are well captured through separate models. This can be achieved via clustering trajectories based on entire trajectory, but this will produce very large number of clusters or large number of outliers due to the wide variation in the different modes of human motion. 230 As a result, in each cluster, we would have very few examples to train our prediction models on. Therefore as a solution to the above stated problem we cluster the trajectories based only on the enter/exit zones. As illustrated in Fig. 3 this approach works reasonably well at separating different modes of human motion. Even with clustering based on entry and exit points, the way that a person moves 235 through the environment and how they are influenced by neighbours will vary considerably. For instance consider the clusters represented in green and blue in Fig. 3 (b). The way that an intersecting trajectory travelling straight up in the scene, from bottom left towards up left, will affect green and blue clusters differently because of their different exit zones. It’s general human nature to try to avoid collisions while keeping the 240 expected heading direction. Therefore entry/exit zones based clustering is sufficient to 1 We consider short trajectories to be those with length less than the time period that we are considering (40 frames) where as fragmented trajectories are trajectories which have discontinuities between 2 consecutive frames due to noise in the tracking process. 13 capture this. 4.1. Quantitative results Similar to Alahi et al. (2016) we report prediction accuracy with the following 3 error metrics. Let n be the number of trajectories in the testing set, xpred be the i,t 245 predicted position for the trajectory i at tth time instance, and xobs i,t be the respective observed positions then, 1. Average displacement error (ADE): TP pred n P ADE = 2 (xpred − xobs i,t ) i,t i=1 t=Tobs +1 n(Tpred − (Tobs + 1)) . (14) . (15) 2. Final displacement error (FDE) : n q P F DE = i=1 obs 2 (xpred i,Tpred − xi,Tpred ) n 3. Average non-linear displacement error (n-ADE): The average displacement error for the non-linear regions of the trajectory, n P n − ADE = TP pred i=1 t=Tobs +1 n P pred 2 I(xpred − xobs i,t ) i,t )(xi,t TP pred i=1 t=Tobs +1 , (16) I(xpred i,t ) where, I(xpred i,t ) =     1 if    0 o.w d2 yi,t dx2i,t 6= 0. (17) In all experiments we have observed the trajectory (and it’s neighbours) for 20 frames and predicted the trajectory for the next 20 frames. Compared to Alahi et al. (2016), which has considered sequences of 20 frames total length, we are considering 250 more lengthy sequences (with a total of 40 frames) as in Baccouche et al. (2011) the authors have shown that LSTM models tend to generate more accurate results with lengthy sequences. 14 In the experimental results, tabulated in Tab 1, we compare our prediction model with the state-of-the-art. As the baseline models we implemented Social Force (SF) 255 model from Yamaguchi et al. (2011) and Social LSTM (S-LSTM) model given in Alahi et al. (2016). For S-LSTM model a local neighbourhood of size 32px was considered and the hyper-parameters were set according to Alahi et al. (2016). In order to make direct comparisons with Alahi et al. (2016), the hidden state dimensions of encoders and decoders of all OUR models were set to be 300 hidden units. 260 For the SF model, preferred speed, destination, and social grouping factors are used to model the agent behaviour. When predicting the destination, a linear support vector machine was trained with the ground truth destination areas detected in the Sec. 3.2. In order to evaluate the strengths of the proposed model, we compare this combined attention model (OURcmb ) and two variations on our proposed approach: 1) OURsft , 265 which ignore the neighbouring trajectories and considers only the soft attention component derived from the trajectory of the person of interest when making predictions; and 2) OURsc which omits the clustering stage such that only a single model (using combined attention weights) is learnt. Metirc Dataset SF S-LSTM OURsc OURsft OURcmb GC 3.364 1.990 1.878 2.041 1.096 EIF 3.124 1.524 1.392 1.685 0.986 GC 5.808 4.519 4.317 5.277 3.011 EIF 3.909 2.510 2.345 3.089 1.311 GC 3.983 1.781 1.701 2.304 0.985 EIF 3.394 2.398 2.098 2.415 0.901 ADE FDE n-ADE Table 1: Quantitative results. In all the methods forecast trajectories are of length 20 frames. The first 2 rows represents the Average displacement error, rows 3 to 4 are for Final displacement error and the final 2 rows are for Average non-linear displacement error. The proposed model outperforms the SF model and S-LSTM model in both datasets. 270 The error reduction is more evident in the GC dataset where there are multiple source and sink positions, different crowd motion patterns are present and motion paths are heavily crowded. Comparing the results of OURsc (proposed approach without clus15 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) Figure 4: Qualitative results: Given (in green), Ground Truth (in Blue), Neighbouring (in purple) and Predicted trajectories from OURcmb model (in red), from S-LSTM model (in yellow), from SF model (in orange). (a) to (l) accurate predictions and (m) to (o) some erroneous predictions tering) against the S-LSTM model we can see that regardless of the clustering process the proposed combined attention architecture is capable of improving the trajectory 275 prediction. For all the measured error metrics OURsc has outperformed S-LSTM and (OURsft ) verifying that it is important to preserve historical data for both the pedestrian of interest as well as the neighbours. Secondly those results show that hard wired 16 weights act as a good approximation of neighbours influence. When comparing results of OURcmb against OURsc it is evident that the cluster280 ing process has partitioned the trajectories based on these different semantics and via utilising separate models for each cluster, we were able to generate more accurate predictions. The combined attention model was capable of learning how the neighbours influence the current trajectory and how this impact varies under different neighbourhood locations. 285 Furthermore, we would like to point out that, because of the separate model learning process we were able to predict the final destination positions with more precision compared to baseline models where they do not consider the environmental configurations of the unstructured scene. While the proposed approach does not explicitly model the environment, a certain amount of environmental information is inherently encoded 290 in the entry and exit locations, which the proposed approach is able to leverage. 4.2. Qualitative results In Fig. 4 we show prediction results of the S-LSTM model, SF model and our combined attention model (OURcmb ) on the GC dataset. It should be noted that our model generates better predictions in heavily crowded areas. As we are learning a sep- 295 arate model for each cluster, the prediction models are able to learn different patterns of influences from neighbouring pedestrians. For instance in the 1st and 3rd column we demonstrate how the model adapts in order to avoid collisions. In the last row of Fig. 4 we show some failure cases. The reason for such deviations from the ground truth were mostly due to sudden changes in destination. Even though these trajectories do 300 not match the ground truth, the proposed method still generates plausible trajectories. For instance, in Fig. 4 (m) and Fig. 4 (o) the model moves side ways to avoid collusion with the neighbours in the left and right directions. Three example scenarios that illustrates the advantage of attending to all the hidden states within that particular context are shown in Fig. 5. The first row shows an exam- 305 ple where the pedestrian of interest exhibits 2 modes of motion (walking and running) within the same trajectory. In the second row we have an example where the previous context of the pedestrian of interest is useful in tthe final prediction. Third row shows 17 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 5: Example scenarios: Columns (left to right): first observation of the trajectory; half way through; last observation prior to prediction; prediction from the respective models. 1st row shows an example where the pedestrian exhibits 2 modes of motion (walking and running) within the same trajectory. 2nd row shows an example where the previous context of the pedestrian is useful in prediction. 3rd row shows an example where the pedestrian is moving as part of a group. Colours: Given trajectory (in green), Ground Truth (in Blue), Neighbouring (in purple) and Predicted trajectories from OURcmb model (in red), from S-LSTM model (in yellow), from OURsc model (in cyan). In order to preserve visibility without occlusions we have plotted only the closest 2 neighbours in each direction. an example where pedestrian exhibits a group motion. The first three columns show the progression of the motion from the start of the trajectory (first column) to the point 310 directly before the prediction is made (third column). The given trajectory of the person of interest is shown in green and the neighbouring trajectories are shown in purple. In order to preserve visibility without occlusions we have plotted only the closest 2 neighbours in each direction. Finally in the fourth column we have presented the respective predictions from OURcmb model (in red), from S-LSTM model (in yellow), 315 from OURsc model (in cyan). The ground truth observations are shown in blue. When considering the example shown in Fig. 5 (a)-(d) the pedestrian of interest exhibits a running behaviour when entering the scene (Fig. 5 (a)). Then at half way through her trajectory she shifts her behaviour to walking (Fig. 5 (b)). Therefore at 18 the time of the prediction (Fig. 5 (c)), the hidden states of the baseline model will be 320 dominated by the walking behaviour as it is the most recent behaviour of this particular pedestrian. Therefore the predictions generated by the S-LSTM model are erroneous. But as we are attending to all the previous hidden states before predicting the future sequence, the multi model nature of that pedestrian’s motion has been captured by the OURsc and OURcmb models. 325 Another example scenario where the proposed model out performs it’s baseline is shown in Fig. 5 (e)-(h). When deciding upon whether to give way or to cut through the pedestrian group, models OURsc and OURcmb outperforms the S-LSTM model. While attending to the historical data in 5 (f) we can see the behaviour of the pedestrian of interest under a similar context. Therefore the proposed models can anticipate that 330 the preferred behaviour of the pedestrian of interest under such context is giving way to the others. But in the S-LSTM model as it’s attending only to the immediate preceding hidden states and those long range dependencies are not captured. In the example shown in Fig. 5 (i)-(l) we illustrate the importance of considering the entire history of the neighbours. The way that the neighbours affect to a pedestrian dur- 335 ing group motion vastly differs to the impact group motion has on a pedestrian walking alone. For example pedestrians moving as part of a group tend to walk at a similar velocity, keeping small distances between themselves, stopping or turing together; where as pedestrians walking alone try to keep a safe distance between themselves and their neighbours to avoid collisions. These notions can be quickly captured while observing 340 the neighbourhood history. Therefore the predictions generated by both OURsc and OURcmb models outperforms the baseline S-LSTM model which only considered the immediate preceding hidden state of the neighbours when generating the predictions. When comparing the predictions from OURsc model against OURcmb model it is evident that the more spatial context specific predictions are generated by OURcmb 345 as it has been specifically trained on the examples from that particular spatial region. Therefore it anticipates the motion of a particular pedestrian more accurately. Still in all the example scenarios OURsc is shown to be capable of generating acceptable predictions compared to the baseline model, showing that the proposed combined attention mechanism is capable of generating more accurate and realistic trajectories than 19 350 the current state-of-the-art. 5. Abnormal behaviour detection The proposed framework can be directly applied for detecting abnormal pedestrian behaviour. A naive approach would be to predict the trajectory for the period of Tobs+1 to Tpred while observing the same trajectory over this time period and measuring the 355 deviation between the observed and the predicted trajectories. If the deviation is greater than a threshold, then an abnormality can be said to have occurred. However due to the adaptive nature of deep neural networks, abnormal behaviours such as: i) sudden turns and changes in walking directions; and ii) trajectories with abnormal velocities; may not be classified as abnormal events. 360 We observe that the hidden states of the LSTM encoder decoder framework hold vital information which is used to model the walking behaviour of the pedestrian of interest. Hence, if his or her behaviour is abnormal then the hidden state values for that pedestrian should be distinct from those of a normal pedestrian. With that intuition we randomly selected 500 trajectories from the Grand Central 365 dataset and predicted the trajectories for those pedestrians. The trajectories were hand labeled for abnormal behaviour, considering sudden turns and changes in walking direction and abnormal velocities as the set of abnormal behaviours. The dataset consists of 445 normal trajectories and 55 abnormal trajectories. Then we extracted the encoded hidden states (h(t) = [h(1) , . . . , h(Tobs ) ]) for the given trajectory for that pedestrian and 370 the hidden states used for decoding (s(t) = [s(Tobs+1 ) , . . . , s(Tpred ) ]). The resultant hidden states are passed through DBSCAN to detect outliers. With the proposed approach we detected 441 trajectories as being normal and 59 trajectories as abnormal. The resultant detections are given in Table 2. Analysing the classification results, we see that false alarms are mainly due to be- 375 haviours that are erroneously detected as abnormal being uncommon in the database. Cases such as people changing direction to buy tickets, and passengers wandering in the free area are detected as abnormal due to the fact that they are not significantly present in the subset of trajectories selected for this task. 20 Ground Truth Abnormal Normal Abnormal 47 12 Normal 8 433 Total 55 445 Predicted Table 2: Abnormal Event detection with the proposed algorithm: This approach has detected 47 out of 55 ground truth abnormal events We compare this approach to the naive approach given above. It is evident that 380 some abnormal trajectories are misclassified as normal behaviour due to its lack of deviation from the observed trajectory. See Tab. 3 Ground Truth Abnormal Normal Abnormal 29 24 Normal 26 421 Total 55 445 Predicted Table 3: Abnormal event detection with naive approach: This approach has detected only 29 out of 55 ground truth abnormal events Some examples of detected abnormal events are shown in Fig. 6. The first row shows the abnormal behaviour detected due to sudden change of moving direction. Even though, in the examples shown (d) and (e), there isn’t a significant deviation be385 tween the predicted path and the observed path, our abnormal event detection approach has accurately classified the event due to the sudden circular turn in the trajectory in (d) and abnormal velocity in (e). 6. Conclusion In this paper we have proposed a novel neural attention based framework to model 390 pedestrian flow in a surveillance setting. We extend the classical encoder-decoder 21 (b) (a) (c) (e) (d) Figure 6: Abnormal event detections: (a)-(c) abnormal behaviour detected due to sudden change of moving direction. Abnormal behaviour due to sudden circular turn (d) and abnormal velocity in (e) framework in sequence to sequence modelling to incorporate both soft attention as well as hard-wired attention. This has a major positive impact when handling longer trajectories in heavily cluttered neighbourhoods. The hand-crafted hard-wired attention weights approximate the neighbour’s influence and make the application of atten395 tion models pursuable for real world scenarios with large number of neighbours. We tested our proposed model in two challenging publicly available surveillance datasets and demonstrated state-of-the-art performance. Our new neural attention framework exhibited a stronger ability to accurately predict pedestrian motion, even in the presence of multiple source and sink positions and with high crowd densities observed. 400 Furthermore, we have shown how the proposed approach can support abnormal event detection through hidden state clustering. 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DeepConf: Automating Data Center Network Topologies Management with Machine Learning arXiv:1712.03890v1 [cs.NI] 11 Dec 2017 Christopher Streiffer∗ , Huan Chen∗† , Theophilus Benson+ , Asim Kadav‡ Duke University∗ UESTC, China† Brown University+ NEC Labs‡ ABSTRACT In recent years, many techniques have been developed to improve the performance and efficiency of data center networks. While these techniques provide high accuracy, they are often designed using heuristics that leverage domain-specific properties of the workload or hardware. In this vision paper, we argue that many data center networking techniques, e.g., routing, topology augmentation, energy savings, with diverse goals actually share design and architectural similarity. We present a design for developing general intermediate representations of network topologies using deep learning that is amenable to solving classes of data center problems. We develop a framework, DeepConf, that simplifies the processing of configuring and training deep learning agents that use the intermediate representation to learns different tasks. To illustrate the strength of our approach, we configured, implemented, and evaluated a DeepConf-agent that tackles the data center topology augmentation problem. Our initial results are promising — DeepConf performs comparably to the optimal. 1. INTRODUCTION Data center networks (DCN) are a crucial and important part of the Internet’s ecosystem. The performance of these DCNs impact a wide variety of services from web browsing and videos to Internet of Things. The poor performance of these DCNs can result in as much as $4 million in lost revenue [1]. Motivated by the importance of these networks, the networking community has explored techniques for improving and managing the performance of the data center network topology by: (1) designing better routing or traffic engineering algorithms [6, 8, 13, 10], (2) improving performance of a fixed topology by adding a limited number of flexible links [33, 11, 17, 16], and (3) removing corrupted and underutilized links from the topology to save energy and improve performance [18, 14, 35]. Regardless of the approach, these topology-oriented techniques have three things in common: (1) Each is formalized as an optimization problem. (2) Due to the impracticality of scalably solving these optimizations, greedy heuristics are employed to create approximate solutions. (3) Each heuristic is intricately tied to the application patterns and does not generalize across novel patterns. Existing domain-specific heuristics provide suboptimal performance and are often limited to specific scenarios. Thus as a community, we are forced to revisit and redesign these heuristics when the application pattern or network details changes – even a minor change. For example, while c-through [33] and FireFly [17] solve broadly identical problems, they leverage different heuristics to account for low-level differences. In this paper, we articulate our vision for replacing domain-specific rule-based heuristics for topology management with a more general machine learning-based (ML) model that quickly learns optimal solutions to a class of problems while adapting to changes in the application patterns, network dynamics, and low-level network details. Unlike recent attempts that employ ML to learn point solutions, e.g., cluster scheduling [23] or routing [9], in this paper, we present a general framework, called DeepConf, that simplifies the process of designing ML models for a broad range of DCN topology problems and eliminates the challenges associated with efficiently training new models. The key challenges in designing DeepConf are: (1) tackling the dichotomy that exists between deep learning’s requirements for large amounts of supervised data and the unavailability of these required datasets, and (2) designing a general, but highly accurate, deep learning model that efficiently generalizes to learning a broad array of data center problems ranging from topology management and routing to energy savings. The key insight underlying DeepConf is that intermediate features generated from the parallel convolutional layers using network data, e.g., traffic matrix, allows us to generate an intermediate representation of the network’s state that enables learning a broad range of data center problems. Moreover, while labeled production data crucial for machine learning is unavailable, empir- ical studies [21] show that modern data center traffic is higly predictable and thus amenable to offline learning with network simulators and historical traces. DeepConf builds on this insight by using reinforcement learning (RL), an unsupervised technique, that learns through experience and makes no assumptions on how the network works. Rather, they are trained through the use of a reward signal which “guides” them towards an optimal solution and thus do not require real world data, and, instead, they can be trained using simulators. The DeepConf framework provides a predefined RL model with the intermediate representation, a specific design for configuring this model to address different problems, an optimized simulator to enable efficient learning, and an SDN-based platform for capturing network data and reconfiguring the network. In this paper, we make the following contributions: • We present a novel RL-based SDN architecture for developing and training deep ML models for a broad range of DCN tasks. • We design a novel input feature extraction for DCNs for developing different ML models over this intermediate representation of network state. • We implemented a DeepConf-agent tackling the topology augmentation problem and evaluated it on representative topologies [15, 4] and traces [3], showing that our approach performs comparable to optimal. 2. RELATED WORK Our work is motivated by the recent success of applying machine learning and RL algorithms to computer games and robotic planning [25, 31, 32]. The most closely related work [9] applies RL to packet routing. Unlike [9], DeepConf tackles the topology augmentation problem and explores the use of deep networks as function approximators for RL. Existing applications of machine learning to data centers focus on improving cluster scheduling [23] and more recently by Google to optimize Power Usage Effectiveness(PUE) [2]. In this vision paper, we take a different stance and focus on identifying a class of equivalent data center management operations, namely topology management and configuration, that are amenable to a common machine learning approach and design a modular system that enables different agents to interoperate over a network. 3. BACKGROUND This section provides an overview of data center networking challenges and solutions and provides background on our reinforcement learning methods. Ethernet Fabric Optical Circuit Optical Switch Figure 1: One optical switch with k=4 Fat Tree Topology. 3.1 Data Center Networks Data center networks introduce a range of challenges from topology design and routing algorithms to VM placement and energy saving techniques. Data centers support a large variety of workloads and applications with time-varying bandwidth requirements. This variance in bandwidth requirements leads to hotspots at varying locations and at different points in time. To support these arbitrary bandwidth requirements, data center operators can employ non-blocking topologies; however, non-blocking topologies are prohibitively costly. Instead, these operators employ a range of techniques ranging from hybrid architectures [33, 11, 17, 16], traffic engineering algorithms [6, 8, 13, 10], and energy saving techniques [18, 28]. Below, we describe these techniques and illustrate common designs. Augmented Architectures: This class of approaches build on the intuition that at any given point in time, there are only a small number of hotspots (or congested links). Thus, there is no need to build an expensive topology that supports full bisection (eliminating all potential points of congestion). Instead, the existing topology can be augmented with a small number of links which can be added ondemand and moved to the location of the hotspot or congested links. These approaches augment the data center’s ethernet network with a small number of optical [33, 11], wireless [16], or free optics [17]. 1 For example, Figure 1 shows a traditional Fat-Tree topology augmented with an optical switch – as was proposed by Helios [11]. These proposals argue for monitoring the traffic, using an Integer Linear Program (ILP) or heuristic to detect the hotspots and place the flexible links at the location with these hotspots. Unfortunately, moving these flexible links incurs a large switching time during which the links are not operational. These intelligent and efficient algorithms are developed to effectively detect hotspots and efficiently place links. Traffic Engineering: Orthogonal approaches [6, 8, 13, 10] focus on routing. Instead of changing the topologies, these approaches change the mapping of flows to paths within a fixed topology. These proposals, also, argue for monitoring traffic and detecting hotspots. Instead of changing topologies, these techniques move a subset of 1 The number of augmented links is significantly smaller than the number of data center’s permanent links. flows from congested links to un-congested links. Energy Savings Data centers are notorious for their energy usage [14]. To address this, researchers have proposed techniques to improve energy efficiency by detecting periods of low utilization and selectively turning off links [18, 28]. These proposals argue for monitoring traffic, detecting low utilization, and powering-down links in portions of the data center with low utilizations. A key challenge with these techniques is to turn on the powered-down links before demands rise. Taking a step back, these techniques roughly follow the same design and operate in three steps (1) gather network traffic matrix, (2) run an ILP to predict heavy (or low) usage, and (3) perform a specific action on a subset of the network. The actions range from augmenting flexible links, turning off links, or moving traffic. In all situations, the ILP does not scale to a large network and a domain-specific heuristic is often used in its place. 3.2 Reinforcement Learning Reinforcement learning (RL) algorithms learn through experience with a goal towards maximizing rewards. Unlike supervised learning where algorithms train over labels, RL algorithms learn by interacting with an environment such as a game or a network simulator. In traditional RL, an agent interacts with an environment over a number of discrete time steps. Hence, at each time step t, the agent in a world observes a state st in order to select an action at from a possible action set A. The agent is guided by a policy, π, which is a function that maps state st to actions at . The agent receives a reward rt for each action and transitions to the next state st+1 . The goal of the agent is maximizing the total reward. This process continues until the agent reaches a final state or time limit, after which the environment is reset and a new training episode is played. After a number of training episodes, the agent learns to pick actions that maximize the rewards and can learn to handle unexpected states. RL is effective and has been successfully used to model robotics, game bots, etc. The goal of commonly used policy-based RL is to find a policy, π, that maximizes the cumulative reward and converges to a theoretical optimal policy. In deep policy-based methods, a neural network computes a policy distribution π(at \|st ; θ), where θ represents the set of parameters of the function. Deep networks as function approximators is a recent development and other learning methods can be used. We now describe the REINFORCE and actor-critic policy methods which represent different methods to score the policy J(θ). REINFORCE methods use gradient ascent on E[Rt ], P∞ [34] i where Rt = i=0 γ rt+i is the accumulated reward starting from time step t and discounted at each step by γ ∈ (0, 1], the discount factor. The REINFORCE method, which is the Monte-Carlo method, updates θ using the gradient ∇θ log π(at \|st ; θ)Rt , which is an unbiased estimator of ∇θ E[Rt ]. The value function is computed as V π (st ) = E[Rt \|st ] which is the expected return for following the policy π in state st . This method provides actions with high returns but suffers from highvariance of gradient estimates. Asynchronous Advantage Actor Critic (A3C): A3C [24] improves REINFORCE performance by operating asynchronously and by using a deep network to approximate the policy and value faction. A3C uses the actor-critic method which additionally computes a critic function that approximates the value function. A3C, as used by us, uses a network with two convolutional layers followed by a fully connected layer. Each hidden layer is followed by a nonlinearity function (ReLU). A softmax layer which approximates the policy function and a linear layer to output an estimate of the value function V (st ; θ) together constitute the output of this network. Asynchronous gradient descent using multiple agents is used to train the network and this improves the training speed. A central server (similar to a parameter server) coordinates the parallel agents – each agent calculates the gradients and sends the updates to the server after a fixed number of steps, or when a final state is reached. Furthermore, following each update, the central server propagates new weights to the agents to achieve a consensus on the policy values. There is a cost function with each deep network (policy and value). Using two loss functions has found to improve convergence and produce better-regularized models. The policy cost function is given as: fπ (θ) = log π (at \|st ; θ) (Rt − V (st ; θt ))+βH (π (st ; θ)) (3.1) where θt represents the values of the parameters θ at Pk−1 time t, Rt = i=0 γ i rt+i + γ k V (st+k ; θt ) is the estimated discounted reward.H (π (st ; θ)) is used to favor exploration and its strength is controlled by the factor β. The cost function for the estimated value function is: 2 fv (θ) = (Rt − V (st ; θ)) (3.2) Additionally, we augment our A3C model to learn current states apart from accumulating rewards for good configurations using GAE [30]. The deep network, that replaces the transition matrix as the function approximator learns the value of the given state and the policy of the given state. The model uses GAE to compute the value of a given state that not only returns the reward for the model for the given policy decision but also rewards the model for estimating the value of the state. This helps to guide the model to learn the states instead of just maximizing rewards. 4. VISION Training Harness Network Simulator DeepConf Agent DeepConf DeepConf …… Agent Agent DeepConf Abstraction Layer SDN Controller …… Network Topology Figure 2: DeepConf Architecture Our vision is to automate a subset of data center management and operational tasks by leveraging DeepRL. At a high-level, we anticipate the existence of several DeepRL agents, each trained for a specific set of tasks e.g. traffic-engineering, energy-savings, or topologyaugmentations. Each agent will run as an application atop an SDN controller. The use of SDN provides the agents with an interface for gathering their required network state and a mechanism for enforcing their actions. For example, DeepConf should be able to assemble the traffic matrix by polling the different devices within the network, compute a decision for how the optical switch should be configured to best accommodate the current network load, and reconfigure the network. At a high-level, DeepConf’s architecture consists of three components (Figure 2): the network simulator to enable offline training of the DeepRL agents, the DeepConf abstraction layer to facilitate communication between the DeepRL agents and the network, and the DeepRL agents, called DeepConf-agents, which encapsulate data center functionality. Applying learning: The primary challenges in applying machine learning to network problems are (1) the deficiency of training data pertaining to operator and network behavior and (2) the lack of models and loss functions that can accurately model the problem and generalize to unexpected situations. This shortage presents a roadblock for using supervised-based approaches for training ML models. To address this issue, DeepConf uses RL where the model is trained by exploring different network states and environments generated by the surplus of simulators available in the network community. Coupled with the wide availability of network job traces, this allows for DeepConf to learn a highly generalizable policy. DeepConf Abstraction Layer: Today’s SDN controllers expose a primitive interface with low-level information. The DeepConf applications will instead require highlevel models and interfaces. For example, our agents will require interfaces that provide control over paths rather than over flow table entries. While emerging approaches [19] argue for similar interfaces, these approaches do not provide a sufficiently rich set of interfaces for the broad range of agents we expect to support and do not provide composition primitives for safely combining the output from the different agents. Moreover, existing composition operators [27, 20, 12] assume that the different SDN applications (or DeepConf-agent in our case) are generating non-conflicting actions – hence these operators can not tackling conflict actions. SDN Composition approaches [29, 7, 26] that do tackle conflicting actions, require significant rewrite of the SDNApp which we are unable to do because DeepConfagent are rewritten within the DeepRL paradigm. More concretely, we require high-layer SDN abstractions that enable to RLAgents to more easily learn and act of the network. Additionally, we require novel composition operators that can reason about and tackle conflicting actions generated by RLAgents. Domain-specific Simulators: (Low hanging fruit) Existing SDN research leverages a popular emulation platform, Mininet, which fails to scale to large experiments. A key requirement for employing DeepRL is to have efficient and scalable simulators that replays traces and enables learning from these traces. We extend flowbased simulators to model the various dimensions that are required to train our models. To improve efficiency, we explore techniques that partition the simulation and enables reuse of results — in essence, to enable incremental simulations. In addressing our high-level vision and developing solutions to the above challenges, there are several highlevel goals that a production-scale system must address: (1) our techniques must generalize across topologies, traffic matrixes, and a range of operational settings, e.g., link failures; (2) our techniques must be as accurate and efficient as existing state-of-the-art techniques; and (3) our solutions must incur low operational overheads, e.g., minimizing optical switching time or TCAM utilization. 5. DESIGN In this section, we provide a broad description of how to define and structure existing data center network management techniques as RL tasks, then describe the methods for training the resulting DeepConf-agents. 5.1 DeepConf Agents In defining each new DeepConf-agent, there are four main functions that a developer must specify: state space, action space, learning model, and reward function. The action space and reward are both specific to the management task being performed and are, in turn, unique to the agent. Respectively, they express the set of actions an agent can take during each step and the reward for the actions taken. The state space and learning models are more general and can be shared and reused across different DeepConf-agents. This is because of the fundamental similarities shared between the data center management problems, and because the agents are specifically designed for data centers. In defining a DeepConf agent for the topology augmentation problem, we (1) define state-spaces specific to the topology, (2) design a reward function based on application level metrics, and (3) define actions that correlate to activating/de-activating links. State Space: In general, the state space consists of two types of data – each reflecting the state of the network at a given point in time. First, the general network state that all DeepConf-agents require: the network’s traffic matrix (TM) which contains information on the flows which will executed during the last t seconds of the simulation. Second, a DeepConf-agent specific state-space that captures the impact of the actions on the network. For example, for the topology augmentation problem, this would be the network topology – note, the actions change the topology. Whereas for the traffic engineering problem, this would be a mapping of flows to paths – note, the actions change a flow’s routes. Learning Model: Our learning model utilizes a Convolutional Neural Network (CNN) to compute policy decisions. The exact model for a DeepConf-agent depends on the number of state-spaces used as input. In general, the model will have as many CNN-blocks as there are state spaces – one CNN-block for each state space. The output of these blocks are concatenated together and input into two fully connected layers, followed by a softmax output layer. For example, for the topology-augmentation problem, as observed in Figure 4, our DeepConf-agent has two CNN blocks to operate on both the topology and the TM states spaces in parallel. This allows for the lower CNN layers to perform feature extraction on the input spatial data, and for the fully connected layers to assemble these features in a meaningful way. 5.2 Network Training To train a DeepConf-agent, we run the agent against a network simulator. The interaction between the simulator (described in Section 7) and RL agent can be described as follows (Figure 3): (1) The DeepConf agent receives state si from the simulator at training step i. (2) The DeepConf-agent uses the state information to make a policy decision about the network, and returns the selected actions to the simulator. For the DeepConfagent for the topology augmentation problem, called the Augmentation-Agent, the actions are the links to activate and hence a new topology. (3) If the topology (5) Get reward Simulator Flows Network Topology (4) Update route (3) Add links Control Interface (1) Read state DeepConf Agent (2) Decision Figure 3: DeepConf-agent model training. changes, the simulator re-computes the paths for the active flows. (4) The simulator executes the flows for t seconds. (5) The simulator returns the reward ri and state si+1 to the DeepConf-agent, and the process restarts. Initialization During the initial training phase, we force the model to explore the environment by randomizing the selection process — the probability that an action is picked corresponds to the value of the index which represents the action. For instance, with the AugmentationAgent, link i has probability wi of being selected. As the model becomes more familiar with the states and corresponding values, the model will better formulate its policy decision. At this point, the model will associate a higher probability with the links it believes to have a higher reward, which will cause these links to be selected at a higher frequency. This methodology allows for the model to reinforce its decisions about links, while the randomization helps the model avoid local-minima. Learning Optimization: To improve the efficiency of learning, the RL agent maintains a log containing the state, policy decision, and corresponding reward. The RL agent performs experience replay after n simulation steps. During replay, the log is unrolled to compute the policy loss across the specified number of steps using Equation 3.1. The agent is trained using Adam stochastic optimization [22], with an initial learning rate of 10−4 and a learning rate decay factor of 0.95. We found that a smaller learning rate and low decay helped the model better explore the environment and form a more optimal policy. 6. USE CASE: AUGMENTATION AGENT More formally defined, in the topology augmentation problem the data center consists of a fixed hierarchical topology and an optical switch, which connects all the top-of-rack switches. While the optical switch is physically connected to all ToR switches, unfortunately, the optical switch can only support a limited number of active links. Given this limitation, the rules for the topology problem are defined as: • The model must select n links to activate at a given step during the simulation. • The model receives a reward based on the link utilization and the flow duration. 12 DeepConf Optimal Flow Completion Time (s) 10 • All flows are routed using equal-cost multi-path routing (ECMP). State Space: The agent specific state space is the network topology, which is represented by a sparse matrix where entries within the cells correspond to active links within the network. Action Space: The RL agent interacts with the environment by adding links between edge switches. The action space for the model, therefore, corresponds to the different possible link combinations and is represented as an n dimensional vector. The values within the vector correspond to a probability distribution, where wi is equal to the probability of link i being the optimal pick for the given input state s. The model selects the highest n values from this distribution as the links that should be added to the network topology. Reward: The goal of the model can be summarized as: (1) Maximize link utilization and (ii) Minimize the average flow-completion time. With this in mind, we formulate our reward function as: X X bf (6.1) R(Θ, s, i) = tf 6 4 2 0 Figure 4: The CNN model utilized by the RL agent. • The model collects the reward on a per-link basis after t seconds of simulation. 8 FatTree Topology VL2 Figure 5: Median Flow Completion Time. simulator using a large scale map-reduce traces from Facebook [3]. We evaluate two state-of-the-art clos-style data center topologies: K=4 Fat-tree [5] and VL2 [15]. In our analysis, we focus on flow completion time (FCT) a metric which captures the duration between the first and last packet of a flow. We augment both topologies by adding an optical switch with four links. Here we compare DeepConf against the optimal solution derived from a linear program — Note: this optimal solution can not be solved with larger topologies [8]. Learning Results: The training results demonstrate that the RL agent learns to optimize its policy decision to increase the total reward received across each episode. We observed that the loss decreases as training increases, with the largest decrease occurs during the initial training episodes, a result consistent with the learning rate decay factor employed during training. Performance Results: The results, Figure 5, show that DeepConf performs comparable with optimal [33, 11] across representative topologies and workloads. Thus, our system is able to learn a solution that’s close to the optimal across a range of topologies. Takeaway We believe these initial results are promising, and that more work is required in order to understand and improve the performance of DeepConf. f ∈F l∈f Where F represents all active and completed flows during the previous iteration step, l represents the links used by flow f , bf represents the number of bytes transferred during the step time, and tf represents the total duration of the flow. The purpose of this reward function is to reward for high link utilization but penalize for long lasting flows. The design of this function has the effect of guiding the model towards completing large flows within a smaller period of time. 7. EVALUATION In this section, we analyze DeepConf under realistic workload with representative topologies. Experiment Setup We evaluate DeepConf on a trace driven flow-level 8. DISCUSSION We now discuss open questions: Learning to generalize: In order to avoid over-fitting to a specific topology, we train our agent over a large number of simulator configurations. DeepRL agents need to be trained and evaluated on many different platforms to avoid being overtly specific to few networks and correctly handle unexpected scenarios. Solutions that employ machine learning to address network problems using simulators need to be cognizant of these issues when deciding the training data. Learning new reward functions: DeepRL methods need appropriate reward functions to ensure that they optimize for the correct goals. 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arXiv:1507.06792v2 [stat.ME] 31 Mar 2017 Efficient Estimation for Diffusions Sampled at High Frequency Over a Fixed Time Interval Nina Munkholt Jakobsen Michael Sørensen∗ Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Ø Denmark munkholt@math.ku.dk Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Ø Denmark michael@math.ku.dk April 3, 2017 Abstract Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (in-fill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variance-mixtures, where the mixing distribution generally depends on the full sample path of the diffusion process over the observation time interval. Utilising the concept of stable convergence, we also obtain the more easily applicable result that for a suitable data dependent normalisation, the estimators converge in distribution to a standard normal distribution. The theory is illustrated by a simulation study comparing an efficient and a non-efficient estimating function for an ergodic and a non-ergodic model. Key words: Approximate martingale estimating functions, discrete time sampling of diffusions, in-fill asymptotics, normal variance-mixtures, optimal rate, random Fisher information, stable convergence, stochastic differential equation. Running title: Efficient Estimation for High Frequency SDE Data. 1 1 Introduction Diffusions given by stochastic differential equations find application in a number of fields where they are used to describe phenomena which evolve continuously in time. Some examples include agronomy (Pedersen, 2000), biology (Favetto and Samson, 2010), finance (Merton, 1971; Vasicek, 1977; Cox et al., 1985; Larsen and Sørensen, 2007) and neuroscience (Ditlevsen and Lansky, 2006; Picchini et al., 2008; Bibbona et al., 2010). While the models have continuous-time dynamics, data are only observable in discrete time, thus creating a demand for statistical methods to analyse such data. With the exception of some simple cases, the likelihood function is not explicitly known, and a large variety of alternate estimation procedures have been proposed in the literature, see e.g. Sørensen (2004) and Kessler et al. (2012). Parametric methods include the following. Maximum likelihood-type estimation, primarily using Gaussian approximations to the likelihood function, was considered by Prakasa Rao (1983), Florens-Zmirou (1989), Yoshida (1992), Genon-Catalot and Jacod (1993), Kessler (1997), Jacod (2006), Gloter and Sørensen (2009) and Uchida and Yoshida (2013). Analytical expansions of the transition densities were investigated by Aït-Sahalia (2002, 2008) and Li (2013), while approximations to the score function were studied by Bibby and Sørensen (1995), Kessler and Sørensen (1999), Jacobsen (2001, 2002), Uchida (2004), and Sørensen (2010). Simulation-based likelihood methods were developed by Pedersen (1995), Roberts and Stramer (2001), Durham and Gallant (2002), Beskos et al. (2006, 2009), Golightly and Wilkinson (2006, 2008), Bladt and Sørensen (2014), and Bladt et al. (2016). A large part of the parametric estimators proposed in the literature can be treated within the framework of approximate martingale estimating functions, see the review in Sørensen (2012). In this paper, we derive easily verified conditions on such estimating functions that imply rate optimality and efficiency under a high frequency asymptotic scenario, and thus contribute to providing clarity and a systematic approach to this area of statistics. Specifically, the paper concerns parametric estimation for stochastic differential equations of the form dXt = a(Xt ) dt + b(Xt ; θ) dWt , (1.1) where (Wt )t≥0 is a standard Wiener process. The drift and diffusion coefficients a and b are deterministic functions, and θ is the unknown parameter to be estimated. The drift function a needs not be known, but as examples in this paper show, knowledge of a can be used in the construction of estimating functions. For ease of exposition, Xt and θ are both assumed to be one-dimensional. The extension of our results to a multivariate parameter is straightforward, and it is expected that multivariate diffusions can be treated in a similar way. For n ∈ N, we consider observations (Xt0n , Xt1n , . . . , Xtnn ) in the time interval [0, 1], at discrete, equidistant time-points tin = i/n, i = 0, 1, . . . , n. We investigate the high frequency scenario where n → ∞. The choice of the time-interval [0, 1] is not restrictive since results generalise to other compact intervals by suitable rescaling of the drift and diffusion coef2 ficients. The drift coefficient does not depend on any parameter, because parameters that appear only in the drift cannot be estimated consistently in our asymptotic scenario. It was shown by Dohnal (1987) and Gobet (2001) that under the asymptotic scenario con√ sidered here, the model (1.1) is locally asymptotic mixed normal with rate n and random asymptotic Fisher information !2 Z 1 ∂θ b(X s ; θ) I(θ) = 2 ds. (1.2) b(X s ; θ) 0 √ Thus, a consistent estimator θ̂n is rate optimal if n(θ̂n − θ0 ) converges in distribution to a non-degenerate random variable as n → ∞, where θ0 is the true parameter value. The estimator is efficient if the limit may be written on the form I(θ0 )−1/2 Z, where Z is standard normal distributed and independent of I(θ0 ). The concept of local asymptotic mixed normality was introduced by Jeganathan (1982), and is discussed in e.g. Le Cam and Yang (2000, Chapter 6) and Jacod (2010). Estimation for the model (1.1) under the high frequency asymptotic scenario described above was considered by Genon-Catalot and Jacod (1993, 1994). These authors proposed estimators based on a class of contrast functions that were only allowed to depend on the −1/2 n ; θ) and ∆ n ). The estimators observations and the parameter through b2 (Xti−1 (Xtin − Xti−1 n were shown to be rate optimal, and an efficient contrast function was identified. Dohnal (1987) gave estimators for particular cases of the model (1.1). Apart from one instance, these estimators are not of the type investigated by Genon-Catalot and Jacod (1993, 1994), but all apart from one are covered by the theory in the present paper. In this paper, we investigate estimators based on the extensive class of approximate martingale estimating functions Gn (θ) = n X n ; θ) g(∆n , Xtin , Xti−1 i=1 n ; θ) \| with ∆n = 1/n, where the real-valued function g(t, y, x; θ) satisfies that Eθ (g(∆n , Xtin , Xti−1 κ n ) is of order ∆ for some κ ≥ 2. Estimators are obtained as solutions to the estimatXti−1 n ing equation Gn (θ) = 0 and are referred to as Gn -estimators. Exact martingale estimating functions, where Gn (θ) is a martingale, constitute a particular case that is not covered by the theory in Genon-Catalot and Jacod (1993, 1994). An example is the maximum likeli√ hood estimator for the Ornstein-Uhlenbeck process with a(x) = −x and b(x; θ) = θ, for which g(t, y, x; θ) = (y − e−t x)2 − 21 θ(1 − e−2t ). A simpler example of an estimating function for the same Ornstein-Uhlenbeck process that is covered by our theory, but is not of the Genon-Catalot & Jacod-type, is given by g(t, y, x; θ) = (y − (1 − t)x)2 − θt. The class of approximate martingale estimating functions was also studied by Sørensen (2010), who considered high frequency observations in an increasing time interval for a model like (1.1) where also the drift coefficient depends on a parameter. Specifically, the observation times were tin = i∆n with ∆n → 0 and n∆n → ∞. Simple conditions on g for rate optimality and efficiency were found under the infinite horizon high frequency 3 asymptotics. To some extent, the methods of proof in the present paper are similar to those in Sørensen (2010). However, while ergodicity of the diffusion process played a central role in that paper, this property is not needed here. Another important difference is that expansions of a higher order are needed in the present paper, which complicates the proofs considerably. Furthermore, the theory in the current paper requires a more complicated version of the central limit theorem for martingales, and we need the concept of stable convergence in distribution, in order to obtain practically applicable convergence results. First, we establish results on existence and uniqueness of consistent Gn -estimators. We √ show that n(θ̂n − θ0 ) converges in distribution to a normal variance-mixture, which implies rate optimality. The limit distribution may be represented by the product W(θ0 )Z of independent random variables, where Z is standard normal distributed. The random variable W(θ0 ) is generally non-degenerate, and depends on the entire path of the diffusion process over the time-interval [0, 1]. Normal variance-mixtures were also obtained as the asymptotic distributions of the estimators of Genon-Catalot and Jacod (1993). These distributions appear as limit distributions in comparable non-parametric settings as well, e.g. when estimating integrated volatility (Jacod and Protter, 1998; Mykland and Zhang, 2006) or the squared diffusion coefficient (Florens-Zmirou, 1993; Jacod, 2000). Rate optimality is ensured by the condition that ∂y g(0, x, x; θ) = 0 (1.3) for all x in the state space of the diffusion process, and all parameter values θ. Here ∂y g(0, x, x; θ) denotes the first derivative of g(0, y, x; θ) with respect to y evaluated in y = x. The same condition was found in Sørensen (2010) for rate optimality of an estimator of the parameter in the diffusion coefficient, and it is one of the conditions for small ∆-optimality; see Jacobsen (2001, 2002). Due to its dependence on (X s ) s∈[0,1] , the limit distribution is difficult to use for statistical applications, such as constructing confidence intervals and test statistics. Therefore, we b that converges in probability to W(θ0 ). Using the stable convergence construct a statistic W √ n in distribution of n(θ̂n − θ0 ) towards W(θ0 )Z, we derive the more easily applicable result √ b n−1 (θ̂n − θ0 ) converges in distribution to a standard normal distribution. that n W The additional condition that ∂2y g(0, x, x; θ) = Kθ ∂θ b2 (x; θ) b4 (x; θ) (1.4) (Kθ , 0) for all x in the state space, and all parameter values θ, ensures efficiency of Gn estimators. The same condition was obtained by Sørensen (2010) in his infinite horizon scenario for efficiency of estimators of parameters in the diffusion coefficient. It is also identical to a condition given by Jacobsen (2002) for small ∆-optimality. The identity of the conditions implies that examples of approximate martingale estimating functions which are rate optimal and efficient in our asymptotic scenario may be found in Jacobsen (2002) and Sørensen (2010). In particular, estimating functions that are optimal in the sense of Godambe and Heyde (1987) are rate optimal and efficient under weak regularity conditions. 4 The paper is structured as follows: Section 2 presents definitions, notation and terminology used throughout the paper, as well as the main assumptions. Section 3 states and discusses our main results, while Section 4 presents a simulation study illustrating the results. Section 5 contains main lemmas used to prove the main theorem, and proofs of the main theorem and the lemmas. Appendix A consists of auxiliary technical results, some of them with proofs. 2 2.1 Preliminaries Model and Observations Let (Ω, F ) be a measurable space supporting a real-valued random variable U, and an independent standard Wiener process W = (Wt )t≥0 . Let (Ft )t≥0 denote the filtration generated by U and W. Consider the stochastic differential equation dXt = a(Xt ) dt + b(Xt ; θ) dWt , X0 = U , (2.1) for θ ∈ Θ ⊆ R. The state space of the solution is assumed to be an open interval X ⊆ R, and the drift and diffusion coefficients, a : X → R and b : X × Θ → R, are assumed to be known, deterministic functions. Let (Pθ )θ∈Θ be a family of probability measures on (Ω, F ) such that X = (Xt )t≥0 solves (2.1) under Pθ , and let Eθ denote expectation under Pθ . Let tin = i∆n with ∆n = 1/n for i ∈ N0 , n ∈ N. For each n ∈ N, X is assumed to be sampled at times tin , i = 0, 1, . . . , n, yielding the observations (Xt0n , Xt1n , . . . , Xtnn ). Let Gn,i denote the σ-algebra generated by the observations (Xt0n , Xt1n , . . . , Xtin ), with Gn = Gn,n . 2.2 Polynomial Growth In the following, to avoid cumbersome notation, C denotes a generic, strictly positive, realvalued constant. Often, the notation Cu is used to emphasise that the constant depends on u in some unspecified manner, where u may be, e.g., a number or a set of parameter values. Note that, for example, in an expression of the form Cu (1 + \|x\|Cu ), the factor Cu and the exponent Cu need not be equal. Generic constants Cu often depend (implicitly) on the unknown true parameter value θ0 , but never on the sample size n. A function f : [0, 1] × X2 × Θ → R is said to be of polynomial growth in x and y, uniformly for t ∈ [0, 1] and θ in compact, convex sets, if for each compact, convex set K ⊆ Θ there exist constants C K > 0 such that sup t∈[0,1], θ∈K \| f (t, y, x; θ)\| ≤ C K (1 + \|x\|C K + \|y\|C K ) for x, y ∈ X. pol Definition 2.1. C p,q,r ([0, 1] × X2 × Θ) denotes the class of real-valued functions f (t, y, x; θ) which satisfy that 5 j (i) f and the mixed partial derivatives ∂it ∂y ∂kθ f (t, y, x; θ), i = 0, . . . , p, j = 0, . . . , q and k = 0, . . . , r exist and are continuous on [0, 1] × X2 × Θ. (ii) f and the mixed partial derivatives from (i) are of polynomial growth in x and y, uniformly for t ∈ [0, 1] and θ in compact, convex sets. pol pol pol pol Similarly, the classes C p,r ([0, 1] × X × Θ), Cq,r (X2 × Θ), Cq,r (X × Θ) and Cq (X) are defined for functions of the form f (t, x; θ), f (y, x; θ), f (y; θ) and f (y), respectively. Note that in Definition 2.1, differentiability of f with respect to x is never required. For the duration of this paper, R(t, y, x; θ) denotes a generic, real-valued function defined on [0, 1] × X2 × Θ, which is of polynomial growth in x and y uniformly for t ∈ [0, 1] and θ in compact, convex sets. The function R(t, y, x; θ) may depend (implicitly) on θ0 . Functions R(t, x; θ), R(y, x; θ) and R(t, x) are defined correspondingly. The notation Rλ (t, x; θ) indicates that R(t, x; θ) also depends on λ ∈ Θ in an unspecified way. 2.3 Approximate Martingale Estimating Functions Definition 2.2. Let g(t, y, x; θ) be a real-valued function defined on [0, 1]×X2 ×Θ. Suppose the existence of a constant κ ≥ 2, such that for all n ∈ N, i = 1, . . . , n, θ ∈ Θ, n ). n ; θ) \| Xtn (2.2) = ∆κn Rθ (∆n , Xti−1 Eθ g(∆n , Xtin , Xti−1 i−1 Then, the function Gn (θ) = n X n ; θ) g(∆n , Xtin , Xti−1 (2.3) i=1 is called an approximate martingale estimating function. In particular, when (2.2) is satisfied with Rθ (t, x) ≡ 0, (2.3) is referred to as a martingale estimating function. By the Markov property of X, it follows that if Rθ (t, x) ≡ 0, then (Gn,i )1≤i≤n defined by Gn,i (θ) = i X g(∆n , Xtnj , Xtnj−1 ; θ) j=1 is a zero-mean, real-valued (Gn,i )1≤i≤n -martingale under Pθ for each n ∈ N. The score function of the observations (Xt0n , Xt1n , . . . , Xtnn ) is a martingale estimating function under weak regularity conditions, and an approximate martingale estimating function can be viewed as an approximation to the score function. A Gn -estimator θ̂n is essentially obtained as a solution to the estimating equation Gn (θ) = 0. A more precise definition is given in the following Definition 2.3. Here we make the ωdependence explicit by writing Gn (θ, ω) and θ̂n (ω). Definition 2.3. Let Gn (θ, ω) be an approximate martingale estimating function as defined in Definition 2.2. Put Θ∞ = Θ ∪ {∞} and let Dn = {ω ∈ Ω \| Gn (θ, ω) = 0 has at least one solution θ ∈ Θ} . 6 A Gn -estimator θ̂n (ω) is any Gn -measurable function Ω → Θ∞ which satisfies that for Pθ0 almost all ω, θ̂n (ω) ∈ Θ and Gn (θ̂n (ω), ω) = 0 if ω ∈ Dn , while θ̂n (ω) = ∞ if ω < Dn . For any Mn , 0, the estimating functions Gn (θ) and MnGn (θ) yield identical estimators of θ and are therefore referred to as versions of each other. For any given estimating function, it is sufficient that there exists a version of the function which satisfies the assumptions of this paper, in order to draw conclusions about the resulting estimators. In particular, we can multiply by a function of ∆n . 2.4 Assumptions We make the following assumptions about the stochastic differential equation. Assumption 2.4. The parameter set Θ is a non-empty, open subset of R. Under the probability measure Pθ , the continuous, (Ft )t≥0 -adapted Markov process X = (Xt )t≥0 solves a stochastic differential equation of the form (2.1), the coefficients of which satisfy that pol a(y) ∈ C6 (X) pol b(y; θ) ∈ C6,2 (X × Θ) . and The following holds for all θ ∈ Θ. (i) For all y ∈ X, b2 (y; θ) > 0. (ii) There exists a real-valued constant Cθ > 0 such that for all x, y ∈ X, \|a(x) − a(y)\| + \|b(x; θ) − b(y; θ)\| ≤ Cθ \|x − y\| . (iii) U has moments of any order. The global Lipschitz condition, Assumption 2.4.(ii), ensures that a unique solution X exists. The Lipschitz condition and (iii) imply that supt∈[0,1] Eθ (\|Xt \|m ) < ∞ for all m ∈ N. Assumption 2.4 is very similar to the corresponding Condition 2.1 of Sørensen (2010). However, an important difference is that in the current paper, X is not required to be ergodic. Here, law of large numbers-type results are proved by what is, in essence, the convergence of Riemann sums. We make the following assumptions about the estimating function. Assumption 2.5. The function g(t, y, x; θ) satisfies (2.2) for some κ ≥ 2, thus defining an approximate martingale estimating function by (2.3). Moreover, pol g(t, y, x; θ) ∈ C3,8,2 ([0, 1] × X2 × Θ) , and the following holds for all θ ∈ Θ. (i) For all x ∈ X, ∂y g(0, x, x; θ) = 0. 7 (ii) The expansion g(∆, y, x; θ) = g(0, y, x; θ) + ∆g(1) (y, x; θ) + 12 ∆2 g(2) (y, x; θ) + 16 ∆3 g(3) (y, x; θ) + ∆4 R(∆, y, x; θ) (2.4) holds for all ∆ ∈ [0, 1] and x, y ∈ X, where g( j) (y, x; θ) denotes the j0 th partial derivative of g(t, y, x; θ) with respect to t, evaluated in t = 0. Assumption 2.5.(i) was referred to by Sørensen (2010) as Jacobsen’s condition, as it is one of the conditions for small ∆-optimality in the sense of Jacobsen (2001), see Jacobsen (2002). The assumption ensures rate optimality of the estimators in this paper, and of the estimators of the parameters in the diffusion coefficient in Sørensen (2010). The assumptions of polynomial growth and existence and boundedness of all moments serve to simplify the exposition and proofs, and could be relaxed. 2.5 The Infinitesimal Generator pol For λ ∈ Θ, the infinitesimal generator Lλ is defined for all functions f (y) ∈ C2 (X) by Lλ f (y) = a(y)∂y f (y) + 21 b2 (y; λ)∂2y f (y) . pol For f (t, y, x, θ) ∈ C0,2,0,0 ([0, 1] × X2 × Θ), let Lλ f (t, y, x; θ) = a(y)∂y f (t, y, x; θ) + 21 b2 (y; λ)∂2y f (t, y, x; θ) . (2.5) Often, the notation Lλ f (t, y, x; θ) = Lλ ( f (t; θ))(y, x) is used, so e.g. Lλ ( f (0; θ))(x, x) means Lλ f (0, y, x; θ) evaluated in y = x. In this paper the infinitesimal generator is particularly useful because of the following result. Lemma 2.6. Suppose that Assumption 2.4 holds, and that for some k ∈ N0 , pol a(y) ∈ C2k (X) , pol b(y; θ) ∈ C2k,0 (X × Θ) and pol f (y, x; θ) ∈ C2(k+1),0 (X2 × Θ) . Then, for 0 ≤ t ≤ t + ∆ ≤ 1 and λ ∈ Θ, Eλ ( f (Xt+∆ , Xt ; θ) \| Xt ) = k X ∆i i=0 i! Liλ f (Xt , Xt ; θ) + ∆ Z u1 Z uk Z ··· 0 0 0 Eλ Lk+1 f (X , X ; θ) \| X duk+1 · · · du1 t+u t t k+1 λ where, furthermore, Z ∆ Z u1 Z ··· 0 0 0 uk Eλ Lk+1 f (X , X ; θ) \| X duk+1 · · · du1 = ∆k+1 Rλ (∆, Xt ; θ) . t+u t t k+1 λ 8 The expansion of the conditional expectation in powers of ∆ in the first part of the lemma corresponds to Lemma 1 in Florens-Zmirou (1989) and Lemma 4 in Dacunha-Castelle and Florens-Zmirou (1986). It may be proven by induction on k using Itô’s formula, see, e.g., the proof of Sørensen (2012, Lemma 1.10). The characterisation of the remainder term follows by applying Corollary A.5 to Lk+1 λ f , see the proof of Kessler (1997, Lemma 1). For concrete models, Lemma 2.6 is useful for verifying the approximate martingale property (2.2) and for creating approximate martingale estimating functions. In combination with (2.2), the lemma is key to proving the following Lemma 2.7, which reveals two important properties of approximate martingale estimating functions. Lemma 2.7. Suppose that Assumptions 2.4 and 2.5 hold. Then g(0, x, x; θ) = 0 and g(1) (x, x; θ) = −Lθ (g(0, θ))(x, x) for all x ∈ X and θ ∈ Θ. Lemma 2.7 corresponds to Lemma 2.3 of Sørensen (2010), to which we refer for details on the proof. 3 Main Results Section 3.1 presents the main theorem of this paper, which establishes existence, uniqueness and asymptotic distribution results for rate optimal estimators based on approximate martingale estimating functions. In Section 3.2 a condition is given, which ensures that the rate optimal estimators are also efficient, and efficient estimators are discussed. 3.1 Main Theorem The final assumption needed for the main theorem is as follows. Assumption 3.1. The following holds Pθ -almost surely for all θ ∈ Θ. (i) For all λ , θ, 1 Z 0 b2 (X s ; θ) − b2 (X s ; λ) ∂2y g(0, X s , X s ; λ) ds , 0 , (ii) 1 Z 0 ∂θ b2 (X s ; θ)∂2y g(0, X s , X s ; θ) ds , 0 , (iii) 1 Z 0 2 b4 (X s ; θ) ∂2y g(0, X s , X s ; θ) ds , 0 . 9 Assumption 3.1 can be difficult to check in practice because it involves the full sample path of X over the interval [0, 1]. It requires, in particular, that for all θ ∈ Θ, with Pθ -probability one, t 7→ b2 (Xt ; θ) − b2 (Xt ; λ) is not Lebesgue-almost surely zero when λ , θ. As noted by Genon-Catalot and Jacod (1993), this requirement holds true (by the continuity of the function) if, for example, X0 = U is degenerate at x0 , and b2 (x0 ; θ) , b2 (x0 ; λ) for all θ , λ. For an efficient estimating function, Assumption 3.1 reduces to conditions on X with no further conditions on the estimating function, see the next section. Specifically, the conditions involve only the squared diffusion coefficient b2 (x; θ) and its derivative ∂θ b2 . Theorem 3.2. Suppose that Assumptions 2.4, 2.5 and 3.1 hold. Then, (i) there exists a consistent Gn -estimator θ̂n . Choose any compact, convex set K ⊆ Θ with θ0 ∈ int K, where int K denotes the interior of K. Then, the consistent Gn estimator θ̂n is eventually unique in K, in the sense that for any Gn -estimator θ̃n with Pθ0 (θ̃n ∈ K) → 1 as n → ∞, it holds that Pθ0 (θ̂n , θ̃n ) → 0 as n → ∞. (ii) for any consistent Gn -estimator θ̂n , it holds that √ D n(θ̂n − θ0 ) −→ W(θ0 )Z . (3.1) The limit distribution is a normal variance-mixture, where Z is standard normal distributed, and independent of W(θ0 ) given by 1 Z W(θ0 ) = 0 1 4 2 b (X s ; θ0 ) 1 Z 0 2 ∂2y g(0, X s , X s ; θ0 ) !1/2 ds . (3.2) 2 2 1 2 ∂θ b (X s ; θ0 )∂y g(0, X s , X s ; θ0 ) ds (iii) for any consistent Gn -estimator θ̂n , bn = − W  1/2 n  1 X  2  n ; θ̂n ) g (∆n , Xtin , Xti−1  ∆n i=1 n X (3.3) n ; θ̂n ) ∂θ g(∆n , Xtin , Xti−1 i=1 P b n −→ W(θ0 ), and satisfies that W √ D bn−1 (θ̂n − θ0 ) −→ nW N(0, 1) . The proof of Theorem 3.2 is given in Section 5.1. 10 Dohnal (1987) and Gobet (2001) showed local asymptotic mixed normality with rate so Theorem 3.2 establishes rate optimality of Gn -estimators. √ n, Observe that the limit distribution in Theorem 3.2.(ii) generally depends on not only the unknown parameter θ0 , but also on the concrete realisation of the sample path t 7→ Xt over [0, 1], which is only partially observed. Note also that a variance-mixture of normal distributions can be very different from a Gaussian distribution. It can be much more heavytailed and even have no moments. Theorem 3.2.(iii) is therefore important as it yields a standard normal limit distribution, which is more useful in practical applications. 3.2 Efficiency Under the assumptions of Theorem 3.2, the following additional condition ensures efficiency of a consistent Gn -estimator. Assumption 3.3. Suppose that for each θ ∈ Θ, there exists a constant Kθ , 0 such that for all x ∈ X, ∂2y g(0, x, x; θ) = Kθ ∂θ b2 (x; θ) . b4 (x; θ) Dohnal (1987) and Gobet (2001) showed that the local asymptotic mixed normality property holds within the framework considered here with random Fisher information I(θ0 ) given by (1.2). Thus, a Gn -estimator θ̂n is efficient if (3.1) holds with W(θ0 ) = I(θ0 )−1/2 , and the following Corollary 3.4 may easily be verified. Corollary 3.4. Suppose that the assumptions of Theorem 3.2 and Assumption 3.3 hold. Then, any consistent Gn -estimator is also efficient. It follows from Theorem 3.2 and Lemma 5.1 that if Assumption 3.3 holds, and if Gn is normalized such that Kθ = 1, then √ where 1 D bn2 (θ̂n − θ0 ) −→ N(0, 1) , nI n X bn = 1 n ; θ̂n ). I g2 (∆n , Xtin , Xti−1 ∆n i=1 It was noted in Section 2.3 that not necessarily all versions of a particular estimating function satisfy the conditions of this paper, even though they lead to the same estimator. Thus, an estimating function is said to be efficient, if there exists a version which satisfies the conditions of Corollary 3.4. The same goes for rate optimality. Assumption 3.3 is identical to the condition for efficiency of estimators of parameters in the diffusion coefficient in Sørensen (2010), and to one of the conditions for small ∆-optimality given in Jacobsen (2002). 11 Under suitable regularity conditions on the diffusion coefficient b, the function ḡ(t, y, x; θ) = ∂θ b2 (x; θ) 2 2 (y − x) − tb (x; θ) b4 (x; θ) (3.4) yields an example of an efficient estimating function. The approximate martingale property (2.2) can be verified by Lemma 2.6. When adapted to the current framework, the contrast functions investigated by GenonCatalot and Jacod (1993) have the form Un (θ) = n 1X 2 −1/2 n ; θ), ∆ n ) , f b (Xti−1 (Xtin − Xti−1 n n i=1 for functions f (v, w) satisfying certain conditions. For the contrast function identified as efficient by Genon-Catalot and Jacod, f (v, w) = log v + w2 /v. Using that ∆n = 1/n, it is P n ; θ) then seen that their efficient contrast function is of the form Ūn (θ) = ni=1 ū(∆n , Xtin , Xti−1 with ū(t, y, x; θ) = t log b2 (x; θ) + (y − x)2 /b2 (x; θ) and ∂θ ū(t, y, x; θ) = −ḡ(t, y, x; θ). In other words, it corresponds to a version of the efficient approximate martingale estimating function given by (3.4). The same contrast function was considered by Uchida and Yoshida (2013) in the framework of a more general class of stochastic differential equations. A problem of considerable practical interest is how to construct estimating functions that are rate optimal and efficient, i.e. estimating functions satisfying Assumptions 2.5.(i) and 3.3. Being the same as the conditions for small ∆-optimality, the assumptions are, for example, satisfied for martingale estimating functions constructed by Jacobsen (2002). As discussed by Sørensen (2010), the rate optimality and efficiency conditions are also satisfied by Godambe-Heyde optimal approximate martingale estimating functions. Consider martingale estimating functions of the form g(t, y, x; θ) = a(x, t; θ)∗ f (y; θ) − φtθ f (x; θ) , where a and f are two-dimensional, ∗ denotes transposition, and φtθ f (x; θ) = Eθ ( f (Xt ; θ) \| X0 = x). Suppose that f satisfies appropriate (weak) conditions. Let ā be the weight function for which the estimating function is optimal in the sense of Godambe and Heyde (1987), see e.g. Heyde (1997) or Sørensen (2012, Section 1.11). It follows by an argument analogous to the proof of Theorem 4.5 in Sørensen (2010) that the estimating function with g(t, y, x; θ) = tā(x, t; θ)∗ [ f (y; θ) − φtθ f (x; θ)] satisfies Assumptions 2.5.(i) and 3.3, and is thus rate optimal and efficient. As there is a simple formula for ā (see Section 1.11.1 of Sørensen (2012)), this provides a way of constructing a large number of efficient estimating functions. The result also holds if φtθ f (x; θ) and the conditional moments in the formula for ā are suitably approximated by the help of Lemma 2.6. 12 Remark 3.5. Suppose for a moment that the diffusion coefficient of (2.1) has the form b2 (x; θ) = h(x)k(θ) for strictly positive functions h and k, with Assumption 2.4 satisfied. This holds true, e.g., for a number of Pearson diffusions, including the (stationary) Ornstein-Uhlenbeck and square root processes. (See Forman and Sørensen (2008) for more on Pearson diffusions.) Then I(θ0 ) = ∂θ k(θ0 )2 /(2k2 (θ0 )). In this case, under the assump√ tions of Corollary 3.4, an efficient Gn -estimator θ̂n satisfies that n(θ̂n − θ0 ) −→ Y in distribution where Y is normal distributed with mean zero and variance 2k2 (θ0 )/∂θ k(θ0 )2 , i.e. the limit distribution is not a normal variance-mixture depending on (Xt )t∈[0,1] . Note also that when b2 (x; θ) = h(x)k(θ) and Assumption 3.3 holds, then Assumption 3.1 is satisfied when, e.g., ∂θ k(θ) > 0 or ∂θ k(θ) < 0. ◦ 4 Simulation study This section presents a simulation study illustrating the theory in the previous section. An efficient and an inefficient estimating function are compared for two models, an ergodic and a non-ergodic model. For both models the limit distributions of the consistent estimators are non-degenerate normal variance-mixtures. First, consider the stochastic differential equation dXt = −2Xt dt + (θ + Xt2 )−1/2 dWt , X0 = 0, (4.1) where θ ∈ (0, ∞) is an unknown parameter. The solution X is ergodic with invariant probability density proportional to exp −2θx2 − x4 θ + x2 , x ∈ R. The process satisfies Assumption 2.4. We compare the two estimating functions given by n X Gn (θ) = g(∆n , X , X tin n ti−1 ; θ) and Hn (θ) = i=1 n X n ; θ) h(∆n , Xtin , Xti−1 i=1 where g(t, y, x; θ) = (y − (1 − 2t)x)2 − (θ + x2 )−1 t h(t, y, x; θ) = (θ + x2 )10 (y − (1 − 2t)x)2 − (θ + x2 )9 t . Both g and h satisfy Assumptions 2.5 and 3.1. Moreover, g satisfies the condition for efficiency, while h is not efficient. Let WG (θ0 ) and WH (θ0 ) be given by (3.2), that is 1 Z WG (θ0 ) = − Z 1 2 0 1 !−1/2 1 ds and WH (θ0 ) = − (θ0 + X s2 )2 !1/2 2(θ0 + 0 1 Z 0 X s2 )18 ds . (4.2) (θ0 + X s2 )8 ds Numerical calculations and simulations were done in R 3.1.3 (R Core Team, 2014). First, m = 104 trajectories of the process X given by (4.1) were simulated over the time-interval 13 ● ● ● 4 2 0 −2 −4 ● ● ● −2 0 2 4 4 ● ●● ●● 0 2 4 ● ● ●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● 2 0 ● ●● ● ● ● −4 −4 2 0 −2 −2 ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● −4 4 −4 ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● −2 −4 −2 0 2 ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● −4 −2 0 2 4 −4 −2 0 2 4 bG,n (left) and Z bH,n (right) to the N(0, 1) distribution in the Figure 1: QQ-plots comparing Z case of the ergodic model (4.1) for n = 103 (above) and n = 104 (below). [0, 1] with θ0 = 1. These simulations were performed using the Milstein scheme as implemented in the R-package sde (Iacus, 2014) with step size 10−5 . The simulations were subsampled to obtain samples sizes of n = 103 and n = 104 . Let θ̂G,n and θ̂H,n denote estimates of θ obtained by solving the equations Gn (θ) = 0 and Hn (θ) = 0 numerically, on bG,n and W b H,n were calculated by (3.3). the interval [0.01, 1, 99]. Using these estimates, W For n = 103 , θ̂H,n could not be computed for 30 of the m = 104 sample paths. For n = 104 , and for the efficient estimator θ̂G,n there were no problems. Figure 1 shows QQ-plots of bG,n = Z √ b −1 (θ̂G,n − θ0 ) nW G,n and bH,n = Z √ b −1 (θ̂H,n − θ0 ) , nW H,n compared with a standard normal distribution, for n = 103 and n = 104 respectively. These QQ-plots suggest that, as n goes to infinity, the asymptotic distribution in Theorem 3.2.(iii) becomes a good approximation faster in the efficient case than in the inefficient case. 14 0.25 5 0.20 4 0.15 3 0.10 2 0.00 0.05 1 0 1.4 1.6 1.8 2.0 2.2 2.4 2.6 5 10 15 eG Figure 2: Approximation to the densities of WG (θ0 ) (left) and WH (θ0 ) (right) based on W e and WH in the case of the ergodic model (4.1). Inserting θ0 = 1 into (4.2), the integrals in these expressions may be approximated by Riemann sums, using each of the simulated trajectories of X (with sample size n = 104 eG and W e H to for maximal accuracy). This method yields a second set of approximations W the realisations of the random variables WG (θ0 ) and WH (θ0 ), presumed to be more accurate bG,104 and W b H,104 as they utilise the true parameter value. The density function in R than W was used (with default arguments) to compute an approximation to the densities of WG (θ0 ) eG and W eH. and WH (θ0 ), using the approximate realisations W It is seen from Figure 2 that the distribution of WH (θ0 ) is much more spread out than the distribution of WG (θ0 ). This corresponds well to the limit distribution in Theorem 3.2.(ii) being more spread out in the inefficient case than in the efficient case. Along the same lines, √ √ Figure 3 shows similarly computed densities based on n(θ̂G,n − θ0 ) and n(θ̂H,n − θ0 ) for n = 104 , which may be considered approximations to the densities of the normal variancemixture limit distributions in Theorem 3.2.(ii). These plots also illustrate that the limit distribution of the inefficient estimator is more spread out than that of the efficient estimator. Now, consider the stochastic differential equation dXt = 2Xt dt + (θ + Xt2 )−1/2 dWt , X0 = 0. (4.3) For this model, the solution X is not ergodic, but again Assumption 2.4 holds. We compare the two estimating functions given by g(t, y, x; θ) = (y − (1 + 2t)x)2 − (θ + x2 )−1 t h(t, y, x; θ) = (θ + x2 )10 (y − (1 + 2t)x)2 − (θ + x2 )9 t . For both g and f Assumptions 2.5 and 3.1 hold, and g is efficient, while h is not. 15 0.25 0.20 0.15 0.10 0.05 0.00 −20 −10 0 10 20 √ √ Figure 3: Estimated densities of n(θ̂G,n − θ0 ) (solid curve) and n(θ̂H,n − θ0 ) (dashed curve) for n = 104 in the case of the ergodic model (4.1). Simulations were carried out in the same manner as for the ergodic model. In the nonergodic case, an estimator was again found for every sample path when the efficient estimating function given by g was used. For the inefficient estimating function given by h, there was no solution to the estimating equation (in [0.01, 1.99]) in 14% of the samples for n = 104 and in 39 % of the samples for n = 103 . Figure 4 shows QQ-plots of √ √ bG,n = n W b −1 (θ̂G,n − θ0 ) and Z bH,n = n W b −1 (θ̂H,n − θ0 ) compared with a standard normal Z G,n H,n distribution, for n = 103 and n = 104 respectively. These QQ-plots indicate that in the non-ergodic case there is a slightly slower convergence to the asymptotic distribution in Theorem 3.2.(iii) for the efficient estimating function, and a considerably slower convergence for the inefficient estimating function, when compared to the ergodic case. 16 −4 ● ● ● ●● 4 2 ● ● −2 0 2 −4 −2 0 2 4 ● ● ● 4 ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●● 4 2 2 0 0 −2 −2 ● −4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● −4 ● ● −4 ● ● 0 ● −2 −2 0 2 ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● −2 0 2 4 −4 −2 0 2 4 bG,n (left) and Z bH,n (right) to the N(0, 1) distribution in the Figure 4: QQ-plots comparing Z case of the non-ergodic model (4.3) for n = 103 (above) and n = 104 (below). 17 5 Proofs Section 5.1 states three main lemmas needed to prove Theorem 3.2, followed by the proof of the theorem. Section 5.2 contains the proofs of the three lemmas. 5.1 Proof of the Main Theorem In order to prove Theorem 3.2, we use the following lemmas, together with results from Jacod and Sørensen (2012), and Sørensen (2012, Section 1.10). Lemma 5.1. Suppose that Assumptions 2.4 and 2.5 hold. For θ ∈ Θ, let Gn (θ) = n X n ; θ) g(∆n , Xtin , Xti−1 i=1 sq Gn (θ) n 1 X 2 n ; θ) = g (∆n , Xtin , Xti−1 ∆n i=1 and A(θ; θ0 ) = B(θ; θ0 ) = Z 1 2 0 Z 1 2 1 1 0 b2 (X s ; θ0 ) − b2 (X s ; θ) ∂2y g(0, X s , X s ; θ) ds b2 (X s ; θ0 ) − b2 (X s ; θ) ∂2y ∂θ g(0, X s , X s ; θ) ds 1 Z 1 2 C(θ; θ0 ) = − Z 1 2 0 1 0 ∂θ b2 (X s ; θ)∂2y g(0, X s , X s ; θ) ds b4 (X s ; θ0 ) + 1 2 2 2 b2 (X s ; θ0 ) − b2 (X s ; θ) ∂2y g(0, X s , X s ; θ) ds . Then, (i) the mappings θ 7→ A(θ; θ0 ), θ 7→ B(θ; θ0 ) and θ 7→ C(θ; θ0 ) are continuous on Θ (Pθ0 -almost surely) with A(θ0 ; θ0 ) = 0 and ∂θ A(θ; θ0 ) = B(θ; θ0 ). (ii) for all t ∈ [0, 1], [nt] P 1 X n ; θ0 ) \| Xtn Eθ0 g(∆n , Xtin , Xti−1 −→ 0 √ i−1 ∆n i=1 (5.1) [nt] 2 P 1 X n ; θ0 ) \| Xtn Eθ0 g(∆n , Xtin , Xti−1 −→ 0 i−1 ∆n i=1 (5.2) [nt] P 1 X 4 n , Xtn ; θ0 ) \| Xtn E g (∆ , X −→ 0 θ n t 0 i i−1 i−1 ∆2n i=1 (5.3) and [nt] P 1 X n ; θ0 ) \| Xtn Eθ0 g2 (∆n , Xtin , Xti−1 −→ i−1 ∆n i=1 t Z 1 2 0 2 b4 (X s ; θ0 ) ∂2y g(0, X s , X s ; θ0 ) ds . (5.4) 18 (iii) for all compact, convex subsets K ⊆ Θ, P sup \|Gn (θ) − A(θ; θ0 )\| −→ 0 θ∈K P sup \|∂θ Gn (θ) − B(θ; θ0 )\| −→ 0 θ∈K P sq sup Gn (θ) − C(θ; θ0 ) −→ 0 . θ∈K Lemma 5.2. Suppose that Assumptions 2.4 and 2.5 hold. Then, for all t ∈ [0, 1], [nt] P 1 X n ; θ0 )(Wtn − Wtn ) \| Ftn Eθ0 g(∆n , Xtin , Xti−1 −→ 0 . √ i i−1 i−1 ∆n i=1 (5.5) Lemma 5.3. Suppose that Assumptions 2.4 and 2.5 hold, and let Yn,t [nt] 1 X n ; θ0 ) . = √ g(∆n , Xtin , Xti−1 ∆n i=1 Then the sequence of processes (Yn )n∈N given by Yn = (Yn,t )t∈[0,1] converges stably in distribution under Pθ0 to the process Y = (Yt )t∈[0,1] given by Z t b2 (X s ; θ0 )∂2y g(0, X s , X s ; θ0 ) dBs . Yt = √1 2 0 Here B = (Bs ) s≥0 denotes a standard Wiener process, which is defined on a filtered extension (Ω0 , F 0 , (Ft0 )t≥0 , P0θ0 ) of (Ω, F , (Ft )t≥0 , Pθ0 ), and is independent of (U, W). D st We denote stable convergence in distribution under Pθ0 as n → ∞ by −→. Proof of Theorem 3.2. Let a compact, convex subset K ⊆ Θ with θ0 ∈ int K be given. The functions Gn (θ), A(θ, θ0 ), B(θ, θ0 ), and C(θ, θ0 ) were defined in Lemma 5.1. By Lemma 5.1.(i) and (iii), P Gn (θ0 ) −→ 0 and P sup \|∂θ Gn (θ) − B(θ, θ0 )\| −→ 0 θ∈K (5.6) with B(θ0 ; θ0 ) , 0 by Assumption 3.1.(ii), so Gn (θ) satisfies the conditions of Theorem 1.58 in Sørensen (2012). Now, we show (1.161) of Theorem 1.59 in Sørensen (2012). Let ε > 0 be given, and let B̄ε (θ0 ) and Bε (θ0 ), respectively, denote closed and open balls in R with radius ε > 0, centered at θ0 . The compact set K\Bε (θ0 ) does not contain θ0 , and so, by Assumption 3.1.(i), A(θ, θ0 ) , 0 for all θ ∈ K\Bε (θ0 ) with probability one under Pθ0 . 19 Because inf θ∈K\ B̄ε (θ0 ) \|A(θ, θ0 )\| ≥ inf θ∈K\Bε (θ0 ) \|A(θ, θ0 )\| > 0 Pθ0 -almost surely, by the continuity of θ 7→ A(θ, θ0 ), it follows that ! Pθ0 inf \|A(θ, θ0 )\| > 0 = 1 . θ∈K\ B̄ε (θ0 ) Consequently, by Theorem 1.59 in Sørensen (2012), for any Gn -estimator θ̃n , Pθ0 θ̃n ∈ K\ B̄ε (θ0 ) → 0 as n → ∞ . (5.7) for any ε > 0. By Theorem 1.58 in Sørensen (2012), there exists a consistent Gn -estimator θ̂n , which is eventually unique, in the sense that if θ̄n is another consistent Gn -estimator, then Pθ0 θ̂n , θ̄n → 0 as n → ∞ . (5.8) Suppose that θ̃n is any Gn -estimator which satisfies that Pθ0 θ̃n ∈ K → 1 as n → ∞ . Combining (5.7) and (5.9), it follows that Pθ0 θ̃n ∈ B̄ε (θ0 ) → 1 as n → ∞, (5.9) (5.10) so θ̃n is consistent. Using (5.8), Theorem 3.2.(i) follows. To prove Theorem 3.2.(ii), recall that ∆n = 1/n, and observe that by Lemma 5.3, √ D st nGn (θ0 ) −→ S (θ0 ) (5.11) where S (θ0 ) = 1 Z 0 √1 b2 (X s ; θ0 )∂2 y g(0, X s , X s ; θ0 ) dB s , 2 and B = (Bs ) s∈[0,1] is a standard Wiener process, independent of (U, W). As X is then also independent of B, S (θ0 ) is equal in distribution to C(θ0 ; θ0 )1/2 Z, where Z is standard normal distributed and independent of (Xt )t∈[0,1] . Note that by Assumption 3.1.(iii), the distribution of C(θ0 ; θ0 )1/2 Z is non-degenerate. Let θ̂n be a consistent Gn -estimator. By (5.6), (5.11) and properties of stable convergence (e.g. (2.3) in Jacod (1997)), √     nGn (θ0 ) Dst  S (θ0 )    −→   . ∂θ Gn (θ0 ) B(θ0 ; θ0 ) 20 Stable convergence in distribution implies weak convergence, so an application of Theorem 1.60 in Sørensen (2012) yields √ D n(θ̂n − θ0 ) −→ −B(θ0 , θ0 )−1 S (θ0 ) . (5.12) The limit is equal in distribution to W(θ0 )Z, where W(θ0 ) = −B(θ0 , θ0 )−1C(θ0 ; θ0 )1/2 and Z is standard normal distributed and independent of W(θ0 ). This completes the proof of Theorem 3.2.(ii). Finally, Lemma 2.14 in Jacod and Sørensen (2012) is used to write √ √ √ n(θ̂n − θ0 ) = −B(θ0 ; θ0 )−1 nGn (θ0 ) + n\|θ̂n − θ0 \|εn (θ0 ) , where the last term goes to zero in probability under Pθ0 . By the stable continuous mapping theorem, (5.12) holds with stable convergence in distribution as well. Lemma 5.1.(iii) may P b n −→ be used to conclude that W W(θ0 ), so Theorem 3.2.(iii) follows from the stable version of (5.12) by application of standard results for stable convergence. 5.2 Proofs of Main Lemmas This section contains the proofs of Lemmas 5.1, 5.2 and 5.3 in Section 5.1. A number of technical results are utilised in the proofs, these results are summarised in Appendix A, some of them with a proof. pol Proof of Lemma 5.1. First, note that for any f (x; θ) ∈ C0,0 (X×Θ) and any compact, convex subset K ⊆ Θ, there exist constants C K > 0 such that \| f (X s ; θ)\| ≤ C K (1 + \|X s \|C K ) for all s ∈ [0, 1] and θ ∈ int K. With probability one under Pθ0 , for fixed ω, C K (1+\|X s (ω)\|C K ) is a continuous function and therefore Lebesgue-integrable over [0, 1]. Using this method of constructing integrable upper bounds, Lemma 5.1.(i) follows by the usual results for continuity and differentiability of functions given by integrals. In the rest of this proof, Lemma A.3 and (A.7) are repeatedly used without reference. First, inserting θ = θ0 into (A.1), it is seen that [nt] [nt] X 1 X P 3/2 n ; θ0 ) \| Xtn n ; θ0 ) −→ 0 Eθ0 g(∆n , Xtin , Xti−1 = ∆ R(∆n , Xti−1 √ n i−1 ∆n i=1 i=1 [nt] [nt] X 2 1 X P 3 n ; θ0 ) \| Xtn n ; θ0 ) −→ 0 , Eθ0 g(∆n , Xtin , Xti−1 = ∆n R(∆n , Xti−1 i−1 ∆n i=1 i=1 proving (5.1) and (5.2). Furthermore, using (A.1) and (A.3), n X P n ; θ) \| Xtn Eθ0 g(∆n , Xtin , Xti−1 −→ A(θ; θ0 ) i−1 i=1 21 n X P n ; θ) \| Xtn −→ 0 , Eθ0 g2 (∆n , Xtin , Xti−1 i−1 i=1 so it follows from Lemma A.1 that point-wise for θ ∈ Θ, P Gn (θ) − A(θ; θ0 ) −→ 0 . (5.13) Using (A.3) and (A.5), [nt] 1 X n ; θ) \| Xtn Eθ0 g2 (∆n , Xtin , Xti−1 i−1 ∆n i=1 Z t 2 2 P −→ 12 b4 (X s ; θ0 ) + 21 b2 (X s ; θ0 ) − b2 (X s ; θ) ∂2y g(0, X s , X s ; θ) ds 0 and [nt] P 1 X 4 n , Xtn ; θ) \| Xtn E g (∆ , X −→ 0 , θ n t 0 i i−1 i−1 ∆2n i=1 completing the proof of Lemma 5.1.(ii) when θ = θ0 is inserted, and yielding P sq Gn (θ) − C(θ; θ0 ) −→ 0 (5.14) point-wise for θ ∈ Θ by Lemma A.1, when t = 1 is inserted. Also, using (A.2) and (A.4), n X P n ; θ) \| Xtn −→ B(θ; θ0 ) Eθ0 ∂θ g(∆n , Xtin , Xti−1 i−1 i=1 n X P 2 n n ; θ) −→ 0 . Eθ0 ∂θ g(∆n , Xtin , Xti−1 \| Xti−1 i=1 Thus, by Lemma A.1, also P ∂θ Gn (θ) − B(θ; θ0 ) −→ 0 , (5.15) j point-wise for θ ∈ Θ. Finally, recall that ∂y g(0, x, x; θ) = 0 for j = 0, 1. Then, using Lemmas A.7 and A.8, it follows that for each m ∈ N and compact, convex subset K ⊆ Θ, there exist constants Cm,K > 0 such that for all θ, θ0 ∈ K and n ∈ N, Eθ0 \|(Gn (θ) − A(θ; θ0 )) − (Gn (θ0 ) − A(θ0 ; θ0 ))\|2m ≤ Cm,K \|θ − θ0 \|2m Eθ0 \|(∂θ Gn (θ) − B(θ; θ0 )) − (∂θ Gn (θ0 ) − B(θ0 ; θ0 ))\|2m ≤ Cm,K \|θ − θ0 \|2m sq Eθ0 \|(Gn (θ) − C(θ; θ0 )) − sq (Gn (θ0 ) − C(θ ; θ0 ))\| 0 2m ≤ Cm,K \|θ − θ \| 0 2m (5.16) . By Lemma 5.1.(i), the functions θ 7→ Gn (θ) − A(θ; θ0 ), θ 7→ ∂θ Gn (θ) − B(θ; θ0 ) and θ 7→ sq Gn (θ) −C(θ, θ0 ) are continuous on Θ. Thus, using Lemma A.9 together with (5.13), (5.14), (5.15) and (5.16) completes the proof of Lemma 5.1.(iii). 22 Proof of Lemma 5.2. The overall strategy in this proof is to expand the expression on the left-hand side of (5.5) in such a manner that all terms either converge to 0 by Lemma A.3, or are equal to 0 by the martingale properties of stochastic integral terms obtained by use of Itô’s formula. By Assumption 2.5 and Lemma 2.7, the formulae g(0, y, x; θ) = 12 (y − x)2 ∂2y g(0, x, x; θ) + (y − x)3 R(y, x; θ) g(1) (y, x; θ) = g(1) (x, x; θ) + (y − x)R(y, x; θ) may be obtained. Using (2.4) and (5.17), n ; θ0 )(Wtn − Wtn ) \| Ftn Eθ0 g(∆n , Xtin , Xti−1 i i−1 i−1 2 2 1 n n n n n ) \| Ftn = Eθ0 2 (Xti − Xti−1 ) ∂y g(0, Xti−1 , Xti−1 ; θ0 )(Wtin − Wti−1 i−1 3 n n n n n n n + Eθ0 (Xti − Xti−1 ) R(Xti , Xti−1 ; θ0 )(Wti − Wti−1 ) \| Fti−1 n , Xtn ; θ0 )(Wtn − Wtn ) \| Ftn + ∆n Eθ0 g(1) (Xti−1 i i−1 i−1 i−1 n ) \| Ftn n n n n n + ∆n Eθ0 (Xti − Xti−1 )R(Xti , Xti−1 ; θ0 )(Wti − Wti−1 i−1 2 n n n n n + ∆ Eθ0 R(∆n , Xti , Xti−1 ; θ0 )(Wti − Wti−1 ) \| Fti−1 . (5.17) (5.18) Note that n n n n , Xtn ; θ0 )Eθ ∆n g(1) (Xti−1 0 Wti − Wti−1 \| Fti−1 = 0 , i−1 and that by repeated use of the Cauchy-Schwarz inequality, Lemma A.4 and Corollary A.5, C 3 n \| ) n ) R(Xtn , Xtn ; θ0 )(Wtn − Wtn ) \| Ftn ≤ ∆2nC(1 + \|Xti−1 Eθ0 (Xtin − Xti−1 i i i−1 i−1 i−1 C n \| ) n )R(Xtn , Xtn ; θ0 )(Wtn − Wtn ) \| Ftn ≤ ∆2nC(1 + \|Xti−1 ∆n Eθ0 (Xtin − Xti−1 i i i−1 i−1 i−1 C n \| ) n ; θ0 )(Wtn − Wtn ) \| Ftn ≤ ∆5/2 ∆2n Eθ0 R(∆n , Xtin , Xti−1 n C(1 + \|Xti−1 i i−1 i−1 for suitable constants C > 0, with [nt] 1 X m/2 P C n \| ) −→ 0 ∆n C(1 + \|Xti−1 √ ∆n i=1 for m = 4, 5 by Lemma A.3. Now, by (5.18), it only remains to show that [nt] P 1 X 2 2 n , Xtn ; θ0 )Eθ n − Xtn ) (Wtn − Wtn ) \| Ftn ∂y g(0, Xti−1 (X −→ 0 . √ t 0 i i i−1 i−1 i−1 i−1 ∆n i=1 Applying Itô’s formula with the function 2 n ) (w − wtn ) f (y, w) = (y − xti−1 i−1 23 (5.19) n , conditioned on (Xtn , Wtn ) = (xtn , wtn ), it follows that to the process (Xt , Wt )t≥ti−1 i−1 i−1 i−1 i−1 2 n ) (Wtn − Wtn ) (Xtin − Xti−1 i i−1 Z Z tn i n n (X s − Xti−1 )(W s − Wti−1 )a(X s ) ds + =2 n ti−1 n ti−1 Z +2 + Z tin n ti−1 n ti n ti−1 tin n )b(X s ; θ0 ) ds + 2 (X s − Xti−1 Z tin n ti−1 2 n )b (X s ; θ0 ) ds (W s − Wti−1 n )(W s − Wtn )b(X s ; θ0 ) dW s (X s − Xti−1 i−1 (5.20) 2 n ) dW s . (X s − Xti−1 By the martingale property of the Itô integrals in (5.20), 2 n ) (Wtn − Wtn ) \| Ftn Eθ0 (Xtin − Xti−1 i i−1 i−1 Z tn i n )(W s − Wtn )a(X s ) \| Ftn =2 Eθ0 (X s − Xti−1 ds i−1 i−1 n ti−1 + Z +2 tin n ti−1 Z 2 n )b (X s ; θ0 ) \| Ftn Eθ0 (W s − Wti−1 ds i−1 tin n ti−1 n )b(X s ; θ0 ) \| Xtn Eθ0 (X s − Xti−1 ds . i−1 Using the Cauchy-Schwarz inequality, Lemma A.4 and Corollary A.5 again, Z tin n ti−1 C n \| ), n )(W s − Wtn )a(X s ) \| Ftn ds ≤ C∆2n (1 + \|Xti−1 Eθ0 (X s − Xti−1 i−1 i−1 and by Lemma 2.6 n n n ; θ0 ) , n )b(X s ; θ0 ) \| Xtn Eθ0 (X s − Xti−1 = (s − ti−1 )R(s − ti−1 , Xti−1 i−1 so also Z tin n ti−1 C n )b(X s ; θ0 ) \| Xtn n \| ). Eθ0 (X s − Xti−1 ds ≤ C∆2n (1 + \|Xti−1 i−1 Now Z tn [nt] i 1 X 2 n n n )(W s − Wtn )a(X s ) \| Ftn ∂y g(0, Xti−1 , Xti−1 ; θ0 ) Eθ0 (X s − Xti−1 ds √ i−1 i−1 n ∆n i=1 ti−1 Z tn [nt] i 1 X 2 n , Xtn ; θ0 ) n )b(X s ; θ0 ) \| Xtn ∂y g(0, Xti−1 + √ Eθ0 (X s − Xti−1 ds i−1 i−1 n ∆n i=1 ti−1 ≤ ∆3/2 n C [nt] X P C n , Xtn ; θ0 ) (1 + \|Xtn \| ) −→ 0 ∂2y g(0, Xti−1 i−1 i−1 i=1 24 (5.21) by Lemma A.3, so by (5.19) and (5.21), it remains to show that Z tn [nt] i 1 X 2 P 2 n )b (X s ; θ0 ) \| Ftn n n ∂y g(0, Xti−1 , Xti−1 ; θ0 ) Eθ0 (W s − Wti−1 ds −→ 0 . √ i−1 n ∆n i=1 ti−1 Applying Itô’s formula with the function 2 n )b (y; θ0 ) , f (y, w) = (w − wti−1 and making use of the martingale properties of the stochastic integral terms, yields Z tn i 2 n )b (X s ; θ0 ) \| Ftn Eθ0 (W s − Wti−1 ds i−1 n ti−1 = tin Z n ti−1 + + s Z n ti−1 Z 1 2 Z n ) \| Ftn Eθ0 a(Xu )∂y b2 (Xu ; θ0 )(Wu − Wti−1 du ds i−1 tin n ti−1 tin n ti−1 s Z n ti−1 Z s n ti−1 n ) \| Ftn Eθ0 b2 (Xu ; θ0 )∂2y b2 (Xu ; θ0 )(Wu − Wti−1 du ds i−1 n Eθ0 b(Xu ; θ0 )∂y b2 (Xu ; θ0 ) \| Fti−1 du ds . Again, by repeated use of the Cauchy-Schwarz inequality and Corollary A.5, Z tn i 5/2 2 C 2 n \| )(∆ + ∆ n )b (X s ; θ0 ) \| Ftn ds ≤ C(1 + \|Xti−1 Eθ0 (Wtin − Wti−1 n ). n i−1 n ti−1 Now Z tn [nt] i 1 X 2 2 n )b (X s ; θ0 ) \| Ftn n , Xtn ; θ0 ) Eθ0 (W s − Wti−1 ds ∂y g(0, Xti−1 √ i−1 i−1 n ∆n i=1 ti−1 [nt] 3/2 X P C 2 n , Xtn ; θ0 ) C(1 + \|Xtn \| ) −→ 0 , ≤ ∆n + ∆ n ∂2y g(0, Xti−1 i−1 i−1 i=1 thus completing the proof. Proof of Lemma 5.3. The aim of this proof is to establish that the conditions of Theorem IX.7.28 in Jacod and Shiryaev (2003) hold, by which the desired result follows directly. For all t ∈ [0, 1], [ns] [nt] 1 X 1 X n ; θ 0 ) \| Xt n n , Xtn ; θ0 ) \| Xtn sup √ Eθ0 g(∆n , Xtin , Xti−1 g(∆ , X ≤ E √ θ n t 0 i i−1 i−1 i−1 s≤t ∆n i=1 ∆n i=1 and since the right-hand side converges to 0 in probability under Pθ0 by (5.1) of Lemma 5.1, so does the left-hand side, i.e. condition 7.27 of Theorem IX.7.28 holds. From (5.2) and (5.4) of Lemma 5.1, it follows that for all t ∈ [0, 1], [nt] 2 1 X 2 n ; θ0 ) \| Xtn n n n Eθ0 g (∆n , Xtin , Xti−1 − E g(∆ , X , X ; θ ) \| X θ0 n ti ti−1 0 ti−1 i−1 ∆n i=1 25 P Z −→ 1 2 0 t 2 b4 (X s ; θ0 ) ∂2y g(0, X s , X s ; θ0 ) ds , establishing that condition 7.28 of Theorem IX.7.28 is satisfied. Lemma 5.2 implies condition 7.29, while the Lyapunov condition (5.3) of Lemma 5.1 implies the Lindeberg condition 7.30 of Theorem IX.7.28 in Jacod and Shiryaev (2003), from which the desired result now follows. Theorem IX.7.28 contains an additional condition 7.31. This condition has the same form n n , where N = (Nt )t≥0 is any bounded as (5.5), but with Wtin − Wti−1 replaced by Ntin − Nti−1 martingale on (Ω, F , (Ft )t≥0 , Pθ0 ), which is orthogonal to W. However, since (Ft )t≥0 is generated by U and W, it follows from the martingale representation theorem (Jacod and Shiryaev, 2003, Theorem III.4.33) that every martingale on (Ω, F , (Ft )t≥0 , Pθ0 ) may be written as the sum of a constant term and a stochastic integral with respect to W, and therefore cannot be orthogonal to W. A Auxiliary Results This section contains a number of technical results used in the proofs in Section 5.2. Lemma A.1. (Genon-Catalot and Jacod, 1993, Lemma 9) For i, n ∈ N, let Fn,i = Ftin (with Fn,0 = F0 ), and let Fn,i be an Fn,i -measurable, real-valued random variable. If n X P Eθ0 (Fn,i \| Fn,i−1 ) −→ F n X and i=1 P 2 Eθ0 (Fn,i \| Fn,i−1 ) −→ 0 , i=1 for some random variable F, then n X P Fn,i −→ F . i=1 Lemma A.2. Suppose that Assumptions 2.4 and 2.5 hold. Then, for all θ ∈ Θ, (i) n ; θ) \| Xtn Eθ0 g(∆n , Xtin , Xti−1 i−1 2 2 2 n ; θ0 ) − b (Xtn ; θ) ∂ g(0, Xtn , Xtn ; θ) + ∆ R(∆n , Xtn ; θ) , = 21 ∆n b2 (Xti−1 y n i−1 i−1 i−1 i−1 (A.1) (ii) n ; θ) \| Xtn Eθ0 ∂θ g(∆n , Xtin , Xti−1 i−1 2 2 2 1 n n ; θ) ∂ ∂θ g(0, Xtn , Xtn ; θ) = 2 ∆n b (Xti−1 ; θ0 ) − b (Xti−1 y i−1 i−1 2 2 n ; θ)∂ g(0, Xtn , Xtn ; θ) + ∆ R(∆n , Xtn ; θ) , − 21 ∆n ∂θ b2 (Xti−1 y n i−1 i−1 i−1 26 (A.2) (iii) n ; θ) \| Xtn Eθ0 g2 (∆n , Xtin , Xti−1 i−1 2 2 2 2 2 1 n ; θ0 ) − b (Xtn ; θ) n ; θ0 ) + n , Xtn ; θ) (A.3) b (X ∂ = 21 ∆2n b4 (Xti−1 g(0, X t t y 2 i−1 i−1 i−1 i−1 n ; θ) , + ∆3n R(∆n , Xti−1 (iv) 2 n ; θ) , n n ; θ) = ∆2n R(∆n , Xti−1 \| Xti−1 Eθ0 ∂θ g(∆n , Xtin , Xti−1 (A.4) n ; θ) . n ; θ) \| Xtn = ∆4n R(∆n , Xti−1 Eθ0 g4 (∆n , Xtin , Xti−1 i−1 (A.5) (v) Proof of Lemma A.2. The formulae (A.1), (A.2) and (A.3) are implicitly given in the proofs of Sørensen (2010, Lemmas 3.2 & 3.4). To prove the two remaining formulae, note first that using (2.5), Assumption 2.5.(i) and Lemma 2.7, Liθ0 (g4 (0; θ))(x, x) = 0 , Liθ0 (g3 (0, θ)g(1) (θ))(x, x) = 0 , i = 1, 2, 3 i = 1, 2 Lθ0 (g (0, θ)g (θ) )(x, x) = 0 2 (1) 2 Lθ0 (g3 (0, θ)g(2) (θ))(x, x) = 0 Lθ0 (∂θ g(0, θ)2 )(x, x) = 0 . The verification of these formulae may be simplified by using e.g. the Leibniz formula for the n’th derivative of a product to see that partial derivatives are zero when evaluated in y = x. These results, as well as Lemmas 2.6 and 2.7, and (A.8) are used without reference in the following. 2 n ; θ) n \| Xti−1 Eθ0 ∂θ g(∆n , Xtin , Xti−1 2 n ; θ) \| Xtn = Eθ0 ∂θ g(0, Xtin , Xti−1 i−1 (1) n ; θ)∂θ g n ; θ) \| Xtn + 2∆n Eθ0 ∂θ g(0, Xtin , Xti−1 (Xtin , Xti−1 i−1 2 n n n + ∆n Eθ0 R(∆n , Xti , Xti−1 ; θ) \| Xti−1 2 2 2 n , Xtn ; θ) + ∆n Lθ (∂θ g(0, θ) )(Xtn , Xtn ) + ∆ R(∆n , Xtn ; θ) = ∂θ g(0, Xti−1 n 0 i−1 i−1 i−1 i−1 (1) n n n n n + 2∆n ∂θ g(0, Xti−1 , Xti−1 ; θ)∂θ g (Xti−1 , Xti−1 ; θ) + ∆n R(∆n , Xti−1 ; θ) n ; θ) , = ∆2n R(∆n , Xti−1 proving (A.4). Similarly, n ; θ) \| Xtn Eθ0 g4 (∆n , Xtin , Xti−1 i−1 27 n ; θ) \| Xtn = Eθ0 g4 (0, Xtin , Xti−1 i−1 (1) n ; θ)g n ; θ) \| Xtn + 4∆n Eθ0 g3 (0, Xtin , Xti−1 (Xtin , Xti−1 i−1 (1) 2 n ; θ)g n ; θ) \| Xtn + 6∆2n Eθ0 g2 (0, Xtin , Xti−1 (Xtin , Xti−1 i−1 (2) n ; θ)g n ; θ) \| Xtn + 2∆2n Eθ0 g3 (0, Xtin , Xti−1 (Xtin , Xti−1 i−1 (1) 3 n ; θ)g n ; θ) \| Xtn + 4∆3n Eθ0 g(0, Xtin , Xti−1 (Xtin , Xti−1 i−1 (1) (2) n ; θ)g n ; θ)g n ; θ) \| Xtn + 6∆3n Eθ0 g2 (0, Xtin , Xti−1 (Xtin , Xti−1 (Xtin , Xti−1 i−1 (3) n ; θ)g n ; θ) \| Xtn (Xtin , Xti−1 + 23 ∆3n Eθ0 g3 (0, Xtin , Xti−1 i−1 n ; θ) \| Xtn + ∆4n Eθ0 R(∆n , Xtin , Xti−1 i−1 4 4 1 2 2 n , Xtn ; θ) + ∆n Lθ (g (0; θ))(Xtn , Xtn ) + ∆ L (g (0; θ))(Xtn , Xtn ) = g4 (0, Xti−1 0 2 n θ0 i−1 i−1 i−1 i−1 i−1 (1) 3 n , Xtn ; θ) n , Xtn ) + 4∆n g (0, Xtn , Xtn ; θ)g (Xti−1 + 61 ∆3n L3θ0 (g4 (0; θ))(Xti−1 i−1 i−1 i−1 i−1 3 2 3 (1) n , Xtn ) + 2∆ L (g (0; θ)g n , Xt n ) + 4∆2n Lθ0 (g3 (0; θ)g(1) (θ))(Xti−1 (θ))(Xti−1 n θ0 i−1 i−1 2 3 2 (1) (1) n , Xt n ) n , Xtn ; θ) + 6∆ Lθ (g (0; θ)g n , Xtn ; θ)g (θ)2 )(Xti−1 (Xti−1 + 6∆2n g2 (0, Xti−1 n 0 i−1 i−1 i−1 (2) 3 3 (2) n , Xtn ; θ)g n , Xtn ; θ) + 2∆ Lθ (g (0; θ)g n , Xtn ) + 2∆2n g3 (0, Xti−1 (Xti−1 (θ))(Xti−1 n 0 i−1 i−1 i−1 3 (1) n , Xtn ; θ) n , Xtn ; θ)g (Xti−1 + 4∆3n g(0, Xti−1 i−1 i−1 (2) (1) n , Xtn ; θ) n , Xtn ; θ)g n , Xtn ; θ)g (Xti−1 (Xti−1 + 6∆3n g2 (0, Xti−1 i−1 i−1 i−1 (3) n , Xtn ; θ) n , Xtn ; θ)g + 23 ∆3n g3 (0, Xti−1 (Xti−1 i−1 i−1 n ; θ) + ∆4n R(∆n , Xti−1 n ; θ) , = ∆4n R(∆n , Xti−1 which proves (A.5). Lemma A.3. Let x 7→ f (x) be a continuous, real-valued function, and let t ∈ [0, 1] be given. Then Z t [nt] X P n ) −→ ∆n f (Xti−1 f (X s ) ds . 0 i=1 Lemma A.3 follows easily by the convergence of Riemann sums. Lemma A.4. Suppose that Assumption 2.4 holds, and let m ≥ 2. Then, there exists a constant Cm > 0, such that for 0 ≤ t ≤ t + ∆ ≤ 1, Eθ0 \|Xt+∆ − Xt \|m \| Xt ≤ Cm ∆m/2 1 + \|Xt \|m . (A.6) Corollary A.5. Suppose that Assumption 2.4 holds. Let a compact, convex set K ⊆ Θ be given, and suppose that f (y, x; θ) is of polynomial growth in x and y, uniformly for θ in K. Then, there exist constants C K > 0 such that for 0 ≤ t ≤ t + ∆ ≤ 1, Eθ0 (\| f (Xt+∆ , Xt , θ)\| \| Xt ) ≤ C K 1 + \|Xt \|C K 28 for all θ ∈ K. Lemma A.4 and Corollary A.5, correspond to Lemma 6 of Kessler (1997), adapted to the present assumptions. For use in the following, observe that for any θ ∈ Θ, there exist constants Cθ > 0 such that ∆n [nt] X n ) ≤ C θ ∆n Rθ (∆n , Xti−1 [nt] X C n \| θ , 1 + \|Xti−1 i=1 i=1 so it follows from Lemma A.3 that for any deterministic, real-valued sequence (δn )n∈N with δn → 0 as n → ∞, δn ∆n [nt] X P n ) −→ 0 . Rθ (∆n , Xti−1 (A.7) i=1 Note that by Corollary A.5, it holds that under Assumption 2.4, Eθ0 (R (∆, Xt+∆ , Xt ; θ) \| Xt ) = R(∆, Xt ; θ) . (A.8) Lemma A.6. Suppose that Assumption 2.4 holds, and that the function f (t, y, x; θ) satisfies that pol f (t, y, x; θ) ∈ C1,2,1 ([0, 1] × X2 × Θ) with f (0, x, x; θ) = 0 (A.9) for all x ∈ X and θ ∈ Θ. Let m ∈ N be given, and let Dk( · ; θ, θ0 ) = k( · ; θ) − k( · ; θ0 ). Then, there exist constants Cm > 0 such that 2m Eθ0 D f (t − s, Xt , X s ; θ, θ0 ) Z t 2m 2m−1 ≤ Cm (t − s) Eθ0 D f1 (u − s, Xu , X s ; θ, θ0 ) du (A.10) s Z t 2m + Cm (t − s)m−1 Eθ0 D f2 (u − s, Xu , X s ; θ, θ0 ) du s for 0 ≤ s < t ≤ 1 and θ, θ0 ∈ Θ, where f1 and f2 are given by f1 (t, y, x; θ) = ∂t f (t, y, x; θ) + a(y)∂y f (t, y, x; θ) + 12 b2 (y; θ0 )∂2y f (t, y, x; θ) f2 (t, y, x; θ) = b(y; θ0 )∂y f (t, y, x; θ) . Furthermore, for each compact, convex set K ⊆ Θ, there exists a constant Cm,K > 0 such that Eθ0 \|D f j (t − s, Xt , X s ; θ, θ0 )\|2m ≤ Cm,K \|θ − θ0 \|2m for j = 1, 2, 0 ≤ s < t ≤ 1 and all θ, θ0 ∈ K. 29 Proof of Lemma A.6. A simple application of Itô’s formula (when conditioning on X s = x s ) yields that for all θ ∈ Θ, Z t Z t f (t − s, Xt , X s ; θ) = f1 (u − s, Xu , X s ; θ) du + f2 (u − s, Xu , X s ; θ) dWu (A.11) s s under Pθ0 . By Jensen’s inequality, it holds that for any k ∈ N, Z Z t k   t 0 jk k−1 0 j   du Eθ0 D f j (u − s, Xu , X s ; θ, θ ) D f j (u − s, Xu , X s ; θ, θ ) du  ≤ (t − s) Eθ0  s s (A.12) for j = 1, 2, and by the martingale properties of the second term in (A.11), the BurkholderDavis-Gundy inequality may be used to show that Z Z t 2m  m!  t  0 Eθ0  . D f2 (u − s, Xu , X s ; θ, θ ) dWu  ≤ Cm Eθ0 D f2 (u − s, Xu , X s ; θ, θ0 )2 du s s (A.13) Now, (A.11), (A.12) and (A.13) may be combined to show (A.10). The last result of the lemma follows by an application of the mean value theorem. Lemma A.7. Suppose that Assumption 2.4 holds, and let K ⊆ Θ be compact and convex. Assume that f (t, y, x; θ) satisfies (A.9) for all x ∈ X and θ ∈ Θ, and define Fn (θ) = n X n ; θ) . f (∆n , Xtin , Xti−1 i=1 Then, for each m ∈ N, there exists a constant Cm,K > 0, such that Eθ0 Fn (θ) − Fn (θ0 ) 2m ≤ Cm,K \|θ − θ0 \|2m en (θ) = ∆−1 for all θ, θ0 ∈ K and n ∈ N. Define F n F n (θ), and suppose, moreover, that the functions h1 (t, y, x; θ) = ∂t f (t, y, x; θ) + a(y)∂y f (t, y, x; θ) + 21 b2 (y; θ0 )∂2y f (t, y, x; θ) h2 (t, y, x; θ) = b(y; θ0 )∂y f (t, y, x; θ) h j2 (t, y, x; θ) = b(y; θ0 )∂y h j (t, y, x, θ) satisfy (A.9) for j = 1, 2. Then, for each m ∈ N, there exists a constant Cm,K > 0, such that en (θ) − F en (θ0 ) 2m ≤ Cm,K \|θ − θ0 \|2m E θ0 F for all θ, θ0 ∈ K and n ∈ N. 30 Proof of Lemma A.7. For use in the following, define, in addition to h1 , h2 and h j2 , the functions h j1 (t, y, x; θ) = ∂t h j (t, y, x; θ) + a(y)∂y h j (t, y, x; θ) + 21 b2 (y; θ0 )∂2y h j (t, y, x; θ) h j21 (t, y, x; θ) = ∂t h j2 (t, y, x; θ) + a(y)∂y h j2 (t, y, x; θ) + 12 b2 (y; θ0 )∂2y h j2 (t, y, x; θ) h j22 (t, y, x; θ) = b(y; θ0 )∂y h j2 (t, y, x; θ) for j = 1, 2, and, for ease of notation, let 0 n 0 n H n,i j (u; θ, θ ) = Dh j (u − ti−1 , Xu , Xti−1 ; θ, θ ) for j ∈ {1, 2, 11, 12, 21, 22, 121, 122, 221, 222}, where Dk( · ; θ, θ0 ) = k( · ; θ) − k( · ; θ0 ). Recall that ∆n = 1/n. First, by the martingale properties of ∆n n Z X i=1 r 0 n,i 0 n ,tn ] (u)H 1(ti−1 2 (u; θ, θ ) dWu , i the Burkholder-Davis-Gundy inequality is used to establish the existence of a constant Cm > 0 such that    2m  m n Z tn n Z tn  X   2 X  i i n,i n,i  0 0 2 Eθ0  ∆n H2 (u; θ, θ ) dWu  ≤ Cm Eθ0  ∆n H2 (u; θ, θ ) du  . tn tn i=1 i=1 i−1 i−1 Now, using also Ito’s formula, Jensen’s inequality and Lemma A.6,   2m  n  X  0  n ; θ, θ ) Eθ0  ∆n D f (∆n , Xtin , Xti−1  i=1     2m  2m  n Z tn n Z tn  X  X   i i  ≤ Cm Eθ0  ∆n H1n,i (u; θ, θ0 ) du  + Cm Eθ0  ∆n H2n,i (u; θ, θ0 ) dWu  n n i=1 ti−1 i=1 ti−1 Z n   2m  m n n Z tn  ti X   2 X  i n,i n,i  0 0 2 ≤ Cm ∆n Eθ0  H1 (u; θ, θ ) du  + Cm Eθ0  ∆n H2 (u; θ, θ ) du  n n i=1 ti−1 i=1 ti−1      2m  m  Z tn n   1 Z tin X     i 1     n,i n,i    0 0 2 Eθ0  ≤ Cm ∆2m+1 n   ∆n tn H1 (u; θ, θ ) du  + Eθ0  ∆n tn H2 (u; θ, θ ) du  i−1 i−1 i=1  Z Z n n n  t t X n,i  n,i  i i 2m 0 2m 0 2m (A.14) Eθ0 \|H (u; θ, θ )\| du + Eθ0 \|H (u; θ, θ )\| du ≤ Cm ∆n  ≤ Cm,K \|θ n ti−1 i=1 − θ0 \|2m ∆2m n , 1 n ti−1 2 thus   2m  n  X  0 2m −2m 0  ≤ Cm,K \|θ − θ0 \|2m n ; θ, θ ) Eθ0 \|DFn (θ, θ )\| = ∆n Eθ0  ∆n D f (∆n , Xtin , Xti−1  i=1 31 for all θ, θ0 ∈ K and n ∈ N. In the case where also h j and h j2 satisfy (A.9) for all x ∈ X, θ ∈ Θ and j = 1, 2, use Lemma A.6 to write Eθ0 \|H1n,i (u; θ, θ0 )\|2m Z u n,i n 2m−1 ≤ Cm (u − ti−1 ) Eθ0 \|H11 (v; θ, θ0 )\|2m dv n ti−1 + Cm (u − u Z n m−1 ti−1 ) n ti−1 Z n 2m−1 ≤ Cm (u − ti−1 ) u n ti−1 n,i Eθ0 \|H11 (v; θ, θ0 )\|2m dv u Z + Cm (u − n m−1 ti−1 ) + Cm (u − n m−1 ti−1 ) n,i Eθ0 \|H12 (v; θ, θ0 )\|2m dv n ti−1 u Z n ti−1  Z  (v − tn )2m−1 i−1 v n ti−1  Z  n m−1 (v − t ) i−1 Eθ0 v n ti−1 Eθ0 2m n,i H121 (w; θ, θ0 ) 2m n,i H122 (w; θ, θ0 )   dw dv   dw dv n 2m n 3m ≤ Cm,K \|θ − θ0 \|2m (u − ti−1 ) + (u − ti−1 ) , and similarly obtain n 2m n 3m Eθ0 \|H2n,i (u; θ, θ0 )\|2m ≤ Cm,K \|θ − θ0 \|2m (u − ti−1 ) + (u − ti−1 ) . Now, inserting into (A.14),   2m  n  X  0  n ; θ, θ ) Eθ0  ∆n D f (∆n , Xtin , Xti−1  i=1 Z n  X n,i Z tin 2m 0 2m ≤ Cm,K ∆n Eθ0 \|H1 (u; θ, θ )\| du +  i=1 n ti−1 ≤ Cm,K \|θ − θ0 \|2m ∆2m n n Z X tin n ti−1 i=1 tin n ti−1  n,i  0 2m Eθ0 \|H2 (u; θ, θ )\| du n 2m n 3m (u − ti−1 ) + (u − ti−1 ) du 5m ≤ Cm,K \|θ − θ0 \|2m ∆4m , n + ∆n and, ultimately, Eθ0 \|DF̃n (θ, θ0 )\|2m   2m  n  X  0  n , Xtn ; θ, θ ) D f (∆ , X = Eθ0  ∆−1 n t n i i−1  i=1   2m  n  X  0  ∆n  n , Xtn ; θ, θ ) = ∆−4m D f (∆ , X n t n Eθ0  i i−1   i=1 0 2m ≤ Cm,K \|θ − θ \| (1 + ∆n ) ≤ Cm,K \|θ − θ \| . 0 2m pol Lemma A.8. Suppose that Assumption 2.4 is satisfied. Let f ∈ C0,1 (X × Θ). Define Z 1 F(θ) = f (X s ; θ) ds 0 32 and let K ⊆ Θ be compact and convex. Then, for each m ∈ N, there exists a constant Cm,K > 0 such that for all θ, θ0 ∈ K, Eθ0 \|F(θ) − F(θ0 )\|2m ≤ Cm,K \|θ − θ0 \|2m . Lemma A.8 follows from a simple application of the mean value theorem. Lemma A.9. Let K ⊆ Θ be compact. Suppose that Hn = (Hn (θ))θ∈K defines a sequence (Hn )n∈N of continuous, real-valued stochastic processes such that P Hn (θ) −→ 0 point-wise for all θ ∈ K. Furthermore, assume that for some m ∈ N, there exists a constant Cm,K > 0 such that for all θ, θ0 ∈ K and n ∈ N, Eθ0 Hn (θ) − Hn (θ0 ) 2m ≤ Cm,K \|θ − θ0 \|2m . (A.15) Then, P sup \|Hn (θ)\| −→ 0 . θ∈K Proof of Lemma A.9. (Hn (θ))n∈N is tight in R for all θ ∈ K, so, using (A.15), it follows from Kallenberg (2002, Corollary 16.9 & Theorem 16.3) that the sequence of processes (Hn )n∈N is tight in C(K, R), the space of continuous (and bounded) real-valued functions on K, and thus relatively compact in distribution. Also, for all d ∈ N and (θ1 , . . . , θd ) ∈ K d ,     0  Hn (θ1 )  .  D  .   ..  −→  ..  ,     Hn (θd ) 0 D so by Kallenberg (2002, Lemma 16.2), Hn −→ 0 in C(K, R) equipped with the uniform D metric. Finally, by the continuous mapping theorem, supθ∈K \|Hn (θ)\| −→ 0 , and the desired result follows. Acknowledgement We are grateful to the referees for their insightful comments and suggestions that have improved the paper. Nina Munkholt Jakobsen was supported by the Danish Council for Independent Research \| Natural Science through a grant to Susanne Ditlevsen. Michael Sørensen was supported by the Center for Research in Econometric Analysis of Time Series funded by the Danish National Research Foundation. The research is part of the Dynamical Systems Interdisciplinary Network funded by the University of Copenhagen Programme of Excellence. 33 References Aït-Sahalia, Y. (2002). 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Evaluation strategies for monadic computations Tomas Petricek Computer Laboratory University of Cambridge United Kingdom tomas.petricek@cl.cam.ac.uk Monads have become a powerful tool for structuring effectful computations in functional programming, because they make the order of effects explicit. When translating pure code to a monadic version, we need to specify evaluation order explicitly. Two standard translations give call-by-value and call-by-name semantics. The resulting programs have different structure and types, which makes revisiting the choice difficult. In this paper, we translate pure code to monadic using an additional operation malias that abstracts out the evaluation strategy. The malias operation is based on computational comonads; we use a categorical framework to specify the laws that are required to hold about the operation. For any monad, we show implementations of malias that give call-by-value and call-by-name semantics. Although we do not give call-by-need semantics for all monads, we show how to turn certain monads into an extended monad with call-by-need semantics, which partly answers an open question. Moreover, using our unified translation, it is possible to change the evaluation strategy of functional code translated to the monadic form without changing its structure or types. 1 Introduction Purely functional languages use lazy evaluation (also called call-by-need) to allow elegant programming with infinite data structures and to guarantee that a program will not evaluate a diverging term unless it is needed to obtain the final result. However, reasoning about lazy evaluation is difficult thus it is not suitable for programming with effects. An elegant way to embed effectful computations in lazy functional languages, introduced by Moggi [16] and Wadler [26], is to use monads. Monads embed effects in a purely functional setting and explicitly specify the evaluation order of monadic (effectful) operations. Wadler [26] gives two ways of translating pure programs to a corresponding monadic version. One approach leads to a call-by-value semantics, where effects of function arguments are performed before calling a function. However, if an argument has an effect and terminates the program, this may not be appropriate if the function can successfully complete without using the argument. The second approach gives a call-by-name semantics, where effects are performed only if the argument is actually used. However, this approach is also not always suitable, because an effect may be performed repeatedly. Wadler leaves an open question whether there is a translation that would correspond to call-by-need semantics, where effects are performed only when the result is needed, but at most once. The main contribution of this paper is an alternative translation of functional code to a monadic form, parameterized by an operation malias. The translation has the following properties: • A single translation gives monadic code with either call-by-name or call-by-value semantics, depending on the definition of malias (Section 2). When used in languages such as Haskell, it is possible to write code that is parameterized by the evaluation strategy (Section 4.1). J. Chapman and P. B. Levy (Eds.): Fourth Workshop on Mathematically Structured Functional Programming (MSFP 2012). EPTCS 76, 2012, pp. 68–89, doi:10.4204/EPTCS.76.7 Tomas Petricek 69 • The translation can be used to construct monads that provide the call-by-need semantics (Section 4.2), which partly answers the open question posed by Wadler. Furthermore, for some monads, it is possible to use parallel call-by-need semantics, where arguments are evaluated in parallel with the body of a function (Section 4.3). • The malias operation has solid foundations in category theory. It arises from augmenting a monad structure with a computational semi-comonad based on the same functor (Section 3). We use this theory to define laws that should be obeyed by malias implementations (Section 2.2). This paper was inspired by work on joinads [19], which introduced the malias operation for a similar purpose. However, operations with the same type and similar laws appear several times in the literature. We return to joinads in Section 4.4 and review other related work in Section 5. 1.1 Translating to monadic code We first demonstrate the two standard options for translating purely functional code to monadic form. Consider the following two functions that use pureLookupInput to read some configuration property. Assuming the configuration is already loaded in memory, we can write the following pure computation1 : chooseSize :: Int → Int → Int chooseSize new legacy = if new > 0 then new else legacy resultSize :: Int resultSize = chooseSize (pureLookupInput "new_size") (pureLookupInput "legacy_size") The resultSize function reads two different configuration keys and chooses one of them using chooseSize. When using a language with lazy evaluation, the call pureLookupInput "legacy_size" is performed only when the value of "new_size" is less than or equal to zero. To modify the function to actually read configuration from a file as opposed to performing in-memory lookup, we now use lookupInput which returns IO Int instead of the pureLookupInput function. Then we need to modify the two above functions. There are two mechanical ways that give different semantics. Call-by-value. In the first style, we call lookupInput and then apply monadic bind on the resulting computation. This reads both of the configuration values before calling the chooseSize function, and so arguments are fully evaluated before the body of a function as in the call-by-value evaluation strategy: chooseSizecbv :: Int → Int → IO Int chooseSizecbv new legacy = return (if new > 0 then new else legacy) resultSizecbv :: IO Int resultSizecbv = do new ← lookupInput cbv "new_size" legacy ← lookupInput cbv "legacy_size" chooseSize cbv new legacy 1 Examples are written in Haskell and can be found at: http://www.cl.cam.ac.uk/~tp322/papers/malias.html Evaluation strategies for monadic computations 70 In this version of the translation, a function of type A → B is turned into a function A → M B. For example, the chooseSizecbv function takes integers as parameters and returns a computation that returns an integer and may perform some effects. When calling a function in this setting, the arguments may not be fully evaluated (the functional part is still lazy), but the effects associated with obtaining the value of the argument happen before the function call. For example, if the call lookupInput cbv "new_size" read a file and then returned 1024, but the operation lookupInput cbv "legacy_size" caused the program to crash because a specified key was not present in a configuration file, then the entire program would crash. Call-by-name. In the second style, we pass unevaluated computations as arguments to functions. This means we call lookupInput to create an effectful computation that will read the input, but the computation is then passed to chooseSize, which may not need to evaluate it: chooseSizecbn :: IO Int → IO Int → IO Int chooseSizecbn new legacy = do newVal ← new if newVal > 0 then new else legacy resultSizecbn :: IO Int resultSizecbn = chooseSize cbn (lookupInput cbn "new_size") (lookupInput cbn "legacy_size") The translation turns a function of type A → B into a function M A → M B. This means that the chooseSize cbn function takes a computation that performs the I/O effect and reads information from the configuration file, as opposed to taking a value whose effects were already performed. Following the mechanical translation, chooseSizecbn returns a monadic computation that evaluates the first argument and then behaves either as new or as legacy, depending on the obtained value. When the resulting computation is executed, the computation which reads the value of the "new_size" key may be executed repeatedly. First, inside the chooseSizecbn function and then repeatedly when the result of this function is evaluated. In this particular example, we can easily change the code to perform the effect just once, but this is not generally possible for computations obtained by the call-by-name translation. 2 Abstracting evaluation strategy The translations demonstrated in the previous section have two major problems. Firstly, it is not easy to switch between the two – when we introduce effects using monads, we need to decide to use one or the other style and changing between them later on involves rewriting of the program and changing types. Secondly, even in the IO monad, we cannot easily implement a call-by-need strategy that would perform effects only when a value is needed, but at most once. 2.1 Translation using aliasing To solve these problems, we propose an alternative translation. We require a monad m with an additional operation malias that abstracts out the evaluation strategy and has a type m a → m (m a). The term aliasing refers to the fact that some part of effects may be performed once and their results shared in multiple monadic computations. The translation of the previous example using malias looks as follows: Tomas Petricek 71 chooseSize :: IO Int → IO Int → IO Int chooseSize new legacy = do newVal ← new if newVal > 0 then new else legacy resultSize :: IO Int resultSize = do new ← malias (lookupInput "new_size") legacy ← malias (lookupInput "legacy_size") chooseSize new legacy The types of functions and access to function parameters are translated in the same way as in the call-byname translation. The chooseSize function returns a computation IO Int and its parameters also become computations of type IO Int. When using the value of a parameter, the computation is evaluated using monadic bind (e.g. the line newVal ← new in chooseSize). The computations passed as arguments are not the original computations as in the call-by-name translation. The translation inserts a call to malias for every function argument (and also for every letbound value, which can be encoded using lambda abstraction and application). The computation returned by malias has a type m (m a), which makes it possible to perform the effects at two different call sites: • When simulating the call-by-value strategy, all effects are performed when binding the outer monadic computation before a function call. • When simulating the call-by-name strategy, all effects are performed when binding the inner monadic computation, when the value is actually needed. These two strategies can be implemented by two simple definitions of malias. However, by delegating the implementation of malias to the monad, we make it possible to implement more advanced strategies as well. We discuss some of them later in Section 4. We keep the translation informal until Section 2.3 and discuss the malias operation in more detail first. Implementing call-by-name. To implement the call-by-name strategy, the malias operation needs to return the computation specified as an argument inside the monad. In the type m (m a), the outer m will not carry any effects and the inner m will be the same as the original computation: malias :: m a → m (m a) malias m = return m From the monad laws (see Figure 1), we know that applying monadic bind to a computation created from a value using return is equivalent to just passing the value to the rest of the computation. This means that the additional binding in the translation does not have any effect and the resulting program behaves as the call-by-name strategy. A complete proof can be found in Appendix A. Implementing call-by-value. Implementing the call-by-value strategy is similarly simple. In the returned computation of type m (m a), the computation corresponding to the outer m needs to perform all the effects. The computation corresponding to the inner m will be a computation that simply returns the previously computed value without performing any effects: malias :: m a → m (m a) malias m = m >>= (return ◦ return) Evaluation strategies for monadic computations 72 Functor with unit and join: unit :: a → m a map :: m a → (a → b) → m b join :: m (m a) → m a Definition using bind (>>=) and return: return :: a → m a >>= :: m a → (a → m b) → m b Join laws (unit and map laws omitted): Monad laws about bind and return: return a >>= f ≡ f a m >>= return ≡ m (m >>= f ) >>= g ≡ m >>= (λ x → f x >>= g) join ◦ map join = join ◦ join join ◦ map unit = id = join ◦ unit join ◦ map (map f ) = map f ◦ join Figure 1: Two equivalent ways of defining monads with monad laws In Haskell, the second line could be written as liftM return m. The liftM operation represents the functor associated with the monad. This means that binding on the returned computation performs all the effects, obtains a value v and returns a computation return v. When calling a function that takes an argument of type m a, the argument passed to it using this implementation of malias will always be constructed using the return operation. Hence the resulting behaviour is equivalent to the original call-by-value translation. Detailed proof can be found in Appendix A. 2.2 The malias operation laws In order to define a reasonable evaluation strategy, we require the malias operation to obey a number of laws. The laws follow from the theoretical background that is discussed in Section 3, namely from the fact that malias is the cojoin operation of a computational semi-comonad. The laws that relate malias to the monad are easier to write in terms of join, map and unit than using the formulation based on >>= and return. For completeness, the two equivalent definitions of monads with the monad laws are shown in Figure 1. Although we do not show it, one can be easily defined in terms of the other. The required laws for malias are the following: map (map f ) ◦ malias map malias ◦ malias malias ◦ unit join ◦ malias = = = = malias ◦ (map f ) malias ◦ malias unit ◦ unit id (naturality) (associativity) (computationality) (identity) The first two laws follow from the fact that malias is a cojoin operation of a comonad. The naturality law specifies that applying a function to a value inside a computation is the same as applying the function to a value inside an aliased computation. The associativity law specifies that aliasing an aliased computation is the same as aliasing a computation produced by an aliased computation. The computationality law is derived from the fact that the comonad defining malias is a computational comonad with unit as one of the components. The law specifies that aliasing of a pure computation creates a pure computation. Finally, the identity law relates malias with the monadic structure, by requiring that join is a left inverse of malias. Intuitively, it specifies that aliasing a computation of type m a and then joining the result returns the original computation. All four laws hold for the two implementations of malias presented in the previous section. We prove that the laws hold for any monad using the standard monad laws. The proofs can be found in Appendix B. We discuss the intuition behind the laws in Section 2.4 and describe their categorical foundations in Section 3. The next section formally presents the translation algorithm. Tomas Petricek 73 JxKcbn Jλ x.eKcbn Je1 e2 Kcbn Jlet x = e1 in e2 Kcbn = = = = x unit (λ x.JeKcbn ) bind Je1 Kcbn (λ f . f Je2 Kcbn ) (λ x.Je2 Kcbn ) Je1 Kcbn Figure 2: Wadler’s call-by-name translation of λ calculus. JxKcbv Jλ x.eKcbv Je1 e2 Kcbv Jlet x = e1 in e2 Kcbv = = = = unit x unit (λ x.JeKcbv ) bind Je1 Kcbv (λ f . bind Je2 Kcbv (λ x. f x)) bind Je1 Kcbv (λ x.Je2 Kcbv ) Figure 3: Wadler’s call-by-value translation of λ calculus. 2.3 Lambda calculus translation The call-by-name and call-by-value translations given in Section 1.1 were first formally introduced by Wadler [26]. In this section, we present a similar formal definition of our translation based on the malias operation. For our source language, we use a simply-typed λ calculus with let-binding: e ∈ Expr e ::= x \| λ x.e \| e1 e2 \| let x = e1 in e2 τ ∈ Type τ ::= α \| τ1 → τ2 The target language of the translation is identical but for one exception – it adds a type scheme M τ representing monadic computations. The call-by-name and call-by-value translation of the lambda calculus are shown in Figure 2 and Figure 3, respectively. In the translation, we write >>= as bind. The translation of types and typing judgements are omitted for simplicity and can be found in the original paper [26]. Our translation, called call-by-alias, is presented below. The translation has similar structure to Wadler’s call-by-name translation, but it inserts malias operation in the last two cases: Jα Kcba Jτ1 → τ2 Kcba = α = M Jτ1 Kcba → M Jτ2 Kcba JxKcba Jλ x.eKcba Je1 e2 Kcba Jlet x = e1 in e2 Kcba = = = = x unit (λ x.JeKcba ) bind Je1 Kcba (λ f .bind (malias Je2 Kcba ) f ) bind (malias Je1 Kcba ) (λ x.Je2 Kcba ) Jx1 : τ1 , . . . , xn : τn ⊢ e : τ Kcba = x1 : M Jτ1 Kcba , . . . , xn : M Jτn Kcba ⊢ JeKcba : M Jτ Kcba The translation turns user-defined variables of type τ into variables of type M τ . A variable access x is translated to a variable access, which now represents a computation (that may have effects). A lambda expression is turned into a lambda expression wrapped in a pure monadic computation. The two interesting cases are application and let-binding. When translating function application, we bind on the computation representing the function. We want to call the function f with an aliased computation as an argument. This is achieved by passing the translated argument to malias and then applying bind again. The translation of let-binding is similar, but slightly simpler, because it does not need to use bind to obtain a function. Evaluation strategies for monadic computations 74 The definition of J−Kcba includes a well-typedness-preserving translation of typing judgements. The details of the proof can be found in Appendix C. In the translation, let-binding is equivalent to (λ x.e2 ) e1 , but we include it to aid the intuition and to simplify motivating examples in the next section. 2.4 The meaning of malias laws Having provided the translation, we can discuss the intuition behind the malias laws and what they imply about the translated code. The naturality law specifies that malias is a natural transformation. Although we do not give a formal proof, we argue that the law follows from the parametricity of the malias type signature and can be obtained using the method described by Voigtländer [25]. Effect conservation. The meaning of associativity and identity can be informally demonstrated by treating effects as units of information. Given a computation type involving a number of occurrences of m, we say that there are effects (or information) associated with each occurrence of m. The identity law specifies that join ◦ malias does not lose effects – given a computation of type m a, the malias operation constructs a computation of type m (m a). This is done by splitting the effects of the computation between two monadic computations. Requiring that applying join to the new computation returns the original computation means that all the effects of the original computation are preserved, because join combines the effects of the two computations. The associativity law specifies how the effects should be split. When applying malias to an aliased computation of type m (m a), we can apply the operation to the inner or the outer m. Underlining second aliasing, the following two computations should be equal: m (m (m a)) = m (m (m a)). The law forbids implementations that split the effects in an asymmetric way. For instance, if m a represents a step-wise computation that can be executed by a certain number of steps, malias cannot execute the first half of steps and return a computation that evaluates the other half. The proportions associated with individual m values would be 1/4, 1/4 and 1/2 on the left and 1/2, 1/4 and 1/4 on the right-hand side. Semantics-preserving transformations. The computationality and identity (again) laws also specify that certain semantics-preserving transformations on the original source code correspond to equivalent terms in the code translated using the call-by-alias transformation. The source transformations corresponding to computationality and identity, respectively, are the following: let f = λ x.e1 in e2 ≡ e2 [ f ← λ x.e1 ] let x = e in x ≡ e The construction of the lambda function in the first equation is the only place where the translation inserts unit. The computationality law can be applied when a lambda abstraction appears in a position where malias is inserted. Thanks to the first monad law (Figure 1), the value assigned to f in the translation is unit (λ x.e1 ), which is equivalent to the translation of a term after the substitution. In the second equation, the left-hand side is translated as bind (malias e) id, which is equivalent to join (malias e), so the equation directly corresponds to the identity law. Discussion of completeness. The above discussion, together with the theoretical foundations introduced in the next section, supports the claim that our laws are necessary. We do not argue that our set of laws is complete – for instance, we might want to specify that aliasing of an already aliased computation has no effect, which is difficult to express in an equational form. Tomas Petricek 75 However, the definition of completeness, in this context, is elusive. One possible approach that we plan to investigate in future work is to expand the set of semantics-preserving transformations that should hold, regardless of an evaluation strategy. 3 Computational semi-bimonads In this section, we formally describe the structure that underlies a monad having a malias operation as described in the previous section. Since the malias operation corresponds to an operation of a comonad associated with a monad, we first review the definitions of a monad and a comonad. Monads are wellknown structures in functional programming. Comonads are dual structures to monads that are less widespread, but they have also been used in functional programming and semantics (Section 5): Definition 1. A monad over a category C is a triple (T, η , µ ) where T : C → C is a functor, η : IC → T is a natural transformation from the identity functor to T , and µ : T 2 → T is a natural transformation, such that the following associativity and identity conditions hold, for every object A: • µA ◦ T µA = µA ◦ µTA • µA ◦ ηTA = idTA = µA ◦ T ηA Definition 2. A comonad over a category C is a triple (T, ε , δ ) where T : C → C is a functor, ε : T → IC is a natural transformation from T to the identity functor, and δ : T → T 2 is a natural transformation from T to T 2 , such that the following associativity and identity conditions hold, for every object A: • T δA ◦ δA = δTA ◦ δA • εTA ◦ δA = idTA = T εA ◦ δA In functional programming terms, the natural transformation η corresponds to unit :: a → m a and the natural transformation µ corresponds to join :: m (m a) → m a. A comonad is a dual structure to a monad – the natural transformation ε corresponds to an operation counit :: m a → a and δ corresponds to cojoin :: m a → m (m a). An equivalent formulation of comonads in functional programming uses an operation cobind :: m a → (m a → b) → m b, which is dual to >>= of monads. A simple example of a comonad is the product comonad. The type m a stores the value of a and some additional state S, meaning that TA = A × S. The ε (or counit) operation extracts the value A ignoring the additional state. The δ (or cojoin) operation duplicates the state. In functional programming, the product comonad is equivalent to the reader monad TA = S → A. In this paper, we use a special variant of comonads. Computational comonads, introduced by Brookes and Geva [5], have an additional operation γ together with laws specifying its properties: Definition 3. A computational comonad over a category C is a quadruple (T, ε , δ , γ ) where (T, ε , δ ) is a comonad over C and γ : IC → T is a natural transformation such that, for every object A, • εA ◦ γA = idA • δA ◦ γA = γTA ◦ γA . A computational comonad has an additional operation γ which has the same type as the η operation of a monad, that is a → m a. In the work on computational comonads, the transformation γ turns an extensional specification into an intensional specification without additional computational information. Evaluation strategies for monadic computations 76 In our work, we do not need the natural transformation corresponding to counit :: m a → a. We define a computational semi-comonad, which is a computational comonad without the natural transformation ε and without the associated laws. The remaining structure is preserved: Definition 4. A computational semi-comonad over a category C is a triple (T, δ , γ ) where T : C → C is a functor, δ : T → T 2 is a natural transformation from T to T 2 and γ : IC → T is a natural transformation from the identity functor to T , such that the following associativity and computationality conditions hold, for every object A: • T δA ◦ δA = δTA ◦ δA • δA ◦ γA = γTA ◦ γA . Finally, to define a structure that models our monadic computations with the malias operation, we combine the definition of a monad and computational semi-comonad. We require that the two structures share the functor T and that the natural transformation η : IC → T of a monad coincides with the natural transformation γ : IC → T of a computational comonad. Definition 5. A computational semi-bimonad over a category C is a quadruple (T, η , µ , δ ) where (T, η , µ ) is a monad over a category C and (T, δ , η ) is a computational semi-comonad over C , such that the following additional condition holds, for every object A: • µA ◦ δA = idTA The definition of computational semi-bimonad relates the monadic and comonadic parts of the structure using an additional law. Given an object A, the law specifies that taking TA to T 2 A using the natural transformation δA of a comonad and then back to TA using the natural transformation µA is identity. 3.1 Revisiting the laws The laws of computational semi-bimonad as defined in the previous section are exactly the laws of our monad equipped with the malias operation. In this section, we briefly review the laws and present the category theoretic version of all the laws demonstrated in Section 2.2. We require four laws in addition to the standard monad laws (which are omitted in the summary below). A diagrammatic demonstration is shown in Figure 4. For all objects A and B of C and for all f : A → B in C : T 2 f ◦ δA T δA ◦ δA δA ◦ η A µA ◦ δA = = = = δB ◦ T f δTA ◦ δA ηTA ◦ ηA idTA (naturality) (associativity) (computationality) (identity) The naturality law follows from the fact that δ is a natural transformation and so we did not state it explicitly in Definition 5. However, it is one of the laws that are translated to the functional programming interpretation. The associativity law is a law of comonad – the other law in Definition 2 does not apply in our scenario, because we only work with semi-comonad that does not have natural transformation ε (counit). The computationality law is a law of a computational comonad and finally, the identity law is the additional law of computational semi-bimonads. Tomas Petricek TA δA Tf T 2A T2 f 77 // // T B δB T 2B (naturality) TA δA δA T 2A // T 2 A T δA δTA // T 3 A (associativity) A ηA ηA // TA δA TA ηTA // T 2 A (computationality) TA❉ idTA ❉❉ ❉❉ ❉❉ δA ❉"" T 2A // TA ③<< ③ µA ③③ ③③ ③③ (identity) Figure 4: Diagramatic representation of the four additional properties of semi-bimonads 4 Abstracting evaluation strategy in practice In this section, we present several practical uses of the malias operation. We start by showing how to write monadic code that is parameterized over the evaluation strategy and then consider expressing callby-need in this framework. Then we also briefly consider parallel call-by-need and the relation between malias and joinads and the docase notation [19]. 4.1 Parameterization by evaluation strategy One of the motivations of this work is that the standard monadic translations for call-by-name and callby-value produce code with different structure. Section 2.3 gave a translation that can be used with both of the evaluation strategies just by changing the definition of the malias operation. In this section, we make one more step – we show how to write code parameterized by evaluation strategy. We define a monad transformer [13] that takes a monad and turns it into a monad with malias that implements a specific evaluation strategy. Our example can then be implemented using functions that are polymorphic over the monad transformer. We continue using the previous example based on the IO monad, but the transformer can operate on any monad. As a first step, we define a type class named MonadAlias that extends Monad with the malias operation. To keep the code simple, we do not include comments documenting the laws: class Monad m ⇒ MonadAlias m where malias :: m a → m (m a) Next, we define two new types that represent monadic computations using the call-by-name and call-byvalue evaluation strategy. The two types are wrappers that make it possible to implement two different instances of MonadAlias for any underlying monadic computation m a: newtype CbV m a = CbV {runCbV :: m a} newtype CbN m a = CbN {runCbN :: m a} The snippet defines types CbV and CbN that represent two evaluation strategies. Figure 5 shows the implementation of three type classes for these two types. The implementation of the Monad type class is the same for both types, because it simply uses return and >>= operations of the underlying monad. The implementation of MonadTrans wraps a monadic computation m a into a type CbV m a and CbN m a, respectively. Finally, the instances of the MonadAlias type class associate the two implementations of malias (from Section 2.1) with the two data types. Evaluation strategies for monadic computations 78 instance Monad m ⇒ Monad (CbV m) where return v = CbV (return v) (CbV a) >>= f = CbV (a >>= (runCbV ◦ f )) instance Monad m ⇒ Monad (CbN m) where return v = CbN (return v) (CbN a) >>= f = CbN (a >>= (runCbN ◦ f )) instance MonadTrans CbV where lift = CbV instance MonadTrans CbN where lift = CbN instance Monad m ⇒ MonadAlias (CbV m) where malias m = m >>= (return ◦ return) instance Monad m ⇒ MonadAlias (CbN m) where malias m = return m Figure 5: Instances of Monad, MonadTrans and MonadAlias for evaluation strategies Example. Using the previous definitions, we can now rewrite the example from Section 1.1 using generic functions that can be executed using both runCbV and runCbN. Instead of implementing malias for a specific monad such as IO a, we use a monad transformer t that lifts the monadic computation to either CbV IO a or to CbN IO a. This means that all functions will have constraints MonadTrans t, specifying that t is a monad transformer, and MonadAlias (t m), specifying that the computation implements the malias operation. In Haskell, this can be succinctly written using constraint kinds [3], that make it possible to define a single constraint EvalStrategy t m that combines both of the conditions2 : type EvalStrategy t m = (MonadTrans t, MonadAlias (t m)) Despite the use of the type keyword, the identifier EvalStrategy actually has a kind Constraint, which means that it can be used to specify assumptions about types in a function signature. In our example, the constraint EvalStrategy t IO denotes monadic computations based on the IO monad that also provide an implementation of MonadAlias: chooseSize :: EvalStrategy t IO ⇒ t IO Int → t IO Int → t IO Int chooseSize new legacy = do newVal ← new if newVal > 0 then new else legacy resultSize :: EvalStrategy t IO ⇒ t IO Int resultSize = do new ← malias $ lift (lookupInput "new_size") legacy ← malias $ lift (lookupInput "legacy_size") chooseSize new legacy Compared to the previous version of the example, the only significant change is in the type signature of the two functions. Instead of passing computations of type IO a, they now work with computations t IO a 2 Constraint kinds are available in GHC 7.4 and generalize constraint families proposed by Orchard and Schrijvers [18] Tomas Petricek 79 with the constraint EvalStrategy t IO. The result of lookupInput is still IO Int and so it needs to be lifted into a monadic computation t IO Int using the lift function. The return type of the resultSize computation is parameterized over the evaluation strategy t. This means that we can call it in two different ways. Writing runCbN resultSize executes the function using call-by-name and writing runCbV resultSize executes it using call-by-value. However, it is also possible to implement a monad transformer for call-by-need. 4.2 Implementing call-by-need strategy In his paper introducing the call-by-name and call-by-value translations for monads, Wadler noted: “It remains an open question whether there is a translation scheme that corresponds to call-by-need as opposed to call-by-name” [26]. We do not fully answer this question, however we hope to contribute to the answer. In particular, we show how to use the mechanism described so far to turn certain monads into an extended versions of such monads that provide the call-by-need behaviour. In the absence of effects, the call-by-need strategy is equivalent to the call-by-name strategy, with the only difference in that performance characteristics. In the call-by-need (or lazy) strategy, a computation passed as an argument is evaluated at most once and the result is cached afterwards. The caching of results needs to be done in the malias operation. This cannot be done for any monad, but we can define a monad transformer similar to the ones presented in the previous section. In particular, we use Svenningsson’s package [22], which defines a transformer version of the ST monad [11]. As documented in the package description, the transformer should be applied only to monads that yield a single result. Combining lazy evaluation with non-determinism is a more complex topic that has been explored by Fischer et al. [6]. newtype CbL s m a = CbL {unCbL :: STT s m a} Unlike CbV and CbN, the CbL type is not a simple wrapper that contains a computation of type m a. Instead, it contains a computation augmented with some additional state. The state is used for caching the values of evaluated computations. The type STT s m a represents a computation m a with an additional local state “tagged” with a type variable s. The use of a local state instead of e.g. IO means that the monadic computation can be safely evaluated even as part of purely functional code. The tags are used merely to guarantee that state associated with one STT computation does not leak to other parts of the program. Implementing the Monad and MonadTrans instances follows exactly the same pattern as instances of other transformers given in Figure 5. The interesting work is done in the malias function of MonadAlias: instance Monad m ⇒ MonadAlias (CbL s m) where malias (CbL marg) = CbL $ do r ← newSTRef Nothing return (CbL $ do rv ← readSTRef r case rv of Nothing → marg >>= λ v → writeSTRef r (Just v) >> return v Just v → return v) The malias operation takes a computation of type m a and returns a computation m (m a). The monad transformer wraps the underlying monad inside STT, so the type of the computation returned by malias is equivalent to a type STT s m (STT s m a). Evaluation strategies for monadic computations 80 The fact that both outer and inner STT share the same tag s means that they operate on a shared state. The outer computation allocates a new reference and the inner computation can use it to access and store the result computed previously. The allocation is done using the newSTRef function, which creates a reference initialized to Nothing. In the returned (inner) computation, we first read the state using readSTRef . If the value was computed previously, then it is simply returned. If not, the computation evaluates marg, stores the result in a reference cell and then returns the obtained value. Discussion. After implementing runCbL function (which can be done easily using runST), the example in Section 4.1 can be executed using the CbL type. With the call-by-need semantics, the program finally behaves as desired: if the value of the "new_size" key is greater than zero, then it reads it only once, without reading the value of the "legacy_size" key. The value of "legacy_size" key is accessed only if the value of the "new_size" key is less than zero. Showing that the above definition corresponds to call-by-need formally is beyond the scope of this paper. However, the use of STT transformer adds a shared state that keeps the evaluated values, which closely corresponds to Launchbury’s environment-based semantics of lazy evaluation [9]. We do not formally prove that the malias laws hold for the above implementation, but we give an informal argument. The naturality law follows from parametricity. Computations considered in the associativity law are of type m (m (m a)); both sides of the equation create a computation where the two outer m computations allocate a new reference (where the outer points to the inner) and the single innermost m actually triggers the computation. The computationality law holds, because aliasing of a unit computation cannot introduce any effects. Finally, the left-hand side of identity creates a computation that allocates a new reference that is encapsulated in the returned computation and cannot be accessed from elsewhere, so no sharing is added when compared with the right-hand side. 4.3 Parallel call-by-need strategy In this section, we consider yet another evaluation strategy that can be implemented using our scheme. The parallel call-by-need strategy [2] is similar to call-by-need, but it may optimistically start evaluating a computation sooner, in parallel with the main program. When carefully tuned, the evaluation strategy may result in a better performance on multi-core CPUs. We present a simple implementation of the malias operation based on the monad for deterministic parallelism by Marlow et al. [14]. By translating purely functional code to a monadic version using our translation and the Par monad, we get a program that attempts to evaluate arguments of every function call in parallel. In practice, this may introduce too much overhead, but it demonstrate that parallel call-by-need strategy also fits with our general framework. Unlike the previous two sections, we do not define a monad transformer that can embody any monadic computation. For example, performing IO operations in parallel might introduce non-determinism. Instead, we implement malias operation directly for computations of type Par a: instance MonadAlias Par where malias m = spawn m >>= return ◦ get The implementation is surprisingly simple. The function spawn creates a computation Par (IVar a) that starts the work in background and returns a mutable variable (I-structure) that will contain the result when the computation completes. The inner computation that is returned by malias calls a function get :: IVar a → Par a that waits until IVar has been assigned a value and then returns it. Tomas Petricek 81 Using the above implementation of malias, we can now translate purely functional code to a monadic version that uses parallel call-by-need instead of the previous standard evaluation strategies (that do not introduce any parallelism). For example, consider a naive Fibonacci function: fibSeq n \| n 6 1 = n fibSeq n = fibSeq (n − 1) + fibSeq (n − 2) The second case calls the + operator with two computations as arguments. The translated version passes these computations to malias, which starts executing them in parallel. The monadic version of the + operator then waits until both computations complete. If we translate only the second case and manually add a case that calls sequential version of the function for inputs smaller than 30, we get the following: fibPar n \| n < 30 = return (fibSeq n) fibPar n = do n1 ← malias $ fibPar (n − 1) n2 ← malias $ fibPar (n − 2) liftM2 (+) n1 n2 Aside from the first line, the code directly follows the general translation mechanism described earlier. Arguments of a function are turned to monadic computations and passed to malias. The inner computations obtained using monadic bind are then passed to a translated function. Thanks to the manual optimization that calls fibSeq for smaller inputs, the function runs nearly twice as fast on a dual-core machine3 . Parallel programming is one of the first areas where we found the malias operation useful. We first considered it as part of joinads, which are discussed in the next section. Discussion. Showing that the above implementation actually implements parallel call-by-need could be done by relating our implementation of malias to the multi-thread transitions of [2]. Informally, each computation that may be shared is added to the work queue (using spawn) when it occurs on the righthand side of a let binding, or as an argument in function application. The queued work evaluates in parallel with the main program and the get function implements sharing, so the semantics is lazy. As previously, the naturality law holds thanks to parametricity. To consider other laws, we need a formal model that captures the time needed to evaluate computations. We assume that evaluating a computation created by unit takes no time, but all other computations take non-zero time. Moreover, all spawned computations start executing immediately (i.e. the number of equally fast threads is unlimited). In the associativity law, the left-hand side returns a computation that spawns the actual work and then spawns a computation that waits for its completion. The right-hand side returns a computation that schedules a computation, which then schedules the actual work. In both cases, the actual work is started immediately when the outer m computation is evaluated, and so they are equivalent. The computationality law holds, because a unit computation evaluates in no time. Finally, the left-hand side of identity returns a computation that spawns the work and then waits for its completion, which is semantically equivalent to just running the computation. 4.4 Simplifying Joinads Joinads [20, 19] were designed to simplify programming with certain kinds of monadic computations. Many monads, especially from the area of concurrent or parallel programming provide additional oper3 When called with input 37 on a Core 2 Duo CPU, the sequential version runs in 8.9s and the parallel version in 5.1s. Evaluation strategies for monadic computations 82 ations for composing monadic computations. Joinads identify three most common extensions: parallel composition, (non-deterministic) choice and aliasing. Joinads are abstract computations that form a monad and provide the three additional operations mentioned above. The work on joinads also introduces a syntactic extension for Haskell and F# that makes it easier to work with these classes of computations. For example, the following snippet uses the Par monad to implement a function that tests whether a predicate holds for all leafs of a tree in parallel: all :: (a → Bool) → Tree a → Par Bool all p (Leaf v) = return (p v) all p (Node left right) = docase (all p left, all p right) of (False, ?) → return False (?, False) → return False (allL, allR) → return (allL ∧ allR) The docase notation intentionally resembles pattern matching and has similar semantics as well. The first two cases use the special pattern ? to denote that the value of one of the computations does not have to be available in order to continue. When one of the sub-branches returns False, we know that the overall result is False and so we return immediately. Finally, the last clause matches if neither of the two previous are matched. It can only match after both sub-trees are processed. Similarly to do notation, the docase syntax is desugared into uses of the joinad operations. The choice between clauses is translated using the choice operator. If a clause requires the result of multiple computations (such as the last one), the computations are combined using parallel composition. If a computation, passed as an argument, is accessed from multiple clauses, then it should be evaluated only once and the clauses should only access aliased computation. This motivation is similar to the one described in this article. Indeed, joinads use a variant of the malias operation and insert it automatically for all arguments of docase. This is very similar to how the translation presented in this paper uses malias. If aliasing was not done automatically behind the scenes, we would have to write: all p (Leaf v) = return (p v) all p (Node left right) = do l ← malias (all p left) r ← malias (all p right) docase (l, r) of (False, ?) → return False (?, False) → return False (allL, allR) → return (allL ∧ allR) One of the limitations of the original design of joinads is that there is no one-to-one correspondence between the docase notation and what can be expressed directly using the joinad operations. This is partly due to the automatic aliasing of arguments, which inserts malias only at very specific locations. We believe that integrating a call-by-alias translation, described in this article, in a programming language could resolve this situation. It would also separate the two concerns – composition of computations using choice and parallel composition (done by joinads) and automatic aliasing of computations that allows sharing of results as in call-by-need or parallel call-by-need. Tomas Petricek 5 83 Related work Most of the work that directly influenced this work has been discussed throughout the paper. Most importantly, the question of translating pure code to a monadic version with call-by-need semantics was posed by Wadler [26]. To our knowledge, this question has not been answered before, but there is various work that either uses similar structures or considers evaluation strategies from different perspectives. Monads with aliasing. Numerous monads have independently introduced an operation of type m a → m (m a). The eagerly combinator of the Orc monad [10] causes computation to run in parallel, effectively implementing the parallel call-by-need evaluation strategy. The only law that relates eagerly to operations of a monad is too restrictive to allow the call-by-value semantics. A monad for purely functional lazy non-deterministic programming [6] uses a similar combinator share to make monadic (non-deterministic) computations lazy. However, the call-by-value strategy is inefficient and the call-by-name strategy is incorrect, because choosing a different value each time a non-deterministic computation is accessed means that generate and test pattern does not work. The share operation is described together with the laws that should hold. The HNF law is similar to our computationality. However, the Ignore law specifies that the share operation should not be strict (ruling out our call-by-value implementation of malias). A related paper [4] discusses where share needs to be inserted when translating lazy non-deterministic programs to a monadic form. The results may be directly applicable to make our translation more efficient by inserting malias only when required. Abstract computations and comonads. In this paper, we extended monads with one component of a comonadic structure. Although less widespread than monads, comonads are also useful for capturing abstract computations in functional programming. They have been used for dataflow programming [23], array programming [17], environment passing, and more [8]. In general, comonads can be used to describe context-dependent computations [24], where cojoin (natural transformation δ ) duplicates the context. In our work, the corresponding operation malias splits the context (effects) in a particular way between two computations. We only considered basic monadic computations, but it would be interesting to see how malias interacts with other abstract notions of computations, such as applicative functors [15], arrows [7] or additive monads (the MonadPlus type-class). The monad for lazy non-deterministic programming [6], mentioned earlier, implements MonadPlus and may thus provide interesting insights. Evaluation strategies. One of the key results of this paper is a monadic translation from purely functional code to a monadic version that has the call-by-need semantics. We achieve that using the monad transformer [13] for adding state. In the absence of effects, call-by-need is equivalent to call-by-name, but it has been described formally as a version of λ -calculus by Ariola and Felleisen [1]. This allows equational reasoning about computations and it could be used to show that our encoding directly corresponds to call-by-need, similarly to proofs for other strategies in Appendix A. The semantics has been also described using an environment that models caching [9, 21], which closely corresponds to our actual implementation. Considering the two basic evaluation strategies, Wadler [27] shows that call-by-name is dual to callby-value. We find this curious as the two definitions of malias in our work are, in some sense, also dual or symmetric as they associate all effects with the inner or the outer monad of the type m (m a). Furthermore, the duality between call-by-name and call-by-value can be viewed from a logical perspective Evaluation strategies for monadic computations 84 thanks to the Curry-Howard correspondence. We believe that finding a similar logical perspective for our generalized strategy may be an interesting future work. Finally, the work presented in this work unifies monadic call-by-name and call-by-value. In a nonmonadic setting, a similar goal is achieved by the call-by-push-value calculus [12]. The calculus is more fine-grained and strictly separates values and computations. Using these mechanisms, it is possible to encode both call-by-name and call-by-value. It may be interesting to consider whether our computations parameterized over evaluation strategy (Section 4.1) could be encoded in call-by-push-value. 6 Conclusions We presented an alternative translation from purely functional code to monadic form. Previously, this required choosing either the call-by-need or the call-by-value translation and the translated code had different structure and different types in both cases. Our translation abstracts the evaluation strategy into a function malias that can be implemented separately providing the required evaluation strategy. Our translation is not limited to the above two evaluation strategies. Most interestingly, we show that certain monads can be automatically turned into an extended version that supports the call-by-need strategy. This answers part of an interesting open problem posed by Wadler [26]. The approach has other interesting applications – it makes it possible to write code that is parameterized by the evaluation strategy and it allows implementing a parallel call-by-need strategy for certain monads. Finally, we presented the theoretical foundations of our approach using a model described in terms of category theory. We extended the monad structure with an additional operation based on computational comonads, which were previously used to give intensional semantics of computations. In our setting, the operation specifies the evaluation order. The categorical model specifies laws about malias and we proved that the laws hold for call-by-value and call-by-name strategies. Acknowledgements. The author is grateful to Sebastian Fischer, Dominic Orchard and Alan Mycroft for inspiring comments and discussion, and to the latter two for proofreading the paper. We are also grateful to anonymous referees for detailed comments and to Paul Blain Levy for shepherding of the paper. The work has been partly supported by EPSRC and through the Cambridge CHESS scheme. References [1] Zena M. Ariola & Matthias Felleisen (1997): The Call-By-Need Lambda Calculus. Journal of Functional Programming 7, pp. 265–301, doi:10.1017/S0956796897002724. [2] Clem Baker-Finch, David J. King & Phil Trinder (2000): An Operational Semantics for Parallel Lazy Evaluation. In: Proceedings of ICFP 2000, doi:10.1145/351240.351256. [3] Max Bolingbroke (2011): Constraint Kinds for GHC (Unpublished manuscript). Available at http://blog. omega-prime.co.uk/?p=127. [4] B. Braßel, S. Fischer, M. Hanus & F. Reck (2011): Transforming Functional Logic Programs into Monadic Functional Programs. In: Proceedings of WFLP 2010, LNCS 6559, pp. 30–47, doi:10.1.1.157.4578. [5] Stephen Brookes & Shai Geva (1992): Computational Comonads and Intensional Semantics. In: Applications of Categories in Computer Science: Proceedings of the LMS Symposium, 177, pp. 1–44, doi:10.1.1. 45.4952. [6] Sebastian Fischer, Oleg Kiselyov & Chung-chieh Shan (2011): Purely Functional Lazy Non-Deterministic Programming. Journal of Functional Programming 21(4–5), pp. 413–465, doi:10.1145/1631687.1596556. Tomas Petricek 85 [7] John Hughes (1998): Generalising Monads to Arrows. Science of Computer Programming 37, pp. 67–111, doi:10.1016/S0167-6423(99)00023-4. [8] Richard B. Kieburtz (1999): Codata and Comonads in Haskell (Unpublished manuscript). [9] John Launchbury (1993): A Natural Semantics for Lazy Evaluation. In: Proceedings of POPL 1993, pp. 144–154, doi:10.1145/158511.158618. [10] John Launchbury & Trevor Elliott (2010): Concurrent orchestration in Haskell. In: Proceedings of Haskell Symposium, Haskell 2010, pp. 79–90, doi:10.1145/1863523.1863534. [11] John Launchbury & Simon Peyton Jones (1995): State in Haskell. In: Proceedings of the LISP and Symbolic Computation Conference, LISP 1995, doi:10.1007/BF01018827. [12] Paul Blain Levy (2004): Call-By-Push-Value. Springer. 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In: Proceedings of FLOPS 2010, pp. 56–71, doi:10.1007/978-3-642-12251-4_6. [19] Tomas Petricek, Alan Mycroft & Don Syme: Extending Monads with Pattern Matching. In: Proceedings of Haskell Symposium, Haskell 2011, doi:10.1145/2096148.2034677. [20] Tomas Petricek & Don Syme (2011): Joinads: A Retargetable Control-Flow Construct for Reactive, Parallel and Concurrent Programming. In: Proceedings of PADL 2011, doi:10.1007/978-3-642-18378-2_17. [21] S. Purushothaman & Jill Seaman (1992): An Adequate Operational Semantics for Sharing in Lazy Evaluation. In: Proceedings of the 4th European Symposium on Programming, doi:10.1007/3-540-55253-7_ 26. [22] Josef Svenningsson (2011): The STMonadTrans package. Available at http://hackage.haskell.org/ package/STMonadTrans-0.3.1. [23] Tarmo Uustalu & Varmo Vene (2006): The Essence of Dataflow Programming. Lecture Notes in Computer Science 4164, pp. 135–167, doi:10.1007/11894100_5. [24] Tarmo Uustalu & Varmo Vene (2008): Comonadic Notions of Computation. Electron. Notes Theor. Comput. Sci. 203, pp. 263–284, doi:10.1016/j.entcs.2008.05.029. [25] Janis Voigtländer (2009): Free Theorems Involving Type Constructor Classes: Functional Pearl. In: Proceedings of ICFP 2009, ICFP 209, ACM, New York, NY, USA, pp. 173–184, doi:10.1145/1596550.1596577. [26] Philip Wadler (1990): Comprehending Monads. In: Proceedings of Conference on LISP and Functional Programming, LFP 1990, ACM, New York, NY, USA, pp. 61–78, doi:10.1145/91556.91592. [27] Philip Wadler (2003): Call-By-Value Is Dual to Call-By-Name. In: Proceedings of ICFP, ICFP 2003, pp. 189–201, doi:10.1145/944705.944723. Evaluation strategies for monadic computations 86 Je1 e2 Kcba = bind Je1 Kcba (λ f .bind (malias Je2 Kcba ) f )) = bind Je1 Kcba (λ f .bind (unit Je2 Kcba ) f )) = bind Je1 Kcba (λ f . f Je2 Kcba ) = bind Je1 Kcbn (λ f . f Je2 Kcbn ) = Je1 e2 Kcbn (translation) (definition) (left identity) (induction hypothesis) (translation) Jlet x = e1 in e2 Kcba = bind (malias Je1 Kcba ) (λ x.Je2 Kcba ) = bind (unit Je1 Kcba ) (λ x.Je2 Kcba ) = (λ x.Je2 Kcba ) Je1 Kcba = (λ x.Je2 Kcbn ) Je1 Kcbn = Jlet x = e1 in e2 Kcbn (translation) (definition) (left identity) (induction hypothesis) (translation) Figure 6: Proving that using an appropriate malias is equivalent to call-by-name. A Equivalence proofs In this section, we show that our translation presented in Section 2.3 can be used to implement standard call-by-name and call-by-value. We prove that using an appropriate definition of malias from Section 2.1 gives the same semantics as the standard translations described by Wadler [26]. Call-by-name. The translation of types is the same for our translation and the call-by-name translation. In addition, the rules for translating variable access and lambda functions are also the same. This means that we only need to prove that our translation of let-binding and function application are equivalent. When implementing call-by-name using our translation, we use the following definition of malias: malias m = unit m Using this definition and the left identity monad law, we can now show that our call-by-alias translation is equivalent to the translation from Figure 2. The Figure 6 shows the equations for function application and let-binding. Call-by-value. Proving that appropriate definition of malias gives a term that corresponds to the one obtained using standard call-by-value translation is more difficult. In the call-by-value translation, functions are translated to a type τ1 → M τ2 , while our translation produces functions of type M τ1 → M τ2 . As a reminder, the definition of malias that gives the call-by-value behaviour is the following: malias m = bind m (unit ◦ unit) To prove that the two translations are equivalent, we show that the following invariant holds: when using the above definition of malias and our call-by-alias translation, a monadic computation of type M τ that is assigned to a variable x always has a structure unit xv where xv is a variable of type τ . The sketch of the proof is shown in Figure 7. We write e1 ∼ = e2 to mean that the expression e1 in the call-by-alias translation corresponds to an expression e2 in call-by-value translation. This means that expressions of form unit xv translate to values x, variable declarations of xv (in lambda abstraction and let-binding) translate to declarations of x and all function values of type M τ1 → M τ2 become τ1 → M τ2 . Tomas Petricek 87 JxKcba = unit xv ∼ = x = JxKcbv (assumption) (correspondence) (translation) Je1 e2 Kcba = bind Je1 Kcba (λ f .bind (malias Je2 Kcba ) f ) = bind Je1 Kcba (λ f .bind (bind Je2 Kcba (unit ◦ unit)) f ) = bind Je1 Kcba (λ f .bind Je2 Kcba (λ xv .bind (unit (unit xv )) f )) = bind Je1 Kcba (λ f .bind Je2 Kcba (λ xv . f (unit xv ))) ∼ = bind Je1 Kcbv (λ f .bind Je2 Kcbv (λ x. f x)) = Je1 e2 Kcbv (translation) (definition) (associativity) (left identity) (correspondence) (translation) Jlet x = e1 in e2 Kcba = bind (malias Je1 Kcba ) (λ x.Je2 Kcba ) = bind (bind Je1 Kcba (unit ◦ unit)) (λ x.Je2 Kcba ) = bind Je1 Kcba (λ xv .bind (unit (unit xv )) (λ x.Je2 Kcba )) = bind Je1 Kcba (λ xv .(λ x.Je2 Kcba ) (unit xv )) ∼ = bind Je1 Kcbv (λ x.Je2 Kcbv ) = Jlet x = e1 in e2 Kcbv (translation) (definition) (associativity) (left identity) (correspondence) (translation) Figure 7: Proving that using an appropriate malias is equivalent to call-by-value. B Proofs for two implementations In this section, we prove that the two implementations of malias presented in Section 2.1 obey the malias laws. We use the formulation of monads based on a functor with additional operations join and unit. The proof relies on the following laws that hold about join and unit: map (g ◦ f ) unit ◦ f join ◦ map (map f ) join ◦ map join join ◦ unit = = = = = (map g) ◦ (map f ) map f ◦ unit map f ◦ join join ◦ join join ◦ map unit = id (functor) (natural unit) (natural join) (assoc join) (identity) The first law follows from the fact that map corresponds to a functor. The next two laws hold because unit and join are both natural transformations. Finally, the last two laws are additional laws that are required to hold about monads (a precise definition can be found in Section 3). The proofs that the two definitions of malias (implementing call-by-name and call-by-value strategies) are correct can be found in Figure 8 and Figure 9, respectively. The proofs use the above monad laws. We use a definition malias ≡ map unit, which is equivalent to the definition in Appendix A. The figures include proofs for the four additional malias laws as defined in Section 2.2: naturality, associativity, computationality and identity. Evaluation strategies for monadic computations 88 map (map f ) ◦ malias = map (map f ) ◦ unit = unit ◦ map f = malias ◦ map f (definition) (naturality) (definition) map malias ◦ malias = map unit ◦ unit = unit ◦ unit = malias ◦ malias (definition) (natural unit) (definition) malias ◦ unit = unit ◦ unit (definition) join ◦ malias = join ◦ unit = id (definition) (identity) Figure 8: Call-by-name definition of malias obeys the laws map (map f ) ◦ malias = map (map f ) ◦ map unit = map (map f ◦ unit) = map (unit ◦ f ) = map unit ◦ map f = malias ◦ map f (definition) (functor) (natural unit) (functor) (definition) map malias ◦ malias = map (map unit) ◦ (map unit) = map (map unit ◦ unit) = map (unit ◦ unit) = map unit ◦ map unit = malias ◦ malias (definition) (functor) (natural unit) (functor) (definition) malias ◦ unit = map unit ◦ unit = unit ◦ unit (definition) (natural unit) join ◦ malias = join ◦ map unit = id (definition) (identity) Figure 9: Call-by-value definition of malias obeys the laws Tomas Petricek C 89 Typing preservation proof In this section, we show that our translation preserves typing. Given a well-typed term e of type τ , the translated term JeKcba is also well-typed and has a type Jτ Kcba . In the rest of this section, we write J−K for J−Kcba . To show that the property holds, we use induction over the typing rules, using the fact that Jτ K = M τ ′ for some τ ′ . The inductive construction of the typing derivation follows the following rules: JΓ, x : τ ⊢ x : τ K = JΓK, x : M Jτ K ⊢ x : M Jτ K JΓ, x : τ1 ⊢ e : τ2 K JΓ ⊢ λ x.e : τ1 → τ2 K = JΓK, x : M Jτ1 K ⊢ JeK : M Jτ2 K JΓK ⊢ unit (λ x.JeK) : M (M Jτ1 K → M Jτ2 K) JΓ ⊢ e1 : τ1 → τ2 K JΓ ⊢ e2 : τ1 K JΓ ⊢ e1 e2 : τ2 K = JΓK ⊢ Je1 K : M (M Jτ1 K → M Jτ2 K) Γ ⊢ Je2 K : M Jτ1 K Γ ⊢ bind Je1 K (λ f .bind (malias Je2 K) f )) : M Jτ2 K JΓ ⊢ e1 : τ1 K JΓ, x : τ1 ⊢ e2 : τ2 K JΓ ⊢ let x = e1 in e2 : τ2 K = JΓK ⊢ Je1 K : M Jτ1 K JΓK, x : M Jτ1 K ⊢ Je2 K : M Jτ2 K JΓK ⊢ bind (malias Je1 K) (λ x.Je2 K) : M Jτ2 K	6
Relevant change points in high dimensional time series Holger Dette, Josua Gösmann Ruhr-Universität Bochum Fakultät für Mathematik arXiv:1704.04614v2 [math.ST] 3 Nov 2017 44780 Bochum, Germany e-mail: holger.dette@rub.de josua.goesmann@rub.de November 7, 2017 Abstract This paper investigates the problem of detecting relevant change points in the mean vector, say µt = (µ1,t , . . . , µd,t )T of a high dimensional time series (Zt )t∈Z . While the recent literature on testing for change points in this context considers hypotheses for the equality of the means (1) (2) µh and µh before and after the change points in the different components, we are interested in a null hypothesis of the form (1) (2) H0 : \|µh − µh \| ≤ ∆h for all h = 1, . . . , d where ∆1 , . . . , ∆d are given thresholds for which a smaller difference of the means in the hth component is considered to be non-relevant. This formulation of the testing problem is motivated by the fact that in many applications a modification of the statistical analysis might not be necessary, if the differences between the parameters before and after the change points in the individual components are small. This problem is of particular relevance in high dimensional change point analysis, where a small change in only one component can yield a rejection by the classical procedure although all components change only in a non-relevant way. We propose a new test for this problem based on the maximum of squared and integrated CUSUM statistics and investigate its properties as the sample size n and the dimension d both converge to infinity. In particular, using Gaussian approximations for the maximum of a large number of dependent random variables, we show that on certain points of the boundary of the null hypothesis a standardised version of the maximum converges weakly to a Gumbel distribution. This result is used to construct a consistent asymptotic level α test and a multiplier bootstrap procedure is proposed, which improves the finite sample performance of the test. The finite sample properties of the test are investigated by means of a simulation study and we also illustrate the new approach investigating data from hydrology. Keywords: high dimensional time series, change point analysis, CUSUM, relevant changes, precise hypotheses, physical dependence measure AMS Subject Classification: 62M10, 62F05, 62G10 1 1 Introduction In the context of high dimensional time series it is typically unrealistic to assume stationarity. A simple form of non-stationarity, which is motivated by financial time series, where large panels of asset returns routinely display break points, is to assume structural breaks at different times (the change points) in the individual components. One goal of statistical inference is to correctly estimate these “change points” such that the original data can be partitioned into shorter stationary segments. This field is called change point analysis in the statistical literature and since the seminal papers of Page (1954, 1955) numerous authors have worked on the problem of detecting structural breaks or change points in various statistical models [see Aue and Horváth (2013) for a recent review]. There exists in particular an enormous amount of literature on testing for and estimating the location of a change in the mean vector µt = (µ1,t , . . . , µd,t )T = E [Zt ] of a multivariate time series (Zt )nt=1 [see Chu et al. (1996), Horváth et al. (1999), Kirch et al. (2015) among others]. A common feature in these references consists in the fact that the dimension, say d, of the time series is fixed. High dimensional change point problems, where the dimension d increases with sample size have only been recently considered in the literature [see Bai (2010), Zhang et al. (2010), Horváth and Hus̆ková (2012) and Enikeeva and Harchaoui (2014), Jirak (2015a), Cho and Fryzlewicz (2015) and Wang and Samworth (2016) among others]. Some of this work uses information across the coordinates in order to detect smaller changes than could be observed in any individual component series. In the simplest case of one structural break in each component many authors attack the problem of detecting the change point by means of hypothesis testing. For example, Jirak (2015a) investigates the hypothesis of no structural break in a high-dimensional time series by testing the hypotheses H0 : µh,1 = µh,2 = . . . = µh,n for all h = 1, . . . , d, (1.1) where µh,t denotes the h-th component of the mean vector µt of the random variable Zt (t = 1, . . . , n). The alternative is then formulated (in the simplest case of one structural break) as (1) H1 : µh = µh,1 = µh,2 = · · · = µh,2 6= (1.2) µh,kh +1 = µh,kh +2 = · · · = µh,n = (2) µh for at least one h ∈ {1, . . . , d}, where kh ∈ {1, . . . , n} denotes the (unknown) location of the change point in the h-th component. While - even under sparsity assumptions - the detection of small changes in each component is a very challenging problem, a modification of the statistical analysis (such as prediction) might not be necessary if the actual size of change is small. For example, in risk management situations, one is interested in fitting a suitable model for forecasting Value at Risk from data after the last change point [see e.g. Wied (2013)], but in practice, small changes in the parameter are perhaps not very interesting because they do not yield large changes in the Value at Risk. The forecasting quality might only improve slightly, but this benefit could be negatively overcompensated by transaction costs, in particular in high-dimensional portfolios. Moreover, even, if the null hypothesis (1.1) is not rejected, it is difficult to quantify the statistical uncertainty for the subsequent statistical analysis 2 (conducted under the assumption of stationarity), as there is no control about the type II error in this case. The present work is motivated by these observations and proposes a test for the null hypothesis of no relevant change point in a high dimensional context, that is (1) (2) (1.3) (1) (2) (1.4) H0,∆ : \|µh − µh \| ≤ ∆h for all h = 1, . . . , d versus HA,∆ : \|µh − µh \| > ∆h for at least one h ∈ {1, . . . , d}, (1) (2) where µh and µh are the parameters before and after the change point in the h-th component and ∆1 , . . . , ∆d are given thresholds for which a smaller difference of the means in the h-th component is considered as non-relevant. The problem of testing for a relevant difference between (one dimensional) means of two samples has been discussed by numerous authors mainly in the field of biostatistics (see Wellek (2010) for a recent review). In particular testing relevant hypotheses avoids the consistency problem as mentioned in Berkson (1938), that is: Any consistent test will detect any arbitrary small change in the parameters if the sample size is sufficiently large. Dette and Wied (2016) considered relevant hypotheses in the context of change point problems for general parameters, but did not discuss the high dimensional setup, where the dimension increases with sample size. In this case the statistical problems are completely different. The alternative approach requires the specification of the thresholds ∆h > 0, and this has to be carefully discussed and depends on the specific application. We also note that the hypotheses (1.3) contain the classical hypotheses (1.1), which are obtained by simply choosing ∆h = 0 for all h = 1, . . . , d. Nevertheless we argue that from a practical point of view it might be very reasonable to think about this choice more carefully and to define the size of change in which one is really interested. In particular it is often known that ∆h 6= 0 although one is testing “classical hypotheses” of the form (1.1) and (1.2). Moreover, a decision of no relevant structural break at a controlled type I error can be easily achieved by interchanging the null hypothesis and alternative in (1.3), i.e. considering the hypotheses versus e 0,∆ : \|µ(1) − µ(2) \| > ∆h for at least one h ∈ {1, . . . , d} H h h (1) (2) e HA,∆ : \|µ − µ \| ≤ ∆h for all h = 1, . . . , d. h h (1.5) (1.6) In this paper we propose for the first time a test for the hypotheses of a relevant structural break in any of the components of a high dimensional time series. The basic ideas are explained in Section 2 (without going into any technical details), where we propose to calculate for any component the integral of the squared CUSUM process and reject the null hypotheses whenever the maximum of these integrals (calculated with respect to all components) is large. In order to obtain critical values for this test we derive in Section 3 the asymptotic distribution of an appropriately standardized version of the maximum as the sample size and the dimension converge to infinity. We also provide several auxiliary results, which are of own interest, and investigate the case where the maximum is only calculated over a subset of the components. These results are then used in Section 4 to prove that the proposed test yields to a valid statistical inference, i.e. it is a consistent and asymptotic 3 level α test. It turns out that - in contrast to the classical change point problem - the analysis of the test for no relevant structural breaks is substantially harder as the null hypothesis does not correspond to a stationary process (non-relevant changes in the means are allowed). Section 5 is devoted to the investigation of a multiplier block bootstrap procedure. In particular we prove that the quantiles generated by this resampling method also yield to a consistent asymptotic level α test. The finite sample properties of the new test are investigated in Section 6, where we also illustrate our approach analysing a data example from hydrology. Finally some of the technical details are deferred to Section 7. 2 Relevant changes in high dimensions - basic principles In this Section we explain the basic idea of our approach to test for a relevant change in at least one component of the mean vector of a high dimensional time series. For the sake of a transparent representation we try to avoid technical details at this stage and refer to the subsequent sections, where we present the basic assumptions and mathematical details establishing the validity of the proposed method. Throughout this paper we consider an array of real valued random variables {Zj,h }j∈Z,h∈N such that Zj,h = µj,h + Xj,h , (2.1) where µj,h ∈ R for all j ∈ Z, h ∈ N and {Xj,h }j∈Z,h∈N denotes an array of centered and real valued random variables, which implies µj,h = E [Zj,h ] for all j ∈ Z, h ∈ N. It follows from the assumptions made in Section 3 that for each fixed d ∈ N the time series {(Xj,1 , . . . , Xj,d )T }j∈Z (2.2) is stationary. Suppose that Z1 = (Z1,1 , . . . , Z1,d )T , . . . , Zn = (Zn,1 , . . . , Zn,d )T ∈ Rd are d-dimensional observations from the array {Zj,h }j∈Z,h∈N and assume that for each component h ∈ {1, . . . , d} there exists an unknown constant th ∈ (0, 1), such that (1) µh = µj,h , j = 1, . . . , bnth c, (2) (2.3) µh = µj,h , j = bnth c + 1, . . . , n, where bxc = sup{z ∈ Z \| z ≤ x} denotes the larges integer smaller or equal than x. In this case the random variables {Zj,h }h=1,...,d;j=1,...,n also depend on n, i.e. they form a triangular array, but for the sake of readability, we will suppress this dependence in our notation. (1) (2) We define ∆µh = µh −µh as the unknown difference between the means in the h-th component before and after the change point th . Note, that in the case ∆µh 6= 0 the actual location kh = bnth c of the change point depends on the sample size n, which is a common assumption in the literature 4 on change point problems to perform asymptotic inference. It simply ensures that the number of observations before and after the change point is growing proportional to n. For each h with ∆µh = 0 the observable process {Zj,h }j∈Z is stationary and to avoid misunderstandings we set th = 1/2, whenever ∆µh = 0. The reader should notice that in this case the actual value is of no interest in the following discussion. With this notation the hypotheses in (1.3) can be rewritten as versus H0,∆ : \|∆µh \| ≤ ∆h for all h ∈ {1, . . . , d} (2.4) HA,∆ : \|∆µh \| > ∆h for some h ∈ {1, . . . , d}. (2.5) To develop an appropriate test statistic for these hypotheses we rely on the widely used concept of CUSUM-statistics. For each component h ∈ {1, . . . , d} we consider the corresponding CUSUMprocess for the h-th component defined by bnsc bnsc n 1X bnsc X n − bnsc X bnsc Un,h (s) = Zj,h − 2 Zj,h = Zj,h − 2 n n n2 n j=1 j=1 j=1 Under the assumptions stated in Section 3 it is shown in Section 7.1 that Z 1 3 2 E Un,h (s)ds = ∆µ2h + o(1) (th (1 − th ))2 0 as n → ∞ and therefore our considerations will be based on the statistic Z 1 3 2 U2n,h (s)ds, M̂n,h = 2 (t̂h (1 − t̂h )) 0 n X Zj,h . (2.6) j=bnsc+1 (2.7) (2.8) where t̂h denotes an appropriate estimator for the unknown location th of the structural break, that will be precisely defined in Section 3.2. The null hypothesis H0,h : \|∆µh \| ≤ ∆h (2.9) of no relevant change in the h-th component will be rejected for large values of M̂2n,h , and in order to determine a critical value we introduce the normalization √ n (∆) T̂n,h = (2.10) M̂2n,h − ∆2h , τ̂h σ̂h ∆h where σ̂h denotes an estimator for the unknown long-run variance (see Section 3.3 for a precise definition), and the quantity τ̂h is a function of the estimate of the change point defined by q 2 1 + 2t̂h (1 − t̂h ) √ τ̂h = τ (t̂h ) := , (2.11) 5t̂h (1 − t̂h ) which arises due to the integral in equation (2.8). It will be shown in Section 7.3.2 that for a fixed component h ∈ {1, . . . , d} (∆) D T̂n,h =⇒ N (0, 1) 5 as n → ∞, (2.12) D if ∆h = ∆µh , where the symbol =⇒ represents weak convergence of a random variable. Moreover monotonicity arguments show that the test, which rejects the null hypothesis (2.9) of a relevant (∆) change point in the mean of the h-th component, whenever T̂n,h exceeds the (1 − α)-quantile of the standard normal distribution, is a consistent asymptotic level α test. In order to construct a test for the hypotheses H0,∆ versus HA,∆ in (2.4) and (2.5) of a relevant change point in at least one of the components of a high dimensional time series we propose a simultaneous test, which rejects the null hypothesis for large values of the statistic d (∆) max T̂n,h . h=1 Note that a similar approach has been investigated by Jirak (2015a), who considered the “classical” change point problem in high dimension, that is H0,class : \|∆µh \| = 0 for all h ∈ {1, . . . , d} versus (2.13) HA,class : \|∆µh \| > 0 for some h ∈ {1, . . . , d}. (∆) In this case the weak convergence (2.12) does not hold (in fact T̂n,h = oP (1) under H0,class ) and a different statistic has to be considered. As it is well known that the (adjusted) maximum of standard normal distributed random variables converges weakly to a Gumbel distribution if they exhibit an appropriate dependence structure [see for example Berman (1964))], it is reasonable to consider the following d (∆) (2.14) Td,n = ad max T̂n,h − bd h=1 for the high-dimensional change point problem (2.4), where ad and bd denote appropriate sequences, such that the left hand side converges weakly at specific points of the “boundary” of the parameter space corresponding to the null hypothesis. To make these arguments more precise, define the sequences p 2 log d, (2.15) ad = p p bd = ad − log(4π log d)/2ad = 2 log d − log(4π log d)/(2 2 log d), (2.16) and note that the parameter spaces corresponding to the null hypothesis (2.4) and the alternative (2.5) are given by n o H0 = (x, y) ∈ Rd × Rd \|xh − yh \| ≤ ∆h ∀h ∈ {1, . . . , d} and HA = R2d \ H0 , respectively (here x = (x1 , . . . , xd )T , y = (y1 , . . . , yd )T ) denote d-dimensional vectors). Define for k = 0, . . . , d the “(d − k)-dimensional boundary of the hypotheses (2.4) and (2.5)” by n o Ak = (x, y) ∈ R2d \|xh − yh \| ≤ ∆h ∀ h with equality for precisely k components ⊆ H0 . (2.17) For the case d = 2, we illustrate this decomposition of the null hypothesis parameter space in Figure 1. In fact, a large part of this paper is devoted to prove the following statements (under appropriate assumptions - see Theorem 4.1 in Section 4): 6 Figure 1: Decomposition of the parameter space corresponding to the null hypothesis in (2.4) into the sets A0 , A1 and A2 for the case d = 2 . (1) If (µ(1) , µ(2) ) ∈ Ad , the weak convergence D Td,n =⇒ G holds as both n, d → ∞, where G denotes a Gumbel distribution, i.e. P (G ≤ x) = e−e −x . (2.18) (1) (2) (2) If (µ(1) , µ(2) ) ∈ Ak and there exists a constant C, such that \|µh − µh \| ≤ C < ∆h , whenever (1) (2) \|µh − µh \| = 6 ∆h and limd→∞ k/d = c ∈ (0, 1], we have D Td,n =⇒ G + 2 log c as n, d → ∞. (3) If (µ(1) , µ(2) ) ∈ Ak and limd→∞ k/d = 0 we have D Td,n =⇒ −∞ as n, d → ∞. Let g1−α denote the (1 − α) quantile of the Gumbel distribution, then it follows from these considerations that the test which rejects H0,∆ in favour of the alternative hypothesis HA,∆ , whenever Td,n > g1−α , 7 (2.19) is an asymptotic level α test. More precisely, it follows (under appropriate assumptions stated below) that PH0,∆ Td,n > g1−α −→    α α0    0 if (µ(1) , µ(2) ) ∈ Ad if (µ(1) , µ(2) ) ∈ Ak and limd→∞ k/d = c ∈ (0, 1] (2.20) if (µ(1) , µ(2) ) ∈ Ak and limd→∞ k/d = 0 as n, d → ∞, where 0 ≤ α0 ≤ α. Additionally the test is consistent (see Theorem 4.3). In the following sections we will make these arguments more rigorous. Moreover, in order to improve the finite sample approximation of the nominal level we also introduce a multiplier bootstrap procedure and prove its consistency in Section 5. 3 Asymptotic properties In this Section we provide the theoretical background for the test suggested in Section 2. We begin introducing some notations and assumptions. After stating the main assumptions we provide in Section 3.2 the asymptotic theory for an analogue of the statistic Td,n defined in (2.14), where the centering in (2.10) is performed by the ”true” squared differences \|∆µh \|2 and the estimates of the variances σh and the locations of the change points th have been replaced by their ”true” counterparts. In Section 3.3 we introduce estimators for the locations of the structural breaks (which may occur at a different location in each component) and investigate their consistency properties. These are then used to define appropriate variance estimators (note that variances have to be estimated in the case of changes in the mean). Finally, we consider in Section 3.4 the asymptotic distribution of the maximum of the statistics (2.10) where again centering is performed by \|∆µh \|2 instead of ∆2h . These results will then be used in the subsequent Section 4 to investigate the statistical properties of the the test (2.19). Throughout this paper we will use the following notation and symbols. Let x ∧ y and x ∨ y define the minimum and maximum of two real numbers x and y, respectively. For an appropriately integrable random variable Y and q ≥ 1, let kY kq = E [\|Y \|q ]1/q denote the Lq -norm. By the symbols D P =⇒ and −→ we denote weak convergence and convergence in probability, respectively. Moreover, we use the notation xn . yn , whenever the inequality xn ≤ C · yn , holds for some constant C > 0 which does not depend on the sample size and dimension and whose actual value is of no further interest. Due to its frequent appearance, G will always represent a (standard) Gumbel distribution defined by (2.18). In the high dimensional setup the dimension d converges to infinity with n → ∞. Recall the definition of model (2.1) and assume that the time series {Xj,h }j∈Z,h∈N forms a physical system [see e.g. Wu (2005)], that is Xj,h = gh (εj , εj−1 , . . . ) , (3.1) where {εj }j∈Z is a sequence of i.i.d. random variables with values in some measure space S such that for each h ∈ N the function gh : SN → R is measurable. Note that it follows from (3.1) that 8 the time series defined in (2.2) is stationary. Let ε00 be an independent copy of ε0 and define 0 Xj,h = gh (εj , εj−1 , . . . , ε1 , ε00 , ε−1 , . . . ). (3.2) 0 is used to quantify the (temporal) dependence The distance between Xj,h and its counterpart Xj,h of the physical system, and for this purpose we introduce the coefficients 0 ϑj,h,p := kXj,h − Xj,h kp , (3.3) which measure the influence of ε0 on the random variable Xj,h . Let us also define some additional parameters. For h, i ∈ N φh,i (j) = Cov(X0,h , Xj,i ) = Cov(Z0,h , Zj,i ) denotes the covariance function of the h-th and i-th component at lag j Accordingly, the autocovariance function for the h-th component is given by φh (j) = φh,h (j) = Cov(X0,h , Xj,h ) and we obtain the well-known representations X X φh (j) (3.4) γh,i = φh,i (j) and σh2 = j∈Z j∈Z for the long-run covariance and long-run variance, respectively. If we have σh , σi > 0, we can additionally define the long-run correlations by ρh,i = γh,i σh σi (3.5) and it will be crucial for our work, that these quantities become sufficiently small with an increasing temporal distance \|h − i\|. This will be precisely formulated in the following section. 3.1 Assumptions Operating in a high-dimensional framework usually needs stronger assumptions than those for the finite-dimensional case. Mainly, we need uniform dependence and moment conditions among all components to exclude extreme cases and to ensure, that the unknown parameters can be estimated accurately. In the high-dimensional setup considered in this paper the number of parameters th , σh grows together with the dimension d, since even under the null hypothesis of no relevant change points each component exhibits its own variance and change point structure. The precise assumptions made in this paper are the following. Assumption 3.1 (temporal assumptions) Suppose that there exist constants δ ∈ (0, 1) and σ− > 0 such that for some p > 4 the physical dependence coefficients ϑj,h,p and long run variances σh defined in (3.3) and (3.4), respectively, satisfy (T1) suph∈N ϑj,h,p . δ j , There exists constants 0 < σ− ≤ σ+ such that for all h ∈ N 9 (T2) σ− ≤ inf h∈N σh ≤ σ+ . Assumption 3.2 (spatial assumptions) The dimension d increases with the sample size at a polynomial rate, i.e. we assume that for constants C1 and D (S1) d = C1 nD , where the exponent D satisfies (S2) 0 < D < min{p/2 − 2, p/2 − B(p/2 + 1) − 1}, and p refers to the Lp -norm k · kp used to measure the physical dependence in Assumption 3.1. Here B ∈ (0, 1) denotes a constant used to control the bandwidth of an variance estimator, that will be defined in Section 3.4. Further we assume for the long-run correlations in (3.5) (S3) sup\|h−i\|>1 \|ρh,i \| ≤ ρ+ < 1, (S4) \|ρh,i \| ≤ C2 log(\|h − i\| + 2)−2−η , where ρ+ > 0, η > 0 and C2 > 0 denote global constants. Assumption 3.3 (moment assumptions) Suppose, that there exists a positive sequence (Md )d∈N and constants C3 > 1 and C4 > 0, such that (M1) maxdh=1 E [exp(\|X1,h \|/Md )] ≤ C3 (M2) Md ≤ C4 nm with m < 3/8. Assumption 3.4 (location of the change points) There exists a constant t ∈ (0, 1/2), such that for all h ∈ N the unknown locations bnth c of the structural breaks satisfy (C1) t ≤ th ≤ 1 − t. Let us give a brief explanation for the Assumptions 3.1 - 3.4. The temporal Assumptions (T1) and (T2) define the temporal dependence structure and bounds for the long-run variance. Further (T1) implies the existence of the quantities σh , γh,i and ρh,i defined in (3.4) and (3.5). Condition (S3) and (S4) refer to the spatial dependence and are only needed to derive the desired extreme value convergence. Assumption (S1) gives a relation between the number of observations and its dimension, while (S2) is a slightly technical assumption, which enables reasonable estimations of the variance σh and the locations th of the structural breaks. For a proof of the uniform consistency of the estimates for the latter quantity we must further rely on Assumption (C1), which makes the change points identifiable and is a common assumption in the literature. Roughly speaking, it simply ensures to have enough data before and after the change point in each component. Assumptions (S1) and (S2) together with n → ∞ directly imply d → ∞. It is also worth mentioning, that (S1) can be replaced by d . nD , if one additionally supposes that d → ∞. 10 The moment Assumptions (M1) and (M2) are required for a Gaussian approximation and are satisfied, if {Xj,h }j∈Z,h∈{1,...,d} is stationary with respect to the index j and for each h ∈ N the random variable X1,h is sub-Gaussian with parameters vh , Vh > 0, i.e. E [exp(λX1,h )] ≤ Vh exp(λ2 vh ), for all λ ∈ R, where the constants (vh )h=1,...,d and (Vh )h=1,...,d satisfy d max vh < n3/4 and h=1 max Vh ≤ C. h∈N √ for some constant C > 0. Choosing Md = maxdh=1 vh we directly obtain condition (M2). Condition (M1) follows by a straightforward calculation, that is d d h=1 h=1 2 max E [exp(\|X1,h \|/Md )] ≤ 2 max Vh evh /Md ≤ 2Ce < ∞. 3.2 Asymptotic properties - known variances and locations In this section we assume that the locations th of the changes and the long-run variances σh are known. Recalling the approximation (2.7) we define Z 1 1 (3.6) U2 (s)ds M2n,h = (th (1 − th ))2 0 n,h and investigate the asymptotic properties of the maximum of the statistics √ n Tn,h = M2n,h − ∆µ2h , τh σh \|∆µh \| (3.7) where τh = τ (th ) and the function τ is defined by (2.11). Note that Tn,h is the analogue of the (∆) statistic T̂n,h , where the thresholds ∆h , estimates t̂h and σ̂h have been replaced by the unknown quantities ∆µh , th and σh , respectively. Our first result shows that an appropriate standardized version of the maximum of the statistics Tn,h converges weakly to a Gumbel distribution. The proof is complicated and we indicate the main steps in this section deferring the more technical arguments to the appendix - see Section 7. Theorem 3.5 Assume that the Assumptions 3.1 - 3.4 are satisfied and that additionally there exist constants C` , Cu (independent of h and d) such that the inequalities C` ≤ \|∆µh \| ≤ Cu hold for all h = 1, . . . , d . Then d D ad max Tn,h − bd =⇒ G h=1 as d, n → ∞, where the sequences ad and bd are defined in (2.15) and (2.16), respectively. Proof of Theorem 3.5 (main steps). Observing the definition (3.6) and (3.7) we obtain by a straightforward calculation the representation Z 1 √ 3 n (1) (2) 2 2 (3.8) Tn,h = Un,h (s)ds − µh (s, th )ds = Tn,h + Tn,h , 2 (th (1 − th )) τh σh \|∆µh \| 0 11 where µh (s, th ) = (th ∧ s − sth ) (µh,1 − µh,2 ) (1) (3.9) (2) and the statistics Tn,h and Tn,h are defined by (1) Tn,h (2) Tn,h √ Z 1 3 n = (Un,h (s) − µh (s, th ))2 ds, (th (1 − th ))2 τh σh \|∆µh \| 0 √ Z 1 6 n = µh (s, th )(Un,h (s) − µh (s, th ))ds. (th (1 − th ))2 τh σh \|∆µh \| 0 For the following discussion, we introduce the additional notation √ Z 1 6 n (2) T̄n,h = µh (s, th ) (Un,h (s) − E [Un,h (s)]) ds, (th (1 − th ))2 τh σh \|∆µh \| 0 (3.10) (3.11) (3.12) (1) Our first auxiliary result shows that the first term Tn,h in the decomposition (3.8) is asymptotically negligible and is proved in Section 7.2.1. Lemma 3.6 If the assumptions of Theorem 3.5 are satisfied, we have d (1) P ad max Tn,h −→ 0 h=1 as d, n → ∞, where the sequence ad is defined in (2.15). (2) By Lemma 3.6 it suffices now to deal with the term Tn,h . The next step in the proof of Theorem (2) 3.5 consists in a (uniform) approximation of the distribution of the maximum of the statistics T̄n,h by of the distribution of the maximum of (dependent) Gaussian random variables. The proof of the following result is given in Section 7.2.2, where we make use of new developments on Gaussian approximations for maxima of sums of random variables [see Chernozhukov et al. (2013) and Zhang and Cheng (2016). Lemma 3.7 If the Assumptions 3.1 - 3.4 are satisfied, there exists a Gaussian distributed random vector N = (N1 , . . . , Nd )T with mean E [N ] = 0 and covariance matrix (Σh,i )dh,i=1 , such that d d (2) sup P max T̄n,h ≤ x − P max Nh ≤ x = o(1) x∈R h=1 h=1 for d, n → ∞. Moreover, the entries of the matrix Σ are bounded by \|Σh,i \| ≤ \|ρh,i \|, where ρh,i are the long-run correlations defined in (3.5). By Lemma 3.7 it suffices to establish the desired limiting distribution for the maximum of a Gaussian distributed vector. Nowadays, this is a well-understood area of mathematics and we can rely on results of Berman (1964), who originally examined the behavior of the maximum of dependent Gaussian random variables. A straightforward adaption of these arguments shows that the sequence of random variables {Ni }i∈N constructed in Lemma 3.7 satisfies 12 d D ad max Nh − bd =⇒ G (3.13) h=1 as d → ∞, where the sequences ad and bd are defined in (2.15) and (2.16), respectively. Now, combining Lemma 3.7 and (3.13) directly leads to d D (2) ad max T̄n,h − bd =⇒ G. h=1 By the assumption \|∆µh \| ≤ Cu it follows that maxdh=1 (µh (s, th ) − E [Un,h (s)]) = O(n−1 ), which yields d D (2) ad max Tn,h − bd =⇒ G. (3.14) h=1 (1) Due to Tn,h ≥ 0, we obtain the inequalities d d d d (1) (2) (2) ad max Tn,h − bd ≤ ad max Tn,h − bd ≤ ad max Tn,h + ad max Tn,h − bd , h=1 h=1 h=1 h=1 which together with Lemma 3.6 yields the assertion of Theorem 3.5. In the next step we will replace the unknown quantities th and σh in (3.7) by suitable estimators, say t̂h and σ̂h , and obtain the statistics √ n T̂n,h = (3.15) M̂2n,h − ∆µ2h , h ∈ {1, . . . , d} . τ̂h σ̂h \|∆µh \| (∆) We emphasize again that the statistics T̂n,h do not coincide with the statistics T̂n,h in (2.10), which (1) (2) are actually used in the test (2.19) except in the case where ∆µ2h = \|µh − µh \|2 = ∆2h for all h = 1, . . . , d. Thus centering is still performed with respect to the unknown difference of the means before and after the change points. In the following two subsections we give a precise definition of the two estimators and derive an analogue of Theorem 3.5 in the case of estimated change points and variances. 3.3 Estimation of long-run variances and change point locations Determining the relative locations th of the structural breaks and constructing an estimator for the long-run variances σh for all components h ∈ {1, . . . , d} is a rather difficult task in a high dimensional setting. A further difficulty in the problem of testing for relevant structural breaks consists in the fact that even under the null hypothesis there may appear structural breaks in the mean and the corresponding process is not stationary. Therefore in contrast of testing the “classical” hypotheses in (2.13) the construction of a suitable variance estimator is not trivial. A P standard long-run variance estimator in terms of i≤βn φ̂h (i) for an increasing sequence {βn }n∈N and appropriate estimators φ̂h (i) of the auto-covariance from the full sample may not be consistent due to possible changes of the mean. 13 Following Jirak (2015a) we define for each component h = 1, . . . , d the sets Dh,1 := {Zj,h \| 1 ≤ j ≤ bnth c} and Dh,2 := {Zj,h \| bnth c < j ≤ n} , which are the observations before and after the (unknown) change point bnth c, respectively. Since these points are usually unknown, we need to estimate them and for this purpose we propose the common estimator bnsc t̂h := argmax Un,h (s) = argmax X s∈(t,1−t) j=1 s∈(t,1−t) n Zj,h − bnsc X Zj,h . n (3.16) j=1 The following Lemma shows that these estimators are uniformly consistent with respect to all components, where a change point exists. Its proof follows by an adaption of Corollary 3.1 in Jirak (2015a), and is therefore omitted. Lemma 3.8 Let Sd = {1 ≤ h ≤ d \| \|∆µh \| = 0} . (3.17) denote the set of all components h ∈ {1, . . . , d}, where the corresponding time series {Zj,h }j∈Z is stationary, and define µ?d = minc \|∆µh \|. h∈Sd (3.18) Suppose that Assumptions 3.1 - 3.4 are satisfied and assume further that the condition lim sup d,n→∞ log n = 0. µ?d n (3.19) is satisfied. Then for a sufficiently small constant C > 0 it holds that maxc \|t̂h − th \| = oP (n−C ), h∈Sd Moreover, if additional the condition µ?d ≥ C` holds for some constant C` > 0 it follows that maxc \|t̂h − th \| = oP (n−1/2 ). h∈Sd Roughly speaking condition (3.19) guarantees that the decreasing sequence µ?d does not converge too fast to 0 if the dimension of the time series converge to infinity. Otherwise it is not possible to identify the (relative) locations of all change points simultaneously. In view of Lemma 3.8 it is reasonable to estimate the unknown sets Dh,1 and Dh,2 by bh,1 : = Zj,h \| 1 ≤ j ≤ n max{S t̂h , t} , D (3.20) bh,2 : = Zj,h \| n − n max{S(1 − t̂h ), t} < j ≤ n , D 2 and σ̂ 2 be the respectively, where S ∈ (0, 1) denotes a user-specified separation constant. Let σ̂h,1 h,2 bh,1 and D bh,2 , respectively. Then we standard long-run variance estimators based on the samples D 14 2 and σ̂ 2 to estimate the long run variances σ 2 , for example can use any convex combination of σ̂h,1 h,2 h 2 + σ̂ 2 ). To simplify the technical arguments in Section 7 we consider a truncated σ̂h2 = 21 (σ̂h,1 h,2 version, that is 2 2 σ̂h2 = min s2+ , max s2− , ( 12 (σ̂h,1 + σ̂h,2 ) . (3.21) where 0 < s− and s+ are a sufficiently small and large constant, respectively. The following statement is a straightforward implication of the results in Section 3 of Jirak (2015a) and yields the consistency of these estimators (uniformly with respect to the spatial component). Lemma 3.9 Suppose that Assumptions 3.1 - 3.4 are satisfied and additionally that there exists a constant C` > 0 such that the inequality C` ≤ \|∆µh \| hold for all h = 1, . . . , d. Then we have d max \|σ̂h − σh \| = op n−η h=1 for a sufficiently small constant η > 0. 3.4 Weak convergence Equipped with Lemmas 3.8 and 3.9 we are now able to state the main result of this section, which will be proved in Section 7.2.3. Theorem 3.10 If the assumptions of Theorem 3.5 are satisfied, then the statistics T̂n,h defined in (3.15) satisfy d D Td,n = ad max T̂n,h − bd =⇒ G (3.22) h=1 as d, n → ∞, where the sequences ad and bd are defined in (2.15) and (2.16), respectively. (∆) Recall again that the statistics T̂n,h and T̂n,h in (2.10) do not coincide in general. Thus Theorem 3.10 does not show that the test (2.19) is an asymptotic level α test because it does not cover all parameter configurations of the the null hypothesis (2.4). However, if the vector (µ(1) , µ(2) ) is an (∆) element of the set Ad defined in (2.17) we have T̂n,h = T̂n,h and it follows from this result that the probability of rejection converges to α, that is lim P(µ(1) ,µ(2) )∈Ad Td,n > g1−α = α. d,n→∞ We conclude this section with a result, which can be used to control the type I error of the test (2.19) for other values of the vector (µ(1) , µ(2) ). Corollary 3.11 Let {Md }d∈N be an increasing sequence of subsets of {1, . . . , d} (as d, n → ∞). If the assumptions of Theorem 3.5 hold, then   G if limd→∞ \|Md \|/d = 1  D ad max T̂n,h − bd =⇒ G + 2 log c if limd→∞ \|Md \|/d = c for some c ∈ (0, 1),  h∈Md  −∞ if limd→∞ \|Md \|/d = 0. Irrespective of the sequence {Md }d∈N , the bound lim sup P ad max T̂n,h − bd > x ≤ P (G > x) d,n→∞ h∈Md is valid for all x ∈ R. 15 (3.23) 4 Relevant changes in high dimensional time series Recall the problem of testing the hypotheses of a relevant change defined in (2.4) and (2.5). We propose to reject the null hypothesis of no relevant change in any component of the high dimensional mean vector, whenever the inequality (2.19) holds, that is Td,n > g1−α , where the test statistic Td,n is defined in (2.14) and g1−α denotes the (1 − α) quantile of the Gumbel distribution. The following two results make the discussion at the end of Section 2 rigorous and show that the test introduced in (2.19) defines in fact a consistent and asymptotic level α test. Theorem 4.1 Suppose that the Assumptions 3.1 - 3.4 are satisfied, α ∈ (0, 1 − e−1 ] and that there exist constants ∆− , ∆+ such that the thresholds ∆h satisfy the inequalities 0 < ∆− ≤ ∆h ≤ ∆+ < ∞ (4.1) for all h = 1, . . . , d. Then, under the null hypothesis H0,∆ of no relevant change, it follows lim sup P (Td,n > g1−α ) ≤ α. (4.2) d,n→∞ Moreover, let Bd = {h ∈ {1, . . . , d} \| ∆h = \|∆µh \|} and assume further that ∆h − \|∆µh \| ≥ C∆ > 0 for all h ∈ Bdc , for some constant C∆ > 0, then, under H0,∆ , we have   G if limd→∞ \|Bd \|/d = 1,    D Td,n =⇒ G + 2 log c if limd→∞ \|Bd \|/d = c for c ∈ (0, 1),     −∞ if limd→∞ \|Bd \|/d = 0. (4.3) (4.4) Remark 4.2 Condition (4.1) is actually not a very strong restriction since the thresholds ∆h are defined by the user. Nevertheless, the condition is crucial since we use the factor 1/∆h as a (∆) normalisation in the statistics T̂n,h defined in (2.10). Note that under the null hypothesis (2.4) the inequality ∆h ≤ ∆+ is equivalent to \|∆µh \| ≤ Cu , which was one of the assumptions in Theorem 3.10. Consequently the assertion of Theorem 4.1 follows from Theorem 3.10 in the case where \|∆µh \| = ∆h for all h ∈ {1, . . . , d}. However, in the general case the proof of Theorem 4.1 is more complicated and deferred to Section 7.3.1, where we also handle the case \|∆µh \| < ∆h . In the following result we investigate the consistency of the new test. Interestingly it requires less assumptions than Theorem 4.1. Theorem 4.3 Suppose that the Assumptions (T1), (T2) and (C1) hold. Then under the alternative hypothesis HA,∆ of at least one relevant change point we have P Td,n −→ ∞, as d, n → ∞, which in particular gives lim P (Td,n > g1−α ) = 1. d,n→∞ 16 If the test (2.19) rejects the null hypothesis H0,∆ in (2.4) we conclude (at a controlled type I error) that there is at least one component with a relevant change in mean. As there could exist relevant changes in several components, the next step in the statistical inference is the identification of the set Rd = {1 ≤ h ≤ d \| \|∆µh \| > ∆h } , (4.5) of all components with a relevant change. Note that the hypotheses in (2.4) and (2.5) are equivalent to H0,∆ : Rd = ∅ versus HA,∆ : Rd 6= ∅ . In light of Theorem 4.1 and 4.3 a natural estimator for this set is therefore given by n o b d (α) = 1 ≤ h ≤ d \| T̂ (∆) > g1−α /ad + bd . R (4.6) n,h The following theorem provides a consistency result of this estimate. Theorem 4.4 Suppose that the Assumptions 3.1 - 3.4 hold and assume additionally that there exist two constants 0 < C < 1/2, Cu > 0 such that nC min (\|∆µh \| − ∆h ) = ∞ and h∈Rd max \|∆µh \| ≤ Cu . h∈Rd (4.7) Then, the set estimator defined in (4.6) satisfies for α ∈ (0, 1 − e−1 ] b d (α) = 1. lim P Rd ⊂ R (4.8) Moreover we have the following lower bound b d (α) ≥ 1 − α. lim inf P Rd = R (4.9) d,n→∞ d,n→∞ 5 Bootstrap The testing procedure introduced in the previous sections is based on the weak convergence of the maximum of appropriately standardized statistics to a Gumbel distribution, and it is well known that the speed of convergence in limit theorems of this type is rather slow. As a consequence the approximation of the nominal level of the test (2.19) for finite sample sizes may not be accurate. A common way to improve the performance of the test, is to obtain the critical values from an appropriate bootstrap procedure. In the context of testing for relevant change points the construction of an appropriate resampling procedure is not obvious as - in contrast to classical change point problems - the parameter space under the null hypothesis is rather large. In particular it will be necessary to simulate the distribution of the statistic Td,n in case \|∆µh \| = ∆h for all h ∈ {1, . . . , d} such that one can replace the quantile of the Gumbel distribution by the corresponding quantile of the bootstrap statistics. A further problem is to mimic the dependence of the underlying times series, which we will address employing a Gaussian multiplier bootstrap, where observations are block-wise multiplied with independent Gaussian random variables [see Künsch (1989) or Lahiri (2003)]. 17 To handle the problem of potential change points under the null hypothesis (2.4) of no relevant changes, observations from blocks in a neighborhood of estimated change points will not be used in the estimate. Furthermore, components without a change point will be ignored when the bootstrap statistic is constructed. We begin describing this idea in more detail and show in Theorem 5.5, that the bootstrap statistic converges weakly to a Gumbel distribution as well. In the sequel we assume without loss of generalization that n = KL and will split the sample into L blocks of length K, and additionally L ∼ n` and K ∼ n1−` for ` ∈ (0, 1). (5.1) We obtain the following quantities, which control the number of blocks before and after the estimated change point. b − = sup{` ∈ N \| `K + K/2 ≤ t̂h n}, L h + b Lh = inf{` ∈ N \| `K − K/2 ≥ t̂h n}, (5.2) where t̂h denotes the estimator of the location th of the change in the h-th component defined in Section 3.4. The corresponding sample means are given by b− KL 1 1 Xh − Zj,h and Z̄h+ = Z̄h = − b b+ ) KL K(L − L h j=1 h KL X Zj,h , (5.3) b + +1 j=K L h which can be used to define an estimator for the unknown amount of change ∆µh = µh,1 − µh,2 in the h-th component, that is ∆b µh = Z̄h− − Z̄h+ . Moreover, these estimators can also be used to  −   Zj,h − Z̄h Zbj,h = 0   + Zj,h − Z̄h (5.4) define the “mean corrected” sample b− , for j ≤ K L h b− < j ≤ K L b+ , for K L h h + b , for j > K L h . (5.5) Finally, we define blocking variables (that are sums with respect to the different blocks) as Vb`,h (k) = `K X Zbj,h I{j ≤ k} (5.6) j=(`−1)K+1 and introduce the notation `K X Vb`,h = Vb`,h (n) = bj,h . Z j=(`−1)K+1 From the representation (5.5) we directly obtain, that blocks near to the estimator t̂h are ignored, i.e. we have b − + 1, . . . , L b + }. Vb`,h = 0 if ` ∈ {L h h 18 Further denote by Zn = σ(Zj,h \| 1 ≤ j ≤ n, 1 ≤ h ≤ d) the σ-field generated by the sample (Zj,h )1≤j≤n,1≤h≤d and by {ξ` }`∈N a sequence of i.i.d. standard Gaussian random variables, which is independent of Zn . Now we consider a multiplier version of the CUSUM-process from the L blocks, that is (L) Un,h (s) L L `=1 `=1 1X b bnsc X b = ξ` V`,h (bnsc) − 2 ξ` V`,h (n), n n and introduce the bootstrap integral statistics (for each component) √ Z 1 6 n (L) U (s)k(s, t̂ )ds · I \|∆b µh \| > n−1/4 Bn,h = h n,h 2 sbh τ (t̂h )(t̂h (1 − t̂h )) 0 + I \|∆b µh \| ≤ n−1/4 · bd , (5.7) (5.8) where k(x, y) = x ∧ y − xy denotes the covariance kernel of the standard Brownian bridge and L ŝ2h = 1X 2 ξ` · σ̂h L `=1 is the variance estimate from the bootstrap sample. In an analogous manner to the previous sections, we define a normalized maximum of the bootstrap statistics Bn,h by d Bd,n = ad max Bn,h − bd , (5.9) h=1 where the normalizing sequences ad and bd are given by (2.15) and (2.16), respectively. Remark 5.1 Let us give a brief heuristic explanation, why (5.8) and (5.9) define an appropriate bootstrap statistic. Our basic aim is to mimic the distribution of the test statistics Td,n on the “0dimensional boundary” of the null hypothesis H0,∆ , i.e. in case of \|∆µh \| = ∆h for all h = 1, . . . , d. µh (s, th ) Note, that we have = sign(∆µh )k(s, th ) and it was outlined in Section 3, that in this ∆h setting the representation √ Z 1 3 n Tn,h = (Un,h (s) − µh (s, th ))2 ds, (th (1 − th ))2 τh σh \|∆µh \| 0 √ Z 1 6 n + µh (s, th )(Un,h (s) − µh (s, th ))ds (th (1 − th ))2 τh σh \|∆µh \| 0 holds, where by Lemma 3.6 the first summand on the right-hand side is of order oP (1). The component-wise bootstrap statistic Bn,h is supposed to imitate the second summand in this decomposition. However, this approach is only sensible for all components h, that actually contain a change and for this reason we introduce the indicator function I \|∆b µh \| > n−1/4 . To be more precise, we will show in the Appendix that d Bd,n = ad max Bn,h − bd ≈ ad maxc Bn,h − bd h=1 h∈Sd 19 as d, n → ∞, where the set Sd is defined in (3.17). The statistic Bd,n will then be used to generate bootstrap quantiles for the statistic Td,n . In order to prove that this is a valid approach we require the following additional assumptions. Assumption 5.2 (assumptions for the bootstrap) For the constants p, D in Assumption 3.2 and the exponent ` in (5.1) assume that (B1) D < min{(1 − `)(p/2 − 2), `(p/4 − 1)} with ` > 3/4 and p > 8, (B2) limn→∞ log n K(µ∗d )2 = 0, where µ∗d is defined in (3.18). Assumption (B1) is a rather technical condition relating the dimension d ∼ nD , the number of blocks L = n` and the constant p, which was initially introduced in Assumption 3.1. Assumption (B2) is only a restriction for the monotone decreasing sequence µ∗d = minh∈Sdc \|∆µh \|, that is not allowed to decrease arbitrarily fast. We are now ready to state the main results of this section. Our first lemma shows, that we are able to identify the set of stable components Sd correctly. Lemma 5.3 If Assumptions 3.1 - 3.4 and Assumption 5.2 are satisfied, then P ad max Bn,h − bd −→ 0, h∈Sd where the set Sd is the set of all components with no change point defined in (3.17) and the sequences ad and bd are defined in (2.15) and (2.16), respectively. Theorem 5.4 If Assumptions 3.1 - 3.4 and Assumption 5.2 are satisfied, then   G if limd→∞ \|Sdc \|/d = 1,    D ad maxc Bn,h − bd =⇒ G + 2 log c if limd→∞ \|Sdc \|/d = c for c ∈ (0, 1),  h∈Sd    −∞ if limd→∞ \|Sdc \|/d = 0 conditional on Zn in probability, where the sequences ad and bd are defined in (2.15) and (2.16), respectively. Finally, the representation Bd,n = max ad max Bn,h − bd , ad maxc Bn,h − bd , h∈Sd h∈Sd Lemma 5.3 and Theorem 5.4 directly yield the following main result of this section. Theorem 5.5 If Assumptions 3.1 - 3.4 and Assumption 5.2 are satisfied, then   G if limd→∞ \|Sdc \|/d = 1,    D Bd,n =⇒ max{G + 2 log c, 0} if limd→∞ \|Sdc \|/d = c for c ∈ (0, 1),     0 if limd→∞ \|Sdc \|/d = 0 conditional on Zn in probability. 20 (5.10) ∗ Remark 5.6 In the following let g1−α denote the (1 − α)-quantile of the distribution of the bootstrap statistic Bd,n and define the bootstrap test by rejecting the null hypothesis (2.4) in favour of (2.5), whenever ∗ Td,n > g1−α , (5.11) where the statistics Td,n is defined in (2.14). If the alternative hypothesis of at least one relevant P change point holds, Theorem 4.3 shows Td,n −→ ∞, which due to Theorem 5.5 directly yields consistence of the bootstrap test. Under the null hypothesis, we consider different cases. Recall the definition of the sets Bd = {h ∈ {1, . . . , d} \| \|∆µh \| = ∆h }, Sd = {h ∈ {1, . . . , d} \| \|∆µh \| = 0}, where we always have Bd ⊂ Sdc ⊂ {1, . . . , d}. A combination of Theorem 4.1 and Theorem 5.5 now shows, that the rule in (5.11) gives an asymptotic level α test, whenever the limit limd→∞ \|Sdc \|/d exists. 6 Finite sample properties In this section we examine the finite sample properties of the asymptotic and bootstrap test by means of a small simulation study and illustrate its application in an example. 6.1 Simulation study The results of the previous section demonstrate that a test which rejects the null hypothesis in (1.3) for large values of the statistic Td,n defined in (2.14) is consistent and has asymptotic level α, provided that the critical values are either chosen by the asymptotic theory or estimated by the bootstrap procedure introduced in Section 5. It turns out that a bias correction, which does not change the asymptotic properties, yields substantial improvements of the finite sample properties of the asymptotic and bootstrap test. To be precise, recall the decomposition in (3.8), that is (1) (2) Tn,h = Tn,h + Tn,h , (1) (2) where Tn,h and Tn,h are defined in (3.10) and (3.11), respectively. It is shown in Section 3, that the (1) (2) maximum of the terms Tn,h is of order op (1), while the maximum of the terms Tn,h (appropriately (1) standardized) converges weakly to a Gumbel distribution. However, Tn,h is always nonnegative, which may lead to a non negligible bias in applications, in particular when the sample size is small relative to the dimension. To solve this problem for the asymptotic test (2.19), note that we have h i (1) E Tn,h = √ 3σh n(th (1 − th ))2 τh \|∆µh \| Z 0 1 σh (1 + o(1)) k(s, s)ds(1 + o(1)) = √ 2 n(th (1 − th ))2 τh \|∆µh \| 21 and therefore we subtract the term σ̂h √ 2 n(t̂h (1 − t̂h ))2 τ̂h ∆h (∆) from each statistic T̂n,h defined in (2.10). Similarly, a bias correction is also suggested for the bootstrap test in Section 5. Recall that the Bootstrap statistic is already constructed to mimic the (2) distribution of the statistic Tn,h . Consequently we use √ Z 1 2 3 n (L) ds U (s) n,h sbh τ̂h (t̂h (1 − t̂h ))2 ∆h 0 (1) to mimic the distribution of Tn,h and we add it (for each h) to the statistic Bn,h defined in (5.8), (∆) while the statistics T̂n,h in (2.10) are left unchanged. To guarantee a stable long-run variance estimation, we replaced the standard long-run variance estimator used in the theory of Section 3.3, by an estimator using the Bartlett kernel [see Newey and West (1987)], that is (for component h) ( X 1 − \|x\| if \|x\| ≤ 1 j φ̂h (0) + k φ̂h (j) with k(x) = βn 0 if \|x\| > 1 . 1≤\|j\|≤βn In order to have a more conservative test we use the maximum of the two variance estimates based bh,1 and D bh,2 defined in (3.20) as the final variance estimation. on the sets D We will focus on two scenarios with independent innovations in model (2.1), i.e. (I) Xj,h ∼ N (0, 1) i.i.d. (II) Xj,h ∼ Exp(1) i.i.d. and on two models of dependent data, an ARMA-process and a MA-process, defined by (III) Xj,h = 0.2Xj−1,h − 0.3Xj−2,h − 0.4Yj,h + 0.8Yj−1,h , P −3 where Yk,h = εk + 19 i=1 i εk−i,h and εk,h ∼ N (0, 1) i.i.d. (IV) Xj,h = εj,h + 1 10 29 P k −3 εj−k,h , where εk,h ∼ N (0, 1) i.i.d. k=1 All innovations are constructed such that Var(Xj,h ) ≈ 1. Throughout this section we assume that different components are independent and we are interested in testing the relevant hypotheses versus H0,∆ : \|∆µh \| ≤ 1 for all h ∈ {1, . . . , d} (6.1) HA,∆ : \|∆µh \| > 1 for at least one h ∈ {1, . . . , d}, (6.2) that is ∆h = 1 for all h ∈ {1, . . . , d}. Power and level of the tests are simulated for d-dimensional vectors with means  0 if j ≤ bnt c h E[Zj,h ] = , h = 1, . . . , d, µ if j > bnth c 22 where different values of the parameter µ are considered and the time of change is th = 1/2. All results presented in this section are based on 1000 simulation runs and the used significance level is always α = 0.05. Further, the constant S involved in (3.20) was fixed to 0.9. In our first example we investigate the finite sample properties of the asymptotic test (2.19) which uses the quantiles of the Gumbel distribution. In Table 1 we display the simulated type I error of this test at the “0-dimensional boundary” of the null hypothesis (that is ∆h = ∆µh = 1 for all h ∈ {1, . . . , d}) for different values of n and d. It is well known that the approximation of the distribution of the maximum of normally distributed random variables by a Gumbel distribution is not very accurate for small samples and therefore we consider relatively large sample sizes and dimensions in order to illustrate the properties of the asymptotic test. The results reflect the asymptotic properties in Section 3. For the independent cases (I) and (II) the approximation of the nominal level is more precise as for the dependent scenarios (III) and (IV), where the test is more conservative. We also mention that the rejection probabilities are increasing with ∆µh as predicted in Section 2 and 4 (these results are not presented for the sake of brevity). n=2000 model (I) (II) (III) (IV) d=500 8.8% 5.4% 4.9% 9.7% d=1000 11.1 % 5.9 % 4.5 % 13 % n=5000 d=500 6% 5% 1.9% 6.4% d=1000 7.5% 5.5% 2.5% 5.2% f n=10000 d=500 6.4% 4% 2.5% 6.2% d=1000 7.4% 5% 4% 5.9% Table 1: Empirical rejection probabilities of the asymptotic test (2.19) under a specific point at the “0-dimensional boundary” of null hypothesis, that is ∆h = ∆µh = 1 for all h ∈ {1, . . . , d}. \|∆µh \| model (I) (I) (II) (II) (III) (III) (IV) (IV) K 1 4 1 4 5 10 2 4 0.9 0.95 1.0 1.05 1.1 2.2% 5.2% 0.6% 0.9% 0% 0.1% 3.3% 5% 4.2% 9.1% 0.9% 3.2% 0% 1.3% 5.9% 10.3% 8.2% 17.3% 2.5% 9.2% 0.6% 4.6% 10.9% 18.4% 14.7% 29.6% 7.8% 20.2% 3.2% 21.6% 21.5% 32.5% 26.9% 47.8% 17.3% 36.6% 13.7% 59.7% 35.8% 48.7% Table 2: Empirical rejection probabilities of the bootstrap test (5.11) for n = d = 100 at specific points in the alternative and null hypothesis (∆h = 1 for all h ∈ {1, . . . , d}). Different block length K in the multiplier bootstrap are considered. Next we analyse the properties of the bootstrap test (5.11), which was developed in Section 5. Here we focus on relatively small sample sizes, that is n = 100, n = 200 and a relative large 23 Figure 2: Simulated power of the Bootstrap test (5.11) for the hypothesis (6.1). The sample size is n = 100 and the dimension d = 100. All differences are given by ∆µh = µ, where the choice µ = 1 corresponds to a point of the “0-dimensional boundary” Ad of the hull hypothesis. Left panel: Solid line: Model (I), dashed line: Model (II). Right panel: Solid line Model (III), dashed line: Model (IV). dimension compared to the sample sizes, that is d = 50, d = 100. It is well known that the multiplier bootstrap is sensitive with respect to the choice of the block length and this dependence is also observed for the bootstrap test proposed here. Exemplarily we show in Table 2 the simulated rejection probabilities for the different models (I) - (IV), different values of K and ∆µh = µ. Here the values \|∆µh \| ≤ 1 correspond to the null hypothesis. For \|∆µh \| = 1 (for all h ∈ {1, . . . , d}) the results of Section 5 predict that at this point the level of the test should be close to α = 0.05. Note that for the case of independent innovations (model (I) and (II)) the choice K = 1 (which means that no blocks are used) leads to the most reasonable results, given by rejection probabilities on the “0-dimensional boundary” Ad of the null hypothesis of 8.2% and 2.5%, respectively, while larger values of K such as K = 4 yield a too large type I error. On the other hand in the dependent models (III) and (IV), the block length needs to be carefully adapted to the time series structure. For model (III) K = 10 seems to be optimal, while for model (IV) choosing K = 2 gives acceptable results. The larger block length for model (III) might be required due to its autoregressive structure. Moreover, inspecting the results in rows 6 and 8 of Table 2 shows that too small values of K lead to a loss of power, while - similar to the first two models - too large values can cause an uncontrolled type I error. Next we display in Figure 2 the simulated power of the bootstrap test for all four models under consideration (where we use the optimal K from Table 2). Note that the rejection probabilities are increasing with ∆µh as predicted by the asymptotic arguments of the previous sections. In the left 24 panel we show the results for the independent scenarios (I) and (II), which are rather similar. On the other hand an inspection of the right panel shows larger difference in the dependent case. The different dependency structures in model (III) and (IV) yield substantial differences in power of the bootstrap test (5.11). We conclude the discussion of the bootstrap test investigating the sample size n = 200. The corresponding results are presented in Table 3 for the dimensions d = 50 and d = 100. \|∆µh \| model (I) (II) (III) (IV) K 1 1 20 2 0.9 0.95 d = 50 1.0 0.5% 0.3% 0% 0% 2.3% 1% 0.6% 0% 5.1% 4.1% 7.6% 9.5% 1.05 1.1 0.9 14.7% 11% 46.6% 47.56% 34.6% 30.4% 94.8% 100% 0.9% 0% 0% 0% 0.95 d = 100 1.0 1.05 1.1 2.3% 1.2% 0.3% 0% 6.9% 3.7% 6.4% 13.1% 18% 10.6% 51% 100% 40.7% 26.6% 95.3% 100% Table 3: Empirical rejection probabilities of the bootstrap test (5.11) at specific points in the alternative and null hypothesis (∆h = 1 for all h ∈ {1, . . . , d}). The sample sitze is n = 200. 6.2 Data example In this section we illustrate in a short example, how the new test can be used in applications. Our dataset is taken from hydrology and consists of average daily flows (m3 /sec) of the river Chemnitz at Goeritzhain (Germany) in the period 1909-2014. This data set has been recently analysed by Sharipov et al. (2016) using a statistical model from functional data analysis. Following these authors we subdivide the data into n = 105 years with d = 365 days per year. To avoid confusion, the reader should note that the German hydrological year starts on the 1st of November. Equipped with our new methodology, we are now able to test if there is a relevant change in the mean of at least one component. To specify the term ’relevant’, we exemplarily set the thresholds for all components to ∆1 = ∆2 = · · · = ∆365 = 0.63, which is close to 10% of the overall mean of the data under consideration. For a significance level of 5% the Bootstrap test defined in Section 5 rejects the null hypothesis of no relevant change for the given thresholds. Moreover, we can also identify the components, where the individual test (∆) ∗ ∗ statistic leads to a rejection, that is ad (T̂n,h − bd ) > g1−α , where g1−α is the (1 − α) quantile of the bootstrap distribution used in (5.11). For the data under consideration we found four components with a relevant change, given by h = 53, 99, 137 and 252 with corresponding estimators nt̂53 = 56, nt̂99 = 70, nt̂137 = 47 and nt̂252 = 41, respectively. This corresponds to the 23th of December 1965, the 7th of February 1979, the 18th of March 1956 and to the 10th of July 1950, respectively. In Figure 3 we display for these cases the time series before and after the year. For example the panel in the first row shows the average annual flow curves before and after 1950 and the other three 25 Figure 3: Average annual flow curves for the river Chemnitz at Goeritzhain for the periods before (gray) and after (black) the estimated year of change. The four figures correspond to different days of the year, where a change point has been localised: 10th of July (first row), 18 of March(second row) 23th of December (third row) and 7th of February (fourth row). 26 years are interpreted separately. In all cases we observe a large difference between both curves close the estimated component (marked by the vertical dashed line). We finally note that the approach of Sharipov et al. (2016) identifies only one change point, namely the year 1965. In contrast our analysis indicates that there might be additional change points in the years 1950, 1956, 1965 and 1979 corresponding to different parts of the hydrological year. Acknowledgments. This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt A1, C1) of the German Research Foundation (DFG). The authors would also like to thank Martina Stein, who typed parts of this paper with considerable technical expertise and Moritz Jirak for extremely helpful discussions. Moreover we are grateful to Andreas Schuhmann and Svenja Fischer from the Institute of Hydrology in Bochum, who provided hydrological data for Section 6.2. 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(Preprint). 7 Technical details Throughout the proofs we will use that Assumption (C1) directly implies the existence of two constants τ− , τ+ , such that the function τ defined in (2.11) satisfies 0 < τ− ≤ τ (th ) ≤ τ+ < ∞ holds for all h ∈ N. 28 7.1 Proof of assertion (2.7) Straightforward calculations yield E [Un,h (s)] = µh (s, th ) + O(n−1 )∆µh n Var (Un,h (s)) = σh2 (s − s2 ) + o(1), (7.1) uniformly with respect to s ∈ [0, 1], where µh (s, th ) is defined in (3.9). An application of Fubini’s theorem now gives i Z 1 hZ 1 2 E U2n,h (s) ds Un,h (s)ds = E 0 0 Z 1 = E (Un,h (s) − µh (s, th ))2 + 2µh (s, th )E [Un,h (s)] − µ2h (s, th ) ds + o(1) 0 Z 1 σh2 (th (1 − th ))2 = µ2h (s, th )ds + o(1) = ∆µh + + o(1). 6n 3 0 7.2 7.2.1 Details in the proof of Theorem 3.5 Proof of Lemma 3.6 Observing the definition (2.11) and Assumptions (T2) and (C1) it easily follows that there exist constants c and C such that the inequalties c < (th (1 − th ))2 σh < C (7.2) hold for all h ∈ {1, . . . , d}. With these inequalities we obtain √ Z 1 3 n d d (1) (Un,h (s) − µh (s, th ))2 ds 0 ≤ ad max Tn,h = ad max h=1 h=1 (th (1 − th ))2 τh σh \|∆µh \| 0 Z 1 d √ . ad max n (Un,h (s) − µh (s, th ))2 ds h=1 0 √ ad d ≤ √ max sup ( n\|Un,h (s) − µh (s, th )\|)2 . n h=1 s∈[0,1] Using (7.1) and maxdh=1 \|∆µh \| ≤ Cu this is further bounded by 2 √ ad d 2 √ max sup n\|Un,h (s) − E [Un,h (s)] \| + o(1) n h=1 s∈[0,1] √ Introducing the notation ed = 2 2 log 2d it follows that the last term can be bounded by the random variable √ ad d e d 2 4 √ max sup n\|Un,h (s) − E [Un,h (s)] \| − + o(1) 4 n h=1 s∈[0,1] and the claim follows from Theorem 2.5 in (Jirak, 2015a) noting that {Un,h (s) − E [Un,h (s)]}s∈[0,1] corresponds to a CUSUM-process under the classical null hypothesis of no change point, that is (1) (2) µh = µh for all h ∈ {1, . . . , d} (note that there is a typo in the original paper, which has been corrected in the arXiv version). 29 7.2.2 Proof of Lemma 3.7 A straightforward calculation yields n Cov(Un,h (s1 ), Un,i (s2 )) = k(s1 , s2 )γh,i + rn,h,i (s1 , s2 ), where k(s1 , s2 ) = s1 ∧s2 −s1 s2 denotes the covariance kernel of a Brownian bridge and the remainder term satisfies √ lim n sup sup \|rn,h,i (s1 , s2 )\| = 0. n→∞ s1 ,s2 ∈[0,1] h,i∈N An application of Fubini’s theorem shows that sign(∆µ ) sign(∆µ )γ τe(t , t ) i h,i h h i (2) (2) Cov T̄n,h , T̄n,i = + rn,h,i , σh σi τ (th )τ (ti ) (7.3) where the function τe is given by 36 τe(t, t ) = (t(1 − t)t0 (1 − t0 ))2 0 Z 1Z 1 0 k(s1 , t)k(s2 , t0 )k(s1 , s2 )ds1 ds2 (7.4) 0 and the remainder term rn,h,i satisfies lim n→∞ √ n sup \|rn,h,i \| = 0. h,i∈N Moreover, for the function τ defined in (2.11) we obtain the representation s Z 1Z 1 6 τ (t) = k(s1 , t)k(s2 , t)k(s1 , s2 )ds1 ds2 , (t(1 − t))2 0 0 and it follows that τ (th )τ (th ) = τe(th , th ). Therefore we obtain as a special case the estimate (2) Var T̄n,h = 1 + rn,h,h . (7.5) Furthermore the representation bnsc n 1X bnsc X Un,h (s) − E [Un,h (s)] = Xj,h =: U0n,h (s) Xj,h − 2 n n j=1 j=1 yields (2) T̄n,h √ Z 1 6 n = µh (s, th )U0n,h (s)ds. (th (1 − th ))2 τh σh \|∆µh \| 0 and the proof can now be performed in two steps: Step 1: For two constants c, C > 0 it holds that d d (2) e ≤ Cn−c , sup P max T̄n,h ≤ x − P max Nh ≤ x x∈R h=1 h=1 (7.6) e is a d-dimensional centered Gaussian distributed random variable with same where N (2) (2) covariance structure as (T̄n,1 , . . . , T̄n,d )T . 30 Step 2: There exist two constants c, C > 0 such that d d e sup P max Nh ≤ x − P max Nh ≤ x ≤ Cn−c , h=1 x∈R h=1 (7.7) where N is a centered d-dimensional Gaussian random variable with covariance matrix Σ = (Σh,i )i,j=1,...,d satisfying \|Σh,i \| ≤ \|ρh,i \| (7.8) for all h, i ∈ {1, . . . , d}. Step 1: At the end of this proof we derive the following representation n 1 X (2) cn,j,h Xj,h , T̄n,h = √ n (7.9) j=1 where the coefficients cn,j,h are uniformly bounded, that is sup \|cn,j,h \| ≤ c0 < ∞ n,j,h∈N Next we apply the Gaussian approximation in Corollary 2.2 of Zhang and Cheng (2016) to the random variables Yn,j,h = cn,j,h Xj,h . For this purpose we check the assumptions of this result. By Assumption (M1) we obtain a sequence (Md0 )d∈N by Md0 = c0 · Md , which still satisfies n d n d max max E exp(\|Yn,j,h \|/Md0 ) ≤ max max E [exp(\|Xj,h \|/Md )] ≤ C0 . j=1 h=1 j=1 h=1 Moreover, Assumption (M2) yields that Md0 . nm with m < 3/8. This means that for sufficiently small b we have m < (3 − 17b)/8 and by Assumption (S1) it follows that d . exp(nb ). Using the identity (3.1) the triangular array {Yn,j,h \| 1 ≤ j ≤ n, 1 ≤ h ≤ d}n∈N exhibits the following structure Yn,j,h = cn,j,h · gh (εj , εj−1 , . . . ) := g̃n,j,h (εj , εj−1 , . . . ). Define the coefficients n ϑYn,j,h,p = max kg̃n,i,h (εi , . . . ) − g̃n,i,h (εi , , . . . , εi−j+1 , ε0i−j , εi−j−1 , . . . )kp , i=1 where ε0i−j is an independent copy of εi−j , and observe the inequality ∞ X j=u d max ϑYn,j,h,p h=1 ≤ ∞ X d n max sup max \|cn,i,h \|ϑj,h,p . c0 h=1 n∈N i=1 j=u ∞ X δj . δu, j=u which holds uniformly with respect to n. By (7.5) there exist constants c1 and c2 , such that d d (2) (2) 0 < c1 < min Var T̄n,h ≤ max Var T̄n,h < c2 h=1 h=1 31 if n is sufficiently large. Since all requirements are met, Corollary 2.2 of Zhang and Cheng (2016) e having the same covariance matrix as the implies the existence of a Gaussian random variable N (2) (2) T vector (T̄n,1 , . . . , T̄n,d ) and satisfying inequality (7.6). Step 2: We choose the random variable N to be d-dimensional centered Gaussian with covariance matrix given by Σh,i = sign(∆µh ) sign(∆µi )γh,i τe(th , ti ) τe(th , ti ) = sign(∆µh ) sign(∆µi )ρh,i , σh σi τ (th )τ (ti ) τ (th )τ (ti ) e the covariance matrix of the where the function τe is defined in equation (7.4). Next denote with Σ e from Step 1. By (7.3) we have vector N d d h,i=1 h,i=1 e h,i \| = max θd := max \|Σh,i − Σ sign(∆µh ) sign(∆µi )γh,i τe(th , ti ) (2) (2) − Cov(Tn,h , Tn,i ) . n−1/2 σh σi τ (th )τ (ti ) and an application of the Gaussian comparison inequality, in Lemma 3.1 of Chernozhukov et al. (2013) gives d d eh ≤ x . θ1/3 max{1, log(d/θd )}2/3 . n−C . sup P max Nh ≤ x − P max N d x∈R h=1 h=1 The proof of Step 2 is now completed observing the bound \|e τ (th , ti )\| ≤ \|τ (ti )\|\|τ (th )\|, which is a consequence of the (generalised) Cauchy-Schwarz inequality. Proof of the representation (7.9). Recall the definition k(s, t) = s ∧ t − st, then 1 Z 0 n−1 Z (i+1)/n 1X k(s, th )Un,h (s)ds = n i=0 = 1 n i/n n bnsc X bnsc X k(s, th ) Xj,h ds Xj,h − n j=1 n−1 n X Z (i+1)/n X i=0 i/n j=1 k(s, th ) (I{j ≤ i} − i/n) Xj,h ds j=1 ! n n−1 Z 1 X X (i+1)/n k(s, th ) (I{j ≤ i} − i/n) ds Xj,h . = n i/n j=1 i=0 (2) Now observe the representation for T̄n,h in (3.12), where µh (s, t) = 2∆µh k(s, t). Define n−1 Z (i+1)/n cn,j,h X 6∆µh := (th (1 − th ))2 τ (th )σh \|∆µh \| i=0 k(s, t) (I{j ≤ i} − i/n) ds, i/n then the representation (7.9) holds, and the proof is completed observing the inequalities n−1 X Z (i+1)/n 6 12 \|cn,j,h \| ≤ 2ds ≤ c0 := . 4 τ− σ− t i=0 i/n τ− σ− t4 32 7.2.3 Proof of Theorem 3.10 As \|∆µh \| > Cu for all h ∈ {1, . . . , d} we have Sd = ∅ and Sdc = {1, . . . , d}, and Lemma 3.8 implies d max \|t̂h − th \| = op (n−1/2 ) (7.10) h=1 Observing that t̂h ∈ [t, 1 − t] it follows that t2h (1 − th )2 − 1 = op (n−1/2 ) 2 2 t̂h (1 − t̂h ) d max h=1 (7.11) Moreover, one easily verifies that the function t → τ (t) defined in (2.11) is Lipschitz-continuous on the interval [t, 1 − t] and therefore we obtain from (7.10) d max \|τ (th ) − τ (t̂h )\| = op (n−1/2 ). (7.12) h=1 Finally, we note that for a sufficiently small constant C > 0 the estimate d max \|σ̂h τ (t̂h ) − σh τ (t̂h )\| = op (n−C ) (7.13) h=1 holds, which is a direct consequence Lemma 3.9 and assertion (7.12). After these preparations we return to the proof of Theorem 3.10. We recall the definition (2.8) and introduce the notation √ n 0 T̂n,h = M̂2n,h − ∆µ2h τ (th )σh \|∆µh \| We will first show the weak convergence D d 0 ad max T̂n,h − bd =⇒ G (7.14) h=1 For a proof of (7.14) we recall the definition of the statistic Tn,h in (3.7) and obtain from Theorem 3.5 D d ad max Tn,h − bd =⇒ G. h=1 With the representation for M̂2n,h and M2n,h in (2.8) and (3.6), respectively, and the notation q̂h = (th (1 − th ))2 /(t̂h (1 − t̂h ))2 it now follows that √ √ n(q̂h − 1)∆µ2h n d d 2 2 0 Mn,h − ∆µh − bd + = Bn,d ad max T̂n,h − bd = ad max q̂h h=1 h=1 τ (th )σh \|∆µh \| τ (th )σh \|∆µh \| where √ d Bn,d := ad max q̂h Tn,h − bd + (q̂h − h=1 1)∆µ2h n . τ (th )σh \|∆µh \| It is easy to see that this term can be bounded by d d ad max q̂h Tn,h − bd − Rn,d ≤ Bn,d ≤ ad max q̂h Tn,h − bd + Rn,d , h=1 h=1 33 where the remainder satisfies d d h=1 h=1 Rn,d = ad max \|q̂h − 1\| max √ n P ∆µ2h −→ 0, τ (th )σh \|∆µh \| which follows observing the inequalities (7.2), (7.11) and the condition C` ≤ \|∆µh \| ≤ Cu . Thus (7.14) follows, if we can establish D d ad max q̂h Tn,h − bd =⇒ G. (7.15) h=1 For a proof of this result we fix x ∈ R and define ud (x) = x/ad + bd . By d d d P 0 ≤ max q̂h max Tn,h ≤ ud (x) ≤ P 0 ≤ max q̂h Tn,h ≤ ud (x) h=1 h=1 h=1 d d h=1 h=1 ≤ P 0 ≤ min q̂h max Tn,h ≤ ud (x) . (7.16) and d d d d d P min q̂h max Tn,h ≥ 0 = P max q̂h Tn,h ≥ 0 = P max q̂h max Tn,h ≥ 0 , h=1 h=1 h=1 h=1 h=1 we obtain d d d d d P max q̂h max Tn,h ≤ ud (x) ≤ P max q̂h Tn,h ≤ ud (x) ≤ P min q̂h max Tn,h ≤ ud (x) . h=1 h=1 h=1 h=1 h=1 (7.17) From (7.11) and d = C1 nD it follows that d P P d ad bd min q̂h − 1 −→ 0 and ad bd max q̂h − 1 −→ 0, h=1 h=1 which due to Slutsky’s theorem directly implies d d −x −x d d P min q̂h max Tn,h ≤ ud (x) −→ e−e , P max q̂h max Tn,h ≤ ud (x) −→ e−e . h=1 h=1 n→∞ h=1 h=1 n→∞ Thus we have established (7.15) and proved (7.14). To complete the proof of Theorem 3.10 note that the assertion (3.22) is equivalent to d −x P max T̂n,h ≤ ud (x) −→ e−e , n→∞ h=1 (7.18) where we use again ud (x) = x/ad + bd . To prove this statement define d Q− d := min h=1 σ̂h τ̂h σ̂h τ̂h d , Q+ , d := max h=1 σh τ (th ) σh τ (th ) − and consider the set Qd := Q+ d − 1 ∨ Qd − 1 ≤ δd , where the involved sequence is given by δd = (log d)−2 . The inequalities (7.2) and the estimate (7.13) yield d d σ̂h τ̂h P Q+ − 1 > δ ≤ P max σ̂ τ̂ − σ τ (t ) > δ c = o(1). − 1 > δ = P max d d h h h h d d h=1 σh τ (th ) h=1 34 c d By a similar argument for the term Q− d − 1 we obtain P (Qd ) → 0. If maxh=1 T̂n,h ≥ 0 holds, we can conclude that √ n 1 d 1 d 0 2 2 max T̂ = max M̂ − ∆µ n,h n,h h h=1 h=1 τ (th )σh \|∆µh \| Q+ Q+ d d √ n 1 d d 0 ≤ max M̂2n,h − ∆µ2h = T̂n,h = − max T̂n,h . h=1 τ (t̂h )σ̂h \|∆µh \| Qd h=1 Therefore the following inequalities hold d d d + 0 0 P 0 ≤ max T̂n,h ≤ ud (x)Q− ≤ P 0 ≤ max T̂ ≤ u (x) ≤ P 0 ≤ max T̂ ≤ u (x)Q n,h d d n,h d d . (7.19) h=1 h=1 h=1 Observing the identity d d 0 P max T̂n,h ≥ 0 = P ∃h : M̂2n,h ≥ ∆µ2h = P max T̂n,h ≥ 0 h=1 h=1 we can derive from (7.19) d d d + 0 0 ≤ P P max T̂n,h ≤ ud (x)Q− max T̂ ≤ u (x) ≤ P ≤ u (x)Q max T̂ n,h d d n,h d d . h=1 h=1 h=1 (7.20) Hence, we directly obtain d d + + c 0 0 P max T̂n,h ≤ ud (x)Qd ≤ P(Qd ) + P max T̂n,h ≤ ud (x)Qd ∩ Qd h=1 h=1 d 0 ≤ ud (x) + ud (x)δd . ≤ o(1) + P max T̂n,h h=1 Now we fix ε > 0 and note that the inequality ud (x − ε) < ud (x) + ud (x)δd < ud (x + ε) holds if n (or equivalently d) is sufficiently large. The weak convergence (7.14) then yields d d −(x+ε ) 0 0 ≤ lim sup P ≤ u (x + ε) = e−e . lim sup P max T̂n,h ≤ ud (x)Q+ max T̂ d n,h d d,n→∞ h=1 d,n→∞ h=1 Using Bonferroni’s inequality we can proceed similarly for the lower bound of (7.20), i.e. d d − − 0 0 lim inf P max T̂n,h ≤ ud (x)Qd ≥ lim inf P max T̂n,h ≤ ud (x)Qd ∩ Qd d,n→∞ d,n→∞ h=1 h=1 d 0 ≥ lim inf P max T̂n,h ≤ ud (x) − ud (x)δd − P (Qcd ) d,n→∞ h=1 d −(x−ε ) 0 ≥ lim inf P max T̂n,h ≤ ud (x − ε) = e−e . d,n→∞ h=1 The assertion (7.18) then follows by ε → 0, which completes the proof of Theorem 3.10. 7.2.4 Proof of Corollary 3.11 At first we consider the case where md := \|Md \| = c · d + o(d) for some constant c ∈ (0, 1] and note that in this case ad ad max T̂n,h − bd = am max T̂n,h − amd bmd + amd bmd − ad bd . h∈Md amd d h∈Md 35 D Theorem 3.10 yields amd maxh∈Md T̂n,h − amd bmd =⇒ G and furthermore we have ad −→ 1 and amd bmd − ad bd −→ 2 log(c) d→∞ amd d→∞ D A short calculation therefore leads to ad maxh∈Md T̂n,h − bd =⇒ G + 2 log(c). The case md = o(d) can be treated similarly. Finally, statement (3.23) is a consequence of the inequality d max T̂n,h ≤ max T̂n,h . h=1 h∈Md 7.3 7.3.1 Proofs of the results in Section 4 Proof of Theorem 4.1 By Assumption (C1) and the definition of t̂h in (3.16), there exists a global constant C(t) > 1 such that (th (1 − th ))2 ≤ C(t). (t̂h (1 − t̂h ))2 (7.21) Recall the definition of the set Sd in (3.17) , choose a constant ∆2− > ζ > 0 and consider the following decomposition of the set {1, . . . , d} \ Sd p Id := {h ∈ {1, . . . , d} \| (∆h − ζ)/ C(t) ≥ \|∆µh \| > 0}, p (7.22) Ed := {h ∈ {1, . . . , d} \| ∆h ≥ \|∆µh \| > (∆h − ζ)/ C(t)}. Using the representation o n (∆) (∆) (∆) Td,n = max ad max T̂n,h − bd , ad max T̂n,h − bd , ad max T̂n,h − bd . h∈Id h∈Sd h∈Ed the first assertion (4.2) follows from the following three statements P (∆) ad max T̂n,h − bd −→ −∞, h∈Sd P (∆) ad max T̂n,h − bd −→ −∞, h∈Id (∆) lim P ad max T̂n,h − bd ≥ g1−α ≤ α. d,n→∞ h∈Ed (7.23) (7.24) (7.25) (∆) Proof of (7.23): Observing the definition of T̂n,h in (2.10) we obtain the inequality √ ad (∆) max T̂n,h h∈S d − bd √ n n 2 ≤ ad max M̂n,h − ad min ∆2h − ad bd . h∈Sd τ (t̂h )σ̂h ∆h h∈Sd τ (t̂h )σ̂h ∆h The first summand of this expression is further bounded by √ √ n ad max M̂2n,h . ad max n · M2n,h , h∈Sd τ (t̂h )σ̂h ∆h h∈Sd 36 (7.26) (7.27) and arguing as in the proof of Lemma 3.6 yields ad maxh∈Sd √ √ summand of (7.26) we can use n ∆2 τ (t̂h )σ̂h ∆h h P n · M2n,h −→ 0. For the second > 0 and ad bd ∼ log d to obtain √ ad min h∈Sd n ∆2h + ad bd −→ ∞, d,n→∞ τ (t̂h )σ̂h ∆h which yields (7.23). Proof of (7.24): By definition of the set Id , we get ∆2h ≥ C(t)∆µ2h + ζ, which leads to √ √ n n (∆) 2 2 ad max T̂n,h − bd ≤ ad max ζ. M̂n,h − C(t)∆µh − bd − ad min h∈Id h∈Id τ (t̂h )σ̂h ∆h h∈Id τ (t̂h )σ̂h ∆h (7.28) For the second summand of the last expression it holds that √ √ ad n ζ & −→ ∞, ad n min h∈Id τ (t̂h )σ̂h ∆h τ+ s+ ∆+ n→∞ (7.29) From inequality (7.21), we obtain M̂2n,h ≤ C(t)M2n,h , which gives the following bound for the first summand of (7.28) √ √ n n 2 2 ad max M̂n,h − C(t)∆µh − bd . ad max M2n,h − ∆µ2h − bd . (7.30) h∈Id τ (t̂h )σ̂h ∆h h∈Id τ (t̂h )σ̂h ∆h Similar to (3.8) we can use the following decomposition √ n (1) (2) M2n,h − ∆µ2h = Sn,h + Sn,h , τ (t̂h )σ̂h ∆h (1) (2) where the quantities Sn,h and Sn,h are given by (1) Sn,h (2) Sn,h √ Z 1 3 n = 2 (Un,h (s) − µh (s, th ))2 ds, 2 th (1 − th ) τ (t̂h )σ̂h ∆h 0 √ Z 1 6 n = 2 µh (s, th )(Un,h (s) − µh (s, th ))ds, th (1 − th )2 τ (t̂h )σ̂h ∆h 0 respectively. Now we have an upper bound for (7.30) given by (1) (2) ad max Sn,h + ad max Sn,h − bd . h∈Id h∈Id (1) (2) Similar as in the proof of (7.23) one easily shows that ad max Sn,h = op (1). In the case that Sn,h ≥ 0 h∈Id we have (2) Sn,h . (3) Sn,h √ Z 1 6 n := 2 µh (s, th )(Un,h (s) − µh (s, th ))ds, th (1 − th )2 τ (th )σh \|∆µh \| 0 37 which gives n o n o (2) (3) (3) ad max Sn,h − bd . ad max max Sn,h , 0 − bd ≤ max ad max Sn,h − bd , −ad bd . h∈Id h∈Id h∈Id Applying Lemma 3.7 yields (3) lim sup P ad max Sn,h − bd > x ≤ P (G > x) h∈Id d,n→∞ for all x ∈ R and consequently the right hand side of (7.30) is of order Op (1). Now (7.24) follows from (7.28) and (7.29). Proof of (7.25): Observing that dh := ∆2h − ∆µ2h ≥ 0 we obtain √ n (∆) ad max T̂n,h − bd ≤ ad max M̂2n,h − ∆µ2h − bd , (7.31) h∈Ed h∈Ed τ (t̂h )σ̂h ∆h 1 )) ≥ 0, As α ∈ 0, 1 − e−1 the quantile of the Gumbel distribution satisfies g1−α = − log(log( 1−α and we can proceed as follows \|∆µh \| (∆) P ad max T̂n,h − bd > g1−α ≤ P max · ad max T̂n,h − bd > g1−α h∈Ed ∆h h∈Ed h∈Ed ≤ P ad max T̂n,h − bd > g1−α . h∈Ed An application of Corollary 3.11 now yields lim sup P ad max T̂n,h − bd > g1−α ≤ P (G > g1−α ) = α, d,n→∞ h∈Ed which gives assertion (7.25) and completes the proof of assertion (4.2). It remains to show assertion (4.4) under the additional assumption of (4.3). Note that under the latter assumption, we can further decompose the set Ed into Ed = (Ed \ Bd ) ∪ Bd and observe that (4.3) yields p Ed \ Bd = {h ∈ {1, . . . , d} \| ∆h − C∆ ≥ \|∆µh \| > (∆h − ζ)/ C(t)}. Again, we can examine both sets separately. For Ed \ Bd we have (∆) ad max T̂n,h − bd ≤ ad max T̂n,h − bd − ad min h∈Ed \Bd h∈Ed \Bd h∈Ed \Bd √ n . τ (t̂h )σ̂h ∆h (7.32) By definition of Ed \ Bd we obtain that the second summand on the right-hand side of (7.32) tends p (in probability) to −∞. Due to the lower bound minh∈Ed \Bd \|∆µh \| > (∆− − ζ)/ C(t), which holds uniformly in d, we can apply Corollary 3.11 to the first summand of the right-hand side of (7.32), which then gives P (∆) ad max T̂n,h − bd −→ −∞. h∈Ed \Bd On the set Bd = Ed we can directly apply Corollary 3.11, so that we obtain (4.4). 38 7.3.2 Proof of Theorem 4.3 It follows from Theorem 3 of Wu (2005) that n 1 bnsc o X D √ =⇒ {σh Ws }s∈[0,1] , Zj,h − E [Zj,h ] n s∈[0,1] (7.33) j=1 where {Ws }s∈[0,1] denotes the (standard) Brownian motion on the interval [0, 1]. The definition of µh (s, th ) in (3.9), E [Un,h (s)] = µh (s, th )(1 + o(1)) (uniformly with respect to s ∈ [0, 1]) and the continuous mapping theorem yield √ D n(Un,h (s) − µh (s, th )) s∈[0,1] =⇒ {σh Bs }s∈[0,1] , (7.34) where {Bs }s∈[0,1] denotes a (standard) Brownian bridge. Observing (3.8) we get √ Z 1 √ 3 n 2 2 (Un,h (s) − µh (s, th ))2 ds n Mn,h − ∆µh = (th (1 − th ))2 0 √ Z 1 6 n + µh (s, th )(Un,h (s) − µh (s, th ))ds. (th (1 − th ))2 0 Statement (7.34) and the continuous mapping theorem imply Z 1 D √ 6 2 2 n Mn,h − ∆µh =⇒ µh (s, th )σh B(s)ds. (th (1 − th ))2 0 It is well known, that the expression on the right-hand side follows a centered normal distribution and a straightforward calculation shows that its variance is given by ∆µ2h τ 2 (th )σh2 . Replacing σh and th by the estimators σ̂h and t̂h we obtain from Lemmas 3.8 and 3.9 the weak convergence √ n 2 D (7.35) M̂n,h − ∆µ2h =⇒ N (0, ∆µ2h ) , τ (t̂h )σ̂h for each (fixed) h ∈ N provided that \|∆µh \| > 0. After these preparations we are ready to prove the consistency of the test (2.19). If the alternative hypothesis HA,∆ is valid, we can fix k ∈ {1, . . . , d}, such that dk := ∆µ2k − ∆2k > 0. From the definition of the test statistic Td,n in (2.14) we obtain √ √ n n (∆) 2 2 Td,n ≥ ad T̂n,k − bd = ad M̂k − ∆µk + dk − bd , τ (t̂k )σk ∆k τ (t̂k )σk ∆k which gives √ √ g n n 1−α 2 2 P (Td,n M̂k − ∆µk > + bd − dk . ad τ (t̂k )σk ∆k τ (t̂k )σk ∆k √ √ Using bd ∼ log d and τ (t̂ )σn ∆ & n leads to k k k √ g1−α n P + bd − dk −→ −∞. ad τ (t̂k )σk ∆k > g1−α ) ≥ P On the other hand we obtain from (7.35) √ n D \|∆µk \| M̂2k − ∆µ2k =⇒ · N (0, 1) , ∆k τ (t̂k )σk ∆k and the assertion of Theorem 4.3 follows. 39 (7.36) (7.37) 7.3.3 Proof of Theorem 4.4 Due to g1−α /ad + bd > 0 we deduce b d (α)) = P min T̂ (∆) > g1−α /ad + bd P(Rd ⊂ R h∈Rd n,h √ n ≥ P min M̂2n,h − ∆2h > g1−α /ad + bd . h∈Rd τ (t̂h )σ̂h \|∆µh \| Using the notation dh = ∆µ2h − ∆2h we get √ √ n n 2 2 b M̂n,h − ∆µh > g1−α /ad + bd − min dh P(Rd ⊂ Rd (α)) > P min h∈Rd h∈Rd τ (t̂h )σ̂h \|∆µh \| τ (t̂h )σ̂h \|∆µh \| √ n = P min T̂n,h > g1−α /ad + bd − min dh h∈Rd h∈Rd τ (t̂h )σ̂h \|∆µh \| √ n = P ad min T̂n,h + bd ≥ g1−α + 2ad bd − ad min dh h∈Rd h∈Rd τ (t̂h )σ̂h \|∆µh \| By assumption (4.7) we have nC min dh ≥ nC min (\|∆µh \| − ∆h ) ∆− −→ ∞, h∈Rd n→∞ h∈Rd which implies (as 1 . τ (t̂h )σ̂h \|∆µh \| . 1) √ ad bd − ad min dh h∈Rd n −→ −∞. τ (t̂h )σ̂h \|∆µh \| d,n→∞ By arguments similar to those in the proofs of Section 3 one can show that for all x ∈ R lim inf P ad min T̂n,h + bd ≥ x ≥ P (−G ≥ x) , d,n→∞ h∈Rd which yields assertion (4.8). For a proof of (4.9) we apply Bonferroni’s inequality, which gives b d (α) = Rd = P R b d (α) ⊂ Rd , Rd ⊂ R b d (α) ≥ 1 − P R b d (α) * Rd − P Rd * R b d (α) . P R b d (α) = o(1) and Theorem 4.1 By the arguments in the previous paragraph we have P Rd * R gives (∆) b lim sup P Rd (α) * Rd = lim sup P maxc T̂n,h > g1−α /ad + bd ≤ α, d,n→∞ h∈Rd d,n→∞ which finishes the proof. 7.4 Proofs of the results in Section 5 To establish the bootstrap results, we recall the definition of the set Sd in (3.17) and we introduce the set n o b− , K L b+ ) , Ld = ∀ h ∈ Sdc : nth ∈ (K L (7.38) h h which represents the event, that the locations of all change points are identified correctly. We need the following basic properties. 40 Lemma 7.1 If the assumptions of Section 3.1 and Assumption 5.2 hold, then d P (i) nC max \|∆b µh − ∆µh \| −→ 0 if C < 1/2, h=1 (ii) P (Lcd ) . nC for a sufficiently small constant C > 0. Proof. For assertion (i) fix ε > 0 and observe b− d P nC max \|∆b µh − ∆µh \| > ε ≤ P nC max d h=1 h=1 KL 1 Xh Z − µ > ε/2 ∩ L j,h h,1 d b− KL h j=1 d 1 n X h=1 b+ ) K(L − L h b h +1 j=K L + P nC max Zj,h − µh,2 > ε/2 ∩ Ld + o(1). (7.39) The first two summands of the right-hand side of (7.39) exhibit the same structure, so we only treat the first of them. Note that on the event Ld , it holds that b − }. b − } = Xj,h I{j ≤ K L (Zj,h − µh,1 ) I{j ≤ K L h h b − ≤ nt̂h Further there exists a constant 0 < C0 < 1, such that the inequalities C0 n ≤ nt − K ≤ K L h hold. This implies − − b b KL L d h X 1 KX Xh 1 −C Z − µ > ε/2 ∩ L ≤ P X > n ε/2 P nC max j,h h,1 d j,h b− b− h=1 K L KL h j=1 h j=1 h=1 d d k n X X ≤ P max Xj,h > C0 n1−C ε/2 . h=1 k=1 j=1 Since 1 − C > 1/2 we obtain by the Fuk-Nagaev inequality (see Lemma E.3 in Jirak (2015b)) for sufficiently large n d k n X X P max Xj,h > C0 n1−C ε/2 . nD h=1 k=1 j=1 n n(1−C)p = nD+(C−1)p+1 ≤ nD−p/2+1 = o(1), where we also used D ≤ p/2 − 2. Assertion (ii) is shown in the proof of Theorem C.12 in Jirak (2015b). Proof of Lemma 5.3. This is a consequence of Lemma 7.1 (i) and the inequality (for any ε > 0) P ad max Bn,h − bd > ε ≤ P max \|∆b µh \| > n−1/4 . h∈Sd h∈Sd 41 7.4.1 Proof of Theorem 5.4 For the proof we introduce the following more simple version of the bootstrap CUSUM-process (L) {Un,h (s)}s∈[0,1] introduced in (5.7) bLsc L X bLsc X b e (L) (s) := 1 b U ξ V − ξ` V`,h , ` `,h n,h n Ln `=1 `=1 where the truncation is not conducted within the blocks. We make also use of the following extra notation √ Z 1 6 n e (L) (s)k(s, th )ds e = U B n,h σh τ (th )(th (1 − th ))2 0 n,h √ Z 1 6 n ∗ e (L) (s)k(s, t̂h )ds e U Bn,h = σh τ (th )(th (1 − th ))2 0 n,h √ Z 1 6 n (L) ∗ Bn,h = Un,h (s)k(s, t̂h )ds 2 σh τ (th )(th (1 − th )) 0 √ Z 1 6 n (L) (I) Un,h (s)k(s, t̂h )ds. Bn,h = 2 sbh τ (t̂h )(t̂h (1 − t̂h )) 0 The theorem’s claim is a direct consequence of the next five lemmas. We will use the notation P\|Zn (·) = P (·\|Zn ). and frequently apply that the implication P (An ) = o(1) ⇒ P\|Zn (An ) = op (1) holds for all sequences of measurable sets {An }n∈N . Lemma 7.2 The weak convergence D e ad maxc Bn,h − bd =⇒ h∈Sd      G if Sdc = {1, . . . , d}, G + 2 log c if \|Sdc \| = c · d + o(d) for c ∈ (0, 1),     0 if \|Sdc \| = o(d) holds conditionally on Zn . Proof. Without loss of generality assume that the sets Sd and Sdc are given by Sdc = {1, . . . , s} and Sd = {s + 1, . . . , d} with s = \|Sdc \|. Let N = (N1 , . . . , Ns )T denote a centered s-dimensional Gaussian vector with covariance matrix Σ = (Σij )di,j=1 defined by Σi,j = γi,j τe(ti , tj ) , σi σj τ (ti )τ (tj ) where the function τe is defined in (7.4). Our aim is to control the (conditional) Kolmogorov-distance e and maxh∈S c Nh . Since the random variables {ξ` } between maxh∈Sdc B `∈N are independent, we n,h d can directly calculate the conditional covariance Cov\|Zn e , B en,i B = n,h L X 36 Vb`,h Vb`,i β`,h β`,i σh σi τ (ti )τ (th )(th (1 − th ))2 (ti (1 − ti ))2 n `=1 42 with the extra notation Z 1 β`,h = 0 bLsc k(s, th ) I{` ≤ bLsc} − ds. L Let θd denote the distance e , B en,i − Σh,i . Cov\|Zn B n,h θd = max 1≤h,i≤s Using the fact that maxh∈Sdc maxL `=1 \|β`,h \| ≤ 1 a straightforward adaption of Lemma E.8 in Jirak (2015a) gives P L max 1≤h,i≤s 36 1 Xb b −δ V > n . L−2 n−C V β β − γ τ e (t , t ) I Ld `,h `,i `,h `,i h,i h i (th (1 − th ))2 (ti (1 − ti ))2 KL `=1 for sufficiently small constants C, δ > 0, where the set Ld is defined in (7.38). Using the lower 2 τ 2 yields bound σh σi τ (th )τ (ti ) ≥ σ− − P C(δ)C . n−C (7.40) for the set o n C(δ) := θd ILd ≤ n−δ . For a sufficiently small C > 0 we have C C C C C −C = o(1). ≤ nC E P\|Zn (LC P P\|Zn (LC d ∪ C(δ) ) = n P Ld ∪ C(δ) d ∪ C(δ) ) ≥ n Now, we can derive the following upper bound d d e ≤ x − P max Nh ≤ x sup P\|Zn max B n,h h=1 h=1 x∈R d d e ≤ sup P\|Zn max Bn,h ≤ x − P max Nh ≤ x ILd ∩C(δ) + OP (n−C ). h=1 h=1 x∈R (7.41) The identities γh,h = σh2 and τ̃ (th ) = τ 2 (th ) give Σh,h = 1 für alle h ∈ {1, . . . , d}, and by Lemma 3.1 in Chernozhukov et al. (2013) on the set Ld ∩ C(δ) we have for the first summand in (7.41) e sup P\|Zn maxc Bn,h ≤ x − P maxc Nh ≤ x ILd ∩C(δ) (7.42) x∈R h∈Sd h∈Sd 1/3 max{1, log(d/θd }2/3 ILd ∩C(δ) ≤ n−C , (7.43) Combining (7.41) and (7.42) yields for the Kolmogorov distance e sup P\|Zn maxc Bn,h ≤ x − P maxc Nh ≤ x = op (1). (7.44) . θd x∈R h∈Sd h∈Sd 43 The proof now follows observing the bound max1≤h,i≤s \|Σh,i \| ≤ max1≤h,i≤s \|ρh,i \| for the covariance matrix of N , which was derived (see the proof of Lemma 3.7) and using adapted scaling sequences (see proof of Corollary 3.11), which yields   G if Sdc = {1, . . . , d},    D ad maxc Nh − bd =⇒ G + 2 log c if \|Sdc \| = c · d + o(d) for c ∈ (0, 1),  h∈Sd    0 if \|Sdc \| = o(d). Lemma 7.3 Conditionally on Zn it holds that P e ∗ −→ e − ad max B 0. ad maxc B n,h n,h c h∈Sd h∈Sd Proof. The covariance kernel k satisfies for all t, t0 , s ∈ [0, 1] \|k(s, t) − k(s, t0 )\| ≤ 2\|t − t0 \|. Due to (th (1 − th ))2 τ (th ) ≥ t4 τ− we derive the bound Z 1 √ 1 e (L) ∗ e e ad maxc Bn,h − maxc Bn,h . ad n maxc Un,h (s) \|k(s, th ) − k(s, t̂h )\|ds h∈Sd h∈Sd h∈Sd 0 σh √ 1 e (L) Un,h (s) maxc \|t̂h − th \|. . ad n maxc max h∈Sd s∈[0,1] σh h∈Sd Choosing C sufficiently it follows from Corollary 3.3 in Jirak (2015a) that for all ε > 0 C P\|Z n maxc \|th − t̂h \| > ε = op (1). h∈Sd (7.45) Theorem 2.5 and 4.4 from the same reference imply the weak convergence √ 1 e (L) ed D ed n maxc max =⇒ G Un,h (s) − h∈Sd s∈[0,1] σh 4 p conditionally on Zn in probability with ed = 2 log(2d). This yields √ −C 1 e (L) P\|Zn ad nn maxc max Un,h (s) > ε = op (1) h∈Sd s∈[0,1] σh for all ε > 0, which completes the proof of Lemma 7.3. Lemma 7.4 Conditionally on Zn it holds that e ∗ − ad max B ∗ −→ 0. ad maxc B n,h n,h c P h∈Sd h∈Sd Proof. Define PL \|Zn (·) = P\|Zn (· ∩ Ld ) with Ld introduced in (7.38). By Lemma 7.1 the claim follows if we can verify ε ∗ ∗ e∗ e ∗ > ε ≤ PL − maxc B = op (1). (7.46) PL maxc Bn,h n,h \|Zn maxc Bn,h − Bn,h > \|Zn h∈Sd h∈Sd h∈Sd ad ad 44 We have the trivial bound ∗ e∗ . e ∗ ≤ max B ∗ − B maxc Bn,h − maxc B n,h n,h n,h c h∈Sd h∈Sd h∈Sd Assumption (T2) and (C1) imply σh ≥ σ− and τ (th )(th (1 − th ))2 ≥ τ− t4 , which yields √ Z 1 6 n (L) ∗ ∗ e (L) (s) k(s, t̂h )ds e Bn,h − Bn,h = U (s) − U n,h n,h σh τ (th )(th (1 − th ))2 0 Z 1 √ (L) e (L) (s) k(s, t̂h )ds . n U (s) − U . √ n,h 0 n,h (L) e (L) (s) ≤ S (1) + S (2) , n max Un,h (s) − U n,h n,h n,h s∈[0,1] where we use the notation bLsc L X 1 X b (1) Sn,h = max √ ξj Vj,h (bnsc) − ξj Vbj,h (n) , n s∈[0,1] j=1 (2) Sn,h = max s∈[0,1] j=1 bLsc bnsc √ − √ n n L n L X ξj Vbj,h (n) j=1 In the proof of Theorem C.4 from Jirak (2015b) it is shown, that ε (1) = op (1) PL \|Z maxc Sn,h > h∈Sd ad for all ε > 0. The second summand can be bounded by L (2) Sn,h = max s∈[0,1] 1 ≤ √ nL bnsc bLscK 1 X b √ − ξj Vj,h (n) K K nL j=1 L X ξj Vbj,h (n) j=1 and it follows again from the above reference PL \|Zn L ε 1 X b maxc √ ξj Vj,h (n) > h∈Sd ad nL = op (1) j=1 for all ε > 0, which finishes the proof of Lemma 7.4 Lemma 7.5 The weak convergence   G if Sdc = {1, . . . , d},    D (I) ad maxc Bn,h − bd =⇒ G + 2 log c if \|Sdc \| = c · d + o(d) for c ∈ (0, 1),  h∈Sd    0 if \|Sdc \| = o(d) holds conditionally on Zn . 45 (7.47) Proof. Combining Lemmas 7.2, 7.3 and 7.4 already gives assertion (7.47) for the random variables ∗ . We use the additional notation Bn,h Q− d = minc h∈Sd sbh τ̂h (t̂h (1 − t̂h ))2 sbh τ̂h (t̂h (1 − t̂h ))2 + und Q = max d h∈Sdc σh τ (th )(th (1 − th ))2 σh τ (th )(th (1 − th ))2 + −2 and the set Qd = Q− d − 1 ∨ Qd − 1 ≤ δd with δd = (log d) . Let ud (x) = x/ad + bd , then we obtain for fixed ε > 0 and d sufficiently large (I) ∗ c P\|Zn maxc Bn,h ≤ ud (x − ε) − P\|Zn (Qd ) ≤ P\|Zn maxc Bn,h ≤ ud (x) h∈Sd h∈Sd ∗ ≤ ud (x + ε) + P\|Zn (Qcd ) . ≤ P\|Zn maxc Bn,h h∈Sd Combining Proposition 3.5 from Jirak (2015a) with assertion (7.45) one can easily verify that P\|Zn (Qcd ) = op (1) and the remainder of the proof can be done analogously to the proof of Theorem 3.10. Lemma 7.6 Conditionally on Zn it holds that (I) P ad maxc Bn,h − ad maxc Bn,h −→ 0. h∈Sd h∈Sd Proof. For fixed ε > 0 we have (I) (I) P ad maxc Bn,h − maxc Bn,h > ε ≤ P ad maxc Bn,h − Bn,h > ε h∈Sd h∈Sd h∈Sd n o −1/4 −1/4 ≤ P maxc I \|∆b µh \| ≤ n > ε = P minc \|∆b µh \| ≤ n . h∈Sd h∈Sd The proof now follows by −1/4 −1/4 −1/4 P minc \|∆b µh \| ≤ n = P minc \|∆b µh \| ≤ n , minc \|∆µh \| ≤ 2n h∈Sd h∈Sd h∈Sd −1/4 −1/4 + P minc \|∆b µh \| ≤ n , minc \|∆µh \| > 2n h∈Sd h∈Sd ≤ P minc \|∆µh \| ≤ 2n−1/4 + P minc \|∆µh \| − minc \|∆b µh \| > n−1/4 h∈Sd h∈Sd h∈Sd −1/4 ≤ P K minc \|∆µh \| ≤ 2 + P maxc \|∆µh − ∆b µh \| > n = o(1), h∈Sd h∈Sd where we also used that K = n1−` and that Assumption 5.2 implies 1 − ` < 1/4 together with lim K minc \|∆µh \| = ∞ . n,d→∞ h∈Sd 46	10
Annals of Mathematical Sciences and Applications Volume 0, Number 0, 1–9, 0000 arXiv:1711.04451v1 [cs.CV] 13 Nov 2017 Visual Concepts and Compositional Voting Jianyu Wang, Zhishuai Zhang, Cihang Xie, Yuyin Zhou, Vittal Premachandran, Jun Zhu, Lingxi Xie, Alan Yuille It is very attractive to formulate vision in terms of pattern theory [26], where patterns are defined hierarchically by compositions of elementary building blocks. But applying pattern theory to real world images is currently less successful than discriminative methods such as deep networks. Deep networks, however, are black-boxes which are hard to interpret and can easily be fooled by adding occluding objects. It is natural to wonder whether by better understanding deep networks we can extract building blocks which can be used to develop pattern theoretic models. This motivates us to study the internal representations of a deep network using vehicle images from the PASCAL3D+ dataset. We use clustering algorithms to study the population activities of the features and extract a set of visual concepts which we show are visually tight and correspond to semantic parts of vehicles. To analyze this we annotate these vehicles by their semantic parts to create a new dataset, VehicleSemanticParts, and evaluate visual concepts as unsupervised part detectors. We show that visual concepts perform fairly well but are outperformed by supervised discriminative methods such as Support Vector Machines (SVM). We next give a more detailed analysis of visual concepts and how they relate to semantic parts. Following this, we use the visual concepts as building blocks for a simple pattern theoretical model, which we call compositional voting. In this model several visual concepts combine to detect semantic parts. We show that this approach is significantly better than discriminative methods like SVM and deep networks trained specifically for semantic part detection. Finally, we return to studying occlusion by creating an annotated dataset with occlusion, called VehicleOcclusion, and show that compositional voting outperforms even deep networks when the amount of occlusion becomes large. Keywords: Pattern theory, deep networks, visual concepts. 1. Introduction It is a pleasure to write an article for a honoring David Mumford’s enormous contributions to both artificial and biological vision. This article ad1 2 dresses one of David’s main interests, namely the development of grammatical and pattern theoretic models of vision [48][26]. These are generative models which represent objects and images in terms of hierarchical compositions of elementary building blocks. This is arguably the most promising approach to developing models of vision which have the same capabilities as the human visual system and can deal with the exponential complexity of images and visual tasks. Moreover, as boldly conjectured by David [27, 17], they suggest plausible models of biological visual systems and, in particular, the role of top-down processing. But despite the theoretical attractiveness of pattern theoretic methods they have not yet produced visual algorithms capable of competing with alternative discriminative approaches such as deep networks. One reason for this is that, due to the complexity of image patterns, it very hard to specify their underlying building blocks patterns (by contrast, it is much easier to develop grammars for natural languages where the building blocks, or terminal nodes, are words). But deep networks have their own limitations, despite their ability to learn hierarchies of image features in order to perform an impressive range of visual tasks on challenging datasets. Deep networks are black boxes which are hard to diagnose and they have limited ability to adapt to novel data which were not included in the dataset on which they were trained. For example, figure (1) shows how the performance of a deep network degrades when we superimpose a guitar on the image of a monkey. The presence of the guitar causes the network to misinterpret the monkey as being a human while also mistaking the guitar for a bird. The deep network presumably makes these mistakes because it has never seen a monkey with a guitar, or a guitar with a monkey in the jungle (but has seen a bird in the jungle or a person with a musical instrument). In short, the deep network is over-fitting to the typical context of the object. But over-fitting to visual context is problematic for standard machine learning approaches, such as deep networks, which assume that their training and testing data is representative of some underlying, but unknown, distribution [37, 38]. The problem is that images, and hence visual context, can be infinitely variable. A realistic image can be created by selecting from a huge range of objects (e.g., monkeys, guitars, dogs, cats, cars, etc.) and placing them in an enormous number of different scenes (e.g., jungle, beach, office) and in an exponential number of different three-dimensional positions. Moreover, in typical images, objects are usually occluded as shown in figure (1). Occlusion is a fundamental aspect of natural images and David has recognized its importance both by proposing a 2.1 D sketch representation 3 Figure 1: Caption: Adding occluders causes deep network to fail. We refer such examples as adversarial context examples since the failures are caused by misusing context/occluder info. Left Panel: The occluding motorbike turns a monkey into a human. Center Panel: The occluding bicycle turns a monkey into a human and the jungle turns the bicycle handle into a bird. Right Panel: The occluding guitar turns the monkey into a human and the jungle turns the guitar into a bird. [28] and also by using it as the basis for a “dead leaves” model for image statistics [18]. Indeed the number of ways that an object can be occluded is, in itself, exponential. For example, if an object has P different parts then we can occlude any subset of them using any of N different occluders yielding O(P N ) possible occluding patterns. In the face of this exponential complexity the standard machine learning assumptions risk breaking down. We will never have enough data both to train, and to test, black box vision algorithms. As in well known many image datasets have confounds which mean that algorithms trained on them rarely generalize to other datasets [4])İnstead we should be aiming for visual algorithms that can learn objects (and other image structures) from limited amounts of data (large, but not exponentially large). In addition, we should devise strategies where the algorithms can be tested on potentially infinite amounts of training data. One promising strategy is to expand the role of adversarial noise [34][10][45], so that we can create images which stress-test algorithms and deliberately target their weaknesses. This strategy is more reminiscent of game theory than the standard assumptions of machine learning (e.g., decision theory). This motivates a research program which we initiate in this paper by addressing three issues. Firstly, we analyze the internal structure of deep networks in order both to better understand them but, more critically, to extract internal representations which can be used as building blocks for pattern theoretic models. We call these representations visual concepts and extract them by analyzing the population codes of deep networks. We quantify their ability for detecting semantic parts of objects. Secondly, we develop 4 a simple compositional-voting model to detect semantic parts of objects by using these building blocks. These models are similar to those in [1]. Thirdly, we stress test these compositional models, and deep networks for comparison, on datasets where we have artificially introduced occluders. We emphasize that neither algorithm, compositional-voting or the deep network, has been trained on the occluded data and our goal is to stress test them to see how well they perform under these adversarial conditions. In order to perform these studies we have constructed two datasets based on the PASCAL3D+ dataset [43]. Firstly, we annotate the semantic parts of the vehicles in this dataset to create the VehicleSemanticPart dataset. Secondly, we create the Vehicle Occlusion dataset by superimposing cutouts of objects from the PASCAL segmentation dataset [8] into images from the VehicleSemanticPart dataset. We use the VehicleSemanticPart dataset to extract visual concepts by clustering the features of a deep network VGG16 [33], trained on ImageNet, when the deep network is shown objects from VehicleSemanticPart. We evaluate the visual concepts, the compositionalvoting mode;s, and the deep networks for detecting semantic parts image with and without occlusion, i.e. VehicleSemanticPart and Vehicle Occlusion respectively. 2. Related Work David’s work on pattern theory [26] was inspired by Grenander’s seminal work [11] (partly due to an ARL center grant which encouraged collaboration between Brown University and Harvard). David was bold enough to conjecture that the top-down neural connections in the visual cortex were for performing analysis by synthesis [27]. Advantages of this approach include the theoretical ability to deal with large numbers of objects by part sharing [46]. But a problem for these types of models was the difficulty in specifying suitable building blocks which makes learning grammars for vision considerably harder than learning them for natural language [35, 36]. For example, researchers could learn some object models in an unsupervised manner, e.g., see [49], but this only used edges as the building blocks and hence ignored most of the appearance properties of objects. Attempts have been made to obtain building blocks by studying properties of image patches [30] and [5] but these have yet to be developed into models of objects. Though promising, these are pixel-based and are sensitive to spatial warps of images and do not, as yet, capture the hierarchy of abstraction that seems necessary to deal with the complexity of real images. 5 Deep networks, on the other hand, have many desirable properties. such as hierarchies of feature vectors (shared across objects) which capture increasingly abstract, or invariant, image patterns as they ascend the hierarchy from the image to the object level. This partial invariance arises because they are trained in a discriminative way so that decisions made about whether a viewed object is a car or not must be based on hierarchical abstractions. It has been shown, see [47][22][44][32], that the activity patterns of deep network features do encode objects and object parts, mainly be considering the activities of individual neurons in the deep networks. Our approach to study the activation patterns of populations of neuron [40] was inspired by the well-known neuroscience debate about the neural code. On one extreme is the concept of grandmother calls described by Barlow [3] which is contrasted to population encoding as reported by Georgopoulos et al. [9]. Barlow was motivated by the finding that comparatively few neurons are simultaneously active and proposed sparse coding as a general principle. In particular, this suggests a form of extreme sparsity where only one, or a few cells, respond to a specific local stimulus (similar to a matched template). As we will see in section (6), visual concepts tend to have this property which is developed further in later work. There are, of course, big differences between studying population codes in artificial neural networks and studying them for real neurons. In particular, as we will show, we can modify the neural network so that the populations representing a visual concept can be encoded by visual concept neurons, which we call vc-neurons. Hence our approach is consistent with both extreme neuroscience perspectives. We might speculate that the brain uses both forms of neural encoding useful for different purposes: a population, or signal representation, and a sparser symbolic representation. This might be analogous to how words can be represented as binary encodings (e.g., cat, dog, etc.) and by continuous vector space encodings which similarity between vectors captures semantic information [25]. Within this picture, neurons could represent a manifold of possibilities which the vc-neurons would quantize into discrete elements. We illustrate the use of visual concepts to build a compositional-voting model inspired by [1], which is arguably the simplest pattern theoretic method. We stress that voting schemes have frequently been used in computer vision without being thought of as examples of pattern theory methods [12][19][24][29]. To stress test our algorithms we train them on un-occluded images and test them on datasets with occlusion. Occlusion is almost always present in natural images, though less frequently in computer vision datasets, and 6 is fundamental to human perception. David’s work has long emphasized its significance [28], [18] and it is sufficiently important for computer vision researchers to have addressed it using deep networks [39]. It introduces difficulties for tasks such as object detection [16] or segmentation [20]. As has already been shown, part-based models are very useful for detecting partially occluded objects [7] [6][21]. 3. The Datasets In this paper we use three related annotated datasets. The first is the vehicles in the PASCAL3D+ dataset [43] and their keypoint interactions. The second is the VehicleSemanticPart dataset which we created by annotating semantic parts on the vehicles in the PASCAL3D+ dataset to give a richer representation of the objects than the keypoints. The third is the Vehicle Occlusion dataset where we occluded the objects in the first two datasets. The vehicles in the PASCAL3D+ dataset are: (i) cars, (ii) airplanes, (iii) motorbikes, (iv) bicycles, (v) buses, and (vi) trains. These images in the PASCAL3D+ dataset are selected from the Pascal and the ImageNet datasets. In this paper we report results only for those images from ImageNet, but we obtain equivalent results on PASCAL, see arxiv paper [40]. These images were supplemented with keypoint annotations (roughly ten keypoints per object) and also estimated orientations of the objects in terms of the azimuth and the elevation angles. These objects and their keypoints are illustrated in figure (2)(upper left panel). We use this dataset to extract the visual concepts, as described in the following section. In order to do this, we first re-scale the images so that the minimum of the height and width is 224 pixels (keeping the aspect ratio fixed). We also use the keypoint annotations to evaluate the ability of the visual concepts to detect the keypoints. But there are typically only 10 keypoints for each object which means they can only give limited evaluation of the visual concepts. In order to test the visual concepts in more detail we defined semantic parts on the vehicles and annotated them to create a dataset called VehicleSemanticParts. There are thirty nine semantic parts for cars, nineteen for motorbikes, ten for buses, thirty one for aeroplanes, fourteen for bicycles, and twenty for trains. The semantic parts are regions of 100 × 100 pixels and provide a dense coverage of the objects, in the sense that almost every pixel on each object image is contained within a semantic part. The semantic parts have verbal descriptions, which are specified in the webpage 7 Figure 2: The figure illustrates the keypoints on the PASCAl3D+ dataset. For cars, there are keypoints for wheels, headlights, windshield corners and trunk corners. Keypoints are specified on salient parts of other objects. Best seen in color. http://ccvl.jhu.edu/SP.html. The annotators were trained by being provided by: (1) a one-sentence verbal description, and (2) typical exemplars. The semantic parts are illustrated in figure (3) where we show a representative subset of the annotations indexed by “A”, “B”, etc. The car annotations are described in the figure caption and the annotations for the other objects are as follows: (I) Airplane Semantic Parts. A: nose pointing to the right, B: undercarriage, C: jet engine, mainly tube-shape, sometimes with highly turned ellipse/circle, D: body with small windows or text, E: wing or elevator tip pointing to the left, F: vertical stabilizer, with contour from top left to bottom right, G: vertical stabilizer base and body, with contour from top left to bottom right. (II) Bicycle Semantic Parts. A: wheels, B: pedal, C: roughly triangle structure between seat, pedal, and handle center, D: seat, E: roughly T-shape handle center. (III) Bus Semantic Parts. A: wheels, B: headlight, C: license plate, D: window (top) and front body (bottom), E: display screen with text, F: window and side body with vertical frame in the middle, G: side windows and side body. (IV) Motorbike Semantic Parts. A: wheels, B: headlight, C: engine, D: seat. (V) Train Semantic Parts. A: front window and headlight, B: headlight, C: part of front window on the left and side body, D: side windows or doors and side body, E: head body on the right or bottom right and background, F: head body on the top left and background.). In order to study occlusion we create the VehicleOcclusion dataset. This consists of images from the VehicleSemanticPart dataset. Then we add 8 Figure 3: This figure illustrates the semantic parts labelled on the six vehicles in the VehicleSemanticParts dataset (remaining panels). For example, for car semantic parts. A: wheels, B: side body, C: side window, D: side mirror and side window and wind shield, E: wind shield and engine hood, F: oblique line from top left to bottom right, wind shield, G: headlight, H: engine hood and air grille, I: front bumper and ground, J: oblique line from top left to bottom right, engine hood. The parts are weakly viewpoint dependent. See text for the semantic parts of the other vehicles. occlusion by randomly superimposing a few (one, two, or three) occluders, which are objects from PASCAL segmentation dataset [8], onto the images. To prevent confusion, the occluders are not allowed to be vehicles. We also vary the size of the occluders, more specifically the fraction of occluded pixels from all occluders on the target object, is constrained to lie in three ranges 0.2−0.4, 0.4−0.6 and 0.6−0.8. To compute these fractions requires estimating the sizes of the vehicles in the VehicleSemanticPart dataset, which can be done using the 3D object models associated to the images in PASCAL3D+. 4. The Visual Concepts We now analyze the activity patterns within deep networks and show that they give rise to visual concepts which correspond to part/subparts of objects and which can be used as building blocks for compositional models. In section (4.2) we obtain the visual concepts by doing K-means clustering followed by a merging stage. Next in section (4.3), we show an alternative unsupervised method which converges to similar visual concepts, but which can be implemented directly by a modified deep network with additional (unsupervised) nodes/neurons. Section (4.4) gives visualization of the visual concepts. Section (4.5) evaluates them for detecting keypoints and semantic parts. 9 Figure 4: Examples for occluded cars with occlusion level 1,5 and 9 respectively. Level 1 means 2 occluders with fraction 0.2-0.4 of the object area occluded, level 5 means 3 occluders with fraction 0.4-0.6 occluded, and level 9 means 4 occluders with fraction 0.6-0.8 occluded. 4.1. Notation We first describe the notation used in this paper. The image is defined on the image lattice and deep network feature vectors are defined on a hierarchy of lattices. The visual concepts are computed at different layers of the hierarchy, but in this paper we will concentrate on the pool-4 layer, of a VGG-16 [33] trained on ImageNet for classification, and only specify the algorithms for this layer (it is trivial to extend the algorithms to other layers). The groundtruth annotations, i.e. the positions of the semantic parts, are defined on the image lattice. Hence we specify correspondence between points of the image lattice and points on the hierarchical lattices. Visual concepts will never be activated precisely at the location of a semantic part, so we specify a neighborhood of allow for tolerance in spatial location. In the following section, we will use more sophisticated neighborhoods which depend on the visual concepts and the semantic parts. An image I is defined on the image lattice L0 by a set of three-dimensional color vectors {Iq : q ∈ L0 }. Deep network feature vectors are computed on a hierarchical set of lattices Ll , where l indicates the layer with Ll ⊂ Ll−1 for l = 0, 1, .... This paper concentrates on the pool-4 layer L4 . We define correspondence between the image lattice and the pool-4 lattice by the mappings π07→4 (.) from L0 to L4 and π47→0 (.) from L4 to L0 respectively. The function π47→0 (.) gives the exact mapping from L4 to L0 (recall the lattices are defined so that L4 ⊂ L0 ). Conversely, π07→4 (q) denotes the closest position at the L4 layer grid that corresponds to q, i.e., π07→4 (q) = arg minp Dist(q, L0 (p)). The deep network feature vectors at the pool-4 layer are denoted by {fp : p ∈ L4 }. These feature vectors are computed by fp = f (Ip ), where the function f is specified by the deep network and Ip is an image patch, a 10 subregion of the input image I, centered on a point π47→0 (p) on the image lattice L0 . The visual concepts {VCv : v ∈ V} at the pool-4 layer are specified by a set of feature vectors {fv : v ∈ V}. They will be learnt by clustering the pool-4 layer feature vectors {fp } computed from all the images in the dataset. The activation of a visual concept VCv to a feature vector fp at p ∈ L4 is a decreasing function of \|\|fv − fp \|\| (see later for more details). The visual concepts will be learnt in an unsupervised manner from a set T = {In : n = 1, ..., N } of images, which in this paper will be vehicle images from PASCAL3D+ (with tight bounding boxes round the objects and normalized sizes). The semantic parts are given by {SPs ∈ S} and are pre-defined for each vehicle class, see section (3) and the webpage http://ccvl.jhu.edu/ SP.html. These semantic parts are annotated on the image dataset {In : n = 1, ..., N } by specifying sets of pixels on the image lattice corresponding to their centers (each semantic part corresponds to an image patch of size 100 × 100). For a semantic part SPs , we define Ts+ to be the set of points q on the image lattices L0 where they have been labeled. From these labels, we compute a set of points Ts− where the semantic part is not present (constrained so that each point in Ts− is at least γ pixels from every point in Ts+ , where γ is chosen to ensure no overlap). We specify a circular neighborhood N (q) for all points q on the image lattices L0 , given by {p ∈ L0 s.t. \|\|p − q\|\| ≤ γth }. This neighborhood is used when we evaluate if a visual concept responds to a semantic part by allowing some uncertainty in the relative location, i.e. we reward a visual concept if it responds near a semantic part, where nearness is specified by the neighborhood. Hence a visual concept at pixel p ∈ L4 responds to a semantic part s at position q provided \|\|fv − fp \|\| is small and π47→0 (p) ∈ N (q). In this paper the neighborhood radius γth is set to be 56 pixels for the visual concept experiments in this section, but was extended to 120 for our late work on voting, see section (5). For voting, we start with this large neighborhood but then learn more precise neighborhoods which localize the relative positions of visual concept responses to the positions of semantic parts. 4.2. Learning visual concepts by K-means We now describe our first method for extracting visual concepts which uses the K-means algorithm. First we extract pool-4 layer feature vectors {fp } using the deep network VGG-16 [33] from our set of training images T . We normalize them to unit norm, i.e. such that \|fp \| = 1, ∀p. 11 Figure 5: Left Panel: The images I are specified on the image lattice L0 and a deep network extracts features on a hierarchy of lattices. We concentrate on the features {fp } at pool-4 lattice L4 . Projection functions π47→0 () and π07→4 () map pixels from L4 to L0 and vice versa. We study visual concepts VCv at layer L4 and relate them to semantic parts SPs defined on the image lattice. Center Panel: the visual concepts are obtained by clustering the normalized feature vectors {fp }, using either K-means or a mixture of Von Mises-Fisher. Right Panel: extracting visual concepts by a mixture of Von Mises-Fisher is attractive since it incorporates visual concepts within the deep network by adding additional visual concept neurons (vc-neurons). Then we use K-means++ [2], to cluster the feature vectors into a set V of visual concepts. Each visual concept VCv has an index v ∈ {1, 2, . . . , \|V\|} and is specified by its clustering center: fv ∈ R512 (where 512 is the dimension of the feature vector at pool-4 of VGG-net.) In mathematical terms, P the K-means algorithm attempts to minimize a cost function F (V, {fv }) = v,p Vp,v \|\|fp − fv \|\|2 , with respect to the assignment variable V and the cluster centers {fv }. The assignment variables V impose hard assignment so that each feature vector fp is assigned to only one visual concept fv , i.e. for each p, Vp,v∗ = 1 if v∗ = argminv \|\|fp − fv \|\|, and Vp,v = 0 otherwise. The K-means algorithm minimizes the cost function F (V, fv }) by minimizing with respect to V and {fv } alternatively. The Kmeans++ algorithm initializes K-means by taking into account the statistics of the data {fp }. We took a random sample of 100 feature vectors per image as input. The number of clusters K = \|V\| is important in order to get good visual concepts and we found that typically 200 visual concepts were needed for each object. We used two strategies to select a good value for K. The first is to specify a discrete set of values for K (64, 128, 256, 512), determine 12 how detection performance depends on K, and select the value of K with best performance. The second is start with an initial set of clusters followed by a merging stage. We use the Davies-Boundin index as a measure for the σk +σm “goodness” of a cluster k. This is given by DB(k) = maxm6=k \|\|f , where k −fl \|\| fk and fm denote the centers of clusters k and m respectively. σk and σm denote the average distance of all data points in clusters k and m to their P respective cluster centers, i.e., σk = n1k p∈Ck \|\|fp − fk \|\|2 , where Ck = {p : Vp,k = 1} is the set of data points that are assigned to cluster k. The DaviesBoundin index take small values for clusters which have small variance of the P feature vectors assigned to them, i.e. where p:Vp,v =1 \|fp − fv \|2 /nv is small P (with nv = p:Vp,v =1 1), but which are also well separated from other visual clusters, i.e., \|fv − fµ \| is large for all µ 6= v. We initialize the algorithm with K clusters (e.g., 256, or 512) rank them by the Davies-Boundin index, and merge them in a greedy manner until the index is below a threshold, see our longer report [40] for more details. In our experiments we show results with and without cluster merging, and the differences between them are fairly small. 4.3. Learning Visual Concepts by a mixture of von Mises-Fisher distributions We can also learn visual concepts by modifying the deep network to include additional vc neurons which are connected to the neurons (or nodes) in the deep network by soft-max layers, see figure (5). This is an attractive alternative to K-means because it allows visual concepts to be integrated naturally within deep networks, enabling the construction of richer architectures which share both signal and symbolic features (the deep network features and the visual concepts). In this formulation, both the feature vectors {fp } and the visual concepts {fv } are normalized so lie on the unit hypersphere. Theoretical and practical advantages of using features defined on the hypersphere are described in [41]. This can be formalized as unsupervised learning where the data, i.e. the feature vectors {fp }, are generated by a mixture of von Mises-Fisher distributions [13]. Intuitively, this is fairly similar to K-means (since K-means relates to learning a mixture of Gaussians, with fixed isotropic variance, and von-Mises-Fisher relates to an isotropic Gaussian as discussed in the next paragraph). Learning a mixture of von Mises-Fisher can be implemented by a neural network, which is similar to the classic result relating K-means to competitive neural networks [14]. 13 The von-Mises Fisher distribution is of form: P (fp \|fv ) = (1) 1 exp{ηfp · fv }, Z(η) where η is a constant, Z(η) is a normalization factor, and fp and fv are both unit vectors. This requires us to normalize the weights of the visual concepts, so that \|fv \| = 1, ∀v, as well as the feature vectors {fp }. Arguably it is more natural to normalize both the feature vectors and the visual concepts (instead of only normalizing the feature vectors as we did in the last section). There ia a simple relationship between von-Mises Fisher and isotropic Gaussian distributions. The square distance term \|\|fp − fv \|\| becomes equal to 2(1 − fp · fv ) if \|\|fp \|\| = \|\|fv \|\| = 1. Hence an isotropic Gaussian distribution on fp with mean fv reduces to von Mises-Fisher if both vectors are required to lie on the unit sphere. Learning a mixture of von Mises-Fisher can be formulated in terms of neural networks where the cluster centers, i.e. the visual concepts, are specified by visual concept neurons, or vc-neurons, with weights {fv }. This can be shown as follows. Recall that Vp,v is the assignment variable between a feature vector fp and each visual concept fv , and denote the set of assignment by Vp = {Vp,v : v ∈ V} During learning this assignment variable Vp is unknown but we can use the EM algorithm, which involving replacing the Vp,v by a distribution qv (p) for the probability that Vp,v = 1. The EM algorithm can be expressed in terms of minimizing the following free energy function with respect to {qv } and {fv }: (2) X X X F({qv }, {fv }) = − { qv (p) log P (fp , Vp \|{fv }) + qv (p) log qv (p)}, p v v The update rules for the assignments and the visual concepts are respectively: (3) exp{ηfp · fvt } qvt = P t µ exp{ηfp · fµ } fvt+1 = fvt − ηqvt fv + η, where η is a Lagrange multiplier to enforce \|fvt+1 \| = 1. Observe that the update for the assignments are precisely the standard neural network soft-max operation where the {fv } are interpreted as the weights of vc-neurons and the neurons compete so that their activity sums to 1 [14]. After the learning algorithm has converged this softmax activation indicates soft-assignment of a feature vector fp to the vc-neurons. This corresponds to the network shown in figure (5). 14 This shows that learning visual concepts can be naturally integrated into a deep network. This can either be done, as we discussed here, after the weights of a deep network have been already learnt. Or alternatively, the weights of the vc neurons can be learnt at the same times as the weights of the deep network. In the former case, the algorithm simply minimizes a cost function: (4) − X v X exp{ηfp · fv } P log exp{ηfp · fν }, ν exp{ηfp · fν } ν which can be obtained from the EM free energy by solving for the {qv } p ·fv } directly in terms of {fv }, to obtain qv = Pexp{ηf , and substituting this ν exp{ηfp ·fν } back into the free energy. Learning the visual concepts and the deep network weights simultaneously is performed by adding this cost function to the standard penalty function for the deep network. This is similar to the regularization method reported in [23]. Our experiments showed that learning the visual concepts and the deep network weights simultaneously risks collapsing the features vectors and the visual concepts to a trivial solution. 4.4. Visualizing the Visual Concepts We visualize the visual concepts by observing the image patches which are assigned to each cluster by K-means. We observe that the image patches for each visual concept roughly correspond to semantic parts of the object classes with larger parts at higher pooling levels, see figure (6). This paper concentrates on visual concepts at the pool-4 layer because these are most similar in scale to the semantic parts which were annotated. The visual concepts at pool-3 layer are at a smaller scale, which makes it harder to evaluate them using the semantic parts. The visual concepts at the pool-5 layer have two disadvantages. Firstly, they correspond to regions of the image which are larger than the semantic parts. Secondly, their effective receptive field sizes which were significantly smaller than their theoretical receptive field size (determined by using deconvolution networks to find which regions of the image patches affected the visual concept response) which meant that they appeared less tight when we visualized them (because only the central regions of the image patches activated them, so the outlying regions can vary a lot). Our main findings are that the pool-4 layer visual concepts are: (i) very tight visually, in the sense that image patches assigned to the same visual 15 Figure 6: This figure shows example visual concepts on different object categories from the VehicleSemanticPart dataset, for layers pool3, pool4 and pool5. Each row visualizes three visual concepts with four example patches, which are randomly selected from a pool of the closest 100 image patches. These visual concepts are visually and semantically tight. We can easily identify the semantic meaning and parent object class. Figure 7: This figure shows that visual concepts typically give a dense coverage of the different parts of the objects, in this case a car. concept tend to look very similar which is illustrated in figure (8) and (ii) give a dense coverage of the objects as illustrated in figure (7). Demonstrating tightness and coverage of the visual concepts is hard to show in a paper, so we have created a webpage http://ccvl.jhu.edu/cluster.html where readers can see this for themselves by looking at many examples. To illustrate visual tightness we show the average of the images and edge maps of the best 500 patches for a few visual concepts, in figure (8), and compare them to the averages for single filters. These averages show that the clusters are tight. 16 (a) pool4 visual concept (b) pool4 single filter Figure 8: This figure shows four visual concept examples (left four) and four single filter examples (right four). The top row contains example image patches, and the bottom two rows contain average edge maps and intensity maps respectively obtained using top 500 patches.Observe that the means of the visual concepts are sharper than the means of the single filters for both average edge maps and intensity maps, showing visual concepts capture patterns more tightly than the single filters. 4.5. Evaluating visual concepts for detecting keypoints and semantic parts We now evaluate the visual concepts as detectors of keypoints and semantic parts (these are used for evaluation only and not for training). Intuitively, a visual concept VCv is a good detector for a semantic part SPs if the visual concept “fires” near most occurrences of the semantic part, but rarely fires in regions of the image where the semantic part is not present. More formally, a visual concept VCv fires at a position p ∈ L4 provided \|fp − fv \| < T , where T is a threshold (which will be varied to get a precision-recall curve). Recall that for each semantic part s ∈ S we have a set of points Ts+ where the semantic part occurs, and a set Ts− where it does not. A visual concept v has a true positive detection for q ∈ Ts+ if minp∈N (q) \|fp −fv \| < T . Similarly it has a false positive if the same condition holds for s ∈ Ts− . As the threshold T varies we obtain a precision-recall curve and report the average precision. We also consider variants of this approach by seeing if a visual concept is able to detect several semantic parts. For example, a visual concept which fires on semantic part “side-window” is likely to also respond to the semantic parts “front-window” and “back-window”, and hence should be evaluated by its ability to detect all three types of windows. Note that if a visual concept is activated for several semantic parts then it will tend to have fairly bad average precision for each semantic part by itself (because the detection for the other semantic parts will count as false positives). 17 We show results for two versions of visual concept detectors, with and without merging indicated by S-VCM and S-VCK respectively. We also implemented two baseline methods for comparison methods. This first baseline uses the magnitude of a single filter response, and is denoted by S-F. The second baseline is strongly supervised and trains a support vector machines (SVM) taking the feature vectors at the pool-4 layer as inputs, notated by SVM-F. The results we report are based on obtaining visual concepts by K-means clustering (with or without merging), but we obtain similar results if we use the von Mises-Fisher online learning approach, 4.5.1. Strategies for Evaluating Visual Concepts. There are two possible strategies for evaluating visual concepts for detection. The first strategy is to crop the semantic parts to 100×100 image patches, {P1 , P2 , · · · , Pn }, and provide some negative patches {Pn+1 , Pn+2 , · · · , Pn+m } (where the semantic parts are not present). Then we can calculate the response of V Cv by calculating Resi = \|\|fi − fv \|\|. By sorting those m + n candidates by the VC responses, we can get a list Pk1 , Pk2 , · · · , Pkm+n in which k1 , k2 , · · · , km+n is a permutation of 1, 2, · · · , m + n. By varying the threshold from Resk1 to Reskm+n , we can calculate the precision and recall curve. The average precision can be obtained by averaging the precisions at different recalls. This first evaluation strategy is commonly used for detection tasks in computer vision, for example to evaluate edge detection, face detection, or more generally object detection. But it is not suitable for our purpose because different semantic parts of object can be visually similar (e.g., the front, side, and back windows of cars). Hence the false positives of a semantic part detector will often be other parts of the same object, or a closely related object (e.g., car and bus semantic parts can be easily confused). In practice, we will want to detect semantic parts of an object when a large region of the object is present. The first evaluation strategy does not take this into account, unless it is modified so that the set of negative patches are carefully balanced to include large numbers of semantic parts from the same object class. Hence we use a second evaluation strategy, which is to crop the object bounding boxes from a set of images belonging to the same object class to make sure they only have limited backgrounds, and then select image patches which densely sample these bounding boxes. This ensures that the negative image patches are strongly biased towards being other semantic parts of the object (and corresponds to the typical situation where we will want to detect semantic parts in practice). It will, however, also introduce image patches which partly overlap with the semantic parts and hence a visual concept 18 will typically respond multiple times to the same semantic part, which we address by using non-maximum suppression. More precisely, at each position a visual concept VCv will have a response \|\|fp − fv \|\|. Next we apply nonmaximum suppression to eliminate overlapping detection results, i.e. if two activated visual concepts have sufficiently similar responses, then the visual concept with weaker response (i.e. larger \|\|fp − fv \|\|) will be suppressed by the stronger one. Then we proceed as for the first strategy, i.e. threshold the responses, calculate the false positives and false negatives, and obtain the precision recall curve by varying the threshold. (A technical issue for the second strategy is that some semantic parts may be close together, so after non-maximum suppression, some semantic parts may be missed since the activated visual concepts could be suppressed by nearby stronger visual concept responses. Then a low AP will be obtained, even though this visual concept might be very for detecting that semantic part). 4.5.2. Evaluating single Visual Concepts for Keypoint detection. To evaluate how well visual concepts detect keypoints we tested them for vehicles in the PASCAL3D+ dataset and compared them to the two baseline methods, see table (1). Visual concepts and single filters are unsupervised so we evaluate them for each keypoint, by by selecting the visual concept, or single filter, which have best detection performance, as measured by average precision. Not surprisingly, the visual concepts outperformed the single filters but were less successful than the supervised SVM approach. This is not surprising, since visual concepts and single filters are unsupervised. The results are fairly promising for visual concepts since for almost all semantic parts we either found a visual concept that was fairly successful at detecting them. We note that typically several visual concepts were good at detecting each keypoint, recall that there are roughly 200 visual concepts but only approximately 10 keypoints. This suggests that we would get better detection results by combining visual concepts, as we will do later in this paper. 4.5.3. Evaluating single Visual Concepts for Semantic Part detection. The richer annotations on VehicleSemanticPart allows us to get better understanding of visual concepts. We use the same evaluation strategy as for keypoints but report results here only for visual concepts with merging S-VC and for single filters S-F. Our longer report [40] gives results for different variants of visual concepts (e.g., without merging, with different values of K, etc.) but there is no significant difference. Our main findings are: (i) that visual concepts do significantly better than single filters, and (ii) for every semantic part there is a visual concept that detects it reasonably 19 Car Bicycle Motorbike 1 2 3 4 5 6 7 mAP 1 2 3 4 5 mAP 1 2 3 4 mAP S-F .86 .44 .30 .38 .19 .33 .13 .38 .23 .67 .25 .43 .43 .40 .26 .60 .30 .22 .35 S-VCM .92 .51 .27 .41 .36 .46 .18 .45 .32 .78 .30 .55 .57 .50 .35 .75 .43 .25 .45 S-VCK .94 .51 .32 .53 .36 .48 .22 .48 .35 .80 .34 .53 .63 .53 .40 .76 .44 .43 .51 SVM-F .97 .65 .37 .76 .45 .57 .30 .58 .37 .80 .34 .71 .64 .57 .37 .77 .50 .60 .56 Bus Train Aeroplane 1 2 3 4 5 6 mAP 1 2 3 4 5 mAP 1 2 3 4 5 mAP S-F .45 .42 .23 .38 .80 .22 .42 .39 .33 .24 .16 .15 .25 .41 .25 .22 .13 .31 .26 S-VCM .41 .59 .26 .29 .86 .51 .49 .41 .30 .30 .28 .24 .30 .21 .47 .31 .16 .34 .30 S-VCK .41 .51 .26 .33 .86 .52 .48 .42 .32 .30 .28 .25 .32 .31 .47 .31 .20 .35 .33 SVM-F .74 .70 .52 .63 .90 .61 .68 .71 .49 .50 .36 .39 .49 .72 .60 .50 .32 .49 .53 Table 1: AP values for keypoint detection on PASCAL3D+ dataset for six object categories. Visual concepts, with S-VCm or without S-VCK merging, achieves much higher results than the single filter baseline S-F. The SVM method SVM-F does significantly better, but this is not surprising since it is supervised. The keypoint number-name mapping is provided below. Cars – 1: wheel 2: wind shield 3: rear window 4: headlight 5: rear light 6: front 7: side; Bicycle –1: head center 2: wheel 3: handle 4: pedal 5: seat; Motorbike – 1: head center 2: wheel 3: handle 4: seat; Bus – 1: front upper corner 2: front lower corner 3: rear upper corner 4: rear lower corner 5: wheel 6: front center; Train – 1: front upper corner 2: front lower corner 3: front center 4: upper side 5: lower side; Aeroplane – 1: nose 2: upper rudder 3: lower rudder 4: tail 5: elevator and wing tip. well. These results, see Table (2), support our claim that the visual concepts give a dense coverage of each object (since the semantic parts label almost every part of the object). Later in this paper, see table (3) (known scale), we compare the performance of SVMs to visual concepts for detecting semantic parts. This comparison uses a tougher evaluation criterion, based on the interSection over union (IOU) [8], but the relative performance of the two methods is roughly the same as for the keypoints. 4.5.4. What do the other visual concepts do?. The previous experiments have followed the “best single visual concept evaluation”. In other words, for each semantic part (or keypoint), we find the single visual concept which best detects that semantic part. But this ignores three issues: (I) A visual concept may respond to more than one semantic parts (which means that its AP for one semantic part is small because the others are treated as false positives). (II) There are far more visual concepts than semantic parts, so our previous evaluations have not told us what all the visual concepts are doing (e.g., for cars there are roughly 200 visual concepts but only 39 semantic parts, so we are only reporting results for twenty percent of the visual concepts). (III) Several visual concepts may be good for detecting the 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 S-F .86 .87 .84 .68 .70 .83 .83 .82 .72 .18 .22 .49 .33 .25 .22 .18 .37 .18 S-VCM .94 .97 .94 .93 .94 .95 .94 .92 .94 .42 .48 .58 .45 .48 .46 .54 .60 .34 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 mAP S-F .28 .13 .16 .14 .30 .20 .16 .17 .15 .27 .19 .25 .18 .19 .16 .07 .07 .36 S-VCM .38 .19 .26 .37 .42 .32 .26 .40 .29 .40 .18 .41 .53 .31 .57 .36 .26 .53 (a) Car 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mAP S-F .32 .42 .08 .61 .58 .42 .49 .31 .23 .09 .07 .20 .12 .29 .29 .07 .09 .28 S-VCM .26 .21 .07 .84 .61 .44 .63 .42 .34 .15 .11 .44 .23 .59 .65 .08 .15 .37 (b) Aeroplane S-F S-VCM 1 2 3 4 5 6 7 8 9 10 11 12 13 mAP .77 .84 .89 .91 .94 .92 .94 .91 .91 .56 .53 .15 .40 .75 .91 .95 .98 .96 .96 .96 .97 .96 .97 .73 .69 .19 .50 .83 (c) Bicycle 1 2 3 4 5 6 7 8 9 10 11 12 mAP S-F .69 .46 .76 .67 .66 .57 .54 .70 .68 .25 .17 .22 .53 S-VCM .89 .64 .89 .77 .82 .63 .73 .75 .88 .39 .33 .29 .67 (d) Motorbike 1 2 3 4 5 6 7 8 9 mAP S-F .90 .40 .49 .46 .31 .28 .36 .38 .31 .43 S-VCM .93 .64 .69 .59 .42 .48 .39 .32 .27 .53 (e) Bus 1 2 3 4 5 6 7 8 9 10 11 12 13 14 mAP S-F .58 .07 .20 .15 .21 .15 .27 .43 .17 .27 .16 .25 .17 .10 .23 S-VCM .66 .50 .32 .28 .24 .15 .33 .72 .36 .41 .27 .45 .27 .47 .39 (f) Train Table 2: The AP values for semantic part detection on VehicleSemanticPart dataset for cars (top), aeroplanes (middle) and bicycle (bottom). We see that visual concepts S-VCm achieves much higher AP than single filter method S-F. The semantic part names are provided in the supplementary material. same semantic part, so combining them may lead to better semantic part detection. In this section we address issues (I) and (II), which are closely related, and leave the issue of combining visual concepts to the next section. We find that some visual concepts do respond better to more than one semantic part, in the sense that their AP is higher when we evaluate them for detecting 21 a small subset of semantic parts. More generally, we find that almost all the visual concepts respond to a small number of semantic parts (one, two, three, or four) while most of the limited remaining visual concepts appear to respond to frequently occuring “backgrounds”, such as the sky in the airplane images. We proceed as follows. Firstly, for each visual concept we determine which semantic part it best detects, calculate the AP, and plot the histogram as shown in Figure 9 (SingleSP). Secondly, we allow each visual concept to select a small subset – two, three, or four – of semantic parts that it can detect (penalizing it if it fails to detect all of them). We measure how well the visual concept detects this subset using AP and plot the same histogram as before. This generally shows much better performance than before. From Figure 9 (MultipleSP) we see that the histogram of APs gets shifted greatly to the right, when taking into account the fact that one visual concept may correspond to one or more semantic parts. Figure 10 shows the percentage of how many semantic parts are favored by each visual concept. If a visual concept responds well to two, or more, semantic parts, this is often because those parts are visually similar, see Figure 11. For example, the semantic parts for car windows are visually fairly similar. For some object classes, particularly aeroplanes and trains, there remain some visual concepts with low APs even after allowing multiple semantic parts. Our analysis shows that many of these remaining visual concepts are detecting background (e.g., the sky for the aeroplane class, and railway tracks or coaches for the train class), see Figure 12. A few other visual concepts have no obvious interpretation and are probably due to limitations of the clustering algorithm and the CNN features. Our results also imply that there are several visual concepts for each semantic part. In other work, in preparation, we show that these visual concepts typically correspond to different subregions of the semantic parts (with some overlap) and that combining them yields better detectors for all semantic parts with a mean gain of 0.25 AP. 4.6. Summary of Visual Concepts This section showed that visual concepts were represented as internal representations of deep networks and could be found either by K-means or by an alternative method where the visual concepts could be treated as hidden variables attached to the deep network. The visual concepts were visually tight, in the sense that image patches assigned to the same visual concept looked similar. Almost all the visual concepts corresponded to semantic parts 22 1 0.7 0.7 0.7 0.6 0.6 0.6 0.4 Percentage 0.8 0.5 0.5 0.4 0.5 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0~0.2 0 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin (a) Car 0.1 0~0.2 0 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin SingleSP MultipleSP (c) Bus SingleSP MultipleSP 0.9 0.7 0.7 0.6 0.6 0.6 Percentage 0.7 Percentage 0.8 0.4 0.5 0.4 0.5 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0 0~0.2 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin SingleSP MultipleSP 0.9 0.8 0.5 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin 1 1 0.8 0 0~0.2 (b) Aeroplane 1 0.9 SingleSP MultipleSP 0.9 0.8 Percentage Percentage SingleSP MultipleSP 0.9 0.8 0 Percentage 1 1 SingleSP MultipleSP 0.9 0.1 0~0.2 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin (e) Bicycle (d) train 0 0~0.2 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin (f) Motorbike Figure 9: The histograms of the AP responses for all visual concepts. For “SingleSP”, we evaluate each visual concept by reporting its AP for its best semantic part (the one it best detects). Some visual concepts have very low APs when evaluated in this manner. For “MultipleSP”, we allow each visual concept to detect a small subset of semantic parts (two, three, or four) and report the AP for the best subset (note, the visual concept is penalized if it does not detect all of them). The APs rise considerably using the MultipleSP evaluation, suggesting that some of the visual concepts detect a subset of semantic parts. The remaining visual concepts with very low APs correspond to the background. 0.6 Car Aeroplane Bus Train Bicycle Motorbike 0.5 Percentage 0.4 0.3 0.2 0.1 0 1 2 3 4 Number of Favored Semantic Parts for Each Visual Concept Figure 10: Distribution of number of semantic parts favored by each visual concept for 6 objects. Most visual concepts like one, two, or three semantic parts. of the objects (one, tow, three, or four), several corresponded to background regions (e.f., sky), but a few visual concepts could not be interpreted. More results and analysis can be found in our longer report [40] and the webpage http://ccvl.jhu.edu/cluster.html. In particular, we observed that visual concepts significantly outper- 23 Figure 11: This figure shows two examples of visual concepts that correspond to multiple semantic parts. The first six columns show example patches from that visual concept, and the last two columns show prototypes from two different corresponding semantic parts. We see that the visual concept in the first row corresponds to “side window” and “front window”, and the visual concept in the second row corresponds to “side body and ground” and “front bumper and ground”. Note that these semantic parts look fairly similar. Figure 12: The first row shows a visual concept corresponding to sky from the aeroplane class, and the second and third rows are visual concepts corresponding to railway track and tree respectively from the train class. formed single filters for detecting semantic parts, but performed worse than supervised methods, such as support vector machines (SVMs), which took the same pool-4 layer features as input. This worse performance was due to several factors. Firstly, supervised approaches have more information so they generally tend to do better on discriminative tasks. Secondly, several visual concepts tended to fire on the same semantic part, though in different locations, suggesting that the visual concepts correspond to subparts of the semantic part. Thirdly, some visual concepts tended to fire on several semantic parts, or in neighboring regions. These findings suggest that visual concepts are suitable as building blocks for pattern theoretic models. We will evaluate this in the next section where we construct a simple compositional-voting model which combines several visual concepts to detect semantic parts. 24 5. Combining Visual Concepts for Detecting Semantic Parts with Occlusion The previous section suggests that visual concepts roughly correspond to object parts/subparts and hence could be used as building blocks for compositional models. To investigate this further we study the precise spatial relationships between the visual concepts and the semantic parts. Previously we only studied whether the visual concepts were active within a fixed size circular window centered on the semantic parts. Our more detailed studies show that visual concepts tend to fire in much more localized positions relative to the centers of the semantic parts. This study enables us to develop a compositional voting algorithm where visual concepts combine together to detect semantic parts. Intuitively the visual concepts provide evidence for subparts of the semantic parts, taking into account their spatial relationships, and we can detect the semantic parts by combining this evidence. The evidence comes both from visual concepts which overlap with the semantic parts, but also from visual concepts which lie outside the semantic part but provide contextual evidence for it. For example, a visual concept which responds to a wheel gives contextual evidence for the semantic part “front window” even though the window is some distance away from the wheel. We must be careful, however, about allowing contextual evidence because this evidence can be unreliable if there is occlusion. For example, if we fail to detect the wheel of the car (i.e. the evidence for the wheel is very low) then this should not prevent us from detecting a car window because the car wheel might be occluded. But conversely, if can detect the wheel then this should give contextual evidence which helps detect the window. A big advantage of our compositional voting approach is that it gives a flexible and adaptive way to deal with context which is particularly helpful when there is occlusion, see figure (13). If the evidence for a visual concept is below a threshold, then the algorithm automatically switches off the visual concept so that it does not supply strong negative evidence. Intuitively “absence of evidence is not evidence of absence”. This allows a flexible and adaptive way to include the evidence from visual concepts if it is supportive, but to automatically switch it off if it is not. We note that humans have this ability to adapt flexibly to novel stimuli, but computer vision and machine learning system rarely do since the current paradigm is to train and test algorithms on data which are presume to be drawn from the same underlying distribution. Our ability to use information flexibly means that our system is in some cases able to detect a semantic part from context even if the part 25 Figure 13: The goal is to detect semantic parts of the object despite occlusion, but without being trained on the occlusion. Center panel: yellow indicates unoccluded part, blue means partially occluded part, while red means fully occluded part. Right panel: Visual concepts (the circles) give evidence for semantic parts. They can be switched off automatically in presence of occlusion (the red circles) so that only the visual concepts that fire (green circles) give positive evidence for the presence of the parts. This robustness to occlusion if characteristic of our compositional model and enables it to outperform fully supervised deep networks. itself is completely occluded and, by analyzing which visual concepts have been switched off, to determine if the semantic part itself is occluded. In later work, this switching off can be extended to deal with the “explaining away” phenomena, highlighted by work in cognitive science [15]. 5.1. Compositional Voting: detecting Semantic Parts with and without Occlusion Our compositional model for detecting semantic parts is illustrated in figure (14). We develop it in the following stages. Firstly, we study the spatial relations between the visual concepts and the semantic parts in greater detail. In the previous section, we only use spatial relations (e.g., is the visual concept activated within a window centered on the semantic part, where the window size and shape was independent of the visual concept or the semantic part). Secondly, we compute the probability distributions of the visual concept responses depending on whether the semantic part is present or nor. This enables us to use the log-likelihood ratio of these two distributions as evidence for the presence of the semantic part. Thirdly, we describe how the evidence from the visual concepts can be combined to vote for a semantic part, taking into account the spatial positions of the parts and allowing visual concepts to switch off their evidence if it falls below threshold (to deal with occlusion). 26 Annotated SP w/ the Best Neighboring VC 1 3 summarization 2 1 42 3 6 7 5 8 Collecting +/- VC Cues for voting SP Prob. Distributions % 𝑇",$ + − 4 quantization 6 + 5 7 loglikelihood % 𝑇",$ − 8 Neighborhood Score Function Figure 14: Illustration of the training phase of one (VCv , SPs ) pair. Only one negative point is shown in each image (marked in a yellow frame in the third column), though the negative set is several times larger than the positive set, see [42] for details. 5.1.1. Modeling the Spatial Relationship between Visual Concepts and Semantic Parts. We now examine the relationship between visual concepts and semantic parts by studying the spatial relationship of each (VCv , SPs ) pair in more detail. Previously we rewarded a visual concept if it responded in a neighborhood N of a semantic part, but this neighborhood was fixed and did not depend on the visual concept or the semantic part. But our observations showed that if a visual concept VCv provides good evidence to locate a semantic part SPs , then the SPs may only appear at a restricted set of positions relative to VCv . For example, if VCv represents the upper part of a wheel, we expect the semantic part (wheel) to appear slightly below the position of VCv . Motivated by this, for each (VCv , SPs ) pair we learn a neighborhood function Nv,s indicating the most likely spatial relationship between VCv and SPs . This is illustrated in figure (14). To learn the neighborhood we measured the spatial statistics for each (VCv , SPs ) pair. For each annotated ground-truth position q ∈ Ts+ we computed the position p? in its L4 neighborhood which best activates VCv , i.e. p? = arg minp∈N(q) kf (Ip ) − fv k, yielding a spatial offset ∆p = p? − q. We normalize these spatial offsets to yield a spatial frequency map Frv,s (∆p) ∈ [0, 1]. The neighborhood Nv,s is obtained by thresholding the spatial frequency map. For more details see [42]. 27 5.1.2. Probability Distributions, Supporting Visual Concepts and Log-likelihood Scores. We use loglikelihood ratio tests to quantify the evidence a visual concept VCv ∈ V can give for a semantic part SPs ∈ S. For each SPs , we use the positive and negative samples Ts+ and Ts− to compute the conditional distributions: d + + Pr min kf (Ip ) − fv k 6 r \| q ∈ Ts , (5) Fv,s (r) = dr p∈Nv,s(q) (6) F− v,s (r) d − = Pr min kf (Ip ) − fv k 6 r \| q ∈ Ts . dr p∈Nv,s(q) We call F+ v,s (r) the target distribution, which specifies the probably activity pattern for VCv if there is a semantic part SPs nearby. If (VCv , SPs ) is a good pair, then the probability F+ v,s (r) will be peaked close to r = 0, (i.e., there will be some feature vectors within Nv,s (q) that cause VCv to activate). The second distribution, F− v,s (r) is the reference distribution which specifies the response of the feature vector if the semantic part is not present. The evidence is provided by the log-likelihood ratio [1]: (7) Λv,s (r) = log F+ v,s (r) + ε F− v,s (r) + ε , where ε is a small constant chosen to prevent numerical instability. In practice, we use a simpler method to first decide which visual concepts are best suited to detect each semantic part. We prefer (VCv , SPs ) pairs for which F+ v,s (r) is peaked at small r. This corresponds to visual concepts which are fairly good detectors for the semantic part and, in particular, those which have high recall (few false negatives). This can be found from our studies of the visual concepts described in section (4). For more details see [42]. For each semantic part we select K visual concepts to vote for it. We report results for K = 45, but good results can be found with fewer. For K = 20 the performance drops by only 3%, and for K = 10 it drops by 7.8%. 5.2. Combining the Evidence by Voting The voting process starts by extracting CNN features on the pool-4 layer, i.e. {fp }) for p ∈ L4 . We compute the log-likelihood evidence Λv,s for each (VCv , SPs ) at each position p. We threshold this evidence and only keep those positions which are unlikely to be false negatives. Then the vote that 28 a visual concepts gives for a semantic part is obtained by a weighted sum of the loglikelihood ratio (the evidence) and the spatial frequency. The final score which is added to the position p + ∆p is computed as: (8) Votev,s (p) = (1 − β) Λv,s (Ip ) + β log Fr(∆p) . U The first term ensures that there is high evidence of VCv firing, and the second term acts as the spatial penalty ensuring that this VCv fires in the right relative position Here we set β = 0.7, and define log Fr(∆p) = −∞ when U Fr(∆p) = 0, and U is a constant. U In order to allow the voting to be robust to occlusion we need this ability to automatically “switch off” some votes if they are inconsistent with other votes. Suppose a semantic part is partially occluded. Some of its visual concepts may still be visible (i.e. unoccluded) and so they will vote correctly for the semantic part. But some of the visual concepts will be responding to the occluders and so their votes will be contaminated. So we switch off votes which fall below a threshold (chosen to be zero) so that failure of a visual concept to give evidence for a semantic part does not prevent the semantic part from being detected – i.e. absence of evidence is not evidence of absence. Hence a visual concept is allowed to support the detection of a semantic part, but not allowed to inhibit it. The final score for detecting a semantic part SPs is the sum of the thresholded votes of its visual concepts: (9) Scores (p) = X max {0, Votev,s (p)}. VCv ∈Vs In practical situations, we will not know the size of the semantic parts in the image and will have to search over different scales. For details of this see [42]. Below we report results if the scale is known or unknown. 5.3. Experiments for Semantic Part Detection with and without Occlusion We evaluate our compositional voting model using the VehicleSemanticParts and the VehicleOcclusion datasets. For these experiments we use a tougher, and more commonly used, evaluation criterion [8], where a detected semantic part is a true-positive if itsIntersection-over-Union (IoU) ratio between the groundtruth region (i.e. the 100 × 100 box centered on the semantic part) 29 Unknown Scale Object S-VCM SVM-VC FR airplane 10.1 18.2 45.3 bicycle 48.0 58.1 75.9 bus 6.8 26.0 58.9 car 18.4 27.4 66.4 motorbike 10.0 18.6 45.6 train 1.7 7.2 40.7 mean 15.8 25.9 55.5 Known Scale VT S-VCM SVM-VC 30.6 18.5 25.9 77.8 61.8 73.8 58.1 27.8 39.6 63.4 28.1 37.9 53.4 34.0 43.8 35.5 13.6 21.1 53.1 30.6 40.4 VT 41.1 81.6 60.3 65.8 58.7 51.4 59.8 Table 3: Detection accuracy (mean AP, %) and scale prediction loss without occlusion. Performance for the deep network FR is not altered by knowing the scale. and the predicted region is greater or equal to 0.5, and duplicate detection is counted as false-positive. In all experiments our algorithm, denoted as VT, is compared to three other approaches. The first is single visual concept detection S-VCm (described in the previous section). The others two are baselines: (I) FasterRCNN, denoted by FR, which trains a Faster-RCNN [31] deep network for each of the six vehicles, where each semantic part of that vehicle is considered to be an “object category”. (II) the Support Vector Machine, denoted by SVM-VC, used in the previous section. Note that the Faster-RCNN is much more complex than the alternatives, since it trains an entire deep network, while the others only use the visual concepts plus limited additional training. We first assume that the target object is not occluded by any irrelevant objects. Results of our algorithm and its competitors are summarized in Table 3. Our voting algorithm achieves comparable detection accuracy to Faster-RCNN, the state-of-the-art object detector, and outperforms it if the scale is known. We observe that the best single visual concept S-VCM performs worse than the support vector machine SVM-VC (consistent with our results on keypoints), but our voting method VT performs much better than the support vector machine. Next, we investigate the case that the target object is partially occluded using the VehicleOcclusion dataset. The results are shown in Table 4. Observe that accuracy drops significantly as the amount of occlusion increases, but our compositional voting method VT outperforms the deep network FR for all levels of occlusion. Finally we illustrate, in figure (15), that our voting method is able to detect semantic parts using context, even when the semantic part itself is 30 2 Occluders 0.2 6 r < 0.4 Object FR VT airplane 26.3 23.2 bicycle 63.8 71.7 bus 36.0 31.3 car 32.9 35.9 motorbike 33.1 44.1 train 17.9 21.7 mean 35.0 38.0 3 Occluders 0.4 6 r < 0.6 FR VT 20.2 19.3 53.8 66.3 27.5 19.3 19.2 23.6 26.5 34.7 10.0 8.4 26.2 28.6 4 Occluders 0.6 6 r < 0.8 FR VT 15.2 15.1 37.4 54.3 18.2 9.5 11.9 13.8 17.8 24.1 7.7 3.7 18.0 20.1 Table 4: Detection accuracy (mean AP, %) when the object is partially occluded and the scale is unknown. Three levels of occlusion are considered. occluded. This is a benefit of our algorithm’s ability to use evidence flexibly and adaptively. 6. Discussion: Visual Concepts and Sparse Encoding Our compositional voting model shows that we can use visual concepts to develop a pattern theoretic models which can outperform deep networks when tested on occluded images. But this is only the starting point. We now briefly describes ongoing work where visual concepts can be used to provide a sparse encoding of objects. The basic idea is to encode an object in terms of those visual concepts which have significant response at each pixel. This gives a much more efficient representation than using the high-dimensional feature vectors (typically 256 or 512). To obtain this encoding we threshold the distance between the feature vectors (at each pixel) and the centers of the visual concepts. As shown in figure (16) we found that a threshold of Tm = 0.7 ensured that most pixels of the objects were encoded on average by a single visual concept. This means that we can approximately encode a feature vector f by specifying the visual concept that is closest to it, i.e. by v̂ = arg minv \|f − fv \| – provided this distance is less than a threshold Tm . More precisely, we can encode a pixel p with feature vectors fp by a binary feature vector bp = (bp1 , ..., bpv , ...) where v indexes the visual concepts and bpv = 1 if \|fp − fv \| < Tm and bpv = 0 otherwise. In our neural network implementation, the bpv correspond to the activity of the vc-neurons (if soft-max is replaced by hard-max). 31 Object: car Actual Case: LOW Occlusion Object: car Actual Case: LOW Occlusion Object: car Actual Case: LOW Occlusion SP: wheel Prediction: LOW Occlusion SP: wheel Prediction: LOW Occlusion SP: wheel Prediction: MOD Occlusion 33 Supporting Visual Concepts 35 Supporting Visual Concepts 22 Supporting Visual Concepts Average Distance: 14.5703 (Low) Average Distance: 19.2344 (Low) Average Distance: 44.4923 (Moderate) Object: car Actual Case: HIGH Occlusion Object: car SP: h-light Prediction: HIGH Occlusion SP: h-light Actual Case: HIGH Occlusion Object: car Prediction: HIGH Occlusion SP: h-light Actual Case: HIGH Occlusion Prediction: LOW Occlusion 25 Supporting Visual Concepts 22 Supporting Visual Concepts 30 Supporting Visual Concepts Average Distance: 73.6239 (High) Average Distance: 75.9826 (High) Average Distance: 22.1090 (Low) Figure 15: This figure shows detection from context: when there are occlusions, the cuesmay come from context which is far away from the part There are several advantages to being able to represent spatial patterns of activity in terms of sparse encoding by visual concepts. Instead of representing patterns as signals f in the high-dimensional feature space we can give them a more concise symbolic encoding in terms of visual concepts, or by thresholding the activity of the vc-neurons. More practically, this leads to a natural similarity measure between patterns, so that two patterns with similar encodings are treated as being the same underlying pattern. This encoding perspective follows the spirit of Barlow’s sparse coding ideas [3] and is also helpful for developing pattern theory models, which are pursuing in current research. it makes it more practical to learning generative models of the spatial structure of image patterns of an object or a semantic part Ss . This is because generating the binary vectors {b} is much easier than generating high-dimensional feature vectors {f }. ItQ can be down, for example, by a simply factorized model P ({bp }\|SPs ) = p P (fp \|SPs ), Q where P (fp \|SPs ) = v P (bv,p \|SPs ), and bv,p = 1 if the v th vc-neuron (or visual concept) is above threshold at position p. It also yields simple distance measures between different image patches which enables one-shot/few-shot 32 Figure 16: This figure shows there is a critical threshold Tm , at roughly 0.7, where most pixels are encoded (i.e. activate) a small number of visual concepts. Below this threshold most pixels have no visual concepts activated. By contrast, above the threshold three of more visual concepts are activated. These results are shown for level four features on the cars (left panel), bikes (center panel) and trains (right panel) in VehicleSemanticPart, and we obtain almost identical plots for the other objects in VehicleSemanticPart. learning, and also to cluster image patches into different classes, such as into different objects and different viewpoints. 7. Conclusion This paper is a first attempt to develop novel visual architectures, inspired by pattern theory, which are more flexible and adaptive than deep networks but which build on their successes. We proceeded by studying the activation patterns of deep networks and extracting visual concepts, which both throws light on the internal representations of deep networks but which can also be used as building blocks for compositional models. We also argued that the complexity of vision, particularly due to occlusions, meant that vision algorithms should be tested on more challenging datasets, such as those where occlusion is randomly added, in order to ensure true generalization. Our studies showed that compositional models outperformed deep networks when both were trained on unoccluded datasets but tested when occlusion was present. 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A para-model agent for dynamical systems arXiv:1202.4707v6 [math.OC] 11 Mar 2018 * 10 years of model-free control methodology * Loı̈c MICHEL Abstract Consider a dynamical system u 7→ x, ẋ = fnl (x, u) where fnl is a nonlinear (convex or nonconvex) function, or a combination of nonlinear functions that can eventually switch. We present, in this preliminary work1 , a generalization of the standard model-free control, that can either control the dynamical system, given an output reference trajectory, or optimize the dynamical system as a derivative-free optimization based ”extremumseeking” procedure. Multiple applications2 are presented and the robustness of the proposed method is studied in simulation. 1 This work is distributed under CC license http://creativecommons.org/licenses/ by-nc-sa/4.0/. Email of the corresponding author : loic.michel54@gmail.com 2 The control of the Epstein frame (described in §3.4) has been experimentally validated and the results have been presented at the French Symposium of Electrical Engineering in Grenoble, Jun. 2016 http://sge2016.sciencesconf.org/. Contents 1 Introduction 1 2 General Principle 2.1 Definition of the closed-loop . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definition of the PMA algorithm . . . . . . . . . . . . . . . . . . . . 2.3 Performances in simulation . . . . . . . . . . . . . . . . . . . . . . . . 1 2 2 4 3 Applications of the Cπ -control 3.1 Motor control in the dq frame . . 3.2 Control of switched non-minimum 3.3 Ballistic and the fire control . . . 3.4 Control of the HIV-1 model . . . 3.5 Control of the Epstein frame . . . . . . . . 6 6 9 16 25 28 4 Derivative-free & ”extremum-seeking” control 4.1 Proposed Cπ -control scheme . . . . . . . . . . . . . . . . . . . . . . . 4.2 Numerical applications . . . . . . . . . . . . . . . . . . . . . . . . . . 35 35 36 5 Concluding remarks 37 2 . . . . . . . . and minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . phase systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction Based on the model-free control methodology, originally proposed by Fliess & Join [1] [2] [3] ten years ago, which is referred to as a self-tuning controller in [4] and which has been widely and successfully applied to many mechanical and electrical processes, the para-model agent (PMA) aims to generalize the model-free control by not only controlling nonlinear system, but also performing an ”extremum-seeking” control. On the one hand, we studied the dynamic performances when controlling generic switched minimum phase, non-minimum phase systems (e.g. [5] [6] [7] [8] [9] [10] [11]) as well as the control of some nonlinear systems taken from applications in physics. On the other hand, we present how the PMA can be used to find the optimum of some classes of nonlinear functions. The proposed para-model agent3 is a simple derivation of the discrete model-free control law [12]. The last progresses result in two contributions: first, the substitution of the computation of the numerical derivatives in the original model-free control approach [3], by an initialization function that makes the controller more robust when stabilizing for example switched processes4 . Then, we propose to extend the properties of the model-free controller to include the extremum seeking control of nonlinear systems without any computation of derivative or gradient. In this case, instead of tracking a working point of nonlinear systems, an appropriate choice of the output reference may stabilize nonlinear systems to their extremum. The paper is structured as follows. Section 2 presents the general principle of the proposed para-model agent. In Section 3, some examples illustrate the control of generic switched linear systems, the control of a three-phase motor, the control of a ballistic fire, the control of a biological system and the control of the measure of magnetic hysteresis in the framework of magnetic materials characterization (this last application is also referred to as the control of nonlinear switched systems). Section 4 presents how the proposed PMA approach can be used as derivative-free optimization / ”extremum-seeking” control. 2 General Principle We consider a nonlinear SISO dynamical system to control: ẋ = fnl (x, u) u 7→ y, y = Cx (1) where fnl is a nonlinear system, the para-model agent is an application (y ∗ , y) 7→ u whose purpose is to control the output y of (1) following an output reference y ∗ . 3 A justification of the proposed name ”para-model” is given in the note of the conclusion. The first steps toward the elaboration of the proposed algorithm were to extend the capabilities of the model-free control regarding the control of switched non-minimum phase systems [13]. 4 1 In simulation, the system (1) is controlled in its ”original formulation” without any modification / linearization. 2.1 Definition of the closed-loop Consider the control scheme depicted in Fig. 1 where Cπ is the proposed PMA controller. Figure 1: Proposed PMA scheme to control or optimize a nonlinear system. 2.2 Definition of the PMA algorithm For any discrete moment tk , k ∈ N∗ , one defines the discrete controller Cπ such that symbolically: Cπ : (y, y ∗ ) 7→ uk = Z 0 t R2 → R i Ki εk−1 d τ uk−1 + Kp (kα e−kβ k − yk−1 ) {z } \| k−1 (2) uik where: y ∗ is the output reference trajectory; Kp and KI are real positive tuning gains; εk−1 = yk∗ − yk−1 is the tracking error; uik = uik−1 + Kp (kα e−kβ k − yk−1 ) is an internal recursive term where kα e−kβ k − yk−1 is the associated exponential tracking error in which kα e−kβ k is an initialization function where kα and kβ are real constants; practically, the integral part is discretized using e.g. Riemann sums. The internal recursion5 on uik is defined such as: uik = uik−1 + Kp (kα e−kβ k − yk−1 ). The set of Cπ -parameters of the controller, that needs to be adjusted by the user, is defined as the set of coefficients {Kp , Ki , kα , kβ }. 5 We refine the definition of the PMA algorithm, for which we aim to optimize the construction; in particular, further investigations concern the study of a direct recursion on uk taking into account the Ki -integration and thus comparing internal recursion (involving uik , uik−1 ) vs external recursion (involving uk , uik−1 )... 2 Practical algorithm A possible algorithmic implementation of the simple definition (2) is given below (for all ii ∈ N∗ ): y_int(ii) = k_alphaexp(-k_betaii); % exp. init. function para_exp_err = y_int(ii-1) - y(ii-1); % exp. tracking error para_stand_err(ii) = y_ref(ii) - y(ii-1); % stand. tracking error para_u(ii) = para_u(ii-1) + Kppara_exp_err; % internal recursion para_G(ii) = Kintpara_stand_err(ii); % def. of the integral part para_tr(ii) = para_tr(ii-1) + h(para_G(ii) + para_G(ii-1))/2; % trapezoidal integration para_u_output = para_u(ii)para_tr(ii); % final product (integrator X internal recursion) where: • ii is the index of the sample in the (optional) vectorized process; • exp is the exponential function; • para exp err is the exponential tracking error; • para stand err is the (standard) output tracking error; • para u is the ”internal” recursion; • para G constitutes the discrete integrator; • para u output is the output of the controller that corresponds to the final product between the discrete integrator and the internal recursive function. and k alpha, k beta, Kp and Kint are real constants. Remark The proposed PMA algorithm could been seen as a ”deformed” integrator Rt since the internal recursion uik multiplies directly the integrator 0 Ki εk−1 d τ . 3 2.3 Performances in simulation 2.3.1 Optimization of the closed-loop Optimizing the performances in simulation means that we want to solve the problem of finding the most appropriate set of Cπ -parameters relating to the minimization of some performances index 6 , that may quantify the dynamical performances of the closed-loop. This problem is thus equivalent to an optimization problem for which any optimization solver can be a priori used. In particular, meta-heuristic solvers or derivative-free optimization solvers are preferred due to the pretty complexity of the closed-loop nonlinear form (in general). We are interested in using the ”Brute Force Optimization” (BFO) solver [14] that is very convenient and efficient to use. Figure 2 illustrates a closed-loop first order system, whose the controlled transient has been BFO-optimized in Fig. 3. Figure 2: Simulation with a set of Cπ -parameters arbitrary fixed to ensure at least asymptotic stability. 2.3.2 Sobol-based sensibility of the controlled transient To investigate the interactions between the Cπ -parameters that influence the minimization of the performances index, we propose a preliminary study of the sensibility of the Cπ -parameters using the Sobol index methodology [15]. Consider a controlled nonlinear system, for which the ISE index is evaluated under strict conditions w.r.t. 6 The classical performance index that are available are IAE, ISE, ITAE, and ITSE. see e.g. (http://www.mathworks.com/matlabcentral/fileexchange/ 18674-learning-pid-tuning-iii--performance-index-optimization/content/html/ optimalpidtuning.html) for a quick review (in the context of PID tuning). 4 Figure 3: Simulation with a set of Cπ -parameters BFO-optimized to minimize a transient performance index. the management of the Cπ -parameters7 , the Sobol-based sensibility is evaluated only during the initialization / transient of the closed-loop. Figure 4 shows that a priori the coefficient kβ does not influence the dynamical performances during the transient. Figure 4: Sobol-based analysis of the ISE index during the closed-loop initialization. 7 case 1 : the value of the evaluated index is bounded to 100 and a tolerance of ±10% is permitted on the Cπ -parameters; case 2 : the value of the evaluated index is not bounded and a tolerance of ±50% is permitted on the Cπ -parameters. The results are very similar between the two cases and the Sobol index for {Kp , Ki , kα , kβ } are respectively {0.3324, 0.3318, 0.3326, 0.0002}. 5 3 3.1 Applications of the Cπ -control Motor control in the dq frame Consider a three-phase motor driven in the dq frame; the motor is supplied by a e voltage source rotating at an ω angular frequency and modeled by a simple RC circuit with an additional voltage source ed that acts as an (internal) disturbance. Figure 5 depicts the proposed model (a single phase is represented) where P (θ) and P i (θ) are respectively the Park and the inverse Park transform. The purpose is to control simultaneously the d and q axis with an a priori unknown disturbance ed considering also that the angular frequency ω is increasing according to the time. Figure 5: Model of the motor in the dq frame including an explicit disturbance ed . The disturbance ed is of the general form:   ed1 = A1 (t) sin(ωd (t)t) ed2 = A2 (t) sin ωd (t)t − ed :=  ed3 = A3 (t) sin ωd (t)t + 2π 3 2π 3 (3) where the amplitude Ai and the angular frequency ωd of each phase i could depend on the time. The control structure is composed of two Cπ controllers: the d axis is ”maintained” close to zero (d∗ denotes the output ref. and dmes , the controlled output) and the q axis tracks a specific reference (q ∗ denotes the output ref. and qmes , the controlled output). The following figures illustrate some cases with different ”behavior” of the disturbance ed . Figure 6 presents the most simple case where A1 = A2 = A3 = Cst and ωd = Cst; in Fig. 7 is considered a time-varying disturbance where A1 = A2 = A3 are increasing according to the time; in Fig. 8, small variations of amplitudes of A1 , A2 and A3 are considered (symbolically, A1 ∼ A2 ∼ A3 ), and finally, Fig. 9 depicts the case where ωd is time-varying only over a short period of time. 6 Figure 6: Control in the dq frame with ”simple” disturbance ed . Figure 7: Control in the dq frame with an increasing amplitude of each component of ed . 7 Figure 8: Control in the dq frame with variation of amplitude of each component of ed . Figure 9: Control in the dq frame with variation of the ωd frequency of ed . 8 3.2 Control of switched non-minimum and minimum phase systems Consider the set Σ of stable linear systems such that Σ = {Σi }, i = 1, 2, · · · n, which are minimum and non-minimum phase systems, and which are considered as unknown in the sense that no explicit model has been identified for control purposes. Assume now that for all systems, there exists an integer p = {1, · · · , 8}, called the switching index, such that during a short time window, we have: ẋ(t) = Ap x(t) + Bp u(t) Σp (u 7−→ y) := (4) y = Cp x(t) where u and y are respectively the input and the output of the system Σp (p is the switching index). The step responses of these p systems are presented Fig. 10. Figure 10: Step responses of each system Σi . Figures 11 12 13 14 present some examples of the application of the Cπ -control under different arbitrary switching sequences that involve both minimum and non-minimum phase systems. The first switching time is t1 , the second is t2 and the third is t3 . Consider now the existence of a delay τ on y that modify (4) such that: ẋ(t) = Ap x(t) + Bp u(t) Σp (u 7−→ y) := (5) y = Cp x(t − τ ) This delay can e.g. simulate the propagation delay inside a sensor network. Figures 15 and 16 present two examples of the application of the Cπ -control under different switching sequences that involve both minimum and non-minimum phase systems. 9 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 11: Switched sequence #1 for t1 = 0.01 s and t2 = 0.05 s. 10 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 12: Switched sequence #2 for t1 = 0.025 s and t2 = 0.072 s. 11 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 13: Switched sequence #3 for t1 = 0.018 s, t2 = 0.035 s and t3 = 0.072 s. 12 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 14: Switched sequence #4 for t1 = 0.35 s, t2 = 0.58 s. 13 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 15: Switched sequence #5 for t1 = 0.015 s, t2 = 0.055 s. An addition of a time-delay occurs at t = 0.06 s. 14 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 16: Switched sequence #6 for t1 = 0.025 s, t2 = 0.072 s. An addition of a time-delay occurs at t = 0.06 s. 15 Discussions These results have been obtained with a single specific set of the Cπ -parameters; to improve the tracking performances, one may consider an on-line adjustment of the Cπ -parameters. Although resonances occur at the instants of switches, the stability of the control is preserved (regarding the studied cases) when switching from the different types of systems. In the same manner, a direct tuning of the Cπ -parameters could damp (ideally, could cancel) the resonant effects. The presented simulation results show that the proposed control law is robust to ”strong” model variations and in particular when the model is a switching nonminimum phase or minimum phase system that include eventually time-delay. Moreover, the proposed control law seems to have the same properties than the original model-free control [2] [3] for which its performances have been successfully verified especially in simulation. 3.3 Ballistic and the fire control If one fire a projectile at an initial angle and an initial speed, then general physics allows calculating how far it will travel... but would it be possible to control the initial speed needed in such manner that the projectile reaches a precise distance? That’s what we propose to do using the Cπ -control. 3.3.1 Ballistic simplified model We define first a simple model of the trajectory w(t) = (wx (t), wz (t)) of a projectile of mass m in the usual frame of reference (Oxz) fired with an initial speed magnitude v0 that makes a fire angle θ with the horizontal reference i.e. v(t) = v0 cos(θ)ex + v0 sin(θ)ez . The origin (0, 0) of the frame reference is considered as the initial position of the projectile. Denote a the acceleration vector and v the speed vector of the projectile. From Newton law, considering the action of the gravity g and the air resistance cv2 , we have: m d2 w(t) = ma = mg − cv2 d t2 (6) with the initial conditions: d wx (t) dt d wz (t) dt = v0 cos θ, (0,0) = v0 sin θ (0,0) Considering no air resistance i.e. c = 0, (6) is simplified: 16 (7)      m d wx (t) =0 dt (8)     m d wz (t) = −mg dt whose solution reads:    wx (t) = v0 cos θ t + cte (9)   wz (t) = v0 sin θ t − 1 gt2 + cte 2 From (9), the range (or the target) of the projectile xd at z = 0 can be easily deduced. We have: d= 3.3.2 2v02 cos θ sin θ . g (10) Ballistic-fire control methodology Proposed strategy Consider by hypothesis that a ”virtual” trajectory w∗ of the projectile hits a target x∗d from an initial speed v0∗ and an initial fire angle θ∗ . Consider now the ”true” projectile to fire with a trajectory w. To hit the target x∗d (at z = 0) from a small initial speed vε , one controls the projectile in such manner that the projectile reaches quickly the initial speed v0∗ and the initial fire angle θ∗ required to hit the specified target x∗d according to (10). During such ”launching” phase, that we define as the ”launching” distance ∆x0 for which the trajectory of the projectile is fully controlled, we start from an initial condition that prevents the projectile to reach its target x∗d i.e. : d wx (t) dt d wz (t) dt = vε cos θε , (0,0) = vε sin θε (11) (0,0) where vε < v0∗ and θε ≤ θ∗ are positive resp. initial speed magnitude and fire angle. Figure 17 illustrates simulation examples of a ”virtual” trajectory (subj. to v0∗ and θ∗ ) and a ”true” trajectory (subj. to vε and θ∗ ) that is not controlled; the simulations of the trajectories considering c 6= 0 are presented in Fig. 17(b). The goal is to accelerate the projectile using a specific mechanical device in such manner that the references v0∗ and θ∗ are reached quickly. We consider therefore controlling the trajectory w(t) of the projectile over the launching distance ∆x0 . 17 (a) Case c = 0 (b) Case c = 0 and c 6= 0 Figure 17: Example of comparison of the uncontrolled ”true” projectile (subj. to vε and θ∗ ) and the ”virtual” trajectory (subj. to v0∗ and θ∗ ), considering c = 0 and c 6= 0. We took c = 0.05, θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 18 Controllable ballistic model To simulate a controllable model of the trajectory w of the projectile, we consider adding an external acceleration force in (6) that represents the mechanical action of the specific mechanical device over the distance ∆x0 . We have: d2 w(t) = mg − cv2 + aext (12) d t2 ext where aext (t) = (aext x (t), az (t)) is equivalent to the external force provided by the mechanical device. Since such device acts only over ∆x0 , then we assume that aext = 0 for all x > ∆x0 . m 3.3.3 Implementation of the Cπ -controller A possible control scheme is to consider controlling the trajectory w that must be ”as close as possible” to the reference w∗ over the distance ∆x0 . Therefore, w is physically measured and the external acceleration aext is driven by the Cπ -controller, through the specific mechanical device. We build a closed-loop that creates a feedback between (2) and (12). We have ”symbolically”, for all x ≤ ∆x0 :  Z t  ext ∗   uk = ak = Ki (wk−1 − wk−1 )d τ    0         m uik−1 + Kp (αe−βk − wk−1 ) {z } k−1 \| uik (13) d2 w(t) = mg − cv2 + uk d t2 Determination of the distance ∆x0 ∗ Since we expect that the projectile is fired from ∆x0 with a speed that is very close to v0∗ (and follows, via the Cπ -control, the same trajectory i.e. w ≈ w∗ over ∆x0 ), then, we propose a possible definition of the theoretical launching distance ∆x∗0 , (considered only over the x axis) that corresponds to the solution in wx of: d wx (t) = v0∗ x (14) dt Geometrically, the theoretical launching distance ∆x∗0 is associated to the speed vx that is reached by the projectile (launched with the initial speed vε < v0∗ ) when vx is close to v0∗ x . 19 3.3.4 Numerical simulations Case ∆x0 > ∆x0 ∗ Consider the simulated ”virtual” trajectory, presented in Fig. 17, as the control reference w∗ ; to simplify, we consider θ as constant. Figure 18 presents the case where the fire is controlled considering c = 0 over ∆x0 = 0.11 m. In particular, Fig. 18(a) presents, at the top, the evolution of the controlled trajectory w in comparison with the reference w∗ , and, at the bottom, the calculated speed d wx /d t in comparison with the initial speed v0 cos θ. Figure 18(b) presents the complete ”true” controlled trajectory in comparison with the virtual trajectory. Figure 19 presents the same simulations in the case c 6= 0. Case ∆x0 ∼ ∆x0 ∗ Consider the simulated ”virtual” trajectory, presented in Fig. 17, as the control reference w∗ ; to simplify, we consider θ as constant. Figure 20 presents the case where the fire is controlled considering c = 0 over ∆x∗0 . In particular, Fig. 20(a) presents, at the top, the evolution of the controlled trajectory w in comparison with the reference w∗ , and, at the bottom, the calculated speed d wx /d t in comparison with the initial speed v0 cos θ. Figure 20(b) presents the complete ”true” controlled trajectory in comparison with the virtual trajectory. Figure 21 presents the same simulations in the case c 6= 0. Discussions These results have been obtained using the same set of the Cπ -parameters. The properties of stabilization of the control law, like in the previous case when dealing with switching systems (§3.2) , seem to be preserved and ensure good tracking performances in particular when considering c = 0 and c 6= 0. Further generalizations would allow using multiple and parallel Cπ -controllers in order to control simultaneous physical quantities. In particular, the speed profile v could be controlled simultaneously with the trajectory w... 20 (a) At the top, controlled trajectory w relating to the reference w∗ ; at the bottom, calculated speed d wx /d t relating to the initial speed v0 cos θ. (b) Complete controlled ”true” trajectory in comparison with the virtual trajectory. Figure 18: Example of controlled trajectory considering c = 0 over ∆x0 = 0.11 m. We took θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 21 (a) At the top, controlled trajectory w relating to the reference w∗ ; at the bottom, calculated speed d wx /d t relating to the initial speed v0 cos θ. (b) Complete controlled ”true” trajectory in comparison with the virtual trajectory. Figure 19: Example of controlled trajectory considering c 6= 0 over ∆x0 = 0.11 m. We took c = 0.05, θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 22 (a) At the top, controlled trajectory w relating to the reference w∗ ; at the bottom, calculated speed d wx /d t relating to the initial speed v0 cos θ. (b) Complete controlled ”true” trajectory in comparison with the virtual trajectory. Figure 20: Example of controlled trajectory considering c = 0 over ∆x∗0 . We took θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 23 (a) At the top, controlled trajectory w relating to the reference w∗ ; at the bottom, calculated speed d wx /d t relating to the initial speed v0 cos θ. (b) Complete controlled ”true” trajectory in comparison with the virtual trajectory. Figure 21: Example of controlled trajectory considering c 6= 0 over ∆x∗0 . We took c = 0.05, θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 24 3.4 Control of the HIV-1 model The problem is to control the predator-prey like model that describes the evolution of the HIV-1 when subjected to an external ”medical agent”. From a mathematical point of view, we study the possibility of controlling the model (15) for which the purpose is to control the output y (corresponding to the viral load) using the double inputs u1 and u2 in such manner that y converges rapidly to zero 8 [16] [17].    ẋ1 = s − dx1 − (1 − u1 )βx1 x3  ẋ2 = (1 − u1 )βx1 x3 − µx2 (15) ẋ3 = (1 − u2 )kx  2 − cx3   y= 0 0 γ x where (mathematically) : d = 0.02, k = 100, s = 10, β = 2.4.10−5 , µ = 0.24, c = 2.4. γ is a scaling factor that allows normalizing the output. Figure 22 presents the evolution of the output y in open-loop when u1 = u2 = 0 i.e. when no medical drug is considered. Figures 23 and 24 illustrate the control of y considering two different ratios between u1 and u2 . Figure 22: Transient response of y considering u1 = u2 = 0. 8 Since we are trying to control this model only from the mathematical point of view, we do not take into account the constraints that are medically imposed. 25 (a) Output y (b) Input u Figure 23: Controlled output y (viral load) in correspondence with u1 = u2 = u. 26 (a) Output y (b) Input u Figure 24: Controlled output y (viral load) in correspondence with u1 = u2 = 12 u. 27 3.5 Control of the Epstein frame The Epstein frame (see Fig. 25 9 ) aims to characterize a magnetic material by determining its B − H hysteresis curve. The principle is to create a magnetic field H inside the material using a ”magnetizing” current iH . The material gives a response to the field H that physically corresponds to the measurable magnetic induction field B. This B field creates a voltage vB through magnetic induction and the quantities vB and iH is a representation of the magnetic hysteresis curve B − H. To describe experimentally the major B − H hysteresis loop, the material under study has to be magnetized using a current iH that is alternative and of enough magnitude in order to describe the magnetic behavior at saturation. The purpose of the control law implementation is to control iH such that the output voltage of the Epstein frame vB remains ”as close as possible” to a desired reference waveform. Figure 25: An experimental Epstein frame to characterize magnetic materials. 3.5.1 Epstein frame control Proposed Cπ -control scheme Consider the control scheme depicted in Fig. 26 where Cπ is the proposed PMA controller. Kin and Kout are positive real gains. Denote fBH the numerical Jiles-Atherton model that is associated to the magnetic hysteresis B − H and fJA is the complete hysteresis to control. 9 Picture taken from Wikipdia http://upload.wikimedia.org/wikipedia/commons/5/51/ Epstein_frame.jpg. 28 Figure 26: Proposed PMA scheme to control the electrical waveforms measured from a magnetic hysteresis. Jiles-Atherton based hysteresis model The Jiles-Atherton model [18] describes a magnetic hysteresis cycle B − H. It reads: 1 Man − M c d Man dM = + dH 1 + c δk − α(Man − M ) 1 + c d H (16) where c, δk, Man , α are physical coefficients well identified from magnetic hysteresis measurement and we assume that the current iH corresponds to the magnetization H i.e. iH ∝ H and the voltage vB corresponds to the derivative of the magnetic induction field response B = µ0 H + JBH (H) (where JBH describes the B − H hysteresis via (16)). Simplified model of the Epstein frame The Epstein frame admits a complex model based on the Jiles-Atherton model that represents all electric phenomena that occur inside the Epstein frame10 . To simplify the model of the Epstein frame to control, we consider controlling a nonlinear function fJA , which represents a modified Jiles-Atherton model. Denote vH = fJA (iH ) the nonlinear dynamical system that describes the B − H hysteresis as a function of iH , and consider controlling directly the hysteresis cycle in such manner that Cπ controls iH in order to get vH = fJA (iH ) as close as possible to a reference waveform. To define the global fJA hysteresis function that is controlled by Cπ , which includes the scaling coefficients needed by the Cπ corrector, consider iH = Kin u, y = Kout vH and a coefficient Ke such that: y = fJA (u) = Kin Ke u + Kout JBH (Kin u). 10 (17) The Epstein frame is equivalent to a transformer and the ”mutual interactions” between primary and secondary coils must be taken into account in addition to the hysteresis behavior. 29 The small coefficient Ke # 10−4 compensates the very small variations of fBH when B is close to Bsat . Such variations may induce time-delays in the response of the Cπ -controller that induce some distortions of the output signal y. The model (17) could be seen as an ”affine” derivation of the original Jiles-Atherton model. The B − H loops, obtained from the Jiles-Atherton model, are depicted in Fig. 27 considering the frequencies 5 Hz, 50 Hz, 500 Hz, 1 kHz and 10 kHz. Since this hysteresis model does not allow H > Hmax , a limitation is necessary to bound the evolution of H, that may occur eventually during the transient of the dynamic stabilization of the control loop (ex. in Fig. 31). . Figure 27: Simulation of the Jiles-Atherton model for different operating frequencies. 3.5.2 Simulation results Figures 28, 29, 30, 31 and 32 depict the input u and the (rescaled) output y of the control loop according to the time. Given a particular operating frequency, for which a particular H − B hysteresis is studied (see Fig. 27), and assuming that the output reference y ∗ is a sinusoid, whose magnitude corresponds to the theoretical Hmax of the H − B hysteresis, different frequencies are considered (5 Hz, 50 Hz, 500 Hz, 1 kHz and 10 kHz) in order to highlight the behavior of the controlled voltage vb when the frequency changes. In particular, high frequencies (e.g. Figures 31 and 32) introduce an important transient response on y due to the fact that the variations of y ∗ are too fast to get an immediate stabilization to the dynamic working point of the hysteresis. The simulation show that the Cπ -controller gives very interesting 30 dynamic performances over a wide range of dynamic working points relating to the operating frequency. An optimization algorithm (see §2.3) has been used to adjust the parameters of the Cπ -controller in such manner that the shape of the output response y is ”as close to” a sine shape11 . Remarks This hysteresis model is composed of three subsystems (the first magnetization branch, the increasing and decreasing branches) that switch depending on the value of dH/dt. When H << Hmax , the switch between the branches may not be smooth and such ”connection” may induce a small transient on y. An illustration is presented in Fig. 33 at a low frequency in comparison with Fig. 30. The closed-loop has been also tested using a triangular shaped output reference y ∗ . Figure 34 depicts the magnitude of the magnetic field H and the corresponding (rescaled) magnetic induction B during the control loop process according to the time. The frequency of 5 Hz has been considered as an example holding the parameters of the simulation with the sine reference at 5 Hz. Figure 28: Simulated u and y signals according to the time at 5 Hz. 11 Remember that the purpose of the optimization procedure is to minimize the tracking error y − y in such manner that ideally y ≡ y ∗ for the particular sine output reference y ∗ . ∗ 31 Figure 29: Simulated u and y signals according to the time at 50 Hz. Figure 30: Simulated u and y signals according to the time at 500 Hz. 32 Figure 31: Simulated u and y signals according to the time at 1 kHz. Figure 32: Simulated u and y signals according to the time at 10 kHz. 33 Figure 33: Simulated u and y signals according to the time at 500 Hz (H << Hmax ). Figure 34: Simulated H and y signals according to the time at 5 Hz. 34 4 Derivative-free & ”extremum-seeking” control To describe how the PMA could be used as a derivative-free optimization (DFO) algorithm (e.g. [19] [20]) or as an ”extremum-seeking” (ES) control scheme (e.g. [21] [22] [23]), we first define each element of the associated control scheme and then, we derive the operating conditions that would allow to minimize nonlinear functions. We assume that it is possible to derive a control scheme such that the PMA can be used to minimize nonlinear functions. 4.1 Proposed Cπ -control scheme Definition of the closed loop Consider the control scheme depicted in Fig. 35 where Cπ is the proposed PMA ”extremum-seeking” controller. Kin and Kout are s positive real gains. We consider either a static nonlinear function fnl (regarding DFO), which does not have any internal dynamical properties, or a nonlinear SISO dynamical system (1) (regarding ES), to minimize. The function Q is e.g. a basic first order transfer function. Figure 35: Proposed PMA scheme to minimize a nonlinear function fnl . Function to control s • Define the fnl static function to optimize such that: s fnl : Rn → R u0 7→ y or consider a nonlinear SISO dynamical system (1). 35 (18) This function represents the ”nonlinear optimization problem”. Currently, the assumption n ≤ 2 is considered and we denote u0x the input variable for n = 1. • Q is a standard linear transfer function (typically of first order), such that: R→R 0 Q : du dt + γ u0 \|k = uk (19) k where γ is a time-constant, chosen in such manner that the step response of Q s in order to is very fast. As presented in the Fig. 35, Q is associated with fnl s to control. provide some minimal dynamical properties regarding the system fnl Obviously, the Q function is not necessary when fnl is already a dynamical function like (1)12 . A single Cπ -controller13 drives a single input of fnl (eventually through Q), as presented in Fig. 35. 4.2 Numerical applications Since the PMA is designed for nonlinear systems and does not contain any derivatives, it is assumed that the ”extremum-seeking” control is possible considering a specific definition of y ∗ in order to reach and stabilize fnl to its minimum. Let us assume that the following (eventually constrained) minimization problem (described for a single variable): min fnl (x), x∈R (x = u0 identically inside the control scheme) (20) is equivalent to the control scheme described in Fig. 35, for which the output reference y ∗ ”follows” the minimum value of fnl . We denote x = xopt the value that gives the minimum of fnl . Results For each case in 1D, are plotted: the difference between two iterations yk , yk−1 and the error between xopt and the evolution of x through the closed-loop. In these cases, all the parameters of Cπ have been set experimentally to give interesting performances but are not optimal (the choice of the y ∗ function may influence the speed of the convergence). The following numerical cases are studied: 12 Last investigations suggest that the linear transfer function Q may be not necessary even for s a fnl function to control. The properties of the para-model algorithm are currently under study considering nonlinear systems that are ”non-dynamical”. 13 To extend this scheme to multi-input variables (n > 2) of fnl , one may consider the use of a Cπ -controller per input variable. 36 • See Fig. 36 regarding the minimization of a 1D convex function such that: min (x − 30)2 x (21) • See Fig. 37 regarding the minimization of a 1D convex function with a minimum that changes according to the time at an unknown instant t1 such that: t ? 1 min (x − 30)2 → min(x − 40)2 x x (22) • See Fig. 38 regarding the minimization of a 1D non convex function such that: min 10(1.5 cos(x) − x) + exp(x − 5) + 100 subj. to : y ≥ 15x − 60 x,y 5 (23) Concluding remarks We presented how the proposed para-model agent14 , as a model-free and derivativefree based controller, can be used to control nonlinear systems or perform optimization / ”extremum-seeking” control. Further investigations include extensive tests and applications to complex systems as well as a complete study of the stability. Acknowledgement The author is sincerely grateful to Dr. Edouard Thomas for his strong guidance and his valuable comments that improved this paper. The author is also sincerely grateful to Olivier Ghibaudo, Ph.D. student at G2Elab-CNRS (Grenoble) France, for having worked on the numerical version of the hysteresis model. 14 Why ”para-model”? Based on the ”model-free” methodology, for which the model is not correlated to the controller in the sense that the controller does not need an explicit definition of the model to be parametrized and can thus rebuild an ultra-local model from measurements, we propose the prefix ”para” to highlight the fact that the agent is ”in close proximity” to the model of the process that he controls. Although no specific information is needed for the PMA control part, some basic information may be needed to configure properly the PMA and the output reference y ∗ in the case of extremum-seeking control approach. The stability study is an essential part that should formalize the proposed PMA approach in order to justify theoretically the operating conditions of the Cπ -control. 37 Figure 36: min (x − 30)2 (initial condition in red spot). x t ? 1 Figure 37: min fnl2 = (x − 30)2 → min fnl1 = (x − 40)2 (initial condition in red spot). x x 38 Figure 38: min 10(1.5 cos(x) − x) + exp(x − 5) + 100 subj. to : y ≥ 15x − 60 (inix,y tial condition in red spot). References [1] M. Fliess, C. Join, M. Mboup and H. Sira-Ramirez, ”Vers une commande multivariable sans modèle”, Conférence internationale francophone d’automatique (CIFA 2006), France, 2006. (available in french at https://arxiv.org/pdf/math/ 0603155.pdf). [2] M. Fliess and C. 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1 Location-Aided Coordinated Analog Precoding for Uplink Multi-User Millimeter Wave Systems Flavio Maschietti‡ , David Gesbert‡ , Paul de Kerret‡ arXiv:1711.03031v1 [eess.SP] 8 Nov 2017 ‡ Communication Systems Department, EURECOM, Sophia-Antipolis, France Email: {flavio.maschietti, david.gesbert, paul.dekerret}@eurecom.fr Abstract Millimeter wave (mmWave) communication is expected to have an important role in next generation cellular networks, aiming to cope with the bandwidth shortage affecting conventional wireless carriers. Using side-information has been proposed as a potential approach to accelerate beam selection in mmWave massive MIMO (m-MIMO) communications. However, in practice, such information is not error-free, leading to performance degradation. In the multi-user case, a wrong beam choice might result in irreducible inter-user interference at the base station (BS) side. In this paper, we consider locationaided precoder design in a mmWave uplink scenario with multiple users (UEs). Assuming the existence of direct device-to-device (D2D) links, we propose a decentralized coordination mechanism for robust fast beam selection. The algorithm allows for improved treatment of interference at the BS side and in turn leads to greater spectral efficiencies. I. I NTRODUCTION The large bandwidths available at mmWave carrier frequencies are expected to help meet the throughput requirements for future mobile networks [1]. Since smaller wavelength signals are more prone to absorption, mmWave communications require beamforming in order to guarantee appropriate link margins and coverage [2], [3]. To this end, m-MIMO techniques [4] are envisioned as high-gain directional antennas with small form factor can be designed for mmWave usage [5]. However, configuring those massive antennas to operate with large bandwidths entails an additional effort. The high cost and power consumption of the radio components impact on the UEs and small BSs, thus limiting the practical implementation of a fully-digital beamforming architecture [1]. Moreover, the large number of antennas at both ends of the radio links would require unfeasible CSI-training overhead to design the precoders. 2 One step towards simplification consists in replacing the fully-digital architecture with a hybrid analog-digital one [6]–[11]. In mixed analog-digital architectures, a low-dimensional digital processor is concatenated with an RF analog beamformer, implemented through phase shifters. Note that while the latter is sufficient to achieve a good part of the overall beamforming gain – through beam steering towards desired spatial directions – the digital stage is essential when processing multiple streams and users. Interestingly, existing works on hybrid architectures typically ignore multi-user interference issues in the analog domain and cope with them in the digital part. For instance, in [12], a procedure is proposed for the downlink transmission, where the analog stage is intended to find the best beam directions for each UE (regardless of multi-user interference), while the digital one applies the conventional Zero-Forcing (ZF) beamformer on the resulting effective channel. The strength of this approach lies in the fact that it is possible to use the existing beam training algorithms for single-user links – such as [13]–[15] – in the analog stage. Such algorithms have been developed bearing in mind the need for fast link establishment in low-latency applications. Nevertheless, the reduced number of digital chains might not always allow to resolve the residual multi-user interference which remains after the analog beamforming stage. In particular, in a mmWave propagation scenario [2], [3], multiple closely located UEs will likely share some common reflectors, causing an alignment of the main path’s angles of arrival at the BS receiver and preventing it from resolving the interference, even at the digital decoding stage. To solve this problem, a principal idea consists in treating interference before it takes place, i.e. the UE side, as is done for example in [16], [17]. Although showing significant performance advantages over the existing solutions, these works assume perfect CSIR for analog beamforming and single-antenna UEs, which might not be realistic in all mmWave contexts [12]. Rather, we are interested in statistically-driven analog beamforming at the UE TX side. In this paper, we point out that simple analog UE beam selection can be designed so as to enable the analog receive beam on the BS side to discriminate for interference. We propose to do this through the help of low-rate side-information at the UEs. Several works can be found in the mmWave literature, where side-information is exploited to improve performance without burdening overhead. Side-information can be obtained from various sources, such as automotive sensors [18], UHF band [19], GNSS [20], or also past multipath fingerprints measurements [21]. We bring forward the idea that position-based side-information can be exploited in order to develop a coordination mechanism between the UEs, so that the interference at the BS side can 3 be treated efficiently through both the analog and digital parts of the receiver, as opposed to relying on the digital part alone. The main intuition is to use coordination to make sure the selected analog beams at the BS convey the full rank of multi-user channels towards the digital part to preserve invertibility. As in some previous work [22], we are interested in establishing a robust form of coordination which accounts for possible noise in the positioning information made available to the UEs. However, [22] targets a single-user scenario only. In the multi-user scenario, the lack of a realtime communication channel prevents the UEs from exchanging instantaneous CSI. We consider, instead, the existence of a low-rate unidirectional D2D channel, allowing communication of GPS-type data. In particular, we consider a hierarchical set-up where higher ranked UEs receive position information from lower ranked ones. The unidirectional aspect and the limitation to position information exchange help keep the D2D overhead much lower than real-time D2D. Our main contributions read as follows: • We formulate the problem of per-user analog precoding with side position information and recast it as a decentralized beam selection problem. • Our algorithm exploits the hierarchical structure of the information, in order to perform robust (with respect to position data noise) interference mitigation at both analog and digital stages. • Under the proposed method, the UEs coordinate to select beams which, while being suboptimal in terms of average power, help attain the full rank condition needed at the BS for interference suppression. II. S YSTEM M ODEL Consider the single-cell uplink multi-user mmWave scenario in Fig. 1. The BS is equipped with NBS 1 antennas to support K UEs with NUE 1 antennas each. The UEs are assumed to reside in a disk of a given radius rcl , which will be used to control inter-UE average distance. Each UE sends one data stream to the BS. We assume that the BS has NRF = K RF chains available, each one connected to all the NBS antennas, assuming a fully-connected hybrid architecture [1]. The u-th UE precodes the data su ∈ C through the analog precoding vector vu ∈ CNUE ×1 . We assume that the UEs have one RF chain each, i.e. UEs are limited to analog beamforming via phase shifters (constant-magnitude elements) [7]. In addition, E[kvu su k2 ] ≤ 1, assuming normalized power constraints. 4 The reconstructed signal after mixed analog/digital combining at the BS can be expressed as follows – assuming no timing and carrier mismatches: x̂ = K X H H WD WRF Hu vu su + WD WRF n (1) u=1 u NBS ×NUE where H ∈ C is the channel matrix from the u-th UE to the BS and n ∈ CNBS ×1 is the thermal noise vector, with zero mean and covariance matrix σn2 INBS . WRF ∈ CNBS ×NRF is, instead, the analog combining matrix, containing the vectors relative to each of the K RF chains (subject to the same hardware constraints as described above), while WD ∈ CK×NRF denotes the digital combining matrix. The received SINR for the u-th UE at the BS is expressed as follows: H \|wDu WRF Hu vu \|2 γu = P u H 2 w w 2 w6=u \|wD WRF H v \| + σñ (2) where wDu ∈ C1×NRF denotes the row of WD related to the u-th UE (one RF chain for each UE), H n for the filtered thermal noise. and where we used the short-hand notation ñ = WD WRF BS UE 1 rcl UE 2 Fig. 1: Scenario example with L = 3 propagation paths, two reflectors, and K = 2 UEs. The UEs are assumed to reside in a disk of radius rcl , which is relatively common in realistic scenarios, e.g. dense UE distribution in a coffee house. In this illustration, two closely located UEs are sharing some reflectors, and paths reflecting on the top reflector arrive quasi-aligned at the BS while originating from distinct UEs. 5 A. Channel Model Unlike the conventional UHF band propagation environment, the mmWave one does not exhibit rich-scattering [2] and is in fact modeled as a geometric channel with a limited number of dominant propagation paths which survive high attenuation. The UE u is thus subject to the channel matrix Hu ∈ CNBS ×NUE , expressed as the sum of L components or contributions [7]: Hu = NBS NUE L 1/2 X α`u aBS (ϑu` )aHUE (φu` ) (3) `=1 where α`u ∼ CN (0, (σ`u )2 ) denotes the complex gain for the `-th path of the u-th UE. Furthermore, we assume that the variances (σ`u )2 , ` ∈ {1, . . . , L}; u ∈ {1, . . . , K} of the paths are such as P u 2 ` (σ` ) = 1, ∀u ∈ {1, . . . , K}. The variables φu` ∈ [0, 2π) and ϑu` ∈ [0, 2π) are the angles of departure (AoDs) and arrival (AoAs) for each contribution, for a given UE u, where one angle pair corresponds to the LoS direction while other might account for the presence of strong reflectors (e.g. buildings, hills) in the environment. The positions of those points of reflection depend on the position of the considered UE (see Fig. 1). We will denote the reflecting points for the u-th UE with Rui , i ∈ {1, . . . , L − 1} in the rest of the paper. The vectors aUE (φu` ) ∈ CNUE ×1 and aBS (ϑu` ) ∈ CNBS ×1 denote the antenna steering vectors at the u-th UE and the BS for the corresponding AoDs φu` and AoAs ϑu` , respectively. We assume to use ULAs at both sides, so that [23]: h iT 1 −iπ cos(φ) −iπ(NUE −1) cos(φ) aUE (φ) = 1, e ,...,e (NUE )1/2 h iT 1 −iπ cos(ϑ) −iπ(NBS −1) cos(ϑ) aBS (ϑ) = 1, e , . . . , e 1/ (NBS ) 2 (4) (5) B. Codebooks for Analog Beams The most recognized method to implement the analog beamformer is through a network of digitally-controlled phase shifters [24] (refer to [1] for alternative architectures). Thus, the phase of each element of the analog beamformer is limited to fixed quantized values, and therefore, the beamforming vectors need to be selected from a finite set (or codebook). We denote the codebooks used for analog beamforming as: VUE = {v1 , . . . , vMUE }, VBS = {w1 , . . . , wMBS } where VUE is assumed to be shared between all the UEs, to ease the notation. (6) 6 For ULAs, a suitable design for the fixed beamforming vectors in the codebook consists in selecting steering vectors over a discrete grid of angles [12], [14]: vp = aUE (φ̄p ), p ∈ {1, . . . , MUE } (7) wq = aBS (ϑ̄q ), q ∈ {1, . . . , MBS } (8) where the angles φ̄p , ∀p ∈ {1, . . . , MUE } and ϑ̄q , ∀q ∈ {1, . . . , MBS } can be chosen according to different strategies, including regular and non-regular sampling of the [0, π] range [22]. Remark 1. Given the one-to-one correspondence between the beamforming vectors in VBS (resp. VUE ), and the grid angles ϑ̄q , ∀q ∈ {1, . . . , MBS } (resp. φ̄p , ∀p ∈ {1, . . . , MUE }), we will make the abuse of notation q ∈ VBS (resp. p ∈ VUE ) to denote the vector wq ∈ VBS (resp. vp ∈ VUE ). III. I NFORMATION M ODEL In this section, we describe the structure of the channel state- and side-information available at both UEs and BS sides. We start with defining the nature of information in an ideal setting before turning to a realistic (noisy) case. Definition 1. The average beam gain matrix Gu ∈ RMBS ×MUE contains the power level associated with each combined choice of analog beam pair between the BS and the u-th UE after averaging over small scale fading. It is defined as: Guq,p =E αu h 2 wqH Hu vp i (9) where the expectation is carried out over the channel coefficients αu = [α1u , α2u , . . . , αLu ] and with Guq,p denoting the (q, p)-element of Gu . Definition 2. The position matrix Pu ∈ R2×(L+1) contains the two-dimensional location coordinates pun = [punx puny ]T for node n, where n indifferently refers to either the BS, the u-th UE or one of the reflectors Rui , i ∈ {1, . . . , L − 1}. It is defined as follows: h i Pu = puBS puR1 . . . puRL−1 puUE (10) We will denote as P the set containing all the position matrices Pu , ∀u ∈ {1, . . . , K}. As shown in [22], the matrix Gu can be expressed as a function of the matrix Pu . We recall here the deterministic relationship that is found between those two matrices. 7 Lemma 1. We can write the average beam gain matrix relative to the u-th UE as follows: Guq,p (Pu ) L X = (σ`u )2 \|LBS (∆u`,q )\|2 \|LUE (∆u`,p )\|2 (11) `=1 where we remind the reader that (σ`u )2 denotes the variance of the channel coefficients α`u and we have defined: u LUE (∆u`,p ) ei(π/2)∆`,p sin((π/2)NUE ∆u`,p ) = u (NUE )1/2 ei(π/2)NUE ∆`,p sin((π/2)∆u`,p ) LBS (∆u`,q ) ei(π/2)∆`,q sin((π/2)NBS ∆u`,q ) = u (NBS )1/2 ei(π/2)NBS ∆`,q sin((π/2)∆u`,q ) 1 1 (12) u (13) and ∆u`,p = (cos(φ̄p ) − cos(φu` )) (14) ∆u`,q = (cos(ϑu` ) − cos(ϑ̄q )) (15) with the angles φu` , ` ∈ {1, . . . , L} and ϑu` , ` ∈ {1, . . . , L} obtained from the position matrix Pu through simple algebra (refer to [22] for more details). A. Distributed Noisy Information Model In the distributed model, each UE u receives its own estimates of the position matrices Pw , ∀w ∈ {1, . . . , K}. We will use the superscript with parenthesis (u) to denote any information known at the u-th UE. In particular, we denote as P̂w,(u) ∈ R2×(L+1) , ∀w ∈ {1, . . . , K} the local information available at the u-th UE about the position matrix Pw . This information is modeled as follows: P̂w,(u) = Pw + Ew,(u) ∀w ∈ {1, . . . , K} where Ew,(u) denotes the following matrix: h i w,(u) w,(u) w,(u) w,(u) Ew,(u) = eBS eR1 . . . eRL−1 eUE (16) (17) containing the random position errors which the u-th UE made in estimating pw n . Such error comes with an arbitrary, yet known, probability density function few,(u) . n Definition 3. We will denote as P̂ (u) , where: P̂ (u) = {P̂1,(u) , . . . , P̂K,(u) } (18) the overall local information available at the u-th UE containing all the estimated position matrices P̂w,(u) , ∀w ∈ {1, . . . , K}. 8 B. Hierarchical Location-Information Exchange The hierarchical (or nested) model is a sub-case of the distributed model in which the u-th UE has access to the estimates of the UEs u + 1, . . . , K. As we will see in the next section, this information structure enables some coordination for just half of the overhead needed in a conventional two-way exchange mechanism. One consequence in particular is that the u-th UE is able to retrieve the beam decisions carried out at (the less informed) UEs u + 1, . . . , K. C. Additional Information In what follows the number of dominant paths, and their average powers (σ`u )2 , ` ∈ {1, . . . , L}; u ∈ {1, . . . , K} are assumed to be known at each UE, based on prior averaged measurements. Likewise, statistical distributions fenw,(u) , ∀u, w are supposed to be quasi-static and as such are supposed to be available to each UE. In other words, the u-th UE is aware of the amount of error in the position estimates which it and other UEs have to cope with. IV. M ULTI -U SER L OCATION -A IDED H YBRID P RECODING In order to maximize the received SNR γ u defined in (2) for each UE, the mutual optimization of both analog and digital components must be taken into account. A common approach consists in decoupling the design, as the analog beamformer can be optimized in terms of long-term channel statistics, whereas the digital one can be made dependent on instantaneous information [25]. A. Uncoordinated Beam Selection We first review here the approach given in [12], where the authors proposed to design the analog beamformers to maximize the received power for each UE, neglecting multi-user interference. Once the analog beamformers are fixed at both UE and BS sides, the design of the digital beamformer at the BS follows the conventional MU-MIMO approach. In this respect, a common choice is to consider ZF combining. Therefore, the digital beamforming matrix WD is the pseudo-inverse of the effective channel matrix H̃ ∈ CNRF ×K , which is defined as follows [23]: WD = H̃H H̃ −1 H̃H (19) where h i h i H H H H̃ = WRF H1 v1 WRF H2 v2 . . . WRF HK vK , and with WRF = w1 w2 . . . wK . 9 When position and path average power information is available, the beam selection (quun ∈ VBS , pun u ∈ VUE ) at the analog stage of the algorithm proposed in [12] – which we will denote as uncoordinated (un) – can be expressed as follows: u (u) ) = argmax R P̂ , q , p (quun , pun u u , ∀u u (20) qu ∈VBS pu ∈VUE where we have defined the single-user rate Ru as [22]: Guqu ,pu (Pu ) u R (P, qu , pu ) = log2 1 + σn2 (21) Equation (20) can be solved through direct search of the maximum in the matrix Gu , derived from P̂ (u) through (11). While being simple to implement, the information at each UE in this method is treated as perfect, although some Bayesian robustization can be introduced [22]. Another limitation of this approach is that each UE solves its own beam selection problem in a way which is independent of other UEs, thus ignoring the possible impairments in terms of interference. We illustrate this effect in Fig. 2, where we plot the mean rate per UE obtained when the analog precoders are chosen through (20), in case of K = 2 UEs, perfect position information, as a function of rcl . As the inter-UE average distance decreases, the performance of this procedure degrades, since the UEs have much more chance to share common best propagation paths (which results in severe interference at the analog stage at the BS). The action of the ZF is noticeable but not sufficient for small cluster radii. In what follows, we consider different flavors of coordination. B. Naive-Coordinated Beam Selection In order to improve performance, we design the analog precoders according to the following figure of merit, which takes into account the average multi-user interference at the analog stage: R(P, q1:K , p1:K ) = K X u=1 log2 1 + P Guqu ,pu (Pu ) w w 2 w Gqu ,pw (P ) + σn (22) Remark 2. We used here the short-hand q1:K , p1:K to denote the indexes q1 , . . . , qK and p1 , . . . , pK , respectively. The hierarchical model allows the u-th UE to predict the beam selected at UEs u + 1, . . . , K. However, for a full coordination, the u-th UE would also need to know the precoding strategies of the more informed UEs, i.e. UE 1, . . . , u − 1, which involves some guessing [26]. 10 14 Mean rate per user [b/s/Hz] 12 10 8 6 4 2 0 Single−User Hybrid Analog 1 3 5 7 9 11 13 Cluster radius [m] 15 17 19 21 Fig. 2: Mean rate per UE vs Cluster radius. The performance degrades sharply as the inter-UE average distance decreases. As a first approximation, the u-th UE can assume that its estimates are perfect (error-free) and global (shared between all the UEs). Since UEs 1, . . . , u − 1 have in fact different estimates, and since such information is not error-free, we call this approach naive-coordinated (nc). The beam indexes (qunc ∈ VBS , pnc u ∈ VUE ) associated to the u-th UE are then found as follows: (u) o o (q̃1:u−1 , qunc , p̃1:u−1 , pnc ) = argmax R P̂ , q , p (23) qu+1:K ,pu+1:K 1:u 1:u u q1 ,...,qu ∈VBS p1 ,...,pu ∈VUE o Remark 3. We make here an abuse of notation. The subscripts qu+1:K , pou+1:K acknowledge for the known strategies at the u-th UE. Those strategies are fixed parameters of the function R. The same notation will be used in the rest of the paper. Remark 4. The u-th UE will use the precoding vector associated to the index pnc u ∈ VUE to reach the BS and will discard the remaining beam indexes q̃1:u−1 , p̃1:u−1 found for the other UEs. Indeed, those indexes only correspond to guesses realized at the u-th UE which do not necessarily correspond to the true beams used for transmission at UEs 1, . . . , u − 1. We have introduced the notation q̃u to denote such beams. 11 C. Statistically-Coordinated Beam Selection The naive-coordinated approach relies on the correctness of the position estimates available at each UE. As a consequence, its performance is expected to degrade in case of GPS inaccuracies or lost location awareness. As the precision of position estimates decreases, the beam selection is expected to have more confidence in long-term statistics alone, to predict the behavior of the UE which are higher ranked in the information chain. In this case, the position estimates are not exploited, and each UE relies on prior statistics to figure out other UEs’ information. We denote the resulting statistically-coordinated (sc) beam indexes relative to the u-th UE as (qusc ∈ VBS , psc u ∈ VUE ), which read as follows: (q̃1:u−1 , qusc , p̃1:u−1 , psc u) h i o o = argmax EP\|rcl Rqu+1:K ,pu+1:K P, q1:u , p1:u (24) q1 ,...,qu ∈VBS p1 ,...,pu ∈VUE This is a long-term optimization which is updated only if prior statistics change. Thus, (24) represents a simple stochastic optimization problem [27] which can be solved through e.g. approximation of the expectation operator (carried out over prior statistics) with Monte-Carlo iterations. D. Robust-Coordinated Beam Selection The UEs have also access to the statistics of their local position estimates. In the previous approach, each UE used prior statistics to guess the precoding strategies of the more informed UEs. This helps in case the local information available at the u-th UE is not accurate enough to gain more knowledge about the UEs 1, . . . , u−1. In the opposite case, local statistical information can be exploited to supplement prior information. In this approach, the UEs look for beam selection strategies which progressively pass from exploiting local information only – in case of perfect local information – to exploiting statistical information only – in case of poor local information. We denote this approach as robust-coordinated (rc). The beams (qurc ∈ VBS , prc u ∈ VUE ) for the u-th UE are obtained through: h i o o R P, q , p (q̃1:u−1 , qurc , p̃1:u−1 , prc ) = argmax E qu+1:K ,pu+1:K 1:u 1:u u P\|P̂ (u) ,rcl (25) q1 ,...,qu ∈VBS p1 ,...,pu ∈VUE Remark 5. Here, the u-th UE considers its locally-available position estimates as imperfect and globally-shared. Also in this case, an approximate solution can be obtained through Monte-Carlo methods, generating possible matrices P according to the (known) distribution P\|P̂ (u) , rcl . 12 We summarize the proposed robust-coordinated beam selection used at the u-th UE in Algorithm 1. In Step 1, the u-th UE retrieves the processing carried out at less informed UEs u + 1, . . . , K. The K-th UE skips this step. In Step 2, beam selection is performed through (22) (an approximation) and (25). Algorithm 1 frc : Robust-Coordinated Beam Selection (u-th UE) INPUT: P̂ (w) , ∀w ∈ {u, . . . , K}, pdf of (P\|P̂ (w) , rcl ), ∀w ∈ {u, . . . , K} Step 1 1: for w = K : u + 1 do 2: o (qw , pow ) = frc (P̂ (w) . The K-th UE skips this decreasing for loop o , qw+1:K , pow+1:K ) 3: end for Step 2 4: return (qurc , prc u ) ← Evaluate (25) . Refer to Algorithm 2 for implementation details Algorithm 2 Implementation details for Step 2 in Algorithm 1 (u-th UE) o o , pou+1 , . . . , poK INPUT: P̂ (u) , pdf of (P\|P̂ (u) , rcl ), qu+1 , . . . , qK 1: for i = 1 : M do . Approximate expectation over (P\|P̂ (w) , rcl ), ∀w with M Monte-Carlo iterations 2: Generate possible position matrices P through sampling over the distribution (P\|P̂ (w) , rcl ) 3: Compute possible gain matrices Ĝw , ∀w ∈ {1, . . . , K} through (11) using the generated P 4: o o , pou+1 , . . . , poK ) S UM R ATE E VAL(Ĝ1 , . . . , ĜK , qu+1 , . . . , qK . Described in Algorithm 3 5: end for 6: Compute the average sum-rate over the Monte-Carlo iterations for all possible beam pairs 7: (qurc , prc u ) ← Indexes relative to the beams achieving maximum average sum-rate 8: The pair of vectors with indexes (qurc , prc u ) is assigned to the u-th UE V. S IMULATION R ESULTS We evaluate here the performance of the proposed algorithms. We consider L = 3 multipath components. A distance of 100 m is assumed from the UEs’ cluster center and the BS. The radius of the UEs’ cluster is set to rcl = 7 m. Both the BS and UEs are equipped with NUE = NBS = 64 antennas (ULA). The number of elements in the beam codebooks is MUE = MBS = 64, with grid angles spaced according to the inverse cosine function so as to guarantee equal gain losses among adjacent angles [22]. All the plotted rates are the averaged – over 10000 Monte-Carlo runs – rates per UE. 13 Algorithm 3 Function evaluating an approximated average sum-rate (22) (u-th UE) o o 1: function S UM R ATE E VAL(Ĝ1 , . . . , ĜK , qu+1 , . . . , qK , pou+1 , . . . , poK ) 2: o o KnownInds = {qu+1 , . . . , qK , pou+1 , . . . , poK } 3: for w = 1 : u − 1 do 4: . The most informed UE 1 skips this loop P v (qw , . . . , qu , pw , . . . , pu ) = max Ĝw . with given KnownInds qw ,pw /( v6=w Ĝqw ,pv + N0 ) 5: KnownInds = {qw , pw } ∪ KnownInds 6: Discard all the other indexes qw+1 , . . . , qu , pw+1 , . . . , pu . Updated set of indexes to be used in the next iteration 7: end for 8: P return Ĝuqu ,pu /( w6=u Ĝw qu ,pw + N0 ) for all possible qu , pu . with given KnownInds 9: end function A. Location Information Model In the simulations, we adopt a uniform bounded error model for location information [20], [22]. In particular, we assume that all the position estimates lie somewhere inside disks centered in the actual positions pun , n ∈ {UE, BS, Rui }; i ∈ {1, . . . , L − 1}; u ∈ {1, . . . , K}. Let S(r) be the two-dimensional closed ball centered at the origin and of radius r, which is S(r) = {υ ∈ w,(u) R2 : kυk ≤ r}. We model the estimation errors en w,(u) in S(rn w,(u) ), where rn as a random variable uniformly distributed is the maximum positioning error for the node n of the w-th UE as seen from the u-th UE. B. Results and Discussion To evaluate the performance, we start with a simple configuration with K = 2 UEs. 1) Strong LoS: In what follows, we consider a stronger (on average) LoS path with respect to the reflected paths, being indeed the prominent propagation driver in mmWave bands [2], [3]. The reflected paths are assumed to have the same average power. The average power of such (1) (2) paths is assumed to be shared across the UEs, i.e. (σ` )2 , = (σ` )2 , ∀` ∈ {1, . . . , L}. In Fig. 3, we consider the performance of the proposed algorithm as a function of the precision of the information available at the less informed UE. In particular, an error radius of 5 m means w,(2) rn = 5 m, ∀w, n. As for the most informed UE, we consider perfect information in Fig. 3a, w,(1) i.e. rn w,(1) = 0 m, ∀w, n, and 3 m of precision in Fig. 3b, i.e. rn = 3 m, ∀w, n. Fig. 3a and Fig. 3b show that both the uncoordinated and the naive approaches degrade fast as the error radius for the less informed UE increases. This is due to the fact that the UEs build their 14 strategies according to their available position estimates, which become unreliable to perform beam selection. In particular, when the precision is less than 6 m, the statistically-coordinated approach – based on statistical information only – behaves better. The robust-coordinated approach outperforms all the other algorithms, being able to discriminate for interference at the BS side, while taking into account the noise present in position information. 9.5 8 Robust−Coordinated Stat.−Coordinated Naive−Coordinated Uncoordinated 7.5 8.5 Mean rate per UE [b/s/Hz] Mean rate per UE [b/s/Hz] 9 8 7.5 7 6.5 6 Robust−Coordinated Stat.−Coordinated Naive−Coordinated Uncoordinated 7 6.5 6 5.5 5 0 2 4 6 8 10 12 14 16 Pos. information precision for less informed UE [m] 18 (a) Most informed UE with perfect information 20 4.5 0 2 4 6 8 10 12 14 16 Pos. information precision for less informed UE [m] 18 20 (b) Most informed UE with 3 m precision Fig. 3: Mean rate per UE vs Position information precision. Strong LoS. 2) LoS Blockage: It is also interesting to observe how the proposed algorithms behave in case of total line-of-sight blockage, while having one stronger reflected path. Fig. 4 compares the proposed algorithms as a function of the error radius for the less informed UE. The most informed UE is assumed to have access to perfect information. The same considerations outlined for the strong LoS case remain valid. It is possible to observe that the uncoordinated approach performs worse than before (with strong LoS), since the UEs choose to use the beams pointing towards the same stronger reflected path. As a consequence, there is much more chance to arrive at the BS with non-distinguishable AoAs. In such cases, coordination between the UEs is essential to combat multi-user interference. VI. C ONCLUSIONS In mmWave communications, multi-user interference has to be handled in the analog stage as well. In this respect, suitable strategies for multi-user interference minimization can be applied in the beam domain through e.g. exploitation of location-dependent information. 15 9.5 Robust−Coordinated Stat.−Coordinated Naive−Coordinated Uncoordinated Mean rate per UE [b/s/Hz] 9 8.5 8 7.5 7 6.5 6 0 2 4 6 8 10 12 14 16 Pos. information precision for less informed UE [m] 18 20 Fig. 4: Mean rate per UE vs Position information precision. LoS blockage. Dealing with the imperfections in location information is not trivial, due to the decentralized nature of the information, which leads to disagreements between the UEs affecting performance. In this work, we introduced a decentralized robust algorithm which aim to select the best precoder for each UE taking both the noise present in location information and multi-user interference in the analog stage into account. Numerical experiments have shown that good performance can be achieved with the proposed algorithm and have confirmed that coordination is essential to counteract inter-UE interference in mmWave multi-user environments. Exploiting Machine Learning tools [28] for solving the proposed algorithms in a more efficient manner is an interesting and challenging problem which we aim to tackle in future studies. ACKNOWLEDGMENTS F. Maschietti, D. Gesbert and P. de Kerret are supported by the ERC under the European Union’s Horizon 2020 research and innovation program (Agreement no. 670896). 16 R EFERENCES [1] R. W. Heath, N. González-Prelcic, S. Rangan, W. Roh, and A. M. Sayeed, “An overview of signal processing techniques for millimeter wave MIMO systems,” IEEE J. Sel. Topics Signal Process., Apr. 2016. [2] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” IEEE J. Sel. 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C ARMA: Collective Adaptive Resource-sharing Markovian Agents Luca Bortolussi Rocco De Nicola Vashti Galpin Stephen Gilmore Saarland University University of Trieste ISTI - CNR IMT Lucca University of Edinburgh University of Edinburgh Jane Hillston Diego Latella Michele Loreti Mieke Massink University of Edinburgh ISTI - CNR Università di Firenze IMT Lucca ISTI - CNR In this paper we present C ARMA, a language recently defined to support specification and analysis of collective adaptive systems. C ARMA is a stochastic process algebra equipped with linguistic constructs specifically developed for modelling and programming systems that can operate in openended and unpredictable environments. This class of systems is typically composed of a huge number of interacting agents that dynamically adjust and combine their behaviour to achieve specific goals. A C ARMA model, termed a collective, consists of a set of components, each of which exhibits a set of attributes. To model dynamic aggregations, which are sometimes referred to as ensembles, C ARMA provides communication primitives that are based on predicates over the exhibited attributes. These predicates are used to select the participants in a communication. Two communication mechanisms are provided in the C ARMA language: multicast-based and unicast-based. In this paper, we first introduce the basic principles of C ARMA and then we show how our language can be used to support specification with a simple but illustrative example of a socio-technical collective adaptive system. 1 Introduction Collective adaptive systems (CAS) typically consist of very large numbers of components which exhibit autonomic behaviour depending on their properties, objectives and actions. Decision-making in such systems is complicated and interaction between their components may introduce new and sometimes unexpected behaviours. CAS are open, in the sense that components may enter or leave the collective at anytime. Components can be highly heterogeneous (machines, humans, networks, etc.) each operating at different temporal and spatial scales, and having different (potentially conflicting) objectives. We are still far from being able to design and engineer real collective adaptive systems, or even specify the principles by which they should operate. CAS thus provide a significant research challenge in terms of both representation and reasoning about their behaviour. The pervasive yet transparent nature of the applications developed in this paradigm makes it of paramount importance that their behaviour can be thoroughly assessed during their design, prior to deployment, and throughout their lifetime. Indeed their adaptive nature makes modelling essential and models play a central role in driving their adaptation. Moreover, the analysis should encompass both functional and non-functional aspects of behaviour. Thus it is vital that we have available robust modelling techniques which are able to describe such systems and to reason about their behaviour in both qualitative and quantitative terms. To move towards this goal, we consider it important to develop a theoretical foundation for collective adaptive systems that would help in understanding their distincN. Bertrand and M. Tribastone (Eds.): QAPL 2015 EPTCS 194, 2015, pp. 16–31, doi:10.4204/EPTCS.194.2 c L. Bortolussi et al. This work is licensed under the Creative Commons Attribution License. L. Bortolussi et al. 17 tive features. In this paper we present C ARMA, a language designed within the QUANTICOL project1 specifically for the specification and analysis of CAS, with the particular objective of supporting quantitive evaluation and verification. C ARMA builds on a long tradition of stochastic process algebras such as PEPA [13], MTIPP [12], EMPA [2], Stochastic π-Calculus [15], Bio-PEPA [5], MODEST [3] and others [11, 4]. It combines the lessons which have been learned from these languages with those learned from developing languages to model CAS, such as SCEL [8] and PALOMA [9], which feature attribute-based communication and explicit representation of locations. SCEL [8] (Software Component Ensemble Language), is a kernel language that has been designed to support the programming of autonomic computing systems. This language relies on the notions of autonomic components representing the collective members, and autonomic-component ensembles representing collectives. Each component is equipped with an interface, consisting of a collection of attributes, describing different features of components. Attributes are used by components to dynamically organise themselves into ensembles and as a means to select partners for interaction. The stochastic variant of SCEL, called StocS [14], was a first step towards the investigation of the impact of different stochastic semantics for autonomic processes, that relies on stochastic output semantics, probabilistic input semantics and on a probabilistic notion of knowledge. Moreover, SCEL has inspired the development of the core calculus AbC [1] that focuses on a minimal set of primitives that defines attribute-based communication, and investigates their impact. Communication among components takes place in a broadcast fashion, with the characteristic that only components satisfying predicates over specific attributes receive the sent messages, provided that they are willing to do so. PALOMA [9] is a process algebra that takes as starting point a model based on located Markovian agents each of which is parameterised by a location, which can be regarded as an attribute of the agent. The ability of agents to communicate depends on their location, through a perception function. This can be regarded as an example of a more general class of attribute-based communication mechanisms. The communication is based on a multicast, as only agents who enable the appropriate reception action have the ability to receive the message. The scope of communication is thus adjusted according to the perception function. A distinctive contribution of the new language is the rich set of communication primitives that are offered. C ARMA supports both unicast and broadcast communication, and locally synchronous, but globally asynchronous communication. This richness is important to enable the spatially distributed nature of CAS, where agents may have only local awareness of the system, yet the design objectives and adaptation goals are often expressed in terms of global behaviour. Representing these rich patterns of communication in classical process algebras or traditional stochastic process algebras would be difficult, and would require the introduction of additional model components to represent buffers, queues and other communication structures. Another feature of C ARMA is the explicit representation of the environment in which processes interact, allowing rapid testing of a system under different open world scenarios. The environment in C ARMA models can evolve at runtime, due to the feedback from the system, and it further modulates the interaction between components, by shaping rates and interaction probabilities. Furthermore the large scale nature of CAS systems makes it essential to support scalable analysis techniques, thus C ARMA has been designed anticipating both a discrete and a continuous semantics in the style of [16]. The focus of this paper is the presentation of the language and its discrete semantics, which are presented in the F U TS style [7]. The structure of the paper is as follows. Section 2 presents the syntax 1 http://www.quanticol.eu 18 C ARMA: Collective Adaptive Resource-sharing Markovian Agents of the language and explains the organisation of a model in terms of a collective of agents that are considered in the context of an environment. In Section 3 we give a detailed account of the semantics, particularly explaining the role of the environment. The use of C ARMA is illustrated in Section 4 where we describe a model of a simple bike sharing system. Some conclusions are drawn in Section 5. 2 C ARMA syntax A C ARMA system consists of a collective (N) operating in an environment (E ). The collective consists of a set of components. It models the behavioural part of a system and is used to describe a set of interacting agents that cooperate to achieve a set of given tasks. The environment models all those aspects which are intrinsic to the context where the agents under consideration are operating. The environment also mediates agent interactions. We let S YS be the set of C ARMA systems S defined by the following syntax: S ::= N in E where N is a collective and E is an environment. The latter provides the global state of the system and governs the interactions in the collective. We let C OL be the set of collectives N which are generated by the following grammar: NkN N ::= C A collective N is either a component C or the parallel composition of two collectives (N k N). A component C can be either the inactive component, which is denoted by 0, or a term of the form (P, γ), where P is a process and γ is a store. A term (P, γ) models an agent operating in the system under consideration: the process P represents the agent’s behaviour whereas the store γ models its knowledge. A store is a function which maps attribute names to basic values. We let: • ATTR be the set of attribute names a, a0 , a1 ,. . . , b, b0 , b1 ,. . . ; • VAL be the set of basic values v, v0 , v1 ,. . . ; • Γ be the set of stores γ, γ1 , γ 0 , . . . i.e. functions from ATTR to VAL. We let C OMP be the set of components C generated by the following grammar: (P, γ) C ::= 0 We let P ROC be the set of processes P, Q,. . . defined by the following grammar: P, Q ::= \| \| \| \| \| \| − act ::= α ? [π]h→ e iσ → − \| α [π]h e iσ nil kill act.P P+Q P\|Q [π]P A \| \| − α ? [π](→ x )σ → − α [π]( x )σ e ::= a \| this.a \| x \| v \| · · · 4 (A = P) π ::= > \| ⊥ \| e1 ./ e2 \| ¬π \| π ∧ π \| · · · − In C ARMA processes can perform four types of actions: broadcast output (α ? [π]h→ e iσ ), broadcast → − → − → − ? input (α [π]( x )σ ), output (α [π]h e iσ ), and input (α [π]( x )σ ). Where: L. Bortolussi et al. 19 • α is an action type in the set of action type ACT T YPE; • π is an predicate; • x is a variable in the set of variables VAR; −· indicates a sequence of elements; • → • σ is an update, i.e. a function from Γ to Dist(Γ) in the set of updates Σ; where Dist(Γ) is the set of probability distributions over Γ. The admissible communication partners of each of these actions are identified by the predicate π. This is a predicate on attribute names. Note that, in a component (P, γ) the store γ regulates the behaviour of P. Primarily, γ is used to evaluate the predicate associated with an action in order to filter the possible synchronisations involving process P. In addition, γ is also used as one of the parameters for computing the actual rate of actions performed by P. The process P can change γ immediately after the execution of an action. This change is brought about by the update σ . The update is a function that when given a store γ returns a probability distribution over Γ which expresses the possible evolutions of the store after the action execution. − The broadcast output α ? [π]h→ e iσ models the execution of an action α that spreads the values result− ing from the evaluation of expressions → e in the local store γ. This message can be potentially received by any process located at components whose store satisfies predicate π. This predicate may contain references to attribute names that have to be evaluated under the local store. These references are prefixed by the special name this. For instance, if loc is the attribute used to store the position of a component, action − α ? [distance(this.loc, loc) ≤ L]h→ v iσ potentially involves all the components located at a distance that is less than or equal to a given threshold L. The broadcast output is non-blocking. The action is executed even if no process is able to receive the values which are sent. Immediately after the execution of an action, the update σ is used to compute the (possible) effects of the performed action on the store of the hosting component where the output is performed. − To receive a broadcast message, a process executes a broadcast input of the form α ? [π](→ x )σ . This → − action is used to receive a tuple of values v sent with an action α from a component whose store − − satisfies the predicate π[→ v /→ x ]. The transmitted values can be part of the predicate π. For instance, α ? [x > 5](x)σ can be used to receive a value that is greater than 5. The other two kinds of action, namely output and input, are similar. However, differently from broadcasts described above, these actions realise a point-to-point interaction. The output operation is blocking, in contrast with the non-blocking broadcast output. Choice and parallel composition are the usual definitions for process algebras. Processes can be guarded so that [π]P behaves as the process P if the predicate π is satisfied. Finally, process kill is used to destroy a component. We assume that this term always occurs under the scope of an action prefix. C ARMA collectives operate in an environment E . This environment is used to model the intrinsic rules that govern, for instance, the physical context where our system is situated. An environment consists of two elements: a global store γg , that models the overall state of the system, and an evolution rule ρ. The latter is a function which, depending on the global store and the current state of the collective, i.e. the configurations of each component in the collective, returns a tuple of functions ε = hµ p , µr , µu i known as the evaluation context where ACT = ACT T YPE ∪ {α ? \|α ∈ ACT T YPE} and: 20 C ARMA: Collective Adaptive Resource-sharing Markovian Agents • µ p : Γ × ACT → [0, 1], expresses the probability to receive a message; • µr : Γ × Γ × ACT → R≥0 , computes the execution rate of an action; • µu : Γ × ACT → Σ × C OL, determines the updates on the environment (global store and collective) induced by the action execution. These functions regulate system behaviour. Function µ p , which takes as parameters the local stores of the two interacting components, i.e. the sender and the receiver, and the action used to interact, returns the probability to receive a message. Function µr computes the rate of an unicast/broadcast output. This function takes as parameter the local store of the component performing the action and the action on which interaction is based. Note that the environment can disable the execution of a given action. This happens when the function µr (resp. µ p ) returns the value 0. Finally, the function µu is used to update the global store and to install a new collective in the system. The function µu takes as parameters the store of the component performing the action together with the action type and returns a pair (σ , N). Within this pair, σ identifies the update on the global store whereas N is a new collective installed in the system. This function is particularly useful for modelling the arrival of new agents into a system. All of these functions are determined by an evolution rule ρ depending on the global store and the actual state of the components in the system. For instance, the probability to receive a given message may depend on the concentration of components in a given state. Similarly, the actual rate of an action may be a function of the number of components whose store satisfies a given property. 3 C ARMA operational semantics In this section we define the operational semantics of C ARMA specifications. This operational semantics · is defined in three stages. First, we introduce the transition relation −−+· that describes the behaviour of · a single component. Second, this relation is used to define the transition relation −−→· which describes · the behaviour of collectives. Finally, the transition relation 7−−→ will be defined to show how C ARMA systems evolve. All these transition relations are defined in the F U TS style [7]. Using this approach, a transition relation is described using a triple of the form (N, `, N ). The first element of this triple is either a component, or a collective, or a system. The second element is a transition label. The third element is a function associating each component, collective, or system with a non-negative number. A non-zero value represents the rate of the exponential distribution characterising the time needed for the execution of the action represented by `. The zero value is associated with unreachable terms. We use the F U TS style semantics because it makes explicit an underlying Action Labelled Markov Chain, which can be simulated with standard algorithms [10] but is nevertheless more compact than Plotkin-style semantics, as the functional form allows different possible outcomes to be treated within a single rule. A complete description of F U TS and their use can be found in [7]. 3.1 Operational semantics of components We use the transition relation + − ε ⊆ C OMP × L AB × [C OMP → R≥0 ] to define the behaviour of a single component. In this relation [C OMP → R≥0 ] denotes the set of functions from C OMP to R≥0 and L AB is L. Bortolussi et al. 21 the set of transition labels ` which are generated by the following grammar: − ` ::= α ? [π]h→ v i, γ Broadcast output \| \| − α ? [π](→ v ), γ → − α [π]h v i, γ Broadcast input Unicast Output \| − α [π](→ v ), γ − τ[α [π]h→ v i, γ] \| − R[α ? [π](→ v ), γ] Broadcast Input Refusal \| Unicast Input Unicast Synchronization The first four labels are associated with the four C ARMA input-output actions and they contain a reference to the action which is performed (α or α ? ), the store of the component where the action is executed (γ), − and the value which is transmitted or received. The transition label τ[α [π]h→ v i, γ] is the one which → − ? is associated with unicast synchronisation. The final label R[α [π]( v ), γ] denotes the case where a component is not able to receive a broadcast output. This arises at the level of the single component either because the associated message has been lost, or because no process is willing to receive that − message. We will observe later in this section that the use of R[α ? [π](→ v ), γ] labels are crucial to handle appropriately dynamic process operators, namely choice and guard. The transition relation + − ε , as formally defined in Table 1 and Table 2, is parametrised with respect to an evaluation context ε. This is used to compute the actual rate of process actions and to compute the probability to receive messages. The process nil denotes the process that cannot perform any action. The transitions which are induced by this process at the level of components can be derived via rules Nil and Nil-F1. These rules respectively say that the inactive process cannot perform any action, and always refuses any broadcast input. Note that, the fact that a component (nil, γ) does not perform any transition is derived from the fact that any label that is not a broadcast input refusal leads to function 0/ (rule Nil). Indeed, 0/ denotes the 0 constant function. Conversely, Nil-F1 states that (nil, γ) can always perform a transition labelled − R[α ? [π](→ v ), γ] leading to [(nil, γ) 7→ 1], where [C 7→ v] denotes the function mapping the component C to v ∈ R≥0 and all the other components to 0. − The behaviour of a broadcast output (α ? [π1 ]h→ e iσ .P, γ) is described by rules B-Out, B-Out-F1 and − B-Out-F2. Rule B-Out states that a broadcast output α ? [π]h→ e iσ can affect components that satisfy 0 2 π = JπKγ . The action rate is determined by the evaluation context ε = hµ p , µr , µu i and, in particular, by the function µr . This function, given a store γ and the kind of action performed, in this case α ? , returns a value in R≥0 . If this value is greater than 0, it denotes the execution rate of the action. However, the evaluation context can disable the execution of some actions. This happens when µr (γ, α ? ) = 0. The possible next local stores after the execution of an action are determined by the update σ . This takes the store γ and yields a probability distribution p = σ (γ) ∈ Dist(Γ). In rule B-Out, and in the rest of the paper, the following notations are used: • let P ∈ P ROC and p ∈ Dist(Γ), (P, p) is a probability distribution in Dist(C OMP) such that:  P ≡ Q\|kill ∧ C ≡ 0  1 p(γ) C ≡ (P, γ) ∧ P 6≡ Q\|kill (P, p)(C) =  0 otherwise • let c ∈ Dist(C OMP) and r ∈ R≥0 , r · c denotes the function C : C OMP → R≥0 such that: C (C) = r · c(C) 2 We let J·Kγ denote the evaluation function of an expression/predicate with respect to the store γ. 22 C ARMA: Collective Adaptive Resource-sharing Markovian Agents − ` 6= R[α ? [π](→ v ), γ] Nil ` Nil-F1 − R[α ? [π](→ v ),γ] (nil, γ) + − ε 0/ (nil, γ) −−−−−−−−+ε [(nil, γ) 7→ 1] JπKγ = π 0 − − J→ e Kγ = → v p = σ (γ) ε = hµ p , µr , µu i − α ? [π 0 ]h→ v i,γ − (α ? [π]h→ e iσ .P, γ) −−−−−−−+ε µr (γ, α ? ) · (P, p) B-Out → − R[β ? [π2 ]( v ),γ] − − (α ? [π1 ]h→ e iσ .P, γ) −−−−−−−−+ε [(α ? [π1 ]h→ e iσ .P, γ) 7→ 1] JπKγ = π 0 B-Out-F1 − − J→ e Kγ = → v − − ` 6= α ? [π 0 ]h→ v i, γ ` 6= R[β ? [π 0 ](→ v 1 ), γ] B-Out-F2 ` − (α ? [π]h→ e iσ .P, γ) + − ε 0/ − − Jπ2 [→ v /→ x ]Kγ2 = π20 γ1 \|= π20 − − p = σ [→ v /→ x ](γ2 ) ε = hµ p , µr , µu i γ2 \|= π1 → − α ? [π1 ]( v ),γ1 − − − (α ? [π2 ](→ x )σ .P, γ2 ) −−−−−−−+ε µ p (γ1 , γ2 , α ? ) · (P[→ v /→ x ], p) − − Jπ2 [→ v /→ x ]Kγ2 = π20 − (α ? [π2 ](→ x )σ .P, γ2 ) γ1 \|= π20 − R[α ? [π1 ](→ v ),γ1 ] −−−−−−−−−+ε γ2 \|= π1 [(α ? [π − − Jπ2 [→ v /→ x ]Kγ2 = π20 ε = hµ p , µr , µu i → − ? 2 ]( x )σ .P, γ2 ) 7→ 1 − µ p (γ1 , γ2 , α )] (γ1 6\|= π20 or γ2 6\|= π1 ) → − α ? [π1 ]( v ),γ1 − (α ? [π2 ](→ x )σ .P, γ2 ) −−−−−−−+ε 0/ B-In-F1 B-In-F2 − − ` 6= α ? [π1 ](→ v ), γ1 ` 6= R[α ? [π1 ](→ v ), γ1 ] B-In-F3 ` → − (α ? [π ]( x )σ .P, γ ) + − 0/ 2 2 ε α 6= β (α ? [π R[β ? [π → − 1 ]( v ),γ1 ] → − − −−−−−−−−+ε [(α ? [π2 ](→ x )σ .P, γ2 ) 7→ 1] 2 ]( x )σ .P, γ2 ) − B-In-F4 Table 1: Operational semantics of components (Part 1) B-In L. Bortolussi et al. 23 Note that, after the execution of an action a component can be destroyed. This happens when the continuation process after the action prefixing contains the term kill. For instance, by applying rule α ? [π1 ]hvi,γ B-Out we have that: (α ? [π1 ]hviσ .(kill\|Q), γ) −−−−−−+ε [0 7→ r]. Rule B-Out-F1 states that a broadcast output always refuses any broadcast input, while B-Out-F2 − − states that a broadcast output can be only involved in labels of the form α ? [π]h→ v i, γ or R[β ? [π2 ](→ v ), γ]. − Transitions related to a broadcast input are labelled with α ? [π1 ](→ v ), γ1 . There, γ1 is the store of the component executing the output, α is the action performed, π1 is the predicate that identifies the − target components, while → v is the sequence of transmitted values. Rule B-In states that a component − (α ? [π2 ](→ x )σ .P, γ2 ) can perform a transition with this label when its store γ2 satisfies the target predicate, − − i.e. γ2 \|= π1 , and the component executing the action satisfies the predicate π2 [→ v /→ x ]. The evaluation context ε = hµ p , µr , µu i can influence the possibility to perform this action. This transition can be performed with probability µ p (γ1 , γ2 , α ? ). Rule B-In-F1 models the fact that even if a component can potentially receive a broadcast message, the message can get lost according to a given probability regulated by the evaluation context, namely 1 − µ p (γ1 , γ2 , α ? ). Rule B-In-F2 models the fact that if a component is not in the set of possible receivers (γ2 6\|= π1 ) or the sender does not satisfy the expected requirements (γ1 6\|= π20 ) then the component cannot − receive a broadcast message. Finally, rules B-In-F3 and B-In-F4 model the fact that (α ? [π2 ](→ x )σ .P, γ2 ) can only perform a broadcast input on action α and that it always refuses input on any other action type β 6= α, respectively. The behaviour of unicast output and unicast input is defined by the first six rules of Table 2. These rules are similar to the ones already presented for broadcast output and broadcast input. The only difference is that both unicast output (Out-F1) and unicast input (In-F1) always refuse any broadcast input with probability 1. The other rules of Table 2 describe the behaviour of other process operators, namely choice P + Q, parallel composition P\|Q, guard and recursion. The term P + Q identifies a process that can behave either as P or as Q. The rule Plus states that the components that are reachable by (P + Q, γ), via a transition that is not a broadcast input refusal, are the ones that can be reached either by (P, γ) or by (Q, γ). In this rule we use C1 ⊕ C2 to denote the function that maps each term C to C1 (C) + C2 (C), for any C1 , C2 ∈ [C OMP → R≥0 ]. At the same time, process P + Q refuses a broadcast input when both the process P and Q do that. This is modelled by Plus-F1, where, for each C1 : C OMP → R≥0 and C2 : C OMP → R≥0 , C1 + C2 denotes the function that maps each term of the form (P + Q, γ) to C1 ((P, γ)) · C2 ((Q, γ)), while any other component is mapped to 0. Note that, differently from rule Plus, when rule Plus-F1 is applied operator + is not removed after the transition. This models the fact that when a broadcast message is refused the choice is not resolved. In P\|Q the two composed processes interleave for all the transition labels except for broadcast input refusal (Par). For this label the two processes synchronise (Par-F1). This models the fact that a message is lost when both processes refuse to receive it. In the rules the following notations are used: • for each component C and process Q we let: 0 C≡0 C\|Q = (P\|Q, γ) C ≡ (P, γ) Q\|C is symmetrically defined. • for each C : C OMP → R≥0 and process Q, C \|Q (resp. Q\|C ) denotes the function that maps each term of the form C\|Q (resp. Q\|C) to C (C), while the others are mapped to 0; • for each C1 : C OMP → R≥0 and C2 : C OMP → R≥0 , C1 \|C2 denotes the function that maps each term of the form (P\|Q, γ) to C1 ((P, γ)) · C2 ((Q, γ)), while the others are mapped to 0. 24 C ARMA: Collective Adaptive Resource-sharing Markovian Agents − − J→ e Kγ = → v JπKγ = π 0 p = σ (γ) ε = hµ p , µr , µu i − α [π 0 ]h→ v i,γ − (α [π]h→ e iσ .P, γ) −−−−−−+ε µr (γ, α) · (P, p) Out Out-F1 → − R[β ? [π2 ]( v ),γ2 ] − − (α [π1 ]h→ e iσ .P, γ1 ) −−−−−−−−−+ε [(α [π1 ]h→ e iσ .P, γ1 ) 7→ 1] − − J→ e Kγ = → v − − ` 6= α [π 0 ]h→ v i, γ ` 6= R[α ? [π 0 ](→ v ), γ] Out-F2 ` − (α [π]h→ e iσ .P, γ) + − ε 0/ JπKγ = π 0 − − Jπ2 [→ v /→ x ]Kγ2 = π20 γ1 \|= π20 − − p = σ [→ v /→ x ](γ2 ) ε = hµ p , µr , µu i γ2 \|= π1 → − α [π1 ]( v ),γ1 − − − (α [π2 ](→ x )σ .P, γ2 ) −−−−−−−+ε µ p (γ1 , γ2 , α) · (P[→ v /→ x ], p) → − R[β ? [π1 ]( v ),γ1 ] − − (α [π2 ](→ x )σ .P, γ2 ) −−−−−−−−−+ε [(α [π2 ](→ x )σ .P, γ2 ) 7→ 1] − − Jπ2 [→ v /→ x ]Kγ2 = π20 (γ1 6\|= π20 or γ2 6\|= π1 ) → − α [π1 ]( v ),γ1 − (α [π2 ](→ x )σ .P, γ2 ) −−−−−−−+ε 0/ ` In-F2 2 (Q, γ) + − ε C2 2 − ` 6= R[α ? [π 0 ](→ v ), γ] ` (P + Q, γ) + − ε C1 ⊕ C2 − R[α ? [π 0 ](→ v ),γ] In-F1 − − ` 6= α [π1 ](→ v ), γ1 ` 6= R[β ? [π1 ](→ v ), γ1 ] In-F3 ` − (α [π ](→ x )σ .P, γ ) + − 0/ ` (P, γ) + − ε C1 In ε Plus − R[α ? [π 0 ](→ v ),γ] (P, γ) −−−−−−−−+ε C1 (Q, γ) −−−−−−−−+ε C2 Plus-F1 − R[α ? [π 0 ](→ v ),γ] (P + Q, γ) −−−−−−−−+ε C1 + C2 ` (P, γ) + − ε C1 ` (Q, γ) + − ε C2 − ` 6= R[α ? [π](→ v ), γ] ` (P\|Q, γ) + − ε C1 \|Q ⊕ P\|C2 − R[α ? [π](→ v ),γ] (P, γ) −−−−−−−−+ε C1 − R[α ? [π](→ v ),γ] (Q, γ) −−−−−−−−+ε C2 − R[α ? [π](→ v ),γ] 4 A=P Par-F1 ` (P, γ) + −ε C − ` 6= R[α ? [π](→ v ), γ] ` ([π]P, γ) + −ε C γ 6\|= π − ` 6= R[α ? [π](→ v ), γ] ` ([π]P, γ) + − ε 0/ (A, γ) + −ε C γ \|= π Guard ` (P, γ) + −ε C ` (P\|Q, γ) −−−−−−−−+ε C1 \|C2 γ \|= π Par Rec − R[α ? [π](→ v ),γ] (P, γ) −−−−−−−−+ε C − R[α ? [π](→ v ),γ] Guard-F1 ([π]P, γ) −−−−−−−−+ε [π]C γ 6\|= π Guard-F2 − R[α ? [π](→ v ),γ] ([π]P, γ) −−−−−−−−+ε [([π]P, γ) 7→ 1] Table 2: Operational semantics of components (Part 2) Guard-F3 L. Bortolussi et al. 25 − α ? [π](→ v ),γ Zero ` (P, γ) −−−−−−→ε − ` 6= R[α ? [π](→ v ), γ] ` ` (P, γ) → −ε N − α ? [π]h→ v i,γ N1 −−−−−−→ε N1o − α [π]h→ v i,γ N1 −−−−−−→ε N1 Comp − α ? [π](→ v ),γ − α ? [π]h→ v i,γ − α ? [π](→ v ),γ N2 −−−−−−→ε N2 B-In-Sync N1 k N2 −−−−−−→ε N1 k N2 − α ? [π]h→ v i,γ N2 −−−−−−→ε N2o k N2i ) ⊕ (N1i − α [π]h→ v i,γ − α [π]h→ v i,γ Comp-B-In − α ? [π](→ v ),γ N2 −−−−−−→ε (N1o N2 −−−−−−→ε N2 N1 ⊕N2 ⊕N1 +⊕N2 − α ? [π](→ v ),γ N1 −−−−−−→ε N1i N1 k N1 −−−−−−→ε N1 (P, γ) −−−−−−−−+ε N2 − α ? [π](→ v ),γ 0→ − ε 0/ (P, γ) + −ε N − R[α ? [π](→ v ),γ] (P, γ) −−−−−−+ε N1 k − α ? [π](→ v ),γ N2 −−−−−−→ε N2i − α [π](→ v ),γ Out-Sync N1 k N2 −−−−−−→ε N1 k N2 ⊕ N1 k N2 B-Sync N2o ) − α [π](→ v ),γ N1 −−−−−−→ε N1 N2 −−−−−−→ε N2 − α [π](→ v ),γ In-Sync N1 k N2 −−−−−−→ε N1 k N2 ⊕ N1 k N2 − τ[α [π]h→ v i,γ] − α [π]h→ v i,γ − α [π](→ v ),γ − τ[α [π]h→ v i,γ] − α [π]h→ v i,γ − α [π](→ v ),γ N1 −−−−−−−→ε N1s N1 −−−−−−→ε N1o N1 −−−−−−→ε N1i N2 −−−−−−−→ε N2s N2 −−−−−−→ε N2o N2 −−−−−−→ε N2i N1 k − τ[α [π]h→ v i,γ] (N1s kN2 )·⊕N1i N2 −−−−−−−→ε ⊕N i i 1 +⊕N2 ⊕ (N1 kN2s )·⊕N2i ⊕N1i +⊕N2i ⊕ (N1o kN2i ) ⊕N1i +⊕N2i ⊕ (N1i kN2o ) ⊕N1i +⊕N2i Sync Table 3: Operational semantics of collective Rule Rec is standard. The behaviour of ([π]P, γ) is regulated by rules Guard, Guard-F1, Guard-F2 and Guard-F3. The first two rules state that ([π]P, γ) behaves exactly like (P, γ) when γ satisfies predicate π. However, in the first case the guard is removed when a transition is performed. In contrast, the guard still remains active after the transition when a broadcast input is refused. This is similar to what we consider for the rule Plus-F1 and models the fact that broadcast input refusals do not remove dynamic operators. In rule Guard-F1 we let [π]C denote the function that maps each term of the form ([π]P, γ) to C ((P, γ))) and any other term to 0, for each C : C OMP → R≥0 . Rules Guard-F2 and Guard-F3 state that no component can be reached from ([π]P, γ) and all the broadcast messages are refused when γ does not satisfy predicate π. 3.2 Operational semantics of collective The operational semantics of a collective is defined via the transition relation → − ε ⊆ C OL × L AB × [C OL → R≥0 ]. This relation is formally defined in Table 3. We use a straightforward adaptation of the notations introduced in the previous section. Rules Zero, Comp-B-In and Comp describe the behaviour of the single component at the level of collective. Rule Zero is similar to rule Nil of Table 1 and states that inactive component 0 cannot perform any action. Rule Comp-B-In states that the result of a broadcast input of a component at the level of collective is obtained by combining (summing) the transition at the level of components − − labelled α ? [π](→ v ), γ with the one labelled R[α ? [π](→ v ), γ]. This value is then renormalised to obtain a probability distribution. There we use ⊕N to denote ∑N∈C OL N (N). The renormalisation guarantees a reasonable computation of broadcast output synchronisation rates (see comments on rule B-Sync below). 26 C ARMA: Collective Adaptive Resource-sharing Markovian Agents Note that each component can always perform a broadcast input at the level of collective. However, we are not able to observe if the message has been received or not. Moreover, thanks to renormalisation, if − α ? [π](→ v ),γ C −−−−−−→ε N then ⊕N = 1, i.e. N is a probability distribution over C OL. Rule Comp simply states that for the single component C 6= 0 all the transition labels that are not a broadcast input, the relation ` ` → − ε coincides with the relation + − ε. Rules B-In-Sync and B-Sync describe broadcast synchronisation. The former states that two collectives N1 and N2 that operate in parallel synchronise while performing a broadcast input. This models the fact that the input can be potentially received by both of the collectives. In this rule we let N1 k N2 denote the function associating the value N1 (N1 ) · N2 (N2 ) with each term of the form N1 k N2 and 0 with − α ? [π](→ v ),γ all the other terms. We can observe that if N −−−−−−→ε N then, as we have already observed for rule Comp-B-In, ⊕N = 1 and N is in fact a probability distribution over C OL. Rule B-Sync models the synchronisation consequent of a broadcast output performed at the level of a collective. For each N1 : C OL → R≥0 and N2 : C OL → R≥0 , N1 ⊕ N2 denotes the function that maps each term N to N1 (N) + N2 (N). − At the level of collective a transition labelled α ? [π]h→ v i, γ identifies the execution of a broadcast output. When a collective of the form N1 k N2 is considered, the result of these kinds of transitions must be computed (in the F U TS style) by considering: − α ? [π]h→ v i,γ • the broadcast output emitted from N1 , obtained by the transition N1 −−−−−−→ε N1o − α ? [π](→ v ),γ • the broadcast input received by N1 , obtained by the transition N1 −−−−−−→ε N1i − α ? [π]h→ v i,γ • the broadcast output emitted from N2 , obtained by the transition N2 −−−−−−→ε N2o − α ? [π](→ v ),γ • the broadcast input received by N2 , obtained by the transition N2 −−−−−−→ε N2i Note that the first synchronises with the last to obtain N1o k N2i , while the second synchronises with the third to obtain N1i k N2o . The result of such synchronisations are summed to model the race condition between the broadcast outputs performed within N1 and N2 respectively. We have to remark that above N1o (resp. N2o ) is 0/ when N1 (resp. N2 ) is not able to perform any broadcast output. Moreover, the label of a broadcast synchronisation is again a broadcast output. This allows further synchronisations in a derivation. Finally, it is easy to see that the total rate of a broadcast synchronisation is equal to the total rate of broadcast outputs. This means that the number of receivers does not affect the rate of a broadcast that is only determined by the number of senders. Rules Out-Sync, In-Sync and Sync control the unicast synchronisation. Rule Out-Sync states that a collective of the form N1 k N2 performs a unicast output if this is performed either in N1 or in N2 . This is rendered in the operational semantics as an interleaving rule, where for each N : C OL → R≥0 , N k N2 denotes the function associating N (N1 ) with each collective of the form N1 k N2 and 0 with all other collectives. Rule In-Sync is similar to Out-Sync. However, it considers unicast input. Finally, rule Sync regulates the unicast synchronisations and generates transitions with labels of the − − form τ[α [π]h→ v i, γ]. This is the result of a synchronisation between transitions labelled α [π](→ v ), γ, i.e. → − an input, and α [π]h v i, γ, i.e. an output. − v i, γ]), unicast output In rule Sync, Nks , Nko and Nki denote the result of synchronisation (τ[α [π]h→ → − → − (α [π]h v i, γ) and unicast input (α [π]( v ), γ) within Nk (k = 1, 2), respectively. The result of a transition − labelled τ[α [π]h→ v i, γ] is therefore obtained by combining: • the synchronisations in N1 with N2 : N1s k N2 ; L. Bortolussi et al. 27 − α ? [π]h→ v i,γ ρ(γg , N) = ε = hµr , µ p , µu i N −−−−−−→ε N µu (γg , α ? ) = (σ , N 0 ) − α ? [π]h→ v i,γ Sys-B N in (γg , ρ) 7−−−−−−→ N k N 0 in (σ (γg ), ρ) − τ[α [π]h→ v i,γ] ρ(γg , N) = ε = hµr , µ p , µu i N −−−−−−−→ε N µu (γg , α) = (σ , N 0 ) − τ[α [π]h→ v i,γ] Sys N in (γg , ρ) 7−−−−−−−→ N k N 0 in (σ (γg ), ρ) Table 4: Operational Semantics of Systems. • the synchronisations in N2 with N1 : N1 k N2s ; • the output performed by N1 with the input performed by N2 : N1o k N2i ; • the input performed by N1 with the output performed by N2 : N1i k N2o . To guarantee a correct computation of synchronisation rates, the first two addendi are renormalised by considering inputs performed in N2 and N1 respectively. This, on one hand, guarantees that the total rate − − of synchronisation τ[α [π]h→ v i, γ] does not exceed the output capacity, i.e. the total rate of α [π]h→ v i, γ in N1 and N2 . On the other hand, since synchronisation rates are renormalised during the derivation, it also ensures that parallel composition is associative [7]. 3.3 Operational semantics of systems The operational semantics of systems is defined via the transition relation → 7− ⊆ S YS × L AB × [S YS → R≥0 ] that is formally defined in Table 4. Only synchronisations are considered at the level of systems. The first rule is Sys-B. This rule states that a system of the form N in (γg , ρ) can perform a broadcast output when the collective N, under the environment evaluation ε = hµr , µ p , µu i = ρ(γg , N), can evolve − at the level of collective with the label α ? [π]h→ v i, γ to N . After the transition, the global store is updated and a new collective can be created according to function µu . In rule Sys-B the following notations are used. For each collective N2 , N : C OL → R≥0 , S : S YS → R≥0 and p ∈ Dist(Γ) we let N in (p, ρ) denote the function mapping each system N in (γ, ρ) to N (N) · p(γ). The second rule is Sys that is similar to Sys-B and regulates unicast synchronisations. 4 C ARMA at work In this section we will use C ARMA to model a bike sharing system [6, 17]. These systems are a recent, and increasingly popular, form of public transport in urban areas. As a resource-sharing system with large numbers of independent users altering their behaviour due to pricing and other incentives, they are a simple instance of a collective adaptive system, and hence a suitable case study to exemplify the C ARMA language. The idea in a bike sharing system is that bikes are made available in a number of stations that are placed in various areas of a city. Users that plan to use a bike for a short trip can pick up a bike at a suitable origin station and return it to any other station close to their planned destination. One of the major issues in bike sharing systems is the availability and distribution of resources, both in terms of available bikes at the stations and in terms of available empty parking places in the stations, where users will park the bikes after using them. 28 C ARMA: Collective Adaptive Resource-sharing Markovian Agents In our scenario we assume that the city is partitioned in homogeneous zones and that all the stations in the same zone can be equivalently used by any user in that zone. Below, we let {z0 , . . . , zn } be the n zones in the city, each of which contains k parking stations. Each parking station is modelled in C ARMA via a component of the form: ( G\|R , {zone = `, bikes = i, slots = j}) where • zone is the attribute identifying the zone where the parking station is located; • bikes is the attribute used to count the number of available bikes; • slots is the attribute containing the total number of parking slots in the parking station. Processes G and R, which model the procedure to get and return a bike in the parking station, respectively, are defined as follow: 4 G = [bikes > 0] get[zone = this.zone]h•i{bikes ← bikes − 1}.G 4 R = [slots > bikes] ret[zone = this.zone]h•i{bikes ← bikes + 1}.R Process G, when the value of attribute bikes is greater than 0, executes the unicast output with action type get that potentially involves components satisfying the predicate zone = this.zone, i.e. the ones that are located in the same zone3 . When the output is executed the value of the attribute bikes is decreased by one to model the fact that one bike has been retrieved from the parking station. Process R is similar. It executes the unicast output with action type ret that potentially involves components satisfying predicate zone = this.zone. This action can be executed only when there is at least one parking slot available, i.e. when the value of attribute bikes is less than the value of attribute slots. When the output considered above is executed, the value of attribute bikes is increased by one to model the fact that one bike has been returned in the parking station. Users, who can be either bikers or pedestrians, are modelled via components of the form: (Q, {zone = `}) where zone is the attribute indicating where the user is located, while Q models the current state of the user and can be one of the following processes: 4 B = move? [⊥]h•i{zone ← U(z0 , . . . , zn )}.B + stop? [⊥]h•i.W S 4 W S = ret[zone = this.zone](•).P 4 P = go? [⊥]h•i.W S 4 W B = get[zone = this.zone](•).B Process B represents a biker. When a user is in this state (s)he can either move from the current zone to another zone or stop to return the bike to a parking station. These activities are modelled with the 3 Here we use • to denote the unit value. L. Bortolussi et al. 29 10 Min. bikes per Park Max. bikes per Park Avg. bikes per Park Bikes 8 6 4 2 0 0 20 40 60 80 100 Time Figure 1: Simulation of bike scenario. execution of a broadcast output via action types move and stop, respectively. Note that in both of these cases, the predicate used to identify the target of the actions is ⊥, denoting the value false. This means that neither of the two actions actually synchronise with any component (since no component satisfies ⊥). This kind of interaction is used in C ARMA to model spontaneous actions, i.e. actions that render the execution of an activity and that do no require synchronisation. After the broadcast move? the value of attribute zone is updated by randomly selecting the next zone in {z0 , . . . , zn }. With {zone ← U(z0 , . . . , zn )} we denote the update σ such that σ (γ) is the probability distribution giving probability n1 to each store γ[zone ← zi ]. This update models a random movement of the user among the city zones. When process B executes broadcast stop? , it evolves to process W S. This process models a user who is waiting for a parking slot. This process executes an input over ret. This models the fact that the user has found a parking station with an available parking slot in their zone. After the execution of this input process P is executed. The latter component definition models a pedestrian user. The user remains in this state until the spontaneous action go? is performed. After that it evolves to process W B which models a user waiting for a bike. The behaviour of W B is similar to that of W S described above. Using a custom-built prototype simulator, we are able to simulate this modelled scenario. The output on one simulation run is presented in Figure 1. In the graph we show the minimum, average and maximum number of bikes in one zone of the city. We consider a scenario with four zones each containing four parking stations. The total number of users is 150. 5 Conclusions We have presented C ARMA, a new stochastic process algebra for the representation of systems developed in the CAS paradigm. The language offers a rich set of communication primitives, and the use of attributes, captured in a store associated with each component, allows attribute-based communication. For most CAS systems we anticipate that one of the attributes will be the location of the agent and thus it is straightforward to capture systems in which, for example, there is a limited scope of communication, or restriction to only interact with components that are co-located. As demonstrated in the case study presented in Section 4, attributes can also be used to capture the "state" of a component, such as the available number of bikes/slots at a bike station. C ARMA reflects the experience that we have gained through earlier languages such as SCEL [8], its Markovian variants [14] and PALOMA [9]. Compared with SCEL, the representation of knowledge here is more abstract, and not designed for detailed reasoning during the evolution of the model. This 30 C ARMA: Collective Adaptive Resource-sharing Markovian Agents reflects the different objectives of the languages. Whilst SCEL is designed to support the programming of autonomic computing systems, the primary focus of C ARMA is quantitative analysis. In stochastic process algebras such as PEPA, MTIPP and EMPA, data is typically abstracted away, and the influence of data on behaviour is captured only stochastically. When the data is important to differentiate behaviour it must be implicitly encoded in the state of components. In the context of CAS we wish to support attribute-based communication to reflect the flexible and dynamic interactions that occur in such systems. Thus it is not possible to entirely abstract from data. On the other hand, the level of abstraction means that choices within the system will be captured stochastically rather than through the rich policies for reasoning offered by SCEL. We believe that this offers a reasonable compromise between expressiveness and tractability. Another key feature of C ARMA is the inclusion of an explicit environment in which components interact. In PALOMA there was a rudimentary form of environment, termed the perception function but this proved cumbersome to use, and it could not itself be influenced by the behaviour of the components. In C ARMA, in contrast, the environment not only modulates the rates and probabilities related to interactions between components, it can also itself evolve at runtime, due to feedback from the collective. The focus of this paper has been the discrete semantics in the structured operational style of FUTS [7], but in future work we plan to develop differential semantics in the style of [16]. This latter approach will be essential in order to support quantitative analysis of CAS systems of realistic scale, but it may not be possible to encompass the full rich set of language features of C ARMA with such efficient analysis. Further work is needed to investigate this issue, and which language features can be supported for the various forms of quantitative analysis available. Additional work involves the development of an appropriate high-level language for designers of CAS which will be mapped to the process algebra, and hence will enable qualitative and quantitive analysis of CAS during system development by enabling a design workflow and analysis pathway. The intention of this high-level language is not to add to the expressiveness of C ARMA, which we believe to be well-suited to capturing the behaviour of CAS, but rather to ease the task of modelling for users who are unfamiliar with process algebra and similar formal notations. Acknowledgements This work is partially supported by the EU project QUANTICOL, 600708. This research has also been partially funded by the German Research Council (DFG) as part of the Cluster of Excellence on Multimodal Computing and Interaction at Saarland University. References [1] Yehia Abd Alrahman, Rocco De Nicola, Michele Loreti, Francesco Tiezzi & Roberto Vigo (2015): A Calculus for Attribute-based Communication. In: Proceedings of SAC 2015, doi:10.1145/2695664.2695668. To appear. [2] Marco Bernardo & Roberto Gorrieri (1998): A Tutorial on EMPA: A Theory of Concurrent Processes with Nondeterminism, Priorities, Probabilities and Time. Theoretical Computer Science 202(1-2), pp. 1–54, doi:10.1016/S0304-3975(97)00127-8. [3] H.C. Bohnenkamp, P.R. D’Argenio, H. Hermanns & J-P. Katoen (2006): MODEST: A Compositional Modeling Formalism for Hard and Softly Timed Systems. IEEE Trans. Software Eng. 32(10), pp. 812–830, doi:10.1109/TSE.2006.104. L. Bortolussi et al. 31 [4] Luca Bortolussi & Alberto Policriti (2010): Hybrid dynamics of stochastic programs. Theor. Comput. Sci. 411(20), pp. 2052–2077, doi:10.1016/j.tcs.2010.02.008. [5] Federica Ciocchetta & Jane Hillston (2009): Bio-PEPA: A Framework for the Modelling and Analysis of Biological Systems. Theoretical Computer Science 410(33), pp. 3065–3084, doi:10.1016/j.tcs.2009.02.037. [6] Paola De Maio (2009): Bike-sharing: Its History, Impacts, Models of Provision, and Future. Journal of Public Transportation 12(4), pp. 41–56, doi:10.5038/2375-0901.12.4.3. [7] Rocco De Nicola, Diego Latella, Michele Loreti & Mieke Massink (2013): A uniform definition of stochastic process calculi. ACM Comput. Surv. 46(1), p. 5, doi:10.1145/2522968.2522973. [8] Rocco De Nicola, Michele Loreti, Rosario Pugliese & Francesco Tiezzi (2014): A Formal Approach to Autonomic Systems Programming: The SCEL Language. TAAS 9(2), p. 7, doi:10.1145/2619998. 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[13] Jane Hillston (1995): A Compositional Approach to Performance Modelling. CUP. [14] Diego Latella, Michele Loreti, Mieke Massink & Valerio Senni (2014): Stochastically timed predicate-based communication primitives for autonomic computing. In Nathalie Bertrand & Luca Bortolussi, editors: Proceedings Twelfth International Workshop on Quantitative Aspects of Programming Languages and Systems, QAPL 2014, Grenoble, France, 12-13 April 2014., EPTCS 154, pp. 1–16, doi:10.4204/EPTCS.154.1. [15] Corrado Priami (1995): Stochastic π-calculus. The Computer Journal 38(7), pp. 578–589, doi:10.1093/comjnl/38.7.578. [16] Mirco Tribastone, Stephen Gilmore & Jane Hillston (2012): Scalable Differential Analysis of Process Algebra Models. IEEE Transactions on Software Engineering 38(1), pp. 205–219, doi:10.1109/TSE.2010.82. [17] Wikipedia (2013): Bicycle sharing system — Wikipedia, The Free Encyclopedia. Available at http: //en.wikipedia.org/w/index.php?title=Bicycle_sharing_system&oldid=573165089. 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AN INVITATION TO 2D TQFT AND QUANTIZATION OF HITCHIN SPECTRAL CURVES arXiv:1705.05969v1 [math.AG] 17 May 2017 OLIVIA DUMITRESCU AND MOTOHICO MULASE Abstract. This article consists of two parts. In Part 1, we present a formulation of twodimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Necessary backgrounds from Frobenius algebras, topological quantum field theories, and cohomological field theories are reviewed. A result on Frobenius algebra twisted topological recursion is included at the end of Part 1. In Part 2, we explain a geometric theory of quantum curves. The focus is placed on the process of quantization as a passage from families of Hitchin spectral curves to families of opers. To make the presentation simpler, we unfold the story using SL2 (C)-opers and rank 2 Higgs bundles defined on a compact Riemann surface C of genus greater than 1. In this case, quantum curves, opers, and projective structures in C all become the same notion. Background materials on projective coordinate systems, Higgs bundles, opers, and non-Abelian Hodge correspondence are explained. Contents 0. Part 1. 2. 3. 4. 5. 6. Preface: An inspiration from the past 2 1. Topological Quantum Field Theory Frobenius algebras TQFT Cohomological field theory Category of cell graphs 2D TQFT from cell graphs TQFT-valued topological recursion Part 2. Quantization of Higgs Bundles 7. Quantum curves 8. Projective structures, opers, and Higgs bundles 9. Semi-classical limit of SL2 (C)-opers 10. Non-Abelian Hodge correspondence between Hitchin moduli spaces References 5 5 9 13 21 26 29 33 33 36 47 48 50 2010 Mathematics Subject Classification. Primary: 14H15, 14N35, 81T45; Secondary: 14F10, 14J26, 33C05, 33C10, 33C15, 34M60, 53D37. Key words and phrases. Topological quantum field theory; quantum curves; opers; Hitchin moduli spaces; Higgs bundles; Hitchin section; quantization; topological recursion. 1 2 O. DUMITRESCU AND M. MULASE 0. Preface: An inspiration from the past A recent discovery of a cuneiform tablet dated around 350 to 50 B.C.E. suggests that ancient Babylonians must have used geometry of time-momentum space to establish accurate calculations of Jupiter’s orbit [87]. This impressive paper also contains the picture of the tablet which describes the Babylonian’s method of integration. Figure 0.1. The cuneiform tablet used in the analysis of [87]. The orbit of the Jupiter is a graph in coordinate space-time. By considering the graph of momentum of the Jupiter in time-momentum space, Babylonians visualized integral of the momentum by the area underneath the curve, and using a trapezoidal approximation, they actually obtained an estimated value of the integral. This gives a prototype of Newton’s Fundamental Theorem of Calculus. This idea of Babylonians relating geometry of timemomentum space with the analysis of actual orbit of the Jupiter is striking, because it suggests their equal treatment of coordinate space the momentum space. Although it is a stretch, we could imagine the very foundation of symplectic geometry here. Many mathematical cuneiform tablets recording numbers and algebraic calculations have been our source of imagination. The most famous is Plimpton 322 of around 1,800 B.C.E. (see Figure 0.2). It lists 15 Pythagorean numbers in the increasing order of hypotenuse angles from about 45 degrees to 60 degrees [88]. Figure 0.2. Plimton 322 INVITATION TO 2D TQFT AND QUANTIZATION 3 Babylonians seem to have known an algorithm to calculate an approximate value of the square root of any √ number. For example, there is a cuneiform tablet that shows the sexagesimal expansion of 2. Although there have been many speculations for practical purposes of Plimton 322 and mechanisms to come up with the listed numbers, our imagination goes to the surprise of the creator of the tablet. For the pair of numbers (y, z) listed in the second and the third columns, z 2 − y 2 is always a perfect square. Therefore, the square root algorithm terminates in a finite number of steps for these values, and gives an exact answer. They must have found a finiteness in the forest of infinity. This is a strong sentiment that resonates with our mind today. The tablet of Figure 0.1 does not show any geometry. The author of [87] convincingly argues that behind the list of these seemingly meaningless numbers on the tablets, there is a profound geometric investigation of the planetary motion. Our imagination is piqued by the discovery: the interplay between algebra, geometry, and astronomy; and the equal treatment of momentum space and coordinate space. What the feeling of the authors of the tablet would have been, when they were creating it? It must have been a sharp happiness of discovery that mathematics correctly predicts the very nature surrounding us, such as planetary motion. The accuracy of calculations of Babylonians is another astonishment, in addition to the lack of pictures. It is in sharp contrast to Euclid’s Elements, which was written around the same time in Greece. In Elements, we see many beautiful geometric pictures, and proofs. The discovery of axiomatic method culminated in it. This dichotomy, being able to calculate a quantity to a high precision, vs. having a proof of the formula based on a finiteness property, is our very motivation of writing these notes. There are a lot of amazing formulas around us in the direction that we describe. At this moment, we do not have the final understanding of them, yet. In the first part of these lectures, we explore two-dimensional topological quantum field theories formulated in terms of cell graphs. A cell graph of type (g, n) is the 1-skeleton of a cell-decomposition of a compact oriented topological surface of genus g ≥ 0 with n labeled 0-cells, or vertices. Such graphs are called in many different names: ribbon graphs, dessins d’enfants, maps, embedded graphs, etc. We use the terminology “cell graph” indicating the different nature of the graphs on surfaces, which record degeneration of surfaces. Although a cell graph is a 1-dimensional object, the notion of a face is well defined as a 2-cell of the cell decomposition that the cell graph defines on a given surface. The degree of a vertex of a cell graph is the number of incident half-edges. We call a 1-cell an edge of the graph. As explained in our previous lecture notes [30], the number of 1 Cm , where cell graphs of type (0, 1) with the unique vertex of degree 2m is 2m 1 2m Cm = m+1 m is the m-th Catalan number. A generating function ∞ X Cm z = z(x) = m=0 1 x2m+1 of Catalan numbers satisfies an algebraic equation 1 (0.0.1) x=z+ . z The story of [30, Section 2] tells us that enumeration problem of cell graphs of arbitrary type (g, n) is solved by the quantization of the algebraic curve (0.0.1). The key formula 4 O. DUMITRESCU AND M. MULASE for the quantization comes from the Laplace transform of a combinatorial equation [30, Catalan Recursion, Section 2.1]. There, the combinatorial formula is derived by analyzing the effect of edge-contraction operations on cell graphs. The new story we wish to tell in these lectures is that the category of cell graphs carries the information of all two-dimensional topological quantum field theories. In Part 1 of this article, we will present an axiomatic setup of 2D TQFT. The key idea is simple: exact same edge-contraction operations characterize Frobenius algebras. When we contract an edge of a cell graph, two vertices collide. It represents multiplication. When we shrink a loop, since it is a cycle on the topological surface, the process breaks a vertex into two vertices. This is the operation of comultiplication. The topological structure of a cell graph makes these operations dual to one another in the context of Frobenius algebras. We start with defining Frobenius algebras. Topological quantum field theories (TQFT) and cohomological field theories (CohFT) are introduced in the following sections. A category of cell graphs is defined. We explain that this category generates all Frobenius algebras. We then make a connection between cell graphs and 2D TQFT. The theory of topological recursion is a leitmotiv in this article. It is another manifestation of a quantization procedure. Since there are many review articles on topological recursion, we touch the subject only tangentially in this article. Our new result related to this topic is the 2D TQFT-twisted topological recursion, which is explained in the end of Part 1. The cohomology ring of a closed oriented manifold X is a Frobenius algebra. When we consider an even dimensional manifold and only even degree cohomologies (0.0.2) A = H even (X, Q), then we have a commutative Frobenius algebra, which is equivalent to a 2D TQFT. The Gromov-Witten invariants of X of genus 0 determine a quantum deformation of the ring A, known as the big quantum cohomology. This quantum structure then defines a holomorphic object Y , the mirror of X. Thus the holomorphic geometry of Y captures quantum cohomology of X. Gromov-Witten invariants are generalized to arbitrary genera. Here arises a question: Question 0.1. What should be the holomorphic geometry on Y that corresponds to highergenus Gromov-Witten theory of X? If we consider the passage from genus 0 to higher genera Gromov-Witten theory as quantization, then on the side on Y , we need a quantized holomorphic geometry. A good candidate is the D-module theory. The simplest such theory fitting to this context is the notion of quantum curves. The relation between quantum curves and B-model geometry is considered in [2, 19, 17, 27, 28, 30, 49, 50] and others. The key point for this idea to work is when the holomorphic geometry Y corresponding to the Gromov-Witten theory of X is captured by an algebraic or analytic curve. These curves often appear in other areas of mathematics as spectral curves, such as integrable systems, random matrix theory, and Hitchin’s theory of Higgs bundles. A spectral curve naturally comes with a projection to a base curve, making it a covering of the base curve. A quantum curve is a D-module on this base curve, quantizing the spectral curve. Since a CohFT based on the Frobenius algebra H even (X, Q) plays a role similar to Gromov-Witten theory of X, CohFT is a quantization of 2D TQFT. This process consists of two steps of quantizations. The first one is from the classical cohomology ring A to its quantum deformation, which is the mirror geometry. If the mirror is a curve, then the second step should be parallel to the quantization of a spectral curve to a quantum curve. We may be able to understand the reconstruction [36] of CohFT based on a semi-simple INVITATION TO 2D TQFT AND QUANTIZATION 5 Frobenius algebra from the 2D TQFT, that is obtained as the degree 0 part of the cohomology H 0 (Mg,n , Q) of the moduli space of stable curves, analogous to the construction of quantum curves. Part 2 deals with quantum curves. Since the authors have produced a long article [30] explaining the relation between quantum curves and topological recursion, the present article is focused on geometry of the process of quantization of Hitchin spectral curves from a perspective of opers. Instead of providing a general theory of [25, 26, 31], we use SL2 (C) to explain our ideas here. We will show that the classical notion of projective structures on a compact Riemann surface C of genus g ≥ 2 studied by Gunning [51], SL2 (C)-opers, and quantum curves of Hitchin spectral curves, are indeed the exact same notion. A conjecture of Gaiotto [42] relating non-Abelian Hodge correspondence and opers, together with its solution by [26], will be briefly explained in the final section. Acknowledgement. The authors are grateful to the organizers of the Warsaw Advanced School on Topological Quantum Field Theory for providing the opportunity to produce this article. They are in particular indebted to Piotr Sulkowski for constant encouragement and patience. The joint research of authors presented in this article is carried out while they have been staying in the following institutions in the last four years: the American Institute of Mathematics in California, the Banff International Research Station, Institutul de Matematică “Simion Stoilow” al Academiei Romane, Institut Henri Poincaré, Institute for Mathematical Sciences at the National University of Singapore, Kobe University, Leibniz Universität Hannover, Lorentz Center Leiden, Mathematisches Forschungsinstitut Oberwolfach, MaxPlanck-Institut für Mathematik-Bonn, and Osaka City University Advanced Mathematical Institute. Their generous financial support, hospitality, and stimulating research environments are greatly appreciated. The research of O.D. has been supported by GRK 1463 Analysis, Geometry, and String Theory at the Leibniz Universität Hannover, Perimeter Institute for Theoretical Physics in Waterloo, and a grant from MPIM-Bonn. The research of M.M. has been supported by IHÉS, MPIM-Bonn, Simons Foundation, Hong Kong University of Science and Technology, Université Pierre et Marie Curie, and NSF grants DMS-1104734, DMS-1309298, DMS-1619760, DMS-1642515, and NSF-RNMS: Geometric Structures And Representation Varieties (GEAR Network, DMS-1107452, 1107263, 1107367). Part 1. Topological Quantum Field Theory 1. Frobenius algebras Throughout this article, we denote by K a field of characteristic 0. Most of the cases we consider K = Q or K = C. Let A be a finite-dimensional, unital, and associative algebra defined over a field K. A bialgebra comes with an extra set of structures, including a comultiplication. Examples of bialgebras include the group algebra A = K[G] of a finite group G. Its algebra structure is canonically determined by the multiplication rule of G. However, the group algebra has two distinct co-algebra structures. One of which makes K[G] a Frobenius algebra, and the other makes it a Hopf algebra. This difference is reflected in which topological invariants we are dealing with. The Frobenius structure is useful for twodimensional topology, and the Hopf structure is natural for three-dimensional topological invariants. Let us start with defining Frobenius algebras. 6 O. DUMITRESCU AND M. MULASE Definition 1.1 (Frobenius algebras). A finite-dimensional, unital, and associative algebra A over a field K is a Frobenius algebra if there exists a linear map : A −→ K called counit such that for every u ∈ A, (ux) = 0, or (xu) = 0, for all x ∈ A implies that u = 0. Proposition 1.2. If A1 and A2 are Frobenius algebras, then A1 ⊕ A2 and A1 ⊗ A2 are also Frobenius algebras. Let us denote by A the category of Frobenius algebras defined over K. Then C = (A, ⊗, K) is a monoidal category. Proof. Denote by i : Ai −→ K the counit of Ai , i = 1, 2. Then (A1 ⊕ A2 , 1 + 2 ) is a Frobenius algebra. Similarly, 1 · 2 : A1 ⊗ A2 3 u1 ⊗ u2 7−→ 1 (u1 )2 (u2 ) ∈ K makes A1 ⊗ A2 a Frobenius algebra. Example 1.3. The one-dimensional vector space A = K, with the identity map (u) = u, is a simple commutative Frobenius algebra. More generally, by taking the direct sum A = K ⊕n and defining (u1 , . . . , un ) = u1 + · · · + un ∈ K, we construct a semi-simple commutative Frobenius algebra K ⊕n . Example 1.4. The full matrix algebra A = Matn (K) consisting of n × n matrices with (u) = trace(u) is an example of a non-commutative simple Frobenius algebra for n ≥ 2. Example 1.5. As mentioned above, the group algebra A = K[G] of a finite group G is a Frobenius algebra. Here, we define : K[G] −→ K by linearly extending the map whose value for every g ∈ G is given by ( 1 g=1 (g) = 0 g 6= 1. Remark 1.6. If we define (g) = 1 for all g ∈ G, then the group algebra becomes a Hopf algebra. Example 1.7. The cohomology ring H ∗ (M, R) of an oriented compact differential manifold M of dimR M = n with the cup product and : H ∗ (M, R) −→ H n (M, R) = R is a Frobenius algebra. The above example makes us wonder if an analogue of Poincaré pairing exists in a general Frobenius algebra. Indeed, the counterpart is a Frobenius bilinear form defined by (1.0.1) η : A ⊗ A −→ K, η(u, v) = (uv) in terms of the counit : A −→ K. The Frobenius form satisfies the Frobenius associativity (1.0.2) η(uv, w) = η(u, vw), u, v, w ∈ A. The condition for the counit of Definition 1.1 exactly means that the Frobenius form η is non-degenerate. It therefore defines a canonical isomorphism (1.0.3) ∼ λ : A −→ A∗ , hλ(u), vi = η(u, v), u, v ∈ A. INVITATION TO 2D TQFT AND QUANTIZATION 7 The isomorphism introduces a unique comultiplication δ : A −→ A ⊗ A by the following commutative diagram: (1.0.4) δ A λ / A⊗A A∗ m∗ λ⊗λ . / A∗ ⊗ A∗ Here, m∗ : A∗ −→ A∗ ⊗ A∗ is the dual of the multiplication operation m : A ⊗ A −→ A in A. Since we are not assuming the multiplication to be commutative, we define the natural pairing (A∗ ⊗ A∗ ) ⊗ (A ⊗ A) −→ K by observing the order of entities as written. For example, for α, β ∈ A∗ and u, v ∈ A, we calculate (1.0.5) hα ⊗ β, u ⊗ vi = hα, hβ, uivi = hβ, uihα, vi. As the dual of the associative multiplication, the comultiplication δ is coassociative, i.e., it satisfies A8 ⊗ A (1.0.6) δ δ⊗1 & A ⊗8 A ⊗ A. A & δ 1⊗δ A⊗A Note that the identity element 1 ∈ A corresponds to ∈ A∗ by λ, i.e., λ(1) = . This is because η(1, u) = η(u, 1) = (u). We now have the full set of data (A, 1, m, , δ) that defines a bialgebra. The algebra and coalgebra structures satisfy a compatibility condition, as described below. Proposition 1.8. The following diagram commutes: (1.0.7) 1⊗δ A⊗A A⊗ 8 A⊗A /A m δ⊗1 & m⊗1 & / A ⊗ A. 8 δ 1⊗m A⊗A⊗A Remark 1.9. Alternatively, we can define a Frobenius algebra as a bialgebra satisfying (1.0.7), together with a non-degenerate Frobenius form satisfying (1.0.2). Proof. Let he1 , e2 , . . . , er i be a K-basis for A, where r = dimK A. The bilinear form η defines r × r matrices (1.0.8) η = [ηij ], η −1 = [η ij ], ηij := η(ei , ej ). 8 O. DUMITRESCU AND M. MULASE It gives the canonical basis expansion of v ∈ A: X X (1.0.9) v= η(v, ea )η ab eb = η(ea , v)η ba eb . a,b a,b From (1.0.4), we calculate δ(v) = X η(v, ei ej )(η jb eb ) ⊗ (η ia ea ) i,j,a,b = X = X η(vei , ej )(η jb eb ) ⊗ (η ia ea ) i,j,a,b (vei ) ⊗ (η ia ea ) i,a = (m ⊗ 1)(v ⊗ δ(1)), using the pairing convention (1.0.5) and (1.0.10) δ(1) = X η ab ea ⊗ eb . a,b We then have  (1 ⊗ m) ◦ (δ ⊗ 1)(u ⊗ v) = (1 ⊗ m)   X η(u, ei ej )(η jb eb ) ⊗ (η ia ea ) ⊗ v  i,j,a,b = X η(uei , ej )(η jb eb ) ⊗ (η ia ea v) i,j,a,b = X η(uei , ej )(η jb eb ) ⊗ η ia η(ea v, ec )η cd ed i,j,a,b,c,d = X uei ⊗ η ia η(ea , vec )η cd ed i,a,c,d = X (uvec ) ⊗ η cd ed c,d = (m ⊗ 1)(uv ⊗ δ(1)) = δ(uv). Similarly,  (m ⊗ 1) ◦ (1 ⊗ δ)(u ⊗ v) = (m ⊗ 1) u ⊗  X η(v, ei ej )(η jb eb ) ⊗ (η ia ea ) i,j,a,b = X uη(vei , ej )(η jb eb ) ⊗ (η ia ea ) i,j,a,b = X uvei ⊗ η ia ea i,a = (m ⊗ 1)(uv ⊗ δ(1)) = δ(uv). This completes the proof of Proposition 1.8. Remark 1.10. We note that if A is commutative, then the quantities (1.0.11) η ei1 · · · eij , eij+1 · · · en = (ei1 · · · ein ), 1 ≤ j < n, INVITATION TO 2D TQFT AND QUANTIZATION 9 are completely symmetric with respect to permutations of indices. Definition 1.11 (Euler element). The Euler element of a Frobenius algebra A is defined by (1.0.12) e := m ◦ δ(1). In terms of basis, the Euler element is given by X (1.0.13) e= η ab ea eb . a,b The Euler element provides the genus expansion of 2D TQFT, allowing us to calculate higher genus correlation functions from the genus 0 part of the theory. Another application of (1.0.9) is the following formula that relates the multiplication and comultiplication: (1.0.14) (λ(u) ⊗ 1) δ(v) = uv. This is because (λ(u) ⊗ 1) δ(v) = (λ(u) ⊗ 1) X (vη ab ea ) ⊗ eb a,b = X = X η(u, vea )η ab eb a,b η(uv, ea )η ab eb = uv. a,b 2. TQFT In this section, we briefly review TQFT. Although there have been explosive mathematical developments in higher dimensional topological quantum field theories mixing different dimensions during the last decade (see for example, [13, 67] and more recent work inspired by them), we restrict our attention to the two-dimensional speciality in these lectures. From now on, Frobenius algebras we consider are finite-dimensional, unital, associative, and commutative. It has been established (see [1, 16]) that 2D TQFT’s are classified by these types of Frobenius algebras. The axiomatic formulation of conformal and topological quantum field theories was discovered in the 1980s by Atiyah [4] and Segal [89]. A (d−1)-dimensional TQFT is a monoidal functor Z from the monoidal category of (d − 1)-dimensional closed (i.e., compact without boundary) oriented smooth manifolds with oriented d-dimensional cobordism as morphisms, to the monoidal category of finite-dimensional vector spaces defined over a field K. The monoidal structure in the category of (d − 1)-dimensional smooth manifolds is defined by the operation of taking disjoint union, which is a symmetric operation. Disjoint union with the empty set is the identity of this operation. Therefore, we define the monoidal category of K-vector spaces by symmetric tensor products, with the field K serving as the identity operation of tensor product. The functor Z satisfies the non-triviality condition Z(∅) = K, and maps a disjoint union of smooth (d − 1)-dimensional manifolds to a symmetric tensor product of vector spaces. 10 O. DUMITRESCU AND M. MULASE Let us denote by M an oriented closed smooth manifold of dimension d − 1, and by M op the same manifold with the opposite orientation. The functor Z satisfies the duality condition Z(M op ) = Z(M )∗ , where Z(M )∗ is the dual vector space of Z(M ). Suppose we have an oriented bordered d-dimensional smooth manifold N . The boundary ∂N is a smooth manifold of dimension d − 1, and the complement N \ ∂N is an oriented smooth manifold of dimension d. We give the orientation induced from N to its boundary ∂N . The TQFT functor Z then gives an element Z(N ) ∈ Z(∂N ). If N is closed, i.e., ∂N = ∅, then Z(N ) ∈ Z(∅) = K is a number that represents a topological invariant of N . Now consider a bordered oriented smooth manifold N1 with boundary ∂N1 = M1op t M2 , meaning that the two separate boundaries carry different orientations. We choose that M2 is given the induced orientation from N1 , and M1 the opposite orientation. We interpret the situation as N1 giving an oriented cobordism from M1 to M2 . In this case, the functor Z defines an element Z(N1 ) ∈ Z(M1 )∗ ⊗ Z(M2 ), or equivalently, a linear map Z(N1 ) : Z(M1 ) −→ Z(M2 ). Suppose we have another smooth manifold N2 with boundary ∂N2 = M2op t M3 corresponding to a linear map Z(N2 ) : Z(M2 ) −→ Z(M3 ). We can then smoothly glue N1 and N2 along the common boundary component M2 forming a new manifold N = N1 ∪M2 N2 . Clearly ∂N = M1op t M3 , and N gives a cobordism from M1 to M3 . The Atiyah-Segal sewing axiom [4] asserts that (2.0.1) Z(M1 ) k Z(M1 ) Z(N1 ) / Z(M2 ) Z(N2 ) Z(N ) / Z(M3 ) k / Z(M3 ). A d-dimensional TQFT Z defines a topological invariant Z(N ) for each closed d-manifold N . The essence of TQFT is to break N into simpler pieces by cutting along (d − 1)dimensional submanifolds, and represent the invariant Z(N ) from that of simpler pieces. The situation is special for the case of 2D TQFT. First of all, we already know all topological invariants of a closed surface. They are just functions of the genus g of the surface. So there should be no reason for another theory to study them. Since there is only one 1-dimensional connected compact smooth manifold, a 2D TQFT is based on a single vector space (2.0.2) Z(S 1 ) = A, INVITATION TO 2D TQFT AND QUANTIZATION 11 and its tensor products. What else could we gain? The biggest surprise is that a 2D TQFT is further quantized [15, 24, 48, 63]. This process starts with a Frobenius algebra A of (0.0.2). It is quantized to a quantum cohomology, and then further to Gromov-Witten invariants. Construction of spectral curves, and then quantizing them, have a parallelism to this story. This is the story we now explore in this and the next sections, but only to take a snapshot of the converse direction, i.e., from a quantum theory to a classical theory. Let us denote by Σg,m̄,n a connected, bordered, oriented, smooth surface of genus g with m + n boundary circles. This is a surface obtained by removing m + n disjoint open discs from a compact oriented two-dimensional smooth manifold of genus g without boundary. We assume that the boundary of Σg,m̄,n itself is a smooth manifold, hence it is a disjoint union of m + n circles. We give the induced orientation from the surface to n boundary circles, and the opposite orientation to the m boundary circles. The surface Σg,m̄,n gives a cobordism of m circles to n circles. The orientation-preserving diffeomorphism class of such a surface is determined by the genus g and the two number m and n of boundary circles with different orientations. Therefore, the oriented equivalence class of cobordism is also determined by (g, m, n). The TQFT functor Z then assigns to each cobordism Σg,m̄,n a multilinear map def ωg,m̄,n = Z(Σg,m̄,n ) : A⊗m −→ A⊗n , which is completely determined by the label (g, m, n). This is the special situation of the two-dimensionality of TQFT, reflecting the simple topological classification of surfaces. Suppose we have another cobordism Σh,k̄,m of genus h from k circles to m circles, with orientation on the m circles induced from the surface, and the opposite orientation on k circles. We can compose two cobordisms, sewing the m circles of the first cobordism with the m circles of the second. Here, we notice that on each pair of circles, one from the first surface and the other from the second, the orientations are the same, and hence we can put one on top of the other. Therefore, the orientations of the two surfaces are consistent after sewing. This sewing process generates a new surface Σg+h+m−1,k̄,n of genus g + h + m − 1. This is because the m pairs of circles sewed together create m − 1 handles (see Figure 2.1). The sewing axiom of Atiyah-Segal [4] then requires that the functor Z associates the composition of linear maps to the composition of cobordisms. In our situation, the composition of two maps generates a new map ωg,m̄,n ◦ ωh,k̄,m = ωg+h+m−1,k̄,n : A⊗k −→ A⊗n , corresponding to the sewing of cobordisms. Figure 2.1. A string of String Interactions. The physics of a simple model for interacting strings is captured by the sewing axiom of Atiyah-Segal. The sewing procedure can be generalized, allowing partial sewing of the boundary circles of matching orientation. For example, if j ≤ ` and j ≤ m, then gluing only j pairs of circles, 12 O. DUMITRESCU AND M. MULASE we have a composition (2.0.3) ωg,m̄,n ◦ ωh,k̄,` = ωg+h+j−1,k+m−j,n+`−j : A⊗k+m−j −→ A⊗n+`−j . Furthermore, since TQFT does not require cobordism to be given by a connected manifold, we can stack up disjoint union of surfaces and apply partial sewing to create a variety of linear maps from A⊗n to A⊗m . It was Dijkgraaf [16] who noticed the equivalence between 2D TQFTs and commutative Frobenius algebras. We can see the resemblance immediately. On the vector space A, we have operations defined by (2.0.4) 1 = ω0,0̄,1 : K −→ A, m = ω0,2̄,1 : A⊗2 −→ A, = ω0,1̄,0 : A −→ K, δ = ω0,1̄,2 : A −→ A⊗2 , η = ω0,2̄,0 : A⊗2 −→ K. A connected cobordism of m incoming circles and n outgoing circles is classified by the genus g of the surface. It does not depend on the history of how the cobordisms are glued together. Thus the diffeomorphism classes of surfaces after sewing cobordisms tell us relations among the operations of (2.0.4). For example, ω0,1̄,0 ◦ ω0,2̄,1 = ω0,2̄,0 =⇒ ◦ m = η, which is (1.0.1). Associativity of multiplication comes from uniqueness of the topology of Σ0,3̄,1 with one boundary circle carrying the induced orientation and three the opposite orientation. Changing the orientation of each boundary component to its opposite provides coassociativity. In this way A = Z(S 1 ) acquires a bialgebra structure. To assure duality between algebra and coalgebra structures, we need to impose another condition here: nondegeneracy of η = ω0,2̄,0 . Then A becomes a commutative Frobenius algebra. Conversely, if we start with a finite-dimenionsal commutative Frobenius algebra A, then we first construct ω0,m̄,n on the list of (2.0.4). More general maps ωg,m̄,n are constructed by partial sewing (2.0.3). By construction, all these maps are associated with cobordism of circles, and hence determine a 2D TQFT [1]. With the above considerations, we give the following definition. Definition 2.1 (Two-dimensional Topological Quantum Field Theory). Let (A, , λ) be a set of data consisting of a finite-dimensional vector space A over K, a non-trivial linear ∼ map : A −→ K, and an isomorphism λ : A −→ A∗ . A Two-dimensional Topological Quantum Field Theory is a system (A, {ωg,m̄,n }) consisting of linear maps ωg,k̄,` : A⊗k −→ A⊗` , 0 ≤ g, k, `, satisfying the following axioms. • TQFT 1. Symmetry: (2.0.5) ωg,k̄,` : A⊗k −→ A⊗` is symmetric with respect to the Sk -action on the domain. • TQFT 2. Non-triviality: (2.0.6) ω0,1̄,0 = : A −→ K. INVITATION TO 2D TQFT AND QUANTIZATION 13 • TQFT 3. Duality: A⊗m (2.0.7) λ⊗m (A∗ )⊗m ωg,m̄,n (ωg,n̄,m )∗ / A⊗n λ⊗n . / (A∗ )⊗n • TQFT 4. Sewing: (2.0.8) ωg,m̄,n ◦ ωh,k̄,` = ωg+h+j−1,k+m−j,n+`−j : A⊗k+m−j −→ A⊗n+`−j , j ≤ m, j ≤ `. Remark 2.2. As noted above, the vector space A acquires the structure of a commutative Frobenius algebra from these axioms. Conversely, if A is a commutative Frobenius algebra, then ω0,m̄,n of (2.0.4) can be extended to ωg,m̄,n that satisfy the TQFT axioms. 3. Cohomological field theory Recall that for any oriented closed manifold X, its cohomology ring H ∗ (X, K) is a Frobenius algebra defined over K. If we restrict ourselves to even dimensional manifold X and consider only even part of the cohomology, then A = H even (X, K) is a finite-dimensional commutative Frobenius algebra. This Frobenius algebra naturally defines a 2D TQFT. The role of a TQFT in general dimension is to represent a topological invariant of higher dimensional manifolds. We are now seeing that the classical topology of a manifold X is a 2D TQFT. Changing our point of view, we can ask if we start with X, then how do we find the corresponding 2D TQFT? Of course the Frobenius algebra structure automatically determine the unique 2D TQFT as we have seen in Section 2. But then the picture of sewing cobordisms is lost in this algebraic formulation. What is the role of the cobordism of circles in the context of understanding the manifold X? The amazing vision emerged in the early 1990s is that 2D TQFT can be further quantized into Gromov-Witten theory, which produces quantum topological invariants of X that cannot be reduced to classical invariants. This epoch making discovery was one of the decisive moments of the fruitful interaction of string theory and geometry, which is still continuing today. String theory deals with flying strings in a space-time manifold X. The trajectory of a moving string is a curved cylinder embedded in the manifold X, like a duct pipe in the attic. When strings are considered as quantum objects, they can interact one another. For example, Figure 2.1 can be interpreted as a string of string interactions. First, three strings collide in a complicated interaction to produce two strings. The complexity of the interaction is classified as “g = 2” in terms of the genus of the first surface. These two strings created in the first interaction then collide in an even more complicated (i.e., g = 3) interaction to produce four strings in the end. The trajectory of these interactions then forms a bordered surface of genus 6 with 7 boundary components embedded in the space-time X. Models for string theory were originally introduced to understand how the space-time X dictates string interactions. In this process, the importance of Calabi-Yau spaces was recognized, through considerations on the consistency of string theory as a physical theory. By changing the point of view 180◦ , researchers then noticed that a simple model of string interactions could be used to obtain a totally new class of topological invariants of X itself. These new invariants are not necessarily captured by the classical topology, such as (co)homology groups and homotopy groups. We use the terminology of quantum topological invariants for the invariants obtained by string theory considerations. 14 O. DUMITRESCU AND M. MULASE Gromov-Witten theory is a mathematically rigorous interpretation of a model for a quantum string theory. A trajectory of interacting strings in X is the image of a map (3.0.1) f : Σg,m̄,n −→ X from a bordered surface Σg,m̄,n into X. Recall that the fundamental group π1 (X, x0 ) of X is determined by the connectivity of the pointed loop space L(X, x0 ) = C ∞ (S 1 , ∗), (X, x0 ) , which is the moduli space of differentiable maps from S 1 to X that send a reference point ∗ ∈ S 1 to x0 ∈ X. In the same spirit, Gromov-Witten invariants are defined by looking at the classical topological invariants of appropriate moduli spaces of maps from Σg,m̄,n to X. When we have the notion of size of a string, the trajectory Σg,m̄,n has a metric on it. If the ambient manifold X has a geometric structure, such as a symplectic structure, complex structure, or a Kähler structure, then the map f needs to be compatible with the structures of the source and the target. For example, if X is Kähler, then we give a complex structure on Σg,m̄,n that is compatible with the metric it has, and require that f is a holomorphic map. Technical difficulties arise in defining moduli spaces of such maps. Even the moduli spaces are defined, they are often very different objects from the usual differentiable manifolds. We need to extend the notion of manifolds here. Thus identifying reasonable classical topological invariants of these spaces also poses a difficult problem. The simplest scenario is the following: We take X = pt to be just a single point. Of course nobody wants to know the topological structure of a point. We all know it! By exploring the Gromov-Witten theory of a point, we learn the structure of the theory itself. Since we can map anything to a point, a point may not be such a simple object, after all. It is like considering a vacuum in physics. Again, anything can be thrown into a vacuum. A quantum theory of a vacuum is inevitably a rich theory. When X = pt is a point, we consider all incoming and outgoing strings to be infinitesimally small. Thus the surfaces we are considering become closed, and boundary components are just several points identified on them. A metric on a closed surface naturally gives rise to a unique complex structure, making it a compact Riemann surface. And every compact Riemann surface acquires a unique projective algebraic structure, making it a projective algebraic curve. Since we are allowing strings to become infinitesimally small during the interaction, the trajectory may contain a process of an embedded S 1 in the surface shrinking to a point, and then becoming a finite size circle again a moment later. Such a process can be understood in complex algebraic geometry as a nodal singularity of an algebraic curve. Locally, every nodal singularity of a curve is the same as the neighborhood of the origin of a singular algebraic curve {(x, y) ∈ C2 \| xy = 0}. We are talking about an abstract notion of trajectories here, because nothing can move in X = pt. In terms of the map idea (3.0.1), we note that there is a unique map f from any projective algebraic curve to a point, which satisfies any reasonable requirement of maps such as being a morphism of algebraic varieties. Then the moduli space of all maps simply means the moduli space of the source. A stable curve is a projective algebraic curve with only nodal singularities and a finite number of smooth points marked on the curve such that it has only a finite number of algebraic automorphisms fixing each marked point. We recall that the holomorphic automorphism group of P1 , Aut(P1 ) = P SL2 (C), INVITATION TO 2D TQFT AND QUANTIZATION 15 acts on P1 triply transitively. The number of automorphisms fixing up to 2 points is always infinity. Since P SL2 (C) is compact and three-dimensional, if we choose three distinct points on P1 and require the automorphism to fix each of these points, then we have only finitely many choices. Actually in our case, it is unique. For an elliptic curve E, since it is an abelian group, E acts on E transitively. To avoid these translations, we need to choose a point on E. We can naturally identify it as the identity element of the elliptic curve as a group. Automorphisms of an elliptic curve fixing a point then form a finite group. If a compact Riemann surface C has genus g = g(C) ≥ 2, then it is known that the order of the analytic automorphism group is bounded by \|Aut(C)\| ≤ 84(g − 1). This bound comes from hyperbolic geometry. It is easy to see the finiteness. First, we note that universal covering of C is the upper half plane H = {z ∈ C \| Im(z) > 0}, and C is constructed by the quotient ∼ C −→ H/ρ(π1 (C)) through a faithful representation (3.0.2) ρ : π1 (C) −→ P SL2 (R) = Aut(H) of the fundamental group of C into the automorphism group of H. Every holomorphic automorphism of C extends to an automorphism of H. We know that C does not have any non-trivial holomorphic vector field v. If it did, then v would be a differentiable vector field with isolated zeros, and each zero comes with positive index, because locally it is given by z n d/dz. We learn from topology that the sum of the indices of isolated zeros of a vector field on C is equal to χ(C) = 2 − 2g < 0. It is a contradiction. Thus Aut(C) ⊂ P SL2 (C) is a discrete subgroup. Since P SL2 (R) is compact, \|Aut(C)\| is finite. An algebraic curve C is stable if and only if (1) every singularity is nodal, and (2) every irreducible component C 0 of C has a finite number of automorphisms. The second condition means that the total number of smooth marked points and singular points of C that are on C 0 has to be 3 or more if g(C 0 ) = 0, and 1 or more if g(C 0 ) = 1. There is no condition for an irreducible component of genus two or more. For a pair (g, n) of integers g ≥ 0 and n ≥ 1 in the stable range 2g − 2 + n > 0, we denote by Mg,n the moduli space of stable curves of genus g and n smooth marked points. It is a complex orbifold of dimension 3g − 3 + n. Quantum topological invariants of a point X = pt is then realized as classical topological invariants of Mg,n . Alas to this day, we cannot identify the cohomology groups H i (Mg,n , Q) for all values of i, g, n. From the very definition of these moduli spaces, we can construct many concrete cohomology classes on each of Mg,n , called tautological classes. What we do not know is indeed how much more we need to know to determine all of H i (Mg,n , Q). The surprise of Witten’s conjecture [93], proved in [62], is that we can actually explicitly write intersection relations of certain tautological classes for all values of g and n. Among many different proofs available for the Witten conjecture (see for example, [61, 62, 71, 72, 80, 85]), [72, 80] deal with recursive relations among surfaces much in the same spirit of TQFT. These relations are first noticed in [20]. We will discuss these relations in connection to topological recursion later in these lectures. 16 O. DUMITRESCU AND M. MULASE Geometric relations among Mg,n ’s for different values of (g, n) have a simple meaning. Let us denote by Mg,n the moduli space of smooth n-pointed curves. Then the boundary Mg,n \ Mg,n consists of points representing singular curves. Simplest singular stable curve has one nodal singularity, which can be described as collision of two smooth points. Analyzing how these singularities occur via degeneration, we come up with three types of natural morphisms among the moduli spaces Mg,n . They are the forgetful morphisms (3.0.3) π : Mg,n+1 −→ Mg,n which simply erase one of the marked points on a stable curve, and gluing morphisms (3.0.4) gl1 : Mg−1,n+2 −→ Mg,n (3.0.5) gl2 : Mg1 ,n1 +1 × Mg2 ,n2 +1 −→ Mg1 +g2 ,n1 +n2 that construct boundary strata of Mg,n . Under a gluing morphism, we put two smooth points of stable curves together to form a one nodal singularity. The first one gl1 glues two points on the same curve together, and gl2 one each on two curves. The fiber of π at a stable curve (C, p1 , . . . , pn ) ∈ Mg,n is the curve C itself, because another marked point can be placed anywhere on C. Thus π is a universal family of curves parameterized by the base moduli space Mg,n . When we place the extra marked point pn+1 at pi , i = 1, . . . , n, then as the point of Mg,n+1 on the fiber of π, the data represented is a singular curve obtained by joining C itself with a P1 at the location of pi ∈ C, but the two marked points pi , pn+1 are actually placed on the line P1 . Assigning this singular curve to (C, p1 , . . . , pn ) defines a section σi : Mg,n −→ Mg,n+1 , which is a right inverse of π. Geometrically, σi sends (C, p1 , . . . , pn ) to the point pi on the fiber C = π −1 (C, p1 , . . . , pn ). Obviously, π ◦ σi : Mg,n −→ Mg,n is the identity map. Gromov-Witten theory for a Kähler manifold X [15] concerns topological structure of the moduli space Mg,n (X, β) = f : (C, p1 , . . . , pn ) −→ X [f (C)] = β of stable holomorphic maps f from a nodal curve C with n smooth marked points p1 , . . . , pn ∈ C to X such that the homology class of the image f (C) agrees with a prescribed homology class β ∈ H2 (X, Z). Here, stability of a map f is again defined by imposing the finiteness of possible automorphisms. Giving a definition of this moduli space is beyond our scope of this article. We refer to [15]. The moduli space, if defined, should come with natural maps ev i Mg,n (X, β) −−−− → X   φy Mg,n , where the forgetful map φ assigns the stabilization of the source (C, p1 , . . . , pn ) to the map f by forgetting about the map itself, and evi f : (C, p1 , . . . , pn ) −→ X = f (pi ) ∈ X, i = 1, . . . , n, INVITATION TO 2D TQFT AND QUANTIZATION 17 is the value of f at the i-th marked point pi ∈ C. Stabilization means that evey irreducible component of (C, p1 , . . . , pn ) that is not stable is shrunk to a point. If we indeed know the moduli space Mg,n (X, β) and that its cohomology theory behaves as we expect, then we would have ev∗ H ∗ Mg,n (X, β), Q ←−−i−− H ∗ (X, Q)   φ! y H ∗ (Mg,n , Q), where φ! is the Gysin map defined by integration along fiber. If Mg,n (X, β) were a manifold, and the map φ : Mg,n (X, β) −→ Mg,n were a fiber bundle, then Mg,n (X, β) would have been locally a direct product, hence the Gysin map φ! associated with φ would be defined by integrating de Rham cohomology classes of Mg,n (X, β) along fiber of φ. Choose any cohomology classes v1 , . . . vn ∈ H ∗ (X, Q) of X. We then could have defined the GromovWitten invariants by Z X,β GWg,n (v1 , . . . , vn ) := φ! ev1∗ (v1 ) · · · evn∗ (vn ) . Mg,n In general, however, construction of the Gysin map does not work as we hope. This is due to the complicated nature of the moduli space Mg,n (X, β), which often has components of unexpected dimensions. The remedy is to define the virtual fundamental class [Mg,n (X, β)]vir of the expected dimension, and avoid the use of φ! by defining Z X,β (3.0.6) GWg,n (v1 , . . . , vn ) := ev1∗ (v1 ) · · · evn∗ (vn ). [Mg,n (X,β)]vir See [15] for more detail. Now recall that A = H even (X, Q) is a Frobenius algebra. Although Gromov-Witten theory goes through a big black box [Mg,n (X, β)]vir , what we wish is a map Ωg,n : H even (X, Q)⊗n −→ H ∗ (Mg,n , Q) whose integral over Mg,n gives the quantum invariants of X. Then what are the properties that Gromov-Witten invariants should satisfy? Can we list the properties as axioms for the above map Ωg,n so that we can characterize Gromov-Witten invariants? This was one of the motivations of Kontsevich and Manin to introduce CohFT in [63]. As we see below, a 2D TQFT can be obtained as a special case of a CohFT. The amazing relation between TQFT and CohFT, i.e., the reconstruction of CohFT from its restriction to TQFT due to Givental and Teleman [48, 91], plays a key role in many new developments (see for example, [3, 36, 39, 68]), some of which are deeply related with topological recursion. Definition 3.1 (Cohomological Field Theory [63]). Let A be a finite-dimensional, unital, associative, and commutative Frobenius algebra with a basis {e1 , . . . , er }. A Cohomological Field Theory is a system (A, {Ωg,n }) consisting of linear maps (3.0.7) Ωg,n : A⊗n −→ H ∗ (Mg,n , K) defined for (g, n) in the stable range 2g − 2 + n > 0 and satisfying the following axioms: CohFT 0: Ωg,n is Sn -invariant, and Ω0,3 (v1 , v2 , v3 ) = η(v1 v2 , v3 ). CohFT 1: Ωg,n+1 (v1 , . . . , vn , 1) = π ∗ Ωg,n (v1 , . . . , vn ). X gl1∗ Ωg,n (v1 , . . . , vn ) = Ωg−1,n+2 (v1 , . . . , vn , ea , eb )η ab . CohFT 2: a,b 18 O. DUMITRESCU AND M. MULASE CohFT 3: gl2∗ Ωg1 +g2 ,\|I\|+\|J\| (vI , vJ ) = X η ab Ωg1 ,\|I\|+1 (vI , ea ) ⊗ Ωg2 ,\|J\|+1 (vJ , eb ), a,b where I t J = {1, . . . , n} is a disjoint partition of the index set, and the tensor product operation on the right-hand side is performed via the Künneth formula of cohomology rings H ∗ (Mg1 ,n1 +1 × Mg2 ,n2 +1 , K) ∼ = H ∗ (Mg1 ,n1 +1 , K) ⊗ H ∗ (Mg2 ,n2 +1 , K). Remark 3.2. The condition Ω0,3 (v1 , v2 , v3 ) = η(v1 v2 , v3 ) says that the product of the Frobenius algebra is determined by the (0, 3)-value of the Gromov-Witten invariants. Proposition 3.3 (2D TQFT is a CohFT). Every 2D TQFT is a CohFT that takes values in H 0 (Mg,n , K). More precisely, let (A, ωg,m̄,n ) be a 2D TQFT. Then ωg,n = ωg,n̄,0 satisfies the CohFT axioms, by identifying K = H 0 (Mg,n , K). Proof. Since Mg,n is connected, the three types of morphisms (3.0.3), (3.0.4), and (3.0.5) all produce isomorphisms of degree 0 cohomologies. Thus the axioms CohFT 1–3 become (3.0.8) (3.0.9) ωg,n+1 (v1 , . . . , vn , 1) = ωg,n (v1 , . . . , vn ), X ωg,n (v1 , . . . , vn ) = ωg−1,n+2 (v1 , . . . , vn , ea , eb )η ab , a,b (3.0.10) ωg1 +g2 ,\|I\|+\|J\| (vI , vJ ) = X η ab ωg1 ,\|I\|+1 (vI , ea ) · ωg2 ,\|J\|+1 (vJ , eb ). a,b We wish to show that (3.0.8)-(3.0.10) are consequences of the partial sewing axiom (2.0.8) of TQFT under the identification ωg,n = ωg,n̄,0 . Since ω0,0̄,1 = 1, we have ωg,n+1,0 ◦(n+1) ω0,0̄,1 = ωg,n̄,0 = ωg,n : A⊗n −→ K from (2.0.8), which is (3.0.8). Here, ◦(n+1) is the composition taken at the (n + 1)-th slot of the input variables of ωg,n+1,0 . Next, we need to identify ω0,0̄,2 : K −→ A ⊗ A. Since P ω0,0̄,2 = ω0,1̄,2 ◦ ω0,0̄,1 , we see that ω0,0̄,2 (1) = δ(1) = a,b η ab ea ⊗ eb . Denoting by ◦(n+1,n+2) to indicate composition taking at the last two slots of variables, we have (3.0.9) ωg−1,n+2,0 ◦(n+1,n+2) ω0,0̄,2 = ωg,n̄,0 : A⊗n −→ K. If we have two disjoint sets of variables vI and vJ , then we can apply composition of ω0,0̄,2 simultaneously to two different maps. For example, we have ωg1 ,\|I\|+1,0 ⊗ ωg2 ,\|J\|+1,0 ◦ ω0,0̄,2 = ωg1 +g2 ,\|I\|+\|J\|,0 : A⊗(\|I\|+\|J\|) −→ K, which is (3.0.10). Notice that a linear map ωg,m+n : A⊗m ⊗ A⊗n −→ K is equivalent to A⊗m −→ (A∗ )⊗n . Thus we can re-construct a map ωg,m̄,n from ωg,m+n by (3.0.11) A⊗m k A⊗m ωg,m+n / (A∗ )⊗n ωg,m̄,n (λ−1 )⊗n / A⊗n k, / A⊗n INVITATION TO 2D TQFT AND QUANTIZATION 19 ∼ where λ : A −→ A∗ is the isomorphism of (1.0.3). For the case of g = 0, m = 2, and n = 1, (3.0.11) implies that λ−1 ω0,3 (v1 , v2 , · ) = ω0,2̄,1 (v1 , v2 ) = v1 v2 for v1 , v2 ∈ A. Or equivalently, we have (3.0.12) ω0,3 (v1 , v2 , v3 ) = η(v1 v2 , v3 ) = (v1 v2 v3 ). This completes the proof. Conversely, the degree 0 part of the cohomology of a CohFT is a 2D TQFT. Proposition 3.4 (Restriction of CohFT to the degree 0 part of the cohomology ring). Let (A, Ωg,n ) be a CohFT associated with a Frobenius algebra A. The Frobenius algebra A itself defines a unique 2D TQFT (A, ωg,m̄,n ). Denote by r : H ∗ (Mg,n , K) −→ H 0 (Mg,n , K) = K the restriction of the cohomology ring to its degree 0 component, and define (3.0.13) ωg,n = r ◦ Ωg,n : A⊗n −→ K. Then we have the equality of maps (3.0.14) ωg,n = ωg,n̄,0 : A⊗n −→ K for all (g, n) with 2g − 2 + n > 0. In other words, the degree 0 restriction of a CohFT is the 2D TQFT determined by the Frobenius algebra A. Proof. We already know that (A, ωg,m̄,n ) defines a CohFT with values in H 0 (Mg,n , K). We need to show that this CohFT is exactly the degree 0 restriction of the given CohFT we start with. First we extend ωg,n to the unstable range by (3.0.15) ω0,1 = : A −→ K, ω0,2 = η : A⊗2 −→ K. We then note that from (2.0.4) and (3.0.15), we see that (3.0.14) holds for ω0,1 and ω0,2 . The general case of (3.0.14) follows by induction on 3g−3+n, provided that ω0,3 is appropriately defined. This is because (3.0.9) and (3.0.10) are induction formula recursively generating ωg,n from those with smaller values of g and n. Since • Case of (3.0.9): 3g − 3 + n = [3(g − 1) − 3 + (n + 2)] + 1, • Case of (3.0.10): 3(g1 + g2 ) − 3 + \|I\| + \|J\| = [3g1 − 3 + \|I\| + 1] + [3g2 − 3 + \|J\| + 1] + 1, we see that the complexity 3g −3+n is always reduced by 1 in each of the recursive formulas. Finally, we see that since M0,3 is just a point as we have noted above in the discussion of Aut(P1 ), we have ω0,3 = Ω0,3 . Hence (3.0.16) ω0,3 (v1 , v2 , v3 ) = η(v1 v2 , v3 ), which makes (3.0.14) holds for all values of (g, n). This completes the proof. Remark 3.5. The above two propositions show that a 2D TQFT can be defined either just by a commutative Frobenius algebra A, by a system of maps {ωg,m̄,n } satisfying the TQFT axioms, or the degree 0 part of a CohFT. From now on, we use (A, ωg,n ) to denote a 2D TQFT, which is less cumbersome and easier to deal with. Proposition 3.6. The genus 0 values of a 2D TQFT is given by (3.0.17) ω0,n (v1 , . . . , vn ) = (v1 · · · vn ). Proof. This is a direct consequence of CohFT 3 and (1.0.9). 20 O. DUMITRESCU AND M. MULASE One of the original motivations of TQFT [4, 89] is to identify the topological invariant Z(N ) of a closed manifold N . In our current setting, it is defined as (3.0.18) Z(Σg ) := λ−1 (ωg,1 ) for a closed oriented surface Σg of genus g. Here, ωg,1 : A −→ K is an element of A∗ , and ∼ λ : A −→ A∗ is the canonical isomorphism (1.0.3). Proposition 3.7. The topological invariant Z(Σg ) of (3.0.18) is given by Z(Σg ) = (eg ), (3.0.19) where e ∈ A is the Euler element of (1.0.12). Lemma 3.8. We have e := m ◦ δ(1) = λ−1 (ω1,1 ). (3.0.20) Proof. This follows from ω1,1 (v) = X ω0,3 (v, ea , eb )η ab = a,b X η(v, ea eb )η ab = η(v, e) a,b for every v ∈ A. Proof of Proposition 3.7. Since the starting case g = 1 follows from the above Lemma, we prove the formula by induction, which goes as follows: X ωg,1 (v) = ωg−1,3 (v, ea , eb )η ab a,b = X ω0,4 (v, ea , eb , ei )ωg−1,1 (ej )η ab η ij i,j,a,b = X = X η(vea eb , ei )ωg−1,1 (ej )η ab η ij i,j,a,b η(ve, ei )ωg−1,1 (ej )η ij i,j = ωg−1,1 (ve) = ω1,1 (veg−1 ) = η(veg−1 , e) = η(v, eg ). A closed genus g surface is obtained by sewing g genus 1 pieces with one output boundaries to a genus 0 surface with g input boundaries. Since the Euler element is the output of the genus 1 surface with one boundary, we obtain the same result g z }\| { Z(Σg ) = ω0,g (e, . . . , e). Finally we have the following: Theorem 3.9. The value of a 2D TQFT is given by (3.0.21) ωg,n (v1 , . . . , vn ) = (v1 · · · vn eg ). INVITATION TO 2D TQFT AND QUANTIZATION 21 Proof. The argument is the same as the proof of Proposition 3.7: ωg,n (v1 , . . . , vn ) = ω1,n (v1 eg−1 , v2 , . . . , vn ) X = ω0,n+2 (v1 eg−1 , v2 , . . . , vn , ea , eb )η ab a,b = (v1 · · · vn eg ). 4. Category of cell graphs In the original formulation of 2D TQFT, the operations of multiplication and comultiplication are associated with an oriented surface of genus 0 with three boundary circles. In this section, we introduce a category of ribbon graphs, which carries the information of all finite-dimensional Frobenius algebras. To avoid unnecessary confusion, we use the terminology of cell graphs in this article, instead of more common ribbon graphs. Ribbon graphs naturally appear for encoding complex structures of a topological surface (see for example, [62, 75]). Our purpose of using ribbon graphs are for degeneration of stable curves, and we label vertices, instead of faces, of a ribbon graph. Definition 4.1 (Cell graphs). A connected cell graph of topological type (g, n) is the 1-skeleton of a cell-decomposition of a connected closed oriented surface of genus g with n labeled 0-cells. We call a 0-cell a vertex, a 1-cell an edge, and a 2-cell a face, of the cell graph. We denote by Γg,n the set of connected cell graphs of type (g, n). Each edge consists of two half-edges connected at the midpoint of the edge. Remark 4.2. • The dual of a cell graph is a ribbon graph, or Grothendieck’s dessin d’enfant. We note that we label vertices of a cell graph, which corresponds to face labeling of a ribbon graph. Ribbon graphs are also called by different names, such as embedded graphs and maps. • We identify two cell graphs if there is a homeomorphism of the surfaces that brings one cell-decomposition to the other, keeping the labeling of 0-cells. The only possible automorphisms of a cell graph come from cyclic rotations of half-edges at each vertex. Definition 4.3 (Directed cell graph). A directed cell graph is a cell graph for which an arrow is assigned to each edge. An arrow is the same as an ordering of the two half-edges forming an edge. The set of directed cell graphs of type (g, n) is denoted by ~Γg,n . Remark 4.4. A directed cell graph is a quiver. Since our graph is drawn on an oriented surface, a directed cell graph carries more information than its underlying quiver structure. The tail vertex of an arrowed edge is called the source, and the head of the arrow the target, in the quiver language. To label n vertices, we normally use the n-set [n] := {1, 2, . . . , n}. However, it is often easier to use any totally ordered set of n elements for labeling. The main reason we label the vertices of a cell graph is we wish to assign an element of a K-vector space A to each vertex. In this article, we consider the case that a cell graph γ ∈ Γg,n defines a linear map (4.0.1) Γg,n 3 γ : A⊗n −→ K. 22 O. DUMITRESCU AND M. MULASE The set of values of these functions γ can be more general. We discuss some of the general cases in [33]. An effective tool in graph enumeration is edge-contraction operations. Often edge contraction leads to an inductive formula for counting problems of graphs. The same edgecontraction operations acquire algebraic meaning in our consideration. Definition 4.5 (Edge-contraction operations). There are two types of edge-contraction operations applied to cell graphs. ~ = p−→ • ECO 1: Suppose there is a directed edge E i pi in a cell graph γ ∈ Γg,n , ~ in γ, and put connecting the tail vertex pi and the head vertex pj . We contract E the two vertices pi and pj together. We use i for the label of this new vertex, and call it again pi . Then we have a new cell graph γ 0 ∈ Γg,n−1 with one less vertices. In this process, the topology of the surface on which γ is drawn does not change. Thus genus g of the graph stays the same. E p i p j p i Figure 4.1. Edge-contraction operation ECO 1. The edge bounded by two vertices pi and pj is contracted to a single vertex pi . ~ for the edge-contraction operation • We use the notation E ~ : Γg,n 3 γ 7−→ γ 0 ∈ Γg,n−1 . (4.0.2) E ~ in γ ∈ Γg,n at the i-th vertex pi . Since • ECO 2: Suppose there is a directed loop L a loop in the 1-skeleton of a cell decomposition is a topological cycle on the surface, its contraction inevitably changes the topology of the surface. First we look at the half-edges incident to vertex pi . Locally around pi on the surface, the directed loop ~ separates the neighborhood of pi into two pieces. Accordingly, we put the incident L half-edges into two groups. We then break the vertex pi into two vertices, pi1 and pi2 , so that one group of half-edges are incident to pi1 , and the other group to pi2 . ~ upward near at vertex The order of two vertices is determined by placing the loop L pi . Then we name the new vertex on its left by pi1 , and on its right by pi2 . ~ and Let γ 0 denote the possibly disconnected graph obtained by contracting L separating the vertex to two distinct vertices labeled by i1 and i2 . ~ is a loop of handle. We use the • If γ 0 is connected, then it is in Γg−1,n+1 . The loop L ~ to indicate the edge-contraction operation same notation L ~ : Γg,n 3 γ 7−→ γ 0 ∈ Γg−1,n+1 . (4.0.3) L • If γ 0 is disconnected, then write γ 0 = (γ1 , γ2 ) ∈ Γg1 ,\|I\|+1 × Γg2 ,\|J\|+1 , where ( g = g1 + g2 . (4.0.4) I t J = {1, . . . , bi, . . . , n} The edge-contraction operation is again denoted by ~ : Γg,n 3 γ 7−→ (γ1 , γ2 ) ∈ Γg ,\|I\|+1 × Γg ,\|J\|+1 . (4.0.5) L 1 2 INVITATION TO 2D TQFT AND QUANTIZATION 23 p i L p i1 p i2 ~ Figure 4.2. Edge-contraction operation ECO 2. The contracted edge is a loop L ~ of a cell graph. Place the loop so that it is upward near at pi to which L is attached. The vertex pi is then broken into two vertices, pi1 on the left, and pi2 on the right. Half-edges incident to pi are separated into two groups, belonging to two sides of the loop near pi . ~ a separating loop. Here, vertices labeled by I belong to In this case we call L the connected component of genus g1 , and those labeled by J are on the other component of genus g2 . Let (I− , i, I+ ) (reps. (J− , i, J+ )) be the reordering of I t {i} (resp. J t {i}) in the increasing order. Although we give labeling i1 , i2 to the two vertices created by breaking pi , since they belong to distinct graphs, we can simply ~ translates use i for the label of pi1 ∈ γ1 and the same i for pi2 ∈ γ2 . The arrow of L into the information of ordering among the two vertices pi1 and pi2 . Remark 4.6. Let us define m(γ) = 2g − 2 + n for a graph γ ∈ Γg,n . Then every edgecontraction operation reduces m(γ) exactly by 1. Indeed, for ECO 1, we have m(γ 0 ) = 2g − 2 + (n − 1) = m(γ) − 1. The ECO 2 applied to a loop of handle produces m(γ 0 ) = 2(g − 1) − 2 + (n + 1) = m(γ) − 1. For a separating loop, we have +) 2g1 − 2 + \|I\| + 1 2g2 − 2 + \|J\| + 1 2g1 + 2g2 − 4 + \|I\| + \|J\| + 2 = 2g − 2 + n − 1. The motivation for our introduction of directed cell graphs is that we need them when we deal with non-commutative Frobenius algebras. The operation of taking disjoint union is symmetric. Therefore, 2D TQFT inevitably leads to a commutative Frobenius algebra. The advantage of our formalism using directed cell graphs is that we can deal with noncommutative Frobenius algebras and non-symmetric tensor products. For the purpose of presenting the idea of the category of cell graphs as simple as possible, we restrict ourselves to undirected cell graphs in this article. Therefore, we will only recover commutative Frobenius algebras and usual 2D TQFT. A more general theory will be given in [33]. We now introduce the category of cell graphs. The most unusual point we present here is that a morphism between cell graphs is not a cell map. Recall that a cell map f : γ −→ γ 0 from a cell graph γ to another cell graph γ 0 is a topological map between 1-dimensional cell complexes. Thus f sends a vertex of γ to a vertex of γ 0 , and an edge of γ to either an edge or a vertex of γ 0 , keeping the incidence relations. In particular, a cell map is continuous with respect to the topological structure on cell graphs indued from the surface on which they are drawn. 24 O. DUMITRESCU AND M. MULASE Definition 4.7 (Category of cell graphs). The category CG of cell graphs is defined as follows. • The set of objects of CG is the set of all cell graphs: a Γg,n . (4.0.6) Ob(CG) = g≥0,n>0 • A morphism f ∈ Hom(γ, γ 0 ) is a composition of a finite sequence of edge-contraction operations and cell graph automorphisms. In particular, Hom(γ, γ) = Aut(γ). If there is no way to bring γ to γ 0 by consecutive applications of edge-contraction operations and automorphisms, then we define Hom(γ, γ 0 ) = ∅, even though there may be cell maps between them. Remark 4.8. The triple (CG, t, ∅) forms a symmetric monoidal category. Remark 4.9. Automorphisms of a cell graph and ECOs of the first kind are cell maps, but ECO 2 operations are not. When an ECO 2 is involved, a morphism between cell graphs does not have to be a cell map. Even it may not be a continuous map. Example 4.10. A few simple examples of morphisms are given below. Note that vertices are all labeled, and automorphisms are required to keep labeling. (4.0.8) Hom (•, •) = {id}. Hom •, • • = ∅. (4.0.9) E1 E2 Hom(• • •, • •) = {E1 , E2 }. (4.0.7) (4.0.10) (4.0.11) E1 E2 Hom(• • •, •) = {E1 E2 = E2 E1 }. E1 Hom • •, • = {E1 , E2 = σ(E1 )}. E2 (4.0.12) Hom • •, • • = {E1 E2 = E2 E1 }. E1 E2 In (4.0.10), we note that E1 E2 := E1 ◦ E2 is equal to E2 E1 := E2 ◦ E1 , because they both produce the same result • • • → • • → •. The cell graph of the left of (4.0.11) and (4.0.12) has an automorphism σ that interchanges E1 and E2 . Thus as an edge-contraction operation, E2 = E1 ◦ σ = σ(E1 ). Note that there is a 2 : 1 covering cell map for the E1 case of (4.0.11) that sends both edges E1 and E2 on • • to the single loop of • , and E2 the two vertices on the first graph to the single vertex on the second. Since it is not an edge-contraction, this cell map is not a morphism. The morphism of (4.0.12) is not a cell map, since it is not continuous. Let Vect be the category of finite-dimensional K-vector spaces. The triple C = (Vect, ⊗, K) forms a monoidal category. Again for simplicity, we are concerned only with symmetric tensor products in this article, so we consider C a symmetric monoidal category. A Kobject in Vect is a pair (V, : V −→ K) consisting of a vector space V and a linear map INVITATION TO 2D TQFT AND QUANTIZATION 25 : V −→ K. We denote by Vect/K the category of K-objects in Vect. It has the unique final object (K, id : K −→ K). Therefore, C/K = Vect/K, ⊗, (K, id : K −→ K) is again a monoidal category. We denote by Fun(C/K, C/K) (4.0.13) the endofunctor category of the monoidal category C/K, which consists of monoidal functors α : C/K −→ C/K as its objects, and their natural transformations τ as morphisms. Schematically, we have V α(h) W β(h) >K g β(f ) α(f ) f h / β(V ) τ α(V ) <K α(g) τ α(W ) / K. < τ / β(W ) β(g) Here, the triangle on the left shows two objects (V, f : V −→ K) and (W, g : W −→ K) of C/K, and a morphism h between them. The prism shape on the right represents two monoidal endofunctors α and β that assigns V 7−→ α(V ), V 7−→ β(V ) W 7−→ α(W ), W 7−→ β(W ), and a natural transformation τ : α −→ β among them. The final object of Fun(C/K, C/K) is the functor (4.0.14) φ : (V, f : V −→ K) −→ (K, idK : K −→ K) which assigns the final object of the codomain C/K to everything in the domain C/K. With respect to the tensor product and the above functor (4.0.14) as its identity object, the endofunctor category Fun(C/K, C/K) is again a monoidal category. Definition 4.11 (ECO functor, [33]). The ECO functor is a monoidal functor (4.0.15) ω : CG −→ Fun(C/K, C/K) satisfying the following conditions. • The graph • ∈ Γ0,1 consisting of only one vertex and no edge corresponds to the identity functor (4.0.16) ω(•) = id : C/K −→ C/K. • Each graph γ ∈ Γg,n corresponds to a functor (4.0.17) ω(γ) : (V, : V −→ K) 7−→ (V ⊗n , ωV (γ) : V ⊗n −→ K). • Edge-contraction operations correspond to natural transformations. Let us recall the notion of Frobenius object. 26 O. DUMITRESCU AND M. MULASE Definition 4.12 (Frobenius object). Let (C, ⊗, K) be a symmetric monoidal category. A Frobenius object is an object V ∈ Ob(C) together with morphisms m : V ⊗ V −→ V, 1 : K −→ V, δ : V −→ V ⊗ V, : V −→ K, satisfying the following conditions: • (V, m, 1) is a monoid object in C. • (V, δ, ) is a comonoid object in C. We also require the compatibility condition (1.0.7) among morphisms m and δ: id⊗δ V ⊗V V ⊗ 8 V ⊗V /V m δ⊗id & m⊗id δ ' / V ⊗ V. 8 id⊗m V ⊗V ⊗V Since we are considering the monoidal category of K-objects in Vect, there is a priori no notion of 1 in V . The existence of the morphism 1 : K −→ V requires a non-degeneracy condition. The following theorem is proved in [33]. Theorem 4.13 (Generation of Frobenius objects [33]). An object (V, : V −→ K) of C/K is a Frobenius object if ωV (• •) : V ⊗V −→ K defines a non-degenerate symmetric bilinear form on V . 5. 2D TQFT from cell graphs The result of Section 3 tells us that a 2D TQFT can be defined as a system (A, ωg,n ) of linear maps (5.0.1) ωg,n : A⊗n −→ K defined for all values of g ≥ 0 and n ≥ 1, satisfying a set of conditions. The required conditions are the following: First, (A, ωg,n ) is a CohFT for 2g − 2 + n > 0. In addition, we require that ω0,1 = : A −→ K, (5.0.2) ω0,2 = η : A ⊗ A −→ K. In this section we give a different formulation of a 2D TQFT, based on cell graphs and a different set of axioms. Our ultimate goal is to relate 2D TQFT, CohFT, mirror symmetry, topological recursion, and quantum curves. Later in these lectures, we introduce quantum curves. Relations between all these subjects will be discussed elsewhere [33]. Theorem 5.1 (Graph independence [29]). Let (A, : A −→ K) be a Frobenius object under the ECO functor ω of Definition 4.11. Then every connected cell graph γ ∈ Γg,n gives rise to the same map (5.0.3) ωA (γ) : A⊗n 3 v1 ⊗ · · · ⊗ vn 7−→ (v1 · · · vn eg ) ∈ K, where e is the Euler element of (1.0.12). Corollary 5.2 (ECO implies TQFT). Define ωg,n (v1 , . . . , vn ) = ωA (γ)(v1 , . . . , vn ) for every γ ∈ Γg,n . Then {ωg,n } is a 2D TQFT. Proof. Since the value of (5.0.3) is the same as (3.0.21), it is a 2D TQFT. INVITATION TO 2D TQFT AND QUANTIZATION 27 The rest of the section is devoted to proving Theorem 5.1. We first give three examples of graph independence. Lemma 5.3 (Edge-removal lemma). Let γ ∈ Γg,n . • Case 1. There is a disc-bounding loop L in γ. Let γ 0 ∈ Γg,n be the graph obtained by simply removing L from γ. Note that we are not contracting L. • Case 2. The graph γ containts two edges E1 and E2 between two distinct vertices pi and pj that bound a disc. Let γ 0 ∈ Γg,n be the graph obtained by removing E2 . Here again, we are just eliminating E2 . • Case 3. Two loops, L1 and L2 , in γ are attached to the i-th vertex pi . If they are homotopic, then let γ 0 ∈ Γg,n be the graph obtained by removing L2 from γ. In each of the above cases, we have ωA (γ)(v1 , . . . , vn ) = ωA (γ 0 )(v1 , . . . , vn ). (5.0.4) Figure 5.1. Removal of disc-bounding edges. Proof. Case 1. Contracting a disc-bounding loop attached to pi creates (γ0 , γ 0 ) ∈ Γ0,1 ×Γg,n , where γ0 consists of only one vertex and no edges. The natural transformation corresponding to ECO 1 then gives X ωA (γ)(v1 , . . . , vn ) = η(vi , ek e` )η ka η `b ωA (γ0 )(ea )ωA (γ 0 )(v1 , . . . , vi−1 , eb , vi+1 . . . , vn ) a,b,k,` = X η(vi , ek e` )η ka η `b η(1, ea )ωA (γ 0 )(v1 , . . . , vi−1 , eb , vi+1 . . . , vn ) a,b,k,` = X = X η(vi , 1 · e` )η `b ωA (γ 0 )(v1 , . . . , vi−1 , eb , vi+1 . . . , vn ) b,k,` η(vi , e` )η `b ωA (γ 0 )(v1 , . . . , vi−1 , eb , vi+1 . . . , vn ) b,` = ωA (γ 0 )(v1 , . . . , vi−1 , vi , vi+1 . . . , vn ). Case 2. Contracting Edge E1 makes E2 a disc-bounding loop at pi . We can remove it by Case 1. Note that the new vertex is assigned with vi vj . Restoring E1 makes the graph exactly the one obtained by removing E2 from γ. Thus (5.0.4) holds. Case 3. Contracting Loop L1 makes L2 a disc-bounding loop. Hence we can remove it by Case 1. Then restoring L1 creates a graph obtained from γ by removing L2 . Thus (5.0.4) holds. Remark 5.4. The three cases treated above correspond to removing degree 1 and 2 vertices from the dual ribbon graph. 28 O. DUMITRESCU AND M. MULASE Definition 5.5 (Reduced graph). A cell graph is reduced if it does not contain any discbounding loops or disc-bounding bigons. In terms of dual ribbon graphs, the dual of a reduced cell graph has no vertices of degree 1 or 2. We can see from Lemma 5.3, Case 1, that every graph γ ∈ Γ0,1 gives the same map (5.0.5) ωA (γ)(v) = (v). Similarly, Cases 2 and 3 of Lemma 5.3 show that every graph γ ∈ Γ0,2 defines ωA (γ)(v1 , v2 ) = η(v1 , v2 ). This is because we can remove all edges and loops but one that connects the two vertices. Then by the natural transformation corresponding to ECO 1, the value of the assignment ωA (γ) is (v1 v2 ) = η(v1 , v2 ). Proof of Theorem 5.1. We use the induction on m = 2g − 2 + n. The base case is m = −1, or (g, n) = (0, 1), for which the theorem holds by (5.0.5). Assume that (5.0.3) holds for all (g, n) with 2g − 2 + n < m. Now let γ ∈ Γg,n be a cell graph of type (g, n) such that 2n − 2 + n = m. Choose an arbitrary straight edge of γ that connects two distinct vertices, say pi and pj . Then the natural transformation of contracting this edge to γ 0 gives ωA (γ)(v1 , . . . , vn ) = ωA (γ 0 )(v1 , . . . , vi−1 , vi vj , vi+1 . . . , vbj , . . . , vn ) = (v1 . . . vn eg ). If we have chosen an arbitrary loop attached to pi , then its contraction by ECO 2 gives two cases, depending on whether the loop is a loop of handle or a separating loop. For the former case, we have a graph γ 0 , and by appealing to (1.0.9) and (1.0.13), we obtain X ωA (γ)(v1 , . . . , vn ) = η(vi , ek e` )η ka η `b ωA (γ 0 )(v1 , . . . , vi−1 , ea , eb , vi+1 , . . . , vn ) a,b,k,` = X η(vi ek , e` )η ka η `b ωA (γ 0 )(v1 , . . . , vi−1 , ea , eb , vi+1 , . . . , vn ) a,b,k,` = X = X η ka ωA (γ 0 )(v1 , . . . , vi−1 , ea , vi ek , vi+1 , . . . , vn ) a,k η ka (v1 · · · vn eg−1 ea eb ) a,k = (v1 · · · vn eg ). For the case of a separating loop, ECO 2 makes γ → (γ1 , γ2 ), and we have X ωA (γ)(v1 , . . . , vn ) = η(vi , ek e` )η ka η `b ωA (γ1 ) vI− , ea , vI+ ωA (γ2 ) vJ− , eb , vJ+ a,b,k,` ! = X ka `b η(vi , ek e` )η η ea Y g1 vc e ! eb c∈I a,b,k,` Y g2 vd e d∈J ! = X η(vi ek , e` )η ka η `b η Y vc , ea eg1 ! eb c∈I a,b,k,` d∈J ! = X a,k η ka η Y c∈I vc eg1 , ea ! vi ek Y Y d∈J vd eg2 vd eg2 INVITATION TO 2D TQFT AND QUANTIZATION 29 ! = vi Y c∈I vc eg1 Y vd eg2 d∈J g1 +g2 = (v1 · · · vn e ). Therefore, no matter how we apply ECO 1 or ECO 2, we always obtain the same result. This completes the proof. 6. TQFT-valued topological recursion There is a direct relation between a Frobenius algebra and Gromov-Witten theory when A is given by the big quantum cohomology of a target space. Since these Frobenius algebras are usually infinite-dimensional over the ground field, they do not correspond to a 2D TQFT discussed in the previous sections. But for the case that the target space is 0-dimensional, the TQFT indeed captures the whole Gromov-Witten theory. In this section, we present a general framework. Fundamental examples are A = Q, which gives ψ-class intersection numbers on Mg,n , and the center of the group algebra of a finite group A = ZC[G], which produces Gromov-Witten invariants of the classifying space BG. The first example is considered in [29, 30]. The latter case will be discussed elswhere [33]. We wish to solve a graph enumeration problem, where as a graph we consider a cell graph, and each of its vertex is colored by a parameter v ∈ A. We also impose functoriality under the edge-contraction of Definition 4.5 for this coloring. The ECOs reduce the complexity of coloring considerably, because the functoriality makes the coloring process graph independent, as we have shown in the last section. Thus the answer is just the number of graphs times the value (v1 · · · vn eg ) for each topological type (g, n). Here comes the difficulty: there are infinitely many graphs for each topological type, since we allow multiple edges and loops. The standard idea for such counting problem is to appeal to the Laplace transform, which is introduced by Laplace for this particular context of counting an infinite number of objects. Let us denote by (6.0.1) Γg,n (µ1 , . . . , µn ) the set of all cell graphs with labeled vertices of degrees (µ1 , . . . , µn ). Denoting by cd the number of d-cells in a cell-decomposition of a surface of genus g, d = 0, 1, 2, we have n = c0 , Pn µ i=1 i = 2c1 , and 2 − 2g = n − c1 + c2 . Therefore, (6.0.1) is a finite set. Counting processes become easier if we do not have any object with non-trivial automorphism. There are many ways to eliminate automorphism, for example, the minimalistic way of imposing the least possible conditions, or an excessive way to kill automorphisms but for the most objects the conditions are redundant. Actually, it is known that most of the cell graphs counted in (6.0.1) are without any non-trivial automorphisms. Since any possible automorphism induces a cyclic permutation of half-edges incident to each vertex, the easiest way to disallow any automorphism is to assign an outgoing arrow to one of these half-edges, as in [34, 79] (but not as a quiver). An automorphism should preserve the arrowed half-edges, in addition to labeled vertices. We denote by b g,n (µ1 , . . . , µn ) (6.0.2) Γ the set of arrowed cell graphs with labeled vertices of degrees (µ1 , . . . , µn ), and by (6.0.3) b g,n (µ1 , . . . , µn ) Cg,n (µ1 , . . . , µn ) := Γ its cardinality. This number is always a non-negative integer, and C0,1 (2m) = the m-th Catalan number. 2m 1 m+1 m is 30 O. DUMITRESCU AND M. MULASE b g,n (µ1 , . . . , µn ). We use the notation ωA (γ) = ωg,n : A⊗n −→ K of Corollary 5.2 for γ ∈ Γ We are interested in considering the Cg,n (µ1 , . . . , µn )-weighted TQFT Cg,n (µ1 , . . . , µn ) · ωg,n : A⊗n −→ K, (6.0.4) and applying the edge-contraction operations to these maps. We contract the edge of γ that carries the outgoing arrow at the first vertex p1 , then place a new arrow to the half-edge next to the original half-edge with respect to the cyclic ordering induced by the orientation of the surface. If the edge that carries the outgoing arrow at p1 is a loop, then after splitting p1 , we place another arrow to the next half-edge at each of the two newly created vertices, again next to the original loop. From this process we obtain the following counting formula: Proposition 6.1. The 2D TQFT weighted by the number of arrowed cell graphs satisfies the following equation. Cg,n (µ1 , . . . , µn ) · ωg,n (v1 , . . . , vn ) = n X µj Cg,n−1 (µ1 + µj − 2, µ2 , . . . , µ cj , . . . , µn ) · ωg,n−1 (v1 vj , v2 , . . . , vbj , . . . , vn ) j=2 (6.0.5) + X Cg−1,n+1 (α, β, µ2 , . . . , µn ) · ωg−1,n+2 δ(v1 ), v2 , . . . , vn α+β=µ1 −2 + X X α+β=µ1 −2 X η(v1 , ek e` )η ka η `b g1 +g2 =g a,b,k,` ItJ={2,...,n} × Cg1 ,\|I\|+1 (α, µI ) · ωg1 ,\|I\|+1 (ea , vI ) Cg2 ,\|J\|+1 (β, µJ ) · ωg2 ,\|J\|+1 (eb , vJ ) . This is exactly the same formula of [34, 79, 92] multiplied by ωg,n (v1 , . . . , vn ) = (v1 . . . vn eg ). Let us now consider a Frobenius algebra twisted topological recursion. To simplify the notation, we adopt the following way of writing: X (6.0.6) (f ⊗ h) ◦ δ = η(•, ek e` )η ka η `b f (ea )h(eb ), a,b,k,` where f, h : A −→ K are linear functions on A. First let us review the original topological recursion of [38] defined on a spectral curve Σ, which is just a disjoint union of r copies of open discs. Let U be the unit disc centered at 0 of the complex z-line. We choose r sets of functions (xα , yα ), α = 1, . . . , r, defined on U with Taylor expansions (6.0.7) 2 xα (z) = z + ∞ X k aα,k z , k=3 yα (z) = z + ∞ X bα,k z k , k=2 and a meromorphic 1-form (Cauchy kernel) dz dz − + ωα z−a z−b on U , where a, b ∈ U , and ωα is a holomorphic 1-form on U . Since each xα : U −→ C is a 2 : 1 map, we have an involution on U that keeps the same xα -value: (6.0.9) σα : U −→ U, xα σα (z) = xα (z). (6.0.8) ωαa−b (z) = INVITATION TO 2D TQFT AND QUANTIZATION 31 To avoid confusion, we label r copies of U by U1 , . . . , Ur , and consider the functions (xα , yα ) to be defined on Uα . The topological recursion is the following recursive equation r I σ (z)−z ωαα (z1 ) 1 X (6.0.10) Wg,n (z1 , . . . , zn ) = 2πi y σ (z) − y (z) dxα (z) α α α ∂U α α=1   No (0,1)  × Wg−1,n+1 z, σα (z), z2 , . . . , zn + X g1 +g2 =g ItJ={2,...,n}  Wg1 ,\|I\|+1 (z, zI )Wg2 ,\|J\|+1 σα (z), zJ   on symmetric meromorphic n-differential forms Wg,n defined on the disjoint union U1n t · · · t Urn for 2g − 2 + n > 0. Here, zI = (zi )i∈I , and “No (0, 1)” in the summation means the partition g = g1 + g2 and the set partition I t J = {2, . . . , n} do not allow g1 = 0 and I = ∅, or g2 = 0 and J = ∅. The integration is performed with respect to z ∈ ∂U . Note that the differential form in the big bracket [ ] in (6.0.10) is a symmetric quadratic differential in the variable z ∈ Uα . The expression 1/dxα (z) is the contraction operator with respect to ∂z ∂ on Uα . Thus the integrand of the recursion becomes a meromorphic the vector field ∂x α ∂z 1-form on Uα in the z-variable, for which the integration is performed. The multiplication σ (z)−z by ωαα (z1 ) is simply the symmetric tensor product with a 1-form proportional to dz1 . The 1-form W0,1 and the 2-form W0,2 are defined separately: (6.0.11) W0,1 (z) := r X yα (z)dxα (z) α=1 is defined on the disjoint union U1 t · · · t Ur . If z ∈ Uα , then W0,1 (z) = yα (z)dxα (z). In the form of (6.0.10), however, W0,1 does not appear anywhere. Similarly, W0,2 is defined by W0,2 (z1 , z2 ) := dz1 ωαz1 −b (z2 ) (6.0.12) if z1 , z2 ∈ Uα . Here, the constant b ∈ Uα does not play any role. This 2-form explicitly appears in the recursion part (the terms in the big bracket [ ]) of the formula. Definition 6.2. A Frobenius algebra twisted topological recursion for Wg,n = Wg,n (z1 , . . . , zn ) : A⊗n −→ K is the following formula: Wg,n (z1 , . . . , zn ; v1 , . . . , vn ) r I X 1 X = Kα z, σα (z), z1 ; ek , e` , v1 2πi α=1 ∂Uα a,b,k,`  (6.0.13)  × Wg−1,n+1 z, σα (z), z2 , . . . , zn ; ea , eb , v2 , . . . , vn )  No (0,1) + X g1 +g2 =g ItJ={2,...,n}  Wg1 ,\|I\|+1 (z, zI ; ea , vI )Wg2 ,\|J\|+1 σα (z), zJ ; eb , vJ  . Here (6.0.14) ωασα (z)−z (z1 )η(ek e` , v1 )η ka η `b Kα z, σα (z), z1 ; ek , e` , v1 = yα σα (z) − yα (z) dxα (z) 32 O. DUMITRESCU AND M. MULASE is the integration-summation kernel. Symbolically we can write (6.0.13) as r I 1 X Kα z, σα (z), z1 Wg,n = 2πi α=1 ∂Uα   (6.0.15) No (0,1) X   × W ◦ δ + W ⊗ W ◦ δ g−1,n+1 g ,\|I\|+1 g ,\|J\|+1 1 2   g1 +g2 =g ItJ={2,...,n} with the usual integration kernel σ (z)−z Kα z, σα (z), z1 := yα ωαα (z1 ) . σα (z) − yα (z) dxα (z) Theorem 6.3. The topological recursion (6.0.13) uniquely determines the (A⊗n )∗ -valued n-linear differential form Wg,n from the initial data W0,2 . If the initial data is given by W0,2 (z1 , z2 ; v1 , v2 ) = W0,2 (z1 , z2 ) · η(v1 , v2 ) for a 2-form (6.0.12), then there exists a solution {Wg,n } of the topological recursion (6.0.10) and a 2D TQFT {ωg,n } such that (6.0.16) Wg,n (z1 , . . . , zn ; v1 , . . . , vn ) = Wg,n (z1 , . . . , zn ) · ωg,n (v1 , . . . , vn ). Proof. The proof is done by induction on m = 2g − 2 + n with the base case m = 0. We assume that (6.0.16) holds for all (g, n) such that 2g − 2 + n < m, and use the value ωg,n = (v1 · · · vn eg ) given by (3.0.21) for the values of (g, n) in the range of induction hypothesis. Then by (6.0.13) and functoriality under ECO 2, we conclude that (6.0.16) also holds for all (g, n) such that 2g − 2 + n = m. Remark 6.4. In comparison to edge-contraction operations, we note that the multiplication case ECO 1 does not seem to have a counterpart in an explicit way. It is actually included in the terms involving g1 = 0, \|I\| = 1 and g2 = 0, \|J\| = 1 in the partition sum, and (1.0.14) is used to change the comultiplication to multiplication. More precisely, if I = {i}, then it gives a term W0,2 (z, zi ; ea , vi )Wg,n−1 σα (z), z2 , . . . , zbi , . . . , zn ; eb , v2 , . . . , vbi , . . . , vn = W0,2 (z, zi )Wg,n−1 σα (z), z2 , . . . , zbi , . . . , zn · ω0,2 (ea , vi )ωg,n−1 (eb , v2 , . . . , vbi , . . . , vn ) in the partition sum, assuming the induction hypothesis. We also note that X η(ek e` , v1 )η ka η `b ω0,2 (ea , vi )ωg,n−1 (eb , v2 , . . . , vbi , . . . , vn ) a,b,k,` = X = X η `b ω0,2 (v1 e` , vi )ωg,n−1 (eb , v2 , . . . , vbi , . . . , vn ) b,` η `b η(v1 vi , e` )ωg,n−1 (eb , v2 , . . . , vbi , . . . , vn ) b,` = ωg,n−1 (v1 vi , v1 , . . . , vbi , . . . , vn ). By taking the Laplace transform of (6.0.5) using the method of [34, 79], we obtain a D (z , . . . , z ) · ω D solution Wg,n = Wg,n 1 n g,n to the topological recursion, where Wg,n is given by INVITATION TO 2D TQFT AND QUANTIZATION 33 [34, (4.14)] with respect to a global spectral curve of [34, Theorem 4.3], and ωg,n is the TQFT corresponding to the Frobenius algebra A. Part 2. Quantization of Higgs Bundles 7. Quantum curves The cohomology ring H ∗ (X, C) of a Kähler variety X is a Z/2Z-graded Z/2Z-commutative Frobenius algebra over C. The genus 0 Gromov-Witten invariants of X define the big quantum cohomology of X, which is a quantum deformation of the cohomology ring. If the mirror symmetry is established for X, then the information of big quantum cohomology of X is supposed to be encoded in holomorphic geometry of a mirror dual space Y . GromovWitten invariants are generalized to all values of (g, n) with 2g − 2 + n > 0, g ≥ 0, n > 0. The question is: Question 7.1. What should be the holomorphic geometry on Y that captures all genera Gromov-Witten invariants of X through mirror symmetry? Since the transition from g = 0 to all values of g ≥ 0 is indeed a quantization, the holomorphic object on Y that should capture higer genera Gromov-Witten invariants of X is a quantum geometry of Y . A naı̈ve guess may be that it should be a D-module that represents Y as its classical limit. Hence the D-module is not defined on Y . Then where does it live? The idea of quantum curves concerns a rather restricted situation, when the mirror geometry Y is captured by an algebraic, or an analytic, curve. Typical examples are the mirror of toric Calabi-Yau orbifolds of three dimensions. Geometry of the mirror Y of a toric Calabi-Yau 3-fold is encoded in a complex curve known as the mirror curve. Another situation is enumeration problems of various Hurwitz-type coverings of P1 , and also many decorated graphs on surfaces. For these examples, although there are no “space” X, the mirror geometry exists, and is indeed a curve. These mirror curves are special cases of more general notion of spectral curves. Besides mirror symmetry, spectral curves appear in theory of integrable systems, random matrix theory, topological recursion, and Hitchin theory of Higgs bundles. A spectral curve Σ has two common features. The first one is that it is a Lagrangian subvariety of a holomorphic symplectic surface. The other is the existence of a projection π : Σ −→ C to another curve C, called a base curve. The quantum curve is a D-module on the base curve C such that its semi-classical limit is the spectral curve realized in the cotangent bundle Σ ⊂ T ∗ C. From the analogy of 2D TQFT and CohFT, we note that a spectral curve is already a quantized object, since it corresponds to quantum cohomology. The even part H even (X, C), which is a commutative Frobenius algebra, is not the one that corresponds to a spectral curve. In this sense, CohFT is a result of two quantizations: the first one from classical cohomology to a big quantum cohomology through n-point Gromov-Witten invariants of genus 0; and the second quantization is the passage from genus 0 to all genera. Remark 7.2. We note that quantum cohomology itself is a Frobenius algebra, though it is not finite dimensional, because it requires the introduction of Novikov ring. The even degree part forms a commutative Frobenius algebra, yet it does not correspond to a 2D TQFT in the way we presented in Part 1. Now let us turn to the topic of Part 2. The prototype of quantization is a Schrödinger equation. Consider a harmonic oscillator of mass 1, energy E, and the spring constant 34 O. DUMITRESCU AND M. MULASE 1/4 in one dimension. It has a geometric description as an elliptical motion of a constant angular momentum in the phase space, or the cotangent bundle of the real axis. Here, the spectral curve is an ellipse 1 2 x + y2 = E 4 in a real symplectic plane. The quantization of this spectral curve is the quantization of the harmonic oscillator, which is a second order stationary Schrödinger equation in one variable: 1 d2 (7.0.1) −~2 2 + x2 − E ψ(x, ~) = 0. dx 4 The quantization we discuss in Part 2 is in complete parallelism to quantization of harmonic oscillator. As a holomorphic symplectic surface, we use (C2 , dx ∧ dy). A plane quadric 1 2 (7.0.2) x − y2 = 1 4 is an example of a spectral curve. In complex coordinates, we identify y = d/dx, ignoring the imaginary unit. An example of quantization of (7.0.2) is a Schrödinger equation ! 1 1 2 d 2 + 1 − ~ − x ψ(x, ~) = 0, (7.0.3) ~ dx 2 4 which is essentially the same as quantum harmonic oscillator equation (7.0.1), and is known as the Hermite-Weber equation. Its solutions are all well studied. Question 7.3. Why do we care this well-known classical differential equation? A surprising answer [28, 30, 34, 79] to this question is that we find the intersection numbers of Mg,n through the asymptotic expansion! First we apply a gauge transformation ! 2 1 2 1 2 d 1 1 e− 4~ x ~ + 1 − ~ − x2 e 4~ x Z(x, ~) dx 2 4 (7.0.4) 2 d d = ~ 2 + ~x + 1 Z(x, ~) = 0, dx dx 1 2 where Z(x, ~) = e− 4~ x ψ(x, ~). Recall the integer valued function Cg,n (µ1 , . . . , µn ) of (6.0.3), and define their generating functions by X Cg,n (µ1 , . . . , µn ) −µ n (7.0.5) Fg,n (x1 , . . . , xn ) := x1 1 · · · x−µ . n µ1 · · · µn µ1 ,...,µn >0 It is discovered in [34] that the derivatives Wg,n = d1 · · · dn Fg,n satisfy the topological recursion based on the spectral curve (0.0.1), which is the semi-classical limit of (7.0.4). With an appropriate adjustment for (g, n) = (0, 1) and (0, 2), we have the following allorder WKB expansion formula [30, 34, 79]: ! X 1 (7.0.6) Z(x, ~) = exp ~2g−2+n Fg,n (x, . . . , x) . n! g,n We find ([34]) that if we change the coordinate from x to t by t+1 t−1 (7.0.7) x = x(t) = + , t−1 t+1 INVITATION TO 2D TQFT AND QUANTIZATION 35 then Fg,n x(t1 ), x(t2 ), . . . , x(tn ) is a Laurent polynomial for each (g, n) with 2g − 2 + n > 0. The coordinate change (7.0.7) is identified in [28] as a normalization of the singular curve (0.0.1) in the Hirzebruch surface P KP1 ⊕ OP1 by a sequence of blow-ups. The highest degree part of this Laurent polynomial is a homogeneous polynomial of degree 6g − 6 + 3n 2di +1 ! n X Y (−1)n ti highest (7.0.8) Fg,n (t1 , . . . , tn ) = 2g−2+n hτd1 · · · τdn ig,n \|2di − 1\|!! , 2 2 i=1 d1 +···+dn =3g−3+n where the coefficients Z (7.0.9) hτd1 · · · τdn ig,n = Mg,n ψ1d1 · · · ψndn are cotangent class intersection numbers on the moduli space Mg,n . Topological recursion is a mechanism to calculate all Fg,n (x1 , . . . , xn ) from the single equation (7.0.3), or equivalently, (7.0.4). Thus the quantum curve (7.0.3) has the information of all intersection numbers (7.0.9). These are the topics discussed in our previous lectures [30]. Although the following topic is not what we deal with in this article, for the moment let us consider a symplectic surface (C∗ × C∗ , d log x ∧ d log y). As a spectral curve, we use the zero locus of the A-polynomial AK (x, y) of a knot defined in [14]. For a given knot K ⊂ S 3 , the SL2 (C)-character variety (7.0.10) Hom π1 (S 3 \ K), SL2 (C) SL2 (C) of the fundamental group of the knot complement determines an algebraic curve in (C∗ )2 defined by AK (x, y) ∈ Z[x, y]. Here, (C∗ )2 , or to be more precise, (C∗ )2 /(Z/2Z), is the SL2 (C)-character variety of the fundamental group of the torus T 2 = S 1 × S 1 , which is the boundary of the knot complement. Now consider a function f (q, n) in 2 variables (q, n) ∈ C∗ × Z+ , and define operators ( (b xf )(q, n) := q n f (q, n) (7.0.11) (b y f )(q, n) := f (q, n + 1), following [41, 44]. These operators satisfy the commutation relation x b · yb = qb y·x b. The procedure of changing x 7−→ x b and y 7−→ yb is the Weyl quantization. Garoufalidis [44] conjectures that there exists a quantization of the A-polynomial such that (7.0.12) bK (b A x, yb; q)JK (q, n) = 0, where JK (q, n) is the colored Jones polynomial of the knot K indexed by the dimension n of the irreducible representation of SL2 (C). Here, the quantization means that the operator bK (b A x, yb; q) recovers the A-poynomial by the restriction bK (x, y; 1) = AK (x, y). A This relations is the semi-classical limit, which provides the initial condition of the WKB analysis. A geometric definition of a quantum curve that arises as the quantization of a Hitchin spectral curve is developed in [31], based on the work of [26] that solves a conjecture of Gaiotto [42, 43]. The WKB analysis of the quantum curve [27, 28, 30, 57, 58] is performed by applying the topological recursion of [38]. The Hermite-Weber differential equation is an example of this geometric theory. Although there have been many speculations [50] of 36 O. DUMITRESCU AND M. MULASE the applicability of the topological recursion to low-dimensional topology, still there is no counterpart of the Hitchin type geometric theory for the case of the quantization of Apolynomials. The appearance of the modularity in this context [22, 60, 95] is a tantalizing phenomenon, on which there has been a great advancement. In the following sections, we unfold a different story of quantum curves. In geometry, there is a process parallel to the passage from a spectral curve (0.0.1) to a quantum curve (7.0.4). This process is a journey from the moduli space of Hitchin spectral curves to the moduli space of opers [25, 31]. The quantization parameter, the Planck constant ~ of (7.0.4), acquires a geometric meaning in this process. We begin the story with finding a coordinate independent description of global differential equations of order 2 on a compact Riemann surface. 8. Projective structures, opers, and Higgs bundles In [27], the authors have given a definition of partial differential equation version of topological recursion for Hitchin spectral curves. When the spectral curve is a double sheeted covering of the base curve, we have shown that this PDE topological recursion produces a quantum curve of the Hitchin spectral curve through WKB analysis. The mechanism is explained in detail in [28, 30]. WKB analysis is certainly one way to describe quantization. Yet the passage from spectral curves to their quantization is purely geometric. This point of view is adopted in [31], based on our work [26] on a conjecture of Gaiotto [42]. Our statement is that the quantization process is a biholomorphic map from the moduli space of Hitchin spectral curves to the moduli space of opers [6]. In this section, we introduce the notion of opers, and construct the biholomorphic map mentioned above. In this passage, we give a geometric interpretation of the Planck constant ~ as a deformation parameter of vector bundles and connections. For simplicity of presentation, we restrict our attention to SL2 (C)-opers. We refer to [26, 31] for more general cases. To deal with linear differential equation of order higher then or equal to 2 globally on a compact Riemann surface C, we need a projective coordinate system. If ω is a global holomorphic or meromorphic 1-form on C, then (8.0.1) (d + ω)f = 0 is a first order linear differential equation that makes sense globally on C. This is because d + ω is a homomorphism from OC to KC , allowing singularities if necessary. Here, KC is the sheaf of holomorphic 1-forms on C. Of course existence of a non-trivial global solution of (8.0.1) is a different matter, because it is equivalent to ω = −d log f . Suppose we have a second order differential equation 2 d − q(z) f (z) = 0 (8.0.2) dz 2 2 d locally on C with a local coordinate z. Clearly dz 2 is not globally defined, in contrast to the exterior differentiation d in (8.0.1). What is the requirement for (8.0.2) to make sense coordinate free, then? Let us find an answer by imposing " # " # d 2 d 2 2 2 (8.0.3) dz − q(z) f (z) = 0 ⇐⇒ dw − q(w) f (w) = 0. dz dw INVITATION TO 2D TQFT AND QUANTIZATION 37 We wish the shape of the equation to be analogous to (8.0.1). Since ω is a 1-form, we impose that q ∈ H 0 (C, KC⊗2 ) is a global quadratic differential on C. Under a coordinate change w = w(z), q satisfies q(z)dz 2 = q(w)dw2 . How should we think about a solution f ? For (8.0.2) to have a coordinate free meeting, we need to identify the line bundle L on C to which f belongs. Let us denote by e−g(w(z)) the transition function of L with respect to the coordinate patch w = w(z). A function g(w) = g w(z) will be determined later. A solution f satisfies the coordinate condition (8.0.4) e−g w(z) f w(z) = f (z). Then " d dz 2 " d dz 2 2 g w(z) 2 g w(z) 0 = dz · e = dz · e = dz 2 · eg w(z) d −g e dz # − q(z) 2 − q(z) 2 f (z) # 2 w(z) e−g w(z) f w(z) f w(z) − dw2 q(w)f (w) 2 d − gw (w)w0 f w(z) − dw2 q(w)f (w) dz " # 2 d d 0 2 = dz 2 · − 2gw (w)w0 − gw (w)w0 − gw (w)w0 f w(z) − dw2 q(w)f (w) dz dz h i 0 2 0 = dz 2 · fw (w)w0 − 2gw (w)w0 fw (w)w0 − gw (w)w0 − gw (w)w0 f (w) = dz 2 · − dw2 q(w)f (w) dw 2 2 dz + fw (w)w00 dz 2 − 2gw (w)fw (w)dw2 = fww (w) dz 0 2 − gw (w)w0 − gw (w)w0 f (w)dz 2 − dw2 q(w)f (w) # " 2 d = dw2 · − q(w) f (w) + fw (w) w00 − 2gw (w)(w0 )2 dz 2 dw 0 2 − gw (w)w0 − gw (w)w0 f (w)dz 2 , where w0 = dw/dz. Therefore, for (8.0.3) to hold, we need (8.0.5) (8.0.6) w00 − 2gw (w)(w0 )2 = 0 0 2 gw (w)w0 − gw (w)w0 = 0. From (8.0.5) we find (8.0.7) gw (w)w0 = 1 w00 . 2 w0 Substitution of (8.0.7) in (8.0.6) yields 00 0 w (z) 1 w00 (z) 2 (8.0.8) sz (w) := − = 0. w0 (z) 2 w0 (z) 38 O. DUMITRESCU AND M. MULASE We thus encounter the Schwarzian derivative sz (w). Since (8.0.7) is equivalent to 0 1 g(w)0 = log(w0 ) , 2 1 0 we obtain g(w) = 2 log w . Here, the constant of integration is 0 because g(z) = 0 when w = z. Then (8.0.4) becomes r dz − 12 log w0 f (w) ⇐⇒ f (z) = f (z) = e f (w), dw −1 which identifies the line bundle to which f belongs: f (z) ∈ KC 2 . We conclude that the coordinate change w = w(z) should satisfy the vanishing of the Schwarzian derivative sz (w) ≡ 0, and the solution f (z) should be considered as a (multivalued) section of the −1 inverse half-canonical KC 2 . The vanishing of the Schwarzian derivative dictates us to use a complex projective coordinate system of C. A holomorphic connection in a vector bundle E on C is a C-linear map ∇ : E −→ KC ⊗E satisfying the Leibniz condition ∇(f s) = f ∇(s) + df ⊗ s for f ∈ OC and s ∈ E. Since C is a complex curve, every connection on C is automatically flat. Therefore, ∇ gives rise to a holonomy representation (8.0.9) ρ : π1 (C) −→ G of the fundamental group π1 (C) of the curve C into the structure group G of the vector bundle E. A flat connection ∇ is irreducible if the image of the holonomy representation (8.0.9) is Zariski dense in the complex algebraic group G. In our case, since G = SL2 (C), this requirement is equivalent that Im(ρ) contains two non-commuting elements of G. The moduli space MdeR of irreducible holomorphic connections (E, ∇) in a G-bundle E has been constructed (see [90]). Definition 8.1 (SL2 (C)-opers). Consider a point (E, ∇) ∈ MdeR consisting of an irreducible holomorphic SL2 (C)-connection ∇ : E −→ E ⊗ KC acting on a vector bundle E. It is an SL2 (C)-oper if there is a line subbundle F ⊂ E such that the connection induces an OC -module isomorphism (8.0.10) ∼ ∇ : F −→ (E/F ) ⊗ KC . The notion of oper is a generalization of projective structures on a Riemann surface. Every compact Riemann surface C amsits a projective structure subordinating the given complex structure (see [51]). For our purpose of quantization of Hitchin spectral curves associated with holomorphic Higgs bundles, let us assume that g(C) ≥ 2 in what follows. When we allow singularities, we can relax this condition and deal with genus 0 and 1 cases. A complex projective coordinate system is a coordinate neighborhood covering [ C= Uα α with a local coordinate zα of Uα such that for every Uα ∩ Uβ , we have a fractional linear transformation aαβ zβ + bαβ aαβ bαβ (8.0.11) zα = , fαβ := ∈ SL2 (C), cαβ dαβ cαβ zβ + dαβ satisfying a cocycle condition [fαβ ][fβγ ] = [fαγ ]. Here, [fαβ ] is the class of fαβ in the projection (8.0.12) 0 −→ Z/2Z −→ SL2 (C) −→ P SL2 (C) −→ 0, INVITATION TO 2D TQFT AND QUANTIZATION 39 which determines the fractional linear transformation. A choice of ± on each Uα ∩ Uβ is an element of H 1 (C, Z/2Z) = (Z/2Z)2g , indicating that there are 22g choices of a lift. We make this choice once and for all, and consider fαβ an SL2 (C)-valued 1-cocycle. Since 1 dzβ , (cαβ zβ + dαβ )2 a transition function for KC is given by the cocycle (cαβ zβ + dαβ )2 on each Uα ∩ Uβ . A dzα = 1 theta characteristic (or a spin structure) KC2 is the line bundle defined by the 1-cocycle (8.0.13) ξαβ = cαβ zβ + dαβ . 1 Here again, we have 22g choices ±ξαβ for a transition function of KC2 . Since we have already made a choice of the sign for fαβ , we have a consistent choice in (8.0.13), as explained below. 1 Thus we see that the choice of the lift (8.0.12) is determined by KC2 . From (8.0.13), we obtain ∂β2 ξαβ = 0. (8.0.14) This property plays an essential role in our construction of global connections on C. First we show that actually (8.0.14) implies (8.0.11). Proposition 8.2 (A condition for projective coordinate). A coordinate system of C with 1 which the second derivative of the transition function of KC2 vanishes is a projective coordinate system. 1 Proof. The condition (8.0.14) means that the transition function of KC2 is a linear polynomial cαβ zβ + dαβ satisfying the cocycle condition. Therefore, dzα 1 = . dzβ (cαβ zβ + dαβ )2 Solving this differential equation, we obtain (8.0.15) za = mαβ (cαβ zβ + dαβ ) − 1/cαβ cαβ zβ + dαβ with a constant of integration mαβ . In this way we find an element of SL2 (C) on each Uα ∩ Uβ . The cocycle condition makes (8.0.15) exactly (8.0.11). The transition function fαβ defines a rank 2 vector bundle E on C whose structure group is SL2 (C). Since fαβ is a constant element of SL2 (C), the notion of locally constant sections of E makes sense independent of the coordinate chart. Thus defining ∇α = d on Uα for each Uα determines a global connection ∇ in E. Suppose we have a projective coordinate system on C. Let E be the vector bundle we have just constructed, and P(E) its projectivization, i.e., the P1 bundle associated with E. Noticing that Aut(P1 ) = P SL2 (C), the local coordinate system {zα } of (8.0.11) is a global section of P(E). Indeed, this section defines a map (8.0.16) zα : Uα −→ P1 , which induces the projective structure of P1 into Uα via pull back. The map (8.0.16) is not a constant, because its derivative dzα never vanishes on the intersection Uα ∩ Uβ . A 40 O. DUMITRESCU AND M. MULASE global section of P(E) corresponds to a line subbundle F of E, such that E is realized as an extension 0 −→ F −→ E −→ E/F −→ 0. The equality 1 za aαβ bαβ zβ = 1 1 cαβ zβ + dαβ cαβ dαβ −1 z shows that a defines a global section of the vector bundle KC 2 ⊗ E, and that the 1 projectivization image of this section is the global section {zα } of P(E) corresponding to F . Here, the choice of the theta characteristic is consistently made so that the ± ambiguity of (8.0.13) and the one in the lift of the fractional linear transformation to fαβ ∈ SL2 (C) cancel. Since this section is nowhere vanishing, it generates a trivial subbundle −1 −1 OC = KC 2 ⊗ F ⊂ KC 2 ⊗ E. 1 −1 Therefore, F = KC2 . Note that det E = OC , hence E/F = KC 2 , and E is an extension 1 −1 0 −→ KC2 −→ E −→ KC 2 −→ 0, 1 −1 determining an element of Ext1 KC 2 , KC2 ∼ = H 1 (C, KC ) ∼ = C. There is a more straightforward way to obtain (8.0.17). (8.0.17) Theorem 8.3 (Projective coordinate systems and opers). Every projective coordinate sys1 tem (8.0.11) determines an oper (E, ∇) of Definition 8.1 with F = KC2 . Proof. Let σαβ = −(d/dzβ )ξαβ ξαβ gαβ : = (8.0.18) = −1 = −cαβ , where ξαβ is defined by (8.0.13). Since σαβ cαβ zβ + dαβ −cαβ −1 = ξαβ (cαβ zβ + dαβ )−1 1 aαβ bαβ zβ −1 , zα cαβ dαβ 1 which follows from aαβ − cαβ zα = aαβ (cαβ zβ + dαβ ) − cαβ (aαβ zβ + bαβ ) 1 = , (cαβ zβ + dαβ ) (cαβ zβ + dαβ ) we find that fαβ and gαβ define the same SL2 (C)-bundle E. The shape of the matrix gαβ immediately shows (8.0.17). Since the connection ∇ in E is simply d on each Uα with respect to fαβ , the differential operator on Uα with respect to the transition function gαβ is given by 1 zα −1 0 0 (8.0.19) ∇α := d =d− dzα . −1 zα 1 1 0 Since the (2, 1)-component of the connection matrix is dzα which is nowhere vanishing, 1 (8.0.20) 1 ∇ F = KC2 −→ E ⊗ KC −→ (E/F ) ⊗ KC ∼ = KC2 given by this component is an isomorphism. This proves that (E, ∇) is an SL2 (C)-oper. In the final step of the proof to show (E, ∇) is an oper, we need that (8.0.20) is an OC -linear homomorphism. This is because we are considering the difference of connections ∇ and ∇\|F in E. More generally, suppose we have two connections ∇1 and ∇2 in the same vector bundle E. Then the Leibniz condition tells us that ∇1 (f s) − ∇2 (f s) = f ∇1 (s) − f ∇2 (s) INVITATION TO 2D TQFT AND QUANTIZATION 41 for f ∈ OC and s ∈ E. Therefore, ∇1 −∇2 : E −→ E⊗KC is an OC -module homomorphism. Although the extension class of (8.0.17) is parameterized by H 1 (C, KC ) = C, the complex structure of E depends only if σαβ = 0 or not. This is because ξαβ σαβ λ−1 ξαβ λ2 σαβ λ (8.0.21) = , −1 −1 ξαβ ξαβ λ−1 λ hence we can normalize σαβ = 0 or σαβ = 1. The former case gives the trivial extension of 1 −1 two line bundles E = KC2 ⊕ KC 2 . Since σαβ = cαβ = 0, the projective coordinate system (8.0.11) is actually an affine coordinate system. Since we are assuming g(C) > 1, there is no affine structure in C. Therefore, only the latter case can happen. And the latter case gives a non-trivial extension, as we will show later. Suppose we have another projective structure in C subordinating the same complex structure of C. Then we can adjust the ± signs of the lift of (8.0.12) and the square root of (8.0.13) so that we obtain the exact same holomorphic vector bundle E of (8.0.17). Since we are dealing with a different coordinate system, the only change we have is reflected in the connection ∇. Thus two different projective structures give rise to two connections in the same vector bundle E. Hence this difference is an OC -linear homomorphism E −→ E ⊗ KC as noted above. This consideration motivates the following. A Higgs bundle of rank r [52, 53] defined on C is a pair (E, φ) consisting of a holomorphic vector bundle E of rank r on C and an OC -module homomorphism φ : E −→ E ⊗ KC . An SL2 (C)-Higgs bundle is a pair (E, φ) of rank 2 with a fixed isomorphism det E = OC and trφ = 0. It is stable if every line subbundle F ⊂ E that is invariant with respect to φ, i.e., φ : F −→ F ⊗ KC , has a negative degree degF < 0. The moduli spaces of stable Higgs bundles are constructed in [90]. We denote by MDol the moduli space of stable holomorphic SL2 (C)-Higgs bundles on C. It is diffeomorphic to the moduli space MdeR of pairs (E, ∇) consisting of an irreducible holomorphic connection in an SL2 (C)-bundle (see [23, 52, 90]). A particular diffeomorphism (8.0.22) ∼ ν : MDol −→ MdeR is the non-Abelian Hodge correspondence, which is explained in Section 10. The total space of the line bundle KC is the cotangent bundle π : T ∗ C −→ C of the curve C. We denote by η ∈ H 0 (T ∗ C, π ∗ KC ) the tautological section η T ∗C = π; ∗ KC / T ∗C KC π / C, which is a holomorphic 1-form on T ∗ C. Since φ is an End(E)-valued holomorphic 1-form on C, its eigenvalues are 1-forms. The set of eigenvalues is thus a multivalued section of KC , and hence a multivalued section of π : T ∗ C −→ C. The image Σ ⊂ T ∗ C of this multivalued section is the Hitchin spectral curve, which defines a ramified covering of C. The formal definition of Hitchin spectral curve Σ is that it is the divisor of zeros in T ∗ C of the characteristic polynomial det(η − π ∗ φ) ∈ π ∗ KC⊗2 . 42 O. DUMITRESCU AND M. MULASE Hitchin fibration [52] is a holomorphic fibration (8.0.23) µH : MDol 3 (E, φ) 7−→ det(η − π ∗ φ) ∈ B, B := H 0 C, KC⊗2 , that defines an algebraically completely integrable Hamiltonian system in MDol . Hitchin 1 notices in [52] that the choice of a spin structure KC2 that we have made allows us to construct a natural section κ : B ,→ MDol . Define 0 0 0 1 1 0 t (8.0.24) X− := , X+ := X− = , H := [X+ , X− ] = . 1 0 0 0 0 −1 These elements generate the Lie algebra hX+ , X− , Hi ∼ = sl2 (C). Lemma 8.4. Let q ∈ B = H 0 (C, KC⊗2 ) be an arbitrary point of the Hitchin base B, and define a Higgs bundle (E0 , φ(q)) consisting of a vector bundle 1 ⊗H 1 −1 = KC2 ⊕ KC 2 (8.0.25) E0 := KC2 and a Higgs field (8.0.26) φ(q) := X− + qX+ = 0 q . 1 0 Then it is a stable SL2 (C)-Higgs bundle. The Hitchin section is defined by (8.0.27) κ : B 3 q 7−→ (E0 , φ(q)) ∈ MDol , which gives a biholomorphic map between B and κ(B) ⊂ MDol . Proof. We first note that X− : E0 −→ E0 ⊗ KC is a globally defined End0 (E0 )-valued 1-form, since it is essentially the constant map 1 (8.0.28) −1 = 1 : KC2 −→ KC 2 ⊗ KC . Similarly, multiplication by a quadratic differential gives −1 3 1 q : KC 2 −→KC2 = KC2 ⊗ KC . Thus φ(q) : E0 −→ E0 ⊗ KC is globally defined as a Higgs field in E0 . The Higgs pair is stable because no line subbundle of E0 is invariant under φ(q), unless q = 0. And when 1 q = 0, the invariant line subbundle KC2 has degree g − 1, which is positive since g ≥ 2. Remark 8.5. Hitchin sections exist for the moduli space of stable G-Higgs bundles for an arbitrary simple complex algebraic group G. The construction utilizes Kostant’s principal three-dimensional subgroup (TDS) [66]. The use of TDS is crucial in our quantization, as noted in [26, 31]. The image κ(B) is a holomorphic Lagrangian submanifold of a holomorphic symplectic space MDol . The holomorphic symplectic structure of MDol is induced from its open dense subspace T ∗ SU(2, C), where SU(2, C) is the moduli space of rank 2 stable bundles of degree 0 on C. Since the codimension of the complement of T ∗ SU(2, C) in MDol is 2, the natural holomorphic symplectic form on the cotangent bundle automatically extends to MDol . Our first step of constructing the quantization of the Hitchin spectral curve Σ is to define ~-connections on C that are holomorphically depending on ~. We use a one-parameter INVITATION TO 2D TQFT AND QUANTIZATION 43 family E of deformations of vector bundles E~ /E / C × H 1 (C, KC ), C × {~} and a C-linear first-order differential operator ~∇~ : E~ −→ E~ ⊗ KC depending holomorphically on ~ ∈ H 1 (C, KC ) ∼ = C for ~ 6= 0. Here, we identify the Planck constant ~ of the quantization as a geometric parameter 1 − 12 1 1 2 ~ ∈ H (C, KC ) = Ext KC , KC ∼ = C, which determines a unique extension 1 −1 0 −→ KC2 −→ E~ −→ KC 2 −→ 0. (8.0.29) This is exactly the same as (8.0.17). The extension E~ is given by a system of transition functions ξαβ ~σαβ (8.0.30) E~ ←→ −1 0 ξαβ on each Uα ∩Uβ . The cocycle condition for the transition functions translates into a condition −1 σαγ = ξαβ σβγ + σαβ ξβγ . (8.0.31) The application of the exterior differentiation d to the cocycle condition ξαγ = ξαβ ξβγ for 1 KC2 yields dξβγ dξαβ dξαγ dzβ ξβγ + ξαβ dzγ = dzγ . dzγ dzβ dzγ 2 = Noticing ξαβ dzβ dzα , we find that σαβ := − (8.0.32) dξαβ = −∂β ξαβ dzβ solves (8.0.31). The negative sign is chosen to relate (8.0.30) and (8.0.11). By the same reason as before, the complex structure of the vector bundle E~ is isomorphic to E1 if ~ 6= 0, and to E0 of (8.0.25) if ~ = 0. The transition function can also be written as (8.0.33) ξαβ ~σαβ 1 0 0 1 = exp log ξαβ exp −~∂β log ξαβ . −1 ξαβ 0 −1 0 0 Therefore, in the multiplicative sense, the extension class is determined by ∂β log ξαβ . Lemma 8.6. The extension class σαβ of (8.0.32) defines a non-trivial extension (8.0.29). 44 O. DUMITRESCU AND M. MULASE Proof. The long exact sequences of cohomologies H 1 (C, C) / H 1 (C, OC ) / H 2 (C, C) k / H 1 (C, O ∗ ) d log / H 1 (C, KC ) C H 1 (C, C∗ ) 0 / H 1 (C, KC ) ∼ c1 = H 2 (C, Z) / H 2 (C, Z) associated with exact sequences of sheaves 0 0 0 /Z 0 /C / OC / KC /0 0 / C∗ / O ∗ d log / KC C /0 0 = /Z /0 d 0 0 show that the class {σαβ } corresponds to the image of {ξab } via the map d log ∗) − −−−→ H 1 (C, KC ). H 1 (C, OC If d log{ξαβ } = 0 ∈ H 1 (C, KC ), then it comes from a class in H 1 (C, C∗ ), which is the moduli space of line bundles with holomorphic connections, as explained in [8]. It leads to a contradiction 1 0 = c1 KC2 = g − 1 > 0, because g(C) ≥ 2. Remark 8.7. Gukov and Sulkowski [50] defines an intriguing quantizability condition for a spectral curve in terms of the algebraic K-group K2 C(Σ) of the function field of spectral curve Σ. They relate the quantizability and Bloch regulators of [8]. The class {σαβ } of (8.0.32) gives a natural isomorphism H 1 (C, KC ) ∼ = C. We identify the deformation parameter ~ ∈ C with the cohomology class {~σαβ } ∈ H 1 (C, KC ) = C. S We trivialize the line bundle KC⊗2 with respect to a coordinate chart C = α Uα , and write q ∈ H 0 (C, KC⊗ 2) as {(q)α } that satisfies the transition relation (8.0.34) 4 (q)α = (q)β ξαβ . The transition function of the trivial extension E0 is given by (8.0.35) H ξαβ = exp(H log ξαβ ). Since X− : E0 −→ E0 ⊗ KC is a globally defined Higgs filed, its local expressions {X− dzα } with respect to a coordinate system satisfies the transition relation (8.0.36) X− dzα = exp(H log ξαβ )X− dzβ exp(−H log ξαβ ) INVITATION TO 2D TQFT AND QUANTIZATION 45 on every Uα ∩ Uβ . The same relation holds for the Higgs field φ(q) as well: (8.0.37) φα (q)dzα = exp(H log ξαβ )φβ (q)dzβ exp(−H log ξαβ ). Theorem 8.8 (Construction of SL2 (C)-opers). On each Uα ∩Uβ define a transition function ξαβ 1 −~∂β log ξαβ ~ (8.0.38) gαβ := exp(H log ξαβ ) exp − ~∂β log ξαβ X+ = , −1 · ξαβ 1 where ∂β = d dzβ , and ~∂β log ξαβ ∈ H 1 (C, KC ). Then ~ } satisfies the cocycle condition • The collection {gαβ ~ ~ ~ gαβ gβγ = gαγ , (8.0.39) which defines the holomorphic vector bundle bundle E~ of (8.0.29). • The local expression 1 (8.0.40) ∇~α (0) := d − X− dza ~ on Uα for ~ 6= 0 defines a global holomorphic connection in E~ , i.e., −1 1 1 ~ ~ (8.0.41) d − X− dzα = gαβ d − X− dzβ gαβ , ~ ~ if and only if the coordinate is a projective coordinate system. We choose one. • With this particular projective coordinate system, every point (E0 , φ(q)) ∈ κ(B) ⊂ MDol of the Hitchin section (8.0.27) gives rise to a one-parameter family of SL2 (C)~ opers E~ , ∇ (q) ∈ MdeR . In other words, the local expression 1 ∇~α (q) := d − φα (q)dzα ~ on every Uα for ~ 6= 0 determines a global holomorphic connection −1 ~ ~ ∇~β (q) gαβ (8.0.43) ∇~α (q) = gαβ (8.0.42) in E~ satisfying the oper condition. • Deligne’s ~-connection ~∇~ (q) : E~ −→ E~ ⊗ KC (8.0.44) interpolates the Higgs pair and the oper, i.e., at ~ = 0, the family (8.0.44) gives the Higgs pair (E, −φ(q)) ∈ MDol , and at ~ = 1 it gives an SL2 (C)-oper E1 , ∇1 (q) ∈ MdeR . • After a suitable gauge transformation depending on ~, the ~ → ∞ limit of the oper ∇~ (q) exists and is equal to ∇~=1 (0). This point corresponds to the C∗ -fixed point on the Hitchin section. Proof. The cocycle condition of gαβ has been established in (8.0.32) and (8.0.33). Proof of (8.0.41) is a straightforward calculation, using the power series expansion of the adjoint action n (8.0.45) e ~A Be −~A ∞ ∞ }\| { X X 1 nz 1 n ~ (adA )n (B) := ~ [A, [A, [· · · , [A, B] · · · ]]]. = n! n! n=0 It follows that −1 ~ ~ gαβ X− gαβ n=0 46 O. DUMITRESCU AND M. MULASE = exp(H log ξαβ ) exp (−~∂β log ξαβ X+ ) X− exp (~∂β log ξαβ X+ ) exp(−H log ξαβ ) = exp(H log ξαβ )X− exp(−H log ξαβ ) − ~∂β log ξαβ H − ~2 (∂β log ξαβ )2 exp(H log ξαβ )X+ exp(−H log ξαβ ). Note that (8.0.14) is equivalent to −1 −2 ∂β ∂β log ξαβ = ∂β ξαβ ∂β ξαβ = −ξαβ (∂β ξαβ )2 = −(∂β log ξαβ )2 , hence to −1 ~ ~ = ∂β log ξαβ H + ~(∂β log ξαβ )2 exp(H log ξαβ )X+ exp(−H log ξαβ ). ∂β gαβ gαβ Therefore, noticing (8.0.36), (8.0.14) is equivalent to −1 −1 1 1 ~ ~ ~ ~ dzβ = exp(H log ξαβ )X− dzβ exp(−H log ξαβ ) + ∂β gαβ gαβ g X− gαβ ~ αβ ~ 1 = X− dzα . ~ The statement follows from Proposition 8.2. To prove (8.0.43), we need, in addition to (8.0.41), the following relation: −1 ~ ~ . (q)β X+ dzβ gαβ (8.0.46) (q)α X+ dzα = gαβ But (8.0.46) is obvious from (8.0.37) and (8.0.38). 1 The line bundle F required in the definition of SL2 (C)-oper is simply KC2 . The isomorphism (8.0.10) is a consequence of (8.0.28). Finally, the gauge transformation of ∇~ (q) by a bundle automorphism " 1 # −2 H ~ (8.0.47) ~− 2 = 1 ~2 on each coordinate neighborhood Uα gives H H 1 1 q (8.0.48) d − φ(q) 7−→ ~− 2 d − φ(q) ~ 2 = d − X− + 2 X+ . ~ ~ ~ This is because H H ~− 2 X− ~ 2 = ~X− and H H 2 2 2 ~− 2 X+ ~ = ~−2 X+ , which follows from the adjoint formula (8.0.45). Therefore, lim ∇~ (q) ∼ d − X− = ∇~=1 (0), ~→∞ where the symbol ∼ means gauge equivalence. This completes the proof of the theorem. The construction theorem yields the following. Theorem 8.9 (Biholomorphic quantization of Hitchin spectral curves). Let C be a compact Riemann surface of genus g ≥ 2 with a chosen projective coordinate system subordinating its complex structure. We denote by MDol the moduli space of stable holomorphic SL2 (C)Higgs bundles over C, and by MdeR the moduli space of irreducible holomorphic SL2 (C)1 connections on C. For a fixed theta characteristic KC2 , we have a Hitchin section κ(B) ⊂ INVITATION TO 2D TQFT AND QUANTIZATION 47 MDol of (8.0.27). We denote by Op ⊂ MdeR the moduli space of SL2 (C)-opers with the 1 condition that the required line bundle is given by F = KC2 . Then the map γ (8.0.49) MDol ⊃ κ(B) 3 (E0 , φ(q)) 7−→ E~ , ∇~ (q) ∈ Op ⊂ MdeR evaluated at ~ = 1 is a biholomorphic map with respect to the natural complex structures induced from the ambient spaces. The biholomorphic quantization (8.0.49) is also C∗ -equivariant. The λ ∈ C∗ action on the Hitchin section is defined by φ 7−→ λφ. The oper corresponding to (E0 , λφ(q)) ∈ κ(B) is d − λ~ φ(q). Proof. The C∗ -equivariance follows from the same argument of the gauge transformation (8.0.47), (8.0.48). The action φ 7−→ λφ on the Hitchin section induces a weighted action B 3 q 7−→ λ2 q ∈ B through µH . Then we have the gauge equivalence via the gauge transformation H − H 2 λ λ 2 λ λ λ2 q d − φ(q) ∼ d − φ(q) = d − X− + 2 X+ . ~ ~ ~ ~ ~ λ ~ H2 : Remark 8.10. In the construction theorem, our use of a projective coordinate system is essential, through (8.0.13). Only in such a coordinate, our particular definition (8.0.42) makes sense. This is due to the vanishing of the second derivative of ξαβ . And as we have seen above, the projective coordinate system determines the origin ∇1 (0) of the space Op of opers. Other opers are simply translation ∇1 (q) from the origin by q ∈ H 0 (C, KC⊗2 ). 9. Semi-classical limit of SL2 (C)-opers A holomorphic connection on a compact Riemann surface C is automatically flat. Therefore, it defines a D-module over C. Continuing the last section’s conventions, let us fix a projective coordinate system on C, and let (E0 , φ(q)) = κ(q) be a point on the Hitchin section of (8.0.27). It uniquely defines an ~-family of opers E~ , ∇~ (q) . In this section, we establish that the ~-connection ~∇~ (q) defines a family of D-modules on C parametrized by B such that the semi-classical limit of the family agrees with the family of spectral curves over B. To calculate the semi-classical limit, let us trivialize the vector bundle E~ on each simply connected coordinate neighborhood Uα with coordinate zα of the chosen projective coordinate system. A flat section Ψα of E~ over Uα is a solution of ~ψ ~ (9.0.1) ~∇α (q)Ψα := (~d − φα (q)) = 0, ψ α ~ Ψ , the function ψ on U satisfies with an appropriate unknown function ψ. Since Ψα = gαβ α β −1 the transition relation (ψ)α = ξαβ (ψ)β . It means that ψ is actually a local section of the −1 line bundle KC 2 . There are two linearly independent solutions of (9.0.1), because q is a holomorphic function on Uα . Since φ(q) is independent of ~ and takes the form 0 q (9.0.2) φ(q) = , 1 0 48 O. DUMITRESCU AND M. MULASE we see that (9.0.1) is equivalent to the second order equation ~2 ψ 00 − qψ = 0 (9.0.3) −1 for ψ ∈ KC 2 . Since we are using a fixed projective coordinate system, the connection ∇~ (q) takes the same form on each coordinate neighborhood Uα . Therefore, the shape of the differential equation of (9.0.3) as an equation for ψ is again the same on every coordinate neighborhood, as we wished to achieve in (8.0.2). This is exactly what we refer to as the quantum curve of the spectral curve det(η − φ(q)) = 0. It is now obvious to calculate the semi-classical limit of the D-module corresponding to ~∇~ (q). Theorem 9.1 (Semi-classical limit of an oper). Under the same setting of Theorem 8.9, let E(q) denote the D-module E~ , ~∇~ (q) associated with the oper of (8.0.49). Then the semiclassical limit of E(q) is the spectral curve Σ ⊂ T ∗ C of φ(q) defined by the characteristic equation det(η − φ(q)) = 0. The semi-classical limit of (9.0.3) is the limit " # 2 1 1 d (9.0.4) lim e− ~ S0 (zα ) ~2 − q e ~ S0 (za ) = y 2 − q, ~→0 dzα where S0 (zα ) is a holomorphic function on Uα so that dS0 = ydzα gives a local trivialization of T ∗ C over Uα . The computation of semi-classical limit is the same as the calculation of the determinant of the connection ~∇~ (q), after taking conjugation by the scalar diagonal 1 matrix e− ~ S0 (zα ) I2×2 , and then take the limit as ~ goes to0. For every ~ ∈ H 1 (C, KC ), the ~-connection E~ , ~∇~ (q) of (8.0.44) defines a global DC module structure in E~ . Thus we have constructed a universal family EC of DC -modules on a given C with a fixed spin structure and a projective structure: ⊃ E~ , ∇~ (q) (9.0.5) EC o C × B × H 1 (C, KC ) o C × {q} × {~} The universal family SC of spectral curves is defined over C × B. (9.0.6) P (KC ⊕ OC ) × B o C ×B o SC o C ×B o = det η − φ(q) ⊃ 0 C × {q}. The semi-classical limit is thus a map of families (9.0.7) EC / SC / C × B. C × B × H 1 (C, KC ) 10. Non-Abelian Hodge correspondence between Hitchin moduli spaces The biholomorphic map (8.0.49) is defined by fixing a projective structure of the base curve C. Gaiotto [42] conjectured that such a correspondence would be canonically constructed through a scaling limit of non-Abelian Hodge correspondence. The conjecture has INVITATION TO 2D TQFT AND QUANTIZATION 49 been solved in [26] for holomorphic Hitchin moduli spaces MDol and MdeR constructed over an arbitrary complex simple and simply connected Lie group G. In this section, we review the main result of [26] for G = SL2 (C) and compare it with our quantization. We denote by E top the topologically trivial complex vector bundle of rank 2 on a compact Riemann surface C of genus g ≥ 2. The correspondence between stability conditions of holomorphic vector bundles on C and PDEs on differential geometric data is used in Narasimhan-Seshadri [83] to obtain topological structures of the moduli space of stable bundles (see also [5, 82]). Extending this classical case, the stability condition for an SL2 (C)-Higgs bundle (E, φ) translates into a system of PDEs, known as Hitchin’s equations, imposed on a set of geometric data [23, 52, 90]. The data we need are a Hermitian fiber metric h on E top , a unitary connection D in E top with respect to h, and a differentiable sl2 (C)-valued 1-form φ on C. In this section we use D for unitary connections to avoid confusion with holomorphic connections we have been using until the last section. Hitchin’s equations are the following system of nonlinear PDEs. ( FD + [φ, φ† ] = 0 (10.0.1) D0.1 φ = 0. Here, FD denotes the curvature 2-form of D, φ† is the Hermitian conjugate of φ with respect to the metric h, and D0,1 is the Cauchy-Riemann part of D defined by the complex structure of the base curve C. D0,1 determines a complex structure in E top , which we simply denote by E. Then φ becomes a holomorphic Higgs field in E because it satisfies the Cauchy-Riemann equation (10.0.1). The pair (E, φ) constructed in this way from a solution of Hitchin’s equations is a stable Higgs bundle. Conversely [90], a stable Higgs bundle (E, φ) gives rise to a unique harmonic Hermitian metric h and the Chern connection D with respect to h so that the data satisfy Hitchin’s equations. The stability condition for the holomorphic Higgs pair (E, φ) is thus translated into (10.0.1). Define a one-parameter family of connections 1 ζ ∈ C∗ . (10.0.2) D(ζ) := · φ + D + ζ · φ† , ζ Then the flatness of D(ζ) for all ζ is equivalent to (10.0.1). The non-Abelian Hodge correspondence [23, 52, 73, 90] is a diffeomorphic correspondence e ∇) e ∈ MdeR . ν : MDol 3 (E, φ) 7−→ (E, Proving the diffeomorphism of these moduli spaces is far beyond of the scope of this article. Here, we only give the definition of the map ν. We start with the solution (D, φ, h) of Hitchin’s equations corresponding to a stable Higgs bundle (E, φ). It induces a family of e in E top by D(ζ = 1)0,1 . Since D(ζ) is flat connections D(ζ). Define a complex structure E 1,0 e := D(ζ = 1) is automatically a holomorphic connection in E. e Stability of (E, φ) flat, ∇ e ∇) e ∈ MdeR . implies that the resulting holomorphic connection is irreducible, hence (E, Since this correspondence goes through the real unitary connection D, the change of the e is not a holomorphic deformation. complex structure of E to that of E Extending the idea of one-parameter family (10.0.2), Gaiotto conjectures: Conjecture 10.1 (Gaiotto [42]). Let (D, φ, h) be the solution of (10.0.1) corresponding to a sable Higgs bundle (E0 , φ(q)) on the SL2 (C)-Hitchin section (8.0.27). Consider the following two-parameter family of connections 1 (10.0.3) D(ζ, R) := · Rφ + D + ζ · Rφ† , ζ ∈ C∗ , R ∈ R+ . ζ 50 O. DUMITRESCU AND M. MULASE Then the scaling limit (10.0.4) lim R→0,ζ→0 ζ/R=~ D(ζ, R) exists for every ~ ∈ C∗ , and forms an ~-family of SL2 (C)-opers. Remark 10.2. (1) The existence of the limit is non-trivial, because the Hermitian metric h blows up as R → 0. (2) Unlike the case of non-Abelian Hodge correspondence, the Gaiotto limit works only for a point in the Hitchin section. Theorem 10.3 ([26]). Gaiotto’s conjecture holds for an arbitrary simple and simply connected complex algebraic group G. Recall that the representation (3.0.2) gives a realization of C from its universal covering space H as C∼ = H ρ π1 (C) . The representation ρ lifts to SL2 (R) ⊂ SL2 (C), and defines a projective structure in C subordinating its complex structure coming from H. This projective structure is what we call the Fuchsian projective structure. Corollary 10.4 (Gaiotto correspondence and quantization [26]). Under the same setting of Conjecture 10.1, the limit oper of (10.0.4) is given by 1 (10.0.5) lim D(ζ, R) = d − φ(q) = ∇~ (q), ~= 6 0, R→0,ζ→0 ~ ζ/R=~ with respect to the Fuchsian projective coordinate system. The correspondence γ (E0 , φ(q)) 7−→ E~ , ∇~ (q) is biholomorphic, unlike the non-Abelian Hodge correspondence. Proof. The key point is that since E0 is made out of KC , the fiber metric h naturally comes from the metric of C itself. Hitchin’s equations (10.0.1) for q = 0 then become a harmonic equation for the metric of C, and its solution is given by the constant curvature hyperbolic metric. This metric in turn defines the Fuchsian projective structure in C. For more detail, we reefer to [25, 26]. 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Olivia Dumitrescu: Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, U.S.A., and Simion Stoilow Institute of Mathematics, Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania E-mail address: dumit1om@cmich.edu Motohico Mulase: Department of Mathematics, University of California, Davis, CA 95616–8633, U.S.A., and Kavli Institute for Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa, Japan E-mail address: mulase@math.ucdavis.edu	0
Containment for Rule-Based Ontology-Mediated Queries Pablo Barceló Gerald Berger Andreas Pieris Center for Semantic Web Research & DCC, University of Chile Institute of Information Systems TU Wien School of Informatics University of Edinburgh arXiv:1703.07994v3 [cs.DB] 19 Apr 2017 pbarcelo@dcc.uchile.cl gberger@dbai.tuwien.ac.at ABSTRACT Following recent work [24, 26, 27, 40], we focus on the case where the ontology is defined by a set of tuple-generating dependencies (tgds), a.k.a. existential rules or Datalog± rules. Handling such OMQs implies new challenges for classical database tasks. Interestingly, some of these challenges are by now well-studied; most notably (a) query evaluation [8, 24, 25, 27]: given an OMQ Q = (S, Σ, q), a database D over S, and a tuple of constants c̄, does c̄ belong to the evaluation of q over every extension of D that satisfies Σ, or, equivalently, is c̄ a certain answer for Q over D? and (b) relative expressiveness [19, 42, 43]: how does the expressiveness of OMQs compare to the one of other query languages? Surprisingly, despite its prominence, no work to date has carried out an in-depth investigation of containment for OMQs based on tgds and UCQs. Query containment is a fundamental static analysis task that amounts to check if the evaluation of a query is always contained in the evaluation of another query. Several database tasks crucially depend on the ability to check query containment; these include, e.g., query optimization, viewbased query answering, querying incomplete databases, integrity checking, and implication of dependencies: cf. [22, 30, 36, 37, 39, 45]. A particularly important instance of the containment problem is the one defined by the class of CQs. It follows from the seminal work of Chandra and Merlin [29] that CQ containment is polynomially equivalent to CQ evaluation, and thus NP-complete. The NP upper bound is not affected if we consider UCQs [54]. This is seen as a positive result for practical applications that rely on UCQ containment, as the input (the two UCQs) is small. In addition, it shows a stark difference with more expressive relational query languages, e.g., relational algebra (or, equivalently, first-order logic), for which containment is undecidable. The main goal of this work is to understand up to which extend the good computational properties of UCQ containment discussed above can be leveraged to the containment problem for OMQs based on tgds and UCQs (simply called OMQs from now on). In particular, we want to understand which classes of tgds guarantee the decidability of the problem, and, whenever this is the case, how can we obtain complexity bounds that are reasonable for practical purposes. We also want to understand what is the exact relationship between OMQ containment and evaluation for such classes. Let us stress that, apart from the traditional applications of containment mentioned above, it has been recently shown that OMQ containment has applications on other important static analysis tasks for OMQs, namely, distribution over components [15], and UCQ rewritability [16]. Many efforts have been dedicated to identifying restrictions on ontologies expressed as tuple-generating dependencies (tgds), a.k.a. existential rules, that lead to the decidability for the problem of answering ontology-mediated queries (OMQs). This has given rise to three families of formalisms: guarded, non-recursive, and sticky sets of tgds. In this work, we study the containment problem for OMQs expressed in such formalisms, which is a key ingredient for solving static analysis tasks associated with them. Our main contribution is the development of specially tailored techniques for OMQ containment under the classes of tgds stated above. This enables us to obtain sharp complexity bounds for the problems at hand, which in turn allow us to delimitate its practical applicability. We also apply our techniques to pinpoint the complexity of problems associated with two emerging applications of OMQ containment: distribution over components and UCQ rewritability of OMQs. 1. apieris@inf.ed.ac.uk INTRODUCTION Motivation and goals. The novel application of knowledge representation tools for handling incomplete and heterogeneous data is giving rise to a new field, recently coined as knowledge-enriched data management [6]. A crucial problem in this field is ontology-based data access (OBDA) [51], which refers to the utilization of ontologies (i.e., sets of logical sentences) for providing a unified conceptual view of various data sources. Users can then pose their queries solely in the schema provided by the ontology, abstracting away from the specifics of the individual sources. In OBDA, one interprets the ontology Σ and the user query q, which is typically a union of conjunctive queries (UCQ), or, equivalently, the expressions defined by the select-project-join-union operators of relational algebra, as two components of one composite query Q = (S, Σ, q), known as ontology-mediated query (OMQ); S is called the data schema, indicating that Q will be posed on databases over S [19]. Therefore, OBDA is often realized as the problem of answering OMQs. 1 The context. As one might expect, when considered in its full generality, i.e., without any restrictions on the set of tgds, the OMQ containment problem is undecidable. To understand, on the other hand, which restrictions lead to decidability, we recall the two main reasons that render the general containment problem undecidable. These are: The above small witness property allows us to devise a simple non-deterministic algorithm, which makes use of query evaluation as a subroutine for checking non-containment of Q1 in Q2 : guess a database D over S of size at most k, and then check if there is a certain answer for Q1 over D that is not a certain answer for Q2 over D. This algorithm allows us to obtain optimal upper bounds for OMQs based on linear and sticky sets of tgds; however, the exact complexity of OMQs based on non-recursive sets of tgds remains open: Undecidability of query evaluation: OMQ evaluation is, in general, undecidable [12], and it can be reduced to OMQ containment. More precisely, OMQ containment is undecidable whenever query evaluation for at least one of the involved languages (i.e., the language of the left-hand or the right-hand side query) is undecidable. • For OMQs based on linear tgds, the problem is in PSpace, and in ΠP 2 if the arity is fixed. The PSpacehardness is shown by reduction from query evaluation [47], while the ΠP 2 -hardness is inherited from [17]. Undecidability of containment for Datalog: decidability of query evaluation does not ensure decidability of query containment. A prime example is Datalog, or, equivalently, the OMQ language based on full tgds. Datalog containment is undecidable [55], and thus, OMQ containment is undecidable if the involved languages extend Datalog. • For OMQs based on sticky sets of tgds, the problem is in coNExpTime, and in ΠP 2 if the arity of the schema is fixed. The coNExpTime-hardness is shown by exploiting the standard tiling problem for the exponential grid, while the ΠP 2 -hardness is inherited from [17]. • Finally, for OMQs based on non-recursive sets of tgds, containment is in ExpSpace and hard for PNEXP , even for fixed arity. The lower bound is shown by exploiting a recently introduced tiling problem [34]. In view of the above observations, we focus on languages that (a) have a decidable query evaluation, and (b) do not extend Datalog. The main classes of tgds, which give rise to OMQ languages with the desirable properties, can be classified into three main families depending on the underlying syntactic restrictions: (i) guarded tgds [24], which contain inclusion dependencies and linear tgds, (ii) non-recursive sets of tgds [35], and (iii) sticky sets of tgds [27]. While the decidability of containment for the above OMQ languages can be established via translations into query languages with a decidable containment problem, such translations do not lead to optimal complexity upper bounds (details are given below). Therefore, the main goal of our paper is to develop specially tailored decision procedures for the containment problem under the OMQ languages in question, and ideally obtain precise complexity bounds. Our second goal is to exploit such techniques in the study of distribution over components and UCQ rewritability of OMQs. We conclude that in all these cases OMQ containment is harder than evaluation, with one exception: the OMQs based on linear tgds over schemas of unbounded arity. Guarded tgds. The OMQ language based on guarded tgds is not UCQ rewritable, which forces us to develop different tools to study its containment problem. Let us remark that guarded OMQs can be rewritten as guarded Datalog queries (by exploiting the translations devised in [9, 43]), for which containment is decidable in 2ExpTime [20]. But, again, the known rewritings are very large [43], and hence the reduction of containment for guarded OMQs to containment for guarded Datalog does not yield optimal upper bounds. To obtain optimal bounds for the problem in question, we exploit two-way alternating parity automata on trees (2WAPA) [32]. We first show that if Q1 and Q2 are guarded OMQs such that Q1 is not contained in Q2 , then this is witnessed over a class of “tree-like” databases that can be represented as the set of trees accepted by a 2WAPA A. We then build a 2WAPA B with exponentially many states that recognizes those trees accepted by A that represent witnesses to non-containment of Q1 in Q2 . Hence, Q1 is contained in Q2 iff B accepts no tree. Since the emptiness problem for 2WAPA is feasible in exponential time in the number of states [32], we obtain that containment for guarded OMQs is in 2ExpTime. A matching lower bound, even for fixed arity schemas, follows from [16]. Similar ideas based on 2WAPA have been recently used to show that containment for OMQs based on expressive description logics (DLs) is in 2ExpTime [16]. In the DL context, schemas consist only of unary and binary relations. Our automata construction, however, is different from the one in [16] for two reasons: (a) we need to deal with higher arity relations, and (b) even for unary and binary relations, our OMQ language allows to express properties that are not expressible by the DL-based OMQ languages studied in [16]. Our contributions. The complexity of OMQ containment for the languages in question is given in Table 1. Using small fonts, we recall the complexity of OMQ evaluation in order to stress that containment is, in general, harder than evaluation. We divide our contributions as follows: Linear, non-recursive and sticky sets of tgds. The OMQ languages based on linear, non-recursive, and sticky sets of tgds share a useful property: they are UCQ rewritable (implicit in [40]), that is, an OMQ can be rewritten into a UCQ. This property immediately yields decidability for their associated containment problems, since UCQ containment is decidable [54]. However, the obtained complexity bounds are not optimal, since the UCQ rewritings are unavoidably very large [40]. To obtain more precise bounds, we reduce containment to query evaluation, an idea that is often applied in query containment; see, e.g., [29, 31, 54]. Consider a UCQ rewritable OMQ language O. If Q1 and Q2 belong to O, both with data schema S, then we can establish a small witness property, which states that noncontainment of Q1 in Q2 can be witnessed via a database over S whose size is bounded by an integer k ≥ 0, the maximal size of a disjunct in a UCQ rewriting of Q1 . For linear tgds, such an integer k is polynomial, but for non-recursive and sticky sets of tgds it is exponential (implicit in [40]). Combining languages. The above complexity results refer to the containment problem relative to a certain OMQ language O, i.e., both queries fall in O. However, it is natural 2 Linear Sticky Arbitrary Arity Bounded Arity PSpace-c ΠP 2 -c PSpace-c NP-c coNExpTime-c ΠP 2 -c ExpTime-c NP-c NEXP Non-recursive Guarded in ExpSpace and P -hard in ExpSpace and PNEXP -hard NExpTime-c NExpTime-c 2ExpTime-c 2ExpTime-c 2ExpTime-c ExpTime-c Table 1: Complexity of OMQ containment – in small fonts, we recall the complexity of OMQ evaluation. to consider the version of the problem where the involved OMQs fall in different languages. Unsurprisingly, if the lefthand side query is expressed in a UCQ rewritable OMQ language (based on linear, non-recursive or sticky sets of tgds), we can use the algorithm that relies on the small witness property discussed above, which provides optimal upper bounds for almost all the considered cases (the only exception is the containment of sticky in non-recursive OMQs over schemas of unbounded arity). Things are more interesting if the ontology of the left-hand side query is expressed using guarded tgds, while the ontology of the right-hand side query is not guarded. By exploiting automata techniques, we show that containment of guarded in non-recursive OMQs is in 3ExpTime, while containment of guarded in sticky OMQs is in 2ExpTime. We establish matching lower bounds, even over schemas of fixed arity, by refining techniques from [31]. size of the UCQs and the maximum arity of the underlying schema, which are typically very small. For such tasks, a single-exponential time procedure is considered to be acceptable, and it is actually the norm in many cases including database and verification problems; see, e.g., [1, 50, 52]. For OMQs based on sticky, non-recursive and guarded sets of tgds, the containment problem becomes coNExpTimecomplete, PNEXP -hard and 2ExpTime-complete, respectively. This means that we require double-exponential time to solve the problem, which is practically not acceptable. Nevertheless, for sticky sets of tgds, the runtime is doubleexponential only in the maximum arity of the schema, while for guarded sets of tgds is double-exponential only in the size of the UCQs and the maximum arity of the schema. This is good news since, as said above, the size of the UCQs and the arity are typically small, and usually UCQs in OMQs are much smaller than the ontologies. For non-recursive sets of tgds, on the other hand, the runtime is double-exponential, not only in the maximum arity, but also in the number of predicates occurring in the ontology. It is unrealistic to assume that the number of predicates occurring in real-life ontologies is small. This fact, together with the fact that the precise complexity of OMQ containment for non-recursive sets of tgds is still open, suggests that a more careful complexity analysis is needed. This is left as an interesting open problem for future work. Applications. Our techniques and results on containment for guarded OMQs can be applied to other important static analysis tasks, in particular, distribution over components and UCQ rewritability. The notion of distribution over components has been introduced in [3], in the context of declarative networking, and it states that the answer to an OMQ Q can be computed by parallelizing it over the (maximally connected) components of the database. If this is the case, then Q can always be evaluated in a distributed and coordination-free manner. The problem of deciding distribution over components for OMQs has been recently studied in [15]. However, the exact complexity of the problem for guarded OMQs has been left open. By exploiting our results on containment, we can show that it is 2ExpTime-complete. It is well-known that the OMQ language based on guarded tgds is not UCQ rewritable. In view of this fact, it is important to study when a given guarded OMQ Q can be rewritten as a UCQ. This has been studied for OMQs based on central Horn DLs [16, 18]. Interestingly, our automata-based techniques for guarded OMQ containment can be adapted to decide in 2ExpTime whether an OMQ based on guarded tgds over unary and binary relations is UCQ rewritable; a matching lower bound is inherited from [16]. Our result generalizes the result that deciding UCQ rewritability for OMQs based on E LHI, one of the most expressive members of the E L-family of DLs, is 2ExpTime-complete [16]. Organization. Preliminaries are in Section 2. In Section 3 we introduce the OMQ containment problem. Containment for UCQ rewritable OMQs is studied in Section 4, and for guarded OMQs in Section 5. In Section 6 we consider the case where the involved queries fall in different languages. In Section 7 we discuss the applications of our results on guarded OMQ containment and we conclude in Section 8. Proofs and additional details can be found in the appendix. 2. PRELIMINARIES Databases and conjunctive queries. Let C, N, and V be disjoint countably infinite sets of constants, (labeled) nulls and (regular) variables (used in queries and dependencies), respectively. A schema S is a finite set of relation symbols (or predicates) with associated arity. We write R/n to denote that R has arity n. A term is a either a constant, null or variable. An atom over S is an expression of the form R(v̄), where R ∈ S is of arity n > 0 and v̄ is an n-tuple of terms. A fact is an atom whose arguments consist only of constants. An instance over S is a (possibly infinite) set of atoms over S that contain constants and nulls, while a database over S is a finite set of facts over S. We may call an instance and a database over S an S-instance and S-database, respectively. Discussion on Applicability. As shown in Table 1, the containment problem for OMQs based on linear sets of tgds is PSpace-complete, and thus can be solved in singleexponential time. This is not a big practical drawback since the containment problem corresponds to a static analysis task. In fact, the runtime is single exponential only in the 3 Ontology-mediated queries. An ontology-mediated query (OMQ) is a triple (S, Σ, q), where S is a schema, Σ is a set of tgds (the ontology), and q is a (U)CQ over S ∪ sch(Σ) (and possibly other predicates), with sch(Σ) the set of predicates occurring in Σ.1 We call S the data schema. Notice that the set of tgds can introduce predicates not in S, which allows us to enrich the schema of the UCQ q. Moreover, the tgds can modify the content of a predicate R ∈ S, or, in other words, R can appear in the head of a tgd of Σ. We have explicitly included S in the specification of the OMQ to emphasize that it will be evaluated over S-databases, even though Σ and q might use additional relational symbols. The semantics of an OMQ is given in terms of certain answers. The certain answers to a UCQ q(x̄) w.r.t. a database D and a set Σ of tgds is the set of tuples: \ {c̄ ∈ dom(I)\|x̄\| \| c̄ ∈ q(I)}. cert (q, D, Σ) = The active domain of an instance I, denoted dom(I), is the set of all terms occurring in I. A conjunctive query (CQ) over S is a formula of the form: q(x̄) := ∃ȳ R1 (v̄1 ) ∧ · · · ∧ Rm (v̄m ) , (1) where each Ri (v̄i ) (1 ≤ i ≤ m) is an atom without nulls over S, each variable mentioned in the v̄i ’s appears either in x̄ or ȳ, and x̄ are the free variables of q. If x̄ is empty, then q is a Boolean CQ. As usual, the evaluation of CQs is defined in terms of homomorphisms. Let I be an instance and q(x̄) a CQ of the form (1). A homomorphism from q to I is a mapping h, which is the identity on C, from the variables that appear in q to the set of constants and nulls C ∪ N such that Ri (h(v̄i )) ∈ I, for each 1 ≤ i ≤ m. The evaluation of q(x̄) over I, denoted q(I), is the set of all tuples h(x̄) of constants such that h is a homomorphism from q to I. We denote by CQ the class of conjunctive queries. A union of conjunctive queries (UCQ) over S is a formula of the form q(x̄) := q1 (x̄) ∨ · · · ∨ qn (x̄), where each qi (x̄) is a CQ of the form (1). The S evaluation of q(x̄) over I, denoted q(I), is the set of tuples 1≤i≤n qi (I). We denote by UCQ the class of union of conjunctive queries. I⊇D,I\|=Σ Consider an OMQ Q = (S, Σ, q). The evaluation of Q over an S-database D, denoted Q(D), is defined as cert (q, D, Σ). It is well-known that cert (q, D, Σ) = q(chase(D, Σ)); see, e.g., [24]. Thus, Q(D) = q(chase(D, Σ)). Ontology-mediated query languages. We write (C, Q) for the OMQ language that consists of all OMQs of the form (S, Σ, q), where Σ falls in the class C of tgds, i.e., C ⊆ TGD (concrete classes of tgds are discussed below), and the query q falls in Q ∈ {CQ, UCQ}. A problem that is quite important for our work is OMQ evaluation, defined as follows: Tgds and the chase procedure. A tuple-generating dependency (tgd) is a first-order sentence of the form: ∀x̄∀ȳ φ(x̄, ȳ) → ∃z̄ ψ(x̄, z̄) , (2) where φ and ψ are conjunctions of atoms without nulls. For brevity, we write this tgd as φ(x̄, ȳ) → ∃z̄ ψ(x̄, z̄) and use comma instead of ∧ for conjoining atoms. Notice that φ can be empty, in which case the tgd is called fact tgd and is written as ⊤ → ∃z̄ ψ(x̄, z̄). We assume that each variable in x̄ is mentioned in some atom of ψ. We call φ and ψ the body and head of the tgd, respectively. The tgd in (2) is logically equivalent to the expression ∀x̄(qφ (x̄) → qψ (x̄)), where qφ (x̄) and qψ (x̄) are the CQs ∃ȳ φ(x̄, ȳ) and ∃z̄ ψ(x̄, z̄), respectively. Thus, an instance I over S satisfies this tgd iff qφ (I) ⊆ qψ (I). We say that an instance I satisfies a set Σ of tgds, denoted I \|= Σ, if I satisfies every tgd in Σ. We denote by TGD the class of (finite) sets of tgds. The chase is a useful algorithmic tool when reasoning with tgds [24, 35, 47, 49]. We start by defining a single chase step. Let I be an instance over a schema S and τ = φ(x̄, ȳ) → ∃z̄ ψ(x̄, z̄) a tgd over S. We say that τ is applicable w.r.t. I if there exists a tuple (ā, b̄) of terms in I such that φ(ā, b̄) holds in I. In this case, the result of applying τ over I with (ā, b̄) ¯ is the instance J that extends I with every atom in ψ(ā, ⊥), ¯ where ⊥ is the tuple obtained by simultaneously replacing each variable z ∈ z̄ with a fresh distinct null not occurring PROBLEM : INPUT : QUESTION : It is well-known that Eval(TGD, CQ) is undecidable; implicit in [12]. This has led to a flurry of activity for identifying syntactic restrictions on sets of tgds that make the latter problem decidable. Such a restriction defines a subclass C of tgds. The known decidable classes of tgds are classified into three main decidability paradigms, which, in turn, give rise to decidable OMQ languages: Guardedness: A tgd is guarded if its body contains an atom, called guard, that contains all the body-variables. Although the chase under guarded tgds does not necessarily terminate, the problem of deciding whether a tuple of constants is a certain answer to a UCQ w.r.t. a database and a set of guarded tgds is decidable. This follows from the fact that the result of the chase has bounded treewidth (see, e.g., [24]). Let G be the class of (finite) sets of guarded tgds. Then: τ,(ā,b̄) in I. For such a single chase step we write I −−−−→ J. Let us assume now that I is an instance and Σ a finite set of tgds. A chase sequence for I under Σ is a sequence: τ0 ,c̄0 Eval(C, Q) An OMQ Q = (S, Σ, q(x̄)) ∈ (C, Q), an S-database D, and c̄ ∈ dom(D)\|x̄\| . Does c̄ ∈ Q(D)? Proposition 1. [24] Eval(G, CQ) and Eval(G, UCQ) are 2ExpTime-complete, and ExpTime-complete for fixed arity. τ1 ,c̄1 I0 −−−→ I1 −−−→ I2 · · · An important subclass of guarded tgds is the class of linear tgds whose body consists of a single atom. We write L for the class of (finite) sets of linear tgds. of chase steps such that: (1) I0 = I; (2) forSeach i ≥ 0, τi is S a tgd in Σ; and (3) i≥0 Ii \|= Σ. We call i≥0 Ii the result of this chase sequence, which always exists. Although the result of a chase sequence is not necessarily unique (up to isomorphism), each such result is equally useful for our purposes, since it can be homomorphically embedded into every other result. Thus, from now on, we denote by chase(I, Σ) the result of an arbitrary chase sequence for I under Σ. Proposition 2. [25, 47] Eval(L, CQ) and Eval(L, UCQ) are PSpace-complete, and NP-complete for fixed arity. 1 OMQs can be defined for arbitrary first-order theories, not only tgds, and first-order queries, not only UCQs [19]. 4 R(x,y), P(y,z) → ∃w T(x,y,w) T(x,y,z) → ∃w S(y,w) PROBLEM : INPUT : QUESTION : R(x,y), P(y,z) → ∃w T(x,y,w) T(x,y,z) → ∃w S(x,w) Whenever O1 = O2 = O, we refer to the containment problem by simply writing Cont(O). In what follows, we establish some simple but fundamental results, which help to better understand the nature of our problem. We first investigate the relationship between evaluation and containment, which in turn allows us to obtain an initial boundary for the decidability of our problem, i.e., we can obtain a positive result only if the evaluation problem for the involved OMQ languages is decidable (e.g., those introduced in the previous section). We then focus on the OMQ languages introduced in Section 2 and observe that, once we fix the class of tgds, it does not make a difference whether we consider CQs or UCQs. In other words, we show that an OMQ in (C, UCQ), where C ∈ {G, L, NR, S}, can be rewritten as an OMQ in (C, CQ). This fact simplifies our later complexity analysis since for establishing upper (resp., lower) bounds it suffices to focus on CQs (resp., UCQs). × (a) (b) R(x,y), P(y,z) → ∃w T(x,y,w) T(x,y,z) → ∃w S(x,w) Figure 1: Stickiness and Marking. Non-recursiveness: A set Σ of tgds is non-recursive (a.k.a. acyclic [35, 48]), if its predicate graph, the directed graph that encodes how the predicates of sch(Σ) depend on each other, is acyclic. Non-recursiveness ensures the termination of the chase, and thus decidability of OMQ evaluation. Let NR be the class of non-recursive (finite) sets of tgds. Then: 3.1 Evaluation vs. Containment As one might expect, OMQ evaluation and OMQ containment are strongly connected. In fact, as we explain below, the former can be easily reduced to the latter. But let us first introduce some auxiliary notation. Consider a database D and a tuple c̄ = (c1 , . . . , cn ) ∈ dom(D)n , where n ≥ 0. We denote by qD,c̄ (x̄), where x̄ = (xc1 , . . . , xcn ), the CQ obtained from the conjunction of atoms occurring in D after replacing each constant c with the variable xc . Consider now an OMQ Q = (S, Σ, q(x̄)) ∈ (C, CQ), where C is some class of tgds, an S-database D, and a tuple c̄ ∈ dom(D)\|x̄\| . It is not difficult to show that Proposition 3. [48] Eval(NR, CQ) and Eval(NR, UCQ) are NExpTime-complete, even for fixed arity. Stickiness: This condition ensures neither termination nor bounded treewidth of the chase. Instead, the decidability of OMQ evaluation is obtained by exploiting query rewriting techniques (more details on query rewriting are given in Section 4). The goal of stickiness is to capture joins among variables that are not expressible via guarded tgds, but without forcing the chase to terminate. The key property underlying this condition can be described as follows: during the chase, terms that are associated (via a homomorphism) with variables that appear more than once in the body of a tgd (i.e., join variables) are always propagated (or “stick”) to the inferred atoms. This is illustrated in Figure 1(a); the left set of tgds is sticky, while the right set is not. The formal definition is based on an inductive marking procedure that marks the variables that may violate the semantic property of the chase described above [27]. Roughly, during the base step of this procedure, a variable that appears in the body of a tgd τ but not in every head-atom of τ is marked. Then, the marking is inductively propagated from head to body as shown in Figure 1(b). Finally, a finite set of tgds Σ is sticky if no tgd in Σ contains two occurrences of a marked variable. Let S be the class of sticky (finite) sets of tgds. Then: c̄ ∈ Q(D) ⇐⇒ (sch(Σ), ∅, qD,c̄ ) ⊆ (sch(Σ), Σ, q). \| {z } \| {z } Q1 Q2 Let O∅ be the OMQ language that consists of all OMQs of the form (S, ∅, q), i.e., the set of tgds is empty, where q is a CQ. It is clear that Q1 ∈ O∅ and Q2 ∈ (C, CQ). Therefore, for every OMQ language O = (C, CQ), where C is a class of tgds, we immediately get that: Proposition 5. Eval(O) can be reduced in polynomial time into Cont(O∅ , O). We now show that the problem of evaluation is reducible to the complement of containment. Let us say that, for technical reasons which will be made clear in a while, we focus our attention on classes C of tgds that are closed under fact tgd extension, i.e., for every set Σ ∈ C, a set obtained from Σ by adding a (finite) set of fact tgds is still in C. This is not an unnatural assumption since every reasonable class of tgds, such as the ones introduced above, enjoy this property. Consider now an OMQ Q = (S, Σ, q(x̄)) ∈ (C, CQ), where C is some class of tgds, an S-database D, and a tuple c̄ ∈ dom(D)\|x̄\| . It is easy to see that Proposition 4. [27] Eval(S, CQ) and Eval(S, UCQ) are ExpTime-complete, and NP-complete for fixed arity. 3. Cont(O1 , O2 ) Two OMQs Q1 ∈ O1 and Q2 ∈ O2 . Does Q1 ⊆ Q2 ? OMQ CONTAINMENT: THE BASICS The goal of this work is to study in depth the problem of checking whether an OMQ Q1 is contained in an OMQ Q2 , both over the same data schema S, or, equivalently, whether Q1 (D) ⊆ Q2 (D) over every (finite) S-database D. In this case we write Q1 ⊆ Q2 ; we write Q1 ≡ Q2 if Q1 ⊆ Q2 and Q2 ⊆ Q1 . The OMQ containment problem in question is defined as follows; O1 and O2 are OMQ languages (C, Q), where C is a class of tgds (e.g., linear, non-recursive, sticky, etc.), and Q ∈ {CQ, UCQ}: c̄ ∈ Q(D) ⇐⇒ (S, Σ⋆D , qc̄⋆ ) 6⊆ (S, ∅, ∃x P (x)), \| {z } \| {z } Q1 Q2 Σ⋆D where is obtained from Σ by renaming each predicate R in Σ into R⋆ ∈ 6 S and adding the set of fact tgds {⊤ → R⋆ (c1 , . . . , ck ) \| R(c1 , . . . , ck ) ∈ D}, 5 3.3 Plan of Attack qc̄⋆ is obtained from q(c̄) by renaming each predicate R into R⋆ 6∈ S, and the predicate P does not occur in S. Indeed, the above equivalence holds since P 6∈ S implies that Q2 (D) = ∅, for every S-database D. Since C is closed under fact tgd extension, Q1 ∈ (C, CQ), while Q2 ∈ O∅ . We write coCont(O1 , O2 ) for the complement of Cont(O1 , O2 ). Hence, for every OMQ language O = (C, CQ), where C is a class of tgds (closed under fact tgd extension), it holds that: We are now ready to proceed with the complexity analysis of containment for the OMQ languages in question. Our plan of attack can be summarized as follows: • We consider, in Section 4, Cont((C, CQ)), for C ∈ {L, NR, S}. These languages enjoy a crucial property, called UCQ rewritability, which is very useful for our purposes. This property allows us to show the following result: if the containment does not hold, then this is witnessed via a “small” database, which in turn allows us to devise simple guess-and-check algorithms. • We then proceed, in Section 5, with Cont((G, CQ)). This OMQ language does not enjoy UCQ rewritability, and the task of establishing a small witness property as above turned out to be challenging. However, we show the following: if the containment does not hold, then this is witnessed via a “tree-shaped” database, which allows us to devise a decision procedure based on two-way alternating parity automata on finite trees. • In Section 6, we study the case where the OMQ containment problem involves two different languages. If the left-hand side language is UCQ rewritable, then we can devise a guess-and-check algorithm by exploiting the above small witness property. The challenging case is when the left-hand side language is (G, CQ), where again we employ techniques based on tree automata. Proposition 6. Eval(O) can be reduced in polynomial time into coCont(O, O∅ ). By definition, O∅ is contained in every OMQ language (C, CQ), where C is a class of tgds. Therefore, as a corollary of Propositions 5 and 6, we obtain an initial boundary for the decidability of OMQ containment: we can obtain a positive result only if the evaluation problem for the involved OMQ languages is decidable. More formally: Corollary 7. Cont(O1 , O2 ) is undecidable if Eval(O1 ) is undecidable or Eval(O2 ) is undecidable. Can we prove the converse of Corollary 7: Cont(O1 , O2 ) is decidable if both Eval(O1 ) and Eval(O2 ) are decidable? The answer to this question is negative. This is due to the fact that containment of Datalog queries is undecidable [55]. Since Datalog queries can be directly encoded in the OMQ language based on the class F of full tgds, i.e., those without existentially quantified variables, we obtain the following: 4. UCQ REWRITABLE LANGUAGES Proposition 8. [55] Cont((F, CQ)) is undecidable. We now focus on OMQ languages that enjoy the crucial property of UCQ rewritability. This result, combined with the fact that Eval(F) is decidable (since the chase under full tgds always terminates), implies that the converse of Corollary 7 does not hold. Proposition 8 also rules out the OMQ languages that are based on classes of tgds that extend F; e.g., the weak versions of the ones introduced in Section 2, called weakly guarded [24], weakly acyclic [35], and weakly sticky [27] that guarantee the decidability of OMQ evaluation.2 The question that comes up concerns the decidability and complexity of containment for the OMQ languages that are based on the non-weak versions of the above classes, i.e., guarded, non-recursive, and sticky. This will be the subject of the next two sections. Definition 1. (UCQ Rewritability) An OMQ language (C, CQ), where C ⊆ TGD, is UCQ rewritable if, for each OMQ Q = (S, Σ, q(x̄)) ∈ (C, CQ) we can construct a UCQ q ′ (x̄) such that Q(D) = q ′ (D) for every S-database D. We proceed to establish our desired small witness property, based on UCQ rewritability. By the definition of UCQ rewritability, for each language O that is UCQ rewritable, there exists a computable function fO from O to the natural numbers such that the following holds: for every OMQ Q = (S, Σ, q(x̄)) ∈ O, and UCQ rewriting q1 (x̄) ∨ · · · ∨ qn (x̄) of Q, it is the case that max1≤i≤n {\|qi \|} ≤ fO (Q), where \|qi \| denotes the number of atoms occurring in qi . Then: 3.2 From UCQs to CQs Before we proceed with the complexity analysis of containment for the OMQ languages in question, let us state the following useful result: Proposition 10. Consider a UCQ rewritable language O, and two OMQs Q ∈ O and Q′ ∈ (TGD, CQ), both with data schema S. If Q 6⊆ Q′ , then there exists an S-database D, where \|D\| ≤ fO (Q), such that Q(D) 6⊆ Q′ (D). Proposition 9. Given an OMQ Q ∈ (C, UCQ), where C ∈ {G, L, NR, S}, we can construct in polynomial time an OMQ Q′ ∈ (C, CQ) such that Q ≡ Q′ . In Proposition 10 we assume that the left-hand side query falls in a UCQ rewritable language, be we do not pose any restriction on the language of the right-hand side query. Thus, we immediately get a decision procedure for Cont(O1 , O2 ) if O1 is UCQ rewritable and Eval(O2 ) is decidable. Given Q1 = (S, Σ1 , q1 (x̄)) ∈ O1 and Q2 = (S, Σ2 , q2 (x̄)) ∈ O2 : The proof of Proposition 9 relies on the idea of encoding boolean operations (in our case the ‘or’ operator) using a set of atoms; this idea has been used in several other works (see, e.g., [14, 21, 41]). Proposition 9 allows us to focus on OMQs that are based on CQs; in fact, Cont((C1 , CQ), (C2 , CQ)) is C-complete, where C1 , C2 ∈ {G, L, NR, S} and C is a complexity class that is closed under polynomial time reductions, iff Cont((C1 , UCQ), (C2 , UCQ)) is C-complete. 1. Guess an S-database D such that \|D\| ≤ fO1 (Q1 ), and a tuple c̄ ∈ dom(D)\|x̄\| ; and 2 2. Verify that c̄ ∈ Q1 (D) and c̄ 6∈ Q2 (D). The idea of those classes is the same: relax the conditions in the definition of the class, so that only those positions that receive null values during the chase are taken into account. We immediately get that: 6 Proposition 14. It holds that \|sch(Σ)\| f(NR,CQ) (S, Σ, q) ≤ \|q\| · max{\|body (τ )\|} . Theorem 11. Cont(O1 , O2 ) is decidable if O1 is UCQ rewritable and Eval(O2 ) is decidable. τ ∈Σ This general result shows that Cont((C, CQ)) is decidable for every C ∈ {L, NR, S}, but it says nothing about its complexity. This will be the subject of the rest of the section. Proposition 14 implies that non-containment for queries that fall in (NR, CQ) is witnessed via a database of at most exponential size. We show next that this bound is optimal: 4.1 Linearity The problem of computing UCQ rewritings for OMQs in (L, CQ) has been studied in [40], where a resolution-based procedure, called XRewrite, has been proposed. This rewriting algorithm accepts a query Q = (S, Σ, q(x̄)) ∈ (L, CQ) and constructs a UCQ rewriting q ′ (x̄) over S by starting from q and exhaustively applying rewriting steps based on resolution. Let us illustrate this via a simple example: Proposition 15. There are sets of (NR, CQ) OMQs n {Qn 1 = (S, Σ1 , q1 )}n>0 R(x, y) → P (y), n {Qn 2 = (S, Σ2 , q2 )}n>0 , n where \|sch(Σn 1 )\| = \|sch (Σ2 )\| = n + 2, such that for every n n−1 S-database D, if Q1 (D) 6⊆ Qn . 2 (D) then \|D\| ≥ 2 Let us now focus on the complexity of Cont((NR, CQ)). The algorithm underlying Theorem 11, together with the exponential bound provided by Proposition 14, implies that coCont((NR, CQ)) is feasible in non-deterministic exponential time with access to a NExpTime oracle, which immediately implies that Cont((NR, CQ)) is in ExpSpace. Unfortunately, the exact complexity of Cont((NR, CQ)) is still an open problem, and we conjecture that is PNEXP -complete; recall that NExpTime ⊆ PNEXP ⊆ ExpSpace. In what follows, we briefly explain how the PNEXP -hardness is obtained. To this end, we exploit a tiling problem that has been recently introduced in [34]. Roughly speaking, an instance of this tiling problem is a triple (m, T1 , T2 ), where m is an integer in unary representation, and T1 , T2 are standard tiling problems for the exponential grid 2n × 2n . The question is whether, for every initial condition w of length m, T1 has no solution with w or T2 has some solution with w. The initial condition w simply fixes the first m tiles of the first row of the grid. We construct in polynomial time two (NR, CQ) queries Q1 and Q2 such that (m, T1 , T2 ) has a solution iff Q1 ⊆ Q2 . The idea is to force every input database to store an initial condition w of length m, and then encode the problem whether Ti has a solution with w into Qi , for each i ∈ {1, 2}. From the above discussion we get that: Example 1. Assume that S = {P, T }. Consider the set Σ consisting of the linear tgds P (x) → ∃y R(x, y), and T (x) → P (x), and the CQ q(x̄) := ∃y(R(x, y) ∧ P (y)). XRewrite will first resolve the atom P (y) in q using the second tgd, and produce the CQ ∃y(R(x, y) ∧ R(x, z)), which is equivalent to the CQ ∃y R(x, y). Then, ∃y R(x, y) will be resolved using the first tgd, and the CQ P (x) will be obtained, which in turn will be resolved using the third tgd in order to produce the CQ T (x). The UCQ rewriting q ′ (x̄) is P (x) ∨ T (x). It is easy to see that, whenever the input OMQ consists of linear tgds, during the execution of XRewrite it is not possible to obtain a CQ that has more atoms than the original one. This is an immediate consequence of the fact that linear tgds have only one atom in their body. Then: Proposition 12. f(L,CQ) (S, Σ, q) ≤ \|q\|. Having the above result in place, it can be shown that the algorithm underlying Theorem 11 guesses a polynomially sized witness to non-containment, and then calls a C-oracle for solving query evaluation under linear OMQs, where C is PSpace in general, and NP if the arity is fixed; these complexity classes are obtained from Proposition 2. Therefore, coCont((L, CQ)) is in PSpace in general, and in ΣP 2 in case of fixed arity. Regarding the lower bounds, Proposition 5 allows us to inherit the PSpace-hardness of Eval(L, CQ); this holds even for constant-free tgds. Unfortunately, in the case of fixed arity, we can only obtain NP-hardness, while Proposition 6 allows to obtain coNP-hardness. Nevertheless, it is implicit in [17] (see the proof of Theorem 9), where the containment problem for OMQ languages based on description logics is considered, that Cont((L, CQ)) is ΠP 2 -hard, even for tgds of the form P (x) → R(x). Then: Theorem 16. Cont((NR, CQ)) is in ExpSpace, and PNEXP -hard. The lower bound holds even if the arity of the schema is fixed and the tgds are without constants. 4.3 Stickiness We now focus on (S, CQ). As shown in [40], given a query (S, Σ, q), there exists an execution of XRewrite that constructs a UCQ rewriting q1 (x̄) ∨ · · · ∨ qn (x̄) over S with the following property: for each i ∈ {1, . . . , n}, if a variable v occurs in qi in more than one atom, then v already occurs in q. This property has been used in [40] to bound the number of atoms that can appear in a single CQ qi . We write T (q) for the set of terms (constants and variables) occurring in q, C(Σ) for the set of constants occurring in Σ, and ar (S) for the maximum arity over all predicates of S. Theorem 13. Cont((L, CQ)) is PSpace-complete, and ΠP 2 -complete if the arity of the schema is fixed. The lower bounds hold even for tgds without constants. Proposition 17. It holds that 4.2 Non-Recursiveness f(S,CQ) ((S, Σ, q)) ≤ \|S\| · (\|T (q)\| + \|C(Σ)\| + 1)\|ar (S)\| . Although the OMQ language (NR, CQ) is not explicitly considered in [40], where the algorithm XRewrite is defined, the same algorithm can deal with (NR, CQ). By analyzing the UCQ rewritings constructed by XRewrite, whenever the input query falls in (NR, CQ), we can establish the following result; here, body(τ ) denotes the body of the tgd τ : Proposition 17 implies that non-containment for (S, CQ) queries is witnessed via a database of at most exponential size. As for (NR, CQ) queries, we can show that this bound is optimal; here, for a set Σ of tgds, we denote by \|\|Σ\|\| the number of symbols occurring in Σ: 7 words, we study the problem for (G, BCQ), where BCQ denotes the class of Boolean CQs. This does not affect the generality of our proof since it is known that Cont((G, CQ)) can be reduced in polynomial time to Cont((G, BCQ)) [16]. Proposition 18. There exists a set of (S, CQ) OMQs n n n 2 {Q = ({S/n}, Σ , q(x̄))}n>0 , where \|\|Σ \|\| ∈ O(n ), such that for every Q = ({S}, Σ′ , q ′ (x̄)) ∈ (TGD, CQ) and {S}-database D, if Qn (D) 6⊆ Q(D) then \|D\| ≥ 2n−2 . A first glimpse. As already said, (G, CQ) is not UCQ rewritable and, therefore, we cannot employ Proposition 10 in order to establish a small witness property as for the languages considered in Section 4. We have tried to establish a small witness property for (G, CQ) by following a different route, but it turned out to be a difficult task. Nevertheless, we can show a tree witness property, which states that non-containment for (G, CQ) is witnessed via a treelike database. This allows us to devise a procedure based on alternating tree automata. Summing up, the proof for the 2ExpTime membership of (G, CQ) proceeds in three steps: We now study the complexity of Cont((S, CQ)). We first focus on schemas of unbounded arity. Proposition 17 implies that the algorithm underlying Theorem 11 runs in exponential time assuming access to a C-oracle, where C is a complexity class powerful enough for solving Eval(S, CQ) and its complement. But, since Eval(S, CQ) is in ExpTime (see Proposition 4), both Eval(S, CQ) and its complement are in NExpTime, and thus, the oracle call is not really needed. Consequently, coCont((C, CQ)) is in NExpTime. A matching lower bound is obtained by a reduction from the standard tiling problem for the exponential grid 2n × 2n . In fact, the same lower bound has been recently established in [15]; however, our result is stronger as it shows that the problem remains hard even if the right-hand side query is a linear OMQ of a simple form – this is also discussed in Section 6, where containment of queries that fall in different OMQ languages is studied. Regarding schemas of fixed arity, Proposition 17 provides a witness for non-containment of polynomial size, which implies that the algorithm underlying Theorem 11 runs in polynomial time with access to an NPoracle. Therefore, coEval(S, CQ) is in ΣP 2 , while a matching lower bound is implicit in [17]. Then: 1. Establish a tree witness property; 2. Encode the tree-like witnesses as trees that can be accepted by an alternating tree automaton; and 3. Construct an automaton that decides Cont((G, CQ)); in fact, we reduce Cont((G, CQ)) into emptiness for two-way alternating parity automata on finite trees. Each one of the above three steps is discussed in more details in the following three sections. 5.1 Tree Witness Property From the above informal discussion, it is clear that treelike databases are crucial for our analysis. Let us make this notion more precise using guarded tree decompositions. A tree decomposition of a database D is a labeled rooted tree T = (V, E, λ), where λ : V → 2dom(D) , such that: (i) for each atom R(t1 , . . . , tn ) ∈ D, there exists v ∈ V such that λ(v) ⊇ {t1 , . . . , tn }, and (ii) for every term t ∈ dom(D), the set {v ∈ V \| t ∈ λ(v)} induces a connected subtree of T . The tree decomposition T is called [U ]-guarded, where U ⊆ V , if, for every node v ∈ V \ U , there exists an atom R(t1 , . . . , tn ) ∈ D such that λ(v) ⊆ {t1 , . . . , tn }. We write root (T ) for the root node of T , and DT (v), where v ∈ V , for the subset of D induced by λ(v). We are now ready to formalize the notion of the tree-like database: Theorem 19. Cont((S, CQ)) is coNExpTime-complete, even if the set of tgds uses only two constants. In the case of fixed arity, it is ΠP 2 -complete, even for constant-free tgds. Clearly, there exists a double-exponential time algorithm for solving Cont((S, CQ)), which might sound discouraging. However, Proposition 17 implies that the runtime is doubleexponential only in the maximum arity of the data schema. 5. GUARDEDNESS We proceed with the problem of containment for guarded OMQs, and we establish the following result: Theorem 20. Cont((G, CQ)) is 2ExpTime-complete. The lower bound holds even if the arity of the schema is fixed, and the tgds are without constants. Definition 2. An S-database D is a C-tree, where C ⊆ D, if there is a tree decomposition T of D such that: The lower bound is immediately inherited from [16], where it is shown that containment for OMQs based on the description logic E LI is 2ExpTime-hard. Recall that a set of E LI axioms can be equivalently rewritten as a constant-free set of guarded tgds using only unary and binary predicates, which implies the lower bound stated in Theorem 20. However, we cannot immediately inherit the desired upper bound since the DL-based OMQ languages considered in [16] are either weaker than or incomparable to (G, CQ). Nevertheless, the technique developed in [16] was extremely useful for our analysis. Actually, our automata-based procedure exploits a combination of ideas from [16, 44]. The rest of this section is devoted to providing a high-level explanation of this procedure. For the sake of technical clarity, we focus on constant-free tgds and CQs, but all the results can be extended to the general case at the price of more involved definitions and proofs. Moreover, for simplicity, we focus on Boolean CQs. In other 1. DT (root (T )) = C and 2. T is [{root (T )}]-guarded. Roughly, whenever a database D is a C-tree, C is the cyclic part of D, while the rest of D is tree-like. Interestingly, for deciding Cont((G, BCQ)) it suffices to focus on databases that are C-trees and \|dom(C)\| depends only on the left-hand side OMQ. Recall that for a schema S we write ar (S) for the maximum arity over all predicates of S. Then: Proposition 21. Let Qi = (S, Σi , qi ) ∈ (G, BCQ), for i ∈ {1, 2}. The following are equivalent: 1. Q1 ⊆ Q2 . 2. Q1 (D) ⊆ Q2 (D), for every C-tree S-database D such that \|dom(C)\| ≤ (ar (S ∪ sch(Σ1 )) · \|q1 \|). 8 for 2WAPA. As usual, given a 2WAPA A, we denote by L(A) the language of A, i.e., the set of labeled trees it accepts. The emptiness problem is defined as follows: given a 2WAPA A, does L(A) = ∅? Thus, given Q1 , Q2 ∈ (G, BCQ), we need to construct a 2WAPA A such that Q1 ⊆ Q2 iff L(A) = ∅. It is well-known that deciding whether L(A) = ∅ is feasible in exponential time in the number of states, and in polynomial time in the size of the input alphabet [32]. Therefore, we should construct A in double-exponential time, while the number of states must be at most exponential. We first need a way to check consistency of labeled trees. It is not difficult to devise an automaton for this task. The fact that (1) ⇒ (2) holds trivially, while (2) ⇒ (1) is shown by using a variant of the notion of guarded unravelling and compactness. Let us clarify that the above result does not provide a decision procedure for Cont((G, BCQ)), since we have to consider infinitely many databases that are Ctrees with \|dom(C)\| ≤ (ar (S ∪ sch(Σ1 )) · \|q1 \|). 5.2 Encoding Tree-like Databases It is generally known that a database D whose treewidth3 is bounded by an integer k can be encoded into a tree over a finite alphabet of double-exponential size in k that can be accepted by an alternating tree automaton; see, e.g., [13]. Consider an alphabet Γ, and let N∗ be the set of finite sequences of natural numbers, including the empty sequence. A Γ-labeled tree is a pair L = (T, λ), where T ⊆ N∗ is closed under prefixes, and λ : T → Γ is the labeling function. The elements of T identify the nodes of L. It can be shown that D and a tree decomposition T of D with width k can be encoded as a Γ-labeled tree L, where Γ is an alphabet of double-exponential size in k, such that each node of T corresponds to exactly one node of L and vice versa. Consider now a C-tree S-database D, and let T be the tree decomposition that witnesses that D is a C-tree. The width of T is at most k = (\|dom(C)\| + ar (S) − 1), and thus, the treewidth of D is bounded by k. Hence, from the above discussion, D and T can be encoded as a Γ-labeled tree, where Γ is of double-exponential size in k. In general, given an Sdatabase D that is a C-tree due to the tree decomposition T , we show that D and T can be encoded as a ΓS,l -labeled tree, with \|dom(C)\| ≤ l and \|ΓS,l \| being double-exponential in ar (S) and exponential in \|S\| and l. Although every C-tree S-database D can be encoded as a ΓS,l -labeled tree, the other direction does not hold. In other words, it is not true that every ΓS,l -labeled tree encodes a C-tree S-database D and its corresponding tree decomposition. In view of this fact, we need the additional notion of consistency. A ΓS,l -labeled tree is called consistent if it satisfies certain syntactic properties – we do not give these properties here since they are not vital in order to understand the high-level idea of the proof. Now, given a consistent ΓS,l -labeled tree L, we can show that L can be decoded into an S-database JLK that is a C-tree with \|dom(C)\| ≤ l. From the above discussion and Proposition 21, we obtain: Lemma 23. Consider a schema S and an integer l > 0. There is a 2WAPA CS,l that accepts a ΓS,l -labeled tree L iff L is consistent. The number of states of CS,l is logarithmic in the size of ΓS,l . Furthermore, CS,l can be constructed in polynomial time in the size of ΓS,l . Now, the crucial task is, given an OMQ Q ∈ (G, BCQ), to devise an automaton that accepts labeled trees which correspond to databases that make Q true. Lemma 24. Let Q = (S, Σ, q) ∈ (G, BCQ). There is a 2WAPA AQ,l , where l > 0, that accepts a consistent ΓS,l labeled tree L iff Q(JLK) 6= ∅. The number of states of AQ,l is exponential in \|\|Q\|\| and l. Furthermore, AQ,l can be constructed in double-exponential time in \|\|Q\|\| and l. The intuition underlying AQ,l can be described as follows. AQ,l tries to identify all the possible ways the CQ q can be mapped to chase(D, Σ), for any C-tree S-database D such that \|dom(C)\| ≤ l. It then arrives at possible ways how the input tree can satisfy Q. These “possible ways” correspond to squid decompositions, a notion introduced in [24] that indicates which part of the query is mapped to the cyclic part C of D, and which to the tree-like part of D. The automaton exhaustively checks all squid decompositions by traversing the input tree and, at the same time, explores possible ways how to match the single parts of the squid decomposition at hand. The automaton finally accepts if it finds a squid decomposition that can be mapped to chase(D, Σ). Having the above automata in place, we can proceed with our main technical result, which shows that Cont(G, BCQ) can be reduced to the emptiness problem for 2WAPA. But let us first recall some key results about 2WAPA, which are essential for our final construction. It is well-known that languages accepted by 2WAPAs are closed under intersection and complement. Given two 2WAPAs A1 and A2 , we write A1 ∩ A2 for a 2WAPA, which can be constructed in polynomial time, that accepts the language L(A1 ) ∩ L(A2 ). Moreover, for a 2WAPA A, we write A for the 2WAPA, which is also constructible in polynomial time, that accepts the complement of L(A). We can now show the following: Lemma 22. Let Qi = (S, Σi , qi ) ∈ (G, BCQ), for i ∈ {1, 2}. The following are equivalent: 1. Q1 ⊆ Q2 . 2. Q1 (JLK) ⊆ Q2 (JLK), for every consistent ΓS,l -labeled tree L, where l = (ar (S ∪ sch(Σ1 )) · \|q1 \|). 5.3 Constructing Tree Automata Having the above result in place, we can now proceed with our automata-based procedure. We make use of twoway alternating parity automata (2WAPA) that run on finite labeled trees. Two-way alternating automata process the input tree while branching in an alternating fashion to successor states, and thereby moving either down or up the input tree; the detailed definition can be found in [11]. Our goal is to reduce Cont((G, BCQ)) to the emptiness problem Proposition 25. Consider Q1 , Q2 ∈ (G, BCQ). We can construct in double-exponential time a 2WAPA A, which has exponentially many states, such that Q1 ⊆ Q2 ⇐⇒ L(A) = ∅. Proof (sketch). Let Qi = (S, Σi , qi ), for i ∈ {1, 2}, and l = (ar (S ∪ sch(Σ1 )) · \|q1 \|). Then A is defined as (CS,l ∩ AQ1 ,l ) ∩ AQ2 ,l . Since ΓS,l has double-exponential size, Lemmas 23 and 24 imply that A can be constructed in 3 Recall that the treewidth of a database D is the minimum width among all possible tree decompositions T = (V, E, λ) of D, while the width of T is defined as maxv∈V {\|λ(v)\|} − 1. 9 The lower bounds hold even if the arity of the schema is fixed. Moreover, for C = L (resp., C ∈ {NR, S}) it holds even for tgds with one constant (resp., without constants). double-exponential time, while it has exponentially many states. Lemma 22 implies that Q1 ⊆ Q2 iff L(A) = ∅. Proposition 25 implies that Cont((G, BCQ)) is in 2ExpTime, and Theorem 20 follows. Thus, there exists a doubleexponential time algorithm for solving Cont((G, CQ)). Interestingly, the runtime is double-exponential only in the size of the CQs and the maximum arity of the schema. This can be obtained by a providing a more refined complexity analysis of the construction of the 2WAPA A in Proposition 25. 6. Upper bounds. The 2ExpTime membership when C = L is an immediate corollary of Theorem 20. This is not true when C ∈ {NR, S} since the right-hand side query is not guarded. But in this case, since (NR, CQ) and (S, CQ) are UCQ rewritable, one can rewrite the right-hand side query as a UCQ, and then apply the machinery developed in Section 5 for solving Cont((G, CQ)). More precisely, given OMQs Q1 ∈ (G, CQ) and Q2 ∈ (C, CQ), where C ∈ {NR, S}, Q1 ⊆ Q2 iff Q1 ⊆ q, where q is a UCQ rewriting of Q2 . Thus, an immediate decision procedure, which exploits the algorithm XRewrite, is the following: COMBINING LANGUAGES In the previous two sections, we studied the containment problem relative to a language O, i.e., both OMQs fall in O. However, it is natural to consider the version of the problem where the involved OMQs fall in different languages. This is the goal of this section. Our analysis proceeds by considering the two cases where the left-hand side (LHS) query falls in a UCQ rewritable OMQ language, or it is guarded. 1. Let q = XRewrite(Q2 ); 2. For each q ′ ∈ q: if Q1 ⊆ q ′ , then proceed; otherwise, reject; and 3. Accept. 6.1 The LHS Query is UCQ Rewritable The above procedure runs in triple-exponential time. The first step is feasible in double-exponential time [40]. Now, for a single CQ q ′ ∈ q (which is a guarded OMQ with an empty set of tgds) the check whether Q1 ⊆ q ′ can be done by using the machinery developed in Section 5, which reduces our problem to checking whether the language of a 2WAPA A is empty. However, it should not be forgotten that q ′ is of exponential size, and thus, the automaton A has double-exponentially many states. This in turn implies that checking whether L(A) = ∅ is in 3ExpTime, as claimed. Although the above algorithm establishes an optimal upper bound for non-recursive OMQs, a more refined analysis is needed for sticky OMQs. In fact, we need a more refined complexity analysis for the problem Cont((G, CQ), UCQ), that is, to decide whether a guarded OMQ is contained in a UCQ. To this end, we provide an automata construction different from the one employed in Section 5, which allows us to establish a refined complexity upper bound for the problem in question. Consider a (G, CQ) query Q, and a UCQ q = q1 ∨ · · · ∨ qn . As usual, we write \|\|Q\|\| and \|\|qi \|\| for the number of symbols that occur in Q and qi , respectively, and we write var ≥2 (qi ) for the set of variables that appear in more than one atom of qi . By exploiting our new automata-based procedure, we show that the problem of checking if Q ⊆ q is feasible in double-exponential time in (\|\|Q\|\| + max1≤i≤n {\|var ≥2 (qi )\|}), exponential time in max1≤i≤n {\|\|qi \|\|}, and polynomial time in n. This result allows us to show that the above procedure establishes 2ExpTime-membership when the right-hand side OMQ is sticky. But first we need to recall the following key properties of the UCQ rewriting q = XRewrite(Q2 ), constructed during the first step of the algorithm: As an immediate corollary of Theorem 11 we obtain the following result: Cont((C1 , CQ), (C2 , CQ)), for C1 6= C2 , C1 ∈ {L, NR, S} and C2 ∈ {L, NR, S, G}, is decidable. By exploiting the algorithm underlying Theorem 11, we establish optimal upper bounds for all the problems at hand with the only exception of Cont((S, CQ), (NR, CQ)). For the latter, we obtain an ExpSpace upper bound, by providing a similar analysis as for Cont((NR, CQ)), while a NExpTime lower bound is inherited from query evaluation by exploiting Proposition 5. It is rather tedious, and not very interesting from a technical point of view, to go through all the containment problems in question4 and explain in details how the exact upper bounds are obtained; we leave this as an exercise to the interested reader. Regarding the matching lower bounds, in most of the cases they are inherited from query evaluation or its complement by exploiting Propositions 5 and 6, respectively. There are, however, some exceptions: • Cont((S, CQ), (L, CQ)) in the case of unbounded arity, where the problem is coNExpTime-hard, even for sets of tgds that use only two constants. This is shown by a reduction from the standard tiling problem for the exponential grid 2n × 2n . • Cont((L, CQ), (S, CQ)) and Cont((S, CQ), (L, CQ)) in the case of bounded arity, where both problems are ΠP 2 -hard even for constant-free tgds; implicit in [17]. 6.2 The LHS Query is Guarded We proceed with the case where the LHS query is guarded, and we show the following result: 1. q consists of double-exponentially many CQs, Theorem 26. Cont((G, CQ), (C, CQ)) is C-complete:   2ExpTime, C ∈ {L, S}, C =  3ExpTime, C = NR. 2. each CQ of q is of exponential size, and 3. for each q ′ ∈ q, var ≥2 (q ′ ) is a subset of the variables of the original CQ that appears in Q2 . 4 There are eighteen different cases obtained by considering all the possible pairs (O1 , O2 ) of OMQ languages, where O1 6= O2 and O1 is UCQ rewritable, and the two cases whether the arity of the schema is fixed or not. By combining these key properties with the complexity analysis performed above, it is now straightforward to show that Cont((G, CQ), (S, CQ)) is in 2ExpTime. 10 has been left open. Our results on containment for guarded OMQs allow us to close this problem. But first we need to recall a key result that semantically characterizes distribution over components. An OMQ Q with data schema S is unsatisfiable if there is no S-database D such that Q(D) 6= ∅. Moreover, for a CQ q, we write co(q) for its components. The next result has been shown in [15]: Lower Bounds. We establish matching lower bounds by refining techniques from [31], where it is shown that containment of Datalog in UCQ is 2ExpTime-complete, while containment of Datalog in non-recursive Datalog is 3ExpTimecomplete; the lower bounds hold for fixed-arity predicates, and constant-free rules. Interestingly, the LHS query can be transformed into a Datalog query such that each rule has a body-atom that contains all the variables, i.e., is guarded. This is achieved by increasing the arity of some predicates in order to have enough positions for all the bodyvariables. However, for each rule, the number of unguarded variables that we need to guard is constant, and thus, the arity of the schema remains constant. We conclude that Cont((G, CQ), (NR, CQ)) is 3ExpTime-hard. Moreover, containment of guarded OMQs in UCQs is 2ExpTime-hard, which in turn allows us to show, by exploiting the construction underlying Proposition 9, that Cont((G, CQ), (L, CQ)) is 2ExpTime-hard, even if the set of linear tgds uses only one constant, while Cont((G, CQ), (S, CQ)) is 2ExpTime-hard, even for tgds without constants. 7. Proposition 27. Let Q = (S, Σ, q(x̄)) ∈ (G, CQ). The following are equivalent: 1. Q distributes over components. 2. Q is unsatisfiable or there exists q̂(x̄) ∈ co(q) such that (S, Σ, q̂(x̄)) ⊆ Q. Checking unsatisfiability can be easily reduced to containment. Thus, the above result, together with Theorem 20, implies that Dist(G, CQ) is in 2ExpTime, while a matching lower bound is implicit in [15]. Then: Theorem 28. Dist(G, CQ) is 2ExpTime-complete. APPLICATIONS Interestingly, our results on Cont((G, CQ)) can be applied to other important static analysis tasks, in particular, distribution over components and UCQ rewritability. Each one of those tasks is considered in the following two sections. 7.2 Deciding UCQ Rewritability Query rewriting is a well-studied method for evaluating OMQs using standard database technology. The key idea is the following: given an OMQ Q = (S, Σ, q(x̄)), combine Σ and q into a new query qΣ (x̄), the so-called rewriting, which can then be evaluated over D yielding the same answer as Q over D, for every S-database D. For this approach to be realistic, though, it is essential that the rewriting is expressed in a language that can be handled by standard database systems. The typical language that is considered in this setting is first-order (FO) queries [28]. Notice, however, that due to Rossman’s Theorem [53], and the fact that OMQs are closed under homomorphisms, FO and UCQ rewritability coincide. Recall that some OMQ languages are UCQ rewritable, such as the ones based on linear, non-recursive and sticky sets of tgds, while others are not, e.g., guarded OMQs. For those languages O that are not UCQ rewritable, it is important to be able to check whether a query Q ∈ O can be rewritten as a UCQ, in which case we say that it is UCQ rewritable. This gives rise to the following fundamental static analysis task for an OMQ language (C, CQ), where C ⊆ TGD: 7.1 Distribution Over Components The notion of distribution over components has been introduced in [3], and it states that the answer to a query can be computed by parallelizing it over the (maximally connected) components of the input database. But let us first make precise what a component is. A set of atoms A is connected if for all c, d ∈ dom(A), there exists a sequence α1 , . . . , αn of atoms in A such that c ∈ dom(α1 ), d ∈ dom(αn ), and dom(αi ) ∩ dom(αi+1 ) 6= ∅, for each i ∈ {1, . . . , n − 1}. We call B ⊆ A a component of A if (i) B is connected, and (ii) for every α ∈ A \ B, B ∪ {α} is not connected.5 Let co(A) be the set of components of A. We are now ready to introduce the notion of distribution over components. Consider an OMQ Q = (S, Σ, q) ∈ (TGD, CQ). We say that Q distributes over components if Q(D) = Q(D1 ) ∪ · · · ∪ Q(Dn ), where co(D) = {D1 , . . . , Dn }, for every S-database D. In this case, Q(D) can be computed without any communication over a network using a distribution where every computing node is assigned some of the components of the database, and every component is assigned to at least one computing node. In other words, Q can be evaluated in a distributed and coordination-free manner; for more details on coordination-free evaluation see [3, 4, 5]. Therefore, it would be quite beneficial if we can decide whether an OMQ distributes over components, and thus, we obtain the following interesting static analysis task: PROBLEM : INPUT : QUESTION : UCQRew(C, CQ) An OMQ Q ∈ (C, CQ). Is it the case that Q is UCQ rewritable? The above problem has been studied in [15], where tight complexity bounds for (L, CQ) and (S, CQ) have been established. However, its exact complexity for guarded OMQs Bienvenu et al. have recently carried out an in-depth study of the above problem for OMQ languages based on central Horn-DLs [16]. One of their main results is that the above problem for the OMQ language based on E LHI, one of the most expressive members of the E L-family of DLs, is 2ExpTime-complete. Interestingly, by adapting the tree automata techniques developed in Section 5, we can generalize the above result: deciding UCQ rewritability for the OMQ language based on guarded tgds over unary and binary relations is in 2ExpTime. Let G2 be the class of (finite) sets of guarded tgds over unary and binary relations. Then: 5 For technical clarity, the notion of component is defined only for sets of atoms that do not contain 0-ary atoms. Theorem 29. UCQRew(G2 , CQ) is 2ExpTime-complete. PROBLEM : INPUT : QUESTION : Dist(C, CQ) An OMQ Q ∈ (C, CQ). Does Q distributes over components? 11 considerably and, in turn, allows us to obtain our desired semantic characterization of UCQ rewritability: Since the lower bound is inherited from [16], we concentrate on the upper bound. As in Section 5, we can focus on BCQs, i.e., it suffices to show that UCQRew(G2 , BCQ) is in 2ExpTime. Our proof proceeds in two steps: Proposition 30. Let Q = (S, Σ, q) ∈ (G2 , BCQ), where q is connected. The following are equivalent: 1. We semantically characterize UCQ rewritability for queries in (G2 , CQ) in terms of a certain boundedness property for the set of C-trees defined in Section 5. 1. Q is UCQ rewritable. 2. There exist k, m ≥ 0 (which depend only on Q) s.t. Q(D) 6= ∅ =⇒ Q(D≤k ) 6= ∅ or Q(D>0 ) 6= ∅ , 2. We extend the techniques developed in Section 5 and construct in double-exponential time a 2WAPA A that has exponentially many states, such that the aforementioned boundedness property does not hold iff L(A) is infinite. (Such an infinity problem for tree automata has been used to obtain the decidability of the boundedness problem for monadic Datalog [32, 56]). for each C-tree S-database D with \|dom(C)\| ≤ 2 · \|q\| and branching degree at most m. The reduction to the infinity problem. We now proceed with step 2, and we explain how the boundedness property established in item (2) of Proposition 30 can be reduced to the infinity problem for 2WAPAs. As in Section 5, we do not reason with C-tree databases directly, but we deal with their encodings as consistent ΓS,l -labeled trees. In fact, using the same ideas as in Lemma 22, we can show by exploiting Proposition 30 that the following are equivalent: Our 2ExpTime upper bound then follows since the infinity problem for a 2WAPA A, i.e., checking if L(A) is infinite, is feasible in exponential time in the number of states, and in polynomial time in the size of the alphabet. This follows from two known results: (a) The 2WAPA A can be converted into an equivalent non-deterministic tree automata B with a single-exponential blow up in the number of states [57], and (b) solving the infinity problem for non-deterministic tree automata is feasible in polynomial time; cf. [56]. It is worth contrasting our proof with the one in [16] for E LHI, which does not make use of the infinity problem for 2WAPA, but applies a different argument based on pumping. This leads to a finer complexity analysis in terms of the size of the different components of the OMQ, but, in our opinion, makes the proof conceptually harder. (i) Q is UCQ rewritable. (ii) There are k, m ≥ 0 such that Q(JLK) 6= ∅ =⇒ Q(JLK≤k ) 6= ∅ or Q(JLK>0 ) 6= ∅ , for every consistent ΓS,l -labeled tree L with l = 2 · \|q\| and whose branching degree is bounded by m. Let us write Boundedness for the property expressed in item (ii) above, which can be reduced to the problem of checking whether some tree language is finite. Let LQ be the set of all ΓS,l -labeled trees L of branching degree at most m such that: (1) Q(JLK) = ∅ and (2) there is some “extension” L′ of L, with branching degree m, such that Q(JL′ K) 6= ∅ and Q(JL′ K>0 ) = ∅. Notice that L′ can increase the depth but not the branching degree of L. It is not difficult to show that Boundedness holds iff LQ is finite. We then devise in double-exponential time a 2WAPA CQ,l , which has exponentially many states, such that LQ = L(CQ,l ). Therefore, the following holds: The semantic characterization. To establish the semantic characterization from step 1, we need to define the notion of distance from the root for an element u in a C-tree database D. Intuitively, this corresponds to the minimal distance between a node that contains u and the root of a tree decomposition T of D that witnesses the fact that D is a C-tree. We do not consider all such tree decompositions, however, but concentrate on a well-behaved subclass, which we call the lean tree decompositions of the C-tree D; the formal definition can be found in [11], as it does not add much to the explanation we provide here. Due to the fact that we focus on unary and binary relations, such lean tree decompositions ensure the invariance of the notion of distance from the root, by severely limiting the level of redundancy allowed in a tree representation of D. Therefore, it does not matter which lean tree decomposition we choose, since in all of them the distance of an element u from the root will be the same. Let D≤k be the subinstance of D induced by the set of elements whose distance from the root is at most k, and let D>k be the subinstance of D induced by the set of elements whose distance from the root is at least k + 1. Another useful notion is the branching degree of a tree decomposition T , that is, the maximum number of child nodes over all nodes of T . Again, lean tree decompositions ensure the invariance of the branching degree. This allows us to define the branching degree of a C-tree database D as the branching degree of a lean tree decomposition that witnesses the fact that D is a C-tree. It follows from [16] that being able to decide containment for the OMQ language (G2 , BCQ) (as we have done in Section 5) allows us to concentrate on connected CQs when deciding UCQ rewritability. This simplifies technicalities Proposition 31. Consider Q ∈ (G2 , BCQ). We can construct in double-exponential time a 2WAPA A, which has exponentially many states, such that Q is UCQ rewritable ⇐⇒ L(A) is finite. Since checking whether L(A) is infinite is feasible in exponential time in the number of states and in polynomial time in the size of the alphabet, Proposition 31 implies that UCQRew(G2 , CQ) is in 2ExpTime, as needed. 8. CONCLUSIONS We have concentrated on the fundamental problem of containment for OMQ languages based on the main decidable classes of tgds. We have also used our techniques to close problems related to distribution over components and UCQ rewritability. We believe that our techniques for solving containment under guarded OMQs can be extended to frontierguarded OMQs, an interesting extension of guardedness [8]. We are also convinced that our solution to the problem of deciding UCQ rewritability of guarded OMQs over unary and binary relations can be extended to guarded (or even frontier-guarded) OMQs over arbitrary schemas. We are currently investigating these challenging problems. 12 APPENDIX PRELIMINARIES Definition of Non-recursiveness In the main body of the paper, we define non-recursive sets of tgds via the notion of predicate graph. Here, we give an alternative definition, based on the well-known notion of stratification, which is more convenient for the combinatorial analysis that we are going to perform in the proof of Proposition 14. Definition 3. Consider a set Σ of tgds. A stratification of Σ is a partition {Σ1 , . . . , Σn }, where n > 0, of Σ such that, for some function µ : sch(Σ) → {0, . . . , n}, the following hold: 1. For each predicate R ∈ sch(Σ), all the tgds with R in their head belong to Σµ(R) , i.e., they belong to the same set of the partition. 2. If there exists a tgd in Σ such that the predicate R appears in its body, while the predicate P appears in its head, then µ(R) < µ(P ). We say that Σ is stratifiable if it admits a stratification. It is an easy exercise to show that the predicate graph of a set Σ of tgds is acyclic iff Σ is stratifiable. Then: Lemma 32. Σ is non-recursive iff Σ is stratifiable. Definition of Stickiness In the main body of the paper, we provide an intuitive explanation of stickiness. Here, we recall the formal definition of sticky sets of tgds, introduced in [27]. Fix a set Σ of tgds; w.l.o.g., we assume that, for every pair (σ, σ ′ ) ∈ Σ × Σ, σ and σ ′ do not share variables. For notational convenience, given an atom α and a variable x occurring in α, pos(α, x) is the set of positions in α at which x occurs; a position P [i] identifies the i-th attribute of the predicate P . The definition of stickiness hinges on the notion of marked variables in a set of tgds. Definition 4. Consider a tgd σ ∈ Σ, and a variable x occurring in the body of σ. We inductively define when x is marked in Σ as follows: 1. If there exists an atom α in the head of σ such that x does not occur in α, then x is marked in Σ; and 2. Assuming that there exists an atom α in the head of σ such that x occurs in α, if there exists σ ′ ∈ Σ (not necessarily different than σ) and an atom β in the body of σ ′ such that (i) α and β have the same predicate and, (ii) each variable in β that occurs at a position of pos(α, x) is marked in Σ, then x is marked in Σ. We are now ready to recall when a set of tgds is sticky: Definition 5. A set Σ of tgds is sticky if, for each σ ∈ Σ, and for each variable x occurring in the body of σ, the following holds: if x is marked in Σ, then x occurs only once in the body of σ. PROOFS OF SECTION 3 Proof of Proposition 5 Consider an OMQ Q = (S, Σ, q(x̄)) ∈ (C, CQ), where C is a class of tgds, an S-database D, and a tuple c̄ ∈ dom(D)\|x̄\| . We show that: c̄ ∈ Q(D) ⇐⇒ (sch(Σ), ∅, qD,c̄ ) ⊆ (sch(Σ), Σ, q) . \| {z } \| {z } Q1 Q2 (⇒) Assume that Q1 6⊆ Q2 . This implies that there exists a sch(Σ)-database D′ , and a tuple t̄ of constants such that t̄ ∈ qD,c̄ (D′ ) and t̄ 6∈ q(chase(D′ , Σ)). Due to the monotonicity of CQs, t̄ ∈ qD,c̄ (chase(D′ , Σ)). Since, by construction, the instance chase(D′ , Σ) satisfies Σ, we conclude that qD,c̄ 6⊆Σ q.6 By exploiting the well-known characterization of CQ containment in terms of the chase, we get that c̄ 6∈ q(chase(D, Σ)), which is equivalent to c̄ 6∈ Q(D), as needed. (⇐) Conversely, assume that c̄ 6∈ Q(D), or, equivalently, c̄ 6∈ q(chase(D, Σ)). This implies that c̄ 6∈ Q2 (D). Observe that c̄ ∈ qD,c̄ (D) holds trivially, which in turn implies that c̄ ∈ Q1 (D). Therefore, Q1 6⊆ Q2 , and the claim follows. 6 This is the standard notation for the fact that qD,c̄ (I) 6⊆ q(I), for every (possibly infinite) instance I that satisfies Σ. 13 Proof of Proposition 9 The construction underlying Proposition 9 relies on the idea of encoding boolean operations (in our case the ‘or’ operator) using a set of atoms; this idea has been exploited in several other works; see, e.g., [14, 21, 41]. Let Q = (S, Σ, q) ∈ (C, UCQ). Our goal is to construct in polynomial time Q′ = (S, Σ′ , q ′ ) ∈ (C, CQ) such that Q ≡ Q′ . We assume, w.l.o.g., that the predicates of S do not appear in the head of a tgd of Σ; we can copy the content of a relation R/k ∈ S into an auxiliary predicate R⋆ /k, using the tgd R(x1 , . . . , xk ) → R⋆ (x1 , . . . , xk ), while staying inside C, and then rename each predicate P in Σ and q with P ⋆ . The set Σ′ consists of the following tgds: 1. For every R/k ∈ S: R(x1 , . . . , xk ) → R′ (x1 , . . . , xk , 1), True(1). These tgds are annotating the database atoms with the truth constant true, indicating that these are true atoms. 2. Assuming that q = ∃ȳ φ(x̄, ȳ), a tgd: True(t) → ∃x̄∃ȳ∃f φ′∧ (x̄, ȳ, f ), ψ(t, f ), where φ′∧ is the conjunction of atoms in φ, after replacing each atom R(v1 , . . . , vk ) with R′ (v1 , . . . , vk , f ), and ψ is the conjunction of atoms Or(t, t, t), Or(t, f, t), Or(f, t, t), Or(f, f, f ). This tgd generates a “copy” of the atoms in q, while annotating them with a null value that represents the truth constant false, indicating that are not necessarily true atoms. Moreover, the truth table of ‘or’ is generated. 3. Finally, for each tgd φ(x̄, ȳ) → ∃z̄ ψ(x̄, x̄) in Σ, a tgd φ′ (x̄, ȳ, w) → ∃z̄ ψ ′ (x̄, z̄, w), where φ′ and ψ ′ are obtained from φ and ψ, respectively, by replacing each atom R(v1 , . . . , vk ) with R′ (v1 , . . . , vk , w). In fact, this is the actual set of tgds Σ, with the difference that the value at the last position of each atom (which indicates whether it is true or false) is propagated to the inferred atoms. Now, assuming that q = q1 ∨ · · · ∨ qn , the CQ q ′ is defined as follows; let x̄ = x1 . . . xn and ȳ = y1 . . . yn+1 : ^ (qi′ [xi ] ∧ Or(yi , xi , yi+1 )) ∧ True(yn+1 )), ∃x̄∃ȳ (False(y1 ) ∧ 1≤i≤n where x̄ and ȳ are fresh variables not in q, and qi′ [xi ] is obtained from qi by replacing each atom R(v1 , . . . , vk ) with R′ (v1 , . . . , vk , xi ). This completes our construction. It is not difficult to show that Q ≡ Q′ , or, equivalently, for every S-database D, q(chase(D, Σ)) = q ′ (chase(D, Σ′ )). The key observation is that in order to satisfy True(yn+1 ) in the CQ q ′ , at least one of the x̄i ’s must be mapped to 1, which means that at least one qi is satisfied by chase(D, Σ). Finally, it is easy to verify that, for each C ∈ {G, L, NR, S}, Σ ∈ C implies Σ′ ∈ C, and Proposition 9 follows. PROOFS OF SECTION 4 Proof of Proposition 10 W ′ ′ We assume that q(x̄) = n i=1 qi (x̄) is a UCQ rewriting of Q. Since, by hypothesis, Q 6⊆ Q , we conclude that q 6⊆ Q , which ′ in turn implies that there exists i ∈ {1, . . . , n} such that qi 6⊆ Q . Let c(x̄) be a tuple of constants obtained by replacing each variable x in x̄ with the constant c(x), and Dqi the S-database obtained from qi after replacing each variable x in qi with the constant c(x). We show that: Lemma 33. c(x̄) 6∈ Q′ (Dqi ). Proof. Since qi 6⊆ Q′ , there exists an S-database D, and a tuple of constants t̄ such that t̄ ∈ qi (D) and t̄ 6∈ Q′ (D). Clearly, there exists a homomorphism h such that h(qi ) ⊆ D and h(x̄) = t̄. Observe also that ρ(Dqi ) ⊆ D, where ρ = h ◦ c−1 . Towards a contradiction, assume that c(x̄) ∈ Q′ (Dqi ). This implies that there exists a homomorphism γ such that γ(q ′ ) ⊆ chase(Dqi , Σ) and γ(ȳ) = c(x̄), where Q′ = (S, Σ, q ′ (ȳ)). It is not difficult to see that there exists an extension ρ′ of ρ such that ρ′ (chase(Dqi , Σ)) ⊆ chase(D, Σ) and ρ′ (x̄) = t̄. Hence, ρ′ (γ(q ′ )) ⊆ chase(D, Σ), which implies that t̄ ∈ q ′ (chase(D, Σ)); thus, t̄ ∈ Q′ (D). But this contradicts the fact that t̄ 6∈ Q′ (D), and the claim follows. Observe that c(x̄) ∈ q(Dqi ), which immediately implies that c(x̄) ∈ Q(Dqi ). Consequently, by Lemma 33, Q(Dqi ) 6⊆ Q′ (Dqi ). The claim follows since, by construction, Dqi is an S-database such that \|Dqi \| ≤ fO (Q). 14 Algorithm 1: The algorithm XRewrite Input: An OMQ Q = (S, Σ, q(x̄)) ∈ (TGD, CQ) Output: A UCQ q ′ (x̄) such that Q(D) = q ′ (D), for every S-database D i := 0; Qrew := {hq, r, ui}; repeat Qtemp := Qrew ; foreach hq, x, ui ∈ Qtemp , where x ∈ {r, f} do foreach σ ∈ Σ do /* rewriting step foreach S ⊆ body (q) such that σ is applicable to S do i := i + 1; q ′ := γS,σ i (q[S/body (σi )]); if there is no (q ′′ , r, ⋆) ∈ Qrew such that q ′ ≃ q ′′ then Qrew := Qrew ∪ {hq ′ , r, ui}; end end /* factorization step foreach S ⊆ body (q) that is factorizable w.r.t. σ do q ′ := γS (q); if there is no (q ′′ , ⋆, ⋆) ∈ Qrew such that q ′ ≃ q ′′ then Qrew := Qrew ∪ {hq ′ , f, ui}; end end end /* query q is now explored Qrew := (Qrew \ {(q, x, u)}) ∪ {(q, x, e)}; end until Qtemp = Qrew ; Qfin := {q \| hq, r, ei ∈ Qrew , and q contains only predicates of S}; return Qfin / / */ The Algorithm XRewrite In view of the fact that the rewriting algorithm XRewrite is heavily used in our complexity analysis, we would like to recall its definition. This algorithm is based on resolution, and thus, before we proceed further, we need to recall the crucial notion of unification. A set of atoms A = {α1 , . . . , αn }, where n > 2, unifies if there exists a substitution γ, called unifier for A, such that γ(α1 ) = · · · = γ(αn ). A most general unifier (MGU) for A is a unifier for A, denoted as γA , such that for each other unifier γ for A, there exists a substitution γ ′ such that γ = γ ′ ◦ γA . Notice that if a set of atoms unify, then there exists a MGU. Furthermore, the MGU for a set of atoms is unique (modulo variable renaming). The algorihtm proceeds by exhaustively applying two steps: rewriting and factorization, which in turn rely on the technical notions of applicability and factorizability, respectively. We assume, w.l.o.g., that tgds and CQs do not share variables. Given a CQ q, a variable x is called shared in q if x is a free variable of q, or it occurs more than once in q. In what follows, we assume, w.l.o.g., that tgds are in normal form, i.e., they have only one atom in the head, and only one occurrence of an existentially quantified variable [27]. We write π∃ (σ) for the position at which the existentially quantified variable of σ occurs; in case σ does not mention an existentially quantified variable, then π∃ (σ) = ε. (Recall that a position P [i] identifies the i-th attribute of a predicate P .) We are now ready to recall applicability and factorizability; in what follows, we write body (q) for the set of atoms occurring in q, and head (σ) for the head-atom of σ. Definition 6. (Applicability) Consider a CQ q and a tgd σ. Given a set of atoms S ⊆ body(q), we say that σ is applicable to S if the following conditions are satisfied: 1. the set S ∪ {head (σ)} unifies, and 2. for each α ∈ S, if the term at position π in α is either a constant or a shared variable in q, then π 6= π∃ (σ). Roughly, whenever σ is applicable to S, this means that the atoms of S may be generated during the chase procedure by applying σ. Therefore, we are allowed to apply a rewriting step (which is essentially a resolution step) that resolves S using σ, i.e., S is replaced by body(σ), and a new CQ that is closer to the input database is obtained. If we start applying rewriting steps blindly, without checking for applicability, then the soundness of the rewriting procedure is not guaranteed. However, it is possible that the applicability condition is not satisfied, but still we should apply a rewriting step. This may happen due to the presence of redundant atoms in a query. For example, given the CQ q = ∃x∃y∃z (R(x, y) ∧ R(x, z)) and the tgd σ = P (u, v) → ∃w R(w, u) 15 the applicability condition fails since the shared variable x in q occurs at the position π∃ (σ) = R[1]. However, q is essentially the CQ q = ∃x∃y R(x, y), and now the applicability condition is satisfied. From the above informal discussion, we conclude that the applicability condition may prevent the algorithm from being complete since some valid rewriting steps are blocked. Because of this reason, we need the so-called factorization step, which aims at converting some shared variables into nonshared variables, and thus, satisfy the applicability condition. In general, this can be achieved by exhaustively unifying all the atoms that unify in the body of a CQ. However, some of these unifications do not contribute in any way to satisfying the applicability condition, and, as a result, many superfluous CQs are generated. It is thus better to apply a restricted form of factorization that generates a possibly small number of CQs that are vital for the completeness of the rewriting algorithm. This corresponds to the identification of all the atoms in the query whose shared existential variables come from the same atom in the chase, and they can be unified with no loss of information. Summing up, the key idea underlying the notion of factorizability is as follows: in order to apply the factorization step, there must exist a tgd that can be applied to its output.7 Definition 7. (Factorizability) Consider a CQ q and a tgd σ. Given a set of atoms S ⊆ body (q), where \|S\| > 2, we say that S is factorizable w.r.t. σ if the following conditions are satisfied: 1. S unifies, 2. π∃ (σ) 6= ε, and 3. there exists a variable x 6∈ var (body (q) \ S) that occurs in every atom of S only at position π∃ (σ). Having the above key notions in place, we are now ready to recall the algorithm XRewrite, which is depicted in Algorithm 1. As said above, the UCQ rewriting of an OMQ q = (S, Σ, q) is computed by exhaustively applying (i.e., until a fixpoint is reached) the rewriting and the factorization steps. Notice that the CQs that are the result of the factorization step, are nothing else than auxiliary queries which are critical for the completeness of the final rewriting, but are not needed in the final rewriting. Thus, during the iterative procedure, the queries are labeled with r (resp., f) in order to keep track which of them are generated by the rewriting (resp., factorization) step. The CQ that is part of the input OMQ, although is not a result of the rewriting step, is labeled by r since it must be part of the final rewriting. Moreover, once the two crucial steps have been exhaustively applied on a CQ q, it is not necessary to revisit q since this will lead to redundant queries. Hence, the queries are also labeled with e (resp., u) indicating that a query has been already explored (resp., is unexplored). Let us now describe the two main steps of the algorithm. In the sequel, consider a triple (q, x, y), where (x, y) ∈ {r, f} × {e, u} (this is how we indicate that q is labeled by x and y), and a tgd σ ∈ Σ. We assume that q is of the form ∃x̄ ϕ(x̄, ȳ). Rewriting Step. For each S ⊆ body (q) such that σ is applicable to S, the i-th application of the rewriting step generates the query q ′ = γS,σ i (q[S/body (σ i )]), where σ i is the tgd obtained from σ by replacing each variable x with xi , γS,σ i is the MGU for the set S ∪ {head (σ i )} (which is the identity on the variables that appear in the body but not in the head of σ i ), and q[S/body (σ i )] is obtained from q be replacing S with body (σ i ). By considering σ i (instead of σ) we basically rename, using the integer i, the variables of σ. This renaming step is needed in order to avoid undesirable clutters among the variables introduced during different applications of the rewriting step. Finally, if there is no (q ′′ , r, ⋆) ∈ Qrew , i.e., an (explored or unexplored) query that is the result of the rewriting step, such that q ′ and q ′′ are the same (modulo bijective variable renaming), denoted q ′ ≃ q ′′ , then (q ′ , r, u) is added to Qrew . Factorization Step. For each S ⊆ body (q) that is factorizable w.r.t. σ, the factorization step generates the query q ′ = γS (q), where γS is the MGU for S. If there is no (q ′′ , ⋆, ⋆) ∈ Qrew , i.e., a query that is the result of the rewriting or the factorization step, and is explored or unexplored, such that q ′ ≃ q ′′ , then (q ′ , f, u) is added to Qrew . Proof of Proposition 14 We assume, w.l.o.g., that the predicates of S do not appear in the head of a tgd of Σ. Since Σ ∈ NR, by Lemma 32, Σ admits a stratification {Σ1 , . . . , Σn } with stratification function µ : sch(Σ) → {0, . . . , n}. Let us briefly explain how the rewriting tree TQ of the OMQ Q = (S, Σ, q) is defined. TQ is a rooted tree with q being its root. The i-th level of TQ consists of the CQs obtained from the CQs of the (i − 1)-th level by applying rewriting steps (see the algorithm XRewrite for details on the rewriting step) using only tgds from Σn−i+1 . It is easy to verify that the CQs of the i-th level contain only predicates P such that µ(P ) < n − i + 1. It is now clear that the n-th level of TQ (i.e., the leaves of TQ ) consists only of CQs obtained during the execution of XRewrite(Q) that contain only predicates of S. Thus, in order to obtained the desired upper bound, it suffices to show that the number of atoms that occur in a CQ that is a leaf of TQ is at most \|q\| · (maxτ ∈Σ {\|body (τ )\|})\|sch(Σ)\| . To this end, let us focus on one branch B of TQ from the root q to a leaf q ′ . Such a branch can be naturally represented as a k-ary forest FQB , where the root nodes are the atoms of q, and whenever an atom α is resolved during the rewriting step using a tgd τ , the atoms of body (τ ), after applying the appropriate MGU, are the child nodes of α. Therefore, to obtain the desired upper bound, it suffices to show that the number of leaves of FQB is at most \|q\| · (maxτ ∈Σ {\|body (τ )\|})\|sch(Σ)\| . By construction, FQB consists, in general, of \|q\| k-ary rooted trees, where k = maxτ ∈Σ {\|body (τ )\|}, of depth n. Hence, the number of leaves of FQB is at most \|q\| · (maxτ ∈Σ {\|body (τ )\|})n . Since n ≤ \|sch(Σ)\|, the claim follows. 7 Let us clarify that for the purposes of the present work we can rely on the naive approach of exhaustively unifying all the atoms that unify in the body of a CQ. However, we would like to be consistent with [40], where the algorithm XRewrite is proposed, and thus, we stick on the slightly more involved notion of factorizability. 16 Proof of Theorem 16 A proof sketch for the coNExpTimeNP upper bound is given in the main body of the paper. We proceed to establish the PNEXP -hardness. Our proof is by reduction from a tiling problem that has been recently introduced in [34], which in turn relies on the standard Exponential Tiling Problem. Let us first recall the latter problem. An instance of the Exponential Tiling Problem is a tuple (n, m, H, V, s), where n, m are numbers (in unary), H, V are subsets of {1, . . . , m} × {1, . . . , m}, and s is a sequence of numbers of {1, . . . , m}. Such a tuple specifies that we desire a 2n × 2n grid, where each cell is tiled with a tile from {1, . . . , m}. H (resp., V ) is the horizontal (resp., vertical) compatibility relation, while s represents a constraint on the initial part of the first row of the grid. A solution to such an instance of the Exponential Tiling Problem is a function f : {0, . . . , 2n − 1} × {0, . . . , 2n − 1} → {1, . . . , m} such that: 1. f (i, 0) = s[i], for each 0 ≤ i ≤ (\|s\| − 1); 2. (f (i, j), f (i + 1, j)) ∈ H, for each 0 ≤ i ≤ 2n − 2 and 0 ≤ j ≤ 2n − 1; and 3. (f (i, j), f (i, j + 1)) ∈ V , for each 0 ≤ i ≤ 2n − 1 and 0 ≤ j ≤ 2n − 2. We will refer to {0, . . . , 2n − 1} × {0, . . . , 2n − 1} as a grid, with the pairs in it being cells. A cell consists of two coordinates, the column-coordinate (for short col-coordinate) and the row-coordinate, and any function on a grid is a tiling. The Exponential Tiling Problem is defined as follows: given an instance T as above, decide whether T has a solution. It is known that this problem is NExpTime-hard (see, e.g., Section 3.2 of [46]). We are now ready to recall the tiling problem introduced in [34], called Extended Tiling Problem (ETP), which is PNEXP hard. An instance of this problem is a tuple (k, n, m, H1 , V1 , H2 , V2 ), where k, n, m are numbers (in unary), and H1 , V1 , H2 , V2 are subsets of {1, . . . , m} × {1, . . . , m}. The question is as follows: is it the case that for every sequence s, where \|s\| = k, of numbers of {1, . . . , m}, (n, m, H1 , V1 , s) has no solution or (n, m, H2 , V2 , s) has a solution? We give a reduction from the ETP to Cont(NR, CQ). More precisely, given an instance T = (k, n, m, H1 , V1 , H2 , V2 ) of the ETP, our goal is to construct in polynomial time two queries Qi = (S, Σi , qi ) ∈ (NR, CQ), for i ∈ {1, 2}, such that T has a solution iff Q1 ⊆ Q2 . Data Schema S The data schema S consists of: • 0-ary predicates Cij , for each i ∈ {0, . . . , k − 1} and j ∈ {1, . . . , m}; the atom Cij indicates that si = j. The Query Q1 The goal of the query Q1 is twofold: (i) to check that the so-called existence property of the input database, i.e., for every i ∈ {0, . . . , k − 1}, there exists at least one atom of the form Cij , is satisfied, and (ii) to check whether (n, m, H1 , V1 , s), where s is the sequence of tilings encoded in the input database, has a solution. To this end, the query Q1 will mention the following predicates: • 0-ary predicate Ci , indicating that there exists at least one atom of the form Cij in the input database. • 0-ary predicate Existence, indicating that the input database enjoys the existence property. • Unary predicate Tilei , for each i ∈ {1, . . . , m}; the atom Tilei (x) states that x is the tile i. • Binary predicate H; the atom H(x, y) encodes the fact that (x, y) ∈ H1 . • Binary predicate V ; the atom V (x, y) encodes the fact that (x, y) ∈ V1 . • 5-ary predicate Ti , for each i ∈ {1, . . . , n}; the atom Ti (x, x1 , x2 , x3 , x4 ) states that x is a 2i × 2i tiling obtained from the 2i−1 × 2i−1 tilings x1 , . . . , x4 – details on the inductive construction of 2i × 2i tilings from 2i−1 × 2i−1 tilings are given below. • Unary predicate Initiali , for each i ∈ {0, . . . , k − 1}; the atom Initiali (x) states that s[i] = x, i.e., the i-th element of the sequence s is x. • Binary predicate Topji , for each i ∈ {1, . . . , n} and j ∈ {0, . . . , k − 1}; the atom Topji (x, y) states that in the 2i × 2i tiling x the tile at position (j, 0) is y. • 0-ary predicate Tiling, indicating that there exists a 2n × 2n tiling that is compatible with the initial tiling s encoded in the input database. • 0-ary predicate Goal, which is derived whenever the predicates Existence and Tiling are derived. Q1 is defined as the query (S, Σ1 , Goal), where Σ1 consists of the following tgds: 17 X1 X2 X2 Y1 Y1 Y2 X3 X4 X4 Y3 Y3 Y4 X3 X4 X4 Y3 Y3 Y4 Z1 Z2 Z2 W1 W1 W2 X1 X2 Y1 Y2 X3 X4 Y3 Y4 Z1 Z2 W1 W2 Z1 Z2 Z2 W1 W1 W2 Z3 Z4 W3 W4 Z3 Z4 Z4 W3 W3 W4 (a) (b) Figure 2: Inductive construction of tilings. • Checking for the existence property of the input database For each i ∈ {0, . . . , k − 1} and j ∈ {1, . . . , m}: Cij → Ci and the tgd that checks for the existence property C0 , . . . , Ck−1 → Existence • Generate the tiles → ∃x1 . . . ∃xm (Tile1 (x1 ), . . . , Tilem (xm )) • Generate the compatibility relations For each (i, j) ∈ H1 : Tilei (x), Tilej (y) → H(x, y) Tilei (x), Tilej (y) → V (x, y) For each (i, j) ∈ V1 : • Generate the 2n × 2n tilings. The key idea is to inductively construct 2i × 2i tilings from 2i−1 × 2i−1 tilings. It is easy to verify that the grid in Figure 2(a) is a 2i × 2i tiling iff the nine subgrids of it, shown in Figure 2(b), are 2i−1 × 2i−1 tilings. This has been already observed in [33], where Datalog with complex values is studied. First, we construct tilings of size 2 × 2 (the base case of the inductive construction): H(x1 , x2 ), H(x3 , x4 ), V (x1 , x3 ), V (x2 , x4 ) → ∃x T1 (x, x1 , x2 , x3 , x4 ) Then, we inductively construct tilings of larger size until we get tilings of size 2n × 2n . This is done using the following tgds. For each i ∈ {2, . . . , n}: Ti−1 (x1 , x11 , x12 , x21 , x22 ), Ti−1 (x2 , x12 , x13 , x22 , x23 ), Ti−1 (x3 , x13 , x14 , x23 , x24 ) Ti−1 (x4 , x21 , x22 , x31 , x32 ), Ti−1 (x5 , x22 , x23 , x32 , x33 ), Ti−1 (x6 , x23 , x24 , x33 , x34 ), Ti−1 (x7 , x31 , x32 , x41 , x42 ), Ti−1 (x8 , x32 , x33 , x42 , x43 ), Ti−1 (x9 , x33 , x34 , x43 , x44 ) → ∃x Ti (x, x1 , x3 , x7 , x9 ) • Extract from the 2n × 2n tilings the tiles at positions (0, 0), (1, 0), . . . , (k − 1, 0). This is done using the following tgds: → Top01 (x, x1 ), Top11 (x, x2 ) → → .. . ℓ−1 Tℓ (x, x1 , x2 , x3 , x4 ), Top0ℓ−1 (x1 , y0 ), · · · , Top2ℓ−1 −1 (x1 , y2ℓ−1 −1 ) → Top02 (x, y0 ), Top12 (x, y1 ) Top22 (x, y0 ), Top32 (x, y1 ) T1 (x, x1 , x2 , x3 , x4 ) T2 (x, x1 , x2 , x3 , x4 ), Top01 (x1 , y0 ), Top11 (x1 , y1 ) T2 (x, x1 , x2 , x3 , x4 ), Top01 (x2 , y0 ), Top11 (x2 , y1 ) Tℓ (x, x1 , x2 , x3 , x4 ), Top0ℓ−1 (x1 , y0 ), · · · k−2ℓ−1 −1 , Topℓ−1 (x1 , yk−2ℓ−1 −1 ) → ℓ−1 Top0ℓ (x, y0 ), · · · , Topℓ2 ℓ−1 Top2ℓ (x, y0 ), · · · −1 (x, y2ℓ−1 −1 ) , Topk−1 (x, yk−2ℓ−1 −1 ), ℓ where ℓ = ⌈log k⌉. Moreover, for each i ∈ {ℓ + 1, . . . , n}: Ti (x, x1 , x2 , x3 , x4 ), Top0i−1 (x1 , y0 ), · · · , Topk−1 i−1 (x1 , yk−1 ) → 18 Top0i (x, y0 ), . . . , Topk−1 (x, yk−1 ) i • Check whether there exists a 2n × 2n tiling that is compatible with the sequence of tilings s For each i ∈ {0, . . . , k − 1} and j ∈ {1, . . . , m}: Cij , Tilej (x) → Initiali (x) and the tgd Top0n (x, y0 ), Initial0 (y0 ), · · · , Topk−1 (x, yk−1 ), Initialk−1 (yk−1 ) → n Tiling • Finally, we have the output tgd Existence, Tiling → Goal This concludes the construction of Q1 . The Query Q2 The goal of the query Q2 is twofold: (i) to check that the so-called uniqueness property of the input database, i.e., for every i ∈ {0, . . . , k − 1}, there exists at most one atom of the form Cij , is satisfied, and (ii) to check whether (n, m, H2 , V2 , s), where s is the sequence of tilings encoded in the input database, has a solution. The query Q2 mentions the same predicates as Q1 , and is defined as (S, Σ2 , Goal), where Σ2 consists of the following tgds: • Checking the uniqueness property For each i ∈ {0, . . . , k − 1} and j, ℓ ∈ {1, . . . , m} with j < ℓ: Cij , Ciℓ → Goal • The rest of Σ2 encodes the tiling problem (n, m, H2 , V2 , s) in exactly the same way as Σ1 encodes (n, m, H1 , V1 , s). This concludes the construction of Q2 . Proof of Proposition 18 The set Σn consists of the following tgds; for brevity, we write x̄ji for xi , xi+1 , . . . , xj :8 S(x1 , . . . , xn ) n i−1 n Pi (x̄i−1 1 , z, x̄i+1 , z, o), Pi (x̄1 , o, x̄i+1 , z, o) P0 (z, . . . , z , z, o) \| {z } → Pn (x1 , . . . , xn ) n → Pi−1 (x̄i−1 1 , z, x̄i+1 , z, o), 1 ≤ i ≤ n, → Ans(z, o), n while q = Ans(0, 1). It can be verified that, for every {S}-database D, Qn (D) 6= ∅ implies that D ⊇ {S(c1 , . . . , cn−2 , 0, 1) \| (c1 , . . . , cn−2 ) ∈ {0, 1}n−2 }, and thus, \|D\| ≥ 2n−2 . Let Q = ({S}, Σ′ , q ′ ), where Σ′ is a set of tgds and q ′ a Boolean CQ, and D an {S}-database. Clearly, Qn (D) 6⊆ Q(D) iff Qn (D) 6= ∅ and Q(D) = ∅. This implies that \|D\| ≥ 2n−2 , and the claim follows. Proof of Theorem 19 The coNExpTime upper bound, as well as the ΠP 2 -hardness in case of fixed-arity predicates, are discussed in the main body of the paper. Here, we show the coNExpTime-hardness. The proof proceeds in two steps: 1. First, we show that Cont((FNR, CQ), (L, UCQ)) is coNExpTime-hard, where FNR denotes the class of full non-recursive sets of tgds, i.e., non-recursive sets of tgds without existentially quantified variables. 2. Then, we reduce Cont((FNR, CQ), (L, UCQ)) to Cont((S, CQ), (L, UCQ)) by showing that (under some assumptions that are explained below) every query in (FNR, CQ) can be rewritten as an (S, CQ) query. By Proposition 9, we immediately get that Cont((S, CQ), (L, CQ)) is coNExpTime-hard, as needed. Step 1: Cont((FNR, CQ), (L, UCQ)) is coNExpTime-hard We show that Cont((FNR, CQ), (L, UCQ)) is coNExpTime-hard, even if we focus on 0-1 queries, that is, queries Q with following property: for every database D, Q(D) = Q(D01 ), where D01 ⊆ D is the restriction of D on the binary domain {0, 1}, i.e., D01 = {R(c̄) ∈ D \| c̄ ⊆ {0, 1}}. The proof is by reduction from the Exponential Tiling Problem, and is a non-trivial adaptation of the one given in [14] for showing that containment of non-recursive Datalog queries is coNExpTime-hard. Theorem 34. Cont((FNR, CQ), (L, UCQ)) is coNExpTime-hard, even for 0-1 queries. Proof. Given an instance T = (n, m, H, V, s) of the Exponential Tiling Problem, we are going to construct a (FNR, CQ) 0-1 query QT = (S, Σ, q) and a (L, UCQ) 0-1 query Q′T = (S, ΣT , qT ) such that T has a solution iff QT 6⊆ Q′T . 8 A similar construction has been used in [40] for showing a lower bound on the size of a CQ in the UCQ rewriting of a (S, CQ) OMQ. 19 Data Schema S The data schema S consists of: • 2n-ary predicates TiledBy i , for each i ≤ m; the atom TiledBy i (x1 , . . . , xn , y1 , . . . , yn ) indicates that the cell with coordinates ((x1 , . . . , xn ), (y1 , . . . , yn )) ∈ {0, 1}n × {0, 1}n is tiled by tile i. Notice that we use n-bit binary numbers to represent a coordinate; this is the key difference between our construction and the one of [14]. The Query QT The goal of the query QT is to assert whether the input database encodes a candidate tiling, i.e., whether the entire grid is tiled, without taking into account the constraints, that is, the compatibility relations and the constraint on the initial part of the first row. To this end, the query QT will mention the following predicates: • Unary predicate Bit; the atom Bit(x) simply says that x is a bit, i.e., x ∈ {0, 1}. • 2n-ary predicate TiledAboveColi , for each i ≤ n; the atom TiledAboveColi (x̄, ȳ) says that for the row-coordinate ȳ there are tiled cells with coordinates (x̄′ , ȳ) for every col-coordinate x̄′ that agrees with x̄ on the first i − 1 bits. In other words, for the row corresponding to ȳ, every column extending the first i − 1 bits of x̄ is tiled. In particular, TiledAboveCol1 (x̄, ȳ) says that the entire row ȳ is tiled. • 2n-ary predicate TiledAboveRowi , for each i ≤ n; the atom TiledAboveRowi (ȳ) says that for every ȳ ′ that agrees with ȳ on the first i − 1 bits, the row ȳ ′ is fully tiled. • n-ary predicate RowTiled; the atom RowTiled(ȳ) says that the row ȳ is fully tiled. • 0-ary predicate AllTiled, which asserts that the entire grid is tiled. • 0-ary predicate Goal, which is derived whenever the predicate AllTiled is derived. QT is defined as the query (S, Σ, Goal), where Σ consists of the following rules: • Generate Bit atoms → Bit(0) → Bit(1). • RowTiled For each j, k ≤ m: TiledBy j (x1 , . . . , xn−1 , 1, y1 , . . . , yn ), TiledBy k (x1 , . . . , xn−1 , 0, y1 , . . . , yn ), Bit(x1 ), . . . , Bit(xn−1 ), Bit(y1 ), . . . , Bit(yn ), Bit(w) → TiledAboveColn (x1 , . . . , xn−1 , w, y1 , . . . , yn ) For each 2 ≤ i ≤ n: TiledAboveColi (x1 , . . . , xi−1 , 1, xi+1 , . . . , xn , y1 , . . . , yn ), TiledAboveColi (x1 , . . . , xi−1 , 0, x′i+1 , . . . , x′n , y1 , . . . , yn ), Bit(wi ), . . . , Bit(wn ) → TiledAboveColi−1 (x1 , . . . , xi−1 , wi , . . . , wn , y1 , . . . , yn ) A row is fully tiled: TiledAboveCol1 (x1 , . . . , xn , y1 , . . . , yn ) → RowTiled(y1 , . . . , yn ) • AllTiled RowTiled(y1 , . . . , yn−1 , 1), RowTiled(y1 , . . . , yn−1 , 0), Bit(w) → TiledAboveRown (y1 , . . . , yn−1 , w) For each 2 ≤ i ≤ n: TiledAboveRowi (y1 , . . . , yi−1 , 1, yi+1 , . . . , yn ), ′ TiledAboveRowi (y1 , . . . , yi−1 , 0, yi+1 , . . . , yn′ ), Bit(wi ), . . . , Bit(wn ) → TiledAboveRowi−1 (y1 , . . . , yi−1 , wi , . . . , wn ) The entire grid is tiled: TiledAboveRow1 (y1 , . . . , yn ) → AllTiled → 20 AllTiled Goal This concludes the construction of the query QT . The Query Q′T Q′T is defined in such a way that Q′T (D) is non-empty exactly when the input database D encodes an invalid tiling, i.e., when one of the constraints on the tiles is violated. The query Q′T will mention the following intensional predicates: • Unary predicate Bit; as above, Bit(x) says that x is a bit. • 2i-ary predicate LastFirsti , for each 1 ≤ i ≤ n; the atom LastFirsti (x1 , . . . , xi , y1 , . . . , yi ) says that (x1 , . . . , xi ) = (1, . . . , 1) and (y1 , . . . , yi ) = (0, . . . , 0). • 2i-ary predicate Succi , for each 1 ≤ i ≤ n; the atom Succi (x̄, ȳ) says that the i-bit binary number ȳ is the successor of the i-bit binary number x̄. • 0-ary predicate Goal. Q′T is defined as the query (S, Σ′ , q ′ ). The set Σ′ consists of the following linear tgds: • Generate Bit atoms: → Bit(0) → Bit(1). • Generate the successor predicates: → → Succ1 (0, 1) LastFirst1 (1, 0). For each 1 ≤ i ≤ n − 1: Succi (x1 , . . . , xi , y1 , . . . , yi ) Succi (x1 , . . . , xi , y1 , . . . , yi ) LastFirsti (x1 , . . . , xi , y1 , . . . , yi ) LastFirsti (x1 , . . . , xi , y1 , . . . , yi ) → Succi+1 (0, x1 , . . . , xi , 0, y1 , . . . , yi ) → Succi+1 (1, x1 , . . . , xi , 1, y1 , . . . , yi ) → Succi+1 (0, x1 , . . . , xi , 1, y1 , . . . , yi ) → LastFirsti+1 (1, x1 , . . . , xi , 0, y1 , . . . , yi ). The UCQ q ′ consists of the following (Boolean) CQs; for brevity, the existential quantifiers in front of the CQs are omitted: • Tile Consistency For each i 6= j ≤ m: TiledBy i (x1 , . . . , xn , y1 , . . . , yn ), TiledBy j (x1 , . . . , xn , y1 , . . . , yn ), Bit(x1 ), . . . , Bit(xn ), Bit(y1 ), . . . , Bit(yn ) • Tile Compatibility For each (i, j) 6∈ V : Succn (x1 , . . . , xn , y1 , . . . , yn ), TiledBy i (w1 , . . . , wn , x1 , . . . , xn ), TiledBy i (w1 , . . . , wn , y1 , . . . , yn ), Bit(w1 ), . . . , Bit(wn ) For each (i, j) 6∈ H: Succn (x1 , . . . , xn , y1 , . . . , yn ), TiledBy i (x1 , . . . , xn , w1 , . . . , wn ), TiledBy i (y1 , . . . , yn , w1 , . . . , wn ), Bit(w1 ), . . . , Bit(wn ) • Tiling of First Row For each j ≤ n, let fj be the function from {1, . . . , n} into {0, 1} such that fj (1) . . . fj (n) is the number j in binary representation, and let k ∈ {1, . . . , m} other than s[j]; recall that s is a sequence of numbers of {1, . . . , m} that represents a constraint on the initial part of the first row of the grid. Then, we have the CQ: TiledBy k (x1 , . . . , xn , z, . . . , z ), Succ1 (z, o) \| {z } n where, for each i ∈ {1, . . . , n}, xi = z if fj (i) = 0, and xi = o if fj (i) = 1. This concludes the definition of the query Q′T . 21 Step 2: Cont((S, CQ), (L, UCQ)) is coNExpTime-hard Our goal is show that every 0-1 query (S, Σ, q) ∈ (F, CQ) can be equivalently rewritten as a 0-1 query (S, Σ′ , q ′ ), where all the tgds of Σ′ are lossless, i.e., all the body-variables appear also in the head, which in turn implies that Σ′ is sticky. Proposition 35. Consider a 0-1 query Q ∈ (F, CQ). We can construct in polynomial time a 0-1 query Q′ ∈ (S, CQ) such that Q ≡ Q′ . Proof. Let Q = (S, Σ, q), and assume that n is the maximum number of variables occurring in the body of a tgd of Σ. We are going to construct in polynomial time a 0-1 query Q′ = (S, Σ′ , q ′ ) ∈ (S, CQ) such that Q ≡ Q′ . The set Σ′ consists of the following tgds: • Initialization Rules We first transform every database atom of the form R(c̄) into an atom R′ (c̄, 0, . . . , 0, 0, 1). This is done as follows: \| {z } n → Bit(0) → Bit(1) and, for each k-ary predicate R ∈ S, we have the lossless tgd R′ (x1 , . . . , xk , 0, . . . , 0) \| {z } R(x1 , . . . , xk ), Bit(x1 ), . . . , Bit(xk ) → n Notice that we can safely force the variables x1 , . . . , xk to take only values from {0, 1} due to the 0-1 property. • Transformation into Lossless Tgds For each tgd σ ∈ Σ of the form R1 (x̄1 ), . . . , Rk (x̄k ) → R0 (x̄0 ) we have the lossless tgd R1′ (x̄1 , 0, . . . , 0), . . . , Rk′ (x̄k , 0, . . . , 0) \| {z } \| {z } n → R0′ (x̄0 , y1 , . . . , yn ), n where, if {v1 , . . . , vℓ }, for ℓ ∈ {1, . . . , n}, is the set of variables occurring in the body of σ (the order is not relevant), then yi = vi , for each i ∈ {1, . . . , ℓ}, and yj = v1 , for each j ∈ {ℓ + 1, . . . , n}. • Finalization Rules Observe that each atom obtained during the chase due to one of the lossless tgds introduced above is of the form R′ (x̄, ȳ), where ȳ ∈ {0, 1}n . If ȳ 6= (0, . . . , 0), then we need to ensure that eventually the atom R′ (x̄, 0, . . . , 0) \| {z } n ′ will be inferred. This is achieved by adding to Σ the following tgds: For each k-ary predicate R occurring in Σ, and for each 1 ≤ i ≤ n, we have the rule: R′ (x1 , . . . , xk , y1 , . . . , yi−1 , 1, yi+1 , . . . , yn ) → R′ (x1 , . . . , xk , y1 , . . . , yi−1 , 0, yi+1 , . . . , yn ). This concludes the definition of Σ′ . The CQ q ′ is defined analogously. More precisely, assuming that q is of the form (the existential quantifiers are omitted) R1 (x̄1 ), . . . , Rk (x̄k ) ′ the CQ q is defined as R1′ (x̄1 , 0, . . . , 0), . . . , Rk′ (x̄k , 0, . . . , 0). \| {z } \| {z } n n It is easy to verify that Σ′ consists of lossless tgds, and thus, Q′ ∈ (S, CQ). It also not difficult to see that, for every database D over S, Q(D01 ) = Q′ (D01 ); thus, by the 0-1 property, Q(D) = Q′ (D), and the claim follows. By Theorem 34 and Proposition 35, we immediately get that Cont((S, CQ), (L, UCQ)) is coNExpTime-hard, as needed. 22 PROOFS OF SECTION 5 Recall that, for the sake of technical clarity, we focus on constant-free tgds and CQs, but all the results can be extended to the general case at the price of more involved definitions and proofs. Moreover, we assume that tgds have only one atom in the head. This does not affect the generality of our proof since every set of guarded tgds can be transformed in polynomial time into a set of guarded tgds with the above property; see, e.g., [24]. Finally, for convenience of presentation, we also assume that the body of a tgd is non-empty, i.e., the body of a tgd is always an atom and not the symbol ⊤. Proof of Proposition 21 Let us start by recalling the key notion of tree decomposition. Notice that the definition of the tree decomposition that we give here is slightly different than the one in the main body of the paper. The reason is because, for convenience of presentation, we prefer to employ a slightly different notation. Definition 8. Let I be an instance. A tree decomposition of I that omits V , where V ⊆ dom(I), is a pair δ = (T , (Xt )t∈T ), where T = (T, E T ) is a tree and (Xt )t∈T a family of subsets of dom(I) (called the bags of the decomposition) such that: 1. For every v ∈ dom(I) \ V , the set {t ∈ T \| v ∈ Xt } is non-empty and connected. 2. For every atom P (s1 , . . . , sn ) ∈ I, there is a t ∈ T such that {s1 , . . . , sn } ⊆ Xt . The width of a tree decomposition δ = (T , (Xt )t∈T ) omitting V is max{\|Xt \| : t ∈ T } − 1. The tree-width of I is the minimum among the widths of all tree decompositions of I that omit V . We call a tree decomposition omitting ∅ simply tree decomposition of I. For v ∈ T , we denote by Iδ (v) the subinstance of I induced by Xv . Notation. We usually denote the strict partial order among the nodes of a tree T of a tree decomposition δ = (T , (Xt )t∈T ) by ≺. Accordingly, we write v w iff v ≺ w or v = w. For brevity, ε will usually denote the root of a tree decomposition at hand. If ambiguities could possibly arise, we shall use subscripts in these notations. Furthermore, when δ is clear from context, we shall omit it from the expression Iδ (v). Let δ = (T , (Xt )t∈T ) be a tree decomposition of I and V ⊆ T . Recall that δ is [V ]-guarded (or guarded except for V ), if for every node v ∈ T \ V , there is an atom P (s1 , . . . , sn ) ∈ I such that Xv ⊆ {s1 , . . . , sn }. A [∅]-guarded tree decomposition of I is simply called guarded tree decomposition. Also recall the crucial notion of C-tree: Definition 9. An S-instance I is a C-tree, where C ⊆ I, if there is a tree decomposition δ = (T , (Xt )t∈T ) of I such that 1. Iδ (ε) = C, i.e., the subinstance of I induced by Xε equals C. 2. δ is guarded except for {ε}. If δ or C is clear from context, we shall often refer to \|dom(C)\| as the diameter of D and to C as the core of D. Remark. The notion of C-tree defined here refers to both instances and databases, i.e., a C-tree may be a (finite) database or an instance. We often do not explicitly mention whether a C-tree at hand is a database or an instance. However, it will be clear from context whether a C-tree is a database or an instance. We proceed to establish the following technical lemma, which in turn allows us to show Proposition 21. It is an adaption of a result in [7] to the case of guarded tgds. Henceforth, for brevity, given a query Q = (S, Σ, q) ∈ (G, BCQ) and an S-database D, we write D \|= Q for the fact that Q(D) 6= ∅. Lemma 36. Let Q = (S, Σ, q) be an OMQ from (G, BCQ). Let D be an S-database and suppose D \|= Q. Then there is a finite S-instance Iˆ such that Iˆ \|= Q and: 1. Iˆ is a C-tree such that \|dom(C)\| ≤ ar (S ∪ sch(Σ)) · \|q\|. 2. There is a homomorphism from Iˆ to D. Before we proceed with its formal proof, let us explain why Proposition 21 is an easy consequence of Lemma 36. The fact that the first item implies the second is trivial. Conversely, suppose that Q1 6⊆ Q2 , which implies that there exists an ˆ where \|dom(C)\| ≤ ar (S ∪ sch(Σ1 )) · \|q1 \|, S-database D such that D \|= Q1 and D 6\|= Q2 . By Lemma 36, there exists a C-tree I, such that Iˆ \|= Q1 . Moreover, there is a homomorphism from Iˆ to D; hence, since Q2 is closed under homomorphisms, it immediately follows that Iˆ 6\|= Q2 . Consequently, the S-database D̂ obtained from Iˆ after replacing each null z with a distinct constant cz is a C-tree such that Q1 (D̂) 6⊆ Q2 (D̂), and Proposition 21 follows. We now proceed with the proof of Lemma 36 which is our main task in this section. Before that, we introduce some additional auxiliary concepts. 23 The Guarded Chase Forest Given a database D and a set Σ of guarded tgds, the guarded chase forest for D and Σ is a forest (whose edges and nodes are labeled) constructed as follows: 1. For each fact R(ā) in D, add a node labeled with R(ā). 2. For each node v labeled with α ∈ chase(D, Σ) and for every atom β resulting from a one-step application of a rule τ ∈ Σ, if α is the image of the guard in this application of τ , then add a node w labeled with β and introduce an arc from v to w labeled with τ . We can assume that the guarded chase forest is always built inductively according to a fixed, deterministic version of the chase procedure. The non-root nodes are then totally ordered by a relation ≺ that reflects their order of generation. Furthermore, we can extend ≺ to database atoms by picking a lexicographic order among them. Notice that one atom can be the label of multiple nodes. Using the order ≺ we can, however, always refer to the ≺-least node. Guarded Unraveling Let I be an instance over S. We say that X ⊆ dom(I) is guarded in I, if there are a1 , . . . , as ∈ dom(I) such that • X ⊆ {a1 , . . . , as } and • there is an R/s ∈ S such that I \|= R(a1 , . . . , as ). A tuple t̄ is guarded in I if the set containing the elements of t̄ is guarded in I. In the following paragraph, we largely follow the notions introduced in [2, 13]. Fix an S-instance I and some X0 ⊆ dom(I). Let Π be the set of finite sequences of the form X0 X1 · · · Xn , where, for i > 0, Xi is a guarded set in I, and, for i ≥ 0, Xi+1 = Xi ∪ {a} for some a ∈ dom(I) \ Xi , or Xi ⊇ Xi+1 . The sequences from Π can be arranged in a tree by their natural prefix order and each sequence π = X0 X1 · · · Xn identifies a unique node in this tree. In this context, we say that a ∈ dom(I) is represented at π whenever a ∈ Xn . Two sequences π, π ′ are a-equivalent, if a is represented at each node on the unique shortest path between π and π ′ . For a represented at π, we denote by [π]a the a-equivalence class of π. The guarded unraveling around X0 is the instance I ∗ over the elements {[π]a \| a is represented at π}, where I ∗ \|= R([π1 ]a1 , . . . , [πn ]an ) ⇐⇒df I \|= R(a1 , . . . , an ) and ∃π ∈ Π, ∀i ∈ {1, . . . , n} : [π]ai = [πi ]ai , for all R/n ∈ S. Lemma 37. For every S-instance I and any X0 ⊆ dom(I), the guarded unraveling I ∗ around X0 is a C-tree over S, where C is the subinstance of I ∗ induced by the elements {[X0 ]a \| a ∈ X0 }. Proof. Let δ = (T , (Xt )t∈T ), where T is the natural tree that arises from ordering the sequences in Π by their prefixes. For π ∈ T , let Xπ := {[π]a \| a is represented at π}. Let ε denote the root of T . We need to show that δ is an appropriate tree decomposition witnessing that I ∗ is a C-tree. First, note that it is clear that I(ε) = {[X0 ]a \| a ∈ X0 } by construction. Let [π]a ∈ dom(I) and consider the set A := {t ∈ T \| [π]a ∈ Xt }. This set is certainly non-empty. Moreover, for t1 , t2 ∈ A, we know that [t1 ]a = [t2 ]a , hence t1 and t2 are a-connected in T . Suppose I ∗ \|= R([π1 ]a1 , . . . , [πn ]an ) for some R/n ∈ S. Then there is a π ∈ T such that [π]ai = [πi ]ai , for all i = 1, . . . , n. Hence, a1 , . . . , an are all represented at π and so {[π1 ]a1 , . . . , [πn ]an } ⊆ Xπ . It remains to show that δ is guarded except for {ε}. Let π 6= ε and consider the set Xt . Since π is a sequence of length greater than one, its last element Y is a guarded set in I. Hence, there are a1 , . . . , as such that Y ⊆ {a1 , . . . , as } and I \|= R(a1 , . . . , as ) for some R/s ∈ S. Let {a1 , . . . , as } \ Y = {b1 , . . . , bm } and define ρ := π · (Y ∪ {b1 }) · (Y ∪ {b1 , b2 }) · · · (Y ∪ {b1 , . . . , bm }). Then I ∗ \|= R([ρ]a1 , . . . , [ρ]as ), as desired. Notice that this lemma implies that the tree-width of I ∗ is bounded by \|X0 \| + ar (S) − 1. We are now ready to prove Lemma 36: Proof of Lemma 36. Let q = ∃ȳ ϕ(ȳ) and µ a homomorphism mapping ϕ(ȳ) to chase(D, Σ). Let R1 (b̄1 ), . . . , Rk (b̄k ) exhaust all facts from D that are the roots of those ≺-least S facts from µ(ϕ(ȳ)) in the guarded chase forest of D and Σ that have an element from dom(D) as argument. Let Gµ := 1≤i≤k {b̄i } and let I ∗ be the unraveling of D around Gµ , regarding all elements from dom(I ∗ ) as labeled nulls. Henceforth, for every a ∈ Gµ , we denote by λa the element [Gµ ]a . We say that λa represents a. Let C be the substructure of I ∗ induced by the set {λa \| a ∈ Gµ }. Notice that I ∗ is an infinite instance that is a C-tree by Lemma 37. We will show later how to get a finite instance from I ∗ that satisfies our constraints. We proceed to show that I ∗ ∪ Σ logically entails q, denoted I ∗ , Σ \|= q: Lemma 38. I ∗ , Σ \|= q. Proof. We will first construct a universal model J of I ∗ and Σ. Recall that an instance U is a universal model of I and Σ, if it can be homomorphically mapped to every model of I ∪ Σ; in particular, it is well-known and easy to prove that chase(I, Σ) is always a universal model of I and Σ. Before constructing J, we introduce some additional notions. In the following, given a guarded set G = {a1 , . . . , ak } in D, a copy of G in I ∗ is a set Γ = {α1 , . . . , αk } which is guarded in I ∗ such 24 that, for i = 1, . . . , k, we have that αi = [πi ]ai for some sequences πi and D \|= R(ai1 , . . . , aim ) iff I ∗ \|= R(αi1 , . . . , αim ) for all R ∈ S and ij ∈ {1, . . . , k}. Copies of guarded tuples are defined accordingly. Consider the structure chase(D, Σ). Let G be a guarded set in D and D ↾ G denote the subinstance of D induced by G. It is well-known and easy to prove that chase(D ↾ G, Σ) is acyclic (cf., e.g., [23]). Henceforth, we loosely call chase(D ↾ G, Σ) the tree attached to G. The model J is constructed as follows. Let J0 be the instance C. Furthermore, for each guarded set G = {a1 , . . . , ak } in D and each copy Γ = {α1 , . . . , αk } of G in I ∗ , construct a new instance JΓ that is isomorphic to the tree attached to G such that (i) the elements ai of G are renamed to αi in JΓ , (ii) dom(J0 ) ∩ dom(JΓ ) = {α1 , . . . , αk }, and (iii) Γ ∩ Θ = dom(JΓ ) ∩ dom(JΘ ), for every copy Θ of G in I ∗ . The model J is the union of J0 and all the JΓ . If a guarded set X in JΓ arises from renaming elements of a guarded set Y in chase(D ↾ G, Σ), we also say that X is a copy of Y in J. Furthermore, the copies of D that are contained in I ∗ (i.e., in J0 ) are also called copies in J. Observe that J is a model of I ∗ by construction. We show that it is a model of Σ. To this end, we show the following claim. Claim 39. Let t̄ be a guarded tuple in J and let q(x̄) be a guarded conjunctive query9 over S ∪ sch(Σ). Suppose t̄ is a copy of s̄ in J, where s̄ is over dom(chase(D, Σ)) and \|t̄\| = \|s̄\|. Then J \|= q(t̄) iff chase(D, Σ) \|= q(s̄). Proof. Suppose J \|= q(t̄). Let {t̄} = {α1 , . . . , αk } be a copy of {s̄} = {a1 , . . . , ak } in J. Since q(x̄) is guarded, there is a Γ ⊇ ({α1 , . . . , αk } ∩ dom(J0 )) such that JΓ \|= q(t̄). Let G ⊇ {s̄} be the guarded set in D of which Γ is a copy in J0 . It clearly holds that chase(D ↾ G, Σ) \|= q(s̄), whence chase(D, Σ) \|= q(s̄) follows. Suppose that chase(D, Σ) \|= q(s̄). Let t̄ = α1 , . . . , αk and s̄ = a1 , . . . , ak and suppose that αi = [πi ]ai (i = 1, . . . , k). The set {a1 , . . . , ak } is guarded in chase(D, Σ). Hence, there is a guarded G ⊇ {a1 , . . . , ak } ∩ dom(D) in D such that chase(D ↾ G, Σ) \|= q(s̄). We show that there is a Γ ⊇ {α1 , . . . , αk } ∩ dom(I ∗ ) which is a copy of G in I ∗ . Suppose G = {b1 , . . . , bl }. Let π = X0 X1 · · · Xm be such that [π]ai = [πi ]ai for all i = 1, . . . , k. For i = 1, . . . , l, define ρi := π · (Xm ∪ {b1 }) · (Xm ∪ {b1 , b2 }) · · · (Xm ∪ {b1 , . . . , bi }). Then bi is represented at ρi . For i = 1, . . . , l, let βi := [ρi ]bi . We claim that Γ := {β1 , . . . , βl } is a copy of G in I ∗ . Let R/s ∈ S and suppose I ∗ \|= R([ρi1 ]bi1 , . . . , [ρis ]bis ). Then we immediately obtain D \|= R(bi1 , . . . , bis ). Conversely, if D \|= R(bi1 , . . . , bis ), let ρ := ρℓ , where ℓ := max{i1 , . . . , is }. Take any j ∈ {i1 , . . . , is }. It is easy to see that ρ and ρj are bj -equivalent. Hence, [ρj ]bj = [ρ]bj and it follows that I ∗ \|= R([ρi1 ]bi1 , . . . , [ρis ]bis ), as required. It follows that Γ is a copy of G in I ∗ and so there is a structure JΓ contained in J that is isomorphic to chase(D ↾ G, Σ) with b1 , . . . , bl respectively renamed to β1 , . . . , βl . Hence, J \|= q(t̄) as required. Now let σ : ϕ(x̄, ȳ) → ∃z̄ α(x̄, z̄) be a guarded rule from Σ. Suppose that J \|= ∃ȳ ϕ(t̄, ȳ). Since every guarded tuple in J is a copy of some guarded tuple in chase(D, Σ), there is an s̄, of which t̄ is a copy, such that chase(D, Σ) \|= ∃ȳ ϕ(s̄, ȳ). Since chase(D, Σ) is a model of Σ, we know that chase(D, Σ) \|= ∃z̄ α(s̄, z̄). It follows that J \|= ∃z̄ α(s̄, z̄) by the above claim, as required. It remains to show that J is universal: Claim 40. J is universal. Proof. It suffices to show that J can be homomorphically mapped to chase(I ∗ , Σ) via a homomorphism η. We let η0 be the homomorphism that maps every element of J0 to itself. It remains to treat the structures JΓ . Consider a copy Γ = {α1 , . . . , αk } in I ∗ of a set G = {b1 , . . . , bk } which is guarded in D. It suffices to show that JΓ can be mapped to chase(I ∗ , Σ). To this end, it we show how to map chase(D ↾ G, Σ) to chase(I ∗ , Σ). We do so by induction on the number of 0 rule applications of chase(D ↾ G, Σ). For the base case, we map D ↾ G to I ∗ as follows. Let ηG (bi ) := αi , for i = 1, . . . , k. Suppose D ↾ G \|= R(bi1 , . . . , bil ) for some R ∈ S and ij ∈ {1, . . . , k}, where j = 1, . . . , l. Recall that Γ is guarded in I ∗ . 0 is indeed a homomorphism Reviewing the construction of I ∗ , it is easy to see that this holds iff I ∗ \|= R(αi1 , . . . , αil ). Hence, ηG i from D ↾ G to I ∗ . The induction step is obvious—we can easily obtain a homomorphism ηG that maps chase k (D ↾ G, Σ) to i chase(I ∗ , Σ). The desired homomorphism ηG is the union of the ηG (i ≥ 0). We then obtain a homomorphism ηΓ from ηG by appropriately renaming the elements from the domain of the latter as we did in the construction of JΓ —which is nothing else than an isomorphic copy of chase(D ↾ G, Σ). Furthermore, each of these homomorphisms maps each element of Γ to itself. The desired homomorphism η that witnesses that J is universal is the union of η0 and the ηΓ . In order to prove I ∗ , Σ \|= q, it remains to show that there is a homomorphism µ̂ that maps q to J. There are guarded S sets G1 , . . . , Gl in D such that µ can be understood to map q to chase( 1≤i≤l (D ↾ Gi ), Σ). By construction, we know S thatS G1 , . . . , Gl can be chosen in such a way that Gµ ⊆ li=1 Gi . Since Σ is guarded, µ can be understood to map q to 1≤i≤l chase(D ↾ Gi , Σ)—assuming that the labeled nulls occurring in these instances are mutually new. Let Cµ := {{b̄1 }, . . . , {b̄k }}. For every X ∈ Cµ , let ΓX := {λb \| b ∈ X}. Notice that ΓX is a copy of X in I ∗ . By construction, all the facts from q that S are mapped via µ to chase(D, Σ) and which have an element from dom(D) in their image under µ are already mapped to X∈Cµ chase(D ↾ X, Σ). For the other facts, the names of the constants in the databases do not matter.10 Let Θ1 , . . . , Θs be arbitrary copies of the sets {G1 , . . . , Gl } \ Cµ in I ∗ . It follows that we can find our desired match µ̂ in the 9 By a guarded conjunctive query we mean here a CQ that contains an atom that contains all the variables occurring in the CQ as argument. 10 Here, it is of course essential to assume constant-free rules. 25 S S S S union of X∈Cµ JΓX and 1≤i≤s JΘi . Notice that X∈Cµ JΓX is isomorphic to X∈Cµ chase(D ↾ X, Σ) with each b ∈ Gµ represented by λb . Now the database I ∗ has the desired form with C being its core. However, I ∗ is infinite. Since I ∗ , Σ \|= q due to Lemma 38, by compactness, there is a finite B̂ ⊆ I ∗ such that B̂, Σ \|= q. Consider a tree decomposition δ = (T , (Xt )t∈T ) witnessing that I ∗ is a C-tree. There is a maximum ℓ such that B̂ contains all the subinstances induced by the bags of depth less or equal ℓ. Let Iˆ be the instance that actually contains all the subinstances induced by the bags of level up to ℓ. Hence, Iˆ is itself a ˆ Σ \|= q, since B̂ ⊆ I. ˆ C-tree and I, Now there is a natural homomorphism mapping Iˆ to D: we simply specify [π]a 7→ a for all a ∈ dom(D). The instance Iˆ is the one we are looking for. Proof of Lemma 22 One can naturally encode instances of bounded tree-width into trees over a finite alphabet such that the alphabet’s size depends only on the tree-width. Our goal here is to appropriately encode C-trees in order to make them accessible to tree automata techniques. Since the tree-width of a C-tree over S depends only on the size of dom(C) and the maximum arity of S, the alphabet of the encoding will depend on the same. Labeled trees. Let Γ be an alphabet and (N \ {0})∗ be the set of finite sequences of positive integers, including the empty sequence ε.11 Let us recall that a Γ-labeled tree is a pair t = (T, µ), where µ : T → Γ is the labeling function and T ⊆ (N \ {0})∗ is closed under prefixes, i.e., x · i ∈ T implies x ∈ T , for all x ∈ (N ∪ {0})∗ and i ∈ (N ∪ {0}). The elements contained in T identify the nodes of t. For i ∈ N \ {0}, nodes of the form x · i ∈ T are the children of x. A path of length n in T from x to y is a sequence of nodes x = x1 , . . . , xn = y such that xi+1 is a child of xi . A branch is a path from the root to a leaf node. For x ∈ T , we set x · i · −1 := x, for all i ∈ N, and x · 0 := x—notice that ε · −1 is not defined. Encoding. Let l ≥ 0 and fix a schema S. Let US,l be the disjoint union of two sets Cl and TS , respectively containing l and 2 · ar (S) elements. The elements from US,l will be called names. Elements from the set Cl will describe core elements, while those of TS will describe the others. Furthermore, neighboring nodes may describe overlapping pieces of the instance. In particular, if one name is used in neighboring nodes, this means that the name at hand refers to the same element—this is why we use 2w elements for the non-root bags. Let KS,l be the finite schema capturing the following information: • For all a ∈ US,l , there is a unary relation Da ∈ KS,l . • For all a ∈ Cl , there is a unary relation Ca ∈ KS,l . n • For each R ∈ S and every n-tuple ā ∈ US,l , there is a unary relation Rā ∈ KS,l . Let ΓS,l := 2KS,l be an alphabet and suppose that D is a (finite) C-tree over S such that \|dom(C)\| ≤ l. Consider a tree decomposition δ = (T , (Xt )t∈T ) witnessing that D is indeed a C-tree and let ε be the root of T . Fix a function f : dom(D) → US,l such that (i) f ↾ dom(C) is injective and (ii) different elements that occur in neighboring bags of δ are always assigned different names from US,l . Using f , we can encode D and δ into a ΓS,l -labeled tree t = (T̂ , µ) such that each node from T corresponds to exactly one node in T̂ and vice versa. For a node v from T , we denote the corresponding node of T by v̂ in the following and vice versa. In this light, the symbols from KS,l have the following intended meaning: • Da ∈ µ(v̂) means that a is used as a name for some element of the bag Xv . • Ca ∈ µ(v̂) indicates that a is used as name for an element of the bag Xv that also occurs in Xε , i.e., a names an element from the core of D. • Rā ∈ µ(v̂) indicates that R holds in D for the elements named by ā in bag Xv . Under certain assumptions, we can decode a ΓS,l -labeled tree t = (T, µ) into a C-tree whose width is bounded by ar (S) − 1. Let names(v) := {a \| Da ∈ µ(v)}. We say that t is consistent, if it satisfies the following properties: 1. For all nodes v it holds that \|names(v)\| ≤ ar (S), except for the root whose number of names are accordingly bounded by l. Furthermore, names(ε) ⊆ Cl . 2. For all Rā ∈ KS,l and all v ∈ T it holds that Rā ∈ µ(v) implies that {ā} ⊆ names(v). 3. For all a ∈ Cl and all v ∈ T it holds that Da ∈ µ(v) iff Ca ∈ µ(v). 4. If Ca ∈ µ(v), then Ca ∈ µ(w) for all w ∈ T on the unique shortest path between v and the root. 5. For all nodes v 6= ε, there is an Rā ∈ KS,l and a node w such that Rā ∈ µ(w), names(v) ⊆ {ā}, and, for all b ∈ names(v), v and w are b-connected. 11 We specify that 0 is included in N as well. 26 Decoding trees. Suppose now that t is consistent. We show how we can decode t into a database JtK which is a C-tree whose diameter is bounded by l. Let a be a name used in t. We say that two nodes v, w of t are a-equivalent if Da ∈ µ(u) for all nodes u on the unique shortest path between v and w. Clearly, a-equivalence defines an equivalence relation and we let [v]a := {(w, a) \| w is a-equivalent to v} and [v]∗a := {w \| (w, a) ∈ [v]a }. The domain of JtK is the set {[v]a \| v ∈ T, a ∈ µ(v)} and, for R/n ∈ S, we define JtK \|= R([v1 ]a1 , . . . , [vn ]an ) ⇐⇒df there is some v ∈ [v1 ]∗a1 ∩ · · · ∩ [vn ]∗an such that Ra1 ,...,an ∈ µ(v). Lemma 41. Let t be a consistent ΓS,l -labeled tree with root node ε. Then JtK is well-defined and a C-tree over S, where C is the subinstance of JtK induced by the set {[ε]a \| a ∈ names(ε)}. Moreover, \|dom(C)\| is bounded by l. Proof. Let t = (T, µ) be a consistent, ΓS,l -labeled tree. The fact that JtK is well-defined is left to the reader. We are going to construct an appropriate decomposition δ = (T , (Xt )t∈T ) for JtK. The tree T has the same structure as t. Furthermore, for v ∈ T , we set Xv := {[v]a \| a ∈ names(v)}. We need to show that δ is indeed a tree decomposition that satisfies the desired properties. Let [v]a ∈ dom(JtK) and consider two nodes v1 , v2 ∈ T such that [v]a ∈ Xv1 and [v]a ∈ Xv2 . Then v1 , v2 ∈ [v]a and so v1 and v2 are a-connected. Hence, w ∈ [v]a for all w ∈ T which lie on the unique shortest path between v1 and v2 . Since a ∈ names(w) for all such w, it follows that [v]a ∈ Xw , and so [v]a is contained in all bags on the unique path between v1 and v2 . Suppose JtK \|= R([v1 ]a1 , . . . , [vn ]an ). Then there is a v ∈ [v1 ]∗a1 ∩ · · · ∩ [vn ]∗an such that Ra1 ,...,an ∈ µ(v). By consistency, {a1 , . . . , an } ⊆ names(v). Moreover, we know that [vi ]ai = [v]ai , for i = 1, . . . , n. It follows that {[v1 ]a1 , . . . , [vn ]an } ⊆ Xv . Now let v ∈ T \ {ε}. By consistency, there is an Ra1 ,...,an ∈ KS,l and a w ∈ T such that names(v) := {ai1 , . . . , ais } ⊆ {a1 , . . . , an } ⊆ names(w), Ra1 ,...,an ∈ µ(w), and v and w are bij -connected for j = 1, . . . , s. By construction, Xv = {[v]ai1 , . . . , [v]ais } and {[w]a1 , . . . , [w]an } ⊆ Xw . The claim follows now since [v]aij = [w]aij for j = 1, . . . , s. It is immediate that \|dom(C)\| is bounded by l. Notation. Given a consistent ΓS,l -labeled tree t = (T, µ) and a label ρ ∈ µ(T ), in order to ease notation we often regard ρ as a database consisting of the facts {R(ā) \| Rā ∈ ρ}. Furthermore, we let names(ρ) := {a \| Da ∈ ρ}. Proof of Lemma 22. The lemma is an easy consequence of Lemma 41 and the fact that, when encoding a C-tree D over S, together with a tree decomposition witnessing that D is a C-tree, into a consistent ΓS,l -labeled tree t, then JtK and D are isomorphic. Roughly, Lemma 22 states that containment among OMQs from (G, BCQ) can be semantically characterized via the decodings of consistent ΓS,l -labeled trees. This makes the problem of deciding containment amenable to tree automata techniques. Proof of Lemma 23 Before proceeding to the proof of Lemma 23, we first introduce the relevant automata model. Automata Techniques For a set of propositional variables X, we denote by B+ (X) the set of Boolean formulas using variables from X, the connectives ∧, ∨, and the constants true, false. Let us now introduce our automata model. Definition 10. A two-way alternating parity automaton (2WAPA) on trees is a tuple A = (S, Γ, δ, s0 , Ω), where S is a finite set of states, Γ an alphabet (the input alphabet of A), δ : S × Γ → B+ (tran(A)) the transition function, where we set tran(A) := {hαis, [α]s \| s ∈ S, α ∈ {−1, 0, ∗}}, s0 ∈ S the initial state, and Ω : S → N the parity condition that assigns to each s ∈ S a priority Ω(s). Elements from tran(A) are called transitions. Intuitively, a transition of the form h0is means that a copy of the automaton should change to state s and stay at the current node. A transition of the form h−1is means that a copy should be sent to the parent node, which is then required to exist, and proceed in state s, while one of the form h∗is means that a copy of the automaton that assumes state s is sent to some child node. The transition [0]s means the same as h0is, while [−1]s means that a copy of the automaton that assumes state s should be sent to the parent node which is there not required to exist at all. Likewise, [∗]s means that a copy of the automaton assuming state s should be sent to all child nodes. W V Notation. We write ✸s for {hαis \| hαis ∈ tran(A), s ∈ S}, ✷s for {[α]s \| [α]s ∈ tran(A), s ∈ S}, and simply s for h0is. Furthermore, for α ∈ N ∪ {−1, ∗}, we define   if α = −1 and x · α ∈ T , {x · α}, Tα (x) := {x · i \| x · i ∈ T, i ∈ N \ {0}}, if α = ∗,  ∅, otherwise. 27 Definition 11. A run of a 2WAPA A = (S, Γ, δ, s0 , Ω) on a Γ-labeled tree (T, η) is a T × S-labeled tree (Tr , ηr ) such that the following holds: 1. ηr (ε) = (ε, s0 ), 2. if y ∈ Tr , ηr (y) = (x, s), and δ(s, η(x)) = ϕ, then there is an I ⊆ tran(A) such that I \|= ϕ holds and the following conditions are satisfied: • If hαis′ ∈ I then there is a node x′ ∈ Tα (x) and a child node y ′ ∈ Tr of y such that ηr (y ′ ) = (x′ , s′ ). • If [α]s′ ∈ I then for all x′ ∈ Tα (x), there is a child node y ′ ∈ Tr of y such that ηr (y ′ ) = (x′ , s′ ). We say that a run (Tr , ηr ) is accepting on A, if on all infinite paths (ε, s0 ), (x1 , s1 ), (x2 , s2 ), . . . in Tr , the maximum priority among Ω(s0 ), Ω(s1 ), Ω(s2 ), . . . that appears infinitely often is even. A accepts a Γ-labeled tree (T, η), if there is an accepting run on (T, η). We denote by L(A) the set of Γ-labeled trees A accepts, i.e., the language accepted by A. Remark. The automaton model defined above resembles that in [58]. However, we explicitly provide transitions that allow the automaton move to the parent node, while the model defined in [58] provides transitions for moving to some neighboring node, including the parent node. Therefore, the automata in [58] offer transitions of the form s, ✸s, and ✷s with their intended meaning as defined above. Using techniques as employed in [57, 58], for a 2WAPA A, one can show that the problem of deciding whether L(A) = ∅ is feasible in exponential time with respect to the number of states of A and in polynomial time with respect to the size of the input alphabet of A. Proof of Lemma 23. We only give an intuitive explanation for the construction of the desired 2WAPA. To check whether a ΓS,l -labeled tree is consistent, we can check each condition for consistency separately by a dedicated 2WAPA and then take the intersection of all of them. Most of the consistency conditions are easy to check. We give here a more detailed verbal explanation for condition (5). A 2WAPA checking this condition can be constructed as follows. At the beginning of its run, the automaton branches universally to all nodes (except the root) in a state whose intended purpose is to find appropriate guards in the input tree for the names available at the current node. To this end, the automaton has to do a reachability analysis on the input tree and store, using exponentially many states in ar (S), the tuple it seeks to guard. By a guard for the node v here, we mean a node w with an Rā ∈ µ(w) such that (i) {ā} contains all the names present at w and (ii) is b-connected to v for all b ∈ names(v). Notice that such a reachability analysis can be easily performed once we have the means to store the information contained in names(v) in a single state. This is, however, possible since for this task we need somewhat O((ar (S)+l)ar (S) ) states, i.e., polynomially many in the size of ΓS,l . Proof of Lemma 24 We first need to introduce some additional auxiliary notions. Strictly Acyclic Queries Let q be a CQ over a schema S. We denote by free(q) the free variables of q; the same notation is used for first-order formulas in general. We can naturally view q as an instance [q] whose domain is the set of variables of q and contains the body atoms of q as facts. In the following, we will often overload notation and write q for both the query q and the instance [q]. The notions of tree-width, acyclicity, etc. then immediately extend to CQs. Given a tree decomposition δ of q (i.e., of [q]), we say that δ is strict, if some bag of δ contains all variables that are free in q (cf. also [38]). Accordingly, q is called strictly acyclic if it has a guarded tree decomposition that is strict. Strictly acyclic queries have the convenient property to be equivalent to guarded formulas of a special form. Recall that the set of guarded formulas over a schema S is built inductively by including all atomic formulas, relativizing quantifiers by atomic formulas, and closing under Boolean connectives. More precisely, all quantifier occurrences have one of the forms ∀ȳ (α(x̄, ȳ) → ϕ) and ∃ȳ (α(x̄, ȳ) ∧ ϕ), such that the free variables of ϕ are among {x̄, ȳ}. We are interested in the guarded formulas that are build up using conjunction and existential quantification; we restrict ourselves to such formulas in the following. We call a formula from this class strictly guarded, if it is of the form ∃ȳ (α(x̄, ȳ)∧ϕ). We explicitly include the case where ȳ is the empty sequence of variables, i.e., if to formulas of the form α(x̄) ∧ ϕ with free(ϕ) ⊆ {x̄}. Notice that every guarded sentence ϕ (i.e., a formula having no free variables) is strictly guarded, since it is equivalent to ∃y (y = y ∧ ϕ). Furthermore, notice that every usual guarded formula that uses only existential quantifiers and conjunction is equivalent to a conjunction of strictly guarded formulas. The following lemma is proved in [38]. Lemma 42. Every strictly acyclic CQ can be rewritten in polynomial time into an equivalent strictly guarded formula that is built up using conjunction and existential quantification only. The converse holds as well. Squid Decompositions Let q be a BCQ over a schema S having n body atoms. An S-cover of q is a BCQ q + that contains all the atoms from q and may additionally contain 2n other body atoms over S. It is pretty straightforward that, for an S-instance I, it holds that I \|= q iff there is an S-cover q + of q such that I \|= q + . 28 Definition 12. Let I be an instance. For V ⊆ dom(I), we say that I is [V ]-acyclic, if it has a guarded tree decomposition that omits V . Definition 13. Let q be a BCQ over S. A squid decomposition of q is a tuple δ = (q + , µ, H, T, V ), where q + is an S-cover of q, µ : var (q + ) → var (q + ) a mapping, V ⊆ var (µ(q + )), and (H, T ) a partition of the atoms µ(q + ) such that • H is the set of atoms of µ(q + ) induced by V , • T = µ(q + ) \ H and T is [V ]-acyclic. Intuitively, a squid decomposition specifies a way how a BCQ can be mapped to an instance that contains some “cyclic parts”—the set H specifies those atoms that are mapped to such cyclic parts, while A declares those atoms that are mapped to the acyclic parts of the instance at hand. We will make this more precise in Lemma 43 below, where we analyze matches in C-trees. Given a CQ q and a set of variables V ⊆ var (q), the V -reduct of q, denoted q V , is the conjunctive query that arises from q by dropping all the existential quantifiers that bind variables in V . Lemma 43. Let J be a C-tree over S and q a BCQ over S. Let (T , (Xt )t∈T ) be a witnessing tree decomposition of J. It holds that J \|= q iff there is a squid decomposition δ = (q + , µ, H, A, V := {x̄}) of q and a homomorphism η : µ(q + ) → J such that 1. C \|= H is witnessed by η, S 2. ε≺v J(v) \|= AV (η(x̄)) is witnessed by η, and 3. there are strictly guarded formulas ϕ1 , . . . , ϕl such that AV (x̄) ≡ ϕ1 ∧ · · · ∧ ϕl . Proof. For the direction from right to left, consider such a given squid decomposition δ and a homomorphism η as in the hypothesis of the lemma. It is immediate that η ◦ µ is a homomorphism mapping q + to J. Since q + is an S-cover of q, we obtain J \|= q as required. For the other direction, suppose that J \|= q is witnessed by a homomorphism θ. For each v ∈ T \ {ε}, let βv be an atom of J such that J(v) \|= βv and βv contains all domain elements from J(v) as arguments. Notice that the βv exist, since δ is guarded except for {ε}. Since θ maps q to J, for each atom α of q, there is a node vα such that θ(α) ∈ J(vα ). Let W be the set of all these nodes and their closure under greatest lower bounds with respect to , excluding the root node ε of T . Consider the set of atoms Q+ := θ(q) ∪ {βv \| v ∈ W }. Notice that at least half of the nodes of W are of the form vα —hence, \|Q+ \| ≤ 3\|q\|. Let q + be a BCQ constructed as follows. Take the conjunction of q and for each βv (a1 , . . . , an ) (v ∈ W ), add an atom βv (x1 , . . . , xn ), where each xi is a newly chosen variable. Then q + is obviously an S-cover of q. Furthermore, by construction, there is a mapping µ : var (q + ) → var (q + ) and an isomorphism η : µ(q + ) → Q+ such that (η ◦ µ)(q + ) = Q+ . Now let H be the greatest set of atoms of µ(q + ) such that η(H) ⊆ J(ε). Moreover, let V := var (H) and A := µ(q + ) \ H. We claim that δ := (q + , µ, H, A, V ) is a squid decomposition of q that satisfies together with η the points mentioned in the statement of the lemma. To see that δ is a squid decomposition of q, the only nontrivial point to prove is that A is indeed [V ]-acyclic. We will prove this below in the course of establishing the third item. The first two items are immediate by construction. We proveS the third item. Suppose V = {x̄} and consider the V -reduct AV (x̄) of A. By construction, the atoms η(A) are contained in ε≺v J(v). Now the set W together with the order T gives rise to a forest consisting of trees T1 , . . . , Tl whose roots are descendants of ε, i.e., the root of T (recall that ε is not contained S S S in W ). Moreover, we have (i) li=1 Ti = W , (ii) Ti ∩ Tj = ∅, for i 6= j, and (iii) v∈Ti Xv ∩ v∈Tj Xv ⊆ dom(C), for i 6= j. S For v ∈ T , let Q+ (v) := {α ∈ Q+ \| J(v) \|= α} and, for i = 1, . . . , l, let Q+ (Ti ) be the set of atoms v∈Ti Q+ (v). Now it is easy to check using the facts stated before that each Q+ (Ti ) is acyclic and, hence, so is η −1 (Q+ (Ti )). Furthermore, denoting by εi the root of Ti , it holds that dom(Q+ (Ti )) ∩ η(V ) ⊆ dom(Q+ (εi ))—indeed, if a ∈ dom(Q+ (v)) ∩ η(V ) for some v εi , then, since εi ≻ ε and a ∈ Xε , it must be the case that a ∈ dom(Q+ (εi )) by connectivity. It follows that the V -reduct of η −1 (Q+ (Ti )) (viewed as Boolean query), henceforth denoted qT+i , is strictly acyclic and is therefore equivalent to a strictly V guarded formula ϕi . Hence, the query AV (x̄) is equivalent to li=1 ϕi . Moreover, it follows that A itself is [V ]-acyclic—notice Vl that A ≡ ∃x̄ i=1 qT+i and that dom(Q+ (Ti )) ∩ dom(Q+ (Tj )) ⊆ η(V ), for i 6= j. Hence, var (qT+i ) ∩ var (qT+j ) ⊆ V , for i 6= j. The claim now follows since every qT+i is acyclic. Derivation trees Let D be an S-database and Σ a set of guarded rules. Let q0 (x̄) be a strictly acyclic query whose free variables are exactly those from x̄ := x1 , . . . , xn and let ā := a1 , . . . , an be a tuple from dom(D). A derivation tree for (ā, q0 (x̄)) with respect to D and Σ is a finite tree T whose nodes are labeled via a function µ with pairs of the form (b1 , . . . , bk ; q(y1 , . . . , yk )), where b1 , . . . , bk are constants from dom(D) and q(y1 , . . . , yk ) is a strictly acyclic query over S ∪ sch(Σ) having exactly y1 , . . . , yk free, such that the following conditions are satisfied: 1. µ(ε) = (ā, q0 (x̄)), where ε is the root node of T . 29 2. If µ(v) = (c1 , . . . , cm ; q(z1 , . . . , zm )) for some node v, then one of the following conditions holds (let c̄ := c1 , . . . , cm and z̄ := z1 , . . . , zm ): (a) v is a leaf node and q(z̄) ≡ β(z̄), for some atomic formula β(z̄) such that D \|= β(c̄). (b) The node v has a successor labeled by (c̄, b̄; p(z̄, ȳ)) and it holds that Σ \|= ∀z̄, ȳ (p(z̄, ȳ) → q(z̄)). (c) The query q(z̄) is logically equivalent to q1 (zi1,1 , . . . , zi1,k1 ) ∧ · · · ∧ ql (zil,1 , . . . , zil,kl ) and v has l successors v1 , . . . , vl respectively labeled by (ci1,1 , . . . , ci1,k1 ; q1 (zi1,1 , . . . , zi1,ki )), . . . , (cil,1 , . . . , ci1,kl ; ql (zil,1 , . . . , zi1,kl )). Lemma 44. Let α(x1 , . . . , xn ) be an atomic formula. (a1 , . . . , an ; α(x1 , . . . , xn )) with respect to D and Σ. Then D, Σ \|= α(a1 , . . . , an ) iff there is a derivation tree for Proof (sketch). Let ā := a1 , . . . , an and x̄ := x1 , . . . , xn . The direction from right to left is an easy induction on the construction of the derivation tree. We sketch the other direction. Consider the guarded chase forest (F, η) for D and Σ, where η is a function labeling the nodes and edges of F. We construct a derivation tree for (ā, α(x̄)) by induction on the number of chase steps required to derive α(ā) from D and Σ. For the base case, if D \|= α(ā), the claim is obvious since we can apply rule 2.(a). Assume that α(ā) is derived using a rule σ: β0 (x̄, ȳ), β1 , . . . , βk → α(x̄), and a homomorphism µ such that µ(x̄) = ā, where β0 (x̄, ȳ) is the guard of σ. If µ({x̄, ȳ}) ⊆ dom(D), the result immediately follows by the induction hypothesis. Otherwise, the image of β0 (x̄, ȳ) under µ contains some labeled nulls as arguments. Assume that all the β1 , . . . , βk contain nulls as their arguments—for those that do not, the induction hypothesis would yield appropriate derivation trees again. Notice that all the nulls occurring in β1 , . . . , βk appear in µ({ȳ}). By construction of F, there is a node v0 that is an ancestor of the nodes having the atoms µ(β0 ), µ(β1 ), . . . , µ(βk ) as labels and which has a label of the form β0 (ā, b̄) which contains no nulls at all as arguments. There is a corresponding atomic formula γ0 (x̄, z̄) whose image under an appropriate homomorphism equals β0 (ā, b̄). Furthermore, there are atoms γ1 , . . . , γl such that dom({γ1 , . . . , γl }) ⊆ {ā, b̄} and Σ \|= β0 (ā, b̄) ∧ γ1 ∧ · · · ∧ γl → ∃ȳ (β0 (ā, ȳ) ∧ β1 ∧ · · · ∧ βk ). Now regard the γi (i = 1, . . . , l) as atomic formulas with free variables among {x̄, z̄}. The formula p(x̄, z̄) := γ0 (x̄, z̄)∧γ1 ∧· · ·∧γl is then a strictly acyclic query that satisfies Σ \|= ∀x̄, z̄ (p(x̄, z̄) → α(x̄)). An application of rule 2.(b) then requires us to find a derivation tree for (ā, b̄; p(x̄, ȳ)), whence an application of rule 2.(c) reduces this task to finding derivation trees for the atoms γ0 , γ1 , . . . , γl and their corresponding tuples of constants. These trees exist by induction hypothesis and we can simply concatenate them appropriately in order to arrive at a derivation tree for (ā, α(x̄)). Given a guarded formula ϕ(x̄) built up from conjunctions and existential quantification, we define the nesting depth of ϕ(x̄), denoted nd(ϕ(x̄)), inductively: • If ϕ(x̄) is an atomic formula, then nd(ϕ(x̄)) := 0. • If ϕ(x̄) = (ψ1 ∧ ψ2 ), then nd(ϕ(x̄)) := max{nd(ψ1 ), nd(ψ2 )}. • If ϕ(x̄) = ∃ȳ (α(x̄, ȳ) ∧ ψ) and ȳ 6= ∅, then nd(ϕ(x̄)) := nd(ψ) + 1. Lemma 45. Let D be a database, Σ a set of guarded rules, and q(x̄) a strictly acyclic conjunctive query. Then D, Σ \|= q(ā) iff there is a derivation tree for (ā, q(x̄)) with respect to D and Σ. Proof (sketch). We again sketch only the direction from left to right. Let ϕ(x̄) be the strictly guarded formula corresponding to q(x̄). We proceed by induction on the nesting depth of ϕ(x̄). If nd(ϕ(x̄)) = 0, then ϕ(x̄) is quantifier free and thus a conjunction of atoms α0 (x̄) ∧ α1 ∧ · · · ∧ αk , where var (αi ) ⊆ {x̄} for i = 1, . . . , k. An application of rule 2.(c) reduces the problem of building a derivation tree for (ā, ϕ(x̄)) to the problem of building corresponding trees for the αi and their corresponding constants from ā. The existence of these trees is guaranteed by Lemma 44. Now suppose that nd(ϕ(x̄)) = n + 1. Let ϕ(x̄) = ∃ȳ (α(x̄, ȳ) ∧ ψ) and ȳ := y1 , . . . , yk . Assume, without loss of generality, that all the bound variables from ϕ(x̄) are pairwise distinct. In the following, we will describe how to construct a derivation tree for (ā, q(x̄)). If D, Σ \|= q(ā), then there is a homomorphism µ mapping each atom of q(x̄) to chase(D, Σ) such that µ(x̄) = ā. Furthermore, µ maps each atom of q(x̄) to a node of the guarded chase forest F of D and Σ. Let αµ (ā, λ1 , . . . , λk ) denote the atom labeling the node of F where α(x̄, ȳ) is mapped to via µ. Let λi1 , . . . , λil exhaust all elements from λ1 , . . . , λk that are not from dom(D) and b̄ := λj1 , . . . , λjm exhaust those from λ1 , . . . , λk that are from dom(D). Let ϕ′ (x̄, yj1 , . . . , yjm ) be the formula ∃yi1 , . . . , yil (α(x̄, ȳ) ∧ ψ). Clearly, Σ \|= ∀x̄, yj1 , . . . , yjm (ϕ′ (x̄, yj1 , . . . , yjm ) → ϕ(x̄)). Hence, we can create a successor of (ā, ϕ(x̄)) that is labeled by (ā, b̄; ϕ′ (x̄, yj1 , . . . , yjm )). Assume now that none of the λ1 , . . . , λk is from dom(D). Furthermore, assume that k ≥ 1, since otherwise we can just simply apply rule 2.(c) to reduce q(x̄) to a conjunction of queries of the desired form. As in the proof of Lemma 44, there is a node v0 in F whose label β0 (ā, b̄) contains only values from dom(D) as arguments and such that v0 is an ancestor of the node labeling αµ (ā, λ1 , . . . , λk ). Furthermore, all the atoms from 30 µ(q(x̄)) that contain an element from λ1 , . . . , λk as argument are also located in the subtree rooted at v0 . Let p be the query that results from deleting all atoms from q(x̄) which are mapped via µ into the subtree rooted at v0 . Notice that p may be empty and has free variables among x̄. Furthermore p is equivalent to a conjunction p1 ∧ · · · ∧ pl of strictly acyclic queries. Let β0 (x̄, z̄) be the atomic formula whose image under an appropriate homomorphism equals β0 (ā, b̄). A similar line of reasoning as in the proof of Lemma 44 shows that there are atomic formulas β1 , . . . , βm such that var (βi ) ⊆ {x̄, z̄} and ∀x̄, z̄ (β0 (x̄, z̄) ∧ β1 ∧ · · · ∧ βm ∧ p → ϕ(x̄)), whence an application of rule 2.(b) and rule 2.(c) reduces the problem of constructing a derivation tree for (ā, ϕ(x̄)) to that of constructing corresponding trees for β0 , . . . , βm and p. Notice that p is a conjunction of strictly guarded formulas of nesting depth at most n. Hence, the induction hypothesis guarantees the existence of such derivation trees. Having the above results in place, it is easy to show the following statement: Lemma 46. Let D be a C-tree over S and Q = (S, Σ, q) an OMQ where Σ is guarded and q a BCQ. Then D \|= Q iff there is a squid decomposition δ = (q + , µ, H, A, V := {x̄}) of q and a homomorphism η : µ(q + ) → chase(D, Σ) such that: 1. F \|= H is witnessed by η, where F is the subinstance of chase(D, Σ) induced by dom(C). 2. There are strictly acyclic queries q1 , . . . , ql such that (a) AV (x̄) ≡ q1 ∧ · · · ∧ ql and (b) for i = 1, . . . , l and free(qi ) = {x̄i }, there are derivation trees for (η(x̄i ), qi ) with respect to D and Σ. Proof. We can easily prove by induction on the number of chase steps that chase(D, Σ) is an F -tree, where F is the subinstance of chase(D, Σ) induced by dom(C). Now the lemma at hand is immediate by combining this fact with Lemma 43 and Lemma 45. We are now ready to proceed with the proof of Lemma 24: Proof of Lemma 24. Lemma 46 will guide the construction of the 2WAPA we are now going to construct. Suppose Q = (S, Σ, q) is an OMQ from (G, BCQ) and let l ≥ 1. We are going to construct a 2WAPA AQ,l = (S, ΓS,l , δ, s0 , Ω) that accepts a consistent ΓS,l -labeled tree t iff JtK \|= Q. In particular, the number of states of AQ,l will be at most exponential in the size of Q and at most polynomial in l, while the construction of AQ,l will be feasible in 2ExpTime. The state set. Let Λ denote the set of all Boolean acyclic queries over S ∪ sch(Σ) that are of size at most 3\|q\|. Notice that each of these queries is equivalent to a strictly guarded formula. Furthermore, assume that Λ is closed under V -reducts, for V ⊆ var (q), provided that they are strictly acyclic as well, i.e., if p ∈ Λ and V ⊆ var (q), then also pV ∈ Λ provided pV is strictly acyclic. For {ā} ⊆ US,l , let Ŝ(ā) := {p(x̄/ā) \| p ∈ Λ, free(p) = {x̄}, \|ā\| = \|x̄\|} and let Ŝ be the union of all the sets Ŝ(ā). Now the set of states S consists of an initial state, denoted s0 , plus the set Ŝ factorized modulo logical equivalence. We denote by [p] the equivalence class of a query p ∈ Ŝ. Furthermore, for a strictly guarded formula ϕ, we may abuse notation and write [ϕ] for the equivalence class of the strictly acyclic query p ∈ Ŝ that is equivalent to ϕ. Notice that the size of S is exponential in the size of Q, since there are only exponentially many CQs of size at most 3\|q\| that are mutually non-equivalent (cf. [10]). The parity condition. We set Ω(s) := 1, for all s ∈ S. This means that only finite trees are accepted. The transition function. In the following, for each ρ ∈ ΓS,l , we denote by Θ̂(ρ) the set of all pairs that are of the form (α1 ∧ · · · ∧ αn , p1 ∧ · · · ∧ pm ) for which there is a squid decomposition of the form (q + , µ, H, T, {x̄}) and a function θ : {x̄} → names(ρ) such that: • H {x̄} (θ(x̄)) ≡ α1 ∧ · · · ∧ αn , where all the αi are relational ground atoms. • T {x̄} (θ(x̄)) ≡ p1 ∧ · · · ∧ pm , where the pi are strictly acyclic queries. Call two pairs (ϕ1 , ψ1 ) and (ϕ2 , ψ2 ) as above equivalent if ϕ1 ≡ ϕ2 and ψ1 ≡ ψ2 . Let Θ(ρ) be the set of equivalence classes under this relation and denote by [ϕ, ψ] the equivalence of a pair (ϕ, ψ) under this relation. Now we fix for each [p] ∈ S \ {s0 } a strictly guarded formula χ[p] that is equivalent to all queries from [p]. Likewise, we fix a function ϑρ : Θ(ρ) → Θ̂(ρ) such that ϑρ ([ϕ, ψ]) ∈ [ϕ, ψ], i.e., which picks a representative for each equivalence class [ϕ, ψ]. Now let ρ ∈ ΓS,l . Specify δ(·, ρ) as follows: 1. For the initial state s0 , set δ(s0 , ρ) := n m ^ _ ^ { [αi ] ∧ [pi ] \| (α1 ∧ · · · ∧ αn , p1 ∧ · · · ∧ pm ) ∈ ϑρ (Θ(ρ))}. i=1 i=1 Intuitively, the automaton selects a squid decomposition where its components are instantiated by names occurring in the root node of the input tree. The automaton tries to verify the single compartments of the squid decomposition, i.e., it tries to match them to the chase expansion of the input database under Σ. 31 2. Let [p] ∈ S \ {s0 }. We define δ([p], ρ) according to a case distinction: (a) Suppose that p ≡ ⊤. Then δ([p], ρ) := true. (b) Suppose χ[p] = ∃ȳ (α(ā, ȳ) ∧ ϕ), where α(ā, ȳ) is an atomic formula (including equality), free(ϕ) ⊆ {ȳ}, and ā exhausts all names occuring in α. If {ā} 6⊆ names(ρ) then δ([p], ρ) := false. Otherwise, _ _ δ([p], ρ) := {[ϕ(ȳ/b̄)] \| ρ \|= α(ā, b̄), {b̄} ⊆ names(ρ)} ∨ ✸[p] ∨ impl(p, ρ), where impl(p, ρ) := {[p1 ] ∧ · · · ∧ [pn ] \| [p1 ], . . . , [pn ] ∈ S \ {s0 }, {b̄} ⊆ names(ρ), p1 ∧ · · · ∧ pn ≡ q, Σ \|= ∀x̄, ȳ (q(ā/x̄, b̄/ȳ) → p(ā/x̄))}. We provide some intuitive explanation for this second case. (a) If p is the empty query, it can be satisfied at any input node and, hence, the automaton accepts unconditionally on this computation branch. (b) Otherwise, we first inspect the strictly guarded formula χ[p] at hand. If the names occurring in the guard α(ā, ȳ) are not present at the current node, it rejects. Otherwise, it tries to satisfy α(ā, ȳ) with all possible assignments for ȳ at the current node and then proceed in state [ϕ(ȳ/b̄)]. Apart from these possibilities, the automaton can decide to move to any neighboring node (i.e., the parent or a child) while remaining in state [p]. This amounts to an exhaustive search of the input tree that tries to satisfy p in the input tree. Furthermore, the automaton may choose to construct derivation trees for p. There, it uses the information provided by Σ in order to find strictly acyclic queries p1 , . . . , pn that imply p. Consequently, it tries to proceed its search with [p1 ], . . . , [pn ]. We shall now briefly comment on the running time needed to construct AQ,l . The interesting part of the construction concerns the transition function δ, in particular point 2.(b) involving impl(p, ρ). We have seen that in the proofs of Lemma 44 and Lemma 45 that there are double-exponentially many candidates for the query q(ā/x̄, b̄/ȳ) that (possibly) implies p(ā/x̄) under Σ. Furthermore, q(ā/x̄, b̄/ȳ) consists of at most exponentially many atoms. Each check whether such a query q at hand implies p requires at most double-exponential time in the size of p. This follows from the well-known fact that checking query implication under a set of guarded rules is feasible in 2ExpTime with respect to the size of the right-hand side query, and in polynomial time with respect to the size of the left-hand side query (cf. [23]), i.e., the data complexity of query answering under guarded tgds is polynomial time. PROOFS OF SECTION 6 Proof of Theorem 26 A proof sketch is given in the main body of the paper. However, the fact that Cont((G, CQ), (S, CQ)) is in 2ExpTime deserves a formal proof. Recall that to establish the latter result we need a more refined complexity analysis of the problem of deciding whether a guarded OMQ is contained in a UCQ; this is discussed in the main body of the paper. In fact, it suffices to show the following result. As in the previous section, we focus on constant-free tgds and CQs, but all the results can be extended to the general case at the price of more involved definitions and proofs. Moreover, we assume that tgds have only one atom in the head. Recall that we write var ≥2 (q) for the variables of q that appear in more than one atom, and we also write var =1 (q) for the variables of q that appear only in one atom. Then: Proposition 47. Consider Q ∈ (G, BCQ) and a Boolean CQ q. The problem of deciding whether Q ⊆ q is feasible in 1. double-exponential time in (\|\|Q\|\| + \|var ≥2 (q)\|); and 2. exponential time in \|var =1 (q)\|. It is easy to verify that the above result, together with the algorithm devised in the main body of the paper, implies that Cont((G, CQ), (S, CQ)) is in 2ExpTime. The rest of this section is devoted to show the above proposition. Our crucial task is, given a CQ q, to devise an automaton that accepts consistent labeled trees which correspond to databases that make q true. Lemma 48. Let q be a Boolean CQ over S. There is a 2WAPA Aq,l , where l > 0, that accepts a consistent ΓS,l -labeled tree t iff JtK \|= q. The number of states of Aq,l is exponential in \|var ≥2 (q)\| and polynomial in (\|var =1 (q)\| + ar (S) + l). Furthermore, Aq,l can be constructed in exponential time. Proof. We are going to construct Aq,l = (S, ΓS,l , δ, s0 , Ω). Let x1 , . . . , xn be the variables of var =1 (q) and fix a total order x1 ≺ x2 ≺ · · · ≺ xn among them. Define the state set S to be S := {sy,θ \| θ : V → US,l , V ⊆ var ≥2 (q), y ∈ var =1 (q) ∪ {♯}}. Notice that \|S\| = O(\|var =1 (q)\| · (ar (S) + l)\|var ≥2 (q)\| ). We set s0 := s♯,∅ , where ∅ denotes the empty substitution. In the following, we treat q as a set of relational atoms and let X = var ≥2 (q). For ρ ∈ ΓS,l and sy,θ ∈ S, define δ(sy,θ , ρ) as follows: 32 • If y = ♯, distinguish the following cases: 1. If there is an atom α ∈ θ(q) such that var (α) ∩ X 6= ∅ and dom(α) ∩ US,l 6⊆ names(ρ), then δ(s♯,θ , ρ) := false. 2. Otherwise, let δ(s♯,θ , ρ) := (W {s♯,η \| η ⊇ θ, ρ \|= ∃x̄ sx1 ,θ , V • Suppose y = xi , for some i = 1, . . . , n. Let αi,θ denote the   sxi+1 ,θ , δ(sxi ,θ , ρ) := true,  ✸s xi ,θ , (η(q) \ θ(q))} ∨ ✸s♯,θ , if X ∩ var (θ(q)) 6= ∅, otherwise. unique atom α ∈ θ(q) such that xi ∈ var (α). Set if ρ \|= ∃x̄ αi,θ and i < n, if ρ \|= ∃x̄ αi,θ and i = n, otherwise. Set the parity condition Ω to be Ω(s) := 1 for all s ∈ S. Intuitively, the automaton works in two passes. The first pass consists of the runs working on states of the form s♯,θ . In this pass, the automaton tries to find an assignment for the variables in the query that appear in at least two distinct atoms. When a candidate assignment θ is found, the automaton changes to state sx1 ,θ which is the beginning of the second pass. A state of the form sxi ,θ means that the assignment θ can be extended to all variables x ≺ xi and, in this state, the automaton tries to extend θ to cover the variable xi . The automaton accepts if it is able to extend the candidate assignment θ to all x1 , . . . , xn . Having the above result in place, we can now reduce the problem in question to the emptiness problem for 2WAPA. Lemma 49. Consider Q ∈ (G, BCQ) and a Boolean CQ q. We can construct in double-exponential time in \|\|Q\|\| and in exponential time in \|\|q\|\| a 2WAPA A, which has exponentially many states in (\|\|Q\|\| + \|var ≥2 (q)\|) and polynomially many states in \|var =1 (q)\|, such that Q ⊆ q ⇐⇒ L(A) = ∅. ′ Proof. Let Q = (S, Σ, q ) and l = ar (S ∪ sch(Σ)) · \|q ′ \|. Then A is defined as: (CS,l ∩ AQ,l ) ∩ Aq,l . It is an easy task to verify that the claim follows from Lemmas 22, 23, 24, and 48. It is clear that Proposition 47 is an easy consequence of Lemma 49. PROOFS OF SECTION 7 Recall that we focus on unary and binary predicates. Moreover, we consider constant-free tgds and CQs, and we assume that tgds have only one atom in the head. Proof of Proposition 30 Basics. Let D be a C-tree of width two. We say that a tree decomposition δ = (T , (Xt )t∈T ) witnessing that D is a C-tree is lean, if it satisfies the following conditions: • The elements from dom(C) occur only in the root of T and its immediate successors. • If w is a child of v in T , then there are unique c, d ∈ dom(D) such that Xv ∩ Xw = {c} and Xw \ Xv = {d}. The element d is called new at w. • It follows from the previous item that every node v 6= ε in T has a unique new element c ∈ dom(D). We additionally require that c appears in the bag of each child of v. Intuitively, C-trees D that have lean tree decomposition represent the actual tree structure of D. It is fairly straightforward to see that every C-tree has a lean tree decomposition. Recall that the Gaifman graph of D is the graph G(D) = (V, E) with V := dom(D) and (a, b) ∈ E if a and b coexist in some atom of D. Given two nodes a, b from G(D), the distance from a to b in G(D), denoted dG(D) (a, b), is the minimum length of a path between a and b, and ∞ if such a path does not exist. For a, b ∈ dom(D), we denote by dδ (a, b) the minimum distance among two nodes of T that respectively have a and b in their bags. We call dδ (a, b) the distance from a to b in δ. Notice that in a tree decomposition δ witnessing that D is a C-tree, any element a ∈ dom(D), if a appears in the bag of v, then it occurs only at v, at v and its children, or at v and its parent. Since furthermore the bag of the root node is uniquely determined by C, each node in the tree has a uniquely determined set of child nodes whose bags are determined by the structure of D alone. Therefore, the following two lemmas follow immediately. Lemma 50. Let δ = (T , (Xt )t∈T ) be a lean tree decomposition witnessing that D is a C-tree. Then dδ (a, b) ≤ dG(D) (a, b) for all a, b ∈ dom(D). 33 Lemma 51. Let δ and δ ′ be two lean tree decompositions witnessing that D is a C-tree. Then dδ (a, b) = dδ′ (a, b) for all a, b ∈ dom(D). In the following, we denote by D≤k the subinstance of D induced by the set of elements whose distance from any a ∈ dom(C) in any lean tree decomposition δ is bounded by k. The subinstance D>k is defined analogously.The branching degree of a lean tree decomposition is the maximum number of child nodes of any node contained in the tree of δ. Notice that two lean tree decompositions of a C-tree D always have the same branching degree; the argument is similar as for the two lemmas above. Hence, we can simply speak about the branching degree of D. Encodings. Recall that a consistent ΓS,l -labeled tree t = (T, µ) encodes information on an S-database D and an appropriate tree decomposition δ of D. It is clear that JtK has a lean tree decomposition, but it is not guaranteed that this is reflected in δ as well. We call (the consistent) t lean, if the tree decomposition δt = (T := (T, E), (Xv )v∈T ) is, where xEy iff y = x · i for some i ∈ N \ {0} and Xv := {[v]a \| a ∈ names(v)}. The following is easy to prove: Lemma 52. There is a 2WAPA on trees LS,l that accepts a consistent ΓS,l -labeled tree iff it is lean. The number of states of LS,l is bounded logarithmically in the size of ΓS,l and LS,l can be constructed in polynomial time in the size of ΓS,l . Let t = (T, µ) be a labeled tree. The branching degree of a node x ∈ T is the cardinality of {i \| x · i ∈ T, i ∈ N \ {0}}; the branching degree of t is the maximum over all branching degrees of its nodes and ∞ is this maximum does not exist. We also say that t is m-ary if the branching degree of t is bounded by m. A node x ∈ T is a leaf node of t if it has branching degree zero. The depth of T is the maximum length among the lengths of all branches and ∞ if this maximum does not exist. Let us remark that the branching degree of the lean ΓS,l -labeled tree t as defined for labeled trees equals the branching degree of JtK as defined above. Lemma 53. Let Q = (S, Σ, q) be an OMQ from (G2 , BCQ). There is an m ≥ 0 such that the following are equivalent: 1. There is an S-database D such that D \|= Q. 2. There is a C-tree D̂ with \|dom(C)\| ≤ 2\|q\| and branching degree at most m such that D̂ \|= Q. Proof. Let l := 2\|q\| and, let AQ,l be the 2WAPA from Lemma 24. Take the intersection of AQ,l with (i) the 2WAPA CS,l from Lemma 23 and (ii) the 2WAPA from Lemma 52 that checks leanness. Call the resulting automaton B. Then B accepts a ΓS,l -labeled tree t iff t is lean and consistent and JtK \|= Q. We let m be the number of states of B and claim that this is the required bound on the branching degree. First of all, notice that the first item of the lemma trivially implies the second independently from the choice of m. For the other direction, suppose that D \|= Q for some S-database D. Then there is a C-tree B such that dom(C) ≤ 2\|q\| and B \|= Q. Being a C-tree, B has a lean tree decomposition δ and the encoding of B together with δ corresponds to a lean ΓS,l -labeled tree t. It follows that t ∈ L(B). By the results of [44], it follows that there is a t′ ∈ L(B) whose branching degree is bounded by the number of states of B, i.e., by m. The tree t′ is lean and consistent, therefore Jt′ K is a C ′ -tree of branching degree at most m for some C ′ ⊆ JtK such that \|dom(C ′ )\| ≤ 2\|q\|. Furthermore, Jt′ K \|= Q, as required. We are now ready to prove Proposition 30: Proof of Proposition 30. We largely follow [16] here. Choose m as in Lemma 53 above. Suppose first that Q is UCQ rewritable. Let p := p1 ∨ · · · ∨ pn be a corresponding UCQ rewriting. Since the query q is connected, we can assume that p is as well. We choose k > max{\|pi \| : i = 1, . . . , n} and suppose that D \|= Q for some C-tree D. Since p is a UCQ rewriting, D \|= pi for some i = 1, . . . , n. Fix a homomorphism µ witnessing that D \|= pi . We distinguish cases. Suppose first that µ(var (pi )) ∩ dom(C) 6= ∅. Since p is connected, it follows D≤k \|= pi by Lemma 50 and so D≤k \|= p. On the other hand, if µ(var (p)) ∩ dom(C) = ∅, then it is also easy to check that D>0 \|= p. For the other direction, suppose that the second item of the proposition’s statement holds, i.e., there is a k ≥ 0 such that for all C-trees D over S with \|dom(C)\| ≤ 2\|q\| and branching degree at most m it holds that D \|= Q implies D≤k \|= Q or D>0 \|= Q. Let Λ be the set of all C-trees such that \|dom(C)\| ≤ 2\|q\| and that have branching degree at most m such that D \|= Q. We regard Λ as a set W of BCQs and regard it as factorized modulo logical equivalence. It is clear that Λ is finite then and we claim that p := n i=1 pi is a UCQ rewriting of Q. We explicitly include the case where Λ is empty, in which case p is equivalent to the empty disjunction ⊥ and there is no database D at all such that D \|= Q. To see that p is indeed a UCQ rewriting of Q, let D be an S-database such that D \|= p. Then there is an i = 1, . . . , n such that D \|= pi . Furthermore, [pi ] \|= Q and so D \|= Q as well, since Q is closed under homomorphisms. Suppose now D \|= Q. We know that there is a C-tree D̂ with \|dom(C)\| ≤ l := 2\|q\| and branching degree at most m such that D̂ \|= Q and—when we regard D̂ as an instance—there is a homomorphism from D̂ to D. Let D′ ⊆ D̂ be a minimal connected subset of D̂ such that D′ \|= Q. ′ ′ D′ is again a C ′ -tree for some C ′ ⊆ D′ . Therefore D≤k \|= Q or D>0 \|= Q. The latter is impossible by minimality of D′ . ′ ′ ′ Hence, D≤k \|= Q and so there is a (logically equivalent) copy of D≤k contained in Λ. Hence, D≤k \|= p, therefore D′ \|= p, and hence D̂ \|= p. Recall that, when D̂ is regarded as an instance, there is a homomorphism from D̂ to D. Therefore, D \|= p. 34 Proof of Proposition 31 Let Q = (S, Σ, q) be an OMQ from (G2 , BCQ) such that q is connected. We are going to show that the desired 2WAPA A can be constructed in 2ExpTime. Notice that, using similar results as in [16], this gives us a decision procedure for deciding UCQRew(G2 , CQ) also for non-connected queries. Let us first introduce some auxiliary notions. 2WAPAs on m-ary trees. A 2WAPA B on m-ary trees is just defined as a 2WAPA, except that its transitions tran(B) are {hkis, [k]s \| −1 ≤ k ≤ m, s ∈ S}, where S is the state set of B. The notion of run is then defined on m-ary trees only and its definition is modified in the obvious way so as to deal with the transitions hkis, [k]s. Intuitively, for k = 1, . . . , m, a transition hkis means that the automaton should move to the k-th child of the current node (which is then required to exist) and assume state s. Correspondingly, [k]s means that the automaton should move to the k-th child and assume state s provided that this k-th child exists at all. We remark that all 2WAPAs constructed in this paper so far can easily be modified to work on m-ary trees as well and we shall assume in the following that they do so. Furthermore, deciding whether L(B) is feasible in exponential time in the number of states of B and in polynomial time in the size of the input alphabet of B (cf. [57]). Let m be as in Proposition 30. In the following, we shall regard all trees mentioned in the following as m-ary and let l := 2\|q\|. Before proceeding to a proof of Proposition 31, we must make the notion of being an “extension” of a labeled tree more precise. Extensions of trees. Let BQ be a 2WAPA that accepts a ΓS,l -labeled tree t iff (i) t is lean and consistent, (ii) JtK \|= Q, and (iii) JtK>0 6\|= Q. Notice that a 2WAPA A>0 Q that accepts a lean and consistent ΓS,l -labeled tree iff JtK>0 6\|= Q can be easily constructed using the construction in Lemma 24. Hence, BQ can be constructed intersecting several 2WAPAs we have already encountered. Let Π be the set of all tuples of the form (s, s′ ), where s and s′ are states of BQ . We define a new alphabet Λ := 2KS,l ∪Π . Notice that Λ is of double-exponential size in the size of Q. For ρ ∈ Λ, we denote by ρ ↾ ΓS,l the restriction of ρ to ΓS,l , that is, ρ ∩ KS,l . The restriction of a Λ-labeled tree t to ΓS,l , denoted t ↾ ΓS,l , is the tree that arises from t when we restrict the label of each node of t to ΓS,l . We say that a Λ-labeled tree is consistent if (i) its restriction to ΓS,l is consistent and (ii) symbols ρ ∈ Λ such that ρ ∩ Π 6= ∅ appear only in leaf nodes of t. Likewise, we say that a consistent t is lean if t ↾ ΓS,l is. The decoding JtK of t is naturally extended to consistent Λ-labeled trees by setting JtK := Jt ↾ ΓS,l K. The following lemma is a straightforward extension of Lemmas 23 and 52. Lemma 54. There are 2WAPAs CΛ and LΛ that respectively accept a Λ-labeled tree iff it is consistent and lean. Both have logarithmically many states in the size of Λ and can be constructed in polynomial time in the size of Λ. Let t be a lean and consistent Λ-labeled tree. We say that t′ is an extension of t if t′ is a ΓS,l -labeled tree that arises from t by attaching ΓS,l -labeled trees to those leaves of t that contain elements from Π. Furthermore, for such nodes, the labels of the corresponding nodes in t′ are those of t restricted to ΓS,l . Definition 14. Let LQ be the set of all lean and consistent Λ-labeled trees t such that JtK 6\|= Q, yet there is an extension t′ of t such that Jt′ K \|= Q and Jt′ K>0 6\|= Q. Lemma 55. LQ is infinite iff Q is not UCQ rewritable. Proof. Suppose LQ is infinite. Since the trees at hand have bounded branching degree, for every k ≥ 0, there is a t ∈ LQ such that JtK is a C-tree (for some C ⊆ JtK) that contains individuals whose distance from any a ∈ dom(C) is greater than or equal to k and JtK 6\|= Q, yet for some extension t′ of t, we have Jt′ K \|= Q but Jt′ K>0 6\|= Q. Suppose now that Q is UCQ rewritable. Let ℓ be such that for all C ′ -trees D (of the appropriate dimensions), D \|= Q implies D≤ℓ \|= Q or D>0 \|= Q. Choose k > ℓ and t, t′ such that (i) t′ is an extension of t, (ii) t has depth greater than k, and (iii) JtK 6\|= Q but Jt′ K \|= Q and Jt′ K>0 6\|= Q. Since Jt′ K \|= Q, we know that Jt′ K≤ℓ \|= Q or Jt′ K>0 \|= Q. The latter is impossible by assumption, the former contradicts the fact JtK 6\|= Q, since k > ℓ. This proves the direction from left to right. The other direction is immediate. We are now ready to establish Proposition 31: Proof of Proposition 31. We are now going to describe the construction of a 2WAPA A such that L(A) = LQ , which will prove the claim by virtue of Lemma 55. This automaton is the intersection of several ones. First of all, we ensure that all the accepted Λ-trees are lean and consistent (cf. Lemma 54). We additionally intersect the automaton with the complement of AQ,l from Lemma 23 (more precisely, the version of it running on Λ-labeled trees) and another automaton DQ = (S, Λ, δ, s0 , Ω) whose construction we shall describe in more detail here. On a high level, DQ will be constructed so as to accept a lean and consistent Λ-labeled tree if and only if there is an extension t′ of t such that BQ accepts t′ . Let Ŝ be the set of states of BQ , δ̂ its transition function, and Ω̂ its parity function. For σ ∈ ΓS,l , let B̂(σ) be the set of tuples (s, s′ ) ∈ Ŝ × (Ŝ ∪ {true}) such that the following holds: • There is a ΓS,l -labeled tree t = (T, η) such that η(ε) = σ and a run tr = (Tr , ηr ) of BQ on t such that 1. ηr (ε) = (ε, s), i.e., tr starts from s;12 12 Strictly speaking, tr is, of course, not a run since it does not start in the initial state. 35 2. s′ = true and tr is accepting on BQ , or there is a node v ∈ Tr such that ηr (v) = (ε, s′ ). Now the set of states of DQ is the same as of BQ , i.e., S := Ŝ. Accordingly, the initial state of DQ is that of BQ . Furthermore Ω(s) := Ω̂(s), for every s ∈ S. Given s ∈ S and ρ ∈ Λ, we let _ δ(s, ρ) := {s′ \| (s, s′ ) ∈ ρ ∩ B̂(ΓS,l ↾ ρ)} ∨ δ̂(s, ΓS,l ↾ ρ). We are going to give an intuitive explanation of this construction in the following. Roughly, a pair (s, s′ ) ∈ B̂(σ) indicates that there is a ΓS,l -labeled tree t and run of BQ on t such that the root of t is labeled with σ, the run starts in state s, and either BQ accepts t, or it traverses the root again at some point, then being in state s′ . The set B̂(σ) can be computed a priori in 2ExpTime; considering that ΓS,l is of double-exponential size in the size of Q, it follows that the collection {B̂(σ)}σ∈ΓS,l can be computed in 2ExpTime. Now the input tree for DQ comes with labels from Π of the form (s, s′ ) in its leaves. These “types” amount to guesses of possible extensions of the input tree. Utilizing the sets B̂(σ), DQ thus explores the possible ways how the given input tree can be extended to a ΓS,l -labeled tree t′ that is accepted by BQ . 9. REFERENCES [1] S. Abiteboul, R. Hull, and V. Vianu. Foundations of Databases. Addison-Wesley, 1995. [2] A. Amarilli, M. Benedikt, P. Bourhis, and M. Vanden Boom. Query answering with transitive and linear-ordered data. 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HYPERBOLICITY AND NEAR HYPERBOLICITY OF QUADRATIC FORMS OVER FUNCTION FIELDS OF QUADRICS arXiv:1609.07100v2 [math.AC] 9 Oct 2017 STEPHEN SCULLY Abstract. Let p and q be anisotropic quadratic forms over a field F of characteristic 6= 2, let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 , and let k denote the dimension of the anisotropic part of q after scalar extension to the function field F (p) of p. We conjecture that dim(q) must lie within k of a multiple of 2s+1 . This can be viewed as a direct generalization of Hoffmann’s Separation theorem. Among other cases, we prove that the conjecture is true if k < 2s−1 . When k = 0, this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism W (F ) → W (F (p)) has dimension divisible by 2s+1 . 1. Introduction Many of the central problems in the algebraic theory of quadratic forms seem to demand an investigation of the behaviour of quadratic forms under scalar extension to function fields of quadrics. This was already apparent from the early beginnings of the subject, with function fields of Pfister quadrics being used in an essential way to prove the fundamental Arason-Pfister Hauptsatz of 1971 ([AP71]). Arason and Pfister’s work led to the foundational papers of Elman-Lam ([EL72]) and Knebusch ([Kne76],[Kne77]), after which properties of function fields of quadrics have been systematically studied and exploited in many of the major developments in the theory; notable examples include the proof of the Milnor conjectures ([Voe03],[OVV07]) and essentially all advances on Kaplansky’s problem concerning the possible u-invariants of fields ([Mer91],[Izh01],[Vis09]). An important problem in this context is the following: Let p and q be anisotropic quadratic forms of dimension ≥ 2 over a field F of characteristic 6= 2, and let F (p) denote the function field of the projective quadric {p = 0}. Under what circumstances does q become isotropic over F (p)? While this problem seems to be rather intractable in general, the extraction of partial information is already enough for non-trivial applications. As a result, it has been of great interest here to identity general constraints coming from the basic invariants of the forms p and q. A key breakthrough in this direction was made in [Hof95], where Hoffmann determined the constraints coming from the simplest invariants of all, namely, the dimensions of p and q. More precisely, Hoffmann’s “Separation theorem” asserts that if dim(q) ≤ 2s < dim(p) for some integer s, then q remains anisotropic over F (p), and this is optimal, in the sense that isotropy can occur in all other cases. An important generalization of Hoffmann’s theorem taking into account one further invariant of p was later given by Karpenko and Merkurjev in [KM03]. The present article is concerned with the following variant of the above question: To what extent can the form q become isotropic over F (p)? Formally, the extent to which a quadratic form is isotropic is measured by its Witt index, i.e., the maximal dimension of a totally isotropic subspace of the vector space on which it is defined. In analogy with 2010 Mathematics Subject Classification. 11E04, 14E05, 14C15. Key words and phrases. Quadratic forms, function fields of quadrics, hyperbolicity, near hyperbolicity, Witt kernels. 1 2 STEPHEN SCULLY Hoffmann’s result, one can ask the following: What constraints do the dimensions of p and q impose on the Witt index iW (qF (p) ) of q after extension to F (p)? Here the results of [Hof95] and [KM03] only go so far. In particular, they are essentially vacuous in the case where dim(p) ≤ dim(q)/2, whereas the problem itself is non-trivial in all dimensions. A similar criticism applies to the main result of [Scu16], which gives a certain refinement the result of Karpenko and Merkurjev. The purpose of this paper is to propose the following conjectural answer to the above problem: Conjecture 1.1. Suppose that p and q are anisotropic quadratic forms of dimension ≥ 2 over F . Let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 , and let k = dim(q) − 2iW (qF (p) ) (i.e., k is the dimension of the anisotropic part of qF (p) ). Then dim(q) = a2s+1 + ǫ for some non-negative integer a and some integer −k ≤ ǫ ≤ k. Again, it is not hard to see that this is optimal, to the extent that there can be no further gaps in the possible values of dim(q) determined by iW (qF (p) ) and dim(p) alone (see Example 1.5 below). In particular, the statement include’s Hoffmann’s theorem: Example 1.2. Suppose that dim(q) ≤ 2s , so that we have separation of dimensions by a power of 2. The reader will easily confirm that, in this case, it is only possible to express dim(q) in the suggested way if k = dim(q), i.e., if q remains anisotropic over F (p). Conjecture 1.1 is vacuously true if k ≥ 2s − 1, so we are interested here in the case where k ≤ 2s − 2. In other words, beyond the Separation theorem, we are looking at the situation in which iW (qF (p)) is “large” (which explains the title of the article). Loosely speaking, the conjecture asserts that the more isotropic q becomes over F (p), the closer its dimension should be to being divisible by 2s+1 . Our main result is the following: Theorem 1.3. Conjecture 1.1 is true when k < 2s−1 . In particular, the conjecture is true in the extreme case where k = 0, which translates as follows: Corollary 1.4. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . If q becomes hyperbolic over F (p), then dim(q) is divisible by 2s+1 . Put another way, Corollary 1.4 says the following: Let W (F (p)/F ) denote the kernel of the natural scalar extension homomorphism W (F ) → W (F (p)) on Witt groups. If q represents an element of W (F (p)/F ), then dim(q) is divisible by 2s+1 . A few low-dimensional cases aside (see, e.g., [Fit83]), this seems to have been unknown, even conjecturally.1. In fact, one can go rather further here, and show that all higher Witt indices of q, except possibly the last, are divisible by 2s+1 in this case (Theorem 3.4). This can be viewed as a precise generalization of a well-known theorem of Fitzgerald ([Fit81]) on the lowdimensional part of W (F (p)/F ) (see 3.B below). Although the proof of the Arason-Pfister Hauptsatz already drew serious attention to the problem of determining Witt kernels of function fields of quadrics, very little progress has been made over the last 40 years, with Fitzgerald’s theorem being among the few highlights. Our computations perhaps raise some new questions here. For example, must q be divisible by an (s + 1)-fold Pfister form in the situation of Corollary 1.4? Conjecturally, this question should have a positive 1Little seems to have been known beyond the fact that dim(q) ≥ 2s+1 in this case, which is an easy consequence of the Cassels-Pfister subform theorem (see [Kne76, Lem. 4.5]). 3 answer in the case where q has degree s + 1, meaning here that q does not represent an element in I s+2 (F ), the (s + 2)-nd power of the fundamental ideal of even-dimensional forms in W (F ). We provide here some further evidence for this claim, including a proof for the case where dim(p) ≤ 16 (Corollary 5.11). While it seems unlikely that such divisibility would hold in general, we are currently unable to provide any counterexample. Theorem 1.3 is a consequence of results of Vishik ([Vis07]) on a certain descent problem for algebraic cycles over function fields of quadrics. The proof is given in §4 below. Beyond this, we also prove Conjecture 1.1 in a number of additional cases; namely, the conjecture is true if (i) 2s+1 − 2 ≤ dim(p) ≤ 2s+1 , (ii) p is a Pfister neighbour, (iii) k ≤ 7, or (iv) dim(q) ≤ 2s+2 + 2s−1 . In the case of (iii) and (iv), we again prove a stronger result at the level of the splitting pattern of q (Theorem 5.3). In the course of treating case (iii), we show that (a stronger version of) our conjecture is implied by a long-standing conjecture of Kahn on the unramified Witt group of a quadric ([Kah95], see §5.B below). We expect that the statement of Conjecture 1.1 is also true in characteristic 2, even if we allow p and q to be degenerate.2 Unfortunately, the methods of the present article rely on the action of mod-2 Steenrod operations on Chow groups of smooth projective varieties, something which not available in characteristic 2 at the present time. Nevertheless, using entirely different methods, we can show that our main results (and more) extend to the case where char(F ) = 2 and q is quasilinear (i.e., diagonalizable). In view of the different nature of the arguments used, this work has been confined to a separate text ([Scu17]). We conclude this introduction with an example which shows that Conjecture 1.1 cannot be improved without stronger hypotheses or permitting the use of additional invariants: Example 1.5. Let p be an anisotropic quadratic form of dimension ≥ 2 over a field E of characteristic 6= 2. Suppose that p is a neighbour of an (s + 1)-fold Pfister form π for some non-negative integer s. Choose a non-negative integer a, a non-negative integer k < 2s , and let ǫ = k − 2l for some integer 0 ≤ l ≤ k/2. Note that we have ǫ + l ≥ 0. Let X = (X1 , . . . , Xa+ǫ+l ) be a set of a + ǫ + l algebraically independent variables over E, and let F = E(X). Let σ be a codimension-l subform of π, and consider the form q = πF ⊗ hX1 , . . . , Xa−1 i ⊥ Xa σF ⊥ hXa+1 , . . . , Xa+ǫ+l i. Since F/E is a purely transcendental extension, πE and σE are anisotropic. It then follows from [EKM08, Cor. 19.6] that q is anisotropic. Note that q has dimension a2s+1 + ǫ. We claim that (qF (p) )an has dimension k. Since l ≤ k/2 < 2s−1 , σ is a neighbour of π. Let τ be its complementary form of dimension l. Since π is a Pfister form, we then have qF (p) = − Xa τ ⊥ hXa+1 , . . . , Xa+ǫ+l i F (p) in W (F (p)). By Hoffmann’s Separation theorem, τE(p) is anisotropic, and since F (p)/E(p) is purely transcendental, [EKM08, Cor. 19.6] then implies that the right-hand side of the above equation is anisotropic. We therefore have that (qF (p) )an ≃ − Xa τ ⊥ hXa+1 , . . . , Xa+ǫ+l i F (p) . In particular, the dimension of (qF (p) )an is equal to dim(τ ) + ǫ + l = 2l + ǫ = k, as claimed. Since the integers k and ǫ in the statement of Conjecture 1.1 must have the same parity, this shows the optimality of the assertion. 2Of course, if q is degenerate, then one can no longer interpret the integer k as the dimension of the anisotropic part of qF (p) . 4 STEPHEN SCULLY Before proceeding, we make the following conventions: Conventions. All fields considered in this paper have characteristic 6= 2, and all quadratic forms are finite-dimensional and non-degenerate. By a variety, we mean an integral separated scheme of finite-type over a field. The field of 2 elements is denoted by F2 . 2. Some preliminary facts and terminology For the remainder of this text, F will denote an arbitrary field of characteristic 6= 2. We assume basic familiarity with the algebraic theory of quadratic forms, and the reader is referred to [EKM08] for all undefined terminology and notation. 2.A. The Knebusch splitting tower of a quadratic form. Let ϕ be a quadratic form over F . Recall the following construction of Knebusch ([Kne76]): Set F0 = F , ϕ0 = ϕan (the anisotropic the kernel of ϕ), and recursively define Fr = Fr−1 (ϕr−1 ) and ϕr = (ϕFr )an , with the understanding that the process stops at the first non-negative integer h(ϕ) for which ϕh(ϕ) is split (i.e., of dimension at most 1). The integer h(ϕ) is called the height of ϕ. By the splitting pattern of ϕ, we will mean the decreasing sequence of integers comprised of the dimensions of the ϕr . For each 0 ≤ r ≤ h(ϕ), we set jr (ϕ) to be the Witt index of ϕ after extension to the field Fr . If ϕ is not split and r ≥ 1, then the integer jr (ϕ) − jr−1 (ϕ) is called the r-th higher Witt index of ϕ, and is denoted ir (ϕ). By [Kne76, Thm. 5.8], the even-dimensional anisotropic forms of height 1 are precisely Nthose which are similar to a Pfister form, i.e., to a form of the shape hha1 , . . . , am ii := m i=1 h1, −ai i. In view of the inductive nature of Knebusch’s construction, it follows that if ϕ is non-split of even dimension, then ϕh(ϕ)−1 is similar to an n-fold Pfister form for some positive integer n. The latter integer is called the degree of ϕ, denoted deg(ϕ). If dim(ϕ) is odd (resp. if ϕ is hyperbolic), then we set deg(ϕ) = 0 (resp. deg(ϕ) = ∞). By a deep result due to Orlov, Vishik and Voevodsky ([OVV07, Thm. 4.3]), deg(ϕ) then coincides with the supremum of the set of all integers d for which ϕ represents an element in the d-th power I d (F ) of the fundamental ideal ideal in the Witt ring W (F ). 2.B. Stable birational equivalence of quadratic forms. Let ψ and ϕ be a pair of anisotropic quadratic forms over F of dimension ≥ 2. We say that ϕ and ψ are stably birationally equivalent if both ϕF (ψ) and ψF (ϕ) are isotropic. For example, an anisotropic Pfister neighbour is stably birationally equivalent to its ambient Pfister form ([EKM08, Rem. 23.11]). More generally, if ψ is similar to a subform of ϕ having codimension less than i1 (ϕ), then ϕ and ψ are stably birationally equivalent (see [EKM08, Lem. 74.1]). Following the previous example, we say in this case that ψ is a neighbour of ϕ. 2.C. Algebraic cycles on quadrics. Given a variety X over a field K, we will write Ch(X) for its total Chow group modulo 2. The group Ch(X) has the natural structure of an F2 -vector space. If L is a field extension of K, then an element of Ch(XL ) is said to be K-rational if it lies in the image of the natural restriction homomorphism Ch(X) → Ch(XL ). Suppose now that X is a smooth projective quadric of dimension n ≥ 1 defined by the vanishing of a quadratic form ϕ over our fixed field F , and let F be an algebraic closure of F . By [EKM08, Prop. 68.1], an F2 -basis of Ch(XF ) is given by the set {li , hi \| 0 ≤ i ≤ [n/2]}, where li (resp. hi ) is the class of an i-dimensional projective linear subspace (resp. a codimension-i hyperplane section) of XF . The following lemma is 5 a basic consequence of the well-know theorem of Springer asserting that odd-degree field extensions do not affect the Witt index of a quadratic form: Lemma 2.1 (see [EKM08, Cor. 72.6]). Let ϕ be a quadratic form of dimension ≥ 2 over F with associated (smooth) projective quadric X, and let 0 ≤ i ≤ [n/2]. If the element li ∈ Ch(XF ) is F -rational, then iW (ϕ) > i. 2.D. The motivic decomposition type and upper motive of a quadric. Given a field K, we will write Chow (K) for the additive category of Grothendieck-Chow motives over K with integral coefficients (as defined in [EKM08, Ch. XII], for example). If X is a smooth projective variety over K, then we will write M (X) for its motive considered as an object of Chow (K). In the special case where X = Spec(K), we simply write Z instead of M (X) (the dependency on the base field is suppressed from the notation). Given an integer i and an object M of Chow (K), we will write M {i} for the i-th Tate twist of M . In particular, Z{i} will denote the Tate motive with shift i in Chow (K). If L is a field extension of K, we write ML to denote the image of an object M in Chow (K) under the natural scalar extension functor Chow (K) → Chow (L). Suppose now that X is a smooth projective quadric of dimension n ≥ 1 defined by the vanishing of a quadratic form ϕ over our fixed field F , and let Λ(n) = {i \| 0 ≤ i ≤ ` [n/2]} {n − i \| 0 ≤ i ≤ [n/2]}. By a result of Vishik (see [Vis04, §§3,4]), any direct summand of M (X) decomposes (in an essentially unique way) into a finite direct sum of indecomposable objects in Chow (F ). If N is a non-zero direct summand of LM (X), then ∼ there exists a unique non-empty subset Λ(N ) ⊆ Λ(n) such that NF = λ∈Λ(N ) Z{λ} L (loc. cit.). In particular, we have Λ M (X) = Λ(n), i.e., M (XF ) ≃ λ∈Λ(n) Z{λ}. Now, if N1 and N2 are distinct indecomposable direct summands of M (X), then Λ(N1 ) and Λ(N2 ) are easily seen to be disjoint ([Vis04, Lem. 4.2]), and so the complete motivic decomposition of X determines in this way a partition of the set Λ(n). This partition is an important discrete invariant of ϕ known as its motivic decomposition type. For the reader’s convenience, we state here the most significant recent advance in the study of this invariant due to Vishik ([Vis11]). Recall first that the upper motive of X is defined as the unique indecomposable direct summand U (X) of M (X) such that 0 ∈ Λ(U (X)), i.e., such that Z is isomorphic to a direct summand of U (X)F . We then have: Theorem 2.2 (see [Vis11, Thm. 2.1]). Let ϕ be an anisotropic quadratic form of dimension ≥ 2 over F with associated (smooth) projective quadric X. Write dim(ϕ) − i1 (ϕ) = 2r1 − 2r2 + · · · + (−1)t−1 2rt for uniquely determined integers r1 > r2 > · · · > rt−1 > rt +1 ≥ 1, and, for each 1 ≤ c ≤ t, set t c−1 X X i−1 ri −1 (−1) 2 + ǫ(c) (−1)j−1 2rj , Dc = i=1 j=l where ǫ(c) = 1 if c is even and ǫ(c) = 0 if c is odd. Then, for any 1 ≤ c ≤ t, we have Dc ∈ Λ(U (X)), i.e., the Tate motive Z{Dc } is isomorphic to a direct summand of U (X)F . 3. Hyperbolicity of quadratic forms over function fields of quadrics The proof of Theorem 1.3 will be given in the next section. Taking this for granted, we give here the basic applications to the study of Witt kernels of function fields of quadrics. We start by recalling the statement of the Cassels-Pfister subform theorem: 6 STEPHEN SCULLY Proposition 3.1 (see [Kne76, Lem. 4.5]). Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F such that qF (p) is hyperbolic. Then, for any non-zero elements a ∈ D(p) and b ∈ D(q), there exists a quadratic form r over F such that q ≃ r ⊥ abp. In particular, dim(p) ≤ dim(q). Taking degrees into account, the dimension inequality can be refined as follows: Corollary 3.2. Let p and q be anisotropic quadratic forms over F such that qF (p) is hyperbolic, and let n = deg(q). Then dim(p) ≤ 2n . Indeed, this follows from the proposition by taking L to be the penultimate entry of the Knebusch splitting tower of q in the following lemma: Lemma 3.3. Let q and p be anisotropic quadratic forms of dimension ≥ 2 over F such that qF (p) is hyperbolic, and let L be a field extension of F . If qL is not hyperbolic, then pL is anisotropic, and (qL )an becomes hyperbolic over L(p). Proof. It is clear that (qL )an becomes hyperbolic over L(p). If pL is isotropic, then L(p) is a purely transcendental extension of L ([EKM08, Prop. 22.9]) and so qL must already be hyperbolic. 3.A. Main result. The statement of Corollary 3.2 can be reformulated as follows: If qF (p) is hyperbolic, then the last higher Witt index of q is equal to 2m for some m ≥ s. The new insight here is that the remaining higher Witt indices are all divisible by 2s+1 : Theorem 3.4. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . If qF (p) is hyperbolic, then ir (q) is divisible by 2s+1 for every 1 ≤ r < h(q). Proof. Since the statement is insensitive to multiplying p and q by non-zero scalars, we can assume that both forms represent 1. We argue by induction on h(q). If h(q) = 1, there is nothing to prove, so assume otherwise. Then, by Lemma 3.3, pF (q) is anisotropic and q1 becomes hyperbolic over F (q)(p). The induction hypothesis therefore implies that ir (q) = ir−1 (q1 ) is divisible by 2s+1 for all 1 < r < h(q). It remains to show that i1 (q) is also divisible by 2s+1 . We begin by showing that i1 (q) > 2s . Suppose, for the sake of contradiction, that this is not the case. Since both p and q represent 1, it follows from the Cassels-Pfister subform theorem (Proposition 3.1 above) that there exists a quadratic form r over F such that q ≃ r ⊥ p. (3.1) We recall the following observation due to Gentile and Shapiro: Lemma 3.5 ([GS78, Lem. 17]). Let L be a field extension of F . In the above situation, either (1) (qL )an ≃ (rL )an ⊥ (pL )an , or (2) (rL )an ≃ (qL )an ⊥ −(pL )an . Proof. Both (1) and (2) are true if we replace the isometry relation with Witt equivalence. In particular, (2) holds if either qL is hyperbolic or (qL )an ⊥ −(pL )an is anisotropic. If not, then pL is anisotropic and (qL )an becomes hyperbolic over L(p) (Lemma 3.3). At the same time, (qL )an and pL represent a common non-zero element of L, and so pL ⊂ (qL )an by the Cassels-Pfister subform theorem. By Witt cancellation, it then follows that (1) holds. We apply this lemma in the case where L = F (q). Note first that since dim(p) > 2s , r has codimension > 2s in q. Since i1 (q) ≤ 2s , it follows from [Vis04, Cor. 4.9] that rF (q) is anisotropic. As pF (q) is also anisotropic, the lemma therefore tells us that either 7 (1) q1 ≃ rF (q) ⊥ pF (q) , or (2) rF (q) ≃ q1 ⊥ −pF (q) . Now (1) cannot hold, since the right-hand side has dimension equal to dim(q), which is larger than dim(q1 ). On the other hand, (2) cannot hold either; indeed, if (2) holds, then dim(q) − dim(q1 ) 2 dim(r) + dim(p) − dim(r) − dim(p) = 2 = dim(p) > 2s , i1 (q) = which contradicts our assumption. We therefore conclude that i1 (q) > 2s . By Karpenko’s theorem on the possible values of the first higher Witt index ([Kar03]), it follows that dim(q) − i1 (q) is divisible by 2s+1 . Now the k = 0 case of Theorem 1.3 tells us that dim(q) is divisible by 2s+1 , and so the same is therefore true of i1 (q). In particular, we get the following more precise version of Corollary 1.4: Corollary 3.6. Let q and p be anisotropic quadratic forms of dimension ≥ 2 over F , let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 , and let n = deg(q). If qF (p) is hyperbolic, then n ≥ s + 1, and: (1) If n = s + 1, then dim(q) = 2s+1 m for some odd integer m. (2) If n ≥ s + 2, then dim(q) is divisible by 2s+2 . Remark 3.7. We remind the reader that, by [OVV07, Thm. 4.3], n = deg(q) coincides with the largest integer d such that [q] ∈ I d (F ). Proof. The inequality n ≥ s + 1 holds by Corollary 3.2. In view of the fact that h(q)−1 n dim(q) = 2 + X 2ir (q), (3.2) r=1 the remaining statements follow immediately from Theorem 3.4. Now, the following statement is well known (see [Vis04, Lem. 6.2] for a proof): Proposition 3.8. Let q be an anisotropic quadratic form over F which is divisible by an m-fold Pfister form for some integer m ≥ 1. Then ir (q) is divisible by 2m for all 1 ≤ r < h(q). Theorem 3.4 therefore raises the following question: Question 3.9. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . If qF (p) is hyperbolic, does it follow that q is divisible by an (s + 1)-fold Pfister form? While it perhaps seems unlikely that this would hold in general, we are unable to provide a counterexample. At the same time, we do expect that the question has a positive answer in the case where deg(q) takes its smallest possible value of s+1. In fact, the hyperbolicity of qF (p) should imply that p is a Pfister neighbour in this case; by [AP71], this would give that q is isometric to the product of an odd-dimensional form and the ambient Pfister form of p. This expectation goes back to Fitzgerald ([Fit83]), and we provide further evidence in its favour in §5 below. In particular, we show that if deg(q) = s + 1 and qF (p) is hyperbolic, then the upper motive of the quadric {p = 0} is a twisted form of a Rost motive (Corollary 5.11). By a result of Karpenko ([Kar01]), this implies that our claim holds if s ≤ 3. Hypothetically, it should imply for all s ([Vis04, Conj. 4.21]). 8 STEPHEN SCULLY 3.B. Fitzgerald’s theorem revisited. The proof of Corollary 3.6 immediately yields the following lower bound for the dimension of an element of the Witt kernel W (F (p)/F ): Corollary 3.10. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 , and let n = deg(q). If qF (p) is hyperbolic, then n ≥ s + 1 and dim(q) ≥ 2n + (h(q) − 1)2s+2 , with equality holding if and only if ir (q) = 2s+1 for all 1 ≤ r < h(q). In particular, we get a satisfying explanation of Fitzgerald’s theorem on the lowdimensional part of W (F (p)/F ): Corollary 3.11 (Fitzgerald, [Fit81, Thm. 1.6]). Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F such that qF (p) is hyperbolic, and let n = deg(q). If dim(p) > 12 (dim(q) − 2n ), then q is similar to a Pfister form. Proof. To say that q is similar to a Pfister form is equivalent to saying that h(q) = 1 ([Kne76, Thm. 5.8]). Let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . Then, by hypothesis, we have dim(q) < 2n + 2dim(p) ≤ 2n + 2s+2 . By Corollary 3.10, it follows that h(q) = 1, as desired. Remark 3.12. As noted by Fitzgerald (see [Fit81, §2]), the inequality of Corollary 3.11 cannot be weakened in general. Here we can clarify the situation further: Let p and q be as in the statement of Corollary 3.10. If we are in the border case where dim(p) = 1 n n s+2 and dim(p) = 2s+1 . 2 (dim(q)−2 ), then our result shows that h(q) = 2, dim(q) = 2 +2 By a theorem of Vishik ([Vis04, Thm. 6.4]), any degree-n form of height ≥ 2 has dimension ≥ 2n + 2n−1 . We therefore have n − 3 ≤ s ≤ n − 1. All three cases can occur. In fact, q should, in this situation, be the product of an (s+1)-fold Pfister form and either an Albert form (if s = n − 3), a 4-dimensional form of non-trivial discriminant (if s = n − 2) or a form of dimension 3 (if s = n − 1) – see [Kah96, §2.4]. This has been shown to be the case when n ≤ 3 (loc. cit.), and can also be proven for n = 4 using more recent results (though this is absent from the literature). The general case remains out of reach at present. 4. Proof of Theorem 1.3 In this section, we give a concrete geometric reformulation of Conjecture 1.1. Theorem 1.3 then follows as a consequence of the results of [Vis07]. We begin by making the following observation: Lemma 4.1. Let d, k and s be positive integers. Assume that d + k is even and that k < 2s . Then the following are equivalent: (1) d = a2s+1 + ǫ for some non-negative integer a and some integer −k ≤ ǫ ≤ k. s (2) d+k 2 = a2 + µ for some non-negative integer a and some integer 0 ≤ µ ≤ k. (3) The binomial coefficient d+k 2 l ( 2s − 2 is even for every k < l ≤ m, where m = 2s−1 if k ≥ 2s−1 if k < 2s−1 . 9 Proof. The equivalence of (1) and (2) is clear. To prove the equivalence of (2) and (3), we need to recall the basic for oddness of binomial coefficients. Let x be a nonP criterion i be its 2-adic expansion. We write π(x) for the (finite) negative integer, and let ∞ x 2 i=0 i set of all non-negative integers i for which xi is non-zero. We then have: Lemma 4.2 (see [Spr98, Lem. 3.4.2]). Let x and y be non-negative integers. Then the binomial coefficient xy is odd if and only if π(y) ⊂ π(x). Returning to the proof of Lemma 4.1, let us write d+k = a2s + µ 2 for some non-negative integer a and some integer 0 ≤ µ < 2s . If 0 ≤ l < 2s , then Lemma 4.2 implies that d+k µ 2 ≡ (mod 2). l l The equivalence (2) ⇔ (3) therefore amounts to the assertion that µ ≤ k if and only if µl is even for every k < l ≤ m. The “only if” implication here is trivial. Conversely, suppose that µl is even for every k < l ≤ m. Since µµ = 1, it follows that either µ ≤ k or µ > m. If µ > m, then (by the definition of m) we either have (i) µ = 2s − 1, or (ii) k < 2s−1 = m < µ. s By Lemma 4.2, 2 −1 is odd for every 0 ≤ l ≤ 2s−1 , so if (i) holds, then we must have l µ s that k = 2 − 1 = m. If (ii) holds, then 2s−1 is even by hypothesis. On the other hand, s−1 s the inequalities 2 < µ < 2 imply in this case that s − 1 ∈ π(µ). By Lemma 4.2, this µ is odd, giving us a contradiction. We conclude that µ ≤ k, and so the means that 2s−1 equivalence of (2) and (3) is proved. Lemma 4.1 reduces Conjecture 1.1 to the problem of checking the parity of certain binomial coefficients. The point now is that these coefficients can be extracted from the action of the Steenrod algebra on the mod-2 Chow ring of the quadric defined by q. For the remainder of this section, let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let Q be the projective quadric of equation q = 0. We fix an algebraic closure F (resp. F (p)) of F (resp. F (p)). For any j ≥ 0, let S j denote the j-th Steenrod operation of cohomological type on the mod-2 Chow ring of a smooth quasi-projective variety. Lemma 4.3. Fix integers 0 ≤ r ≤ [dim(Q)/2] and 0 ≤ j ≤ r. Then the element S j (lr ) ∈ Ch(QF ) is F -rational if and only if the binomial coefficient dim(q) − r − 1 dim(q) − r − 1 − j is even. Proof. By [EKM08, Cor. 78.5], we have dim(q) − r − 1 j S (lr ) = lr−j . j If S j (lr ) is F -rational, then the coefficient here must be even by Lemma 2.1 (because q is anisotropic). The converse is trivial. If Y is a variety over F , then an element α ∈ Ch(YF ) will be called F (p)-rational if αF (p) ∈ Ch(YF (p) ) is F (p)-rational. We recall the following result of Vishik (extended to odd characteristic in [Fin13]): 10 STEPHEN SCULLY Theorem 4.4 ([Vis07, Cor. 3.5], [Fin13, Thm. 1.1, Prop. 2.1]). Let Y be a smooth quasi-projective variety over F , and let α be an F (p)-rational element of Chm (YF ). If the integral Chow group CH(YF ) is torsion-free, then: (1) S j (α) is F -rational for every j > m − [(dim(p) − 1)/2]. (2) If pF (Y ) is not split, then S j (α) is also F -rational for j = m − [(dim(p) − 1)/2]. Applying Theorem 4.4 to the quadric Q, we get the following: Corollary 4.5. Let r = iW (qF (p) ) − 1, and consider the element lr ∈ Ch(QF ). Then: (1) S j (lr ) is F -rational for all j > dim(Q) − r − [(dim(p) − 1)/2]. (2) If pF (q) is not split, then S j (lr ) is also F -rational for j = dim(Q) − r − [(dim(p) − 1)/2] Proof. Since r = iW (qF (p) ) − 1, the element lr is F (p)-rational. As it is represented by a cycle of dimension r, the assertions follow immediately from Vishik’s theorem. Now, as in the statement of Conjecture 1.1, let k = dim(q) − 2iW (qF (p) ) (i.e., let k be the dimension of the anisotropic part of qF (p) ). Corollary 4.6. The binomial coefficient dim(q)+k 2 l is even for all k < l ≤ [(dim(p) − 1)/2] (and also for l = [(dim(p) − 1)/2] + 1 if pF (q) is not split). Proof. To simplify the notation, let n = [(dim(p) − 1)/2]. Let r = iW (qF (p) ) − 1, and let dim(Q) − r − n ≤ j ≤ r. If j = dim(Q) − r − n, assume additionally that pF (q) is not split. Then, by Corollary 4.5, S j (lr ) is F -rational. Since j ≤ r, Lemma 4.3 then implies that the binomial coefficient dim(q) − r − 1 dim(q) − r − 1 − j is even. It now only remains to observe that dim(q) + k , dim(q) − r − 1 = 2 and that dim(Q) − r − n ≤ j ≤ r if and only if k < dim(q) − r − 1 − j ≤ n + 1. This yields Theorem 1.3, as well as some other cases of Conjecture 1.1: Theorem 4.7. Conjecture 1.1 is true in the following cases: (1) The case where k < 2s−1 . (2) The case where 2s+1 − 2 ≤ dim(p) ≤ 2s+1 . (3) The case where p is a Pfister neighbour. Proof. Let p, q, s and k be as in the statement of Conjecture 1.1, and let ( 2s − 2 if k ≥ 2s−1 m= 2s−1 if k < 2s−1 . 11 By Lemma 4.1, the statement of the conjecture holds for the pair (p, q) if and only if dim(q)+k 2 l is even for every k < l ≤ m. Corollary 4.5 shows that this holds in both cases (1) and (2). Finally, suppose that p is a Pfister neighbour with ambient Pfister form π. Since p and π are stably birationally equivalent (see §2.B), we have iW (qF (π) ) = iW (qF (p) ). To prove that the statement of the conjecture holds in this case, we can therefore replace p by π and assume that p is a Pfister form. In particular, we can assume that dim(p) = 2s+1 . The validity of the conjecture in this case therefore follows from its validity in case (2). Further partial results towards Conjecture 1.1 will be given in the next section, but the case where k ≥ 2s−1 remains open in general. From the above arguments, it is easy to see that we have the following reformulation of the conjecture as a refinement of [Vis07, Thm. 3.1] for quadrics. The details are left to the reader: Proposition 4.8. The following are equivalent: (1) Conjecture 1.1 is true. (2) Let p be an anisotropic quadratic form of dimension ≥ 2 over F , and let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . Let Y be a (possibly isotropic) smooth projective F -quadric. If α ∈ Chm (YF ) is F (p)-rational, then S j (α) is F -rational for all j ≥ m − (2s − 2). 5. Additional results In this section, we provide further justification for Conjecture 1.1 beyond Theorem 1.3. First, we prove that the conjecture holds under a certain assumption on the splitting pattern of q. We then explain how the remaining cases would follow if a related, but rather stronger conjecture of Kahn on the unramified Witt ring of a quadric were true. This approach reveals that, in analogy with Theorem 3.4, Conjecture 1.1 should extend to a statement at the level of the splitting pattern of q. We begin by stating this extension. 5.A. A refinement of the main conjecture. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let k = dim(q) − 2iW (qF (p) ) (i.e., k is the dimension of the anisotropic part of qF (p)). By [Kne76, §5], k is equal to the dimension of q or one of its higher anisotropic kernels. We can therefore propose the following: Conjecture 5.1. In the above situation, let 0 ≤ l ≤ h(q) be such that dim(ql ) = k. Suppose that k < 2s , where s is the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . Then, for each 0 ≤ r < l, there exist integers ar , br ≥ 1 and −k ≤ ǫr ≤ k such that (1) dim(qr ) = 2s+1 ar + ǫr . (2) With one possible exception, either ir+1 (q) ≤ 12 (k+ǫr ) or ir+1 (q) = br 2s+1 +ǫr . The exception is where dim(qr ) = 2s+1 − k, in which case r = l − 1 and ir+1 (q) = 2s − k. Remark 5.2. If k = 0 (i.e., if qF (p) is hyperbolic), then l = h(q) − 1, and this is just Theorem 3.4. In general, the exceptional case of (2) corresponds to the case of Theorem 3.4 in which deg(q) = s + 1, so that ih(q) (q) is not divisible by 2s+1 (being equal to 2s ). Observe that, for k < 2s , Conjecture 1.1 correponds to assertion (1) of Conjecture 5.1 in the case where r = 0 (of course, Conjecture 1.1 is vacuously true if k ≥ 2s ). We will prove here the following: Theorem 5.3. Conjecture 5.1 (and hence Conjecture 1.1) is true if either 12 STEPHEN SCULLY (1) k ≤ 7. (2) dim(q) ≤ 2s+2 + k. The first case, in particular, implies that both our conjectures hold if dim(p) ≤ 16. The proof of Theorem 5.3 is given in §5.C below. Case (2) is treated using a motivic argument which in fact gives more (Theorem 5.8). Case (1), on the other hand, is treated by relating Conjecture 5.1 to Theorem 3.4 and the aforementioned conjecture of Kahn. We we now explain, this approach more clearly illustrates the philosophy underlying this paper. 5.B. A conjecture of Kahn. Let ϕ be an anisotropic quadratic form of dimension ≥ 2 over F . Recall that the unramified Witt ring of the projective quadric {ϕ = 0}, denoted Wnr (F (ϕ)), is defined as the subring of W (F (ϕ)) consisting of those classes of anisotropic quadratic forms which have no non-trivial (second) residues at the codimension-1 points of the projective quadric {ϕ = 0} (see [Kah95]). The image of the canonical scalar extension map W (F ) → W (F (ϕ)) trivially lies in Wnr (F (ϕ)), but equality need not hold in general (see, e.g., [KRS98, Cor. 10]). Nevertheless, Kahn has made the following strong conjecture concerning the “low-dimensional” part of Wnr (F (ϕ)): Conjecture 5.4 (Kahn, [Kah95, Conj. 1]). Let ϕ be an anisotropic quadratic form over F and let [τ ] ∈ Wnr (F (ϕ)). If dim(τ ) < 12 dim(ϕ), then τ is defined over F , i.e., there exists a quadratic form σ over F such that τ ≃ σF (ϕ) . If true, this has the following application to Conjecture 5.1: Proposition 5.5. Assume in the situation of Conjecture 5.1 that dim(ql−1 ) > 2k. Then, (qF (p) )an is defined over F provided that Conjecture 5.4 holds whenever dim(τ ) ≤ k and dim(ϕ) ≥ dim(ql−1 ). Proof. Let (Fr ) be the Knebusch splitting tower of q. The form ql is evidently unramified at the codimension-1 points of the quadric {ql−1 = 0}. Since dim(ql ) = k and dim(ql−1 ) > 2k, our hypothesis implies that ql ≃ τFl for some form τ over Fl−1 . If l > 1, then τ is unramified at the codimension-1 points of X = {ql−2 = 0}. Indeed, since ql ≃ τFl , Corollary 3.2 implies that [τ ⊥ −ql−1 ] ∈ I n (Fl−1 ), where 2n ≥ dim(ql−1 ) > 2k. Thus, if x ∈ X (1) , and π is a uniformizer for OX,x , we have (using [EKM08, Lem. 19.14]) ∂x,π ([τ ]) = ∂x,π ([τ ⊥ −ql−1 ]) ∈ ∂x,π I n (Fl−1 ) ⊆ I n−1 (Fl−2 (x)). But dim(τ ) = k, and since 2n−1 > k, the Arason-Pfister Hauptsatz ([AP71]) then implies that ∂x,π ([τ ]) = 0. Again, our hypothesis now gives that ql is defined over Fl−2 . Continuing in this way, we see that ql is defined over F , say ql ≃ σFl . We claim that (qF (p) )an ≃ σF (p) . Since dim(σ) = k = dim((qF (p) )an ), it suffices to show that (q ⊥ −σ)F (p) is hyperbolic. But (q ⊥ −σ)Fl (p) is hyperbolic, and so, by [EKM08, Lem. 7.15], it only remains to observe that Fl (p) is a purely transcendental extension of F (p). But if r < l, then the form qr becomes isotropic over Fr (p) by the very definition of k. Hence Fr+1 (p) = Fr (p)(qr ) is a purely transcendental extension of Fr (p), and so the claim follows by an easy induction. If one assumes another related conjecture of Vishik, then one can go a bit further here (see Remark 5.13 (2)). In any case, the philosophy underlying Conjecture 5.1 is now clear: If qF (p) is “nearly hyperbolic”, then its anisotropic part should typically be defined over F , and the necessary conclusions can be made using Theorem 3.4. In more details: Lemma 5.6. Conjecture 5.1 is true in the case where (qF (p) )an is defined over F . 13 Proof. By an easy induction on l, it suffices to show that if l > 0, then dim(q) and i1 (q) have the prescribed form. Assume that (qF (p) )an ≃ σF (p) for some form σ over F , and let η = (q ⊥ −σ)an . Since both q and σ are anisotropic, there exists an integer 0 ≤ λ ≤ dim(σ) = k, and codimension-λ subforms qe ⊂ q and σ e ⊂ σ such that η ≃ qe ⊥ −e σ. In particular, setting ǫ = 2λ − k, we have dim(q) = dim(η) + ǫ. Now, by construction, η becomes hyperbolic over F (p). Thus, by Corollary 3.6, we have dim(η) = 2s+1 a for some a ≥ 1 (and also deg(η) ≥ s+1). We therefore have dim(q) = 2s+1 a+ǫ. As for the assertion regarding i1 (q), we may assume that i1 (q) > (k + ǫ)/2 = λ, so that qe is a neighbour of q in the sense of §2.B above. This implies, in particular, that q and qe are stably birationally equivalent. If η is similar to an (s + 1)-fold Pfister form, then the Cassels-Pfister subform theorem implies that q is a subform of η, whence η ≃ q ⊥ −σ by Witt cancellation. Since dim(σ) = k < 2s , q is then a neighbour of η, and so i1 (q) = 2s − k by [Vis04, Cor. 4.9]. Otherwise, Theorem 3.4 implies that i1 (η) = 2s+1 b for some b ≥ 1. Again, since dim(e σ ) < 2s , it follows that qe is a neighbour of η. As q is a neighbour of qe, we see that q and η are stably birationally equivalent, and so i1 (q) = i1 (η) + dim(q) − dim(η) = 2s+1 b + ǫ by another application of [Vis04, Cor. 4.9]. This proves the lemma. 5.C. Proof of Theorem 5.3. For the remainder of the paper, we fix the following notation: • q and p are anisotropic quadratic forms of dimension ≥ 2 over F . • Q and P denote the (smooth) projective quadrics defined by the vanishing of q and p, respectively. • s is the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . • k := dim(q) − 2iW (qF (p) ) (i.e., k is the dimension of the anisotropic part of qF (p)). • l is the unique integer in [0, h(q)] such that dim(ql ) = k. • i := iW (qF (p) ). We now proceed with the proof of Theorem 5.3. We begin by showing that Conjecture 5.1 is true provided that 2s+1 − k lies in the splitting pattern of q. To make this more precise, we note the following: Lemma 5.7. Assume, in the situation of Conjecture 5.1, that l ≥ 1 (i.e, that qF (p) is isotropic). Then dim(ql−1 ) = 2N − k for some integer N ≥ s + 1. Proof. Let (Fr ) be the Knebusch splitting pattern of q. Since ql−1 becomes isotropic over Fl−1 (p), and since dim(p) > 2s , Hoffmann’s Separation theorem ([Hof95, Thm. 1]) implies that dim(ql−1 ) > 2s . In particular, dim(ql−1 ) = 2N − m for some integers N ≥ s + 1 and 0 ≤ m < 2N −1 . By definition, we then have i1 (ql−1 ) = 12 (2N − m − k), so that 1 dim(ql−1 ) − i1 (ql−1 ) = (2N − m + k) > 2N −2 (remember that k < 2s ). 2 By Karpenko’s theorem on the possible values of the first Witt index ([Kar03]), it follows that 2N −1 divides m − k. Since both both m and k are strictly less than 2N −1 , this implies that m = k, and so dim(ql−1 ) = 2N − k. Our result is now that Conjecture 5.1 holds in the case where N = s + 1: Theorem 5.8. Assume that l ≥ 1 (i.e., that qF (p) is isotropic). If dim(ql−1 ) = 2s+1 − k, then Conjecture 5.1 is true. Proof. The proof is similar to that of [Scu16, Thm. 4.1], and begins with the following observation regarding the motivic decomposition of the quadric Q: 14 STEPHEN SCULLY Lemma 5.9. In the situation of Theorem 5.8, U (P ){i − 1} is isomorphic to a direct summand of M (Q). Proof. By [Vis04, Thm. 4.15], it suffices to check that for every field extension L of F , we have iW (pL ) > 0 ⇔ iW (qL ) ≥ i. The left to right implication is immediate; indeed, if pL is isotropic, then L(p) is a purely transcendental extension of L, and so iW (qL ) = iW (qL(p) ) ≥ iW (qF (p) ) = i. For the other implication, assume that iW (qL ) ≥ i. If (Fr ) denotes the Knebusch splitting tower of q, then it follows from the very definition of l that the free composite Fl · L is a purely transcendental extension of L (compare the end of the proof of Proposition 5.5). To show that iW (pL ) > 0, it therefore suffices to show that p becomes isotropic over Fl . Note first that, by hypothesis, we have 1 (dim(ql−1 ) − dim(ql )) 2 1 = (2s+1 − k) − (2s+1 − 2k) = 2s . 2 On the other hand, the form ql−1 becomes isotropic over Fl−1 (p). Since dim(p) > 2s , and since Fl = Fl−1 (ql−1 ), the desired assertion thus follows from Izhboldin’s “strong incompressibility” theorem ([Izh00, Thm. 0.2]). dim(ql−1 ) − i1 (ql−1 ) = dim(ql−1 ) − The proof of Theorem 5.8 now proceeds by induction on l, with the case where l = 1 being trivial. Indeed, if l = 1, then dim(q) = 2s+1 − k and i1 (q) = (dim(q) − k)/2 = 2s − k. Assume now that l ≥ 2 (in particular, h(q) ≥ 2). Applying the induction hypothesis to the pair (q1 , pF (q) ), we immediately get that the integers dim(qr ) and ir+1 (q) have the prescribed form for all 1 ≤ r < l. In particular, we have dim(q1 ) = 2s+1 a1 + ǫ1 for some a1 ≥ 1 and −k ≤ ǫ1 ≤ k. Since dim(q) = dim(q1 ) + 2i1 (q), we then have k − ǫ1 1 s . (5.1) i = (dim(q) − k) = 2 a1 + i1 (q) − 2 2 It still remains show that dim(q) and i1 (q) are as claimed. We separate two cases: Case 1. i1 (q) ≤ (k − ǫ1 )/2. In this case, the assertions are clear. Indeed, we have 2s+1 a1 + ǫ1 = dim(q1 ) < dim(q) = dim(q1 ) + 2i1 (q) ≤ dim(q1 ) + (k − ǫ1 ) = 2s+1 a1 + k. Since −k ≤ ǫ1 ≤ k, this shows that dim(q) = 2i+1 a1 + ǫ0 for some −k < ǫ0 ≤ k, and we then have ǫ0 − ǫ1 k + ǫ0 dim(q) − dim(q1 ) = ≤ . i1 (q) = 2 2 2 Case 2. i1 (q) > (k − ǫ1 )/2. In this case, let u denote the smallest non-negative integer such that 2u ≥ i1 (q). By Karpenko’s theorem on the possible values of the first Witt index ([Kar03]), the integer dim(q) − i1 (q) is divisible by 2u . Since dim(q) = dim(q1 ) + 2i1 (q), it follows that 2s+1 a1 + ǫ1 + i1 (q) ≡ 0 (mod 2u ). (5.2) We claim that u > i. Suppose that this is not the case. Then (5.2) implies that i1 (q) = 2u µ − ǫ1 for some integer µ. Now, by hypothesis (and the fact that ǫ1 ≥ −k), we have i1 (q) + ǫ1 > k + ǫ1 k − ǫ1 + ǫ1 = ≥ 0, 2 2 15 and so µ > 0. At the same time, we have 2u µ ≤ 2s . Indeed, since 2u ≥ i1 (q), the inequality 2u µ > 2s would imply that ǫ1 = 2u µ − i1 (q) ≥ 2s , which is not the case (since ǫ1 ≤ k < 2s ). The situation is thus as follows: We have dim(q) − i1 (q) = dim(q1 ) + i1 (q) = 2s+1 a1 + 2u µ, 2u µ 2s . (5.3) 2s+1 a1 where a1 ≥ 1 and 0 < ≤ Now the integer can be written in the form r r r r w 1 2 w−1 2 −2 +···+2 − 2 for some integers r1 > r2 > · · · > rw ≥ s + 1, while 2u µ can be r r w+1 w+2 written as 2 −2 + · · · + (−1)t−1 2rt for unique integers s ≥ rw+1 > rw+2 > · · · > rt−1 > rt + 1 ≥ 1. Equation 5.3 can therefore be re-written as dim(q) − i1 (q) = (2r1 − 2r2 + · · · + 2rw−1 − 2rw ) + 2rw+1 − 2rw+2 + · · · + (−1)t−1 2rt . \| {z } \| {z } 2s+1 a1 2u µ 2u µ 2s Unless = and rw = s + 1, this is precisely the presentation of dim(q) − i1 (q) as an alternating sum of 2-powers appearing in the statement of Vishik’s Theorem 2.2. If 2u µ = 2s and rw = s + 1, then the needed presentation is dim(q) − i1 (q) = (2r1 − 2r2 + · · · + 2rw−1 ) −2s . {z } \| 2s+1 (a1 +1) Either way, see that the Tate motive Z{2s a1 } is isomorphic to a direct summand of U (Q)F . Indeed, this follows by applying Vishik’s Theorem with c = w + 1 in the first case, and with c = w in the second. Let j = i1 (q) − 1 − (k − ǫ1 )/2. Since i1 (q) > (k − ǫ1 )/2, we then have 0 ≤ j < i1 (q). Thus, by [Vis11, Cor. 3.10], U (Q){j} is isomorphic to a direct summand of M (Q). By the preceding discussion, Z{2s a1 + j} is isomorphic to a direct summand of U (Q){j}F . But 2s a1 + j = 2s a1 + i1 (q) − (k − ǫ1 )/2 − 1 = i − 1 by equation (5.1). In view of Lemma 5.9, we thus see that there are two indecomposable direct summands of M (Q) containing the Tate motive Z{i − 1} in their respective decompositions over F , namely, U (Q){j} and U (P ){i − 1}. As result, we must have U (Q){j} ∼ = U (P ){i − 1} (see §2.D above). In particular, we have j = i − 1. Since j < i1 (q), and since i > 0, this implies that i = i1 (q). In other words, it implies that l = 1, which provides us with the needed contradiction to our supposition. We can therefore conclude that u > s, i.e., that i1 (q) > 2s . By (5.2), it follows that i1 (q) = 2s+1 b0 − ǫ1 for some positive integer b0 . Moreover, we then have dim(q) = dim(q1 ) + 2i1 (q) = 2s+1 (a1 + 2b0 ) − ǫ1 . Since −k ≤ ǫ1 ≤ k, this proves the theorem. Let us note that the proof of Theorem 5.8 allows us to extract more information. Recall that a direct summand N in the motive of a smooth projective F -quadric is said to be binary if NF is isomorphic to a direct sum of two Tate motives. We then have: Proposition 5.10. If k < 2s−1 + 2s−2 and dim(ql−1 ) = 2s+1 − k, then the upper motive U (P ) of the quadric P is binary. Proof. By Lemma 5.9, U (P ){i − 1} is isomorphic to a direct summand of M (Q). Since i − 1 = jl (q) − 1, it follows from [Vis11, Thm. 4.13] that N := U (P ){jl−1 (q)} is also isomorphic to a direct summand of M (Q). By an argument identical to that given in [Vis11, Proof of Thm. 7.7], it follows that N , and hence U (P ), is binary provided that ir (q) < il (q) for all r > l. But il (q) = (dim(ql−1 ) − k)/2 = 2s − k > 2s−2 by hypothesis, and 16 STEPHEN SCULLY so it only remains to note that ir (q) ≤ 2s−2 for all r > l. Indeed, if dim(qr−1 ) ≤ 2s−1 , then this is trivial; otherwise, our hypothesis again implies that 2s−1 < dim(qr−1 ) < 2s−1 + 2s−2 and so the claim follows from Hoffmann’s Separation theorem (see [Hof95, Cor. 1]). As conjectured by Vishik (cf. [Vis11, Conj. 4.21]), the upper motive of an anisotropic projective quadric over a field of characteristic 6= 2 should be binary only when its underlying quadratic form is a Pfister neighbour (the converse being a well-known result of Rost ([Ros90, Prop. 4])). Significant evidence for the validity of this assertion has been given in [IV00, §6]. Moreover, in [Kar01], Karpenko showed that Vishik’s conjecture holds for forms of dimension ≤ 16. Applying this to the special case where k = 0 and deg(q) = s+1, we get the following result promised in §3.A above: Corollary 5.11. If qF (p) is hyperbolic and deg(q) = s + 1, then: (1) The upper motive U (P ) of the quadric P is binary. (2) If s ≤ 3 (i.e., dim(p) ≤ 16), there exists an (s + 1)-fold Pfister form π and an odd-dimensional form r such that q ≃ π ⊗ r. Moreover, p is a neighbour of π. Proof. The first statement is the k = 0 case of Proposition 5.10. The second statement then follows from the aforementioned result of Karpenko together with [EKM08, Cor. 23.6]. More precisely, Karpenko showed in [Kar01] that if the upper motive of P is binary, and dim(p) ∈ {3, 5, 9}, then p is a Pfister neighbour. But this implies the same assertion in all dimensions ≤ 16 by [IV00, Thm. 6.1] and [Vis04, Cor. 3.9, Cor. 4.7]. We can now give the proof of Theorem 5.3: Proof of Theorem 5.3. We may assume that l ≥ 1, since otherwise there is nothing to prove. If dim(ql−1 ) = 2s+1 − k, then Conjecture 5.1 holds by Theorem 5.8. Thus, by Lemma 5.7, we can assume that dim(ql−1 ) ≥ 2s+2 − k. Now, if dim(q) ≤ 2s+2 + k, then we obviously have dim(ql−1 ) = 2s+2 − k, and the assertions of Conjecture 5.1 hold trivially. This takes care of case (2) of the theorem. It now remains to treat the case where k ≤ 7. As dim(ql−1 ) ≥ 2s+2 − k, Proposition 5.5 and Lemma 5.6 show that it suffices to know that Kahn’s Conjecture 5.4 holds in the following cases: (i) dim(τ ) ∈ {0, 1}, dim(ϕ) ≥ 7. (ii) dim(τ ) ∈ {2, 3}, dim(ϕ) ≥ 13. (iii) dim(τ ) ∈ {4, 5, 6, 7}, dim(ϕ) ≥ 25. But all these cases of Kahn’s conjecture are already known to be true: for the cases where dim(τ ) ≤ 5, see [Kah95, Thm. 2]; for the case where dim(τ ) = 6, see [Lag99, Thm. principale], and for the case where dim(τ ) = 7, see [IV00, Thm. 3.9]. Corollary 5.12. Conjecture 1.1 holds if dim(q) ≤ 2s+1 + 2s−1 . Proof. If k ≥ 2s−1 , apply 5.3; otherwise, apply Theorem 1.3. Remarks 5.13. (1) Laghribi ([Lag99, Théorème principale]) has shown that if dim(ϕ) > 16, then Conjecture 5.4 also holds for some special classes of 8, 9 and 10-dimensional forms τ . This permits to extend Theorem 5.3 to the case where k ∈ {8, 9, 10} under some additional assumptions on q. We refrain from going into the details here. (2) Conjecturally, the form (qF (p) )an should be defined over F in the situation of Conjecture 5.1 provided that either (i) k < 2s−1 + 2s−2 or (ii) l ≥ 1 and dim(ql−1 ) 6= 2s+1 − k. Indeed, that the second condition should be enough is implicit in our proof of Theorem 5.3. As for the first condition, we may additionally assume in this case that l ≥ 1 and dim(ql−1 ) = 2s+1 − k. By Proposition 5.10, this ensures that the upper motive of P 17 is binary. According to [Vis04, Conj. 4.21], this should in turn imply that p is a Pfister neighbour. By replacing p by its ambient Pfister form, we could then assume that dim(p) = 2s+1 > 2k, and so the claim would follow directly from Conjecture 5.4. Acknowledgements. I would like to thank Alexander Vishik for helpful comments. 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[Vis09] A. Vishik. Fields of u-invariant 2r + 1. Algebra, arithmetic and geometry: in honor of Yu. I. Manin. Vol. II, 661–685, Progr. Math., 270, Birkhäuser Boston, Inc., Boston, MA, 2009. [Vis11] A. Vishik. Excellent connections in the motives of quadrics. Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 1, 183–195. [Voe03] V. Voevodsky. Motivic cohomology with Z/2-coefficients. Publ. Math. Inst. Hautes Études Sci. No. 98 (2003), 59–104. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada E-mail address: stephenjscully@gmail.com	0
Logical Methods in Computer Science Vol. 8 (1:29) 2012, pp. 1–36 www.lmcs-online.org Submitted Published Oct. 4, 2011 Mar. 26, 2012 ON IRRELEVANCE AND ALGORITHMIC EQUALITY IN PREDICATIVE TYPE THEORY ∗ ANDREAS ABEL a AND GABRIEL SCHERER b a Department of Computer Science, Ludwig-Maximilians-University Munich e-mail address: andreas.abel@ifi.lmu.de b Gallium team, INRIA Paris-Rocquencourt e-mail address: gabriel.scherer@gmail.com Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To separate static and dynamic code, several static analyses and type systems have been put forward. We consider Pfenning’s type theory with irrelevant quantification which is compatible with a type-based notion of equality that respects η-laws. We extend Pfenning’s theory to universes and large eliminations and develop its meta-theory. Subject reduction, normalization and consistency are obtained by a Kripke model over the typed equality judgement. Finally, a type-directed equality algorithm is described whose completeness is proven by a second Kripke model. 1. Introduction and Related Work Dependently typed programming languages such as Agda [BDN09], Coq [INR10], and Epigram [MM04] allow the programmer to express in one language programs, their types, rich invariants, and even proofs of these invariants. Besides code executed at run-time, dependently typed programs contain much code needed only to please the type checker, which is at the same time the verifier of the proofs woven into the program. Program extraction takes type-checked terms and discards parts that are irrelevant for execution. Augustsson’s dependently typed functional language Cayenne [Aug99] erases types using a universe-based analysis. Coq’s extraction procedure has been designed by Paulin-Mohring and Werner [PMW93] and Letouzey [Let02] and discards not only types but also proofs. The erasure rests on Coq’s universe-based separation between propositional (Prop) and computational parts (Set/Type). The rigid Prop/Set distinction has the drawback of code duplication: A structure which is sometimes used statically and sometimes dynamically needs to be coded twice, once in Prop and once in Set. An alternative to the fixed Prop/Set-distinction is to let the usage context decide whether a term is a proof or a program. Besides whole-program analyses such as data 1998 ACM Subject Classification: F.4.1. Key words and phrases: dependent types, proof irrelevance, typed algorithm equality, logical relation, universal Kripke model. ∗ Revision and extension of FoSSaCS 2011 conference publication. l LOGICAL METHODS IN COMPUTER SCIENCE c DOI:10.2168/LMCS-8 (1:29) 2012 CC A. Abel and G. Scherer Creative Commons 2 A. ABEL AND G. SCHERER flow, some type-based analyses have been put forward. One of them is Pfenning’s modal type theory of Intensionality, Extensionality, and Proof Irrelevance [Pfe01], later pursued by Reed [Ree03], which introduces functions with irrelevant arguments that play the role of proofs.1 Not only can these arguments be erased during extraction, they can also be disregarded in type conversion tests during type checking. This relieves the user of unnecessary proof burden (proving that two proofs are equal). Furthermore, proofs can not only be discarded during program extraction but directly after type checking, since they will never be looked at again during type checking subsequent definitions. In principle, we have to distinguish “post mortem” program extraction, let us call it external erasure, and proof disposal during type checking, let us call it internal erasure. External erasure deals with closed expressions, programs, whereas internal erasure deals with open expressions that can have free variables. Such free variables might be assumed proofs of (possibly false) equations and block type casts, or (possibly false) proofs of wellfoundedness and prevent recursive functions from unfolding indefinitely. For type checking to not go wrong or loop, those proofs can only be externally erased, thus, the Prop/Set distinction is not for internal erasure. In Pfenning’s type theory, proofs can never block computations even in open expressions (other than computations on proofs), thus, internal erasure is sound. Miquel’s Implicit Calculus of Constructions (ICC) [Miq01a] goes further than Pfenning and considers also parametric arguments as irrelevant. These are arguments which are irrelevant for function execution but relevant during type conversion checking. Such arguments may only be erased in function application but not in the associated type instantiation. Barras and Bernardo [BB08] and Mishra-Linger and Sheard [MLS08] have built decidable type systems on top of ICC, but both have not fully integrated inductive types and types defined by recursion (large eliminations). Barras and Bernardo, as Miquel, have inductive types only in the form of their impredicative encodings, Mishra-Linger [ML08] gives introduction and elimination principles for inductive types by example, but does not show normalization or consistency. While Pfenning’s type theory uses typed equality, ICC and its successors interpret typed expressions as untyped λ-terms up to untyped equality. In our experience, the implicit quantification of ICC, which allows irrelevant function arguments to appear unrestricted in the codomain type of the function, is incompatible with type-directed equality. Examples are given in Section 2.3. Therefore, we have chosen to scale Pfenning’s notion of proof irrelevance up to inductive types, and integrated it into Agda. In this article, we start with the “extensionality and proof irrelevance” fragment of Pfenning’s type theory in Reed’s version [Ree02, Ree03]. We extend it by a hierarchy of predicative universes, yielding Irrelevant Intensional Type Theory IITT (Sec. 2). After specifying a type-directed equality algorithm (Sec. 3), we construct a Kripke model for IITT (Sec. 4). It allows us to prove normalization, subject reduction, and consistency, in one go (Sec. 5). A second Kripke logical relation yields correctness of algorithmic equality and decidability of IITT (Sec. 6). Our models are ready for data types, large eliminations, types with extensionality principles, and internal erasure (Sec. 7). 1 Awodey and Bauer [AB04] give a categorical treatment of proof irrelevance which is very similar to Pfenning and Reed’s. However, they work in the setting of Extensional Type Theory with undecidable type checking, we could not directly use their results for this work. IRRELEVANCE IN TYPE THEORY 3 Contribution and Related Work. We consider the design of our meta-theoretic argument as technical novelty, although it heavily relies on previous works to which we owe our inspiration. Allen [All87] describes a logical relation for Martin-Löf type theory with a countable universe hierarchy. The seminal work of Coquand [Coq91] describes an untyped equality check for the Logical Framework and justifies it by a logical relation for dependent types that establishes subject reduction, normalization, completeness of algorithmic equality, and injectivity of function types in one go. However, his approach cannot be easily extended to a typed algorithmic equality, due to problems with transitivity. Goguen introduces Typed Operational Semantics [Gog94] to construct a Kripke logical relation that simultaneously proves normalization, subject reduction, and confluence for a variant of the Calculus of Inductive Constructions. From his results one can derive an equality check based on reduction to normal form. Goguen also shows how to derive syntactic properties, such as closure of typing and equality under substitution, by a Kripke-logical relation [Gog00]. Harper and Pfenning [HP05] popularize a type-directed equality check for the Logical Framework that scales to extensionality for unit types. They prove completeness of algorithmic equality by a Kripke model on simple types which are obtained by erasure from the dependent types. Erasure is necessary since algorithmic equality cannot be shown transitive before it is proven sound; yet soundness hinges on subject reduction which rests on function type injectivity which in turn is obtained from completeness of algorithmic equality—a vicious cycle. While erasure breaks the cycle, it also prevents types to be defined by recursion on values (so-called large eliminations), a common feature of proof assistants like Agda, Coq, and Epigram. Normalization by evaluation (NbE) has been successfully used to obtain a type-directed equality check based on evaluation in the context of dependent types with large eliminations [ACD07]. In previous work [ACD08], the first author applied NbE to justify a variant of Harper and Pfenning’s algorithmic equality without erasure. However, the meta-theoretic argument is long-winded, and there is an essential gap in the proof of transitivity of the Kripke logical relation. In this work, we explore a novel approach to justify type-directed algorithmic equality for dependent types with predicative universes. First, we show its soundness by a Kripke model built on top of definitional equality. The Kripke logical relation yields normalization, subject reduction, and type constructor injectivity, which also imply logical consistency of IITT. Further, it proves syntactic properties such as closure under substitution, following Goguen’s lead [Gog00]. The semantic proof of such syntactic properties relieves us from the deep lemma dependencies and abundant traps of syntactic meta-theory of dependent types [HP05, AC07]. Soundness of algorithmic equality entails transitivity (which is the stumbling stone), paving the way to show completeness of algorithmic equality by a second Kripke logical relation, much in the spirit of Coquand [Coq91] and Harper and Pfenning [HP05]. This article is a revised and extended version of paper Irrelevance in Type Theory with a Heterogeneous Equality Judgement presented at the conference FoSSaCS 2011 [Abe11]. Unfortunately, the conference version has inherited the above-mentioned gap [ACD08] in the proof of transitivity of the Kripke logical relation. This is fixed in the present article by an auxiliary Kripke model (Section 4). Further, we have dropped the heterogeneous approach to equality in favor of a standard homogeneous one. Heterogeneous equality is not necessary for the style of irrelevance we are embracing here. 4 A. ABEL AND G. SCHERER 2. Irrelevant Intensional Type Theory In this section, we present Irrelevant Intensional Type Theory IITT which features two of Pfenning’s function spaces [Pfe01], the ordinary “extensional” (x : U ) → T and the proof irrelevant (x÷U ) → T . The main idea is that the argument of a (x÷U ) → T function is counted as a proof and can neither be returned nor eliminated on, it can only be passed as argument to another proof irrelevant function or data constructor. Technically, this is realized by annotating variables as relevant, x : U , or irrelevant, x÷U , in the typing context, to confine occurrences of irrelevant variables to irrelevant arguments. Expression and context syntax. We distinguish between relevant (t : u or simply t u) and irrelevant application (t ÷ u). Accordingly, we have relevant (λx : U. T ) and irrelevant abstraction (λx÷U. T ). Our choice of typed abstraction is not fundamental; a bidirectional type-checking algorithm [Coq96] can reconstruct type and relevance annotations at abstractions and applications. Var ∋ x, y, X, Y Sort ∋ s ::= Setk (k ∈ N) Ann ∋ ⋆ ::= ÷ \| : universes annotation: irrelevant, relevant s,s′ Exp ∋ t, u, T, U ::= s \| (x⋆U ) → T sort, (ir)relevant function type ⋆ \| x \| λx⋆U. t \| t u lambda-calculus Cxt ∋ Γ, ∆ ::= ⋄ \| Γ. x⋆T empty, (ir)relevant extension Expressions are considered modulo α-equality, we write t ≡ t′ when we want to stress that t and t′ identical (up to α). Similarly, we consider variables bound in a context to be distinct, and when opening a term binder we will implicitly use α-conversion to add a fresh variable in the context. For technical reasons, namely, to prove transitivity (Lemma 4.3) of the Kripke logical s,s′ relation in Section 4, we explicitly annotate function types (x⋆U ) → T with the sorts s of domain U and s′ of codomain T . We may omit the annotation if it is inessential or determined by the context of discourse. In case T does not mention x, we may write U → T for (x : U ) → T . Sorts. IITT is a pure type system (PTS) with infinite hierarchy of predicative universes Set0 : Set1 : .... The universes are not cumulative. We have the PTS axioms Axiom = {(Seti , Seti+1 ) \| i ∈ N} and the rules Rule = {(Seti , Setj , Setmax(i,j) ) \| i, j ∈ N}. As is customary, we will write the side condition (s, s′ ) ∈ Axiom just as (s, s′ ) and likewise (s1 , s2 , s3 ) ∈ Rule just as (s1 , s2 , s3 ). IITT is a full and functional PTS, which means that for all s1 , s2 there is exactly one s3 such that (s1 , s2 , s3 ). There is no subtyping, so that types—and thus, sorts—are unique up to equality. A proof of sort unicity might relieve us from the sort annotation in function types, however, we obtain sort discrimination too late in our technical development (Lemma 5.10). Substitutions. Substitutions σ are maps from variables to expressions. We require that the domain dom(σ) = {x \| σ(x) 6= x} is finite. We write id for the identity substitution and [u/x] for the singleton substitution σ such that σ(x) := u and σ(y) := y for y 6= x. Substitution extension (σ, u/x) is formally defined as σ ⊎ [u/x]. Capture avoiding parallel substitution of σ in t is written as juxtaposition tσ. IRRELEVANCE IN TYPE THEORY 5 Contexts. Contexts Γ feature two kinds of bindings, relevant (x : U ) and irrelevant (x ÷ U ) ones. The intuition, implemented by the typing rules below, is that only relevant variables are in scope in an expression. Resurrection Γ÷ turns all irrelevant bindings (x ÷ T ) into the corresponding relevant ones (x : T ) [Pfe01]. It is the tool to make irrelevant variables, also called proof variables, available in proofs. The generalization Γ⋆ shall mean Γ÷ if ⋆ = ÷, and just Γ otherwise. We write Γ.∆ for the concatenation of Γ and ∆; herein, we suppose dom(Γ) ∩ dom(∆) = ∅. Primitive judgements of IITT. The following three judgements are mutually inductively defined by the rules given below and in Figure 1. ⊢Γ Γ ⊢t:T Γ ⊢ t = t′ : T Context Γ is well-formed. In context Γ, expression t has type T . In context Γ, t and t′ are equal expressions of type T . Derived judgements. To simplify notation, we introduce the following four abbreviations: Γ Γ Γ Γ ⊢t÷T ⊢ t = t′ ÷ T ⊢T ⊢ T = T′ iff iff iff iff Γ÷ ⊢ t : T, Γ ⊢ t ÷ T and Γ ⊢ t′ ÷ T, Γ ⊢ T : s for some s, Γ ⊢ T = T ′ : s for some s. Γ ⊢ t ⋆ T may mean Γ ⊢ t : T or Γ ⊢ t ÷ T , depending on the value of placeholder ⋆; same for Γ ⊢ t = t′ ⋆ T . We sometimes write Γ ⊢ t, t′ ⋆ T to abbreviate the conjunction of Γ ⊢ t ⋆ T and Γ ⊢ t′ ⋆ T . The notation Γ ⊢ T, T ′ is to be understood similarly. 2.1. Rules. Our rules for well-typed terms Γ ⊢ t : T extend Reed’s rules [Ree02] to PTS style. There are only 6 rules; we shall introduce them one-by-one. Variable rule. Only relevant variables can be extracted from the context. ⊢Γ (x : U ) ∈ Γ Γ ⊢x:U There is no variable rule for irrelevant bindings (x ÷ U ) ∈ Γ, in particular, the judgement x÷U ⊢ x : U is not derivable. This essentially forbids proofs to appear in relevant positions. Abstraction rule. Relevant and irrelevant functions are introduced analogously. Γ. x⋆U ⊢ t : T s,s′ Γ ⊢ (x⋆U ) → T s,s′ Γ ⊢ λx⋆U. t : (x⋆U ) → T To check a relevant function λx : U. t, we introduce a relevant binding x : U into the context and continue checking the function body t. In case of an irrelevant function λx÷U. t, we proceed with an irrelevant binding x ÷ U . This means that an irrelevant function cannot computationally depend on its argument—it is essentially a constant function. In particular, λx÷U. x is never well-typed. s,s′ As a side condition, we also need to check that the introduced function type (x⋆U ) → T is well-sorted; the rule is given below. 6 A. ABEL AND G. SCHERER Application rule. Γ ⊢ t : (x⋆U ) → T Γ ⊢u⋆U ⋆ Γ ⊢ t u : T [u/x] This rule uses our overloaded notations for bindings ⋆, that can be specialized into two different instances for relevant and irrelevant applications. For relevant functions, we get the ordinary dependently-typed application rule: Γ ⊢ t : (x : U ) → T Γ ⊢u:U Γ ⊢ t u : T [u/x] When applying an irrelevant function, we resurrect the context before checking the function argument. Γ ⊢ t : (x÷U ) → T Γ÷ ⊢ u : U Γ ⊢ t ÷ u : T [u/x] This means that irrelevant variables become relevant and can be used in u. The intuition is that the application t ÷ u does not computationally depend on u, thus, u may refer to any variable, even the “forbidden ones”. One may think of u as a proof which may refer to both ordinary and proof variables. For example, let Γ = f : (y÷U ) → U . Then the irrelevant η-expansion λx÷U. f ÷ x is well-typed in Γ, with the following derivation: Γ. x ÷ U ⊢ f : (y÷U ) → U Γ. x : U ⊢ x : U Γ. x ÷ U ⊢ f ÷ x : U Γ ⊢ λx ÷ U. f ÷ x : (x÷U ) → U Observe how the status of x changes for irrelevant to relevant when we check the argument of f . Sorting rules. These are the “Axioms” and the “Rules” of PTSs to form types. ⊢Γ (s, s′ ) Γ ⊢ s : s′ Γ ⊢ U : s1 Γ. x⋆U ⊢ T : s2 s1 ,s2 Γ ⊢ (x⋆U ) → T : s3 (s1 , s2 , s3 ) The rule for irrelevant function type formation follows Reed [Ree02]. Γ ⊢ U : s1 Γ. x÷U ⊢ T : s2 Γ ⊢ U : s1 Γ. x : U ⊢ T : s2 (s1 , s2 , s3 ) s1 ,s2 Γ ⊢ (x÷U ) → T : s3 It states that the codomain of an irrelevant function cannot depend relevantly on the function argument. This fact is crucial for the construction of our semantics in Section 4. Note that it rules out polymorphism in the sense of Barras and Bernado’s Implicit Calculus of Constructions ICC∗ [BB08] and Mishra-Linger and Sheard’s Erasure Pure Type Systems EPTS [MLS08]; the type (X÷Set0 ) → (x : X) → X is ill-formed in IITT, but not in ICC∗ or EPTS. In EPTS, there is the following rule: (s1 , s2 , s3 ) s1 ,s2 Γ ⊢ (x÷U ) → T : s3 It allows the codomain T of an irrelevant function to arbitrarily depend on the function argument x. This is fine in an erasure semantics, but incompatible with our typed semantics in the presence of large eliminations; we will detail the issues in examples 2.3 and 2.8. IRRELEVANCE IN TYPE THEORY 7 Another variant is Pfenning’s rule for irrelevant function type formation [Pfe01]. Γ ⊢ U ÷ s1 Γ. x÷U ⊢ T : s2 (s1 , s2 , s3 ) s1 ,s2 Γ ⊢ (x÷U ) → T : s3 It allows the domain of an irrelevant function to make use of irrelevant variables in scope. It does not give polymorphism, e. g., (X÷Set0 ) → (x : X) → X is still ill-formed. However, (X÷Set0 ) → (x÷X) → X would be well-formed. It is unclear how the equality rule for irrelevant function types would look like—it is not given by Pfenning [Pfe01]. The rule Γ ⊢ U = U ′ ÷ s1 Γ. x÷U ⊢ T = T ′ : s2 (s1 , s2 , s3 ) s1 ,s2 s1 ,s2 Γ ⊢ (x÷U ) → T = (x÷U ′ ) → T ′ : s3 would mean that any two irrelevant function types are equal as long as their codomains are equal—their domains are irrelevant. This is not compatible with our typed semantics and seems a bit problematic in general.2 Type conversion rule. We have typed conversion, thus, strictly speaking, IITT is not a PTS, but a Pure Type System with Judgemental Equality [Ada06]. Γ ⊢t:T Γ ⊢ T = T′ Γ′ ⊢ t : T ′ Equality. Figure 1 recapitulates the typing rules and lists the rules to derive context wellformedness ⊢ Γ and equality Γ ⊢ t = t′ : T . Equality is the least congruence over the βand η-axioms. Since equality is typed we can extend IITT to include an extensional unit type (Section 7). Let us inspect the congruence rule for application: Γ ⊢ t = t′ : (x⋆U ) → T Γ ⊢ u = u′ ⋆ U Γ ⊢ t ⋆ u = t′ ⋆ u′ : T [u/x] In case of relevant functions (⋆ = :) we obtain the usual dependently-typed application rule of equality. Otherwise, we get: Γ ⊢ t = t′ : (x÷U ) → T Γ÷ ⊢ u : U Γ ⊢ t ÷ u = t′ ÷ u′ : T [u/x] Γ ÷ ⊢ u′ : U Note that the arguments u and u′ to the irrelevant functions need to be well-typed but not related to each other. This makes precise the intuition that t and t′ are constant functions. 2.2. Simple properties of IITT. In the following, we prove two basic invariants of derivable IITT-judgements: The context is always well-formed, and judgements remain derivable under well-formed context extensions (weakening). Lemma 2.1 (Context well-formedness). (1) If ⊢ Γ. x : U. Γ′ then Γ ⊢ U . (2) If Γ ⊢ t : T or Γ ⊢ t = t′ : T then ⊢ Γ. Proof. By a simple induction on the derivations. 2This is why Reed [Ree02] differs from Pfenning. 8 A. ABEL AND G. SCHERER ⊢Γ Context well-formedness. ⊢Γ Γ ⊢T ⊢ Γ. x⋆T ⊢⋄ Γ ⊢t:T Typing. ⊢Γ (s, s′ ) Γ ⊢ s : s′ ⊢Γ Γ ⊢ U : s1 Γ. x⋆U ⊢ T : s2 (s1 , s2 , s3 ) s1 ,s2 Γ ⊢ (x⋆U ) → T : s3 (x : U ) ∈ Γ Γ ⊢x:U s,s′ Γ. x⋆U ⊢ t : T Γ ⊢ (x⋆U ) → T s,s′ Γ ⊢ λx⋆U. t : (x⋆U ) → T Γ ⊢ t : (x⋆U ) → T Γ ⊢u⋆U ⋆ Γ ⊢ t u : T [u/x] Γ ⊢t:T Γ ⊢ T = T′ Γ′ ⊢ t : T ′ Γ ⊢ t = t′ : T Equality. Computation (β) and extensionality (η). Γ. x⋆U ⊢ t : T Γ ⊢u⋆U ⋆ Γ ⊢ (λx⋆U. t) u = t[u/x] : T [u/x] s,s′ Γ ⊢ t : (x⋆U ) → T s,s′ Γ ⊢ t = λx⋆U. t ⋆ x : (x⋆U ) → T Equivalence rules. Γ ⊢t:T Γ ⊢t=t:T Compatibility rules. Γ ⊢ t = t′ : T Γ ⊢ t′ = t : T Γ ⊢ U = U ′ : s1 s1 ,s2 Γ ⊢ t1 = t2 : T Γ ⊢ t2 = t3 : T Γ ⊢ t1 = t3 : T Γ. x⋆U ⊢ T = T ′ : s2 s1 ,s2 Γ ⊢ (x⋆U ) → T = (x⋆U ′ ) → T ′ : s3 Γ ⊢ U = U ′ : s1 Γ. x⋆U ⊢ T : s2 (s1 , s2 , s3 ) Γ. x⋆U ⊢ t = t′ : T s1 ,s2 Γ ⊢ λx⋆U. t = λx⋆U ′ . t′ : (x⋆U ) → T Γ ⊢ t = t′ : (x⋆U ) → T Γ ⊢ u = u′ ⋆ U Γ ⊢ t ⋆ u = t′ ⋆ u′ : T [u/x] Conversion rule. Γ ⊢ t = t′ : T Γ ⊢ T = T′ Γ ⊢ t = t′ : T ′ Figure 1: Rules of IITT IRRELEVANCE IN TYPE THEORY 9 It should be noted that we only prove the most basic well-formedness statements here. One would expect that Γ ⊢ t : T or Γ ⊢ t = t′ : T also implies Γ ⊢ T , or that Γ ⊢ t = t′ : T implies Γ ⊢ t : T . This is true—and we will refer to these implications as syntactic validity—but this cannot be proven without treatment of substitution, due to the typing rule for application, which requires substitution in the type, and due to the equality rule for a β-redex, which uses substitution in both term and type. Therefore, syntactic validity is delayed until Section 4 (Corollary 4.17), where substitution will be handled by semantic, rather than syntactic, methods. Weakening. We can weaken a context Γ by adding bindings or making irrelevant bindings relevant. Formally, we have an order on binding annotations, which is the order induced by : ≤ ÷, and we define weakening by monotonic extension. A well-formed context ⊢ ∆ extends a well-formed context ⊢ Γ, written ∆ ≤ Γ, if and only if: ∀x ∈ dom(Γ), (x ⋆1 U ) ∈ Γ =⇒ (x ⋆2 U ) ∈ ∆ with ⋆1 ≤ ⋆2 . Note that this allows to insert new bindings or relax existing ones at any position in Γ, not just at the end. Lemma 2.2 (Weakening). Let ∆ ≤ Γ. (1) If ⊢ Γ.Γ′ and dom(∆) ∩ dom(Γ′ ) = ∅ then ⊢ ∆.Γ′ . (2) If Γ ⊢ t : T then ∆ ⊢ t : T . (3) If Γ ⊢ t = t′ : T ′ then ∆ ⊢ t = t′ : T . Proof. Simultaneously by induction on the derivation. Let us look at some cases: Case ⊢Γ (s, s′ ) Γ ⊢ s = s : s′ By assumption ⊢ ∆, thus ∆ ⊢ s = s : s′ . Case (x : U ) ∈ Γ ⊢ Γ Γ ⊢x=x:U Since ∆ ≤ Γ we have (x : U ) ∈ ∆, thus ∆ ⊢ x = x : U . Case Γ ⊢ U = U ′ : s1 Γ. x⋆U ⊢ T : s2 Γ. x⋆U ⊢ t = t′ : T s1 ,s2 Γ ⊢ λx⋆U. t = λx⋆U ′ . t′ : (x⋆U ) → T W. l. o. g., x 6∈ dom(∆). By (1) and definition of context weakening, ∆ ≤ Γ implies ∆. x⋆U ≤ Γ. x⋆U , so all premises can be appropriately weakened by induction hypothesis. 2.3. Examples. Example 2.3 (Relevance of types). sionality principle. ⊢Γ Γ ⊢ 1 : Seti 3 We can extend IITT by a unit type 1 with exten- ⊢Γ Γ ⊢ () : 1 3Example suggested by a reviewer of this paper. Γ ⊢t:1 Γ ⊢ t′ : 1 Γ ⊢ t = t′ : 1 10 A. ABEL AND G. SCHERER Typed equality allows us to equate all inhabitants of the unit type. As a consequence, the Church numerals over the unit type all coincide, e. g., Γ ⊢ λf : 1 → 1. λx : 1. x = λf : 1 → 1. λx : 1. f x : (1 → 1) → 1 → 1. In systems with untyped equality, like ICC∗ and EPTS, these terms erase to untyped Church-numerals λf λx.x and λf λx. f x and are necessarily distinguished. If we trade the unit type for Bool or any other type with more than one inhabitant, the two terms become different in IITT. This means that in IITT, types are relevant, and we need to reject irrelevant quantification over types like in (X÷Set0 ) → (X → X) → X → X. In IITT, the polymorphic types of Church numerals are (X : Seti ) → (X → X) → X → X. Example 2.4 (Σ-types). IITT can be readily extended by weak Σ-types. Γ ⊢ U : s1 Γ. x⋆U ⊢ T : s2 (s1 , s2 , s3 ) Γ ⊢ (x⋆U ) × T : s3 Γ ⊢ u⋆U Γ ⊢ t : T [u/x] Γ ⊢ (x⋆U ) × T Γ ⊢ (u, t) : (x⋆U ) × T Γ ⊢ p : (x⋆U ) × T Γ. x⋆U. y : T ⊢ v : V Γ ⊢ let (x, y) = p in v : V Γ ⊢u⋆U Γ ⊢ t : T [u/x] Γ. x⋆U. y : T ⊢ v : V Γ ⊢ (x⋆U ) × T Γ ⊢ (let (x, y) = (u, t) in v) = v[u/x][t/y] : V Additional laws for equality could be considered, like commuting conversions, or the identity (let (x, y) = p in (x, y)) = p. The relevant form (x : U ) × T admits a strong version with projections fst and snd and full extensionality p = (fst p, snd p) : (x : U ) × T . However, strong irrelevant Σ-types (x÷U ) × T are problematic because of the first projection: Γ ⊢ p : (x÷U ) × T Γ ⊢ fst p ÷ U With our definition of Γ ⊢ u ÷ U as Γ÷ ⊢ u : U , this rule is misbehaved: it allows us get hold of an irrelevant value in a relevant context. We could define a closed function π1 : (x÷U ) × 1 → U , and composing it with ( , ()) : (x÷U ) → (x÷U ) × 1 would give us an identity function of type (x÷U ) → U which magically makes irrelevant things relevant and IITT inconsistent. In this article, we will not further consider strong Σ-types with irrelevant components; we leave the in-depth investigation to future work. Example 2.5 (Squash type). The squash type \|\|T \|\| was first introduced in the context of NuPRL [CAB+ 86]; it contains exactly one inhabitant iff T is inhabited. Semantically, one obtains \|\|T \|\| from T by equating all of T ’s inhabitants. In IITT, we can define \|\|T \|\| as internalization of the irrelevance modality, as already suggested by Pfenning [Pfe01]. The IRRELEVANCE IN TYPE THEORY 11 first alternative is via the weak irrelevant Σ-type. \|\| \|\| : Seti → Seti \|\|T \|\| := ( ÷T ) × 1 [] [x] : (x÷T ) → \|\|T \|\| := (x, ()) sqelim (T : Seti ) (P : \|\|T \|\| → Setj ) (f : (x÷T ) → P [x]) (t : \|\|T \|\|) : Pt := let (x, ) = t in f ÷ x It is not hard to see that \|\| \|\| is a monad. All canonical inhabitants of \|\|T \|\| are definitionally equal: Γ ⊢ t, t′ ÷ T Γ ⊢ [t] = [t′ ] : \|\|T \|\| This is easily shown by expanding the definition of [ ] and using the congruence rule for pairs with an irrelevant first component. However, we cannot show that all inhabitants of \|\|T \|\| are definitionally equal, because of the missing extensionality principles for weak Σ. Thus, the second alternative is to add the squash type to IITT via the rules: Γ ⊢ T : Seti Γ ⊢ \|\|T \|\| : Seti Γ ⊢t÷T Γ ⊢ [t] : \|\|T \|\| Γ ⊢ t, t′ : \|\|T \|\| Γ ⊢ t = t′ : \|\|T \|\| Γ ⊢ t : \|\|T \|\| Γ. x÷T ⊢ v : V Γ ⊢ let [x] = t in v : V Γ ⊢t÷T Γ. x÷T ⊢ v : V Γ ⊢ (let [x] = [t] in v) = v[t/x] : V Our model (Section 4) is ready to interpret these rules, as well as normalization-by-evaluation inspired models [ACP11]. Example 2.6 (Subset type). The subset type {x : U \| T } is definable from Σ and squash as (x : U ) × \|\|T \|\|. To discuss the next example, we consider a further extension of IITT by Leibniz equality and natural numbers: a ≡ b : Seti for A : Seti and a, b : A refl : a ≡ a for A : Seti and a : A Nat : Seti 0, 1, . . . : Nat +, ∗ : Nat → Nat → Nat. Example 2.7 (Composite). 4 Let the set of composite numbers {4, 6, 8, 9, 10, 12, 14, 15, . . . } be numbers that are the product of two natural numbers ≥ 2. Composite = {n : Nat \| (k : Nat) × (l : Nat) × (n ≡ (k + 2) ∗ (l + 2))} Most composite numbers have several factorizations, and thanks to irrelevance the specific composition is ignored when handling composite numbers. For instance, 12 as product of 3 and 4 is not distinguished from the 12 as product of 2 and 6. (12, [(1, (2, refl))]) = (12, [(0, (4, refl))]) : Composite. 4Example suggested by reviewer. 12 A. ABEL AND G. SCHERER Example 2.8 (Large eliminations). 5 The ICC∗ [BB08] or EPTS [MLS08] irrelevant function type (x ÷ A) → B allows x to appear relevantly in B. This extra power raises some issues with large eliminations. Consider T : Bool → Set0 T true = Bool → Bool T false = Bool t = λF : (b÷Bool) → (T b → T b) → Set0 . λg : (F ÷ false (λx : Bool. x)) → Bool. λa : F ÷ true (λx : Bool → Bool.λy : Bool. x y). g a. The term t is well-typed in ICC∗ + T because the domain type of g and the type of a are βη-equal after erasure (−)∗ of type annotations and irrelevant arguments: (F ÷ false (λx : Bool. x))∗ = F (λxx) =βη F (λxλy. x y) = (F ÷ true (λx : Bool → Bool.λy : Bool. x y))∗ While a Curry view supports this, it is questionable whether identity functions at different types should be viewed as one. It is unclear how a type-directed equality algorithm (see Sec. 3) should proceed here; it needs to recognize that x : Bool is equal to λy : Bool. x y : Bool → Bool. This situation is amplified by a unit type 1 with extensional equality. When we change T true to 1 and the type of a to F ÷ true (λx : 1. ()) then t should still type-check, because λx. () is the identity function on 1. However, η-equality for 1 cannot be checked without types, and a type-directed algorithm would end up checking (successfully) x : Bool for equality with () : 1. This algorithmic equality cannot be transitive, because then any two booleans would be equal. Summarizing, we may conclude that the type of F bears trouble and needs to be rejected. IITT does this because it forbids the irrelevant b in relevant positions such as T b; ICC∗ lacks T altogether. Extensions of ICC∗ should at least make sure that b is never eliminated, such as in T b. Technically, T would have to be put in a separate class of recursive functions, those that actually compute with their argument. We leave the interaction of the three different function types to future research. 3. Algorithmic Equality The algorithm for checking equality in IITT is inspired by Harper and Pfenning [HP05]. Like theirs, it is type-directed, but we are using the full dependent type and not an erasure to simple types (which would anyway not work due to large eliminations). We give the algorithm in form of judgements and rules in direct correspondence to a functional program. Algorithmic equality is meant to be used as part of a type checking algorithm. It is the algorithmic counterpart of the definitional conversion rule; in particular, it will only be called on terms that are already know to be well-typed – in fact, types that are well-sorted. We rely on this precondition in the algorithmic formulation. Algorithmic equality consists of three interleaved judgements. A type equality test checks equality between two types, by inspecting their weak head normal forms. Terms found inside dependent types are reduced and the resulting neutral terms are compared by structural equality. The head variable of such neutrals provides type information that is 5Inspired by discussions with Ulf Norell during the 11th Agda Implementers’ Meeting. IRRELEVANCE IN TYPE THEORY 13 then used to check the (non-normal) arguments using type-directed equality, by reasoning on the (normalized) type structure to perform η-expansions on product types. After enough expansions, a base type is reached, where structural equality is called again, or a sort, at which we use type equality. Informally, the interleaved reductions are the algorithmic counterparts of the β-equality axiom, the type and structural equalities account for the compatibility rules, and typedirected equality corresponds to the η-equality axiom. The remaining equivalence rules are emergent global properties of the algorithm. Weak head reduction. Weak head normal forms (whnfs) are given by the following grammar: Whnf ∋ a, b, f, A, B, F Wne ∋ n, N s,s′ ::= s \| (x⋆U ) → T \| λx⋆U. t \| n ::= x \| n ⋆ u whnf neutral whnf Weak head evaluation t ց a and active application f @⋆ u ց a are functional relations given by the following rules. tցf f @⋆ u ց a t ⋆u ց a aցa t[u/x] ց a (λx⋆U. t) @⋆ u ց a n @⋆ u ց n ⋆ u Instead of writing the propositions t ց a and P [a] we will sometimes simply write P [↓t]. Similarly, we might write P [f @⋆ u] instead of f @⋆ u ց a and P [a]. In rules, it is understood that the evaluation judgement is always an extra premise, never an extra conclusion. Algorithmic equality is given as type equality, structural equality, and type-directed equality, which are mutually recursive. The equality algorithm is only invoked on wellformed expressions of the correct type. Type equality. Type equality ∆ ⊢ A ⇐⇒ A′ , for weak head normal forms, and ∆ ⊢ T ⇐⇒ b ′ T , for arbitrary well-formed types, checks that two given types are equal in their respective contexts. ∆ ⊢ N ←→ b N′ : T ∆ ⊢ ↓T ⇐⇒ ↓T ′ ∆ ⊢ T ⇐⇒ b T′ ∆ ⊢ N ⇐⇒ N ′ ∆ ⊢ U ⇐⇒ b U′ ∆ ⊢ s ⇐⇒ s ∆. x : U ⊢ T ⇐⇒ b T′ s,s′ s,s′ ∆ ⊢ (x⋆U ) → T ⇐⇒ (x⋆U ′ ) → T ′ Note that when invoking structural equality on neutral types N and N ′ , we do not care which type T is returned, since we know by well-formedness that N and N ′ must have the same sort. Structural equality. Structural equality ∆ ⊢ n ←→ n′ : A and ∆ ⊢ n ←→ b n′ : T checks the neutral expressions n and n′ for equality and at the same time infers their type, which is returned as output. ∆ ⊢ n ←→ b n′ : T ∆ ⊢ n ←→ n′ : ↓T (x : T ) ∈ ∆ ∆ ⊢ x ←→ b x:T ∆ ⊢ n ←→ n′ : (x : U ) → T ∆ ⊢ u ⇐⇒ b u′ : U ∆ ⊢ n u ←→ b n′ u′ : T [u/x] ∆ ⊢ n ←→ n′ : (x÷U ) → T ∆ ⊢ n ÷ u ←→ b n′ ÷ u′ : T [u/x] 14 A. ABEL AND G. SCHERER Type-directed equality. Type-directed equality ∆ ⊢ t ⇐⇒ t′ : A and ∆ ⊢ t ⇐⇒ b t′ : T checks ′ terms t and t for equality and proceeds by the structure of the supplied type, to account for η. ∆ ⊢ t ⇐⇒ t′ : ↓T ∆ ⊢ t ⇐⇒ b t′ : T ∆. x⋆U ⊢ t ⋆ x ⇐⇒ b t′ ⋆ x : T ∆ ⊢ t ⇐⇒ t′ : (x⋆U ) → T ∆ ⊢ T ⇐⇒ b T′ ∆ ⊢ ↓t ←→ b ↓t′ : T ∆ ⊢ T ⇐⇒ T ′ : s ∆ ⊢ t ⇐⇒ t′ : N Note that in the but-last rule we do not check that the inferred type T of ↓t equals the ascribed type N . Since algorithmic equality is only invoked for well-typed t, we know that this must always be the case. Skipping this test is a conceptually important improvement over Harper and Pfenning [HP05]. Due to dependent typing, it is not obvious that algorithmic equality is symmetric and transitive. For instance, consider symmetry in case of application: We have to show that ∆ ⊢ n′ u′ ←→ b n u : T [u/x], but using the induction hypothesis we obtain this equality only at type T [u′ /x]. To conclude, we need to convert types, which is only valid if we know that u and u′ are actually equal. Thus, we need soundness of algorithmic equality to show its transitivity. Soundness w. r. t. declarative equality requires subject reduction, which is not trivial, due to its dependency on function type injectivity. In the next section (4), we construct by a Kripke logical relation which gives us subject reduction and soundness of algorithmic equality (Section 5), and, finally, symmetry and transitivity of algorithmic equality. A simple fact about algorithmic equality is that the inferred types are unique up to syntactic equality (where we consider α-convertible expressions as identical). Also, they only depend on the left hand side neutral term n. Lemma 3.1 (Uniqueness of inferred types). (1) If ∆ ⊢ n ←→ n1 : A1 and ∆ ⊢ n ←→ n2 : A2 then A1 ≡ A2 . (2) If ∆ ⊢ n ←→ b n1 : T1 and ∆ ⊢ n ←→ b n2 : T2 then T1 ≡ T2 . Extending structural equality to irrelevance, we let ∆÷ ⊢ n ←→ n : A ∆÷ ⊢ n′ ←→ n′ : A ∆ ⊢ n ←→ n′ ÷ A ′ and analogously for ∆ ⊢ n ←→ b n ÷ T. 4. A Kripke Logical Relation for Soundness In this section, we construct a Kripke logical relation in the spirit of Goguen [Gog00] and Vanderwaart and Crary [VC02] that proves weak head normalization, function type injectivity, and subject reduction plus syntactical properties like substitution in judgements and syntactical validity. As an important consequence, we obtain soundness of algorithmic equality w. r. t. definitional equality. This allows us to establish that algorithmic equality on well-typed terms is a partial equivalence relation. IRRELEVANCE IN TYPE THEORY 15 4.1. An Induction Measure. Following Goguen [Gog94] and previous work [ACD08], we first define a semantic universe hierarchy Ui whose sole purpose is to provide a measure for defining a logical relation and proving some of its properties. The limit Uω corresponds to the proof-theoretic strength or ordinal of IITT. We denote sets of expressions by A, B and functions from expressions to sets of expressions by F. Let Ab = {t \| ↓t ∈ A} denote the closure of A by weak head expansion. The b f @ u ∈ F(u)}. dependent function space is defined as Π A F = {f ∈ Whnf \| ∀u ∈ A. By recursion on i ∈ N we define inductively sets Ui ⊆ Whnf × P(Whnf) as follows [ACD08, Sec. 5.1]: (N, Wne) ∈ Ui (Setj , Seti ) ∈ Axiom (Setj , \|Uj \|) ∈ Ui cj b (T [u/x], F(u)) ∈ U (U, A) ∈ Ubi ∀u ∈ A. (Seti , Setj , Setk ) ∈ Rule ((x⋆U ) → T, Π A F) ∈ Uk Herein, Ubi = {(T, A) \| (↓T, A) ∈ Ui } and \|Uj \| = {A \| (A, A) ∈ Uj for some A}. Only interested in computational strength, we treat relevant and irrelevant function spaces alike— at the level of predicates A, irrelevance is anyhow not observable, only by relations as given later. The induction measure A ∈ Seti shall now mean the minimum height of a derivation of (A, A) ∈ Ui for some A. Note that due to universe stratification, A ∈ Seti is smaller than Seti ∈ Setj . 4.2. A Kripke Logical Relation. Let ∆ ⊢ t :=: t′ ⋆ T stand for the conjunction of the propositions • ∆ ⊢ t ⋆ T and ∆ ⊢ t′ ⋆ T , and • ∆ ⊢ t = t′ ⋆ T . By induction on A ∈ s we define two Kripke relations ∆ ⊢ A s A′ : s ∆ ⊢ a s a′ : A. b and the generalization to ⋆. For better readability, together with their respective closures s the clauses are given in rule form meaning that the conclusion is defined as the conjunction of the premises. ∀ and =⇒ are meta-level quantification and implication, respectively. ∆ ⊢ N :=: N ′ : s ∆ ⊢ N s N′ : s ∆ ⊢ n :=: n′ : N ∆ ⊢ n s n′ : N ⊢∆ (s, s′ ) ∆ ⊢ s s s : s′ b U ′ : s1 ∆ ⊢U s b u′ ⋆ U =⇒ Γ ⊢ T [u/x] s b T ′ [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s ,s 1 2 ∆ ⊢ (x⋆U ) → T :=: (x⋆U ′ ) → T ′ : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T s (x⋆U ′ ) → T ′ : s3 (s1 , s2 , s3 ) b u′ ⋆ U =⇒ Γ ⊢ f ⋆ u s b f ′ ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s s,s′ ∆ ⊢ f :=: f ′ : (x⋆U ) → T s,s′ ∆ ⊢ f s f ′ : (x⋆U ) → T 16 A. ABEL AND G. SCHERER tցa T ցA ∆ ⊢T =A ∆ ⊢t=a:A ∆ ⊢ t′ = a′ : A ∆ ⊢ a s a′ : A ∆ ⊢ t :=: t′ : T b t′ : T ∆ ⊢ts ∆÷ ⊢ a s a : A ∆÷ ⊢ a′ s a′ : A ∆ ⊢ a s a′ ÷ A b t:T ∆÷ ⊢ t s t ′ ց a′ b t′ : T ∆ ÷ ⊢ t′ s b t′ ÷ T ∆ ⊢ts It is immediate that the logical relation contains only well-typed and definitionally equal terms. We will demonstrate that it is also closed under weakening and conversion, symmetric and transitive. Lemma 4.1 (Weakening). (1) If ∆ ⊢ a s a′ : A and Γ ≤ ∆ then there exists a derivation of Γ ⊢ a s a′ : A with the same height. b t′ : T . (2) Analogously for ∆ ⊢ t s Proof. By induction on A ∈ s and T ∈ s, resp. Lemma 4.2 (Type conversion). (1) If Γ ⊢ A s A′ : s then Γ ⊢ a s a′ : A iff Γ ⊢ a s a′ : A′ . b T ′ : s then Γ ⊢ t s b t′ : T iff Γ ⊢ t s b t′ : T ′ . (2) If Γ ⊢ T s Proof. Simultaneously induction in A ∈ s and T ∈ s, resp. We show the “if” direction, the “only if” follows analogously. The interesting case is the one of functions. Case b U ′ : s1 ∆ ⊢U s b u′ ⋆ U =⇒ Γ ⊢ T [u/x] s b T ′ [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T :=: (x⋆U ′ ) → T ′ : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T s (x⋆U ′ ) → T ′ : s3 b u′ ⋆ U =⇒ Γ ⊢ f ⋆ u s b f ′ ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s s,s′ ∆ ⊢ f :=: f ′ : (x⋆U ) → T s,s′ ∆ ⊢ f s f ′ : (x⋆U ) → T s,s′ First, ∆ ⊢ f :=: f ′ : (x⋆U ′ ) → T ′ , holds because of the conversion rule for typing and b u′ ⋆ U ′ and show Γ ⊢ f ⋆ u s b equality. Now assume arbitrary Γ ≤ ∆ and Γ ⊢ u s ′ ⋆ ′ ′ ′ b u ⋆ U , thus, f u : T [u/x]. By induction hypothesis on U ∈ s1 we have Γ ⊢ u s ⋆ ′ ⋆ ′ b Γ ⊢ f u s f u : T [u/x] by assumption. By induction hypothesis on T [u/x] ∈ s2 we b f ′ ⋆ u′ : T ′ [u/x]. obtain Γ ⊢ f ⋆ u s b T : s. Lemma 4.3 (Symmetry and Transitivity). Let ∆ ⊢ T s b t′ : T then ∆ ⊢ t′ s b t : T. (1) If ∆ ⊢ t s b t2 : T and ∆ ⊢ t2 s b t3 : T then ∆ ⊢ t1 s b t3 : T . (2) If ∆ ⊢ t1 s IRRELEVANCE IN TYPE THEORY 17 Proof. We generalize the two statements to whnfs ∆ ⊢ A s A : s and prove all four statements simultaneously by induction in A ∈ s and T ∈ s, resp. Case Let us look at the case for functions. b U : s1 ∆ ⊢U s b u′ ⋆ U =⇒ Γ ⊢ T [u/x] s b T [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T :=: (x⋆U ) → T : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T s (x⋆U ) → T : s3 Case Symmetry: b u′ ⋆ U =⇒ Γ ⊢ f ⋆ u s b f ′ ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s ′ ∆ ⊢ f :=: f : (x⋆U ) → T ∆ ⊢ f s f ′ : (x⋆U ) → T b u⋆U To show ∆ ⊢ f ′ s f : (x⋆U ) → T , assume arbitrary Γ ≤ ∆ and Γ ⊢ u′ s ′ ⋆ ′ ⋆ ′ b and show Γ ⊢ f u s f u : T [u /x]. By induction hypothesis on U ∈ s2 , with b U : s1 , we have Γ ⊢ u s b u′ ⋆ U , thus, Γ ⊢ f ⋆ u s b f ′ ⋆ u′ : T [u/x] weakened Γ ⊢ U s b u ⋆ U, by assumption. Using symmetry and transitivity on U we obtain Γ ⊢ u s b thus, Γ ⊢ T [u/x] s T [u/x] : s2 . By induction hypothesis on T [u/x] ∈ s2 we apply b f ⋆ u : T [u/x], and since Γ ⊢ T [u/x] s b T [u′ /x] : s2 symmetry to obtain Γ ⊢ f ′ ⋆ u′ s we conclude by type conversion (Lemma 4.2). Case Transitivity: b u′ ⋆ U =⇒ Γ ⊢ f1 ⋆ u s b f2 ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s s,s′ ∆ ⊢ f1 :=: f2 : (x⋆U ) → T s,s′ ∆ ⊢ f1 s f2 : (x⋆U ) → T b u′ ⋆ U =⇒ Γ ⊢ f2 ⋆ u s b f3 ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s s,s′ ∆ ⊢ f2 :=: f3 : (x⋆U ) → T s,s′ ∆ ⊢ f2 s f3 : (x⋆U ) → T s,s′ s,s′ We wish to prove that ∆ ⊢ f1 s f3 : (x⋆U ) → T . We get ∆ ⊢ f1 :=: f3 : (x⋆U ) → T b u′ ⋆ U , immediately by transitivity of definitional equality. Given Γ ≤ ∆ and Γ ⊢ u s b f3 ⋆ u′ : T [u/x]. we need to show that Γ ⊢ f1 ⋆ u s b u ⋆ U , which b : U is a PER by induction hypothesis, we have Γ ⊢ u s As Γ ⊢ s ⋆ ⋆ ′ b b b f 3 ⋆ u′ : entails f1 u s f2 u : T [u/x]. From Γ ⊢ u s u ⋆ U also have Γ ⊢ f2 ⋆ u s b f3 ⋆ u′ : T [u/x] by transitivity at T [u/x]. T [u/x], which allows to conclude Γ ⊢ f1 ⋆ u s Case Now, we consider function spaces: 18 A. ABEL AND G. SCHERER Case Transitivity: b U2 : s 1 ∆ ⊢ U1 s b u′ ⋆ U1 =⇒ Γ ⊢ T1 [u/x] s b T2 [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U1 ) → T1 :=: (x⋆U2 ) → T2 : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U1 ) → T1 s (x⋆U2 ) → T2 : s3 b U3 : s 1 ∆ ⊢ U2 s b u′ ⋆ U2 =⇒ Γ ⊢ T2 [u/x] s b T3 [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U2 ) → T2 :=: (x⋆U3 ) → T3 : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U2 ) → T2 s (x⋆U3 ) → T3 : s3 b U3 : s1 By transitivity we have ∆ ⊢ (x⋆U1 ) → T1 :=: (x⋆U3 ) → T3 : s3 and ∆ ⊢ U1 s by induction hypothesis on s1 . Note that this is where the arrow sort annotations are useful. Without them we would b U2 : s1 not know that the sorts in both derivations are equal. We could have ∆ ⊢ U1 s ′ ′ b and ∆ ⊢ U2 s U3 : s1 for apparently unrelated s1 and s1 , and would therefore be unable to use transitivity. b u′ ⋆ U1 , we need to show that Γ ⊢ T1 [u/x] s b T3 [u′ /x] : s3 . Given Γ ≤ ∆ and Γ ⊢ u s b at type U is a PER by induction hypothesis, we have Γ ⊢ u s b u⋆U1 , from which As s b b U2 : s1 – we can deduce Γ ⊢ T1 [u/x] s T2 [u/x] : s2 . By conversion using ∆ ⊢ U1 s ′ b u ⋆ U2 , which implies Γ ⊢ T2 [u/x] s b T3 [u′ /x] : s2 . weakened at Γ – we have Γ ⊢ u s This allows us to conclude by transitivity at type s2 . In the following we show that the variables are in the logical relation, i. e., ∆ ⊢ x s x : ∆(x) for well-formed contexts ∆. As usual, this statement has to be generalized to neutrals n to be proven inductively. b n′ ⋆T . Lemma 4.4 (Into the logical relation). Let T ∈ s. If ∆ ⊢ n :=: n′ ⋆T then ∆ ⊢ n s Proof. By induction on T ∈ s. Case N ∈ s and ∆ ⊢ n :=: n′ ⋆ N . Then ∆ ⊢ n s n′ ⋆ N by cases on ⋆, unfolding definitions. Case s ∈ s′ and ∆ ⊢ N :=: N ′ ⋆ s. Then ∆ ⊢ N s N ′ ⋆ s by cases on ⋆. Case (x⋆U ) → T ∈ s3 and ∆ ⊢ n :=: n′ ⋆0 (x⋆U ) → T . First, the case for ⋆0 = :. We have ∆ ⊢ n :=: n′ : (x⋆U ) → T . Assume arbitrary b u′ ⋆ U , which yields Γ ⊢ u :=: u′ ⋆ U and Γ ⊢ T [u/x] s b T [u/x] : s2 . Γ ≤ ∆ and Γ ⊢ u s ⋆ ′ ⋆ ′ By weakening, Γ ⊢ n u :=: n u : T [u/x], thus, by induction hypothesis, Γ ⊢ n ⋆ u s n′ ⋆ u′ : T [u/x], q.e.d. The case for ⋆0 = ÷ proceeds analogously. IRRELEVANCE IN TYPE THEORY 19 b to substitutions, by 4.3. Validity in the Model. We now extend our logical relation s induction on the destination context. b σ′ : Γ b σ ′ (x) ⋆ U σ ∆ ⊢σs ∆ ⊢ σ(x) s b σ′ : ⋄ b σ ′ : Γ. x⋆U ∆ ⊢σs ∆ ⊢σs b for terms. This relation inherits weakening from s We then define the context ( Γ), type (Γ T = T ′ ) and term (Γ relations, by induction on the length of contexts. Γ ⋄ Γ U Γ. x⋆U Γ T = T′ : s Γ T = T′ Γ t = t′ : T ) validity T =T Γ T Γ (Γ T unless T = s) b b t′ σ ′ : T σ ∆ ⊢ σ s σ ′ : Γ =⇒ ∆ ⊢ tσ s Γ t=t:T ′ Γ t=t :T Γ t:T Because of its asymmetric definition, the logical relation on substitutions may not be a PER in general, but it is for valid contexts. b : Γ is symmetric Lemma 4.5 (Substitution relation is a PER). If Γ, then ∆ ⊢ s and transitive. ∀∆, σ, σ ′ , Proof. By induction on Γ. We demonstrate symmetry for the case b σ′ : Γ b σ ′ (x) ⋆ U σ ∆ ⊢σs ∆ ⊢ σ(x) s Γ. x⋆U . b σ ′ : Γ. x⋆U ∆ ⊢σs b σ : Γ, and by symmetry of s b for terms (Lemma 4.3), By induction hypothesis, ∆ ⊢ σ ′ s ′ b σ(x) ⋆ U σ. We instantiate Γ b U σ ′ : s and conclude ∆ ⊢ σ (x) s U to ∆ ⊢ U σ s b σ(x) ⋆ U σ ′ by conversion (Lemma 4.2). ∆ ⊢ σ ′ (x) s = : T is symmetric and transitive. Lemma 4.6 (Validity is a PER). The relation Γ b for substitutions and conversion with ∆ ⊢ T σ s b Proof. Symmetry requires symmetry of s ′ ′ T σ : s , similar as in Lemma 4.5. We demonstrate transitivity in detail. Given Γ t1 = t2 : T and Γ t2 = t3 : T we show Γ t1 = t3 : T . Clearly, Γ and Γ T or T = s by one of our two assumptions. b σ ′ : Γ and show ∆ ⊢ t1 σ s b t3 σ ′ : T σ. By Lemma 4.5, Assume arbitrary ∆ ⊢ σ s b σ : Γ, thus ∆ ⊢ t1 σ s b t2 σ : T σ. Also, ∆ ⊢ t2 σ s b t3 σ ′ : T σ which entails our goal ∆ ⊢σs b by transitivity of s (Lemma 4.3). s1 ,s2 s′ ,s′ 1 2 Lemma 4.7 (Function type injectivity is valid). If Γ (x⋆U ) → T = (x⋆U ′ ) → T′ ′ ′ ′ ′ ′ T = T : s2 . then s1 = s1 and s2 = s2 and Γ U = U : s1 and Γ. x⋆U ′ ′ 1 ,s2 1 ,s2 b σ ′ : Γ. We have ∆ ⊢ (x⋆U σ) s→ b (x⋆U ′ σ ′ ) s→ Proof. Assume arbitrary ∆ ⊢ σ s Tσ s b U σ : s1 —note that sorts T ′ σ ′ : s3 , thus by definition s1 = s′1 and s2 = s′2 and ∆ ⊢ U ′ σ ′ s b and since ∆, σ, σ ′ are closed and therefore invariant by substitution. By symmetry of s, ′ were arbitrary, we have Γ U = U : s1 . b u′ ⋆ U ′ σ and let ρ = (σ, u/x) and ρ′ = (σ ′ , u′ /x). Further, assume arbitrary ∆ ⊢ u s b ρ′ : Γ. x⋆U ′ . We have Note that w. l. o. g., x 6∈ dom(Γ) and x 6∈ FV(U ′ ) and ∆ ⊢ ρ s b T ′ ρ′ : s2 and since ρ, ρ′ were arbitrary, Γ. x⋆U ′ T = T ′ : s2 . ∆ ⊢ Tρ s 20 A. ABEL AND G. SCHERER b id : Γ. Γ then ⊢ Γ and Γ ⊢ id s Lemma 4.8 (Context satisfiable). If Proof. By induction on Γ. The ⋄ case is immediate. In the Γ. x⋆U case, given Γ Γ U Γ. x⋆U we can use inference b id : Γ Γ. x⋆U ⊢ id s b id(x) ⋆ U id Γ. x⋆U ⊢ id(x) s . b id : Γ. x⋆U Γ. x⋆U ⊢ id s b id : Γ, we obtain the first premise by weakening From the induction hypothesis Γ ⊢ id s b It also yields Γ ⊢ U id : sid for some s by definition of Γ of s. U . Using induction hypothesis, ⊢ Γ, this entails ⊢ Γ. x⋆U . Further, Γ. x⋆U ⊢ x = x ⋆ U , and since trivially b x⋆U , by the Lemma 4.4. This concludes Γ. x⋆U ⊢ x ←→ x⋆U , we can derive Γ. x⋆U ⊢ x s b id(x) ⋆ U id. the second premise Γ. x⋆U ⊢ id(x) s We can now show that every equation valid in the model is derivable in IITT. Theorem 4.9 (Completeness of IITT rules). If Γ t = t′ : T then both Γ ⊢ t : T and ′ ′ Γ ⊢ t : T and Γ ⊢ t = t : T and Γ ⊢ T . b t′ : T , which entails Γ ⊢ t, t′ : T and Γ ⊢ t = Proof. Using Lemma 4.8 we obtain Γ ⊢ t s ′ t : T . Analogously, since our assumption entails Γ T by definition, we get Γ ⊢ T . 4.4. Fundamental theorem. We prove a series of lemmata which constitute parts of the fundamental theorem for the Kripke logical relation. Lemma 4.10 (Resurrection). If b σ ′ : Γ÷ . ∆÷ ⊢ σ ′ s b σ ′ : Γ then ∆÷ ⊢ σ s b σ : Γ÷ and Γ and ∆ ⊢ σ s Proof. By induction on Γ, the interesting case being b σ′ : Γ b σ ′ (x) ⋆ U σ ∆ ⊢σs ∆ ⊢ σ(x) s . b σ ′ : Γ. x⋆U ∆ ⊢σs b σ : (Γ÷ . x : U ). By induction hypothesis ∆÷ ⊢ σ s b σ : Γ÷ , and First, we show ∆÷ ⊢ σ s ÷ b σ(x) : U σ. This immediately entails our goal. by definition, ∆ ⊢ σ(x) s b σ ′ : (Γ÷ . x : U ), observe that Γ U , hence ∆ ⊢ U σ s b For the second goal ∆÷ ⊢ σ ′ s b σ ′ (x) : U σ to U σ ′ U σ ′ : s for some sort s. Thus, we can cast our hypothesis ∆ ⊢ σ ′ (x) s and conclude analogously. Corollary 4.11. If Γ⋆ b σ ′ : Γ then ∆ ⊢ uσ s b uσ ′ ⋆ U σ. u : U and ∆ ⊢ σ s b Proof. In case ⋆ = : it holds by definition, but we need resurrection for ⋆ = ÷. If ∆ ⊢ σ s ′ ÷ ÷ ÷ b σ : Γ , so from Γ σ : Γ, then by resurrection (Lemma 4.10) we have ∆ ⊢ σ s u:U ÷ ÷ ′ ′ ′ b b we deduce ∆ ⊢ uσ s uσ : U σ. Analogously we get ∆ ⊢ uσ s uσ : U σ which we cast b uσ ′ : U σ. to ∆÷ ⊢ uσ ′ s Lemma 4.12 (Validity of β-reduction). Γ. x⋆U t : T Γ⋆ u : U Γ (λx⋆U. t) ⋆ u = t[u/x] : T [u/x] IRRELEVANCE IN TYPE THEORY 21 b ρ′ : Γ Proof. Γ is contained in the first hypothesis Γ u ⋆ U . Then, given ∆ ⊢ ρ s ⋆ ′ ′ b t(ρ , uρ /x) : T (ρ, uρ/x) and also ∆ ⊢ T [u/x]ρ s b we need to show ∆ ⊢ (λx⋆U. t)ρ uρ s ′ T [u/x]ρ : s for some s (the latter to get Γ T [u/x]). Let σ = (ρ, uρ/x) and σ ′ = (ρ′ , uρ′ /x). From the second hypothesis and Cor. 4.11 we get b uρ′ ⋆ U ρ, which gives ∆ ⊢ σ s b σ ′ : Γ. x⋆U . By instantiating the first hypothesis ∆ ⊢ uρ s b tσ ′ : T σ, and also (from the premise Γ. x⋆U T ) ∆ ⊢ T σ = T σ ′ , which we get ∆ ⊢ tσ s gives Γ T [u/x]. b tσ ′ we get the desired ∆ ⊢ (λx⋆U. t)ρ ⋆ uρ s b tσ ′ : T σ, as s b is Finally, from ∆ ⊢ tσ s ⋆ closed by weak head expansion to well-typed ∆ ⊢ (λx⋆U. t)ρ uρ : T σ. Lemma 4.13 (Validity of η). Γ Γ t : (x⋆U ) → T t = λx⋆U. t ⋆ x : (x⋆U ) → T b Proof. Γ and Γ (x⋆U ) → T are direct consequences of our hypothesis. Given ∆ ⊢ ρ s ′ ⋆ ′ b ρ : Γ, we need to show ∆ ⊢ tρ s (λx⋆U. t x)ρ : ((x⋆U ) → T )ρ. W. l. o. g., x is not free in the domain nor range of substitutions ρ and ρ′ , thus with t′ := tρ, t′′ := tρ′ , U ′ := U ρ, U ′′ := b λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ . U ρ′ , T ′ := T ρ and T ′′ := T ρ′ it is sufficient to show ∆ ⊢ t′ s ′ ′ ′ ′ b u′ ⋆ U ′ , we show ∆′ ⊢ t′ ⋆ u s b First, given (∆ , u, u ) such that ∆ ≤ ∆ and ∆ ⊢ u s ′′ ′′ ⋆ ⋆ ′ ′ ′ b (λx⋆U . t x) u : T [u/x]. Our hypothesis Γ t : (x⋆U ) → T entails ∆ ⊢ tρ s tρ : ′ ′′ b ((x⋆U ) → T )ρ, that is to say ∆ ⊢ t s t : (x⋆U ′ ) → T ′ . This logical relation at a b t′′ ⋆ u′ : T ′ [u/x], which function type, when instantiated to (∆′ , u, u′ ), gives us ∆′ ⊢ t′ ⋆ u s weak-head expands to the desired goal. Second, we show ∆ ⊢ t′ :=: λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ . • Γ ⊢ t′ : (x⋆U ′ ) → T ′ is a simple consequence of our hypothesis Γ t : (x⋆U ) → T . • Γ ⊢ λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ has the following proof: Γ t : (x⋆U ) → T =====′′======′′======′′ ∆ ⊢ t : (x⋆U ) → T ∆. x⋆U ′′ ⊢ t′′ : (x⋆U ′′ ) → T ′′ weak (∆. x⋆U ′′ )⋆ ⊢ x : U ′′ var ∆. x⋆U ′′ ⊢ t′′ ⋆ x : T ′′ Γ (x⋆U ) → T ========′′======′′ ∆ ⊢ (x⋆U ) → T ∆ ⊢ λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′′ ) → T ′′ ∆ ⊢ λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ conv • ∆ ⊢ t′ = λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ . The η-rule of definitional equality gives us ∆ ⊢ t′′ = λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′′ ) → T ′′ . From Γ (x⋆U ) → T we can convert it to the type (x⋆U ′ ) → T ′ , and then conclude by transitivity using ∆ ⊢ t′ = t′′ : (x⋆U ′ ) → T ′ , which is a direct consequence of Γ t : (x⋆U ) → T . Lemma 4.14 (Validity of function equality). Γ Γ U = U′ Γ. x⋆U t = t′ : T (λx⋆U. t) = (λx⋆U ′ . t′ ) : (x⋆U ) → T Proof. Again Γ and Γ (x⋆U ) → T are simple consequences of our hypotheses. Given ′ b ∆ ⊢ ρ s ρ : Γ (w. l. o. g., x is not free in ρ, ρ′ domain or range), we need to show 22 A. ABEL AND G. SCHERER b (λx⋆U ′ ρ′ . t′ ρ′ ) : ((x⋆U ρ) → T ρ). We will skip the proof of ∆ ⊢ ∆ ⊢ (λx⋆U ρ. tρ) s (λx⋆U ρ. tρ) :=: (λx⋆U ′ ρ′ . t′ ρ′ ) : ((x⋆U ρ) → T ρ), as it is similar to the corresponding part of the η-validity lemma. b u′ ⋆ U ρ, we have to show that Given (∆′ , u, u′ ) such that ∆′ ≤ ∆ and ∆′ ⊢ u s ′ ⋆ ′ ′ ′ ′ ⋆ ′ b (λx⋆U ρ . t ρ ) u : T ρ[u/x]. Let σ = (ρ, u/x) and σ ′ = (ρ′ , u′ /x). ∆ ⊢ (λx⋆U ρ. tρ) u s b u′ ⋆ U ρ, we have ∆′ ⊢ σ s b σ ′ : Γ. x⋆U ρ. Instantiating the second As we supposed ∆′ ⊢ u s ′ ′ ′ hypothesis with ∆ , σ, σ therefore gives us ∆ ⊢ tσ = t′ σ ′ : T σ, which can also be written b t′ ρ′ [u′ /x] : T ρ[u/x], which is weak-head expansible to our goal. ∆′ ⊢ tρ[u/x] s Lemma 4.15 (Validity of irrelevant application). Γ t = t′ : (x÷U ) → T Γ÷ u : U Γ t ÷ u = t′ ÷ u′ : T [u/x] Γ÷ u′ : U b ρ′ : Γ and show ∆ ⊢ tρ ÷ uρ s b t′ ρ′ ÷ u′ ρ′ : T (ρ, uρ/x). Proof. Assume arbitrary ∆ ⊢ ρ s b u′ ρ′ ÷ U ρ, which means ∆÷ ⊢ By the first hypothesis, it is sufficient to show ∆ ⊢ uρ s ÷ ′ ′ ′ ′ b b b uρ s uρ : U ρ and ∆ ⊢ u ρ s u ρ : U ρ. By Resurrection (Lemma 4.10), ∆÷ ⊢ ρ s b uρ : U ρ from the second hypothesis. Analogously, we obtain ρ : Γ÷ , hence ∆÷ ⊢ uρ s ÷ ′ ′ ′ ′ ′ b u ρ : U ρ from the third hypothesis which we can cast to U ρ by virtue of ∆ ⊢ uρ s Γ U which we get from Γ (x÷U ) → T by Lemma 4.7. Theorem 4.16 (Fundamental theorem of logical relations). (1) If ⊢ Γ then Γ. (2) If Γ ⊢ t : T then Γ t : T . (3) If Γ ⊢ t = t′ : T then Γ t = t′ : T . Proof. By induction on the derivation. As a simple corollary we obtain syntactic validity, namely that definitional equality implies well-typedness and well-typedness implies well-formedness of the involved type. This lemma could have been proven purely syntactically, but the syntactic proof requires a sequence of carefully arranged lemmata like context conversion, substitution, functionality, and inversion on types [HP05, AC07]. Our “sledgehammer” semantic argument is built into the Kripke logical relation, in the spirit of Goguen [Gog00]. Corollary 4.17 (Syntactic validity). (1) If Γ ⊢ t : T then Γ ⊢ T . (2) If Γ ⊢ t = t′ : T then Γ ⊢ t : T and Γ ⊢ t′ : T . Proof. By the fundamental theorem, Γ ⊢ t = t′ : T implies Γ Thm. 4.9 implies Γ ⊢ t, t′ : T and Γ ⊢ T . t = t′ : T , which by IRRELEVANCE IN TYPE THEORY 23 5. Meta-theoretic Consequences of the Model Construction In this section, we explicate the results established by the Kripke model. 5.1. Admissibility of Substitution. Goguen [Gog00] observes that admissibility of substitution for the syntactic judgements can be inherited from the Kripke logical relation, which is closed under substitution by its very definition. To show that the judgements of IITT are closed under substitution we introduce relations Γ ⊢ σ : Γ′ for substitution typing and Γ ⊢ σ = σ ′ : Γ′ for substitution equality which are given inductively by the following rules: Γ ⊢ σ : Γ′ ⊢Γ Γ ⊢σ:⋄ Γ′ ⊢ U Γ ⊢ σ(x) ⋆ U σ ′ Γ ⊢ σ : Γ . x⋆U Γ ⊢ σ = σ ′ : Γ′ Γ′ ⊢ U Γ ⊢ σ(x) = σ ′ (x) ⋆ U σ ⊢Γ Γ ⊢ σ = σ′ : ⋄ Γ ⊢ σ = σ ′ : Γ′ . x⋆U Substitution typing and equality are closed under weakening. Semantically, substitutions are explained by environments. We define substitution validity as follows, again in rule form but not inductively: Γ Γ′ b b σ ′ ρ′ : Γ ′ Γ σ=σ: ∀∆ ρ s : Γ. ∆ σρ s Γ σ : Γ′ Γ σ = σ ′ : Γ′ Lemma 5.1 (Fundamental lemma for substitutions). (1) If Γ ⊢ σ : Γ′ then Γ σ : Γ′ . (2) If Γ ⊢ σ = σ ′ : Γ′ then Γ σ = σ ′ : Γ′ . Γ′ ρ′ Proof. We demonstrate 2 by induction on Γ ⊢ σ = σ ′ : Γ′ . Case ⊢Γ Γ ⊢ σ = σ′ : ⋄ b σ ′ ρ′ : ⋄ trivially for any We have Γ by Thm. 4.16 and ⋄ trivially. Also, ∆ ⊢ σρ s ′ b ρ : Γ. ∆ ⊢ρs Case Γ ⊢ σ = σ ′ : Γ′ Γ′ ⊢ U Γ ⊢ σ(x) = σ ′ (x) ⋆ U σ Γ ⊢ σ = σ ′ : Γ′ . x⋆U ′ We have Γ and Γ by induction hypothesis and Γ′ U by Thm. 4.16, thus, ′ b ρ′ : Γ and show ∆ ⊢ σρ s b σ ′ ρ′ : Γ′ . x⋆U . Γ . x⋆U . Now assume arbitrary ∆ ⊢ ρ s b σ ′ ρ′ : Γ′ follows by induction hypothesis. The second subgoal ∆ ⊢ First, ∆ ⊢ σρ s ′ ′ b (σρ)(x) s (σ ρ )(x) ⋆ U σρ is just an instance of the second induction hypothesis. Theorem 5.2 (Substitution and functionality). (1) If Γ ⊢ σ : Γ′ and Γ′ ⊢ t : T then Γ ⊢ tσ : T σ. (2) If Γ ⊢ σ : Γ′ . and Γ′ ⊢ t = t′ : T then Γ ⊢ tσ = t′ σ : T σ. (3) If Γ ⊢ σ = σ ′ : Γ′ . and Γ′ ⊢ t : T then Γ ⊢ tσ = tσ ′ : T σ. (4) If Γ ⊢ σ = σ ′ : Γ′ . and Γ′ ⊢ t = t′ : T then Γ ⊢ tσ = t′ σ ′ : T σ. 24 A. ABEL AND G. SCHERER Proof. We demonstrate 4, the other cases are just variations of the theme. First, from b σ ′ : Γ′ by the fundamental lemma for substitutions Γ ⊢ σ = σ ′ : Γ′ we get Γ ⊢ σ s b id : Γ. Now, by the fundamental (Lemma 5.1), using the identity environment Γ ⊢ id s ′ ′ ′ b t σ : T σ, which entails our goal Γ ⊢ tσ = theorem on Γ ⊢ t = t : T we obtain Γ ⊢ tσ s ′ ′ t σ : T σ by Thm. 4.9. 5.2. Context conversion. Context equality ⊢ Γ = Γ′ is defined inductively by the rules ⊢ Γ = Γ′ Γ ⊢ U = U′ . ⊢⋄=⋄ ⊢ Γ. x⋆U = Γ′ . x⋆U ′ All declarative judgements are closed under context conversion. This fact is easy to prove by induction over derivations, but we get it as just a special case of substitution. Lemma 5.3 (Identity substitution). If ⊢ Γ = Γ′ then Γ ⊢ id = id : Γ′ . Proof. By induction on ⊢ Γ = Γ′ . Case ⊢ Γ = Γ′ Γ ⊢ U = U′ ⊢ Γ. x⋆U = Γ′ . x⋆U ′ By induction hypothesis and weakening, Γ. x⋆U ⊢ id = id : Γ′ . Also, Γ. x⋆U ⊢ x = x ⋆ U and by conversion Γ. x⋆U ⊢ x = x ⋆ U ′ . Together, Γ. x⋆U ⊢ id = id : Γ′ . x⋆U ′ . Theorem 5.4 (Context conversion). Let ⊢ Γ′ = Γ. (1) If Γ ⊢ t : T then Γ′ ⊢ t : T . (2) If Γ ⊢ t = t′ : T then Γ′ ⊢ t = t′ : T . Proof. By Thm. 5.2 with Γ′ ⊢ id = id : Γ. U′ As a consequence, context equality is symmetric and transitive (we can trade Γ ⊢ U = for Γ′ ⊢ U = U ′ ). Thus, context conversion can be applied in the other direction as well. 5.3. Inversion, injectivity, and type unicity. A condition for the decidability of type checking is the ability to invert typing derivations. The proof requires substitution. Lemma 5.5 (Inversion). (1) If Γ ⊢ x : T then (x : U ) ∈ Γ for some U with Γ ⊢ U = T . (2) If Γ ⊢ λx⋆U. t : T then Γ. x⋆U ⊢ t : T ′ for some T ′ with Γ ⊢ (x⋆U ) → T ′ = T . (3) If Γ ⊢ t ⋆ u : T then Γ ⊢ t : (x⋆U ) → T ′ and Γ ⊢ u ⋆ U for some U, T ′ with Γ ⊢ T ′ [u/x] = T . (4) If Γ ⊢ s : T then there is (s, s′ ) ∈ Axiom such that Γ ⊢ s′ = T . s1 ,s2 (5) If Γ ⊢ (x⋆U ) → T ′ : T then Γ ⊢ U : s1 and Γ. x⋆U ⊢ T ′ : s2 , and for some s3 we have Γ ⊢ s3 = T and (s1 , s2 , s3 ) ∈ Rule. Proof. Each by induction on the typing derivation. Remark 5.6. The need for inversion during type checking is the only good reason to have separate typing rules and not simply define typing Γ ⊢ t : T as the diagonal Γ ⊢ t = t : T of equality. While by a logical relation argument we will obtain a suitable inversion result for Γ ⊢ (x⋆U ) → T = (x⋆U ) → T —the famous function type injectivity (Theorem 5.7)— it seems hard to get something similar for application t u. IRRELEVANCE IN TYPE THEORY 25 Injectivity for function types w. r. t. typed equality is known to be tricky. It is connected to subject reduction and required for many meta-theoretic results. We harvest it from our Kripke model. s1 ,s2 s′ ,s′ 1 2 Theorem 5.7 (Function type injectivity). If Γ ⊢ (x⋆U ) → T = (x⋆U ′ ) → T ′ : s3 then ′ ′ ′ ′ s1 = s1 and s2 = s2 and Γ ⊢ U = U : s1 and Γ. x⋆U ⊢ T = T : s2 . b Proof. This follows from Lemma 4.7. Or we can prove it directly as follows: Since Γ ⊢ id s s1 ,s2 s ,s 1 2 b (x⋆U ′ ) → T ′ : s3 which by id : Γ we have by the fundamental theorem Γ ⊢ (x⋆U ) → T s ′ ′ ′ b U : s1 and Γ ⊢ U = U ′ : s1 . Since inversion yields first s1 = s1 and s2 = s2 and Γ ⊢ U s b x⋆U , we also obtain Γ. x⋆U ⊢ T s b T ′ : s2 and conclude Γ. x⋆U ⊢ T = T ′ : s2 . Γ. x⋆U ⊢ x s From the inversion lemma we can prove uniqueness of types, since we are dealing with a functional PTS, and we have function type injectivity. Theorem 5.8 (Type unicity). If Γ ⊢ t : T and Γ ⊢ t : T ′ then Γ ⊢ T = T ′ . Proof. By induction on t, using inversion. 5.4. Normalization and Subject Reduction. An immediate consequence of the model construction is that each term has a weak head normal form and that typing and equality is preserved by weak head normalization. Theorem 5.9 (Normalization and subject reduction). If Γ ⊢ t : T then t ց a and Γ ⊢ t = a : T. b t : T which by definition contains a derivation Proof. By the fundamental theorem, Γ ⊢ t s of Γ ⊢ t = ↓t : T . 5.5. Consistency. Importantly, not every type is inhabited in IITT, thus, it can be used as a logic. A prerequisite is that types can be distinguished, which follows immediately from the construction of the logical relation. Lemma 5.10 (Type constructor discrimination). Neutral types, sorts and function types are mutually unequal. (1) Γ ⊢ N 6= s. (2) Γ ⊢ N 6= (x⋆U ) → T . (3) Γ ⊢ s = s′ implies s ≡ s′ . (4) Γ ⊢ s 6= (x⋆U ) → T . Proof. By the fundamental theorem applied to the identity substitution. For instance, assuming Γ ⊢ N = s : s′ we get Γ ⊢ N s s : s′ but this is a contradiction to the definition of s. From normalization and type constructor discrimination we can show that not every type is inhabited. Theorem 5.11 (Consistency). X : Set0 6 ⊢ t : X. 26 A. ABEL AND G. SCHERER Proof. Let Γ = (X : Set0 ). Assuming Γ ⊢ t : X, we have Γ ⊢ a : X for the whnf a of t. We invert on the typing of a. By Lemma 5.10, X cannot be equal to a function type or sort, thus, a can neither be a λ nor a function type nor a sort, it can only be neutral. The only variable X must be in the head of a, but since X is not of function type, it cannot be applied. Thus, a ≡ X and Γ ⊢ X : X, implying Γ ⊢ X = Set0 by inversion (Lemma 5.5). This is in contradiction to Lemma 5.10! 5.6. Soundness of Algorithmic Equality. Soundness of the equality algorithm is a consequence of subject reduction. Theorem 5.12 (Soundness of algorithmic equality). (1) Let ∆ ⊢ t, t′ : T . If ∆ ⊢ t ⇐⇒ b t′ : T then ∆ ⊢ t = t′ : T . ′ (2) Let ∆ ⊢ n, n : T . If ∆ ⊢ n ←→ b n′ : U then ∆ ⊢ n = n′ : U and ∆ ⊢ U = T . Proof. Generalize the theorem to all six algorithmic equality judgments and prove it by induction on the algorithmic equality derivation. Since we have subject reduction, the proof proceeds mechanically, because each algorithmic rule corresponds, modulo weak head normalization, to a declarative rule. Case ∆ ⊢ T : s and ∆′ ⊢ T ′ : s and ∆ ⊢ ↓T ⇐⇒ ↓T ′ ∆ ⊢ T ⇐⇒ b T′ ′ By induction hypothesis, ∆ ⊢ ↓T = ↓T : s. By subject reduction ∆ ⊢ T = ↓T : s and ∆ ⊢ T ′ = ↓T ′ : s. By transitivity ∆ ⊢ T = T ′ : s. Case ∆ ⊢ T ⇐⇒ b T′ ∆ ⊢ T ⇐⇒ T ′ : s By induction hypothesis, ∆ ⊢ T = T ′ : s. 5.7. Symmetry and Transitivity of Algorithmic Equality. Since algorithmic equality is sound for well-typed terms, it is also symmetric and transitive. Lemma 5.13 (Type and context conversion in algorithmic equality). Let ⊢ ∆ = ∆′ . (1) If ∆ ⊢ A, A′ and ∆ ⊢ A ⇐⇒ A′ then ∆′ ⊢ A ⇐⇒ A′ . (2) If ∆ ⊢ n, n′ : A and ∆ ⊢ n ←→ n′ : A then ∆′ ⊢ n ←→ n′ : A′ for some A′ with ∆ ⊢ A = A′ . (3) If ∆ ⊢ t, t′ : A and ∆ ⊢ t ⇐⇒ t′ : A and ∆ ⊢ A = A′ then ∆′ ⊢ t ⇐⇒ t′ : A′ . Proof. By induction on the derivation of algorithmic equality, where we extend the statements to ←→ b and ⇐⇒ b accordingly. (1) Type equality. Case ∆ ⊢ U ⇐⇒ b U′ ∆. x⋆U ⊢ T ⇐⇒ b T′ ∆ ⊢ (x⋆U ) → T ⇐⇒ (x⋆U ′ ) → T ′ By inversion, ∆ ⊢ U, U ′ and by induction hypothesis, ∆′ ⊢ U ⇐⇒ b U ′ . Again by ′ ′ inversion, ∆. x⋆U ⊢ T and ∆. x⋆U ⊢ T , yet by soundness of algorithmic equality, ∆ ⊢ U = U ′ , hence ∆. x⋆U ⊢ T ′ by context conversion. Further, ⊢ ∆. x⋆U = IRRELEVANCE IN TYPE THEORY 27 ∆′ . x⋆U . Thus, we can apply the other induction hypothesis to obtain ∆′ . x⋆U ⊢ T ⇐⇒ b T ′ , which finally yields ∆′ ⊢ (x⋆U ) → T ⇐⇒ (x⋆U ′ ) → T ′ . (2) Structural equality. Case (x : T ) ∈ ∆ ∆ ⊢ x ←→ b x:T Since ⊢ ∆ = ∆′ , there is a unique (x : T ′ ) ∈ ∆′ with ∆ ⊢ T = T ′ . Hence, ∆′ ⊢ x ←→ b x : T ′. Case Type-directed equality. Case ∆ ⊢ t, t′ : T and ∆ ⊢ T = T ′ and T ցA ∆ ⊢ t ⇐⇒ t′ : A ∆ ⊢ t ⇐⇒ b t′ : T By normalization, T ′ ց A′ , and subject reduction ∆ ⊢ A = T = T ′ = A′ . Since by conversion, ∆ ⊢ t, t′ : A, by induction hypothesis ∆′ ⊢ t ⇐⇒ t′ : A′ . Thus, ∆′ ⊢ t ⇐⇒ b t′ : T ′ . Case ∆ ⊢ (x⋆U ) → T = A′ and ∆. x⋆U ⊢ t ⋆ x ⇐⇒ b t′ ⋆ x : T ∆ ⊢ t ⇐⇒ t′ : (x⋆U ) → T By injectivity A′ ≡ (x⋆U ′ ) → T ′ with ∆ ⊢ U = U ′ and ∆. x⋆U ⊢ T = T ′ . Since ⊢ ∆. x⋆U = ∆′ . x⋆U ′ , by induction hypothesis we have ∆′ . x⋆U ′ ⊢ t ⋆ x ⇐⇒ b t′ ⋆ x : ′ ′ ′ ′ ′ T . We conclude ∆ ⊢ t ⇐⇒ t : (x⋆U ) → T . Lemma 5.14 (Algorithmic equality is transitive). Let ⊢ ∆ = ∆′ . In the following, let the terms submitted to algorithmic equality be well-typed. (1) If ∆ ⊢ n1 ←→ b n2 : T and ∆′ ⊢ n2 ←→ b n3 : T ′ then ∆ ⊢ n1 ←→ b n3 : T and ′ ∆ ⊢T =T . (2) If ∆ ⊢ t1 ⇐⇒ b t2 : T and ∆′ ⊢ t2 ⇐⇒ b t3 : T ′ and ∆ ⊢ T = T ′ then ∆ ⊢ t1 ⇐⇒ b t3 : T . ′ (3) If ∆ ⊢ T1 ⇐⇒ b T2 : s and ∆ ⊢ T2 ⇐⇒ b T3 : s then ∆ ⊢ T1 ⇐⇒ b T3 : s Proof. We extend these statements to ←→ and ⇐⇒ and prove them simultaneously by induction on the first derivation. Case ∆′ ⊢ n2 ←→ b n3 : T ′ ∆ ⊢ n1 ←→ b n2 : T ∆ ⊢ n1 ⇐⇒ n2 : N ∆′ ⊢ n2 ⇐⇒ n3 : N ′ By induction hypothesis ∆ ⊢ n1 ←→ b n3 : T , hence, ∆ ⊢ n1 ⇐⇒ n3 : N . Case ∆ ⊢ N1 ←→ b N2 : T ∆ ⊢ N2 ←→ b N3 : T ′ ∆ ⊢ N1 ⇐⇒ N2 ∆ ⊢ N2 ⇐⇒ N3 Analogously. 28 A. ABEL AND G. SCHERER Case s1 ,s2 ∆ ⊢ n1 ←→ n2 : (x : U ) → T ∆ ⊢ u1 ⇐⇒ b u2 : U ∆ ⊢ n1 u1 ←→ b n2 u2 : T [u1 /x] s′ ,s′ 1 2 ∆′ ⊢ n2 ←→ n3 : (x : U ′ ) → T′ ∆′ ⊢ u2 ⇐⇒ b u3 : U ′ ∆′ ⊢ n2 u2 ←→ b n3 u3 : T ′ [u2 /x] s1 ,s2 s1 ,s2 By induction hypothesis we have ∆ ⊢ n1 ←→ n3 : (x : U ) → T and ∆ ⊢ (x : U ) → s′ ,s′ 1 2 T = (x : U ′ ) → T ′ which gives in particular s1 = s′1 , s2 = s′2 , and ∆ ⊢ U = U ′ : s1 by function type injectivity (Thm. 5.7). By induction hypothesis we can then deduce ∆ ⊢ u1 ⇐⇒ b u3 : U , and therefore conclude ∆ ⊢ n1 u1 ←→ n3 u3 : T [u1 /x]. Case ∆ ⊢ U1 ⇐⇒ b U2 : s 1 s1 ,s2 ∆. x⋆U1 ⊢ T1 ⇐⇒ b T2 : s2 s1 ,s2 ∆ ⊢ (x⋆U1 ) → T1 ⇐⇒ (x⋆U2 ) → T2 : s3 ∆ ⊢ U2 ⇐⇒ b U3 : s 1 ∆. x⋆U2 ⊢ T2 ⇐⇒ b T3 : s2 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U2 ) → T2 ⇐⇒ (x⋆U3 ) → T3 : s2 We get ∆ ⊢ U1 ⇐⇒ b U3 : s1 by transitivity. To also get ∆. x⋆U1 ⊢ T1 ⇐⇒ b T3 : s2 we need ⊢ ∆. x⋆U2 = ∆. x⋆U1 , but this stems from ∆ ⊢ U1 ⇐⇒ b U2 : s1 by soundness of algorithmic equality. Theorem 5.15. The algorithmic equality relations are PERs on well-typed expressions. Proof. By Lemma 5.14 and an analogous proof of symmetry. 6. A Kripke Logical Relation for Completeness The only open issues in the meta-theory of IITT are completeness and termination of algorithmic equality. In parts, completeness has been established in the last section already, namely, we have shown injectivity and discrimination for type constructors. What is missing is injectivity and discrimination for neutrals, e. g., if ∆ ⊢ n u = n′ u′ : T ′ then necessarily ∆ ⊢ n = n′ : (x : U ) → T and ∆ ⊢ u = u′ : U , plus ∆ ⊢ T [u/x] = T ′ . In untyped λcalculus, this is an instance of Boehm’s theorem [Bar84]. We follow Coquand [Coq91] and Harper and Pfenning [HP05] and prove it by constructing a second Kripke logical relation, c , for completeness which is very similar to the first one, s, but at base types additionally requires algorithmic equality to hold. After proving the fundamental lemma again, we know that definitionally equal terms are also algorithmically so. As a consequence, equality is decidable in IITT, and so is type checking. IRRELEVANCE IN TYPE THEORY 29 6.1. Another Kripke Logical Relation. Again, by induction on A ∈ s we define two Kripke relations ∆ ⊢ A c A′ : s ∆ ⊢ a c a′ : A. c and the generalization to ⋆. This time, however, together with their respective closures c at base types we will additionally require algorithmic equality to hold, more precisely, the relation ∆ ⊢ t :⇐⇒: t′ : T which stands for the conjunction of the propositions • ∆ ⊢ t : T and ∆ ⊢ t′ : T , and • ∆ ⊢ t ⇐⇒ b t′ : T . Note that by soundness of algorithmic equality, :⇐⇒: implies :=:. Again, we allow ourselves rule notation for the defining clauses of c . ∆ ⊢ n :⇐⇒: n′ : N ∆ ⊢ n c n′ : N ∆ ⊢ N :⇐⇒: N ′ : s ∆ ⊢ N c N′ : s ⊢∆ (s, s′ ) ∆ ⊢ s c s : s′ c U ′ : s1 ∆ ⊢U c c u′ ⋆ U =⇒ Γ ⊢ T [u/x] c c T ′ [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u c s1 ,s2 s1 ,s2 ′ ′ ∆ ⊢ (x⋆U ) → T :=: (x⋆U ) → T : s3 s ,s s ,s 1 2 1 2 ∆ ⊢ (x⋆U ) → T c (x⋆U ′ ) → T ′ : s3 (s1 , s2 , s3 ) c u′ ⋆ U =⇒ Γ ⊢ f ⋆ u c c f ′ ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u c s,s′ ∆ ⊢ f :=: f ′ : (x⋆U ) → T s,s′ ∆ ⊢ f c f ′ : (x⋆U ) → T ∆ ⊢ ↓t c ↓t′ : ↓T ∆ ⊢ t :=: t′ : T c t′ : T ∆ ⊢tc ∆÷ ⊢ a c a : A ∆÷ ⊢ a′ c a′ : A ∆ ⊢ a c a′ ÷ A c t:T ∆÷ ⊢ t c c t′ : T ∆ ÷ ⊢ t′ c c t′ ÷ T ∆ ⊢tc This logical relation contains only well-typed and definitionally equal terms. It is symmetric, transitive, and closed under weakening and type conversion. The proofs are in analogy to those of Section 4, which are relying on the fact that the underlying relation :=: is a Kripke PER and closed under type conversion. The relation :⇐⇒: underlying c has the same properties, thanks to soundness of algorithmic equality. Note that in the definition of ∆ ⊢ f c f ′ : (x⋆U ) → T we did not require f and f ′ to be algorithmically equal. This would hinder the proof of the fundamental theorem for c , since algorithmic equality is not closed under application by definition—it will follow from the fundamental theorem, though. In the next lemma we shall prove that f and f ′ are algorithmically equal if they are related by c . The name Escape Lemma was coined by Jeffrey Sarnat [SS08]. Lemma 6.1 (Escape from the logical relation). Let ∆ ⊢ A c A′ : s (1) ∆ ⊢ A ⇐⇒ A′ . c t′ : A then ∆ ⊢ t ⇐⇒ t′ : A. (2) If ∆ ⊢ t c (3) If ∆ ⊢ n ←→ n′ ⋆ A and ∆ ⊢ n = n′ ⋆ A then ∆ ⊢ n c n′ ⋆ A. 30 A. ABEL AND G. SCHERER c T′ : s Corollary 6.2. Let ∆ ⊢ T c (1) ∆ ⊢ T ⇐⇒ b T ′. c t′ : T then ∆ ⊢ t ⇐⇒ (2) If ∆ ⊢ t c b t′ : T . c n′ ⋆ T . (3) If ∆ ⊢ n ←→ b n′ ⋆ T and ∆ ⊢ n = n′ ⋆ T then ∆ ⊢ n c The corollary is a direct, non-inductive consequence of the lemma, so we can use it in the proof of the lemma, quoted as “IH”. Proof of the lemma. Simultaneously by induction on A :⇐⇒: A′ : s. Case ∆ ⊢ N c N ′ : s. Case 1. ∆ ⊢ N ⇐⇒ N ′ by assumption. Case 2. We have ∆ ⊢ ↓t ←→ ↓t′ : , thus ∆ ⊢ t ⇐⇒ t′ : N . Case 3. First, consider ⋆ = :. If ∆ ⊢ n = n′ : N and ∆ ⊢ n ←→ n′ : N then ∆ ⊢ n ⇐⇒ n′ : N and trivially ∆ ⊢ n c n′ : N . Then, take ⋆ = ÷. Note that if ∆÷ ⊢ n = n : N and ∆÷ ⊢ n ←→ n : N then ∆÷ ⊢ n ⇐⇒ n : N and ∆÷ ⊢ n c n : N . This implies that if ∆ ⊢ n = n′ ÷ N and ∆ ⊢ n ←→ n′ ÷ N then ∆ ⊢ n ⇐⇒ n′ ÷ N and ∆ ⊢ n c n′ ÷ N . Case ∆ ⊢ s c s : s′ . Case 1. Clearly, ∆ ⊢ s ⇐⇒ s. c T ′ : s. Then ∆ ⊢ T ⇐⇒ Case 2. Let ∆ ⊢ T c b T ′ by IH 1, thus ∆ ⊢ T ⇐⇒ T ′ : s ′ Case 3. For ⋆ = : let ∆ ⊢ N ←→ N : s. By inversion, ∆ ⊢ N ←→ b N ′ : T for some T . ′ ′ Then ∆ ⊢ N ⇐⇒ N and ∆ ⊢ N c N : s by definition. Considering ⋆ = ÷, it is sufficient to observe that ∆÷ ⊢ N ←→ N : s implies ∆÷ ⊢ N ⇐⇒ N and ∆÷ ⊢ N c N : s by definition. Case ∆ ⊢ (x⋆U ) → T c (x⋆U ′ ) → T ′ : s3 . Case 1. Similar to 2. c t′ : (x⋆U ) → T . It is sufficient to show ∆. x⋆U ⊢ Case 2. By assumption, ∆ ⊢ t c ⋆ ′ ⋆ c U ′ : s1 , which includes ∆ ⊢ U , we have ∆. x⋆U ⊢ t x ⇐⇒ b t x : T . Since ∆ ⊢ U c c t′ ⋆ x : ↓T via x = x ⋆ U . Since also ∆. x⋆U ⊢ x ←→ b x ⋆ U , we obtain ∆. x⋆U ⊢ t ⋆ x c IH 3, ∆. x⋆U ⊢ x c x ⋆ U . IH 2 then entails our goal. Case 3. First, the case for ⋆ = :. We reuse variable ⋆ for a different irrelevance marker. We have ∆ ⊢ n ←→ n′ : (x⋆U ) → T . Assume arbitrary Γ :⇐⇒: Γ ≤ ∆ and c u′ ⋆ U , which yields Γ ⊢ u = u′ ⋆ U and Γ ⊢ T [u/x] c c T [u′ /x] : Seti . In Γ ⊢uc ′ case ⋆ = : we have to apply IH 2 for Γ ⊢ u ⇐⇒ u : ↓U . Otherwise, we obtain directly Γ ⊢ n ⋆ u ←→ n′ ⋆ u′ : ↓(T [u/x]). By IH 3, Γ ⊢ n ⋆ u c n′ ⋆ u′ : ↓(T [u/x]). The case for ⋆ = ÷ proceeds analogously. b we extend c c to substitutions and define the semantic validity judgements In analogy to s c Γ and Γ c t : T and Γ c t = t′ : T based on c c . Since by the escape lemma, c x : ∆(x), we have Γ ⊢ id c c id : Γ for c Γ. Finally, we reprove the fundamental ∆ ⊢xc theorem: c ). Theorem 6.3 (Fundamental theorem for c c (1) If ⊢ Γ then Γ. (2) If Γ ⊢ t : T then Γ c t : T . (3) If Γ ⊢ t = t′ : T then Γ c t = t′ : T . IRRELEVANCE IN TYPE THEORY 31 6.2. Completeness and Decidability of Algorithmic Equality. Derivations of algorithmic equality can now be obtained by escaping from the logical relation. Theorem 6.4 (Completeness of algorithmic equality). Γ ⊢ t = t′ : T implies Γ ⊢ t ⇐⇒ b t′ : T . c id : Γ, we have Γ ⊢ t c c t′ : T by the fundamental theorem, and Proof. Since Γ ⊢ id c conclude with Lemma 6.1.2. Termination of algorithmic equality is a consequence of completeness. When invoking the algorithmic equality check ∆ ⊢ t ⇐⇒ b t′ : T on two well-typed expressions ∆ ⊢ t, t′ : T we know by completeness that t and t′ are related to themselves, i. e., ∆ ⊢ t ⇐⇒ b t : T and ∆ ⊢ t′ ⇐⇒ b t′ : T . This means that t, t′ , and T are weakly normalizing by the strategy the equality algorithm implements: reduce to weak head normal form and recursively continue with the subterms. Running the equality check on t and t′ performs, if successful, exactly the same reductions, and if it fails, at most the same reductions in t, t′ , and T . Hence, testing equality on well-typed terms always terminates. This argument has been applied in previous work to untyped equality [AC07]. Here, we apply it to typed equality; it is an alternative to Goguen’s technique of proving termination for typed equality from strong normalization [Gog05], which, in our opinion, does not scale to dependently-typed equality. Lemma 6.5 (Termination of algorithmic equality). Let ⊢ ∆. (1) Type equality. (a) Let ∆ ⊢ A, A′ . If D :: ∆ ⊢ A ⇐⇒ A and ∆ ⊢ A′ ⇐⇒ A′ then the query ∆ ⊢ A ⇐⇒ A′ terminates. (b) Let ∆ ⊢ T, T ′ . If D :: ∆ ⊢ T ⇐⇒ b T and ∆ ⊢ T ′ ⇐⇒ b T ′ then the query ∆ ⊢ T ⇐⇒ b ′ T terminates. (2) Structural equality. Let ∆ ⊢ n : T and ∆ ⊢ n′ : T ′ . (a) If D :: ∆ ⊢ n ←→ n : A and ∆ ⊢ n′ ←→ n′ : A′ then the query ∆ ⊢ n ←→ n′ : ? terminates. If successfully, it returns A and we have ∆ ⊢ A = T = T ′ = A. (b) If D :: ∆ ⊢ n ←→ b n : T and ∆ ⊢ n′ ←→ b n′ : T ′ then the query ∆ ⊢ n ←→ b n′ : ? ′ terminates. If successfully, it returns T and we have ∆ ⊢ T = T . (3) Type-directed equality. (a) Let ∆ ⊢ t, t′ : A. If D :: ∆ ⊢ t ⇐⇒ t : A and ∆ ⊢ t′ ⇐⇒ t′ : A then the query ∆ ⊢ t ⇐⇒ t′ : A terminates. (b) Let ∆ ⊢ t, t′ : T . If D :: ∆ ⊢ t ⇐⇒ b t : T and ∆ ⊢ t′ ⇐⇒ b t′ : T then the query ′ ∆ ⊢ t ⇐⇒ b t : T terminates. Proof. Simultaneously by induction on derivation D. (1) Type equality. Case A = A′ = s. The query ∆ ⊢ A ⇐⇒ A′ terminates successfully. Case A = (x⋆U ) → T and A′ = (x⋆U ′ ) → T ′ . First, the query ∆ ⊢ U ⇐⇒ b U ′ runs. By induction hypothesis, it terminates. If it fails, the whole query fails. Otherwise, the query ∆. x⋆U ⊢ T ⇐⇒ b T ′ is run. By induction hypothesis on ∆. x⋆U ⊢ T ⇐⇒ b T ′ ′ and ∆. x⋆U ⊢ T ⇐⇒ b T ′ , the query terminates. Case A = N and A′ = N ′ neutral. By induction hypothesis on ∆ ⊢ N ←→ b N :T and ∆ ⊢ N ′ ←→ b N ′ : T ′ , the query ∆ ⊢ N ←→ b N ′ : ? terminates. Hence, the query ∆ ⊢ N ⇐⇒ N ′ terminates. 32 A. ABEL AND G. SCHERER Case Weak head normal forms A, A′ not covered by previous cases: the query ∆ ⊢ A ⇐⇒ A′ fails immediately, since there is no applicable algorithmic type equality rule. Case The query ∆ ⊢ T ⇐⇒ b T ′ first invokes weak head normalization on T and T ′ . Both terminate since ∆ ⊢ T ⇐⇒ b T , which implies T ց A, and analogously T ′ ց A′ ′ ′ since ∆ ⊢ T ⇐⇒ b T by assumption. Then, the query ∆ ⊢ A ⇐⇒ A′ is run, which terminates by induction hypothesis on ∆ ⊢ A ⇐⇒ A and ∆ ⊢ A′ ⇐⇒ A′ . (2) Structural equality. Case n = n′ = x. The query ∆ ⊢ n ←→ b n′ : ? terminates successfully, returning type ∆(x). Since ⊢ ∆, by inversion (Lemma 5.5) ∆ ⊢ T = T ′ = ∆(x). Case Neutral relevant application for ∆ ⊢ n u : T0 and ∆ ⊢ n′ u′ : T0′ . ∆ ⊢ n ←→ n : (x : U ) → T ∆ ⊢ u ⇐⇒ b u:U ∆ ⊢ n u ←→ b n u : T [u/x] ∆ ⊢ n′ ←→ n′ : (x : U ′ ) → T ′ ∆ ⊢ u′ ⇐⇒ b u′ : U ′ ∆ ⊢ n′ u′ ←→ b n′ u′ : T ′ [u′ /x] The query ∆ ⊢ n u ←→ b n′ u′ : ? first invokes query ∆ ⊢ n ←→ n′ : ?. By induction hypothesis on ∆ ⊢ n ←→ n : (x : U ) → T and ∆ ⊢ n′ ←→ n′ : (x : U ′ ) → T ′ the query terminates. If it fails the whole query fails. Otherwise it returns a type A in weak head normal form, which is identical to (x : U ) → T by uniqueness of inferred types (Lemma 3.1). Further, ∆ ⊢ (x : U ) → T = (x : U ′ ) → T ′ , and by function type injectivity (Thm. 5.7), ∆ ⊢ U = U ′ and ∆. x : U ⊢ T = T ′ . Thus, we can invoke the induction hypothesis on ∆ ⊢ u ⇐⇒ b u : U and ∆ ⊢ u′ ⇐⇒ b u′ : U ′ ′ ′ (cast from ∆ ⊢ u ⇐⇒ b u : U , Lemma 5.13) to infer that the second subquery ∆ ⊢ u ⇐⇒ b u′ : U terminates. If this one is successful, then by soundness of algorithmic equality, ∆ ⊢ u = u′ : U , which implies ∆ ⊢ T [u/x] = T ′ [u′ /x]. Case Neutral irrelevant application with typing ∆ ⊢ n : (x÷U1 ) → T1 ∆ ⊢ u ÷ U1 ÷ ∆ ⊢ n u : T1 [u/x] and algorithmic self-equality ∆ ⊢ n ←→ n : (x÷U ) → T ∆ ⊢ n ÷ u ←→ b n ÷ u : T [u/x] ∆ ⊢ u′ ÷ U1′ ∆ ⊢ n′ : (x÷U1′ ) → T1′ ∆ ⊢ n′ ÷ u′ : T1′ [u′ /x] ∆ ⊢ n′ ←→ n′ : (x÷U ′ ) → T ′ ∆ ⊢ n′ ÷ u′ ←→ b n′ ÷ u′ : T ′ [u′ /x] The query ∆ ⊢ n ÷ u ←→ b n′ ÷ u′ : ? invokes query ∆ ⊢ n ←→ n′ : ?, which terminates by induction hypothesis. If successfully, then ∆ ⊢ (x÷U1 ) → T1 (x÷U ) → T = (x÷U ′ ) → T ′ = (x÷U1′ ) → T1′ . By function type injectivity, ∆ ⊢ U1 = U = U ′ = U1′ and ∆. x÷U ⊢ T1 = T = T ′ = T1′ . By conversion ∆ ⊢ u = u′ ÷ U , thus, ∆ ⊢ T1 [u/x] = T [u/x] = T ′ [u′ /x] = T1′ [u′ /x]. Case In all other cases, the query ∆ ⊢ n ←→ b n′ : ? fails immediately. Case The query ∆ ⊢ n ←→ n : ? spawns subquery ∆ ⊢ n ←→ b n′ : ? which terminates ′ by induction hypothesis on ∆ ⊢ n ←→ b n : T and ∆ ⊢ n ←→ b n′ : T ′ . If successfully, it returns type T , and since T ց A, the original query also terminates, returning A. (3) Type-directed equality. Case Function type ∆ ⊢ t, t′ : (x⋆U ) → T . The query ∆ ⊢ t ⇐⇒ t′ : (x⋆U ) → T spawns subquery ∆. x⋆U ⊢ t ⋆ x ⇐⇒ b t′ ⋆ : T . Since ∆. x⋆U ⊢ t ⋆ x, t′ ⋆ x : T and the IRRELEVANCE IN TYPE THEORY 33 subquery terminates by induction hypothesis on ∆. x⋆U ⊢ t ⋆ x ⇐⇒ b t ⋆ x : T and ′ ⋆ ′ ⋆ ∆. x⋆U ⊢ t x ⇐⇒ b t x : T. Case Sort ∆ ⊢ T, T ′ : s. The query ∆ ⊢ T ⇐⇒ T ′ : s calls ∆ ⊢ T ⇐⇒ b T ′ , which terminates by induction hypothesis on ∆ ⊢ T ⇐⇒ b T and ∆ ⊢ T ′ ⇐⇒ b T ′. Case Neutral type N . t′ ց n ′ ∆ ⊢ n′ ←→ b n′ : T ′ tցn ∆ ⊢ n ←→ b n:T ∆ ⊢ t ⇐⇒ t : N ∆ ⊢ t′ ⇐⇒ t′ : N ′ The query ∆ ⊢ t ⇐⇒ t : N first weak head normalizes t and t′ . By assumption, t ց n and t′ ց n′ , so this terminates. The subquery ∆ ⊢ n ←→ b n′ : ? terminates by induction hypothesis. Thus, the whole query terminates. Case If A is neither a function type, a sort, or a neutral type, the query ∆ ⊢ t ⇐⇒ t′ : A fails immediately. Case The query ∆ ⊢ t ⇐⇒ b t′ : T first weak head normalizes T which terminates since T ց A by assumption. Then it calls ∆ ⊢ t ⇐⇒ t′ : A which terminates by induction hypothesis. Theorem 6.6. If ∆ ⊢ t : T and ∆ ⊢ t′ : T then the query ∆ ⊢ t ⇐⇒ b t′ : T terminates. Proof. From the lemma by completeness of algorithmic equality. Thus we have shown that algorithmic equality is correct, i. e., sound, complete, and terminating. Together, this entails decidability of equality in IITT. Theorem 6.7 (Decidability of IITT). (1) Γ ⊢ t = t′ : T is decidable. (2) Γ ⊢ t : T is decidable. Proof. Decidability of equality follows from soundness (Thm. 5.12), completeness (Thm. 6.4), and termination (Thm. 6.6). Decidability of typing follows from decidability of type conversion, weak head normalization, and function type injectivity, using inversion (Lemma 5.5) on typing derivations. Any reasonable type inference algorithm will do. 7. Extensions Data types and recursion. The semantics of IITT is ready to cope with inductive data types like the natural numbers and the associated recursion principles. Recursion into types, aka known as large elimination, is also accounted for since we have universes and a semantics which does not erase dependencies (unlike Pfenning’s model [Pfe01]). Types with extensionality principles. One purpose of having a typed equality algorithm is to handle η-laws that are not connected to the shape of the expression (like η-contraction for functions) but to the shape of the type only. Typically these are types T with at most one inhabitant, i. e., the empty type, the unit type, singleton types or propositions.6 For such T we have the η-law Γ ⊢ t, t′ : T Γ ⊢ t = t′ : T 6Some care is necessary for the type of Leibniz equality [Abe09, Wer08]. 34 A. ABEL AND G. SCHERER which can only be checked in the presence of type T . Realizing such η-laws gives additional “proof” irrelevance which is not covered by Pfenning’s irrelevant quantification (x÷U ) → T . Internal erasure. Terms u÷U in irrelevant position are only there to please the type checker, they are ignored during equality checking. This can be inferred from the substitution principle: If Γ. x÷U ⊢ T and Γ ⊢ u, u′ ÷ U , then Γ ⊢ T [u/x] = T [u′ /x]; the type T has the same shape regardless of u, u′ . Hence, terms like u serve the sole purpose to prove some proposition and could be replaced by a dummy • immediately after type-checking. Internal erasure can be realized by making Γ ⊢ t ÷ T a judgement (as opposed to just a notation for Γ÷ ⊢ t : T ) and adding the rule Γ ⊢ t÷T . Γ ⊢•÷T The rule states that if there is already a proof t of T , then • is a new proof of T . This preserves provability while erasing the proof terms. Conservativity of this rule can be proven as in joint work of the author with Coquand and Pagano [ACP11]. 8. Conclusions We have extended Pfenning’s notion of irrelevance to a type theory IITT with universes that accommodates types defined by recursion. We have constructed a Kripke model s that shows soundness of IITT, yielding normalization, subject reduction and consistency, plus syntactical properties of the judgements of IITT. A second Kripke logical relation c has proven correctness of algorithmic equality and, thus, decidability of IITT. Integrating irrelevance and data types in dependent type theory does not seem without challenges. We have succeeded to treat Pfenning’s notion of irrelevance, but our proof does not scale directly to parametric function types, a stronger notion of irrelevant function types called implicit quantification by Miquel [Miq01b].7 Two more type theories build on Miquel’s calculus [Miq01a], Barras and Bernardo’s ICC∗ [BB08] and Mishra-Linger and Sheard’s Erasure Pure Type Systems (EPTS) [MLS08], but none has offered a satisfying account of large eliminations yet. Miquel’s model [Miq00] features data types only as impredicative encodings. For irrelevant, parametric, and recursive functions to coexist it seems like three different function types are necessary, e. g., in the style of Pfenning’s irrelevance, extensionality and intensionality. We would like to solve this puzzle in future work, not least to implement high-performance languages with dependent types. Acknowledgments. The first author thanks Bruno Barras, Bruno Bernardo, Thierry Coquand, Dan Doel, Hugo Herbelin, Conor McBride, Ulf Norell, and Jason Reed for discussions on irrelevance in type theory. Work on a previous paper has been carried out while he was invited researcher at PPS, Paris, in the INRIA πr 2 team headed by Pierre-Louis Curien and Hugo Herbelin. The second author acknowledges financial support by the École Normale Superiéure de Paris for his internship at the Ludwig-Maximilians-Universität München from May to September 2011. We thank the two anonymous referees, who suggested changes and examples which significantly improved the presentation, and the patience of the editors waiting for our revisions. 7A function argument is parametric if it is irrelevant for computing the function result while the type of the result may depend on it. In Pfenning’s notion, the argument must also be irrelevant in the type. 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A Method for Reducing the Severity of Epidemics by Allocating Vaccines According to Centrality Krzysztof Drewniak kdrewniak@utexas.edu Joseph Helsing JosephHelsing@my.unt.edu arXiv:1407.7288v1 [cs.SI] 27 Jul 2014 ABSTRACT One long-standing question in epidemiological research is how best to allocate limited amounts of vaccine or similar preventative measures in order to minimize the severity of an epidemic. Much of the literature on the problem of vaccine allocation has focused on influenza epidemics and used mathematical models of epidemic spread to determine the effectiveness of proposed methods. Our work applies computational models of epidemics to the problem of geographically allocating a limited number of vaccines within several Texas counties. We developed a graph-based, stochastic model for epidemics that is based on the SEIR model, and tested vaccine allocation methods based on multiple centrality measures. This approach provides an alternative method for addressing the vaccine allocation problem, which can be combined with more conventional approaches to yield more effective epidemic suppression strategies. We found that allocation methods based on in-degree and inverse betweenness centralities tended to be the most effective at containing epidemics. Categories and Subject Descriptors J.3 [Life and Medical Sciences]: Health; G.2.2 [Discrete Mathematics]: Graph Theory—Graph algorithms General Terms Algorithms, Experimentation, Performance Keywords Computational epidemiology, health informatics, vaccine distribution, centrality measures 1. INTRODUCTION One long-standing question in epidemiological research is how best to allocate limited amounts of vaccine or similar preventative measures in order to minimize the severity of an epidemic. Discovering a way to reduce the severity of epidemics would benefit society by not only preventing loss Armin R Mikler mikler@cs.unt.edu of life, but also reducing the overall disease burden of those epidemics. These burdens, or societal costs, can be seen in reduced economic productivity due to widespread sickness, or increased strain on health infrastructure. Much of the literature on the problem of vaccine allocation has focused on influenza epidemics and has used mathematical models of epidemic spread to determine the effectiveness of proposed methods [12, 13, 17, 22, 24]. Most investigations of vaccine allocation performed their analyses on geographically large scales and focused on determining which subpopulations should be prioritized for vaccination. Since previous research focuses on large-scale, subpopulation-based models, our work explores alternative approaches to the question of vaccine allocation. Much work has been devoted to the development of computational models for disease simulation. Mikler et. al. have developed multiple stochastic models of disease spread in a population, which use the standard SEIR(S) model [15, 19]. Some of these models, such as their Global Stochastic Cellular Automata model [16], can be adapted to the problem of vaccine allocation. Simulation techniques relevant to our work have been developed by Indrakanti [10], who implemented a SEIR-based framework for simulating epidemic spread within a county. Our work applies computational models of epidemics to the problem of geographically allocating a limited number of vaccines within several Texas counties. We developed a graph-based, stochastic model for epidemics that is based on the SEIR model and the work of Mikler et. al. [16]. Various centrality measures have been proposed over the years [18, 2, 3, 7, 4], mainly in the field of social network analysis, which provide means for determining which nodes in a graph are the most important. Our model was then used to investigate various centrality-based vaccine allocation strategies, in which vaccines are allocated to various census blocks within the county in order of their centrality measure score. This approach provides an alternative method for addressing the vaccine allocation problem, which can be combined with more conventional approaches to yield more effective epidemic suppression strategies. In this paper, we present the results of several experiments with centrality-based vaccine allocation strategies. These experiments were performed on graphs constructed from the census data of several Texas counties, and have yielded results that indicate directions for further study. 2. BACKGROUND 2.1 Vaccine Allocation Several authors have previously addressed the problem of vaccine allocation. Medlock and Galvani have investigated the question of which groups should be prioritized for influenza vaccination throughout the United States in the event of a shortage [13]. They developed a model, parameterized with real-world data, for analyzing various vaccination strategies under multiple effectiveness criteria. Medlock and Galvani found that optimal epidemic control could be achieved by targeting adults between the ages of 30 and 39 and children, and that current CDC vaccine allocation recommendations were suboptimal under all considered measures. Tuite et. al. used similar methods to investigate vaccine allocation strategies in Canada during a H1N1 epidemic [22]. They found that targeting high-risk groups, such as people with chronic medical conditions, and then targeting age groups, such as children, that are the most likely to develop complications after influenza infection will cause the greatest reduction in the number of deaths and serious illnesses caused by an influenza epidemic. Similar work was performed by Matrajt and Longini [12]. In their work, Matrajt and Longini attempted to determine whether the optimal vaccine allocation strategy varies with the state of the epidemic. They developed an SIR-based model to analyze the spread of influenza among and between several groups of people in idealized countries, and used criteria similar to those of Medlock and Galvani to ascertain the effectiveness of their methods. The authors determined that the demographic structure of the population in an area significantly affects the optimal allocation strategy. They also found that, at the beginning of an epidemic, those groups that are the most likely to transmit influenza should be targeted immediately, while the groups that are most vulnerable should be targeted after a point before, but near, the peak of the epidemic. The work of Mylius et. al modeled influenza epidemics and found that individuals that are most at risk for complications should be targeted if additional vaccines become available during an epidemic [17]. Mylius et. al. also found that schoolchildren should be prioritized for vaccination at the beginning of an epidemic, as they are heavily involved in disease transmission. Their conclusions bear a close resemblance to those of Matrajt and Longini. The problem of large-scale vaccine allocations were addressed by [12, 13, 17, 22]. They used different models to reach their conclusions, but used similar metrics, such as the reduction in the predicted number of deaths from a disease after the vaccination program, to assess the significance of their results. None of these works addressed the question of vaccine allocation at the level of counties or other similar geographic divisions, nor did they address the vaccine allocation problem from a geographic perspective. Another approach to the vaccine allocation problem, which we extend in our work, can be found in the work of Johnson [11]. Johnson’s approach, unlike those of many other researchers, focuses on vaccine allocation in the small scale. Johnson generated synthetic social network graphs and used various measures of centrality to determine which individuals within those graphs should be vaccinated. Her results showed that the optimal vaccination strategy depends on the structure of the social network it is applied to. Our work adapts Johnson’s methods to the problem of allocating vaccinations within a county. 2.2 Centrality and its Applications The notion of centrality has been used in various fields, most notably in the study of social networks. Multiple centrality measures exist, all of which allow the vertices of a graph to be ranked in order of their importance. Out-Degree and in-degree centralities were first defined by Nieminen [18]. In those measures, the centrality c of a node n in a graph G = (V, E), where V is the set of nodes in the graph, u and v are nodes in V, E is the set of edges, and \|V \| is the number of nodes is dn (1) c= \|V \| − 1 where dn is the out-degree or in-degree of n. Eigenvector centrality, first proposed by Bonacich [2], of a graph G = (V, E) can be calculated by representing G as an adjacency matrix A, where ( 1 if (u, v) ∈ E Auv = (2) 0 if (u, v) 6∈ E Then, the eigenvector centrality x is defined as Ax = λx (3) where λ is the largest eigenvalue of A. Eigenvector centrality is useful because it gives a higher centrality score to a highdegree node that is connected to other high-degree nodes than to a high-degree node connected to low-degree nodes [3]. Another important measure of centrality is betweenness centrality. Betweenness centrality, first described by Freeman [7], is defined as follows, for a node n in a graph G = (V, E), where dst (n) is the number of shortest paths from s ∈ V to t ∈ V that include n, and dst is the number of shortest paths from s to t where the shortest path between two nodes is the path where the sum of the weights of its edges is minimized X dst (n) (4) dst s6=n6=t∈V Betweenness centrality gives higher centrality scores to nodes that are most likely to be involved in the transmission of information throughout a graph [4]. Centrality has been applied to multiple epidemiological questions. Rothenberg et. al. applied various centrality measures to a social network graph generated from CDC data on the spread of HIV in Colorado Springs [20]. Rothenberg analyzed the collected data under multiple centrality measures and found that all of the measures identified all but one of the HIV cases as non-central. Their work also noted several differences in response patterns to the CDC questionnaire between people with high centrality and people with low centrality under all measures, which were closely correlated. Similar work was also performed by Christley et. al. [6]. Christley generated synthetic social networks and performed SIR-based simulations in order to study the applicability of centrality to the problem of identifying individuals at high risk of HIV infection. They found that degree centrality performed no worse than other centrality measures, such as betweenness, but noted that the results might not be valid for larger datasets. Centrality has also played a key role in Johnson’s work, which was discussed previously [11]. Johnson investigated how vaccinating those individuals identified as central in a social network would affect the spread of disease in that network. Johnson found that vaccination strategies based on centrality measures were an effective means of mitigating outbreaks. Related work was performed by Rustam, who applied centrality to the spread of viruses and worms in computer networks [21]. He found that nodes with high centrality scores have a large amount of influence on the spread of a virus, especially when those nodes are rated central by multiple measures. 2.3 Computational Simulation of Epidemics Several methods for simulating epidemics computationally are based on the SEIR model of epidemics [15, 19, 16, 10]. In the SEIR model, the population is divided into four classes: susceptible individuals, who can become infected; exposed or latent individuals, who have been infected but are not capable of spreading the infection; infectious individuals, who can spread the disease to susceptible individuals; and recovered individuals, who can no longer be infected. Mikler et. al. proposed a SEIR-based model that uses a stochastic cellular automaton [16]. The concepts used in their model are applicable to a wide range of epidemic simulations. Agent-based and metapopulation models are the two dominant methods for computational epidemic simulation [1]. In agent-based models, each individual within the population is simulated. Such models, which are often implemented using cellular automata [16, 15, 8], are useful because they provide insights into the progression of the disease. However, they are impractical when large populations need to be simulated due to space required for information about each agent in the simulation. Metapopulation models, on the other hand, break the population into subpopulations and then simulate the interactions between and within the subpopulations [10, 23]. Such models sacrifice some of the precision offered by agent-based modeling in exchange for the ability to simulate large populations. Our work uses a metapopulation-based model, where each census block constitutes a subpopulation. This helps to overcome the excessive computational resources that would be required to simulate hundreds of thousands of individuals. A closely related model that we have adapted for our own work has been developed by Indrakanti [10]. Indrakanti developed a county-level epidemic simulator, which uses census blocks as the base unit of simulation. The model allows for contacts to be generated between any two census blocks, and uses an interaction coefficient to determine the likelihood of contact. The model was used to conduct several experiments which found that epidemics with higher infectivity (likelihood of spread during a contact) reached their peak percentage of infected individuals earlier. It also discovered that the distribution of population between census blocks had a significant effect on the spread of disease. 3. METHODS 3.1 Representing Counties The United States Census Bureau provides geographic data on US counties. They subdivide counties into census blocks at the finest granularity, which are typically bounded by roads, streets, or water features. Being that they may be varying sizes and shapes, rural census blocks are often significantly larger than urban ones. Additionally, the population of a census block may vary greatly as there are many cencus-blocks that contain zero population, i.e. farmland, greenspace, or bodies of water, and others that contain highdesntiy residential structures, i.e. appartment buildings, that may have several hundred people [5]. This geographic data, along with the population of each census block, was obtained for multiple Texas counties, specifically Rockwall, Hays, and Denton County, and was stored in a PostGIS database. The centroid of each census block, which is the average of the points defining the census block, was precomputed and stored within the database. After the census block data was obtained, several methods for representing a county as a graph with census block nodes were investigated. Rockwall County was used for these investigations due to its small size, allowing centrality measures to be computed quickly. All of these representations were based on various methods of constructing a graph G = (V, E) to represent a county, where V , the set of nodes, was the set of populated census blocks in the county. Our first approach to representing counties as a graph, which was ineffective, was a representation where E = {(u, v) \| u, v ∈ V, u 6= v, ST Touches(u, v)}. The ST Touches() function, which determines whether two geographic entities are adjacent, was provided by PostGIS. This representation used undirected edges. Another of our early attempts was an approach where E = {(u, v) \| u, v ∈ V, u 6= v, ST Distance(Centroid(u), v) < r} for various constant values of r, where the ST Distance() function was used to compute the shortest distance between the centroid of u and the block v. The Centroid() function retrieves the centroid of a block from the database. A variation of this approach where the r value for each node u ∈ V was lowered by multiplying it by 1− Population(u) 1000 (5) on the assumption that the population of highly-populated blocks was more likely to interact within the block than that of sparsely-populated blocks was also investigated. All of these approaches were ineffective and gave unrealistic results, such as the identification of a large rural block with a population of 5 as the most central block within Rockwall County. Therefore, they were rejected in favor of a representation based on weighted edges. The results of our investigations led us to use the following method to represent US counties in our experiments. In our final model, the county was represented as a directed graph G = (V, E), where V was the set of populated census blocks. The set of edges E was equal to {(u, v, Wu→v \| u ∈ V, v ∈ V, u 6= v}, where Wu→v was the weight of the edge from u to v. The weights were computed by the following formula, which was adapted from the work of Mikler et. al.[16]: Wu→v = Population(u) × Population(v) 10000 × ST Distance(Centroid(u), v) + 0.00001 (6) This definition assigns higher weights to blocks that are more likely to interact, a fact that is used in our simulator. The multiplication by 1/10000 was included to offset the behavior of ST Distance(), which returns distance as real value less than 1. If the distances were left unscaled, the division could have produced weights outside of the range that can be accurately represented by real values in a computer. The addition of 0.00001 was used to address cases where the centroid of block u is located within block v, which ordinarily results in a distance of 0, from producing an error. Once these weights were computed, the graphs were represented using the NetworkX library [9]. 3.2 Centrality Measures We modified the centrality measures described in section 2.2 to work with our representation. Code from [9] was used as a basis for our work. Out-degree and in-degree centralities were trivially adapted; the out-degree and in-degree of a node were redefined to be the sum of the weights of the outedges or in-edges, respectively, instead of a count of those edges. The implementation of eigenvector centrality in NetworkX already allowed for weighted edges, so it was used without modification. Since that implementation of eigenvector centrality used only the out-edges for its computations, a variation of the measure, eigenvector-in centrality, was also tested where the in-edges were used instead. Betweenness centrality was the most heavily modified measure. Since betweenness centrality relies on shortest paths, the reciprocals of all edge weights were taken before centrality computation, so that the shortest path between nodes s and t is the one that the disease is most likely to spread across. Afterwards, the betweenness centrality scores were computed by a variation of Brandes’s algorithm [4], where Dijkstra’s algorithm was applied to each node in parallel as opposed to sequentially. This modification was found to have no effect on the results of the centrality computation. The centrality measure described above was reported as inverse betweenness centrality. Finally, we defined random centrality, which is not a centrality measure, for use as a baseline. In random centrality, each node is assigned a random real value in [0, 1), which is used as its centrality score. The results of all centrality calculations except random centrality, which was re-computed before each set of simulations, were saved to avoid expensive recomputation when multiple experiments were performed on the same data. 3.3 Graph-Based Stochastic Epidemic Simulation A stochastic epidemic simulator based on the SEIR model and the work of [10] was developed for use with our representation of counties. The simulation began by constructing the graph G = (V, E) as described above, and computing all centrality measures that were to be tested. Each node n ∈ V was labeled with the following attributes: \|Nn \|, the total Figure 1: An illustration of the grouping of census blocks that have been split by the simulator. population of the block, \|Sn \| = \|Nn \|, the number of susceptibles, and \|En \|, \|In \|, \|Rn \| = 0, where \|En \| is the number of latent (exposed) individuals in the block, \|In \| the number of infectious, and \|Rn \| the number of recovered individuals. Throughout the simulation, \|Sn + En + In + Rn \| = \|Nn \|. In addition to these counts, each node was labeled with a mapping ME,n = {0 7→ E0,n , 1 7→ E1,n . . . Tlp 7→ ETlp ,n }, where Tlp is the latent period and Em,n is the number of people in block n with 0 ≤ m ≤ Tlp days remaining until they transition to the next simulation state. In this mapping, Tlp X Em,n = \|En \|. A similar mapping was also produced for m=0 the infectious period, MI,n = {0 7→ I0,n . . . Tip 7→ ITip ,n , where Tip is the infectious period. For this mapping, 0 ≤ m ≤ Tip , similarly to ME,n . These attributes are illustrated in Figure 1. For the purposes of experimentation, 20 simulation sets were performed in parallel, with the random number generator reseeded for each set. Each set consisted of multiple simulations with varying values of available vaccine and centrality measures. The simulations were performed on a server with four six-core Intel Xeon E7540 processors and 256 GB of RAM. At the beginning of each set, a proportion p of the census blocks were chosen randomly for initial infection, which remained constant throughout all simulations in the set. The percentage of the population infected initially was varied between experiments. Vaccines were distributed to the entire population of each chosen census until the total supply of vaccines was exhausted. Census blocks were selected for vaccination in a decreasing order based on their centrality. The centrality measure used to allocate vaccines was varied between experiments so that the measures could be compared for effectiveness. The set of population (blocks) that was initially targeted for infection was held constant as the centrality measure and percent of population that was able to receive vaccination were varied. This allowed us to more accurately determine which centrality measure or measures were most effective under varying conditions. In experiments involving the prevention scenario, vaccines were distributed before the initial infection, while they were dis- Contact rate Transmissibility Mobility Latent period (Tlp ) Infectious period (Tip ) Percentage of blocks to infect (p) 20 0.05 0.99 2 days 3 days 1% Table 1: Parameters kept constant throughout all experiments. These parameters serve to characterize the disease being simulated. Figure 2: An illustration of the infection process used by the simulator. tributed six simulated days after initial infection in the intervention scenario. The value of six days was chosen because at the beginning of day six the initial cases would have transitioned to the recovered state. After the initial infections, which were recorded as day zero and placed those affected into ETE ,n , the simulation was executed until no latent or infectious individuals remained in the population. During each day, after any scheduled vaccine distribution, each block’s ME and MI is updated by setting ME = {i ∈ ME 7→ ME (i + 1), Tlp 7→ 0}, and similarly for MI . After these updates, the census blocks are iterated over. For each block n , i contacts are simulated, where i is In multiplied by the contact rate, 20 contacts/day, which was taken from [10]. For each contact, a random real value in [0, 1) is generated. If this number is greater than or equal to the mobility parameter, which was 0.99, the contact takes place within n, otherwise, the contact takes place within a different census block. The mobility parameter expresses the likelihood that an individual will contact someone who resides in a census block that the individual does not reside in. In the case of non-local contacts, i.e. contacts made between agents in separate blocks, the external block is chosen by the following method. First, the weights (given by Equation 6) of the out-edges of the originating block are normalized by dividing by the out-degree of the block and then sorted in descending order. These values were precomputed once as an optimization. A random target value in [0, 1) is generated, and the list of weights is summed element by element until the sum is greater than or equal to the target value. The node t whose associated weight causes the summation to terminate is chosen as the block for contact. This process is illustrated by Figure 2. Once the block that will be contacted is chosen, a diseasespreading contact occurs if a random real number in [0, 1) is less than the transmissibility parameter, which was 0.05. For non-local contacts, this test is performed before block selection to speed up the simulation significantly. When the contact occurs, one person is, if possible, removed from St , the susceptible population of the target block t, and added to Et , the latent population. The new infection is also added to MEt (Tlp ), which ensures that the newly-infected individual remains in the latent period for Tlp days before becoming infectious. After all the contacts are generated, the SEIR states are updated. For each block n ∈ V , MEn (0) people are removed from En , and added to MIn (Tip ) and In . Similarly, MIn (0) people are transferred from In to Rn . At the conclusion of the updates, the number of people throughout the county in each SEIR state and the percentage of the population in that state are reported. In addition, the number of people who are infected, that is, either latent or infectious, is reported, along with the associated percentage. 3.4 Experiments and Parameters In all of the experiments performed, the parameters listed in Table 1 were used. These parameters were not chosen to reflect an extant disease, and were adapted from the work of Indrakanti [10]. Experiments were performed on Rockwall, Hays, and Denton counties using two epidemic scenarios for each county. In the prevention scenario, vaccines were distributed before any infection took place, and 5 percent the population of the blocks that were selected for infection was initially infected. In the intervention scenario 50 percent of the population of the infected blocks were infected initially, and at the beginning of the 6th day of the simulation vaccines were distributed. These scenarios were intended to simulate a naturally-occurring epidemic of a disease such as influenza or a random, mass-exposure bio-terror attack using perhaps smallpox, respectively. Each combination of county and scenario received its own run of 20 simulation sets. Within each simulation set, the following experiments were performed. The initially infected blocks were held constant for all of these experiments within a set. The amount of vaccine available was varied between 30 percent, 50 percent, 75 percent, and 90 percent of the population of the county. Out-degree, in-degree, eigenvector, eigenvector-in, inverse betweenness, and random centralities were tested. 4. RESULTS In Tables 2 and 3, we present the average peak infected percentage from our experiments. This data was obtained by first finding the maximum percent infected for each simulation. Then, the maximums of each of the 20 simulations for each experiment were arithmetically averaged. The maximum percentage infected at any given time indicates the severity of the epidemic and, consequently, the strain that epidemic places on health resources, such as hospitals. These averages, along with their corresponding standard deviations, are reported in Table 2 for the prevention scenario, and Table 3 for the intervention scenario. These results have been rounded to four decimal places to avoid the appearance Figure 5: 200 most central blocks in Hays County according to multiple centrality measures. Figure 3: Average total number of infected individuals vs. number of vaccines for Denton County intervention scenario experiments, log scale. of false precision. We also obtained the total number of infected individuals for each simulation execution. These totals allow us to determine the severity of infections that do not result in outbreaks. These results were averaged together on a perexperiment basis and plotted on a log scale. The plot for an intervention scenario in Denton County is presented in Figure 3. We also plotted the average percentage of the population that was in each state during every day of the simulation for each experiment to better visualize the progress of the simulated epidemics. Such a plot for a prevention scenario in Hays County at 50 percent vaccination is included as Figure 4. We used maps generated by QuantumGIS, which highlighted approximately the top 10 percent most central blocks according to multiple centrality measures, to ensure that our model was producing realistic results. These maps were also used as an aid in the analysis of our results. A map of Hays County, showing the 200 most central blocks according outdegree, in-degree, and inverse betweenness centralities, is included as Figure 5. This map was created by obtaining the list of the 200 most central census blocks under each centrality measure and marking them with distinct colors. Blocks that were central according to multiple centrality measures received their own colors. 5. DISCUSSION We found that in-degree and inverse betweenness centralities tended to be the most effective at containing epidemics. At low vaccination levels, such as 30 percent and 50 percent, where almost all of the population becomes infected under all vaccination strategies, the peak infected percentage was used to compare the effectiveness of various strategies. Lowering the number of infected people at one time reduces strain on public health infrastructure, allowing the cases that exist to be treated more effectively. At higher vaccination levels, where no significant peak occurred, the state totals, along with the plots of the course of the epidemics, were used to determine effectiveness. The peaks were not used to analyze vaccination effectiveness at the 75 percent and 90 percent intervention experiments, as the peaks were found to reflect the degree to which the disease had spread before the vaccines were distributed, and to have little relation to the effectiveness of allocation strategies. These analyses allowed us to determine which of the tested vaccination strategies was most effective at containing the disease. In all but one of our experiments, at least one of in-degree or inverse betweenness had lower peaks than the control, which was utilizing a random vaccination strategy. Neither measure had a lower peak significantly more times than the other, which shows than inverse betweenness and in-degree centralities are both effective in various situations. In-degree was significantly more effective in the intervention scenario, while inverse betweenness was more effective in the prevention scenario. In-degree centrality allocated vaccines to those blocks that were the most vulnerable, that is, the most likely to be contacted. Because it shielded vulnerable blocks from the disease, in-degree was more effective at containing epidemics that had already begun to spread throughout the county at the time of vaccination. Inverse betweenness centrality, however, allocated vaccines to those blocks most likely to be involved in disease transmission. This allocation % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random No vaccination 59.5177 (0.5823) No vaccination 56.9851 (1.1604) No vaccination 62.3313 (0.4791) Prevention 30% (23501) 31.8124 (0.3003) 29.2634 (0.3717) 32.5062 (0.3980) 29.9352 (0.4541) 29.1990 (0.2862) 29.7944 (0.4934) 30% (47132) 32.6743 (0.3588) 26.1614 (0.5565) 28.1819 (0.2369) 27.7457 (0.3377) 28.0340 (0.4022) 29.3217 (0.3645) 30% (198784) 34.9600 (0.1829) 32.2106 (0.1637) 34.4785 (0.1946) 32.1597 (0.1371) 32.3480 (0.1617) 32.3165 (0.2183) Scenario: Rockwall 50% (39168) 12.9924 (0.2802) 9.6473 (0.2685) 13.9375 (0.3912) 12.5988 (0.3685) 10.5134 (0.2083) 10.5293 (0.2976) Hays County 50% (78553) 13.8937 (0.1090) 9.3640 (0.1403) 11.1222 (0.1744) 10.8385 (0.1924) 9.9216 (0.1894) 10.5042 (0.2072) Denton County 50% (331307) 14.4588 (0.0759) 13.1199 (0.0911) 15.3266 (0.0855) 13.8856 (0.1084) 12.8719 (0.1070) 13.0035 (0.0921) County 75% (58752) 0.0207 (0.0079) 0.0181 (0.0086) 1.4525 (0.7278) 1.1630 (0.6695) 0.0220 (0.0106) 0.0184 (0.0135) 90% (70503) 0.0073 (0.0030) 0.0051 (0.0051) 0.0079 (0.0038) 0.0077 (0.0037) 0.0068 (0.0030) 0.0060 (0.0081) 75% (117830) 0.0208 (0.0066) 0.0172 (0.0071) 0.0183 (0.0058) 0.0190 (0.0061) 0.0175 (0.0057) 0.0244 (0.0115) 90% (141396) 0.0053 (0.0029) 0.0059 (0.0036) 0.0052 (0.0023) 0.0050 (0.0025) 0.0062 (0.0025) 0.0095 (0.0094) 75% (496960) 0.0196 (0.0028) 0.0178 (0.0040) 0.0374 (0.0359) 0.0212 (0.0039) 0.0190 (0.0024) 0.0186 (0.0086) 90% (596353) 0.0059 (0.0014) 0.0057 (0.0020) 0.0058 (0.0014) 0.0058 (0.0014) 0.0057 (0.0016) 0.0061 (0.0038) Table 2: For the prevention scenario, average peak infected percentage with standard deviation in parentheses. Lowest values in boldface. Because there were no vaccines to be distributed, there is only one value for no vaccination. % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random No vaccination 59.3418 (1.0607) No vaccination 58.0324 (1.8390) No vaccination 62.8987 (0.5339) Intervention Scenario: Rockwall County 30% (23501) 50% (39168) 75% (58752) 32.0758 (0.5265) 13.5754 (0.4445) 1.7544 (0.5848) 29.4420 (0.5808) 10.1374 (0.3663) 1.8280 (0.6144) 32.8470 (0.7541) 15.2428 (0.4611) 2.0908 (0.3910) 30.7960 (0.7225) 13.8876 (0.4673) 1.8606 (0.4803) 29.3326 (0.7046) 10.6855 (0.4030) 1.7150 (0.5744) 30.5434 (0.7600) 11.9360 (0.3728) 1.7970 (0.6138) Hays County 30% (47132) 50% (78553) 75% (117830) 32.3683 (0.7623) 13.4622 (0.3566) 1.8578 (0.6219) 27.5758 (0.9813) 10.2484 (0.4134) 1.9364 (0.6338) 28.9617 (0.8530) 12.1266 (0.4515) 1.8978 (0.6402) 28.4758 (0.7967) 11.9360 (0.4113) 1.8689 (0.6336) 28.2601 (0.9398) 9.8329 (0.4272) 1.8171 (0.6087) 29.0436 (1.0581) 11.3709 (0.6273) 1.9226 (0.6457) Denton County 30% (198784) 50% (331307) 75% (496960) 34.8273 (0.4330) 14.5756 (0.1599) 1.7676 (0.3085) 32.8160 (0.2959) 13.5322 (0.1470) 1.8390 (0.3245) 34.6035 (0.2898) 15.4285 (0.1743) 1.7687 (0.3033) 32.3856 (0.2001) 14.1328 (0.1577) 1.7384 (0.3112) 32.2367 (0.2579) 13.0264 (0.1090) 1.7361 (0.3060) 33.0462 (0.4129) 13.6822 (0.1478) 1.8266 (0.3223) 90% (70503) 1.6049 (0.5464) 1.6352 (0.5571) 1.6432 (0.5579) 1.6393 (0.5607) 1.6143 (0.5672) 1.6320 (0.5323) 90% (141396) 1.7342 (0.5743) 1.7479 (0.5707) 1.7367 (0.5627) 1.7339 (0.5832) 1.7230 (0.5756) 1.7459 (0.5652) 90% (596353) 1.6589 (0.2921) 1.6654 (0.3033) 1.6604 (0.2939) 1.6528 (0.2914) 1.6519 (0.2921) 1.6691 (0.3046) Table 3: For the intervention scenario, average peak infected percentage with standard deviation in parentheses. Lowest values in boldface. Because there were no vaccines to be distributed, there is only one value for no vaccination. Figure 4: The average percentage of the population that was latent, infectious, or infected (either latent or infectious) v. time for Hays County prevention scenario (50 percent vaccination). Points with a value of 0 were omitted from the graph. strategy was more effective in the prevention scenario because it blocked off likely transmission paths, which would have allowed the epidemic to spread quickly. Both of these targeting methods appear to be effective at reducing the severity of epidemics at the county level, though they are effective in different scenarios. This conclusion agrees with that of Matrajt and Longini, who found that vaccinating the vulnerable and those likely to transmit the disease was an optimal strategy at the national level [12]. Out-degree centrality was consistently the least effective method of vaccination, with peaks higher than those of random vaccination in most of the 30 percent and 50 percent vaccination experiments. This resulted from out-degree centrality’s tendency to allocate vaccines to high-population areas that are likely to spread the disease once it reaches them, regardless of their probability of infection. This result is supported by Figure 5, which shows out-degree targeting mainly small, urban blocks. Figure 5 also shows that out-degree centrality selected different blocks than the effective measures, which had many blocks in common. At 75 percent vaccination, at least one of in-degree and inverse betweenness was more effective at halting disease spread than random vaccination in all experiments. Similar to the 30 percent and 50 percent levels of vaccination, both measures were the most effective about half the time. In-degree centrality tended to fall slightly faster than inverse betweenness at the beginning of the simulation, though inverse betweenness was more effective at stopping disease spread near the end of the simulation, a results that agrees with those of Matrajt and Longini [12]. Effectively no disease spread occurred under all measures, at 90 percent vaccination. As at previous vaccination levels, in-degree and inverse betweenness were each most effective in half of the experiments. The differences in effectiveness in the prevention and intervention scenarios were much less significant than at lower vaccination levels. The lack of significant disease spread at 90 percent was a consequence of herd immunity, the primary factor influencing disease spread at high vaccination levels [14]. Due to the initial unvaccinated period, epidemics in the intervention scenario at 90 percent vaccination tended to last longer than those in the prevention scenario. Eigenvector-based measures, especially eigenvector-in centrality, which targeted vulnerable blocks that were likely to spread the disease to other vulnerable blocks, showed an interesting trend: their effectiveness increased with county size and the prevalence of urban areas. In fact, in the prevention scenario at 30 percent vaccination in Denton County, eigenvector-in centrality was the most effective, followed by inverse betweenness. In Rockwall County, which is small and sparsely populated, these measures were the worst performers. At 75 percent vaccination in Rockwall, eigenvector and eigenvector-in centralities produced epidemic curves similar to those observed at lower vaccination levels, which did not occur for the other measures. In contrast, eigenvector-based measures, especially eigenvector-in centrality, were more effective in Hays and Denton counties, which are larger and more heavily urbanized. In several experiments, eigenvectorin centrality outperformed random vaccination, though they were much less reliable than in-degree and inverse betweenness. Eigenvector-in centrality’s effectiveness increased in urbanized counties because eigenvector-based measures favor central blocks that are connected to other central blocks. Clusters of vulnerable blocks, which eigenvector-in centrality selects, were prevalent in counties such as Hays and Denton than in Rockwall, leading to the increased effectiveness of that centrality measure. However, as seen in Figure 3, eigenvector-based measures were still generally ineffective compared to other options. In-degree centrality, at 30 percent and 50 percent vaccination, caused the longest delay in the peak of the epidemic. This result arose because contacts within in-degree central blocks, which were more likely to occur than those within non-central ones, did not cause disease spread because those blocks were vaccinated. This resulted in the disease taking longer to spread because it was forced into less probable paths. In previous experiments with our simulator, we received unusual data, which we do not represent here. Specifically, random vaccination was reported as the most effective method for both the intervention and prevention scenarios in Denton County. These results were most likely a result of the stochastic nature of our simulator. It is possible, however, that this result was caused by in-degree centrality’s preference for the large number of highly-populated blocks in Denton County, which left large areas of the county unvaccinated. Some of the limitations of our work are our assumptions that the population is homogenous within each block, that contact rates between census blocks are approximated by Equation 6, and that the individual contact rate is constant. Another limitation is that we are only testing one isolated county at a time. These assumptions do not reflect reality, but are common approximations that have been used in computational models of epidemics such as the Global Stochastic Cellular Automata model [16]. Future work will primarily focus on addressing the limitations described above. Methods for simulating non-homogenous populations [19, 10] will be incorporated into our model. Alternative centrality measures and more accurate weighting formulas will be investigated. We will also attempt to determine what methods would produce results that surpass random vaccination in cases that are currently outliers. 6. REFERENCES [1] M. Ajelli, B. Gonçalves, D. Balcan, V. Colizza, H. Hu, J. J. Ramasco, S. Merler, and A. Vespignani. Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models. BMC infectious diseases, 10:190, Jan. 2010. [2] P. Bonacich. Factoring and weighting approaches to status scores and clique identification. 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Adaptive Bernstein–von Mises theorems in Gaussian white noise arXiv:1407.3397v3 [math.ST] 19 Dec 2016 Kolyan Ray∗ Abstract We investigate Bernstein–von Mises theorems for adaptive nonparametric Bayesian procedures in the canonical Gaussian white noise model. We consider both a Hilbert space and multiscale setting with applications in L2 and L∞ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets based on the posterior distribution. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved. AMS 2000 subject classifications: Primary 62G20; secondary 62G15, 62G08. Keywords and phrases: Bayesian inference, posterior asymptotics, adaptation, credible set, confidence set. Contents 1 Introduction 2 2 Statistical setting 2.1 Function spaces and the white noise model . . . . . . . . . . . . . . . . . . . 2.2 Weak Bernstein–von Mises phenomena . . . . . . . . . . . . . . . . . . . . . . 2.3 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 7 8 3 Bernstein–von Mises results 3.1 Empirical and hierarchical Bayes in `2 . . . . . . . . . . . . . . . . . . . . . . 3.2 Slab and spike prior in L∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 11 4 Applications 4.1 Adaptive credible sets in `2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Adaptive credible bands in L∞ . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13 15 5 Posterior independence of the credible sets 16 ∗ Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands. E-mail: k.m.ray@math.leidenuniv.nl Most of this work was completed during the author’s PhD at the University of Cambridge. This work was supported by UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 and the European Research Council under ERC Grant Agreement 320637. 1 6 Simulation example 19 7 Proofs 7.1 Proofs of weak BvM results in `2 (Theorems 3.1 and 3.2) 7.2 Proof of weak BvM result in L∞ (Theorem 3.5) . . . . . . 7.3 Credible sets . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Posterior independence of the credible sets . . . . . . . . . 7.5 Remaining proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 23 25 27 30 38 8 Technical facts and results 8.1 Results for `2 -setting . . . . . . . . . 8.2 Results for L∞ -setting . . . . . . . . 8.3 Wavelets . . . . . . . . . . . . . . . . 8.4 Weak convergence . . . . . . . . . . 8.5 Results on empirical and hierarchical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 39 41 43 44 45 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bayes procedures . . . . . . . . . . Introduction A key aspect of statistical inference is uncertainty quantification and the Bayesian approach to this problem is to use the posterior distribution to generate a credible set, that is a region of prescribed posterior probability (often 95%). This can be considered an advantage of the Bayesian approach since Bayesian credible sets can be computed by simulation. In particular, the Bayesian generates a number of posterior draws and then keeps a prescribed fraction of the draws, discarding the remainder which are considered “extreme” in some sense. From a frequentist perspective, key questions are whether such a method has a theoretical justification and what is an effective rule for determining which draws to discard. A natural approach is to characterize such draws using a geometric notion, in particular by considering a minimal ball in some metric. In finite dimensions, the Euclidean distance has a clear interpretation as the natural measure of size. However in infinite dimensions such a notion is less clear-cut: the L2 metric is the natural generalization of the Euclidean norm, but lacks a clear visual interpretation, while L∞ can be easily visualized but is more difficult to treat mathematically. From the Bayesian perspective of simulating credible sets, the practitioner ultimately seeks a practical and effective rule for sorting through posterior draws and such geometric interpretations can be viewed as somewhat artificial impositions. The aim of this article is therefore to study possible geometric choices of credible sets that behave well from a frequentist asymptotic perspective. (n) Consider data Y (n) arising from some probability distribution Pf , f ∈ F. We place a prior distribution Π on F and study the behaviour of the posterior distribution Π(· \| Y (n) ) (n) under the frequentist assumption Y (n) ∼ Pf0 for some non-random true f0 ∈ F as the data size or quality n → ∞. From such a viewpoint, the theoretical justification for posterior based inference using any (Borel) credible set in finite dimensions is provided by the Bernstein–von Mises (BvM) theorem (see [33, 50]). This deep result establishes mild conditions on the prior under which the posterior is approximately a normal distribution centered at an efficient estimator of the true parameter. It thus provides a powerful tool to study the asymptotic 2 behaviour of Bayesian procedures and justifies the use of Bayesian simulations for uncertainty quantification. A BvM in infinite-dimensions fails to hold in even very simple cases. Freedman [20] showed that in the basic conjugate `2 sequence space setting with both Gaussian priors and data, the BvM does not hold for `2 -balls centered at the posterior mean – see also the related contributions [17, 28, 34]. The resulting message is that despite their intuitive interpretation, credible sets based on posterior draws using an `2 -based selection procedure do not behave as in classical parametric models. Recently, Castillo and Nickl [11, 12] have established fully infinite-dimensional BvMs by considering weaker topologies than the classical √ Lp spaces. Their focus lies on considering spaces which admit 1/ n-consistent estimators and where Gaussian limits are possible, unlike Lp -type loss. Credible regions selected using these different geometries are shown to behave well, generating asymptotically exact frequentist confidence sets. In this paper, we explore this approach in practice via both theoretical results for adaptive priors, as well as by numerical simulations. We consider an empirical Bayes, a hierarchical Bayes and a multiscale Bayes approach. Before going into more abstract detail, it is useful to consider an example from [12] to numerically illustrate this approach in practice. Suppose that we observe Y1 , ..., Yn i.i.d. observations from an unknown density f0 on [0, 1]. We take a simple histogram prior Π, L f =2 L −1 2X hk 1ILk , ILk = (k2−L , (k + 1)2−L ], k ≥ 0, (1.1) k=0 L where the hk are drawn from a D(1, ..., 1)-Dirichlet distribution on the unit simplex in R2 . Here we ignore adaptation issues and select L = Ln based on the smoothness of the true function. Letting {ψlk : l ≥ 0, k = 0, ..., 2l − 1} denote the standard Haar wavelets and wl = l1/2+ for > 0 small, consider the multiscale credible ball (1.2) Cn = f : max wl−1 \|hf − fˆn , ψlk i\| ≤ Rn n−1/2 , k,l≤Ln where fˆn denotes the posterior mean and Rn = R(Y1 , ..., Yn ) is chosen such that Π(Cn \| Y1 , ..., Yn ) = 0.95. By Proposition 1 of [12], Pf0 (f0 ∈ Cn ) → 0.95 as n → ∞, whereas no such result is available for the L∞ -credible ball. Due to the conjugacy of the Dirichlet distribution with multinomial sampling, the posterior distribution can be computed straightforwardly and Rn can be easily obtained by simulation. For convenience we take f0 to be a Laplace distribution with location parameter 1/2 and scale parameter 5 that is truncated to [0, 1], that is f0 (x) ∝ e−5\|x−1/2\| 1[0,1] (x) with f0 ∈ H2s ([0, 1)) for s < 3/2. In Figure 1, we plotted the true density (solid black) and the posterior mean (red) in the cases n = 1000, 2000, 5000, 10000. We generated 100,000 posterior draws and plotted the 95% closest to the posterior mean in the M(w) sense (grey) to simulate Cn . We also used the posterior draws to generate a 95% credible band in L∞ by estimating Qn satisfying Π(f : \|\|f − fˆn \|\|∞ ≤ Qn \| Y ) = 0.95 and then plotting fˆn ± Qn (dashed black). We see that the L∞ diameter of Cn is strictly greater than that of the L∞ -credible band, with this difference particularly marked at the peak of the density. However, the diameter of Cn is spatially heterogeneous and has greatest width at the peak, whilst having smaller width around points where the true density is more regular. In all cases, Cn contains the true f0 , whereas the L∞ confidence band has more difficulty capturing the peak. 3 Figure 1: Credible sets based on the Dirichlet prior with the true density function (solid black), the posterior mean (red), a 95% credible band in L∞ (dashed black) and the set Cn given in (1.2) (grey). We have n = 1000, 2000, 5000 and 10000 respectively. The main message of this numerical example is that simulating the credible set Cn , which uses a slightly different geometry, yields a set that does not look particularly strange in practice and in fact resembles an L∞ credible band. Both approaches are methodologically similar, the only difference being the rule for discarding posterior draws. From a theoretical point of view, the difference between the two sets is far more significant, with Cn yielding exact coverage statements at the expense of unbounded L∞ diameter. It is however possible to improve upon the naive implementation of such sets to also obtain the optimal L∞ diameter (see Proposition 1 of [12] and related results below). Modifying the geometry in such a way to obtain an exact coverage statement therefore comes at little additional cost from a practitioner’s perspective. Nonparametric priors typically contain tuning or hyper parameters, and it is a key challenge to study procedures that select these parameters automatically in a data-driven manner. This avoids the need to make unreasonably strong prior assumptions about the unknown parameter of interest, since incorrect calibration of the prior can lead to suboptimal performance (see e.g. [30]). It therefore makes sense to use an automatic procedure, unless a practitioner is particularly confident that their prior correctly captures the fine details of the unknown parameter, such as its level of smoothness or regularity. Adaptive procedures are widely used in practice, with hyper parameters commonly selected using a hyperprior or an empirical Bayes method. In the case of Gaussian white noise, a number of Bayesian procedures have been shown to be rate adaptive over common smoothness classes (see for example [27, 29, 42]). Most such frequentist analyses restrict attention to obtaining contraction rates and do not 4 study coverage properties of credible sets. The focus of this paper is therefore to investigate nonparametric BvMs for adaptive priors, with the goal of studying the coverage properties of credible sets. In the case of Gaussian white noise, there has been recent work [30, 34] circumventing the need for a BvM by explicitly studying the coverage properties of certain specific credible sets. Of particular relevance is a nice recent paper by Szabó et al. [48], where the authors use an empirical Bayes approach combined with scaling up the radius of `2 -balls to obtain adaptive confidence sets under a so-called polished tail condition. Their approach relies on explicit prior computations and provides an alternative to the more abstract point of view taken here. One of our principal goals is exact coverage statements and this seems more difficult to obtain using such an explicit approach. Since adaptive confidence sets do not exist in full generality, we also require self-similarity conditions on the true parameter to exclude certain “difficult” functions [24],[26],[6]. In particular, we shall consider the procedure of [48] in Section 3.1 and obtain exact coverage statements under the self-similarity condition introduced there. We note other work dealing with BvM results in the nonparametric setting. Leahu [34] has studied the impact of prior smoothness on the existence of BvM theorems in the conjugate Gaussian sequence space model. Bickel and Kleijn [3], Castillo [8], Rivoirard and Rousseau [45] and Castillo and Rousseau [13] provide sufficient conditions for BvMs for semiparametric functionals. For the case of finite-dimensional posteriors with increasing dimension, see Ghosal [22] and Bontemps [4] for the case of regression or Boucheron and Gassiat [5] for discrete probability distributions. Much of the approach taken here can equally be applied to other statistical settings such as sparsity and inverse problems [43], but we restrict to the nonparametric regime for ease of exposition. Since our focus lies on BvM results and coverage statements and this changes little conceptually, we omit such generalizations to maintain mathematical clarity. 2 2.1 Statistical setting Function spaces and the white noise model We use the usual notation Lp = Lp ([0, 1]) for p-times Lebesgue integrable functions and denote by `p the usual sequence spaces. We consider the canonical white noise model, which is equivalent to the fixed design Gaussian regression model with known variance. For f ∈ L2 = L2 ([0, 1]), consider observing the trajectory (n) dYt 1 = f (t)dt + √ dBt , n t ∈ [0, 1], (2.1) where dB is a standard white noise. By considering the action of an orthonormal basis {eλ }λ∈Λ on (2.1), it is statistically equivalent to consider the Gaussian sequence space model (n) Yλ 1 ≡ Yλ = fλ + √ Zλ , n λ ∈ Λ, (2.2) where the (Zλ )λ∈Λ are i.i.d. standard normal random variables and the unknown parameter of interest f = (fλ )λ∈Λ is assumed to be in `2 . We denote by Pf0 or P0 the law of Y arising from (2.2) under the true function f0 . In the following, Λ will represent either a Fourier-type basis or a wavelet basis. In the `2 -setting, (2.2) can be interpreted purely in sequence form with Λ = N and we do not need to associate to it a time index t ∈ [0, 1]. 5 In L∞ we consider a multiscale approach so that Λ = {(j, k) : j ≥ 0, k = 0, ..., 2j − 1}. In particular, we consider an S-regular (S ≥ 0) wavelet basis of L2 ([0, 1]), {ψlk : l ≥ J0 − 1, k = 0, ..., 2l − 1}, with J0 ∈ N. For notational simplicity, denote the scaling function φ by the first wavelet ψ(J0 −1)0 . We consider either periodized wavelets or boundary corrected wavelets (see e.g. [38] for more details). Moreover, in certain applications we require in addition that the wavelets satisfy a localization property sup J0 −1 2X J0 /2 \|φJ0 k (x)\| ≤ c(φ)2 < ∞, sup x∈[0,1] k=0 j −1 2X \|ψjk (x)\| ≤ c(ψ)2j/2 < ∞, (2.3) x∈[0,1] k=0 j ≥ J0 (see Section 8.3 for more discussion). The sequence model (2.2) corresponds to estimating the wavelet coefficients flk = hf, ψlk i, for all (l, k) ∈ Λ, since any function f ∈ L2 generates such a wavelet sequence. Conversely, any such sequence P(flk ) generates the wavelet series of a function (or distribution if the sequence is not in `2 ) (l,k) flk ψlk . For s, δ ≥ 0, define the Sobolev spaces at the logarithmic level: ( ) ∞ X s,δ 2 k 2s (log k)−2δ \|fk \|2 < ∞ . H s,δ ≡ H2 := f ∈ `2 : \|\|f \|\|s,2,δ := k=1 From this we recover the usual definition of the Sobolev spaces H s ≡ H2s = H2s,0 and by duality we define for s > 0, H2−s := (H2s )∗ . By standard Hilbert space duality arguments, we can consider `2 as a subspace of H2−s and can similarly define the logarithmic spaces for s < 0 and δ ≥ 0 using the above series definition. In the `2 -setting we shall classify smoothness via the Sobolev hyper rectangles for β ≥ 0: ) ( Q(β, R) = f ∈ `2 : sup k 2β+1 fk2 ≤ R . k≥1 In the L∞ ([0, 1])-setting we consider multiscale spaces: for a monotone increasing sequence w = (wl )l≥1 with wl ≥ 1, define ( ) 1 M = M(w) = x = (xlk ) : \|\|x\|\|M(w) := sup max \|xlk \| < ∞ l≥0 wl k (for further references to multiscale statistics see [12]). A separable closed subspace is obtained by considering the restriction 1 M0 = M0 (w) = x ∈ M(w) : lim max \|xlk \| = 0 , l→∞ wl k that is those (weighted) sequences in M(w) that converge to 0. Note that M contains the space `2 , since \|\|x\|\|M ≤ \|\|x\|\|`2 as wl ≥ 1. In this setting, we consider norm-balls in the Besov β spaces B∞∞ ([0, 1]), H(β, R) = {f = (flk )(l,k)∈Λ : \|flk \| ≤ R2−l(β+1/2) , ∀(l, k) ∈ Λ}. β We recall that B∞∞ ([0, 1]) = C β ([0, 1]), the classical Hölder (-Zygmund in the case β ∈ N) spaces. For more details on these embeddings and identifications see [38]. 6 Whether an `2 -white noise defines a tight random element of M0 (w) depends √ on the weighting sequence (wl ). Recall that we call a sequence (wl )l≥1 admissible if wl / l % ∞ as l → ∞ [12]. Let Z = {Zλ = hZ, eλ i : λ ∈ Λ}, where Zλ ∼ N (0, 1) i.i.d., denote the Gaussian white noise in (2.2). We have from [11, 12] that for δ > 1/2 and (wl ) an admissible sequence, −1/2,δ Z defines a tight Gaussian Borel random variable on H2 and M0 (w) respectively, which we denote Z. In view of this tightness, we can consider (2.1) as a Gaussian shift model: 1 Y(n) = f + √ Z, n √ −1/2,δ where the above inequality is in the H2 - or M0 (w)-sense. Since n(Y(n) − f ) = Z in −1/2,δ H2 or M0 (w), it immediately follows that Y(n) is an efficient estimator of f in either norm. Among the two classes {H2s,δ }s∈R,δ≥0 and {M0 (w)}w of spaces considered, one can show that s = −1/2, δ > 1/2 and admissibility of w determine the minimal spaces where the law of the `2 -white noise Z is tight (see [11, 12] for further discussion). We therefore focus attention on these spaces since they provide the threshold for which a weak convergence approach can −1/2,δ work. For convenience, we denote H ≡ H(δ) ≡ H2 . We further denote the law of Z in H or M0 (w) by N as appropriate. 2.2 Weak Bernstein–von Mises phenomena Due to the continuous embeddings `2 ⊂ H and `2 ⊂ M0 (w), any Borel probability measure on `2 yields a tight Borel probability measure on H and M0 (w). Consider a prior Π on `2 and let Πn = Π(· \| Y (n) ) denote the posterior distribution based on data (2.2). For S a vector space and z ∈ S, consider the map τz : S → S given by √ τz : f 7→ n(f − z). Let Πn ◦τY−1 (n) denote the image measure of the posterior distribution (considered as a measure on H or M0 (w)) under the map τY(n) . Thus for any Borel set B arising from these topologies, √ Πn ◦ τY−1 n(f − Y(n) ) ∈ B \| Y (n) ), (n) (B) = Π( √ so that we can more intuitively write Πn ◦ τY−1 n(f − Y(n) ) \| Y (n) ), where L(f \| Y (n) ) (n) = L( denotes the law of f under the posterior. For convenience, we metrize the weak convergence of probability measures via the bounded Lipschitz metric (defined in Section 8.4). Recalling that we denote by N the law of the white noise Z in (2.2) as an element of S, we define the notion of nonparametric BvM. Definition 1. Consider data generated from (2.2) under a fixed function f0 and denote by P0 the distribution of Y (n) . Let βS be the bounded Lipschitz metric for weak convergence of probability measures on S. We say that a prior Π satisfies a weak Bernstein-von Mises phenomenon in S if, as n → ∞, √ E0 βS (Πn ◦ τY−1 n(f − Y(n) ) \| Y (n) ), N ) → 0. (n) , N ) = E0 βS (L( Here S is taken to be one of H(δ) for δ > 1/2, H −s for s > 1/2 or M0 (w) for (wl )l≥1 an admissible sequence. 7 The weak BvM says that the (scaled and centered) posterior distribution asymptotically looks like an infinite-dimensional Gaussian distribution in some ‘weak’ sense, quantified via the bounded Lipschitz metric (8.9). Weak convergence in S implies that these two probability measures are approximately equal on certain classes of sets, whose boundaries behave smoothly with respect to the measure N (see Sections 1.1 and 4.1 of [11]). 2.3 Self-similarity The study of adaptive BvM results naturally leads to the topic of adaptive frequentist confidence sets. It is known that confidence sets with radius of optimal order over a class of submodels nested by regularity that also possess honest coverage do not exist in full generality (see [26, 40] for recent references). We therefore require additional assumptions on the parameters to be estimated and so consider self-similar functions, whose regularity is similar at both small and large scales. Such conditions have been considered in Giné and Nickl [24], Hoffmann and Nickl [26] and Bull [6] and ensure that we remove those functions whose norms (measuring smoothness) are difficult to estimate and which statistically look smoother than they actually are. We firstly consider the `2 -type self-similarly assumption found in Szabó et al. [48]. Definition 2. Fix an integer N0 ≥ 2 and parameters ρ > 1, ε ∈ (0, 1). We say that a function f ∈ Q(β, R) is self-similar if dρN e X fk2 ≥ εRN −2β for all N ≥ N0 . k=N We denote the class of self-similar elements of Q(β, R) by QSS (β, R, ε). This condition says that each block (fN , ..., fdρN e ) of consecutive components contains at least a fixed fraction (in the `2 -sense) of the size of a “typical” element of Q(β, R), so that the signal looks similar at all frequency levels (see [48, 39, 40] for further discussion). The parameters N0 and ρ affect the results of this article through the sample size at which the asymptotic results take effect, that is n → ∞ implicitly implies statements of the form “for n ≥ n0 large enough”, where n0 depends on N0 and ρ. For this reason, the impact of N0 and ρ is not explicitly mentioned below and one may simply treat these constants as fixed (e.g. N0 = 2 and ρ = 2). The lower bound in Definition 2 can be slightly weakened to permit for example logarithmic deviations from N −2β . However since this results in additional technicality whilst adding little extra insight, we do not pursue such a generalization here. It is possible to consider a weaker self-similarity condition using a strictly frequentist approach [40], though this has not been explored in the Bayesian setting and it is unclear whether our P approach extends in such a way. Let Kj (f ) = k hf, φjk iφjk denote the wavelet projection at resolution level j. In L∞ we consider Condition 3 of Giné and Nickl [24], which can only be slightly relaxed [6]. Definition 3. Fix a positive integer j0 . We say that a function f ∈ H(β, R) is self-similar if there exists a constant ε > 0 such that \|\|Kj (f ) − f \|\|∞ ≥ ε2−jβ for all j ≥ j0 . We denote the class of self-similar elements of H(β, R) by HSS (β, R, ε). 8 In particular, since f ∈ H(β, R), we have that \|\|Kj (f ) − f \|\|∞ 2−jβ for all j ≥ j0 . What we really require is that there is at least one significant coefficient at the level log2 ((n/ log n)1/(2β+1) ) that the posterior distribution can detect. However, this level depends also on unknown constants in practice (see proof of Proposition 4.5) and so we require a statement for all (sufficiently large) resolution levels as in Definition 3. See Giné and Nickl [24] and also Bull [6] for further discussion about this condition. 3 3.1 Bernstein–von Mises results Empirical and hierarchical Bayes in `2 We continue the frequentist analysis of the adaptive priors studied in [29, 48, 47] in `2 . For α > 0 define the product prior on the `2 -coordinates by the product measure Πα = ∞ O N (0, k −2α−1 ), k=1 so that the coordinates are independent. A draw from this distribution will be Πα -almost 0 surely in all Sobolev spaces H2α for α0 < α. The posterior distribution corresponding to Πα is given by ∞ O n 1 Πα (· \| Y ) = N Yk , 2α+1 . (3.1) k 2α+1 + n k +n k=1 Hβ If f0 ∈ and α = β, it has been shown [2, 7, 30] that the posterior contracts at the minimax rate of convergence, while if α 6= β, then strictly suboptimal rates are achieved. Since the true smoothness β is generally unknown, two data-driven procedures have been considered in [29]. The empirical Bayes procedure consists of selecting the smoothness parameter by using a likelihood-based approach. Namely, we consider the estimate α̂n = argmax `n (α), (3.2) α∈[0,an ] where an → ∞ is any sequence such that an = o(log n) as n → ∞ and ∞ n 1X n2 2 log 1 + 2α+1 − 2α+1 `n (α) = − Y 2 k k +n k k=1 is the marginal log-likelihood for α in the joint model (f, Y ) in the Bayesian setting (relative to the infinite product measure ⊗∞ k=1 N (0, 1)). The quantity an is needed to uniformly control the finite dimensional projections of the empirical Bayes procedure to establish a parametric BvM (Theorem 7.2). The posterior distribution is defined via the plug-in procedure Πα̂n (· \| Y ) = Πα (· \| Y ) \|α=α̂n . If there exist multiple maxima to (3.2), then any of them can be selected. A fully Bayesian approach is to put a hyperprior on the parameter α. This yields the hierarchical prior distribution Z ∞ ΠH = λ(α)Πα dα, 0 where λ is a positive Lebesgue density on (0, ∞) satisfying the following assumption (Assumption 2.4 of [29]). 9 Condition 1. Assume that for every c1 > 0, there exists c2 ≥ 0, c3 ∈ R, with c3 > 1 if c2 = 0 and c4 > 0 such that for α ≥ c1 , −c3 c−1 exp (−c2 α) ≤ λ(α) ≤ c4 α−c3 exp (−c2 α) . 4 α The exponential, gamma and inverse gamma distributions satisfy Condition 1 for example. Knapik et al. [29] showed that both these procedures contract to the true parameter adaptively at the (almost) minimax rate, uniformly over Sobolev balls, and a similar result holds for Sobolev hyper rectangles. Both procedures satisfy weak BvMs in the sense of Definition 1. Theorem 3.1. Consider the empirical Bayes procedure described above. For every β, R > 0 and s > 1/2, we have sup E0 βH −s (Πα̂n ◦ τY−1 , N ) → 0 f0 ∈Q(β,R) as n → ∞. Moreover, for δ > 2 we have the (slightly) stronger convergence sup f0 ∈QSS (β,R,ε) E0 βH(δ) (Πα̂n ◦ τY−1 , N ) → 0 as n → ∞. Theorem 3.2. Consider the hierarchical Bayes procedure described above, where the prior density λ satisfies Condition 1. For every β, R > 0 and s > 1/2, we have sup f0 ∈Q(β,R) −1 E0 βH −s (ΠH n ◦ τY , N ) → 0 as n → ∞. Moreover, for δ > 2 we have the (slightly) stronger convergence sup f0 ∈QSS (β,R,ε) −1 E0 βH(δ) (ΠH n ◦ τY , N ) → 0 as n → ∞. The requirement of self-similarity for a weak BvM in H(δ) could conceivably be relaxed, but such an assumption is natural since it is anyway needed for the construction of adaptive confidence sets in Section 4.1. It is not clear whether this is a fundamental limit or a technical artefact of the proof. The condition δ > 2 is also required for technical reasons. Whilst minimax optimality is clearly desirable from a theoretical frequentist perspective, it may be too stringent a goal in our context. Using a purely Bayesian point of view, we derive an analogous result to Doob’s almost sure consistency result. Specifically, a weak BvM holds in H(δ) for prior draws, almost surely under both the empirical Bayes and hierarchical priors. For this, it is sufficient to show that prior draws are self-similar almost surely. Proposition 3.3. The parameter f0 is self-similar in the sense of Definition 2, Πα -almostsurely for any α > 0. Consequently, Πα̂n and ΠH satisfy a weak BvM in H(δ) for δ > 2, Πα -a.s., α > 0, and ΠH -a.s. respectively. In particular, f satisfies Definition 2 with smoothness α and parameters ρ > 1 and ε = ε(α, ρ, R) > 0 sufficiently small and random N0 sufficiently large, Πα -almost surely. As a simple corollary to Theorems 3.1 and 3.2, we have that the rescaled posteriors merge weakly 10 (with respect to weak convergence on H(δ)) in the sense of Diaconis and Freedman [19]. By Proposition 2.1 of [41], we immediately have that the unscaled posteriors merge weakly with respect to the `2 -topology since they are both consistent [29]. However, in the case of bounded Lipschitz functions (rather than the full case of continuous and bounded functions), we can improve this result to obtain a rate of convergence. Corollary 3.4. For every β, R > 0, s > 1/2 and δ > 2, we have sup f0 ∈Q(β,R) −1 −1 E0 βH −s (ΠH n ◦ τY , Πα̂n ◦ τY ) → 0 sup f0 ∈QSS (β,R,ε) −1 −1 E0 βH(δ) (ΠH n ◦ τY , Πα̂n ◦ τY ) → 0 as n → ∞. In particular, for S = H −s or H(δ) as above, Z L H . sup u d(Πn − Πα̂n ) = oP0 √ n u:\|\|u\|\|BL ≤L S 3.2 Slab and spike prior in L∞ Consider the slab and spike prior, whose frequentist contraction rate has been analyzed in Castillo and van der Vaart [15], Hoffmann et al. [27] and Castillo et al. [14]. The assumptions in [27] ensure that prior draws are very sparse and only very few coefficients are fitted. We therefore modify the prior slightly so that the prior automatically fits the first few coefficients of the signal without any thresholding. This ensures that the posterior will have a rough approximation of the signal before fitting wavelet coefficients more sparsely at higher resolution levels. This makes sense from a practical point of view by preventing overly sparse models and is in fact necessary from a theoretical perspective (see Proposition 3.7). Let Jn = blog n/ log 2c be such that n/2 < 2Jn ≤ n and define some strictly increasing sequence j0 = j0 (n) → ∞ such that j0 (n) < Jn . For the low resolutions j ≤ j0 (n) we fit a simple product prior where we draw the fjk ’s independent from a bounded density g that is strictly positive on R. For the middle resolution levels j0 (n) < j ≤ Jn , the fjk ’s are drawn independently from the mixture Πj (dx) = (1 − wj,n )δ0 (dx) + wj,n g(x)dx, n−K ≤ wj,n ≤ 2−j(1+θ) , for some K > 0 and θ > 1/2. All coefficients at levels j > Jn are set to 0. Since this is a product prior, one can sample from the posterior by sampling from each component separately (using either an MCMC scheme or explicit expressions depending on the choice of density g). We have a weak BvM in the multiscale space M0 (w), where the rate at which the admissible sequence (wl ) diverges depends on the how many coefficients we automatically √ fit in the prior via the sequence j0 (n). Recall that a sequence (wl )l≥1 is admissible if wl / l % ∞. Theorem 3.5. Consider the slab and spike prior defined above with lower threshold given by the strictly increasing sequence j0 (n) → ∞. The posterior distribution satisfies a weak BvM in M0 (w) in the sense of Definition 1, that is for every β, R > 0, sup f0 ∈H(β,R) E0 βM0 (w) (Πn ◦ τY−1 , N ) → 0 √ as n → ∞, for any admissible sequence (wl ) satisfying wj0 (n) / log n % ∞. 11 √ Note that in the limiting case wl = l, we recover j0 (n) ' log n, so that the prior automatically fits the same fixed fraction of the full 2Jn ' n coefficients. Since we consider only admissible sequences, the fraction of coefficients that the prior fits automatically is asymptotically vanishing. An alternative way to consider this result √ is in reverse: based on a desired rate in practice, we prescribe an admissible sequence wl = lul , where ul is some divergent sequence, and then pick j0 (n) appropriately. Taking j0 (n) to grow more slowly than any power of log n means (wl ) must grow faster than any power of l, resulting in a greater than logarithmic down-weighting of the wavelet coefficients in M(w). It may therefore be more appropriate to take j0 (n) a power of log n, which yields the following specific case. Corollary 3.6. Consider the slab and spike prior defined above with lower threshold j0 (n) ' 1 (log n) 2+1 for some > 0. Then it satisfies a weak BvM in M0 (w) in the sense of Definition 1, that is for every β, R > 0, sup f0 ∈H(β,R) E0 βM0 (w) (Πn ◦ τY−1 , N ) → 0 as n → ∞ for the admissible sequence wl = l1/2+ ul , where ul is any (arbitrarily slowly) diverging sequence. While the requirement to fit the first few coefficients of the prior is mild and of practical use in nonparametrics, it is naturally of interest to study the behaviour of the posterior distribution with full thresholding, that is when j0 (n) ≡ 0, which we denote by Π0 . In general √ however, the full posterior contracts to the truth at a rate strictly slower than 1/ n in M(w), √ so that a n-rescaling of the posterior cannot converge weakly to a limit. This holds even for self-similar functions. Proposition 3.7. Let (wl ) be any admissible sequence. Then for any β, R > 0, there exists ε = ε(β, R, ψ) > 0 and f0 ∈ HSS (β, R, ε) such that along some subsequence (nm ), E0 Π0 (\|\|f − Y\|\|M(w) ≥ Mnm nm −1/2 \| Y (nm ) ) → 1 for all Mn → ∞ sufficiently slowly. Consequently, for such an f0 , a weak BvM in M0 (w) in the sense of Definition 1 cannot hold. It is particularly relevant that Proposition 3.7 applies to self-similar parameters since a major application of the weak BvM is the construction of adaptive credible regions with √ good frequentist properties under self-similarity (see Proposition 4.5). On the level of a nrescaling as in Definition 1, the rescaled posterior distribution asymptotically puts vanishingly small probability mass on any given M(w)-ball infinitely often. p This occurs because the posterior selects non-zero coordinates by thresholding at the level log n/n rather than √ the √ required 1/ n (Lemma 1 of [27]). The weighting sequence (wl ) regularizes the extra log n factor at high frequencies, but does not do so at low frequencies. This√is the reason that the weighting sequence (wl ) depends explicitly on the thresholding factor log n in Theorem 3.5. It seems that using such an adaptive scheme on low frequencies of the signal causes the weak BvM to fail. This prior closely resembles the frequentist practice of wavelet thresholding, where such a phenomenon has also been observed. For example, Giné and Nickl [23] require similar (though stronger) assumptions on the number of coefficients that need to be fitted automatically to obtain a central limit theorem for the distribution function of the hard thesholding wavelet estimator in density estimation (Theorem 8 of [23]). 12 4 4.1 Applications Adaptive credible sets in `2 We propose credible sets from the hierarchical or empirical Bayes procedures, which we show are adaptive frequentist confidence sets for self-similar parameters. We consider the natural Bayesian approach of using the quantiles of the posterior distribution to obtain a credible set of prescribed posterior probability. By considering sets whose geometry is amenable to the space H(δ), the weak BvM P implies that such credible sets are asymptotically confidence sets. −1 −2δ f 2 . For a given significance level 0 < γ < 1, Recall that \|\|f \|\|2H(δ) = ∞ k=1 k (log k) k consider the credible set √ Cn = f : \|\|f − Y\|\|H(δ) ≤ Rn / n , (4.1) where Rn = Rn (Y, γ) is chosen such that Πα̂n (Cn \|Y ) = 1 − γ or ΠH (Cn \|Y ) = 1 − γ. Since the empirical and hierarchical Bayes procedures both satisfy a weak BvM in H(δ), we have from Theorem 1 of [11] that in both cases Pf0 (f0 ∈ Cn ) → 1 − γ and Rn = OP0 (1) as n → ∞, so that Cn is asymptotically an exact frequentist confidence set (of unbounded `2 -diameter). We control the diameter of the set using either the estimator α̂n or the posterior median as a smoothness estimate, and then use the standard frequentist approach of undersmoothing. In the first case, consider n o p √ C̃n = f : \|\|f − Y\|\|H(δ) ≤ Rn / n, \|\|f − fˆn \|\|H α̂n −n ≤ C log n , (4.2) where fˆn is the posterior mean, Rn is chosen as in Cn , n (chosen possibly data dependently) satisfies r1 /(log n) ≤ n ≤ (r2 / log n) ∧ (α̂n /2) for some 0 < r1 ≤ r2 ≤ ∞ and C > 1/r1 . 0 The undersmoothing by n is necessary since the posterior assigns probability 1 to H α for α0 < α̂n , while probability 0 to H α̂n itself. Geometrically, C̃n is the intersection of two `2 -ellipsoids, Cn and an H α̂n −n -norm ball. For a typical element f in C̃n , the size of the low frequency coordinates of f are determined by Cn , while the smoothness condition in C̃n acts to regularize the elements of Cn (which are typically not in `2 ) by shrinking the higher frequencies. Proposition 4.1. Let 0 < β1 ≤ β2 < ∞, R ≥ 1 and ε > 0. Then the confidence set C̃n given in (4.2) satisfies sup f0 ∈QSS (β,R,ε) β∈[β1 ,β2 ] Pf0 (f0 ∈ C̃n ) − (1 − γ) → 0 as n → ∞. For every β ∈ [β1 , β2 ], uniformly over f0 ∈ QSS (β, R, ε), 0 1/(4β+2) Πα̂n (C̃n \| Y ) = 1 − γ + OP0 n−C n for some C 0 > 0 independent of β, R, ε, N0 , ρ, while the `2 -diameter satisfies for δ > 2, \|C̃n \|2 = OP0 n−β/(2β+1) (log n)(2δβ+1/2)/(2β+1) . 13 The logarithmic correction in the definition of H(δ) that is required for a weak BvM causes the (log n)2δβ/(2β+1) penalty (which is O((log n)2δ ) uniformly over β ≥ 0); this is the price required for using a plug-in approach in H(δ). The remaining (log n)1/(4β+2) factor arises due to the second constraint in C̃n , where the H α̂n -radius must be taken sufficiently large to ensure C̃n has sufficient posterior probability. While the second constraint in (4.2) reduces the credibility below 1 − γ, Proposition 4.1 shows that this credibility loss is very small. The Bayesian approach takes care of this automatically since the posterior concentrates on a much more regular set than `2 . This is corroborated empirically by numerical evidence (see Figure 3), which shows that the credibility of the set C̃n rapidly approaches 1 − γ as n increases. Remark 4.2. A naive interpretation of Cn yields a credible set that is far too large, having unbounded `2 -diameter, with the additional constraint in C̃n needed to regularize the set. In actual fact the posterior does this regularization automatically with Cn being “almost optimal”. Proposition 4.1 could be rewritten for Cn with exact credibility Πα̂n (Cn \| Y ) = 1 − γ and `2 diameter satisfying 2δβ+1/2 0 1/(4β+2) − β Πα̂n (f ∈ Cn : \|\|f − fˆn \|\|2 ≤ Cn 2β+1 (log n) 2β+1 \| Y ) = 1 − γ + OP0 n−C n , for some C, C 0 > 0. In view of this, the sets Cn and C̃n are essentially the same from the point of view of the posterior, with Cn having exact credibility for finite n and correct `2 -diameter asymptotically and C̃n having the reverse. In particular, the finite time credibility “gap” for either having too large radius in Cn or smaller than 1 − γ credibility for C̃n is of the same size. Moreover, the above statement holds without the need for a self-similarity assumption, which is possible since the confidence set does not strictly have optimal diameter. The same notion also holds for Cn arising from the hierarchical Bayes procedure. Remark 4.3. By Lemma 8.2 the empirical Bayes posterior mean fˆn satisfies \|\|fˆn − Y\|\|H(δ) = √ oP0 (1/ n) and so is an efficient estimator of f0 in H(δ). Consequently one can substitute Y with fˆn in the definitions of Cn and C̃n . Replacing the estimate α̂n with the median αnM of the marginal posterior distribution λn (·\|Y ) yields a fully Bayesian analogue. To obtain the necessary undersmoothing over a target range [β1 , β2 ], we consider the shifted estimator β̂n = αnM − (C + 1)/ log n, where C = C(R, β2 , ε, ρ) = maxβ1 ≤β≤β2 C(R, β, ε, ρ) is the constant appearing in Lemma 8.7 (which can be explicitly computed). Consider n o p √ C̃n0 = f : \|\|f − Y\|\|H ≤ Rn / n, \|\|f − fˆn \|\|H β̂n ≤ Mn log n , (4.3) where fˆn is the posterior mean, Mn → ∞ grows more slowly than any polynomial and Rn is chosen as in Cn . Taking Cn arising from the hierarchical Bayesian procedure ΠH , C̃n0 is a “fully Bayesian” object. We have an analogue of Proposition 4.1. Proposition 4.4. Let 0 < β1 ≤ β2 < ∞, R ≥ 1 and ε > 0. Then the confidence set C̃n0 given in (4.3) satisfies sup f0 ∈QSS (β,R,ε) β∈[β1 ,β2 ] Pf0 (f0 ∈ C̃n0 ) − (1 − γ) → 0 14 as n → ∞. For every β ∈ [β1 , β2 ], uniformly over f0 ∈ QSS (β, R, ε), ΠH (C̃n0 \| Y ) = 1 − γ + oP0 (1) , while the `2 -diameter satisfies for δ > 2, \|C̃n0 \|2 = OP0 n−β/(2β+1) (log n)(2δβ+1/2)/(2β+1) . 4.2 Adaptive credible bands in L∞ We provide a fully Bayesian construction of adaptive credible bands using the slab and spike prior. The posterior median f˜ = (f˜n,lk )(l,k)∈Λ (defined coordinate-wise) takes the form of a thresholding estimator (c.f. [1]), which we use to identify significant coefficients. This has the advantage of both simplicity and interpretability and also provides a natural Bayesian approach for this coefficient selection. Such an approach was used by Kueh [31] to construct an asymptotically honest (i.e. uniform in the parameter space) adaptive frequentist confidence set on the sphere using needlets. In that article, the coefficients are selected based on the empirical wavelet coefficients with the thresholds selected conservatively using Bernstein’s inequality. In contrast, we use a Bayesian approach to automatically select the thresholding quantile constants that then yields exact coverage statements. Let √ Dn = {f : \|\|f − Y\|\|M(w) ≤ Rn / n}, (4.4) where Rn = Rn (Y, γ) is chosen such that Π(Dn \| Y ) = 1 − γ. We then define the data driven width of our confidence band r Jn 2l −1 X log n X vn σn,γ = σn,γ (Y ) = sup 1{f˜lk 6=0} \|ψlk (x)\|, (4.5) n x∈[0,1] l=0 k=0 where (vn ) is any (possibly data-driven) sequence such that vn → ∞. Under a local selfsimilarity type condition as in Kueh [31], one could possibly remove the supremum in (4.5) to obtain a spatially adaptive procedure. However, we restrict attention to more global self-similarity conditions here for simplicity. Since we consider wavelets satisfying (2.3), we have r r l −1 Jn 2X Jn X p log n log n X σn,γ ≤ vn sup \|ψlk (x)\| ≤ C(ψ)vn 2l/2 ≤ C 0 vn log n < ∞ a.s., n x∈[0,1] n l=0 k=0 l=0 for all n and γ ∈ (0, 1). Let πmed denote the projection onto the non-zero coordinates of the posterior median and in a slight abuse of notation set l πmed (Y )(x) = Jn 2X −1 X Ylk 1{f˜lk 6=0} ψlk (x), l=0 k=0 where we recall Ylk = R1 ψlk (t)dY (t). Consider the set √ Dn = {f : \|\|f − Y\|\|M(w) ≤ Rn / n, \|\|f − πmed (Y )\|\|∞ ≤ σn,γ (Y )}, 0 (4.6) where Rn is as in (4.4). This involves a two-stage procedure: we firstly calculate the required M(w)-radius Rn and then use the posterior median to select the coefficients deemed significant. 15 Proposition 4.5. Let 0 < β1 ≤ β2 < ∞, R ≥ 1 and ε > 0. Consider the slab and spike prior defined√ above with threshold j0 (n) → ∞ and let (wl ) be any admissible sequence that satisfies wj0 (n) / log n % ∞. Then the confidence set Dn given in (4.6), using the choice (wl ) and σn,γ (Y ) defined in (4.5) for vn → ∞, satisfies sup f0 ∈HSS (β,R,ε) β∈[β1 ,β2 ] Pf0 (f0 ∈ Dn ) − (1 − γ) → 0 as n → ∞. For every β ∈ [β1 , β2 ], uniformly over f0 ∈ HSS (β, R, ε), Π(Dn \| Y ) = 1 − γ + oP0 (1), while the L∞ -diameter satisfies \|Dn \|∞ = OP0 (n/ log n)−β/(2β+1) vn . Under self-similarity, Dn has radius equal to the minimax rate in L∞ up to some factor vn that can be taken to diverge arbitrarily slowly, again mirroring a frequentist undersmoothing penalty. The choice of the posterior median is for simplicity and can be replaced by any other suitable thresholding procedure, for example directly using the posterior mixing probabilities between the atom at zero and the continuous density component. One could also consider other alternatives to σn,γ that simultaneously control the L∞ norm of the credible set whilst also preserving coverage and credibility. A similar construction β̂n -ball, to the credible sets in Section 4.1 could also be pursued by intersecting Dn with a B∞1 where β̂n is a suitable estimate of the smoothness. Alternatively, in view of Remark 4.2, one can also show that ! ! wjn (β) −β/(2β+1) Π f ∈ Dn : \|\|f − Tn \|\|∞ ≤ p n (log n)(β+1)/(2β+1) Y = 1 − γ + oP0 (1) , jn (β) where Tn is an efficient estimator of f0 in M that is alsoprate-optimal in L∞ (e.g. (8.5) or (8.6)) and 2jn ∼ (n log n)1/(2β+1) . The factor wjn (β) / jn (β) can be made to diverge arbitrarily slowly by the prior choice of j0 (n). 5 Posterior independence of the credible sets −1/2,δ As shown above, the spaces H(δ) = H2 and M(w) yield credible sets with good frequentist properties. However, given the different geometries proposed, it is of interest to compare them to more classical credible sets. Consider the `2 -ball studied in [20, 48] (though without the blow-up factor of the latter) Cn`2 = {f : \|\|f − fˆn \|\|2 ≤ Q̃n (α̂n , γ)}, (5.1) where Q̃n (α̂n , γ) is selected such that Πα̂n (Cn`2 \| Y ) = 1 − γ. Since the posterior variance of Πα (· \| Y ) is independent of the data, the radius Q̃n (α̂n , γ) depends only on the data through α̂n . By Theorem 1 of [20] we have Q̃n (α, γ) = Qn n−α/(2α+1) , where Qn → Q > 0. Numerical examples of C̃n and Cn`2 are displayed in Section 6. Given the similarity of C̃n and Cn`2 in Figures 2 and 4, a natural question (voiced for example in [10, 39, 49]) is to what 16 extent these sets actually differ, both in theory and practice. From a purely geometric point of view these sets can be considered as infinite-dimensional ellipsoids with differing orientations. From a Bayesian perspective, an intriguing question is to what degree the decision rules on which these credible sets are based differ with respect to the posterior. For simplicity, we centre C̃n at the posterior mean fˆn , which we can do by Remark 4.3. Theorem 5.1. Suppose an in (3.2) satisfies in addition an ≤ log n/(6 log n log n). Then the (1 − γ)-H(δ)-credible ball C̃n defined in (4.2) and the (1 − γ)-`2 -credible ball Cn`2 defined in (5.1) are asymptotically independent under the empirical Bayes posterior, that is as n → ∞, Πα̂n (C̃n ∩ Cn`2 \| Y ) = Πα̂n (C̃n \| Y )Πα̂n (Cn`2 \| Y ) + oP0 (1) = (1 − γ)2 + oP0 (1) uniformly over f0 ∈ Q(β, R). The first equality above also holds with C̃n replaced by Cn or Cn`2 replaced by the blown-up `2 -credible ball studied in [48]. Moreover, the above statement also holds for the hierarchical Bayes posterior with C̃n replaced by the (1 − γ)-H(δ)-credible ball C̃n0 given in (4.3) and Cn`2 replaced by the corresponding hierarchical Bayes `2 -credible set. Theorem 5.1 says that the Bayesian decision rules leading to the construction of C̃n and Cn`2 are fundamentally unrelated - one contains asymptotically no information about the other. Although we can conclude that C̃n and (blown-up) Cn`2 are frequentist confidence sets with similar properties, they express completely different aspects of the posterior. Note that this is not simply an artefact of the prior choice since the equivalent prior credible sets are not independent under the prior despite its product structure. An alternative interpretation is to consider Bayesian tests based on the credible regions, which have optimal frequentist properties. In this context, the two tests screen different and unrelated features. While both of these approaches are valid, both for the frequentist and the Bayesian, Theorem 5.1 says that neither of these constructions can be reduced to the other. The H(δ)-credible sets are principally determined by the low frequencies (k ≤ kn , where kn → ∞ in the proof), whereas the `2 -credible sets are driven by the high frequencies (k > kn ). The product structure of the posterior asymptotically decouples these two regimes yielding the independence statement. In particular the H(δ)-norm down-weights the higher order frequencies enough that one is dealing with a close to finite-dimensional model. Such a result is unlikely to hold for arbitrary priors, unless there is some degree of posterior independence between the frequency ranges driving the different credible sets (though less independence than a full product posterior is necessary). The numerical simulations in Figure 3 corroborate Theorem 5.1 very closely, indicating that this result provides a good finite sample approximation to the posterior behaviour. The posterior draws plotted in Section 6 are approximately drawn from the posterior distribution conditioned to the respective credible sets. Corollary 5.2 quantifies how close these draws are in terms of the total variation distance \|\| · \|\|T V . `2 Cn n Corollary 5.2. Let ΠC̃ α̂n (· \| Y ), Πα̂n (· \| Y ) denote the posterior distribution conditioned to the sets C̃n , Cn`2 respectively. Then as n → ∞, `2 Cn n \|\|ΠC̃ α̂n (· \| Y ) − Πα̂n (· \| Y )\|\|T V = γ + oP0 (1). Proof. Each conditional distribution consists of the posterior distribution restricted to the relevant credible set and normalized by the same factor (1 − γ). The two distributions are 17 therefore identical on their intersection and so the total variation distance equals ! 1 Πα̂n (C̃n ∩ (Cn`2 )c \| Y ) Πα̂n (C̃nc ∩ Cn`2 \| Y ) 1 2γ(1 − γ) + oP0 (1) = γ + oP0 (1). + = 2 2 1−γ Πα̂n (C̃n \| Y ) Πα̂n (Cn`2 \| Y ) Turning to the L∞ -setting, P for mathematical convenience let us consider the slightly stronger Besov norm kf kB∞1 = l 2l/2 maxk \|hf, ψlk i\| as in Hoffmann et al. [27]. This norm 0 0 0 is closely related to the k · k∞ -norm via the Besov space embbedings B∞1 ⊂ L∞ ⊂ B∞∞ (Chapter 4.3 of [25]). Define ∞ DnL = {f : \|\|f − Tn \|\|B∞1 ≤ Qn (γ)}, 0 (5.2) (2) where Tn = Tn is given by (8.6) and is an efficient estimator of f0 in M that is also rate∞ optimal in L∞ and Qn (γ) is selected such that Π(DnL \| Y ) = 1 − γ. The choice of Tn is not essential, but it is convenient to select an estimator that can simultaneously act as the ∞ centering for both Dn and DnL . We take the density g in Π to be Gaussian to simplify certain computations. Analogous results to those in H(δ) then hold. Theorem 5.3. Consider the slab and spike prior Π with lower threshold j0 (n) → ∞ and let 2 g be the density of √ the Gaussian distribution N (0, τ ). Let (wl ) be any admissible sequence satisfying wj0 (n) / log n % ∞ as n → ∞. Then the (1 − γ)-M(w)-credible ball Dn defined in ∞ (4.6) and the (1 − γ)-L∞ -credible ball DnL defined in (5.2) are asymptotically independent under the posterior, that is as n → ∞, ∞ Π(Dn ∩ DnL ∞ \| Y ) = Π(Dn \| Y )Π(DnL \| Y ) + oP0 (1) = (1 − γ)2 + oP0 (1) uniformly over f0 ∈ H(β, R). In particular the choice of j0 (n) in Corollary 3.6 satisfies the conditions of Theorem 5.3 √ since then wj0 (n) ' un log n, where un can be made to diverge arbitrarily slowly. L∞ Corollary 5.4. Consider the same conditions as in Theorem 5.3 and let ΠDn (· \| Y ), ΠDn (· \| ∞ Y ) denote the posterior distribution conditioned to the sets Dn , DnL respectively. Then as n → ∞, L∞ \|\|ΠDn (· \| Y ) − ΠDn (· \| Y )\|\|T V = γ + oP0 (1). Heuristics for an extension to density estimation The proofs of Theorems 5.1 and 5.3 presented here rely on the independence of the coordinates in the Gaussian white noise model. This model can be viewed as an idealized version of other more concrete statistical models, being mathematically more tractable. In view of the extension of the nonparametric BvM to density estimation in [12], let us briefly discuss a heuristic of what we might expect in this setting. Suppose we observe Y1 , ..., Yn i.i.d. observations from an unknown density f0 on [0, 1]. Assume that f0 is uniformly bounded away from 0 and that f0 ∈ C β ([0, 1]), where 1/2 < β ≤ 1. Consider the simple histogram prior Π specified in (1.1), where we again ignore adaptation issues and select L = Ln → ∞ based on the smoothness of f0 . Such a prior has been shown 18 to contract optimally in L∞ by Castillo [9] and to satisfy a weak BvM in M0 by Castillo and Nickl [12]. Let (ψlk ) denote the Haar wavelet basis on [0, 1] and M the related multiscale space for a suitable admissible sequence (wl ). Further let πA denote the projection onto the elements of the Haar wavelet basis with resolution level contained in A. One can show that for suitable sequences jn,1 , jn,2 → ∞ satisfying 2jn,1 2jn,2 , √ Π(f : \|\|f − Tn \|\|M ≤ Rn / n, \|\|f − Tn \|\|∞ ≤ Q̄n \| Y ) √ = Π(f : kπ≤jn,1 (f − Tn )kM ≤ (Rn + o(1))/ n, kπ≥jn,2 (f − Tn )k∞ ≤ Q̄n + δn \|Y ) + oP0 (1), √ where Tn is a suitable centering, Rn / n and Q̄n are the (1 − γ)-quantiles of the respective credible sets and δn is selected small enough to only change the credibility of the latter set by oP0 (1). Using the conjugacy of the Dirichlet distribution with multinomial sampling, the posterior distribution for (hk ) is D(N1 + 1, ..., NL + 1), where Nk = \|{Yi : Yi ∈ ILk }\|. Observe that the law of flk = hf, ψlk i under the posterior depends principally on the observations falling within supp(ψlk ) = [k2−l , (k + 1)2−l ]. Unlike the Gaussian white noise model, there is dependence across the posterior wavelet coefficients due to the dependence within the Dirichlet distribution and the constraint that the number of observations sums to n. The k · kM -norm in the above display is the weighted maximum of {flk : l ≤ jn,1 }. If jn,1 does not grow too fast, this consists of relatively few “large sample” averages. Heuristically, this term behaves like a central order statistic, being driven by the average sample behaviour. On the contrary, the k·k∞ -term is determined by the largest coefficients at each resolution level l ≥ jn,2 . Since 2jn,1 2jn,2 , these can be seen to behave more like extreme order statistics, being the maximum of many almost independent “small samples” (at least relative to the frequencies l ≤ jn,1 ). Even though order statistics depend, by definition, on all observations, central and extreme order statistics asymptotically depend on the observations in orthogonal ways and become stochastically independent (c.f. Chapter 21 of [50]). One might therefore hope that the two norms in the previous display are asymptotically independent in the sense of Theorems 5.1 and 5.3. An alternative way to understand why the wavelet coefficients at a given resolution level may be considered “almost independent” under the posterior is via Poissonization. It is wellknown that density estimation is asymptotically equivalent to Poisson intensity estimation [37, 44], where one observes a Poisson process with intensity measure nf0 . Equivalently, the Poisson experiment corresponds to observing a Poisson random variable N with expectation n and then independently of N observing Y1 , ..., YN i.i.d. with density f0 . In this framework, the dependence induced by the number of observations summing to n is removed, meaning that the variables (N1 , ..., NL ) defined above are fully independent. The remaining dependence is due to the Dirichlet distribution and becomes negligible as the number of bins Ln → ∞. Since this equivalence is asymptotic in nature, one should expect such a heuristic to manifest itself also asymptotically. 6 Simulation example We now apply our approach in a numerical example. Following on from the example of the −1/2,δ M(w)-based credible set (1.2), we now consider the space H2 . Consider the Fourier sine 19 basis ek (x) = √ 2 sin(kπx), k = 1, 2, ..., and define the true function f0,k = hf0 , ek i2 = k −3/2 sin(k) so that the true smoothness is β = 1. We consider realisations of the data (2.2) at levels n = 500 and 2000 and use the empirical Bayes posterior distribution. We plotted the true f0 (black), the posterior mean (red) and an approximation to the credible sets (grey). To simulate the `2 credible balls Cn`2 given in (5.1), we sampled 2000 curves from the posterior distribution and kept the 95% closest in the `2 sense to the posterior mean and plotted them (grey). We performed the same approach to obtain the full H(δ)-credible set Cn given in (4.1) and then plotted the full adaptive confidence set C̃n given in (4.2) with C = 1 and n = 1/ log n. We also present the approximate credibility of C̃n by considering the fraction of the simulated √ curves from the ˆ posterior that satisfy the extra constraint of C̃n that \|\|f − fn \|\|H α̂n −n ≤ log n. This is given in Figure 2. While the true `2 and H(δ) credible balls are unbounded in L∞ , the posterior draws can be shown to be bounded in L∞ explaining the boundedness of the plots. Sampling from the posterior (and thereby implicitly intersecting the sets Cn`2 and C̃n with the posterior support) seems the natural approach for the Bayesian. Indeed those elements that constitute the “roughest” or least regular elements of the credible sets are not seen by the posterior, that is they have little or no posterior mass (see Lemma 8.3). The posterior contains significantly more information than merely the `2 or H(δ) norm of the parameter of interest, as can be seen by it assigning mass 1 to a strict subset of `2 . For further discussion on plotting such credible sets see [10, 36, 49]. For a given set of 2000 posterior draws, we also computed the credibility of C̃n at a chosen significance level and the credibility of the posterior draws falling in both C̃n and Cn`2 . This latter quantity has value (1 − γ)2 + oP0 (1) by Theorem 5.1. We repeated this 20 times and the average values are presented in Figure 3. The posterior distribution appears to have some difficulty visually capturing the resulting function at its peak. In fact the credible sets do “cover the true function”, but do so in an `2 rather than an L∞ -sense. Indeed, any `2 -type confidence ball will be unresponsive to highly localized pointwise features since they occur on a set of small Lebesgue measure (as in this case). Similar reasoning also explains the performance of the posterior mean at this point. The posterior mean estimates the Fourier coefficients of f0 and hence estimates the true function in an `2 -sense via its Fourier series. In Section 5 it was shown that the two approaches behave very differently theoretically and the numerical results in Figure 3 match this theory very closely. It appears that the two methods do indeed use different rejection criteria in practice resulting in different selection outcomes. The visual similarity between the `2 and H(δ)-credible balls in Figure 2 is therefore a result of the posterior draws themselves looking similar, rather than the methods performing identically. We note that already by n = 500, C̃n has the correct credibility so that the high frequency smoothness constraint is satisfied with posterior probability virtually equal to 1 (c.f. Proposition 4.1). C̃n is therefore an actual credible set for reasonable (finite) sample sizes rather than a purely asymptotic credible set. The posterior distribution already strongly regularizes the high frequencies so that the posterior draws are very regular with high probability. This can be quantitatively seen by the rapidly decaying variance term of the posterior distribution (3.1). This is indeed the case in the simulation, where the credibility gap is negligible, thereby 20 Figure 2: Empirical Bayes credible sets for the Fourier sine basis with the true curve (black) and the empirical Bayes posterior mean (red). The left panels contain the `2 credible ball Cn`2 given in (5.1) and the right panels contain the set C̃n given in (4.2). From top to bottom, n = 500, 2000 and α̂n = 1.29, 1.01, with the right-hand side each having credibility 95%. demonstrating that most of the posterior draws already satisfy the smoothness constraint in C̃n . We repeat the same simulation using the same true function f0,k = k −3/2 sin(k), but with basis equal to the singular value decomposition (SVD) of the Volterra operator (c.f. [30]): √ ek (x) = 2 cos((k − 1/2)πx), k = 1, 2, ... and plot this in Figure 4 for n = 1000.and plot this in Figure 4 for n = 1000. Unlike Figure 2, the resulting function has no “spike” and so both credible sets have no trouble visually capturing the true function (though one should remember that these are `2 rather than L∞ type credible sets). We now illustrate √ the multiscale approach using the slab and spike prior with lower threshold j0 (n) = log n, plotting the true function (solid black) and posterior mean (red) at levels n = 200, 500. We have used Haar wavelets, set g to be N (0, 1/2) and have taken prior weights wj,n = min(n−1 , 2−5.5j ), corresponding to K = 1 and θ = 5. For n = 200 and 500, we have fitted one scaling function plus 28 − 1 = 255 and 29 − 1 = 511 wavelet coefficients respectively (i.e. 2Jn +1 − 1). We again sampled 2000 curves from the posterior distribution and plotted the 95% closest to the posterior mean in the M(w) sense (grey) to simulate Dn in (4.4). We also used the posterior draws to generate a 95% credible band in L∞ by estimating ∞ Q̄n (0.05) and then plotting DnL in (5.2) (dashed black). Finally we computed local 95% credible intervals at every point x ∈ [0, 1] and joined these to form a credible band (dashed blue). This is given in Figure 5. 21 Chosen significance Credibility of C̃n Credibility of C̃n ∩ Cn`2 Expected credibility of C̃n ∩ Cn`2 0.95 0.9500 0.9020 0.9025 Chosen significance Credibility of C̃n Credibility of C̃n ∩ Cn`2 Expected credibility of C̃n ∩ Cn`2 0.95 0.9500 0.9025 0.9025 n=500 0.90 0.85 0.8999 0.8499 0.8102 0.7220 0.8100 0.7225 n=2000 0.90 0.85 0.9000 0.8500 0.8095 0.7226 0.8100 0.7225 0.80 0.8000 0.6406 0.6400 0.80 0.8000 0.6409 0.6400 Figure 3: Table showing the average credibility of C̃n , the average credibility of the posterior draws falling in both sets and the expected value of the latter (from Theorem 5.1). Figure 4: Empirical Bayes credible sets for the Volterra SVD basis with the true curve (black) and the empirical Bayes posterior mean (red) for n = 1000 and α̂n = 1.07. The left and right panels contain the `2 credible ball Cn`2 given in (5.1) and C̃n (credibility 95%) given in (4.2) respectively. We see from Figure 5 that each posterior draw consists of a rough approximation of the signal via frequencies j ≤ j0 (n) with a few “spikes” from the high frequencies; the rather unusual shape is a reflection of the prior choice. It is worth noting that the posterior draws are bounded in L∞ since the posterior contracts rate optimally to the truth in L∞ [27]. We see that the L∞ diameter of Dn is strictly greater than that of the L∞ -credible bands, though this only manifests itself in a few places. The size of the L∞ -bands is driven by the size of the spikes, which are few in a number but occur in every posterior draw, resulting in seemingly very wide credible bands. On the contrary, the local credible intervals ignore the spikes since less than 5% of the draws have a spike at any given point, resulting in much tighter bands. The dashed blue lines in effect correspond to the 95% L∞ -band from a prior fitting exclusively the low frequencies j ≤ j0 (n), which is a non-adaptive prior modelling analytic smoothness. This dramatically oversmoothes the truth resulting in far too narrow credible bands and is highly dangerous since it is known that oversmoothing the truth can yield zero coverage [30, 34]. 22 Figure 5: Slab and spike credible sets with the true curve (black), posterior mean (red), a 95% credible band in L∞ (dashed black), pointwise 95% credible intervals (dashed blue) and the set Dn given in (4.4) (grey). We have n = 200, 500 respectively. 7 Proofs In what follows denote by πj the projection onto either Vj = span{ek : 1 ≤ k ≤ j} or Vj = span{ψlk : 0 ≤ l ≤ j, k = 0, ..., 2l −1} depending on whether we are considering a Fouriertype basis or a wavelet basis. Similarly define π>j to be the projection onto span{ek : k > j} or Vj = span{ψlk : l > j, k = 0, ..., 2l − 1}. 7.1 Proofs of weak BvM results in `2 (Theorems 3.1 and 3.2) √ To prove a weak BvM we need to show that the posterior contracts at rate 1/ n to the truth in the relevant space and that the finite-dimensional projections of the rescaled posterior converge weakly to those of the normal law N (see Theorem 8 of [11] for more discussion). The latter condition is implied by a classical parametric BvM in total variation. Recall that α̂n is the maximum marginal likelihood estimator defined in (3.2). Theorem 7.1. For every β, R > 0 and Mn → ∞, we have sup f0 ∈Q(β,R) E0 Πα̂n (f : \|\|f − f0 \|\|S ≥ Mn Ln n−1/2 \|Y ) → 0, where S = H(δ) or H −s for s > 1/2. If S = H(δ) then Ln = (log n)3/2 (log log n)1/2 ; if in addition f0 ∈ QSS (β, R, ε), then the rate improves to Ln = 1 for δ ≥ 2. If S = H −s for s > 1/2, then Ln = 1. Proof. This contraction result is proved in the same manner as Theorem 2 in [29], with suitable modifications for the different norms used. In the case S = H(δ), self-similarity is needed to obtain a sharp upper bound on the behaviour of α̂n , which is required to bound the posterior bias. Theorem 7.2. The finite dimensional projections of the empirical Bayes procedure satisfy a parametric BvM, that is for every finite dimensional subspace V ⊂ `2 , sup f0 ∈Q(β,R) E0 \|\|Πα̂n (·\|Y ) ◦ TY−1 − NV (0, I)\|\|T V → 0, where πV denotes the projection onto V and Tz : f 7→ 23 √ nπV (f − z). Proof. Without loss of generality, let V = span{ek : 1 ≤ k ≤ J}. Using Pinsker’s inequality and that α̂n ∈ [0, an ] by the choice (3.2), \|\|Πα̂n (· \| Y ) ◦ TY−1 − N (0, IJ )\|\|2T V ≤ sup \|\|Πα (· \| Y ) ◦ TY−1 − N (0, IJ )\|\|2T V α∈[0,an ] ≤ sup KL(Πα (· \| Y ) ◦ TY−1 , N (0, IJ )), α∈[0,an ] where KL denotes the Kullback-Leibler divergence. Using the exact formula for the KullbackLeibler divergence between two Gaussian measures on RJ , KL(Πα (· \| Y ) ◦ TY−1 , N (0, IJ )) " J # J J X X k 4α+2 Yk2 1 X n n −J = +n + log 2 k 2α+1 + n (k 2α+1 + n)2 k 2α+1 + n ≤ . 1 2n k=1 J X k=1 2α+2 J n k=1 k 2α+1 + k 4α+2 Yk2 + k=1 J 4α+3 max Y 2 . n 1≤k≤J k Since E0 max1≤k≤J Yk2 = O(1) for fixed J and α ≤ an = o(log n) by the choice of an , the result follows. Proof of Theorem 3.1. Fix η > 0, let S denote H −s or H(δ) as appropriate and set Π̃α̂n = Πα̂n ◦ τY−1 . By the triangle inequality, βS (Π̃α̂n , N ) ≤ βS (Π̃α̂n , Π̃α̂n ◦ πj−1 ) + βS (Π̃α̂n ◦ πj−1 , N ◦ πj−1 ) + βS (N ◦ πj−1 , N ), for some j > 0. Using the contraction result of Theorem 7.1 and following the argument of Theorem 8 of [11], we deduce that the E0 -expectation of the first term is smaller that η/3 for sufficiently large j, uniformly over the relevant function class (in the case of H(δ) the result holds for all δ > 2 - we recall from the proof of that theorem that if the required contraction is established in H(δ 0 ), then the required tightness argument holds in H(δ) for any δ > δ 0 ). A similar result holds for the third term. For the middle term, note that the total variation distance dominates the bounded Lipschitz metric. For fixed j, we thus have that for n large enough, E0 βS (Π̃α̂n ◦ πj−1 , N ◦ πj−1 ) ≤ E0 \|\|Πα̂n (·\|Y ) ◦ TY−1 − NV (0, I)\|\|T V ≤ η/3, using Theorem 7.2 with V = Vj . A similar situation holds true for the hierarchical Bayesian prior. Theorem 7.3. Suppose that the prior density λ satisfies Condition 1. Then for every β, R > 0 and Mn → ∞, we have sup E0 ΠH f : \|\|f − f0 \|\|S ≥ Mn Ln n−1/2 \|Y → 0, f0 ∈Q(β,R) where S = H(δ) or H −s for s > 1/2. If S = H(δ) then Ln = (log n)3/2 (log log n)1/2 ; if in addition f0 ∈ QSS (β, R, ε), then the rate improves to Ln = 1 for δ ≥ 2. If S = H −s for s > 1/2, then Ln = 1. 24 Proof. This result is proved in the same manner as Theorem 3 in [29], with suitable modifications arising as in the proof of Theorem 7.1. Theorem 7.4. The finite dimensional projections of the hierarchical Bayesian procedure satisfy a parametric BvM, that is for every finite dimensional subspace V ⊂ `2 , sup f0 ∈Q(β,R) E0 \|\|ΠH (·\|Y ) ◦ TY−1 − NV (0, I)\|\|T V → 0, where πV denotes the projection onto V and Tz : f 7→ √ nπV (f − z). Proof. Again let V = span{ek : 1 ≤ k ≤ J}. Using Fubini’s theorem and that the total variation distance is bounded by 1, \|\|ΠH (· \| Y ) ◦ TY−1 − N (0, IJ )\|\|T V Z ∞ Z 1 λ(α \| Y )dΠα (· \| Y ) ◦ TY−1 (x)dα − dN (0, IJ )(x) = 2 RJ 0 Z Z 1 ∞ ≤ λ(α \| Y ) dΠα (· \| Y ) ◦ TY−1 (x)dα − dN (0, IJ )(x) dα 2 0 RJ Z ∞ = λ(α \| Y )\|\|Πα (· \| Y ) ◦ TY−1 − N (0, IJ )\|\|T V dα 0 Z αn −1 λ(α \| Y )dα ≤ sup \|\|Πα (· \| Y ) ◦ TY − N (0, IJ )\|\|T V 0<≤α≤αn 0 Z ∞ + λ(α \| Y )dα, αn where αn is defined in Section 8.5. The first term is oP0 (1) by the same argument as in the proof of Theorem 7.2 and the second term is oP0 (1) by the proof of Theorem 3 of [29]. Since the total variation distance is bounded, convergence in P0 -probability is equivalent to convergence in L1 (P0 ). Proof of Theorem 3.2. The proof is exactly the same as that of Theorem 3.1, using Theorems 7.3 and 7.4 instead of Theorems 7.1 and 7.2. 7.2 Proof of weak BvM result in L∞ (Theorem 3.5) Following Theorem 3.1 of [27], define the sets o n p Jn (γ) = (j, k) ∈ Λ : \|f0,jk \| > γ log n/n for γ > 0. In what follows, we denote by S the support of the prior draw, that is the set of non-zero coefficients of f = (fjk )(j,k)∈Λ drawn from the prior. We require the following contraction result. Theorem 7.5. Consider the slab and spike prior defined in Section 3.2 with lower threshold given by the strictly increasing sequence j0 (n) → ∞. Then for every 0 < βmin ≤ βmax , R > 0 and Mn → ∞, we have sup f0 ∈H(β,R) E0 Π(f : \|\|f − f0 \|\|M(w) ≥ Mn n−1/2 \| Y ) → 0 uniformly over β ∈ [βmin , βmax ], where (wl ) is any admissible sequence satisfying wj0 (n) ≥ √ c log n for some c > 0. 25 Proof of Theorem 7.5. Fix η > 0. Consider the event An = {S c ∩ Jn (γ) = ∅} ∩ {S ∩ Jnc (γ) = ∅} ∩ { max (j,k)∈Jn (γ) \|f0,jk − fjk \| ≤ γ p (log n)/n}. (7.1) By Theorem 3.1 of [27], there exist constants 0 < γ < γ < ∞ (independent of β and R) such that sup E0 Π(Acn \| Y ) . n−B , (7.2) f0 ∈∪β∈[βmin ,βmax ] H(β,R) for some B = B(βmin , βmax , R) > 0 (this follows since the probabilities of the complements of each of the events constituting An satisfy the above bound individually). We then have the following decomposition for some D = D(η) > 0 large enough to be specified later, E0 Π \|\|f − f0 \|\|M ≥ Mn n−1/2 \| Y ≤ E0 Π {\|\|f − f0 \|\|M ≥ Mn n−1/2 } ∩ {\|\|πj0 (f − f0 )\|\|M ≤ Dn−1/2 } ∩ An \| Y (7.3) −1/2 −1/2 + E0 Π {\|\|f − f0 \|\|M ≥ Mn n } ∩ {\|\|πj0 (f − f0 )\|\|M > Dn } ∩ An \| Y + E0 Π(Acn \| Y ). Note that the first term on the right-hand side of (7.3) is bounded by E0 Π({\|\|π>j0 (f − f0 )\|\|M ≥ (Mn − D)n−1/2 } ∩ An \| Y ). Combining this with (7.2), we can upper bound the right hand side of (7.3) by E0 Π({\|\|π>j0 (f − f0 )\|\|M ≥ M̃n n−1/2 } ∩ An \| Y ) + E0 Π(\|\|πj0 (f − f0 )\|\|M > Dn−1/2 \| Y ) + o(1), (7.4) where M̃n = Mn −D → ∞ as n → ∞. We bound the two remaining terms in (7.4) separately. For the first term in (7.4), we can proceed as in the proof of Theorem 3.1 of [27]. By the definition of the Hölder ball H(β, R), there exists Jn (β) such that 2Jn (β) ≤ k(n/ log n)1/(2β+1) for some constant k > 0 such that Jn (γ) ⊂ {(j, k) : j ≤ Jn (β), k = 0, ..., 2j − 1} and sup sup wl−1 max \|f0,lk \| ≤ f0 ∈H(β,R) l>Jn (β) k R2−Jn (β)(β+1/2) 1 p ≤ C(β, R) √ . n Jn (β) Consider now the frequencies j0 < l ≤ Jn (β). On the event An , we have that r log n γ 1 1 1 max \|flk − f0,lk \| ≤ γ ≤ √ , sup wj 0 n c n j0 <l≤Jn (β) wl k √ since wj0 (n) ≥ c log n by hypothesis. We thus have that on the event An , \|\|π>j0 (f − f0 )\|\|M = O(n−1/2 ) for any f0 ∈ H(β, R), which proves that the first term in (7.4) is 0 for n sufficiently large. Consider now the second term in (7.4). We shall use the approach of [9] using the moment generating function to control the low frequency terms. coordinates we Qj0 Recall that on these Π have the simple product prior Π(dx1 , ..., dxj0 ) = k=1 g(xi )dxi . Let E (· \| Y ) denote the 26 expectation with respect to the posterior measure. Following Lemma 1 of [9], we have the subgaussian bound √ 2 E0 EΠ (et n(flk −Ylk ) \| Y ) ≤ Cet /2 for some some C > 0. Using this and proceeding as in the proof of Theorem 4 of [12] yields ! √ √ nE0 EΠ \|\|πj0 (f − Y )\|\|M \| Y = E0 EΠ sup l−1/2 max n\|flk − Ylk \| \| Y ≤ C, j≤j0 k for some C > 0. By Markov’s inequality and then the triangle inequality, the second term in (7.4) is then bounded by √ √ C n n E0 EΠ \|\|πj0 (f − f0 )\|\|M \| Y ≤ E0 EΠ \|\|πj0 (Y − f0 )\|\|M \| Y + D D D (7.5) C E0 \|\|Z\|\|M + . ≤ D D By Proposition 2 of [12] and the fact that (wl ) is an admissible sequence, the first term in (7.5) is also bounded by C 0 /D for some C 0 > 0. Taking D = D(η) > 0 sufficiently large, (7.5) can be then made smaller than η/2. Proof of Theorem 3.5. Fix η > 0 and denote Π̃n = Πn ◦ τY−1 . By the triangle inequality, uniformly over the relevant class of functions, βM0 (Π̃n , N ) ≤ βM0 (Π̃n , Π̃n ◦ πj−1 ) + βM0 (Π̃n ◦ πj−1 , N ◦ πj−1 ) + βM0 (N ◦ πj−1 , N ), √ for fixed j > 0. Since we have a 1/ n-contraction rate in M for the posterior from Theorem 7.5, we can make the E0 -expectation of the first term smaller than η/3 by taking j sufficiently large, again using the arguments of Theorem 8 of [11]. We recall from the proof of that theorem that if the required contraction is established in M(w) for an admissible sequence (wl ), then the required tightness argument holds in M0 (w) for any admissible (wl ) such that wl /wl % ∞. A similar result holds for the third term. For the middle term, note that j0 (n) ≥ j for n large enough. For such n, the projected prior onto the first j coordinates is a simple product prior which satisfies the usual conditions of the parametric BvM, namely it has a density that is positive and continuous at the true (projected) parameter (see Chapter 10 of [50] for more details). Since the total variation distance dominates the bounded Lipschitz metric, this completes the proof. 7.3 Credible sets `2 confidence sets Proof of Proposition 4.1. By Lemma 8.1 and the definition of C̃n , we have √ sup \|P0 (f0 ∈ C̃n ) − P0 (\|\|f0 − Y\|\|H ≤ Rn / n)\| → 0 f0 ∈QSS (β,R,ε) β∈[β1 ,β2 ] as n → ∞, so that it is sufficient to show that the second probability in the above display tends to 1 − γ, uniformly over the relevant self-similar Sobolev balls. This follows directly by Theorem 1 of [11] (an examination of that proof shows that the convergence holds uniformly 27 over the parameter space as long as the weak BvM itself holds uniformly, as is the case here), thereby establishing the required coverage statement. Since n ≥ r1 / log n and C > 1/r1 in the definition (4.2) of C̃n , applying Lemma 8.3 (with η = 1) yields the inequality p 1 Πα̂n \|\|f − fˆn \|\|H α̂n −n ≥ C log n Y . exp −C 0 (log n)n 4α̂n +2 , where C 0 > 0 does not depend on α̂n . Since by Lemma 8.6 we have α̂n ≤ β for large enough n with P0 -probability tending to 1, the right-hand side is bounded by a multiple of exp(−C 00 (log n)n1/(4β+2) ) with the same probability. The completes the credibility statement. Let f1 , f2 ∈ C̃n and set g = f1 − f2 . Picking Jn ∼ [n/(log n)2δ−1 ]1/(1+2α̂n −2n ) yields \|\|g\|\|22 = ∞ X 2 \|gk \| = k=1 Jn X kk −1 2δ−2δ (log k) \|gk \| + k=1 2δ 2 \|\|g\|\|2H(δ) ∞ X k 2(α̂n −n )−2(α̂n −n ) \|gk \|2 k=Jn +1 −2(α̂n −n ) Jn \|\|g\|\|2H α̂n −n ≤ Jn (log Jn ) + = OP0 Jn (log Jn )2δ n−1 + Jn−2(α̂n −n ) (log n) 4δ(α̂n −n )+1 2(α̂ −n ) − 1+2α̂n −2 n n (log n) 1+2α̂n −2n , = OP0 n where the constants do not depend on g. Since \|α̂n − β\| = OP0 (1/ log n) by Lemma 8.6 and n = O(1/ log n) by assumption, some straightforward computations yield that \|\|g\|\|22 = OP0 (n−2β/(2β+1) (log n)(4δβ+1)/(2β+1) ) as n → ∞. Proof of Proposition 4.4. The proof follows in the same way as that of Proposition 4.1, using Lemma 8.7 and an analogue of Lemma 8.1. The only difference is for the credibility statement, where we no longer have an exponential inequality like Lemma 8.3. However, arguing as in [29] with the H β̂n -norm instead of the `2 -norm, one can show that under self-similarity the √ posterior contracts about the posterior mean at rate M̃n log n for any M̃n → ∞ (see also Theorem 1.1 of [49]). It then follows that the second constraint in (4.3) is satisfied with credibility 1 − oP0 (1). L∞ confidence bands Proof of Proposition 4.5. By Lemma 8.4, it suffices to prove all the results on the event Bn defined in (8.1). We firstly establish the diameter of the confidence set. Recall that πmed denotes the projection onto the non-zero coordinates of the posterior median and for a set of coordinates E, let πE denote the projection onto span(E). Taking f1 , f2 ∈ Dn and setting 2Jn (β) ' (n/ log n)1/(2β+1) , we have on Bn , \|\|f1 − f2 \|\|∞ ≤ \|\|f1 − πmed (Y )\|\|∞ + \|\|f2 − πmed (Y )\|\|∞ r Jn (β) 2l −1 X X log n ≤ 2 sup vn \|ψlk (x)\| n x∈[0,1] l=0 k=0 r r Jn (β) log n X l/2 2Jn (β) log n 0 ≤ C(ψ)vn 2 ≤ C vn = OP0 n n l=0 28 log n n β 2β+1 ! vn . We now establish asymptotic coverage. Split f0 = πJn (γ) (f0 ) + πJnc (γ) (f0 ). Since πJnc (γ) ◦ πmed (Y ) = 0 on Bn , we can write \|\|f0 − πmed (Y )\|\|∞ ≤ \|\|πmed (f0 − Y )\|\|∞ + \|\|(id − πmed ) ◦ πJn (γ) (f0 )\|\|∞ + \|\|πJnc (γ) (f0 )\|\|∞ , (7.6) where id denotes the identity operator. For the third term in (7.6), note that since f0 ∈ H(β, R), \|\|πJnc (γ) (f0 )\|\|∞ ≤ ∞ X 2l/2 l=0 max k:(l,k)∈Jnc (γ) Jn (β) ≤ X r 2 l/2 l=0 γ \|hf0 , ψlk i\| (7.7) β X log n 2β+1 log n −lβ 2 ≤ C(β, R) + . n n l>Jn (β) For the second term in (7.6), we note that any indices remaining satisfy (l, k) ∈ Jnc (γ 0 ) and so by the same reasoning as above, this term is also O((log n/n)β/(2β+1) ). By the proof of Proposition 3 of [26], we have that for f0 ∈ HSS (β, R, ε), sup \|hf0 , ψlk i\| ≥ d(b, R, β, ψ)2−j(β+1/2) . (l,k):l≥j ˜ Let J˜n (β) be such that 2 (n/ log n)1/(2β+1) ≤ 2Jn (β) ≤ (n/ log n)1/(2β+1) , where = (b, R, β, ψ) > 0 is small enough so that d/β+1/2 > γ 0 . Using this yields r r log n d(b, R, β, ψ) log n sup \|hf0 , ψlk i\| ≥ > γ0 . β+1/2 n n (l,k):l≥J˜n (β) We therefore have that on the event Bn , there exists (l0 , k 0 ) with l0 ≥ J˜n (β) such that f˜l0 k0 6= 0 and a non-zero coefficient therefore appears in the definition (4.5) of σn,γ . We can thus lower bound s r β log n 2J˜n (β) log n log n 2β+1 0 σn,γ ≥ vn sup \|ψl0 k0 (x)\| ≥ c(ψ)vn = c vn . (7.8) n x∈[0,1] n n Now, since vn → ∞ as n → ∞, we have from (7.7) and the remark after it that for sufficiently large n (depending on β and R), the last two terms in (7.6) satisfy \|\|(id − πmed ) ◦ πJn (γ) (f0 )\|\|∞ + \|\|πJnc (γ) (f0 )\|\|∞ ≤ C log n n β 2β+1 ≤ σn,γ /2. For the first term in (7.6) we recall that on Bn , the posterior median only picks up coefficients (l, k) with l ≤ Jn (β) ≤ Jn . Therefore on this event, X \|\|πmed (f0 − Y )\|\|∞ ≤ sup \|f0,lk − Ylk \|\|ψlk (x)\| x∈[0,1] (l,k):f˜lk 6=0 r ≤ C(ψ) log n n X (l,k):l≤Jn (β) 29 2 l/2 ≤C 0 log n n 1 2β+1 . Using the lower bound (7.8), we deduce that on Bn , \|\|πmed (f0 − Y )\|\|∞ ≤ σn,γ (Y )/2 for n large enough, uniformly over f0 ∈ HSS (β, R). Combining all of the above yields that Bn ⊂ {\|\|f0 − πmed (Y )\|\|∞ ≤ σn,γ }. We therefore conclude that √ P0 (f0 ∈ Dn ) = P0 ({\|\|f0 − Y\|\|M(w) ≤ Rn / n} ∩ {\|\|f0 − πmed (Y )\|\|∞ ≤ σn,γ } ∩ Bn ) + o(1) √ = P0 ({\|\|f0 − Y\|\|M(w) ≤ Rn / n} ∩ Bn ) + o(1) = 1 − γ + o(1), √ where we have used that P0 (Bn ) → 1 and that P0 (\|\|f0 − Y\|\|M(w) ≤ Rn / n) → 1 − γ by Theorem 5 of [12]. Noting finally that both of these probabilities converge uniformly over the relevant self-similar Sobolev balls, the coverage statement also holds uniformly as required. For the credibility statement it suffices to show that the second constraint in (4.6) is satisfied with posterior probability tending to 1. Again using that \|\|πmed (f0 − Y )\|\|∞ ≤ σn,γ (Y )/2 on Bn as well as (7.8), we have that uniformly over f0 ∈ H(β, R), E0 Π(f : \|\|f − πmed (Y )\|\|∞ ≥ σn,γ \| Y ) ≤ E0 Π(f : \|\|f − f0 \|\|∞ ≥ σn,γ /2 \| Y ) + E0 Π(f : \|\|f0 − πmed (Y )\|\|∞ ≥ σn,γ /2 \| Y ) ≤ E0 Π(f : \|\|f − f0 \|\|∞ ≥ c0 vn ((log n)/n)β/(2β+1) /2 \| Y ) + P0 (Bnc ) → 0 since the posterior contracts at rate (log n/n)β/(2β+1) by Theorem 3.1 of [27]. 7.4 Posterior independence of the credible sets Proof of Theorem 5.1. We first consider the fixed-regularity prior Πα with α ∈ [0, an ], re(α) placing the sets C̃n and Cn`2 respectively by the (1 − γ)-H(δ)-credible ball Cn and the (α,`2 ) (1 − γ)-`2 -credible ball Cn for Πα (· \| Y ) (i.e. (4.1) and (5.1) for Πα (· \| Y ) rather than Πα̂n (· \| Y )). By the definition of the posterior distribution (3.1), we can write a posterior draw f ∼ Πα (· \| Y ) as ∞ X 1 √ f − fˆn,α = ζ e , 2α+1 + n k k k k=1 where ζk ∼ N (0, 1) are independent and fˆn,α is the posterior mean. Let kn → ∞ be some sequence satisfying kn = o(n1/(4α+2) ). We shall prove the result by showing that the H(δ)credible ball is determined by the frequencies k ≤ kn , while the `2 -credible ball is determined by the frequencies k ≥ kn . We therefore decompose both credible balls according to the threshold kn . By Lemma 1 of [32], which can be adapted to the case D = ∞, and some elementary computations, we have the following exponential inequalities for any x ≥ 0:   r ∞ 2 X ζk C(δ) x x  P ≥ log kn + + ≤ e−x , (7.9) kn kn k(log k)2δ (log kn )2δ k=kn +1 P kn X ! p ζk2 ≥ kn + 2 kn x + 2x ≤ e−x , (7.10)  1 √ 1 − 23/2 n− 4α+2 x  ≤ e−x , 4α (7.11) k=1  P ∞ X k=kn +1 ζk2 k 2α+1 + 2α n ≤ n− 2α+1 30 where C(δ) < ∞ for δ > 1/2. Moreover, note that P kn X k=1 ζk2 x ≤ n (k 2α+1 + n)k(log k)2δ−1 ! p ≤ P ζ12 /2 ≤ x ≤ x/π, (7.12) using that ζ1 is standard normal. Define the event   ( ) kn ∞  X X ζk2 3C(δ)  2 Ãn,α = ≤ ∩ ζk ≤ 5kn  k(log k)2δ (log kn )2δ−1  k=kn +1 k=1   ∞  X  2α ζk2 1 − 2α+1 n ∩ ≥   k 2α+1 + n 8an k=kn +1 (k ) n X ζk2 1 ∩ ≥ . δ−1/2 (k 2α+1 + n)k(log k)2δ−1 n(log k ) n k=1 1/(2an +1) in (7.11) and Setting x = kn in the inequalities (7.9) and (7.10), x = 2−9 a−2 n n x = (log kn )−(δ−1/2) in (7.12) yields sup Πα (Ãcn,α \| Y ) ≤ 2e−kn + e−2 −9 a−2 n1/(2an +1) n √ + (log kn )−(2δ−1)/4 / π → 0 α∈[0,an ] as n → ∞, since an ≤ log n/(6 log log n) by assumption. We have that on Ãn,α , ∞ X 2α ζk2 1 − 2α+1 5kn n ≤ \|\|f − fˆn,α \|\|22 ≤ + , 8an n k 2α+1 + n k=kn +1 k n X ζk2 1 3C(δ) ˆn,α \|\|2 ≤ \|\|f − f ≤ + . H(δ) 2α+1 (k + n)k(log k)2δ n(log kn )2δ−1 n(log kn )δ−1/2 k=1 √ (α) Recall that Cn has radius equal to Rn / n, where Rn (Y, γ) →P0 R(γ) > 0 by Theorem (α,` ) 1 of [11]. Similarly, Cn 2 has radius Qn n−α/(2α+1) , where Qn → Q > 0 by Theorem 1 of [20]. Using these facts, the above bounds and the definition of kn , the probability 31 (α) (α,`2 ) Πα (Cn ∩ Cn \| Y ) equals o n 2α R2 \|\|f − fˆn \|\|2H(δ) ≤ n , \|\|f − fˆn \|\|22 ≤ Q2n n− 2α+1 ∩ Ãn Y + o(1) n X kn ζk2 1 2 −(2δ−1) = Πα ≤ R + O (log k ) , n n n (k 2α+1 + n)k(log k)2δ k=1 ∞ X 2α 1 ζk2 − 2α+1 − 2α+1 2 ∩ Ãn Y + o(1) n ≤ Qn + O kn n k 2α+1 + n Πα k=kn +1 1 ∞ ζk2 ζk2 Rn2 + o(1) X Q2n + o(n− 4α+2 ) = Πα ≤ , ≤ Y + o(1) 2α n k 2α+1 + n (k 2α+1 + n)k(log k)2δ n 2α+1 k=1 k=kn +1 ! kn X ζk2 Rn2 + o(1) = Πα ≤ Y n (k 2α+1 + n)k(log k)2δ k=1   1 ∞ − 4α+2 2 2 X ζk Q + o(n )  ≤ n Y + o(1), × Πα  2α 2α+1 k +n n 2α+1 X kn k=kn +1 where in the last line we have used the independence of the coordinates under the posterior. Using again the inequalities (7.9)-(7.12), the final line equals Πα \|\|f − fˆn,α \|\|2H(δ) ≤ (Rn2 + o(1))/n Y 2α 1 × Πα \|\|f − fˆn,α \|\|22 ≤ Q2n + o(n− 4α+2 ) n− 2α+1 Y + o(1) (2) =: Π(1) α,n × Πα,n + o(1). Since supα∈[0,an ] Πα (Ãcn,α \| Y ) → 0, the previous display holds uniformly over α ∈ [0, an ] so that we have shown sup (2) Πα (Cn(α) ∩ Cn(α,`2 ) \| Y ) − Π(1) α,n × Πα,n = o(1). (7.13) α∈[0,an ] For the full empirical Bayes posterior, note that the second constraint in (4.2) is satisfied with posterior probability 1 − oP0 (1) uniformly over f0 ∈ Q(β, R) by the proof of Proposition 4.1, so that it suffices to prove the theorem with Cn in (4.1) instead of C̃n . Since an = o(log n), we can take kn → ∞ such that kn = o(n1/(4an +2) ), from which also kn = o(n1/(4α+2) ) for all α ∈ [0, an ], the interval over which α̂n ranges. By (7.13), it therefore remains to check that (1) (2) Πα̂n ,n , Πα̂n ,n →P0 1 − γ as n → ∞. For this we require a finer understanding of the posterior behaviour of the norms in (1) the second to last display. Consider firstly Πα̂n ,n . By Theorem 3.1 and Lemma 8.2, Πα̂n satisfies a weak BvM in H(δ) with centering fˆn = fˆn,α̂ instead of Y. By Theorem 1 of [11], n Rn →P0 Φ̃−1 (1 − γ) > 0, where Φ̃ is defined via Φ̃(t) = N (kZkH(δ) ≤ t) and we recall N is (1) the law of the white noise Z as an element of H(δ). Note that Πα̂n ,n equals N (kZkH(δ) √ ˆ ≤ Rn + o(1)) + O sup \|Πα̂n ( n\|\|f − fn,α̂n \|\|H(δ) ≤ t\|Y ) − N (kZkH(δ) ≤ t)\| . t≥0 32 Since Φ̃ is strictly monotone and continuous, the first term equals Φ̃(Rn + o(1)) = (1 − γ) + oP0 (1). Using that H(δ)-norm balls form a uniformity class for N (see the proof of Theorem 1 of [11]), the second term is oP0 (1) as required. (2) We now turn our attention to Πα̂n ,n . By Theorem 1 of [29], supf0 ∈Q(β,R) P0 (α̂n ∈ [αn , αn ]) → 1, where αn , αn are defined in Section 8.5. By Lemma 1(i) of [29], αn ≥ β − C/ log n ≥ β/2 for n large enough. Since also α̂n ≤ an by the choice (3.2), supf0 ∈Q(β,R) P0 (α̂n ∈ [β/2, an ]) → 1 and we may therefore restrict α̂n to this interval. By Theorem 1 of Freedman [20], we have that for f ∼ Πα (·\|Y ), p kf − fˆn,α k22 = Cn,α + Dn,α Zn,α , R∞ R∞ 4α+1 2α where Cn,α n 2α+1 → Cα = 0 (1 + u2α+1 )−1 du, Dn,α n 2α+1 → Dα = 2 0 (1 + u2α+1 )−2 du, Zn,α has mean 0, variance 1 and Zn,α →d N (0, 1) as n → ∞. We wish to make these three convergence statements uniform over α ∈ [β/2, an ]. The first two statements essentially follow from [20] upon keeping careful track of the remainder terms. Let gα (u) = 1/(1 + u2α+1 ). 2α P∞ 2α+1 + n) and n 2α+1 /(k 2α+1 + n) = h g (kh ), where First note that Cn,α = n α n k=1 1/(k 2α hn = n−1/(2α+1) . Using these yields that for any L > 0, \|Cn,α n 2α+1 − Cα \| is bounded above by L/hn X Z hn gα (khn ) − gα (u)du + 0 k=1 ∞ X L Z ∞ hn gα (khn ) + gα (u)du. L k=L/h+1 The second and third terms are easily bounded by L−2α /(2α). The first term is just the error RL when approximating 0 gα with its (right) Riemann sum with L/h points, which is bounded 1 1 by kgα0 kL∞ [0,L] L2 /(2L/hn ). One can show that kgα0 kL∞ [0,∞) = α1− 2α+1 (α+1)1+ 2α+1 /(2α+1). Substituting this into the bound for the Riemann sum and optimizing the three terms over L gives that the previous display is bounded by a multiple of 1 α 1 1 (2α)2− 2α+1 (2α + 2)1+ 2α+1 2α + 1 2α ! 2α+1 2α hn2α+1 . It can be checked that over the range 0 < β/2 ≤ α ≤ an → ∞, the above display without 2α the hn2α+1 term is maximized at an for n large enough, whereupon it can be bounded by 2α − 2α a constant multiple of an . The continuous function α 7→ hn2α+1 = n (2α+1)2 has a single minimum on [0, ∞) occurring at α = 1/2, is strictly decreasing on [0, 1/2], strictly increasing on [1/2, ∞) and attains its maximal value of 1 at α = 0, ∞. Since we consider only the region α ≥ β/2, it follows that the maximum will occur at an for n large enough, depending only on β. We have therefore shown that the previous display is bounded above by log n − 2an 2 (2a +1) n C(β)an n ≤ C(β) exp log an − → 0, 4an where the inequality holds for n large enough (depending on an ) and the convergence to zero 2α follows since an ≤ log n/(6 log n log n) by assumption. In conclusion, supα∈[β/2,an ] \|Cn,α n 2α+1 − 4α+1 Cα \| → 0. Identical computations yield that supα∈[β/2,an ] \|Dn,α n 2α+1 − Dα \| → 0. 33 It remains only to show the uniformity of the convergence in distribution, which is based on the central limit theorem (Theorem 1 of [20]). Under the posterior, −1/2 Zn,α = Dn,α ∞ X k=1 1 k 2α+1 + n 2 (ζn,k − 1) =: ∞ X Xn,k , k=1 where ζn,k ∼ N (0, 1) are independent and −1/2 2 Xn,k = Dn,α (ζn,k − 1)/(k 2α+1 + n) are also independent. Recalling the exact definition of Dn,α from Theorem 1 of [20], Dn,α = P Pn1/(2α+1) −2 1 −2+ 1 r 2α+1 + n)−2 ≥ 1 2α+1 . Letting λ = E\|ζ 2 2 ∞ n = 2n r k=1 (k k=1 n,k − 1\| , 2 ∞ X  −3/2 E\|Xn,k \|3 ≤ Dn,α λ3  n1/(2α+1) X k=1 k=1 1  1 + n3 X k −6α−3  k>n1/(2α+1) 1 1 ≤ 2(2n2− 2α+1 )3/2 λ3 n−3+ 2α+1 = 25/2 λ3 n− 4α+2 . Let Fn,α and Φ denote the cdfs of Zn,α and the standard normal distribution respectively. By the Berry-Esseen theorem for infinite arrays (Theorem 3.2 of [21] with summability matrix pn,k ≡ 1), there exists a universal constant C0 such that kFn,α − Φk∞ ≤ C0 ∞ X 1 E\|Xn,k \|3 ≤ 25/2 C0 λ3 n− 4α+2 , k=1 which implies supα∈[β/2,an ] kFn,α − Φk∞ → 0 since an = o(log n). We have shown that p 2α+1/2 2α kf − fˆn,α k22 = (Cα + o(1))n− 2α+1 + ( Dα + o(1))n− 2α+1 Zn,α , where all o(1) terms and the convergence in distribution are uniform over α ∈ [β/2, an ]. We (2) consequently see that the remainder term in the posterior probability Πα,n only changes this (2) probability by o(1) for any α ∈ [β/2, an ] and so Πα̂n ,n = (1 − γ) + oP0 (1). (2) Proof of Theorem 5.3. Throughout we will write Tn instead of Tn for convenience, where  ˆ if j ≤ j0 (n),  fn,jk (2) Tn,jk = Yjk 1{f˜jk 6=0} if j0 (n) < j ≤ blog n/ log 2c.   0 if blog n/ log 2c < j, and fˆn denotes the posterior mean of the slab and spike procedure. Since the second constraint in (4.6) is satisfied with posterior probability 1 − oP0 (1) uniformly over f0 ∈ H(β, R) by the proof of Proposition 4.5, it suffices to prove the theorem p with Dn in (4.4) instead of Dn . Let f0 ∈ H(β, R) and jn → ∞ satisfy jn ≤ j0 (n), wjn ≤ j0 (n) and wj2n 2jn = o(2j0 (n) ). Similarly to Theorem 5.1, we shall decompose both credible balls according to the threshold jn . For this we must understand the typical sizes of the projections of f − Tn in both norms under the posterior. 34 Consider firstly k · kM . For the frequencies l > j0 (n), \|\|π>j0 (n) (f −Tn )\|\|M ≤ \|\|π>j0 (n) (f −f0 )\|\|M +\|\|π>j0 (n) (f0 −Y)\|\|M +\|\|π>j0 (n) (Y−Tn )\|\|M . (7.14) p The third term is OP0 (wj−1 (log n)/n) by the proof of Lemma 8.5. For the first term, on (n) 0 the event An defined in (7.1), max wl−1 max \|flk − f0,lk \| ≤ l>j0 (n) k max wl−1 max \|flk − f0,lk \| + max wl−1 max \|f0,lk \| k k l>Jn (β) p p −1 −1 ≤ wj0 (n) (log n)/n + wJn (β) (log n)/n p = O(wj−1 (log n)/n). (n) 0 j0 (n)<l≤Jn (β) For the second term in (7.14), E0 \|\|π>j0 (n) (f0 − Y)\|\|M(w) p j0 (n) √ E0 \|\|π>j0 (n) (Z)\|\|M(√l) = O ≤ wj0 (n) n ! p j0 (n) √ wj0 (n) n using that E0 \|\|Z\|\|M(√l) is finite by Proposition 2 of [12]. Combining these yields Π(f : \|\|π>j0 (n) (f − Tn )\|\|M = O(wj−1 0 (n) p (log n)/n) \| Y ) = 1 − oP0 (1) since j0 (n) . log n. (2) Recall that by definition Tn,lk equals the posterior mean for l ≤ j0 (n) (see (8.6)), so that τ2 (2) flk − Tn,lk \|Ylk ∼ N 0, , 0 ≤ l ≤ j0 (n), 1 + nτ 2 under the posterior. For √ (ζlk ) i.i.d. standard Gaussians, we have the well-known bound E max0≤k<2l \|ζlk \| ≤ C l for some universal constant C. Applying the Borell-SudakovTsirelson inequality [35] to the maximum at level l yields that for M > 0 large enough, p Π(f :k(π>jn − π>j0 (n) )(f − Tn )kM ≥ M jn wj−1 n−1/2 \|Y ) n √ 1 M jn τ √ =P max ζlk ≥ max √ wj n n jn <l≤j0 (n) wl 0≤k<2l 1 + nτ 2 ! √ √ j0 (n) X M jn wl 1 + nτ 2 √ ≤ P max \|ζlk \| − E max \|ζlk \| > − E max \|ζlk \| wjn nτ 0≤k<2l 0≤k<2l 0≤k<2l l=jn +1  !2  √ j0 (n) X w j 0 l √n − C l ≤ Ce−c jn → 0. ≤2 exp −c M wj n l l=jn +1 Combining this with the above inequalities gives p Π(f : \|\|π>jn (f − Tn )\|\|M = O(wj−1 (log n)/n) \| Y ) = 1 − oP0 (1). 0 (n) (7.15) We now show that the k · kM -norm of the remaining frequencies j ≤ jn is of strictly larger size with high probability. For any un → 0, √ Π(f : \|\|πjn (f − Tn )\|\|M ≤ w0 un n−1/2 \| Y ) ≤ Π(f : \|ζ00 \| ≤ cun \| Y ) ≤ 2ceun / 2π = o(1). (7.16) 35 √ In particular, taking un log n/wj0 (n) gives the result. Turn now to k · kB∞1 0 . For f ∈ Dn , \|\|πjn (f − Tn )\|\|B∞1 = 0 jn X 2l/2 max \|flk − Tn,lk \| ≤ k l=0 jn X l=0 Rn 2l/2 wl √ = OP0 n wjn 2jn /2 √ n ! . (7.17) Note that Π(f : kπ>jn (f − Tn )kB∞1 ≤ c2 0 j0 (n)/2 p j0 (n)/n \| Y ) ≤ P max 0≤k<2j0 (n) ζj0 (n)k p ≥ c j0 (n) . 0 (7.18) Using again the Borell-Sudakov-Tsirelson inequality as above gives that the right-hand side 00 is O(2−c j0 (n) ) for c > 0 small enough. Combining (7.15)-(7.18), we have shown that for some √ C1 , ..., C4 > 0 and un → 0 such that un log n/wj0 (n) , √ n log n √ , Ān := \|\|π>jn (f − Tn )\|\|M ≤ C1 wj0 (n) n \|\|πjn (f − Tn )\|\|B∞1 0 un kπjn (f − Tn )kM ≥ C2 √ , n wj 2jn /2 , ≤ C3 n√ n kπ>jn (f − Tn )kB∞1 0 p j0 (n)2j0 (n)/2 o √ ≥ C4 n satisfies Π(Ān \|Y ) = 1 − oP0 (1). √ Write δn := wjn 2jn /2 / n. Note that by the previous display and the assumptions on the p √ ∞ L j (n) 0 growth of jn → ∞, the radius Q̄n (γ) of the credible set Dn satisfies Q̄n & j0 (n)2 / n δn with P0 -probability tending to one. Using the independence of the different coordinates ∞ under the posterior, the probability Π(Dn ∩ DnL \| Y ) equals √ Π \|\|f − Tn \|\|M ≤ Rn / n, \|\|f − Tn \|\|∞ ≤ Q̄n ∩ Ān \| Y + oP0 (1) n p √ = Π \|\|πjn (f − Tn )\|\|M ≤ (Rn + O(wj−1 log n))/ n, (n) 0 o ≤ Q̄n + O(δn ) ∩ Ān \| Y + oP0 (1) \|\|π>jn (f − Tn )\|\|B∞1 0 √ = Π f : \|\|πjn (f − Tn )\|\|M ≤ (Rn + o(1))/ n \| Y × Π f : \|\|π>jn (f − Tn )\|\|B∞1 ≤ Q̄n + O(δn ) \| Y + oP0 (1). 0 Again using that Π(Ān \|Y ) = 1 − oP0 (1), the final line equals √ Π f : \|\|f − Tn \|\|M ≤ (Rn + o(1))/ n × Π f : \|\|f − Tn \|\|B∞1 ≤ Q̄n + O(δn ) \| Y + oP0 (1). 0 (7.19) We now check that the above product has asymptotically the correct posterior probability. By the same argument as in Theorem 5.1 (replacing H(δ) by M), the first probabilityP equals (1 − γ) + oP0 (1). Setting Ml = 2l/2 max0≤k<2l \|flk − Tn,lk \|, we have kf − Tn kB∞1 = l Ml , 0 where the (Ml ) are independent due to the product structure of the posterior. We can therefore write the posterior density of kf − Tn kB∞1 as hn ∗ Gn , where Mj0 (n) has density hn 0 P and l6=j0 (n) Ml has probability distribution Gn , both supported on [0, ∞). 36 Using standard extreme value theory, we can establish the limiting distribution of Mj0 (n) . For Zi ∼ N (0, 1) independent, we have a−1 m (max1≤i≤m \|Zi \| − bm ) converges in distribution to the standard Gumbel distribution (i.e. distribution function F (x) = exp(−e−x )), where p 1 log(4π log 2m) 1 √ am = √ , bm = 2 log 2m − +O log m 2 log 2m 2 2 log 2m 2 (bm is the solution to 2m2 = πb2m ebm ). This follows from Theorem 10.5.2(c) and Example 10.5.3 of [18] with only minor modifications due to the absolute values within the maximum (intuitively it is the same as the maximum of 2m standard Gaussians). Moreover, by Pólya’s Theorem (p. 265 of [16]) the convergence of the distribution functions is uniform: sup \|(Φ(am x + bm ) − Φ(−am x − bm ))m − exp(−e−x )\| → 0, (7.20) x∈R where we recall \|Zi \| has distribution function Φ(x) − Φ(−x). Recall that Mj0 (n) is the sum of i.i.d. (rescaled) Gaussians under the posterior. Using that Q̄n is the (1 − γ)-posterior quantile for kf − Tn kB∞1 0 , \|Π(f :\|\|f − Tn \|\|B∞1 ≤ Q̄n + O(δn ) \| Y ) − (1 − γ)\| 0 X ≤ Π Q̄n − cδn ≤ Ml ≤ Q̄n + cδn Y l Z Q̄n +cδn Z ∞ hn (x − y)dGn (y)dx = Q̄n −cδn Z 0 ∞ Z Q̄n −y+cδn hn (z)dGn (y) = 0 Q̄n −y−cδn ≤ sup P(Mj0 (n) ∈ [t − cδn , t + cδn ]). t≥0 Using the limiting distribution of Mj0 (n) , (7.20) and that the maximum of the standard Gumbel density function is e−1 , the last probability equals ! 2j0 (n)/2 τ max \|ζj0 (n)k \| ∈ [t − cδn , t + cδn ] P √ 1 + nτ 2 0≤k<2j0 (n) ! √ 2 1 + nτ = P a−1 max \|ζ \| − b2j0 (n) ∈ a−1 [t − b2j0 (n) − cδn , t − b2j0 (n) + cδn ] 2j0 (n) 0≤k<2j0 (n) j0 (n)k 2j0 (n) 2j0 (n)/2 τ ! √ 1 + nτ 2 −1 ≤ P Gumbel(0, 1) ∈ a2j0 (n) j (n)/2 [t − b2j0 (n) − cδn , t − b2j0 (n) + cδn ] + o(1) 20 τ r p jn −j0 (n) nj0 (n) 0 −1 2 + o(1) → 0, ≤ c0 e−1 δ + o(1) = c e w j (n)2 n j 0 n 2j0 (n) by the choice of jn . In conclusion, we have shown that the second probability in (7.19) equals (1 − γ) + oP0 (1). This completes the proof. 37 7.5 Remaining proofs Proof of Proposition 3.3. Fix ρ > 1, let ε = ε(α, ρ, R) < (1 − ρ−2α )/(2αR) be sufficiently PdρN e small so that ε ∈ (0, 1) and consider the events Aα,N = { k=N fk2 < εRN −2α }. By a simple PdρN e integral comparison we have that k=N k −2α−1 ≥ (2α)−1 N −2α (1 − ρ−2α ), so that under the conditional prior,   dρN e X Πα (Aα,N ) = P  k −2α−1 gk2 < εRN −2α  k=N   1 k −2α−1 (gk2 − 1) < εRN −2α − ≤ P N −2α (1 − ρ−2α ) 2α k=N   dρN e X k −2α−1 (gk2 − 1) < −ε0 N −2α  , ≤ P dρN e X k=N where the gk ’s are i.i.d. standard normal random variables and ε0 > 0 (by the choice of ε). By (4.2) of Lemma 1 of [32] we have the exponential inequality    1/2 dρN e dρN e X X √   k −4α−2  P k −2α−1 (gk2 − 1) ≤ −2  x ≤ e−x . k=N k=N PdρN e For N ≥ 2, again by an integral comparison we have that k=N k −4α−2 ≤ C(α)N −4α−1 . Using this and letting x = M N , the exponential inequality becomes   dρN e X √ k −2α−1 (gk2 − 1) ≤ −C 0 (α) M N −2α  ≤ e−M N . P k=N √ Taking M sufficiently small so that C 0 (α) M < ε0 , we obtain that Πα (Aα,N ) ≤ e−M N . Since this sequence is summable in N , the result follows from the first Borel-Cantelli Lemma. √ Proof of Proposition 3.7. Under the law P0 , nE0 \|\|Y − f0 \|\|M(w) = E0 \|\|Z\|\|M(w) < ∞ by Proposition 2 of [12]. By the triangle inequality it therefore suffices to show the conclusion of Proposition 3.7 with Y replaced by f0 . Rewrite the multiscale indices Λ = {(l, k) : l ≥ 0, k = 0, ..., 2l − 1} in increasing lexicographic order, so that Λ = {(lm , km ) : m ∈ N}, where lm = i, if 2i ≤ m < 2i+1 , i = 0, 1, 2, ..., km = m − 2i , if 2i ≤ m < 2i+1 , i = 0, 1, 2, ... Consider a strictly increasing subsequence (nm )m≥1 of N such that (log nm )/wl2m → ∞ as m → ∞ (such a subsequence can be constructed for any admissible (wl ) since wl % ∞). Define a function f0 ∈ `2 via its wavelet coefficients p hf0 , ψlm km i = r log nm /nm , 38 where r ≤ γ for γ the value given in the proof of Theorem 7.5. Since r 2 lm (β+1/2) β+1/2 \|hf0 , ψlm km i\| ≤ rm log nm , nm we can ensure f0 is in any given Hölder ball H(β, R) by letting r be sufficiently small and taking the subsequence nm to grow fast enough. Consider now a further subsequence, removing terms corresponding to one index per resolution level, say (l, kl ) (i.e. removing terms with indices m = 2l + kl , l = 0, 1, 2, ..., from the above subsequence), and set 0 \|hf0 , ψlkl i\| = R2−l(β+1/2) . Using the Besov space embedding L∞ ⊂ B∞∞ , \|\|Kj (f ) − f \|\|∞ ≥ C(ψ) max 2l/2 max \|hf0 , ψlk i\| l>j k ≥ C(ψ)2(j+1)/2 \|hf0 , ψ(j+1)kj+1 i\| = C(ψ)R2−β 2−jβ = ε(β, R, ψ)2−jβ , thereby establishing that f0 ∈ HSS (β, R, ε). Let An denote the event defined in (7.1). We have that on Anm , the posterior distribution 0 Π (· \| Y (nm ) ) assigns the (lm , km ) coordinate to the Dirac mass component of the distribution. Consequently, by the choice of (nm ), E0 Π0 ( \|\|f − f0 \|\|M ≤ Mnm nm −1/2 \| Y (nm ) ) = E0 Π0 ({\|\|f − f0 \|\|M ≤ Mnm nm −1/2 } ∩ Anm \| Y (nm ) ) + o(1) ≤ E0 Π0 ({\|flm km − f0,lm km \| ≤ Mnm wlm nm −1/2 } ∩ Anm \| Y (nm ) ) + o(1) p = E0 Π0 ({r log nm /nm ≤ Mnm wlm n−1/2 } ∩ Anm \| Y (nm ) ) + o(1) m p ≤ E0 Π0 (r log nm /wlm ≤ Mnm \| Y (nm ) ) + o(1) = o(1) √ for any sequence Mn such that Mnm = o(wl−1 log nm ) as m → ∞. m 8 8.1 Technical facts and results Results for `2 -setting The following two lemmas describe the behaviour of the posterior mean of the empirical Bayes procedure. The first says that the posterior mean is a consistent estimator of f0 in a sequence of Sobolev norms with data driven exponent. In particular, we are interested in the case n → 0 when the Sobolev exponent tends to the true smoothness β. Note that we require n strictly positive since the posterior mean is itself not an element of H α̂n . The −1/2,δ second says that fˆn is an efficient estimator of f0 in H2 . Both proofs are similar to that of Theorem 2 of [29] and are thus omitted. Lemma 8.1. Let fˆn denote the posterior mean of the empirical Bayes procedure and let n > 0. Then for every β, R > 0 and Mn → ∞, we have sup P0 \|\|fˆn − f0 \|\|H α̂n −n ≥ Mn → 0 f0 ∈QSS (β,R,ε) as n → ∞. 39 Lemma 8.2. Let fˆn denote the posterior mean of the empirical Bayes procedure. Then for f0 ∈ QSS (β, R, ε), δ > 1 and as n → ∞, √ \|\|fˆn − Y\|\|H(δ) = oP0 (1/ n). We have an exponential inequality which measures posterior spread in a variety of Sobolev norms. Since for fixed α, the posterior only depends on the data through the posterior mean fˆn,α , the following probabilities are independent of the observed data Y . Lemma 8.3. Let fˆn,α denote the posterior mean of Πα (· \| Y ). Then for any 0 ≤ s < α and any η > 0, 2(α−s) 1 − 2α+1 2 ˆ Πα f : \|\|f − fn,α \|\|H s ≥ (1 + η) 1 + n Y 2(α − s) η 1 1/4 1/(4α+2) √ . ≤ e exp − 1+ n 2(α − s) 24 Proof. For f ∼ Πα (· \| Y ) we can use the explicit form of the posterior mean in (3.1) to write \|\|f − fˆn,α \|\|2H s = ∞ X k=1 k 2s ζ 2, +n k k 2α+1 where the ζk ∼ N (0, 1) are independent. Letting tn = n1/(2α+1) and using standard tail bounds, ∞ h i X EΠ \|\|f − fˆn,α \|\|2H s \| Y = k 2s 1 X 2s X −2(α−s)−1 ≤ k + k k 2α+1 + n n k=1 k≤tn k>tn 2(α−s) 1 ≤ 1+ n− 2α+1 . 2(α − s) The posterior variance of \|\|f − fˆn,α \|\|2H s is given by ν2 = 2 ∞ X k=1 4(α−s)+1 2 X 4s X −4(α−s)−2 k 4s ≤ 2 k + k ≤ 3n− 2α+1 . 2 + n) n (k 2α+1 k≤tn k>tn Combining the above with the exponential inequality for χ2 -squared random variables found in Proposition 6 of [46], we have ! ∞ √ X k 2s 1/4 −x/ 8 2 e e ≥P (ζ − 1) ≥ νx k 2α+1 + n k k=1 √ − 4(α−s)+1 2(α−s) 1 − 2α+1 2 ˆ 4α+2 ≥ Πα f : \|\|f − fn,α \|\|H s ≥ 1 + n + 3n x Y . 2(α − s) √ Taking x = (η/ 3)[1 + 1/(2α − 2s)]n1/(4α+2) gives the desired result. 40 8.2 Results for L∞ -setting To prove Proposition 4.5 we need to understand the behaviour of the posterior median under the law P0 . Lemma 8.4. Let f˜ = f˜n denote the posterior median (defined coordinate-wise) of the slab and spike prior. Then the event Bn = {f˜lk = 0 ∀(l, k) ∈ Jnc (γ)} ∩ {f˜lk 6= 0 ∀(l, k) ∈ Jn (γ 0 )} √ ∩ { n\|Ylk − f0,lk \| ≤ (8l log 2 + a log n)1/2 ∀l ≤ Jn , ∀k = 0, ..., 2l − 1} (8.1) satisfies inf f0 ∈H(β,R) P0 (Bn ) → 1 as n → ∞, for some constants 0 < γ < γ 0 < ∞ and a > 0. Proof. We show that the P0 -probability of each of these events individually tends to 1. For the first event {f˜lk = 0 ∀(l, k) ∈ Jnc (γ)} ⊇ {Π(flk = 0 \| Y ) ≥ 1/2 ∀(l, k) ∈ Jnc (γ)} ⊇ {Π(flk = 0 ∀(l, k) ∈ Jnc (γ)) ≥ 1/2} = {Π(S ∩ Jnc (γ) = ∅) ≥ 1/2}. By Lemma 1 of [27] the P0 -probability of this last event tends to 1 for some γ > 0 as n → ∞. Consider the third event, √ Ωn = { n\|Ylk − f0,lk \| ≤ (8l log 2 + a log n)1/2 ∀l ≤ Jn , ∀k = 0, ..., 2l − 1}, which by (41) of [27] (or the Borell-Sudakov-Tsireslon inequality [35]) satisfies P0 (Ωcn ) → 0. We shall lastly show that Ωn ⊂ {f˜lk 6= 0 ∀(l, k) ∈ Jn (γ 0 )}, (8.2) which then completes the proof. Consider firstly the case f0,lk ∈ Jn (γ 0 ) with f0,lk > 0. Write Π(flk ≤ 0 \| Y ) = Π(flk = 0 \| Y ) + Π(flk < 0 \| Y ). (8.3) By the proof of Lemma 1 of [27], we have that on the event Ωn and for sufficiently large 0 2 γ 0 , the first posterior probability in (8.3) is bounded above by a multiple of nK+1/2−(γ ) /8 . Again on the event Ωn , we use (42) of [27] to bound the second term via Π(flk < 0 \| Y ) = ≤ wjn R∞ −n (x−Ylk )2 2 g(x)dx −∞ e 2 −n (x−Y ) lk 2 R0 g(x)dx + (1 − wj,n ) −∞ e R −√nYlk − 1 v2 \|\|g\|\|∞ −∞ e 2 dv √ √ = C nΦ̄( nYlk ), 1/2 a(π/n) wjn (8.4) where Φ̄ = 1 − Φ with Φ the distribution function of a standard normal variable. On Ωn , we have for l ≤ Jn , s r r 2Jn log 2 + 12 log n log n log n 0 Ylk = (Ylk − f0,lk ) + f0,lk ≥ − +γ ≥δ n n n 41 for some δ = δ(γ 0 ) > 0 that can be made arbitrarily large by taking γ 0 large enough. Thus applying the standard tail bounds for Φ̄ we have that the right-hand side of (8.4) is bounded above by a multiple of √ 1 1 2 p √ n n2−2δ − 12 δ 2 log n nΦ̄(δ log n) ≤ √ e = C(δ) √ . δ 2π log n log n Combining the above results, we have that for sufficiently large γ 0 (and hence δ), (8.3) is bounded above by a constant times n−B for some B > 0, uniformly over the positive coefficients in Jn (γ 0 ). In particular, the posterior median satisfies f˜lk > 0 for all (l, k) ∈ Jn (γ 0 ) with flk > 0 and n large enough. The case f0,lk < 0 is dealt with similarly, thereby proving (8.2). A simultaneous estimator in M(w) and L∞ It may be of interest to obtain an efficient estimator of f0 in M(w) that is also an element of L∞ , unlike Y. Letting f˜ = f˜n and fˆn denote the posterior median and mean of the slab and spike procedure respectively, define the estimators  if j ≤ j0 (n),  Yjk (1) Tn,jk = Yjk 1{f˜jk 6=0} if j0 (n) < j ≤ blog n/ log 2c. (8.5)   0 if blog n/ log 2c < j, (2) Tn,jk  ˆ  fn,jk = Yjk 1{f˜jk 6=0}   0 if j ≤ j0 (n), if j0 (n) < j ≤ blog n/ log 2c. (8.6) if blog n/ log 2c < j. Lemma 8.5. Consider the slab and spike prior Π with lower threshold j0 (n) → ∞ satisfying j0 (n) = o(log n) and let (wl ) be any admissible sequence satisfying wj0 (n) = o(nv ) for any (i) v > 0. Then the estimators Tn , i = 0, 1, defined in (8.5) and (8.6) satisfy for some M 0 > 0 and any Mn → ∞, √ sup P0 (\|\|Tn(i) − f0 \|\|M(w) ≥ Mn / n) → 0, f0 ∈H(β,R) sup f0 ∈H(β,R) P0 (\|\|Tn(i) − f0 \|\|∞ ≥ M 0 (log n/n)β/(2β+1) ) → 0 (i) as n → ∞. Moreover, \|\|Tn − Y\|\|M(w) = oP0 (n−1/2 ), uniformly over f0 ∈ H(β, R). (i) Proof. Since Y is an efficient estimator of f0 in M, if suffices to establish \|\|Tn − Y\|\|M = (i) oP0 (n−1/2 ) to show that Tn is also an efficient estimator of f0 in M. Consider firstly j > j0 (n), where the estimators coincide, and let Jn (β) be as in the proof of Theorem 7.5. On the event Bn defined in (8.1) and following the proof of Theorem 7.5, we have \|\|π>j0 (n) (Tn(i) − Y)\|\|M ≤ max j0 (n)<l≤Jn (β) wl−1 max \|Ylk 1{f˜lk =0} \| + max wl−1 max \|f0,lk \| k wl−1 l≥Jn (β) k max max (\|Ylk − f0,lk \| + \|f0,lk \|) + o(n−1/2 ) k:(l,k)∈Jnc (γ) p ≤ Cwj−1 (log n)/n + o(n−1/2 ) = o(n−1/2 ) (n) 0 ≤ j0 (n)<l≤Jn (β) 42 by the choice of j0 (n). (1) For j ≤ j0 (n) and i = 1, we trivially have \|\|πj0 (n) (Tn − Y)\|\|M = 0. Consider now i = 2. Arguing as in Theorem 2 of [12] and using the conditions on j0 (n), one obtains the √ uniform bound E0 EΠ [\|\| nπj0 (n) (f − Y)\|\|1+ M(w) \| Y ] ≤ C() for > 0 small enough. From the weak convergence of Π(· \| Y ) ◦ τY−1 towards N and a uniform integrability argument (via the √ moment bound), it follows as in Theorem 10 of [11] that nπj0 (n) (EΠ (f \|Y ) − Y) → EN = 0 in M0 (w) in probability, which implies the result. For L∞ we have on Bn , Jn (β) 2l/2 max \|Ylk − f0,lk \|1{(l,k)∈Jn (γ)} + \|f0,lk \|1{(l,k)∈Jnc (γ)} X \|\|π>j0 (n) (Tn(i) − f0 )\|\|∞ . k l=j0 (n)+1 ∞ X 2l/2 max \|f0,lk \| + k l=Jn (β)+1 r Jn (β)/2 .2 log n + R2−Jn (β)β . n max \|Ylk − f0,lk \| . 2 j0 (n)/2 log n n β 2β+1 . Now j0 (n) \|\|πj0 (n) (Tn(1) − f0 )\|\|∞ . X r 2 l/2 k l=0 (1) log n . n log n n β 2β+1 , (2) thereby proving the second statement for Tn . For Tn , using the convergence of the posterior mean to Y in M0 for j ≤ j0 (n) shown above, j0 (n) \|\|πj0 (n) (Tn(2) − Tn(1) )\|\|∞ . X 2l/2 max \|fˆn,lk − Ylk \| k l=0   j0 (n) = oP0  X l=0 8.3 l/2 2 w √ l  = oP0 n 2j0 (n)/2 wj0 (n) √ n ! = oP0 log n n β 2β+1 ! . Wavelets Let us briefly recall the notion of periodized and boundary corrected wavelets and discuss condition (2.3). Let φ, ψ denote a scaling and corresponding wavelet function on R satisfying X X sup \|φ(x − k)\| < ∞, sup \|ψ(x − k)\| < ∞. (8.7) x∈R k∈Z x∈R k∈Z Examples include Meyer wavelets (see Section 2 in [38] for other choices). Consider firstly the periodic case. As usual define the dilated and translated wavelet at resolution level j and scale position k/2j by φjk (x) = 2j/2 φ(2j x − k), ψjk (x) = 2j/2 ψ(2j x − k) for j, k ∈ Z. Periodize the wavelet functions via X X per φper φjk (x + m), ψjk (x) = ψjk (x + m), x ∈ [0, 1] jk (x) = m∈Z m∈Z 43 per J0 − for j = 0, 1, ... and k = 0, ..., 2j − 1. Then the wavelet system {φper J0 k , ψjm : k = 0, ..., 2 1, m = 0, ..., 2j − 1, j = J0 , J0 + 1, ...} forms an orthonormal wavelet basis of L2 ((0, 1]) and satisfies (2.3) due to (8.7). In the case of R, an orthonormal basis of Vj = span{(φjk )k }, j ≥ J0 , can be obtained by taking 2j−J0 dilations of the orthonormal basis (φJ0 k )k of a basic resolution space VJ0 . In the case of boundary corrected wavelets, the analogous orthonormal basis of the basic resolution space VJ0 , J0 ∈ N, contains 2J0 elements and consists of 3 components. At resolution level J0 , t J0 /2 φlef t (2J0 x), a basis consists of 3 components. Firstly, N left edge functions φlef J0 k (x) = 2 k t k = 0, ..., N − 1, where φlef is a modification of φ that remains bounded and has compact k J0 /2 φright (2J0 x), k = 0, ..., N − 1, support. Secondly, N right edge functions φright J0 k (x) = 2 k with the same properties. Thirdly, 2J0 − 2N interior functions, that are the usual translates of dilations of φ defined on R, that is φJ0 k for k = N, ..., 2J0 − N − 1, which we note are all J0 supported in the interior of [0, 1]. Writing for convenience {φbc J0 k : k = 0, ..., 2 − 1} instead lef t right of {φJ0 k , φJ0 k0 , φJ0 m : k = 0, ..., N − 1, k 0 = 0, ..., N − 1, m = N, ..., 2J0 − N − 1}, we have the first part of (2.3) J0 −1 2X \|φbc J0 k (x)\| ≤2 J0 /2 ≤2 J0 /2 k=0 N t max \|\|φlef k \|\|∞ 0≤k<N C(N, φ) + 2 J0 /2 +2 J0 /2 N \|\|∞ max \|\|φright k 0≤k<N + 2J0X −N −1 2J0 /2 \|φ(2J0 x − k)\| k=N 0 C (φ), (8.8) where we have used that N is fixed and that the original wavelet function on R satisfies (8.7). Starting at resolution level J0 with the usual dilated wavelets on R, ψJ0 k (x) = 2J0 /2 ψ(2J0 x− k), 2J0 ≥ N , it is possible to construct corresponding boundary wavelet functions t 0 J0 − N − 1}. , ψJright {ψJlef 0 , ψJ0 m : k = 0, ..., N − 1, k = 0, ..., N − 1, m = N, ..., 2 0k 0k For j ≥ J0 , we can then define the dilates of the boundary wavelets in the usual way: right (2j−J0 x). (x) = 2(j−J0 )/2 ψJright ψjk 0k t j−J0 lef t (2 x), (x) = 2(j−J0 )/2 ψJlef ψjk 0k lef t right This yields the required wavelets at resolution level j, namely {ψjk , ψjk , ψjm : k = 0 0 j 0, ..., N − 1, k = 0, ..., N − 1, m = N, ..., 2 − N − 1}, which for convenience we write as bc : k = 0, ..., 2j − 1}. Arguing as in (8.8) and again using (8.7) gives the second part of {ψjk (2.3). 8.4 Weak convergence For µ and ν probability measures on a metric space (S, d), define the bounded Lipschitz metric by Z βS (µ, ν) = sup u(s)(dµ(s) − dν(s)) , (8.9) u:\|\|u\|\|BL ≤1 S \|\|u\|\|BL = sup \|u(s)\| + s∈S \|u(s) − u(t)\| . d(s, t) s,t∈S:s6=t sup βS metrizes the weak convergence of probability distributions, that is random variables Xn →d X converge in distribution in (S, d) if and only if βS (L(Xn ), L(X)) → 0, where −1/2,δ L(X) denotes the law of X. In particular, we shall consider the choices S = H(δ) = H2 or S = H −s for s > 1/2 in `2 and S = M0 (w) for {wl }l≥1 an admissible sequence in L∞ . 44 8.5 Results on empirical and hierarchical Bayes procedures Let us recall some definitions and results from [29, 48] that appear in proofs elsewhere. Define hn : (0, ∞) → [0, ∞) to be hn (α) = ∞ 2 log k X n2 k 2α+1 f0,k 1 + 2α n1/(2α+1) log n k=1 (k 2α+1 + n)2 (8.10) and for 0 < l < L define the bounds αn = inf{α > 0 : hn (α) > l} ∧ p log n, αn = inf{α > 0 : hn (α) > L(log n)2 }. The behaviour of the empirical Bayes estimator α̂n defined in (3.2) is contained in Lemma 3.11 of [48], which is summarized below for convenience. Lemma 8.6 (Szabó et al.). Fix βmax > 0. For any 0 < β ≤ βmax and R ≥ 1, there exist constants K1 and K2 such that P0 (β − K1 / log n ≤ α̂n ≤ β + K2 / log n) → 1 uniformly over f0 ∈ QSS (β, R, ε). 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11 Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design H. Boyer, F. Miranville, D. Bigot, S. Guichard, I. Ingar, A. P. Jean, A. H. Fakra, D. Calogine and T. Soubdhan University of La Reunion, Physics and Mathematical Engineering for Energy and Environment Laboratory, LARGE - GéoSciences and Energy Lab., University of Antilles et de la Guyanne, France 1. Introduction The aim of this paper is to briefly recall heat transfer modes and explain their integration within a software dedicated to building simulation (CODYRUN). Detailed elements of the validation of this software are presented and two applications are finally discussed. One concerns the modeling of a flat plate air collector and the second focuses on the modeling of Trombe solar walls. In each case, detailed modeling of heat transfer allows precise understanding of thermal and energetic behavior of the studied structures. Recent decades have seen a proliferation of tools for building thermal simulation. These applications cover a wide spectrum from very simplified steady state models to dynamic simulation ones, including computational fluid dynamics modules (Clarke, 2001). These tools are widely available in design offices and engineering firms. They are often used for the design of HVAC systems and still subject to detailed research, particularly with respect to the integration of new fields (specific insulation materials, lighting, pollutants transport, etc.). 2. General overview of heat transfer and airflow modeling in CODYRUN software 2.1 Thermal modeling This part is detailed in reference (Boyer, 1996). With the conventional assumptions of isothermal air volume zones, unidirectional heat conduction and linearized exchange coefficients, nodal analysis integrating the different heat transfer modes (conduction, convection and radiation) achieve to establish a model for each constitutive thermal zone of the building (one zone being a room of a group of rooms with same thermal behaviour). For heat conduction in walls, it results from electrical analogy that the nodal method leads to the setting up of an electrical network as shown in Fig. 1, the number of nodes depending on the number of layers and spatial discretization scheme : Tsi T1 T2 T3 Tse Fig. 1. Example of associated electrical network associated to wall conduction 228 Evaporation, Condensation and Heat Transfer The physical model of a room (or group of rooms) is then obtained by combining the thermal models of each of the walls, windows, air volume, which constitute what CODYRUN calls a zone. To fix ideas, the equations are of the type encountered below: C si C se dTsi dt dTse dt C ai = hci (Tai − Tsi ) + hri (Trm − Tsi ) + K (Tse − Tsi ) + ϕswi = hce (Tae − Tse ) + hre (Tsky − Tse ) + K (Tsi − Tse ) + dTai dt Nw ∑ hci S j (Tai − Tsi ( j ) + c m (Tae − Tai ) = ϕswe (1) (2) (3) j =1 0 = Nw ∑ hri A j (Tsi ( j ) − Trm ) (4) j =1 Equations of type (1) and (2) correspond to energy balance of nodes inside and outside surfaces. Nw denote the number of walls of the room, and correspond to the following figure: ϕlwi ϕswi ϕ ci ϕ lwe ϕswe ϕce Fig. 2. Wall boundary conditions ϕ being heat flux density, lw, sw, and c indices refers to long wave radiation, short wave radiation and convective exchanges. Indices e and i are for exterior and indoor. Equation (3) comes from the thermo-convective balance of the dry-bulb inside air temperature Tai, taking into account an air flow m through outside (Tae) to inside. (4) represents the radiative equilibrium for the averaged radiative node temperature, Trm. The generic approach application implies a building grid-construction focus. Following the selective model application logic developed (Boyer, 1999), an iterative coupling process is implemented to manage multi-zone projects. From the first software edition, some new models were implemented. They are in relation with the radiosity method, radiative zone coupling (short wavelength, though window glass) or the diffuse-light reduction by close solar masks (Lauret, 2001). Another example of CODYRUN evolution capacity could be illustrated by a specific study using nodal reduction algorithms (Berthomieu, 2003). To resolve it, a new implementation was done: the system was modified into a canonic form (in state space). Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 229 From a user utilization point of view, this thermal module exit is mainly linked to the discretizated elements temperatures, the surfacic heat flows, zone specific comfort indices, energy needs, etc. 2.2 Airflow modeling Thermal and airflow modules are coupled to each other thanks to an iterative process, the coupling variables being the air mass flows. It is useful to implement the inter zones flows [i , j ] then qualifies the mass transfer from the zone . The terms m with the help of a matrix m i to the zone j, as it is depicted . Fig. 3. Interzonal airflow rates matrix Then, a pressure model for calculating previous airflow rates is set and takes into account wind effect and thermal buoyancy. To date, the components are integrated ventilation vents, small openings governed by the equation of crack flow, as : = Cd S (ΔP)n m (5) S is aperture surface, Cd and n (typically 0.5- 0.67) depends on airflow regime and type of openings and ΔP the pressure drop. Therefore, the following scheme can be applied for encountered cases. Outdoor Indoor Indoor Indoor Tai (j) Tai (k) Tae Tai (i) Wind z z Pz(j) Pz(i) Fig. 4. Exterior and interior small openings Taking into account the inside air volume incompressibility hypothesis, the mass weight balance should be nil. Thus, in the established system, called pressure system, the unknowns are the reference pressure for each of the zones. Mechanical ventilation is simply taken into account through its known airflow values. In this way, ventilation inlets are not taken into detailed consideration which could have been the case owing to their flowpressure characteristics. 230 Evaporation, Condensation and Heat Transfer ⎧ i=N i,1) + m vmc ( 1 ) = 0 ⎪ ∑ m( ⎪ i =0,i ≠1 ⎪ i=N ⎪ i,2) + m vmc ( 2 ) = 0 ⎪ ∑ m( ⎪ i = 0,i ≠ 2 ⎨ ⎪ ... ⎪ ⎪ ⎪i = N − 1 ⎪ i,N) + m vmc ( N ) = 0 m( ⎪ ∑ ⎩ i =0 (6) m vmc ( k ) is the flow extracted from zone k (by vents) and N the total number of zones. After the setting up of this non linear system, its resolution is performed using a variant of the Newton-Raphson (under relaxed method). This airflow model has recently been supplemented by a CO2 propagation model in buildings (Calogine, 2010). 2.3 Humidity transfers For decoupling reasons, zones dry temperatures, as well as inter zone airflows, have been previously calculated. At this calculation step, for a given building, each zone's specific humidity evolution depends on the matricial equation : Ch .r = A h .r + Bh (7) According to Duforestel’s model, an improvement of the hydric model (Lucas, 2003) was given by the use of a hygroscopic buffer. If we consider N as the number of zones, a linear system of dimension 2N is obtained and a finite difference scheme is used for numerical solution. 3. Some elements of CODYRUN validation 3.1 Inter model comparison A large number of comparisons on specific cases were conducted with various softwares, whether CODYBA, TRNSYS for thermal part, with AIRNET, BUS, for airflow calculation or even CONTAM, ECOTECT and ESP. To verify the correct numerical implementation of models, we compared the simulation results of our software with those from other computer codes (Soubdhan, 1999). The benchmark for inter-comparison software is the BESTEST procedure (Judkoff, 1983), (Judkoff, 1995). This procedure has been developed within the framework of Annex 21 Task 12 of the Solar Heating and Cooling Program (SHC) by the International Energy Agency (IEA). This tests the software simulation of the thermal behavior of building by simulating various buildings whose complexity grows gradually and comparing the results with other simulation codes (ESP DOE2, SERI-RES, CLIM2000, TRNSYS, ...). The group's collective experience has shown that when a program showed major differences with the so-called reference programs, the underlying reason was a bug or a faulty algorithm. Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 231 3.1.1 IEA BESTEST cases The procedure consists of a series of buildings carefully modeled, ranging progressively from stripped, in the worst case, to the more realistic. The results of numerical simulations, such as energy consumed over the year, annual minimum and maximum temperatures, peak power demand and some hourly data obtained from the referral programs, are a tolerance interval in which the software test should be. This procedure was developed using a number of numerical simulation programs for thermal building modeling, considered as the state of the art in this field in the US and Europe. After correction of certain parts of CODYRUN, an example output from this confrontation is the following concerns and energy needs for heating over a year, for which the results of our application are clearly appropriate. Fig. 5. Exemple of BESTEST result The procedure requires more than a hundred simulations and cases treated with CODYRUN showed results compatible with most referral programs, except for a few. We were thus allowed to reveal certain errors, such as the algorithm for calculating the transmittance of diffuse radiation associated with double glazing and another in the distribution of the incident radiation by the windows inside the envelope. As in the reference codes, two points are to remember: the majority of errors came from a poor implementation of the code and all the efficiency of the procedure BESTEST is rechecked, errors belonging to modules that have been used for years. Concerning airflow validation, IEA multizone airflow test cases describe several theoretical and comparative test cases for the airflow including multizone cases. It was developed within the International Energy Agency (IEA) programs: Solar Heating and Cooling (SHC) Task 34 and Energy Conservation in Building and Community Systems (ECBCS) Annex 43. The tests are designed for testing the capability of building energy simulation programs to predict the ventilation characteristics including wind effect, buoyancy and mechanical fan. CODYRUN passed all the test cases. 232 Evaporation, Condensation and Heat Transfer 3.1.2 TN51 airflow case In (Orme, 1999), a test case composed of a building with three storeys is described and first published in (Tuomala, 1995). All boundary conditions are imposed (wind, external temperature) and indoor conditions are constant, i.e. in steady state. Fig. 6. TN 51 Test case Next figure shows solution of 3 references codes, i.e. COMIS, CONTAM93 and BREEZE. CODYRUN’s results were added on the figure extracted from the paper (Orme, 1999). Fig. 7. Intermodel airflow comparison Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 233 As it can be seen, in terms of numerical results, CODYRUN gives nearly the same values as the other codes, little differences being linked to numerical aspects as algorithms or convergence criteria used. 3.2 Experimental confrontation As part of measurement campaigns on dedicated cells and real homes, a set of elements of models have been to face action (and for some improved), mostly concerning thermal and humidity aspects. Not limited to, some of these aspects are presented in the articles (Lauret, 2001) (Lucas, 2001) and (Mara, 2001), (Mara, 2002). 4. Two applications of the software 4.1 Air solar collector An air solar collector is used to heat air from solar irradiation. More precisely, it allows to convert solar energy (electromagnetic form) into heat (Brownian motion). This type of system can be used to heat buildings or to preheat air for drying systems. Air solar collector is constituted by an absorber encapsulated into an isolated set-up (Fig. 8). To avoid heat loss by the incoming flux side, a glass is located above the absorber, letting a gap between them. This glass allowing to catch long wavelength emitted by the set-up when its temperature rises-up. Air temperature improvement is mostly done by air convection at the absorber interface. This heat is then carried by air in the building or in the drying system. Fig. 8. Air solar collector principle, adapted from (Sfeir, 1984) The enclosure is not strictly speaking a building, but the generic nature of the software allows to study this device. This is a zone (within the captation area), a glazed windows, exterior walls and an inner wall of absorber on which it is possible to indicate that solar radiation is incident. It should be noted that this is part of a class simulation exercise allowing students to learn collector physics basis. 234 Evaporation, Condensation and Heat Transfer 4.1.1 Collector modeling and description of the study It is assumed that before initial moment, there is no solar irradiation and the collector is at thermal equilibrium, i.e. outside air temperature is the set-up temperature. At t=0, an irradiation flux Dh is applied without interactions with the outside air temperature. In a simplified approach, a theoretical study is easily obtained. Some equations below are used to understand the functioning of the collector. In steady state, the incoming irradiation flux reaching the absorber is fully converted and transmitted to the air (assumed to uniform temperature): ρasC asV dTai as (Tai - Tae ) = τ 0 Dh - K pS(Tai - Tae ) - mC dt (8) Where Kp is the thermal conductance (W.m-2.K-1) of each wall through which losses can pass, m dot is a constant air mass flow through the collector, Tai is the inside air temperature, Tae is the outside air temperature, and ρas (1.2 kg.m-3) and Cas (1000 J.kg-1.K-1) are respectively the air thermal conductivity and density. By knowing initial conditions (at t=0, Tai=Tae), the solution of equation (8) appears: Tai ( t ) = Tae + τ 0 Dh K (1− e − t K / ρ as C as V ) (9) The power delivered by the collector is given by as (Tai - Tae ) Pu = mC (10) The studied collector is horizontal, the irradiated face has a surface of 1m², and an air gap thickness of 0.1m. The meteorological file is constituted of a constant outside air temperature (25°C) and a constant direct horizontal solar irradiation of 500W (between 6 am and 7 pm). It is assumed (and further verified) that the steady state is reached at the end of the day. Concerning the building description file, the collector is composed by 1 zone, 3 boundaries (1 for the upper face, 1 for the lower face, and 1 for all lateral faces), and 4 components (glass, lateral walls, lower wall and absorber). Walls are made - from inside to outside - of wood (1cm), polystyrene (5cm) and wood (1cm). Wood walls have a short wave absorptivity of α = 0.3. The glass type is a single layer ( τ 0 = 0.85), and the absorber is a metallic black sheet of 2mm (α = 0.95). The solar irradiation absorption model of the collector is a specific one. This has to be chosen in CODYRUN, because traditionally in buildings the solar irradiation doesn’t impact directly an internal wall (absorber), but the floor. Even if this solar air heater is considered as a very simple one, the following figure gives some details about heat exchanges in the captation area. Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 235 Fig. 9. Solar irradiation absorption process. 4.1.2 Simulation results in the case of forced convection in air gaps An air flow of 1 m3.h-1 is set. After running simulations, the following graph is obtained for two identical days : Fig. 10. Evolution of air temperature at the output of the solar collector with forced convection (1m3.h-1). 236 Evaporation, Condensation and Heat Transfer Simulation results are consistent with simplified study, but not exactly identical. In fact, CODYRUN model is more precise than those use in eq. 8: thermal inertia of walls are taken into account, superficial exchanges are detailed, exchange coefficients are depending on inclination, glass transmittance is depending on solar incidence, … With a low air flow rate of 1 m3.h-1, the collector output power is about 22W and its efficiency about 4.5%. A solution to improve efficiency is to increase air flow rate (see Fig. 11), in link with the fact that conductive thermal losses are decreasing when internal temperature is decreasing. Increase of flow rate leads to a lower air output temperature (and also to change convection exchange coefficients). In some cases like some drying systems where a minimal air output temperature should be needed, this can be problematic. In these cases, CODYRUN can help to choose the best compromise between air output temperature and collector power or efficiency. Fig. 11. Evolution of collector outputs regarding air flow rate. Many other simulations could be conducted, in particular in the case of natural convection. Some of these cases could be applied to air pre-heating in case of passive building design, as for next application. 4.2 Trombe wall modeling 4.2.1 Trombe-Michel wall presentation A Trombe-Michel wall (often called Trombe wall), is a passive heating system invented and patented (in 1881) by Edward Morse. It takes its name from two French people who popularized it in 1964, one engineer and one architect, respectively Félix Trombe and Jacques Michel. A Trombe-Michel wall allows to store solar energy and distribute heat following simple physical principles. This system is composed of a vertical wall, submitted to the sunshine, which absorb and store the radiative energy as thermal energy by inertia. To improve its absorptivity, the wall can be paint in a dark color. Stored energy rises-up the wall Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 237 temperature, inducing some heat transfer such as conduction (into the wall from outdoor to indoor side), radiation and convection (both with in/outdoor). To avoid outdoor losses, a glazing, which is not transmissive to ultraviolet and far infrared, is located few centimeters in front of the wall (outdoor side). So the glazing reduces outside losses by convection and traps wall's radiation due to its temperature (far infrared emission) into the building. But it also slightly reduces the solar incoming flux, because some of it is reflected or absorbed. Nota: Considering the glazing, the sunshine insulation (etc.), it is not easy to choose the optimal set-up configuration; this kind of situations are typical cases where CODYRUN simulations allow to find out. Heat transfer through the wall is done by conduction. According to the wall size and its composition, a thermal delay appears between heat absorption and emission. This characteristic allows to configure specifically the system in function of the needs. Considering the thermal delay, conductive transfer-mode is convenient to heat by night, but not by day. To palliate this lack, a faster transfer mode is brought by a variant of the Trombe-Michel wall, called recycling wall (Fig. 12). The recycling wall is obtained by holes addition at the top and the bottom of the wall. This variant allows to get a natural air flow, transferring energy by sensitive enthalpy between the system side of the wall (also called Trombe side) and the room. Indeed, as long as the glass is far enough of the wall, the temperature difference between them induces some air-convective movement, initiating the global air-circulation, so allowing heat distribution. airstream absorber convection window radiation Fig. 12. Trombe-Michell recycling wall set-up (Sfeir, 1981) Several variants of the Trombe-Michel wall can be used. These one usually evolve in function of the climatic need and can, for example, be defined by: the holes size and presence (or not), the characteristics of the storage wall (e.g. thickness, density, presence of fluid circulation or phase change materials, etc. ) or the glazing type (e.g. simple or double, treated surface, noble gas, etc.). Some technical indications about these considerations are given by Mazria (Mazria, 1975). Though two levels experimental design, (Zalewski, 1997) shows that external glazing emissivity, window type and wall absorptivity mainly characterize the system. Trombe-Michel system (and its variants) inspired numerous publications about temperate climate and proved their efficiency as passive heating system. By extension, there is also a lot of studies about similar system, as solar chimney (Ong, 2003), (Awbi, 2003). 238 Evaporation, Condensation and Heat Transfer There are several values usually used to describe Trombe-Michel wall functioning: • Transmitted energy through the wall by conduction, • Transmitted energy through the holes by sensitive enthalpy, • The system efficiency, defined as the rate between the energy received into the room over the total energy received by the North wall (South hemisphere) in the meantime. • The FGS (solar benefits fraction), defined as the rate between the benefits from the Trombe-Michel system and the energy needed by the room without it, for a room temperature of 22°C. Nota: Even if the FGS refers to active heating systems, it allows to compare easily passive heating systems between them and to other installations. 4.2.2 Presentation of Trombe-Michel wall simulations under CODYRUN This numerical study is resolved by software simulations. The case study case presented here takes place in Antananarivo, Madagascar (Indian Ocean). The aim of the simulations conducted was to help this developing country to face low temperatures in classrooms during winter season. The word 'zone', used previously, is understood as CODYRUN vocable, correspond usually to one room. Into the actual description, isothermal air assumption is made into the capitation area. This consideration is taken as first approximation and can be possibly modified later to improve the model (e.g. replaced by linear gradient hypothesis, etc.). According to the bibliography, one of the main issues mentioned is about the convection coefficient into the Trombe side of the wall. Because of the system successively laminar and turbulent, convection correlation such as - Nu = f(Gr) – cannot be used. In this case, (Zalewski, 1997) says that the laminar coefficient evolve between 2 and 2.3 W.m-2.K-1 and the turbulent one evolve between 2.25 and 3.75 W.m-2.K-1. So, in a first approach, we approximate the temperature-evolving value of the convection coefficient as the average of the turbulent and laminar values (2.9 W.m-2.K-1 ). Considering thermo-circulation, the CODYRUN's airflow module allows to calculate air flows through wall's holes. This consideration and the case study description, the two areas (the Trombe system and the room) are studied coupled, which is a right physical representation of the reality. Results validity are certified by the fact that each module of CODYRUN, and their combinations, has been validated by comparison with experimental results, reference codes and BESTEST procedures. This multi-model code structure (Boyer, 1996) can be efficiently exploited by studies such as this one, where it is necessary to choose the convection coefficients by area or by wall, to mesh slightly the walls to calculate precisely the conductive fluxes, to choose a solar gains repartition model or equivalent sky temperature, etc. CODYRUN also creates numerous output simulation variables, in addition to the one related to the dry temperature area, such as sensible power outputs from mass exchange through wall holes, conductive fluxes through the wall, enthalpic zone balance and the PMV (Predictive Mean Vote, according to Gagge) of interior zone to quantify comfort conditions. It is also possible to explore the conductive flux through the Trombe-Michel wall glass. Simulation results are presented with meteorological data of Antananarivo coming from TRNSYS 16.0 software and its TMY2 documentation data file. According to this one, daily Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 239 sunshine insulation annual average is about 5.5 kWh.m-2. The one on the North and East walls are respectively 2.23 and 2.67 kWh.m-2. Modified in function of the current albedo (0.3), the previous values become 3 and 3.46 kWh.m-2. The simulated building represents a residence unit based on a geometric cell description (9 m² parallelepiped square base ground). Moreover, materials, thermo-physics characteristics and wall type are typical from Madagascar. The room is supposed to be constantly occupied by two persons and have a usual profile for intern lighting. This area is supposed to exchange air with the outside at a rate of 1 volume per hour. Finally, the system area is of 2 m2 and is located on the North wall (South hemisphere). 4.2.3 Simulation Passive heating oriented-simulations are made the first day of July (Julian calender days 181 and following). In a first time, the objective is to explain the set-up functioning. So, some hourly based analyses will be conducted. Finally, the last experiment will compare the same Trombe-Michel wall with and without the recycling variant. 4.2.3.1 Inverse thermo-circulation evidence By night, over some system specific conditions (open holes) and in function of previous day insulations (e.g. low insulation), the flow evolution can sometime be the symmetric of the one by day (Fig. 13). In this configuration, the system functioning is reversed, cooling the room instead of heating it. Between points 19 and 32 (i.e. 7 pm to 8 am next day), the Trombe-area temperature is lower than the room temperature, inducing the inverse thermocirculation. This physical process can be illustrated by the negative part of the sensitive aeraulic benefits for the inside area. Even if the flux values by night are low, it is important to note the following point : the curves are obtained by assuming a sky temperature equal to the outside air temperature. This assumption reduces the impact of the system/sky radiation transfer, so involves an error. This error could be reduced by taking a sky temperature value more realistic, so by considering the night-sky wide wavelength emission and the high glass emissivity (0.9). A better approximation can be done by a sky temperature model such as Tsky=Tae– K, where K=6. Fig. 13. Dry temperatures and sensible power balance through holes. 4.2.3.2 Night hole closing system Hole closing system strategy deserve a specific comment. This one can be done nicely from flow measurement, but a simpler and passive way to do it also exists. This one consists in a 240 Evaporation, Condensation and Heat Transfer plastic foil, only attached on one side and occulting the holes preventing from inverse thermo-circulation. Into CODYRUN, the closure system is designed by the “little opening” component added to a hourly profile (i.e. 24 opening percentages), where the hole is closed by night and opened by day. However, the thermo-circulation (by system inertia) continues during the two first hours of the evening, so the profile includes this pattern too. Because of the small gap with the previous curves (Fig. 13), the results are firstly surprising. An explanation can be done by the small thermal gradient during the considered night. According to Mazria (Mazria, 1975) and Bansal (Bansal, 1994), recycling variant are only useful into cold climates (winter averages: -1 °C to -7 °C). So, simulation results show the minor impact of the hole control in this study case. In other words, it seems that the mass flow have a very low sensibility to the opening state. This hypothesis is rejoined by another author (Zalewski, 1997) who have substantively the same conclusion for a North France located case. 4.2.3.3 Global Trombe wall functioning To obtain the full power delivered by the wall, airflow and conductive heat transfer have to be summed. By night, it appears a constantly positive heat flux inducing a rise-up of the room temperature (Fig. 14). In the morning (e.g. 9am first day), the global flux becomes negative. A full curve interpretation is approached by airflow and conductive heat flux analyze through the wall (right part of Fig. 14). Early morning, the global heat flux inversion can be explained by highly negative conductive heat flux resulting certainly from the heat flux night inversion (around 9 to 14, so a delay of 4 hours). Drawn curves of the incoming heat flux through the wall confirm this hypothesis. As foreseeable (without simulation tools), the curves show the alternate functioning of the conductive and sensible airflow input power (the wall characteristics setting the shifting period). By night, the conductive heat flux through the wall rises-up the room temperature, whereas by day this action is done by the heat flux associated to air displacement, physically expressed by the natural air movement which is a priori synchronous with the sunlight distribution profile. The previous report allows to ask about the holes necessity. In fact, even if there is no heat transfer improvement, the main interest comes from the heat distribution and its timeevolving modulation into the building. Fig. 14. Global and specific fluxes induced by the Trombe wall into the room. To increase the room input power (i.e. the global heat flux), it could be simple to rise-up the wall surface and/or to add some reflectors to raise the incoming sunlight intensity. Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 241 However, in every cases the system have to be simulated for summer conditions. In order to reduce the external loss (illustrated by negative heat fluxes), a double glazing could be used, even if it reduces the incoming sunlight intensity (transmission loss). For reference, double glazing can be easily made from simple glazing. Another solution consists in isolating the indoor side of the wall. 4.2.3.4 Trombe-Michel wall and recycling variant comparison The comparison, in the meantime, of the wall's holes presence (or not), expresses a gap between the global heat fluxes. The values corresponding to the recycling variant are shifted to the left compared to the classic Trombe-Michel wall (Fig. 15). The input power delivered with the holes is slightly superior, which confirms the hypothesis of low impact on section 4.2.3.2. A piece of explanation (checkable by simulation) can be given by the thermal gradient between the room and the Trombe system, the decrease of this one reducing the conductive transfer through the glazing. Predicted Mean Votes (PMV) curves (of the right of Fig. 15) express the comfort sensation which evolve between -3 and +3. By comparison, it can be affirmed that the holes improve the comfort sensation. This improvement is related to the previous comment (about the superior input power amount which led to a rise-up of the dry temperature), and to the higher indoor side wall temperatures. Fig. 15. Input power and PMV comparisons 4.2.4 Trombe wall simulation exercise conclusion This example of CODYRUN use allows to find some interesting results about TrombeMichel wall configurations for the specific location of Antananarivo. Results such as the interest of the recycling variant (which is almost not influenced by hole closing procedure), the time-repartition (and impact) of the airflow and conductive heat flow on the room temperature, (etc.) have been found. Considering the case study, much more informations could be deduced from other well designed simulations. The considerable amount of knowledge found was only deduced from simulations, so only following a numerical approach. From this consideration and the results precision, a good estimation about CODYRUN quality and level of development can be done. 5. Conclusion This paper describes the models, validation elements and two applications of a general software simulation of building envelopes, CODYRUN. Many studies and applications have been conducted using this code and illustrate the benefits of this specific development. 242 Evaporation, Condensation and Heat Transfer Conducted and sustained for over fifteen years, the specific developments made around CODYRUN led our team to be owner of a building simulation environment and develop connex themes such as validation or application to large scale in building architectural requirements. This evolving environment has enabled us to drive developments in methodological areas (Mara, 2001) (modal reduction, sensitivity analysis, coupling with genetic algorithms (Lauret, 2005), neural networks, meta models,...) or related to technological aspects (solar masks, integration of split-system, taking into account the radiant barriers products, ...). This approach allows a detailed understanding of the physical phenomena involved, the use of this application in teaching and capitalization within a work tool of a research team. Accompanied by the growth of computer capabilities, the subject of high environmental quality in buildings, our field becomes more complex by the integration of aspects other than those originally associated with the thermal aspects (environmental quality, including lighting, pollutants, acoustics, ...). Simultaneously, it is also essential to achieve a better efficiency in the transfer of knowledge and operational tools. 6. References Awbi H. (2003),Ventilation of buildings, Spon Press, ISBN 0415270-55-3. Balcomb J.D. & al. (1997), Passive solar heating of buildings, in Solar Architecture, Proceedings of the aspen Energy Forum, Ann Arbor Science Publishers Inc., Aspen, Colorado, 1977. Bansal N.K., Hauser G., Minke G. (1994). Passive Building design. A handbook of natural climatic control, Elsevier, ISBN 0-444-81745-X, Amsterdam (Netherlands) Berthomieu T., Boyer H. 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arXiv:1507.06739v5 [math.ST] 30 Nov 2016 Selective inference with a randomized response Xiaoying Tian, Jonathan Taylor∗ Department of Statistics Stanford University Sequoia Hall Stanford, CA 94305, USA e-mail: xtian@stanford.edu jonathan.taylor@stanford.edu Abstract: Inspired by sample splitting and the reusable holdout introduced in the field of differential privacy, we consider selective inference with a randomized response. We discuss two major advantages of using a randomized response for model selection. First, the selectively valid tests are more powerful after randomized selection. Second, it allows consistent estimation and weak convergence of selective inference procedures. Under independent sampling, we prove a selective (or privatized) central limit theorem that transfers procedures valid under asymptotic normality without selection to their corresponding selective counterparts. This allows selective inference in nonparametric settings. Finally, we propose a framework of inference after combining multiple randomized selection procedures. We focus on the classical asymptotic setting, leaving the interesting high-dimensional asymptotic questions for future work. AMS 2000 subject classifications: Primary 62M40; secondary 62J05. Keywords and phrases: selective inference, nonparametric, differential privacy. 1. Introduction Tukey (1980) promoted the use of exploratory data analysis to examine the data and possibly formulate hypotheses for further investigation. Nowadays, many statistical learning methods allow us to perform these exploratory data analyses, based on which we can posit a model on the data generating distribution. Since this model is not given a priori, classical statistical inference will not provide valid tests that control the Type-I errors. Selective inference seeks to address this problem, see Lee et al. (2013a), Lockhart et al. (2014), Lee & Taylor (2014), Fithian et al. (2014). Loosely speaking, there are two stages in selective inference. The first is the selection stage that explores the data and formulates a plausible model for the data distribution. Then we enter the inference stage that seeks to provide valid inference under the selected model which is proposed after inspecting the data. Inference under different models have been studied, notably the Gaussian families Lee et al. (2013a), Tian et al. (2015), Lee & Taylor (2014) as well as other exponential families Fithian et al. (2014). In this work, we consider selective inference in a general setting that include nonparametric settings. In addition, we introduced the use of randomized response in ∗ Supported in part by NSF grant DMS 1208857 and AFOSR grant 113039. 1 Tian and Taylor/Selective inference with a randomized response 2 model selection. A most common example of randomized model selection is probably the practice of data splitting. Assuming independent sampling, we can divide the data into two subsets, using the first for model selection and the second subset for inference. Though not emphasized, this split is often random. Hence, data splitting can be thought of as a special case of randomized model selection. To motivate the use of randomized selection and introduce the inference problem that ensues, we consider the following example. 1.1. A first example Publication bias, (also called the “file drawer effect” by Rosenthal (1979)) is a bias introduced to scientific literature by failure to report negative or non-confirmatory results. We formulate the problem in the simple example below. Example 1 (File drawer problem). Let n X̄n = 1X Xi,n n i=1 be the sample mean of a sample of n iid draws from Fn in a standard triangular array. We set µn = EFn [X1,n ] and assume EFn [(X1,n − µn )2 ] = 1. Suppose that we are interested in discovering positive effects and would only report the sample mean if it survives the file drawer effect, i.e. n1/2 X̄n > 2. (1) Then what is the “correct” p-value to report for an observation X̄n,obs that exceeds the threshold? If we have Gaussian family, namely Fn = N (µn , 1), then the distribution of X̄n surviving the file drawer effect (1) is a truncated Gaussian distribution. We also call this distribution the selective distribution. Formally, its survival function is 1 1/2 P (t) = P X̄n > t\|n X̄n > 2 , X̄n ∼ N µn , n 1/2 1 − Φ n (t − µn ) = 1 − Φ(2 − n1/2 µn ) where Φ is the CDF of an N (0, 1) random variable. Therefore, we get a pivotal quantity 1 − Φ n1/2 (X̄n,obs − µn ) ∼ Unif(0, 1), P (X̄n,obs ) = 1 − Φ(2 − n1/2 µn ) (2) 1 1/2 n X̄n,obs > 2, Xn,obs ∼ N µn , n The pivotal quantity in (2) allows us to construct p-values or confidence intervals for Gaussian families. When the distributions Fn ’s are not normal distributions, central limit theorem states that the sample mean X̄n is asymptotically normal when Fn Tian and Taylor/Selective inference with a randomized response 3 has second moments. Thus a natural question is whether the pivotal quantity in (2) is asymptotically Unif(0, 1) when Xi,n does not come from a normal distribution? The following lemma provides a negative answer to this question in the case when Fn is a translated Bernoulli distribution that has a negative mean. Essentially when the selection event n1/2 X̄n > 2 becomes a rare event with vanishing probability, the pivotal quantity in (2) no longer converges to Unif(0, 1). We defer the proof of the lemma to the appendix. Lemma 1. If Xi,n takes values in {−1.5, 0.5}, with P (Xi,n = −1.5) = P (Xi,n = 0.5) = 0.5. Thus µn = −0.5. Then the pivot in (2) does not converge to Unif(0, 1) P (X̄n ) 6→ Unif(0, 1), for the X̄n ’s surviving the file drawer effect (1). Randomized selection circumvents this problem. In the following, we propose a randomized version of the “file drawer problem”. Example 2 (File drawer problem, randomized ). We assume the same setup of a triangular array of observations Xi,n as in Example 1. But instead of reporting X̄n when it survives the file drawer effect (1), we independently draw ω ∼ G, and only report X̄n if n1/2 X̄n + ω > 2. (3) Note that the selection event is different from that in (1) in that we randomize the sample mean before checking whether it passes the threshold. In this case, if Fn = N (µn , 1), the survival function of X̄n is 1 1/2 ×G P (t) = P X̄n > t\|n X̄n + ω > 2 , (X̄n , ω) ∼ N µn , n = P Z > n1/2 (t − µn )\|Z + ω > 2 − n1/2 µn , (Z, ω) ∼ N (0, 1) × G. (4) To compute the exact form of P (t), we have to compute the convolution of N (0, 1) and G which has explicit forms for many distributions G. Moreover, when G is Logistic or Laplace distribution, we have P (X̄n,obs ) → Unif(0, 1), as long as Fn has centered exponential moments in a fixed neighbourhood of 0. The convergence is in fact uniform for −∞ < µn < ∞. For details, see Lemma 10 in Section 5.2. The only difference between these two examples is the randomization in selection. After selection, we need to consider the conditional distribution for inference, which conditions on the selection event. If we denote by F∗n the distribution used for selective inference, we have in Example 1, 1{n1/2 X̄n >2} dF∗n (X̄n ) = . dFn PFn (n1/2 X̄n > 2) (5) Tian and Taylor/Selective inference with a randomized response 4 We also call the ratio between F∗n and Fn the selective likelihood ratio. In this case, the selective likelihood ratio is simply a restriction to the X̄n ’s that survives the file drawer effect. We observe that √ √ nX̄n = nµn + Z, Z ∼ N (0, 1), which leads to three scenarios for selection. • µn > δ > 0, for some δ > 0. √ In this case, the dominant term for selection is nµn , and since we have a big positive effect, we would always report the sample mean X̄n when n is big. This corresponds to the selection event having probability tending to 1 and the selective likelihood ratio goes to 1 as well. In this case, there is very little selection bias, and the original law is a good approximation to the selective distribution for valid inference. • µn < −δ < 0, for some δ > 0. √ In this case, the dominant term is also nµn , but in the negative direction. As n → ∞, the selection probability vanishes and the selective likelihood becomes degenerate. We almost never report the sample mean in this scenario, but in the rare event where we do, by no means can we use the original distribution for inference. • −δ < n1/2 µn < δ, for some δ > 0. This corresponds to local alternatives. In this case, the selective likelihood neither converges to 1 or becomes degenerate. Rather, it becomes an indicator function of a half interval. Proper adjustment is needed for valid inference in this case. It is in the second scenario that pivotal quantity (2) will not converge to Unif(0, 1). Different distributions will have different behaviors in the tail. Since the conditioning event n1/2 X̄n > 2 becomes a large-deviations event, we cannot expect it to behave like the normal distribution in the tail. On the other hand, in Example 2, if we denote by F̃∗n the law for selective inference, we have Ḡ 2 − n1/2 (X̄n − µn ) − n1/2 µn dF̃∗n Ḡ(2 − n1/2 X̄n ) (X̄n ) = = dFn EFn (Ḡ(2 − n1/2 X̄n )) EFn Ḡ 2 − n1/2 (X̄n − µn ) − n1/2 µn (6) R∞ where Ḡ(t) = t G(du) is the survival function of G. When µn < −δ < 0 for some δ > 0, and G is the Laplace or Logistic distribution so that Ḡ has an exponential tail, the dominant term exp(n1/2 µn ) in both the numerator and the denominator will cancel out, making the selective likelihood ratio properly behaved in this difficult scenario. It turns out that this selective likelihood ratio is fundamental to formalizing asymptotic properties of selective inference procedures. Its behavior determines not only the asymptotic convergence of the pivotal quantities like in (4), but also whether consistent estimation of the population parameters is possible with large samples. Again in the negative mean scenario where µn < −δ < 0, the sample mean X̄n surviving the non-randomized “file drawer effect” cannot be a consistent estimator for the underlying means µn because it will always be positive. But if X̄n is reported as in Tian and Taylor/Selective inference with a randomized response 5 Example 2, it will be consistent for µn even if µn is negative and bounded away from 0. For detailed discussion, see Section 3. In general, the behavior of the selective likelihood ratio can be used to study the asymptotic properties of selective inference procedures. We study consistent estimation and weak convergence for selective inference procedures in Section 3 and Section 5 respectively. We are especially inspired by the field of differential privacy (c.f. Dwork et al. (2014) and references therein) to study the use of randomization in selective inference. Privatized algorithms purposely randomize reports from queries to a database in order to allow valid interactive data analysis. To our understanding, our results are the first results related to weak convergence in privatized algorithms, as most guarantees provided in the differentially private literature are consistency guarantees. Some other asymptotic results in selective inference have also been considered in Tibshirani et al. (2015), Tian & Taylor (2015), though these have a slightly different flavor in that they marginalize over choices of models. We conclude this section with some more examples. 1.2. Linear regression Consider the linear regression framework with response y ∈ Rn , and feature matrix X ∈ Rn×p , with X fixed. We make a homoscedasticity assumption that Cov [y\|X] = σ 2 I, with σ 2 considered known. Of interest is µ = E(y\|X), a functional of F = F(X) the conditional law of y given X. When F is a Gaussian distribution, exact selective tests have been proposed for different selection procedures Tibshirani (1996), Taylor et al. (2014), Tian et al. (2015). Removing the Gaussian distribution on F, Tian & Taylor (2015) showed that the same tests are asymptotically valid under some conditions. Randomized selection in this setting is a natural extension of these works. Fithian et al. (2014) proposed to use a subset of data for model selection, which yields a significant increase in power. In this work, we study general randomized selection procedures. Consider the following example. Due to the sparsity of the solution of LASSO Tibshirani (1996) 1 β̂λ (y) = argmin ky − Xβk22 + λ · kβk1 , β∈Rp 2 a small subset of variables can be chosen for which we want to report p-values or confidence intervals. This problem has been studied in Lee et al. (2013a). However, instead of using the original response y to select the variables, we can independently draw ω ∼ Q and choose the variables using y ∗ = y + ω. Specifically, we choose subset E by solving 1 β̂λ (y, ω) = argmin ky ∗ − Xβk22 + λ · kβk1 , p 2 β∈R y ∗ = y + ω, (7) Tian and Taylor/Selective inference with a randomized response 6 and take E = supp(β̂λ (y, ω)). In Section 4.2.2, we discuss how to carry out inference after this selection procedure, with much increased power. We also discuss the reason behind this increase in Section 4.2. 1.3. Nonparametric selective inference All the previous works on selective inference assume a parametric model like the Gaussian family or the exponential family. In this work, we allow selective inference in a non-parametric setting. Consider the following examples. Suppose in a classification problem, we observe independent samples, iid (xi , yi ) ∼ F, (xi , yi ) ∈ Rp × {0, 1}. with fixed p. This problem is non-parametric if we do not assume any parametric structure for F and are simply interested in some population parameters of the distribution F. In Section 5, we developed asymptotic theory to construct an asymptotically valid test for the population parameters of interest. More details can be found in Section 5.4.1. Also consider a multi-group problem where a response x is measured on p treatment groups. A special case is the two-sample problem where there are two groups. It is of interest to form a confidence interval for the effect size in the “best” treatment group. This arises often in medical experiments where multiple treatments are performed and we are interested to discover whether one of the treatment has a positive effect. The fact we have chosen to report the “best” treatment effect exposes us to selection bias and multiple testing issues Benjamini & Hochberg (1995), and therefore calls for adjustment after selection. Benjamini & Stark (1996) have considered the parametric setting iid where xj ∼ N (µj , σ 2 ) for each group. Suppose for robustness, it is of interest to report the median effect size instead of the mean (assuming responses are not symmetric). Then without any assumptions on the distribution of the measurements, this also becomes a nonparametric problem. But we can apply the theory in Section 5 to cope with this problem, for details, see Section 5.3. 1.4. Outline of the paper There are three main advantages of applying randomization for selective inference, • Consistent estimation under the selective distribution • Increase in power for selective tests • Weak convergence of selective inference procedures In the following sections, Section 2 gives the setup of selective inference and introduced selective likelihood ratio, which is the key for studying consistent estimation and weak convergence of selective inference procedures. Section 4 focuses on linear regression models with different randomization schemes, demonstrating the increase in power. Section 5 proposes an asymptotic test for the nonparametric settings. Theorem 9 proves that the central limit theorem holds under the selective distribution with mild conditions. Applications to the two examples in Section 1.3 are discussed. This is a Tian and Taylor/Selective inference with a randomized response 7 result for fixed dimension p. Finally, Section 6 discusses the possibility of extending our work to the setting, when multiple selection procedures are performed on different randomizations of the original data. One application is selective inference after cross validation for the square-root LASSO Belloni et al. (2011). 2. Selective Likelihood Ratio We first review some key concepts of selective inference. Our data D lies in some measurable space (D, F), with unknown sampling distribution D ∼ F. Selective inference seeks a reasonable probability model M – a subset of the probability measures on (D, F), and carry out inference in M . Central to our discussion is a selection algorithm, a set-valued map b:D→Q Q (8) where Q is loosely defined as being made up of “potentially interesting statistical questions”. For instance, in the linear regression setting, D = Rn , our data D = y and we have a fixed feature matrix X ∈ Rn×p . The unknown sampling distribution is F = L(y\|X), the conditional law of y given X. b might be all linear regression modA reasonable candidate for the range of Q els indexed by subsets of {1, . . . , p} with known or unknown variance. For any selected subset of variables E, we carry out selective inference within the model M = {N (XE βE , σ 2 I), βE ∈ R\|E\| }. Since we use the data to choose the model M , it is only fair to consider the conditional distribution for inference, b D\|M ∈ Q(D), D ∼ F. Therefore, we seek to control the selective Type-I error: b ≤α PM,H0 (reject H0 \| M ∈ Q) (9) b and H0 ⊂ M is the where M is the selected family of distributions in the range of Q null hypothesis. Selective intervals for parametric models M can then be constructed by inverting such selective hypothesis tests, though only the one-parameter case has really been considered to date. 2.1. Randomized selection Randomized selection is a natural extension of the framework above. We enlarge our probability space to include some element of randomization. Specifically, let H denote an auxiliary probability space and Q is a probability measure on H. A randomized selection algorithm is then simply b∗ : D × H → Q. Q Tian and Taylor/Selective inference with a randomized response 8 Note the randomization is completely under the control of the data analyst and hence Q will be fully known. This is an extension of the non-randomized selective inference framework in the sense that we can take Q to be the Dirac measure at 0. Many choices b∗ are natural extensions of Q, b which we will see in many examples. of Q Randomized selective inference is simply based on the law F∗ , which we also call the selective distribution, b∗ (D, ω), (D, ω) ∼ F × Q. D\|M ∈ Q (10) Note that although randomization is incorporated into selection, inference is still carried out using the original data D, after adjusting for the selection bias by considering the conditional distribution F∗ . Similar to the selective inference we defined above, we seek to control the selective Type-I error, b∗ ) ≤ α. PF∗ (reject H0 ) = PM,H0 (reject H0 \|M ∈ Q (11) Moreover, we also want to achieve good estimation, which makes EF∗ ((θ̂(y) − θ(F))2 ) (12) small. b∗ . But In Sections 3 to 5, we will discuss concrete examples of D, D, F and Q before that we first introduce the selective likelihood ratio, which is a crucial quantity in studying the selective distribution F∗ . 2.2. Selective likelihood ratio Selective likelihood ratio provides a way of connecting the original distribution F and its selective counterpart F∗ . It is easy to see from (10) that the selective distribution is simply a restriction of the (D, ω)’s such that model M will be selected. Thus F∗ is absolutely continuous with respect to F, and the selective likelihood ratio is W(M ; D) dF∗ (D) = = `F (D) ∀ F ∈ M, dF EF (W(M ; D)) n o b∗ (D, ω) . W(M ; D) = Q ω : M ∈ Q (13) The numerator in `F (D) is the restriction of (D, ω), integrated over the randomizations ω, and the denominator is simply a normalizing constant. One implication of the selective likelihood ratio is that for distributions F in parametric families, their selective counterparts may have the same parametric structure. 2.2.1. Exponential families One commonly used parametric family is the exponential family. Assume that F = Fθ is an exponential family with natural parameter space Θ and D = Rn and the data Tian and Taylor/Selective inference with a randomized response 9 D = y. Its density with respect to the reference measure dF0 is, dFθ (y) = exp{θT T (y) − ψ(θ)}, dF0 θ ∈ Θ. (14) Through the relationship in (13) we conclude, for any randomization scheme, the law F∗M,θ is another exponential family. Formally, Lemma 2. If Fθ belongs to the exponential family in (14), then for any randomized b∗ , the selective distribution is also an exponential family, selection procedure Q dF∗M,θ (y) ∝ W(M ; y) exp{θT T (y) − ψ(θ)}, dF0 θ ∈ Θ. with the same sufficient statistic T (y) and natural parameters θ. Furthermore, to test H0j : θj = 0, we consider the following law, Tj (y) \| T−j (y), y ∼ F∗M,θ . (15) The first claim of the lemma is quite straight-forward using the relationship in (13). The second claim is a Lehmann–Scheffe (c.f. Chapter 4.4 in Lehmann (1986)) construction which was proposed in Fithian et al. (2014), to construct tests for one of the natural parameters treating the others as nuisance parameters. For detailed construction of such tests in the linear regression setting, see Section 4. 3. Consistent Estimation After Model Selection In this section, we leave the parametric setup and consider general models M . In particular, we study the consistency of estimators under the selective distribution for arbitrary models. We first introduce the framework of asymptotic analysis under the selective model. Then we state conditions for consistent estimation in Lemma 3 and conclude with examples. For any model M , which is a collection of distributions, we define its corresponding selective model, which is the collection of corresponding selective distributions, ∗ ∗ ∗ dF (D) = `F (D), F ∈ M , (16) M = F : dF b∗ }. Selecwhere `F (D) is the selective likelihood ratio for the selection event {M ∈ Q ∗ tive inference is carried out under the selective model M . In order to make meaningful asymptotic statements, we consider a sequence of ranb∗n )n≥1 and models (Mn )n≥1 with each Mn in the domized selection procedures (Q ∗ b range of Qn . Often, we are interested in some population parameter θn , which can be thought be as a functional of the distribution Fn ∈ Mn , θn : Mn → R. Tian and Taylor/Selective inference with a randomized response 10 b∗ , which already incorporates the It is worth pointing out that Mn is selected by Q n statistical questions we are interested in. In this sense, Mn is chosen a posteriori. The selected model Mn∗ does not change our target of inference, it merely changes the distribution under which such inference should be carried out. In other words, if θn is the mean parameter, we are interested in the underlying mean of Fn , not F∗n . We might have a good estimator θ̂n : D → R for θn (Fn ) under Fn , namely i h EFn (θ̂n − θn (Fn ))2 → 0. θ̂n is a consistent estimator if our model Mn is given a priori. But as we use data select Mn , what really cares about is its performance under the selective distribution F∗n . Will this estimator still be consistent under the selective distribution F∗n ? Formally, we say an estimator θ̂n is uniformly consistent in Lp for θn (Fn ) under the sequence (Mn )n≥1 if lim sup sup kθ̂n − θn (Fn )kLp (Fn ) → 0. n Fn ∈Mn Similarly, we say that θ̂n is uniformly consistent in probability for the functional θn (Fn ) under the sequence (Mn )n≥1 if for every > 0 there exists δ() > 0 such that for all δ ≥ δ() lim sup sup Fn (\|θ̂n − θn (Fn )\| > δ) ≤ . n Fn ∈Mn The following lemma states the conditions for consistency of θ̂n under the sequence of corresponding selective models (Mn∗ )n≥1 , b∗n , Mn )n≥1 of randomized selection procedures Lemma 3. Consider a sequence (Q and models. Suppose the selective likelihood ratios satisfies, for some p > 1, lim sup sup k`Fn kLp (Fn ) < C. n (17) Fn ∈Mn Then for any sequence of estimators θ̂n uniformly consistent for θn (Fn ) in Lα , it is also uniformly consistent for θn (Fn ) in Lγ under (Mn∗ )n≥1 , γ ≤ α/q, p1 + 1q = 1. Further, if θ̂n is uniformly consistent for θn in probability, then θ̂n is uniformly consistent for θn in probability under the sequence (Mn∗ )n≥1 . Proof. Let ∆n = θ̂n − θn (Fn ). To prove the first assertion note that for any F∗n ∈ Mn∗ Z k∆n kLγ (F∗n ) = \|∆n \|γ `Fn (y)Fn (dy) Dn ≤ k\|∆n \|γ kLq (Fn ) k`Fn (y)kLp (Fn ) = k∆n kγLγq (Fn ) k`Fn (y)kLp (Fn ) ≤ k∆n kγLα (Fn ) k`Fn (y)kLp (Fn ) ≤ Ck∆n kγLα (Fn ) Tian and Taylor/Selective inference with a randomized response 11 For any δ > 0, F∗n (\|∆n \| Z > δ) = Dn 1{\|∆n \| > δ}`Fn (y)Fn (dy) 1/q ≤ [Fn (\|∆n \| > δ)] k`Fn kLp (F) 1/q ≤ C [Fn (\|∆n \| > δ)] . We illustrate the application of Lemma 3 through our “file drawer effect” examples in Section 1.1. 3.1. Revisit the “file drawer problem” First we note that in Example 1 and 2, we observe data Dn = (X1,n , . . . , Xn,n ), with Xi,n ∼ Fn . The randomized selection in Example 2 can be realized as ( √ report p-values for X̄n , if nX̄n + ω > 2, ∗ b Q (Dn , ω) = √ do nothing, if nX̄n + ω ≤ 2, where we independently draw ω ∼ G. By law of large numbers, we easily see that if we always report X̄n , it will be an unbiased estimator for µn . However, since we only observe the sample means surviving the file drawer effect. Will X̄n still be consistent for µn ? In the most difficult scenario discussed in Section 1.1, where µn < −δ < 0 for some δ > 0, X̄n cannot be a consistent estimator for µn in Example 1. This is easy to see as Example 1 will only report positive sample means. A remarkable feature of randomized selection is that consistent estimation of the population parameters is possible even when the selection event has vanishing probabilities. In fact, the following lemma states that when G is a Logistic distribution, X̄n is consistent for µn after the randomized file drawer effect in Example 2. Lemma 4. Suppose as in Example 2, we observe a triangular array with Xi,n ∼ Fn . Fn has mean µn = µ < 0. If we draw ω ∼ Logistic(κ), where κ is the scale of the Logistic distribution. Then the sample means X̄n surviving the “randomized” file drawer effect are consistent for µ, √ p X̄n → µ, conditional on nX̄n + ω > 2. if Fn has moment generating function in a neighbourhood of 0. Namely, ∃a > 0, such that EFn [exp (a \|Xi,n − µn \|)] ≤ C. Before we prove the lemma, we want to point out that although the selection √ procedure in Example 2 is different from that in Example 1 because of randomization, nµn is still the dominant term in selection. Note that √ √ √ nX̄n + ω = nµn + n(X̄n − µn ) + ω. Tian and Taylor/Selective inference with a randomized response 12 √ Since both n(X̄n − µn ) and ω are Op (1) random variables, the dominant term √ nµn → −∞, would ensure that the selection event has vanishing probabilities in Example 2 as well. Thus it is particularly impressive that Example 2 gives consistent estimation where Example 1 cannot. The proof of Lemma 4 is deferred to the appendix. We also verified this theory of consistent estimation through simulations. Figure 1 shows the empirical distributions of the sample mean X̄n after the file drawer effect in Example 1 or the “randomized” file drawer effect in Example 2. They are marked with “blue” colors or “red” colors respectively. We set the true underlying mean to be µn = µ = −1 and mark it with the dotted vertical line in Figure 1. The upper panel Figure 1a is simulated with n = 100 and the lower panel Figure 1b is simulated with n = 250. We notice that in both simulations, the sample mean in Example 1 concentrates around the thresholding boundary, which is positive. Thus, these sample means can not be possibly for the underlying mean µ = −1. However, the existence of randomization allows us to report negative sample means. As a result, the sample mean in Example 2 will be consistent for µ = −1. We see that as we increase sample size n, the sample means concentrates closer to µ = −1. 4. Inference in linear regression models In the linear regression setting, we assume a fixed feature matrix X ∈ Rn×p , and observe the response vector D = y ∈ Rn . We assume the noises are normally distributed. There are two ways to parametrize a linear model, and both belong to some exponential family. Now we introduce the selected model, n o Msel (E) = N (XE βE , σ 2 I) : βE ∈ R\|E\| , E ⊂ {1, . . . , p} (18) with σ 2 known or unknown or the saturated model, Msat = N (µ, σ 2 I) : µ ∈ Rn (19) with known variance. Now we consider some randomized selection procedures and inference after selection. 4.1. Data splitting and data carving In the introduction, we introduced data splitting Cox (1975) as a special case of randomized selective inference. In Fithian et al. (2014), the term data carving was introduced to demonstrate that data splitting is inadmissible. In data splitting (and data carving) inference makes most sense in the selected model Msel (E), hence we should b as returning a subset E of variables selected. think of Q Let us formalize this notion in our notation. Let Q be some measure on assignments b a selection algorithm defined on datasets of any of n data points into groups and Q size. The distribution Q determines a randomized selective inference procedure with b∗ , an algorithm applied to subsets of the original data set. In this selection algorithm Q 13 Tian and Taylor/Selective inference with a randomized response X̄ after file drawer effect 90 80 4.0 70 3.5 60 3.0 50 2.5 40 2.0 30 1.5 20 1.0 10 0.5 0 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 X̄ after randomized file drawer effect 4.5 0.0 0.2 0.4 0.0 −1.2 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.2 0.4 (a) n = 100 X̄ after file drawer effect 120 X̄ after randomized file drawer effect 7 6 100 5 80 4 60 3 40 2 20 0 -1.2 1 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0 −1.2 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 (b) n = 250 Fig 1: Empirical distributions of sample means X̄n in Example 1 and Example 2, with original or randomized file drawer effect. For the randomization, we draw ω ∼ Logistic(κ), with κ = 0.5. Tian and Taylor/Selective inference with a randomized response 14 case, it is easy to see that D W(E; y) = W(Msel (E); y) ∝ X ω qω · 1{Msel (E)∈Q(y b 1 (y,ω))} where qω is the mass assigned to assignment ω by Q. Multiple assignments or splits considered in Meinshausen et al. (2009), Meinshausen & Bühlmann (2010) can be formalized in a similar fashion. We can construct UMPU tests for βE in the selected model Msel (E) by using Lemma 2, (also see Fithian et al. (2014)). We note that in Fithian et al. (2014) the authors conditioned unnecessarily on the split ω, and we would expect that aggregating over splits would yield a more powerful procedure. However, there are two disadvantages with this randomization scheme. First, it is computationally difficult to aggregate over all random splits. Second, it seems difficult to consider the saturated model Msat for inference, which is more robust to model misspecifications. To overcome those difficulties, we introduce other randomization schemes below. 4.2. Additive noise and more powerful tests Our second randomization scheme in linear regression involves additive noise. Specifically, we draw ω ∼ Q and use the randomized response y ∗ (y, ω) = y + ω for selection In this case, we can consider both the selected model Msel,E and the saturated model Msat . Per Lemma 2, we can perform valid inference for βE in Msel,E or linear functionals of µ in Msat . One major advantage of using a randomized response y ∗ for selective inference is that these procedures yield much more powerful tests, at a small cost of on the quality of the selected models. In other words, small amount of randomization is cause a small loss in the model selection stage, but we gain much more power in the inference stage. The reason for increased power can be explained by a notion called leftover Fisher Information first introduced in Fithian et al. (2014). Since selective inference is essentially inference under the selective distribution F∗n , the Fisher Information under F∗n would determine how efficient the selective tests are. In the saturated model with ∗ Gaussian noise Msat , y−µ σ 2 is the score statistic and its variance under Fn is exactly the leftover Fisher Information (a similar relationship holds in the selected model Msel,E ). Lemma 5 gives a lower bound on this leftover Fisher Information when the randomization noise Q = N (0, γ 2 I). Lemma 5. For either Msat or Msel (E), if we use Gaussian randomization noise Q = b ∗ ) = Q(y b + ω), then the leftover Fisher N (0, γ 2 ), and the selection is based on Q(y information is bounded below by (1 − τ )I(θ), τ = σ 2 /(σ 2 + γ 2 ), and I(θ) is the non-selective Fisher information for θ in Msat or Msel (E). The parameters θ depend on which of the two models we are considering. 15 Tian and Taylor/Selective inference with a randomized response b ∗ Proof. In the saturated model Msat , the score statistic is V = y−µ σ 2 . Since Q(y ) is measurable with respect to y ∗ , h i b ∗ ) ≥ Var [V \| y ∗ ] = 1 Var [y \| y ∗ ] . Var V \| Q(y σ4 Since y and y ∗ = y + ω are both normal distributions with covariance matrices, Cov [y, y ∗ ] = σ 2 I, Var [y ∗ ] = (γ 2 + σ 2 )I, we have the leftover Fisher Information h i b ∗ ) ≥ 1 Var [y \| y ∗ ] Var V \| Q(y σ4 4 1 σ 1 = 4 (σ 2 I − 2 I) = 2 (1 − τ )I = (1 − τ )I(µ). 2 σ γ +σ σ In the selected model Msel,E , the score statistic is V = T XE (y−XE βE ) . σ2 Similarly, h i T b ∗ ) ≥ 1 Var XE Var V \| Q(y y \| y∗ 4 σ 1 σ4 1 2 T T T = 4 σ (XE XE ) − 2 (X XE ) = 2 (1 − τ )XE XE = (1 − τ )I(βE ). σ γ + σ2 E σ When there is no randomization γ = 0, we potentially have no leftover Fisher information. This corresponds to a very rare selection event. However after randomization, even with very extreme selection, there is always leftover Fisher information, which makes the selective tests more powerful. Consider the following examples. 4.2.1. Revisit the “file drawer problem” In Example 1 and Example 2, if we assume Fn = N (µ, 1), they are a special case of the linear regression model, with the feature matrix X = 1, the all ones vector. In this case, nX̄n is the score statistic, and its variance under the selective distribution is the Fisher information. Lemma 5 states that the leftover Fisher information is lower bounded by n(1 − τ ) if we draw randomize using Gaussian variables, Q = N (0, γ 2 ), τ = 1/(1 + γ 2 ). Moreover, the increase in leftover Fisher information with randomization is not specific to Gaussian randomizations. For example, in Figure 1 when we use Logistic randomization, we also observe that under the selective distribution with randomization, X̄n has a much bigger variance than without randomization. As discussed above, this variance multiplied by n2 is exactly the leftover Fisher information, which explains why selective procedures after randomization will have better performances than without. We investigate the relationship between the leftover Fisher information and the length of confidence intervals constructed by inverting the pivot in (4). Specifically, 16 Tian and Taylor/Selective inference with a randomized response 1.0 1.0 0.5 0.5 0.0 0.0 0.5 0.5 1.0 1.0 1.5 2.0 0.2 1.5 Gaussian randomization Nominal 0.1 0.0 0.1 Observed X̄ 0.2 0.3 Logistic randomization Nominal 0.4 (a) Gaussian added noise 2.0 0.2 0.1 0.0 0.1 Observed X̄ 0.2 0.3 0.4 (b) logistic added noise Fig 2: Selective confidence intervals for different added noise in Example 2, after observing a reported sample mean, we want to report confidence intervals for the underlying mean µ. Figure 2 demonstrates the selective intervals (solid lines) after (3) with ω being either Gaussian or Logistic noises. The sample size n = 100. Unlike the nominal confidence intervals (dashed lines), the selective intervals are valid with 90% coverage for the underlying mean. Since Lemma 3 gives a lower bound of (1 − τ )I(µ), we would intuitively expect the selective confidence intervals to be 1/(1 − τ ) the length of the nominal intervals. This is verified in Figure 2a, when we observe really negative sample means. (The sample means can be negative because we added randomization.) On the other hand, for Logistic randomization√in Figure 2b, the intervals are slightly wider than the nominal intervals around the 2/ n, but narrow to roughly the nominal size on both sides of the truncation point. This indicates that added logistic noise might preserve more information than Gaussian additive noise. Both additive noises improve significantly over a non-randomization scheme (c.f. Figure 3 in Fithian et al. (2014)). Of course, the increase in power and shortening of selective confidence intervals does not come without a price. Because we select with a randomized response, we are likely to select a worse model. But the trade-off between model quality and power is highly in favor of randomization. See the following example. 4.2.2. Linear regression with added noise Back to the general setup of linear regression models, we select a model by solving LASSO with the randomized response y ∗ = y + ω and return the active set E of the solution (as in (7)). Then per Lemma 2, we can construct valid selective tests in both Msat and Msel (E). For instance, in Msel (E), we can construct tests for the hypothesis H0j : βj = 0, j ∈ E based on the law, η T y AE (y + ω) ≤ bE , PE\j y, (y, ω) ∼ N (XE βE , σ 2 I) × Q, βj = 0, (20) † T where η = (XE ) ej , ej is the j-th column of the identity matrix, PE\j is the projection matrix onto the column space of E but orthogonal to η, and AE , bE are the appropriate Tian and Taylor/Selective inference with a randomized response 0.5 Type II error 0.4 17 Data splitting Data carving Additive noise 0.3 0.2 0.1 0.0 1.0 0.9 0.8 0.7 Probability of screening 0.6 0.5 Fig 3: Comparison of inference in additive noise randomization vs. data carving. matrix and vector corresponding to LASSO selection. This is a UMPU test due to the Lehmann–Scheffe construction (Fithian et al. 2014) and controls the selective Type-I error (11). Although, we cannot compute the explicit forms of (20), the selection events in (20) are polyhedrons and thus a hit-and-run or Hamiltonian Monte Carlo algorithm Pakman & Paninski (2012) can be used for sampling. Figure 3 compares inference in the additive Gaussian noise scheme to the data carving procedure proposed in Fithian et al. (2014) as well as data splitting. In Msel (E), the probability of screening (i.e. selecting E including all the nonzero β’s) is a surrogate for the quality of the model. As additive noise uses a different randomization scheme than data splitting and data carving, we vary the amount of randomization used in each scheme and match on the probability of screening. Thus Figure 3 is like an ROC curve for the trade-off between model quality and power of tests. The x-axis goes in the direction of increased randomization, with the left most point corresponding to no randomization at all. We see even with a small randomization that barely affects model selection, we can substantially lower the Type-II error from 0.2 to less than 0.05. The trade-off is highly in favor of (small) randomization. We see in Figure 3 that additive noise lowers the Type-II error by almost half than data carving for the same screening probability and they both clearly dominate data splitting. For the concrete setup of the simulation, see Chapter 7 of Fithian et al. (2014). 5. Weak convergence and selective inference for statistical functionals In the nonparametric setting, we assume a triangular array of data, Dn = (d1,n , . . . , dn,n ), iid and di,n ∼ Fn . When Fn = F, it is the special case of independent sampling. We are interested in some functional of the distribution µn = µ(Fn ). Associated with µn is our statistic T which is a linearizable statistic (Chung et al. 2013). iid Definition 6 (Linearizable statistic). Suppose di,n ∼ Fn , we call T a linearizable Tian and Taylor/Selective inference with a randomized response 18 statistic for µn = µ(Fn ) if for any sample size n, n T (Dn ) = 1X ξi,n + R, n i=1 p E [ξi,n ] = µn ∈ R , ξi,n = ξ(di,n ), Cov [ξi,n ] = Σn ∈ R (21) p×p . 1 where ξ a function of the data and R is bounded with probability 1, R = op (n− 2 ) under F. We use the slight abuse of notations to denote ξi,n as iid random variables as well. Throughout this section, we assume the dimension p is fixed. We are interested in establishing a pivotal quantity for Tn = T (Dn ) like (4) in Example 2 where Tn is the sample mean after the randomized “file drawer effect”. It turns out we have an exact pivotal quantity if Tn is normally distributed. To lighten notation, we suppress the script n in the following lemma, which is a finite sample result valid for any n. We prove the lemma in Section 7. Lemma 7. If the statistic T is normally distributed from N (µ, Σ n ) and the model M b∗ (T, ω), where ω ∼ Q. Then for any contrast η, is selected by randomized selection Q b∗ , we have which could depend on the outcome of selection Q R∞ Q(t; Vη ) · exp −n(t − η T µ)2 /2ση2 dt F∗ ηT T T P (T ; η µ, Σ) = R ∞ ∼ Unif(0, 1) (22) Q(t; Vη ) · exp(−n(t − η T µ)2 /2ση2 ) dt −∞ where ση2 = η T Ση, Q(t, Vη ) = Q Vη = I− 1 Σηη T ση2 T b∗ t · Ση/ση2 + Vη , ω ω:M ∈Q . Remark 8. In selected models Msel,E , the selection is often made not only based on (T, ω), but also other statistic of the data, which we call the null statistic N . Thus b∗ (T, N ), ω }. To make notation the selection event should be expressed as {M ∈ Q simpler, we exclude such possibilities. But a slightly modified pivot where we replace Q(t; Vη ) with Q(t; Vη , N ) in (22) and integrate over N , is still Unif(0, 1) distributed. Note that Lemma 7 provides a valid pivotal quantity for any randomized selection b∗ and any randomization noise Q provided that T is normally distributed. procedure Q In fact, Lemma 7 does not require T to be a linearizable statistic. In some sense, the lemma is a reformulation (after rescaling) of the selective tests constructed in the linear regression model with additive noises (see Section 4.2.2). For example, in the selected model Msel,E , to test the hypothesis H0j : βj = 0, j ∈ E, we consider the law (20). After introducing the null statistic N = PE⊥ y, the pivot in (22) is in fact the CDF transform of this law, taking T = PE y, Σ = nσ 2 PE , and the selection event b∗ ((T, N ), ω)} to be the affine selection event defined in (20). With simple {M ∈ Q −2 calculation, it is easy to see Vη = (PE − kηk η T η)y = PE\η y, which we condition on in both (22) and (20). Tian and Taylor/Selective inference with a randomized response 19 Of course the pivot in (22) is very difficult to compute explicitly, and we need to use sampling schemes like in (20). But in a nutshell, P (T ; η T µ, Σ) is simply a CDF transform of the law Σ T ∗ b η T \| Vη , M ∈ Q (T, ω), (T, ω) ∼ N µ, × Q. (23) n After introducing the null statistic, Lemma 7 is agnostic to the selected model Msel,E , where µ = XE βE or the saturated model Msat , where the parameter is simply µ. The nuances between the two models in terms of sampling is that the saturated model condition on N (treating it as part of Vη ), but selected model integrate over N . Lemma 7 is written with T implicitly being the approximate average of n i.i.d variables, hence the distribution N (µ, Σ n ). Linearizable statistics are of particular interest as they converge to N (µ, Σ ) due to central limit theorem. In the following, we seek to esn tablish conditions under which the pivot P (T ; µ, Σ) will be asymptotically Unif(0, 1). 5.1. Selective central limit theorem In other work on asymptotics of selective inference Tian & Taylor (2015), Tibshirani et al. (2015), the setup considered is usually the saturated model Msat . These works b∗ . In considered asymptotics of selective inference marginalized over the range of Q contrast, we consider the convergence for any particular selected model Mn , under b∗n }. Specifically, we allow weak the conditional law of the selection event {Mn ∈ Q convergence of the pivot in (22) in the sequence of selected models (Mn )n≥1 . As explained above, selected models integrate over the null statistics while saturated models condition on those, thus the selective tests should have more power provided that the selected model is believable. In the saturated model, our result provides a finer measure of convergence than in Tian & Taylor (2015). On the other hand, Tian & Taylor (2015) allows high-dimensional setting in some cases while we consider fixed dimension p. Similar to the asymptotic setting in Section 3, we consider the convergence of P (Tn ; η T µn , Σn ) under a sequence of models (Mn )n≥1 selected by a sequence of b∗n )n≥1 . (Tn )n≥1 is a sequence of linearizable statistics defined selection procedures (Q in Definition 6, with asymptotic mean µn and asymptotic covariance matrix Σnn . It turns out that in this setting, the selective likelihood ratio `Fn again plays an important role in the convergence of the pivot. Recall that with randomized selection b∗ (Tn , ω), the selective likelihood is Q W(Tn ; Mn ) , EFn [W(Tn ; Mn )] n o b∗ (Tn , ω) W(Tn ; Mn ) = Q ω : Mn ∈ Q n `Fn (Tn ; Mn ) = (24) It will √ be convenient to rewrite the likelihood ratio in terms of the normalized vector Zn = n(Tn − µn ) `¯Fn (Zn ) = `F (n−1/2 Zn + µn ). (25) as well as the pivot (22) P̄Fn (Zn ) = P (n−1/2 Zn + µn ; ηnT µn , Σn ). (26) 20 Tian and Taylor/Selective inference with a randomized response Our approach is basically a comparison of how the pivot will behave under Fn and its Gaussian counterpart Φn = N (µ(Fn ), Σ(Fn )). Specifically, it is a modification of the proof of Theorem 1.1 of Chatterjee (2005), modified to allow for the fact the derivatives of the pivot and the likelihood are not required to be uniformly bounded. Given a norm Ω on Rp , define n o r/k λΩ ∂ k f (s) exp(−rΩ(s)) : 1 ≤ k ≤ r , (27) r (f ) = sup s∈Rp where ∂ k denotes the k-fold differentiation with respect to the p-dimensional vector s, k · k denotes element wise maximum. Now we state our selective central limit theorem, which we prove in Section 7. Theorem 9 (Selective central limit theorem). Suppose the statistics Tn = T (Dn ) are linearizable statistics according to Definition 6. We also assume the norms Ω : Rp → R are such that for each f ∈ {P̄n , `¯Fn , `¯Φn }, it satisfies sup λΩ 3 (f ) ≤ C1 . (28) Fn ∈Mn Moreover, assume ξi,n has uniformly bounded moment generating function in some neighbourhood of 0. Namely, ∃a > 0, such that sup sup EFn (exp(akξi,n − µ(Fn )k1 )) ≤ C2 . (29) n≥1 Fn ∈Mn Furthermore, we assume lim sup n1/2 · n b∗n ] b∗n ] − P(Φ ×Q) [Mn ∈ Q P(Fn ×Q) [Mn ∈ Q n ≤ C3 . b∗ ] P(Φ ×Q) [Mn ∈ Q n (30) n Then, for any g with uniformly bounded derivatives up to third order EF∗n g P (Tn ) − Z 0 1 g(x)dx ≤ n−1/2 K(g, C1 , C2 , C3 , p), n ≥ n0 (31) where K depends only on the bounds on the derivatives of g, the constants C1 , C2 , C3 and the dimension p. Thus the convergence is uniform in (Mn )n≥1 for models satisfying (28), (29) and (30). Theorem 9 provides a finite sample bound on the convergence of the pivot P (Tn ). Since we allow g to be functions with uniformly bounded derivatives up to the third order, (31) implies convergence of P (Tn ) to Unif(0, 1) under F∗n . In the following examples, we show how to verify conditions (28), (29) and (30). 5.2. Revisit the “file drawer problem” In Examples 1 and 2, we considered only reporting an interval or a p-value about µn when n1/2 X̄n > 2 or n1/2 X̄n + ω > 2. This is an example where we do not really Tian and Taylor/Selective inference with a randomized response 21 select a model, but rather select only a proportion of the data to report. The selective distribution simply refers to the law of the reported sample means, which pass the threshold. The data we observe is Dn = (X1,n,...,Xn,n ) with the linearizable statistic Tn simply being the sample mean X̄n . Example 1 corresponds to the degenerate randomization of adding 0 to X̄n . Work of Tian & Taylor (2015) show that in order for the corresponding pivot to converge weakly we can take, for ∆ < 0 fixed n o 3 Mn = F : EF [X̄n ] > n−1/2 ∆, EF [Xi,n ]<∞ . (32) That is, X̄n will satisfy a selective CLT when the population mean is not too negative. On the other hand, in Example 2, the pivot in (22) is of the form, R∞ √ 2 Ḡ(2 − nt)e−n(t−µn ) /2 dz X̄n , (33) P (X̄n ) = R ∞ √ Ḡ(2 − nt)e−n(t−µn )2 /2 dz −∞ and likelihood `Fn (X̄n ) is defined in (6). When G is the Logistic noise, then condition (28) and (30) can be verified. Formally, we have the following lemma whose proof we defer to the appendix, Lemma 10. If G = Logistic(κ), with κ being the scale parameter, then if centered Xi,n ’s have moment generating functions in the neighbourhood of zero, then the pivot P (X̄n ) is asymptotically Unif(0, 1). In other words, with Logistic randomization noise, we can take the sequence of models to be Mn = Fn : EFn exp a \|X1,n − µn \| < ∞ , for some a > 0. (34) Requiring exponential moments is stricter than the third moment condition in (32), but we would have a stronger conclusion, namely weak convergence uniformly over all µn ’s. 5.3. Two-sample median problem In the two-sample median problem, we have two treatment groups from which we take iid iid measurements, x1i ∼ F1 and x2i ∼ F2 ; for simplicity of notation, we assume we observe n samples from each group, and drop n in the subscript. We will report the bigger median from this group in the non-randomized setting. Exact formulation of randomized selection will be discussed below. Suppose our underlying distribution is F = F1 ×F2 . Let µ = (µ1 , µ2 ) is the population median of the two groups, and T = (T1 , T2 ) is the sample median. The well-known result by Bahadur (1966) states that the sample median is a linearizable statistic for the median when the CDF of the distribution F has positive density f , and f 0 is bounded iid in a neighbourhood of the population median m. Formally, if xi ∼ F , then the sample median n 1 X 1{xi > m} − 1/2 + Rn , (35) T (x1 , . . . , xn ) = m + n i=1 F 0 (m) Tian and Taylor/Selective inference with a randomized response 22 with R = O(n−3/4 log n) with probability 1. b∗ reports Our (randomized) selection algorithm Q ( P (T ; µ1 , Σ), if T1 > T2 + n−1/2 ω P (T ; µ2 , Σ), if T1 ≤ T2 + n−1/2 ω, where ω ∼ Q and Σ = diag( 41 f1 (µ1 )−2 , 14 f2 (µ2 )−2 ) is a diagonal matrix. f1 , f2 are the densities of F1 and F2 . Without loss of generality, we suppose M1 is selected, i.e. the first group is the “best” group. We choose the randomization noise Q to be a Logistic(κ) with mean 0 and κ is the scale, and let Gκ be the CDF. The resulting pivot for µ1 is R∞ √ √ Gκ ( nt − nT2 ) · exp(−n(t − µ1 )2 /2σ12 ) dt 1 T1 , σ12 = . P (T ; µ1 , Σ) = R ∞ √ √ 2 2 4f1 (µ1 )2 G ( nt − nT2 ) · exp(−n(t − µ1 ) /2σ1 ) dt −∞ κ This pivot strikes a similarity with √ the pivot in (33) for Example 2 with the truncation threshold 2 being replaced by nT2 and plugging in the appropriate means and variances of the medians. A result similar to Lemma 10 can be established, which ensures convergence of the pivot uniformly for any underlying medians (µ1 , µ2 ). In order to construct the above pivot, we need knowledge of the variance σ12 . Without selection, there are natural estimates of this variance. One may ask, how will inference be affected if we plug this estimate into our pivot? We revisit this question in Section 5.5. 5.4. Affine selection events In this section, we discuss the special case of affine selection events (regions). This combined with the asymptotic result in Theorem 9 applies to more general settings. In particular, it allows us to approximate non-affine regions. For a concrete example, see Section 5.4.1. We drop the subscript n where possible to simplify notations. Suppose for our model b∗ } can be deM , the selection is based on (T, ω), and the selection event {M ∈ Q scribed as √ { nAM T + ω ∈ KM }, where the affine matrix AM ∈ Rd×p and KM is a region in Rd . Many examples of nonrandomized selective inference can be expressed in this way (c.f. Lee et al. (2013b), Taylor et al. (2014), Lee & Taylor (2014), Fithian et al. (2015)). In this section, we provide conditions under which Theorem √9 can be applied. We again normalize T to be Z = n(T − µ), then the selection event can be rewritten as {AM (Z + ∆) + ω ∈ KM }, (36) √ where nµ = ∆, Z converges to N (0, Σ). Suppose ω ∼ Q, which has distribution function G. Then we introduce some conditions on the selection region KM and the added noise distribution G, Tian and Taylor/Selective inference with a randomized response 23 Lower bound: We assume there is some norm h, such that Z G(dw) ≥ C − exp − inf h(w) , ∀θ ∈ Rd . w∈KM −θ KM −θ Smoothness: Suppose G has density g, we assume the first 3 derivatives of g are integrable, Z k∂ j g(w)kdw ≤ Cj , j = 0, 1, 2, 3 Rd where the norm on the left-hand side is the maximum element-wise of the partial derivatives. The above two conditions essentially require G to be differentiable and have heavier tails than (or equal to) exponential tails. In fact we prove that the lower bound and smoothness conditions ensure that (28) are satisfied under the local alternatives introduced below. Definition 11 (Local alternatives). For the sequence of selected model (Mn )n≥1 , we define the local alternatives of radius of B to be the set all sequences (µn )n≥1 , such that √ dh (0, KMn − AMn ∆) ≤ B, ∆ = nµn where dh (·, ·) is the distance induced by the norm h. The notion of local alternatives is natural in the asymptotic setting as we expect even a small effect size will be more prominent when we collect more and more data. Formally, we have the following lemma, whose proof is deferred to the appendix. Lemma 12. Suppose G, KM satisfy the lower bound and smoothness conditions, then condition (28) are satisfied under the local alternatives. Now, we are left to verify conditions (29) and (30). Condition (29) is essentially a moment condition on the centered statistics ξi,n − µn , which we have to assume. Condition (30) can be verified using the well known results in multivariate CLT (see Gotze (1991)). To be rigorous, we state the following lemma, which we also prove in the appendix. Lemma 13. If Fn is such that the centered statistics ξi,n −µn have finite third moments, then under the local alternatives, condition (30) is satisfied. To summarize, Lemma 12 and Lemma 13 state that if G has integrable derivatives and exponential tails, then the pivot in (22) converges to Unif(0, 1) uniformly for F∗n so long as Fn ’s are such that ξi,n − µn have exponential moments in a neighbourhood of 0. Unlike the sample mean and sample median examples, the pivot is difficult to compute explicitly in this case. However, as we discuss in the beginning of Section 5, the pivot is essentially the CDF transform of the conditional law (23), which we can sample from. As discussed above, we can just take ω to be from a Logistic distribution. Now we apply the above theory to logistic regression. Tian and Taylor/Selective inference with a randomized response 24 5.4.1. Example: randomized logistic lasso iid Suppose we observe independent samples, di = (yi , xi ) ∼ F, where yi ’s are binary observations and xi ∈ Rp . The ordinary logistic regression solves the following problem, β̄ = argminβ∈Rp `(β) " n # X = argminβ∈Rp − yi log π(xi β) + (1 − yi ) log(1 − π(xi β)) , (37) i=1 where π(x) = exp(x)/(1 + exp(x)). This is a nonparametric setting as we do not assume any parametric structure for F. The randomized logistic lasso adds an `1 penalty, a randomization term and a small quadratic term, 1 1 β̂ = argminβ∈Rp √ `(β) + ω T β + kΛβk1 + √ kβk22 , n 2 n (38) iid where ωj ∼ Logistic(κ) is the perturbation to the gradient and Λ is a diagonal matrix which introduces (possibly) unequal feature weights, κ controls the amount of randomization added. The addition of the quadratic term ensures that (38) is strictly convex, thus has a unique solution. A similar formulation for linear regression has been proposed in Meinshausen & Bühlmann (2010). Selective inference in this setting has not been considered before. Without the Gaussian assumptions Lee et al. (2013a) does not apply. The parametric setting of this problem has been discussed in Fithian et al. (2014), but computation of the selective tests are mostly infeasible for general X. Finally, the asymptotic result by Tian & Taylor (2015) does not apply here as the framework require exactly affine selection regions, which is not the case in this setting. Suppose the solution to (38) has nonzero entry set E, then our target of inference ∗ βE , the unique population minimizer which satisfies T ∗ EF [XE (y − π(XE βE ))] = 0. (39) ∗ )) with independently samNote that a parametric model yi \|xi ∼ Bernoulli(π(xi,E βE ∗ pled xi ’s will have βE satisfying (39). But we by no means assume such an underlying ∗ distribution. Rather, for any well-behaved distribution F, βE can be thought of of a statistical functional of the underlying distribution F, depending on the outcome of selection E. Selective inference in this setting is carried out conditioned on (E, sE ), the active set and its signs. We first introduce the following notations, exp(XE βE ) , WE (βE ) = diag(πE (βE )(1 − πE (βE ))), 1 + exp(XE βE ) 1 T 1 T QE (βE ) = XE WE (βE )XE , CE (βE ) = X−E WE (βE )XE , n n DE (βE ) = CE (βE )Q−1 E (βE ) πE (βE ) = 25 Tian and Taylor/Selective inference with a randomized response where X is the feature matrix, and XE , X−E is the columns corresponding to the active set and inactive set respectively. By law of large numbers, we have p def ∗ ∗ QE (βE ) → EF QE (βE ) = Q, p def p def ∗ ∗ CE (βE ) → EF CE (βE ) = C, ∗ DE (βE ) → CQ−1 = D. (40) Now we introduce our linearizable statistics and show that the conditioning event (E, sE ) can be expressed as affine regions of these statistics. Lemma 14. Suppose E is the active set of the solution of (38), and we denote " n # X β̄E = argmin − yi log π(xi,E βE ) + (1 − yi ) log(1 − π(xi,E βE )) βE ∈RE i=1 as the unpenalized MLE restricted to the selected variables E. ∗ The following statistic T is linearizable with asymptotic mean (βE , ρ) and variance Σ/n, β̄E + R, T = 1 T n X−E y − πE (β̄E ) ∗ )) . Morewhere R = op (n−1/2 ) is a small residual, and ρ = E xTi,−E (yi − π(xi,E βE over, √ the selection event {Ê, zÊ = (E, sE )} can be characterized as the affine region { nAM T + BM ω ≤ bM }, where       −SE 0 SE Q−1 0 −SE Q−1 ΛE sE I−E  , BM =  D −I−E  , bM = λ−E − DΛE sE  , AM =  0 0 −I−E −D I−E λ−E + DΛE sE where I−E denotes the identity matrix of n − \|E\| dimensions and ΛE , Λ−E denote the active block and the inactive block of Λ respectively and λ is the diagonal elements of Λ, SE = diag(sE ). The proof of this lemma is also deferred to the appendix. Thus using Lemma 12 and Lemma 13, we can conclude under local alternatives, the pivot (22) converges to Unif(0, 1). To test H0j : βj∗ = 0, we take η = ej , and sample ∗ √ Σ βE η T T \| Vη , nAM T + BM ω ≤ bM , (T, ω) ∼ N , × G, ρ n ∗ where ρ = E xTi,−E (yi − π(xi,E βE )) . Since ρ is the nuisance parameters for testing H0j , j ∈ E, the conditional law above will not depend on its value. A hit-and-run algorithm for sampling this law can be implemented. Moreover, recent development by Tian et al. (2016), Harris et al. (2016) propose more general and efficient sampling schemes for this law. For details, see for example Chapter 3.2 Tian et al. (2016) where the sampling scheme for this very example is considered and simulation results are provided. In Lemma 14, we assume the covariance matrix Σ is known. In applications, we can bootstrap it. But is it valid to plug in the bootstrap estimate of Σ? Tian and Taylor/Selective inference with a randomized response 26 5.5. Plugging in variance estimates In Section 5.3 we derived quantities that were asymptotically pivotal for the best median, up to an unknown variance. In the sample median case, by (35), the variance of the sample median is approximately [4nf (m)2 ]−1 , where f (m) is the PDF evaluated at the median m. A simple consistent estimator for f (m) is to take 1/2 ± √1n quantiles an and bn , then 2 f (m) ≈ √ (41) n(bn − an ) is consistent for f (m) based on which we get a consistent estimator for σ12 . More generally, computing the pivot (22) requires knowledge of Σ. In practice, we usually do not have prior knowledge of the variance Σ and need a consistent estimate for Σ. We might use a bootstrap or jackknife estimator. When p is fixed, the bootstrap estimator is consistent and thus we get a consistent estimator Σ̂. Lemma 3 states that under moment conditions on the likelihood, Σ̂ will be consistent for Σ under F∗n as well, justifying the plug-in estimator of Σ. Figure 4 is some simulation results for the two-sample medians problem. In each case, we take the sample size for each treatment group to be 500, and generate the noise from a skewed distribution N (0, 1) + 0.5Exp(1). We standardize it such that the noise has median 0 under the null hypothesis. We use additive logistic noise with scale κ = 0.8 for randomization. The better group is decided using the randomized sample median, and selective inference is carried out. In Figure 4a, the pivot with plugin variance estimate σ b in (41) is plotted under both the null hypothesis H0 : µbetter = 0 and the HA : µbetter > √1n . The pivot has reasonable power even for identifying local alternatives. The pivot is almost exactly Unif(0, 1) under the null hypothesis with the sample size n = 500. In fact, it is very close at a relatively small sample size n = 50 justifying the application of asymptotics in the nonparametric setting. Figure 4b further illustrates the difference in the unselective v.s. selective distribution and its convergence to its theoretical limit. We see that there is a clear shift in selective distribution that calls for adjustment for the selection. For sample size n = 500, the empirical selective distribution converges to our theoretical distribution. 6. Multiple Randomizations of the Data Most of the examples above focus on a single randomization ω on the data, which we use for model selection. We naturally want to extend it to multiple randomizations, and multiple randomized selections, which will collectively suggest a model for inference. In this section, we allow multiple randomizations in a possibly sequential fashion and discuss how inference can be carried out. 27 Tian and Taylor/Selective inference with a randomized response 1.0 ECDF for the pvalues, two-sample median, n=500 null alternative Better median histogram, n=500 selected pdf median, unselected median, selected 6 5 0.8 4 0.6 3 0.4 2 0.2 0.0 0.0 1 0.2 0.4 0.6 0.8 0 0.3 1.0 0.2 0.1 0.0 0.1 0.2 median values 0.3 0.4 0.5 (a) Null and alternative pivot for the “bet- (b) Selective v.s. unselective distribution, ter” median and theoretical PDF Fig 4: Asymptotic distribution of the median for the selected group 6.1. Selective inference after cross-validation Consider the case where we first choose a regularization parameter by cross-validation, and then fit the square-root LASSO problem Belloni et al. (2011) at this parameter, β̂λ (y; X) = argmin ky − Xβk2 + λkβ\|1 , (42) β where λ is picked from a fixed grid Λ = [λ1 , . . . , λk ]. The discussion below is not specific to selection by square-root LASSO. The model selected by cross-validated square-root LASSO involves two steps of selection. We denote by yCV the response for selecting the randomization parameter, and yselect the response vector for fitting the square-root LASSO at the selected regularization parameter λ. Both vectors are randomized version of the original vector y. Inference after cross validation requires combining two steps of randomized selection. Consider the following procedure. First, we randomize y to get the vector yCV and yselect ∼ N (y, σ12 I) yinter \|y, X yCV \|yinter , y, X yselect \|yinter , y, X ∼ 2 ∼ N (yinter , σ2,CV I) (43) 2 N (yinter , σ2,select I). Note the intermediate vector yinter is introduced convenience of sampling. The above is just one of the plausible randomization schemes. After having randomized, we select λ with K-fold cross-validation using yCV : λ̂ = λ̂(yCV , X) = argminλ∈Λ CVK (yCV , X, λ) (44) where CVK (y, X, λ) is the usual K-fold cross-validation score with coefficients estimated by the square-root LASSO. Alternatively, one could compute the cross-validation score using the OLS estimators of the selected variables. Note that we have left implicit Tian and Taylor/Selective inference with a randomized response 28 the randomization that splits observations into groups. That is λ̂ in (44) above is a function of (yCV , X, ω) where ω is a random partition of {1, . . . , n} into K groups. When we sample yCV below, we redraw ω each time. The subset of variables and signs is selected using the square-root LASSO with response yselect : n o Ê(yCV , yselect , X) = j : β̂λ̂(yCV ,X),j 6= 0 , (45) zÊ (yCV , yselect , X) = sign(β̂λ̂(yCV ,X) ). After seeing the selected variables Ê, we perform inference in the selected model Msel (Ê). Since Msel (Ê), we will still have an exponential family after selection. Per Lemma 2, we sample from the following law, L XjT y λ̂(yCV , X) = λ, (Ê, zÊ ) = (E, zE ), PE\j y . The additional conditioning on the signs are for computational reasons. In fact, recent development in Harris et al. (2016) proposes sampling schemes that overcome these difficulties, so that we do not need to condition on this additional information. To sample from the above law, we use a Gibbs-type sampler, which iterate over y, yinter , yCV and yselect , conditional on the other three and the selection event. It includes the following steps. Sampling yCV Using the conditional independence of yCV and yselect given yinter , we have n o 2 L yCV yinter , yselect , y, λ̂(yCV , X) = λ = N (yinter , σ2,CV I) λ̂(yCV , X) = λ . This is the computational bottleneck, as we do not have good description for the selection event for cross validation. A brute-force sampling scheme will be computationally expensive, as we need to refit the model over a grid of λ’s. Thus, we do not update yCV too often. Sampling yselect The conditional independence of yselect and yCV given yinter implies, L yselect yinter , yCV , y, (Ê, zÊ )(yCV , yselect , X) = (E, zE ), λ̂(yCV , X) = λ n o 2 = N (yinter , σ2,select I) \| (Êλ , zÊ,λ )(yselect , X) = (E, zE ) Tian et al. (2015) has given an explicit description of the selection event {Êλ , zÊ,λ = (E, zE )}. Thus hit-and-run sampling provides a tractable sampling scheme. Sampling yinter This is a simple step. Because the selection event is based on yCV and yselect , we have L yinter y, yselect , yCV  !−1  1 y + σ21 yCV + σ2 1 yselect σ12 1 1 1 2,CV 2,select . , + 2 + 2 =N  1 1 1 σ12 σ2,CV σ2,select + + 2 2 σ σ σ2 1 2,CV 2,select 29 Tian and Taylor/Selective inference with a randomized response Sampling y This is also simple with our randomization scheme. Note that y is conditionally independent of yselect and yCV given yinter , 1 1 1 σ 2 XE βE + σ12 yinter , 1 1 2 σ + 2 2 σ σ1 L y yinter , yCV , yselect = L y yinter = N 1 + 2 σ1 −1 ! Since we condition on PE\j y, we essentially take y and project out the update on the space orthogonal to that of Xj . A chain that iterates through the above four steps will give us samples from the desired distribution for inference. 6.2. Collaborative selective inference One of the motivations of the reusable holdout described in Dwork et al. (2014) is that it allows a data analyst to repeatedly query a database yet still be able to approximately estimate expectations even after asking many questions about the data. Another version of this model may be that several groups wish to model the same data and then, as a consortium, decide on a final model and be able to approximately estimate expectations in this final model. We might call this collaborative selective inference. Formally, suppose each of L groups has its own preferred method of model selecbl )1≤l≤L . We assume there is a central “data” tion, encoded as selection procedures (Q bank that decides what “data” each group is allowed to see. We express this is as a sequence of randomization schemes (yl∗ )1≤l≤L . Formally, this is equivalent to enlarging the probability space to D × B with measure F × B and fixing a function y ∗ (y, ω) = (y1∗ (y, ω), . . . , yl∗ (y, ω)). It may be desirable to choose the law of y ∗ \|y so that the coordinates are conditionally independent given y, though it is not necessary. bl (y ∗ ) ∈ σ(y ∗ ) and convene Now suppose that the L groups choose models M̂l∗ = Q l l to discuss what the best model is M . For every choice of L models (M1 , . . . , ML ) and final model M , the following selective distribution can be used for valid selective inference b ∗ dF∗ B(ω : ∩L l=1 Ql (yl (y, ω)) = Ml ) (y) = . (46) b ∗ = Ml ) dF (F × B)(∩L Q l=1 l When the yl∗ ’s are conditionally independent given y then it is clear that b ∗ B(∩L l=1 Ql (y (y, ω)) = Ml ) = L Y l=1 bl (y ∗ (y, ω)) = Ml ). B(Q l It is possible that the consortium has beforehand decided on an algorithm that will choose a best model automatically, determined by some function S(M1 , . . . , ML ). In this case, one should use the selective distribution dF∗ B(ω : S(M1∗ (y, ω), . . . , ML∗ (y, ω)) = M ) (y) = dF (F × B)(S(M1∗ , . . . , ML∗ ) = M ) (47) Tian and Taylor/Selective inference with a randomized response 30 When the models in question are parametric, perhaps Gaussian distributions, and the randomization is additive Gaussian noise the central data bank can explicitly lower bound the leftover information by ∗ Var(y\|y1∗ , . . . , yL ). This quantity is expressible in terms of the marginal variance of y and the central data bank’s noise generating distribution for y ∗ (y, ω) = (y + ω1 , . . . , y + ωL ). By maintaining a lower bound on the above quantity, the central data bank can maintain a minimum prescribed information in the data for final estimation and/or inference. In a sequential setting, where valid inference is desired at each step, maintaining a lower bound may involve releasing noisier and noisier versions of y. Sampling under this scheme seems quite difficult, and we leave it as an area of interesting future research. 7. Proof 7.1. Proof of Theorem 9 To prove Theorem 9, we first prove the following lemma, which might be of independent interest. Lemma 15.√Suppose Tn is a linearizable statistic for µn = µ(Fn ) as defined in (21). Let Zn = n(Tn − µn ) ∈ Rp and a function f : Rp → R with finite λΩ 3 (f ) for some norm Ω on Rp . Moreover, if Fn has finite centered exponential moments in a neighbourhood of zero. Then 1 −2 \|EFn [f (Zn )] − EΦn [f (Zn )] \| ≤ C(p)λΩ , 3 (f )n n ≥ n0 for some n0 ≥ 1, where C(p) is a constant only dependent on the dimension. Lemma 15 can be seen as an extension of the result by Chatterjee (2005) in the sense that the author in Chatterjee (2005) established result for the case Ω = 0. The proof is also an adaptation of the technique in Chatterjee (2005). Pn Proof. Without loss of generality, we assume Tn = n1 i=1 ξi,n and the residual is 0. First, we define the normalizing operator. For any S ∈ Rn×p , n 1 X N (S) = √ S[i] ∈ Rp , n i=1 where S[i] is the i-th row of S. We also define for any n and 0 ≤ k ≤ n,  0   0  ξ1,n − µ(Fn ), ξ1,n − µ(Fn ), ..     ..     ., .,  0   0  ξk−1,n − µ(Fn ), ξ  − µ(F ), n  k−1,n  0   −     Sn,k =  ξk,n − µ(Fn ),  Sn,k =  0  ξk+1,n − µ(Fn ), ξk+1,n − µ(Fn ),         .. ..     ., ., ξn,n − µ(Fn ) ξn,n − µ(Fn ) Tian and Taylor/Selective inference with a randomized response iid 31 iid 0 where ξi,n ∼ Fn with mean µ(Fn ) and variance Σ(Fn ) and ξi,n ∼ N (µ(Fn ), Σ(Fn )). − − Let Fn,k . Fn,k denote the distribution of Sn,k and Sn,k ’s respectively. Note Fn,k and − Fn,k are determined by Fn . For simplicity of notation we only distinguish the two − distributions by Fn,k and Fn,k , avoiding verbose notations of S, e.g. EFn,k [S] = EFn,k [Sn,k ]. It is then easy to see Zn = N (Sn,0 ). Now by telescoping: EFn [f ] − EΦn [f ] = EFn,0 [f ◦ N (S)] − EFn,n [f ◦ N (S)] n X ≤ EFn,i−1 [f ◦ N (S)] − EF − [f ◦ N (S)] + EF − [f ◦ N (S)] − EFn,i [f ◦ N (S)] . n,i n,i i=1 Let ∂i be the derivative with respect to the i-th row S[i]. Using Taylor’s expansion − at Sn,k , we have EFn,i [f ◦ N ] − EF − [f ◦ N ] n,i i 1 h 1 T = √ EF − [∂i f ◦ N (S)] 0 + Tr EF − ∂i2 f ◦ N (S) · Σ(Fn ) + Rn,i n,i n n n,i where the precise form of the Taylor remainder Rn,i depends on realizing the laws Fn,i − and Fn,i on the same probability space. In order to not introduce new notation, we have avoided explicitly writing out this construction, directing readers to Chatterjee (2005) for details. Nevertheless, 1 Ω − 23 0 0 3 √ \|Rn,i \| ≤ c1 (p)[λ3 (f ) · n ]EF − exp Ω(N (S)) + Ω(ξi,n ) kξi,n k1 n,i n 0 where ξi,n are centered version of ξi,n and c1 is some dimension dependent constant. Let C(Ω) be the constant s.t Ω(·) ≤ C(Ω)k · k1 , C(Ω) only depends on the dimension p. Thus, using the independence of the ξi,n ’s, 1 0 0 3 EF − exp Ω(N (S)) + √ Ω(ξi,n ) kξi,n k1 n,i n !# " 0 C(Ω)kξi,n k1 0 3 √ . ≤EF − [exp (C(Ω)kN (S)k1 )] · EFn kξi,n k1 exp n,i n Now we bound these two expectations. By the exponential moment condition (29) and Lemma 17, it is easy to conclude the first term is bounded by lim sup EFn [exp (C(Ω)kZn k1 )] ≤ c2 (p). n 0 The second expectation is bounded by γ, an upper bound on the third moment of ξi,n , " 0 lim sup EFn kξi,n k31 exp n 0 C(Ω)kξi,n k1 √ n !# ≤ γ, 32 Tian and Taylor/Selective inference with a randomized response Thus it is not hard to see 3 −2 \|Rn,i \| ≤ c1 (p)c2 (p)γλΩ . 3 (f )n Notice the first and second order terms in EFn,i [f ◦ N ] − EF − [f ◦ N ] cancel with n,i those in EFn,i−1 [f ◦ N ] − EF − [f ◦ N ], and therefore we have, n,i EFn [f ] − EΦn [f ] ≤ n X (Rn,i + R̃n,i ) i=1 where R̃n,i is the remainder of EFn,i−1 [f ◦ N ] − EF − [f ◦ N ]. With a similar argument n,i 3 −2 , \|R̃n,i \| ≤ c1 (p)c2 (p)γλΩ 3 (f )n and summing over n terms, we have the conclusion of the lemma. Now we prove the main theorem, Theorem 9. Proof. First, notice that per Lemma 7, we have EΦ∗n [g ◦ P (Tn )] = the selective likelihood ratio, it is easy to see, R1 0 g(x)dx. Using EF∗n [g(P (Tn ))] = EFn [g(P (Tn ))`Fn (Tn )] The same equation holds for Φn = Φ(Fn ), thus we have EF∗n [g(P (Tn ))] − EΦ∗n [g(P (Tn ))] ≤ \|EFn [g(P (Tn ))`Φn (Tn )] − EΦn [g(P (Tn ))`Φn (Tn )]\| + (48) \|EFn [g(P (Tn ))`Fn (Tn )] − EFn [g(P (Tn ))`Φn (Tn )]\| We need to bound both terms. Recall the notation P n and `¯Fn for the normalized statistic Zn . If we let f = g(P̄n ) · `¯Φn , then per Lemma 15, we have \|EFn [g(P̄n ) · `¯Φn ] − EΦn [g(P̄n ) · `¯Φn ]\| ≤ 2C(p) · C1 n−1/2 , where we use the bound in condition (28). Now we replace `Φn with `Fn in the second term. With some algebra, we can bound it by h i b∗ (Tn , ω) PFn ×Q M ∈ Q h i , EFn [g(P (Tn ))`Fn (Tn )] 1 − b∗ (Tn , ω) PΦn ×Q M ∈ Q which in turn is bounded by C(g) b∗ ] − P(Φ ×Q) [M ∈ Q b∗ ] P(Fn ×Q) [M ∈ Q n ≤ C(g)C3 n−1/2 , b∗ ] P(Φ ×Q) [M ∈ Q n per condition (30) and C(g) is a bound on g. Tian and Taylor/Selective inference with a randomized response 33 7.2. Proof of Lemma 7 Proof. Let φµ, n1 Σ denote the density for N (µ, n1 Σ) and T = 1 T ση2 Ση·(η T )+Vη , we see ∗ that Q(η T T, Vη ) is W(M ; T ) in (13). Thus under the selective law F , the distribution of T has density proportional to φµ, n1 Σ (T ) · Q(η T T, Vη ). Since η T T ⊥ Vη under F, we can factorize φµ, n1 Σ (T ) into the product of densities of η T T and Vη . Thus conditioning on Vη , the density of η T T is proportional to n(η T T − η T µ)2 exp − · Q(η T T, Vη ). 2ση2 Therefore, the pivot in (22) is the survival function of η T T under F∗ and is distributed as Unif(0, 1). Moreover, we note the distribution does not depend on the conditioned value of Vη , thus P (T ; η T µ, Σ) in (22) is Unif(0, 1). References Bahadur, R. R. (1966), ‘A note on quantiles in large samples’, The Annals of Mathematical Statistics 37(3), 577–580. Belloni, A., Chernozhukov, V. & Wang, L. 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(2009), ‘p-values for high-dimensional regression’, Journal of the American Statistical Association 104(488), 1671–1681. URL: http://amstat.tandfonline.com/doi/abs/10.1198/jasa.2009.tm08647 Pakman, A. & Paninski, L. (2012), ‘Exact hamiltonian monte carlo for truncated multivariate gaussians’, arXiv:1208.4118 [stat] . arXiv: 1208.4118. URL: http://arxiv.org/abs/1208.4118 Rosenthal, R. (1979), ‘The file drawer problem and tolerance for null results.’, Psychological bulletin 86(3), 638. Taylor, J., Lockhart, R., Tibshirani, R. J. & Tibshirani, R. (2014), ‘Post-selection adaptive inference for least angle regression and the lasso’, arXiv:1401.3889 [stat] . URL: http://arxiv.org/abs/1401.3889 Tian, X., Bi, N. & Taylor, J. (2016), ‘Magic: a general, powerful and tractable method for selective inference’, arXiv preprint arXiv:1607.02630 . Tian, X., Loftus, J. R. & Taylor, J. E. (2015), ‘Selective inference with unknown variance via the square-root lasso’, arXiv preprint arXiv:1504.08031 . Tian, X. & Taylor, J. (2015), ‘Asymptotics of selective inference’, arXiv preprint arXiv:1501.03588 . Tibshirani, R. (1996), ‘Regression shrinkage and selection via the lasso’, Journal of the Royal Statistical Society: Series B 58(1), 267–288. Tibshirani, R. J., Rinaldo, A., Tibshirani, R. & Wasserman, L. (2015), ‘Uniform asymptotic inference and the bootstrap after model selection’, arXiv preprint arXiv:1506.06266 . Tukey, J. W. (1980), ‘We need both exploratory and confirmatory’, The American Statistician 34(1), 23–25. Tian and Taylor/Selective inference with a randomized response 35 Appendix A: Proof of Lemma 1 √ Proof. First we normalize the sample mean as Z = n(X̄n + 0.5) and rewrite the pivot as 1√ 1 − Φ(Z) n, Pe(Z) = √ , Z >2+ 2 1 − Φ(2 + 21 n) and Φ is the CDF of the standard normal distribution. As n → √ ∞, we can use Mills ratio to approximate the normal tail. Specifically, denote bn = 12 n + 2, 1 − Φ(Z) bn 1 ≈ exp (bn + Z)(bn − Z) 1 − Φ(bn ) Z 2 (49) bn 1 = exp − (bn − Z)2 exp [−bn (Z − bn )] , Z > bn . Z 2 We study the behavior of −bn (Z − bn ) for Z > bn . By studying its distribution, we p will also see that Z − bn → 0, for Z > bn , thus the term 1 bn p exp − (bn − Z)2 → 1, as n → ∞. Rn = Z 2 Now we study the distribution of bn (Z − bn ) conditioning on Z > bn . Since X̄n is a translation of a binomial distribution divided by n, we can rewrite Z in terms of a Binomial distribution, which will be useful for calculating the conditional distribution of bn (Z − bn ). Specifically, Z= 2Sn − n √ , n 1 Sn ∼ Bin(n, ). 2 Thus for t ≥ 0, 2Sn − n √ − bn P (bn (Z − bn ) > t) = P bn n √ √ 3 t n =P Sn > n + n + = 4 2bn √ √ t n 1 4 n− n− 2bn X i=0 n 1 · n i 2 To study the conditional distribution P (bn (Z − bn ) > t \| Z > bn ), we essentially need to study the ratio of two partial sums of binomial coefficients. Note for any n, m ∈ Z+ , n > m we have n m m−1 = n n − m+1 m Noticing that k n−k+1 ≤ m n−m+1 , for any k ≤ m, thus Pm−1 n i Pi=0 ≤ m n i=0 i m . n−m+1 36 Tian and Taylor/Selective inference with a randomized response Now let m = 14 n − √ n, and use the above inequality j = Pm−j Pi=0 m n i n i=0 i ≤ m n−m+1 √ t n 2bn times, we have j . Therefore we have, P (bn (Z − bn ) > t \| Z > bn ) = ≤ P (bn (Z − bn ) > t) P (bn (Z − bn ) > 0) √ 1+ t√4 − n n → exp[− log(3)t] √ 3 n + n + 1 4 1 4n (50) p We can draw two conclusions from (50). First, conditional on Z > bn , Z − bn → 0, which implies the first term in the pivot approximation (49) Rn → 1. Moreover, (50) shows that the overshoot bn (Z − bn ) is not Exp(1) distributed in the limit. In fact, we can conclude its limit (if existed) is strictly stochastically dominated by an Exp(1). Thus, exp [−bn (Z − bn )] 6→ Unif(0, 1), and hence the pivot does not converge to Unif(0, 1). Appendix B: Proof of Lemma 14 Proof. We first prove that T is in fact a linearizable statistic. Since β̄E is the restricted MLE, we see that 1 T XE y − πE (β̄E ) n 1 T ∗ ∗ [y − πE (βE )] + Q(βE − β̄E ) + R1 , = XE n 0= ∗ ∗ − β̄E ) + R̃1 , where R̃1 = op (n−1/2 ) is the residual ) − Q)(βE where R1 = (QE (βE ∗ ∗ ) from its from the Taylor’s expansion at βE . R1 = op (n−1/2 ) since deviations QE (βE −1/2 ∗ asymptotic mean should be Op (n ) and β̄E − βE = op (1). Thus, we can deduce β̄E = 1 −1 T ∗ ∗ Q XE [y − πE (βE )] + βE + Q−1 R1 . n Similarly, 1 T 1 T 1 ∗ T ∗ X−E y − πE (XE β̄E ) = X−E [y−πE (βE )]− DXE [y−πE (βE )]+op (n−1/2 ) n n n Thus we can conclude that T is a linearizable statistic with T ∗ QXi,E (yi − π(xi,E βE )) ξi,n = . T ∗ ∗ Xi,−E (yi − π(xi,E βE )) − DXi,E (yi − π(xi,E βE )) Tian and Taylor/Selective inference with a randomized response 37 Now we rewrite the selection event in terms of (T, ω). Using the KKT conditions of (38), √ β̂E T X (y − πE (β̂E )) = n(ω + Λz) + 0 (51) sE β̂E ≥ 0, ku−E k∞ < 1, , sE = sign(β̂E ) and u−E is the subgradient for the inactive sE u−E variables. Using a Taylor expansion on the β̂E as well, we see that 1 1 T 0 Q X [y − πE (β̂E )] = + (β̄E − β̂E ) + op (n−1/2 ). T C X [y − π ( β̄ )] n n E E −E where z = Plugging in the equalities in the KKT conditions, we will have, 1 β̂E = β̄E − √ Q−1 (ωE + ΛE zE ) + op (n−1/2 ) n 1 T 1 T X [y − πE (β̂E )] = X−E [y − πE (β̄E )] + C(β̄E − β̂E ) + op (n−1/2 ) n −E n 1 1 T [y − πE (β̄E )] + √ D(ωE + ΛE zE ) + op (n−1/2 ) = X−E n n Using the inequalities in the KKT conditions, we have the selection event is {AM T + BM ω ≤ bM } with AM , BM and bM defined in the lemma. Appendix C: Proofs related to Logistic noise Throughout the article, logistic noise has played an important role in all the examples. The following lemma on the tail behavior of the logistic distribution is crucial to all the proofs with added logistic noise. Let G be the CDF of Logistic(κ), with κ being the scale parameter. g is the PDF of G. G(w) = eκw , 1 + eκw g(w) = κe−κ\|w\| 1 + e−κ\|w\| 2 . Lemma 16. The following lower bounds hold, Ḡ(κw) ≥ 1 −κ\|w\| e , 2 g(w) ≥ 1 −κ\|w\| e . 4 (52) For k = 0, 1, 2, 3, . . . : ∂k ek e−κ\|w\| . g(w) ≤ κk+1 C ∂wk ek ’s are universal constants. where C (53) Tian and Taylor/Selective inference with a randomized response 38 Proof. We can write ( g(w) = κe−κw h0 (e−κw ) w > 0 κeκw h0 (eκw ) w≤0 where h0 (x) = (1 + x)−2 . For j ≥ 1, define hj (x) = x · h0j−1 (x). By induction, I claim that for each j, hj is rational such that the polynomial in the numerator is of order 2 less than the denominator, and the denominator polynomial is bounded below by 1. Hence, hj ’s are bounded on the interval [0, 1]. Now, it is not hard to see that ( Pk −(−κ)k+1 j=0 cj,k hj (e−κw )e−κw w > 0 ∂k g(w) = P k ∂wk κk+1 j=0 cj,k hj (eκw )eκw w≤0 for universal cj,k ’s and k = 0, 1, 2, . . . . Now we state the following lemmas which are foundations of the proofs of various lemmas in the article. Lemma 17. Assume Tn is a decomposable statistic and ξi,n has mean 0, variance σ 2 , and centered exponential moments in a neighbourhood of zero, i.e satisfies condition √ (29). Denote Zn = n(Tn − µn ), then 2 2 κ σ E [exp (κZn )] → exp , for κ > 0. 2 √ Lemma 18. In Example 2, if we normalize the sample mean Zn = n(X̄n − µn ), we can rewrite the selective likelihood ratio and the pivot as √ Ḡ(2 − Zn − nµn ) , `¯Fn (Zn ) = √ EFn Ḡ(2 − Zn − nµn ) and R∞ P (Zn ) = n RZ ∞ −∞ √ nµn ) exp(−t2 /2) dt . √ Ḡ(2 − t − nµn ) exp(−t2 /2) dt Ḡ(2 − t − Then for any Fn with finite centered exponential moment in a neighbourhood of zero, we have for k = 0, 1, 2, 3 ∂k ¯ `F (Z) ≤ C1 exp[κ\|Z\|], ∂Z k n ∂k P (Z) ≤ C1 . ∂Z k (54) for some C1 only depending on κ. Proof of Lemma 17. Pn Proof. Without loss of generality, we assume Tn = n1 i=1 ξi,n . Since Fn has centered exponential moments in a neighbourhood of zero, it is each to see n κ E [exp (κZn )] = M √ n Tian and Taylor/Selective inference with a randomized response 39 exists as long as √κn < a. M (·) is the moment generating function of ξi,n − µn , M (t) = EFn [exp(t(ξi,n − µ))]. Therefore, √1 t= n κ log [M (κt)] lim n log M √ = lim n→∞ t→0 t2 n κ2 M 00 (κt) κ2 σ 2 = lim = . t→0 2M (κt) 2 To derive the equality, we used M 00 (0) = Var [ξi,n ] = σ 2 and M 0 (0) = 0, M (0) = 1. Proof of Lemma 18. Proof. Noticing the lower bound in (52), we have i 1 h i √ √ 1 h √ E Ḡ(2 − Zn − nµn ) ≥ E e(−κ\|2−Zn − nµn \|) ≥ E e(−κ\|2−Zn \|−κ n\|µn \|) 2 2 On the other hand, using the upper bounds in (53), we have for k = 1, 2, 3, √ ek−1 eκ\|2−Z\| ek−1 e−κ\|2−Z− nµn \| κk C ∂k ¯ κk C h i ≤ 2 ` (Z) ≤ 2 √ F n ∂Z k E e(−κ\|2−Z\|) E e(−κ\|2−Z\|−κ n\|µn \|) Since x−1 is convex on the positive axis, it is hard to see h i 1 ≤ E e(κ\|2−Z\|) ≤ e2κ E eκZ + e−κZ . (−κ\|2−Z\|) E e Thus using Lemma 17, we know E e±κZ → exp(κ2 /2). Thus, we conclude sup n ∂k ¯ `F (Z) ≤ C1 exp[κ \|Z\|], ∂Z k n k = 1, 2, 3. (55) √ To verify the above inequality for k = 0. Notice that for µn ≥ 0, Ḡ(2 − Z − nµn ) ≤ Ḡ(2 − Z). Thus the denominator of `¯Fn (Z) is bounded below using the argument above. For µ < 0, √ √ √ Ḡ(2 − Z − nµn ) ≤ exp(−κ(2 − Z − nµn ) ≤ exp(−κ n \|µn \| + κ \|2 − Z\|). √ The term exp(−κ n \|µn \|) cancels with the one in the denominator, thus (55) holds for k = 0 as well. Analogously, similar bounds can be derived for the derivatives of P (Z) as well, thus we have the conclusion of the lemma. C.1. Proof of Lemma 4 and Lemma 10 . The proof of Lemma 4 is a simple application of Lemma 17 and Lemma 18. Tian and Taylor/Selective inference with a randomized response 40 Proof. By law of large numbers, we know that X̄n is consistent for µ unselectively. Thus, using the result by Lemma 3, we only need to verify that the selective likelihood is integrable in Lq . For simplicity, we take q = 2. First notice from (54) that the selective likelihood ratio is bounded by a multiple of exp[κ\|Z\|]. Then by Lemma 17, lim sup EFn `Fn (X̄)2 ≤ 2C12 exp(2κ2 ). n The proof of Lemma 10 uses results in Lemma 18 and Lemma 15 Proof. It follows simply from (54) that condition (28) are satisfied with the norm function Ω simply being the absolute value function. Therefore, we only need to verify (30). Note for µn = µ < 0 √ √ PFn ×Q [ nXn + ω > 2] − PΦ×Q [ nXn + ω > 2] √ PΦ×Q [ nXn + ω > 2] √ √ EFn Ḡ(2 − Z − nµ) − EΦ Ḡ(2 − Z − nµ) = √ EΦ Ḡ(2 − Z − nµ) √ √ √ exp(−κ nµ)EFn Ḡ(2 − Z − nµ) − EΦ Ḡ(2 − Z − nµ) ≤2 EΦ [exp(−κ\|2 − Z\|)] √ √ ≤2EΦ [exp(2κ + 2\|Z\|)] · C exp(−κ nµ) exp(κ nµ)n−1/2 ≤2C exp(κ2 )n−1/2 , √ Ω The second to last inequality uses Lemma 15 √ and the fact that λ3 (Ḡ) ≤ exp(κ nµ). For µn = µ ≥ 0, the denominator PΦ×Q [ nXn + ω > 2] is bounded below, and Ḡ has bounded derivatives. Therefore, a simple application of the Berry-Esseen Theorem will suffice. Appendix D: Proofs related to affine selection regions D.1. Proof of Lemma 12 The quantity that appears in both the pivot and the selective likelihood ratio is Z Q(z; ∆) = P(A(z + ∆) + ω ∈ K) = G(dw), K−A(z+∆) where ω ∼ G. The associated selective likelihood in terms of z is `F (z; ∆) = R Q(z; ∆) . Q(t; ∆)F(dt) Rq We first rewrite the pivot in terms of U . R∞ Q(t; Lz, ∆) exp(−t2 /2ση2 ) dt ηT z P (z; ∆) = R ∞ , Q(t; Lz, ∆) exp(−t2 /2ση2 ) dt −∞ (56) Tian and Taylor/Selective inference with a randomized response 41 where we use the slight abuse of notation for the one dimensional function Q(t; Lz, ∆) 1 Σηη T ση2 1 Q(t; Lu, ∆) = P(t · 2 AΣη + ALu + A∆ + w ∈ K) ση Z = G(dw). L=I− K−t· σ12 AΣη−ALu−A∆ η We first establish a lower bound on EF [Q(Z; ∆)] = PFn ×Q [A(Z + ∆) + ω ∈ K] under the local alternatives. Lemma 19. If we assume the lower bound condition, then under the local alternatives with radius B, i.e. dh (0, K − A∆) ≤ B, we have Z Q(u; ∆)F(du) ≥ C − C(Φ, h) · e−B , Rp where C(Φ, h) is a constant only depending on the normal distribution Φ = N (0, Σ) and the norm h in the local alternatives condition. Proof. We first see that the lower bound condition gives the following lower bound. Z − Q(u; ∆) = G(dw) ≥ C exp − inf h(w) . (57) w∈K−A(u+∆) K−A(u+∆) Consider Z Z Q(u; ∆)F(du) ≥ C − exp − inf h(w) F(du) w∈K−A(∆+u) Rp Rp Z = C− exp − inf h(w − Au) F(du) w∈K−A∆ Rp Z ≥ C− exp − inf h(w) + h(−Au) F(du) w∈K−A∆ Rp Z = C − · exp − inf h(w) e−h(Au) F(du) w∈K−A∆ Rp Z − =C e−h(Au) F(du) · e−B . Rp Finally, since the exp(−h(Au)) has uniformly bounded derivatives up to the third order, we have Z Z exp(−h(Au))F(du) → exp(−h(Au))Φ(du) Rp Rp as Z → N (0, Σ) in distribution. Let C(Φ, h) = have the conclusion of the lemma. R Rp exp(−h(Au))Φ(du), and we will Tian and Taylor/Selective inference with a randomized response 42 The following lemmas establish the bounds on the derivatives for the likelihood function `F and the pivot P (u; ∆). Lemma 12 is easily obtained using Lemma 20 and Lemma 21 below. Lemma 20. Suppose the smoothness and the lower bound conditions are satisfied, then for local alternatives with radius B, ∂α `F (z; ∆) ≤ C ∗ (B, Φ, h), ∂z α Φ = N (0, Σ). (58) Proof. The smoothness condition implies the following upper bound. For a multi-index α, we have Z ∂α ∂α Q(z; ∆) = α g(w)dw ∂z α ∂z K−A(∆+z) Z ∂α g(w − Az)dw. = α K−A∆ ∂z Therefore, from the smoothness condition, ∂α def Q(z; ∆) ≤ C(A)Cα = Cα+ (A). ∂z α (59) This combined with Lemma 19 gives the conclusion of the lemma. Next, we derive the exponential bounds on the derivatives of the pivot P (z; ∆) with respect to z. Lemma 21. Assuming the conditions of Lemma 12, for a multi-index α up to the order of 3, ∂α P (z; ∆) ≤ Cα∗ · eα·Lip(h)kALzk2 , ∂z α where the norm on the left is the element-wise maximum and C is independent of (z, ∆) and Lip(h) is the Lipschitz constant of h with respect to `2 norm. Proof. To get a lower bound on the denominator, note (57) 1 − Q(t; Lz, ∆) ≥ C exp − inf h(w − t · 2 AΣη) w∈K−ALz−A∆ ση ≥ C − exp − inf h(w) − \|t\|h(AΣη/ση2 ) . w∈K−ALz−A∆ Therefore, the denominator will be lower bounded by Z ∞ 1 √ Q(t; Lz, ∆) exp(−t2 /2ση2 )dt 2π −∞ − 2 2 2 ≥C exp h(AΣη/ση ) ση /2 · exp − inf h(w) w∈K−ALz−A∆ ≥C − exp h(AΣη/ση2 )2 ση2 /2 e−B · e−h(ALz) . 43 Tian and Taylor/Selective inference with a randomized response On the other hand, the upper bound (59) ensures, Z ∂α 1 √ Q(t; Lz, ∆) exp(−t2 /2ση2 )dt ≤ Cα+ (A). 2π R ∂z α Note the derivatives of the pivot will be a polynomial in terms of the form, R∞ ∂ Q(t; Lz, ∆) exp(−t2 /2ση2 )dt ηT z α R∞ Q(t; Lz, ∆) exp(−t2 /2ση2 )dt −∞ and therefore, it is easy to get the conclusion of the lemma. D.2. Proof of Lemma 13 Using Lemma 19 and the following lemma, we can easily prove Lemma 13. √ Lemma 22. Let Zn = n(Tn − µn ) ∈ Rp , and Fn has finite third moments γ. Moreover, suppose the randomization noise ω ∈ Q, a probability measure on Rd . Then for any sequence of sets (Un )n≥1 , Un ⊆ Rp × Rd , we have 1 \|PFn ×Q [(Zn , ω) ∈ Un ] − PΦn ×Q [(Zn , ω) ∈ Un ]\| ≤ C3 γn− 2 , where Φn = N (µ(Fn ), Σ(Fn )) and C3 is a constant depending only on p. Proof of Lemma 22 uses the well known results of Berry-Esseen Theorem. A multivariate extension can be found in Gotze (1991). Proof. For each ω, we denote Un (ω) = {Z ∈ Rp : (Z, ω) ∈ Un } ⊆ Rp . Thus the difference in the two probabilities is \|PFn ×Q [(Z, ω) ∈ Un ] − PΦn ×Q [(Z, ω) ∈ Un ]\| ≤EQ [\|PFn [Z ∈ Un (ω)] − PΦ [Z ∈ Un (ω)]\|] ≤ sup \|Fn (U ) − Φn (U )\| < C3 γn−1/2 , U ∈Rp where C3 only depends on the dimension p. The last inequality is a direct application of equation (1.5) in Gotze (1991).	10
arXiv:1204.2394v5 [math.AC] 26 Feb 2013 ON INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS OF LOCAL COHOMOLOGY MODULES MAJID RAHRO ZARGAR AND HOSSEIN ZAKERI Abstract. Let (R, m) be a commutative Noetherian local ring and let M be an Rmodule which is a relative Cohen-Macaulay with respect to a proper ideal a of R and set n := ht M a. We prove that injdimM < ∞ if and only if injdimHn a (M ) < ∞ and that injdimHn a (M ) = injdimM − n. We also prove that if R has a dualizing complex n and Gid R M < ∞, then Gid R Hn a (M ) < ∞ and Gid R Ha (M ) = Gid R M − n. More- over if R and M are Cohen-Macaulay, then it is proved that Gid R M < ∞ whenever Gid R Hn a (M ) < ∞. Next, for a finitely generated R-module M of dimension d, it is proved d d that if KM c is Cohen-Macaulay and Gid R Hm (M ) < ∞, then Gid R Hm (M ) = depth R−d. The above results have consequences which improve some known results and provide characterizations of Gorenstein rings. 1. introduction Throughout this paper, R is a commutative Noetherian ring, a is a proper ideal of R and M is an R-module. For a prime ideal p of R, the residue class field Rp /pRp is denoted by k(p). For each non-negative integer i, let Hia (M ) denotes the i-th local cohomology module of M with respect to a; see [1] for its definition and basic results. Also, we use injdimR M to denote the usual injective dimension of M . The notion of Gorenstein injective module was introduced by E.E. Enochs and O.M.G. Jenda in [4]. The class of Gorenstein injective modules is greater than the class of injective modules; but they are same classes whenever R is a regular local ring. The Gorenstein injective dimension of M , which is denoted by Gid R M , is defined in terms of resolutions of Gorenstein injective modules. This notion has been used in [3, 15, 21] and has led to some interesting results. Notice that Gid R M ≤ injdimR M and the equality holds if injdimR M < ∞. The principal aim of this paper is to study the injective (resp. Gorenstein injective) dimension of certain R-modules in terms of injective (resp. Gorenstein injective) dimension of its local cohomology modules at support a. In this paper we will use the concept of relative Cohen-Macaulay modules which has been studied in [7] under the title of cohomologically complete intersections. The organization of this paper is as follows. In section 2, among other things, we prove, in 2.5, that if M is relative Cohen-Macaulay with respect to a, then injdimM and injdimHht M a (M ) are a simultaneously finite and there is an equality injdimHaht M a (M ) = injdimM − ht M a. Then, 2000 Mathematics Subject Classification. 13D05, 13D45. Key words and phrases. Injective dimension, Gorenstein injective dimension, Local cohomology, Gorenstein ring, Relative Cohen-Macaulay module. 1 2 M.R. ZARGAR AND H. ZAKERI as a corollary, we obtain a characterization of Gorenstein rings. Next, in 2.10, for all n ≥ 0 and any p ∈ Supp (M ), we establish a comparison between the Bass numbers of HnpRp (Mp ) dim (R/p) (M ) whenever (R, m) is a homomorphic image of a Gorenstein ring and and Hn+ m M is finitely generated. In section 3, we first prove some basic properties about Gorenstein injective dimension of a module. In particular, Proposition 3.6 indicates that Gorenstein injective dimension is a refinement of the injective dimension. As a main result, in Theorem 3.8 we establish a Gorenstein injective version of 2.5. Indeed, it is proved that if, in addition to the hypothesis of 2.5, R has a dualizing complex, then Gid R M < ∞ implies Gid R Hna (M ) < ∞ and the converse holds whenever R and M are Cohen-Macaulay. This theorem has consequences which recover some interesting results that have currently been appeared in the literature. As a first corollary of 3.8, we deduce that Gid R Hnm (M ) = Gid R M − n, wherever M is a Cohen-Macaulay module over the Cohen-Macaulay local ring (R, m) and dim M = n. This corollary improves the main result [15, Theorem 3.10](see the explanation which is offered before 3.9). As a second corollary, we obtain a characterization of Gorenstein local rings which recovers [21, Theorem 2.6]. As a main result, it has been shown in [15, Theorem 3.10] that if R and M are Cohen-Macaulay with dim M = d and Gid R Hdm (M ) < ∞, then Gid R Hdm (M ) = dim R − d. In 3.12, we will use the canonical module of a module to improve the above result without Cohen-Macaulay assumption on R and M . This result provides some characterizations of Gorenstein local rings. 2. local cohomology and injective dimension The starting point of this section is the next proposition, which plays essential role in the proof of Theorems 2.5 and 3.8. Proposition 2.1. Let n be a non-negative integer and let N be an a-torsion R-module. Suppose that Hia (M ) = 0 for all i 6= n. Then Ext iR (N, Hna (M )) ∼ = Ext i+n R (N, M ) for all i ≥ 0. Proof. First we notice that Hom R (N, M ) = Hom R (N, Γa (M )). Hence, in view of [14, Theorem 10.47], we have the Grothendieck third quadrant spectral sequence with E2p,q = Ext pR (N, Hqa (M )) =⇒ Ext p+q R (N, M ). p Now, since Hqa (M ) = 0 for all q 6= n, E2p,q = 0 for all q 6= n. Therefore, this spectral sequence collapses in the column q = n; and hence one gets, for all i ≥ 0, the isomorphism Ext iR (N, Hna (M )) ∼ = Ext i+n R (N, M ), as required. The following corollary, which is an immediate consequence of 2.1, determines the Bass numbers µi (p, Hna (M )) := vdim k(p) Ext iRp (k(p), HnaRp (Mp )) of the local cohomology module Hna (M ). INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 3 Corollary 2.2. Let n and M be as in 2.1. Then, for all p ∈ V(a), µi (p, Hna (M )) = µi+n (p, M ) for each i ≥ 0. Definition 2.3. We say that a finitely generated R-module M is relative Cohen Macaulay with respect to a if there is precisely one non-vanishing local cohomology module of M with respect to a. Clearly this is the case if and only if grade (a, M ) = cd (a, M ), where cd (a, M ) is the largest integer i for which Hia (M ) 6= 0 and grade (a, M ) is the least integer i such that Hia (M ) 6= 0. Observe that the above definition provides a generalization of the concept of CohenMacaulay modules. Also, notice that the notion of relative Cohen-Macaulay modules is connected with the notion of cohomologically complete intersection ideals which has been studied in [7] and has led to some interesting results. Furthermore, such modules have been studied in [8] over certain rings. Remark 2.4. Let M be a relative Cohen-Macaulay module with respect to a and let cd (a, M ) = n. Then, in view of [1, Theorems 6.1.4, 4.2.1, 4.3.2], it is easy to see that Supp Hna (M ) = Supp (M/aM ) and ht M a = grade (a, M ), where ht M a = inf{ dim Rp Mp \| p ∈ Supp (M/aM ) }. The following theorem, which is one of the main results of this section, provides a comparison between the injective dimensions of a relative Cohen-Macaulay module and its non-zero local cohomology module. Here we adopt the convention that the injective dimension of the zero module is to be taken as −∞. Theorem 2.5. Let (R, m) be local and let n be a non-negative integer such that Hia (M ) = 0 for all i 6= n. (i) If injdimM < ∞, then injdimHna (M ) < ∞. (ii) The converse holds whenever M is finitely generated. Furthermore, if M is non-zero finitely generated, then injdimHna (M ) = injdimM − n. Proof. (i). Let s := injdimM < ∞. We may assume that Hna (M ) 6= 0; and hence s − n ≥ 0. Therefore, in view of 2.2, µi+(s−n) (p, Hna (M )) = 0 for all p ∈ Spec (R) and for all i > 0; so that injdimHna (M ) ≤ s − n. (ii). Suppose that M is finitely generated. We first notice that Hna (M ) = 0 if and only if M = aM ; and this is the case if and only if M = 0. Therefore we may assume that Hna (M ) 6= 0. Suppose that t := injdimHna (M ) < ∞. Then there exists a prime ideal q of R such that µt (q, Hna (M )) 6= 0. Hence, by 2.2, µt+n (q, M ) 6= 0. Next we show that µt+n+i (p, M ) = 0 for all p ∈ Spec (R) and for all i > 0. Assume the contrary. Then there exists a prime ideal p of R such that µt+n+j (p, M ) 6= 0 for some j > 0. Let r := dim R/p. Then, by [11, §18, Lemma 4], we have µt+n+j+r (m, M ) 6= 0. Hence, by 2.2, µj+t+r (m, Hna (M )) 6= 0 which is a contradiction in view of the choice of t. Therefore, injdimM ≤ t + n. The final assertion is a consequence of (i) and (ii). 4 M.R. ZARGAR AND H. ZAKERI Next, we provide an example to show that if R is non-local, then Theorem 2.5(ii) is no longer true. Also, in 3.11, we present two examples which show that 2.5(ii) and 2.5(i), respectively, are no longer true without the finiteness and the relative Cohen-Macaulayness assumptions on M . Example 2.6. Suppose that R is a non-local Artinian ring with injdimR = ∞. Let L Max (R) = {m1 , ..., mn }. Then, in view of [17, Exercise 8.49], we have R = m∈Max (R) Γm (R). Now, since the injective dimension of R is infinite, there exists a maximal ideal mt of R L such that the injective dimension of Γmt (R) is infinite. Set M := ER (R/ms ) Γmt (R), where ms ∈ Max (R) with ms 6= mt . Then M is a finitely generated R-module with infinite injective dimension and Hims (M ) = 0 for all i 6= 0; but Γms (M ) is injective. It is well-known that if (R, m, k) is a d-dimensional local ring, then R is Gorenstein if and only if R is Cohen-Macaulay and Hd (R) ∼ = ER (k). The following corollary, which recovers m this fact, is an immediate consequence of 2.5. Corollary 2.7. Let (R, m) be local and let R be relative Cohen-Macaulay with respect to a. Then R is Gorenstein if and only if injdimHaht R a (R) is finite. In particular, if x = x1 , ..., xn is an R-sequence for some non-negative integer n, then R is Gorenstein if and only if injdimHn(x) (R) is finite. The following proposition, which is needed in the proof of 3.8, provides an explicit minimal injective resolution for the non-zero local cohomology module of a relative Cohen-Macaulay module. Proposition 2.8. Suppose that M is relative Cohen-Macaulay with respect to a and that n = cd (a, M ). Then 0 −→ Hna (M ) −→ M µn (p, M )E(R/p) −→ p∈V(a) is a minimal injective resolution for M µn+1 (p, M )E(R/p) −→ · · · p∈V(a) Hna (M ). Furthermore, Ass R Hna (M ) = {p ∈ V(a)\| µn (p, M ) 6= 0}. Proof. Let d−1 d0 dn−1 dn dn+1 0 −→ M −→ E0 (M ) −→ · · · −→ En−1 (M ) −→ En (M ) −→ En+1 (M ) −→ · · · be a minimal injective resolution for M . If there exists a prime ideal p in V(a) with µn−1 (p, M ) 6= 0, then depth Rp Mp ≤ n − 1. On the other hand, since p ∈ Supp (M/aM ), 2.4 implies that HnaRp (Mp ) 6= 0. Therefore n = grade Rp (aRp , Mp ) ≤ depth Rp Mp ≤ n − 1 which is a contradiction. It follows that Γa (E n−1 (M )) = 0; and hence we obtain the minimal injective resolution 0 −→ Hna (M ) −→ Γa (En (M )) −→ Γa (En+1 (M )) −→ · · · for Hna (M ). Now, we may use this resolution to complete the proof. INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 5 The following elementary lemma, which is needed in the proof of the next theorem, can be proved by using a minimal free resolution for M and the concept of localization. Lemma 2.9. Let (R, m, k) be local and let M be finitely generated. Then, for any prime R ideal p of R, vdim k(p) Tor i p (k(p), Mp ) ≤ vdim k Tor R i (k, M ) for all i ≥ 0. The next theorem provides a comparison of Bass numbers of certain local cohomology modules. Theorem 2.10. Suppose that (R, m, k) is a local ring which is a homomorphic image of a Gorenstein local ring and that M is finitely generated. Let n, m be non-negative integers. Then µm (p, Hn (M )) ≤ µm (m, Hn+dim R/p (M )) for all p ∈ Spec (R). p m Proof. Let (R′ , m′ ) be a Gorenstein local ring of dimension n′ for which there exists a surjective ring homomorphism f : R′ → R. Let p be a prime ideal of R and set p′ = f −1 (p). Now Rp′ ′ is a Gorenstein local ring and dim R′ /p′ = dim R/p. Since R′ is Gorenstein, we have dim Rp′ ′ = dim R′ − dim R′ /p′ . In view of [1, Exercise 11.3.1] there is, for each j ∈ Z, j an Rp -isomorphism Ext jR′ (Mp , Rp′ ′ ) ∼ = (Ext R′ (M, R′ ))p . Also, by the Local Duality Thep′ n′ −n−t orem [1, Theorem 11.2.6], we have HnpRp (Mp ) ∼ (Mp , Rp′ ′ ), ERp (k(p)) = Hom Rp Ext R ′ p′ n′ −n−t ∼ as Rp -modules, where t := dim R/p, and Hn+t (M, R′ ), ER (k)). It m (M ) = Hom R (Ext R′ therefore follows that n n′ −n−t m ∼ Ext m (Mp , Rp′ ′ ), ERp (k(p)))) Rp (k(p), HpRp (Mp )) = Ext Rp (k(p), Hom Rp (Ext R′ ′ p ′ p n −n−t ∼ Hom R (Tor R (Mp , R′ ′ )), ER (k(p))). = m (k(p), Ext ′ p Rp ′ p p and n+t Ext m R (k, Hm (M )) n′ −n−t ∼ (M, R′ ), ER (k))) = Ext m R (k, Hom R (Ext R′ ′ n −n−t R ∼ Hom R (Tor (k, Ext ′ (M, R′ )), ER (k)). = m R Since by 2.9 R ′ ′ n −n−t n −n−t vdim k(p) (Tor mp (k(p), Ext R (Mp , Rp′ ′ ))) ≤ vdim k (Tor R (M, R′ ))), one ′ m (k, Ext R′ p′ may use the above isomorphisms to complete the proof. It is known as Bass’s conjecture that if a local ring admits a finitely generated module of finite injective dimension, then it is a Cohen-Macaulay ring. For the proof of this fact the reader is referred to [12] and [13]. In the next corollary we shall use this fact and the concept of a generalized Cohen-Macaulay module. Recall that, over a local ring (R, m), a finitely generated module of positive dimension is a generalized Cohen-Macaulay module if Him (M ) is finitely generated for all 0 ≤ i < dim M . Corollary 2.11. Let the situation be as in 2.10. Then the following statements hold. dim R/p (M ) for all prime ideals p of R and for (i) injdimRp HnpRp (Mp ) ≤ injdimR Hn+ m any n ≥ 0. (ii) If M is generalized Cohen-Macaulay with dimension d such that Hdm (M ) is injective, then Mp is Gorenstein, in the sense of [18], for all p ∈ Supp (M ) \ {m}. 6 M.R. ZARGAR AND H. ZAKERI Proof. (i) is clear by 2.10. (ii) Let p ∈ Supp (M ) \ {m}. By [1, Exercise 9.5.7], Mp is Cohen-Macaulay and dim Mp + dim R/p = dim M . Hence, in view of (i) and 2.5, we have injdimMp = dim Mp . Therefore, by [18, Theorem 3.11], [2, Theorem 3.1.17] and Bass’s conjecture, Mp is Gorenstein. 3. local cohomology and gorenstein injective dimension We first recall some definitions that we will use in this section. Definition 3.1. Following [4], an R-module M is said to be Gorenstein injective if there exists a Hom (Inj, −) exact exact sequence · · · → E1 → E0 → E 0 → E 1 → · · · of injective R-modules such that M = Ker (E 0 → E 1 ). We say that an exact sequence 0 → M → G0 → G1 → G2 → · · · of R-modules and R-homomorphisms is a Gorenstein injective resolution for M , if each Gi is Gorenstein injective. We say that Gid R M ≤ n if and only if M has a Gorenstein injective resolution of length n. If there is no shorter resolution, we set Gid R M = n. Dually, an R-module M is said to be Gorenstein projective if there is a Hom (−, P roj) exact exact sequence · · · → P1 → P0 → P 0 → P 1 → · · · of projective R-modules such that M = Ker (P 0 → P 1 ). Similarly, one can define the Gorenstein projective dimension, Gpd R M , of M . Definition 3.2. For a local ring R admitting the dualizing complex DR , we denote by KM the canonical module of an R-module M , which is defined to be KM = Hd−n (RHom R (M, DR )), where d = dim R and n = dim M . Note that if R is Cohen-Macaulay, then KR coincides with the classical definition of the canonical module of R which is denoted by ωR . Definition 3.3. Let R be a Cohen-Macaulay local ring of Krull dimension d which admits a canonical module ωR . Following [4], let J0 (R) be the class of R-modules M which satisfies the following conditions. (i) Ext iR (ωR , M ) = 0 , for all i > 0. (ii) Tor R i (ωR , Hom R (ωR , M )) = 0, for all i > 0. (iii) The natural map ωR ⊗R Hom R (ωR , M ) → M is an isomorphism. This class of R-modules is called the Bass class. Definition 3.4. Following [19], let a and b be ideals of R. We set W (a, b) = { p ∈ Spec (R) \| an ⊆ p + b for some integer n > 0}. For an R-module M , Γa,b (M ) denotes a submodule of M consisting of all elements of M with support in W (a, b), that is, Γa,b (M ) = { x ∈ M \| Supp (Rx) ⊆ W (a, b)}. INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 7 The following lemma has been proved in [10, Lemma 4.2] and is of assistance in the proof of Proposition 3.6. Lemma 3.5. Let (R, m, k) be local. Then Ext iR (ER (k), M ) ∼ = Ext iR̂ (ER̂ (k), M ⊗R R̂) = Ext iR (E(k), M ⊗R R̂) ∼ for all i ≥ 0. Let (R, m, k) be local and let M be a non-zero non-injective R-module of finite Gorenstein injective dimension. It was shown in [5, Corollary 4.4] that if Ext iR (E, M ) = 0 for all i ≥ 0 and all indecomposable injective R-modules E 6= ER (k), then Gid R M = sup{ i \| Ext iR (ER (k), M ) 6= 0}. Our next proposition, which is concerned with this result, indicates that Gorenstein injective dimension is a refinement of the injective dimension. However we will use 3.6 and 3.7 to prove the main theorem 3.8. Proposition 3.6. Let (R, m, k) be local and let M be non-zero with Gid R M < ∞. If either M is finitely generated or Artinian, then Gid R M = sup{ i \| Ext iR (ER (k), M ) 6= 0}. Proof. First assume that M is finitely generated. Then, by [3, Theorem 3.24], Gid R M = c. Now, since R b is complete and M c is finitely generated as an R-module, b Gid Rb M the proof of [6, Proposition 2.2] in conjunction with [5, Corollary 4.4 ] implies that c = sup{ i \| Ext i (E b (R/ b m), c) 6= 0}. b M Gid Rb M b R R Therefore we can use 3.5 to complete the proof. Next, we consider the case where M is Artinian. By [15, Lemma 3.6 ] and [1, Exercise b m))) b 8.2.4], Gid R (M ) = Gid b (M ) and, by [3, Theorem 4.25], Gfd b (Hom b (M, E b (R/ = R R R R Gid Rb (M ), where, for an R-module X, Gfd R (X), denotes the Gorenstein flat dimension of b m)) b b is finitely generated as an R-module, X. Therefore, since Hom Rb (M, ERb (R/ in view of [3, Theorem 4.24] and [3. Theorem 1.10] we have the first equality in the next display b m))) b Gfd Rb (Hom Rb (M, ERb (R/ b m)), b 6= 0} b R) = sup{ i\| Ext iRb (Hom Rb (M, ERb (R/ = sup{ i\| Ext iR (ER (k), M ) 6= 0}. The last equality follows from [10, Theorem 4.3], because ER (k) and M are Artinian. Lemma 3.7. Let (R, m) be a Cohen-Macaulay local ring and let M be finitely generated. Suppose that x ∈ m is both R-regular and M -regular. Then the following statements are equivalent. (i) Gid R M < ∞. (ii) Gid R/xR M/xM < ∞. Furthermore, Gid R/xR M/xM = Gid R M − 1. Proof. In view of [3, Theorem 3.24], we can assume that R is complete; and hence it admits a canonical module ωR . 8 M.R. ZARGAR AND H. ZAKERI (i)⇒(ii). It follows from [3, Proposition 3.9] that Gid R M/xM < ∞. Therefore we can use [4, Proposition 10.4.22], [11, p.140,lemma 2] and [2, Theorem 3.3.5], to see that Gid R/xR M/xM < ∞. (ii)⇒(i). By [4, Proposition 10.4.23], M/xM ∈ J0 (R/xR). Since ωR/xR ∼ = ωR /xωR , in i R view of [11, p.140, lemma 2] we have Ext R (ωR , M/xM ) = Tor i (ωR , Hom R (ωR , M/xM )) = 0 for all i > 0 and ωR /xωR ⊗R/xR Hom R (ωR , M/xM ) ∼ = M/xM . Now, using the exact x sequence Ext iR (ωR , M ) −→ Ext iR (ωR , M ) −→ Ext iR (ωR , M/xM ) and Nakayama’s lemma, we deduce that Ext iR (ωR , M ) = 0 for all i > 0. Thus we have the exact sequence (3.1) x 0 −→ Hom R (ωR , M ) −→ Hom R (ωR , M ) −→ Hom R (ωR , M/xM ) −→ 0. Now, we may use (3.1) and similar arguments as above to see that Tor iR (ωR , Hom R (ωR , M )) = ∼ M. 0 for all i > 0. Also, in view of [2, Lemma 3.3.2], we can see that ωR ⊗R Hom R (ωR , M ) = Therefore by [4, Proposition 10.4.23], Gid R M < ∞. The final assertion is an immediate consequence of [3, Theorem 3.24]. Theorem 3.8, which is a Gorenstein injective version of 2.5, is one of the main results of this section. As we will see, this theorem has consequences which recover some interesting results that have currently been appeared in the literature. Here we adopt the convention that the Gorenstein injective dimension of the zero module is to be taken as −∞. Theorem 3.8. Suppose that the local ring (R, m) has a dualizing complex and let n be a non-negative integer such that Hia (M ) = 0 for all i 6= n. (i) If Gid R M < ∞, then Gid R Hna (M ) < ∞. (ii) The converse holds whenever R and M are Cohen-Macaulay. Furthermore, if M is non-zero finitely generated with finite Gorenstein injective dimension, then Gid R Hna (M ) = Gid R M − n. Proof. (i) Notice that if Hna (M ) = 0, then there is nothing to prove. So, we may assume that Hna (M ) 6= 0. Hence, by [20, Lemma 1.1], we have n ≤ d, where d = Gid R M . Let d0 d−1 dn−1 d1 dd−1 dn+1 dn 0 −→ M −→ G0 −→ G1 −→ · · · −→ Gn−1 −→ Gn −→ Gn+1 −→ · · · −→ Gd−1 −→ Gd −→ 0 be a Gorenstein injective resolution for M . By applying the functor Γa (−) on this exact sequence, we obtain the complex Γa (d−1 ) Γa (d0 ) Γa (d1 ) 0 −→ Γa (M ) −→ Γa (G0 ) −→ Γa (G1 ) −→ · · · −→ Γa (Gn−1 ) Γa (dn−1 ) −→ Γa (dn ) Γa (Gn ) −→ Γa (Gn+1 ) Γa (dn+1 ) −→ · · · −→ Γa (Gd−1 ) Γa (dd−1 ) −→ Γa (Gd ) −→ 0 in which, by [15, Theorem 3.2], Γa (Gi ) is Gorenstein injective for all 0 ≤ i ≤ d. If n = 0 the result is clear. So suppose that n > 0. Now, since by [20, Lemma 1.1] each Gi is Γa -acyclic for all i, we may use [1, Exercise 4.1.2 ] in conjunction with our assumption on local cohomology modules of M to obtain the following two exact sequences Γa (d−1 ) Γa (d0 ) 0 −→ Γa (M ) −→ Γa (G0 ) −→ · · · −→ Γa (Gn−1 ) and Γa (dn−1 ) −→ Γa (Gn ) −→ Γa (Gn ) −→ 0 Im Γa (dn−1 ) INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 9 0 −→ Im Γa (dn ) = Ker Γa (dn+1 ) ֒→ Γa (Gn+1 ) −→ · · · −→ Γa (Gd−1 ) −→ Γa (Gd ) −→ 0. But, by assumption, Γa (M ) = 0. Therefore, by using the first above exact sequence and n ) [4, Theorem 10.1.4], we see that ImΓΓa (G is Gorenstein injective. Notice that Hna (M ) = n−1 ) a (d n ker Γa(d ) Im Γ (dn−1 ) . Therefore, patching the second above long exact sequence together with the a exact sequence 0 −→ Hna (M ) −→ Γa (Gn ) Γa (Gn ) −→ −→ 0, Im Γa (dn−1 ) ker Γa (dn ) yields the following long exact sequence 0 −→ Hna (M ) −→ Γa (Gn ) −→ Γa (Gn+1 ) −→ · · · −→ Γa (Gd−1 ) −→ Γa (Gd ) −→ 0. Im Γa (dn−1 ) Hence, Gid R Hna (M ) ≤ Gid R M − n. (ii) Suppose that R and M are Cohen-Macaulay. Since H0m (E(R/m)) = E(R/m) and, for any non-maximal prime ideal p of R, H0m (E(R/p)) = 0, we may apply 2.8 to see that Him (Hna (M )) = Hn+i m (M ) for all i ≥ 0. Therefore, we can use the Cohen-Macaulayness of M to deduce that Him (Hna (M )) =  0 if i 6= dim M/aM Hd (M ) if i = dim M/aM, m where d = dim M . Hence, by using part(i) for Hna (M ), we have Gid R Hdm (M ) < ∞. Now, we proceed by induction on d to show that Gid R M is finite. The case d = 0 is clear. Let d > 0 and assume that the result has been proved for d − 1. Suppose that x ∈ m is both R-regular and M -regular. Then one can use the induced exact sequence d d 0 −→ Hd−1 m (M/xM ) −→ Hm (M ) −→ Hm (M ) −→ 0 and [3, Proposition 3.9] to see that Gid R (Hd−1 m (M/xM )) is finite. Hence, by inductive hypothesis, Gid R M/xM is finite. Therefore, since, in view of [9, Corollary 6.2], R admits a canonical module, one can use the same argument as in the proof of 3.7(i)⇒(ii) to deduce that Gid R/xR M/xM < ∞. Therefore Gid R M is finite by 3.7. Now the result follows by induction. For the final assertion, let M be non-zero finitely generated with Gid R M = s < ∞. Then, by part(i), we have Gid R Hna (M ) ≤ s − n. If Gid R Hna (M ) < s − n, Then, in view s−n of [3, Theorem 3.6], we deduce that Ext R (E(k), Hna (M )) = 0. Hence, by Proposition 2.1, Ext sR (E(k), M ) = 0 which is a contradiction by 3.6. Therefore, Gid R Hna (M ) = Gid R M − n. Let (R, m) be a local ring. As a main theorem, it was proved in [15, Theorem 3.10] that if R and M are Cohen-Macaulay with dim M = n and Gid R Hnm (M ) < ∞, then Gid R Hnm (M ) = dim R − n. Notice that if Gid R Hnm (M ) < ∞, then, in view of [15, Lemma 3.6], 3.8(ii) and [3, Theorem 3.24], we have depth R = Gid R M . Therefore, the next corollary, which is established without the assumption that Gid R Hnm (M ) < ∞, recovers [15, Theorem 3.10]. Another improvement of the above result will be given in 3.12. 10 M.R. ZARGAR AND H. ZAKERI Corollary 3.9. Let (R, m) be a Cohen-Macaulay local ring and let M be Cohen-Macaulay of dimension n. Then Gid R Hnm (M ) = Gid R M − n. b is a Cohen-Macaulay R-module b of dimension n. By Proof. First we notice that M ⊗R R n b using [15, Lemma 3.6] and [1, Exercise 8.2.4], we have Gid R H (M ) = Gid b Hnb (M ⊗R R). m R m c are simultaneously finite. Therefore Also, in view of [3, Theorem 3.24], Gid R M and Gid Rb M we may assume that R is complete; and hence it has a dualizing complex. Now, one can use 3.8 to obtain the assertion. In [21, Theorem 2.6] a characterization of a complete Gorenstein local ring R, in terms of Gorenstein injectivity of the top local cohomology module of R, is given. The next corollary together with 2.7 recover that characterization. Corollary 3.10. Let (R, m) be a Cohen-Macaulay local ring which has a dualizing complex. Then the following conditions are equivalent. (i) R is Gorenstein. (ii) Gid R Hna (R) < ∞ for any ideal a of R such that R is relative Cohen-Macaulay with respect to a and that ht R a = n. (iii) Gid R Hna (R) < ∞ for some ideal a of R such that R is relative Cohen-Macaulay with respect to a and that ht R a = n. Proof. The implication (i)⇒(ii) follows from 2.7 and the implication (ii)⇒(iii) is clear. The implication (iii)⇒(i) follows from 3.8(ii) and [3, Proposition 3.11]. Concerning the above corollary, we notice that if Hna (R) is Artinian, then it is not needed to impose the hypothesis that R has a dualizing complex. Therefore [21, Theorem 2.6] follows from 3.10 without the completeness assumption on R. Next, as promised before, we provide examples to show that if M is not finitely generated or M is not relative Cohen-Macaulay, then 2.5(ii) and 2.5(i), respectively, are no longer true. Examples 3.11. (i). Let (R, m) be a Gorenstein local ring with dim R ≥ 2 such that Rp is not regular for some non-maximal prime ideal p of R (for example, one can take R= K[[X,Y,Z]] (X 2 ) and p = (x, y)R, where K is a field). Then one can use [3, Theorem 3.14] to see that there exists a non-zero Gorenstein injective Rp -module Mp which is neither injective nor finitely generated. Hence, by [4, Proposition 10.1.2 ], injdimRp Mp = ∞; so that injdimR Mp = ∞. Now, we notice that, for all x ∈ m − p, injdimR Mx = ∞ because (Mx )pRx ∼ = Mp . It is easy to check that Him (Mx ) = 0 for all i and that Mx is not finitely generated as an R-module. Set N = Mx ⊕ ER (k). Then injdimN = ∞, but Γm (N ) is injective. This example shows that, in 2.5(ii), the finiteness assumption on M is required. (ii). Let R = k[[x, y]]/(xy), where k is a field. Then R is a 1-dimensional complete Gorenstein local ring. Let m be the maximal ideal of R and let J = (y)R. In view of [1, Theorem 8.2.1] and [21, Corollary 2.10], H1J (R) is a non-zero Gorenstein injective R-module. Note that ΓJ (R) 6= 0. Now, we show that H1J (R) is not injective. If H1J (R) were injective, then Hom R (H1J (R), ER (R/m)) = Rn for some positive integer n. Therefore, by using [19, INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 11 Theorem 5.11] and [21, Lemma 3.1], we get an R-isomorphism ψ : Rn = Hom R (H1J (R), ER (R/m)) → Γm,J (R) = J. Now, ψ(xRn ) = xJ = 0 which is a contradiction. Hence, by [4, Proposition 10.1.2], we have injdimH1J (R) = ∞. Therefore, by using the exact sequences 0 −→ ΓJ (R) −→ R −→ R/ΓJ (R) −→ 0 and 0 −→ R/ΓJ (R) −→ Ry+(xy) −→ H1J (R) −→ 0, we achieve injdimΓJ (R) = ∞. As a mentioned just above 3.9, the next theorem is an improvement of [15, Theorem 3.10]. Indeed, we will use the canonical module of a module to prove the above result without assuming that R and M are Cohen-Macaulay. Notice that if M is Cohen-Macaulay, then KM is Cohen-Macaulay. But the converse does not hold in general; see for example [16, Lemma 1.9] and [16, Theorem 1.14]. Theorem 3.12. Assume that (R, m) is local, and M is non-zero finitely generated of dimension d. Then the following statements hold. (i) Gid R Hda (M ) = Gpd Rb ΓmR,a b R b (KM c). d d (ii) If KM c is Cohen-Macaulay and Gid R Hm (M ) < ∞, then Gid R Hm (M ) = depth R−d. Proof. (i) By [1, Theorem 7.1.6], Hda (M ) is Artinian. Therefore, by use of [1, Theorem 4.3.2] and [19, Theorem 5.11], we have Hda (M ) = 0 if and only if ΓmR,a b R b (KM c) = 0. Hence, we may assume that Hda (M ) 6= 0. Now, by [15, Lemma 3.6], Gid R Hda (M ) = Gid Rb Hda (M ) and, by [3, Theorem 4.25], Gid Rb Hda (M ) = Gfd Rb Hom Rb (Hda (M ), ER (k)). Therefore, since b Hom b (Hd (M ), ER (k)) is finitely generated as an R-module, one can use [3, Theorem 4.24] R a and [19, Theorem 5.11] to establish the result. d (ii) First notice that ΓmR,m b R b (KM b (KM c) = KM c. Hence, by part(i), Gid R Hm (M ) = Gpd R c). d b− Therefore, in view of [3, Proposition 2.16] and [3, Theorem 1.25], Gid R H (M ) = depth R m depth KM c. Now, one can use [16, Lemma 1.9(c)] to complete the proof. The following corollary is a generalization of the main result [21, Theorem 2.6]. Corollary 3.13. Assume that (R, m) is local with dim R = d and that KRb is CohenMacaulay. Then the following statements are equivalent. (i) R is Gorenstein. (ii) injdimR Hdm (R) < ∞. (iii) Gid R Hdm (R) < ∞. Proof. The implication (i)⇒(ii) follows from 2.5 while the implication (ii)⇒(iii) is clear. (iii)⇒(i). By 3.12(ii), we have Gid R Hdm (R) = depth R − dim R; and hence R is CohenMacaulay. Now one can use 3.9 to obtain the assertion. 12 M.R. ZARGAR AND H. ZAKERI Corollary 3.14. Let (R, m) be local with dim R = d ≤ 2. Then the following statements are equivalent. (i) R is Gorenstein. (ii) Gid R Hdm (R) < ∞. (iii) Hda (M ) is Gorenstein injective for all finitely generated R–modules M and for all ideals a of R. Proof. Let M be a non-zero finitely generated R-module. Then, by [1, Theorem 7.1.6], Hda (M ) and Hdm (R) are Artinian. Therefore, in view of [15, Lemma 3.5], we may assume that R is complete. Since, by [16, Lemma 1.9], KR is Cohen-Macaulay, (i)⇔(ii) follows immediately from 3.13. The implication (iii)⇒(i) is clear and the implication (i)⇒(iii) follows from [21, Corollary 2.10]. Acknowledgements. The authors would like to thank Alberto Fernandez Boix and the referee for careful reading of manuscript and helpful comments. References [1] M.P. Brodmann and R.Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998. [2] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993. [3] L.W. Christensen, H-B. Foxby, and H. Holm, Beyond totally reflexive modules and back, In: Noetherian and Non-Noetherian Perspectives, edited by M. Fontana, S-E. Kabbaj, B. Olberding and I. Swanson, Springer Science+Business Media, LLC, New York, 2011, 101–143. [4] E.E. Enochs and O.M.G. Jenda, Relative homological algebra, de Gruyter, Berlin, 2000. [5] E.E. Enochs and O.M.G. Jenda, Gorenstein injective dimension and Tor-depth of modules, Arch. Math, 72 (1999) 107–117. [6] R. Fossum, H-B. Foxby, P. Griffith and I. Reiten, Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, Publ. Math. I.H.E.S, 45 (1975) 193–215. [7] M. Hellus and P. Schenzel, On cohomologically complete intersections, Journal of Algebra, 320 (2008) 3733–3748. [8] M. Jahangiri, and A. Rahimi, Relative Cohen-Macaulayness and relative unmixedness of bigraded modules, to appear in Commutative Algebra. [9] T. Kawasaki, On Macaulayfication of Noetherian schemes, Trans. Amer. Math. Soc., 352 (2000) 2517– 2552. [10] B. Kubik, M.J. Leamerb and S. Wagstaff, Homology of artinian and Matlis reflexive modules, I, J. Pure Appl. Algebra, 215 (2011) 2486–2503. [11] H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1992. [12] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S. 42 (1973) 47–119. [13] P. Roberts, Le theoreme d’intersection, C. R. Acad. Sci. Paris Ser. I, 304 (1987) 177–180. [14] J.J. Rotman, An introduction to homological algebra, Second ed., Springer, New York, 2009. [15] R. Sazeedeh, Gorenstein injective of the section functor, Forum Mathematicum, 22 (2010) 1117–1127. [16] P. Schenzel, On the use of local cohomoloy in algebra and geometry, in: Lectures at the Summmer school of Commutative Algebra and Algebraic Geometry, Birkhä, Basel, Ballaterra, 1996. [17] R.Y. Sharp, Steps in commutative algebra, Second Edition, Cambridge University Press, Cambridge, 2000. INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 13 [18] R.Y. Sharp, Gorenstein Modules, Mathematische Zeitschrift, 115 (1970) 117–139. [19] R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra, 213 (2009) 582–600. [20] S. Yassemi, A generalization of a theorem of Bass, Comm. Algebra, 35 (2007) 249–251. [21] T. Yoshizawa, On Gorenstein injective of top local cohomology modules, Proc. Amer. Math. Soc, 140 (2012) 1897–1907. M.R. Zargar and H. Zakeri, Faculty of mathematical sciences and computer, Kharazmi University, 599 Taleghani Avenue, Tehran 15618, Iran E-mail address: zargar9077@gmail.com E-mail address: zakeri@tmu.ac.ir M.R. Zargar, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395–5746, Tehran, Iran.	0
Distributed Compression of Graphical Data Payam Delgosha and Venkat Anantharam arXiv:1802.07446v1 [cs.IT] 21 Feb 2018 Department of Electrical Engineering and Computer Sciences University of California, Berkeley {pdelgosha, ananth} @ berkeley.edu February 22, 2018 Abstract In contrast to time series, graphical data is data indexed by the nodes and edges of a graph. Modern applications such as the internet, social networks, genomics and proteomics generate graphical data, often at large scale. The large scale argues for the need to compress such data for storage and subsequent processing. Since this data might have several components available in different locations, it is also important to study distributed compression of graphical data. In this paper, we derive a rate region for this problem which is a counterpart of the Slepian–Wolf Theorem. We characterize the rate region when the statistical description of the distributed graphical data is one of two types – a marked sparse Erdős–Rényi ensemble or a marked configuration model. Our results are in terms of a generalization of the notion of entropy introduced by Bordenave and Caputo in the study of local weak limits of sparse graphs. 1 Introduction Nowadays, storing combinatorically structured data is of great importance in many applications such as the internet, social networks and biological data. For instance, a social network could be presented as a graph where each node models an individual and each edge stands for a friendship. Also, vertices and edges can carry marks, e.g. the mark of a vertex represents its type, and the mark of an edge represents its shared information. Due to the sheer amount of such data, compressing it has drawn attention, see e.g. [CS12], [Abb16], [DA17]. As the data is not always available in one location, it is important to also consider distributed compression of graphical data. Traditionally, distributed lossless compression is modeled using two (or more) correlated stationary and ergodic processes representing the components of the data at the individual locations. In this case, the rate region is given by the Slepian–Wolf Theorem [CT12]. We adopt an analogous framework, namely that two correlated marked random graphs on the same vertex set are presented to two encoders which then individually compress their data such that a third party can recover both realizations from the two compressed representations, with a vanishing probability of error in the asymptotic limit of data size. We characterize the compression rate region for two scenarios, namely, a marked sparse Erdős–Rényi ensemble and a marked configuration model. We employ the framework of local weak convergence, also called the objective method, as a counterpart for marked graphs of the notion of stochastic processes [BS01, AS04, AL07]. Our characterization is best understood in terms of a generalization of a measure of entropy introduced by Bordenave and Caputo, which we call the BC entropy [BC14]. It turns out that the BC entropy captures the per–vertex growth rate of the Shannon entropy for the ensembles 1 we study in this paper. This motivates it as a natural measure governing the asymptotic compression bounds. The paper is organized as follows. In Section 2 we introduce the notation and formally state the problem. Sections 3 and 4 give a brief introduction to the objective method and the BC entropy, mostly specialized for the examples we study. Finally, in Section 5, we characterize the rate region for the scenarios we present in Section 2. 2 Notations and Problem Statement The set of real numbers is denoted by R. For an integer n, [n] denotes the set {1, 2, . . . , n}. For a probability distribution P , H(P ) denotes its Shannon entropy. Also, for a random variable X, we denote by H(X) its P Shannon entropy. For a positive integer N and a sequence of positive integers {ai }1≤i≤k such that ai ≤ N , we define N N! := . {ai }1≤i≤k a1 ! . . . ak !(N − a1 − · · · − ak )! For sequences of reals an and bn we write an = O(bn ) if, for some constant C ≥ 0, we have \|an \| ≤ C\|bn \| for n large enough. Furthermore, we write an = o(bn ) if an /bn → 0 as n → ∞. We denote by 1 [A] the indicator of the event A. For a probability distribution P , X ∼ P denotes that the random variable X has law P . Throughout the paper logarithms are to the natural base. A marked graph with edge mark set Ξ and vertex mark set Θ is a graph where each edge carries a mark in Ξ and each vertex carries a mark in Θ. We assume that all graphs are simple unless otherwise stated. Also, we assume that all edge and vertex mark sets are finite. For two vertices v and w in a graph G, v ∼G w denotes that v and w are adjacent in G. Let G be a marked graph on a finite vertex set with edges and vertices carrying marks in the sets Ξ and Θ, respectively. We denote the edge mark count vector of G by m ~ G = {mG (x)}x∈Ξ , where mG (x) is the number of edges in G carrying mark x. Furthermore, we denote the vertex mark count vector of G by ~uG = {uG (θ)}θ∈Θ , where uG (θ) denotes the number of vertices in G with mark θ. Additionally, − → for a graph G on the vertex set [n], we denote the degree sequence of G by dgG = {dgG (1), . . . , dgG (n)} where dgG (i) denotes the degree of vertex i. For a degree sequence d~ = (d(1), . . . , d(n)) and an integer k, we define ~ := \|{1 ≤ i ≤ n : d(i) = k}. ck (d) (1) Also, for two degree sequences d~ = (d(1), . . . , d(n)) and d~′ = (d′ (1), . . . , d′ (n)), and two integers k and l, we define ~ d~′ ) := \|{1 ≤ i ≤ n : d(i) = k, d′ (i) = l}\|. ck,l (d, (2) (n) Given a degree sequence d~ = (d(1), . . . , d(n)), we let G ~ denote the set of simple unmarked graphs G d on the vertex set [n] such that dgG (i) = d(i) for 1 ≤ i ≤ n. Throughout this paper, we assume that Ξ1 and Ξ2 are two fixed and finite sets of edge marks. Moreover, Θ1 and Θ2 are two fixed and finite vertex mark sets. For i ∈ {1, 2} and an integer n, let (n) Gi be the set of marked graphs on the vertex set [n] with edge and vertex mark sets Ξi and Θi , (n) (n) respectively. For two graphs G1 ∈ G1 and G2 ∈ G2 , G1 ⊕ G2 denotes the superposition of G1 and G2 which is a marked graph defined as follows: a vertex 1 ≤ v ≤ n in G1 ⊕ G2 carries the mark (θ1 , θ2 ) where θi is the mark of v in Gi . Furthermore, we place an edge in G1 ⊕ G2 between vertices v and w if there is an edge between them in at least one of G1 of G2 , and mark this edge (x1 , x2 ), where, for 1 ≤ i ≤ 2, xi is the mark of the edge (v, w) in Gi if it exists and ◦i otherwise. Here ◦1 and ◦2 are auxiliary marks not present in Ξ1 ∪ Ξ2 . Note that G1 ⊕ G2 is a marked graph with edge and vertex 2 mark sets Ξ1,2 := (Ξ1 ∪ {◦1 }) × (Ξ2 ∪ {◦2 }) \ {(◦1 , ◦2 )} and Θ1,2 := Θ1 × Θ2 , respectively. We use the terminology jointly marked graph to refer to a marked graph with edge and vertex makr sets Ξ1,2 and (n) Θ1,2 respectively. With this, let G1,2 denote the set of jointly marked graphs on the vertex set [n]. Moreover, for i ∈ {1, 2}, we say that a graph is in the i–th domain if edge and vertex marks come from Ξi and Θi , respectively. For a jointly marked graph G1,2 and 1 ≤ i ≤ 2, the i–th marginal of G1,2 , denoted by Gi , is the marked graph in the i–th domain obtained by projecting all vertex and edge marks onto Ξi and Θi , respectively, followed by removing edges with mark ◦i . Note that any jointly marked graph G1,2 is uniquely determined by its marginals G1 and G2 , because G1,2 = G1 ⊕ G2 . Given an edge mark count m ~ = {m(x)}x∈Ξ1,2 , for x1 ∈ Ξ1 ∪ {◦1 }, with an abuse of notation we define X m(x1 ) := (x′1 ,x′2 )∈Ξ1,2 m((x′1 , x′2 )). (3) : x′1 =x1 In a similar fashion, we define m(x2 ) for x2 ∈ Ξ2 ∪ {◦2 }. Likewise, given a vertex mark count vector ~u = {u(θ)}θ∈Θ1,2 , and for θ1 ∈ Θ1 and θ2 ∈ Θ2 , we define u(θ1 ) = X u((θ1 , θ2′ )) u(θ2 ) = X u((θ1′ , θ2 )). (4) θ1′ ∈Θ1 θ2′ ∈Θ2 (n) (n) Assume that we have a sequence of random graphs G1,2 ∈ G1,2 , drawn according to some ensemble distribution. Additionally, assume that there are two encoders who want to compress realizations of such jointly marked graphs in a distributed fashion. Namely, the i–th encoder, 1 ≤ i ≤ 2, has only (n) (n) access to the i–th marginal Gi . We assume that the encoders know the distribution of G1,2 . (n) (n) (n) (n) Definition 1. An hn, L1 , L2 i code is a tuple of functions (f1 , f2 , g (n) ) for each n such that (n) fi (n) : Gi and (n) → [Li ] (n) (n) i ∈ {1, 2}, (n) g (n) : [L1 ] × [L2 ] → G1,2 . (n) (n) The probability of error for this code corresponding to the ensemble of G1,2 , which is denoted by Pe , is defined as (n) (n) (n) (n) (n) Pe(n) := P g (n) (f1 (G1 ), f2 (G2 )) 6= G1,2 . Now we define our achievability criterion. Definition 2. A rate tuple (α1 , R1 , α2 , R2 ) ∈ R4 is said to be achievable for the above scenario if there (n) (n) is a sequence of hn, L1 , L2 i codes such that (n) lim sup n→∞ log Li − (αi n log n + Ri n) ≤0 n i ∈ {1, 2}, (5) (n) and also Pe → 0. The rate region R ∈ R4 is defined as follows: for fixed α1 and α2 , if there are (m) (m) sequences R1 and R2 with limit points R1 and R2 in R, respectively, such that for each m, the (m) (m) rate tuple (α1 , R1 , α2 , R2 ) is achievable, then we include (α1 , R1 , α2 , R2 ) in the set R. In this paper, we characterize the above rate region for the following two sequences of ensembles: 3 The Erdős–Rényi ensemble: Assume that nonnegative real numbers ~p = {px }x∈Ξ1,2 together with a probability distribution ~ q = {qθ }θ∈Θ1,2 are given such that for all x1 ∈ Ξ1 ∪ {◦1 } and x2 ∈ Ξ2 ∪ {◦2 }, we have X X p(x′1 ,x′2 ) > 0. (6) p(x′1 ,x′2 ) > 0 and (x′ ,x′ )∈Ξ1,2 1 2 x′2 =x2 (x′ ,x′ )∈Ξ1,2 1 2 x′1 =x1 (n) For an integer n large enough, we define the probability distribution G(n; p~, ~q) on G1,2 as follows: for each pair of vertices 1 ≤ i < j ≤ n, the edge (i, j) is present Pin the graph and has mark x ∈ Ξ1,2 with probability px /n, and is not present with probability 1 − x∈Ξ1,2 px /n. Furthermore, each vertex in the graph is given a mark θ ∈ Θ1,2 with probability qθ . The choice of edge and vertex marks is done independently. The configuration model ensemble: Assume that a fixed integer ∆ > 0 and a probability distribution ~r = {rk }∆ k=0 supported on the set {0, . . . , ∆} are given, such that r0 < 1. Moreover, assume that probability distributions ~γ = {γx }x∈Ξ1,2 and ~q = {qθ }θ∈Θ1,2 on the sets Ξ1,2 and Θ1,2 , respectively, are given. We assume that for all x1 ∈ Ξ1 ∪ {◦1 } and x2 ∈ Ξ2 ∪ {◦2 }, we have X X γ(x′1 ,x′2 ) > 0. (7) γ(x′1 ,x′2 ) > 0 and (x′1 ,x′2 )∈Ξ1,2 x′2 =x2 (x′1 ,x′2 )∈Ξ1,2 x′1 =x1 Furthermore, let d~(n) = {d(n) (1), . . . , d(n) (n)} be P a sequence of degree sequences such Pnthat for all n n and 1 ≤ i ≤ n, we have d(n) (i) ≤ ∆ and also i=1 d(n) (i) is even. Let mn := ( i=1 d(n) (i))/2. Additionally, if for 0 ≤ k ≤ ∆, ck (d~(n) ) denotes the number of 1 ≤ i ≤ n such that d(n) (i) = k, we assume that for some constant K > 0, ∆ X \|ck (d~(n) ) − nrk \| ≤ Kn1/2 . (8) k=0 (n) Now, we define the law G(n; d~(n) , ~γ , ~ q ) on G1,2 for n large enough as follows. First, we pick an unmarked graph on the vertex set [n] uniformly at random among the set of graphs G with maximum degree ∆ − → such that for each 0 ≤ k ≤ ∆, ck (dgG ) = ck (d~(n) ).1 Then, we assign i.i.d. marks with law ~γ on the edges and i.i.d. marks with law ~q on the vertices. As we will discuss in Section 3 below, the sequence of Erdős–Rényi ensembles defined above converges in the local weak sense to a marked Poisson Galton Watson tree. Moreover, the configuration model ensemble converges in the same sense to a marked Galton Watson process with degree distribution ~r. We will characterize the achievability rate regions in Section 5 in terms of these limiting objects for the above two sequences of ensembles. This will turn out to be best understood in terms of a measure of entropy discussed in Section 4 below. 3 The framework of Local Weak Convergence In this section, we discuss the framework of local weak convergence mainly in the context of the Erdős– Rényi and configuration model ensembles discussed in Section 2. For a general discussion, the reader is referred to [BS01, AS04, AL07]. 1 The fact that each degree is bounded to ∆, r < 1 and the sum of degrees is even implies that d ~(n) is a graphic 0 sequence for n large enough. This is, for instance, a consequence of Theorem 4.5 in [BC14]. 4 Let Ξ and Θ be finite mark sets. A marked graph G with edge and vertex mark sets Ξ and Θ respectively together with a distinguished vertex o, is called a rooted marked graph and is denoted by (G, o). For a rooted marked graph (G, o) and integer h ≥ 1, (G, o)h denotes the h neighborhood of o, i.e. the subgraph consisting of vertices with distance no more than h from o. Note that (G, o)h is connected by definition. Two connected rooted marked graphs (G1 , o1 ) and (G2 , o2 ) are said to be isomorphic if there is a vertex bijection between the two graphs that maps o1 to o2 , preserves adjacencies and also preserves vertex and edge marks. With this, we denote the isomorphism class corresponding to a rooted marked graph (G, o) by [G, o]. We simply use [G, o]h as a shorthand for [(G, o)h ]. Let G∗ (Ξ, Θ) denote the set of isomorphism classes [G, o] of connected rooted marked graphs on a countable vertex set with edge and vertex marks coming from the sets Ξ and Θ, respectively. It can be shown that G∗ (Ξ, Θ) can be turned into a separable and complete metric space [AL07]. For a probability distribution µ on G∗ (Ξ, Θ), let deg(µ) denote the expected degree at the root in µ. For a finite marked graph G and a vertex v in G, let G(v) denote the connected component of v. With this, if v is a vertex chosen uniformly at random in G, we define U (G) be the law of [G(v), v], which is a probability distribution on G∗ (Ξ, Θ). (n) Let G1,2 be a random jointly marked graph with law G(n; p~, ~q) and let vn be a vertex chosen uniformly at random in the set [n]. A simple Poisson approximation implies that Dx (vn ), the number of edges adjacent to vn with mark x ∈ Ξ1,2 , converges in distribution to a Poisson random variables with mean px as n goes to infinity. Moreover, {Dx (vn )}x∈Ξ1,2 are asymptotically mutually independent. A similar argument can be repeated for any other vertex in the neighborhood of vn . Also, it can be (n) shown that the probability of having cycles converges to zero. In fact, the structure of (G1,2 , vn )h converges in distribution to a rooted marked Poisson Galton Watson tree with depth h. ER More precisely, let (T1,2 , o) be a rooted jointly marked tree defined as follows. First, the mark of the root is chosen from distribution ~ q . Then, for x ∈ Ξ1,2 , we independently generate Dx with law Poisson(px ). We then add Dx many edges with mark x to the root o. For each offspring, we repeat the same procedure independently, i.e. choose its mark and edges with each mark from the corresponding ER Poisson distribution. Recursively repeating this, we get a connected jointly marked tree T1,2 rooted at ER o, which has possibly countably infinitely many vertices. Let µ1,2 denote the law of the isomorphism ER class [T1,2 , o]. Note that µER 1,2 is a probability distribution on G∗ (Ξ1,2 , Θ1,2 ). The above discussion (n) ER implies that [G1,2 , vn ]h converges in distribution to [T1,2 , o]h . In fact, even a stronger statement can (n) be proved. More precisely, if we consider the sequence of random graphs G1,2 independently on a (n) joint probability space, U (G1,2 ) converges weakly to µER 1,2 with probability one. With this, we say (n) that, almost surely, µER 1,2 is the local weak limit of the sequence G1,2 , where the term “local” stands for looking at a fixed depth neighborhood of a typical node. ER With the above construction, for 1 ≤ i ≤ 2, let TiER be the i–th marginal of T1,2 . Moreover, let ER ER ER µi be the law of [Ti (o), o]. Therefore, µi is a probability distribution on G∗ (Ξi , Θi ). Similarly, (n) one can see that, almost surely, µER is the local weak limit of the sequence Gi . i CM A similar picture also holds for the configuration model. More precisely, let (T1,2 , o) be a rooted jointly marked random tree constructed as follows. First, we generate the degree of the root with law ~r. Then, for each offspring w of o, we independently generate the offspring count of w with law r~′ = {rk′ }∆−1 k=0 defined as (k + 1)rk+1 , 0 ≤ k ≤ ∆ − 1, rk′ = E [X] where X has law ~r. We continue this process recursively, i.e. for each vertex other than the root, we independently generate its offspring count with law r′ . The distribution ~r′ is called the sized biased distribution, and takes into account the fact that each node other than the root has an extra edge on 5 top of it, and hence its degree should be biased in order to get the correct degree distribution ~r. Then, CM for each vertex and edge existing in the graph T1,2 , we generate marks independently with laws ~q CM CM and ~γ , respectively. Let µ1,2 be the law of [T1,2 , o]. Moreover, for 1 ≤ i ≤ 2, let µCM be the law of i (n) CM (n) CM ~ [T (o), o]. It can be shown that if G has law G(n; d , ~γ , ~q) then, almost surely, µ is the local i 1,2 1,2 (n) G1,2 , (n) Gi , µCM i for 1 ≤ i ≤ 2. weak limit of and is the local weak limit of Given Ξ and Θ, not all probability distributions on G∗ (Ξ, Θ) can appear as the local weak limit of a sequence. In fact, the condition that all vertices have the same chance of being chosen as the root for a finite graph manifests itself as a certain stationarity condition at the limit called unimodularity [AL07]. 4 The BC entropy In this section, we discuss a notion of entropy for probability distributions on a space of rooted marked graphs. This is a marked version of the entropy defined by Bordenave and Caputo in [BC14], which was defined for probability distributions on the space of rooted (unmarked) graphs. To distinguish it from the Shannon entropy, we call this notion of entropy the BC entropy. Let Ξ and Θ be finite mark sets and let µ be a probability distribution on G∗ (Ξ, Θ). Moreover, let m ~ (n) and ~u(n) be sequences of edge and vertex mark counts, respectively, such that for all x ∈ Ξ, (n) 2m (x)/n converges to the expected number of edges with mark x connected to the root in µ, and for all θ ∈ Θ, u(n) (θ)/n converges to the probability of the mark of the root in µ being θ. Let (n) Gm (µ, ǫ) be the set of graphs G on the vertex set [n] with edge and vertex marks in Ξ and Θ, ~ (n) ,~ u(n) respectively, such that m ~G =m ~ (n) , ~uG = ~u(n) , and U (G) is in the ball around µ with radius ǫ with respect to the Lévy–Prokhorov distance [Bil13]. P Definition 3. If an := x∈Ξ m(n) (x), define (n) Σ(µ, ǫ) := lim sup (µ, ǫ)\| − an log n log \|Gm ~ (n) ,~ u(n) n n→∞ Σ(µ, ǫ) := lim inf (n) (µ, ǫ)\| log \|Gm ~ (n) ,~ u(n) − an log n n n→∞ Note that both Σ(µ, ǫ) and Σ(µ, ǫ) decrease as ǫ decreases. Therefore, we may define the upper and lower BC entropies as Σ(µ) := limǫ↓0 Σ(µ, ǫ) and Σ(µ) := limǫ↓0 Σ(µ, ǫ). If Σ(µ) = Σ(µ), we denote the common value by Σ(µ) and call it the BC entropy of µ. Using similar techniques as in the proof of Theorem 1.2 in [BC14], one can show that Σ(µ) and ~ (n) and ~u(n) , and Σ(µ) = Σ(µ) for all µ Σ(µ) do not depend on the specific choice of the sequences m with positive expected degree of the root. Now, we connect the asymptotic behavior of the entropy of the ensembles defined in Section 2 to (n) ER ER the P BC entropy of their local weak limits. Assume that G1,2 has law G(n; p~, ~q). Let d1,2 := deg(µ1,2 ) = x∈Ξ1,2 px . Moreover, we use the following notational conventions for xi ∈ Ξi and θi ∈ Θi , 1 ≤ i ≤ 2. px1 := X p(x1 ,x′2 ) px2 := x′2 ∈Ξ2 ∪{◦2 } qθ1 := X X p(x′1 ,x2 ) x′1 ∈Ξ1 ∪{◦1 } qθ2 := q(θ1 ,θ2′ ) X θ1′ ∈Θ1 θ2′ ∈Θ2 6 q(θ1′ ,θ2 ) (9) P q , it can be easily verified For 1 ≤ i ≤ 2, let dER := deg(µER i i ) = xi ∈Ξi pxi . If Q = (Q1 , Q2 ) has law ~ that with s(x) defined as x2 − x2 log x for x > 0 and zero for x = 0, we have   ER X d 1,2 (n) n log n + n H(Q) + s(px ) + o(n) H(G1,2 ) = 2 x∈Ξ1,2 ! X dER (n) 1 s(px1 ) + o(n) H(G1 ) = n log n + n H(Q1 ) + 2 x1 ∈Ξ1 ! X dER (n) 2 n log n + n H(Q2 ) + H(G2 ) = s(px2 ) + o(n) 2 x ∈Ξ 2 (10a) (10b) (10c) 2 Using a generalization of Theorem 1.3 in [BC14], it can be seen that the coefficient of n in the above ER ER 3 equations are Σ(µER 1,2 ), Σ(µ1 ) and Σ(µ2 ), respectively. (n) Similarly, for the configuration model, let G be distributed according to G(n; d~(n) , ~γ , ~q) and let 1,2 X be a random variable with law ~r. Moreover, let Γk = (Γk1 , Γk2 ), 1 ≤ k ≤ ∆, be an i.i.d. sequence distributed according to ~γ . With this, let X1 := X X 1 Γk1 6= ◦1 k=1 X2 := X X 1 Γk2 6= ◦2 . (11) k=1 Note that for 1 ≤ i ≤ 2, Xi is basically the distribution of the degree of the root in µCM . If dCM 1,2 := i CM CM CM deg(µ1,2 ) and, for 1 ≤ i ≤ 2, di := deg(µi ), it can be seen that (see Appendix A for the details) dCM 1,2 n log n + n − s(dCM 1,2 ) + H(X) − E [log X!] 2 dCM 1,2 H(Γ) + o(n) + H(Q) + 2 CM d (n) H(G1 ) = 1 n log n + n − s(dCM 1 ) + H(X1 ) − E [log X1 !] 2 dCM + H(Q1 ) + 1 H(Γ1 \|Γ1 6= ◦1 ) + o(n) 2 dCM (n) H(G2 ) = 2 n log n + n − s(dCM 2 ) + H(X2 ) − E [log X2 !] 2 dCM + H(Q2 ) + 2 H(Γ2 \|Γ2 6= ◦2 ) + o(n) 2 (n) H(G1,2 ) = (12a) (12b) (12c) CM CM Also, it can be seen that the coefficients of n in the above equations are Σ(µCM 1,2 ), Σ(µ1 ) and Σ(µ2 ), respectively. CM If µ1,2 is any of the two distributions µER 1,2 or µ1,2 , and µ1 and µ2 are its marginals, we define the conditional BC entropies as Σ(µ2 \|µ1 ) := Σ(µ1,2 ) − Σ(µ1 ) and Σ(µ1 \|µ2 ) := Σ(µ1,2 ) − Σ(µ2 ). 5 Main Results Now, we are ready to state our main result, which is to characterize the rate region in Definition 2. In the following, for pairs of reals (α, R) and (α′ , R′ ), we write (α, R) ≻ (α′ , R′ ) if either α > α′ , or α = α′ and R > R′ . We also write (α, R) (α′ , R′ ) if either (α, R) ≻ (α′ , R′ ) or (α, R) = (α′ , R′ ). 7 CM Theorem 1. Assume µ1,2 is either of the two distributions µER 1,2 or µ1,2 defined in Section 4. Then, if R is the rate region for the sequence of ensembles corresponding to µ1,2 defined in Section 2, a rate tuple (α1 , R1 , α2 , R2 ) ∈ R if and only if (α1 , R1 ) ((d1,2 − d2 )/2, Σ(µ1 \|µ2 )) (13a) (α2 , R2 ) ((d1,2 − d1 )/2, Σ(µ2 \|µ1 )) (13b) (α1 + α2 , R1 + R2 ) (d1,2 /2, Σ(µ1,2 )) (13c) where d1,2 = deg(µ1,2 ), d1 = deg(µ1 ) and d2 = deg(µ2 ). We prove the achievability for the Erdős–Rényi case and the configuration model in Sections 5.1 and 5.2, respectively. Afterwards, we prove the converse for the two cases in Sections 5.3 and 5.4, respectively. Before this, we state the following general lemma used in the proofs, whose proof is straightforward using Stirling’s approximation. Lemma 1. Assume that a positive integer k and sequences of integers an and bn1 , . . . , bnk are given. Pk 1. If an /n → a > 0 and for each 1 ≤ i ≤ k, bni /n → bi ≥ 0 where a = i=1 bi , we have ! bi an 1 . = aH log lim n→∞ n a 1≤i≤k {bni }1≤i≤k 2. If an / n 2 → 1 and bni /n → bi ≥ 0, we have an {bn i }1≤i≤k log lim n→∞ where s(x) is defined to be 5.1 x 2 − x 2 − P k n i=1 bi n log n = k X s(2bi ), i=1 log x for x > 0 and 0 if x = 0. Proof of Achievability for the Erdős–Rényi case Here we show that a rate tuple (α1 , R1 , α2 , R2 ) is achievable for the Erdős–Rényi ensemble if it satisfies the following ER ER ER (α1 , R1 ) ≻ ((dER 1,2 − d2 )/2, Σ(µ1 \|µ2 )) (14a) ER ER ER ((dER 1,2 − d1 )/2, Σ(µ2 \|µ1 )) ER (dER 1,2 /2, Σ(µ1,2 )) (14b) (α2 , R2 ) ≻ (α1 + α2 , R1 + R2 ) ≻ (14c) Note that if a rate tuple (α′1 , R1′ , α′2 , R2′ ) satisfies the weak inequalities (13a)–(13c) then, for any ǫ > 0, (α′1 , R1′ + ǫ, α′2 , R2′ + ǫ) satisfies the above strict inequalities. As we show below, this implies that (α′1 , R1′ + ǫ, α′2 , R2′ + ǫ) is achievable. Hence, after sending ǫ → 0, we get (α′1 , R1′ , α′2 , R2′ ) ∈ R. We show that any (α1 , R1 , α2 , R2 ) satisfying (14a)–(14c) is achievable by employing a random (n) binning method. More precisely, for i ∈ {1, 2}, we set Li = ⌊exp(αi n log n + Ri n)⌋ and for each (n) (n) (n) Gi ∈ Gi , we assign fi (Gi ) uniformly at random in the set [Li ] and independent of everything else. To describe our decoding scheme, we first need to define some notation. Let M(n) denote the set of edge count vectors m ~ = {m(x)}x∈Ξ1,2 such that X \|m(x) − npx /2\| ≤ n2/3 . x∈Ξ1,2 8 Moreover, let U (n) denote the set of vertex mark count vectors ~u = {u(θ)}θ∈Θ1,2 such that X \|u(θ) − nqθ \| ≤ n2/3 . θ∈Θ1,2 (n) (n) (n) Furthermore, we define Gp~,~q to be the set of graphs H1,2 ∈ G1,2 such that m ~ H (n) ∈ M(n) and 1,2 (n) (n) (n) (n) ~uH (n) ∈ U (n) . Upon receiving (i, j) ∈ [L1 ] × [L2 ], we form the set of graphs H1,2 ∈ Gp~,~q such that 1,2 (n) (n) (n) (n) (n) (n) (n) f1 (H1 ) = i and f2 (H2 ) = j, where H1 and H2 are the marginals of H1,2 . If this set has only one element, we output this element as the decoded graph; otherwise, we report an error. (n) In what follows, assume that G1,2 is a random graph with law G(n; p~, ~q). We consider the following four error events corresponding to the above scheme (n) := {G1,2 ∈ / Gp~,~q }, (n) := {∃H1,2 6= G1,2 : fi (Hi ) = fi (Gi ), i ∈ {1, 2}}, (n) := {∃H2 (n) := {∃H1 E1 E2 E3 E4 (n) (n) (n) (n) (n) (n) 6= G2 (n) 6= G1 (n) (n) : G1 ⊕ H2 (n) : H1 (n) (n) (n) (n) (n) ∈ Gp~,~q , f2 (H2 ) = f2 (G2 )}, (n) ∈ Gp~,~q , f1 (H1 ) = f1 (G1 )}. ⊕ G2 (n) (n) (n) (n) (n) (n) (n) (n) (n) (n) (n) Note that outside the above four events, the decoder successfully decodes the input graph G1,2 . (n) P(E1 ) Using Chebyshev’s inequality, for some κ > 0 we have as n goes to infinity. Moreover, using the union bound, we have ≤ κn−1/3 , which converges to zero (n) (n) ≤ P E2 \|Gp~,~q \| (n) (n) L1 L2 . (15) (n) (n) ~ H (n) and ~uH (n) are in the sets M(n) and Note that for each graph H1,2 ∈ Gp~,~q , the mark count vectors m 1,2 1,2 U (n) , respectively. Additionally, we have \|M(n) \| ≤ (2n2/3 )\|Ξ1,2 \| and \|U (n) \| ≤ (2n2/3 )\|Θ1,2 \| . Therefore, (n) \|Gp~,~q \| ≤ (2n2/3 )(\|Ξ1,2 \|+\|Θ1,2 \|) max A1 (m, ~ ~u), (16) (n) m∈M ~ u ~ ∈U (n) where A1 (m, ~ ~u) := n {u(θ)}θ∈Θ1,2 ! n 2 {m(x)}x∈Ξ1,2 ! . Now, let m ~ (n) and ~u(n) be sequences in M(n) and U (n) , respectively. Then, for all x ∈ Ξ1,2 and θ ∈ Θ1,2 , we have m(n) (x)/n → px /2 and u(n) (θ)/n → qθ . Thereby, using Lemma 1, we have P log A1 (m ~ (n) , ~u(n) ) − ( x∈Ξ1,2 m(n) (x)) log n lim n→∞ n X = H(~q) + s(px ) = Σ(µER 1,2 ). x∈Ξ1,2 Substituting this into (16) and using the fact that \|m(n) (x) − npx /2\| ≤ n2/3 , we have (n) lim n→∞ log \|Gp~,~q \| − n dER 1,2 2 n 9 log n = Σ(µER 1,2 ). (17) Substituting this into (15), we have lim sup 1 (n) log P E2 n (n) ≤ lim sup log \|Gp~,~q \| − n dER 1,2 2 log n − nΣ(µER 1,2 ) n dER n( 1,2 2 − α1 − α2 ) log n + n(Σ(µER 1,2 ) − R1 − R2 ) n (n) (n) n(α1 + α2 ) log n + n(R1 + R2 ) − log L1 L2 . + lim sup n + lim sup The first term is nonpositive due to (17), the second term is strictly negative due to the assumption (n) (n) (14c), and the third term is nonpositive due to our choice of L1 and L2 . Consequently, the RHS is (n) strictly negative, which implies that P(E2 ) → 0. (n) (n) (n) (n) (n) (n) Now, we show that P(E3 \ E1 ) vanishes. In order to do so, for H1 ∈ G1 , define S2 (H1 ) := (n) (n) (n) (n) (n) {H2 ∈ G2 : H1 ⊕ H2 ∈ Gp~,~q }. Using the union bound, we have (n) (n) ≤ P E3 \ E1 ≤ X (n) (n) (n) P(G1,2 = H1,2 ) (n) (n) L2 (n) H1,2 ∈Gp ~,~ q 1 (n) L2 max (n) (n) H1,2 ∈Gp ~,~ q (n) \|S2 (H1 )\| (18) (n) (n) \|S2 (H1 )\|. It can be shown that (See Appendix B) max (n) lim sup (n) (n) H1,2 ∈Gp ~,~ q (n) ER dER 1,2 − d1 log n 2 n n→∞ where H1 (n) log \|S2 (H1 )\| − n (19) ER ≤ Σ(µER 2 \|µ1 ), (n) is the first marginal of H1,2 . Substituting in (18), we get n 1 (n) (n) lim sup log P E3 \ E1 ≤ lim sup n ER dER 1,2 −d1 2 (n) ER log n + nΣ(µER 2 \|µ1 ) − log L2 n ER dER 1,2 −d1 2 ER − α2 ) log n + n(Σ(µER 2 \|µ1 ) − R2 ) n (n) nα2 log n + nR2 − log L2 . + lim sup n ≤ lim sup n( (20) Note that the first term is strictly negative due to the assumption (14b), while the second term (n) (n) (n) is nonpositive due to our way of choosing L2 . This means that P(E3 \ E1 ) goes to zero as n (n) (n) goes to infinity. Similarly, P(E4 \ E1 ) converges to zero. This means that there exists a sequence of deterministic codebooks with vanishing probability of error, which completes the proof of achievability. 5.2 Proof of Achievability for the Configuration model Our achievability proof for this case is very similar in nature to that for the Erdős–Rényi case, with the modifications discussed below. 10 ~ = ck (d~(n) ) Let D(n) be the set of degree sequences d~ with entries bounded by ∆ such that ck (d) (n) for all 0 ≤ k ≤ ∆. Moreover, redefine M to be the set of mark count vectors m ~ such that P P P 2/3 , where recall that mn = ( ni=1 d(n) (i))/2. We x∈Ξ1,2 \|m(x) − mn γx \| ≤ n x∈Ξ1,2 m(x) = mn and (n) use the same as in the previous section, i.e. the set of vertex mark count vectors ~u P definition for U such that θ∈Θ1,2 \|u(θ) − nqθ \| ≤ n2/3 . In what follows, let X be a random variable with law ~r, X1 and X2 defined as in (11) and Γ = (Γ1 , Γ2 ) a random variable with law ~γ . − → (n) (n) We define W (n) to be the set of graphs H1,2 ∈ G1,2 such that: (i) dgH (n) ∈ D(n) , (ii) m ~ H (n) ∈ M(n) , 1,2 1,2 (iii) ~uH (n) ∈ U (n) , (iv) for all 0 ≤ l ≤ k ≤ ∆, recalling the notation in (2), we have 1,2 − → − → \|ck,l (dgH (n) , dgH (n) ) − nP (X = k, X1 = l) \| ≤ n2/3 , 1,2 1 (21) and (v), for all 0 ≤ l ≤ k ≤ ∆, we have − → − → \|ck,l (dgH (n) , dgH (n) ) − nP (X = k, X2 = l) \| ≤ n2/3 . 1,2 2 (22) We employ a similar random binning framework as in Section 5.1. For decoding, upon receiving (n) (n) (n) (n) (n) a pair (i, j), we form the set of graphs H1,2 ∈ W (n) such that f1 (H1 ) = i and f2 (H2 ) = j. If this set has only one element, we output it as the source graph; otherwise, we output an indication of (n) error. In order to prove the achievability, we consider the four error events Ei , 1 ≤ i ≤ 4, exactly as (n) those in the previous section, with Gp~,~q being replaced with W (n) . (n) (n) It can be shown that if G ∼ G(n; d~(n) , ~γ , ~q), the probability of G ∈ W (n) goes to one as n goes 1,2 1,2 (n) to infinity (see Lemma 4 in Appendix A). Therefore, P(E1 ) goes to zero. (n) To show that P(E2 ) vanishes, similar to the analysis in Section 5.1, we find an asymptotic upper bound for log \|W (n) \|. By only considering the conditions (i), (ii) and (iii) in the definition of W (n) , we have n (n) log \|W (n) \| ≤ log + log \|Gd~(n) \| {ck (d~(n) )}∆ k=0 mn 2/3 \|Ξ1,2 \| + log (2n ) max (23) (n) {m(x)}x∈Ξ1,2 m∈M ~ n . + log (2n2/3 )\|Θ1,2 \| max (n) {u(θ)} u ~ ∈U θ∈Θ1,2 By assumption, we have r0 < 1, hence dCM 1,2 > 0. The condition (8) together with Lemma 3 then implies that (n) d log \|G ~(n) \| − n dCM 1,2 2 log n = −s(dCM 1,2 ) − E [log X!] . n Using this together with Lemma 1 for the other terms in (23), we have lim n→∞ dCM log \|W (n) \| − n 1,2 2 log n ≤ −s(dCM lim sup 1,2 ) + H(X) n n→∞ dCM 1,2 + H(Γ) + H(Q) − E [log X!] = Σ(µCM 1,2 ), 2 where Γ and Q are random variables with law ~γ and ~q, respectively. 11 (24) (n) (n) (n) (n) Now, in order to show that P(E3 \E1 ) vanishes, we prove a counterpart for (19). For H1 ∈ G1 , (n) (n) (n) (n) (n) (n) we define S2 (H1 ) to be the set of graphs H2 ∈ G2 such that H1 ⊕ H2 ∈ W (n) . Then, it can be shown that (see Appendix C) (n) max (n) lim sup H1,2 ∈W (n) (n) log \|S2 (H1 )\| − n n n→∞ (n) Then, similar to (20), this shows that P(E3 This completes the proof of achievability. 5.3 CM dCM 1,2 − d1 log n 2 (25) CM ≤ Σ(µCM 2 \|µ1 ). (n) (n) \ E1 ) vanishes. The proof for P(E4 (n) \ E1 ) is similar. Proof of the Converse for the Erdős–Rényi case In this section, we show that every rate tuple (α1 , R1 , α2 , R2 ) ∈ R for the Erdős–Rényi scenario must satisfy the conditions (13a)–(13c). By definition, for a rate tuple (α1 , R1 , α2 , R2 ) ∈ R, there exist (m) (m) (m) (m) sequences R1 and R2 such that for each m, (α1 , R1 , α2 , R2 ) is achievable, and besides we have (m) (m) (m) (m) R1 → R1 and R2 → R2 . If we show that (α1 , R1 , α2 , R2 ) satisfies (13a)–(13c) for each m, it is easy to see that (α1 , R1 , α2 , R2 ) must also satisfy the same inequalities. Therefore, it suffices to show that any achievable rate tuple satisfies (13a)–(13c). For this, take an achievable rate tuple (α1 , R1 , α2 , R2 ) together with a corresponding sequence of (n) (n) (n) (n) hn, L1 , L2 i codes (f1 , f2 , g (n) ). By definition, we have (n) lim sup − (αi n log n + Ri n) ≤0 n log Li n→∞ i ∈ {1, 2}, (26) (n) (n) and also the error probability Pe goes to zero as n goes to infinity. Now, we define the set A(n) ⊆ G1,2 as (n) (n) (n) (n) (n) (n) (n) (n) A(n) := Gp~,~q ∩ {H1,2 ∈ G1,2 : g (n) (f1 (H1 ), f2 (H2 )) = H1,2 }, (27) (n) where Gp~,~q was defined in Section 5.1. In fact, A(n) is the set of “typical” graphs with respect to the (n) (n) Erdős–Rényi model that are successfully decoded by the code (f1 , f2 , g (n) ). In the following, let (n) (n) q ) be distributed according to the Erdős–Rényi model. Moreover, let PER be the law G1,2 ∼ G (n) (n; p~, ~ (n) (n) (n) (n) (n) (n) (n) of G1,2 , i.e. for H1,2 ∈ G1,2 , PER (H1,2 ) := P(G1,2 = H1,2 ). With this, we define a random variable (n) G̃1,2 (n) whose distribution is the conditional distribution of G1,2 , conditioned on lying in A(n) , i.e. ( (n) (n) (n) PER (H1,2 )/πn H1,2 ∈ A(n) (n) (n) (28) P G̃1,2 = H1,2 = 0 o.t.w. (n) (n) (n) where πn := P G1,2 ∈ A(n) is the normalizing factor. Additionally, let P̃ER be the law of G̃1,2 . If, (n) for i ∈ {1, 2}, M̃i (n) (n) denotes fi (G̃i ), we have (n) (n) log L1 + log L2 (n) (n) (n) (n) ≥ H(M̃1 ) + H(M̃2 ) ≥ H(M̃1 , M̃2 ) (29) (n) = H(G̃1,2 ), (n) where the last equality follows from the fact that, by definition, G̃1,2 takes values among the graphs (n) that are successfully decoded, and hence is uniquely identified given M̃1 12 (n) and M̃2 . (n) (n) (n) Now, we find a lower bound for H(G̃1,2 ). For doing so, note that for H1,2 ∈ G1,2 and n large enough, we have   ! P X X p n x∈Ξ1,2 px x (n) (n) − log PER (H1,2 ) = − mH (n) (x) log − − mH (n) (x) log 1 − 1,2 1,2 n n 2 x∈Ξ1,2 x∈Ξ1,2 X uH (n) (θ) log qθ . − θ∈Θ1,2 1,2 (n) Gp~,~q , (n) H1,2 (n) Gp~,~q On the other hand, due to the definition of if ∈ then, for all x ∈ Ξ1,2 have px px n − n2/3 ≤ mH (n) (x) ≤ n + n2/3 , and 1,2 2 2 nqθ − n2/3 ≤ uH (n) (θ) ≤ nqθ + n2/3 . (30) and θ ∈ Θ1,2 , we 1,2 Substituting these in (30) and using the inequality log(1 − x) ≤ −x which holds for x ∈ (0, 1), for n large enough, we have  P X px X px n x∈Ξ1,2 px (n) (n) n − n2/3 (log n − log px ) +  − log PER (H1,2 ) ≥ n + n2/3  − 2 2 n 2 x∈Ξ1,2 x∈Ξ1,2 X (nqθ − n2/3 ) log qθ . − θ∈Θ1,2 Using P x∈Ξ1,2 px = dER 1,2 and simplifying the above, we realize that there exists a constant c > 0 that (n) (n) does not depend on n or H1,2 , such that for all H1,2 ∈ A(n) , we have (n) (n) − log PER (H1,2 ) ≥ n X px X px X dER 1,2 qθ log qθ − cn2/3 log n log n − n log px + n −n 2 2 2 x∈Ξ1,2 x∈Ξ1,2 θ∈Θ1,2 dER 1,2 2/3 =n log n + nΣ(µER log n. 1,2 ) − cn 2 Now, if (n) G̃1,2 (31) is the random variable defined in (28), we have (n) H(G̃1,2 ) = − (n) X (n) (n) (n) P̃ER (H1,2 ) log P̃ER (H1,2 ) (n) H1,2 ∈A(n) = log πn − 1 πn (n) X (n) (n) (n) PER (H1,2 ) log PER (H1,2 ). (n) H1,2 ∈A(n) (n) Note that since the probability of error of the above code vanishes, i.e. Pe (n) (n) → 0, and P G1,2 ∈ Gp~,~q → (n) (n) 1, we have πn → 1 as n → ∞. On the other hand, with probability one, we have G̃1,2 ∈ Gp~,~q . Also, P (n) (n) by the definition of πn , we have H (n) ∈A(n) PER (H1,2 ) = πn . Thereby, employing the bound (31), we 1,2 have (n) H(G̃1,2 ) − n lim inf n→∞ n dER 1,2 2 13 log n ≥ Σ(µER 1,2 ). (32) Now, using the assumption (26) together with the bound (29), we have (n) 0 ≥ lim sup n→∞ (n) ≥ lim inf (n) log L1 + log L2 − (α1 + α2 )n log n − n(R1 + R2 ) n H(G̃1,2 ) − n n→∞ dER 1,2 2 log n − nΣ(µER n 1,2 ) + lim inf n→∞ n dER 1,2 2 log n + nΣ(µER 1,2 ) − (α1 + α2 )n log n − n(R1 + R2 ) . n (33) The first term is nonnegative due to (32). Consequently, ER d − α − α log n + n(Σ(µER n 1,2 1 2 1,2 ) − R1 − R2 ) 2 . 0 ≥ lim inf n→∞ n (34) ER Note that this is impossible unless α1 + α2 ≥ dER 1,2 /2. Furthermore, if α1 + α2 = d1,2 , it must be the ER ER case that R1 + R2 ≥ Σ(µ1,2 ). But this is precisely (13c) for µ1,2 = µ1,2 . Now, we turn to showing (13a). We have (n) log L1 (n) (n) (n) ≥ H(M̃1 ) ≥ H(M̃1 \|M̃2 ) (n) (n) (n) (n) (n) (n) = H(G̃1 , M̃1 \|M̃2 ) − H(G̃1 \|M̃1 , M̃2 ) (a) (n) (n) = H(G̃1 \|M̃2 ) (b) (n) (35) (n) ≥ H(G̃1 \|G̃2 ) (n) (n) = H(G̃1,2 ) − H(G̃2 ). (n) (n) where (a) uses the facts that M̃1 (n) M̃2 we Now, we (n) (n) can unambiguously determine G̃1,2 and (n) find an upper bound for H(G̃2 ). Note hence (n) and also, since G̃1,2 ∈ A(n) , given M̃1 is a function of G̃1 (n) G̃1 . Also, (b) uses data processing inequality. (n) that since G̃1,2 ∈ A(n) with probability one, we have (n) (n) H(G̃2 ) ≤ log \|A2 \|, where (n) A2 (n) (n) (n) := {H2 (n) (n) ∈ G2 (n) (n) : H1 and (n) ⊕ H2 (n) (36) (n) ∈ A(n) for some H1 (n) ∈ G1 }. (n) But for H1,2 := H1 ⊕ H2 ∈ A2 , since A2 ⊆ Gp~,~q , by definition we have that for all x ∈ Ξ1,2 and all θ ∈ Θ1,2 , X X \|uH (n) (θ) − nqθ \| ≤ n2/3 . \|mH (n) (x) − npx /2\| ≤ n2/3 and x∈Ξ1,2 1,2 1,2 θ∈Θ1,2 P Moreover, for x2 ∈ Ξ2 and θ2 ∈ Θ2 , we have mH (n) (x2 ) = x1 ∈Ξ1 ∪{◦1 } mH (n) ((x1 , x2 )) and uH (n) (θ2 ) = 1,2 2 2 P θ1 ∈Θ1 mH (n) ((θ1 , θ2 )). Using this in the above and using triangle inequality, we realize that for 1,2 (n) H2 (n) (n) (n) (n) ∈ A2 , we have m ~ H (n) ∈ M2 and ~uH (n) ∈ U2 , where M2 is the set of edge mark count 2 2 P (n) vectors m ~ such that x2 ∈Ξ2 \|m(x2 ) − npx2 /2\| ≤ n2/3 , and U2 is the set of vertex mark count vectors P ~u such that θ2 ∈Θ2 \|u(θ2 ) − nqθ2 \| ≤ n2/3 . Consequently, (n) \|A2 \| ≤ (2n2/3 )(\|Ξ2 \|+\|Θ2 \|) max (n) m∈M ~ 2 n 2 {m(x2 )}x2 ∈Ξ2 14 max (n) u ~ ∈U2 n . {u(θ2 )}θ2 ∈Θ2 (n) Using Lemma 1 and the definition of M2 (n) log \|A2 \| − n lim sup n n→∞ dER 2 2 (n) and U2 log n above, with Q = (Q1 , Q2 ) ∼ ~q, we get ≤ H(Q2 ) + X s(px2 ) = Σ(µER 2 ). x2 ∈Ξ2 Substituting into (36), we get (n) log H(G̃2 ) − n lim sup n n→∞ dER 2 2 log n ≤ Σ(µER 2 ). Using this together with (32) and substituting into (35) we get (n) lim inf log L1 − n ER dER 1,2 −d2 2 n n→∞ log n ER ER ER ≥ Σ(µER 1,2 ) − Σ(µ2 ) = Σ(µ1 \|µ2 ). Using a similar method as in (33) and (34), this implies (13a). The proof of (13b) is similar. This completes the proof of the converse for the Erdős–Rényi case. 5.4 Proof of the Converse for the configuration model The proof of the converse for the configuration model is similar to that for the Erdős–Rényi model presented in the previous section. Take an achievable rate tuple (α1 , R1 , α2 , R2 ) together with a (n) (n) (n) (n) sequence of hn, L1 , L2 i codes (f1 , f2 , g (n) ) achieving this rate tuple. Moreover, redefine the set A(n) to be (n) (n) (n) (n) (n) (n) (n) A(n) := W (n) ∩ {H1,2 ∈ G1,2 : g (n) (f1 (H1 ), f2 (H2 )) = H1,2 }, (37) (n) where the set W (n) was defined in Section 5.2. Now, let G1,2 ∼ G(n; d~(n) , ~γ , ~q) be distributed according (n) to the configuration model ensemble, and let G̃1,2 ∈ A(n) have the distribution obtained from that of (n) G1,2 (n) A (n) by conditioning on it lying in the set A(n) . Note that the normalizing constant πn := P(G1,2 ∈ (n) (n) ) goes to 1 as n → ∞ since P(G1,2 ∈ W (n) ) → 1 and the error probability of the code, Pe , (n) (n) (n) (n) vanishes. Moreover, let PCM and P̃CM be the laws of G1,2 and G̃1,2 , respectively. In the following, we show that dCM (n) H(G̃1,2 ) − n 1,2 2 log n ≥ Σ(µCM (38) lim inf 1,2 ), n→∞ n and (n) dCM H(G̃2 ) − n 22 log n lim sup ≤ Σ(µCM (39) 2 ). n n→∞ The rest of the proof is then identical to that of the previous section, so we only focus on proving the above two statements. − → (n) (n) For (38), note that for H1,2 ∈ G1,2 such that dgH (n) ∈ D(n) , where D(n) was defined in Section 5.2, 1,2 we have (n) (n) − log PCM (H1,2 ) = log n {ck (d~(n) )}∆ k=0 (n) d + log \|G ~(n) \| − 15 X x∈Ξ1,2 mH (n) (x) log γx − 1,2 X θ∈Θ1,2 uH (n) (θ) log qθ . 1,2 (n) Now, if H1,2 ∈ W (n) , using the definition of W (n) we realize that there exists a constant c > 0 such that X X n (n) (n) (n) +log \|G \|− − log PCM (H1,2 ) ≥ log m γ log γ − nqθ log qθ −cn2/3 =: Kn . n x x d~(n) {ck (d~(n) )}∆ k=0 x∈Ξ1,2 θ∈Θ1,2 (n) (n) Note that the right hand side is a constant independent of H1,2 and is denoted by Kn . Since G̃1,2 falls in (n) W (n) with probability one, this means that H(G̃1,2 ) ≥ log πn + Kn . But πn → 1 as n → ∞. Therefore, using the assumption (8) together with (24) from Section 5.2 and also the fact that mn /n → dCM 1,2 /2, we realize that (n) H(G̃1,2 ) − n lim inf n→∞ n dCM 1,2 2 log n ≥ H(X) − s(dCM 1,2 ) − E [log X!] + dCM 1,2 H(Γ) + H(Q), 2 where X ∼ ~r, Γ ∼ ~γ and Q ∼ ~ q . Note that the right hand side is precisely Σ(µCM 1,2 ), hence we have proved (38). (n) (n) (n) (n) (n) In order to show (39), note that H(G̃2 ) ≤ log \|A2 \| where A2 consists of graphs H2 ∈ G2 (n) (n) (n) (n) (n) (n) such that for some H1 ∈ G1 , we have H1 ⊕ H2 ∈ A(n) . Since A(n) ⊆ W (n) , for all H2 ∈ A2 we have X X \|mH (n) (x2 ) − mn γx2 \| ≤ n2/3 and (40) \|uH (n) (θ2 ) − nqθ2 \| ≤ n2/3 . 2 x2 ∈Ξ2 2 θ2 ∈Θ2 − → (n) (n) On the other hand, the condition (22) implies that dgH (n) ∈ D2 where D2 denotes the set of degree 2 sequences d~ of size n with elements bounded by ∆ such that ~ − nP (X2 = k) \| ≤ (∆ + 1)n2/3 \|ck (d) ∀0 ≤ k ≤ ∆, where X2 is the random variable defined in (11). Consequently, we have P (n) log \|A2 \| ≤ (n) log \|D2 \| + (n) max log \|G ~ \| d (n) ~ d∈D + 2 max (n) H2 (n) log ∈A2 x2 ∈Ξ2 (41) mH (n) (x2 ) 2 {mH (n) (x2 )}x2 ∈Ξ2 2 n + max log . (n) (n) {uH (n) (θ2 )}θ2 ∈Θ2 H2 ∈A2 (42) 2 (n) Note that (41) implies that \|D2 \| ≤ (2(∆ + 1)n2/3 )∆+1 maxd∈D (n) ~ 2 implies 1 (n) lim sup log \|D2 \| ≤ H(X2 ). n n→∞ n . ~ ∆ {ck (d)} k=0 Therefore, Lemma 1 (43) > 0. Hence, using Lemma 3 in On the other hand, the assumptions r0 < 1 and (7) imply that dCM 2 Appendix A we have lim sup (n) d maxd∈D (n) log \|G ~ ~ \|−n 2 (n) log n n n→∞ Moreover, if H2 dCM 2 2 ≤ −s(dCM 2 ) − E [log X2 !] . (n) is a sequence in A2 , from (40), for all x2 ∈ Ξ2 , we have lim P n→∞ mH (n) (x2 ) 2 ′ x′2 ∈Ξ2 mH2(n) (x2 ) =P γx 2 x′2 ∈Ξ2 16 γx′2 = P (Γ2 = x2 \|Γ2 6= ◦2 ) , (44) where Γ = (Γ1 , Γ2 ) has law ~γ . Additionally, we have 1 X dCM mH (n) (x2 ) = 2 . n→∞ n 2 2 lim x2 ∈Ξ2 Thereby, from Lemma 1, we have 1 lim sup max log n→∞ n H2(n) ∈A2(n) P x2 ∈Ξ2 mH (n) (x2 ) 2 {mH (n) (x2 )}x2 ∈Ξ2 2 ≤ dCM 2 H(Γ2 \|Γ2 6= ◦2 ). 2 (45) Finally, as we have uH (n) (θ2 )/n → qθ2 for all θ2 ∈ Θ2 , another usage of Lemma 1 implies that 2 lim sup n→∞ 1 n max log ≤ H(Q2 ), n H2(n) ∈A2(n) {uH (n) (θ2 )}θ2 ∈Θ2 (46) 2 where Q = (Q1 , Q2 ) has law ~ q . Now, combining (43), (44), (45) and (46) and substituting into (42), (n) (n) and also using the bound H(G̃2 ) ≤ log \|A2 \|, we realize that (n) lim sup n→∞ H(G̃2 ) − n n dCM 2 2 log n ≤ H(X2 ) − s(dCM 2 ) − E [log X2 !] + dCM 2 H(Γ2 \|Γ2 6= ◦2 ) + H(Q2 ). 2 But the right hand side is precisely Σ(µCM 2 ). This completes the proof of (39). As was mentioned before, the rest of the proof is identical to that in the previous section. 6 Conclusion We gave a counterpart of the Slepian–Wolf Theorem for graphical data, employing the framework of local weak convergence. We derived the rate region for two graph ensembles, namely an Erdős–Rényi model and a configuration model. A Asymptotic behavior of the entropy of the configuration model Here, we prove (12a)–(12c). Before this, we set some notation and state some general lemmas. In what follows, we employ the definitions of the sets W (n) and D(n) from Section 5.2. Moreover, Γ = (Γ1 , Γ2 ) and Q = (Q1 , Q2 ) be random variables with laws ~γ and ~q, respectively. Let β1 := P(Γ1 6= ◦1 ) and Γ̃1 (n) be a random variable on Ξ1 with the law of Γ1 conditioned on Γ1 6= ◦1 . Let F1,2 be a simple unmarked (n) (n) graph chosen uniformly at random from the set ∪ ~ (n) G . By definition, G ∼ G(n; d~(n) , ~γ , ~q) is d∈D obtained from (n) F1,2 d~ 1,2 by adding independent edge and vertex marks according to the laws of ~γ and ~q (n) (n) respectively. Let F1 be obtained from F1,2 by independently removing each edge with probability (n) (n) 1 − β1 . Then, G1 can be thought of as obtained from F1 by adding independent vertex and edge (n) (n) (n) (n) marks with the laws of Q1 and Γ̃1 , respectively. Hence, we may consider G1,2 , F1,2 , G1 and F1 on a joint probability space. It is straightforward to see the following. 17 Lemma 2. Assume X is an integer valued random variable taking value in {0, . . . , ∆} and 0 ≤ ǫ ≤ 1. Let {Yi }i≥1 be a sequence of i.i.d. Bernoulli random variables with P (Yi = 1) = ǫ. Define the random PX variable X1 to be i=1 Yi and X2 := X − X1 . Then, we have X H(X1 , X2 ) = H(X) + E [X] H(Y1 ) − E log . X1 The following lemma which is a direct consequence of Theorem 4.5 and Corollary 4.6 in [BC14], is useful in the asymptotic analysis of the count of the graphs with a given degree sequence. Lemma 3. Given an integer ∆, assume that Y is an integer random variable bounded by ∆ such that (n) d := E [Y ] > 0. Moreover, assume that for each n, ~a(n) = (1), . . . , a(n) (n)) is a degree sequence P(a n (n) (i) is even and, for 0 ≤ k ≤ ∆, of length n with entries bounded by ∆ such that bn := i=1 a (n) ck (~a )/n → P(Y = k). Then, we have (n) log \|G~a(n) \| − n→∞ n lim where s(d) := d 2 − d 2 bn 2 log n = −s(d) − E [log Y !] , log d. (n) (n) / W (n) ) ≤ κn−1/3 for some constant κ > 0. q ), we have P(G1,2 ∈ Lemma 4. If G1,2 ∼ G(n; d~(n) , ~γ , ~ Proof. For this, we note the following. Condition (i) in the definition of W (n) always holds for a (n) realization G1,2 . Chebyshev’s inequality implies that conditions (ii) and (iii) hold with probability at least 1 − κ1 n−1/3 , for some κ1 > 0. To show (iv), fix k and l and for 1 ≤ i ≤ n, let Yi be the indicator Pn − → − → of dgG(n) (i) = k and dgG(n) (i) = l. With Y := i=1 Yi , we have ck,l (dgG(n) , dgG(n) ) = Y . Note that 1,2 1 (n) 1,2 1 (n) an edge of G1,2 exists in G1 if its mark is not of the form (◦1 , x2 ), which happens with probability β1 . Therefore, i h h i dg (n) (i) (n) F1,2 E Yi \|F1,2 = 1 dgF (n) (i) = k β1l (1 − β1 )k−l . 1,2 l Consequently, i h k l (n) (n) ~ β (1 − β1 )k−l . E Y \|F1,2 = ck (d ) l 1 Since this is a constant, it is also equal to E [Y ]. Now, if sk,l := P (X = k, X1 = l), we have sk,l = rk kl β1l (1 − β1 )k−l . Hence, the assumption (8) implies that k l β (1 − β1 )k−l . (47) \|E [Y ] − nsk,l \| ≤ n1/2 l 1 (n) Furthermore, since edge marks are chosen independently, conditioned on F1,2 , if i and j are nonadjacent (n) (n) vertices in F1,2 , then Yi are Yj are independent, conditioned on F1,2 . As a result, if I denotes the set 18 (n) of (i, j) with 1 ≤ i 6= j ≤ n such that i and j are not adjacent in F1,2 , we have n i h i X h (n) (n) E Yi2 \|F1,2 + E Y 2 \|F1,2 = i=1 ≤n+ X (i,j)∈I / X 1≤i6=j≤n i h (n) E Yi Yj \|F1,2 i h i h X (n) (n) E Yi Yj \|F1,2 E Yi Yj \|F1,2 + (i,j)∈I ≤ n + 2mn + X i h (n) E Yi Yj \|F1,2 X i i h h (n) (n) E Yi \|F1,2 E Yj \|F1,2 (i,j)∈I (a) = n + 2mn + (i,j)∈I X ≤ n + 2mn + 1≤i6=j≤n i i h h (n) (n) E Yi \|F1,2 E Yj \|F1,2 i2 h (n) = n + 2mn + E Y \|F1,2 , (n) where (a) uses the fact that conditioned on F1,2 , the random variables Yi and Yj are independent for 1/2 (i, j) ∈ I.h From (8), and κ2 := ∆(∆ + 1)/4 is a constant. As we saw i we have \|mn − nd1,2 /2\| ≤ κ2 n (n) above, E Y \|F1,2 = E [Y ]. Therefore, we have Var Y ≤ κ3 n for some κ3 > 0. This together with (47) and Chebyshev’s inequality implies that the condition (iv) holds with probability at least 1 − κ4 n−1/3 , for some κ4 > 0. Similarly, the same statement holds for the condition (v). Now we show (12a)–(12c). In the following, with X ∼ ~r and X1 and X2 defined as in (11), let (n) B1,2 be the set of pairs of degree sequences d~ and ~δ with n elements bounded by ∆ such that for all ~ ~δ) − nP(X1 = k, X ′ = l)\| ≤ n2/3 , where X ′ := X − X1 . Moreover, let B (n) be 0 ≤ k, l ≤ ∆, \|ck,l (d, 1 1 1 ~ be the set of degree ~ ~δ) ∈ B (n) . For d~ ∈ B (n) , let B (n) (d) the set of d~ such that for some ~δ, we have (d, 1 1,2 2\|1 ~ ~δ) ∈ B (n) . sequences ~δ such that (d, 1,2 (n) (n) Now, we show (12a). Since G1,2 is formed by adding independent vertex and edge marks to F1,2 , we have (n) (n) H(G1,2 ) = log \|D(n) \| + log \|Gd~(n) \| + mn H(Γ) + nH(Q). ∆K 1/2 From (8), we have \|mn − ndCM . Moreover, we have E [X] > 0. Consequently, using 1,2 /2\| ≤ 2 n 1 (n) Lemma 3 and the fact that n log \|D \| → H(X), we get (12a). (n) We now turn to showing (12b). Since the expected number of the edges in F1 is ndCM 1 /2, we have dCM (n) (n) H(G1 ) = H(F1 ) + n 1 H(Γ1 \|Γ1 6= ◦1 ) + nH(Q1 ). (48) 2 (n) (n) With this, we focus on H(F1 ). With En being the indicator of G1,2 ∈ / W (n) , we have (n) (n) H(F1 ) = H(F1 \|En = 0)P(En = 0) (n) +H(F1 \|En = 1)P(En = 1). From Lemma 4, we have (n) (n) H(F1 \|En = 1)P(En = 1) ≤ (H(F1,2 ) + mn H(β1 ))κn−1/3 , 19 (49) (n) where H(β1 ) denotes the binary entropy of β1 . Note that as we discussed above, H(F1 ) = O(n log n). Thereby, the RHS of the above is o(n). On the other hand, by the definition of W (n) , if E = 0, − → (n) (n) (n) (n) dgF (n) ∈ B1 . Therefore, H(F1 \|E = 0) ≤ log B1 + maxd∈B \|. The assumption r0 < 0 (n) log \|G ~ d~ 1 1 together with (7) imply that dCM > 0. Hence, using Lemma 3 together with (49), 1 (n) H(F1 ) − n lim sup n (n) Now, let F̃1 note that dCM 1 2 log n ≤ −s(dCM 1 ) + H(X1 ) − E [log X1 !] . (50) (n) (n) be the unmarked graph consisting of the edges removed from F1,2 to obtain F1 (n) (n) (n) (n) and (n) H(F1 ) = H(F1 , F̃1 ) − H(F̃1 \|F2 ) (51) (n) (n) (n) = H(F1,2 ) + mn H(β1 ) − H(F̃1 \|F1 ) − → → (n) − Note that, conditioned on E = 0, we have dgF̃ (n) ∈ B2\|1 (dgF (n) ). Moreover, the assumption (7) 1 1 CM together with r0 < 0 imply that dCM > 0. Hence, using a similar method in proving (50), we 1,2 − d1 have (n) (n) H(F̃1 \|F1 ) − n lim sup n CM dCM 1,2 −d1 2 log n CM ≤ −s(dCM 1,2 − d1 ) +H(X1′ \|X1 ) − E [log X1′ !] . (n) Using this together with the asymptotic of H(F1,2 ) which was derived above in showing (12a) and substituting into (51), followed by a simplification using Lemma 2, we get (n) H(F1 ) − n lim inf n dCM 1 2 log n (52) ≥ −s(dCM 1 ) + H(X1 ) − E [log X1 !] . This together with (50) and (48) completes the proof of (12b). The proof of (12c) is similar. (n) (n) Bounding \|S2 (H1 )\| for the Erdős–Rényi case B (n) (n) (n) Note that for G1,2 ∈ G1,2 and H2 (n) (n) ∈ G2 , if G1 (n) ⊕ H2 (n) ∈ Gp~,~q , we have m ~ G(n) ⊕H (n) ∈ M(n) and 1 2 ~uG(n) ⊕H (n) ∈ U (n) . On the other hand, for fixed m ~ ∈ M(n) and ~u ∈ U (n) , if for all x1 ∈ Ξ1 we have 1 2 (n) m(x1 ) = mG(n) (x1 ) and for all θ1 ∈ Θ1 we have u(θ1 ) = uG(n) (θ1 ), then the number of H2 1 1 m ~ G(n) ⊕H (n) = m ~ and ~uG(n) ⊕H (n) = ~u is at most 1 2 A2 (m, ~ ~u) := 1 Y x1 ∈Ξ1 such that 2 m(x1 ) {m(x1 , x2 )}x2 ∈Ξ2 ∪{◦2 } !! × !  P Y − x1 ∈Ξ1 m(x1 ) × {m(◦1 , x2 )}x2 ∈Ξ2 θ ∈Θ n 2 1 1 ! u(θ1 ) , {u(θ1 , θ2 )}θ2 ∈Θ2 where we have used the notational conventions in (3) and (4). Consequently, we have max (n) (n) G1,2 ∈Gp ~,~ q (n) (n) \|S2 (G1 )\| ≤ \|M(n) \|\|U (n) \| max A2 (m, ~ ~u) (n) m∈M ~ u ~ ∈U (n) ≤ (2n2/3 )(\|Ξ1,2 \|+\|Θ1,2 \|) max A2 (m, ~ ~u). (n) m∈M ~ u ~ ∈U (n) 20 (53) Now, if m ~ (n) and ~u(n) are sequences in M(n) and U (n) , respectively, for all x ∈ Ξ1,2 we have (n) m (x)/n → px /2. Furthermore, for all x1 ∈ Ξ1 and θ1 ∈ Θ1 , we have m(n) (x1 )/n → px1 /2 and ~ (n) and ~u(n) , with Q = (Q1 , Q2 ) u(n) (θ1 )/n → qθ1 . As a result, using Lemma 1, for any such sequences m having law ~ q we have P log A2 (m ~ (n) , ~u(n) ) − ( x2 ∈Ξ2 m(n) (◦1 , x2 )) log n lim n→∞ n ! X X px p(x1 ,x2 ) 1 = s(p◦1 ,x2 ) + H 2 px 1 x2 ∈Ξ2 ∪{◦2 } x2 ∈Ξ2 x1 ∈Ξ1 ! X qθ1 ,θ2 qθ1 H + qθ1 θ2 ∈Θ2 θ1 ∈Θ1 X X = H(Q2 \|Q1 ) + s(px ) − s(px1 ) x∈Ξ1,2 = x1 ∈Ξ1 ER Σ(µER 2 \|µ1 ), where the second inequality follows by rearranging the terms and using the definition of s(.). Using the fact that \|m(n) (◦1 , x2 ) − np◦1 ,x2 /2\| ≤ n2/3 , log An (m ~ (n) , ~u(n) ) − n lim n→∞ n ER dER 1,2 −d1 2 log n ER = Σ(µER 2 \|µ1 ). This together with (53) implies (19). C (n) (n) Bounding \|S2 (H1 )\| for the configuration model (n) (n) (n) Here, we find an upper bound for maxH (n) \|S2 (H1 )\| and use it to show (25). Fix G1,2 ∈ W (n) and 1,2 (n) assume H2 (n) (n) (n) (n) (n) (n) ∈ S2 (G1 ). With H1,2 := G1 ⊕ H2 , let H̃2 (n) be the subgraph of H1,2 consisting of (n) edges not present in G1 . Employing the notation of Appendix A, by the definition of the set W (n) , − → → (n) (n) − (n) we have dgH̃ (n) ∈ B2\|1 (dgG(n) ). Therefore, we can think of H1,2 as constructed from G1 by adding a 2 1 − → (n) graph to G1 with degree sequence dgH̃ (n) , markings its edges, adding second domain marks to edges 2 (n) in G1 , and also adding second domain marks to vertices. Consequently, max (n) H1,2 ∈W (n) (n) (n) log \|S2 (H1 )\| ≤ + max log (n) m∈M ~ + max log u∈U (n) ~ max (n) G1,2 ∈W (n) → (n) − log \|B2\|1 (dgG(n) )\| + 1 mn − x1 ∈Ξ1 m(x1 ) {m(◦1 , x2 )}x2 ∈Ξ2 u(θ1 ) {u(θ1 , θ2 )}θ2 ∈Θ2 ! Y θ1 ∈Θ1 log \|G~δ \| G 1 ! P (n) max → (n) (n) − G1,2 ∈W (n) ,~ δ∈B2\|1 (dg (n) ) Y x1 ∈Ξ1 m(x1 ) {m(x1 , x2 )}x2 ∈Ξ2 ! (n) The definition of B2\|1 implies that → 1 (n) − max log \|B2\|1 (dgH (n) )\| = H(X − X1 \|X1 ). (n) n→∞ n H 1 (n) 1,2 ∈W lim 21 (54) CM Note that the assumption (7) together with r0 < 1 implies that dCM > 0. Therefore, Lemma 3 1,2 − d1 in Appendix A implies that lim sup max n→∞ (n) G1,2 ∈W (n) (n) δ log \|G~ \| − n CM dCM 1,2 −d1 2 log n n CM ≤ −s(dCM 1,2 − d1 ) − E [log(X − X1 )!] . − → ~ δ∈B(dg (n) ) G 1 Furthermore, a usage of Lemma 1 implies that the third and the fourth terms in (54) divided by n dCM converge to 1,2 γ and Q has law ~q. 2 H(Γ2 \|Γ1 ) and H(Q), respectively, where Γ = (Γ1 , Γ2 ) has law ~ Putting these together, we have (n) (n) lim maxG(n) ∈W (n) log \|S2 (G1 )\| − n 1,2 n→∞ CM dCM 1,2 −d1 2 n log n CM = −s(dCM 1,2 − d1 ) + H(X − X1 \|X1 ) −E [log(X − X1 )!] + dCM 1,2 H(Γ2 \|Γ1 ) + H(Q2 \|Q1 ). 2 CM Using Lemma 2 and rearranging, this is precisely equal to Σ(µCM 2 \|µ1 ), which completes the proof of (25). Acknowledgments The authors acknowledge support from the NSF grants ECCS-1343398, CNS- 1527846, CCF-1618145, the NSF Science & Technology Center grant CCF- 0939370 (Science of Information), and the William and Flora Hewlett Foundation supported Center for Long Term Cybersecurity at Berkeley. References [Abb16] Emmanuel Abbe. Graph compression: The effect of clusters. In Communication, Control, and Computing (Allerton), 2016 54th Annual Allerton Conference on, pages 1–8. IEEE, 2016. [AL07] David Aldous and Russell Lyons. Processes on unimodular random networks. Electron. J. Probab, 12(54):1454–1508, 2007. [AS04] David Aldous and J Michael Steele. The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on discrete structures, pages 1–72. Springer, 2004. [BC14] Charles Bordenave and Pietro Caputo. Large deviations of empirical neighborhood distribution in sparse random graphs. Probability Theory and Related Fields, pages 1–73, 2014. [Bil13] Patrick Billingsley. Convergence of probability measures. John Wiley & Sons, 2013. [BS01] Itai Benjamini and Oded Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6:no. 23, 13 pp. (electronic), 2001. [CS12] Yongwook Choi and Wojciech Szpankowski. Compression of graphical structures: Fundamental limits, algorithms, and experiments. IEEE Transactions on Information Theory, 58(2):620–638, 2012. 22 [CT12] Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012. [DA17] Payam Delgosha and Venkat Anantharam. Universal lossless compression of graphical data. In Information Theory (ISIT), 2017 IEEE International Symposium on, pages 1578–1582. IEEE, 2017. 23	7
REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS arXiv:1308.5285v1 [math.AC] 24 Aug 2013 KUEI-NUAN LIN AND CLAUDIA POLINI A BSTRACT. In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M ) onto it, where M is the direct sum of powers of m. We compute a Gröbner basis of the ideal defining R(M ). It turns out that the normal domain R(M ) is a Koszul algebra and from this we deduce that in many instances R(I) is a Koszul algebra as well. 1. I NTRODUCTION In this paper we investigate the Rees algebra R(I) = R[It] as well as the special fiber ring F(I) = R(I) ⊗ k of an ideal I in a standard graded algebra R over a field k. These objects are important to commutative algebraists because they encode the asymptotic behavior of the ideal I and to algebraic geometers because their projective schemes define the blowup and the special fiber of the blowup of the scheme Spec(R) along V (I). One of the central problems in the theory of Rees rings is to describe R(I) and F(I) in terms of generators and relations (see for instance [24, 29, 10, 28, 26, 17, 18, 15, 11, 6, 13, 14]). This is a challenging quest which is open for most classes of ideals, even three generated ideals in a polynomial ring in two variables (see for instance [2, 1, 22]). The goal is to find an ideal A in a polynomial ring S = R[T1 , . . . , Ts ] so that R(I) = S/A. If the ideal I is generated by forms of the same degree, then these forms define rational maps between projective spaces and the special fiber ring and the Rees ring describe the image and the graph of such rational maps, respectively. By computing the defining equations of these algebras, one is able to exhibit the implicit equations of the graph and of the variety parametrized by the map. This classical and difficult problem in elimination theory has also been studied in applied mathematics, most notably in modeling theory, where it is known as the implicitization problem (see for instance [3, 4, 5, 9]). If the ideal I is not generated by forms of the same degree, one can consider the truncation of I past its generator degree. In this paper we treat truncations of complete intersection ideals in a polynomial ring. More precisely, let R = k[x1 , . . . , xn ] be a polynomial ring over a field k with homogeneous maximal ideal m, let f1 , . . . , fr be a homogeneous regular sequence in R AMS 2010 Mathematics Subject Classification. Primary 13A30; Secondary 13H15, 13B22, 13C14, 13C15, 13C40. The first author was partially supported by an AWM-NSF mentor travel grant. The second author was partially supported by NSF grant DMS-1202685 and NSA grant H98230-12-1-0242. Keywords: Elimination theory, Gröbner basis, Koszul algebras, Rees algebra, Special fiber ring. 1 2 K. N. LIN AND C. POLINI of degrees d1 ≥ . . . ≥ dr , let d ≥ d1 be an integer, and write ai = d − di . The truncation I = (f1 , . . . , fr )≥d of the complete intersection (f1 , . . . , fr ) in degree d is the R-ideal generated by P the forms in (f1 , . . . , fr ) of degree at least d. In other words, I = (f1 , . . . , fr ) ∩ md = ri=1 mai fi . The Cohen-Macaulayness of the Rees algebra of such ideals was previously studied in [12], where the authors show that R(I) is Cohen-Macaulay for all d > D and they give a sharp estimate for D. However, the defining equations of R(I) were unknown. In this paper we describe them explicitly for all d when r = 2 and for d ≥ d1 + d2 when r ≥ 3. Furthermore we prove that the Rees ring and the special fiber ring are Koszul algebras for d ≥ d1 + d2 and for d ≥ d1 + d2 − 1 if r = 2. To determine the defining equations of R(I) we map another Rees ring R(M ) onto R(I), where M is the module ma1 ⊕ . . . ⊕ mar . Our aim then becomes to find the defining ideal of R(M ) and the kernel Q, 0 → Q −→ R(M ) −→ R(I) → 0 . The problem of computing the implicit equation of R(M ) is interesting in its own right and it was previously addressed in [23], where the relation type of R(M ) was computed. We solve it in Section 2. It turns out that R(M ) and F(M ) are normal domains whose defining ideals have a Gröbner basis of quadrics; hence, they are Koszul algebras. For r = 2, the kernel Q is a height one prime ideal of the normal domain R(M ); therefore it is a divisorial ideal of R(M ). Our goal is then reduced to explicitly describing ideals that represent the elements in the divisor class group of R(M ). The approach is very much inspired by [21] and [20]. For r = 2 and d ≥ d1 + d2 − 1 or r ≥ 3 and d ≥ d1 + d2 , the ideal I has a linear presentation and Q turns out to be a linear ideal in the T ′ s. Using this we prove that the defining ideal of R(I) has a quadratic Gröbner basis and hence R(I) is a Koszul algebra as well. 2. T HE B LOWUP ALGEBRAS OF DIRECT SUMS OF POWERS OF THE MAXIMAL IDEAL Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k with homogeneous maximal ideal m. Let 0 ≤ a1 ≤ . . . ≤ ar be integers. Write a = a1 , . . . , ar . In this section, we will describe explicitly the Rees algebra and the special fiber ring of the module M = Ma = ma1 ⊕ · · · ⊕ mar in terms of generators and relations and we will prove that they are Koszul normal domains. We will end the section with a study of the divisor class group of the blowup algebras of M . Definition 2.1. Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k. Let a be a positive integer, write Ja and Ja′ for the two sets of multi-indices in (N ∪ {0})n−1 and Nn−1 , respectively, that are defined as follows Ja = {j = (jn−1 , . . . , j1 ) \| 0 ≤ j1 ≤ . . . ≤ jn−1 ≤ a} , Ja′ = {j = (jn−1 , . . . , j1 ) \| 1 ≤ j1 ≤ . . . ≤ jn−1 ≤ a} . REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 3 Write xa,j and xa,j,s for the monomials x a,j = n Y j −ji−1 xi i i=1 with j ∈ Ja , j0 = 0, jn = a and xa,j,s = xs a,j x x1 with j ∈ Ja′ , 1 ≤ s ≤ n . The Rees algebra R(M ) of M is the subalgebra R(M ) = R[{xal ,j tl \| 1 ≤ l ≤ r, j ∈ Jal }] ⊂ R[t1 , . . . , tr ] of the polynomial ring R[t1 , . . . , tr ], while the special fiber ring F(M ) is the subalgebra of R(M ) F(M ) = k[{xal ,j tl \| 1 ≤ l ≤ r, j ∈ Jal ] ⊂ R(M ) . To find a presentation of these algebras we consider the polynomial rings T = Ta = k[{Tl,j \| 1 ≤ l ≤ r, j ∈ Jal }] S = Sa = R ⊗k Ta = Ta [x1 , . . . , xn ] in the new variables Tl,j and the epimorphisms of algebras (1) φ : S → R(M ) ψ : T → F(M ) defined by φ(Tl,j ) = ψ(Tl,j ) = xal ,j tl . Notice that ψ is the restriction of φ to T . We can assume that the ai ’s are all positive because if ai = 0 for 1 ≤ i ≤ s with s ≤ r then the Rees algebra R(M ) is isomorphic to a polynomial ring over the Rees algebra of mas+1 ⊕ · · · ⊕ mar R(⊕rl=1 mal ) ∼ = R(⊕rl=s+1 mal )[t1 , · · · , ts ] , and likewise for the special fiber ring. Furthermore, we can treat simultaneously the special fiber ring and the Rees algebra since R(⊕rl=1 mal ) ∼ = F(m ⊕ (⊕rl=1 mal )) and F(M ) = R(M )/mR(M ) . Definition 2.2. Let τ be the lexicographic order on a set of multi-indices in (N ∪ {0})n , i.e. p > q if the first nonzero entry of p− q is positive. Let 1 ≤ a1 ≤ . . . ≤ ar be integers. Write a = a1 , . . . , ar . Order the set of multi-indices {(l, j) \| 1 ≤ l ≤ r, j ∈ Ja′l } by τ . Write Tl,j,s for the variable Tl,jn−1 ,...,js ,js−1 −1,...,j1 −1 . P +n−2 (1) Let Ba be the n × ( rl=1 aln−1 ) matrix whose entry in the s-row and the (l, j)-column is the variable Tl,j,s with 1 ≤ s ≤ n, 1 ≤ l ≤ r, and j ∈ Ja′l . 4 K. N. LIN AND C. POLINI (2) Let Ca be the n × (1 + Pr l=1 al +n−2 ) n−1  matrix x1  .. Ca =  . xn   Ba  . Example 2.3. For instance if n = 3, r = 2 and a = 1, 2 then Ba is the 3 × 4 matrix   T1,1,1 T2,1,1 T2,2,1 T2,2,2 Ba =  T1,1,0 T2,1,0 T2,2,0 T2,2,1  . T1,0,0 T2,0,0 T2,1,0 T2,1,1 Write L for the kernel of the epimorphism φ defined in (1). Notice that φ(Tl,j,s ) = xa,j,s tl = xs a,j x1 x tl hence one can easily deduce the inclusion I2 (Ca ) ⊂ L . Our goal is to show that the above inclusion is an equality. In order to establish this claim it suffices to show that I2 (Ca ) is a prime ideal of dimension at most n + r = dim R(M ) (see for instance [27, 2.2]). Theorem 2.4. Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k with homogeneous maximal ideal m. Let 1 ≤ a1 ≤ . . . ≤ ar be integers and let M = ma1 ⊕ · · · ⊕ mar . The Rees algebra and the special fiber ring of M are Koszul normal domains. Furthermore, R(M ) = S/I2 (Ca ) F(M ) = T /I2 (Ba ) , where a, Ca , and Ba are as in Definition 2.2. An important step in the proof of Theorem 2.4 is to show that the set of 2 by 2 minors of Ca forms a Gröbner basis for I2 (Ca ). In the next lemma we will show much more. Indeed, the set of 2×2 minors of any submatrix Da of Ca forms a Gröbner basis for I2 (Da ). Notice that τ induces an ordering of the variables of T . With respect to this ordering we consider the reverse lexicographic order on the monomials in the ring T , which we also call τ . Remark 2.5. Let a′ = 1, a1 , . . . , ar and denote with Ta′ the polynomial ring associated with the sequence a′ . Notice that Ta′ /I2 (Ba′ ) ∼ = Sa /I2 (Ca ) and the matrix Ca is equal (after changing the name of the variables) to the matrix Ba′ . Lemma 2.6. Adopt assumption 2.2 and let Da be any submatrix of Ca with the same number of rows. The set of 2 × 2 minors of Da forms a Gröbner basis for I2 (Da ) with respect to τ . We use a general strategy to compute the Gröbner bases that we outline below. REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 5 Strategy 2.7. Let R be a polynomial ring over a field with a fixed monomial order ” > ”. Let D be a collection of 2 × 2 minors of a matrix with entries in R. The Buchberger criterion (see for instance [8, 15.8]) states that a generating set D of an ideal is a Gröbner basis for the ideal if the S-polynomials (or S-pairs) of any two elements of D reduces to zero module D. The following strategy will be used to show that the remainders of the S-pairs of the elements of D reduces to zero modulo D (see for instance [7, Section 2.9, Definition 1 and Theorem 3]). Let ad − bc and hg − ef be two elements of D. To compute the S-polynomial S(ad − bc, hg − ef ) we can assume, for instance, that h = a and ad > bc and ag > ef because otherwise the leading terms of the two polynomials are relatively prime and therefore S(ad − bc, ag − ef ) reduces to zero modulo D (see for instance [7, Section 2.9, Proposition 4]). We consider the matrix   d b 0 M = c a f . G e g We obtain the following equality by computing the determinant of M in two ways d (2) a f e g −g d b c a =b c f G g −f d b . G e Notice that the left hand side of equation (2) is our S-polynomial S(ag − ef, ad − bc). If by a suitable choice of G ∈ R, the polynomials cg − f G and de − bG are in D and each term on the right hand side has order strictly less than adg, then the S-pair S(ag − ef, ad − bc) has remainder zero modulo D and in particular it reduces to zero modulo D (see for instance [7, Section 2.9, Lemma 2]). The choice of the polynomial G and the matrix M depends on the polynomials we start with as we will explain later. Proof of Lemma 2.6 Because of Remark 2.5 it is enough to prove the statement for Da any submatrix of Ba′ with the same number of rows. Let D denote the set of 2 × 2 minors of Da . We use the strategy described above to show that the S-polynomials of any two elements of D reduces to zero module D. To simplify notation, we will use (l, j, s) to represent the entry Tl,j,s in the matrix Da . The leading terms of two elements of D h1 = (l1 , i, s) (l2 , j, s) (l1 , i, t) (l2 , j, t) and h2 = (l3 , p, u) (l4 , q, u) , (l3 , p, v) (l4 , q, v) where 1 ≤ s < t ≤ n, (l2 , j, s) > (l1 , i, s), 1 ≤ u < v ≤ n, and (l4 , q, u) > (l3 , p, u) are (l1 , i, s)(l2 , j, t) and (l3 , p, u)(l4 , q, v). Notice that Tl,j,s > Tl,j,t if s < t. These will be relatively prime unless (l1 , i, s) = (l3 , p, u), or (l1 , i, s) = (l4 , q, v), or (l2 , j, t) = (l3 , p, u), or (l2 , j, t) = (l4 , q, v). Since (l1 , i, s) = (l4 , q, v) and (l2 , j, t) = (l3 , p, u) are symmetric, it suffices to consider three cases. Case 1: Set (l1 , i, s) = a (l2 , j, s) = f (l1 , i, s) = a (l3 , p, s) = c h1 = and h2 = , (l1 , i, t) = e (l2 , j, t) = g (l1 , i, u) = b (l3 , p, u) = d 6 K. N. LIN AND C. POLINI where we can assume s < u ≤ t and l1 ≤ l2 ≤ l3 . We use equation (2) and the matrix M of Strategy 2.7 with G = (l3 , p, t). The S-polynomial S(h1 , h2 ) reduces to zero modulo D because b and e are smaller than any other entries of the two matrices defining h1 and h2 . Case 2: Set h1 = (l1 , i, s) = a (l2 , j, s) = e (l1 , i, t) = f (l2 , j, t) = g and h2 = (l3 , p, u) = d (l1 , i, u) = c (l3 , p, s) = b (l1 , i, s) = a , where u < s < t and l3 ≤ l1 ≤ l2 . We use equation (2) and the matrix M of Strategy 2.7 with G = (l2 , j, u). The S-polynomial S(h1 , h2 ) reduces to zero modulo D because b and f are smaller than any other entries of the two matrices defining h1 and h2 . Case 3: Set h1 = (l1 , i, s) = d (l2 , j, s) = b (l1 , i, t) = c (l2 , j, t) = a and h2 = (l3 , p, u) = g (l2 , j, u) = e (l3 , p, t) = f (l2 , j, t) = a , where we can assume u ≤ s < t and l1 ≤ l3 ≤ l2 . We use equation (2) and the matrix M of Strategy 2.7 with G = (l1 , i, u). The S-polynomial S(h1 , h2 ) reduces to zero modulo D because c and f are smaller than any other entries of the two matrices defining h1 and h2 . Corollary 2.8. Adopt assumptions 2.2 and let Da be any submatrix of Ca with the same number of rows. The ideal I2 (Da ) is prime in Sa . Proof. Write S = Sa . Notice that by Lemma 2.6 the variable u ∈ S appearing in the first column and the last row of the matrix Da does not divide any element in the generating set of inτ I2 (Da ). Hence u is regular on S/I2 (Da ). After localizing at u the ideal I2 (Da )u is isomorphic to an ideal generated by variables, which is a prime ideal in the ring Su . Thus I2 (Da ) is a prime ideal in the ring S. Corollary 2.9. Adopt assumptions 2.2. The initial ideal inτ (I2 (Ba ) is generated by the monomials Tl1 ,i,s Tl2 ,j,t with (l1 , i) <τ (l2 , j) and s < t. Proof. The assertion follows from Lemma 2.6. Proof of Theorem 2.4 We first show that for any sequence of r positive integers a = a1 , . . . , ar , the Rees algebra of M is defined by the ideal of minors I2 (Ca ), R(M ) = S/I2 (Ca ). Let L be the kernel of the epimorphism φ defined in (1). Recall that I2 (Ca ) ⊂ L, where the first ideal is prime, according to Corollary 2.8, and the second ideal has dimension n + r, according to [27, 2.2]. Hence to show that equality holds it will be enough to prove that the dimension of I2 (Ca ) is at most n + r. By Remark 2.5, Ta′ /I2 (Ba′ ) ∼ = Sa /I2 (Ca ). Hence it will be enough to prove that the dimension of I2 (Ba′ ) is at most n + r which is equivalent to show that the dimension of I2 (Ba ) REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 7 is at most n + r − 1. As ht I2 (Ba ) = ht inτ (I2 (Ba )), we can compute the dimension of inτ (I2 (Ba ). Let U be the set of r + n − 1 variables U = {{Tl,0,··· ,0 }1≤l≤r , Tr,ar ,··· ,ar , Tr,ar ,...,ar ,0, . . . , Tr,ar ,0,··· ,0 } . Consider the prime p ∈ Spec(T ) generated by all the variables of T that are not in U . We claim that p is a minimal prime over inτ (I2 (Ba ). It is clear that the image of inτ (I2 (Ba )) in the quotient ring T /p is zero. To show the claim, consider the prime ideal p′ ⊂ p obtained by deleting one variable Tl,j , with j ∈ Jal , from the generating set of p. If l < r, then Tl,j Tr,0 ∈ inτ (I2 (Ba ) \ p′ . Hence the image of inτ (I2 (Ba )) in the quotient ring T /p′ is not zero. If l = r, then 0 < ju ≤ . . . ≤ ji < ar for some u, i with 1 ≤ u ≤ i ≤ n − 1. Thus we can assume that Tl,j = Tr,ar ,...,ar ,ji ,...,ju ,0,...,0 6∈ p′ . Therefore the element Tl,j appears at least in two columns of the matrix Ba , namely the columns corresponding to the sequences r, ar , . . . , ar , ji , . . . , ju , 1, . . . , 1 and r, ar , . . . , ar , ji + 1, . . . , ju + 1, 1, . . . , 1. 2 ∈ in (I (B ) \ p′ . Again the image of in (I (B )) in the quotient ring T /p′ is not zero. Hence Tl,j τ 2 a τ 2 a Hence dim T /I2 (Ba ) = dim T /inτ (I2 (Ba )) = dim T /p = \|U \| = r + n − 1 as claimed, where the second equality follows as I2 (Ba ) is a prime ideal (see for instance [19]). From the above follows that for any sequence of r positive integers a = a1 , . . . , ar the special fiber ring of M is defined by the ideal of minors I2 (Ba ), indeed F(M ) = k ⊗R R(M ) = k ⊗R S/I2 (Ca ) = T /I2 (Ba ). For any sequence a the ideals I2 (Ca ) and I2 (Ba ) have a Gröbner basis of quadrics according to Lemma 2.6, hence both the Rees algebra and the special fiber ring of M are Koszul domains. Normality follows because F(M ) is a direct summand of R(M ) which in turn is a direct summand of R[t1 , . . . , tr ]. The latter claim can be easily seen once we consider R(M ) as a Nr+1 −graded R[t1 , . . . , tr ]-algebra. In the rest of this section we study the divisor class group of the normal domain A = Ta /I2 (Ba ) for any sequence a. Definition 2.10. Let K be the A-ideal generated by all the variables appearing in the first row of Ba , i.e. all the variables Tl,j with 1 ≤ l ≤ r and j ∈ Ja′l . Theorem 2.11. The divisor class group Cl(A) is cyclic generated by K. Proof. To compute the divisor class group we use Nagata’s Theorem: If W ⊂ A is a multiplicatively closed set, then there is an exact sequence of Abelian groups 0 −→ U −→ Cl(A) −→ Cl(AW ) −→ 0 , 8 K. N. LIN AND C. POLINI where U is the subgroup of Cl(A) generated by {[p] \| p a height one prime ideal with p ∩ W 6= ∅} . i i \| i ∈ Z}. Notice Tr,ar 6∈ I2 (Ba ). Hence = Tr,a We use the above theorem with W = {Tr,a r r ,··· ,ar Tr,ar is regular on T /I2 (Ba ). After localizing at Tr,ar the ideal I2 (Ba )Tr,ar is isomorphic to an ideal generated by variables, and the ring ATr,ar is a polynomial ring, hence factorial. Thus Cl(AW ) = 0 and Cl(A) = U . Now we will show that U is cyclic generated by K. Notice that [K] ∈ U because K is a prime ideal of height one containing Tr,ar . Clearly, Tr,ar ∈ K. To show that A/K is a domain of dimension dim A − 1, let R′ = k[x2 , . . . , xn ] be the polynomial ring over k in n − 1 variables, let m′ be its homogeneous maximal ideal, and let M ′ = m′a1 ⊕ · · · ⊕ m′ar . The claim follows by ∼ F(M ′ ) and F(M ′ ) is a domain of dimension n − 1 + r − 1 = dim A − 1. Theorem 2.4 as A/K = Let P be the A-ideal generated by all the variables Tr,j with j ∈ Jar . Clearly, P is prime. Indeed, ∼ F(Ma′ ) with a′ = if r = 1, then A/P = k; if r > 1, then, according to Theorem 2.4, A/P = a1 , . . . , ar−1 . Furthermore, if r > 1, P has height one as dim A/P = dim F(Ma′ ) = dim A − 1. Next we show that every prime ideal p in A containing Tr,ar contains either K or P . Assume that K 6⊂ p, then there exists a variable Tl,i 6= Tr,ar with i ∈ Ja′l that is not in p. Recall that the set of multi indices Jar is ordered by τ . For all j ∈ Jar we show that Tr,j ∈ p by descending induction on Jar . The base case is trivial since Tr,ar ∈ p. Assume Tr,j 6= Tr,ar . Then there exists a multi-index s ∈ Jar with s > j such that the equality Tr,j Tl,i = Tr,s Tl,i,t holds in A for some integer t ≥ 2. Notice that Tl,i,t is well defined because i ∈ Ja′l . By induction Tr,s ∈ p, thus Tr,j ∈ p since p is prime and Tl,i 6∈ p. Hence P ⊂ p. If r = 1 the above inclusion implies that ht p > 1 and hence U is cyclic generated by K. If r > 1, then U is generated by P and K. We conclude by showing that [P ] = ar [K], or equivalently, P = (Tr,ar ) :A K ar . Since both ideals have height one and P is prime it is enough to prove the inclusion P ⊂ (Tr,ar ) :A K ar . The latter follows from the equation (3) Tr,j K ar −j1 ∈ (Tr,ar ) . We prove equation (3) using descending induction on j1 with 0 ≤ j1 ≤ ar . The base case is trivial since j1 = ar implies Tr,j = Tr,ar . If j1 < ar , then the variable Tr,j appears in a row s of Ba with s > 1. Thus for any element λ ∈ K there exists β ∈ S such that the equality Tr,j λ = Tr,jn−1 ,··· ,js+1 ,js +1,··· ,j1 +1 β holds in A. Now the claim follows by induction. According to Theorem 2.11 the classes of the divisorial ideals K (δ) and P (δ) , the δ-th symbolic power of K and P respectively, constitute Cl(A). In the next theorem we exhibit a monomial REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 9 generating set for K (δ) for δ ≥ 1. As in [20] and [25] we identify K (δ) with a graded piece of A. We put a new grading on the ring T , Deg(Tl,j ) = j1 . Notice that I2 (Ba ) is an homogeneous ideal with respect to this grading. Thus Deg induces a grading on A. Let A≥δ be the ideal generated by all monomials m in A with Deg(m) ≥ δ. Theorem 2.12. The δ-th symbolic power of K, K (δ) , equals the monomial ideal A≥δ . Proof. One proceeds as in [21, 1.5]. 3. T HE B LOWUP ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS In this section our goal is to compute explicitly the defining equations of the blowup algebras of truncations of complete intersections. Assumptions 3.1. Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k with homogeneous maximal ideal m. Let r be an integer with 1 ≤ r ≤ n. Let f1 , . . . , fr be a homogeneous regular sequence in R of degree d1 ≥ . . . ≥ dr . Let d ≥ d1 be an integer, write ai = d − di . Let I be the truncation of the complete intersection (f1 , . . . , fr ) in degree d, i.e. the R-ideal generated by {mai fi \| 1 ≤ i ≤ r} , I = (f1 , . . . , fr )≥d = (ma1 f1 , . . . , mar fr ). Theorem 3.2. Adopt assumptions 3.1 and write M = ma1 ⊕ . . . ⊕ mar . There is a short exact sequence 0 → Q → R(M ) → R(I) → 0 where Q is a prime ideal of height r − 1 in the normal domain R(M ). Proof. The natural map ζ M −→ (u1 , . . . , ur ) 7→ I = ma1 f1 + . . . + mar fr → 0 u1 f1 + . . . + ur fr induces a surjection on the level of Rees algebras Ψ : R(M ) −→ R(I) and Ker Ψ is a prime of height r − 1 since R(M ) and R(I) are domains and dim R(M ) = n + r = dim R(I) + r − 1. 10 K. N. LIN AND C. POLINI Definition 3.3. Adopt assumptions 3.1. Let a be the sequence a1 , . . . , ar and let S be the polynomial ring Sa . Consider the algebra epimorphism χ obtained by the composition of the two algebra epimorphisms φ : S −→ R(M ) Ψ : R(M ) −→ R(I) and with φ as in (1) and let A be the S-ideal defined by the short exact sequence χ 0 → A −→ S −→ R(I) → 0 . Let τ be the order on Tl,j defined in Section 2. Notice that, through the algebra epimorphism χ, the order τ induces an order on the set C = xal ,j fl \| 1 ≤ l ≤ r, j ∈ Jal of generators of I. Let ϕ be the presentation matrix of I over R with respect to C. In the following remark we give a resolution of I when r = 2. In particular, we describe explicitly ϕ. Remark 3.4. To compute a resolution of I = (f1 , f2 )≥d we first truncate the Koszul complex of f1 , f2 in degree d. (1) If d2 ≤ d ≤ d1 + d2 we obtain the short exact sequence: ρ ζ 0 −→ R(−d1 − d2 + d) −→ M −→ I −→ 0 , −f2 where ζ is the natural surjection described in Theorem 3.2 and ρ = . Set ϕ′′ to f1 −f2 be the map that rewrites in terms of the k-basis B of M ordered with the order f1 induced by τ . The Eagon-Northcott complex gives us an R-resolution F• of M , while the complex G• that is trivial everywhere except in degree 0 gives us an R-resolution of R(−d1 − d2 + d). The map ρ can be trivially lifted to a morphism of complexes ρ• : G• −→ F• that is trivial in positive degree and is ϕ′′ in degree zero. (2) If d ≥ d1 + d2 + 1 we obtain the short exact sequence: ρ ζ 0 −→ md−d1 −d2 −→ M −→ I −→ 0 , −f2 where ζ is the natural surjection described in Theorem 3.2 and ρ = . Write E f1 for the k-bases of md−d1 −d2 ordered with the order induced by τ . Set ϕ′′ to be the map −f2 that rewrites E in terms of the k-basis B of M ordered with the order induced f1 by τ . From the Eagon-Northcott complex we obtain R-resolutions F• and G• of M and md−d1 −d2 , respectively. The map ρ can be lifted to a morphism of complexes ρ• : G• −→ F• with ϕ′′ in degree zero. REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 11 In both cases the mapping cone C(ρ• ) is a non-minimal free resolution of I. In particular, the presentation matrix of I with respect to C is ϕ = [ϕ′ , ϕ′′ ] where ϕ′ is the matrix presenting M with respect to B. In the following remark we describe explicitly ϕ for any r when d ≥ d1 + d2 . Remark 3.5. Adopt assumptions 3.1. To compute the presentation matrix for I = (f1 , . . . , fr )≥d V we need to truncate the Koszul complex K• (f1 , . . . , fr ) = Re1 ⊕ . . . ⊕ Rer . If d ≥ d1 + d2 then d ≥ di + dj for any 1 ≤ i < j ≤ r and we have: M . . . −→ m d−di −dj ei ∧ ej −→ M = 1≤i<j≤r r M mai −→ I −→ 0 . i=1 md−di −dj , Write B and Ei,j for the k-bases of M and of respectively, ordered with the order induced ′′ by τ . Set ϕ to be the map that rewrites Ei,j (−fj ei + fi ej ) in terms of B for all 1 ≤ i < j ≤ r. The presentation matrix of I with respect to C is ϕ = [ϕ′ , ϕ′′ ] where ϕ′ is the matrix presenting M with respect to B. Definition 3.6. Adopt assumptions 3.1. Let h1 , . . . , ht ∈ S be the homogeneous polynomials obtained by the matrix multiplication [T ]ϕ′′ , where ϕ′′ is the matrix described in Remark 3.4 and 3.5. Think of S as a naturally bigraded ring with deg xi = (1, 0) and deg Tj = (0, 1). If r = 2 and (δ, 1) with δ = d2 −a1 = d1 −a2 = d1 +d2 −d ≥ d1 ≤ d ≤ d1 +d2 , then t = 1 and h1 has bidegree X σi,j + n − 1 0. For any r, if d ≥ d1 + d2 then t = with σi,j = d − di − dj ≥ 0 and the n−1 1≤i<j≤r hk ’s have bidegree (0, 1). Proposition 3.7. Adopt assumptions 3.1, 3.6 with d ≥ d1 +d2 then (h1 , . . . , ht )R(M ) is a non-zero prime ideal of height ≥ r − 1. Proof. Let 1 ≤ i < j ≤ r. Write X λk xdi ,k fi = and fj = X αk xdj ,k k∈Jdj k∈Jdi Let σi,j = d − di − dj ≥ 0. As in Definition 2.1, denote the elements of the basis Ei,j of mσi,j with xσi,j ,t,s , t ∈ Jσ′ i,j . Since we have a one to one correspondence between the hk ’s and the elements of Ei,j , we write hk as hσi,j ,t,s . Let H be the S-ideal generated by {hσi,j ,t,s \| 1 ≤ i < j ≤ r, t ∈ Jσ′ i,j , 1 ≤ s ≤ n}. We obtain X X λk xai ,k+t,s αk xaj ,k+t,s , and xσi,j ,t,s fj = xσi,j ,t,s fi = k∈Jdi k∈Jdj hence hσi,j ,t,s = X k∈Jdi λk Ti,k+t,s − X k∈Jdj αk Tj,k+t,s . 12 K. N. LIN AND C. POLINI Let Hj be the S-ideal generated by {hσ1,l ,t,s \| 2 ≤ l ≤ j, t ∈ Jσ′ 1,l , 1 ≤ s ≤ n}. We show by induction on j with 2 ≤ j ≤ r that the ideal Hj R(M ) is prime of height ≥ j − 1. Let j = 2. Notice that for each t ∈ Jσ′ 1,2 , the column [hσ1,2 ,t,1 , . . . , hσ1,2 ,t,n ]tr is a linear combination of columns of P +n−2 +n−2 Ca . Write Ea for the n × (1 + rq=1 aqn−1 ) matrix obtained by Ca by substituting σ1,2n−1 columns with [hσ1,2 ,t,1 , . . . , hσ1,2 ,t,n ]tr , t ∈ Jσ′ 1,2 . The two S-ideals I2 (Ca ) + H2 and I2 (Ea ) + H2 are equal. The ideal H2 R(M ) is prime according to Corollary 2.8 as S/(I2 (Ea ) + H2 ) ∼ = T ′ /I2 (Da ) for some polynomial ring T ′ and Da a suitable submatrix of Ca with n rows. Now degree considerations show that hσ1,2 ,t,s 6∈ I2 (Ca ), hence H2 R(M ) 6= 0. Thus the ideal H2 R(M ) is prime of height at least one. Let 2 ≤ l ≤ j and assume by induction that Hj−1 R(M ) is prime of height ≥ j − 2, we show Hj R(M ) is prime of height ≥ j − 1. For each t ∈ Jσ′ 1,l , the column [hσ1,l ,t,1 , . . . , hσ1,l ,t,n ]tr P +n−2 is a linear combination of columns of Ca . Write Ea for the n × (1 + rq=1 aqn−1 ) matrix Pj σ1,l +n−2 tr columns with [hσ1,l ,t,1 , . . . , hσ1,l ,t,n ] , 2 ≤ l ≤ j obtained by Ca by substituting l=2 n−1 ′ and t ∈ Jσ1,l . The two S-ideals I2 (Ca ) + Hl and I2 (Ea ) + Hl are equal. The ideal Hj R(M ) is prime according to Corollary 2.8 as S/(Ij (Ea ) + Hj ) ∼ = T ′ /I2 (Da ) for some polynomial ring T ′ and Da a suitable submatrix of Ca with n rows. Notice that for each t ∈ Jσ′ 1,j , the column [hσ1,j ,t,1 , . . . , hσ1,j ,t,n ]tr is a linear combination of a subset of the columns {(1, t) \| t ∈ Ja′1 } and {(j, t) \| t ∈ Ja′j } of Ca , while the column [hσ1,l ,t,1 , . . . , hσ1,l ,t,n ]tr is a linear combination of a subset of the columns {(1, t) \| t ∈ Ja′1 } and {(l, t) \| t ∈ Ja′l } of Ca with 2 ≤ l ≤ j − 1. Hence degree considerations show that hσ1,j ,t,s 6∈ (Hj−1 , I2 (Ca )), hence Hj R(M ) 6= 0. Thus the ideal Hj R(M ) is prime of height ≥ j − 1. Using the same argument one can show that HR(M ) is a prime ideal and its height is at least r − 1 as HR(M ) ⊃ Hr R(M ) . Remark 3.8. If d ≥ d1 +d2 −1, then the ideal I = (f1 , . . . , fr )≥d has a linear resolution. The Rees algebra of linearly presented ideals of height 2 has been described explicitly in terms of generators and relations in [18] under the additional assumption that I is perfect. However, if r < n, the truncations of codimension r complete intersections are never perfect. For large d, the Rees algebra R(I) is Cohen-Macaulay as shown in [12]. But the defining equations of R(I) were unknown, we give them explicitly in Theorem 3.9. If r = 2 we prove that for d ≥ d1 + d2 − 1 the Rees ring R(I) is a Koszul domain (see Corollary 3.13). If r ≥ 3, we prove that R(I) is a Koszul domain for d ≥ d1 + d2 (see Corollary 3.13). In addition in [16] we study the depth and regularity of the blowup algebras of I. REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 13 Theorem 3.9. Adopt assumptions 3.1 and 3.6 with d ≥ d1 + d2 then R(I) = R(Ma )/(h1 , . . . , ht ) = Sa /(I2 (Ca ), h1 , . . . , ht ) . Proof. Adopt the notation of the proof of Proposition 3.7. Let H be the S-ideal generated by {hσi,j ,t,s \| 1 ≤ i < j ≤ r, t ∈ Jσ′ , 1 ≤ s ≤ n}. According to Remark 3.5, we have H ⊂ A. Hence HR(M ) ⊂ AR(M ) where the first ideal is a non-zero prime ideal of height ≥ r − 1 by Proposition 3.7 and the second one has height r − 1. Assume r = 2. In the following theorem we express the Rees algebra of I as defined by a divisor on the Rees algebra of the module M = ma1 ⊕ ma2 that we computed explicitly in the previous section. Indeed, for r = 2 the prime ideal Q ∈ Spec(R(M )) of Theorem 3.2 gives rise to an element of the divisor class Cl(R(M )). This group has been studied explicitly in Theorem 2.11: it is cyclic generated by the prime ideal L, where L be the R(M )-ideal generated by all the variables appearing in the first row of Ca . In the next theorem we identify for which s the ideal L(s) is isomorphic to Q. Definition 3.10. Let L be the R(M )-ideal generated by all the variables appearing in the first row of Ca . We will use the convention that the (symbolic) power of any element or ideal with nonpositive exponent is one or the unit ideal, respectively. Theorem 3.11. Adopt assumptions 3.1, 3.6, and 3.10 with r = 2 then xδ1 AR(M ) = (h1 , . . . , ht )L(δ) . In particular, (1) If d1 ≤ d ≤ d1 + d2 − 1 then the R(M )-ideals xδ1 AR(M ) and h1 L(δ) are equal and the bigraded R(M )-modules A and L(δ) (0, −1) are isomorphic. (2) If d ≥ d1 + d2 then R(I) = Sa /(I2 (Ca ), h1 , . . . , ht ) . Proof. Notice that the first statement follows from (1) and (2) and (2) has been proven in Theorem 3.9. Thus it will be enough to prove (1). Write h = h1 . One proceeds as in [21, 1.11]. For clarity we rewrite part of the proof here since there are some minor differences and in [21, 1.11] the integer δ was assumed to be ≥ 2. Degree considerations show that x1 is not in I2 (Ca ). The ideal I2 (Ca ) is prime, so xδ1 is also not in I2 (Ca ). The second assertion in (1) follows from the first as xδ1 has bi-degree (δ, 0) and h1 has bi-degree (δ, 1). Write to mean image in R(M ). We prove the equality xδ1 AR(M ) = hL(δ) by showing that A = (h̄/x̄δ1 )L(δ) , where the fraction is taken in the quotient field Q of R(M ). 14 K. N. LIN AND C. POLINI Notice that (x̄i1 ) :Q L(i) = (x̄1 , . . . , x̄n )i . This follows as in the proof of claim (1.12) in [21, 1.11]. Furthermore h̄ ∈ mδ = (x̄δ1 ) :Q L(δ) . Thus, h̄L(δ) ⊆ x̄δ1 R(M ). Define D to be the ideal (h̄/x̄δ1 )L(δ) of R(M ). At this point, we see that the ideal D is either zero or divisorial. To show that D is not zero and to establish the equality A = D, it suffices to prove that A ⊆ D, because A is a height one prime ideal of R(M ). This is the only part where the argument differs from [21, 1.11]. Notice that h̄ ∈ D as x̄1 ∈ L. For every w ∈ m, one has Iw = (f1 , f2 )w . Therefore, R[It]w = R[(f1 , f2 )t]w and we obtain (h̄)w = Aw . It follows that h̄ 6= 0 and Aw ⊆ Dw . The rest of the proof follows as in [21, 1.11]. Corollary 3.12. Adopt assumptions 3.1, 3.6, and 2.10. (a) If d ≥ d1 + d2 then F(I) = F(Ma )/(h1 , . . . , ht ) = Ta /(I2 (Ba ), h1 , . . . , ht ) . (b) If r = 2 and d1 ≤ d ≤ d1 + d2 − 1 then F(I) = Ta /K with K∼ = K (δ) (−1). Proof. The proof follows from Theorem 3.9 and Theorem 3.11 and the fact that F(I) = k ⊗ R(I). Also the same argument as in [21, 4.2] shows the isomorphism K ∼ = K (δ) (−1). Corollary 3.13. Adopt assumptions 3.1. If d ≥ d1 + d2 or if r = 2 and d = d1 + d2 − 1, then R(I) and F(I) are Koszul algebras. Proof. We will prove the statement for the Rees ring. The same proof works for the special fiber ring using Corollary 3.12. If d ≥ d1 + d2 , then R(I) = S/(I2 (Ca ), h1 , . . . , ht ) according to Theorem 3.9, and according to the proof of Corollary 2.8 the latter ring is isomorphic to T ′ /I2 (Da ) for some polynomial ring T ′ and Da a suitable submatrix of Ca with n rows. But this ring is Koszul as it has a Gröbner of quadrics by Lemma 2.6. 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1 Anti-jamming Communications Using Spectrum Waterfall: A Deep Reinforcement Learning Approach arXiv:1710.04830v1 [cs.IT] 13 Oct 2017 Xin Liu, Yuhua Xu, Member, IEEE, Luliang Jia, Student Member, IEEE, Qihui Wu, Senior Member, IEEE, and Alagan Anpalagan, Senior Member, IEEE Abstract—This letter investigates the problem of anti-jamming communications in dynamic and unknown environment through on-line learning. Different from existing studies which need to know (estimate) the jamming patterns and parameters, we use the spectrum waterfall, i.e., the raw spectrum environment, directly. Firstly, to cope with the challenge of infinite state of raw spectrum information, a deep anti-jamming Q-network is constructed. Then, a deep anti-jamming reinforcement learning algorithm is proposed to obtain the optimal anti-jamming strategies. Finally, simulation results validate the the proposed approach. The proposed approach is relying only on the local observed information and does not need to estimate the jamming patterns and parameters, which implies that it can be widely used various anti-jamming scenarios. Index Terms—Anti-jamming, Deep Q-Network, Deep Reinforcement Learning I. I NTRODUCTION Anti-jamming is always an active research topic, as wireless transmissions are naturally vulnerable to jamming attacks. The mainstream anti-jamming techniques includes Frequency Hopping Spread Spectrum (FHSS) and Direct-Sequence Spread Spectrum (DSSS) [1]. Recently, to address the interactions between the legitimate users and the jammers, game theory has been widely applied in the literature [2]–[7]. In methodology, these approaches need to know the jamming strategies, which implies that the legitimate users are required to estimate the jamming patterns and parameters from the observed environment. However, with the rapid development of artificial intelligence and universal software radio peripheral (USRP) [8], the jammers can easily create dynamic and intelligent jamming attacks. As a consequence, there are two limitations with regard to estimation-based anti-jamming communications: i) there may be information loss for unknown jamming patterns, and ii) if the intelligent jammer switches its strategies dynamically and rapidly, it is not possible to track and react it in real time. Thus, it is challenging and interesting to investigate antiX. Liu is with the College of Information Science and Engineering, Guilin University of Technology, Guilin 541004, China. (email:leo nanjing@126.com). Y. Xu and L. Jia are with the College of Communication Engineering, PLA Army Engineering University, Nanjing 210007, China. (e-mail: yuhuaenator@gmail.com;jiallts@163.com). Qihui Wu is with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China (e-mail: wuqihui2014@sina.com). Alagan Anpalagan is with the Department of Electrical and Computer Engineering, Ryerson University, Toronto, Canada (alagan@ee.ryerson.ca). jamming communication approaches in dynamic and unknown environment. To overcome the above limitations, a promising way is to design new anti-jamming approaches that utilize the raw environmental information, which is known as spectrum waterfall [9], without estimating jamming patterns and parameters. These kind of anti-jamming approaches would avoid information loss and adapt to the dynamic environment, as can be expected. In addition, online learning is an effective way to solve the decision problems in dynamic environment. The widely used technique is Q-learning [10], which has been used in anti-jamming problems [2], [3]. Unfortunately, Q-learning is not able to deal with the raw environmental information directly because of the infinite state of the environment. Motivated by the deep reinforcement learning technique for learning successful control policies from raw video data in [11], we investigate the anti-jamming problem in unknown and dynamic environment. First, the raw spectrum information is defined as the state of the environment to avoid losing the jammer information as much as possible. Then, a deep anti-jamming Q-network (DAQN) is constructed to realize the direct processing of raw spectrum information. Finally, a deep anti-jamming reinforcement learning algorithm (DARLA) is proposed. Simulation results show that the proposed DARAL achieves the best anti-jamming strategies in various scenarios. The main contributions are summarized as follows. • • Based on the deep reinforcement learning technique, a smart anti-jamming communication scheme is proposed. In particular, the raw spectrum information is defined as a state, which describes the detail features of jammer more accurately. The proposed algorithm is relying only on the locally observed information and does not need to estimate the jamming patterns and parameters the jammer in advance, i.e., it is model-free, which can be widely used in various anti-jamming scenarios. Note that the most related work is [12], which also adopted deep reinforcement learning to investigate the anti-jamming problems. The main differences in this work are as follows: i) the environment state is presented by extracting features of signal-to-interference-plus-noise ratio (SINR) and primary user occupancy in [12], while it is presented by the raw spectrum information in this work, and ii) it requires the jammer to have the same channel-slot transmission structure 2 known as spectrum waterfall [9], as shown in Fig. 2. Taking the swept jamming as example, we can accurately determine the frequency range and intensity (color) of jamming at the next moment by looking at the thermal chart, which also means we can determine the anti-jamming strategy accordingly. In the unknown and dynamic environment, we do not consider estimation-based anti-jamming strategies. Instead, define St as the environment state, and then consider a dynamic decision problem in which the agent (anti-jamming user) interacts with an environment through a sequence of observations of environment (St ), actions (at ) and rewards (rt ). Specifically, an action a can be a combination decisions of frequency, power, coding schemes, spread spectrum, and other kinds of anti-jamming decisions, e.g., a = (f, p) represents the combination actions of frequency (f ) and power (p). The rewards associated with the actions and environment is defined as: R(a) − λδ β(a, S) ≥ βth (a) r(a, S) = , (2) 0 β(a, S) < βth (a) Fig. 1. System model. Fig. 2. Thermodynamic chart of various jamming pattern. with the users in [12]. On the contrary, this requirement does not hold in our work, which makes the proposed approach more general. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION We consider the transmission of one user (a transmitterreceiver pair) against one or several jammers, as shown in Fig. 1. The agent, which is disposed at the receiving end, sends anti-jamming strategies to the transmitter through a reliable control link. Jammers may adopt fixed, random, or possibly intelligent jamming patterns. However, we do not analyze the specific jamming models, but obtain the optimal anti-jamming strategies based on the raw spectrum information. While the receiver receives the desired signal, the agent continuously senses the whole communication bands and stores the sensed values. Denote the spectrum vector as Pt = {pt,1 , pt,2 , · · · , pt,N }, where pt,n is the power of frequency n at time t and N is the number of sampling points in frequency space. In order to sufficiently use history spectrum information, a two-dimensional matrix, which describes timefrequency features of spectrum environment, is expressed as:   pt−1,1 Pt−1  Pt−2   pt−2,1    St =  .  =  .  ..   ..  Pt−M pt−M,1 pt−1,2 pt−2,2 .. . ··· ··· .. . pt−1,N pt−2,N .. . pt−M,2 ··· pt−M,N     . (1)  It is noted that St contains all the spectrum information until time t, as M tends to infinity. However, the difficulty of the decision optimization problem is significantly increased with the increase of M . Therefore, M can take an appropriate value, which would be determined by the time-varying characteristics of the spectrum environment. To illustrate the rationality of using St as the basis of antijamming decision-making, we give the thermodynamic charts of the St matrix of several common jamming patterns, also where R(a) is the bit rate when the action a is selected, λ is the cost when action changes, δ is an indication of action change (δ = 1 if at 6= at−1 ; δ = 0 if at = at−1 ), β(a, S) is the received signal to interference plus noise ratio (SINR) in state S with action a. βth (a) is the required SINR threshold for successful transmission. Note that R(a) and βth (a) are modeled as a function of action a, the reason is as follows: the bit rate and SINR requirements change for different anti-jamming strategies, such as forward error correction and spread spectrum schemes. Then, the goal of the agent is to select anti-jamming actions in a fashion that maximizes cumulative future reward Rt = P∞ i i=0 γ rt+i+1 , where γ is the discount factor. One way of achieving this goal is to compute the following optimal actionvalue (also known as Q) function [10]: Q∗ (S, a) = max E {Rt \| St = S, at = a, π}, π (3) where the anti-jamming policy π = P (a \| S) refers to a probability distribution over the actions. Based on the Bellman equation, n o ∗ ′ ′ Q∗ (S, a) = E r + γ max Q (S , a ) \| S, a . (4) ′ a III. A NTI - JAMMING C OMMUNICATION S CHEME The traditional Q-learning is unable to cope with the antijamming problem described in section II, as the state space size of St is almost infinite. In order to solve this problem, a deep anti-jamming Q-network (DAQN) is constructed to address the interactive decision-making problem with raw spectrum information input, which contains decision network and update network, as shown in Fig. 3 and Fig. 4 respectively. We use a deep convolutional neural network (CNN) to approximate the optimal action-value as shown in decision network, where the input state St is represented by a thermal chart of M ×N pixels. After the processing of two convolutional layers and two fully connected layers, the output is the estimated Q function, where K is the size of action space. At last, the 3 Fig. 3. Decision network of the DAQN. Algorithm 1: Deep Anti-jamming Reinforcement Learning Algorithm (DARLA) Initialize : Set D = ∅, ǫ = 1, Set θ with random weights, Sense initial environment S1 . For t = 1, T do With probability ǫ, select a random action at otherwise, select at = arg max Q(St , a; θ) a Fig. 4. Updating network of the DAQN. decision layer outputs the corresponding action based on the estimated Q function. However, reinforcement learning is known to be unstable or even to diverge when a nonlinear function approximator [11], such as the neural network is used to represent the Q function. The main reason is correlation during the learning process. The idea of experience replay is adopted to address these instabilities as shown in the update network. To perform experience replay, we store the agent’s experiences et = (St , at , rt , St+1 ) at each time-step t in data set Dt = (e1 , · · · et ). When the experience pool is big enough, we construct target values r + γ max Q(S′ , a′ ) by randomly choosing elements in a a′ uniform distribution (S, a, r, S′ ) ∼ U (D), which reduces the correlation during sequential observation. The Q-learning update at iteration i uses the following loss function: i h 2 Li (θi ) = E(S,a,r,S′ )∼U(D) (yi − Q(S, a; θi )) , (5) where θi is the parameter of Q-network at iteration i and yi = r + γ max Q(S′ , a′ ; θi−1 ) is target value computed by a′ Q-network parameter θi−1 with greedy strategy. By assuming that yi is the expected output of CNN with network weight θi when the input is S, we calculate the difference between real output Q(S, a; θi ) and target value yi to determine the update of network parameters. Differentiating the loss function with respect to the weights, we arrive at the following gradient: ∇θi Li (θi ) = E(S,a,r,S′ ) [(yi − Q(S, a; θi )) ∇θi Q(S, a; θi )] . (6) According to the gradient descent algorithm, the network weight θi is updated according to (6). Although there are two Execute action at and compute rt and observe St+1 Store transitions (St , a, r, St+1 ) in D If Sizeof (D) > N (Enough amount of transitions) Sample random minibatch of transistions (S, a, r, S′ ) from D Compute yi = r + γ max Q(S′ , a′ ; θ) a′ Compute gradient based on Eq.(6) and Update θ End If Calculate ǫ = max(0.1, ǫ − ∆ǫ) End For CNN networks with different weights, as shown in Fig. 4, the actual implementation requires only one CNN network, as the computing of target values and the updating of network weights are in different stages. The algorithm for anti-jamming communication based on deep reinforcement learning is presented in Algorithm 1. IV. N UMERICAL R ESULTS AND D ISCUSSIONS In the simulation setting, the user and the jammer combat with each other in a frequency band of 20MHz, where the frequency resolution of spectrum sensing is 100kHz. The user performs a full band sensing every 1ms and retains the spectrum data within the 200ms. Hence, the size of matrix St is 200 × 200. The bandwidth of user signal is 4MHz, and the center frequency is allowed to change in each 10ms with the step of 2MHz, which means K = 9. Both signal and jamming are raised cosine waveform with roll-off factor α = 0.3, in which jamming power is 30dBm and signal power is 0dBm. The demodulation threshold βth at all frequency is set to be 10dB, and the cost of action change λ is set to be 0.2R(a). Four kinds of jamming patterns are given for simulation: i) Sweep jamming (sweep speed is 1GHz/s); ii) Comb jamming (three fixed frequency signals at 2MHz, 10MHz, and 18MHz); 4 1 0.9 Normalized throughput 0.8 0.7 0.6 0.5 0.4 0.3 Sweep Jamming Comb Jamming Random Jamming 0.2 0.1 0 0 0.5 1 Iteration 1.5 2 4 x 10 Fig. 5. Normalized throughput under different jamming patterns. and (c) respectively, and the converging states are given in Fig. 6(d), (e), and (f) respectively. These states that contain time-frequency information can clearly reflect the past actions of user and jamming. Taking sweep jamming as an example, at the beginning of the learning procedure, the user adopts randomized action as it is unfamiliar with environment (the locations of rectangular blocks are randomly distributed), and after convergence, the frequency is properly changed before the jamming arrives (the rectangular blocks are distributed according to the slashes). With regard to the intelligent jamming, since the probability distribution of user actions is the basis for jammer to release jamming, the best strategy for user is that the probability of each action is almost identical. The simulation results in Fig. 7 show the probabilities of each action being selected during the learning procedure, which is consistent with our analysis. V. C ONCLUSION Fig. 6. Environmental states at initial and convergent stages under different jamming patterns. 0.25 Probability 0.2 0.15 In this letter, we investigated the anti-jamming problem in unknown and dynamic environment. Aiming at employing the waterfall spectrum information directly, we constructed a deep anti-jamming Q-network to handle the complex interactive decision-making problem with infinite number of states. Then, a deep anti-jamming reinforcement learning algorithm was proposed. Using the proposed learning algorithm, the user is able to learn the best anti-jamming strategy by constantly trying various actions and sensing the spectrum environment. Simulation results in various scenarios are presented to validate the proposed anti-jamming communication approach. Future work on designing multi-user deep anti-jamming reinforcement learning algorithms is ongoing. R EFERENCES 0.1 0.05 0 10 8 200 6 150 4 100 2 Action index 50 0 0 Iteration/100 Fig. 7. Probability of user actions during learning. iii) Random jamming (frequency is randomly changed every 20ms with the step of 4MHz); iv) Intelligent jamming (the jammer continuously observes the probability that the user signal appears at each frequency point, and chooses the largest one as jamming channel). For all Jamming patterns, the instantaneous bandwidth of the jamming is set to be 4MHz. The normalized average throughput of legitimate user under different jamming patterns is given in Fig. 5. It is shown that the anti-jamming ability of user has been improved significantly with the proposed DARLA learning. Especially in the case of comb jamming, the normalized throughput is close to one after convergence, which indicates that the jamming is almost completely avoided. Environmental states at initial stages under sweep, comb and random jamming patterns are given in Fig. 6(a), (b), [1] L. 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arXiv:1706.03568v1 [cs.DS] 12 Jun 2017 Monitoring of Domain-Related Problems in Distributed Data Streams∗ Pascal Bemmann Felix Biermeier Jan Bürmann Arne Kemper Till Knollmann Steffen Knorr Nils Kothe Alexander Mäcker Manuel Malatyali Friedhelm Meyer auf der Heide Sören Riechers Johannes Schaefer Jannik Sundermeier Heinz Nixdorf Institute & Computer Science Department Paderborn University, Germany {pbemmann, felixbm, jbuerman, kempera, tillk, stknorr, nkothe, amaecker, malatya, fmadh, soerenri, jschaef, janniksu} @mail.uni-paderborn.de Abstract Consider a network in which n distributed nodes are connected to a single server. Each node continuously observes a data stream consisting of one value per discrete time step. The server has to continuously monitor a given parameter defined over all information available at the distributed nodes. That is, in any time step t, it has to compute an output based on all values currently observed across all streams. To do so, nodes can send messages to the server and the server can broadcast messages to the nodes. The objective is the minimisation of communication while allowing the server to compute the desired output. We consider monitoring problems related to the domain Dt defined to be the set of values observed by at least one node at time t. We provide randomised algorithms for monitoring Dt , (approximations of) the size \|Dt \| and the frequencies of all members of Dt . Besides worst-case bounds, we also obtain improved results when inputs are parameterised according to the similarity of observations between consecutive time steps. This parameterisation allows to exclude inputs with rapid and heavy changes, which usually lead to the worst-case bounds but might be rather artificial in certain scenarios. ∗ This work was partially supported by the German Research Foundation (DFG) within the Priority Program “Algorithms for Big Data” (SPP 1736) and by the Federal Ministry of Education and Research (BMBF) as part of the poject “Resilience by Spontaneous Volunteers Networks for Coping with Emergencies and Disaster” (RESIBES), (grant no 13N13955 to 13N13957). 1 1 Introduction Consider a system consisting of a huge amount of nodes such as a distributed sensor network. Each node continuously observes its environment and measures information such as temperature, pollution or similar parameters. Given such a system, we are interested in aggregating information and continuously monitoring properties describing the current status of the system at a central server. To keep the server’s information up to date, the server and the nodes can communicate with each other. In sensor networks, however, the amount of such communication is particularly crucial, as communication translates to energy consumption, which determines the overall lifetime of the network due to limited battery capacities. Therefore, algorithms aim at minimizing the communication required for monitoring the respective parameter at the server. One very basic parameter is the domain of the system defined to be the values currently observed across all nodes. We consider different notions related to the domain and propose algorithms for monitoring the domain itself, (approximations of) its size and (approximations of) the frequencies of values comprising the domain, respectively. Each of these parameters can provide useful information, e.g. the information about the (approximated) frequency of each value allows to approximate very precisely the histogram of the observed values, and this allows to determine (approximations of) several functions of the input, e.g. heavy hitters, quantiles, top-k, frequency moments or threshold problems. 1.1 Model and Problems We consider the continuous distributed monitoring setting, introduced by Cormode, Muthukrishnan, and Yi in [1], in which there are n distributed nodes, each uniquely identified by an identifier (ID) from the set {1, . . . , n}, connected to a single server. Each node observes a stream of values over time and at any discrete time step t node i observes one value vit ∈ {1, . . . , ∆}. The server is asked to, at any point t in time, compute an output f (t) which depends on the ′ values vit (for t′ ≤ t, and i = 1, . . . , n) observed across all distributed streams up to the current time step t. The exact definition of f (·) depends on the concrete problems under consideration, which are defined in the section below. For the solution of these problems, we are usually interested in approximation algorithms. An ε-approximation of f (t) is an output f˜(t) of the server such that (1 − ε)f (t) ≤ f˜(t) ≤ (1 + ε)f (t). We call an algorithm that, for each time step, provides an ε-approximation with probability at least 1 − δ, an (ε, δ)approximation algorithm. To be able to compute the output, the nodes and the server can communicate with each other by exchanging single cast messages or by broadcast messages sent by the server and received by all nodes. Both types of communication are instantaneous and have unit cost per message. That is, sending a single message to one specific node incurs cost of one and so does one broadcast message. Each message has a size of O(log ∆ + log n + log log 1δ ) bits and will usually, besides a constant number of control bits, consist of a value from {1, . . . , ∆}, a node ID and an identifier to distinguish between messages of different instances of an algorithm applied in parallel (as done when using standard probability amplification techniques). Having a broadcast channel is an extension to [1], which was originally proposed in [2] and afterwards applied in [7, 8]. For ease of presentation, we assume that not only the server can send 2 broadcast messages, but also the nodes. This changes the communication cost only by a factor of at most two, as a broadcast by a node can always be implemented by a single cast message followed by a broadcast of the server. Between any two time steps we allow a communication protocol to take place, which may use polylogarithmic O(logc n) rounds, for some constant c. The optimisation goal is the minimisation of the communication cost, given by the number of exchanged messages, required to monitor the considered problem. 1.1.1 Monitoring of Domain-Related Functions. In this paper, we consider the monitoring of different problems related to the domain of the network. The domain at time t is defined as Dt := {v ∈ {1, . . . , ∆} \| ∃i with vit = v}, the set of values observed by at least one node at time t. We study the following three problems related to the domain: • Domain Monitoring. At any point in time, the server needs to know the domain of the system as well as a representative node for each value of the domain. Formally, monitor Dt = {v1 , . . . , v\|Dt \| } ⊆ {1, . . . , ∆}, at any point t in time. Also, maintain a sequence Rt = (j1 , . . . , j∆ ) of nodes such that for all observed values v ∈ Dt a representative i is determined with jv = i and vit = v. For each value v ∈ / Dt which is not observed, no representative is given and jv = nil. • Frequency Monitoring. For each v ∈ Dt monitor the frequency \|Ntv \| of nodes in Ntv := {i ∈ {1, . . . , n} \| vit = v} that observed v at t, i.e. the number of nodes currently observing v. • Count Distinct Monitoring. Monitor \|Dt \|, i.e. the number of distinct values observed at time t. We provide an exact algorithm for the Domain Monitoring Problem and (ε, δ)approximations for the Frequency and Count Distinct Monitoring Problem. 1.2 Our Contribution P For the Domain Monitoring Problem, an algorithm which uses Θ( t∈T \|Dt \|) messages on expectation for T time steps is given in Section 2. This is asymptotically optimal in the worst-case in which Dt ∩ Dt+1 = ∅ holds for all t ∈ T . We also provide an algorithm and an analysis based on the minimum possible number R∗ of changes of representatives for a given input. It exploits situations where Dt ∩ Dt+1 6= ∅ and uses O(log n · R∗ ) messages on expectation. For an (ε,δ)-approximation of the Frequency Monitoring Problem for T time P steps, we first provide an algorithm using Θ( t∈T \|Dt \| ε12 log \|Dδt \| ) messages on expectation in Section 3. We then improve this bound for instances in which observations between consecutive steps have a certain similarity. That is, for inputs fulfilling the property that for all v ∈ {1, . . . , ∆} and some σ ≤ 1/2, the number of nodes observing v does not change by a factor larger than σ between consecutive time steps, we provide an algorithm that uses an expected amount of O(\|D1 \|(max(δ, σ)T + 1) ε12 log \|Dδ1 \| ) messages. In Section 4, we provide an algorithm using Θ(T · ε12 log δ1 ) messages on expectation for the Count Distinct Monitoring Problem for T time steps. For instances which exhibit a certain similarity an algorithm is presented which monitors the problem using log(n)·R∗ 1 Θ (1 + T · max{2σ, δ}) \|Dt \|·ε2 log δ messages on expectation. 3 1.3 Related Work The basis of the model considered in this paper is the continuous monitoring model as introduced by Cormode, Muthukrishnan and Yi in [1]. In this model, there is a set of n distributed nodes each observing a stream given by a multiset of items in each time step. The nodes can communicate with a central server, which in turn has the task to continuously, at any time t, compute a function f defined over all data observed across all streams up to time t. The goal is to design protocols aiming at the minimisation of the number of bits communicated between the nodes and the server. In [1], the monitoring of several functions is studied in their (approximate) threshold variants, in which the server has to output 1 if f ≥ τ and 0 if f ≤ (1 − ε)τ given τ and ε. Precisely, algorithms P , for p m where mi denotes the frequency of for the frequency moments Fp = i i item i for p = 0, 1, 2 are given. F1 represents the simple sum of all items received so far and F0 the number of distinct items received so far. Since the introduction of the model, monitoring of several functions has been studied such as the monitoring of frequencies and ranks by Huang, Yi and Zhang in [5]. The frequency of an item i is defined to be the number of occurrences of i across all streams up to the current time. The rank of an item i is the number of items smaller than i observed in the streams. Frequency moments for any p > 2 are considered by Woodruff and Zhang in [9]. A variant of the Count Distinct Monitoring Problem is considered by Gibbons and Tirthapura in [4]. The authors study a model in which each of two nodes receives a stream of items and at the end of the streams a server is asked to compute F0 based on both streams. A main technical ingredient is the use of so called public coins, which, once initialized at the nodes, provide a way to let different nodes observe identical outcomes of random experiments without further communication. We will adopt this technique in Section 4. Note that the previously mentioned problems are all defined over the items received so far, which is in contrast to the definition of monitoring problems which we are going to consider and which are all defined only based on the current time step. This fact has the implication that in our problems the monitored functions are no longer monotone, which makes its monitoring more complicated. Concerning monitoring problems in which the function tracked by the server only depends on the current time step, there is also some previous work to mention. In [6], Lam, Liu and Ting study a setting in which the server needs to know, at any time, the order type of the values currently observed. That is, the server needs to know which node observes the largest value, second largerst value and so on at time t. In [10], Yi and Zhang consider a system only consisting of one node connected to the server. The node continuously observes a d-dimensional vector of integers from {1, . . . , ∆}. The goal is to keep the server informed about this vector up to some additive error per component. In [3], Davis, Edmonds and Impagliazzo consider the following resource allocation problem: n nodes observe streams of required shares of a given resource. The server has to assign, to each node, in each time step, a share of the resource that is as least as large as the required share. The objective is then given by the minimization of communication necessary for adapting the assignment of the resource over time. 4 2 The Domain Monitoring Problem We start by presenting an algorithm to solve the Domain Monitoring Problem for a single time step. We analyse the communication cost using standard worstcase analysis and show tight bounds. By applying the algorithm for each time step, we then obtain tight bounds for monitoring the domain for any T time steps. The basic idea of the protocol as given in Algorithm 1 is quite simple: Applied at a time t with a value v ∈ {1, . . . , ∆}, the server gets informed whether v ∈ Dt holds or not. To do so, each node i with vit = v essentially draws a value from a geometric distribution and then those nodes having drawn the largest such value send broadcast messages. By this, one can show that on expectation only a constant number of messages is sent. Furthermore, if applied with v = nil, the server can decide whether v ′ ∈ Dt for all v ′ ∈ {1, . . . , ∆} at once with Θ(\|Dt \|) messages on expectation. To this end, for each v ′ ∈ {1, . . . , ∆} independently, the nodes i with vit = v ′ drawing the largest value from the geometric distribution send broadcast messages. In the presentation of Algorithm 1, we assume that vit = v is always true if v = nil. Also, in order to apply it to a subset of nodes, we assume that each node maintains a value statusi ∈ {0, 1} and only nodes i take part in the protocol for which statusi = status holds. Algorithm 1 ConstantResponse(v, status) [for fixed time t] 1. Each node i for which statusi = status and (v 6= nil ⇒ vit = v) hold, draws a value ĥi from a geometric distribution with success probability p := 1/2. 2. Let hi = min{log n, ĥi }. 3. Node i broadcasts its value in round log n−hi unless a node i′ with vit = vit′ has broadcasted before. We have the following lemma, which bounds the expected communication cost of Algorithm 1 and has already appeared in a similar way in [8] (Lemma III.1). Lemma 2.1. Applied for a fixed time t, ConstantResponse(v, 1) uses Θ(1) messages on expectation if v 6= nil and Θ(\|Dt \|) otherwise. Proof. First consider the case where v 6= nil. Regarding the expected communication of ConstantResponse(v, 1) we introduce some notation. Let Xi be a {0, 1}-random variable P indicating whether the node i ∈ Ntv sends a message to the server, and X := Xi . According to the algorithm a sensor i sends a message if and only if its height hi matches the round specified for that height and no other sensor i′ has sent its value beforehand. We obtain Pr [Xi = 1] = Pr [∃r ∈ {1, . . . , log n} : hi = r ∧ ∀i′ ∈ Ntv \ {i} : hi′ ≤ r] nv −1 log Xn 1 1 1− r . ≤ 2r 2 r=1 We know that E[Xi ] = Pr [Xi = 1] and thus v E[X] ≤ n · log Xn r=1 1 2r 5 nv −1 1 1− r . 2 nv −1 has only one extreme point and Observing that f (r) = nv · 21r 1 − 21r f (r) ≤ 2 for all r ∈ [0, log(n)], we use the integral test for convergence to obtain nv −1 nv −1 Z log n 1 1 1 v E[X] ≤ n · 1− r 1− r ≤n dr + 2 2 2r 2 0 r=1 " nv #log n 1 1 1 1− r +2≤ + 2 < 4. ≤ ln (2) 2 ln (2) v log Xn 1 2r 0 For the case v = nil we can apply the same argumentation independently for each value v ∈ Dt . This concludes the proof of the lemma. In order to solve the domain monitoring problem for T time steps, the server proceeds as follows: In each step t the server calls ConstantResponse(nil, 1) to identify all values belonging to Dt as well as a valid sequence Rt . By the previous lemma we then have an overall communication cost Pof Θ(\|Dt \|) for each time step t. For monitoring T time steps, the cost is Θ( t∈T \|Dt \|). This is asymptotically optimal in the worst-case P since on instances where Dt ∩Dt+1 = ∅ for all t, any algorithm has cost Ω( t∈T \|Dt \|). Theorem 2.2. Using ConstantResponse(v, P 1), the Domain Monitoring Problem for T time steps can be solved using Θ( t∈T \|Dt \|) messages on expectation. A Parameterised Analysis Despite the optimality of the result, the strategy of computing a new solution from scratch in each time step seems unwise and the analysis does not seem to capture the essence of the problem properly. It often might be the case that there are some similarities between values observed in consecutive time steps and particularly, that Dt ∩ Dt+1 6= ∅. In this case, there might be the chance to keep a representative for several consecutive time steps, which should be exploited. Due to these observations we next define a parameter describing this behavior and provide a parameterised analysis. To this end, we consider the number of component-wise differences in the sequences of nodes Rt−1 and Rt and call this difference the number of changes of representatives in time step t. Let R∗ denote the minimum possible number of changes of representatives (over all considered time steps T ). The formal description of our algorithm is given in Algorithm 2. Roughly speaking, the algorithm defines, for each value v, phases, where a phase is defined as a maximal time interval during which there exists one node observing value v throughout the entire interval. Whenever a node being a representative for v changes its observation, it informs the server so that a new representative can be chosen (from those observing v throughout the entire phase, which is indicated by statusi = 1). If no new representative is found this way, the server tries to find a new representative among those observing v and for which statusi = 0 and ends the current phase. Additionally, if a node observes a value v at time t for which v ∈ / Dt , a new representative is determined among these nodes. Note that this requires each node to store Dt at any time t and hence a storage of O(∆). 6 Algorithm 2 DomainMonitoring (Node i) 1. Define statusi := 1. 2. If at some time t, vit 6= vit−1 , then 2.1. If vit ∈ / Dt−1 , set statusi = 0 and apply ConstantResponse(vit , 0). t 2.2. If vi ∈ Dt−1 , set statusi = 0. Additionally inform server in case i ∈ Rt−1 . 2.3. If server starts a new phase for v = vit , set statusi = 1. (Server) [Initialisation] Call ConstantResponse(nil, 1) to define D0 and for each v ∈ D0 choose a representative uniformly at random from all nodes which have sent v. [Maintaining Dt and Rt at time t] Start with Dt = Dt−1 and Rt = Rt−1 and apply the following rules: • [Current Phase, (try to) find new representative] If informed by representative of a value v ∈ Dt−1 , 1) Call ConstantResponse(v, 1). 2) If node(s) respond(s), choose new representative among the responding sensors uniformly at random. 3) Else call ConstantResponse(v, 0). End current phase for v and, if there is no response, delete v from Dt and the respective representative from Rt . • [If ConstantResponse(v, 0) leads to received message(s), start new phase] Start a new phase for value v if message from an application of ConstantResponse(v, 0) (by Step 3) initialised by the server or initialised in Step 2.1. by a node) is received. Add or replace respective representative in Rt by choosing a node uniformly at random from those responding to ConstantResponse(v, 0). Theorem 2.3. DomainMonitoring as described in Algorithm 2 solves the Domain Monitoring Problem using O(log n·R∗ ) messages on expectation, where R∗ denotes the minimum possible number of changes of representatives. S Proof. We consider each value v ∈ t Dt separately. Let Nt1 ,t2 := {i \| vit = v ∀t1 ≤ t ≤ t2 } denote the set of nodes that observe the value v at each point in time t with t1 ≤ t ≤ t2 . Consider a fixed phase for v and let t1 and t2 be the points in time where the phase starts and ends, respectively. A phase only ends in Step 3), hence there was no response from ConstantResponse(v, 1), for v we can associate a cost of which implies Ntv1 ,t2 = ∅. Thus, to each phase S at least one to R∗ and this holds for each v ∈ t Dt . Therefore, R∗ is at least the overall number of phases of all values. Next we analyze the expected cost of Algorithm 2 during the considered phase for v. Let w.l.o.g. Nt1 := Nt1 ,t1 = {1, 2, . . . , k}. With respect to the fixed phase, only nodes in Nt1 can communicate and the communication is bounded by the number of changes of the representative for v during the phase. Let t′i be the first time after t1 at which node i does not observe v. Let the nodes be sorted such that i < j implies t′i ≥ t′j . Let a1 , . . . , am be the nodes Algorithm 2 chooses as representatives in the considered phase. We want to show that E[m] = O(log k). To this end, partition the set of time steps t′i into groups Gi . Intuitively, Gi represents the time steps in which the nodes 7 continuously observe value v since time t1 and the size of the initial set of nodes that observed v is halved i times. Formally, Gi contains all time steps tℓi−1 +1 , . . . , tℓi (where ℓ−1 := 0 for convenience) such that ℓi is the largest integer fulfilling \|Nt1 ,t′ℓ \| ∈ (k/2i+1 , k/2i ]. i Let Si be the numberPof changes of representatives in time steps belonging log k to Gi . We have E[m] = i=0 E[Si ]. Consider a fixed Si . Let Ej be the event that the j-th representative chosen in time steps belonging to Gi is the first one k ⌋ . Observe that as soon as this happens, the with an index in 1, . . . , ⌊ 2i+1 respective representative will be the last one chosen in a time step belonging to group Gi . Now, since the algorithm chooses a new representative uniformly at random from the index set 1, . . . , ⌊ 2ki ⌋ , the probability that it chooses a representative k from 1, . . . , ⌊ 2i+1 ⌋ is at least 1/2 except for the first representative of v, where it might be slightly smaller due to rounding errors. Ej occurs only if the first j−2 j − 1 representatives were each not chosen from this set, i.e. Pr [Ej ] ≤ 12 . P P P j = O(1). Hence, E[Si ] = j E[Si \|Ej ] · Pr[Ej ] ≤ j j · ( 12 )j−2 = j 2j−2 3 The Frequency Monitoring Problem In this section we design and analyse an algorithm for the Frequency Monitoring Problem, i.e. to output (an approximation) of the number of nodes currently observing value v. We start by considering a single time step and present an algorithm which solves the subproblem to output the number of nodes that observe v within a constant multiplicative error bound. Afterwards, and based on this subproblem, a simple sampling algorithm is presented which solves the Frequency Monitoring Problem for a single time step up to a given (multiplicative) error bound and with demanded error probability. While in the previous section we used the algorithm ConstantResponse with the goal to obtain a representative for a measured value, in this section we will use the same algorithm to estimate the number of nodes that measure a certain value v. Observe that the expected maximal height of the geometric experiment increases with a growing number of nodes observing v. We exploit this fact and use it to estimate the number of nodes with value v, while still expecting constant communication cost only. For a given a time step t and a value v ∈ Dt , we define an algorithm ConstantFactorApproximation as follows: We apply ConstantResponse(v, 1) with statusi = 1 for all nodes i. If the server receives the first response in communication round r ≤ log n, the algorithm outputs ñvconst = 2r as the estimation for \|Ntv \|. We show that we compute a constant factor approximation with constant probability. Then we amplify this probability using multiple executions of the algorithm and taking the median (of the executions) as a final result. Lemma 3.1. The algorithm ConstantFactorApproximation estimates the number \|Ntv \| of nodes observing the value v at time t up to a factor of 8, i.e. ñvconst ∈ [\|Ntv \|/8, \|Ntv \| · 8] with constant probability. Proof. Let nv be the number of nodes currently observing value v, i.e. nv := \|Ntv \|. Recall that the probability for a single node to draw height h is Pr[hi = h] = 21h , if h < log n, and Pr[hi = h] = 22h , if h = log n. Hence, Pr[hi ≥ h] = 1 for all h ∈ {1, . . . , log n}. 2h−1 8 We estimate the probability of the algorithm to fail, by analysing the cases that ñvconst is larger than log nv + 3 or smaller than log nv − 3. We start with the first case and by applying a union bound we obtain: Pr[∃i : hi > log nv + 3] ≤ Pr[∃i : hi ≥ ⌈log nv ⌉ + 3] ⌈log nv ⌉+2 1 1 v =n · ≤ . 2 4 For the latter case we bound the probability that each node has drawn a height strictly smaller than log nv − 3 by Y Pr[hi < ⌈log nv ⌉ − 3] Pr[∀i : hi < log nv − 3] ≤ i = 1− 1 2⌈log nv ⌉−4 nv ≤ 8 1− v n nv ≤ 1 . e8 Thus, the probability that we compute an 8-approximation is bounded by v n v hi Pr ≤ 2 ≤ 8n = 1 − Pr[∃i : hi > log nv + 3] + Pr[∀i : hi < log nv − 3] 8 1 1 > 0.7 + ≥1− 4 e8 We apply an amplification technique to boost the success probability to arbitrary 1 − δ ′ using Θ(log δ1′ ) parallel executions of the ConstantFactorApproximation algorithm and choose the median of the intermediate results as the final output. Corollary 3.2. Applying Θ log δ1′ independent, parallel instances of ConstantFactorApproximation, we obtain a constant factor approximation of \|Ntv \| with success probability at least 1 − δ ′ using Θ log δ1′ messages on expectation. 1 Proof. Choose d = 45 2 ln δ ′ to be the number of copies of the algorithm and return the median of the intermediate results. Let Ij be the indicator variable for the event that the j-th experiment does not result in an 8-approximation. By Lemma 3.1 the failure probability can be upper bounded by a constant, i.e. Pr [Ij ] ≤ 0.3. Hence, using a Chernoff bound, the probability that at least half of the experiments do meet the required approximation factor of 8 is     d d X X 2 1 · 0.3 · d Ij ≥ 1 + Pr  Ij ≥ d ≤ Pr  2 3 j=1 j=1 2 1 · 3 ·0.3·d ≤ e −( 3 ) 2 2 2 45 = e− 45 ·d = e− 45 · 2 ln 1 δ′ = δ′ . Observe that if at least half of the intermediate results are within the demanded error bound, so is the median. Thus, the algorithm produces an 8-approximation of \|Ntv \| with success-probability of at least 1 − δ ′ , concluding the proof. 9 To obtain an (ε, δ)-approximation, in Algorithm 3 we first apply the ConstantFactorApproximation algorithm to obtain a rough estimate of \|Ntv \|. It is used to compute a probability p, which is broadcasted to the nodes, so that every node observing value v sends a message with probability p. Since the ConstantFactorApproximation result ñvconst in the denominator of p is close to \|Ntv \|, the number of messages sent on expectation is independent of \|Ntv \|. The estimated number of nodes observing v is then given by the number of responding nodes n̄v divided by p, which, on expectation, results in \|Ntv \|. Algorithm 3 EpsilonFactorApprox(v ∈ Dt , ε, δ) (Node i) 1. Receive p from the server. 2. Send a response message with probability p. [for fixed time t] (Server) 1. Set δ ′ := δ3 2. Call ConstantFactorApproximation(v, δ ′ ) to obtain ñvconst . 24 1 3. Broadcast p = min 1, ε2 ñv · ln δ′ . const v 4. Receive n̄ messages. 5. Compute and output estimated number of nodes in Ntv as ñv = n̄v /p. Lemma 3.3. The algorithm EpsilonFactorApprox as given in Algorithm 3 provides an (ε,δ)-approximation of \|Ntv \|. Proof. The algorithm obtains a constant factor approximation ñvconst with probability 1 − δ ′ . The expected number of messages is E [n̄v ] = nv · p. We start by estimating the conditional probability that more than (1+ε) nv p responses are sent under the condition that ñvconst ≤ 8nv and p < 1. In this case we have 3 1 1 24 · ln ′ ≥ 2 v · ln ′ , p= 2 v ε ñconst δ ε n δ hence using a Chernoff bound it follows p1 := Pr [n̄v ≥ (1 + ε)nv p \|ñvconst ≤ 8nv ∧ p < 1 ] ≤ e− ε2 3 nv · ε23nv ·ln 1 δ′ = δ′. Likewise the probability that less than (1 − ε) nv p messages are sent under the condition that ñvconst ≤ 8nv and p < 1 is p2 := Pr [n̄v ≤ (1 − ε)nv p \|ñvconst ≤ 8nv ∧ p < 1 ] ≤ e− ε2 2 nv · ε23nv ·ln 1 δ′ 3 1 ≤ e− 2 ln δ′ < δ ′ . Next consider the case that ñvconst > 8nv and p < 1 holds. Using nv v v v v v Pr [ñconst > 8n ] ≤ Pr ñconst > 8n ∨ ñconst < ≤ δ′ 8 and pi · Pr [ñvconst ≤ 8nv ] ≤ pi for i ∈ {1, 2}, Pr [(1 − ε)nv p < n̄v < (1 + ε)nv p \|p < 1 ] ≥ 1 − (Pr [ñvconst > 8nv ] + (p1 + p2 )) ≥ 1 − 3δ ′ = 1 − δ. 10 For the last case p = 1, we have Pr [(1 − ε)nv p < n̄v < (1 + ε)nv p \|p ≥ 1 ] = 1, by using n̄v = nv . Now, Pr [(1 − ε)nv p < n̄v < (1 + ε)nv p] ≥ 1 − δ directly follows. Lemma 3.4. Algorithm EpsilonFactorApprox as given in Algorithm 3 uses Θ( ε12 log δ1 ) messages on expectation. Proof. Recall that each of the nv nodes sends a message with probability p, leading to nv · p messages on expectation. First assume that the constant factor approximation was successful, i.e. n81 ≤ ñvconst ≤ 8n1 . If p < 1, we have 24 1 24 · 8 1 1 1 . nv · p = nv 2 v · ln ′ ≤ · ln = Θ log ε ñconst δ ε2 δ′ ε2 δ If p = 1, by definition ε2 ñ24 · ln δ1′ ≥ 1, hence ñvconst = O ε12 · log δ1′ . Thus, v const nv p ≤ 8ñvconst p = O ε12 · log δ1′ . For the case constant factor approximation was not successful, that the 1 1 v note that Pr ñvconst < 8·2 holds analogously to the calculation in ≤ e2i+3 in 1 v Lemma 3.1. Also, for ñvconst ≥ 8·2 n and p < 1, we have i 24 1 1 1 . nv p ≤ 8 · 2i · ñvconst · 2 v · ln = 2i · Θ 2 log ε ñconst δ ε δ Similarly, for p = 1, we have nv p ≤ 8 · 2i · ñvconst = 2i · Θ ε12 log δ1 as in this case, ñvconst = O ε12 · log δ1′ . Hence, we can conclude 1 v 1 1 v v · Pr ñconst ≥ n E [n̄ ] ≤ Θ 2 log ε δ 8 ∞ X 1 1 1 v 1 i+1 v v · 2 · Θ Pr n ≤ ñ < n log + const 8 · 2i+1 8 · 2i ε2 δ i=0 ! ! ∞ ∞ X X i+3 2i+1 1 1 ≤Θ ≤ Θ 2 log 2i+1−2 1+ 1+ 2i+3 ε δ e i=0 i=0 ! ∞ X 1 1 1 1 ≤ Θ 2 log 2−i = Θ 2 log 1+ . ε δ ε δ i=0 1 1 log ε2 δ Theorem 3.5. There exists an algorithm that provides an (ε,δ)-approximation for the Frequency Monitoring Problem for T time steps with an expected number P \|Dt \| 1 messages. of Θ t∈T \|Dt \| ε2 log δ Proof. In every time step t we first identify Dt by applying ConstantResponse using Θ (\|Dt \|) messages on expectation. On every value v ∈ Dt we then perform algorithm EpsilonFactorApprox(v,ε, \|Dδt \| ), resulting in an amount of Θ \|Dt \| ε12 log \|Dδt \| messages on expectation for a single time step, while 0 \|δ achieving a probability (using a union bound) of 1 − \|D \|D0 \| = 1 − δ that in one time step the estimations for every v are ε-approximations. Applied for each of the T time steps, we obtain a bound as claimed. 11 A Parameterised Analysis Applying EpsilonFactorApprox in every time step is a good solution in worst case scenarios. But if we assume that the change in the set of nodes observing a value is small in comparison to the size of the set, we can do better. We extend the EpsilonFactorApprox such that in settings where from one time step to another only a small fraction σ of nodes change the value they measure, the amount of communication can be reduced, while the quality guarantees remain intact. We define σ such that ∀t : σ ≥ v v \|Nt−1 \ Ntv \| + \|Ntv \ Nt−1 \| . v \|Nt \| Note that this also implies that Dt = Dt−1 holds for all time steps t, i.e. the set of measured values stays the same over time. The extension is designed so that compared to EpsilonFactorApprox, also in settings with many changes the solution quality and message complexity asymptotically does not increase. The idea is the following: For a fixed value v, in a first time step EpsilonFactorApprox is executed (defining a probability p in Step 3 of Algorithm 3). In every following time step, up to 1/δ consecutive time steps, nodes that start or stop measuring a value v send a message to the server with the same probability p, while nodes that do not observe a change in their value remain silent. In every time step t, the server uses the accumulated messages from the first time step and all messages from nodes that started measuring v in time steps 2 . . . t, while subtracting all messages from nodes that stopped measuring v in the time steps 2 . . . t. This accumulated message count is then used similarly as in EpsilonFactorApprox to estimate the total number of nodes observing v in the current time step. The algorithm starts again if a) 1/δ time steps are over, so that the probability of a good estimation remains good enough, or b) the sum of estimated nodes to start/stop measuring value v is too large. The latter is done to ensure that the message probability p remains fitting to the number of nodes, ensuring a small amount of communication, while guaranteeing an (ε, δ)-approximation. − Let n+ t , nt be the number of nodes that start measuring v in time step t or − v v v v that stop measuring it, respectively, i.e. n+ t = \|Nt \ Nt−1 \|, nt = \|Nt−1 \ Nt \|, + − and n̄t and n̄t the number of them that sent a message to the server in time − step t. In the following we call nodes contributing to n+ t and nt entering and leaving, respectively. Lemma 3.6. For any v ∈ D1 , the algorithm ContinuousEpsilonApprox provides an (ε,δ)-approximation of \|Ntv \|. Proof. By the same arguments as in Lemma 3.3, we obtain an (ε,δ ′ )-approximation of n1 . In any further time step we compute our estimate over the sum of all received messages (n̄1 , arrivals and departures). If too many nodes change their measured value, we redo a complete estimation of the nodes in Ntv . Recall that ñt is the random variable giving the estimated number of nodes − n̄+ n̄− by the algorithm, and ñ+ t = p , ñt = p are the random variables giving the estimated arrivals and departures in that time step. We look at Ptany time step − t > 1 where the restart criteria are not met: Since ñt = ñ1 + i=2 ñ+ i − ñi and the linearity of expectation, for any time t ≥ 1 we can use a Chernoff bound as in Lemma 3.3 to show that the estimation is an (ε, δ ′ )-approximation. 12 Algorithm 4 ContinuousEpsilonApprox(v, ε, δ) (Node i) 1. If t = 1, take part in EpsilonFactorApprox called in Step 2 by the server. 2. If t > 1, broadcast a message with probability p if vit−1 = v ∧ vit 6= v or vit−1 6= v ∧ vit = v. (Server) 1. Set δ ′ := δ 2 . 2. Set t := 1 and run EpsilonFactorApprox(v, ε/3, δ) to obtain n̄1 , p. 3. Output ñ1 = n̄p1 . 4. Repeat at the beginning of every new time step t > 1: 4.1. Receive messages from nodes changing the observed value to obtain − n̄+ t and n̄t . P Pt t + − 4.2. Break if t ≥ 1/δ or /p ≥ n̄1 /2. i=1 n̄i + i=1 n̄i Pt P t − /p. 4.3. Output ñt = n̄1 + i=1 n̄+ i − i=1 n̄i 5. Go to Step 2. Using a union bound on the fail probability of up to 1/δ time steps, we get a 1 − δ1 · δ ′ = 1 − δ probability of having a correct estimation in any time step. 1 1 1 Lemma 3.7. For a fixed value v and T ′ = min{ 2σ , δ }, σ ≤ 2 , time steps, 1 1 ContinuousEpsilonApprox uses Θ ε2 log δ messages on expectation. Proof. The message complexity depends on the initial size \|N1v \| and on the number of nodes leaving and entering N v in those time steps, which is bounded by σ. If EpsilonFactorApprox obtained a correct probability p in Step 1, i.e. p = Θ( n11 ), the expected number of messages (in case p < 1) is    ′  T T′ X X 1 1 −  = E n̄1 +  p=Θ E n̄+ n̄t p = Θ i + n̄i n n 1 1 t=1 i=2   T′ X − n+ p ≤ (n1 + T ′ σn1 ) p = n1 + i + ni i=2 1 1 = n1 (1 + T ′ σ) · 24 · 2 v ln ′ ε ñconst δ 1 1 1 2 σ · 1/ε log . =Θ 1 + min , 2σ δ δ Considering the case where EpsilonFactorApprox estimated wrong, the message complexity could increase greatly if the probability p is too large for the actual number of nodes (i.e. an underestimation leads to high message complexity). But the probability to misestimate by some constant factor (which would increase the message complexity by that factor) decreases exponentially in this factor (as shown in Lemma 3.4 for EpsilonFactorApprox), leaving 13 the expected number of messages to be Θ 1 1 Θ ε2 log δ . 1 + min 1 1 2σ , δ σ · 1 ε2 · log 1δ = Theorem 3.8. There exists an (ε,δ)-approximation algorithm for the Frequency Monitoring Problem for T consecutive time steps which uses an amount of Θ \|D1 \| (1 + T · max{2σ, δ}) ε12 log \|Dδ1 \| messages on expectation, if σ ≤ 1/2. Proof. The algorithm works by first applying ConstantResponse(nil,1) to obtain D1 and then applying ContinuousEpsilonApprox(v, ε, δ/\|D1 \|) for every v ∈ D1 . By Lemma 3.6 we know that in every time step and for all v ∈ D1 , the frequency of v is approximated up to a factor of ε with probability 1−δ/\|D1\|. 1 1 We divide the T time steps into intervals of size T ′ = min{ 2σ , δ } and perform ContinuousEpsilonApprox on each of them for every value v ∈ D1 . There are ⌈ TT′ ⌉ ≤ 1+T ·max{2σ, δ} such intervals. For each of those, by Lemma 3.7 we \|D1 \| 1 1 2 messages on expectation for each v ∈ need Θ 1 + min{ 2σ , δ }σ · 1/ε log δ D1 . This yields a complexity of Θ \|D1 \| (1 + T · max{2σ, δ}) ε12 log \|Dδ1 \| due to 1 1 1 , δ }σ ≤ 2σ · σ = Θ(1). Using a union bound over the fail probability for min{ 2σ 1 \|δ every v ∈ D1 , a success probability of at least 1 − \|D \|D1 \| = 1 − δ follows. By Theorem 3.5, trivially repeating the single step algorithm EpsilonFac \|D1 \| 1 messages on expectation for T (betorApprox needs Θ T \|D1 \| ε2 log δ cause the number of nodes in Ntv for any v ∈ D1 is at least N1v /2 in every time step of that interval). Hence, the number of messages sent when using ContinuousEpsilonApprox is reduced in the order of max{2σ, δ}. 4 The Count Distinct Monitoring Problem In this section we present an (ε,δ)-approximation algorithm for the Count Distinct Monitoring Problem. The basic approach is similar to the one presented in the previous section for monitoring the frequency of each value. That is, we first estimate \|Dt \| up to a (small) constant factor and then use the result to define a protocol for obtaining an (ε, δ)-approximation. If we could assume that, at any fixed time t, each value was observed by at most one node, it would be possible to solve this problem with expected communication cost of O( ε12 log δ1 ) (per time step t and per value v ∈ Dt ) using the same approach as in the previous section. Since this assumption is generally not true, we aim at simulating such behaviour that for each value in the domain only one random experiment is applied. We apply the concept of public coins, which allows nodes measuring the same value to observe identical outcomes of their random experiments. To this end, nodes have access to a shared random string R of fully independent and unbiased bits. This can be achieved by letting all nodes use the same pseudorandom number generator with a common starting seed, adding a constant number of messages to the bounds proven below. We assume that the server sends a new seed in each phase by only loosing at most a constant factor in the amount of communication used. However, we can drop this assumption by checking whether there are nodes that changed their value such that only in 14 rounds in which there are changes new public randomness is needed. The formal description of the algorithm for a constant factor and an ε-approximation are given in Algorithm 5 and Algorithm 6, respectively. We consider the access of the public coin to behave as follows: Initialised with a seed, a node accesses the sequence of random bits R bitwise, i.e. after reading the j’th bit, the node next accesses bit j + 1. Observe the crucial fact that as long as each node accesses the exact same number of bits, each node observes the exact same random bits simultaneously. Algorithm 5 essentially works as follows: In a first step, each node draws a number from a geometrical distribution using the public coin. By this, all nodes observing the same value v obtain the same height hv . In the second step we apply the strategy as in the previous section to reduce communication if lots of nodes observe the same value: Each node i draws a number gi from a geometrical distribution without using the public coin. Afterwards, all nodes with the largest height gi among those with the largest height hv broadcast their height hv . Algorithm 5 ConstantFactorApproximation [for fixed time t] (Node i, observes value v = vi ) 1. Draw a random number hv as follows: Consider the next ∆ · log n random bits b1 , . . . , b∆·log n from R. Let h be the maximal number of bits bv·log n+1 , . . . , bv·log n+1+h that equal 0. Define hv := min{h, log n}. 2. Let gi′ be a random value drawn from a geometric distribution with successprobability p = 1/2 and define gi = min(gi′ , log n) (without accessing public coins). 3. Broadcast drawn height hv in round r = log2 n − (hv − 1) · log n − gi unless a node i′ has broadcasted before. (Server) 1. Receive a broadcast message containing height h in round r. 2. Output dˆt = 2h . Note that only (at most n) nodes that observe value v with hv = maxv′ hv′ may send a message in Algorithm 5. Now, all nodes observing the same value observe the same outcome of their random experiments determining hv . Hence, by a similar reasoning as in Lemma 3.1, one execution of the algorithm uses O(1) messages on expectation. Using the algorithm given in Algorithm 5 and applying the same idea as in the previous section, we obtain an (ε, δ)-approximation as given in Algorithm 6: Each node tosses a coin with a success probability depending on the constant factor approximation (for which we have a result analogous to Corollary 3.2). Again, all nodes use the public coin so that all nodes observing the same value obtain the same outcome of this coin flip. Afterwards, those nodes which have observed a success apply the same strategy as in the previous section, that is, they draw a random value from a geometric distribution, and the nodes having the largest height send a broadcast. 15 Algorithm 6 EpsilonFactorApprox [for fixed time t] (Node i) 1/δ 1. Flip a coin with success probability p = 2−q = c log , q ∈ N as follows: ε2 dˆt Consider the next ∆ · q random bits b1 , . . . b∆·q . The experiment is successful if and only if all random bits bv·q+1 , . . . , bv·q+q equal 0. The node deactivates (and does not take part in Steps 2. and 3.) if the experiment was not successful. 2. Draw a random value h′i from a geometric distribution and define hi = min(h′i , log n) (without accessing public coins). 3. Node i broadcasts its value in round log n−hi unless a node i′ with vit = vit′ has broadcasted before. (Server) 1. Let St be the set of received values. 2. Output d˜t := \|St \|/p Using arguments analogous to Lemmas 3.3 and 3.4 and applying EpsilonFactorApprox for T time steps, we obtain the following theorem. Theorem 4.1. There exists an (ε, δ)-approximation algorithm for the Count Distinct Monitoring Problem for T time steps using O(T · ε12 log δ1 ) messages on expectation. A Parameterised Analysis In this section we consider the problem for multiple time steps and parameterise the analysis with respect to instances in which the domain does not change arbitrarily between consecutive time steps. Recall that for monitoring the frequency from a time step t − 1 to the current time step t, all nodes that left and all nodes that entered toss a coin to estimate the number of changes. However, to identify that a node observes a value which was not observed in the previous time step, the domain has to be determined exactly. We apply the following idea instead: For each value v ∈ {1, . . . , ∆} we flip a (public) coin. We denote the set of values with a successful coin flip as the sample. Afterwards, the algorithm only proceeds on the values of the sample, i.e. in cases in which a node observes a value with a successful coin flip and no node observed this value in previous time steps, this value contributes to the estimate d˜+ t at time t. Regarding the (sample) of nodes that leave the set of observed values, the DomainMonitoring algorithm is applied to identify which (sampled) values are not observed any longer (and thus contribute to d˜− t ). Analogous to Lemma 3.6, we have the following lemma. Lemma 4.2. ContinuousEpsilonApprox achieves an (ε,δ)-approximation of \|Dt \| in any time step t. For the number of messages, we argue based on the previous section. However, in addition the DomainMonitoring algorithm is applied. Observe that the size of the domain changes by at most n/2, and consider the case that this number of nodes observed the same value v. The expected cost (where the expectation is taken w.r.t. whether v is within the sample) is O(log n · R∗ · p) = n·R∗ 1 O log \|Dt \|ε2 log δ . Similar to Theorem 3.8, we then obtain the following theorem. 16 Algorithm 7 ContinuousEpsilonApprox(ε, δ) 1. Compute δ ′ = 2 δ 2 2. Broadcast a new seed value for the public coin. 3. Compute an (ε, δ ′ )-approximation d˜1 of \|D1 \| using Algorithm 6. Furthermore, obtain the success-probability p. 4. Repeat for each time step t > 1: 4.1. Each node i applies Algorithm 2 if the observed value vi is in the sample set. Let dˆ− t be the number of values (in sample set) which number of nodes that join the sample. left the domain and dˆ+ t the P Pt ˆ− t ˜ ˜ 4.2. Server computes dt = d1 + i=2 dˆ+ i /p − i=2 di /p. Pt ˜+ Pt ˜− 4.3. Break if t = 1/δ or d + d /p exceeds d˜1 /2. i=2 i i=2 i 5. Set t = 1 and go to Step 2. Theorem 4.3. ContinuousEpsilonApprox provides and (ε, δ)-approximation for the Count Distinct Monitoring Problem for T time steps using an log(n)·R∗ 1 amount of Θ (1 + T · max{2σ, δ}) \|Dt \|·ε2 log δ messages on expectation, if σ ≤ 1/2. References [1] Graham Cormode, S. Muthukrishnan, and Ke Yi. Algorithms for distributed functional monitoring. In Proceedings of the 19th Annual ACMSIAM Symposium on Discrete Algorithms (SODA ’08), pages 1076–1085. SIAM, 2008. [2] Graham Cormode, S. Muthukrishnan, and Ke Yi. Algorithms for Distributed Functional Monitoring. ACM Transactions on Algorithms, 7(2):21:1–21:20, 2011. [3] Sashka Davis, Jeff Edmonds, and Russell Impagliazzo. Online Algorithms to Minimize Resource Reallocations and Network Communication. In Proceedings of the 9th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th International Conference on Randomization and Computation (APPROX’06/RANDOM ’06), volume 4110 of Lecture Notes in Computer Science, pages 104–115. Springer, 2006. [4] Phillip B. Gibbons and Srikanta Tirthapura. Estimating simple functions on the union of data streams. In Proceedings of the 13th annual ACM Symposium on Parallel Algorithms and Architectures (SPAA ’01), pages 281–291. ACM, 2001. [5] Zengfeng Huang, Ke Yi, and Qin Zhang. Randomized algorithms for tracking distributed count, frequencies, and ranks. In Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS ’12), pages 295–306. ACM, 2012. 17 [6] Tak Wah Lam, Chi-Man Liu, and Hing-Fung Ting. Online Tracking of the Dominance Relationship of Distributed Multi-dimensional Data. In Proceedings of the 8th International Workshop on Approximation and Online Algorithms (WAOA ’10), volume 6534 of Lecture Notes in Computer Science, pages 178–189. Springer, 2010. [7] Alexander Mäcker, Manuel Malatyali, and Friedhelm Meyer auf der Heide. Online Top-k-Position Monitoring of Distributed Data Streams. In Proceedings of the 2015 IEEE International Parallel and Distributed Processing Symposium (IPDPS ’15), pages 357–364. IEEE, 2015. [8] Alexander Mäcker, Manuel Malatyali, and Friedhelm Meyer auf der Heide. On Competitive Algorithms for Approximations of Top-k-Position Monitoring of Distributed Streams. In Proceedings of the 2016 IEEE International Parallel and Distributed Processing Symposium (IPDPS ’16), pages 700–709. IEEE, 2016. [9] David P. Woodruff and Qin Zhang. Tight bounds for distributed functional monitoring. In Proceedings of the 44th Symposium on Theory of Computing (STOC ’12), pages 941–960. ACM, 2012. [10] Ke Yi and Qin Zhang. Multidimensional online tracking. ACM Transactions on Algorithms, 8(2):12, 2012. 18	8
Inverse of a Special Matrix and Application Thuan Nguyen arXiv:1708.07795v1 [cs.DM] 25 Aug 2017 School of Electrical and Computer Engineering, Oregon State University, Corvallis, OR, 97331 Email: nguyeth9@oregonstate.edu Abstract—The matrix inversion is an interesting topic in algebra mathematics. However, to determine an inverse matrix from a given matrix is required many computation tools and time resource if the size of matrix is huge. In this paper, we have shown an inverse closed form for an interesting matrix which has much applications in communication system. Base on this inverse closed form, the channel capacity closed form of a communication system can be determined via the error rate parameter α. Keywords: Inverse matrix, convex optimization, channel capacity. I. M ATRIX C ONSTRUCTION In Wireless communication system or Free Space Optical communication system, due to the shadow effect or the turbulent of environment, the channel conditions can be flipped from “good” to “bad” or “bad” to “good” state such as Markov model after the transmission time σ [1] [2]. For simple intuition, in “bad” channel, a signal will be transmitted incorrectly and in “good” channel, the signal is received perfectly. Suppose a system has total n channels, the “good” channel is noted as “1” and “bad” channel is “0”, respectively, the transmission time between transmitter and receiver is σ, the probability the channel is flipped after the transmission time σ is α. We note that if the system using a binary code such as On-Off Keying in Free Space Optical communication, then the flipped probability α is equivalent to the error rate. Consider a simple case for n = 2, suppose that at the beginning, both channel is “good” channel, the probability of system has both of channels are “good” after transmission time σ, for example, is (1−α)2 . Let call Aij is the probability of system from the state has i − 1 “good” channels and n − i + 1 “bad” channels transfers to state has j − 1 “good” and n − j + 1 “bad” channels. Obviously that 1 ≤ i ≤ n + 1 and 1 ≤ j ≤ n + 1. For example, the transition matrix A2 and A3 for n = 2 and n = 3 are constructed respectively as follows: (1 − α)2  A2 = α(1 − α) α2   (1 − α)3  (1 − α)2 α A3 =  (1 − α)α2 α3 2α(1 − α) (1 − α)2 + α2 2α(1 − α) 3(1 − α)2 α 2(1 − α)α2 + (1 − α)3 2(1 − α)2 α + α3 3α2 (1 − α)  α2 α(1 − α) . (1 − α)2 3α2 (1 − α) 2(1 − α)2 α + α3 2(1 − α)α2 + (1 − α)3 3(1 − α)2 α given by Proposition 2. Moreover, this matrices are obviously central symmetric matrix. Proposition 1. For n channels system, the transition matrix An has size (n + 1) × (n + 1) and all entries An ij in row i column j will be established by An ij = s=max(i−j,0) n+1−i s i−1 j−i+2s n−(j−i+2s) α (1 − α) Proof. From the definition, Anij is the probability from state has i − 1 “good” channels or i − 1 bit “1” transfer to state has j − 1 “good” channels or j − 1 bit “1”. Therefore, suppose s is the number channels in i − 1 “good” channels that is flipped to “bad” channels after the transmission time σ and 0 ≤ s ≤ i − 1. Thus, to maintain j − 1 “good” channels after the time σ, the number of “bad” channels in n + 1 − i “bad” channels must be flipped to “good” channels is: (j − 1) − ((i − 1) − s) = j − i + s Therefore, the total number of channels are flipped their state after transmission time σ is: s + (j − i + s) = j − i + 2s and the total number of channels that preserves their state after transmission time σ is n−(j −i+2s). However, 0 ≤ s ≤ i−1. Similarly, the number of “bad” channels in n + 1 − i “bad” channels must be flipped to “good” channels should be in 0 ≤ j − i + s ≤ n + 1 − i. Hence: ( max s = min(n + 1 − j; i − 1) min s = max(0; i − j) Therefore, An ij can be determined by below form: An ij = j − i + s s=min(n+1−j,i−1) X s=max(i−j,0) n+1−i s i−1 j−i+2s n−(j−i+2s) α (1 − α) Proposition 2. All the entries of inverse matrix A−1 n given in Proposition 1 can be determined via original transition matrix  A n for ∀ α 6= 1/2. α3  (1 − α)α2  . (1 − α)2 α (1 − α)3 These transition matrices are obviously size (n+1)×(n+1) since the number of “good” channels can achieve n+1 discrete values from 0, 1, . . . , n. Moreover, these class matrices have several interesting properties: (1) all entries in matrix An can be determined by Proposition 1; (2) the inverse of matrix An is s=min(n+1−j,i−1) j − i + s X An −1 ij = (−1)i+j An (1 − 2α)n ij Due to the pages limitation, we will show the detailed proof at the end of this paper. To illustrate our result, an example of the inverse matrix A2 are shown as follows: A−1 2  (1 − α)2 1  = −α(1 − α) (1 − 2α)2 α2 −2α(1 − α) (1 − α)2 + α2 −2α(1 − α)  α2 −α(1 − α) . (1 − α)2 Next, base on the existence of inverse matrix closed form, we will show that a capacity closed form for a discrete memory-less channel can be established. We note that in [3], the authors said that haven’t closed form for channel capacity problem. However, with our approach, the closed form can be established for a wide range of channel with error rate α is small. II. O PTIMIZE SYSTEM CAPACITY A discrete memoryless channel is characterized by a channel matrix A ∈ Rm×n with m and n representing the numbers of distinct input (transmitted) symbols xi , i = 1, 2, . . . , m, and output (received) symbols yj , j = 1, 2, . . . , n, respectively. The matrix entry Aij represents the conditional probability that given a symbol xi is transmitted, the symbol xj is received. Let p = (p1 , p2 , . . . , pm )T be the input probability mass vector, where pi denotes the probability of transmitting symbol xi , then the probability mass vector of output symbols q = (q1 , q2 , . . . , qn )T = AT p, where qi denotes the probability of receiving symbol yi . For simplicity, we only consider the case n = m such that the number of transmitted input patterns is equal the number of received input patterns. The mutual information between input and output symbolsis: I(X; Y ) = H(Y ) − H(Y \|X), where H(Y ) = H(Y \|X) = j=n X qj j=1 m X n X − log qj pi Aij log Aij . i=1 j=1 Thus, the mutual information function can be written as: I(X; Y ) = − j=n X (AT p)j log (AT p)j + m X n X pi Aij log Aij , i=1 j=1 j=1 where (AT p)j denotes the jth component of the vector q = (AT p). The capacity C of a discrete memoryless channel associated with a channel matrix A puts a theoretical maximum rate that information can be transmitted over the channel [3]. It is defined as: C = max I(X; Y ). p (1) Therefore, finding the channel capacity is to find an optimal input probability mass vector p such that the mutual information between the input and output symbols is maximized. For a given channel matrix A, I(X; Y ) is a concave function in p [3]. Therefore, maximizing I(X; Y ) is equivalent to minimizing −I(X; Y ), and the capacity problem can be cast as the following convex problem: Minimize: m X n n X X pi Aij log Aij (AT p)j log (AT p)j − i=1 j=1 j=1 Subject to: ( pi 0 1T p = 1 Optimal numerical values of p∗ can be found efficiently using various algorithms such as gradient methods [4] [5]. However, in this paper, we try to figure out the closed form for optimal distribution p via KKT condition. The KKT conditions state that for the following canonical optimization problem: Problem Miminize: f (x) Subject to: gi (x) ≤ 0, i = 1, 2, . . . n, hj (x) = 0, j = 1, 2, . . . , m, construct the Lagrangian function: L(x, λ, ν) = f (x) + n X i=1 λi gi (x) + m X νj hj (x), (2) j=1 then for i = 1, 2, . . . , n, j = 1, 2, . . . , m, the optimal point x∗ must satisfy:   gi (x∗ ) ≤ 0,    ∗   hj (x ) = 0, dL(x,λ,ν) (3) \|x=x∗ ,λ=λ∗ ,ν=ν ∗ = 0, dx   ∗ ∗  λi xi = 0,    λ∗ ≥ 0. i Our transition matrix that is already established in previous part can represent as a channel matrix. In the optical transmission, for example, the transmission bits are denoted by the different levels of energy, for example, in On-Off Keying code bit “1” and “0” is represented by high and low power level. This energy is received by a photo diode and converse directly to the voltage for example. However, these photo diode work base on the aggregate property when collecting all the incident energy, that said, if two channels transmit a bit “1” then the photo diode will receive the same energy “2” even though this energy comes from a different pair of channels. Therefore, the received signal is completely dependent to the number of bits “1” in transmission side. Hence, in receiver side, the photo diode will recognize n + 1 states 0, 1, 2, . . . , n. From this property, the transition matrix A is the previous section is exactly the system channel matrix. The channel capacity of system, therefore, is determined as an optimization problem in (1). Next, we will show that the above optimization problem can be solved efficiently by KKT condition. We note that our method can establish the closed form for general channel matrix and then the results are applied to special matrix An . First, we try to optimize directly with input distribution p, however, the KKT condition for input distribution is too complicated to construct the first derivation. On the other hand, base on the existence of inverse channel matrix, the output variable is more suitable to work with KKT condition since. Due to 0 ≤ qj ≤ 1, the Lagrange function from (2) with output variable q is: L(qj , λj , νj ) = I(X, Y ) + j=n X j=n X qj λj + ν( qj − 1) j=1 j=1 Using KKT conditions, at optimal point qj∗ , λ∗j , ν ∗ :  ∗ qj ≥ 0   Pj=n ∗     j=1 qj = 1   dI(X, Y ) ν ∗ − λ∗j − =0 dqj∗    λ∗ ≥ 0     j∗ ∗ λj qj = 0 P Because 0 ≤ pi ≤ 1, i = 1, . . . , (n) and ni=1 pi = 1, so Pi=n always exist pi > 0. From qj = i=1 pi Aij with ∀ Aij > 0, we can see clearly that qj > 0 with ∀qj or qj∗ > 0 with ∀qj∗ . Therefore with fifth condition, λ∗j = 0 with ∀λ∗j . Then, we have simplified KKT conditions: Pj=n ∗  j=1 qj = 1 dI(X, Y ) =0 ν ∗ − dqj∗ The derivations are determined by: j=n i=n dI(X, Y ) X −1 X Aij log Aij − (1 + log qj ) Aji = dqj j=1 i=1 Let call: i=n X A−1 ji j=n X Aij log Aij = Kj From the second KKT simplified condition, we can compute ∀ qj∗ : ∗ qj∗ = 2Kj −ν −1 And finally: ∗ Due to the channel matrix is a closed form of α, the optimal input vector p and output vector q also is a function of α. However, we note that since the KKT condition works directly to the output variable q, the optimal input p can be invalid pi > 1 or pi < 0. In next step, our simulations shown that for n ≤ 10 and α ≤ 0.2, both output and input vector are valid. That said, our approach will be worked with a good system where the error probability α is small. In case of the invalid optimal input vector, the upper bound of channel capacity, of course, will be established. III. C ONCLUSION In this paper, our contributions are twofold: (1) establish an inverse closed form for a class of channel matrix based on the error probability α; (2) figure out the closed form for channel matrix with small error rate α and determine the upper bound system capacity for a high error rate channel. R EFERENCES [1] Jeff McDougall and Scott Miller. Sensitivity of wireless network simulations to a two-state markov model channel approximation. In Global Telecommunications Conference, 2003. GLOBECOM’03. IEEE, volume 2, pages 697–701. IEEE, 2003. [2] Hong Shen Wang and Nader Moayeri. Finite-state markov channel-a useful model for radio communication channels. IEEE transactions on vehicular technology, 44(1):163–171, 1995. [3] Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012. [4] Michael Grant, Stephen Boyd, and Yinyu Ye. Cvx: Matlab software for disciplined convex programming, 2008. [5] Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. j=1 i=1 A PPENDIX Next, using derivation of I(X,Y) at qj = qj∗ and last KKT condition: ν ∗ = Kj − (1 + log qj ∗ ) Hence: qj∗ = 2Kj −ν ∗ −1 Next, using first KKT simplified condition, we have the sum of all output states is 1. j=n X 2Kj −ν ∗ −1 =1 j=1 2 ν∗ = j=n X 2 Kj −1 j=1 Therefore, ν ∗ can be figured out by: ∗ ∗ pT = q T A−1 ij ν = log j=n X j=1 2Kj −1 Proof for Proposition 2. Proof. To simplify our notation, the “good” and “bad” channel are represented by bit “1” and “0”, respectively. Next, we will use the definition to show that: An An −1 = I If matrix A∗n is constructed by A∗nij = (−1)i+j An ij , then we need to show that: An A∗n = B = (1 − 2α)n I Firstly, we note that the An ij and An ∗ij is only different by sign of the first index (−1)i+j . Therefore, Bij which is computed by product of row i in matrix An ij and column j in matrix An ∗ij , can be computed by: Bij = k=n+1 X k=1 An ik A∗nkj Note that the An ik is the probability from state i “good” channels (with i − 1 bit “1” and n − i + 1 bit “0”) to medium state has k “good” channels (with k − 1 bit “1” and n − k + 1 bit “0”). Moreover, if the sign is ignored, then An ∗kj also is the probability going from state k to state j, too. However, n the state k includes C k−1 sub-states which have a same number of “good” and “bad” channels. For example with n = 2, state k = 2 includes two sub-states that contains one “good” and one “bad” channels are “10” an “01”. Therefore, the total number of sub-states while k runs from 1 to n is P k=n+1 n C k−1 = 2n sub-states. Let compute Bij by divided k=1 into two subsets: Compute Bij for i = j: This means that Bii is the sum of the probability from state i − 1 bit “1” go to states has k − 1 bit “1” then come back to state has i − 1 bit “1”. In 2n sub-states, we can divide back to n + 1 categories by the number of different position between i and k. • If all the bit in i and k are the same, then the probability is: ! ! C n n (1 − α)2n (1 − α)n (1 − α)n = C 0 0 • If all the bit in i and k different at only one position, then the probability is: ! n (1 − α)2(n−1) (1 − α)2 C 1 • If all the bit in i and k different at only two positions, then the probability is: ! n C (1 − α)2(n−2) (1 − α)2×2 2 • If all the bit in i and k different at all positions, then the probability is: ! C n (1 − α)2n n Therefore, Bii can be determined by the probability of all n + 1 categories such as: ! n 2t C α (1−α)2n−2t = ((1 − α)2 − α2 )n = (1−2α)n Bii = t t=0 t=n X A∗nkj into two subsets: Compute Bij for i6= j: Let divide k + j is odd and A∗nkj < 0 or k + j is even and A∗nkj > 0, Pk=n respectively. Therefore, Bij = k=1 An ik A∗nkj also is distributed into positive or negative subsets. Next, we will show that the positive subset in Bij is equal the negative subset then Bij = 0 for i 6= j. Indeed, suppose that state i with i − 1 bit “1” go to state k1 and then to back to state j with j −1 bit “1” and Bik1 is positive value. Next, we will show that existence a state k2 such that Bik2 is negative value and Bik1 = −Bik2 . Let call s is the number of positions where state i and j have a same bit. Obviously that s ≤ n − 1 due to i 6= j. For example if n = 4 and i = 1111 and j = 0001, we have s = 1 because i and j share a same bit “1” in the positions fourth. Suppose that an arbitrary state k1 are picked, we will show how to chose the state k2 with Bik1 = −Bik2 . Consider two follows cases: • If (n − s) is odd. k2 is constructed by maintain s position of k1 where i and j have same bit and flip bit in the n − s rest positions. • If (n − s) is even. k2 is constructed by maintain s + 1 position of k1 where s position are i and j have a same bit and one position where i and j have a different bit, next n − s − 1 rest positions will be flipped. Note that since s ≤ n − 1 then we are able to flip n − s − 1 rest positions. We obviously can see that k1 and k2 satisfied the probability condition \|Bik1 \| = \|Bik2 \| due to the number of flipped bit between i and k1 equals the number of flipped bit between k2 and j and the number of flipped bit between j and k1 equals the number of flipped bit between k2 and i. Next, we will prove that k1 and k2 make Bik1 and Bik2 in different subsets. Indeed, call number of bit “1” in k1 is b1 , number of bit “1” in k2 is b2 , number of bit “1” in s bit same of i and j is bs , respectively. Therefore, the number of bit “1” of k1 in (n − s) rest positions is (k1 − ks ), the number of bit “1” of k2 in (n − s) rest positions is (k2 − ks ). • If (n − s) is odd. Since all bit in (n − s) rest positions of k1 is flipped to create k2 , then total number of bit “1” in n − s bit of k1 and k2 is (k1 − ks + k2 − ks = n − s) is odd. So, (k1 + k2 ) should be an odd number. That said (k1 − k2 ) is odd or (k1 + j) − (k2 + j) is odd. Therefore, Bik1 and Bik2 bring the contradict sign. • If (n − s) is even. Because, we fix one more position to create k2 , then number of flipped bit (n−s−1) is odd number. If one more bit is fixed in k1 is “0”, we have a same result with case (n − s) is odd. If fixed bit is “1”, similarly in first case (k1 − ks − 1) + (k2 − ks − 1) = n − s − 1 is odd number, therefore (k1 + k2) is odd number. That said (k1 − k2 ) is odd or (k1 + j) − (k2 + j) is odd. Therefore, Bik1 and Bik2 bring the contradict sign. Therefore, the state k2 always can be created from a random state k1 and Bik1 and Bik2 bring a contradict sign. That said for i 6= j, Bij = 0. Therefore: B = (1 − 2α)n I The Proposition 2, therefore, are proven.	7
AN INTRODUCTION TO PRESENTATIONS OF MONOID ACTS: QUOTIENTS AND SUBACTS arXiv:1709.08916v1 [math.GR] 26 Sep 2017 CRAIG MILLER AND NIK RUŠKUC Abstract. The purpose of this paper is to introduce the theory of presentations of monoids acts. We aim to construct ‘nice’ general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an M -act A and a subact B of A, on the one hand we construct presentations for B and the Rees quotient A/B using a presentation for A, and on the other hand we derive a presentation for A from presentations for B and A/B. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a subact B has finite complement in an M -act A. We show that if M is a finitely generated monoid and B is finitely presented, then A is finitely presented. We also show that if M belongs to a wide class of monoids, including all finitely presented monoids, then the converse also holds. 1. Introduction The concept of presentations is significant within many areas of algebra. Finite presentability of acts was first studied by P. Normak in 1977 [11], and is a fundamental finiteness condition for the theory of monoid acts (see [9]). The related notion of coherency for monoids was introduced by V. Gould in 1992 [6], and has since been intensively studied by several authors (see [7], [8]). Finite presentability of acts also plays a key role in the monoid properties of being right Noetherian [11] and being completely right pure [5]. However, there has not yet been developed a systematic theory of presentations for acts over monoids. This paper is concerned with introducing such a theory through considering presentations for two of the most basic constructions: quotients and subacts. A follow-on article will deal with various product constructions of acts. The paper is structured as follows. In Section 2, we collect some basic definitions and facts about acts. In Section 3, we introduce the notions of presentations and finite presentability for a monoid act, provide various examples of act presentations, and record several results which will be of vital importance in the rest of the paper. In the remainder of the paper, we study presentations for various constructions. Typically we first obtain a general presentation for a construction and then derive corollaries regarding finite presentability. Section 4 is concerned 2010 Mathematics Subject Classification. 20M30, 20M05. 1 with presentations of Rees quotients. Before moving to presentations of subacts in general in Section 6, we first discuss presentations of unions of subacts in Section 5. Section 6 splits into two parts; in the first part we construct a general presentation for a subact, and in the second part we study a specific case where the subact has finite complement. 2. Preliminaries The theory of monoid acts is essentially the theory of representations of monoids by transformations. A monoid is a semigroup with an identity. One of the most common and universal ways of defining monoids is by means of presentations, and we briefly review the basics here. Let Z be an alphabet. We denote by Z ∗ the monoid of all words in Z. A monoid presentation is a pair hZ \| P iMon, where P ⊆ Z ∗ × Z ∗ . A monoid M is said to be defined by the monoid presentation hZ \| P iMon if M∼ = Z ∗ /ρ, where ρ is the smallest congruence on Z ∗ containing P . Thus we can identify M with Z ∗ /ρ, so that the elements of M are the ρ-classes of words from Z ∗ . To put it differently, each word w ∈ Z ∗ represents an element of M. Let u, v ∈ Z ∗ . We say that v is obtained from u by an application of a relation from P if u = pqr and v = pq ′ r, where p, r ∈ Z ∗ and (q, q ′) ∈ P . We say that u = v is a consequence of P if either u and v are identical words or if there exists a sequence u = w1 , w2 , . . . , wk = v where each wi+1 is obtained from wi by an application of a relation from R. We have the following basic fact: Lemma 2.1. Let M be a monoid defined by a presentation hZ \| P iMon, and let u, v ∈ Z ∗ . Then u = v holds in M if and only u = v is a consequence of R. Let M be a monoid with identity 1. An M-act is a non-empty set A together with a map A × M → A, (a, m) 7→ am such that a(mn) = (am)n and a1 = a for all a ∈ A and m, n ∈ M. For instance, M itself is an M-act via right multiplication. A subset B of an M-act A is a subact of A if bm ∈ B for all b ∈ B, m ∈ M. Note that the right ideals of M are precisely the subacts of the M-act M. A subset U of an M-act A is a generating set for A if for any a ∈ A, there exist u ∈ U, m ∈ M such that a = um. We write A = hUi if U is a generating set for A. An M-act A is said to be finitely generated (resp. cyclic) if it has a finite (resp. one-element) generating set. Note that a right ideal of M can be generated by a set as an M-act or as a semigroup. We introduce the convention that ‘generate’ will always mean ‘generate as an M-act’. For M-acts A and B, a map θ : A → B is an M-homomorphism if (am)θ = (aθ)m for all a ∈ A, m ∈ M. If θ is also bijective, then it is an M-isomorphism, and we write A ∼ = B. 2 An equivalence relation ρ on A is an (M-act) congruence on A if (a, b) ∈ ρ implies (am, bm) ∈ ρ for all a, b ∈ A and m ∈ M. For a congruence ρ on an M-act A, the quotient set A/ρ = {[a] : a ∈ A} becomes an M-act by defining [a]m = [am]. Given an M-act A and a subact B of A, we define the Rees congruence ρB on A by aρB b ⇐⇒ a = b or a, b ∈ B for all a, b ∈ A. We denote the quotient act A/ρB by A/B and call it the Rees quotient of A by B. We usually identify the ρB -class {a} ∈ A/B with a for each a ∈ A \ B, and denote the ρ-class B ∈ A/B by 0. For an M-act A and X ⊆ A × A, we denote by hXicg the smallest congruence on A containing X. A congruence ρ on an M-act A is finitely generated if there exists a finite subset X ⊆ A × A such that ρ = hXicg . Let A be an M-act and let X ⊆ A × A. We introduce the notation X = X ∪ {(u, v) : (v, u) ∈ X}, which will be used throughout the paper. For a, b ∈ A, an X-sequence connecting a and b is any sequence a = p1 m1 , q1 m1 = p2 m2 , q2 m2 = p3 m3 , . . . , qk mk = b, where (pi , qi ) ∈ X and mi ∈ M for 1 ≤ i ≤ k. We now provide the following useful lemma (see [9, Section 1.4] for a proof): Lemma 2.2. Let M be a monoid, let A be an M-act, let X ⊆ A × A and let a, b ∈ A. Then (a, b) ∈ hXicg if and only if either a = b or there exists an X-sequence connecting a and b. A generating set U for an M-act A is a basis of A if for any a ∈ A, there exist unique u ∈ U and m ∈ M such that a = um. An M-act A is said to be free if it has a basis. We have the following structure theorem for free acts. Proposition 2.3. [9, Theorem 1.5.13]. An M-act A is free if and only if it is isomorphic to a disjoint union of M-acts all of which are M-isomorphic to M. This leads to the following explicit construction of a free act. Construction 2.4. Let M be a monoid. Let X be a non-empty set and consider the set X × M. With the operation (x, m)n = (x, mn) for all (x, m) ∈ X × M and n ∈ M, the set X × M is a free M-act with basis X. We denote this M-act by FX,M , although we will usually just write FX . We will also usually write x · m for (x, m) and x for (x, 1). 3 Proposition 2.5. [9, Theorem 1.5.15]. Let A be an M-act and let F be a free M-act with basis X. If φ is any map from X to A, then there exists a unique M-homomorphism θ : F → A such that θ\|X = φ. Further, if Xφ is a generating set for A, then θ is surjective. Corollary 2.6. For any M-act A, there exists a free M-act F such that A is a homomorphic image of F . 3. Presentations of monoid acts Let M be a monoid. An (M-act) presentation is a pair hX \| Ri, where X is a non-empty set and R ⊆ FX × FX is a relation on the free M-act FX . An element x of X is called a generator, while an element (u, v) of R is called a (defining) relation, and is usually written as u = v. An M-act A is said to be defined by the presentation hX \| Ri if A is M-isomorphic to the quotient act FX /ρ, where ρ = hRicg is the smallest congruence on FX containing R. Let A be an M-act and θ : A → FX /ρ an M-isomorphism, where ρ = hRicg . We say an element w ∈ FX represents an element a ∈ A if aθ = [w]ρ . In the context of presentations, we write w1 ≡ w2 if w1 and w2 are equal in FX , and w1 = w2 if they represent the same element of A. Remark 3.1. Let A be an M-act and let X be any generating set for A. By Proposition 2.5, there exists a surjective M-homomorphism θ : FX → A, so we have that A ∼ = FX /ker θ by the First Isomorphism Theorem. Therefore, A is defined by the presentation hX \| Ri where R is any relation which generates ker θ. Hence, every M-act can be defined by a presentation. Definition 3.2. Let hX \| Ri be a presentation and let w1 , w2 ∈ FX . We say that the relation w1 = w2 is a consequence of R if w1 ≡ w2 or there is an R-sequence connecting w1 and w2 . We say that w2 is obtained from w1 by an application of a relation from R if there exists an R-sequence with only two distinct terms connecting w1 and w2 . The next lemma follows immediately from Lemma 2.2. Lemma 3.3. Let hX \| Ri be a presentation, let A be the M-act defined by hX \| Ri, and let w1 , w2 ∈ FX . Then w1 = w2 in A if and only if w1 = w2 is a consequence of R. Let M be a monoid, let A be an M-act generated by a set Y , and let φ : X → Y be a surjective map. Let θ : FX → A be the unique M-homomorphism extending φ, and let R be a subset of FX × FX . We say that A satisfies R (with respect to φ) if for each (u, v) ∈ R, we have uθ = vθ; that is, R ⊆ ker θ. Note that the M-act defined by a presentation hX \| Ri satisfies R. From the definition of an act defined by a presentation and Lemma 2.2, we have: 4 Proposition 3.4. Let M be a monoid, let A be an M-act generated by a set X, and let R ⊆ FX × FX . Then hX \| Ri is a presentation for A if and only if the following conditions hold: (1) A satisfies R; (2) if w1 , w2 ∈ FX such that A satisfies w1 = w2 , then w1 = w2 is a consequence of R. The next fact follows from Proposition 2.5 and the Third Isomorphism Theorem for acts. Proposition 3.5. Let A be an M-act defined by a presentation hX \| Ri, let B be an M-act, and let φ : X → B be a map onto a generating set for B. If the generators Xφ of B satisfy all the relations from R, then there exists a surjective M-homomorphism ψ : A → B. Definition 3.6. A finite presentation is a presentation hX \| Ri where X and R are finite. An M-act A is finitely presented if it can be defined by a finite presentation. Note that a right ideal of a monoid M may be finitely presented as an M-act or as a semigroup. When we say that a right ideal is ‘finitely presented’, we will always mean as an M-act. Example 3.7. (1) The free M-act FX is defined by the finite presentation hX \| i. In particular, if X is finite, then FX is finitely presented. (2) For any monoid M, the M-act M is finitely presented, since M is a free M-act with basis {1}. The following results are specialisations of well-known facts from general algebra. They essentially reflect the fact that congruence-generation is an algebraic closure operator. See, for instance, Section 1.5 and Theorem 2.5.5 in [3] for more details. Proposition 3.8. Let M be a monoid, let A be an M-act defined by a finite presentation hX \| Ri, and let Y be another finite generating set for A. Then A can be defined by a finite presentation in terms of the generators Y . Proposition 3.9. Let M be a monoid, and let A be a finitely presented M-act with a presentation hX \| Si where X is finite and S is infinite. Then there exists a finite subset S ′ ⊆ S such that A is defined by the finite presentation hX \| S ′i. Corollary 3.10. [11, Theorem 2]. Let M be a monoid and let A be a cyclic M-act. Then A is finitely presented if and only if A is isomorphic to a quotient act of M by a finitely generated right congruence on M. Let M be a monoid with a generating set S and let A be an M-act with a generating set X. It can be easily proved, using Proposition 3.4, that the following are all presentations for A: (1) hA \| a · m = am (a ∈ A, m ∈ M)i; 5 (2) hX \| x · m = y · n (x, y ∈ X, m, n ∈ M, xm = yn)i; (3) hA \| a · s = as (a ∈ A, s ∈ S)i. Given the presentation in (3), we immediately have the following: Lemma 3.11. If M is a finitely generated monoid and A is a finite M-act, then A is finitely presented. If M is a non-finitely generated monoid, however, then finite M-acts are not necessarily finitely presented, as the following example demonstrates. Example 3.12. Let M = X ∗ be a free monoid with X infinite and consider the trivial M-act A = {0}. Now A is defined by the presentation h0 \| 0 · x = 0 (x ∈ X)i. If A were finitely presented, then it could be defined by a finite presentation P = h0 \| 0 · x = 0 (x ∈ X0 )i, where X0 is a finite subset of X. But for x 6∈ X0 , the relation 0 · x = 0 is clearly not a consequence of the relations of P , so A is not finitely presented. Remark 3.13. One may be tempted to think that the trivial M-act being finitely presented is equivalent to M being finitely generated. However, the trivial M-act is in fact finitely presented for a much larger class of monoids M. For example, if M is a monoid with a left zero z, it can be easily proved that the trivial M-act {0} is defined by the finite presentation h0 \| 0 = 0 · zi. The following lemma provides a necessary and sufficient condition for the trivial act to be finitely presented. Lemma 3.14. Let M be a monoid. Then the trivial M-act {0} is finitely presented if and only if there exists a finitely presented M-act A which contains a zero. Proof. The direct implication is obvious. For the converse, let A be an M-act with a zero 0, and suppose that A is defined by a finite presentation hX \| Ri where 0 ∈ X. We define a finite set R′ = {0 · m = 0 · n : (x · m, y · n) ∈ R for some x, y ∈ X}, and claim that {0} is defined by the presentation h0 \| R′i. We need to show that for any m ∈ M, the relation 0 · m = 0 is a consequence of R′ . Let m ∈ M. Since 0 · m = 0 holds in A, it is a consequence of R, so we have an R-sequence connecting 0 · m and 0. Now, replacing every x ∈ X appearing in this R-sequence with 0, we obtain an R′ -sequence connecting 0 · m and 0, so 0 · m = 0 is a consequence of R′ . Tietze transformations (for acts) provide a method for yielding a new presentation for a monoid act from a known presentation. Given a presentation hX \| Ri for an M-act A, the elementary Tietze transformations are: 6 (T1) adding new relations ui = vi , i ∈ I, to hX \| Ri, providing that each ui = vi is a consequence of R; (T2) deleting relations ui = vi , i ∈ I, from R, providing that each ui = vi is a consequence of R \ {ui = vi : i ∈ I}; (T3) adding new generating elements yi , i ∈ I, and new relations yi = wi , i ∈ I, to hX \| Ri, for any wi ∈ FX ; (T4) if hX \| Ri has relations xi = wi , i ∈ I, where xi ∈ X and wi ∈ FX ′ where X ′ = X \ {xi : i ∈ I}, then deleting each xi from X, deleting each xi = wi from R, and replacing all remaining appearances of xi with wi . Proposition 3.15. Two presentations define the same M-act if and only if one can be obtained from the other by a finite number of applications of elementary Tietze transformations. Remark 3.16. The proof of Proposition 3.15 is essentially the same as the proof for its analogue in group theory or semigroup theory. To see an idea of the proof, one may consult [13, Theorem 2.5]. Corollary 3.17. Let M be a monoid, and let A be an M-act defined by a presentation hX \| Si where X is finite and S is infinite. Then A is finitely presented if and only if there exists a finite subset S ′ ⊆ S such that every relation from S is a consequence of S ′ . Proof. Suppose that A is finitely presented. By Proposition 3.9, there exists a finite subset S ′ ⊆ S such that A is defined by a finite presentation hX \| S ′ i. Therefore, since every relation from S holds in A, it must be a consequence of S ′ . Conversely, suppose that there exists a finite subset S ′ ⊆ S such that every relation from S is a consequence of S ′ . Using Tietze transformations, we can delete every relation from S \ S ′ . By Proposition 3.15, we have that A is defined by the finite presentation hX \| S ′i. 4. Rees quotients Let M be a monoid, let A be an M-act and let B be a subact of A. Recall that the Rees quotient A/B is the quotient act resulting from the Rees congruence ρB on A given by aρB b ⇐⇒ a = b or a, b ∈ B for all a, b ∈ A. We shall identify the ρB -class {a} ∈ A/B with a for each a ∈ A\B, and denote the ρ-class B ∈ A/B by 0. The purpose of this section is, on the one hand, to construct a presentation for A/B using a presentation for A and a generating set for B, and on the other hand, to derive a presentation for A using presentations for B and A/B. These general presentations will give rise to corollaries pertaining to finite presentability. Let X be any generating set for A. We now give a presentation for A/B in terms of the generators (X \ B) ∪ {0}. 7 Theorem 4.1. Let M be a monoid. Let A be an M-act defined by a presentation hX \| Ri, let B be a subact of A generated by Y , and let h0 \| Si be a presentation for the trivial M-act {0}. For each y ∈ Y , choose wy ∈ FX such that y = wy holds in A, and let R′ = R ∪ {y = wy : y ∈ Y }. We now define the sets R1 = {(u, v) ∈ R : u represents an element of A \ B}, R2 = {u = 0 : u ∈ FX\B ∩ L(X, B), (u, v) ∈ R′ for some v ∈ FX∪Y }, where L(X, B) denotes the set of elements of FX which represent elements of B. Then A/B is defined by the presentation hX \ B, 0 \| R1, R2 , Si. Proof. It is clear that A/B satisfies R1 , R2 and S. Let X ′ = X \ B, and let w1 , w2 ∈ FX ′ ∪{0} such that w1 = w2 holds in A/B. By Proposition 3.4, we just need to show that w1 = w2 is a consequence of R1 , R2 and S. If w1 represents an element of A \ B, then w1 = w2 is a consequence of R1 . Suppose w1 represents 0 in A/B. We claim that w1 = 0 is a consequence of R1 ∪ R2 ∪ S. If w1 ∈ F0 , then w1 = 0 is a consequence of S. Now assume that w1 ∈ FX ′ . By Proposition 3.15, we have that A is defined by the presentation hX, Y \| R′i. Choose w1′ ∈ FY such that w1 = w1′ holds in A. Then w1 = w1′ is a consequence of R′ , so there exists an R′ -sequence w1 = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = w1′ , where (pi , qi ) ∈ R′ and mi ∈ M for 1 ≤ i ≤ k. Note that for i ∈ {1, . . . , k}, if pi represents an element of A \ B, then pi+1 ∈ FX ′ . Therefore, since p1 ∈ FX ′ and pk represents an element of B (pk = qk in A and qk ∈ FY ), we may choose i minimal such that pi ∈ FX ′ and pi represents an element of B. We then have that w1 = pi mi is a consequence of R1 , and we obtain 0 · mi from pi mi by an application of a relation from R2 . Now, since 0 · mi = 0 is a consequence of S, we deduce that w1 = 0 is a consequence of R1 , R2 and S. This proves the claim. Exactly the same argument proves that w2 = 0 is a consequence of R1 ∪ R2 ∪ S, and hence so is w1 = 0 = w2 , as required. Corollary 4.2. Let M be a monoid, let A be a finitely presented M-act and let B be a finitely generated subact of A. Then A/B is finitely presented if and only if the trivial M-act is finitely presented (which includes all finitely generated monoids). Proof. If A/B is finitely presented, then it follows from Lemma 3.14 that the trivial M-act is finitely presented, since A/B contains a zero. The converse follows immediately from Theorem 4.1. We now turn to our second aim in this section: assembling a presentation for A from those for a subact and the Rees quotient. So, let M be a monoid, let A be an M-act and let B be a subact of A. Let X be a generating set for B and let Y be a generating set for A/B, and let Y ′ = Y \ {0}. Note that if A \ B is a subact of A, the element 0 must belong to Y , and so Y = Y ′ ∪ {0}. If A \ B is not a 8 subact of A/B, then there exist elements w ∈ FY ′ that represent 0 in A/B, and the set Y need not contain 0. We shall now give a presentation for A in terms of the generators X ∪ Y ′ . Theorem 4.3. Let M be a monoid, let A be an M-act and let B be a subact of A. Let hX \| Ri and hY \| Si be presentations for B and A/B respectively, and let Y ′ = Y \ {0}. If A \ B is not a subact of A/B, for each w ∈ FY ′ that represents 0 in A/B choose αw ∈ FX such that w = αw in A, and also fix one of them and denote it by z. We now define the sets S1 = {(u, v) ∈ S : u represents an element of A \ B}; S2 = {u = αu : (u, v) ∈ S for some v ∈ FY , u ∈ FY ′ , u = 0 in A/B} ∪ {z = αz }. Then A is defined by the presentation hX, Y ′ \| R, S1, S2 i. Proof. We first claim that if an element w ∈ FY ′ represents an element of B in A, then there exists w ′ ∈ FX such that w = w ′ is a consequence of relations from S1 and S2 . Indeed, we have that w = z holds in A/B, so w = z is a consequence of S; that is, there exists an S-sequence w = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = z, where (pi , qi ) ∈ S and mi ∈ M for 1 ≤ i ≤ k. If all (pi , qi ) ∈ S1 , then w = αz is a consequence of S1 and z = αz . Otherwise, we take (pi , qi ) ∈ S \ S1 with i minimal, so w = pi mi is a consequence of S1 , and we obtain αpi mi from pi mi by an application of a relation from S2 . We shall now show that A is defined by the presentation hX, Y ′ \| R, S1 , S2 i. It is clear that A satisfies R, S1 and S2 . Let w1 , w2 ∈ FX∪Y ′ be such that w1 = w2 in A. If w1 represents an element of A \ B, then w1 = w2 is a consequence of S1 . Now suppose that w1 represents an element of B. Using the above claim, if necessary, we have w1′ , w2′ ∈ FX such that w1 = w1′ and w2 = w2′ are consequences of S1 and S2 (if wi ∈ FX , simply let wi ≡ wi′ ). But then w1′ = w2′ holds in B, so it is a consequence of R. Hence, we have that w1 = w2 is a consequence of R, S1 and S2 . Corollary 4.4. Let M be a monoid, let A be an M-act and let B be a subact of A. If B and A/B are finitely presented, then A is finitely presented. 5. Unions In this section we consider presentations for unions of acts. A union of acts can be of one of two types: disjoint or amalgamated. An amalgamated union of Macts is a union of a family of M-acts intersecting pairwise in a common subact. We only consider the union of two acts, although the results of this section can easily be generalised to any finite number of acts. Throughout the section we aim to prove our results in the general setting where C = A ∪ B is an M-act with 9 A and B subacts, and A ∩ B is potentially non-empty. Each of those results will typically have an immediate corollary for disjoint unions, which we state separately immediately after. The main purpose of the section is to explore under what conditions we have C = A∪B is finitely generated (resp. finitely presented) if and only if A and B are finitely generated (resp. finitely presented), and to provide interesting examples to demonstrate that this does not occur in general. We begin by considering finite generation. Lemma 5.1. Let C = A ∪ B be an M-act with A and B subacts of C. If A and B are finitely generated, then C is finitely generated. Proof. If A = hXi and B = hY i, then C = hX ∪ Y i. In the following example, we show that the converse to Lemma 5.1 does not hold in general by constructing a finitely generated monoid M and right ideals A and B of M such that C = A ∪ B is finitely generated (in fact, finitely presented) but neither A nor B are finitely generated. Example 5.2. Let M = {a, b}∗ . Let X = {ai b : i ≥ 0} and Y = {bi a : i ≥ 0}, and let A and B be the right ideals generated by X and Y respectively. It is clear that A and B are not finitely generated. We have that C = A ∪ B is generated by the set {a, b} and is free with respect to this generating set, so C is finitely presented. Lemma 5.3. Let C = A ∪ B be an M-act with A and B subacts of C, and suppose that A ∩ B is either empty or finitely generated. If C is finitely generated, then A and B are finitely generated. Proof. If A ∩ B = ∅, let U = ∅; otherwise, let A ∩ B = hUi where U is finite. Suppose that C = hZi. Let Y = Z \ B and let X = Y ∪ U. Let a ∈ A. If a ∈ A \ B, then a = ym for some y ∈ Y and m ∈ M. If a ∈ A ∩ B, then a = um for some u ∈ U and m ∈ M. Therefore, we have that A = hXi. Hence, if Z is finite, A is finitely generated, and by symmetry so is B. Corollary 5.4. Let A and B be disjoint M-acts. Then A ∪ B is finitely generated if and only if A and B are finitely generated. We now turn our attention to finite presentability. We begin by giving a general presentation for C = A ∪ B, and we then immediately derive a corollary that gives a sufficient condition for C to be finitely presented. Theorem 5.5. Let C = A ∪ B be an M-act with A and B subacts of C. Let A and B have presentations hX \| Ri and hY \| Si respectively. If A ∩ B 6= ∅, let U be a generating set for A ∩ B; otherwise, let U = ∅. For each u ∈ U, choose ρX (u) ∈ FX and ρY (u) ∈ FY which both represent u in C, and define a set T = {ρX (u) = ρY (u) : u ∈ U}. 10 Then C is defined by the presentation hX, Y \| R, S, T i. Proof. Let w1 , w2 ∈ FX∪Y such that w1 = w2 in C. If w1 , w2 ∈ FX , then w1 = w2 is a consequence of R. If w1 , w2 ∈ FY , then w1 = w2 is a consequence of S. Suppose now that w1 ∈ FX and w2 ∈ FY . Let c = um, with u ∈ U and m ∈ M, be the element of A ∩ B that both w1 and w2 represent. Since w1 = ρX (u)m holds in A, it is a consequence of R, and likewise w2 = ρY (u)m is a consequence of S. We also obtain ρY (u)m from ρX (u)m by an application of a relation from T . Therefore, we have that w1 = w2 is a consequence of R, S and T . Corollary 5.6. Let C = A ∪ B be an M-act with A and B subacts of C, and suppose that A ∩ B is either empty or finitely generated. If A and B are finitely presented, then C is finitely presented. The converse to Corollary 5.6 does not hold in general. Recall that in Example 5.2 we showed that there exists a monoid M and M-acts A and B such that C = A ∪ B is finitely presented but neither A nor B are finitely generated. We now present a more striking example: Example 5.7. There exist a monoid M and finitely generated right ideals A and B of M such that A ∩ B is finitely generated and C = A ∪ B is finitely presented but neither A nor B are finitely presented. Let M be the monoid defined by the presentation ha, b, s, t \| abi a = aba, bai b = bab, sa = a, tb = b (i ≥ 2)iMon . We have a complete rewriting system RM on X = {a, b, s, t} consisting of the rules abi a → aba, bai b → bab, sa → a, tb → b (i ≥ 2), and this yields the following set of normal forms for M: X ∗ \ X ∗ ({abi a, bai b : i ≥ 2} ∪ {sa, tb})X ∗ ; that is, the set of all words in X which do not contain as a subword the left-hand side of one of the rewriting rules. For more information on rewriting systems, one may consult [2] for instance. Let A and B be the right ideals of M generated by {a, t} and {b, s} respectively. From the monoid presentation for M, we see that A is defined by the infinite presentation ha, t \| a · bi a = a · ba (i ≥ 2)i. If A were finitely presented, then it could be defined by a presentation P = ha, t \| a · bi a = a · ba (2 ≤ i ≤ k)i. But if i > k, then the relation a · bi a = a · ba cannot be a consequence of the relations of P , since there do not exist m, n ∈ M such that bi a = mn in M and (a · m, a · ba) ∈ P ; in other words, no relation of P can be applied to a · bi a. 11 Therefore, A is not finitely presented. Similarly, we have that B is not finitely presented. We also that A ∩ B = ha, bi. It is clear from the monoid presentation for M that C = A ∪ B is generated by the set {s, t} and is free with respect to this generating set, so hence C is finitely presented. We now turn to consider conditions for when C = A ∪ B being finitely presented implies that the components A and B are both finitely presented. Theorem 5.8. Let C = A ∪ B be an M-act with A and B subacts of C, and suppose that A ∩ B is either empty or finitely presented. If C is finitely presented, then both A and B are finitely presented. Proof. It clearly suffices to show that A is finitely presented. If A ∩ B = ∅, let U = ∅; otherwise, let A ∩ B be defined by a finite presentation hU \| Si. Suppose C is defined by a finite presentation hZ \| Ri. Let Y = Z \ B and X = Y ∪ U. As in the proof of Lemma 5.3, we have that A = hXi. Also, let Y ′ = Z ∩ B. For each w ∈ FZ such that w represents an element of A∩B, choose ρU (w) ∈ FU which represents the same element of A ∩ B. Also, for each u ∈ U, choose wu ∈ FZ such that wu represents u. We now define the following sets: R1 = {(u, v) ∈ R : u, v ∈ FY }; R2 = {u = wu : u ∈ U, wu ∈ FY }; R3 = {u = ρU (u) : u ∈ FY , (u, v) ∈ R for some v ∈ FY ′ }. Note that the set R2 may be empty (if wu ∈ FY ′ for every u ∈ U); however, this does not affect the argument that follows. We make the following claim: Claim. If an element w ∈ FY represents an element of A ∩ B, then there exists w ′ ∈ FU such that w = w ′ is a consequence of relations from R1 , R2 and R3 . Proof. Let w ∈ FY represent an element c ∈ A ∩ B. Now c = um for some u ∈ U and m ∈ M. Since w = wu m holds in C, it is a consequence of R, so there exists an R-sequence w = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = wu m, where (pi , qi ) ∈ R and mi ∈ M for 1 ≤ i ≤ k. If each (pi , qi ) ∈ R1 , then w = wu m is a consequence of R1 , and we obtain u · m from wu m by an application of a relation from R2 . Otherwise, there exists i minimal such that pi ∈ FY and qi ∈ FY ′ , so w = pi mi is a consequence of R1 , and we obtain ρU (pi )mi from pi mi by an application of a relation from R3 . Returning to the proof of Theorem 5.8, we shall show that A is defined by the finite presentation hX \| R1 , R2 , R3 , Si. Let w1 , w2 ∈ FX such that w1 = w2 holds in A. If w1 represents an element of A \ B, then w1 = w2 is a consequence of R1 . Suppose w1 represents an element of A ∩ B. Using the claim above, if necessary, we have w1′ , w2′ ∈ FU such that w1 = w1′ and w2 = w2′ are consequences of R1 , R2 and R3 (if wi ∈ FU , simply let 12 wi ≡ wi′ ). Since w1′ = w2′ holds in A ∩ B, we have that w1′ = w2′ is a consequence of S. Therefore, w1 = w2 is a consequence of R1 , R2 , R3 and S. Corollary 5.9. Let A and B be disjoint M-acts. Then A ∪ B is finitely presented if and only if A and B are finitely presented. We now investigate how finite presentability of C = A∪B affects the intersection A ∩ B. Proposition 5.10. Let C = A ∪ B be an M-act with A and B subacts of C, and suppose that A ∩ B is non-empty. If C is finitely presented and A and B are finitely generated, then A ∩ B is finitely generated. Proof. Let A and B be generated by finite sets X and Y respectively. Since C is finitely presented, it can be defined by a finite presentation hX, Y \| Ri. For any w ∈ FX , let w denote the element of A which w represents, and define U = {u : u ∈ FX , (u, v) ∈ R for some v ∈ FY } ⊆ A ∩ B. Let c ∈ A ∩ B. Choose w1 ∈ FX and w2 ∈ FY which both represent the element c. Since w1 = w2 holds in C, it is a consequence of R, so there exists an R-sequence w1 = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = w2 , where (pi , qi ) ∈ R and mi ∈ M for 1 ≤ i ≤ k. Now, there exists i ∈ {1, . . . , k} such that pi ∈ FX and qi ∈ FY , and we have that c = pi mi , so c ∈ hUi. Hence, we have that A ∩ B = hUi, so A ∩ B is finitely generated. Corollary 5.11. Let C = A ∪ B be an M-act with A and B finitely presented subacts of C and A ∩ B non-empty. Then C is finitely presented if and only if A ∩ B is finitely generated. Remark 5.12. In the categorical sense, the disjoint union of acts is a coproduct and the amalgamated union of acts is a pushout. In the category of groups, the coproduct is called the free product and the pushout is called the free product with amalgamation. Notice the similarity between Corollary 5.11 and a well-known result, due to G. Baumslag, which states that for two finitely presented groups G1 and G2 such that H = G1 ∩ G2 is a group, the amalgamated free product G = G1 ∗H G2 is finitely presented if and only if H is finitely generated; see [1, Chapter 6] for more details. Given the results concerning finite presentability in this section, the following two questions arise: Do there exist monoids M and M-acts A, B and C with C = A ∪ B, such that: (1) A and B are finitely presented but C is not finitely presented? (2) A and B are finitely presented, while A ∩ B is finitely generated but not finitely presented? 13 In the following, we exhibit examples which provide positive answers to both of the above two questions. Example 5.13. There exists a monoid M and finitely presented right ideals A and B of M such that C = A ∪ B is not finitely presented. Let M be the monoid defined by the presentation ha, b, c \| aci a = bci−1 b (i ≥ 2)iMon . We have a complete rewriting system RM on X = {a, b, s, t} consisting of the rules aci a → bci−1 b (i ≥ 2), and this yields the following set of normal forms for M: X ∗ \ (X ∗ {aci a : i ≥ 2}X ∗ ). Let A and B be the right ideals of M generated by {a} and {b} respectively. We have that A and B are free M-acts and hence finitely presented. Let X = {aci a : i ≥ 2}. It is clear from the monoid presentation for M that X generates A∩B, and that this is a minimal generating set for A ∩ B, so A ∩ B is not finitely generated. It now follows from Corollary 5.11 that C = A ∪ B is not finitely presented. Example 5.14. There exists a monoid M and finitely presented right ideals A and B of M such that A ∩ B is finitely generated but not finitely presented. Let M be the monoid defined by the presentation ha, b, c \| a2 = a, cab = ab, abi a = aba (i ≥ 2)iMon. We have a complete rewriting system RM on X = {a, b, c} consisting of the rules a2 → a, cab → ab, abi a → aba (i ≥ 2), and this yields the following set of normal forms for M: X ∗ \ X ∗ ({abi a : i ≥ 2} ∪ {a2 , cab})X ∗ . Let A and B be the right ideals of M generated by {a} and {c} respectively. From the monoid presentation for M, we see that B is a free M-act (and hence finitely presented), that A is defined by the infinite presentation ha \| a · a = a, a · bi a = a · ba (i ≥ 2)i, and that A ∩ B is defined by the infinite presentation hy \| y · bi a = y · a (i ∈ N)i, where y represents ab. We claim that A is also defined by the finite presentation ha \| a · a = ai. Indeed, for any i ≥ 2, we have a · bi a = (a · a)bi a ≡ a · abi a ≡ a · aba ≡ (a · a)ba ≡ a · ba. It can be shown that A ∩ B is not finitely presented using a similar argument to the one in Example 5.7. 14 Note that since A and B are finitely presented and A ∩ B is finitely generated, Corollary 5.11 implies that C = A ∪ B is finitely presented. In fact, it is easy to see that C is defined by the finite presentation ha, c \| a · a = a, a · b = c · abi. 6. Subacts In this section we consider presentations for subacts of monoid acts. In the first part of the section we construct a general (infinite) presentation for a subact of a monoid act. From this presentation we obtain a method for finding ‘nicer’ presentations in special situations. We note that for general monoids M, finitely generated subacts of finitely presented M-acts are not necessarily finitely presented. In the second part of the section, we shall consider a particular case where we have a subact B with finite complement in an M-act A; we say that B is large in A and A is a small extension of B. This was motivated by the analogous concept of ‘large subsemigroups’ within semigroup theory; see [12] for more details. In particular, it is shown there that various finiteness properties, including finite generation and finite presentability, are inherited by both large subsemigroups and small extensions of semigroups. Given these results, it is natural to ask whether similar results hold in the setting of monoid acts. We shall show that, for finitely generated monoids M, finite generation is inherited by both large subacts and small extensions, and finite presentability is also inherited by small extensions. Somewhat surprisingly, though, there exist finitely generated monoids M for which large subacts of finitely presented M-acts are not necessarily finitely presented. We shall show, however, that there is a large class of monoids for which finite presentability is inherited by large subacts. Let M be a monoid, let A be an M-act defined by a presentation hX \| Ri, and let B be a subact of A generated by a set Y . We seek a presentation for B in terms of the generators Y . For each y ∈ Y , we choose wy ∈ FX which represents y, and let ψ : FY → FX be the unique M-homomorphism extending y 7→ wy . We call ψ the representation map. For an element w ∈ FX which represents an element of B, we have w = x · m for some x ∈ X and m ∈ M, and xm = yn for some y ∈ Y, n ∈ M, so w = (yψ)n holds in B. Therefore, we have a map φ : L(X, B) → FY , where L(X, B) denotes the set of all elements of FX which represent elements of B, satisfying (wφ)ψ = w in A for all w ∈ L(X, B). We call φ a rewriting map. Note that the existence of φ follows from the Axiom of Choice. We now state our first result of this section, giving a presentation for B, which has analogues within group and semigroup theory; see [10, Theorem 2.6] and [4, Theorem 2.1] for more details. 15 Theorem 6.1. Let M be a monoid. Let A be an M-act defined by a presentation hX \| Ri and let B be a subact of A generated by Y . For each y ∈ Y , we choose wy ∈ FX which represents y. Let ψ be the representation map and let φ be a rewriting map, and define the following sets of relations: R1 = {y = wy φ : y ∈ Y }; R2 = {(wm)φ = (wφ)m : w ∈ L(X, B), m ∈ M}; R3 = {(um)φ = (vm)φ : (u, v) ∈ R, m ∈ M, um ∈ L(X, B)}. Then B is defined by the presentation hY \| R1 , R2 , R3 i. Proof. We first show that B satisfies R1 , R2 and R3 . This amounts to showing that uψ = vψ holds in A for each u = v in R1 , R2 and R3 . For each y ∈ Y , we have yψ ≡ wy = (wy φ)ψ holds in A, since wy ∈ L(X, B). For any w ∈ L(X, B) and m ∈ M, we have ((wm)φ)ψ = wm = ((wφ)ψ)m ≡ ((wφ)m)ψ holds in A. Finally, for any (u, v) ∈ R, m ∈ M such that um ∈ L(X, B), we have ((um)φ)ψ = um = vm = ((vm)φ)ψ holds in A. We now claim that for any w ∈ FY , we have that w = (wψ)φ is a consequence of R1 and R2 . Indeed, we have w ≡ y · m for some y ∈ Y and m ∈ M, so wψ ≡ wy m. We obtain (wy φ)m from w by an application of the relation y = wy φ, and since wy ∈ L(X, B), we have that (wψ)φ = (wy φ)m is a relation from R2 . Now let w1 , w2 ∈ FY be such that w1 = w2 holds in B. Since w1 ψ = w2 ψ holds in A, it is a consequence of R, so we have an R-sequence w1 ψ = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = w2 , where (pi , qi ) ∈ R and mi ∈ M for 1 ≤ i ≤ k. For each i ∈ {1, . . . , k}, we have pi mi ∈ L(X, B), so (w1 ψ)φ = (w2 ψ)φ is a consequence of the relations (pi mi )φ = (qi mi )φ of R3 . Finally, since w1 = (w1 ψ)φ and (w2 ψ)φ = w2 are consequences of R1 and R2 , we conclude that w1 = w2 is a consequence of R1 , R2 and R3 . Remark 6.2. The presentation from Theorem 6.1 has the disadvantage that it always has infinitely many relations if M is an infinite monoid, and neither the rewriting map nor the set L(X, B) have been defined constructively. However, the result does give a method for finding ‘nice’ presentations for subacts in certain cases. Given an M-act A defined by a presentation hX \| Ri and a subact B of A, this method consists of the following: (1) finding a generating set Y for B; (2) finding a rewriting map φ : L(X, B) → FY ; (3) finding a set S ⊆ FY × FY of relations which hold in B and imply the relations of the presentation P given in Theorem 6.1. 16 Using Tietze transformations, we can add S to P (S must be a consequence of the relations of P since these are defining relations for B) and then remove the remaining relations (since they are consequences of S). Hence, by Proposition 3.15, we have that B is defined by the presentation hY \| Si. For the remainder of this section we shall be considering large subacts. Recall that a subact B of an M-act is said to be large in A, and A is said to be a small extension of B, if the set A \ B is finite. We shall investigate how similar an Mact A and a large subact B of A are with regard to finite generation and finitely presentability. We begin by considering finite generation. Lemma 6.3. Let M be a monoid, let A be an M-act and let B be a large subact of A. If B is finitely generated, then A is finitely generated. Proof. If B is generated by a set X, then A is generated by X ∪ (A \ B). In the following, we show that the converse to Lemma 6.3 does not hold for monoids in general, but it does however hold for all groups and all finitely generated monoids. Example 6.4. Let M = X ∗ with X infinite. Let I = X + , so I is a large subact of the cyclic M-act M. Clearly X is a minimal generating set for I, so I is not finitely generated. Lemma 6.5. Let M be a group, let A be an M-act and let B be a large subact of A. If A is finitely generated, then B is finitely generated. Proof. Since A is the disjoint union of its subacts B and A \ B, it follows from Corollary 5.4 that B is finitely generated if A is finitely generated. The following result provides a generating set for a large subact. Proposition 6.6. Let M be a monoid generated by a set Z, let A be an M-act generated by a set X, and let B be a large subact of A. Define a set S = {am ∈ B : a ∈ A \ B, m ∈ Z}, and let Y = (X ∩ B) ∪ S. Then B is generated by the set Y . Proof. Since Y ⊆ B and B is a subact of A, we have hY i ⊆ B. Let b ∈ B. If b = xm for some x ∈ X ∩ B, m ∈ M, then b ∈ hY i. Otherwise, b = xm1 . . . mk for some x ∈ X \ B, mi ∈ Z. Let s be minimal such that xm1 . . . ms ∈ B, and let a = xm1 . . . ms−1 . Then a ∈ A \ B and ams ∈ B, so ams ∈ S. Therefore, we have b = (ams )ms+1 . . . mk ∈ hSi ⊆ hY i. Hence, we have that B = hY i. Corollary 6.7. Let M be a finitely generated monoid, let A be an M-act and let B be a large subact of A. If A is finitely generated, then B is finitely generated. 17 We have shown that, for groups and finitely generated monoids M, finite generation is inherited by both large subacts and small extensions. We now turn our attention to finite presentability. For a monoid M, there are two questions relating to large subacts that arise: (1) Is every small extension of every finitely presented M-act finitely presented? (2) Is every large subact of every finitely presented M-act finitely presented? We first show that the property every small extension of every finitely presented M-act is finitely presented, is equivalent to another monoid property, and from this we immediately derive as a corollary that finitely generated monoids M admit a positive answer to question (1). Proposition 6.8. The following are equivalent for a monoid M: (1) every finite M-act is finitely presented; (2) every small extension of every finitely presented M-act is finitely presented. Proof. (1) ⇒ (2). Let A be a small extension of a finitely presented M-act B. We have that A/B is finite and hence finitely presented by assumption. Since B and A/B are finitely presented, it follows that A is finitely presented by Corollary 4.4. (2) ⇒ (1). Let A be a finite M-act. Choose a finitely presented M-act B disjoint from A. We have that A ∪ B is a small extension of B, so it is finitely presented by assumption. Hence, by Corollary 5.9, we have that A is finitely presented. Corollary 6.9. Let M be a finitely generated monoid. Then every small extension of every finitely presented M-act is finitely presented. We now turn to consider which monoids M give a positive answer to question (2); that is, every large subact of every finitely presented M-act is finitely presented. We first present an example which reveals that there exist finitely generated monoids which do not possess this property, and then we show that there exists a large class of monoids for which the property holds. Example 6.10. Let M be the monoid defined by the presentation ha, b \| abi a = aba (i ≥ 2)iMon . Let I = M \ {1, a}, so I is a large subact of the finitely presented M-act M. Now I = hb, a2 , abi is a disjoint union of hbi, ha2 i and habi. Clearly hbi and ha2 i are free and hence finitely presented. Letting y = ab, we have that hyi is defined by the presentation hy \| y · bi a = y · a (i ∈ N)i. We saw in Example 5.14 that hyi is not finitely presented. It hence follows from Corollary 5.9 that I is not finitely presented. Lemma 6.11. Let M be a group, let A be an M-act and let B be a large subact of A. If A is finitely presented, then B is finitely presented. 18 Proof. Since A is the disjoint union of its subacts B and A \ B, it follows from Corollary 5.9 that B is finitely presented if A is finitely presented. Definition 6.12. A monoid M is right coherent if every finitely generated subact of every finitely presented M-act is finitely presented. Examples of right coherent monoids include groups, Clifford monoids, semilattices, the bicyclic monoid, free commutative monoids and free monoids [8]. Since for any finitely generated monoid M, a large subact of a finitely generated M-act is finitely generated, we have the following result: Lemma 6.13. Let M be a finitely generated right coherent monoid, let A be an M-act and let B be a large subact of A. If A is finitely presented, then B is finitely presented. Before stating our final result of this section, we first introduce a technical definition. Let M be a monoid with a presentation hZ \| P iMon, and let A be an M-act with a presentation hX \| Ri. For a word w in Z ∗ , let w denote the element of M which w represents. We say an element x · w ∈ FX,Z ∗ represents an element a ∈ A if x · w ∈ FX,M represents a ∈ A. Theorem 6.14. Let M be a finitely presented monoid, let A be an M-act and let B be a large subact of A. If A is finitely presented, then B is finitely presented. Proof. We shall prove this result using the method based on Theorem 6.1 and outlined in Remark 6.2. Let M be defined by the presentation hZ \| P iMon, where Z and P are finite. Suppose A is defined by the finite presentation hX \| Ri. We define the finite set S = {am ∈ B : a ∈ A \ B, m ∈ Z}, and let Y = (X ∩ B) ∪ S. We have that B = hY i by Proposition 6.6. Let W denote the set of elements of FX,Z ∗ which represent elements of B. We define a map θ : W → FY,M as follows. For u ∈ W , we have u = x · w for some x ∈ X and w ∈ Z ∗ . If x ∈ B, let uθ = x · w. Suppose x ∈ A \ B. Now w = m1 . . . mk where mi ∈ Z. Let s be minimal such that x · m1 . . . ms represents an element of B, say b, and let uθ = b · ms+1 . . . mk . Let L(X, B) denote the set of elements of FX,M which represent elements of B. For each m ∈ M, choose an element wm ∈ Z ∗ which represents m. We now have a well-defined rewriting map φ : L(X, B) → FY , x · m 7→ (x · wm )θ. 19 Note that for any x ∈ X ∩ B, we have x ≡ xφ. Now, for each y ∈ S, we have y = ay my for some ay ∈ A \ B and my ∈ Z. Choose uy ∈ FX which represents ay in A, so (uy my )φ = y holds in B. We now define the following sets of relations: S1 = {uφ = vφ : (u, v) ∈ R, u ∈ L(X, B)}; S2 = {b · w = c · z : b, c ∈ S, w and z are suffixes of p and q respectively for some (p, q) ∈ P, b · w = c · z holds in B}. Since R, S and P are finite, we have that S1 and S2 are finite. We now make the following claim: Claim. Let x ∈ X and w, w ′ ∈ Z ∗ such that w = w ′ holds in M and x·w represents an element of B. Then (x · w)θ = (x · w ′)θ is a consequence of S2 . Proof. If x ∈ B, then (x · w)θ ≡ x · w ≡ x · w ′ ≡ (x · w ′ )θ. Suppose now that x ∈ A \ B. Since w = w ′ is a consequence of P , it is clearly sufficient to consider the case where w ′ is obtained from w by a single application of a relation from P , so let w = pqr and w ′ = pq ′ r where p, r ∈ Z ∗ and (q, q ′) ∈ P . There are three cases. Case 1: x · p represents an element of B. Since q = q ′ in M, we have (x · w)θ ≡ ((x · p)θ)qr ≡ ((x · p)θ)q ′ r ≡ (x · w ′ )θ. Case 2: x · pq represents an element of A \ B. Now r = m1 . . . mk where mi ∈ Z. Let s be minimal such that x · pqm1 . . . ms represents an element of B, say b. Since x · pq and x · pq ′ represent the same element of A, we have (x · w)θ ≡ b · ms+1 . . . mk ≡ (x · w ′ )θ. Case 3: x · p represents an element of A \ B and x · pq represents an element of B. Now q = m1 . . . mk and q ′ = n1 . . . nl where mi , ni ∈ Z. Let s be minimal such that x · pm1 . . . ms represents an element of B, say b, and let t be minimal such that x · pn1 . . . nt represents an element of B, say c. We have that (x · w)θ ≡ (b · ms+1 . . . mk )r = (c · nt+1 . . . nl )r ≡ (x · w ′ )θ, using an application of a relation from S2 . Returning to the proof of Theorem 6.14, we shall show that B is defined by the finite presentation hY \| S1 , S2 i. We need to show that the relations R1 , R2 and R3 of the presentation for B given in Theorem 6.1 are consequences of S1 and S2 . That is, we show that for any y ∈ S, w ∈ L(X, B) and m ∈ M, and (u, v) ∈ R, n ∈ M such that un ∈ L(X, B), the relations y = (uy my )φ, (wm)φ = (wφ)m and (un)φ = (vn)φ are consequences of S1 and S2 . 20 Let y ∈ S. We have that uy = x · m for some x ∈ X \ B and m ∈ M. Since wm my = wmmy holds in M, we have that y ≡ (x · wm my )θ = (x · wmmy )θ ≡ (uy my )φ is a consequence of S2 by the above claim. Now let u = x · n ∈ L(X, B) and m ∈ M. We have that (um)φ ≡ (x · wnm )θ and (uφ)m ≡ (x · wn wm )θ. Since wnm = wn wm holds in M, we have that (um)φ = (uφ)m is a consequence of S2 by the above claim. Finally, let (u, v) ∈ R and n ∈ M such that un ∈ L(X, B). Suppose first that u ∈ L(X, B). We have that (un)φ = (uφ)n and (vn)φ = (vφ)n are consequences of S2 , and we obtain (vφ)n from (uφ)n by an application of a relation from S1 . Therefore, (un)φ = (vn)φ is a consequence of S1 and S2 . Suppose now that u represents an element of A \ B. We have that u = x · m and v = x′ · m′ for some x, x′ ∈ X and m, m′ ∈ M. Now (un)φ = (x · wm wn )θ and (vn)φ = (x′ · wm′ wn )θ are consequences of S2 by the above claim. We have that wn = m1 . . . mk where mi ∈ Z. Let s be minimal such that x · wm m1 . . . ms represents an element of B, say b. Since x · wm and x′ · wm′ represent the same element of A, we have (x · wm wn )θ ≡ b · ms+1 . . . mk ≡ (x′ · wm′ wn )θ. Therefore, we have that (un)φ = (vn)φ is a consequence of S2 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] G. Baumslag. Topics in Combinatorial Group Theory. Birkhauser, 1993. R. Book and F. Otto. String rewriting systems. Springer-Verlag, 1993. S. Burris and H. Sankappanavar. A Course in Universal Algebra. Springer-Verlag, 1981. C. Campbell, E. Robertson, N. Ruskuc and R. Thomas. Reidermeister-Schreier type rewriting for semigroups. Semigroup Forum, 51:47-62, 1995. V. Gould. Completely right pure monoids. Proc. Royal Irish Acad., 87A:73-82, 1987. V. Gould. Coherent monoids. J. Australian Math. Soc., 53:166-182, 1992. V. Gould, M. Hartmann and N. Ruskuc. Free monoids are coherent. Proc. Edinburgh Math. Soc., 60:127-131, 2017. V. Gould and M. Hartmann. Coherency, free inverse monoids and related free algebras. Math. Proc. Cambridge Phil. Soc., to appear. M. Kilp, U. Knauer and A. Mikhalev. Monoids, Acts and Categories. Walter de Gruyter, 2000. W. Magnus, A. Karrass and D. Solitar. Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Courier Corporation, 2004. P. Normak. On Noetherian and finitely presented acts (in Russian). Tartu Ul. Toimetised, 431:37-46, 1977. N. Ruskuc. On large subsemigroups and finiteness conditions of semigroups. Proc. London Math. Soc., 76:383-405, 1998. N. Ruskuc. Semigroup Presentations. PhD Thesis, University of St Andrews, 1995. 21 School of Mathematics and Statistics, University of St Andrews, St Andrews, Scotland, UK E-mail address: {cm30,nik.ruskuc}@st-andrews.ac.uk 22	4
On the Behaviours Produced by Instruction Sequences under Execution arXiv:1106.6196v2 [cs.PL] 11 Jun 2012 J.A. Bergstra and C.A. Middelburg Informatics Institute, Faculty of Science, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands J.A.Bergstra@uva.nl,C.A.Middelburg@uva.nl Abstract. We study several aspects of the behaviours produced by instruction sequences under execution in the setting of the algebraic theory of processes known as ACP. We use ACP to describe the behaviours produced by instruction sequences under execution and to describe two protocols implementing these behaviours in the case where the processing of instructions takes place remotely. We also show that all finite-state behaviours considered in ACP can be produced by instruction sequences under execution. Keywords: instruction sequence, remote instruction processing, instruction sequence producible process 1 Introduction The concept of an instruction sequence is a very primitive concept in computing. It has always played a central role in computing because of the fact that execution of instruction sequences underlies virtually all past and current generations of computers. It happens that, given a precise definition of an appropriate notion of an instruction sequence, many issues in computer science can be clearly explained in terms of instruction sequences. A simple yet interesting example is that a program can simply be defined as a text that denotes an instruction sequence. Such a definition corresponds to an empirical perspective found among practitioners. In theoretical computer science, the meaning of programs usually plays a prominent part in the explanation of many issues concerning programs. Moreover, what is taken for the meaning of programs is mathematical by nature. On the other hand, it is customary that practitioners do not fall back on the mathematical meaning of programs in case explanation of issues concerning programs is needed. They phrase their explanations from an empirical perspective. An empirical perspective that we consider appealing is the perspective that a program is in essence an instruction sequence and an instruction sequence under execution produces a behaviour that is controlled by its execution environment in the sense that each step performed actuates the processing of an instruction by the execution environment and a reply returned at completion of the processing determines how the behaviour proceeds. This paper concerns the behaviours produced by instruction sequences under execution as such and two issues relating to the behaviours produced by instruction sequences under execution, namely the issue of implementing these behaviours in the case where the processing of instructions takes place remotely and the issue of the extent to which the behaviours considered in process algebra can be produced by instruction sequences under execution. Remote instruction processing means that a stream of instructions to be processed arises at one place and the processing of that stream of instructions is handled at another place. This phenomenon is increasingly encountered. It is found if loading the instruction sequence to be executed as a whole is impracticable. For instance, the storage capacity of the execution unit is too small or the execution unit is too far away. Remote instruction processing requires special attention because the transmission time of the messages involved in remote instruction processing makes it hard to keep the execution unit busy without intermission. In the literature on computer architecture, hardly anything can be found that contributes to a sound understanding of the phenomenon of remote instruction processing. As a first step towards such an understanding, we give rigorous descriptions of two protocols for remote instruction processing at a level of abstraction that captures the underlying essence of the protocols. One protocol is very simple, but makes no effort keep the execution unit busy without intermission. The other protocol is more complex and is directed towards keeping the execution unit busy without intermission. It is reminiscent of an instruction pre-fetching mechanism as found in pipelined processors (see e.g. [26]), but its range of application is not restricted to pipelined instruction processing. The work presented in this paper belongs to a line of research which started with an attempt to approach the semantics of programming languages from the perspective mentioned above. The first published paper on this approach is [7]. That paper is superseded by [8] with regard to the groundwork for the approach: program algebra, an algebraic theory of single-pass instruction sequences, and basic thread algebra, an algebraic theory of mathematical objects that represent in a direct way the behaviours produced by instruction sequences under execution.1 The main advantages of the approach are that it does not require a lot of mathematical background and that it is more appealing to practitioners than the main approaches to programming language semantics: the operational approach, the denotational approach and the axiomatic approach. For an overview of these approaches, see e.g. [32]. The work presented in this paper is based on the instruction sequences considered in program algebra and the representation of the behaviours produced by instruction sequences under execution considered in basic thread algebra. It is rather awkward to describe and analyse the behaviours of this kind using al1 In [8], basic thread algebra is introduced under the name basic polarized process algebra. 2 gebraic theories of processes such as ACP [3,6], CCS [27,31] and CSP [23,29]. However, the objects considered in basic thread algebra can be viewed as representations of processes as considered in process algebra. This allows for the protocols for remote instruction processing to be described using ACP or rather ACPτ , an extension of ACP which supports abstraction from internal actions. Process algebra is an area of the study of concurrency which is considered relevant to computer science, as is witnesses by the extent of the work on algebraic theories of processes such as ACP, CCS and CSP in theoretical computer science. This strongly hints that there are programmed systems whose behaviours can be taken for processes as considered in process algebra. Therefore, it is interesting to know to which extent the behaviours considered in process algebra can be produced by programs under execution, starting from the perception of a program as an instruction sequence. In this paper, we will show that, by apposite choice of instructions, all finite-state processes can be produced by instruction sequences (provided that the cluster fair abstraction rule, see e.g. Section 5.6 of [24], is valid). The instruction sequences considered in program algebra are single-pass instruction sequences, i.e. finite or infinite sequences of instructions of which each instruction is executed at most once and can be dropped after it has been executed or jumped over. Program algebra does not provide a notation for programs that is intended for actual programming: programs written in an assembly language are finite instruction sequences for which single-pass execution is usually not possible. We will also show that all finite-state processes can as well be produced by programs written in a program notation which is close to existing assembly languages. Instruction sequences under execution may make use of services provided by their execution environment such as counters, stacks and Turing tapes. The use operators added to basic thread algebra in e.g. [12] can be used to describe the behaviours produced by instruction sequences under execution that make use of services. Interesting is that instruction sequences under execution that make use of services may produce infinite-state processes. On that account, we will make precise what processes are produced by instruction sequences under execution that make use of services provided by their execution environment. As a continuation of the work on a new approach to programming language semantics mentioned above, the notion of an instruction sequence was subjected to systematic and precise analysis using the groundwork laid earlier. This led among other things to expressiveness results about the instruction sequences considered and variations of the instruction sequences considered (see e.g. [12,18,20,21,36]). Instruction sequences are under discussion for many years in diverse work on computer architecture, as witnessed by e.g. [4,22,25,30,33,34,35,39,41], but the notion of an instruction sequence has never been subjected to any precise analysis before. As another continuation of the work on a new approach to programming language semantics mentioned above, selected issues relating to well-known subjects from the theory of computation and the area of computer architecture were rigorously investigated thinking in terms of instruction sequences 3 (see e.g. [14,15,16,17,19]). The subjects from the theory of computation, namely the halting problem and non-uniform computational complexity, are usually investigated thinking in terms of a common model of computation such as Turing machines and Boolean circuits (see e.g. [1,28,38]). The subjects from the area of computer architecture, namely instruction sequence performance, instruction set architectures and remote instruction processing, are usually not investigated in a rigorous way at all. The general aim of the work in both continuations mentioned is to bring instruction sequences as a theme in computer science better into the picture. The work presented in this paper forms a part of the last mentioned continuation. This paper is organized as follows. The body of the paper consists of three parts. The first part (Sections 2–7) concerns the behaviours produced by instruction sequences under execution as such and includes surveys of program algebra, basic thread algebra and the algebraic theory of processes known as ACP. The second part (Sections 8–10) concerns the issue of implementing these behaviours in the case where the processing of instructions takes place remotely and includes rigorous descriptions of two protocols for remote instruction processing. The third part (Sections 11–14) concerns the issue of the extent to which the behaviours considered in process algebra can be produced by instruction sequences under execution and includes the result that, by apposite choice of instructions, all finite-state processes can be produced by instruction sequences. This paper consolidates material from [11,13,14]. 2 Program Algebra In this section, we review PGA (ProGram Algebra). The starting-point of program algebra is the perception of a program as a single-pass instruction sequence. The concepts underlying the primitives of program algebra are common in programming, but the particular form of the primitives is not common. The predominant concern in the design of program algebra has been to achieve simple syntax and semantics, while maintaining the expressive power of arbitrary finite control. In PGA, it is assumed that a fixed but arbitrary set A of basic instructions has been given. The intuition is that the execution of a basic instruction may modify a state and produces a reply at its completion. The possible replies are the Boolean values T and F. PGA has the following primitive instructions: – – – – – for each a ∈ A, a plain basic instruction a; for each a ∈ A, a positive test instruction +a; for each a ∈ A, a negative test instruction −a; for each l ∈ N, a forward jump instruction #l; a termination instruction !. We write I for the set of all primitive instructions of PGA. On execution of an instruction sequence, these primitive instructions have the following effects: 4 – the effect of a positive test instruction +a is that basic instruction a is executed and execution proceeds with the next primitive instruction if T is produced and otherwise the next primitive instruction is skipped and execution proceeds with the primitive instruction following the skipped one — if there is no primitive instructions to proceed with, inaction occurs; – the effect of a negative test instruction −a is the same as the effect of +a, but with the role of the value produced reversed; – the effect of a plain basic instruction a is the same as the effect of +a, but execution always proceeds as if T is produced; – the effect of a forward jump instruction #l is that execution proceeds with the l-th next instruction of the program concerned — if l equals 0 or there is no primitive instructions to proceed with, inaction occurs; – the effect of the termination instruction ! is that execution terminates. PGA has the following constants and operators: – for each u ∈ I, an instruction constant u ; – the binary concatenation operator ; ; – the unary repetition operator ω . We assume that there is a countably infinite set of variables which includes x, y, z. Terms are built as usual. We use infix notation for concatenation and postfix notation for repetition. A closed PGA term is considered to denote a non-empty, finite or eventually periodic infinite sequence of primitive instructions.2 The instruction sequence denoted by a closed term of the form t ; t′ is the instruction sequence denoted by t concatenated with the instruction sequence denoted by t′ . The instruction sequence denoted by a closed term of the form tω is the instruction sequence denoted by t concatenated infinitely many times with itself. Some simple examples of closed PGA terms are a;b;c, a ; (b ; c)ω . +a ; #2 ; #3 ; b ; ! , On execution of the instruction sequence denoted by the first term, the basic instructions a, b and c are executed in that order and after that inaction occurs. On execution of the instruction sequence denoted by the second term, the basic instruction a is executed first, if the execution of a produces the reply T, the basic instruction b is executed next and after that execution terminates, and if the execution of a produces the reply F, inaction occurs. On execution of the instruction sequence denoted by the third term, the basic instruction a is executed first, and after that the basic instructions b and c are executed in that order repeatedly forever. Closed PGA terms are considered equal if they represent the same instruction sequence. The axioms for instruction sequence equivalence are given in Table 1. In this table, n stands for an arbitrary positive natural number. The term tn , 2 An eventually periodic infinite sequence is an infinite sequence with only finitely many distinct suffixes. 5 Table 1. Axioms of PGA (x ; y) ; z = x ; (y ; z) (xn )ω = xω xω ; y = xω (x ; y)ω = x ; (y ; x)ω PGA1 PGA2 PGA3 PGA4 where t is a PGA term, is defined by induction on n as follows: t1 = t and tn+1 = t ; tn . The unfolding equation xω = x ; xω is derived as follows: xω = (x ; x)ω by PGA2 = x ; (x ; x)ω by PGA4 = x ; xω by PGA2 . Each closed PGA term is derivably equal to a term in canonical form, i.e. a ω term of the form t or t ; t′ , where t and t′ are closed PGA terms in which the repetition operator does not occur. For example: (a ; b)ω ; c ; ! = a ; (b ; a)ω , +a ; (#4 ; b ; (−c ; #5 ; !)ω )ω = +a ; #4 ; b ; (−c ; #5 ; !)ω . The initial models of PGA are considered its standard models. Henceforth, we restrict ourselves to the initial model I PGA of PGA in which: – the domain is the set of all non-empty, finite and eventually periodic infinite sequences over the set I of primitive instructions; – the operation associated with ; is concatenation; – the operation associated with ω is the operation ω defined as follows: • if F is a finite sequence over I, then F ω is the unique eventually periodic infinite sequence F ′ such that F concatenated n times with itself is a proper prefix of F ′ for each n ∈ N; • if F is an eventually periodic infinite sequence over I, then F ω is F . In the sequel, we use the term instruction sequence for the elements of the domain of I PGA , and we denote the interpretations of the constants and operators of PGA in I PGA by the constants and operators themselves. I PGA is loosely called the initial model of PGA because all initial models of PGA are isomorphic, i.e. there exist bijective homomorphism between them (see e.g. [37,40]). 3 Basic Thread Algebra In this section, we review BTA (Basic Thread Algebra). BTA is an algebraic theory of mathematical objects that represent in a direct way the behaviours produced by instruction sequences under execution. The objects concerned are called threads. In BTA, it is assumed that a fixed but arbitrary set A of basic actions, with tau ∈ / A, has been given. Besides, tau is a special basic action. We write Atau for 6 Table 2. Axiom of BTA x E tau D y = x E tau D x T1 A ∪ {tau}. A thread performs basic actions in a sequential fashion. Upon each basic action performed, a reply from an execution environment determines how it proceeds. The possible replies are the Boolean values T and F. Performing tau, which is considered performing an internal action, always leads to the reply T. Although BTA is one-sorted, we make this sort explicit. The reason for this is that we will extend BTA with an additional sort in Section 13. BTA has one sort: the sort T of threads. To build terms of sort T, it has the following constants and operators: – the inaction constant D : T; – the termination constant S : T; – for each a ∈ Atau , the binary postconditional composition operator E a D : T × T → T. We assume that there are infinitely many variables of sort T, including x, y, z. Terms of sort T are built as usual. We use infix notation for the postconditional composition operators. We introduce basic action prefixing as an abbreviation: a ◦ t, where a ∈ Atau and t is a term of sort T, abbreviates t E a D t. The thread denoted by a closed term of the form t E a D t′ will first perform a, and then proceed as the thread denoted by t if the reply from the execution environment is T and proceed as the thread denoted by t′ if the reply from the execution environment is F. The threads denoted by D and S will become inactive and terminate, respectively. Some simple examples of closed BTA terms are a ◦ (S E b D D) , (b ◦ S) E a D D . The first term denotes the thread that first performs basic action a, next performs basic action b, if the reply from the execution environment on performing b is T, after that terminates, and if the reply from the execution environment on performing b is F, after that becomes inactive. The second term denotes the thread that first performs basic action a, if the reply from the execution environment on performing a is T, next performs the basic action b and after that terminates, and if the reply from the execution environment on performing a is F, next becomes inactive. BTA has only one axiom. This axiom is given in Table 2. Using the abbreviation introduced above, axiom T1 can be written as follows: x E tau D y = tau ◦ x. Notice that each closed BTA term denotes a thread that will become inactive or terminate after it has performed finitely many actions. Infinite threads can be described by guarded recursion. A guarded recursive specification over BTA is a set of recursion equations E = {X = tX \| X ∈ V }, where V is a set of variables of sort T and each tX is a BTA term of the form D, S or t E a D t′ with t and t′ that contain only variables 7 Table 3. RDP, RSP and AIP hX\|Ei = htX \|Ei if X = tX ∈ E RDP E ⇒ X = hX\|Ei if X ∈ V(E) RSP V n≥0 πn (x) = πn (y) ⇒ x = y π0 (x) = D πn+1 (S) = S πn+1 (D) = D πn+1 (x E a D y) = πn (x) E a D πn (y) AIP P0 P1 P2 P3 from V . We write V(E) for the set of all variables that occur in E. We are only interested in models of BTA in which guarded recursive specifications have unique solutions, such as the projective limit model of BTA presented in [5]. A simple example of a guarded recursive specification is the one consisting of following two equations: x = x EaD y , y = y EbD S . The x-component of the solution of this guarded recursive specification is the thread that first performs basic action a repeatedly until the reply from the execution environment on performing a is F, next performs basic action b repeatedly until the reply from the execution environment on performing b is F, and after that terminates. For each guarded recursive specification E and each X ∈ V(E), we introduce a constant hX\|Ei of sort T standing for the X-component of the unique solution of E. We write htX \|Ei for tX with, for all Y ∈ V(E), all occurrences of Y in tX replaced by hY \|Ei. The axioms for the constants for the components of the solutions of guarded recursive specifications are RDP (Recursive Definition Principle) and RSP (Recursive Specification Principle), which are given in Table 3. RDP and RSP are actually axiom schemas in which X stands for an arbitrary variable, tX stands for an arbitrary BTA term, and E stands for an arbitrary guarded recursive specification over BTA. Side conditions are added to restrict what X, tX and E stand for. The equations hX\|Ei = htX \|Ei for a fixed E express that the constants hX\|Ei make up a solution of E. The conditional equations E ⇒ X = hX\|Ei express that this solution is the only one. RDP and RSP are means to prove closed terms that denote the same infinite thread equal. We introduce AIP (Approximation Induction Principle) as an additional means to prove closed terms that denote the same infinite thread equal. AIP is based on the view that two threads are identical if their approximations up to any finite depth are identical. The approximation up to depth n of a thread is obtained by cutting it off after it has performed n actions. AIP is also given in Table 3. Here, approximation up to depth n is phrased in terms of the unary projection operator πn : T → T. The axioms for the projection operators are axioms P0–P3 in Table 3. P1–P3 are actually axiom schemas in which a stands for arbitrary basic action and n stands for an arbitrary natural number. We write BTA+REC for BTA extended with the constants for the components of the solutions of guarded recursive specifications, the projection operators and the axioms RDP, RSP, AIP and P0–P3. 8 The minimal models of BTA+REC are considered its standard models.3 Recall that a model of an algebraic theory is minimal iff all elements of the domains associated with the sorts of the theory can be denoted by closed terms. Henceforth, we restrict ourselves to the minimal models of BTA+REC. We assume that a minimal model MBTA+REC of BTA+REC has been given. In the sequel, we use the term thread for the elements of the domain of MBTA+REC , and we denote the interpretations of constants and operators in MBTA+REC by the constants and operators themselves. Let T be a thread. Then the set of states or residual threads of T , written Res(T ), is inductively defined as follows: – T ∈ Res(T ); – if T ′ E a D T ′′ ∈ Res(T ), then T ′ ∈ Res(T ) and T ′′ ∈ Res(T ). Let T be a thread and let A′ ⊆ Atau . Then T is regular over A′ if the following conditions are satisfied: – Res(T ) is finite; – for all T ′ , T ′′ ∈ Res(T ) and a ∈ Atau , T ′ E a D T ′′ ∈ Res(T ) implies a ∈ A′ . We say that T is regular if T is regular over Atau . For example, the x-component of the solution of the guarded recursive specification consisting of the following two equations: x=a◦y , y = (c ◦ y) E b D (x E d D S) has five states and is regular over any A′ ⊆ Atau for which {a, b, c, d} ⊆ A′ . We will make use of the fact that being a regular thread coincides with being a component of the solution of a finite guarded recursive specification in which the right-hand sides of the recursion equations are of a restricted form. A linear recursive specification over BTA is a guarded recursive specification E = {X = tX \| X ∈ V } over BTA, where each tX is a term of the form D, S or Y E a D Z with Y, Z ∈ V . Proposition 1. Let T be a thread and let A′ ⊆ Atau . Then T is regular over A′ iff there exists a finite linear recursive specification E over BTA in which only basic actions from A′ occur such that T is a component of the solution of E. Proof. The implication from left to right is proved as follows. Because T is regular, Res(T ) is finite. Hence, there are finitely many threads T1 , . . . , Tn , with T = T1 , such that Res(T ) = {T1 , . . . , Tn }. Now T is the x1 -component of the solution of the linear recursive specification consisting of the following equations:  if Ti = S S xi = D for all i ∈ [1, n] . if Ti = D  xj E a D xk if Ti = Tj E a D Tk 3 A minimal model of an algebraic theory is a model of which no proper subalgebra is a model as well. 9 Table 4. Defining equations for thread extraction operation \|a\| = a ◦ D \|a ; F \| = a ◦ \|F \| \|+a\| = a ◦ D \|+a ; F \| = \|F \| E a D \|#2 ; F \| \|−a\| = a ◦ D \|−a ; F \| = \|#2 ; F \| E a D \|F \| \|#l\| = D \|#0 ; F \| = D \|#1 ; F \| = \|F \| \|#l + 2 ; u\| = D \|#l + 2 ; u ; F \| = \|#l + 1 ; F \| \|!\| = S \|! ; F \| = S Because T is regular over A′ , only basic actions from A′ occur in the linear recursive specification constructed in this way. The implication from right to left is proved as follows. Thread T is a component of the unique solution of a finite linear specification in which only basic actions from A′ occur. This means that there are finitely many threads T1 , . . . , Tn , with T = T1 , such that for every i ∈ [1, n], Ti = S, Ti = D or Ti = Tj E a D Tk for some j, k ∈ [1, n] and a ∈ A′ . Consequently, T ′ ∈ Res(T ) iff T ′ = Ti for some i ∈ [1, n] and moreover T ′ E a D T ′′ ∈ Res(T ) only if a ∈ A′ . Hence, Res(T ) is finite and T is regular over A′ . ⊓ ⊔ Remark 1. A structural operational semantics of BTA+REC and a bisimulation equivalence based on it can be found in e.g. [10]. The quotient algebra of the algebra of closed terms of BTA+REC by this bisimulation equivalence is one of the minimal models of BTA+REC. 4 Thread Extraction In this short section, we use BTA+REC to make mathematically precise which threads are produced by instruction sequences under execution. For that purpose, A is taken such that A ⊇ A is satisfied. The thread extraction operation \| \| assigns a thread to each instruction sequence. The thread extraction operation is defined by the equations given in Table 4 (for a ∈ A, l ∈ N, and u ∈ I) and the rule that \|#l ; F \| = D if #l is the beginning of an infinite jump chain. This rule is formalized in e.g. [12]. Let F be an instruction sequence and T be a thread. Then we say that F produces T if \|F \| = T . For example, a;b;c +a ; #2 ; #3 ; b ; ! +a ; −b ; c ; ! +a ; #2 ; (b ; #2 ; c ; #2)ω produces produces produces produces a◦b◦c◦D, (b ◦ S) E a D D , (S E b D (c ◦ S)) E a D (c ◦ S) , D E a D (b ◦ D) . In the case of instruction sequences that are not finite, the produced threads can be described as components of the solution of a guarded recursive specification. For example, the infinite instruction sequence (a ; +b)ω 10 produces the x-component of the solution of the guarded recursive specification consisting of following two equations: x=a◦y , y = x EbD y and the infinite instruction sequence a ; (+b ; #2 ; #3 ; c ; #4 ; −d ; ! ; a)ω produces the x-component of the solution of the guarded recursive specification consisting of following two equations: x=a◦y , 5 y = (c ◦ y) E b D (x E d D S) . Algebra of Communicating Processes In this section, we review ACPτ (Algebra of Communicating Processes with abstraction). This algebraic theory of processes will among other things be used to make precise what processes are produced by the threads denoted by closed terms of BTA+REC. For a comprehensive overview of ACPτ , the reader is referred to [3,24]. In ACPτ , it is assumed that a fixed but arbitrary set A of atomic actions, with τ, δ ∈ / A, and a fixed but arbitrary commutative and associative function \| : A ∪ {τ } × A ∪ {τ } → A ∪ {δ}, with τ \| e = δ for all e ∈ A ∪ {τ }, have been given. The function \| is regarded to give the result of synchronously performing any two atomic actions for which this is possible, and to give δ otherwise. In ACPτ , τ is a special atomic action, called the silent step. The act of performing the silent step is considered unobservable. Because it would otherwise be observable, the silent step is considered an atomic action that cannot be performed synchronously with other atomic actions. We write Aτ for A ∪ {τ }. ACPτ has the following constants and operators: – – – – – – – – – – for each e ∈ A, the atomic action constant e ; the silent step constant τ ; the inaction constant δ ; the binary alternative composition operator + ; the binary sequential composition operator · ; the binary parallel composition operator k ; the binary left merge operator ⌊⌊ ; the binary communication merge operator \| ; for each H ⊆ A, the unary encapsulation operator ∂H ; for each I ⊆ A, the unary abstraction operator τI . We assume that there are infinitely many variables, including x, y, z. Terms are built as usual. We use infix notation for the binary operators. The precedence conventions used with respect to the operators of ACPτ are as follows: + binds weaker than all others, · binds stronger than all others, and the remaining operators bind equally strong. Let t and t′ be closed ACPτ terms, e ∈ A, and H, I ⊆ A. Intuitively, the constants and operators to build ACPτ terms can be explained as follows: 11 – the process denoted by e first performs atomic action e and next terminates successfully; – the process denoted by τ performs an unobservable atomic action and next terminates successfully; – the process denoted by δ can neither perform an atomic action nor terminate successfully; – the process denoted by t + t′ behaves either as the process denoted by t or as the process denoted by t′ , but not both; – the process denoted by t · t′ first behaves as the process denoted by t and on successful termination of that process it next behaves as the process denoted by t′ ; – the process denoted by t k t′ behaves as the process that proceeds with the processes denoted by t and t′ in parallel; – the process denoted by t ⌊⌊ t′ behaves the same as the process denoted by t k t′ , except that it starts with performing an atomic action of the process denoted by t; – the process denoted by t \| t′ behaves the same as the process denoted by t k t′ , except that it starts with performing an atomic action of the process denoted by t and an atomic action of the process denoted by t′ synchronously; – the process denoted by ∂H (t) behaves the same as the process denoted by t, except that atomic actions from H are blocked; – the process denoted by τI (t) behaves the same as the process denoted by t, except that atomic actions from I are turned into unobservable atomic actions. The operators ⌊⌊ and \| are of an auxiliary nature. They are needed to axiomatize ACPτ . The axioms of ACPτ are given in Table 5. CM2–CM3, CM5–CM7, C1–C4, D1–D4 and TI1–TI4 are actually axiom schemas in which a, b and c stand for arbitrary constants of ACPτ , and H and I stand for arbitrary subsets of A. ACPτ is extended with guarded recursion like BTA. A recursive specification over ACPτ is a set of recursion equations E = {X = tX \| X ∈ V }, where V is a set of variables and each tX is an ACPτ term containing only variables from V . We write V(E) for the set of all variables that occur in E. Let t be an ACPτ term without occurrences of abstraction operators containing a variable X. Then an occurrence of X in t is guarded if t has a subterm of the form e · t′ where e ∈ A and t′ is a term containing this occurrence of X. Let E be a recursive specification over ACPτ . Then E is a guarded recursive specification if, in each equation X = tX ∈ E: (i) abstraction operators do not occur in tX and (ii) all occurrences of variables in tX are guarded or tX can be rewritten to such a term using the axioms of ACPτ in either direction and/or the equations in E except the equation X = tX from left to right. We are only interested models of ACPτ in which guarded recursive specifications have unique solutions, such as the models of ACPτ presented in [3]. For each guarded recursive specification E and each X ∈ V(E), we introduce a constant hX\|Ei standing for the X-component of the unique solution of E. We 12 Table 5. Axioms of ACPτ x·τ =x B1 x · (τ · (y + z) + y) = x · (y + z) B2 x+y = y+x (x + y) + z = x + (y + z) x+x = x (x + y) · z = x · z + y · z (x · y) · z = x · (y · z) x+δ = x δ·x=δ A1 A2 A3 A4 A5 A6 A7 x k y = x ⌊⌊ y + y ⌊⌊ x + x \| y a ⌊⌊ x = a · x a · x ⌊⌊ y = a · (x k y) (x + y) ⌊⌊ z = x ⌊⌊ z + y ⌊⌊ z a · x \| b = (a \| b) · x a \| b · x = (a \| b) · x a · x \| b · y = (a \| b) · (x k y) (x + y) \| z = x \| z + y \| z x \| (y + z) = x \| y + x \| z CM1 CM2 CM3 CM4 CM5 CM6 CM7 CM8 CM9 ∂H (a) = a if a ∈ /H ∂H (a) = δ if a ∈ H ∂H (x + y) = ∂H (x) + ∂H (y) ∂H (x · y) = ∂H (x) · ∂H (y) D1 D2 D3 D4 τI (a) = a if a ∈ /I τI (a) = τ if a ∈ I τI (x + y) = τI (x) + τI (y) τI (x · y) = τI (x) · τI (y) TI1 TI2 TI3 TI4 a\|b=b\|a (a \| b) \| c = a \| (b \| c) δ\|a=δ τ \|a=δ C1 C2 C3 C4 Table 6. RDP, RSP and AIP hX\|Ei = htX \|Ei if X = tX ∈ E RDP E ⇒ X = hX\|Ei if X ∈ V(E) RSP V n≥0 πn (x) = πn (y) ⇒ x = y AIP π0 (a) = δ πn+1 (a) = a π0 (a · x) = δ πn+1 (a · x) = a · πn (x) πn (x + y) = πn (x) + πn (y) πn (τ ) = τ πn (τ · x) = τ · πn (x) PR1 PR2 PR3 PR4 PR5 PR6 PR7 write htX \|Ei for tX with, for all Y ∈ V(E), all occurrences of Y in tX replaced by hY \|Ei. The axioms for the constants for the components of the solutions of guarded recursive specifications are RDP and RSP, which are given in Table 6. RDP and RSP are actually axiom schemas in which X stands for an arbitrary variable, tX stands for an arbitrary ACPτ term, and E stands for an arbitrary guarded recursive specification over ACPτ . Side conditions are added to restrict what X, tX and E stand for. Closed terms of ACPτ extended with constants for the components of the solutions of guarded recursive specifications that denote the same process cannot always be proved equal by means of the axioms of ACPτ together with RDP and RSP. We introduce AIP to remedy this. AIP is based on the view that two processes are identical if their approximations up to any finite depth are identical. The approximation up to depth n of a process behaves the same as that process, except that it cannot perform any further atomic action after n atomic actions have been performed. AIP is given in Table 6. Here, approximation up to depth 13 n is phrased in terms of a unary projection operator πn . The axioms for the projection operators are axioms PR1–PR7 in Table 6. PR1–PR7 are actually axiom schemas in which a stands for arbitrary constants of ACPτ different from τ and n stands for an arbitrary natural number. We write ACPτ +REC for ACPτ extended with the constants for the components of the solutions of guarded recursive specifications, the projection operators, and the axioms RDP, RSP, AIP and PR1–PR7. The minimal models of ACPτ +REC are considered its standard models. Henceforth, we restrict ourselves to the minimal models of ACPτ +REC. We assume that a fixed but arbitrary minimal model MACPτ +REC of ACPτ +REC has been given. From Section 12, we will sometimes assume that CFAR (Cluster Fair Abstraction Rule) is valid in MACPτ +REC . CFAR says that a cluster of silent steps that has exits can be eliminated if all exits are reachable from everywhere in the cluster. A precise formulation of CFAR can be found in [24]. We use the term process for the elements from the domain of MACPτ +REC , and we denote the interpretations of constants and operators in MACPτ +REC by the constants and operators themselves. Let P be a process. Then the set of states or subprocesses of P , written Sub(P ), is inductively defined as follows: – P ∈ Sub(P ); – if e · P ′ ∈ Sub(P ), then P ′ ∈ Sub(P ); – if e · P ′ + P ′′ ∈ Sub(P ), then P ′ ∈ Sub(P ). Let P be a process and let A′ ⊆ Aτ . Then P is regular over A′ if the following conditions are satisfied: – Sub(P ) is finite; – for all P ′ ∈ Sub(P ) and e ∈ Aτ , e · P ′ ∈ Sub(P ) implies e ∈ A′ ; – for all P ′ , P ′′ ∈ Sub(P ) and e ∈ Aτ , e · P ′ + P ′′ ∈ Sub(P ) implies e ∈ A′ . We say that P is regular if P is regular over Aτ . We will make use of the fact that being a regular process over A coincides with being a component of the solution of a finite guarded recursive specification in which the right-hand sides of the recursion equations are linear terms. Linearity of terms is inductively defined as follows: – – – – δ is linear; if e ∈ Aτ , then e is linear; if e ∈ Aτ and X is a variable, then e · X is linear; if t and t′ are linear, then t + t′ is linear. A linear recursive specification over ACPτ is a guarded recursive specification E = {X = tX \| X ∈ V } over ACPτ , where each tX is linear. Proposition 2. Let P be a process and let A′ ⊆ A. Then P is regular over A′ iff there exists a finite linear recursive specification E over ACPτ in which only atomic actions from A′ occur such that P is a component of the solution of E. 14 Proof. The proof follows the same line as the proof of Proposition 1. ⊓ ⊔ Remark 2. Proposition 2 is concerned with processes that are regular over A. We can also prove that being a regular process over Aτ coincides with being a component of the solution of a finite linear recursive specification over ACPτ if we assume that the cluster fair abstraction rule [24] holds in the model MACPτ +REC . However, we do not need this more general result. P , in } and ti1 , . . . , tin are ACPτ We will write i∈S ti , where S = {i1 , . . .P terms, for ti1 + . . . + tin . The convention is that i∈S ti stands for δ if S = ∅. We will often write X for hX\|Ei if E is clear from the context. It should be borne in mind that, in such cases, we use X as a constant. 6 Program-Service Interaction Instructions Recall that, in PGA, it is assumed that a fixed but arbitrary set A of basic instructions has been given. In the sequel, we will make use a version of PGA in which the following additional assumptions relating to A are made: – a fixed but arbitrary finite set F of foci has been given; – a fixed but arbitrary finite set M of methods has been given; – A = {f.m \| f ∈ F , m ∈ M}. Each focus plays the role of a name of some service provided by an execution environment that can be requested to process a command. Each method plays the role of a command proper. Executing a basic instruction of the form f.m is taken as making a request to the service named f to process command m. A basic instruction of the form f.m is called a program-service interaction instruction. Recall that, in BTA, it is assumed that a fixed but arbitrary set A of basic actions has been given. In the sequel, we will make use of a version of BTA in which A = A. A basic action of the form f.m is called a thread-service interaction action. The intuition concerning program-service interaction instructions given above will be made fully precise in Section 7, using ACP. 7 Process Extraction In this section, we use ACPτ +REC to make mathematically precise which processes are produced by threads. For that purpose, A and \| are taken such that the following conditions are satisfied:4 A ⊇ {sf (d) \| f ∈ F , d ∈ M ∪ B} ∪ {rf (d) \| f ∈ F , d ∈ M ∪ B} ∪ {stop, i} 4 As usual, we will write B for the set {T, F}. 15 Table 7. Defining equations for process extraction operation \|S\|c = stop \|D\|c = δ \|T E tau D T ′ \|c = i · i · \|T \|c \|T E f.m D T ′ \|c = sf (m) · (rf (T) · \|T \|c + rf (F) · \|T ′ \|c ) and for all f ∈ F , d ∈ M ∪ B, and e ∈ A: sf (d) \| rf (d) = i , sf (d) \| e = δ if e 6= rf (d) , e \| rf (d) = δ if e 6= sf (d) , stop \| e = δ i\|e= δ . if e 6= stop , Actions of the forms sf (d) and rf (d) are send and receive actions, respectively, stop is an explicit termination action, and i is a concrete internal action. The process extraction operation \| \| assigns a process to each thread. The process extraction operation \| \| is defined by \|T \| = τ{stop} (\|T \|c ), where \| \|c is defined by the equations given in Table 7 (for f ∈ F and m ∈ M). Let P be a process, T be a thread, and F be an instruction sequence. Then we say that T produces P if τ · τI (\|T \|) = τ · P for some I ⊆ A, and we say that F produces P if \|F \| produces P . Notice that two atomic actions are involved in performing a basic action of the form f.m: one for sending a request to process command m to the service named f and another for receiving a reply from that service upon completion of the processing. Notice also that, for each thread T , \|T \|c is a process that in the event of termination performs a special termination action just before termination. Abstraction from this termination action yields the process denoted by \|T \|. The process extraction operation preserves the axioms of BTA+REC. Before we make this fully precise, we have a closer look at the axioms of BTA+REC. A proper axiom is an equation or a conditional equation. In Table 3, we do not find proper axioms. Instead of proper axioms, we find axiom schemas without side conditions and axiom schemas with side conditions. The axioms of BTA+REC are obtained by replacing each axiom schema by all its instances. Henceforth, we write α∗ , where α is a valuation of variables in MBTA+REC , for the unique homomorphic extension of α to terms of BTA+REC. Moreover, we identify t1 = t2 and ∅ ⇒ t1 = t2 . Proposition 3. Let E ⇒ t1 = t2 be an axiom of BTA+REC, and let α be a valuation of variables in MBTA+REC . Then \|α∗ (t1 )\| = \|α∗ (t2 )\| if \|α∗ (t′1 )\| = \|α∗ (t′2 )\| for all t′1 = t′2 ∈ E. Proof. The proof is trivial for the axiom of BTA and the axioms RDP and RSP. Using the equation \|πn (T )\|c = π2n (\|T \|c ), the proof is also trivial for the axioms AIP and P0–P3. This equation is easily proved by induction on n and case distinction on the structure of T in both the basis step and the inductive step. ⊓ ⊔ 16 Remark 3. Proposition 3 would go through if no abstraction of the above-mentioned special termination action was made. Notice further that ACPτ without the silent step constant and the abstraction operator, better known as ACP, would suffice if no abstraction of the special termination action was made. 8 A Simple Protocol for Remote Instruction Processing In this section and the next section, we consider two protocols for remote instruction processing. The simple protocol described in this section is presumably the most straightforward protocol for remote instruction processing that can be achieved. Therefore, we consider it a suitable starting-point for the design of more advanced protocols for remote instruction processing – such as the one described in the next section. Before this simple protocol is described, an extension of ACP is introduced to simplify the description of the protocols. The following extension of ACP from [2] will be used: the non-branching conditional operator :→ over B. The expression b :→ p, is to be read as if b then p else δ. The additional axioms for the non-branching conditional operator are T :→ x = x and F :→ x = δ . In the sequel, we will use expressions whose evaluation yields Boolean values instead of the constants T and F. Because the evaluation of the expressions concerned are not dependent on the processes denoted by the terms in which they occur, we will identify each such expression with the constant for the Boolean value that its evaluation yields. Further justification of this can be found in [9, Section 9]. The protocols concern systems whose main components are an instruction stream generator and an instruction stream execution unit. The instruction stream generator generates different instruction streams for different threads. This is accomplished by starting it in different states. The general idea of the protocols is that: – the instruction stream generator generating an instruction stream for a thread T E a D T ′ sends a to the instruction stream execution unit; – on receipt of a, the instruction stream execution unit gets the execution of a done and sends the reply produced to the instruction stream generator; – on receipt of the reply, the instruction stream generator proceeds with generating an instruction stream for T if the reply is T and for T ′ otherwise. In the case where the thread is S or D, the instruction stream generator sends a special instruction (stop or dead) and the instruction stream execution unit does not send back a reply. In this section, we consider a very simple protocol for remote instruction processing that makes no effort to keep the execution unit busy without intermission. 17 In the protocols, the generation of an instruction stream start from the thread produced by an instruction sequence under execution instead of the instruction sequence itself. It follows immediately from the definition of the thread extraction operation that the threads produced by instruction sequences under execution are regular threads. Therefore, we restrict ourselves to regular threads. We write I for the set A∪{stop, dead}. Elements from I will loosely be called instructions. The restriction of the domain of MBTA+REC to the regular threads will be denoted by RT . The functions act , thrt , and thrf defined below give, for each thread T different from S and D, the basic action that T will perform first, the thread with which it will proceed if the reply from the execution environment is T, and the thread with which it will proceed if the reply from the execution environment is F, respectively. The functions act :RT → I, thrt :RT → RT , and thrf :RT → RT are defined as follows: thrf (S) = D , act (S) = stop , thrt (S) = D , thrf (D) = D , thrt (D) = D , act (D) = dead , act (T E a D T ′ ) = a , thrt (T E a D T ′ ) = T , thrf (T E a D T ′ ) = T ′ . The function nxt 0 defined below is used by the instruction stream generator to distinguish when it starts with handling the instruction to be executed next between the different instructions that it may be. The function nxt 0 :I ×RT → B is defined as follows: T if act (T ) = a nxt 0 (a, T ) = F if act (T ) 6= a . For the purpose of describing the simple protocol outlined above in ACPτ , A and \| are taken such that, in addition to the conditions mentioned at the beginning of Section 7, the following conditions are satisfied: A ⊇ {si (d) \| i ∈ {1, 2}, d ∈ I} ∪ {ri (d) \| i ∈ {1, 2}, d ∈ I} ∪ {si (r) \| i ∈ {3, 4}, r ∈ B} ∪ {ri (r) \| i ∈ {3, 4}, r ∈ B} ∪ {j} and for all i ∈ {1, 2}, j ∈ {3, 4}, d ∈ I, r ∈ B, and e ∈ A: si (d) \| ri (d) = j , si (d) \| e = δ if e 6= ri (d) , e \| ri (d) = δ if e 6= si (d) , sj (r) \| rj (r) = j , sj (r) \| e = δ if e 6= rj (r) , e \| rj (r) = δ if e 6= sj (r) , j\|e =δ . Notice that the set B is the set of replies. Let T ∈ RT . Then the process representing the simple protocol for remote instruction processing with regard to thread T is described by 0 ∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 ) , 18 0 where the process ISGT is recursively specified by the following equation: X 0 ISGT = nxt 0 (f.m, T ) :→ f.m∈A 0 0 s1 (f.m) · (r4 (T) · ISGthrt (T ) + r4 (F) · ISGthrf (T ) ) + nxt 0 (stop, T ) :→ s1 (stop) + nxt 0 (dead, T ) :→ s1 (dead) , the process IMTC 0 is recursively specified by the following equation: X IMTC 0 = r1 (d) · s2 (d) · IMTC 0 , d∈I the process RTC 0 is recursively specified by the following equation: X RTC 0 = r3 (r) · s4 (r) · RTC 0 , r∈B the process ISEU 0 is recursively specified by the following equation: X ISEU 0 = r2 (f.m) · sf (m) · (rf (T) · s3 (T) + rf (F) · s3 (F)) · ISEU 0 f.m∈A + r2 (stop) + r2 (dead) · δ and H = {si (d) \| i ∈ {1, 2}, d ∈ I} ∪ {ri (d) \| i ∈ {1, 2}, d ∈ I} ∪ {si (r) \| i ∈ {3, 4}, r ∈ B} ∪ {ri (r) \| i ∈ {3, 4}, r ∈ B} . 0 ISGT is the instruction stream generator for thread T , IMTC 0 is the transmission channel for messages containing instructions, RTC 0 is the transmission channel for replies, and ISEU 0 is the instruction stream execution unit. If we abstract from all communications via the transmission channels, then 0 the process denoted by ∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 ) and the process \|T \| are equal modulo an initial silent step. 0 Theorem 1. For each T ∈ RT , τ · τ{j} (∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 )) denotes the process τ · \|T \|. Proof. Let T ∈ RT . Moreover, let E be a finite linear recursive specification over ACPτ with X ∈ V(E) such that \|T \| is the X-component of the solution of E in MACPτ +REC . By Proposition 2 and the definition of the process extraction operation, it is sufficient to prove that 0 τ · τ{j} (∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 )) = τ · hX\|Ei . By AIP, it is sufficient to prove that for all n ≥ 0: 0 πn (τ · τ{j} (∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 ))) = πn (τ · hX\|Ei) . This is easily proved by induction on n and in the inductive step by case distinction on the structure of T , using the axioms of ACPτ and RDP and in addition the fact that \|T ′ \| ∈ Sub(\|T \|) for all T ′ ∈ Res(T ) and the fact that there exists an bijection between Sub(\|T \|) and V(E). ⊓ ⊔ 19 9 A More Complex Protocol In this section, we consider a more complex protocol for remote instruction processing that makes an effort to keep the execution unit busy without intermission. The specifics of the more complex protocol considered here are that: – the instruction stream generator may run ahead of the instruction stream execution unit by not waiting for the receipt of the replies resulting from the execution of instructions that it has sent earlier; – to ensure that the instruction stream execution unit can handle the runahead, each instruction sent by the instruction stream generator is accompanied with the sequence of replies after which the instruction must be executed; – to correct for replies that have not yet reached the instruction stream generator, each instruction sent is also accompanied with the number of replies received since the last sending of an instruction. This protocol is reminiscent of an instruction pre-fetching mechanism as found in pipelined processors (see e.g. [26]), but its range of application is not restricted to pipelined instruction processing. We write B≤n , where n ∈ N, for the set {u ∈ B∗ \| len(u) ≤ n}.5 It is assumed that a natural number ℓ has been given. The number ℓ is taken for the maximal number of steps that the instruction stream generator may run ahead of the instruction stream execution unit. Whether the execution unit can be kept busy without intermission with the given ℓ depends on the actual execution times of instructions and the actual transmission times over the transmission channels involved. If the execution unit can be kept busy without intermission with the given ℓ, then it is useless to increase ℓ. The set IM of instruction messages is defined as follows: IM = [0, ℓ] × B≤ℓ × I . In an instruction message (n, u, a) ∈ IM: – n is the number of replies that are acknowledged by the message; – u is the sequence of replies after which the instruction that is part of the message must be executed; – a is the instruction that is part of the message. The instruction stream generator sends instruction messages via an instruction message transmission channel to the instruction stream execution unit. We refer to a succession of transmitted instruction messages as an instruction stream. An instruction stream is dynamic by nature, in contradistinction with an instruction sequence. 5 As usual, we write D∗ for the set of all finite sequences with elements from set D and len(σ) for the length of finite sequence σ. Moreover, we write ǫ for the empty sequence, d for the sequence having d as sole element, σσ ′ for the concatenation of finite sequences σ and σ ′ , and tl(σ) for the tail of finite sequence σ. 20 The set SISG of instruction stream generator states is defined as follows: SISG = [0, ℓ] × P(B≤ℓ+1 × RT ) . In an instruction stream generator state (n, R) ∈ SISG : – n is the number of replies that has been received by the instruction stream generator since the last acknowledgement of received replies; – in each (u, T ) ∈ R, u is the sequence of replies after which the thread T must be performed. The functions updpm and updcr defined below are used to model the updates of the instruction stream generator state on producing a message and consuming a reply, respectively. The function updpm : (B≤ℓ × RT ) × SISG → SISG is defined as follows: updpm((u, T ), (n, R)) = (0, (R \ {(u, T )}) ∪ {(uT, thrt(T )), (uF, thrf (T ))}) if act (T ) ∈ A (0, (R \ {(u, T )})) if act (T ) ∈ /A. The function updcr : B × SISG → SISG is defined as follows: updcr (r, (n, R)) = (n + 1, {(u, T ) \| (ru, T ) ∈ R}) . The function sel defined below is used to model the selection of the sequence of replies and the instruction that will be part of the next message produced by the instruction stream generator. The function sel : P(B≤ℓ × RT ) → P(B≤ℓ × RT ) is defined as follows: sel (R) = {(u, T ) ∈ R \| ∀(v, T ′ ) ∈ R • len(u) ≤ len(v)} . Notice that (u, T ) ∈ sel (R) and (v, T ′ ) ∈ R only if len(u) ≤ len(v). By that breadth-first run-ahead is enforced. The performance of the protocol would change considerably if breadth-first run-ahead was not enforced. The set SISEU of instruction stream execution unit states is defined as follows: SISEU = [0, ℓ] × P(B≤ℓ × I) . In an instruction stream execution unit state (n, S) ∈ SISEU : – n is the number of replies for which the instruction stream execution unit still has to receive an acknowledgement; – in each (u, a) ∈ S, u is the sequence of replies after which the instruction a must be executed. The functions updcm and updpr defined below are used to model the updates of the instruction stream execution unit state on consuming a message and producing a reply, respectively. The function updcm : IM × SISEU → SISEU is defined as follows: . . updcm((k, u, a), (n, S)) = (n − k, S ∪ {(tl n−k (u), a)}) .6 21 The function updpr : B × SISEU → SISEU is defined as follows: updpr (r, (n, S)) = (n + 1, {(u, a) \| (ru, a) ∈ S}) . The function nxt defined below is used by the instruction stream execution unit to distinguish when it starts with handling the instruction to be executed next between the different instructions that it may be. The function nxt : I × P(B≤ℓ × I) → B is defined as follows: T if (ǫ, a) ∈ S nxt(a, S) = F if (ǫ, a) ∈ /S. The instruction stream execution unit sends replies via a reply transmission channel to the instruction stream generator. We refer to a succession of transmitted replies as a reply stream. For the purpose of describing the transmission protocol in ACPτ , A and \| are taken such that, in addition to the conditions mentioned at the beginning of Section 7, the following conditions are satisfied: A ⊇ {si (d) \| i ∈ {1, 2}, d ∈ IM} ∪ {ri (d) \| i ∈ {1, 2}, d ∈ IM} ∪ {si (r) \| i ∈ {3, 4}, r ∈ B} ∪ {ri (r) \| i ∈ {3, 4}, r ∈ B} ∪ {j} and for all i ∈ {1, 2}, j ∈ {3, 4}, d ∈ IM, r ∈ B, and e ∈ A: si (d) \| ri (d) = j , si (d) \| e = δ if e 6= ri (d) , e \| ri (d) = δ if e 6= si (d) , sj (r) \| rj (r) = j , sj (r) \| e = δ if e 6= rj (r) , e \| rj (r) = δ if e 6= sj (r) , j\|e =δ . Let T ∈ RT . Then the process representing the more complex protocol for remote instruction processing with regard to thread T is described by ∂H (ISGT k IMTC k RTC k ISEU ) , where the process ISGT is recursively specified by the following equations: ISGT ′ ISG(n,R) ′ = ISG(0,{(ǫ,T )}) , X ′ s1 ((n, u, act (T ))) · ISGupdpm((u,T = ),(n,R)) (u,T )∈sel(R) + X ′ r4 (r) · ISGupdcr (r,(n,R)) r∈B (for every (n, R) ∈ SISG with R 6= ∅) , ′ ISG(n,∅) =j (for every (n, ∅) ∈ SISG ) , 6 . . . As usual, we write i − j for the monus of i and j, i.e. i −j = i−j if i ≥ j and i −j =0 otherwise. As usual, tl n (u) is defined by induction on n as follows: tl 0 (u) = u and tl n+1 (u) = tl(tl n (u)). 22 the process IMTC is recursively specified by the following equation: X IMTC = r1 (d) · s2 (d) · IMTC , d∈IM the process RTC is recursively specified by the following equation: X RTC = r3 (r) · s4 (r) · RTC , r∈B the process ISEU is recursively specified by the following equations: ISEU ′ ISEU(n,S) ′ = ISEU (0,∅) , X ′ = r2 (d) · ISEU updcm(d,(n,S)) d∈IM X ′′ + nxt(f.m, S) :→ sf (m) · ISEU (f,(n,S)) f.m∈A + nxt(stop, S) :→ j + nxt(dead, S) :→ δ (for every (n, S) ∈ SISEU ) , X ′′ ′ ISEU(f,(n,S)) = rf (r) · s3 (r) · ISEU updpr(r,(n,S)) r∈B X ′′ + r2 (d) · ISEU (f,updcm(d,(n,S))) d∈IM (for every (f, (n, S)) ∈ F × SISEU ) , and H = {si (d) \| i ∈ {1, 2}, d ∈ IM} ∪ {ri (d) \| i ∈ {1, 2}, d ∈ IM} ∪ {si (r) \| i ∈ {3, 4}, r ∈ B} ∪ {ri (r) \| i ∈ {3, 4}, r ∈ B} . ISGT is the instruction stream generator for thread T , IMTC is the transmission channel for instruction messages, RTC is the transmission channel for replies, and ISEU is the instruction stream execution unit. The protocol described above has been designed such that, for each T ∈ RT , τ · τ{j} (∂H (ISGT k IMTC k RTC k ISEU )) denotes the process τ · \|T \|. We refrain from presenting a proof of the claim that the protocol satisfies this because this paper is first and foremost a conceptual paper and the proof is straightforward but tedious. The transmission channels IMTC and RTC can keep one instruction message and one reply, respectively. The protocol has been designed in such a way that the protocol will also work properly if these channels are replaced by channels with larger capacity and even by channels with unbounded capacity. Suppose that the transmission times over the transmission channels are small compared with the execution times of instructions. Even then the protocol described in Section 8 will always have to idle for a short time after the execution of an instruction, whereas after an initial phase the protocol described above will never have to idle after the execution of an instruction if the instruction stream generator may run a few steps ahead of the instruction stream execution unit. 23 10 Adaptations of the Protocol In this section, we discuss some conceivable adaptations of the protocol described in Section 9. While we were thinking through the details of that protocol, various variations suggested themselves. The variations discussed below are among the most salient ones. We think they deserve mention. However, their discussion is not in depth. The reason for this is that these variations have not yet been investigated thoroughly. Consider the case where, for each instruction, it is known what the probability is with which its execution leads to the reply T. This might give reason to adapt the protocol described in Section 9. Suppose that the instruction stream generator states do not only keep the sequences of replies after which threads must be performed, but also the sequences of instructions involved in producing those sequences of replies. Then the probability with which the sequences of replies will happen can be calculated and several conceivable adaptations of the protocol to this probabilistic knowledge are possible by mere changes in the selection of the sequence of replies and the instruction that will be part of the next instruction message produced by the instruction stream generator. Among those adaptations are: – restricting the instruction messages that are produced ahead to the ones where the sequence of replies after which the instruction must be executed will happen with a probability ≥ 0.50, but sticking to breadth-first runahead; – restricting the instruction messages that are produced ahead to the ones where the sequence of replies after which the instruction must be executed will happen with a probability ≥ 0.95, but not sticking to breadth-first runahead. At first sight, these adaptations are reminiscent of combinations of an instruction pre-fetching mechanism and a branch prediction mechanism as found in pipelined processors (see e.g. [26]). However, usually branch prediction mechanisms make use of statistics based on recently processed instructions instead of probabilistic knowledge of the kind used in the protocols sketched above. Regular threads can be represented in such a way that it is effectively decidable whether the two threads with which a thread may proceed after performing its first action are identical. Consider the case where threads are represented in the instruction stream generator states in such a way. Then the protocol can be adapted such that no duplication of instruction messages takes place in the cases where the two threads with which a thread possibly proceeds after performing its first action are identical. This can be accomplished by using sequences of elements from B ∪ {∗}, instead of sequences of elements from B, in instruction messages, instruction stream generator states, and instruction stream execution unit states. The occurrence of ∗ at position i in a sequence indicates that the ith reply may be either T or F. The impact of this change on the updates of instruction stream generator states and instruction stream execution unit states is minor. This adaptation is reminiscent of an instruction pre-fetch mechanism 24 as found in pipelined processors that prevents instruction pre-fetches that are superfluous due to identity of branches. 11 Alternative Choice Instructions Process algebra is an area of the study of concurrency which is considered relevant to computer science, as is witnesses by the extent of the work on algebraic theories of processes such as ACP, CCS and CSP in theoretical computer science. This strongly hints that there are programmed systems whose behaviours can be taken for processes as considered in process algebra. Therefore, it is interesting to know to which extent the behaviours considered in process algebra can be produced by programs under execution, starting from the perception of a program as an instruction sequence. In coming sections, we will establish results concerning the processes as considered in ACP that can be produced by instruction sequences under execution. For the purpose of producing processes as considered in ACP, we need a version of PGA with special basic instructions to deal with the non-deterministic choice between alternatives that stems from the alternative composition of processes. Recall that, in PGA, it is assumed that a fixed but arbitrary set A of basic instructions has been given. In the coming sections, we will make use a version of PGA in which the following additional assumptions relating to A are made: – – – – a fixed but arbitrary finite set F of foci has been given; a fixed but arbitrary finite set M of methods has been given; a fixed but arbitrary set AA of atomic actions, with t ∈ / AA, has been given; A = {f.m \| f ∈ F , m ∈ M} ∪ {ac(e1 , e2 ) \| e1 , e2 ∈ AA ∪ {t}}. On execution of a basic instruction ac(e1 , e2 ), first a non-deterministic choice between the atomic actions e1 and e2 is made and then the chosen atomic action is performed. The reply T is produced if e1 is performed and the reply F is produced if e2 is performed. Basic instructions of this kind are material to produce all regular processes by means of instruction sequences. A basic instruction of the form ac(e1 , e2 ) is called an alternative choice instruction. Henceforth, we will write PGAac for the version of PGA with alternative choice instructions. The intuition concerning alternative choice instructions given above will be made fully precise at the end of this section, using ACPτ . It will not be made fully precise using an extension of BTA because it is considered a basic property of threads that they are deterministic behaviours. Recall that we make use of a version of BTA in which A = A. A basic action of the form ac(e1 , e2 ) is called an alternative choice action. Henceforth, we will write BTAac for the version of BTA with alternative choice actions. For the purpose of making precise what processes are produced by the threads denoted by closed terms of BTAac +REC, A and \| are taken such that, in addition to the conditions mentioned at the beginning of Section 7, the following conditions are satisfied: A ⊇ AA ∪ {t} 25 Table 8. Additional defining equation for process extraction operation \|T E ac(e, e′ ) D T ′ \|c = e · \|T \|c + e′ · \|T ′ \|c and for all e, e′ ∈ A: e′ \| e = δ if e′ ∈ AA ∪ {t} . The process extraction operation for BTAac has as defining equations the equations given in Table 7 and in addition the equation given in Table 8. Proposition 3 goes through for BTAac . 12 Instruction Sequence Producible Processes It follows immediately from the definitions of the thread extraction and process extraction operations that the instruction sequences considered in PGA produce regular processes. The question is whether all regular processes are producible by these instruction sequences. In this section, we show that all regular processes can be produced by the instruction sequences with alternative choice instructions. We will make use of the fact that all regular threads over A can be produced by the single-pass instruction sequences considered in PGA. Proposition 4. For each thread T that is regular over A, there exists a PGA instruction sequence F such that F produces T , i.e. \|F \| = T . Proof. By Proposition 1, T is a component of the solution of some finite linear recursive specification E over BTA. There occur finitely many variables X0 , . . . , Xn in E. Assume that T is the X0 -component of the solution of E. Let F be the PGA instruction sequence (F0 ; . . . ; Fn )ω , where Fi is defined as follows (0 ≤ i ≤ n):  !;!;!   #0 ; #0 ; #0     +a ; #3·(j−i)−1 ; #3·(k−i)−2 Fi = +a ; #3·(j−i)−1 ; #3·(n+1−(i−k))−2    +a ; #3·(n+1−(i−j))−1 ; #3·(k−i)−2   +a ; #3·(n+1−(i−j))−1 ; #3·(n+1−(i−k))−2 if if if if if if Xi Xi Xi Xi Xi Xi = = = = = = S∈E D∈E Xj E a D Xk Xj E a D Xk Xj E a D Xk Xj E a D Xk ∈ ∈ ∈ ∈ E E E E ∧i ∧i ∧i ∧i < < ≥ ≥ j j j j ∧i ∧i ∧i ∧i < ≥ < ≥ k k k k. Then F is a PGA instruction sequence such that the interpretation of \|F \| = T . ⊓ ⊔ All regular processes over AA can be produced by the instruction sequences considered in PGAac . Theorem 2. Assume that CFAR is valid in MACPτ +REC . Then, for each process P that is regular over AA, there exists an instruction sequence F in which only basic instructions of the form ac(e, t) occur such that F produces P , i.e. τ · τ{t} (\|\|F \|\|) = τ · P . 26 Proof. By Propositions 1, 2 and 4, it is sufficient to show that, for each finite linear recursive specification E over ACPτ in which only atomic actions from AA occur, there exists a finite linear recursive specification E ′ over BTAac in which only basic actions of the form ac(e, t) occur such that τ ·hX\|Ei = τ ·τ{t} (\|hX\|E ′ i\|) for all X ∈ V(E). Take the finite linear recursive specification E over ACPτ that consists of the recursion equations Xi = ei1 · Xi1 + . . . + eiki · Xiki + e′i1 + . . . + e′ili , where ei1 , . . . , eiki , e′i1 , . . . , e′ili ∈ AA, for i ∈ {1, . . . n}. Then construct the finite linear recursive specification E ′ over BTAac that consists of the recursion equations Xi = Xi1 E ac(ei1 , t) D (. . . (Xiki E ac(eiki , t) D (S E ac(e′i1 , t) D (. . . (S E ac(e′ili , t) D Xi ) . . .))) . . .) for i ∈ {1, . . . n}; and the finite linear recursive specification E ′′ over ACPτ that consists of the recursion equations Zi1 = e′i1 + t · Zi2 , Zi2 = e′i2 + t · Zi3 , .. . Zili = e′ili + t · Xi , Xi = ei1 · Xi1 + t · Yi2 , Yi2 = ei2 · Xi2 + t · Yi3 , .. . Yiki = eiki · Xiki + t · Zi1 , where Yi2 , . . . , Yiki , Zi1 , . . . , Zili are fresh variables, for i ∈ {1, . . . n}. It follows immediately from the definition of the process extraction operation that \|hX\|E ′ i\| = hX\|E ′′ i for all X ∈ V(E). Moreover, it follows from CFAR that τ ·hX\|Ei = τ ·τ{t} (hX\|E ′′ i) for all X ∈ V(E). Hence, τ ·hX\|Ei = τ ·τ{t} (\|hX\|E ′ i\|) for all X ∈ V(E). ⊓ ⊔ For example, assuming that CFAR is valid, the instruction sequence (+ac(r3 (T), t) ; #4 ; +ac(r3 (F), t) ; #5 ; #7; +ac(s4 (T), t) ; #5 ; #9 ; +ac(s4 (F), t) ; #2 ; #9)ω produces the reply transmission channel process RTC of which a guarded recursive specification is given in Section 9. Remark 4. Theorem 2 with “τ · τ{t} (\|\|F \|\|) = τ · P ” replaced by “\|\|F \|\| = P ” can be established if PGA is extended with multiple-reply test instructions, see [11]. In that case, the assumption that CFAR is valid is superfluous. 13 Services and Use Operators An instruction sequence under execution may make use of services. That is, certain instructions may be executed for the purpose of having the behaviour 27 produced by the instruction sequence affected by a service that takes those instructions as commands to be processed. Likewise, a thread may perform certain actions for the purpose of having itself affected by a service that takes those actions as commands to be processed. The processing of an action may involve a change of state of the service and at completion of the processing of the action the service returns a reply value to the thread. The reply value determines how the thread proceeds. The use operators can be used in combination with the thread extraction operation from Section 4 to describe the behaviour produced by instruction sequences that make use of services. In this section, we first review the use operators, which are concerned with threads making such use of services, and then extend the process extraction operation to the use operators. A service H consists of – – – – a set S of states; an effect function eff : M × S → S; a yield function yld : M × S → B ∪ {B}; an initial state s0 ∈ S; satisfying the following condition: ∀m ∈ M, s ∈ S • (yld (m, s) = B ⇒ ∀m′ ∈ M • yld (m′ , eff (m, s)) = B) . The set S contains the states in which the service may be, and the functions eff and yld give, for each method m and state s, the state and reply, respectively, that result from processing m in state s. By the condition imposed on services, once the service has returned B as reply, it keeps returning B as reply. Let H = (S, eff , yld , s0 ) be a service and let m ∈ M. Then the derived service ∂ H, is the service (S, eff , yld , eff (m, s0 )); and of H after processing m, written ∂m the reply of H after processing m, written H(m), is yld (m, s0 ). When a thread makes a request to the service to process m: – if H(m) 6= B, then the request is accepted, the reply is H(m), and the service ∂ proceeds as ∂m H; – if H(m) = B, then the request is rejected and the service proceeds as a service that rejects any request. We introduce the sort S of services. However, we will not introduce constants and operators to build terms of this sort. The sort S, standing for the set of all services, is considered a parameter of the extension of BTA being presented. Moreover, we introduce, for each f ∈ F , the binary use operator /f : T × S → T. The axioms for these operators are given in Table 9. Intuitively, T /f H is the thread that results from processing all actions performed by thread T that are of the form f.m by service H. When a basic action of the form f.m performed by thread T is processed by service H, it is turned into the basic action tau and postconditional composition is removed in favour of basic action prefixing on the basis of the reply value produced. We add the use operators to PGAac as well. We will only use the extension in combination with the thread extraction operation \| \| and define \|F /f H\| = 28 Table 9. Axioms for use operators S /f H = S D /f H = D (x E tau D y) /f H = (x /f H) E tau D (y /f H) (x E g.m D y) /f H = (x /f H) E g.m D (y /f H) if f 6= g ∂ H) if H(m) = T (x E f.m D y) /f H = tau ◦ (x /f ∂m ∂ (x E f.m D y) /f H = tau ◦ (y /f ∂m H) if H(m) = F (x E f.m D y) /f H = tau ◦ D if H(m) = B (x E ac(e1 , e2 ) D y) /f H = (x /f H) E ac(e1 , e2 ) D (y /f H) πn (x /f H) = πn (πn (x) /f H) U1 U2 U3 U4 U5 U6 U7 U8 U9 \|F \| /f H. Hence, \|F /f H\| denotes the thread produced by F if F makes use of H. If H is a service such as an unbounded counter, an unbounded stack or a Turing tape, then a non-regular thread may be produced. In order to extend the process extraction operation to the use operators, we need an extension of ACPτ with action renaming operators ρh , where h:Aτ → Aτ such that h(τ ) = τ . The axioms for action renaming are given in [24]. Intuitively, ρh (P ) behaves as P with each atomic action replaced according to h. We write ρe′ 7→e′′ for the renaming operator ρh with h defined by h(e′ ) = e′′ and h(e) = e if e 6= e′ . For the purpose of extending the process extraction operation to the use operators, A and \| are taken such that, in addition to the conditions mentioned at the beginning of Section 7, with everywhere B replaced by B ∪ {B}, and the conditions mentioned at the end of Section 11, the following conditions are satisfied: A ⊇ {sserv (r) \| r ∈ B ∪ {B}} ∪ {rserv (m) \| m ∈ M} ∪ {stop∗ } and for all e ∈ A, m ∈ M, and r ∈ B ∪ {B}: sserv (r) \| e = δ , e \| rserv (m) = δ , stop \| stop = stop∗ , stop∗ \| e = δ . We also need to define a set Af ⊆ A and a function hf : Aτ → Aτ for each f ∈ F: Af = {sf (d) \| d ∈ M ∪ B ∪ {B}} ∪ {rf (d) \| d ∈ M ∪ B ∪ {B}} ; for all e ∈ Aτ , m ∈ M and r ∈ B ∪ {B}: hf (sserv (r)) = sf (r) , hf (rserv (m)) = rf (m) , V V hf (e) = e if r′ ∈N e 6= sserv (r′ ) ∧ m′ ∈M e 6= rserv (m′ ) . To extend the process extraction operation to the use operators, the defining equation concerning the postconditional composition operators has to be adapted and a new defining equation concerning the use operators has to be 29 Table 10. Adapted and additional defining equations for process extraction operation \|T E f.m D T ′ \|c = sf (m) · (rf (T) · \|T \|c + rf (F) · \|T ′ \|c + rf (B) · δ) \|T /f H\|c = ρstop∗ 7→stop (∂{stop} (∂Af (\|T \|c k ρhf (\|H\|c )))) added. These two equations are given in Table 10, where \|H\|c is the XH component of the solution of X {XH ′ = rserv (m) · sserv (H ′ (m)) · X ∂ H ′ + stop \| H ′ ∈ ∆(H)} , ∂m m∈M where ∆(H) is inductively defined as follows: – H ∈ ∆(H); – if m ∈ M and H ′ ∈ ∆(H), then ∂ ′ ∂m H ∈ ∆(H). The extended process extraction operation preserves the axioms for the use operators. Owing to the presence of axiom schemas with semantic side conditions in Table 9, the axioms for the use operators include proper axioms, which are all of the form t1 = t2 , and axioms that have a semantic side condition, which are all of the form t1 = t2 if H(m) = r. By that, the precise formulation of the preservation result is somewhat complicated. Proposition 5. 1. Let t1 = t2 be a proper axiom for the use operators, and let α be a valuation of variables in MBTA+REC . Then \|α∗ (t1 )\| = \|α∗ (t2 )\|. 2. Let t1 = t2 if H(m) = r be an axiom with semantic side condition for the use operators, and let α be a valuation of variables in MBTA+REC . Then \|α∗ (t1 )\| = \|α∗ (t2 )\| if H(m) = r. Proof. The proof is straightforward. We sketch the proof for axiom U5. By the definition of the process extraction operation, it is sufficient to show that ∂ H)\|c if H(m) = T. In outline, this goes \|(T E f.m D T ′ ) /f H\|c = \|tau ◦ (T /f ∂m as follows: \|(T E f.m D T ′ ) /f H\|c = ρstop∗ 7→stop (∂{stop} (∂Af (sf (m) · (rf (T) · \|T \|c + rf (F) · \|T ′ \|c + rf (B) · δ) k ρhf (\|H\|c )))) ∂ H\|c )))) = i · i · ρstop∗ 7→stop (∂{stop} (∂Af (\|T \|c k ρhf (\| ∂m ∂ c = \|tau ◦ (T /f ∂m H)\| . In the first and third step, we apply defining equations of \| \|c . In the second step, we apply axioms of ACPτ +REC with action renaming, and use that H(m) = T. ⊓ ⊔ Remark 5. Let F be a PGAac instruction sequence and H be a service. Then \|\|F /f H\|\| is the process produced by F if F makes use of H. Instruction sequences that make use of services such as unbounded counters, unbounded stacks or Turing tapes are interesting because they may produce non-regular processes. 30 14 PGLD Programs and the Use of Boolean Registers In this section, we show that all regular processes can also be produced by programs written in a program notation which is close to existing assembly languages, and even by programs in which no atomic action occurs more than once in an alternative choice instruction. The latter result requires programs that make use of Boolean registers. A hierarchy of program notations rooted in PGA is introduced in [8]. One program notation that belongs to this hierarchy is PGLD, a very simple program notation which is close to existing assembly languages. It has absolute jump instructions and no explicit termination instruction. In PGLD, like in PGA, it is assumed that there is a fixed but arbitrary finite set of basic instructions A. The primitive instructions of PGLD differ from the primitive instructions of PGA as follows: for each l ∈ N, there is an absolute jump instruction ##l instead of a forward jump instruction #l. PGLD programs have the form u1 ; . . . ; uk , where u1 , . . . , uk are primitive instructions of PGLD. The effects of all instructions in common with PGA are as in PGA with one difference: if there is no next instruction to be executed, termination occurs. The effect of an absolute jump instruction ##l is that execution proceeds with the l-th instruction of the program concerned. If ##l is itself the l-th instruction, then inaction occurs. If l equals 0 or l is greater than the length of the program, then termination occurs. We define the meaning of PGLD programs by means of a function pgld2pga from the set of all PGLD programs to the set of all closed PGA terms. This function is defined by pgld2pga(u1 ; . . . ; uk ) = (φ1 (u1 ) ; . . . ; φk (uk ) ; ! ; !)ω , where the auxiliary functions φj from the set of all primitive instructions of PGLD to the set of all primitive instructions of PGA are defined as follows (1 ≤ j ≤ k): φj (##l) = #l − j if φj (##l) = #k + 2 − (j − l) if φj (##l) = ! if φj (u) =u if j≤l≤k, 0<l<j, l =0∨l >k , u is not a jump instruction . PGLD is as expressive as PGA. Before we make this fully precise, we introduce a useful notation. Let α is a valuation of variables in I PGA , and let α∗ be the unique homomorphic extension of α to terms of PGA. Then α∗ (t) is independent of α if t is a closed term, i.e. α∗ (t) is uniquely determined by I PGA . Therefore, we write tI PGA for α∗ (t) if t is a closed term. Proposition 6. For each closed PGA term t, there exists a PGLD program p such that \|tI PGA \| = \|pgld2pga(p)I PGA \|. 31 Proof. In [8], a number of functions (called embeddings in that paper) are defined, whose composition gives, for each closed PGA term t, a PGLD program p such that \|tI PGA \| = \|pgld2pga(p)I PGA \|. ⊓ ⊔ Let p be a PGLD program and P be a process. Then we say that p produces P if \|pgld2pga(p)I PGA \| produces P . Below, we will write PGLDac for the version of PGLD in which the additional assumptions relating to A mentioned in Section 11 are made. As a corollary of Theorem 2 and Proposition 6, we have that all regular processes over AA can be produced by PGLDac programs. Corollary 1. Assume that CFAR is valid in MACPτ +REC . Then, for each process P that is regular over AA, there exists a PGLDac program p such that p produces P . We switch to the use of Boolean registers now. First, we describe services that make up Boolean registers. A Boolean register service accepts the following methods: – a set to true method set:T; – a set to false method set:F; – a get method get. We write MBR for the set {set:T, set:F, get}. It is assumed that MBR ⊆ M. The methods accepted by Boolean register services can be explained as follows: – set:T : the contents of the Boolean register becomes T and the reply is T; – set:F : the contents of the Boolean register becomes F and the reply is F; – get : nothing changes and the reply is the contents of the Boolean register. Let s ∈ B ∪ {B}. Then the Boolean register service with initial state s, written BR s , is the service (B ∪ {B}, eff , eff , s), where the function eff is defined as follows (b ∈ B): eff (set:T, b) = T , eff (set:F, b) = F , eff (get, b) = b , eff (m, b) = B if m 6∈ MBR , eff (m, B) = B . Notice that the effect and yield functions of a Boolean register service are the same. Let p be a PGLD program and P be a process. Then we say that p produces P using Boolean registers if (. . . (\|pgld2pga(p)I PGA \| /br:1 BR F ) . . . /br:k BR F ) produces P for some k ∈ N+ . We have that PGLDac programs in which no atomic action from AA occurs more than once in an alternative choice instruction can produce all regular processes over AA using Boolean registers. 32 Theorem 3. Assume that CFAR is valid in MACPτ +REC . Then, for each process P that is regular over AA, there exists a PGLDac program p in which each atomic action from AA occurs no more than once in an alternative choice instruction such that p produces P using Boolean registers. Proof. By the proof of Theorem 2 given in Section 12, it is sufficient to show that, for each thread T that is regular over A, there exist a PGLD program p in which each basic action from A occurs no more than once and a k ∈ N+ such that (. . . (\|pgld2pga(p)I PGA \| /br:1 BR F ) . . . /br:k BR F ) = T . Let T be a thread that is regular over A. We may assume that T is produced by a PGLD program p′ of the following form: +a1 ; ##(3 · k1 + 1) ; ##(3 · k1′ + 1) ; .. . +an ; ##(3 · kn + 1) ; ##(3 · kn′ + 1) ; ##0 ; ##0 ; ##0 ; ##(3 · n + 4) , where, for each i ∈ [1, n], ki , ki′ ∈ [0, n − 1] (cf. the proof of Proposition 2 from [36]). It is easy to see that the PGLD program p that we are looking for can be obtained by transforming p′ : by making use of n Boolean registers, p can distinguish between different occurrences of the same basic instruction in p′ , and in that way simulate p′ . ⊓ ⊔ 15 Conclusions Using the algebraic theory of processes known as ACP, we have described two protocols to deal with the phenomenon that, on execution of an instruction sequence, a stream of instructions to be processed arises at one place and the processing of that stream of instructions is handled at another place. The more complex protocol is directed towards keeping the execution unit busy. In this way, we have brought the phenomenon better into the picture and have ascribed a sense to the term instruction stream which makes clear that an instruction stream is dynamic by nature, in contradistinction with an instruction sequence. We have also discussed some conceivable adaptations of the more complex protocol. The description of the protocols start from the behaviours produced by instruction sequences under execution. By that we abstract from the instruction sequences which produce those behaviours. How instruction streams can be generated efficiently from instruction sequences is a matter that obviously requires investigations at a less abstract level. The investigations in question are an option for future work. We believe that the more complex protocol described in this paper provides a setting in which basic techniques aimed at increasing processor performance, such as pre-fetching and branch prediction, can be studied at a more abstract level than usual (cf. [26]). In particular, we think that the protocol can serve 33 as a starting-point for the development of a model with which trade-offs encountered in the design of processor architectures can be clarified. We consider investigations into this matter an interesting option for future work. The fact that process algebra is an area of the study of concurrency which is considered relevant to computer science, strongly hints that there are programmed systems whose behaviours are taken for processes as considered in process algebra. In that light, we have investigated the connections between programs and the processes that they produce, starting from the perception of a program as an instruction sequence. We have shown that, by apposite choice of basic instructions, all regular processes can be produced by means of instruction sequences as considered in PGA. We have also made precise what processes are produced by instruction sequences under execution that make use of services. 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On the Origin of Deep Learning On the Origin of Deep Learning Haohan Wang haohanw@cs.cmu.edu Bhiksha Raj bhiksha@cs.cmu.edu arXiv:1702.07800v4 [cs.LG] 3 Mar 2017 Language Technologies Institute School of Computer Science Carnegie Mellon University Abstract This paper is a review of the evolutionary history of deep learning models. It covers from the genesis of neural networks when associationism modeling of the brain is studied, to the models that dominate the last decade of research in deep learning like convolutional neural networks, deep belief networks, and recurrent neural networks. In addition to a review of these models, this paper primarily focuses on the precedents of the models above, examining how the initial ideas are assembled to construct the early models and how these preliminary models are developed into their current forms. Many of these evolutionary paths last more than half a century and have a diversity of directions. For example, CNN is built on prior knowledge of biological vision system; DBN is evolved from a trade-off of modeling power and computation complexity of graphical models and many nowadays models are neural counterparts of ancient linear models. This paper reviews these evolutionary paths and offers a concise thought flow of how these models are developed, and aims to provide a thorough background for deep learning. More importantly, along with the path, this paper summarizes the gist behind these milestones and proposes many directions to guide the future research of deep learning. 1 Wang and Raj 1. Introduction Deep learning has dramatically improved the state-of-the-art in many different artificial intelligent tasks like object detection, speech recognition, machine translation (LeCun et al., 2015). Its deep architecture nature grants deep learning the possibility of solving many more complicated AI tasks (Bengio, 2009). As a result, researchers are extending deep learning to a variety of different modern domains and tasks in additional to traditional tasks like object detection, face recognition, or language models, for example, Osako et al. (2015) uses the recurrent neural network to denoise speech signals, Gupta et al. (2015) uses stacked autoencoders to discover clustering patterns of gene expressions. Gatys et al. (2015) uses a neural model to generate images with different styles. Wang et al. (2016) uses deep learning to allow sentiment analysis from multiple modalities simultaneously, etc. This period is the era to witness the blooming of deep learning research. However, to fundamentally push the deep learning research frontier forward, one needs to thoroughly understand what has been attempted in the history and why current models exist in present forms. This paper summarizes the evolutionary history of several different deep learning models and explains the main ideas behind these models and their relationship to the ancestors. To understand the past work is not trivial as deep learning has evolved over a long time of history, as showed in Table 1. Therefore, this paper aims to offer the readers a walk-through of the major milestones of deep learning research. We will cover the milestones as showed in Table 1, as well as many additional works. We will split the story into different sections for the clearness of presentation. This paper starts the discussion from research on the human brain modeling. Although the success of deep learning nowadays is not necessarily due to its resemblance of the human brain (more due to its deep architecture), the ambition to build a system that simulate brain indeed thrust the initial development of neural networks. Therefore, the next section begins with connectionism and naturally leads to the age when shallow neural network matures. With the maturity of neural networks, this paper continues to briefly discuss the necessity of extending shallow neural networks into deeper ones, as well as the promises deep neural networks make and the challenges deep architecture introduces. With the establishment of the deep neural network, this paper diverges into three different popular deep learning topics. Specifically, in Section 4, this paper elaborates how Deep Belief Nets and its construction component Restricted Boltzmann Machine evolve as a trade-off of modeling power and computation loads. In Section 5, this paper focuses on the development history of Convolutional Neural Network, featured with the prominent steps along the ladder of ImageNet competition. In Section 6, this paper discusses the development of Recurrent Neural Networks, its successors like LSTM, attention models and the successes they achieved. While this paper primarily discusses deep learning models, optimization of deep architecture is an inevitable topic in this society. Section 7 is devoted to a brief summary of optimization techniques, including advanced gradient method, Dropout, Batch Normalization, etc. This paper could be read as a complementary of (Schmidhuber, 2015). Schmidhuber’s paper is aimed to assign credit to all those who contributed to the present state of the art, so his paper focuses on every single incremental work along the path, therefore cannot elab2 On the Origin of Deep Learning Year 300 BC 1873 1943 1949 1958 1974 1980 1982 1985 1986 1990 1997 2006 2009 2012 Table 1: Major milestones that will be covered in this paper Contributer Contribution introduced Associationism, started the history of human’s Aristotle attempt to understand brain. introduced Neural Groupings as the earliest models of Alexander Bain neural network, inspired Hebbian Learning Rule. introduced MCP Model, which is considered as the McCulloch & Pitts ancestor of Artificial Neural Model. considered as the father of neural networks, introduced Donald Hebb Hebbian Learning Rule, which lays the foundation of modern neural network. introduced the first perceptron, which highly resembles Frank Rosenblatt modern perceptron. Paul Werbos introduced Backpropagation Teuvo Kohonen introduced Self Organizing Map introduced Neocogitron, which inspired Convolutional Kunihiko Fukushima Neural Network John Hopfield introduced Hopfield Network Hilton & Sejnowski introduced Boltzmann Machine introduced Harmonium, which is later known as Restricted Paul Smolensky Boltzmann Machine Michael I. Jordan defined and introduced Recurrent Neural Network introduced LeNet, showed the possibility of deep neural Yann LeCun networks in practice Schuster & Paliwal introduced Bidirectional Recurrent Neural Network Hochreiter & introduced LSTM, solved the problem of vanishing Schmidhuber gradient in recurrent neural networks introduced Deep Belief Networks, also introduced Geoffrey Hinton layer-wise pretraining technique, opened current deep learning era. Salakhutdinov & introduced Deep Boltzmann Machines Hinton introduced Dropout, an efficient way of training neural Geoffrey Hinton networks 3 Wang and Raj orate well enough on each of them. On the other hand, our paper is aimed at providing the background for readers to understand how these models are developed. Therefore, we emphasize on the milestones and elaborate those ideas to help build associations between these ideas. In addition to the paths of classical deep learning models in (Schmidhuber, 2015), we also discuss those recent deep learning work that builds from classical linear models. Another article that readers could read as a complementary is (Anderson and Rosenfeld, 2000) where the authors conducted extensive interviews with well-known scientific leaders in the 90s on the topic of the neural networks’ history. 4 On the Origin of Deep Learning 2. From Aristotle to Modern Artificial Neural Networks The study of deep learning and artificial neural networks originates from our ambition to build a computer system simulating the human brain. To build such a system requires understandings of the functionality of our cognitive system. Therefore, this paper traces all the way back to the origins of attempts to understand the brain and starts the discussion of Aristotle’s Associationism around 300 B.C. 2.1 Associationism “When, therefore, we accomplish an act of reminiscence, we pass through a certain series of precursive movements, until we arrive at a movement on which the one we are in quest of is habitually consequent. Hence, too, it is that we hunt through the mental train, excogitating from the present or some other, and from similar or contrary or coadjacent. Through this process reminiscence takes place. For the movements are, in these cases, sometimes at the same time, sometimes parts of the same whole, so that the subsequent movement is already more than half accomplished.” This remarkable paragraph of Aristotle is seen as the starting point of Associationism (Burnham, 1888). Associationism is a theory states that mind is a set of conceptual elements that are organized as associations between these elements. Inspired by Plato, Aristotle examined the processes of remembrance and recall and brought up with four laws of association (Boeree, 2000). • Contiguity: Things or events with spatial or temporal proximity tend to be associated in the mind. • Frequency: The number of occurrences of two events is proportional to the strength of association between these two events. • Similarity: Thought of one event tends to trigger the thought of a similar event. • Contrast: Thought of one event tends to trigger the thought of an opposite event. Back then, Aristotle described the implementation of these laws in our mind as common sense. For example, the feel, the smell, or the taste of an apple should naturally lead to the concept of an apple, as common sense. Nowadays, it is surprising to see that these laws proposed more than 2000 years ago still serve as the fundamental assumptions of machine learning methods. For example, samples that are near each other (under a defined distance) are clustered into one group; explanatory variables that frequently occur with response variables draw more attention from the model; similar/dissimilar data are usually represented with more similar/dissimilar embeddings in latent space. Contemporaneously, similar laws were also proposed by Zeno of Citium, Epicurus and St Augustine of Hippo. The theory of associationism was later strengthened with a variety of philosophers or psychologists. Thomas Hobbes (1588-1679) stated that the complex experiences were the association of simple experiences, which were associations of sensations. He also believed that association exists by means of coherence and frequency as its strength 5 Wang and Raj Figure 1: Illustration of neural groupings in (Bain, 1873) factor. Meanwhile, John Locke (1632-1704) introduced the concept of “association of ideas”. He still separated the concept of ideas of sensation and ideas of reflection and he stated that complex ideas could be derived from a combination of these two simple ideas. David Hume (1711-1776) later reduced Aristotle’s four laws into three: resemblance (similarity), contiguity, and cause and effect. He believed that whatever coherence the world seemed to have was a matter of these three laws. Dugald Stewart (1753-1828) extended these three laws with several other principles, among an obvious one: accidental coincidence in the sounds of words. Thomas Reid (1710-1796) believed that no original quality of mind was required to explain the spontaneous recurrence of thinking, rather than habits. James Mill (1773-1836) emphasized on the law of frequency as the key to learning, which is very similar to later stages of research. David Hartley (1705-1757), as a physician, was remarkably regarded as the one that made associationism popular (Hartley, 2013). In addition to existing laws, he proposed his argument that memory could be conceived as smaller scale vibrations in the same regions of the brain as the original sensory experience. These vibrations can link up to represent complex ideas and therefore act as a material basis for the stream of consciousness. This idea potentially inspired Hebbian learning rule, which will be discussed later in this paper to lay the foundation of neural networks. 2.2 Bain and Neural Groupings Besides David Hartley, Alexander Bain (1818-1903) also contributed to the fundamental ideas of Hebbian Learning Rule (Wilkes and Wade, 1997). In this book, Bain (1873) related the processes of associative memory to the distribution of activity of neural groupings (a term that he used to denote neural networks back then). He proposed a constructive mode of storage capable of assembling what was required, in contrast to alternative traditional mode of storage with prestored memories. To further illustrate his ideas, Bain first described the computational flexibility that allows a neural grouping to function when multiple associations are to be stored. With a few hypothesis, Bain managed to describe a structure that highly resembled the neural 6 On the Origin of Deep Learning networks of today: an individual cell is summarizing the stimulation from other selected linked cells within a grouping, as showed in Figure 1. The joint stimulation from a and c triggers X, stimulation from b and c triggers Y and stimulation from a and c triggers Z. In his original illustration, a, b, c stand for simulations, X and Y are outcomes of cells. With the establishment of how this associative structure of neural grouping can function as memory, Bain proceeded to describe the construction of these structures. He followed the directions of associationism and stated that relevant impressions of neural groupings must be made in temporal contiguity for a period, either on one occasion or repeated occasions. Further, Bain described the computational properties of neural grouping: connections are strengthened or weakened through experience via changes of intervening cell-substance. Therefore, the induction of these circuits would be selected comparatively strong or weak. As we will see in the following section, Hebb’s postulate highly resembles Bain’s description, although nowadays we usually label this postulate as Hebb’s, rather than Bain’s, according to (Wilkes and Wade, 1997). This omission of Bain’s contribution may also be due to Bain’s lack of confidence in his own theory: Eventually, Bain was not convinced by himself and doubted about the practical values of neural groupings. 2.3 Hebbian Learning Rule Hebbian Learning Rule is named after Donald O. Hebb (1904-1985) since it was introduced in his work The Organization of Behavior (Hebb, 1949). Hebb is also seen as the father of Neural Networks because of this work (Didier and Bigand, 2011). In 1949, Hebb stated the famous rule: “Cells that fire together, wire together”, which emphasized on the activation behavior of co-fired cells. More specifically, in his book, he stated that: “When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that As efficiency, as one of the cells firing B, is increased.” This archaic paragraph can be re-written into modern machine learning languages as the following: ∆wi = ηxi y (1) where ∆wi stands for the change of synaptic weights (wi ) of Neuron i, of which the input signal is xi . y denotes the postsynaptic response and η denotes learning rate. In other words, Hebbian Learning Rule states that the connection between two units should be strengthened as the frequency of co-occurrences of these two units increase. Although Hebbian Learning Rule is seen as laying the foundation of neural networks, seen today, its drawbacks are obvious: as co-occurrences appear more, the weights of connections keep increasing and the weights of a dominant signal will increase exponentially. This is known as the unstableness of Hebbian Learning Rule (Principe et al., 1999). Fortunately, these problems did not influence Hebb’s identity as the father of neural networks. 7 Wang and Raj 2.4 Oja’s Rule and Principal Component Analyzer Erkki Oja extended Hebbian Learning Rule to avoid the unstableness property and he also showed that a neuron, following this updating rule, is approximating the behavior of a Principal Component Analyzer (PCA) (Oja, 1982). Long story short, Oja introduced a normalization term to rescue Hebbian Learning rule, and further he showed that his learning rule is simply an online update of Principal Component Analyzer. We present the details of this argument in the following paragraphs. Starting from Equation 1 and following the same notation, Oja showed: wit+1 = wit + ηxi y where t denotes the iteration. A straightforward way to avoid the exploding of weights is to apply normalization at the end of each iteration, yielding: wt + ηxi y wit+1 = Pn i 1 ( i=1 (wit + ηxi y)2 ) 2 where n denotes the number of neurons. The above equation can be further expanded into the following form: Pn t w i yx w j yxj wj i + ) + O(η 2 ) wit+1 = i + η( Z Z Z3 P 1 where Z = ( ni wi2 ) 2 . Further, two more assumptions are introduced: 1) η is small. P 1 Therefore O(η 2 ) is approximately 0. 2) Weights are normalized, therefore Z = ( ni wi2 ) 2 = 1. When these two assumptions were introduced back to the previous equation, Oja’s rule was proposed as following: wit+1 = wit + ηy(xi − ywit ) (2) Oja took a step further to show that a neuron that was updated with this rule was effectively performing Principal Component Analysis on the data. To show this, Oja first re-wrote Equation 2 as the following forms with two additional assumptions (Oja, 1982): d t w = Cwit − ((wit )T Cwit )wit d(t) i where C is the covariance matrix of input X. Then he proceeded to show this property with many conclusions from his another work (Oja and Karhunen, 1985) and linked back to PCA with the fact that components from PCA are eigenvectors and the first component is the eigenvector corresponding to largest eigenvalues of the covariance matrix. Intuitively, we could interpret this property with a simpler explanation: the eigenvectors of C are the solution when we maximize the rule updating function. Since wit are the eigenvectors of the covariance matrix of X, we can get that wit are the PCA. Oja’s learning rule concludes our story of learning rules of the early-stage neural network. Now we proceed to visit the ideas on neural models. 8 On the Origin of Deep Learning 2.5 MCP Neural Model While Donald Hebb is seen as the father of neural networks, the first model of neuron could trace back to six years ahead of the publication of Hebbian Learning Rule, when a neurophysiologist Warren McCulloch and a mathematician Walter Pitts speculated the inner workings of neurons and modeled a primitive neural network by electrical circuits based on their findings (McCulloch and Pitts, 1943). Their model, known as MCP neural model, was a linear step function upon weighted linearly interpolated data that could be described as: ( P 1, i wi xi ≥ θ AND zj = 0, ∀j y= 0, otherwise where y stands for output, xi stands for input of signals, wi stands for the corresponding weights and zj stands for the inhibitory input. θ stands for the threshold. The function is designed in a way that the activity of any inhibitory input completely prevents excitation of the neuron at any time. Despite the resemblance between MCP Neural Model and modern perceptron, they are still different distinctly in many different aspects: • MCP Neural Model is initially built as electrical circuits. Later we will see that the study of neural networks has borrowed many ideas from the field of electrical circuits. • The weights of MCP Neural Model wi are fixed, in contrast to the adjustable weights in modern perceptron. All the weights must be assigned with manual calculation. • The idea of inhibitory input is quite unconventional even seen today. It might be an idea worth further study in modern deep learning research. 2.6 Perceptron With the success of MCP Neural Model, Frank Rosenblatt further substantialized Hebbian Learning Rule with the introduction of perceptrons (Rosenblatt, 1958). While theorists like Hebb were focusing on the biological system in the natural environment, Rosenblatt constructed the electronic device named Perceptron that was showed with the ability to learn in accordance with associationism. Rosenblatt (1958) introduced the perceptron with the context of the vision system, as showed in Figure 2(a). He introduced the rules of the organization of a perceptron as following: • Stimuli impact on a retina of the sensory units, which respond in a manner that the pulse amplitude or frequency is proportional to the stimulus intensity. • Impulses are transmitted to Projection Area (AI ). This projection area is optional. • Impulses are then transmitted to Association Area through random connections. If the sum of impulse intensities is equal to or greater than the threshold (θ) of this unit, then this unit fires. 9 Wang and Raj (a) Illustration of organization of a perceptron in (Rosenblatt, 1958) (b) A typical perceptron in modern machine learning literature Figure 2: Perceptrons: (a) A new figure of the illustration of organization of perceptron as in (Rosenblatt, 1958). (b) A typical perceptron nowadays, when AI (Projection Area) is omitted. • Response units work in the same fashion as those intermediate units. Figure 2(a) illustrates his explanation of perceptron. From left to right, the four units are sensory unit, projection unit, association unit and response unit respectively. Projection unit receives the information from sensory unit and passes onto association unit. This unit is often omitted in other description of similar models. With the omission of projection unit, the structure resembles the structure of nowadays perceptron in a neural network (as showed in Figure 2(b)): sensory units collect data, association units linearly adds these data with different weights and apply non-linear transform onto the thresholded sum, then pass the results to response units. One distinction between the early stage neuron models and modern perceptrons is the introduction of non-linear activation functions (we use sigmoid function as an example in Figure 2(b)). This originates from the argument that linear threshold function should be softened to simulate biological neural networks (Bose et al., 1996) as well as from the consideration of the feasibility of computation to replace step function with a continuous one (Mitchell et al., 1997). After Rosenblatt’s introduction of Perceptron, Widrow et al. (1960) introduced a followup model called ADALINE. However, the difference between Rosenblatt’s Perceptron and ADALINE is mainly on the algorithm aspect. As the primary focus of this paper is neural network models, we skip the discussion of ADALINE. 2.7 Perceptron’s Linear Representation Power A perceptron is fundamentally a linear function of input signals; therefore it is limited to represent linear decision boundaries like the logical operations like NOT, AND or OR, but not XOR when a more sophisticated decision boundary is required. This limitation was highlighted by Minski and Papert (1969), when they attacked the limitations of perceptions by emphasizing that perceptrons cannot solve functions like XOR or NXOR. As a result, very little research was done in this area until about the 1980s. 10 On the Origin of Deep Learning (a) (b) (c) (d) Figure 3: The linear representation power of preceptron To show a more concrete example, we introduce a linear preceptron with only two inputs x1 and x2 , therefore, the decision boundary w1 x1 + w2 x2 forms a line in a two-dimensional space. The choice of threshold magnitude shifts the line horizontally and the sign of the function picks one side of the line as the halfspace the function represents. The halfspace is showed in Figure 3 (a). In Figure 3 (b)-(d), we present two nodes a and b to denote to input, as well as the node to denote the situation when both of them trigger and a node to denote the situation when neither of them triggers. Figure 3 (b) and Figure 3 (c) show clearly that a linear perceptron can be used to describe AND and OR operation of these two inputs. However, in Figure 3 (d), when we are interested in XOR operation, the operation can no longer be described by a single linear decision boundary. In the next section, we will show that the representation ability is greatly enlarged when we put perceptrons together to make a neural network. However, when we keep stacking one neural network upon the other to make a deep learning model, the representation power will not necessarily increase. 11 Wang and Raj 3. From Modern Neural Network to the Era of Deep Learning In this section, we will introduce some important properties of neural networks. These properties partially explain the popularity neural network gains these days and also motivate the necessity of exploring deeper architecture. To be specific, we will discuss a set of universal approximation properties, in which each property has its condition. Then, we will show that although a shallow neural network is an universal approximator, deeper architecture can significantly reduce the requirement of resources while retaining the representation power. At last, we will also show some interesting properties discovered in the 1990s about backpropagation, which may inspire some related research today. 3.1 Universal Approximation Property The step from perceptrons to basic neural networks is only placing the perceptrons together. By placing the perceptrons side by side, we get a single one-layer neural network and by stacking one one-layer neural network upon the other, we get a multi-layer neural network, which is often known as multi-layer perceptrons (MLP) (Kawaguchi, 2000). One remarkable property of neural networks, widely known as universal approximation property, roughly describes that an MLP can represent any functions. Here we discussed this property in three different aspects: • Boolean Approximation: an MLP of one hidden layer1 can represent any boolean function exactly. • Continuous Approximation: an MLP of one hidden layer can approximate any bounded continuous function with arbitrary accuracy. • Arbitrary Approximation: an MLP of two hidden layers can approximate any function with arbitrary accuracy. We will discuss these three properties in detail in the following paragraphs. To suit different readers’ interest, we will first offer an intuitive explanation of these properties and then offer the proofs. 3.1.1 Representation of any Boolean Functions This approximation property is very straightforward. In the previous section we have shown that every linear preceptron can perform either AND or OR. According to De Morgan’s laws, every propositional formula can be converted into an equivalent Conjunctive Normal Form, which is an OR of multiple AND functions. Therefore, we simply rewrite the target Boolean function into an OR of multiple AND operations. Then we design the network in such a way: the input layer performs all AND operations, and the hidden layer is simply an OR operation. The formal proof is not very different from this intuitive explanation, we skip it for simplicity. 1. Through this paper, we will follow the most widely accepted naming convention that calls a two-layer neural network as one hidden layer neural network. 12 On the Origin of Deep Learning (a) (b) (c) (d) Figure 4: Example of Universal Approximation of any Bounded Continuous Functions 3.1.2 Approximation of any Bounded Continuous Functions Continuing from the linear representation power of perceptron discussed previously, if we want to represent a more complex function, showed in Figure 4 (a), we can use a set of linear perceptrons, each of them describing a halfspace. One of these perceptrons is shown in Figure 4 (b), we will need five of these perceptrons. With these perceptrons, we can bound the target function out, as showed in Figure 4 (c). The numbers showed in Figure 4 (c) represent the number of subspaces described by perceptrons that fall into the corresponding region. As we can see, with an appropriate selection of the threshold (e.g. θ = 5 in Figure 4 (c)), we can bound the target function out. Therefore, we can describe any bounded continuous function with only one hidden layer; even it is a shape as complicated as Figure 4 (d). This property was first shown in (Cybenko, 1989) and (Hornik et al., 1989). To be specific, Cybenko (1989) showed that, if we have a function in the following form: X f (x) = ωi σ(wiT x + θ) (3) i f (x) is dense in the subspace of where it is in. In other words, for an arbitrary function g(x) in the same subspace as f (x), we have \|f (x) − g(x)\| < where > 0. In Equation 3, σ denotes the activation function (a squashing function back then), wi denotes the weights for the input layer and ωi denotes the weights for the hidden layer. This conclusion was drawn with a proof by contradiction: With Hahn-Banach Theorem and Riesz Representation Theorem, the fact that the closure of f (x) is not all the subspace where f (x) is in contradicts the assumption that σ is an activation (squashing) function. Till today, this property has drawn thousands of citations. Unfortunately, many of the later works cite this property inappropriately (Castro et al., 2000) because Equation 3 is not the widely accepted form of a one-hidden-layer neural network because it does not deliver a thresholded/squashed output, but a linear output instead. Ten years later after this property was shown, Castro et al. (2000) concluded this story by showing that when the final output is squashed, this universal approximation property still holds. Note that, this property was shown with the context that activation functions are squashing functions. By definition, a squashing function σ : R → [0, 1] is a non-decreasing function 13 Wang and Raj Figure 5: Threshold is not necessary with a large number of linear perceptrons. with the properties limx→∞ σ(x) = 1 and limx→−∞ σ(x) = 0. Many activation functions of recent deep learning research do not fall into this category. 3.1.3 Approximation of Arbitrary Functions Before we move on to explain this property, we need first to show a major property regarding combining linear perceptrons into neural networks. Figure 5 shows that as the number of linear perceptrons increases to bound the target function, the area outside the polygon with the sum close to the threshold shrinks. Following this trend, we can use a large number of perceptrons to bound a circle, and this can be achieved even without knowing the threshold because the area close to the threshold shrinks to nothing. What left outside the circle is, in fact, the area that sums to N2 , where N is the number of perceptrons used. Therefore, a neural network with one hidden layer can represent a circle with arbitrary diameter. Further, we introduce another hidden layer that is used to combine the outputs of many different circles. This newly added hidden layer is only used to perform OR operation. Figure 6 shows an example that when the extra hidden layer is used to merge the circles from the previous layer, the neural network can be used to approximate any function. The target function is not necessarily continuous. However, each circle requires a large number of neurons, consequently, the entire function requires even more. This property was showed in (Lapedes and Farber, 1988) and (Cybenko, 1988) respectively. Looking back at this property today, it is not arduous to build the connections between this property to Fourier series approximation, which, in informal words, states that every function curve can be decomposed into the sum of many simpler curves. With this linkage, to show this universal approximation property is to show that any one-hiddenlayer neural network can represent one simple surface, then the second hidden layer sums up these simple surfaces to approximate an arbitrary function. As we know, one hidden layer neural network simply performs a thresholded sum operation, therefore, the only step left is to show that the first hidden layer can represent a simple surface. To understand the “simple surface”, with linkage to Fourier transform, one can imagine one cycle of the sinusoid for the one-dimensional case or a “bump” of a plane in the two-dimensional case. 14 On the Origin of Deep Learning Figure 6: How a neural network can be used to approximate a leaf shaped function. For one dimension, to create a simple surface, we only need two sigmoid functions appropriately placed, for example, as following: f1 (x) = f2 (x) = h 1+ e−(x+t1 ) h 1 + ex−t2 Then, with f1 (x) + f2 (x), we create a simple surface with height 2h from t1 ≤ x ≤ t2 . This could be easily generalized to n-dimensional case, where we need 2n sigmoid functions (neurons) for each simple surface. Then for each simple surface that contributes to the final function, one neuron is added onto the second hidden layer. Therefore, despite the number of neurons need, one will never need a third hidden layer to approximate any function. Similarly to how Gibbs phenomenon affects Fourier series approximation, this approximation cannot guarantee an exact representation. The universal approximation properties showed a great potential of shallow neural networks at the price of exponentially many neurons at these layers. One followed-up question is that how to reduce the number of required neurons while maintaining the representation power. This question motivates people to proceed to deeper neural networks despite that shallow neural networks already have infinite modeling power. Another issue worth attention is that, although neural networks can approximate any functions, it is not trivial to find the set of parameters to explain the data. In the next two sections, we will discuss these two questions respectively. 15 Wang and Raj 3.2 The Necessity of Depth The universal approximation properties of shallow neural networks come at a price of exponentially many neurons and therefore are not realistic. The question about how to maintain this expressive power of the network while reducing the number of computation units has been asked for years. Intuitively, Bengio and Delalleau (2011) suggested that it is nature to pursue deeper networks because 1) human neural system is a deep architecture (as we will see examples in Section 5 about human visual cortex.) and 2) humans tend to represent concepts at one level of abstraction as the composition of concepts at lower levels. Nowadays, the solution is to build deeper architectures, which comes from a conclusion that states the representation power of a k layer neural network with polynomial many neurons need to be expressed with exponentially many neurons if a k − 1 layer structured is used. However, theoretically, this conclusion is still being completed. This conclusion could trace back to three decades ago when Yao (1985) showed the limitations of shallow circuits functions. Hastad (1986) later showed this property with parity circuits: “there are functions computable in polynomial size and depth k but requires exponential size when depth is restricted to k − 1”. He showed this property mainly by the application of DeMorgan’s law, which states that any AND or ORs can be rewritten as OR of ANDs and vice versa. Therefore, he simplified a circuit where ANDs and ORs appear one after the other by rewriting one layer of ANDs into ORs and therefore merge this operation to its neighboring layer of ORs. By repeating this procedure, he was able to represent the same function with fewer layers, but more computations. Moving from circuits to neural networks, Delalleau and Bengio (2011) compared deep and shallow sum-product neural networks. They showed that a function√ that could be expressed with O(n) neurons on a network of depth k required at least O(2 n ) and O((n − 1)k ) neurons on a two-layer neural network. Further, Bianchini and Scarselli (2014) extended this study to a general neural network with many major activation functions including tanh and sigmoid. They derived the conclusion with the concept of Betti numbers, and used this number to describe the representation power of neural networks. They showed that for a shallow network, the representation power can only grow polynomially with respect to the number of neurons, but for deep architecture, the representation can grow exponentially with respect to the number of neurons. They also related their conclusion to VC-dimension of neural networks, which is O(p2 ) for tanh (Bartlett and Maass, 2003) where p is the number of parameters. Recently, Eldan and Shamir (2015) presented a more thorough proof to show that depth of a neural network is exponentially more valuable than the width of a neural network, for a standard MLP with any popular activation functions. Their conclusion is drawn with only a few weak assumptions that constrain the activation functions to be mildly increasing, measurable, and able to allow shallow neural networks to approximate any univariate Lipschitz function. Finally, we have a well-grounded theory to support the fact that deeper network is preferred over shallow ones. However, in reality, many problems will arise if we keep increasing the layers. Among them, the increased difficulty of learning proper parameters is probably the most prominent one. Immediately in the next section, we will discuss the main drive of searching parameters for a neural network: Backpropagation. 16 On the Origin of Deep Learning 3.3 Backpropagation and Its Properties Before we proceed, we need to clarify that the name backpropagation, originally, is not referring to an algorithm that is used to learn the parameters of a neural network, instead, it stands for a technique that can help efficiently compute the gradient of parameters when gradient descent algorithm is applied to learn parameters (Hecht-Nielsen, 1989). However, nowadays it is widely recognized as the term to refer gradient descent algorithm with such a technique. Compared to a standard gradient descent, which updates all the parameters with respect to error, backpropagation first propagates the error term at output layer back to the layer at which parameters need to be updated, then uses standard gradient descent to update parameters with respect to the propagated error. Intuitively, the derivation of backpropagation is about organizing the terms when the gradient is expressed with the chain rule. The derivation is neat but skipped in this paper due to the extensive resources available (Werbos, 1990; Mitchell et al., 1997; LeCun et al., 2015). Instead, we will discuss two interesting and seemingly contradictory properties of backpropagation. 3.3.1 Backpropagation Finds Global Optimal for Linear Separable Data Gori and Tesi (1992) studied on the problem of local minima in backpropagation. Interestingly, when the society believes that neural networks or deep learning approaches are believed to suffer from local optimal, they proposed an architecture where global optimal is guaranteed. Only a few weak assumptions of the network are needed to reach global optimal, including • Pyramidal Architecture: upper layers have fewer neurons • Weight matrices are full row rank • The number of input neurons cannot smaller than the classes/patterns of data. However, their approaches may not be relevant anymore as they require the data to be linearly separable, under which condition that many other models can be applied. 3.3.2 Backpropagation Fails for Linear Separable Data On the other hand, Brady et al. (1989) studied the situations when backpropagation fails on linearly separable data sets. He showed that there could be situations when the data is linearly separable, but a neural network learned with backpropagation cannot find that boundary. He also showed examples when this situation occurs. His illustrative examples only hold when the misclassified data samples are significantly less than correctly classified data samples, in other words, the misclassified data samples might be just outliers. Therefore, this interesting property, when viewed today, is arguably a desirable property of backpropagation as we typically expect a machine learning model to neglect outliers. Therefore, this finding has not attracted many attentions. However, no matter whether the data is an outlier or not, neural network should be able to overfit training data given sufficient training iterations and a legitimate learning algorithm, especially considering that Brady et al. (1989) showed that an inferior algorithm 17 Wang and Raj was able to overfit the data. Therefore, this phenomenon should have played a critical role in the research of improving the optimization techniques. Recently, the studying of cost surfaces of neural networks have indicated the existence of saddle points (Choromanska et al., 2015; Dauphin et al., 2014; Pascanu et al., 2014), which may explain the findings of Brady et al back in the late 80s. Backpropagation enables the optimization of deep neural networks. However, there is still a long way to go before we can optimize it well. Later in Section 7, we will briefly discuss more techniques related to the optimization of neural networks. 18 On the Origin of Deep Learning 4. The Network as Memory and Deep Belief Nets Figure 7: Trade off of representation power and computation complexity of several models, that guides the development of better models With the background of how modern neural network is set up, we proceed to visit the each prominent branch of current deep learning family. Our first stop is the branch that leads to the popular Restricted Boltzmann Machines and Deep Belief Nets, and it starts as a model to understand the data unsupervisedly. Figure 7 summarizes the model that will be covered in this Section. The horizontal axis stands for the computation complexity of these models while the vertical axis stands for the representation power. The six milestones that will be focused in this section are placed in the figure. 4.1 Self Organizing Map The discussion starts with Self Organizing Map (SOM) invented by Kohonen (1990). SOM is a powerful technique that is primarily used in reducing the dimension of data, usually to one or two dimensions (Germano, 1999). While reducing the dimensionality, SOM also retains the topological similarity of data points. It can also be seen as a tool for clustering while imposing the topology on clustered representation. Figure 8 is an illustration of Self Organizing Map of two dimension hidden neurons. Therefore, it learns a two dimension representation of data. The upper shaded nodes denote the units of SOM that are used to 19 Wang and Raj Figure 8: Illustration of Self-Organizing Map represent data while the lower circles denote the data. There is no connection between the nodes in SOM 2 . The position of each node is fixed. The representation should not be viewed as only a numerical value. Instead, the position of it also matters. This property is different from some widely-accepted representation criterion. For example, we compare the case when one-hot vector and one-dimension SOM are used to denote colors: To denote green out of a set: C = {green, red, purple}, one-hot representation can use any vector of (1, 0, 0), (0, 1, 0) or (0, 0, 1) as long as we specify the bit for green correspondingly. However, for a one-dimensional SOM, only two vectors are possible: (1, 0, 0) or (0, 0, 1). This is because that, since SOM aims to represent the data while retaining the similarity; and red and purple are much more similar than green and red or green and purple, green should not be represented in a way that it splits red and purple. One should notice that, this example is only used to demonstrate that the position of each unit in SOM matters. In practice, the values of SOM unit are not restricted to integers. The learned SOM is usually a good tool for visualizing data. For example, if we conduct a survey on the happiness level and richness level of each country and feed the data into a two-dimensional SOM. Then the trained units should represent the happiest and richest country at one corner and represent the opposite country at the furthest corner. The rest two corners represent the richest, yet unhappiest and the poorest but happiest countries. The rest countries are positioned accordingly. The advantage of SOM is that it allows one 2. In some other literature, (Bullinaria, 2004) as an example, one may notice that there are connections in the illustrations of models. However, those connections are only used to represent the neighborhood relationship of nodes, and there is no information flowing via those connections. In this paper, as we will show many other models that rely on a clear illustration of information flow, we decide to save the connections to denote that. 20 On the Origin of Deep Learning to easily tell how a country is ranked among the world with a simple glance of the learned units (Guthikonda, 2005). 4.1.1 Learning Algorithm With an understanding of the representation power of SOM, now we proceed to its parameter learning algorithm. The classic algorithm is heuristic and intuitive, as shown below: Here we use a two-dimensional SOM as example, and i, j are indexes of units; w is weight Initialize weights of all units, wi,j ∀ i, j for t ≤ N do Pick vk randomly Select Best Matching Unit (BMU) as p, q := arg mini,j \|\|wij − vk \|\|22 Select the nodes of interest as the neighbors of BMU. I = {wi,j \|dist(wi,j , wp,q ) < r(t)} Update weights: wi,j = wi,j + P (i, j, p, q)l(t)\|\|wij − vk \|\|22 , ∀i, j ∈ I end for of the unit; v denotes data vector; k is the index of data; t denotes the current iteration; N constrains the maximum number of steps allowed; P (·) denotes the penalty considering the distance between unit p, q and unit i, j; l is learning rate; r denotes a radius used to select neighbor nodes. Both l and r typically decrease as t increases. \|\| · \|\|22 denotes Euclidean distance and dist(·) denotes the distance on the position of units. This algorithm explains how SOM can be used to learn a representation and how the similarities are retained as it always selects a subset of units that are similar with the data sampled and adjust the weights of units to match the data sampled. However, this algorithm relies on a careful selection of the radius of neighbor selection and a good initialization of weights. Otherwise, although the learned weights will have a local property of topological similarity, it loses this property globally: sometimes, two similar clusters of similar events are separated by another dissimilar cluster of similar events. In simpler words, units of green may actually separate units of red and units of purple if the network is not appropriately trained. (Germano, 1999). 4.2 Hopfield Network Hopfield Network is historically described as a form of recurrent3 neural network, first introduced in (Hopfield, 1982). “Recurrent” in this context refers to the fact that the weights connecting the neurons are bidirectional. Hopfield Network is widely recognized because of its content-addressable memory property. This content-addressable memory property is a simulation of the spin glass theory. Therefore, we start the discussion from spin glass. 3. The term “recurrent” is very confusing nowadays because of the popularity recurrent neural network (RNN) gains. 21 Wang and Raj Figure 9: Illustration of Hopfield Network. It is a fully connected network of six binary thresholding neural units. Every unit is connected with data, therefore these units are denoted as unshaded nodes. 4.2.1 Spin Glass The spin glass is physics term that is used to describe a magnetic phenomenon. Many works have been done for a detailed study of related theory (Edwards and Anderson, 1975; Mézard et al., 1990), so in this paper, we only describe this it intuitively. When a group of dipoles is placed together in any space. Each dipole is forced to align itself with the field generated by these dipoles at its location. However, by aligning itself, it changes the field at other locations, leading other dipoles to flip, causing the field in the original location to change. Eventually, these changes will converge to a stable state. To describe the stable state, we first define the total field at location j as X sk sj = oj + ct d2jk k ct where oj is an external field, is a constant that depends on temperature t, sk is the polarity of the kth dipole and djk is the distance from location j to location k. Therefore, the total potential energy of the system is: X X sk PE = sj oj + ct sj (4) d2jk j k The magnetic system will evolve until this potential energy is minimum. 4.2.2 Hopfield Network Hopfield Network is a fully connected neural network with binary thresholding neural units. The values of these units are either 0 or 14 . These units are fully connected with bidirectional weights. 4. Some other literature may use -1 and 1 to denote the values of these units. While the choice of values does not affect the idea of Hopfiled Network, it changes the formulation of energy function. In this paper, we only discuss in the context of 0 and 1 as values. 22 On the Origin of Deep Learning With this setting, the energy of a Hopfield Network is defined as: X X E=− si bi − si sj wi,j i (5) i,j where s is the state of a unit, b denotes the bias; w denotes the bidirectional weights and i, j are indexes of units. This energy function closely connects to the potential energy function of spin glass, as showed in Equation 4. Hopfield Network is typically applied to memorize the state of data. The weights of a network are designed or learned to make sure that the energy is minimized given the state of interest. Therefore, when another state presented to the network, while the weights are fixed, Hopfield Network can search for the states that minimize the energy and recover the state in memory. For example, in a face completion task, when some image of faces are presented to Hopfield Network (in a way that each unit of the network corresponds to each pixel of one image, and images are presented one after the other), the network can calculate the weights to minimize the energy given these faces. Later, if one image is corrupted or distorted and presented to this network again, the network is able to recover the original image by searching a configuration of states to minimize the energy starting from corrupted input presented. The term “energy” may remind people of physics. To explain how Hopfield Network works in a physics scenario will be clearer: nature uses Hopfield Network to memorize the equilibrium position of a pendulum because, in an equilibrium position, the pendulum has the lowest gravitational potential energy. Therefore, whenever a pendulum is placed, it will converge back to the equilibrium position. 4.2.3 Learning and Inference Learning of the weights of a Hopfield Network is straightforward (Gurney, 1997). The weights can be calculated as: X wi,j = (2si − 1)(2sj − 1) i,j the notations are the same as Equation 5. This learning procedure is simple, but still worth mentioning as it is an essential step of a Hopfield Network when it is applied to solve practical problems. However, we find that many online tutorials omit this step, and to make it worse, they refer the inference of states as learning/training. To remove the confusion, in this paper, similar to how terms are used in standard machine learning society, we refer the calculation of weights of a model (either from closed-form solution, or numerical solution) as “parameter learning” or “training”. We refer the process of applying an existing model with weights known onto solving a real-world problem as “inference”5 or “testing” (to decode a hidden state of data, e.g. to predict a label). The inference of Hopfield Network is also intuitive. For a state of data, the network tests that if inverting the state of one unit, whether the energy will decrease. If so, the 5. “inference” is conventionally used in such a way in machine learning society, although some statisticians may disagree with this usage. 23 Wang and Raj network will invert the state and proceed to test the next unit. This procedure is called Asynchronous update and this procedure is obviously subject to the sequential order of selection of units. A counterpart is known as Synchronous update when the network first tests for all the units and then inverts all the unit-to-invert simultaneously. Both of these methods may lead to a local optimal. Synchronous update may even result in an increasing of energy and may converge to an oscillation or loop of states. 4.2.4 Capacity One distinct disadvantage of Hopfield Network is that it cannot keep the memory very efficient because a network of N units can only store memory up to 0.15N 2 bits. While a network with N units has N 2 edges. In addition, after storing M memories (M instances of data), each connection has an integer value in range [−M, M ]. Thus, the number of bits required to store N units are N 2 log(2M + 1) (Hopfield, 1982). Therefore, we can safely draw the conclusion that although Hopfield Network is a remarkable idea that enables the network to memorize data, it is extremely inefficient in practice. As follow-ups of the invention of Hopfield Network, many works are attempted to study and increase the capacity of original Hopfield Network (Storkey, 1997; Liou and Yuan, 1999; Liou and Lin, 2006). Despite these attempts made, Hopfield Network still gradually fades out of the society. It is replaced by other models that are inspired by it. Immediately following this section, we will discuss the popular Boltzmann Machine and Restricted Boltzmann Machine and study how these models are upgraded from the initial ideas of Hopfield Network and evolve to replace it. 4.3 Boltzmann Machine Boltzmann Machine, invented by Ackley et al. (1985), is a stochastic with-hidden-unit version Hopfield Network. It got its name from Boltzmann Distribution. 4.3.1 Boltzmann Distribution Boltzmann Distribution is named after Ludwig Boltzmann and investigated extensively by (Willard, 1902). It is originally used to describe the probability distribution of particles in a system over various possible states as following: Es F (s) ∝ e− kT where s stands for the state and Es is the corresponding energy. k and T are Boltzmann’s constant and thermodynamic temperature respectively. Naturally, the ratio of two distribution is only characterized by the difference of energies, as following: r= Es2 −Es1 F (s1 ) = e kT F (s2 ) which is known as Boltzmann factor. 24 On the Origin of Deep Learning Figure 10: Illustration of Boltzmann Machine. With the introduction of hidden units (shaded nodes), the model conceptually splits into two parts: visible units and hidden units. The red dashed line is used to highlight the conceptual separation. With how the distribution is specified by the energy, the probability is defined as the term of each state divided by a normalizer, as following: Esi e− kT psi = P − Esj kT je 4.3.2 Boltzmann Machine As we mentioned previously, Boltzmann Machine is a stochastic with-hidden-unit version Hopfield Network. Figure 10 introduces how the idea of hidden units is introduced that turns a Hopfield Network into a Boltzmann Machine. In a Boltzmann Machine, only visible units are connected with data and hidden units are used to assist visible units to describe the distribution of data. Therefore, the model conceptually splits into the visible part and hidden part, while it still maintains a fully connected network among these units. “Stochastic” is introduced for Boltzmann Machine to be improved from Hopfield Network regarding leaping out of the local optimum or oscillation of states. Inspired by physics, a method to transfer state regardless current energy is introduced: Set a state to State 1 (which means the state is on) regardless of the current state with the following probability: 1 p= ∆E 1 + e− T where ∆E stands for the difference of energies when the state is on and off, i.e. ∆E = Es=1 − Es=0 . T stands for the temperature. The idea of T is inspired by a physics process that the higher the temperature is, the more likely the state will transfer6 . In addition, the probability of higher energy state transferring to lower energy state will be always greater than the reverse process7 . This idea is highly related to a very popular optimization 6. Molecules move faster when more kinetic energy is provided, which could be achieved by heating. 7. This corresponds to Zeroth Law of Thermodynamics. 25 Wang and Raj algorithm called Simulated Annealing (Khachaturyan et al., 1979; Aarts and Korst, 1988) back then, but Simulated Annealing is hardly relevant to nowadays deep learning society. Regardless of the historical importance that the term T introduces, within this section, we will assume T = 1 as a constant, for the sake of simplification. 4.3.3 Energy of Boltzmann Machine The energy function of Boltzmann Machine is defined the same as how Equation 5 is defined for Hopfield Network, except that now visible units and hidden units are noted separately, as following: X X X X X E(v, h) = − vi bi − hk bk − vi vj wij − vi hk wik − hk hl wk,l i k i,j i,k k,l where v stands for visible units, h stands for hidden units. This equation also connects back to Equation 4, except that Boltzmann Machine splits the energy function according to hidden units and visible units. Based on this energy function, the probability of a joint configuration over both visible unit the hidden unit can be defined as following: p(v, h) = P e−E(v,h) −E(m,n) m,n e The probability of visible/hidden units can be achieved by marginalizing this joint probability. For example, by marginalizing out hidden units, we can get the probability distribution of visible units: P −E(v,h) e p(v) = P h −E(m,n) m,n e which could be used to sample visible units, i.e. generating data. When Boltzmann Machine is trained to its stable state, which is called thermal equilibrium, the distribution of these probabilities p(v, h) will remain constant because the distribution of energy will be a constant. However, the probability for each visible unit or hidden unit may vary and the energy may not be at their minimum. This is related to how thermal equilibrium is defined, where the only constant factor is the distribution of each part of the system. Thermal equilibrium can be a hard concept to understand. One can imagine that pouring a cup of hot water into a bottle and then pouring a cup of cold water onto the hot water. At start, the bottle feels hot at bottom and feels cold at top and gradually the bottle feels mild as the cold water and hot water mix and heat is transferred. However, the temperature of the bottle becomes mild stably (corresponding to the distribution of p(v, h)) does not necessarily mean that the molecules cease to move (corresponding to each p(v, h)). 4.3.4 Parameter Learning The common way to train the Boltzmann machine is to determine the parameters that maximize the likelihood of the observed data. Gradient descent on the log of the likelihood 26 On the Origin of Deep Learning function is usually performed to determine the parameters. For simplicity, the following derivation is based on a single observation. First, we have the log likelihood function of visible units as X X l(v; w) = log p(v; w) = log e−Ev,h − log e−Em,n m,n h where the second term on RHS is the normalizer. Now we take the derivative of log likelihood function w.r.t w, and simplify it, we have: X ∂l(v; w) ∂E(m, n) ∂E(v, h) X p(m, n) =− p(h\|v) + ∂w ∂w ∂w m,n h ∂E(v, h) ∂E(m, n) + Ep(m,n) ∂w ∂w where E denotes expectation. Thus the gradient of the likelihood function is composed of two parts. The first part is expected gradient of the energy function with respect to the conditional distribution p(h\|v). The second part is expected gradient of the energy function with respect to the joint distribution over all variable states. However, calculating these expectations is generally infeasible for any realistically-sized model, as it involves summing over a huge number of possible states/configurations. The general approach for solving this problem is to use Markov Chain Monte Carlo (MCMC) to approximate these sums: = − Ep(h\|v) ∂l(v; w) = − < si , sj >p(hdata \|vdata ) + < si , sj >p(hmodel \|vmodel ) (6) ∂w where < · > denotes expectation. Equation 6 is the difference between the expectation value of product of states while the data is fed into visible states and the expectation of product of states while no data is fed. The first term is calculated by taking the average value of the energy function gradient when the visible and hidden units are being driven by observed data samples. In practice, this first term is generally straightforward to calculate. Calculating the second term is generally more complicated and involves running a set of Markov chains until they reach the current models equilibrium distribution, then taking the average energy function gradient based on those samples. However, this sampling procedure could be very computationally complicated, which motivates the topic in next section, the Restricted Boltzmann Machine. 4.4 Restricted Boltzmann Machine Restricted Boltzmann Machine (RBM), originally known as Harmonium when invented by Smolensky (1986), is a version of Boltzmann Machine with a restriction that there is no connections either between visible units or between hidden units. Figure 11 is an illustration of how Restricted Boltzmann Machine is achieved based on Boltzmann Machine (Figure 10): the connections between hidden units, as well as the connections between visible units are removed and the model becomes a bipartite graph. With this restriction introduced, the energy function of RBM is much simpler: X X X E(v, h) = − vi bi − hk bk − vi hk wik (7) i k 27 i,k Wang and Raj Figure 11: Illustration of Restricted Boltzmann Machine. With the restriction that there is no connections between hidden units (shaded nodes) and no connections between visible units (unshaded nodes), the Boltzmann Machine turns into a Restricted Boltzmann Machine. The model now is a a bipartite graph. 4.4.1 Contrastive Divergence RBM can still be trained in the same way as how Boltzmann Machine is trained. Since the energy function of RBM is much simpler, the sampling method used to infer the second term in Equation 6 becomes easier. Despite this relative simplicity, this learning procedure still requires a large amount of sampling steps to approximate the model distribution. To emphasize the difficulties of such a sampling mechanism, as well as to simplify followup introduction, we re-write Equation 6 with a different set of notations, as following: ∂l(v; w) = − < si , sj >p0 + < si , sj >p∞ ∂w (8) here we use p0 to denote data distribution and p∞ to denote model distribution. Other notations remain unchanged. Therefore, the difficulty of mentioned methods to learn the parameters is that it requires potentially “infinitely” many sampling steps to approximate the model distribution. Hinton (2002) overcame this issue magically, with the introduction of a method named Contrastive Divergence. Empirically, he found that one does not have to perform “infinitely” many sampling steps to converge to the model distribution, a finite k steps of sampling is enough. Therefore, Equation 8 is effectively re-written into: ∂l(v; w) = − < si , sj >p0 + < si , sj >pk ∂w Remarkably, Hinton (2002) showed that k = 1 is sufficient for the learning algorithm to work well in practice. Carreira-Perpinan and Hinton (2005) attempted to justify Contrastive Divergence in theory, but their derivation led to a negative conclusion that Contrastive Divergence is a 28 On the Origin of Deep Learning Figure 12: Illustration of Deep Belief Networks. Deep Belief Networks is not just stacking RBM together. The bottom layers (layers except the top one) do not have the bi-directional connections, but only connections top down. biased algorithm, and a finite k cannot represent the model distribution. However, their empirical results suggested that finite k can approximate the model distribution well enough, resulting a small enough bias. In addition, the algorithm works well in practice, which strengthened the idea of Contrastive Divergence. With the reasonable modeling power and a fast approximation algorithm, RBM quickly draws great attention and becomes one of the most fundamental building blocks of deep neural networks. In the following two sections, we will introduce two distinguished deep neural networks that are built based on RBM/Boltzmann Machine, namely Deep Belief Nets and Deep Boltzmann Machine. 4.5 Deep Belief Nets Deep Belief Networks is introduced by Hinton et al. (2006)8 , when he showed that RBMs can be stacked and trained in a greedy manner. Figure 12 shows the structure of a three-layer Deep Belief Networks. Different from stacking RBM, DBN only allows bi-directional connections (RBM-type connections) on the top one layer while the following bottom layers only have top-down connections. Probably a better way to understand DBN is to think it as multi-layer generative models. Despite the fact that DBN is generally described as a stacked RBM, it is quite different from putting one RBM on the top of the other. It is probably more appropriate to think DBN as a one-layer RBM with extended layers specially devoted to generating patterns of data. Therefore, the model only needs to sample for the thermal equilibrium at the topmost layer and then pass the visible states top down to generate the data. 8. This paper is generally seen as the opening of nowadays Deep Learning era, as it first introduces the possibility of training a deep neural network by layerwise training 29 Wang and Raj 4.5.1 Parameter Learning Parameter learning of a Deep Belief Network falls into two steps: the first step is layer-wise pre-training and the second step is fine-tuning. Layerwise Pre-training The success of Deep Belief Network is largely due to the introduction of the layer-wised pretraining. The idea is simple, but the reason why it works still attracts researchers. The pre-training is simply to first train the network component by component bottom up: treating the first two layers as an RBM and train it, then treat the second layer and third layer as another RBM and train for the parameters. Such an idea turns out to offer a critical support of the success of the later finetuning process. Several explanations have been attempted to explain the mechanism of pre-training: • Intuitively, pre-training is a clever way of initialization. It puts the parameter values in the appropriate range for further fine-tuning. • Bengio et al. (2007) suggested that unsupervised pre-training initializes the model to a point in parameter space which leads to a more effective optimization process, that the optimization can find a lower minimum of the empirical cost function. • Erhan et al. (2010) empirically argued for a regularization explanation, that unsupervised pretraining guides the learning towards basins of attraction of minima that support better generalization from the training data set. In addition to Deep Belief Networks, this pretraining mechanism also inspires the pretraining for many other classical models, including the autoencoders (Poultney et al., 2006; Bengio et al., 2007), Deep Boltzmann Machines (Salakhutdinov and Hinton, 2009) and some models inspired by these classical models like (Yu et al., 2010). After the pre-training is performed, fine-tuning is carried out to further optimize the network to search for the parameters that lead to a lower minimum. For Deep Belief Networks, there are two different fine tuning strategies dependent on the goals of the network. Fine Tuning for Generative Model Fine-tuning for a generative model is achieved with a contrastive version of wake-sleep algorithm (Hinton et al., 1995). This algorithm is intriguing for the reason that it is designed to interpret how the brain works. Scientists have found that sleeping is a critical process of brain function and it seems to be an inverse version of how we learn when we are awake. The wake-sleep algorithm also has two steps. In wake phase, we propagate information bottom up to adjust top-down weights for reconstructing the layer below. Sleep phase is the inverse of wake phase. We propagate the information top down to adjust bottom-up weights for reconstructing the layer above. The contrastive version of this wake-sleep algorithm is that we add one Contrastive Divergence phase between wake phase and sleep phase. The wake phase only goes up to the visible layer of the top RBM, then we sample the top RBM with Contrastive Divergence, then a sleep phase starts from the visible layer of top RBM. Fine Tuning for Discriminative Model The strategy for fine tuning a DBN as a discriminative model is to simply apply standard backpropagation to pre-trained model 30 On the Origin of Deep Learning Figure 13: Illustration of Deep Boltzmann Machine. Deep Boltzmann Machine is more like stacking RBM together. Connections between every two layers are bidirectional. since we have labels of data. However, pre-training is still necessary in spite of the generally good performance of backpropagation. 4.6 Deep Boltzmann Machine The last milestone we introduce in the family of deep generative model is Deep Boltzmann Machine introduced by Salakhutdinov and Hinton (2009). Figure 13 shows a three layer Deep Boltzmann Machine (DBM). The distinction between DBM and DBN mentioned in the previous section is that DBM allows bidirectional connections in the bottom layers. Therefore, DBM represents the idea of stacking RBMs in a much better way than DBN, although it might be clearer if DBM is named as Deep Restricted Boltzmann Machine. Due to the nature of DBM, its energy function is defined as an extension of the energy function of an RBM (Equation 7), as showed in the following: E(v, h) = − X i vi bi − N X X hn,k bn,k − n=1 k X i,k vi wik hk − N −1 X X hn,k wn,k,l hn+1,l n=1 k,l for a DBM with N hidden layers. This similarity of energy function grants the possibility of training DBM with constrative divergence. However, pre-training is typically necessary. 4.6.1 Deep Boltzmann Machine (DBM) v.s. Deep Belief Networks (DBN) As their acronyms suggest, Deep Boltzmann Machine and Deep Belief Networks have many similarities, especially from the first glance. Both of them are deep neural networks originates from the idea of Restricted Boltzmann Machine. (The name “Deep Belief Network” 31 Wang and Raj seems to indicate that it also partially originates from Bayesian Network (Krieg, 2001).) Both of them also rely on layerwise pre-training for a success of parameter learning. However, the fundamental differences between these two models are dramatic, introduced by how the connections are made between bottom layers (un-directed/bi-directed v.s. directed). The bidirectional structure of DBM grants the possibility of DBM to learn a more complex pattern of data. It also grants the possibility for the approximate inference procedure to incorporate top-down feedback in addition to an initial bottom-up pass, allowing Deep Boltzmann Machines to better propagate uncertainty about ambiguous inputs. 4.7 Deep Generative Models: Now and the Future Deep Boltzmann Machine is the last milestone we discuss in the history of generative models, but there are still much work after DBM and even more to be done in the future. Lake et al. (2015) introduces a Bayesian Program Learning framework that can simulate human learning abilities with large scale visual concepts. In addition to its performance on one-shot learning classification task, their model passes the visual Turing Test in terms of generating handwritten characters from the worlds alphabets. In other words, the generative performance of their model is indistinguishable from human’s behavior. Being not a deep neural model itself, their model outperforms several concurrent deep neural networks. Deep neural counterpart of the Bayesian Program Learning framework can be surely expected with even better performance. Conditional image generation (given part of the image) is also another interesting topic recently. The problem is usually solved by Pixel Networks (Pixel CNN (van den Oord et al., 2016) and Pixel RNN (Oord et al., 2016)). However, given a part of the image seems to simplify the generation task. Another contribution to generative models is Generative Adversarial Network (Goodfellow et al., 2014), however, GAN is still too young to be discussed in this paper. 32 On the Origin of Deep Learning 5. Convolutional Neural Networks and Vision Problems In this section, we will start to discuss a different family of models: the Convolutional Neural Network (CNN) family. Distinct from the family in the previous section, Convolutional Neural Network family mainly evolves from the knowledge of human visual cortex. Therefore, in this section, we will first introduce one of the most important reasons that account for the success of convolutional neural networks in vision problems: its bionic design to replicate human vision system. The nowadays convolutional neural networks probably originate more from the such a design rather than from the early-stage ancestors. With these background set-up, we will then briefly introduce the successful models that make themselves famous through the ImageNet Challenge (Deng et al., 2009). At last, we will present some known problems of the vision task that may guide the future research directions in vision tasks. 5.1 Visual Cortex Convolutional Neural Network is widely known as being inspired by visual cortex, however, except that some publications discuss this inspiration briefly (Poggio and Serre, 2013; Cox and Dean, 2014), few resources present this inspiration thoroughly. In this section, we focus on the discussion about basics on visual cortex (Hubel and Wiesel, 1959), which lays the ground for further study in Convolutional Neural Networks. The visual cortex of the brain, located in the occipital lobe which is located at the back of the skull, is a part of the cerebral cortex that plays an important role in processing visual information. Visual information coming from the eye, goes through a series of brain structures and reaches the visual cortex. The parts of the visual cortex that receive the sensory inputs is known as the primary visual cortex, also known as area V1. Visual information is further managed by extrastriate areas, including visual areas two (V2) and four (V4). There are also other visual areas (V3, V5, and V6), but in this paper, we primarily focus on the visual areas that are related to object recognition, which is known as ventral stream and consists of areas V1, V2, V4 and inferior temporal gyrus, which is one of the higher levels of the ventral stream of visual processing, associated with the representation of complex object features, such as global shape, like face perception (Haxby et al., 2000). Figure 14 is an illustration of the ventral stream of the visual cortex. It shows the information process procedure from the retina which receives the image information and passes all the way to inferior temporal gyrus. For each component: • Retina converts the light energy that comes from the rays bouncing off of an object into chemical energy. This chemical energy is then converted into action potentials that are transferred onto primary visual cortex. (In fact, there are several other brain structures involved between retina and V1, but we omit these structures for simplicity9 .) 9. We deliberately discuss the components that have connections with established technologies in convolutional neural network, one who is interested in developing more powerful models is encouraged to investigate other components. 33 Wang and Raj Figure 14: A brief illustration of ventral stream of the visual cortex in human vision system. It consists of primary visual cortex (V1), visual areas (V2 and V4) and inferior temporal gyrus. • Primary visual cortex (V1) mainly fulfills the task of edge detection, where an edge is an area with strongest local contrast in the visual signals. • V2, also known as secondary visual cortex, is the first region within the visual association area. It receives strong feedforward connections from V1 and sends strong connections to later areas. In V2, cells are tuned to extract mainly simple properties of the visual signals such as orientation, spatial frequency, and colour, and a few more complex properties. • V4 fulfills the functions including detecting object features of intermediate complexity, like simple geometric shapes, in addition to orientation, spatial frequency, and color. V4 is also shown with strong attentional modulation (Moran and Desimone, 1985). V4 also receives direct input from V1. • Inferior temporal gyrus (TI) is responsible for identifying the object based on the color and form of the object and comparing that processed information to stored memories of objects to identify that object (Kolb et al., 2014). In other words, IT performs the semantic level tasks, like face recognition. Many of the descriptions of functions about visual cortex should revive a recollection of convolutional neural networks for the readers that have been exposed to some relevant technical literature. Later in this section, we will discuss more details about convolutional neural networks, which will help build explicit connections. Even for readers that barely 34 On the Origin of Deep Learning have knowledge in convolutional neural networks, this hierarchical structure of visual cortex should immediately ring a bell about neural networks. Besides convolutional neural networks, visual cortex has been inspiring the works in computer vision for a long time. For example, Li (1998) built a neural model inspired by the primary visual cortex (V1). In another granularity, Serre et al. (2005) introduced a system with feature detections inspired from the visual cortex. De Ladurantaye et al. (2012) published a book describing the models of information processing in the visual cortex. Poggio and Serre (2013) conducted a more comprehensive survey on the relevant topic, but they didn’t focus on any particular subject in detail in their survey. In this section, we discuss the connections between visual cortex and convolutional neural networks in details. We will begin with Neocogitron, which borrows some ideas from visual cortex and later inspires convolutional neural network. 5.2 Neocogitron and Visual Cortex Neocogitron, proposed by Fukushima (1980), is generally seen as the model that inspires Convolutional Neural Networks on the computation side. It is a neural network that consists of two different kinds of layers (S-layer as feature extractor and C-layer as structured connections to organize the extracted features.) S-layer consists of a number of S-cells that are inspired by the cell in primary visual cortex. It serves as a feature extractor. Each S-cell can be ideally trained to be responsive to a particular feature presented in its receptive field. Generally, local features such as edges in particular orientations are extracted in lower layers while global features are extracted in higher layers. This structure highly resembles how human conceive objects. C-layer resembles complex cell in the higher pathway of visual cortex. It is mainly introduced for shift invariant property of features extracted by S-layer. 5.2.1 Parameter Learning During parameter learning process, only the parameters of S-layer are updated. Neocogitron can also be trained unsupervisedly, for a good feature extractor out of S-layers. The training process for S-layer is very similar to Hebbian Learning rule, which strengthens the connections between S-layer and C-layer for whichever S-cell shows the strongest response. This training mechanism also introduces the problem Hebbian Learning rule introduces, that the strength of connections will saturate (since it keeps increasing). The solution was also introduced by Fukushima (1980), which was introduced with the name “inhibitory cell”. It performed the function as a normalization to avoid the problem. 5.3 Convolutional Neural Network and Visual Cortex Now we proceed from Neocogitron to Convolutional Neural Network. First, we will introduce the building components: convolutional layer and subsampling layer. Then we assemble these components to present Convolutional Neural Network, using LeNet as an example. 35 Wang and Raj Figure 15: A simple illustration of two dimension convolution operation. 5.3.1 Convolution Operation Convolution operation is strictly just a mathematical operation, which should be treated equally with other operations like addition or multiplication and should not be discussed particularly in a machine learning literature. However, we still discuss it here for completeness and for the readers who may not be familiar with it. Convolution is a mathematical operation on two functions (e.g. f and g) and produces a third function h, which is an integral that expresses the amount of overlap of one function (f ) as it is shifted over the other function (g). It is described formally as the following: Z ∞ f (τ )g(t − τ )dτ h(t) = −∞ and denoted as h = f ? g. Convolutional neural network typically works with two-dimensional convolution operation, which could be summarized in Figure 15. As showed in Figure 15, the leftmost matrix is the input matrix. The middle one is usually called a kernel matrix. Convolution is applied to these matrices and the result is showed as the rightmost matrix. The convolution process is an element-wise product followed by a sum, as showed in the example. When the left upper 3×3 matrix is convoluted with the kernel, the result is 29. Then we slide the target 3 × 3 matrix one column right, convoluted with the kernel and get the result 12. We keep sliding and record the results as a matrix. Because the kernel is 3 × 3, every target matrix is 3 × 3, thus, every 3 × 3 matrix is convoluted to one digit and the whole 5 × 5 matrix is shrunk into 3 × 3 matrix. (Because 5 − (3 − 1) = 3. The first 3 means the size of the kernel matrix. ) One should realize that convolution is locally shift invariant, which means that for many different combinations of how the nine numbers in the upper 3 × 3 matrix are placed, the convoluted result will be 29. This invariant property plays a critical role in vision problem because that in an ideal case, the recognition result should not be changed due to shift or rotation of features. This critical property is used to be solved elegantly by Lowe (1999); Bay et al. (2006), but convolutional neural network brought the performance up to a new level. 5.3.2 Connection between CNN and Visual Cortex With the ideas about two dimension convolution, we further discuss how convolution is a useful operation that can simulate the tasks performed by visual cortex. 36 On the Origin of Deep Learning (a) Identity kernel (b) Edge detection kernel (c) Blur kernel (d) Sharpen kernel (e) Lighten kernel (f) Darken kernel (g) Random kernel 1 (h) Random kernel 2 Figure 16: Convolutional kernels example. Different kernels applied to the same image will result in differently processed images. Note that there is a 91 divisor applied to these kernels. The convolution operation is usually known as kernels. By different choices of kernels, different operations of the images could be achieved. Operations are typically including identity, edge detection, blur, sharpening etc. By introducing random matrices as convolution operator, some interesting properties might be discovered. Figure 16 is an illustration of some example kernels that are applied to the same figure. One can see that different kernels can be applied to fulfill different tasks. Random kernels can also be applied to transform the image into some interesting outcomes. Figure 16 (b) shows that edge detection, which is one of the central tasks of primary visual cortex, can be fulfilled by a clever choice of kernels. Furthermore, clever selection of kernels can lead us to a success replication of visual cortex. As a result, learning a meaningful convolutional kernel (i.e. parameter learning) is one of the central tasks in convolutional neural networks when applied to vision tasks. This also explains that why 37 Wang and Raj Figure 17: An illustration of LeNet, where Conv stands for convolutional layer and Sampling stands for SubSampling Layer. many well-trained popular models can usually perform well in other tasks with only limited fine-tuning process: the kernels have been well trained and can be universally applicable. With the understanding of the essential role convolution operation plays in vision tasks, we proceed to investigate some major milestones along the way. 5.4 The Pioneer of Convolutional Neural Networks: LeNet This section is devoted to a model that is widely recognized as the first convolutional neural network: LeNet, invented by Le Cun et al. (1990) (further made popular with (LeCun et al., 1998a)). It is inspired from the Neocogitron. In this section, we will introduce convolutional neural network via introducing LeNet. Figure 17 shows an illustration of the architecture of LeNet. It consists of two pairs of Convolutional Layer and Subsampling Layer and is further connected with fully connected layer and an RBF layer for classification. 5.4.1 Convolutional Layer A convolutional layer is primarily a layer that performs convolution operation. As we have discussed previously, a clever selection of convolution kernel can effectively simulate the task of visual cortex. Convolutional layer introduces another operation after convolution to assist the simulation to be more successful: the non-linearity transform. Considering a ReLU (Nair and Hinton, 2010) non-linearity transform, which is defined as following: f (x) = max(0, x) which is a transform that removes the negative part of the input, resulting in a clearer contrast of meaningful features as opposed to other side product the kernel produces. Therefore, this non-linearity grants the convolution more power in extracting useful features and allows it to simulate the functions of visual cortex more closely. 38 On the Origin of Deep Learning 5.4.2 Subsampling Layer Subsampling Layer performs a simpler task. It only samples one input out every region it looks into. Some different strategies of sampling can be considered, like max-pooling (taking the maximum value of the input), average-pooling (taking the averaged value of input) or even probabilistic pooling (taking a random one.) (Lee et al., 2009). Sampling turns the input representations into smaller and more manageable embeddings. More importantly, sampling makes the network invariant to small transformations, distortions, and translations in the input image. A small distortion in the input will not change the outcome of pooling since we take the maximum/average value in a local neighborhood. 5.4.3 LeNet With the two most important components introduced, we can stack them together to assemble a convolutional neural network. Following the recipe of Figure 17, we will end up with the famous LeNet. LeNet is known as its ability to classify digits and can handle a variety of different problems of digits including variances in position and scale, rotation and squeezing of digits, and even different stroke width of the digit. Meanwhile, with the introduction of LeNet, LeCun et al. (1998b) also introduces the MNIST database, which later becomes the standard benchmark in digit recognition field. 5.5 Milestones in ImageNet Challenge With the success of LeNet, Convolutional Neural Network has been shown with great potential in solving vision tasks. These potentials have attracted a large number of researchers aiming to solve vision task regarding object recognition in CIFAR classification (Krizhevsky and Hinton, 2009) and ImageNet challenge (Russakovsky et al., 2015). Along with this path, several superstar milestones have attracted great attentions and has been applied to other fields with good performance. In this section, we will briefly discuss these models. 5.5.1 AlexNet While LeNet is the one that starts the era of convolutional neural networks, AlexNet, invented by Krizhevsky et al. (2012), is the one that starts the era of CNN used for ImageNet classification. AlexNet is the first evidence that CNN can perform well on this historically difficult ImageNet dataset and it performs so well that leads the society into a competition of developing CNNs. The success of AlexNet is not only due to this unique design of architecture but also due to the clever mechanism of training. To avoid the computationally expensive training process, AlexNet has been split into two streams and trained on two GPUs. It also used data augmentation techniques that consist of image translations, horizontal reflections, and patch extractions. The recipe of AlexNet is shown in Figure 18. However, rarely any lessons can be learned from the architecture of AlexNet despite its remarkable performance. Even more unfortunately, the fact that this particular architecture of AlexNet does not have a wellgrounded theoretical support pushes many researchers to blindly burn computing resources 39 Wang and Raj Figure 18: An illustration of AlexNet to test for a new architecture. Many models have been introduced during this period, but only a few may be worth mentioning in the future. 5.5.2 VGG In the blind competition of exploring different architectures, Simonyan and Zisserman (2014) showed that simplicity is a promising direction with a model named VGG. Although VGG is deeper (19 layer) than other models around that time, the architecture is extremely simplified: all the layers are 3 × 3 convolutional layer with a 2 × 2 pooling layer. This simple usage of convolutional layer simulates a larger filter while keeping the benefits of smaller filter sizes, because the combination of two 3×3 convolutional layers has an effective receptive field of a 5 × 5 convolutional layer, but with fewer parameters. The spatial size of the input volumes at each layer will decrease as a result of the convolutional and pooling layers, but the depth of the volumes increases because of the increased number of filters (in VGG, the number of filters doubles after each pooling layer). This behavior reinforces the idea of VGG to shrink spatial dimensions, but grow depth. VGG is not the winner of the ImageNet competition of that year (The winner is GoogLeNet invented by Szegedy et al. (2015)). GoogLeNet introduced several important concepts like Inception module and the concept later used by R-CNN (Girshick et al., 2014; Girshick, 2015; Ren et al., 2015), but the arbitrary/creative design of architecture barely contribute more than what VGG does to the society, especially considering that Residual Net, following the path of VGG, won the ImageNet challenge in an unprecedented level. 5.5.3 Residual Net Residual Net (ResNet) is a 152 layer network, which was ten times deeper than what was usually seen during the time when it was invented by He et al. (2015). Following the path VGG introduces, ResNet explores deeper structure with simple layer. However, naively 40 On the Origin of Deep Learning Figure 19: An illustration of Residual Block of ResNet increasing the number of layers will only result in worse results, for both training cases and testing cases (He et al., 2015). The breakthrough ResNet introduces, which allows ResNet to be substantially deeper than previous networks, is called Residual Block. The idea behind a Residual Block is that some input of a certain layer (denoted as x) can be passed to the component two layers later either following the traditional path which involves convolutional layers and ReLU transform succession (we denote the result as f (x)), or going through an express way that directly passes x there. As a result, the input to the component two layers later is f (x) + x instead of what is typically seen as f (x). The idea of Residual Block is illustrated in Figure 19. In a complementary work, He et al. (2016) validated that residual blocks are essential for propagating information smoothly, therefore simplifies the optimization. They also extended the ResNet to a 1000-layer version with success on CIFAR data set. Another interesting perspective of ResNet is provided by (Veit et al., 2016). They showed that ResNet behave behaves like ensemble of shallow networks: the express way introduced allows ResNet to perform as a collection of independent networks, each network is significantly shallower than the integrated ResNet itself. This also explains why gradient can be passed through the ultra-deep architecture without being vanished. (We will talk more about vanishing gradient problem when we discuss recurrent neural network in the next section.) Another work, which is not directly relevant to ResNet, but may help to understand it, is conducted by Hariharan et al. (2015). They showed that features from lower layers are informative in addition to what can be summarized from the final layer. ResNet is still not completely vacant from clever designs. The number of layers in the whole network and the number of layers that Residual Block allows identity to bypass are still choices that require experimental validations. Nonetheless, to some extent, ResNet has shown that critical reasoning can help the development of CNN better than blind 41 Wang and Raj experimental trails. In addition, the idea of Residual Block has been found in the actual visual cortex (In the ventral stream of the visual cortex, V4 can directly accept signals from primary visual cortex), although ResNet is not designed according to this in the first place. With the introduction of these state-of-the-art neural models that are successful in these challenges, Canziani et al. (2016) conducted a comprehensive experimental study comparing these models. Upon comparison, they showed that there is still room for improvement on fully connected layers that show strong inefficiencies for smaller batches of images. 5.6 Challenges and Chances for Fundamental Vision Problems ResNet is not the end of the story. New models and techniques appear every day to push the limit of CNNs further. For example, Zhang et al. (2016b) took a step further and put Residual Block inside Residual Block. Zagoruyko and Komodakis (2016) attempted to decrease the depth of network by increasing the width. However, incremental works of this kind are not in the scope of this paper. We would like to end the story of Convolutional Neural Networks with some of the current challenges of fundamental vision problems that may not able to be solved naively by investigation of machine learning techniques. 5.6.1 Network Property and Vision Blindness Spot Convolutional Neural Networks have reached to an unprecedented accuracy in object detection. However, it may still be far from industry reliable application due to some intriguing properties found by Szegedy et al. (2013). Szegedy et al. (2013) showed that they could force a deep learning model to misclassify an image simply by adding perturbations to that image. More importantly, these perturbations may not even be observed by naked human eyes. In other words, two objects that look almost the same to human, may be recognized as different objects by a well-trained neural network (for example, AlexNet). They have also shown that this property is more likely to be a modeling problem, in contrast to problems raised by insufficient training. On the other hand, Nguyen et al. (2015) showed that they could generate patterns that convey almost no information to human, being recognized as some objects by neural networks with high confidence (sometimes more than 99%). Since neural networks are typically forced to make a prediction, it is not surprising to see a network classify a meaningless patter into something, however, this high confidence may indicate that the fundamental differences between how neural networks and human learn to know this world. Figure 20 shows some examples from the aforementioned two works. With construction, we can show that the neural networks may misclassify an object, which should be easily recognized by the human, to something unusual. On the other hand, a neural network may also classify some weird patterns, which are not believed to be objects by the human, to something we are familiar with. Both of these properties may restrict the usage of deep learning to real world applications when a reliable prediction is necessary. Even without these examples, one may also realize that the reliable prediction of neural networks could be an issue due to the fundamental property of a matrix: the existence of null space. As long as the perturbation happens within the null space of a matrix, one may be able to alter an image dramatically while the neural network still makes the 42 On the Origin of Deep Learning (a) (b) (c) (d) (e) (f) (g) (h) Figure 20: Illustrations of some mistakes of neural networks. (a)-(d) (from (Szegedy et al., 2013)) are adversarial images that are generated based on original images. The differences between these and the original ones are un-observable by naked eye, but the neural network can successfully classify original ones but fail adversarial ones. (e)-(h) (from (Nguyen et al., 2015)) are patterns that are generated. A neural network classify them into (e) school bus, (f) guitar, (g) peacock and (h) Pekinese respectively. misclassification with high confidence. Null space works like a blind spot to a matrix and changes within null space are never sensible to the corresponding matrix. This blind spot should not discourage the promising future of neural networks. On the contrary, it makes the convolutional neural network resemble the human vision system in a deeper level. In the human vision system, blind spots (Gregory and Cavanagh, 2011) also exist (Wandell, 1995). Interesting work might be seen about linking the flaws of human vision system to the defects of neural networks and helping to overcome these defects in the future. 5.6.2 Human Labeling Preference At the very last, we present some of the misclassified images of ResNet on ImageNet Challenge. Hopefully, some of these examples could inspire some new methodologies invented for the fundamental vision problem. Figure 21 shows some misclassified images of ResNet when applied to ImageNet Challenge. These labels, provided by human effort, are very unexpected even to many other humans. Therefore, the 3.6% error rate of ResNet (a general human usually predicts with error rate 5%-10%) is probably hitting the limit since the labeling preference of an annotator is harder to predict than the actual labels. For example, Figure 21 (a),(b),(h) are labeled as a tiny part of the image, while there are more important contents expressed by the image. On the other hand, Figure 21 (d) (e) are annotated as the background of the image while that image is obviously centering on other object. 43 Wang and Raj (a) flute (b) guinea pig (c) wig (d) seashore (e) alp (f) screwdriver (g) comic book (h) sunglass Figure 21: Some failed images of ImageNet classification by ResNet and the primary label associated with the image. To further improve the performance ResNet reached, one direction might be to modeling the annotators’ labeling preference. One assumption could be that annotators prefer to label an image to make it distinguishable. Some established work to modeling human factors (Wilson et al., 2015) could be helpful. However, the more important question is that whether it is worth optimizing the model to increase the testing results on ImageNet dataset, since remaining misclassifications may not be a result of the incompetency of the model, but problems of annotations. The introduction of other data sets, like COCO (Lin et al., 2014), Flickr (Plummer et al., 2015), and VisualGenome (Krishna et al., 2016) may open a new era of vision problems with more competitive challenges. However, the fundamental problems and experiences that this section introduces should never be forgotten. 44 On the Origin of Deep Learning 6. Time Series Data and Recurrent Networks In this section, we will start to discuss a new family of deep learning models that have attracted many attentions, especially for the tasks on time series data, or sequential data. The Recurrent Neural Network (RNN) is a class of neural network whose connections of units form a directed cycle; this nature grants its ability to work with temporal data. It has also been discussed in literature like (Grossberg, 2013) and (Lipton et al., 2015). In this paper, we will continue to offer complementary views to other surveys with an emphasis on the evolutionary history of the milestone models and aim to provide insights into the future direction of coming models. 6.1 Recurrent Neural Network: Jordan Network and Elman Network As we have discussed previously, Hopfield Network is widely recognized as a recurrent neural network, although its formalization is distinctly different from how recurrent neural network is defined nowadays. Therefore, despite that other literature tend to begin the discussion of RNN with Hopfield Network, we will not treat it as a member of RNN family to avoid unnecessary confusion. The modern definition of “recurrent” is initially introduced by Jordan (1986) as: If a network has one or more cycles, that is, if it is possible to follow a path from a unit back to itself, then the network is referred to as recurrent. A nonrecurrent network has no cycles. His model in (Jordan, 1986) is later referred to as Jordan Network. For a simple neural network with one hidden layer, with input denoted as X, weights of hidden layer denoted as wh and weights of output layer denoted as wy , weights of recurrent computation denoted as wr , hidden representation denoted as h and output denoted as y, Jordan Network can be formulated as ht = σ(Wh X + Wr y t−1 ) y = σ(Wy ht ) A few years later, another RNN was introduced by Elman (1990), when he formalized the recurrent structure slightly differently. Later, his network is known as Elman Network. Elman network is formalized as following: ht = σ(Wh X + Wr ht−1 ) y = σ(Wy ht ) The only difference is that whether the information of previous time step is provided by previous output or previous hidden layer. This difference is further illustrated in Figure 22. The difference is illustrated to respect the historical contribution of these works. One may notice that there is no fundamental difference between these two structures since yt = Wy ht , therefore, the only difference lies in the choice of Wr . (Originally, Elman only introduces his network with Wr = I, but more general cases could be derived from there.) Nevertheless, the step from Jordan Network to Elman Network is still remarkable as it introduces the possibility of passing information from hidden layers, which significantly improve the flexibility of structure design in later work. 45 Wang and Raj (a) Structure of Jordan Network (b) Structure of Elman Network Figure 22: The difference of recurrent structure from Jordan Network and Elman Network. 6.1.1 Backpropagation through Time The recurrent structure makes traditional backpropagation infeasible because of that with the recurrent structure, there is not an end point where the backpropagation can stop. Intuitively, one solution is to unfold the recurrent structure and expand it as a feedforward neural network with certain time steps and then apply traditional backpropagation onto this unfolded neural network. This solution is known as Backpropagation through Time (BPTT), independently invented by several researchers including (Robinson and Fallside, 1987; Werbos, 1988; Mozer, 1989) However, as recurrent neural network usually has a more complex cost surface, naive backpropagation may not work well. Later in this paper, we will see that the recurrent structure introduces some critical problems, for example, the vanishing gradient problem, which makes optimization for RNN a great challenge in the society. 6.2 Bidirectional Recurrent Neural Network If we unfold an RNN, then we can get the structure of a feedforward neural network with infinite depth. Therefore, we can build a conceptual connection between RNN and feedforward network with infinite layers. Then since through the neural network history, bidirectional neural networks have been playing important roles (like Hopfield Network, RBM, DBM), a follow-up question is that what recurrent structures that correspond to the infinite layer of bidirectional models are. The answer is Bidirectional Recurrent Neural Network. 46 On the Origin of Deep Learning Figure 23: The unfolded structured of BRNN. The temporal order is from left to right. Hidden layer 1 is unfolded in the standard way of an RNN. Hidden layer 2 is unfolded to simulate the reverse connection. Bidirectional Recurrent Neural Network (BRNN) was invented by Schuster and Paliwal (1997) with the goal to introduce a structure that was unfolded to be a bidirectional neural network. Therefore, when it is applied to time series data, not only the information can be passed following the natural temporal sequences, but the further information can also reversely provide knowledge to previous time steps. Figure 23 shows the unfolded structure of a BRNN. Hidden layer 1 is unfolded in the standard way of an RNN. Hidden layer 2 is unfolded to simulate the reverse connection. Transparency (in Figure 23) is applied to emphasize that unfolding an RNN is only a concept that is used for illustration purpose. The actual model handles data from different time steps with the same single model. BRNN is formulated as following: ht1 = σ(Wh1 X + Wr1 ht−1 1 ) ht2 = σ(Wh2 X + Wr2 ht+1 2 ) y = σ(Wy1 ht1 + Wy2 ht2 ) where the subscript 1 and 2 denote the variables associated with hidden layer 1 and 2 respectively. With the introduction of “recurrent” connections back from the future, Backpropagation through Time is no longer directly feasible. The solution is to treat this model as a combination of two RNNs: a standard one and a reverse one, then apply BPTT onto each of them. Weights are updated simultaneously once two gradients are computed. 6.3 Long Short-Term Memory Another breakthrough in RNN family was introduced in the same year as BRNN. Hochreiter and Schmidhuber (1997) introduced a new neuron for RNN family, named Long Short-Term Memory (LSTM). When it was invented, the term “LSTM” is used to refer the algorithm 47 Wang and Raj that is designed to overcome vanishing gradient problem, with the help of a special designed memory cell. Nowadays, “LSTM” is widely used to denote any recurrent network that with that memory cell, which is nowadays referred as an LSTM cell. LSTM was introduced to overcome the problem that RNNs cannot long term dependencies (Bengio et al., 1994). To overcome this issue, it requires the specially designed memory cell, as illustrated in Figure 24 (a). LSTM consists of several critical components. • states: values that are used to offer the information for output. ? input data: it is denoted as x. ? hidden state: values of previous hidden layer. This is the same as traditional RNN. It is denoted as h. ? input state: values that are (linear) combination of hidden state and input of current time step. It is denoted as i, and we have: it = σ(Wix xt + Wih ht−1 ) (9) ? internal state: Values that serve as “memory”. It is denoted as m • gates: values that are used to decide the information flow of states. ? input gate: it decides whether input state enters internal state. It is denoted as g, and we have: g t = σ(Wgi it ) (10) ? forget gate: it decides whether internal state forgets the previous internal state. It is denoted as f , and we have: f t = σ(Wf i it ) (11) ? output gate: it decides whether internal state passes its value to output and hidden state of next time step. It is denoted as o and we have: ot = σ(Woi it ) (12) Finally, considering how gates decide the information flow of states, we have the last two equations to complete the formulation of LSTM: mt =g t it + f t mt−1 ht =ot mt (13) (14) where denotes element-wise product. Figure 24 describes the details about how LSTM cell works. Figure 24 (b) shows that how the input state is constructed, as described in Equation 9. Figure 24 (c) shows how 48 On the Origin of Deep Learning (a) LSTM “memory” cell (b) Input data and previous hidden state form into input state (c) Calculating input gate and forget gate (d) Calculating output gate (e) Update internal state (f) Output and update hidden state Figure 24: The LSTM cell and its detailed functions. 49 Wang and Raj input gate and forget gate are computed, as described in Equation 10 and Equation 11. Figure 24 (d) shows how output gate is computed, as described in Equation 12. Figure 24 (e) shows how internal state is updated, as described in Equation 13. Figure 24 (f) shows how output and hidden state are updated, as described in Equation 14. All the weights are parameters that need to be learned during training. Therefore, theoretically, LSTM can learn to memorize long time dependency if necessary and can learn to forget the past when necessary, making itself a powerful model. With this important theoretical guarantee, many works have been attempted to improve LSTM. For example, Gers and Schmidhuber (2000) added a peephole connection that allows the gate to use information from the internal state. Cho et al. (2014) introduced the Gated Recurrent Unit, known as GRU, which simplified LSTM by merging internal state and hidden state into one state, and merging forget gate and input gate into a simple update gate. Integrating LSTM cell into bidirectional RNN is also an intuitive follow-up to look into (Graves et al., 2013). Interestingly, despite the novel LSTM variants proposed now and then, Greff et al. (2015) conducted a large-scale experiment investigating the performance of LSTMs and got the conclusion that none of the variants can improve upon the standard LSTM architecture significantly. Probably, the improvement of LSTM is in another direction rather than updating the structure inside a cell. Attention models seem to be a direction to go. 6.4 Attention Models Attention Models are loosely based on a bionic design to simulate the behavior of human vision attention mechanism: when humans look at an image, we do not scan it bit by bit or stare at the whole image, but we focus on some major part of it and gradually build the context after capturing the gist. Attention mechanisms were first discussed by Larochelle and Hinton (2010) and Denil et al. (2012). The attention models mostly refer to the models that were introduced in (Bahdanau et al., 2014) for machine translation and soon applied to many different domains like (Chorowski et al., 2015) for speech recognition and (Xu et al., 2015) for image caption generation. Attention models are mostly used for sequence output prediction. Instead of seeing the whole sequential data and make one single prediction (for example, language model), the model needs to make a sequential prediction for the sequential input for tasks like machine translation or image caption generation. Therefore, the attention model is mostly used to answer the question on where to pay attention to based on previously predicted labels or hidden states. The output sequence may not have to be linked one-to-one to the input sequence, and the input data may not even be a sequence. Therefore, usually, an encoder-decoder framework (Cho et al., 2015) is necessary. The encoder is used to encode the data into representations and decoder is used to make sequential predictions. Attention mechanism is used to locate a region of the representation for predicting the label in current time step. Figure 25 shows a basic attention model under encoder-decoder network structure. The representation encoder encodes is all accessible to attention model, and attention model only selects some regions to pass onto the LSTM cell for further usage of prediction making. 50 On the Origin of Deep Learning Figure 25: The unfolded structured of an attention model. Transparency is used to show that unfolding is only conceptual. The representation encoder learns are all available to the decoder across all time steps. Attention module only selects some to pass onto LSTM cell for prediction. Therefore, all the magic of attention models is about how this attention module in Figure 25 helps to localize the informative representations. To formalize how it works, we use r to denote the encoded representation (there is a total of M regions of representation), use h to denote hidden states of LSTM cell. Then, the attention module can generate the unscaled weights for ith region of the encoded representation as: βit = f (ht−1 , r, {αjt−1 }M j=1 ) where αjt−1 is the attention weights computed at the previous time step, and can be computed at current time step as a simple softmax function: exp(βit ) αit = PM t j exp(βj ) Therefore, we can further use the weights α to reweight the representation r for prediction. There are two ways for the representation to be reweighted: • Soft attention: The result is a simple weighted sum of the context vectors such that: t r = M X αjt cj j • Hard attention: The model is forced to make a hard decision by only localizing one region: sampling one region out following multinoulli distribution. 51 Wang and Raj (a) Deep input architecture (b) Deep recurrent architecture (c) Deep output architecture Figure 26: Three different formulations of deep recurrent neural network. One problem about hard attention is that sampling out of multinoulli distribution is not differentiable. Therefore, the gradient based method can be hardly applied. Variational methods (Ba et al., 2014) or policy gradient based method (Sutton et al., 1999) can be considered. 6.5 Deep RNNs and the future of RNNs In this very last section of the evolutionary path of RNN family, we will visit some ideas that have not been fully explored. 6.5.1 Deep Recurrent Neural Network Although recurrent neural network suffers many issues that deep neural network has because of the recurrent connections, current RNNs are still not deep models regarding representation learning compared to models in other families. Pascanu et al. (2013a) formalizes the idea of constructing deep RNNs by extending current RNNs. Figure 26 shows three different directions to construct a deep recurrent neural network by increasing the layers of the input component (Figure 26 (a)), recurrent component (Figure 26 (b)) and output component (Figure 26 (c)) respectively. 52 On the Origin of Deep Learning 6.5.2 The Future of RNNs RNNs have been improved in a variety of different ways, like assembling the pieces together with Conditional Random Field (Yang et al., 2016), and together with CNN components (Ma and Hovy, 2016). In addition, convolutional operation can be directly built into LSTM, resulting ConvLSTM (Xingjian et al., 2015), and then this ConvLSTM can be also connected with a variety of different components (De Brabandere et al., 2016; Kalchbrenner et al., 2016). One of the most fundamental problems of training RNNs is the vanishing/exploding gradient problem, introduced in detail in (Bengio et al., 1994). The problem basically states that for traditional activation functions, the gradient is bounded. When gradients are computed by backpropagation following chain rule, the error signal decreases exponentially within the time steps the BPTT can trace back, so the long-term dependency is lost. LSTM and ReLU are known to be good solutions for vanishing/exploding gradient problem. However, these solutions introduce ways to bypass this problem with clever design, instead of solving it fundamentally. While these methods work well practically, the fundamental problem for a general RNN is still to be solved. Pascanu et al. (2013b) attempted some solutions, but there are still more to be done. 53 Wang and Raj 7. Optimization of Neural Networks The primary focus of this paper is deep learning models. However, optimization is an inevitable topic in the development history of deep learning models. In this section, we will briefly revisit the major topics of optimization of neural networks. During our introduction of the models, some algorithms have been discussed along with the models. Here, we will only discuss the remaining methods that have not been mentioned previously. 7.1 Gradient Methods Despite the fact that neural networks have been developed for over fifty years, the optimization of neural networks still heavily rely on gradient descent methods within the algorithm of backpropagation. This paper does not intend to introduce the classical backpropagation, gradient descent method and its stochastic version and batch version and simple techniques like momentum method, but starts right after these topics. Therefore, the discussion of following gradient methods starts from the vanilla gradient descent as following: θt+1 = θt − η5tθ where 5θ is the gradient of the parameter θ, η is a hyperparameter, usually known as learning rate. 7.1.1 Rprop Rprop was introduced by Riedmiller and Braun (1993). It is a unique method even studied back today as it does not fully utilize the information of gradient, but only considers the sign of it. In other words, it updates the parameters following: θt+1 = θt − ηI(5tθ > 0) + ηI(5tθ < 0) where I(·) stands for an indicator function. This unique formalization allows the gradient method to overcome some cost curvatures that may not be easily solved with today’s dominant methods. This two-decade-old method may be worth some further study these days. 7.1.2 AdaGrad AdaGrad was introduced by Duchi et al. (2011). It follows the idea of introducing an adaptive learning rate mechanism that assigns higher learning rate to the parameters that have been updated more mildly and assigns lower learning rate to the parameters that have been updated dramatically. The measure of the degree of the update applied is the `2 norm of historical gradients, S t = \|\|51θ , 52θ , ... 5tθ \|\|22 , therefore we have the update rule as following: θt+1 = θt − St where is small term to avoid η divided by zero. 54 η 5t + θ On the Origin of Deep Learning AdaGrad has been showed with great improvement of robustness upon traditional gradient method (Dean et al., 2012). However, the problem is that as `2 norm accumulates, the fraction of η over `2 norm decays to a substantial small term. 7.1.3 AdaDelta AdaDelta is an extension of AdaGrad that aims to reducing the decaying rate of learning rate, proposed in (Zeiler, 2012). Instead of accumulating the gradients of each time step as in AdaGrad, AdaDelta re-weights previously accumulation before adding current term onto previously accumulated result, resulting in: (S t )2 = β(S t−1 )2 + (1 − β)(5tθ )2 where β is the weight for re-weighting. Then the update rule is the same as AdaGrad: θt+1 = θt − St η 5t + θ which is almost the same as another famous gradient variant named RMSprop10 . 7.1.4 Adam Adam stands for Adaptive Moment Estimation, proposed in (Kingma and Ba, 2014). Adam is like a combination momentum method and AdaGrad method, but each component are re-weighted at time step t. Formally, at time step t, we have: ∆tθ =α∆t−1 + (1 − α)5tθ θ (S t )2 =β(S t−1 )2 + (1 − β)(5tθ )2 η θt+1 =θt − t ∆t S + θ All these modern gradient variants have been published with a promising claim that is helpful to improve the convergence rate of previous methods. Empirically, these methods seem to be indeed helpful, however, in many cases, a good choice of these methods seems only to benefit to a limited extent. 7.2 Dropout Dropout was introduced in (Hinton et al., 2012; Srivastava et al., 2014). The technique soon got influential, not only because of its good performance but also because of its simplicity of implementation. The idea is very simple: randomly dropping out some of the units while training. More formally: on each training case, each hidden unit is randomly omitted from the network with a probability of p. As suggested by Hinton et al. (2012), Dropout can be seen as an efficient way to perform model averaging across a large number of different neural networks, where overfitting can be avoided with much less cost of computation. 10. It seems this method never gets published, the resources all trace back to Hinton’s slides at http://www.cs.toronto.edu/t̃ijmen/csc321/slides/lecture slides lec6.pdf 55 Wang and Raj Because of the actual performance it introduces, Dropout soon became very popular upon its introduction, a lot of work has attempted to understand its mechanism in different perspectives, including (Baldi and Sadowski, 2013; Cho, 2013; Ma et al., 2016). It has also been applied to train other models, like SVM (Chen et al., 2014). 7.3 Batch Normalization and Layer Normalization Batch Normalization, introduced by Ioffe and Szegedy (2015), is another breakthrough of optimization of deep neural networks. They addressed the problem they named as internal covariate shift. Intuitively, the problem can be understood as the following two steps: 1) a learned function is barely useful if its input changes (In statistics, the input of a function is sometimes denoted as covariates). 2) each layer is a function and the changes of parameters of below layers change the input of current layer. This change could be dramatic as it may shift the distribution of inputs. Ioffe and Szegedy (2015) proposed the Batch Normalization to solve this issue, formally following the steps: n 1X µB = xi n i=1 n 1X 2 σB = (xi − µB )2 n i=1 x i − µB x̂i = σB + yi =σL x̂i + µL where µB and σB denote the mean and variance of that batch. µL and σL two parameters learned by the algorithm to rescale and shift the output. xi and yi are inputs and outputs of that function respectively. These steps are performed for every batch during training. Batch Normalization turned out to work very well in training empirically and soon became popularly. As a follow-up, Ba et al. (2016) proposes the technique Layer Normalization, where they “transpose” batch normalization into layer normalization by computing the mean and variance used for normalization from all of the summed inputs to the neurons in a layer on a single training case. Therefore, this technique has a nature advantage of being applicable to recurrent neural network straightforwardly. However, it seems that this “transposed batch normalization” cannot be implemented as simple as Batch Normalization. Therefore, it has not become as influential as Batch Normalization is. 7.4 Optimization for “Optimal” Model Architecture In the very last section of optimization techniques for neural networks, we revisit some old methods that have been attempted with the aim to learn the “optimal” model architecture. Many of these methods are known as constructive network approaches. Most of these methods have been proposed decades ago and did not raise enough impact back then. Nowadays, with more powerful computation resources, people start to consider these methods again. 56 On the Origin of Deep Learning Two remarks need to be made before we proceed: 1) Obviously, most of these methods can trace back to counterparts in non-parametric machine learning field, but because most of these methods did not perform enough to raise an impact, focusing a discussion on the evolutionary path may mislead readers. Instead, we will only list these methods for the readers who seek for inspiration. 2) Many of these methods are not exclusively optimization techniques because these methods are usually proposed with a particularly designed architecture. Technically speaking, these methods should be distributed to previous sections according to the models associated. However, because these methods can barely inspire modern modeling research, but may have a chance to inspire modern optimization research, we list these methods in this section. 7.4.1 Cascade-Correlation Learning One of the earliest and most important works on this topic was proposed by Fahlman and Lebiere (1989). They introduced a model, as well as its corresponding algorithm named Cascade-Correlation Learning. The idea is that the algorithm starts with a minimum network and builds up towards a bigger network. Whenever another hidden unit is added, the parameters of previous hidden units are fixed, and the algorithm only searches for an optimal parameter for the newly-added hidden unit. Interestingly, the unique architecture of Cascade-Correlation Learning grants the network to grow deeper and wider at the same time because every newly added hidden unit takes the data together with outputs of previously added units as input. Two important questions of this algorithm are 1) when to fix the parameters of current hidden units and proceed to add and tune a newly added one 2) when to terminate the entire algorithm. These two questions are answered in a similar manner: the algorithm adds a new hidden unit when there are no significant changes in existing architecture and terminates when the overall performance is satisfying. This training process may introduce problems of overfitting, which might account for the fact that this method is seen much in modern deep learning research. 7.4.2 Tiling Algorithm Mézard and Nadal (1989) presented the idea of Tiling Algorithm, which learns the parameters, the number of layers, as well as the number of hidden units in each layer simultaneously for feedforward neural network on Boolean functions. Later this algorithm was extended to multiple class version by Parekh et al. (1997). The algorithm works in such a way that on every layer, it tries to build a layer of hidden units that can cluster the data into different clusters where there is only one label in one cluster. The algorithm keeps increasing the number of hidden units until such a clustering pattern can be achieved and proceed to add another layer. Mézard and Nadal (1989) also offered a proof of theoretical guarantees for Tiling Algorithm. Basically, the theorem says that Tiling Algorithm can greedily improve the performance of a neural network. 57 Wang and Raj 7.4.3 Upstart Algorithm Frean (1990) proposed the Upstart Algorithm. Long story short, this algorithm is simply a neural network version of the standard decision tree (Safavian and Landgrebe, 1990) where each tree node is replaced with a linear perceptron. Therefore, the tree is seen as a neural network because it uses the core component of neural networks as a tree node. As a result, standard way of building a tree is advertised as building a neural network automatically. Similarly, Bengio et al. (2005) proposed a boosting algorithm where they replace the weak classifier as neurons. 7.4.4 Evolutionary Algorithm Evolutionary Algorithm is a family of algorithms uses mechanisms inspired by biological evolution to search in a parameter space for the optimal solution. Some prominent examples in this family are genetic algorithm (Mitchell, 1998), which simulates natural selection and ant colony optimization algorithm (Colorni et al., 1991), which simulates the cooperation of an ant colony to explore surroundings. Yao (1999) offered an extensive survey of the usage of evolution algorithm upon the optimization of neural networks, in which Yao introduced several encoding schemes that can enable the neural network architecture to be learned with evolutionary algorithms. The encoding schemes basically transfer the network architecture into vectors, so that a standard algorithm can take it as input and optimize it. So far, we discussed some representative algorithms that are aimed to learn the network architecture automatically. Most of these algorithms eventually fade out of modern deep learning research, we conjecture two main reasons for this outcome: 1) Most of these algorithms tend to overfit the data. 2) Most of these algorithms are following a greedy search paradigm, which will be unlikely to find the optimal architecture. However, with the rapid development of machine learning methods and computation resources in the last decade, we hope these constructive network methods we listed here can still inspire the readers for substantial contributions to modern deep learning research. 58 On the Origin of Deep Learning 8. Conclusion In this paper, we have revisited the evolutionary path of the nowadays deep learning models. We revisited the paths for three major families of deep learning models: the deep generative model family, convolutional neural network family, and recurrent neural network family as well as some topics for optimization techniques. This paper could serve two goals: 1) First, it documents the major milestones in the science history that have impacted the current development of deep learning. These milestones are not limited to the development in computer science fields. 2) More importantly, by revisiting the evolutionary path of the major milestone, this paper should be able to suggest the readers that how these remarkable works are developed among thousands of other contemporaneous publications. Here we briefly summarize three directions that many of these milestones pursue: • Occam’s razor: While it seems that part of the society tends to favor more complex models by layering up one architecture onto another and hoping backpropagation can find the optimal parameters, history says that masterminds tend to think simple: Dropout is widely recognized not only because of its performance, but more because of its simplicity in implementation and intuitive (tentative) reasoning. From Hopfield Network to Restricted Boltzmann Machine, models are simplified along the iterations until when RBM is ready to be piled-up. • Be ambitious: If a model is proposed with substantially more parameters than contemporaneous ones, it must solve a problem that no others can solve nicely to be remarkable. LSTM is much more complex than traditional RNN, but it bypasses the vanishing gradient problem nicely. Deep Belief Network is famous not due to the fact the they are the first one to come up with the idea of putting one RBM onto another, but due to that they come up an algorithm that allow deep architectures to be trained effectively. • Widely read: Many models are inspired by domain knowledge outside of machine learning or statistics field. Human visual cortex has greatly inspired the development of convolutional neural networks. Even the recent popular Residual Networks can find corresponding mechanism in human visual cortex. Generative Adversarial Network can also find some connection with game theory, which was developed fifty years ago. We hope these directions can help some readers to impact more on current society. More directions should also be able to be summarized through our revisit of these milestones by readers. Acknowledgements Thanks to the demo from http://beej.us/blog/data/convolution-image-processing/ for a quick generation of examples in Figure 16. 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An evolutionary approach to the identification of Cellular Automata based on partial observations Witold Bołt†∗ , Jan M. Baetens∗ and Bernard De Baets∗ † Systems arXiv:1508.05752v1 [cs.NE] 24 Aug 2015 ∗ KERMIT, Research Institute, Polish Academy of Sciences, Warsaw, Poland Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Ghent, Belgium Abstract—In this paper we consider the identification problem of Cellular Automata (CAs). The problem is defined and solved in the context of partial observations with time gaps of unknown length, i.e. pre-recorded, partial configurations of the system at certain, unknown time steps. A solution method based on a modified variant of a Genetic Algorithm (GA) is proposed and illustrated with brief experimental results. I. I NTRODUCTION CAs present an attractive and effective modelling technique for a variety of problems. In order to use CAs in a practical modelling task, one needs to understand the underlying rules, relevant to the given phenomenon, and translate them into a CA local rule. Additionally, the state space, tessellation and neighborhood structure need to be pinned down beforehand. This narrows the application area for CAs, since there are problems for which it is hard to manually design a proper local rule. In some cases only the initial and final states of the system are known (e.g. [1]–[3]). Such problems motivate the research on automated CA identification. Various methods have been used, including genetic algorithms (GAs) [4]–[7], genetic programming [8]–[10], gene expression programming [11], ant intelligence [12], machine learning [13], as well as direct search/construction approaches [14]–[17]. Existing methods can be divided into two main groups. Firstly, methods for solving specific, global problems. An example of such a problem is majority classification in which one only knows the initial condition and the desired outcome. Secondly, methods that exploit the entire time series of configurations, where it is assumed that all configurations are known. Only limited research efforts have been devoted to problems involving identification based on partial information [4]. The main goal of the research presented in this paper is to develop methods capable of automated CA identification in case of partial information. The paper is organized as follows. In Section II we start with introducing basic definitions and presenting some well-known facts on CAs. Section III holds the formal definition of the CA identification problem, while in Section IV we reformulate this problem as an optimization task. In Section V the evolutionary algorithm for solving the identification problem is presented. The paper is concluded by Section VI which presents initial results of computational experiments. An introduction to the methods presented in this paper, and a simpler formulation of the discussed algorithm can be found in [18]. 978-1-4799-7492-4/15/$31.00 c 2015 IEEE II. P RELIMINARIES We start by defining a CA. In this paper we will concentrate on 1D, deterministic CAs with a symmetric neighborhood. Let r ∈ N and fA : {0, 1}2 r+1 → {0, 1} be any function, then for N > 0 we define the N –cell global CA rule AN : {0, 1}N → {0, 1}N as: AN (. . . , si , . . .) = (. . . , fA (si−r , . . . , si+r ), . . .), (1) using periodic boundary conditions, i.e. for any i ∈ Z it holds that si+N = si . The function fA used in this definition will be referred to as a local rule, and the integer r will be referred to as the radius of the neighborhood. Any local rule can be uniquely defined by a lookup table (LUT) that lists all of the possible arguments together with the corresponding function values. It is assumed that the arguments are listed in a lexicographic order. The general form of such a LUT in the case of radius r = 1 is shown in Table I. TABLE I. LUT OF LOCAL RULE R = (l8 , l7 , l6 , l5 , l4 , l3 , l2 , l1 )2 111 110 101 100 011 010 001 000 l8 l7 l6 l5 l4 l3 l2 l1 The LUT can be used to enumerate local rules, as the coefficients li can be treated as P8digits in the binary representation of an integer R, i.e. R = i=1 li 2i−1 . Clearly this extends to higher values of the radius. Due to the fact that the ordering of arguments in the LUT is fixed, only the second row needs to be stored, such that a LUT may be represented as a binary vector. The length of such a vector is 22 r+1 . With {0, 1}∗ we will denote the S set of all binary sequences of finite length, i.e. {0, 1}∗ = N >0 {0, 1}N . The function A : {0, 1}∗ → {0, 1}∗ , satisfying A(X) = AN (X) if and only if X ∈ {0, 1}N , where each of the global rules AN is defined with the same local rule fA , will be referred to as a generalized global rule of a CA. Such functions will be frequently used throughout this paper, therefore we will simply refer to them as global rules or rules. In this paper a CA will be identified in terms of its global rule, and by referring to a CA we therefore always refer to its global rule in this generalized sense. Note that rule A is uniquely defined by a given local rule fA , but the opposite is not true. For a given rule A we may find different local rules defining it. Fact 1 highlights the relationship between different local rules defining the same CA. Fact 1. Two local rules f : {0, 1}2 r+1 → {0, 1} and g : {0, 1}2 u+1 → {0, 1}, u ≤ r, define the same CA if and only if it holds: f (s1 , . . . , s2 r+1 ) = g(sr−u+1 , . . . , sr+u+1 ), 2 r+1 for any (s1 , . . . , s2 r+1 ) ∈ {0, 1} (2) . Example 1. Let g : {0, 1} → {0, 1} be defined by g(s) = s and f : {0, 1}3 → {0, 1} be defined by f (s1 , s2 , s3 ) = s2 s3 + s2 (1 − s3 ). We can see that for any s1 , s3 ∈ {0, 1} it holds that f (s1 , s2 , s3 ) = g(s2 ) = s2 , and thus f and g define the same CA rule, which happens to be the identity rule. For a given neighborhood radius r, Ar denotes the set of all CAs that can be expressed with the use of a local rule with a neighborhood of radius r. CAs belonging to A1 are referred to as Elementary CAs (ECAs), and form the most commonly studied class of 2–state CAs [19]. Fig. 1. Space-time diagram of ECA 150 Two important properties of the sets Ar are underlined in Fact 2. 2 r+1 Fact 2. For any r ≥ 0, Ar ⊂ Ar+1 and \|Ar \| = 22 . Let A be a CA, X ∈ {0, 1}M for some M and T > 0. The finite sequence of vectors given by: (X, A(X), A2 (X)), . . . , AT −1 (X)), where At denotes the t–th application of the rule A, will be referred to as the space-time diagram containing T time steps. Each of the elements of a space-time diagram will be referred to as a configuration of the CA, while the first element will be referred to as the initial configuration. If t = 0, 1, . . . , T − 1 and m = 1, . . . , M , then At (X)[m] refers to the state of the m–th cell in the t–th element of the space-time diagram. Example 2. We consider an ECA defined by local rule 150. The LUT of ECA 150 is shown in Table II. TABLE II. LUT OF ECA 150 111 110 101 100 011 010 001 000 1 0 0 1 0 1 1 0 Figure 1 depicts a space-time diagram of ECA 150, starting from a random initial configuration. Following a common convention, the space-time diagram is visualized as a bitmap in which every row corresponds to a configuration at specific time step. The first row in the image is the initial configuration. State one is drawn as a black pixel, while white pixel corresponds to state zero. III. P ROBLEM STATEMENT belonging to the set {0, 1}. Additionally, let the first row I[1] ∈ {0, 1}M . Such an array I will be referred to as an observation. If an observation I does not contain the symbol ?, we refer to it as spatially complete. The first row I[1] is assumed to represent the initial configuration of a CA, and row I[n] for n > 1 represents the configuration at time step τn . It is assumed that τn < τn+1 . Let I be an observation. The number C(I) = #{I[n, m] 6= ?} will be referred to as the number of completely observed states. In our case, for any observation I it holds that C(I) > 0. For each observation I, we define the set com(I) that contains all of the spatially complete observations I 0 , satisfying I 0 [n, m] = I[n, m] for all n, m such that I[n, m] 6= ?. Example 3. Let observation I be given by: 0 I= 0 1 Then the set com(I) is   0 1 0 0 com(I) =  1 1 1 ? 1 0 1 . ? given by: 0 0 1 0 1 , 0 0 1 , 0 1 1 1  0 1 0 0 1 0  0 1 1 , 0 1 1 . 1 1 0 1 1 1  As can be easily counted, C(I) = 7. In this section we define the identification problem. The formulation presented below is based on the concept of an observation of a space-time diagram, which is assumed to be incomplete, i.e. it contains only partial information on the states of the CA. We will say that a CA A fits the observation I if and only if there exists an I 0 ∈ com(I) and a sequence of natural numbers (τn ) such that τn < τn+1 and for any n ∈ {1, 2, . . . , N − 1} it holds: Aτn (I 0 [1]) = I 0 [n + 1]. (3) Formally, we assume that the states of a system, which is assumed to be an unknown CA, were observed at certain, unknown time steps. Let I be an N × M array containing symbols belonging to the set {0, 1, ?}, where the symbols 0 and 1 denote valid states, while ? denotes an unknown state Proposition 3. Rule A fits the observation I if and only if there exist an I 0 ∈ com(I) and a sequence of natural numbers (tn ) such that tn ≤ tn+1 and for any n ∈ {1, 2, . . . , N − 1} it holds: Atn (I 0 [n]) = I 0 [n + 1]. (4) The sequence (τn ) in the definition of fitting, corresponds to the time steps in the CA evolution (which are assigned to the rows of the observation), while the sequence (tn ) in Proposition 3 refers to the number of missing time frames between two P consecutive rows in the observed diagram. Obn viously τn = i=1 ti . In practice, it is useful to be able to use more than one observation for the identification. Therefore, we will consider observation sets I containing a finite number of observations. For simplicity, we assume that the elements of I are numbered, i.e. I = {I1 , . . . , I\|I\| }. We will say that rule A fits the observation set I, if it fits all of the observations in the set. Note that for the sake of simplicity we will write C(I) to express the number of observed P states in all of the observations belonging to I, i.e. C(I) = I∈I C(I). Additionally, we will write M (I) to denote the total number of columns in all of P the observations belonging to I, i.e. M (I) = I∈I MI where MI is the number of columns of observation I. For a non-empty observation set I, the set R(I) will denote all CA rules that fit the observation set I. The identification problem is defined as finding the elements of the set R(I) based on I. In practice, our goal will be limited to finding at least one of the elements of R(I) ∩ Ar for some r > 0. The problem can also be seen from the machine learning perspective in which the observation set is a training set, from which we try to learn and build a set of rules based on this knowledge. The following fact will be used in the design of the identification algorithm, to simplify calculations. Informally, it could be expressed by understanding the observation set I as a set of conditions that the rule needs to meet. Having fewer conditions, it becomes more likely to find solutions meeting those conditions. Fact 4. Let I be an observation set, and let I 0 ⊂ I. Then R(I) ⊂ R(I 0 ). Since we consider only finite observation sets, we know that for every observation set I there exists a T > 0 such that, if a solution exists, and (tIn ) is the time gap sequence of observation I ∈ I, then 1 ≤ tIn ≤ T , for every n. In the construction of the solution algorithm, we will assume that an upper-bound for T is known. IV. CA I DENTIFICATION AS AN OPTIMIZATION PROBLEM The identification problem, defined in Section III, can be formulated as an optimization problem, which in turn enables the use of evolutionary search methods. We start with an auxiliary definition. Let a, b ∈ {0, 1, ?}M be some vectors. We define the distance between a and b as: X dist(a, b) = \|ai − bi \|. (5) ai ,bi ∈{0,1} We assume that if there is no i such that ai 6= ? and bi 6= ? then dist(a, b) = 0. Therefore dist(a, b) = 0 6⇒ a = b. Assume that I is a set of observations of some unknown CA belonging to Ar , i.e. R(I) ∪ Ar 6= ∅. Let A be a CA, and for every I ∈ I, let (τnI ) be a strictly increasing sequence of natural numbers. As a start, we define the error measure EI (A, (τiI )), which measures how well a given CA A fits the observation set I, assuming that τiI is the time step of the i–th row in observation I. The measure EI is defined as: EI (A, (τiI )) = I −1 X NX I dist(Aτn (I[1]), I[n + 1]), (6) I∈I n=1 where NI is the number of rows of observation I ∈ I. The following fact is an direct consequence of the definition of the identification problem. Fact 5. A ∈ R(I) if and only if there exists a sequence (τiI ) such that EI (A, (τiI )) = 0. Note that in the case when I = {I} we will write EI instead of E{I} . Let (ti ) be a sequence of natural numbers, and let A be a A CA rule. Observation I¯(t defined as: i) ( I[n, m], if I[n, m] 6= ?, A I¯(t [n, m] = A i) Atn−1 (I¯(t [n − 1])[m], if I[n, m] = ?, i) will be referred to as the A–completion of I with time gaps A (ti ). Note that any observation I satisfies I[1] = I¯(t [1] for i) any A, (ti ). Fact 6. I¯A ∈ com(I). (ti ) Example 4. Assume that CA A is ECA 150 with LUT given by Table II and local rule f150 . Let (ti )2i=1 = (1, 2). We consider A the observation I defined in Example 3 and compute I¯(t . i) 0 I= 0 1 1 ? 1 0 1 ? 0 1 0 A 0 1 1 I¯(t = ) i 1 1 0 A The calculation is as follows. Firstly we compute I¯(t [2, 2]. i) Since t1 = 1 we simply apply the rule to the first row of A I, i.e. I¯(t [2, 2] = f150 (I[1, 1], I[1, 2], I[1, 3]) = 1. Since i) A t2 = 2, to find I¯(t [3, 3], we first need to compute one i) additional configuration by evaluating the rule on configuration A A I¯(t [2]. It is easy to check that A(I¯(t [2]) = (0, 0, 0), and thus i) i) A I¯(ti ) [3, 3] = f150 (0, 0, 0) = 0. Based on Proposition 3, we define an alternative error eI (A, (tI )) that will turn out to be more useful in measure E i the construction of the solution algorithm. Assuming that (tIi ) eI is a sequence of natural numbers representing time gaps, E is defined as: I −1 I X NX A A eI (A, (tIi )) = ¯ E dist Atn I¯(t I ) [n] , I(tI ) [n + 1] . i i I∈I n=1 (7) A eI without using the Since I¯(t ∈ com(I), we can express E i) function dist as: eI (A, (tIi )) = E I −1 X NX I A ¯A \|Atn I¯(t I ) [n] − I(tI ) [n + 1]\|. i I∈I n=1 i (8) Example 5. We refer again to observation I, CA A and (ti ) used in Example 4 and we compute the error measuresPEI and eI . Let us start with EI . Following the fact that τn = n ti , E i=1 we get (τi ) = (1, 3). The error measure EI can be computed easily by evolving A, starting from the initial configuration I[1] and comparing the results with the values in I, for entries not occupied by ?. Starting from the top: A(I[1]) = (1, 1, 1). Since τ1 = 1 we compare the outcome with the second row of I. As we see, I[2, 1] = 0 6= 1 has an incorrect value, I[2, 2] =? so it does not contribute to the error and I[2, 3] = 1 which is a correct value. Since τ2 = 3 we should further evolve A three times, starting from A(I[1]), but since A((1, 1, 1)) = (1, 1, 1), we can simply compare I[3] with (1, 1, 1) and see that no errors occur. Summing up, the total error is: EI (A, (τi )) = 1. T NI −1 possibilities, which holds a substantial computational burden. Due to this, even in the case of partial observations, we follow the approach described above and treat the time steps independently. The only difference that we introduce is that if for given n, few different candidate values for tIn lead to the same, minimal value of the pairwise error, one of those candidates is being selected randomly. Such an approach, is a stochastic overestimation of the error, i.e. the calculated value will never be lower than the actual error. Additionally, if a given CA is a solution to the problem, recalculating the approximate error measure multiple times increases the probability of finding the exact value, which is found by taking the minimum of all of the obtained results. Such an approach turned out to be sufficient in the discussed context. V. E VOLUTIONARY ALGORITHM eI , by taking pairs of rows Similarly, we find the value of E A A I¯(t [n] and I[n+1] and comparing the results of Atn (I¯(t [n]) i) i) and I[n + 1]. The error in the first pair of rows is the same as in the case of EI . For the second pair the initial condition is A I¯(t [2] = (0, 1, 1), and since A(0, 1, 1) = (0, 0, 0) and since i) A((0, 0, 0)) = (0, 0, 0), we do not further evaluate A. We compare (0, 0, 0) with I[3], which yields 2 incorrect values. eI (A, (ti )) = 3. Summing up, the total error is E Having stated the identification problem as an optimization problem in this section, we describe its solution using an evolutionary algorithm based on the classical GA. In order to follow the GA approach, we need to define the individuals’ representation, the population structure, a fitness function for ranking the individuals, but also the selection procedure for reproduction, and finally the cross-over and mutation operators. Formally, also halting conditions need to be formulated. eI is expressed by the The relation between EI and E following proposition. A. Representation of individuals and population structure Proposition 7. Let A be a CA rule and I an observation set. There exists a strictly increasing sequence (τiI ) of natural numbers, such that EI (A, (τiI )) = 0 if and only if there exists eI (A, (tI )) = 0. a sequence (tIi ) of natural numbers such that E i As a consequence of Proposition 7, the identification problem can be expressed mathematically as the minimization of e Note that this is only possible due to the assumption that E. observation set I contains partial space-time diagrams of some unknown CA. In a more general setting, where the observations could have a more complex origin, such a simplification is not possible. As mentioned earlier, we consider the case where the upper bound for the time gaps is known. Using this knowledge, we eI independently of the selection of define the error measure E (tIi ) as: eI (A) = min E eI (A, (tIi )). E (9) (tIi ) 1≤tIi ≤T Note that the minimum in (9) is always defined, since there is a finite number of possibilities for the choice of tIn . Additionally, note that for a spatially complete observation I, the choice of tIn is independent of the choice of tIm for any n 6= m, and for observations I and J, the choice of (tIi ) is independent from the choice of (tJi ). Consequently, to find the eI in the case of a spatially complete observation value of E P set, we need to examine at most I∈I T (NI − 1) sequences of time gap lengths. (tIi ) In the general case, the choices of the values of are dependent on each other, and thus in order to find the exact value of the error measure we need to examine all of the Here, the individuals that make up the population are CAs, encoded through the LUT of their local rules, which is possible since the LUT of any CA A ∈ Ar can be represented as a bitstring of length 22 r+1 . We assume that the population consists of CA belonging to Ar , for some r > 0. We consider populations of P > 0 individuals. By P i we denote the population of the i–th generation of the GA. The population P 1 is the initial population, and is constructed by randomly selecting P bit-strings. Populations P i for i > 1 are the outcomes of applying the genetic operators, according to the rules described in the remainder of this section. B. Fitness function The fitness function is directly related to the error measure eI defined by (9). Although Proposition 7 states that the E error measures given by (6) and (9) can be used interchangeably, preliminary experiments showed that the later results in efficient and convergent algorithm, while suboptimal results were obtained using the measure given by (6). This follows from the fact that the error in row n is affected by errors appearing in rows 2, . . . , n − 1. As we know from the research on dynamical properties of CAs, small initial perturbations can strongly affect the final system state [20]. For that reason, it eI with a GA as compared to EI . is easier to optimize E 2 r+1 be a LUT of some local rule which Let L ∈ {0, 1}2 defines a CA A. Then fitI (L) denotes the fitness of A, and is defined as: eI (A). fitI (L) = C(I) − M (I) − E (10) The fitness function takes integer values from 0 up to C(I) − M (I), i.e. there are finitely many possible values of the fitness function. The goal of the GA is to maximize fitness, and a CA with a maximal fitness value is a solution of the identification problem. From the above, it is clear that if C(I) − M (I) is close to zero, solving the problem is infeasible, since the number of possible values is very small and the population is not able to gradually increase its fitness. Additionally, if C(I) = M (I), then the problem is trivial because every CA is a solution. The fitness defined by (10) has proven to work effectively, but the computing time needed for its evolution becomes unacceptable if the observation set is large. Therefore, during the evolution, to estimate the value of fitI we use fitI 0 for some non-empty subset I 0 ⊂ I. We start by randomly selecting elements for the subset I 0 . Subsequently, but before evolving a new population we replace one of the elements in the subset I 0 with a randomly selected observation from I. Due to Fact 4 we are sure that such an approach does not result in reducing the solution set. C. Selection operator Having defined the fitness function, we can define the selection operator, which is responsible for selecting the parent individuals that will be used to produce the next generation. We use a random selection method where the selection probability of a given individual is proportional to its fitness. Individuals are selected with replacement, i.e. individuals might be selected multiple times for reproduction. G. Halting conditions The algorithm evolves by generating populations according to the procedure described above until a maximum, predefined number of populations Λ was evolved or, if a CA that fits the observation set was discovered. As mentioned in Subsection V-B during the evolution, the fitness fitI is approximated by fitI 0 for some I 0 ( I, which is effective for selection, but can not be used in the halting condition since fitI 0 (A) = C(I 0 ) − M (I 0 ) does not imply fitI (A) = C(I) − M (I). Therefore, for the individual A with the highest value fitI 0 (A), we additionally calculate fitI (A) and base the halting condition on it, i.e. the algorithm stops as soon an element is found. VI. R ESULTS OF EXPERIMENTS By means of our experiments we verified to what extent the partiality of observations affects the efficiency of the GA in terms of the number of GA iterations required to find a solution. We concentrated on two ECAs: 150 and 180, with LUTs given in Table II and III, respectively. TABLE III. LUT OF ECA 180 111 110 101 100 011 010 001 000 1 0 1 0 1 0 1 0 D. Cross-over operator To produce offspring, we select two parents according to the procedure described in Subsection V-C. A uniform crossover operator is used, i.e. if L1 , L2 denote parents, the outcome of the cross-over operator is a vector Lc with values that are randomly selected from L1 and L2 , i.e. P(Lc [i] = L1 [i]) = P(Lc [i] = L2 [i]) = 0.5. E. Mutation operator Finally, the offspring individual is mutated. A simple bitflip mutation is being used, i.e. for every position of the vector a decision is made whether or not the value should be flipped, with pf being the probability of flipping the value. The expected number of flipped positions in the population is P pf 22 r+1 . F. Elite survival After evolving a new population, the elite survival procedure is applied. Our experiments proved that such an approach is required to reach convergence. The procedure is implemented by a deterministic selection of PE P fittest individuals from the previous population used to replace randomly selected individuals in the newly evolved one. Including this elite survival process can dramatically increase the performance of the algorithm, though there are cases where such an approach causes the population to progress towards a local optimum. To overcome this, we apply a simple, adaptive procedure that deactivates elite survival in cases when the maximum fitness value of the population remained constant for more than Noff generations. The elite survival procedure is again switched on after a predefined number of Non generations, or if the maximum fitness improved. In this experiment, the GA evolution is based on observation sets IA (k) for k = {0, 1, . . . , 150} and ECA A ∈ {150, 180}. The integer k will be referred to as the problem number. The observation set IA (0) is a set of Ω > 0 observations obtained from Ω different, random initial conditions common for both A, by selecting subsequent configurations of ECA A generating time gaps of random length from 1 to T . The set IA (k) for k > 0 is built from observations belonging to IA (k − 1) by modifying them in such a way that π = 2000 randomly selected, completely observed entries are replaced by “?”. In other words, by increasing k the effect of spatial partiality is increased. As a result of such 150 a procedure we obtained a series of observation sets IA (k) k=0 , for which it holds C(IA (k)) − C(IA (k + 1)) = π. The identification algorithm was then executed for each of the obtained observation sets. Given that the family of ECAs contains only 256 members, the identification problem would be relatively easy to tackle, so we set the radius r = 2, i.e. the population contains local rules with radius r = 2 represented as bit-strings of length 32. Without this modification the algorithm is able to find a solution in a few iterations, by examining the entire search space. In order to account for the stochastic nature of the GA, the experiment is repeated L > 0 times for each r, k. The values of of the GA parameters used in our experiment setup are shown in Table IV. The results vary significantly depending on the rule in question, which is not surprising since the dynamics of ECAs 150 and 180 is different. The normalized Maximum Lyapunov Exponent (nMLE) [21]–[23] of the former is the highest among param r pf P PE T C Ω π s Λ L value 2 0.01 512 32 10 69 64 2000 8 5000 20 PARAMETERS USED IN THE EXPERIMENT description rule radius probability of flipping 1-bit in mutation number of individuals in population elite size bound for the time gap length number of rows / columns in each observation number of observations number of cells being removed from each observation set number of samples for fitness approximation maximal number of the GA populations number of repetitions of the GA per rule no. of successful GA executions TABLE IV. 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 problem number k no. of successful GA executions (a) ECA 150 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 problem number k (b) ECA 180 Fig. 3. Fig. 2. Number of successful GA executions Space-time diagram of ECA 180 all of the ECAs, and thereby this CA’s behavior may be considered complex. In contrast, the nMLE of ECA 180 is only approximately 0.48, which hints that, in some sense, the behavior of this ECA is simpler than the one displayed by ECA 150. The differences in the overall dynamical complexity of these two CAs can be acknowledged by examining their spacetime diagrams, which are depicted in Fig. 1 and Fig. 2. To understand the performance of the GA, we first checked for which k the algorithm was able to find a solution (Fig. 3). When comparing the plot for ECA 150 with the one for ECA 180, it is clear that the identification problem turned out to be much more challenging for ECA 150. Indeed, for this ECA, the algorithm was effective only if the number removed observation elements was smaller than 50 π, whereas it mostly failed when more spatial partiality was added. Besides, even for k close to 0, not all of the GA executions were successful. In contrast, identifying ECA 180 was always possible for k < 120, but for k > 120 we see a sudden drop in the performance. Note that in both cases, for k = 150 a solution was easily found, since for this setting the problem is trivial, i.e. almost all CAs can be considered a solution. The above results suggest that, depending on the dynamical characteristics of the CA in question, the maximum allowable number of missing elements in the observations differs. Further research is undertaken to better understand the link between the identifiability and dynamics of CAs. Figure 4 depict the minimum, average and maximum number of GA iterations among the runs resulting in a solution for ECA 150 and ECA 180, respectively. In the case of ECA 180, we see that the efforts needed for finding a solution grows as k increases, up to the point where it becomes impossible. Furthermore, we see that in most cases the difference between maximal and minimal values is relatively low. In the case of ECA 150, the results are much less stable. The differences between maximal and minimal values are substantial, and the efforts needed to find the solution do not steadily grows with the growing spatial partiality. The only similarity between the two CAs seems to be in the fact that there exists some critical k beyond which the problem becomes impossible to solve. S UMMARY In this paper we introduced the identification problem of CAs in the context of partial observations. An evolutionary algorithm for tackling the problem was presented, and its performance was verified for the two ECAs. The initial experiments suggest that the difficulty of the identification problem is somehow linked to the dynamical complexity of the CAs. The problem and solution algorithm presented in this paper, should be considered as one of the first steps in identifying CAs from data originating from real-world phenomenon observations. Unavoidably, such observations will be somehow incomplete in the sense that it is impossible to continuously track the involved processes. R EFERENCES [1] S. Al-Kheder, J. Wang, and J. Shan, “Cellular automata urban growth model calibration with genetic algorithms,” in Proc. Urban Remote Sensing Joint Event, 2007. IEEE, 2007, pp. 1–5. [2] E. Sapin, L. Bull, and A. Adamatzky, “Genetic approaches to search for computing patterns in cellular automata,” IEEE Comput. Intell. 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The Cross-Quantilogram: Measuring Quantile arXiv:1402.1937v2 [math.ST] 21 Jan 2018 Dependence and Testing Directional Predictability between Time Series∗ Heejoon Han† Oliver Linton‡ Tatsushi Oka§ Yoon-Jae Whang¶ March 14, 2016 Abstract This paper proposes the cross-quantilogram to measure the quantile dependence between two time series. We apply it to test the hypothesis that one time series has no directional predictability to another time series. We establish the asymptotic distribution of the cross-quantilogram and the corresponding test statistic. The limiting distributions depend on nuisance parameters. To construct consistent confidence intervals we employ a stationary bootstrap procedure; we establish consistency of this bootstrap. Also, we consider a self-normalized approach, which yields an asymptotically pivotal statistic under the null hypothesis of no predictability. We provide simulation studies and two empirical applications. First, we use the cross-quantilogram to detect predictability from stock variance to excess stock return. Compared to existing tools used in the literature of stock return predictability, our method provides a more complete relationship between a predictor and stock return. Second, we investigate the systemic risk of individual financial institutions, such as JP Morgan Chase, Morgan Stanley and AIG. Keywords: Quantile, Correlogram, Dependence, Predictability, Systemic risk. ∗ We thank a Co-Editor, Jianqing Fan, an Associate Editor and three anonymous referees for constructive comments. Han’s work was supported by the National Research Foundation of Korea (NRF2013S1A5A8021502). Linton’s work was supported by Cambridge INET and the ERC. Oka’s work was supported by Singapore Academic Research Fund (FY2013-FRC2-003). Whang’s work was supported by the SNU Creative Leading Researcher Grant. † Department of Economics, Sungkyunkwan University, Seoul, Republic of Korea. ‡ Faculty of Economics, University of Cambridge, Cambridge, UK. § Department of Economics, National University of Singapore, Singapore. ¶ Department of Economics, Seoul National University, Seoul, Republic of Korea. 1 1 Introduction Linton and Whang (2007) introduced the quantilogram to measure predictability in different parts of the distribution of a stationary time series based on the correlogram of “quantile hits”. They applied it to test the hypothesis that a given time series has no directional predictability. More specifically, their null hypothesis was that the past information set of the stationary time series {yt } does not improve the prediction about whether yt will be above or below the unconditional quantile. The test is based on comparing the quantilogram to a pointwise confidence band. This contribution fits into a long literature of testing predictability using signs or rank statistics, including the papers of Cowles and Jones (1937), Dufour et al. (1998), and Christoffersen and Diebold (2002). The quantilogram has several advantages compared to other test statistics for directional predictability. It is conceptually appealing and simple to interpret. Since the method is based on quantile hits it does not require moment conditions like the ordinary correlogram and statistics like the variance ratio that are derived from it, Mikosch and Starica (2000), and so it works well for heavy tailed series. Many financial time series have heavy tails, see, e.g., Mandelbrot (1963), Fama (1965), Rachev and Mittnik (2000), Embrechts et al. (1997), Ibragimov et al. (2009), and Ibragimov (2009), and so this is an important consideration in practice. Additionally, this type of method allows researchers to consider very long lags in comparison with regression type methods, such as Engle and Manganelli (2004). There have been a number of recent works either extending or applying this methodology. Davis and Mikosch (2009) have introduced the extremogram, which is essentially the quantilogram for extreme quantiles, and Davis et al. (2012) has provided the inference methods based on bootstrap and permutation for the extremogram. See also Davis et al. (2013). Li (2008, 2012) has introduced a Fourier domain version of the quantilogram while 1 Hong (2000) has used a Fourier domain approach for test statistics based on distributions. Further development in the Fourier domain approach has been made by Hagemann (2013) and Dette et al. (2015). See also Li (2014) and Kley et al. (2016). The quantilogram has recently been applied to stock returns and exchange rates, Laurini et al. (2008) and Chang and Shie (2011). Our paper addresses three outstanding issues with regard to the quantilogram. First, the construction of confidence intervals that are valid under general dependence structures. Linton and Whang (2007) derived the limiting distribution of the sample quantilogram under the null hypothesis that the quantilogram itself is zero, in fact under a special case of that where the process has a type of conditional heteroskedasticity structure. Even in that very special case, the limiting distribution depends on model specific quantities. They derived a bound on the asymptotic variance that allows one to test the null hypothesis of the absence of predictability (or rather the special case of this that they work with). Even when this model structure is appropriate, the bounds can be quite large especially when one looks into the tails of the distribution. The quantilogram is also useful in cases where the null hypothesis of no predictability is not thought to be true - one can be interested in measuring the degree of predictability of a series across different quantiles. We provide a more complete solution to the issue of inference for the quantilogram. Specifically, we derive the asymptotic distribution of the quantilogram under general weak dependence conditions, specifically strong mixing. The limiting distribution is quite complicated and depends on the long run variance of the quantile hits. To conduct inference we propose the stationary bootstrap method of Politis and Romano (1994) and prove that it provides asymptotically valid confidence intervals. We investigate the finite sample performance of this procedure and show that it works well. We also provide R code that carries out the computations efficiently.1 We also define a self-normalized version of the statistic for testing the null hypothesis that the quantilogram is zero, following Lobato (2001). This statistic has an asymptotically pivotal distribution, 1 This can be found at http://www.oliverlinton.me.uk/research/software. 2 under the null hypothesis, whose critical values have been tabulated so that there is no need for long run variance estimation or even bootstrap. Second, we develop our methodology inside a multivariate setting and explicitly consider the cross-quantilogram. Linton and Whang (2007) briefly mentioned such a multivariate version of the quantilogram but they provided neither theoretical results nor empirical results. In fact, the cross-correlogram is a vitally important measure of dependence between time series: Campbell, Lo, and MacKinlay (1997), for example, use the cross autocorrelation function to describe lead lag relations between large stocks and small stocks. We apply the cross-quantilogram to the study of stock return predictability; our method provides a more complete picture of the predictability structure. We also apply the cross-quantilogram to the question of systemic risk. Our theoretical results described in the previous paragraph are all derived for the multivariate case. Third, we explicitly allow the cross-quantilogram to be based on conditional (or regression) quantiles (Koenker and Basset, 1978). Using conditional quantiles rather than unconditional quantiles, we measure directional dependence between two time-series after parsimoniously controlling for the information at the time of prediction.2 Moreover, we derive the asymptotic distribution of the cross-quantilogram that are valid uniformly over a range of quantiles. The remainder of the paper is as follows: Section 2 introduces the cross-quantilogram and Section 3 discusses its asymptotic properties. For consistent confidence intervals and hypothesis tests, we define the bootstrap procedure and introduce the self normalized test statistic. Section 4 considers the partial cross-quantilogram and gives a full treatment of its behavior in large samples. In Section 5 we report results of some Monte Carlo simulations to 2 Our analysis includes the cross-quantilogram based on unconditional quantiles as a special case. In this case, the cross-quantilogram is shown to be a functional of the empirical copula introduced by Ruschendorf (1976) and Deheuvels (1979) as some nonparametric measures of dependence, such as Spearman’s rho and Kendall’s tau. In this special case, the asymptotic results for the empirical copula, which are found in Stute (1984), Fermanian et al. (2004) and Segers (2012) among others, can apply for the cross-quantilogram. Generally, however, the cross-quantilogram here differs from the empirical copula process and needs different treatment for analyzing its properties. 3 evaluate the finite sample properties of our procedures. In Section 6 we give two applications: we investigate stock return predictability and system risk using our methodology. Appendix contains all the proofs. We use the following notation: The norm k · k denotes the Euclidean norm, i.e., kzk = Pd for z = (z1 , . . . , zd )⊤ ∈ Rd and the norm k · kp indicates the Lp norm of a d × 1 P random vector z, given by kzkp = ( dj=1 E\|zj \|p )1/p for p > 0. Let 1[·] be the indicator ( 2 1/2 j=1 zj ) function taking the value one when its argument is true, and zero otherwise. We use R, Z and N to denote the set of all real numbers, all integers and all positive integers, respectively. Let Z+ = N ∪ {0}. 2 The Cross-Quantilogram Let {(yt , xt ) : t ∈ Z} be a strictly stationary time series with yt = (y1t , y2t )⊤ ∈ R2 and (1) (d ) xt = (x1t , x2t ) ∈ Rd1 × Rd2 , where xit = [xit , . . . , xit i ]⊤ ∈ Rdi with di ∈ N for i = 1, 2. We use Fyi \|xi (·\|xit ) to denote the conditional distribution function of the series yit given xit with density function fyi \|xi (·\|xit ), and the corresponding conditional quantile function is defined as qi,t (τi ) = inf{v : Fyi \|xi (v\|xit ) ≥ τi } for τi ∈ (0, 1), for i = 1, 2. Let T be the range of quantiles we are interested in evaluating the directional predictability. For simplicity, we assume that T is a Cartesian product of two closed intervals in (0, 1), that is T ≡ T1 × T2 , where Ti = [τ i , τ i ] for some 0 < τ i < τ i < 1.3 We consider a measure of serial dependence between two events {y1t ≤ q1,t (τ1 )} and {y2,t−k ≤ q2,t−k (τ2 )} for an arbitrary pair of τ = (τ1 , τ2 )⊤ ∈ T and for an integer k. In the literature, {1[yit ≤ qi,t (·)]} is called the quantile-hit or quantile-exceedance process for i = 1, 2. The cross-quantilogram is defined as the cross-correlation of the quantile-hit processes 3 E [ψτ1 (y1t − q1,t (τ1 ))ψτ2 (y2,t−k − q2,t−k (τ2 ))] ρτ (k) = q q , 2 2 E ψτ1 (y1t − q1,t (τ1 )) E ψτ2 (y2,t−k − q2,t−k (τ2 )) (1) It is straightforward to extend the results to a more general case, e.g. the case for which T is the union of a finite number of disjoint closed subsets of (0, 1)2 . 4 for k = 0, ±1, ±2, . . . , where ψa (u) ≡ 1[u < 0] − a. The cross-quantilogram captures serial dependence between the two series at different conditional quantile levels. In the special case of a single time series, the cross-quantilogram becomes the quantilogram proposed by Linton and Whang (2007). Note that it is well-defined even for processes {(y1t , y2t )}t∈N with infinite moments. Like the quantilogram, the cross-quantilogram is invariant to any strictly monotonic transformation applied to both series, such as the logarithmic transformation.4 To construct the sample analogue of the cross-quantilogram based on observations {(yt , xt )}Tt=1 , we first estimate conditional quantile functions. In this paper, we consider the linear quantile regression model proposed by Koenker and Bassett (1978) for simplicity and let qi,t (τi ) = x⊤ it βi (τi ) with a di × 1 vector of unknown parameters βi (τi ) for i = 1, 2. To estimate the parameters β(τ ) ≡ [β1 (τ1 )⊤ , β2 (τ2 )⊤ ]⊤ , we separately solve the following minimization problems: β̂i (τi ) = arg min βi ∈Rdi T X t=1 ̺τi yit − x⊤ it βi , where ̺a (u) ≡ u(a − 1[u < 0]). Let β̂(τ ) ≡ [β̂1 (τ1 )⊤ , β̂2 (τ2 )⊤ ]⊤ and q̂i,t (τi ) = x⊤ it β̂i (τi ) for i = 1,2. The sample cross-quantilogram is defined by PT ψτ1 (y1t − q̂1,t (τ1 ))ψτ2 (y2,t−k − q̂2,t−k (τ2 )) qP , T 2 (y − q̂ (τ )) 2 (y ψ ψ − q̂ (τ )) 1,t 1 2,t−k 2 t=k+1 τ1 1t t=k+1 τ2 2,t−k ρ̂τ (k) = qP T t=k+1 (2) for k = 0, ±1, ±2, . . . . Given a set of conditional quantiles, the cross-quantilogram considers dependence in terms of the direction of deviation from conditional quantiles and thus measures the directional predictability from one series to another. This can be a useful del h When one is interested in measuring serial dependence events {q1,t (τ1 ) ≤ y1t ≤ q1,t (τ1 )} l two l h between h l h and {q2,t−k (τ2 ) ≤ y2,t−k ≤ q2,t−k (τ2 )} for arbitrary τ1 , τ1 and τ2 , τ2 , one can use an alternative version of the cross-quantilogram that is defined by replacing ψτi (yit − qi,t (τi )) in (1) with ψ[τ l ,τ h ] (yit − qi,t ( τil , τih )) = 1[qi,t (τil ) < yit < qi,t τih ] − τih − τil . i i 4 For example, if τ1 = [0.9, 1.0] and τ2 = [0.4, 0.6] , the alternative version measures dependence between an event that y1t is in a high range and an event that y2,t−k is in a mid-range. In some cases, such an alternative version could be easier to interpret and therefore be useful. The inference procedure provided in this paper is also valid for the alternative version of the cross-quantilogram. See the working paper version of this paper for an empirical application using the alternative version. 5 scriptive device. By construction, ρ̂τ (k) ∈ [−1, 1] with ρ̂τ (k) = 0 corresponding to the case of no directional predictability. The form of the statistic generalizes to the l dimensional multivariate case and the (i, j)th entry of the corresponding cross-correlation matrices Γτ̄ (k) is given by applying (2) for a pair of variables (yit , xit ) and (yjt−k , xjt−k ) and a pair of con⊤ ditional quantiles (q̂i,t (τi ), q̂j,t−k (τj ))) for τ̄ = (τ1 , . . . , τl ) . The cross-correlation matrices ⊤ possess the usual symmetry property Γτ̄ (k) = Γτ̄ (−k) when τ1 = · · · = τd . Suppose that τ ∈ T and p are given. One may be interested in testing the null hypothesis H0 : ρτ (1) = · · · = ρτ (p) = 0 against the alternative hypothesis that ρτ (k) 6= 0 for some k ∈ {1, . . . , p}. This is a test for the directional predictability of events up to p lags {y2,t−k ≤ q2,t−k (τ2 ) : k = 1, . . . , p} for {y1t ≤ q1,t (τ1 )}. For this hypothesis, we can use the P (p) Box-Pierce type statistic Q̂τ = T pk=1 ρ̂2τ (k). In practice, we recommend to use the BoxP (p) Ljung version Q̌τ ≡ T (T + 2) pk=1 ρ̂2τ (k)/(T − k) which had small sample improvements in our simulations. On the other hand, one may be interested in testing a stronger null hypothesis, i.e. the absence of directional predictability over a set of quantiles: H0 : ρτ (1) = · · · = ρτ (p) = 0, ∀τ ∈ T , against the alternative hypothesis that ρτ (k) 6= 0 for some (k, τ ) ∈ {1, . . . , p} × T with p fixed. In this case, we can use the sup-version test statistic sup Q̂τ(p) τ ∈T = sup T τ ∈T p X ρ̂2τ (k). k=1 (p) Note that the portmanteau test statistic Q̂τ for a specific quantile is a special case of the sup-version test statistic. 3 Asymptotic Properties We next present the asymptotic properties of the sample cross-quantilogram and related test statistics. Since these quantities contain non-smooth functions, we employ techniques widely used in the literature on quantile regression, see Koenker and Bassett (1978) and Pollard 6 (1991) among others. Define yt,k = (y1t , y2,t−k )⊤ , xt,k = (x1t , x2,t−k ), qt,k (τ ) = [q1,t (τ1 ), q2,t−k (τ2 )]⊤ and q̂t,k (τ ) = [q̂1,t (τ1 ), q̂2,t−k (τ2 )]⊤ and let {yt,k ≤ qt,k (τ )} = {y1t ≤ q1 (τ1 \|x1t ), y2,t−k ≤ q2 (τ2 \|x2t−k )} and (k) Fy\|x (·\|xt,k ) = P (yt,k ≤ ·\|xt,k ) for t = k + 1, . . . , T and for some finite integer k > 0. We (k) use ∇G(k) (τ ) to denote ∂/∂vE[Fy\|x (vt,k \|xt,k )] evaluated at vt,k = qt,k (τ ), where vt,k = ⊤ ⊤ di [x⊤ (i = 1, 2). Let d0 = 1 + d1 + d2 . 1t v1 , x2,t−k v2 ] for vi ∈ R Assumption A1. {(yt , xt )}t∈Z is strictly stationary and strong mixing with coefficients {αj }j∈Z+ that P 2s−2 ν/(2s+ν) satisfy ∞ αj < ∞ for some integer s ≥ 3 and ν ∈ (0, 1). For each j=0 (j + 1) (j) (1) (d ) i = 1, 2, E\|xit \|2s+ν < ∞ for all j = 1, . . . , di , given xit = [xit , . . . , xit i ]⊤ . A2. The conditional distribution function Fyi \|xi (·\|xit ) has continuous densities fyi \|xi (·\|xit ), which is uniformly bounded away from 0 and ∞ at qi,t (τi ) uniformly over τi ∈ Ti , for i = 1, 2 and for all t ∈ Z. A3. For any ǫ > 0 there exists a ν(ǫ) such that supτi ∈Ti sups:\|s\|≤ν(ǫ) \|fyi \|xi (qi,t (τi )\|xit ) − fyi \|xi (qi,t (τi ) + s\|xit )\| < ǫ for i = 1, 2 and for all t ∈ Z. (k) A4. For every k ∈ {1, . . . , p}, the conditional joint distribution Fy\|x (·\|xt,k ) has the condi(k) tional density fy\|x (·\|xt,k ), which is bounded uniformly in the neighborhood of quantiles of interest, and also has a bounded, continuous first derivative for each argument uniformly in the neighborhood of quantiles of interest and thus ∇G(k) (τ ) exists over τ ∈ T . A5. For each i = 1, 2, there exist positive definite matrices Mi and Di (τi ) such that PT P ⊤ −1 (a) plimT →∞ T −1 Tt=1 xit x⊤ it = Mi and (b) plimT →∞ T t=1 fyi \|xi (qi,t (τi )\|xit )xit xit = Di (τi ) uniformly in τi ∈ Ti . Assumption A1 imposes the mixing rate used in Andrews and Pollard (1994) and a moment condition on regressors, while allowing for the dependent variables to be processes 7 with infinite moments. For a strong mixing process, ρτ (k) → 0 as k → ∞ for all τ ∈ (0, 1). Assumption A2 ensures that the conditional quantile function given xit is uniquely defined while allowing for dynamic misspecification, or P (yit ≤ qi,t (τi )\|Fit ) 6= τi given some information set Fit containing all “relevant” information available at t for i = 1, 2. In the absence of dynamic misspecification, which is assumed in Hong et al. (2009) under their null hypothesis, the analysis becomes substantially simple because each hit-process {ψτi (y − qi,t (τi ))} is a sequence of iid Bernoulli random variables. As Corradi and Swanson (2006) discuss, however, results under correct dynamic specification crucially rely on an appropriate choice of the information set; specification search for the information set based on pre-testing may have a nontrivial impact on inference. Thus, Assumption A2 is appropriate for the purpose of testing directional predictability given a particular information set xit . Assumption A3 implies that the densities are smooth in some neighborhood of the quantiles of interest. Assumption A4 ensures that the joint distribution of (x1t , x2t−k ) is continuously differentiable. Assumption A5 is standard in the quantile regression literature. To describe the asymptotic behavior of the cross-quantilogram, we define a set of d0 dimensional mean-zero Gaussian process {Bk (τ ) : τ ∈ [0, 1]2 }pk=1 with covariance-matrix function for k, k ′ ∈ {1, . . . , p} and for τ, τ ′ ∈ T , given by ⊤ k′ ′ ′ Ξkk′ (τ, τ ) ≡ E[Bk (τ )B (τ )] = ∞ X l=−∞ ⊤ cov ξl,k (τ ), ξ0,k′ (τ ′ ) , ⊤ ⊤ where ξt,k (τ ) = (1[yt,k ≤ qt,k (τ )], x⊤ 1t 1[y1t ≤ q1,t (τ1 )], x2t 1[y2t ≤ q2,t (τ2 )]) for t ∈ Z. Define ⊤ ⊤ ⊤ B(p) (τ ) = [B1 (τ ) , . . . , Bp (τ ) ] as the d0 p-dimensional zero-mean Gaussian process with the covariance-matrix function denoted by Ξ(p) (τ, τ ′ ) for τ, τ ′ ∈ T . We use ℓ∞ (T ) to denote the space of all bounded functions on T equipped with the uniform topology and (ℓ∞ (T ))p to denote the p-product space of ℓ∞ (T ) equipped with the product topology. Let the notation “⇒” denote the weak convergence due to Hoffman-Jorgensen in order to handle the measurability issues, although outer probabilities and expectations are not used explicitly in this 8 paper for notational simplicity. See Chapter 1 of van der Vaart and Wellner (1996) for a comprehensive treatment of weak convergence in non-separable metric spaces. The next theorem establishes the asymptotic properties of the cross-quantilogram. Theorem 1 Suppose that Assumptions A1-A5 hold for some finite integer p > 0. Then, in the sense of weak convergence of the stochastic process in (ℓ∞ (T ))p we have: √ (p) T ρ̂τ(p) − ρτ(p) ⇒ Λτ(p) B(p) (τ ), (p) ⊤ ⊤ (3) ⊤ where ρ̂τ ≡ [ρ̂τ (1), . . . , ρ̂τ (p)] and Λτ = diag(λτ 1 , . . . , λτ p ) with λτ,k = p 1 τ1 (1 − τ1 )τ2 (1 − τ2 )    1 −∇G(k) (τ )[D1−1 (τ1 ), D2−1 (τ2 )]⊤   . (4) Under the null hypothesis that ρτ (1) = · · · = ρτ (p) = 0 for every τ ∈ T , it follows that sup Q̂τ(p) ⇒ sup kΛτ(p) B(p) (τ )k2 , τ ∈T (5) τ ∈T by the continuous mapping theorem. 3.1 3.1.1 Inference Methods The Stationary Bootstrap The asymptotic null distribution presented in Theorem 1 depends on nuisance parameters. We suggest to estimate the critical values by the stationary bootstrap of Politis and Romano (1994). The stationary bootstrap is a block bootstrap method with blocks of random lengths. The stationary bootstrap resample is strictly stationary conditional on the original sample. Let {Li }i∈N denote a sequence of iid random block lengths having the geometric distribution with a scalar parameter γ ≡ γT ∈ (0, 1): P ∗(Li = l) = γ(1 − γ)l−1 for each positive 9 integer l, where P ∗ denotes the conditional probability given the original sample. We assume that the parameter γ satisfies the following growth condition: √ Assumption A6. T ν/2(2s+ν)(s−1) γ + ( T γ)−1 → 0 as T → ∞, where s and ν are defined in Assumption A1. We need the condition that γ = o(T −ν/2(2s+ν)(s−1) ) for the purpose of establishing uniform convergence over the subset T of [0, 1]2 , given the moment conditions on regressors under Assumption A1. This condition can be relaxed when regressors are uniformly bounded because γ = o(1) when s = ∞. Let {Ki }i∈N be a sequence of iid random variables, which have the discrete uniform distribution on {k + 1, . . . , T } and are independent of both the original data and {Li }i∈N . Ki +Li −1 representing the blocks of length Li starting with the We set BKi ,Li = {(yt,k , xt,k )}t=K i Ki -th pair of observations. The stationary bootstrap procedure generates the bootstrap ∗ samples {(yt,k , x∗t,k )}Tt=k+1 by taking the first (T − k) observations from a sequence of the resampled blocks {BKi ,Li }i∈N . In this notation, when t > T , (yt,k , xt,k ) is set to be (yjk , xjk ), where j = k + (t mod (T − k)) and (yk,k , xk,k ) = (yt,k , xt,k ), where mod denotes the modulo operator.5 Using the stationary bootstrap resample, we estimate the parameter β(τ ) by solving the minimization problem: β̂1∗ (τ1 ) = arg min β1 ∈Rd1 T X t=k+1 ∗ ∗ ̺τ1 (y1t − x∗⊤ 1t β1 ) and β̂2 (τ2 ) = arg min β2 ∈Rd2 T −k X t=1 ∗ ̺τ2 (y2t − x∗⊤ 2t β2 ). ∗ Then the conditional quantile function given the stationary bootstrap resample, qi,t (τi ) ≡ ∗ ∗⊤ ∗ ∗ ∗⊤ ∗⊤ ⊤ x∗⊤ it βi (τi ), is estimated by q̂i,t (τi ) ≡ xit β̂i (τi ) for each i = 1, 2. Define β̂ (τ ) = [β̂1 (τ1 ), β̂2 (τ2 )] ∗ ∗ ∗ ∗ (τ2 )]⊤ . We construct β̂ ∗ (τ ) (τ1 ), q2,t−k (τ2 )]⊤ and q∗t,k (τ ) = [q1,t and let q̂∗t,k (τ ) = [q̂1,t (τ1 ), q̂2,t−k by using (T − k) bootstrap observations, while β̂(τ ) is based on T observations, but the 5 For any positive integers a and b, the modulo operation a mod b is equal to the remainder, on division of a by b. 10 difference of sample sizes is asymptotically negligible given the finite lag order k. The cross-quantilogram based on the stationary bootstrap resample is defined as follows: ρ̂∗τ (k) PT ∗ ∗ ∗ ∗ ψτ1 (y1t − q̂1,t (τ1 ))ψτ2 (y2,t−k − q̂2,t−k (τ2 )) qP . T ∗ ∗ ∗ ∗ 2 2 t=k+1 ψτ1 (y1t − q̂1,t (τ1 )) t=k+1 ψτ2 (y2,t−k − q̂2,t−k (τ2 )) = qP T t=k+1 We consider the stationary bootstrap to construct a confidence interval for each statistic of p cross-quantilograms {ρ̂τ (1), . . . , ρ̂τ (p)} for a finite positive integer p and subsequently construct a confidence interval for the omnibus test based on the p statistics. To maintain the original dependence structure, we use (T −p) pairs of observations {[(yt,1 , xt,1 ), . . . , (yt,p , xt,p )]}Tt=p+1 to resample the blocks of random lengths. (p)∗ Given a vector cross-quantilogram ρ̂τ , we define the omnibus test based on the sta(p)∗ tionary bootstrap resample as Q̂τ (p)∗ = T (ρ̂τ (p) ⊤ (p)∗ − ρ̂τ ) (ρ̂τ (p) − ρ̂τ ). The following theorem shows the validity of the stationary bootstrap procedure for the cross-quantilogram. We use the concept of weak convergence in probability conditional on the original sample, which is denoted by “⇒∗ ”, see van der Vaart and Wellner (1996, p. 181). Theorem 2 Suppose that Assumption A1-A6 hold. Then, in the sense of weak convergence conditional on the sample we have: √ (p)∗ (p) (p) ⇒∗ Λτ B(p) (τ ) (a) T ρ̂τ − ρ̂τ in probability; (b) Under the null hypothesis that ρτ (1) = · · · = ρτ (p) = 0 for every τ ∈ T , (p)∗ (p) sup P sup Q̂τ ≤ z − P sup Q̂τ ≤ z →p 0. ∗ z∈R τ ∈T τ ∈T In practice, repeating the stationary bootstrap procedure B times, we obtain B sets (p)∗ ⊤ of cross-quantilograms and {ρ̂τ,b = [ρ̂∗τ,b (1), . . . , ρ̂∗τ,b (p)] }B b=1 and B sets of omnibus tests (p)∗ (p)∗ (p)∗ (p) ⊤ (p)∗ (p) {Q̂τ,b }B b=1 with Q̂τ,b = T (ρ̂τ,b − ρ̂τ ) (ρ̂τ,b − ρ̂τ ). For testing jointly the null of no directional predictability, a critical value, c∗Q,α , corresponding to a significance level α is 11 (p)∗ given by the (1 − α)100% percentile of B test statistics {supα∈T Q̂α,b }B b=1 , that is, c∗Q,α (p)∗ = inf c : P sup Q̂τ,b ≤ c ≥ 1 − α . ∗ τ ∈T For the individual cross-quantilogram, we pick up percentiles (c∗1k,α , c∗2k,α ) of the bootstrap √ √ ∗ ∗ T (ρ̂∗τ,b (k) − ρ̂τ (k)) ≤ c∗2k,α ) = distribution of { T (ρ̂∗τ,b (k) − ρ̂τ (k))}B b=1 such that P (c1k,α ≤ 1 − α, in order to obtain a 100(1 − α)% confidence interval for ρτ (k) given by [ρ̂τ (k) + T −1/2 c∗1k,α, ρ̂τ (k) + T −1/2 c∗2k,α ]. In the following theorem, we provide a power analysis of the omnibus test statistic (p) supτ ∈T Q̂τ when we use a critical value c∗Q,α . We consider fixed and local alternatives. The fixed alternative hypothesis against the null of no directional predictability is H1 : ρτ (k) 6= 0 for some (τ, k) ∈ T × {1, . . . , p}, (6) and the local alternative hypothesis is given by √ H1T : ρτ (k) = ζ/ T for some (τ, k) ∈ T × {1, . . . , p}, (7) where ζ is a finite non-zero constant. Thus, under the local alternative, there exists a p × 1 (p) vector ζτ (p) (p) (p) such that ρτ = T −1/2 ζτ with ζτ having at least one non-zero element for some τ ∈T. We consider the asymptotic power of a test for the directional predictability over a range of quantiles with multiple lags in the following theorem; however, the results can be applied to test for a specific quantile or a specific lag order. The following theorem shows that the √ cross-quantilogram process has non-trivial local power against the T -local alternatives. Theorem 3 Suppose that Assumptions A1-A6 hold. Then: (a) Under the fixed alternative in (6), (p) ∗ lim P sup Q̂τ > cQ,α → 1. T →∞ τ ∈T 12 (b) Under the local alternative in (7) (p) (p) (p) 2 (p) ∗ lim P sup Q̂τ > cQ,α = P sup kΛτ B (τ ) + ζτ k ≥ cQ,α , T →∞ τ ∈T τ ∈T (p) where cQ,α = inf{c : P (supτ ∈T kΛτ B(p) (τ )k2 ≤ c)) ≥ 1 − α}. 3.1.2 The Self-Normalized Cross-Quantilogram We use recursive estimates to construct a self-normalized cross-quantilogram. The selfnormalized approach was proposed by Lobato (2001) and was recently extended by Shao (2010) to a class of asymptotically linear test statistics.6 The self-normalized approach has a tight link with the fixed-b asymptotic framework proposed by Kiefer et al. (2000).7 The self-normalized statistic has an asymptotically pivotal distribution whose critical values have been tabulated so that there is no need for long run variance estimation or even bootstrap. As discussed in section 2.1 of Shao (2010), the self-normalized and the fixed-b approach have better size properties, compared with the standard approach involving a consistent asymptotic variance estimator, while it may be asymptotically less powerful under local alternatives (see Lobato (2001) and Sun et al. (2008) for instance). Given a subsample {(yt , xt )}st=1 , we can estimate sample quantile functions by solving minimization problems β̂i,s (τi ) = arg min βi ∈Rdi s X t=1 ̺τi yit − x⊤ it βi , for i = 1, 2. Let q̂i,t,s (τi ) = x⊤ it β̂i,s (τi ). We consider the minimum subsample size s larger than [T ω], where ω ∈ (0, 1) is an arbitrary small positive constant. The trimming parameter, ω, 6 Kuan and Lee (2006) apply the approach to a class of specification tests, the so-called M tests, which are based on the moment conditions involving unknown parameters. Chen and Qu (2015) propose a procedure for improving the power of the M test, by dividing the original sample into subsamples before applying the self-normalization procedure. 7 The fixed-b asymptotic has been further studied by Bunzel et al. (2001), Kiefer and Vogelsang (2002, 2005), Sun et al. (2008), Kim and Sun (2011) and Sun and Kim (2012) among others. 13 is necessary to guarantee that the quantiles estimators based on subsamples have standard asymptotic properties and plays a different role to that of smoothing parameters in long-run variance estimators. Our simulation study suggests that the performance of the test is not sensitive to the trimming parameter. A key ingredient of the self-normalized statistic is an estimate of cross-correlation based on subsamples: Ps ψτ1 (y1t − q̂1,t,s (τ1 ))ψτ2 (y2,t−k − q̂2,t−k,s (τ2 )) qP , s 2 2 t=k+1 ψτ1 (y1t − q̂1,t,s (τ1 )) t=k+1 ψτ2 (y2,t−k − q̂2,t−k,s (τ2 )) ρ̂τ,s (k) = qP s t=k+1 (p) ⊤ for [T ω] ≤ s ≤ T . For a finite integer p > 0, let ρ̂τ,s = [ρ̂τ,s (1), . . . , ρ̂τ,s (p)] . We construct an outer product of the cross-quantilogram using the subsample V̂τ,p = T −2 T X s=[T ω] (p) s2 ρ̂τ,s − ρ̂τ(p) (p) ρ̂τ,s − ρ̂τ(p) ⊤ . We can obtain the asymptotically pivotal distribution using V̂τ,p as the asymptotically random normalization. For testing the null of no directional predictability, we define the selfnormalized omnibus test statistic ⊤ −1 (p) Ŝτ(p) = T ρ̂τ(p) V̂τ,p ρ̂τ . (p) The following theorem shows that Ŝτ is asymptotically pivotal. To distinguish the process used in the following theorem from the one used in the previous section, let {B̄(p) (·)} denote a p-dimensional, standard Brownian motion on (ℓ([0, 1]))p equipped with the uniform topology. Theorem 4 Suppose that Assumptions A1-A5 hold. Then, for each τ ∈ T , Ŝτ(p) →d B̄(p) (1) where V̄(p) = R1 ω ⊤ V̄(p) −1 B̄(p) (1), ⊤ {B̄(p) (r) − r B̄(p) (1)}{B̄(p) (r) − r B̄(p) (1)} dr. 14 The joint test based on finite multiple quantiles can be constructed in a similar manner, while the extension of the self-normalized approach to a range of quantiles is not obvious. The asymptotic null distribution in the above theorem can be simulated and a critical value, cS,α , corresponding to a significance level α is tabulated by using the (1 − α)100% percentile of the simulated distribution.8 In the theorem below, we consider a power function of the (p) self-normalized omnibus test statistic, P (Ŝτ > cS,α ). For a fixed τ ∈ T , we consider a fixed alternative H1 : ρτ (k) 6= 0 for some k ∈ {1, . . . , p}, (8) √ H1T : ρτ (k) = ζ/ T for some k ∈ {1, . . . , p}, (9) and a local alternative (p) where ζ is a finite non-zero scalar. This implies that there exists a p-dimensional vector ζτ (p) (p) such that ρτ = T −1/2 ζτ (p) with ζτ having at least one non-zero element. Theorem 5 (a) Suppose that the fixed alternative in (8) and Assumptions A1-A5 hold. Then, lim P Ŝτ(p) > cS,α → 1. T →∞ (b) Suppose that the local alternative in (9) is true and Assumptions A1-A5 hold. Then, (p) lim P Ŝτ > cS,τ = P B̄(p) (1) + (Λτ(p) ∆τ(p) )−1 ζτ(p) T →∞ (p) (p) (p) ⊤ ⊤ V (p) −1 where ∆τ is a d0 p × d0 p matrix with ∆τ (∆τ ) ≡ Ξ(p) (τ, τ ). 8 We provide the simulated critical values in our R package. 15 (p) (p) (p) −1 (p) ≥ cS,α , B̄ (1) + (Λτ ∆τ ) ζτ 4 The Partial Cross-Quantilogram We define the partial cross-quantilogram, which measures the relationship between two events {y1t ≤ q1,t (τ1 )} and {y2,t−k ≤ q2,t−k (τ2 )}, while controlling for intermediate events between t and t − k as well as whether some state variables exceed a given quantile. Let zt ≡ [ψτ3 (y3t − ⊤ q3,t (τ3 )), . . . , ψτl (ylt − ql,t (τl ))] be an (l − 2) × 1 vector for l ≥ 3, where qi,t (τi ) = x⊤ it βi (τi ) for τi and a di × 1 vector xit (i = 3, . . . , l), and zt may include the quantile-hit processes based on some of the lagged predicted variables {y1,t−1 , . . . , y1,t−k }, the intermediate predictors {y2,t−1 , . . . , y1,t−k−1} and some state variables that may reflect some historical events up to t.9 ⊤ For simplicity, we present the results for a single set of quantiles τ̄ = (τ1 , . . . , τl ) and a single lag k, although the results can be extended to the case of a range of quantiles and multiple lags in an obvious way. To ease the notational burden in the rest of this section, we consider the case for which a lag k = 0 without loss of generality and suppress the ⊤ ⊤ dependence on k. Let ȳt = [y1t , . . . , ylt ]⊤ and x̄t = [x⊤ 1t , . . . , xlt ] . We introduce the correlation matrix of the hit processes and its inverse matrix i h ⊤ and Pτ̄ = Rτ̄−1 , Rτ̄ = E ht (τ̄ )ht (τ̄ ) where an l × 1 vector of the hit process is denoted by ht (τ̄ ) = [ψτ1 (y1t − q1,t (τ1 )), . . . , ψτl (ylt − ⊤ ql,t (τl ))] . For i, j ∈ {1, . . . , l}, let rτ̄ ,ij and pτ̄ ,ij be the (i, j) element of Rτ̄ and Pτ̄ , re√ spectively. Notice that the cross-quantilogram is rτ̄ ,12 / rτ̄ ,11 rτ̄ ,22 , and the partial crossquantilogram is defined as ρτ̄ \|z = − √ pτ̄ ,12 . pτ̄ ,11 pτ̄ ,22 9 In principle, the intermediate predictors and state variables do not need to be transformed into quantile hits. As emphasized earlier, however, one of the main advantages of considering qauntile hits is its applicability to more general time series, being robust to the existence of moments. If needed, it is straightforward to extend the results here to the case of the original variables in zt with additional moment conditions. We thank an anonymous referee for pointing this out. 16 The partial cross-correlation also has a form ρτ̄ \|z = δ s τ1 (1 − τ1 ) , τ2 (1 − τ2 ) where δ is a scalar parameter defined in the following regression: ψτ1 (y1t − q1,t (τ1 )) = δψτ2 (y2t − q2,t (τ2 )) + γ ⊤ zt + ut , with a (l − 2) × 1 vector γ and an error term ut . Thus, testing the null hypothesis of ρτ̄ \|z = 0 can be viewed as testing predictability between two quantile hits with respect to information z̄ as in Granger causality test based on the regression form (Granger, 1969). By choosing relevant variables z̄, one can use ρτ̄ \|z for the purpose of testing Granger causality (Pierce and Haugh, 1977). See also Hong et al. (2009) for testing Granger causality in tail distribution. To obtain the sample analogue of the partial cross-quantilogram, we first construct a vector of hit processes, ĥt (τ̄ ), by replacing the population conditional quantiles in ht (τ̄ ) by the sample analogues {q̂1,t (τ1 ), . . . , q̂l,t (τl )}. Then, we obtain the estimator for the correlation matrix and its inverse as R̂τ̄ = T 1X ⊤ ĥt (τ̄ )ĥt (τ̄ ) and P̂τ̄ = R̂τ̄−1 , T t=1 which leads to the sample analogue of the partial cross-quantilogram ρ̂τ̄ \|z = − p p̂τ̄ ,12 , p̂τ̄ ,11 p̂τ̄ ,22 (10) where p̂τ̄ ,ij denotes the (i, j) element of P̂τ̄ for i, j ∈ {1, . . . , l}. In Theorem 6 below, we show that ρ̂τ̄ \|z asymptotically follows a normal distribution, while the asymptotic variance depends on nuisance parameters as in the previous section. To address the issue of the nuisance parameters, we may employ the stationary bootstrap or the 17 self-normalization technique. For the bootstrap, we can use pairs of variables {(ȳt , x̄t )}Tt=1 to generate the stationary bootstrap resample {(ȳt∗ , x̄∗t )}Tt=1 and then obtain the stationary bootstrap version of the partial cross-quantilogram, denoted by ρ̂∗τ̄ \|z , using the formula in (10). When we use the self-normalized test statistics, we estimate the partial cross-quantilogram ρτ̄ ,s\|z based on the subsample up to s, recursively and then use V̂τ̄ \|z = T −2 T X s=[T ω] s2 ρ̂τ̄ ,s\|z − ρ̂τ̄ ,T \|z 2 , to normalize the cross-quantilogram, thereby obtaining the asymptotically pivotal statistics. To obtain the asymptotic results, we impose the following conditions on the conditional distribution function Fyi \|xi (·\|xit ) and its density function fyi \|xi (·\|xit ) of each pair of additional variables (yit , xit ) for i = 1, . . . , l and on the pairwise joint distribution Fij (v1 , v2 \|xit , xjt ) ≡ P (yit ≤ v1 , yjt ≤ v2 \|xit , xjt ) for (v1 , v2 ) ∈ R2 . Assumption A7. (a) {(ȳt , x̄t )}t∈Z is a strictly stationary and strong mixing sequence satisfying the condition in Assumption A1; (b) The conditions in Assumption A2 and A3 hold for the Fyi \|xi (·\|xit ) and fyi \|xi (·\|xit ) at the relevant quantile for t = 1, . . . , T , for i = 1, . . . , l; (c) Fij (·\|xit , xjt ) satisfies the condition in Assumption A4 and there exists a vector ⊤ ∇r Gij ≡ ∂/∂br E[Fij (x⊤ it b1 , xjt b2 \|xit , xjt )] evaluated at (b1 , b2 ) = (βi (τi ), βi (τj )) for (r, i, j) ∈ {1, 2}×{1, . . . , l}2 ; (d) There exist positive definite matrices Mi and Di (τi ) as in Assumption A5 for i = 1, . . . , l. Assumption A7(a) requires the same weak dependence property as in Assumption A1. Assumptions A7(b)-(c) ensure the smoothness of the marginal conditional distribution, marginal density function and the joint distribution of each pair (yit , yjt ) given (xit , xjt ) for 1 ≤ i, j ≤ l. Assumption A7(d) is used to derive a Bahadur representation of q̂it (τi ) for i = 1, . . . , l. We now state the asymptotic properties of the partial cross-quantilogram and the related 18 inference methods. Theorem 6 (a) Suppose that Assumption A7 holds. Then, √ T (ρ̂τ̄ \|z − ρτ̄ \|z ) →d N(0, στ̄2\|z ), for each τ̄ ∈ [0, 1]l , where στ̄2\|z = ξτ̄ t = − X 1≤i,j≤l i6=j and λτ̄ i = P P∞ l=−∞ cov(ξτ̄ l , ξτ̄ 0 ) with pτ̄ ,1i pτ̄ ,2j ψτi (yit − qi,t (τi ))ψτj (yjt − qj,t (τj )) + 1≤j≤l j6=i l X i=1 −1 λ⊤ τ̄ i Di (τi ) xit ψτi (yit − qi,t (τi )), (pτ̄ ,1i pτ̄ ,2j + pτ̄ ,2i pτ̄ ,1j ) ∇1 Gij . (b) Suppose that Assumption A6 and A7 hold. Then, sup P ∗ ρ̂∗τ̄ \|z ≤ s − P ρ̂τ̄ \|z ≤ s →p 0, s∈R for each τ̄ ∈ [0, 1]l . (c) Suppose that Assumption A7 holds. Then, under the null hypothesis that ρτ̄ \|z = 0, we have √ T ρ̂τ̄ \|z 1/2 V̂τ̄ \|z for each τ̄ ∈ [0, 1]l . →d nR 1 ω B(1) {B(1) − rB(r)}2dr o1/2 , We can show that the partial cross-quantilogram has non-trivial local power against a √ sequence of T -local alternatives, applying the similar arguments used in Theorem 3 and Theorem 5, and thus we omit the details. 19 5 Monte Carlo Simulation We investigate the finite sample performance of our test statistics. We adopt the following simple VAR model with covariates and consider two data generating processes for the error terms. y1t = 0.1 + 0.3y1,t−1 + 0.2y2,t−1 + 0.3z1t + u1t y2t = 0.1 + 0.2y2,t−1 + 0.3z2t + u2t , where zit ∼ iid χ2 (3)/3 for i = 1, 2. DGP1: (u1t , u2t )⊤ ∼ iid N (0, I2 ) where I2 is a 2 ×2 identity matrix. We let (u1t , u2t , z1t , z2t ) be mutually independent. DGP2:       u1t   σ1t 0   ε1t    =  ε2t 0 1 u2t 2 2 where (ε1t , ε2t )⊤ ∼ iid N (0, I2) and σ1t = 0.1+0.2u21,t−1+0.2σ1,t−1 +u22,t−1 . We let (ε1t , ε2t , z1t , z2t ) be mutually independent. The sample cross-quantilogram defined in (2) adopts conditional quantiles q̂it (τi ) = ⊤ ⊤ ⊤ x⊤ by quantile regression of the above it β̂i (τi ). We first estimate β(τ ) ≡ [β1 (τ1 ) , β2 (τ2 ) ] VAR model, where x1t = (1, y1,t−1, y2,t−1 , z1t )⊤ and x2t = (1, y2,t−1 , z2t )⊤ and then obtain the sample cross-quantilogram using q̂it (τi ) = x⊤ it β̂i (τi ). Under DGP1, there is no predictability from the event {y2,t−k ≤ q2,t−k (τ2 )} to the event {y1t ≤ q1t (τ1 )} for all quantiles τ1 and τ2 , because Pr [y1t ≤ q1t (τ1 ) \| y2,t−k , x2,t−k ] = Pr [u1t ≤ Φ−1 (τ1 )] = τ1 for all t ≥ 1 and τ1 ∈ (0, 1), where Φ denotes the standard normal cdf. Under DGP2, (u1t ) is defined as the GARCH-X process, where its conditional variance is the GARCH(1,1) process with an exogenous covariate. The GARCH-X process is 20 commonly used for modeling volatility of economic or financial time series in the literature, see Han (2015) and references therein. Under DGP2, there exists predictability from 2 {y2,t−k ≤ q2,t−k (τ2 )} to {y1t ≤ q1t (τ1 )} through σ1t for all quantiles (τ1 ,τ2 ) ∈ (0, 1)2 , except the case τ1 = 0.5 because the conditional distribution of u1t given x1t is symmetric around 0.10 5.1 Results Based on the Bootstrap Procedure We first examine the finite-sample performance of the Box-Ljung test statistics based on the stationary bootstrap procedure. To save space, only the results for the case where τ1 = τ2 are reported here because the results for the cases where τ1 6= τ2 are similar. The Box-Ljung (p) test statistics Q̂τ are based on ρ̂τ (k) for τi = 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9 or 0.95 and k = 1, 2, . . . , 5. Tables 1 and 2 report empirical rejection frequencies of the Box-Ljung test statistics based on the bootstrap critical values at the 5% level. The sample sizes considered are T =500, 1,000 and 2,000. The number of simulation repetitions is 1,000. The bootstrap critical values are based on 1,000 bootstrapped replicates. The tuning parameter γ is set to be 0.01.11 In general, our simulation results in Tables 1-3 show that the test has reasonably good size and power performance in finite samples. Table 1 reports the simulation results for the DGP1, which show the size performance. The rejection frequencies are close to 0.05 in mid quantiles, while the test tends to slightly under-reject in low and high quantiles. Table 2 reports the simulation results for the DGP2, which show the power performance. Except for the median, the rejection frequencies approach one as the sample size increases, which shows that our test is consistent. As expected, the rejection frequencies are close to 10 To see this, note that the conditional distribution of u1t given x1t has median zero because Pr(u1t ≤ 0 \| x1t ) = Pr(σ1t ε1t ≤ 0 \| x1t ) = Pr(ε1t ≤ 0 \| x1t ) = Pr(ε1t ≤ 0) = 0.5 and likewise Pr(u1t ≥ 0 \| x1t ) = 0.5. Therefore, letting Ft = (y2,t−k , x2,t−k ), Pr (y1t < q1,t (0.5) \| Ft ) = Pr (u1t < 0 \| Ft ) = Pr (ε1t < 0 \| Ft ) = 0.5. This implies that there is no predictability from {y2,t−k ≤ q2,t−k (τ2 )} to {y1t ≤ q1,t (τ1 )} at τ1 = 0.5 under DGP2. 11 Recall that 1/γ indicates the average block length. We tried different values for γ including one chosen by the data dependent rule suggested by Politis and White (2004) and the results are still similar particularly for a large sample. The details of the data dependent rule is explained in Section 6. 21 0.05 at the median because there is no predictability at the median under the DGP2 (see Footnote 10 for an explanation). Next, we examine the finite-sample performance of the sup-version of the Box-Ljung test (p) statistic supτ ∈T Q̂τ over a range of quantiles.12 The simulation results in Table 3 show that (p) the sup-version test statistic supτ ∈T Q̂τ also has reasonably good finite sample performance, though it tends to under-reject under DGP1. For DGP2, the rejection frequencies approach one as the sample size increases. 5.2 Results for the Self-Normalized Statistics (p) We also examine the performance of the self-normalized version of Q̂τ under the same setup as above. We fix the trimming constant ω to be 0.1.13 The number of repetitions is 3,000. The empirical sizes of the test are reported in Table 4, where the underlying process is the VAR model with DGP1. The test generally under-rejects under the null hypothesis (DGP1), while at the extreme quantiles (τ = 0.05 or 0.95) the test slightly over-rejects in the small sample (T = 500). This finding is not very surprising because the self-normalized statistic is based on subsamples and at the extreme quantiles there are effectively not enough observations to compute the test statistic accurately. Using the GARCH-X process of DGP2, we obtain empirical powers and present the results in Table 5. With a one-period lag (p = 1), the self-normalized quantilogram at τ1 , τ2 ∈ {0.1, 0.2, 0.8, 0.9} rejects the null by about 23.0-30.0%, 64.3-68.3% and 91.7-94.0% for sample sizes 500, 1,000 and 2,000, respectively. In general, the rejection frequencies increase as the sample size increases, decline as the maximum number of lags p increases, and are not sensitive to the choice of the trimming value. Our results suggest that the selfnormalized statistics may have lower power in finite samples compared with the test statistics based on the stationary bootstrap procedure, see Lobato (2001) for a similar finding. 12 Due to computational burden, we compute the Box-Ljung test statistic as a maximum over nine quantile levels τi = 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9 and 0.95. 13 We also considered 0.03 and 0.05 for ω and the results are similar to those for ω = 0.1. 22 6 Empirical Studies 6.1 Stock Return Predictability We apply the cross-quantilogram to detect directional predictability from an economic state variable to stock returns. The issue of stock return predictability has been very important and extensively investigated in the literature; see Lettau and Ludvigson (2010) for an extensive review. A large literature has considered predictability of the mean of stock return. The typical mean return forecast examines whether the mean of an economic state variable is helpful in predicting the mean of stock return (mean-to-mean relationship). Recently, Cenesizoglu and Timmermann (2008) considered whether the mean of an economic state variable is helpful in predicting different quantiles of stock returns representing left tail, right tail or shoulders of the return distribution. The cross-quantilogram adds one more dimension to analyze predictability compared with the linear quantile regression, and so it provides a more complete picture on the relationship between a predictor and stock returns. Moreover, we can consider very large lags in the framework of the quantilogram. We use daily data from 3 Jan. 1996 to 29 Dec. 2006 with sample size 2,717.14 Stock returns are measured by the log price difference of the S&P 500 index and we employ stock variance as the predictor. The stock variance is treated as an estimate of equity risk in the literature. The risk-return relationship is an important issue in the finance literature; see Lettau and Ludvigson (2010) for an extensive review. The cross-quantilogram can provide a more complete relationship from risk to return, which cannot be examined using existing methods. To measure stock variance, we use the realized variance given by the sum of squared 5-minute returns.15 The autoregressive coefficient for stock variance is estimated to be 0.68 and the unit root hypothesis is clearly rejected. The sample mean and median of stock returns are 0.0003 and 0.0005, respectively. 14 The working paper version of this paper provides the results using the monthly data previously analyzed in Goyal and Welch (2008). 15 The realized variance is obtained from ‘Oxford-Man Institute’s realised library’. 23 (p) In Figures 1-3, we provide the sup-type test statistic supτ ∈T Q̂τ , the cross-quantilogram (p) ρ̂τ (k) and the portmanteau test Q̂τ (we use the Box-Ljung versions throughout) to detect directional predictability from stock variance, representing risk, to stock return. In each graph, we show the 95% bootstrap confidence intervals for no predictability based on 1,000 bootstrapped replicates. The tuning parameter 1/γ is chosen by adapting the rule suggested by Politis and White (2004) (and later corrected in Patton et al. (2009)).16 Since it is for univariate data, we apply it separately to each time series and define γ as the average value. (p) We first examine the sup-version Box-Ljung test statistic supτ ∈T Q̂τ and the results are provided in Figure 1. We consider low and high ranges of quantiles. For the low range, we set T = [0.1, 0.3] and τi = 0.1 + 0.02k for k = 0, 1, · · · , 10. For the high range, we set T = [0.7, 0.9] and τi = 0.7 + 0.02k for k = 0, 1, · · · , 10. In each range, there are eleven different values of τi and we let τ1 = τ2 in calculating ρ̂τ (k) for simplicity. Figure 1 clearly shows that there exists predictability from stock variance to stock return in each range. (p) Next we investigate the cross-quantilogram ρ̂τ (k) and the portmanteau test Q̂τ for different quantile points in Figures 2(a)-3(b). For the quantiles of stock return q1 (τ1 ), we consider τ1 = 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9 and 0.95. For the quantiles of stock variance q2 (τ2 ), we consider τ2 = 0.1 and 0.9. Figures 2(a) and 2(b) are for the case when the stock variance is in the low quantile, i.e. τ2 = 0.1. The cross-quantilograms ρ̂τ (k) for τ1 = 0.05, 0.1, 0.2 and 0.3 are negative and significant for many lags. For example, in case of τ1 = 0.05, it means that when risk is very low, it is less likely to have a large negative loss. On the other hand, the cross-quantilograms for τ1 = 0.7, 0.8, 0.9 and 0.95 is positive and significant for many lags. For example, in case of τ1 = 0.95, it means that when risk is very low, it is less likely to have a large positive gain. However, the cross-quantilogram for τ1 = 0.5 is mostly insignificant, which means that risk is not helpful in predicting whether stock return is located below or above its median. Figure 2(b) shows that the Box-Ljung test statistics are mostly significant except for τ1 = 0.5. 16 Specifically, 1/γ̂ = (2Ĝ2 /D̂SB )1/3 T 1/3 where D̂SB = 2ĝ 2 (0). The definitions of ĝ and Ĝ are given on page 58 of Politis and White (2004). 24 Figures 3(a) and 3(b) are for the case when stock variance is in the high quantile, i.e. τ2 = 0.9. Compared to the previous case of τ2 = 0.1, the cross-quantilograms have similar trends but much larger absolute values. For τ1 = 0.05, the cross-quantilogram ρ̂τ (1) is −0.193, which implies that when risk is higher than its 0.9 quantile, there is an increased likelihood of having a very large negative loss in the next day. For τ1 = 0.95, the cross-quantilogram ρ̂τ (1) is 0.188, which implies that when risk is high (higher than its 0.9 quantile), there is an increased likelihood of having a very large positive gain in the next day. The crossquantilogram for τ1 = 0.5 is mostly insignificant and the Box-Ljung test statistics in Figure 3(b) are mostly significant except for τ1 = 0.5. The results in Figures 1-3 show that stock variance is helpful in predicting stock return and detailed features depend on each quantile of stock variance and stock return. When stock variance is in high quantile, the absolute value of the cross-quantilogram is higher and the cross-quantilogram is significantly different from zero for larger lags. Our results exhibit a more complete relationship between risk and return and additionally show how the relationship changes for different lags. 6.2 Systemic Risk The Great Recession of 2007-2009 has motivated researchers to better understand systemic risk—the risk that the intermediation capacity of the entire financial system can be impaired, with potentially adverse consequences for the supply of credit to the real economy. One approach to measure systemic risk is measuring co-dependence in the tails of equity returns of an individual financial institution and the financial system.17 Prominent examples include the work of Adrian and Brunnermeier (2011), Brownlees and Engle (2012) and White et al. (2012). Since the cross-quantilogram measures quantile dependence between time series, we apply it to measure systemic risk. 17 Bisias et al. (2012) categorize the current approaches to measuring systemic risk along the following lines: 1) tail measures, 2) contingent claims analysis, 3) network models, and 4) dynamic stochastic macroeconomic models. 25 We use the daily CRSP market value weighted index return as the market index return as in Brownlees and Engle (2012). We consider returns on JP Morgan Chase (JPM), Morgan Stanley (MS) and AIG as individual financial institutions. As in Brownlees and Engle (2012), JPM, MS and AIG belong to the Depositories group, the Broker-Dealers group and the Insurance group, respectively. We investigate the cross-quantilogram ρ̂τ (k) between an individual institution’s stock return and the market index return for k = 60 and τ1 = τ2 = 0.05. In each graph, we show the 95% bootstrap confidence intervals for no quantile dependence based on 1,000 bootstrapped replicates. The sample period is from 24 Feb. 1993 to 31 Dec. 2014 with sample size 5,505.18 The data including the financial crisis from 2007 and 2009 might not be suitable to be viewed as a strictly stationary sequence and hence may not fit into our theoretical framework.19 Nevertheless, we provide the empirical results because it would be practically interesting to consider a sample period that includes the recent crisis and post-crisis.20 In Figure 4, each graph in the left column shows the cross-quantilogram from each individual institution to the market. The cross-quantilograms are positive and generally significant for large lags. The cross-quantilogram from JPM to the market reaches its peak (0.146) at k = 12 and declines steadily afterwards. This means that it takes about two weeks for the systemic risk from JPM to reach its peak once JPM is in distress. From MS to the market, the cross-quantilogram reaches its peak (0.127) at k = 2. From AIG to the market, the cross-quantilogram reaches its peak (0.127) at k = 17. When AIG is in distress, the systemic risk from AIG takes a longer time (about three weeks) to reach its peak. When an individual financial institution is in distress, each institution makes an influence on the market in a different way. Each graph in the right column of Figure 4 shows the cross-quantilograms from the 18 The stock return series of Morgan Stanley are available from 24 Feb. 1993. The stock return series of individual financial institutions are obtained from Yahoo Finance. 19 A rigorous treatment of nonstationary time series in our context is a challenging issue and will be reported in a future work. 20 The results for the sample period from 24 Feb. 1993 to 29 Dec. 2006 are also available from the authors upon request. 26 market to an individual institution. The cross-quantilogram for this case is a measure of an individual institution’s exposure to system wide distress and therefore it is similar to the stress tests performed by individual institutions. From the market to each institutions, the cross-quantilogram at k = 1 is relatively low for JPM (0.062) and MS (0.073) while it is higher for AIG (0.104). Overall, when the market is in distress, each institution is influenced by its impact in a different way. But the cross-quantilogram reaches its peak at k = 2 for all cases. The cross-quantilograms at k = 2 are 0.135, 0.131 and 0.139 for JPM, MS and AIG, respectively. As shown in Figure 4, the cross-quantilogram is a measure for either an institution’s systemic risk or an institution’s exposure to system wide distress. Compared to existing methods, one important feature of the cross-quantilogram is that it provides in a simple manner how such a measure changes as the lag k increases. For example, White et al. (2012) adopt an additional impulse response function within the multivariate and multiquantile framework to consider tail dependence for a large k. Moreover, another feature of the cross-quantilogram is that it does not require any modeling. For example, the approach by Brownlees and Engle (2012) is based on the standard multivariate GARCH model and it requires the modeling of the entire multivariate distribution. Next, we apply the partial cross-quantilogram to examine the systemic risk after controlling for an economic state variable. Following Adrian and Brunnermeier (2011) and Bedljkovic (2010), we adopt the VIX index as the economic state variable. Since the VIX index itself is highly persistent and can be modeled as an integrated process, we instead use the VIX index change, the first difference of the VIX index level, as the state variable. For the quantile of the state variable, i.e. τ3 in (10), we let τ3 = 0.95 because a low quantile of a stock return is generally associated with a rapid increase of the VIX index. Figure 5 shows that the partial cross-quantilograms are still significant in some cases even if their values are generally lower than the values of the cross-quantilograms in Figure 4. This indicates that there still remains systemic risk from an individual institution after 27 controlling for an economic state variable. These significant partial cross-quantilograms will be of interest for the management of the systemic risk of an individual financial institution. 7 Conclusion We have established the limiting properties of the cross-quantilogram in the case of a finite number of lags. Hong (1996) established the properties of the Box-Pierce statistic in the case that p = pn → ∞ : after a location and scale adjustment the statistic is asymptotically normal, see also Hong et. al. (2009) for a related work. No doubt our results can be extended to accommodate this case, although in practice the desirability of such a test is questionable, and the chi-squared type limit in our theory may provide better critical values for even quite long lags. The cross-quantilogram is easy to compute and the bootstrap confidence intervals appear to represent modest enlargements of the Bartlett intervals in the series that we examined. The statistic shows the cross dependence structure of the time series in a granular fashion that is more informative than the usual methods. 28 Appendix In appendix, we use C, C1 , C2 , . . . to denote generic positive constants without further clarification. Appendix A. Asymptotic Results of Cross-Quantilogram Lemma A1 Let {zt }t∈Z be a strict stationary, strong mixing sequence of Rd -valuedPrandom variables for some integer d ≥ 1 with strong mixing coefficients {αj }j∈Z+ satisfying ∞ j=0 (j + ν/(2s+ν) 1)2s−2 αj ∞. Then, for some integer s ≥ 2 and ν ∈ (0, 1). Suppose that E[z1 ] = 0 and kz1 k2s+ν < E T X t=1 zt 2s 1−s kz1 k2s ≤ T s C kz1 k2s 2s+ν . 2+ν + T Proof. See Supplemental Material. We define the process indexed by τ ∈ T : T o 1 X n (k) 1[yt,k ≤ qt,k (τ )] − E[Fy\|x (qt,k (τ )\|xt,k )] . Vt,k (τ ) := √ T t=k+1 Also, define a di × 1 vector of random variables indexed by τi ∈ Ti for each i = 1, 2: T 1 X Wi,T (τi ) := √ xit ψτi (yit − qi,t (τi )) . T t=1 The below lemma shows the stochastic equicontinuity of the processes defined above, using a similar argument in Bai (1996). Proposition A1 Suppose Assumption A1-A5 hold. Let k ∈ {1, . . . , p} and P2 define metrics ′ ′ ′ ′ ′ ρi (τi , τi ) = \|τi − τi \| for τi , τi ∈ Ti (i = 1, 2) and a metric ρ(τ, τ ) = i=1 ρi (τi , τi ) for ′ τ, τ ∈ T . Then, (a) VT,k (τ ) is stochastically equicontinuous on (T , ρ); (b) Wi,T (τi ) is stochastically equicontinuous on (Ti , ρi ) for each i = 1, 2. Proof. See Supplemental Material. Because of the importance of the result, we present the central limit theorem for strong mixing sequence in the lemma below. The proof can be found in Corollary 5.1 of Hall and Heyde (1980) or Rio (1997, 2013) among others. A.1 Lemma A2 Suppose that the strict stationary sequence {zt }t∈Z satisfies the strong mixing P ς/(2+ς) condition with E[z1 ] = 0 and E\|z1 \|2+ς < ∞ for some ς ∈ (0, ∞), while ∞ < j=1 αj PT −1/2 2 2 2 2 ∞. Then, limT →∞ E[(T ∈ [0, ∞). If σ > 0, then t=1 zt ) ] = σ for some σ PT −1 −1/2 d σ T t=1 zt → N(0, 1). Define a d0 × 1 vector BT,k (τ ) = [VT,k (τ ), W1,T (τ1 )⊤ , W2,T (τ2 )⊤ ]⊤ for τ ∈ T and k = 1, . . . , p. The following proposition shows the weak convergence of the process {BT,k (τ ) : τ ∈ T }pk=1. Proposition A2 Suppose Assumptions A1-A5 hold. Then, [BT,1 (·), . . . , BT,p (·)]⊤ ⇒ [B1 (·), . . . , Bp (·)]⊤ . Proof. Proposition A1 shows that [BT,1 (·), . . . , BT,p (·)]⊤ is stochastic equicontinuous. Thus, it remains to establish convergence of the finite dimensional distributions. By the CramerWold device, it suffices to show J X j=1 θj p X k=1 d (j) 2 κ⊤ , → N 0, σθ,κ k BT,k τ for any {θj ∈ R}Jj=1 , {κk ∈ Rd }pk=1 , {τ (j) ∈ [0, 1]2 }Jj=1, and J ≥ 1, where 2 σθ,κ = J X J X j=1 j ′ =1 θj θj ′ p p X X ′ (j) , τ (j ) )κk′ . κ⊤ k Ξk,k ′ (τ (A-1) k=1 k ′ =1 The original time-series is a stationary sequence satisfying the strong mixing condition in Assumption A1 and a measurable transformation involving lagged variables satisfies the same mixing condition if the lag order is finite. Hence, the central limit theorem for strong-mixing sequences in Lemma A2 shows that the convergence in distribution to the normal law with the finite variance. Therefore, we establish the weak convergence. Let v = (v1 , v2 ) ∈ Rd1 × Rd2 and vt,k = (v1,t , v2,t−k )⊤ ∈ R2 with vi,t = x⊤ it vi for i = 1, 2 and for t = 1, . . . , T . Define T o 1 X n (k) −1/2 −1/2 1[yt,k ≤ qt,k (τ ) + T vt,k ] − E[Fy\|x (qt,k (τ ) + T vt,k \|xt,k )] , VT,k (τ, v) := √ T t=k+1 and T 1 X Wi,T (τi , vi ) := √ xit 1[yit ≤ qi,t (τi ) + T −1/2 vi,t ] − Fyi \|xi (qi,t (τi ) + T −1/2 vi,t \|xit ) . T t=1 Proposition A3 Suppose Assumption A1-A5 hold. Then, A.2 (a) supτ ∈T supv∈VM \|VT,k (τ, v) − VT,k (τ )\| = op (1) for every M > 0; (b) supτi ∈Ti supvi ∈Vi,M kWi,T (τi , vi ) − Wi,T (τi )k = op (1) for every M > 0 and i = 1, 2, where VM = V1,M × V2,M with Vi,M = {vi ∈ Rdi : kvi k ≤ M} for i = 1, 2. Proof. See Supplemental Material. Proposition A4 Suppose Assumption A1-A5 hold. Then, for i = 1, 2 √ T {β̂i (τi ) − βi (τi )} = −Di−1 (τi )Wi,T (τi ) + op (1), uniformly in τ ∈ Ti . Proof. See Supplemental Material. The below lemma shows that the limiting behavior of the cross-quantilogram process reflects the contributions of estimation errors due to the estimation of the conditional quantile function. Proposition A5 Suppose that Assumption A1-A5 hold. Then, for each k ∈ {1, . . . , p}, √ √ VT,k (τ ) + ∇G(k) (τ )⊤ T {β̂(τ ) − β(τ )} p + op (1), T {ρ̂τ (k) − ρτ (k)} = τ1 (1 − τ1 )τ2 (1 − τ2 ) uniformly in τ ∈ T . P Proof. Let γ̂τ,k = T −1 Tt=k+1 ψτ1 (y1t − q̂1,t (τ1 ))ψτ2 (y2,t−k − q̂2,t−k (τ2 )) and γτ,k = E[ψτ1 (y1t − q1,t (τ1 ))ψτ2 (y2,t−k − q2,t−k (τ2 ))]. Using a similar argument in Lemma 2.1 of Arcones (1998), P we can show supτi ∈Ti \|T −1/2 Tt=1 ψτi (yit − q̂i,t (τi ))\| = op (1) for i = 1, 2, because xit includes a constant term. It follows that, uniformly in τ ∈ T , T −1 T X t=1 ψτ2i (yit − q̂i,t (τi )) = τi (1 − τi ) + op (1), for i = 1, 2, (A-2) and √ T (γ̂τ,k − γτ,k ) = T −1/2 T X (k) 1[yt,k ≤ q̂t,k (τ )] − E Fy\|x (qt,k (τ )\|xt,k ) + op (1). t=k+1 Define VM = {v ≡ (v1 , v2 ) ∈ Rd1 × Rd2 : maxi=1,2 kvi k ≤ M} for some M > 0 and let ⊤ ⊤ vt,k = (x⊤ 1t v1 , x2,t−k v2 ) . Then, Proposition A3 implies T −1/2 T X (k) 1[yt,k ≤ qt,k (τ ) + T −1/2 vt,k ] − E Fy\|x (qt,k (τ )\|xt,k ) t=k+1 = VT,k (τ ) + √ (k) (k) T E Fy\|x (qt,k (τ ) + T −1/2 vt,k \|xt,k ) − Fy\|x (qt,k (τ )\|xt,k ) + op (1), A.3 uniformly in (τ, v) ∈ T × VM for any M > 0. Also, the mean-value theorem implies √ (k) (k) T E[Fy\|x (qt,k (τ ) + T −1/2 vt,k \|xt,k ) − Fy\|x (qt,k (τ )\|xt,k )] = ∇G(k) (τ )⊤ v + o(1) uniformly in (τ, v) ∈ T × VM . Thus, for any M > 0, sup (τ,v)∈T ×VM \|RT (τ, v)\| = op (1), (A-3) where RT (τ, v) :=T −1/2 T X (k) 1[yt,k ≤ qt,k (τ ) + T −1/2 vt,k ] − E Fy\|x (qt,k (τ )\|xt,k ) t=k+1 − VT,k (τ ) + ∇G(k) (τ )⊤ v . Let ǫ be an arbitrary positive constant. Proposition A2 √ and A4 imply that there exists a constant M > 0 such that P (supτ ∈T kβ̂(τ ) − β(τ )k > M/ T ) < ǫ for a sufficiently large T . It follows that there exists an M > 0 such that √ sup P sup RT (τ, T {β̂(τ ) − β(τ )}) > ǫ < ǫ + P \|RT (τ, v)\| > ǫ , τ ∈T (τ,v)∈T ×VM for a sufficiently large T . Thus, (A-3) yields √ √ T (γ̂τ,k − γτ,k ) = VT,k (τ ) + ∇G(k) (τ )⊤ T {β̂(τ ) − β(τ )} + op (1), uniformly in τ ∈ T . This together with (A-2) yields the desired result. Proof of Theorem 1. For each i = 1, 2, Proposition A4 yields an asymptotic linear √ −1 approximation, T {β̂i (τi ) − βi (τ √i )} = −Di (τi )Wi,T (τi )⊤+ op (1) uniformly in τi ∈ Ti , which with Proposition A5 shows that T {ρ̂τ (k) − ρτ (k)} = λτ,k BT,k (τ )+op (1) uniformly in τ ∈ T . For a finite p > 0, we have √ (p) T ρ̂τ(p) − ρτ(p) = Λτ(p) BT (τ ) + op (1), (A-4) uniformly in τ ∈ T . The desired result is obtained from Proposition A2 with the continuous mapping theorem. Appendix B. Stationary Bootstrap A positive integer valued, possibly infinite random variable µ is said to be a stopping time with respect to a filtration {Fn , n ≥ 1} if {µ = n} ∈ Fn , ∀n ∈ N. Given random block lengths {Li }i∈N under the stationary bootstrap, define N = inf{i ∈ N : L1 + · · · + Li ≥ n}. Then, N is a stopping time with respect to {σ(L1 , . . . , Li ) : 1 ≤ i ≤ n}. In the following lemma, we present a moment inequality using ideas found in the literature on the sopped random walk process. See Gut (2009) for a comprehensive treatment. Lemma B1 Let {zt }t∈Z be a strict stationary, strong mixing sequence of Rd -valued random A.4 variables for some integer d ≥ 1 with strong mixing coefficients {αj }j∈Z+ satisfying 2s−2 ν/(2s+ν) αj P∞ j=0 (j + 1) for some integer s ≥ 2 and ν ∈ (0, 1). Suppose that kz1 k2s+ν < ∞ and a stationary bootstrap resample, {zt∗ }Tt=1 , from {ztP }Tt=1 satisfies Assumption A6 with the sample Pk+l−1 ∗ zt∗ . Then, size T > 0. Define Sk,l = t=k zt and Sk,l = k+l−1 t=k E where S̃k,l = ∗ S1,T Pk+l−1 t=k −E ∗ ∗ S1,T 2s ∞ n X s πl E S̃1,l ≤ C (T γ) 2s + E S̃1,T l=1 2s o , (zt − Ezt ) for k, l ∈ N. Proof. See Supplemental Material. Lemma B2 Suppose that the same conditions assumed in Lemma B1 hold. Then, 2s s−1 ∗ ∗ ≤ T s C kz1 k2s kz1 k2s E S1,T − E ∗ S1,T 2+ν + γ 2s+ν , for a sufficiently large T . Proof. See Supplemental Material. We now turn to the asymptotic results of cross-quantilogram based on the stationary bootstrap. Define V∗T,k (τ ) and W∗i,T (τi ) T 1 X ∗ √ 1[yt,k ≤ q∗t,k (τ )] − 1[yt,k ≤ qt,k (τ )] := T t=k+1 T 1 X ∗ ∗ √ := xit ψτi yit∗ − qi,t (τi ) − xit ψτi (yit − qi,t (τi )) T t=k+1 for each i = 1, 2. The lemma below shows the stochastic equicontinuity of the processes, V∗T,k (·) and W∗i,T (·), unconditional on the original sample. Proposition B1 Suppose Assumption A1-A6 hold. Let k ∈ {1, . . . , p} and define metrics ρi (·, ·) for i = 1, 2 and a metric ρ(·, ·) as in Proposition A1. Then, (a) V∗T,k (τ ) is stochastically equicontinuous on (T , ρ); (b) W∗i,T (τi ) is stochastically equicontinuous on (Ti , ρi ) for each i = 1, 2. Proof. See Supplemental Material. Let B∗T,k (τ ) = [V∗T,k (τ ), W∗1,T (τ1 )⊤ , W∗2,T (τ2 )⊤ ]⊤ for (k, τ ) ∈ {1, . . . , p} × T and define (p)∗ BT,k (τ ) := [B∗T,1 (τ ), . . . , B∗T,p (τ )]⊤ . As a norm that introduces the topology of (ℓ∞ (T ))pd0 , we use supτ ∈T k · k defined on (ℓ∞ (T ))pd0 , so that supτ ∈T kf (τ )k for any f ∈ (ℓ∞ (T ))pd0 . A.5 Let BL1 be the set of all Lipschitz continuous, real-valued functions on (ℓ∞ (T ))pd0 with a Lipschitz constant bounded by 1. We prove the following proposition by modifying the argument used in Theorem 2 of Galvao et. al. (2014), where the approach of van der Vaart and Wellner (1996, Theorem 2.9.6) is extended for the dependent process but their setup differs from the one here. Proposition B2 Suppose Assumptions A1-A6 hold. Then, (p)∗ sup E ∗ h(BT ) − E h(B(p) ) →p 0. h∈BL1 Proof. Let δ > 0. Given the compact set T in [0, 1]2 , there exists a finite partition (j) (j) {T (j) }Jj=1 such that max1≤j≤J supτ ′ ,τ ′′ ∈T (j) kτ ′′ −τ ′ k ≤ δ. Pick up τ (j) ≡ (τ1 , τ2 )⊤ ∈ T (j) for j = 1, . . . , J and let Πδ be a map from T to {τ (j) }Jj=1 so that Πδ (τ ) = τ (j) if τ ∈ T (j) . Define (p)∗ (p)∗ (p)∗ BT ◦ Πδ and B(p) ◦ Πδ as the stochastic processes on T , given by BT ◦ Πδ (τ ) = BT (Πδ (τ )) and B(p) ◦ Πδ (τ ) = B(p) (Πδ (τ )) for τ ∈ T . It follows from the triangle inequality that, for any h ∈ BL1 , (p)∗ (p)∗ (p)∗ E ∗ h(BT ) − E h(B(p) ) ≤ E ∗ h(BT ) − E ∗ h(BT ◦ Πδ ) (p)∗ + E ∗ h(BT ◦ Πδ ) − E h(B(p) ◦ Πδ ) + E h(B(p) ◦ Πδ ) − E h(B(p) ) . (A-5) (A-6) (A-7) It suffices to show that (A-5) - (A-7) are op (1) uniformly in h ∈ BL1 . We first consider (A-5). We have (p)∗ (p)∗ (p)∗ (p)∗ ∗ ∗ E sup E h(BT ) − E h(BT ◦ Πδ ) ≤ E sup h(BT ) − h(BT ◦ Πδ ) . h∈BL1 h∈BL1 (p)∗ (p)∗ ∗ Let IT,δ,ǫ := 1[supτ ∈T kBT (τ ) − BT ◦ Πδ (τ )k > ǫ] for ǫ > 0. Proposition B1 implies that (p)∗ (p)∗ ∗ ] < ǫ for every ǫ > 0. Also suph∈BL1 \|h(BT ) − h(BT ◦ Πδ )\| ≤ 2 limδ↓0 limT →∞ E[IT,δ,ǫ because the range of a function h is [−1, 1]. It follows that (p)∗ (p)∗ ∗ lim lim E sup h(BT ) − h(BT ◦ Πδ ) · IT,δ,ǫ ≤ 2ǫ. δ↓0 T →∞ (p)∗ h∈BL1 (p)∗ (p)∗ (p)∗ Since suph∈BL1 \|h(BT ) − h(BT ◦ Πδ )\| ≤ supτ ∈T kBT (τ ) − BT ◦ Πδ (τ )k, we have (p)∗ (p)∗ ∗ E sup h(BT ) − h(BT ◦ Πδ ) · (1 − IT,δ,ǫ) ≤ ǫ. h∈BL1 (p)∗ (p)∗ Thus, limδ↓0 limT →∞ E[suph∈BL1 \|h(BT )−h(BT ◦Πδ )\|] ≤ 3ǫ. An application of the Markov inequality yields that (A-5) is op (1) uniformly in h ∈ BL1 . A.6 (p)∗ Next we shall show that suph∈BL1 \|E ∗ [h(BT ◦ Πδ )] − E[h(B(p) ◦ Πδ )]\| →p 0 for any (p)∗ δ > 0. It suffices to show that {BT (τ (j) )}Jj=1 →d {B(p) (τ (j) )}Jj=1 conditional on the original sample, sequence. To this end, we use the Cramer-Wold device Pp every PJfor almost ⊤ ∗ (j) for some {θj ∈ R}Jj=1 and {κk ∈ Rd }pk=1. Let and consider k=1 κk BT,k τ j=1 θj P P P P (j) ∗ (j) ), where ξt,k (·) is defined in ) and vt = Jj=1 θj pk=1 κ⊤ vt∗ = Jj=1 θj pk=1 κ⊤ k ξt,k (τ k ξt,k (τ ∗ Section 3 and ξt,k (·) is its bootstrap counterpart. Then, we can write J X θj j=1 p X ∗ κ⊤ k BT,k k=1 τ (j) =T −1/2 T X (vt∗ − vt ). t=k+1 As discussed in Proposition A2, {vt }t∈N is a stationary time-series satisfying Assumption 1. As shown in p. 1237 of Kunsch (1989), the moment and strong-mixing assumption imposed on the original time series implies the condition imposed on the forth joint cumulant in (8) of Politis and Romano (1994). Hence, Theorems 1 and 2 of Politis and Romano (1994) 2 2 imply that the bootstrap estimate of the variance converges to σθ,κ in probability, where σθ,κ is defined in (A-1), and that we obtained the distribution convergence conditional on the original sample. Finally, consider (A-7). The process B(p) is uniformly continuous on T , which withthe dominated convergence theorem yields that limδ↓0 suph∈BL1 E h(B(p) ◦ Πδ ) − E h(B(p) ) = 0. Hence, we obtain the desired conclusion. ∗ ∗ ∗ ∗ For v = (v1 , v2 ) ∈ Rd1 × Rd2 , let vt,k = (v1,t , v2,t−k )⊤ with vi,t = x∗⊤ it vi for i = 1, 2. Define V∗T,k (τ, v) := T −1/2 T X ∗ ∗ 1[yt,k ≤ q∗t,k (τ ) + T −1/2 vt,k ] − 1[yt,k ≤ qt,k (τ ) + T −1/2 vt,k ] , t=k+1 and W∗i,T (τi , vi ) := T −1/2 T X ∗ ∗ ∗ xit ψτi yit∗ − qi,t (τi ) − T −1/2 vi,t − xit ψτi yit − qi,t (τi ) − T −1/2 vi,t . t=k+1 Proposition B3 Suppose Assumption A1-A6 hold. Then, (a) supτ ∈T supv∈VM \|V∗T,k (τ, v) − V∗T,k (τ )\| = op (1) for every M > 0; (b) supτi ∈Ti supvi ∈Vi,M kW∗i,T (τi , vi ) − W∗i,T (τi )k = op (1) for every M > 0 and i = 1, 2, where VM = V1,M × V2,M with Vi,M = {vi ∈ Rdi : kvi k ≤ M} for i = 1, 2. Proof. See Supplemental Material. Proposition B4 Suppose Assumption A1-A6 hold. Then, for i = 1, 2, √ T {β̂i∗ (τi ) − βi (τi )} = 1 −Di−1 (τi ) √ T A.7 T X t=k+1 ∗ x∗it ψτi (yit∗ − qi,t (τi )) + op (1), uniformly in τi ∈ Ti . Proof. A similar argument used in Proposition A4 completes the proof and thus the details are omitted. Proposition B5 Suppose that Assumption A1-A6 hold. Then, for each k ∈ {1, . . . , p}, √ √ V∗T,k (τ ) + ∇G(k) (τ )⊤ T {β̂ ∗ (τ ) − β̂(τ )} ∗ p + op (1), T {ρ̂τ (k) − ρ̂τ (k)} = τ1 (1 − τ1 )τ2 (1 − τ2 ) uniformly in τ ∈ T . P ∗ ∗ ∗ ∗ ∗ Proof. Let γ̂τ,k = T −1 Tt=k+1 ψτ1 (y1t − q̂1,t (τ1 ))ψτ2 (y2,t−k − q̂2,t−k (τ2 )). Using a similar arguP ment used to show Lemma 2.1 of Arcones (1998), we can show supτi ∈Ti \|T −1/2 Tt=1 ψτi (yit∗ − ∗ q̂i,t (τi ))\| = op (1) for i = 1, 2. It follows that T −1 T X t=k+1 ψτ2i (yit∗ − q̂it∗ (τi )) = τi (1 − τi ) + op (1), for i = 1, 2, and √ T ∗ γ̂τ,k − γ̂τ,k = T −1/2 T X ∗ 1[yt,k ≤ q̂∗t,k (τ )] − 1[yt,k ≤ q̂t,k (τ )] + op (1), t=k+1 uniformly in τi ∈ Ti and τ ∈ T , respectively. As in Proposition A5, we can show T √ (k) 1 X √ 1[yt,k ≤ q̂t,k (τ )]−E Fy\|x (qt,k (τ )xt,k ) = VT,k (τ )+∇G(k) (τ )⊤ T {β̂(τ )−β(τ )}+oP (1), T t=k+1 uniformly in τ ∈ T . A similar argument used in Proposition A5 together with Proposition B3 and A3 yields that, uniformly in τ ∈ T , T h i 1 X ∗ (k) √ 1[yt,k ≤ q̂∗t,k (τ )] − E Fy\|x (qt,k (τ )\|xt,k ) T t=k+1 = VT,k (τ )∗ + VT,k (τ ) √ + ∇G(k) (τ )⊤ T {β̂ ∗ (τ ) − β(τ )} + op (1). √ √ ∗ It follows that T (γ̂τ,k − γ̂τ,k ) = V∗T,k (τ ) + ∇G(k) (τ )⊤ T {β̂ ∗ (τ ) − β̂(τ )} + op (1) uniformly in τ ∈ T . Thus, we obtained the desired result. √ (p)∗ ∗(p) (p) Proof of Theorem 2. (a) Define the processes ĜT (τ ) := T (ρ̂τ − ρ̂τ ) and G(p) (τ ) := (p) Λτ B(p) (τ ) for τ ∈ T and for an integer p > 0. Let g BL1 denote the set of all Lipschitz ∞ p continuous, real-valued functions on (ℓ (T )) with a Lipschitz constant bounded by 1. It A.8 suffices to show that (p)∗ sup E ∗ h(GT ) − E h(G(p) ) →p 0. g1 h∈BL (p)∗ (p) (p)∗ Let GT (τ ) := Λτ BT (τ ). We can write (p)∗ (p)∗ (p)∗ sup E ∗ h(ĜT ) − E h(G(p) ) ≤ sup E ∗ h(ĜT ) − E ∗ h(GT ) g1 h∈BL g1 h∈BL (p)∗ + sup E ∗ h(GT ) − E h(G(p) ) . g1 h∈BL √ Propositions A4 and B4 imply that T {β̂i∗ (τi )−β̂i (τi )} = −Di−1 (τi )W∗1,T (τi )+op(1) uniformly in τi ∈ Ti for each i = 1, 2. It follows from Proposition B3 that √ ∗ T {ρ̂∗τ (k) − ρ̂τ (k)} = λ⊤ τ,k BT,k (τ ) + op (1), uniformly in τ ∈ T , where λτ,k is defined in (4). This leads to (p)∗ (p)∗ sup ĜT (τ ) − GT (τ ) = op (1). τ ∈T (p)∗ (p)∗ (p)∗ (p)∗ This implies that suph∈BL g1 \|h(ĜT )−h(GT )\| = op (1), because suph∈BL g1 \|h(ĜT )−h(GT )\| ≤ (p)∗ (p)∗ supτ ∈T kĜT (τ ) − GT (τ )k. It follows from the dominated convergence theorem that (p)∗ (p)∗ ∗ ∗ limT →∞ E suph∈BL g1 \|E [h(ĜT )] − E [h(GT )]\| = 0. An application of the Markov inequal(p)∗ (p)∗ ∗ ∗ ity shows that suph∈BL g1 \|E [h(ĜT )] − E [h(GT )]\| = op (1). (p) Under Assumption A4 and A5, Λτ is bounded uniformly in τ ∈ T , we have (p)∗ (p)∗ sup E ∗ h(GT ) − E h(G(p) ) ≤ C1 sup E ∗ g(BT ) − E g(B(p) ) , g∈BL1 g1 h∈BL where the right-hand side is negligible in probability from Proposition B2. Hence, we obtain the desired result. (b) From the continuous mapping theorem, the result in (a) of this theorem yields the desired result. See Theorem 10.8 of Kosorok (2007) for a general argument. Proof of Theorem 3. As shown in (A-4), under both fixed and local alternatives, √ (p) T ρ̂τ(p) − ρτ(p) = Λτ(p) BT (τ ) + op (1) (p) (p) uniformly in τ ∈ T , and it follows from Theorem 1 that Λτ BT (τ ) = OP (1) uniformly in τ ∈T. (p) (a) Under the fixed alternative, there is some τ ∈ T such that ρτ is some non-zero √ (p) √ (p) (p) (p) constant and then T ρ̂τ = Λτ BT (τ ) + T ρτ + op (1) uniformly in τ ∈ T . This im(p) (p) plies that, under the fixed alternative, supτ ∈T Q̂τ = T supτ ∈T kρτ k2 (1 + op (1)). Thus, (p) supτ ∈T Q̂τ →p ∞ under the fixed alternative, whereas the critical value c∗Q,τ is bounded in A.9 (p) probability from Theorem 2. Therefore, limT →∞ P (supτ ∈T Q̂τ > c∗Q,τ ) = 1. Therefore, our test is shown to be consistent under the fixed alternative. √ (p) (p) (p) (b) Under the local alternative, we can write ρτ = ζτ / T , where ζτ is a p-dimensional constant vector, at least one of elements is non-zero. Thus, we have (p) Q̂τ(p) = kΛτ(p) BT (τ ) + ζτ(p) k2 + oP (1), uniformly in τ ∈ T . From Theorem 1 and the continuous mapping theorem, sup Q̂τ(p) ⇒ sup kΛτ(p) B(p) (τ ) + ζτ(p)k2 . τ ∈T τ ∈T (p)∗ Also, Theorem 2 implies supτ ∈T Q̂τ desired result follows. (p) ⇒∗ supτ ∈T kΛτ B(p) (τ )k2 in probability. Thus, the Appendix C. Self-Normalized Cross-Quantilogram Lemma C1 Let {zt }t∈Z be a strict stationary, strong mixing sequence of Rd -valuedPrandom variables for some integer d ≥ 1 with strong mixing coefficients {αj }j∈Z+ satisfying ∞ j=0 (j + ν/(2s+ν) 1)2s−2 αj ∞. Then, for some integer s ≥ 2 and ν ∈ (0, 1). Suppose that E[z1 ] = 0 and kz1 k2s+ν < E sup r∈[0,1] [T r] X t=1 zt 2s ≤ CT s z1 2s 2+ν + T 1−s z1 2s+ν . Proof. The desired result follows from Theorem 6.3 and Annexes C of Rio (2013) as in Lemma A1. We define the process indexed by r ∈ [0, 1] [T r] o 1 X n (k) 1[yt,k ≤ qt,k (τ )] − E[Fy\|x (qt,k (τ )\|xt,k )] , V̄T,k,τ (r) := √ T t=k+1 and [T r] 1 X W̄i,T,τi (r) := √ xit {1[yit ≤ qi,t (τi )] − τi } , T t=1 for each i = 1, 2. The following proposition shows the stochastic equicontinuity of the processes defined above. Proposition C1 Suppose Assumption A1-A5 hold. Let k ∈ {1, . . . , p} and define metrics ρ̄(r, r ′ ) = \|r ′ − r\| for r, r ′ ∈ [0, 1]. Then, (a) V̄T,k,τ (r) is stochastically equicontinuous on ([0, 1], ρ̄). A.10 (b) W̄i,T,τi (r) is stochastically equicontinuous on ([0, 1], ρ̄) for each i = 1, 2. Proof. See Supplemental Material. Define a d0 × 1 vector B̄T,k,τ (r) = [V̄T,k,τ (r), W̄1,T,τ1 (r)⊤ , W̄2,T,τ2 (r)⊤ ]⊤ . for r ∈ [0, 1] and k = 1, . . . , p. The following proposition shows the weak convergence of the process {B̄T,k,τ (r) : r ∈ [0, 1]}pk=1 . Proposition C2 Suppose Assumptions A1-A5 hold. Then, B̄T,1,τ (·), . . . , B̄T,p,τ (·) ⊤ ⊤ ⇒ B̄1,τ (·), . . . , B̄p,τ (·) . ⊤ Proof. Proposition C1 establishes the stochastic equicontinuity of B̄T,1,τ (·), . . . , B̄T,p,τ (·) and it suffices to show convergence of the finite dimensional distributions. Since the finite dimensional convergences can be shown by a similar argument used in Proposition A2, we omit the details. For v = (v1 , v2 ) ∈ Rd1 × Rd2 , we define [T r] h io 1 X n (k) 1[yt,k ≤ qt,k (τ ) + T −1/2 vt,k ] − E Fy\|x qt,k (τ ) + T −1/2 vt,k \|xt,k , V̄T,k,τ (r, v) := √ T t=k+1 and [T r] 1 X W̄i,T,τi (r, vi ) := √ xit 1[yit ≤ qi,t (τi ) + T −1/2 vi,t ] − Fyi \|xi qi,t (τi ) + T −1/2 vi,t \|xit . T t=k+1 Proposition C3 Suppose Assumption A1-A5 hold. Then, (a) supω≤r≤1 supv∈VM \|V̄T,k,τ (r, v) − V̄T,k,τ (r)\| = op (1) for every M > 0; (b) supω≤r≤1 supvi ∈Vi,M kW̄i,T,τi (r, vi ) − W̄i,T,τi (r)k = op (1) for every M > 0 and for i = 1, 2, where VM = V1,M × V2,M with Vi,M = {vi ∈ Rdi : kvi k ≤ M} for i = 1, 2. Proof. See Supplemental Material. Proposition C4 Suppose Assumption A1-A5 hold. Then, for i = 1, 2 and for each τi ∈ Ti , √ T {β̂i,[T r] (τi ) − βi (τi )} = −Di−1 (τi )r −1 W̄i,T,τi (r) + op (1), uniformly in r ∈ [ω, 1]. Proof. The proof follows the line of Proposition A4 with Proposition C3(b). Hence, we omit the details. A.11 Proposition C5 Suppose Assumption A1-A5 hold. Then, for each (k, τ ) ∈ {1, . . . , p} × T , √ √ r −1 V̄T,k,τ (r) + ∇G(k) (τ )⊤ T {β̂[T r] (τ ) − β(τ )} p T ρ̂τ,[T r] (k) − ρτ (k) = + op (1), τ1 (1 − τ1 )τ2 (1 − τ2 ) ⊤ ⊤ ⊤ uniformly in r ∈ [ω, 1], where β̂[T r] = (β̂1,[T r] , β̂2,[T r] ) . Proof. A similar argument used in Proposition A5 with Proposition C3(a) yields the desired result and thus we omit the detail. Proof of Theorem 4. Proposition C4 and C5 imply that, for each (k, τ ) ∈ {1, . . . , p} × T , √ T ρ̂τ,[T r] (k) − ρτ (k) = r −1 λ⊤ τ,k B̄T,k,τ (r) + op (1), uniformly in r ∈ [ω, 1]. It follows that in r ∈ [ω, 1]. This implies √ (p) (p) (p) (p) T (ρ̂τ,[T r] − ρτ ) = r −1 Λτ B̄T,τ (r) + op (1) uniformly n o [T r] (p) (p) (p) (p) √ ρ̂τ,[T r] − ρ̂τ,T = Λτ(p) B̄T,τ (r) − r B̄T,τ (1) + op (1), T (p) (p) (p) uniformly in r ∈ [ω, 1]. From Proposition C2, {Λτ (B̄T,τ (r) − r B̄T,τ (1)) : r ∈ [ω, 1]} weakly (p) (p) (p) converges to {Λτ (B̄τ (r) − r B̄τ (1)) : r ∈ [ω, 1]}, which is equivalent in distribution to (p) a p × 1 vector of the Brownian bridge process {∆τ (B̄(p) (r) − r B̄(p) (1)) : r ∈ [ω, 1]} with (p) (p) ∆τ (∆τ )⊤ ≡ Ξ(p) (τ, τ ),and thus it follows from the continuous mapping theorem that √ (p) T ρ̂τ,T , V̂τ,p →d ∆τ(p) B̄(p) (1), ∆τ(p) V̄(p)(∆τ(p) )⊤ . (p) Thus, we obtain Ŝτ →d B̄(p) (1)⊤ (V̄(p) )−1 B̄(p) (1). This completes the proof. Proof of Theorem 5. Under both fixed and local alternative, the argument used in Theorem 4 gives √ (p) (p) T ρ̂τ,[T r] − ρτ(p) = r −1 Λτ(p) B̄T,τ (r) + op (1), (p) (p) (p) (p) ⊤ thereby yielding V̂τ,p ⇒ (Λτ ∆τ )V̄(p) (Λτ ∆τ ) . √ (p) √ (p) (p) (p) (a) Under the fixed alternative, we have T ρ̂τ,T = Λτ B̄T,τ (1) + T ρτ + op (1), where the right-hand side diverges in probability as T → ∞. Since the critical value we use is finite in probability from Theorem 4, we obtain result. √ (p)the desired (p) (p) (p) (b) Under the local alternative, T ρ̂τ,T = Λτ B̄T,τ (1) + ξτ + op (1). It follows that Ŝτ(p) →d B̄(p) (1) + (Λτ(p) ∆τ(p) )−1 ξτ(p) ⊤ V̄(p) This completes the proof. A.12 −1 (p) B̄ (1) + (Λτ(p) ∆τ(p) )−1 ξτ(p) . Appendix D. Partial Cross-Quantilogram For 1 ≤ i, j ≤ l, let 1ij = 1[yit ≤ qi,t (τi ), yjt ≤ qj,t (τj )] and define VT,ij T T 1 X 1 X =√ (1ij − E [1ij ]) and Wi,T = √ xit ψτi (yit − qi,t (τi )) . T t=1 T t=1 Proof of Theorem 6. We first consider (a). The correlation matrix Rτ̄ is symmetric and R̂τ̄−1 − Rτ̄−1 = −R̂τ̄−1 (R̂τ̄ − Rτ̄ )Rτ̄−1 . It follows that vec(P̂τ̄ − Pτ̄ ) = −Pτ̄ ⊗ P̂τ̄ vec(R̂τ̄ − Rτ̄ ), which implies l l X X √ √ pτ̄ ,1i p̂τ̄ ,2j T (r̂τ̄ ,ij − rτ̄ ,ij ). T (p̂τ̄ ,12 − pτ̄ ,12 ) = − i=1 j=1 Following the line of proof of Theorem 1, we can show P̂τ̄ = Pτ̄ + op (1) and also have √ T (r̂τ̄ ,ii − rτ̄ ,ii ) = op (1) for i = 1, . . . , l, from argument in Lemma 2.1 of Arcones (1998). Thus, we have X √ √ pτ̄ ,1i pτ̄ ,2j T (r̂τ̄ ,ij − rτ̄ ,ij ) + op (1). T (p̂τ̄ ,12 − pτ̄ ,12 ) = − 1≤i,j≤l i6=j Proposition A5 implies √ √ T (r̂τ̄ ,ij − rτ̄ ,ij ) = VT,ij + ∇1 G⊤ ij T {β̂i (τi ) − βi (τi )} √ + ∇2 G ⊤ ij T {β̂j (τj ) − βj (τj )} + op (1), for 1 ≤ i, j ≤ l with i 6= j. Since VT,ij = VT,ji and ∇2 Gij = ∇1 Gji for 1 ≤ i, j ≤ l, √ T (p̂τ̄ ,12 − pτ̄ ,12 ) = − X 1≤i,j≤l i6=j pτ̄ ,1i pτ̄ ,2j VT,ij − l X i=1 √ λ⊤ τ̄ i T {β̂i (τi ) − βi (τi )} + op (1), where λτ̄ i is defined in Theorem 6. Proposition A4 implies √ T (p̂τ̄ ,12 − pτ̄ ,12 ) = − X pτ̄ ,1i pτ̄ ,2j VT,ij + l X −1 λ⊤ τ̄ i Di (τi ) Wi,T + op (1). i=1 1≤i,j≤l i6=j The asymptotic normality can be established by using the central limit theorem for mixing random vectors. The proofs of (b) and (c) are similar to those of Theorems 2 and 4, respectively, and thus we omit the details. A.13 Appendix E. Tables and Figures Table 1. (size) Empirical rejection frequency of the Box-Ljung test statistic the bootstrap procedure (VAR with DGP1 and the nominal level 5%) Quantiles (τ1 = τ2 ) T p 0.05 0.10 0.20 0.30 0.50 0.70 0.80 0.90 500 1 0.051 0.025 0.037 0.045 0.040 0.043 0.043 0.033 2 0.017 0.032 0.043 0.072 0.068 0.060 0.057 0.036 3 0.011 0.022 0.051 0.073 0.066 0.055 0.050 0.032 4 0.007 0.022 0.047 0.062 0.059 0.057 0.046 0.026 5 0.009 0.025 0.035 0.052 0.051 0.052 0.054 0.027 1000 1 0.033 0.030 0.037 0.048 0.047 0.039 0.037 0.052 2 0.018 0.037 0.045 0.051 0.043 0.046 0.052 0.041 3 0.011 0.031 0.049 0.056 0.044 0.054 0.045 0.028 4 0.013 0.027 0.049 0.053 0.041 0.055 0.041 0.022 5 0.007 0.022 0.044 0.040 0.044 0.040 0.036 0.021 2000 1 0.038 0.034 0.040 0.034 0.034 0.048 0.050 0.034 2 0.028 0.025 0.043 0.035 0.045 0.051 0.050 0.035 3 0.023 0.033 0.031 0.045 0.050 0.045 0.042 0.029 4 0.017 0.023 0.042 0.052 0.038 0.036 0.038 0.025 5 0.009 0.025 0.038 0.038 0.035 0.035 0.034 0.019 (p) Q̂τ based on 0.95 0.047 0.012 0.010 0.008 0.006 0.042 0.015 0.006 0.008 0.006 0.054 0.024 0.018 0.016 0.014 Notes: The first and second columns report the sample size T and the number of lags p for the Box(p) Ljung test statistics Q̂τ , respectively. The rest of columns show empirical rejection frequencies based on bootstrap critical values at the 5% significance level. The tuning parameter γ is set to be 0.01. A.14 (p) Table 2. (power) Empirical rejection frequency of the Box-Ljung test statistic Q̂τ based on the bootstrap procedure (VAR with DGP2 (GARCH-X process)) Quantiles (τ1 = τ2 ) T p 0.05 0.10 0.20 0.30 0.50 0.70 0.80 0.90 0.95 500 1 0.361 0.701 0.722 0.383 0.042 0.383 0.713 0.684 0.362 2 0.303 0.610 0.584 0.257 0.063 0.231 0.589 0.589 0.300 3 0.270 0.541 0.491 0.202 0.053 0.174 0.467 0.515 0.246 4 0.230 0.451 0.403 0.172 0.058 0.126 0.378 0.447 0.208 5 0.203 0.393 0.344 0.134 0.060 0.115 0.314 0.386 0.177 1000 1 0.751 0.948 0.942 0.638 0.048 0.619 0.951 0.952 0.760 2 0.708 0.916 0.912 0.425 0.046 0.431 0.908 0.932 0.712 3 0.651 0.877 0.845 0.322 0.052 0.315 0.849 0.897 0.651 4 0.589 0.838 0.784 0.255 0.048 0.250 0.778 0.854 0.596 5 0.537 0.801 0.716 0.203 0.042 0.190 0.714 0.809 0.563 2000 1 0.969 0.999 0.999 0.905 0.044 0.923 0.999 0.998 0.974 2 0.965 1.000 0.999 0.808 0.053 0.817 0.999 1.000 0.979 3 0.959 1.000 0.997 0.688 0.053 0.673 0.998 1.000 0.967 4 0.944 1.000 0.990 0.585 0.047 0.573 0.994 0.999 0.957 5 0.930 1.000 0.982 0.510 0.037 0.485 0.987 0.997 0.938 Notes: Same as Table 1. A.15 Table 3. Empirical Rejection Frequencies of the sup-version of the Box-Ljung test statistic (p) supτ ∈T Q̂τ based on the bootstrap procedure (VAR with DGP1/DGP2 and the nominal level 5%) T p DGP1 (size) DGP2 (power) 500 1 0.004 0.624 2 0.007 0.460 3 0.008 0.356 4 0.008 0.265 5 0.009 0.221 1000 1 0.004 0.976 2 0.011 0.946 3 0.006 0.895 4 0.003 0.825 5 0.007 0.765 2000 1 0.012 1.000 2 0.015 1.000 3 0.020 1.000 4 0.020 1.000 5 0.017 0.999 Notes: The first and second columns report the sample size T and the number of lags p for the sup(p) version of the Box-Ljung test statistic supτ ∈T Q̂τ , respectively. The sup-version test statistic is the Box-Ljung test statistic maximized over nine quantiles τi = 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9and 0.95. The third and fourth columns show empirical rejection frequencies based on bootstrap critical values at the 5% significance level. The tuning parameter γ is set to be 0.01. A.16 Table 4. (size) Empirical Rejection Frequencies of the Self-Normalized Statistics (VAR with DGP1 and the nominal level: 5%) Quantiles (τ1 = τ2 ) T p 0.05 0.10 0.20 0.30 0.50 0.70 0.80 0.90 0.95 500 1 0.043 0.000 0.000 0.007 0.003 0.013 0.007 0.000 0.047 2 0.090 0.010 0.007 0.003 0.003 0.003 0.000 0.003 0.127 3 0.130 0.007 0.000 0.007 0.003 0.000 0.003 0.000 0.143 4 0.150 0.007 0.000 0.000 0.000 0.000 0.000 0.000 0.167 5 0.187 0.003 0.000 0.000 0.000 0.000 0.000 0.000 0.177 1000 1 0.010 0.013 0.010 0.013 0.020 0.003 0.007 0.003 0.007 2 0.023 0.007 0.000 0.007 0.000 0.003 0.003 0.007 0.037 3 0.040 0.003 0.010 0.000 0.007 0.003 0.007 0.000 0.047 4 0.043 0.000 0.007 0.000 0.007 0.003 0.003 0.000 0.047 5 0.047 0.000 0.007 0.000 0.000 0.000 0.003 0.000 0.053 2000 1 0.013 0.030 0.017 0.017 0.033 0.013 0.020 0.017 0.027 2 0.007 0.000 0.007 0.007 0.027 0.010 0.027 0.017 0.020 3 0.017 0.000 0.003 0.003 0.013 0.010 0.003 0.003 0.013 4 0.013 0.000 0.003 0.000 0.010 0.007 0.003 0.000 0.013 5 0.010 0.003 0.003 0.000 0.007 0.003 0.000 0.000 0.017 Notes: The first and second columns report the sample size T and the number of lags p for the (p) test statistics Q̂τ , respectively. The rest of columns show empirical rejection frequencies given simulated critical values at 5% significance level. The trimming value ω is set to be 0.1. Table 5. (power) Empirical Rejection Frequencies of the Self-Normalized (VAR with DGP2: GARCH-X process) Quantiles (τ1 = τ2 ) T p 0.05 0.10 0.20 0.30 0.50 0.70 0.80 0.90 500 1 0.067 0.230 0.297 0.077 0.007 0.150 0.300 0.253 2 0.030 0.070 0.113 0.033 0.000 0.037 0.113 0.077 3 0.047 0.010 0.043 0.010 0.000 0.017 0.023 0.020 4 0.063 0.007 0.023 0.000 0.000 0.010 0.013 0.003 5 0.120 0.003 0.007 0.003 0.000 0.003 0.003 0.000 1000 1 0.347 0.643 0.683 0.313 0.010 0.323 0.673 0.663 2 0.153 0.523 0.527 0.177 0.020 0.180 0.543 0.463 3 0.063 0.300 0.347 0.090 0.010 0.097 0.377 0.283 4 0.033 0.210 0.223 0.050 0.000 0.037 0.243 0.153 5 0.047 0.097 0.133 0.030 0.000 0.023 0.127 0.097 2000 1 0.757 0.917 0.923 0.663 0.030 0.693 0.940 0.920 2 0.577 0.873 0.917 0.513 0.013 0.540 0.883 0.863 3 0.427 0.787 0.860 0.400 0.007 0.397 0.800 0.810 4 0.270 0.680 0.807 0.323 0.017 0.297 0.740 0.680 5 0.197 0.567 0.700 0.223 0.003 0.213 0.680 0.590 Notes: Same as Table 4. A.17 Statistics 0.95 0.050 0.010 0.023 0.050 0.080 0.317 0.157 0.063 0.017 0.020 0.707 0.577 0.390 0.250 0.163 Sup−version for low range Sup−version for high range 1400 1800 1600 1200 1400 Portmanteau Portmanteau 1000 800 600 1200 1000 800 600 400 400 200 0 200 0 10 20 30 Lag 40 50 60 0 0 10 (p) 20 30 Lag 40 50 60 Figure 1. Sup-version Box-Ljung test statistic supτ ∈T Q̂τ for each lag p to detect directional predictability from stock variance to stock return. For the low range, we set T = [0.1, 0.3]and τi = 0.1 + 0.02kfor k = 0, 1, . . . , 10. We let τ1 = τ 2 for ρ̂τ (k).For the high range, we set T = [0.7, 0.9] and τi = 0.7 + 0.02k for k = 0, 1, . . . , 10. The dashed lines are the 95% bootstrap confidence intervals centred at the null hypothesis. A.18 τ = 0.05 τ = 0.1 Quantilogram 1 Quantilogram 1 0.2 0.2 0.2 0 0 0 −0.2 0 20 40 τ = 0.3 60 −0.2 0 1 20 40 τ = 0.5 60 −0.2 0.2 0 0 0 0 20 40 τ = 0.8 60 −0.2 0 20 40 τ = 0.9 60 −0.2 0.2 0 0 0 20 40 60 −0.2 0 20 40 Lag 60 20 40 τ = 0.95 60 20 60 1 0.2 0 0 1 0.2 −0.2 20 40 τ = 0.7 1 0.2 −0.2 0 1 0.2 1 Quantilogram τ = 0.2 1 60 −0.2 0 40 Lag Lag Figure 2(a). The sample cross-quantilogram ρ̂τ (k) for τ2 = 0.1 to detect directional predictability from stock variance to stock return. Bar graphs describe sample cross-quantilograms and lines are the 95% bootstrap confidence intervals centred at zero. τ = 0.05 τ = 0.1 Portmanteau 1 500 0 0 20 40 τ = 0.3 60 Portmanteau 1 1000 1000 500 500 0 0 1 Portmanteau τ = 0.2 1 20 40 τ = 0.5 60 60 500 0 0 20 40 60 0 20 40 τ = 0.95 1 60 0 20 60 1 200 50 20 40 τ1 = 0.8 20 40 τ = 0.7 400 100 0 0 1 150 500 0 0 60 0 0 20 40 τ = 0.9 1 60 0 1000 1000 500 500 0 0 Lag 20 40 Lag (p) 60 0 40 Lag Figure 2(b). Box-Ljung test statistic Q̂τ for each lag p and quantile τ using ρ̂τ (k) with τ2 = 0.1. The dashed lines are the 95% bootstrap confidence intervals centred at zero. A.19 τ = 0.05 τ = 0.1 Quantilogram 1 Quantilogram 1 0.2 0.2 0.2 0 0 0 −0.2 0 20 40 τ = 0.3 60 −0.2 0 1 20 40 τ = 0.5 60 −0.2 0.2 0 0 0 0 20 40 τ = 0.8 60 −0.2 0 20 40 τ = 0.9 60 −0.2 0.2 0 0 0 20 40 60 −0.2 0 20 40 Lag 60 20 40 τ = 0.95 60 20 60 1 0.2 0 0 1 0.2 −0.2 20 40 τ = 0.7 1 0.2 −0.2 0 1 0.2 1 Quantilogram τ = 0.2 1 60 −0.2 0 40 Lag Lag Figure 3(a). The sample cross-quantilogram ρ̂τ (k) with τ2 = 0.9 to detect directional predictability from stock variance to stock return. Same as Figure 1(a). τ = 0.05 τ = 0.1 Portmanteau 1 1000 1000 500 500 0 20 40 τ = 0.3 60 0 0 Portmanteau 1 Portmanteau 1 1500 1000 0 τ = 0.2 1 2000 20 40 τ = 0.5 60 60 0 0 20 40 τ = 0.9 1 60 0 1000 2000 2000 500 1000 1000 0 20 40 0 20 40 τ = 0.95 1 60 0 20 60 500 50 0 60 1000 100 20 40 τ1 = 0.8 20 40 τ = 0.7 1 150 0 0 1 500 0 0 60 0 0 Lag 20 40 Lag (p) Figure 3(b). Box-Ljung test statistic Q̂τ Same as Figure 1(b). 60 0 40 Lag for each lag p and quantile τ using ρ̂τ (k) with τ2 = 0.9. A.20 Market to JPM 0.2 0.15 0.15 Quantilogram Quantilogram JPM to Market 0.2 0.1 0.05 0 −0.05 −0.1 0.1 0.05 0 −0.05 0 20 40 −0.1 60 0 20 Lag 0.15 0.15 0.1 0.05 0 −0.05 0.1 0.05 0 −0.05 0 20 40 −0.1 60 0 20 Lag AIG to Market 60 Market to AIG 0.2 0.15 0.15 Quantilogram Quantilogram 40 Lag 0.2 0.1 0.05 0 −0.05 −0.1 60 Market to MS 0.2 Quantilogram Quantilogram MS to Market 0.2 −0.1 40 Lag 0.1 0.05 0 −0.05 0 20 40 60 −0.1 Lag 0 20 40 60 Lag Figure 4. The sample cross-quantilogram ρ̂τ (k). Bar graphs describe sample cross-quantilograms and lines are the 95% bootstrap confidence intervals centred at zero. A.21 Market to JPM 0.2 0.15 0.15 Partial Quantilogram Partial Quantilogram JPM to Market 0.2 0.1 0.05 0 −0.05 −0.1 0 20 40 0.1 0.05 0 −0.05 −0.1 60 0 20 Lag 0.15 0.15 0.1 0.05 0 −0.05 0 20 40 0.1 0.05 0 −0.05 −0.1 60 0 20 Lag 0.15 0.15 0.1 0.05 0 −0.05 20 60 Market to AIG 0.2 Partial Quantilogram Partial Quantilogram AIG to Market 0 40 Lag 0.2 −0.1 60 Market to MS 0.2 Partial Quantilogram Partial Quantilogram MS to Market 0.2 −0.1 40 Lag 40 60 0.1 0.05 0 −0.05 −0.1 Lag 0 20 40 60 Lag Figure 5. The sample partial cross-quantilogram ρ̂τ̄ \|z (k). Bar graphs describe sample partial cross-quantilograms and lines are the 95% bootstrap confidence intervals centred at zero. A.22 References Adrian, T. and M.K. Brunnermeier (2011) CoVaR, Tech. rep., Federal Reserve Bank of New York, Staff Reports. Andrews, D.W.K. and D. 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arXiv:1301.6890v1 [math.AC] 29 Jan 2013 TIGHT CLOSURE WITH RESPECT TO A MULTIPLICATIVELY CLOSED SUBSET OF AN F -PURE LOCAL RING RODNEY Y. SHARP Abstract. Let R be a (commutative Noetherian) local ring of prime characteristic that is F -pure. This paper studies a certain finite set I of radical ideals of R that is naturally defined by the injective envelope E of the simple R-module. This set I contains 0 and R, and is closed under taking primary components. For a multiplicatively closed subset S of R, the concept of tight closure with respect to S, or S-tight closure, is discussed, together with associated concepts of S-test element and S-test ideal. It is shown that an ideal a of R belongs to I if and only if it is the S ′ -test ideal of R for some multiplicatively closed subset S ′ of R. When R is complete, I is also ‘closed under taking test ideals’, in the following sense: for each proper ideal c in I, it turns out that R/c is again F -pure, and if g and h are the unique ideals of R that contain c and are such that g/c is the (tight closure) test ideal of R/c and h/c is the big test ideal of R/c, then both g and h belong to I. The paper ends with several examples. 0. Introduction This paper is concerned with a (commutative Noetherian) local ring R having maximal ideal m and prime characteristic p; the Frobenius homomorphism f : R −→ R, for which f (r) = r p for all r ∈ R, will play a central rôle. Let us (temporarily) use Rf to denote the Abelian group R considered as an (R, R)-bimodule with r1 · r · r2 = r1 rr2p for all r, r1 , r2 ∈ R. The ring R is said to be F -pure if and only if, for every R-module M, the map αM : M −→ Rf ⊗R M for which αM (m) = 1 ⊗ m, for all m ∈ M, is injective. This property can be reformulated in terms of certain left modules over the skew polynomial ring R[x, f ] associated to R and f in the indeterminate x over R, which we refer to as the Frobenius skew polynomial ring over R. Recall that R[x, f ] is, as a left R-module, freely generated by (xi )i∈N0 (we use N0 (respectively N) to denote the set (respectively positive) integers), and so consists of all polynomials Pnof non-negative i i=0 ri x , where n ∈ N0 and r0 , . . . , rn ∈ R; however, its multiplication is subject to the rule xr = f (r)x = r p x for all Lr∞∈ R. Note that R[x, f ] can benconsidered as a positively-graded ring R[x, f ] = n=0 R[x, f ]n , with R[x, f ]n = Rx for n ∈ N0 . If we endow Rxn with its natural structure as an (R, R)-bimodule (inherited from its being a graded component of R[x, f ]), then Rxn is isomorphic (as (R, R)-bimodule) Date: June 22, 2017. 2010 Mathematics Subject Classification. Primary 13A35, 16S36, 13E05, 13E10, 13H05; Secondary 13J10. Key words and phrases. Commutative Noetherian ring, local ring, prime characteristic, Frobenius homomorphism, tight closure, test element, big test element, test ideal, big test ideal, excellent ring, Frobenius skew polynomial ring, F -pure ring, complete local ring. 1 2 RODNEY Y. SHARP to R viewed as a left R-module in the natural way and as a right R-module via f n , the nth iterate of the Frobenius ring homomorphism. In particular, Rx ∼ = Rf as (R, R)-bimodules. A left R[x, f ]-module G is said to be x-torsion-free precisely when xg = 0, where g ∈ G, implies that g = 0. We can say that R is F -pure if and L only if,n for every R-module M, the graded left R[x, f ]-module R[x, f ] ⊗R M = n∈N0 Rx ⊗R M is x-torsion-free. We shall also use the following alternative characterization of F -purity. 0.1. Theorem ([9, Theorem 3.2]). The local ring (R, m) is F -pure if and only if the R-module structure on ER (R/m), the injective envelope of the simple R-module, can be extended to an x-torsion-free left R[x, f ]-module structure. In the paper [7], some useful properties of x-torsion-free left R[x, f ]-modules were developed, and it is appropriate to recall some of them at this point. L The graded two-sided ideals of R[x, f ] are just the subsets of the form n∈N0 an xn , where (an )n∈N0 is an ascending sequence of ideals of R. (Of course, such a sequence a0 ⊆ a1 ⊆ · · · ⊆ an ⊆ · · · is eventually stationary.) Let H be a left R[x, f ]-module. An R[x, f ]-submodule of H is said to be a special annihilator submodule of H if it has the form annH (A) := {h ∈ H : θh = 0 for all θ ∈ A} for some graded two-sided ideal A of R[x, f ]. We shall use A(H) to denote the set of special annihilator submodules of H. The graded annihilator gr-annR[x,f ] H of H is defined to be the largestL graded twosided ideal of R[x, f ] that annihilates H. Note that, if gr-annR[x,f ] H = n∈N0 an xn , then a0 = (0 :R H), which we shall sometimes refer to as the R-annihilator of H. We shall use I(H) (or IR (H) when it is desirable to specify the ring R) to denote the set of R-annihilators of R[x, f ]-submodules of H. 0.2. The Basic Correspondence ([7, §1, §3]). Let G be an x-torsion-free left module over R[x, f ]. (i) By [7, Lemma 1.9], the members of I(G) are all radical ideals of R; they are referred to as the G-special R-ideals;L in fact, the graded annihilator of an R[x, f ]-submodule L of G is equal to n∈N0 axn , where a = (0 :R L). (ii) By [7, Proposition 1.11], there is an order-reversing bijection, Θ : A(G) −→ I(G) given by Θ : N 7−→ gr-annR[x,f ] N ∩ R = (0 :R N). The inverse bijection, Θ−1 : I(G) −→ A(G), also order-reversing, is given by Θ−1 : b 7−→ annG (bR[x, f ]). (iii) By [7, Theorem 3.6 and Corollary 3.7], the set of G-special R-ideals is precisely the set of all finite intersections of prime G-special R-ideals (provided one includes the empty intersection, R, which corresponds under the bijection of part (ii) to the zero special annihilator submodule of G). In symbols, I(G) = {p1 ∩ · · · ∩ pt : t ∈ N0 and p1 , . . . , pt ∈ I(G) ∩ Spec(R)} . (iv) By [7, Corollary 3.11], when G is Artinian as an R-module, the sets I(G) and A(G) are both finite. S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 3 Thus, if (R, m) is F -pure and we endow ER (R/m) with a structure as x-torsionfree left R[x, f ]-module (this is possible, by Theorem 0.1), then the resulting set I(ER (R/m)) is finite. In fact, rather more can be said. 0.3. Theorem ([9, Corollary 4.11]). Suppose that (R, m) is F -pure and local. Then the left R[x, f ]-module R[x, f ] ⊗R ER (R/m) is x-torsion-free, and, furthermore, the set I(R[x, f ] ⊗R ER (R/m)) of (R[x, f ] ⊗R ER (R/m))-special R-ideals, is finite. In fact, for any x-torsion-free left R[x, f ]-module structure on ER (R/m) that extends its R-module structure (and such exist, by Theorem 0.1), we have I(R[x, f ] ⊗R ER (R/m)) ⊆ I(ER (R/m)), and the latter set is finite. It turns out that, in the situation of Theorem 0.3, all the minimal prime ideals of R belong to I(R[x, f ] ⊗R ER (R/m)), and the smallest ideal of positive height in I(R[x, f ] ⊗R ER (R/m)) is the big test ideal of R, that is, the ideal generated by all big test elements for R. (A reminder about big test elements is included in the next section.) In short, to each F -pure local ring (R, m) of characteristic p, there is naturally associated a finite set of radical ideals I(R[x, f ] ⊗R ER (R/m)) of R, and the smallest member of positive height in this set has significance for tight closure theory. The purpose of this paper is to address the following question: what can be said about the other members of I(R[x, f ] ⊗R ER (R/m))? We shall show that, for each multiplicatively closed subset S of R, one can define reasonable concepts of S-tight closure, S-test element and S-test ideal; it turns out that an ideal b of R is a member of I(R[x, f ] ⊗R ER (R/m)) if and only if it is the S-test ideal of R for some choice of multiplicatively closed subset S of R. We shall also show that, when R is complete, I(R[x, f ] ⊗R ER (R/m)) is ‘closed under taking (big) test ideals’, in the following sense: it turns out that, for each proper ideal c ∈ I(R[x, f ] ⊗R ER (R/m)), the ring R/c is again F -pure, and the set I(R[x, f ] ⊗R ER (R/m)) has among its membership the unique ideals g and h of R such that g, h ⊇ c and g/c = τe(R/c), the big test ideal of R/c, and h/c = τ (R/c), the test ideal of R/c. I am grateful to Mordechai Katzman for helpful discussions about the material in this paper. 1. Internal S-tight closure 1.1. Notation. From this point onwards in the paper, R will denote a commutative Noetherian ring of prime characteristic p. We shall only assume that R is local when this is explicitly stated; then, the notation ‘(R, m)’ will denote that m is the maximal ideal of R. As in tight closure theory, we use R◦ to denote the complement in R of the union of the minimal prime ideals of R. The Frobenius homomorphism on R will always be denoted by f . We shall use Φ (or ΦR when it is desirable to specify which ring is being considered) to denote the functor R[x, f ] ⊗R • from the category of R-modules (and all R-homomorphisms) to the category of all N0 -graded left R[x, f ]-modules (and all homogeneous R[x, f ]-homomorphisms). For an R-module M, we shall identify Φ(M) 4 RODNEY Y. SHARP L n with n∈N0 Rx ⊗R M, and (sometimes) identify its 0th component R ⊗R M with M, in the obvious ways. For n ∈ Z, we shall denote the nth component of a Z-graded module L by Ln . Throughout the paper, S will denote a multiplicatively closed subset of R. (We require that each multiplicatively closed subset of R contains 1.) Also throughout the paper, H will denote a left R[x, f ]-module and G will denote an x-torsion-free left R[x, f ]-module. 1.2. Definitions. We define the internal S-tight closure of zero in H, denoted ∆S (H) (or ∆SR (H)), to be the R[x, f ]-submodule of H given by ∆S (H) = {h ∈ H : there exists s ∈ S with sxn h = 0 for all n ≫ 0} . Note that if the left R[x, f ]-module H is Z-graded, then ∆S (H) is a graded submodule. Let M be an R-module, and consider Φ(M), as in 1.1. Now ∆S (Φ(M)) is a graded R[x, f ]-submodule of Φ(M); we refer to the 0th component of ∆S (Φ(M)) as the Stight closure of 0 in M, or the tight closure with respect to S of 0 in M, and denote ∗,S it by 0∗,S when it is clear what M is). M (or by 0 ∗,S Thus 0M is the set of all elements m of M for which there exists s ∈ S such that, ∗,R◦ is the usual for all n ≫ 0, we have sxn ⊗ m = 0 in Rxn ⊗R M. In particular, 0M tight closure of 0 in M. (See M. Hochster and C. Huneke [2, §8].) Now let N be an R-submodule of M. The inverse image of 0∗,S M/N under the natural epimorphism M −→ M/N is defined to be the S-tight closure of N in M, and is ∗,S ∗,S denoted by NM or N ∗,S . Thus NM is the set of all elements m of M for which there exists s ∈ S such that, for all n ≫ 0, the element sxn ⊗ m of Rxn ⊗R M belongs to ∗,R◦ the image of Rxn ⊗R N in Rxn ⊗R M. Note that NM is the usual tight closure of N in M. (See Hochster–Huneke [2, §8].) I am grateful to the referee for pointing out that the S-tight closure of N in M is the tight closure of N in M with respect to C in the sense of Hochster–Huneke [2, Definition (10.1)], where C is {sR : s ∈ S}, the family of principal ideals generated by elements of S, directed by reverse inclusion ⊇. The S-tight closure of a in R is referred to simply as the S-tight closure of a and is denoted by a∗,S . The fact that there is a homogeneous R[x, f ]-isomorphism M n R/a[p ] R[x, f ] ⊗R (R/a) ∼ = n∈N0 n (where the right-hand side has the left R[x, f ]-module structure for which x(r+a[p ] ) = n+1 r p + a[p ] for all r ∈ R) enables one to conclude that n n a∗,S = r ∈ R : there exists s ∈ S with sr p ∈ a[p ] for all n ≫ 0 . ◦ Thus a∗,R is the usual tight closure a∗ of a. (See Hochster–Huneke [2, §3].) 1.3. Examples. In the situation of Definition 1.2, let S, T be multiplicatively closed subsets of R with S ⊆ T . Then clearly ∆S (H) ⊆ ∆T (H). (i) Note that {1} is a multiplicatively closed subset of R, and ∆{1} (H) = {h ∈ H : xn h = 0 for all n ≫ 0} = {h ∈ H : there exists n ∈ N0 with xn h = 0}, S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 5 the x-torsion submodule Γx (H) of H. Thus Γx (H) ⊆ ∆S (H) and therefore (0 :R ∆S (H)) ⊆ (0 :R Γx (H)). (ii) Let M be an R-module and let h, n ∈ N0 . Endow Rxn and Rxh with their natural structures as (R, R)-bimodules (inherited from their being graded components of R[x, f ]). Then there is an isomorphism of (left) R-modules φ : ∼ = Rxn+h ⊗R M −→ Rxn ⊗R (Rxh ⊗R M) for which φ(rxn+h ⊗m) = rxn ⊗(xh ⊗m) for all r ∈ R and m ∈ M. One can use isomorphisms like that described in the above paragraph to see that ∗,S ∗,S ∆S (R[x, f ] ⊗R M) = 0∗,S M ⊕ 0Rx⊗R M ⊕ · · · ⊕ 0Rxn ⊗R M ⊕ · · · . 1.4. Notation. We define MS (G), for the x-torsion-free left R[x, f ]-module G, to be the set of minimal members (with respect to inclusion) of the set {p ∈ Spec(R) ∩ I(G) : p ∩ S 6= ∅} of prime G-special R-ideals that meet S. T When MS (G) is finite, we shall set b := p∈MS (G) p, although we shall write bS,G for b when it is desirable to indicate S and G; in that case, it follows from 0.2(iii) that b is the smallest member of I(G) that meets S (and, in particular, b is contained in every other member of I(G) that meets S). (In the special case where MS (G) = ∅, we interpret b as R, the intersection of the empty family of prime ideals of R.) 1.5. Proposition. Consider the x-torsion-free left R[x, f ]-module G. The set MS (G) (see 1.4) is finite if and only if (0 :R ∆S (G)) ∩ S 6= ∅. When these conditions are satisfied, and b denotes the intersection of the prime ideals in the finite set MS (G), then ∆S (G) = annG (bR[x, f ]) and (0 :R ∆S (G)) = b. Proof. (⇐) Set c := (0 :R ∆S (G)) and assume that there exists s ∈ c ∩ S. Since G is x-torsion-free and ∆S (G) is an R[x, f ]-submodule of G, it follows from 0.2(i) that c is radical and gr-annR[x,f ] ∆S (G) = cR[x, f ]. Now annG (cR[x, f ]) ⊆ annG ((sR)R[x, f ]) ⊆ ∆S (G) ⊆ annG (cR[x, f ]), and so annG (cR[x, f ]) = ∆S (G). Note that c ∈ I(G). If c = R, then ∆S (G) = 0, so that MS (G) is empty because a p ∈ I(G) ∩ Spec(R) with p ∩ S 6= ∅ must satisfy annG (pR[x, f ]) ⊆ ∆S (G) = 0, and this leads to a contradiction to 0.2(ii). We therefore suppose that c 6= R. Let c = p1 ∩ · · · ∩ pt be the minimal primary decomposition of the (radical) ideal c. By [7, Theorem 3.6 and Corollary 3.7], the prime ideals p1 , . . . , pt all belong to I(G); they all meet S. Since Spec(R) satisfies the descending chain condition, each member of {p′ ∈ Spec(R) ∩ I(G) : p′ ∩ S 6= ∅} contains a member of MS (G). In particular, each of p1 , . . . , pt contains a member of MS (G). Now let p ∈ MS (G). Since p meets S, we must have that annG (pR[x, f ]) ⊆ ∆S (G) = annG (cR[x, f ]). It now follows from the inclusion-reversing bijective correspondence of 0.2(ii) that p ⊇ c, so that p contains one of p1 , . . . , pt . We can therefore conclude that p1 , . . . , pt are precisely the minimal members of {p ∈ I(G) ∩ Spec(R) : p ∩ S 6= ∅}. 6 RODNEY Y. SHARP Therefore MS (G) is finite. (⇒) Assume that MS (G) is finite. Then b is the smallest member of I(G) that meets S. Consequently, annG (bR[x, f ]) ⊆ ∆S (G). Let g ∈ ∆S (G), so that there L exist s′ ∈ S and n0 ∈ N0 such that s′ xn g = 0 for all n ≥ n0 . Thus g ∈ annG ( n≥n0 Rs′ xn ) =: J, a special annihilator submodule of G. Let b′ be the Gspecial R-ideal that corresponds toL this special annihilator submodule (in the bijective correspondence of 0.2(ii)). Since n≥n0 Rs′ xn ⊆ gr-annR[x,f ] J = b′ R[x, f ], we have s′ ∈ b′ , so that b′ ∩ S 6= ∅. Therefore b′ ⊇ b, and g ∈ J = annG (b′ R[x, f ]) ⊆ annG (bR[x, f ]). Therefore ∆S (G) ⊆ annG (bR[x, f ]). We conclude that ∆S (G) = annG (bR[x, f ]), and that (0 :R ∆S (G)) = (0 :R annG (bR[x, f ])) = b. All the claims in the statement of the proposition have now been proved. 1.6. Definition. An S-test element for R is an element s ∈ S such that, for every R-module M and every j ∈ N0 , the element sxj annihilates 1 ⊗ m ∈ (Φ(M))0 for every m ∈ 0∗,S M . The ideal of R generated by all the S-test elements for R is called the S-test ideal of R, and denoted by τ S (R). One of the aims of this paper is to show that S-test elements for R exist when R is F -pure and local. This is a suitable point at which to remind the reader of some of the classical concepts related to tight closure test elements. 1.7. Reminder. Recall that a test element for modules for R is an element c ∈ R◦ such that, for every finitely generated R-module M and every j ∈ N0 , the element cxj annihilates 1 ⊗ m ∈ (Φ(M))0 for every m ∈ 0∗M . The phrase ‘for modules’ is inserted because Hochster and Huneke have also considered a concept of a test element for ideals for R, which is defined to be an element c ∈ R◦ such that, for every cyclic Rmodule M and every j ∈ N0 , the element cxj annihilates 1 ⊗ m ∈ (Φ(M))0 for every m ∈ 0∗M . When R is reduced and excellent, the concepts of test element for modules and test element for ideals for R coincide: see [2, Discussion (8.6) and Proposition (8.15)]. T Hochster and Huneke define the test ideal τ (R) of R to be M (0 :R 0∗M ), where the intersection is taken over all finitely generated R-modules M. In [2, Proposition (8.23)(b)] they show that, if R has a test element for modules, then τ (R) is the ideal generated by the test elements for modules, and τ (R) ∩ R◦ is the set of test elements for modules. 1.8. Reminder. Recall also that a big test element for R is an element c ∈ R◦ such that, for every R-module M and every j ∈ N0 , the element cxj annihilates 1 ⊗ m ∈ (Φ(M))0 for every m ∈ 0∗M . We let τe(R) denote the ideal generated by all big test elements for R, and call this the big test ideal of R. Note that an R◦ -test element for R, in the sense of Definition 1.6, is just a big test element for R. Recall that an injective cogenerator of R is an injective R-module E with the property that, for every R-module M and every non-zero element m ∈ M, there exists an R-homomorphism φ : M −→ E such that φ(m) 6= 0. As R is Noetherian, L m∈Max(R) ER (R/m), where Max(R) denotes the set of maximal ideals of R, is one injective cogenerator of R. S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 7 L 1.9. Reminders. Let E := m∈Max(R) ER (R/m), an injective cogenerator of R. (i) Recall from Hochster–Huneke [2, Definition (8.19)] that the finitistic tight S ∗ closure of 0 in E, denoted by 0∗fg E , is defined to be M 0M , where the union is taken over all finitely generated R-submodules M of E. It was shown in [2, Proposition (8.23)(d)] that τ (R) = (0 :R 0∗fg E ). ∗fg ∗ (ii) It is conjectured that (0 :R 0E ) = (0 :R 0E ). This conjecture is known to be true (a) if R is an excellent Gorenstein local ring (see K. E. Smith [12, p. 48]); (b) if R is the localization of a finitely generated N0 -graded algebra over an F -finite field K (of characteristic p and having K as its component of degree 0) at its unique homogeneous maximal ideal (see G. Lyubeznik and K. E. Smith [5, Corollary 3.4]); (c) if R is a Cohen–Macaulay local ring which is Gorenstein on its punctured spectrum (see Lyubeznik–Smith [6, Theorem 8.8]); or (d) if (R, m) is local and an isolated singularity (see Lyubeznik–Smith [6, Theorem 8.12]). (iii) It was shown in [10, Theorem 3.3] that if R has a big test element, then the ◦ big test ideal τe(R) of R is equal to (0 :R ∆R (Φ(E))), and the set of big test ◦ elements for R is (0 :R ∆R (Φ(E))) ∩ R◦ . 2. Existence of S-test elements in F -pure local rings Theorem 0.3 was proved by means of an ‘embedding theorem’ [9, Theorem 4.10]. Similar embedding theorems were established in [8, Theorem 3.5] and [10, Theorem 3.2]. In this paper, the ideas underlying those theorems are going to be pursued further, and so we begin with three remarks and a lemma that can be viewed as addenda to [10, §2]. The notation in this section is as described in 1.1. e denote the graded companion of H described in [10, Example 2.1. Remark. Let H S (H), so that (0 : ∆S (H)) e = ∆^ e = (0 :R ∆S (H)). 2.1]. It is easy to check that ∆S (H) R 2.2. Remark. Let (H (λ) )λ∈Λ be a non-empty family of Z-graded left R[x, f ]-modules, Q and consider the graded product ′λ∈Λ H (λ) of the H (λ) , described in [8, Lemma 2.1] and [10, Example 2.2]. It is routine to check that, if there ideal d0 of R such Q′ is an (λ) H = d0 . that (0 :R ∆S (H (λ) )) = d0 for all λ ∈ Λ, then 0 :R ∆S λ∈Λ L 2.3. Remark. Assume that the left R[x, f ]-module H = n∈Z Hn is Z-graded; let t ∈ Z. Denote by H(t) the result of application of the tth shift functor to H; this is described in [10, Example 2.3]. It is clear that ∆S (H(t)) = ∆S (H)(t), so that (0 :R ∆S (H(t))) = (0 :R ∆S (H)). L 2.4. Lemma. Let b, h ∈ N and let W := n≥b Wn be a graded left R[x, f ]-module. Let (gi )i∈I be a family of arbitrary elements of Wb . Consider the h-place extension exten(W ; (gi )i∈I ; h) of W by (gi )i∈I , defined in [9, Definition 4.5] and [10, §2]. Then (0 :R ∆S (exten(W ; (gi )i∈I ; h))) = (0 :R ∆S (W )). Proof. This can be proved by making obvious modifications to the proof, presented in ◦ ◦ [10, Proposition 2.4(i)], that (0 :R ∆R (exten(W ; (gi )i∈I ; h))) = (0 :R ∆R (W )). 8 RODNEY Y. SHARP The above remarks and lemma are helpful for use in the proof of some of the claims in the following Embedding Theorem. 2.5. Embedding Theorem. (See [10, Theorem 3.2].) Let E be an injective Lcogenerator of R. Assume that there exists an N0 -graded left R[x, f ]-module H = n∈N0 Hn such that H0 is R-isomorphic to E. Let M be an R-module. Then there is a family L(n) n∈N0 of N0 -graded left R[x, f ]modules, where L(n) is an n-place extension of the −nth shift of a graded product of copies of H (for each n ∈ N0 ), for which there exists a homogeneous R[x, f ]monomorphism Y′ M L(n) =: K. (Rxi ⊗R M) −→ ν : Φ(M) = i∈N0 n∈N0 Consequently, (0 :R ∆S (H)) = (0 :R ∆S (K)) ⊆ (0 :R ∆S (Φ(M))). Furthermore, if H is x-torsion-free, then so too is Φ(M), and I(Φ(M)) ⊆ I(H) and (0 :R ∆S (Φ(M))) ∈ I(H). Proof. The existence of K and ν with the stated properties were proved in [10, Theorem 3.2]. The existence of the R[x, f ]-monomorphism ν shows that (0 :R ∆S (K)) ⊆ (0 :R ∆S (Φ(M))), while Remarks 2.2 and 2.3 and Lemma 2.4 show that (0 :R ∆S (K)) = (0 :R ∆S (H)). Now suppose H is x-torsion-free. It follows from [8, Lemmas 2.3 and 2.8] that K is x-torsion-free and that I(K) = I(H). The existence of the R[x, f ]-monomorphism ν shows that Φ(M) is x-torsion-free and R[x, f ]-isomorphic to an R[x, f ]-submodule of K; therefore I(Φ(M)) ⊆ I(K) = I(H). Finally, (0 :R ∆S (Φ(M))) ∈ I(Φ(M)). 2.6. Theorem. Suppose that the local ring (R, m) is F -pure. Then R has an S-test element. In more detail, set E := ER (R/m). Recall that bS,Φ(E) denotes the intersection of all the minimal members of the set {p ∈ Spec(R) ∩ I(Φ(E)) : p ∩ S 6= ∅} (see 1.4). Then S ∩bS,Φ(E) is (non-empty and) equal to the set of S-test elements for R. Furthermore, bS,Φ(E) = (0 :R ∆S (Φ(E))). Proof. By Theorem 0.3, the set I(Φ(E)) of Φ(E)-special R-ideals is finite. Consequently, MS (Φ(E)) is finite, and Proposition 1.5 shows that (0 :R ∆S (Φ(E)))∩S 6= ∅. We use the Embedding Theorem 2.5 with Φ(E) playing the rôle of H. We conclude that, for every R-module M, we have (0 :R ∆S (Φ(E))) ⊆ (0 :R ∆S (Φ(M))). Thus (0 :R ∆S (Φ(E))) is precisely the set of elements of R that annihilate ∆S (Φ(M)) for every R-module M. Since ∆S (Φ(M)) is an R[x, f ]-submodule of Φ(M) (for each R-module M), it follows that (0 :R ∆S (Φ(E))) ∩ S (which is non-empty) is the set of S-test elements for R. It also follows from Proposition 1.5 that ∆S (Φ(E)) = annΦ(E) (bS,Φ(E) R[x, f ]) and (0 :R ∆S (Φ(E))) = bS,Φ(E) . S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 9 In our applications of these ideas, we shall frequently take S to be the complement in R of the union of finitely many prime ideals. In that case, a little more can be said. 2.7. S Lemma. Let A be a commutative ring and let p1 , . . . , pn ∈ Spec(A). Set T := A \ ni=1 pi , and let a be an ideal of A such that a ∩ T 6= ∅. Then a can be generated by elements of a ∩ T . Proof. Let a′ be the ideal of A generated by a ∩ T . Then a ⊆ (a ∩ T ) ∪ (a ∩ (A \ T )) ⊆ a′ ∪ p1 ∪ · · · ∪ pn . Since a ∩ T 6= ∅, we have, for every i ∈ {1, . . . , n}, that a 6⊆ pi . Therefore, by the Prime Avoidance Theorem (in the form given in [4, Theorem 81]), we must have a ⊆ a′ . Sn 2.8. Corollary. Suppose that the local ring (R, m) is F -pure, and S = R \ i=1 pi , where p1 , . . . , pn ∈ Spec(R). It was shown in Theorem 2.6 that R has an S-test element. Set E := ER (R/m). The S-test ideal of R, that is, the ideal of R generated by all S-test elements for R, is bS,Φ(E) , the smallest member of I(Φ(E)) that meets S. In symbols, τ S (R) = bS,Φ(E) . In particular, the big test ideal τe(R) is the smallest member of I(Φ(E)) of positive height. (We interpret the height of the improper ideal R as ∞.) Proof. We saw in Theorem 2.6 that the set S ∩ bS,Φ(E) is non-empty and equal to the set of S-test elements for R. By Lemma 2.7, the ideal bS,Φ(E) can be generated by elements of S ∩ bS,Φ(E) . For the final claim, take S = R◦ and note that a proper ideal of R has positive height if and only if it meets R◦ . We shall actually use variations of the Embedding Theorem 2.5 in Proposition 2.10 below. 2.9. Definitions. Suppose that (R, m) is local and F -pure; set E := ER (R/m). Let M be an R-module. S (i) We define the finitistic S-tight closure 0∗fg,S of 0 in M to be N 0∗,S M N , where the union is taken over all finitely generated submodulesTN of M. (ii) We define the finitistic S-test ideal τ fg,S (R) of R to be L (0 :R 0∗,S L ), where the intersection is taken over all finitely generated R-modules L. 2.10. Proposition. Suppose that (R, m) is local and F -pure; set E := ER (R/m). (i) For every R-module M, we have I(Φ(M)) ⊆ I(Φ(E)) and (0 :R ∆S (Φ(E))) ⊆ (0 :R ∆S (Φ(M))) ∈ I(Φ(E)). ∗fg,S (ii) The ideal (0 :R 0E ) annihilates (0 :R 0∗,S L ) for every finitely generated Rmodule L. ∗fg,S (iii) We have τ fg,S (R) = (0 :R 0E ), and this ideal belongs to I(Φ(E)). ∗,S (iv) For every R-module M, we have (0 :R 0∗,S E ) ⊆ (0 :R 0M ). (v) We have bS,Φ(E) = (0 :R 0∗,S E ). Consequently, when S is the complement in R of the union of finitely many prime ideals, then the S-test ideal τ S (R) of R is equal to (0 :R 0∗,S E ). 10 RODNEY Y. SHARP Proof. (i) This follows from the Embedding T Theorem 2.5, with H taken as Φ(E). (ii) By Krull’s Intersection Theorem, n∈N mn L = 0. We can therefore express the zero submodule of L as the intersection of a countable family (Qi )i∈N of irreducible submodules of finite colength. (A submodule of L is said to be irreducible if it is proper and cannot be expressed as the intersection of two strictly larger submodules.) Note that ER (L/Qi ) = E for all i ∈ N. Q The R-monomorphism Λ0 : L −→ i∈N L/Qi for which λ0 (g) = (g + Qi )i∈N for all g ∈ L can be extended to a homogeneous R[x, f ]-homomorphism ! Y′ M Y′ M Rxn ⊗R (L/Qi ) Φ(L/Qi ) = Rxn ⊗R L −→ λ : Φ(L) = n∈N0 i∈N i∈N n∈N0 whose restriction to the nth component of Φ(L), for n ∈ N0 , satisfies λ(rxn ⊗ g) = (rxn ⊗ (g + Qi ))i∈N for all r ∈ R and g ∈ L. It is clear that the R-monomorphism Q ∗,S λ0 (that is, the component of degree 0 of λ) maps 0∗,S L into i∈N 0L/Qi . But L/Qi is R-isomorphic to a finitely generated submodule of E, and so 0∗,S L/Qi is annihilated by ∗fg,S ∗fg,S (0 :R 0E ) (for all i ∈ N). It follows that 0∗,S ). LT is annihilated by (0 :R 0E ∗fg,S ∗,S (iii) By part (ii), we have (0 :R 0E ) ⊆ L (0 :R 0L ), where the intersection is ∗fg,S taken over all finitely generated R-modules L. Thus (0 :R 0E ) ⊆ τ fg,S (R). On the S other hand, by definition, τ fg,S (R) annihilates N 0∗,S N , where the union is taken over ∗fg,S all finitely generated submodules N of E; therefore τ fg,S (R) ⊆ (0 :R 0E ). For each finitely generated R-module L, ∗,S ∗,S ∆S (Φ(L)) = 0∗,S L ⊕ 0Rx⊗R L ⊕ · · · ⊕ 0Rxn ⊗R L ⊕ · · · , T by Example 1.3(ii), so that (0 :R ∆S (Φ(L))) = n∈N0 (0 :R 0∗,S Rxn ⊗R L ). Note that Rxn ⊗R L is a finitely generated R-module, for each n ∈ N0 . It therefore follows that T T ∗,S S L (0 :R ∆ (Φ(L))) = L (0 :R 0L ), where in both cases the intersection is taken over generated R-modules L. T all finitely fg,S S Therefore τ (R) is equal to the above ideal L (0 :R ∆ (Φ(L))), and the latter is in I(Φ(E)) because each (0 :R ∆S (Φ(L))) is (by part (i)) and I(Φ(E)) is closed under taking arbitrary intersections (by [7, Corollary 1.12]). (iv) By [10, Lemma 3.1], there is a family of graded left R[x, f ]-modules H (λ) λ∈Λ , with each H (λ) equal to Φ(E), and a homogeneous R[x, f ]-homomorphism Y′ H (λ) µ : Φ(M) −→ λ∈Λ such that its component µ0 of degree 0 is a monomorphism. Since µn (sxn ⊗ m) = sxn µ0 (m) for all m ∈ M, s ∈ S and n ∈ N0 , it follows that the R-monomorphism ∗,S ∗,S µ0 maps 0∗,S M into a direct product of copies of 0E . Therefore (0 :R 0E ) annihilates 0∗,S M . Note that this is true for each R-module M. (v) It now follows from part (iv) and the fact (see Example 1.3(ii)) that ∗,S ∗,S ∆S (Φ(E)) = 0∗,S E ⊕ 0Rx⊗R E ⊕ · · · ⊕ 0Rxn ⊗R E ⊕ · · · S S S,Φ(E) that (0 :R 0∗,S , by Theorem 2.6. E ) = (0 :R ∆ (Φ(E))). But (0 :R ∆ (Φ(E))) = b S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 11 When S is the complement in R of the union of finitely many prime ideals, it follows from Corollary 2.8 that τ S (R) = bS,Φ(E) . 2.11. Remarks. Suppose that (R, m) is local and F -pure; set E := ER (R/m). (i) In the special case in which S is taken to be R◦ , Proposition 2.10(v) reduces ◦ to the (probably well-known) result that the big test ideal τe(R) = τ R (R) of R is equal to (0 :R 0∗E ). (ii) In the special case in which S is taken to be R◦ , the first part of Proposition 2.10(iii) reduces to (a special case of) a result of Hochster–Huneke [2, Proposition (8.23)(d)] about the test ideal τ (R): ◦ τ (R) = τ fg,R (R) = (0 :R 0∗fg,R ) = (0 :R 0∗fg E E ). ◦ We have seen that, over an F -pure local ring (R, m), the set I(Φ(E)) (where E := ER (R/m)) of radical ideals includes the test ideal τ (R), the big test ideal ◦ ◦ τe(R) = bR ,Φ(E) = τ R (R) of R and the S-test ideal, for each multiplicatively closed subset S of R which is the complement in R of the union of finitely many prime ideals. It is natural to ask whether every member of the finite set I(Φ(E)) occurs as the S ′ -test ideal for some multiplicatively closed subset S ′ of R. In Theorem 2.12 below, we shall answer this question in the affirmative. 2.12. Theorem. Suppose that the local ring (R, m) is F -pure, and set E := ER (R/m). Let a ∈ I(Φ(E)). Then there exists a multiplicatively closed subset S of R such that a is the S-test ideal of R. Moreover, S can be taken to be the complement in R of the union of finitely many prime ideals. Proof. If a = R, then we can take S = {1} or S = R \ m. We therefore assume henceforth in this proof that a is proper. Let p1 , . . . , pt be the (distinct) associated prime ideals of a; recall from [7, Theorem 3.6 and Corollary 3.7] that they all belong to I(Φ(E)). Let T be the set of all prime members of I(Φ(E)) which neither contain, nor are contained in, any of p1 , . . . , pt . Let q1 , . . . , qu be the maximal members of the set of prime ideals in I(Φ(E)) that are properly contained in one of p1 , . . . , pt , and let U := {q1 , . . . , qu }. (Observe that T and/or U could S be empty; for example, both are empty if a = 0.) Set S := R \ q∈T ∪U q. Our aim is to show that a is the S-test ideal τ S (R) of R. It follows from Corollary 2.8 that τ S (R) = bS,Φ(E), the intersection of the minimal members of the set of prime ideals in I(Φ(E)) that meet S. Let p ∈ I(Φ(E)) ∩ Spec(R). Then, since I(Φ(E)) and therefore T and U are finite, p ∩ S = ∅ if and only if p is contained in some q ∈ T ∪ U. Let i ∈ {1, . . . , t}. Then pi meets S, or else pi ⊆ qj for some j ∈ {1, . . . , u}, and as qj is properly contained in one of p1 , . . . , pt , this would lead to a contradiction to the minimality of the primary decomposition a = p1 ∩ · · · ∩ pt . Furthermore, pi must be a minimal member of the set J := {p ∈ I(Φ(E)) ∩ Spec(R) : p ∩ S 6= ∅} , for otherwise there would exist p ∈ J with p ⊂ pi , so that p would be contained in one of q1 , . . . , qu and therefore disjoint from S. This shows that p1 , . . . , pt are all associated primes of bS,Φ(E) . To complete the proof, it is enough for us to show that there is no other associated prime of this ideal. So suppose that p ∈ ass bS,Φ(E) \ {p1 , . . . , pt } and seek a contradiction. Then p must contain, or be contained in, pi for some i ∈ {1, . . . , t} (or else it would be in T and 12 RODNEY Y. SHARP disjoint from S); if p ⊂ pi , then p would be contained in qj for some j ∈ {1, . . . , t} and so would be disjoint from S; if p ⊃ pi , then p could not be a primary component of the radical ideal bS,Φ(E) . Thus each possibility leads to a contradiction. Therefore ass bS,Φ(E) = {p1 , . . . , pt } and bS,Φ(E) = a. 3. The complete case In this section we shall concentrate on the case where (R, m) is local, F -pure and complete. 3.1. Theorem. Suppose (R, m) is local, F -pure and complete. Set E := ER (R/m). Let c ∈ I(Φ(E)) with c 6= R. In the light of Theorem 2.12, letSp1 , . . . , pw be prime ideals of R for which the multiplicatively closed subset S = R \ w i=1 pi of R satisfies c = τ S (R). Set J := ∆S (Φ(E)), a graded left R[x, f ]-module. ∗,S ∗,S (i) We have J = 0∗,S E ⊕ 0Rx⊗R E ⊕ · · · ⊕ 0Rxn ⊗R E ⊕ · · · . (ii) When we regard J as a graded left (R/c)[x, f ]-module in the natural way, it is x-torsion-free and has IR/c (J) = {g/c : g ∈ I(Φ(E)) : g ⊇ c} . (iii) The 0th component J0 of J is (0 :E c); as R/c-module, this is isomorphic to ER/c ((R/c)/(m/c)). (iv) The ring R/c is F -pure. (v) We have I(ΦR/c (J0 )) ⊆ IR/c (J), so that d : d is an ideal of R with d ⊇ c and d/c ∈ I(ΦR/c (J0 )) ⊆ I(ΦR (E)). Proof. Set R := R/c. (i) It follows from Example 1.3(ii) that ∗,S ∗,S ∆S (Φ(E)) = 0∗,S E ⊕ 0Rx⊗R E ⊕ · · · ⊕ 0Rxn ⊗R E ⊕ · · · . (ii) Since gr-annR[x,f ] ∆S (Φ(E)) = cR[x, f ], we see that c annihilates ∆S (Φ(E)), and so the latter inherits a structure as an x-torsion-free left R[x, f ]-module. As the R[x, f ]-submodules of J are exactly the R[x, f ]-submodules of Φ(E) contained in J, the claim about IR (J) is clear. (iii) Note that c = bS,Φ(E) = τ S (R), and that, by Proposition 2.10(v), this is the ∗,S R-annihilator of 0∗,S E . Since R is complete, we can conclude that 0E = (0 :E c), by Matlis duality (see, for example, [11, p. 154]). (iv),(v) Let N be an R-module. Use the Embedding Theorem 2.5 over the ring R (with J playing the rôle of H) to deduce that ΦR (N) is x-torsion-free and I(ΦR (N)) ⊆ IR (J). These are true for each R-module N, and, in particular, for J0 . It follows that R is F -pure. The final claim follows from the description of IR (J) given in part (ii). 3.2. Corollary. Suppose that (R, m) is local, F -pure and complete. Denote ER (R/m) by E. Let c ∈ I(Φ(E)) with c 6= R. Denote R/c by R, and note that R is F pure, by Theorem 3.1(iv). Let T be a multiplicatively closed subset of R which is the complement in R of the union of finitely many prime ideals. (i) If h denotes the unique ideal of R that contains c and is such that h/c = τ fg,T (R), the finitistic T -test ideal of R, then h ∈ I(Φ(E)). S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 13 (ii) In particular, if h′ denotes the unique ideal of R that contains c and is such that h′ /c = τ (R), the test ideal of R, then h′ ∈ I(Φ(E)). (iii) If g denotes the unique ideal of R that contains c and is such that g/c = τ T (R), the T -test ideal of R, then g ∈ I(Φ(E)). (iv) In particular, if g′ denotes the unique ideal of R that contains c and is such that g′ /c = τe(R), the big test ideal of R, then g′ ∈ I(Φ(E)). Proof. Use the notation of Theorem 3.1. Note that, as R-module, J0 is the injective envelope of the simple R-module. (i) By Proposition 2.10(iii), we have τ fg,T (R) ∈ I(ΦR (J0 )). The result therefore follows from Theorem 3.1(v). ◦ (ii) This is a special case of part (i): take T = R . (iii) By Proposition 2.10(v), we have τ T (R) ∈ I(ΦR (J0 )). The result therefore follows from Theorem 3.1(v). ◦ (iv) This is a special case of part (iii): take T = R . The remainder of the paper is devoted to the provision of some examples of the above ideas. 3.3. Example. Let K be an algebraically closed field of characteristic p, and assume that p ≥ 5 and that p ≡ 1 (mod 3). Let R′ = K[[X, Y, Z]], where X, Y, Z are independent indeterminates, and a = (X 3 + Y 3 + Z 3 ) ∈ Spec(R′ ). By Huneke [3, Examples 4.7 and 4.8], R := R′ /a is F -pure, and the test ideal τ (R) = m. Because R is Gorenstein and excellent, the test ideal τ (R) is equal to the big test ideal τe(R), by 1.9(ii)(a). This means that m must be the smallest ideal in I(Φ(E)) of positive height, so that I(Φ(E)) ∩ Spec(R) = {0, m} . 3.4. Reminder. Suppose that (R, m) is local and F -pure, and set E = ER (R/m). In the case where R is an (F -pure) homomorphic image of an F -finite regular local ring, Janet Cowden Vassilev showed in [13, §3] that there exists a strictly ascending chain 0 = τ0 ⊂ τ1 ⊂ · · · ⊂ τt ⊂ τt+1 = R of radical ideals of R such that, for each i = 0, . . . , t, the reduced local ring R/τi is F -pure and its test ideal is exactly τi+1 /τi . If R is complete, all of τ0 , τ1 , . . . , τt and all their associated primes belong to I(Φ(E)) (by Corollary 3.2(ii) and [7, Theorem 3.6 and Corollary 3.7]). 3.5. Lemma. Assume that (R, m) is local, F -pure and complete. Set E = ER (R/m). (i) There is a strictly ascending chain 0 = τ0 ⊂ τ1 ⊂ · · · ⊂ τt ⊂ τt+1 = R of radical ideals of R such that, for each i = 0, . . . , t, the reduced local ring R/τi is F -pure and its test ideal is τi+1 /τi . We call this the test ideal chain of R. All of τ0 = 0, τ1 , · · · , τt , and all their associated primes, belong to I(Φ(E)). (ii) There is a strictly ascending chain 0 = τe0 ⊂ τe1 ⊂ · · · ⊂ τew ⊂ τew+1 = R of radical ideals in I(Φ(E)) such that, for each i = 0, . . . , w, the reduced local ring R/e τi is F -pure and its big test ideal is τei+1 /e τi . We call this the big test ideal chain of R. All of τe0 = 0, τe1 , · · · , τew , and all their associated primes, belong to I(Φ(E)). Note. We have not assumed that R is F -finite in part (i). If the conjecture mentioned in 1.9(ii) turns out to be true, then the big test ideal chain and the test ideal chain of R would coincide. 14 RODNEY Y. SHARP Proof. (i) We know from Theorem 0.1 that E can be given a structure as an x-torsionfree left R[x, f ]-module (that extends its R-module structure). It therefore follows from work of the present author in [8, Corollary 3.8] that, in this complete case, the test ideal chain of R exists. By Corollary 3.2(ii), all the terms in the test ideal chain of R belong to I(Φ(E)), and all the associated prime ideals of these ideals also belong to I(Φ(E)), by [7, Theorem 3.6 and Corollary 3.7]. (ii) Let c ∈ I(Φ(E)) with c 6= R. By Theorem 3.1(iv), the ring R/c is F -pure. By Corollary 3.2(iv), if g′ denotes the unique ideal of R that contains c and is such that g′ /c = τe(R/c), then g′ ∈ I(Φ(E)). One can therefore construct τe1 (R), τe2 (R), . . . successively until some τet+1 (R) = R, when the process stops. Use Corollary 3.2(iv) and [7, Theorem 3.6 and Corollary 3.7] again to complete the proof. We can now use some of Vassilev’s computations in [13, §3] to give some examples. 3.6. Examples. Let K be an algebraically closed field of characteristic p. In these examples, X, Y, Z, W denote independent indeterminates over K, and x, y, z, w denote the natural images of X, Y, Z, W (respectively) in R′ /a = R for appropriate choices of R′ and a proper ideal a of R′ . (i) As in Vassilev [13, Example 3.12(1)], take R′ = K[[X, Y, Z]], a = (XY, XZ, Y Z) and R = R′ /a. For this R, the test ideal chain is 0 ⊂ m ⊂ R. Since R is an isolated singularity, we have τ (R) = τe(R), by 1.9(ii)(d). Therefore m is the smallest ideal of positive height in I(Φ(E)). Thus in this case, I(Φ(E)) ∩ Spec(R) = {(x, y), (x, z), (y, z), m} . (ii) As in Vassilev [13, Example 3.12(2)], take R′ = K[[X, Y, Z, W ]], a = (XY Z, XY W, XZW, Y ZW ) and R = R′ /a. For this R, the test ideal chain is 0 ⊂ (xy, xz, xw, yz, yw, zw) ⊂ m ⊂ R. It therefore follows from Lemma 3.5(i) that { (x, y), (x, z), (x, w), (y, z),(y, w), (z, w), (x, y, z), (x, y, w), (x, z, w), (y, z, w), m} ⊆ I(Φ(E)) ∩ Spec(R). (iii) As in Vassilev [13, Example 3.12(3)], take R′ = K[[X, Y, Z]], a = (XY, Y Z), and R = R′ /a. For this R, the test ideal chain is 0 ⊂ m ⊂ R. Because R is an isolated singularity, τ (R) = τe(R), by 1.9(ii)(d). This means that m must be the smallest ideal in I(Φ(E)) of positive height, so that I(Φ(E)) ∩Spec(R) = {(x, z), (y), m} . (iv) As in Vassilev [13, Example 3.12(4)], take R′ = K[[X, Y, Z, W ]], a = (XY, ZW ) and R = R′ /a. For this R, the test ideal chain is 0 ⊂ (xy, xz, xw, yz, yw, zw) ⊂ m ⊂ R, so that {(x, z), (x, w), (y, z), (y, w), (x, y, z), (x, y, w), (x, z, w), (y, z, w), m} ⊆ I(Φ(E)) ∩ Spec(R) by Lemma 3.5(i). S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 15 References [1] R. Fedder, ‘F -purity and rational singularity’, Transactions Amer. Math. Soc. 278 (1983) 461–480. [2] M. Hochster and C. Huneke, ‘Tight closure, invariant theory and the Briançon-Skoda Theorem’, J. Amer. Math. Soc. 3 (1990) 31–116. [3] C. Huneke, ‘Tight closure, parameter ideals and geometry’, Six lectures on commutative algebra, Eds. J. Elias, J. M. Giral, R. M. Miró-Roig and S. Zarzuela, Progress in Mathematics 166 (Birkhäuser, Basel, 1998), pp. 187–239. [4] I. Kaplansky, Commutative rings (Allyn and Bacon, Boston, 1970). [5] G. Lyubeznik and K. E. Smith, ‘Strong and weak F -regularity are equivalent for graded rings’, American J. Math. 121 (1999) 1279–1290. [6] G. Lyubeznik and K. E. Smith, ‘On the commutation of the test ideal with localization and completion’, Transactions Amer. Math. Soc. 353 (2001) 3149–3180. [7] R. Y. Sharp, ‘Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure’, Transactions Amer. Math. Soc. 359 (2007) 4237–4258. [8] R. Y. Sharp, ‘Graded annihilators and tight closure test ideals’, J. Algebra 322 (2009) 3410– 3426. [9] R. Y. Sharp, ‘An excellent F -pure ring of prime characteristic has a big tight closure test element’, Transactions Amer. Math. Soc. 362 (2010) 5455–5481. [10] R. Y. Sharp, ‘Big tight closure test elements for some non-reduced excellent rings’, J. Algebra 349 (2012) 284-316. [11] D. W. Sharpe and P. Vámos, Injective modules, Cambridge Tracts in Mathematics and Mathematical Physics 62 (Cambridge University Press, Cambridge, 1972). [12] K. E. Smith, ‘Tight closure of parameter ideals’, Inventiones math. 115 (1994) 41–60. [13] J. C. Vassilev, ‘Test ideals in quotients of F -finite regular local rings’, Transactions Amer. Math. Soc. 350 (1998) 4041–4051. School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom E-mail address: R.Y.Sharp@sheffield.ac.uk	0
Power Beacon-Assisted Millimeter Wave Ad Hoc Networks Xiaohui Zhou, Jing Guo, Salman Durrani, and Marco Di Renzo arXiv:1703.06611v2 [cs.IT] 6 Sep 2017 Abstract Deployment of low cost power beacons (PBs) is a promising solution for dedicated wireless power transfer (WPT) in future wireless networks. In this paper, we present a tractable model for PB-assisted millimeter wave (mmWave) wireless ad hoc networks, where each transmitter (TX) harvests energy from all PBs and then uses the harvested energy to transmit information to its desired receiver. Our model accounts for realistic aspects of WPT and mmWave transmissions, such as power circuit activation threshold, allowed maximum harvested power, maximum transmit power, beamforming and blockage. Using stochastic geometry, we obtain the Laplace transform of the aggregate received power at the TX to calculate the power coverage probability. We approximate and discretize the transmit power of each TX into a finite number of discrete power levels in log scale to compute the channel and total coverage probability. We compare our analytical predictions to simulations and observe good accuracy. The proposed model allows insights into effect of system parameters, such as transmit power of PBs, PB density, main lobe beam-width and power circuit activation threshold on the overall coverage probability. The results confirm that it is feasible and safe to power TXs in a mmWave ad hoc network using PBs. Index Terms Wireless communications, wireless power transfer, millimeter wave transmission, power beacon, stochastic geometry. X. Zhou, J. Guo and S. Durrani are with the Research School of Engineering, College of Engineering and Computer Science, The Australian National University, Canberra, ACT 2601, Australia (Emails: {xiaohui.zhou, jing.guo, salman.durrani}@anu.edu.au). M. Di Renzo is with the Laboratoire des Signaux et Systèmes, Centre National de la Recherche Scientifique, CentraleSupélec, University Paris Sud, Université Paris-Saclay, 91192 Gif-sur-Yvette, France (Email: marco.direnzo@lss.supelec.fr). 1 I. I NTRODUCTION Wireless power transfer (WPT) can prolong the lifetime of low-power devices in the network and is currently in the spotlight as a key enabling technology in future wireless communication networks [1]–[3]. Compared to energy harvesting from ambient energy sources, e.g., solar, wind or ambient radio frequency (RF) sources, which may change rapidly with time, location and weather conditions, WPT has a significant advantage of being always available and controllable [1]. There are currently two main approaches to WPT: (i) simultaneous information and power transfer (SWIPT) and (ii) power beacon (PB) based approach. While SWIPT, which proposes to extract the information and power from the same signal, has been the subject of intense research in the academic community [1], [4], [5], industry has preferred to adopt the PB approach. In this approach, low cost PBs, which do not require backhaul links, are deployed to provide dedicated power transfer in wireless networks. For example, the Cota Tile is a PB designed to wirelessly charge devices like smartphones in a home environment and was showcased at the 2017 Consumer Electronics Show (CES) [6]. There are two key challenges in the application of PBs to wider networks. The first challenge is the lack of tractable models for analysis and design of such networks. Although simulations can be used in this regard, exhaustive simulation of every possible scenario of interest will be extremely time-consuming and onerous. Hence, it is important to explore tractable models for PB-assisted communications in wireless networks. The second challenge is the use of practical models for WPT, which capture realistic aspects of WPT. For instance, WPT receivers (RXs) can only harvest power if the incident received power is greater than the power circuit activation threshold (typically around −20 dBm [1]). Similarly, WPT transmitters (TXs) have to adhere to maximum transmit power constraints due to safety considerations. Hence, it is important to adopt a realistic and practical model for WPT. A. Related Work Microwave (below 6 GHz) systems: Recently, the investigation of PBs has drawn attention in the literature from different aspects. For point-to-point or point-to-multipoint communication systems, the resource allocation for PB-assisted system was considered in [7], [8], where the authors mainly aimed at finding 2 the optimum time ratio for power transfer (PT) and information transmission (IT). In [9], the authors studied the PB-assisted network in the context of physical layer security, where an energy constrained source is powered by a dedicated PB. For large scale networks, some papers have characterized the performance of PB-assisted communications using stochastic geometry, which is a powerful mathematical tool to provide tractable analysis by incorporating the randomness of users. Specifically, the feasibility of PB deployment in a cellular network, under the outage constraint at the base station, was investigated in [10], where cellular users are charged by PBs for uplink transmission. By considering that the secondary TX is charged by the primary user in a cognitive network, the authors derived the spatial throughput for the secondary network in [11]. Adaptively directional PBs were proposed for sensor network in [12] and the authors found the optimal charging radius for different sensing tasks. In [13], three WPT schemes were proposed to select the PB for charging in a device-to-device-aided cognitive cellular network. The authors in [14] formulated the total outage probability in a PB-assisted ad hoc network by including the energy harvesting sensitivity into the analysis. Note that all the aforementioned works considered the conventional microwave frequency band, i.e., below 6 GHz. MmWave systems: Millimeter wave (mmWave) communication, which aims to use the spectrum band typically around 30 GHz, is emerging as a key technology for the fifth generation systems [15]. Considerable advancements have already been made in the understanding, modelling and analysis of mmWave communication using stochastic geometry [16]–[19]. From the prior work, we can summarize two distinctive features of mmWave communication: (i) owing to the smaller wavelength, mmWave allows a large number of antenna arrays with directional beamforming to be equipped at the TX and RX; (ii) since the mmWave propagation is susceptible to blockage, it causes the large difference for path-loss and fading characteristics between line of sight (LOS) and non light of sight (NLOS) environment. MmWave communication can be beneficial for WPT since both technologies inherently operate over short distances and the narrow beams in mmWave communication can focus the transmit power. Very recently, some papers have used stochastic geometry to analyse mmWave SWIPT networks [20], [21]. The statistics of the aggregate received power from PBs in a mmWave ad hoc network were studied in 3 our preliminary work in [22]. To the best of our knowledge, the study of a PB-assisted mmWave network using stochastic geometry, taking into account realistic and practical WPT and mmWave characteristics such as building blockages, beamforming, power circuit activation threshold, maximum harvested power and maximum transmit power, is not available in the literature. B. Our Approach and Contributions In this paper, we consider a PB-assisted wireless ad hoc network under mmWave transmission where TXs adopt the harvest-then-transmit protocol, i.e., they harvest energy from the aggregate RF signal transmitted by PBs and then use the harvested energy to transmit the information to their desired RXs. Both the PT and IT phases are carried out using antenna beamforming under the mmWave channel environment, which is subjected to building blockages. Using tools from stochastic geometry, we develop a tractable analytical framework to investigate the power coverage probability, the channel coverage probability and the total coverage probability at a reference RX taking a mmWave three-state propagation model and multislope bounded path-loss model into account. In the proposed framework, the power coverage probability is efficiently and accurately computed by numerical inversion using the closed-form expression for the Laplace transform of the aggregate received power1 at the typical TX. The novel contributions of this paper are summarized as follows: • We adopt a realistic model of wirelessly powered TXs by taking into consideration (i) the power circuit activation threshold, which accounts for the minimum aggregate received power required to activate the energy harvesting circuit, (ii) the allowed maximum harvested power, which accounts for the saturation of the energy harvesting circuit and (iii) the maximum transmit power, which accounts for the safety regulation and the electrical rating of the antenna circuit. • For tractable analysis of the channel coverage probability and the total coverage probability, we propose to discretize the transmit power of each TX into a finite number of discrete power levels in the log scale. Using this approximation, we derive the channel coverage probability and the total 1 In this paper, we use the Laplace transform of a random variable to denote the Laplace transform of the distribution of a random variable for brevity. 4 coverage probability at the typical RX. Comparison with simulation results shows that the model, with only 10 discrete levels for the transmit power of TXs, has good accuracy in the range of 5%-10%. • Based on our proposed model, we investigate the impact of varying important system parameters (e.g., transmit power of PB, PB density, allowed maximum harvested power, directional beamforming parameters etc.) on the network performance. These trends are summarized in Table V. • We investigate the feasibility of using PBs to power up TXs while providing an acceptable performance for IT towards RXs in mmWave ad hoc network. Our results show that under practical setups, for PB transmit power of 50 dBm and TXs with a maximum transmit power between 20 − 40 dBm, which are practical and safe for human exposure, the total coverage probability is around 90%. C. Notation and Paper Organization The following notation is used in this paper. Pr(·) indicates the probability measure and E[·] denotes the expectation operator. j is the imaginary number and Re[·] denotes the real part of a complex numR∞ R∞ ber. Γ(x) = 0 tx−1 exp(−t)dt is the complete gamma function and Γ(a, x) = x ta−1 exp(−t)dt is R 1 tb−1 (1−t)c−b−1 Γ(c) dt is the the upper incomplete gamma function, respectively. 2 F1 (a, b; c; z) = Γ(b)Γ(c−b) (1−tz)a 0 Gaussian hypergeometric function. fX (x) and FX (x) denotes the probability density function (PDF) and the cumulative distribution function (CDF) of a random variable X. LX (s) = E[exp(−sX)] denotes the Laplace transform of a random variable X. A list of the main mathematical symbols employed in this paper is given in Table I. The rest of the paper is organized as follows: Section II describes the system model and assumptions. Section III focuses on the PT phase of the system and derives the power coverage probability. Section IV details the IT phase, which covers the analysis of transmit power statistics and channel coverage probability. Section V summaries the total coverage probability. Section VI presents the results and the effect of the system parameters on the network performance. Finally, Section VII concludes the paper. 5 TABLE I: Summary of Main Symbols Used in the Paper. Symbol Definition Symbol Definition φp PB PPP Pp PB transmit power φt TX PPP Pt TX transmit power φn t nth level TX PPP Ptn nth level TX transmit power λp Density of PB PPP kn Portion of TXs at the nth level λt Density of TX PPP N Number of battery levels λn t Density of nth level TX PPP w Step size of each battery level d0 Length of desired TX-RX link σ2 Noise power rmin Radius of the LOS region η Power conversion efficiency rmax Exclusion radius of the OUT region ρ Time switching parameter αL LOS link path-loss exponent γPT Power circuit activation threshold αN NLOS link path-loss exponent P1max Allowed maximum harvested power at active TX gL LOS link channel fading P2max Maximum transmit power of active TX gN NLOS link channel fading γTR SINR threshold m Nakagami-m fading parameter PP cov Power coverage probability min Gmax p , Gp , θp PB beamforming parameters PC cov Channel coverage probability min Gmax t , Gt , θt TX beamforming parameters Pcov Total coverage probability min Gmax r , Gr , θr RX beamforming parameters II. S YSTEM M ODEL We consider a two-dimensional mmWave wireless ad hoc network, where TXs are first wirelessly charged by PBs and then they transmit information to RXs. The locations of PBs are modeled as a homogeneous Poisson point process (PPP) φp in R2 with constant node density λp . TXs are assumed to be randomly independently deployed and their locations are modeled as a homogeneous PPP φt with node density λt . For each TX, it has a desired RX located at a distance d0 in a random direction. Throughout the paper, we use Xi to denote both the random location as well as the ith TX itself, Yi to denote both the location and the corresponding ith RX and Zi to denote both the location and the ith PB, respectively. Note that we assume the indoor-to-outdoor penetration loss is high. Therefore, all the PBs, TXs and RXs can be regarded as outdoor devices. 6 A. Power Transfer and Information Transmission Model We assume that each PB has access to a dedicated power supply (e.g., a battery or power grid) and transmits with a constant power Pp . Time is divided into slots and let T denote one time slot. Each TX adopts the harvest-then-transmit protocol to perform PT and IT. Specifically, each time slot T is divided into two parts with ratio ρ ∈ (0, 1): in the first ρT seconds TX harvests energy from the RF signal transmitted by PBs and stores the energy in an ideal (infinite capacity) battery2 . In the remaining (1 − ρ)T seconds, TXs use all the harvested energy to transmit information to their desired RXs. Hence, there is no interference between the PT and IT stages. We make the following assumptions for realistic modelling of PT: • Different from previous works [10], [21], [24], where energy harvesting activation threshold is not considered and the devices can harvest power from any amount of incident power, we assume that the TX can scavenge energy if and only if the instantaneous aggregate received power from all PBs is greater than a power circuit activation threshold γPT . If this condition is met, then the TX is called an active TX. Otherwise, the TX will be inactive and will not scavenge any energy from the PBs. • Once the energy harvesting circuit is activated, the harvested power at the active TX is assumed to be linearly proportional to the aggregate received power with power conversion efficiency η. Due to the saturation of the energy harvesting circuit, the harvested power at the active TX cannot exceed a maximum level denoted as P1max [25]. In addition, the active TX cannot transmit information with a power greater than P2max because of the safety regulation and the electrical rating of the antenna circuit [26]. B. MmWave Blockage Model Under outdoor mmWave transmissions, each link between the PB and the TX (i.e., PB-TX link) or between the TX and the RX (i.e., TX-RX link) is susceptible to building blockages due to their high diffraction and penetration characteristics [16]. In this work, we adopt the state-of-the-art three-state blockage model as in [17], [27], where each PB-TX or TX-RX link can be in one of the following three 2 In this work, we do not consider the impact of battery imperfections [23]. 7 states: (i) the link is in LOS state if no blockage exists, (ii) the link is in NLOS state if blockage exists and (iii) the link is in outage (OUT) state if the link is too weak to be established. Given that the PB-TX or TX-RX link has a length of r, the probabilities pLOS (·), pNLOS (·) and pOUT (·) of it being in LOS, NLOS and OUT states, respectively, are pOUT (r) = u(r − rmax ); pNLOS (r) = u(r − rmin ) − u(r − rmax ); (1) pLOS (r) = 1 − u(r − rmin ), where u(·) denotes the unit step function, rmin is the radius of the LOS region and rmax is the exclusion radius of the OUT region3 , as illustrated in Fig. 1. The values of rmin and rmax depend on the propagation scenario and the mmWave carrier frequency [17]. Moreover, the communication link between TX and its desired RX is assumed to be always in LOS state. C. MmWave Channel Model It has been shown by the measurements that mmWave links experience different channel conditions under LOS, NLOS and OUT states [30]. Thus, we consider the following path-loss plus block fading channel model. For the path-loss, we adopt and modify a multi-slope path-loss model [31] and define the path-loss of PB-TX or TX-RX link with a propagation distance     1,        r−αL ,  l(r) =    βr−αN ,        ∞, of r as follows 06r<1 1 6 r < rmin , (2) rmin 6 r < rmax rmax 6 r where the first condition is added to avoid the singularity as r → 0, αL denotes the path-loss exponent for the link in LOS state, αN denotes the path-loss exponent for the link in NLOS state (2 6 αL 6 αN ), 3 Note that the two-state blockage model in [28], [29], which does not consider the OUT region, can be considered as a special case of the three-state blockage model with rmax = ∞. 8 the path-loss of the link in OUT state is assumed to be infinite [17] and the continuity in the multi-slope αN −αL path-loss model is maintained by introducing the constant β , rmin [31]. As for the fading, the link under LOS state is assumed to experience Nakagami-m fading, while the link under NLOS state is assumed to experience Rayleigh fading4 . Furthermore, both the LOS and the NLOS links experience additive white Gaussian noise (AWGN) with variance σ 2 . However, under the PT phase, the AWGN power is too small to be harvested by TXs. Hence, we ignore it in the PT phase. D. Beamforming Model To compensate the large path-loss in mmWave band, directional beamforming is necessary for devices [33]. In this work, we consider that mmWave antenna arrays perform directional beamforming at all PBs, TXs and RXs. Similar to [16], [17], the actual antenna array pattern can be approximated by a sectorized gain pattern which is given by Ga (θ) =     Gmax \|θ\| ≤ a ,    Gmin a , θa 2 , (3) otherwise is the main lobe antenna gain, where subscript a = p for PB, a = t for TX and a = r for RX, Gmax a is the side lobe antenna gain, θ ∈ [−π, π) is the angle off the boresight direction and θa is the main Gmin a lobe beam-width. Note that, as shown in Section VI-C, this model can be easily related to specific array geometries, such as an N element uniform planar or linear or circular array [18]. The main beam at the PBs are assumed to be randomly and independently oriented with respect to each other and uniformly distributed in [−π, π). Given a sufficient density of the PBs, this simple strategy ensures that the aggregate received power from PBs at different locations in the network is roughly on the same order and avoids the need for channel estimation and accurate beam alignment. In addition, it has been shown in [33] that the random directional beamforming can perform reasonably well given that more than one users need to be served. 4 We do not consider shadowing but it can be included using the composite fading model in [32]. 9 PB-TX gain Gij Fig. 1: Illustration of mmWave blockage model. TX-RX gain Dij k Gain Gk Probability pk Gain Dk Probability qk 1 max Gmax p Gt θp θt 4π 2 max Gmax t Gr θt θr 4π 2 2 min Gmax p Gt θp (2π−θt ) 4π 2 min Gmax t Gr θt (2π−θr ) 4π 2 3 max Gmin p Gt (2π−θp )θt 4π 2 max Gmin t Gr (2π−θt )θr 4π 2 4 min Gmin p Gt (2π−θp )(2π−θt ) 4π 2 min Gmin t Gr (2π−θt )(2π−θr ) 4π 2 TABLE II: Probability Mass Function of Gij and Dij . Let Gij be the effective antenna gain on the link from the ith PB to the jth TX. Under sectorization, Gij is a discrete random variable with probability pk = Pr(Gij = Gk ) and k ∈ {1, 2, 3, 4}, where its distribution is summarized in Table II. With regards to TX and RX, we assume that each TX points its main lobe towards its desired RX max and the directly. Therefore, the effective antenna gain of the desired TX-RX link is D0 = Gmax t Gr orientation of the beam of the interfering TX is uniformly distributed in [−π, π). Let Dij (i 6= j) be the effective antenna gain on the link from the ith TX to the jth RX. Similar to Gij , Dij is a discrete random variable with probability qk = Pr(Dij = Dk ), where its distribution is given in Table II. E. Metrics In this paper, we are interested in the PB-assisted mmWave wireless ad hoc network in terms of the total coverage probability for RXs (i.e., the probability that a RX can successfully receive the information from its TX after the TX successfully harvests energy from PBs). Based on the system model described above, the success of this event has to satisfy two requirements, which are: • The corresponding TX is in power coverage. Due to the random network topology and the fading channels, the aggregate received power from all PBs is a random variable. If the aggregate received power at a TX is greater than the power circuit activation threshold, the energy harvesting circuit is active and this TX can successfully harvest energy from PBs. As a result, the TX is under power coverage and IT then takes place. 10 • The RX is in channel coverage. The instantaneous transmit power for each active TX depends on its random received power. RX can receive the information from its desired TX (i.e., in channel coverage) if the signal-to-interference-plus-noise ratio (SINR) at the RX is above a certain threshold. By leveraging the Laplace transform of the aggregate received power at a typical TX and the interference at a typical RX, we compute the power coverage probability and channel coverage probability in the following sections. In the subsequent analysis, we condition on having a reference RX Y0 at the origin (0, 0) and its associated TX X0 located at a distance d0 away at (d0 , 0). According to Slivnyak’s theorem, the conditional distribution is the same as the original one for the rest of the network [34]. III. P OWER T RANSFER In this section, we focus on the PT phase of the system. We analyze the aggregate received power at a reference TX from all PBs and find the power coverage probability at the corresponding RX. Since the power harvested from the noise is negligible, the instantaneous aggregate received power at the typical TX X0 from all the PBs can be expressed as PPT = Pp X Gi0 gi0 l(ri ), (4) Zi ∈φp where Pp is the PB transmit power, Gi0 is the effective antenna gain between Zi and X0 , gi0 is the fading power gain between the ith PB Zi and the typical TX X0 , which follows the gamma distribution (under Nakagami-m fading assumption) if the PB-TX link is in LOS state and exponential distribution (under Rayleigh fading distribution) if the PB-TX link is in NLOS state. l(ri ) is the path-loss function given in (2) and ri = \|Zi − X0 \| is the Euclidean length of the PB-TX link between Zi and X0 . Using (4), the power coverage probability is defined as follows. Definition 1: The power coverage probability is the probability that the aggregate received power at the typical TX is higher than the power circuit activation threshold γPT . It can be expressed as PPcov (γPT ) = Pr(PPT > γPT ). (5) Remark 1: Analytically characterizing the power coverage probability in (5) is a challenging open problem in the literature. Generally, it is not possible to obtain a closed-form power coverage probability 11 because of the randomness in the antenna gain, mmWave channels and locations of PBs. The closed-form expression only exists under the unbounded path-loss model with α = 4 and Rayleigh fading for all links, which is shown to be Lévy distribution [34]. To overcome this problem, some works [20], [28], [35] employed the Gamma scaling method. This approach involves introducing a dummy Gamma random variable with parameter N 0 to reformulate the original problem. However, the approach can sometimes lead to large errors with finite N 0 value. Other works adopted the Gil-Pelaez inversion theorem [36] . This approach involves one fold integration and is only suitable for the random variable with a simple Laplace transform. If the Laplace transform is even moderately complicated, this method is not very efficient even if the Laplace transform is in closed-form. In this work, we adopt a numerical inversion method, which is easy to compute, if the Laplace transform of a random variable is in closed-form, and provides controllable error estimation. Following [37], [38], the CDF of the aggregate received power PPT is given as 1 FPPT (x) = 2πj = 1 2πj Z a+j∞ a−j∞ Z a+j∞ a−j∞ LFPPT (s) exp(sx)ds (6a) LPPT (s) exp(sx)ds. s (6b) where (6a) is obtained according to the Bromwich integral [39] and (6b) follows from probability theory that LPPT (s) = sLFPPT (s). Using the trapezoidal rule and the Euler summation, the above integral can be transformed into a finite sum. Therefore, we can express the power coverage probability as B C+b 2−B exp( A2 ) X B X (−1)c LPPT (s) P Pcov (γPT ) =1 − Re , γPT b D s c c=0 b=0 where Re[·] is the real part operator, s = A+j2πc , 2γPT (7) LPPT (s) is the Laplace transform of PPT , Dc = 2 (if c = 0) and Dc = 1 (if c = 1, 2, ..., C + b). A, B and C are positive parameters used to control the estimation accuracy. From (7), the key parameter in order to obtain the power coverage probability is LPPT (s). By the definition of Laplace transform of a random variable, we express LPPT (s) in closed-form in the following theorem. Theorem 1: Following the system model in Section II, the Laplace transform of the aggregate received 12 power at the typical TX from all the PBs in a mmWave ad hoc network is 4 Y −αL 2 LPPT (s) = exp πλp rmin pk mm (m+srmin Pp Gk )−m−1 + πλp pk (sPp Gk )δL (Ξ1 (1) − Ξ1 (rmin )) k=1 + πλp pk sPp Gk β (Ξ2 (rmin ) − Ξ2 (rmax )) + πλp pk (sPp Gk β)δN (Ξ3 (rmin ) − Ξ3 (rmax )) , 2 + αN (8) where mm (r−αL sPp Gk )−δL −m αL Γ(1 + m) mrαL Ξ1 (r) = , (9) 2 F1 1 + m,m + δL ;1 + m + δL ;− (2 + mαL )Γ(m) sPp Gk r2 Ξ2 (r) = α , (10) r N + sPp Gk β (r−αN sPp Gk β)−δN −1 rαN β −1 αN Ξ3 (r) = , sPp Gk β(2 + αN ) −2(r +sPp Gk β)2 F1 1, δN +1; 2+δN ; − rαN + sPp Gk β sPp Gk (11) and Γ(·) is the complete gamma function, function, δL , 2 αL and δN , 2 F1 (·, ·; ·; ·) is the Gaussian (or ordinary) hypergeometric 2 . αN Proof: See Appendix A. By substituting (8) into (7), we can compute the power coverage probability. As shown in Theorem 1, the Laplace transform of PPT is in closed-form; hence, PPcov (γPT ) is just a summation over a finite number of terms. Following the selection guideline of parameters A, B and C in [38], we can achieve a stable numerical result by carefully choosing them. Before ending this section, we validate the analysis for the power coverage probability. Fig. 2 plots the power coverage probability versus power circuit activation threshold. The simulation results are generated by averaging over 108 Monte Carlo simulation runs. We set A = 24, B = 20 and C = 30 in order to achieve an estimation error of 10−10 . The other system parameters follow Table IV. From the figure, we can see that the analytical results match perfectly with the simulation results, which demonstrates the accuracy of the proposed approach. Fig. 2 also shows that the power coverage probability increases with the density of PBs, because the aggregate received power at TX increases as the PB density increases. IV. I NFORMATION T RANSMISSION In this section we focus on the IT phase between the TX and RX. We assume that the TX uses all the harvested energy in the IT phase. As indicated in Section II-A, the transmit power of an active TX is a 13 Power coverage probability 1 =100/km2 p 0.8 =50/km2 p =10/km2 p 0.6 Simulation 0.4 0.2 0 -20 -10 0 10 Power circuit activation threshold 20 PT 30 (dBm) Fig. 2: Power coverage probability versus power circuit activation threshold γPT for different PB densities. Other system parameters follow Table IV. random variable which depends on its harvested power. Hence, we first evaluate the transmit power for an active TX. Then, we calculate the channel coverage probability at the reference RX. Note that the derived channel coverage probability is in fact a conditional probability, which is conditioned on the reference TX-RX link being active. A. Transmit Power and Locations of Active TX Using the PT assumptions in Section II-A, the instantaneous transmit power for each active TX is    ρ  η 1−ρ PPT , min(η −1 P1max , 1−ρ P max ) > PPT > γPT ηρ 2 Pt = , (12)   ρ 1−ρ  min( P max , P max ), P > min(η −1 P max , P max ) 1−ρ 1 2 PT 1 ηρ 2 where 0 6 η 6 1 is the power conversion efficiency. Note that the first condition in (12) comes from the fact that the received power at an active TX must be greater than γPT . For the second condition in (12), P1max is the maximum harvested power at an active TX when the energy harvesting circuit is saturated and P2max is the maximum transmit power for an active TX. Thus, the second condition caps the transmit power by the allowed maximum harvested power constraint or the maximum transmit power constraint. The following remarks discuss the modelling challenges and proposed solution for characterizing Pt . Remark 2: To the best of our knowledge, the closed-form expression for the PDF of Pt is very difficult to obtain. This is because Pt and PPT are correlated and the closed-form CDF of PPT is not available 14 according to Section III. In the literature, some papers [21], [40] have proposed to use the average harvested power as the transmit power for each TX. However, this does not always lead to accurate results. Hence, inspired from the approach in [41], we propose to discretize Pt in (12) into a finite number of levels. We show that this approximation allows tractable computation of the channel coverage probability. The accuracy of this approximation depends on the number of levels. Our results in Section VI-A show that if we discretize the power level in the log scale, a reasonable level of accuracy is reached with as little as 10 levels. Remark 3: From (12), we can see that Pt depends on PPT . Hence, the motivation for discretizing Pt in the log scale comes from looking into two important measures of PPT , the skewness and the kurtosis. The skewness and the kurtosis describe the shape of the probability distribution of PPT . As presented in [22], the distribution of the aggregate received power is skewed to the right with a heavy tail, because both the skewness and the kurtosis of PPT are much greater than 0 for most cases. Therefore, most of the TXs will be at the lowest power level if we discretize Pt in linear scale. Hence, we discretize the power level in the log scale. This improves the accuracy of the approximation. Let N + 1 and w denote the total number of levels and the step size of each level, respectively. P2max )−γPT min(η −1 P1max , 1−ρ ηρ dBm. We further define kn as the portion of TXs They are related by w = N whose Pt is at the nth level, i.e., kn = Pr ((nw + γPT ) dBm 6 PPT < ((n + 1) w + γPT ) dBm) for n = {0, 1, 2, ..., N − 1} and kN = Pr(PPT > min(η −1 P1max , 1−ρ P max )). Combining with the power coverage ηρ 2 probability derived in Section III, we can express kn as     PPcov ((nw + γPT )dBm) − PPcov (((n + 1) w + γPT ) dBm) , kn =    PP (min(η −1 P max , 1−ρ P max )), cov 1 ηρ 2 n = {0, 1, 2, ..., N − 1} . (13) n=N The above expression allows us to determine the portion of TXs whose Pt is at the nth level. The transmit power for the active TX at the nth level is nw+γPT −30 ρ n 10 Pt = η 10 W. 1−ρ (14) The next step is to decide how to model the locations of the TXs whose Pt is at the nth level. This is discussed in the remark below. 15 Remark 4: In general, the location and the transmit power of an active TX are correlated, i.e., a TX has higher chance to be activated and transmits with a larger power, if its location is closer to a PB. However, it is not easy to identify and fit a spatial point process with local clustering to model the location of active TXs [14], [42]. In this paper, for analytical tractability, we assume that the location and the transmit power of an active TX are independent, i.e., a TX in φt can have a transmit power of Ptn with probability kn independently of other TXs. Therefore, using the thinning theorem, we interpret the active TX at the nth level as an independent homogeneous PPP with node density λnt = kn λt , denoted as φnt . The accuracy of this approximation will be validated in Section VI-A. B. Channel Coverage Probability Given that the desired TX is active, the instantaneous SINR at the reference RX, Y0 , is given as SIN R = P PX0 D0 h0 l(d0 ) , 2 Xi ∈φactive PXi Di0 hi0 l(Xi ) + σ (15) where h0 and hi0 denote the fading power gains on the reference link and the ith interference link respectively, D0 and Di0 denote the beamforming antenna gain at the RX from its reference TX and the ith interfering TX respectively and σ 2 is the AWGN power. PX0 and PXi are the transmit power for the reference TX and the active TX Xi , respectively. Using (15), the power coverage probability is defined as follows. Definition 2: The channel coverage probability is the probability that the SINR at the reference RX is above a threshold γTR and can be expressed as PC cov (γTR ) = Pr(SIN R > γTR ). (16) Remark 5: It is possible to employ the numerical inversion method in Section III to find the channel coverage probability. In doing so, the Laplace transform of the term IX +σ 2 PX0 D0 h0 l(d0 ) is required. This Laplace tranform cannot be expressed in closed-form because of the random variables PX0 and h0 in the denominator. Although it is still computable, it leads to greater computation complexity. Consequently, we employ the reference link power gain (RLPG) based method in [38] to efficiently find the channel coverage 16 probability. The basic principle of this approach is to first find the conditional outage probability in terms of the CDF of the reference links fading power gain and then remove the conditioning on the fading power gains and locations of the interferers, respectively. In order to apply this method, the reference TX-RX link is assumed to undergo Nakagami-m fading with integer m. The result for the conditional channel coverage probability is presented in the following proposition. Proposition 1: Following the system model in Section II, the conditional channel coverage probability at the reference RX in a mmWave ad hoc network is PC cov (γTR ) = where IX = PN P n=0 Xi ∈φn t N m−1 X X (−s)l dl kn 2 (s) , L I +σ X l P (γ l! ds P ) PT cov n=0 l=0 Ptn Di0 hi0 l(Xi ) and s = (17) mγTR . Ptn D0 l(d0 ) Proof: See Appendix B. (17) needs the Laplace transform of the interference plus noise. Using stochastic geometry, we can derive it and the result is shown in the following corollary. Corollary 1: Following the system model in Section II and the discretization assumption in Section IV-A, the Laplace transform of the aggregate interference plus noise at the reference RX in a mmWave ad hoc network is LIX +σ2 (s) = N Y 4 Y −αL n 2 exp πλnt rmin qk mm (m+srmin Pt Dk )−m−1 + πλnt qk (sPtn Dk )δL (Ξ01 (1) − Ξ01 (rmin )) n=0k=1 +πλnt qk sPtn Dk β (Ξ02 (rmin ) − Ξ02 (rmax )) + πλnt qk (sPtn Dk β)δN (Ξ03 (rmin ) − Ξ03 (rmax )) exp(−sσ 2 ), 2 + αN (18) where mm (r−αL sPtn Dk )−δL −m αL Γ(1 + m) mrαL = , (19) 2 F1 1 + m,m + δL ;1 + m + δL ;− (2 + mαL )Γ(m) sPtn Dk r2 Ξ02 (r) = α , (20) r N + sPtn Dk β (r−αN sPtn Dk β)−δN −1 n rαN β −1 0 αN n Ξ3 (r) = sPt Dk β(2 + αN ) −2(r +sPt Dk β)2 F1 1, δN +1; 2+δN ; − n . rαN + sPtn Dk β sPt Dk Ξ01 (r) (21) Proof: Following the definition of Laplace transform, we have LIX +σ2 (s) =EIX [exp(−s(IX + σ 2 ))] = EIX [exp(−sIX )] exp(−sσ 2 ) = LIX (s) exp(−sσ 2 ), (22) 17 where the Laplace transform of the aggregate interference can be expressed as    N X X LIX (s) =EIX [exp(−sIX )] = EDi0 ,hi0 ,φnt exp −s Ptn Di0 hi0 l(Xi ) n=0 Xi ∈φn t = N Y n=0    EDi0 ,hi0 ,φnt exp −s X Ptn Di0 hi0 l(Xi ) . (23) Xi ∈φn t Then, following the same steps as the proof of Laplace transform of aggregate received power in Appendix A, we can find the expectation in (23) and arrive at the result in (18). The Laplace transform shown in Corollary 1 is in closed-form. Substituting (18) into (17), we can easily compute the conditional channel coverage probability. Note that (17) requires higher order derivatives of the Laplace transform of the interference plus noise dl 2 (s), L dsl IX +σ which can be yielded in closed-form using chain rules and changing variables. For brevity, the details are omitted here. V. T OTAL C OVERAGE P ROBABILITY As discussed in Section II-E, the event that the information can be successfully delivered to RX has two requirements, i.e., satisfying power coverage and channel coverage. Based on our definition, the total coverage probability is Pcov (γPT , γTR ) = Pr(TX is in power coverage & RX is in channel coverage) = Pr(TX is in power coverage) Pr(RX is in channel coverage \| TX is in power coverage) Combining our analysis presented in Section III and IV, we have Pcov (γPT , γTR ) =PPcov (γPT )PC cov (γTR ) N m−1 X X (−s)l dl = LI +σ2 (s)kn , l! dsl X n=0 l=0 where s = mγTR , Ptn D0 l(d0 ) (24) LIX +σ2 (s) is given in Corollary 1, kn is presented in (13), which is determined by the power coverage probability. The key metrics are summarized in Table III. VI. R ESULTS In this section, we first validate the proposed model and then discuss the design insights provided by the model. Unless stated otherwise, the values of the parameters summarized in Table IV are used. The 18 TABLE III: Summary of the Analytical Model for PB-assisted mmWave Ad Hoc Networks. Performance metrics General form Key factor(s) Power coverage probability (7) LPPT (s) in (8) Channel coverage probability (17) PP cov (γPT ) in (7) & LIX +σ 2 (s) in (18) Total coverage probability (24) PP cov (γPT ) in (7) & LIX +σ 2 (s) in (18) TABLE IV: Parameter Values. Parameter Value Parameter Value Parameter Value Parameter Value λp 50 /km2 m 5 αL 2 ρ 0.5 λt 100 /km2 min Gmax p , Gp , θp [20 dB, −10 dB, 30o ] αN 4 η 0.5 d0 20 m min Gmax t , Gt , θt [10 dB, −10 dB, 45o ] Pp 40 dBm γPT -20 dBm rmin 100 m min Gmax r , Gr , θr [10 dB, −10 dB, 45o ] P1max 20 dBm γTR 30 dBm rmax 200 m σ2 -30 dBm P2max 30 dBm N 10 chosen values are consistent with the literature in mmWave and WPT [1], [16], [17]. Note that the values of rmin and rmax correspond to 28 GHz mmWave carrier frequency [16]. We mainly focus on illustrating the results for total coverage probability and channel coverage probability. As for the power coverage probability, it will be explained within the text. Table V summarizes the impact of varying the important system parameters5 , i.e., SINR threshold γTR , PB density λp , TX density λt , PB transmit power Pp , radius of the LOS region rmin , power circuit activation threshold γPT , the beam-width of the main lobe of TX θt , RX’s main lobe beam-width θr , allowed maximum harvested power at active TX P1max , time switching parameter ρ and TX maximum transmit power P2max on the three network performance metrics. In Table V, ↑, ↓ and - denote increase, decrease and unrelated, respectively. ↑↓ represents that the performance metric first increases then decreases with the system parameter. Please note that the trends in Table V originate from the analysis of the numerical results, which is presented in detail in the following subsections. 5 Note that the trends reported in Table V remain the same for a two-state blockage model. 19 TABLE V: Effect of Important System Parameters. Parameter Power coverage probability Channel coverage probability Total coverage probability Increasing γTR - ↓ ↓ Increasing λp ↑ ↓↑ ↑ Increasing λt - ↓ ↓ Increasing Pp ↑ ↑ ↑ Increasing rmin ↑ ↑ ↑ Increasing γPT ↓ ↑ ↓ Increasing θt and θr ↑ ↓ ↓ Increasing P1max - ↑↓ ↑↓ Increasing ρ - ↑ ↑ Increasing P2max - ↓ ↓ A. Model Validation In this section, we validate the proposed model for the channel coverage probability and the total coverage probability. Fig. 3 plots the channel coverage probability and the total coverage probability for a reference RX against SINR threshold for different densities of PBs and TXs. The analytical results are obtained using Proposition 1 and (24) with 10 discrete levels for Pt . The simulation results are generated by averaging over 108 Monte Carlo simulation runs and do not assume any discretization of power levels. From the figure, we can see that our analytical results provide a good approximation to the simulation. The small gap between them comes from two reasons: (i) discretization of the power levels, as discussed in Remark 3, and (ii) ignorance of the correlation between the location and the transmit power of active TX, as discussed in Remark 4. From Fig. 3, we can see that the gap between the simulation and the analytical results is smaller, when γTR is higher. At γTR = 30 dBm, which is a typical SINR threshold, the relative errors between the proposed model and the simulation results for both channel coverage probability and total coverage probability are between 5% to 10%. This validates the use of 10 discrete levels for Pt , which provides good accuracy. Insights: Comparing the four cases for the different PB and TX densities, Fig. 3 shows that: (i) The channel coverage probability decreases while the total coverage probability increases as PB density 1 1 0.8 0.8 Coverage probability Coverage probability 20 0.6 0.4 0.2 0 10 Channel Total Simulation 20 30 SINR threshold 40 TR 50 0.6 0.4 0.2 0 10 60 (dBm) 0.8 0.8 Coverage probability Coverage probability 1 0.6 0.4 Channel Total Simulation 20 30 SINR threshold 40 TR 50 (dBm) (c) λp = 50 /km2 , λt = 250 /km2 . 30 40 TR 50 60 (dBm) (b) λp = 10 /km2 , λt = 100 /km2 . 1 0 10 20 SINR threshold (a) λp = 50 /km2 , λt = 500 /km2 . 0.2 Channel Total Simulation 60 0.6 0.4 0.2 0 10 Channel Total Simulation 20 30 SINR threshold 40 TR 50 60 (dBm) (d) λp = 10 /km2 , λt = 50 /km2 . Fig. 3: Channel coverage probability and total coverage probability versus SINR threshold γTR . The PB density is 50 and 10 per km2 and the TX density is 500, 100, 250 and 50 per km2 . increases. As the PB density increases, the aggregate received power at TX increases as well as the number of active TXs. Therefore, interfering power received by the RX is higher and the channel coverage probability decreases. However, the total coverage probability increases because the power coverage probability increases with the PB density. (ii) When the PB density is low, the TXs are very likely to be inactive and the total coverage probability is dominated by the power coverage probability. When the PB density is high, the TXs are very likely to be active. Hence, the interference is strong and the channel coverage probability dominates the total coverage probability. (iii) For the same PB density, both the total coverage probability and the channel coverage probability are higher, when the TX density is lower. This is because more interfering TXs exist if TX density increases. 21 B. Effect of PB Transmit Power Fig. 4(a) illustrates the effect of PB transmit power Pp on the total coverage probability and channel coverage probability, with different radius of the LOS region rmin = 50m, 100m. The simulation results are also plotted in the figure, which are averaged over 108 Monte Carlo simulation runs. The accuracy is between 3% to 8%, which again validates the proposed model. Hence, in the subsequent figures in the paper we only show the analytical results and discuss the insights. Fig. 4(b) plots the total coverage probability against the transmit power of PB. We also plot an asymptotic result when Pp approaches infinity. This result is obtained as follows. As Pp approaches infinity, if one or more PBs fall into the LOS or NLOS region of a TX, this TX will be active and transmit with a ρ P1max , P2max ). Hence, the asymptotic power coverage probability is equivalent to power of Pt = min( 1−ρ the probability that at least one PB falls into the LOS or NLOS region of the TX, which is given by 2 lim PPcov = 1 −exp −πλp rmax . Pp →∞ (25) The asymptotic conditional channel coverage probability and the asymptotic total coverage probability can be found by (17) and (24) respectively with the portion of TXs at the nth level as     0, n = {0, 1, 2, ..., N − 1} . lim kn = Pp →∞    PPcov , n = N (26) From the figure, we can see that the analytical and asymptotic results converge as Pp gets large, which validates the derivation of the asymptotic results. In addition, in Fig. 4(b), we have marked the safe RF exposure region with a PB transmit power less than 51 dBm, equivalently power density smaller than 10 W/m2 at 1 m from the PB [26]. We will discuss in detail later in the feasibility study in Section VI-E. Insights: Fig. 4(a) shows that: (i) The channel coverage probability first slightly decreases and then increases with the increase of Pp . This can be explained as follows. At first, both the transmit power of the desired TX and the number of interfering TX increase with Pp . The interplay of this two factors results in the slightly decreasing trend for the channel coverage probability. As Pp further increases, the increase in the number of interfering TX is negligible, while the transmit power of the desired TX continues to 22 1 Total coverage probability 1 Coverage probability Channel coverage probability 0.8 Total coverage probability 0.6 rmin=50m 0.4 rmin=100m Simulation (P ccov) 0.2 Simulation (P cov) 0 30 35 40 45 Asymptotic total coverage probability=0.9953 0.8 0.6 0.4 0.2 50 PB transmit power Pp (dBm) (a) Channel coverage and total coverage with different rmin . Analytical Asymptotic 0 30 Safe level of RF exposure 40 50 Unsafe level of RF exposure 60 70 80 PB transmit power Pp (dBm) (b) Total coverage and asymptotic total coverage. Fig. 4: Coverage probabilities versus PB transmit power Pp . increase, which leads to the increase of the channel coverage probability. (ii) The total coverage probability increases as PB transmit power Pp increases. When Pp is small, the desired TX might not receive enough power to activate the IT process. So the total coverage probability is small and is limited by the power coverage probability. When Pp is large, the channel coverage probability becomes the dominant factor in determining the total coverage probability. Hence, eventually the channel coverage probability and total coverage probability curves merge. (iii) The total coverage probability increase with rmin , because more PBs falls into the LOS region and the path-loss is less severe, which improves the power coverage probability. The benefit of increasing the radius of the LOS region is less significant for the channel coverage probability. C. Effect of Directional Beamforming at PB, TX and RX Fig. 5 plots the total coverage probability and channel coverage probability against the power circuit activation threshold of TX for different beamforming parameters at TX and RX, i.e., [20 dB, −10 dB, 30o ] and [10 dB, −10 dB, 45o ]. Insights: Fig. 5 shows that, for both sets of beamforming parameters, as the power circuit activation threshold γPT increases, the channel coverage probability is always increasing, while the total coverage probability stays roughly the same at first and then decreases. This can be explained as follows. When γPT 23 1 Channel coverage probability 0.9 0.8 Total coverage probability 0.7 o TX, RX [20, -10, 30 ] TX, RX [10, -10, 45 o] 0.6 -30 Total coverage probability Coverage probability 1 Np =4, 9, 16 0.9 1 0.8 0.95 0.7 0.9 0.6 0.85 4 6 8 N =4 10 p Np=9 0.5 Np=16 0.4 -25 -20 -15 Power circuit activation threshold -10 PT (dBm) 5 10 15 20 Number of TX and RX antenna elements Nt , N r Fig. 5: Channel coverage probability and total coverage probability Fig. 6: Total coverage probability versus the numbers of antenna versus power circuit activation threshold γPT with different TX and elements at TX and RX Nt and Nr with different numbers of RX beamforming parameters. antenna elements at PB. increases, the power coverage probability decreases. The reduction in the number of active TXs improves the channel coverage probability. With regards to the total coverage probability, its trend is determined by the interplay of channel coverage probability and power coverage probability. At first, the drop in power coverage is relatively small as shown in Fig. 2; so the total coverage probability is almost unchanged. After a certain point, the power coverage probability drops a lot, which mainly governs the total coverage probability. Hence, the total coverage probability decreases later on. Comparing the curves for the different beamforming parameters, we can see that TX and RX with [20 dB, −10 dB, 30o ] gives a higher total coverage probability in the low power circuit activation threshold region. This is because a narrower main lobe beam-width gives a larger main lobe gain and makes less interfering TXs fall into its main lobe which results in higher channel coverage probability. However, the total coverage probability is limited by the power coverage probability when γPT is large. Impact of number of antenna elements: The beamforming model adopted in this paper can be related to any specific array geometry by substituting the appropriate values for the three beamforming parameters. For instance, a uniform planar square array with half-wavelength antenna element spacing can be used min at the PBs, TXs and RXs. The values for the main lobe antenna gain Gmax a , side lobe antenna gain Ga and main lobe beamwidth θa depend on the number of the antenna elements Na and can be calculated 24 by using the equations below [18]: √ Gmax = Na , Gmin a a √ √ √ Na − 2π3 Na sin( 2√N3 a ) 3 √ √ , θa = √ , = √ 3 3 Na Na − 2π sin( 2√Na ) (27) where subscript a = p for PB, a = t for TX and a = r for RX. Fig. 6 plots the total coverage probability versus the numbers of antenna elements at the TX and RX Nt and Nr with different PB antenna element number Np . The figure shows that the total coverage probability increases with the numbers of antenna elements at the TX and RX, which agrees with our previous findings. However, under our considered system parameters, the total coverage probability stays roughly the same after having more than about 15 TX and RX antenna elements, as the side lobe antenna gain and the main lobe beamwidth stay almost constant with further increase in the number of antenna elements. In addition, the number of antenna elements at the PB does not significantly impact the total coverage probability. D. Effect of Allowed Maximum Harvested Power at TX Fig. 7 plots the total coverage probability and channel coverage probability against the allowed maximum harvested power of TX P1max for different time switching ratios 0.2, 0.5 and 0.8. Note that both the time switching ratio and the allowed maximum harvested power do not affect the power coverage probability. Insights: Fig. 7 shows that the channel coverage probability and the total coverage probability both first increase with P1max , then decrease. The rise of the channel coverage probability is because the possible transmit power of the desired TX increases with its allowed maximum harvested power. However, as P1max further increases, the accumulated harvested energy during the PT phase is higher and the transmit power of other active TX also goes up. As a result, the interfering power received at the RX is higher and the channel coverage probability decreases. The channel coverage probability will converge to a constant value as P1max increases even further, because the maximum transmit power of active TX has limited the channel performance. Comparing the curves for different ρ, we can see that for a given maximum harvested power of TX P1max , increasing ρ improves the coverage probabilities. When ρ is higher, more energy is captured during 25 1 1 Coverage probability Coverage probability Pp =50 dBm 0.8 =0.2 =0.5 =0.8 0.6 0.4 0.2 Channel coverage probability Total coverage probability 0 -20 -10 0 10 20 30 0.9 0.8 Pp =30 dBm 0.7 0.6 Channel coverage probability Total coverage probability 0.5 20 Maximum harvested power Pmax (dBm) 1 25 30 35 40 Maximum TX transmit power Pmax (dBm) 2 Fig. 7: Channel coverage probability and total coverage probability Fig. 8: Channel coverage probability and total coverage probability versus allowed maximum harvested power P1max with different time versus maximum TX transmit power P2max for different PB transmit switching ratios. power with the allowed maximum harvested power of TX being 50 dBm. the PT phase. Therefore, the transmit power of active TX is now limited by the maximum transmit power P2max . As a result, the channel coverage probability and total coverage probability converge and do not vary much with the changes in the allowed maximum harvested power. E. Feasibility of PB-assisted mmWave Wireless Ad hoc Networks Finally, we investigate the feasibility of PB-assisted mmWave wireless ad hoc network. Fig. 8 is a plot of the total coverage probability and channel coverage probability versus maximum TX transmit power P2max with varied PB transmit power, 50 dBm and 30 dBm. To better highlight the impact of the maximum transmit power at TX, we have set P1max equal to 50 dBm which is much higher than P2max . From the figure, we can see that the channel coverage probability and total coverage probability do not change much with the considered maximum TX transmit power, which means that the probability mass function (PMF) of the transmit power for the desired TX remains almost the same. Note that the power coverage probability is independent of the maximum transmit power of TX. Insight: From Fig. 8, the total coverage probability and the channel coverage probability are around 90% if Pp is 50 dBm. If PB transmits with a constant power of 50 dBm, the power density at a distant of 1 m from the PB is 7.95 W/m2 . This power density is smaller than 10 W/m2 , which is the permissible 26 safety level of human exposure to RF electromagnetic fields based on IEEE Standard. Under this safety regulation, the maximum permissible PB transmit power would be 51 dBm. We have marked this value in Fig. 4(b). From Fig. 4(b), we can see that the total coverage probability with a PB transmit power less than 51 dBm can be up to 93.4% of the maximum system performance, as given by the asymptotic analysis in Section VI-B, based on our considered system parameters. The results in Fig. 4(b) and Fig. 8 show that PB-assisted mmWave ad hoc networks are feasible under practical network setup. VII. C ONCLUSION In this paper, we have presented an approximate yet accurate model for PB-assisted mmWave wireless ad hoc networks, where TXs harvest energy from all PBs and then use the harvested energy to transmit information to their corresponding RXs. We first obtained the Laplace transform of the aggregate received power at the TX to compute the power coverage probability. Then, the channel coverage probability and total coverage probability were formulated based on discretizing the transmit power of TXs into a finite number of levels. The simulation results confirmed the accuracy of the proposed model. The results have shown that the total coverage probability improves by increasing the transmit power of PB, narrowing the main lobe beam-width and decreasing the maximum harvested power at the TX. Our results also showed that PB-assisted mmWave ad hoc network is feasible under realistic setup conditions. Future work can consider extensions to other MAC protocols such as carrier-sense multiple access (CSMA) [43], [44] and optimal allocation of the transmit power of an active TX. 27 A PPENDIX A P ROOF OF T HEOREM 1 Following the definition of Laplace transform, the Laplace transform of the aggregate received power can be expressed as    LPPT (s) = EPPT [exp(−sPPT )] = Eφp ,Gi0 ,gi0 exp −sPp X Gi0 gi0 l(ri ) Zi ∈φp " !# = Eφp ,Gi0 ,gi0 exp −sPp X " Eφp ,Gi0 ,gi0 exp −sPp Gi0 gi0 l(ri ) 06ri <1 !# X Gi0 gi0 l(ri ) 16ri <rmin " !# × Eφp ,Gi0 ,gi0 exp −sPp X Gi0 gi0 l(ri ) rmin 6ri <rmax Z πZ 1 EGi0 ,gi0 [1 − exp(−sPp Gi0 gi0 )]λp rdrdθ = exp − −π 0 {z } \| A1 Z π Z rmin −αL EGi0 ,gi0 [1−exp(−sPp Gi0 gi0 r )]λp rdrdθ × exp − −π 1 \| {z } A2 Z π Z rmax −αN EGi0 ,gi0 [1−exp(−sPp Gi0 gi0 βr )]λp rdrdθ . × exp − −π rmin {z } \| (28) A3 The first term A1 is evaluated as follows A1 = exp (−πλp (1 − EGi0 ,gi0 [exp(−sPp Gi0 gi0 )])) Z ∞ = exp −πλp 1 − EGi0 exp(−sPp Gi0 g)fgL (g)dg 0 = exp −πλp + πλp mm EGi0 (m + sPp Gi0 )−m ! 4 X = exp −πλp + πλp mm (m + sPp Gk )−m pk , (29) k=1 where we use the fact that the link in LOS state experiences Nakagami-m fading with fgL (g) = mm g m−1 exp(−mg) . Γ(m) The second term A2 is evaluated as follows −αL 2 A2 = exp πλp EGi0 ,gi0 [1 − exp(−sPp Gi0 gi0 )] − πλp rmin EGi0 ,gi0 1 − exp(−srmin Pp Gi0 gi0 ) h i δL δL − πλp EGi0 ,gi0 (sPp Gi0 ) gi0 γ(1 − δL , sPp gi0 Gi0 ) h i δL −αL +πλp EGi0 ,gi0 (sPp Gi0 )δL gi0 γ(1 − δL , sPp gi0 Gi0 rmin ) (30a) 28 m = exp πλp −πλp m 4 X −m (m + sPp Gk ) 2 pk −πλp rmin k=1 + πλp 4 X (sPp Gk )δL k=1 4 X m + 4 X −αL 2 πλp rmin mm (m + srmin Pp Gk )−m pk k=1 −δL−m m m (sPp Gk ) αL Γ(1+m) pk 2 F1 1 + m,m + δL ;1 + m + δL ;− (2 + mαL )Γ(m) sPp Gk −αL mm (rmin sPp Gk )−δL−m αL Γ(1+m) (2 + mαL )Γ(m) k=1 αL rmin m pk , × 2 F1 1 + m,m + δL ;1 + m + δL ;− sPp Gk − πλp (sPp Gk )δL (30b) where (30a) follows from changing variables and integration by parts and (30b) is obtained after taking the expectation over gL then Gi0 . Similarly, the third term A3 can be worked out by taking the expectation over gN , which has a PDF as fgN (h) = exp(−g). The details are omitted for sake of brevity. Finally, the Laplace transform in Theorem 1 is obtained by substituting A1 , A2 and A3 into (28). A PPENDIX B P ROOF OF P ROPOSITION 1 By substituting (15) into (16), we can express the conditional channel coverage probability as ! PX0 D0 h0 l(d0 ) C Pcov (γTR ) = Pr P > γTR 2 Xi ∈φactive PXi Di0 hi0 l(Xi ) + σ ! PX0 D0 h0 l(d0 ) ≈ Pr PN P (31a) > γTR n 2 P D i0 hi0 l(Xi ) + σ t n=0 Xi ∈φn t γTR (IX + σ 2 ) γTR (IX + σ 2 ) = Pr h0 > = EPX0 ,IX 1 − Fh0 , (31b) PX0 D0 l(d0 ) PX0 D0 l(d0 ) P P where approximation in (31a) comes from our power level discretization, IX = N Ptn Di0 hi0 l(Xi ) n=0 Xi ∈φn t and Fh0 (·) is the CDF of the fading power gain on the reference TX-RX link. Since the desired link is assumed to experience Nakagami-m fading with integer m, the CDF of h0 has a nice form, which is P 1 l Fh0 (h) = 1 − m−1 l=0 l! (mh) exp(−mh). Hence, we can re-write (31b) as "m−1 l # 2 2 X1 γ (I + σ ) γ (I + σ ) TR X TR X PC m exp −m cov (γTR ) =EPX0 ,IX l! 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SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 arXiv:1512.02289v2 [math.RA] 5 Aug 2016 ANTHONY BAK AND ALEXEI STEPANOV Abstract. In 2012 the second author obtained a description of the lattice of subgroups of a Chevalley group G(Φ, A), containing the elementary subgroup E(Φ, K) over a subring K ⊆ A provided Φ = Bn , Cn or F4 , n > 2, and 2 is invertible in K. It turns out that this lattice is a disjoint union of “sandwiches” parameterized by subrings R such that K ⊆ R ⊆ A. In the current article a similar result is proved for Φ = Cn , n > 3, and 2 = 0 in K. In this setting one has to introduce more sandwiches, namely, sandwiches which are parameterized by form rings (R, Λ) such that K ⊆ Λ ⊆ R ⊆ A. The result we get generalizes Ya. N. Nuzhin’s theorem of 2013 concerning the root systems Φ = Bn , Cn , n > 3, where the same description of the subgroup lattice is obtained, but under the condition that A is an algebraic extension of a field K. Introduction Throughout this paper K, R and A will denote commutative rings. Let G = GP (Φ, ) denote a Chevalley–Demazure group scheme with a reduced irreducible root system Φ and weight lattice P . If the weight lattice P is not important, we leave it out of the notation. Denote by E(A) = EP (Φ, A) the elementary subgroup of G(A), i. e. the subgroup generated by all elementary root unipotent elements xα (t), α ∈ Φ, t ∈ A. Let K be a subring of A. We study the lattice L = L E(K), G(A) of subgroups of G(A), containing E(K). The standard description of L is called a sandwich classification theorem. It states that for each H ∈ L there exists a unique subring R between K and A suchthat H lies between E(R) and its normalizer NA (R) in G(A). The lattice L E(R), NA (R) of all subgroups of G(A) that lie between E(R) and NA (R) is called a standard sandwich. Thus, the sandwich classification theorem holds iff L is the disjoint union of all standard sandwiches. In [17] the second author proved the sandwich classification theorem provided that Φ is doubly laced and 2 is invertible in K. In this article we consider the symplectic case of rank n > 3 with 2 = 0 in A, in particular, we always assume that Φ = Cn . In this situation we show that the sandwich classification theorem, as formulated above, does not hold. Let R be a subring of A, containing K. Recall [5] that an additive subgroup Λ of R is called a (symplectic) form parameter if it contains 2R and is closed under multiplication by ξ 2 for all ξ ∈ R. If 2 = 0, the set R2 = {ξ 2 \| ξ ∈ R} is a subring of R, and a form parameter Λ in R is just an R2 -submodule of R. Let Ep2n (R, Λ) denote the subgroup of Sp2n (R) generated by all root unipotents xα (µ) and xβ (λ), where α is a short root and µ ∈ R and β is a long root and λ ∈ Λ. Suppose now that Λ ⊇ K. Then clearly Ep2n (R, Λ) > Ep2n (K). But one can check that if Λ 6= R ′ ′ then Ep2n (R, Λ) is not contained in any standard sandwich L Ep2n (R ), NA (R ) such that K ⊆ R′ . This shows that L cannot be a union of standard sandwiches, unless we enlarge our standard sandwiches or introduce additional ones. It turns out that the latter approach is the correct one. It is analogous to that followed by the first author in [3] and by E. Abe and The work of the second named author under this publication is supported by Russian Science Foundation, grant N.14-11-00297. 1 2 ANTHONY BAK AND ALEXEI STEPANOV K.Suzuki in [2, 1] in order to provide enough sandwiches to classify subgroups of Sp2n (R), which are normalized by Ep2n (R). If A and K are field and A is algebraic over K, then the sandwich classification theorem was obtained by Ya. N. Nuzhin in [14, 15]. In [15] he considered the case of doubly laced root systems in characteristic 2 and G2 in characteristics 2 and 3. In all these cases sandwiches were parameterized by pairs; each pair consists of a subring and an additive subgroup, satisfying certain properties. Let F be a field of characteristic 2. Then there are injections Sp2n (F ) ֌ SO2n+1 (F ) ֌ Sp2n (F ), which turn to isomorphisms if F is perfect, see [16] Theorem 28, Example (a) after this theorem, and Remark before Theorem 29. Note that in this case the Chevalley groups GP (Cn , F ) and GP (Bn , F ) do not depend on the weight lattice P ([16], Exercise after ′ Corollary 5 to Theorem 4 ). Therefore, the lattice L E(Φ, K), G(Φ, F ) embeds to the lattice L Ep2n (F2 ), Sp2n (F ) for Φ = Bn , Cn and a subring K of F , and its description follows from the main result of the current article. Actually, the injections above are constructed in [16] on the level of Steinberg groups. Since over a field K2 (Φ, F ) is generated by Steinberg symbols, they induce the injections of Chevalley groups. Over a ring A there is no appropriate description of K2 available, therefore we can not use the above arguments. Moreover, in the ring case the group GP (Φ, A) depends on P . By these reasons, the adjoint group of type Cn and groups of type Bn over rings will be considered in a subsequent article. We state now our main result. Following [5], we call a pair (R, Λ) consisting of a ring R and a form parameter Λ in R a form ring. Theorem 1. Let A denote a (commutative) ring such 2 = 0 in A. Let K be a subring of A. If H is a subgroup of Sp2n (A) containing Ep2n (K), n > 3, then there is a unique form ring (R, Λ) such that K ⊆ Λ ⊆ R ⊆ A and Ep2n (R, Λ) 6 H 6 NA (R, Λ), where NA (R, Λ) is the normalizer of Ep2n (R, Λ) in Sp2n (A). Let L(R, Λ) = L Ep2n (R, Λ), NA (R, Λ) denote the lattice of all subgroups H of Sp2n (A) such that Ep2n (R, Λ) 6 H 6 NA (R, Λ). From now on, L(R, Λ) is what we shall mean by a standard sandwich. In view of Theorem 1 it is natural to study the lattice structure of L(R, Λ). By definition Ep2n (R, Λ) is normal in NA (R, Λ). Therefore the lattice structure of L(R, Λ) is the same as that of the quotient group NA (R, Λ)/ Ep2n (R, Λ). The lattice L(R, Λ) contains an important subgroup Sp2n (R, Λ), which is called a Bak symplectic group, whose definition will be recalled in Section 1. A group will be called quasi-nilpotent if it is a direct limit of nilpotent subgroups, i. e. the group has a directed system of nilpotent subgroups whose colimit is the group itself. In the following result we use the notion of Bass–Serre dimension of a ring introduced by the first author in [6, § 4]. Recall that the Krull (or combinatorial) dimension of a topological space is the supremum of the lengths of proper chains of nonempty closed irreducible subsets. Bass–Serre dimension d = BS-dim R of a ring R is the smallest integer such that the maximal spectrum Max R is a union of a finite number of irreducible Noetherian subspaces of Krull dimension not greater than d. If there is no such integer, then BS-dim R = ∞. Theorem 2. Let A denote a (commutative) ring and let R be a subring of A and (R, Λ) a form ring. Suppose n > 2. 1. Sp2n (R, Λ) = Sp2n (R) ∩ NA (R, Λ). 2. Sp2n (R, Λ) is normal in NA (R, Λ). SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 3 3. NA (R, Λ)/ Sp2n (R, Λ) is abelian. 4. Sp2n (R, Λ)/ Ep2n (R, Λ) is quasi-nilpotent. 5. Let R0 denotes the subring of R, generated by all elements ξ 2 such that ξ ∈ R. If Bass–Serre dimension of R0 is finite, then Sp2n (R, Λ)/ Ep2n (R, Λ) is nilpotent. In particular, the sandwich quotient group NA (R, Λ)/ Ep2n (R, Λ) is quasi-nilpotent by abelian or nilpotent by abelian if BS-dim R0 < ∞. Although Theorem 1 is proven under the assumptions 2 = 0 and n > 3, we do not invoke these assumptions until the proof of the theorem in Section 6. In particular, Theorem 2 is proven without these assumptions. Of course, we always assume that n > 2. Notation. Let H be a group. For two elements x, y ∈ H we write [x, y] = xyx−1 y −1 for their commutator and xy = y −1xy for the y-conjugate of x. For subgroups X, Y 6 H we let X Y denote the normal closure of X in the subgroup generated by X and Y , while [X, Y ] stands for the mixed commutator group generated by X and Y . By definition it is the group generated by all commutators [x, y] such that x ∈ X and y ∈ Y . The commutator subgroup [X, X] of X will be also denoted by D(X) and we set D k (X) = D k−1 (X), D k−1(X) . Recall that a group X is called perfect if D(X) = X. The identity matrix is denoted by e as well as the identity element of a Chevalley group. We denote by eij the matrix with 1 in position (ij) and zeroes elsewhere. The entries of a matrix g are denoted by gij . For the entries of the inverse matrix we use abbreviation (g −1)ij = gij′ . The transpose of the matrix g is denoted by g T, thus (g T )ij = gji . 1. The symplectic group The symplectic group Sp2n (R), its elementary root unipotent elements, and its elementary subgroup Ep2n (R) will be recalled below. The groups Sp2n (R, Λ) and their elementary subgroups Ep2n (R, Λ) will be defined in Section 2. Since the groups Sp2n (R, Λ) are not in general algebraic (in fact, Sp2n (R, Λ) is algebraic iff Λ = R or Λ = {0}), it is convenient to work with the standard matrix representation of Sp2n (R). On the other hand, we want to use the notions of parabolic subgroup, unipotent radical, etc. from the theory of algebraic groups. Thus, to simplify the exposition, we define below these notions directly in terms of the matrix representation we use. Following Bourbaki [9] we view the root system Cn and its set of fundamental roots Π in the following way. Let Vn denote n-dimensional euclidean space with the orthonormal basis ε1 , . . . , εn . Let Cn = {±εi ± εj \| 1 6 i < j 6 n} ∪ {±2εk \| 1 6 k 6 n}, Π = {αi = εi − εi+1 , where i = 1, . . . , n − 1, αn = 2εn }. The elements ±εi ± εj are called short roots and the elements ±2εk long roots. From the perspective of algebraic groups, we want the standard Borel subgroup of Sp2n (R) to be the group of all upper triangular matrices in Sp2n (R). Accordingly, we take the following matrix description of Sp2n (R). Let J denote the n × n-matrix with 1 in each antidiagonal 0 J position and zeroes elsewhere. Let F = −J 0 . Then, the group Sp2n (R) is the subgroup of GL2n (R) consisting of all matrices which preserve the bilinear form whose matrix is F . In other words, Sp2n (R) = {g ∈ GL2n (R) \| g T F g = F }. Let I = (1, . . . , n, −n, . . . , −1) denote the linearly ordered set whose linear ordering is obtained by reading from left to right. We enumerate the rows and columns of the matrices 4 ANTHONY BAK AND ALEXEI STEPANOV of GL2n (R) by indexes from I. Thus the position of a coordinate of a matrix in GL2n (R) is denoted by a pair (i, j) ∈ I × I. In the language of algebraic groups the set of all diagonal matrices in Sp2n (R) is a maximal torus. Let e denote the 2n × 2n identity matrix. The following matrices are elementary root unipotent elements of Sp2n (R) with respect to the torus above: xεi −εj (ξ) = Tij (ξ) = T−j,−i (−ξ) = e + ξeij − ξe−j,−i, xεi +εj (ξ) = Ti,−j (ξ) = Tj,−i (ξ) = e + ξei,−j + ξej,−i , x−εi −εj (ξ) = T−i,j (ξ) = T−j,i (ξ) = e + ξe−i,j + ξe−j,i , x2εk (ξ) = Tk,−k (ξ) = e + ξek,−k , x−2εk (ξ) = T−k,k (ξ) = e + ξe−k,k , where ξ ∈ R, 1 6 i, j, k 6 n, i 6= j. Note that the subscripts (i, j), (i, j), (i, −j), (j, −i), (−j, −i), (k, −k), and (−k, k) on the T above all belong to the set C̃n = I × I \ {(k, k) \| k ∈ I} of nondiagonal positions of a matrix from Sp2n (R) and exhaust C̃n . There is a surjective map p : C̃n → Cn defined by the following rule. p(i, j) = p(−j, −i) = εi − εj ; p(i, −j) = p(j, −i) = εi + εj ; p(−i, j) = p(−j, i) = −εi − εj ; p(k, −k) = 2εk ; p(−k, k) = −2εk ; where ξ ∈ R, 1 6 i, j, k 6 n, i 6= j. With this notation the correspondence between elementary symplectic transvections Tij (ξ) and root elements xα (ξ) looks as follows. Tij (ξ) = xp(i,j) (− sign(ij)ξ) for all i 6= j ∈ I. Note that p maps symmetric (with respect to the antidiagonal) positions to the same root. Therefore, for (ij) ∈ I and ξ ∈ R we have Ti,j (ξ) = T−j,−i (− sign(ij)ξ). The root subgroup scheme Xα is defined by Xα (R) = {xα (ξ) \| ξ ∈ R}. The scheme Xα is naturally isomorphic to Ga , i. e. xα (ξ)xα (µ) = xα (ξ + µ) for all ξ, µ ∈ R. The following commutator formulas are well known in matrix language, cf. [8, § 3]. They are special cases of the Chevalley commutator formula in the algebraic group theory. c = 2π/3; [xα (λ), xβ (µ)] = xα+β (±λµ), if α + β ∈ Φ, αβ c = π/2; [xα (λ), xβ (µ)] = xα+β (±2λµ), if α + β ∈ Φ, αβ c = 3π/4; [xα (λ), xβ (µ)] = xα+β (±λµ)xα+2β (±λµ2 ), if α + β, α + 2β ∈ Φ, αβ [xα (λ), xβ (µ)] = e, if α + β ∈ / Φ ∪ {0} In our proofs we make frequent use of the parabolic subgroup P1 of Sp2n . In the matrix language above it is defined as follows: SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 5 P1 (R) = {g ∈ Sp2n R \| gi1 = g−1−i = 0 ∀i 6= 1} −1 The definition above of Sp2n (R) shows that g11 = g−1,−1 for any matrix g ∈ P1 (R). The unipotent radical U1 of P1 is the subgroup generated by all root subgroups T1i such that i 6= 1. The Levi subgroup L1 (R) of P1 (R) consists of all g ∈ P1 (R) such that g1i = g−i,−1 = 0 for all i 6= 1. As a group scheme it is isomorphic to Gm × Sp2n−2 . 2. Bak symplectic groups The Bak symplectic group Sp2n (R, Λ) is the particular case of the Bak general unitary group, where the involution is trivial and the symmetry λ = −1. The main references for the definition and the structure of the general unitary group is the book [5] and the paper [8] by N. Vavilov and the first author. In this section we recall definitions and simple properties to be used in the sequel. Let R be a commutative ring. An additive subgroup Λ of R is called a symplectic form parameter in R, if it contains 2R and is closed under multiplication by squares, i. e. µ2 λ ∈ Λ for all µ ∈ R and λ ∈ Λ. Define Sp2n (R, Λ) as the subgroup of Sp2n (R) consisting of all matrices preserving the quadratic form which takes values in R/Λ and is defined by the matrix ( 00 J0 ). Let ∗ denote the involution on the matrix ring Mn (R) given by the formula a∗ = JaT J. Note that this is the reflection of a matrix with respect to the antidiagonal. Define Mn (R, Λ) = {a ∈ Mn (R) \| a = a∗ , an−k+1 k ∈ Λ for all k = 1, . . . , n}. Write a matrix of degree 2n in the block form ( ac db ), where a, b, c, d ∈ Mn (R). It follows from [8, Lemma 2.2] and its proof that under the notation above we have the following formula. a b ∗ ∗ ∗ ∗ Lemma 2.1. Sp2n (R, Λ) = \| a d − c b = e and c a, d b ∈ Mn (R, Λ) . c d It is easy to check that Mn (R, Λ) is a form parameter in the ring Mn (R) with involution ∗, corresponding to the symmetry λ = −1. The minimal form parameter in Mn (R) with the same involution and symmetry is Mn (R, 2R); it is denoted by Minn (R). Let M̄n (R, Λ) denote the additive group Mn (R, Λ)/ Minn (R). Lemma 2.2. The group M̄n (R, Λ) has a natural structure of a left Mn (R)-module under the operation a ◦ b̄ = aba∗ mod Minn (R), where a ∈ Mn (R), b̄ ∈ M̄n (R, Λ), and b is a preimage of b̄ in Mn (R, Λ). By abuse of notation for a ∈ Mn (R) and b ∈ Mn (R, Λ) we shall write a ◦ b instead of a ◦ (b + Minn (R)). It is easy to check that an elementary root unipotent element xα (ξ) ∈ Sp2n (R) belongs to Sp2n (R, Λ) if and only if α is a short root or ξ ∈ Λ. Denote by Ep2n (R, Λ) the group generated by all such elements: Ep2n (R, Λ) = hxα (ξ) \| α ∈ Cnshort & ξ ∈ R ∨ α ∈ Cnlong & ξ ∈ Λi. 3. Subgroups generated by elementary root unipotents In this section we assume that n > 3. Let H be a subgroup of Sp2n (A), containing Ep2n (K). The following lemma shows that we can uncouple a short root element from a long one inside H. 6 ANTHONY BAK AND ALEXEI STEPANOV Lemma 3.1. Let α, β ∈ Cn be a short and a long root, respectively, such that α + β is not a root. Let g = xα (µ)xβ (λ), where λ, µ ∈ A. If Ep2n (K)g 6 H (e. g. g ∈ H), then each factor of g belongs to H. Proof. Since n > 3, there exists a short root γ such that γ + α is a short root and γ + β is not a root. Then, Xβ (A) commutes with Xα (A) and Xγ (A), hence [g, xγ (1)] = xα+γ (±µ) ∈ H. Conjugating this element by an appropriate element from the Weyl group over K and taking the inverse if necessary, we get xα (µ) ∈ H. It follows that Ep2n (K)xβ (λ) 6 H. Now, take a short root δ such that β+δ is a root. Then, [xδ (1), xβ (λ)] = xδ+β (±λ)x2δ+β (±λ) ∈ H. Notice that the root δ + β is short whereas 2δ + β is long. As in the first paragraph of the proof one concludes that xδ+β (±λ) ∈ H, hence x2δ+β (±λ) ∈ H. Again, using the action of the Weyl group and taking inverse if necessary, one shows that xβ (λ) ∈ H. Put Pα (H) = {t ∈ A \| xα (t) ∈ H}. Since the Weyl group acts transitively on the set of roots of the same length, it is easy to see that Pα (H) = Pβ (H) if \|α\| = \|β\|. Let R = RH = Pα (H) for any short root α, and let Λ = ΛH = Pβ (H) for any long root β. Lemma 3.2. With the above notation (R, Λ) is a form ring and K ⊆ Λ ⊆ R ⊆ A. Proof. Clearly, Pα (H) is an additive subgroup of A. Since n > 3, there are two short roots α, α′ such that α + α′ also is short. The commutator formula [xα (λ), xα′ (µ)] = xα+α′ (±λµ) shows that R is a ring. Now, let α, α′ be short roots which are orthogonal in Vn and such that β = α + α′ ∈ Φ is a long root. Then [xα (µ), xα′ (1)] = xβ (±2µ). If µ ∈ R, then this element belongs to H. This proves that 2R ⊆ Λ. Finally, we show that Λ is closed under multiplication by squares in R. Let g = [xβ (λ), x−α (µ)] = xα′ (±λµ)xβ−2α (±λµ2 ) If λ ∈ Λ and µ = 1, then g ∈ H, and Lemma 3.1 shows that xα′ (±λ) ∈ H. Therefore, Λ ⊆ R. On the other hand, if µ ∈ R and λ ∈ Λ, then g also lies in H, and by Lemma 3.1 we have xβ−2α (±λµ2 ) ∈ H. This shows that Λ is stable under multiplication by squares of elements of R. If H is a subgroup of Sp2n (A) containing Ep2n (K), then (RH , ΛH ) is called the form ring associated with H. 4. The normalizer Let (R, Λ) be a form subring of a ring A such that 1 ∈ Λ. In this section we develop properties of the normalizer NA (R, Λ) of the group Ep2n (R, Λ) in Sp2n (A) and prove Theorem 2. Here we do not assume that n > 3. By a result of Bak and Vavilov [7, Theorem 1.1] we know that Ep2n (R, Λ) is normal in Sp2n (R, Λ), thus Sp2n (R, Λ) 6 NA (R, Λ). First, we show that the quotient NA (R, Λ)/ Sp2n (R, Λ) is abelian. Lemma 4.1. Let g ∈ Sp2n (A). If Ep2n (R, Λ)g 6 GL2n (R) then gij gkl ∈ R for all i, j, k, l ∈ I. P Proof. To begin we express a matrix unit ejk as a linear combination ξm a(m) for some (m) ξm ∈ R and a ∈ Ep2n (R, Λ). Note that the set of such linear combinations is closed under multiplication. Suppose j 6= k and let i 6= ±j, ±k. Then ejk = (Tji (1) − e)(Tik (1) − e) and ejj = ejk ekj . It follows that if Ep2n (R, Λ)g 6 GL2n (R), then g −1 ejk g ∈ Mn (R). Thus SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 7 gij′ gkl ∈ R for all i, j, k, l ∈ I (recall that gij′ = (g −1 )ij ). The conclusion of the lemma follows since the entries of g −1 coincide with the entries of g up to sign and a permutation. The next proposition describes the normalizer in the case A = R. Proposition 4.2. NR (R, Λ) = Sp2n (R, Λ). Proof. Let g = ( ac db ) ∈ NR (R, Λ). Since g ∈ Sp2n (R), then d∗ a − b∗ c = e. We have to prove that c∗ a, d∗ b ∈ Mn (R, Λ). Since h = ( 0e Je ) ∈ Ep2n (R, Λ), we have e − aJc∗ aJa∗ −1 ∈ Ep2n (R, Λ) f = ghg = −cJc∗ e + cJa∗ It follows that the matrices −(cJc∗ )∗ (e − aJc∗ ) and (e + cJa∗ )∗ (aJa∗ ) belong to Mn (R, Λ). Since cJc∗ and aJa∗ are automatically in Mn (R, Λ), we have that cJc∗ aJc∗ , aJc∗ aJa∗ ∈ Mn (R, Λ). Modulo Minn (R) we can write (cJ)◦(c∗a), (aJ)◦(c∗ a) ∈ M̄n (R, Λ). By Lemma 2.2 M̄n (R, Λ) is an Mn (R) module, therefore c∗ a + Minn (R) = J(d∗ a − b∗ c)J ◦ (c∗ a) = (Jd∗ ) ◦ (aJ) ◦ (c∗ a) − (Jb∗ ) ◦ (cJ) ◦ (c∗ a) ∈ M̄n (R, Λ). The proof that d∗ b ∈ Mn (R, Λ) is essentially the same. Corollary 4.3. [NA (R, Λ), NA (R, Λ)] 6 Sp2n (R, Λ). Proof. If g, h ∈ NA (R, Λ) and f = [g, h], then by Lemma 4.1 for all indexes p, q we have fpq = n X i,j,k=1 ′ ′ gpi hij gjk hkq = n X ′ (gpi gjk )(h′ij hkq ) ∈ R i,j,k=1 Therefore, f ∈ Sp2n (R) ∩ NA (R, Λ) = NR (R, Λ) = Sp2n (R, Λ). The first 3 items of Theorem 2 are already proved. Our next goal is to show that the main theorem of [12] implies that Sp2n (R, Λ)/ Ep2n (R, Λ) is nilpotent, provided that R0 has finite Bass–Serre dimension. This will imply the rest of Theorem 2. Recall that R0 denotes the subring of R generated by all elements ξ 2 such that ξ ∈ R. Lemma 4.4. If R0 is semilocal and 1 ∈ Λ, then Sp2n (R, Λ) = Ep2n (R, Λ). Proof. Since R is integral over R0 , it is a direct limit of R0 -subalgebras R′ ⊆ R such that R′ is module finite and integral over R0 . By the first theorem of Cohen–Seidenberg, each R′ is semilocal. Thus by [4, Lemma 4] Sp2n (R′ , Λ ∩ R′ ) = Ep2n (R′ , Λ ∩ R′ ) for each R′ . Since Sp2n and Ep2n commute with direct limits, it follows that Sp2n (R, Λ) = Ep2n (R, Λ). Lemma 4.5. If R is a finitely generated Z-algebra then so is R0 . Proof. Since 2ξ = (ξ + 1)2 − ξ 2 − 1, we have 2R ⊆ Q R0 . Let S be a finite set of generators 2 for R as a Z-algebra. We set S0 = {s \| s ∈ S} ∪ {2 s∈S ′ s \| S ′ ⊆ S} and denote by R1 the Z-subalgebra generated by S0 . Clearly, 2R ⊆ R1 . The map ξ 7→ ξ 2 is a ring epimorphism R/2R → R0 /2R, therefore R0 /2R is generated by the images of generators of R. It follows that R0 /2R = R1 /2R, hence R0 = R1 is finitely generated. 8 ANTHONY BAK AND ALEXEI STEPANOV Proof of Theorem 2. Since NR (R, Λ) = NA (R, Λ) ∩ Sp2n (R), the first statement follows from Proposition 4.2, the second is a particular case of [7, Theorem 1.1], and the third one coincides with Corollary 4.3. Recall that Sp02n (R, Λ) = ∩ϕ Ker ϕ, where ϕ ranges over all group homomorphisms Sp2n (R, Λ) → Sp2n (R̃, Λ̃′ )/ Ep2n (R̃, Λ̃) induced by form ring morphisms (R, Λ) → (R̃, Λ̃) such that R̃0 is semilocal. By Lemma 4.4 Sp02n (R, Λ) = Sp2n (R, Λ). Since R is integral over R0 , it is a direct limit of R0 -subalgebras R′ ⊆ R such that R′ is module finite over R0 . By [12][Theorem 3.10] the group Sp02n (R′ , Λ ∩ R′ )/ Ep2n (R′ , Λ ∩ R′ ) is nilpotent, provided that R0 has finite Bass–Serre dimension. Since Sp2n and Ep2n commute with direct limits, this proves (5). Any commutative ring is a direct limit of finitely generated Z-algebras. Let R = inj lim R(i) , where i ranges over some index set I and each R(i) is a finitely generated Z-algebra. Then (i) R0 = inj lim R0 and Sp2n (R, Λ ∩ R)/ Ep2n (R, Λ) = inj lim Sp(R(i) , Λ ∩ R(i) )/ Ep2n (R(i) , Λ ∩ (i) R(i) ). By Lemma 4.5 each R0 is a finitely generated Z-algebra. Therefore, this ring has finite Krull dimension and hence finite Bass–Serre dimension. By (5) each group Sp(R(i) , Λ ∩ R(i) )/ Ep2n (R(i) , Λ ∩ R(i) ) is nilpotent which implies item (4). The following statement is crucial for the proof of our main theorem. For Noetherian rings it is almost immediate consequence of Theorem 2. Lemma 4.6. Let g ∈ Sp2n (A), n > 3. If Ep2n (R, Λ)g 6 NA (R, Λ), then g ∈ NA (R, Λ). Moreover, Ep2n (R, Λ) is a characteristic subgroup of NA (R, Λ). Proof. Let θ be either an automorphism of NA (R, Λ) or an automorphism of Sp2n (A) such that Ep2n (R, Λ)θ 6 NA (R, Λ) (we denote by hθ the image of an element h ∈ NA (R) under the action of θ). Since n > 3, the Chevalley commutator formula (see section 1) implies that the group Ep2n (R, Λ) is perfect. By Corollary 4.3 we have Ep2n (R, Λ)θ = [Ep2n (R, Λ), Ep2n (R, Λ)]θ 6 [NA (R, Λ), NA (R, Λ)] 6 Sp2n (R, Λ). We shall prove that hθ ∈ Ep2n (R, Λ) for any h ∈ Ep2n (R, Λ). Write h as a product of elementary root unipotents xα1 (s1 ) · · · xαm (sm ). Let R′ denote the Z-subalgebra of R generated by all si ’s, and let Λ′ denote the form parameter of R′ generated by those sj for which αj is a long root. Clearly h ∈ Ep2n (R′ , Λ′) and Ep2n (R′ , Λ′ ) is a finitely generated group. Let R′′ denote the R′ -algebra generated by all entries of the matrices y θ , where y ranges over all generators of Ep2n (R′ , Λ′ ). Let Λ′′ = Λ ∩ R′′ . The inclusion Ep2n (R, Λ)θ 6 Sp2n (R, Λ) shows that R′′ ⊆ R. Note that Ep2n (R′ , Λ′ )θ 6 Sp2n (R′′ , Λ′′ ) by the choice of R′′ . Since R′′ is a finitely generated R′ -algebra, it is a finitely generated Z-algebra. By Lemma 4.5 R0′′ is a finitely generated Z-algebra. Therefore, it has finite Krull dimension and hence finite Bass–Serre dimension. Thus by Theorem 2(5), the kth commutator subgroup D k Sp2n (R′′ , Λ′′ ) equals to Ep2n (R′′ , Λ′′ ) for some positive integer k. Now, since Ep2n (R′ , Λ′) is perfect, it is equal to D k Ep2n (R′ , Λ′ ). It follows that Ep2n (R′ , Λ′ )θ = D k Ep2n (R′ , Λ′ )θ 6 D k Sp2n (R′′ , Λ′′) = Ep2n (R′′ , Λ′′ ). In particular, hθ ∈ Ep2n (R′′ , Λ′′ ) 6 Ep2n (R, Λ). Thus, Ep2n (R, Λ) is invariant under θ. If θ is an automorphism of NA (R) this means that Ep2n (R, Λ) is a characteristic subgroup of NA (R). If θ is an inner automorphism defined by g ∈ G(A), then the statement we proved is the first assertion of the lemma. SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 9 The following straightforward corollary shows that the normalizers of all subgroups of the sandwich L Ep2n (R, Λ), NA (R, Λ) lie in that sandwich. Corollary 4.7. For any H 6 NA (R, Λ) containing Ep2n (R, Λ) its normalizer is contained in NA (R, Λ). In particular, the group NA (R, Λ) is self normalizing. 5. Inside a parabolic subgroup Let H be a subgroup of Sp2n (A), normalized by Ep2n (K). Denote by (R, Λ) the form ring, associated with H. In the proof of the following lemma we keep Q using the ordering on the index set I, defined in section 1. For example, the product −1 j=2 means that j = 2, . . . , n, −n, . . . , −1. We assume that the order of factors agrees with the ordering on I. Lemma 5.1. If g ∈ U1 (R) and Ep2n (K)g ∈ H, then g ∈ Ep2n (R, Λ). Q Proof. Let g = −1 j=2 T1j (µj ). We have to prove that µj ∈ R for any j 6= −1 and µ−1 ∈ Λ. Let h be the smallest element from I such that µj 6= 0. We proceed by going down induction on h. If h = −1, then g consists of a single factor and there is nothing to prove. If h = −2, then the result follows from Lemma 3.1. Q Now, let h 6 −2. Denote by i the successor of h in I. Then, [Th,i (−1), g] = T1,i (µh ) T1,j (ξh ). j>i By induction hypothesis µh ∈ R. The element T1h (−µh )g satisfies the conditions of the lemma. Again by induction hypothesis it belongs to Ep2n (R, Λ). Thus, g ∈ Ep2n (R, Λ). It is known that in a Chevalley group the unipotent radicals of two opposite standard parabolic subgroups span the elementary group (see e. g. [18, Lemma 2.1]). The next lemma shows that this holds for the parabolic subgroup P1 of Sp2n (R, Λ) as well. Lemma 5.2. The set {T1i (µ), Ti1 (µ), T±1 ∓1 (λ) \| i 6= ±1, µ ∈ R, λ ∈ Λ} generates the elementary group Ep2n (R, Λ). Proof. If i 6= ±j, ±1 then Tij (µ) = [Ti1 (µ), T1j (1)]. On the other hand, for λ ∈ Λ we have T−ii (λ) = [T−11 (µ), T1i (1)]T−i1 (−λ). Lemma 5.3. Let H be a subgroup of Sp2n (A), containing Ep2n (K) and let (R, Λ) be a form ring, associated with H. Suppose that g commutes with a long root subgroup Xγ (K) and Ep2n (K)g ∈ H. Then g ∈ NA (R, Λ). Proof. Without loss of generality we may assume that γ = 2ε1 is the maximal root. Then g belongs to the standard parabolic subgroup P1 , corresponding to the simple root α1 = ε1 −ε2 , ′ ′ in other words, gi1 = g−1i = g−11 = 0 for all i 6= ±1. Moreover, g11 = g−1−1 = g11 = g−1−1 = 2 µ, where µ = 1. Then g = ab for some b ∈ U1 (A) and a ∈ L1 (A) (a and b are not necessarily in H). For any d ∈ U1 (K) the element dg belongs to H ∩ U1 (A). By Lemma 5.1, dg ∈ Ep2n (R, Λ). If d = T1i (1), then the above inclusion implies that µgij ∈ R for all i 6= 1 and all j ∈ I. It follows that T1i (1)a and Ti1 (1)a belong to Ep2n (R, Λ). On the other hand, a commutes with root subgroups X±γ (K). By the previous lemma we conclude that a normalizes Ep2n (R, Λ). Now, we have Ep2n (K)b 6 Ep2n (R, Λ)g ∈ H and by Lemma 5.1 b ∈ Ep2n (R, Λ). Thus, g ∈ NA (R, Λ) as required. 10 ANTHONY BAK AND ALEXEI STEPANOV 6. Proof of Theorem 1 In this section we assume that 2 = 0 in K. Let G be a Chevalley group with a not simply laced root system, e. g. G = Sp2n . Recall that in this case a short root unipotent element h is called a small unipotent element, see [11]. The terminology reflects the fact that the conjugacy class of h is small. In our settings a small unipotent element is conjugate to Tij (µ), where i 6= ±j and µ ∈ R. The following lemma was obtained over a field by Golubchik and Mikhalev in [10], Gordeev in [11] and Nesterov and Stepanov in [13]. Lemma 6.1. Let Φ = Bn , Cn , F4 and let R be a ring such that 2 = 0. Let α be a long root g and g ∈ G(Φ, R). If h ∈ G(R) is a small unipotent element, then Xα (R)h commutes with Xα (R). g Proof. The identity with constants [Xα (R)h , Xα (R)] = {1} is inherited by subrings and quotient rings. Any commutative ring with 2 = 0 is a quotient of a polynomial ring over F2 which is a subring of a field of characteristic 2. The next lemma is the last ingredient for the proof of Theorem 1. It follows from the normal structure of the general unitary group GU2n (R, Λ) obtained by Bak and Vavilov at the middle of 1990-s. Since this result has not been published yet, we give a prove of a very simple special case of it. Lemma 6.2. A normal subgroup N of Ep2n (R, Λ), containing a root element Tij (1), coincides with Ep2n (R, Λ). Proof. First, suppose that i 6= ±j. Take k 6= ±i, ±j. Then, [Tij (1), Tjk (ξ)] = Tik (µ) ∈ N. Since the Weyl group acts transitively on the set of all short roots, we have Tlm (µ) ∈ N for all l 6= ±m and µ ∈ R. Further, Tl,−m (−λ)[Tlm (1), Tm,−m (λ)] = Tl,−l (λ) ∈ N for all λ ∈ Λ and l ∈ I. Now, let i = m, j = −m and l 6= ±m. Put λ = 1 in the latter commutator identity. Then, Tl,−m (1)Tl,−l (1) ∈ N. By transitivity of the Weyl group we know that Tl,−l (1) ∈ N, therefore Tl,−m (1) ∈ N. By the first paragraph of the proof, N contains all the generators of Ep2n (R, Λ). Now we are ready to prove Theorem 1. The idea of the proof is the same as for the main result of [17]. Proof of Theorem 1. Let (R, Λ) be the form subring associated with H. Put h = T12 (1) and x = T1,−1 (λ), where λ ∈ Λ. Take two arbitrary elements a, b from Ep2n (R, Λ) and consider agb the element c = xh ∈ H. By Lemma 6.1 this element commutes with a long root subgroup and by Lemma 5.3 c ∈ NA (R). Rewrite c in the form b c = g −1 (a−1 h−1 a)g(bxb−1 )g −1(a−1 ha)g Since b ∈ Ep2n (R, Λ), the element bcb−1 is in NA (R). Fix a and let b and λ vary. The subgroup generated by bxb−1 is normal in Ep2n (R, Λ). By Lemma 6.2 it must coincide with −1 −1 −1 Ep2n (R, Λ). Thus, Ep2n (R, Λ)g (a h a)g ∈ NA (R), and by Lemma 4.6 (a−1 h−1 a)g ∈ NA (R). Again, elements of the form a−1 h−1 a, as a ranges over Ep2n (R, Λ), generate a normal subgroup in Ep2n (R, Λ). The minimal normal subgroup of Ep2n (R, Λ) containing h must be equal to Ep2n (R, Λ) by Lemma 6.2. Therefore, Ep2n (R, Λ)g ∈ NA (R). By Lemma 4.6 one has g ∈ NA (R), which completes the proof. SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 11 References [1] E. Abe, Normal subgroups of Chevalley groups over commutative rings, Contemp. Math. 83 (1989), 1–17. [2] E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tohoku Math. J. 28 (1976), no. 2, 185–198. [3] A. Bak, The stable structure of quadratic modules, Ph.D. thesis, Columbia Univ., Columbia, USA, 1969. , Odd dimension surgery groups of odd torsion groups vanish, Topology 14 (1975), [4] no. 4, 367–374. , K-theory of forms, Ann. of Math. Stud. 98, Princeton Univ. Press, Princeton [5] N.J., 1981. , Nonabelian K-theory: The nilpotent class of K1 and general stability, K-Theory [6] 4 (1991), 363–397. [7] A. Bak and N. A. Vavilov, Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35–47. [8] , Structure of hyperbolic unitary groups I: Elementary subgroups, Algebra Colloq 7 (2000), no. 2, 159–196. [9] N. Bourbaki, Elements of mathematics. Lie groups and Lie algebras. Chapters 4-6, Springer-Verlag, Berlin-Heidelberg-New York, 2008. [10] I. Z. Golubchik and A. V. Mikhalev, Generalized group identities is classical groups, J. Soviet Math. 27 (1984), no. 4, 2902–2918. [11] N. L. Gordeev, Freedom in conjugacy classes of simple algebraic groups and identities with constants, St.Petersburg Math. J. 9 (1998), no. 4, 709–723. [12] R. Hazrat, Dimension theory and nonstable K1 of quadratic modules, K-Theory 27 (2002), no. 4, 293–328. [13] V. V. Nesterov and A. V. Stepanov, Identity with constant in a Chevalley group of type F4 , St.Petersburg Math. J. 21 (2010), no. 5, 819–823. [14] Ya. N. Nuzhin, Groups contained between groups of Lie type over different fields, Algebra Logic 22 (1983), 378–389. , Intermediate subgroups in the Chevalley groups of type Bl , Cl , F4 , and G2 over [15] the nonperfect fields of characteristic 2 and 3, Sib. Math. J. 54 (2013), no. 1, 119–123. [16] R. G. Steinberg, Lectures on Chevalley groups, Dept. of Mathematics, Yale University, 1968. [17] A. V. Stepanov, Subring subgroups in Chevalley groups with doubly laced root systems, J. Algebra 362 (2012), 12–29. [18] A. V. Stepanov, Elementary calculus in Chevalley groups over rings, J. Prime Research in Math. 9 (2013), 79–95. Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany E-mail address: bak@mathematik.uni-bielefeld.de St.Petersburg State University and St.Petersburg Electrotechnical University E-mail address: stepanov239@gmail.com	4
Joint Modelling of Location, Scale and Skewness Parameters of the Skew Laplace Normal Distribution Fatma Zehra Doğru1* and Olcay Arslan2 1 Giresun University, Faculty of Economics and Administrative Sciences, Department of Econometrics, 28100 Giresun/Turkey fatma.dogru@giresun.edu.tr 2 Ankara University, Faculty of Science, Department of Statistics, 06100 Ankara/Turkey oarslan@ankara.edu.tr Abstract In this article, we propose joint location, scale and skewness models of the skew Laplace normal (SLN) distribution as an alternative model for joint modelling location, scale and skewness models of the skewt-normal (STN) distribution when the data set contains both asymmetric and heavy-tailed observations. We obtain the maximum likelihood (ML) estimators for the parameters of the joint location, scale and skewness models of the SLN distribution using the expectation-maximization (EM) algorithm. The performance of the proposed model is demonstrated by a simulation study and a real data example. Keywords: EM algorithm, joint location, scale and skewness models, ML, SLN, SN. 1. Introduction There are many remarkable and tractable methods for modeling the mean. In practice, modelling the dispersion will be of direct interest in its own right, to identify the sources of variability in the observations (Smyth and Verbyla (1999)). In recent years, joint mean and dispersion models have been used for modeling heteroscedastic data sets. For instance, Park (1966) proposed a log linear model for the variance parameter and described the Gaussian model using a two stage process to estimate the parameters. Harvey (1976) examined the maximum likelihood (ML) estimation of the location and scale effects and also proposed a likelihood ratio test for heteroscedasticity. Aitkin (1987) proposed the modelling of variance heterogeneity in normal regression analysis. Verbyla (1993) estimated the parameters of the normal regression model under the log linear dependence of the variances on explanatory variables using the restricted ML. Engel and Huele (1996) represented an extension of the response surface approach to Taguchi type experiments for robust design by accommodating generalized linear modeling. Taylor and Verbyla (2004) introduced joint modelling of location and scale parameters of the t distribution. Lin and Wang (2009) proposed a robust approach for joint modelling of mean and scale parameters for longitudinal data. Lin and Wang (2011) studied Bayesian inference for joint modelling of location and scale parameters of the t distribution for longitudinal data. Wu and Li (2012) explored the variable selection for joint mean and dispersion models of the inverse Gaussian distribution. Li and Wu (2014) proposed joint modelling of location and scale parameters of the skew normal (SN) (Azzalini (1985, 1986)) distribution. Zhao and Zhang (2015) proposed variable selection of varying dispersion student-t regression models. Recently, Li et al. (2017) proposed variable selection in joint location, scale and skewness models of the SN distribution and Wu et al. (2017) explored variable selection in joint location, scale and skewness models of the STN distribution. The skew exponential power distribution was proposed by Azzalini (1986) to deal with both skewness and heavy-tailedness, simultaneously. Its properties and inferential aspects were studied by DiCiccio and Monti (2004). Gómez et al. (2007) studied the skew Laplace normal (SLN) distribution that is a special case of the skew exponential power distribution. This distribution has wider range of skewness and also more applicable than the SN distribution. In literature, skewness and heavy-tailedness are modelled by using STN distribution for joint location, scale and skewness models. However, the 1 STN distribution has an extra parameter that is the degrees of freedom parameter. Since this parameter should be estimated along with the other parameters, it may be computationally more exhaustive in practice. Therefore, in this paper, we propose to model joint location, scale and skewness models of the SLN distribution as an alternative model for the joint location, scale and skewness models of the STN distribution to model both skewness and heavy-tailedness in the data. The rest of the paper is designed as follows. In Section 2, we give some properties of the SLN distribution. In Section 3, we introduce joint location, scale and skewness models of the SLN distribution. In Section 4, we give the ML estimation of the proposed joint location, scale and skewness model using the EM algorithm. In Section 5, we provide a simulation study to show the performance of the proposed model. In Section 6, modeling applicability of the proposed model is illustrated by using a real data set. The paper is finalized with a conclusion section. 2. Skew Laplace normal distribution Let 𝑌 be a SLN distributed random variable (𝑌 ∼ 𝑆𝐿𝑁(𝜇, 𝜎 2 , 𝜆)) with the location parameter 𝜇 ∈ ℝ, scale parameter 𝜎 2 ∈ (0, ∞) and the skewness parameter 𝜆 ∈ ℝ. The probability density function (pdf) of 𝑌 is given as 𝑓(𝑦) = 2𝑓𝐿 (𝑦; 𝜇, 𝜎)Φ (𝜆 𝑦−𝜇 ), 𝜎 (1) where 𝑓𝐿 (𝑦; 𝜇, 𝜎) represents the pdf of Laplace distribution with 𝑓𝐿 (𝑦; 𝜇, 𝜎) = 1 −\|𝑦−𝜇\| 𝑒 𝜎 2𝜎 and Φ is the cumulative distribution function of the standard normal distribution. Figure 1 displays the plots of the pdf of the SLN distribution for 𝜇 = 0, 𝜎 = 1 and different values of 𝜆. Figure 1. Examples of the SLN pdf for 𝜇 = 0, 𝜎 = 1 and different skewness parameter values of 𝜆. Let the random variables 𝑍 ∼ 𝑆𝑁(0,1, 𝜆) and 𝑉 with the pdf 𝑓𝑉 (𝑣) = 𝑣 −3 exp(−(2𝑣 2 )−1 ), 𝑣 > 0 be two independent random variables. Then, the random variable 𝑌 ∼ 𝑆𝐿𝑁(𝜇, 𝜎 2 , 𝜆) has the following scale mixture form 2 𝑌 =𝜇+𝜎 𝑍 . 𝑉 (2) Further, using the stochastic representation of the SN (Azzalini (1986, p. 201) and Henze (1986, Theorem 1)) distributed random variable 𝑍, the stochastic representation of the random variable 𝑌 is obtained as 𝑌 = 𝜇 +𝜎( 𝜆\|𝑍1 \| √𝑉 2 (𝑉 2 + 𝜆2 ) + 𝑍2 √𝑉 2 + 𝜆2 (3) ), where 𝑍1 ∼ 𝑁(0,1) and 𝑍2 ∼ 𝑁(0,1) are independent random variables. This stochastic representation will give the following hierarchical representation of the SLN distribution. Let 𝑈 = √𝑉 −2 (𝑉 2 + 𝜆2 )\|𝑍1 \|. Then, 𝜎𝜆𝑢 𝜎2 , ), 𝑣 2 + 𝜆2 𝑣 2 + 𝜆2 𝑣 2 + 𝜆2 𝑈\|𝑣 ∼ 𝑇𝑁 ((0, ) ; (0, ∞)) , 𝑣2 𝑉 ∼ 𝑓𝑉 (𝑣) = 𝑣 −3 exp(−(2𝑣 2 )−1 ), 𝑌\|𝑢, 𝑣 ∼ 𝑁 (𝜇 + (4) where 𝑇𝑁(∙) shows the truncated normal distribution. The hierarchical representation will allow us to carry on the parameter estimation using the EM algorithm. Using this hierarchical representation the joint pdf of 𝑌, 𝑈 and 𝑉 can be written as 2 1 −2 1 𝑣 2 (𝑦 − 𝜇)2 𝜆(𝑦 − 𝜇) 2 )−1 ) 𝑓(𝑦, 𝑢, 𝑣) = 𝑣 exp(−(2𝑣 exp {− ( + − (𝑢 ) )}. 𝜋𝜎 2 𝜎2 𝜎 (5) Next we will turn our attention to the conditional distribution of 𝑈 given 𝑌 and 𝑉. Taking the integral of (5) over 𝑈, we obtain the joint pdf of 𝑌 and 𝑉 as 2 1⁄2 −2 𝑣 2𝑠2 𝑓(𝑦, 𝑣) = ( 2 ) 𝑣 exp (−(2𝑣 2 )−1 − ) Φ(𝜆𝑠) , 𝜋𝜎 2 (6) where 𝑠 = (𝑦 − 𝜇)⁄𝜎. Then, dividing (5) by (6) yields the following conditional density function of 𝑈 given the others 𝑓(𝑢\|𝑦, 𝑣) = 1 √2𝜋 exp {− (𝑢 − 𝜆𝑠)2 } Φ(𝜆𝑠). 2 (7) It is clear from the density given in (7) that 𝑈 and 𝑉 are conditionally independent. Therefore, the distribution of 𝑈\|𝑌 = 𝑦 is 𝑈\|𝑌 = 𝑦 ∼ 𝑇𝑁((𝜆𝑠, 1); (0, ∞)). (8) Further, after dividing (6) by (1), we get the following conditional density function of 𝑉 given 𝑌 2 𝑣 2 𝑠 2 \|𝑦 − 𝜇\| 𝑓(𝑣\|𝑦) = √ 𝑣 −2 exp {−(2𝑣 2 )−1 − + }. 𝜋 2 𝜎 3 (9) Now, we are ready to give the following proposition. The proof of this proposition can be easily done using the conditional pdfs given above. Proposition 1. Using the hierarchical representation given in (4), we have the following conditional expectations 𝜎 , \|𝑦 − 𝜇\| Φ(𝜆𝑠) 𝐸(𝑈\|𝑦) = 𝜆𝑠 + , 𝜙(𝜆𝑠) 𝐸(𝑈 2 \|𝑦) = 1 + 𝜆𝑠𝐸(𝑈\|𝑦) . (10) 𝐸(𝑉 2 \|𝑦) = (11) (12) Note that these conditional expectations will be used in the EM algorithm given in Section 4. 3. Joint location, scale and skewness models of the SLN distribution In this study, we consider the following joint location, scale and skewness models of the SLN distribution 𝑦𝑖 ∼ 𝑆𝐿𝑁(𝜇𝑖 , 𝜎𝑖2 , 𝜆𝑖 ), 𝑖 = 1,2, … , 𝑛 𝜇𝑖 = 𝒙𝑇𝑖 𝜷 , log 𝜎𝑖2 = 𝒛𝑇𝑖 𝜸 , 𝑇 { 𝜆𝑖 = 𝒘 𝑖 𝜶 , (13) 𝑇 𝑇 where 𝑦𝑖 is the 𝑖𝑡ℎ observed response, 𝒙𝑖 = (𝑥𝑖1 , … , 𝑥𝑖𝑝 ) , 𝒛𝑖 = (𝑧𝑖1 , … , 𝑧𝑖𝑞 ) and 𝒘𝑖 = (𝑤𝑖1 , … , 𝑤𝑖𝑟 )𝑇 𝑇 are observed covariates corresponding to 𝑦𝑖 , 𝜷 = (𝛽1 , … , 𝛽𝑝 ) is a 𝑝 × 1 vector of unknown parameters 𝑇 in the location model, and 𝜸 = (𝛾1 , … , 𝛾𝑞 ) is a 𝑞 × 1 vector of unknown parameters in the scale model and 𝜶 = (𝛼1 , … , 𝛼𝑟 )𝑇 is a 𝑟 × 1 vector of unknown parameters in the skewness model. These covariate vectors 𝒙𝑖 , 𝒛𝑖 and 𝒘𝑖 are not needed to be identical. 4. ML estimation of joint location, scale and skewness models of the SLN distribution Let (𝑦𝑖 , 𝒙𝑖 , 𝒛𝑖 ), 𝑖 = 1,2, … , 𝑛, be a random sample from model given in (13). Let 𝜽 = (𝜷, 𝜸, 𝜶). Then, the log-likelihood function of 𝜽 based on the observed data is written as 𝑛 𝑛 𝑛 1 \|𝑦𝑖 − 𝒙𝑇𝑖 𝜷\| ℓ(𝜽) = − ∑ 𝒛𝑇𝑖 𝜸 − ∑ + ∑ log Φ(𝜅𝑖 ), 𝑇 2 𝑒 𝒛𝑖 𝜸⁄2 𝑖=1 where 𝜅𝑖 = 𝒘𝑇𝑖 𝜶 (𝑦𝑖 −𝒙𝑇 𝑖 𝜷) 𝑇 ⁄ 𝑒 𝒛𝑖 𝜸 2 𝑖=1 (14) 𝑖=1 . The ML estimator of 𝜽 can be found by maximizing the equation (14). We can see that the direct maximization of this function does not seem very tractable, so numerical algorithms may be needed to approximate the possible maximizer of this function. Since, the SLN distribution has a scale mixture form, the EM algorithm (Dempster et al. (1977)) can be implemented to obtain the ML estimator for 𝜽. To simplify the steps of the EM algorithm, we will use the stochastic representation of the SLN distribution given in (3). 4 Let 𝑉 and 𝑈 be the latent variables. Using the hierarchical representation given in (4), or the model (13) we get the following hierarchical representation 𝑇 𝑌𝑖 \|𝑢𝑖 , 𝑣𝑖 ∼ 𝑁 (𝒙𝑇𝑖 𝜷 + 𝑒 𝒛𝑖 𝜸⁄2 (𝒘𝑇𝑖 𝜶)𝑢𝑖 2 𝑣𝑖2 + (𝒘𝑇𝑖 𝜶) 𝑇 , 𝑒 𝒛𝑖 𝜸 2) , 𝑣𝑖2 + (𝒘𝑇𝑖 𝜶) 2 𝑣𝑖2 + (𝒘𝑇𝑖 𝜶) 𝑈𝑖 \|𝑣𝑖 = 1 ∼ 𝑇𝑁 ((0, ) ; (0, ∞)) , 𝑣𝑖2 −1 (15) 𝑣𝑖 ∼ 𝑓(𝑣𝑖 ) = 𝑣𝑖−3 exp (−(2𝑣𝑖2 ) ). Let 𝒖 = (𝑢1 , … , 𝑢𝑛 ) and 𝒗 = (𝑣1 , … , 𝑣𝑛 ) be the missing data and (𝒚, 𝒖, 𝒗) be the complete data, where 𝒚 = (𝑦1 , … , 𝑦𝑛 ). Then, using the hierarchical representation given in (15), the complete data loglikelihood function of 𝜽 can be written as 𝑛 2 1 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 2 −1 ℓ𝑐 (𝛉; 𝒚, 𝒖, 𝒗) = ∑ {− log 𝜋 − 𝒛𝑇𝑖 𝜸 − 2 log 𝑣𝑖 − (2𝑣𝑖2 ) − ( 𝑣𝑖 𝑇 2 2 𝑒 𝒛𝑖 𝜸 𝑖=1 +𝑢𝑖2 −2 2 𝒘𝑇𝑖 𝜶 𝑇 𝑒 𝒛𝑖 𝜸⁄2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝑢𝑖 + (𝒘𝑇𝑖 𝜶) 𝑇 𝑒 𝒛𝑖 𝜸 2 (16) (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) )}. To obtain the ML estimator of 𝜽, we have to maximize (16). However, the estimators obtained from this maximization will be dependent on the latent variables. Thus, to handle this latency problem, we have to take the conditional expectation of the complete data log-likelihood function given the observed data 𝑦𝑖 𝑛 1 −1 𝐸(ℓ𝑐 (𝛉; 𝒚, 𝒖, 𝒗)\|𝑦𝑖 ) = ∑ {− log 𝜋 − 𝒛𝑇𝑖 𝜸 − 2𝐸(log 𝑉𝑖 \|𝑦𝑖 ) − 𝐸 ((2𝑉𝑖2 ) \| 𝑦𝑖 ) 2 𝑖=1 2 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) − ( 𝐸(𝑉𝑖2 \|𝑦𝑖 ) + 𝐸(𝑈𝑖2 \|𝑦𝑖 ) 𝑇𝜸 𝒛 2 𝑒 𝑖 −2 𝒘𝑇𝑖 𝜶 𝑇 𝑒 𝒛𝑖 𝜸⁄2 2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝐸(𝑈𝑖 \|𝑦𝑖 ) + (𝒘𝑇𝑖 𝜶) 𝑇 𝑒 𝒛𝑖 𝜸 2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) )}. (17) The conditional expectations 𝐸(𝑉𝑖2 \|𝑦𝑖 ), 𝐸(𝑈𝑖 \|𝑦𝑖 ) and 𝐸(𝑈𝑖2 \|𝑦𝑖 ) in (17) can be calculated using the conditional expectations given in (10)-(12). Note that since the other conditional expectations are not related to the parameters, we do not calculate them. Let 𝑇 𝑣̂𝑖 = 𝐸(𝑉𝑖2 \|𝑦𝑖 ) = 𝑒 𝒛𝑖 𝜸̂⁄2 \|𝑦𝑖 − ̂\| 𝒙𝑇𝑖 𝜷 (18) , Φ(𝜅̂ 𝑖 ) , 𝜙(𝜅̂ 𝑖 ) = 𝐸(𝑈𝑖2 \|𝑦) = 1 + 𝜅̂ 𝑖 𝑢̂1𝑖 , 𝑢̂1𝑖 = 𝐸(𝑈𝑖 \|𝑦𝑖 ) = 𝜅̂ 𝑖 + (19) 𝑢̂2𝑖 (20) ̂ where, 𝜅̂ 𝑖 = 𝒘𝑇𝑖 𝜶 ̂ (𝑦𝑖 −𝒙𝑇 𝑖 𝜷) 𝑇 ̂⁄ 𝑒 𝒛𝑖 𝜸 2 . Then, using these conditional expectations in (17) we get the following objective function to be maximized with respect to 𝜽 5 𝑛 2 1 1 (𝑦 − 𝒙𝑇 𝜷) ̂ ) = ∑ {− log 𝜋 − 𝒛𝑇𝑖 𝜸 − ( 𝑖 𝑇 𝑖 𝑄(𝜽; 𝜽 𝑣̂𝑖 + 𝑢̂2𝑖 2 2 𝑒 𝒛𝑖 𝜸 𝑖=1 −2 2 𝒘𝑇𝑖 𝜶 𝑇 𝑒 𝒛𝑖 𝜸⁄2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝑢̂1𝑖 + (𝒘𝑇𝑖 𝜶) 𝑇 𝑒 𝒛𝑖 𝜸 2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) )}. (21) Now, the steps of the EM algorithm will be as follows: EM algorithm: 1. Take initial value for 𝜽(0) = (𝜷(0) , 𝜸(0) , 𝜶(0) ). 2. E-Step: Given the observed data and the current parameter values, find the conditional expectation of the complete data log-likelihood function given in (16). This corresponds to calculating the conditional expectations given in (18)-(20). This step will be carried on as follows. Compute the following conditional expectations for the 𝑘 = 0,1,2, … iteration 𝑣̂𝑖 (𝑘) = ̂ (𝑘) ) 𝐸(𝑉𝑖2 \|𝑦𝑖 , 𝜽 𝑇 (𝑘) ⁄ 2 = 𝑒 𝒛𝑖 𝜸̂ ̂ (𝑘) \| \|𝑦𝑖 − 𝒙𝑇𝑖 𝜷 (22) , (𝑘) (𝑘) 𝑢̂1𝑖 ̂ (𝑘) = 𝐸(𝑈𝑖 \|𝑦𝑖 , 𝜽 )= (𝑘) 𝜅̂ 𝑖 (𝑘) ̂ (𝑘) ) = 1 𝑢̂2𝑖 = 𝐸(𝑈 2 \|𝑦, 𝜽 (𝑘) where, 𝜅̂ 𝑖 ̂ (𝑘) = 𝒘𝑇𝑖 𝜶 + Φ (𝜅̂ 𝑖 ) , (𝑘) 𝜙 (𝜅̂ 𝑖 ) (𝑘) (𝑘) + 𝜅̂ 𝑖 𝑢̂1𝑖 , (23) (24) ̂ (𝑘) ) (𝑦𝑖 −𝒙𝑇 𝑖𝜷 𝑇 ̂(𝑘) ⁄2 𝑒 𝒛𝑖 𝜸 . ̂ ) and maximize it with respect to 𝜽 to obtain 3. M-Step: Use these conditional expectations in 𝑄(𝜽; 𝜽 new estimates. This maximization step yields the following formulation to update the new estimates. The (𝑘 + 1)𝑡ℎ parameter estimates can be computed using −1 ̂ (𝑘+1) = 𝜽 ̂ (𝑘) + (−𝐻(𝜽 ̂ (𝑘) )) 𝜽 where 𝐺(𝜽) = ̂) 𝜕𝑄(𝜽;𝜽 and 𝐻(𝜽) = 𝜕𝜽 ̂ (𝑘) ), 𝐺(𝜽 ̂) 𝜕2 𝑄(𝜽;𝜽 𝜕𝜽𝜕𝜽𝑇 (25) . 4. Repeat E and M steps until the convergence is satisfied. Remark. For the detail expressions of 𝐺(𝜽) and 𝐻(𝜽) see Appendix. 5. Simulation study In this section, we give a simulation study to show the performance of the proposed location, scale and skewness models of the SLN distribution in terms of mean squared error (MSE). The MSE is given with the following formula 𝑁 ̂ (𝜃̂) = 𝑀𝑆𝐸 1 2 ∑(𝜃̂𝑗 − 𝜃) , 𝑁 𝑗=1 6 where 𝜃 is the true parameter value, 𝜃̂𝑗 is the estimate of 𝜃 for the 𝑗𝑡ℎ simulated data and 𝜃̅ = 1 𝑁 ∑ 𝜃̂ . 𝑁 𝑗=1 𝑗 All simulation studies are conducted as 𝑁 = 1000 times. We set the sample sizes as 50, 100, 150 and 200. Note that the simulation study and real data example are performed using MATLAB R2015b. For all numerical calculations, the convergence rule is taken as 10−6. The data are generated from the following location, scale and skewness models of the SLN distribution 𝑦𝑖 ∼ 𝑆𝐿𝑁(𝜇𝑖 , 𝜎𝑖2 , 𝜆), 𝑖 = 1,2, … , 𝑛 𝜇𝑖 = 𝒙𝑇𝑖 𝜷 , log 𝜎𝑖2 = 𝒛𝑇𝑖 𝜸 , 𝑇 { 𝜆𝑖 = 𝒘𝑖 𝜶. Here, all covariate vectors 𝒙𝑖 , 𝒛𝑖 and 𝒘𝑖 are independently generated from uniform distribution 𝑈(−1,1). To carry out the simulation study, we take the following two cases for true parameter values: Case I: 𝛽0 = (0, −1, −1)𝑇 , 𝛾0 = (0, −1, −1)𝑇 and 𝛼0 = (0, −1, −1)𝑇 , Case II: 𝛽0 = (0,1,1)𝑇 , 𝛾0 = (0,1,1)𝑇 and 𝛼0 = (0,1,1)𝑇 , Case III: 𝛽0 = (1,1,0,0,1)𝑇 , 𝛾0 = (0.7,0.7,0,0,0.7)𝑇 and 𝛼0 = (0.5,0.5,0,0,0.5)𝑇 . The simulation results are summarized in Tables 1, 2 and 3. These tables include the mean of the estimators and the values of MSE. From these tables, we observe the followings. The proposed EM algorithm is working accurately for estimating the parameters. When the sample sizes increase, the values of MSE decrease. Table 1. Mean of the estimators and the values of MSE for the different sample sizes for the Case I. Model Location Model Scale Model Skewness Model 𝑛 𝛽0 𝛽1 𝛽2 𝛾0 𝛾1 𝛾2 𝛼0 𝛼1 𝛼2 50 Mean MSE 0.0010 0.0433 -1.0068 0.0624 -0.9950 0.0643 -0.0655 0.0905 -1.0595 0.3300 -1.1024 0.3445 -0.0527 0.5398 -1.5780 1.7814 -1.6140 2.0557 100 Mean 0.0020 -0.9983 -0.9987 -0.0384 -1.0576 -1.0281 0.0035 -1.1981 -1.2066 MSE 0.0161 0.0243 0.0256 0.0369 0.1376 0.1272 0.0600 0.2234 0.2339 150 Mean -0.0002 -0.9987 -1.0077 -0.0278 -1.0264 -1.0118 -0.0034 -1.1213 -1.1157 MSE 0.0088 0.0143 0.0150 0.0228 0.0829 0.0843 0.0296 0.1113 0.1141 200 Mean -0.0006 -0.9965 -0.9987 -0.0223 -1.0088 -1.0060 -0.0016 -1.0852 -1.0754 MSE 0.0061 0.0109 0.0117 0.0161 0.0594 0.0629 0.0190 0.0727 0.0652 Table 2. Mean of the estimators and the values of MSE for the different sample sizes for the Case II. Model Location Model Scale Model Skewness Model 𝑛 𝛽0 𝛽1 𝛽2 𝛾0 𝛾1 𝛾2 𝛼0 𝛼1 𝛼2 50 Mean 0.0059 1.0046 1.0036 -0.0794 1.0957 1.0562 -0.0130 1.5993 1.5928 MSE 0.0417 0.0623 0.0661 0.0935 0.3421 0.3332 0.4643 1.9956 1.7924 100 Mean 0.0035 1.0040 1.0095 -0.0369 1.0380 1.0357 0.0045 1.2105 1.2117 7 150 MSE 0.0142 0.0247 0.0265 0.0346 0.1282 0.1252 0.0635 0.2636 0.2621 Mean -0.0020 0.9956 1.0085 -0.0209 1.0281 1.0193 -0.0009 1.1210 1.1148 200 MSE 0.0088 0.0158 0.0138 0.0224 0.0796 0.0783 0.0285 0.1035 0.1075 Mean 0.0018 1.0034 1.0036 -0.0239 1.0200 1.0130 -0.0013 1.0717 1.0802 MSE 0.0067 0.0115 0.0105 0.0162 0.0600 0.0580 0.0203 0.0672 0.0705 Table 3. Mean of the estimators and the values of MSE for the different sample sizes for the Case III. Model Location Model Scale Model Skewness Model 𝑛 𝛽0 𝛽1 𝛽2 𝛽3 𝛽4 𝛾0 𝛾1 𝛾2 𝛾3 𝛾4 𝛼0 𝛼1 𝛼2 𝛼3 𝛼4 50 Mean 1.0035 1.0018 0.0117 -0.0151 0.9971 0.5812 0.8583 -0.0148 -0.0303 0.8151 1.3635 1.2163 0.0499 -0.0369 1.3361 100 MSE 0.1734 0.2093 0.2129 0.2219 0.2102 0.1404 0.5997 0.6155 0.5936 0.5718 3.2188 3.1849 1.8253 1.6726 3.4589 Mean 0.9631 1.0057 -0.0181 -0.0006 1.0053 0.6667 0.7411 -0.0024 -0.0136 0.7538 0.7983 0.7769 0.0156 -0.0245 0.7471 150 MSE 0.0900 0.0709 0.0806 0.0784 0.0767 0.0598 0.1412 0.1715 0.1621 0.1561 0.4941 0.4341 0.1861 0.2151 0.3766 Mean 0.9596 0.9966 0.0045 0.0037 0.9958 0.6913 0.7145 0.0170 -0.0168 0.7044 0.6819 0.6210 -0.0059 -0.0103 0.6317 200 MSE 0.0508 0.0415 0.0473 0.0447 0.0372 0.0387 0.0955 0.0963 0.0951 0.0920 0.1866 0.1381 0.0807 0.0764 0.1347 Mean 0.9808 1.0012 0.0091 -0.0095 1.0081 0.6792 0.7226 -0.0147 -0.0110 0.7309 0.6067 0.5894 0.0033 -0.0041 0.5827 MSE 0.0356 0.0296 0.0294 0.0301 0.0287 0.0240 0.0685 0.0612 0.0665 0.0621 0.0975 0.0831 0.0420 0.0499 0.0608 6. Real data example The Martin Marietta data set includes the relationship of the excess rate of returns of the Marietta Company and an index for the excess rate of return for the New York Exchange (CRSP). These rate of returns for the company and the CRSP index were determined monthly over a period of five years. This data set used by Butler et al. (1990) for modelling a simple linear regression with Gaussian errors. Azzalini and Capitanio (2003) analyzed this data set for modelling the linear regression model when the errors have the skew t distribution. Also, Taylor and Verbyla (2004) examined this data set for joint modelling of location and scale parameters of the t distribution. We display the scatter plot of the data set and the histogram of the Martin Marietta excess returns. Since the skewness coefficient of Martin Marietta excess returns is 2.9537 and also according to the Figure 2 (b), we can say that it will be more suitable to model this data set with a joint location, scale and skewness models of a skew distribution. Figure 2. (a) Scatter plot of the data set. (b) Histogram of the Martin Marietta excess returns. In this article, we analyze this data set to illustrate the applicability of the joint location, scale and skewness models of SLN distribution over the joint location, scale and skewness models of the STN distribution. For the comparison of the models, we use the values of the Akaike information criterion 8 (AIC) (Akaike (1973)), the Bayesian information criterion (BIC) (Schwarz (1978)), and the efficient determination criterion (EDC) (Bai et al. (1989)). These criteria have the following form ̂) + 𝑚𝑐𝑛 , −2ℓ(𝛉 where ℓ(∙) represents the maximized log-likelihood, 𝑚 is the number of free parameters to be estimated in the model and 𝑐𝑛 is the penalty term. Here, we take 𝑐𝑛 = 2 for AIC, 𝑐𝑛 = log(𝑛) for BIC and 𝑐𝑛 = 0.2√𝑛 for EDC. We give the estimation results in Table 4 for all models. This table contains the estimates, bootstrap standard errors (BSEs) (Efron and Tibshirani (1993)) of estimates based on 500 random samples, the log-likelihood, and the values of AIC, BIC, and EDC. Note that we take the heteroscedastic t model results given in Taylor and Verbyla (2004) as initial values for the parameters of location and scale models. Also, we set 𝛼0 = 𝛼1 = 0 as initial values for the parameters of skewness model and the degrees of freedom parameter 3.75. In Figure 3, we show the scatter plot of the data set with the fitted regression lines obtained from the joint location, scale and skewness models of the SLN distribution and the joint location, scale and skewness models of the STN distribution. We observe that the joint location, scale and skewness models of the SLN distribution has better fit than the location, scale and skewness models of the STN distribution according to the information criteria and also Figure 3. Table 4. Estimation results for Martin Marietta data set. Location model Scale model Skewness model Information Criteria 𝛽0 𝛽1 𝛾0 𝛾1 𝛼0 𝛼1 ̂) ℓ(𝛉 AIC BIC EDC Skew t Normal Estimate BSE 0.0289 -0.0349 1.1387 0.4888 0.4418 -5.7905 15.8670 25.1552 1.6808 0.5614 33.9531 19.3303 75.9986 -139.9872 -125.3268 -118.3268 9 Skew Laplace Normal Estimate BSE 0.0227 -0.0267 0.3683 0.7344 0.3234 -5.8282 11.8023 17.2765 1.0923 0.4040 9.0474 13.3093 76.9986 -139.9971 -127.4311 -121.4311 Figure 3. The scatterplot of the data set with the fitted regression lines obtained from joint location, scale and skewness models of the SN and SLN distributions. 7. Conclusions In this study, we have proposed the joint location, scale and skewness models of the SLN distribution as an alternative to the joint location, scale and skewness models of the STN distribution. We have obtained the ML estimates via the EM algorithm. We have provided a simulation study to show the estimation performance of the proposed model. We have observed from simulation results that the parameters can be accurately estimated. We have given a real data application to test the applicability of the proposed model and also to compare with the joint, location, scale and skewness models of the STN distribution. We have seen from real data example results that the joint location, scale and skewness models of the SLN distribution gives better fit than the joint, location, scale and skewness models of the STN distribution. 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Appendix Using the objective function given in (21), we obtain the score function 𝐺(𝜽) = ̂) 𝑇 𝜕𝑄(𝜽; 𝜽 = (𝐺1𝑇 (𝜷), 𝐺2𝑇 (𝜸), 𝐺3𝑇 (𝜶)) , 𝜕𝜽 where 𝑛 𝐺1 (𝜷) = ∑ 𝑖=1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒙𝑖 𝑇 𝑒 𝒛𝑖 𝜸 𝑛 𝑛 𝑖=1 𝑖=1 𝑛 (𝑣̂𝑖 + 2 (𝒘𝑇𝑖 𝜶) ) − ∑ 𝑖=1 2 (𝒘𝑇𝑖 𝜶)𝒙𝑖 𝑢̂1𝑖 𝑇 𝑒 𝒛𝑖 𝜸⁄2 , 𝑛 1 1 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒛𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒛𝑖 𝑢̂1𝑖 2 𝑇 𝐺2 (𝜸) = − ∑ 𝒛𝑖 + ∑ (𝑣 ̂ + 𝜶) ) − ∑ , (𝒘 𝑖 𝑖 𝑇 𝑇 2 2 2 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸⁄2 𝑖=1 11 𝑛 𝐺3 (𝜶) = ∑ 𝑛 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒘𝑖 𝑢̂1𝑖 −∑ 𝑇 𝑒 𝒛𝑖 𝜸⁄2 𝑖=1 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒘𝑖 𝑇 𝑒 𝒛𝑖 𝜸 𝑖=1 , and observed Fisher information matrix ̂ ) 𝜕 2 𝑄(𝜽; 𝜽 ̂ ) 𝜕 2 𝑄(𝜽; 𝜽 ̂) 𝜕 2 𝑄(𝜽; 𝜽 𝜕𝜷𝜕𝜷𝑇 𝜕𝜷𝜕𝜸𝑇 𝜕𝜷𝜕𝜶𝑇 2 2 2 2 ̂) ̂ ) 𝜕 𝑄(𝜽; 𝜽 ̂ ) 𝜕 𝑄(𝜽; 𝜽 ̂) 𝜕 𝑄(𝜽; 𝜽 𝜕 𝑄(𝜽; 𝜽 𝐻(𝜽) = = , 𝑇 𝑇 𝑇 𝑇 𝜕𝜽𝜕𝜽 𝜕𝜸𝜕𝜷 𝜕𝜸𝜕𝜸 𝜕𝜸𝜕𝜶 ̂ ) 𝜕 2 𝑄(𝜽; 𝜽 ̂ ) 𝜕 2 𝑄(𝜽; 𝜽 ̂) 𝜕 2 𝑄(𝜽; 𝜽 𝑇 𝑇 𝑇 [ 𝜕𝜶𝜕𝜷 𝜕𝜶𝜕𝜸 𝜕𝜶𝜕𝜶 ] where 𝑛 𝑛 𝑖=1 𝑛 𝑖=1 2 ̂) 𝜕 2 𝑄(𝜽; 𝜽 𝒙𝑖 𝒙𝑇𝑖 (𝒘𝑇𝑖 𝜶) = − ∑ 𝑇 𝑣̂𝑖 − ∑ 𝒙𝑖 𝒙𝑇𝑖 , 𝑇 𝜕𝜷𝜕𝜷𝑇 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸 𝑛 ̂) 𝜕 𝑄(𝜽; 𝜽 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒙𝑖 𝒛𝑇𝑖 (𝒘𝑇𝑖 𝜶)𝒙𝑖 𝒛𝑇𝑖 𝑢̂1𝑖 2 𝑇 = − ∑ (𝑣 ̂ + 𝜶) ) + ∑ , (𝒘 𝑖 𝑖 𝑇 𝑇 𝜕𝜷𝜕𝜸𝑇 2 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸⁄2 2 𝑖=1 𝑛 𝑛 𝑖=1 𝑛 𝑖=1 𝑖=1 ̂) 𝜕 2 𝑄(𝜽; 𝜽 𝒙𝑖 𝒘𝑇𝑖 𝑢̂1𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒙𝑖 𝒘𝑇𝑖 = − ∑ + 2 ∑ , 𝑇 𝑇 𝜕𝜷𝜕𝜶𝑇 𝑒 𝒛𝑖 𝜸⁄2 𝑒 𝒛𝑖 𝜸 𝑛 ̂) 𝜕 𝑄(𝜽; 𝜽 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒛𝑖 𝒙𝑇𝑖 (𝒘𝑇𝑖 𝜶)𝒛𝑖 𝒙𝑇𝑖 𝑢̂1𝑖 2 𝑇 = −∑ (𝑣̂𝑖 + (𝒘𝑖 𝜶) ) + ∑ , 𝑇 𝑇 𝜕𝜸𝜕𝜷𝑇 2 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸⁄2 2 𝑖=1 𝑛 𝑖=1 𝑛 2 ̂) 𝜕 𝑄(𝜽; 𝜽 1 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒛𝑖 𝒛𝑇𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒛𝑖 𝒛𝑇𝑖 𝑢̂1𝑖 2 𝑇 = − ∑ (𝑣 ̂ + 𝜶) ) + ∑ , (𝒘 𝑖 𝑖 𝑇 𝑇 𝜕𝜸𝜕𝜸𝑇 2 4 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸⁄2 2 𝑖=1 𝑛 𝑛 𝑖=1 2 ̂) 𝜕 2 𝑄(𝜽; 𝜽 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒛𝑖 𝒘𝑇𝑖 𝑢̂1𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒛𝑖 𝒘𝑇𝑖 = − ∑ + ∑ , 𝑇 𝑇 𝜕𝜸𝜕𝜶𝑇 2 𝑒 𝒛𝑖 𝜸⁄2 𝑒 𝒛𝑖 𝜸 𝑖=1 𝑛 𝑖=1 𝑛 ̂) 𝜕 𝑄(𝜽; 𝜽 𝒘𝑖 𝒙𝑇𝑖 𝑢̂1𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒘𝑖 𝒙𝑇𝑖 = −∑ + 2∑ , 𝑇 𝑇 𝜕𝜶𝜕𝜷𝑇 𝑒 𝒛𝑖 𝜸⁄2 𝑒 𝒛𝑖 𝜸 2 𝑖=1 𝑛 𝑖=1 𝑛 2 ̂) 𝜕 𝑄(𝜽; 𝜽 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒘𝑖 𝒛𝑇𝑖 𝑢̂1𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒘𝑖 𝒛𝑇𝑖 =− ∑ +∑ , 𝑇 𝑇 𝜕𝜶𝜕𝜸𝑇 2 𝑒 𝒛𝑖 𝜸⁄2 𝑒 𝒛𝑖 𝜸 2 𝑖=1 𝑛 𝑖=1 2 ̂) 𝜕 2 𝑄(𝜽; 𝜽 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒘𝑖 𝒘𝑇𝑖 = − ∑ . 𝑇 𝜕𝜶𝜕𝜶𝑇 𝑒 𝒛𝑖 𝜸 𝑖=1 12	10
1 Investigations of a Robotic Testbed with Viscoelastic Liquid Cooled Actuators arXiv:1711.01649v2 [cs.SY] 8 Mar 2018 Donghyun Kim, Junhyeok Ahn, Orion Campbell, Nicholas Paine, and Luis Sentis, Abstract—We design, build, and thoroughly test a new type of actuator dubbed viscoelastic liquid cooled actuator (VLCA) for robotic applications. VLCAs excel in the following five critical axes of performance: energy efficiency, torque density, impact resistence, joint position and force controllability. We first study the design objectives and choices of the VLCA to enhance the performance on the needed criteria. We follow by an investigation on viscoelastic materials in terms of their damping, viscous and hysteresis properties as well as parameters related to the longterm performance. As part of the actuator design, we configure a disturbance observer to provide high-fidelity force control to enable a wide range of impedance control capabilities. We proceed to design a robotic system capable to lift payloads of 32.5 kg, which is three times larger than its own weight. In addition, we experiment with Cartesian trajectory control up to 2 Hz with a vertical range of motion of 32 cm while carrying a payload of 10 kg. Finally, we perform experiments on impedance control and mechanical robustness by studying the response of the robotics testbed to hammering impacts and external force interactions. Index Terms—Viscoelastic liquid cooled actuator, Torque feedback control, Impedance control. I. I NTRODUCTION ERIES elastic actuators (SEAs) [1] have been extensively used in robotics [2], [3] due to their impact resistance and high-fidelity torque controllability. One drawback of SEAs is the difficulty that arises when using a joint position controller due to the presence of the elastic element in the drivetrain. To remedy this problem the addition of dampers has been previously considered [4]–[6]. However, incorporating mechanical dampers makes actuators bulky and increases their mechanical complexity. One way to avoid this complexity is to employ elastomers instead of metal springs. Using a viscoelastic material instead of combined spring-damper systems enables compactness [7] and simplified drivetrains [8]. However, it is difficult to achieve high bandwidth torque control due to the nonlinear behavior of elastomers. To address this difficulty, [9] models the force-displacement curve of elastomer using a “standard linear model.” The estimated elastomer force is employed in a closed-loop force controller. Unfortunately, the hysteresis in the urethane elastomer destabilized the system at frequencies above 2 Hz. In contrast our controllers achieve a bandwidth of 70 Hz. The study on [10] accomplishes reasonably good torque control performance, but the range of torques is small to ensure that the elastomer operates in the linear region; our design and control methods described here achieve more than an order of magnitude higher range of torques with high fidelity tracking. To sufficiently address the nonlinear behavior of elastomers, which severely reduce force control performance, we empiri- S cally analyze various viscoelastic materials with a custom-built elastomer testbed. We measure each material’s linearity, creep, compression set, and damping under preloaded conditions, which is a study under-documented in the academic literature. To achieve stable and accurate force control, we study various feedback control schemes. In a previous work, we showed that the active passivity obtained from motor velocity feedback [11] and model-based control such as disturbance observer (DOB) [12] play an essential role in achieving high-fidelity force feedback control. Here, we analyze the phase margins of various feedback controllers and empirically show their operation in the new actuators. We verify the stability and accuracy of our controllers by studying impedance control and impact tests. To test our new actuator, we have designed a two degree-offreedom (DOF), robotic testbed, shown in Fig. 5. It integrates two of our new actuators, one in the ankle, and another in the knee, while restricting motions to the sagittal plane. With the foot bolted to the floor for initial tests, weight plates can be loaded on the hip joint to serve as an end-effector payload. We test operational space control to show stable and accurate operational space impedance behaviors. We perform dynamic motions with high payloads to showcase another important aspect of our system, which is its cooling system aimed at significantly increasing the power of the robot. The torque density of electric motors is often limited by sustainable core temperature. For this reason, the maximum continuous torque achieved by these motors can be significantly enhanced using an effective cooling system. Our previous study [13] analyzed the improvements on achievable power based on thermal data of electric motors and proposed metrics for design of cooling systems. Based on the metrics from that study, we chose a 120 W Maxon EC-max 40, which is expected to exert 3.59 times larger continuous torque when using the proposed liquid cooling system. We demonstrate the effectiveness of liquid cooling by exerting 860N continous force during 5 min and 4500N peak force during 0.5s while keeping the core temperatures below 115◦C, which is much smaller than the maximum, 155◦C. We accurately track fast motions of 2 Hz while carrying a 10 kg payload for endurance tests. In addition we perform heavy lift tests with a payload of 32.5 kg keeping the motor temperatures under 80◦C. The main contribution of this paper is the introduction of a new viscoelastic liquid cooled actuator and a thorough study of its performance and its use on a multidof testbed. We demonstrate that the use of liquid cooling and the elastomer significantly improve joint position controllability and power density over traditional SEAs. More concretely, we 1) design 2 a new actuator, dubbed the VLCA, 2) extensively study viscoelastic materials, 3) extensively analyze torque feedback controllers for VLCAs, and 4) examine the performance in a multidof prototype. II. BACKGROUND Existing actuators can be characterized using four criteria: power source (electric or hydraulic), cooling type (air or liquid), elasticity of the drivetrain (rigid or elastic), and drivetrain type (direct, harmonic drive, ball screw, etc.) [14], [15]. One of the most powerful and common solutions is the combination of hydraulic, liquid-cooling, rigid and direct drive actuation. This achieves high power-to-weight and torque-to-weight ratios, joint position controllability, and shock tolerance. Existing robots that use this type of actuators include Atlas, Spot, Big Dog, and Wildcat of Boston Dynamics, BLEEX of Berkeley [16], and HyQ of IIT [17]. However, hydraulics are less energy efficient primarily because they require more energy transformations [18]. Typically, a gasoline engine or electric motor spins a pump, which compresses hydraulic fluid, which is modulated by a hydraulic servo valve, which finally causes a hydraulic piston to apply a force. Each stage in this process incurs some efficiency loss, and the total losses can be very significant. The combination of electric, air-cooled, rigid, and harmonic drive actuators are other widely used actuation types. Some robots utilizing these actuator types include Asimo of Honda, HRP2,3,4 of AIST [19], HUBO of KAIST [20], REEM-C of PAL Robotics, JOHNNIE and LOLA of Tech. Univ. of Munich [21], [22], CHIMP of CMU [23], Robosimian of NASA JPL [24], and more. These actuators have precise position control and high torque density. For example, LOLA’s theoretical knee peak torque-density (129N m/kg) is comparable to ours (107N m/kg), although they did not validate their number experimentally and their max speed is roughly 2/3 of our max speed [22]. Compared to us, low shock tolerance, low fidelity force sensing, and low efficiency gearboxes are common drawbacks of these type of actuators. According to Harmonic Drive AGs catalog, the efficiency of harmonic drives may be as poor as 25% and only increases above 80% when optimal combinations of input shaft speed, ambient temperature, gear ratio, and lubrication are present. Conversely, the efficiency of our VLCA is consistently above 80% due to the use of a ball screw mechanism. [25] used liquid cooling for electric, rigid, harmonic drive actuators to enhance continuous power-to-weight ratio. The robots using this type of actuation include SCHAFT and Jaxon [26]. These actuators share the advantages and disadvantages of electric, rigid, harmonic drive actuators, but have a significant increase of the continuous power output and torque density. One of our studies [13], indicates a 2x increase in sustained power output by retrofitting an electric motor with liquid cooling. Other published results indicate a 6x increase in torque density through liquid cooling [14], [27], though such performance required custom-designing a motor specifically for liquid cooling. In our case we use an off-theshelf electric motor. In contrast with our design, these actuators do not employ viscoelastic materials reducing their mechanical robustness and high quality force sensing and control. Although the increased power density achieved via liquid cooling amplifies an electric actuator’s power, the rigid drivetrain is still vulnerable to external impacts. To increase impact tolerance, many robots (e.g. Walkman and COMAN of IIT [28], Valkyrie of NASA [29], MABEL and MARLO in UMich [30], [31], and StarlETH of ETH [32]) adopt electric, air-cooled, elastic, harmonic drive actuators. This type of actuation provides high quality force sensing, force control, impact resistance, and energy efficiency. However, precise joint position control is difficult because of the elasticity in the drivetrain and the coupled effect of force feedback control and realtime latencies [33]. Low efficiency originating from the harmonic drives is another drawback. As an alternative to harmonic drives, ball screws are great drives for mechanical power transmission. SAFFiR, THOR, and ESCHER of Virginia Tech [34]–[36], M2V2 of IHMC [37], Spring Flamingo of MIT [38], Hume of UT Austin [11], and the X1 Mina exoskeleton of NASA [39] use electric, air-cooled, elastic, ball-screw drives. These actuators show energy efficiency, good power and force density, low noise force sensing, high fidelity force controllability, and low backlash. Compared to these actuators our design significantly reduces the bulk of the actuator and increases its joint position controllability. There are some other actuators that have special features such as the electric actuators used in MIT’s cheetah [40], which allow for shock resistance through a transparent but backlash-prone drivetrain. However, the lack of passive damping limits the joint position controllability of these type of actuators compared to us. III. VISCOELASTIC MATERIAL CHARACTERIZATION The primary driver for using elastomers instead of metal springs is to benefit from their intrinsic damping properties. However, the mechanical properties of viscoelastic materials can be difficult to predict, thus making the design of an actuator based on these materials a challenging endeavor. The most challenging aspect of incorporating elastomers into the structural path of an actuator is in estimating or modeling their complex mechanical properties. Elastomers possess both hysteresis and strain-dependent stress, which result in nonlinear force displacement characteristics. Additionally, elastomers also exhibit time-varying stress-relaxation effects when exposed to a constant load. The result of this effect is a gradual reduction of restoration forces when operating under a load. A third challenge when using elastomers in compression is compression set. This phenomenon occurs when elastomers are subjected to compressive loads over long periods of time. An elastomer that has been compressed will exhibit a shorter free-length than an uncompressed elastomer. Compression set is a common failure mode for o-rings, and in our application, it could lead to actuator backlash if not accounted for properly. To address these various engineering challenges we designed experiments to empirically measure the following four properties of our viscoelastic springs: 1) force versus displacement, 2) stress relaxation, 3) compression set, and 4) 3 E-stop Load cell Displacement sensor Elastic material EtherCAT-based embedded control system Belt drive BLDC motor 2500 2000 1500 Force (N) 1000 Ball screw drive (a) Viscoelastic material Testbed 0 -500 Spring steel Viton 75A Polyurethane 80A Polyurethane 90A EPDM 80A Reinforced Silicon 70A Buna- N 90A Silicone 90A -1000 Reinforced silicone 70A Viton 75A Buna-N 90A Polyurethane 80A Polyurethane 90A Spring steel 500 EPDM 80A -1500 -2000 0 1 2 3 4 5 -2500 6 -5 Compression set (%) -4 -3 -2 -1 0 1 2 3 Dispacement (m) 5 10 -4 (c) Force vs Displacement curve Phase (deg) Force (N) Magnitude (dB) (b) Complession set 4 time (sec) (d) Stess relaxation Increasing system bandwidth Spring steel Viton 75A Buna-N 90A Polyurethane 90A Frequency (Hz) (e) Dynamic response of four elastomer Fig. 1. Viscoelastic material test. (a) The elastomer testbed is designed and constructed to study various material properties of candidate viscoelastic materials. (b) We measured each elastomers free length both before and after they were placed in the preloaded testbed. (c) A strong correlation between material hardness and the materials stiffness can be observed. An exception to this correlation is the fabric reinforced silicone which we hypothesize had increased stiffness due to the inelastic nature of its reinforcing fabric. Nonlinear effects such as hysteresis can also be observed in this plot. (d) We command a rapid change in material displacements and then measured the materials force change versus time for 300 seconds. Note that the test of reinforced silicone 70A is omitted due to its excessive stiffness. (d) Although the bandwidths of the four responses are different, their damping ratios (signal peak value) are relatively constant, which implies different damping. frequency response, which will be used to characterize each material’s effective viscous damping. We built a viscoelastic material testbed, depicted in Fig. 1(a), to measure each of these properties. We selected and tested the seven candidate materials that are listed in Table I. The dimension of the tested materials are fairly regular, with 46mm diameter and 27mm thickness. A. Compression set Compression set is the reduction in length of an elastomer after prolonged compression. The drawback of using materials with compression set in compliant actuation is that the materials must be installed with larger amounts of preload forces to avoid the material sliding out of place during usage. To measure this property, we measured each elastomers free length both before and after the elastomer was placed in the preloaded testbed. The result of our compression set experiments are summarized in Table I. B. Force versus displacement In the design of compliant actuation, it is essential to know how much a spring will compress given an applied force. This displacement determines the required sensitivity of a spring-deflection sensor and also affects mechanical aspects of the actuator such as usable actuator range of motion and clearance to other components due to Poisson ratio expansion. In this experiment, we identify the force versus displacement curves for the various elastomer springs. Experimental data for all eight springs as shown in Fig 1(b). Note that there is a disagreement between our empirical measurements and the analytic model relating stiffness to hardness, i.e. the Gent’s relation shown in [41]. This mismatch arises because in our experiments the materials are preloaded whereas the analytical models assume unloaded materials. 4 Materials Compression Linearity set (%) (R-square) Linear stiffness (N/mm) Spring steel 0 0.996 860.8 Polyurethane 90A 2 0.992 8109 Preloaded elastic modulus (N/mm) Material damping (N s/m) Creep (%) Material Cost ($) 0 0 - 112.5 16000 15.3 19.40 Reinforced silicone 70A 2.7 0.978 57570 798.7 242000 - 29.08 Buna-N 90A 2.8 0.975 11270 156.4 29000 25 51.47 Viton 75A 4 0.963 2430 33.7 9000 30.14 105.62 Polyurethane 80A 4.5 0.993 2266 31.4 4000 16.8 19.40 EPDM 80A 6.48 0.939 6499 90.2 16000 23.4 35.28 Silicone 90A - 0.983 12460 172.9 37000 10.7 29.41 TABLE I S UMMARY OF VISCOELASTIC MATERIALS C. Stress relaxation E. Selection of Polyurethane 90A Stress-relaxation is an undesirable property in compliant actuators for two reasons. First, the time-varying force degrades the quality of the compliant material as a force sensor. When a material with significant stress-relaxation properties is used, the only way to accurately estimate actuator force based on deflection data is to model the effect and then pass deflection data through this model to obtain a force estimate. This model introduces complexity and more room for error. The second reason stress-relaxation can be problematic is that it can lead to the loss of contact forces in compression-based spring structures. The experiment for stress relaxation is conducted as follow: 1) enforce a desired displacement to a material, 2) record the force data over time from the load cell, 3) subtract the initially measured force from all of the force data. Empirically measured stress-relaxation properties for each of the materials are shown in Fig. 1 (c), which represents force offsets as time goes under the same displacement enforced. Note that each material shows different initial force due to the different stiffness and each initial force data is subtracted in the plot. A variety of other experiments were conducted to strengthen our analysis and are summarized in Table I. Based on these results, Polyurethane 90A appears to be a strong candidate for viscoelastic actuators based on its high linearity (0.992), low compression set (2%), low creep (15%), and reasonably high damping (16000 N s/m). It is also the cheapest of the materials and comes in the largest variety of hardnesses and sizes. D. Dynamic response In regards to compliant actuation, the primary benefit of using an elastomer spring is its viscous properties, which can characterize the dynamic response of an actuator in series with such a component. To perform this experiment, we generate motor current to track an exponential chirp signal, testing frequencies between 0.001Hz and 200Hz. Given the inputoutput relation of the system, we can fit a second order transfer function to the experimental data to obtain an estimate of the system’s viscous properties. However, this measure also includes the viscoelastic testbed’s ballscrew drive train friction (Fig. 1(a)). To quantify the elastomer spring damping independently of the damping of the testbed drive train, the latter (8000 N s/m) was first characterized using a metal spring, and then subtracted from subsequent tests of the elastomer springs to obtain estimates for the viscous properties of the elastomer materials. Fig. 1(d) shows the frequency response results for current input and force output of three different springs, while controlling the damping ratio. The elastomers have higher stiffness than the metal spring, hence their natural frequencies are higher. IV. V ISCOELASTIC L IQUID C OOLED ACTUATION The design objectives of the VLCA are 1) power density, 2) efficiency, 3) impact tolerance, 4) joint position controllability, and 5) force controllability. Compactness of actuators is also one of the critical design parameters, which encourage us to use elastomers instead of metal springs and mechanical dampers. Our previous work [13] shows a significant improvement in motor current, torque, output power and system efficiency for liquid cooled commercial off-the-shelf (COTS) electric motors and studied several Maxon motors for comparison. As an extension of this previous work, in this new study we studied COTS motors and their thermal behavior models and selected the Maxon EC-max 40 brushless 120 W (Fig. 2(e)), with a custom housing designed for the liquid cooling system (Fig. 2(h)). The limit of continuous current increases by a factor of 3.59 when liquid convection is used for cooling the motor. Therefore, a continuous motor torque of 0.701 N · m is theoretically achievable. Energetically, this actuator is designed to achieve 366 W continuous power and 1098W short-term power output with an 85% ball screw efficiency (Fig. 2(b)) since short-term power is generally three time larger than continuous power. With the total actuator mass of 1.692 kg, this translates into a continuous power of 216W/kg and a short-term power of 650W/kg. The liquid pump, radiator, and reservoir are products of Swiftech which weight approximately 1kg. By combining convection liquid cooling, high power brushless DC (BLDC) motors, and a high-efficiency ball screw, we aim to surpass existing electric actuation technologies with COTS motors in terms of power density. In terms of controls, a common problem with conventional SEAs is their lack of physical damping at their mechanical output. As a result, active damping must be provided from torque produced by the motor [42]. However, the presence of 5 (e) BLDC Motor (a) Timing belt transmission (b) Ball screw drive (c) Load cell (d) Actuator output (f) Quadrature encoder (g) Temperature sensor (h) Liquid cooling jacket Opposite side Motor part Rubber part Load part (i) Tube connector (j) Polyuretane elastomer (k) Compliance deflection sensor (l) Mechanical ground pivot (m) Quadrature encoder (deflection) ground Fig. 2. Viscoelastic Liquid Cooled Actuator. The labels are explanatory. In addition, the actuator contains five sensors: a load cell, a quadrature encoder for the electric motor, a temperature sensor, and two elastomer deflection sensors. One of the elastomer deflection sensors is absolute and the other one is a quadrature encoder. The quadrature encoder gives high quality velocity data of the elastomer deflection. signal latency and derivative signal filtering limit the amount by which this active damping can be increased, resulting in SEA driven robots achieving only relatively low output impedances [33] and thus operating with limited joint position control accuracy and bandwidth. Our VLCA design incorporates damping directly into the compliant element itself, reducing the requirements placed on active damping efforts from the controller. The incorporation of passive damping aims to increase the output impedance while retaining compliance properties, resulting in higher joint position control bandwidth. The material properties we took into consideration will be introduced in Section III. The retention of a compliant element in the VLCA drive enables the measurement of actuator forces based on deflection. The inclusion of a load cell (Fig. 2(c)) on the actuators output serves as a redundant force sensor and is used to calibrate the force displacement characteristics of the viscoelastic element. Mechanical power is transmitted when the motor turns a ball nut via a low-loss timing belt and pulley (Fig. 2 (a)), which causes a ball screw to apply a force to the actuator’s output (Fig. 2(d)). The rigid assembly consisting of the motor, ball screw, and ball nut connects in series to a compliant viscoelastic element (Fig. 2(j)), which connects to the mechanical ground of the actuator (Fig. 2(k)). When the actuator applies a force, the reaction force compresses the viscoelastic element. The viscoelastic element enables the actuator to be more shock tolerant than rigid actuators yet also maintain high output impedance due to the inherent damping in the elastomer. V. ACTUATOR F ORCE F EEDBACK C ONTROL To demonstrate various impedance behaviors in operational space, robots must have a stable force controller. Stable and accurate operational space control (OSC) is not trivial to achieve because of the bandwidth interference between outer position feedback control (OSC) and inner torque feedback control [11]. Since stable torque control is a critical component for a successful OSC implementation, we extensively study various force feedback controls. Jm (kg m2 ) bm (N m s) mr (kg) br (N s/m) kr (N/m) 3.8e−5 2.0e−4 1.3 2.0e4 5.5e6 TABLE II ACTUATOR PARAMETERS The first step in this analysis is to identify the actuator dynamics. The transfer functions of the reaction force sensed in the series elastic actuators (elastomer deflection) are well explained in [43]. When the actuator output is fixed, the transfer function from the motor current input to the elastomer deflection is given by Px = xr ηkτ Nm = , (1) 2 + m )s2 + (b N 2 + b )s + k im (Jm Nm r m m r r where η, kτ , Nm , and im are the ball screw efficiency, the torque constant of a motor, the speed reduction ratio of the motor to the ball screw, and the current input for the motor, respectively. The equations follow the nomenclature in Fig. 3(a). We can find η, kτ , and Nm in data sheets, which are 0.9, 0.0448 N · m/A, and 3316 respectively. The gear ratio of the drivetrain is computed by dividing the speed reduction of pulleys (2.111) with lead length of the ball screw (0.004m) using the equation 2π × 2.111/0.004. However, we need to experimentally identify kr , br , Jm , and bm . We infer kr by dividing the force measurement from the load cell by the elastomer deflection. The other parameters are estimated by comparing the frequency response of the model and experimental data. The frequency response test is done with the ankle actuator while prohibiting joint movement with a load and an offset force command. The results are presented in Fig. 3 with solid gray lines. Note that the dotted gray lines are the estimated response from the transfer function (measured elastomer force/ input motor force) using the parameters of Table. II. The estimated response and experimental result match closely with one another, implying that the parameters we found are close to the actual values. We also study the frequency response for different load masses to understand how the dynamics changes as the joint moves. When 10kg is attached to the end of link, the reflected 6 (a) VCLA -149 o 27.5 Hz Fig. 3. Frequency response of VLCA. Gray solid lines are experimental data and the other lines are estimated response with the model using empirically parameters. (b) Open-loop PDm PIDm PDf PDm + DOB 60 Magnitude (dB) Experiment result 40 20 0 -20 -40 0 Phase margin -45 47.6 -90 41.0 1) Proportional (P) + Derivative (Df ) using velocity signal obtained by a low-pass derivative filtered elastomer deflection 2) Proportional (P) + Derivative (Dm ) using motor velocity signal measured by a quadrature encoder connected to a motor axis The second controller (PDm ) has benefits over the first one (PDf ) with respect to sensor signal quality. The velocity of motor is directly measured by a quadrature encoder rather than low-pass filtered elastomer deflection data, which is relatively noisy and lagged. In addition, Fig. 4 shows that the phase margin of the second controller (47.6) is larger than the first one (17.1). To remove the force tracking error at low frequencies, we consider two options: augmenting the controller either with integral control or with a DOB on the PDm controller. To compare the two controllers, we analyzed the phase margins of all the mentioned controllers. First, we chose to focus on the location where the sensor data returns in order to address the time delay of digital controllers (Fig. 4 (a) and (c)). Next, we have to compute the open-loop transfer function for each closed loop system. For example, the PDf controller’s closed loop transfer function is kr Px kp (Fr − e−T s Fk ) + Fr − kd,f Qd e−T s Fk , N (2) where Fk , Fr , T , and Qd are the measured force from a elastomer deflection, a reference force, a time delay, a low pass derivative filter, respectively.For convenience, we use N instead of the multiplication of three terms, ηkτ Nm . When 34.0 -225 10 (c) 42.8 17.7 -135 -180 mass to the actuator varies from 1500kg to 2500kg because the length of the effective moment arm changes depending on joint position. In Fig. 3(b), the bode plots are presented and the response is not significantly different than the fixed output case. Therefore, we design and analyze the feedback controller based on the fixed output dynamics. For the force feedback controller, we first compare two options, which we have used in our previous studies [11], [12]: Fk = kr -3dB Phase (deg) Inf. mass 2500 kg 2000 kg 1500 kg 0 10 1 10 2 Frequency (Hz) VCLA kr s Fig. 4. Stability analysis of controllers. Phase margins of each controllers and open-loop system are presented. gathering the term with e−T s of Eq. (2), we obtain Fk kr Px (Kp + 1)/N = . Fr 1 + e−T s kr Px (Kp + Kd,f Qd )/N (3) Then, the open-loop transfer function of the closed system with the time delay is open PPD = kr Px (Kp + Kd,f Qd )/N. f (4) We can apply the same method for the PIDm and PDm +DOB controllers. The transfer function of PIDm , which is presented in Fig. 4(c), is 1 kr Px (Fr − e−T s Fk )(Kp + Ki ) + Fr Fk = N s (5) Fk −T s − Kd,m e sNm . kr Then it becomes Fk kr Px (Kp + Ki /s + 1)/N = . Fr 1 + e−T s Px (kr (Kp + Ki /s) + Kd sNm )/N (6) When we apply a DOB instead of integral control, we need the inverse of the plant. In our case, the plant of the DOB is PDm , which is similar to Eq. (6) except that Ki and e−T s are omitted: kr Px (Kp + 1) PPDm (= Pc ) = . (7) N + Px (kr Kp + Kd,m sNm ) 7 Fig. 5. Robotic testbed. Our testbed consists of two VLCAs at the ankle and the knee. The foot of the testbed is fixed on the ground. The linkages are designed to vary the maximum peak torques and velocities depending on postures. As the joint positions change, the ratios between ball screw velocities (L̇0,1 ) and joint velocities (q̇0,1 ) also change because of effective lengths of moment arms vary. The linkages are designed to exert more torque when the robot crouches, which is the posture that the gravitational loads on the joints are large. The formulation of PDm including the DOB, which is shown in Fig. 4(c), is Fk = kr Px (Kp + 1)(Fd − e−T s Pc−1 Qτ d Fk ) , (8) (N + e−T s Px (kr Kp + Kd,m sNm )) (1 − Qτ d ) the hip, and the foot is fixed on the ground. With this testbed, we intended to demonstrate coordinated position control with two VLCAs, the viability of liquid cooling on an articulated platform, cartesian position control of a weighted end effector, and verification of a linkage design. The two joints each have a different linkage structure that was carefully designed so that the moment arm accommodates the expected torques and joint velocities as the robot posture changes (Fig. 5). For example, each joint can exert a peak torque of approximately 270 N m and the maximum joint velocity ranges between 7.5 rad/s and 20+ rad/s depending on the mechanical advantage of the linkage along the configurations. The joints can exert a maximum continuous torque of 91 N m at the point of highest mechanical advantage. This posture dependent ratio of torque and velocity is a unique benefit of prismatic actuators. Given cartesian motion trajectories, which are 2nd order Bspline or sinusoidal functions, the centralized controller computes the torque commands with operational space position and velocity, which are updated by the sensed joint position and velocity. The OSC formulation that we use is −1 τ = AJhip (ẍdes + Kp e + Kd ė − J˙hip q̇) + b + g, (11) where A, b, and g represent inertia, coriolis, and gravity joint torque, respectively. ẍdes , e, and ė are desired trajectory acceleration, position and velocity error, respectively. q̇ ∈ R2 is the joint velocity of the robot and τ is the joint torque. Jhip is a jacobian of the hip, which is a 2 × 2 square matrix and assumed to be full-rank. where Qτ d is a second order low-pass filter. Then the transfer function is VII. R ESULTS Fk kr Px (Kp + 1) . = We first conducted various single actuator tests to show Fd N (1 − Qτ d ) + e−T s (N Qτ d + Px (kr Kp + Kd,m sNm )) (9) basic performance such as torque and joint position controllability, continuous and peak torque, and impact resistance. SubThe open-loop transfer function is sequently, we focused on the performance of OSC using the N Qτ d + Px (kr Kp + Kd,m sNm ) open robotic testbed integrated with DOB based torque controllers PPD = (10) m +DOB N (1 − Qτ d ) to demonstrate actuator efficiency and high power motions. open open open open The bode plots of PPD , P , P , and P PDm PIDm PDm +DOB f are presented in Fig. 4(b). The gains (Kp , Kd,m , Ki ) are the A. Single Actuator Tests same as the values that we use in the experiments presented in Fig. 6(a) shows the experimental results of our frequency Section VII-A, which are 4, 15, and 300, respectively. The PDf response testing as well as the estimated response based on controller uses Kd Nm /kr for Kd,f to normalize the derivative the transfer functions. We compare three types of controllers: gain. The cutoff frequency of the DOB is set to 15Hz because PD , PID , and PD + DOB. As we predicted in the m m m this is where the PDm +DOB shows a magnitude trend similar analysis of Section V, the PD + DOB controller shows less m to the integral controller (PIDm ). The results imply that the phase drop and overshoot than PID . The integral control m PDm +DOB controller is more stable than PIDm with respect feedback gain used in the experiment is 300 and the cutoff to phase margin and maximum phase lag. This analysis is also frequency of the DOB’s Q filter is 60Hz, which shows τd experimentally verified in Section VII-A. similar error to the PID controller (Fig. 6(b)). Another test m VI. ROBOTIC T ESTBED We built a robotic testbed shown in Fig. 5. To demonstrate dynamic motion, we implemented an operational space controller (OSC) incorporating the multi-body dynamics of the robot. We designed and built a robotic testbed (Fig. 5) consisting of two VLCAs - one for the ankle (q0 ) and one for the knee (q1 ). The design constrains motion to the sagittal plane, the robot carries 10kg, 23kg, or 32.5kg of weight at presented in Fig. 6(c) also supports the stability and accuracy of torque control. In the test, we command a ramp in joint torque from 1 to 25N m in 0.1s. The sensed torque (blue solid line) almost overlaps the commanded torque (red dashed line). Fig. 6(d) is the result of a joint position control test designed to show that VLCAs have better joint position controllability than SEAs using springs. In the experiment, we use a joint encoder for position control and a motor quadrature encoder for velocity feedback. To compare the VLCAs performance 8 20 10 1.8 2 2.2 2.4 2.6 2.8 -1.8 JPos (rad) Phase (deg) (c) Torque fast response Open (estimated) (estimated) (estimated) -2 -2.4 1.6 1.8 2 2.2 2.4 2.6 70 1 60 0 50 40 -2 -3 2.8 30 0 1 2 3 time (sec) 20 (f) peak force (d) Position fast response Actuator force (N) 80 1000 150 500 100 0 -500 time (sec) 50 actuator force core temperature (w/ liquid) core temperature (w/o liquid) 0 50 100 150 200 250 0 350 300 Motor temperature (oC) Error (dB) (a) Frequency responses of different controllers Frequency (Hz) 90 2 time (sec) Reference 100 -1 (sim.) metal spring (sim.) elastomer (exp.) command (exp.) sensed -2.2 Frequency (Hz) 110 3 0 1.6 Open x 103 4 command sensed Actuator force (N) Magnitude (dB) 5 Motor temperature (oC) Torque (Nm) 30 time (sec) (e) Continous force and core temperature (b) Error magintude and chirp test trajectories Fig. 6. Torque Feedback Control Test. (a) Experimental data and estimated response based on the transfer functions are presented. Estimated response of PD controller is identical to the PD+DOB since DOB theoretically does not change the transfer function. The plot show PD+DOB shows better performance in terms of less overshoot and smaller phase drop near to the natural frequency. (b) We choose integral controller feedback gain that shows similar accuracy of PD+DOB’s. The left is error magnitude of three controllers. PD controller has larger error than the other two controller in the low frequency region. The right is torque trajectories in the time domain. Load cell (solid holding) Load cell (w/ elastomer) Rubber deflection (solid holding) Rubber deflection (w/ elastomer) 1000 Actuator force (N) with that of spring-based SEAs, we present simulation results for a spring-based SEA on the same plot as the experiment result for the VLCA. The green dashed line is the simulated step response of our actuator and the yellow dotted line is the result of the simulation model using the same parameters except the spring stiffness and damping. The spring stiffness was selected to be 11% of the elastomer’s, based on the results of our tests in Section III, and the damping for the spring case was set to 8000 N s/m which only includes the drivetrain friction. The results show a notable improvement in joint position control when using an elastomer instead of a steel spring. Fig. 6(e) shows the continous force and the motor core temperature trend with and without liquid cooling. The observed continous force is 860N and the motor core temperature settles at 115◦C with liquid cooling. Fig. 6(f) is the the result of shortterm torque test. In the experiment, we fix the output of the actuator and command a 31A current for 0.5s. The observed force measured by a loadcell (Fig. 2(c)) is 4500N, which is a little smaller than the theoretically expected value, 5900N. Considering that the estimated core temperature surpassed 107◦C (< 155◦C limit), we expect that the theoretical value is reasonable. Thus, we conclude that the maximum force density of our actuator is larger than 2700N/kg and potentially 3500N/kg. Fig. 7 shows loadcell and elastomer force data from the impact tests. In the tests, we hit the loadcell connected to the ball screw (Fig. 2(c)) with a hammer falling from a constant height while fixing the actuator in two different places to 500 0 95% interval -500 -1000 -2 0 2 4 6 8 10 12 14 time (ms) Fig. 7. Impact test. 83 trials are plotted and estimated with gaussian process. We can see the deflections of the elastomer, which imply that the elastic element absorbes the external impact force. compare the rigid actuator to viscoelastic actuator response. In the rigid scenario, outer case of ballnut, a blue part in Fig. 2, is fixed to exclude the elastomer from the external impact force path. In the second case, we fixed the ground pin of the actuator, which is depicted by a gray part in Fig. 2(l), to see how the elastomers react to the impact. The impact experiment is challenging because the number of data points we can obtain is very small with a 1ms update rate. To overcome the lack of data points, we estimate the mean and variance of 83 trials by gaussian process regression. The results presented in Fig. 7 imply that there is no significant difference in the forces measured by the loadcell in both cases, which is predictable because the elastic element is placed behind the drivetrain. However, the elastomer does Ankle command sensed 20 0 power supply 14 16 18 20 22 24 14 16 18 20 22 24 40 Wm Wb 0.2 Hip position (m) joint electric motor 50 30 0 -0.1 14 16 18 20 22 24 1 Wk Average efficiency 1.5 0.1 Wk / W m Compliant in horizontal direction servo drive -20 Knee Stiff in vertical direction Torque (Nm) 9 1 0.5 0.8 14 16 (a) Impedance control 22 0 0.6 0.4 -20 12 14 16 18 40 12 14 16 18 Hip position (m) 0.1 10 12 10 12 14 16 18 14 16 18 1 time (sec) (b) Operational space impact test 0.9 0.1 0.85 0 0.8 -0.1 2 command 2.5 sensed 0.9 0.75 0.7 0.8 0.65 0.7 0.6 0.6 1.5 time (sec) 1.5 2 2.5 3 3.5 5 (sec) 3 1 0.5 0.3 0.2 0 -0.2 0.5 1 1.5 2 2.5 3 3.5 0 -0.1 0.8 1 1 time (sec) 0.2 1.5 0.5 -0.4 30 10 1 0 24 Wk / W b Ankle 20 -10 10 Knee Torque (Nm) Hitting down 18 time (sec) 2 2.5 -0.05 0 0.05 (c) Fast up and down (1.7 Hz) Fig. 8. Operational Space Impedance Control Test. (a) The robot demonstrates different impedance: stiff in the vertical direction and compliant in the horizontal direction. The high tracking performance of force feedback control results in the overlapped commanded and sensed torques. (b) To show the stability, we hit the weight with a hammer while operating the impedance control. Even under the impact, force control show stable and accurate tracking. (c) The robot demonstrates a 1.7Hz up and down motion while carrying 10kg weight at the hip, and shows a position error of less than 2.5cm. play a significant role in absorbing energy from the impact which is evident from large elastomer deflection in the second case. Thus, the presence of the elastic element mitigates the propagation of an impulse to the link where the actuator grounds. B. Operational Space Impedance Control Fig. 8 shows our OSC experimental tests (Section VI) carrying a 10kg weight. In the first test presented in Fig. 8(a), the commanded behavior is to be compliant in the horizontal direction (x) and to be stiff in the vertical direction (y). When pushing the hip with a sponge in the x direction, the robot smoothly moves back to comply with the push, but it strongly resists the given vertical disturbance to maintain the Fig. 9. Efficiency analysis of the ankle actuator. Efficiencies of mechanical system using electrical power has 3 steps from a power supply to robot joints. The graph shows the ratio of the mechanical power of the ankle joint and the motor power and the ratio of the joint power and power supply’s input power. commanded height. To show the stability of our controller, we also test the response to impacts by hitting the weight with a hammer (Fig. 8(b)). Even when there are sudden disturbances, the torque controllers rapidly respond to maintain good torque tracking performance as shown in Fig. 6(d). Fig. 8(c) shows the tracking performance of our system while following a fast vertical hip trajectory. While traveling 0.3m with 1.7Hz frequency, the hip position errors are bounded by 0.025m. This result demonstrates that our system is capable of stable and accurate OSC, which is challenging because of the bandwidth conflict induced by its cascaded structure. C. Efficiency Analysis Fig. 9 explains the power flow from the power supply to the robot joint. Input current (Ib ) and voltage (Vb ) are measured in the micro-controllers and the product of those two yields the input power from the power supply. θ̇m is measured by the quadrature encoder connected to the motor’s axis (Fig. 2(f)) and τm is computed from kτ im with im measured in the micro-controller. Joint velocity is low-pass derivative filtered joint positions measured at the absolute joint encoders. The torque (τk ) is computed from projecting the load cell data across the linkage’s effective moment arm. In this test, the robot lifts a 23kg load using five different durations to observe efficiency over a range of different speeds and torques. The results are presented in Fig. 9 with the description of three different power measures. The sensed torque data measured by a load cell is noisy; therefore, we compute the average of the drivetrain efficiency for a clearer comparison. The averages are the integrations of efficiency divided by the time durations. Here we only integrate efficiency while the mechanical power is positive, to prevent confounding our results by incorporating the work done by gravity. 7 7.5 8 200 100 0 -100 0 7 joint torque 7.5 mechanical power 8 400 200 0 -200 -400 -600 200 100 0 6.5 7 7.5 Mechanical power (W) Ankle 7 200 -200 6.5 Knee 8 5 4 6.5 Joint torque (Nm) 7.5 8 (a) 2Hz up and down motion Ankle 0 45 40 -1000 35 -2000 -3000 0 0.5 1 1.5 2 2.5 actuator force 3 3.5 4 motor temperature 30 4.5 80 Knee -1000 60 -2000 -3000 40 -4000 0 Power (W) Ankle Knee 6.5 6 1000 Actuator force (N) command joint encoder motor encoder 2.6 2.4 2.2 2 1.8 0.5 1 1.5 2 2.5 3 3.5 4 Motor temperature (oC) Joint position (rad) 10 4.5 total ankle knee 1500 1000 500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time (sec) (b) Heavy weight lift Fig. 10. High power motion experiment. (a) Joint position data from joint encoder and motor encoder are shown. In this experiment, the maximum observed torque of the ankle joint is 250 N m and the maximum observed mechanical power of the knee joint is 310W. (b) The robot lifts by 0.3m a 32.5 kg load during 0.4s. There is still a safety margin with respect to the limits equal to 5900N and 155◦C. The experimental results show that the drivetrain efficiency is approximately 0.89, which means that we lose only a small amount of power in the drivetrain and most of the torque from the motor is delivered to the joint. This high efficiency indicates only minor drivetrain friction, which is beneficial for dynamics-based motion controllers. D. High Power Motion Experiment To demonstrate high power motions such as fast vertical trajectories and heavy payload lifts, we use the motor position control mode, which uses the quadrature encoders attached directly to the motor for feedback. Fig. 10(a) presents the results of a test comprised of 2Hz vertical motion with 0.32 m of travel while carrying a load of 10 kg at the hip. With respect to mechanical power, the knee joint repeatedly exerts 305W, which is close to the predicted constant power (360W). Although the limited range of motion makes it hard to demonstrate continuous mechanical power, these results convincingly support our claim of enhanced continuous power enabled through liquid cooling. Fig. 10(b) presents another test in which the robot lifts a 32.5kg weight. We can see that the robot operates in the safe region (≤ 5900N and ≤ 155◦C) while demonstrating high power motion. 6(e). As we can see, when turning off liquid cooling the temperature rises quickly above safety limits whereas when turning on the cooling we can sustain large payload torques for long periods of time. The use of elastomers versus steel springs has demonstrated a clear improvement on joint position performance as shown in Fig. 6(d). This capability is important to achieve a large range of output joint or Cartesian space impedances. In the future we will explore further reducing the size of our viscoelastic liquid cooled actuators. Maintaining the current compact design structure we can still reduce another significant percentage the bulk of the actuator by exploring new types of bearings, ballnut sizes and piston bearings at the front end of the actuator. We will also explore using different material for the liquid cooling actuator jacket. The current polyoxymethylene material is easily breakable and develops cracks due to the vibrations and impacts of this kind of robotic applications. In the future we will switch to sealed metal chambers for instance. Further in the future we will consider designing our own motor stators and rotors for improved performance. We expect this kind of actuators to make their way into full humanoid robots and high performance exoskeleton devices and we look forward to participate in such interesting future studies. ACKNOWLEDGMENT VIII. C ONCLUDING R EMARKS Overall our main contribution has been on the design and extensive testing of a new viscoelastic liquid cooled actuator for robotics. One of the tests addressed is impedance control in the operational space instead of joint impedance control. It is often the case that humanoid robots require impedance control in the operational space. For instance, controlling the operational space impedance can enable improved locomotion behaviors such as running. Our controllers demonstrate that we can control the impedance in the Cartesian operational space as a potential functionality for future robotic systems. The use of liquid cooling has allowed to sustain high output torque for prolonged times as shown in the experiments of Fig. The authors would like to thank the members of the Human Centered Robotics Laboratory at The University of Texas at Austin for their help and support. This work was supported by the Office of Naval Research, ONR Grant [grant #N000141512507] and NASA Johnson Space Center, NSF/NASA NRI Grant [grant #NNX12AM03G]. R EFERENCES [1] G. A. Pratt and M. M. Williamson, “Series elastic actuators,” in Intelligent Robots and Systems 95. ’Human Robot Interaction and Cooperative Robots’, Proceedings. 1995 IEEE/RSJ International Conference on, 1995, pp. 399–406. [2] B. Henze, M. A. Roa, and C. Ott, “Passivity-based whole-body balancing for torque-controlled humanoid robots in multi-contact scenarios,” The International Journal of Robotics Research, p. 0278364916653815, Jul. 2016. 11 [3] N. Paine, J. S. Mehling, and J. 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1 Surprise Search for Evolutionary Divergence arXiv:1706.02556v1 [cs.NE] 8 Jun 2017 Daniele Gravina, Antonios Liapis, and Georgios N. Yannakakis Abstract—Inspired by the notion of surprise for unconventional discovery we introduce a general search algorithm we name surprise search as a new method of evolutionary divergent search. Surprise search is grounded in the divergent search paradigm and is fabricated within the principles of evolutionary search. The algorithm mimics the self-surprise cognitive process and equips evolutionary search with the ability to seek for solutions that deviate from the algorithm’s expected behaviour. The predictive model of expected solutions is based on historical trails of where the search has been and local information about the search space. Surprise search is tested extensively in a robot maze navigation task: experiments are held in four authored deceptive mazes and in 60 generated mazes and compared against objective-based evolutionary search and novelty search. The key findings of this study reveal that surprise search is advantageous compared to the other two search processes. In particular, it outperforms objective search and it is as efficient as novelty search in all tasks examined. Most importantly, surprise search is faster, on average, and more robust in solving the navigation problem compared to any other algorithm examined. Finally, our analysis reveals that surprise search explores the behavioural space more extensively and yields higher population diversity compared to novelty search. What distinguishes surprise search from other forms of divergent search, such as the search for novelty, is its ability to diverge not from earlier and seen solutions but rather from predicted and unseen points in the domain considered. Index Terms—Surprise search, novelty search, divergent search, deception, fitness-based evolution, maze navigation, NEAT. I. I NTRODUCTION VER the last 50 years, evolutionary computation (EC) has shown vast potential in numerical and behavioural optimization. The most common approach to optimization in artificial evolution is via an objective function, which rewards solutions based on their ‘goodness’ [1], i.e. how close they are to an optimal behaviour (if such a behaviour is known beforehand) or how much they improve a performance metric. The objective function (or fitness function) encapsulates the principle of evolutionary pressure for fitting (adapting) within the environment. Despite the success of such approaches in a multitude of tasks [1], [2], they are challenged in deceptive fitness landscapes [3] where the global optimum is neighboured by low-quality solutions. In such cases, the local search of an objective-based evolutionary algorithm can guide search away from a global optimum and towards local optima. As a general principle, more deceptive problems challenge the design of a corresponding objective function; this paper follows [4] and views deception as the intuitive definition of problem hardness. Many algorithms have been proposed to tackle the problem O All authors are with the Institute of Digital Games, University of Malta, Msida 2080, Malta (e-mail: daniele.gravina@um.edu.mt; antonios.liapis@um.edu.mt; georgios.yannakakis@um.edu.mt) of deception, primarily revolving around diversity preservation [5]–[7], which deters premature convergence while still rewarding proximity to the objective and divergent search [8] which abandons objectives in favour of rewarding diversity in the population. There are, however, problems which lack an easily defined objective — or a gradient to reaching it. For instance, openended evolution studies within artificial life [9] do not have a goal state and instead prioritize e.g. survival [10], [11]. In evolutionary art, music or design, a large body of research in computational creativity and generative systems [12]–[14] focuses on the creative capacity of search rather than on the objectives. In [13], computational creativity is considered on two core properties of a produced solution: value and novelty [13]. Value is the degree to which a solution is of high quality, whereas novelty is the degree to which a solution (or output) is dissimilar to existing examples. While objective-based EC can be seen as a metaphor of searching for value, a divergent EC strategy as novelty search [8], [15] can be considered as a metaphor of searching for novelty. An effective use of both in EC can lead to highly novel and valuable at the same time outcomes [16], thus realizing quality diversity [17]. However, according to [18], novelty and value are not sufficient for the discovery of highly creative and unconventional solutions to problems. While novelty can be considered as a static property, surprise considers the temporal properties of the discovery, an important dimension to assess the creativity of the generated solution [18], [19]. Further studies in general intelligence and decision making [20] support the importance of unexpectedness for problem-solving. Driven by the notion of computational surprise for the purpose of creatively traversing the search space towards unexpected or serendipitous solutions, this paper proposes surprise search for the purposes of divergent evolutionary search. The hypothesis is that searching for unexpected — not merely unseen — solutions is beneficial to EC as it complements our search capacities with highly efficient and robust algorithms beyond the search for objectives or mere novelty. Surprise search is built upon the novelty search [8] paradigm that rewards individuals which differ from other solutions in the same population and a historical archive. Surprise is assumed to arise from a violation of expectations [21]: as such, it is different than novelty which rewards deviation from past and current behaviours. A computational, quantifiable model of surprise must build expectations based on trends in past behaviours, and predict future behaviours from which it must diverge from. In order to create expected behaviours, the algorithm maintains a lineage of where evolutionary search has been. These groups of evolutionary lineages require the right level of locality in the behavioural space — surprise can be inclusive of all behaviours (globally) or merely consider part 2 of all possible behaviours (locally). Any deviation from these stepping stones of search would elicit surprise; alternatively, they can be viewed as serendipitous discovery if the deviation leads to a surprisingly good point in the behavioural space. The findings of this paper suggest that surprise constitutes a powerful drive for computational discovery as it incorporates predictions of an expected behaviour that it attempts to deviate from; these predictions may be based on behavioural relationships in the solution space as well as historical trends derived from the algorithm’s sampling of the domain. This paper builds upon and extends significantly our earlier work which introduced the notion of surprise search [22] and compared its performance against novelty search and objective-based search [23] in the maze navigation testbed of [8]. The current paper extends the preliminary study of [23] by introducing two new maze navigation problems of increased complexity, and analysing the impact of several parameters to the behaviour of surprise search. This paper also includes an extensive comparison between novelty search and surprise search both in the behavioural and the genotypic space. Finally, to further examine how surprise search performs across a wide range of problems, we test how it scales in sixty randomly generated mazes of varying degrees of complexity. The key findings of the paper suggest that surprise search is as efficient as novelty search and both algorithms, unsurprisingly, outperform fitness-based search. Furthermore, surprise search appears to be the most robust algorithm in the four testbed tasks and to be the most successful and fastest algorithm in the 60 randomly generated mazes. While both novelty and surprise search converge to the solution significantly faster than fitnessbased search, surprise search solves the navigation problem faster, on average, and more often than novelty search. The experiments of this paper validate our hypothesis that surprise can be beneficial as a divergent search approach and provide evidence for its supremacy over novelty search in the tasks examined. II. D ECEPTION , D IVERGENT S EARCH AND Q UALITY D IVERSITY This section motivates surprise search by providing a brief overview of the challenges faced by fitness-based approaches when handling deceptive problems and how divergent search has been used to address such challenges. The section concludes with a discussion on the relationship between surprise search and quality diversity algorithms. A. Deception in Evolutionary Computation The term deception in the context of EC was introduced by [3] to describe instances where highly-fit building blocks, when recombined, may guide search away from the global optimum. Since that first mention, the notion of deception (including deceptive problems and deceptive search spaces) has been refined and expanded to describe several problems that challenge evolutionary search for a solution. [4] argues that “the only challenging problems are deceptive”. EC-hardness is often attributed to deception, as well as sampling error [24] and a rugged fitness landscape [25]. In combinatorial optimisation problems, the fitness landscape can affect optimisation when performing local search. Such a search process assumes that there is a high correlation between the fitness of neighbouring points in the search space, and that genes in the chromosome are independent of each other. The latter assumption refers to epistasis [26] which is a factor of GA-hardness: when epistasis is high (i.e. where too many genes are dependent on other genes in the chromosome), the algorithm searches for a unique optimal combination of genes but no substantial fitness improvements are noticed [26]. As noted, epistasis is evaluated from the perspective of the fitness function and thus is susceptible to deception; [27] argue that deceptive functions can not have low epistasis, although fitness functions with high epistasis are not necessarily deceptive. Such approaches are often based on the concepts of correlation, i.e. the degree to which an individual’s fitness score is well correlated to its neighbours’ in the search space, or epistasis, i.e. the degree of interaction among genes’ effects. As noted by [8], most of the factors of EC-hardness originate from the fitness function itself; however, poorly designed genetic operators and poorly chosen evolutionary parameters can exacerbate the problem. B. Divergent Search By definition, the biggest issue of a deceptive problem is premature convergence to local optima, while the global optimum is difficult to reach as the deceptive fitness landscape lead the search away from it. Several approaches have been proposed to counter this behaviour, surveyed by [15]. For instance, speciation [28] and niching [29] are popular diversity maintenance techniques, which enforce local competition among similar solutions. Similarity can be measured on the genotypical level [5], on the fitness scores [6], or on the age of the individuals [7]. Multiple promising directions are favoured by this localised competition, making premature convergence less likely. Alternatives such as coevolution [30] can lead to an arms race that finds a better gradient for search, but can suffer when individuals’ performance is poor or one individual vastly outperforms others [31]. Finally, techniques from multiobjective optimisation can, at least in theory, explore the search space more effectively by evaluating individuals in more than one measures of quality [32] and thus avoid local optima in one fitness function by attempting to improve other objectives; however, multi-objective optimisation can not guarantee to circumvent deception [33]. Divergent search methods differ from previous approaches as they explicitly ignore the objective of the problem they are trying to solve. While the approaches described above provide control mechanisms, modifiers or alternate objectives which complement the gradient search towards a better solution, divergent algorithms such as novelty search [8] motivates exploration of the search space by rewarding individuals that are phenotypically (or behaviourally) different without considering whether they are objectively “better” than others. Novelty search is neither random walk nor exhaustive search, however, as it gives higher rewards to solutions that are different from others in the current and past populations 3 by maintaining a memory of the areas of the search space it has explored via a novelty archive. The archive contains previously found novel individuals, and the highest-scoring individuals in terms of novelty are continuously added to it as evolution carries on. The distance measure which assesses “difference” is based on a behaviour characterization, which is problem-dependent: for a maze navigation task, distance may be calculated on the agents’ final positions or directions [8], [17], for robot locomotion it may be on the position of a robot’s centre of mass [8], for evolutionary art it may be on properties of the images such as brightness and symmetry [34]. C. Quality Diversity A recent trend in evolutionary computation is inspired by a species’ tendency to face a strong competition for survival within its own niche [17]. EC algorithms of this type seek for the discovery of both quality and diversity at the same time, following the traditional approach within computational creativity of seeking outcomes characterized by both quality (or value) and novelty [13]. Such evolutionary algorithms have been named quality diversity algorithms [17] and aim to find a maximally diverse population of highly performing individuals. Examples of such algorithms include novelty search with local competition [35] and MAP-Elites [36] as well as algorithms that constrain the feasible space of solutions — thereby forcing high quality solutions — while searching for divergence such as constrained novelty search [16], [37]. Surprise search can complement the search for diversity and replace other divergent search algorithms commonly used for quality diversity. It can, for instance, be used in combination with local competition instead of novelty search as in [35]. Towards that goal, a recent study employed surprise search for game weapon generation in a constrained fashion [38]. The constrained surprise search algorithm rewarded the generation of surprising weapons — thereby maintaining diversity — with guaranteed high quality imposed by constraints on weapon balance and effectiveness. III. T HE N OTION OF S URPRISE S EARCH This section examines the notion of surprise as a driver of evolutionary search. To this end, we first describe the concept of surprise, we then highlight the differences between surprise and novelty and, finally, we frame surprise search in the context of divergent search. A. What is Surprise? The study of surprise has been central in neuroscience, psychology, cognitive science, and to a lesser degree in computational creativity and computational search. In psychology and emotive modelling studies, surprise defines one of Ekman’s six basic emotions [39]. Within cognitive science, surprise has been defined as a temporal-based cognitive process of the unexpected [21], [40], a violation of a belief [41], a reaction to a mismatch [21], or a response to novelty [14]. In computational creativity, surprise has been attributed to the creative output of a computational process [14], [18]. While variant types and taxonomies of surprise have been suggested in the literature — such as aggressive versus passive surprise [18] — we can safely derive a definition of surprise that is general across all disciplines that study surprise as a phenomenon. For the purposes of this paper we define surprise as the deviation from the expected and we use the notions surprise and unexpectedness interchangeably due to their highly interwoven nature. B. Novelty vs. Surprise Novelty and surprise are different notions by definition as it is possible for a solution to be both novel and/or expected to variant degrees. Following the core principles of Lehman and Stanley [8] and Grace et al. [18], novelty is defined as the degree to which a solution is different from prior solutions to a particular problem. On the other hand, surprise is the degree to which a solution is different from the expected solution to a particular problem. Expectations are naturally based on inference from past experiences; analogously surprise is built on the temporal model of past behaviours. To exemplify the difference between the notions of novelty and surprise, [22] uses a card memory game where cards are revealed, one at a time, to the player who has to predict which card will be revealed next. The novelty of game outcome (i.e. next card) is the highest if all past revealed cards are different. The surprise value of the game outcome in that case is low as the player has grown to expect a new, unseen, card every time. On the other hand, if seen cards are revealed after a while then the novelty of the game outcome decreases, but surprise increases as the game deviates from the expected behaviour which calls for a new card every time. Surprise is a temporal notion as expectations are by nature temporal. Prior information is required to predict what is expected; hence a prediction of the expected [19] is a necessary component for modelling surprise computationally. By that logic, surprise can be viewed as a temporal novelty process or as novelty on the prediction (rather than the behavioural) space. Surprise search maintains a prediction (a gradient) of where novelty has been on the prediction space, which is derived from the behavioural space. In that sense surprise resembles a time derivative of novelty. C. Novelty Search vs. Surprise Search According to Grace et al. [18], novelty and value (i.e. objective in the context of EC) are not sufficient for the discovery of unconventional solutions to problems (or creative outputs) as novelty does not cater for the temporal aspects of discovery. Novelty search rewards divergence from current behaviours (i.e. other individuals of the population) and prior behaviours (i.e. an archive of previously novel solutions) [8]; in this way it provides the necessary stepping stones toward achieving an objective (i.e. value). Surprise, on the other hand, complements the search for novelty as it rewards divergence from the expected behaviour. In other words while novelty search attempts to deviate from previously seen solutions, surprise search attempts to deviate from solutions that are expected to be seen in the future. 4 As discussed above, surprise search must reward an individual’s deviation from the expected behaviour. This goal can be decomposed into two tasks: prediction and deviation. At the highest descriptive level, surprise search uses local information from past generations to predict behaviour(s) of the population in the current generation; observing the behaviours of each individual in the actual population, it rewards individuals that deviate from predicted behaviour(s): this is summarized in Fig. 1. The following sections will describe the models of prediction and deviation, and the parameters which can affect their behaviour. Fig. 1: High-level overview of the surprise search algorithm when evaluating an individual i in a population at generation t. The h previous generations are considered, with respect to k behavioural characteristics per generation, to predict the expected k behaviours of generation t. The surprise score of individual i is the deviation of the behaviour of i from a subset of these k expected behaviours. Highly relevant to this study is the work on computational models of surprise for artificial agents [42], which however does not consider using such a model of surprise for search. Other aspects of unexpectedness such as intrinsic motivation [43] and artificial curiosity [44] have also been modelled. The concepts of novelty within reinforcement learning research are also interlinked to the idea of surprise search [43], [45]. Artificial curiosity and intrinsic motivation differ from surprise search as the latter is based on evolutionary divergent search and motivated by open-ended evolution, similarly to novelty search. Specifically, surprise search does not keep a persistent world model as [44] does; instead it focuses on the current trajectory of search using the latest points of the search space it has explored (and ordering them temporally). Additionally, it rewards deviations from expected behaviours agnostically rather than based on how those deviations improve a world model. This allows surprise search to backtrack and re-visit areas of the search space it has already visited, which is discouraged in both novelty search and curiosity. Inspired by the above arguments and findings in computational creativity, we view surprise for computational search as the degree to which expectations about a solution are violated through observation [18]. Our hypothesis is that if modelled appropriately, surprise may enhance divergent search and complement or even surpass the performance of traditional forms of divergent search such as novelty. The main findings of this paper validate our hypothesis. IV. T HE S URPRISE S EARCH A LGORITHM This section discusses the principles of designing a surprise search algorithm for any task or search space. To realise surprise as a search mechanism, an individual should be rewarded when it deviates from the expected behaviour, i.e. the evaluation of a population in evolutionary search is adapted. This means that surprise search can be applied to any EC method, such as NEAT [28] in the case study of this paper. A. Model of Prediction As shown in Fig. 1, the predictive model uses local information from previous generations to estimate (in a quantitative way) the expected behaviour(s) in the current population. Formally, predicted behaviours (p) are created via eq. (1), where m is the predictive model that uses a degree of local (or global) behavioural information (expressed by k) from h previous generations. Each parameter (m, h, k) may influence the scope and impact of predictions, and are problem-specific both from a theoretical (as they can affect performance of surprise search) and a practical (as certain domains may limit the possible choice of parameters) perspective. p = m(h, k) (1) How much history of prior behaviours (h) should surprise search consider?: In order to predict behaviours in the current population, the predictive model must consider previous generations. In order to estimate behaviours of a population at generation t, the predictive model must find trends in the populations of generations t − 1, t − 2, · · · , t − h. The minimum number of generations to consider in order to observe an evolutionary trend, therefore, is h = 2 (the two prior generations to the one being evaluated). However, behaviours that have performed well in the past could also be included in a surprise archive, similar to the novelty archive of novelty search [8], and subsequently used to make predictions of interesting future behaviours. Such a surprise archive would serve as a more persistent history (h > 2) but considering only the interesting historical behaviours rather than all past behaviours. How local (k) are the behaviours surprise search needs to consider to make a prediction?: Surprise search can consider behavioural trends of the entire population when creating a prediction (global information). In that case, k = 1 and all behaviours are aggregated into a meaningful average metric for each prior generation. The current generation’s expected behaviours are similarly expressed as a single (average) metric; deviation of individuals in the actual population is derived from that single metric. At the other extreme, surprise search can consider each individual in the population and derive an estimated behaviour based on the behaviours of its ancestors in the genotypic sense (parents, grandparents etc.) or behavioural sense (past individuals with the closest behaviour). In this case k = P where P is the size of the population, and the number of predictions to deviate from will similarly be P . Therefore, 5 the parameter k determines the level of prediction locality which can vary from 1 to P ; intermediate values of k split prior populations into a number of population groups using problem-specific criteria and clustering methods. What predictive model (m) should surprise search use?: Any predictive modelling approach can be used to predict a future behaviour, such as a simple linear regression of a number of points in the behavioural space, non-linear extrapolations, or machine learned models. Again, we consider the predictive model, m, to be problem-dependent and contingent on the h and k parameters. B. Model of Deviation To put pressure on unexpected behaviours, we need an estimate of the deviation of an observed behaviour from the expected behaviour (if k = 1) or behaviours. Following the principles of novelty search [8], this estimate is derived from the behaviour space as the average distance to the n-nearest expected behaviours (prediction points). The surprise score s for an individual i in the population is calculated as: n s(i) = 1X ds (i, pi,j ) n j=0 (2) where ds is the domain-dependent measure of behavioural difference between an individual and its expected behaviour, pi,j is the j-closest prediction point (expected behaviour) to individual i and n is the number of prediction points considered; n is a problem-dependent parameter determined empirically (n≤k). C. Important notes The evolutionary dynamics of surprise search are similar to those achieved via novelty search. A temporal window of where the search has been is maintained by looking at the prior behaviours (expressed by h and k), which are used to make a prediction of the expected behaviours. However, surprise search works on a different search space (the prediction space) which is orthogonal to the behavioural space used by novelty, as it deviates from the expected and not from the actual behaviours. Therefore, a new form of divergent search emerges, which looks at previous behaviours only in an implicit way in order to derive the prediction space. A concern could be if surprise search is merely a version of random walk, especially considering that it deviates from predictions which can differ from actual behaviours. Several comparative experiments in section VII show that surprise search is different and far more effective compared to various random benchmark algorithms. The source code of the surprise search algorithm is publicly available here1 . V. M AZE NAVIGATION T EST B ED Inspired by the work of Lehman and Stanley for testing novelty search [8], we use a maze navigation problem as a testbed for surprise search, as it particularly suits the definition 1 http://www.autogamedesign.eu/mazesurprisesearch (a) Neural Network (b) Sensors Fig. 2: Robot controller for the maze navigation task. Fig. 2a shows the network’s inputs and outputs. Fig. 2b shows the layout of the sensors: the six black arrows are rangefinder sensors, and the four blue pie-slice sensors act as a compass towards the goal. of deceptive problem. The maze navigation task is made of a closed two-dimensional maze, which contains a start position and a goal position: the navigation task has a deceptive landscape—which is directly visible to a human observer— due to the several local optima present in the search space of the problem. Cul-de-sacs added in the shortest path to the goal makes the problem more deceptive, as EC must visiting positions with lower fitness scores before reaching the goal, making the problem harder and more deceptive. In this case, navigation is performed by virtual robot controllers with sensors and mechanisms for controlling their direction and speed: the mapping between the two is provided is via an artificial neural network (ANN) evolved via neuroevolution of augmenting topologies [28]. Starting from the mazes introduced in [8], we designed two additional mazes of enhanced complexity and deceptiveness. This section briefly describes the maze navigation problem, the four mazes adopted, and the parameters for the experiment of Section VII. A. The Maze Navigation Task The maze navigation task consists of finding the path from a starting point to a goal in a two-dimensional maze, in a fixed number of simulation steps. The problem becomes harder when mazes include dead-ends and the goal is far away from the starting point. As in [8], the robot has six range sensors to measure its distance from the closest obstacle, plus four range radars that fire if the goal is in their arc (see Fig. 2b). Therefore, the robot’s ANN receives 10 inputs from the sensors and it controls two actuators, i.e. whether to turn or change the speed (see Fig. 2a). Evolving a controller able to successfully navigate a maze is a challenging problem, as EC needs to evolve a complex mapping between the input (sensors) and the output (movement) in an unknown environment. Even if it can be considered a toy problem, it is an interesting testbed as it stands for a general deceptive search space. Two properties have made this environment a canonical test for divergent search (e.g. [8], [17]): the ease of manually designing deceptive mazes and the low computational burden, which enables researchers to run multiple comparative tests among 6 (a) Medium (b) Hard (c) Very hard (d) Extremely hard Fig. 3: The maze testbeds that appear in [8] (Fig. 3a and 3b) and new mazes introduced in [46] and this paper (Fig. 3c and 3d respectively). The filled circle is the robot’s starting position and the empty circle is the goal. The maze size is 300×150 units for medium and 200×200 for the other mazes. algorithms. Furthermore, the generality of the findings can be tested with automatically generated mazes, as in Section VIII. B. Mazes This paper tests the performance of surprise search on four mazes (see Fig. 3), two of which (medium and hard) have been used in [8]. The medium maze (see Fig. 3a) is somewhat challenging as an algorithm should evolve a robot that avoids dead-ends placed alongside the path to the goal. The hard maze (see Fig. 3b) is more deceptive, due to the dead-end at the leftmost part of the maze; an algorithm must search in less promising (less fit) areas of the maze to find the global optimum. For these two mazes we follow the experimental parameters set in [8] and consider a robot successful if it manages to reach the goal within a radius of five units at the end of an evaluation of 400 simulation steps. Beyond the two mazes of [8], two additional mazes (very hard and extremely hard) were designed to test an algorithm’s performance in even more deceptive environments. The very hard maze (see Fig. 3c) is a modification of the hard maze introduced in [46] with more dead ends and winding passages. The extremely hard maze is a new maze (see Fig. 3d) that features a longer and more complex path from start to goal, thereby increasing the deceptive nature of the problem. If we define a maze’s complexity as the shortest path between the start and the goal, complexity increases substantially from medium (240 units), to hard (360 units), to very hard (442 units) and finally to the extremely hard maze (552 units); note that the last three mazes are of equal size. The high problem complexity of the very hard and the extremely hard mazes led us to empirically increase the number of simulation steps for the evaluation of a robot to 500 and 1000 simulation steps, respectively. By increasing the simulation time in the more deceptive mazes we manage to achieve reasonable performances for at least one algorithm examined which allows for a better analysis and comparison. VI. A LGORITHM PARAMETERS FOR M AZE NAVIGATION This section provides details about the general and specific parameters for all the algorithms compared. We primarily test the performance of three algorithms: objective search, novelty search and surprise search, and include three baseline algorithms for comparative purposes. All algorithms use NEAT to evolve a robot controller with the same parameters as in [8], where the maze navigation task and the mazes of Fig. 3 were introduced. Evolution is carried on a population of 250 individuals for a maximum of 300 generations in the medium and hard maze for a fair comparison to results obtained in [8]. However, the number of generations is increased to 1000 for the more deceptive mazes (very hard and extremely hard) to allow us to analyse the algorithms’ behaviour over a longer evolutionary period. The NEAT algorithm uses speciation and recombination, as described in [28]. The specific parameters of all compared algorithms are detailed below. A. Objective search Objective search uses the agent’s proximity to the goal as a measure of its fitness. Following [8], proximity is measured as the Euclidean distance between the goal and the position of the robot at the end of the simulation. This distance does not account for the maze’s topology and walls, and can be deceptive in the presence of dead-ends. B. Novelty Search Novelty search uses the same novelty metric and parameter values as presented in [8]. In particular, the novelty metric is the average distance of the robot from the nearest neighbouring robots among those in the current population and in a novelty archive. Distance in this case is the Euclidean distance between two robot positions at the end of the simulation; this rewards robots ending in positions that no other robot has explored yet. The parameter for the novelty archive (e.g. the initial novelty threshold for inserting individuals to the archive is 6) is as given in [8]. Sensitivity Analysis: While in [8] novelty is calculated as the average distance from the 15 nearest neighbours, the introduction of new mazes in this paper mandates that the n parameter of novelty search is tested empirically. For that purpose we vary n from 5 to 30 in increments of 5 across all mazes and select the n values that yield the highest number of maze solutions (successes) in 50 independent runs of 300 generations for the medium and hard maze, and 1000 generations for the other mazes. If there is more than one n value that yields the highest number of successes then the lowest average evaluations to solve the maze is taken into account as a selection criterion. Figure 5a shows the results obtained by this analysis across all mazes. The best results are indeed obtained with 15 nearest neighbours for the medium and hard maze, as in [8] (49 and 48 successes, respectively). In the very hard maze there is no difference between 10 and 15 in terms of successes (39) but n = 15 yields less evaluations, while in the extremely hard maze n = 10 yields less evaluations and more successes (24) than any other value tested. In summary, reported results in Section VII use n = 15 for the medium, hard and very hard maze, and n = 10 for the extremely hard maze. C. Surprise search Surprise search uses the surprise metric of eq. (2) to reward unexpected behaviours. As with the other algorithms compared, behaviour in the maze navigation domain is expressed Normalized Evaluations 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 5 10 15 20 25 30 n (a) Generation t − 2 (b) Generation t − 1 Normalized Evaluations 7 0.8 0.6 0.4 0.2 20 40 60 80 100 120 140 160 180 200 220 240 1.0 0.8 0.6 0.4 0.2 20 40 60 80 100 120 140 160 180 200 220 240 k (c) Generation t Fig. 4: The key phases of the surprise search algorithm as applied to the maze navigation domain. Surprise search uses a history of two generations (h = 2) and 10 behavioural clusters (k = 10) in this example. Cluster centroids and prediction points are depicted as empty red (light gray in grayscale) and solid blue (dark gray in grayscale) circles, respectively. as the position of the robot at the end of a simulation. The behavioural difference ds in eq. (2) is the Euclidean distance between the robots’ final position and a considered prediction point, p. Following the general formulation of surprise in Section IV-A, the prediction points are a function of a model m that considers k local behaviours of h prior generations. In this comparative study we use the simplest possible prediction model (m) which is a one-step linear regression of two points (h = 2) in the behavioural space. Thus, only the two previous generations are considered when creating prediction points to deviate from in the current generation (see Fig. 4). In the first two generations the algorithm performs mere random search due to a lack of prediction points. The locality (k) of behaviours is determined by the number of behavioural clusters in the population that is obtained by running k-means on the final robot positions. The surprise search algorithm applies k-means clustering at each generation by seeding the initial configuration of the k centroids with the centroids obtained in the previous generation; this seeding process is omitted only in the first generation due to the lack of earlier clusters. This way the algorithm is able to pair centroids in subsequent generations and track their behavioural history. Using the k pairs of centroids of the last two generations, the algorithm creates k prediction points for the current generation through a simple linear projection. Surprise search rewards the behaviour that obtains a surprising outcome given the temporal sequence of the final points of the robot across generations. Surprise search is thus orthogonal to objective and novelty search as it rewards robots that visit areas outside the predicted space(s), without any explicit knowledge of the final goal. It should be noted that for high values of k the k-means algorithm might end up not assigning any data point to a particular cluster; the chance of this happening increases with k and the sparseness of data (in particular in datasets containing outliers) [47]. Further, the seeding initialization procedure we follow for k-means in this domain aims to behaviourally connect centroids across generations so as to enable us to predict and deviate from the expected behaviour in the next generation. The adopted initialization procedure (i.e. inheriting from centroids of the previous generation) does not guarantee (a) Novelty search parameters (b) Surprise search parameters Fig. 5: Sensitivity Analysis: selecting n for novelty search (Fig. 5b), k and n for surprise search (Fig. 5a). Figures depict the average number of evaluations (normalized by the total number of evaluations) obtained out of 50 runs . Error bars represent the 95% confidence interval. that all seeded centroids will be allocated a robot position during the assignment step of k-means, as robot positions (data points) might change drastically from one generation to the next. In the case of surprise search, when an empty cluster appears in the current generation (i.e. in positions where a cluster existed in a past generation but not currently), then its prediction is not updated (i.e. moved) until a final robot position gets close to the empty cluster’s centroid. Predicted centroids that have not been recently updated (due to empty clusters) are still considered when calculating the surprise score, and indirectly act as an archive of earlier predictions. However, this archive is not persistent as the number of ‘archived’ prediction points can increase or decrease during the course of evolution, and depends on k. Sensitivity Analysis: To choose appropriate parameters for k (information locality) and n (number of prediction points) in the prediction and deviation models respectively, a sensitivity analysis is conducted for all mazes. We obtain k empirically by varying its value between 20 and P in increments of 20 for each maze. We also test all k for n = 1 and n = 2 in this paper. As in the sensitivity analysis for novelty search we select the k and n values that yield the highest number of successes in 50 independent runs. If there is more than one k, n combination that yields the highest number of successes we select the combination that solves the maze in the fewest average evaluations. Figure 5b shows the average number of evaluations for all k values tested, for n = 1 and n = 2. It is clear that higher k values result to less evaluations on average. Moreover, it seems that n = 2 leads to better performance in the two more deceptive mazes. Based on the above selection criteria, we pick k = 200 and n = 1 for the medium maze, which gives the highest number of successes (50) and the lowest number of evaluations (16, 364 evaluations on average). For the hard maze we select k = 100 and n = 1, as it is the most robust (49 success) and fastest (23, 214 evaluations on average) among tested values. Following the same procedure k = 200 and n = 2 in the very hard maze, and k = 220 and n = 2 in the extremely hard maze (see Fig. 5b). 300 280 260 240 220 200 1800 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 3 Average Maximum Fitness In this paper we started our investigations with the simplest possible prediction model (m), which is a linear regression, and the shortest possible time window of two generations for the history parameter (h). The impact of the history parameter and the prediction model on the algorithm’s performance is not examined empirically in this paper and remains open to future investigations. We get back to this discussion in Section IX. Average Maximum Fitness 8 The robot maze navigation problem is used to compare the performance of surprise, novelty and objective search. To test the algorithms’ performance, we follow the approach proposed in [48] and compare their efficiency and robustness in all four test bed mazes. We finally analyse some typical examples on both the behavioural and the genotypical space of the generated solutions. All results reported are obtained from 100 independent evolutionary runs; reported significance and corresponding p values are obtained via two-tailed MannWhitney U-test, with a significance level of 5%. A. Efficiency Efficiency is defined as the maximum fitness over time, where fitness is calculated as 300 − d(i); d(i) the Euclidean distance between the final position of robot i and the goal, as in [8]. Figure 6 shows the average maximum fitness across evaluations for each approach for the four mazes. In the medium maze, we can observe that both surprise and novelty search converge after approximately 35,000 evaluations. Even if novelty seems to yield a higher average maximum fitness values than surprise search, the difference is insignificant. Novelty search, on average, obtains a final maximum fitness of 295.84 (σ = 1.47), while surprise search obtains a fitness of 295.93 (σ = 0.72); p > 0.05. By looking at the 95% confidence intervals, it seems that novelty search yields higher average maximum fitness between 240 220 200 1800 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 3 Average Maximum Fitness 300 280 260 240 220 200 1800 25 50 75 100 125 150 175 200 225 250 Evaluations (x103 ) (c) Very hard maze (b) Hard maze Average Maximum Fitness (a) Medium maze D. Other baseline algorithms VII. S URPRISE S EARCH IN AUTHORED D ECEPTIVE M AZES 260 Evaluations (x10 ) Evaluations (x10 ) Three more baseline algorithms are included for comparative purposes. Random search is a baseline proposed in [8] that uses a uniformly-distributed random value as the fitness function of an individual. The other two baselines are variants of surprise search that test the impact of the predictive model. Surprise search (random), SSr , selects k random prediction points (pi,j in eq. 2) within the maze following a uniform distribution, and tests how surprise search would perform with a highly inaccurate predictive model. Surprise search (no prediction), SSnp , uses the current generation’s actual clusters as its prediction points (pi,j in eq. 2), thereby, omitting the prediction phase of the surprise search algorithm. SSnp uses real data (cluster centroids) from the current generation rather than predicted data regarding the current generation, and tests how the algorithm performs divergent search from real data. Note that SSnp is reminiscent of novelty search, except that it uses deviation from cluster centroids (not points) and does not use a novelty archive. The same parameter values (k and n) are used for these variants of surprise search. 300 280 300 280 260 240 220 200 1800 25 50 75 100 125 150 175 200 225 250 Evaluations (x103 ) (d) Extremely hard maze Fig. 6: Efficiency (average maximum fitness) comparison for the four mazes in Fig. 3. The graphs depict the evolution of fitness over the number of evaluations. Values are averaged across 100 runs of each algorithm and the error bars represent the 95% confidence interval of the average. 7500 and 25,000 evaluations. This difference is due to the predictions that surprise search tries to deviate from. Early during evolution, two consecutive generations may have robots far from each other, lead to distant and erratic predictions. Eventually, we have a convergence and the predictions become more consistent, allowing surprise search to solve the maze. Both objective search and SSnp seem fairly efficient to solve the maze; however, they are not able to find the goal in all the runs. The random baselines, instead, perform poorly and show very little improvement as evolution progresses. The baselines’ performance proves that surprise search is different from a random walk and that the prediction model positively affects the performance of the algorithm. In a more deceptive test, the hard maze, we can see from Fig. 6b that novelty and surprise perform much better than all other algorithms; differences in efficiency between novelty and surprise search are not significant. Surprise search and novelty search find the goal in 99 and 93 out of 100 runs respectively, SSnp finds the solution in 61 runs, while for the rest of the baselines the success rate is far below. It is interesting to note that objective search reaches a high fitness score around 260 at the very beginning of the evolutionary process, and then it stops to improve. This is due to the dead-end at the upper right corner of the maze (Fig. 3b), which prevents the algorithm from discovering the global optimum. In order to discover the global optimum, in fact, the algorithm needs to explore the least fit areas of the search space, such as the bottom-right corner of the maze. In the very hard maze, objective search never finds the 100 80 80 Successes 100 60 40 20 60 40 20 00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 3 00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 3 Evaluations (x10 ) Evaluations (x10 ) (a) Medium maze (b) Hard maze 100 100 80 80 Successes Successes solution in 100 runs; Fig. 6c shows that this algorithm is not able to reach the goal because, as in the hard maze, it reaches the left-most dead end and is unable to bypass that local optimum. Unsurprisingly, the random and SSR baselines also perform poorly. On the other hand, novelty search finds the solution in 85 out of 100 runs, while surprise search finds a solution in 99 runs. Interestingly, SSnp finds 88 solutions out of 100 runs, similar to novelty search. This can be explained by looking at how SSnp is implemented. Its behaviour is quite similar to novelty search as it merely uses the local behaviours of the current generation; the key difference is that these behaviours are clustered in SSnp . In such a complex maze surprise search seems to handle maze deceptiveness in a better way; it obtains a final maximum fitness of 295.77 (σ = 3.59) which is higher (but not significantly) than that of novelty search (292.15; σ = 9.85); p > 0.05. From the confidence intervals of Fig. 7c, it appears that surprise search is performing better or significantly better than novelty search after 50, 000 evaluations until the end of the run. In the most deceptive (extremely hard) maze, objective search, random and SSr do not find any solution in 100 runs, performing poorly in terms of efficiency. This is not surprising as all of these algorithms perform consistently poorly in all but the simplest mazes. Novelty search and surprise search find the solution 48 and 67 times, respectively, while the SSnp obtains 39 solutions. As can be seen in Fig. 6d, surprise search yields a higher maximum fitness after 150,000 evaluations, with a final maximum fitness of 287.68 (σ = 12.16). Another way of estimating efficiency is the effort it takes an algorithm to find a solution. In this case, surprise clearly manages to be more advantageous. In the medium maze surprise search manages to find the goal, on average, in 16, 084 evaluations (σ = 11, 588) which is faster than novelty (19, 814; σ = 15, 441) and significantly faster than objective search (48, 186; σ = 23, 590) and SSnp (26, 452; σ = 21, 249). We observe the same comparative advantage in the hard maze as surprise search solves the problem in 23, 566 evaluations on average (σ = 15, 925) whereas novelty search, SSnp , and objective search solve it in 28, 493 (σ = 19, 939), 47, 550 (σ = 25, 524) and 73, 643 (σ = 7, 542) evaluations, respectively. Most importantly surprise search is significantly faster (p < 0.05) than novelty search in the more deceptive problems: on average surprise search finds the solution in 76, 261 evaluations (σ = 52, 385) in the very hard maze and in 154, 794 evaluations (σ = 84, 733) in the extremely hard maze, whereas novelty search requires 115, 600 evaluations (σ = 81, 091) and 178, 045 evaluations (σ = 86, 410), respectively. Furthermore surprise search is significantly faster (p < 0.01) than SSnp , which requires 117, 560 (σ = 74, 430) and 200, 190 (σ = 75, 569) evaluations in the very hard and the extremely hard maze, respectively. The findings from the above experiments indicate that, in terms of maximum fitness obtained, surprise search is comparable to novelty search and far more efficient than objective search in deceptive domains. We can further argue that the deviation from the predictions (which are neither random nor omitted) is beneficial for surprise search as indicated by the performances of SSr and SSnp . The performance of this Successes 9 60 40 40 20 20 00 60 25 50 75 100 125 150 175 200 225 250 Evaluations (x103 ) (c) Very hard maze 00 25 50 75 100 125 150 175 200 225 250 Evaluations (x103 ) (d) Extremely hard maze Fig. 7: Robustness comparison for the four mazes in Fig. 3. The graphs depict the evolution of algorithm successes in solving the maze problem over the number of evaluations. baseline appears to be similar to novelty search, especially in harder mazes; this is not surprising as SSnp is conceptually similar to novelty search, as noted in Section VI-D. It is also clear that, on average, surprise search finds the solution faster than any other algorithm in all mazes. B. Robustness Robustness is defined as the number of successes obtained by the algorithm across time (i.e. evaluations). In Figure 7 we compare the robustness of each approach across the four mazes, collected from 100 runs. In the medium maze (Fig. 7a), surprise search is more successful than novelty search in the first 20,000 evaluations; moreover, surprise search finds, on average, the 100 solutions in fewer evaluations compared to the other approaches. As noticed in the previous section, in the first 20,000 evaluations novelty search has a comparable or higher efficiency in Fig. 6a; this points to the fact that while some individuals in surprise search manage to reach the goal, others do not get as close to it as in novelty search. On the other hand, objective search fails to find the goal in 29 runs, because of the several dead-ends present in this maze. The control algorithm SSnp finds the goal 93 times out of 100, but it’s slower compared to novelty and surprise search. Few solutions are found by the baseline random search and SSr , and they are significantly slower than the other approaches. Fig. 7b shows that, in the hard maze, novelty search attains more successes than surprise search in the first 10,000 evaluations but the opposite is true for the remainder of the evolutionary progress. As in the previous maze, this behaviour is not reflected in the efficiency graph (Fig. 6b): this 10 can be explained by how surprise search evolves individuals, as they change their distance to the goal more abruptly, while novelty search evolves behaviours in smooth incremental steps. On the other hand, SSnp finds fewer solutions in this maze, 62 out of 100. Finally, the deceptive properties of this maze are exemplified by the poor performance of objective search and the two random baselines. The capacity of surprise search is more evident in the very hard maze (see Fig. 7c) where the difference in terms of robustness becomes even larger between surprise and novelty search. While in the first 50,000 evaluations novelty and surprise search attain a comparable number of successes, the performance of surprise search is boosted for the remainder of the evolutionary run. Ultimately, surprise search solves the very hard maze in 99 out of 100 times in just 160,000 evaluations whereas novelty search manages to obtain 85 solutions by the end of the 250,000 evaluations. With 88 solutions out of 100, SSnp performs similarly to novelty search in this maze but is generally slower compared to surprise search. Objective search and the two random baselines, as expected, do not succeed in solving the maze. Similarly, in the extremely hard maze (see Fig. 7d) the benefits of surprise search over the other algorithms are quite apparent. While surprise and novelty obtain a similar number of successes in the first 100,000 evaluations, surprise search obtains more successes in the remaining evaluations of the run. At the end of 250,000 evaluations in the most deceptive map examined, surprise search finds solutions in 67 runs versus 48 runs of novelty search. SSnp finds 39 solutions and it is generally slower than novelty and surprise search. As in the very hard maze, the remaining algorithms fail to find a single solution to this maze. C. Analysis As an additional comparison between surprise and novelty search, we study the behavioural and genotypical characteristics of these two approaches. The behavioural space is presented in a number of typical runs collected from the four mazes, while the genotypical space is inspected through the metrics computed from the final ANNs evolved by these two algorithms. Objective search and the other baselines are not further analysed in this section to emphasise on comparisons between surprise and novelty search. 1) Behavioural Space: Typical Examples: Table I shows pairs of typical surprise and novelty search runs for each of the four mazes; in all examples illustrated the maze is solved at different number of evaluations as indicated at the captions of the images. The typical runs are shown as heatmaps which represent the aggregated distribution of the robots’ final positions throughout all evaluations. Moreover, we report the entropy (H) of those positions as a measure of the populations’ spatial diversity in the maze. Surprise search seems to explore more uniformly the space, as revealed by the final positions depicted in the heatmaps. The corresponding H values further support this claim, especially in the more deceptive mazes. 2) Genotypic Space: Table II contains a set of metrics that characterize the final ANNs evolved by surprise and novelty search obtained from all four mazes, which quantify aspects of genomic complexity and genomic diversity. For genomic complexity we consider the number of connections and the number of hidden nodes of the final ANNs evolved, while genomic diversity is measured as the average pairwise distance of the final ANNs evolved. This distance is computed with the compatibility metric, a linear combination of disjoint and excess genes and weight difference, as defined in [28]. As noted in [8], novelty search tends to evolve simpler networks in terms of connections when compared to objective search. Surprise search, on the other hand, seems to generate significantly more densely connected ANNs than novelty search (based on the number of connections). It also evolves slightly larger ANNs than novelty search (based on the number of hidden nodes). Most importantly, surprise search yields population diversity — as expressed by the compatibility metric [8] — that is significantly higher than novelty search. This difference seems to be mostly due to the disjoint factor, which counts the number of mismatching genes between two genomes, depending on whether their genes are within the innovation numbers of the other genome [28]. This suggests that ANNs evolved with surprise search are more diverse in terms of evolutionary history. In the more deceptive mazes, differences in genomic complexity and diversity become significantly larger. In the very hard maze the average number of connections for surprise search grows to 101.94 (σ = 52.45) while novelty search evolves ANNs with 42.03 connections (σ = 14.53), on average; the number of hidden nodes used by surprise search is significantly larger (6.94; σ = 3.62) compared to novelty search. Moreover the diversity metric (compatibility) is around three times that of novelty search. A similar trend can be noticed in the extremely hard maze, where again surprise search evolves denser, larger and more diverse ANNs. As mentioned earlier, handling more complex and larger ANNs has a direct impact on the computational cost of surprise search since it takes more time to simulate new networks across generations. It should be noted that creating larger networks does not imply that this behaviour is beneficial, it is however an indication that surprise search operates differently to novelty search. VIII. S URPRISE S EARCH IN G ENERATED M AZES In the previous sections we showed the power of surprise search in four selected instances of deceptive problems. While surprise search outperforms novelty and objective search both in terms of efficiency and robustness in four human-designed mazes, an important concern is whether these results are general enough across a broader set of problems. In order to assess how surprise search generalises in any maze navigation task, we follow the methodology presented in [49] and test the performance of surprise, novelty and objective search as well as the baselines across numerous mazes generated through an automated process. Moreover, the parameters of k and n which were fine-tuned for the problem at hand in each maze of Section VII are now kept the same, enabling us to observe if a particular parameter setup for surprise search can perform well in unseen problems of varying complexity. 11 TABLE I: Behavioural Space. Typical successful runs solved after a number of evaluations (E) across the four mazes examined. Heatmaps illustrate the aggregated numbers of final robot positions across all evaluations. Note that white space in the maze indicates that no robot visited that position. The entropy (H ∈ [0, 1]) of visited positions is also reported and is calculated as P follows: H = (1/logC) i {(vi /V )log(vi /V )}; where vi is the number of robot visits in a position i, V is the total number of visits and C is the total number of discretized positions (cells) considered in the maze. Medium Maze (E = 25, 000) Novelty Surprise H = 0.63 H = 0.67 Hard Maze (E = 25, 000) Novelty Surprise H = 0.61 H = 0.67 Very Hard Maze (E = 75, 000) Novelty Surprise H = 0.63 H = 0.69 Extremely Hard Maze (E = 75, 000) Novelty Surprise H = 0.64 H = 0.68 TABLE II: Genotypic Space. Metrics of genomic complexity and diversity of the final ANNs evolved using NEAT, averaged across successful runs. Values in parentheses denote standard deviations. Maze Medium Hard Very Hard Extremely Hard Algorithm Surprise Novelty Surprise Novelty Surprise Novelty Surprise Novelty Genomic Complexity Connections Hidden Nodes 33.76 (15.08) 2.46 (1.53) 29.08 (6.10) 2.2 (1.0) 52.34 (28.57) 3.84 (2.66) 32.55 (9.84) 2.48 (1.29) 101.94 (52.45) 6.94 (3.62) 42.03 (14.53) 3.27 (1.90) 158.16 (89.65) 10.63 (5.85) 41.79 (11.77) 3.25 (1.53) Compatibility 42.52 (19.80) 32.55 (7.90) 73.24 (36.82) 39.35 (12.05) 160.31 (67.01) 56.53 (19.66) 260.52 (113.08) 54.05 (14.44) Genomic Diversity Disjoint Weight Difference 27.40 (16.27) 1.22 (0.25) 24.97 (7.05) 1.09 (0.26) 51.60 (27.76) 1.25 (0.24) 31.06 (11.32) 1.19 (0.28) 121.56 (57.11) 1.26 (0.22) 46.65 (19.25) 1.16 (0.27) 207.97 (107.2) 1.31 (0.23) 45.30 (14.55) 1.15 (0.24) Excess 11.44 (11.45) 4.28 (3.41) 17.86 (20.99) 4.71 (4.66) 34.96 (37.34) 6.38 (7.71) 48.62 (54.20) 5.28 (4.93) A. Experiment Description To compare the capabilities in navigation policies of surprise, novelty and objective search in increasingly complex maze problems, we test their performance against 60 randomly generated mazes. These mazes are created by a recursive division algorithm [50], which starts from an empty maze and divides it into two areas by adding a vertical or a horizontal wall with a randomly located hole in it. This process is repeated until no areas can be further subdivided, because doing so would make the maze untraversable or because a maximum number of subdivisions is reached. In this experiment, the starting position and the ending position of the maze have been fixed in the lower left and upper right corner respectively, while the generated mazes have a number of subdivisions chosen randomly between 2 and 6. These values have been chosen empirically to avoid generating mazes that are too easy (solvable by all three methods in few generations) or impossible to solve (because of too many subdivisions). Examples of the mazes generated are shown in Fig. 8. The parameters of surprise search and novelty search are fixed based on well-performing setups with mazes of Section VII: surprise search uses k = 200 and n = 2 (used in the very hard maze) and novelty uses n = 15 (used in medium, hard and very hard mazes). Each generated maze was tested 50 times for each of three methods, measuring the number of successes (i.e. once the agent reaches the goal) in each maze. The number of simulation timesteps is set to 200 and the number of generations to 600. 2 subdivisions 3 subdivisions 4 subdivisions 5 subdivisions 6 subdivisions Fig. 8: Maze generator: Sample generated mazes (200x200 units) created via recursive division, showing the starting location (black circle) and the goal location (white circle). B. Results As a first analysis on the results obtained on the 60 mazes, we focus on which of the evolutionary approaches finds strictly more successes. Table III shows that surprise search has more successes than novelty search in 40% of mazes, while novelty achieves more successes than surprise search in 8% of the generated mazes. Comparing the results of these two approaches against objective search, surprise search outperforms objective search in more mazes (56%) than novelty search (40%). If we look at the baselines, SSnp reaches comparable performance to novelty search, but surprise search remains the most successful algorithm, as it outperforms SSnp in 45% of the considered mazes. Finally, SSr and random search evidently perform poorly compared to the other approaches. An important question to put to the test is how novelty search and surprise search perform with respect to maze deceptiveness. Intuitively, we can say that the deceptiveness (or difficulty) of the maze can be determined by the number of successes obtained by objective search: the more deceptive 12 Fig. 9: Linear regression: relation between the failures of objective search and successes of surprise and novelty search. Fig. 10: Robustness: algorithm successes in solving all the generated mazes over the number of evaluations for each considered method. the maze, the more often objective search would fail to find a solution. Figure 9 shows the number of successes obtained by novelty and surprise search against the number of failures obtained by objective search. Unsurprisingly, surprise search and novelty search find mazes where objective search failed more difficult as well, since their successes are highly correlated with the failures of objective search (adjusted R2 > 0.85 for each method, p < 0.001). Looking at the trends of the linear regression lines, surprise search constantly achieves a greater number of successes, based on the intercept values of the linear regression models which are significantly different according to an ANCOVA test (p < 0.05). Based on the angle of the linear regression line, it also seems that surprise search scales better for more deceptive mazes. As a final analysis, we report the robustness obtained by aggregating all the runs of the 60 generated mazes for each approach, i.e. a total of 3000 runs. From Fig. 10 we can observe that surprise search is faster, on average, than novelty and objective search in reaching the goal from 112, 500 evaluations onward (p < 0.05). Surprise, novelty and objective search require on average 71, 865 (σ = 63, 007), 76, 225 (σ = 64, 961) and 84, 398 (σ = 65, 081) evaluations for each success, respectively. As Fig. 10 shows, some maze problems are easy to solve as all three methods fare similarly in the first 20, 000 evaluations but as the problems become more complex, surprise and novelty search become faster than objective search and eventually surprise search surpasses novelty search in terms of successes. Furthermore, surprise search shows a significant improvement compared to its baseline variants. Respectively, SSnp , SSr and random search obtain 75, 531 (σ = 63, 820), 118, 668 (σ = 53934) and 123, 558 (σ = 50, 568) evaluations; all results are significantly different from the performance of surprise search (p < 0.05). IX. D ISCUSSION This paper identified the notion of surprise, i.e. deviation from expectations, as an alternative measure of divergence to the notion of novelty and presented a general framework for incorporating surprise in evolutionary search in Section IV. In order to highlight the differences between surprise search and other divergent search techniques (such as novelty search) or baselines (such as random search or search with inaccurate predictions), an experiment in the robot maze navigation testbed was carried out and comparisons between algorithms were made on several dimensions. The key findings of these experiments suggest that surprise search yields comparable efficiency to novelty search and it outperforms objective search. Moreover it finds solutions faster and more often than any other algorithm considered. In a broader range of mazes, generated via recursive subdivision, surprise search was also shown to be more robust and generalise well, as it had more successes than novelty search in 40% of generated mazes. The comparative advantages of surprise search over novelty search are inherent to the way the algorithm searches, attempting to deviate from predicted unseen behaviours instead of prior seen behaviours. Compared to novelty search, surprise search may also deviate from expected behaviours that exist in areas that have been visited in the past by the algorithm. The novelty archive operates in a similar fashion; however it contains positions (instead of prediction points) and these positions are always considered for the calculation of the novelty score. In surprise search, instead, the prediction points are derived from clusters that characterise areas in the behavioural space. The findings in the maze navigation experiments show a clear difference between novelty and surprise, both behaviourally and genotypically. Surprise search has greater exploratory capabilities, which are more obvious in later generations. Surprise search also creates genotypically diverse populations, with larger and denser ANNs. It is likely this combination of diverse populations, larger and denser networks and a higher spatial diversity that gives surprise search its advantage over novelty search. Finally, the comparative analysis of surprise search against random search suggests that surprise search is not random search. Clearly it outperforms random search in efficiency and robustness. Furthermore, the poor performance of the two surprise search variants — employing random predictions and omitting predictions — suggests that the prediction of expected behaviour is beneficial for divergent search. By now we have enough evidence for the benefits of surprise search and enough findings suggesting that surprise search is a different and more robust algorithm compared to novelty search in this domain. Furthermore, through our analysis, we have identified qualitative characteristics of the algorithm that gave us critical insights on the way the algorithm operates. However, we still lack empirical evidence on the reasons the algorithm manages to perform that well compared to other divergent search algorithms. An intuition by the anonymous reviewers of a previous paper about the comparative benefits of surprise search is that the algorithm allows search to revisit areas in the behavioural space. Such a behaviour, in contrast, is penalised in novelty search. This difference in how the two algorithms operate leads to the assumption that surprise search is more willing to revisit points in the behaviour space — in a form of backtracking or cyclical manner. As a result of this shifting selection pressure in the behavioural space a different strategy is adopted every time a particular area is revisited as each time the ANN controller is different and potentially larger. Such an algorithmic behaviour appears to be beneficial for search and might explain why ANNs get significantly larger in surprise search. 13 TABLE III: Successes: Percentage of generated mazes for which the algorithm in the row has a strictly greater number of successes than the algorithm in the column. The last row and the last column are respectively the average of each column and the average of each row. Objective Novelty Surprise SSnp SSr Random Total Objective 40 56 51 1 0 29.6 Novelty 15 40 35 1 0 19 Surprise 5 8 11 1 0 5 More importantly than a performance comparison between algorithms, however, is the introduction of surprise as a drive for divergent search. As will be discussed in Section X, the general framework of surprise search can be used with other behaviour characterizations and in other domains. Moreover, while some properties such as the model of prediction are inherent to surprise search, properties such as the clustering of behaviours as well as findings regarding the ability of surprise search to backtrack can be re-used in other divergent search algorithms; examples include a variant of novelty search which considers neighboring cluster centroids rather than neighboring individuals in the behavioural space, or a novelty archive which is pruned (and reduces in size) over time to allow backtracking. We can only hypothesise that the way surprise search operates may result in increased evolvability [51], which is an individual’s capacity to generate future phenotypic variation, or alternatively, the potential for further evolution. Naturally, all these hypotheses need to be tested empirically in future studies as outlined in the next section. SSnp 6 20 45 1 0 16.5 SSr 71 75 78 78 20 62.8 Generation t − 2 Random 78 81 85 85 36 71.8 Total 35 44.8 60.8 52 8 4 - Generation t − 1 Generation t Fig. 11: Behaviour Characterization: The key phases of the surprise search algorithm on the maze navigation task, characterizing behaviour via 200 samples of robot positions over time. Surprise search uses a history of two generations (h = 2) and 10 clusters (k = 10) in this example (for the sake of visualization, only one cluster is shown). Robot trails are depicted as green lines. Cluster centroids in generations t − 2 and t−1 as well as their predictions are depicted, respectively, as red, dark red and blue lines. X. E XTENSIONS AND F UTURE W ORK While this study already offers evidence for the advantages of surprise as a form of divergent search, further work on several directions needs to be performed. Behaviour Characterization: The surprise search algorithm currently characterises a behaviour merely as a point (i.e. the final robot position on the maze after the simulation time elapses). While such a decision was made in order to compare our findings against the initial results of [8], we currently have no evidence suggesting that surprise search would be able to generalise well in behaviours that are characterised by higher dimensions (e.g. a robot trail). To envision the behaviour of surprise search in a high dimensional space, Fig. 11 shows a possible implementation for surprise search with robot trails, sampled over time. Following the implementation described in Section VI-C, we show the three key phases of the algorithm: the robots’ trails are clustered at generation t − 2 via kmeans (see Fig. 11a), then at generation t − 1 we seed the clustering algorithm with the ones computed in the previous generation and we find the new trajectories’ centroids (Fig. 11b). Finally the prediction at generation t is computed via linear interpolation of each point on the computed (centroid) trajectories at generations t−2 and t−1 (Fig. 11c). Preliminary experiments have shown that predictions of robot trails do not affect the performance of surprise search compared to results in this paper; future studies, however, should investigate Ht−2 at generation t−2 Ht−1 at generation t−1 Ht at generation t Fig. 12: Deviation: Surprise search using heatmaps, at generation t. The first two heatmaps are computed in the last two generations by using the final robot positions, Ht−2 and Ht−1 . Using linear interpolation, the difference Ht−1 − Ht−2 is computed and applied to Ht−1 to derive the predicted current population’s Ht . The surprise score penalizes a robot if its position (green point) is on a high concentration cell on the predicted heatmap Ht . the impact of higher dimensional behaviours across several domains, especially on how they affect the predictive model of the algorithm. Deviation: The current algorithm allows for any degree and type of deviation from the expected behaviour. Inspired by novelty search, this paper only investigated a linear deviation from expectations — i.e. the further a behaviour is from the prediction the better. There exist, however, several ways of computing deviation in a non-linear or probabilistic fashion, e.g. as per [18]. In a maze navigation environment for instance, 14 we can alternatively consider a non-distance-based deviation by using heatmaps of the chosen behaviour characterization. Figure 12 shows the key phases of this implementation: in generation t − 2 and t − 1 we compute the heatmaps Ht−2 and Ht−1 which map the final positions of all robots (k = 1), and we use a linear interpolation to compute the predicted heatmap at generation t. The surprise score is then computed by mapping the individual’s position on the predicted heatmap: 1 − Ht (x, y), where x, y is the final position of the robot mapped onto the heatmap. Complexity and generality: To a degree, the experiment with 60 generated mazes tests how generalizable surprise search is without explicit parameter tuning. Since results from that experiment indicated that surprise search scales better to more deceptive problems, we need to further test the algorithm’s potential within the maze navigation domain through more deceptive and complex environments. The capacity of surprise search will then need to be tested in other domains such as robot locomotion or procedural content generation. As an example, the potential of surprise search has been explored for the generation of unexpected weapons in a first person shooter game [38]. In that work, the considered behaviour characterization is a weapon tester agent’s death location: as we have described above, we can employ a heatmap as a probability distribution of the population’s behaviour and predict the next generation’s heatmap by means of linear interpolation of each singular cell. Surprise search has shown its capacity to generate feasible and diverse content, thereby achieving quality diversity. Surprise search has also been successfully implemented to evolve soft robot morphologies [52] constrained by a fixed lattice. The goal of this experiment is to create robots able to travel as far as possible from a starting position, within a number of simulation steps. In this task, surprise search was shown to be as efficient as novelty search; moreover, it evolved more diverse morphologies. When computing behavioural distance for this problem, the surprise score is computed by predicting an entire robot trace in all simulation steps, based on past behaviours (traces). This further supports the claim that surprise is unaffected by the dimensionality of the behaviour characterization. The experiments presented in this paper, in [38] and [52] already demonstrate the generalizability of surprise search across three rather diverse domains: maze navigation, game content generation and robot design. Model of expected behaviour: When it comes to designing a model of expected behaviour there are two key aspects that need to be considered: how much prior information the model requires and how is that information used to make a prediction. In this paper the prediction of behaviour is based on the simplest form of 1-step predictions via a linear regression. This simple predictive model shows the capacity of surprise search given its performance advantages over novelty search. However we can envision that better results can be achieved if machine learned or non-linear predicted models are built on more prior information (h > 2). A possible way of considering more extensive history is to apply linear interpolation of past centroids over time. It is thus possible to compute a 3-dimensional line over the three dimensions considered (x, y, t) and compute the next predicted position over the line by taking the interpolated position at generation t. Linear regression can be easily replaced by a quadratic or cubic regression or even an artificial neural network model or a support vector machine. Prediction locality of surprise search: The algorithm presented in this paper allows for various degrees of prediction locality. We define prediction locality as the amount of local information considered by surprise search to make a prediction. This is expressed by k which is left as a variable for the algorithm designer. Prediction locality can be derived from the behavioural space (as in this paper) but also on the genotypic space. Future investigations should investigate the effect of locality for surprise search. Experiments in this paper (see Fig. 5b) already showcase that algorithm performance is not sensitive with respect to k (as long as k is sufficiently high for the problem at hand). Clustering: Surprise search, in the form presented here, requires some form of behavioural clustering. While k-means was investigated in the experiments of this paper for its simplicity and popularity, any clustering algorithm is applicable. Comparative studies between approaches need to be investigated, including different ways of dealing with (or taking advantage of) empty clusters. XI. C ONCLUSIONS In this paper, we argue that surprise is a concept that can be exploited for evolutionary divergent search, we provide a general definition of the algorithm that follows the principles of searching for surprise and we test the idea in a maze navigation task. Results show that surprise search has clear advantages over other forms of evolutionary divergent search, i. e. novelty search, and outperforms traditional fitness search in deceptive problems. In particular, surprise search has shown a comparable efficiency to novelty search and, most importantly, to be more successful and faster in finding the goal. Moreover, a detailed analysis of the behaviours and the genomes evolved by surprise search has revealed a more diverse population and a higher exploratory capacity. 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Journal of Automated Reasoning manuscript No. (will be inserted by the editor) Mechanized semantics for the Clight subset of the C language arXiv:0901.3619v1 [cs.PL] 23 Jan 2009 Sandrine Blazy · Xavier Leroy the date of receipt and acceptance should be inserted later Abstract This article presents the formal semantics of a large subset of the C language called Clight. Clight includes pointer arithmetic, struct and union types, C loops and structured switch statements. Clight is the source language of the CompCert verified compiler. The formal semantics of Clight is a big-step operational semantics that observes both terminating and diverging executions and produces traces of input/output events. The formal semantics of Clight is mechanized using the Coq proof assistant. In addition to the semantics of Clight, this article describes its integration in the CompCert verified compiler and several ways by which the semantics was validated. Keywords The C programming language · Operational semantics · Mechanized semantics · Formal proof · The Coq proof assistant 1 Introduction Formal semantics of programming languages—that is, the mathematical specification of legal programs and their behaviors—play an important role in several areas of computer science. For advanced programmers and compiler writers, formal semantics provide a more precise alternative to the informal English descriptions that usually pass as language standards. In the context of formal methods such as static analysis, model checking and program proof, formal semantics are required to validate the abstract interpretations and program logics (e.g. axiomatic semantics) used to analyze and reason about programs. The verification of programming tools such as compilers, type-checkers, static analyzers and program verifiers is another area where formal semantics for the languages involved is a prerequisite. While This work was supported by Agence Nationale de la Recherche, grant number ANR-05-SSIA-0019. S. Blazy ENSIIE, 1 square de la Résistance, 91025 Evry cedex, France E-mail: Sandrine.Blazy@ensiie.fr X. Leroy INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France E-mail: Xavier.Leroy@inria.fr 2 formal semantics for realistic languages can be defined on paper using ordinary mathematics [31, 16, 7], machine assistance such as the use of proof assistants greatly facilitates their definition and uses. For high-level programming languages such as Java and functional languages, there exists a sizeable body of mechanized formalizations and verifications of operational semantics, axiomatic semantics, and programming tools such as compilers and bytecode verifiers. Despite being more popular for writing systems software and embedded software, lower-level languages such as C have attracted less interest: several formal semantics for various subsets of C have been published, but only a few have been mechanized. The present article reports on the definition of the formal semantics of a large subset of the C language called Clight. Clight features most of the types and operators of C, including pointer arithmetic, pointers to functions, and struct and union types, as well as all C control structures except goto. The semantics of Clight is mechanized using the Coq proof assistant [10, 4]. It is presented as a big-step operational semantics that observes both terminating and diverging executions and produces traces of input/output events. The Clight subset of C and its semantics are presented in sections 2 and 3, respectively. The work presented in this paper is part of an ongoing project called CompCert that develops a realistic compiler for the C language and formally verifies that it preserves the semantics of the programs being compiled. A previous paper [6] reports on the development and proof of semantic preservation in Coq of the front-end of this compiler: a translator from Clight to Cminor, a low-level, imperative intermediate language. The formal verification of the back-end of this compiler, which generates moderately optimized PowerPC assembly code from Cminor is described in [28]. Section 4 describes the integration of the Clight language and its semantics within the CompCert compiler and its verification. Formal semantics for realistic programming languages are large and complicated. This raises the question of validating these semantics: how can we make sure that they correctly capture the expected behaviors? In section 5, we argue that the correctness proof of the CompCert compiler provides an indirect but original way to validate the semantics of Clight, and discuss other approaches to the validation problem that we considered. We finish this article by a discussion of related work in section 6, followed by future work and conclusions in section 7. Availability The Coq development underlying this article can be consulted on-line at http://compcert.inria.fr . Notations [x, y[ denotes the semi-open interval of integers {n ∈ Z \| x ≤ n < y}. For functions returning “option” types, ⌊x⌋ (read: “some x”) corresponds to success with return value x, and 0/ (read: “none”) corresponds to failure. In grammars, a∗ denotes 0, 1 or several occurrences of syntactic category a, and a? denotes an optional occurrence of syntactic category a. 2 Abstract syntax of Clight Clight is structured into expressions, statements and functions. In the Coq formalization, the abstract syntax is presented as inductive data types, therefore achieving a deep embedding of Clight into Coq. 3 Signedness: Integer sizes: Float sizes: signedness ::= Signed \| Unsigned intsize ::= I8 \| I16 \| I32 floatsize ::= F32 \| F64 Types: τ ::= int(intsize,signedness) \| float(floatsize) \| void \| array(τ ,n) \| pointer(τ ) \| function(τ ∗ , τ ) \| struct(id, ϕ ) \| union(id, ϕ ) \| comp pointer(id) Field lists: ϕ ::= (id, τ )∗ Fig. 1 Abstract syntax of Clight types. 2.1 Types The abstract syntax of Clight types is given in figure 1. Supported types include arithmetic types (integers and floats in various sizes and signedness), array types, pointer types (including pointers to functions), function types, as well as struct and union types. Named types are omitted: we assume that typedef definitions have been expanded away during parsing and type-checking. The integral types fully specify the bit size of integers and floats, unlike the C types int, long, etc, whose sizes are left largely unspecified in the C standard. Typically, the parser maps int and long to size I32, float to size F32, and double to size F64. Currently, 64-bit integers and extended-precision floats are not supported. Array types carry the number n of elements of the array, as a compile-time constant. Arrays with unknown sizes (τ [] in C) are replaced by pointer types in function parameter lists. Their only other use in C is within extern declarations of arrays, which are not supported in Clight. Functions types specify the number and types of the function arguments and the type of the function result. Variadic functions and unprototyped functions (in the style of Ritchie’s pre-standard C) are not supported. In C, struct and union types are named and compared by name. This enables the definition of recursive struct types such as struct s1 { int n; struct * s1 next;}. Recursion within such types must go through a pointer type. For instance, the following is not allowed in C: struct s2 { int n; struct s2 next;}. To obviate the need to carry around a typing environment mapping struct and union names to their definitions, Clight struct and union types are structural: they carry a local identifier id and the list ϕ of their fields (names and types). Bit-fields are not supported. These types are compared by structure, like all other Clight types. In structural type systems, recursive types are traditionally represented with a fixpoint operator µα .τ , where α names the type µα .τ within τ . We adapt this idea to Clight: within a struct or union type, the type comp pointer(id) stands for a pointer type to the nearest enclosing struct or union type named id. For example, the structure s1 defined previously in C is expressed by struct(s1, (n, int(I32, signed))(next, comp pointer(s1))) 4 Expressions: a ::= id \|n \|f \| sizeof(τ ) \| op1 a \| a1 op2 a2 \| a \| a.id \| &a \| (τ )a \| a1 ? a2 : a3 Unary operators: op1 ::= - \| ~ \| ! Binary operators: op2 ::= + \| - \| \| / \| % \| << \| >> \| & \| \| \| ^ \| < \| <= \| > \| >= \| == \| != variable identifier integer constant float constant size of a type unary arithmetic operation binary arithmetic operation pointer dereferencing field access taking the address of type cast conditional expressions arithmetic operators bitwise operators relational operators Fig. 2 Abstract syntax of Clight expressions Incorrect structures such as s2 above cannot be expressed at all, since comp_pointer let us refer to a pointer to an enclosing struct or union, but not to the struct or union directly. Clight does not support any of the type qualifiers of C (const, volatile, restrict). These qualifiers are simply erased during parsing. The following operations over types are defined: sizeof(τ ) returns the storage size, in bytes, of type τ , and field offset(id, ϕ ) returns the byte offset of the field named id in a struct whose field list is ϕ , or 0/ if id does not appear in ϕ . The Coq development gives concrete definitions for these functions, compatible with the PowerPC ABI [48, chap. 3]. Typically, struct fields are laid out consecutively and padding is inserted so that each field is naturally aligned. Here are the only properties that a Clight producer or user needs to rely on: – Sizes are positive: sizeof(τ ) > 0 for all types τ . – Field offsets are within the range of allowed byte offsets for their enclosing struct: if field offset(id, ϕ ) = ⌊δ ⌋ and τ is the type associated with id in ϕ , then [δ , δ + sizeof(τ )[ ⊆ [0, sizeof(struct id′ ϕ )[. – Different fields correspond to disjoint byte ranges: if field offset(idi , ϕ ) = ⌊δi ⌋ and τi is the type associated with id i in ϕ and id 1 6= id 2 , then [δ1 , δ1 + sizeof(τ1 )[ ∩ [δ2 , δ2 + sizeof(τ2 )[ = 0. / – When a struct is a prefix of another struct, fields shared between the two struct have the same offsets: if field offset(id, ϕ ) = ⌊δ ⌋, then field offset(id, ϕ .ϕ ′) = ⌊δ ⌋ for all additional fields ϕ ′ . 2.2 Expressions The syntax of expressions is given in figure 2. All expressions and their sub-expressions are annotated by their static types. In the Coq formalization, expressions a are therefore pairs (b, τ ) of a type τ and a term b of an inductive datatype determining the kind and 5 Statements: Switch cases: s ::= skip \| a1 = a2 \| a1 = a2 (a∗ ) \| a(a∗ ) \| s1 ;s2 \| if(a) s1 else s2 \| switch(a) sw \| while(a) s \| do s while(a) \| for(s1 ,a2 ,s3 ) s \| break \| continue \| return a? sw ::= default : s \| case n : s;sw empty statement assignment function call procedure call sequence conditional multi-way branch “while” loop “do” loop “for” loop exit from the current loop next iteration of the current loop return from current function default case labeled case Fig. 3 Abstract syntax of Clight statements. arguments of the expression. In this paper, we omit the type annotations over expressions, but write type(a) for the type annotating the expression a. The types carried by expressions are necessary to determine the semantics of type-dependent operators such as overloaded arithmetic operators. The following expressions can occur in left-value position: id, a, and a. id. Within expressions, only side-effect free operators of C are supported, but not assignment operators (=, +=, ++, etc) nor function calls. In Clight, assignments and function calls are presented as statements and cannot occur within expressions. As a consequence, all Clight expressions always terminate and are pure: their evaluation performs no side effects. The first motivation for this design decision is to ensure determinism of evaluation. The C standard leaves evaluation order within expressions partially unspecified. If expressions can contain side-effects, different evaluation orders can lead to different results. As demonstrated by Norrish [36], capturing exactly the amount of nondeterminism permitted by the C standard complicates a formal semantics. It is of course possible to commit on a particular evaluation order in a formal semantics for C. (Most C compiler choose a fixed evaluation order, typically right-to-left.) This is the approach we followed in an earlier version of this work [6]. Deterministic side-effects within expressions can be accommodated relatively easily with some styles of semantics (such as the big-step operational semantics of [6]), but complicate or even prevent other forms of semantics. In particular, it is much easier to define axiomatic semantics such as Hoare logic and separation logic if expressions are terminating and pure: in this case, syntactic expressions can safely be used as part of the logical assertions of the logic. Likewise, abstract interpretations and other forms of static analysis are much simplified if expressions are pure. Most static analysis and program verification tools for C actually start by pulling assignments and function calls out of expressions, and only then perform analyses over pure expressions [9, 13, 42, 8, 1, 17]. 6 dcl ::= (τ id)∗ Variable declarations: name and type F ::= τ id(dcl1 ) { dcl2 ; s } (dcl1 = parameters, dcl2 = local variables) Internal function definitions: External function declarations: Fe ::= extern τ id(dcl) Functions: Fd ::= F \| Fe internal or external P ::= dcl;Fd∗ ;main = id Programs: global variables, functions, entry point Fig. 4 Abstract syntax of Clight functions and programs. Some forms of C expressions are omitted in the abstract syntax but can be expressed as syntactic sugar: array access: a1 [a2 ] indirect field access: a->id sequential “and”: a1 && a2 sequential “or”: a1 \|\| a2 ≡ ≡ ≡ ≡ (a1 + a2 ) (a.id) a1 ? (a2 ? 1 : 0) : 0 a1 ? 1 : (a2 ? 1 : 0) 2.3 Statements Figure 3 defines the syntax of Clight statements. All structured control statements of C (conditional, loops, Java-style switch, break, continue and return) are supported, but not unstructured statements such as goto and unstructured switch like the infamous “Duff’s device” [12]. As previously mentioned, assignment a1 = a2 of an r-value a2 to an l-value a1 , as well as function calls, are treated as statements. For function calls, the result can either be assigned to an l-value or discarded. Blocks are omitted because block-scoped variables are not supported in Clight: variables are declared either with global scope at the level of programs, or with function scope at the beginning of functions. The for loop is written for(s1 , a2 , s3 ) s, where s1 is executed once at the beginning of the loop, a2 is the loop condition, s3 is executed at the end of each iteration, and s is the loop body. In C, s1 and s3 are expressions, which are evaluated for their side effects. In Clight, since expressions are pure, we use statements instead. (However, the semantics requires that these statements terminate normally, but not by e.g. break.) A switch statement consists in an expression and a list of cases. A case is a statement labeled by an integer constant (case n) or by the keyword default. Contrary to C, the default case is mandatory in a Clight switch statement and must occur last. 2.4 Functions and programs A Clight program is composed of a list of declarations for global variables (name and type), a list of functions (see figure 4) and an identifier naming the entry point of the program (the main function in C). The Coq formalization supports a rudimentary form of initialization for global variables, where an initializer is a sequence of integer or floating-point constants; we omit this feature in this article. Functions come in two flavors: internal or external. An internal function, written τ id(dcl1 ) { dcl2 ; s }, is defined within the language. τ is the return type, id the name of 7 the function, dcl1 its parameters (names and types), dcl2 its local variables, and s its body. External functions extern τ id(dcl) are merely declared, but not implemented. They are intended to model “system calls”, whose result is provided by the operating system instead of being computed by a piece of Clight code. 3 Formal semantics for Clight We now formalize the dynamic semantics of Clight, using natural semantics, also known as big-step operational semantics. The natural semantics observe the final result of program execution (divergence or termination), as well as a trace of the invocations of external functions performed by the program. The latter represents the input/output behavior of the program. Owing to the restriction that expressions are pure (section 2.2), the dynamic semantics is deterministic. The static semantics of Clight (that is, its typing rules) has not been formally specified yet. The dynamic semantics is defined without assuming that the program is well-typed, and in particular without assuming that the type annotations over expressions are consistent. If they are inconsistent, the dynamic semantics can be undefined (the program goes wrong), or be defined but differ from what the C standard prescribes. 3.1 Evaluation judgements The semantics is defined by the 10 judgements (predicates) listed below. They use semantic quantities such as values, environments, etc, that are summarized in figure 5 and explained later. G, E ⊢ a, M ⇐ ℓ (evaluation of expressions in l-value position) G, E ⊢ a, M ⇒ v (evaluation of expressions in r-value position) G, E ⊢ a∗ , M ⇒ v∗ (evaluation of lists of expressions) t G, E ⊢ s, M ⇒ out, M ′ (execution of statements, terminating case) t G, E ⊢ sw, M ⇒ out, M ′ (execution of the cases of a switch, terminating case) t G ⊢ Fd(v∗ ), M ⇒ v, M ′ (evaluation of function invocations, terminating case) T G, E ⊢ s, M ⇒ ∞ (execution of statements, diverging case) T G, E ⊢ sw, M ⇒ ∞ (execution of the cases of a switch, diverging case) T G ⊢ Fd(v∗ ), M ⇒ ∞ ⊢ P⇒B (evaluation of function invocations, diverging case) (execution of whole programs) Each judgement relates a syntactic element to the result of executing this syntactic element. For an expression in l-value position, the result is a location ℓ: a pair of a block identifier b and a byte offset δ within this block. For an expression in r-value position and for a function application, the result is a value v: the discriminated union of 32-bit integers, 64-bit floating-point numbers, locations (representing the value of pointers), and the special value undef representing the contents of uninitialized memory. Clight does not support assignment between struct or union, nor passing a struct or union by value to a function; therefore, struct and union values need not be represented. Following Norrish [36] and Huisman and Jacobs [21], the result associated with the execution of a statement s is an outcome out indicating how the execution terminated: either normally by running to completion or prematurely via a break, continue or return statement. 8 Block references: b ∈Z Memory locations: ℓ ::= (b, δ ) byte offset δ (a 32-bit integer) within block b Values: v ::= int(n) \| float( f ) \| ptr(ℓ) \| undef integer value (n is a 32-bit integer) floating-point value ( f is a 64-bit float) pointer value undefined value Statement outcomes: out ::= Normal \| Continue \| Break \| Return \| Return(v) Global environments: G ::= (id 7→ b) ×(b 7→ Fd) map from global variables to block references and map from function references to function definitions Local environments: E ::= id 7→ b map from local variables to block references Memory states: M ::= b 7→ (lo,hi, δ 7→ v) map from block references to bounds and contents Memory quantities: κ ::= int8signed \| int8unsigned \| int16signed \| int16unsigned \| int32 \| float32 \| float64 vν ::= int(n) \| float( f ) I/O values: continue with next statement go to the next iteration of the current loop exit from the current loop function exit function exit, returning the value v I/O events: ν ::= id( vν ∗ 7→ vν ) name of external function, argument values, result value Traces: t ::= ε \| ν .t T ::= ε \| ν .T finite traces (inductive) finite or infinite traces (coinductive) Program behaviors: B ::= terminates(t,n) \| diverges(T ) termination with trace t and exit code n divergence with trace T Operations over memory states: alloc(M,lo,hi) = (M′ ,b) Allocate a fresh block of bounds [lo,hi[. free(M,b) = M′ Free (invalidate) the block b. load(κ ,M,b,n) = ⌊v⌋ Read one or several consecutive bytes (as determined by κ ) at block b, offset n in memory state M. If successful return the contents of these bytes as value v. store(κ ,M,b,n,v) = ⌊M′ ⌋ Store the value v into one or several consecutive bytes (as determined by κ ) at offset n in block b of memory state M. If successful, return an updated memory state M′ . Operations over global environments: funct(G,b) = ⌊b⌋ Return the function definition Fd corresponding to the block b, if any. symbol(G,id) = ⌊b⌋ Return the block b corresponding to the global variable or function name id. globalenv(P) = G Construct the global environment G associated with the program P. initmem(P) = M Construct the initial memory state M for executing the program P. Fig. 5 Semantic elements: values, environments, memory states, statement outcomes, etc Most judgements are parameterized by a global environment G, a local environment E, and an initial memory state M. Local environments map function-scoped variables to references of memory blocks containing the values of these variables. (This indirection through memory is needed to allow the & operator to take the address of a variable.) These blocks are allocated at function entry and freed at function return (see rule 32 in figure 10). Likewise, the global environment G associates block references to program-global variables and functions. It also records the definitions of functions. The memory model used in our semantics is detailed in [29]. Memory states M are modeled as a collection of blocks separated by construction and identified by integers b. Each block has lower and upper bounds lo, hi, fixed at allocation time, and associates values 9 Expressions in l-value position: E(id) = b or (id ∈ / Dom(E) and symbol(G,id) = ⌊b⌋) G,E ⊢ a,M ⇒ ptr(ℓ) (1) G,E ⊢ id,M ⇐ (b,0) G,E ⊢ a,M ⇐ (b, δ ) (2) G,E ⊢ a,M ⇐ ℓ type(a) = struct(id′ , ϕ ) field offset(id, ϕ ) = ⌊δ ′ ⌋ G,E ⊢ a.id,M ⇐ (b, δ + δ ) (3) ′ G,E ⊢ a,M ⇐ ℓ type(a) = union(id′ , ϕ ) (4) G,E ⊢ a.id,M ⇐ ℓ Expressions in r-value position: G,E ⊢ n,M ⇒ int(n) (5) G,E ⊢ f ,M ⇒ float( f ) (6) G,E ⊢ sizeof(τ ),M ⇒ int(sizeof(τ )) (7) G,E ⊢ a,M ⇐ ℓ G,E ⊢ &a,M ⇒ ptr(ℓ) G,E ⊢ a1 ,M ⇒ v1 (8) G,E ⊢ a,M ⇒ v G,E ⊢ a1 ,M ⇒ v1 (9) loadval(type(a),M′ ,ℓ) = ⌊v⌋ G,E ⊢ a,M ⇐ ℓ eval unop(op1 ,v1 ,type(a1 )) = ⌊v⌋ (10) G,E ⊢ op1 a1 ,M ⇒ v G,E ⊢ a2 ,M1 ⇒ v2 eval binop(op2 ,v1 ,type(a1 ),v2 ,type(a2 )) = ⌊v⌋ (11) G,E ⊢ a1 op2 a2 ,M ⇒ v G,E ⊢ a1 ,M ⇒ v1 is true(v1 ,type(a1 )) G,E ⊢ a2 ,M ⇒ v2 (12) G,E ⊢ a1 ? a2 : a3 ,M ⇒ v2 G,E ⊢ a1 ,M ⇒ v1 is false(v1 ,type(a1 )) G,E ⊢ a3 ,M ⇒ v3 (13) G,E ⊢ a1 ? a2 : a3 ,M ⇒ v3 G,E ⊢ a,M ⇒ v1 cast(v1 ,type(a), τ ) = ⌊v⌋ G,E ⊢ (τ )a,M ⇒ v (14) Fig. 6 Natural semantics for Clight expressions to byte offsets δ ∈ [lo, hi[. The basic operations over memory states are alloc, free, load and store, as summarized in figure 5. Since Clight expressions are pure, the memory state is not modified during expression evaluation. It is modified, however, during the execution of statements and function calls. The corresponding judgements therefore return an updated memory state M ′ . They also produce a trace t of the external functions (system calls) invoked during execution. Each such invocation is described by an input/output event ν recording the name of the external function invoked, the arguments provided by the program, and the result value provided by the operating system. In addition to terminating behaviors, the semantics also characterizes divergence during the execution of a statement or of a function call. The treatment of divergence follows the coinductive natural approach of Leroy and Grall [30]. The result of a diverging execution is the trace T (possibly infinite) of input/output events performed. In the Coq specification, the judgements of the dynamic semantics are encoded as mutually inductive predicates (for terminating executions) and mutually coinductive predicates (for diverging executions). Each defining case of each predicate corresponds exactly to an inference rule in the conventional, on-paper presentation of natural semantics. We show most of the inference rules in figures 6 to 12, and explain them in the remainder of this section. 10 Access modes: µ ::= By value(κ ) \| By reference \| By nothing access by value access by reference no access Associating access modes to Clight types: A (int(I8,Signed)) A (int(I8,Unsigned)) A (int(I16,Signed)) A (int(I16,Unsigned)) A (int(I32, )) A (pointer( )) = = = = = = By By By By By By value(int8signed) A (array( , )) value(int8unsigned) A (function( , )) value(int16signed) A (struct)( , )) value(int16unsigned) A (union)( , )) value(int32) A (void) value(int32) = = = = = By By By By By reference reference nothing nothing nothing Accessing or updating a value of type τ at location (b, δ ) in memory state M: loadval(τ ,M,(b, δ )) loadval(τ ,M,(b, δ )) loadval(τ ,M,(b, δ )) storeval(τ ,M,(b, δ ),v) storeval(τ ,M,(b, δ ),v) = = = = = load(κ ,M,b, δ ) ⌊(b, δ )⌋ 0/ store(κ ,M,b, δ ,v) 0/ if A (τ ) = By if A (τ ) = By if A (τ ) = By if A (τ ) = By otherwise value(κ ) reference nothing value(κ ) Fig. 7 Memory accesses. 3.2 Evaluation of expressions Expressions in l-value position The first four rules of figure 6 illustrate the evaluation of an expression in l-value position. A variable id evaluates to the location (b, 0), where b is the block associated with id in the local environment E or the global environment G (rule 1). If an expression a evaluates (as an r-value) to a pointer value ptr(ℓ), then the location of the dereferencing expression a is ℓ (rule 2). For field accesses a. id, the location ℓ = (b, δ ) of a is computed. If a has union type, this location is returned unchanged. (All fields of a union share the same position.) If a has struct type, the offset of field id is computed using the field_offset function, then added to δ . From memory locations to values The evaluation of an l-value expression a in r-value position depends on the type of a (rule 8). If a has scalar type, its value is loaded from memory at the location of a. If a has array type, its value is equal to its location. Finally, some types cannot be used in r-value position: this includes void in C and struct and union types in Clight (because of the restriction that structs and unions cannot be passed by value). To capture these three cases, figure 7 defines the function A that maps Clight types to access modes, which can be one of: “by value”, with a memory quantity κ (an access loads a quantity κ from the address of the l-value); “by reference” (an access simply returns the address of the l-value); or “by nothing” (no access is allowed). The loadval and storeval functions, also defined in figure 7, exploit address modes to implement the correct semantics for conversion of l-value to r-value (loadval) and assignment to an l-value (storeval). Expressions in r-value position Rules 5 to 14 of figure 6 illustrate the evaluation of an expression in r-value position. Rule 8 evaluates an l-value expression in an r-value context. The expression is evaluated to its location ℓ. From this location, a value is deduced using 11 ε G,E ⊢ skip,M ⇒ Normal,M (15) ε G,E ⊢ break,M ⇒ Break,M (16) ε G,E ⊢ continue,M ⇒ Continue,M (17) ε ⇒ Return,M (18) G,E ⊢ (return 0),M / G,E ⊢ a,M ⇒ v (19) ε G,E ⊢ (return ⌊a⌋),M ⇒ Return(v),M G,E ⊢ a1 ,M ⇐ ℓ storeval(type(a1 ),M,ℓ,v) = ⌊M′ ⌋ G,E ⊢ a2 ,M ⇒ v ε (20) G,E ⊢ (a1 = a2 ),M ⇒ Normal,M′ t t 2 out,M2 G,E ⊢ s2 ,M1 ⇒ 1 Normal,M1 G,E ⊢ s1 ,M ⇒ (21) t .t 1 2 G,E ⊢ (s1 ;s2 ),M ⇒ out,M2 t G,E ⊢ s1 ,M ⇒ out,M′ out 6= Normal (22) t G,E ⊢ (s1 ;s2 ),M ⇒ out,M′ Fig. 8 Natural semantics for Clight statements (other than loops and switch statements) the loadval function described above. By rule 9, &a evaluates to the pointer value ptr(ℓ) as soon as the l-value a evaluates to the location ℓ. Rules 10 and 11 describe the evaluation of unary and binary operations. Taking binary operations as an example, the two argument expressions are evaluated and their values v1 , v2 are combined using the the eval_binop function, which takes as additional arguments the types τ1 and τ2 of the arguments, in order to resolve overloaded and type-dependent operators. To give the general flavor of eval_binop, here are the cases corresponding to binary addition: τ1 τ2 v1 int( ) int( ) int(n1 ) float( ) float( ) float( f1 ) ptr(τ ) int( ) ptr(b, δ ) int( ) ptr(τ ) int(n) otherwise v2 int(n2 ) float( f2 ) int(n) ptr(b, δ ) eval binop(+, v1 , τ1 , v2 , τ2 ) ⌊int(n1 + n2 )⌋ ⌊float( f1 + f2 )⌋ ⌊ptr(b, δ + n × sizeof(τ ))⌋ ⌊ptr(b, δ + n × sizeof(τ ))⌋ 0/ The definition above rejects mixed arithmetic such as “int + float” because the parser that generates Clight abstract syntax (described in section 4.1) never produces this: it inserts explicit casts from integers to floats in this case. However, it would be easy to add cases dealing with mixed arithmetic. Likewise, the definition above adds two single precision floats using double-precision addition, in violation of the ISO C standard. Again, it would be easy to recognize this case and perform a single-precision addition. Rules 12 and 13 define the evaluation of conditional expressions a1 ? a2 : a3 . The predicates is_true and is_false determine the truth value of the value of a1 , depending on its type. At a float type, float(0.0) is false and any other float value is true. At an int or ptr type, int(0) is false and int(n) (n 6= 0) and ptr(ℓ) values are true. (The null pointer is represented as int(0).) All other combinations of values and types are neither true nor false, causing the semantics to go wrong. Rule 14 evaluates a cast expression (τ )a. The expression a is evaluated, and its value is converted from its natural type type(a) to the expected type τ using the partial function cast. This function performs appropriate conversions, truncations and sign-extensions between integers and floats. We take a lax interpretation of casts involving pointer types: if the 12 Outcome updates (at the end of a loop execution): loop Break ❀ Normal loop loop Return ❀ Return Return(v) ❀ Return(v) while loops: G,E ⊢ a,M ⇒ v is false(v,type(a)) ε G,E ⊢ a,M ⇒ v is true(v,type(a)) t G,E ⊢ s,M ⇒ out,M′ (23) G,E ⊢ (while(a) s),M ⇒ Normal,M loop out ❀ out ′ (24) t G,E ⊢ (while(a) s),M ⇒ out′ ,M′ G,E ⊢ a,M ⇒ v is true(v,type(a)) t1 t2 G,E ⊢ s,M ⇒ (Normal \| Continue),M1 G,E ⊢ (while(a) s),M1 ⇒ out′ ,M2 (25) t .t 1 2 G,E ⊢ (while(a) s),M ⇒ out′ ,M2 for loops: s1 6= skip t 1 G,E ⊢ s1 ,M ⇒ Normal,M1 t 2 G,E ⊢ (for(skip,a2 ,s3 ) s),M1 ⇒ out,M2 (26) t .t 1 2 G,E ⊢ (for(s1 ,a2 ,s3 ) s),M ⇒ out,M2 G,E ⊢ a2 ,M ⇒ v is false(v,type(a2 )) ε G,E ⊢ (for(skip,a2 ,s3 ) s),M ⇒ Normal,M G,E ⊢ a2 ,M ⇒ v is true(v,type(a2 )) t 1 out1 ,M1 G,E ⊢ s,M ⇒ (27) loop out1 ❀ out (28) t G,E ⊢ (for(skip,a2 ,s3 ) s),M ⇒ out,M1 G,E ⊢ a2 ,M ⇒ v is true(v,type(a2 )) t1 t2 G,E ⊢ s,M ⇒ (Normal \| Continue),M1 G,E ⊢ s3 ,M1 ⇒ Normal,M2 t 3 out,M3 G,E ⊢ (for(skip,a2 ,s3 ) s),M2 ⇒ (29) t .t .t 2 3 out,M3 G,E ⊢ (for(skip,a2 ,s3 ) ),M 1 ⇒ Fig. 9 Natural semantics for Clight loops source and destination types are both either pointer types or 32-bit int types, any pointer or integer value can be converted between these types without change of representation. However, the cast function fails when converting between pointer types and float or small integer types, for example. 3.3 Statements and function invocations, terminating case The rules in figure 8 define the execution of a statement that is neither a loop nor a switch statement. The execution of a skip statement yields the Normal outcome and the empty trace (rule 15). Similarly, the execution of a break (resp. continue) statement yields the Break (resp. Continue) outcome and the empty trace (rules 16 and 17). Rules 18–19 describe the execution of a return statement. The execution of a return statement evaluates the argument of the return, if any, and yields a Return outcome and the empty trace. Rule 20 executes an assignment statement. An assignment statement a1 = a2 evaluates the l-value a1 to a location ℓ and the r-value a2 to a value v, then stores v at ℓ using the storeval function of figure 7, producing the final memory state M ′ . We assume that the types of a1 and a2 are identical, therefore no implicit cast is performed during assignment, unlike in C. (The Clight parser described in section 4.1 inserts an explicit cast on the r-value 13 Function calls: G,E ⊢ a f un ,M ⇒ ptr(b,0) G,E ⊢ aargs ,M ⇒ vargs t G ⊢ Fd(vargs ),M ⇒ vres ,M′ funct(G,b) = ⌊Fd⌋ type of fundef(Fd) = type(a f un ) (30) t G,E ⊢ a f un (aargs ),M ⇒ vres ,M′ G,E ⊢ a,M ⇐ ℓ G,E ⊢ a f un ,M ⇒ ptr(b,0) G,E ⊢ aargs ,M ⇒ vargs t funct(G,b) = ⌊Fd⌋ type of fundef(Fd) = type(a f un ) G ⊢ Fd(vargs ),M ⇒ vres ,M1 storeval(type(a),M1 ,ptr(ℓ),vres ) = ⌊M2 ⌋ (31) t G,E ⊢ a = a f un (aargs ),M ⇒ vres ,M2 Compatibility between values, outcomes and return types: Normal,void # undef Return, void # undef Return(v), τ # v when τ 6= void Function invocations: F = τ id(dcl1 ) { dcl2 ; s } alloc vars(M,dcl 1 + dcl2 ,E) = (M1 ,b∗ ) bind params(E,M1 ,dcl1 ,vargs ) = M2 t G,E ⊢ s,M2 ⇒ out,M3 out, τ # vres (32) t G ⊢ F(vargs ),M ⇒ vres ,free(M3 ,b∗ ) Fe = extern τ id(dcl) ν = id( vargs ,vres ) ν (33) G ⊢ Fe(vargs ),M ⇒ vres ,M Fig. 10 Natural semantics for function calls a2 when necessary.) Note that storeval fails if a1 has a struct or union type: assignments between composite data types are not supported in Clight. The execution of a sequence of two statements starts with the execution of the first statement, thus yielding an outcome that determines whether the second statement must be executed or not (rules 21 and 22). The resulting trace is the concatenation of both traces originating from both statement executions. The rules in figure 9 define the execution of while and for loops. (The rules describing the execution of dowhile loops resemble the rules for while loops and are omitted in this paper.) Once the condition of a while loop is evaluated to a value v, if v is false, the execution of the loop terminates normally, with an empty trace (rules 23 and 27). If v is true, the loop body s is executed, thus yielding an outcome out (rules 24, 25, 28 and 29). If out is Normal or Continue, the whole loop is re-executed in the memory state modified by the first execution of the body. In s, the execution of a continue statement interrupts the current execution of the loop body and triggers the next iteration of s. If out is Break, the loop terminates normally; if out is Return, the loop terminates prematurely with the same outcome (rules 24 loop and 28). The ❀ relation models this evolution of outcomes after the premature end of the execution of a loop body. Rules 26–29 describe the execution of a for(s1 , a2 , s3 ) s loop. Rule 26 executes the initial statement s1 of a for loop, which must terminate normally. Then, the loop with an empty initial statement is executed in a way similar to that of a while loop (rules 27–29). If the body s terminates normally or by performing a continue, the statement s3 is executed before re-executing the for loop. As in the case of s1 , it must be the case that s3 terminates normally. 14 t T G,E ⊢ s1 ,M ⇒ ∞ T G,E ⊢ s2 ,M1 ⇒ ∞ G,E ⊢ s1 ,M ⇒ Normal,M1 (34) T (35) t.T G,E ⊢ s1 ;s2 ,M ⇒ ∞ G,E ⊢ s1 ;s2 ,M ⇒ ∞ G,E ⊢ a,M ⇒ v T G,E ⊢ s,M ⇒ ∞ is true(v,type(a)) (36) T G,E ⊢ (while(a) s),M ⇒ ∞ G,E ⊢ a,M ⇒ v is true(v,type(a)) t G,E ⊢ s,M ⇒ (Normal \| Continue),M1 T G,E ⊢ (while(a) s),M1 ⇒ ∞ (37) t.T G,E ⊢ (while(a) s),M ⇒ ∞ G,E ⊢ a f un ,M ⇒ ptr(b,0) funct(G,b) = ⌊Fd⌋ G,E ⊢ aargs ,M ⇒ vargs type of fundef(Fd) = type(a f un ) T G ⊢ Fd(vargs ),M ⇒ ∞ (38) T G,E ⊢ a f un (aargs ),M ⇒ ∞ F = τ id(dcl1 ) { dcl2 ; s } bind params(E,M1 ,dcl1 ,vargs ) = M2 alloc vars(M,dcl1 + dcl2 ,E) = (M1 ,b∗ ) T G,E ⊢ s,M2 ⇒ ∞ (39) T G ⊢ F(vargs ),M ⇒ ∞ Fig. 11 Natural semantics for divergence (selected rules) We omit the rules for switch(a) sw statements, which are standard. Based on the integer value of a, the appropriate case of sw is selected, and the corresponding suffix of sw is executed like a sequence, therefore implementing the “fall-through” behavior of switch cases. A Break outcome for one of the cases terminates the switch normally. The rules of figure 10 define the execution of a call statement a f un (aargs ) or a = a f un (aargs ). The expression a f un is evaluated to a function pointer ptr(b, 0), and the reference b is resolved to the corresponding function definition Fd using the global environment G. This function definition is then invoked on the values of the arguments aargs vres as per the judgment G ⊢ Fd(vargs ), M ⇒ t, M ′ . If needed, the returned value vres is then stored in the location of the l-value a (rules 30 and 31). The invocation of an internal Clight function F (rule 32) allocates the memory required for storing the formal parameters and the local variables of F, using the alloc_vars function. This function allocates one block for each variable id : τ , with lower bound 0 and upper bound sizeof(τ ), using the alloc primitive of the memory model. These blocks initially contain undef values. Then, the bind_params function iterates the storeval function in order to initialize formal parameters to the values of the corresponding arguments. The body of F is then executed, thus yielding an outcome (fourth premise). The return value of F is computed from this outcome and from the return type of F (fifth premise): for a function returning void, the body must terminate by Normal or Return and the return value is undef; for other functions, the body must terminate by Return(v) and the return value is v. Finally, the memory blocks b∗ that were allocated for the parameters and local variables are freed before returning to the caller. A call to an external function Fe simply generates an input/output event recorded in the trace resulting from that call (rule 33). 15 G = globalenv(P) M = initmem(P) t symbol(G,main(P)) = ⌊b⌋ funct(G,b) = ⌊ f ⌋ G ⊢ f (nil),M ⇒ int(n),M′ (40) ⊢ P ⇒ terminates(t,n) G = globalenv(P) symbol(G,main(P)) = ⌊b⌋ M = initmem(P) funct(G,b) = ⌊ f ⌋ T G ⊢ f (nil),M ⇒ ∞ (41) ⊢ P ⇒ diverges(T ) Fig. 12 Observable behaviors of programs 3.4 Statements and function invocations, diverging case Figure 11 shows some of the rules that model divergence of statements and function invocations. As denoted by the double horizontal bars, these rules are to be interpreted coinductively, as greatest fixpoints, instead of the standard inductive interpretation (smallest fixpoints) used for the other rules in this paper. In other words, just like terminating executions correspond to finite derivation trees, diverging executions correspond to infinite derivation trees [30]. A sequence s1 ; s2 diverges either if s1 diverges, or if s1 terminates normally and s2 diverges (rules 34 and 35). Likewise, a loop diverges either if its body diverges, or if it terminates normally or by continue and the next iteration of the loop diverges (rules 36 and 37). A third case of divergence corresponds to an invocation of a function whose body diverges (rules 38 and 39). 3.5 Program executions Figure 12 defines the execution of a program P and the determination of its observable behavior. A global environment and a memory state are computed for P, where each global variable is mapped to a fresh memory block. Then, the main function of P is resolved and applied to the empty list of arguments. If this function invocation terminates with trace t and result value int(n), the observed behavior of P is terminates(t, n) (rule 40). If the function invocation diverges with a possibly infinite trace T , the observed behavior is diverges(T ) (rule 41). 4 Using Clight in the CompCert compiler In this section, we informally discuss how Clight is used in the CompCert verified compiler [27, 6, 28]. 4.1 Producing Clight abstract syntax Going from C concrete syntax to Clight abstract syntax is not as obvious as it may sound. After an unsuccessful attempt at developing a parser, type-checker and simplifier from scratch, we elected to reuse the CIL library of Necula et al. [33]. CIL is written in OCaml and provides the following facilities: 16 1. A parser for ISO C99 (plus GCC and Microsoft extensions), producing a parse tree that is still partially ambiguous. 2. A type-checker and elaborator, producing a precise, type-annotated abstract syntax tree. 3. A simplifier that replaces many delicate constructs of C by simpler constructs. For instance, function calls and assignments are pulled out of expressions and lifted to the statement level. Also, block-scoped variables are lifted to function scope or global scope. 4. A toolkit for static analyses and transformations performed over the simplified abstract syntax tree. While conceptually distinct, (2) and (3) are actually performed in a single pass, avoiding the creation of the non-simplified abstract syntax tree. Thomas Moniot and the authors developed (in OCaml) a simple translator that produces Clight abstract syntax from the output of CIL. Much information produced by CIL is simply erased, such as type attributes and qualifiers. struct and union types are converted from the original named representation to the structural representation used by Clight. String literals are turned into global, initialized arrays of characters. Finally, constructs of C that are unsupported in Clight are detected and meaningful diagnostics are produced. The simplification pass of CIL sometimes goes too far for our needs. In particular, the original CIL transforms all C loops into while(1) { ... } loops, sometimes inserting goto statements to implement the semantics of continue. Such CIL-inserted goto statements are problematic in Clight. We therefore patched CIL to remove this simplification of C loops and natively support while, do and for loops CIL is an impressive but rather complex piece of code, and it has not been formally verified. One can legitimately wonder whether we can trust CIL and our hand-written translator to preserve the semantics of C programs. Indeed, two bugs in this part of CompCert were found during testing: one that we introduced when adding native support for for loops; another that is present in the unmodified CIL version 1.3.6, but was corrected since then. We see two ways to address this concern. First, we developed a pretty-printer that displays Clight abstract syntax tree in readable, C concrete syntax. This printer makes it possible to conduct manual reviews of the transformations performed by CIL. Moreover, experiment shows that re-parsing and re-transforming the simplified C syntax printed from the Clight abstract syntax tree reaches a fixed point in one iteration most of the time. This does not prove anything but nonetheless instills some confidence in the approach. A more radical way to establish trust in the CIL-based Clight producer would be to formally verify some of the simplifications performed. A prime candidate is the simplification of expressions, which transforms C expressions into equivalent pairs of a statement (performing all side effects of the expression) and a pure expression (computing the final value). Based on initial experiments on a simple “while” language, the Coq verification of this simplification appears difficult but feasible. We leave this line of work for future work. 4.2 Compiling Clight The CompCert C compiler is structured in two parts: a front-end compiler translates Clight to an intermediate language called Cminor, without performing any optimizations; a backend compiler generates PowerPC assembly code from the Cminor intermediate representation, performing good register allocation and a few optimizations. Both parts are composed of multiple passes. Each pass is proved to preserve semantics: if the input program P has observable behavior B, and the pass translates P to P′ without reporting a compile-time error, then the output program P′ has the same observable behavior B. The proofs of semantic 17 preservation are conducted with the Coq proof assistant. To facilitate the proof, the compiler passes are written directly in the specification language of Coq, as pure, recursive functions. Executable Caml code for the compiler is then generated automatically from the functional specifications by Coq’s extraction facility. The back-end part of CompCert is described in great detail in [28]. We now give an overview of the front-end, starting with a high-level overview of Cminor, its target intermediate language. (Refer to [28, section 4] for detailed specifications of Cminor.) Cminor is a low-level imperative language, structured like Clight into expressions, statements, and functions. A first difference with Clight is that arithmetic operators are not overloaded and their behavior is independent of the static types of their operands: distinct operators are provided for integer arithmetic and floating-point arithmetic. Conversions between integers and floats are explicit. Likewise, address computations are explicit in Cminor, as well as individual load and store operations. For instance, the C expression a[x] where a is a pointer to int is expressed as load(int32, a +i x i 4), making explicit the memory quantity being addressed (int32) as well as the address computation. At the level of statements, Cminor has only 5 control structures: if-then-else conditionals, infinite loops, block-exit, early return, and goto with labeled statements. The exit n statement terminates the (n + 1) enclosing block statements. Within Cminor functions, local variables can only hold scalar values (integers, pointers, floats) and they do not reside in memory. This makes it easy to allocate them to registers later in the back-end, but also prohibits taking a pointer to a local variable like the C operator & does. Instead, each Cminor function declares the size of a stack-allocated block, allocated in memory at function entry and automatically freed at function return. The expression addrstack(n) returns a pointer within that block at constant offset n. The Cminor producer can use this block to store local arrays as well as local scalar variables whose addresses need to be taken. To translate from Clight to Cminor, the front-end of CompCert C therefore performs the following transformations: 1. Resolution of operator overloading and materialization of all type-dependent behaviors. Based on the types that annotate Clight expressions, the appropriate flavors (integer or float) of arithmetic operators are chosen; conversions between ints and floats, truncations and sign-extensions are introduced to reflect casts; address computations are generated based on the types of array elements and pointer targets; and appropriate memory chunks are selected for every memory access. 2. Translation of while, do and for loops into infinite loops with blocks and early exits. The break and continue statements are translated as appropriate exit constructs. 3. Placement of Clight variables, either as Cminor local variables (for local scalar variables whose address is never taken), sub-areas of the Cminor stack block for the current function (for local non-scalar variables or local scalar variables whose address is taken), or globally allocated memory areas (for global variables). In the first version of the front-end, developed by Zaynah Dargaye and the authors and published in [6], the three transformations above were performed in a single pass, resulting in a large and rather complex proof of semantic preservation. To make the proofs more manageable, we split the front-end in two passes: the first performs transformations (1) and (2) above, and the second performs transformation (3). A new intermediate language called C#minor was introduced to connect the two passes. C#minor is similar to Cminor, except that it supports a & operator to take the address of a local variable. Accordingly, 18 the semantics of C#minor, like that of Clight, allocates one memory block for each local variable at function entrance, while the semantics of Cminor allocates only one block. To account for this difference in allocation patterns, the proof of semantic preservation for transformation (3) exploits the technique of memory injections formalized in [29, section 5.4]. It also involves nontrivial reasoning about separation between memory blocks and between sub-areas of a block. The proof requires about 2200 lines of Coq, plus 800 lines for the formalization of memory injections. The proof of transformations (1) and (2) is more routine: since the memory states match exactly between the original Clight and the generated C#minor, no clever reasoning over memory states, blocks and pointers is required. The Coq proof remains relatively large (2300 lines), but mostly because many cases need to be considered, especially when resolving overloaded operators. 5 Validating the Clight semantics Developing a formal semantics for a real-world programming language is no small task; but making sure that the semantics captures the intended behaviors of programs (as described, for example, by ISO standards) is even more difficult. The smallest mistake or omission in the rules of the semantics can render it incomplete or downright incorrect. Below, we list a number of approaches that we considered to validate a formal semantics such as that of Clight. Many of these approaches were prototyped but not carried to completion, and should be considered as work in progress. 5.1 Manual reviews The standard way to build confidence in a formal specification is to have it reviewed by domain experts. The size of the semantics for Clight makes this approach tedious but not downright impossible: about 800 lines of Coq for the core semantics, plus 1000 lines of Coq for dependencies such as the formalizations of machine integers, floating-point numbers, and the memory model. The fact that the semantics is written in a formal language such as Coq instead of ordinary mathematics is a mixed blessing. On the one hand, the typechecking performed by Coq guarantees the absence of type errors and undefined predicates in the specification, while such trivial errors are common in hand-written semantics. On the other hand, domain experts might not be familiar with the formal language used and could prefer more conventional presentations as e.g. inference rules. (We have not yet found any C language expert who is comfortable with Coq, while several of them are fluent with inference rules.) Manual transliteration of Coq specifications into LATEX inference rules (as we did in this paper) is always possible but can introduce or (worse) mask errors. Better approaches include automatic generation of LATEX from formal specifications, like Isabelle/HOL and Ott do [35, 43]. 5.2 Proving properties of the semantics The primary use of formal semantics is to prove properties of programs and meta-properties of the semantics. Such proofs, especially when conducted on machine, are effective at revealing errors in the semantics. For example, in the case of strongly-typed languages, type 19 soundness proofs (showing that well-typed programs do not go wrong) are often used for this purpose. In the case of Clight, a type soundness proof is not very informative, since the type system of C is coarse and unsound to begin with: the best we could hope for is a subject reduction property, but the progress property does not hold. Less ambitious sanity checks include “common sense” properties such as those of the field_offset function mentioned at end of section 2.1, as well as determinism of evaluation, which we obtained as a corollary of the verification of the CompCert compiler [28, sections 2.1 and 13.3]. 5.3 Verified translations Extending the previous approach to proving properties involving two formal semantics instead of one, we found that proving semantics preservation for a translation from one language to another is effective at exposing errors not only in the translation algorithm, but also in the semantics of the two languages involved. If the translation “looks right” to compiler experts and the semantics of the target language has already been debugged, such a proof of semantic preservation therefore generates confidence in the semantics of the source language. In the case of CompCert, the semantics of the Cminor intermediate language is smaller (300 lines) and much simpler than that of Clight; subsequent intermediate languages in the back-end such as RTL are even simpler, culminating in the semantics of the PPC assembly language, which is a large but conceptually trivial transition function [28]. The existence of semantic-preserving translations between these languages therefore constitutes an indirect validation of their semantics. Semantic preservation proofs and type soundness proofs detect different kinds of errors in semantics. For a trivial example, assume that the Clight semantics erroneously interprets the + operator at type int as integer subtraction. This error would not invalidate an hypothetical type soundness proof, but would show up immediately in the proof of semantic preservation for the CompCert front-end, assuming of course that we did not commit the same error in the translations nor in the semantics of Cminor. On the other hand, a type soundness proof can reveal that an evaluation rule is missing (this shows up as failures of the progress property). A semantic preservation proof can point out a missing rule in the semantics of the target language but not in the semantics of the source language, since it takes as hypothesis that the source program does not go wrong. 5.4 Testing executable semantics Just like programs, formal specifications can be tested against test suites that exemplifies expected behaviors. An impressive example of this approach is the HOL specification of the TCP/IP protocol by Sewell et al. [5], which was extensively validated against network traces generated by actual implementations of the protocol. In the case of formal semantics, testing requires that the semantics is executable: there must exist an effective way to determine the result of a given program in a given initial environment. The Coq proof assistant does not provide efficient ways to execute a specification written using inductive predicates such as our semantics for Clight. (But see [11] for ongoing work in this direction.) As discussed in [3], the eauto tactic of Coq, which performs Prolog-style resolution, can sometimes be used as the poor man’s logic interpreter to execute inductive predicates. However, the Clight semantics is too large and not syntax-directed enough to render this approach effective. 20 On the other hand, Coq provides excellent facilities for executing specifications written as recursive functions: an interpreter is built in the Coq type-checker to perform conversion tests; Coq 8.0 introduced a bytecode compiler to a virtual machine, speeding up the evaluation of Coq terms by one order of magnitude [15]; finally, the extraction facility of Coq can also be used to generate executable Caml code. The recommended approach to execute a Coq specification by inductive predicates, therefore, is to define a reference interpreter as a Coq function, prove its equivalence with the inductive specification, and evaluate applications of the function. Since Coq demands that all recursive functions terminate, these interpretation functions are often parameterized by a nonnegative integer counter n bounding the depth of the evaluation. Taking the execution of Clight statements as an example, the corresponding interpretation function is of the shape exec stmt(W, n, G, E, M, s) = Bottom(t) \| Result(t, out, M ′ ) \| Error where n is the maximal recursion depth, G, E, M are the initial state, and s the statement to execute. The result of execution is either Error, meaning that execution goes wrong, or Result(t, out, M ′ ), meaning that execution terminates with trace t, outcome out and final memory state M ′ , or Bottom(t), meaning that the maximal recursion depth was exceeded after producing the partial trace t. To handle the non-determinism introduced by input/output operations, exec_stmt is parameterized over a world W : a partial function that determines the result of an input/output operation as a function of its arguments and the input/output operation previously performed [28, section 13.1]. The following two properties characterize the correctness of the exec_stmt function with respect to the inductive specification of the semantics: t G, E ⊢ s, M ⇒ out, M ′ ∧ W \|= t ⇔ ∃n, exec stmt(W, n, G, E, M, s) = Result(t, out, M) T G, E ⊢ s, M ⇒ ∞ ∧ W \|= T ⇔ ∀n, ∃t, exec stmt(W, n, G, E, M, s) = Bottom(t) ∧ t is a prefix of T Here, W \|= t means that the trace t is consistent with the world W , in the sense of [28, section 13.1]. See [30] for detailed proofs of these properties in the simpler case of call-by-value λ calculus without traces. The proof of the second property requires classical reasoning with the axiom of excluded middle. We are currently implementing the approach outlined above, although it is not finished at the time of this writing. Given the availability of the CompCert verified compiler, one may wonder what is gained by using a reference Clight interpreter to run tests, instead of just compiling them with CompCert and executing the generated PowerPC assembly code. We believe that nothing is gained for test programs with well-defined semantics. However, the reference interpreter enables us to check that programs with undefined semantics do go wrong, while the CompCert compiler can (and often does) turn them into correct PowerPC code. 5.5 Equivalence with alternate semantics Yet another way to validate a formal semantics is to write several alternate semantics for the same language, using different styles of semantics, and prove logical implications between them. In the case of the Cminor intermediate language and with the help of Andrew Appel, we developed three semantics: (1) a big-step operational semantics in the style of the Clight semantics described in the present paper [27]; (2) a small-step, continuation-based semantics 21 [2, 28]; (3) an axiomatic semantics based on separation logic [2]. Semantics (1) and (3) were proved correct against semantics (2). Likewise, for Clight and with the help of Keiko Nakata, we prototyped (but did not complete yet) three alternate semantics to the big-step operational semantics presented in this paper: (1) a small-step, continuation-based semantics; (2) the reference interpreter outlined above; (3) an axiomatic semantics. Proving the correctness of a semantics with respect to another is an effective way to find mistakes in both. For instance, the correctness of an axiomatic semantics against a big-step operational semantics without traces can be stated as follows: if {P}s{Q} is a valid Hoare triple, then for all initial states G, E, M satisfying the precondition P, either the statement s diverges (G, E ⊢ s, M ⇒ ∞) or it terminates (G, E ⊢ s, M ⇒ out, M ′ ) and the outcome out and the final state G, E, M ′ satisfy postcondition Q. The proof of this property exercises all cases of the big-step operational semantics and is effective at pointing out mistakes and omissions in the latter. Extending this approach to traces raises delicate issues that we have not solved yet. First, the axiomatic semantics must be extended with ways for the postconditions Q to assert properties of the traces generated by the execution of the statement s. A possible source of inspiration is the recent work by Hoare and O’Hearn [20]. Second, in the case of a loop such as {P} while(a) s {Q}, we must not only show that the loop either terminates or diverges without going wrong, as in the earlier proof, but also prove the existence of the corresponding traces of events. In the diverging case, this runs into technical problems with Coq’s guardedness restrictions on coinductive definitions and proofs. In the examples given above, the various semantics were written by the same team and share some elements, such as the memory model and the semantics of Clight expressions. Mistakes in the shared parts will obviously not show up during the equivalence proofs. Relating two independently-written semantics would provide a more convincing validation. In our case, an obvious candidate for comparison with Clight is the Cholera semantics of Norrish [36]. There are notable differences between our semantics and Cholera, discussed in section 6, but we believe that our semantics is a refinement of the Cholera model. A practical issue with formalizing this intuition is that Cholera is formalized in HOL while our semantics is formalized in Coq. 6 Related work Mechanized semantics for C The work closest to ours is Norrish’s Cholera project [36], which formalizes the static and dynamic semantics of a large subset of C using the HOL proof assistant. Unlike Clight, Cholera supports side effects within expressions and accounts for the partially specified evaluation order of C. For this purpose, the semantics of expressions is given in small-step style as a non-deterministic reduction relation, while the semantics of statements is given in big-step style. Norrish used this semantics to characterize precisely the amount of non-determinism allowed by the C standard [37]. Also, the memory model underlying Cholera is more abstract than that of Clight, leaving unspecified a number of behaviors that Clight specifies. Tews et al [46, 47] developed a denotational semantics for a subset of the C++ language. The semantics is presented as a shallow embedding in the PVS prover. Expressions and statements are modeled as state transformers: functions from initial states to final states plus value (for expressions) or outcome (for statements). The subset of C++ handled is close to our Clight, with a few differences: side effects within expressions are allowed (and treated using a fixed evaluation order); the behavior of arithmetic operations in case of overflow is 22 not specified; the goto statement is not handled, but the state transformer approach could be extended to do so [45]. Using the Coq proof assistant, Giménez and Ledinot [14] define a denotational semantics for a subset of C appropriate as target language for the compilation of the Lustre synchronous dataflow language. Owing to the particular shape of Lustre programs, the subset of C does not contain general loops nor recursive functions, but only counted for loops. Pointer arithmetic is not supported. As part of the Verisoft project [40], the semantics of a subset of C called C0 has been formalized using Isabelle/HOL, as well as the correctness of a compiler from C0 to DLX assembly language [26, 44, 41]. C0 is a type-safe subset of C, close to Pascal, and significantly smaller than Clight: for instance, there is no pointer arithmetic, nor break and continue statements. A big-step semantics and a small-step semantics have been defined for C0, the latter enabling reasoning about non-terminating executions. Paper and pencil semantics for C Papaspyrou [39] develops a monadic denotational semantics for most of ISO C. Non-determinism in expression evaluation is modeled precisely. The semantics was validated by testing with the help of a reference interpreter written in Haskell. Nepomniaschy et al. [34] define a big-step semantics for a subset of C similar to Pascal: it supports limited uses of goto statements, but not pointer arithmetic. Abstract state machines have been used to give semantics for C [16] and for C# [7]. The latter formalization is arguably the most complete (in terms of the number of language features handled) formal semantics for an imperative language. Other examples of mechanized semantics Proof assistants were used to mechanize semantics for languages that are higher-level than C. Representative examples include [23, 25] for Standard ML, [38] for a subset of OCaml, and [24] for a subset of Java. Other Java-related mechanized verifications are surveyed in [18]. Many of these semantics were validated by conducting type soundness proofs. Subsets of C Many uses of C in embedded or critical applications mandate strict coding guidelines restricting programmers to a “safer” subset of C [19]. A well-known example is MISRA C [32]. MISRA C and Clight share some restrictions (such as structured switch statements with default cases at the end), but otherwise differ significantly. For instance, MISRA C prohibits recursive functions, but permits all uses of goto. More generally, the restrictions of MISRA C and related guidelines are driven by software engineering considerations and the desire for tool-assisted checking, while the restrictions of Clight stem from the desire to keep its formal semantics manageable. Several tools for static analysis and deductive verification of C programs use simplified subsets of C as intermediate representations. We already discussed the CIL intermediate representation [33]. Other examples include the Frama-C intermediate representation [8], which extends CIL’s with logical assertions, and the Newspeak representation [22]. CIL is richer than Clight and accurately represents all of ISO C plus some extensions. Newspeak is lower-level than Clight and targeted more towards static analysis than towards compilation. 7 Conclusions and future work In this article, we have formally defined the Clight subset of the C programming language and its dynamic semantics. While there is no general agreement on the formal semantics of 23 the C language, we believe that Clight is a reasonable proposal that works well in the context of the formal verification of a compiler. We hope that, in the future, Clight might be useful in other contexts such as static analyzers and program provers and their formal verification. Several extensions of Clight can be considered. One direction, discussed in [29], is to relax the memory model so as to model byte- and bit-level accesses to in-memory data representations, as is commonly done in systems programming. Another direction is to add support for some of the C constructs currently missing, in particular the goto statement. The main issue here is to formalize the dynamic semantics of goto in a way that lends itself well to proofs. Natural semantics based on statement outcomes can be extended with support for goto by following the approach proposed by Tews [45], but at the cost of nearly doubling the size of the semantics. Support for goto statements is much easier to add to transition semantics based on continuations, as the Cminor semantics exemplifies [28, section 4]. However, such transition semantics do not lend themselves easily to proving transformations of loops such as those performed by the front-end of CompCert (transformation 2 in section 4.2). 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CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS arXiv:1011.2083v4 [math.GR] 10 Mar 2016 MANOJ K. YADAV Abstract. In 1904, Issai Schur proved the following result. If G is an arbitrary group such that G/ Z(G) is finite, where Z(G) denotes the center of the group G, then the commutator subgroup of G is finite. A partial converse of this result was proved by B. H. Neumann in 1951. He proved that if G is a finitely generated group with finite commutator subgroup, then G/ Z(G) is finite. In this short note, we exhibit few arguments of Neumann, which provide further generalizations of converse of the above mentioned result of Schur. We classify all finite groups G such that \|G/ Z(G)\| = \|γ2 (G)\|d , where d denotes the number of elements in a minimal generating set for G/ Z(G). Some problems and questions are posed in the sequel. 1. Introduction In 1951, Neumann [18, Theorem 5.3] proved the following result: If the index of Z(G) in G is finite, then γ2 (G) is finite, where Z(G) and γ2 (G) denote the center and the commutator subgroup of G respectively. He mentioned [19, End of page 237] that this result can be obtained from an implicit idea of Schur [23], and his proof also used Schur’s basic idea. However there is no mention of this fact in [18] in which Schur’s paper is also cited. In this note, this result will be termed as ‘the Schur’s theorem’. Neumann also provided a partial converse of the Schur’s theorem [18, Corollary 5.41] as follows: If G is finitely generated by k elements and γ2 (G) is finite, then G/ Z(G) is finite, and bounded by \|G/ Z(G)\| ≤ \|γ2 (G)\|k . Our first motivation of writing this note is to exhibit an idea of Neumann[18, page 179] which proves much more than what is said above on converse of the Schur’s theorem. We quote the text here (with a minor modification in the notations): “Let G be generated by g1 , g2 , . . . , gk . Then Z(G) = ∩κ=k κ=1 CG (gκ ); for, an element of G lies in the centre if and only if it (is permutable) commutes with all the generators of G. If G is an FC-group (group whose all conjugacy classes are of finite length), then \|G : CG (gκ )\| is finite for 1 ≤ κ ≤ k, and Z(G), as intersection of a finite set of subgroups of finite index, also has finite index. The index of the intersection of two subgroups does not exceed the product of the indices of the subgroups: hence in this case one obtains an upper bound for the index of the centre, namely \|G : Z(G)\| ≤ Πκ=k κ=1 \|G : CG (gκ )\|.” Just a soft staring at the quoted text for a moment or two suggests the following. The conclusion does not require the group G to be FC-group. It only requires the finiteness of the conjugacy classes of the generating elements. If a generator of the group G is contained in Z(G), one really does not need to count it. Thus the argument works perfactly well even if G is 2010 Mathematics Subject Classification. Primary 20F24, 20E45. Key words and phrases. commutator subgroup, Schur’s theorem, class-preserving automorphism. 1 CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 2 generated by infinite number of elements, all but finite of them lie in the center of G. Thus the following result holds true. Theorem A. Let G be an arbitrary group such that G/ Z(G) is finitely generated by x1 Z(G), x2 Z(G), . . . , xt Z(G) and the conjugacy class of xi in G is of finite length for 1 ≤ i ≤ t. Then G G/ Z(G) is finite. Moreover \|G/ Z(G)\| ≤ Πti=1 \|xG i \| and γ2 (G) is finite, where xi denotes the conjugacy class of xi in G. Neumann’s result [18, Corollary 5.41] was reproduced by Hilton [12, Theorem 1]. It seems that Hilton was not aware of Neumann’s result. This lead two more publications [21] and [24] dedicated to proving special cases of Theorem A. Converse of the Schur’s theorem is not true in general as shown by infinite extraspecial pgroups, where p is an odd prime. It is interesting to know that example of such a 2-group also exists, which is mentioned on page 238 (second para of Section 3) of [19]. It is a central product of infinite copies of quaternion groups of order 8 amalgamated at the center of order 2. Our second motivation of writing this note is to provide a modification of an innocent looking result of Neumann [20, Lemma 2], which allows us to say little more on converse of the Schur’s theorem. A modified version of this lemma is the following. Lemma 1.1. Let G be an arbitrary group having a normal abelian subgroup A such that the index of CG (A) in G is finite and G/A is finitely generated by g1 A, g2 A, . . . , gr A, where \|giG \| < ∞ for 1 ≤ i ≤ r. Then G/Z(G) is finite. This lemma helps proving the first three statements of the following result. Theorem B. For an arbitrary group G, G/ Z(G) is finite if any one of the following holds true: (i) Z2 (G)/ Z(Z2 (G)) is finitely generated and γ2 (G) is finite. (ii) G/ Z(Z2 (G)) is finitely generated and G/(Z2 (G)γ2 (G)) is finite. (iii) γ2 (G) is finite and Z2 (G) ≤ γ2 (G). (iv) γ2 (G) is finite and G/ Z(G) is purely non-abelian. Our final motivation is to provide a classification of all groups G upto isoclinism (see Section 3 for the definition) such that \|G/ Z(G)\| = \|γ2 (G)\|d is finite, where d denotes the number of elements in a minimal generating set for G/ Z(G), discuss example in various situations and pose some problems. We conclude this section with fixing some notations. For an arbitrary group G, by Z(G), Z2 (G) and γ2 (G) we denote the center, the second center and the commutator subgroup of G respectively. For x ∈ G, [x, G] denotes the set {[x, g] \| g ∈ G}. Notice that \|[x, G]\| = \|xG \|, where xG denotes the conjugacy class of x in G. If [x, G] ⊆ Z(G), then [x, G] becomes a subgroup of G. For a subgroup H of G, CG (H) denotes the centralizer of H in G and for an element x ∈ G, CG (x) denotes the centralizer of x in G. 2. Proofs We start with the proof of Lemma 1.1, which is essentialy same as the one given by Neumann. Proof of Lemma 1.1. Let G/A be generated by g1 A, g2 A, . . . , gr A for some gi ∈ G, where 1 ≤ i ≤ r < ∞. Let X := {g1 , g2 , . . . , gr } and A be generated by a set Y . Then G = hX ∪ Y i and Z(G) = CG (X) ∩ CG (Y ). Notice that CG (A) = CG (Y ). Since CG (A) is of finite index, CG (Y ) is also of finite index in G. Also, since \|giG \| < ∞ for 1 ≤ i ≤ r, CG (X) is of finite index in G. Hence the index of Z(G) in G is finite and the proof is complete. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 3 Proof of Theorem A can be made quite precise by using Lemma 1.1. Proof of Theorem A. Taking A = Z(G) in Lemma 1.1, it follows that G/ Z(G) is finite. Moreover, \|G/ Z(G)\| = \|G/ ∩ti=1 CG (xi )\| ≤ Πti=1 \|G : CG (xi )\| = Πti=1 \|[xi , G]\| = Πti=1 \|xG i \|. That γ2 (G) is finite now follows from the Schur’s theorem. For the proof of Theorem B we need the following result of Hall [9] and the subsequent proposition. Theorem 2.1. If G is an arbitrary group such that γ2 (G) is finite, then G/ Z2 (G) is finite. Explicit bounds on the order of G/ Z2 (G) were first given by Macdonald [14, Theorem 6.2] and later on improved by Podoski and Szegedy [22] by showing that if \|γ2 (G)/(γ2 (G) ∩ Z(G))\| = n, then \|G/ Z2 (G)\| ≤ nc log2 n with c = 2. Proposition 2.2. Let G be an arbitrary group such that γ2 (G) is finite and G/ Z(G) is infinite. Then G/ Z(G) has an infinite abelian group as a direct factor. Proof. Since γ2 (G) is finite, by Theorem 2.1 G/ Z2 (G) is finite. Thus Z2 (G)/ Z(G) is infinite. Again using the finiteness of γ2 (G), it follows that the exponent of Z2 (G)/ Z(G) is finite. Hence by [6, Theorem 17.2] Z2 (G)/ Z(G) is a direct sum of cyclic groups. Let G/ Z2 (G) be generated by x1 Z2 (G), . . . , xr Z2 (G) and H := hx1 , . . . , xr i. Then it follows that modulo Z(G), H ∩ Z2 (G) is finite. Thus we can write Z2 (G)/ Z(G) = hy1 Z(G)i × · · · × hys Z(G)i × hys+1 Z(G)i × · · · , such that (H ∩ Z2 (G)) Z(G)/ Z(G) ≤ hy1 Z(G)i × · · · × hys Z(G)i. It now follows that the infinite abelian group hys+1 Z(G)i × · · · is a direct factor of G/ Z(G), and the proof is complete. We are now ready to prove Theorem B. Proof of Theorem B. Since γ2 (G) is finite, it follows from Theorem 2.1 that G/ Z2 (G) is finite. Now using the fact that Z2 (G)/ Z(Z2 (G)) is finitely generated, it follows that G/ Z(Z2 (G)) is finitely generated. Take Z(Z2 (G)) = A. Then notice that A is a normal abelian subgroup of G such that the index of CG (A) in G is finite, since Z2 (G) ≤ CG (A). Hence by Lemma 1.1, G/ Z(G) is finite, which proves (i). Again take Z(Z2 (G)) = A and notice that Z2 (G)γ2 (G) ≤ CG (A). (ii) now directly follows from Lemma 1.1. If Z2 (G) ≤ γ2 (G), then Z2 (G) is abelian. Thus (iii) follows from (i). Finally, (iv) follows from Proposition 2.2. This completes the proof of the theorem. We conclude this section with an extension of Theorem A in terms of conjugacy classpreserving automorphisms of given group G. An automorphism α of an arbitrary group G is called (conjugacy) class-preserving if α(g) ∈ g G for all g ∈ G. We denote the group of all class-preserving automorphisms of G by Autc (G). Notice that Inn(G), the group of all inner automorphisms of G, is a normal subgroup of Autc (G) and Autc (G) acts trivially on the center of G. A detailed survey on class-preserving automorphisms of finite p-groups can be found in [25]. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 4 Let G be the group as in the statement of Theorem A. Then G is generated by x1 , x2 , . . . , xt along with Z(G). Since Autc (G) acts trivially on the center of G, it follows that (2.3) \| Autc (G)\| ≤ Πti=1 \|xG i \| as there are only \|xG i \| choices for the image of each xi under any class-preserving automorphism. G Since \|xi \| is finite for each xi , 1 ≤ i ≤ t, it follows that \| Autc (G)\| ≤ Πti=1 \|xG i \| is finite. We have proved the following result of which Theorem A is a corollary, because \|G/ Z(G)\| = \| Inn(G)\| ≤ \| Autc (G)\|. Theorem 2.4. Let G be an arbitrary group such that G/ Z(G) is finitely generated by x1 Z(G), x2 Z(G), . . . , xt Z(G) and the conjugacy class of xi in G is of finite length for 1 ≤ i ≤ t. Then Autc (G) is finite. Moreover \| Autc (G)\| ≤ Πti=1 \|xG i \| and γ2 (G) is finite. Proof of Theorem A is also reproduced using IA-automorphisms (automorphisms of a group that induce identity on the abelianization) in [7, Theorem 2.3]. Proof goes on the same way as in the case of class-preserving automorphisms. 3. Groups with maximal central quotient We start with the following concept due to Hall [8]. For a group X, the commutator map aX : X/ Z(X) × X/ Z(X) → γ2 (X) given by aX (x1 Z(X), x2 Z(X)) = [x1 , x2 ] is well defined. Two groups K and H are said to be isoclinic if there exists an isomorphism φ of the factor group K̄ = K/ Z(K) onto H̄ = H/ Z(H), and an isomorphism θ of the subgroup γ2 (K) onto γ2 (H) such that the following diagram is commutative a K̄ × K̄ −−−G−→ γ2 (K)     φ×φy yθ a H̄ × H̄ −−−H−→ γ2 (H). The resulting pair (φ, θ) is called an isoclinism of K onto H. Notice that isoclinism is an equivalence relation among groups. The following proposition (also see Macdonald’s result [14, Lemma 2.1]) is important for the rest of this section. Proposition 3.1. Let G be a group such that G/ Z(G) is finite. Then there exists a finite group H isoclinic to the group G such that Z(H) ≤ γ2 (H). Moreover if G is a p-group, then H is also a p-group. Proof. Let G be the given group. Then by Schur’s theorem γ2 (G) is finite. Now it follows from a result of Hall [8] that there exists a group H which is isoclinic to G and Z(H) ≤ γ2 (H). Since \|γ2 (G)\| = \|γ2 (H)\| is finite, Z(H) is finite. Hence, by the definition of isoclinism, H is finite. Now suppose that G is a p-groups. Then it follows that H/ Z(H) as well as γ2 (H) are p-groups. Since Z(H) ≤ γ2 (H), this implies that H is a p-group. For an arbitrary group G with finite G/ Z(G), we have (3.2) \|G/ Z(G)\| ≤ \|γ2 (G)\|d , where d = d(G/ Z(G)). For simplicity we say that a group G has Property A if G/ Z(G) is finite and equality holds in (3.2) for G. We are now going to classify, upto isoclinism, all groups G having Property A. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 5 Let G be an arbitrary group having Property A. Then by Proposition 3.1 there exists a finite group H isoclinic to G and, by the definition of isoclinism, H has Property A. Thus for classifying all groups G, upto isoclinism, having Property A, it is sufficient to classify all finite group with this property. Let us first consider non-nilpotent finite groups. For such group we prove the following result in [5] Theorem 3.3. There is no non-nilpotent group G having Property A. So we only need to consider finite nilpotent groups. Since a finite nilpotent group is a direct product of it’s Sylow p-subgroups, it is sufficient to classify finite p-groups admitting Property A. Obviously, all abelian groups admit Property A. Perhaps the simplest examples of non-abelian groups having Property A are finite extraspecial p-groups. The class of 2-generated finite capable nilpotent groups with cyclic commutator subgroup also admits Property A. A group G is said ∼ H/ Z(H). Isaacs [13, Theorem 2] proved: to be capable if there exists a group H such that G = Let G be finite and capable, and suppose that γ2 (G) is cyclic and that all elements of order 4 in γ2 (G) are central in G. Then \|G/ Z(G)\| ≤ \|γ2 (G)\|2 , and equality holds if G is nilpotent. So G admits Property A, if G is a nilpotent group as in this statement and G is 2-generated. A complete classification of 2-generated finite capable p-groups of class 2 is given in [15]. Motivated by finite extraspecial p-groups, a more general class of groups G admitting Property A can be constructed as follows. For any positive integer m, let G1 , G2 , . . . , Gm be 2-generated finite p-groups such that γ2 (Gi ) = Z(Gi ) ∼ = X (say) is cyclic of order q for 1 ≤ i ≤ m, where q is some power of p. Consider the central product (3.4) Y = G1 ∗X G2 ∗X · · · ∗X Gm of G1 , G2 , . . . , Gm amalgamated at X (isomorphic to cyclic commutator subgroups γ2 (Gi ), 1 ≤ i ≤ m). Then \|Y \| = q 2m+1 and \|Y / Z(Y )\| = q 2m = \|γ2 (Y )\|d(Y ) , where d(Y ) = 2m is the number of elements in any minimal generating set for Y . Thus Y has Property A. Notice that in all of the above examples, the commutator subgroup is cyclic. Infinite groups having Property A can be easily obtained by taking a direct product of an infinite abelian group with any finite group having Property A. We now proceed to showing that any finite p-group G of class 2 having Property A is isoclinic to a group Y defined in (3.4). Let x ∈ Z2 (G) for a group G. Then, notice that [x, G] is a central subgroup of G. We have the following simple but useful result. Lemma 3.5. Let G be an arbitrary group such that Z2 (G)/ Z(G) is finitely generated by x1 Z(G), x2 Z(G), . . . , xt Z(G) such that exp([xi , G]) is finite for 1 ≤ i ≤ t. Then \| Z2 (G)/ Z(G)\| = t Y exp([xi , G]). i=1 Proof. By the given hypothesis exp([xi , G]) is finite for all i such that 1 ≤ i ≤ t. Suppose that exp([xi , G]) = ni . Since [xi , G] ⊆ Z(G), it follows that [xni i , G] = [xi , G]ni = 1. Thus xni i ∈ Z(G) and no smaller power of xi than ni can lie in Z(G), which implies that the order of xi Z(G) is Qt ni . Since Z2 (G)/ Z(G) is abelian, we have \| Z2 (G)/ Z(G)\| = i=1 exp([xi , G]). CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 6 Let Φ(X) denote the Frattini subgroup of a group X. The following result provides some structural information of p-groups of class 2 admitting Property A. Proposition 3.6. Let H be a finite p-group of class 2 having Property A and Z(H) = γ2 (H). Then (i) γ2 (H) is cyclic; (ii) H/ Z(H) is homocyclic; (iii) [x, H] = γ2 (H) for all x ∈ H − Φ(H); (iv) H is minimally generated by even number of elements. Proof. Let H be the group given in the statement, which is minimally generated by d elements x1 , x2 . . . , xd (say). Since Z(H) = γ2 (H), it follows that H/ Z(H) is minimally generated by x1 Z(H), x2 Z(H), . . . , xd Z(H). Thus by the identity \|H/ Z(H)\| = \|γ2 (H)\|d , it follows that order of xi Z(H) is equal to \|γ2 (H)\| for all 1 ≤ i ≤ d. Since the exponent of H/ Z(H) is equal to the exponent of γ2 (H), we have that γ2 (H) is cyclic and H/ Z(H) is homocyclic. Now by Qt Lemma 3.5, \|γ2 (H)\|d = \|H/ Z(H)\| = i=1 exp([xi , H]). Since [xi , H] ⊆ γ2 (H), this implies that [xi , H] = γ2 (H) for each i such that 1 ≤ i ≤ d. Let x be an arbitrary element in H − Φ(H). Then the set {x} can always be extended to a minimal generating set of H. Thus it follows that [x, H] = γ2 (H) for all x ∈ H − Φ(H). This proves first three assertions. For the proof of (iv), we consider the group H̄ = H/Φ(γ2 (H)). Notice that both H as well as H̄ are minimallay generated by d elements. Since [x, H] = γ2 (H) for all x ∈ H − Φ(H), it follows that for no x ∈ H − Φ(H), x̄ ∈ Z(H̄), where x̄ = xΦ(γ2 (H)) ∈ H̄. Thus it follows that Z(H̄) ≤ Φ(H̄). Also, since γ2 (H) is cyclic, γ2 (H̄) is cyclic of order p. Thus it follows that H̄ is isoclinic to a finite extraspecial p-group, and therefore it is minimally generated by even number of elements. Hence H is also minimally generated by even number of elements. This completes the proof of the proposition. Using the definition of isoclinism, we have Corollary 3.7. Let G be a finite p-group of class 2 admitting Property A. Then γ2 (G) is cyclic and G/ Z(G) is homocyclic. We need the following important result. Theorem 3.8 ([3], Theorem 2.1). Let G be a finite p-group of nilpotency class 2 with cyclic center. Then G is a central product either of two generator subgroups with cyclic center or two generator subgroups with cyclic center and a cyclic subgroup. Theorem 3.9. Let G be a finite p-group of class 2 having Property A. Then G is isoclinic to the group Y , defined in (3.4), for suitable positive integers m and n. Proof. Let G be a group as in the statement. Then by Proposition 3.1 there exists a finite p-group H isoclinic to G such that Z(H) = γ2 (H). Obviously H also satisfies \|H/ Z(H)\| = \|γ2 (H)\|d , where d denotes the number of elements in any minimal generating set of G/ Z(G). Then by Proposition 3.6, γ2 (H) = Z(H) is cyclic of order q = pn (say) for some positive integer n, and H/ Z(H) is homocyclic of exponent q and is of order q 2m for some positive integer m. Since Z(H) = γ2 (H) is cyclic, it follows from Theorem 3.8 that H is a central product of 2-generated CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 7 groups H1 , H2 , . . . , Hm . It is easy to see that γ2 (Hi ) = Z(Hi ) for 1 ≤ i ≤ m and \|γ2 (H)\| = q. This completes the proof of the theorem. We would like to remark that Theorem 3.9 is also obtained in [26, Theorem 11.2] as a consequence on study of class-preserving automorphisms of finite p-group. But we have presented here a direct proof. Now we classify finite p-groups of nilpoency class larger than 2. Consider the metacylic groups E D r+t r r+s t (3.10) K := x, y \| xp = 1, y p = xp , [x, y] = xp , where 1 ≤ t < r and 0 ≤ s ≤ t (t ≥ 2 if p = 2) are non-negative integers. Notice that the nilpotency class of K is at least 3. Since K is generated by 2 elements, it follows from (2.3) that \| Autc (K)\| ≤ \|γ2 (K)\|2 = p2r . It is not so difficult to see that \| Inn(K)\| = \|K/ Z(K)\| = p2r . Since Inn(K) ≤ Autc (K), it follows that \| Autc (K)\| = \| Inn(K)\| = \|γ2 (K)\|2 = \|γ2 (K)\|d(K) (That Autc (G) = Inn(G), is, in fact, true for all finite metacylic p-groups). Thus K admits Property A. Furthermore, if H is any 2-generator group isoclinic to K, then it follows that H admits Property A. For a finite p-groups having Property A, there always exists a p-group H isoclinic G such that \|H/ Z(H)\| = \|γ2 (H)\|d , where d = d(H). The following theorem now classifies, upto isoclinism, all finite p-groups G of nilpotency class larger than 2 having Property A. Theorem 3.11 (Theorem 11.3, [26]). Let G be a finite p-group of nilpotency class at least 3. Then the following hold true. (i) If \|G/ Z(G)\| = \|γ2 (G)\|d , where d = d(G), then d(G) = 2; (ii) If \|γ2 (G)/γ3 (G)\| > 2, then \|G/ Z(G)\| = \|γ2 (G)\|d if and only if G is a 2-generator group with cyclic commutator subgroup. Furthermore, G is isoclinic to the group K defined in (3.10) for suitable parameters; (iii) If \|γ2 (G)/γ3 (G)\| = 2, then \|G/ Z(G)\| = \|γ2 (G)\|d if and only if G is a 2-generator 2-group of nilpotency class 3 with elementary abelian γ2 (G) of order 4. It is clear that the groups G occuring in Theorem 3.11(iii) are isoclinic to certain groups of order 32. Using Magma (or GAP), one can easily show that such groups of order 32 are SmallGroup(32,k) for k = 6, 7, 8 in the small group library. We conclude this section with providing some different type of bounds on the central quotient of a given group. If \|γ2 (G) Z(G)/ Z(G)\| = n is finite for a group G, then it follows from [22, Theorem 1] that \|G/ Z2 (G)\| ≤ n2 log2 n . Using this and Lemma 3.5 we can also provide an upper bound on the size of G/ Z(G) in terms of n, the rank of Z2 (G)/ Z(G) and exponents of certain sets of commutators (here these sets are really subgroups of G) of coset representatives of generators of Z2 (G)/ Z(G) with the elements of G. This is given in the following theorem. Theorem 3.12. Let G be an arbitrary group. Let \|γ2 (G) Z(G)/ Z(G)\| = n is finite and Z2 (G)/ Z(G) is finitely generated by x1 Z(G), x2 Z(G), . . . , xt Z(G) such that exp([xi , G]) is finite for 1 ≤ i ≤ t. Then Yt exp([xi , G]). \|G/ Z(G)\| ≤ n2 log2 n i=1 4. Problems and Examples Theorem B provides some conditions on a group G under which G/ Z(G) becomes finite. It is interesting to solve CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 8 Problem 1. Let G be an arbitrary group. Provide a set C of optimal conditions on G such that G/ Z(G) is finite if and only if all conditions in C hold true. As we know that there is no finite non-nilpotent group G admitting Property A. Since Inn(G) ≤ Autc (G), it is interesting to consider Problem 2. Classify all non-nilpotent finite group G such that \| Autc (G)\| = \|γ2 (G)\|d , where d = d(G). A much stronger result than Theorem 3.3 is known in the case when the Frattini subgroup of G is trivial. This is given in the following theorem of Herzog, Kaplan and Lev [10, Theorem A] (the same result is also proved independently by Halasi and Podoski in [11, Theorem 1.1]). Theorem 4.1. Let G be any non-abelian group with trivial Frattini subgroup. Then \|G/ Z(G)\| < \|γ2 (G)\|2 . The following result with the assertion similar to the preceding theorem is due to Isaacs [13]. Theorem 4.2. If G is a capable finite group with cyclic γ2 (G) and all elements of order 4 in γ2 (G) are central in G, then \|G : Z(G)\| ≤ \|γ2 (G)\|2 . Moreover, equality holds if G is nilpotent. So, there do exist nilpotent groups with comparatively small central quotient. A natural problem is the following. Problem 3. Classify all finite p-groups G such that \|G : Z(G)\| ≤ \|γ2 (G)\|2 . Let G be a finite nilpotent group of class 2 minimally generated by d elements. Then it Qd follows from Lemma 3.5 that \|G/ Z(G)\| ≤ i=1 exp([xi , G]), which in turn implies (4.3) \|G/ Z(G)\| ≤ \|exp(γ2 (G))\|d . Problem 4. Classify all finite p-groups G of nilpotency class 2 for which equality holds in (4.3). Now we discuss some examples of infinite groups with finite central quotient. The most obvious example is the infinite cyclic group. Other obvious examples are the groups G = H × Z, where H is any finite group and Z is the infinite cyclic group. Non-obvious examples include finitely generated F C-groups, in which conjugacy class sizes are bounded, and certain Cernikov groups. We provide explicit examples in each case. Let Fn be the free group on n symbols and p be a prime integer. Then the factor group Fn /(γ2 (Fn )p γ3 (Fn )) is the required group of the first type, where γ2 (Fn )p = hup \| u ∈ γ2 (Fn )i. Now let H = Z(p∞ ) × A be the direct product of quasi-cyclic (Prüfer) group Z(p∞ ) and the cyclic group A = hai of order p, where p is a prime integer. Now consider the group G = H ⋊ B, the semidirect product of H and the cyclic group B = hbi of order p with the action by xb = x for all x ∈ Z(p∞ ) and ab = ac, where c is the unique element of order p in Z(p∞ ). This group is suggested by V. Romankov and Rahul D. Kitture through ResearchGate, and is a Cernikov group. The following problem was suggested by R. Baer in [1]. Problem 5. Let A be an abelian group and Q be a group. Obtain necessary and sufficient conditions on A and G so that there exists a group Q with A ∼ = Z(G) and Q ∼ = G/A. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 9 This problem was solved by Baer himself for an arbitrary abelian group A and finitely generated abelian group G. Moskalenko [16] solved this problem for an arbitrary abelian group A and a periodic abelian group G. He [17] also solved this problem for arbitrary abelian group A and a non-periodic abelian group G such that the rank of G/t(G) is 1, where t(G) denotes the tortion subgroup of G. If this rank is more than 1, then he solved the problem when A is a torsion abelian group. For a given group Q, the existence of a group G such that Q ∼ = G/ Z(G) has been studied extensively under the theme ‘Capable groups’. However, to the best of our knowledge, Problem 5 has been poorly studied in full generality. Let us restate a special case of this problem in a little different setup. A pair of groups (A, Q), where A is an arbitray abelian group and Q is an arbitrary group, is said to be a capable pair if there exists a group G such that A ∼ = Z(G) and Q ∼ = G/ Z(G). So, in our situation, the following problem is very interesting. Problem 6. Classify capable pairs (A, Q), where A is an infinite abelian group and Q is a finite group. Finally let us get back to the situation when G is a group with finite γ2 (G) but infinite G/ Z(G). The well known examples of such type are infinite extraspecial p-groups. Other class of examples can be obtained by taking a central product (amalgamated at their centers) of infinitely many copies of a 2-generated finite p-group of class 2 such that γ2 (H) = Z(H) is cyclic of order q, where q is some power of p. Notice that both of these classes consist of groups of nilpotency class 2. Now if we take G = X × H, where X is an arbitrary group with finite γ2 (X) and H is with finite γ2 (H) and infinite H/ Z(H), then γ2 (G) is finite but G/ Z(G) is infinite. So we can construct nilpotent groups of arbitrary class and even non-nilpotent group with infinite central quotient and finite commutator subgroup. A non-nilpotent group G is said to be purely non-nilpotent if it does not have any non-trivial nilpotent subgroup as a direct factor. With the help of Rahul D. Kitture, we have also been able to construct purely non-nilpotent groups G such that γ2 (G) is finite but G/ Z(G) is infinite. Let H be an infinite extra-special p-group. Then we can always find a field Fq , where q is some power of a prime, containing all pth roots of unity. Now let K be the special linear group sl(p, Fq ), which is a non-nilpotent group having a central subgroup of order p. Now consider the group G which is a central product of H and K amalgamated at Z(H). Then G is a purely non-nilpotent group with the required conditions. It will be interesting to see more examples of this type which do not occur as a central product of such infinite groups of nilpotency class 2 with non-nilpotent groups. By Proposition 2.2 we know that for an arbitrary group G with finite γ2 (G) but infinite G/ Z(G), the group G/ Z(G) has an infinite abelian group as a direct factor. Further structural information is highly welcome. Problem 7. Provide structural information of the group G such that γ2 (G) is finite but G/ Z(G) is infinite? References [1] R. Baer, Groups with preassigned central and central quotient group, Trans. Amer. Math. Soc. 44 (1938), 387-412. [2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 10 [3] J. M. Brady, R. A. Bryce, and J. Cossey, On certain abelian-by-nilpotent varieties, Bull. Austral. Math. Soc. 1 (1969), 403 - 416. [4] Y. Cheng, On finite p-groups with cyclic commutator subgroup, Arch. Math. 39 (1982), 295-298. [5] S. Dolfi, E. Pacifici and M. K. Yadav, Bounds for the index of the centre in non-nilpotent groups, preprint. [6] L. Fuchs, Infinite abelian groups, Vol. I Pure and Applied Mathematics, Vol 36 Academic Press, Yew York - London, 1970. [7] D. K. Gumber and H. Kalra, On the converse of a theorem of Schur, Arch. Math. (Basel) 101 (2013), 17-20. [8] P. Hall, The classification of prime power groups, Journal für die reine und angewandte Mathematik 182 (1940), 130-141. [9] P. Hall, Finite-by-nilpotent groups, Proc. Cambridge Phil. Soc. 52 (1956), 611-616. [10] M. Herzog, G. Kaplan and A. Lev, The size of the commutator subgroup of finite groups, J. Algebra 320 (2008), 980-986. [11] Z. Halasi and K. Podoski, Bounds in groups with trivial Frattini subgroups, J. Algebra 319 (2008), 893-896. [12] P. Hilton, On a theorem of Schur, Int. J. Math. Math. 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Soc. 21 (Series A) (1976), 467-472. [21] P. Niroomand, The converse of Schur’s theorem, Arch. Math. 94 (2010), 401-403. [22] K. Podoski and B. Szegedy, Bounds for the index of the centre in capable groups, Proc. Amer. Math. Soc. 133 (2005), 3441-3445. [23] I. Schur, Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, Journal für die reine und angewandte Mathematik 127 (1904), 20-50. [24] B. Sury, A generalization of a converse of Schur’s theorem, Arch. Math. 95 (2010), 317-318. [25] M. K. Yadav, Class-preserving automorphisms of finite p-groups: a survey, Groups St Andrews - 2009 (Bath), LMS Lecture Note Ser. 388 (2011), 569-579. [26] M. K. Yadav, Class-preserving automorphisms of finite p-groups II, Israel J. Math. 209 (2015), 355-396. DOI: 10.1007/s11856-015-1222-4 School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad - 211 019, INDIA E-mail address: myadav@hri.res.in	4
arXiv:1709.00410v2 [cs.NE] 10 Jan 2018 Visual art inspired by the collective feeding behavior of sand–bubbler crabs Hendrik Richter HTWK Leipzig University of Applied Sciences Faculty of Electrical Engineering and Information Technology Postfach 301166, D–04251 Leipzig, Germany. Email: hendrik.richter@htwk-leipzig.de. January 11, 2018 Abstract Sand–bubblers are crabs of the genera Dotilla and Scopimera which are known to produce remarkable patterns and structures at tropical beaches. From these pattern–making abilities, we may draw inspiration for digital visual art. A simple mathematical model is proposed and an algorithm is designed that may create such sand–bubbler patterns artificially. In addition, design parameters to modify the patterns are identified and analyzed by computational aesthetic measures. Finally, an extension of the algorithm is discussed that may enable controlling and guiding generative evolution of the art–making process. 1 Introduction Among the various forms of inspiration from biology for digital visual art, swarms and other types of collective animal behavior are considered to be particularly promising of showing algorithmic ideas and templates that may have the potential to create non–trivial patterns of aesthetic value. Examples are ant– and ant–colony–inspired visual art [12, 25], but our imagination has also been stimulated by the pattern– making of flies [1] and other swarms, flocks, or colonies [15, 22]. In this paper, the collective behavior of another family of animals provides a source of inspiration: crabs of the genera Dotilla and Scopimera, commonly called sand–bubblers. Sand–bubbler crabs are known for their burrow–orientated feeding behavior that produces radial patterns of regular shapes and designs in the wet sand of tropical beaches [2, 3, 7]. The patterns consist of tiny balls that are placed in curves or spirals, straight or bent lines, which finally form overall structures, thus producing astonishing works of natural art [4]. The creation of structures starts at receding tide. The patterns grow in complexity and scale as time goes by until the returning tide cleans them off, in fact resetting the art–making process. This pattern–making behavior, its visual results, and the potentials of thus inspired algorithms producing digital visual art are our topics. We approach these topics by discussing three interconnected main themes in this paper. The first is to describe an aspect of animal behavior observed in nature (the feeding pattern of a colony of sand–bubbler crabs) by economic mathematical means in order to make it accessible for an algorithmic process producing two–dimensional visual art. This goes along with identifying design parameters that allow the patterns to vary within the restriction that they should generally resemble structures as found in nature. The second theme is to study the image–producing process by computational aesthetic measures. Here, the focus is not on numerically evaluating an overall artistic value. There are some strong arguments that such an evaluation is rather elusive and may not really be achievable free from ambiguity, see also the discussion in Sec. 3. Instead, the focus is put on how some aesthetic measures, namely Benford’s law measure, Ross, 1 Figure 1: Feeding patterns of sand–bubbler crabs, observed by the author at Tanjung Beach, Langkawi, Malaysia, in March 2017. Ralph and Zong’s bell–curve measure and a fractal dimension measure, scale to the design parameters. A third theme is to use these measures to guide and control the art–making process. This is done by updating the design parameters while the algorithmic process is running. By employing the relationships between aesthetic measures and design parameters for updating the parameters, the measures can be changed in an intended way. Hence, this theme studies the possibilities of including generative development, which takes up some ideas from evolving and evolutionary art. The paper is structured as follows. Sec. 2 reviews the behavior of sand–bubbler crabs, particularly the burrow–orientated feeding process. Inspired by these behavioral patterns, Sec. 3 presents an algorithm to create visual art, studies computational aesthetic measures and generative development, and finally shows examples of art pieces thus created. Concluding remarks and an outlook end the paper. 2 Behavior of sand–bubbler crabs Sand–bubbler crabs are tiny crustacean decapods of the genera Dotilla and Scopimera in the family Dotillidae that dwell on tropical sandy beaches across the Indo–Pacific region. They permanently inhabit intertidal environments as they adapted to the interplay between high and low tide. Sand–bubblers are surface deposit feeders. Thus, their habitat must be periodically covered by water for their food supply to restore. They are also air–breathers. Thus, the habital zone needs to be periodically uncovered from water for giving them time for feeding. Observing these periodic changes, the crabs dig burrows in the sand where they seek shelter and hide during high tide. In low tide (and daylight) they swarm out in search for food. They pick up sand grains with their pincers and eat the microscopic organic particles that are left over by the receding tide and coat the grains. After they have cleaned the sand grains of eatable components, they form them into small balls, also called pseudo–faecal pellets, and throw them behind. Supposedly, they do so in order to avoid searching the same sand twice. As high tide comes back in, they retreat to their burrows, seal them to trap air, and wait for the next tidal cycle. By this process of feeding, and particularly by molding sand into pellets and distributing them, sand– bubbler crabs create intricate patterns in the sand that remain there until the next high tide washes them away, see Fig. 1 that shows such feeding patterns as observed by the author at Tanjung Beach, Langkawi, Malaysia in March 2017. The patterns in the sand start from a center point, have a radial orientation and comprise of small sand balls along a tiny path. The patterns vary in complexity depending on how many crabs there are contributing, on how long the last high tide is away, and on several other factors discussed below. The patterns in the sand are starting point and inspiration for an algorithm producing two–dimensional visual art. Before presenting such an algorithm in Sec. 3, it may be interesting to explore the feeding process and the pattern–making a bit more deeply. A main feature of the lifestyle of sand–bubblers is their burrow–orientated feeding behavior [2, 3, 5, 7]. 2 This means that the entrance of the burrow is the center of the feeding range and the feeding process is highly structured. Basically, such a structured and stereotype behavior can be found generally among the 14 species of Dotilla and 17 species of Scopimera currently recognized, but there is variety in the details of the feeding. This, in turn, offers to vary the art–making algorithm in order to achieve different graphic effects. The feeding starts from the burrow entrance, where the crab sets out to progress sideways along a straight line radiating from the burrow. While moving along, it sorts sand from organic–rich fine particles which are scraped off and subsequently ingested. The crab only feeds on the upper millimeter or two of the sand. The residual larger nutrient–deficient sand grains are molded into ball–like globules, but not digested, which motivates calling them pseudo–faecal pellets. When a pellet has reached a certain diameter, which scales to the size of the crab, it is moved along the body and pushed away behind the animal. After moving along the straight line radiating from the burrow and ejecting pellets for a certain distance, the crab steps aside, turns around and moves back toward the burrow entrance while continuing with feeding and pellet ejecting. When reaching the entrance, its turns around again, and works the next line (see also Fig. 2 for a schematic description). The pellets are placed in such a way that a straight path to the burrow entrance always remains clear. This is essential for the crab to retreat to the burrow in case of external disturbance or danger. Consequently, the feeding progresses radially line by line while the lines (which are also called trenches in the biological literature) rotate around the burrow entrance just like the hands of a clock. In fieldwork it has been observed that the rotation may be clockwise or anti–clockwise with approximately the same frequency [17, 20]. By such a rotation of the feeding trench, an angular sector of a circle in the sand is excavated, cleared of eatable components, and amassed with pellets. The angular sector of uninterrupted feeding may vary; an average of 60◦ has been reported, but also that occasionally a full circle has been completed [2, 13, 17], see also the right–hand side of Fig. 1. Furthermore, it has been reported that there is a fairly close relationship between the maximum length of a trench and the size of the crab. Such a relationship is plausible as the maximum length together with the crab size scales to the maximum time needed for the crab to seek shelter in the burrow, if an external disturbance requires such an escape. For one species of sand–bubblers, Dotilla wichmanni, even a measurement of the maximum length has been recorded with the body length of eight to nine crabs [17]. Furthermore, another stereotype of pattern has been identified as sometimes the distribution of pellets along the trench follows a rhythmic pattern where the crab regularly suspends pellet ejecting and “concentric rings” emerge, as shown for Dotilla clepsydrodactylus [20]. In view of the structured burrow–orientated feeding behavior of sand–bubbler crabs as described above, it seems not surprising that regular patterns in the sand emerge. Ideally, a well–ordered collection of full circular areas with the burrow entrance in the center should appear that are filled with radial strings of equally distanced pellets. However, looking at patterns from nature (see, for instance, Fig. 1) reveals them as irregular in several respects. There are reasons for these irregularities. A first reason is that for a colony of sand–bubblers the coordinates of the burrow entrances are randomly distributed across the beach where they are dwelling. There is almost always a certain minimum distance between two burrow entrances and the number of burrows for a given square of beach follows a Poisson distribution, as shown for the species Dotilla fenestrata [11]. A second reason is that the orientation of the first trench (and hence the overall orientation of the angular sector) is not different from a random distribution [17, 20]. There is no indication that the orientation is related to the slope of the sand surface, or any compass direction or the sun’s position. Another reason, for instance reported for the species Scopimera inflata [7], is that a crab may move a short distance away from the entrance of the burrow before starting to feed, which results in a gap between burrow and trench. The width of the gap appears to be randomly distributed around the crab’s size. A further reason is that for a given burrow (and therefore a crab of given size) the trenches have not all the same length, even if the feeding is undisturbed [11]. There is a weak correlation between trench length and feeding time, with the trenches growing longer the longer the crab feeds, but generally also trench length must be regarded as to follow a random distribution. Finally, a crab may be externally disturbed at any time, for instance by intruders such as other crabs or human observers, or eating enemies such as shorebirds. 3 y j=36 s Cs j = 4 j=2 Cs J B J C B J^ BMB j = 1 J J Cs s rk J Cs M J Cs Pi j =5P θ1 PP D P JC u x D T (xi , yi ) > j = 6 DD Figure 2: Algorithmically generating a pellet location with the burrow coordinates (xi , yi )T , the trench angle θ1 of the (lower branch of the) first trench and the radial coordinate rk ; see also the first term of the right–hand side of Eq. (1). The arrows indicate the feeding direction of a sand–bubbler crab. The outbound trench j = 3 depicts a radial string of six equally distanced pellets, while for j = 1 there is one pellet; for the other trenches the pellets are not shown. As sand–bubblers are extremely perceptive to movement, in such a case the crab rapidly escapes to the burrow. It may reemerge a little later and resume eating. The feeding may or may not continue along the trench last used before escape. It has been convincingly argued by field experiments that sand–bubblers (for instance the species Dotilla wichmanni) must possess remarkable orientation abilities that allow them to resume the direction along which they were feeding, even in the absence of visual clues such as a partly finished trench [17]. Whether or not a crab uses these abilities in a given situation, however, also appears to follow a random distribution. Lastly, a crab may interrupt its feeding even if there is no evident external trigger, for instance to rest or to restore its water supply, which is done in the burrow. To sum it up: The exact position of each pellet follows from the structure of the burrow–oriented feeding behavior, but is subject to a substantial degree of random. If we reframe the organized and sequential placement of a sufficiently large number of pellets as a dynamic process, the pattern generation could be interpreted as a random walk substantially biased by the feeding structure. This interplay between determinism and random may be one of the reasons why the patterns in the sand appear non–trivial and of some aesthetic value. Finally, we may note that apart from feeding, there are at least two further aspects of the sand–bubbler’s lifestyle that might be interesting as inspiration for visual art, burrow–digging and interaction with other crabs. These aspects could be addressed in future work. 3 3.1 Design of art–making algorithms Generating patterns from pellet distribution The feeding behavior of sand–bubbler crabs as described in Sec. 2 is now used to design templates for algorithmically generating two–dimensional visual art. With respect to the regular and irregular aspects of the sand patterns, the art–making algorithm is also comprised of deterministic and random elements. Describing the algorithm starts with the notion that every pellet has a location (xijk , yijk )T in a two– dimensional plane. The index (ijk) refers to the k–th pellet (k = 1, 2, 3, . . . , Kij ) belonging to the j–th trench (j = 1, 2, 3, . . . , Ji ) of the i–th burrow (i = 1, 2, 3, . . . , I). The pellet location is algorithmically specified by 2 ) N (µijk , σijk xijk xi + rk · cos (θj ) = + , (1) 2 ) yijk yi + rk · sin (θj ) N (µijk , σijk where (xi , yi )T are the coordinates of the i–th burrow, θj is the trench angle belonging to the j–th trench, and rk is the radial coordinate of the k–th pellet. For each burrow, the maximum number of pellets Kij remains to be specified for a given i and j; the same applies to the maximum number of trenches Ji . Fig. 2 unpacks the description and shows an example with one burrow and six trenches (three outbound and three returning). 4 Algorithm 1 Generate patterns from pellet distribution Set maximum number of burrows I for i=1 to I do Generate burrow coordinates (xi , yi )T Set maximum number of trenches Ji for j=1 to Ji do Generate trench angle θj Set maximum number of pellets Kij for k=1 to Kij do Generate pellet radial coordinate rk 2 Set mean µijk and variance σijk Calculate pellet location (xijk , yijk )T by Eq. (1) Return location and store for subsequent coloring and visualization end for end for end for The first term of the right–hand side of Eq. (1) can be seen as to represent the deterministic aspect of the pellet ejecting that characterizes the burrow–orientated feeding behavior of sand–bubblers. The second term represents the random aspect with shifting the position by a realization of a random variable 2 . Accordingly, every pellet location (x , y )T has normally distributed with mean µijk and variance σijk ijk ijk three deterministic parameters: burrow coordinates (xi , yi )T , trench angle θj and radial coordinate rk ; 2 of the normal distribution from which the and two stochastic parameters: mean µijk and variance σijk random shift in position is realized. Eq. (1) not only specifies the location of a pellet, it can also be used to generate patterns consisting of pellets, trenches and burrows, see Algorithm 1. Therefore, for each burrow coordinates (xi , yi )T , one sequence of trench angles (θ1 , θ2 , . . . , θJi ) and Ji sequences of pellet radial coordinates (r1 , r2 , . . . , rKij ) are defined. A major guideline for defining these parameter sequences is the intention to replicate characteristic features of the sand–bubbler pattern as discussed in Sec. 2. These patterns can be labeled according to the graphic effects they convey and subsequently taken as templates to create two–dimensional visual art works. In the numerical experiments reported in Sec. 3.4 the following templates are used: (i) random trench length (RTL), (ii) growing trench length (GTL), (iii) concentric rings (CCR), and (iv) burrow–to–trench gaps (BTG). Each template varies the art–making algorithm and replicates a specific pattern found in nature. For random trench length (RTL), the maximal number of trenches Ji is a random integer uniformly distributed on [1, Jmax ]. The first and the last trench angle (θ1 and θJi ) are chosen as realizations of a uniform random variable distributed on [0, 2π]; the remaining trench angles are calculated from the number Ji to be equidistant between θ1 and θJi . Each trench has a length `j that is a realization of a random variable normally distributed on [µT , σT2 ]. From the length `j , the pellet radial coordinates are calculated to be equidistant with distance dj . Growing trench length (GTL) is similar to RTL (and uses the same general design parameters), but the trench length `j is calculated by a normal distribution where the mean µT grows with trench j by a growth rate ∆µT . Thus, the first trench has mean µT , while the Ji –th trench has mean µT + (Ji − 1)∆µT . The variance σT2 is constant. Concentric rings (CCR) again have trenches with constant mean and variance but do not always have equidistant pellet radial coordinates as there are λ gaps of width gλ where no pellets are placed. Finally, burrow–to–trench gaps (BTG) start with a gap of width g0 in pellet placement. Tab. 1 summarizes the design parameters. Algorithm 1 specifies the location of the pellets and hence the structure of the patterns. In nature, the patterns are monochromatic as the pellets all have the color of the sand they are built from, see Fig. 1. The artistic interpretation of the patterns suggests using different colors for making them more visually appealing and also conveying additional information. As the algorithm imposes a chronological order on pellet placement by a forward counting of burrows, trenches and finally the pellets themselves, a possible 5 Table 1: Design parameters and default values for generating patterns from pellet distribution, see Algorithm 1. The parameters of RTL are general and also apply to the other templates. Template Design parameter Symbol Default value RTL Number of burrows I 3 T (and Burrow coordinates (xi , yi ) N (0, 7) general) Maximum number of trenches Jmax 50 Pellet distance dj 0.25 Mean random shifting µijk −1, 0, 1 2 Variance random shifting σijk 0.3, 0.8 Mean trench length µT 25 Variance trench length σT2 1 GTL Grow rate ∆µT 2 CCR Number of gaps λ 3 Gap width gλ 4 BTG Burrow gap width g0 8 design is to use a colorization that reflects this order. Alternatively, each burrow can be associated with a specific color and the order of the pellet placement can be characterized by shades or brightness of this color. In addition, the burrows using the same template (see Tab. 1) may take colors that are similar. 3.2 Analysis by computational aesthetic measures Next, the templates and their design parameters are analyzed by computational aesthetic measures. This analysis aims at studying two questions. (i) To what extend do the measures vary over the templates and parameters? This goes along with assessing how sensitive these relationships are. (ii) Which templates and/or parameters are most suitable to guide and control the art–making process toward desired values of the aesthetic measures? Three computational aesthetic measures are studied, Benford’s law measure (BFL), Ross, Ralph and Zong’s bell curve measure (RRZ) and a fractal dimension measure (FRD), which were chosen because a recent study reported them to be weakly correlated [6]. All measures are calculated for the images with 512 × 512 pixels. The Benford’s law measure BFL [6, 21] is known for scaling to the naturalness of an image. It is calculated by taking the distribution of the luminosity of an image and comparing it to the Benford’s law distribution: BFL = 1 − dtotal /dmax (2) P9 where dtotal = i=1 (HI (i) − HB (i)), dmax = 1.398 is the maximum possible value of dtotal , HB = (0.301, 0.176, 0.125, is the Benford’s law distribution and HI is the normalized sorted 9–bin histogram of the luminosity of the image. The luminosity of the image is calculated for each pixel of the image by taking the weighted sum of the red (R), green (G) and blue (B) values: lum = 0.2126 · R + 0.7152 · G + 0.0722 · B. Ross, Ralph and Zong’s bell–curve measure RRZ [6, 23] is a measure of color gradient normality. It compares the color gradients of the image to a normal (bell-curve or Gaussian) distribution. Therefore, the color gradient of the colors red, green and blue are calculated for each pixel, summarized and normalized by a detection threshold, see [6, 23] for details. Hence, we obtain a distribution with mean µ and variance σ 2 . The distribution is discretized into 100 bins to obtain a discrete color gradient distribution pi . For µ and σ 2 calculated from the observed color gradient distribution, an expected discrete Gaussian distribution qi is computed. Finally, the Kullback–Leibler divergence between these two distributions gives the bell–curve measures X RRZ = pi log (pi /qi ). (3) i 6 Figure 3: Aesthetic measures as a function of design parameters: number of burrows I, maximum number of trenches Jmax and pellet distance dj . RTL (blue), GTL (magenta), CCR (red), BTG (green); solid lines, 2 = 0.3, dotted lines, σ 2 = 0.8. σijk ijk Note that in deviation from [6, 23], the Kullback–Leibler divergence in Eq. (3) is not amplified by the factor 1000. The third aesthetic measure studied is the fractal dimension FRD [6, 24] of the image. Following empirical studies and an argument by Spehar et al. [24], a fractal image of dimension 1.3 ≤ d ≤ 1.5 is most preferred by human evaluation, compared to images that have a fractal dimension outside this range. Thus, a fractal dimension d ≈ 1.35 is considered to be most desirable and the fractal aesthetic measure FRD = max (0, 1 − \|1.35 − d\|) (4) can be defined, where d is the fractal dimension of the image calculated by box–counting. Fig. 3 shows the measures for the templates RTL, GTL, CCR and BTG for selected design parameters 2 . The results are means over 100 images generated and two values of the variance of the random shifting σijk by Algorithm 1 and the design parameters listed in Tab. 1. The design parameter number of burrows I is varied over 2 ≤ I ≤ 10, the maximum number of trenches Jmax over 20 ≤ Jmax ≤ 100 and the pellet distance dj over 0.05 ≤ dj ≤ 1. The default values of these parameters all lie within the range studied. 7 Furthermore, as the aesthetic measures are calculated from the RGB values of the pixels, the colors of the image have a profound effect on the values of the measures. As discussed in Sec. 3.1, the colorization of the pellets is also a subject of algorithmic design. Thus, to counter the effect of a given colorization and for not singularizing particular colors, the results are calculated as means over 70 hue values randomly selected. From the results in Fig. 3, we see that the different templates (RTL, GTL, CCR, BTG) generally produce characteristic curves over the design parameters for the aesthetic measures BFL and RRZ. Also, the two values of random shifting (which can be interpreted as noise levels) are clearly distinct. The general trend is that BFL and RRZ decrease with increasing number of burrows I and increasing maximum number of trenches Jmax , while there is an increase with increasing pellet distance dj . An interesting exception is the template CCR which for varying pellet distance dj behaves differently and crosses the curves of the other templates. The results for the aesthetic measures FRD are slightly different. For FRD, the templates RTL and BTG (blue and green lines) have almost the same results and for all templates, different noise levels have only negligible effect. A possible explanation is that the difference between RTL and BTG is only that in the images there is a circle free from pellets around the burrow entrance for BTG. The overall structure of the pellet placement is the same, which is recognized by the calculation of the fractal dimension accounting mainly for density of geometrical objects but less for their layout. The same may apply for varying the noise level. Based on these results assessing how sensitive the relationships between design parameters and aesthetic measures are, some conclusions can be drawn about possible ways to guide and control the art– making algorithm. Basically, there are two options. One is to switch to a particular template but keep the values of design parameters, another is to keep the template but change the values of design parameters. (In fact, it would also be thinkable to change both template and parameter value at the same time, but this may appear rather contrived.) For instance, to increase the aesthetic measure BFL, employing the template BTG is suitable, while RRZ is largest for RTL (but not very clearly) and FRD for GTL. Another possibility to raise BFL is to decrease the maximum number of trenches Jmax . The same is also useful to increase RRZ, particularly for the template CCR, but not for FRD. There are more subtle schemes to manipulate the measures, but it is also clear that the three aesthetic measures studied here react differently, yet even contradictory to these manipulations. With this in mind, next section discusses possible designs of such manipulations and their effects on the art–making process. 3.3 Guiding and controlling generative evolution Main features of generative and evolving digital art are that the creating process develops over time, is guided and controlled by the algorithmic design, and thus gains functional autonomy to produce complete art works [10]. A frequently used design for guidance and control is to implement a feedback, for instance via evaluating the works by using a measure of their aesthetic value [18, 22]. Such ideas are, for instance, implemented by employing evolutionary algorithms for finding “optimal” values of the aesthetic measures, which has shown to create interesting works of art [1, 6, 14, 22]. Algorithm 1 given in Sec. 3.1 does not really, in itself, evoke such a perspective of autonomously guiding and controlling generative evolution. The algorithm is suitable for producing visual art works. It also depends on selecting templates and design parameters (see Tab. 1), which enables modifying the works by changing the selection. Thus, promoting generative evolution essentially requires measuring the aesthetic value of the algorithmically generated patterns and designing a feedback. Of course, it is possible to evaluate the pattern by interaction with humans. However, there are some issues with interactive evaluation [8, 9, 22]. The first is that humans evaluating art are usually much slower in doing so than computers algorithmically generating the works, which creates the proverbial “fitness bottleneck”. A second reason is that human evaluation may change over time, for instance caused by fatigue, but also by boredom, which may bias the results towards superficial novelty rather than overall quality. A third reason is that human evaluation of art is always conditioned by personal and cultural “taste”. For this reason, it is no real help to distribute the evaluation to a larger group of human evaluators. In such a case, either the tastes of different subgroups become visible (as for instance shown for evaluating 8 Algorithm 2 Guide and control pattern–making Set up look–up table with template and design parameters that increase computational aesthetic measures Calculate expected aesthetic measures for the look–up table Select random template and take default design parameter while Termination criterion not met do Generate burrow by Algorithm 1 Calculate aesthetic measures if Measure < Expectation then Select randomly increasing template or design parameter else Keep template and parameter end if end while the beauty of abstract paintings [19]), or generally the averaging effect of asking a large group of people about their aesthetic choices is reproduced, but arguably “the unique kind of vision expected from artists” [8] is not achieved. An alternative are computational aesthetic measures [6, 8, 9, 19, 21], which enable an automated rating without human interference or supervision. Sec. 3.2 discussed such measures for evaluating sand–bubbler patterns. These aesthetic measures of two–dimensional visual art are frequently calculated by analyzing the (spatial) distribution of colored pixels (or groups of pixels). In other words, computational aesthetic measures use properties such as pixel–wise color information to deduce an overall rating of an image. However, there is a significant conceptual gap between measuring computational image properties and aesthetic beauty. Apart from the measures considered in Sec. 3.2, there is a huge number of measures that try to bridge this gap, see e.g. [9, 6, 16, 19] for an overview. Some of them are strongly correlated, while others address different aspects of the color distribution. Moreover, recent studies [6, 19] suggested that it seems rather difficult to establish stable correlations between beauty ratings by humans and image properties, which casts some doubts on whether an evaluation of the aesthetic value of visual art based on computational aesthetic measures is really objective, meaningful and feasible. These studies have also shown that computational aesthetic measures account more for certain visual effects (or “visual styles” [6]) than for aesthetic value in general. For instance, Benford’s law measure (BFL) assigns high values for images that have a grainy texture, while Ross, Ralph and Zong’s bell curve measure (RRZ) likes distinct color progression and the fractal dimension measure (FRD) values low colorfulness highly; see den Heijer & Eiben [6] for a discussion and comparison of 7 computational aesthetic measures. In the following, we study an alternative approach for guiding and controlling generative evolution that tries to address the desire to promote certain visual effects, see Algorithm 2. It utilizes the relationships between design parameters and aesthetic measures discussed in Sec. 3.2. Therefore, a look–up–table is set up that lists templates and design parameters that increase aesthetic measures. After generating the pattern for each single burrow, an aesthetic measure is calculated and compared to an “expected value” recorded before, see Fig. 3. If the value is below the expectation, a measure–increasing template or design parameter is activated. If not, the generation process continues unaltered. Which template or parameter is selected is due to chance. These steps are repeated until a termination criterion is met. In this implementation, the pattern generation ends if a maximum number of burrows are placed in the image. (In the rare case that no measure–increasing template or parameter is available, the setting is also kept.) An interesting feature of such an algorithmic design is that it immediately allows to control and intervene while the image is being created. In other words, the control does not wait until the image is completed, but uses feedback for intervening during the process creating the art works. Fig. 4 shows the aesthetic measures BFL, RRZ and FRD for Algorithm 2. The experimental setup is the same as for the templates, see Fig. 3. Again, the 9 Figure 4: Aesthetic measures for guiding and controlling pattern–making. The measures for the templates (RTL, GLT, CCR, BTG) with the default values of the design parameters (see Tab. 1) are compared to the measures obtained by Algorithm 2. GCB means using the measure BFL to guide and control, GCR uses RRZ, GCF uses FRD. results are averages over 100 images and 70 random hue values. The aesthetic measures for the templates in Tab. 1 are compared to using the measures BFL (GCB ), RRZ (GCR ) and FRD (GCF ) for guiding and controlling the art–making process. Apart from the results of the measures that are obtained by using the same measures to guide and control, we also record the results of the other measures to examine cross–effects (GCR and GCF for BFL, GCB and GCF for RRZ and GCB and GCR for FRD). The results in Fig. 4 indicate that using templates and design parameters is mostly suitable to manipulate the aesthetic measures in an intended way. For BFL we obtain that the measure for GCB is increased as compared to the templates RTL, GTL and CCR, while the images do contain a mix of all templates. For FRD, we even find an increase in GCF compared to all templates. However, for RRZ we get a rather strong decrease in GCR . A possible explanation is that the measure RRZ accounts for color gradient normality, while the effect of Algorithm 2 is basically in changing the selection of templates and design parameters but only indirectly in colorization. Hence, the measure RRZ is poorly sensitive to such a changed selection. These speculations are supported by looking at cross–effects as we see that also the measures that are not used for guidance and control show similar results. This additionally allows to conjecture that for images such as the feeding patterns the aesthetic measures are more correlated than initially assumed. Next, we look at the visual results produced by both algorithms1 . 3.4 Examples of art works Fig. 5 shows a gallery of nine art works generated by Algorithm 1 with the parameter values given in Tab. 1. Looking at these images, we can see that they all display structures that have similarity to natural sand– bubbler patterns. However, colorization of the actual choices of the random–dependent design parameters also offers diversity. In addition, another six images generated by Algorithm 2 are shown in Fig. 6. Although there is a substantial degree of similarity between these images and the images generated by Algorithm 1, it is worth observing some subtle differences. For instance, it is noticeable that typically the images produced by Algorithm 2 are more widespread and grainier, see for instance Fig. 6 upper left and lower middle, which may be attributed to the effect of promoting the visual effects associated with the measure BFL. However, at least in the opinion of the author, there was no real success in provoking a distinct color progression or a particularly low colorfulness, as supposedly associated with RRZ and FRD. More analytic work is needed to clarify the relationships between aesthetic measures and visual effects for a given image–producing algorithm. 1 See https://feit-msr.htwk-leipzig.de/sandbubblerart/ for further images and videos. 10 Figure 5: Gallery with examples of sand–bubbler inspired art works generated by Algorithm 1. 4 Concluding remarks and outlook An algorithm to create visual art has been presented that draws inspiration from the collective feeding behavior of sand–bubbler crabs. Thus, the paper exemplifies another instance of an art–making process based on mathematical models of animal behavior observed in nature. Apart from the algorithmic design and its artistic output, we mainly studied the algorithmic process, particularly how computational aesthetic measures scale to the design parameters and how these measures can be used for controlling and guiding generative evolution. The interest in analyzing the algorithmic process of digital art–making is from both a computational and an artistic point of view. The rationale for the latter is that art–theoretically and following an argument of Philip Galanter [8, 10], in generative art the artistic work in itself is not seen as important as the artistic process. In a reminiscence to the art movement of “truth to material”, generative art may focus on “truth to process”. A possible interpretation states that it is far less interesting to just ask if a human observer likes a particular piece of artificial art. What is much more interesting is to analyze the creation procedure of generative art and study the interplay between algorithmic design (and design parameters) and the works thus created. Not only is the question of whether or not an artificially created image is more or less aesthetic from a human point of view merely opening up an inexhaustible topic 11 Figure 6: Gallery with examples of sand–bubbler inspired art works generated by Algorithm 2. of debate and might be not answerable, this question may actually not be very useful for advancing our understanding of what constitutes artistic beauty and how it can be algorithmically (re–)created. In this spirit the galleries of art works shown in Figs. 5 and 6 could be rather seen as accompanying the analytic results, and not the other way around. The analytic and visual results given here only use a small subset of the design space opened up by Algorithm 1 and 2. Future work could further explore this design space. For instance, the images only contain pellets of a given size and color. 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ICA based on the data asymmetry arXiv:1701.09160v1 [math.ST] 31 Jan 2017 P. Spurek, J. Tabor Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Cracow, Poland P. Rola Department of Mathematics of the Cracow University of Economics, Rakowicka 27, 31-510 Cracow, Poland M. Ociepka Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Cracow, Poland Abstract Independent Component Analysis (ICA) - one of the basic tools in data analysis - aims to find a coordinate system in which the components of the data are independent. Most of existing methods are based on the minimization of the function of fourth-order moment (kurtosis). Skewness (third-order moment) has received much less attention. In this paper we present a competitive approach to ICA based on the Split Gaussian distribution, which is well adapted to asymmetric data. Consequently, we obtain a method which works better than the classical approaches, especially in the case when the underlying density is not symmetric, which is a typical situation in the color distribution in images. Keywords: ICA, Split Normal distribution, skewness. 1. Introduction Independent component analysis (ICA) is one of the most popular methods of data analysis and preprocessing. Historically, Herault and Jutten [1] Email addresses: przemyslaw.spurek@ii.uj.edu.pl,jacek.tabor@ii.uj.edu.pl (P. Spurek, J. Tabor), przemyslaw.rola@outlook.com (P. Rola) Preprint submitted to Pattern Recognition February 1, 2017 seem to be the first (around 1983) to have addressed the problem of ICA to separate mixtures of independent signals. In signal processing ICA is a computational method for separating a multivariate signal into additive subcomponents and has been applied in magnetic resonance [2], MRI [3, 4], EEG analysis [5, 6, 7], fault detection [8], financial time series [9] and seismic recordings [10]. Moreover, it is hard to overestimate the role of ICA in pattern recognition and image analysis; its applications include face recognition [11, 12], facial action recognition [13], image filtering [14], texture segmentation [15], object recognition [16, 17], image modeling [18], embedding graphs in pattern-spaces [19, 20], multi-label learning [21] and feature extraction [22]. The calculation of ICA is discussed in several papers [23, 24, 25, 26, 27, 28, 29], where the problem is given various names, in particular it is also called “source separation problem”. ICA is similar in many aspects to principal component analysis (PCA). In PCA we look for an orthonormal change of basis so that the components are not linearly dependent (uncorrelated). ICA can be described as a search for the optimal basis (coordinate system) in which the components are independent. Let us now, for the readers convenience, describe how the ICA works. Data is represented by the random vector x and the components as the random vector s. The aim is to transform the observed data x into maximally independent components s with respect to some measure of independence. Typically we use a linear static transformation W , called the transformation matrix, combined with the formula s = W x. Most ICA methods are based on the maximization of non-Gaussianity. This follows from the fact that one of the theoretical foundations of ICA is given by the dual view at the Central Limit Theorem [30], which states that the distribution of the sum (average or linear combination) of N independent random variables approaches Gaussian as N → ∞. Obviously if all source variables are Gaussian, ICA will not work. The classical measure of non-Gaussianity is kurtosis (the forth central moment), which can be both positive or negative. Random variables that have a negative kurtosis are called subgaussian, and those with the positive one are called supergaussian. Supergaussian random variables have typically a “spiky” pdf with heavy tails, i.e. the pdf is relatively large at zero and at large values of the variable, while being small for intermediate values (ex. the Laplace distribution). Typically non-Gaussianity is measured by the absolute value of kurtosis (the square of kurtosis can also be used). Thus many methods of finding independent components are based on fitting a 2 Figure 1: Comparison of images separation by our method (ICASG ), with FastICA and ProDenICA. density with similar kurtosis as the data, and consequently are very sensitive to the existence of outliers. Moreover, typically data sets are bounded, and therefore the credible estimation of tails is not easy. Another problem with these methods, is that they usually assume that the underlying density is symmetric, which is rarely the case. In our work we introduce and explore a new approach ICASG , based on the asymmetry of the data, which can be measured by the third central moment (skewness). Any symmetric data, in particular gaussian, has skewness equal to zero. Negative values of skewness indicate the data skewed to the left and the positive ones indicate the data skewed to the right1 . Consequently, 1 By skewed to the left, we mean that the left tail is long relative to the right tail. Similarly, skewed to the right means that the long tail is on the right-hand side. 3 skewness is a natural measure of non-Gaussianity. In our approach, instead of approximating the data by product of densities with heavy tails, we approximate it by a product of asymmetric densities (so called Split Gaussians). Contrary to classical approaches which consider third or fourth central moment, our algorithm is based on second moments. This is a consequence of the fact that Split Gaussian distributions arise from merging two opposite halves of normal distributions in their common mode (for more information see Section 4). Therefore we use only second order moments to describe skewness in dataset, and therefore we obtain an effective ICA method which is resistant to outliers. Figure 2: MLE estimation for image histograms with respect to Logistic and Split Gaussian distributions. The results of classical ICA and ICASG in the case of image separation (for more detail comparison we refer to Section 6) is presented in Fig. 1. In the experiment we mixed two images (see Fig. 1 a) by adding and subtracting them (see Fig. 1 b). Our approach gives essentially better results than the classical FastICA approach, compare Fig. 1 c) to Fig. 1 d) and Fig. 1 f) to Fig. 1 g). In the case of classical ICA we can see artifacts in background, which means that the method does not separate signal properly. On the other hand, ProDenICA and ICASG almost perfectly recovered images, compare Fig. 1 c) to Fig. 1 e) and Fig. 1 f) to Fig. 1 h). In general, ICASG in most cases gives better results than other ICA methods, see Section 6, while its numerical complexity lies below the methods 4 Figure 3: Comparison between our approach and classical ICA in the case of resistance on outliers. which obtain comparable results, that is ProDenICA and PearsonICA. This is caused in particular by the fact that asymmetry is more common than heavy tails in real data sets – we performed the symmetry test by using R package lawstat [31] with 5 percent confidence ratio, and it occurred that all image datasets we used in our paper have asymmetric densities. We also verified it in the case of density estimation of our images. We found optimal parameters of Logistic and Split Gaussian distributions and compared the values of MLE function in Fig. 2. As we see, in most cases Split Gaussian distribution fits the data better than the Logistic one. Summarizing the results obtained in the paper, our method works better than classical approaches for asymmetric data, and is more resistant to outliers (see Example 1.1). Example 1.1. We consider the data with heavy tails (a sample from the Logistic distribution) and skewed ones (a sample from the Split Normal distribution). We added to the data outliers uniformly generated from rectangle [min(X1 ) − sd(X1 ), max(X1 ) + sd(X1 )] × [min(X2 ) − sd(X2 ), max(X2 ) + sd(X2 )], where sd(Xi ) is a standard deviation of the i-th coordinate of X. In Fig. 3 we present how the absolute value of the Tucker’s congruence coefficient (the similarity measure of extracted factors, see Section 6) is changing when we add the outliers. As we see, ICASG is more stable and deals better with outliers in the 5 Figure 4: Logistic, Split Normal and Classical Gaussian distribution fitted to data with heavy tails and skew one. data, which follows from the fact that classical ICA typically depends on the moments of order four, while our approach uses moments of order two. This paper is arranged as follows. In the second section, we discuss related works. In the third, the theoretical background of our approach to ICA is presented. We introduce a cost function which uses the General Split Gaussian distribution and show that it is enough to minimize it respectively to only two parameters: vector m ∈ Rd and d×d matrix W . We also calculate the gradient of the cost function, which is necessary for the efficient use in the minimization procedure. The last section describes numerical experiments. The effects of our algorithm are illustrated on simulated and real datasets. 2. Related works Various ICA methods were discussed in [23, 24, 25, 26, 27, 28]. Herault and Jutten seem to be the first who introduced the ICA around 1983. They proposed an iterative real-time algorithm based on a neuro-mimetic architecture, which nevertheless, can show the lack of convergence in a number of cases [32]. It is worth mentioning that in their framework, higher-order statistics were not introduced explicitly. Giannakis et al. [33] addressed the 6 issue of identifiability of ICA in 1987 using third-order cumulants. However, the resulting algorithm required an exhaustive search. Lacoume and Ruiz [34] sketched a mathematical approach to the problem using higher-order statistics, which can be interpreted as a measure of fitting independent components. Cardoso [35, 36] focused on the algebraic properties of the fourth-order cumulants (kurtosis) what is still a popular approach [37]. Unfortunately kurtosis has some drawbacks in practice, when its value has to be estimated from a measured sample. The main problem is that kurtosis can be very sensitive to the outliers. Its value may depend on only a few observations in the tails of the distribution. In high-dimensional problems, where separation process contains PCA (for dimension reduction), whitening (for scale normalization), and standard ICA this effect is called a small sample size problem [38, 39]. This is caused by the fact that for the high-dimensional data sets ICA algorithms tend to extract the independent features simply by the projections that isolate single or very few samples (outliers). To address the difficulty random pursuit and locality pursuit methods were applied [39]. Another commonly used solution is to use skewness [40, 41, 42, 43] instead of kurtosis. Unfortunately, skewness has received much less attention than kurtosis, and consequently methods based on skewness are usually not well theoretically justified. One of the most popular ICA method dedicated to the skew data is PearsonICA [44, 45], which minimizes mutual information using a Pearson [46] system-based parametric model. The model covers a wide class of source distributions including skewed distributions. The Pearson system is defined by the differential equation f 0 (x) = (a1 x − a0 )f (x) , b0 + b1 x + b2 x 2 where a0 , a1 , b0 , b1 and b2 are the parameters of the distribution. The parameters of the Pearson system can be estimated using the method of moments. Therefore such algorithms have strong limitations connected with the optimization procedure. The main problems are number of parameters which have to be fitted and numerical efficiency of the minimization procedure. An important measure of fitting independent components is given by negentropy [47]. FastICA [48], one of the most popular implementations of ICA, uses this approach. Negentropy is based on the information-theoretic 7 quantity of (differential) entropy. This concept leads to the mutual information which is the natural information-theoretic measure of the independence of random variables. Consequently, one can use it as the criterion for finding the ICA transformation [28, 49]. It can be shown that minimization of the mutual information is roughly equivalent to maximization of negentropy and it is easier to estimate since we do not need additional parameters. ProDenICA [50, 51] is based not on a single nonlinear function, but on an entire function space of candidate nonlinearities. In particular, the method works with the functions in a reproducing kernel Hilbert space, and make use of the “kernel trick” to search over this space efficiently. The use of a function space makes it possible to adapt to a variety of sources and thus makes ProDenICA algorithms more robust to varying source distributions. A somewhat similar approach to ICA is based on the maximum likelihood estimation [27]. It is closely connected to the infomax principle since the likelihood is proportional to the negative of mutual information. In recent publications, the maximum likelihood estimation is one of the mot popular [24, 52, 53, 54, 55, 56, 57] approaches to ICA. Maximum likelihood approach needs the source pdf. In the classical ICA it is common to use the superGaussian logistic density or other heavy tails distributions. In this paper we present ICASG , a method which joins the positive aspects of classical ICASG approaches with recent ones like ProDenICA or Pearson ICA. First of all we use a General Split Gaussian distribution, which uses second order moments to describe skewness in dataset, and therefore is relatively robust to noise or outliers. The GSG distribution can be fitted by minimizing a simple function, which depends on only two parameters m ∈ Rd , W ∈ M(Rd ), see Theorem 5.1. Moreover we calculate its gradient, and therefore we can use numerically efficient gradient type algorithms, see Theorem 5.2. 3. Theoretical justification Let us describe the idea2 behind ICA [30]. Suppose that we have a random vector X in Rd which is generated by the model with the density F . Then it is well-known that components of X are independent iff there exist onedimensional densities f1 , . . . , fd ∈ DR , where by DR we denote the set of 2 In fact it is one of the possible approaches, as there are many explanations which lead to similar formula. 8 Figure 5: Logistic and General Split Normal distributions fitted to data with heavy tails and skew ones. densities on R, such that F (x) = f1 (x1 ) · . . . · fd (xd ), for x = (x1 , . . . , xd ) ∈ Rd . Now suppose that the components of X are not independent, but that we know (or suspect) that there is a basis A (we put W = A−1 ) such that in that base the components of X become independent. This may be formulated in the form F (x) = det(W ) · f1 (ω1T (x − m)) · . . . · fd (ωdT (x − m)) for x ∈ Rd , 9 (3.1) where ωiT (x − m) is the i-th coefficient of x − m (the basis is centered in m) in the basis A (ωi denotes the i-th column of W ). Observe, that for a fixed family of one-dimensional densities F ⊂ DR , the set of all densities given by (3.1) for fi ∈ F, forms an affine invariant set of densities. Thus, if we want to find such a basis that components become independent, we need to search for a matrix W and one-dimensional densities such that the approximation F (x) ≈ det(W ) · f1 (ω1T (x − m)) · . . . · fd (ωdT (x − m)), for x ∈ Rd , is optimal. However, before proceeding to practical implementations, we need to precise: 1. how to measure the above approximation, 2. how to deal with data X, since we do not have the density, 3. how to work with the family of all possible densities. The answer to the first point is simple and is given by the Kullback-Leibler divergence, which is defined to be the integral: Z ∞ p(x) dx, DKL (P kQ) = p(x) log q(x) −∞ where p and q denote the densities of P and Q. This can be written as DKL (P kQ) = h(P ) − M LE(P, Q), where h is the classical Shannon entropy. Thus to minimize the KullbackLeibler divergence, we can equivalently maximize the MLE. This is helpful, since for a discrete data X we have nice estimator of the LE (likelihood estimation): 1 X LE(X, Q) = ln(q(x)). \|X\| x∈X Thus we arrive at the following problem. Problem [reduced]. Let X be a data set. Find an unmixing matrix W , center m, and densities f1 , . . . , fd ∈ DR so that the value LE(X, f1 , . . . , fd , m, W ) = P 1 ln(f1 (ω1T (x − m)) . . . fd (ωdT (x − m))) + ln(det(W )) = \|X\| 1 \|X\| x∈X d P P ln(fi (ωiT (x − m))) + ln(det(W )) i=1 x∈X 10 is maximized. However, there is still a problem with the last point, as the search over the space of all densities DR is not feasible. Thus, we naturally have to reduce our search to a subclass of all densities F (which should be parametrized by a finite amount of parameters). Problem [final]. Let X ⊂ Rd be a data set and F ⊂ DR be a set of densities. Find an unmixing matrix W , center m, and densities f1 , . . . , fd ∈ F so that the value d 1 XX ln(fi (ωiT (x − m))) + ln(det(W )) \|X\| i=1 x∈X is maximized. It may seem that the most natural choice is Gaussian densities. However, this is not the case as Gaussian densities are affine invariant, and therefore do not “prefer” any fixed choice of coordinates3 . In other words we have to choose a family of densities which is distant from Gaussian ones. In the classical ICA approach it is common to use the super-Gaussian logistic distribution: f (x; µ, s) = e x−µ s s 1+e 1 2 x−µ sech . = x−µ 2 4s 2s s The main difference between the gaussian and super-gaussian is the existence of the heavy tails. This can be also viewed as the difference in the fourth moments. However, such a choice leads to some negative consequences, namely the model is very sensitive to outliers. Moreover, if the data is not-symmetric, the approximation could not give the expected results, as the model consists only of symmetric densities. The idea behind this paper was to choose the model of densities which wouldn’t have the two above disadvantages. So, instead of choosing the family which differs from the Gaussians by the size of tail (fourth moment), we chose a family which would allow estimation of asymmetric densities – Split Gaussian distribution [58]. 3 In fact one can observe that the choice of gaussian densities leads to PCA, if we restrict to the case of orthonormal bases 11 Figure 6: Level sets of the General Split Normal distribution with different parameters. Example 3.1. In Fig. 4 and Fig. 5 we present a comparison between the Logistic and the Split Normal distribution in 1d and 2d respectively. In experiments we use the classical skew dataset Lymphoma [59, 60] and the classical heavy tails dataset Australian athletes [61]. In the case of heavy tails both methods work nice, since real dataset represent heavy tails which are not symmetric and the skew model is able to detect it. On the other hand, in the case of skew data Split Normal gives essentially better results. 4. Split Gaussian distribution In this section we present our density model. A natural direction for extending the normal distribution is the introduction of some skewness, and several proposals have indeed emerged, both in the univariate and multivariate case, see [62, 63, 64]. One of the most popular approaches is the Split Normal (SN) distribution, or the Split Gaussian (SG) distribution [58]. In our paper we use a generalization of this model, which we call the General Split Normal (GSN) distribution. 12 We start from the one-dimensional case. After that we present a possible generalization of this definition to the multidimensional setting, which corresponds with the formula (3.1). Contrary to the Split Gaussian distribution, we skip the assumption of the orthogonality of coordinates (often called principal components), and obtain an ICA model. 4.1. One-dimensional case The density of the one-dimensional Split Gaussian distribution is given by the formula for x ≤ m, c · exp[− 2σ1 2 (x − m)2 ], 2 2 SN (x; m, σ , τ ) = 1 2 c · exp[− 2τ 2 σ2 (x − m) ], for x > m, q where c = π2 σ −1 (1 + τ )−1 . As we see the split normal distribution arises from merging two opposite halves of two probability density functions of normal distributions in their common mode. In general the use of the Split Gaussian distribution (even in 1D) allows to fit data with better precision (from the likelihood function point of view). In 1982 John [65] showed that the likelihood function can be expressed in an intensive form, in which the scale parameters σ and τ are a function of the location parameter m (see Theorem 3.1 proved by [64]). Thanks to this theorem we can maximize the likelihood function numerically with respect to a single parameter m only. The rest of parameters are explicitly given by simple formulas. 4.2. Multidimensional Split Gaussian distribution A natural generalization of the univariate split normal distribution to the multivariate settings was presented by [64]. Roughly speaking, authors assume that a vector x ∈ Rd follows the multivariate Split Normal distribution, if its principal components are orthogonal and follow the one-dimensional Split Normal distribution. Definition 4.1 (Definition 2.2. [64]). A density of the multivariate Split Normal distribution is given by SNd (x; m, Σ, τ ) = d Y SN (ωjT (x − m); 0, σj2 , τj2 ), j=1 13 where ωj is the eigenvector corresponding to the j-th largest eigenvalue in the spectral decomposition of Σ = W AW T and m = [m1 , . . . , md ]T , A = diag(σ12 , . . . , σd2 ) and τ = [τ12 , . . . , τd2 ]. One can easily observe that the principal components ωjT x are independent. For this generalization a similar theorem, like in the one-dimensional case, is valid. We can extract the maximum likelihood estimation by maximizing the function with respect to two parameters m ∈ Rd and W ∈ Md (R) where columns of W are orthonormal vectors (Md (R) denotes the set of d-dimensional square matrices). We may use this theorem for numerical maximization of the likelihood function w.r.t. m and W . Unfortunately, the optimization process on Stiefel manifold (the set of orthogonal matrices) studied by [66] is numerically ineffective and requires additional tools. This problem can be omitted by using Eulerian angles described by [67]. In the two-dimensional case, W is explicitly parametrized as π π cos(θ) sin(θ) W = , − <θ≤ . − sin(θ) cos(θ) 2 2 In such a case we can straightforwardly apply standard numerical optimization algorithm. Both of these solutions can be applied. Nevertheless, unnatural assumption of the orthogonality of principal components causes two negative effects: the optimization process is time consuming and the model with the restriction that the coordinates are orthogonal can not accommodate data as good as the general one. Therefore, in this article we use more flexible model – the General Split Normal [68] distribution: Definition 4.2. A density of the multivariate General Split Normal distribution is given by 2 2 GSNd (x; m, W, σ , τ ) = det(W ) d Y SN (ωjT (x − m); 0, σj2 , τj2 ), j=1 where ωj is the j-th column of non-singular matrix W , m = (m1 , . . . , md )T , σ = (σ1 , . . . , σd ) and τ = (τ1 , . . . , τd ). 14 Figure 7: Comparison between fitting Gaussian, Split Gaussian and General Split distribution on [59, 60]. Observe that, contrary to Split Gaussian, General Split Gaussian does not have orthogonal basis. Our model is a natural generalization of the multivariate Split Normal distribution proposed in [64] (see Definition 4.1) and is given in the form formulated by (3.1) for the set of Split Gaussian densities. Clearly every Split Normal distribution is a General Split Normal distribution. The above generalization is flexible and allows to fit data with greater precision, see Fig. 7. The level sets of the GSN distribution with different parameters are presented in Fig. 6. We skip the constraints of orthogonality of the principal components. Consequently, we can apply the standard optimization procedure directly. In the next section we discuss how to fit data in our model. 5. Maximum likelihood estimation In the previous section we introduced the GSN distribution. Now we show how to use the likelihood estimation in our setting. As it was mentioned, we have to maximize the likelihood function with respect to four parameters. In the case of the General Split Normal distribution (contrary to the classical Gaussian one) we do not have explicit formulas and consequently we heave to solve the optimization problem. In the first subsection, we reduce our problem to the simpler one by introducing the function l. Minimization of l is equivalent to maximization of the likelihood function. In the second subsection we present how to minimize our function by using the gradient method. 15 5.1. Optimization problem The density of the GSN distribution depends on four parameters m ∈ Rd , W ∈ M(Rd ), σ ∈ Rd , τ ∈ Rd . We can find them by minimizing the simpler function, which depends on only m ∈ Rd and W ∈ M(Rd ). Other parameters are given by explicit formulas. Theorem 5.1. Let x1 , . . . , xn be given. Then the likelihood maximized w.r.t. σ and τ is −3n/2 dn/2 d Y 1 2n gj (m, W ) , (5.1) L̂(X; m, W ) = 2 πe \|det(W )\| 3 j=1 where 1/3 1/3 gj (m, W ) = s1j + s2j , s1j = P [ωjT (xi − m)]2 , Ij = {i = 1, . . . , n : ωjT (xi − m) ≤ 0}, i∈Ij s2j = P i∈Ijc [ωjT (xi − m)]2 , Ijc = {i = 1, . . . , n : ωjT (xi − m) > 0}, and the maximum likelihood estimators of σj2 and τj are σ̂j2 (m, W ) = 1 2/3 s g (m, W ), n 1j j τ̂j (m, W ) = s2j s1j 1/3 . Proof. See Appendix 8. Thanks to the above theorem, instead of looking for the maximum of the likelihood function, it is enough to obtain the maximum of the simpler function (5.1) which depends on two parameters m ∈ Rd and W ∈ M(Rd ) l(X; m, W ) = d Y 1 2 \|det(W )\| 3 gj (m, W ), (5.2) j=1 where ωj stands for the j-th column of matrix W . Consequently, maximization of (5.1) is equivalent to minimization of (5.2), see the following corollary. Corollary 5.1. Let X ⊂ Rd , m ∈ Rd , W ∈ M(Rd ) be given, then argmax L̂(X; m, W ) = argmin l(X; m, W ). m,W m,W 16 Figure 8: Results of image separation with the uses of various ICA algorithms. 5.2. Gradient One of the possible methods of optimization is the gradient method. Since the minimum of l is equal to the minimum of ln(l), in this subsection we 17 calculate the gradient of ln(l). Before we prove suitable Theorem 5.2, we recall the following lemma. Lemma 5.1. Let A = (aij )1≤i,j≤d be a differentiable map from real numbers to d × d matrices then ∂det(A) = adjT (A)ij , (5.3) ∂aij where adj(A) stands for the adjugate of A, i.e. the transpose of the cofactor matrix. Proof. By the Laplace expansion detA = d P (−1)i+j aij Mij where Mij is the j=1 minor of the entry in the i-th row and j-th column. Hence ∂detA = (−1)i+j Mij = adjT (A)ij . ∂aij Now we are ready to calculate gradient of our cost function. Theorem 5.2. Let X ⊂ Rd , m = (m1 , . . . , md )T ∈ Rd , W = (ωij )1≤i,j≤d non T ) ∂ ln l(X;m,W ) singular be given. Then ∇m ln l(X; m, W ) = ∂ ln l(X;m,W , . . . , , ∂m1 ∂md where d P T P T P ∂ ln l(X;m,W ) 1 1 −1 2ωj (xi − m)ωjk + 2 2ωj (xi − m)ωjk . = 1 1 2 ∂mk 3 +s 3 j=1 s1j 2j 3 3s1j i∈Ij Moreover, ∇W ln l(X; m, W ) = ∂ ln l(X;m,W ) ∂ωpk −2 + 31 s2p3 P i∈Ipc = − 23 (ω −1 )Tpk + h 3 3s2j i∈Ijc ∂ ln l(X;m,W ) ∂ωpk 1 1 1 3 +s 3 s1p 2p i 2 1 −3 s 3 1p , where 1≤p,k≤d P 2ωpT (xi − m)(xik − mk )+ i∈Ip T 2ωp (xi − m)(xik − mk ) , and s1j = P [ωjT (xi − m)]2 , Ij = {i = 1, . . . , n : ωjT (xi − m) ≤ 0}, i∈Ij s2j = P i∈Ijc [ωjT (xi − m)]2 , Ijc = {i = 1, . . . , n : ωjT (xi − m) > 0}. 18 Proof. See Appendix 9. Thanks to the above theorem we can use gradient descent, a first-order optimization algorithm. To find a local minimum of the cost function ln(l) using gradient descent, one takes steps proportional to the negative of the gradient of the function at the current point. If instead one takes steps proportional to the positive of the gradient, one approaches a local maximum of that function, see Algorithm 1. Algorithm 1 : Input data set X Initial conditions initialization of mean vector m = mean(X) initialization of matrix W = cov(X) Gradient algorithm obtain new values of m and V by applying gradient method for function log(l) (see formula 5.1): (m, W ) = argmin log(l(X; m̄, W̄ )), m̄,W̄ where ∇m ln l(X; m, W ) ∇W ln l(X; m, W ) are given by Theorem 5.2 calculate σ ∈ Rd and τ ∈ Rd by using Theorem 5.1 Return value return optimal ICA basis (m, W ). At the end of this section we present comparison of computational efficiency between ICASG and various ICA methods, see Fig. 9. In our experiment we consider the classical image separation problem, where we mixed two images by adding and subtracting them. We use ten pairs of images. Each pair was scaled to different sizes. In Fig. 9 we present mean value of computation time. FastICA, Infomax and JADE are the most effective but 19 Figure 9: Comparison of computational efficiency between ICASG and various ICA methods. do not solve the problem of image separation sufficiently well, see Tab. 1. On the other hand, the ProDenICA which gives comparable result to ICASG , is much slower. 6. Experiments and analysis To compare our method to classical ones we use Tucker’s congruence coefficient [69] (uncentered correlation) defined by Pd Cr(s, s̄) = qP d i=1 si s̄i qP d 2 i=1 si . 2 i=1 s̄i Its values range between −1 and +1. It can be used to study the similarity of extracted factors across different samples. Generally, a congruence coefficient of 0.9 indicates a high degree of factor similarity, while a coefficient of 0.95 20 or higher indicates that the factors are virtually identical. In the case of ICA methods multiplying by the scalar any of the sources do not change results. Therefore the sign of congruence coefficient is not important and we can compare absolute value of Tucker’s congruence. We evaluate our method in the context of images, sound, hyperspectral unmixing and EEG data. For comparison we use R package ica [70], PearsonICA [71], ProDenICA [72], tsBSS [73]. The most popular method used in practice is FastICA [48, 74] algorithm, which uses negentropy. In this context we can use three different functions to estimate neg-entropy: logcosh, exp and kurtosis. We also compare our method with algorithm using Information-Maximization (Infomax) approach [49]. Similarly to FastICA we consider three possible nonlinear functions: hyperbolic tangent, logistic and extended Infomax. We also consider algorithm which uses Joint Approximate Diagonalization of Eigenmatrices (JADE) proposed by Cardoso and Souloumiac’s [75, 75, 74]. One of the most popular ICA methods dedicated for skew data is PearsonICA [44, 45], which minimizes mutual information using a Pearson [46] system-based parametric model. Another model we consider is ProDenICA [50, 51], which is based not on a single nonlinear function, but on an entire function space of candidate nonlinearities. In particular, the method works with the functions in a reproducing kernel Hilbert space, and make use of the kernel trick to search over this space efficiently. We also compare our method with FixNA [76], method for blind source separation problem. 6.1. Separation of images One of the most popular application of ICA is the separation of images. In our experiments we use four images from the USC-SIPI Image Database of size 256 × 256 pixels (4.1.01, 4.1.06, 4.1.02, 4.1.03) and eight of size 512 × 512 pixels (4.2.04, 4.2.02, boat.512, elaine.512, 5.2.10, 5.2.08, 5.3.01, 4.2.03). We also use 8 images from the Berkeley Segmentation Dataset of size 482 × 321 with indexes (#119082, #42049, #43074, #38092, #157055, #220075, #295087, #167062). We make random pairs of above images and use them 1 1 as a source signal, combined by the mixing matrix A = . From 1 −1 practical point of view, we simply obtain two new images by adding and dividing sources pictures. Our goal is to reconstruct original images by using only the knowledge about mixed ones. The visualization of this process we 21 ICASG 4.1.01 4.1.02 4.1.06 4.1.03 4.2.04 5.2.10 4.2.02 5.2.08 boat.512 5.3.01 elaine.512 4.2.03 119082 157055 42049 220075 43074 295087 38092 167062 -0.9818 0.992 -0.9609 0.5664 -0.5034 0.2893 0.2305 0.5717 0.3593 0.4316 0.5874 -0.0226 0.9987 0.389 -0.7493 0.4359 -0.7371 -0.3997 -0.5949 0.3255 FastICA logcosh exp 0.5481 -0.5457 0.6696 0.6644 -0.4297 -0.4297 0.2062 0.2062 0.0506 0.0528 -0.0719 -0.0749 -0.0376 0.0203 0.1037 -0.0625 0.0351 0.0314 0.0078 0.0138 0.32 0.32 -0.3196 -0.3196 0.5736 0.5736 -0.3619 -0.3619 0.3009 0.3028 -0.5087 -0.5154 0.0344 0.0323 -0.048 -0.0458 0.0555 0.0564 -0.0025 -0.0041 kurtosis -0.5485 0.6707 -0.4296 0.2057 -0.0499 0.0709 -0.0017 -0.0097 -0.056 -0.0262 -0.32 0.3201 0.5731 -0.3618 -0.299 0.503 0.0429 -0.0566 0.031 0.0425 Infomax JADE PearsonICA tanh tangent logistic 0.548 -0.5484 -0.548 -0.5492 -0.5308 0.6695 0.6705 0.6695 0.6726 0.6696 -0.4297 -0.4297 -0.4296 -0.4296 -0.4297 0.2061 0.206 0.2058 0.2058 -0.2062 0.0505 -0.0512 0.0508 0.0397 0.3123 -0.0717 0.0727 -0.0722 -0.057 -0.4275 0.0377 0.0265 0.0061 -0.0093 -0.1228 -0.1039 -0.0773 -0.0285 0.0086 -0.2913 0.0343 -0.0449 0.0298 0.0356 -0.1046 0.0091 -0.008 0.0164 0.007 0.1061 0.32 -0.32 0.32 0.32 -0.32 -0.3196 0.3199 -0.3196 -0.3202 -0.3195 0.5737 0.5733 0.5735 0.5735 -0.032 -0.3619 -0.3619 -0.3619 -0.3619 0.0046 -0.3005 -0.3031 -0.3007 -0.2898 0.2596 0.5074 0.5168 0.5081 0.4789 0.4838 0.0348 0.0404 0.0342 0.0324 0.0891 -0.0484 -0.0541 -0.0478 -0.0459 -0.1035 -0.0553 0.041 0.0557 0.0375 0.0535 0.0021 0.0241 -0.0029 0.0306 0.0011 ProDenICA FixNA -0.0013 -0.0981 0.4297 0.207 -0.3164 0.4334 0.1282 -0.3091 -0.0461 0.0486 0.0287 -0.048 0.5744 0.3619 0.0421 -0.0645 0.3925 0.4015 0.4036 0.7404 0.5503 -0.6761 0.0148 0.0127 0.1461 -0.1979 0.1235 -0.2931 0.3175 -0.5303 0.2282 -0.2554 0.3695 -0.2446 0.142 -0.1839 0.2458 -0.2406 0.2614 -0.5495 Table 1: The Tucker’s congruence coefficient measure between original images and results of different ICA algorithms. present in Fig. 8. The results of this experiment are presented in Tab. 1 where we exhibit Tucker’s congruence coefficients. In the case of the Tucker’s congruence coefficient measure almost in all situation we obtain better results. The ICASG method essentially better recovers original signals. In Fig. 8 we can sow that ICASG almost perfectly recovers source signal. 6.2. Cocktail-party problem In this subsection we compare our method with classical ones in the case of cocktail-party problem. Imagine that you are in a room where two people are speaking simultaneously. You have two microphones, which you hold in different locations. The microphones give you two recorded time signals, which we could interpret as mixed signal x. Each of these recorded signals is a weighted sum of the speech signals emitted by the two speakers, which we denote by s. The cocktail-party problem is to estimate the two original speech signals. In our experiments we use signal obtained by mixing synthetic sources4 4 We use signals from http://research.ics.aalto.fi/ica/cocktail/cocktail_en. cgi. 22 ICASG source source source source source source source source source source source source source source source source 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 0.1597 0.7739 0.1388 0.9435 0.1985 0.8453 0.232 0.7679 0.1728 0.9424 0.15 0.7417 0.1036 0.908 0.1166 0.8212 logcosh 0.1097 0.7705 0.0899 0.9075 0.079 0.8887 0.0989 0.7798 0.0989 0.9245 0.0404 0.7129 0.0839 0.9016 0.1153 0.8136 FastICA exp 0.1096 0.7713 0.0899 0.9076 0.0791 0.8882 0.0989 0.7799 0.099 0.9243 0.0404 0.7134 0.084 0.9015 0.1156 0.8116 kurtosis 0.1101 0.7672 0.0908 0.898 0.079 0.8889 0.099 0.7793 0.0988 0.9256 0.0402 0.707 0.0839 0.9019 0.1145 0.8195 tanh 0.1097 0.7705 0.0899 0.9074 0.079 0.8887 0.0989 0.7798 0.0989 0.9246 0.0404 0.7132 0.0839 0.9019 0.1152 0.8141 Infomax tangent 0.11 0.7685 0.0899 0.907 0.079 0.8892 0.0989 0.7798 0.0989 0.925 0.0404 0.7125 0.0839 0.9019 0.1148 0.8174 logistic 0.1097 0.7705 0.0899 0.9074 0.079 0.8887 0.0989 0.7798 0.0989 0.9246 0.0404 0.7129 0.0839 0.9017 0.1153 0.8138 JADE PearsonICA ProDenICA FixNA 0.1101 0.7704 0.0908 0.9075 0.079 0.8898 0.099 0.7801 0.0989 0.9245 0.0402 0.7124 0.0839 0.9014 0.1155 0.8165 0.1097 0.7704 0.0899 0.9075 0.0789 0.8898 0.0989 0.7801 0.0989 0.9245 0.0404 0.7124 0.084 0.9014 0.1149 0.8165 0.1412 0.9998 0.0984 0.9989 0.0843 0.8459 0.1153 0.9344 0.0963 0.9729 0.0567 0.9998 0.093 0.9999 0.1427 0.9996 0.109 0.7751 0.0907 0.8988 0.0791 0.8882 0.0989 0.7796 0.0987 0.9273 0.0402 0.7099 0.0836 0.9056 0.1147 0.8176 Table 2: The Tucker’s congruence coefficient measure between original sound and results of different ICA algorithms in the case of cocktail-party problem. 1 1 (similar as before we use mixing matrix A = ). Comparison between 1 −1 methods we present in Tab. 2. In the case of cocktail-party problem our method recovers sources signal better then classical methods. 6.3. Hyperspectral Unmixing Independent component analysis has been recently applied into hyperspectral unmixing as a result of its low computation time and its ability to perform without prior information. However, when applying ICA for hyperspectral unmixing, the independence assumption in the ICA model conflicts with the abundance sum-to-one constraint and the abundance nonnegative constraint in the linear mixture model, which affects the hyperspectral unmixing accuracy. Nevertheless, ICA was recently applied in this area [77, 78]. In this subsection we apply simple example which shows that our method can by used for spectral data. Urban data [79, 80, 81] is one of the most widely used hyperspectral datasets used in the hyperspectral unmixing study. Each image has 307 × 307 pixels, each of which corresponds to a 2 × 2 m area. In this image, there are 210 wavelengths ranging from 400 nm to 2500 nm, resulting in a spectral resolution of 10 nm. After the channels 1–4, 76, 87, 101–111, 136–153 and 198–210 are removed (due to dense water vapor and atmospheric effects), there remain 162 channels (this is a common preprocess for hyperspectral unmixing analyses). There is ground truth [79, 80, 81], which contains 4 channels: #1 Asphalt, #2 Grass, #3 Tree and #4 Roof. 23 ICASG #1 Asphalt #2 Grass #3 Tree #4 Roof 0.6774 -0.7784 0.7267 0.6666 logcosh 0.2859 -0.2746 0.2338 -0.4256 FastICA exp 0.2864 -0.2605 0.2717 0.4279 kurtosis -0.2595 -0.2798 -0.2547 0.4167 Infomax PearsonICA ProDenICA tanh tangent logistic -0.2972 -0.2954 -0.2972 0.20978 0.4928 -0.2814 -0.2816 -0.2814 -0.2412 -0.4323 0.2441 0.2354 0.2442 0.2482 -0.5961 -0.4244 0.4301 -0.4244 0.4193 -0.6128 Table 3: The Tucker’s congruence coefficient measure between reference layers and results of different ICA algorithms in the case of the urban data set. Figure 10: Congruence distance between layers obtain by different ICA algorithms and the closest reference channel. A highly mixed area is cut from the original data set in this experiment (similar example was showed in [77]), with the size of 200 × 150 pixels. In our experiment we apply various ICA methods and report the Tucker’s congruence coefficient measure between each layer and the closest reference channel, see Fig. 10. ICASG and ProDenICA give layers which contain more information then the other approaches. Distance between four best channels to the reference ones we present in Tab. 3. 24 6.4. EEG At the end of this section we present how our method works in the case of EEG signals. In this context, ICA is applied to many different task like eye movements, blinks, muscle, heart and line noise e.t.c.. In this experiment we concentrate on eye movement and blink artifacts. Our goal here is to demonstrate that our method is capable of finding artifacts in real EEG data. However, we emphasize that it does not provide a complete solution to any of these practical problems. Such a solution usually entails a significant amount of domain-specific knowledge and engineering. Nevertheless, from these preliminary results with EEG data, we believe that the method presented in this paper provides a reasonable solution for signal separation, which is simple and effective enough to be easily customized for a broad range of practical problems. For EEG analysis, the rows of the input matrix x are the EEG signals recorded at different electrodes, the rows of the output data matrix s = W x are time courses of activation of the ICA components, and the columns of the inverse matrix, W , give the projection strengths of the respective components onto the scalp sensors. One EEG data set used in the analysis was collected from 40 scalp electrodes (see Fig. 11 a)). The second and the third are located very near to eye and can be understood as a base (we can use them for removing eye blinking artifacts). In Fig. 11 b) we present signals obtained by ICASG . The scale of this figure is large but we can find the data which have spikes exactly in the same place as the two base signals (see Fig. 11 c)). After removing selected signal and going back to the original situation we obtain signal (see Fig. 11 d)) without eye blinking artifacts (compare Fig. 11 a) with Fig. 11 d)). 7. Conclusion In our work we introduce and explore a new approach to ICA which is based on the asymmetry of the data. Roughly speaking in our approach, instead of approximating the data by product of densities with heavy tails, we approximate it by a product of asymmetric densities – the Split Gaussian distribution. Contrary to classical approaches which consider third or fourth central moment, our algorithm in practice is based on second moments. This is a consequence of the fact that Split Gaussian distributions arise from merging two opposite halves of normal distributions in their common mode. 25 Figure 11: Results of ICASG in the case of EEG data. Therefore we use only second order moments to describe skewness in dataset, and therefore we obtain an effective ICA method which is resistant to outliers. We verified our approach on images, sound and EEG data. In the case of source signal reconstructing our approach gives essentially better results (better recover original signals). The main reason is such that kurtosis is very sensitive to the outliers and that the asymmetry of the data is more popular than heavy tails in real data sets. 8. Appendix A Proof of Theorem 5.1. Let X = {x1 , . . . , xn }. We write zi = W (xi − m), zij = ωjT (xi − m), 26 for observation i, where i = 1, . . . , n and coordinates j = 1, . . . , d. Let us consider the likelihood function, i.e. n Q L(X; m, W, σ, τ ) = GSNd (xi ; m, W, σ, τ ) i=1 n Q d Q SN (ωjT (xi − m); 0, σj2 , τj2 ) i=1 j=1 n Q −n Q h d n Q d = c1 \|det(W )\| σj (1 + τj ) exp − = \|det(W )\| j=1 where c1 = q d 2 π i=1 j=1 1 2 z (1{zij ≤0} 2σj2 ij i + τj−2 1{zij >0} ) , . Now we take the log-likelihood function, i.e. ln(L(X; m, W, σ, τ )) n Q −n P i d n P d h σj (1 + τj ) = ln c1 \|det(W )\| + − 2σ1 2 zij2 (1{zij ≤0} + τj−2 1{zij >0} ) j j=1 i=1 j=1 n Q −n d d P −2 P 2 σ −2 P 2 σj zij = ln c1 \|det(W )\| σj (1 + τj ) zij + τj2 − 12 j j=1 j=1 i∈Ijc i∈Ij n Q d d −n P 1 = ln c1 \|det(W )\| σj (1 + τj ) s1j + τ12 s2j . − 2σ 2 j=1 j=1 j j We fix m, W and maximize the log-likelihood function over τ and σ. In such a case we have to solve the following system of equations ∂ ln(L(X;m,W,σ,τ )) ∂σj = − σnj + σj−3 (s1j + τj−2 s2j ) = 0, ∂ ln(L(X;m,W,σ,τ )) ∂τj n = − 1+τ + j s2j τj3 σj2 = 0, for j = 1, . . . , d. By simple calculations we obtain the expressions for the estimators 1/3 s2j 2 1 2/3 σ̂j (m, W ) = n s1j gj (m, W ), τ̂j (m, W ) = . s1j Substituting it into the log-likelihood function, we get dn Q d 2 3 −n dn 2 1 n √ 2 g (m, W ) e− 2 L̂(m, W ) = π \|det(W )\| · n j j=1 dn − 3n2 d 2 Q 1 = 2n . g (m, W ) 2 j πe \|det(W )\| 3 j=1 27 9. Appendix B Proof of Theorem 5.2. Let us start with the partial derivative of ln(l) with respect to m. We have 1 1 d d d 3 +s 3 ) P P P ∂(s1j ∂ ln(gj (m,W )) ∂ ln l(X;m,W ) 2j 1 1 ∂s1j 1 ∂s2j 1 = = . 1 1 2 ∂m + 2 ∂m 1 1 ∂mk ∂mk ∂mk k k 3 +s 3 j=1 s1j 2j j=1 Now, we need ∂s1j ∂mk = P i∈Ij ∂s1j ∂mk ∂s2j , ∂mk and ∂[ωjT (xi −m)]2 ∂mk = 3 +s 3 j=1 s1j 2j 3 3s1j 3 3s2j therefore 2ωjT (xi − m) P i∈Ij ∂ωjT (xi −m) ∂mk = P −2ωjT (xi − m)ωjk . i∈Ij Analogously we get ∂s2j ∂mk = P i∈Ijc −2ωjT (xi − m)ωjk . Hence ∂ ln l ∂mk = d P −1 1 1 3 +s 3 j=1 s1j 2j 1 2 P 3 3s1j i∈Ij 2ωjT (xi − m)ωjk + 1 2 P 3 3s2j i∈Ijc 2ωjT (xi − m)ωjk . Now we calculate the partial derivative of ln l(X; m, W ) with respect to the matrix W . We have ∂ ln l(X;m,W ) ∂ωpk 2 = ∂ ln \|det(W )\|− 3 ∂ωpk + d P j=1 ∂ ln(gj (m,W )) . ∂ωpk To calculate the derivative of the determinant we use Jacobi’s formula (see Lemma 5.1). Hence 2 5 2 ∂ ln(det(W )− 3 ) ) 2 3 = det(W ) det(W )− 3 ∂det(W = − 23 det(W )−1 adjT (W )pk − ∂ωpk 3 ∂ωpk 1 det(W )(W −1 )Tpk = − 23 (ω −1 )Tpk , = − 32 det(W ) where (ω −1 )Tpk is the element in the p-th row and k-th column of the matrix (W −1 )T . Now we calculate 1 1 3 +s 3 ) ∂(s1j ∂ ln(gj (m,W )) 2j 1 1 1 ∂s1j 1 ∂s2j = 1 1 ∂ωpk = 1 1 + 2 ∂ωpk , 2 ∂ω ∂ωpk pk 3 +s 3 s1j 2j 3 +s 3 s1j 2j 28 3 3s1j 3 3s2j where ∂s1j ∂ωpk ( = 0, P P i∈Ij ∂[ωjT (xi −m)]2 ∂ωpk = P 2ωjT (xi − m) i∈Ij ∂ωjT (xi −m) ∂ωpk = if j = 6 p 2ωpT (xi − m)(xik − mk ), if j = p i∈Ip and xik is the k-th element of the vector xi . 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Infinitesimal topological generators and quasi non-archimedean topological groups arXiv:1505.00415v2 [math.GR] 14 Mar 2016 Tsachik Gelander∗and François Le Maître† Abstract We show that connected separable locally compact groups are infinitesimally finitely generated, meaning that there is an integer n such that every neighborhood of the identity contains n elements generating a dense subgroup. We generalize a theorem of Schreier and Ulam by showing that any separable connected compact group is infinitesimally 2-generated. Inspired by a result of Kechris, we introduce the notion of a quasi non-archimedean group. We observe that full groups are quasi non-archimedean, and that every continuous homomorphism from an infinitesimally finitely generated group into a quasi non-archimedean group is trivial. We prove that a locally compact group is quasi non-archimedean if and only if it is totally disconnected, and provide various examples which show that the picture is much richer for Polish groups. In particular, we get an example of a Polish group which is infinitesimally 1-generated but totally disconnected, strengthening Stevens’ negative answer to Problem 160 from the Scottish book. 1 Introduction One of the simplest invariants one can come up with for a topological group G is its topological rank t(G), that is, the minimum number of elements needed to generate a dense subgroup of G. For this invariant to have a chance to be ﬁnite, one needs to assume that G is separable since every ﬁnitely generated group is countable. The oldest result in this topic is probably due to Kronecker in 1884 [Kro31], and says that an n-tuple (a1 , ..., an ) of real numbers projects down to a topological generator of Tn = Rn /Zn if and only if (1, a1 , ..., an ) is Q-linearly independent. Since such n-tuples always exist, the topological rank of the compact connected abelian group Tn is equal to one. Using this result, it is then an amusing exercise to show that t(Rn ) = n+1. This means that as a topological group, Rn remembers its vector space dimension. In particular, we see that the topological rank sometimes contains useful information (another somehow similar instance of this phenomenon was recently discovered by the second-named author for full groups, see [LM14]). In order to deal with general connected Lie groups, it is useful to introduce the following deﬁnition. ∗ Research supported by the ISF, Moked grant 2095/15. Research supported by the Interuniversity Attraction Pole DYGEST and Projet ANR-14-CE25-0004 GAMME. † 1 Definition 1.1. The infinitesimal rank of a topological group G is the minimum n ∈ N ∪ {+∞} such that every neighborhood of the identity in G contains n elements g1 , ..., gn which generate a dense subgroup of G. We denote it by tI (G). One can easily show that tI (Rn ) = t(Rn ) = n + 1. This is relevant for the study of the inﬁnitesimal rank of real Lie groups because if we know that the Lie algebra of a connected Lie group G is generated (as a Lie algebra) by n elements, then using the fact fact that tI (R) = 2 we can deduce that tI (G) 6 2n (see the well-known Lemma 4.2). For various classes of connected Lie groups one can say more. In particular this is the situation for compact connected Lie groups, in which case a uniform result is useful in view of the important role of compact groups in the structure theory of locally compact groups. Auerbach showed that for every compact connected Lie group G, one has tI (G) 6 2 [Aue34]. Moreover, he could show that the set of pairs of topological generators of G has full measure, which then led Schreier and Ulam to the following general result, for which we will include a short proof. Theorem 1.2 (Schreier-Ulam, [SU35]). Let G be a connected compact metrisable group. Then almost every pair of elements of G generates a dense subgroup of G. In particular, tI (G) = t(G) = 2 for any non-abelian such G. Note that there is a vast area between metrizable compact groups and separable ones; R for instance (T1 ) is compact separable, but not metrizable. Conversely, being separable is a minimal assumption for a group to have ﬁnite topological rank. Going back to the n-torus, Kronecker’s result admits a far-reaching generalization due to Halmos and Samelson, which settles the abelian case. Theorem 1.3 (Halmos-Samelson, [HS42, Corollary]). The topological rank of every connected separable compact abelian group is equal to one. Our ﬁrst result is a generalization of Schreier and Ulam’s Theorem to the separable compact case. One of the diﬃculties is that we cannot use their Haar measure argument anymore. Theorem 1.4 (see Theorem 3.1). Let G be a separable compact connected group. If G is non-abelian, then tI (G) = 2, and if G is abelian, then tI (G) = 1. The fact that the topological rank t(G) = 2 was proved by Hoﬀmann and Morris [HM90, Theorem. 4.13]. Note that it follows that tI (G) = t(G) for separable compact connected group. It is natural to ask if the following stronger statement is true: Question 1.5. Let G be a separable compact connected group. Is the set of pairs in G which topologically generate G necessarily of full measure in G × G? Say that a topological group is infinitesimally finitely generated if it has ﬁnite inﬁnitesimal rank. Using the previous result, and the solution to Hilbert’s ﬁfth problem, we establish the following result, which was noted by Schreier and Ulam in the abelian case (see the last paragraph of [SU35]). Theorem 1.6 (see Theorem. 4.1). Let G be a separable locally compact group. Then G is infinitesimally finitely generated if and only if G is connected. 2 Let us now leave the realm of locally compact groups for a moment and discuss the class of Polish groups, i.e. separable topological groups whose topology admits a compatible metric. These groups abound in analysis, for instance the unitary group of a separable Hilbert space or the group of measure-preserving transformations of a standard probability space are Polish groups. Moreover, they form a robust class of groups, e.g. every countable product of Polish groups is Polish (see [Gao09] for other properties of this ﬂavour). Recall that a locally compact group is Polish if and only if it is second-countable. It is not hard to show that Theorem 1.6 fails for Polish groups: for instance RN is connected but not topologically ﬁnitely generated, in particular it is not inﬁnitesimally ﬁnitely generated. The question of the converse is more interesting, even for the following weaker property. Definition 1.7. A topological group G is infinitesimally generated if every neighbourhood of the identity generates G. Clearly every connected group is inﬁnitesimally generated, and every inﬁnitesimally ﬁnitely generated group is inﬁnitesimally generated. Moreover it follows from van Dantzig’s theorem that every inﬁnitesimally generated locally compact group is connected. Question 1.8 (Mazur’s Problem 160 [Mau81]). Must an inﬁnitesimally generated Polish group be connected? In [Ste86], Stevens exhibited the ﬁrst examples of inﬁnitesimally generated Polish group which are totally disconnected. We show that her examples actually have inﬁnitesimal rank 2 and then provide the following stronger negative answer to Question 1.8. Theorem 1.9 (see Theorem. 5.15). There exists a Polish group of infinitesimal rank 1 which is totally disconnected. Let us now introduce the quasi non-archimedean property which is a strong negation of being inﬁnitesimally ﬁnitely generated. Definition 1.10. A topological group is quasi non-archimedean if for every neighborhood of the identity U in G and every n ∈ N, there exists a neighborhood of the identity V such that for every g1 , ..., gn ∈ V , the group generated by g1 , ..., gn is contained in U. Remark. If we switch the quantiﬁers and ask for a V which works for every n ∈ N, it is not hard to see that the deﬁnition then becomes that of a non-archimedean topological group (i.e. admitting a basis of neighborhoods of the identity made of open subgroups). Our inspiration for the above deﬁnition comes from the following result of Kechris: every continuous homomorphism from an inﬁnitesimally ﬁnitely generated group into a full group is trivial (see the paragraph just before Section (E) of Chapter 4 in [Kec10]). We upgrade this by showing that every full group is quasi non-archimedean, and that any continuous homomorphism from an inﬁnitesimally ﬁnitely generated group into a quasi non-archimedean group is trivial (see Proposition 5.5). For locally compact groups, we obtain the following characterisation. Theorem 1.11 (see Theorem. 5.8). Let G be a separable locally compact group. Then G is quasi non-archimedean if and only if G is totally disconnected. 3 Note that full groups are connected and at the same time quasi non-archimedean Polish groups. Moreover, we show that every quasi non-archimedean Polish groups embeds into a connected quasi non-archimedean Polish group (see Proposition 5.4.). We also provide examples of totally disconnected Polish groups which are quasi nonarchimedean, but not non-archimedean (see Corollary. 5.12). On the other hand Theorem 1.9 ensures us that there are totally disconnected Polish groups which are not quasi nonarchimedean. The paper is organised as follows. In Section 2, we prove some basic results on topological generators. In Section 3, we show that separable connected compact groups are inﬁnitesimally 2-generated. Section 4 is devoted to the proof that every connected separable locally compact group is inﬁnitesimally ﬁnitely generated. In Section 5 we introduce quasi non-archimedean groups and study their basic properties. We also give numerous examples, and show that a separable locally compact group is totally disconnected if and only if it is quasi non-archimedean. Finally, a Polish group into which no non-discrete locally compact group can embed is built in Section 6, where we also ask three questions raised by this work. Acknowledgements. We warmly thank Pierre-Emmanuel Caprace for coming up with the terminology “inﬁnitesimally generated”, as well as for useful conversations around this topic. We also thank Yves de Cornulier for his helpful remarks on a ﬁrst version of the paper. 2 Basic results about topological generators We collect some results which will serve us in the preceding sections. Proposition 2.1. Let G be a connected locally compact group. Suppose that K is a profinite normal subgroup of G such that G/K is infinitesimally finitely generated. Then tI (G) = tI (G/K). Moreover, if ḡ1 , . . . , ḡk topologically generate the group G/K, and g1 , . . . , gk are arbitrary respective lifts in G, then g1 , . . . , gk topologically generate G. The proof of Proposition 2.1 will rely on the following two lemmas: Lemma 2.2. Let H be a connected locally compact group and f : H ։ L a finite covering map. Let {l1 , . . . lk } be a topological generating set for L and pick hi ∈ f −1 (li ), i = 1, . . . , k arbitrarily. Then h1 , . . . , hk topologically generate H. Proof. Set F = hh1 , . . . , hk i. Note that f , being a ﬁnite cover, is a closed map, and hence f (F ) is closed in L. Since l1 , . . . , lk ∈ f (F ) we have f (F ) = L. Thus Ker f · F = H. Hence by the Baire category theorem F has a non-empty interior. Since H is connected, this implies that F = H. Lemma 2.3. Let G be a (connected) locally compact group admitting a pro-finite normal subgroup K ⊳G such that L = G/K is a Lie group. Then G is an inverse limit G = lim Lα ←− of finite (central) extensions Lα of L. Although the Lemma holds without the assumption that G is connected, since it signiﬁcantly simplify the proof while being suﬃcient for our needs, we will prove it only under the connectedness assumption. 4 Proof. Given k ∈ K, the image of the orbit map G → K, g 7→ gkg −1 is at the same time connected, since G is connected and the map is continuous, and totally disconnected as the image lies in K. It follows that it is constant and k is central. Since k is arbitrary we deduce that K is central in G. In particular every subgroup of K is normal in G. Let Kα be a net of open subgroups in K with trivial intersection. Then K = lim K/Kα ←− and G = lim G/Kα . Set Lα = G/Kα and note that as Kα is open in K, it is of ﬁnite ←− index there. Thus the map Lα → L is a ﬁnite covering. Proof of Proposition 2.1. Let G and K be as in Proposition 2.1. By Lemma 2.3, G = lim Lα is an inverse limit of ﬁnite covers Lα of L = G/K. Let ḡ1 , . . . , ḡk be topological ←− generators of L, let g1 , . . . , gk be arbitrary lifts in G and denote by giα the projection of gi in Lα , for every i, α. By Lemma 2.2, hgiα : i = 1, . . . , ki is dense in Lα . That is, the group hgi : i = 1, . . . , ki projects densely to all Lα . This implies that it is dense in G. 3 Connected compact separable groups are infinitesimally 2-generated Our aim in this section is to prove the following result. Recall that t(G) is the minimal number of topological generators of G, while tI (G) is the minimal n ∈ N such that every neighborhood of the identity contains n elements which topologically generate G. Theorem 3.1. Let G be a separable compact connected group. Then tI (G) = t(G) = 2 if G is nonabelian and tI (G) = t(G) = 1 if G is abelian. 3.1 The metrizable case The case where G is metrizable was proved by Schreier and Ulam [SU35]. Recall that a compact group is metrizable if and only if it is ﬁrst countable. In that case one can show that almost every pair of elements in G (or a single element if G is abelian) topologically generates G. Let us give a short argument for Theorem 1.2 in that case. First recall that by the Peter–Weyl theorem, G is an inverse limit of compact Lie groups and, being ﬁrst countable, the limit is over a countable net. Since a countable intersection of full measured sets is of full measure, it is enough to prove the analog statement for compact connected Lie groups. Note also that, in complete generality, a subgroup H ≤ G is dense if and only if • H ∩ G′ is dense in G′ , where G′ is the commutator subgroup in G, and • HG′ is dense in G. A connected compact Lie group G is reductive, hence an almost direct product of its commutator G′ with its centre Z. Moreover G′ is connected and a ﬁnite cover of G/Z which is a semisimple group of adjoint type. In view of Lemma 2.2, we deduce: Lemma 3.2. Let G be a connected compact Lie group and H ≤ G a subgroup. Then H is dense in G if and only if both HG′ and HZ(G) are dense in G. Therefore, for a pair (x, y) ∈ G2 to generate a dense subgroup, it is suﬃcient if • the projections of x, y to G/Z generate a dense subgroup in G/Z, and 5 • the projection of x to G/G′ generates a dense subgroup in G/G′ . It is easy to check that both conditions are satisﬁed with Haar probability 1 (cf. [Gel08, Lem. 1.4 and Lem. 1.10]). 3.2 Proof of Theorem 3.1 in the general case Let us ﬁrst deal with abelian groups. By the Halmos–Samelson theorem, whenever G is connected compact separable abelian, one has t(G) = 1. From their result, we deduce the following consequence. Lemma 3.3. Let G be a compact connected separable abelian group. Then tI (G) = 1. Proof. By the Halmos–Samelson theorem, we may and do pick g ∈ G such that hgi is dense in G. Let U be a neighborhood of the identity in G, and ﬁx n ∈ Z \ {0} such that g n ∈ U. Then since hg n i has ﬁnite index in hgi, its closure hg n i has ﬁnite index in hgi = G. Since G is connected, we must have hg n i = G. Now suppose that G is any connected compact group. By the Peter–Weil theorem G = lim Gα is an inverse limit of compact connected Lie groups Gα . ←− Observe that a surjective map f : G1 ։ G2 between groups always satisﬁes f (G′1 ) = G′2 and f (Z(G1 )) ⊂ Z(G2 ), while for reductive Lie groups we also have f (Z(G1 )) = Z(G2 ). Since every connected compact Lie group is reductive, hence the product of its centre and its commutator, we deduce that the same hold for general compact connected groups, i.e. Z(G) = lim Z(Gα ), G′ = lim G′α and G = Z(G)G′ . ←− ←− Moreover, since the G′α are semisimple, and in particular perfect, G′ is also perfect, i.e. G′ = G′′ . It follows that if Γ ≤ G′ is dense, then Γ′ is dense in G′ . Hence we have: Claim. Suppose that a, b ∈ G′ topologically generate G′ , and h ∈ Z(G) is an element whose image mod G′ topologically generates G/G′ . Then ah and b topologically generate G. Suppose from now on that G is separable. Then every quotient of G is also separable. Now G/G′ is connected and abelian, so we have tI (G/G′ ) = 1 by Lemma 3.3. Since Z(G) surjects onto G/G′ , every identity neighbourhood in G contains a central element h whose image in G/G′ generate a dense subgroup. Thus we are left to show that tI (G′ ) = 2. The centre of G′ is totally disconnected since it can be written as Z(G′ ) = lim Z(G′α ), and every G′α has ﬁnite center. In view of Proposition 2.1 we may ←− thus suppose that G′ is center-free. In order to simplify notations, let us suppose below that G itself is center-free. Note that: Lemma 3.4. A center-free connected compact group is a direct product of simple Lie groups. Q Thus, G is of the form G = α∈I Sα with Sα being connected adjoint simple Lie group. Q Lemma 3.5. The group G = α∈I Sα is separable (if and) only if Card(I) ≤ 2ℵ0 . 6 Let us explain the ‘only if’ side. The other direction will follow once we show that Card(I) ≤ 2ℵ0 implies that G has a two generated dense subgroup. Suppose by way of contradiction that Card(I) > 2ℵ0 . Since there are only countably many isomorphism types of (compact adjoint) simple Lie groups, we deduce that there is some compact simple Lie group S and a cardinality κ > 2ℵ0 such that G admits a factor isomorphic to S κ . However by the cardinals version of the pigeon hole principal, if D ⊂ S κ is a countable subset, there must be two factors S1 , S2 of S κ such that the projection of D to S1 × S2 lies in the diagonal. In particular, D cannot be dense, conﬁrming the desired contradiction. Thus, we may suppose below that Card(I) ≤ 2ℵ0 . Definition 3.6. Let S1 , S2 be two groups, F a set and fi : F → Si , i = 1, 2 two maps. We shall say that the maps f1 and f2 are isomorphically related if there is an isomorphism φ : S1 → S2 such that the following diagram is commutative F f1 − → S1 ց f2 ↓ φ S2 Q Q Lemma 3.7. Let ki=1 Si (k ≥ 2) be a product of simple groups and let R < ki=1 Si be a proper subgroup that projects onto every Si . Then there are two factors Si , Sj , i 6= j such that the restrictions of the quotient maps R → Si and R → Sj are isomorphically related. Q Proof. For a subset J ⊂ {1, . . . , k} let us denote SJ = i∈J Si and RJ = ProjSJ (R). Let J ⊂ {1, . . . , k} be a minimal subset such that RJ is a proper subgroup of SJ . By our assumption J exists and satisﬁes \|J\| > 1. We claim that \|J\| = 2. To see this, we may reorder the indices so that J = {1, . . . , j} and suppose by way of contradiction that j ≥ 3. This, together with the minimality of J, implies that for every g ∈ S1 and every 1 < i ≤ j there is an element in RJ whose ﬁrst coordinate is g and whose i’th coordinate is 1. However, multiplying commutators of elements as above, forcing that at each coordinate 1 < i ≤ j at least one of the elements we use is trivial, and using the fact that S1 is perfect, we deduce that S1 ≤ RJ . In the same way we get that Si ≤ RJ for all i ∈ J contradicting the assumption that RJ is proper in SJ . Thus J = {1, 2}. Moreover, as Si is simple and normal, and RJ projects onto Si , while RJ is proper in SJ , it follows that RJ ∩ Si is trivial, for i = 1, 2. Therefore, the restriction of the projection from RJ to Si is an isomorphism, for i = 1, 2, hence the maps R → S1 and R → S2 are isomorphically related. Assembling together isomorphic factors of G, we may decompose G as a direct product Q G = n∈N Gn where diﬀerent n’s correspond to non-isomorphic simple Lie factors Sn , and each Gn is of the form Gn = SnIn with \|In \| ≤ 2ℵ0 . For x ∈ G we shall denote by xαn , α ∈ In its corresponding coordinates. Two elements x, y ∈ G generate a dense subgroup if and only if for any ﬁnite Q set of indices F = {(ni , αi )} the projections of x and y generate a dense subgroup in F Snαii . In view of Lemma 3.7 we have: Corollary 3.8. Two elements x, y ∈ G generate a dense subgroup in G if and only if • hxαn , ynαi is dense in Snα for every n ∈ N and every α ∈ In , and • For every n, m ∈ N and α ∈ In , β ∈ Im the (projection) maps {x, y} → Snα , {x, y} → β Sm are not isomorphically related. 7 Thus, in order to prove the theorem, we should explain how to pick (xαn , ynα) arbitrarily close to the identity in Snα , for every n ∈ N, α ∈ In , which generate a dense subgroup of Sn , such that for every pair (n, α) 6= (m, β) there is no isomorphism Sn → Sm taking β (xαn , ynα ) to (xβm , ym ). Since for n 6= m there is no isomorphism between Sn and Sm we should only consider the case n = m. Let S = Sn . Recall that there is an identity neighbourhood (a Zassenhaus neighbourhood) Ω ⊂ S on which the logarithm is well deﬁned, and for x, y ∈ Ω the group hx, yi is dense in S if log(x) and log(y) generate the Lie algebra Lie(S), (see [Kur49]). Thus we may pick some open set U1 × U2 ⊂ Ω2 such that for all (x, y) ∈ U1 × U2 the group hx, yi is dense in S (cf. [GŻuk02]). Furthermore, we may pick U1 , U2 arbitrarily close to the identity. Since Out(S) is ﬁnite and S is compact, we may also suppose, by taking U1 and U2 suﬃciently small, that for (x, y) ∈ U1 × U2 , if f ∈ Aut(S) satisﬁes (f (x), f (y)) ∈ U1 × U2 , then f is inner. Finally, since the orbit of each (x, y) ∈ U1 × U2 under the action of S on S × S by conjugation, is dim(S) dimensional, while dim(U1 × U2 ) = 2 dim(S), we can pick a section S transversal to the orbits foliation, inside U1 × U2 . Clearly the cardinality of that section is 2ℵ0 , hence we may imbed In inside this section. This imbedding yields a choice of (xα , y α) ∈ S ⊂ U1 × U2 for every α ∈ In , and we have that each pair (xα , y α ) generates a dense subgroup in S, and no two pairs are isomorphically related. This completes the proof of Theorem 3.1. 4 Local generators and locally compact connected groups Recall that a topological group G is infinitesimally finitely generated if tI (G) < +∞, that is, if there exists n ∈ N such that every neighborhood of the identity contains n topological generators for G. Similarly, we say that G is topologically finitely generated if t(G) < ∞, that is, if G admits a dense ﬁnitely generated subgroup. Last but not least, a topological group is infinitesimally generated if it is generated by every neighborhood of the identity. Note that if a group is inﬁnitesimally ﬁnitely generated, then it is also topologically ﬁnitely generated and inﬁnitesimally generated. Our main goal in this section is to prove the following result. Theorem 4.1. Let G be a separable connected locally compact group. Then G is infinitesimally finitely generated. As already noted, the separability condition is necessary for a group to be topologically ﬁnitely generated. The connectedness assumption is also necessary (for local generation) since otherwise G/G◦ , the group of connected components, is non-trivial and by the van Dantzig’s theorem admits a base of identity neighbourhood consisting of open compact subgroups. In particular, if O ≤ G/G◦ is a proper open subgroup, its pre-image in G cannot contain a topological generating set. Let us start by dealing with the nicest connected locally compact groups: Lie groups. For these, one can use the Lie algebra to produce topological generators. The following lemma is well known, but we include a proof for the reader’s convenience. Lemma 4.2 (Folklore). Let G be a connected Lie group, and let g be its Lie algebra. Suppose that g is generated as a Lie algebra by X1 , ..., Xn . Then every neighborhood of 8 the identity in G contains 2n elements g1 , ..., g2n which generate a dense subgroup in G. In particular, G is infinitesimally finitely generated. Proof. Let V be a neighborhood of the identity in G. Let U be a small enough neighborhood of 0 in g such that exp : U → G is a homeomorphism onto its image, and exp(U) ⊆ V . Fix 2n elements Y1 , ..., Y2n of U such that for every i ∈ {1, ..., n}, {Y2i , Y2i+1 } generates a dense subgroup of RXi . For all i ∈ {1, ..., 2n}, let gi = exp(Yi ), we will show that these elements topologically generate G. Let H be the closed subgroup generated by the set {gi }2n i=1 . Note that for all i ∈ {1, ..., n}, the group H contains exp(RXi ) since the restriction of the exponential map to RXi is a continuous group homomorphism and RXi is topologically generated by Y2i and Y2i+1 which are mapped to g2i ∈ H and g2i+1 ∈ H. Furthermore, H is a Lie group by Cartan’s theorem, and since H contains every exp(RXi ) the Lie algebra h of H contains every Xi , so h = g. Because G is connected, we get that G = H. We deduce from the above lemma that for any connected Lie group, tI (G) 6 2 dim(G). Better bounds on tI (G) can be deduced from the analysis in [BG03, BGSS06, Gel08]. Let now G be a general connected locally compact group. Recall the celebrated Gleason-Yamabe theorem (cf. [Kap71, Page 137]): Theorem 4.3. (Gleason–Yamabe) Let G be a connected locally compact group. Then there is a compact normal subgroup K ⊳ G such that G/K is a Lie group. Since G is connected, G/K is a connected Lie group and hence tI (G/K) is ﬁnite. However, K may not be connected. Example 4.4. (1) (The solenoid) For every n, let Tn be a copy of the circle group {z ∈ C : \|z\| = 1}, and whenever m divides n let fn,m : Tn → Tm be the n/m sheeted cover fn,m (z) = z n/m . Let T = lim Tn be the inverse limit group. Then T is connected, abelian ←− and locally compact, but admits no connected co-Lie subgroups. (2) Similarly, as SL2 (R) is homotopic to a circle, we can deﬁne, for every n ∈ N, Gn as the n sheeted cover of SL2 (R). Then whenever m divides n there is a canonical covering morphisms ψn,m : Gn → Gm , and we may let G be the inverse limit G = lim Gn . Then ←− G is a connected locally compact group which admits no nontrivial connected compact normal subgroups. In order to prove Theorem 4.1, we need one last elementary lemma. Lemma 4.5. Let G be a topological group and N a normal subgroup. Then tI (G) ≤ tI (N) + tI (G/N). In particular if G/N and N are infinitesimally finitely generated then so is G. Proof of Theorem 4.1. Let G be a connected separable locally compact group. Let K ⊳ G be a compact normal subgroup such that G/K is a Lie group (see Theorem 4.3), and let K ◦ be its identity connected component. Then K ◦ is characteristic in K and therefore normal in G. Being a closed subgroup of a locally compact separable group, K ◦ is separable [CI77]. Let H = G/K ◦ and K t = K/K ◦ . By the isomorphism theorem, G/K ∼ = H/K t . t Note that K is a pro-ﬁnite group, hence by Proposition 2.1 H is inﬁnitesimally ﬁnitely generated. By Theorem 3.1, K ◦ is inﬁnitesimally ﬁnitely generated, hence, by Lemma 4.5, G is inﬁnitesimally ﬁnitely generated. 9 5 Quasi non-archimedean groups A topological group is non-archimedean if it has a basis of neighborhoods of the identity made of open subgroups. Equivalently, every neighborhood of the identity V contains a smaller neighborhood of the identity U such that the group generated by U is contained in V , which is the same as requiring that for every n ∈ N and g1 , ..., gn ∈ U, the group generated by g1 , ..., gn is contained in V . The deﬁnition that follows is obtained by switching two quantiﬁers in the above condition. Definition 5.1. Say a topological group G is quasi non-archimedean if for all n ∈ N and all neighborhood of the identity V ⊆ G, there exists a neighborhood of the identity U ⊆ V such that for all g1 , ..., gn ∈ U, the group generated by g1 , ..., gn is contained in V . Clearly every non-archimedean group is also quasi non-archimedean. Let us give right away the motivating example for this deﬁnition. We ﬁx a standard probability space (X, µ), that is, a probability space which is isomorphic to the interval [0, 1] with its Borel σ-algebra and the Lebesgue-measure. A Borel bijection T of X is called a non-singular automorphism if for all measurable A ⊆ X, one has µ(A) = 0 if and only if µ(T −1(A)) = 0. The group of all these automorphisms is denoted by Aut∗ (X, µ), two such automorphisms being identiﬁed if they coincide on a full measure set. We then deﬁne the uniform metric du on Aut∗ (X, µ) by: for all T, U ∈ Aut∗ (X, µ), du (T, U) = µ({x ∈ X : T (x) 6= U(x)}). This is a complete metric, though far from being separable (e.g. the group S1 acts freely on itself, yielding an uncountable discrete subgroup of (Aut∗ (X, µ), du )). But among closed subgroups of (Aut∗ (X, µ), du ), full groups are separable. Full groups are invariants of orbit equivalence attached to nonsingular actions of countable groups on (X, µ): given a non-singular action of a countable group Γ on (X, µ), its full group [RΓ ] is the group of all T ∈ Aut∗ (X, µ) such that for every x ∈ X, T (x) ∈ Γ · x. Since every subgroup of a quasi non-archimedean group is quasi non-archimedean for the induced topology, the following result implies that full groups are quasi nonarchimedean. Theorem 5.2 (Kechris). Aut∗ (X, µ) is quasi non-archimedean for the uniform metric. Proof. Deﬁne the support of T ∈ Aut∗ (X, µ) to be the set of all x ∈ X such that T (x) 6= x. Note that du (idX , T ) is precisely the measure of the support of T . Let ǫ > 0 and n ∈ N, and consider the open ball U := Bdu (idX , ǫ). Suppose that g1 , ..., gn belong to V := Bdu (idX , ǫ/n), and let A be the reunion of their supports. By assumption, A has measure less than ǫ. Then the group generated by g1 , ..., gn is contained in the group of elements supported in A, which is itself a subset of U = Bdu (idX , ǫ). The following proposition shows that the class of quasi non-archimedean groups satisﬁes basically the same closure properties as the class of non-archimedean groups. Proposition 5.3. The class of quasi non-archimedean groups is closed under taking subgroups (with the induced topology), products and quotients. 10 The next proposition highlights the main diﬀerence between non-archimedean and quasi non-archimedean groups. From now on, we will restrict ourselves to the narrower but well-behaved class of Polish groups, that is, separable groups whose topology admits a compatible complete metric, e.g. full groups for the uniform topology, the group Aut∗ (X, µ) endowed with the weak topology, or the unitary group of a separable Hilbert space endowed with the strong operator topology. Let us point out that a locally compact group is Polish if and only if it is second-countable (see [Kec95, Thm. 5.3]). Recall that if G is a Polish group and (X, µ) is a standard (non-atomic) probability space, then the group L0 (X, µ, G) of measurable maps from X to G is a Polish group for the topology of convergence in measure, two such maps being identiﬁed if they coincide on a full measure set. A basis of neighborhoods of the identity for this topology is given by the sets Ũǫ = {f ∈ L0 (X, µ, G) : µ({x ∈ X : f (x) 6∈ U}) < ǫ}, where U is an open neighborhood of the identity in G and ǫ > 0. The Polish group L0 (X, µ, G) enjoys the two following nice properties (see e.g. [Kec10, Chap. 19]) . • G embeds into L0 (X, µ, G) via constant maps. • L0 (X, µ, G) is connected, in fact contractible. Proposition 5.4. Let G be a quasi non-archimedean Polish group. Then L0 (X, µ, G) is quasi non-archimedean. In particular any quasi non-archimedean Polish group embeds in a connected quasi non-archimedean Polish group. Proof. Let Ũǫ = {f : µ({x ∈ X : f (x) 6∈ U}) < ǫ} be a basic neighborhood of the identity in L0 (X, µ, G). Let n ∈ N and V be a corresponding neighborhood of the identity witnessing that G is quasi non-archimedean. Consider the following open neighborhood of the identity in L0 (X, µ, G): Ṽǫ/n = {f : µ({x ∈ X : f (x) 6∈ V }) < ǫ/n}. Then if we let f1 , ..., fn ∈ Ṽǫ,n , the reunion of the sets {x ∈ X : fi (x) 6∈ V } has measure less than ǫ. By the deﬁnition of V and U the group generated by f1 , ..., fn is a subset of Ũǫ . Remark. Since non-archimedean groups are totally disconnected, the above proposition implies there are a lot more quasi non-archimedean groups than the non-archimedean ones. The following proposition is inspired by Section (D) of Chapter 4 in [Kec10], where it is shown that any continuous homomorphism from an inﬁnitesimally ﬁnitely generated group into a full group is trivial. Proposition 5.5. Any continuous homomorphism from an infinitesimally finitely generated group into a quasi non-archimedean group is trivial. Proof. Let ϕ : G → H be such a morphism, let V be any neighborhood of the identity in H, and let n = tI (G). Then there is a neighborhood of the identity U in H such that any subgroup of H generated by n elements of U is contained in V . Since ϕ is continuous, ϕ−1 (U) is a neighborhood of the identity in G, and because tI (G) = n we may ﬁnd g1 , ..., gn ∈ ϕ−1 (U) which generate a dense subgroup in G. Then the closure of ϕ(G) coincides with the closure of the group generated by ϕ(g1 ), ..., ϕ(gn ) ∈ U, which by assumption is contained in V . So ϕ(G) is contained in the closure of any neighborhood of the identity in H, and since H is Hausdorﬀ this means that ϕ is trivial. 11 Corollary 5.6. The only topological group which is both infinitesimally finitely generated and quasi non-archimedean is the trivial group. Corollary 5.7. Every continuous homomorphism from a connected separable locally compact group into (Aut∗ (X, µ), du ) is trivial. Proof. Since every connected separable locally compact group is inﬁnitesimally ﬁnitely generated by Theorem 4.1 and Aut∗ (X, µ) is quasi non-archimedean by Theorem 5.2, the previous proposition readily applies. As a consequence of the previous proposition and Theorem 1.6, we have the following interesting characterizations of connectedness and total disconnectedness for locally compact separable groups. Theorem 5.8. Let G be a locally compact separable group. Then the following hold: (1) G is connected if and only if G is infinitesimally finitely generated. (2) G is totally disconnected if and only if G is quasi non-archimedean. Proof. If G is not connected but inﬁnitesimally ﬁnitely generated, then G/G0 must also be inﬁnitesimally ﬁnitely generated, which is impossible by van Dantzig’s theorem. The converse is provided by Theorem 1.6. If G is totally disconnected, then G is non-archimedean by van Dantzig’s theorem. But this implies that G is quasi non-archimedean. For the converse, suppose G is quasi nonarchimedean. Then G0 also is, but then by (1) and Proposition 5.5 it must be trivial. Remark. As was pointed out by Caprace and Cornulier, one can prove (2) more directly. Indeed, if G0 is non-trivial then it admits a non-trivial one-parameter subgroup which is in particular inﬁnitesimally 2-generated, contradicting Proposition 5.5. This actually gives a proof that a locally compact group is quasi non-archimedean if and only if it is totally disconnected, regardless of its separability. Let us now give an example of a totally disconnected Polish group which is quasi non-archimedean, but not non-archimedean. This class of examples was introduced by Tsankov [Tsa06, Sec. 5], using work of Solecki [Sol99]. We denote by S∞ the group of all permutations of the integers, equipped with its Polish topology of pointwise convergence. Recall that every non-archimedean Polish group arises as a closed subgroup of S∞ (see [BK96, Thm. 1.5.1]). Here, the groups that we will consider are subgroups of S∞ , but equipped with a Polish topology which reﬁnes the topology of pointwise convergence. Definition 5.9. A lower semi-continuous submeasure on N is a function λ : P(N) → [0, +∞] such that the following hold: • λ(∅) = 0; • for all n ∈ N, we have 0 < λ({n}) < +∞; • for all A ⊆ B ⊆ N, we have λ(A) 6 λ(B); • (subadditivity) for all A, B ⊆ N, we have λ(A ∪ B) 6 λ(A) + λ(B); • (lower semi-continuity) for every increasing sequence (Ak )k∈N of subsets of N, we S have λ( k∈N Ak ) = limk∈N λ(Ak ). 12 We associate to every lower semi-continuous submeasure λ on N a subgroup of S∞ , denoted by Sλ , deﬁned by Sλ = {σ ∈ S∞ : λ(supp σ \ {0, ..., n}) → 0 [n → +∞]}, where supp σ = {n ∈ N : σ(n) 6= n}. Note that since µ({0, ..., n}) < +∞ for every n ∈ N, the support of every σ ∈ Sλ has ﬁnite measure. Also, if λ is actually a measure, then Sλ = {σ ∈ S∞ : λ(supp σ) < +∞}; furthermore if λ is a probability measure then Sλ = S∞ . The group Sλ is equipped with a natural left-invariant metric dλ analogous to the uniform metric on Aut∗ (X, µ) deﬁned by dλ (σ, σ ′ ) = λ({n ∈ N : σ(n) 6= σ ′ (n)}). Note that the condition λ(supp σ \ {0, ..., n}) → 0 ensures that the countable group of permutations of ﬁnite support is dense in Sλ which is thus separable. It is a theorem of Tsankov that Sλ is actually a Polish group. The following result is a straightforward adaptation of Theorem 5.2, replacing du by dλ . Proposition 5.10. Let λ be a lower semi-continuous submeasure on N. Then Sλ is quasi non-archimedean. It is easily checked that the topology of Sλ reﬁnes the topology induced by S∞ , so that Sλ is always totally disconnected, and that the open subgroups of Sλ separate points from the identity (in particular, Sλ is not locally generated). The following example shows that it can furthermore fail to be non-archimedean. Note that this is just a particular case of a more general phenomenon: one can actually characterize when the topology fails to be zero-dimensional1 (see [Tsa06, Thm. 5.3]; the example below is taken from [Mal15, Cor. 4]). Example 5.11. Consider the measure λ on N deﬁned by X1 . λ(A) = n n∈A Then Sλ is not a non-archimedean group. Indeed, if we ﬁx ǫ > 0 and N ∈ N, we can ﬁnd a ﬁnite family (Ai )N i=1 of disjoint subsets of N such that for every i ∈ {1, ..., N}, ǫ < λ(Ai ) < ǫ. For all i ∈ {1, ..., N}, let σi be a permutation whose support is equal 2 Q to Ai . Then σ = N i=1 σi is at distance at least Nǫ/2 from the identity, so that the ball of radius ǫ around the identity generates a group which contains elements arbitrarily far away from the identity. Corollary 5.12. There exists a totally disconnected Polish group which is quasi nonarchimedean, but not non-archimedean. Let us now give examples of totally disconnected Polish group which are inﬁnitesimally ﬁnitely generated. To do this, we will upgrade a result of Stevens [Ste86] who showed the existence of totally disconnected inﬁnitesimally generated Polish groups: we will show that her examples are actually inﬁnitesimally ﬁnitely generated. This will be a consequence of the following general statement, which also implies the well-known fact that R has a dense Gδ of pairs of topological generators. 1 A topology is zero-dimensional if it has a basis made of clopen sets. 13 Theorem 5.13. Let G be an abelian Polish group which contains the group Z[1/2] of dyadic rationals as a dense subgroup, and assume furthermore that in the topology induced by G on Z[1/2], we have 21n → 0 [n → +∞]. Then for every g0 ∈ Z[1/2], the set of h ∈ G such that hg, hi generate a dense subgroup of G is dense. In particular there is a dense Gδ set of couples of topological generators of G in G2 and so G is infinitesimally finitely generated with infinitesimal rank at most 2. Proof. Let g0 ∈ Z[1/2]. In order for a couple (g0 , h) ∈ G2 to generate a dense subgroup of G, it suﬃces for the closed subgroup they generate to contain 21n for every n ∈ N, for 2 the group Z[1/2] is dense in G. T So the set T := {h ∈ G : hg0 , hi = G} may be written as a countable intersection T = n∈N Tn , where 1 Tn := h ∈ G : n ∈ hg0 , hi . 2 Since G is Polish the set Tn is Gδ , so we only need to show that Tn is dense. To this end, ﬁx ǫ > 0, n ∈ N, and let h0 ∈ Z[1/2]. We want to ﬁnd h ∈ Tn such that d(h, h0 ) < ǫ, where d is a ﬁxed compatible metric on G. Write g0 = 2km1 and h0 = 2km2 , where k1 , k2 ∈ Z and m ∈ N. We will ﬁnd β ∈ N such β 1 that for all N ∈ N, if h = h0 + 2m+N , then hg0 , hi contains 2N+m , so that in particular it 1 1 if and only if we can contains 2n as soon as N + m > n. The group hg0 , hi contains 2N+m 1 ﬁnd u, v ∈ Z such that ug0 + vh = 2m+N . This condition can be rewritten as 2N k1 u + (2N k2 + β)v = 1. So we want to ﬁnd β ∈ N such that for all N ∈ N we have that 2N k1 and 2N k2 + β are relatively prime. Let us furthermore ask that β is odd, so that we only have to make sure that every odd prime divisor of k1 does not divide 2N k2 + β. Let p1 , ..., pk list the odd primes which divide both k1 and k2 , while pk+1, ..., pl are the odd primes which divide k1 but not k2 . Then it is easily checked that β = (2 + p1 · · · pk )pk+1 · · · pl works: for all i 6 k we have that β is invertible modulo pi and pi divides k2 so that 2N k2 + β is not divisible by pi , while for k < i 6 l, β is null modulo pi while 2N k2 is invertible so that 2N k2 + β is not divisible by pi . β 1 But then, since 2m+N tends to zero as N tends to +∞, we also have 2m+N → 0 β [N → +∞]. Then, as explained before, the group generated by g := g0 and h := h0 + 2m+N 1 contains 2m+N , so that (g, h) ∈ Tn , while 0 = d(g, g0) < ǫ and d(h, h0 ) < ǫ if N was chosen large enough. So every Tn is a dense subset of G, which ends the proof since this furthermore shows that the set of couples generating a dense subgroup of G is dense in G2 and this set has to be a Gδ . Let us now apply the previous theorem and describe Steven’s examples of Polish groups which are inﬁnitesimally ﬁnitely generated but totally disconnected. These groups arise as a Polishable subgroups of the real line, constructed by taking a completion of the dyadic rationals with respect to a well chosen norm which makes 2−n have much bigger norm than usually. We ﬁx a biinﬁnite sequence of positive real number (ri )i∈Z such that ri → 0[i → +∞] and for all i ∈ Z, we have ri+1 6 ri 6 2ri+1 . Then one can deﬁne the following group 14 norm2 k·k on the ring Z[1/2] of dyadic rationals: for every x ∈ Z[1/2], ( n ) n X X kxk := inf \|ai \| ri : x = ai 2−i , ai ∈ Z, n ∈ N . i=−n i=−n It is easy to check that this deﬁnes a group norm on Z[1/2] which reﬁnes the usual norm. Using the fact that ri 6 2ri+1 , one can easily show that for all x ∈ Z[1/2], ( n ) n X X kxk = inf \|ai \| ri : x = ai 2−i , ai ∈ {−1, 0, 1}, n ∈ N . i=−n i=−n In particular, we see that for all n ∈ N, we have k2−n k = rn so that 2−n → 0 as n → +∞. Let Z[1/2] k·k denote the completion of Z[1/2] with respect to this norm. Since this norm reﬁnes the usual norm, Z[1/2] k·k is a subgroup of R. Stevens explicitely described the k·k k·k elements of R belonging to Z[1/2] and showed that the group Z[1/2] is inﬁnitesimally generated [Ste86, Thm. 2.1 (ii)], and we see that Theorem 5.13 strengthens this because k·k it implies that Z[1/2] is inﬁnitesimally 2-generated. To obtain totally disconnected examples, we need another result of Stevens stating that the following are equivalent (see [Ste86, Thm. 2.2]): P (i) i∈N ri = +∞, (ii) k·k is not equivalent to \|·\| when restricted to Z[1/2], (iii) Q ∩ Z[1/2] (iv) Z[1/2] k·k k·k = Z[1/2], is totally disconnected, Remark. Note that every subgroup G of R which is not equal to R has to be totally disconnected for the induced topology, since its complement is dense in R so that the sets of the form ]r, +∞[ for r ∈ R \ G are clopen in G. In particular, if G is a proper subgroup of R equipped with a topology which reﬁnes the usual topology of R then G is totally disconnected. So conditions (ii) and condition (iii) clearly imply condition (iv). P So suppose further that i∈N ri = +∞ (e.g. take ri = 1 if i 6 0 and ri = 1i otherwise). k·k Then we see that Z[1/2] is a totally disconnected Polish group which has inﬁnitesimal rank at most 2. Moreover since this group is a subgroup of the real line endowed with a ﬁner topology (see [Ste86, Thm. 2.1]) it cannot be monothetic, so its inﬁnitesimal rank is actually equal to 2. Corollary 5.14. There exists a totally disconnected Polish group which has infinitesimal rank 2, in particular there is a totally disconnected Polish group which is not quasi nonarchimedean. Remark. Note that one can see directly that Stevens’ groups are not quasi non-archimedean, even for n = 1. Indeed, if U is a neighborhood of 0 not containing 1, then if V is another neighborhood of 0 there is some N ∈ N such that 1/2N ∈ V , but the group generated by 1/2N contains 1 hence it is not a subset of U. 2 A norm on an abelian group is a function \|·\| : G → [0, +∞) such that for any x, y ∈ G, \|x + y\| 6 \|x\| + \|y\|, and \|x\| = \|−x\|. 15 We know that while the group of the reals has inﬁnitesimal rank 2, its quotient S1 = R/Z has inﬁnitesimal rank 1. The same is true of Stevens’ examples, which is going to yield the following result. Theorem 5.15. There exists a totally disconnected Polish group which has infinitesimal rank 1. Proof. Let G be a totally disconnected Polish group obtained by Stevens’ construction from a sequence ; then G is a proper subgroup of R containing Z[1/2] as a dense subgroup. Observe that Z is a discrete subgroup of G and we may thus form the Polish group G̃ := G/Z. The group G̃ is a proper dense subgroup of S1 = R/Z, so S1 \ G̃ is thus dense in S1 . Let A = p([0, 1/2[) where p : R → R/Z is the usual projection, then for all g ∈ S1 \ G̃ the set (g + A) ∩ G̃ is clopen in G̃. Moreover since S1 \ G̃ is dense in S1 the family of sets ((g + A) ∩ G̃)g∈S1 \G̃ separates points in G̃, so G̃ is totally disconnected. Furthermore, we have by Theorem 5.13 that there is a dense Gδ of h ∈ G such that the group generated by 1 and h is dense in G. Since 1 ∈ Z and G̃ = G/Z we conclude that there is a dense Gδ of h ∈ G̃ which generate a dense subgroup in G̃, in particular G̃ has inﬁnitesimal rank 1. We don’t know an example of a totally disconnected Polish group which is inﬁnitesimally generated and quasi non-archimedean. Moreover, we want to stress out that all the examples we know of Polish groups which are quasi non-archimedean actually fail the property even for n = 1, so it would be very interesting to have examples having a “non-QNA rank” greater than 1. 6 Further remarks and questions Let us point out how one can easily build Polish groups into which no non-discrete locally compact group can embed. Lemma 6.1. Let Γ be a countable discrete group without elements of finite order. Then every monothetic subgroup of L0 (X, µ, Γ) is infinite discrete. In particular, no nontrivial compact group embeds into L0 (X, µ, Γ). Proof. Given 1 6= f ∈ L0 (X, µ, Γ), ﬁnd A ⊆ X non-null and γ ∈ Γ \ {1} such that f↾A is constant equal to γ. By assumption, for all n ∈ Z \ {0}, the support of f n contains A, and so hf i is discrete. Theorem 6.2. Let Γ be a countable discrete group without elements of finite order, and let G be a separable locally compact group. Then every continuous morphism G → L0 (X, µ, Γ) factors through a discrete group. Proof. Let G0 be the connected component of the identity. Because L0 (X, µ, Γ) is quasi non-archimedean, π factors through G/G0 by Proposition 5.5 and Theorem 1.6. Then by van Dantzig’s theorem and the previous lemma, the kernel of the later map contains an open subgroup of G/G0 , hence it factors through a discrete group. Question 6.3. Is there a Polish group without any locally compact closed subgroup? 16 The group L0 (X, µ, G) was originally introduced to show that every Polish group embeds into a connected group, and we saw that being quasi non-archimedean is somehow opposite to being connected. Because L0 (X, µ, G) can be quasi non-archimedean, one may ask whether every Polish group embeds into a connected not quasi non-archimedean group. The isometry group of the Urysohn space answers this question — it is universal for Polish groups, connected (see [Mel06, Mel10] for stronger versions of these results as well as background on the Urysohn space) and cannot be quasi non-archimedean by universality. However, the same question can be asked replacing not quasi non-archimedean by inﬁnitesimally ﬁnitely generated. It seems to be open wether the isometry group of the Urysohn space is inﬁnitesimally ﬁnitely generated (it is topologically 2-generated by a result of Solecki, see [Sol05]). Question 6.4. Is the isometry group of the Urysohn space inﬁnitesimally 2-generated? Let us end this paper by mentioning a question related to ample generics. A Polish group G has ample generics if the diagonal conjugacy action of G onto Gn has a comeager orbit for every n ∈ N (see [KR07]). It has been recently discovered that there exists Polish groups with ample generics which are not non-archimedean (see[KLM15] and [Mal15]). These examples arise either as full groups or as groups of the form Sλ , which are quasi non-archimedean groups by Theorem 5.2 and Proposition 5.10. This motivates the following question. Question 6.5. Is there a Polish group which has ample generics, but which is not quasi non-archimedean? References [Aue34] H. Auerbach. Sur les groupes linéaires bornés (iii). Studia Mathematica, 5(1):43– 49, 1934. [BG03] E. Breuillard and T. Gelander. On dense free subgroups of Lie groups. J. Algebra, 261(2):448–467, 2003. [BGSS06] Emmanuel Breuillard, Tsachik Gelander, Juan Souto, and Peter Storm. Dense embeddings of surface groups. Geom. Topol., 10:1373–1389, 2006. [BK96] Howard Becker and Alexander S. Kechris. The descriptive set theory of Polish group actions, volume 232 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1996. [CI77] W. W. Comfort and G. L. Itzkowitz. Density character in topological groups. Math. Ann., 226(3):223–227, 1977. 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Proceedings of Machine Learning Research vol 65:1–58, 2017 Towards Instance Optimal Bounds for Best Arm Identification arXiv:1608.06031v2 [cs.LG] 24 May 2017 Lijie Chen Jian Li Mingda Qiao CHENLJ 13@ MAILS . TSINGHUA . EDU . CN LIJIAN 83@ MAIL . TSINGHUA . EDU . CN QMD 14@ MAILS . TSINGHUA . EDU . CN Institute for Interdisciplinary Information Sciences (IIIS), Tsinghua University, Beijing, China. Abstract In the classical best arm identification (Best-1-Arm) problem, we are given n stochastic bandit arms, each associated with a reward distribution with an unknown mean. Upon each play of an arm, we can get a reward sampled i.i.d. from its reward distribution. We would like to identify the arm with the largest mean with probability at least 1 − δ, using as few samples as possible. The problem has a long history and understanding its sample complexity has attracted significant attention since the last decade. However, the optimal sample complexity of the problem is still unknown. Recently, Chen and Li (2016) made an interesting conjecture, called gap-entropy conjecture, concerning the instance optimal sample complexity of Best-1-Arm. Given a Best-1-Arm instance I (i.e., a set of arms), let µ[i] denote the ith largest mean and ∆[i] = µ[1] − µ[i] denote the corPn −2 responding gap. H(I) = i=2 ∆[i] denotes the complexity of the instance. The gap-entropy conjecture states that for any instance I, Ω H(I) · ln δ −1 + Ent(I) is an instance lower bound, where Ent(I) is an entropy-like term determined by the gaps, and there is a δ-correct algorithm for −1 . We note Best-1-Arm with sample complexity O H(I) · ln δ −1 + Ent(I) + ∆−2 ln ln ∆ [2] [2] −2 −1 that Θ ∆[2] ln ln ∆[2] is necessary and sufficient to solve the two-arm instance with the best and second best arms. If the conjecture is true, we would have a complete understanding of the instance-wise sample complexity of Best-1-Arm (up to constant factors). In this paper, we make significant progress towards a complete resolution of the gap-entropy conjecture. For the upper bound, we provide a highly nontrivial algorithm which requires −1 −1 O H(I) · ln δ −1 + Ent(I) + ∆−2 ln ln ∆ polylog(n, δ ) [2] [2] samples in expectation for any instance I. For the lower bound, we show that for any Gaussian Best-1-Arm instance with gaps of the form 2−k , any δ-correct monotone algorithm requires at least Ω H(I) · ln δ −1 + Ent(I) samples in expectation. Here, a monotone algorithm is one which uses no more samples (in expectation) on I ′ than on I, if I ′ is a sub-instance of I obtained by removing some sub-optimal arms. Keywords: best arm identification, instance optimality, gap-entropy 1. Introduction The stochastic multi-armed bandit is one of the most popular and well-studied models for capturing the exploration-exploitation tradeoffs in many application domains. There is a huge body c 2017 L. Chen, J. Li & M. Qiao. C HEN L I Q IAO of literature on numerous bandit models from several fields including stochastic control, statistics, operation research, machine learning and theoretical computer science. The basic stochastic multi-armed bandit model consists of n stochastic arms with unknown distributions. One can adaptively take samples from the arms and make decision depending on the objective. Popular objectives include maximizing the cumulative sum of rewards, or minimizing the cumulative regret (see e.g., Cesa-Bianchi and Lugosi (2006); Bubeck et al. (2012)). In this paper, we study another classical multi-armed bandit model, called pure exploration model, where the decision-maker first performs a pure-exploration phase by sampling from the arms, and then identifies an optimal (or nearly optimal) arm, which serves as the exploitation phase. The model is motivated by many application domains such as medical trials Robbins (1985); Audibert and Bubeck (2010), communication network Audibert and Bubeck (2010), online advertisement Chen et al. (2014), crowdsourcing Zhou et al. (2014); Cao et al. (2015). The best arm identification problem (Best-1-Arm) is the most basic pure exploration problem in stochastic multiarmed bandits. The problem has a long history (first formulated in Bechhofer (1954)) and has attracted significant attention since the last decade Audibert and Bubeck (2010); Even-Dar et al. (2006); Mannor and Tsitsiklis (2004); Jamieson et al. (2014); Karnin et al. (2013); Chen and Li (2015); Carpentier and Locatelli (2016); Garivier and Kaufmann (2016). Now, we formally define the problem and set up some notations. Definition 1.1 Best-1-Arm: We are given a set of n arms {A1 , . . . , An }. Arm Ai has a reward distribution Di with an unknown mean µi ∈ [0, 1]. We assume that all reward distributions are Gaussian distributions with unit variance. Upon each play of Ai , we get a reward sampled i.i.d. from Di . Our goal is to identify the arm with the largest mean using as few samples as possible. We assume here that the largest mean is strictly larger than the second largest (i.e., µ[1] > µ[2] ) to ensure the uniqueness of the solution, where µ[i] denotes the ith largest mean. Remark 1.2 Some previous algorithms for Best-1-Arm take a sequence (instead of a set) of n arms as input. In this case, we may simply assume that the algorithm randomly permutes the sequence at the beginning. Thus the algorithm will have the same behaviour on two different orderings of the same set of arms. Remark 1.3 For the upper bound, everything proved in this paper also holds if the distributions are 1-sub-Gaussian, which is a standard assumption in the bandit literature. On the lower bound side, we need to assume that the distributions are from some family parametrized by the means and satisfy certain properties. See Remark D.4. Otherwise, it is possible to distinguish two distributions using 1 sample even if their means are very close. We cannot hope for a nontrivial lower bound in such generality. The Best-1-Arm problem for Gaussian arms was first formulated in Bechhofer (1954). Most early works on Best-1-Arm did not analyze the sample complexity of the algorithms (they proved their algorithms are δ-correct though). The early advances are summarized in the monograph Bechhofer et al. (1968). For the past two decades, significant research efforts have been devoted to understanding the optimal sample complexity of the Best-1-Arm problem. On the lower bound Mannor and Tsitsiklis Pn side,−2 (2004) proved that any δ-correct algorithm for Best-1-Arm takes Ω( i=2 ∆[i] ln δ−1 ) samples in expectation. In fact, their result is an instance-wise lower bound (see Definition 1.6). Kaufmann et al. 2 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION P −1 (2015) also provided an Ω( ni=2 ∆−2 [i] ln δ ) lower bound for Best-1-Arm, which improved the constant factor in Mannor and Tsitsiklis (2004). Garivier and Kaufmann (2016) focused on the asymptotic sample complexity of Best-1-Arm as the confidence level δ approaches zero (treating the gaps as fixed), and obtained a complete resolution of this case (even for the leading constant).1 Chen and Li (2015) showed that for each n there exists a Best-1-Arm instance with n arms that Pn −2 require Ω ln ln n samples, which further refines the lower bound. ∆ i=2 [i] The algorithms for Best-1-Arm have also been significantly improved in the last two decades Even-Dar et al. (2002); Gabillon et al. (2012); Kalyanakrishnan et al. (2012); Karnin et al. (2013); Jamieson et al. (2014); Chen and Li (2015); Garivier and Kaufmann (2016). Karnin et al. (2013) obtained an upper bound of Xn −1 −1 . O + ln δ ln ln ∆ ∆−2 [i] [i] i=2 The same upper bound was obtained by Jamieson et al. (2014) using a UCB-type algorithm called lil’UCB. Recently, the upper bound was improved to Xn −1 −1 −2 −1 + ln ln ∆ , n) + ln δ ln ln min(∆ O ∆−2 ∆ [2] [2] [i] [i] i=2 by Chen and Li (2015). There is still a gap between the best known upper and lower bound. To understand the sample complexity of Best-1-Arm, it is important to study a special case, which we term as SIGN-ξ. The problem can be viewed as a special case of Best-1-Arm where there are only two arms, and we know the mean of one arm. SIGN-ξ will play a very important role in our lower bound proof. Definition 1.4 SIGN-ξ: ξ is a fixed constant. We are given a single arm with unknown mean µ 6= ξ. The goal is to decide whether µ > ξ or µ < ξ. Here, the gap of the problem is defined to be ∆ = \|µ − ξ\|. Again, we assume that the distribution of the arm is a Gaussian distribution with unit variance. In this paper, we are interested in algorithms (either for Best-1-Arm or for SIGN-ξ) that can identify the correct answer with probability at least 1 − δ. This is often called the fixed confidence setting in the bandit literature. Definition 1.5 For any δ ∈ (0, 1), we say that an algorithm A for Best-1-Arm (or SIGN-ξ) is δcorrect, if on any Best-1-Arm (or SIGN-ξ) instance, A returns the correct answer with probability at least 1 − δ. 1.1. Almost Instance-wise Optimality Conjecture It is easy to see that no function f (n, δ) (only depending on n and δ) can serve as an upper bound of the sample complexity of Best-1-Arm (with n arms and confidence level 1 − δ). Instead, the sample complexity depends on the gaps. Intuitively, the smaller the gaps are, the harder the instance is (i.e., more samples are required). Since the gaps completely determine an instance (for Gaussian arms with unit variance, up to shifting), we use ∆[i] ’s as the parameters to measure the sample complexity. 1. In contrast, our work focus on the situation that both δ and all gaps are variables that tend to zero. In fact, if we let the gaps (i.e., ∆[i] ’s) tend to 0 while maintaining δ fixed, their lower bound is not tight. 3 C HEN L I Q IAO Now, we formally define the notion of instance-wise lower bounds and instance optimality.For algorithm A and instance I, we use TA (I) to denote the expected number of samples taken by A on instance I. Definition 1.6 (Instance-wise Lower Bound) For a Best-1-Arm instance I and a confidence level δ, we define the instance-wise lower bound of I as L(I, δ) := inf TA (I). A:A is δ-correct for Best-1-Arm We say a Best-1-Arm algorithm A is instance optimal, if it is δ-correct, and for every instance I, TA (I) = O(L(I, δ)). Now, we consider the Best-1-Arm problem from the perspective of instance optimality. Unfortunately, even for the two-arm case, no instance optimal algorithm may exist. In fact, Farrell (1964) showed that for any δ-correct algorithm A for SIGN-ξ, we must have lim inf ∆→0 TA (I) = Ω(1). ∆−2 ln ln ∆−1 This implies that any δ-correct algorithm requires ∆−2 ln ln ∆−1 samples in the worst case. Hence, the upper bound of ∆−2 ln ln ∆−1 for SIGN-ξ is generally not improvable. However, for a particular SIGN-ξ instance I∆ with gap ∆, there is an δ-correct algorithm that only needs O(∆−2 ln δ−1 ) samples for this instance, implying L(I∆ , δ) = Θ(∆−2 ln δ−1 ). See Chen and Li (2015) for details. Despite the above fact, Chen and Li (2016) conjectured that the two-arm case is the only obstruction toward an instance optimal algorithm. Moreover, based on some evidence from the previous work Chen and Li (2015), they provided an explicit formula and conjecture that L(I, δ) can be expressed by the formula. Interestingly, the formula involves an entropy term (similar entropy terms also appear in Afshani et al. (2009) for completely different problems). In order to state Chen and Li’s conjecture formally, we define the entropy term first. Definition 1.7 Given a Best-1-Arm instance I and k ∈ N, let X Gk = {i ∈ [2, n] \| 2−(k+1) < ∆[i] ≤ 2−k }, Hk = ∆−2 [i] , i∈Gk and pk = H k / X j Hj . We can view {pk } as a discrete probability distribution. We define the following quantity as the gap entropy of instance I: X 2 Ent(I) = pk ln p−1 k . k∈N:Gk 6=∅ Remark 1.8 We choose to partition the arms based on the powers of 2. There is nothing special about the constant 2, and replacing it by any other constant only changes Ent(I) by a constant factor. Conjecture 1.9 (Gap-Entropy Conjecture (Chen and Li, 2016)) There is an algorithm for Best1-Arm with sample complexity −1 ln ln ∆ O L(I, δ) + ∆−2 [2] , [2] 2. Note that it is exactly the Shannon entropy for the distribution defined by {pk }. 4 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION for any instance I and δ < 0.01. And we say such an algorithm is almost instance-wise optimal for Best-1-Arm. Moreover, Xn −1 L(I, δ) = Θ . · ln δ + Ent(I) ∆−2 [i] i=2 Remark 1.10 As we mentioned before, the term ∆−2 ln ln ∆−1 is sufficient and necessary for distinguishing the best and the second best arm, even though it is not an instance-optimal bound. The gap entropy conjecture states that modulo this additive term, we can obtain an instance optimal algorithm. Hence, the resolution of the conjecture would provide a complete understanding of the sample complexity of Best-1-Arm (up to constant factors). All the previous bounds for Best-1-Arm agree with Conjecture 1.9, i.e., existing upper (lower) bounds are no smaller (larger) the conjectured bound. See Chen and Li (2016) for details. 1.2. Our Results In this paper, we make significant progress toward the resolution of the gap-entropy conjecture. On the upper bound side, we provide an algorithm that almost matches the conjecture. Theorem 1.11 There is a δ-correct algorithm for Best-1-Arm with expected sample complexity Xn −1 −2 −1 −1 O · polylog(n, δ ) . ln ln ∆ · ln δ + Ent(I) + ∆ ∆−2 [2] [2] i=2 [i] P −1 + Ent(I) in Conjecture 1.9. For the Our algorithm matches the main term ni=2 ∆−2 [i] · ln δ additive term (which is typically small), we lose a polylog(n, δ−1 ) factor. In particular, for those instances where the additive term is polylog(n, δ−1 ) times smaller than the main term, our algorithm is optimal. On the lower bound side, despite that we are not able to completely solve the lower bound, we do obtain a rather strong bound. We need to introduce some notations first. We say an instance is discrete, if the gaps of all the sub-optimal arms are of the form 2−k for some positive integer k. We say an instance I ′ is a sub-instance of an instance I, if I ′ can be obtained by deleting some sub-optimal arms from I. Formally, we have the following theorem. Theorem 1.12 For any discrete instance I, confidence level δ < 0.01, and any δ-correct algorithm A for Best-1-Arm, there exists a sub-instance I ′ of I such that Xn −1 TA (I ′ ) ≥ c · , · ln δ + Ent(I) ∆−2 [i] i=2 where c is a universal constant. We say an algorithm is monotone, if TA (I ′ ) ≤ TA (I) for every I ′ and I such that I ′ is a subinstance of I. Then we immediately have the following corollary. Corollary 1.13 For any discrete instance I, and confidence level δ < 0.01, for any monotone δ-correct algorithm A for Best-1-Arm, we have that Xn −1 TA (I) ≥ c · , · ln δ + Ent(I) ∆−2 [i] i=2 where c is a universal constant. We remark that all previous algorithms for Best-1-Arm have monotone sample complexity bounds. The above corollary also implies P that if an algorithm has amonotone sample complexn −2 −1 + Ent(I) on all discrete instances. ity bound, then the bound must be Ω i=2 ∆[i] · ln δ 5 C HEN L I Q IAO 2. Related Work SIGN-ξ and A/B testing. In the A/B testing problem, we are asked to decide which arm between the two given arms has the larger mean. A/B testing is in fact equivalent to the SIGN-ξ problem. It is easy to reduce SIGN-ξ to A/B testing by constructing a fictitious arm with mean ξ. For the other direction, given an instance of A/B testing, we may define an arm as the difference between the two given arms and the problem reduces to SIGN-ξ where ξ = 0. In particular, our refined lower bound for SIGN-ξ stated in Lemma 4.1 also holds for A/B testing. Kaufmann et al. (2015); Garivier and Kaufmann (2016) studied the limiting behavior of the sample complexity of A/B testing as the confidence level δ approaches to zero. In contrast, we focus on the case that both δ and the gap ∆ tend to zero, so that the complexity term due to not knowing the gap in advance will not be dominated by the ln δ−1 term. Best-k-Arm. The Best-k-Arm problem, in which we are required to identify the k arms with the k largest means, is a natural extension of Best-1-Arm. Best-k-Arm has been extensively studied in the past few years Kalyanakrishnan and Stone (2010); Gabillon et al. (2011, 2012); Kalyanakrishnan et al. (2012); Bubeck et al. (2013); Kaufmann and Kalyanakrishnan (2013); Zhou et al. (2014); Kaufmann et al. (2015); Chen et al. (2017), and most results for Best-k-Arm are generalizations of those for Best-1Arm. As in the case of Best-1-Arm, the sample complexity bounds of Best-k-Arm depend on the gap parameters of the arms, yet the gap of an arm is typically defined as the distance from its mean to either µ[k+1] or µ[k] (depending on whether the arm is among the best k arms or not) in the context of Best-k-Arm problem. The Combinatorial Pure Exploration problem, which further generalizes the cardinality constraint in Best-k-Arm (i.e., to choose exactly k arms) to general combinatorial constraints, was also studied Chen et al. (2014, 2016); Gabillon et al. (2016). PAC learning. The sample complexity of Best-1-Arm and Best-k-Arm in the probably approximately correct (PAC) setting has also been well studied in the past two decades. For Best-1-Arm, the tight worst-case sample complexity bound was obtained by Even-Dar et al. (2002); Mannor and Tsitsiklis (2004); Even-Dar et al. (2006). Kalyanakrishnan and Stone (2010); Kalyanakrishnan et al. (2012); Zhou et al. (2014); Cao et al. (2015) also studied the worst case sample complexity of Best-k-Arm in the PAC setting. 3. Preliminaries Throughout the paper, I denotes an instance of Best-1-Arm (i.e., I is a set of arms). The arm with the largest mean in I is called the optimal arm, while all other arms are sub-optimal. We assume that every instance has a unique optimal arm. Ai denotes the arm in I with the i-th largest mean, unless stated otherwise. The mean of an arm A is denoted by µA , and we use µ[i] as a shorthand notation for µAi (i.e., the i-th largest mean in an instance). Define ∆A = µ[1] − µA as the gap of arm A, and let ∆[i] = ∆Ai denote the gap of arm Ai . We assume that ∆[2] > 0 to ensure the optimal arm is unique. We partition the sub-optimal arms into different groups based on their gaps. For each k ∈ N, group Gk is defined as Ai : ∆[i] ∈ 2−(k+1) , 2−k . For brevity, let G≥k and G≤k denoted S∞ Sk −2 i=k Gi and i=1 Gi respectively. The complexity of arm Ai is defined as ∆[i] , while the comPn plexity of instance I is denoted by H(I) = i=2 ∆−2 [i] (or simply H, if the instance is clear from P −2 the context). Moreover, Hk = A∈Gk ∆A denotes the total complexity of the arms in group Gk . 6 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION (Hk )∞ k=1 naturally defines a probability distribution on N, where the probability of k is given by pk = Hk /H. The gap-entropy of the instance I is then denoted by X Ent(I) = pk ln p−1 k . k Here and in the following, we adopt the convention that 0 ln 0−1 = 0. 4. A Sketch of the Lower Bound 4.1. A Comparison with Previous Lower Bound Techniques We briefly discuss the novelty of our new lower bound technique, and argue why the previous techniques are not sufficient to obtain our result. To obtain a lower bound on the sample complexity of Best-1-Arm, all the previous work Mannor and Tsitsiklis (2004); Chen et al. (2014); Kaufmann et al. (2015); Garivier and Kaufmann (2016) are based on creating two similar instances with different answers, and then applying the change of distribution method (originally developed in Kaufmann et al. (2015)) to argue that a certain number of samples are necessary to distinguish such two instances. The idea was further refined by Garivier and Kaufmann (2016). They formulated a max-min game between the algorithm and some instances (with different answers than the given instance) created by an adversary. The value of the game at equilibrium would be a lower bound of the samples one requires to distinguish the current instance and several worst adversary instances. However, we notice that even in the two-arm case, one cannot prove the Ω(∆−2 ln ln ∆−1 ) lower bound by considering only one max-min game to distinguish the current instance from other instance. Roughly speaking, the ln ln ∆−1 factor is due to not knowing the actual gap ∆, and any lower bound that can bring out the ln ln ∆−1 factor should reflect the union bound paid for the uncertainty of the instance. In fact, for the Best-1-Arm problem with n arms, the gap entropy Ent(I) term exists for a similar reason (not knowing the gaps). Hence, any lower bound proof for Best1-Arm that can bring out the Ent(I) term necessarily has to consider the uncertainty of current instance as well (in fact, the random permutation of all arms is the kind of uncertainty we need for the new lower bound). In our actual lower bound proof, we first obtain a very tight understanding of the SIGN-ξ problem (Lemma 4.1).3 Then, we provide an elegant reduction from SIGN-ξ to Best-1-Arm, by embedding the SIGN-ξ problem to a collection of Best-1-Arm instances. 4.2. Proof of Theorem 1.12 Following the approach in Chen and Li (2015), we establish the lower bound by a reduction from SIGN-ξ to discrete Best-1-Arm instances, together with a more refined lower bound for SIGNξ stated in the following lemma. Lemma 4.1 Suppose δ ∈ (0, 0.04), m ∈ N and A is a δ-correct algorithm for SIGN-ξ. P is a probability distribution on {2−1 , 2−2 , . . . , 2−m } defined by P (2−k ) = pk . Ent(P ) denotes the Shannon entropy of distribution P . Let TA (µ) denote the expected number of samples taken by A when it runs on an arm with distribution N (µ, 1) and ξ = 0. Define αk = TA (2−k )/4k . Then, m X pk αk = Ω(Ent(P ) + ln δ−1 ). k=1 3. Farrell’s lower bound Farrell (1964) is not sufficient for our purpose. 7 C HEN L I Q IAO It is well known that to distinguish the normal distribution N (2−k , 1) from N (−2−k , 1), Ω(4k ) samples are required. Thus, αk = TA (2−k )/4k denotes the ratio between the expected number of samples taken by A and the corresponding lower bound, which measures the “loss” due to not knowing the gap in advance. Then Lemma 4.1 can be interpreted as follows: when the gap is drawn from a distribution P , the expected loss is lower bounded by the sum of the entropy of P and ln δ−1 . We defer the proof of Lemma 4.1 to Appendix D. Now we prove Theorem 1.12 by applying Lemma 4.1 and an elegant reduction from SIGN-ξ to Best-1-Arm. Proof [Proof of Theorem 1.12] Let c0 be the hidden constant in the big-Ω in Lemma 4.1, i.e., m X pk αk ≥ c0 · (Ent(P ) + ln δ−1 ). k=1 We claim that Theorem 1.12 holds for constant c = 0.25c0 . Suppose towards a contradiction that A is a δ-correct (for some δ < 0.01) algorithm for Best-1Arm and I = {A1 , A2 , . . . , An } is a discrete instance, while for all sub-instance I ′ of I, TA (I ′ ) < c · H(I)(Ent(I) + ln δ−1 ). Recall that H(I) and Ent(I) denote the complexity and entropy of instance I, respectively. Construct a distribution of SIGN-ξ instances. Let nk be the number of arms in I with gap 2−k , and m be the greatest integer such that nm > 0. Since I is discrete, the complexity of instance I is given by m X H(I) = 4k nk . k=1 −1 −2 −m }. Moreover, Let pk = 4k nk /H(I). Then (pk )m k=1 defines a distribution P on {2 , 2 , . . . , 2 the Shannon entropy of distribution P is exactly the entropy of instance I, i.e., Ent(P ) = Ent(I). Our goal is to construct an algorithm for SIGN-ξ that violates Lemma 4.1 on distribution P . A family of sub-instances of I. Let U = {k ∈ [m] : nk > 0} be the set of “types” of arms that are present in I. We consider the following family of instances obtained from I. For S ⊆ U , define IS as the instance obtained from I by removing exactly one arm of gap 2−k for each k ∈ S. Note that IS is a sub-instance of I. Let S denote U \ S, the complement of set S relative to U . For S ⊆ U and k ∈ S, let τkS denote the expected number of samples taken on all the nk arms with gap 2−k when A runs on IS . Define αSk = 4−k τkS /nk . We note that 4k αSk is the expected number of samples taken on every arm with gap 2−k in instance IS .4 We have the following inequality: XX XX X 4k nk αSk = τkS ≤ TA (IS ) < c · 2\|U \| H(I)(Ent(I) + ln δ−1 ). (1) S⊆U k∈S S⊆U k∈S S⊆U The second step holds because the lefthand side only counts part of the samples taken by A. The last step follows from our assumption and the fact that IS is a sub-instance of I. 4. Recall that a Best-1-Arm algorithm is defined on a set of arms, so the arms with identical means in the instance cannot be distinguished by A. See Remark 1.2 for details. 8 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Construct algorithm Anew from A. Now we define an algorithm Anew for SIGN-ξ with ξ = 0. Given an arm A, we first choose a set S ⊆ U uniformly at random from all subsets of U . Recall that µ[1] denotes the mean of the optimal arm in I. Anew runs the following four algorithms A1 through A4 in parallel: 1. Algorithm A1 simulates A on IS ∪ {µ[1] + A}. 2. Algorithm A2 simulates A on IS ∪ {µ[1] + A}. 3. Algorithm A3 simulates A on IS ∪ {µ[1] − A}. 4. Algorithm A4 simulates A on IS ∪ {µ[1] − A}. More precisely, when one of the four algorithms requires a new sample from µ[1] + A (or µ[1] − A), we draw a sample x from arm A, feed µ[1] + x to A1 and A2 , and then feed µ[1] − x to A3 and A4 . Note that the samples taken by the four algorithms are the same up to negation and shifting. Anew terminates as soon as one of the four algorithms terminates. If one of A1 and A2 identifies µ[1] + A as the optimal arm, or one of A3 and A4 identifies an arm other than µ[1] − A as the optimal arm, Anew outputs “µA > 0”; otherwise it outputs “µA < 0”. Clearly, Anew is correct if all of A1 through A4 are correct, which happens with probability at least 1 − 4δ. Note that since 4δ < 0.04, the condition of Lemma 4.1 is satisfied. Upper bound the sample complexity of Anew . The crucial observation is that when µA = −2−k and k ∈ S, A1 effectively simulates the execution of A on IS\{k} . In fact, since all arms are Gaussian distributions with unit variance, the arm µ[1] + A is the same as an arm with gap 2−k in the original Best-1-Arm instance. Recall that the number of samples taken on each of the arms with S\{k} . Therefore, the expected number of samples taken on A gap 2−k in instance IS\{k} is 4k αk S\{k} is upper bounded by 4k αk .5 Likewise, when µA = −2−k and k ∈ S, A2 is equivalent to the execution of A on IS\{k} , and thus the expected number of samples on A is less than or equal to S\{k} 4k αk . Analogous claims hold for the case µA = +2−k and algorithms A3 and A4 as well. It remains to compute the expected loss of Anew on distribution P and derive a contradiction to Lemma 4.1. It follows from a simple calculation that   m X X X X 1 S\{k}  S\{k} pk αk ≤ αk pk · \|U \|  + αk 2 k=1 k∈U S⊆U :k∈S S⊆U :k∈S = = ≤ 1 2\|U \|−1 1 2\|U \|−1 X X S\{k} pk αk k∈U S⊆U :k∈S X X 4k nk · αS H(I) k S⊆U k∈S 2\|U \| · c · (Ent(I) + ln δ−1 ) < c0 (Ent(P ) + ln(4δ)−1 ). 2\|U \|−1 5. Recall that if A1 terminates after taking T samples from µ[1] + A, the number of samples taken by Anew on A is also T (rather than 4T ). 9 C HEN L I Q IAO The first step follows from our discussion on algorithm Anew . The third step renames the variables and rearranges the summation. The last line applies (1). This leads to a contradiction to Lemma 4.1 and thus finishes the proof. 5. Warmup: Best-1-Arm with Known Complexity To illustrate the idea of our algorithm for Best-1-Arm, we consider the following Pn simplified yet still non-trivial version of Best-1-Arm: the complexity of the instance, H(I) = i=2 ∆−2 [i] , is given, yet the means of the arms are still unknown. 5.1. Building Blocks We introduce some subroutines that are used throughout our algorithm. Uniform sampling. The first building block is a uniform sampling procedure, Unif-Sampl(S, ε, δ), which takes 2ε−2 ln(2/δ) samples from each arm in set S. Let µ̂A be the empirical mean of arm A (i.e., the average of all sampled values from A). It obtains an ε-approximation of the mean of each arm with probability 1 − δ. The following fact directly follows by the Chernoff bound. Fact 5.1 Unif-Sampl(S, ε, δ) takes O(\|S\|ε−2 ln δ−1 ) samples. For each arm A ∈ S, we have Pr [\|µ̂A − µA \| ≤ ε] ≥ 1 − δ. We say that a call to procedure Unif-Sampl(S, ε, δ) returns correctly, if \|µ̂A − µA \| ≤ ε holds for every arm A ∈ S. Fact 5.1 implies that when \|S\| = 1, the probability of returning correctly is at least 1 − δ. Median elimination. Even-Dar et al. (2002) introduced the Median Elimination algorithm for the PAC version of Best-1-Arm. Med-Elim(S, ε, δ) returns an arm in S with mean at most ε away from the largest mean. Let µ[1] (S) denote the largest mean among all arms in S. The performance guarantees of Med-Elim is formally stated in the next fact. Fact 5.2 Med-Elim(S, ε, δ) takes O(\|S\|ε−2 ln δ−1 ) samples. Let A be the arm returned by Med-Elim. Then Pr[µA ≥ µ[1] (S) − ε] ≥ 1 − δ. We say that Med-Elim(S, ε, δ) returns correctly, if it holds that µA ≥ µ[1] (S) − ε. Fraction test. Procedure Frac-Test(S, clow , chigh , θ low , θ high , δ) decides whether a sufficiently large fraction (compared to thresholds θ low and θ high ) of arms in S have small means (compared to thresholds clow and chigh ). The procedure randomly samples a certain number of arms from S and estimates their means using Unif-Sampl. Then it compares the fraction of arms with small means to the thresholds and returns an answer accordingly. The detailed implementation of Frac-Test is relegated to Appendix A, where we also prove the following fact. Fact 5.3 Frac-Test(S, clow , chigh , θ low , θ high , δ) takes O (ε−2 ln δ−1 ) · (∆−2 ln ∆−1 ) samples, where ε = chigh − clow and ∆ = θ high − θ low . With probability 1 − δ, the following two claims hold simultaneously: 10 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION • If Frac-Test returns True, \|{A ∈ S : µA < chigh }\| > θ low \|S\|. • If Frac-Test returns False, \|{A ∈ S : µA < clow }\| < θ high \|S\|. We say that a call to procedure Frac-Test returns correctly, if both the two claims above hold; otherwise the call fails. Elimination. Finally, procedure Elimination(S, dlow , dhigh , δ) eliminates the arms with means smaller than threshold dlow from S. More precisely, the procedure guarantees that at most a 0.1 fraction of arms in the result have means smaller than dlow . On the other hand, for each arm with mean greater than dhigh , with high probability it is not eliminated. We postpone the pseudocode of procedure Elimination and the proof of the following fact to Appendix A. Fact 5.4 Elimination(S, dlow , dhigh , δ) takes O(\|S\|ε−2 ln δ−1 ) samples in expectation, where ε = dhigh − dlow . Let S ′ denote the set returned by Elimination(S, dlow , dhigh , δ). Then with probability at least 1 − δ/2, \|{A ∈ S ′ : µA < dlow }\| ≤ 0.1\|S ′ \|. Moreover, for each arm A ∈ S with µA ≥ dhigh , we have Pr A ∈ S ′ ≥ 1 − δ/2. We say that a call to Elimination returns correctly if both \|{A ∈ S ′ : µA < dlow }\| ≤ 0.1\|S ′ \| and A1 (S) ∈ S ′ hold; otherwise the call fails. Here A1 (S) denotes the arm with the largest mean in set S. Fact 5.4 directly implies that procedure Elimination returns correctly with probability at least 1 − δ. 5.2. Algorithm Now we present our algorithm for the special case that the complexity of the instance is known in advance. The Known-Complexity algorithm takes as its input a Best-1-Arm instance I, the complexity H of the instance, as well as a confidence level δ. The algorithm proceeds in rounds, and maintains a sequence {Sr } of arm sets, each of which denotes the set of arms that are still considered as candidate answers at the beginning of round r. Roughly speaking, the algorithm eliminates the arms with Ω(εr ) gaps at the r-th round, if they constitute a large fraction of the remaining arms. Here εr = 2−r is the accuracy parameter that we use in round r. To this end, Known-Complexity first calls procedures Med-Elim and Unif-Sampl to obtain µ̂ar , which is an estimation of the largest mean among all arms in Sr up to an O(εr ) error. After that, Frac-Test is called to determine whether a large proportion of arms in Sr have Ω(εr ) gaps. If so, Frac-Test returns True, and then Known-Complexity calls the Elimination procedure with carefully chosen parameters to remove suboptimal arms from Sr . The following two lemmas imply that there algorithm for Best-1-Arm that matches is a δ-correct −1 −2 the instance-wise lower bound up to an O ∆[2] ln ln ∆[2] additive term.6 6. Lemma 5.6 only bounds the number of samples conditioning on an event that happens with probability 1 − δ, so the algorithm may take arbitrarily many samples when the event does not occur. However, Known-Complexity can be transformed to a δ-correct algorithm with the same (unconditional) sample complexity bound, using the “parallel simulation” technique in the proof of Theorem 1.11 in Appendix C. 11 C HEN L I Q IAO Algorithm 1: Known-Complexity(I, H, δ) Input: Instance I with complexity H and risk δ. Output: The best arm. S1 ← I; Ĥ ← 4096H; for r = 1 to ∞ do if \|Sr \| = 1 then return the only arm in Sr ; ; εr ← 2−r ; δr ← δ/(10r 2 ); ar ← Med-Elim(Sr , 0.125εr , 0.01); µ̂ar ← Unif-Sampl({ar }, 0.125εr , δr ); if Frac-Test(S r , µ̂ar −1.75εr , µ̂ar − 1.125εr , 0.3, 0.5, δr ) then δr′ ← \|Sr \|ε−2 r /Ĥ δ; Sr+1 ← Elimination(Sr , µ̂ar − 0.75εr , µ̂ar − 0.625εr , δr′ ); else Sr+1 ← Sr ; end Lemma 5.5 For any Best-1-Arm instance I and δ ∈ (0, 0.01), Known-Complexity(I, H(I), δ) returns the optimal arm in I with probability at least 1 − δ. Lemma 5.6 For any Best-1-Arm instance I and δ ∈ (0, 0.01), conditioning on an event that happens with probability 1 − δ, Known-Complexity(I, H(I), δ) takes −1 ln ln ∆ O H(I) · (ln δ−1 + Ent(I)) + ∆−2 [2] [2] samples in expectation. 5.3. Observations We state a few key observations on Known-Complexity, which will be used throughout the analysis. The proofs are exactly identical to those of Observations A.3 through A.5 in Appendix A. The following observation bounds the value of µ̂ar at round r, assuming the correctness of Unif-Sampl and Med-Elim. Observation 5.7 If Unif-Sampl returns correctly at round r, µ̂ar ≤ µ[1] (Sr ) + 0.125εr . Here µ[1] (Sr ) denotes the largest mean of arms in Sr . If both Unif-Sampl and Med-Elim return correctly, µ̂ar ≥ µ[1] (Sr ) − 0.25εr . The following two observations bound the thresholds used in Frac-Test and Elimination by applying Observation 5.7. = µ̂ar − 1.75εr and chigh = µ̂ar − 1.125εr denote the Observation 5.8 At round r, let clow r r two thresholds used in Frac-Test. If Unif-Sampl returns correctly, chigh ≤ µ[1] (Sr ) − εr . If both r low Med-Elim and Unif-Sampl return correctly, cr ≥ µ[1] (Sr ) − 2εr . Observation 5.9 Let dlow = µ̂ar − 0.75εr and dhigh = µ̂ar − 0.625εr denote the two thresholds r r used in Elimination. If Unif-Sampl returns correctly, dhigh ≤ µ[1] (Sr ) − 0.5εr . If both Med-Elim r and Unif-Sampl return correctly, dlow ≥ µ (S ) − ε . r [1] r r 12 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION 5.4. Correctness We define E as the event that all calls to procedures Unif-Sampl, Frac-Test, and Elimination return correctly. We will prove in the following that Known-Complexity returns the correct answer with probability 1 conditioning on E, and Pr [E] ≥ 1 − δ. Note that Lemma 5.5 directly follows from these two claims. Event E implies correctness. It suffices to show that conditioning on E, Known-Complexity never removes the best arm, and the algorithm eventually terminates. Suppose that A1 ∈ Sr . Observation 5.9 guarantees that at round r, the upper threshold used by Elimination is smaller than or equal to µ[1] (Sr ) − 0.5εr < µ[1] . By Fact 5.4, the correctness of Elimination guarantees that A1 ∈ Sr+1 . It remains to prove that Known-Complexity terminates conditioning on E. Define rmax := maxGr 6=∅ r. Suppose r ∗ is the smallest integer greater than rmax such that Med-Elim returns correctly at round r ∗ .7 By Observation 5.9, the lower threshold in Elimination is greater than or equal to µ[1] − εr∗ . The correctness of Elimination implies that \|Sr∗ +1 \|−1 = \|Sr∗ +1 ∩G≤rmax \| ≤ \|Sr∗ +1 ∩G<r∗ \| = \|{A ∈ Sr∗ +1 : µA < µ[1] −εr∗ }\| < 0.1\|Sr∗ +1 \|. It follows that \|Sr∗ +1 \| = 1. Therefore, the algorithm terminates either before or at round r ∗ + 1. E happens with high probability. We first note that at round r, the probability that either Unif-Sampl or Frac-Test fails (i.e., returns incorrectly) is at most 2δr . By a union bound, the probability that at least one call to Unif-Sampl or Frac-Test returns incorrectly is upper bounded by ∞ X r=1 ∞ X δ < δ/2. 2δr = 5r 2 r=1 It remains to bound the probability that Elimination fails at some round, yet procedures Unif-Sampl and Frac-Test are always correct. Define P (r, Sr ) as the probability that, given the value of Sr at the beginning of round r, at least one call to Elimination returns incorrectly in round r or later, yet Unif-Sampl and Frac-Test always return correctly. We prove by induction that for any Sr that contains the optimal arm A1 , P (r, Sr ) ≤ δ Ĥ where M (r, Sr ) := \|Sr ∩ G≤r−2 \| and C(r, Sr ) := 128C(r, Sr ) + 16M (r, Sr )ε−2 , r ∞ X \|Sr ∩ Gi \| i+1 X ε−2 j + j=r i=r−1 rmax X+1 (2) ε−2 i . i=r The details of the induction are postponed to Appendix E. 7. Med-Elim returns correctly with probability at least 0.99 in each round, so r ∗ is well-defined with probability 1. 13 C HEN L I Q IAO Observe that M (1, I) = 0 and C(1, I) = ∞ X \|Sr ∩ Gi \| ∞ X rmax X+1 \|Sr ∩ Gi \|4i + 4rmax ∞ 16 X ≤ 3 X i=0 A∈Sr ∩Gi Pr [E] ≥ 1 − P (1, S1 ) − 4i i=1 i=0  Therefore we conclude that 4j + j=1 i=0 16 ≤ 3 i+1 X !  −2  ∆−2 ≤ A + ∆[2] 32 H(I). 3 δ 2 δ δ 128C(1, I) + 16M (1, I)ε−2 − 1 2 Ĥ δ 32H δ ≥ 1 − 128 · · − ≥ 1 − δ, 4096H 3 2 which completes the proof of correctness. Here the first step applies a union bound. The second step follows from inequality (2), and the third step plugs in C(1, I) ≤ 32H(I)/3 and Ĥ = 4096H. ≥1− 5.5. Sample Complexity As in the proof of Lemma 5.5, we define E as the event that all calls to procedures Unif-Sampl, Frac-Test, and Elimination return correctly. We prove that Known-Complexity takes −1 ln ln ∆ O H(I)(ln δ−1 + Ent(I)) + ∆−2 [2] [2] samples in expectation conditioning on E. Samples taken by Unif-Sampl and By Facts 5.1 and 5.3, procedures Unif-Sampl Frac-Test. −1 −2 −1 −2 and Frac-Test take O εr ln δr = O εr (ln δ + ln r) samples in total at round r. In the proof of correctness, we showed that conditioning on E, the algorithm does not terminate before or at round k (for k ≥ rmax + 1) implies that Med-Elim fails between round rmax + 1 and round k − 1, which happens with probability at most 0.01k−rmax −1 . Thus for k ≥ rmax + 1, the expected number of samples taken by Unif-Sampl and Frac-Test at round k is upper bounded by −1 O 0.01k−rmax −1 · ε−2 (ln δ + ln k) . k Summing over all k = 1, 2, . . . yields the following upper bound: rX max k=1 −1 ε−2 + ln k) + k (ln δ ∞ X −1 0.01k−rmax −1 · ε−2 + ln k) k (ln δ k=rmax +1 −1 −1 . ln δ + ln ln ∆ =O 4rmax (ln δ−1 + ln rmax ) = O ∆−2 [2] [2] Here the first step holds since the first summation is dominated by the last term (k = rmax ), while the second one is dominated by the firstj term (k = k rmax + 1). The second step follows from the −1 observation that rmax = maxGr 6=∅ r = log2 ∆[2] . 14 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Samples taken by Med-Elim and Elimination. By Facts 5.2 and 5.4, Med-Elim and Elimination (if called) take H −2 −2 ′ −2 −1 O(\|Sr \|εr ) + O(\|Sr \|εr ln(1/δr )) = O \|Sr \|εr ln δ + ln \|Sr \|ε−2 r samples in total at round r. We upper bound the number of samples by a charging argument. For each round i, define ri as the largest integer r such that \|G≥r \| ≥ 0.5\|Si \|.8 Then we define   0, j < ri , Ti,j = H −2  ln δ−1 + ln , j ≥ ri εi \|Gj \|εi−2 as the number of samples that each arm in G Pj is charged at round i. We prove in Appendix E that for any i, j \|Gj \|Ti,j is an upper bound on the number of samples taken by Med-Elim and Elimination at the i-th round. Moreover, the expected number of samples that each arm in group Gj is charged is upper bounded by !! X H E[Ti,j ] = O ε−2 . ln δ−1 + ln j \|Gj \|ε−2 j i Note that Hk = P A∈Gk  O X i,j −2 ∆−2 A = Θ(\|Gk \|εk ). Therefore, Med-Elim and Elimination take   ! H  −2 \|G \|ε j j j   X H  Hj ln δ−1 + ln = O Hj j = O H(I) ln δ−1 + Ent(I) \|Gj \|E[Ti,j ] = O  X \|Gj \|ε−2 j ln δ−1 + ln samples in expectation conditioning on E. In total, algorithm Known-Complexity takes −1 −1 + O H(I) ln δ−1 + Ent(I) ln δ + ln ln ∆ O ∆−2 [2] [2] −1 ln ln ∆ =O H(I) ln δ−1 + Ent(I) + ∆−2 [2] [2] samples in expectation conditioning on E. This proves Lemma 5.6. 5.6. Discussion In the Known-Complexity algorithm, knowing the complexity H in advance is crucial to the efficient allocation of confidence levels (δr′ ’s) to different calls of Elimination. When H is unknown, 8. Note that \|G≥0 \| = n − 1 ≥ 0.5\|Si \| and \|G≥r \| = 0 < 0.5\|Si \| for sufficiently large r, so ri is well-defined. 15 C HEN L I Q IAO our approach is to run an elimination procedure similar to Known-Complexity with a guess of H. The major difficulty is that when our guess is much smaller than the actual complexity, the total confidence that we allocate will eventually exceed the total confidence δ. Thus, we cannot assume in our analysis that all calls to the Elimination procedure are correct. We present our ComplexityGuessing algorithm for the Best-1-Arm problem in Appendix A. References Peyman Afshani, Jérémy Barbay, and Timothy M Chan. Instance-optimal geometric algorithms. In Foundations of Computer Science, 2009. FOCS’09. 50th Annual IEEE Symposium on, pages 129–138. IEEE, 2009. Jean-Yves Audibert and Sébastien Bubeck. Best arm identification in multi-armed bandits. In COLT-23th Conference on Learning Theory-2010, pages 13–p, 2010. Robert E Bechhofer. A single-sample multiple decision procedure for ranking means of normal populations with known variances. The Annals of Mathematical Statistics, pages 16–39, 1954. Robert Eric Bechhofer, Jack Kiefer, and Milton Sobel. 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Optimal pac multiple arm identification with applications to crowdsourcing. In Proceedings of the 31st International Conference on Machine Learning (ICML-14), pages 217–225, 2014. 18 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Organization of the Appendix The appendix contains the proofs of our main results. In Section A, we present our algorithm for Best-1-Arm along with a few useful observations. In Section B and Section C, we prove the correctness and the sample complexity of our algorithm, thus proving Theorem 1.11. We present the complete proof of Theorem 1.12 in Section D. Finally, Section E contains the complete proofs of Lemma 5.5 and Lemma 5.6. Appendix A. Upper Bound A.1. Building Blocks We start by presenting the missing implementation and performance guarantees of our subroutines Frac-Test and Elimination. Fraction test. Recall that on input (S, clow , chigh , θ low , θ high , δ), procedure Frac-Test decides whether a sufficiently large fraction (with respect to θ low and θ high ) of arms in S have means smaller than the thresholds clow and chigh . The pseudocode of Frac-Test is shown below. Algorithm 2: Frac-Test(S, clow , chigh , θ low , θ high , δ) Input: An arm set S, thresholds clow , chigh , θ low , θ high , and confidence level δ. ε ← chigh − clow ; ∆ ← θ high − θ low ; m ← (∆/6)−2 ln(2/δ); cnt ← 0; for i = 1 to m do Pick A ∈ S uniformly at random; µ̂A ← Unif-Sampl({A}, ε/2, ∆/6); if µ̂A < (clow + chigh )/2 then cnt ← cnt + 1; end if cnt/m > (θ low + θ high )/2 then return True; else return False; Now we prove Fact 5.3. Fact 5.3 (restated) Frac-Test(S, clow , chigh , θ low , θ high , δ) takes O((ε−2 ln δ−1 ) · (∆−2 ln ∆−1 )) samples, where ε = chigh − clow and ∆ = θ high − θ low . With probability 1 − δ, the following two claims hold simultaneously: • If Frac-Test returns True, \|{A ∈ S : µA < chigh }\| > θ low \|S\|. • If Frac-Test returns False, \|{A ∈ S : µA < clow }\| < θ high \|S\|. Proof The first claim directly follows from Fact 5.1 and m · O(ε−2 ln ∆−1 ) = O((ε−2 ln δ−1 ) · (∆−2 ln ∆−1 )). It remains to prove the contrapositive of the second claim: \|{A ∈ S : µA < clow }\| ≥ θ high \|S\| implies Frac-Test returns True, and \|{A ∈ S : µA < chigh }\| ≤ θ low \|S\| implies Frac-Test returns False. 19 C HEN L I Q IAO Suppose \|{A ∈ S : µA < clow }\| ≥ θ high \|S\|. Then in each iteration of the for-loop, it holds that µA < clow with probability at least θ high . Conditioning on µA < clow , by Fact 5.1 we have µ̂A ≤ µA + ε/2 < clow + ε/2 = (clow + chigh )/2 with probability at least 1 − ∆/6. Thus, the expected increment of counter cnt is lower bounded by θ high (1 − ∆/6) ≥ θ high − ∆/6. Thus, cnt/m is the mean of m i.i.d. Bernoulli random variables with means greater than or equal to θ high − ∆/6. By the Chernoff bound, it holds with probability 1 − δ/2 that cnt/m ≥ θ high − ∆/6 − ∆/6 > (θ low + θ high )/2. An analogous argument proves cnt/m < (θ low + θ high )/2 with probability 1 − δ/2, given \|{A ∈ S : µA < chigh }\| ≤ θ low \|S\|. This completes the proof. Elimination. We implement procedure Elimination by repeatedly calling Frac-Test to determine whether a large fraction of the remaining arms have means smaller than the thresholds. If so, we uniformly sample the arms, and eliminate those with low empirical means. Algorithm 3: Elimination(S, dlow , dhigh , δ) Input: An arm set S, thresholds dlow , dhigh , and confidence level δ. Output: Arm set after the elimination. S1 ← S; dmid ← (dlow + dhigh )/2; for r = 1 to +∞ do δr ← δ/(10 · 2r ); if Frac-Test(Sr , dlow , dmid , 0.05, 0.1, δr ) then high − dmid )/2, δ ); µ̂ ← Unif-Sampl(S r , (d r Sr+1 ← A ∈ Sr : µ̂A > (dmid + dhigh )/2 ; else return Sr ; end We prove Fact 5.4 in the following. Fact 5.4 (restated) Elimination(S, dlow , dhigh , δ) takes O(\|S\|ε−2 ln δ−1 ) samples in expectation, where ε = dhigh − dlow . Let S ′ be the set returned by Elimination(S, dlow , dhigh , δ). Then we have Pr[\|{A ∈ S ′ : µA < dlow }\| ≤ 0.1\|S ′ \|] ≥ 1 − δ/2. Moreover, for each arm A ∈ S with µA ≥ dhigh , we have Pr[A ∈ S ′ ] ≥ 1 − δ/2. 20 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Proof Let ε = dhigh − dlow . To bound the number of samples taken by Elimination, we note that the number of samples taken in the r-th iteration is dominated by that taken by Unif-Sampl, O(\|Sr \|ε−2 ln δr−1 ). It suffices to show that \|Sr \| decays exponentially (in expectation); a direct summation over all r proves the sample complexity bound. We fix a particular round r. Suppose Frac-Test returns correctly (which happens with probability at least 1 − δr ) and the algorithm does not terminate at round r. Then by Fact 5.3, it holds that \|{A ∈ Sr : µA < dmid }\| > 0.05\|Sr \|. For each A ∈ Sr with µA < dmid , it holds with probability 1 − δr that µ̂A < µA + (dhigh − dmid )/2 < dmid + (dhigh − dmid )/2 = (dmid + dhigh )/2. Note that δr = δ/(10 · 2r ) ≤ 0.1. Thus, at most a 0.1 fraction of arms in {A ∈ Sr : µA < dmid } would remain in Sr+1 in expectation. It follows that conditioning on the correctness of Frac-Test at round r, the expectation of \|Sr+1 \| is upper bounded by 0.05\|Sr \| · δr + 0.95\|Sr \| ≤ 0.05\|Sr \|/10 + 0.95\|Sr \| = 0.955\|Sr \|. Moreover, even if Frac-Test returns incorrectly, which happens with probability at most 0.1, we still have \|Sr+1 \| ≤ \|Sr \|. Therefore, E[\|Sr+1 \|] ≤ 0.9 · 0.955E[\|Sr \|] + 0.1E[\|Sr \|] < 0.96E[\|Sr \|]. A simple induction yields E[\|Sr \|] ≤ 0.96r−1 \|S\|. Then the sample complexity of Elimination is upper bounded by ! ∞ ∞ X X r−1 −1 −2 −1 −2 0.96 (ln δ + r) E[\|Sr \|]ε ln δr = O \|S\|ε r=1 r=1 = O \|S\|ε−2 ln δ−1 . Then we proceed to the proof of the second claim. Let E denote the event that all calls to procedure Frac-Test returns correctly. By Fact 5.3 and a union bound, ∞ X Pr [EA ] ≥ 1 − δr ≥ 1 − δ/2. r=1 Conditioning on event E, if the algorithm terminates and returns Sr at round r, Fact 5.3 implies that \|{A ∈ Sr : µA < dlow }\| < 0.1\|Sr \|. This proves the second claim. Finally, fix an arm A ∈ S with µA > dhigh . Define EA as the event that every call to Frac-Test returns correctly in the algorithm, and \|µ̂A − µA \| < (dhigh − dmid )/2 in every round. By Facts 5.1 and 5.3, ∞ X Pr [EA ] ≥ 1 − 2δr ≥ 1 − δ/2. r=1 Then in each round r, it holds conditioning on EA that µ̂A ≥ µA − (dhigh − dmid )/2 > dhigh − (dhigh − dmid )/2 = (dmid + dhigh )/2. Thus, with probability 1 − δ/2, A is never removed from Sr . 21 C HEN L I Q IAO A.2. Overview As shown in Section 5, we can solve Best-1-Arm using −1 ln ln ∆ O H · (Ent + ln δ−1 ) + ∆−2 [2] [2] P samples, if we know in advance the complexity of the instance, i.e., H = ni=2 ∆−2 [i] . The value of H is essential for allocating appropriate confidence levels to different calls of Elimination and achieving the near-optimal sample complexity. When H is unknown, our strategy is to guess its value. The major difficulty with our approach is that when our guess, Ĥ, is much smaller than the actual complexity H, the total confidence that we allocate will exceed the total confidence δ. To prevent this from happening, we maintain the total confidence that we have allocated so far, and terminate the algorithm as soon as the sum exceeds δ.9 After that, we try a guess that is a hundred times larger. As we will see later, the most challenging part of the analysis is to ensure that our algorithm does not return an incorrect answer when Ĥ is too small. We also keep track of the number of samples that have been taken so far. Roughly speaking, when the number exceeds 100Ĥ , we also terminate the algorithm and try the next guess of Ĥ. This simplifies the analysis by ensuring that the number of samples we take for each guess grows exponentially, and thus it suffices to bound the number of samples taken on the last guess. A.3. Algorithm Algorithm Entropy-Elimination takes an instance of Best-1-Arm, a confidence δ and a guess of complexity Ĥt = 100t . It either returns an optimal arm (i.e., “accept” Ĥt ) or reports an error indicating that the given Ĥt is much smaller than the actual complexity (i.e., “reject” Ĥt ). Throughout the algorithm, we maintain Sr , Hr and Tr for each round r. Sr denotes the collection of arms that are still under consideration at the beginning of round r. We say that an arm is removed (or eliminated) at round r, if it is in Sr \ Sr+1 . Roughly speaking, Hr is an estimate of the total complexity of arms in group G1 , G2 , . . . , Gr . When this quantity exceeds our guess Ĥt , Entropy-Elimination directly rejects (i.e., returns an error). Tr is an upper bound on the number of samples taken by Med-Elim and Elimination10 before round r. As mentioned before, we also terminate the algorithm when Tr exceeds 100Ĥt . Intuitively, this prevents Entropy-Elimination from taking too many samples on small guesses of H, which gives rise to an inferior sample complexity. In each round of Entropy-Elimination, we first call Med-Elim to obtain a near-optimal arm ar . Then we use Unif-Sampl to estimate the mean of ar , denoted by µ̂ar . After that, we call Frac-Test with appropriate parameters to find out whether a considerable fraction of arms in Sr have gaps larger than εr . If so, we call procedure Elimination and update the value of Hr+1 accordingly. Note that we set the thresholds {θr } of Frac-Test such that the intervals [θr−1 , θr ] are disjoint. In particular, this property is essential for proving Lemma B.6 in the analysis of the correctness of the algorithm. Our algorithm for Best-1-Arm guesses the complexity of the instance and invokes EntropyElimination to check whether the guess is reasonable. If Entropy-Elimination reports an error, 9. For ease of analysis, we actually use δ 2 instead of δ in the algorithm. 10. As we will see later, the analysis of the sample complexity of Med-Elim and Elimination are different from the other two procedures. 22 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Algorithm 4: Entropy-Elimination(I, δ, Ĥt ) Input: Instance I, confidence δ and a guess of complexity Ĥt = 100t . Output: The best arm, or an error indicating the guess is wrong. S1 ← I; H1 ← 0; T1 ← 0; θ0 ← 0.3; c ← log4 100; for r = 1 to ∞ do if \|Sr \| = 1 then return the only arm in Sr ; εr ← 2−r ; δr ← δ/(50r 2 t2 ); 2 δr′ ← (4\|Sr \|ε−2 r /Ĥ)δ ; −1 −2 δ/Ĥ ln \|S \|ε ; Tr+1 ← Tr + \|Sr \|ε−2 r r t r if (Hr + 4\|Sr \|ε−2 r ≥ Ĥt ) or (Tr+1 ≥ 100Ĥt ) then return error; ar ← Med-Elim(Sr , 0.125εr , 0.01); µ̂ar ← Unif-Sampl({ar }, 0.125εr , δr ); θr ← θr−1 + (ct − r)−2 /10; if Frac-Test(Sr , µ̂ar − 1.75εr , µ̂ar − 1.125εr , δr , θr−1 , θr ) then Hr+1 ← Hr + 4\|Sr \|ε−2 r ; Sr+1 ← Elimination(Sr , µ̂ar − 0.75εr , µ̂ar − 0.625εr , δr′ ); else Sr+1 ← Sr ; Hr+1 ← Hr ; end 23 C HEN L I Q IAO we try a guess that is a hundred times larger. Otherwise, we return the arm chosen by EntropyElimination. Algorithm 5: Complexity-Guessing Input: Instance I and confidence δ. Output: The best arm. for t = 1 to ∞ do Ĥt ← 100t ; Call Entropy-Elimination(I, δ, Ĥt ); if Entropy-Elimination does not return an error then return the arm returned by Entropy-Elimination; end A.4. Observations We start with a few simple observations on Entropy-Elimination that will be used throughout the analysis. We first note that Entropy-Elimination lasts O(t) rounds on guess Ĥt , and our definition of θr ensures that all θr are in [0.3, 0.5]. Observation A.1 The for-loop in Entropy-Elimination(I, δ, Ĥt ) is executed at most ct times, where c = log4 100. Proof When r ≥ ct − 1, ct−1 Hr + 4\|Sr \|ε−2 = Ĥt . r ≥4·4 Thus Entropy-Elimination rejects at the if-statement. Observation A.2 For all t ≥ 1 and 1 ≤ r ≤ ct − 1, 0.3 ≤ θr−1 ≤ θr ≤ 0.5. Proof Clearly θr ≥ θ0 = 0.3. Moreover, r ∞ X 1 X −2 −2 (ct − k) /10 ≤ 0.3 + θr = θ0 + k ≤ 0.5. 10 k=1 k=1 The following observation bounds the value of µ̂ar at round r, conditioning on the correctness of Unif-Sampl and Med-Elim. Observation A.3 If Unif-Sampl returns correctly at round r, µ̂ar ≤ µ[1] (Sr ) + 0.125εr . Here µ[1] (Sr ) denotes the largest mean of arms in Sr . If both Unif-Sampl and Med-Elim return correctly, µ̂ar ≥ µ[1] (Sr ) − 0.25εr . 24 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Proof By definition, µar ≤ µ[1] (Sr ). When Unif-Sampl({ar }, 0.125εr , δr ) returns correctly, it holds that µ̂ar ≤ µar + 0.125εr ≤ µ[1] + 0.125εr . When both Med-Elim and Unif-Sampl are correct, µar ≥ µ[1] (Sr ) − 0.125εr , and thus µ̂ar ≥ µar − 0.125εr ≥ µ[1] (Sr ) − 0.25εr . The following two observations bound the thresholds used in Frac-Test and Elimination by applying Observation A.3. = µ̂ar − 1.75εr and chigh = µ̂ar − 1.125εr denote the Observation A.4 At round r, let clow r r two thresholds used in Frac-Test. If Unif-Sampl returns correctly, chigh ≤ µ[1] (Sr ) − εr . If both r low Med-Elim and Unif-Sampl return correctly, cr ≥ µ[1] (Sr ) − 2εr . Proof Observation A.3 implies that when Unif-Sampl is correct, chigh ≤ µ[1] (Sr ) + 0.125εr − 1.125εr = µ[1] (Sr ) − εr r and when both Med-Elim and Unif-Sampl return correctly, clow ≥ µ[1] (Sr ) − 0.25εr − 1.75εr = µ[1] (Sr ) − 2εr . r Observation A.5 Let dlow = µ̂ar − 0.75εr and dhigh = µ̂ar − 0.625εr denote the two thresholds r r used in Elimination. If Unif-Sampl returns correctly, dhigh ≤ µ[1] (Sr ) − 0.5εr . If both Med-Elim r low and Unif-Sampl return correctly, dr ≥ µ[1] (Sr ) − εr . Proof By the same argument, we have dhigh ≤ µ[1] (Sr ) + 0.125εr − 0.625εr = µ[1] (Sr ) − 0.5εr r when Unif-Sampl returns correctly, and dlow ≥ µ[1] (Sr ) − 0.25εr − 0.75εr = µ[1] (Sr ) − εr r when both Med-Elim and Unif-Sampl are correct. 25 C HEN L I Q IAO Appendix B. Analysis of Correctness B.1. Overview We start with a high-level overview of the proof of our algorithm’s correctness. We first define a good event on which we condition in the rest of the analysis. Let E1 be the event that in a particular run of Complexity-Guessing, all calls of procedure Unif-Sampl and Frac-Test return correctly. Recall that δr , the confidence of Unif-Sampl and Frac-Test, is set to be δ/(50r 2 t2 ) in the r-th round of iteration t. By a union bound, Pr[E1 ] ≥ 1 − 2 ∞ X ∞ X δ/(50t2 r 2 ) = 1 − 2δ(π 2 /6)2 /50 ≥ 1 − δ/3. t=1 r=1 The δ-correctness of our algorithm is guaranteed by the following two lemmas. The first lemma states that Entropy-Elimination accepts a guess Ĥt and returns correctly with high probability when Ĥt is sufficiently large. The second lemma guarantees that Entropy-Elimination rejects a guess Ĥt when Ĥt is significantly smaller than H, the actual complexity. More precisely, we define the following two thresholds: tmax = ⌊log100 H⌋ − 2 and t′max = log100 H(Ent + ln δ−1 )δ−1 + 2. The precise statements of the two lemmas are shown below. Lemma B.1 With probability 1 − δ/3 conditioning on event E1 , Complexity-Guessing halts before or at iteration t′max and it never returns a sub-optimal arm between iteration tmax + 1 and t′max . Lemma B.2 With probability 1 − δ/3 conditioning on event E1 , Complexity-Guessing never returns a sub-optimal arm in the first tmax iterations. Lemma B.1 and Lemma B.2 directly imply the following theorem. Theorem B.3 Complexity-Guessing is a δ-correct algorithm for Best-1-Arm. Proof Recall that Pr[E1 ] ≥ 1 − δ/3. It follows directly from Lemma B.1 and Lemma B.2 that with probability 1 − δ, Entropy-Elimination accepts at least one of Ĥ1 , Ĥ2 , . . . , Ĥt′max . Moreover, when Entropy-Elimination accepts, it returns the optimal arm. Therefore, Complexity-Guessing is δ-correct. B.2. Useful Lemmas To analyze our algorithm, it is essential to bound the probability that a specific guess Ĥt gets rejected by Entropy-Elimination. We hope that this probability is high when Ĥt is small (compared to the true complexity H), while it is reasonably low when Ĥt is large enough. It turns out to be useful to consider the following procedure P obtained from Entropy-Elimination by removing the if-statement that checks whether Hr + 4\|Sr \|ε−2 r ≥ Ĥt and Tr+1 ≥ 100Ĥt . In other 26 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION words, the modified procedure P never rejects, regardless the value of Ĥt . Note that r, the number of rounds, may exceed ct in P, which leads to invalid values of θr . In this case, we simply assume that the thresholds used in Frac-Test are 0.3 and 0.5 respectively, and the following analysis still works. Define random variable H∞ and T∞ to be the final estimation of the complexity and the number of samples at the end of P. More precisely, if P terminates at round r ∗ , then H∞ and T∞ are defined as Hr∗ and Tr∗ , respectively. Note that there is a natural mapping from an execution of P to an execution of EntropyElimination. In particular, if both H∞ < Ĥt and T∞ < 100Ĥt hold in an execution of procedure P, then Entropy-Elimination accepts in the corresponding run. Therefore, we may upper bound the probability of rejection by establishing upper bounds of H∞ and T∞ . The following two lemmas bound the expectation of H∞ and T∞ conditioning on the event that Elimination always returns correctly. Lemma B.4 E[H∞ \|all Elimination return correctly] ≤ 256H. Lemma B.5 Suppose Ĥt ≥ H. E[T∞ \|all Elimination return correctly] ≤ 16(H(Ent + ln δ−1 + ln(Ĥt /H))). Note that it is crucial for the two lemmas above that all Elimination are correct. The following lemma gives an upper bound on the probability that some call of Elimination returns incorrectly. Lemmas B.4 through B.6 together can be used to upper bound the probability of rejecting a guess Ĥ. In the statement of Lemma B.6, we abuse the notation a little bit by assuming A1 ∈ G∞ and ∆−2 [1] = +∞. Lemma B.6 Suppose that s ∈ {2, 3, . . . , n} and r ∗ ∈ N ∪ {∞} satisfy As−1 ∈ Gr∗ . When Entropy-Elimination runs on parameter Ĥt < ∆−2 [s−1] , the probability that there exists a call of procedure Elimination that returns incorrectly before round r ∗ is upper bounded by ! n X −2 ∆[i] δ2 /Ĥt . 3000s i=s The proofs of the three lemmas above are shown below. Proof [Proof of Lemma B.4] In the following analysis, we always implicitly condition on the event that all Elimination return correctly. Define H(r, S) as the expectation of H∞ −Hr at the beginning of the r-th kround of Entropy-Elimination, when the current set of arms is Sr = S. Let rmax denote j log2 ∆−1 [2] . Define C(r, S) = ∞ X \|S ∩ Gi \| i+1 X ε−2 j + j=r i=r−1 rmax X+1 ε−2 i i=r and M (r, S) = \|S ∩ G≤r−2 \|. We prove by induction on r that H(r, S) ≤ 128C(r, S) + 16M (r, S)ε−2 r . (3) We start with the base case at round rmax + 2. Recall that clow and dlow denote the lower r r threshold of Frac-Test and Elimination in round r respectively. For all r ≥ rmax + 2, if Med-Elim 27 C HEN L I Q IAO returns correctly at round r (which happens with probability 0.99), according to Observation A.4 and Observation A.5, we have dlow ≥ clow ≥ µ[1] − 2εr ≥ µ[1] − 2−(rmax +1) ≥ µ[2] . r r Since Frac-Test returns correctly (contioning on E1 ) and \|{A ∈ Sr : µA ≤ clow r }\| ≥ \|{A ∈ Sr : µA ≤ µ[2] }\| = \|Sr \| − 1 ≥ 0.5\|Sr \| ≥ θr \|Sr \| (the last step applies Observation A.2), Frac-Test must return True and Elimination will be called. Since we assume that all calls of Elimination return correctly, we have \|Sr+1 \| − 1 = \|{A ∈ Sr+1 : µA ≤ µ[2] }\| ≤ \|{A ∈ Sr+1 : µA ≤ dlow r }\| ≤ 0.1\|Sr+1 \|, which guarantees that Sr+1 only contains the optimal arm and the algorithm will return correctly in the next round. Let r0 denote the first round after round rmax + 2 (inclusive) in which Med-Elim returns correctly. Then according to the discussion above, we have Pr[r0 = r] ≤ 0.01r−rmax −2 for all r ≥ rmax + 2. Thus it follows from a direct summation on possible values of r0 that ∞ X H(rmax + 2, S) ≤ ≤ ≤ r=rmax +2 ∞ X Pr[r0 = r] · 4\|S\|ε−2 r r−rmax −2 4\|S\|ε−2 r 0.01 r=rmax +2 8\|S\|ε−2 rmax +2 ≤ 16M (rmax + 2, S)ε−2 rmax +2 , which proves the base case. Before proving the induction step, we note the following fact: for r = 1, 2, . . . , rmax + 1, C(r, S) − C(r + 1, S) = = ∞ X i=r−1 ∞ X \|S ∩ Gi \| i+1 X ε−2 j − j=r −2 \|S ∩ Gi \|ε−2 r + εr ∞ X i=r \|S ∩ Gi \| i+1 X −2 ε−2 j + εr j=r+1 (4) i=r−1 = (\|S ∩ G≥r−1 \| + 1)ε−2 r . Suppose inequality (3) holds for r + 1. Consider the following three cases of the execution of Entropy-Elimination in round r. Let Ncur = \|S ∩ Gr−1 \| and Nbig = \|S ∩ G≥r \|. For brevity, let Nsma denote M (r, S) in the following. We have Nsma + Ncur + Nbig = \|S\| − 1. Note that Sr+1 is the set of arms that survive round r. Case 1: Med-Elim returns correctly and Frac-Test returns True. 28 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION According to the induction hypothesis, the expectation of H∞ − Hr in this case can be bounded by: H(r + 1, Sr+1 ) + 4\|S\|ε−2 r −2 ≤128C(r + 1, Sr+1 ) + 16M (r + 1, Sr+1 )ε−2 r+1 + 4\|S\|εr −2 ≤128C(r + 1, S) + 16[(Nsma + Ncur )/10] · (4ε−2 r ) + 4\|S\|εr −2 =128[C(r, S) − (Ncur + Nbig + 1)ε−2 r ] + (6.4Nsma + 6.4Ncur + 4\|S\|)εr =128C(r, S) + (10.4Nsma − 117.6Ncur − 124Nbig − 124)ε−2 r ≤128C(r, S) + 10.4Nsma ε−2 r . Here the third line follows from the fact that Sr+1 ⊆ S and C(r+1, S) is monotone in S. Moreover, the correctness of the Elimination procedure implies that M (r + 1, Sr+1 ) ≤ (Nsma + Ncur )/10. The fourth line applies identity (4). Case 2: Med-Elim returns correctly and Frac-Test returns False. Since Frac-Test is always correct (conditioning on E1 ) and it returns False, Fact 5.3, Observation A.2 and Observation A.4 together imply Nsma ≤ θr \|S\| ≤ \|S\|/2. Thus Nsma ≤ \|S\| − Nsma = Ncur + Nbig + 1. As Elimination is not called in this round, the expectation of H∞ − Hr in this case can be bounded by H(r + 1, S) ≤128C(r + 1, S) + 16M (r + 1, S)ε−2 r+1 −2 ≤128[C(r, S) − (Ncur + Nbig + 1)ε−2 r ] + 64(Nsma + Ncur )εr =128C(r, S) + (64Nsma − 64Ncur − 128Nbig − 128)ε−2 r ≤ 128C(r, S). Here the last step follows from 64Nsma −64Ncur −128Nbig −128 ≤ 64(Nsma −Ncur −Nbig −1) ≤ 0. Case 3: Med-Elim returns incorrectly. In this case, the worst scenario happens when we add 4\|S\|ε−2 r to the complexity Hr , but no arms are eliminated. Then the expectation of H∞ − Hr in this case can be bounded by H(r + 1, S) + 4\|S\|ε−2 r −2 ≤128C(r + 1, S) + 16M (r + 1, S)ε−2 r+1 + 4\|S\|εr −2 ≤128[C(r, S) − (Ncur + Nbig + 1)ε−2 r ] + [64(Nsma + Ncur ) + 4\|S\|]εr −2 =128C(r, S) + (68Nsma − 60Ncur − 124Nbig − 124)ε−2 r ≤ 128C(r, S) + 68Nsma εr . Recall that Case 3 happens with probability at most 0.01. Thus we have: H(r, S) ≤ 0.01 128C(r, S) + 68M (r, S)ε−2 + 0.99 128C(r, S) + 10.4M (r, S)ε−2 r r ≤ 128C(r, S) + 16M (r, S)ε−2 r . 29 C HEN L I Q IAO The induction is completed. Note that (3) directly implies our bound: E [H∞ \|all Elimination return correctly] =H(1, S) ≤ 128C(1, S) + 16M (1, S)   i+1 ∞ X X 4j  \|S ∩ Gi \| ·  =128 j=0 i=0 ≤256 ∞ X 4i+1 \|S ∩ Gi \| i=0 ≤256 ∞ X X ∆−2 A = 256H. i=0 A∈S∩Gi Then we prove Lemma B.5, which is restated below. Lemma B.5. (restated) Suppose Ĥt ≥ H. E[T∞ \|all Elimination return correctly] ≤ 16(H(Ent + ln δ−1 + ln(Ĥt /H))). Proof [Proof of Lemma B.5] Recall that T∞ is the sum of ! −1 −2 H \|S \|ε Ĥ r r t ln = \|Sr \|ε−2 \|Sr \|ε−2 δ + ln δ−1 + ln r r ln H \|Sr \|εr−2 Ĥt (5) for all round r. T∞ serves as an upper bound on the expected number of samples taken by Med-Elim and Elimination (up to a constant factor). Before the technical proof, we discuss the intuition of our analysis. In order to bound T∞ , we attribute each term in (5) to a specific subset of arms. For simplicity, r we assume for now that this term is just \|Sr \|ε−2 r = 4 \|Sr \|. Roughly speaking, we “charge” a cost of r −2 εr = 4 to each arm in group G≥r . We expect that \|G≥r \| is at least a constant times \|Sr \|, so that the number of samples (i.e., 4r \|Sr \|) can be covered by the total charges. Then the analysis reduces to calculating the total cost that each arm is charged. Fix an arm A ∈ Gr′ for some r ′ . As described ′ above, A is charged 4r in round r (1 ≤ r ≤ r ′ ), and thus the total charge is bounded by 4r , which is the actual complexity of A. Now we start the formal proof. Consider the execution of procedure P on Ĥt . We define a collection of random variables {Ti,j : i, j ≥ 1}, where Ti,j corresponds to the cost we charge each arm in Gj at round i. For each i, let ri denote the largest integer such that \|G≥ri \| ≥ 0.5\|Si \|. Note that such an ri always exists, as \|G≥1 \| = \|S1 \| ≥ 0.5\|Si \| and \|G≥r \| = 0 for sufficiently large r. We define Ti,j as  0, j < ri , Ti,j = −2 Ĥ H −1 t εi ln \|G \|ε−2 + ln δ + ln H , j ≥ ri . j i Note that this slightly differs from the proof idea described above: Ti,j might be positive when i > j (i.e., we may not always charge G≥i in round i). In fact, the charging argument described 30 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION in the proof idea works only if, ideally, all calls of Med-Elim are correct. Since actually some Med-Elim may return incorrectly, we have to slightly modify the charging method. Nevertheless, we will show that this difference only incurs a reasonably small cost in expectation. We first claim that X T∞ ≤ 2 \|Gj \| · Ti,j . (6) i,j In other words, the total costh we charge is indeed an upper bound on Ti∞ . Note that the contribution −1 + ln(Ĥ /H) , while its contribution to the ln(H/(\|Sr \|ε−2 of round i to T∞ is \|Si \|ε−2 t r )) + ln δ i right-hand side of (6) is X X −2 −1 )) + ln δ + ln( Ĥ /H) ln(H/(\|G \|ε 2 \|Gj \| · Ti,j = 2 \|Gj \| · ε−2 t j i i j j h i −2 −1 ln(H/(\|S \|ε )) + ln δ + ln( Ĥ /H) ≥ 2\|G≥ri \| · ε−2 r t r i h i −2 −1 ln(H/(\|S \|ε )) + ln δ + ln( Ĥ /H) . ≥ \|Si \|ε−2 r t r i Then identity (6) directly follows from a summation on i. Then we bound the expectation of each Ti,j . When i ≤ j, we have the trivial bound ! Ĥt H −1 −2 . + ln δ + ln ln E[Ti,j ] ≤ εi H \|Gj \|ε−2 i When i > j, we bound the probability that Ti,j > 0. By definition, Ti,j > 0 if and only if ri ≤ j, where ri is the largest integer that satisfies \|G≥ri \| ≥ 0.5\|Si \|. It follows that Ti,j > 0 only if \|G≥j+1 \| < 0.5\|Si \|. Observe that in order to have \|G≥j+1 \| < 0.5\|Si \|, Med-Elim must return incorrectly between round j + 1 and round i − 1. In fact, suppose towards a contradiction that Med-Elim is correct in round k ∈ [j + 1, i − 1]. Then we have \|G≥j+1 \| ≥ \|G≥k \| ≥ \|Sk+1 ∩ G≥k \| > 0.5\|Sk+1 \| ≥ 0.5\|Si \|, a contradiction. Here the third step is due to the fact that when Elimination returns correctly at round k, the fraction of arms in Sk+1 with gap greater than 2−k is less than 0.1. Note that for each specific round, the probability that Med-Elim returns incorrectly is at most 0.01. Thus, the probability that Ti,j > 0 for i > j is upper bounded by 0.01i−j−1 . Therefore, ! Ĥt H −1 i−j−1 −2 + ln δ + ln . E[Ti,j ] ≤ 0.01 εi ln H \|Gj \|εi−2 It remains to sum up the upper bounds of E[Ti,j ] to yield our bound of E[T∞ ]. X X X \|Gj \| · E[Ti,j ] = 2 \|Gj \| · E[Ti,j ] + 2 \|Gj \| · E[Ti,j ]. E[T∞ ] ≤ 2 i,j i≤j 31 i>j C HEN L I Q IAO Here the first part can be bounded by 2 X i≤j j XX Ĥt H + ln δ−1 + ln \|Gj \| · E[Ti,j ] ≤ 2 \|Gj \| · 4 ln i \|Gj \|4 H j i=1 ! X Ĥt H + ln δ−1 + ln ≤4 \|Gj \| · 4j ln j \|Gj \|4 H j ! X H Ĥ t ≤4 Hj ln + Hj ln δ−1 + Hj ln Hj /4 H j ! Ĥt . ≤ 8H Ent + ln δ−1 + ln H i ! The second part can be bounded similarly. 2 X i>j ∞ X X H Ĥt \|Gj \| · E[Ti,j ] ≤ 2 0.01 \|Gj \| · 4 ln + ln δ−1 + ln i \|Gj \|4 H j i=j+1 ! X H Ĥt ≤4 \|Gj \| · 4j ln + ln δ−1 + ln j \|Gj \|4 H j ! X Ĥ H t + Hj ln δ−1 + Hj ln ≤4 Hj ln Hj /4 H j ! Ĥt . ≤ 8H Ent + ln δ−1 + ln H i−j−1 i ! In fact, the crucial observation for both the two inequalities above is that the summation decreases exponentially as i becomes farther away from j. The lemma directly follows. Finally, we prove Lemma B.6, which is restated below. Recall that we abuse the notation a little bit by assuming A1 ∈ G∞ and ∆−2 [1] = +∞. Lemma B.6. (restated) Suppose that s ∈ {2, 3, . . . , n} and r ∗ ∈ N ∪ {∞} satisfy As−1 ∈ Gr∗ . When Entropy-Elimination runs on parameter Ĥt < ∆−2 [s−1] , the probability that there exists a call of procedure Elimination that returns incorrectly before round r ∗ is upper bounded by ! n X δ2 /Ĥt . ∆−2 3000s [i] i=s Proof [Proof of Lemma B.6] Recall that As−1 ∈ Gr∗ . Suppose As ∈ Gr′ . Suppose that we are at the beginning of round r of Entropy-Elimination and the subset of arms that have not been removed is Sr = S. Moreover, we assume that the optimal arm, A1 , is still in Sr . Let P (r, S) denote the probability that some call of procedure Elimination returns incorrectly in round r, r + 1, . . . , r ∗ − 1. 32 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION As in the proof of Lemma B.4, we bound P (r, S) by induction using the potential function method. Define ′ +2 rX i+1 r′ X X −2 ε−2 εj + (s − 1) \|S ∩ Gi \| C(r, S) = j j=r j=r i=r−1 and M (r, S) = \|S ∩ G≤r−2 \|. Then it holds that for 1 ≤ r ≤ r ′ + 1, ′ C(r, S) − C(r + 1, S) = r X −2 −2 \|S ∩ Gi \|ε−2 r + (s − 1)εr ≥ (\|S ∩ G≥r−1 \| + 1)εr . i=r−1 We prove by induction that 2 P (r, S) ≤ 128C(r, S) + 16M (r, S)ε−2 δ /Ĥ. r (7) We first prove the base case at round r ′ + 2. If r ′ + 2 ≥ r ∗ , the bound holds trivially. Otherwise, we consider the ratio α = \|Sr′ +2 ∩ {As , As+1 , . . . , An }\|/\|Sr′ +2 \|, which is the fraction of arms at round r ′ + 2 that are strictly worse than As−1 . Let r0 be the first round after r ′ + 2 (inclusive) in which Med-Elim returns correctly. If Frac-Test returns False in round r0 , according to Fact 5.3 and the correctness of Frac-Test conditioning on event E1 , we have α ≤ θr0 . Consequently, in each of the following rounds (say, round r > r0 ), Frac-Test always returns False since α ≤ θr0 ≤ θr−1 , and Elimination will never be called before round r ∗ . Note that it is crucial that the threshold interval of Frac-Test in diffrent rounds are disjoint. For the other case, suppose Frac-Test returns True and we call Elimination in round r0 . Then after that, assuming Elimination returns correctly, the fraction of arms worse than As−1 will be smaller than 0.1. It also follows that Elimination will never be called after round r0 . Therefore, Elimination is called at most once between round r ′ + 2 and r ∗ − 1, and it can only be called at round r0 . Note that ′ for r ≥ r ′ + 2, Pr[r0 = r] ≤ 0.01r−r −2 . A direct summation on all possible values of r0 yields ′ P (r + 2, S) ≤ ∗ −1 rX Pr[r0 = r] · δr′ r=r ′ +2 = ∗ −1 rX ′ 2 0.01r−r −2 · 4\|S\|ε−2 r δ /Ĥ r=r ′ +2 ∞ X 2 ≤ 4\|S\|ε−2 δ / Ĥ 0.01k 4k ′ r +2 k=0 ≤ 2 5\|S\|ε−2 r ′ +2 δ /Ĥ. ′ ′ ′ Note that C(r ′ +2, S) = (s−1)ε−2 r ′ +2 , M (r +2, S) = \|S ∩G≤r \| and \|S\| ≤ \|S ∩G≤r \|+(s−1). Thus 2 P (r ′ + 2, S) ≤ 5(\|S ∩ G≤r′ \| + s − 1)ε−2 r ′ +2 δ /Ĥ which proves the base case. 2 ≤ 128C(r ′ + 2, S) + 16M (r ′ + 2, S)ε−2 ′ r +2 δ /Ĥ, 33 C HEN L I Q IAO Then we proceed to the induction step. Again, we consider whether Med-Elim returns correctly and whether Frac-Test returns True. Let Ncur = \|S ∩ Gr−1 \| and Nbig = \|S ∩ G≥r \|. Again, we denote M (r, S) by Nsma for brevity. Note that Sr+1 is the set of arms that survive round r. Case 1: Med-Elim returns correctly and Frac-Test returns True. In this case, Elimination is called with confidence level δr′ . Then the conditional probability that some Elimination returns incorrectly in this case is bounded by P (r + 1, Sr+1 ) + δr′ 2 −2 ≤ 128C(r + 1, Sr+1 ) + 16M (r + 1, Sr+1 )ε−2 δ /Ĥ r+1 + 4\|S\|εr −2 ≤ 128C(r + 1, S) + 64(Nsma + Ncur )ε−2 δ2 /Ĥ r /10 + 4\|S\|εr 2 −2 = 128C(r, S) − 128(Ncur + Nbig + s − 1)ε−2 δ /Ĥ r + (6.4Nsma + 6.4Ncur + 4\|S\|)εr ≤[128C(r, S) + 10.4M (r, S)εr−2 ]δ2 /Ĥ. Case 2: Med-Elim returns correctly and Frac-Test returns False. Since Frac-Test returns False, according to Fact 5.3 and Observation A.4, we have Nsma ≤ \|S\| − Nsma = Ncur + Nbig + 1. Then the conditional probability in this case is bounded by 2 P (r + 1, S) ≤ [128C(r + 1, S) + 16M (r + 1, S)ε−2 r+1 ]δ /Ĥ −2 2 ≤ [128C(r, S) − 128(Ncur + Nbig + s − 1)ε−2 r + 64(Nsma + Ncur )εr ]δ /Ĥ 2 ≤ [128C(r, S) + (64Nsma − 64Ncur − 128Nbig − 128(s − 1))ε−2 r ]δ /Ĥ 2 ≤ 128C(r, S)ε−2 r δ /Ĥ. Here the last step follows from 64Nsma − 64Ncur − 128Nbig − 128(s − 1) ≤ 64(Nsma − Ncur − Nbig − 1) ≤ 0. Case 3: Med-Elim returns incorrectly. 2 In this case, the worst scenario is that we call Elimination with confidence δr′ ≤ 4\|S\|ε−2 r δ /Ĥ, yet no arms are removed. So the conditional probability in this case is bounded by 2 P (r + 1, S) + 4\|S\|ε−2 r δ /Ĥ 2 −2 ≤ 128C(r + 1, S) + 16M (r + 1, S)ε−2 δ /Ĥ r+1 + 4\|S\|εr −2 −2 2 ≤[128C(r, S) − 128(Ncur + Nbig + s − 1)ε−2 r + 64(Nsma + Ncur )εr + 4(Nsma + Ncur + Nbig + 1)εr ]δ /Ĥ 2 ≤[128C(r, S) + (68Nsma − 60Ncur − 124Nbig − 124)ε−2 r ]δ /Ĥ ≤ 128C(r, S) + 68M (r, S)ε−2 δ2 /Ĥ. r Recall that Case 3 happens with probability at most 0.01. Thus we have: 2 2 P (r, S) ≤ 0.01 128C(r, S) + 68M (r, S)ε−2 δ /Ĥ + 0.99 128C(r, S) + 10.4M (r, S)ε−2 δ /Ĥ r r ≤ 128C(r, S) + 16M (r, S)ε−2 δ2 /Ĥ. r 34 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION The induction is completed. It follows from (7) that   ′ +2 rX i+1 r′ X X  δ2 /Ĥ ε−2 ε−2 \|Gi \| P (1, S) ≤ 128  j j + (s − 1) r′ ≤ 128 (4/3) " X i+1 \|Gi \|4 i=0 n X ≤ 128 (16/3) ≤ 3000s j=1 j=1 i=0 " + (4/3)(s − 1)4 r′ ∆−2 [i] + (64/3)(s − 1)4 i=s n X i=s r ′ +2 ∆−2 [i] ! # # δ2 /Ĥ δ2 /Ĥ δ2 /Ĥ. B.3. Proof of Lemma B.1 Recall that tmax = ⌊log100 H⌋ − 2 and t′max = ⌈log100 [H(Ent + ln δ−1 )δ−1 ]⌉ + 2. We restate and prove Lemma B.1 in the following. Lemma B.1. (restated) With probability 1 − δ/3 conditioning on event E1 , Complexity-Guessing halts before or at iteration t′max and it never returns a sub-optimal arm between iteration tmax + 1 and t′max . The high-level idea of the proof is to construct three other “good events” E2 , E3 and E4 . We show that each event happens with high probability conditioning on E1 . Moreover, events E1 through E4 together imply the desired event. Proof Recall that tmax = ⌊log100 H⌋ − 2 and t′max = ⌈log100 [H(Ent + ln δ−1 )δ−1 ]⌉ + 2. Let E2 denote the following event: for all t such that t ≥ tmax + 1 and Ĥt < 1003 H, Entropy-Elimination either rejects or outputs the optimal arm. Since Ĥtmax +1 = 100tmax +1 ≥ H/10000, there are at most log100 [1003 H/(H/10000)] + 1 = 6 different values of such Ĥt . For each Ĥt , the probability of returning a sub-optimal arm is bounded by the probability that the optimal arm is deleted, which is in turn upper bounded by δ2 as a corollary of Lemma B.9 proved in the following section. Thus, by a union bound, Pr[E2 \|E1 ] ≥ 1 − 6δ2 . Let E3 be the event that for all Ĥt such that t ≤ t′max and Ĥt ≥ 1003 H (or equivalently, ⌈log100 H⌉ + 3 ≤ t ≤ t′max ), Entropy-Elimination never returns an incorrect answer. In fact, in order for Entropy-Elimination to return incorrectly, some call of Elimination must be wrong. Thus we may apply Lemma B.6 to bound the probability of E3 . Specifically, we apply Lemma B.6 with 35 C HEN L I Q IAO s = 2. Then we have t′max Pr[E3 \|E1 ] ≥ 1 − ≥1− ≥1− 3000s X t=⌈log100 H⌉+3 ∞ X P n −2 i=s ∆[i] Ĥt δ2 6000H 2 δ 100t t=⌈log100 H⌉+3 ∞ X 6000 2 δ ≥ 100k k=3 1 − δ2 /100. Here the third step is due to the simple fact that 100⌈log 100 H⌉ ≥ H. Finally, let E4 denote the event that when Entropy-Elimination runs on Ĥt′max , no Elimination is wrong and the algorithm finally accepts. In order to bound the probability of the last event, we simply apply Markov inequality based on Lemma B.4 and Lemma B.5. Let E0 be the event that no Elimination is wrong when Entropy-Elimination runs on Ĥt′max . Then we have Pr[E4 \|E1 ] ≥ Pr[E0 \|E1 ] − E[H∞ \|E0 ] Ĥt′max − E[T∞ \|E0 ] 100Ĥt′max h i 16H Ent + ln δ−1 + ln(Ĥt′max /H) 256H − + ln δ−1 )δ−2 1003 H(Ent + ln δ−1 )δ−2 256 2 16 Ent + 3 ln δ−1 + ln(1002 (Ent + ln δ−1 )) 2 2 δ δ − ≥1−δ − 1002 1003 (Ent + ln δ−1 ) ≥ 1 − 2δ2 . ≥ 1 − δ2 − 1002 H(Ent Note that conditioning on events E1 through E4 , Entropy-Elimination never outputs an incorrect answer between iteration tmax + 1 and t′max . Moreover, our algorithm terminates before or at iteration t′max . The lemma directly follows from a union bound and the observation that for all δ ∈ (0, 0.01), 6δ2 + δ2 /100 + 2δ2 ≤ δ/3. Remark B.7 The last part of the proof implies a more general fact: for fixed Ĥt , EntropyElimination accepts with probability at least 1 − δ2 − 256H Ĥt − 16H(Ent + ln δ−1 + ln(Ĥt /H)) 100Ĥt . B.4. Mis-deletion of Arms We prove Lemma B.2 in the following. Again, our analysis in this subsection conditions on event E1 , which guarantees that all calls of Frac-Test and Unif-Sampl in Entropy-Elimination are correct. The high-level idea of the proof is to show that a large proportion of arms will not be accidentally removed before they have contributed a considerable amount to the total complexity. Formally, we define the mis-deletion of arms as follows. 36 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Definition B.8 An arm A ∈ Gr is mis-deleted in a particular run of Entropy-Elimination, if A is deleted before or at round r − 1. In particular, the optimal arm is mis-deleted if it is deleted in any round. The following lemma bounds the probability that a certain collection of arms are all mis-deleted. Lemma B.9 For a fixed collection of k arms, the probability that all of them are mis-deleted is at most δ2k . Proof Let S = {A1 , A2 , . . . , Ak } be a fixed set of k arms. (Here we temporarily drop the convention that Ai denotes the arm with the i-th largest mean.) For each Ai , let Eibad denote the event that Ai is mis-deleted, and let ri denote the group that contains Ai (i.e., Ai ∈ Gri ). By definition, µAi ≥ µ[1] − εri . We start by proving the following fact: suppose Elimination is called with confidence level δr′ in round r. Then the probability that all arms in S are mis-deleted in round r simultaneously is bounded by δ′ kr . We assume that r < ri for all i = 1, 2, . . . , k. Otherwise, if r ≥ ri for some i, then Ai cannot be mis-deleted in round r, since the definition of mis-deletion requires that r < ri . To analyze the behaviour of Elimination, we recall that each run of Elimination consists of several stages. (Here we use the term “stage” for an iteration of Elimination, while the term for Entropy-Elimination is “round”.) In each stage, procedure Unif-Sampl is called at line 6 to estimate the means of the arms that have not been eliminated. Let ribad denote the stage in which Ai gets deleted. Recall that dhigh is the upper threshold used in Elimination in round r. According to Observar tion A.5, dhigh ≤ µ[1] (Sr ) − 0.5εr = µ[1] (Sr ) − 2−(r+1) ≤ µ[1] − εri ≤ µAi . r Here the third step follows from our assumption that r < ri . In order for Elimination to eliminate an arm Ai with mean greater than dhigh in stage ribad , the Unif-Sampl subroutine must return an incorrect estimation for Ai (i.e., \|µ̂Ai − µAi \| > (dhigh − dmid )/2), which happens with probability bad . Since the samples taken on different arms are independent, the events that at most δr′ / 10 · 2ri Unif-Sampl returns incorrect estimates for different arms are also independent, and it follows that Qk ′ bad r bad . the probability that each arm Ai is removed at stage ri is bounded by i=1 δr / 10 · 2 i Therefore, the probability that all the k arms in S are mis-deleted in Elimination is upper bounded by ∞ X ∞ X r1bad =1 r2bad =1 = k Y i=1 ≤ k Y ··· ∞ Y k X rkbad =1 i=1 bad δr′ /(10 · 2ri )  ∞ X bad   δr′ / 10 · 2ri  ribad =1 k δr′ = δ′ r . i=1 Then we start with the proof of the lemma. Suppose that we are at the beginning of round r. m arms among S are still in Sr , while the sum of confidence levels allocated in the previous rounds 37 C HEN L I Q IAO Pr−1 ′ is δ′ (i.e., δ′ = i=1 δi ). Let P (r, δ′ , m) denote the probability that all the m remaining arms are mis-deleted in the future. We prove by induction that P (r, δ′ , m) ≤ (δ2 − δ′ )m . (8) Recall that the number of rounds that Entropy-Elimination lasts is bounded by ct according to Observation A.1. Thus when r = ⌈ct⌉ + 1, we have P (r, δ′ , m) = 0. Observe that δ′ never exceeds δ2 according to the behaviour of Entropy-Elimination. Therefore (8) holds for the base case. Now we proceed to the induction step. If Elimination is not called in round r, by induction hypothesis we have P (r, δ′ , m) ≤ P (r + 1, δ′ , m) ≤ (δ2 − δ′ )m , which proves inequality (8). If, on the other hand, Elimination is called with confidence δr′ . We observe that by the claim we proved above, the probability that exactly j arms among the m arms m ′j are mis-deleted is at most δ r . Thus by a simple summation, j ′ P (r, δ , m) ≤ m X m j=0 j j δ′ r ·P (r +1, δ′ +δr′ , m−j) ≤ m X m j=0 j j δ′ r (δ2 −δ′ −δr′ )m−j = (δ2 −δ′ )m , which completes the induction step. Finally, the lemma directly follows from (8) by plugging in r = 1, δ′ = 0 and m = k. Remark B.10 Let Eibad denote the event that Ai is mis-deleted. Note that although the events {Eibad } are not independent, we can still obtain an exponential bound (i.e., δ2k ) on the probability that k such events happen simultaneously. We call such events quasi-independent to reflect this property. Formally, a collection of n events {Ei }ni=1 are δ-quasi-independent, if for all 1 ≤ k ≤ n and 1 ≤ a1 < a2 < · · · < ak ≤ n, we have Pr[Ea1 ∩ Ea2 ∩ · · · ∩ Eak ] ≤ δk . Then the collection of events {Eibad } are δ2 -quasi-independent. The following lemma proves a generalized Chernoff bound for quasi-independent events. Lemma B.11 Suppose v1 , v2 , . . . , vn > 0. {Yi }ni=1 is a collection of random variables, where the support of Yi is {0, vi }. Moreover, the collection of events {YP i = vi } are δ-quasi-independent. Let (S1 , S2 , . . . , Sm ) be a partition of {1, 2, . . . , n} such that j∈Si vj ≤ 1 for all i. Define P 1 Pm δ Pn Xi = j∈Si Yj . Let X = m i=1 Xi and p = m i=1 vi . Then for all q ∈ (p, 1), Pr[X ≥ q] ≤ e−mD(q\|\|p) , where D(x\|\|y) = x ln(x/y) + (1 − x) ln[(1 − x)/(1 − y)] is the relative entropy function. 38 T OWARDS I NSTANCE O PTIMAL B OUNDS Proof Let pi = δ P j∈Si vj . Then p = 1 m Pm i=1 pi . FOR B EST A RM I DENTIFICATION For t > 0, we have Pr[X ≥ q] = Pr[etmX ≥ etmq ] ≤ E[etmX ] . etmq To bound E[etmX ], we consider a collection of independent random P variables Ỹ1 , Ỹ2 , . . . , Ỹn defined by Pr[Ỹi = vi ] = δ and Pr[Ỹi = 0] = 1 − δ. Define X̃i = j∈Si Ỹj for i = 1, 2, . . . , m, 1 Pm tmX can be written as and X̃ = m i=1 X̃i . Note that each term in the Taylor expansion of e Ql α i=1 Ynl , where l ≥ 0, (n1 , n2 , . . . , nl ) ∈ {1, 2, . . . , n}l , and α = tl /(l!) > 0. The correspondQ ing term in etmX̃ is then α li=1 Ỹnl . Let U = \|{ni : i ∈ {1, 2, . . . , l}}\| denote the set of distinct numbers among n1 , n2 , . . . , nl . We have " l # # " l l l Y Y Y Y vn l = E vnl ≤ δ\|U \| Ỹnl . Ynl = Pr[Yi = vi for all i ∈ U ] · E i=1 i=1 i=1 i=1 Summing over all terms in the expansion yields m i h i Y h E etX̃i . E etmX ≤ E etmX̃ = i=1 Here the last step holds since {X̃i } are independent. Note that since X̃i ∈ [0, 1], it follows from Jensen’s inequality that i i h h E etX̃i ≤ E et X̃i + 1 − X̃i = pi et + 1 − pi . Then tmX E e Pm ≤ m Y (pi et + 1 − pi ) ≤ (pet + 1 − p)m . i=1 1 Recall that p = m i=1 pi . Here the last step follows from Jensen’s inequality and the concavity t of ln(e x + 1 − x) for t > 0. By setting t = ln q(1−p) p(1−q) , we have Pr[X ≥ q] ≤ t m pe + 1 − p E[etmX ] ≤ = e−mD(q\|\|p) . etmq etq The following lemma states that if a collection of arms with a considerable amount of total complexity are not mis-deleted, Entropy-Elimination rejects Ĥ. Lemma B.12 S is a set of sub-optimal arms with complexity H(S) > Ĥ. Let r ∗ = maxA∈S log2 ∆−1 A . If in a particular run of Entropy-Elimination, no arm in S is mis-deleted and there exists an arm A∗ outside S with µA∗ ≥ maxA∈S µA such that A∗ is not deleted in the first r ∗ − 1 rounds, then Ĥ is rejected in that run. 39 C HEN L I Q IAO Proof Suppose S = {A1 , A2 , . . . , Ak } and Ai ∈ Gri . Without loss of generality, µA1 ≤ µA2 ≤ · · · ≤ µAk . By definition of r ∗ , we have r ∗ = max1≤i≤k ri = rk . According to our assumption, both Ak and A∗ are not deleted in the first r ∗ − 1 rounds. Thus Entropy-Elimination does not accept in the first r ∗ rounds. Suppose for contradiction that Ĥ is not rejected by Entropy-Elimination in a particular run. Define R = {r ∈ [1, r ∗ − 1] : Elimination is called in round r}. Let N1 = {i ∈ [k] : ∃r ∈ R, r ≥ ri } and N2 = [k] \ N1 . For each i ∈ N1 , since Ai is not mis-deleted, Ai ∈ Sri . Define ri′ = min{r ∈ R : r ≥ ri } as the first round after ri (inclusive) in which Elimination is called. . It follows that Ai ∈ Sri′ . At round ri′ of Entropy-Elimination, Hri′ +1 is set to Hri′ + 4\|Sri′ \|ε−2 r′ i . It follows that Hr∗ is at least the total cost = ε−2 Therefore we can “charge” Ai a cost of 4ε−2 ri′ +1 ri′ P . that arms in N1 are charged, i∈N1 ε−2 ri′ +1 For each i ∈ N2 , we have Ai ∈ Sri and Sri = Sr∗ . Thus it holds that \|Sr∗ \| ≥ \|N2 \|. When the if-statement in Entropy-Elimination is checked in round r ∗ , we have Hr ∗ + 4\|Sr∗ \|ε−2 r∗ ≥ X i∈N1 ε−2 ri′ +1 + N2 ε−2 r ∗ +1 ≥ k X i=1 ε−2 ri +1 ≥ k X ∆−2 Ai = H(S) > Ĥ. i=1 Here the second step follows from ri′ ≥ ri and r ∗ ≥ ri , while the third step follows from ∆Ai ≥ 2−(ri +1) . Therefore Entropy-Elimination rejects in round r ∗ , a contradiction. B.5. Proof of Lemma B.2 Lemma B.2 is restated below. Recall that tmax = ⌊log100 H⌋ − 2. Lemma B.2. (restated) With probability 1 − δ/3 conditioning on event E1 , Complexity-Guessing never returns a sub-optimal arm in the first tmax iterations. The high-level idea of the proof is simple. For each Ĥt , we identify a collection of near-optimal “crucial arms”. By Lemma B.9, the probability that all “crucial arms” are mis-deleted is small, thus we may assume that at least one crucial arm survives. This crucial arm serves as A∗ in Lemma B.12. Then according to Lemma B.12, in order for Entropy-Elimination to accept Ĥt , it must mis-delete a collection of “non-crucial” arms with a significant fraction of complexity. The probability of this event can also be bounded by using the generalized Chernoff bound proved in Lemma B.11. The major technical difficulty is the choice of “crucial arms”. We deal the following three cases separately: (1) Ĥt is greater than ∆−2 [2] , the complexity of the arm with the second largest mean; (2) −2 −2 Ĥt is between ∆−2 [s] and ∆[s−1] for some 3 ≤ s ≤ n; and (3) Ĥt is smaller than ∆[n] . We bound the probability that the lemma is violated in each case, and sum them up using a union bound. Proof [Proof of Lemma B.2] Case 1: ∆−2 [2] ≤ Ĥt ≤ Ĥtmax . We first deal with the case that Ĥt is relatively large. We partition the sequence of sub-optimal arms A2 , A3 , . . . , An into contiguous B1 , B2 , . . . , Bm such that the total complexity in each P blocks −2 −2 −2 block Bi , denoted by H(Bi ) = A∈Bi ∆A , is between ∆[2] and 3∆[2] . To construct such a partition, we append arms to the current block one by one from A2 to An . When the complexity 40 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION of the current block exceeds ∆−2 [2] , we start with another block. Clearly, the complexity of each . Note that the last block may have a complexity less than resulting block is upper bounded by 2∆−2 [2] . In that case, we simply merge it into the second last block. As a result, the total complexity ∆−2 [2] i i h h −2 −2 −2 , 3m∆ . It follows that H ∈ m∆ , 3∆ of every block is in ∆−2 [2] . [2] [2] [2] ≤ Ĥt < For brevity, let B≤i denote B1 ∪B2 ∪· · ·∪Bi and B<i = B≤i−1 . Since H(B1 ) = ∆−2 [2] H = H(B≤m ), therehexists a unique integer i k ∈ [2, m] that satisfies H(B<k ) ≤ Ĥt < H(B≤k ). −2 −2 Then we have Ĥt ∈ (k − 1)∆[2] , 3k∆[2] . Since B≤k contains at least k arms, it follows from Lemma B.9 that with probability 1 − δ2k , at least one arm in B≤k is not mis-deleted. Recall that by Lemma B.12, Entropy-Elimination accepts Ĥt only if either of the following two events happens: (a) no arm in B≤k survives, which happens with probability δ2k ; (b) a collection of arms among B>k with total complexity of at least H(B>k ) − Ĥ are mis-deleted. −2 For i = 2, 3, . . . , n, define vi = ∆−2 [i] /(3∆[2] ) and Yi = vi · I[Ai is mis-deleted]. For i = 1, 2, . . . , m, Xi is defined as Xi = X Aj ∈Bi Yj = X ∆−2 A −2 · I[A is mis-deleted]. 3∆ [2] A∈B i In other words, Xi is the total complexity of the arms in block Bi that are mis-deleted, divided by a −2 constant 3∆−2 [2] . Recall that H(Bi ) ≤ 3∆[2] , so Xi is between 0 and 1. Let X= n m X 1 1 X ∆−2 Xi = −2 [i] · I[Ai is mis-deleted] m 3m∆ [2] i=2 i=1 denote the mean of these random variables. Since the events of mis-deletion of arms are δ2 -quasiindependent, we may apply Lemma B.11. Note that −2 n n Hδ2 δ2 X δ2 X ∆[i] 2 = p= vi = −2 −2 ≤ δ . m m 3∆ 3m∆ [2] [2] i=2 i=2 Here the last step applies H ≤ 3m∆−2 [2] . On the other hand, conditioning on event (b) (i.e., a collection of arms with total complexity H(B>k ) − Ĥ are mis-deleted), we have X= n X 1 ∆−2 [i] · I[Ai is mis-deleted] 3m∆−2 [2] i=2 −2 (m − k)∆−2 H(B>k ) − Ĥ [2] − 3k∆[2] ≥ ≥ 3m∆−2 3m∆−2 [2] [2] ≥ m − 4k m − 12m/10000 1 ≥ ≥ . 3m 3m 6 −2 Here the third step follows from H(B>k ) ≥ (m − k)∆−2 [2] and Ĥ ≤ 3k∆[2] . The last line holds since −2 k∆−2 [2] ≤ Ĥ ≤ Ĥtmax ≤ H/10000 ≤ 3m∆[2] /10000, 41 C HEN L I Q IAO which implies k ≤ 3m/10000. According to Lemma B.11, we have Pr[X ≥ 1/6] ≤ exp −mD 1/6\|\|δ2 1 5 m 5m ln = exp − ln 2 − 6 6δ 6 6(1 − δ2 ) ≤ (6δ2 )m/6 · (6/5)5m/6 ≤ δm/6 . Recall that D(x\|\|y) stands for the relative entropy function. The last step follows from 6δ ·(6/5)5 ≤ 1. Therefore, i h (9) Pr Entropy-Elimination accepts Ĥt ≤ δ2k + δm/6 . i h It remains to apply a union bound to (9) for all values of Ĥt in ∆−2 , Ĥ t max . Recall that k ≥ 2, [2] and the ratio between different guesses Ĥt is at least 100. It follows that the values of k are distinct for different values of Ĥt , and thus the sum of the first term, δ2k , can be bounded by ∞ X k=2 δ2k = δ4 ≤ 2δ4 . 1 − δ2 For the second term, we note that the number of guesses Ĥt between ∆−2 [2] and Ĥtmax is at most m l H −2 − 1 ≤ log100 (3m) − 1. tmax − log100 ∆−2 [2] + 1 ≤ log100 H − 2 − log 100 ∆[2] + 1 = log100 ∆−2 [2] In particular, if m < 1002 /3, no Ĥt will fall into [∆−2 [2] , tmax ]. Thus we focus on the nontrivial case m ≥ 1002 /3. Then the sum of the second term δm/6 can be bounded by 2 /18 δm/6 · (log100 (3m) − 1) ≤ δ100 , since δm/6 · (log100 (3m) − 1) decreases on [1002 /3, +∞) for δ ∈ (0, 0.01). Finally, we have i h 2 , H ] ≤ 2δ4 + δ100 /18 ≤ 3δ4 . Pr Entropy-Elimination accepts Ĥt for some Ĥt ∈ [∆−2 t max [2] −2 Case 2: ∆−2 [s] ≤ Ĥ < ∆[s−1] for some 3 ≤ s ≤ n. In this case, Ĥ is between the complexity of As−1 and As . Our goal is to prove an upper bound of δΩ(s) on the probability of returning a sub-optimal arm for each specific s. Summing Pover all s yields a bound on the total probability. Our analysis depends on the ratio between Ĥ and ni=s ∆−2 [i] , the complexity of arms that are worse than As . Intuitively, when Ĥ is greater than the sum (Case 2-1), the contribution of the arms worse than As to the complexity is negligible. Thus we have to rely on the fact that the s − 1 arms with the largest means will not be mis-deleted simultaneously with high probability. On the other hand, when Ĥ is significantly smaller than the sum (Case 2-2), we may apply the same analysis as in Case 1. Finally, if the value of Ĥ is between the two cases (Case 2-3), it suffices to prove a relatively loose bound, since the number of possible values is small. 42 T OWARDS I NSTANCE O PTIMAL B OUNDS Case 2-1: Ĥ > 300000s FOR B EST A RM I DENTIFICATION Pn −2 i=s ∆[i] . In this case, our guess Ĥ is significantly larger than the total complexity of As , As+1 , . . . , An , yet Ĥ is smaller than the complexity of any one among the remaining arms. Thus intuitively, in order to reject Ĥ, Entropy-Elimination should not mis-delete all the first s − 1 arms. More formally, we have the following fact: in order for Entropy-Elimination to return a sub-optimal arm, it must delete A1 along with at least s − 3 arms among A2 , A3 , . . . , As−1 before round r ∗ , where r ∗ r ∗ +1 ≥ ∆−2 is the group that contains As−1 . In fact, since 4ε−2 r∗ = 4 [s−1] ≥ Ĥt , Entropy-Elimination ∗ terminates before or at round r . If A1 is not deleted before round r ∗ , Entropy-Elimination can only return A1 as the optimal arm, which is correct. If less than s−3 arms among A2 , A3 , . . . , As−1 are deleted before round r ∗ , for example Ai and Aj are not deleted (2 ≤ i < j ≤ s − 1), then both of them are contained in Sr∗ . It follows that Entropy-Elimination does not return before round r ∗ . We first bound the probability that A1 is deleted before round r ∗ . In order for this to happen, some Elimination must return incorrectly. By Lemma B.6, the probability of this event is upper bounded by ! n X δ2 /Ĥt . ∆−2 3000s [i] i=s In fact, we have a more general fact: the probability that a fixed set of k arms among {A2 , A3 , . . . , As−1 } together with A1 are deleted before round r ∗ is bounded by ! ! n n X X δ2(k+1) /Ĥt . ∆−2 δ2 /Ĥt · δ2k = 3000s ∆−2 3000s [i] [i] i=s i=s The proof follows from combining the two inductions in the proof of Lemma B.6 and Lemma B.9, and we omit it here. Since {A2 , A3 , . . . , As−1 } contains s − 2 subsets of size s − 3, the probability that Entropy-Elimination returns an incorrect answer on a particular guess Ĥt is at most ! n X −2 ∆[i] δ2(s−2) /Ĥt . (s − 2) · 3000s i=s It remains to apply a union bound on all Ĥt that fall into this case. Recall that Ĥt > 300000s Pn −2 i=s ∆[i] and Ĥt grows exponentially in t at a rate of 100. Thus the total probability is upper bounded by P n −2 ∞ ∞ 2(s−2) δ2(s−2) (s − 2) X 3000s ∆ X i=s [i] 1 2(s−2) δ (s − 2) δ (s − 2). = = P n −2 k+1 100 99 100k · 300000s i=s ∆[i] k=0 k=0 P Case 2-2: Ĥ < ni=s ∆−2 [i] /(78s). In this case, we apply the technique in the proof of Case 1. We partition the sequence As , As+1 , . . . , An i h −2 −2 into m consecutive blocks B1 , B2 , . . . , Bm such that H(Bi ) ∈ ∆[s] , 3∆[s] . Let B≤i denote Pn −2 B1 ∪ B2 ∪ · · · ∪ Bi . Since H(B1 ) = ∆−2 i=s ∆[i] /(78s) < H(B≤m ), there [s] ≤ Ĥ < exists a unique integer i k ∈ [2, m] such that H(B<k ) ≤ Ĥ < H(B≤k ). It follows that Ĥ ∈ h −2 −2 (k − 1)∆[s] , 3k∆[s] . By Lemma B.12, in order for Entropy-Elimination to accept Ĥ, one of the following two events happens: (a) Entropy-Elimination mis-deletes all arms in B≤k ∪ {A1 , A2 , . . . , As−1 }; (b) the total 43 C HEN L I Q IAO complexity of mis-deleted arms among B>k is greater than H(B>k ) − Ĥ. Since B≤k contains at least k arms, by Lemma B.9, the probability of event (a) is bounded by δ2(s+k−1) . Again, we bound the probability of event (b) using the generalized Chernoff bound in Lemma B.11. −2 For each i = s, s + 1, . . . , n, define vi = ∆−2 [i] /(3∆[s] ) and Yi = vi · I[Ai is mis-deleted]. Define random variables {Xi : i ∈ {1, 2, . . . , m}} as Xi = X Yj = Aj ∈Bi 1 X −2 ∆A · I[A is mis-deleted]. 3∆−2 [s] A∈B i Since H(Bi ) ≤ 3∆−2 [s] , Xi is between 0 and 1. Let X= n m X 1 X 1 ∆−2 Xi = [i] · I[Ai is mis-deleted] −2 m 3m∆ [s] i=s i=1 denote the mean of these random variables. Since the events {Yi = vi } are δ2 -quasi-independent, we may apply Lemma B.11. We have n p= H(B≤m )δ2 δ2 X vi = ≤ δ2 . −2 m 3m∆ [s] i=s Here the last step applies H(B≤m ) ≤ 3m∆−2 [s] . On the other hand, conditioning on event (b) (i.e., a collection of arms in B>k with total complexity H(B>k ) − Ĥ are mis-deleted), we have n X 1 · I[Ai is mis-deleted] ∆−2 X= −2 3m∆[s] i=s [i] −2 (m − k)∆−2 H(B>k ) − Ĥ [s] − 3k∆[s] ≥ ≥ 3m∆−2 3m∆−2 [s] [s] ≥ m − 4m/(26s) 1 m − 4k ≥ ≥ . 3m 3m 6 . The last line holds and Ĥ ≤ 3k∆−2 Here the third step follows from H(B>k ) ≥ (m − k)∆−2 [s] [s] since −2 k∆−2 [s] ≤ Ĥ ≤ H(B≤m )/(78s) ≤ m∆[s] /(26s), which implies k ≤ m/(26s). By Lemma B.11, we have Pr[X ≥ 1/6] ≤ δm/6 , and thus the probability that Entropy-Elimination return an incorrect answer on Ĥt is bounded by δ2(s+k−1) + δm/6 . It remains to apply a union bound on all valus of Ĥt that fall into this case. Since k ≥ 2 and the values of k are distinct, the sum of the first term is bounded by ∞ X k=2 δ2(s+k−1) = δ2s+2 ≤ 2δ2s+2 . 1 − δ2 44 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION For the second term, note that the number of different values of Ĥt between ∆−2 [s] and H(B≤m )/(78s) is bounded by i h log100 H(B≤m )/(78s)/∆−2 [s] + 1 ≤ log 100 [m/(26s)] + 1. Pn −2 i=s ∆[i] /(78s) In particular, if m < 26s, no Ĥt will fall into this case. So in the following we focus on the nontrivial case that m ≥ 26s. Since The sum of the second term is at most δm/6 (log100 [m/(26s)] + 1) ≤ δ13s/3 ≤ δ2s+2 . Here the first step follows from the fact that δm/6 (log100 [m/(26s)] + 1) decreases on [26s, +∞) for all δ ∈ (0, 0.01) and s ≥ 3. The second step follows from s ≤ 3. Therefore, the total probability that Entropy-Elimination returns incorrectly in this sub-case is bounded by 2δ2s+2 + δ2s+2 ≤ 3δ2s+2 . Pn P −2 Case 2-3: Ĥ ∈ [ ni=s ∆−2 i=s ∆[i] ]. [i] /(78s), 300000s In this case, we simply bound the probability of returning an incorrect answer by the probability that at least s − 2 arms in {A1 , A2 , . . . , As−1 } are mis-deleted, which is in turn bounded by (s − 1)δ2(s−2) according to Lemma B.9. As in the argument of Case 2-1, suppose that two arms Ai and Aj (1 ≤ i < j ≤ s − 1) are not mis-deleted. Let r ∗ be the group that contain As−1 . Then both Ai r ∗ +1 ≥ ∆−2 and Aj are contained in Sr∗ . However, as 4ε−2 r∗ = 4 [s−1] ≥ Ĥt , Entropy-Elimination will ∗ reject in round r , which implies that Entropy-Elimination will never return a sub-optimal arm. Note that at most log100 300000s + 1 ≤ 2 log100 s + 5 = log10 s + 5 1/(78s) different values of Ĥ fall into this case. Therefore, the total probability is bounded by δ2(s−2) (log10 s + 5)(s − 1). Combining Case 2-1 through Case 2-3 yields the following bound: the probability that Entropy−2 Elimination outputs an incorrect answer for some 3 ≤ s ≤ n and Ĥ ∈ [∆−2 [s] , ∆[s−1] ) is at most n X 1 2(s−2) 2s+2 2(s−2) δ (s − 2) + 3δ +δ (log10 s + 5)(s − 1) 99 s=3 n X 6 2(s−2) s − 2 + 3δ + (log10 s + 5)(s − 1) δ = 99 ≤ s=3 ∞ X δ2(s−2) (log10 s + 6)(s − 1) s=3 ∞ X 2 ≤δ 0.012(s−3) (log10 s + 6)(s − 1) ≤ 20δ2 . s=3 Case 3: Ĥt < ∆−2 [n] . 45 = C HEN L I Q IAO Finally, we turn to the case that Ĥ is smaller than ∆−2 [n] . In this case, Complexity-Guessing always rejects. Suppose An ∈ Gr∗ . Then in the first r ∗ − 1 rounds of Entropy-Elimination, Frac-Test always returns False. Thus no elimination is done before round r ∗ . Since Ĥt < ∆−2 [n] ≤ ∗ 4ε−2 r ∗ , Entropy-Elimination directly rejects when checking the if-statement at round r . Case 1 through Case 3 together directly imply the lemma, as 3δ4 + 20δ2 < δ/3 for all δ ∈ (0, 0.01). Appendix C. Analysis of Sample Complexity Recall that E1 is the event that all calls of Frac-Test and Unif-Sampl in Entropy-Elimination return correctly. We bound the sample complexity of our algorithm using the following two lemmas. Lemma C.1 Conditioning on E1 , the expected number of samples taken by Med-Elim and Elimination in Complexity-Guessing is O(H(Ent + ln δ−1 )). Lemma C.2 Conditioning on E1 , the expected number of samples taken by Unif-Sampl and Frac-Test in Complexity-Guessing is −1 −1 O(∆−2 [2] ln ln ∆[2] polylog(n, δ )). The two lemmas above directly imply the following theorem. Theorem C.3 Conditioning on E1 , the expected sample complexity of Complexity-Guessing is −1 −1 polylog(n, δ ) . ln ln ∆ O H(ln δ−1 + Ent) + ∆−2 [2] [2] Theorems B.3 and C.3 together imply that Complexity-Guessing is a δ-correct algorithm for Best-1-Arm, and its expected sample complexity is −1 −1 polylog(n, δ ) ln ln ∆ O H(ln δ−1 + Ent) + ∆−2 [2] [2] conditioning on an event which happens with probability at least 1 − δ. However, to prove Theorem 1.11, we need a δ-correct algorithm with the desired sample complexity in expectation (not conditioning on another event). In the following, we prove Theorem 1.11 using a parallel simulation trick developed in Chen and Li (2015). Proof [Proof of Theorem 1.11] Given an instance I of Best-1-Arm and a confidence level δ, we define a collection of algorithms {Ak : k ∈ N}, where Ak simulates Complexity-Guessing on instance I and confidence level δk = δ/2k . Then we construct the following algorithm A: • A runs in iterations. In iteration t, for each k such that 2k−1 divides t, A simulates Ak until Ak requires a sample from some arm A. A draws a sample from A, feeds it to Ak , and continue simulating Ak until it requires another sample. After that, A temporarily suspends Ak . • When some algorithm Ak terminates, A also terminates and returns the same answer. 46 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION We first note that if all algorithms in {Ak } are correct, A eventually returns the correct answer. Recall that Ak is a δ/2k -correct for Best-1-Arm. Thus by a simple union bound, A is P∞ algorithm k correct with probability 1 − k=1 δ/2 = 1 − δ, thus proving that A is δ-correct. It remains to bound the sample complexity of A. According to Theorem C.3, there exist constants C and m, along with a collection of events {Ek }, such that for each k, Pr[Ek ] ≥ 1 − δk , and the expected number of samples taken by Ak conditioning on Ek is at most i h −1 m m −1 (ln n + ln δ ) ln ln ∆ C H · (ln δk−1 + Ent) + ∆−2 k [2] [2] i h −1 m −1 m (ln n + (k ln δ ) ) ln ln ∆ ≤C H · (k ln δ + Ent) + ∆−2 [2] [2] ≤km · T (I). i h −1 m m −1 (ln n + ln δ ) , the desired samln ln ∆ Here T (I) denotes C H · (ln δ−1 + Ent) + ∆−2 [2] [2] ple complexity. The first step follows from the fact that ln δk−1 = ln δ−1 + k ≤ k ln δ−1 for δ < 0.01. Since different algorithms in {Ak } take independent samples, the events {Ek } are independent. Define random variable σ as the minimum number such that event Eσ happens. Then it follows that Pr[σ = k] ≤ Pr[E 1 ∩ E 2 ∩ · · · ∩ E k−1 ] ≤ k−1 Y δi ≤ 0.01k−1 . i=1 Let Tk denote the number of samples taken by Ak if it is allowed to run indefinitely (i.e., A does not terminate). Conditioning on σ = k, we have E[Tk ] ≤ km · T (I). Moreover, A terminates before or at iteration 2k−1 km · T (I). It follows that the number of samples taken by A is bounded by ∞ ∞ X X 2−(i−1) ≤ 2k km · T (I). ⌊2k−1 km · T (I)/2i−1 ⌋ ≤ 2k−1 km · T (I) i=1 i=1 Thus the expected sample complexity of A is bounded by ∞ X Pr[σ = k] · 2k km · T (I) k=1 ≤ ∞ X 0.01k−1 · 2k km · T (I) k=1 ≤100T (I) ∞ X 0.02k km = O(T (I)). k=1 Therefore, A is a δ-correct algorithm for Best-1-Arm with expected sample complexity of −1 −1 O(H · (ln δ−1 + Ent) + ∆−2 [2] ln ln ∆[2] polylog(n, δ )). We conclude the section with the proofs of Lemmas C.1 and C.2. 47 C HEN L I Q IAO Proof [Proof of Lemma C.1] Suppose that Complexity-Guessing terminates after iteration t0 . According to Entropy-Elimination, for each 1 ≤ t ≤ t0 , the algorithm takes O(Ĥt ) = O(100t ) samples in Med-Elim and Elimination when it runs on Ĥt . As 100t grows exponentially in t, it suffices to bound the expectation of the last term, namely 100t0 . Let t∗ = ⌈log100 H(Ent + ln δ−1 ) + 3⌉. We first show that when t ≥ t∗ , Entropy-Elimination accepts Ĥt with constant probability. According to Remark B.7, the probability that EntropyElimination rejects Ĥt is upper bounded by 256H Ĥt + H(Ent + ln δ−1 + ln(Ĥt /H)) 100Ĥt ≤ 256H Ĥt /20 ≤ 1/200. + 3 100 H 100Ĥt The first step follows from the following two observations. First, as Ĥt ≥ Ĥt∗ ≥ 1003 H(Ent + ln δ−1 ), we have H(Ent + ln δ−1 ) ≤ 100−3 Ĥt . Second, since Ĥt /H ≥ 1003 and x ≥ 100 ln x 1 holds for all x ≥ 106 , we have H ln(Ĥt /H) ≤ H · (Ĥt /H) = Ĥt /100. 100 Therefore, the probability that t0 equals t∗ + k is bounded by 200−k for all k ≥ 1. It follows from a simple summation on all possible t0 that ∞ X 100t Pr[t0 = t] E 100t0 = ≤ t=1 t∗ X t 100 · 1 + t=1 ∞ X ∗ +k 100t · 200−k k=1 t∗ = O H(Ent + ln δ−1 ) . = O 100 Proof [Proof of Lemma C.2] When Entropy-Elimination runs on guess Ĥt , Unif-Sampl takes −1 O(ε−2 r ln δr ) samples in the r-th round, while the number of samples taken by Frac-Test is −1 −2 −1 −3 ln(θr − θr−1 )−1 = O ε−2 O ε−2 . r ln δr (θr − θr−1 ) r ln δr (θr − θr−1 ) As the second term dominates the first, we focus on the complexity of Frac-Test in the following analysis. Recall that εr = 2−r , δr = δ/(50r 2 t2 ) ≥ δ2 /(r 2 t2 ) and θr − θr−1 = (ct − r)−2 /10. For each t, suppose r ranges from 1 to rmax , then the complexity at iteration t is bounded by rX max −1 −3 ε−2 r ln δr (θr − θr−1 ) r=1 rX max ≤2 4r (ln δ−1 + ln r + ln t)[(ct − r)−2 /10]−3 r=1 rX max ≤2000 4r (ln δ−1 + ln r + ln t)(ct − r)6 r=1 rmax =O(4 (ln δ−1 + ln t)(ct − rmax )6 ) 48 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION The last step follows from the observation that the last term dominates the summation, and the fact ln rmax = O(ln t) due to Observation A.1. Let random variable t0 denote the last t in the execution of Complexity-Guessing. As in the proof of Lemma C.1, we define t∗ = ⌈log100 H(Ent + ln δ−1 ) + 3⌉. We have also shown that Pr[t ≥ t0 + k] ≤ 200−k for all k ≥ 1. Thus, the expected complexity incurred after iteration t∗ can be bounded by the complexity at iteration t∗ . When t < log100 ∆−2 [2] , it follows from rmax ≤ ct − 1 that the complexity is O(4ct (ln δ−1 + ln t)) = O(100t (ln δ−1 + ln t)). Summing over t = 1, 2, . . . , log100 ∆−2 [2] yields log100 ∆−2 [2] X −1 + ln ln ∆−1 100t (ln δ−1 + ln t) = O(∆−2 [2] )). [2] (ln δ t=1 Clearly this term is bounded by the desired complexity. −2 −1 ∗ When log100 ∆−2 [2] ≤ t ≤ t , we choose rmax = log 2 ∆[2] = log4 ∆[2] . Note that in fact the algorithm may not always terminate before or at round rmax . However, since the probability that the algorithm lasts rmax + k rounds is bounded by O(100−k ), the contribution of those rounds to total complexity is also dominated. Thus we have t0 X O(4rmax (ln δ−1 + ln t)(ct − rmax )6 ) t=log100 ∆−2 [2] = t0 X 6 −1 + ln t)(ct − log4 ∆−2 O(∆−2 [2] ) ) [2] (ln δ t=log100 ∆−2 [2] −2 6 −2 −1 ∗ ∗ ) (ln δ + ln t )(ct − log ∆ · O ∆ =O t∗ − log100 ∆−2 4 [2] [2] [2] )7 (ln δ−1 + ln t∗ )(ct∗ − log4 ∆−2 =O ∆−2 [2] [2] −2 −1 7 −1 ) + ln Ent + ln δ ) (ln δ + ln ln H)(ln(H/∆ =O ∆−2 [2] [2] −1 −1 −1 7 (ln δ + ln ln n)(ln n + ln δ ) ln ln ∆ =O ∆−2 [2] [2] −1 −1 polylog(n, δ ) . ln ln ∆ =O ∆−2 [2] [2] The fourth step follows from O(ln t∗ ) = O(ln ln(H(Ent + ln δ−1 ))) = O(ln ln H + ln ln Ent + ln ln ln δ−1 ), while the last two terms are dominated by ln δ−1 + ln ln H. The fifth step follows from the simple observation that H/∆−2 [2] ≤ n and Ent = O(ln ln n). 49 C HEN L I Q IAO Appendix D. Lower Bound In this section, we prove Lemma 4.1. We restate it here for convenience. Lemma 4.1. (restated) Suppose δ ∈ (0, 0.04), m ∈ N and A is a δ-correct algorithm for SIGN-ξ. P is a probability distribution on {2−1 , 2−2 , . . . , 2−m } defined by P (2−k ) = pk . Ent(P ) denotes the Shannon entropy of distribution P . Let TA (µ) denote the expected number of samples taken by A when it runs on an arm with distribution N (µ, 1) and ξ = 0. Define αk = TA (2−k )/4k . Then, m X pk αk = Ω(Ent(P ) + ln δ−1 ). k=1 D.1. Change of Distribution We introduce a lemma that is essential for proving the lower bound for SIGN-ξ in Lemma 4.1, which is a special case of (Kaufmann et al., 2015, Lemma 1). In the following, KL stands for the Kullback-Leibler divergence, while D(x\|\|y) = x ln(x/y)+(1−x) ln[(1−x)/(1−y)] is the relative entropy function. Lemma D.1 (Change of Distribution) Let A be an algorithm for SIGN-ξ. Let A and A′ be two instances of SIGN-ξ (i.e., two arms). PrA and PrA′ (EA and EA′ ) denote the probability law (expectation) when A runs on instance A and A′ respectively. Random variable τ denotes the number of samples taken by the algorithm. For all event E in Fσ , where σ is a stopping time with respect to the filtration {Ft }, we have ′ EA [τ ]KL(A, A ) ≥ D Pr[E] Pr′ [E] . A A D.2. Proof of Lemma 4.1 We start with an overview of our proof of Lemma 4.1. For each k, we consider the number of samples taken by Algorithm A when it runs on an arm with mean 2−k . We first show that with high probability, this number is between Ω(4k ) and O(4k αk ). Then we apply Lemma D.1 to show that the same event happens with probability at least e−αk when the input is an arm with mean zero. Since the probability of an event is at most 1, we would like to bound the sum of e−αk by 1, yet the problem is that the events for different k may not be disjoint. avoid this difficulty, we carefully PTo m select P a collection of disjoint events denoted by S. We bound k=1 e−dαk (for appropriate constant d) by k∈S e−αk based on the way Pwe construct S. After that, we use the “change of distribution” argument (Lemma D.1) to bound k∈S e−αk by 1. As a result, we have the following inequality for appropriate constant M , which is reminiscent of Kraft’s inequality in coding theory. m X e−dαk ≤ M . (10) k=1 Once we obtain (10), the desired bound directly follows from a simple calculation. Proof [Proof of Lemma 4.1] Fix m ∈ N. Recall that all arms are normal distributions with a standard deviation of 1 and ξ is always equal to zero. 4k αk is the expected number of samples taken 50 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION by A on an arm A with distribution N (2−k , 1). It is well-known that to distinguish N (2−k , 1) from N (−2−k , 1) with confidence level δ, Ω(4k ln δ−1P ) samples are required in expectation. Therefore, −1 −1 we have αk = Ω(ln δ ) for P all k. It follows that m k=1 pk αk = Ω(ln δ ). m It remains to prove that k=1 pk αk = Ω(Ent(P )) for all 0.04-correct algorithms. For each µ ∈ R, let Prµ and Eµ denote the probability and expectation when A runs on an arm with mean µ (i.e., N (µ, 1)). Define random variable τA as the number of samples taken by A. Let c = 1/64. Let Ek denote the event that A outputs “µ > 0” and τA ∈ [4k c, 16 · 4k αk ]. The following lemma gives a lower bound of Pr0 [Ek ]. Lemma D.2 1 Pr[Ek ] ≥ e−αk . 0 4 Our second step is to choose a collection of disjoint events from {Ek : 1 ≤ k ≤ m}. We have the following lemma. Lemma D.3 There exists a set S ⊆ {1, 2, . . . , m} such that: • {Ek : k ∈ S} is a collection of disjoint events. P Pm −dα −αk for universal constants d and M independent of m and A. k ≤ M • k∈S e k=1 e It follows that m X e−dαk ≤ M k=1 X e−αk ≤ 4M k∈S X k∈S Pr[Ek ] = 4M . 0 Here the first two steps follow from Lemma D.3 and Lemma D.2, respectively. The last step follows from the fact that {Ek : k ∈ S} is a disjoint collection of events. Finally, for a distribution P on {2−1 , 2−2 , . . . , 2−m } defined by P (2−k ) = pk , we consider the following optimization problem with variables α1 , α2 , . . . , αm : minimize m X pk αk k=1 subject to m X e−dαk ≤ 4M k=1 P −dαk = The method of Lagrange multipliers yields that the minimum value is obtained when m k=1 e 1 4M and e−dαk is proportional to pk . It follows that αk = − ln(4M pk ) and consequently d m X k=1 m pk αk ≥ 1 1X pk (ln(4M )−1 + ln p−1 ) = (Ent(P ) − ln(4M )) . k d d k=1 Note that d and M are constants independent of m, distribution P and algorithm A. This completes the proof. 51 C HEN L I Q IAO D.3. Proofs of Lemma D.2 and Lemma D.3 Finally, we prove the two technical lemmas. Proof [Proof of Lemma D.2] Recall that our goal is to lower bound Pr0 [Ek ]. We first show that Pr2−k [Ek ] ≥ 1/2 and then prove the desired lower bound by applying change of distribution. Recall that Ek = (A outputs µ > 0) ∧ (τA ∈ [4k c, 16 · 4k αk ]). We have h i h i Pr [Ek ] ≥ Pr [A outputs µ > 0] − Pr τA > 16 · 4k αk − Pr τA < 4k c 2−k 2−k 2−k 2−k h i ≥ 1 − 0.04 − 1/16 − Pr τA < 4k c 2−k h i k ≥ 0.8 − Pr τA < 4 c . 2−k Here the second step follows fromMarkov’sinequality and the fact that E2−k [τA ] = 4k αk . It remains to show that Pr2−k τA < 4k c ≤ 0.3. Suppose towards a contradiction this does not hold. Then we consider the algorithm A′ that simulates A in the following way: if A terminates within 4k c samples, A′ outputs the same answer; otherwise A′ outputs nothing. Let PrA′ ,µ denote the probability when A′ runs on an arm of mean µ. Moreover, let Ekbad denote the event that the output is “µ > 0”. Then we have i h i h Pr [Ekbad ] = Pr Ekbad ∧ τA < 4k c ≥ Pr τA < 4k c − 0.04 > 0.26. 2−k 2−k A′ ,2−k On the other hand, when we run A′ on an arm with mean −2−k , we have i i h h Pr Ekbad ≤ Pr Ekbad ≤ 0.04. −2−k A′ ,−2−k Since A′ never takes more than 4k c samples, it follows from Lemma D.1 that 2c = 4k c · KL(N (2−k , 1), N (−2−k , 1)) ≥ EA′ ,2−k [τA′ ] · KL(N (2−k , 1), N (−2−k , 1)) bad bad Pr [Ek ] ≥D Pr [Ek ] A′ ,2−k A′ ,−2−k ≥ D(0.26\|\|0.04) ≥ 0.2, which leads to a contradiction as c = 1/64. In the following, we lower bound Pr0 [Ek ] using change of distribution. Note that D Pr [Ek ] Pr[Ek ] ≤ 4k αk · KL(N (2−k , 1), N (0, 1)) 2−k 0 ≤ 4k αk · Let θk = e−αk /4. We have D(1/2\|\|θk ) = 1 −k 2 2 = αk /2. 2 1 1 1 1 ln ≥ ln = αk /2. 2 4θk (1 − θk ) 2 4θk 52 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Since we have shown Pr2−k [Ek ] ≥ 1/2, the two inequalities above imply 1 Pr[Ek ] ≥ θk = e−αk . 0 4 Proof [Proof of Lemma D.3] We map each event Ek to an interval Ik = [log4 (4k c) + 3, log4 (16 · 4k αk ) + 3] = [k, k + log4 αk + 5]. By construction, two events Ei and Ej are disjoint if and only if their corresponding intervals, Ii and Ij , are disjoint. We construct a subset of {1, 2, . . . , m} using the following greedy algorithm: • Sort (1, 2, . . . , m) into a list (l1 , l2 , . . . , lm ) such that αl1 ≤ αl2 ≤ · · · ≤ αlm . • While the list is not empty, we add the first element x in the list into set S. Let Sx = {y : y is in the current list, and Ix ∩ Iy 6= ∅}. We remove all elements in Sx from the list. Note that the way we construct S ensures that {Ek : k ∈ S} is indeed a disjoint collection of events, which proves the first part of the lemma. Moreover, {Sk : k ∈ S} is a partition of {1, 2, . . . , m}. Thus we have m X XX e−dαj . (11) e−dαk = k=1 k∈S j∈Sk It suffices to bound j∈Sk e−dαj by M e−αk for appropriate constants d and M . Summing over all k yields the desired bound m X X e−dαk ≤ e−αk . P k=1 k∈S According to our construction of S, for all j ∈ Sk we have αj ≥ αk . For each integer l ≥ ⌊log4 αk ⌋, we consider the values of j such that log4 αj ∈ [l, l + 1). Recall that the interval corresponding to event Ek is Ik = [k, k + log4 αk + 5]. In order for Ij to intersect Ik , we must have j ∈ [k − log4 αj − 5, k + log4 αk + 5]. Since log4 αj < l + 1, j must be contained in [k − l − 6, k + log4 αk + 5], and thus there are at most (log4 αk + l + 12) such values of j. Recall that since Ik = [k, k + log4 αk + 5] is nonempty, we have αk ≥ 4−5 . In the following calculation, we assume for simplicity that αk ≥ 1 for all k, since it can be easily verified that the contribution of the terms with αk < 1 (i.e., l = −5, −4, . . . , −1) is a constant, and thus can be 53 C HEN L I Q IAO covered by a sufficiently large constant M in the end. Then we have X e−dαj ≤ j∈Sk ≤ ∞ X X l=⌊log4 αk ⌋ j∈Sk ∞ X exp(−dαj )I[log4 αj ∈ [l, l + 1)] exp(−d4l )(log4 αk + l + 12) l=⌊log4 αk ⌋ = (log4 αk + 12) ∞ X l=⌊log 4 αk ⌋ exp −d4l + ∞ X l=⌊log4 αk ⌋ l exp −d4l (12) = (log4 αk + 12) · O(exp(−dαk )) + O(exp(−dαk ) · log4 αk ) ≤ M (log4 αk + 12) exp(−dαk ) = M exp(−dαk + ln log4 αk + ln 12) ≤ M e−αk . The first step rearranges the summation based on the value of l. The second step follows from the observation that Sk contains at most log4 αk + l + 12 values of j corresponding to each l. The fourth step holds since both summations decrease double-exponentially, and thus can be bounded by their respective first terms. Then we find a sufficiently large constant M (which depends on d) to cover the hidden constant in the big-O notation. Finally, the last step holds for sufficiently large d. In fact, we first choose d according to the last step, and then find the appropriate constant M . Clearly the choice of M and d is independent of the value of m and the algorithm A. Remark D.4 Recall that all distributions are assumed to be Gaussian distributions with a fixed variance of 1. In fact, our proof of Lemma 4.1 only uses the following property: the KL-divergence between two distributions with mean µ1 and µ2 is Θ((µ1 − µ2 )2 ). Note that this property is indeed essential to the “change of distribution” argument in the proof of Lemma D.2. In general, suppose U is a set of real numbers and D = {Dµ : µ ∈ U } is a family of distributions with the following two properties: (1) the mean of distribution Dµ is µ; (2) KL(Dµ1 , Dµ2 ) ≤ C(µ1 − µ2 )2 for fixed constant C > 0. Then Lemma 4.1 also holds for distributions from D. For instance, suppose D = {B(1, µ) : µ ∈ [1/2 − ε, 1/2 + ε]}, where ε ∈ (0, 1/2) is a constant and B(1, µ) denotes the Bernoulli distribution with mean µ. Since KL(B(1, p), B(1, q)) ≤ (p − q)2 (p − q)2 ≤ , q(1 − q) 1/4 − ε2 4 . It follows that Lemma 4.1 1 − 4ε2 also holds for Bernoulli distributions with means sufficiently away from 0 and 1. distribution family D satisfies the condition above with C = Appendix E. Missing Proofs in Section 5 In this section, we present the technical details in the proofs of Lemma 5.5 and Lemma 5.6. These are essentially identical to the proofs of Lemmas B.4 and B.5, which either use a potential function or apply a charging argument. 54 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION E.1. Proof of Lemma 5.5 Proof [Proof of Lemma 5.5 (continued)] Recall that P (r, Sr ) is defined as the probability that, given the value of Sr at the beginning of round r, at least one call to Elimination returns incorrectly at round r or later rounds, while Unif-Sampl and Frac-Test always return correctly. We prove inequality (2) by induction: for any Sr that contains the optimal arm A1 , P (r, Sr ) ≤ where C(r, Sr ) := δ 128C(r, Sr ) + 16M (r, Sr )ε−2 , r Ĥ ∞ X \|Sr ∩ Gi \| ε−2 j + rmax X+1 ε−2 i , i=r j=r i=r−1 and i+1 X M (r, Sr ) := \|Sr ∩ G≤r−2 \|. Note that if \|Sr \| = 1, the algorithm directly terminates at round r, and the inequality clearly holds. Thus, we assume \|Sr \| ≥ 2 in the following. Base case. We prove the base case r = rmax + 2, where rmax = maxGr 6=∅ r. Note that C(r, S) = 0 and M (r, S) = \|S\| − 1 for r = rmax + 2 and any S ⊆ I with A1 ∈ S. Let random variable r ∗ be the smallest integer greater than or equal to r, such that Med-Elim is correct at round r ∗ . Note that for k ≥ r, Pr [r ∗ = k] ≤ 0.01k−r . We claim that conditioning on r ∗ = k, if Elimination is correct between round r and round k, the algorithm will terminate at round k + 1. Consequently, the probability that Elimination fails in some round is bounded by the probabilityP that it fails between round r and k. This allows us to upper bound the conditional probability by ki=r δi′ . Now we prove the claim. By Observation 5.8, the lower threshold used in Frac-Test at round k, denoted by clow k , is greater than or equal to µ[1] − εk . Since k ≥ r = rmax + 2, \|{A ∈ S : µA < clow k }\| ≥ \|{A ∈ S : µA < µ[1] −εk }\| = \|S∩G≤k−1 \| ≥ \|S∩G≤rmax +1 \| = \|S\|−1 ≥ 0.5\|S\|. Thus by Fact 5.3, Frac-Test is guaranteed to return True in round k, and the algorithm calls Elimination. Then, by Observation 5.9, it holds that dlow k ≥ µ[1] −0.5εk . Assuming that Elimination returns correctly at round k, the set returned by Elimination, denoted by Sk+1 , satisfies \|{A ∈ Sk+1 : µA < dlow k }\| < 0.1\|Sk+1 \|, which implies \|Sk+1 \| − 1 ≤ \|Sk+1 ∩ G≤k \| = \|{A ∈ Sk+1 : µA < µ[1] − 0.5εk }\| < 0.1\|Sk+1 \|. Thus we have \|Sk+1 \| = 1, which proves the claim. Summing over all possible k yields that the probability that Elimination returns incorrectly is upper bounded by ∞ X k=r Pr [r ∗ = k] k X δj′ ≤ j=r ∞ X 0.01k−r j=r ∞ X \|Sr \|ε−2 r δ k=r 4 ≤ · 3 ≤ k X \|Sj \|ε−2 j Ĥ 2\|Sr \|ε−2 r δ Ĥ Ĥ δ 0.01k−r · 4k−r k=r ≤ δ Ĥ 55 128C(r, Sr ) + 16M (r, Sr )ε−2 . r C HEN L I Q IAO Inductive step. Assuming that the inequality holds for r + 1 and all Sr+1 that contains A1 , we bound the probability P (r, Sr ). We first note that both C and M are monotone in the following sense: C(r, S) ≤ C(r, S ′ ) and M (r, S) ≤ M (r, S ′ ) for S ⊆ S ′ . Moreover, we have C(r, Sr ) − C(r + 1, Sr ) = ∞ X −2 −2 \|Sr ∩ Gi \|ε−2 r + εr = εr (\|Sr ∩ G≥r−1 \| + 1). (13) i=r−1 We consider the following three cases separately: • Case 1. Med-Elim is correct and Frac-Test returns True. • Case 2. Med-Elim is correct and Frac-Test returns False. • Case 3. Med-Elim is incorrect. Let P1 through P3 denote the conditional probability of the event that Elimination fails at some round while Unif-Sampl and Frac-Test are correct in Case 1 through Case 3. Upper bound P1 . Assuming that Frac-Test returns True, procedure Elimination will be called at round r. By a union bound, we have P1 ≤ P (r + 1, Sr+1 ) + δr′ , where Sr+1 is the set of arms returned by Elimination. According to the inductive hypothesis, the monotonicity of C, and identity (13), P (r + 1, Sr+1 ) ≤ ≤ = δ Ĥ δ Ĥ δ Ĥ 128C(r + 1, Sr+1 ) + 16M (r + 1, Sr+1 )ε−2 r+1 128C(r + 1, Sr ) + 64M (r + 1, Sr+1 )ε−2 r 128C(r, Sr ) + ε−2 r (64M (r + 1, Sr+1 ) − 128\|Sr ∩ G≥r−1 \| − 128) . (14) ≤ µ[1] − εr . If Elimination returns correctly at round r, we have By Observation 5.9, dlow r M (r+1, Sr+1 ) = \|{A ∈ Sr+1 : µA < µ[1] −εr }\| ≤ \|{A ∈ Sr+1 : µA < dlow r }\| < 0.1\|Sr+1 \| ≤ 0.1\|Sr \|. For brevity, let Nsma , Ncur and Nbig denote \|Sr ∩ G≤r−2 \|, \|Sr ∩ Gr−1 \| and \|Sr ∩ G≥r \|, respectively. Note that \|Sr \| = Nsma + Ncur + Nsma + 1. Then we have P1 ≤ P (r + 1, Sr+1 ) + δr′ δ ≤ 128C(r, Sr ) + ε−2 r (\|Sr \| + 64M (r + 1, Sr+1 ) − 128\|Sr ∩ G≥r−1 \| − 128) Ĥ δ 128C(r, Sr ) + ε−2 ≤ r (7.4(Nsma + Ncur + Nbig + 1) − 128(Ncur + Nbig + 1)) Ĥ δ 128C(r, Sr ) + 7.4ε−2 ≤ r Nsma . Ĥ 56 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Upper bound P2 . Since Frac-Test returns True, procedure Elimination is not called. Then P2 ≤ P (r + 1, Sr+1 ) = P (r + 1, Sr ). By inequality (14), P2 ≤ ≤ ≤ δ Ĥ δ Ĥ δ 128C(r, Sr ) + ε−2 r (64M (r + 1, Sr ) − 128\|Sr ∩ G≥r−1 \| − 128) 128C(r, Sr ) + ε−2 r (64(Nsma + Ncur ) − 128(Ncur + Nbig + 1)) δ 128C(r, Sr ) + 64ε−2 · 128C(r, Sr ). r (Nsma − Ncur − Nbig − 1) ≤ Ĥ Ĥ Here the last step holds since by Observation 5.8, clow ≥ µ[1] − 2εr , and thus Frac-Test returns r False implies that Nsma = \|Sr ∩ G≤r−2 \| = \|{A ∈ Sr : µA < εr−1 }\| < 0.5\|Sr \| = (Nsma + Ncur + Nbig + 1)/2. Upper bound P3 . By (14) and M (r + 1, Sr+1 ) ≤ M (r + 1, Sr ), we have P3 ≤ P (r + 1, Sr+1 ) + δr′ δ 128C(r, Sr ) + ε−2 ≤ r (64M (r + 1, Sr ) − 128\|Sr ∩ G≥r−1 \| − 128 + \|Sr \|) Ĥ δ = 128C(r, Sr ) + ε−2 r (64(Nsma + Ncur ) − 128(Ncur + Nbig ) − 128 + (Nsma + Ncur + Nbig + 1)) Ĥ δ 128C(r, Sr ) + 65ε−2 ≤ r Nsma . Ĥ Recall that Case 3 happens with probability at most 0.01, and Nsma = \|Sr ∩G≤r−2 \| = M (r, Sr ). Therefore, we obtain the following bound on P (r, Sr ), which finishes the proof. δ δ 128C(r, Sr ) + 65ε−2 128C(r, Sr ) + 7.4ε−2 r Nsma + 0.99 · r Nsma Ĥ Ĥ δ 128C(r, Sr ) + 16ε−2 ≤ r Nsma Ĥ δ 128C(r, Sr ) + 16M (r, Sr )ε−2 . ≤ r Ĥ P (r, Sr ) ≤ 0.01 · E.2. Proof of Lemma 5.6 Proof [Proof of Lemma 5.6 (continued)] Recall that for each round i, ri is defined as the largest integer r such that \|G≥r \| ≥ 0.5\|Si \|, and   0, j < ri , Ti,j = H −2  , j ≥ ri ln δ−1 + ln εi \|Gj \|εi−2 is the number of samples that each arm in Gj is charged at round i. 57 C HEN L I Q IAO H We first show that j \|Gj \|Ti,j is an upper bound on + ln , the num\|Si \|ε−2 i ber of samples taken by Med-Elim and Elimination at round i. Recall that \|G≥ri \| ≥ 0.5\|Si \|. By definition of Ti,j , X X H −2 −1 \|Gj \|Ti,j = \|Gj \|εi ln δ + ln \|Gj \|εi−2 j j≥ri H −2 −1 ≥ \|G≥ri \|εi ln δ + ln \|Si \|ε−2 i H 1 −2 −1 . ln δ + ln ≥ \|Si \|εi 2 \|Si \|ε−2 i P Then we prove the upper bound on i E[Ti,j ], the expected number of samples that each arm in Gj is charged. For i ≤ j + 1, we have the straightforward bound H −2 −1 E[Ti,j ] ≤ εi ln δ + ln . (15) \|Gj \|ε−2 i \|Si \|ε−2 i P ln δ−1 For i ≥ j + 2, we note that Ti,j is non-zero only if j ≥ ri , which implies that \|G≥j+1 \| < 0.5\|Si \|. We claim that this happens only if Med-Elim fails between round j + 2 and round i − 1, which happens with probability at most 0.01i−j−1 . In fact, suppose Med-Elim is correct at some round k, ≥ µ[1] − 2εk and dlow ≥ µ[1] − εk , where j + 2 ≤ k ≤ i − 1. By Observations 5.8 and 5.9, clow k k low low where c and d are the two lower thresholds used in Frac-Test and Elimination. If Frac-Test returns False, by Fact 5.3, we have \|Sk ∩ G<k−1 \| = {A ∈ Sk : µA < µ[1] − 2εk } ≤ {A ∈ Sk : µA < clow k } < 0.5\|Sk \|. Since Sk+1 = Sk in this case, it follows that \|Sk+1 ∩ G<k−1 \| < 0.5\|Sk+1 \|. If Frac-Test returns True and the algorithm calls Elimination, by Fact 5.4, \|Sk+1 ∩ G<k \| = \|{A ∈ Sk+1 : µA < µ[1] − εk }\| ≤ \|{A ∈ Sk+1 : µA < dlow k }\| < 0.1\|Sk+1 \|. In either case, we have \|Sk+1 ∩ G≥k−1 \| > 0.5\|Sk+1 \|, and thus, \|G≥j+1 \| ≥ \|G≥k−1 \| ≥ \|Sk+1 ∩ G≥k−1 \| > 0.5\|Sk+1 \| ≥ 0.5\|Si \|, which contradicts \|G≥j+1 \| < 0.5\|Si \|. Therefore, for i ≥ j + 2, we have H −2 −1 E[Ti,j ] = Pr [Ti,j > 0] · εi ln δ + ln \|Gj \|εi−2 H −2 i−j−1 −1 . ≤ 0.01 · εi ln δ + ln \|Gj \|ε−2 i By (15) and (16), a direct summation gives X i E[Ti,j ] = O ε−2 j H ln δ−1 + ln \|Gj \|ε−2 j 58 !! . (16)	8
Combinatorial Multi-Armed Bandit with General Reward Functions arXiv:1610.06603v3 [cs.LG] 1 Feb 2017 Wei Chen∗ Wei Hu† Fu Li‡ Jian Li§ Yu Liu¶ Pinyan Luk Abstract In this paper, we study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our framework enables a much larger class of reward functions such as the max() function and nonlinear utility functions. Existing techniques relying on accurate estimations of the means of random variables, such as the upper confidence bound (UCB) technique, do not work directly on these functions. We propose a new algorithm called stochastically dominant confidence bound (SDCB), which estimates the distributions of underlying random variables and their stochastically dominant confidence bounds.√We prove that SDCB can achieve O(log T ) distribution-dependent regret and Õ( T ) distribution-independent regret, where T is the time horizon. We apply our results to the K-MAX problem and expected utility maximization problems. In particular, for K-MAX, we provide the first polynomial-time √ approximation scheme (PTAS) for its offline problem, and give the first Õ( T ) bound on the (1 − ǫ)approximation regret of its online problem, for any ǫ > 0. 1 Introduction Stochastic multi-armed bandit (MAB) is a classical online learning problem typically specified as a player against m machines or arms. Each arm, when pulled, generates a random reward following an unknown distribution. The task of the player is to select one arm to pull in each round based on the historical rewards she collected, and the goal is to collect cumulative reward over multiple rounds as much as possible. In this paper, unless otherwise specified, we use MAB to refer to stochastic MAB. MAB problem demonstrates the key tradeoff between exploration and exploitation: whether the player should stick to the choice that performs the best so far, or should try some less explored alternatives that may provide better rewards. The performance measure of an MAB strategy is its cumulative regret, which is defined as the difference between the cumulative reward obtained by always playing the arm with the largest expected reward and the cumulative reward achieved by the learning strategy. MAB and its variants have been extensively studied in the literature, with classical √ results such as tight Θ(log T ) distribution-dependent and Θ( T ) distribution-independent upper and lower bounds on the regret in T rounds [19, 2, 1]. An important extension to the classical MAB problem is combinatorial multi-armed bandit (CMAB). In CMAB, the player selects not just one arm in each round, but a subset of arms or a combinatorial ∗ Microsoft Research, email: weic@microsoft.com. The authors are listed in alphabetical order. Princeton University, email: huwei@cs.princeton.edu. ‡ The University of Texas at Austin, email: fuli.theory.research@gmail.com. § Tsinghua University, email: lapordge@gmail.com. ¶ Tsinghua University, email: liuyujyyz@gmail.com. k Shanghai University of Finance and Economics, email: lu.pinyan@mail.shufe.edu.cn. † 30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain. object in general, referred to as a super arm, which collectively provides a random reward to the player. The reward depends on the outcomes from the selected arms. The player may observe partial feedbacks from the selected arms to help her in decision making. CMAB has wide applications in online advertising, online recommendation, wireless routing, dynamic channel allocations, etc., because in all these settings the action unit is a combinatorial object (e.g. a set of advertisements, a set of recommended items, a route in a wireless network, and an allocation between channels and users), and the reward depends on unknown stochastic behaviors (e.g. users’ click through behaviors, wireless transmission quality, etc.). Therefore CMAB has attracted a lot of attention in online learning research in recent years [12, 8, 22, 15, 7, 16, 18, 17, 23, 9]. Most of these studies focus on linear reward functions, for which the expected reward for playing a super arm is a linear combination of the expected outcomes from the constituent base arms. Even for studies that do generalize to non-linear reward functions, they typically still assume that the expected reward for choosing a super arm is a function of the expected outcomes from the constituent base arms in this super arm [8, 17]. However, many natural reward functions do not satisfy this property. For example, for the function max(), which takes a group of variables and outputs the maximum one among them, its expectation depends on the full distributions of the input random variables, not just their means. Function max() and its variants underly many applications. As an illustrative example, we consider the following scenario in auctions: the auctioneer is repeatedly selling an item to m bidders; in each round the auctioneer selects K bidders to bid; each of the K bidders independently draws her bid from her private valuation distribution and submits the bid; the auctioneer uses the first-price auction to determine the winner and collects the largest bid as the payment.1 The goal of the auctioneer is to gain as high cumulative payments as possible. We refer to this problem as the K-MAX bandit problem, which cannot be effectively solved in the existing CMAB framework. Beyond the K-MAX problem, many expected utility maximization (EUM) problems are studied in stochastic P optimization literature [27, 20, 21, 4]. The problem can be formulated as maximizing E[u( i∈S Xi )] among all feasible sets S, where Xi ’s are independent random variables and u(·) is a utility function. For example, Xi could be the random delay of edge ei in a routing graph, S is a routing path in the graph, and the objective is maximizing the utility obtained from any routing path, and typically the shorter the delay, the larger the utility. The utility function u(·) is typically nonlinear to model risk-averse or risk-prone behaviors of users (e.g. a concave utility function is often used to model risk-averse behaviors). The non-linear utility function makes the objective function much more complicated: in particular, it is no longer a function of the means of the underlying random variables Xi ’s. When the distributions of Xi ’s are unknown, we can turn EUM into an online learning problem where the distributions of Xi ’s need to be learned over time from online feedbacks, and we want to maximize the cumulative reward in the learning process. Again, this is not covered by the existing CMAB framework since only learning the means of Xi ’s is not enough. In this paper, we generalize the existing CMAB framework with semi-bandit feedbacks to handle general reward functions, where the expected reward for playing a super arm may depend more than just the means of the base arms, and the outcome distribution of a base arm can be arbitrary. This generalization is non-trivial, because almost all previous works on CMAB rely on estimating the expected outcomes from base arms, while in our case, we need an estimation method and an analytical tool to deal with the whole distribution, not just its mean. To this end, we turn the problem into estimating the cumulative distribution function (CDF) of each arm’s outcome distribution. We use stochastically dominant confidence bound (SDCB) to obtain a distribution that stochastically dominates the true distribution with high probability, and hence √ we also name our algorithm SDCB. We are able to show O(log T ) distribution-dependent and Õ( T ) distribution-independent regret bounds in T rounds. Furthermore, we propose a more efficient algorithm called Lazy-SDCB, which first executes a discretization step √ and then applies SDCB on the discretized problem. We show that Lazy-SDCB also achieves Õ( T ) distribution-independent regret bound. Our regret bounds are tight with respect to their dependencies on T (up to a logarithmic factor for distribution-independent bounds). To make our scheme work, we make a few reasonable assumptions, including boundedness, monotonicity and Lipschitz-continuity2 of the reward function, and independence among base arms. We apply our algorithms to the K-MAX and EUM problems, and provide efficient solutions with 1 We understand that the first-price auction is not truthful, but this example is only for illustrative purpose for the max() function. 2 The Lipschitz-continuity assumption is only made for Lazy-SDCB. See Section 4. 2 concrete regret bounds. Along the way, we also provide the first polynomial time approximation scheme (PTAS) for the offline K-MAX problem, which is formulated as maximizing E[maxi∈S Xi ] subject to a cardinality constraint \|S\| ≤ K, where Xi ’s are independent nonnegative random variables. To summarize, our contributions include: (a) generalizing the CMAB framework to allow a general reward function whose expectation may depend on the entire distributions of the input random variables; (b) proposing the SDCB algorithm to achieve efficient learning in this framework with near-optimal regret bounds, even for arbitrary outcome distributions; (c) giving the first PTAS for the offline K-MAX problem. Our general framework treats any offline stochastic optimization algorithm as an oracle, and effectively integrates it into the online learning framework. Related Work. As already mentioned, most relevant to our work are studies on CMAB frameworks, among which [12, 16, 18, 9] focus on linear reward functions while [8, 17] look into nonlinear reward functions. In particular, Chen et al. [8] look at general non-linear reward functions and Kveton et al. [17] consider specific non-linear reward functions in a conjunctive or disjunctive form, but both papers require that the expected reward of playing a super arm is determined by the expected outcomes from base arms. The only work in combinatorial bandits we are aware of that does not require the above assumption on the expected reward is [15], which is based on a general Thompson sampling framework. However, they assume that the joint distribution of base arm outcomes is from a known parametric family within known likelihood function and only the parameters are unknown. They also assume the parameter space to be finite. In contrast, our general case is non-parametric, where we allow arbitrary bounded distributions. Although in our known finite support case the distribution can be parametrized by probabilities on all supported points, our parameter space is continuous. Moreover, it is unclear how to efficiently compute posteriors in their algorithm, and their regret bounds depend on complicated problem-dependent coefficients which may be very large for many combinatorial problems. They also provide a result on the K-MAX problem, but they only consider Bernoulli outcomes from base arms, much simpler than our case where general distributions are allowed. There are extensive studies on the classical MAB problem, for which we refer to a survey by Bubeck and Cesa-Bianchi [5]. There are also some studies on adversarial combinatorial bandits, e.g. [26, 6]. Although it bears conceptual similarities with stochastic CMAB, the techniques used are different. Expected utility maximization (EUM) encompasses a large class of stochastic optimization problems and has been well studied (e.g. [27, 20, 21, 4]). To the best of our knowledge, we are the first to study the online learning version of these problems, and we provide a general solution to systematically address all these problems as long as there is an available offline (approximation) algorithm. The K-MAX problem may be traced back to [13], where Goel et al. provide a constant approximation algorithm to a generalized version in which the objective is to choose a subset S of cost at most K and maximize the expectation of a certain knapsack profit. 2 Setup and Notation Problem Formulation. We model a combinatorial multi-armed bandit (CMAB) problem as a tuple (E, F , D, R), where E = [m] = {1, 2, . . . , m} is a set of m (base) arms, F ⊆ 2E is a set of subsets of E, D is a probability distribution over [0, 1]m , and R is a reward function defined on [0, 1]m × F . The arms produce stochastic outcomes X = (X1 , X2 , . . . , Xm ) drawn from distribution D, where the i-th entry Xi is the outcome from the i-th arm. Each feasible subset of arms S ∈ F is called a super arm. Under a realization of outcomes x = (x1 , . . . , xm ), the player receives a reward R(x, S) when she chooses the super arm S to play. Without loss of generality, we assume the reward value to be nonnegative. Let K = maxS∈F \|S\| be the maximum size of any super arm. Let X (1) , X (2) , . . . be an i.i.d. sequence of random vectors drawn from D, where X (t) = (t) (t) (X1 , . . . , Xm ) is the outcome vector generated in the t-th round. In the t-th round, the player (t) chooses a super arm St ∈ F to play, and then the outcomes from all arms in St , i.e., {Xi \| i ∈ St }, are revealed to the player. According to the definition of the reward function, the reward value in the t-th round is R(X (t) , St ). The expected reward for choosing a super arm S in any round is denoted by rD (S) = EX∼D [R(X, S)]. 3 We also assume that for a fixed super arm S ∈ F , the reward R(x, S) only depends on the revealed outcomes xS = (xi )i∈S . Therefore, we can alternatively express R(x, S) as RS (xS ), where RS is a function defined on [0, 1]S .3 A learning algorithm A for the CMAB problem selects which super arm to play in each round based on the revealed outcomes in all previous rounds. Let StA be the super arm selected by A to maximize the expected cumulative reward in T rounds, which in the t-th round.4 The goal hP i is PT T (t) A is E , St ) = t=1 E rD (StA ) . Note that when the underlying distribution D is t=1 R(X known, the optimal algorithm A∗ chooses the optimal super arm S ∗ = argmaxS∈F {rD (S)} in every round. The quality of an algorithm A is measured by its regret in T rounds, which is the difference between the expected cumulative reward of the optimal algorithm A∗ and that of A: ∗ RegA D (T ) = T · rD (S ) − T X t=1 E rD (StA ) . For some CMAB problem instances, the optimal super arm S ∗ may be computationally hard to find even when the distribution D is known, but efficient approximation algorithms may exist, i.e., an α-approximate (0 < α ≤ 1) solution S ′ ∈ F which satisfies rD (S ′ ) ≥ α · maxS∈F {rD (S)} can be efficiently found given D as input. We will provide the exact formulation of our requirement on such an α-approximation computation oracle shortly. In such cases, it is not fair to compare a CMAB algorithm A with the optimal algorithm A∗ which always chooses the optimal super arm S ∗ . Instead, we define the α-approximation regret of an algorithm A as ∗ RegA D,α (T ) = T · α · rD (S ) − T X t=1 E rD (StA ) . As mentioned, almost all previous work on CMAB requires that the expected reward rD (S) of a super arm S depends only on the expectation vector µ = (µ1 , . . . , µm ) of outcomes, where µi = EX∼D [Xi ]. This is a strong restriction that cannot be satisfied by a general nonlinear function RS and a general distribution D. The main motivation of this work is to remove this restriction. Assumptions. Throughout this paper, we make several assumptions on the outcome distribution D and the reward function R. Assumption 1 (Independent outcomes from arms). The outcomes from all m arms are mutually independent, i.e., for X ∼ D, X1 , X2 , . . . , Xm are mutually independent. We write D as D = D1 × D2 × · · · × Dm , where Di is the distribution of Xi . We remark that the above independence assumption is also made for past studies on the offline EUM and K-MAX problems [27, 20, 21, 4, 13], so it is not an extra assumption for the online learning case. Assumption 2 (Bounded reward value). There exists M > 0 such that for any x ∈ [0, 1]m and any S ∈ F , we have 0 ≤ R(x, S) ≤ M . Assumption 3 (Monotone reward function). If two vectors x, x′ ∈ [0, 1]m satisfy xi ≤ x′i (∀i ∈ [m]), then for any S ∈ F , we have R(x, S) ≤ R(x′ , S). Computation Oracle for Discrete Distributions with Finite Supports. We require that there exists an α-approximation computation oracle (0 < α ≤ 1) for maximizing rD (S), when each Di (i ∈ [m]) has a finite support. In this case, Di can be fully described by a finite set of numbers (i.e., its support {vi,1 , vi,2 , . . . , vi,si } and the values of its cumulative distribution function (CDF) Fi on the supported points: Fi (vi,j ) = PrXi ∼Di [Xi ≤ vi,j ] (j ∈ [si ])). The oracle takes such a representation of D as input, and can output a super arm S ′ = Oracle(D) ∈ F such that rD (S ′ ) ≥ α · maxS∈F {rD (S)}. 3 SDCB Algorithm [0, 1]S is isomorphic to [0, 1]\|S\| ; the coordinates in [0, 1]S are indexed by elements in S. Note that StA may be random due to the random outcomes in previous rounds and the possible randomness used by A. 3 4 4 Algorithm 1 SDCB (Stochastically dominant confidence bound) 1: Throughout the algorithm, for each arm i ∈ [m], maintain: (i) a counter Ti which stores the number of times arm i has been played so far, and (ii) the empirical distribution D̂i of the observed outcomes from arm i so far, which is represented by its CDF F̂i 2: // Initialization 3: for i = 1 to m do 4: // Action in the i-th round 5: Play a super arm Si that contains arm i 6: Update Tj and F̂j for each j ∈ Si 7: end for 8: for t = m + 1, m + 2, . . . do 9: // Action in the t-th round 10: For each i ∈ [m], let Di be a distribution whose CDF Fi is Fi (x) = ( max{F̂i (x) − 1, q 3 ln t 2Ti , 0}, 0≤x<1 x=1 11: Play the super arm St ← Oracle(D), where D = D1 × D2 × · · · × Dm 12: Update Tj and F̂j for each j ∈ St 13: end for We present our algorithm stochastically dominant confidence bound (SDCB) in Algorithm 1. Throughout the algorithm, we store, in a variable Ti , the number of times the outcomes from arm i are observed so far. We also maintain the empirical distribution D̂i of the observed outcomes from arm i so far, which can be represented by its CDF F̂i : for x ∈ [0, 1], the value of F̂i (x) is just the fraction of the observed outcomes from arm i that are no larger than x. Note that F̂i is always a step function which has “jumps” at the points that are observed outcomes from arm i. Therefore it suffices to store these discrete points as well as the values of F̂i at these points in order to store the whole function F̂i . Similarly, the later computation of stochastically dominant CDF Fi (line 10) only requires computation at these points, and the input to the offline oracle only needs to provide these points and corresponding CDF values (line 11). The algorithm starts with m initialization rounds in which each arm is played at least once5 (lines 27). In the t-th round (t > m), the algorithm consists of three steps. First, it calculates for each i ∈ [m] a distribution Di whose CDF Fi is obtained by lowering the CDF F̂i (line 10). The second step is to call the α-approximation oracle with the newly constructed distribution D = D1 × · · · × Dm as input (line 11), and thus the super arm St output by the oracle satisfies rD (St ) ≥ α·maxS∈F {rD (S)}. Finally, the algorithm chooses the super arm St to play, observes the outcomes from all arms in St , and updates Tj ’s and F̂j ’s accordingly for each j ∈ St . The idea behind our algorithm is the optimism in the face of uncertainty principle, which is the key principle behind UCB-type algorithms. Our algorithm ensures that with high probability we have Fi (x) ≤ Fi (x) simultaneously for all i ∈ [m] and all x ∈ [0, 1], where Fi is the CDF of the outcome distribution Di . This means that each Di has first-order stochastic dominance over Di .6 Then from the monotonicity property of R(x, S) (Assumption 3) we know that rD (S) ≥ rD (S) holds for all S ∈ F with high probability. Therefore D provides an “optimistic” estimation on the expected reward from each super arm. √ Regret Bounds. We prove O(log T ) distribution-dependent and O( T log T ) distributionindependent upper bounds on the regret of SDCB (Algorithm 1). 5 Without loss of generality, we assume that each arm i ∈ [m] is contained in at least one super arm. We remark that while Fi (x) is a numerical lower confidence bound on Fi (x) for all x ∈ [0, 1], at the distribution level, Di serves as a “stochastically dominant (upper) confidence bound” on Di . 6 5 We call a super arm S bad if rD (S) < α · rD (S ∗ ). For each super arm S, we define ∆S = max{α · rD (S ∗ ) − rD (S), 0}. Let FB = {S ∈ F \| ∆S > 0}, which is the set of all bad super arms. Let EB ⊆ [m] be the set of arms that are contained in at least one bad super arm. For each i ∈ EB , we define ∆i,min = min{∆S \| S ∈ FB , i ∈ S}. Recall that M is an upper bound on the reward value (Assumption 2) and K = maxS∈F \|S\|. Theorem 1. A distribution-dependent upper bound on the α-approximation regret of SDCB (Algorithm 1) in T rounds is 2 X 2136 π 2 M K ln T + + 1 αM m, ∆i,min 3 i∈EB and a distribution-independent upper bound is √ 93M mKT ln T + π2 + 1 αM m. 3 The proof of Theorem 1 is given in Appendix A.1. The main idea is to reduce our analysis on general reward functions satisfying Assumptions 1-3 to the one in [18] that deals with the summation reward P function R(x, S) = i∈S xi . Our analysis relies on the Dvoretzky-Kiefer-Wolfowitz inequality [10, 24], which gives a uniform concentration bound on the empirical CDF of a distribution. Applying Our Algorithm to the Previous CMAB Framework. Although our focus is on general reward functions, we note that when SDCB is applied to the previous CMAB framework where the expected reward depends only on the means of the random variables, it can achieve the same regret bounds as the previous combinatorial upper confidence bound (CUCB) algorithm in [8, 18]. Let µi = EX∼D [Xi ] be arm i’s mean outcome. In each round CUCB calculates (for each arm i) an upper confidence bound µ̄i on µi , with the essential property that µi ≤ µ̄i ≤ µi +Λi holds with high probability, for some Λi > 0. In SDCB, we use Di as a stochastically dominant confidence bound of Di . We can show that µi ≤ EYi ∼Di [Yi ] ≤ µi + Λi holds with high probability, with the same interval length Λi as in CUCB. (The proof is given in Appendix A.2.) Hence, the analysis in [8, 18] can be applied to SDCB, resulting in the same regret bounds.We further remark that in this case we do not need the three assumptions stated in Section 2 (in particular the independence assumption on Xi ’s): the summation reward case just works as in [18] and the nonlinear reward case relies on the properties of monotonicity and bounded smoothness used in [8]. 4 Improved SDCB Algorithm by Discretization In Section 3, we have shown that our algorithm SDCB achieves near-optimal regret bounds. However, that algorithm might suffer from large running time and memory usage. Note that, in the t-th round, an arm i might have been observed t − 1 times already, and it is possible that all the observed values from arm i are different (e.g., when arm i’s outcome distribution Di is continuous). In such case, it takes Θ(t) space to store the empirical CDF F̂i of the observed outcomes from arm i, and both calculating the stochastically dominant CDF Fi and updating F̂i take Θ(t) time. Therefore, the worst-case space usage of SDCB in T rounds is Θ(T ), and the worst-case running time is Θ(T 2 ) (ignoring the dependence on m and K); here we do not count the time and space used by the offline computation oracle. In this section, we propose an improved algorithm Lazy-SDCB which reduces the worst-case √ √ mem3/2 ory usage and running time to O( T ) and O(T ), respectively, while preserving the O( T log T ) distribution-independent regret bound. To this end, we need an additional assumption on the reward function: Assumption 4 (Lipschitz-continuous reward function). There exists C > 0 such that for any S ∈ F ′ ′ m ′ ′ and P any x, x ′ ∈ [0, 1] , we have \|R(x, S) − R(x , S)\| ≤ CkxS − xS k1 , where kxS − xS k1 = \|x − x \|. i i i∈S 6 Algorithm 2 Lazy-SDCB with known time horizon Input: time √ horizon T 1: s ← ⌈ T ⌉ 1 j=1 [0, s ], 2: Ij ← j−1 j ( s , s ], j = 2, . . . , s 3: Invoke SDCB (Algorithm 1) for T rounds, with the following change: whenever observing an outcome x (from any arm), find j ∈ [s] such that x ∈ Ij , and regard this outcome as sj Algorithm 3 Lazy-SDCB without knowing the time horizon 1: q ← ⌈log2 m⌉ 2: In rounds 1, 2, . . . , 2q , invoke Algorithm 2 with input T = 2q 3: for k = q, q + 1, q + 2, . . . do 4: In rounds 2k + 1, 2k + 2, . . . , 2k+1 , invoke Algorithm 2 with input T = 2k 5: end for We first describe the algorithm when the time horizon T is known in advance. The algorithm is summarized in Algorithm 2. We perform a discretization on the distribution D = D1 × · · · × Dm to obtain a discrete distribution D̃ = D̃1 × · · · × D̃m such that (i) for X̃ ∼ D̃, X̃1 , . . . , X̃m are also mutually independent, and (ii) √ every D̃i is supported on a set of equally-spaced values { 1s , 2s , . . . , 1}, where s is set to be ⌈ T ⌉. Specifically, we partition [0, 1] into s intervals: I1 = s−1 s−1 [0, 1s ], I2 = ( 1s , 2s ], . . . , Is−1 = ( s−2 s , s ], Is = ( s , 1], and define D̃i as Pr [X̃i = j/s] = X̃i ∼D̃i Pr [Xi ∈ Ij ] , Xi ∼Di j = 1, . . . , s. For the CMAB problem ([m], F , D, R), our algorithm “pretends” that the outcomes are drawn from D̃ instead of D, by replacing any outcome x ∈ Ij by js (∀j ∈ [s]), and then applies SDCB to the problem ([m], F , D̃, R). Since each D̃i has a known support { 1s , 2s , . . . , 1}, the algorithm only needs to maintain the number of occurrences of each support value in order to obtain the empirical CDF of all the observed outcomes from arm i. Therefore, all the operations in a round can be done √ using O(s) = O( T ) time and space, and the total time and space used by Lazy-SDCB are O(T 3/2 ) √ and O( T ), respectively. The discretization parameter s in Algorithm 2 depends on the time horizon T , which is why Algorithm 2 has to know T in advance. We can use the doubling trick to avoid the dependency on T . We present such an algorithm (without knowing T ) in Algorithm 3. It is easy to see that Algorithm 3 has the same asymptotic time and space usages as Algorithm 2. √ Regret Bounds. We show that both Algorithm 2 and Algorithm 3 achieve O( T log T ) distribution-independent regret bounds. The full proofs are given in Appendix B. Recall that C is the coefficient in the Lipschitz condition in Assumption 4. Theorem 2. Suppose the time horizon T is known in advance. Then the α-approximation regret of Algorithm 2 in T rounds is at most 2 √ √ π + 1 αM m. 93M mKT ln T + 2CK T + 3 Proof Sketch. The regret consists of two parts: (i) the regret for the discretized CMAB problem ([m], F , D̃, R), and (ii) the error due to discretization. We directly apply Theorem 1 for the first part. For the second part, a key step is to show \|rD (S) − rD̃ (S)\| ≤ CK/s for all S ∈ F (see Appendix B.1). Theorem 3. For any time horizon T ≥ 2, the α-approximation regret of Algorithm 3 in T rounds is at most √ √ 318M mKT ln T + 7CK T + 10αM m ln T. 7 5 Applications We describe the K-MAX problem and the class of expected utility maximization problems as applications of our general CMAB framework. The K-MAX Problem. In this problem, the player is allowed to select at most K arms from the set of m arms in each round, and the reward is the maximum one among the outcomes from the selected arms. In other words, the set of feasible super arms is F = S ⊆ [m] \|S\| ≤ K , and the reward function is R(x, S) = maxi∈S xi . It is easy to verify that this reward function satisfies Assumptions 2, 3 and 4 with M = C = 1. Now we consider the corresponding offline K-MAX problem of selecting at most K arms from m independent arms, with the largest expected reward. It can be implied by a result in [14] that finding the exact optimal solution is NP-hard, so we resort to approximation algorithms. We can show, using submodularity, that a simple greedy algorithm can achieve a (1 − 1/e)-approximation. Furthermore, we give the first PTAS for this problem. Our PTAS can be generalized to constraints other than the cardinality constraint \|S\| ≤ K, including s-t simple paths, matchings, knapsacks, etc. The algorithms and corresponding proofs are given in Appendix C. Theorem 4. There exists a PTAS for the offline K-MAX problem. In other words, for any constant ǫ > 0, there is a polynomial-time (1 − ǫ)-approximation algorithm for the offline K-MAX problem. We thus can apply our SDCB√algorithm to the K-MAX bandit problem and obtain O(log T ) distribution-dependent and Õ( T ) distribution-independent regret bounds according to Theorem 1, √ or can apply Lazy-SDCB to get Õ( T ) distribution-independent bound according to Theorem 2 or 3. Streeter and Golovin [26] study an online submodular maximization problem in the oblivious adversary model. In particular, their result can cover the stochastic K-MAX bandit problem as a special √ case, and an O(K mT log m) upper bound on the (1 − 1/e)-regret can be shown. While the techniques in [26] can√only give a bound on the (1 − 1/e)-approximation regret for K-MAX, we can obtain the first Õ( T ) bound on the (1 − ǫ)-approximation regret for any constant ǫ > 0, using our PTAS as the offline oracle. Even when we use the simple greedy algorithm as the oracle, our experiments show that SDCB performs significantly better than the algorithm in [26] (see Appendix D). Expected Utility Maximization. Our framework can also be applied to reward functions of the P utility function. The corresponding offline form R(x, S) = u( i∈S xi ), where u(·) is an increasing P problem is to maximize the expected utility E[u( i∈S xi )] subject to a feasibility constraint S ∈ F . Note that if u is nonlinear, the expected utility may not be a function of the means of the arms in S. Following the celebrated von Neumann-Morgenstern expected utility theorem, nonlinear utility functions have been extensively used to capture risk-averse or risk-prone behaviors in economics (see e.g., [11]), while linear utility functions correspond to risk-neutrality. Li and Deshpande [20] obtain a PTAS for the expected utility maximization (EUM) problem for several classes of utility functions (including for example increasing concave functions which typically indicate risk-averseness), and a large class of feasibility constraints (including cardinality constraint, s-t simple paths, matchings, and knapsacks). Similar results for other utility functions and feasibility constraints can be found in [27, 21, 4]. In the online problem, we can apply our algorithms, using their PTASs as the offline oracle. Again, we can obtain the first tight regret bounds on the (1 − ǫ)-approximation regret for any ǫ > 0, for the class of online EUM problems. Acknowledgments Wei Chen was supported in part by the National Natural Science Foundation of China (Grant No. 61433014). Jian Li and Yu Liu were supported in part by the National Basic Research Program of China grants 2015CB358700, 2011CBA00300, 2011CBA00301, and the National NSFC grants 61033001, 61361136003. The authors would like to thank Tor Lattimore for referring to us the DKW inequality. 8 References [1] Jean-Yves Audibert and Sébastien Bubeck. Minimax policies for adversarial and stochastic bandits. In COLT, pages 217–226, 2009. [2] Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multiarmed bandit problem. 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In Section A.1.1, we review the L1 distance between two distributions and present a property of it. In Section A.1.2, we review the DvoretzkyKiefer-Wolfowitz (DKW) inequality, which is a strong concentration result for empirical CDFs. In Section A.1.3, we prove some key technical lemmas. Then we complete the proof of Theorem 1 in Section A.1.4. A.1.1 The L1 Distance between Two Probability Distributions For simplicity, we only consider discrete distributions with finite supports – this will be enough for our purpose. Let P be a probability distribution. For any x, let P (x) = PrX∼P [X = x]. We write P = P1 ×P2 × · · · × Pn if the (multivariate) random variable X ∼ P can be written as X = (X1 , X2 , . . . , Xn ), where X1 , . . . , Xn are mutually independent and Xi ∼ Pi (∀i ∈ [n]). For two distributions P and Q, their L1 distance is defined as X L1 (P, Q) = \|P (x) − Q(x)\|, x where the summation is taken over x ∈ supp(P ) ∪ supp(Q). The L1 distance has the following property. It is a folklore result and we provide a proof for completeness. Lemma 1. Let P = P1 ×P2 ×· · ·×Pn and Q = Q1 ×Q2 ×· · ·×Qn be two probability distributions. Then we have n X L1 (Pi , Qi ). (1) L1 (P, Q) ≤ i=1 Proof. We prove (1) by induction on n. When n = 2, we have XX L1 (P, Q) = \|P (x, y) − Q(x, y)\| x = x ≤ = y XX y XX x X y \|P1 (x)P2 (y) − Q1 (x)Q2 (y)\| (\|P1 (x)P2 (y) − P1 (x)Q2 (y)\| + \|P1 (x)Q2 (y) − Q1 (x)Q2 (y)\|) P1 (x) x X y \|P2 (y) − Q2 (y)\| + = 1 · L1 (P2 , Q2 ) + 1 · L1 (P1 , Q1 ) = 2 X X y Q2 (y) X x \|P1 (x) − Q1 (x)\| L1 (Pi , Qi ). i=1 Here the summation is taken over x ∈ supp(P1 ) ∪ supp(Q1 ) and y ∈ supp(P2 ) ∪ supp(Q2 ). Suppose (1) is proved for n = k − 1 (k ≥ 3). When n = k, using the results for n = k − 1 and n = 2, we get L1 (P, Q) ≤ k−2 X i=1 L1 (Pi , Qi ) + L1 (Pk−1 × Pk , Qk−1 × Qk ) 10 ≤ = k−2 X L1 (Pi , Qi ) + L1 (Pk−1 , Qk−1 ) + L1 (Pk , Qk ) i=1 k X L1 (Pi , Qi ). i=1 This completes the proof. A.1.2 The DKW Inequality Consider a distribution D with CDF F (x). Let F̂n (x) be the empirical CDF of n i.i.d. samples Pn X1 , . . . , Xn drawn from D, i.e., F̂n (x) = n1 i=1 1{Xi ≤ x} (x ∈ R).7 Then we have: Lemma 2 (Dvoretzky-Kiefer-Wolfowitz inequality [10, 24]). For any ǫ > 0 and any n ∈ Z+ , we have 2 Pr sup F̂n (x) − F (x) ≥ ǫ ≤ 2e−2nǫ . x∈R h i Note that for any fixed x ∈ R, from the Chernoff bound we have Pr F̂n (x) − F (x) ≥ ǫ ≤ 2 2e−2nǫ . The DKW inequality states a stronger guarantee that the Chernoff concentration holds simultaneously for all x ∈ R. A.1.3 Technical Lemmas The following lemma describes some properties of the expected reward rP (S) = EX∼P [R(X, S)]. ′ be two probability distributions over Lemma 3. Let P = P1 × · · · × Pm and P ′ = P1′ × · · · × Pm m ′ ′ [0, 1] . Let Fi and Fi be the CDFs of Pi and Pi , respectively (i = 1, . . . , m). Suppose each Pi (i ∈ [m]) is a discrete distribution with finite support. (i) If for any i ∈ [m], x ∈ [0, 1] we have Fi′ (x) ≤ Fi (x), then for any super arm S ∈ F , we have rP ′ (S) ≥ rP (S). (ii) If for any i ∈ [m], x ∈ [0, 1] we have Fi (x) − Fi′ (x) ≤ Λi (Λi > 0), then for any super arm S ∈ F , we have X rP ′ (S) − rP (S) ≤ 2M Λi . i∈S Proof. It is easy to see why (i) is true. If we have Fi′ (x) ≤ Fi (x) for all i ∈ [m] and x ∈ [0, 1], then for all i, Pi′ has first-order stochastic dominance over Pi . When we change the distribution from Pi into Pi′ , we are moving some probability mass from smaller values to larger values. Recall that the reward function R(x, S) has a monotonicity property (Assumption 3): if x and x′ are two vectors in [0, 1]m such that xi ≤ x′i for all i ∈ [m], then R(x, S) ≤ R(x′ , S) for all S ∈ F . Therefore we have rP (S) ≤ rP ′ (S) for all S ∈ F . Now we prove (ii). Without loss of generality, we assume S = {1, 2, . . . , n} (n ≤ m). Let P ′′ = ′′ P1′′ × · · · × Pm be a distribution over [0, 1]m such that the CDF of Pi′′ is the following: max{Fi (x) − Λi , 0}, 0 ≤ x < 1, ′′ Fi (x) = (2) 1, x = 1. It is easy to see that Fi′′ (x) ≤ Fi′ (x) for all i ∈ [m] and x ∈ [0, 1]. Thus from the result in (i) we have (3) rP ′ (S) ≤ rP ′′ (S). 7 We use 1{·} to denote the indicator function, i.e., 1{H} = 1 if an event H happens, and 1{H} = 0 if it does not happen. 11 Let supp(Pi ) = {vi,1 , vi,2 , . . . , vi,si } where 0 ≤ vi,1 < · · · < vi,si ≤ 1. Define PS = P1 × P2 × · · · × Pn , and define PS′ and PS′′ similarly. Recall that the reward function R(x, S) can be written as RS (xS ) = RS (x1 , . . . , xn ). Then we have rP ′′ (S) − rP (S) X X RS (x1 , . . . , xn )PS (x1 , . . . , xn ) RS (x1 , . . . , xn )PS′′ (x1 , . . . , xn ) − = x1 ,...,xn x1 ,...,xn = X x1 ,...,xn ≤ X x1 ,...,xn RS (x1 , . . . , xn ) · (PS′′ (x1 , . . . , xn ) − PS (x1 , . . . , xn )) M · \|PS′′ (x1 , . . . , xn ) − PS (x1 , . . . , xn )\| = M · L1 (PS′′ , PS ), where the summation is taken over xi ∈ {vi,1 , . . . , vi,si } (∀i ∈ S). Then using Lemma 1 we obtain X rP ′′ (S) − rP (S) ≤ M · L1 (Pi′′ , Pi ). (4) i∈S ′′ Now we give an upper bound on L1 (Pi′′ , Pi ) for each i. Let Fi,j = Fi (vi,j ), Fi,j = Fi′′ (vi,j ), and ′′ Fi,0 = Fi,0 = 0. We have L1 (Pi′′ , Pi ) = si X j=1 = = si X j=1 si X j=1 \|Pi′′ (vi,j ) − Pi (vi,j )\| ′′ ′′ (Fi,j − Fi,j−1 ) − (Fi,j − Fi,j−1 ) (5) ′′ ′′ (Fi,j − Fi,j ) − (Fi,j−1 − Fi,j−1 ) . ′′ ′′ In fact, for all 1 ≤ j < si , we have Fi,j − Fi,j ≥ Fi,j−1 − Fi,j−1 . To see this, consider two cases: ′′ ′′ = Fi,j−1 = • If Fi,j < Λi , then we have Fi,j−1 ≤ Fi,j < Λi . By definition (2) we have Fi,j ′′ ′′ 0. Thus Fi,j − Fi,j = Fi,j ≥ Fi,j−1 = Fi,j−1 − Fi,j−1 . ′′ ′′ • If Fi,j ≥ Λi , then by definition (2) we have Fi,j − Fi,j = Λi ≥ Fi,j−1 − Fi,j−1 . Therefore (5) becomes L1 (Pi′′ , Pi ) = sX i −1 j=1 ′′ ′′ ′′ (Fi,j − Fi,j ) − (Fi,j−1 − Fi,j−1 ) + (1 − 1) − (Fi,si −1 − Fi,s ) i −1 ′′ ′′ = Fi,si −1 − Fi,s + Fi,si −1 − Fi,s i −1 i −1 ′′ = 2 Fi,si −1 − Fi,s i −1 (6) ≤ 2Λi , where the last inequality is due to (2). We complete the proof of the lemma by combining (3), (4) and (6): X X rP ′ (S) − rP (S) ≤ rP ′′ (S) − rP (S) ≤ M · L1 (Pi′′ , Pi ) ≤ 2M Λi . i∈S i∈S The following lemma is similar to Lemma 1 in [18]. We will use some additional notation: • For t ≥ m + 1 and i ∈ [m], let Ti,t be the value of counter Ti right after the t-th round of SDCB. In other words, Ti,t is the number of observed outcomes from arm i in the first t rounds. 12 • Let St be the super arm selected by SDCB in the t-th round. Lemma 4. Define an event in each round t (m + 1 ≤ t ≤ T ): s ( ) X 3 ln t Ht = 0 < ∆St ≤ 4M · . 2Ti,t−1 (7) i∈St Then the α-approximation regret of SDCB in T rounds is at most " T # X π2 E + 1 αM m. 1{Ht }∆St + 3 t=m+1 Proof. Let Fi be the CDF of Di . Let F̂i,l be the empirical CDF of the first l observations from arm i. For m + 1 ≤ t ≤ T , define an event s ( ) 3 ln t Et = there exists i ∈ [m] such that sup F̂i,Ti,t−1 (x) − Fi (x) ≥ , 2Ti,t−1 x∈[0,1] which means that the empirical CDF F̂i is not close enough to the true CDF Fi at the beginning of the t-th round. Recall that we have S ∗ = argmaxS∈F {rD (S)} and ∆S = max{α · rD (S ∗ ) − rD (S), 0} (S ∈ F ). We bound the α-approximation regret of SDCB as RegSDCB D,α (T ) = T X t=1 =E " E [α · rD (S ∗ ) − rD (St )] ≤ m X # ∆St + E t=1 " where ¬Et is the complement of event Et . T X t=m+1 T X E [∆St ] t=1 # 1{Et }∆St + E " T X t=m+1 # (8) 1{¬Et }∆St , We separately bound each term in (8). (a) the first term The first term in (8) can be trivially bounded as "m # m X X E α · rD (S ∗ ) ≤ m · αM. ∆St ≤ (9) t=1 t=1 (b) the second term By the DKW inequality we know that for any i ∈ [m], l ≥ 1, t ≥ m + 1 we have # " r 3 ln t 3 ln t ≤ 2e−2l· 2l = 2e−3 ln t = 2t−3 . Pr sup F̂i,l (x) − Fi (x) ≥ 2l x∈[0,1] Therefore E " T X t=m+1 # 1{Et } ≤ ≤ T m X t−1 X X t=m+1 i=1 l=1 T m X t−1 X X t=m+1 i=1 l=1 ≤ 2m ≤ T X t−2 t=m+1 π2 m, 3 13 Pr " 2t−3 F̂i,j,l − Fi,j ≥ r 3 ln t 2l # and then the second term in (8) can be bounded as " T # X π2 π2 E 1{Et }∆St ≤ m · (α · rD (S ∗ )) ≤ αM m. 3 3 t=m+1 (10) (c) the third term q 3 ln t We fix t > m and first assume ¬Et happens. Let ci = 2Ti,t−1 for each i ∈ [m]. Since ¬Et happens, we have F̂i,Ti,t−1 (x) − Fi (x) < ci ∀i ∈ [m], x ∈ [0, 1]. (11) Recall that in round t of SDCB (Algorithm 1), the input to the oracle is D = D1 × · · · × Dm , where the CDF Fi of Di is 0 ≤ x < 1, max{F̂i,Ti,t−1 (x) − ci , 0}, Fi (x) = (12) 1, x = 1. From (11) and (12) we know that Fi (x) ≤ Fi (x) ≤ Fi (x) + 2ci for all i ∈ [m], x ∈ [0, 1]. Thus, from Lemma 3 (i) we have rD (S) ≤ rD (S) ∀S ∈ F , (13) and from Lemma 3 (ii) we have rD (S) ≤ rD (S) + 2M X 2ci i∈S ∀S ∈ F . (14) Also, from the fact that the algorithm chooses St in the t-th round, we have rD (St ) ≥ α · max{rD (S)} ≥ α · rD (S ∗ ). (15) S∈F From (13), (14) and (15) we have α · rD (S ∗ ) ≤ α · rD (S ∗ ) ≤ rD (St ) ≤ rD (St ) + 2M X 2ci , i∈St which implies ∆St ≤ 4M X ci . i∈St P Therefore, when ¬Et happens, we always have ∆St ≤ 4M i∈St ci . In other words, s ) ( X 3 ln t . ¬Et =⇒ ∆St ≤ 4M 2Ti,t−1 i∈St This implies {¬Et , ∆St > 0} =⇒ ( 0 < ∆St ≤ 4M X i∈St s 3 ln t 2Ti,t−1 ) = Ht . Hence, the third term in (8) can be bounded as " T # " T # " T # X X X E 1{¬Et }∆St = E 1{¬Et , ∆St > 0}∆St ≤ E 1{Ht }∆St . t=m+1 t=m+1 t=m+1 Finally, by combining (8), (9), (10) and (16) we have " T # X π2 SDCB RegD,α (T ) ≤ E 1{Ht }∆St + + 1 αM m, 3 t=m+1 completing the proof of the lemma. 14 (16) A.1.4 Finishing the Proof of Theorem 1 Lemma 4 is very similar to Lemma 1 in [18]. We now apply the counting argument in [18] to finish the proof of Theorem 1. hP i T From Lemma 4 we know that it remains to bound E t=m+1 1{Ht }∆St , where Ht is defined in (7). Define two decreasing sequences of positive constants 1 = β0 >β1 > β2 > . . . α1 > α2 > . . . such that limk→∞ αk = limk→∞ βk = 0. We choose {αk } and {βk } as in Theorem 4 of [18], which satisfy ∞ √ X βk−1 − βk 6 ≤1 (17) √ αk k=1 and ∞ X αk k=1 βk < 267. For t ∈ {m + 1, . . . , T } and k ∈ Z+ , let ( 2 ln T αk 2MK ∆St mk,t = +∞ (18) ∆St > 0, ∆St = 0, and Ak,t = {i ∈ St \| Ti,t−1 ≤ mk,t }. Then we define an event Gk,t = {\|Ak,t \| ≥ βk K}, which means “in the t-th round, at least βk K arms in St had been observed at most mk,t times.” Lemma 5. In the t-th round (m + 1 ≤ t ≤ T ), if event Ht happens, then there exists k ∈ Z+ such that event Gk,t happens. Proof. Assume that Ht happens and that none of G1,t , G2,t , . . . happens. Then \|Ak,t \| < βk K for all k ∈ Z+ . Let A0,t = St and Āk,t = St \ Ak,t for k ∈ Z+ ∪{0}. It is easy to see Āk−1,t ⊆ Āk,t for all k ∈ Z+ . Note that limk→∞ exists N ∈ Z+ such that Āk,t = St for all k ≥ N , and then S∞mk,t = 0. Thus there we have St = k=1 Āk,t \ Āk−1,t . Finally, note that for all i ∈ Āk,t , we have Ti,t−1 > mk,t . Therefore ∞ X i∈St X 1 p = Ti,t−1 k=1 ∞ X ∞ X i∈Āk,t \Āk−1,t X 1 p ≤ Ti,t−1 k=1 ∞ X X i∈Āk,t \Āk−1,t 1 √ mk,t ∞ Āk,t \ Āk−1,t \|Ak−1,t \ Ak,t \| X \|Ak−1,t \| − \|Ak,t \| = = √ √ √ mk,t mk,t mk,t k=1 k=1 k=1 ∞ X 1 \|St \| 1 + \|Ak,t \| √ =√ −√ m1,t mk+1,t mk,t k=1 ∞ X K 1 1 <√ + βk K √ −√ m1,t mk+1,t mk,t = k=1 ∞ X (βk−1 − βk )K = . √ mk,t k=1 15 Note that we assume Ht happens. Then we have s X X √ 1 3 ln t p ∆St ≤ 4M · ≤ 2M 6 ln T · 2Ti,t−1 Ti,t−1 i∈S i∈S t t ∞ ∞ X √ √ X (βk−1 − βk )K βk−1 − βk = · ∆St ≤ ∆St , 6 < 2M 6 ln T · √ √ mk,t αk k=1 k=1 where the last inequality is due to (17). We reach a contradiction here. The proof of the lemma is completed. By Lemma 5 we have T X t=m+1 ∞ T X X 1{Ht }∆St ≤ 1{Gk,t , ∆St > 0}∆St . k=1 t=m+1 For i ∈ [m], k ∈ Z+ , t ∈ {m + 1, . . . , T }, define an event Gi,k,t = Gk,t ∧ {i ∈ St , Ti,t−1 ≤ mk,t }. Then by the definitions of Gk,t and Gi,k,t we have 1{Gk,t , ∆St > 0} ≤ Therefore T X t=m+1 1{Ht }∆St ≤ 1 X 1{Gi,k,t , ∆St > 0}. βk K i∈EB ∞ T X X X 1{Gi,k,t , ∆St > 0} i∈EB k=1 t=m+1 ∆St . βk K B B B For each arm i ∈ EB , suppose i is contained in Ni bad super arms Si,1 , Si,2 , . . . , Si,N . Let ∆i,l = i ∆Si,l B (l ∈ [Ni ]). Without loss of generality, we assume ∆i,1 ≥ ∆i,2 ≥ . . . ≥ ∆i,Ni . Note that 2 = 0. Then we ∆i,Ni = ∆i,min . For convenience, we also define ∆i,0 = +∞, i.e., αk 2MK ∆i,0 have T X t=m+1 ≤ ≤ = 1{Ht }∆St Ni T ∞ X X X X i∈EB k=1 t=m+1 l=1 Ni ∞ T X X X X i∈EB k=1 t=m+1 l=1 Ni ∞ T X X X X i∈EB k=1 t=m+1 l=1 = B 1{Gi,k,t , St = Si,l } B 1{Ti,t−1 ≤ mk,t , St = Si,l } ( 1 Ti,t−1 ≤ αk Ni X ∞ T l X X X X i∈EB k=1 t=m+1 l=1 j=1 ≤ ≤ ≤ Ni X ∞ T l X X X X i∈EB k=1 t=m+1 l=1 j=1 Ni X Ni ∞ T X X X X i∈EB k=1 t=m+1 l=1 j=1 Ni ∞ T X X X X i∈EB k=1 t=m+1 j=1 ∆St βk K ( 2M K ∆i,l ( 1 αk 2M K ∆i,j−1 2 ( 1 αk 2M K ∆i,j−1 2 ( 2M K ∆i,j−1 2 1 αk 1 αk 2M K ∆i,j−1 2 2 ∆i,l βk K B ln T, St = Si,l ) ∆i,l βk K B ln T, St = Si,l ) ∆i,l βk K ln T, St = B Si,l ) ∆i,j βk K 2 ln T, St = B Si,l ) ∆i,j βk K ln T ) ln T < Ti,t−1 ≤ αk 2M K ∆i,j 2 ln T < Ti,t−1 ≤ αk 2M K ∆i,j 2 ln T < Ti,t−1 ≤ αk 2M K ∆i,j ln T < Ti,t−1 ≤ αk 16 2M K ∆i,j 2 ∆i,j βk K ≤ Ni ∞ X X X αk i∈EB k=1 j=1 ∞ X αk 2M K ∆i,j 2 ln T − αk 2M K ∆i,j−1 2 Ni X X ! 1 1 ln T · − 2 =4M K 2 βk ∆i,j ∆i,j−1 i∈EB j=1 k=1 ! Ni X X 1 1 ≤1068M 2K ln T · ∆i,j , 2 − ∆2 ∆ i,j i,j−1 j=1 2 ln T ! ! ∆i,j βk K ∆i,j i∈EB where the last inequality is due to (18). Finally, for each i ∈ EB we have Ni X 1 1 − 2 ∆2i,j ∆i,j−1 j=1 ! ∆i,j = ≤ 1 ∆i,Ni 1 ∆i,Ni + N i −1 X j=1 + Z 1 (∆i,j − ∆i,j+1 ) ∆2i,j ∆i,1 ∆i,Ni 2 1 dx x2 1 − ∆i,Ni ∆i,1 2 . < ∆i,min = It follows that T X t=m+1 1{Ht }∆St ≤ 1068M 2K ln T · X i∈EB 2 ∆i,min = M 2K X 2136 ln T. ∆i,min (19) i∈EB Combining (19) with Lemma 4, the distribution-dependent regret bound in Theorem 1 is proved. To prove the distribution-independent bound, we decompose T X t=m+1 1{Ht }∆St = T X t=m+1 ≤ ǫT + PT 1{Ht , ∆St ≤ ǫ}∆St + T X t=m+1 T X 1{Ht }∆St into two parts: 1{Ht , ∆St > ǫ}∆St t=m+1 (20) 1{Ht , ∆St > ǫ}∆St , t=m+1 where ǫ > 0 is a constant to be determined. The second term can be bounded in the same way as in the proof of the distribution-dependent regret bound, except that we only consider the case ∆St > ǫ. Thus we can replace (19) by T X t=m+1 1{Ht , ∆St > ǫ}∆St ≤ M 2 K It follows that T X t=m+1 Finally, letting ǫ = q T X i∈EB ,∆i,min >ǫ 2136 2136 ln T ≤ M 2 Km ln T. ∆i,min ǫ 1{Ht }∆St ≤ ǫT + M 2 Km 2136M 2 Km ln T T t=m+1 X (21) 2136 ln T. ǫ , we get √ √ 1{Ht }∆St ≤ 2 2136M 2KmT ln T < 93M mKT ln T . Combining this with Lemma 4, we conclude the proof of the distribution-independent regret bound in Theorem 1. 17 Algorithm 4 CUCB [8, 18] 1: For each arm i, maintain: (i) µ̂i , the average of all observed outcomes from arm i so far, and (ii) Ti , the number of observed outcomes from arm i so far. 2: // Initialization 3: for i = 1 to m do 4: // Action in the i-th round 5: Play a super arm Si that contains arm i, and update µ̂i and Ti . 6: end for 7: for t = m + 1, m + 2, . . . do 8: // Action in the t-th q round ln t 9: µ̄i ← min{µ̂i + 32T , 1} i ∀i ∈ [m] 10: Play the super arm St ← Oracle(µ̄), where µ̄ = (µ̄1 , . . . , µ̄m ). 11: Update µ̂i and Ti for all i ∈ St . 12: end for A.2 Analysis of Our Algorithm in the Previous CMAB Framework We now give an analysis of SDCB in the previous CMAB framework, following our discussion in Section 3. We consider the case in which the expected reward only depends on the means of the random variables. Namely, rD (S) only depends on µi ’s (i ∈ S), where µi is arm i’s mean outcome. In this case, we can rewrite rD (S) as rµ (S), where µ = (µ1 , . . . , µm ) is the vector of means. Note that the offline computation oracle only needs a mean vector as input. We no longer need the three assumptions (Assumptions 1-3) given in Section 2. In particular, we do not require independence among outcome distributions of all arms (Assumption 1). Although we cannot write D as D = D1 × · · · × Dm , we still let Di be the outcome distribution of arm i. In this case, Di is the marginal distribution of D in the i-th component. We summarize the CUCB algorithm [8, 18] in Algorithm 4. It maintains the empirical mean µ̂i of the outcomes from each arm i, and stores the number of observed outcomes from arm i in a variable Ti . In each round, it calculates an upper confidence bound (UCB) µ̄i of µi , Then it uses the UCB vector µ̄ as the input to the oracle, and plays the super arm output by the oracle. In the t-th round (t > m), each UCB µ̄i has the key property that s 3 ln t µi ≤ µ̄i ≤ µi + 2 (22) 2Ti,t−1 holds with high probability. (Recall q that Ti,t−1 is the value of Ti after t − 1 rounds.) To see this, ln t with high probability (by Chernoff bound), and then (22) note that we have \|µi − µ̂i \| ≤ 2T3i,t−1 follows from the definition of µ̄i in line 9 of Algorithm 4. We prove that the same property as (22) also holds for SDCB. Consider a fixed t > m, and let D = D1 × · · · × Dm be the input to the oracle in the t-th round of SDCB. Let νi = EYi ∼Di [Yi ]. We can think that SDCB uses the mean vector ν = (ν1 , . . . , νm ) as the input to the oracle used by CUCB. We now show that for each i, we have s 3 ln t µi ≤ ν i ≤ µi + 2 (23) 2Ti,t−1 with high probability. To show (23), we first prove the following lemma. Lemma 6. Let P and P ′ be two distributions over [0, 1] with CDFs F and F ′ , respectively. Consider two random variables Y ∼ P and Y ′ ∼ P ′ . (i) If for all x ∈ [0, 1] we have F ′ (x) ≤ F (x), then we have E[Y ] ≤ E[Y ′ ]. (ii) If for all x ∈ [0, 1] we have F (x) − F ′ (x) ≤ Λ (Λ > 0), then we have E[Y ′ ] ≤ E[Y ] + Λ. 18 Proof. We have E[Y ] = Z 1 x dF (x) = (xF (x)) 0 1 0 − Z 1 0 F (x) dx = 1 − Z 1 F (x) dx. 0 Similarly, we have ′ E[Y ] = 1 − Then the lemma holds trivially. Z 1 F ′ (x) dx. 0 Now we prove (23). According to the DKW inequality, with high probability we have s 3 ln t ≤ Fi (x) ≤ Fi (x) Fi (x) − 2 2Ti,t−1 (24) for all i ∈ [m] and x ∈ [0, 1], where Fi is the CDF of Di used in round t of SDCB, and Fi is the CDF of Di . Suppose (24) holds for all i, x, then qfor any i, the two distributions Di and Di ln t satisfy the two conditions in Lemma 6, with Λ = 2 2T3i,t−1 ; then from Lemma 6 we know that q ln t µi ≤ νi ≤ µi + 2 2T3i,t−1 . Hence we have shown that (23) holds with high probability. The fact that (23) holds with high probability means that the mean of Di is also a UCB of µi with the same confidence as in CUCB. With this property, the analysis in [8, 18] can also be applied to SDCB, resulting in exactly the same regret bounds. B Missing Proofs from Section 4 B.1 Analysis of the Discretization Error The following lemma gives an upper bound on the error due to discretization. Refer to Section 4 for the definition of the discretized distribution D̃. Lemma 7. For any S ∈ F , we have \|rD (S) − rD̃ (S)\| ≤ CK . s To prove Lemma 7, we show a slightly more general lemma which gives an upper bound on the discretization error of the expectation of a Lipschitz continuous function. Lemma 8. Let g(x) be a Lipschitz continuous function on P [0, 1]n such that for any x, x′ ∈ [0, 1]n , ′ ′ ′ we have \|g(x) − g(x )\| ≤ Ckx − x k1 , where kx − x k1 = ni=1 \|xi − x′i \|. Let P = P1 × · · · × Pn be a probability distribution over [0, 1]n . Define another distribution P̃ = P̃1 × · · ·× P̃n over [0, 1]n as follows: each P̃i (i ∈ [n]) takes values in { 1s , 2s , . . . , 1}, and Pr [X̃i = j/s] = Pr [Xi ∈ Ij ] , X̃i ∼P̃i Xi ∼Pi j ∈ [s], s−1 s−1 where I1 = [0, 1s ], I2 = ( 1s , 2s ], . . . , Is−1 = ( s−2 s , s ], Is = ( s , 1]. Then EX∼P [g(X)] − EX̃∼P̃ [g(X̃)] ≤ C ·n . s (25) Proof. Throughout the proof, we consider X = (X1 , . . . , Xn ) ∼ P and X̃ = (X̃1 , . . . , X̃n ) ∼ P̃ . Let vj = j s (j = 0, 1, . . . , s) and pi,j = Pr[X̃i = vj ] = Pr[Xi ∈ Ij ] We prove (25) by induction on n. 19 i ∈ [n], j ∈ [s]. (1) When n = 1, we have E[g(X1 )] = X j∈[s],p1,j >0 p1,j · E g(X1 ) X1 ∈ Ij . (26) Since g is continuous, for each j ∈ [s] such that p1,j > 0, there exists ξj ∈ [vj−1 , vj ] such that E [g(X1 )\|X1 ∈ Ij ] = g(ξj ) From the Lipschitz continuity of g we have \|g(vj ) − g(ξj )\| ≤ C\|vj − ξj \| ≤ C\|vj − vj−1 \| = C . s Hence E[g(X1 )] − E[g(X̃1 )] = X p1,j · E[g(X1 )\|X1 ∈ Ij ] − X p1,j · g(ξj ) − j∈[s],p1,j >0 = j∈[s],p1,j >0 ≤ ≤ = j∈[s],p1,j >0 X p1,j · \|g(ξj ) − g(vj )\| X p1,j · j∈[s],p1,j >0 j∈[s],p1,j >0 C . s X X j∈[s],p1,j >0 p1,j · g(vj ) p1,j · g(vj ) C s This proves (25) for n = 1. (ii) Suppose (25) is correct for n = 1, 2, . . . , k − 1. Now we prove it for n = k (k ≥ 2). We define two functions on [0, 1]k−1 : h(x1 , . . . , xk−1 ) = EXk [g(x1 , . . . , xk−1 , Xk )] and h̃(x1 , . . . , xk−1 ) = EX̃k [g(x1 , . . . , xk−1 , X̃k )]. For any fixed x1 , . . . , xk−1 ∈ [0, 1], the function g(x1 , . . . , xk−1 , x) on x ∈ [0, 1] is Lipschitz continuous. Therefore from the result for n = 1 we have h(x1 , . . . , xk−1 ) − h̃(x1 , . . . , xk−1 ) ≤ 20 C s ∀x1 , . . . , xk−1 ∈ [0, 1]. Then we have E[g(X)] − E[g(X̃)] = EX1 ,...,Xk−1 [E[g(X)\|X1 , . . . , Xk−1 ]] − E[g(X̃)] = EX1 ,...,Xk−1 [h(X1 , . . . , Xk−1 )] − E[g(X̃)] ≤ EX1 ,...,Xk−1 [h(X1 , . . . , Xk−1 )] − EX1 ,...,Xk−1 [h̃(X1 , . . . , Xk−1 )] + EX1 ,...,Xk−1 [h̃(X1 , . . . , Xk−1 )] − E[g(X̃)] h i ≤ EX1 ,...,Xk−1 h(X1 , . . . , Xk−1 ) − h̃(X1 , . . . , Xk−1 ) + EX1 ,...,Xk−1 ,X̃k [g(X1 , . . . , Xk−1 , X̃k )] − E[g(X̃)] h i C + EX̃k E[g(X1 , . . . , Xk−1 , X̃k )\|X̃k ] − E[g(X̃1 , . . . , X̃k−1 , X̃k )\|X̃k ] ≤ EX1 ,...,Xk−1 s h i C ≤ + EX̃k E[g(X1 , . . . , Xk−1 , X̃k )\|X̃k ] − E[g(X̃1 , . . . , X̃k−1 , X̃k )\|X̃k ] s X C = + pk,j · E[g(X1 , . . . , Xk−1 , vj )] − E[g(X̃1 , . . . , X̃k−1 , vj )] . s j∈[s],pk,j >0 (27) For any j ∈ [s], the function g(x1 , . . . , xk−1 , vj ) on (x1 , . . . , xk−1 ) ∈ [0, 1] uous. Then from the induction hypothesis at n = k − 1, we have E[g(X1 , . . . , Xk−1 , vj )] − E[g(X̃1 , . . . , X̃k−1 , vj )] ≤ k−1 C(k − 1) s is Lipschitz contin- ∀j ∈ [s]. (28) From (27) and (28) we have E[g(X)] − E[g(X̃)] ≤ C + s X j∈[s],pk,j >0 pk,j · C(k − 1) s C(k − 1) C + s s Ck . = s = This concludes the proof for n = k. Now we prove Lemma 7. Proof of Lemma 7. We have rD (S) = EX∼D [R(X, S)] = EX∼D [RS (XS )] = EXS ∼DS [RS (XS )], where XS = (Xi )i∈S and DS = (Di )i∈S . Similarly, we have rD̃ (S) = EX̃S ∼D̃S [RS (X̃S )]. According to Assumption 4, the function RS defined on [0, 1]S is Lipschitz continuous. Then from Lemma 8 we have \|rD (S) − rD̃ (S)\| = EXS ∼DS [RS (XS )] − EX̃S ∼D̃S [RS (X̃S )] ≤ This completes the proof. 21 C·K C · \|S\| ≤ . s s B.2 Proof of Theorem 2 Proof of Theorem 2. Let S ∗ = argmaxS∈F {rD (S)} and S̃ ∗ = argmaxS∈F {rD̃ (S)} be the optimal super arms in problems ([m], F , D, R) and ([m], F , D̃, R), respectively. Suppose Algorithm 2 selects super arm St in the t-th round (1 ≤ t ≤ T ). Then its α-approximation regret is bounded as Alg. 2 RegD,α (T ) = T · α · rD (S ∗ ) − T X E [rD (St )] t=1 T X = T · α rD (S ∗ ) − rD̃ (S̃ ∗ ) + E [rD̃ (St ) − rD (St )] + t=1 ≤ T · α (rD (S ∗ ) − rD̃ (S ∗ )) + T X t=1 T · α · rD̃ (S̃ ∗ ) − T X E [rD̃ (St )] t=1 ! Alg. 1 (T ). E [rD̃ (St ) − rD (St )] + RegD̃,α where the inequality is due to rD̃ (S̃ ∗ ) ≥ rD̃ (S ∗ ). Then from Lemma 7 and the distribution-independent bound in Theorem 1 we have 2 √ CK CK π Alg. 2 +T · + 93M mKT ln T + + 1 αM m RegD,α (T ) ≤ T · α · s s 3 2 √ CKT π (29) ≤2· + 93M mKT ln T + + 1 αM m s 3 2 √ √ π ≤ 93M mKT ln T + 2CK T + + 1 αM m. 3 √ √ Here in the last two inequalities we have used α ≤ 1 and s = ⌈ T ⌉ ≥ T . The proof is completed. B.3 Proof of Theorem 3 Proof of Theorem 3. Let n = ⌈log2 T ⌉. Then we have 2n−1 < T ≤ 2n . If n ≤ q = ⌈log2 m⌉, then T ≤ 2m and the regret in T rounds is at most 2m · αM . The regret bound holds trivially. Now we assume n ≥ q + 1. Using Theorem 2, we have Alg. 3 (T ) RegD,α Alg. 3 ≤ RegD,α (2n ) Alg. 2 q = RegD,α (2 ) + n−1 X 2 k RegAlg. D,α (2 ) k=q 2 ≤ RegAlg. D,α (2m) ≤ 2m · αM + + n−1 X Alg. 2 k RegD,α (2 ) k=q n−1 X k=q 2 √ √ π k k k + 1 αM m 93M mK · 2 ln 2 + 2CK 2 + 3 2 n−1 X√ √ π n−1 k ≤ 2αM m + 93M mK ln 2 + 2CK · 2 + (n − 1) · + 1 αM m 3 k=1 2 √2n √ π n−1 + 2CK · √ + + 3 (n − 1) · αM m ≤ 93M mK ln 2 3 2−1 √ 2 √ 2T π ≤ 93M mK ln T + 2CK · √ + 3 log2 T · αM m + 3 2−1 22 Algorithm 5 Greedy-K-MAX 1: S ← ∅ 2: for i = 1 to K do 3: k ← argmaxj∈[m]\S rD (S ∪ {j}) 4: S ← S ∪ {k} 5: end for Output: S √ √ ≤ 318M mKT ln T + 7CK T + 10αM m ln T. C The Offline K-MAX Problem In this section, we consider the offline K-MAX problem. Recall that we have m independent random variables {Xi }i∈[m] . Xi follows the discrete distribution Di with support {vi,1 , . . . , vi,si } ⊂ [0, 1], and D = D1 × · · · × Dm is the joint distribution of X = (X1 , . . . , Xm ). Let pi,j = Pr[Xi = vi,j ]. Define rD (S) = EX∼D [maxi∈S Xi ] and OPT = maxS:\|S\|=K rD (S). Our goal is to find (in polynomial time) a subset S ⊆ [m] of cardinality K such that rD (S) ≥ α · OPT (for certain constant α). First, we show that rD (S) can be calculated in polynomial time given any S ⊆ [m]. Let S = {iS 1 , i2 , . . . , in }. Note that for X ∼ D, maxi∈S Xi can only take values in the set V (S) = i∈S supp(Di ). For any v ∈ V (S), we have Pr max Xi = v X∼D i∈S = Pr [Xi1 = v, Xi2 ≤ v, . . . , Xin ≤ v] X∼D + Pr [Xi1 < v, Xi2 = v, Xi3 ≤ v, . . . , Xin ≤ v] (30) X∼D + ··· + Pr [Xi1 < v, . . . , Xin−1 < v, Xin = v]. X∼D Since Xi1 , . . . , Xin are mutually independent, each probability appearing in (30) can be calculated in polynomial time. Hence for any v ∈ V (S), PrX∼D [maxi∈S Xi = v] can be calculated in polynomial time using (30). Then rD (S) can be calculated by X v · Pr max Xi = v rD (S) = v∈V (S) X∼D i∈S in polynomial time. C.1 (1 − 1/e)-Approximation We now show that a simple greedy algorithm (Algorithm 5) can find a (1 − 1/e)-approximate solution, by proving the submodularity of rD (S). In fact, this is implied by a slightly more general result [13, Lemma 3.2]. We provide a simple and direct proof for completeness. Lemma 9. Algorithm 5 can output a subset S such that rD (S) ≥ (1 − 1/e) · OPT. Proof. For any x ∈ [0, 1]m , let fx (S) = maxi∈S xi be a set function defined on 2[m] . (Define fx (∅) = 0.) We can verify that fx (S) is monotone and submodular: • Monotonicity. For any A ⊆ B ⊆ [m], we have fx (A) = maxi∈A xi ≤ maxi∈B xi = fx (B). • Submodularity. For any A ⊆ B ⊆ [m] and any k ∈ [m] \ B, there are three cases (note that maxi∈A xi ≤ maxi∈B xi ): (i) If xk ≤ maxi∈A xi , then fx (A ∪ {k}) − fx (A) = 0 = fx (B ∪ {k}) − fx (B). 23 (ii) If maxi∈A xi < xk ≤ maxi∈B xi , then fx (A ∪ {k}) − fx (A) = xk − maxi∈A xi > 0 = fx (B ∪ {k}) − fx (B). (iii) If xk > maxi∈B xi , then fx (A ∪ {k}) − fx (A) = xk − maxi∈A xi ≥ xk − maxi∈B xi = fx (B ∪ {k}) − fx (B). Therefore, we always have fx (A ∪ {k}) − fx (A) ≥ fx (B ∪ {i}) − fx (B). The function fx (S) is submodular. For any S ⊆ [m] we have rD (S) = s2 s1 X X j1 =1 j2 =1 ··· sm X f(v1,j1 ,...,vm,jm ) (S) m Y pi,ji . i=1 jm =1 Since each set function f(v1,j1 ,...,vm,jm ) (S) is monotone and submodular, rD (S) is a convex combination of monotone submodular functions on 2[m] . Therefore, rD (S) is also a monotone submodular function. According to the classical result on submodular maximization [25], the greedy algorithm can find a (1 − 1/e)-approximate solution to maxS⊆[m],\|S\|≤K {rD (S)}. C.2 PTAS Now we provide a PTAS for the K-MAX problem. In other words, we give an algorithm which, given any fixed constant 0 < ε < 1/2, can find a solution S of cardinality \|K\| such that rD (S) ≥ (1 − ε) · OPT in polynomial time. We first provide an overview of our approach, and then spell out the details later. 1. (Discretization) We first transform each Xi to another discrete distribution X̃i , such that all X̃i ’s are supported on a set of size O(1/ε2 ). 2. (Computing signatures) For each Xi , we can compute from X̃i a signature Sig(X P i ) which is a vector of size O(1/ε2 ). For a set S, we define its signature Sig(S) to be i∈S Sig(Xi ). We show that if two sets S1 and S2 have the same signature, their objective values are close (Lemma 12). 3. (Enumerating signatures) We enumerate all possible signatures (there are polynomial number of them when treating ε as a constant) and try to find the one which is the signature of a set of size K, and the objective value is maximized. C.2.1 Discretization We first describe the discretization step. We say that a random variable X follows the Bernoulli distribution B(v, q) if X takes value v with probability q and value 0 with probability 1 − q. For any discrete distribution, we can rewrite it as the maximum of a set of Bernoulli distributions. Definition 1. Let X be a discrete random variable with support {v1 , v2 , . . . , vs }(v1 < v2 < · · · < vs ) and Pr[X = vj ] = pj . We define a set of independent Bernoulli random variables {Zj }j∈[s] as ! pj Zj ∼ B vj , P . j ′ ≤j pj ′ We call {Zj } the Bernoulli decomposition of Xi . Lemma 10. For a discrete distribution X and its Bernoulli decomposition {Zj }, maxj {Zj } has the same distribution with X. Proof. We can easily see the following: Pr[max{Zj } = vi ] = Pr[Zi = vi ] j = P pi i′ ≤i pi′ 24 Y Pr[Zi′ = 0] i′ >i Y h>i 1− P ph h′ ≤h p h′ ! Algorithm 6 Discretization 1: We first run Greedy-K-MAX to obtain a solution SG and let W = rD (SG ). 2: for i = 1 to m do 3: Compute the Bernoulli decomposition {Zi,j }j of Xi . 4: for all Zi,j do 5: Create another Bernoulli variable Z̃i,j as follows: 6: if vi,j > W/ε then ε 7: Let Z̃i,j ∼ B W ε , E[Zi,j ] W (Case 1) 8: else Zi,j 9: Let Z̃i,j = ⌊ εW ⌋εW (Case 2) 10: end if 11: end for 12: Let X̃i = maxj {Z̃ij } 13: end for = P pi i′ ≤i pi′ Y h>i Hence, Pr[maxj {Zj } = vi ] = Pr[X = vi ] for all i ∈ [s]. P h′ ≤h−1 ph′ P = pi . h′ ≤h ph′ Now, we describe how to construct the discretization X̃i of Xi for all i ∈ [m]. The pseudocode can be found in Algorithm 6. We first run Greedy-K-MAX to obtain a solution SG . Let W = rD (SG ). By Lemma 9, we know that W ≥ (1 − 1/e)OPT. Then we compute the Bernoulli decomposition {Zi,j }j of Xi . For each Zi,j , we create another Bernoulli variable Z̃i,j as follows: Recall that vi,j is the nonzero possible value of Zij . We distinguish two cases. Case 1: If vi,j > W/ε, then we let ε , E[Z ] . It is easy to see that E[Z̃ij ] = E[Zij ]. Case 2: If vi,j ≤ W/ε, then we Z̃i,j ∼ B W i,j ε W Zi,j ⌋εW. We note that more than one Z̃ij ’s may have the same support, and all Z̃ij ’s let Z̃i,j = ⌊ εW are supported on DS = {0, εW, 2εW, . . . , W/ε}. Finally, we let X̃i = maxj {Z̃ij }, which is the discretization of Xi . Since X̃i is the maximum of a set of Bernoulli distributions, it is also a discrete distribution supported on DS. We can easily compute Pr[X̃i = v] for any v ∈ DS. Now, we show that the discretization only incurs a small loss in the objective value. The key is to show that we do not lose much in the transformation from Zi,j ’s to Z̃i,j ’s. We prove a slightly more general lemma as follows. Lemma 11. Consider any set of Bernoulli variables {Zi ∼ B(ai , pi )}1≤i≤n . Assume that E[maxi∈[n] Zi ] < cW, where c is a constant such that cε < 1/2. For each Zi , we create a Bernoulli variable Z̃i in the same way as Algorithm 6. Then the following holds: E[max Zi ] ≥ E[max Z̃i ] ≥ E[max Zi ] − (2c + 1)εW. Proof. Assume a1 is the largest among all ai ’s. If a1 < W/ε, all Z̃i are created in Case 2. In this case, it is obvious to have that E[max Zi ] ≥ E[max Z̃i ] ≥ E[max Zi ] − εW. If a1 ≥ W/ε, the proof is slightly more complicated. Let L = {i \| ai ≥ W/ε}. We prove by induction on n (i.e., the number of the variables) the following more general claim: X E[max Zi ] ≥ E[max Z̃i ] ≥ E[max Zi ] − εW − c εai pi . (31) i∈L Consider the base case n = 1. The lemma holds immediately in Case 1 as E[Z1 ] = E[Z˜1 ]. Assuming the lemma is true for n = k, we show it also holds for n = k + 1. Recall we have Z̃1 ∼ B( W ε , εE[Z1 ]/W). Thus E[max Zi ] − E[max Z̃i ] =a1 p1 + (1 − p1 )E[max Zi ] − a1 p1 − (1 − εE[Z1 ]/W)E[max Z̃i ] i≥1 i≥1 i≥2 25 i≥2 ≥(1 − p1 )E[max Z̃i ] − (1 − εE[Z1 ]/W)E[max Z̃i ] i≥2 i≥2 =(εa1 p1 /W − p1 )E[max Z̃i ] ≥ 0, i≥2 where the first inequality follows from the induction hypothesis and the last from a1 ≥ W/ε. The other direction can be seen as follows: E[max Z̃i ] − E[max Zi ] =a1 p1 + (1 − εE[Z1 ]/W)E[max Z̃i ] − (a1 p1 + (1 − p1 )E[max Zi ]) i≥1 i≥1 i≥2 i≥2 X εai pi ≥(1 − εE[Z1 ]/W)E[max Zi ] − (1 − p1 )E[max Zi ] − εW − c i≥2 i≥2 ≥(−εE[Z1 ]/W)E[max Zi ] − εW − c i≥2 ≥ − εW − c X X i∈L\{1} εai pi i∈L\{1} εai pi , i∈L where the last inequality holds since E[maxi≥2 Zi ] ≤ cW. This finishes the proof of (31). P Now, we show that i∈L ai pi ≤ 2W. This can be seen as follows. First, we can see from Markov inequality that Pr[max Zi > W/ε] ≤ cε. Q Equivalently, we have i∈L (1 − pi ) ≥ 1 − cε. Then, we can see that W≥ X ai i∈L X Y 1X (1 − pj )pi ≥ (1 − cε) ai p i ≥ ai p i . 2 j<i i∈L i∈L Plugging this into (31), we prove the lemma. Corollary 1. For any set S ⊆ [m], suppose E[maxi∈S Xi ] < cW, where c is a constant such that cε < 1/2. Then the following holds: E[max Xi ] ≥ E[max X̃i ] ≥ E[max Xi ] − (2c + 1)εW. i∈S i∈S i∈S C.2.2 Signatures For each Xi , we have created its discretization X̃i = maxj {Z̃ij }. Since X̃i is a discrete distribution, we can define its Bernoulli decomposition {Yij }j∈[h] where h = \|DS\|. Suppose Yij ∼ B(jεW, qij ). Now, we define the signature of Xi to be the vector Sig(Xi ) = (Sig(Xi )1 , . . . , Sig(Xi )h ) where 4 − ln (1 − qij ) ε ln(1/ε4 ) Sig(Xi )j = min , · j ∈ [h]. ε4 /m ε4 /m m For any set S, define its signature to be Sig(S) = X Sig(Xi ). i∈S Define the set SG of signature vectors to be all nonnegative h-dimensional vectors, where each coordinate is an integer multiple of ε4 /m and at most m ln(1/ε4 ). Clearly, the size of SG is 2 h−1 O mε−4 log(h/ε2 ) = Õ(mO(1/ε ) ), which is polynomial for any fixed constant ε > 0 (recall h = \|DS\| = O(1/ε2 )). Now, we prove the following crucial lemma. Lemma 12. Consider two sets S1 and S2 . If Sig(S1 ) = Sig(S2 ), the following holds: E[max X̃i ] − E[max X̃i ] ≤ O(ε)W. i∈S1 i∈S2 26 Algorithm 7 PTAS-K-MAX 1: U ← ∅ 2: for all signature vector sg ∈ SG do 3: Find a set S such that \|S\| = K and Sig(S) = sg 4: if rD (S) > rD (U ) then 5: U ←S 6: end if 7: end for Output: U Proof. Suppose {Yij }j∈[h] is the Bernoulli decomposition of X̃i . For any set S, we define Yk (S) = maxi∈S Yik (it is the max of a set of Bernoulli distributions). Q It is not hard to see that Yk (S) has a Bernoulli distribution B(kεW, pk (S)) with pk (S) = 1 − i∈S (1 − qik ). As Sig(S1 ) = Sig(S2 ), we have that Y Y \|pk (S1 ) − pk (S2 )\| = \| (1 − qik ) − (1 − qik )\| i∈S1 = exp i∈S2 X i∈S1 ≤ 2ε4 ! ln(1 − qik ) − exp ∀k ∈ [h]. X i∈S2 ! ln(1 − qik ) Noticing maxi∈S X̃i = maxk Yk (S), we have that E[max X̃i ] − E[max X̃i ] = E[max Yk (S1 )] − E[max Yk (S2 )] k k i∈S2 i∈S1 ! X W \|pk (S1 ) − pk (S2 )\| ≤ ε k ≤4hε3 W = O(ε)W where the first inequality follows from Lemma 1. For any signature vector sg, we associate to it a set of random variables {Bk ∼ B(kεW, 1 − e−sgk )}hk=1 .8 Define the value of sg to be Val(sg) = E[maxk∈[h] Bk ]. Corollary 2. For any feasible set S with Sig(S) = sg, \|E[maxi∈S X̃i ] − Val(sg)\| ≤ O(ε)W. Moreover, combining with Corollary 1, we have that \|E[maxi∈S Xi ] − Val(sg)\| ≤ O(ε)W. C.2.3 Enumerating Signatures Our algorithm enumerates all signature vectors sg in SG. For each sg, we check if we can find a set S 2 of size K such that Sig(S) = sg. This can be done by a standard dynamic program in Õ(mO(1/ε ) ) ′ ′ time as follows: We use Boolean variable R[i][j][sg ] to represent whether signature vector sg ∈ SG can be dominated by i variables in set {X1 , . . . , Xj }. The dynamic programming recursion is R[i][j][sg′ ] = R[i][j − 1][sg′ ] ∧ R[i − 1][j − 1][sg′ − Sig(Xj )]. If the answer is yes (i.e., we can find such S), we say sg is a feasible signature vector and S is a candidate set. Finally, we pick the candidate set with maximum rD (S) and output the set. The pseudocode can be found in Algorithm 7. Now, we are ready to prove Theorem 4 by showing Algorithm 7 is a PTAS for the K-MAX problem. Proof of Theorem 4. Suppose S ∗ is the optimal solution and sg∗ is the signature of S ∗ . By Corollary 2, we have that \|OPT − Val(sg∗ )\| ≤ O(ε)W. 8 It is not hard to see the signature of maxk∈[h] Bk is exactly sg. 27 Algorithm 8 Online Submodular Maximization [26] 1: Let A1 , A2 , . . . , AK be K instances of Exp3 2: for t = 1, 2, . . . do 3: // Action in the t-th round 4: for i = 1 to K do 5: Use Ai to select an arm at,i ∈ [m] 6: end for SK 7: Play the super arm St ← i=1 {at,i } 8: for i = 1 to K do Si−1 Si 9: Feed back ft ( j=1 {at,j }) − ft ( j=1 {at,j }) as the payoff Ai receives for choosing at,i 10: end for 11: end for When Algorithm 7 is enumerating sg∗ , it can find a set S such that Sig(S) = sg∗ (there exists at least one such set since S ∗ is one). Therefore, we can see that \|E[max Xi ] − E[max∗ Xi ]\| ≤ \|Val(sg∗ ) − max Xi \| + \|Val(sg∗ ) − E[max∗ Xi ]\| ≤ O(ε)W. i∈S i∈S i∈S i∈S Let U be the output of Algorithm 7. Since W ≥ (1 − 1/e)OPT, we have rD (U ) ≥ rD (S) = E[maxi∈S Xi ] ≥ (1 − O(ε))OPT. The running time of the algorithm is polynomial for a fixed constant ε > 0, since the number of signature vectors is polynomial and the dynamic program in each iteration also runs in polynomial time. Hence, we have a PTAS for the K-MAX problem. Remark. In fact, Theorem 4 can be generalized in the following way: instead of the cardinality constraint \|S\| ≤ K, we can have more general combinatorial constraint on the feasible set S. As long as we can execute line 3 in Algorithm 7 in polynomial time, the analysis wound be the same. Using the same trick as in [20], we can extend the dynamic program to a more general class of combinatorial constraints where there is a pseudo-polynomial time for the exact version9 of the deterministic version of the corresponding problem. The class of constraints includes s-t simple paths, knapsacks, spanning trees, matchings, etc. D Empirical Comparison between the SDCB Algorithm and Online Submodular Maximization on the K-MAX Problem We perform experiments to compare the SDCB algorithm with the online submodular maximization algorithm in [26], on the K-MAX problem. Online Submodular Maximization. First we briefly describe the online submodular maximization problem considered in [26] and the algorithm therein. At the beginning, an oblivious adversary sets a sequence of submodular functions f1 , f2 , . . . , fT on 2[m] , where ft will be used to determine the reward in the t-th round. In the t-th round, if the player selects a feasible super arm St , the reward will be ft (St ). This model covers the K-MAX problem as an instance: suppose (t) (t) X (t) = (X1 , . . . , Xm ) ∼ D is the outcome vector sampled in the t-th round, then the func(t) tion ft (S) = maxi∈S Xi is submodular and will determine the reward in the t-th round. We summarize the algorithm in Algorithm 8. It uses K copies of the Exp3√algorithm (see [3] for an introduction). For the K-MAX problem, Algorithm 8 achieves an O(K mT log m) upper bound on the (1 − 1/e)-approximation regret. Setup. We set m = 9 and K = 3, i.e., there are 9 arms in total and it is allowed to select at most 3 arms in each round. We compare the performance of SDCB/Lazy-SDCB and the online submodular maximization algorithm on four different distributions. Here we use the greedy algorithm Greedy-K-MAX (Algorithm 5) as the offline oracle. 9 In the exact version of a problem, we ask for a feasible set S such that total weight of S is exactly a given target value B. For example, in the exact spanning tree problem where each edge has an integer weight, we would like to find a spanning tree of weight exactly B. 28 Let Xi ∼ Di (i = 1, . . . , 9). We consider the following distributions. For all of them, the optimal super arm is S ∗ = {1, 2, 3}. • Distribution 1: All Di ’s have the same support {0, 0.2, 0.4, 0.6, 0.8, 1}. For i ∈ {1, 2, 3}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.1 and Pr[Xi = 1] = 0.5. For i ∈ {4, 5, 6, . . . , 9}, Pr[Xi = 0] = 0.5 and Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = Pr[Xi = 1] = 0.1. • Distribution 2: All Di ’s have the same support {0, 0.2, 0.4, 0.6, 0.8, 1}. For i ∈ {1, 2, 3}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.1 and Pr[Xi = 1] = 0.5. For i ∈ {4, 5, 6, . . . , 9}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.12 and Pr[Xi = 1] = 0.4. • Distribution 3: All Di ’s have the same support {0, 0.2, 0.4, 0.6, 0.8, 1}. For i ∈ {1, 2, 3}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.1 and Pr[Xi = 1] = 0.5. For i ∈ {4, 5, 6}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.12 and Pr[Xi = 1] = 0.4. For i ∈ {7, 8, 9}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.16 and Pr[Xi = 1] = 0.2. • Distribution 4: All Di ’s are continuous distributions on [0, 1]. For i ∈ {1, 2, 3}, Di is the uniform distribution on [0, 1]. For i ∈ {4, 5, 6, . . . , 9}, the probability density function (PDF) of Xi is 1.2 x ∈ [0, 0.5], f (x) = 0.8 x ∈ (0.5, 1]. These distributions represent several different scenarios. Distribution 1 is relatively “easy” because the suboptimal arms 4-9’s distribution is far away from arms 1-3’s distribution, whereas distribution 2 is “hard” since the distribution of arms 4-9 is close to the distribution of arms 1-3. In distribution 3, the distribution of arms 4-6 is close to the distribution of arms 1-3’s, while arms 7-9’s distribution is further away. Distribution 4 is an example of a group of continuous distributions for which Lazy-SDCB is more efficient than SDCB. We use SDCB for distributions 1-3, and Lazy-SDCB (with known time horizon) for distribution 4. Figure 1 shows the regrets of both SDCB and the online submodular maximization algorithm. We plot the 1-approximation regrets instead of the (1 − 1/e)-approximation regrets, since the greedy oracle usually performs much better than its (1 − 1/e)-approximation guarantee. We can see from Figure 1 that our algorithms achieve much lower regrets in all examples. 29 1000 SDCB Online Submodular Maximization 900 450 SDCB Online Submodular Maximization 800 400 700 350 600 Regret Regret 300 500 400 250 200 300 150 200 100 100 0 50 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 10000 Time 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 8000 9000 10000 Time (a) Distribution 1 (b) Distribution 2 600 600 Lazy-SDCB Online Submodular Maximization 500 500 400 400 Regret Regret SDCB Online Submodular Maximization 300 300 200 200 100 100 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 10000 Time 0 1000 2000 3000 4000 5000 6000 7000 Time (c) Distribution 3 (d) Distribution 4 Figure 1: Regrets of SDCB/Lazy-SDCB and Algorithm 8 on the K-MAX problem, for distributions 1-4. The regrets are averaged over 20 independent runs. 30	8
Published as a conference paper at ICLR 2018 C ONTINUOUS A DAPTATION VIA M ETA -L EARNING IN N ONSTATIONARY AND C OMPETITIVE E NVIRONMENTS Maruan Al-Shedivat∗ CMU Trapit Bansal UMass Amherst arXiv:1710.03641v2 [cs.LG] 23 Feb 2018 Igor Mordatch OpenAI Yura Burda OpenAI Ilya Sutskever OpenAI Pieter Abbeel UC Berkeley A BSTRACT The ability to continuously learn and adapt from limited experience in nonstationary environments is an important milestone on the path towards general intelligence. In this paper, we cast the problem of continuous adaptation into the learning-to-learn framework. We develop a simple gradient-based meta-learning algorithm suitable for adaptation in dynamically changing and adversarial scenarios. Additionally, we design a new multi-agent competitive environment, RoboSumo, and define iterated adaptation games for testing various aspects of continuous adaptation. We demonstrate that meta-learning enables significantly more efficient adaptation than reactive baselines in the few-shot regime. Our experiments with a population of agents that learn and compete suggest that meta-learners are the fittest. 1 I NTRODUCTION Recent progress in reinforcement learning (RL) has achieved very impressive results ranging from playing games (Mnih et al., 2015; Silver et al., 2016), to applications in dialogue systems (Li et al., 2016), to robotics (Levine et al., 2016). Despite the progress, the learning algorithms for solving many of these tasks are designed to deal with stationary environments. On the other hand, real-world is often nonstationary either due to complexity (Sutton et al., 2007), changes in the dynamics or the objectives in the environment over the life-time of a system (Thrun, 1998), or presence of multiple learning actors (Lowe et al., 2017; Foerster et al., 2017a). Nonstationarity breaks the standard assumptions and requires agents to continuously adapt, both at training and execution time, in order to succeed. Learning under nonstationary conditions is challenging. The classical approaches to dealing with nonstationarity are usually based on context detection (Da Silva et al., 2006) and tracking (Sutton et al., 2007), i.e., reacting to the already happened changes in the environment by continuously fine-tuning the policy. Unfortunately, modern deep RL algorithms, while able to achieve super-human performance on certain tasks, are known to be sample inefficient. Nevertheless, nonstationarity allows only for limited interaction before the properties of the environment change. Thus, it immediately puts learning into the few-shot regime and often renders simple fine-tuning methods impractical. A nonstationary environment can be seen as a sequence of stationary tasks, and hence we propose to tackle it as a multi-task learning problem (Caruana, 1998). The learning-to-learn (or meta-learning) approaches (Schmidhuber, 1987; Thrun & Pratt, 1998) are particularly appealing in the few-shot regime, as they produce flexible learning rules that can generalize from only a handful of examples. Meta-learning has shown promising results in the supervised domain and have gained a lot of attention from the research community recently (e.g., Santoro et al., 2016; Ravi & Larochelle, 2016). In this paper, we develop a gradient-based meta-learning algorithm similar to (Finn et al., 2017b) and suitable for continuous adaptation of RL agents in nonstationary environments. More concretely, our agents meta-learn to anticipate the changes in the environment and update their policies accordingly. While virtually any changes in an environment could induce nonstationarity (e.g., changes in the physics or characteristics of the agent), environments with multiple agents are particularly challenging ∗ Correspondence: maruan.alshedivat.com. Work done while MA and TB interned at OpenAI. 1 Published as a conference paper at ICLR 2018 due to complexity of the emergent behavior and are of practical interest with applications ranging from multiplayer games (Peng et al., 2017) to coordinating self-driving fleets Cao et al. (2013). Multi-agent environments are nonstationary from the perspective of any individual agent since all actors are learning and changing concurrently (Lowe et al., 2017). In this paper, we consider the problem of continuous adaptation to a learning opponent in a competitive multi-agent setting. To this end, we design RoboSumo—a 3D environment with simulated physics that allows pairs of agents to compete against each other. To test continuous adaptation, we introduce iterated adaptation games—a new setting where a trained agent competes against the same opponent for multiple rounds of a repeated game, while both are allowed to update their policies and change their behaviors between the rounds. In such iterated games, from the agent’s perspective, the environment changes from round to round, and the agent ought to adapt in order to win the game. Additionally, the competitive component of the environment makes it not only nonstationary but also adversarial, which provides a natural training curriculum and encourages learning robust strategies (Bansal et al., 2018). We evaluate our meta-learning agents along with a number of baselines on a (single-agent) locomotion task with handcrafted nonstationarity and on iterated adaptation games in RoboSumo. Our results demonstrate that meta-learned strategies clearly dominate other adaptation methods in the few-shot regime in both single- and multi-agent settings. Finally, we carry out a large-scale experiment where we train a diverse population of agents with different morphologies, policy architectures, and adaptation methods, and make them interact by competing against each other in iterated games. We evaluate the agents based on their TrueSkills (Herbrich et al., 2007) in these games, as well as evolve the population as whole for a few generations—the agents that lose disappear, while the winners get duplicated. Our results suggest that the agents with meta-learned adaptation strategies end up being the fittest. Videos that demonstrate adaptation behaviors are available at https://goo.gl/tboqaN. 2 R ELATED W ORK The problem of continuous adaptation considered in this work is a variant of continual learning (Ring, 1994; 1997) and is related to lifelong (Thrun & Pratt, 1998; Silver et al., 2013) and neverending (Mitchell et al., 2015) learning. Life-long learning systems aim at solving multiple tasks sequentially by efficiently transferring and utilizing knowledge from already learned tasks to new tasks while minimizing the effect of catastrophic forgetting (McCloskey & Cohen, 1989). Neverending learning is concerned with mastering a fixed set of tasks in iterations, where the set keeps growing and the performance on all the tasks in the set keeps improving from iteration to iteration. The scope of continuous adaptation is narrower and more precise. While life-long and never-ending learning settings are defined as general multi-task problems (Silver et al., 2013; Mitchell et al., 2015), continuous adaptation targets to solve a single but nonstationary task or environment. The nonstationarity in the former two problems exists and is dictated by the selected sequence of tasks. In the latter case, we assume that nonstationarity is caused by some underlying dynamics in the properties of a given task in the first place (e.g., changes in the behavior of other agents in a multiagent setting). Finally, in the life-long and never-ending scenarios the boundary between training and execution is blurred as such systems constantly operate in the training regime. Continuous adaptation, on the other hand, expects a (potentially trained) agent to adapt to the changes in the environment at execution time under the pressure of limited data or interaction experience between the changes1 . Nonstationarity of multi-agent environments is a well known issue that has been extensively studied in the context of learning in simple multi-player iterated games (such as rock-paper-scissors) where each episode is one-shot interaction (Singh et al., 2000; Bowling, 2005; Conitzer & Sandholm, 2007). In such games, discovering and converging to a Nash equilibrium strategy is a success for the learning agents. Modeling and exploiting opponents (Zhang & Lesser, 2010; Mealing & Shapiro, 2013) or even their learning processes (Foerster et al., 2017b) is advantageous as it improves convergence or helps to discover equilibria of certain properties (e.g., leads to cooperative behavior). In contrast, each episode in RoboSumo consists of multiple steps, happens in continuous time, and requires learning a good intra-episodic controller. Finding Nash equilibria in such a setting is hard. Thus, fast adaptation becomes one of the few viable strategies against changing opponents. 1 The limited interaction aspect of continuous adaptation makes the problem somewhat similar to the recently proposed life-long few-shot learning (Finn et al., 2017a). 2 Published as a conference paper at ICLR 2018 Policy ... τφ τθ ... T φi−1 φi+1 φi τi−1 τi τi+1 Ti−1 Ti Ti+1 Trajectory + + ... deterministic (a) Loss ... Intermediate steps φ θ (b) stochastic gradient (c) Fig. 1: (a) A probabilistic model for MAML in a multi-task RL setting. The task, T , the policies, π, and the trajectories, τ , are all random variables with dependencies encoded in the edges of the given graph. (b) Our extended model suitable for continuous adaptation to a task changing dynamically due to non-stationarity of the environment. Policy and trajectories at a previous step are used to construct a new policy for the current step. (c) Computation graph for the meta-update from φi to φi+1 . Boxes represent replicas of the policy graphs with the specified parameters. The model is optimized via truncated backpropagation through time starting from LTi+1 . Our proposed method for continuous adaptation follows the general meta-learning paradigm (Schmidhuber, 1987; Thrun & Pratt, 1998), i.e., it learns a high-level procedure that can be used to generate a good policy each time the environment changes. There is a wealth of work on meta-learning, including methods for learning update rules for neural models that were explored in the past (Bengio et al., 1990; 1992; Schmidhuber, 1992), and more recent approaches that focused on learning optimizers for deep networks (Hochreiter et al., 2001; Andrychowicz et al., 2016; Li & Malik, 2016; Ravi & Larochelle, 2016), generating model parameters (Ha et al., 2016; Edwards & Storkey, 2016; Al-Shedivat et al., 2017), learning task embeddings (Vinyals et al., 2016; Snell et al., 2017) including memory-based approaches (Santoro et al., 2016), learning to learn implicitly via RL (Wang et al., 2016; Duan et al., 2016), or simply learning a good initialization (Finn et al., 2017b). 3 M ETHOD The problem of continuous adaptation in nonstationary environments immediately puts learning into the few-shot regime: the agent must learn from only limited amount of experience that it can collect before its environment changes. Therefore, we build our method upon the previous work on gradient-based model-agnostic meta-learning (MAML) that has been shown successful in the fewshot settings (Finn et al., 2017b). In this section, we re-derive MAML for multi-task reinforcement learning from a probabilistic perspective (cf. Grant et al., 2018), and then extend it to dynamically changing tasks. 3.1 A PROBABILISTIC VIEW OF MODEL - AGNOSTIC META - LEARNING (MAML) Assume that we are given a distribution over tasks, D(T ), where each task, T , is a tuple: T := (LT , PT (x) , PT (xt+1 \| xt , at ) , H) (1) LT is a task-specific loss function that maps a trajectory, τ := (x0 , a1 , x1 , R1 , . . . , aH , xH , RH ) ∈ T , to a loss value, i.e., LT : T 7→ R; PT (x) and PT (xt+1 \| xt , at ) define the Markovian dynamics of the environment in task T ; H denotes the horizon; observations, xt , and actions, at , are elements (typically, vectors) of the observation space, X , and action space, A, respectively. The loss of a PH trajectory, τ , is the negative cumulative reward, LT (τ ) := − t=1 Rt . The goal of meta-learning is to find a procedure which, given access to a limited experience on a task sampled from D(T ), can produce a good policy for solving it. More formally, after querying K trajectories from a task T ∼ D(T ) under policy πθ , denoted τθ1:K , we would like to construct a new, task-specific policy, πφ , that would minimize the expected subsequent loss on the task T . In particular, MAML constructs parameters of the task-specific policy, φ, using gradient of LT w.r.t. θ: K 1 X φ := θ − α∇θ LT τθ1:K , where LT τθ1:K := LT (τθk ), and τθk ∼ PT (τ \| θ) K k=1 3 (2) Published as a conference paper at ICLR 2018 Algorithm 1 Meta-learning at training time. Algorithm 2 Adaptation at execution time. input Distribution over pairs of tasks, P(Ti , Ti+1 ), learning rate, β. 1: Randomly initialize θ and α. 2: repeat 3: Sample a batch of task pairs, {(Ti , Ti+1 )}n i=1 . 4: for all task pairs (Ti , Ti+1 ) in the batch do 5: Sample traj. τθ1:K from Ti using πθ . Compute φ = φ(τθ1:K , θ, α) as given in (7). 6: 7: Sample traj. τφ from Ti+1 using πφ . 8: end for 9: Compute ∇θ LTi ,Ti+1 and ∇α LTi ,Ti+1 using τθ1:K and τφ as given in (8). 10: Update θ ← θ + β∇θ LT (θ, α). 11: Update α ← α + β∇α LT (θ, α). 12: until Convergence output Optimal θ∗ and α∗ . input A stream of tasks, T1 , T2 , T3 , . . . . 1: Initialize φ = θ. 2: while there are new incoming tasks do 3: Get a new task, Ti , from the stream. 4: Solve Ti using πφ policy. 1:K 5: While solving Ti , collect trajectories, τi,φ . 1:K ∗ ∗ 6: Update φ ← φ(τi,φ , θ , α ) using importance-corrected meta-update as in (9). 7: end while Policy parameter space We call (2) the adaptation update with a step α. The adaptation update is parametrized by θ, which we optimize by minimizing the expected loss over the distribution of tasks, D(T )—the meta-loss: min ET ∼D(T ) [LT (θ)] , where LT (θ) := Eτθ1:K ∼PT (τ \|θ) Eτφ ∼PT (τ \|φ) LT (τφ ) \| τθ1:K , θ (3) θ where τθ and τφ are trajectories obtained under πθ and πφ , respectively. In general, we can think of the task, trajectories, and policies, as random variables (Fig. 1a), where φ is generated from some conditional distribution PT (φ\| θ, τ1:k ). The meta-update(2) is equivalent to PK 1 2 assuming the delta distribution, PT (φ \| θ, τ1:k ) := δ θ − α∇θ K k=1 LT (τk ) . To optimize (3), we can use the policy gradient method (Williams, 1992), where the gradient of LT is as follows: " " ## K X k ∇θ LT (θ) = Eτθ1:K ∼PT (τ \|θ) LT (τφ ) ∇θ log πφ (τφ ) + ∇θ log πθ (τθ ) (4) τφ ∼PT (τ \|φ) k=1 The expected loss on a task, LT , can be optimized with trust-region policy (TRPO) (Schulman et al., 2015a) or proximal policy (PPO) (Schulman et al., 2017) optimization methods. For details and derivations please refer to Appendix A. 3.2 C ONTINUOUS ADAPTATION VIA META - LEARNING In the classical multi-task setting, we make no assumptions about the distribution of tasks, D(T ). When the environment is nonstationary, we can see it as a sequence of stationary tasks on a certain timescale where the tasks correspond to different dynamics of the environment. Then, D(T ) is defined by the environment changes, and the tasks become sequentially dependent. Hence, we would like to exploit this dependence between consecutive tasks and meta-learn a rule that keeps updating the policy in a way that minimizes the total expected loss encountered during the interaction with the changing environment. For instance, in the multi-agent setting, when playing against an opponent that changes its strategy incrementally (e.g., due to learning), our agent should ideally meta-learn to anticipate the changes and update its policy accordingly. In the probabilistic language, our nonstationary environment is equivalent to a distribution of tasks represented by a Markov chain (Fig. 1b). The goal is to minimize the expected loss over the chain of tasks of some length L: " L # X min EP(T0 ),P(Ti+1 \|Ti ) LTi ,Ti+1 (θ) (5) θ 2 i=1 Grant et al. (2018) similarly reinterpret adaptation updates (in non-RL settings) as Bayesian inference. 4 Published as a conference paper at ICLR 2018 Here, P(T0 ) and P(Ti+1 \| Ti ) denote the initial and the transition probabilities in the Markov chain of tasks. Note that (i) we deal with Markovian dynamics on two levels of hierarchy, where the upper level is the dynamics of the tasks and the lower level is the MDPs that represent particular tasks, and (ii) the objectives, LTi ,Ti+1 , will depend on the way the meta-learning process is defined. Since we are interested in adaptation updates that are optimal with respect to the Markovian transitions between the tasks, we define the meta-loss on a pair of consecutive tasks as follows: h i 1:K 1:K ∼P LTi ,Ti+1 (θ) := Eτi,θ Eτi+1,φ ∼PTi+1 (τ \|φ) LTi+1 (τi+1,φ ) \| τi,θ ,θ (6) Ti (τ \|θ) 1:K The principal difference between the loss in (3) and (6) is that trajectories τi,θ come from the current task, Ti , and are used to construct a policy, πφ , that is good for the upcoming task, Ti+1 . Note that even though the policy parameters, φi , are sequentially dependent (Fig. 1b), in (6) we always start from the initial parameters, θ 3 . Hence, optimizing LTi ,Ti+1 (θ) is equivalent to truncated backpropagation through time with a unit lag in the chain of tasks. To construct parameters of the policy for task Ti+1 , we start from θ and do multiple4 meta-gradient steps with adaptive step sizes as follows (assuming the number of steps is M ): φ0i := θ, τθ1:K ∼ PTi (τ \| θ), m−1 1:K m−1 LT φm := φ − α ∇ τ , m = 1, . . . , M − 1, m−1 m i i i (7) φi i,φ i −1 1:K φi+1 := φM − αM ∇φM −1 LTi τi,φ M −1 i i i where {αm }M m=1 is a set of meta-gradient step sizes that are optimized jointly with θ. The computation graph for the meta-update is given in Fig. 1c. The expression for the policy gradient is the same as in (4) but with the expectation is now taken w.r.t. to both Ti and Ti+1 : ∇θ,α LTi ,Ti+1 (θ, α) = " " ## K X (8) k 1:K E τi,θ LTi+1 (τi+1,φ ) ∇θ,α log πφ (τi+1,φ ) + ∇θ log πθ (τi,θ ) ∼PT (τ \|θ) i τi+1,φ ∼PTi+1 (τ \|φ) k=1 More details and the analog of the policy gradient theorem for our setting are given in Appendix A. Note that computing adaptation updates requires interacting with the environment under πθ while computing the meta-loss, LTi ,Ti+1 , requires using πφ , and hence, interacting with each task in the sequence twice. This is often impossible at execution time, and hence we use slightly different algorithms at training and execution times. Meta-learning at training time. Once we have access to a distribution over pairs of consecutive tasks5 , P(Ti−1 , Ti ), we can meta-learn the adaptation updates by optimizing θ and α jointly with a gradient method, as given in Algorithm 1. We use πθ to collect trajectories from Ti and πφ when interacting with Ti+1 . Intuitively, the algorithm is searching for θ and α such that the adaptation update (7) computed on the trajectories from Ti brings us to a policy, πφ , that is good for solving Ti+1 . The main assumption here is that the trajectories from Ti contain some information about Ti+1 . Note that we treat adaptation steps as part of the computation graph (Fig. 1c) and optimize θ and α via backpropagation through the entire graph, which requires computing second order derivatives. Adaptation at execution time. Note that to compute unbiased adaptation gradients at training time, we have to collect experience in Ti using πθ . At test time, due to environment nonstationarity, we usually do not have the luxury to access to the same task multiple times. Thus, we keep acting according to πφ and re-use past experience to compute updates of φ for each new incoming task (see Algorithm 2). To adjust for the fact that the past experience was collected under a policy different from πθ , we use importance weight correction. In case of single step meta-update, we have: K 1 X πθ (τ k ) φi := θ − α ∇θ LTi−1 (τ k ), τ 1:K ∼ PTi−1 (τ \| φi−1 ), (9) K πφi−1 (τ k ) k=1 3 This is due to stability considerations. We find empirically that optimization over sequential updates from φi to φi+1 is unstable, often tends to diverge, while starting from the same initialization leads to better behavior. 4 Empirically, it turns out that constructing φ via multiple meta-gradient steps (between 2 and 5) with adaptive step sizes tends yield better results in practice. 5 Given a sequences of tasks generated by a nonstationary environment, T1 , T2 , T3 , . . . , TL , we use the set of all pairs of consecutive tasks, {(Ti−1 , Ti )}L i=1 , as the training distribution. 5 Published as a conference paper at ICLR 2018 (a) (b) (c) Fig. 2: (a) The three types of agents used in experiments. The robots differ in the anatomy: the number of legs, their positions, and constraints on the thigh and knee joints. (b) The nonstationary locomotion environment. The torques applied to red-colored legs are scaled by a dynamically changing factor. (c) RoboSumo environment. where πφi−1 and πφi are used to rollout from Ti−1 and Ti , respectively. Extending importance weight correction to multi-step updates is straightforward and requires simply adding importance weights to each of the intermediate steps in (7). 4 E NVIRONMENTS We have designed a set of environments for testing different aspects of continuous adaptation methods in two scenarios: (i) simple environments that change from episode to episode according to some underlying dynamics, and (ii) a competitive multi-agent environment, RoboSumo, that allows different agents to play sequences of games against each other and keep adapting to incremental changes in each other’s policies. All our environments are based on MuJoCo physics simulator (Todorov et al., 2012), and all agents are simple multi-leg robots, as shown in Fig. 2a. 4.1 DYNAMIC First, we consider the problem of robotic locomotion in a changing environment. We use a six-leg agent (Fig. 2b) that observes the absolute position and velocity of its body, the angles and velocities of its legs, and it acts by applying torques to its joints. The agent is rewarded proportionally to its moving speed in a fixed direction. To induce nonstationarity, we select a pair of legs of the agent and scale down the torques applied to the corresponding joints by a factor that linearly changes from 1 to 0 over the course of 7 episodes. In other words, during the first episode all legs are fully functional, while during the last episode the agent has two legs fully paralyzed (even though the policy can generate torques, they are multiplied by 0 before being passed to the environment). The goal of the agent is to learn to adapt from episode to episode by changing its gait so that it is able to move with a maximal speed in a given direction despite the changes in the environment (cf. Cully et al., 2015). Also, there are 15 ways to select a pair of legs of a six-leg creature which gives us 15 different nonstationary environments. This allows us to use a subset of these environments for training and a separate held out set for testing. The training and testing procedures are described in the next section. 4.2 C OMPETITIVE Our multi-agent environment, RoboSumo, allows agents to compete in the 1-vs-1 regime following the standard sumo rules6 . We introduce three types of agents, Ant, Bug, and Spider, with different anatomies (Fig. 2a). During the game, each agent observes positions of itself and the opponent, its own joint angles, the corresponding velocities, and the forces exerted on its own body (i.e., equivalent of tactile senses). The action spaces are continuous. Iterated adaptation games. To test adaptation, we define the iterated adaptation game (Fig. 3)—a game between a pair of agents that consists of K rounds each of which consists of one or more fixed length episodes (500 time steps each). The outcome of each round is either win, loss, or draw. The agent that wins the majority of rounds (with at least 5% margin) is declared the winner of the game. There are two distinguishing aspects of our setup: First, the agents are trained either via pure self-play or versus opponents from a fixed training collection. At test time, they face a new opponent from a testing collection. Second, the agents are allowed to learn (or adapt) at test time. In particular, an 6 To win, the agent has to push the opponent out of the ring or make the opponent’s body touch the ground. 6 Published as a conference paper at ICLR 2018 Round 1 Round 2 Round 3 Round K version 1 version 2 version 3 version K Agent: Episodes: Opponent: Fig. 3: An agent competes with an opponent in an iterated adaptation games that consist of multi-episode rounds. The agent wins a round if it wins the majority of episodes (wins and losses illustrated with color). Both the agent and its opponent may update their policies from round to round (denoted by the version number). agent should exploit the fact that it plays against the same opponent multiple consecutive rounds and try to adjust its behavior accordingly. Since the opponent may also be adapting, the setup allows to test different continuous adaptation strategies, one versus the other. Reward shaping. In RoboSumo, rewards are naturally sparse: the winner gets +2000, the loser is penalized for -2000, and in case of a draw both opponents receive -1000 points. To encourage fast learning at the early stages of training, we shape the rewards given to agents in the following way: the agent (i) gets reward for staying closer to the center of the ring, for moving towards the opponent, and for exerting forces on the opponent’s body, and (ii) gets penalty inversely proportional to the opponent’s distance to the center of the ring. At test time, the agents continue having access to the shaped reward as well and may use it to update their policies. Throughout our experiments, we use discounted rewards with the discount factor, γ = 0.995. More details are in Appendix D.2. Calibration. To study adaptation, we need a well-calibrated environment in which none of the agents has an initial advantage. To ensure the balance, we increased the mass of the weaker agents (Ant and Spider) such that the win rates in games between one agent type versus the other type in the non-adaptation regime became almost equal (for details on calibration see Appendix D.3). 5 E XPERIMENTS Our goal is to test different adaptation strategies in the proposed nonstationary RL settings. However, it is known that the test-time behavior of an agent may highly depend on a variety of factors besides the chosen adaptation method, including training curriculum, training algorithm, policy class, etc. Hence, we first describe the precise setup that we use in our experiments to eliminate irrelevant factors and focus on the effects of adaptation. Most of the low-level details are deferred to appendices. Video highlights of our experiments are available at https://goo.gl/tboqaN. 5.1 T HE SETUP Policies. We consider 3 types of policy networks: (i) a 2-layer MLP, (ii) embedding (i.e., 1 fully-connected layer replicated across the time dimension) followed by a 1-layer LSTM, and (iii) RL2 (Duan et al., 2016) of the same architecture as (ii) which additionally takes previous reward and done signals as inputs at each step, keeps the recurrent state throughout the entire interaction with a given environment (or an opponent), and resets the state once the latter changes. For advantage functions, we use networks of the same structure as for the corresponding policies and have no parameter sharing between the two. Our meta-learning agents use the same policy and advantage function structures as the baselines and learn a 3-step meta-update with adaptive step sizes as given in (7). Illustrations and details on the architectures are given in Appendix B. Meta-learning. We compute meta-updates via gradients of the negative discounted rewards received during a number of previous interactions with the environment. At training time, meta-learners interact with the environment twice, first using the initial policy, πθ , and then the meta-updated policy, πφ . At test time, the agents are limited to interacting with the environment only once, and hence always act according to πφ and compute meta-updates using importance-weight correction (see Sec. 3.2 and Algorithm 2). Additionally, to reduce the variance of the meta-updates at test time, the agents store the experience collected during the interaction with the test environment (and the corresponding importance weights) into the experience buffer and keep re-using that experience 7 Published as a conference paper at ICLR 2018 Total episodic reward Back two legs Middle two legs Front two legs Policy + adaptation method 1000 1000 1000 500 500 0 500 0 0 1 3 5 7 MLP MLP + PPO-tracking MLP + meta-updates LSTM LSTM + PPO-tracking LSTM + meta-updates RL2 1 3 5 7 Consecutive episodes 1 3 5 7 Fig. 4: Episodic rewards for 7 consecutive episodes in 3 held out nonstationary locomotion environments. To evaluate adaptation strategies, we ran each of them in each environment for 7 episodes followed by a full reset of the environment, policy, and meta-updates (repeated 50 times). Shaded regions are 95% confidence intervals. Best viewed in color. to update πφ as in (7). The size of the experience buffer is fixed to 3 episodes for nonstationary locomotion and 75 episodes for RoboSumo. More details are given in Appendix C.1. Adaptation baselines. We consider the following three baseline strategies: (i) naive (or no adaptation), (ii) implicit adaptation via RL2 , and (iii) adaptation via tracking (Sutton et al., 2007) that keeps doing PPO updates at execution time. Training in nonstationary locomotion. We train all methods on the same collection of nonstationary locomotion environments constructed by choosing all possible pairs of legs whose joint torques are scaled except 3 pairs that are held out for testing (i.e., 12 training and 3 testing environments for the six-leg creature). The agents are trained on the environments concurrently, i.e., to compute a policy update, we rollout from all environments in parallel and then compute, aggregate, and average the gradients (for details, see Appendix C.2). LSTM policies retain their state over the course of 7 episodes in each environment. Meta-learning agents compute meta-updates for each nonstationary environment separately. Training in RoboSumo. To ensure consistency of the training curriculum for all agents, we first pre-train a number of policies of each type for every agent type via pure self-play with the PPO algorithm (Schulman et al., 2017; Bansal et al., 2018). We snapshot and save versions of the pretrained policies at each iteration. This lets us train other agents to play against versions of the pre-trained opponents at various stages of mastery. Next, we train the baselines and the meta-learning agents against the pool of pre-trained opponents7 concurrently. At each iteration k we (a) randomly select an opponent from the training pool, (b) sample a version of the opponent’s policy to be in [1, k] (this ensures that even when the opponent is strong, sometimes an undertrained version is selected which allows the agent learn to win at early stages), and (c) rollout against that opponent. All baseline policies are trained with PPO; meta-learners also used PPO as the outer loop for optimizing θ and α parameters. We retain the states of the LSTM policies over the course of interaction with the same version of the same opponent and reset it each time the opponent version is updated. Similarly to the locomotion setup, meta-learners compute meta-updates for each opponent in the training pool separately. A more detailed description of the distributed training is given in Appendix C.2. Experimental design. We design our experiments to answer the following questions: • When the interaction with the environment before it changes is strictly limited to one or very few episodes, what is the behavior of different adaptation methods in nonstationary locomotion and competitive multi-agent environments? • What is the sample complexity of different methods, i.e., how many episodes is required for a method to successfully adapt to the changes? We test this by controlling the amount of experience the agent is allowed to get form the same environment before it changes. 7 In competitive multi-agent environments, besides self-play, there are plenty of ways to train agents, e.g., train them in pairs against each other concurrently, or randomly match and switch opponents each few iterations. We found that concurrent training often leads to an unbalanced population of agents that have been trained under vastly different curricula and introduces spurious effects that interfere with our analysis of adaptation. Hence, we leave the study of adaptation in naturally emerging curricula in multi-agent settings to the future work. 8 Published as a conference paper at ICLR 2018 Agent: Opponent: 0.6 Win rate RL2 Ant Ant LSTM + PPO-tracking Opponent: 0.6 Bug 0.2 0.2 0 25 Agent: 0.8 Win rate Spider 0.4 0.2 50 75 100 0.0 0 25 50 75 Consecutive rounds RL2 Bug Opponent: 0.8 Ant 100 0 LSTM + PPO-tracking Opponent: 0.6 0.6 0.4 0.4 0.4 0.2 0 25 Agent: 50 75 25 50 75 Consecutive rounds RL2 Ant Bug 0.2 0.2 0.2 75 100 0 25 50 75 Consecutive rounds 25 100 Spider 100 50 75 100 LSTM + meta-updates Opponent: 0.6 0.4 50 Opponent: 0 0.4 25 75 LSTM + meta-updates LSTM + PPO-tracking Opponent: 0.6 100 0.4 0 50 0.2 0 Spider Opponent: 0.6 100 25 0.8 Bug 0.6 0.2 Win rate Opponent: 0.6 0.4 0.4 LSTM + meta-updates 0 25 50 Spider 75 100 Fig. 5: Win rates for different adaptation strategies in iterated games versus 3 different pre-trained opponents. At test time, both agents and opponents started from versions 700. Opponents’ versions were increasing with each consecutive round as if they were learning via self-play, while agents were allowed to adapt only from the limited experience with a given opponent. Each round consisted of 3 episodes. Each iterated game was repeated 100 times; shaded regions denote bootstrapped 95% confidence intervals; no smoothing. Best viewed in color. Additionally, we ask the following questions specific to the competitive multi-agent setting: • Given a diverse population of agents that have been trained under the same curriculum, how do different adaptation methods rank in a competition versus each other? • When the population of agents is evolved for several generations—such that the agents interact with each other via iterated adaptation games, and those that lose disappear while the winners get duplicated—what happens with the proportions of different agents in the population? 5.2 A DAPTATION IN THE FEW- SHOT REGIME AND SAMPLE COMPLEXITY Few-shot adaptation in nonstationary locomotion environments. Having trained baselines and meta-learning policies as described in Sec. 5.1, we selected 3 testing environments that corresponded to disabling 3 different pairs of legs of the six-leg agent: back, middle, and front legs. The results are presented on Fig. 4. Three observations: First, during the very first episode, the meta-learned initial policy, πθ? , turns out to be suboptimal for the task (it underperforms compared to other policies). However, after 1-2 episodes (and environment changes), it starts performing on par with other policies. Second, by the 6th and 7th episodes, meta-updated policies perform much better than the rest. Note that we use 3 gradient meta-updates for the adaptation of the meta-learners; the meta-updates are computed based on experience collected during the previous 2 episodes. Finally, tracking is not able to improve upon the baseline without adaptation and sometimes leads to even worse results. Adaptation in RoboSumo under the few-shot constraint. To evaluate different adaptation methods in the competitive multi-agent setting consistently, we consider a variation of the iterated adaptation game, where changes in the opponent’s policies at test time are pre-determined but unknown to the 9 Published as a conference paper at ICLR 2018 Ant vs Ant Win rate 0.6 Bug vs Bug 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0 25 50 75 0.2 0 Spider vs Spider 0.6 25 50 75 Episodes per round 0.2 0 25 50 RL2 LSTM + PPO-tracking LSTM + meta-updates No adaptation 75 Fig. 6: The effect of increased number of episodes per round in the iterated games versus a learning opponent. agents. In particular, we pre-train 3 opponents (1 of each type, Fig. 2a) with LSTM policies with PPO via self-play (the same way as we pre-train the training pool of opponents, see Sec. 5.1) and snapshot their policies at each iteration. Next, we run iterated games between our trained agents that use different adaptation algorithms versus policy snapshots of the pre-trained opponents. Crucially, the policy version of the opponent keeps increasing from round to round as if it was training via self-play8 . The agents have to keep adapting to increasingly more competent versions of the opponent (see Fig. 3). This setup allows us to test different adaptation strategies consistently against the same learning opponents. The results are given on Fig. 5. We note that meta-learned adaptation strategies, in most cases, are able to adapt and improve their win-rates within about 100 episodes of interaction with constantly improving opponents. On the other hand, performance of the baselines often deteriorates during the rounds of iterated games. Note that the pre-trained opponents were observing 90 episodes of self-play per iteration, while the agents have access to only 3 episodes per round. Sample complexity of adaptation in RoboSumo. Meta-learning helps to find an update suitable for fast or few-shot adaptation. However, how do different adaptation methods behave when more experience is available? To answer this question, we employ the same setup as previously and vary the number of episodes per round in the iterated game from 3 to 90. Each iterated game is repeated 20 times, and we measure the win-rates during the last 25 rounds of the game. The results are presented on Fig. 6. When the number of episodes per round goes above 50, adaptation via tracking technically turns into “learning at test time,” and it is able to learn to compete against the self-trained opponents that it has never seen at training time. The meta-learned adaptation strategy performed near constantly the same in both few-shot and standard regimes. This suggests that the meta-learned strategy acquires a particular bias at training time that allows it to perform better from limited experience but also limits its capacity of utilizing more data. Note that, by design, the meta-updates are fixed to only 3 gradient steps from θ? with step-sizes α? (learned at training), while tracking keeps updating the policy with PPO throughout the iterated game. Allowing for meta-updates that become more flexible with the availability of data can help to overcome this limitation. We leave this to future work. 5.3 E VALUATION ON THE POPULATION - LEVEL Combining different adaptation strategies with different policies and agents of different morphologies puts us in a situation where we have a diverse population of agents which we would like to rank according to the level of their mastery in adaptation (or find the “fittest”). To do so, we employ TrueSkill (Herbrich et al., 2007)—a metric similar to the ELO rating, but more popular in 1-vs-1 competitive video-games. In this experiment, we consider a population of 105 trained agents: 3 agent types, 7 different policy and adaptation combinations, and 5 different stages of training (from 500 to 2000 training iterations). First, we assume that the initial distribution of any agent’s skill is N (25, 25/3) and the default distance that guarantees about 76% of winning, β = 4.1667. Next, we randomly generate 1000 matches between pairs of opponents and let them adapt while competing with each other in 100-round iterated adaptation games (states of the agents are reset before each game). After each game, we 8 At the beginning of the iterated game, both agents and their opponent start from version 700, i.e., from the policy obtained after 700 iterations (PPO epochs) of learning to ensure that the initial policy is reasonable. 10 Published as a conference paper at ICLR 2018 record the outcome and updated our belief about the skill of the corresponding agents using the TrueSkill algorithm9 . The distributions of the skill for the agents of each type after 1000 iterated adaptation games between randomly selected players from the pool are visualized in Fig. 7. TrueSkill 40 Sub-population: Ants Sub-population: 40 Bugs 30 30 30 20 20 20 MLP LSTM MLP no adaptation Sub-population: 40 LSTM PPO-tracking MLP meta-updates RL2 Spiders LSTM Fig. 7: TrueSkill for the top-performing MLP- and LSTM-based agents. TrueSkill was computed based on outcomes (win, loss, or draw) in 1000 iterated adaptation games (100 consecutive rounds per game, 3 episodes per round) between randomly selected pairs of opponents from a population of 105 pre-trained agents. There are a few observations we can make: First, recurrent policies were dominant. Second, adaptation via RL2 tended to perform equally or a little worse than plain LSTM with or without tracking in this setup. Finally, agents that meta-learned adaptation rules at training time, consistently demonstrated higher skill scores in each of the categories corresponding to different policies and agent types. Proportion (%) Finally, we enlarge the population from 105 to 1050 agents by duplicating each of them 10 times and evolve it (in the “natural selection” sense) for several generations as follows. Initially, we start with a balanced population of different creatures. Next, we randomly match 1000 pairs of agents, make them play iterated adaptation games, remove the agents that lost from the population and duplicate the winners. The same process is repeated 10 times. The result is presented in Fig 8. We see that many agents quickly disappear form initially uniform population and the meta-learners end up dominating. Policy + adaptation 100 75 50 25 0 MLP MLP + PPO-tracking MLP + meta-updates LSTM LSTM + PPO-tracking LSTM + meta-updates RL2 Spiders Bugs Ants 0 1 2 3 4 5 6 Creature generation (#) 7 8 9 10 Fig. 8: Evolution of a population of 1050 agents for 10 generations. Best viewed in color. 6 C ONCLUSION AND F UTURE D IRECTIONS In this work, we proposed a simple gradient-based meta-learning approach suitable for continuous adaptation in nonstationary environments. The key idea of the method is to regard nonstationarity as a sequence of stationary tasks and train agents to exploit the dependencies between consecutive tasks such that they can handle similar nonstationarities at execution time. We applied our method to nonstationary locomotion and within a competitive multi-agent setting. For the latter, we designed the RoboSumo environment and defined iterated adaptation games that allowed us to test various aspects of adaptation strategies. In both cases, meta-learned adaptation rules were more efficient than the baselines in the few-shot regime. Additionally, agents that meta-learned to adapt demonstrated the highest level of skill when competing in iterated games against each other. The problem of continuous adaptation in nonstationary and competitive environments is far from being solved, and this work is the first attempt to use meta-learning in such setup. Indeed, our meta-learning algorithm has a few limiting assumptions and design choices that we have made mainly due to computational considerations. First, our meta-learning rule is to one-step-ahead update of the 9 We used an implementation from http://trueskill.org/. 11 Published as a conference paper at ICLR 2018 policy and is computationally similar to backpropagation through time with a unit time lag. This could potentially be extended to fully recurrent meta-updates that take into account the full history of interaction with the changing environment. Additionally, our meta-updates were based on the gradients of a surrogate loss function. While such updates explicitly optimized the loss, they required computing second order derivatives at training time, slowing down the training process by an order of magnitude compared to baselines. Utilizing information provided by the loss but avoiding explicit backpropagation through the gradients would be more appealing and scalable. 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Machine learning, 8(3-4):229–256, 1992. Chongjie Zhang and Victor R Lesser. Multi-agent learning with policy prediction. In AAAI, 2010. 14 Published as a conference paper at ICLR 2018 A D ERIVATIONS AND THE POLICY GRADIENT THEOREM In this section, we derive the policy gradient update for MAML as give in (4) as well as formulate and equivalent of the policy gradient theorem (Sutton et al., 2000) in the learning-to-learn setting. Our derivation is not bound to a particular form of the adaptation update. In general, we are interested in meta-learning a procedure, fθ , parametrized by θ, which, given access to a limited experience on a task, can produce a good policy for solving it. Note that fθ is responsible for both collecting the initial experience and constructing the final policy for the given task. For example, in case of MAML (Finn et al., 2017b), fθ is represented by the initial policy, πθ , and the adaptation update rule (4) that produces πφ with φ := θ − α∇θ LT (τθ1:K ). More formally, after querying K trajectories, τθ1:K , we want to produce πφ that minimizes the expected loss w.r.t. the distribution over tasks: h i L(θ) := ET ∼D(T ) Eτθ1:K ∼pT (τ \|θ) Eτφ ∼pT (τ \|φ) LT (τφ ) \| τθ1:K (10) Note that the inner-most expectation is conditional on the experience, τθ1:K , which our meta-learning procedure, fθ , collects to produce a task-specific policy, πφ . Assuming that the loss LT (τθ1:K ) is linear in trajectories, and using linearity of expectations, we can drop the superscript 1 : K and denote the trajectory sampled under φθ for task Ti simply as τθ,i . At training time, we are given a finite sample of tasks from the distribution D(T ) and can search for θ̂ close to optimal by optimizing over the empirical distribution: θ̂ := argmin L̂(θ), where L̂(θ) := θ N h i 1 X Eτθ,i ∼pTi (τ \|θ) Eτφ,i ∼pTi (τ \|φ) [LTi (τφ,i ) \| τθ,i ] (11) N i=1 We re-write the objective function for task Ti in (11) more explicitly by expanding the expectations: h i LTi (θ) := Eτθ,i ∼pTi (τ \|θ) Eτφ,i ∼pTi (τ \|φ) [LTi (τφ,i ) \| τθ,i ] = Z (12) LTi (τφ,i ) PTi (τφ,i \| φ) PTi (φ \| θ, τθ,i ) PTi (τθ,i \| θ) dτφ,i dφ dτθ,i Trajectories, τφ,i and τθ,i , and parameters φ of the policy πφ can be thought as random variables that we marginalize out to construct the objective that depends on θ only. The adaptation update rule (4) assumes the following PTi (φ \| θ, τθ,i ): ! K 1 X k LTi (τθ,i ) (13) PTi (φ \| θ, τθ,i ) := δ θ − α∇θ K k=1 Note that by specifying PTi (φ \| θ, τθ,i ) differently, we may arrive at different meta-learning algorithms. After plugging (13) into (12) and integrating out φ, we get the following expected loss for task Ti as a function of θ: h i LTi (θ) = Eτθ,i ∼pTi (τ \|θ) Eτφ,i ∼pTi (τ \|φ) [LTi (τφ,i ) \| τθ,i ] = ! Z K (14) 1 X k LTi (τφ,i ) PTi τφ,i \| θ − α∇θ LTi (τθ,i ) PTi (τθ,i \| θ) dτφ,i dτθ,i K k=1 The gradient of (14) will take the following form: Z ∇θ LTi (θ) = [LTi (τφ,i ) ∇θ log PTi (τφ,i \| φ)] PTi (τφ,i \| φ) PTi (τθ,i \| θ) dτ dτθ,i + Z [LTi (τ ) ∇θ log PTi (τθ,i \| θ)] PTi (τ \| φ) PTi (τθ,i \| θ) dτ dτθ,i (15) 1:K where φ = φ(θ, τθ,i ) as given in (14). Note that the expression consists of two terms: the first term is the standard policy gradient w.r.t. the updated policy, πφ , while the second one is the policy 1:K gradient w.r.t. the original policy, πφ , that is used to collect τθ,i . If we were to omit marginalization 15 Published as a conference paper at ICLR 2018 1:K of τθ,i (as it was done in the original paper (Finn et al., 2017b)), the terms would disappear. Finally, the gradient can be re-written in a more succinct form: " " ## K X 1:K ∇θ LTi (θ) = Eτθ,i log πθ (τk ) (16) ∼PT (τ \|θ) LTi (τ ) ∇θ log πφ (τ ) + ∇θ i τ ∼PTi (τ \|φ) k=1 The update given in (16) is an unbiased estimate of the gradient as long as the loss LTi is simply the sum of discounted rewards (i.e., it extends the classical REINFORCE algorithm (Williams, 1992) to meta-learning). Similarly, we can define LTi that uses a value or advantage function and extend the policy gradient theorem Sutton et al. (2000) to make it suitable for meta-learning. Theorem 1 (Meta policy gradient theorem). For any MDP, gradient of the value function w.r.t. θ takes the following form: ∇θ VTθ (x0 ) = " # X φ X ∂πφ (a \| x) φ Eτ1:K ∼pT (τ \|θ) dT (x) QT (a, x) + ∂θ (17) x a " ! # K X ∂ X log πθ (τk ) πφ (a \| x0 )QφT (a, x0 ) , Eτ1:K ∼pT (τ \|θ) ∂θ a k=1 where dφT (x) is the stationary distribution under policy πφ . Proof. We define task-specific value functions under the generated policy, πφ , as follows: "H # X φ t VT (x0 ) = Eτ ∼pT (τ \|φ) γ RT (xt ) \| x0 , t=k " QφT (x0 , a0 ) = Eτ ∼pT (τ \|φ) H X t=k # (18) t γ RT (xt ) \| x0 , a0 , where the expectations are taken w.r.t. the dynamics of the environment of the given task, T , and the policy, πφ . Next, we need to marginalize out τ1:K : ## " "H X t θ VT (x0 ) = Eτ1:K ∼pT (τ \|θ) Eτ ∼pT (τ \|φ) γ RT (xt ) \| x0 , (19) t=k and after the gradient w.r.t. θ, we arrive at: ∇θ VTθ (x0 ) = # " X ∂πφ (a \| x0 ) φ ∂QφT (a, x0 ) Eτ1:K ∼pT (τ \|θ) QT (a, x0 ) + πφ (a \| x0 ) + ∂θ ∂θ a " K ! # X X ∂ φ Eτ1:K ∼pT (τ \|θ) log πθ (τk ) πφ (a \| x0 )QT (a, x0 ) , ∂θ a (20) k=1 where the first term is similar to the expression used in the original policy gradient theorem (Sutton et al., 2000) while the second one comes from differentiating trajectories τ1:K that depend on θ. Following Sutton et al. (2000), we unroll the derivative of the Q-function in the first term and arrive at the following final expression for the policy gradient: ∇θ VTθ (x0 ) = " # X φ X ∂πφ (a \| x) φ Eτ1:K ∼pT (τ \|θ) dT (x) QT (a, x) + ∂θ (21) x a " ! # K X ∂ X Eτ1:K ∼pT (τ \|θ) log πθ (τk ) πφ (a \| x0 )QφT (a, x0 ) ∂θ a k=1 Remark 1. The same theorem is applicable to the continuous setting with the only changes in the distributions used to compute expectations in (17) and (18). In particular, the outer expectation in (17) should be taken w.r.t. pTi (τ \| θ) while the inner expectation w.r.t. pTi+1 (τ \| φ). 16 Published as a conference paper at ICLR 2018 A.1 M ULTIPLE ADAPTATION GRADIENT STEPS All our derivations so far assumed single step gradient-based adaptation update. Experimentally, we found that the multi-step version of the update often leads to a more stable training and better test time performance. In particular, we construct φ via intermediate M gradient steps: φ0 := θ, τθ1:K ∼ PT (τ \| θ), φm := φm−1 − αm ∇φm−1 LT τφ1:K , m−1 φ := φM −1 − αM ∇φM −1 LT τφ1:K M −1 m = 1, . . . , M − 1, (22) where φm are intermediate policy parameters. Note that each intermediate step, m, requires interacting with the environment and sampling intermediate trajectories, τφ1:K m . To compute the policy gradient, we need to marginalize out all the intermediate random variables, πφm and τφ1:K m , m = 1, . . . , M . The objective function (12) takes the following form: LT (θ) = Z LT (τ ) PT (τ \| φ) PT φ \| φM −1 , τφ1:K dτ dφ× M −1 M −2 Y PT m=1 τφ1:K m+1 \|φ m+1 (23) PT φ m+1 \|φ m , τφ1:K m m+1 dτφ1:K × m+1 dφ PT τ 1:K \| θ dτ 1:K Since PT φm+1 \| φm , τφ1:K at each intermediate steps are delta functions, the final expression m for the multi-step MAML objective has the same form as (14), with integration taken w.r.t. all intermediate trajectories. Similarly, an unbiased estimate of the gradient of the objective gets M additional terms: " " ## M −1 K X X k M −1 ∇θ LT = E{τ 1:K ∇θ log πφm (τφm ) , (24) ,τ LT (τ ) ∇θ log πφ (τ ) + m } φ m=0 m=0 k=1 where the expectation is taken w.r.t. trajectories (including all intermediate ones). Again, note that at training time we do not constrain the number of interactions with each particular environment and do rollout using each intermediate policy to compute updates. At testing time, we interact with the environment only once and rely on the importance weight correction as described in Sec. 3.2. 17 Published as a conference paper at ICLR 2018 surrogate loss surrogate loss surrogate loss at Vt at Vt at Vt MLP MLP LSTM LSTM LSTM LSTM 64 x 64 64 x 64 64 x 64 64 x 64 xt xt (a) MLP (b) LSTM 64 x 64 xt 64 x 64 dt-1 rt-1 (c) RL2 Fig. 9: Policy and value function architectures. B A DDITIONAL DETAILS ON THE ARCHITECTURES The neural architectures used for our policies and value functions are illustrated in Fig. 9. Our MLP architectures were memory-less and reactive. The LSTM architectures had used a fully connected embedding layer (with 64 hidden units) followed by a recurrent layer (also with 64 units). The state in LSTM-based architectures was kept throughout each episode and reset to zeros at the beginning of each new episode. The RL2 architecture additionally took reward and done signals from the previous time step and kept the state throughout the whole interactions with a given environment (or opponent). The recurrent architectures were unrolled for T = 10 time steps and optimized with PPO via backprop through time. C C.1 A DDITIONAL DETAILS ON META - LEARNING AND OPTIMIZATION M ETA - UPDATES FOR CONTINUOUS ADAPTATION Our meta-learned adaptation methods were used with MLP and LSTM policies (Fig. 9). The metaupdates were based on 3 gradient steps with adaptive step sizes α were initialized with 0.001. There are a few additional details to note: 1. θ and φ parameters were a concatenation of the policy and the value function parameters. 2. At the initial stages of optimization, meta-gradient steps often tended to “explode”, hence we clipped them by values norms to be between -0.1 and 0.1. 3. We used different surrogate loss functions for the meta-updates and for the outer optimization. For meta-updates, we used the vanilla policy gradients computed on the negative discounted rewards, while for the outer optimization loop we used the PPO objective. C.2 O N PPO AND ITS DISTRIBUTED IMPLEMENTATION As mentioned in the main text and similar to (Bansal et al., 2018), large batch sizes were used to ensure enough exploration throughout policy optimization and were critical for learning in the competitive setting of RoboSumo. In our experiments, the epoch size of the PPO was set 32,000 episodes and the batch size was set to 8,000. The PPO clipping hyperparameter was set to = 0.2 and the KL penalty was set to 0. In all our experiments, the learning rate (for meta-learning, the learning rate for θ and α) was set to 0.0003. The generalized advantage function estimator (GAE) (Schulman et al., 2015b) was optimized jointly with the policy (we used γ = 0.995 and λ = 0.95). To train our agents in reasonable time, we used a distributed implementation of the PPO algorithm. To do so, we versioned the agent’s parameters (i.e., kept parameters after each update and assigned it a version number) and used a versioned queue for rollouts. Multiple worker machines were generating rollouts in parallel for the most recent available version of the agent parameters and were pushing them into the versioned rollout queue. The optimizer machine collected rollouts from the queue and made a PPO optimization steps (see (Schulman et al., 2017) for details) as soon as enough rollouts were available. 18 Published as a conference paper at ICLR 2018 We trained agents on multiple environments simultaneously. In nonstationary locomotion, each environment corresponded to a different pair of legs of the creature becoming dysfunctional. In RoboSumo, each environment corresponded to a different opponent in the training pool. Simultaneous training was achieved via assigning these environments to rollout workers uniformly at random, so that the rollouts in each mini-batch were guaranteed to come from all training environments. D D.1 A DDITIONAL DETAILS ON THE ENVIRONMENTS O BSERVATION AND ACTION SPACES Both observation and action spaces in RoboSumo continuous. The observations of each agent consist of the position of its own body (7 dimensions that include 3 absolute coordinates in the global cartesian frame and 4 quaternions), position of the opponent’s body (7 dimensions), its own joint angles and velocities (2 angles and 2 velocities per leg), and forces exerted on each part of its own body (6 dimensions for torso and 18 for each leg) and forces exerted on the opponent’s torso (6 dimensions). All forces were squared and clipped at 100. Additionally, we normalized observations using a running mean and clipped their values between -5 and 5. The action spaces had 2 dimensions per joint. Table 1 summarizes the observation and action spaces for each agent type. Table 1: Dimensionality of the observation and action spaces of the agents in RoboSumo. Agent Coordinates Observation space Self Opponent Velocities Forces Coordinates Forces Action space Ant 15 14 78 7 6 8 Bug Spider 19 23 18 22 114 150 7 7 6 6 12 16 Note that the agents observe neither any of the opponents velocities, nor positions of the opponent’s limbs. This allows us to keep the observation spaces consistent regardless of the type of the opponent. However, even though the agents are blind to the opponent’s limbs, they can sense them via the forces applied to the agents’ bodies when in contact with the opponent. D.2 S HAPED REWARDS In RoboSumo, the winner gets 2000 reward, the loser is penalized for -2000, and in case of draw both agents get -1000. In addition to the sparse win/lose rewards, we used the following dense rewards to encourage fast learning at the early training stages: • Quickly push the opponent outside. The agent got penalty at each time step proportional to exp{−dopp } where dopp was the distance of the opponent from the center of the ring. • Moving towards the opponent. Reward at each time step proportional to magnitude of the velocity component towards the opponent. • Hit the opponent. Reward proportional to the square of the total forces exerted on the opponent’s torso. • Control penalty. The l2 penalty on the actions to prevent jittery/unnatural movements. D.3 R O B O S U M O CALIBRATION To calibrate the RoboSumo environment we used the following procedure. First, we trained each agent via pure self-play with LSTM policy using PPO for the same number of iterations, tested them one against the other (without adaptation), and recorded the win rates (Table 2). To ensure the balance, we kept increasing the mass of the weaker agents and repeated the calibration procedure until the win rates equilibrated. 19 Published as a conference paper at ICLR 2018 Table 2: Win rates for the first agent in the 1-vs-1 RoboSumo without adaptation before and after calibration. E Masses (Ant, Bug, Spider) Ant vs. Bug Ant vs. Spider Bug vs. Spider Initial (10, 10, 10) Calibrated (13, 10, 39) 25.2 ± 3.9% 50.6 ± 5.6% 83.6 ± 3.1% 51.6 ± 3.4% 90.2 ± 2.7% 51.7 ± 2.8% A DDITIONAL DETAILS ON EXPERIMENTS E.1 AVERAGE WIN RATES Table 3 gives average win rates for the last 25 rounds of iterated adaptation games played by different agents with different adaptation methods (win rates for each episode are visualized in Figure 5). Table 3: Average win-rates (95% CI) in the last 25 rounds of the 100-round iterated adaptation games between different agents and different opponents. The base policy and value function were LSTMs with 64 hidden units. Adaptation Strategy LSTM + PPO-tracking LSTM + meta-updates Agent Opponent Ant Ant Bug Spider 24.9 (5.4)% 21.0 (6.3)% 24.8 (10.5)% 30.0 (6.7)% 15.6 (7.1)% 27.6 (8.4)% 44.0 (7.7)% 34.6 (8.1)% 35.1 (7.7)% Bug Ant Bug Spider 33.5 (6.9)% 28.6 (7.4)% 45.8 (8.1)% 26.6 (7.4)% 21.2 (4.2)% 42.6 (12.9)% 39.5 (7.1)% 43.7 (8.0)% 52.0 (13.9)% Spider Ant Bug Spider 40.3 (9.7)% 38.4 (7.2)% 33.9 (7.2)% 48.0 (9.8)% 43.9 (7.1)% 42.2 (3.9)% 45.3 (10.9)% 48.4 (9.2)% 46.7 (3.8)% E.2 RL2 T RUE S KILL RANK OF THE TOP AGENTS Rank Agent TrueSkill rank* Bug+LSTM-meta - 1 0.50 0.59 0.69 0.70 0.70 0.89 0.93 0.94 0.97 0.97 1 2 3 4 5 Bug + LSTM-meta Ant + LSTM-meta Bug + LSTM-track Ant + RL2 Ant + LSTM 31.7 30.8 29.1 28.6 28.4 Ant+LSTM-meta - 2 0.41 0.50 0.60 0.62 0.62 0.85 0.90 0.92 0.96 0.95 Bug+LSTM-track - 3 0.31 0.40 0.50 0.51 0.51 0.78 0.84 0.88 0.93 0.92 Ant+LSTM-RL2 - 4 0.30 0.38 0.49 0.50 0.50 0.77 0.83 0.87 0.92 0.91 Ant+LSTM - 5 0.30 0.38 0.49 0.50 0.50 0.77 0.83 0.87 0.92 0.91 6 7 8 9 10 Bug + MLP-meta Ant + MLP-meta Spider + MLP-meta Spider + MLP Bug + MLP-track 23.4 21.6 20.5 19.0 18.9 Bug+MLP-meta - 6 0.11 0.15 0.22 0.23 0.23 0.50 0.60 0.65 0.75 0.74 Ant+MLP-meta - 7 0.07 0.10 0.16 0.17 0.17 0.40 0.50 0.56 0.66 0.65 Spd+MLP-meta - 8 0.06 0.08 0.12 0.13 0.13 0.35 0.44 0.50 0.61 0.60 Spd+MLP - 9 0.03 0.04 0.07 0.08 0.08 0.25 0.34 0.39 0.50 0.49 Bug+MLP-track - 10 0.03 0.05 0.08 0.09 0.09 0.26 0.35 0.40 0.51 0.50 2 3 4 5 6 7 8 9 10 * The rank is a conservative estimate of the skill, r = µ − 3σ, to ensure that the actual skill of the agent is higher with 99% confidence. 1 Table 4 & Fig. 10: Top-5 agents with MLP and LSTM policies from the population ranked by TrueSkill. The heatmap shows a priori win-rates in iterated games based on TrueSkill for the top agents against each other. Since TrueSkill represents the belief about the skill of an agent as a normal distribution (i.e., with two parameters, µ and σ), we can use it to infer a priori probability of an agent, a, winning against 20 Published as a conference paper at ICLR 2018 its opponent, o, as follows (Herbrich et al., 2007): ! µa − µo 1 x P (a wins o) = Φ p , where Φ(x) := 1 + erf √ 2 2 2β 2 + σa2 + σo2 (25) The ranking of the top-5 agents with MLP and LSTM policies according to their TrueSkill is given in Tab. 1 and the a priori win rates in Fig. 10. Note that within the LSTM and MLP categories, the best meta-learners are 10 to 25% more likely to win the best agents that use other adaptation strategies. E.3 I NTOLERANCE TO LARGE DISTRIBUTIONAL SHIFTS Continuous adaptation via meta-learning assumes consistency in the changes of the environment or the opponent. What happens if the changes are drastic? Unfortunately, the training process of our meta-learning procedure turns out to be sensitive to such shifts and can diverge when the distributional shifts from iteration to iteration are large. Fig. 11 shows the training curves for a meta-learning agent with MLP policy trained against versions of an MLP opponent pre-trained via self-play. At each iteration, we kept updating the opponent policy by 1 to 10 steps. The meta-learned policy was able to achieve non-negative rewards by the end of training only when the opponent was changing up to 4 steps per iteration. # of opponent updates per iteration 1 step 4 steps 7 steps 10 steps Avg. reward per episode 2000 1000 0 −1000 −2000 −3000 0 500 1000 Iteration 1500 Fig. 11: Reward curves for a meta-learning agent trained against a learning opponent. Both agents were Ants with MLP policies. At each iteration, the opponent was updating its policy for a given number of steps using self-play, while the meta-learning agent attempted to learn to adapt to the distributional shifts. For each setting, the training process was repeated 15 times; shaded regions denote 90% confidence intervals. 21	2
A TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFi arXiv:1702.02046v1 [cs.CE] 6 Feb 2017 Xuyu Wang, Auburn University, Auburn, AL Chao Yang, Auburn University, Auburn, AL Shiwen Mao, Auburn University, Auburn, AL Breathing signal monitoring can provide important clues for human’s physical health problems. Comparing to existing techniques that require wearable devices and special equipment, a more desirable approach is to provide contact-free and long-term breathing rate monitoring by exploiting wireless signals. In this paper, we propose TensorBeat, a system to employ channel state information (CSI) phase difference data to intelligently estimate breathing rates for multiple persons with commodity WiFi devices. The main idea is to leverage the tensor decomposition technique to handle the CSI phase difference data. The proposed TensorBeat scheme first obtains CSI phase difference data between pairs of antennas at the WiFi receiver to create CSI tensor data. Then Canonical Polyadic (CP) decomposition is applied to obtain the desired breathing signals. A stable signal matching algorithm is developed to find the decomposed signal pairs, and a peak detection method is applied to estimate the breathing rates for multiple persons. Our experimental study shows that TensorBeat can achieve high accuracy under different environments for multi-person breathing rate monitoring. General Terms: Health sensing, breathing rate estimation, tensor decomposition, stable roommate matching, commodity WiFi, channel state information, phase difference 1. INTRODUCTION It is estimated that 100 million Americans suffer chronic health conditions such as lung disorders, diabetes, and heart disease [Boric-Lubeke and Lubecke 2002]. About three-fourths of the total US healthcare cost is spent on dealing with these health conditions. To reduce such costs, there is an increasing demand for longterm health monitoring in indoor environments. By tracking vital signs such as breathing and heart beats, the patient’s physical health can be timely evaluated and meaningful clues for medical problems can be provided [Rashidi and Cook 2013]. For example, monitoring breathing signals can help identify sleep disorders or anomalies for patients, as well as decreasing sudden infant death syndrome (SIDS) for sleeping infants [Hunt and Hauck 2006]. Traditional approaches for monitoring vital signs require patients to wear special devices, such as a capnometer [Mogue and Rantala 1988] to estimate breathing rate, or a pulse oximeter [Shariati and Zahedi 2005] on the finger to track heart beats. Recently, smartphones are used to estimate breathing rate by employing the built-in gyroscope, accelerometer [Aly and Youssef 2016], and microphone [Ren et al. 2015], and for physical activity recognition using accelerometer [Tao et al. 2016]. This requires the patient to place a smartphone near-by and wear sensors in the monitoring environment. Moreover, the readily available smartphone sensors such as accelerometer and gyroscope This work is supported in part by the U.S. NSF under Grants CNS-1247955 and CNS-1320664, and by the Wireless Engineering Research and Education Center (WEREC) at Auburn University. Author’s addresses: X. Wang, C. Yang, and S. Mao, Department of Electrical and Computer Engineering, Auburn University, Auburn, AL 36849-5201 USA. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. c YYYY ACM. 2157-6904/YYYY/01-ARTA $15.00 DOI: http://dx.doi.org/10.1145/0000000.0000000 ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:2 X. Wang et al. can only monitor breathing rate for single person. Such existing approaches could be expensive, inconvenient to use, and annoying even for a short period of time. An alternative approach is in need to provide a contact-free and long-term breathing monitoring at low costs. Recently, several RF based systems for vital signs tracking are proposed, which employ wireless signal to monitor breathing-induced chest movements and are mainly based on radar and WiFi techniques. For radar based vital signals monitoring, several techniques, such as the Doppler radar [Droitcour et al. 2009] and ultra-wideband radar [Salmi and Molisch 2011], are used to track vital signs, which require special hardware operated at high frequency and at a high cost. Moreover, the Vital-Radio system employs a frequency modulated continuous wave (FMCW) radar to track breathing and heart rates [Adib et al. 2015]; it requires a customized hardware using a wide bandwidth from 5.46 GHz to 7.25 GHz. For WiFi based vital signs monitoring, mmVital [Yang et al. 2016] can exploit the received signal strength (RSS) of 60 GHz millimeter wave (mmWave) signals for breathing and heart rate estimation. mmVital also operates with a larger bandwidth of about 7 GHz, and uses a customized hardware with a mechanical rotator. Another technique UbiBreathe uses WiFi RSS for breathing rate estimation, which, however, requires the device be placed in the line of sight (LOS) path between the transmitter and the receiver [Abdelnasser et al. 2015]. Compared with RSS, channel state information (CSI) provides fine-grained channel information in the physical layer, which can now be read from modified device drivers for several off-the-shelf WiFi network interface cards (NIC), such as the Intel WiFi Link 5300 NIC [Halperin. et al. 2010] and the Atheros AR9580 chipset [Xie et al. 2015]. Moreover, CSI includes both amplitude and phase values at the subcarrier-level for orthogonal frequency-division multiplexing (OFDM) channels, which is a more stable and accuracy representation of channel characteristics than RSS, including the non-LOS (NLOS) components for small-scale fading. Recently, the authors in [Liu et al. 2015] use the CSI amplitude data to monitor breathing and heart signals, which requires the person to remain in the sleeping mode. However, the measured CSI phase data has not been fully exploited in prior works, largely due to random phase fluctuation resulting from asynchronous times and frequencies of the transmitter and receiver. For multiple person breathing monitoring, because the reflected components in the received signal are from the chests of multiple persons, each moves slightly due to breathing and the movements are independent. Thus, vital signs monitoring and estimation for multiple persons still remains a challenging and open problem. In this paper, we propose to utilize CSI phase difference data between antenna pairs to monitor the breathing rates of multiple persons. First, we show that when the person is in a stationary state, such as standing, sitting, or sleeping, the CSI phase difference data is highly stable in consecutively received packets, which can be leveraged for extracting the small, periodic breathing signal hidden in the received WiFi signal. In fact, phase difference is more robust than amplitude, which usually exhibits large fluctuations because of the attenuation over the link distance, obstacles, and the multipath effect. Moreover, the phase difference data captures and preserves the periodicity of breathing, when the wireless signal is reflected from the patients’ chests. To extract the weak breathing signal, and more important, to distinguish among multiple persons, we propose to employ a tensor decomposition method to handle the phase difference data [Papalexakis et al. 2016; Luo et al. 2015; Hu et al. 2014]. We create the CSI tensor data by increasing the dimension of CSI data from one to three, which can be used to effectively separate different breathing signals in different clusters. We present a system termed TensorBeat, Tensor decomposition for estimating multiple persons breathing Beats, by exploiting CSI phase difference data. TensorBeat ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:3 operates as follows. First, it obtains 60 CSI phase difference data from antenna pairs 1 and 2, and 2 and 3, at the receiver. Next, a data preprocessing procedure is applied to the measured phase difference data, including data calibration and Hankelization. In the data calibration phase, the direct current (DC) component and high frequency noises are removed. In the Hankelization phase, a two dimensional Hankel matrix is created based on the calibrated phase difference data from every subcarrier, and the rank of the Hankel matrix is analyzed. Then, we adopt Canonical Polyadic (CP) decomposition for estimating multiple persons breathing signs, and prove the uniqueness of the proposed CSI tensor. After CP decomposition, we obtain twice amount of breathing signals, which, however, are randomly indexed. We thus design a stable signal matching algorithm (for the stable roommate problem [Irving 1985]) to identify the decomposed signal pairs for each person. Finally, we combine the decomposed signals in each pair and employ a peak detection method to estimate the breathing rate for each person. We implement TensorBeat on commodity 5 GHz WiFi devices and verify its performance with five persons over six months in different indoor environments, such as a computer laboratory, a through-wall scenario, and a long corridor. The results show that the proposed TensorBeat system can achieve high accuracy and high success rates for multiple persons breathing rates estimation. Moreover, we demonstrate the robustness of the proposed TensorBeat system for monitoring multiple persons’ breathing beats under a wide range of environmental parameters. The main contributions of this paper are summarized as follows. (1) We theoretically and experimentally verify the feasibility of leveraging CSI phase difference for breathing monitoring. In particular, we analyze the measured phase errors in detail and demonstrate that phase difference data is stable and can be used to extract breathing signs. To the best of our knowledge, we are the first to leverage phase difference for multiple persons breathing rate estimation. (2) We are also the first to apply tensor decomposition for RF sensing based vital signs monitoring. We use the phase difference data to create a CSI tensor for all subcarrier at the three antennas of the WiFi receiver. We then incorporate CP decomposition to obtain the desired breathing signals. A stable signal matching algorithm is developed to match the decomposed signals for each person, while a peak detection method is used to estimate multiple persons’ breathing rates. (3) We prototype the TensorBeat system with commodity 5 GHz WiFi devices and demonstrate its superior performance in different indoor environments with extensive experiments. The results show that the proposed TensorBeat system can achieve very high accuracy and high success rates for multiple persons breathing rate estimation. The remainder of this paper is organized as follows. The preliminaries and phase difference analysis are provided in Section 2. We present the TensorBeat system design and performance analysis in Section 3 and verify its performance with extensive experiments in Section 4. We provide the related work in Section 5. Section 6 concludes this paper. 2. PRELIMINARIES AND PHASE DIFFERENCE INFORMATION 2.1. Tensor Decomposition Preliminaries A tensor is considered as a multidimensional array [Kolda and Bader 2009]. The dimensions of the tensor are called as modes, and the order of the tensor is the number of the modes. For example, the N -order tensor is a N -mode tensor. Moreover, It is noticed that a first-order tensor is a vector, a second-order tensor is a matrix, and ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:4 X. Wang et al. a third-order tensor is a cubic structure. Higher-order tensors with (N ≥ 3) have a wide range of applications such as data mining, brain data analysis, recommendation systems, wireless communications, computer vision, and healthcare and medical applications [Papalexakis et al. 2016]. For higher-order tensors, they face various computational challenging because of the exponential increase in time and space complexity with the orders increase of tensors. This leads to the curse of dimensionality. Fortunately, tensor decomposition as one powerful tool is leveraged for alleviating the curve by decomposing high-order tensors into a limited number of factors. Also, it can obtain hidden feature components, thus extracting physical insight of higher-order tensors. Two main tensor decompositions are tucker decomposition and CP decomposition [Kolda and Bader 2009]. We consider CP decomposition for multiple persons breathing rate estimation because it can easily obtain the unique solution [Kolda and Bader 2009]. On the other hand, we will provide some necessary definitions and equations of tensor decomposition, which can be used for our proposed algorithm. Definition 2.1. (Frobenius Norm of a Tensor). The Frobenius norm of a tensor χ ∈ KI1 ×I2 ×···×IN is the square root of the sum of the squares of all its elements, which is defined by v u I1 I2 IN X uX X t kχkF = (1) x2i1 ,i2 ···iN . ··· i1 =1 i2 =1 iN =1 where K stands for R or C. Definition 2.2. (Kronecker Product). The Kronecker product of matrics A ∈ KI×J and B ∈ KM×N is denoted as A ⊗ B. The result is an (IM ) × (JN ) matrix, which is defined by   a11 B a12 B . . . a1J B a21 B a22 B . . . a2J B  A⊗B =  . (2) .. .. ..   ... . . .  aI1 B aI2 B . . . aIJ B Definition 2.3. (Khatri-Rao Product). The Khatri-Rao product of A ∈ KI×J and B ∈ KM×J is denoted as A ⊙ B. It is the column-wise Kronecker product with the size (IM ) × J, which is defined by (3) A ⊙ B = [a1 ⊗ b1 , a2 ⊗ b2 , · · · , aJ ⊗ bJ ]. I×J Definition 2.4. (Hadamard product). The Hadamard product of A ∈ K and B ∈ KI×J is denoted as A ∗ B. It is the elementwise matrix product with the size I × J, which is defined by   a11 b11 a12 b12 . . . a1J b1J a21 b21 a22 b22 . . . a2J b2J  A∗B = . (4) .. .. ..   ... . . .  aI1 bI1 aI2 bI2 . . . aIJ bIJ 2.2. Channel State Information Preliminaries OFDM is an effective wireless transmission technique widely used in many wireless systems, including WiFi (such as IEEE 802.11 a/g/n) and LTE [Wang et al. 2016; Xu et al. 2014]. The OFDM system partitions the wireless channel into multiple ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:5 orthogonal subcarriers, where data is transmitted over all the subcarriers by using the same modulation and coding scheme (MCS) to combat frequency selective fading. With modified device driver for off-the-shelf NICs, such as the Intel 5300 NIC [Halperin. et al. 2010] and the Atheros AR9580 chipset [Xie et al. 2015], the CSI data can be extracted, which represents fine-grained physical (PHY) information. Moreover, CSI captures rich wireless channel characteristics such as shadowing fading, distortion, and the multipath effect. The WiFi OFDM channel can be regarded as a narrowband flat fading channel, which can be expressed in the frequency domain as ~ =H·X ~ +N ~, Y (5) ~ denote the received and transmitted wireless signal vectors, respecwhere Y~ and X ~ is the additive white Gaussian noise, and H represents the channel frequency tively, N ~ and X. ~ response, which can be estimated from Y Although the WiFi OFDM system can use 56 subcarriers for data transmission on a 20 MHz channel, the Intel 5300 NIC device driver can only provide CSI for 30 out of the 56 subcarriers using the channel bonding technique. The channel frequency response of subcarrier i, denoted by Hi , is a complex value, given as Hi = Ii + jQi = \|Hi \| exp (j 6 Hi ), (6) where Ii and Qi are the in-phase component and quadrature component, respectively; \|Hi \| and 6 Hi are the amplitude and phase response of subcarrier i, respectively. For indoor environments with multipath components, the channel frequency response of subcarrier i, Hi , can also be written as Hi = K X rk · e−j2πfi τk , (7) k=0 where K is the number of multipath components, and rk and τk are the attenuation and propagation delay on the kth path, respectively. Traditionally, the multipath components are regarded as harmful for wireless communications, since they cause the delay spread (requires guard times) and large fluctuation of received wireless signal (harder to demodulate). For indoor localization systems, multiple signals will be received from a single transmission, including one LOS signal and many reflected signals. It is a challenging problem to detect the LOS signal from the mixed multipath components, which is indicative of the direction of the transmitter [Yang et al. 2013; Wang et al. 2014]. In this paper, however, we take a different view and show that the multiple signals reflected from the chests of multiple persons can be useful for estimating their breathing rates simultaneously. 2.3. Phase Difference Information As discussed, we exploit phase difference information for breathing rate estimation. We verify that the phase difference values between two adjacent antennas are stable for consecutively received packets in this section. In fact, the extracted phase information from the Intel 5300 NIC is high random and cannot be used for breathing monitoring. This is because of the asynchronous times and frequencies of transmitter and receiver NICs. Recently, two effective techniques are proposed for CSI phase calibration, to remove the unknown random components in CSI phase data. The first technique is to take a linear transformation for the CSI phase data over all the subcarirers [Qian et al. 2014; Wang et al. 2016; Wang et al. 2015]. The other technique is to use the phase difference between two adjacent antennas in the 2.4 GHz band, and to ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:6 X. Wang et al. remove the measured average of phase difference for LOS recognition [Wu et al. 2015]. It can be seen that these techniques only obtain the stable phase information and phase difference data with a zero mean, respectively, but none of these are useful for breathing rate estimation. To prove the stability of measured CSI phase difference in the 5 GHz band, we write b i , as [Speth et al. 1999] the measured phase of subcarrier i, denoted as 6 H 6 b i = 6 Hi + (λp + λs )mi + λc + β + Z, H (8) where Hi is the true phase of CSI data, mi is the subcarrier index of subcarrier i, β is the initial phase offset at the phase-locked loop (PLL), Z is the measurement environment noise, and λp , λs and λc are the phase errors from the packet boundary detection (PBD), the sampling frequency offset (SFO), and central frequency offset (CFO), respectively [Speth et al. 1999], which are given by   λp = 2π ∆t N′ (9) λ = 2π( T T−T ) TTus n  s λc = 2π∆f Ts n, 6 where ∆t is the packet boundary detection delay, N is the FFT size, T ′ and T are the sampling periods from the receiver and the transmitter, respectively, Tu is the length of the data symbol, Ts is the total length of the data symbol and the guard interval, n is the sampling time offset for the current packet, and ∆f is the center frequency ′ difference between the transmitter and receiver. In fact, the values of ∆t, T T−T , n, ∆f , and β in (8) and (9) are unknown, and the values of λp , λs , and λc can be different for different packets. Thus, we cannot obtain the true phase 6 Hi of CSI data from measured phase values. However, the measured phase difference on subcarrier i is stable, which can be employed for breathing rate estimation. Since the three antennas (radios) of the Intel 5300 NIC are on the same NIC, they use the same system clock and the same downconverter frequency. The measured CSI phases on subcarrier i from two adjacent antennas have the same λp , λs , λc , and mi . The phase difference can be computed as b i = ∆6 Hi + ∆β + ∆Z, ∆6 H (10) where ∆6 Hi is the true phase difference of subcarrier i, ∆β is the unknown difference in phase offsets, which is a constant [Gjengset et al. 2014], and ∆Z is the noise difference. Since in (10), the random values ∆t, ∆f , and n are all removed, the phase difference becomes more stable for back-to-back received packets. As an example, we plot in Fig. 1 the phase differences (marked as red dots) and the single antenna phases (marked as gray crosses) read from the 3rd subcarrier for 500 consecutively received packets. It can be seen that the single antenna phase is nearly uniformly distributed between 0◦ and 360◦ . However, all the phase difference data concentrate in a small sector between 330◦ and 340◦ , which is significantly more stable than phase data. Breathing rate estimation for multiple persons is a challenging problem, because the reflected components in the received signal are from the chests of multiple persons, each moves slightly due to breathing and the movements are independent. Thus, the peak-to-peak detection method cannot be effective for detecting the multiple breathing signals from the received signal. The aggregated breathing signal from multiple persons is not a clearly periodic signal anymore. Fig. 2 shows the detected breathing signals for one person (the upper plot) and three persons (the lower plot). We can see that for one person, the breathing signal exhibits a noticeable periodicity. So the breathing rate can be estimated by peak detection after removing the noise. However, the aggregated breathing signal of three persons does not show noticeable periodicity for ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:7 90 20 120 15 60 10 150 30 5 180 0 330 210 300 240 270 Fig. 1. The phase differences (marked by red dots) and the single antenna phases (marked by gray crosses) of the 3rd subcarrier for 500 back-to-back packets, plotted in the polar coordinate system. packet 400 to 600. Traditional FFT based methods can transform the received signal from the time domain to the frequency domain to estimate the breathing frequencies from multiple persons. Fig. 3 shows the breathing rate estimation for one person (the upper plot) and three persons (the lower plot) with the FFT method. We can see that the estimated frequency for one person is 0.2 Hz, which is almost the same as the true breathing rate. However, for three-person breathing rate estimation, the FFT curve only has two peaks, and the estimated breathing rates are much less accurate. In particular, the third peak cannot be estimated. This is because FFT based methods require a larger window size to improve the frequency resolution. We show that the proposed tensor decomposition based method is highly effective for multi-person breathing rate estimation in the following section. 3. THE TENSORBEAT SYSTEM 3.1. TensorBeat System Architecture The main idea of the proposed TensorBeat system is to estimate multi-person breathing rates by employing a tensor decomposition method. To obtain CSI tensor data, we first create a two dimensional Hankel matrix with phase difference data from backto-back received packets extracted from each subcarrier at each antenna. Then, by leveraging the phase differences from the 60 subcarriers, i.e., that between antennas 1 and 2, and between antennas 2 and 3, we can construct the third dimension of the CSI tensor data. The TensorBeat system will then leverage the created CSI tensor to estimate multi-person breathing signs. Our approach is motivated by two observations. First, for stationary modes of a person, such as standing, sitting, or sleeping, CSI phase difference from consecutively received packets is highly stable. It can thus be useful for extracting the periodic breathing signals. Second, the tensor decomposition method can effectively estimate multi-person breathing beats. We create the CSI tensor data by increasing the dimension of CSI data, from one dimension to three dimensions. The higher dimension CSI data is helpful to effectively separate ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:8 X. Wang et al. Breathing Curve of One Person 0.8 0.7 0.6 0.5 0 100 200 300 400 500 600 500 600 Breathing Curve of Three Persons -1 -1.5 -2 0 100 200 300 400 Packet Index FFT Magnitude FFT Magnitude Fig. 2. Detected breathing signals for one person (the upper plot) and three persons (the lower plot). Breathing Rate Estimation for One Person 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.7 Breathing Rate Estimation for Three Persons 40 20 0 0 0.1 0.2 0.3 0.4 0.5 Frequency (Hz) Fig. 3. Breathing rate estimation for one person (the upper plot) and three persons (the lower plot) based on FFT. different breathing signals by forming different clusters. This strategy is similar to the kernel method in traditional machine learning, such as support vector machine (SVM) [Wu et al. 2007] or multiple hidden layers in deep learning [LeCun et al. 2015; Wang et al. 2015; Wang et al. 2017]. As shown in Fig. 4, the TensorBeat system consists of four main modules: Data Extraction, Data Preprocessing, CP Decomposition, Signal Matching, and Breathing Rate Estimation. For Data Extraction, TensorBeat obtains 60 CSI phase difference data, 30 between antennas 1 and 2, and 30 between antennas 2 and 3, at the receiver with an off-the-shelf WiFi device. The Data Preprocessing module includes data calibration and Hankelization. Data calibration is implemented to remove the DC component and high frequency noises. Hankelization is to create a two dimensional Hankel matrix with phase difference data from each subcarrier for back-to-back received packets. The rank of the constructed Hankel matrix is then analyzed. We next apply CP decomposition to estimate multiple persons’ breathing signals, and prove the uniqueness of the proposed CSI tensor. For Signal Matching, we first compute the autocorrelation function of the decomposed signals, and incorporate a stable roommate matching algorithm to identify the decomposed signal pairs for each person, where a preference ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:9 )&1&#$,(3,2.(''-4 )&1&#/01,&.12!"#$%&'(#)++(,(-.(# /01,&.12- '; 6. &, ,( ,' < < < )&1&# &56,&12- 3&.8(1' < $#)(.2:32'12< '; 6. &, ,( ,' 7&-8(59&12- 7&-8(5# :&1,0 =#͑ ͑ ͑ ͑ ͑ = !?@ !> ͑͑͑͑͑͑͑͑͑͑ !4-&5#B&1.%-4#A542,1%: D,(&1%-4#/'1:&12!4-&5#C;'2- !1&65(#@22::&1(# B&1.%-4 $(&8#)(1(.12- !?@ +> ͑͑͑͑͑͑ A;12.2,,(5&12- ͑͑͑͑͑͑ !> +@ Fig. 4. The TensorBeat system architecture. list is computed with the dynamic time warping (DTW) values of the autocorrelation signals. For Breathing Rate Estimation, we combine the decomposed signals in each pair and use the peak detection method to compute the breathing rate for each person. In the remainder of this section, we present the design and analysis of each module of the TensorBeat system in detail. 3.2. Data Preprocessing 3.2.1. Data Calibration. We use a 20 Hz sampling rate to obtain 60 CSI phase difference data, 30 between antennas 1 and 2, and 30 between antennas 2 and 3, at the receiver with an off-the-shelf WiFi device at 5 GHz for data extraction. Then, data calibration is applied to remove the DC component and high frequency noises. Because the DC component is also considered as a kind of signal, which may affect CSI tensor decomposition, TensorBeat adopts the Hampel filter to remove the DC component. Unlike traditional data calibration approaches that only remove the high frequency noise, we use the Hampel Filter for detrending the original CSI phase difference data to remove DC component. In fact, the Hampel Filter, which is set as a large sliding window with 150 samples wide and a small threshold of 0.001, is firstly used to extract the basic trend of the original data. Then, the detrended data is generated by subtracting the basic trend data from the original data. We also utilize the Hampel Filter to reduce ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:10 X. Wang et al. Original Phase Difference -0.5 -1 -1.5 0 100 200 300 400 500 600 500 600 Calibrated Phase Difference 0.2 0 -0.2 -0.4 0 100 200 300 400 Packet Index Fig. 5. Data calibration: an example. the high frequency noise by using a sliding window of 6 samples wide and a threshold of 0.01. Fig. 5 presents an example of data calibration. We can see that the original phase differences of all the subcarriers have both a DC component and high frequency noises. With the proposed data calibration approach, it can be seen that the DC components are readily removed and all the subcarriers demonstrate a similar calibrated signal over the 600 packet range with low noise. Such calibrated signal will then be used for estimating the breathing rates of multiple persons. 3.2.2. Hankelization. After data calibration, we obtain the CSI phase difference data matrix with a dimension of (number of packets × number of subcarriers). We then employ a Hankelization method to transform the large CSI matrix into a CSI tensor by expanding the packets into an additional dimension [Lathauwer 2011]. Specifically, we rearrange the signals of each subcarrier into a 2-D Hankel matrix, so that the signals from all the 60 subcarriers can be considered as a 3-Dimensional tensor. Define Hr as the constructed Hankel matrix with the size I × J for subcarrier r, which is created by mapping N packets onto the Hankel matrix with N = I +J −1. We consider the Hankel matrix with size I = J = N2+1 . We thus obtain the Hankel matrix Hr for subcarrier r, as   hr (0) hr (1) . . . hr ( N2+1 − 1)  hr (1) hr (2) . . . hr ( N2+1 )    Hr =  (11) , .. .. .. ..   . . . . hr ( N2+1 − 1) hr ( N2+1 ) . . . hr (N − 1) where hr (i) is the calibrated phase difference data from subcarrier r for packet i. In our experiments, we set N = 599 and I = J = 300. To determine the number of components needed for CSI tensor decomposition, we provide the following theorem for estimating R breathing signals. T HEOREM 3.1. If there are R breathing signals in an indoor monitoring environment, the constructed Hankel matrix H r for subcarrier r has a rank of 2R when noise is negligible. P ROOF. When analyzing signal data structure, we assume the noise is negligible. Moreover, let the ith breathing signal be represented as Si (t) = Ai cos(wi t + ϕi ). The ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:11 observed signal from a subcarrier can be represented by [Cichocki et al. 2015] Y (t) = i=R X Ki Si (t) = i=R X (12) K̂i cos(wi t + ϕi ), i=1 i=1 where Ki is the coefficient for breathing signal i and the new coefficient K̂i = Ki Ai . The ith component of Y (t), K̂i cos(wi t + ϕi ), can be decomposed using Euler’s formula. We have K̂i K̂i exp(j(wi t + ϕi )) + exp(j(−wi t − ϕi )) 2 2 K̂i K̂i exp(jϕi ) exp(jwi t) + exp(−jϕi ) exp(−jwi t). (13) = 2 2 Each breathing signal can be separated into two exponential signals with different coefficients. Combining all the R breathing signals, we have ! R X K̂i K̂i Y (t) = exp(jϕi ) exp(jwi t) + exp(−jϕi ) exp(−jwi t) 2 2 i=1 K̂i cos(wi t + ϕi ) = = 2R X K̃i Zit , (14) i=1 where the updated signal Zit is denoted as Zit = exp(±jwi t), and K̃i = K̂2i exp(±jϕi ) is its coefficient. For packets received at discrete times, we represent the received P2R n signal as Y (n) = i=1 K̃i Zi . Note that the combined signal can be considered as an exponential polynomial with 2R different exponential terms. Map signal Y (n) for n = 1, 2, · · · , N into a Hankel matrix with size I = J = N2+1 , we have  P  N +1 P2R P2R −1 2R K̃i Zi1 · · · K̃i Zi 2 K̃i Zi0 i=1 i=1 i=1  P N +1  P2R P2R 2R   2 1 2 ···   i=1 K̃i Zi i=1 K̃i Zi i=1 K̃i Zi Hr =  (15) . . . .   .. .. . · · · .   N +1 N +1 P2R P2R P2R N −1 2 −1 2 K̃ Z · · · K̃ Z K̃ Z i i i i i i i=1 i=1 i=1 We can see that the Hankel matrix can be decomposed with Vandermonde decomposition [Lathauwer 2011], as T H r = V r · diag(K̃1 , K̃1 , · · · , K̃2R ) · Ṽ r , where the Vandermode matrices V r ∈ K  1  Z1  .. V r = Ṽ r =  .  N +1 2 −1 Z1 N +1 2 ×2R and Ṽ r ∈ K 1 Z2 .. . N +1 2 −1 Z2 ··· ··· ··· N +1 2 ×2R 1 Z2R .. . N +1 2 −1 · · · Z2R (16) are given by    .  (17) Because a Vandermode matrix is full rank, which is obtained by different poles, the rank of the Hankel matrix generated by R breathing signals is 2R. According to Theorem 3.1, 2R signal components is required to separate the R breathing signals. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:12 X. Wang et al. Next we consider the influence of measurement noise on the Hankel matrix H r . Because of noise, the Hankel matrix Hr is actually a full-rank matrix. However, Theorem 3.1 shows that the rank of the combined breathing signal is 2R, meaning that the first 2R weighted decomposed components are much stronger than the remaining ones as long as the signal to noise ratio (SNR) is not very low. This shows that the Hankel matrix structure can be used to effectively separate breathing signals from white noise. Actually, the different signals will be well denoised and separated by using tensor decomposition, as to be discussed in Section 3.3. 3.3. Canonical Polyadic Decomposition Once the CSI tensor is ready, we apply CP decomposition to estimate multiple persons’ breathing signals. With CP decomposition, the CSI tensor data can be approximated as the sum of 2R rank-one tensors according to Theorem 3.1. Denote χ ∈ KI×J×K as a third-order CSI tensor, which can be obtained by the sum of three-way outer products as [Kolda and Bader 2009; Papalexakis et al. 2016] χ≈ 2R X (18) ar ◦ b r ◦ cr , r=1 where ar , br , cr are the vectors at the rth position for the first, second, and third dimension, respectively, and 2R is the number of decomposition components, which is the approximation rank of the tensor based on CP decomposition [Sun and Kumar 2014; Sun and Kumar 2015]. Their outer product is defined by for all i, j, k. (ar ◦ br ◦ cr )(i, j, k) = ar (i)br (j)cr (k), (19) I×2R We consider factor matrices A = [a1 , a2 , · · · , a2R ] ∈ K , B = [b1 , b2 , · · · , b2R ] ∈ KJ×2R , and C = [c1 , c2 , · · · , c2R ] ∈ KK×2R as the combination of vectors from rankone components. Moreover, define X(1) ∈ KI×JK , X(2) ∈ KJ×IK , and X(3) ∈ KK×IJ as 1-mode, 2-mode, and 3-mode matricization of CSI tensor χ ∈ KI×J×K , respectively, which are obtained by fixing one mode and arranging the slices of the rest of the modes into a long matrix [Kolda and Bader 2009]. Then, we can write the three matricized forms as X(1) ≈ A(C ⊙ B)T , (20) T (21) T (22) X(2) ≈ B(C ⊙ A) , X(3) ≈ C(B ⊙ A) , where ⊙ denotes the Khatri-Rao product. When the number of components 2R is given, we apply the Alternating Least Squares (ALS) algorithm, the most widely used algorithm for CP decomposition [Kolda and Bader 2009]. To decompose the CSI tensor, we minimize the square sum of the differences between the CSI tensor χ and the estimated tensor. min A,B,C χ− 2R X 2 ar ◦ b r ◦ cr r=1 . (23) F Note that (23) is not convex. However, the ALS algorithm can effectively solve the problem by fixing two of the factor matrices, to reduce the problem to a linear least squares problem with the third factor matrix as variable. If we fix B and C, we can rewrite problem (23) as min X(1) − A(C ⊙ B)T A 2 F . (24) ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:13 We can derive the optimal solution to problem (24) as A = X(1) [(C ⊙ B)T ]† . Applying the property of pseudoinverse of the Khatri-Rao product, it follows that A = X(1) (C ⊙ B)(C T C ∗ B T B)† , (25) where ∗ denotes the Hadamard product. This equation only requires computing the pseudoinverse of a 2R × 2R matrix rather than a JK × 2R matrix. Note that R is much smaller than J and K, thus the computing complexity can be greatly reduced. Similarly, we can obtain the optimal solutions for B and C as B = X(2) (C ⊙ A)(C T C ∗ AT A)† T T † C = X(3) (B ⊙ A)(B B ∗ A A) . (26) (27) Applying ALS to CP decomposition, we obtain matrices A, B, and C. To guarantee the effectiveness of the decomposed components, we next examine the uniqueness of CP decomposition. The basic theorem on the uniqueness of CP decomposition is given in [Kolda and Bader 2009], which is provided in the following. FACT 1. For tensor χ with rank L, if kA + kB + kC ≥ 2L + 2, then the CP decomposition of χ is unique, where kA , kB , and kC denote the k-rank of matrix A, B, C, respectively. Here k-rank means the maximum value k such that any k columns are linearly independent [Kolda and Bader 2009]. Based on Fact 1, we have the following theorem for the CSI tensor. T HEOREM 3.2. For the proposed CSI tensor χ with rank 2R, the CP decomposition of χ is unique. P ROOF. The proposed CSI tensor χ is created by K Hankel matrix, where the rth Hankel matrix H r is rank-2R according to Theorem 3.1. Thus, for the k-rank of the matrices A and B, we have kA = 2R and kB = 2R. On the other hand, because phase differences of subcarriers between antennas 1 and 2, and antennas 2 and 3 are independent, the k-rank of matrix C has kC ≥ 2. Thus, the expression is kA + kB + kC ≥ 2R + 2R + 2 = 2(2R) + 2, which satisfies the conditions in Theorem 1. This proofs the theorem. Theorem 3.2 indicates that the CP decomposition of the created CSI tensor is unique, which can be used to effectively estimate multiple breathing rates. In the proposed TensorBeat system, we leverages the matrix A = [a1 , a2 , · · · , a2R ] as decomposed signals S1 , S2 , · · · , S2R . For example, Fig. 6 shows the results of CP decomposition for CSI tensor data from three persons (R = 3). We can see that there are six signals. Moreover, signals 1 and 2 are similar, signals 3 and 5 are similar, and signals 4 and 6 are similar. This is because CP decomposition cannot guarantee that similar signals are located in adjacent locations (i.e., the output signals are randomly indexed). Thus, we need to identify the signal pairs among the decomposed signals for each person, which will be addressed in Section 3.4. 3.4. Signal Matching Algorithm The CP decomposition of CSI tensor data yields 2R decomposed signals, i.e., S1 , S2 , · · · , S2R , which, however, are randomly indexed. In this section, we propose a signal matching algorithm to pair the two similar decomposed signals that belong to the same person. The main idea is to leverage the autocorrelation to strengthen the periodicity of decomposed signals and use the Dynamic Time Warping (DTW) method to compute the similarity value for any pair of signals. Finally, we apply the stableroommate matching algorithm to pair the decomposed signals for each person, using ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:14 X. Wang et al. 0.2 6 0 5 4 -0.2 0 3 100 2 200 Packet Index 300 Signal Index 1 Fig. 6. CP decomposition results for a CSI tensor of three persons. 2 6 0 -2 0 5 4 100 3 200 300 400 Packet Index 2 500 600 Signal Index 1 Fig. 7. Autocorrelation of the decomposed breathing signals. the DTW values as the closeness metric. We introduce the proposed signal matching algorithm in the following. 3.4.1. Autocorrelation and Dynamic Time Warping. After CP decomposition of CSI tensor data, we first compute the autocorrelation function of the 2R decomposed signals to strengthen their periodicity. We evaluate the autocorrelation function of the decomposed signals for two reasons. The first is that the autocorrelation of a decomposed signal can increase the data length, which helps to improve the accuracy of the peak detection. Second, because the decomposed signals have phase shift and nonalignment, using the autocorrelation of decomposed signals can reduce such shifts and strengthen the periodicity of the decomposed signals. Fig. 7 shows the autocorrelation of the decomposed breathing signals produced by CP decomposition. We can see that each autocorrelation signal exhibits a more obvious periodicity than that of the original decomposition signals. Moreover, the data length is increased from 300 to 600. Furthermore, we employ the DTW approach to measure the distance between any pair of autocorrelation signals, which is different from the Euclidean distance method that computes the sum of distances from each value on one curve to the corresponding ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:15 value on the other curve. Moreover, the Euclidean distance method believes that two autocorrelation signals with the same length are different as long as one of them has a small shift. However, DTW can automatically identify these shifts and provide the similar distance measurement between two autocorrelation signals by aligning the corresponding time series, thus overcoming the limitation of the Euclidean distance method. With the autocorrelation signals, we design the DTW method for measuring their pairwise distance. Given two autocorrelation signals and a cost function, the DTW method seeks an alignment by matching each point in the first autocorrelation signal to one or more points in the second signal, thus minimizing the cost function for all points [Wang and Katabi 2013; Salvador and Chan 2004; Salvador and Chan 2007]. To reduce the computational complexity of DTW, we apply downsampling to the two autocorrelation signals, which leads to a reduced number of packets N ′ . Then, consider two downsampled autocorrelation signals Pi = [Pi (0), Pi (1), · · · , Pi (N ′ − 1)] and Pj = [Pj (0), Pj (1), · · · , Pj (N ′ − 1)], we need to find a warping path W = [w1 , w2 , · · · , wL ], where L is the length of the path, and the lth element of the warping path is wl = (ml , nl ), where m and n are the packet index for the two downsampled autocorrelation signals. The objective is to minimize the total cost function by implementing the non-linear mapping between two downsampled autocorrelation signals Pi and Pj . The formulated problem is given by min L X kPi (ml ) − Pj (nl )k (28) l=1 s.t. (m1 , n1 ) = (0, 0) (mL , nL ) = (N ′ − 1, N ′ − 1) ml ≤ ml+1 ≤ ml + 1 nl ≤ nl+1 ≤ nl + 1. (29) (30) (31) (32) The objection function is to minimize the distance between two downsampled autocorrelation signals. The first and second constraints are boundary constraints, which require that the warping path starts at Pi (0) and Pj (0) and ends at Pi (N ′ − 1) and Pj (N ′ − 1). This can guarantee all points of the two downsampled autocorrelation signals are used for measuring their distance, thus avoiding to use only local data. Furthermore, the third and fourth constraints are monotonic and marching constraints, which require that there be no cycles for wi and wj in the warping path and the path is increased with the maximum 1 at each step. We apply dynamic programming to solve problem (28), to obtain the minimum distance warping path between two downsampled autocorrelation signals. We consider a two-dimensional cost matrix C with size N ′ × N ′ , whose element C(ml , nl ) is the minimum distance warping path for two downsampled autocorrelation signals Pi = [Pi (0), Pi (1), · · · , Pi (ml )] and Pj = [Pj (0), Pj (1), · · · , Pj (nl )]. We design the recurrence equation in dynamic programming as follows. C(ml , nl ) = kPi (ml ) − Pj (nl )k + min [C(ml − 1, nl ), C(ml , nl − 1), C(ml − 1, nl − 1)]. (33) By filling all elements of the cost matrix C, the value C(N ′ − 1, N ′ − 1) can be computed as the DTW value between the two downsampled autocorrelation signals. The time complexity is O(N ′2 ). Fig. 8 shows the DTW results for downsampled autocorrelation signals 4 and 6 (the upper plot), and downsampled autocorrelation signals 4 and 3 (the N lower plot), where we set the downsampling number of packets as N ′ = 10 = 60. It can be seen that downsampled autocorrelation signals 4 and 6 have a smaller DTW value (i.e., 3.65) than downsampled autocorrelation signals 4 and 3 (i.e., 13.7). That is, ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:16 X. Wang et al. DTW Value with 3.6500 3 2 1 0 0 10 20 30 40 50 60 50 60 DTW Value with 13.7013 3 2 1 0 0 10 20 30 40 Packet Index Fig. 8. DTW results for downsampled autocorrelation signals 4 and 6 (the upper plot), and downsampled autocorrelation signal 4 and 3 (the lower plot), respectively. signals 4 and 6 are more similar, and more likely to belong to the same person. We also find that the downsampled autocorrelation signals have a high similarity in the center than that on the boundary, and it can reduce the phase shift values. Thus, the DTW value is a good measure of the distance between two downsampled autocorrelation signals. We need to compute the DTW values for all the downsampled autocorrelation signal pairs, which are then used in stable roommate matching. 3.4.2. Stable Roommate Matching. Since the CP decomposed signals are randomly indexed (see Fig. 7), we need to identify the pair for each person. With the DTW values for all downsampled autocorrelation signal pairs, we can model this problem as a stable roommate matching problem [Irving 1985; Feng et al. 2015b; Feng et al. 2015a]. There are a group of 2R signals, and each signal maintains a preference list of all other signals in the group, where the preference value for another signal is the inverse of the corresponding DTW value (i.e., distance). The problem is to pair the signals, such that there is no such a pair of signals that both of them have a more desired selection than their current selection, i.e., to find a stable matching [Irving 1985; Feng et al. 2015b; Feng et al. 2015a]. The proposed signal matching algorithm is presented in Algorithm 1. We first compute the autocorrelation of all decomposed signals. Then each autocorrelation signal populates its preference list with other autocorrelation signals according to the DTW values. The stable roommate matching algorithm is executed in two steps. In step 1, each signal proposes to other signals according to its preference list. If a signal m receives a proposal from another signal n, we implement the following strategy: (i) signal m rejects signal n if it has a better proposal from another signal; (ii) signal m accepts signal n’s proposal if it is better than all other proposals that signal m currently holds. Moreover, signal n stops to propose when its proposal is accepted, while it needs to continue to propose to other signals if being rejected. This strategy is implemented in step 1 of the signal matching algorithm, where we use f inish f lag to mark whether the current signal num is accepted or not. Moreover, variables accept num and propose num are used to record the current signal’s proposed number and proposing number, respectively. Also, variable scan num is used to record the current scanning signal number. After completing step 1, every signal holds a proposal or one signal has been rejected by other signals (this case hardly happens in TensorBeat, because the CP decomposition produces two very similar signals with high probability for each ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:17 ALGORITHM 1: Signal Matching Algorithm Input: Decomposed signals: S1 , S2 , · · · , S2R . Output: Matched signal pairs. Compute autocorrelation of all decomposed signals; Compute the DTW values for every pair of autocorrelation signals; Each autocorrelation signal sets its preference list using the DTW values; //step1; for signal num = 1 : 2R do Set f inish f lag = 0; Set scan num = signal num; while f inish f lag = 0 do if the proposal is the first one then Proposing signal’s propose num=the current choice; Set f inish f lag = 1; Proposed signal’s accept num = scan num; else if the signal prefers the former proposal then Reject the current proposal symmetrically; Propose to the next choice; else Accept the current proposal; Reject the former proposal symmetrically; scan num = proposed signal’s accept num; Propose to the next choice; end end end end for signal num = 1 : 2R do Reject signals that have less than accept num in every preference list symmetrically; end //step2; signal num = 1; while signal num < 2R + 1 do if propose num = accept num then signal num=signal num+1; else Let p1 be a signal whose preference list contains more than one element; while p sequence is not cyclic do qi = the second preference of pi ’s current list; pi+1 = the last preference of qi ’s current list; end Denote ps as the first element in the p sequence to be repeated and r as the length of the circle; for i = 1 : r do Reject matching (qs+i−2 , ps+i−1 ) symmetrically; end signal num = 1; end end Obtain signal matching pairs based on all processed preference lists; person). Then, we need to delete some elements in all the preference lists based on the following method, which is that if signal m is the first on signal n’s list , then signal n ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:18 X. Wang et al. is the last on signal m’s list. For the proposed algorithm, every signal can reject signals that have less than accept num in its preference list symmetrically (reject each other). An example is shown in Fig. 6. According to the DTW values, signals 1, 2, 3, 4, 5, and 6 have their preference lists as (2, 3, 5, 6, 4), (1, 3, 5, 6, 4), (5, 1, 2, 6, 4), (6, 5, 3, 2, 1), (3, 2, 1, 4, 6), and (4, 5, 3, 2, 1), respectively. When step 1 is executed, we have: Signal 1 proposes to 2, and signal 2 holds 1; Signal 2 proposes 1, and signal 1 holds; Signal 3 proposes to signal 5, and signal 5 holds; Signal 4 proposes to signal 6, and signal 6 holds; Signal 5 proposes to signal 3, and signal 3 holds; Signal 6 proposes to signal 4, and signal 4 holds. It is easy to find three pairs (1,2), (3,5), and (4,6). Although most of decomposed signals are paired in step 1, step 2 will still be necessary for the more challenging cases of much more breathing signals and NLOS environments. In step 2, we consider the reduced preference lists, where some of the lists have more than one signals. By implementing step 2, we can reduce the preference lists such that each signal only holds one proposal. The main idea is that we need to find some all-or-nothing cycles and symmetrically delete signals in the cycle sequence by rejecting the first and last choice pairs. The signal in the cycle accepts the secondary choice, thus obtaining a stable roommate matching. To find all-or-nothing cycles, let p1 be a signal with a preference list that contains more than one element, and generate the sequences such that qi = the second preference of pi ’s current list, and pi+1 = the last preference of qi ’s current list. After the cycle sequence generation, denote ps as the first element in the p sequence to be repeated. Then, we reject matching (qs + i − 2, ps + i − 1) for i = 1 to r symmetrically, where r is the length of the cycle. Finally, we can obtain signal matching pairs based on all processed preference lists. The computational complexity of Algorithm 1 is O(R2 ), because steps 1 and 2 each has a complexity of O(R2 ), respectively. 3.5. Breathing Rate Estimation 3.5.1. Signal Fusion and Autocorrelation. After obtaining the outcomes of the signal matching algorithm, TensorBeat next applies peak detection to estimate the breathing rates for multiple persons. Comparing to the FFT method, a higher resolution in the time domain can be achieved. To implement peak detection, we first need to combine the decomposed signal pairs for each person into a single signal, by taking the average of the signal pairs. Averaging can decrease the variance of the decomposed signals while preserving the same period. For example, Fig. 9 shows the fusion results based on the outcome of the signal matching algorithm, where three smoothly decomposed signals with different periods are obtained. To strengthen the accuracy of peak detection, we compute the autocorrelation function again for every fused signal. Fig. 10 shows the autocorrelation of the three fused signals. It can be seen that the length of data is increased from 300 to 600 and the number of peaks of every signal are also increased, which help to improve the estimation accuracy. 3.5.2. Peak Detection. Although breathing signal is generated by the small periodic chest movement of inhaling and exhaling, the phase difference data can effectively capture the breathing rate. Traditionally, estimation of breathing rates is achieved with FFT based methods. However, the FFT approach may have limited accuracy, because the frequency resolution of breathing signals is based on the window size of FFT. When the window size becomes larger, the accuracy will be higher, but the time domain resolution will be reduced. Also see Figs. 2 and 3 for the limitation of the FFT based approach for the multi-person scenario. Therefore, we leverage peak detection instead in TensorBeat system to achieve accurate breathing rate estimation for each of autocorrelations of fused signals. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:19 0.2 3 0 -0.2 0 2 50 100 Signal Index 150 200 Packet Index 250 300 1 Fig. 9. Fusion results based on the outcomes of the signal matching algorithm. 5 3 0 -5 0 2 100 200 300 Signal Index 400 Packet Index 500 600 1 Fig. 10. Autocorrelation of fused signals. For peak detection, the traditional method based on amplitude needs to detect the fake peak, which is not a real peak but has larger values than its two immediate neighboring points. To avoid the fake peak, a large moving window can be used to identify the real peak based on the maximum breathing periodicity. This method is not robust, which requires adjusting the window size. In TensorBeat, we only consider a smaller moving window of 7 samples wide. This is because we leverage the Hankel matrix and CP decomposition to smooth out the breathing curves, which hardly contains any fake peaks. Then, for the ith autocorrelation curve of fused signal, we seek all the peaks by determining whether or not the medium of the 7 samples in the moving window is the maximum value. Finally, we consider the median of all peak-to-peak intervals as the final period of the ith breathing signal, which is denoted as Ti . Finally, the estimated breathing rates can be computed as fi = 60/Ti , for i = 1 to R. 4. EXPERIMENTAL STUDY 4.1. Experiment Configuration In this section, we validate the TensorBeat performance with an implementation with 5 GHz Wi-Fi devices. To obtain 5 GHz CSI data, we use a desktop computer and a Dell ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:20 X. Wang et al. laptop as access point and mobile device, respectively, both of which are equipped with an Intel 5300 NIC. We use the desktop computer instead of the commodity routers, because none is equipped with the Intel 5300 NIC. The operating system is Ubuntu destop 14.04 LTS OS for both the access point and the mobile device. The PHY is the IEEE 802.11n OFDM system with QPSK modulation and 1/2 coding rate. Moreover, the access point is set in the monitor model and the distance between its two adjacent antennas is approximately 2.68 cm, which is half of the wavelength of 5GHz WiFi. Also, the mobile device is set in the injection model and uses one antenna to transmit data. Moreover, we use omnidirectional antennas for both the receiver and transmitter to estimate breathing signs beats. With the packet injection technique with LORCON version 1, we can obtain 5 GHz CSI data from the three antennas of the receiver. Our experimental study is with up to five persons over a period of six months. The experimental scenarios include a computer laboratory, a through-wall scenario, and a corridor, as shown in Fig. 11. The first scenario is within a 4.5 × 8.8 m2 laboratory, where both single person and multi-person breathing rate estimation experiments are conducted. There are lots of tables and desktop computers crowded in the laboratory, which block parts of the LOS paths and form a complexity radio propagation environment. The second setup is a through-wall environment, where single person breathing rate estimation is tested due to the relatively weaker signal reception. The person is on the transmitter side, and the receiver is behind a wall in this experiment. The third scenario is a long corridor of 20 m, where the maximum distance between the receiver and transmitter is 11 m in the experiment. This scenario is still considered for single person breathing rate monitoring. We use a NEULOG Respiration to record the ground truths for single person breathing rates. The single person breathing rates estimation can be easily implemented by removing the signal matching algorithm, because there are only two decomposed signals after CP decomposition in this case. For muti-person breathing rate estimation in the first scenario, all persons participating in the experiment record their breathing rates by using a metronome smartphone application with 1 bpm accuracy at the same time. We consider five persons are stationary for LOS and NLOS environments for breathing monitoring. Moreover, there are no other persons in the breathing measurement area. For multi-person breathing rate estimation, we need to define a proper metric for evaluating TensorBeat’s performance. For R estimated breathing rates [f1 , f2 , ...fR ], the ith breathing rate estimation error, Ei , is defined as Ei = fi − fˆi , for i = 1, 2, · · · , R, (34) where fˆi is the ground truth of the ith breathing rate. We also define a new metric termed success rate, denoted as SR, which is defined as SR = N {maxi {Ei } < 2bpm} × 100%, N {E} (35) where N {maxi {Ei } < 2bpm} means the number of repeated experiments of the maximum breathing rate error less than 2 bpm, and N {E} is the number of repeated experiments. We adopt the success rate metric because there are weak signals for multi-person experiments in indoor experiments at different locations, and sometimes a breathing signal may not be successfully detected [Wang et al. 2016]. 4.2. Performance of Breathing Estimation In Fig. 12, we present the cumulative distribution functions (CDF) of the estimation errors for single person breathing rate detection for three different experiment scenarios. We can see that for TensorBeat, high estimation accuracy of breathing rates ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:21 '()+#,--.(/01/+,-2)-,0/, &!&# !"# %!$# 3.4,.5.(/01/+,-2)-,0*/, $!$# Fig. 11. Experimental setup: computer laboratory, through-wall, and long corridor scenarios. 1 0.8 Laboratory Corridor Through-wall CDF 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Estimation Error (bpm) Fig. 12. Performance of single person breathing rate estimation in the computer laboratory, through-wall, and long corridor scenarios. can be achieved in all the three scenarios. The maximum estimation error is less than 0.9 bpm. Moreover, it is noticed that 50% of the tests for the computer laboratory experiment have errors less than aboout 0.19 bpm, while the tests for the corridor and through-wall scenarios have errors less than approximately 0.25 bpm and 0.35 bpm, respectively. Thus, the performances in the laboratory setting is better than that in the corridor and through-wall scenarios. This is because the laboratory has a smaller space and the breathing signal is stronger than that of other two cases with larger attenuation due to the long distance and the wall. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:22 X. Wang et al. 1 0.8 CDF 0.6 One Person Two Persons Three Persons Four Persons Five Persons 0.4 0.2 0 0 0.5 1 1.5 2 Estimation Error (bpm) Fig. 13. Performance of breathing rate estimation for different number of persons (computer laboratory). Fig. 13 presents the performance of breathing rate estimation for different number of persons. It is noticed that higher accuracy is achieved for the single person test, where approximately 96% of the test data have an estimation error less than 0.5 bpm. The five-person test has the worse performance, where approximately 62% of the test data have an estimation error less than 0.5 bpm. Moreover, we fine that the performances of the two-person and three-person tests are similar, both of which can have an error smaller than 0.5 bpm for 93% of the test data. Generally, when the number of persons is increased, the performance of breathing rate estimation gets worse. In fact, when the number of breathing signals is increased, the distortion of the mixed received signal will become larger, thus leading to high estimation errors. Fig. 14 plots the success rates for different number of persons. We find that although the success rate for one person is the highest, there are still few of test data that cannot obtain high accuracy breathing rates estimation. These test data should come from different locations in the indoor environments, where parts of the received signals are severely distorted. In fact, we find that low phase difference usually occurs when the SNR is low. On the other hand, we can see that breathing rate estimation for two persons also has a high success rate, because the probability for two persons to have exactly the same breathing rate is very low. When the number of persons is increased, the chance of getting two close breathing rates becomes higher. Even in this case, TensorBeat can still effectively separate them with a high success probability. With the increase of the number of persons, the success rate for TensorBeat system decreases. The reason is that each breathing rate is more likely to cover each other and the strength of the received signal becomes lower. From Fig. 14, we can see that the success rate is about 82.4% when the number of persons is five. Fig. 15 shows the success rate for different sampling rates. In this experiment, there are four persons and the window size is set to 30 s. From Fig. 15, we can see that with the increase of the number of sampling rates, the success rate is also increased. It is noticed that the success rates for 5 Hz and 30 Hz are approximately 70% and 90%, respectively. As the sampling rate is increased, the length of the data for CP decomposition is increased for the 30 s window size case, which helps to improve the estimation accuracy. Furthermore, we find that the performance becomes stable when the sampling rate exceeds 20 Hz, indicating that a sampling rate of 20 Hz is sufficient ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:23 100 Success Rate (%) 80 60 40 20 0 1 2 3 4 5 The Number of Persons Fig. 14. Success rates for different number of persons (computer laboratory). 100 Success Rate (%) 80 60 40 20 0 5 10 15 20 25 30 Sampling Rate (Hz) Fig. 15. Success rates for different sampling rates (computer laboratory). for CP decomposition. Thus, we set the sampling rate to 20 Hz for the TensorBeat experiments. Fig. 16 plots the success rates for different window sizes. This experiment is for the computer laboratory scenario with four persons and the sampling rate is set to 20 Hz. From Fig. 16, we can see that the success rate is greatly increased by increasing the window size of the Hankel matrix from 15 s to 30 s. This is because Hankelizaiton will take half of the data to smooth the phase difference signal, which reduces the resolution in the time domain. Thus, we need to increase the window size to improve the estimation accuracy. Furthermore, the change of success rate is small for window sizes from 30 s to 45 s. Thus, we select the window size of 30 s for the TensorBeat experiments. Finally we examine the impact of LOS and NLOS scenarios. The success rates are plotted in Fig. 17. In this experiment, we consider the challenge condition of the NLOS scenario, where all the persons stay on the LOS path between the transmitter and receiver, i.e., they form a straight line and block each other. From Fig. 15, we find ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:24 X. Wang et al. 100 Success Rate (%) 80 60 40 20 0 15 20 25 30 35 40 45 The Size of Window (s) Fig. 16. Success rates for different window sizes (computer laboratory). 100 LOS NLOS Success Rate (%) 80 60 40 20 0 2 3 4 5 The Number of Persons Fig. 17. Success rates for (i) when multiple persons form a line in the LOS path between the transmitter and receiver; (ii) when multiple persons are in scattered around (computer laboratory). that the performances for LOS and NLOS are nearly the same for the cases of two or three persons, where high estimation accuracy can be achieved. This is due to the WiFi multipath effect, which is regarded harmful in general but becomes helpful in breathing rate estimation when tensor decomposition is used. The breathing signal of every person can still be captured at the receiver from the phase difference data. However, when the number of the persons is further increased, the success rate will decrease quickly. In fact, the strength of the breathing signals for some persons will become too weak to be detected when there are too many people blocking each other. 5. RELATED WORK This work is closely related to sensor based and RF signal based vital signs monitoring, as well as CSI based indoor localization and human activity recognition, which are discussed in the following. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:25 Sensor-based vital signs monitoring with wearable and smart devices employs the special hardware attached the person body to monitor breathing and heart rate data. The capnometer can measure carbon dioxide (CO2) concentrations in respired gases, which are employed to monitor patients breathing rate in hospital. However, it is uncomforted for patients to wear them, which are thus leveraged in clinical environments [Mogue and Rantala 1988]. Photoplethysmography (PPG) is an optical technique to monitor the blood volume changes in the tissues by measuring light absorption changes, thus requiring the sensors attached to persons finger such as pulse oximeters [Shariati and Zahedi 2005]. In addition, smartphone can use the camera to detect light changes from the video frames which can extract the PPG signal to monitor the heart rate [Scully et al. 2010]. Moreover, the smartphones can also estimate breathing rate by using the built-in accelerometer, gyroscope [Aly and Youssef 2016] and microphone [Ren et al. 2015], which require persons to put smartphones near-by and wear sensors for breathing monitoring. However, these techniques based on sensors cannot be applied for remote monitoring vital signs. RF based systems for vital signs monitoring use wireless signals to track the breathing-induced chest change of a person, which are mainly based on radar and WiFi techniques. For radar based vital signals monitoring, Vital-Radio employs frequency modulated continuous wave (FMCW) radar to estimate breathing and heart rates, even for two person subjects in parallel [Adib et al. 2015]. But the system requires a custom hardware with a large bandwidth from 5.46 GHz to 7.25 GHz. For WiFi based vital signs monitoring, UbiBreathe system employ WiFi RSS for breathing rate monitoring, which, however, requires the device placed in the line of sight path between the transmitter and the receiver for estimating the breathing rate [Abdelnasser et al. 2015]. Moreover mmVital based on RSS can use 60 GHz millimeter wave (mmWave) signal for breathing and heart rates monitoring with the larger bandwidth about 7GHz, which cannot monitor the longer distance and require high gain directional antennas for the transmitter and the receiver [Yang et al. 2016][Gong et al. 2010]. Recently, the authors leverage the amplitudes of CSI data to monitoring vital signs [Liu et al. 2015]. This work is mainly to track the vital signs when a person is sleeping, which is limited for monitoring a maximum of two persons at the same time. In additional to vital signs monitoring, recently, CSI based sensing systems have also been used for indoor localization and human activity recognition [Cushman et al. 2016]. CSI-based fingerprinting systems have been proposed to obtain high localization accuracy. FIFS is the first work to uses the weighted average of CSI amplitude values over multiple antennas for indoor localization [Xiao et al. 2012]. To exploit the diversity among the multiple antennas and subcarriers, DeepFi leverage 90 CSI amplitude data from the three antennas with a deep autoencoder network for indoor localization [Wang et al. 2015]. Also, PhaseFi leverages calibrated CSI phase data for indoor localization based on deep learning [Wang et al. 2015]. Different from CSI-based fingerprinting techniques, SpotFi system leverages a super-resolution algorithm to estimate the angle of arrival (AoA) of multipath components for indoor localization based on CSI data from three antennas [Kotaru et al. 2015]. On the other hand, E-eyes system leverages CSI amplitude values for recognizing household activities such as washing dishes and taking a shower [Wang et al. 2014]. WiHear system employs specialized directional antennas to measure CSI changes from lip movement for determining spoken words [Wang et al. 2014]. CARM system considers a CSI based speed model and a CSI based activity model to build the correlation between CSI data dynamics and a given human activity [Wang et al. 2015]. Although CSI based sensing are effective for indoor localization and activity recognitions, there are few works for using CSI phase difference data to detect multiple persons behaviors at the same time. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:26 X. Wang et al. The TensorBeat system is motivated by these interesting prior works. To the best of our knowledge, we are the first to leverage CSI phase difference data for multiple persons breathing rate estimation. We are also the first to employ tensor decomposition for RF sensing based vital signs monitoring, which can be also employed for indoor localization and human activity recognition. 6. 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SAMPLE-LEVEL CNN ARCHITECTURES FOR MUSIC AUTO-TAGGING USING RAW WAVEFORMS Taejun Kim1 , Jongpil Lee2 , Juhan Nam2 arXiv:1710.10451v2 [cs.SD] 14 Feb 2018 1 School of Electrical and Computer Engineering, University of Seoul, 2 Graduate School of Culture Technology, KAIST, ktj7147@uos.ac.kr, {richter, juhannam}@kaist.ac.kr ABSTRACT Recent work has shown that the end-to-end approach using convolutional neural network (CNN) is effective in various types of machine learning tasks. For audio signals, the approach takes raw waveforms as input using an 1-D convolution layer. In this paper, we improve the 1-D CNN architecture for music auto-tagging by adopting building blocks from state-of-the-art image classification models, ResNets and SENets, and adding multi-level feature aggregation to it. We compare different combinations of the modules in building CNN architectures. The results show that they achieve significant improvements over previous state-of-the-art models on the MagnaTagATune dataset and comparable results on Million Song Dataset. Furthermore, we analyze and visualize our model to show how the 1-D CNN operates. Index Terms— convolutional neural networks, music auto-tagging, raw waveforms, multi-level learning 1. INTRODUCTION Time-frequency representations based on short-time Fourier transform, often scaled in a log-like frequency such as melspectrogram, are the most common choice of input in the majority of state-of-the-art music classification algorithms [1, 2, 3, 4, 5]. The 2-dimentional input represents acoustically meaningful patterns well but requires a set of parameters, such as window size/type and hop size, which may have different optimal settings depending on the type of input signals. In order to overcome the problem, there have been some efforts to directly use raw waveforms as input particularly for convolutional neural networks (CNN) based models [6, 7]. While they show promising results, the models used large filters, expecting them to replace the Fourier transform. Recently, Lee et. al. [8] addressed the problem using very small filters and successfully applied the 1D CNN to the music autotagging task. Inspired from the well-known VGG net that uses very small size of filters such as 3 × 3, [9], the sample-level CNN model was configured to take raw waveforms as input and have filters with such small granularity. A number of techniques to further improve performances of CNNs have appeared recently in image domain. He et. al. introduced ResNets which includes skip connections that enables a very deep CNN to be effectively trained and makes gradient propagation fluent [10]. Using the skip connections, they could successfully train a 1001-layer ResNet [11]. Hu et. al proposed SENets [12] which includes a building block called Squeeze-and-Excitation (SE). Unlike other recent approaches, the block concentrates on channel-wise information, not spatial. The SE block adaptively recalibrates feature maps using a channel-wise operation. Most of the techniques were developed in the field of computer vision but they are not fully adopted for music classification tasks. Although there were a few approaches to readily apply them to audio domain [7, 13]. They used 2D representations as input [13] or used large filters for the first 1D convolutional layer [7]. On the other hand, some methods are concerned with overall architecture of the model rather than designing a fine-grained building block [2, 14, 15, 16, 17]. Specifically, multi-level feature aggregation combines several hidden layer representations for final prediction [2, 14]. They significantly improved the performance in music auto-tagging by taking different levels of abstractions of tag labels into account. In this paper, we explore the building blocks of advanced CNN architectures, ResNets and SENets, based on the sample-level CNN for music auto-tagging. Also, we observe how the multi-level feature aggregation affects the performance. The results show that they achieve significant improvements over previous state-of-the-art models on the MagnaTagATune dataset and comparable results on Million Song Dataset. Furthermore, we analyze and visualize our model built with the SE blocks to show how the 1D CNN operates. The results show that the input signals are processed in a different manner depending on the level of layers. 2. ARCHITECTURES All of our models are based on the sample-level 1D CNN model [8], which is constructed with the basic block shown in Figure 1(b). Every filter size of the convolution layers is fixed raw waveform strided conv 19683 × 128 1D convolutional block ×9 2187× 128 729 × 256 243 × 256 81 × 256 FC ×2 9 × 256 3 × 256 1 × 512 512 global max pooling global max pooling 27 × 256 multi-level feature aggregation 6551 × 128 256 Conv1D Conv1D BatchNorm BatchNorm relu relu Conv1D Conv1D BatchNorm BatchNorm 59049 × 1 MaxPool relu relu MaxPool T×C GlobalAvgPool 1×C FC 1×αC relu Dropout Dropout Conv1D Conv1D BatchNorm BatchNorm T×C FC T×C 1×C GlobalAvgPool sigmoid relu Scale 1×C FC MaxPool T×C 1×αC relu FC T×C 1×C sigmoid Scale 256 T×C 50 relu MaxPool tag prediction (a) Overview of the architecture (b) Basic block [8] (c) SE block (d) Res-n block (e) ReSE-n block Fig. 1. The proposed architecture for music auto-tagging. (a) The models consist of a strided convolutional layer, 9 blocks, and two fully-connected (FC) layers. The outputs of the last three blocks are concatenated and then used as input of the last two FC layers. Output dimensions of each block (or layer) are denoted inside of them (temporal×channel). (b-e) The 1D convolutional building blocks that we evaluate. to three. The differences between the sample-level CNN and ours are the use of advanced building blocks and multi-level feature aggregation. In this section, we describe the details. 2.1. 1D convolutional building blocks 2.1.1. SE block We utilize the SE block from SENets to increase representational power of the basic block. As shown in Figure 1(c), we simply attached the SE block to the basic block. The SE block recalibrates feature maps from the basic block through two operations. One is squeeze operation that aggregates a global temporal information into channel-wise statistics using global average pooling. The operation reduces the temporal dimensionality (T ) to one, averaging outputs from each channel. The other is excitation operation that adaptively recalibrates feature maps of each channel using the channel-wise statistics from the squeeze operation and a simple gating mechanism. The gating mechanism consists of two fully-connected (FC) layers that compute nonlinear interactions among channels. Finally, the original outputs from the basic block are rescaled by channel-wise multiplication between the feature map and the sigmoid activation of the second FC layer. Unlike the original SE block in SENets, our excitation operation does not form a bottleneck. On the contrary, we expand the channel dimensionality (C) to αC at the first FC layer, and then reduce the dimensionality back to C at the second layer. We set the amplifying ratio α to be 16, after a grid search with α = [2−3 , 2−2 , ..., 26 ]. 2.1.2. Res-n block Inspired by skip connections from ResNets, we modified the basic block by adding a skip connection as shown in Figure 1(d). Res-n denotes that the block uses n convolutional layers where n is one or two. Specifically, Res-2 is a block that has the additional layers denoted by the dotted line in Figure 1(d), and Res-1 is a block that has a skip connection only. When the block uses two convolutional layers (Res-2), we add a dropout layer (with a drop ratio of 0.2) between two convolutions to avoid overfitting. This technique was firstly introduced at WideResNets [18]. 2.1.3. ReSE-n block The ReSE-n block is a combination of the SE and Res-n blocks as shown in Figure 1(e). n denotes the number of convolutional layers in the block, where n is also one or two. A dropout layer is inserted when n is two. 2.2. Multi-level feature aggregation Fig. 1(a) shows the multi-level feature aggregations that we configured. The outputs of the last three blocks are concate- Table 1. AUCs of CNN architectures on MTAT. “multi” and “no multi” indicates if the multi-level feature aggregation is used or not. † denotes using a weight decay of 10−4 . MTAT Block Basic [8] SE Res-1 Res-2 ReSE-1 ReSE-2 multi no multi 0.9077 0.9111 0.9037 0.9098 0.9053 0.9113† 0.9055 0.9083 0.9048 0.9061 0.9066 0.9102† nated and then delivered to the FC layers. Before the concatenation, temporal dimensions of the outputs are reduced to one by a global max pooling. Unlike [2], the concatenation occurs while training the CNN and the average pooling over the whole audio clip (i.e. 29 second long), which followed by the global max pooling, is not included. 3. EXPERIMENTS 3.1. Datasets We evaluated the proposed architectures on two datasets, MagnaTagATune (MTAT) dataset [19] and Million Song Dataset (MSD) annotated with the Last.FM tags [20]. We split and filtered both of the datasets, following the previous work [5, 6, 8]. We used the 50 most frequent tags. All songs are trimmed to 29 seconds long, and resampled to 22050Hz as needed. The song is divided into 10 segments of 59049 samples. To evaluate the performance of music autotagging which is a multi-class and multi-label classification task, we computed the Area Under the Receiver Operating Characteristic curve (AUC) for each tag and computed the average across all 50 tags. During the evaluation, we average predictions across all segments. 3.2. Implementation details All the networks were trained using SGD with Nesterov momentum of 0.9 and mini-batch size 23. The initial learning rate is set to 0.01, decayed by a factor of 5 when a validation loss is on a plateau. None of the regularizations are used on MSD. A dropout layer of 0.5 was inserted before the last FC layer on MTAT. For all building blocks, we evaluated either with or without the multi-level feature aggregation. Since the training for MSD takes much time longer than MTAT, we explored the architectures mainly on MTAT, and then trained the two best models on MSD. Code and models built with TensorFlow and Keras are available at the link1 . 1 https://github.com/tae-jun/resemul Table 2. AUCs of state-of-the-art models on MTAT and MSD. † denotes that the model used an ensemble of three. Model Bag of multi-scaled features [3] End-to-end [6] Transfer learning [4] Persistent CNN [21] Time-Frequency CNN [22] Timbre CNN [23] 2D CNN [5] CRNN [1] Multi-level & multi-scale [2] SampleCNN multi-features [14] SampleCNN [8] SE [This work] ReSE [This work] MTAT 0.8980 0.8815 0.8800 0.9013 0.9007 0.8930 0.8940 0.9017† 0.9064† 0.9055 0.9111 0.9113 MSD 0.8510 0.8620 0.8878† 0.8842 0.8812 0.8840 0.8847 4. RESULTS AND DISCUSSION 4.1. Comparison of the architectures Table 1 summarizes the evaluation results of compared CNN architectures on the MTAT dataset. They show that the SE block is more effective than the Res-n blocks, increasing the performance of the basic block for all cases. In the Res-n block, only adding the skip connection to the basic block (Res-1) actually decreases the performance. The combination of the SE and the Res-2 improves it slightly more. However, a training time of the ReSE-2 is 1.8 times longer than the basic block whereas the SE block only 1.08 times longer. Thus, if the training or prediction time of the models is important, the SE model will be preferred to the ReSE-2. The effect of the multi-level aggregation is valid for the majority of the models. We obtained two best results in Table 1 by using the multi-level aggregation. 4.2. Comparison with state-of-the-arts Table 2 compares previous state-of-the-art models in music auto-tagging with our best models, the SE block and ReSE2 block, each with multi-level aggregation. On the MTAT dataset, our best models outperform all the previous results. On MSD, they are not the best but are comparable to the second-tier. 5. ANALYSIS OF EXCITATION To lay the groundwork for understanding how 1D CNNs operate, we analyze the sigmoid activations of excitations in the SE blocks at different levels graphically and quantitatively. In this section, we observe how the SE blocks recalibrate channels, depending on which level they exist. The blocks used for the analysis are from the SE model using the multi-level Table 3. Co-occurrence matrix of the tags used in Figure 2 1st block classical metal dance classical 704 0 1 metal 0 166 0 dance 1 0 153 5th (mid) block std 0.05 0.04 0.03 0.02 1 2 3 4 5 6 7 8 9 block level 9th (last) block Fig. 2. Visualization of the sigmoid activations of excitations in the SE model. The channel index was sorted by the average of the activations. feature aggregation and they were trained on MTAT. The activations were extracted from its test set. The activations were averaged over all segments separately for each tag. 5.1. Graphical analysis For this analysis, we chose three tags, classical, metal, and dance that are not similar to each other as shown in Table 3. Figure 2 shows the average sigmoid activations in the SE blocks for the songs with the three tags. The different levels of activations indicate that the SE blocks process input audio differently depending on the tag (or genre) of the music. That is, every block in Figure 2 fires different patterns of activations for each tag at a specific channel. This trend is strongest at the first block (top), weakest at the mid block (middle), and becomes stronger again at the last block (bottom). This trend is somewhat different from what are observed in the image domain [12], where the exclusiveness of average excitation for input with different labels are monotonically increasing along the layers. Specifically, the first block fires high activations for classical, low ones for dance, and even lower ones for metal for the majority of the channels. On the other hand, the activations of the last block vary depending on the tags. For example, the activations of metal are high at some channels but low at the others, which makes the activations noisy even though they are sorted. We can interpret this result as follows. The first block normalizes the loudness of the audios because the block fires high activations for classical music, which tend to have small volume, and low activa- Fig. 3. Standard deviations (std) of the activations of excitations across all tags along each layer. tions for metal music, which tend to have large volume. Also, the middle block processes common features among them as they have similar levels of activations. Finally, the noisy exclusiveness in the last block indicates that they effectively discriminate the music with different tags. 5.2. Quantitative analysis We assure the exclusiveness trend by measuring standard deviations of the activations across all tags at every level. Figure 3 shows that the higher the standard deviation is, the more the block responses to the song differently according to its tag. The result shows that the standard deviation is highest at the first block, it drops and stays low up to the 5th block and then increases gradually until the last block. That is, the four lower blocks except the the bottom one (2 to 5) tend to handle general features whereas the four upper blocks (6 to 9) tend to progressively more discriminative features. 6. CONCLUSION We proposed 1D convolutional building blocks based on the previous work, the sample-level CNN, ResNets, and SENets. The ReSE block, which is a combination of the three models, showed the best performance. Also, the multi-level feature aggregation showed improvements on the majority of the building blocks. Through the experiments, we obtained stateof-the-art performance on the MTAT dataset and high-ranked results on MSD. In addition, we analyzed the activations of excitation in SE model to understand the effect. With this analysis, we could observe that the SE blocks process nonsimilar songs exclusively and how the different levels of the model process the songs in a different manner. 7. REFERENCES [1] Keunwoo Choi, György Fazekas, Mark Sandler, and Kyunghyun Cho, “Convolutional recurrent neural networks for music classification,” in ICASSP. IEEE, 2017, pp. 2392–2396. [2] Jongpil Lee and Juhan Nam, “Multi-level and multiscale feature aggregation using pretrained convolutional neural networks for music auto-tagging,” IEEE Signal Processing Letters, vol. 24, no. 8, pp. 1208–1212, 2017. [3] Sander Dieleman and Benjamin Schrauwen, “Multiscale approaches to music audio feature learning,” in International Society of Music Information Retrieval Conference (ISMIR), 2013, pp. 116–121. 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On the linear quadratic problem for systems with time reversed Markov jump parameters and the duality with filtering of Markov jump linear systems arXiv:1611.07558v1 [cs.SY] 22 Nov 2016 Daniel Gutierrez and Eduardo F. Costa Abstract— We study a class of systems whose parameters are driven by a Markov chain in reverse time. A recursive characterization for the second moment matrix, a spectral radius test for mean square stability and the formulas for optimal control are given. Our results are determining for the question: is it possible to extend the classical duality between filtering and control of linear systems (whose matrices are transposed in the dual problem) by simply adding the jump variable of a Markov jump linear system. The answer is positive provided the jump process is reversed in time. I. I NTRODUCTION In this note we study a class of systems whose parameters are driven by a time reversed Markov chain. Given a time horizon ℓ and a standard Markov chain {η(t), t = 0, 1, . . .} taking values in the set {1, 2, . . . , N }, we consider the process θ(t) = η(ℓ − t), 0 ≤ t ≤ ℓ, (1) and the system ( x(t + 1) = Aθ(t) x(t) + Bθ(t) u(t), Φ: x(0) = x0 , 0 ≤ t ≤ ℓ − 1, (2) where, as usual, x represents the state variable of the system and u is the control variable. These systems may be encountered in real world problems, specially when a Markov chain interacts with the system parameters via a first in last out queue. An example consists of drilling sedimentary rocks whose layers can be modelled by a Markov chain from bottom to top as a consequence of their formation process. The first drilled layer is the last formed one. Another example is a DC-motor whose brush is grind by a machine subject to failures, leaving a series of imprecisions on the brush width that can be described by a Markov chain, so that the last failure will be the first to affect the motor collector depending on how the brush is installed. One of the most remarkable features of system Φ is that it provides a dual for optimal filtering of standard Markov jump linear systems (MJLS). In fact, if we consider a quadratic cost functional for system Φ with linear state feedback, leading to an optimal control problem [4] that we call time reversed Markov jump linear quadratic problem (TRM-JLQP), then we show that the solution is identical to the gains of the linear minimum mean square estimator This work was supported by FAPESP, CNPQ and Capes. Authors are with the Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Brasil dgutip@icmc.usp.br, efcosta@icmc.usp.br (LMMSE) formulated in [8], [9], with time-reversed gains and transposed matrices. In perspective with existing duality relations, the one obtained here is a direct generalization of the well known relation between control and filtering of linear time varying systems as presented for instance in [10, Table 6.1], or also in [3], [16], [17] in different contexts. As for MJLS, the duality between control and filtering have been considered e.g. in [2], [5], [7], [8], [12], [13], [15], while purely in the context of standard MJLS, thus leading to more complex relations involving certain generalized coupled Riccati difference equations. Here, the duality follows naturally from the simple reversion of the Markov chain given in (1), with no extra assumptions nor complex constructions. Another interesting feature of Φ is that the variable E{x(t)x(t)′ · 1{θ(t)=i} }, which is commonly used in the literature of MJLS, [1], [8], [11], [14], evolves along time t according to a time-varying linear operator, as shown in Remark 1, in a marked dissimilarity with standard MJLS. This motivated us to employ X(k), the conditioned second moment of x(k), leading to time-homogeneous operators. The contents of this note are as follows. We present basic notation in Section II. In Section III we give the recursive equation describing X, which leads to a stability condition involving the spectral radius of a time-homogeneous linear operator. In Section IV, we formulate and solve the TRMJLQ problem, following a proof method where we decompose X into two components as to handle θ that are visited with zero probability. The duality with the LMMSE then follows in a straightforward manner, as presented in Section V. Concluding remarks are given in Section VI. II. N OTATION AND THE SYSTEM SETUP Let ℜn be the n-dimensional euclidean space and ℜm,n be the space formed by real matrices of dimension m by n. We write C m,n to represent the Hilbert space composed of N real matrices, that is Y = (Y1 , . . . , YN ) ∈ C m,n , where Yi ∈ ℜm,n , i = 1, . . . , N . The C n,m equipped Pspace N ′ with the inner product hY, Zi = Tr(Y i Zi ), where i=1 Tr(·) is the trace operator and the superscript ′ denotes the transpose, is a Hilbert space. The inner product induces the norm kY k = hY, Y i1/2 . If n = m, we write simply C n . The mathematical operations involving elements of C n,m , are used in element-wise fashion, e.g. for Y and Z in C n we have Y Z = (Y1 Z1 , . . . , YN ZN ), where Yi Zi is the usual matrix multiplication. Similarly, for a set of scalars α = (α1 , . . . , αN ) ∈ C 1 we write αY = (α1 Y1 , . . . , αN YN ). Regarding the system setup, it is assumed throughout the paper that x0 ∈ ℜn is a random variable with zero mean satisfying E{x0 x′0 } = ∆. We have x(t) ∈ ℜn and u(t) ∈ ℜm . The system matrices belong to given sets A, E ∈ C n , B ∈ C n,m , C ∈ C s,n and D ∈ C s,m with Ci′ Di = 0 and Di′ Di > 0 for each i = 1, . . . , N . We write πi (t) = Pr(θ(t) = i), where Pr(·) is the probability measure; π(t) is considered as an element of C 1 , that is, π(t) = (π1 (t), . . . , πN (t)). πi stands for the limiting distribution of the Markov chain η when it exists, in such a manner that πi = limℓ→∞ πi (0). Also, we denote by P = [pij ], i, j = 1, . . . , N the transition probability matrix of the Markov chain η, so that for any t = 1, . . . , ℓ, Pr(θ(t − 1) = j \| θ(t) = i) = = Pr(η(ℓ − t + 1) = j \| η(ℓ − t) = i) = pij . UZ,i (Y ) = VZ,i (Y ) = Di (Y ) = pij Zj Yj Zj′ j=1 Zi′ Di (Y N X )Zi , (3) pji Yj . OF SYSTEM Φ Let E{·} be the expected value of a random variable. We consider the conditioned second moment of x(t) defined by Xi (t) = E{x(t)x(t)′ \| θ(t) = i}, t = 0, 1, . . . . (4) Lemma 3.1: Consider the system Φ with u(t) = 0 for each t. The conditioned second moment X(t) ∈ C n is given by X(0) = (∆, . . . , ∆) and X(t + 1) = UA (X(t)), t = 0, 1, . . . ℓ − 1. (5) Proof: For a fixed, arbitrary i ∈ {0, . . . , N }, note that Xi (0) = E{x0 x′0 \| θ(0) = i} = E{x0 x′0 } = ∆. From (2), (4) and the total probability law we obtain: Xi (t + 1) = E{Aθ(t) x(t)x(t)′ A′θ(t) \| θ(t + 1) = i} = N X ′ E{Aθ(t) x(t)x(t) A′θ(t) = E{Γ(η(ℓ), . . . , η(ℓ − t))\|η(ℓ − t) = j, η(ℓ − t − 1) = i} · Prob(η(ℓ − t) = j \| η(ℓ − t − 1) = i} = E{Γ(η(ℓ), . . . , η(ℓ − t))\|η(ℓ − t) = j}pij = E{Γ(θ(0), . . . , θ(t)) \| θ(t) = j}pij , then by replacing Γ with Aθ (t)x(t)x(t)′ A′θ(t) and applying the above in (6) yields Xi (t + 1) = · 1θ(t)=j \| θ(t + 1) = i}. N X pij Aj Xj (t)A′j = UA,i (X(t)), j=1 which completes the proof. Remark 1: Let W (t) ∈ C n , t = 0, . . . , ℓ be given by Wi (t + 1) = N X pij j=1 πi (t + 1) Aj Wj (t)A′j . πj (t) j=1 In order to compute the right hand side of (6), we need the following standard Markov chain property: for any function (8) Note that the Markov chain measure appears explicitly, leading to a time-varying mapping from W (t) to W (t + 1). The only exception is when the Markov chain is reversible, in which case the facts that πj pji = πi pij and that the Markov chain starts with the invariant measure (by definition) yield πi (t + 1) πi = pji , = pij πj (t) πj in which case W evolves exactly as in a standard MJLS. The following notion is adapted from [8, Chapter 3]. Definition 3.1: We say that the system Φ with u(t) = 0 is mean square stable (MS-stable), whenever lim E kx(ℓ)k2 = 0. ℓ→∞ This is equivalent to say that the variable X(ℓ) converges to zero as ℓ goes to infinity, leading to the following result. Theorem 3.1: The system Φ with u(t) = 0 is MS-stable if and only if the spectral radius of UA is smaller than one. IV. T HE TRM-JLQ PROBLEM Let the output variable y given by y(ℓ) = Eθ(ℓ) x(ℓ) and y(t) = Cθ(t) x(t) + Dθ(t) u(t), t = 0, . . . , ℓ − 1. The TRM-JLQ consists of minimizing the mean square of y with ℓ stages, as usual in jump linear quadratic problems, ) ( ℓ X 2 (9) min E ky(t)k . u(0),...,u(ℓ−1) (6) (7) This variable is commonly encountered in the majority of papers dealing with (standard) MJLS. However, calculations similar to that in Lemma 3.1 lead to pij j=1 III. P ROPERTIES E{Γ(θ(0), . . . , θ(t)) · 1θ(t)=j \| θ(t + 1) = i} = E{Γ(η(ℓ), . . . , η(ℓ − t)) · 1η(ℓ−t)=j \| η(ℓ − t − 1) = i} Wi (t) = E{x(t)x(t)′ · 1θ(t)=i }. No additional assumption is made on the Markov chain, yielding a rather general setup that includes periodic chains, important for the duality relation given in Remark 2. We shall deal with linear operators UZ , VZ , D : C n → C n . We write the i-th element of UZ (Y ) by UZ,i (Y ) and similarly for the other operators. For each i = 1, . . . , N , we define: N X Γ : θ(0), . . . , θ(t) → ℜn,n we have t=0 Regarding the information structure of the problem, we assume that θ(t) is available to the controller, that the control is in linear state feedback form, u(t) = Kθ(t) (t)x(t), t = 0, 1, . . . ℓ − 1, (10) where K(t) ∈ C m,n is the decision variable, and that one should be able to compute the sequence K(0), . . . , K(ℓ − 1) prior to the system operation, that is, K(t) is not a function of the observations (x(s), θ(s)), 0 ≤ s ≤ ℓ. The conditioned second moment X for the closed loop system is of much help in obtaining the solution. The recursive formula for X follows by a direct adaptation of Lemma 3.1, by replacing A ∈ C n with its closed loop version Ai (t) = Ai + Bi Ki (t). (11) Then, V t (X) = hP (t), Xi (19) op and K (t) = M (t), t = 0, . . . , ℓ. Proof: We apply the dynamic programming approach for the costs defined in (16) and the system in (12), whose state is the variable X := X(t). It can be checked that U is the adjoint operator of V, and consequently V t (X) = min hπ(t)Q(t), Xi + hP (t + 1), UA(t) (X)i, K(t) = min hπ(t)Q(t) + A(t)′ D(P (t + 1))A(t), Xi. (20) K(t) Lemma 4.1: The conditioned second moment X(t) ∈ C is given by X(0) = (∆, . . . , ∆) and X(t + 1) = UA(t) (X(t)), t = 0, 1, . . . , ℓ − 1. In what follows, for brevity we denote Q(ℓ) = π(ℓ)E ′ E, Q(t) = C ′ C + K(t)′ D′ DK(t), n N t = 0, . . . , ℓ − 1. t=0 Proof: The mean square of the terminal cost, y(ℓ) is: E{ky(ℓ)k2 } = N X ′ E{x(ℓ)′ Eθ(ℓ) Eθ(ℓ) x(ℓ) · 1θ(ℓ)=i } i=1 = N X (14) i=1 Now, by a calculation similar as above leads to ′ ′ Substituting (14) and (15) into (9) we obtain (13). Let us denote the gains attaining (13) by K op (t). From a dynamic programming standpoint, we introduce value functions V t : C n → ℜ by: V ℓ = hπ(ℓ)Q(ℓ), X(ℓ)i and for t = ℓ − 1, ℓ − 2, . . . , 0, ( ℓ ) X t (16) hπ(τ )Q(τ ), X(τ )i , V (X) = min τ =t where X(t) = X and X(τ ), τ = t + 1, . . . , ℓ, satisfies (12). Theorem 4.1: Define P (t) ∈ C n and M (t) ∈ C m,n , t = 0, . . . , ℓ − 1, as follows. Let P (ℓ) = π(ℓ)E ′ E and for each t = ℓ − 1, . . . , 0 and i = 1, . . . , N , compute: if πi (t) = 0, Mi (t) = 0 and Pi (t) = 0, Ri (t) = (Bi′ Di (P (t + 1))Bi + πi (t)Di′ Di ), Pi (t) = πi (t)Ci′ Ci + A′i Di (P (t + 1))Ai − A′i Di (P (t + 1))Bi Ri (t)−1 Bi′ Di (P (t is zero irrespectively of K(t). First, note that for i ∈ Nt we have πi (t)Qi (t)XN i = 0. Second, for i ∈ Nt one can check that πj (t + 1) = 0 for all j such that pji > 0, so that Pj (t + 1) = 0. This yields Di (P (t + 1)) = 0. Bringing these facts together and recalling that by construction XN i = 0 for all i ∈ / Nt , we evaluate (21) By substituting (21) into (20) we write V t (X) = min hπ(t)Q(t) + A(t)′ D(P (t + 1))A(t), XP i. K(t) X Tr (πi (t)Qi (t) = min {i∈N / t} + Ai (t)′ Di (P (t + 1))Ai (t))XPi . (22) By expanding some terms and after some algebra to complete the squares, we have X Tr (πi (t)Ci′ Ci + A′i Di (P (t + 1))Ai V t (X) = min K(t) {i∈N / t} +(Ki (t) − Oi (t))′ (Bi′ Di (P (t + 1))Bi + πi (t)Di′ Di ) ·(Ki (t) − Oi (t)) − Oi (t)′ (Bi′ Di (P (t + 1))Bi + πi (t)Di′ Di )Oi (t))XP i , (23) where Oi (t) = Mi (t) as given in (17). This makes clear that the minimal cost is achieved by setting Kiop (t) = Mi (t), op i∈ / Nt . Now, by replacing Ki (t) with Ki (t) in (23), X Tr Pi (t)XP V t (X) = i {i∈N / t} else (if πi (t) > 0), Mi (t) = Ri (t)−1 Bi′ Di (P (t + 1))Ai , hπ(t)Q(t) + A(t)′ D(P (t + 1))A(t), XN i K(t) ′ E{ky(t)k } = hπ(t)(C C + K(t) D DK(t)), X(t)i. (15) K(t),...,K(ℓ−1) We write X = X + XP , where XN is such that XN i = 0 for any i ∈ / Nt , and in a similar fashion XPi = 0 for i ∈ Nt . We now show that the term hπ(t)Q(t) + A(t)′ D(P (t + 1))A(t), XN i = 0. Tr(πi (ℓ)Ei′ Ei Xi (ℓ) = hπ(ℓ)E ′ E, X(ℓ)i. 2 Nt = {i : πi (t) = 0}. (12) Lemma 4.2: The TRM-JLQ problem can be formulated as ( ℓ ) X (13) min hπ(t)Q(t), X(t)i . K(0),...,K(ℓ−1) Let us decompose X as follows; let the set of states θ having zero probability of being visited at time t be denoted by (17) with Pi (t) as given in (18), i ∈ / Nt . Finally, by choosing Pi (t) = 0, i ∈ Nt , we write X X Tr Pi (t)XN + Tr Pi (t)XP V t (X) = i , i {i∈N / t} + 1))Ai . (18) P {i∈Nt } N = hP (t), X i + hP (t), X i = hP (t), Xi, which completes the proof. V. T HE DUALITY BETWEEN THE TRM-JLQ LMMSE FOR STANDARD MJLS AND THE We consider the LMMSE for standard MJLS as presented in [6], [9]. The problem consists of finding the sequence of sets of gains K f (t), t = 0, . . . , ℓ, that minimizes the covariance of the estimation error z̃(t) = ẑ(t) − z(t) when the estimate is given by a Luenberger observer in the form f ẑ(t + 1) = Aη(t+1) ẑ(t) + Kη(t+1) (t)(y(t) − Lη(t) ẑ(t)), where y(t) is the output of the MJLS   z(t + 1) = Fη(t+1) z(t) + Gη(t+1) ω(t) y(t) = Lη(t+1) z(t) + Hη(t+1) ω(t)   z(0) = z0 , (24) and ω(t) and z0 are i.i.d. random variables satisfying E{ω(t)} = 0, E{ω(t)ω(t)′ } = I and E{z0 z0′ } = Σ. Moreover, it is assumed that Li Hi′ = 0 and Hi Hi′ > 0. We write υi (t) = Prob(η(t) = i), i = 1, . . . , N , so that it is the time-reverse of π, υ(t) = π(ℓ − t). Note that we are considering the same problem as in [6], [9], though our notation is slightly different: here we assume that (y(t), η(t + 1)) is available for the filter to obtain ẑ(t + 1) and the system matrices are indexed by η(t+1), while in the standard formulation (y(t), η(t)) are observed at time t and the system matrices are indexed by η(t). This “time shifting” in η avoids a cluttering in the duality relation. Along the same line, instead of writing the filter gains as a function of the variable Yi (t) = E{z̃(t)z̃(t)′ · 1{η(t+1)=i} }, given by the coupled Riccati difference equation [9, Equation 24] Yi (t + 1) = N X j=1 pji Fj Yj (t)Fj′ + υj (t)Gj G′j − Fj Yj (t)L′j · Lj Yj (t)L′j + υj (t)Hj Hj′ −1 Lj Yj (t)Fj′ whenever υi (t) > 0 and Yi (t + 1) = 0 otherwise, in this note we use the variable Si (t) = E{z̃(t)z̃(t)′ · 1{η(t)=i} } defined in [7], leading to Y (t) = D(S(t)). Replacing this in the above equation, after some algebraic manipulation one obtains [7, Equation 8]: Si (t + 1) = υi (t)Gi G′i + Fi Di (S(t))Fi′ − Fi Di (S(t))L′i · (Li Fi (S(t))L′i + υi (t)Hi Hi′ )−1 Li Di (S(t))Fi′ , (25) whenever υi (t) > 0 and Si (t + 1) = 0 otherwise, with initial condition Si (0) = E{z̃(0)z̃(0)′ · 1{η(0)=i} } = υi (0)Σ. The optimal gains are given for t = 0, . . . , ℓ by −1 Kif (t) = Fi Di (S(t))L′i (Li Di (S(t))L′i + υi (t)Hi Hi′ ) (26) whenever υi (t) > 0 and Kif (t) = 0 otherwise. The duality relations between the filtering and control problems are now evident by direct comparison between (18) and (25). Fi , Li , Gi and Hi are replaced with A′i , Bi′ , Ci′ and Di′ , respectively. Moreover, comparing the initial conditions of the coupled Riccati difference equations, we see Σ replaced with E ′ E. Also, we note that P (0), P (1), . . . , P (ℓ) are equivalent to S(ℓ), S(ℓ−1), . . . , S(0), with a similar relation op for the gains Kif and Ki . The Markov chains driving the filtering and control systems are time-reversed one to each other. Remark 2: Time-varying parameters can be included both in standard MJLS and in Φ by augmenting the Markov state as to describe the pair (θ, t), 1 ≤ θ ≤ N , 0 ≤ t ≤ ℓ, and considering a suitable matrix P of higher dimension N × (ℓ + 1). Although this reasoning leads to a matrix P of high dimension, periodic and sparse, it is useful to make clear that our results are readily adaptable to plants whose matrices are in the form Aθ(t) (t). Either by this reasoning or by re-doing all computations given in this note for timevarying plants, we obtain the following generalization of [10, Table 6.1]. FILTERING of MJLS CONTROL of Φ Fi (t) Li (t) Gi (t) Hi (t) A′i (t) Bi′ (t) Ci′ (t) Di′ (t) Kif (t) Si (0) = υi (0)Σ Si (t) ηi (t) Ki (ℓ − t) Pi (ℓ) = πi (ℓ)Ei′ Ei Pi (ℓ − t) θi (ℓ − t) op ′ TABLE I S UMMARY OF THE F ILTERING /C ONTROL D UALITY. t = 0, . . . , ℓ. VI. C ONCLUDING REMARKS We have presented an operator theory characterization of the conditional second moment X, an MS stability test and formulas for the optimal control of system Φ. The results have exposed some interesting relations with standard MJLS. For system Φ it is fruitful to use the true conditional second moment X whereas for standard MJLS one has to resort to the variable W given in (7) to obtain a recursive equation similar to the ones expressed in the Lemmas 3.1 and 4.1. Moreover, these classes of systems are equivalent if and only if the Markov chain is revertible, as indicated in Remark 1. The solution of the TRM-JLQ problem is given in Theorem 4.1 in the form of a coupled Riccati equation that can be computed backwards prior to the system operation, as usual in linear quadratic problems for linear systems. The result beautifully extends the classic duality between filtering and control into the relations expressed in Table 1. R EFERENCES [1] E. F. Costa A. N. 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Offline and Online Models of Budget Allocation for Maximizing Influence Spread arXiv:1508.01059v3 [cs.DS] 12 Mar 2018 Noa Avigdor-Elgrabli∗ Gideon Blocq† Iftah Gamzu‡ Ariel Orda§ Abstract The research of influence propagation in social networks via word-of-mouth processes has been given considerable attention in recent years. Arguably, the most fundamental problem in this domain is the influence maximization problem, where the goal is to identify a small seed set of individuals that can trigger a large cascade of influence in the network. While there has been significant progress regarding this problem and its variants, one basic shortcoming of the underlying models is that they lack the flexibility in the way the overall budget is allocated to different individuals. Indeed, budget allocation is a critical issue in advertising and viral marketing. Taking the other point of view, known models allowing flexible budget allocation do not take into account the influence spread in social networks. We introduce a generalized model that captures both budgets and influence propagation simultaneously. For the offline setting, we identify a large family of natural budget-based propagation functions that admits a tight approximation guarantee. This family extends most of the previously studied influence models, including the well-known Triggering model. We establish that any function in this family implies an instance of a monotone submodular function maximization over the integer lattice subject to a knapsack constraint. This problem is known to admit an optimal 1 − 1/e ≈ 0.632-approximation. We also study the price of anarchy of the multi-player game that extends the model and establish tight results. For the online setting, in which an unknown subset of agents arrive in a random order and the algorithm needs to make an irrevocable budget allocation in each step, we develop a 1/(15e) ≈ 0.025-competitive algorithm. This setting extends the celebrated secretary problem, and its variant, the submodular knapsack secretary problem. Notably, our algorithm improves over the best known approximation for the latter problem, even though it applies to a more general setting. ∗ Yahoo Labs, Haifa, Israel. Email: noaa@oath.com. Technion, Haifa, Israel. Email: gideon@alumni.technion.ac.il. This work was done when the author was an intern at Yahoo Labs, Haifa, Israel. ‡ Amazon, Israel. Email: iftah.gamzu@yahoo.com. This work was done when the author was affiliated with Yahoo Labs, Haifa, Israel. § Technion, Haifa, Israel. Email: ariel@ee.technion.ac.il. † 1 Introduction The study of information and influence propagation in societies has received increasing attention for several decades in various areas of research. Recently, the emergence of online social networks brought forward many new questions and challenges regarding the dynamics by which information, ideas, and influence spread among individuals. One central algorithmic problem in this domain is the influence maximization problem, where the goal is to identify a small seed set of individuals that can trigger a large word-of-mouth cascade of influence in the network. This problem has been posed by Domingos and Richardson [13, 36] in the context of viral marketing. The premise of viral marketing is that by targeting a few influential individuals as initial adopters of a new product, it is possible to trigger a cascade of influence in a social network. Specifically, those individuals are assumed to recommend the product to their friends, who in turn recommend it to their friends, and so on. The influence maximization problem was formally defined by Kempe, Kleinberg and Tardos [26, 27]. In this setting, we are given a social network graph, which represents the individuals and the relationships between them. We are also given an influence function that captures the expected number of individuals that become influenced for any given subset of initial adopters. Given some budget b, the objective is to find a seed set of b initial adopters that will maximize the expected number of influenced individuals. Kempe et al. studied several operational models representing the step-by-step dynamics of propagation in the network, and analyzed the influence functions that are derived from them. While there has been significant progress regarding those models and related algorithmic issues, one shortcoming that essentially has not been treated is the lack of flexibility in the way that the budget is allocated to the individuals. Indeed, budget allocation is a critical factor in advertising and viral marketing. This raises some concerns regarding the applicability of current techniques. Consider the following scenario as a motivating example. A new daily deals website is interested in increasing its exposure to new audience. Consequently, it decides to provide discounts to individuals who are willing to announce their purchases in a social network. The company has several different levels of discounts that it can provide to individuals to incentivize them to better communicate their purchases, e.g., making more enthusiastic announcements on their social network. The company hopes that those announcements will motivate the friends of the targeted individuals to visit the website, so a word-of-mouth process will be created. The key algorithmic question for this company is whom should they offer a discount, and what amount of discounts should be offered to each individual. Alon et al. [1] have recently identified the insufficiency of existing models to deal with budgets. They introduced several new models that capture issues related to budget distribution among potential influencers in a social network. One main caveat in their models is that they do not take into account the influence propagation that happens in the network. The main aspect of our work targets this issue. 1.1 Our results We introduce a generalized model that captures both budgets and influence propagation simultaneously. Our model combines design decisions taken from both the budget models [1] and propagation models [26]. The model interprets the budgeted influence propagation as a two-stage process consisting of: (1) influence related directly to the budget allocation, in which the seed set of targeted individuals influence their friends based on their budget, and (2) influence resulting from a secondary word-of-mouth process in the network, in which no budgets are involved. Note that the two stages give rise to two influence functions whose combination designates the overall influence function of the model. We study our model in both offline and online settings. An offline setting. We identify a large family of natural budget-based propagation functions that admits a tight approximation guarantee. Specifically, we establish sufficient properties for the two influence functions 1 mentioned above, which lead to a resulting influence function that is both monotone and submodular. It is important to emphasize that the submodularity of the combined function is not the classical set-submodularity, but rather, a generalized version of submodularity over the integer lattice. Crucially, when our model is associated with such an influence function, it can be interpreted as a special instance of a monotone submodular function maximization over the integer lattice subject to a knapsack constraint. This problem is known to have an efficient algorithm whose approximation ratio of 1 − 1/e ≈ 0.632, which is best possible under P 6= NP assumption [40]. We then focus on social networks scenario, and introduce a natural budget-based influence propagation model that we name Budgeted Triggering. This model extends many of the previously studied influence models in networks. Most notably, it extends the well-known Triggering model [26], which in itself generalizes several models such as the Independent Cascade and the Linear Threshold models. We analyze this model within the two-stage framework mentioned above, and demonstrate that its underlying influence function is monotone and submodular. Consequently, we can approximate this model to within a factor of 1 − 1/e. We also consider a multi-player game that extends our model. In this game, there are multiple players, each of which is interested to spend her budget in a way that maximizes her own network influence. We establish that the price of anarchy (PoA) of this game is equal to 2. This result is derived by extending the definition of a monotone utility game on the integer lattice [32]. Specifically, we show that one of the conditions of the utility game can be relaxed, while still maintaining a PoA of at most 2, and that the refined definition captures our budgeted influence model. An online setting. In the online setting, there is unknown subset of individuals that arrive in a random order. Whenever an individual arrives, the algorithm learns the marginal influence for each possible budget assignment, and needs to make an irrevocable decision regarding the allocation to that individual. This allocation cannot be altered later on. Intuitively, this setting captures the case in which there is an unknown influence function that is partially revealed with each arriving individual. Similarly to before, we focus on the case that the influence function is monotone and submodular. Note that this setting is highly-motivated in practice. As observed by Seeman and Singer [37], in many cases of interest, online merchants can only apply marketing techniques on individuals who have engaged with them in some way, e.g., visited their online store. This gives rise to a setting in which only a small unknown sample of individuals from a social network arrive in an online fashion. We identify that this setting generalizes the submodular knapsack secretary problem [5], which in turn, extends the well-known secretary problem [14]. We develop a 1/(15e) ≈ 0.025-competitive algorithm for the problem. Importantly, our results not only apply to a more general setting, but also improve the best known competitive bound for the former problem, which is 1/(20e) ≈ 0.018, due to Feldman, Naor and Schwartz [17]. 1.2 Related work Models of influence spread in networks are well-studied in social science [22] and marketing literature [18]. Domingos and Richardson [13, 36] were the first to pose the question of finding influential individuals who will maximize adoption through a word-of-mouth effect in a social network. Kempe, Kleinberg and Tardos [26, 27] formally modeled this question, and proved that several important models have submodular influence functions. Subsequent research have studied extended models and their characteristics [29, 34, 6, 24, 8, 39, 15, 37, 28, 12], and developed techniques for inferring influence models from observable data [20, 19, 31]. Influence maximization with multiple players has also been considered in the past [6, 21, 25]. Kempe et al. [26], and very recently, Yang et al. [43], studied propagation models that have a similar flavor to our budgeted setting. We like to emphasize that there are several important distinctions between their models and ours. Most importantly, their models assume a strong type of fractional diminishing returns property that our integral model does not need to satisfy. Therefore, their models cannot capture the scenarios we describe. The reader may refer to the cited papers and the references therein for a broader review of the literature. 2 Alon et al. [1] studied models of budget allocation in social networks. As already mentioned, our model follows some of the design decisions in their approach. For example, their models support constraints on the amount of budget that can be assigned to any individual. Such constraints are motivated by practical marketing conditions set by policy makers and regulations. Furthermore, their models focus on a discrete integral notion of a budget, which is consistent with common practices in many organizations (e.g., working in multiplications of some fixed value) and related simulations [38]. We include those considerations in our model as well. Alon et al. proved that one of their models, namely, the budget allocation over bipartite influence model, admits an efficient (1 − 1/e)-approximation algorithm. This result was extended by Soma et al. [40] to the problem of maximizing a monotone submodular function over the integer lattice subject to a knapsack constraint. The algorithm for the above problems is a reminiscent of the algorithm for maximizing a monotone submodular set function subject to a knapsack constraint [41]. Note that none of those papers have taken into consideration the secondary propagation process that occurs in social networks. The classical secretary problem was introduced more than 50 years ago (e.g., [14]). Since its introduction, many variants and extension of that problem have been proposed and analyzed [30, 2, 3, 4]. The problem that is closest to the problem implied from our online model is the submodular knapsack secretary problem [5, 23, 17]. An instance of this problem consists of a set of n secretaries that arrive in a random order, each of which has some intrinsic cost. An additional ingredient of the input is a monotone submodular set function that quantifies the value gained from any subset of secretaries. The objective is to select a set of secretaries of maximum value under the constraint that their overall cost is no more than a given budget parameter. Note that our model extends this setting by having a more general influence function that is submodular over the integer lattice. Essentially, this adds another layer of complexity to the problem as we are not only required to decide which secretaries to select, but we also need to assign them budgets. 2 Preliminaries We begin by introducing a very general budgeted influence propagation model. This model will be specialized later when we consider the offline and online settings. In our model, there is a set of n agents and an influence function f : Nn → R+ . Furthermore, there is a capacity vector c ∈ Nn+ and a budget B ∈ N+ . Our objective is to compute a budget assignment to the agents b ∈ Nn , which maximizes the influence f (b). The vector b must Pn (1) respect the capacities, that is, 0 ≤ bi ≤ ci , for every i ∈ [n], (2) respect the total budget, namely, i=1 bi ≤ B. In the following, we assume without loss of generality that each ci ≤ B. We primarily focus on influence functions that maintain the properties of monotonicity and submodularity. A function f : Nn → R+ is called monotone if f (x) ≤ f (y) whenever x ≤ y coordinate-wise, i.e., xi ≤ yi , for every i ∈ [n]. The definition of submodularity for functions over the integer lattice is a natural extension of the classical definition of submodularity over sets (or boolean vectors): Definition 2.1. A function f : Nn → R+ is said to be submodular over the integer lattice if f (x) + f (y) ≥ f (x ∨ y) + f (x ∧ y), for all integer vectors x and y, where x ∨ y and x ∧ y denote the coordinate-wise maxima and minima, respectively. Specifically, (x ∨ y)i = max{xi , yi } and (x ∧ y)i = min{xi , yi }. In the remainder of the paper, we abuse the term submodular to describe both set functions and functions over the integer lattice. We also make the standard assumption of a value oracle access for the function f . A value oracle for f allows us to query about f (x), for any vector x. The question of how to compute the function f in an efficient (and approximate) way has spawned a large body of work in the context of social networks (e.g., [26, 10, 9, 33, 11, 7]). Notice that for the classical case of sets, the submodularity condition implies that f (S) + f (T ) ≥ f (S ∪ T ) + f (S ∩ T ), for every S, T ⊆ [n], and the monotonicity property implies that f (S) ≤ f (T ) if S ⊆ T . 3 An important distinction between the classical set setting and the integer lattice setting can be seen when we consider the diminishing marginal returns property. This property is an equivalent definition of submodularity of set functions, stating that f (S ∪ {i}) − f (S) ≥ f (T ∪ {i}) − f (T ), for every S ⊆ T and every i ∈ / T. However, this property, or more accurately, its natural extension, does not characterize submodularity over the integer lattice, as observed by Soma et al. [40]. For example, there are simple examples of a submodular function f for which f (x + χi ) − f (x) ≥ f (x + 2χi ) − f (x + χi ) does not hold. Here, χi is the characteristic vector of the set {i}, so that x + kχi corresponds to an update of x by adding an integral budget k to agent i. Note that a weaker variant of the diminishing marginal returns does hold for submodular functions over the integer lattice. Lemma 2.2 (Lem 2.2 [40]). Let f be a monotone submodular function over the integer lattice. For any i ∈ [n], k ∈ N, and x ≤ y, it follows that f (x ∨ kχi ) − f (x) ≥ f (y ∨ kχi ) − f (y) . 3 An Offline Model In this section, we study the offline version of the budgeted influence propagation model. As already noted, we consider budgeted influence propagation to be a two-stage process consisting of (1) direct influence related to the budget assignment, followed by (2) influence related to a propagation process in the network. In particular, in the first stage, the amount of budget allocated to an individual determines her level of effort and success in influencing her direct friends. This natural assumption is consistent with previous work [1]. Then, in the second stage, a word-of-mouth propagation process takes place in which additional individuals in the network may become affected. Note that the allocated budgets do not play role at this stage. We identify a large family of budget-based propagation functions that admit an efficient solution. Specifically, we first identify sufficient properties of the influence functions of both stages, which give rise to a resulting (combined) influence function that is monotone and submodular. Consequently, our model can be interpreted as an instance of a monotone submodular function maximization over the integer lattice subject to a knapsack constraint. This problem is known to have an efficient (1 − 1/e)-approximation [40], which is best possible under the assumption that P 6= NP. This NP-hardness bound of 1 − 1/e already holds for the special case of maximum coverage [16, 1]. We subsequently develop a natural model of budgeted influence propagation in social networks that we name Budgeted Triggering. This model generalizes many settings, including the well-known Triggering model. Note that the Triggering model already extends several models used to capture the spread of influence in networks, like the Independent Cascade, Linear Threshold, and Listen Once models [26]. We demonstrate that the influence function defined by this model is monotone and submodular, and thus, admits an efficient (1 − 1/e)-approximation. Technically, we achieve this result by demonstrating that the two-stage influence functions that underlie this model satisfy the sufficient properties mentioned above. Finally, we study an extension of the Budgeted Triggering model to a multi-player game. In this game, there are multiple self-interested players (e.g., advertisers), each of which is interested to spend her budget in a way that maximizes her own network influence. We establish that the price of anarchy (PoA) of the game is exactly 2. In fact, we prove that this result holds for a much more general type of games. Maehara et al. [32] recently defined the notion of a monotone utility game on the integer lattice, and demonstrated that its PoA is at most 2. Their utility game definition does not capture our multi-player game. We show that one of the conditions in their game definition can be relaxed while still maintaining the same PoA. Crucially, this relaxed definition captures our model. 4 3.1 A two-stage influence composition The two-stage process can be formally interpreted as a composition of two influence functions, f = h◦g. The first function g : Nn → {0, 1}n captures the set of influenced agents for a given budget allocation, while the second function h : {0, 1}n → R+ captures the overall number (or value) of influenced agents, given some seed agent set for a propagation process. In particular, the influenced agents of the first stage are the seed set for the second stage. We next describe sufficient conditions for the functions g and h which guarantee that their composition is monotone and submodular over the integer lattice. Note that we henceforth use notation related to sets and their binary vector representation interchangeably. Definition 3.1. A function g : Nn → {0, 1}n is said to be coordinate independent if it satisfies g(x ∨ y) ≤ g(x) ∨ g(y), for any x, y ∈ Nn . Definition 3.2. A function g : Nn → {0, 1}n is said to be monotone if g(x) ≤ g(y) coordinate-wise whenever x ≤ y coordinate-wise. Many natural influence functions are coordinate independent. One such example is the family of functions in which the output vector is a coordinate-wise disjunction over a set of n vectors, each of which captures the independent influence implied by some agent. Specifically, the ith vector in the disjunction is the result of some function fi : N → {0, 1}n indicating the affected agents as a result of any budget allocation assigned only to agent i. We are now ready to prove our composition lemma. Lemma 3.3. Given a monotone coordinate independent function g : Nn → {0, 1}n and a monotone submodular function h : {0, 1}n → R+ , the composition f = h ◦ g : Nn → R+ is a monotone submodular function over the integer lattice. Proof. The coordinate independence properties of g and the monotonicity of h imply that h(g(x) ∨ g(y)) ≥ h(g(x ∨ y)). In addition, from the monotonicity of g we know that g(x ∧ y) ≤ g(x) and g(x ∧ y) ≤ g(y). Thus, together with the monotonicity of h, we get that h(g(x) ∧ g(y)) ≥ h(g(x ∧ y)). Utilizing the above results, we attain that f is submodular since f (x) + f (y) = h(g(x)) + h(g(y)) ≥ h(g(x) ∨ g(y)) + h(g(x) ∧ g(y)) ≥ h(g(x ∨ y)) + h(g(x ∧ y)) = f (x ∨ y) + f (x ∧ y) , where the first inequality is by the submodularity of h. We complete the proof by noting that f is monotone since both g and h are monotone. Formally, given x ≤ y then it follows that f (x) = h(g(x)) ≤ h(g(y)) = f (y) by h’s monotonicity and since g(x) ≤ g(y) by g’s monotonicity. As a corollary of the lemma, we get the following theorem. 5 Theorem 3.4. Given a monotone coordinate independent function g : Nn → {0, 1}n and a monotone submodular function h : {0, 1}n → R+ , there is a (1 − 1/e)-approximation algorithm for maximizing the influence function f = h ◦ g : Nn → R+ under capacity constraints c ∈ Nn+ and a budget constraint B ∈ N+ , whose running time is polynomial in n, B, and the query time of the value oracle for f . Proof. We know by Lemma 3.3 that f is monotone and submodular. Consequently, we attain an instance of maximizing a monotone submodular function over the integer lattice subject to a knapsack constraint. Soma et al. [40] recently studied this problem, and developed a (1 − 1/e)-approximation algorithm whose running time is polynomial in n, B, and the query time for the value oracle of the submodular function. 3.2 The budgeted triggering model We now focus on social networks, and introduce a natural budget-based influence model that we call the Budgeted Triggering model. This model consists of a social network, represented by a directed graph G = (V, E) with n nodes (agents) and a set E of directed edges (relationships between agents). In addition, there is a function f : Nn → R+ that quantifies the influence of any budget allocation b ∈ Nn to the agents. The concrete form of f strongly depends on the structure of the network, as described later. The objective is to find a budget allocation b that maximizes the numberPof influenced nodes, while respecting the feasibility constraints: (1) bi ≤ ci , for every node i ∈ V , and (2) i∈V bi ≤ B. For ease of presentation, we begin by describing the classic Triggering model [26]. Let N (v) be the set of neighbors of node v in the graph. The influence function implied by a Triggering model is defined by the following simple process. Every node v ∈ V independently chooses a random triggering set T v ⊆ N (v) among its neighbors according to some fixed distribution. Then, for any given seed set of nodes, its influence value is defined as the result of a deterministic cascade process in the network which works in steps. In the first step, only the selected seed set is affected. At step ℓ, each node v that is still not influenced becomes influenced if any of its neighbors in T v became influenced at time ℓ − 1. This process terminates after at most n rounds. Our generalized model introduces the notion of budgets into this process. Specifically, the influence function in our case adheres to the following process. Every node v independently chooses a random triggering vector tv ∈ N\|N (v)\| according to some fixed distribution. Given a budget allocation b ∈ Nn , the influence value of that allocation is the result of the following deterministic cascade process. In the first step, every node v that was allocated a budget bv > 0 becomes affected. At step ℓ, every node v that is still not influenced becomes influenced if any of its neighbors u ∈ N (v) became influenced at time ℓ − 1 and bu ≥ tvu . One can easily verify that the Triggering model is a special case of our Budgeted Triggering model, where the capacity vector c = 1n , and each tvu = 0 if u ∈ T v , and tvu = B + 1, otherwise. Intuitively, the triggering vectors in our model capture the amount of effort that is required from each neighbor of some agent to affect her. Of course, the underlying assumption is that the effort of individuals correlates with the budget they receive. As an example, consider the case that a node v selects a triggering value tvu = 1 for some neighbor u. In this case, u can only influence v if it receives a budget of at least 1. However, if v selects a value tvu = 0 then it is enough that u becomes affected in order to influence v. In particular, it is possible that u does not get any budget but still influences v after it becomes affected in the cascade process. Given a budget allocation b, the value of the influence function f (b) is the expected number of nodes influenced in the cascade process, where the expectation is taken over the random choices of the model. Formally, let σ be some fixed choice of the triggering vectors of all nodes (according to the model distribution), and let Pr(σ) be the probability of this outcome. Let fσ (b) be the (deterministic) number of Pnodes influenced when the triggering vectors are defined by σ and the budget allocation is b. Then, f (b) = σ Pr(σ) · fσ (b). 6 Theorem 3.5. There is a (1 − 1/e)-approximation algorithm for influence maximization under the Budgeted Triggering model whose running time is polynomial in n, B, and the query time of the value oracle for the influence function. Proof. Consider an influence function f : Nn → R+ resulting from the Budgeted Triggering model. We next show that the function f is monotone and submodular over the integer lattice. As a result, our model can be interpreted as an instance of maximizing a monotone submodular function over the integer lattice subject to a knapsack constraint, which admits an efficient (1 − 1/e)-approximation. Notice that it is sufficient to prove that each (deterministic) function fσ is monotone submodular function over the integer lattice. This follows as f is a non-negative linear combination of all fσ . One can easily validate that submodularity and monotonicity are closed under non-negative linear combinations. Consider some function fσ : Nn → R+ . For the purpose of establishing that fσ is monotone and submodular, we show that fσ can be interpreted as a combination of a monotone coordinate independent function gσ : Nn → {0, 1}n , and a monotone submodular function hσ : {0, 1}n → R+ . The theorem then follows by utilizing Lemma 3.3. We divide the diffusion process into two stages. In the first stage, we consider the function gσ , which given a budget allocation returns (the characteristic vector of) the set S of all the nodes that were allocated a positive budget along with their immediate neighbors that were influenced according to the Budgeted Triggering model. Formally, gσ (b) = S , v : bv > 0 ∪ u : ∃v ∈ N (u), bv > 0, bv ≥ tuv . In the second stage, we consider the function hσ that receives (the characteristic vector of) S as its seed set, and makes the (original) Triggering model interpretation of the vectors. Specifically, the triggering set of each node v is considered to be T v = {u : tvu = 0}. Intuitively, the function gσ captures the initial budget allocation step and the first step of the propagation process, while the function hσ captures all the remaining steps of the propagation. Observe that fσ = hσ ◦ gσ by our construction. Also notice that hσ is trivially monotone and submodular as it is the result of a Triggering model [26, Thm. 4.2]. Therefore, we are left to analyze the function gσ , and prove that it is monotone and coordinate independent. The next claim establishes these properties, and completes the proof of the theorem. Claim 3.6. The function gσ is monotone and coordinate independent. Proof. Let x, y ∈ Nn , and denote w = x ∨ y. We establish coordinate independence by considering each influenced node in gσ (w) separately. Recall that gσ (w) consist of the union of two sets {v : wv > 0} and {u : ∃v ∈ N (u), wv > 0, wv ≥ tuv }. Consider a node v for which wv > 0. Since wv = max{xv , yv }, we know that at least one of {xv , yv } is equal to wv , say wv = xv . Hence, v ∈ gσ (x). Now, consider a node u ∈ gσ (w) having wu = 0. It must be the case that u is influenced by one of its neighbors v. Clearly, wv > 0 and wv ≥ tuv . Again, we can assume without loss of generality that wv = xv , and get that u ∈ gσ (x). This implies that for each v ∈ gσ (x ∨ y), either v ∈ gσ (x) or v ∈ gσ (y), proving coordinate independence, i.e., gσ (x ∨ y) ≤ gσ (x) ∨ gσ (y). We prove monotonicity in a similar way. Let x ≤ y. Consider a node v ∈ gσ (x) for which xv > 0. Since yv ≥ xv > 0, we know that v ∈ gσ (y). Now, consider a node u ∈ gσ (x) having xu = 0. There must be a node v ∈ N (u) such that xv > 0, and xv ≥ tuv . Accordingly, we get that yv ≥ tuv , and hence, u ∈ gσ (y). This implies that gσ (x) ≤ gσ (y), which completes the proof. 3.3 A multi-player budgeted influence game We now focus on a multi-player budgeted influence game. In the general setting of the game, which is i M formally defined by the tuple (M, (Ai )M i=1 , (f )i=1 ), there are M self-interested players, each of which needs 7 to decide how to allocate its budget B i ∈ N+ among n agents. Each player has a capacity vector ci ∈ Nn+ that bounds the amount of budget she may allocate to every agent. The budget assignment of player i is denoted The strategy of player i is feasible if it respects the constraints: by bi ∈ Nn+ , and is referred to as its strategy. P (1) bij ≤ cij , for every j ∈ [n], and (2) nj=1 bij ≤ B i . Let Ai be the set of all feasible strategies for player i. Note that we allow mixed (randomized) strategies. Each player has an influence function f i : NM ×n → R+ that designates her own influence (payoff) in the game. Specifically, f i (b) is the payoff of player i for the budget allocation b of all players. This can also be written as f i (bi , b−i ), where the strategy of i is bi and the strategies of all the other players are marked as b−i . Note that the goal of each player is to maximize PM i her own influence, given her feasibility constraints and the strategies of other players. Let F (b) = i=1 f (b) be the social utility of all players in the game. One of the most commonly used notions in game theory is Nash equilibrium (NE) [35]. This notion translates to our game as follows: A budget allocation b is said to be in a NE if f i (bi , b−i ) ≥ f i (b̃i , b−i ), for every i and b̃i ∈ Ai . Monotone utility game on the integer lattice. We begin by studying a monotone utility game on the integer lattice, and establish that its PoA is no more than 2. Later on, we demonstrate that our Budgeted Triggering model can be captured by this game. Utility games were defined for submodular set functions by Vetta [42], and later extended to submodular functions on the integer lattice by Maehara et al. [32]. We build on the latter work, and demonstrate that one of the conditions in their utility game definition, namely, the requirement that the submodular function satisfies component-wise concavity, can be neglected. Note that component-wise concavity corresponds to the diminishing marginal returns property, which does not characterize submodularity over the integer lattice, as noted in Section 2. Therefore, removing this constraint is essential for proving results for our model. We refine the definition of a monotone utility game on the integer lattice [32], so it only satisfies the following conditions: (U1) F (b) is a monotone submodular function on the integer lattice. P i (U2) F (b) ≥ M i=1 f (b). (U3) f i (b) ≥ F (bi , b−i ) − F (0, b−i ), for every i ∈ [M ]. Theorem 3.7. The price of anarchy of the monotone utility game designated by U1-U3 is at most 2. 1 M Proof. Let b∗ = (b1∗ , . . . , bM ∗ ) be the social optimal budget allocation, and let b = (b , . . . , b ) be a budget i 1 i allocation in Nash equilibrium. Let b̃ = (b∗ , . . . , b∗ , 0, . . . , 0) be the optimal budget allocation restricted to the first i players. Notice that F (b∗ ) − F (b) ≤ F (b∗ ∨ b) − F (b) = M X F (b̃i ∨ b) − F (b̃i−1 ∨ b) i=1 ≤ M X F (bi∗ ∨ bi , b−i ) − F (bi , b−i ) , i=1 where the first inequality is due to the monotonicity of F , the equality holds by a telescoping sum, and the last inequality is due to submodularity of F . Specifically, submodularity implies that inequality since F (bi∗ ∨ bi , b−i ) + F (b̃i−1 ∨ b) ≥ F (b̃i ∨ b) + F (bi , b−i ) . 8 Now, observe that F (bi , b−i ) + F (bi∗ , b−i ) ≥ F (bi∗ ∨ bi , b−i ) + F (bi∗ ∧ bi , b−i ) ≥ F (bi∗ ∨ bi , b−i ) + F (0, b−i ) . Here, the first inequality holds by the submodularity of F , while the last inequality follows from the monotonicity of F . Consequently, we derive that F (b∗ ) − F (b) ≤ M X F (bi∗ , b−i ) − F (0, b−i ) ≤ M X i=1 i=1 f i (bi∗ , b−i ) ≤ M X f i (bi , b−i ) ≤ F (b) , i=1 where the second inequality is by condition (U3) of the utility game, the third inequality holds since b is a Nash equilibrium, and the last inequality is by condition (U2) of the utility game. This completes the proof as F (b∗ ) ≤ 2F (b). A multi-player Budgeted Triggering model. We extend the Budgeted Triggering model to a multi-player setting. As before, we have a social network, represented by a directed graph G = (V, E), such that every node v has an independent random triggering vector tv ∈ N\|N (v)\| . Each player i has a budget B i ∈ N+ , and a function f i : NM ×n → R+ that quantifies her influence, given the budget allocation of all players. The objective of each player i is to find a budget allocation bi , given the budget allocations of other players, that maximizes the number of nodes that she influences, while respecting the feasibility constraints: (1) bij ≤ cij , P for every node j ∈ V , and (2) j∈V bij ≤ B. The process in which nodes become affected is very similar to that in Budgeted Triggering, but needs some refinement for the multi-player setting. We follow most design decisions of Bharathi et al. [6]. Specifically, whenever a player influences a node, this node is assigned the color of that player. Once a node become influenced, its color cannot change anymore. If two or more players provide positive budgets to the same node, then the node is given the color of the player that provided the highest budget. In case there are several such players, the node is assigned a color uniformly at random among the set of players with the highest budget assignment. If a node u becomes influenced at step ℓ, it attempts to influence each of its neighbors. If the activation attempt from u to its neighbor v succeeds, which is based on the triggering vector of v, then v becomes influenced with the same color as u at step ℓ + Tuv , assuming that it has not been influenced yet. All Tuv ’s are independent positive continuous random variables. This essentially prevents simultaneous activation attempts by multiple neighbors. PM i Lemma 3.8. The social function F (b) = i=1 f (b) is a monotone submodular function on the integer lattice. Proof. We prove this lemma along similar lines to those in the proof of Theorem 3.5, which attains to the single-player scenario. Let σ be some fixed choice of triggering vectors of all nodes and all the activation times Tuv . We also assume that σ encodes other random decisions in the model, namely, all tie-breaking choices related to equal (highest) budget assignments for nodes. Let Fσ (b) be the deterministic number of P influenced nodes for the random choices σ and the budget allocation b, and note that F (b) = σ Pr(σ)·Fσ (b). Similar to Theorem 3.5, it is sufficient to prove that Fσ is monotone submodular function on the integer lattice. Again, we view the social influence as a two-stage process. In the first step, we consider a function Gσ : NM ×n → {0, 1}n that given the budget allocation of all players returns a set S of immediate influenced nodes. Formally, Gσ (b) = S , v : ∃i, biv > 0 ∪ u : ∃v ∈ N (u), ∃i, biv > 0, biv ≥ tuv . 9 In the second stage, we consider the function Hσ that receives S as its seed set, and makes the original Triggering model interpretation of the vectors, that is, it sets each T v = {u : tvu = 0}. Notice that the fact that there are multiple players at this stage does not change the social outcome, i.e., the number of influenced nodes, comparing to a single-player scenario. The only difference relates to the identity of the player that affects every node. This implies that Hσ is monotone and submodular as its result is identical to that of the original (single-player) Triggering model [26]. Observe that Fσ = Hσ ◦ Gσ by our construction. Therefore, by Lemma 3.3, we are left to establish that the function Gσ is monotone and coordinate independent. The next claim proves that. Claim 3.9. The function Gσ is monotone and coordinate independent. We prove this claim using almost identical line of argumentation to that in Claim 3.6. Let x, y ∈ NM ×n , and denote w = x ∨ y. We establish coordinate independence by considering every affected node in Gσ (w) separately. Recall that Gσ (w) consist of the union of two sets {v : ∃i, wvi > 0} and {u : ∃v ∈ N (u), ∃i, wvi > 0, wvi ≥ tuv }. Consider a node v that has some player i with wvi > 0. Since wvi = max{xiv , yvi }, we know that at least one of {xiv , yvi } is equal to wvi , say wvi = xiv . Hence, v ∈ Gσ (x), since in particular, player i competes on influencing v. Now, consider a node u ∈ Gσ (w) with wui = 0, for all i. It must be the case that u is influenced by one of its neighbors v. Clearly, there exists some player i such that wvi > 0 and wvi ≥ tuv . Again, we can assume without loss of generality that wvi = xiv , and get that u ∈ Gσ (x), since in particular, player i competes on influencing u via v. This implies that for each v ∈ Gσ (x ∨ y), either v ∈ Gσ (x) or v ∈ Gσ (y), proving coordinate independence, i.e., Gσ (x ∨ y) ≤ Gσ (x) ∨ Gσ (y). We prove monotonicity in a similar way. Let x ≤ y. Consider a node v ∈ Gσ (x) such that there is a player i for which xiv > 0. Since yvi ≥ xiv > 0, we know that player i competes on influencing v, and thus, v ∈ Gσ (y). Now, consider a node u ∈ Gσ (x) with xiu = 0, for all i. There must be a node v ∈ N (u) and a player i such that xiv > 0 and xiv ≥ tuv . Accordingly, we get that yvi ≥ tuv . Therefore, player i also competes on influencing u via v, and thus, u ∈ Gσ (y). This implies that Gσ (x) ≤ Gσ (y), which completes the proof. Theorem 3.10. The Budgeted Triggering model with multiple players has a PoA of exactly 2. Proof. We begin by demonstrating that the model satisfies conditions U1-U3 of the monotone utility game on the integer lattice. As a result, we can apply Theorem 3.7 to attain an upper bound of 2 on the PoA of the model. Notice that condition (U1) holds by Lemma 3.8. Also, condition (U2) trivially holds by the definition of the social function F . For the purpose of proving that the model satisfies condition (U3), let σ be some fixed choice of triggering vectors of all nodes and all the activation times Tuv . We also assume that σ encodes other random decisions in the model, namely, all tie-breaking choices related to equal (highest) budget assignments for nodes. Let Fσ (b) be the deterministic number of influenced nodes for the random choices σ and the budget allocation b. Finally, let fσi (b) be the deterministic number of nodes influenced by player i for the random choices σ and the budget allocation b. We next argue that fσi (b) ≥ Fσ (bi , b−i ) − Fσ (0, b−i ), for any σ. Notice that this implies condition (U3) since X X f i (b) = Pr(σ)fσi (b) ≥ Pr(σ) Fσ (bi , b−i ) − Fσ (0, b−i ) = F (bi , b−i ) − F (0, b−i ) . σ σ We turn to prove the above argument. Notice that it is sufficient to focus only on cases that Fσ (bi , b−i ) > Fσ (0, b−i ), since otherwise, the argument is trivially true as fσi (b) ≥ 0. We concentrate on all nodes u that are not influenced by any player when the mutual strategy is (0, b−i ), but became influenced for a strategy (bi , b−i ). We claim that all those nodes must be assigned the color of player i. It is easy to verify that 10 increasing the budget assignment of a player to any node can only negatively affect other players, that is, they may only influence a subset of the nodes. This follows as all the activation results are deterministically encoded in the choices σ, so adding a competition can only make the outcome worse, i.e., players may not affect a node that they previously did. This implies the claim. As a result, fσi (b) ≥ Fσ (bi , b−i ) − Fσ (0, b−i ). This completes the proof that the model is an instance of the monotone utility game on the integer lattice, and thus, has a PoA of at most 2. We proceed by proving the tightness of the PoA result. We show there is an instance of the multi-player Budget Triggering model whose PoA is 2N/(N + 1). Notice that as N → ∞, the lower bound on the PoA tends to 2. This instance has been presented in a slightly different context by He and Kempe [25, Proposition 1]. Concretely, the input graph is a union of a star with one center and N leaves, and N additional (isolated) nodes. The triggering vectors are selected from a degenerate distribution that essentially implies that each activated node also activates all of its neighbors. Every player has one unit of budget. One can easily verify that the solution in which all players assign their unit budget to the center of the star is a NE. This follows since the expected payoff for each player is (N + 1)/N , while unilaterally moving the budget to any other node leads to a payoff of 1. However, the strategy that optimizes the social utility is to place one unit of budget at the center of the star graph, and the remaining budget units at different isolated nodes. 4 An Online Model We study the online version of the budgeted influence propagation model. This setting can capture scenarios in which the social influences in a network are known in advance, but the (subset of) agents that will arrive and their order is unknown. The input for this setting is identical to that of the offline variant with the exception that the n agents arrive in an online fashion. This intuitively means that we do not know the monotone submodular influence function f : Nn → R+ in advance, but rather, it is revealed to us gradually with time. More specifically, upon the arrival of the ith agent, we can infer the (constrained) function fi , which quantifies the influence of f for the set of the first i agents, while fixing the budget of all other agents to 0. Note that we also learn the maximum budget ci that can be allocated to agent i whenever she arrives. For every arriving agent i, the algorithm needs to make an irrevocable decision regarding the amount of budget bi allocated to that agent without knowing the potential contribution of future arriving agents. As mentioned in the introduction, this problem is a generalization of the classical secretary problem. This immediately implies that any online algorithm preforms very poorly under an unrestricted adversarial arrival of the agents. We therefore follow the standard assumption that the agents and their influence are fixed in advanced, but their order of arrival is random. Note that the overall influence of some budget allocation to the agents is not affected by the arrival order of the agents. We analyze the performance of our algorithm, ON, using the competitive analysis paradigm. Note that competitive analysis focuses on quantifying the cost that online algorithms suffer due to their complete lack of knowledge regarding the future, and it does not take into account computational complexity. Let OPT be an optimal algorithm for the offline setting. Given an input instance I for the problem, we let OPT(I) and ON(I) be the influence values that OPT and ON attain for I, respectively. We say that ON is c-competitive if inf I E[ON(I)]/OPT(I) ≥ c, where E[ON(I)] is the expected value taken over the random choices of the algorithm and the random arrival order of the agents. We like to note that our algorithm and its analysis are inspired by the results of Feldman et al. [17] for the submodular knapsack secretary problem. However, we make several novel observations and identify some interesting structural properties that enable us to simultaneously generalize and improve their results. Also note that in the interests of expositional simplicity, we have not tried to optimize the constants in our analysis. 11 Theorem 4.1. There is an online randomized algorithm that achieves 1/(15e) ≈ 0.025-competitive ratio for the budgeted influence maximization problem. Proof. Recall that an instance of the online budgeted influence maximization problem consists of a set of n agents that arrive in a random order, a budget constraint B ∈ N+ , capacity constraints c ∈ Nn+ , and a monotone submodular influence function over the integer lattice f : Nn → R+ . We begin by describing the main component of our algorithm. This component is built to address the case that the contribution of each agent is relatively small with respect to the optimal solution. That is, even when one assigns the maximum feasible budget to any single agent, the contribution of that agent is still small compared to the optimum. We refer to this component as light influence algorithm (abbreviated, LI). This component will be later complemented with another component, derived from the classical secretary algorithm, to deal with highly influential agents. Let ha1 , a2 , . . . , an i be an arbitrary fixed ordering of the set of agents. This is not necessarily the arrival order of the agents. Algorithm light influence, formally described below, assumes that each agent ai is assigned a uniform continuous random variable ti ∈ [0, 1) that determines its arrival time. Note that this assumption does not add restrictions on the model since one can create a set of n samples of the uniform distribution from the range [0, 1) in advance, and assign them by a non-decreasing order to each arriving agent (see, e.g., the discussion in [17]). The algorithm begins by exploring the first part of the agent sequence, that is, the agents in L = {ai : ti ≤ 1/2}. Note that it does not allocate any budget to those agents. Let bL be an optimal (offline) budget allocation for the restricted instance that only consists of the agents in L, and let f (bL ) be its influence value. Furthermore, let f (bL )/B be a lower bound on the average contribution of each unit of budget in that solution. The algorithm continues by considering the remainder of the sequence. For each arriving agent, it allocates a budget of k if the increase in the overall influence value is at least αkf (bL )/B, for some fixed α to be determined later. That is, the average influence contribution of an each unit of budget is (up to the α-factor) at least as large as the average unit contribution in the optimal solution for the first part. If there are several budget allocations that satisfy the above condition then the algorithm allocates the maximal amount of budget that still satisfies the capacity and budget constraints. Prior to formally describing our algorithm, we like to remind that χi corresponds to the characteristic vector of ai , i.e., (χi )i = 1 and (χi )j = 0 for every j 6= i. Accordingly, if b ∈ Nn is a budget allocation vector in which the ith coordinate represents the allocation to agent ai , and bi = 0, then the allocation b ∨ kχi corresponds to an update of b by adding a budget k to agent ai . We say that the marginal value of assigning k units of budget to ai is f (b ∨ kχi ) − f (b), and the marginal value per unit is (f (b ∨ kχi ) − f (b))/k. Having described our main component, we are now ready to complete the description of our algorithm. As already , we randomly combine algorithm LI with the classical algorithm for the secretary problem. Specifically, algorithm LI is employed with probability 5/8 and the classical secretary algorithm with probability 3/8. This latter algorithm assigns a maximal amount of budget to a single agent ai to attain an influence value of f (ci χi ). The algorithm selects ai by disregarding the first n/e agents that arrive, and then picking the first agent whose influence value is better than any of the values of the first n/e agents. This optimal algorithm is known to succeed in finding the single agent with the best influence with probability of 1/e [14]. 4.1 Analysis We begin by analyzing the performance guarantee of algorithm LI, and later analyze the complete algorithm. Let OPT∗ = [OPT∗1 , . . . , OPT∗n ] be the optimal budget allocation for a given instance, and let OPTL be the L ∗ budget allocation for the agents in L, that is, OPTL i = OPTi whenever i ∈ L and OPTi = 0, otherwise. R R Similarly, OPT is the budget allocation for the agents in R = [n] \ L, i.e., OPTi = OPT∗i for i ∈ R, and 12 Algorithm 1: Light Influence (LI) Input : an online sequence of n agents, a budget constraint B ∈ N+ , capacity constraints c ∈ Nn+ , a monotone submodular function f : Nn → R+ , a parameter α ∈ R+ Output: A budget allocation b b ← (0, 0, . . . , 0) f (bL ) ← value of the optimal budget allocation for agents in L = {ai : ti ≤ 1/2} for every agent ai such that ti ∈ (1/2, 1] do P Ki ← k ≤ ci : f (b ∨ kχi ) − f (b) ≥ αkf (bL )/B ∪ k + j6=i bj ≤ B if Ki 6= ∅ then k ← maxk {Ki } b ← b ∨ kχi end end return b Algorithm 2: Online Influence Maximization Input : an online sequence of n agents, a budget constraint B ∈ N+ , capacity constraints c ∈ Nn+ , a monotone submodular function f : Nn → R+ , a parameter α ∈ R+ Output: A budget allocation b r ← random number in [0, 1] if r ∈ [0, 3/8] then b ← run the classical secretary algorithm with (n, B, c, f ) else if r ∈ (3/8, 1] then b ← run algorithm LI with (n, B, c, f, α) end return b OPTR / R. Recall that algorithm LI attends to the case in which no single agent has a significant i = 0 for i ∈ influence contribution compared to the optimal value. More formally, let β = maxi f (ci χi )/f (OPT∗ ) be the ratio between the maximal contribution of a single agent and the optimal value. Lemma 4.2. If α ≥ 2β then f (b) ≥ min{αf (OPTL )/2, f (OPTR ) − αf (OPT∗ )}. Proof. We prove this lemma by bounding the expected influence value of the algorithm in two cases and taking the minimum of them: Case I: Algorithm LI allocates a budget of more than B/2 units. We know that the algorithm attains a value of at least αf (bL )/B from each allocated budget unit by the selection rule f (b ∨ ki χi ) − f (b) ≥ αkf (bL )/B. Hence, the total influence of this allocation is at least f (b) > αf (bL ) αf (OPTL ) B αf (bL ) · = ≥ . 2 B 2 2 Case II: Algorithm LI allocates at most B/2 budget units. We utilize the following lemma proven in [40, Lem 2.3]. Lemma 4.3. Let f be a monotone submodular function over the integer lattice. For arbitrary x, y, X f (x ∨ y) ≤ f (x) + f (x ∨ yi χi ) − f (x) . i∈[n]: yi >xi 13 This lemma applied to our case implies that f (b ∨ OPTR ) ≤ f (b) + X i∈[n]: OPTR i >bi f (b ∨ OPTR i χi ) − f (b) . (1) We consider two sub-cases: Subcase A: There is ℓ ∈ [n] such that OPTR ℓ > B/2. Clearly, there can only be one agent ℓ having this ∗ property. One can easily validate that f (b ∨ OPTR ℓ χℓ ) − f (b) ≤ β · f (OPT ) by the definition of β and R Lemma 2.2. Now, consider any agent i 6= ℓ with OPTi > bi . The reasonP that the optimal solution allocated more budget to i than our algorithm cannot be the lack of budget since i bi < B/2 and OPTR i < B/2. Hence, it must be the case that f (bL ) f (b ∨ OPTR i χi ) − f (b) < α , (2) B OPTR i by the selection rule of the algorithm. Note that b in the above equation designates the budget allocation at the time that the agent ai was considered and not the final allocation. However, due to the weak version of marginal diminishing returns that was described in Lemma 2.2, the inequality also holds for the final allocation vector. As a result, f (OPTR ) ≤ f (b ∨ OPTR ) ≤ f (b) + f (b ∨ OPTR ℓ χℓ ) − f (b) + X i∈[n]\{ℓ}: OPTR i >bi f (b ∨ OPTR i χi ) − f (b) f (bL ) B · ≤ f (b) + βf (OPT∗ ) + α B 2 α , ≤ f (b) + f (OPT∗ ) · β + 2 where the first inequality follows due to the monotonicity of f , and the third inequality uses the sub-case assumption that there is one agent that receives at least half of the overall budget in OPTR , and thus, P R R ∗ i6=ℓ OPTi ≤ B/2. Recall that α ≥ 2β, and thus, f (b) ≥ f (OPT ) − αf (OPT ). Subcase B: OPTR i ≤ B/2, for every i ∈ [n]. The analysis of this sub-case follows the same argumentation of the previous sub-case. Notice that for every agent i ∈ [n] such that OPTR i > bi , we can apply inequality (2). Consequently, we can utilize inequality (1), and get that f (OPTR ) ≤ f (b ∨ OPTR ) ≤ f (b) + X i∈[n]: OPTR i >bi f (bL ) f (b ∨ OPTR ·B , i χi ) − f (b) ≤ f (b) + α B which implies that f (b) ≥ f (OPTR ) − αf (OPT∗ ). Recall that we considered some arbitrary fixed ordering of the agents ha1 , a2 , . . . , an i that is not necessary their arrival order. Let wi the marginal contribution of agent ai to the optimal value when calculated according to this order. Namely, let OPT∗<i = [OPT∗1 , . . . , OPT∗i−1 , 0, . . . , 0] be the allocation giving the same budget as OPT∗ for every agent aj with j < i, and 0 for the rest, and define wi = f (OPT∗<i ∨OPT∗i χi )−f (OPT<i ). This point of view allow us to associate fixed parts of the optimal value to the agents in a way that is not affected by their order of arrival. Let Xi be a random indicator for the event that ai ∈ L, and let W = Pn i=1 wi Xi . Let α ≥ 2β to be determined later. 14 By the weak version of marginal in Lemma 2.2, it holds that f (OPTL ) ≥ Pn diminishing returns specified R ∗ W , and similarly, f (OPT ) ≥ i=1 wi (1 − Xi ) = f (OPT ) − W . Using this observation, in conjunction with Lemma 4.2, we get that f (b) ≥ min{αW/2, f (OPT∗ )·(1−α−W/f (OPT∗ ))}. Let Y = W/f (OPT∗ ), and observe that f (b) ≥ f (OPT∗ ) · min{αY /2, 1 − α − Y } . (3) Note that Y ∈ [0, 1] captures the ratio between the expected optimum value associated with the agents in L and the (overall) optimum value. We continue to bound the expected value of f (b) by proving the following lemma. Lemma 4.4. Let α = 2/5 and assume that β ≤ 1/5, then, E[f (b)] ≥ p 2 f (OPT∗ ) · 1− β . 20 Proof. By assigning α = 2/5 to the bound in inequality 3, we obtain that Y 3 ∗ , −Y . f (b) ≥ f (OPT ) · min 5 5 Notice that the expected value of f (b) is ∗ E[f (b)] ≥ f (OPT ) Z 3 5 ′ [Pr(Y ≤ γ)] · min 0 γ 3 , − γ dγ , 5 5 since [Pr(Y ≤ γ)]′ is the probability density function of Y . Now, observe that we can split the integral range into two parts ∗ E[f (b)] ≥ f (OPT ) f (OPT∗ ) ≥ 5 Z 1 2 [Pr(Y ≤ γ)] 0 Z ′γ 1 2 5 ∗ dγ + f (OPT ) Z 3 5 1 2 ′ [Pr(Y ≤ γ)] [Pr(Y ≤ γ)]′ γdγ. 3 − γ dγ 5 (4) 0 To bound Pr(Y ≤ γ), we use Chebyshev’s inequality, while noting that E[Y ] = n X wi E[Xi ]/f (OPT∗ ) = W/(2f (OPT∗ )) = 1/2 , i=1 since E[Xi ] = 1/2 and W = f (OPT∗ ). Now, 1 β Var[Y ] Pr Y − ≤ 2 , ≥c ≤ 2 c2 4c where the last inequality follows from [17, Lem B.5]. For completeness, the proof of this lemma appears as Lemma A.1 in the Appendix. Now, observe that Y is symmetrically distributed around 1/2, and therefore, Pr(Y ≤ 12 − c) = Pr(Y ≥ 21 + c) ≤ β/(8c2 ). This implies that for every γ ≤ 1/2, Pr(Y ≤ γ) ≤ 15 β . 8( 21 − γ)2 Note that we cannot simply plug this upper bound on the cumulative distribution function into inequality (4). R 1/2 The fact that Y is symmetrically distributed around 1/2 implies that 0 [Pr(Y ≤ γ)]′ dγ = 1/2, and this does hold with this bound. To bypass this issue, and maintain the later constrain, we decrease the integration range. One can easily verify that #′ Z 1−√β " 2 β 1 dγ = , 1 2 2 8( 2 − γ) 0 and as a result, we can infer that Z 1 2 ′ [Pr[Y ≤ γ]] γdγ ≥ 0 Z √ 1− β 2 0 " β 1 8( 2 − γ)2 #′ γdγ. Specifically, this inequality holds since we essentially considered the worst distribution (from an algorithms analysis point of view) by shifting probability from higher values of Y to smaller values (note that multiplication by γ). The proof of the lemma now follows since #′ Z 1−√β " 2 β f (OPT∗ ) γdγ E[f (b)] ≥ 5 8( 12 − γ)2 0 √ ∗ Z 1−2 β β · f (OPT ) γ = dγ 1 20 ( 2 − γ)3 0 1−√β 2 4γ − 1 β · f (OPT∗ ) = 20 (1 − 2γ)2 0 ∗ p 2 f (OPT ) · 1− β = . 20 Recall that β = maxi f (ci χi )/f (OPT∗ ). We next consider two cases depending on the value of β. When β > 1/5, our algorithm executes the classical secretary algorithm with probability 3/8. This algorithm places a maximal amount of budget on the agent having maximum influence, maxi f (ci χi ), with probability 1/e. Consequently, 3 β · f (OPT∗ ) 3f (OPT∗ ) f (OPT∗ ) E[f (b)] ≥ · > > . 8 e 40e 15e When β ≤ 1/5, we know that our algorithm executes the classical secretary algorithm with probability 3/8, and algorithm LI with probability 5/8. 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P 2 P 2 P w V ar[ i wi Xi ] i wi V ar[Xi ] = = 2 i i ∗ V ar[Y ] = ∗ ∗ 2 2 f (OPT ) f (OPT ) 4f (OPT ) P maxi wi · i wi maxi wi · f (OPT∗ ) β = ≤ . ≤ ∗ ∗ 2 2 4f (OPT ) 4f (OPT ) 4 19	8
arXiv:1403.6920v3 [math.AC] 10 Apr 2014 GRÖBNER BASES OF BALANCED POLYOMINOES JÜRGEN HERZOG, AYESHA ASLOOB QURESHI AND AKIHIRO SHIKAMA Abstract. We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a squarefree Gröbner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced. Introduction Polyominoes are, roughly speaking, plane figures obtained by joining squares of equal size edge to edge. Their appearance origins in recreational mathematics but also has been subject of many combinatorial investigations including tiling problems. A connection of polyominoes to commutative algebra has first been established by the second author of this paper by assigning to each polyomino its ideal of inner minors, see [14]. This class of ideals widely generalizes the ideal of 2-minors of a matrix of indeterminates, and even that of the ideal of 2-minors of two-sided ladders. It also includes the meet-join ideal of plane distributive lattices. Those classical ideals have been extensively studied in the literature, see for example [2], [3] and [7]. Typically one determines for such ideals their Gröbner bases, determines their resolution and computes their regularity, checks whether the rings defined by them are normal, Cohen-Macaulay or Gorenstein. A first step in this direction for the inner minors of a polyomino (also called polyomino ideals) has been done by Qureshi in the afore mentioned paper. Very recently those convex polyominoes have been classified in [5] whose ideal of inner minors is linearly related or has a linear resolution. For some special polyominoes also the regularity of the ideal of inner minors is known, see [8]. In this paper, for balanced polyominoes, we provide some positive answers to the questions addressed before. To define a balanced polyomino, one labels the vertices of a polyomino by integer numbers in a way that row and column sums are zero along intervals that belong to the polyomino. Such a labeling is called admissible. There is a natural way to attach to each admissible labeling of a polyomino P a binomial. The given polyomino is called balanced if all the binomials arising from admissible labelings belong to the ideal of inner minors IP of P. It turns out that P is balanced if and only if IP coincides with the lattice ideal determined by P, 1991 Mathematics Subject Classification. 13C05, 05E40, 13P10. Key words and phrases. polyominoes, Gröbner bases, lattice ideals. This paper was partially written during the visit of the second and third author at Universität Duisburg-Essen, Campus Essen. The second author wants to thank the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy for supporting her. The third author wants to thank Professor Hibi who made his visit to Essen possible. 1 see Proposition 1.2. This is the main observation of Section 1 where we provide the basic definitions regarding polyominoes and their ideals of inner minors. An important consequence of Proposition 1.2 is the result stated in Corollary 1.3 which asserts that for any balanced polyomino P, the ideal IP is a prime ideal and that its height coincides with the number of cells of P. It is conjectured in [14] by the second author of this paper that IP is a prime ideal for any simple polyomino P. A polyomino is called simple if it has no ‘holes’, see Section 1 for the precise definition. We expect that simple polyominoes are balanced. This would then imply Qureshi’s conjecture. In Section 2 we identify the primitive binomials of a balanced polyomino (Theorem 2.1) and deduce from this that for a balanced polyomino P the ideal IP has a squarefree initial ideal for any monomial order. This then implies, as shown in Corollary 2.3, that the residue class ring of IP is a normal Cohen-Macaulay domain. Finally, in Section 3 we show that all row or column convex, and all tree-like polyminoes are simple and balanced. This supports our conjecture that simple polyominoes are balanced. 1. The ideal of inner minors of a polyomino In this section we introduce polyominoes and their ideals of inner minors. Given a = (i, j) and b = (k, l) in N2 we write a ≤ b if i ≤ k and j ≤ l. The set [a, b] = {c ∈ N2 : a ≤ c ≤ b} is called an interval, and an interval of the from C = [a, a + (1, 1)] is called a cell (with left lower corner a). The elements of C are called the vertices of C, and the sets {a, a+(1, 0)}, {a, a+(0, 1)}, {a+(1, 0), a+(1, 1)} and {a + (0, 1), a + (1, 1)} the edges of C. Let P be a finite collection of cells of N2 , and let C and D be two cells of P. Then C and D are said to be connected, if there is a sequence of cells C = C1 , . . . , Cm = D of P such that Ci ∩ Ci+1 is an edge of Ci for i = 1, . . . , m − 1. If in addition, Ci 6= Cj for all i 6= j, then C is called a path (connecting C and D). The collection of cells P is called a polyomino if any two cells of P are connected, see Figure 1. Figure 1. A polyomino Let P be a polyomino, and let K be a field. We denote by S the polynomial over K with variables xij with (i, j) ∈ V (P). Following [14] a 2-minor xij xkl − xil xkj ∈ S is called an inner minor of P if all the cells [(r, s), (r + 1, s + 1)] with i ≤ r ≤ k − 1 and j ≤ s ≤ l − 1 belong to P. In that case the interval [(i, j), (k, l)] is called an 2 inner interval of P. The ideal IP ⊂ S generated by all inner minors of P is called the polyomino ideal of P. We also set K[P] = S/IP . Let P be a polyomino. An interval [a, b] with a = (i, j) and b = (k, l) is called a horizontal edge interval of P if j = l and the sets {r, r + 1} for r = i, . . . , k − 1 are edges of cells of P. Similarly one defines vertical edge intervals of P. According to [14], an integer value function α : V (P) → Z is called admissible, if for all maximal horizontal or vertical edge intervals I of P one has X α(a) = 0. a∈I −3 −1 −2 5 −2 2 −2 0 1 −4 4 0 3 0 0 −2 −1 0 3 −1 2 −2 Figure 2. An admissible labeling In Figure 2 an admissible labeling of the polyomino displayed Figure 1 is shown. Given an admissible labeling α we define the binomial fα = Y xaα(a) − a∈V (P) α(a)>0 Y xa−α(a) , a∈V (P) α(a)<0 Let JP be the ideal generated by the binomials fα where α is an admissible labeling of P. It is obvious that IP ⊂ JP . We call a polyomino balanced if for any admissible labeling α, the binomial fα ∈ IP . This is the case if and only if IP = JP . L Consider the free abelian group G = (i,j)∈V (P) Zeij with basis elements eij . To any cell C = [(i, j), (i + 1, j + 1)] of P we attach the element bC = eij + ei+1,j+1 − ei+1,j − ei,j+1 in G and let Λ ⊂ G be the lattice spanned by these elements. Lemma 1.1. The elements bC form a K-basis of Λ and hence rankZ Λ = \|P\|. Moreover, Λ is saturated. In other words, G/Λ is torsionfree. Proof. We order the basis elements eij lexicographically. Then the lead term of bC is eij . This shows that the elements bC are linearly independent and hence form a Z-basis of Λ. We may complete this basis of Λ by the elements eij for which (i, j) is not a left lower corner of a cell of P to obtain a basis of G. This shows that G/Λ is free, and hence torsionfree. The lattice ideal IΛ attached to the lattice Λ is the ideal generated by all binomials fv = Y xvaa − Y a∈V (P) va <0 a∈V (P) va >0 with v ∈ Λ. 3 xa−va Proposition 1.2. Let P be a balanced polyomino. Then IP = IΛ . Proof. The assertion follows once we have shown that for any v ∈ Λ there exists an admissible labeling α of P such that va = α(a) for all a ∈ V (P). Indeed, since the elements bC ∈ Λ form a Z-basis of Λ, there exist integers zC ∈ Z such that P P v = C zC bC . We set α = C∈P zC αC where for C = [(i, j), (i + 1, j + 1)], αC ((k, l)) =    1, if (k, l) = (i, j) or (k, l) = (i + 1, j + 1), −1, if (k, l) = (i + 1, j) or (k, l) = (i, j + 1),   0, otherwise. Then α(a) = va for all a ∈ V (P). Since each αC is an admissible labeling of P and since any linear combination of admissible labelings is again an admissible labeling, the desired result follows. Corollary 1.3. If P is a balanced polyomino, then IP is a prime ideal of height \|P\|. Proof. By Proposition 1.2, IP = IΛ and by Lemma 1.1, Λ is saturated. It follows that IP is a prime ideal, see [9, Theorem 7.4]. Next if follows from [12, Corollary 2.2] (or [9, Proposition 7.5]) that height IP = rankZ Λ. Hence the desired conclusion follows from Lemma 1.1. Let P be a polyomino and let [a, b] an interval with the property that P ⊂ [a, b]. According to [14], a polyomino P is called simple, if for any cell C not belonging to P there exists a path C = C1 , C2 , . . . , Cm = D with Ci 6∈ P for i = 1, . . . , m and such that D is not a cell of [a, b]. It is conjectured in [14] that IP is a prime ideal if P is simple. There exist examples of polyominoes for which IP is a prime ideal but which are not simple. Such an example is shown in Figure 3. On the other hand, we conjecture that a polyomino is simple if and only it is balanced. This conjecture implies Qureshi’s conjecture on simple polyominoes. Figure 3. Not simple but prime 2. Primitive binomials of balanced polyominoes The purpose of this section is to identify for any balanced polyomino P the primitive binomials in IP . This will allow us to show that the initial ideal of IP is a squarefree monomial ideal for any monomial order. The primitive binomials in P are determined by cycles. A sequence of vertices C = a1 , a2 , . . . , am in V (P) with am = a1 and such that ai 6= aj for all 1 ≤ i < j ≤ m − 1 is a called a cycle in P if the following conditions hold: (i) [ai , ai+1 ] is a horizonal or vertical edge interval of P for all i = 1, . . . , m − 1; 4 (ii) for i = 1, . . . , m one has: if [ai , ai+1 ] is a horizonal interval of P, then [ai+1 , ai+2 ] is a vertical edge interval of P and vice versa. Here, am+1 = a2 . a1 a2 a5 a9 a7 a8 a3 a1 a2 a4 a3 a4 Figure 4. A cycle and a non-cycle in P It follows immediately from the definition of a cycle that m − 1 is even. Given a cycle C, we attach to C the binomial (m−1)/2 fC = Y (m−1)/2 xa2i−1 − i=1 Y xa2i i=1 Theorem 2.1. Let P be a balanced polyomino. (a) Let C be a cycle in P. Then fC ∈ IP . (b) Let f ∈ IP be a primitive binomial. Then there exists a cycle C in P such that each maximal interval of P contains at most two vertices of C and f = ±fC . Proof. (a) Let C = a1 , a2 , . . . , am be the cycle in P. We define the labeling α of P by setting α(a) = 0 if a 6∈ C and α(ai ) = (−1)i+1 for i = 1, . . . , m, and claim that α is an admissible labeling of P. To see this we consider a maximal horizontal edge interval I of P. If I ∩ C = ∅, then α(a) = 0 for all a ∈ I. On the other hand, if I ∩ C = 6 ∅, then there exist integers i such that ai , ai+1 ∈ I (where ai+1 = a1 if i = m − 1), and P no other vertex of I belongs C. It follows that a∈I α(a) = 0. Similarly, we see that P a∈I α(a) = 0 for any vertical edge interval. It follows form the definition of α that fC = fα , and hence since P is balanced it follows that fC ∈ IP . (b) Let f ∈ IP be a primitive binomial. Since P is balanced and f is irreducible, [14, Theorem 3.8(a)] implies that there exists an admissible labeling α of P such that Y Y f = fα = xaα(a) − xa−α(a) . a∈V (P) α(a)>0 a∈V (P) α(a)<0 Choose a1 ∈ V (P) such that α(a1 ) > 0. Let I1 be the maximal horizontal edge interval with a1 ∈ I1 . Since α is admissible, there exists some a2 ∈ I1 with α(a2 ) < 0. Let I2 be the maximal vertical edge interval containing a2 . Then similarly as before, there exists a3 ∈ I2 with α(a3 ) > 0. In the next step we consider the maximal horizontal edge interval containing and a3 and proceed as before. Continuing in this way we obtain a sequence a1 , a2 , a3 . . . . , of vertices of P such that α(a1 ), α(a2 ), α(a3 ), . . . is a sequence with alternating signs. Since V (P) is a finite set, there exist a number m such that ai 6= aj for all 1 ≤ i < j ≤ m and am = ai for some i < m. If follows that α(am ) = α(ai ) which implies that m − i is even. Then the sequence C = ai , ai+1 , . . . am is a cycle in P, and hence by (a), fC ∈ IP . For any binomial g = u − v we set g (+) = u and g (−) = v. Now if i is odd, then (+) (−) (+) fC divides f (+) and fC divides f (−) , while if i is even, then fC divides f (−) and (−) fC divides f (+) . Since f is primitive, this implies that f = ±fC , as desired. 5 Corollary 2.2. Let P be a balanced polyomino. Then IP admits a squarefree initial ideal for any monomial order. Proof. By Corollary 1.3, IP is a prime ideal, since P is a balanced polyomino. This implies that IP is a toric ideal, see for example [4, Theorem 5.5]. Now we use the fact (see [10, Lemma 4.6] or [6, Corollary 10.1.5]) that the primitive binomials of a toric ideal form a universal Gröbner basis. Since by Theorem 2.1, the primitive binomials of IP have squarefree initial terms for any monomial order, the desired conclusion follows. Corollary 2.3. Let P be a balanced polyomino. Then K[P] is a normal CohenMacaulay domain of dimension \|V (P)\| − \|P\|. Proof. A toric ring whose toric ideal admits a squarefree initial ideal is normal by theorem of Sturmfels [10, Chapter 8], and by a theorem of Hochster ([1, Theorem 6.3.5]) a normal toric ring is Cohen–Macaulay. Since P is balanced, we know from L Proposition 1.2 that IP = IΛ where Λ is the lattice in (i,j)∈V (P) Zeij spanned by the elements bC = eij + ei+1,j+1 − ei+1,j − ei,j+1 where C = [(i, j), (i + 1, j + 1)] is a cell of P. By [9, Proposition 7.5], the height of IΛ is equal to the rank of Λ. Thus we see that height IP = \|P\|. It follows that the Krull dimension of K[P] is equal to \|V (P)\| − \|P\|, as desired. 3. Classes of balanced polyominoes In this section we consider two classes of balanced polyominoes. As mentioned in Section 1 one expects that any simple polyomino is balanced. In this generality we do not yet have a proof of this statement. Here we want to consider only two special classes of polyominoes which are simple and balanced, namely the row and column convex polyominoes, and the tree-like polyominoes. Let P be a polyomino. Let C = [(i, j), (i + 1, j + 1)] be a cell of P. We call the vertex a = (i, j) the left lower corner of C. Let C1 and C2 be two cells with left lower corners (i1 , j1 ) and (i2 , j2 ), respectively. We say that C1 and C2 are in horizontal (vertical) position if j1 = j2 (i1 = i2 ). The polyomino P is called row convex if for any two horizontal cells C1 and C2 with lower left corners (i1 , j) and (i2 , j) and i1 < i2 , all cells with lower left corner (i, j) with i1 ≤ i ≤ i2 belong to P. Similarly one defines column convex polyominoes. For example, the polyomino displayed in Figure 5 is column convex but not row convex. Figure 5. Column convex but not row convex A neighbor of a cell C in P is a cell D which shares a common edge with C. Obviously, any cell can have at most four neighbors. We call a cell of P a leaf, if 6 it has an edge which does not has a common vertex with any other cell. Figure 6 illustrates this concept. Figure 6. A polyomino with three leaves The polyomino P is called tree-like each subpolyomino of P has a leaf. The polyomino displayed in Figure 6 is not tree-like, because it contains a subpolyomino which has no leaf. On the other hand, Figure 7 shows a tree-like polyomino. Figure 7. A tree-like polyomino A free vertex of P is a vertex which belongs to exactly one cell. Notice that any leaf has two free vertices. We call a path of cells a horizontal (vertical) cell interval, if the left lower corners of the path form a horizontal (vertical) edge interval. Let C be a leaf, and let I by the maximal cell interval to which C belongs, and assume that I is a horizontal (vertical) cell interval. Then we call C a good leaf, if for one of the free vertices of C the maximal horizontal (vertical) edge interval which contains it has the same length as I. We call a leaf bad if it is not good, see Figure 8. Theorem 3.1. Let P be a row or column convex, or a tree-like polyomino. Then P is balanced and simple. good bad good Figure 8. Bad and good leaves 7 Proof. Let P be a tree-like polyomino. We first show that P is balanced. Let α be an admissible labeling of P. We have to show that fα ∈ IP . To prove this we first show that P has a good leaf. If \|P\| = 1, then the assertion is trivial. We may assume that \|P\| ≥ 2. Let ni = \|{C ∈ P : deg C = i}\|. Observe that C∈P deg C = n1 + 2n2 + 3n3 + 4n4 , where deg C denotes the number of neighbor cells of P. Let P be any polyomino with cells C1 , . . . , Cn . Associated to P, is the so-called connection graph on vertex [n] with the edge set {{i, j} : E(Ci ) ∩ E(Cj ) 6= ∅}. It is easy to see that connection graph of a tree-like polyomino is a tree. Therefore, using some elementary facts from graph theory, we obtain that P n1 + 2n2 + 3n3 + 4n4 = 2(\|P\| − 1) = 2(n1 + n2 + n3 + n4 − 1). This implies that n1 = n3 + 2n4 + 2. Let g(P) be the number of good leaves in P and b(P) be the number of bad leaves in P. It is obvious that n1 = g(P) + b(P). Then we have (1) g(P) = n3 + 2n4 + 2 − b(P). Next, we show that b(P) ≤ n3 . Suppose that C is a bad leaf in P and I the unique maximal cell interval to which C belongs. Let DC be the end cell of the interval I. Observe that C 6= DC . We claim that deg DC = 3. Indeed, since C is bad, the length of the maximal intervals containing the two free vertices of C is bigger than the length of the interval I. See Figure 9 where the cells belonging to I are marked with dots and E is the cell next to DC which does not belong to P. Since E ∈ / P, the cells D1 and D2 belong to P. Suppose D3 6∈ P. Since P is a polyomino there exists a path C connecting D1 and DC . Since E, D3 6∈ P the path C (which is a subpolyomino of P) does not have a leaf, contradicting the assumption that P is tree-like. Therefore, D3 ∈ P. Similarly one shows that D4 ∈ P. This shows that deg DC = 3. D3 D1 E D4 D2 Figure 9. Moreover, DC cannot be the end cell of any other cell interval, because D3 and D4 are neighbors of DC . Thus we obtain a one-to-one correspondence between the bad leafs C and the cells DC as defined before. It follows that n3 ≤ b(P). Therefore, by (1) we obtain g(P) = n3 + 2n4 + 2 − b(P) ≥ 2n4 + 2 ≥ 2. 8 Thus, we have at least 2 good end cell for every tree-like polyomino. Now we show P is balanced. Indeed, let α be an admissible labeling of P. We want to show that fα ∈ IP . Let C be a good leaf of P with free vertices a1 and a2 . Then α(a1 ) = −α(a2 ). If α(ai ) = 0 for i = 1, 2, then α restricted to P ′ is an admissible labeling of P ′ , where P ′ is obtained from P by removing the cell C from P. Inducting on the number of cells we may then assume that fα ∈ IP ′ . Since IP ′ ⊂ IP , we are done is this case. Assume now that α(a1 ) 6= 0. We proceed by induction on \|α(a1)\|. Note that α(a2 ) 6= 0, too. Since C is a good leaf, we may assume that the maximal interval [a2 , b] to which a2 belongs has the same length as the cell interval [C, DC ] and α(a1 ) > 0. Then α(a2 ) < 0 and hence, since α is admissible, there exists a c ∈ [a2 , b] with α(c) > 0. The cells of the interval [c, a1 ] all belong to [C, DC ]. Therefore, g = xc xa1 − xd xa2 ∈ IP . Here d is the vertex as indicated in Figure 10. a2 c b DC C d a1 Figure 10. Notice that the labeling β of P defined by β1 (e) =    α(e) − 1, if e = a1 or e = c, α(e) + 1, if e = a2 or e = d,   α(e), elsewhere. is admissible and \|β(a1 )\| < \|α(a1 )\|. By induction hypothesis we have that fβ ∈ IP . Then the following relation (2) fα − (fα(+) /xc xa1 )g = (xa2 xd )fβ gives that fα ∈ IP , as well. Next, we show that P is simple by applying induction on number of cells. The assertion is trivial if P consists of only one cell. Suppose now \|P\| > 1, and let D be a cell not belonging to P and I an interval such V (P) ⊂ I. Since P is tree-like, P admits a leaf cell C. Let P ′ be the polyomino which is obtained from P by removing the cell C. Then P ′ is again tree-like and hence is simple by induction. Therefore, since D 6∈ P ′ , there exists a path D ′ : D = D1 , D2 , · · · , Dm of cells with Di 6∈ P ′ for all i and such that Dm is a border cell of I. We let D = D ′ , if Di 6= C for all i. Note that C 6= D1 , Dm . Suppose Di = C for some i with 1 < i < m. Since C a leaf, it follows that the cells C1 , C2 , C3 , C4 and C5 as shown in Figure 11 do not belong to P. Since Di = C and since D ′ is a path of cells it follows that Di−1 ∈ {C1 , C3 , C5 } and Di+1 ∈ {C1 , C3 , C5 }. If Di−1 = C1 and Di+1 = C3 , then we let D : D1 , . . . , Di−1 , C2 , Di+1 , . . . , Dm , 9 C1 C2 C C3 C5 C4 Figure 11. and if Di−1 = C1 and Di+1 = C5 , then we let D : D1 , . . . , Di−1 , C2 , C3 , C4 , Di+1 , . . . , Dm . Similarly one defines D in all the other cases that my occur for Di−1 and Di+1 , so that in any case D ′ can be replaced by the path of cells D which does not contain any cell of P c and connects D with the border of I. This shows that P is simple. Now we show that if P is row or column convex then P is balanced and simple. We may assume that P be column convex. We first show that P is balanced. For this part of the proof we follow the arguments given in [14, Proof of Theorem 3.10]. Let C1 = [C1 , Cn ] be the left most column interval of P and a be lower left corner of C1 . Let α be an admissible labeling for P. If α(a) = 0, then α restricted to P ′ is an admissible labeling of P ′ , where P ′ is obtained from P by removing the cell C1 from P. By applying induction on the number of cells we have that fα ∈ IP ′ ⊂ IP , and we are done is this case. Now we may assume that α(a) 6= 0. We may assume that α(a) > 0. By following the same arguments as in case of tree-like polyominoes, it suffice to show that there exist an inner interval [(i, j), (k, l)] with i < k and j < l such that α((i, j))α((k, l)) > 0 or α((k, j))α((i, l)) > 0. Let [a, b] and [a, c] be the maximal horizontal and vertical edge intervals of P containing a. Then there exist e ∈ [a, b] and f ∈ [a, c] such that α(e), α(f ) < 0 because α is admissible. Let [f, g] be the maximal horizontal edge interval of P which contains f . Then there exists a vertex h ∈ [f, g] such that α(h) > 0. If size([f, g]) ≤ size([a, b]), then column convexity of P gives that [a, h] is an inner interval of P and α(a)α(h) > 0. Otherwise, if size([f, g]) ≥ size([a, b]), then again by using column convexity of P, xa xq − xe xf ∈ IP where q is as shown in following figure. By following the same arguments as in case of tree-like polyominoes, we conclude that fα ∈ IP . Now we will show that P is simple. We may assume that V (P) ⊂ [(k, l), (r, s)] with l > 0. Let C 6∈ P be a cell with with lower left corner a = (i, j), and consider the infinite path C of cells where C = C0 , C1 , . . . and where the lower left corner of Ck is (i, k) for k = 0, 1, . . .. If C ∩ P = ∅, then C is connected to C0 . On the other hand, if C ∩ P = 6 ∅, then, since P is column convex, there exist integers k1 , k2 with l ≤ k1 ≤ k2 ≤ s such that Ck ∈ P if and only if k1 ≤ k ≤ k2 . Since C 6∈ P it follows that j < k1 or j > k2 . In the first case, C is connected to C0 and in the second case C is connected to Cs . 10 Corollary 3.2. Let P be a row or column convex, or a tree-like polyomino. Then K[P] is a normal Cohen–Macaulay domain. 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Jürgen Herzog, Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany E-mail address: juergen.herzog@uni-essen.de Ayesha Asloob Qureshi, The Abdus Salam International Center of Theoretical Physics, Trieste, Italy E-mail address: ayesqi@gmail.com Akihiro Shikama, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 5600043, Japan E-mail address: a-shikama@cr.math.sci.osaka-u.ac.jp 11	0
arXiv:1802.02568v1 [cs.CV] 7 Feb 2018 V I S E R: Visual Self-Regularization Hamid Izadinia∗ University of Washington Pierre Garrigues Flickr, Yahoo Research izadinia@cs.uw.edu garp@yahoo-inc.com Abstract problem. However, image recognition remains an area of active research. ImageNet is indeed biased towards single objects appearing in the middle of the image, which is in contrast with the photos we take with our mobile phones that typically contain a range of objects that appear in context. Also, the list of object categories in ImageNet is a subset of the lexical database WordNet [29]. This makes ImageNet biased towards certain categories such as breeds of dogs, and does not match the scope of more general image recognition tasks such as object detection and localization in context. In this work, we propose the use of large set of unlabeled images as a source of regularization data for learning robust visual representation. Given a visual model trained by a labeled dataset in a supervised fashion, we augment our training samples by incorporating large number of unlabeled data and train a semi-supervised model. We demonstrate that our proposed learning approach leverages an abundance of unlabeled images and boosts the visual recognition performance which alleviates the need to rely on large labeled datasets for learning robust representation. To increment the number of image instances needed to learn robust visual models in our approach, each labeled image propagates its label to its nearest unlabeled image instances. These retrieved unlabeled images serve as local perturbations of each labeled image to perform Visual Self-Regularization (V I S E R). To retrieve such visual self regularizers, we compute the cosine similarity in a semantic space defined by the penultimate layer in a fully convolutional neural network. We use the publicly available Yahoo Flickr Creative Commons 100M dataset as the source of our unlabeled image set and propose a distributed approximate nearest neighbor algorithm to make retrieval practical at that scale. Using the labeled instances and their regularizer samples we show that we significantly improve object categorization and localization performance on the MS COCO and Visual Genome datasets where objects appear in context. Datasets such as MS COCO [25] or Visual Genome [21] have been constructed such that photos are typically composed of multiple objects appearing at a variety of positions and scales. They provide a more realistic benchmark for image recognition systems that are intended for consumer photography products such as Flickr or Google Photos. MS COCO currently contains 300K images and 80 object categories, whereas Visual Genome contains 100K images and thousands of object categories. CNNs are also showing the best performance on these datasets [34, 21]. As training deep neural networks requires a large amount of data and the size of MS COCO and Visual Genome is an order of magnitude smaller than ImageNet, the CNN weights are initialized using the weights of a model that was originally trained on ImageNet. In this paper we focus on improving image recognition performance on MS COCO and Visual Genome. The labels in MS COCO and Visual Genome are obtained via crowdsourcing platforms such as Amazon Mechanical Turk. Hence it is time-consuming and expensive to obtain additional labels. However, we have access to huge quantities of unlabeled or weakly labeled images. For example, the Yahoo Flickr Creative Commons 100M dataset (YFCC) [40] is comprised of a hundred million Flickr photos with userprovided annotations such as photo tags, titles, or descriptions. 1. Introduction Image recognition has rapidly progressed in the last five years. It was shown in the ground-breaking work of [22] that deep convolutional neural networks (CNNs) are extremely effective at recognizing objects and images. The development of deeper neural networks with over a hundred layers has kept improving performance on the ImageNet dataset [7], and we have arguably achieved human performance on this task [35]. These developments have become mainstream and may lead to the perception of image recognition as a solved In this paper, we present a simple yet effective semisupervised learning algorithm that is able to leverage labeled and unlabeled data to improve classification accuracy on the MS COCO and Visual Genome datasets. We first train a fully convolutional network using the multi-labeled data ∗ Work was done while the author was an intern at Flickr, Yahoo Research. 1 Figure 1. The t-SNE [27] map of the whole set of images (including MS COCO and YFCC images) labeled as ‘Bus’ category after applying our proposed V I S E R approach. Can you guess whether green or blue background correspond to the human annotated images of MS COCO dataset? Answer key: blue: MS COCO, green: YFCC (e.g. MS COCO or Visual Genome). Then, we retrieve for each training sample the nearest samples in YFCC using the cosine similarity in a semantic space of the penultimate layer in the trained fully convolutional network. We call these Regularizer samples which can be considered as real perturbed samples compared to the Gaussian noise perturbation considered in virtual adversarial training [31]. Having access to a large set of unlabeled data is critical for finding representative regularizer samples for each training instance. For making this approach practical at scale, we propose an approximate distributed algorithm to find the images with semantically similar attention activation. We then fine-tune the network using the labeled instances and Regularizer samples. Our experimental results show that we significantly improve performance over previous methods where models are trained using only the labeled data. We also demonstrate how our approach is applicable to object-in-context retrieval. 2. Related work The recognition and detection of objects that appear “in context” is an active area of research. The most common benchmarks for this task are the PASCAL VOC [10] and MS COCO datasets. Deep convolutional neural networks have been shown to provide optimal performance in this setting with state-of-the-art performance results for object detection in [34]. It has recently been shown in [33, 38, 9, 3, 42] that it is possible to accurately classify and localize objects using training data that does not contain any object bounding box information. We refer to this training data that does not contain the location information of the object as weaklysupervised. The size of labeled "objects in context" datasets is typically small. For example, MS COCO has around 300,000 images and Visual Genome has over 100,000 images. However, we have access to large amounts of unlabeled web images. The Yahoo Flickr Creative Commons 100M dataset has one hundred million images that have user annotations such as tags, titles, and description. There has been some recent efforts to leverage this user annotation to build object classifiers. For instance, [18] proposes a noise model that is able to better capture the uncertainty in the user annotations and improve the classification performance. It is shown in [19] that it is possible to learn state-of-the-art image features when training a convolutional neural network from a random initialization using user annotations as target labels. In [12], the authors also train deep neural networks from scratch and use the output layers as classifiers directly. However, classifier performance is lower when training on noisy data. Contrary to these approaches, we propose a form of curriculum learning [4] where we first train a model on a small set of clean data, and then augment the training set by mining instances from a large set of unlabeled images. While it is shown that by making small perturbations to the input it is possible to make adversarial examples which can fool machine learning models [39, 23, 24], adversarial examples can be used as a means for data augmentation to improve the regularization capability of the deep models. Our method is related to adversarial training techniques [13, 31, 30] in the sense that additional training instances with small perturbations are created and added to the training data. However, in contrast to those methods, we retrieve real adversarial examples from a large set of person person, sports ball, tennis racket 3 sports ball tennis racket High 64 128 256 512 Conv(3,3,512) 512 Low 1024 Conv(3,3,512) 2048 N N Noisy-Or Conv(3,3,256) Conv(3,3,64) Conv(3,3,128) Figure 2. We use a Fully Convolutional Network to simultaneously categorize images and localize the objects of interest in a single forward pass. The last layer of the network produces an tensor of N heatmaps for localizing objects where each corresponds to one of the Nth object. The green areas correspond to regions with high probability for the object produced by our network. unlabeled images. Our examples are thus real image instances which possess high correlation to the labeled data in the semantic space determined by the penultimate layer of the neural network after the first phase of training. Such instances usually correspond to large perturbations in the input space but follow the natural distribution of the data which is analogous to the adversarial perturbations. We call our retrieved image instances as Regularizer and show that the Regularizer instances can be used to re-train the model and further improve performance. other modalities. For example, in [8] the performance of several nearest neighbor methods is examined on the image captioning task. By conducting extensive experiments, the results of [8] have shown that nearest neighbor approaches can perform as good as state-of-the-art methods for image captioning. Semi-supervised learning is the class of algorithms where classifiers are trained using labeled and unlabeled data. A number of approaches have been proposed in this setting such as Naive Bayes and EM algorithm [32], ensemble methods [5] and propagating labels based on similarity as in [43]. In our case the size of the unlabeled set is three orders of magnitude larger than the size of the labeled set. Existing methods are therefore impractical, and we propose a simple method to propagate labels using a nearest neighbor search. The metric is the cosine distance in the space defined by the penultimate layer of the fully convolutional neural network after it has been trained on the clean dataset. We argue that the size of our unlabeled set is critical in order for the label propagation to work effectively, and we propose approximations using MapReduce to make the search practical [6]. Most recent developments in image recognition have been driven by optimizing performance on the ImageNet dataset. However, images in this dataset have a bias for single objects appearing in the center of the image. In order to increase performance on photos where multiple objects may appear at different scales and position, we adopt a fully convolutional neural network architecture inspired by [26]. Each fully connected layer is replaced with a convolutional layer. Hence, the output of the network is a H × W × N tensor where the width and height depend on the input image size and N is the number of object classes. For each object class the corresponding heatmap provides information about the object’s location as illustrated in Figure 2. In our experiments we use the base architecture of VGG16 [36] shown in Figure 2. Large-scale nearest neighbor search is commonly used in the computer vision community for a variety of tasks such as scene completion [16], image editing with the PatchMatch algorithm [2], or image annotation with the TagProp algorithm [15]. Techniques such as TagProp [15] have been proposed to transfer tags from labeled to unlabeled images. In this work we take advantage of the powerful image representation from a deep neural network to transfer labels as well as regularize training. Similarly labels can be propagated using semantic segmentation [14]. This method is applied on ImageNet which has a bias towards a single object appearing in the center of the image. We focus here on images where objects appear in context. 3.2. Multiple instance learning for multilabel classification Nearest neighbor search has also been shown to be successful in other computer vision applications which involve 3. Proposed Method 3.1. Fully Convolutional Network Architecture We are given a set of annotated images A = {(xi , yi )}i=1...n , where xi is an image and yi = (yi1 , . . . , yiN ) ∈ {0, 1}N is a binary vector determining which object category labels are present in xi . Let f l be the object heatmap for the lth label in the final layer of the network. The probability at location j is given by applying a sigmoid unit to the logits f l , e.g. plj = σ(fjl ). We do not have access to the location information of the objects since we are in a weakly labeled setting. Therefore, to compute the probability score for the lth object category to appear at the jth location, we incorporate a multiple instance learning approach with Noisy-OR operation [28, 41, 11]. The probability for label l is given by Equation 1. Also, for learning the parameters of the FCN, we use stochastic gradient descent to minimize the cross-entropy loss L formalized in Equation 2. Y pl = 1 − (1 − plj ). (1) L= N X j −y l log pl − (1 − y l ) log (1 − pl ) (2) l=1 3.3. Visual Self-Regularization It has been observed that deep neural networks are vulnerable to adversarial examples [39]. Let x be an image and η a small perturbation such that kηk∞ ≤ . If the perturbation is aligned with the gradient of the loss function η = sign(∇x L) which is the most discriminative direction in the image space, then the output of the network may change dramatically, even though the perturbed image x̃ = x + η is virtually indistinguishable from the original. Goodfellow et al. suggest that this is due to the linear nature of deep neural networks. They also show that augmenting the training set with adversarial examples results in regularization similar to dropout[13]. In Virtual Adversarial Training [31] the perturbation is produced by maximizing the smoothness of the local model distribution around each data point. This method does not require the labels for data perturbations and can also be used in semi-supervised learning. The virtual adversarial example is the point in an ball around the datapoint that maximally perturbs the label distribution around that point as measured by the Kullback-Leibler divergence η = arg min KL[p(y\|x, θ) \|\| p(y\|x + r, θ)]. (3) r:krk2 ≤ We propose to draw perturbations from a large dataset of unlabeled images U whose cardinality is much higher than A. For each example x, we use the example x̃ that is nearby in the space defined by the penultimate layer in our fully convolutional network. This layer contains spatial and semantic information about the objects present in the image, and therefore x and x̃ have similar semantics and composition while they may be far away in pixel space. We consider the cosine similarity metric to find samples which are close to each other in the feature space and for efficiency we compute the dot product of the L2 normalized feature vectors. Let θ denote the optimal parameters found after minimizing the cross-entropy loss using the training data in A, and fθ (x) be the L2 normalized feature vector obtained from the penultimate layer of our network(Conv(1,1,2048)). The similarity between two images x and x0 is then computed by their dot product S(x, x0 ) = fθ (x)T fθ (x0 ). For each training sample (xi , yi ) in A, we find the most similar item in U x̃i = arg max S(xi , x), (4) x∈U Algorithm 1 Distributed Regularizer Sample Search function M AP(k, x) . k: sample index in U, x: image data Compute network output layer fθ (x) Compute similarities with samples in A: si = fθ (x)T fθ (xi ), ∀i = 1 . . . n Sort s by descending similarity values si1 ≥ si2 . . . ≥ sin for l = 1 to km do EMIT il , (k, sil ) end for end function function R EDUCE(k, v) . k: sample index in A, v: Iterator over (sample index in U, similarity score) tuples Let v = ((i1 , c1 ), (i2 , c2 ) . . . Sort v by descending similarity values cj1 ≥ cj2 . . . ... for l = 1 to kr do EMIT k, ijl , cjl end for end function and transfer the labels from xi , to generate a new, Real Adversarial (Regularizer), training sample (x̃i , yi ). Similar to adversarial and virtual adversarial training, our method improves the classification performance. We interpret our sample perturbation as a form of adversarial training where additional examples are sampled from a similar semantic distribution as opposed to noise. We also used the perturbation of each labeled sample in the gradient direction (similar to adversarial training) to find the nearest neighbor in unlabeled set and observed similar performance. Therefore, in this paper our focus is on using the labeled samples for finding Regularizer instances to improve performance. 3.4. Large scale approximate regularizer sample search In our experiments we use the YFCC dataset as our set of unlabeled images. Since it contains 100 million images, an exhaustive nearest neighbor search is impractical. Hence we use the MapReduce framework [6] to find approximate nearest neighbors in a distributed fashion. Our approach is outlined in Algorithm 1. We first pre-compute the feature representations fθ (xi ) for xi ∈ A. The size of A for datasets such as MS COCO or Visual Genome is small enough that it is possible for each mapper to load a copy into memory. A mapper then iterates over samples x in U, computes the feature representation fθ (x) and its inner product with the pre-computed features in A. It emits tuples for the top km matches that are keyed by the index in A, and also contain the index in U and similarity score. After the shuffling phase, the reducers can select for each sample in A the kr closest samples in U. We use km = 1000 and kr = 10. We are able to run the search in a few hours, with the majority of Figure 3. Top regularizer examples from unlabeled YFCC dataset (row 2-6) that are retrieved for multi-label image queries in several of the MS COCO categories (first row). the time being in the mapper phase where we compute the image feature representation. Note that our method does not guarantee that we can retrieve the nearest neighbor for each sample in A. Indeed, if for a sample xi there exists km samples xj such that fθ (xj )T fθ (x̃i ) ≥ fθ (xi )T fθ (x̃i ), then the algorithm will output either no nearest neighbor or another sample in U. However we found our approximate method to work well in practice. 4. Experiments 4.1. Semi-Supervised Multilabel Object Categorization and Localization We use the MS COCO [25] and Visual Genome [21] datasets as our source of clean training data as well as for evaluating our algorithms. MS COCO has 80 object categories and is a common benchmark for evaluating object detectors and classifiers in images where objects appear in context. The more recent Visual Genome dataset has annotations for a larger number of categories than MS COCO. Applying our proposed method on the Visual Genome dataset is important to understand whether the algorithm scales to a larger number of categories, as it is ultimately important to recognize thousands of object classes in real world applications. All images for both datasets come from Flickr. In all experiments we only use the image labels for training our models and discard image captions and bounding box annotations. For the MS COCO dataset we use the standard split used in [25] for training and evaluating the models. The training set contains 82,081 images and validation set has 40,504 images. For the Visual Genome dataset we only use object category annotations for images. The images are labeled as a positive instance for each object if the area ratio of the bounding box with regards to the image area is more than 0.025. We only consider the 1,432 object categories for which there are at least 80 image instances in the training set. The Visual Genome test set is the intersection of Visual Genome with the MS COCO validation set which is comprised of 17,471 images. We use the remaining 90,606 images for training our models. As for the source of unlabeled images, we use the tie bottle umbrella refrigerator truck remote Figure 4. Object localization comparison between “FCN,N-OR”(mid row) and “FCN,N-OR,V I S E R”(last row). YFCC dataset [40] and discard the images that are present in Visual Genome or MS COCO. The data is 14TB and is stored in the Hadoop Distributed File System. We use the TensorFlow software library [1] to implement our neural networks and conduct our experiments. To conduct the distributed nearest-neighbor search, we use a CPU cluster. We use VGG16 architecture pre-trained for the ImageNet classification task as our base network. We resize images to 500×500. Our initial learning rate is 0.01 and we apply the 0.1 decay factor for adapting the learning rate two times during the training after 20K and 40K mini-batches. We run stochastic gradient descent for 60K iterations with mini-batches of size 15 which corresponds to 11 epochs. We conduct our experiments on the object classification and point-based object localization tasks. As for the object evaluation metric, we use the mean Average Precision (AP) metric where we first compute the precision for each class and then take the average over classes. For evaluating our object localization, we use the point localization metric introduced in [33], where the location for a given class is given by the location with maximum response in the corresponding object heatmap. The location is considered correct if it is located inside the bounding box associated with that class. Similar to [33] we use a tolerance of 18 pixels. Tables 1 and 2 summarize our results with the mean AP for the classification and localization tasks on the MS COCO and Visual Genome datasets. We compare our performance with three state-of-the-art methods for object localization and classification [33, 38, 3]. In [33], to handle the uncertainty in object localization, the last fully connected layers of the network are considered as convolution layers and a maxpooling layer is used to hypothesize the possible location of the object in the images. In contrast, we use Noisy-OR as our pooling layer. In [38], a multi-scale fully convolutional neural network called ProNet is proposed that aims to zoom into promising object-specific boxes for localization and classification. We compare against the different variants of ProNet with chain and tree cascades. Our method uses a single fully convolutional network, is simpler and has a lighter architecture as compared to ProNet. In all tables ‘FullyConn’ refers to the standard VGG16 architecture while ’FullyConv’ refers to the fully convolutional version of our network (see Figure 2). The Noisy-OR loss is abbreviated as ’N-OR’, and we denote our algorithm with V I S E R. We can see in Table 1 that our proposed algorithm reaches 50.64% accuracy in the object localization task on the MS COCO dataset which is more than a 4% boost over [38] and a 9.5% boost over [33]. Also, without doing any regularization and by only using Noisy OR (N-OR) paired with a fully convolutional network, we obtain higher localization accuracy than Oquab et al. [33] and different variants of ProNet [38]. In the object classification task, our proposed V I S E R approach outperforms other state-of-the-art baselines of [38, 33] by a margin of more than 4.5% and gains an accuracy of 75.48% for the MS COCO dataset. In addition, other variants of [38] are less accurate than our fully convolutional network architecture with Noisy-OR pooling (‘FullyConv, NOR’). This result is consistent with the results we obtained in the object localization task. While the method of [3] obtains competitive performance on the MS COCO localization task, our method outperforms it in the classification task by a large margin of 21.4%. A recent method [9] obtains classification and localization accuracy of 80.7% and 53.4% respectively using the deeper ResNet [17] architecture. Hence it is not directly comparable with ProNet [38], [3] and our method which use the VGG [36] as base network architecture. In addition our proposed method has a label propagation step which produces a large set of labeled images with object level localization in “object in context” scenes and can be used in Table 1. Mean AP for classification and localization tasks on the MS COCO dataset (higher is better). Table 2. Mean AP for classification and localization tasks on the Visual Genome dataset (higher is better). Method Classification Localization Method Classification Localization Oquab et al. [33] ProNet (proposal) [38] ProNet (chain cascade) [38] ProNet (tree cascade) [38] Bency et al. [3] 62.8 67.8 69.2 70.9 54.1 41.2 43.5 45.4 46.4 49.2 FullyConn FullyConv,N-OR FullyConv,N-OR,AT [13] FullyConv,N-OR,VAT [31] FullyConv,N-OR,V I S E R 9.94 12.35 13.96 13.95 14.82 – 7.55 9.05 9.06 9.74 FullyConn FullyConv,N-OR FullyConv,N-OR,AT [13] FullyConv,N-OR,VAT [31] FullyConv,N-OR,V I S E R 66.68 72.52 74.38 74.30 75.48 – 47.47 49.75 49.42 50.64 other learning methods. Also, the method of [9] is based on a new pooling mechanism while our method proposes a better regularization for training ConvNets using a large scale set of unlabeled images in a semi-supervised setting and therefore is orthogonal to [9]. We also perform an ablation study and compare against other forms of regularization using our fully convolutional network architecture with Noisy-OR pooling (‘FullyConv, N-OR’). In Table 1 and 2, we compare three forms of regularization: adversarial training (‘AT’) [13], virtual adversarial training (‘VAT’) [31], and our proposed Visual Self-Regularization (V I S E R) using the YFCC dataset as source of unlabeled images. To conclude, our proposed approach outperforms state-ofthe-art methods as well as several baselines by a substantial margin in object classification and localization tasks according to the results shown in Table 1 and Table 2. Hence, the regularization mechanism of our proposed method results in a performance boost compared to the other forms of adversarial example data augmentation. We show that visual self-regularizers (V I S E R) make our learning robust to noise and provides better generalization capabilities. 4.2. Object-in-Context Retrieval To qualitatively evaluate V I S E R, we show several examples of the Regularizer instances retrieved using our approach in Figure 3. For each of the labeled images shown in the first row of Figure 3, we show the top 5 retrieved images. As we can see, the unlabeled images retrieved by our approach have high similarity with the queried labeled image. Furthermore, most of the objects in the labeled images also appear in the retrieved images. This observation qualitatively demonstrates the effectiveness of our label propagation approach. It is worth mentioning that Figure 3 shows that the relative location of the objects in the retrieved images is fairly consistent with that of the query images. This suggests that our simultaneous categorization and localization approach can also be used for propagating bounding box annotations. Figure 1 shows the results of our V I S E R approach on the ‘Bus’ category. We visualize the t-SNE [27] map of the whole set of images labeled as ‘Bus’ which includes instances from both the labeled images in the MS COCO and unlabeled instances from the YFCC dataset. To produce the t-SNE visualization we take the output of the penultimate layer of our network as explained in Section 3. We L2 normalize the feature vectors to compute the pairwise cosine similarity between images using a dot product. We visualize the t-SNE map using a grid [20]. A different background color (blue vs. green) is assigned to images depending on whether they are from the labeled or unlabeled set. Notice that it is challenging to determine the color corresponding to each dataset as photos are from a similar domain. The images with a blue background belong to the MS COCO dataset and the images with a green background belong to the YFCC dataset. This visualization reveals that there are many images in the large unlabeled web resources that can potentially be used to populate the fully annotated dataset with more examples. This is a step forward for improving object categorization as well as decreasing human effort for supervision. Figure 6 demonstrates the qualitative performance of “FCN,N-OR,V I S E R” for multi-label localization. We visualize the object localization score maps where the localization regions with high probability are shown in green. We also display the localized objects using red dots. The score maps show that our approach can accurately localize small and big objects even in extreme cases where a big portion of the object is occluded. In the first row of Figure 6 ‘dog’ and ‘laptop’ are localized quite accurately while they are largely occluded and truncated. Similarly, the third row shows the accurate localization of a ‘chair’ although it appears in a small region of the image and is largely occluded. When there are multiple instances of an object category, such as ‘person’ in the second row, ‘potted plant’ in the third row, and ‘car’ in the sixth row, all regions corresponding to these instances get a high score. The failure cases of our approach are distinguished via red boxes in Figure 6. For instance, the ‘skateboard’ in row 6 is localized around the region close to the person’s leg. In row 8, although the ‘backpack’ region gets a high score map, it fails to contain the highest peak and thus the localization metric considers it as a mistake. 0.95 0.80 0.65 0.50 0.35 0.20 Cross entropy Dropout Adversarial training VAT VISER(ours) 0.05 Figure 5. Generalization comparison on a synthetic dataset between proposed V I S E R , dropout, adversarial training, and virtual adversarial training (VAT). Training samples are shown with black borders and rest of instances are test set. Each plot demonstrates the contour of the p(y = 1\|x, θ), from p = 0 (blue) to p = 1 (red). Table 3. Classification error on test synthetic dataset (lower is better). Error(%) cross entropy dropout [37] AT [13] VAT [31] VISER 9.244±0.651 9.262±0.706 8.960±0.849 8.940±0.393 8.508±0.493 We show several examples of the localization score maps produced by “FCN,N-OR” and “FCN,N-OR,V I S E R” in Figure 4. By comparing the localized regions in green, we see that “FCN,N-OR,V I S E R” can locate both small and big objects more accurately. For example, in localizing small objects such as ‘tie’, ‘bottle’ and ‘remote’, the peak of the localization region produced by “FCN,N-OR” has a large distance with the correct location of the object while “FCN,N-OR,V I S E R” localizes these objects precisely. Also, “FCN,N-OR” fails to be as accurate as “FCN,N-OR,V I S E R” in localizing big objects such as ‘umbrella’ and ‘refrigerator’. 4.3. Classification on Synthetic Data In order to evaluate the ability of our algorithm to leverage unlabeled data to regularize learning, we generate a synthetic two-class dataset with a multimodal distribution. The dataset contains 16 training instances (each class has 8 modes with random mean and covariance for each mode and 1 random sample per mode is selected), 1000 unlabeled and 1000 test samples. We linearly embed the data in 100 dimensions. Since the data has different modes, we can mimic the object categorization task where each object category appears in a variety of shapes and poses, each of which can be considered as a mode in the distribution. We use a multi layer neural network with two fully connected layers of size 100, each followed by a ReLU activation and optimized via the cross-entropy loss. We compare the generalization behavior of V I S E R with the following regularization methods: dropout [37], adversarial training [13], and virtual adversarial training (VAT) [31]. The contour visualization of the estimated model distribution is shown in Figure 5. We can see that both adversarial training and virtual adversarial training are vulnerable to the location of the training sample of each mode. These regularization techniques can learn a good boundary of the class when the training instance is at the center of the mode, but they over-smooth the boundary whenever the training instance is off-center. However, our proposed V I S E R sampling from unlabeled data learns a better local class distribution as adversarial samples follow the true distribution of the data and are less biased to the training instances. The dropout technique is also learning a good regularization, but it is less smooth at the boundaries of the local modes. Table 3 summarizes the misclassification error on test data over 50 independent runs on the synthetic dataset. 5. Conclusion and Future Work In this paper we have presented a simple yet effective method to leverage a large unlabeled dataset in addition to a small labeled dataset to train more accurate image classifiers. Our semi-supervised learning approach is able to find Regularizer examples from a large unlabeled dataset. We have achieved significant improvements on the MS COCO and Visual Genome datasets for both the classification and localization tasks. The performance of our approach could be further improved in future work by incorporating user provided data such as ‘tags’. Also, having access to a large set of unlabeled data is fairly common in other domains and hence we believe our approach could be applicable beyond visual recognition. References [1] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous systems. 2015. 6 [2] C. Barnes, E. Shechtman, A. Finkelstein, and D. Goldman. Patchmatch: A randomized correspondence algorithm for structural image editing. ACM TOG, 2009. 3 [3] A. J. Bency, H. Kwon, H. Lee, S. Karthikeyan, and B. Manjunath. Weakly supervised localization using deep feature maps. In ECCV, 2016. 2, 6, 7 Figure 6. Localization results of the proposed model on MS COCO validation set. 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Article Antenna Arrays for Line-Of-Sight Massive MIMO: Half Wavelength Is Not Enough Daniele Pinchera , Marco Donald Migliore, Fulvio Schettino and Gaetano Panariello DIEI and ELEDIA@UNICAS, University of Cassino and Southern Lazio, via G. Di Biasio 43, 03043 Cassino, Italy; mdmiglio@unicas.it (M.D.M.); schettino@unicas.it (F.S.); panariello@unicas.it (G.P.) Correspondence: pinchera@unicas.it; Tel.: +39-0776-299-4348 arXiv:1705.06804v2 [cs.IT] 1 Sep 2017 Received: 22 July 2017; Accepted: 8 August 2017; Published: 10 August 2017 Abstract: The aim of this paper is to analyze the array synthesis for 5 G massive MIMO systems in the line-of-sight working condition. The main result of the numerical investigation performed is that non-uniform arrays are the natural choice in this kind of application. In particular, by using non-equispaced arrays, we show that it is possible to achieve a better average condition number of the channel matrix and a significantly higher spectral efficiency. Furthermore, we verify that increasing the array size is beneficial also for circular arrays, and we provide some useful rules-of-thumb for antenna array design for massive MIMO applications. These results are in contrast to the widely-accepted idea in the 5 G massive MIMO literature, in which the half-wavelength linear uniform array is universally adopted. Keywords: MIMO; antenna arrays; line-of-sight propagation; millimeter wave propagation; 5 G mobile communication 1. Introduction Massive MIMO is going to become one of the key technologies of the incoming next generation 5 G wireless systems [1–12]. In particular, the use of frequencies above 6 GHz seems to be particularly appealing, because of two main reasons: first of all, the availability of wide contiguous frequency bands to exploit for communications; second, the relative compactness of the resulting arrays [12–15]. If we briefly focus on this second aspect, we would observe that the use of a frequency of the order of 60 GHz would mean a wavelength λ of 5 mm; thus, a 100λ array would be just half a meter in size. The use of mm-length frequencies is not, unfortunately, without drawbacks. The well-known Friis formula for wireless connections shows a power attenuation that increases with the square of the wavelength, which significantly reduces the area that can be covered by a transmitting antenna. Furthermore, the propagation of the waves at mm-wave frequencies is much different from the propagation at lower frequencies, resembling more a quasi-optical connection between the terminals: the rich scattering condition is really difficult to realize, and the Line-of-Sight (LoS) component becomes dominant (if not the only present) [16,17]. This fact has posed some questions on the actual possibility to achieve the so-called “favorable propagation” in the Multi-User MIMO (MU-MIMO) channel matrix, but both measurements and simulations have confirmed the advantages of massive MIMO also in the extreme LoS condition [18–24]. One of the key points to understand why massive MIMO keeps on working with a very good spectral efficiency also in this case is that the massive antenna array, because of its huge size, does not work in the far-field condition, so that the wave fronts are not planar, but spherical: we could be able to distinguish between two terminals positioned at the same angle with respect to the Base Station (BS) by means of their different distances [25–27]. Electronics 2017, 6, 57; doi:10.3390/electronics6030057 www.mdpi.com/journal/electronics Electronics 2017, 6, 57 2 of 20 Among the many aspects of massive MIMO systems, the optimization of the radiation system has been the object of relatively little research, and equispaced half-wavelength arrays are commonly considered for massive MIMO antennas. However, as pointed out above, massive MIMO antennas work in the near-field condition. This characteristic does not seem to have been exploited in the literature. A further point to be better explored is the reduction of the cost and the complexity of the overall antenna, which is critically related to the number of radiating elements. Accordingly, a fundamental consideration is now in order: is it possible to exploit this near-field behavior with a fixed complexity of the antenna in terms of the number of elements without affecting, or hopefully increasing, the performance? This paper will try to answer this question, showing that optimization of the antenna array using non-uniform arrays in the case of linear antennas and properly equispaced arrays in circular antennas can allow a significant improvement of the communication system performance. 2. Antennas and Propagation Model In this paper, we will use a simplified model for antennas and propagation; in particular, we will consider isotropic radiating elements; we will neglect the effect of mutual coupling and polarization, and we will consider a simple 2D propagation model, i.e., BS antennas and user terminals will lie on the same plane. Furthermore, we will consider a fixed frequency f and the corresponding wavelength λ. These assumptions have a two-fold reason; first, the model we will discuss, and the achieved results, could be easily reproduced by any interested reader; second, the simplified model will allow us to remove the aspects that play a minor role in the definition of the channel matrix (polarization and mutual coupling), focusing only on the aspects, like the antenna positioning, that play the main role in the overall system performance. In particular, we will consider a linear antenna array of N identical elements, with the antennas aligned along the x axis, whose abscissas are stored in the vectors xs = [ xs,1 ; · · · ; xs,N ]. The position of the K terminal antennas will be instead described by the vectors xt = [ xt,1 ; · · · ; xt,K ] and yt = [yt,1 ; · · · ; yt,K ] (see Figure 1). The transfer function between the n-th antenna of the BS and the k-th terminal will be achieved as: e− jβRn,k hn,k = γ (1) Rn,k where γ is an inessential multiplicative constant, β = 2π/λ is the free space wavenumber and: Rn,k = q ( xs,n − xt,k )2 + y2t,k (2) The MU-MIMO channel matrix can be thus built as: H = [ h1 · · · h K ] (3) where the column vector hk represents the channel response of the k-th terminal. This channel matrix can be properly normalized, according to the guidelines in [20] before performing its singular value decomposition or calculating the spectral efficiency (thus making the constant γ inessential). It has also to be underlined that the use of Equation (1) allows us to take into account the different power received by the antennas at the base station because of their different distance from the terminals, as well as the spherical shape of the wavefronts. Electronics 2017, 6, 57 3 of 20 Figure 1. Massive MIMO system scenario. Circles: base station antennas; squares: terminals. 3. User Correlation Analysis From this section on, we will choose f = 60 GHz as the working frequency and a base station array of 200 elements. The choice of considering a fixed number of antennas at the BS has been made just to compare different antenna geometries with the same system complexity. In order to get better insight into the effect of the antenna shape on the system performance, we will start considering the simple case of only two terminals, and we will calculate the correlation coefficient between the two columns of H, as: χ1,2 = \|h1 · h2∗ \| . \|\|h1 \|\| \|\|h2 \|\| (4) The use of this performance metric is due to the fact that it provides a simple way to check the effectiveness of a maximal ratio combining approach at the BS; furthermore, as will be clear in the following, it allows us to provide a simple and effective graphical description of the correlation between user terminal positions. Let us first consider the case of an equispaced array, with the inter-element distance d = λ/2; in Figure 2, it is possible to see the case in which the two terminals are aligned along the y axis, and have only different distances (r1 and r2 ) from the BS. In the left subplot, we can see an image map representing the correlation coefficient of Equation (4) for variable r1 and r2 , while in the right subplot, we can see the plot of the correlation coefficient for some fixed values of r2 . This figure confirms that it is possible to distinguish between two users sharing the same angular position with respect to the center of the BS, but this feature can be exploited only when one terminal is very close to the BS with respect to the other. It has to be underlined that this behavior has been obtained when the two terminals lie on the y axis; for other directions, it is possible to obtain different plots, showing a slightly increased correlation, not reported here for the lack of space. Let us now consider an equispaced array with a larger inter-element distance, say d = 2λ. From an antenna point of view, increasing the inter element distance, without changing the number of elements, increases the far-field distance of the array, which is conventionally calculated as: r FAR = 2[( N − 1)d]2 . λ (5) Electronics 2017, 6, 57 4 of 20 This means that an increase of the inter-element distance of a four factor increases the far field distance of a 16 factor. In Figure 3, we can see the image map and the plot of the correlation for the larger array. It is clear that we are now allowed to easily distinguish two users that share the same angular position with respect to the BS, much more than in the previous case. Figure 2. Two terminal correlation analysis for the λ/2 equispaced array. (Left) Image map with the correlation level for variable r1 and r2 ; (right): plot of the correlation coefficient for some fixed values of r2 . Figure 3. Two terminal correlation analysis for the 2λ equispaced array. (Left) Image map with the correlation level for variable r1 and r2 ; (right): plot of the correlation coefficient for some fixed values of r2 . The previous result has shown that the increase of the inter-element distance seems to have a beneficial effect on the capability of decorrelating users. To confirm this feature, let us see what happens if we consider the first terminal in a fixed position with respect to the array, and we move the second terminal on the xy plane. In Figure 4, we can see the case of a λ/2 equispaced array: apart from an angular sector around the broadside direction of the array, the correlation is very low, thus confirming the very good results, in terms of favorable propagation, found by other researchers. If we instead employ the 2λ equispaced array, we obtain the result depicted in Figure 5: apart from the broadside direction, which shows a slightly lower correlation level with respect to the λ/2 case, there are other directions in space that present a non-negligible correlation. These regions are related to the unavoidable grating lobes, appearing when we consider equispaced elements with an inter-element distance greater than half-wavelength. Clearly, the presence of grating lobes causes ambiguities and must be avoided. As an example, let us imagine a communication system with just two user terminals: when user Terminal A is in the angular position of a grating lobe relative to user Terminal B, it could be very difficult to distinguish them. Electronics 2017, 6, 57 5 of 20 Figure 4. Image map of the correlation coefficient when the first terminal is fixed in the position marked with the circle and the second user is moved on the plane for the λ/2 equispaced array. Figure 5. Image map of the correlation coefficient when the first terminal is fixed in the position marked with the circle and the second user is moved on the plane for the 2λ equispaced array. We would like to specify that in an array not working in the far field region, we could not rigorously talk of grating lobes, but for the sake of simplicity, we will keep on using this definition for the rest of the manuscript. According to the correlation analysis, it seems that the possible advantage of distinguish aligned user terminals could be negated by the effect of grating lobes. These results suggest that a non-equispaced array, not affected by grating lobes, could get the advantages of a larger size of the array, without the drawbacks of the grating lobes, but to confirm this hypothesis, we are first required to analyze what happens with multiple terminals. 4. Multi User Channel Matrix Let us now consider a number K of terminals, uniformly randomly displaced in an angular sector ϕ ∈ [− ϕs , ϕs ], with a minimum distance rmin and a maximum distance rmax from the center of the BS array. For all of the simulations that will be discussed in this section, we will use ϕs = π/3, rmin = 5 m and rmax = 200 m; the choice of these parameters has been done in order to simulate a sectorized coverage of a large planar area, like a plaza or a park. It has to be underlined that differently from [28], we will consider a fixed number of antennas at the BS, in order to compare systems with the same complexity (and cost). Electronics 2017, 6, 57 6 of 20 In order to better emphasize the possibility to discriminate different users according to their distance, we will employ “Normalization 1” from [20]. In this way, all of the columns of the channel matrix H will have the same norm, thus removing the unbalance of channel attenuation between users due to their different relative distance from the BS. M = 1000 different random scenarios have been considered in the numerical simulations. As a first test, in Figure 6, we show as a function of the inter-element distance d and the number of users K the average condition number of the channel matrix, defined as: ρdB = 20 ∗ log10 σ1 σK (6) where σ1 and σK are the greatest and smallest singular values of the channel matrix. The choice of the condition number is due to the fact that it provides a direct indication of the orthogonality of the columns of the channel matrix, and it is related to the stability of its inversion [29], needed in many multi-user MIMO transmission schemes [30]. Figure 6. Mean condition number of the channel matrix as a function of the inter-element distance d and the number of users K. The main advantage in using the condition number is that the presented results are independent of the actual SNR at the receiver; it is indeed true that the condition number provides only a rough indication of the channel quality, since it gives only the spread of the singular values and not their actual distribution, but its value is strongly related with the performances of MIMO precoders and detectors [31–33]. We see that the average condition number is an increasing function of the number of users, but it is interesting to note that (besides some oscillations), it is also a decreasing function of the inter-element distance d at the base station, confirming the advantages of a larger array at the BS foreseen in the previous paragraph. A brief discussion of the oscillations is now needed. Taking as a reference the K = 100 curve of Figure 6, we can see that the condition number increases up to about d = 0.7λ, then it decreases up to d = 1.1λ, then it increase again up to d = 1.3λ, and so on. As we have seen in Figure 5, the presence of grating lobes implies that a number of angular regions of the scenario should be avoided for an effective communication. The number of grating lobes that affect the results is indeed an increasing discrete function of d, while the reduction of the beamwidth and the correlation among users sharing the same angle ϕ with respect to the BS are instead a continuous decreasing function of d. These two effects together contribute to the aforementioned oscillating/decreasing behavior seen in Figure 6. Electronics 2017, 6, 57 7 of 20 It is important to recall that a very low condition number is necessary to apply the Maximal Ratio Combining (MRC) algorithm at the BS, and a low condition number is also necessary to correctly calculate the matrix inversion needed to efficiently use Zero Forcing (ZF) at the BS [11,34]. To get better insight into this result, let us focus on the case of K = 50; in Figure 7, we show the distribution of the condition number as a function of the distance d (each column of the matrix in the image map is a histogram of the condition number). The increase of d from λ/2 to 2λ reduces the average value of the condition number of about 15 dB, and also, its spreading decreases. Figure 7. Distribution of the condition number as a function of the distance d for K = 50. Each column of the matrix in the image map is a histogram of the condition number. With reference to the two extreme cases of d = λ/2 and d = 2λ, let us now analyze the distribution of the singular values for K = 50. In Figures 8 and 9, we have the image map of the distribution of the singular values, as well as the plot of their average values. The larger spacing improves in particular the lower singular values, leading to a much more stable inversion. Electronics 2017, 6, 57 8 of 20 Figure 8. Singular values analysis for d = λ/2 and K = 50. (Left) Image map of the distribution of each one of the 50 singular values; (right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. It has to be underlined that the results achieved in this paragraph hold also when we do not consider the normalization. Since we would just see a slightly larger spreading of the singular values for all of the considered cases, we do not report these results because of the limited space of the paper. Figure 9. Singular values analysis for d = 2λ and K = 50. (Left) Image map of the distribution of each one of the 50 singular values. (Right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. 5. Exploiting Sparse Arrays In the previous section, we have shown that increasing the size of the array is beneficial to the overall MU-MIMO channel matrix condition number: even if we have to deal with the effect of grating lobes, the possibility to discriminate sources that share a similar angular direction, as well as the reduced width of the main beam lead to a better conditioned matrix. It is trivial to recognize that, if we could get rid of the grating lobes, we should expect a further improvement of the performances. To this aim, we would have two different possibilities: in the first Electronics 2017, 6, 57 9 of 20 case, we could employ a sub-array architecture [35], but we should also deal with the reduced scanning capability of our array (and hence, the dimension of the sector we could cover); the second possibility would be to employ a non-equispaced, sparse array. The synthesis of sparse arrays is one of the most studied antenna topics in the last 50 years, and a very large number of methods and algorithms has been developed for linear and planar arrays [36–47]. Unfortunately, none of these algorithms can be directly applied to an MU-MIMO system: our aim is to obtain the antenna positions, within a given range, that allow the best performances of a multi-user scenario, and this problem cannot be directly translated into a classical antenna pattern synthesis problem. To solve this issue, we propose a novel paradigm for antenna array synthesis, inspired by the results presented in the previous paragraph: we will be seeking for the antenna positions providing the best performances on a relatively large set of user terminal scenarios. We could, for example, optimize the antenna array to provide the lowest average condition number of the channel matrix, or we could look for the antenna configuration that guarantees the lowest 0.99-quantile of the condition number. Once we have defined our optimization objective, we could use an evolutionary search method to achieve the sought antenna positions. Actually, the direct application of this approach could be very ineffective, since the number of unknowns we have to deal with is of the order of the number of antennas, and evolutionary algorithms are known to require a computational time that increases exponentially with the number of unknowns [48]. A much more convenient, yet sub-optimal, choice is to define a proper parametrization of the antenna positions, thus reducing the search space of the evolutionary algorithm to a small number of parameters and leading to a reasonable computational time. A simple and effective parametrization of the antenna positions can be found using Tchebyshev polynomials. If we are looking for a symmetric array, the position of the n-th antenna xs,n could be found has: ! ! P xs,n = L 1− ∑ αp p =1 with: L= P un + ∑ α p T2p+1 (un ) (7) p =1 d0 2( N − 1) (8) where Tp (·) is the Tchebyshev polynomial of order p, [α1 , · · · , α P ] is the set of P parameters defining the position of the antenna elements, d0 is the desired average inter-element distance, N is the overall number of antennas and [u1 ; · · · ; u N ] is an equispaced ordered sampling of the interval [−1, 1]. Not all of the combinations of [α1 ; · · · ; α P ] would lead to a monotonically-increasing vector of positions xs in the range [− L, L]; for instance, it is easy to demonstrate that if P = 1, any value of α1 in the range: − 1 1 < α1 < 8 4 (9) would lead to a set of positions in the range [− L, L] satisfying the monotonicity constraint, but for higher values of P, it would be non-trivial to achieve a formula like Equation (9) in closed form. In the remaining part of the paper, we will focus on the relatively easy case of P = 1; this choice is justified by the results of some numerical tests, which have shown only a marginal improvement when using P > 1. In particular, values of α1 < 0 would lead to antenna arrays with a higher density of elements at the extrema, while values of α1 > 0 would lead to antenna arrays with a higher density of elements at the center. Following the guidelines discussed before, we will consider now a parametric analysis of the performances, in terms of average condition number, on a set of 1000 configurations of K = 50 user terminals, as a function of α1 . The result is depicted in Figure 10: apart from values very close to α1 = 0, corresponding to the equispaced array, we achieve a significant advantage in the mean condition number for both positive and negative values of α1 (with a slight advantage for the latter). Electronics 2017, 6, 57 10 of 20 Figure 10. Average condition number as a function of α1 for a number of users K = 50 and d0 = 2λ. To get better insight into this surprising result and to better understand the effect of the non-linear spacing on the array performances, we will consider the same analysis developed in the previous two paragraphs on a test array obtained with α = −0.03 with an average inter-element distance d0 = 2λ. The choice of this particular value of α1 is driven by the desire of reducing the spreading of the inter-element distance, in particular in this case, we have a minimum inter-element distance (at the edges) of 1.53λ and a maximum inter-element distance (at the center) of 2.24λ. In Figure 11, we can see that the capability of the array to distinguish terminals sharing the same angular position with respect to the BS has been marginally improved with respect to the case of Figure 3. The difference with respect to the corresponding equispaced array is instead much clearer from the analysis of Figure 12: now, the effect of the grating lobes is replaced by some sectors of the plane that show a lower (but non-null) correlation coefficient. Figure 11. Two terminal correlation analysis for the 2λ non-equispaced array with α1 = −0.03. (Left) Image map with the correlation level for variable r1 and r2 ; (right) plot of the correlation coefficient for some fixed values of r2 . Electronics 2017, 6, 57 11 of 20 Figure 12. Image map of the correlation coefficient when the first terminals are fixed in the position marked with the circle and the second user is moved on the plane for the 2λ non-equispaced array with α1 = −0.03. In Figure 13, we have instead depicted the effect of the non-linear spacing for a variable number of the average inter-element distance d and the number of users K: the oscillatory behavior of Figure 6 has now disappeared, and the condition number is now a monotonic function of the number of terminals and the average inter-element distance. The improvement with respect to the equispaced array is even more significant when the number of user terminals is higher. It is worth underlining that the use of a non-equispaced array is beneficial even when using an inter-element distance only slightly higher than half-wavelength, so to achieve a performance improvement, we are not required to employ large array sizes. It is also interesting to see that the spreading of the condition number is reduced when using a non-equispaced array: the result in Figure 14, where the case with K = 50 is analyzed, shows a conditioning significantly better than the one presented in the equispaced case of Figure 7, for any average inter-element spacing. Finally, also the analysis of the distribution of the singular values for the K = 50 case confirms the advantages of the non-equispaced array (see Figure 15). Figure 13. Mean condition number of the channel matrix as a function of the average inter-element distance d and the number of users K for a non-equispaced array with α1 = −0.03. Electronics 2017, 6, 57 12 of 20 Figure 14. Distribution of the condition number as a function of the average distance d for K = 50 and the sparse array with α1 = −0.03. Each column of the matrix in the image map is a histogram of the condition number. Figure 15. Singular values analysis for d0 = 2λ, α1 = −0.03 and K = 50. (Left) Image map of the distribution of each one of the 50 singular values; (right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. It is important to underline that, whatever the optimization approach used to define the position of the elements in the sparse array, those positions are fixed: their choice has to be done only once, when designing the array. It is also worth underlining that in the simulations shown, no smart terminal selection techniques [18] have been employed. This means that we can achieve an even better conditioned channel matrix by the application of these approaches. 6. Circular Arrays In the previous sections, we have considered linear BS arrays for a sectoral coverage; if the terminals are displaced in a circular area, it could be convenient to use a circular array. The increase of the antenna dimension, without changing the number of active elements, can be very beneficial also in this case. A circular array does not show grating lobes, so it will not be necessary to consider non-equispaced positioning of the array elements on the circle, but the increase in size would allow a better discrimination of the users, because of the increased far field distance of the array. Electronics 2017, 6, 57 13 of 20 To better compare the results of the present section to the results of the previous ones, we will always consider N = 200 radiating elements, displaced on a circle of radius R = Nd/(2π ) where d is the (approximate) inter-element distance. For the sake of compactness of the paper, we will skip the correlation analysis, and we will only focus on the condition number analysis of the MU-MIMO channel matrix. In particular, we will consider a set of 1000 different propagation scenarios, where a variable number K of randomly-positioned user terminals is displaced between a distance rmin = 5 m and rmax = 200 m from the center of the BS antenna. In Figure 16, we can see the average condition number that we can obtain for a variable inter-element distance d and a variable number of users K. This result confirms the importance of increasing the size of the antenna array as much as possible if we are interested in improving the conditioning of the channel matrix. It is also worth noting that this result is surprisingly very similar to the result obtained with the non-equispaced antenna array of Figure 13. In Figure 17, we instead depict the distribution of the condition number for a variable antenna distance d for the specific case K = 50; even in this case, it is surprising to see the behavior of the condition number that is very similar to the result depicted in Figure 14. Finally, in Figures 18 and 19, we depict the distribution of the singular values for the two specific cases of d = λ/2 and d = 2λ, confirming the advantage of an increased antenna array dimension. Figure 16. Mean condition number of the channel matrix as a function of the average inter-element distance d and the number of users K for a circular array. Figure 17. Distribution of the condition number as a function of the average distance d for the circular array with K = 50. Each column of the matrix in the image map is a histogram of the condition number. Electronics 2017, 6, 57 14 of 20 Figure 18. Singular values analysis for the circular array with d = λ/2 and K = 50. (Left) Image map of the distribution of each one of the 50 singular values. (Right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. Figure 19. Singular values analysis for the circular array with d = 2λ and K = 50. (Left) Image map of the distribution of each one of the 50 singular values; (right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. 7. Evaluating the Spectral Efficiency Up to this point, we have focused on the singular values of the channel matrix and its condition number. As stated before, this choice provides results independent of the SNR at the receiver. A further figure of merit that is often used in MIMO system analysis is spectral efficiency. In this section, we show how the better conditioning of the channel matrix influences the spectral efficiency of massive MIMO systems. In particular, we will evaluate the average maximum achievable rate using a Zero Forcing (ZF) receiver [49], considering a set of 1000 different scenarios for the user terminals: K CZF = ∑ k =1 log 1 + ρ N [(H H H)−1 ]kk (10) where ρ is the average SNR at the receivers and [A]ij takes the (i, j)-entry of the matrix A. In Figure 20, we compare the performances of the equispaced array, with inter-element distances (λ/2, λ, 2λ), with the performances of the non-equispaced array, calculated with a α1 = −0.03 and average inter-element distance of (λ, 2λ); the SNR at the receiver is 5 dB. It is worth noting that Electronics 2017, 6, 57 15 of 20 the maximum spectral efficiency is increased from C = 146.1 bit/s/Hz/cell for K = 45 of the λ/2 equispaced array to C = 162.6 bit/s/Hz/cell for K = 55 for the 2λ non equispaced array. We can obtain an average spectral efficiency gain of about 11% just positioning the radiating element of the BS in an optimal fashion. In Figure 21, we present the same results, but comparing only the λ/2 equispaced array and the 2λ non-equispaced array. In this plot, we also represent, by means of two colored areas, the spreading of the spectral efficiency: the areas represent a range of a standard deviation with respect to the mean. It is interesting to compare the “top” border of these two areas, which gives us an indication of the performances achievable by a smart user scheduling scheme: the advantage of the non-equispaced architecture is again clear. As a final remark, in our numerical tests, we have also considered the Dirty Paper Coding (DPC) capacity [30], which is calculated supposing the use of the optimum receiver. In this case, the results are stable with respect to the inter-element distance and marginally depend on the use of non-equispaced architectures. The reason for this fact is that only the greatest singular values of the channel matrix affect the DPC, and from the plots in Figures 8, 9 and 15, we can see that the distribution of the elements of the BS array influences mostly the distribution of the smallest singular values. Figure 20. Average maximum achievable Zero Forcing (ZF) rate as a function of the number of user terminals K. Electronics 2017, 6, 57 16 of 20 Figure 21. Average maximum achievable ZF rate as a function of the number of user terminals K; the areas represent the range of the standard deviation with respect to the mean. Actually, communication schemes that achieve the DPC capacity are very hard to implement in the massive MIMO framework, where linear processing (like ZF) is preferred for computing reasons, and for this reason, we decided to show only the ZF performances. 8. Some Considerations and Antenna Array Design Guidelines In this section, we would like to sum up some design consideration that can be inferred by the shown results. All of the simulations have been performed focusing on arrays with N = 200 radiating elements, but we have also performed other simulations, with a different number of radiators, confirming the advantages of sparse arrays for massive MIMO applications. We have indeed chosen a very simple antenna model for the simulations (an array of isotropic elements without mutual coupling), but in the examples reported in the paper, the inter-element spacing is always larger than 0.5 wavelengths, and in some cases, it is much larger. For such inter-element distances, it is well known that mutual coupling usually plays a minor role. Anyway, we have performed some numerical tests including the mutual coupling (not reported here for lack of space), and we found that there was no significant modification of the distribution of the singular values. Similarly, the element pattern and the polarization will provide a different amplification of the signal received by the terminals, but the overall impact on the singular values distribution would be small. For these reasons, we preferred the use of the architecture of isotropic elements without mutual coupling, providing more easily repeatable results by interested readers. We would also like to underline that the proposed increase in size of the array can have an effect on the overall cost of the antenna system, but this increment would not be significant when dealing with mm-wave frequencies. It is indeed true that the optimization of the BS antenna that we propose has to be realized only when setting up the system; no optimizations/antenna selections are required during operation. On the contrary, the use of non-equispaced architectures, instead of equispaced ones, should not have, in general, a direct impact on the overall cost or complexity of the communication system. The only relevant effect that should be taken into account in an architecture trade-off is the difficulty of employing modular BS arrays (i.e., antenna arrays divided in equal modules of a fixed number of elements): in this case, it could be advantageous to design non-equispaced modules for the BS array and placing them in a non-regular fashion, but the discussion of this possibility goes beyond the aims of this paper. Electronics 2017, 6, 57 17 of 20 A final consideration is now needed. In the numerical tests shown, we have considered the LoS component only; even if this is a reasonable assumption, we could argue if the shown results hold also when a non-LoS propagation component is present. Some preliminary simulations show that the advantage of using larger, non-equispaced arrays is still present, but smaller. Anyway, the analysis of this case deserves particular attention and will be the main topic of a forthcoming work. Summing up, we would like to list some rules-of-thumb for designing massive MIMO arrays that work in LoS propagation. 1. 2. 3. The antenna array geometry needs to be matched to the expected user terminal displacement. For sectoral coverage, a linear array is preferable, while for circular coverage, a circular antenna works better, since a linear array is not capable of discriminating between two terminals in specular positions with respect to the array. Given the number of radiating elements, and hence transceivers, that can be used in the MIMO system, the BS antenna array should be as large as possible. Once the maximum dimension is used, in a linear array, the radiating elements should be possibly arranged in a non-equispaced fashion, by using for instance the positions provided by Equation (7). 9. Conclusions In this paper, we have discussed the impact of the array on the performances of a massive MIMO system. In particular, we have analyzed the LoS case, showing that increasing the size of the array can be very beneficial to the condition number of the MU-MIMO channel matrix: since the BS antenna is working in the near-field region of the array, a larger element distance allows one to exploit the spherical shape of the wavefronts. We have also shown that using a non-equispaced array, we are able to further improve the resulting condition number, since by getting rid of the grating lobes, we are able to achieve a significant lowering of the average condition number also for small increases of the antenna array dimension. Such an improvement in the conditioning of the channel matrix results in an improvement in the spectral efficiency of the massive MIMO systems when using linear processing of the signals. We have also analyzed the results achievable by means of circular arrays and a full circle coverage, finding results perfectly aligned with the ones of the non-equispaced linear array. It is worth underlining that all of the performed analyses have been done considering a fixed BS complexity. It is trivial to understand that with a proper sparse array design, we could reduce the number of antennas at the BS, achieving the same average conditioning of the channel matrix for a desired number of user terminals. It is fundamental to underline that the improvement that can be achieved by optimizing the antenna array geometry has a minimum impact on the overall system cost and does not increase the computational effort of the communication schemes in any way. As a development of the present work, we are working on the design of more complex planar and conformal antenna array geometries; we are also working towards the use of a ray-tracing simulator, in order to check the achieved results when the line of sight is the dominant, but not the only, propagation term present. 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On Reachable Sets of Hidden CPS Sensor Attacks arXiv:1710.06967v1 [cs.SY] 19 Oct 2017 Carlos Murguia and Justin Ruths Abstract— For given system dynamics, observer structure, and observer-based fault/attack detection procedure, we provide mathematical tools – in terms of Linear Matrix Inequalities (LMIs) – for computing outer ellipsoidal bounds on the set of estimation errors that attacks can induce while maintaining the alarm rate of the detector equal to its attack-free false alarm rate. We refer to these sets to as hidden reachable sets. The obtained ellipsoidal bounds on hidden reachable sets quantify the attacker’s potential impact when it is constrained to stay hidden from the detector. We provide tools for minimizing the volume of these ellipsoidal bounds (minimizing thus the reachable sets) by redesigning the observer gains. Simulation results are presented to illustrate the performance of our tools. I. I NTRODUCTION There has recently been significant interest and work in the broad area of security of cyber-physical systems (CPS), see for example [1]-[8]. This topic investigates the properties of conventional control systems in the presence of adversarial disturbances. Control theory has shown great ability to robustly deal with disturbances and uncertainties [9]. However, adversarial attacks raise all-new issues due to the aggressive and strategic nature of the disturbances that attackers might inject into the system. This paper focuses on attack detection and attack capabilities in CPSs. A majority of the work on attack detection leverages the established literature of fault detection [1],[2],[10],[11]. A fault detection approach uses an estimator to forecast the evolution of the system dynamics. When the residual (the difference between what is measured and the estimation), or some function of the residual, is larger than a predetermined threshold, an alarm is raised. Arguably the most insidious attacks are those that occur without our knowledge. Fault detectors impose limits on the attacker, if the attacker aims to avoid being identified. Beyond retooling these existing methods for the new attack detection context, a fundamental question is: given a chosen fault detection approach, how does this method constrain the influence of an attacker? More specifically, what is an attacker able to accomplish when a system employs certain fault detection procedure? Different methodologies exist for evaluating the impact of attacks. Most of the existing work uses some measure This work was supported by the National Research Foundation (NRF), Prime Minister’s Office, Singapore, under its National Cybersecurity R&D Programme (Award No. NRF2014NCR-NCR001-40) and administered by the National Cybersecurity R&D Directorate. C. Murguia is with the Engineering Systems and Design Pillar, Singapore University of Technology and Design. J. Ruths is with the Departments of Mechanical and Systems Engineering, University of Texas at Dallas. emails: murguia_rendon@sutd.edu.sg & jruths@utdallas.edu. of state (or state estimate) deviation. In [2], the authors identify that if the attacker can take advantage of the zero dynamics of a (noise-free) input-output system, it can modify the system dynamics without reflecting its influence in the residual variables. This type of attacks are stealthy to any fault detector. A number of groups have studied the system response when the attacks are constrained by the detector. An important distinction between the collection of existing work – and the work discussed here – is the definition of how the attacker is constrained. We suggest the following terminology. While the term stealthy attack is used very broadly, we suggest that this refer to the zero-dynamics case, as discussed in [2], because these attacks do not propagate to the residual. Some work has investigated the case of system response due to what we here call zero-alarm attacks, i.e., attacks such that the detector threshold is never crossed [12]-[16]. Because real systems (with noise) always have a nonzero rate of false alarms raised by the detector, this attack model yields a relatively obvious attack signature because the alarms stop as soon as the attack starts. Other papers identify attacks that mimic the false alarm rate, thus making the alarm rate during the attack very close to the false alarm rate before the attack started [17],[18]. These attacks we call hidden attacks because although they do change the distribution of the residual, these changes are hidden from the way the detector evaluates the distribution. A majority of this work uses state bounds or steady-state limits to quantify the impact that an attacker can have. The exceptions to this are [17],[18], which quantify the reachable set of states and estimation errors when driven by the attack input. This paper fuses several of these successful lines of research with a more strict interpretation of hidden attacks. The papers [17],[18] consider hidden attacks, however, they permit the alarm rate to change by a small value; the attacker capabilities that are derived are associated with this small deviation rather than the full scope of allowable attacks. Here, we fix the alarm rate exactly to study true hidden attacks (i.e., alarm rate exactly equal to the false alarm rate), and characterize the reachable sets on the estimation error dynamics associated with this broader definition of possible attack vectors. In this work, we characterize the hidden reachable sets that the attacker can induce through manipulation of sensor data. Because in general, it is quite difficult to compute these sets exactly, for given system dynamics and attack detection scheme, we derive ellipsoidal bounds on the hidden reachable sets using Linear Matrix Inequalities (LMIs) [19]. Then, we provide synthesis tools for minimizing these bounds (minimizing thus the hidden reachable set) by properly redesigning the detectors. This builds off of our previous work in [18]. The strict interpretation of hidden attacks requires more direct handling of the effect of noise. To derive finite ellipsoidal bounds, we introduce the notion of p-probable reachable sets, which provides a nested set of ellipsoidal bounds based on the probability of the driving random sequences taking certain values. Because the derivation of the reachable set of states from the reachable set of estimation errors is captured in [18] (for a class of observer-based output feedback controllers), and similar techniques can be used in this paper, we report here only on estimation error reachable sets. Note that the problem formulation in this paper, while seemingly similar, requires an entirely different characterization from [18]. II. S YSTEM D ESCRIPTION & ATTACK D ETECTION We study LTI stochastic systems of the form: ( x(tk+1 ) = F x(tk ) + Gu(tk ) + v(tk ), y(tk ) = Cx(tk ) + η(tk ), (1) with sampling time-instants tk , k ∈ N, state x ∈ Rn , measured output y ∈ Rm , control input u ∈ Rl , matrices F , G, and C of appropriate dimensions, and i.i.d. multivariate zero-mean Gaussian noises v ∈ Rn and η ∈ Rm with covariance matrices R1 ∈ Rn×n , R1 ≥ 0 and R2 ∈ Rm×m , R2 ≥ 0, respectively. The initial state x(t1 ) is assumed to be a Gaussian random vector with covariance matrix R0 ∈ Rn×n , R0 ≥ 0. The processes v(tk ), k ∈ N and η(tk ), k ∈ N and the initial condition x(t1 ) are mutually independent. It is assumed that (F, G) is stabilizable and (F, C) is detectable. At the time-instants tk , k ∈ N, the output of the process y(tk ) is sampled and transmitted over a communication network. The received output ȳ(tk ) is used to compute control actions u(tk ) which are sent back to the process, see Fig. 1. The complete control-loop is assumed to be performed instantaneously, i.e., the sampling, transmission, and arrival time-instants are supposed to be equal. In this paper, we focus on attacks on sensor measurements. That is, in between transmission and reception of sensor data, an attacker may replace the signals coming from the sensors to the controller, see Fig. 1. After each transmission and reception, the attacked output ȳ takes the form: ȳ(tk ) := y(tk ) + δ(tk ) = Cx(tk ) + η(tk ) + δ(tk ), (2) where δ(tk ) ∈ Rm denotes additive sensor attacks. Denote xk := x(tk ), uk := u(tk ), vk := v(tk ), ȳk := ȳ(tk ), ηk := η(tk ), and δk := δ(tk ). Using this new notation, the attacked system is written as follows xk+1 = F xk + Guk + vk , (3) ȳk = Cxk + ηk + δk . A. Observer In order to estimate the state of the process, we use the following Luenberger observer [20] (4) x̂k+1 = F x̂k + Guk + L ȳk − C x̂k , with estimated state x̂k ∈ Rn , x̂1 = E[x(t1 )], where E[ · ] denotes expectation, and observer gain matrix L ∈ Rn×m . Define the estimation error ek := xk − x̂k . Given the system Fig. 1. Cyber-physical system under attacks on the sensor measurements. dynamics (3) and the observer (4), the estimation error is governed by the following difference equation ek+1 = F − LC ek − Lηk − Lδk + vk . (5) The pair (F, C) is detectable; hence, the observer gain L can be selected such that (F − LC) is Schur. Moreover, under detectability of (F, C), if there are no attacks (i.e., δk = 0), where 0 denotes the zero matrix of appropriate dimensions, the covariance matrix Pk := E[ek eTk ] converges to steady state in the sense that limk→∞ Pk = P exists, see [21]. For a given L and δk = 0, it can be verified that the asymptotic covariance matrix P = limk→∞ Pk is given by the solution P of the following Lyapunov equation: (F − LC)P (F − LC)T − P + R1 + LR2 LT = 0. (6) It is assumed that the system has reached steady state before an attack occurs. B. Residuals and Hypothesis Testing In this manuscript, we characterize the effect that output injection attacks can induce in the system with being detected by fault detection techniques. The main idea behind fault detection theory is the use of an estimator to forecast the evolution of the system. If the difference between what it is measured and the estimation is larger than expected, there may be a fault in or attack on the system. Although the notion of residuals and model-based detectors is now routine in the fault detection literature, the primary focus has been on detecting and isolating failures that have known signatures in the degradation of measurement quality, i.e., faults with specific structures. Now, in the context of an intelligent adversarial attacker for which there is no known attack signature, new challenges arise to understand the effect that an adaptive intruder can have on the system without being detected. In this paper, we use the linear observer introduced in the previous section as our estimator. Define the residual sequence rk , k ∈ N, as rk := ȳk − C x̂k = Cek + ηk + δk , (7) which evolves according to the difference equation: ek+1 = F − LC ek − Lηk − Lδk + vk , rk = Cek + ηk + δk . If there are no attacks, the steady state mean of rk is E[rk+1 ] = CE[ek+1 ] + E[ηk+1 ] = 0m×1 , and its asymptotic covariance matrix is given by T Σ := E[rk+1 rk+1 ] = CP C T + R2 . (8) (9) (10) It is assumed that Σ ∈ Rm×m is positive definite. For this residual, we identify two hypotheses to be tested: H0 the normal mode (no attacks) and H1 the faulty mode (with faults/attacks). Then, we have E[rk ] = 0m×1 , E[rk ] 6= 0m×1 , or H0 : H1 : T E[rk rk ] = Σ, E[rk rkT ] 6= Σ, where 0m×1 denotes an m-dimensional vector composed of zeros only. In this manuscript, we use the chi-squared procedure for examining the residual and subsequently detecting attacks. C. Distance Measure and Chi-squared Procedure The input to any detection procedure is a distance measure zk ∈ R, k ∈ N, i.e., a measure of how deviated the estimator is from the sensor measurements. We employ distance measures any time we test to distinguish between H0 and H1 . The chi-squared test uses a quadratic form on the residual as distance measure to test for substantial variations in mean and variance of the error between the measured output and the estimate. Consider the residual sequence rk , (8), and its covariance matrix Σ, (10). The chisquared procedure is defined as follows. Chi-squared procedure: If zk := rkT Σ−1 rk > α, k̃ = k. (11) Design parameter: threshold α ∈ R>0 . Output: alarm time(s) k̃. Thus, the procedure is designed so that alarms are triggered if zk exceeds the threshold α. The normalization by Σ−1 makes setting the value of the threshold α system independent. This quadratic expression leads to a sum of the squares of m normally distributed random variables which implies that the distance measure zk follows a chi-squared distribution with m degrees of freedom, see, e.g., [22] for details. D. False Alarms The occurrence of an alarm in the chi-squared procedure when there are no attacks to the CPS is referred to as a false alarm. The threshold α must be selected to fulfill a desired false alarm rate A∗ . Let A ∈ [0, 1] denote the false alarm rate of the chi-squared procedure defined as the expected proportion of observations which are false alarms, i.e., A := pr[zk ≥ α], where pr[·] denotes probability, see [23] and [24]. Proposition 1 [13]. Assume that there are no attacks on the system and consider the chi-squared procedure (11) with residual rk ∼ N (0, Σ) and threshold α ∈ R>0 . Let −1 ∗ (·, ·) denotes the α = α∗ := 2P−1 ( m 2 , 1 − A ), where P inverse regularized lower incomplete gamma function (see [22]), then A = A∗ . III. H IDDEN R EACHABLE S ETS In this section, we provide tools for quantifying (for given L) and minimizing (by selecting L) the impact of the attack sequence δk on the estimation error ek when the chisquared procedure is used for attack detection. To quantify the effect of attacks, we need to introduce some measure of impact. However, because malicious adversaries may launch any arbitrary attack, we need a measure which can capture all possible trajectories that the attacker can induce in the estimation error dynamics, given how it accesses the dynamics (i.e., through residual variables by tampering with sensor measurements). We propose to use the reachable set of the attack sequence δk as our measure of impact. We are interested in attacks that do not change the false alarm rate of the detector A, i.e., Ā = A, where Ā denotes the alarm rate under the attacker’s action. This class of attacks is what we refer to as hidden attacks and the trajectories that hidden attacks can induce in the system are referred to as hidden reachable sets. In this section, we provide tools based on Linear Matrix Inequalities (LMIs) for computing outer ellipsoidal bounds on the hidden reachable sets induced by the attack sequence δk given the system dynamics, the chi-squared procedure, the noise, and the false alarm rate A. A. Attack Model and Hidden Reachable Sets We assume that the attacker has perfect knowledge of the system dynamics, the observer, measurements, and detection procedure (chi-squared). It is further assumed that all the sensors can be compromised by the attacker at each time step (the case where not all the sensors are attacked is left as future work). By considering this strong, worst-case attacker, we are able to construct an upper bound on the abilities of the attacker. Consider the estimation error dynamics (8), the residual sequence rk = Cek + ηk + δk , and the distance measure 1 1 (12) zk = \|\|Σ− 2 rk \|\|2 = \|\|Σ− 2 (Cek + ηk + δk )\|\|2 , − 21 denotes the symmetric squared root matrix of where Σ Σ−1 . The set of feasible attack sequences that the opponent can launch while satisfying Ā = A can be written as the following constrained control problem on δk : ( δk ∈ R m ek satisfies (8), and 1 pr[\|\|Σ− 2 (Cek + ηk + δk )\|\|2 > α ] = A, ) , (13) for k ∈ N. We are interested in the error trajectories that the attacker can induce in the system restricted to satisfy (13). Note that, as long as Ā = A, the attacker may induce any arbitrary random sequence δk . This and the fact that vk and ηk are Gaussian (thus having infinite support) imply that deterministic reachable sets induced by δk and the noise sequences are generally unbounded. To overcome this obstacle, we introduce the notion of p-probable hidden 1 reachable sets Rpα . Define ζk := Σ− 2 (Cek + ηk + δk ) and note that the estimation error dynamics (8) can be written in terms of ζk as: 1 ek+1 = F ek − LΣ 2 ζk + vk , (14) 1 ζk = Σ− 2 (Cek + ηk + δk ). For given false alarm rate A and probability p ∈ (0, 1), the p-probable hidden reachable set of the attack sequence δk in (14), Rpα , is defined as the set of ek ∈ Rn , k ∈ N that can be reached from the origin e1 = 0 due to the the attacker’s action δk restricted to satisfy Ā = A and 2 pr[\|\|ζk \|\|2 ≤ ζ̄p ] = pr[kvk k ≤ v̄p ] = p, (15) for some constants ζ̄p , v̄p ∈ R>0 , i.e., p n e1 = 0, . (16) Rα := ek ∈ R ek , δk , vk satisfy (13)-(15), By restricting the probabilities in (15), we are delimiting the support of the attack and noise sequences to compact sets. Then, the p-probable hidden reachable sets correspond to the trajectories of the system when the driving random sequences are restricted to satisfy Ā = A and (15). For delimited vk and δk , we can characterize reachable sets using deterministic tools. In general, it is analytically intractable to compute Rpα exactly. Instead, using LMIs, for some positive definite matrix Pαp ∈ Rn×n , we derive outer ellipsoidal bounds of the form Eαp := {ek ∈ Rn \|eTk Pαp ek ≤ 1} containing Rpα . Remark 1 Note, from (14), that if for some k = k ∗ , ek∗ 6= 0 and ρ[F ] > 1, where ρ[·] denotes spectral radius, then \|\|ek \|\| diverges to infinity as k → ∞ for any non-stabilizing ζk . That is, Rpα is unbounded if the system is open-loop unstable. If ρ[F ] ≤ 1, then \|\|ek \|\| may or may not diverge to infinity depending on algebraic and geometric multiplicities of the eigenvalues with unit modulus of F (a known fact from stability of LTI systems), see [21] for details. Given Remark 1, in what follows, we consider open-loop stable systems (ρ[F ] < 1). The following result is used to compute the ellipsoidal bounds Eαp . Lemma 1 [25] Let ξk ∈ Rn , ξ1 = 0, Vk := ξkT Pξk , for some positive definite matrix P ∈ Rn×n , and ωkT ωk ≤ ω̄, ω̄ ∈ R>0 . If there exists a constant b ∈ (0, 1) such that 1−b T Vk+1 − bVk − ωk ωk ≤ 0, ∀ k ∈ N, (17) ω̄ T then, Vk = ξk Pξk ≤ 1. B. Case 1: p ∈ [0, 1 − A] Because the attack sequence is restricted to satisfy (13), we start computing the ellipsoidal bounds corresponding to p = 1 − A, i.e., Eα1−A . It is easy to verify using Lemma 1 that Eαp ⊆ Eα1−A for p ∈ [0, 1 − A] because ζ̄p ≤ ζ̄1−A = α and v̄p ≤ v̄1−A in (15); i.e., all p-probable ellipsoidal bounds for p ∈ [0, 1 − A] lie within the 1 − A-probable ellipsoidal bound. It follows that Rpe ⊆ Eα1−A for p ∈ [0, 1−A], i.e., for p in this interval, we only need to compute the ellipsoidal bound corresponding to p = 1−A. Characterizing p-probable sets for small p values is of little interest because they do not provide a informative bound on system trajectories (since the smaller p is, the more trajectories lie outside the p-probable ellipsoidal bound). We work with the data available in this setting, namely the number of alarms raised by the detector, to bound the most informative p = 1 − A probable reachable set; in Case 2, we extend these results for larger p values. Theorem 1 For given system matrix F , observer gain L, residual covariance matrix Σ, and false alarm rate A, consider the set R1−A in (16). If there exists a positive α definite matrix P ∈ Rn×n and b ∈ (0, 1) satisfying the following matrix inequality:   bP FTP 0 0 00 1 PF P P −PLΣ 2 0 0   1−b  0 0 0 0 P  ≥ 0; ω̄ I  (18) 1 1−b  0 −Σ 2 LT P 0 0 0   ω̄ I  0 0 0 0 I 0 0 0 0 0 0I 1−A 1−A for ω̄ = α + v̄1−A ; then, Rα ⊆ Eα with Pα1−A = P, i.e., the (1 − A)-probable hidden reachable set is contained in the ellipsoid Eα1−A = {ek ∈ Rn \|eTk Pα1−A ek ≤ 1}. Proof : For a positive definite matrix P ∈ Rn×n , consider the function Vk := eTk Pek , then, from (16), inequality (17) takes the form:   1 bP − F T PF F T PLΣ 2 −F T P 1 1 1 1 T 2 2 Σ 2 LT P  ϑk = −ϑTk  Σ 2 LT PF 1−b ω̄ I − Σ L1 PLΣ 1−b 2 −PF PLΣ ω̄ I − P T =: −ϑk Qe ϑk ≤ 0, where ϑ := (eTk , ζkT , vkT )T . The above inequality is satisfied if and only if Qe ≥ 0. Matrix Qe can be written as the Schur complement of a higher dimensional matrix Q′e ; hence, Qe ≥ 0 ↔ Q′e ≥ 0, i.e., Qe ≥ 0 ↔   bP 0 0 F T P 00 1  0 1−b I 0 −Σ 2 LT P 00 ω̄   1−b  0 (19) I P 00 0 ′  ≥ 0. ω̄  Qe :=  1  2 P 00 PF −PLΣ P  0 0 0 0 I 0 0 0 0 0 0I Finally, inequality (18) follows from (19) by a simple reordering of rows and columns.The result follows now from Lemma 1 by taking Pα1−A = P and ω̄ = α + v̄1−A . The result in Theorem 1 provides a tool for computing ellipsoidal bounds on R1−A . To make the bounds most α useful, we next construct ellipsoids with minimal volume, i.e., the tightest possible ellipsoid bounding R1−A . In this α case, we have to minimize det P −1 subject to (18) (because det P −1 is proportional to the volume of eTk Pek = 1). This is formally stated in the following corollary of Theorem 1, see [19] for further details. Corollary 1 For given matrices (F, L, Σ), false alarm rate A, and b ∈ (0, 1), the solution P of the following convex optimization: ( minP − log det P, (20) s.t. P > 0 and (18), for ω̄ = α + v̄1−A , minimizes the volume of the ellipsoid Eα1−A (with Pα1−A = P) bounding R1−A . α See [26] for an example of how to solve (26) using YALMIP. As we now move toward redesigning L to minimize the ellipsoids, we note that as \|\|L\|\| → 0, the volume of Eα1−A goes to zero because the attack-dependent term in 1 (14), LΣ 2 ζk , vanishes. In other words, without any other considered criteria, the observer gain leading to the minimum volume ellipsoid is trivially given by L = 0. While this is effective at eliminating the impact of the attacker, it implies that we discard the observer altogether and, therefore, forfeit any ability to build a reliable estimate of the system state. If we impose a performance criteria that the observer must satisfy in the attack-free case (e.g., convergence speed, noiseoutput gain, and minimum asymptotic variance), it has to be added into the minimization problem (26) so as to minimize the volume of Eα1−A while still achieving the observer performance in the attack-free case. For completeness, in the following proposition, we provide an LMI criteria for ensuring that the H∞ gain from the noise to the residual rk in (8) is less than or equal to some γ ∈ R>0 . Then, using this criteria and Theorem 1, we provide a synthesis tool for minimizing the volume of Eα1−A while ensuring a desired H∞ performance in the attack-free case. Proposition 2 For given matrices (F, C, L), if there exist a positive definite matrix P ∈ Rn×n and constant γ ∈ R>0 satisfying the following matrix inequality:   P 0 0 (F − LC)T P C T  0 γ2I 0 −LT P I    2  0 0 γ I P 0   ≥ 0, (21)  P(F − LC) −PL P P 0  C I 0 0 I then, the H∞ gain from the noise νk := (ηkT , vkT )T to the residual rk = Cek + ηk of the estimation error dynamics (8) is less than or equal to γ. P) bounding R1−A and guarantees that the H∞ gain from α the noise νk = (ηkT , vkT )T to the residual rk of (8) is less than or equal to γ in the attack-free case. Proof : This follows from Theorem 1, Proposition 2, and the linearizing change of variables M = PL. C. Case 2: p ∈ (1 − A, 1] Note that, for p ∈ (1 − A, 1], ζ̄p > ζ̄1−A = α according 2 to (15). Then, we can write ζ̄p = α + ǫp and pr[kζk k ≤ α + ǫp ] = 1 − A + ap, for some ǫp ∈ (0, ∞) and ap ∈ (0, A]. To be able to compute ellipsoidal bounds, the constant ǫp corresponding to a given probability 1−A+ap is required. If ǫp is available, we can restrict ζk to compact sets as in Case 1. Note, however, that the distribution of the attack sequence δk (and thus the one of ζk ) is generally unknown. Actually, the attacker may induce any arbitrary (and possibly) nonstationary random sequence ζk in (14) as long as Ā = A. Nevertheless, we can obtain bounds on ǫp using Markov’s inequality [22] to link the statistical properties of ζk with ǫp . This is stated in the following proposition. Proposition 3 Denote Mk := E[ζk ζkT ] and µk := E[ζk ]. For given false alarm rate A, probability p = 1 − A + ap , and ap ∈ (0, A), the following is satisfied:  2  (23a)  pr[kζk k ≤ α + ǫp ] ∈ [1 − A + ap , 1], tr[Mk ] + µTk µk   − α. for all ǫp ≥ ǫp := A − ap (23b) 2 The proof of Proposition 2 is omitted here due to the page limit. However, this is a standard result and details about the proof can be found in, e.g., [9] and references therein. In the following corollary of Theorem 1 and Proposition 2, we formulate the optimization problem for designing the observer gain L such that the volume of the ellipsoid Eα1−A is minimized and a desired H∞ performance is achieved in the attack-free case. Corollary 2 For given system matrices (F, C), residual covariance matrix Σ, false alarm rate A, b ∈ (0, 1), and γ ∈ R>0 , if there exist matrices P ∈ Rn×n and M ∈ Rn×m solution to the following convex optimization:  minP,M − log det P,         bP F T P 0 0 00   1  PF  P P −M Σ 2 0 0     1−b   0 P I 0 0 0  ω̄  s.t. P > 0,  ≥ 0, and  1  T 1−b   0 −Σ 2 M 0 I 0 0  ω̄   0 0 0 0 I0  0 0 0 0 0 I        P 0 0 F T P − CT M T CT   2 T   0 γ I 0 −M I     0 0 γ2I P 0    ≥ 0,     PF − M C −M P P 0  C I 0 0 I (22) for ω̄ = α + v̄1−A ; then, the observer gain L = P −1 M minimizes the volume of the ellipsoid Eα1−A (with Pα1−A = Proof : The probability pr[kζk k ≤ α + ǫp ] can be written as 2 2 pr[kζk k ≤ α + ǫp ] = 1 − pr[kζk k > α + ǫp ]. Then, using Markov’s inequality [22], we can write the following 2 E[kζk k ] pr[kζk k2 > α + ǫp ] = 1 − pr[kζk k2 ≤ α + ǫp ] ≤ . α + ǫp Therefore, if ǫp satisfies E[kζk k2 ]/(α + ǫp ) ≤ A − ap , then 2 2 pr[kζk k > α+ǫp ] ≤ A−ap and hence pr[kζk k ≤ α+ǫp ] ∈ [1 − A + ap , 1]. The expectation of the quadratic form ζkT ζk 2 is given by E[kζk k ] = E[ζkT ζk ] = tr[Mk ] + µTk µk [22]; 2 then, E[kζk k ]/(α + ǫp ) ≤ A − ap is satisfied for all ǫp ≥ ǫp with ǫp as defined in (23b), and the assertion follows. Using Proposition 3, for given false alarm rate A and probability p = 1 − A + ap ∈ (1 − A, 1], ap ∈ (0, A), we can characterize p-probable hidden reachable sets, R̃pα , 2 by using the lower bounds on pr[kζk k ≤ α + ǫp ] and ǫp in (23). Specifically, for p > 1 − A, the set R̃pα of the sequence δk is defined as the set of ek ∈ Rn that can be reached from e1 = 0 restricted to satisfy Ā = A and pr[kvk k2 ≤ v̄p ] = 1 − A + ap and (24) ǫp = ǫp → pr[kζk k2 ≤ α + ǫp ] ∈ [1 − A + ap , 1], for some constant v̄p ∈ R>0 and ǫp as defined in (24), i.e., R̃pα := e k ∈ Rn e1 = 0, ek , ζk , vk , satisfy (13)-(14),(24), . (25) Remark 2 For a p-probable reachable set, we select ap such that p = 1 − A + ap , then determine ǫp using (23b). Note that, because the attacker can induce an attack sequence with Fig. 2. 1−A Ellipsoid Eα for different values of false alarm rate A. Fig. 3. The improvement in the (1 − A)-probable hidden reachable set 1−A ellipsoidal bound Eα , for A = 0.01, through application of Corollary 2 to design the optimal observer gain. arbitrarily large covariance Mk and mean µk , the lower bound on ǫp , ǫp , in (23b) can be made arbitrarily large for any ap . Therefore, if δk (and thus ζk ) is only restricted to satisfy Ā = A, the opponent can induce arbitrarily large reachable sets R̃pα . Remark 2 implies that if we only monitor the alarms raised by the detector, the attacker can inject arbitrarily large signals in the residual sequence rk without changing the alarm rate. Consequently, the sets R̃pα can be made arbitrarily large for arbitrarily small ap . If we place additional assumptions on the attacker, namely that the mean and covariance of the attack sequence ζk are finite, the reachable sets will be bounded by Proposition 3. In particular, if we assume the attacker maintains the mean and covariance of the attack-free scenario, i.e., E[ζk ] = µk = 0 and E[ζk ζkT ] = Mk = Im , m − α. Hence, if in addition to imposing Ā = then ǫp = A−a p A, the attack is restricted to keep the statistical properties of ζk in the attack-free case, i.e., µk = 0 and Mk = Im , the reachable sets R̃pα are bounded for each ap ∈ (0, A) (because ǫp is bounded); and therefore, in this case, we can compute ellipsoidal bounds on R̃pα . This additional assumption could be enforced by adding detectors that identify anomalies in the sample mean and sample covariance of the residual. Such detectors would force the attacker to avoid arbitrarily large attack values in order to avoid detection by these additional mean and covariance detectors. As before, we characterize, for some positive definite matrix P̃αp ∈ Rn×n , outer ellipsoidal bounds of the form Ẽαp := {ek ∈ Rn \|eTk P̃αp ek ≤ 1} containing R̃pα . The results corresponding to Theorem 1, and Corollary 1 for Case 1 are stated in the following corollary. Corollary 3 For given false alarm rate A, probability p = m − α, 1 − A + ap , ap ∈ (0, A), threshold ǫp = ǫp = A−a p p and matrices (F, L, Σ), consider the set R̃α in (25). Then, for given b ∈ (0, 1), if there exists a matrix P ∈ Rn×n solution of the following convex optimization: ( minP − log det P, (26) s.t. P > 0 and (18), for ω̄ = α + ǫp + v̄p ; then, R̃pα ⊆ Ẽαp (with P̃αp = P) and Ẽαp has minimum volume, i.e., the p-probable hidden reachable set R̃pα is contained in the minimum volume ellipsoid Eαp = {ek ∈ Rn \|eTk Pαp ek ≤ 1}. p Fig. 4. Ellipsoidal bound Ẽα for different values of ap obtained using Corollary 3. A result for redesigning the observer gain for minimizing the volume of the above ellipsoids, as in Corollary 2 for Case 1, can be stated in a similar manner as the corollary above; however, this is omitted here due to the page limit. IV. S IMULATION E XPERIMENTS Consider the closed-loop system (3)-(4) with matrices:  0.84 0.23 0.07  F = , G = ,C= 10 ,   −0.47 0.12 0.23      1.16 0.45 −0.11 L= , R1 = , (27) −0.69 −0.11 0.45        R = 1 0 , R = 1, Σ = 3.26.  0 2 01 We start with Case 1. Using Proposition 2, the observer gain L is designed such that the H∞ gain from the noise to the residual rk of (8) is less than or equal to γ = 1.86 in the attack-free case. Consider the false alarm rates A = {0.01, 0.05, 0.10, 0.20} and the corresponding α = {6.63, 3.84, 2.70, 1.64}, obtained using Proposition 1. The 2 thresholds v̄1−A in (15) are computed such that pr[kvk k ≤ v̄1−A ] = 1−A. Because the entries on the diagonal of R1 are 2 equal and vk ∼ N (0, R1 ), the random sequence kvk k , k ∈ N follows a gamma distribution, Γ(κ, θ), with shape parameter κ = 1 and scale parameter θ = 0.90, see [22]. It follows that, for these A, v̄1−A = {4.14, 2.69, 2.07, 1.44}. For these values of v̄1−A and α, in Figure 2, we depict the ellipsoidal bounds Eα1−A on the (1 − A)-probable hidden reachable sets obtained using Theorem 1 and Corollary 1. Next, for R1−A α A = 0.01, using Corollary 2, we redesign the observer gain L to minimize the volume of Eα1−A while maintaining the H∞ performance below γ = 1.86. The obtained optimal ellipsoidal bound, Eα1−A , is depicted in Figure 3 for the optimal observer gain L = (0.1272, −0.0160)T . For Case 2, let A = 0.05, p = 1 − A + ap , ap = {0.01, 0.03}, and L as in (27); then, the corresponding v̄p are v̄p = {2.8970, 3.5208} and the ǫp , computed through (23), are given by ǫp = {21.16, 46.16}. In Figure 4, we show the ellipsoidal bounds Ẽαp on the reachable sets R̃pα obtained using Corollary 3. Remark 3 Many numerical results considering hidden attacks with different distributions are presented in the accompanying paper [27] (Section 4). Also, extensive Monte-Carlo simulations showing the tightness of the bounds presented here are given in [27]. V. C ONCLUSION In this paper, for a class of discrete-time LTI systems subject to sensor/actuator noise, we have provided tools for quantifying and minimizing the negative impact of sensor attacks on the estimation error dynamics performance given how the opponent accesses the dynamics (i.e., through the controller by tampering with sensor measurements). We have proposed to use the reachable set as a measure of the impact of an attack given a chosen detection method. For given system dynamics and attack detection scheme, we have derived ellipsoidal bounds on these reachable sets using LMIs. Then, we have provided synthesis tools for minimizing these bounds (minimizing thus the reachable sets) by properly redesigning the detectors. R EFERENCES [1] A. Cárdenas, S. Amin, Z. Lin, Y. Huang, C. Huang, and S. 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arXiv:1710.11354v1 [cs.CV] 31 Oct 2017 Spatio-temporal interaction model for crowd video analysis Neha Bhargava Indian Institute of Technology Bombay India Subhasis Chaudhuri Indian Institute of Technology Bombay India neha@ee.iitb.ac.in sc@ee.iitb.ac.in Abstract scopic modeling methods. They also present a critical survey on crowd event detection. Julio et al. cover various vision techniques applicable to crowd analysis such as tracking, density estimation, and computer simulation [3]. Zhan et al. discuss various vision based techniques used in crowd analysis. They also discuss crowd analysis from the perspective of different disciplines − psychology, sociology and computer graphics [4]. At the top level, the techniques used in crowd motion analysis can be divided into two major classes − holistic and particle based. The holistic methods consider crowd as a single entity and analyze the overall behavior. These methods fail to provide much insight at an individual or intermediate level. On the other hand, particle based methods consider crowd as a collection of individuals. But their performance degrades with the increase in crowd density due to occlusion and tracking problems. The analysis at intermediate level i.e. at group level might provide more insights at individual and overall levels. We present an unsupervised approach to analyze crowd at various levels of granularity − individual, group and collective. We also propose a motion model to represent the collective motion of the crowd. The model captures the spatio-temporal interaction pattern of the crowd from the trajectory data captured over a time period. Furthermore, we also propose an effective group detection algorithm that utilizes the eigenvectors of the interaction matrix of the model. We also show that the eigenvalues of the interaction matrix characterize various group activities such as being stationary, walking, splitting and approaching. The algorithm is also extended trivially to recognize individual activity. Finally, we discover the overall crowd behavior by classifying a crowd video in one of the eight categories. Since the crowd behavior is determined by its constituent groups, we demonstrate the usefulness of group level features during classification. Extensive experimentation on various datasets demonstrates a superlative performance of our algorithms over the state-of-the-art methods. We believe that a moderately dense crowd consists of various groups which form a primary entity of a crowd [6, 7] whereas a highly dense crowd can be considered to form a single group and a highly sparse crowd might have groups with cardinality of one. Together, they guide the overall behavior of the crowd and individually influence the actions of the members. Therefore, group level analysis and hence group detection becomes important in crowd analysis. We define a group as a set of individuals (agents) having some sort of interactions e.g. the group members are walking together. Spatial proximity is also necessary to form a group; if there are agents with a similar motion pattern but are far away from each other, they do not form a group as per our definition. Each group has its own set of goals that leads to various interaction patterns among the members of the group. The collective behavior of these constituent groups identifies the global crowd behavior which can vary from a highly structured to a completely unstructured pat- Understanding human behavior at an individual level, at a group level and at a crowd level in different scenarios has always attracted the researchers. The variability and complexity in the behavior make it a highly challenging task. However, this decade is witnessing a huge interest of researchers in the area of crowd motion analysis due to its various applications in surveillance, safety, public place management, hazard prevention, and virtual environments. This interest has resulted in many interesting papers in the area. We are aware of at least four survey papers on the subject of crowd analysis that indicate the amount of attention, it has drawn in this and the previous decade [1],[2],[3],[4]. The latest survey paper [1] by Chang et al. encapsulates the recent works published after 2009, covering topics of motion pattern segmentation, crowd behavior and anomaly detection. Thida et al. [2] provide a review on macroscopic and micro1 (a) Uniform crowd (d) Walking (b) Mixed crowd (c) Stationary group (e) Approaching (f) Splitting Figure 1: (a) and (b) give examples of structured and unstructured crowd. Output of the proposed algorithm: (c) - (f) show groups with different activities: Standing (St), Walking (W), Splitting (Sp) and Approaching (A). Tracklets for some of the agents over past few frames are also shown. Each color represents a detected group (Best viewed in color). The videos are from BEHAVE [5] and CUHK [6] datasets. tern. In case of a structured crowd, for example − marching of soldiers, all groups are in coordination and share the same goal (see Fig.1a); whereas in an unstructured crowd, for example − at railway station or at a shopping complex, there are multiple groups with different goals (see Fig.1b). We are interested in understanding these different types of crowd behaviors at various levels by exploiting motion information of individuals. The paper makes the following contributions: strates the effectiveness of the algorithm. 3. We also demonstrate how the activities can be classified at three different levels − at atomic (individual) level, at group level and at crowd level. The eigenvalues of the interaction matrix characterize various group and individual activities − Fig 1c-1f show examples of activities at group level. At crowd level, we employ group level features to identify the behavior of the crowd. We classify the crowd videos in one of the 8 categories as defined by [6] and demonstrate its performance in terms of classification accuracy. 1. A framework is proposed to model the collective motion of the crowd by a first order dynamical system. The model captures the interaction patterns among the individuals. Although, the proposed model does not capture any possible nonlinear relations, its usefulness for short-term analysis has been verified experimentally. We also provide an optimization formulation for the estimation of the interaction matrix under the constraints of spatial proximity, temporal continuity and sparsity of inter-agent relationship. The remaining part of the paper is organized as follows. Next section reviews the related literature. Section 2 explains the proposed mathematical formulation followed by group detection algorithm in Section 3. Detection of group activity and atomic activity is discussed in Section 4. We look at crowd video classification in Section 5. The experimental results are presented in Section 6 followed by conclusions in Section 7. 2. Since the interaction matrix is learned from the trajectory data, it captures the spatio-temporal patterns present among the agents. We observe that the eigenvectors of the interaction matrix reflect the spatio-temporal patterns. Thus, we propose a spectral clustering [8] based algorithm to identify the groups present in the scene. Extensive experimentation on various datasets demon- 1. Related Work There are numerous research papers in the challenging and interesting area of crowd behavior analysis. There are several holistic approaches (e.g. [9], [10], [11], [12], [13]) as well as particle based algorithms (e.g. [14], [15], [7], [16], [17]) in the literature. Holistic methods analyze crowd as a single en2 2. Mathematical Formulation tity and ignore individuals or groups. In many papers, a dense crowd is considered analogous to fluid and hence concepts from fluid mechanics are applied for analysis. Mehran et al. in [9] present streakline representation of crowd flow for behavior analysis. Solmaz et al. recognize crowd behaviors such as bottlenecks, fountainheads, lanes, arches and blocks through stability analysis of a dynamical system [10]. Benabbas et al. detect motion patterns and events in the crowded scenes by modeling motion and velocity at each spatial location [11]. In [12], Lin et al. find a coherent motion regions in the video by generating thermal energy field. We define a group as a set of agents having spatial proximity and some sort of interaction. In general, such interactions are complex and non-linear in nature. We approximate these interactions locally in time by a first order dynamical model. Note that we refer by agent an individual entity (represented by a point to be tracked) in the crowd. 2.1. Proposed Interaction Model We model the collective relationship among the agents by a first order affine system. Our hypothesis is based on the intuition that each agent takes into consideration (a) the movement of other agents present nearby and (b) her/his desired goal, while taking the next step. To capture these two intuitions, our model relates the next position of each agent to the current positions of all the agents including herself/himself. Let x(k ) = [ x1 (k), x2 (k), ..., x N (k )] T , then The agent based approaches analyze each individual or group to discover the global behavior. Solera et al. propose correlation clustering based group detection which uses socially constrained features. Shao et al. introduce a collective transition prior in [6] and represent each group by a Markov chain. They define interesting group descriptors which proved to be useful in group state analysis and crowd classification. In [15], Sethi and Chowdhury propose a phase space algorithm to identify pairwise correlation between the motion patterns. Ge et al. find groups by hierarchical clustering based on pairwise velocities and distance [18], [7]. Zhou et al. find groups by using coherent filtering [16]. They propose a coherent neighbor invariance property which characterizes coherently moving individuals. Sochman et al. [19] infer groups based on social force model [14]. They define a pairwise group activity confidence to identify groups. Srikrishnan and Chaudhuri in [20] define a linear cyclic pursuit based framework for collective motion modelling with the goal of short-term prediction. But they do not explore group detection and there is no analysis of crowd behavior. In the interesting work of [17], they consider group detection as a clustering problem and learn a socially meaningful pairwise affinity under Structural SVM framework. x ( k + 1) = [ A \| a ] x(k) = A0 x0 (k ) 1 (1) where N is the total number of agents, A ∈ R N × N , A0 ∈ R N ×( N +1) , a ∈ R N ×1 is the bias, x0 (k) ∈ R( N +1)×1 and xi (k) ∈ R is the location of the ith agent at time instant k along the x-axis. We call A as the interaction matrix which captures the evolution of an agent as a function of all agents present in the scene. Note that A has no assumption on its form and entries. It need not be symmetric i.e. agent i may not depend on agent j in the same way as agent j depends on agent. For example, consider a case where agent i is stationary and agent j approaches him/her. Since their behaviors are not symmetric with respect to each other, we assume that it implies aij 6= a ji . In this paper, it is assumed that the motions along x and y directions are independent and hence can be analyzed independently. The corresponding model along y direction is y(k + 1) = By(k) + b. In the rest of the paper, we discuss the solution for matrix A noting this fact that the same process is also carried out for B. In the end, the outputs from both the models are combined appropriately to get the final output. We expect matrices A and B to be dependent on crowd motion. Since crowd behavior might change with time, the interaction matrix is time varying in nature, which we represent as Ak where k is a time instant. Assuming A has N independent eigenvectors, the general solution Most of the particle based algorithms compute pairwise velocity and spatial cues to find the groups hierarchically. They do not model spatio-temporal patterns of the agents collectively which might capture more complex interactions. Additionally, most of the methods assume a constant velocity motion model which is not valid for many scenarios. To address these limitations in the paper, we propose to model motion trajectories collectively instead of individually or pairwise. Also instead of relying on spatio-temporal information directly (which is prone to noise) for group detection, we use spectral clustering to identify groups. 3 to Eq.(1) is given as N x(k) = ∑ {ci λik ei + di i =1 λ i 6 =1 (λik − 1) e }+ λi − 1 i A and a with each incoming frame as interaction patterns may change over the time. In addition, sudden changes in these interactions are unlikely. Therefore it is desired that the entries of A and a do not change drastically in consecutive time instants − we assume them to be varying smoothly over time. We incorporate this constraint by minimizing l2 norm of the difference between the current matrix A0k and the previous estimate at (k − 1)th instant. Furthermore for crowded scenes, it is unlikely that an agent’s motion depends on all the agents present in the neighborhood. We capture this sparse relationship in A0k by minimizing l1 norm of A0k . Adding these constraints to the cost function, the final formulation at kth time instant becomes: N ∑ (ci + kdi )ei , i =1 λ i =1 (2) where λi is the ith eigenvalue, ei is the corresponding normalized eigenvector, ci and di are the corresponding constant coefficients that depend on the initial condition and a respectively. Different values of λi and ei generate various motion patterns for an agent. These patterns can be associated to different motion tracks generated by an agent while walking, approaching, splitting or being stationary. For example, an agent is stationary if λ1 = 1 and d1 = 0 at location c1 e1 or an agent is moving with a constant speed if λ1 = 1 and d1 6= 0. Hence, this more generalized model is appropriate for modeling temporally localized complex motions. Â0k j k N (n + i ) − x j ∑ ∑ \|x actual j pred A0k ∈R N ×( N +1) n 1 k 2 \|\|A0k Xkk− − L − Xk− L+1 \|\|2 (4) where Xi ∈ R N +1× L contains the positions of all N agents from ith to jth frames concatenated together with an appended row of ones to account for the bias, A0k−1 is the estimate at the previous frame and r1 and r2 are appropriate regularization parameters. Note that we will use A0 instead of A0k for notational convenience. One requires at least L ≥ ( N + 1) past positions to solve the Eq. 4. Therefore the interaction pattern is assumed to remain constant over L frames. Hence we want L to be small enough to capture the shortterm linear relationship among the agents. A large N (in crowded scenes) leads to two major problems: (i) longer trajectories (i.e. higher L) are required to learn the interaction matrix A0 as L ≥ N + 1 which may not be available and (ii) the interaction may not remain constant over past L positions for high values of L as discussed before and we would like to keep L ≤ 30 as discussed in the previous section. To address these problems, we identify spatial neighbors of each agent separately and learn only the corresponding entries in the matrix (one row at a time). The neighborhood is defined as follows − the agent p is a neighbor to the agent q if dist(p, q) < Rp . The value of Rp is decided so as to satisfy the constraint L ≤ 30. The intuition for enforcing the neighborhood criteria is that it is unlikely that far away agents influence the motion of an agent. The advantage is that the shorter trajectories are now sufficient as the number of entries of A0 to be learned are lesser. Note that we estimate matrix A0 in a row-wise manner where the ith row has number of entries to be estimated as equal to one more than We use an average k-step prediction error as a measure to test the validity of the proposed model on real videos. Fig. 2a shows average errors for different step size prediction on videos from BEHAVE and CUHK datasets, each curve corresponding to a different video. The k-step prediction error at any time instant n is calculated as follows: 1 kN min o + r1 \|\|A0k − A0k−1 \|\|22 + r2 \|\|A0k \|\|1 , 2.2. Validation of the Model En (k ) = = arg (n + i )\|, (3) i =1 j =1 It may be noted that matrix A is estimated from the latest video frames up to n and then Eq. 1 is used to pred obtain x j . The k-step prediction error for the video is obtained by averaging En (k ) over all the frames of the video. As expected, error increases with k but with a marginal increment. We observe that, for both the databases, prediction is quite valid up to 1-1.5 seconds (about 30 frames). Since the model assumes that the interaction remains same over L frames, Fig 2a suggests that one can select L upto 30 frames without introducing much error. These error plots show that the proposed model is suitable for short-term analysis, which is the underlying theme of the proposed algorithm. 2.3. Estimation of Interaction Model Parameters The matrix A and vector a at any time instant are learned from the immediate past trajectory data of all the agents in a least squares framework. We update 4 (a) On videos from BEHAVE dataset Error (in pixels) 30 20 10 0 5 10 15 20 25 30 35 40 45 50 45 50 (b) On videos from CUHK dataset Error (in pixels) 30 20 10 0 5 10 15 20 25 30 35 40 Number of frames predicted (k) (a) (b) Figure 2: (a) Illustration of suitability of the proposed model: Average k-step prediction error for sample videos from BEHAVE and CUHK datasets, each curve corresponds to a different video. (b) Neighborhood criteria: Spatial neighborhoods around agents p and r are represented as circles around them. There are a total of 20 agents in the scene out of which only 8 are neighbors of p. Estimation of elements of a row of A corresponding to agent p, considering all agents present in the scene requires 2.5 × 20 = 50 previous video frames (assuming L = 2.5N). While the use of neighborhood constraint reduces this to 2.5 ∗ 9 ≈ 23 frames. the number of the neighbors of agent i. Further, there could be an agent within the spatial proximity of another agent but there may not be any interaction between them. Hence it is required that the corresponding entry in the matrix A0 should be zero. This is enforced by adding sparsity constraint in Eq. 4. We use L1General package developed by Schmidt [21] for solving L1-regularization problems. For an illustration, see Fig.2b. There are a total of N = 20 agents present in the scene. Estimation of the row of matrix A corresponding to agent p requires 50 previous frames (assuming L = 2.5N) whereas the neighborhood based estimation reduces this to 23. Also consider a case where agents p and r interact with each other but are not within the spatial proximity owing to neighborhood constraint. The interaction is captured when intersection of neighborhoods of p and r has at least one interacting agent, in this case its q who is in the spatial proximity of both. Let the eigenvector matrix contain all the eigenvectors column-wise. We define a mapping for ith agent as f ( xi ) : xi ∈ R → zi = (ei1 , ei2 , . . . , eir ) T ∈ Rr×1 where e ji is the ith entry of jth eigenvector of interaction matrix A and r is the number of significant eigenvalues. A clustering algorithm is applied on the points {zi }, ∀i = 1, 2, . . . , N to identify the groups. The clustering algorithm runs on the components of eigenvectors, therefore this algorithm falls in the category of spectral clustering [8]. Since the number of groups is unknown, we apply a threshold based clustering. The adaptive threshold used for the ith point is c\|zi \|, where \|zi \| is its magnitude and c is found empirically. For example, all the agents within the distance of c\|z1 \| from z1 will form a group with agent 1. In this way, all the groups are obtained. We consider only significant eigenvectors (with \|λ\| ≥ 0.90) of A for group detection since the response from the eigenvectors with \|λ\| < 0.9 dies down to an insignificant level within the period of L frames. It may be noted that this group detection algorithm remains the same in the case where A does not have N independent eigenvectors. In such a case, the clustering algorithm runs on generalized eigenvectors. 3. Group Detection In this section, we discuss an algorithm for identifying the groups present in the scene. As seen from Eq. 2, the general solution is a linear combination of eigenvectors at any time instant k. Notice that if the corresponding entries of any two rows of the eigenvector matrix are similar, the corresponding agents form a group. This group information is not available from the position vector alone at a particular time instant x(k) because temporal evolution is also an important factor in deciding the groups. Since the eigenvectors are learned from the trajectories collectively, it encapsulates spatio-temporal evolution of the agents and hence can be exploited for group detection. 4. Group Activity Identification While the eigenvectors identify the groups, the eigenvalues can be used to determine the activity of a group. We employ the same model mentioned in Eq. 1 for the group g to estimate its interaction matrix Ag and ag . We do not use the submatrix formed by the agents of the group g in the previously learned 0 matrix A0 = [A\|a] to get Ag = [Ag \|ag ]. This is to get a refined matrix for the group and avoid any possi5 this corresponds to j = 1. Initially the two agents were standing together and then the second agent starts moving away from the first one leading to split of the group. ble interference from the outside agents in the estimag g g tion. Let xg (k) = [ x1 (k), x2 (k ), . . . , x M (k)] T , where M g is the cardinality of the group g and xi (k) is the position of the ith agent of the group at time instant k. To learn matrix Ag’ at kth time instant, we define a similar optimization framework as follows, where the second term enforces temporal continuity in the activity but unlike Eq. 4, there is no need for sparsity constraint as, by definition, all agents in a group interact. Therefore, g’ Âk = arg g’ min n Ak ∈R M×( M+1) g’ g’ + λ\|\|Ak − Ak−1 \|\|22 This group activity detection method is dependent on eigenvalues and hence sensitive to perturbations in the measurements. To address this, we define threshold bands for crucial values of eigenvalues. For example, if 0.995 < µ < 1.005, we consider µ to be 1 and if µ < 0.5 then it is considered as 0. 4.1. Atomic Activity Detection g’ 1 k 2 \|\|Ak Xkk− − L − Xk− L+1 \|\|2 o This algorithm is now extended to identification of individual’s activity as follows. Let x (k ) denotes position of an agent at time k, then (5) x (k + 1) = µx (k) + b Assuming Ag to be again diagonalizable, the general solution is similar as given in Eq. 2. The velocity vector v(k) for the group g can be written as M v(k) = ∑ i =1 {ci (µi − 1)µik−1 + di µik−1 }ui , (7) The velocity v(k) is as follows: v ( k ) = (1 − µ ) µ k −1 x (0 ) + µ k −1 b (6) (8) Note that there is no longer a activity called splitting as one needs at least two agents to define it. We identify following activities based on the value of µ: where \|µ1 \| ≥ \|µ2 \| . . . ≥ \|µ M \| are the eigenvalues of Ag . Since some of the coefficients ci and di could be zero, let µ j be the largest eigenvalue for which at least one of the coefficients ci or di is non-zero. Now we state how different values of µ j characterize various activities: 1. Stationary: An agent is stationary if \|µ\| = 0 at the location given by b. It is also stationary when \|µ\| = 1 and b = 0. 2. Stopping: 0 < \|µ\| < 1 indicates that the agent is stopping soon. 1. Stationary: A group is stationary if \|µ j \| = 0 indicating all the eigenvalues (with at least one nonzero coefficient) to be zero. That corresponds to zero velocity vector and hence the agents are stationary. In the illustration shown in Fig. 3(a), the deciding eigenvalue is µ2 which is 0. The two agents are stationary at locations 140 and 120 respectively. 3. Walking: An agent is walking if \|µ\| > 1. Further, an agent is walking with a constant velocity if \|µ\| = 1 and b 6= 0. Note that the group detection and activity recognition algorithms run in x and y directions independently and results need to be combined together. For group detection, a group is formed only if it is formed in both the directions. For example, let Zx = [1, 1, 2, 1] and Zy = [2, 1, 2, 2] be the label vectors (indicating assigned group number for all the four agents) obtained along x and y directions respectively. It says that agents {1,2,4} form a group along x direction while {1,3,4} form a group along y axis. Combining both the labels will result in the final label vector as Z = [1, 2, 3, 1] i.e. out of 4 agents, 1 and 4 are grouped together while agents 2 and 3 are singleton groups. To identify the final group activity from the two separate group activity estimates along x and y directions, we merge the two decisions according to the following priority sequence − Splitting>Walking>Approaching>Stationary. For example, 2. Approaching: A group has an approaching members if \|µ j \| < 1 as limk→∞ v(k ) → 0. In the example shown in Fig. 3(b), j = 2. One agent is stationary at 120 while the other agent starts from the location 100 and approaches to the first one. 3. Walking: If \|µ j \| = 1 then the group is walking with a constant velocity of d j u j . In Fig. 3(c), both the agents walk together and deciding eigenvalue corresponds to j = 2. Note that we do not discriminate between walking and running in this work. 4. Splitting: A group has a tendency for divergence if \|µ j \| > 1 as limk→∞ v(k) → ∞. In Fig. 3(d), 6 150 145 125 µ1 = 1, d1 = 0 µ2* = 0 120 125 µ1 = 1, d1 = 0 µ2* = 0.9, d2 ≠ 0 x(k) 100 115 140 110 115 135 130 50 105 100 110 0 95 125 90 105 120 115 0 150 120 85 5 10 100 0 k 5 10 k 80 0 µ1 = 1, d1 = 0 µ2* = 1, d2 ≠ 0 5 10 k −50 15 −100 0 µ1* >1, c1 ≠ 0 5 10 k Figure 3: Illustration of group activity - Stationary, Approaching, Walking and Splitting respectively from the estimated model parameters for a group consisting of two members. Eigenvalue with ∗ is the activity deciding eigenvalue. See the text for details. if a group has splitting and approaching activities in x and y directions respectively, the final group activity is splitting. We employ group level features that cover lowlevel details such as motion information to high-level information such as group activities. The features are described as follows: 5. Crowd Video Classification 1. Group density (GD): It is the ratio of number of groups by the total number of agents in the scene. A low value of GD indicates highly structured crowd. For example, GD for a group of marching soldiers is small whereas a mixed crowd has a higher group density. Having group level information in hand, we can use them in identifying the overall crowd behavior. Ability to identify crowd behavior enables crowd management systems to design and manage public places effectively to ensure safety and smooth operation. The overall crowd behavior is determined by how each group behaves. Depending on the synchronization among the groups, the behavior of crowd varies from being structured to unstructured. In this section, we define group level features that are useful for crowd video classification. We classify crowd videos into 8 classes as defined by [6]. The dataset containing 474 video clips covers a variety of videos. The eight classes are as follows: 2. Histogram of λmax : The histogram has three bins which are λmax ≥ 1, λmax < 1 and λmax = 0, where λmax is the largest eigenvalue of the interaction matrix for a group (µ j from the last section). The value at a particular bin is the number of groups in a scene having λmax as defined by that bin. Left skewed histogram i.e. towards λmax ≥ 1 indicates moving crowd whereas right skewed histogram suggests more or less stationary crowd. C1 : Mixed crowd C2 : Well organized crowd following mainstream: 3. Histogram of motion direction: The motion direction of each member of a group is calculated from its trajectory data and the mean direction is assigned to the group. This histogram has eight bins covering 0◦ to 360◦ with a bin size of 45◦ . The bin value is the number of groups falling in that particular bin. The uniform histogram indicates a mixed crowd whereas a skewed histogram indicates directionality in the crowd movement. C3 : Not well organized crowd following any mainstream C4 : Crowd merge C5 : Crowd split C6 : Crowd crossing in opposite directions Since the analysis is conducted independently in x and y directions; we get two histograms for λmax , leading to final feature vector of length 1 + 2 × 3 + 8 = 15. C7 : Intervened escalator traffic C8 : Smooth escalator traffic 7 Table 1: Performance comparison of different group detection algorithms on CUHK dataset. We use random forest (RF) as a classifier [22]. It consists of a multitude of decision trees that are trained from randomly sampled subsets of training dataset (bootstrap aggregating). This bootstrapping increases the performance by reducing the variance of the classifier. Also the split at each node of a tree is decided by m features selected randomly out of n features where m << n. We use RF to classify a crowd video by training it with the above mentioned features. The classification results are discussed in next section. NMI Purity RI CF [16] 0.66 0.71 0.67 CT [6] 0.69 0.72 0.69 Proposed 0.86 0.90 0.85 for each video where we compare the proposed algorithm with other methods and the ground truth instead of manually deciding on the instants when the performance has to be evaluated. We use Normalized Mutual Information (NMI) [26], Purity [27] and Rand Index (RI) [28] which are widely used for evaluation of clustering algorithms. Table 1 shows the comparison on these measures. It is quite evident from the table that the performance of the proposed algorithm far surpasses those of [6] and [16]. Fig. 4 demonstrates a visual comparison for different scenarios. Since Zhou et al. in [16] find coherent motion patterns at one time and then update them over time, it is sensitive to tracking errors and has the possibility of accumulation of errors if any frame has tracking error. Shao et al. [6] assign every agent to a collective transition prior. They have spatial proximity constraint only at the initial time instant which might not be effective as time progresses. Their algorithm groups all the agents moving in the same direction giving less importance to their spatial relationships. This can be observed from the output figures in column (b) of Fig. 4. Further in 4th row, a person with red hat is moving faster than the group behind him but CT and CF fail to capture this difference in velocity while the proposed algorithm could capture it. The groups in last row have small changes in their directions of movement which is again not captured by these two methods while the proposed method detects such small changes. We also compare the proposed group detection algorithm with the method of [17] on the videos VEIIG, student003 and eth. To compare with [17], we also use G-MITRE precision P and recall R as proposed by them. Table 2 shows the quantitative results that indicate an improved performance by the proposed method. The proposed algorithm outperforms these stateof-the-art methods because it is more robust to tracking errors since we extract groups from the eigenvectors rather than directly using the tracklets. It is quite evident from Fig. 4 where the tracklets for various agents are marked with different colors to indicate the group they belong to, that the proposed algorithm is able to detect agents in a group much better than the 6. Experiments and Results In this section, we discuss the performance of the proposed algorithms for group detection, group activity recognition and crowd video classification. We have tested our algorithms on various publicly available datasets containing real videos. We first discuss these datasets followed by performance evaluation of the proposed algorithms. 6.1. Datasets We tested our algorithms on different videos from various datasets contributed by several researchers namely CUHK [6], BEHAVE [5], BIWI Walking Pedestrians [23], Crowds-By-Example (CBE) [24] and Vittorio Emanuele II Gallery (VEIIG) [25]. CUHK dataset is a comprehensive crowd video dataset containing 474 video clips covering various crowd behaviors with varying crowd density. BEHAVE dataset has video clips with low crowd density and covering various group activities. BIWI dataset contains two low density crowd videos (namely eth and hotel). CBE has a medium density crowd video (student003) recorded outside a university. These datasets collectively cover a large variety of crowd videos. 6.2. Group Detection We tested group detection algorithm on all the 474 videos from CUHK dataset and 3 video clips (having duration of more than 10 minutes in total) from BEHAVE dataset. In case of videos from CUHK dataset, we restricted our algorithm to run only on those data that have sufficiently long tracks, since some of the clips are too short to accommodate for an analysis of a large number of agents. We compared the proposed algorithm with other methods on these selected agents. The ground truth for CUHK dataset was obtained manually. We compare the proposed algorithm for group detection with state-of-the-art methods by Shao et al. [6] and Zhou et al. [16]. For quantitative analysis on CUHK videos, we randomly select two time instants 8 Table 2: Performance comparison of the proposed group detection with [17] . BIWI eth CBE student003 VEIIG Baseline [17] P R 72.4 ± 4.4 65.2 ± 3.4 59.9 ± 2.9 53.5 ± 6.8 49.2 ± 9.9 34.4 ± 6.7 [17] P R 91.8 ± 1.2 94.2 ± 0.9 81.7 ± 0.2 82.5 ± 0.2 84.12 ± 0.6 84.11 ± 0.5 other existing methods. Also the proposed algorithm yields NMI = 0.92, Purity = 0.94 and RI = 0.93 on video clips from BEHAVE dataset whereas the corresponding measures for [6] and [16] have very low values (e.g. Purity for CF is only 0.35). It shows that these methods do not perform well in videos of a sparse crowd whereas the proposed method can also handle a sparse crowd effectively. Proposed P R 95.78 96.15 77.58 85.90 82.97 84.70 for Class 4 (Class Merge) is difficult, while the rest of the classes could be categorized quite easily using the proposed method. The OOB error, which indicates the generalized error, converges to a value 30% as shown in Fig. 7b. The importance plots, which show the significance of each group level feature in the classification are shown in Fig. 7c. It shows that the group density and histogram of eigenvalues are important for classification. 6.3. Group Activity Recognition We use BEHAVE and CUHK datasets for testing the algorithm for group activity identification. Here, we have excluded the clips containing other activities such as fight. We compared the activity results with the ground truth at regular intervals. Table 8 shows the confusion matrix for the proposed algorithm on BEHAVE dataset. The algorithm gives an accuracy of 70% for Walking and Stationary activities whereas it is less for the other two activities. We observed that the algorithm gets confused between these two activities. We suspect that the confusion is due to the fact that Splitting and Approaching are more abrupt in the motion dynamics than Walking and Stationary, which results in a poorer estimate of eigenvalues over the window of L frames. In CUHK dataset, since groups in most of the videos are walking, we obtain an accuracy of 85%. Some of the qualitative results on the videos from BEHAVE and CUHK dataset are given in Fig. 5 and Fig. 6, respectively. 7. Conclusions In this work, we presented a framework for analysis of medium dense crowd videos at various levels. We proposed a first order dynamical system to model agent trajectories collectively and subsequently demonstrated the effectiveness of this interaction model for group detection. We also show how eigenvalues of the model characterize group activities. We then showed the effectiveness of group level features in crowd video classification. Our algorithm assumes the availability of tracks which itself is a challenge in many crowded videos due to occlusion and other tracking problems. As a next goal, we aspire to define a unified framework where the proposed model and a tracker work together to improve each other’s performance in crowded videos by incorporating group interaction cues. 6.4. Crowd Video Classification References Since we update the interaction model with each incoming frame as explained in Section 6, we collect group level features at regular intervals. From each class, we randomly pick 70% feature vectors to train the classifier and the remaining for testing. As discussed before, we use random forest as a classifier with n = 17 and m = 4. We run the classifier 100 times with random splits of dataset for training and testing. To avoid over-fitting, the training data and testing data do not contain features from the same video. The average accuracy obtained is around 74%, an improvement over [6] where the reported accuracy is 70%. The confusion matrix is shown in Fig. 7a. From this figure, it is seen that classification of the crowd [1] T. Li, H. Chang, M. Wang, B. Ni, R. Hong, S. Yan, Crowded scene analysis: A survey, IEEE Transactions on Circuits and Systems for Video Technology 25 (3) (2015) 367–386. [2] M. Thida, Y. L. Yong, P. Climent-Pérez, H.-l. Eng, P. Remagnino, A literature review on video analytics of crowded scenes, in: Intelligent Multimedia Surveillance, Springer, 2013, pp. 17–36. [3] J. S. J. Junior, S. Musse, C. 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Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers∗ arXiv:1608.01700v3 [cs.DS] 3 Nov 2016 Sara Ahmadian† Chaitanya Swamy† Abstract We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {Li }i∈F and a set D of clients located in a common metric space {c(i, j)}i,j∈F ∪D , and bounds k, m. A feasible solution is a pair S ⊆ F, σ : D 7→ S ∪ {out} , where σ specifies the client assignments, such that \|S\| ≤ k, \|σ −1 (i)\| ≥ Li for all i ∈ S, and \|σ −1 (out)\| ≤ m.P In the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, the objective is to minimize i∈S maxj∈σ−1 (i) c(i, j), and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize maxi∈S maxj∈σ−1 (i) c(i, j). We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the nonoutlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-ofradii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the \|S\| ≤ k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplierpreserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds. 1 Introduction Clustering is an ubiquitous problem that arises in many applications in different fields such as data mining, machine learning, image processing, and bioinformatics. Many of these problems involve finding a set S of at most k “cluster centers”, and an assignment σ mapping an underlying set D of data points located in some metric space {c(i, j)} to S, to minimize some objective P function; examples include the k-center (minimize maxj∈D P c(σ(j), j)) [21, 22], k-median (minimize j∈D c(σ(j), j)) [10, 23, 26, 7], and min-sumof-radii (minimize i∈S maxj:σ(j)=i c(i, j)) [16, 12] problems. Viewed from this perspective, clustering problems can often be viewed as facility-location problems, wherein an underlying set of clients that require service need to be assigned to facilities that provide service in a cost-effective fashion. Both clustering and facility-location problems have been extensively studied in the Computer Science and Operations Research literature; see, e.g., [28, 30] in addition to the above references. We consider clustering problems with (non-uniform) lower-bound requirements on the cluster sizes, and where a bounded number of points may be designated as outliers and left unclustered. One motivation for considering lower bounds comes from an anonymity consideration. In order to achieve data privacy, [29] proposed an anonymization problem where we seek to perturb (in a specific way) some of (the attributes of) ∗ A preliminary version [3] appeared in the Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP), 2016. † {sahmadian,cswamy}@uwaterloo.ca. Dept. of Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1. Supported in part by NSERC grant 327620-09 and an NSERC Discovery Accelerator Supplement award. 1 the data points and then cluster them so that every cluster has at least L identical perturbed data points, thus making it difficult to identify the original data from the clustering. As noted in [2, 1], this anonymization problem can be abstracted as a lower-bounded clustering problem where the clustering objective captures the cost of perturbing data. Another motivation comes from a facility-location perspective, where (as in the case of lower-bounded facility location), the lower bounds model that it is infeasible or unprofitable to use services unless they satisfy a certain minimum demand (see, e.g., [27]). Allowing outliers enables one to handle a common woe in clustering problems, namely that data points that are quite dissimilar from any other data point can often disproportionately (and undesirably) degrade the quality of any clustering of the entire data set; instead, the outlier-version allows one to designate such data points as outliers and focus on the data points of interest. Formally, adopting the facility-location terminology, our setup is as follows. We have a set F of facilities with lower bounds {Li }i∈F and a set D of clients located in a common metric space {c(i, j)}i,j∈F ∪D , and bounds k, m. A feasible solution chooses a set S ⊆ F of at most k facilities, and assigns each client j to a facility σ(j) ∈ S, or designates j as an outlier by setting σ(j) = out P so that \|σ −1 (i)\| ≥ Li for all i ∈ S, and −1 \|σ (out)\| ≤ m. We consider two clustering objectives: minimize i∈S maxj:σ(j)=i c(i, j), which yields the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, and minimize maxi∈S maxj:σ(j)=i c(i, j), which yields the lower-bounded k-supplier with outliers (LBkSupO) problem. (k-supplier denotes the facility-location version of k-center; the latter typically has F = D.) We refer to the non-outlier versions of the above problems (i.e., where m = 0) as LBkSR and LBkSup respectively. Our contributions. We obtain the first results for clustering problems with non-uniform lower bounds and outliers. We develop various techniques for tackling these problems using which we obtain constantfactor approximation guarantees for LBkSRO and LBkSupO. Note that we need to ensure that none of the three types of hard constraints involved here—at most k clusters, non-uniform lower bounds, and at most m outliers—are violated, which is somewhat challenging. We obtain an approximation factor of 12.365 for LBkSRO (Theorem 2.8, Section 2.2), which improves to 3.83 for the non-outlier version LBkSR (Theorem 2.7, Section 2.1). These also constitute the first approximation results for the min-sum-of-radii objective when we consider: (a) lower bounds (even uniform bounds) but no outliers (LBkSR); and (b) outliers but no lower bounds. Previously, an O(1)-approximation was known only in the setting where there are no lower bounds and no outliers (i.e., Li = 0 for all i, m = 0) [12]. For the k-supplier objective (Section 3), we obtain an approximation factor of 5 for LBkSupO (Theorem 3.2), and 3 for LBkSup (Theorem 3.1). These are the first approximation results for the k-supplier problem with non-uniform lower bounds. Previously, [1] obtained approximation factors of 4 and 2 respectively for LBkSupO and LBkSup for the special case of uniform lower bounds and when F = D. Complementing our approximation bounds, we prove a factor-3 hardness of approximation for LBkSup (Theorem 3.3), which shows that our approximation factor of 3 is optimal for LBkSup. We also show (Appendix C) that LBkSupO is equivalent to the k-center version of the problem (where F = D). Our techniques. Our main technical contribution is an O(1)-approximation algorithm for LBkSRO (Section 2.2). Whereas for the non-outlier version LBkSR (Section 2.1), one can follow an approach similar to that of Charikar and Panigrahi [12] for the min-sum-of-radii problem without lower bounds or outliers, the presence of outliers creates substantial difficulties whose resolution requires various novel ingredients. As in [12], we view LBkSRO as a k-ball-selection (k-BS) problem of picking k suitable balls (see Section 2) and consider its LP-relaxation (P2 ). Let OPT denote its optimal value. Following the Jain-Vazirani (JV) template for k-median [23], we move to the version where we may pick any number of balls but incur a fixed cost of z for each ball we pick. The dual LP (D2 ) has αj dual variables for the clients, which “pay” for (i, r) pairs (where (i, r) denotes the ball {j ∈ D : c(i, j) ≤ r}). For LBkSR (where m = 0), as observed 2 in [12], it is easy to adapt the JV primal-dual algorithm for facility location to handle this fixed-cost version of k-BS: we raise the αj s of uncovered clients until all clients are covered by some fully-paid (i, r) pair (see PDAlg). This yields a so-called Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm: if F P is the primal solution constructed, then 3 j αj can pay for cost(F ) + 3\|F \|z; hence, by varying z, one can find two solutions F1 , F2 for nearby values of z, and combine them to extract a low-cost k-BS-solution. The presence of outliers in LBkSRO significantly complicates things. The natural adaptation of the primal-dual algorithm is to now stop when at least \|D\| − m clients are covered by fully-paid (i, r) pairs. But now, the dual objective involves a −m · γ term, where γ = maxj αj , which potentially cancels the dual contribution of (some) clients that pay for the last fully-paid (i, r) pair, say f . Consequently, we do not obtain anPLMP-approximation: if F is the primal solution we construct, we can only say that (loosely speaking) 3( j αj − m · γ) pays for cost(F \ f ) + 3\|F \ f \|z (see Theorem 2.9 (ii)). In particular, this means that even if the primal-dual algorithm returns a solution with k pairs, its cost need not be bounded, an artifact that never arises in LBkSR (or k-median). This in turn means that by combining the two solutions F1 , F2 found for z1 , z2 ≈ z1 , we only obtain a solution of cost O(OPT + z1 ) (see Theorem 2.14). Dealing with the case where z1 = Ω(OPT ) is technically the most involved portion of our algorithm (Section 2.2.2). We argue that in this case the solutions F1 , F2 (may be assumed to) have a very specific structure: \|F1 \| = k + 1, and every F2 -ball intersects at most one F1 -ball, and vice versa. We utilize this structure to show that either we can find a good solution in a suitable neighborhood of F1 and F2 , or F2 itself must be a good solution. We remark that the above difficulties (i.e., the inability to pay for the last “facility” and the ensuing complications) also arise in the k-median problem with outliers. We believe that our ideas also have implications for this problem and should yield a much-improved approximation ratio for this problem. (The current approximation ratio is a large (unspecified) constant [13].) For the k-supplier problem, LBkSupO, we leverage the notion of skeletons and pre-skeletons defined by [15] in the context of capacitated k-supplier with outliers, wherein facilities have capacities instead of lower bounds limiting the number of clients that can be assigned to them. Roughly speaking, a skeleton F ⊆ F ensures there is a low-cost solution (F, σ). A pre-skeleton satisfies some of the properties of a skeleton. We show that if F is a pre-skeleton, then either F is a skeleton or F ∪ {i} is a pre-skeleton for some facility i. This allows one to find a sequence of facility-sets such that at least one of them is a skeleton. For a given set F , one can check if F admits a low-cost assignment σ, so this yields an O(1)-approximation algorithm. Related work. There is a wealth of literature on clustering and facility-location (FL) problems (see, e.g., [28, 30]); we limit ourselves to the work that is relevant to LBkSRO and LBkSupO. The only prior work on clustering problems to incorporate both lower bounds and outliers is by Aggarwal et al. [1]. They obtain approximation ratios of 4 and 2 respectively for LBkSupO and LBkSup with uniform lower bounds and when F = D, which they consider as a means of achieving anonymity. They also consider an alternate cellular clustering (CellC) objective and devise an O(1)-approximation algorithm for lower-bounded CellC, again with uniform lower bounds, and mention that this can be extended to an O(1)-approximation for lower-bounded CellC with outliers. More work has been directed towards clustering problems that involve either outliers or lower bounds (but not both), and here, clustering with outliers has received more attention than lower-bounded clustering problems. Charikar et al. [11] consider (among other problems) the outlier-versions of the uncapacitated FL, k-supplier and k-median problems. They devise constant-factor approximations for the first two problems, and a bicriteria approximation for the k-median problem with outliers. They also proved a factor-3 approximation hardness result for k-supplier with outliers. This nicely complements our factor-3 hardness result for k-supplier with lower bounds but no outliers. Chen [13] obtained the first true approximation for k-median with outliers via a sophisticated combination of the primal-dual algorithm for k-median and 3 local search that yields a large (unspecified) O(1)-approximation. As remarked earlier, the difficulties that we overcome in designing our 12.365-approximation for LBkSRO are similar in spirit to the difficulties that arise in k-median with outliers, and we believe that our techniques should also help and significantly improve the approximation ratio for this problem. Cygan and Kociumaka [15] consider the capacitated ksupplier with outliers problem, and devise a 25-approximation algorithm. We leverage some of their ideas in developing our algorithm for LBkSupO. Lower-bounded clustering and FL problems remain largely unexplored and are not well understood. Besides LBkSup (which has also been studied in Euclidean spaces [17]), another such FL problem that has been studied is lower-bounded facility location (LBFL) [24, 20], wherein we seek to open (any number of) facilities (which have lower bounds) and assign each client j to an open facility σ(j) so as to miniP mize j∈D c(σ(j), j). Svitkina [31] obtained the first true approximation for LBFL, achieving an O(1)approximation; the O(1)-factor was subsequently improved by [4]. Both results apply to LBFL with uniform lower bounds, and can be adapted to yield O(1)-approximations to the k-median variant (where we may open at most k facilities). We now discuss work related to our clustering objectives, albeit that does not consider lower bounds or outliers. Doddi et al. [16] introduced the k-clustering min-sum-of-diameters (kSD) problem, which is closely related to the k-clustering min-sum-of-radii (kSR) problem: the kSD-cost is at least the kSR-cost, and at most twice the kSR-cost. The former problem is somewhat better understood than the latter one. Whereas the kSD problem is APX-hard even for shortest-path metrics of unweighted graphs (it is NP-hard to obtain a better than 2 approximation [16]), the kSR problem is only known to be NP-hard for general metrics, and its complexity for shortest-path metrics of unweighted graphs is not yet settled with only a quasipolytime (exact) algorithm known [18]. On the positive side, Charikar and Panigrahi [12] devised the first (and current-best) O(1)-approximation algorithms for these problems, obtaining approximation ratios of 3.504 and 7.008 for the kSR and kSD problems respectively, and Gibson et al. [18] obtain a quasi-PTAS for the kSR problem when F = D. Various other results are known for specific metric spaces and when F = D, such as Euclidean spaces [19, 8] and metrics with bounded aspect ratios [18, 6]. The k-supplier and k-center (i.e., k-supplier with F = D) objectives have a rich history of study. Hochbaum and Shmoys [21, 22] obtained optimal approximation ratios of 3 and 2 for these problems respectively. Capacitated versions of k-center and k-supplier have also been studied: [25] devised a 6approximation for uniform capacities, [14] obtained the first O(1)-approximation for non-uniform capacities, and this O(1)-factor was improved to 9 in [5]. Finally, our algorithm for LBkSRO leverages the template based on Lagrangian relaxation and the primal-dual method to emerge from the work of [23, 9] for the k-median problem. 2 Minimizing sum of radii with lower bounds and outliers Recall that in the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, we have a facility-set F and client-set D located in a metric space {c(i, j)}i,j∈F ∪D, lower bounds {Li }i∈F , and bounds k and m. A feasible solution is a pair S ⊆ F, σ : D 7→ S ∪ {out} , where σ(j) ∈ S indicates that j is assigned −1 to facility σ(j), and σ(j) = out designates j as an outlier, such P that \|σ (i)\| ≥ Li for all i ∈ S, and −1 \|σ (out)\| ≤ m. The cost of such a solution is cost(S, σ) := i∈S ri , where ri := maxj∈σ−1 (i) c(i, j) denotes the radius of facility i; the goal is to find a solution of minimum cost. We use LBkSR to denote the non-outlier version where m = 0. It will be convenient to consider a relaxation of LBkSRO that we call the k-ball-selection (k-BS) problem, which focuses on selecting at most k balls centered at facilities of minimum total radius. More precisely, let B(i, r) := {j ∈ D : c(i, j) ≤ r} denote the ball of clients centered at i with radius r. Let S cmax = maxi∈F ,j∈D c(i, j). Let Li := {(i, r) : \|B(i, r)\| ≥ Li }, and L := i∈F Li . The goal in k-BS is to 4 P S find a set F ⊆ L with \|F \| ≤ k and D \ (i,r)∈F B(i, r) ≤ m so that cost(F ) := (i,r)∈F r is minimized. (When formulating the LP-relaxation of the k-BS-problem, we equivalently view L as containing only pairs of the form (i, c(i, j)) for some client j, which makes L finite.) It is easy to see that any LBkSRO-solution yields a k-BS-solution of no greater cost. The key advantage of working with k-BS is that we do not explicitly consider the lower bounds (they are folded into the Li s) and we do not require the balls B(i, r) for (i, r) ∈ F to be disjoint. While a k-BS-solution F need not directly translate to a feasible LBkSROsolution, one can show that it does yield a feasible LBkSRO-solution of cost at most 2 · cost(F ). We prove a stronger version of this statement in Lemma 2.1. In the following two sections, we utilize this relaxation to devise the first constant-factor approximation algorithms for for LBkSR and LBkSRO. To our knowledge, our algorithm is also the first O(1)-approximation algorithm for the outlier version of the min-sum-of-radii problem without lower bounds. We consider an LP-relaxation for the k-BS-problem, and to round a fractional k-BS-solution to a good integral solution, we need to preclude radii that are much larger than those used by an (integral) optimal solution. We therefore “guess” the t facilities in the optimal solution with the largest radii, and Otheir 2t radii, where t ≥ 1 is some constant. That is, we enumerate over all O (\|F\| + \|D\|) S choices F = {(i1 , r1 ), . . . , (it , rt )} of t (i, r) pairs from L. For each such selection, we set D0 = D \ (i,r)∈F O B(i, r), L0 = {(i, r) ∈ L : r ≤ min(i,r)∈F O r} and k 0 = k − \|F O \|, and run our k-BS-algorithm on the modified k-BS-instance (F, D0 .L0 , c, k 0 , m) to obtain a k-BS-solution F . We translate F ∪ F O to an LBkSROsolution, and return the best of these solutions. The following lemma, and the procedure described therein, is repeatedly used to bound the cost of translating F ∪ F O to a feasible LBkSRO-solution. We call pairs (i, r), (i0 , r0 ) ∈ F × R≥0 non-intersecting, if c(i, i0 ) > r + r0 , and intersecting otherwise. Note that B(i, r) ∩ B(i0 , r0 ) = ∅ if (i, r) and (i0 , r0 ) are non-intersecting. For a set P ⊆ F × R≥0 of pairs, define µ(P ) := {i ∈ F : ∃r s.t. (i, r) ∈ P }. Lemma 2.1. Let F O ⊆ L, and D0 , L0 , k 0 be as defined above. Let F ⊆ L be a k-BS-solution for the k-BS-instance (F, D0 , L0 , c, kS0 , m). Suppose for each i ∈ µ(F ), we have a radius ri0 ≤ maxr:(i,r)∈F r and U ⊆ L. Then there exists a feasible such that the pairs in U := i∈µ(F ) (i, ri0 ) are non-intersecting P LBkSRO-solution (S, σ) with cost(S, σ) ≤ cost(F ) + (i,r)∈F O 2r. Proof. Pick a maximal subset P ⊆ F O to add to U such that all pairs in U 0 = U ∪ P are non-intersecting. For each (i, r) ∈ F O \ P , define κ(i, r) to be some intersecting pair (i0 , r0 ) ∈ U 0 . Define S = µ(U 0 ). Assign each client j to σ(j) ∈ S as follows. If j ∈ B(i, r) for some (i, r) ∈ U 0 , set σ(j) = i. Note that U 0 ⊆ L, so this satisfies the lower bounds for all i ∈ S. Otherwise, if j ∈ B(i, r) for some (i, r) ∈ F , set σ(j) = i. Otherwise, if j ∈ B(i, r) for some (i, r) ∈ F O \ P and (i0 , r0 ) = κ(i, r), set σ(j) = i0 . Any remaining unassigned client is not covered by the balls corresponding to pairs in F ∪ F O . There are at most m such clients, and we set σ(j) = out for each such client j. Thus (S, σ) is a feasible LBkSRO-solution. For any i ∈ S and j ∈ σ −1 (i) either j ∈ B(i, r) for some (i, r) ∈ F ∪ U 0 , or j ∈ B(i0 , r0 ) where P κ(i0 , r0 ) = (i, r) ∈ U 0 , in which case c(i, j) ≤ r + 2r0 . So cost(S, σ) ≤ cost(F ) + (i,r)∈F O 2r. 2.1 Approximation algorithm for LBkSR We now present our algorithm for the non-outlier version, LBkSR, which will introduce many of the ideas underlying our algorithm for LBkSRO described in Section 2.2. Let O∗ denote the cost of an optimal solution to the given LBkSR instance. As discussed above, for each selection of (i1 , r1 ), . . . , (it , rt ) of t pairs, we do the following. We set S D0 = D \ tp=1 B(ip , rp ), L0 = {(i, r) ∈ L : r ≤ R∗ := minp=1,...,t rp }, k 0 = k − t, and consider the k-BS-problem of picking at most k 0 pairs from L0 whose corresponding balls cover D0 incurring minimum cost (but our algorithm k-BSAlg will return pairs from L). We consider the following natural LP-relaxation 5 (P1 ) of this problem, and its dual (D1 ). X r · yi,r min (P1 ) max X yi,r ≥ 1 ∀j ∈ D0 s.t. (D1 ) X αj − z ≤ r ∀(i, r) ∈ L0 (2) j∈B(i,r)∩D0 (i,r)∈L0 :j∈B(i,r) X αj − k 0 · z j∈D0 (i,r)∈L0 s.t. X yi,r ≤ k 0 α, z ≥ 0. (1) (i,r)∈L0 y ≥ 0. If (P1 ) is infeasible then we discard this choice of t pairs and move on to the next selection. So we assume (P1 ) is feasible in the remainder of this section. Let OPT denote the common optimal value of (P1 ) and (D1 ). As in the JV-algorithm for k-median, we Lagrangify constraint (1) and consider the unconstrained problem where we do not bound the number of pairs we may pick, but we incur a fixed cost z for each pair (i, r) that we pick (in addition to r). It is easy to adapt the JV primal-dual algorithm for facility location [23] to devise a simple Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for this problem (see PDAlg and Theorem 2.3). We use this LMP algorithm within a binary-search procedure for z to obtain two solutions F1 and F2 with \|F1 \| ≤ k < \|F2 \|, and show that these can be “combined” to extract a k-BSsolution F of cost at most 3.83 · OPT + O(R∗ ) (Lemma A.4). This combination step is more involved than in k-median. The main idea here is to use the F2 solution as a guide to merge some F1 -pairs. We cluster the F1 pairs around the F2 -pairs and setup a covering-knapsack problem whose solution determines for each F2 -pair (i, r), whether to “merge” the F1 -pairs clustered around (i, r) or select all these F1 -pairs (see step B2). Finally, we add back the pairs (i1 , r1 ), . . . (it , rt ) selected earlier and apply Lemma 2.1 to obtain an LBkSR-solution. As required by Lemma 2.1, to aid in this translation, our k-BS-algorithm returns, along with F , a suitable radius rad(i) for every facility i ∈ µ(F ). This yields a (3.83 + )-approximation algorithm (Theorem 2.7). While our approach is similar to the one in [12] for the min-sum-of-radii problem without lower bounds (although our combination step is notably simpler), an important distinction that arises is the following. In the absence of lower bounds, the ball-selection problem k-BS is equivalent to the min-sum-of-radii problem, but (as noted earlier) this is no longer the case when we have lower bounds since in k-BS we do not insist that the balls we pick be disjoint. Consequently, moving from overlapping balls in a k-BS-solution to an LBkSR-solution incurs, in general, a factor-2 blowup in the cost (see Lemma 2.1). It is interesting that we are able to avoid this blowup and obtain an approximation factor that is quite close to the approximation factor (of 3.504) achieved in [12] for the min-sum-of-radii problem without lower bounds. We now describe our algorithm in detail and proceed to analyze it. We describe a slightly simpler (6.183 + )-approximation algorithm below (Theorem 2.2). We sketch the ideas behind the improved approximation ratio at the end of this section and defer the details to Appendix A. Algorithm 1. Input: An LBkSR-instance I = F, D, {Li }, {c(i, j)}, k , parameter > 0. Output: A feasible solution (S, σ). A1. Let t = min k, 1 . For each set F O ⊆ L with \|F O \| = t, do the following. S A1.1. Set D0 = D \ (i,r)∈F O B(i, r), L0 = {(i, r) ∈ L : r ≤ R∗ = min(i,r)∈F O r}, and k 0 = k − t. O 0 0 0 0 A1.2. If (P1 ) is infeasible, then reject this guess and move to the next set F . If D 6= ∅, run k-BSAlg(D , L , k , ) to obtain F, {rad(i)}i∈F ; else set (F, rad) = (∅, ∅). A1.3. Apply the procedure in Lemma 2.1 taking ri0 = rad(i) for all i ∈ µ(F ) to obtain (S, σ). A2. Among all the solutions (S, σ) found in step A2, return the one with smallest cost. 6 Figure 1: An example of stars formed by F1 and F2 where F1 = {u1 , u2 , . . . , u11 } and F2 = {v1 , v2 , . . . , v6 } depicted by squares and circles, respectively. Algorithm k-BSAlg(D 0 , L0 , k0 , ). Output: F ⊆ L with \|F \| ≤ k 0 , a radius rad(i) for all i ∈ µ(F ). B1. Binary search for z. B1.1. Set z1 = 0 and z2 = 2k 0 cmax . For p = 1, 2, let(Fp , {radp (i)}, αp ) ← PDAlg(D0 , L0 , zp ), and let kp = \|Fp \|. If k1 ≤ k 0 , stop and return F1 , {rad1 (i)} . We prove in Theorem 2.3 that k2 ≤ k 0 ; if k2 = k 0 , stop and return F2 , {rad2 (i)} . z1 +z2 B1.2. Repeat the following until z2 − z1 ≤ δz = OPT 3n , where n = \|F\| + \|D\|. Set z = 2 . Let 0 0 0 (F, {rad(i)}, α) ← PDAlg(D , L , z). If \|F \| = k , stop and return F, {rad(i)} ; if \|F \| > k 0 , update z1 ← z and (F1 , rad1 , α1 ) ← (F, rad, α), else update z2 ← z and (F2 , rad2 , α2 ) ← (F, rad, α). B2. Combining F1 and F2 . Let π : F1 7→ F2 be any map such that (i0 , r0 ) and π(i0 , r0 ) intersect ∀(i0 , r0 ) ∈ F1 . (This exists since every j ∈ D0 is covered by B(i, r) for some (i, r) ∈ F2 .) Define star Si,r = π −1 (i, r) for all (i, r) ∈ F2 (see Fig. 1). Solve the following covering-knapsack LP. X P P min xi,r (2r + (i0 ,r0 )∈Si,r 2r0 ) + (1 − xi,r ) (i0 ,r0 )∈Si,r r0 (C-P) (i,r)∈F2 s.t. X xi,r + \|Si,r \|(1 − xi,r ) ≤ k, 0 ≤ xi,r ≤ 1 ∀(i, r) ∈ F2 . (i,r)∈F2 Let x∗ be an extreme-point optimal solution to (C-P). The variable x(i,r) has the following interpretation. If x∗i,r = 0, then we select all pairs in Si,r . Otherwise, if Si,r 6= ∅, we pick a pair in (i0 , r0 ) ∈ Si,r , and include (i0 , 2r + r0 + max(i00 ,r00 )∈Si,r \{(i0 ,r0 )} 2r00 ) in our solution. the radius of i0 to 2r + r0 + S Notice that by expanding 00 00 00 max(i00 ,r00 )∈Si,r \{(i0 ,r0 )} 2r , we cover all the clients in (i00 ,r00 )∈Si,r B(i , r ). Let F 0 be the resulting set of pairs. B3. If cost(F2 ) ≤ cost(F ), return (F2 , rad2 ), else return F 0 , {rad1 (i)}i∈µ(F 0 ) . Algorithm PDAlg(D 0 , L0 , z). Output: F ⊆ L, radius rad(i) for all i ∈ µ(F ), dual solution α. P1. Dual-ascent phase. Start with αj = 0 for all j ∈ D0 , D0 as the set of active clients, and the set T of tight pairs initialized to ∅. We repeat the following until all clients become inactive: we raise the αj s of all active clients uniformly until constraint (2) becomes tight for some (i, r); we add (i, r) to T and mark all active clients in B(i, r) as inactive. P2. Pruning phase. Let TI be a maximal subset of non-intersecting pairs in T picked by a greedy algorithm that scans pairs in T in non-increasing order of radius. Note that for each i ∈ µ(TI ), there is exactly one pair (i, r) ∈ TI . We set rad(i) = r, and ri = max {c(i, j) : j ∈ B(i0 , r0 ), (i0 , r0 ) ∈ T, r0 ≤ r, (i0 , r0 ) intersects (i, r) ((i0 , r0 ) could be (i, r))}. Let F = {(i, ri )}i∈µ(TI ) . Return F , {rad(i)}i∈µ(TI ) , and α. Analysis. We prove the following result. Theorem 2.2. For any > 0, Algorithm 1 returns a feasible LBkSR-solution of cost at most 6.1821 + O() O∗ in time nO(1/) . We firstP prove that PDAlg is an LMP 3-approximation algorithm, i.e., its output (F, α) satisfies cost(F )+ 3\|F \|z ≤ 3 j∈D0 αj (Theorem 2.3). Utilizing this, we analyze k-BSAlg, in particular, the output of the combination step B2, and argue that k-BSAlg returns a feasible solution of cost at most 6.183 + O() · OPT + O(R∗ ) (Theorem 2.5). For the right choice of F O , combining this with Lemma 2.1 yields Theorem 2.2. 7 Theorem 2.3. Suppose PDAlg(D0 , L0 , z) returns (F, {rad(i)}, α). Then (i) the balls corresponding to F cover D0 , P (ii) cost(F ) + 3\|F \|z ≤ 3 j∈D0 αj ≤ 3(OPT + k 0 z), (iii) (i, rad(i)) i∈µ(F ) ⊆ L0 , is a set of non-intersecting pairs, and rad(i) ≤ ri ≤ 3R∗ ∀i ∈ µ(F ), (iv) if \|F \| ≥ k 0 then cost(F ) ≤ 3 · OPT ; if \|F \| > k 0 , then z ≤ OPT . (Hence, k2 ≤ k 0 in step B1.1.) Proof. We prove parts (i)—(iii) first. Note that (i, rad(i)) i∈µ(F ) is TI (by definition). Consider a client j ∈ D0 and let (i0 , r0 ) denote the pair in T that causes j to become inactive. Then there must be a pair (i, r) ∈ TI that intersects (i0 , r0 ) such that r ≥ r0 (we could have (i, r) = (i0 , r0 )). Since by definition ri ≥ c(i, j), j ∈ B(i, ri ). Also, c(i, i0 ) ≤ r + r0 . So if j is the client that determines ri , then ri = c(i, j) ≤ c(i0 , i) + c(i, j) ≤ 2r0 + r ≤ 3r ≤ 3R∗ . All pairs in TI are tight and non-intersecting. So for every i ∈ µ(F ), there must be some j ∈ B(i, rad(i)) ∩ D0 with c(i, j) = rad(i), so rad(i) ≤ ri . Since \|F \| = \|TI \|, X X X X (ri + 3z) ≤ 3αj ≤ 3(OPT + k 0 z). cost(F ) + 3\|F \|z = (3r + 3z) = 3αj ≤ (i,r)∈TI (i,r)∈TI (i,r)∈TI j∈B(i,r)∩D0 j∈D0 The last inequality above follows since (α, z) is a feasible solution to (D1 ). Rearranging the bound yields 3(\|F \|−k 0 )z ≤ 3·OPT −cost(F ), so when \|F \| ≥ k 0 , we have cost(F ) ≤ 3 · OPT , and when \|F \| > k 0 , we have z ≤ OPT . Recall that in step B1.1, k2 is the number of pairs returned by PDAlg for z = 2k 0 cmax . So the last 0 0 statement follows Psince OPT ≤ k0 cmax , as all balls in L have radius at most cmax and any feasible solution to (P1 ) satisfies (i,r)∈L0 yi,r ≤ k . Let F, {rad(i)} = k-BSAlg(D0 , L0 , k 0 ). If k-BSAlg terminates in step B1, then cost(F ) ≤ 3 · OPT due to part (ii) of Theorem 2.3, so assume otherwise. Let a, b ≥ 0 be such that ak1 + bk2 = k 0 , a + b = 1. Let C1 = cost(F1 ) and C2 = cost(F2 ). Recall that (F1 , rad1 , α1 ) and (F2 , rad2 , α2 ) are the outputs of PDAlg for z1 and z2 respectively. Claim 2.4. We have aC1 + bC2 ≤ (3 + )OPT . Proof. By part (ii) of Theorem 2.3, we have C1 +3k1 z1 ≤ 3(OPT +k 0 z1 ) and C2 +3k2 z2 ≤ 3(OPT +k 0 z2 ). Combining these, we obtain aC1 +bC2 ≤ 3OPT +3k 0 (az1 +bz2 )−3(ak1 z1 +bk2 z2 ) ≤ 3(OPT +k 0 z2 )−3k 0 z2 +3ak1 δz ≤ (3+)OPT . The second inequality follows since 0 ≤ z2 − z1 ≤ δz . 0 , k 0 ) returns a feasible solution F, {rad(i)} with cost(F ) ≤ 6.183 + Theorem 2.5. k-BSAlg(D0 , L O() · OPT + O(R∗ ) where (i, rad(i))}i∈µ(F ) ⊆ L0 is a set of non-intersecting pairs. Proof. The radii {rad(i)}i∈µ(F ) are simply the radii obtained from some execution of PDAlg, so (i, rad(i)) i∈µ(F ) ⊆ L0 and comprises non-intersecting pairs. If k-BSAlg terminates in step B1, we have argued a better bound on cost(F ). If not, and we return F2 , the cost incurred is C2 . Otherwise, we return the solution F 0 found in step B2. Since (C-P) has only one constraint in addition to the bound constraints 0 ≤ xi,r ≤ 1, the extreme-point x∗ has at most one fractional P optimal solution ∗ component, and if it has a fractional component, then (i,r)∈F2 xi,r + \|Si,r \|(1 − x∗i,r ) = k 0 . For any (i, r) ∈ F2 with x∗i,r ∈ {0, 1}, the number of pairs we include is exactly x∗i,r + \|Si,r \|(1 − x∗i,r ), and the total cost of these pairs is at most the contribution to the objective function of (C-P) from the x∗i,r and (1 − x∗i,r ) 8 terms. If x∗ has a fractional component (i0 , r0 ) ∈ F2 , then x∗i0 ,r0 + \|Si0 ,r0 \|(1 − x∗i0 ,r0 ) is a positive integer. Since we include at most one pair for (i0 , r0 ), this implies that \|F 0 \| ≤ k 0 . The cost of the pair we include is at most 15R∗ , since all (i, r) ∈ F1 ∪ F2 satisfy r ≤ 3R∗ . Therefore, cost(F 0 ) ≤ OPT C-P + 15R∗ . Also, OPT C-P ≤ 2bC2 + (2b + a)C1 = 2bC2 + (1 + b)C1 , since setting xi,r = b for all (i, r) ∈ F2 yields a feasible solution to (C-P) of this cost. So when we terminate in step B3, we return a solution F with cost(F ) ≤ min{C2 , 2bC2 + (1 + b)C1 + 15R∗ }. The following claim (Claim 2.6) shows that min{C2 , 2bC2 + (1 + b)C1 } ≤ 2.0607(aC1 + bC2 ) for all a, b ≥ 0 with a + b = 1. Combining this with Claim 2.4 yields the bound in the theorem. Claim 2.6. min{C2 , 2bC2 + (1 + b)C1 } ≤ ( 3b2b+1 )(aC1 + bC2 ) ≤ 2.0607(aC1 + bC2 ) for all a, b ≥ 0 −2b+1 such that a + b = 1. Proof. Since the minimum is less than any convex combination, min(C2 , 2bC2 + bC1 + C1 ) ≤ = 3b2 − b 1−b C2 + 2 (2bC2 + bC1 + C1 ) 2 3b − 2b + 1 3b − 2b + 1 (1 − b)(1 + b) b2 + b b+1 C1 + 2 C2 = 2 ((1 − b)C1 + bC2 ) 2 3b − 2b + 1 3b − 2b + 1 3b − 2b + 1 Since a = 1 − b, the first inequality in the claim follows.√ 3 The expression 3b2b+1 ≈ 2.0607, which yields is maximized at b = −1 + 2, and has value 1 + 2√ −2b+1 2 the second inequality in the claim. Now we have all the ingredients needed for proving the main theorem of this section. Proof of Theorem 2.2. It suffices to show that when the selection F O = {(i1 , r1 ), . . . (it , rt )} in step A1 corresponds to the t facilities in an optimal solution with largest radii, we obtain the desired approximation ∗ bound. In this case, if t = k, then F O is an optimal solution. Otherwise, t ≥ 1 , so we have R∗ ≤ Ot ≤ O∗ P and OPT ≤ O∗ − tp=1 rp . Combining Theorem 2.5 and Lemma 2.1 then yields the theorem. Improved approximation ratio. The improved approximation ratio comes from a better way of combining F1 and F2 in step B2. The idea is that we can ensure that the dual solutions α1 and α2 are componentwise quite close to each other by setting δz in the binary-search procedure to be sufficently small. Thus, we may essentially assume that if T1,I , T2,I denote the tight pairs yielding F1 , F2 respectively, then every pair in T1,I intersects some pair in T2,I , because we can augment T2,I to include non-intersecting pairs of T1,I . This yields dividends when we combine solutions as in step B2, because we can now ensure that if π(i0 , r0 ) = (i, r), then the pairs of T2,I and T1,I yielding (i, r) and (i0 , r0 ) respectively intersect, which yields an improved bound on ci,i0 . This yields an improved approximation of 3.83 for the combination step (Lemma A.4), and hence for the entire algorithm (Theorem 2.7); we defer the details to Appendix A. Theorem 2.7. For any > 0, our algorithm returns a feasible LBkSR-solution of cost at most (3.83 + O())O∗ in time nO(1/) . 2.2 Approximation algorithm for LBkSRO We now build upon the ideas in Section 2.1 to devise an O(1)-approximation algorithm for the outlier version LBkSR. The high-level approach is similar to the one in Section 2.1. We again “guess” the t (i, r) pairs F O corresponding to the facilities with largest radii in an optimal solution, and consider the modified k-BS-instance (D0 , L0 , k 0 , m) (where D0 , L0 , k 0 are defined as before). We design a primal-dual algorithm for the Lagrangian relaxation of the k-BS-problem where we are allowed to pick any number of pairs from 9 L0 (leaving at most m uncovered clients) incurring a fixed cost of z for each pair picked, utilize this to obtain two solutions F1 and F2 , and combine these to extract a low-cost solution. However, the presence of outliers introduces various difficulties both in the primal-dual algorithm and in the combination step. We consider the following LP-relaxation of the k-BS-problem and its dual (analogous to (P1 ) and (D1 )). X X max αj − k 0 · z − m · γ (D2 ) min r · yi,r (P2 ) j∈D0 (i,r)∈L0 s.t. X yi,r + wj ≥ 1 ∀j ∈ D0 s.t. αj − z ≤ r ∀(i, r) ∈ L0 (3) j∈B(i,r)∩D0 (i,r)∈L0 :j∈B(i,r) X X yi,r ≤ k 0 , (i,r)∈L0 X αj ≤ γ wj ≤ m ∀j ∈ D0 α, z, γ ≥ 0. j∈D0 y, w ≥ 0. As before, if (P2 ) is infeasible, we reject this guess; so we assume (P2 ) is feasible in the remainder of this section. Let OPT denote the optimal value of (P2 ). The natural modification of the earlier primal-dual algorithm PDAlg is to now stop the dual-ascent process when the number of active clients is at most m and set γ = maxj∈D0 αj . This introduces the significant complication that one may not be able to pay for the (r + z)-cost of non-intersecting tight pairs selected in the pruning phase by the dual objective value P j∈D0 αj − m · γ, since clients with αj = γ may be needed to pay for both the r + z-cost of the last tight pair f = (if , rf ) but their contribution gets canceled by the −m · γ term. This issue affects us in various guises. First, we no longer obtain an LMP-approximation for the unconstrained problem since we have to account for the (r + z)-cost of f separately. Second, unlike Claim 2.4, given solutions F1 and F2 obtained via binary search for z1 , z2 ≈ z1 respectively with \|F2 \| ≤ k 0 < \|F1 \|, we now only obtain a fractional k-BS-solution of cost O(OPT + z1 ). While one can modify the covering-knapsack-LP based procedure in step B2 of k-BSAlg to combine F1 , F2 , this only yields a good solution when z1 = O(OPT ). The chief technical difficulty is that z1 may however be much larger than OPT . Overcoming this obstacle requires various novel ideas and is the key technical contribution of our algorithm. We design a second combination procedure that is guaranteed to return a good solution when z1 = Ω(OPT ). This requires establishing certain structural properties for F1 and F2 , using which we argue that one can find a good solution in the neighborhood of F1 and F2 . We now detail the changes to the primal-dual algorithm and k-BSAlg in Section 2.1 and analyze them to prove Theorem 2.18, which states the performance guarantee we obtain for the modified k-BSAlg. As before, for the right guess of F O , combining this with Lemma 2.1 immediately yields the following result. Theorem 2.8. There exists a 12.365 + O() -approximation algorithm for LBkSRO that runs in time nO(1/) for any > 0. Modified primal-dual algorithm PDAlgo (D 0 , L0 , z). This is quite similar to PDAlg (and we again return pairs from L). We stop the dual-ascent process when there are at most m active clients. We set γ = maxj∈D0 αj . Let f = (if , rf ) be the last tight pair added to the tight-pair set T , and Bf = B(if , rf ). We sometimes abuse notation and S use (i, r) to also denote the singleton set {(i, r)}. For a set P of (i, r) pairs, define uncov(P ) := D0 \ (i,r)∈P B(i, r). Note that \|uncov(T \ f )\| > m ≥ \|uncov(T )\|. Let Out be a set of m clients such that uncov(T ) ⊆ Out ⊆ uncov(T \ f ). Note that αj = γ for all j ∈ Out. The pruning phase is similar to before, but we only use f if necessary. Let TI be a maximal subset of non-intersecting pairs picked by greedily scanning pairs in T \ f in non-increasing order of radius. For i ∈ µ(TI ), set rad(i) to be the unique r such that (i, r) ∈ TI , and let ri be the smallest radius ρ such that B(i, ρ) ⊇ B(i0 , r0 ) for every (i0 , r0 ) ∈ T \ f such that r0 ≤ rad(i) and (i0 , r0 ) intersects (i, rad(i)). Let F 0 = {(i, ri )}i∈µ(TI ) . If uncov(F 0 ) ≤ m, set F = F 0 . If uncov(F 0 ) > m and ∃i ∈ µ(F 0 ) such 10 that c(i, if ) ≤ 2R∗ , then increase ri so that B(i, ri ) ⊇ Bf and let F be this updated F 0 . Otherwise, set F = F ∪ f and rif = rad(if ) = rf . We return (F, f, Out, {rad(i)}i∈µ(F ) , α, γ). The proof of Theorem 2.9 dovetails the proof of Theorem 2.3. Theorem 2.9. Let (F, f, Out, {rad(i)}, α, γ) = PDAlgo (D0 , L0 , z). Then (i) uncov(F ) ≤ m, P (ii) cost(F \ f ) + 3\|F \ f \|z − 3R∗ ≤ 3( j∈D0 αj − mγ) ≤ 3(OPT + k 0 z), (iii) (i, rad(i)) i∈µ(F ) ⊆ L0 , is a set of non-intersecting pairs, and rad(i) ≤ ri ≤ 3R∗ ∀i ∈ µ(F ), (iv) if \|F \ f \| ≥ k 0 then cost(F ) ≤ 3 · OPT + 4R∗ , and if \|F \ f \| > k 0 then z ≤ OPT . Proof. We first prove parts (i)–(iii). Let F 0 = {(i, ri0 )}i∈µ(TI ) be the set of pairs obtained from the set TI in the pruning phase. By the same argument as in the proof of Theorem 2.3, we have ri0 ≤ 3rad(i) ≤ 3R∗ for all i ∈ µ(TI ), and uncov(F 0 ) ⊆ uncov(T \ f ). If we return F = F 0 , then \|uncov(F )\| ≤ m by definition. If uncov(F 0 ) > m and we increase the radius of some i ∈ µ(F 0 ) with c(i, if ) ≤ 2R∗ , then we have ri ≤ max{ri0 , 3R∗ } ≤ 3R∗ and uncov(F ) ⊆ uncov(T ), so \|uncov(F )\| ≤ m. If f ∈ F , then we again have uncov(F ) ⊆ uncov(T ). This proves part (i). P The above argument shows that cost(F \ f ) ≤ i∈µ(TI ) 3 · rad(i) + 3R∗ . All pairs in TI are tight and non-intersecting and \|F \ f \| = \|TI \|. Also, Out ⊆ uncov(T \ f ) ⊆ uncov(TI ). (Recall that \|Out\| = m and αj = γ for all j ∈ Out.) So X X cost(F \ f ) + 3\|F \ f \|z − 3R∗ ≤ (3 · rad(i) + 3z) = 3αj i∈µ(TI ) i∈µ(TI ) j∈B(i,rad(i))∩D0 X X X ≤3 αj − αj = 3 αj − mγ ≤ 3(OPT + k 0 z). (4) j∈D0 j∈Out j∈D0 The last inequality follows since (α, γ, z) is a feasible solution to (D2 ). This proves part (ii). Notice that (i, rad(i)) i∈µ(F ) is TI if f ∈ / F , and TI + f otherwise. In the latter case, we know that ∗ c(i, if ) > 2R for all i ∈ µ(TI ), so f does not intersect (i, rad(i)) for any i ∈ µ(TI ). Thus, all pairs in (i, rad(i)) i∈µ(F ) are non-intersecting. The claim that rad(i) ≤ ri for all i ∈ µ(F ) follows from exactly the same argument as that in the proof of Theorem 2.3. Part (iv) follows from part (ii) and (4). The bound on cost(F P ) follows from part (ii) since that cost(F ) ≤ cost(F \ f ) + R∗ . Inequality (4) implies that \|F \ f \|z ≤ i∈µ(TI ) (rad(i) + z) ≤ OPT + k 0 z, and so z ≤ OPT if \|F \ f \| > k 0 . Modified algorithm k-BSAlgo (D 0 , L0 , k0 , ). As before, we use binary search to find solutions F1 , F2 and extract a low-cost solution from these. The only changes to step B1 are as follows. We start with z1 = 0 and z2 = 2nk 0 cmax ; for this z2 , we argue below that PDAlgo returns at most k 0 pairs. We stop o 0 when z2 − z1 ≤ δz := OPT 3n2n . We do not stop even if PDAlg returns a solution (F, . . .) with \|F \| = k for 2 some z = z1 +z 2 , since Theorem 2.9 is not strong enough to bound cost(F ) even when this happens!. If 0 \|F \| > k , we update z1 ← z and the F1 -solution; otherwise, we update z2 ← z and the F2 -solution. Thus, we maintain that k1 = \|F1 \| > k 0 , and k2 = \|F2 \| ≤ k 0 . Claim 2.10. When z = z2 = 2nk 0 cmax , PDAlgo returns at most k 0 pairs. Proof. Let (F, f, out, {rad(i)}i∈µ(F ) , α, γ) be the output of PDAlgo for this z. Let T be the sight of tight pairs after the dual-ascent process. Observe that γ ≥ 2k 0 cmax , since for any tight pair (i, r) ∈ T , we have 11 P P that nγ ≥ j∈B(i,r)∩D0 αj ≥ z. We have j∈D0 αj − mγ ≤ OPT + k 0 z ≤ k 0 cmax + k 0 z. On the other hand, since uncov(T \ f ) \ out 6= ∅ and αj = γ for all j ∈ uncov(T \ f ), we also have the lower bound X X αj − mγ ≥ αj + γ ≥ \|F \ f \|z + γ. j∈D0 i∈µ(F \f ) j∈B(i,rad(i))∩D0 So if \|F \| > k 0 , we arrive at the contradiction that γ ≤ k 0 cmax . The main change is in the way solutions F1 , F2 are combined. We adapt step B2 to handle outliers (procedure A in Section 2.2.1), but the key extra ingredient is that we devise an alternate combination procedure B (Section 2.2.2) that returns a low-cost solution when z1 = Ω(OPT ). We return the better of the solutions output by the two procedures. We summarize these changes at the end in Algorithm k-BSAlgo (D0 , L0 , k 0 , ) and state the approximation bound for k-BSAlgo (Theorem 2.18). Combining this with Lemma 2.1 (for the right selection of t (i, r) pairs) immediately yields Theorem 2.8. We require the following continuity lemma, which is essentially Lemma 6.6 in [12]; we include a proof in Appendix B for completeness. Lemma 2.11. Let (Fp , . . . , αp , γ p ) = PDAlgo (D0 , L0 , zp ) for p = 1, 2, where 0 ≤ z2 − z1 ≤ δz . Then, 1 2 n 1 2 n 0 kα P j − αj k∞ ≤p2 δz and \|γ −n γ \| ≤ 2 δz . Thus, if (3) is tight for some (i, r) ∈ L in one execution, then j∈B(i,r)∩D0 αj ≥ r + z1 − 2 δz for p = 1, 2. 2.2.1 Combination subroutine A (F1 , rad1 ), (F2 , rad2 ) As in step B2, we cluster the F1 -pairs around F2 -pairs in stars. However, unlike before, some (i0 , r0 ) ∈ F1 may remain unclustered and and we may not pick (i0 , r0 ) or some pair close to it. Since we do not cover all clients covered by F1 , we need to cover a suitable number of clients from uncov(F1 ). We again setup an LP to obtain a suitable collection of pairs. Let ucp denote uncov(Fp ) and Dp := D0 \ ucp for p = 1, 2. Let π : F1 → F2 ∪ {∅} be defined as follows: for each (i0 , r0 ) ∈ F1 , if (i0 , r0 ) ∈ F1 intersects some F2 -pair, pick such an intersecting (i, r) ∈ F2 and set π(i0 , r0 ) = (i, r); otherwise, set π(i0 , r0 ) = ∅. In the latter case, (i0 , r0 ) is unclustered, and B(i0 , r0 ) ⊆ uc2 . Define Si,r = π −1 (i, r) for all (i, r) ∈ F2 . Let Q = π −1 (∅). Let {uc1 (i, r)}(i,r)∈F2 be a partition of uc1 ∩ D2 such that uc1 (i, r) ⊆ uc1 ∩ B(i, r) for all (i, r) ∈ F2 . Similarly, let {uc2 (i0 , r0 )}(i0 ,r0 )∈F1 be a partition of uc2 ∩ D1 such that uc2 (i0 , r0 ) ⊆ uc2 ∩ B(i0 , r0 ) for all (i0 , r0 ) ∈ F1 . We consider the following 2-dimensional covering knapsack LP. X X P P min xi,r (2r + (i0 ,r0 )∈Si,r 2r0 ) + (1 − xi,r ) (i0 ,r0 )∈Si,r r0 + qi0 ,r0 · r0 (2C-P) (i0 ,r0 )∈Q (i,r)∈F2 X s.t. xi,r + \|Si,r \|(1 − xi,r ) + (i,r)∈F2 (1 − xi,r )\|uc1 (i, r)\| + qi0 ,r0 ≤ k (5) (i0 ,r0 )∈Q (i,r)∈F2 X X X (1 − qi0 ,r0 )\|uc2 (i0 , r0 )\| ≤ m − \|uc1 ∩ uc2 \| (6) (i0 ,r0 )∈Q 0 ≤ xi,r ≤ 1 ∀(i, r) ∈ F2 , 0 ≤ qi0 ,r0 ≤ 1 ∀(i0 , r0 ) ∈ Q. The interpretation of the variable xi,r is similar to before. If xi,r = 0, or xi,r = 1, Si,r 6= ∅, we proceed as in step B2 (i.e., select all pairs in Si,r , or pick some (i0 , r0 ) ∈ Si,r and expand its radius suitably). But if xi,r = 1, Si,r = ∅, then we may also pick (i, r) (see Theorem 2.14). Variable qi0 ,r0 indicates if we pick (i0 , r0 ) ∈ F1 . The number of uncovered clients in such a solution is at most \|uc1 ∩ uc2 \| + (LHS of (6)), and (6) enforces that this is at most m. 12 Let (x∗ , q ∗ ) be an extreme-point optimal solution to (2C-P). The number of fractional components in is at most the number of tight constraints from (5), (6). We exploit this to round (x∗ , q ∗ ) to an integer solution (x̃, q̃) of good objective value (Lemma 2.13), and then use (x̃, q̃) to extract a good set of pairs as sketched above (Theorem 2.14). Recall that k1 = \|F1 \|, k2 = \|F2 \|. Let a, b ≥ 0 be such that ak1 + bk2 = k 0 , a + b = 1. Let C1 = cost(F1 ) and C2 = cost(F2 ). (x∗ , q ∗ ) Lemma 2.12. The following hold. (i) aC1 + bC2 ≤ (3 + )OPT + 4R∗ + 3z1 , (ii) OPT2C-P ≤ 2bC2 + (1 + b)C1 . Proof. Part (i) follows easily from part (ii) of Theorem 2.9 and since cost(Fp ) ≤ cost(Fp \ fp ) + R∗ for p = 1, 2. So we have C1 + 3(k1 − 1)z1 ≤ 3(OPT + k 0 z1 ) + 4R∗ and C2 + 3(k2 − 1)z2 ≤ 3(OPT + k 0 z2 ) + 4R∗ . Combining these, we obtain aC1 + bC2 ≤ 3OPT + 3k 0 (az1 + bz2 ) − 3(ak1 z1 + bk2 z2 ) + 3(az1 + bz2 ) + 4R∗ ≤ 3(OPT + k 0 z2 ) − 3k 0 z2 + 3ak1 δz + 3z1 + 3bδz + 4R∗ ≤ (3 + )OPT + 4R∗ + 3z1 . The second inequality follows since 0 ≤ z2 − z1 ≤ δz . For part (ii), we claim that setting xi,r = b for all (i, r) ∈ F2 , and qi0 ,r0 = a for all (i0 , r0 ) ∈ Q yields 0 a feasible solution to (2C-P). The P LHS of (5) evaluates to ak1 + bk2 , which is exactly k . The first term on the LHS of (6) evaluates to a (i,r)∈F2 \|uc1 (i, r)\| = a\|uc1 ∩ D2 \| = a\|uc1 \ uc2 \| since {uc1 (i, r)}(i,r)∈F2 is a partition of uc1 ∩ D2 . Similarly, the second term on the LHS of (6) evaluates to at most b\|uc2 ∩ D1 \| = b\|uc2 \ uc1 \|. So we have (LHS of (6)) + \|uc1 ∩ uc2 \| = a\|uc1 \| + b\|uc2 \| ≤ m since \|uc1 \|, \|uc2 \| ≤ m. The objective value of this solution is 2bC2 + 2bC1 + (1 − b)C1 = 2bC2 + (1 + b)C1 . Let P = {(i, r) ∈ F2 : Si,r = ∅}. Lemma 2.13. (x∗ , q ∗ ) can be rounded to a feasible integer solution (x̃, q̃) to (2C-P) of objective value at most OPT2C-P + O(R∗ ). Proof. Let S be the set of fractional components of (x∗ , q ∗ ). As noted earlier, \|S\| is at most the number of tight constraints from (5), (6). Let X X l∗ := x∗i,r + \|Si,r \|(1 − x∗i,r ) + qi∗0 ,r0 (i0 ,r0 )∈S∩Q (i,r)∈S∩F2 denote the contribution of the fractional components of (x∗ , q ∗ ) to the LHS of (5). Note that if (5) is tight, then l∗ must be an integer. For a vector v = (vj )j∈I where I is some index-set, let dve denote dvj e j∈I . We round (x∗ , q ∗ ) as follows. • If l∗ ≥ 2 or \|S\| ≤ 1 or \|S ∩ (F2 \ P)\| ≥ 1, set (x̃, q̃) = d(x∗ , q ∗ )e. • Otherwise, we set x̃i,r = x∗i,r , q̃i0 ,r0 = qi∗0 ,r0 for all the integer-valued coordinates. We set the fractional component with larger absolute coefficient value on the LHS of (6) equal to 1 and the other fractional component to 0. 13 We prove that (x̃, q̃) is a feasible solution to (2C-P). Note that (6) holds for (x̃, q̃) since we always have X X (1 − q̃i0 ,r0 )\|uc2 (i0 , r0 )\| (1 − x̃i,r )\|uc1 (i, r)\| + (i0 ,r0 )∈Q (i,r)∈F2 ≤ X (1 − x∗i,r )\|uc1 (i, r)\| + X (1 − qi∗0 ,r0 )\|uc2 (i0 , r0 )\|. (i0 ,r0 )∈Q (i,r)∈F2 Clearly, the contribution to the LHS of (5) from the components not in S is the same in both (x̃, q̃) and (x∗ , q ∗ ). Let l denote the contribution from (x̃, q̃) to the LHS of (5) from the components in S. Clearly, l is an integer. If l∗ ≥ 2, then l = 2. If \|S\| ≤ 1, then l = 1. If l∗ ≥ 1, then in these cases the LHS of (5) evaluated at (x̃, q̃) is at most the LHS of (5) evaluated at (x∗ , q ∗ ). If l∗ < 1 and \|S\| ≤ 1 (so l = 1), then since l∗ is fractional, we know that (5) is not tight for (x∗ , q ∗ ). So despite the increase in LHS of (5), we have that (5) holds for (x̃, q̃). If \|S\| = 2 and \|S ∩ (F2 \ P)\| ≥ 1, then we actually have l∗ > 1 and l = 2. Again, since l∗ is fractional, we can conclude that (x̃, q̃) satisfies (5) despite the increase in LHS of (5). Finally, suppose l∗ < 2, \|S\| P P = 2, and S ∩ (F2 \ P) = ∅. Then the contribution from S to the LHS of (5) is x + (i,r)∈S∩F2 i,r (i0 ,r0 )∈S∩Q qi0 ,r0 , and at most one of the components in S is set to 1 in (x̃, q̃). So l = 1, ∗ ∗ and either l ≤ l or l < 1, and in both cases (5) holds for (x̃, q̃). To bound the objective value of (x̃, q̃), notice that compared to (x∗ , q ∗ ), the solution (x̃, q̃) pays extra only for the components that are rounded up. There are at most two such components, and their objectivefunction coefficients are bounded by 15R∗ , so the objective value of (x̃, q̃) is at most OPT2C-P + 30R∗ . Theorem 2.14. The integer solution (x̃, q̃) returnedby Lemma 2.13 yields a solution F, {rad(i)} to i∈µ(F ) ∗ the k-BS-problem with cost(F ) ≤ 6.1821 + O() (OPT + z1 ) + O(R ) where (i, rad(i)) i∈µ(F ) ⊆ L0 is a set of non-intersecting pairs. Proof. Unlike in step B2 of k-BSAlg, we will not simply pick a subset of pairs of F1 and expand their radii. We will sometimes need to pick pairs from F2 in order to ensure that we have at most m outliers, but we need to be careful in doing so because we also need to find suitable radii for the facilities we pick so that we obtain non-intersecting pairs. We first construct F 00 as follows. If q̃i0 ,r0 = 1, we include (i0 , r0 ) ∈ F 00 and set rad(i0 ) = rad1 (i0 ). If x̃i,r = 0, we include all pairs in Si,r in F 00 and set rad(i0 ) = rad1 (i0 ) for all (i0 , r0 ) ∈ Si,r . If x̃i,r = 1 and Si,r 6= ∅, we pick a pair in (i0 , r0 ) ∈ Si,r , and include (i0 , 2r + r0 + max(i00 ,r00 )∈Si,r \{(i0 ,r0 )} 2r00 ) in F 00 . We set rad(i0 ) = rad1 (i0 ). Now we initialize F 0 = F 00 and consider all (i, r) ∈ P with x̃i,r = 1. If (i, r) does not intersect any (i0 , r0 ) ∈ F 00 then we add (i, r) to F 0 , and set rad(i) = rad2 (i). Otherwise, if (i, r) intersects some (i0 , r0 ) ∈ F 00 , then we replace (i0 , r0 ) ∈ F 0 with (i0 , r0 + 2r). We have thus ensured that (i, rad(i)) i∈µ(F 0 ) ⊆ L0 and consists of non-intersecting pairs. Note that in all the cases above, the total cost of the pairs we include when we process some q̃i0 ,r0 or x̃i,r term is at most the total contribution to the objective function from the q̃i0 ,r0 term, or the x̃i,r and 1 − x̃i,r terms. Therefore, cost(F 0 ) is at most the objective value of (x̃, q̃). Finally, we argue that \|uncov(F 0 )\| ≤ m. We have \|uncov(F 0 )\| ≤ \|uc1 ∩ uc2 \| + \|uncov(F 0 ) ∩ D1 \| + \|uncov(F 0 ) ∩ D2 ∩ uc1 \|. Observe that for every client j ∈ uncov(F 0 ) ∩ D1 and every 0 , r0 ) (i0 , r0 ) ∈ F1 such that j ∈ B(i0 , r0 ), it must be that (i0 , r0 ) ∈ Q and q̃i0 ,r0 = P0. It follows that j ∈ uc2 (i 0 0 0 0 for some (i , r ) ∈ Q with q̃i0 ,r0 = 0. Therefore, \|uncov(F ) ∩ D1 \| ≤ (i0 ,r0 )∈Q (1 − q̃i0 ,r0 )\|uc2 (i , r0 )\|. Similarly, for every j ∈ uncov(F 0 ) ∩ D2 ∩ uc1 and every (i, r) ∈ F2 such that j ∈ B(i, r), we must have (i, r) ∈ P and x̃i,r P = 0; hence, j ∈ uc1 (i, r) for some (i, r) ∈ P with x̃i,r = 0. Therefore, 0 \|uncov(F ) ∩ D2 ∩ uc1 \| ≤ (i,r)∈P (1 − x̃i,r )\|uc1 (i, r)\|. Thus, since (x̃, q̃) is feasible, constraint (6) implies that \|uncov(F 0 )\| ≤ m. We return (F2 , rad2 ) if cost(F2 ) ≤ cost(F 0 ), and F 0 , {rad(i)}i∈µ(F 0 ) otherwise. Combining the above bound on cost(F 0 ) with part (ii) of Lemma 2.12 and Lemma 2.13, we obtain that the cost of the 14 solution returned is at most min C2 , 2bC2 + (1 + b)C1 + 30R∗ ≤ 2.0607 aC1 + bC2 + 30R∗ ≤ 2.0607 (3 + )OPT + 4R∗ + 3z1 + 30R∗ ≤ (6.1821 + 3)(OPT + z1 ) + 39R∗ . The first inequality follows from Claim 2.6, and the second follows from part (i) of Lemma 2.12. 2.2.2 Subroutine B (F1 , f1 , Out 1 , rad1 , α1 , γ 1 ), (F2 , f2 , Out 2 , rad2 , α2 , γ 2 ) Subroutine A in the previous section yields a low-cost solution only if z1 = O(OPT ). We complement subroutine A by now describing a procedure that returns a good solution when z1 is large. We assume in this section that z1 > (1 + )OPT . Then \|F1 \ f1 \| ≤ k 0 (otherwise z1 ≤ OPT by part (iv) of Theorem 2.9), so \|F1 \ f1 \| ≤ k 0 < \|F1 \|, which means that k1 = k 0 + 1 and f1 ∈ F1 . Hence, αj1 = γ 1 for all j ∈ Bf1 ∩ D0 . First, we take care of some simple cases. If there exists (i, r) ∈ F1 \f1 such that \|uncov F1 \{f1 , (i, r)}∪ (i, r + 12R∗ ) \| ≤ m, then set F = F1 \ {f1 , (i, r)} ∪ (i, r + 12R∗ ). We have cost(F ) = cost(F1 \ f1 ) + 12R∗ ≤ 3 · OPT + 15R∗ (by part (ii) of Theorem 2.9). If there exist pairs (i, r), (i0 , r0 ) ∈ F1 such that c(i, i0 ) ≤ 12R∗ , take r00 to be the minimum ρ ≥ r such that B(i0 , r0 ) ⊆ B(i, ρ) and set F = ∗ ∗ F1 \ {(i, r), (i0 , r0 )} ∪ (i, r00 ). We have cost(F ) ≤ cost(F1 \ f1 ) + 13R ≤ 3 · OPT + 16R . In both cases, we return F, {rad1 (i)}i∈µ(F ) . So we assume in the sequel that neither of the above apply. In particular, all pairs in F1 are wellP separated. Let AT = {(i, r) ∈ L0 : j∈B(i,r)∩D0 αj1 ≥ r + z1 − 2n δz } and AD = {j ∈ D0 : αj1 ≥ γ 1 − 2n δz }. By Lemma 2.11, AT includes the tight pairs of PDAlgo (D0 , L0 , zp ) for both p = 1, 2, and Out 1 ∪ Out 2 ⊆ AD. Since the tight pairs T2 used for building solution F2 are almost tight in (α1 , γ 1 , z1 ), we swap them in and swap out pairs from F1 one by one while maintaining a feasible solution. Either at some point, we will be able to remove f , which will give us a solution of size k 0 , or we will obtain a bound on cost(F2 ). The following lemma is our main tool for bounding the cost of the solution returned. Lemma 2.15. Let F ⊆ L0 , and let TF = (i, ri0 ) i∈µ(F ) where ri0 ≤ r for each (i, r) ∈ F . Suppose S TF ⊆ AT and pairs in TF are non-intersecting. If \|F \| ≥ k 0 and \|AD \ (i,r)∈F B(i, r))\| ≥ m then cost(TF ) ≤ (1 + )OPT . Moreover, if \|F \| > k 0 then z1 ≤ (1 + )OPT . S Proof. Let Out F be a subset of exactly m of clients from AD \ (i,r)∈F B(i, r). Since the pairs in TF are P P non-intersecting and almost tight, i∈µ(F ) (ri0 + z1 ) ≤ j∈D0 \Out F (αj1 + 2n δz ), so X i∈µ(F ) (ri0 +z1 ) ≤ X j∈D0 (αj1 +2n δz )−m(γ 1 −2n δz ) ≤ X αj1 −mγ 1 +(m+\|D0 \|)2n δz ≤ (1+)OPT +k 0 z1 j∈D0 where the last inequality follows since (α1 , γ 1 , z1 ) is a feasible solution to (D2 ). So cost(TF ) ≤ (1+)OPT if \|TF \| = \|F \| ≥ k 0 , and z1 ≤ (1 + )OPT if \|F \| > k 0 . Define a mapping ψ : F2 → F1 \ f1 as follows. Note that any (i, r) ∈ F2 may intersect with at most one F1 -pair: if it intersects (i0 , r0 ), (i00 , r00 ) ∈ F1 , then we have c(i0 , i00 ) ≤ 12R∗ . First, for each (i, r) ∈ F2 that intersects with some (i0 , r0 ) ∈ F1 , we set ψ(i, r) = (i0 , r0 ). Let M ⊆ F2 be the F2 -pairs mapped by ψ this way. For every (i, r) ∈ F2 \ M , we arbitrarily match (i, r) with a distinct (i0 , r0 ) ∈ F1 \ ψ(M ). We claim that ψ is in fact a one-one function. Lemma 2.16. Every (i, r) ∈ F1 \ f1 intersects with at most one F2 -pair. 15 Proof. Suppose two pairs (i1 , r1 ), (i2 , r2 ) ∈ F2 intersect with a common pair (i, r) ∈ F1 \ f1 . Let T1,I be the tight pairs corresponding to F1 \ f1 obtained from (the pruning phase of) PDAlgo (D0 , L0 , z1 ). Let (i, rad1 (i)) ∈ T1,I be the tight pair corresponding to (i, r). Let (i1 , rad2 (i1 )), (i2 , rad2 (i2 )) be the tight pairs corresponding to (i1 , r1 ), (i2 , r2 ) obtained from PDAlgo (D0 , L0 , z2 ). Let F 00 = F1 \ {f1 , (i, r)} ∪ (i, r + 12R∗ ). We show that either z1 ≤ OPT or \|uncov(F 00 )\| ≤ m, both of which lead to a contradiction. Define F 0 = F1 \ {f1 , (i, r)} ∪ {(i1 , r1 ), (i2 , r2 )}, so \|F 0 \| = k + 1. Consider the set TF 0 = T1,I \ {(i, rad1 (i))} ∪ {(i1 , rad2 (i1 )), (i2 , rad2 (i2 ))}. Since (i1 , rad2 (i1 ) and (i2 , rad2 (i2 )) are non-intersecting and they do not intersect with any pair in TS1,I \ (i, rad1 (i)), the pairs in TF 0 are non-intersecting. Also, TF 0 ⊆ AT . If \|AD ∩ uncov(F 0 )\| = \|AD \ (i0 ,r0 )∈F 0 B(i0 , r0 )\| ≥ m, then z1 ≤ OPT by Lemma 2.15. Otherwise, note that every client in B(i1 , r1 ) ∪ B(i2 , r2 ) is at distance at most r + 2 max{r1 , r2 } ≤ r + 6R∗ from i. So we have uncov(F 00 ) ⊆ uncov(F ) ∪ Bf1 ⊆ AD and uncov(F 00 ) ⊆ uncov(F 0 ). So \|uncov(F 00 )\| ≤ \|AD ∩ uncov(F 0 )\| ≤ m. Let F20 be the pairs (i, r) ∈ F2 such that if (i0 , r0 ) = ψ(i, r), then r0 < r. Let P = F20 ∩ M and Q = F20 \ M . For every (i0 , r0 ) ∈ ψ(Q) and j ∈ B(i0 , r0 ), we have j ∈ uncov(F2 ) ⊆ AD (else (i0 , r0 ) would lie in ψ(M )). Starting with F = F1 \ f1 , we iterate over (i, r) ∈ F20 and do the following. Let (i0 , r0 ) = ψ(i, r). If (i, r) ∈ P , we update F ← F \ (i0 , r0 ) ∪ (i, r + 2r0 ) (so B(i, r + 2r0 ) ⊇ B(i0 , r0 )), else we update F ← F \ (i0 , r0 ) ∪ (i, r). Let TF = {(i, rad1 (i))}(i,r)∈F ∩F1 ∪ {(i, rad2 (i))}(i,r)∈F \F1 . Note that \|F \| = k 0 and uncov(F ) ⊆ AD at all times. Also, since (i, r) intersects only (i0 , r0 ), which we remove when (i, r) is added, we maintain that TF is a collection of non-intersecting pairs and a subset of AT ⊆ L0 . This process continues until \|uncov(F )\| ≤ m, or when all pairs of F20 are swapped in. In the former case, we argue that cost(F ) is small and return F, {rad1 (i)}(i,r)∈F ∩F1 ∪ {rad2 (i)}(i,r)∈F \F1 . In the latter case, we show that cost(F20 ), and hence cost(F2 ) is small, and return (F2 , rad2 ). Lemma 2.17. (i) If the algorithm stops with \|uncov(F )\| ≤ m, then cost(F ) ≤ (9 + 3)OPT + 18R∗ . (ii) If case (i) does not apply, then cost(F2 ) ≤ (3 + 3)OPT + 9R∗ . (iii) The pairs corresponding to the radii returned are non-intersecting and form a subset of L0 . Proof. Part (iii) follows readily from the algorithm description and the discussion above. Consider part (i). Let (i, r) ∈ F20 be the last pair scanned by the algorithm before it terminates, and (i0 , r0 ) = ψ(i, r). Let F 0 be the set F before the last iteration. So F 0 = F \ (i, r + 2r0 ) ∪ (i0 , r0 ) if (i, r) ∈ P , and F 0 = F \ (i, r) ∪ (i0 , r0 ) if (i, r) ∈ Q. Note that r + 2r0 ≤ 9R∗ . Since uncov(F 0 ) ⊆ AD and \|uncov(F 0 )\| > m, by Lemma 2.15, we have cost(TF 0 ) ≤ (1 + )OPT . For all (i, r) ∈ F1 , we have r ≤ 3rad1 (i) (since f1 ∈ F1 ). For all but at most one (i, r) ∈ F2 , we have r ≤ 3rad2 (i) and for the one possible exception, we have r ≤ 3R∗ . Therefore, cost(F ) ≤ cost(F 0 ∩ F1 ) + cost(F 0 \ F1 ) + 9R∗ ≤ 3 · cost(TF 0 ) + 3R∗ + 2 · cost(F1 \ F 0 ) + 9R∗ ≤ 3(1 + )OPT + 3R∗ + 2(3 · OPT + 3R∗ ) + 9R∗ = 9 + 3)OPT + 18R∗ . P The second inequality above follows since cost(F 0 ∩ F1 ) ≤ (i,r)∈F 0 ∩F1 3rad1 (i) and cost(F 0 \ F1 ) ≤ P ∗ 0 (i,r)∈F 0 \F1 3rad2 (i) + 3R + 2cost(F1 \ F ). Forpart (ii), Lemma 2.15 shows that cost(TF ) ≤ (1 + )OPT , and so cost(F20 ) + cost F1 \ (f1 ∪ ψ(F20 )) ≤ 3 · cost(TF ) + 3R∗ . Now cost(F2 ) = cost(F20 ) + cost(F2 \ F20 ) ≤ cost(F20 ) + cost ψ(F2 \ F20 ) = cost(F20 ) + cost F1 \ (f1 ∪ ψ(F20 ) ≤ 3(1 + ) · OPT + 3R∗ where the first inequality follows by the definition of F20 . 16 Algorithm k-BSAlgo (D 0 , L0 , k0 , ). Output: F ⊆ L with \|F \| ≤ k 0 , a radius rad(i) for all i ∈ µ(F ). C1. Binary search. Let (F1 , rad1 , . . .) = PDAlgo (D0 , L0 , 0). If \|F1 \| ≤ k 0 pairs, return (F1 , rad1 ). Else perform binary-search in the range [0, ncmax ] to find z1 , z2 with 0 ≤ z2 − z1 ≤ δz = OPT 3n2n such that letting (Fp , fp , Out p , radp , αp , γ p ) = PDAlg(D0 , L0 , zp ) for p = 1, 2, we have \|F2 \| ≤ k 0 < \|F1 \|. C2. Let FA , {radA (i)}i∈µ(FA ) = A (F1 , rad1 ), (F2 , rad2 ) (Section 2.2.1). If \|F1 \ f1 \| > k 0 , return (FA , radA ). C3. If ∃(i, r) ∈ F1 \f1 such that \|uncov F1 \{f1 , (i, r)}∪(i, r +12R∗ ) \| ≤ m, then set F = F1 \{f1 , (i, r)}∪(i, r + 0 0 12R∗ ). If ∃(i, r), (i0 , r0 ) ∈ F1 such that c(i, i0 ) ≤ 12R∗ , let r00 be the minimum ρ ≥ r such that B(i ,r ) ⊆ 0 0 00 B(i, ρ); set F = F1 \ {(i, r), (i , r )} ∪ (i, r ). If either of the above apply, return F, {rad1 (i)}i∈µ(F ) . C4. Let FB , {radB (i)}i∈µ(FB ) be the output of subroutine B (Section 2.2.2). C5. If cost(FA ) ≤ cost(FB ), return (FA , radA ), else return (FB , radB ). Theorem 2.18. k-BSAlgo (D0 , L0 , k 0 ) returns a solution (F, rad) with cost(F ) ≤ 12.365 + O() · OPT + O(R∗ ) where (i, rad(i)) i∈µ(F ) ⊆ L0 comprises non-intersecting pairs. Proof. This follows essentially from Theorem 2.14 and Lemma 2.17. When z1 ≤ (1 + ) · OPT , Theorem 2.14 yields the above bound on cost(FA ). Otherwise, if none of the cases in step C3 apply, then Lemma 2.17 bounds cost(FB ). In the boundary cases, when we terminate in step C1 or C3, we have cost(F ) ≤ cost(F1 \ f1 ) + cost(f1 ) + 12R∗ , which is at most the expression in the theorem due to part (ii) of Theorem 2.9. 3 Minimizing the maximum radius with lower bounds and outliers The lower-bounded k-supplier with outliers (LBkSupO) problem is the min max-radius version of LBkSRO. The input and the set of feasible solutions are the same as in LBkSRO: the input is an instance I = F, D, {Li }, {c(i, j)}, k 0 , m , and a feasible solution is S ⊆ F, σ : D 7→ S ∪ {out} with \|S\| ≤ k, \|σ −1 (i)\| ≥ Li for all i ∈ S, and \|σ −1 (out)\| ≤ m. The cost of (S, σ) is now maxi∈S maxj∈σ−1 (i) c(i, j). The special case where m = 0 is called the lower-bounded k-supplier (LBkSup) problem, and the setting where D = F is often called the k-center version. Let τ ∗ denote the optimal value; note that there are only polynomially many choices for τ ∗ . As is common in the study of min-max problems, we reduce the problem to a “graphical” instance, where given some value τ , we try to find a solution of cost O(τ ) or deduce that τ ∗ > τ . We construct a bipartite unweighted graph Gτ = Vτ = D ∪ Fτ , Eτ ), where Fτ = {i ∈ F : \|B(i, τ )\| ≥ Li }, and Eτ = {ij : c(i, j) ≤ τ, i ∈ Fτ , j ∈ D}. Let dist τ (i, j) denote the shortest-path distance in Gτ between i and j, so c(i, j) ≤ distτ (i, j) · τ . We say that an assignment σ : D 7→ Fτ ∪ {out} is a distance-α assignment if dist τ (j, σ(j)) ≤ α for every client j with σ(j) 6= out. We call such an assignment feasible, if it yields a feasible LBkSupO-solution, and we say that Gτ is feasible if it admits a feasible distance-1 assignment. It is not hard to see that given F ⊆ Fτ , the problem of finding a feasible distance-α-assignment σ : D 7→ F ∪ {out} in Gτ (if one exists) can be solved by creating a network-flow instance with lower bounds and capacities. Observe that an optimal solution yields a feasible distance-1 assignment in Gτ ∗ . We devise an algorithm that for every τ , either finds a feasible distance-α assignment in Gτ for some constant α, or detects that Gτ is not feasible. This immediately yields an α-approximation algorithm since the smallest τ for which the algorithm returns a feasible LBkSupO-solution must be at most τ ∗ . We obtain Theorems 3.1 and 3.2 via this template. Theorem 3.1. There is a 3-approximation algorithm for LBkSup. Theorem 3.2. There is a 5-approximation algorithm for LBkSupO. 17 We complement our approximation results via a simple hardness result (Theorem 3.3) showing that our approximation factor for LBkSup is tight. We also show that LBkSupO is equivalent to the k-center version (i.e., where F = D) of the problem (Appendix C); a similar equivalence is known to hold for the capacitated versions of k-supplier and k-center with outliers [15]. Theorem 3.3. It is NP-hard to approximate LBkSup within a factor better than 3, unless P = N P . Proof. The result is shown via a reduction from set cover problem. Suppose we have a set cover instance 0 with set U = [n] of elements and collection S = ∪np=1 {Sp } of subsets of U, and we want to know if there exists k subsets of U in S that cover all elements of U. Let j1 , j2 , · · · , jn represent the elements and i1 , i2 , · · · , in0 represent subsets of U in S. Construct an LBkSup instance I with client set D = ∪np=1 {jp }, 0 facility set F = ∪nq=1 {iq }, define c(jp , iq ) for jp ∈ D, iq ∈ F to be 1 if p ∈ Sq , 3 otherwise, and let Li = 1 for each i ∈ F. Suppose there exists a collection F of k subsets in S that cover all elements. First, remove any set i in F , if i does not cover an element that is not covered by F \ i. Let σ : D → F be defined for element j to be some set in F that covers j. Since each set i in F covers at least one element that is not covered by F \ i, \|σ −1 (i)\| ≥ 1, so (F, σ) is a feasible solution to I with radius 1. If no collection of k subsets of U in S covers all elements, then there does not exist k facilities in F that all elements are at distance at most 1 from them, so optimal solution of I has cost at least 3. Therefore, it is NP-hard to approximate LBkSup with a factor better than 3 as otherwise the algorithm can be used to answer the decision problem. Finding a distance-3 assignment for LBkSup. Consider the graph Gτ ∗ . Note that there exists an optimal center among the neighbors of each client in G. Moreover, two clients at distance at least 3 are served by two distinct centers. These insights motivate the following algorithm. Let N (v) denote the neighbors of vertex v in the given graph Gτ . Find a maximal subset Γ of clients with distance at least 3 from each other. If \|Γ\| > k or there exists a client j with N (j) = ∅, then return Gτ is not feasible. For each j ∈ Γ, let ij denote the center S in N (j) with minimum lower bound. If there exists a feasible distance-3 assignment σ of clients to F = j∈Γ {ij }, return σ, otherwise return Gτ is not feasible. The following lemma yields Theorem 3.1. Lemma 3.4. The above algorithm finds a feasible distance-3 assignment in Gτ if Gτ is feasible. Proof. Let σ ∗ : D 7→ F ∗ be a feasible distance-1 assignment in Gτ . So F ∗ ⊆ Fτ and every client has a non-empty neighbor set. Since each client in Γ has to be served by a distinct center in F ∗ , \|Γ\| ≤ \|F ∗ \| ≤ k. For each client j ∈ Γ, let i∗j = σ ∗ (j). Note that i∗j ∈ N (j), so Lij ≤ Li∗j by the choice of ij , and every client in σ ∗−1 (i∗j ) is at distance at most 3 from ij . We show that there is a feasible distance-3 assignment σ : D 7→ F . For each j ∈ Γ, we assign all clients in σ ∗−1 (i∗j ) to ij . As argued above this satisfies the lower bound of ij . For any unassigned client j, let j 0 ∈ Γ be a client at distance at most 2 from j (which must exist by maximality of Γ). We assign j to ij 0 . Finding a distance-5 assignment for LBkSupO. The main idea here is to find a set F ⊆ Fτ of at most k centers that are close to the centers in F ∗ ⊆ Fτ for some feasible distance-1 assignment σ ∗ : D 7→ F ∗ ∪ {out} in Gτ . The non-outlier clients of (F ∗ , σ ∗ ) are close to F , so there are at least \|D\| − m clients close to F . If centers in F do not share a neighbor in Gτ , then clients in N (i) can be assigned to i for each i ∈ F to satisfy the lower bounds. We cannot check if F satisfies the above properties, but using an idea similar to that in [15], we will find a sequence of facility sets such that at least one of these sets will have the desired properties when Gτ is feasible. 18 Definition 3.5. Given the bipartite graph Gτ , a set F ⊆ F is called a skeleton if it satisfies the following properties. (a) (Separation property) For i, i0 ∈ F , i 6= i0 , we have dist τ (i, i0 ) ≥ 6; (b) There exists a feasible distance-1 assignment σ ∗ : D 7→ F ∗ ∪ {out} in Gτ such that • (Covering property) For all i∗ ∈ F ∗ , dist τ (i∗ , F ) ≤ 4, where dist τ (i∗ , F ) = mini∈F dist τ (i∗ , i). • (Injection property) There exists f : F 7→ F ∗ such that dist τ (i, f (i)) ≤ 2 for all i ∈ F . If F satisfies the separation and injection properties, it is called a pre-skeleton. Note that if F ⊆ Fτ is a skeleton or pre-skeleton, then Gτ is feasible. Suppose F ⊆ Fτ is a skeleton and satisfies the properties with respect to a feasible distance-1 assignment (F ∗ , σ ∗ ). The separation property ensures that the neighbor sets of any two locations i, i0 ∈ F are disjoint. The covering property ensures that F ∗ is at distance at most 4 from F , so there are at least \|D\| − m clients at distance at most 5 from F . Finally, the injection and separation properties together ensure that \|F \| ≤ k since no two locations in F can be mapped to the same location in F ∗ . Thus, if F is a skeleton, then we can obtain a feasible distance-5 assignment σ : D 7→ F ∪ {out}. Lemma 3.6. Let F be a pre-skeleton in Gτ . Define U = {i ∈ Fτ : dist τ (i, F ) ≥ 6} and let i = arg maxi0 ∈U \|N (i0 )\|. Then, either F is a skeleton, or F ∪ {i} is a pre-skeleton. Proof. Suppose F is not a skeleton and F ∪ {i} is not a pre-skeleton. Let σ ∗ : D 7→ F ∗ ∪ {out} be a feasible distance-1 assignment in Gτ such F satisfies the injection property with respect to (F ∗ , σ ∗ ). Let f : F 7→ F ∗ be the mapping given by the injection property. Since F ∪ {i} is not a pre-skeleton and dist τ (i, F ) ≥ 6, this implies that dist τ (i, F ∗ ) > 2, and hence, dist τ (i, F ∗ ) ≥ 4 as Gτ is bipartite. This means that all clients in N (i) are outliers in (F ∗ , σ ∗ ). Moreover, since F is not a skeleton, there exists a center i∗ ∈ F ∗ with dist τ (i∗ , F ) > 4, and so dist(i∗ , F ) ≥ 6. Therefore, i∗ ∈ U . By the choice of i, we know that \|N (i)\| ≥ \|N (i∗ )\|. Now consider F 0 = F ∗ \{i∗ }∪{i}, and define σ 0 : D 7→ F 0 ∪{out} as follows: σ 0 (j) = σ ∗ (j) for all j ∈ / N (i) ∪ N (i∗ ), σ 0 (j) = i for all j ∈ N (i), and σ 0 (j) = out for all j ∈ N (i∗ ). Note that the F covers as many clients as F ∗ , and so σ 0 : D 7→ F 0 ∪ {out} is another feasible distance-1 assignment. But this yields a contradiction since F ∪ {i} now satisfies the injection property with respect to (F 0 , σ 0 ) as certified by the function f 0 : F → F 0 defined by f 0 (s) = f (s) for s ∈ F , f 0 (i) = i. If Gτ is feasible, then ∅ is a pre-skeleton. A skeleton can have size at most k. 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A Improved Approximation Ratio for LBkSR We now describe in detail the changes to algorithm k-BSAlg and its analysis leading to Theorem 2.7. First, we set δz = OPT 3n2n in the binary-search procedure (step B1); note that the binary search still takes polynomial time. By Lemma 2.11 (specialized to the non-outlier setting), we have kα1 − α2 k∞ ≤ 2n δz , which implies 21 that every (i, r) ∈ T1 ∪ T2 is almost tight with respect to (αp , zp ) for p = 1, 2. To obtain the improved guarantee, we construct the mapping π : F1 7→ F2 , and hence, our stars, based on whether pairs (i0 , rad1 (i0 )) and (i, rad2 (i)) intersect for i0 ∈ µ(F1 ), i ∈ µ(F2 ). To ensure that every (i0 , r0 ) ∈ F1 belongs to some star, we first modify F2 and T2,I by including non-intersecting pairs from T1,I (which are almost tight in (α2 , z2 )). We consider pairs in F1 in arbitrary order. For each (i, r) ∈ F1 , if (i, rad1 (i)) does not intersect any pair in T2,I , we add (i, rad1 (i)) to T2,I , add (i, r) to F2 , and set rad2 (i) = rad1 (i). We continue this process until all pairs in F1 are scanned or \|F2 \| = k 0 . Lemma A.1. If \|F2 \| = k 0 after the above process, then F2 is a feasible k-BS solution with cost(F2 ) ≤ (3 + )OPT , and T2,I ⊆ L0 is a set of non-intersecting pairs. Proof. All clients in D0 are covered by balls corresponding to the F2 -pairs since this holds even before any 0 pairs P are added to F2 . It is clear Using Lemma 2.11, we P that T12,I ⊆nL and consists P of non-intersecting pairs. have (î,r̂)∈T2,I (r̂ + z1 ) ≤ j∈D0 αj + 2 δz \|T2,I \|, so (î,r̂)∈T2,I r̂ ≤ 1 + 3 OPT . For every (i, r) ∈ F2 P we have r ≤ 3rad2 (i), so cost(F2 ) ≤ (î,r̂)∈T2,I 3r̂ ≤ (3 + )OPT . So if \|F2 \| = k 0 after the above preprocessing, we simply return (F2 , rad2 ). Otherwise, we combine solutions F1 and F2 using an LP similar to (C-P). We construct a map π : F1 → F2 similar to before, but with the small modification that we set π(i0 , r0 ) = (i, r) only if (i0 , rad1 (i0 )) intersects with (i, rad2 (i)). Due to our preprocessing, π is well-defined. As before, let star Si,r = π −1 (i, r) for each (i, r) ∈ F2 . Figure 2: Old combination method. Si,r = {(i1 , r1 ), (i2 , r2 ), (i3 , r3 )} Figure 3: New combination method. Si,r = {(i1 , r1 ), (i2 , r2 ), (i3 , r3 )} The LP again has an indicator variable xi,r . If xi,r = 0,Pwe select all pairs in Si,r . Otherwise, if Si,r 6= ∅, we select a pair (i0 , r0 ) ∈ Si,r and include i0 , 2rad2 (i) + (i00 ,r00 )∈Si,r 4rad1 (i00 ) in our solution; note that S the corresponding ball covers all clients in (i00 ,r00 )∈Si,r B(i00 , r00 ). So we consider the following LP. min X xi,r 2rad2 (i) + P (i0 ,r0 )∈S P 0 0 4rad (i ) + (1 − x ) 3rad (i ) 0 0 1 i,r 1 (i ,r )∈Si,r i,r (i,r)∈F2 s.t. X xi,r + \|Si,r \|(1 − xi,r ) ≤ k, (i,r)∈F2 22 0 ≤ xi,r ≤ 1 ∀(i, r) ∈ F2 . (C-P’) Let x∗ be an extreme point of (C-P’). Let F 0 be the pairs obtained by picking the pairs corresponding to dx∗ e as described above. Since x∗ has at most one fractional component, it follows as before that \|F 0 \| ≤ k 0 . 0 As before, we return F , {rad(i)}µ(i)∈F 0 or P (F2 , {rad2 (i)}), whichever has lower cost. P 0 = 0 Let C10 = rad (i) and C 1 2 (i,r)∈F1 (i0 ,r0 )∈F2 rad2 (i ). The following claims are analogous to Claims 2.4 and 2.6. Claim A.2. We have aC10 + bC20 ≤ 1 + 3 OPT . Proof. Using Lemma 2.11, we have aC10 + bC20 = a X (i,r)∈F1 rad1 (i) + b X rad2 (i) ≤ a · X j∈D0 (i,r)∈F2 X αj1 − k1 z1 + b · k2 2n δz + αj1 − k2 z2 j∈D0 · OPT 3 j∈D0 X = αj1 − k 0 · z1 + · OPT ≤ 1 + OPT . 3 3 0 ≤ X (aαj1 + bαj1 ) − (ak1 + bk2 ) · z1 + j∈D Claim A.3. min{3C20 , 2bC20 + (3 + b)C10 } ≤ such that a + b = 1. 3(b+3) (aC10 3b2 −2b+3 + bC20 ) ≤ 3.83(aC10 + bC20 ) for all a, b ≥ 0 Proof. Since the minimum is less than any convex combination, min(3C20 , 2bC20 + bC10 + 3C10 ) ≤ = = 3b2 + b −3b + 3 (3C20 ) + 2 (2bC20 + bC10 + 3C10 ) 3b2 − 2b + 3 3b − 2b + 3 3b(b + 3) 3(1 − b)(b + 3) 0 (C1 ) + 2 C0 3b2 − 2b + 3 3b − 2b + 3 2 3(b + 3) ((1 − b)C10 + bC20 ). 3b2 − 2b + 3 Since a = 1 − b, the first inequality in the claim follows. The expression 3b3(b+3) 2 −2b+3 is maximized at √ √ 3 b = −3 + 2 3, and has value 8 (5 + 3 3) ≈ 3.8235, which yields the second inequality in the claim. Lemma A.4. The cost of the solution (F, {rad(i)}) returned by the above combination subroutine is at most (3.83 + O())OP T + O(R∗ ) where {(i, rad(i))}i∈µ(F ) ⊆ L0 is a set of non-intersecting pairs. Proof. First note that {rad(i)} correspond to {rad2 (i)} if F = F2 and {rad(i)} ⊆ {rad1 (i)} if F = F 0 , so in both cases it consists of non-intersecting pairs from L0 . The cost of the pair included in F 0 corresponding to a fractional component of x∗ is at most 7R∗ as each radp (i) is bounded by R∗ for p ∈ {1, 2}. Since x∗ has at most one fractional component, cost(F 0 ) ≤ OPT C-P’ + 7R∗ . Also, OPT C-P’ ≤ 2bC20 + (4b + 3a)C10 = 2bC20 + (3 + b)C10 , since setting xi,r = b for all (i, r) ∈ F2 yields a feasible solution to (C-P’) of this cost. Therefore, cost(F ) ≤ min{3C20 , 2bC20 + (b + 3)C10 + 7R∗ }, which is at most 3.83(aC10 + bC20 ) + 7R∗ by Claim A.3. Combining this with Claim A.2 yields the bound in the lemma. Proof of Theorem 2.7. It suffices to show that when the selection F O = {(i1 , r1 ), . . . (it , rt )} in step A1 corresponds to the t facilities in an optimal solution with largest radii, we obtain the desired approximation ∗ bound. In this case, if t = k, then F O is an optimal solution; otherwise, we have R∗ ≤ Ot ≤ O∗ and P OPT ≤ O∗ − tp=1 rp . Combining Lemma A.4 and Lemma 2.1 then yields the theorem. 23 B Proof of Lemma 2.11 We abbreviate PDAlgo (D0 , L0 , z) to PDAlgo (z). We use x− to denote a quantity infinitesimally smaller than x. Consider the dual-ascent phase of PDAlgo for z1 and z2 . First, suppose that m = 0. Sort clients with respect to their αj0 = min(αj1 , αj2 ) value. Let this ordering be α10 ≤ α20 ≤ · · · ≤ αn0 . We prove by induction that \|αj1 − αj2 \| ≤ 2j−1 δz . For the base case, assume without loss of generality that αj0 = αj1 , and let (i, r) be the tight pair that 0 caused j to become inactive in PDAlgo (z1 ). Consider time point t = Pα1 in the two executions. By definition − o all clients are active at time t in PDAlg (z2 ). So the contribution j∈B(i,r)∩D0 αj of clients to the LHS of (3) at time t− is at least as much as their contribution in PDAlgo (z1 ) at time t− . Therefore, we can increase α1 by at most δz beyond time t in PDAlgo (z2 ) as z2 − z1 = δz . Suppose we have shown that for all clients j = 1, 2, · · · , ` − 1 (where ` ≥ 2), Now consider client ` and let (i, r) be the tight pair that makes ` inactive at time α`0 in PDAlgo (zp ), where p ∈ {1, 2}. Consider time point t = α`0 in both executions. By definition, all clients j > ` are still active at time t− in both o executions PDAlgo (z1 ) and PPDAlg (z2 ). (They might become inactive at time t but can not become inactive earlier.) The contribution j∈B(i,r)∩D0 αj of clients to the LHS of (3) in the execution other than p at time P j−1 δ . The values of z in the t− is at least their contribution in PDAlgo (zp ) at time t− minus `−1 z j=1 2 two executions differs by at most δ , so in the execution other than p, α can grow beyond t by at most z ` P`−1 j−1 2 )δz ≤ 2` δz . (1 + j=1 P Now if we consider a tight pair (i, r) in one of the execution, the value of RHS and LHS of j∈B(i,r) αj ≤ P r + z for the other execution can differ by at most (1 + nj=1 2j−1 )δz ≤ 2n δz . Now consider the case where m > 0. Note that in this case, we can assume that we have the execution for m = 0, pick the first time at which there are at most m active clients, i.e., time γ in PDAlgo , and set αj = γ for every active client at this time point. Let γ 0 = min(γ 1 , γ 2 ), suppose γ 0 = γp , where p ∈ {1, 2}. Note that by time γ 0 + 2n δz , all pairs that are tight in the p-th execution by time γ 0 are also tight in the other execution. So the number of active clients after this time point is at most m. Therefore \|γ 1 −γ 2 \| ≤ 2n δz . C Equivalence of lower-bounded k-supplier with outliers and lower-bounded k-center with outliers Let LBkCentO denote the special case of LBkSupO where F = D. In this section, we show that if there exists an α-approximation for LBkCentO, then there exists an α-approximation for LBkSupO. Let I = (k, F, D, c, L, m) be an instance of LBkSupO with N = \|F\| + 1 and \|D\| = n. Define an instance I 0 = (k 0 , D0 , c0 , L0 , m0 ) as follows: let k 0 = k and D0 = (D × {1, 2, · · · , N }) ∪ F. Let c0 ((j, p), i) = c(j, i) for each j ∈ D, p ∈ [N ], i ∈ F, and let c0 be the metric completion of these distances (i.e., c0 (q, q 0 ) is the shortest-path distance between q and q 0 with respect to these distances for q, q 0 ∈ D0 ). Define L0i = N Li for i ∈ F and L0(j,p) = N (n + 1), and let m0 = N · m + (N − 1). Clearly I 0 can be constructed from I in polynomial time. The lower-bounds for (j, p), j ∈ D, p ∈ [N ] are set so that L0(j,p) < \|D0 \|, so (j, p) cannot be opened as a center in any feasible solution to I 0 . Let OP T (I 0 ) denote the value of optimal solution of I 0 and OP T (I) denote the value of optimal solution of I. We claim that OP T (I 0 ) ≤ OP T (I). Let (F ∗ , σ ∗ ) denote an optimal solution of I. Let solution (F̂ , σ̂) for I be constructed as follows: let F̂ = F ∗ , for each p ∈ [N ], define σ(q) = i for q = (j, p) if σ ∗ (j) = i, and σ(q) = out otherwise. Note that since there are at most m outliers in solution (F ∗ , σ ∗ ) then there are at most N m + \|F\| = N m + (N − 1) outliers in (F̂ , σ̂). Clearly the radius of the opened centers is the same as before, so OP T (I 0 ) ≤ OP T (I). Now suppose there exists an α-approximation algorithm A for LBkCentO problem. Use A to generate 24 a solution (F̂ , σ̂) for I 0 with maximum radius R. As noted above, we have F̂ ⊆ F. We construct a solution (F̂ , σ 0 ) for I of maximum radius at most R using Algorithm 1. Algorithm 1 Constructing a feasible assignment σ 0 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: Construct network N = (V, E) where V = {s, t} ∪ D ∪ F̂ and E = {si : i ∈ F̂ } ∪ {ij : i ∈ F̂ , j ∈ D, c(i, j) ≤ r} ∪ {jt : j ∈ D}. Set lij = 0, uij = ∞ for each ij ∈ E, i ∈ F̂ , j ∈ D. Set lsi = Li , usi = ∞ for each si ∈ E, i ∈ F̂ . Set ljt = 0, ujt = 1 for each jt ∈ E, j ∈ D. Let f ← max-flow(N ) respecting lower-bounds (l) and upper-bounds (u) on edges. if value of f is ≥ n − m then, set σ 0 (j) = i if fjt = 1 and fij = 1 for i ∈ F̂ . set σ 0 (j) = out if fjt = 0. return f . return σ 0 = ∅. Lemma C.1. Solution (F̂ , σ 0 ) is a feasible solution to I with maximum radius at most R, where σ 0 is the output of Algorithm 1. P Proof. Consider any set S ⊆ F̂ . There are at least Pi∈S N Li clients in D0 assigned to S. Since there are P i∈S N Li −(N −1) at most N − 1 facilities among D0 , there are at least > i∈S Li − 1 clients at distance at N P most R from S. So there are P at least i∈S Li clients in neighbor set of S in N . It follows that every s-t cut in N has capacity at least i∈F̂ Li , so there exists a flow f that satisfies the lower-bounds and upper-bounds on the edges. It remains to show that value of f is at least \|D\| − m. If there is an incoming edge to a client in N , then a flow of 1 can be sent through j. So we want to bound the number of clients with no incoming edge in N . If any copy of client j is served by some facility in the solution (F̂ , σ̂) then j is at distance at most R from some facility in F̂ . Since there are at most N m + (N − 1) outliers in (F̂ , σ̂), there are at most N m+(N −1) < m + 1 clients with no incoming edge in N . N Since algorithm A is an α-approximation algorithm, wehave R ≤ α · OP T (I 0 ) ≤ αOP T (I). 25	8
Quintic algebras over Dedekind domains and their resolvents arXiv:1511.03162v1 [math.NT] 10 Nov 2015 Evan O’Dorney November 11, 2015 This is an addendum to [4], which classified quadratic, cubic, and quartic rings over a Dedekind domain. 1 A coordinate-free description of resolvents Let Q be a quintic ring over a Dedekind domain R, and let L = Q/R. Our first task is to generalize the notion of a sextic resolvent, developed by Bhargava in [2] in the case R = Z. Following the approach of [6] and the present author’s senior thesis, we expect the resolvent to consist of a rank-5 lattice M (to be thought of as S/R, where S is a sextic ring) with two linear maps relating certain multilinear expressions in L and M . The orientation map θ—which relates the top exterior powers of L and M —is easy to guess. The discriminant of an R-algebra T naturally lies in (Λtop (T ))⊗−2 . Just as the equality Disc Q = Disc C between the discriminants of a quartic ring and its cubic resolvent(s) suggests an identification of the top exterior powers of the two rings, so the relation Disc S = (16 Disc Q)3 (Bhargava’s (33) of [2]) linking the discriminants of a quintic ring and its sextic resolvent(s) suggests an isomorphism θ : Λ5 M →(Λ4 L)⊗3 . The second piece of data—that which contains the 40 integers that actually parametrize resolvents over Z—is slightly trickier to work out. Bhargava presents it as a map φ from L to Λ2 M (equivalently, from Λ2 M ∗ to L∗ ), but this does not have the correct properties in our situation. The correct construction, foreshadowed somewhat by the mysterious constant factor in Bhargava’s fundamental resolvent ((28) in [2]), is to take a map φ : Λ4 L ⊗ L → Λ2 M. Finally, we must find the fundamental relations that link φ and θ to the ring structure. Just as Lemma 9 of [1] provided the inspiration for Bhargava’s coordinate-free description of resolvents of a quartic ring ([1], section 3.9), so we begin at Lemma 4(a), which, after eliminating the references to S5 -closure, states that 1 φ(y) φ(x) φ(y) φ(x) Pfaff − Pfaff = 1 ∧ y ∧ x ∧ z ∧ yz. φ(x) φ(z) φ(x) −φ(z) 2 The Pfaffians are to be interpreted by writing φ(x), etc., as a 5 × 5 skew-symmetric matrix with regard to any convenient basis (i.e. viewing it as a skew bilinear form on Λ2 M , once a generator of Λ4 L is fixed. Then we paste together four of these to make a 10 × 10 skew-symmetric matrix and take the Pfaffian. This is a clever way to manufacture certain degree-5 integer polynomials in the 40 coefficients of φ. To re-express them in a way that is coordinate-free (and applicable in characteristic 2), we consider two preliminary multilinear constructions. 1 1.1 The quadratic map µ 7→ µ Let V be a 5-dimensional vector space over a field K (which we will soon take to be Frac R). We examine the constructions that can be made starting with elements of Λ2 V . We have a bilinear map ∧ : Λ2 V × Λ2 V → Λ4 V . However, the most fundamental map from Λ2 V to Λ4 V is not the bilinear map ∧ but the quadratic map from which it arises. It is defined by ! n X X vi ∧ wi = vi ∧ wi ∧ vj ∧ wj . (1) i=1 1≤i<j≤n It is not hard to prove that this is well defined. Note that if char K 6= 2, then µ can be described more simply by 1 µ = µ ∧ µ. 2 Moreover, the bilinear map ∧ can always be recovered from • via µ ∧ ν = (µ + ν) − µ − ν . 1.2 (2) The contraction µ(α, β) The second construction takes one element µ ∈ Λ2 V and two elements α, β ∈ Λ4 V and outputs an element of a suitable one-dimensional vector space as follows. First, the perfect pairing ∧ : Λ4 V × V → Λ5 V allows us to identify α and β as elements of Λ5 V ⊗ V ∗ . These have a wedge product α ∧ β ∈ Λ2 (Λ5 V ⊗ V ∗ ) ∼ = (Λ5 V )⊗2 ⊗ Λ2 V ∗ . We now use the duality between Λ2 V ∗ and Λ2 V , described explicitly by (f ∧ g)(v ∧ w) = f v · gw − f w · gv, to obtain an element µ(α, β) ∈ (Λ5 V )⊗2 . 1.3 The definition We are now ready to state the definition of a sextic resolvent. Definition 1.1. Let Q be a quintic ring over a Dedekind domain R, and let L = Q/R. A resolvent for Q consists of a rank-5 lattice M and a pair of linear maps φ : Λ4 L ⊗ L → Λ2 M and θ : Λ5 M →(Λ4 L)⊗3 satisfying the identity θ⊗2 [φ(λ1 x)(φ(λ2 y) , φ(λ3 z) ] = λ1 λ22 λ23 (x ∧ y ∧ z ∧ yz) (3) where x, y, z ∈ L and λi ∈ Λ4 L are formal variables. The resolvent is called numerical if θ is an isomorphism. Note that the expression within square brackets lies in (Λ5 M )⊗2 ; applying θ⊗2 , one ends up in (Λ4 L)⊗6 which is where the right-hand side also resides. It should also be remarked that the product yz is carried out in Q; translating the lifts ỹ, z̃ by constants in R simply changes the product ỹ z̃ by multiples of ỹ, z̃, and 1, thereby not changing the product y ∧ z ∧ yz. 2 2 Resolvent to ring Our first task is to show that the resolvent maps φ and θ uniquely encode the multiplication data of the ring Q. Theorem 2.1. Let L and M be lattices over R of ranks 4 and 5 respectively, and let φ : Λ4 L ⊗ L → Λ2 M and θ : Λ5 M →(Λ4 L)⊗3 be maps. There is a quintic ring Q with a quotient map Q/R ∼ = L, unique up to isomorphism, such that (M, φ, θ) is a resolvent of Q. Proof. Let (e1 , e2 , e3 , e4 ) be a basis for L, by which we mean that there is a decomposition L = a1 e1 ⊕ · · · ⊕ a4 e4 for some fractional ideals ai of R. To place a ring structure on the module −1 Q = L ⊕ R, it is then necessary to choose the coefficient ckij ∈ ak a−1 i aj of ek in the product ei ej . We allow k = 0, with the conventions e0 = 1 and a0 = R. On the other hand, allowing i = 0 or j = 0 gives no useful information. Hence the ring structure is given by the 40 coefficients ckij , 1 ≤ i ≤ j ≤ 4, 0 ≤ k ≤ 5. Some of these coefficients are immediately determined by the resolvent. For instance, if {i, j, k, ℓ} is a permutation of {1, 2, 3, 4}, and ǫ = ±1 its sign, then −6 ⊗2 ckij = −ǫe−1 [φ(eℓ etop )(φ (ei etop ), φ (ej etop )], top · eℓ ∧ ei ∧ ej ∧ ei ej = −ǫetop · θ (4) where etop = e1 ∧ e2 ∧ e3 ∧ e4 = ǫ · ei ∧ ej ∧ ek ∧ eℓ is the natural generator of Λ4 L. This determines the values of all ckij where i, j, and k are nonzero and distinct. Likewise, the following expressions are determined, for i, j, k distinct: j cjii = ǫe−1 top · eℓ ∧ ei ∧ (ei + ek ) ∧ ei (ei + ek ) − cik j k ckik − cjij = ǫe−1 top · eℓ ∧ ei ∧ (ej + ek ) ∧ ei (ej + ek ) − cik + cij ciii − cjij − ckik = ǫe−1 top · eℓ ∧ (ei + ek ) ∧ (ei + ej ) ∧ (ei + ej )(ei + ek ) (5) − cijk + cjik + ckij + cjii + ckii + (cjkj − ciki ) + (ckjk − ciji ). The reader familiar with ring parametrizations will recognize the left-hand sides of (4) and (5) as the linear expressions in the ckij that are invariant under translations ei 7→ ei + ti (ti ∈ a−1 i ) of the ring basis elements. If we normalize our basis so that, say, c112 = c212 = c334 = c434 = 0, then all the ckij are now uniquely determined, except for the c0ij . The c0ij can be computed by comparing the coefficients of k in (ei ej )ek and ei (ej ek ) for any k 6= i, yielding formula (22) of [2]: 4 X c0ij = (crjk ckri − crij ckrk ). r=1 The theorem is now reduced to three verifications. −1 1. That all ckij belong to the correct ideals ak a−1 i aj . This is routine. 2. ThatP the c0ij are well defined, and more generally that the associative law holds on the ring Q = ai ei that we have just constructed. This is a family of integer polynomial identities in the 40 free coefficients of φ in the chosen basis; as such, it was proved in the course of Bhargava’s parametrization of quintic rings over Z. 3. That the original maps φ and θ indeed form a resolvent of Q, i.e. that the identity (3) holds. This can probably also be proved by appeal to results over Z, but here a direct proof is not difficult. We can assume that λ1 = λ2 = λ3 = etop and x is a basis element eℓ , since the equation (3) is linear in those variables. We can also assume that each of y and z is a basis element or a sum of two different basis elements, since (3) is quadratic in those variables. Now we have a finite set of cases, some of which are the relations (4) and (5), and the rest of which will be reduced to them using the following properties of the underlying multilinear operations: 3 Lemma 2.2. Let V be a 5-dimensional vector space, and let µ, ν, ξ ∈ Λ2 V and α ∈ Λ4 V . Then (a) µ(µ ∧ ν, α) = ν(µ , α) (b) µ(µ , α) = 0 (c) ν(µ , µ ∧ ξ) = −ξ(µ , µ ∧ ν). Proof. Calculation, although only (a) need be checked directly, as (b) follows by setting µ = ν and (c) by the derivation ν(µ , µ ∧ ξ) = µ(µ ∧ ν, µ ∧ ξ) = −µ(µ ∧ ξ, µ ∧ ν) = −ξ(µ , µ ∧ ν). Now we return to proving θ⊗2 [φ(etop x)(φ(etop y) , φ(etop z) ] = e5top (eℓ ∧ y ∧ z ∧ yz) (6) for x = eℓ and y, z ∈ {ei }i ∪ {ei + ej }i<j . The cases where eℓ does not appear in y or z are all subsumed by the definitions (4) and (5), with one exception: the expression for cjii is not visibly symmetric under switching k and ℓ. This can be seen by writing cjii = ǫe−1 top (eℓ ∧ ei ∧ (ei + ek ) ∧ ei (ei + ek ) − eℓ ∧ ei ∧ ek ∧ ei ek ) = ǫe−5 top φ(eℓ )(φ (ei ), φ (ei + ek )) − φ(eℓ )(φ (ei ), φ (ek )) = ǫe−5 top φ(eℓ )(φ (ei ), φ (ei ) + φ(ei ) ∧ φ(ek ) + φ (ek )) − φ(eℓ )(φ (ei ), φ (ek )) = ǫe−5 top φ(eℓ )(φ (ei ), φ(ei ) ∧ φ(ek )) and using Lemma 2.2(c). It remains to dispose of the cases where eℓ does appear in y or z. The key is to use Lemma 2.2(a) to reduce the case (x, x + y, z) of (6) to the cases (x, y, z) and (y, x, z). The details are left to the reader. Remark. Over Z, assuming that θ is an isomorphism, the resolvent devolves into the basis representation of φ. This has 40 independent entries which can be arranged into a quadruple of 5 × 5 skew-symmetric matrices, representing the values φ(x) (as x runs through a basis) as skew bilinear forms on M ∗ . The coefficients ckij of the ring we have constructed are certain degree-5 polynomials in these 40 entries which are easily identified with the formulas given in (21) of [2]. Thus our definition of resolvent is compatible with Bhargava’s (Definition 10), which justifies our invocation of his computations in our situation, despite the dissimilarities of the definitions. 2.1 The sextic ring It ought to be remarked that, given any resolvent (L, M, φ, θ), the rank-6 lattice M ⊕ R also picks up a canonical ring structure, whose structure coefficients dkij are integer polynomials in the coefficients of φ of degree 12 (for k 6= 0) and 24 (for k = 0). As the construction given by Bhargava in [2], Section 6 works without change over a Dedekind domain, we will not discuss it further. 3 3.1 Constructing resolvents Resolvents over a field Let K be a field. We will first investigate what sort of family of resolvents a quintic K-algebra has. In the quartic case, it was the trivial ring T = K[x, y, z]/(x, y, z)2 that had a large family, all 4 other rings having a unique resolvent. Here, if a ring has multiple resolvents, it is not necessarily trivial, but as we will see, it is in a sense minimally far from being trivial. The appropriate definition is as follows: Definition 3.1. A quintic algebra Q over K is very degenerate if it has subspaces Q4 ⊆ Q3 , of dimension 4 and 3 respectively, such that Q4 Q3 = 0 (that is, the product of any element of Q4 and any element of Q3 is zero). This implies that Q has a multiplication table × 1 α β γ δ 1 1 α β γ δ α α u 0 0 0 β β 0 0 0 0 γ γ 0 0 0 0 δ δ 0 0 0 0 (7) in which 15 of the 16 non-forced entries are zero. We prove: Theorem 3.2. Every not very degenerate quintic K-algebra has a unique resolvent up to isomorphism. Proof. The first few steps are easy: let M be a K-vector space of dimension 5, and let θ : Λ5 M →(Λ4 L)⊗3 be any isomorphism. So far we have not made any choices. (The choice θ = 0 works only for the trivial ring.) We will first try to construct the map φ = φ(•) , a quadratic map from Λ4 L ⊗ L to Λ4 M . For this purpose we concoct a corollary of (3) that involves only φ . Lemma 3.3. Let V be a 5-dimensional vector space. Let µ ∈ Λ2 M and α, β, γ, δ ∈ Λ4 M . Then µ ∧ α ∧ β ∧ γ ∧ δ = µ(α, β)µ(γ, δ) + µ(α, γ)µ(δ, β) + µ(α, δ)µ(β, γ) in Λ5 (Λ4 V ) ∼ = (Λ5 V )⊗4 . Proof. Write the general µ as u ∧ v + w ∧ x (u, v, w, x ∈ V ) and expand. As a corollary, we get that if (M, φ, θ) is a resolvent of a quintic ring Q, then for all a, b, c, d, e ∈ Λ4 L ⊗ L, Motivated by this, we define for any quintic ring Q the pentaquadratic form F (a, b, c, d, e) = (a ∧ b ∧ c ∧ bc)(a ∧ d ∧ e ∧ de) + (a ∧ b ∧ d ∧ bd)(a ∧ e ∧ c ∧ ec) + (a ∧ b ∧ e ∧ be)(a ∧ c ∧ d ∧ cd) (8) from L5 to (Λ4 L)⊗2 , or equivalently from (Λ4 L ⊗ L)5 to (Λ4 L)⊗12 . We get that for any resolvent (M, φ, θ) of Q, θ⊗4 (φ (a) ∧ φ (b) ∧ φ (c) ∧ φ (d) ∧ φ (e)) = F (a, b, c, d, e). (9) We claim the following: Lemma 3.4. F is identically zero if and only if Q is very degenerate. Proof. We prove that the property of being very degenerate is invariant under base-changing to the algebraic closure K̄ of K; then the lemma can be proved by checking the finitely many quintic algebras over an algebraically closed field (see [3, 5]). Let Q̄ = Q ⊗K K̄ be the corresponding 5 K̄-algebra. Clearly if Q is very degenerate, so is Q̄, so assume that Q̄ is very degenerate. Then the subsets M = {x ∈ Q′ \| dim ker x ≥ 3} and N = {x ∈ Q′ \|M x = 0} are, by reference to the multiplication table (7), vector spaces with dim M = 4 and dim N ∈ {3, 4}. Moreover, because they are canonically defined, they are invariant under the Galois group Gal(K̄/K). This shows that M ∩ Q and N ∩ Q are K-vector spaces of the same dimensions with (M ∩ Q)(N ∩ Q) = 0, so Q is very degenerate. Picking a1 , . . . , a5 ∈ Λ4 L ⊗ L such that F (a1 , a2 , a3 , a4 , a5 ) = f0 6= 0, we get that the five vectors vi = φ (ai ) must form a basis such that θ⊗4 (v1 ∧ v2 ∧ v3 ∧ v4 ∧ v5 ) = f0 . Any such basis is as good as any other—they are all related by elements of SL(∧4 M ), which is canonically isomorphic to SL(M ) (although GL(∧4 M ) ∼ 6 GL(M ) in general). Once the vi are = fixed, there is at most one candidate for the map φ up to SL(M )-equivalence, namely φ (a) = 5 X F (a1 , . . . , âi , a, . . . , a5 ) F (a1 , a2 , a3 , a4 , a5 ) i=1 vi (10) Then the relations φ(x)(φ (ai ), φ (aj )) = x ∧ ai ∧ aj ∧ ai aj , for 1 ≤ i < j ≤ 5, determine the map φ uniquely. So the resolvent map φ, if it exists, must be given by a predetermined formula, or rather by any one of a finite number of such formulas, inasmuch as the ai in (10) can be chosen from the finite set {e1 , e2 , e3 , e4 , e1 + e2 , e1 + e3 , . . . , e3 + e4 } for any basis {e1 , e2 , e3 , e4 } of Λ4 L ⊗ L. It remains to prove that the (M, φ, θ) we have hereby constructed is actually a resolvent; this is a collection of integer polynomial identities, not in a family of free variables as in the previous lemma, but in the coefficients ckij of the given ring Q, which are restricted by the associative law. If Q has nonzero discriminant—the most common case—the theorem can be proved by base-changing to the algebraic closure K̄ and noting that K̄ ⊕5 , the unique nondegenerate quintic K̄-algebra, does have a resolvent (Example 4.1). The general case can be handled by a limiting argument, appealing to the known fact that all quintic K̄-algebras can be deformed to K̄ ⊕5 , at least in characteristic zero (see [3]). 3.2 From field to Dedekind domain Let Q be a quintic ring over a Dedekind domain R. We will assume that Q is not very degenerate and hence that the corresponding K-algebra QK = Q ⊗R K has a unique resolvent (MK , φ, θ). Resolvents of Q are now in bijection with lattices M in the vector space MK such that φ(Λ4 L ⊗ L) ⊆ Λ2 M (11) ⊗3 (12) 5 4 θ(Λ M ) ⊆ (Λ L) . For any resolvent M , note that we must have M∗ ∼ = Λ4 M ⊗ (M 5 )⊗−1 ⊇ φ (Λ4 L ⊗ L) ⊗ (θ((Λ4 L)⊗3 ))⊗−1 . Since Q is not very degenerate, the right-hand side is a lattice of full rank and we may take its dual, which we denote by M0 . Then any resolvent is contained in M0 . Condition (11) is vacuous for M = M0 , since φ(λx)(φ (λ′ y), φ (λ′′ z)) = θ⊗2 (λλ′ λ′′ (x ∧ y ∧ z ∧ yz)) ∈ (θ(Λ3 L))⊗2 6 for all λ′ , λ′′ ∈ Λ3 L and y, z ∈ L. On the other hand, condition (12) is generally not satisfied by M = M0 ; indeed, one readily finds that θ−1 ((Λ4 L)⊗3 ) ⊆ Λ5 M0 using (9), so if M0 is a resolvent, it is numerical. The classification of resolvents is now reduced to a local problem. Any M determines a family of resolvents (Mp , φ, θ) of the quintic algebras Qp over the DVR’s Rp ⊆ K, and conversely an arbitrary choice of resolvents Mp of the Rp can be glued together to form the resolvent T M = p Mp . The choice Mp = M0,p = M0 ⊗ Rp is forced for all but finitely many primes p, namely those dividing the ideal c = [Λ5 M0 : θ−1 ((Λ4 L)⊗3 )] = [(Λ4 L)⊗2 : hF (a, b, c, d, e) : a, b, c, d, e ∈ Li]. (13) In the lucky case that c is the unit ideal, M0 is the only resolvent. This occurs in one important instance: Theorem 3.5. If Q is a maximal quintic ring, that is, is not contained in any strictly larger quintic ring, then Q has a unique resolvent, which is numerical. Proof. Suppose that c were not the unit ideal, so there is a prime p such that p\|F (a, b, c, d, e) for all a, b, c, d, e ∈ L. We will prove that Q is not maximal at p. It is convenient to localize and to assume that R = Rp is a DVR with uniformizer π. Note that Q/pQ, a quintic algebra over R/p, has its associated pentaquadratic form F identically zero, so by Lemma 3.4, it is very degenerate. So Q has an R-basis (1, x, ǫ1 , ǫ2 , ǫ3 ) such that (x, ǫ1 , ǫ2 , ǫ3 )(ǫ1 , ǫ2 , ǫ3 ) ⊆ pR. We claim that the lattice Q′ with basis (1, x, π −1 ǫ1 , π −1 ǫ2 , π −1 ǫ3 either is a quintic ring or is contained in a quintic ring, showing that Q is not maximal. Set M = hπ, x, ǫ1 , ǫ2 , ǫ3 i and N = hπ, πx, ǫ1 , ǫ2 , ǫ3 i. Then Q ⊇ M ⊇ N ⊇ πQ and M N ⊆ πQ. Consider, for any i, j ∈ {1, 2, 3}, the multiplication maps Q/N ǫi h1, xi / N/πQ ǫj hǫ1 , ǫ2 , ǫ3 i ǫi / πQ/πN hπ, πxi / πN /π 2 Q hπǫ1 , πǫ2 , πǫ3 i . These are all linear maps of R/p-vector spaces. Denote by f the composition of the left two maps and by g the composition of the right two. Write f (1) = π(a + bx), where a, b ∈ R/p. Then g(ǫi ) = aπǫi , since xǫi ∈ πQ. Thus g is given in the bases above by the scalar matrix a. But g has rank at most 2, since it factors through the two-dimensional space πQ/πN ; hence a = 0. So N 2 ⊆ πM . Now consider the following multiplication maps: Q/M h1i ǫi / N/πQ hǫ1 , ǫ2 , ǫ3 i ǫj / πM /πN ǫk / π 2 Q/π 2 M π2 hπxi ǫi / π 2 N /π 3 Q π 2 ǫ1 , π 2 ǫ2 , π 2 ǫ3 . Similarly to the previous argument, the composition of the first three maps must be zero, or else the composition of the last three would be a nonzero scalar. Since the images of the first map (as i varies) span N/πQ, the composition of the middle two maps is always zero. There are two cases: (a) The second map is always zero, that is, N 2 ⊆ πN . This implies that π −1 N is a quintic ring, as desired. 7 (b) The third map is always zero, that is, M N ⊆ πM . We get that π −1 ǫi is integral over R (look at the characteristic polynomial of its action on M ), so R[π −1 ǫ1 , π −1 ǫ2 , π −1 ǫ3 ] is finitely generated and thus a quintic ring, as desired. Note that, in this proof, if the resolvent is not unique, then the extension Q′ ) Q has (R/p)3 ⊆ Q′ /Q. So the following stronger theorem holds: Theorem 3.6. If Q is a quintic ring such that the R/p-vector space of congruence classes in π −1 Q/Q whose elements are integral over R has dimension at most 2, for each prime p, then Q has a unique resolvent, which is numerical. 3.3 Bounds on the number of numerical resolvents Finally, we examine bounds on the number of numerical resolvents a not very degenerate quintic ring can have. A lower bound of 1 was proved over Z in [2], Theorem 12; the method is adaptable to our situation, and we do not attempt to sharpen the bound. Instead, let us prove a complementary upper bound in terms of the invariant c of (13). Theorem 3.7. A not very degenerate ring Q has at most Y N (p)5 − 1 n N (p) − 1 p prime, pn kc numerical resolvents, provided that the absolute norms N (p) = \|R/p\| are finite. In particular, a not very degenerate quintic ring over the ring of integers of a number field has finitely many numerical resolvents. Proof. Since all numerical resolvents have index c in M0 , it suffices to bound the number of sublattices of index c in a fixed lattice M0 . By localization we may reduce to the case c = pn , where p is prime. Now a fixed lattice M has (N (p)5 − 1)/(N (p) − 1) sublattices of index p, the kernels of the nonzero linear functionals ℓ : M/pM → R/p mod scaling. A sublattice Mn of index pn has a filtration M0 ( M1 ( · · · ( Mn where the quotients are R/p; given Mi , there are at most (N (p)5 − 1)/(N (p) − 1) possibilities for Mi+1 , giving the claimed bound. 4 Examples Example 4.1. The most fundamental example of a sextic resolvent is as follows. Let Q = R⊕5 , with basis e1 , e2 , . . . , e5 , and let M = R5 with basis f1 , . . . , f5 . Then the map φ(ei ) = fi ∧ (fi−1 + fi+1 ) (indices mod 5), supplemented by the natural orientation θ(ftop ) = e3top , is verified to be a resolvent for Q (indeed the unique one, as Q is maximal). The automorphism group S5 of Q acts on M by the 5-dimensional irreducible representation obtained (in characteristic not 2) by restricting to the image of the exceptional embedding S5 ֒→ S6 the standard representation of S6 , permuting the six vectors fi−2 − fi−1 + fi − fi+1 + fi+2 (1 ≤ i ≤ 5) and f1 + f2 + f3 + f4 + f5 . 8 Example 4.2. For the subring Q = {x1 e1 + · · · + x5 e5 ∈ Z⊕5 : x1 ≡ x2 ≡ x3 ≡ x4 mod p}, the bounding module M0 of Section 3.2 is no longer a resolvent, as can be seen by observing that Q/pQ ∼ = Fp [t, ǫ1 , ǫ2 , ǫ3 ]/h{t2 − t, tǫi , ǫi ǫj }i is very degenerate. We have L = hpe1 , pe2 , pe3 , e5 i and thus Λ4 L = hp3 etop i. One computes that M0 = p(f1 + f4 ), p2 f2 , p2 f3 , p2 f4 , pf5 , and thus c = [Λ5 M0 : θ−1 ((Λ4 L)⊗3 )] = [hp8 ftop i : hp9 ftop i] = p. Consequently a numerical resolvent of Q is a submodule M of index p in M0 having the property that φ(Λ4 L ⊗ L) ⊆ Λ2 M . Writing M as the kernel of some linear functional ℓ : M0 /pM0 → Fp , the condition is that ℓ lies in the kernel of each of the skew-symmetric bilinear forms obtained by reducing φ(x) ∈ Λ2 M0 mod p for all x ∈ Λ4 L ⊗ L). Let f1′ = p(f1 + f4 ), f2′ = p2 f2 , f3′ = p2 f3 , f4′ = p2 f4 , f5′ = pf5 be the basis elements of M0 listed above. We compute φ(p4 etop e1 ) = (pf1′ − f4′ ) ∧ (pf5′ + f2′ ) φ(p4 etop e2 ) = f2′ ∧ (pf1′ − f4′ + f3′ ) φ(p4 etop e3 ) = f3′ ∧ (pf2′ + f4′ ) φ(p3 etop e5 ) = f5′ ∧ (pf1′ ). So, letting f¯i′ denote the basis vector of M0 /pM0 corresponding to fi′ and f¯i′∗ the corresponding vector of the dual basis, we have ℓ ∈ ker(f¯2′ ∧ f¯4′ ) ∩ ker(f¯2′ ∧ f¯3′ ) ∩ ker(f¯3′ ∧ f¯4′ ) = f¯1′∗ , f¯5′∗ . Since ℓ can take any value in the last-named vector space, up to scaling, we get p + 1 numerical resolvents (and, as it turns out, no nonnumerical ones). Example 4.3. The ring Q = Z ⊕ Z ⊕ Z[x, y]/(x, y)2 is a curious example of Theorem 3.6. Although Q is infinitely far from being maximal (Z ⊕ Z ⊕ Z[n−1 x, n−1 y]/(n−2 (x, y)2 ) is a quintic extension ring for any n > 0), the extensions are only in two directions, as it were, and the resolvent is accordingly unique. Example 4.4. The simplest example of a non-numerical resolvent is given by the ring Q = Z + p2 Z⊕5 = {x1 e1 + · · · + x5 e5 ∈ Z⊕5 : x1 ≡ · · · ≡ x5 mod p2 }. We recognize L = p2 L1 , and we set M = p5 M1 , φ = φ1 \|Λ4 L⊗L , and θ = θ1 \|M , where the subscript 1 denotes the corresponding entity in Example 4.1. Here φ(Λ4 L ⊗ L) = φ(p8 Λ4 L1 ⊗ p2 L1 ) = p10 Λ2 M1 = Λ2 M, while θ(Λ5 M ) = θ(p25 Λ5 M1 ) = p25 (Λ4 L1 )⊗3 ( p24 (Λ4 L1 )⊗3 = (p8 Λ4 L1 )⊗3 = (Λ4 L)⊗3 . The ring Q also has a large number of numerical resolvents, including for instance any supermodule of index p over M . 9 Example 4.5. Very degenerate rings. Over a field K, a very degenerate ring has a multiplication table (7) with a single indeterminate entry u = α2 . By changing basis, we can reduce to the case that u is either α, β, or 0, giving three very degenerate rings up to isomorphism; in Mazzola’s nomenclature [3], they are A18 = K ⊕ K[x, y, z]/(x, y, z)2 , A19 = K[x, y, z]/(x3 , xy, y 2 , xz, yz, z 2) A20 = K[x, y, z, w]/(x, y, z, w)4 . Each of these has a large family of resolvents. Utilizing Bhargava’s representation quadruple of 5 × 5 skew-symmetric matrices, the maps        0 0 0 0 ∗ 0 0 0 0 ∗ 0 0 −1 0 ∗ 0 1 0 0 0 0 0 0 ∗ 0 0 1 0 ∗ 0 0 0 0 ∗ −1 0 0 0        0 0 0 0 ∗ , 0 −1 0 0 ∗ , 1 0 0 0 ∗ ,  0 0 0 0        0 0 0 0 1 0 0 0 0 0 0 0 0 0 0  0 0 0 0 ∗ ∗ ∗ −1 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 (where ∗ represents any  0 0 0 0 0 0 1 0  0 −1 0 0  0 0 0 0 ∗ ∗ ∗ −1 element)   ∗ 0 0 ∗    ∗  , 0 1 0 0 ∗ are resolvents for A18 ,   0 0 0 ∗ 0 0 0 0 0 0 0 ∗    0 0 0 ∗  , 1 0 0 0 0 0 0 0 ∗ ∗ 0 0 ∗ ∗ while −1 0 0 0 ∗ 0 0 0 0 0   ∗ 0 −1 ∗    ∗ , 0 0  0 0 ∗ 1 0 0 0 ∗ 0 0 0 0 ∗ 0 0 0 0 0 of φ as a  ∗  ∗   ∗  0 0  ∗  ∗   ∗  0 0 works for A19 . The trivial ring A20 has an even larger family of resolvents, namely those where φ lands in Λ2 N , for any hyperplane N ⊆ M , or in V ∧ M for any 2-plane V , or where θ = 0. Are these all the resolvents? The classification of resolvents of very degenerate rings, even over fields, is an inviting problem which does not readily yield to the φ -based method of Theorem 3.2. References [1] Manjul Bhargava. Higher composition laws. III. The parametrization of quartic rings. Ann. of Math. (2), 159(3):1329–1360, 2004. [2] Manjul Bhargava. Higher composition laws. IV. The parametrization of quintic rings. Ann. of Math. (2), 167(1):53–94, 2008. [3] Guerino Mazzola. Generic finite schemes and Hochschild cocycles. Comment. Math. Helv., 55(2):267–293, 1980. [4] Evan O’Dorney. Rings of small rank over a Dedekind domain. http://arxiv.org/abs/1508.02777 (accessed 11/10/2015). Awaiting publication. [5] Bjorn Poonen. The moduli space of commutative algebras of finite rank. J. Eur. Math. Soc. (JEMS), 10(3):817–836, 2008. [6] Melanie Matchett Wood. Parametrizing quartic algebras over an arbitrary base. Algebra Number Theory, 5(8):1069–1094, 2011. 10	0
arXiv:1710.07177v1 [cs.CL] 19 Oct 2017 Findings of the Second Shared Task on Multimodal Machine Translation and Multilingual Image Description Desmond Elliott School of Informatics University of Edinburgh d.elliott@ed.ac.uk Stella Frank Centre for Language Evolution University of Edinburgh stella.frank@ed.ac.uk Loı̈c Barrault and Fethi Bougares LIUM University of Le Mans first.last@univ-lemans.fr Lucia Specia Department of Computer Science University of Sheffield l.specia@sheffield.ac.uk Abstract We present the results from the second shared task on multimodal machine translation and multilingual image description. Nine teams submitted 19 systems to two tasks. The multimodal translation task, in which the source sentence is supplemented by an image, was extended with a new language (French) and two new test sets. The multilingual image description task was changed such that at test time, only the image is given. Compared to last year, multimodal systems improved, but text-only systems remain competitive. 1 Introduction The Shared Task on Multimodal Translation and Multilingual Image Description tackles the problem of generating descriptions of images for languages other than English. The vast majority of image description research has focused on Englishlanguage description due to the abundance of crowdsourced resources (Bernardi et al., 2016). However, there has been a significant amount of recent work on creating multilingual image description datasets in German (Elliott et al., 2016; Hitschler et al., 2016; Rajendran et al., 2016), Turkish (Unal et al., 2016), Chinese (Li et al., 2016), Japanese (Miyazaki and Shimizu, 2016; Yoshikawa et al., 2017), and Dutch (van Miltenburg et al., 2017). Progress on this problem will be useful for native-language image search, multilingual ecommerce, and audio-described video for visually impaired viewers. The first empirical results for multimodal translation showed the potential for visual context to improve translation quality (Elliott et al., 2015; Hitschler et al., 2016). This was quickly followed by a wider range of work in the first shared task at WMT 2016 (Specia et al., 2016). The current shared task consists of two subtasks: • Task 1: Multimodal translation takes an image with a source language description that is then translated into a target language. The training data consists of parallel sentences with images. • Task 2: Multilingual image description takes an image and generates a description in the target language without additional source language information at test time. The training data, however, consists of images with independent descriptions in both source and target languages. The translation task has been extended to include a new language, French. This extension means the Multi30K dataset (Elliott et al., 2016) is now triple aligned, with English descriptions translated into both German and French. The description generation task has substantially changed since last year. The main difference is that source language descriptions are no longer observed for test images. This mirrors the realworld scenario in which a target-language speaker wants a description of image that does not already have source language descriptions associated with it. The two subtasks are now more distinct because multilingual image description requires the use of the image (no text-only system is possible because the input contains no text). Another change for this year is the introduction of two new evaluation datasets: an extension of the existing Multi30K dataset, and a “teaser” evaluation dataset with images carefully chosen to contain ambiguities in the source language. This year we encouraged participants to submit systems using unconstrained data for both tasks. Training on additional out-of-domain data is underexplored for these tasks. We believe this setting will be critical for future real-world improvements, given that the current training datasets are small and expensive to construct.1 2 2.1 Tasks & Datasets Tasks The Multimodal Translation task (Task 1) follows the format of the 2016 Shared Task (Specia et al., 2016). The Multilingual Image Description Task (Task 2) is new this year but it is related to the Crosslingual Image Description task from 2016. The main difference between the Crosslingual Image Description task and the Multilingual Image Description task is the presence of source language descriptions. In last year’s Crosslingual Image Description task, the aim was to produce a single target language description, given five source language descriptions and the image. In this year’s Multilingual Image Description task, participants received only an unseen image at test time, without source language descriptions. 2.2 En: A group of people are eating noddles. De: Eine Gruppe von Leuten isst Nudeln. Fr: Un groupe de gens mangent des nouilles. Datasets The Multi30K dataset (Elliott et al., 2016) is the primary dataset for the shared task. It contains 31K images originally described in English (Young et al., 2014) with two types of multilingual data: a collection of professionally translated German sentences, and a collection of independently crowdsourced German descriptions. This year the Multi30K dataset has been extended with new evaluation data for the Translation and Image Description tasks, and an additional language for the Translation task. In addition, we released a new evaluation dataset featuring ambiguities that we expected would benefit from visual context. Table 1 presents an overview of the new evaluation datasets. Figure 1 shows an example of an image with an aligned English-German-French description. In addition to releasing the parallel text, we also distributed two types of ResNet-50 visual features 1 All of the data and results are available at http://www. statmt.org/wmt17/multimodal-task.html Figure 1: Example of an image with a source description in English, together with German and French translations. (He et al., 2016) for all of the images, namely the ‘res4 relu’ convolutional features (which preserve the spatial location of a feature in the original image) and averaged pooled features. Multi30K French Translations We extended the translation data in Multi30K dataset with crowdsourced French translations. The crowdsourced translations were collected from 12 workers using an internal platform. We estimate the translation work had a monetary value of e9,700. The translators had access to the source segment, the image and an automatic translation created with a standard phrase-based system (Koehn et al., 2007) trained on WMT’15 parallel text. The automatic translations were presented to the crowdworkers to further simplify the crowdsourcing task. We note that this did not end up being a post-editing task, that is, the translators did not simply copy and paste the suggested translations. To demonstrate this, we calculated text-similarity metric scores between the phrase-based system outputs and the human translations on the training corpus, resulting in 0.41 edit distance (measured using the TER metric), meaning that more than 40% of the words between these two versions do not match. Multi30K 2017 test data We collected new evaluation data for the Multi30K dataset. We sampled new images from five of the six Flickr groups used to create the original Flickr30K dataset using MMFeat (Kiela, 2016)2 . 2 Strangers!, Wild Child, Dogs in Action, Action Photography, and Outdoor Activities. Training set Development set Images Sentences Images Sentences Translation 29,000 29,000 1,014 1,014 Description 29,000 145,000 1,014 5,070 2017 test COCO Images Sentences Images Sentences Translation 1,000 1,000 461 461 Description 1,071 5,355 — Table 1: Overview of the Multi30K training, development, 2017 test, and Ambiguous COCO datasets. Task 1 Task 2 Strangers! 150 154 Wild Child 83 83 remaining 1,071 images were used for the Multilingual Image Description task. We collected five additional independent German descriptions of those images from Crowdflower. Dogs in Action 78 92 Ambiguous COCO Action Photography 238 259 Flickr Social Club 241 263 Everything Outdoor 206 214 Outdoor Activities 4 6 As a secondary evaluation dataset for the Multimodal Translation task, we collected and translated a set of image descriptions that potentially contain ambiguous verbs. We based our selection on the VerSe dataset (Gella et al., 2016), which annotates a subset of the COCO (Lin et al., 2014) and TUHOI (Le et al., 2014) images with OntoNotes senses for 90 verbs which are ambiguous, e.g. play. Their goals were to test the feasibility of annotating images with the word sense of a given verb (rather than verbs themselves) and to provide a gold-labelled dataset for evaluating automatic visual sense disambiguation methods. Altogether, the VerSe dataset contains 3,518 images, but we limited ourselves to its COCO section, since for our purposes we also need the image descriptions, which are not available in TUHOI. The COCO portion covers 82 verbs; we further discarded verbs that are unambiguous in the dataset, i.e. although some verbs have multiple senses in OntoNotes, they all occur with one sense in VerSe (e.g. gather is used in all instances to describe the ‘people gathering’ sense), resulting in 57 ambiguous verbs (2,699 images). The actual descriptions of the images were not distributed with the VerSe dataset. However, given that the ambiguous verbs were selected based on the image descriptions, we assumed that in all cases at least one of the original COCO description (out of the five per image) should contain the ambiguous verb. In cases where more than one description contained the verb, we Group Table 2: Distribution of images in the Multi30K 2017 test data by Flickr group. We sampled additional images from two thematically related groups (Everything Outdoor and Flickr Social Club) because Outdoor Activities only returned 10 new CC-licensed images and Flickr-Social no longer exists. Table 2 shows the distribution of images across the groups and tasks. We initially downloaded 2,000 images per Flickr group, which were then manually filtered by three of the authors. The filtering was done to remove (near) duplicate images, clearly watermarked images, and images with dubious content. This process resulted in a total of 2,071 images. We crowdsourced five English descriptions of each image from Crowdflower3 using the same process as Elliott et al. (2016). One of the authors selected 1,000 images from the collection to form the dataset for the Multimodal Translation task based on a manual inspection of the English descriptions. Professional German translations were collected for those 1,000 English-described images. The 3 http://www.crowdflower.com En: A man on a motorcycle is passing another vehicle. De: Ein Mann auf einem Motorrad fährt an einem anderen Fahrzeug vorbei. Fr: Un homme sur une moto dépasse un autre véhicule. En: A red train is passing over the water on a bridge De: Ein roter Zug fährt auf einer Brücke über das Wasser Fr: Un train rouge traverse l’eau sur un pont. Figure 2: Two senses of the English verb ”to pass” in their visual contexts, with the original English and the translations into German and French. The verb and its translations are underlined. randomly selected one such description to be part of the dataset of descriptions containing ambiguous verbs. This resulted in 2,699 descriptions. As a consequence of the original goals of the VerSe dataset, each sense of each ambiguous verb was used multiple times in the dataset, which resulted in many descriptions with the same sense, for example, 85 images (and descriptions) were available for the verb show, but they referred to a small set of senses of the verb. The number of images (and therefore descriptions) per ambiguous verb varied from 6 (stir) to 100 (pull, serve). Since our intention was to have a small but varied dataset, we selected a subset of a subset of descriptions per ambiguous verb, aiming at keeping 1-3 instances per sense per verb. This resulted in 461 descriptions for 56 verbs in total, ranging from 3 (e.g. shake, carry) to 26 (reach) (the verb lay/lie was excluded as it had only one sense). We note that the descriptions include the use of the verbs in phrasal verbs. Two examples of the English verb “to pass” are shown in Figure 2. In the German translations, the source language verb did not require disambiguation (both German translations use the verb “fährt”), whereas in the French translations, the verb was disambiguated into “dépasse” and “traverse”, respectively. 3 Participants This year we attracted submissions from nine different groups. Table 3 presents an overview of the groups and their submission identifiers. AFRL-OHIOSTATE (Task 1) The AFRLOHIOSTATE system submission is an atypical Machine Translation (MT) system in that the image is the catalyst for the MT results, and not the textual content. This system architecture assumes an image caption engine can be trained in a target language to give meaningful output in the form of a set of the most probable n target language candidate captions. A learned mapping function of the encoded source language caption to the corresponding encoded target language captions is then employed. Finally, a distance function is applied to retrieve the “nearest” candidate caption to be the translation of the source caption. CMU (Task 2) The CMU submission uses a multi-task learning technique, extending the baseline so that it generates both a German caption and an English caption. First, a German caption is generated using the baseline method. After the LSTM for the baseline model finishes producing a German caption, it has some final hidden state. Decoding is simply resumed starting from that final state with an independent decoder, separate vocabulary, and this time without any direct access to the image. The goal is to encourage the model to keep information about the image in the hidden state throughout the decoding process, hopefully improving the model output. Although the model is trained to produce both German and English cap- ID AFRL-OHIOSTATE Participating team Air Force Research Laboratory & Ohio State University (Duselis et al., 2017) CMU Carnegie Melon University (Jaffe, 2017) CUNI Univerzita Karlova v Praze (Helcl and Libovický, 2017) DCU-ADAPT Dublin City University (Calixto et al., 2017a) LIUMCVC Laboratoire d’Informatique de l’Université du Maine & Universitat Autonoma de Barcelona Computer Vision Center (Caglayan et al., 2017) NICT National Institute of Information and Communications Technology & Nara Institute of Science and Technology (Zhang et al., 2017) OREGONSTATE SHEF UvA-TiCC Oregon State University (Ma et al., 2017) University of Sheffield (Madhyastha et al., 2017) Universiteit van Amsterdam & Tilburg University (Elliott and Kádár, 2017) Table 3: Participants in the WMT17 multimodal machine translation shared task. tions, at evaluation time the English component of the model is ignored and only German captions are generated. den state. Each image has one corresponding feature vector, obtained from the activations of the FC7 layer of the VGG19 network, and consist of a 4096D real-valued vector that encode information about the entire image. CUNI (Tasks 1 and 2) For Task 1, the submissions employ the standard neural MT (NMT) scheme enriched with another attentive encoder for the input image. It uses a hierarchical attention combination in the decoder (Libovický and Helcl, 2017). The best system was trained with additional data obtained from selecting similar sentences from parallel corpora and by back-translation of similar sentences found in the SDEWAC corpus (Faaß and Eckart, 2013). The submission to Task 2 is a combination of two neural models. The first model generates an English caption from the image. The second model is a text-only NMT model that translates the English caption to German. LIUMCVC (Task 1) LIUMCVC experiment with two approaches: a multimodal attentive NMT with separate attention (Caglayan et al., 2016) over source text and convolutional image features, and an NMT where global visual features (2048dimensional pool5 features from ResNet-50) are multiplicatively interacted with word embeddings. More specifically, each target word embedding is multiplied with global visual features in an elementwise fashion in order to visually contextualize word representations. With 128-dimensional embeddings and 256-dimensional recurrent layers, the resulting models have around 5M parameters. DCU-ADAPT (Task 1) This submission evaluates ensembles of up to four different multimodal NMT models. All models use global image features obtained with the pre-trained CNN VGG19, and are either incorporated in the encoder or the decoder. These models are described in detail in (Calixto et al., 2017b). They are model IMGW , in which image features are used as words in the source-language encoder; model IMGE , where image features are used to initialise the hidden states of the forward and backward encoder RNNs; and model IMGD , where the image features are used as additional signals to initialise the decoder hid- NICT (Task 1) These are constrained submissions for both language pairs. First, a hierarchical phrase-based (HPB) translation system s built using Moses (Koehn et al., 2007) with standard features. Then, an attentional encoder-decoder network (Bahdanau et al., 2015) is trained and used as an additional feature to rerank the n-best output of the HPB system. A unimodal NMT model is also trained to integrate visual information. Instead of integrating visual features into the NMT model directly, image retrieval methods are employed to obtain target language descriptions of images that are similar to the image described by the source sentence, and this target description information is integrated into the NMT model. A multimodal NMT model is also used to rerank the HPB output. All feature weights (including the standard features, the NMT feature and the multimodal NMT feature) were tuned by MERT (Och, 2003). On the development set, the NMT feature improved the HPB system significantly. However, the multimodal NMT feature did not further improve the HPB system that had integrated the NMT feature. OREGONSTATE (Task 1) The OREGONSTATE system uses a very simple but effective model which feeds the image information to both encoder and decoder. On the encoder side, the image representation was used as an initialization information to generate the source words’ representations. This step strengthens the relatedness between image’s and source words’ representations. Additionally, the decoder uses alignment to source words by a global attention mechanism. In this way, the decoder benefits from both image and source language information and generates more accurate target side sentence. UvA-TiCC (Task 1) The submitted systems are Imagination models (Elliott and Kádár, 2017), which are trained to perform two tasks in a multitask learning framework: a) produce the target sentence, and b) predict the visual feature vector of the corresponding image. The constrained models are trained over only the 29,000 training examples in the Multi30K dataset with a source-side vocabulary of 10,214 types and a target-side vocabulary of 16,022 types. The unconstrained models are trained over a concatenation of the Multi30K, News Commentary (Tiedemann, 2012) parallel texts, and MS COCO (Chen et al., 2015) dataset with a joint source-target vocabulary of 17,597 word pieces (Schuster and Nakajima, 2012). In both constrained and unconstrained submissions, the models were trained to predict the 2048D GoogleLeNetV3 feature vector (Szegedy et al., 2015) of an image associated with a source language sentence. The output of an ensemble of the three best randomly initialized models - as measured by BLEU on the Multi30K development set - was used for both the constrained and unconstrained submissions. SHEF (Task 1) The SHEF systems utilize the predicted posterior probability distribution over the image object classes as image features. To do so, they make use of the pre-trained ResNet-152 (He et al., 2016), a deep CNN based image network that is trained over the 1,000 object categories on the Imagenet dataset (Deng et al., 2009) to obtain the posterior distribution. The model follows a standard encoder-decoder NMT approach using softdot attention as described in (Luong et al., 2015). It explores image information in three ways: a) to initialize the encoder; b) to initialize the decoder; c) to condition each source word with the image class posteriors. In all these three ways, non-linear affine transformations over the posteriors are used as image features. Baseline — Task 1 The baseline system for the multimodal translation task is a text-only neural machine translation system built with the Nematus toolkit (Sennrich et al., 2017). Most settings and hyperparameters were kept as default, with a few exceptions: batch size of 40 (instead of 80 due to memory constraints) and ADAM as optimizer. In order to handle rare and OOV words, we used the Byte Pair Encoding Compression Algorithm to segment words (Sennrich et al., 2016b). The merge operations for word segmentation were learned using training data in both source and target languages. These were then applied to all training, validation and test sets in both source and target languages. In post-processing, the original words were restored by concatenating the subwords. Baseline — Task 2 The baseline for the multilingual image description task is an attention-based image description system trained over only the German image descriptions (Caglayan et al., 2017). The visual representation are extracted from the so-called res4f relu layer from a ResNet-50 (He et al., 2016) convolutional neural network trained on the ImageNet dataset (Russakovsky et al., 2015). Those feature maps provide spatial information on which the model focuses through the attention mechanism. 4 Text-similarity Metric Results The submissions were evaluated against either professional or crowd-sourced references. All submissions and references were pre-processed to lowercase, normalise punctuation, and tokenise the sentences using the Moses scripts.4 The evaluation was performed using MultEval (Clark et al., 2011) with the primary metric of Meteor 4 https://github.com/moses-smt/ mosesdecoder/blob/master/scripts/ 1.5 (Denkowski and Lavie, 2014). We also report the results using BLEU (Papineni et al., 2002) and TER (Snover et al., 2006) metrics. The winning submissions are indicated by •. These are the topscoring submissions and those that are not significantly different (based on Meteor scores) according the approximate randomisation test (with p-value ≤ 0.05) provided by MultEval. Submissions marked with * are not significantly different from the Baseline according to the same test. 4.1 Task 1: English → German 4.1.1 Multi30K 2017 test data Table 4 shows the results on the Multi30K 2017 test data with a German target language. It interesting to note that the metrics do not fully agree on the ranking of systems, although the four best (statistically indistinguishable) systems win by all metrics. All-but-one submission outperformed the text-only NMT baseline. This year, the best performing systems include both multimodal (LIUMCVC MNMT C and UvA-TiCC IMAGINATION U) and text-only (NICT NMTrerank C and LIUMCVC MNMT C) submissions. (Strictly speaking, the UvATiCC IMAGINATION U system is incomparable because it is an unconstrained system, but all unconstrained systems perform in the same range as the constrained systems.) 4.1.2 Ambiguous COCO Table 5 shows the results for the out-of-domain ambiguous COCO dataset with a German target language. Once again the evaluation metrics do not fully agree on the ranking of the submissions. It is interesting to note that the metric scores are lower for the out-of-domain Ambiguous COCO data compared to the in-domain Multi30K 2017 test data. However, we cannot make definitive claims about the difficulty of the dataset because the Ambiguous COCO dataset contains fewer sentences than the Multi30K 2017 test data (461 compared to 1,000). The systems are mostly in the same order as on the Multi30K 2017 test data, with the same four systems performing best. However, two systems (DCU-ADAPT MultiMT C and OREGONSTATE 1NeuralTranslation C) are ranked higher on this test set than on the in-domain Flickr dataset, indicating that they are relatively more robust and possibly better at resolving the ambiguities found in the Ambiguous COCO dataset. 4.2 4.2.1 Task 1: English → French Multi30K Test 2017 Table 6 shows the results for the Multi30K 2017 test data with French as target language. A reduced number of submissions were received for this new language pair, with no unconstrained systems. In contrast to the English→German results, the evaluation metrics are in better agreement about the ranking of the submissions. Translating from English→French is an easier task than English→German systems, as reflected in the higher metric scores. This also includes the baseline systems where English→French results in 63.1 Meteor compared to 41.9 for English→German. Eight out of the ten submissions outperformed the English→French baseline system. Two of the best submissions for English→German remain the best for English→French (LIUMCVC MNMT C and NICT NMTrerank C), the text-only system (LIUMCVC NMT C) decreased in performance, and no UvA-TiCC IMAGINATION U system was submitted for French. An interesting observation is the difference of the Meteor scores between text-only NMT system (LIUMCVC NMT C) and Moses hierarchical phrase-based system with reranking (NICT NMTrerank C). While the two systems are very close for the English→German direction, the hierarchical system is better than the text-only NMT systems in the English→French direction. This pattern holds for both the Multi30K 2017 test data and Ambiguous COCO test data. 4.2.2 Ambiguous COCO Table 7 shows the results for the out-of-domain Ambiguous COCO dataset with the French target language. Once again, in contrast to the English→German results, the evaluation metrics are in better agreement about the ranking of the submissions. The performance of all the models is once again in mostly agreement with the Multi30K 2017 test data, albeit lower. Both DCU-ADAPT MultiMT C and OREGONSTATE 2NeuralTranslation C again perform relatively better on this dataset. 4.3 Task 2: English → German The description generation task, in which systems must generate target-language (German) captions for a test image, has substantially changed since •LIUMCVC MNMT C •NICT NMTrerank C •LIUMCVC NMT C •UvA-TiCC IMAGINATION U UvA-TiCC IMAGINATION C CUNI NeuralMonkeyTextualMT U OREGONSTATE 2NeuralTranslation C DCU-ADAPT MultiMT C CUNI NeuralMonkeyMultimodalMT U CUNI NeuralMonkeyTextualMT C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyMultimodalMT C SHEF ShefClassInitDec C SHEF ShefClassProj C Baseline (text-only NMT) AFRL-OHIOSTATE-MULTIMODAL U BLEU ↑ Meteor ↑ TER ↓ 33.4 31.9 33.2 33.3 30.2 31.1 31.0 29.8 29.5 28.5 29.7 25.8 25.0 24.2 19.3 6.5 54.0 53.9 53.8 53.5 51.2 51.0 50.6 50.5 50.2 49.2 48.9 47.1 44.5 43.4 41.9 20.2 48.5 48.1 48.2 47.5 50.8 50.7 50.7 52.3 52.5 54.3 51.6 56.3 53.8 55.9 72.2 87.4 Table 4: Official results for the WMT17 Multimodal Machine Translation task on the English-German Multi30K 2017 test data. Systems with grey background indicate use of resources that fall outside the constraints provided for the shared task. •LIUMCVC NMT C •LIUMCVC MNMT C •NICT 1 NMTrerank C •UvA-TiCC IMAGINATION U DCU-ADAPT MultiMT C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyTextualMT U UvA-TiCC IMAGINATION C OREGONSTATE 2NeuralTranslation C CUNI NeuralMonkeyMultimodalMT U CUNI NeuralMonkeyTextualMT C CUNI NeuralMonkeyMultimodalMT C SHEF ShefClassInitDec C SHEF ShefClassProj C Baseline (text-only NMT) BLEU ↑ Meteor ↑ TER ↓ 28.7 28.5 28.1 28.0 26.4 27.4 26.6 26.4 26.1 25.7 23.2 22.4 21.4 21.0 18.7 48.9 48.8 48.5 48.1 46.8 46.5 46.0 45.8 45.7 45.6 43.8 42.7 40.7 40.0 37.6 52.5 53.4 52.9 52.4 54.5 52.3 54.8 55.4 55.9 55.7 59.8 60.1 56.5 57.8 66.1 Table 5: Results for the Multimodal Translation task on the English-German Ambiguous COCO dataset. Systems with grey background indicate use of resources that fall outside the constraints provided for the shared task. •LIUMCVC MNMT C •NICT NMTrerank C DCU-ADAPT MultiMT C LIUMCVC NMT C OREGONSTATE 2NeuralTranslation C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyMultimodalMT C CUNI NeuralMonkeyTextualMT C Baseline (text-only NMT) SHEF ShefClassInitDec C SHEF ShefClassProj C BLEU ↑ Meteor ↑ TER ↓ 55.9 55.3 54.1 53.3 51.9 51.0 49.9 50.3 44.3 45.0 43.6 72.1 72.0 70.1 70.1 68.3 67.2 67.2 67.0 63.1 62.8 61.5 28.4 28.4 30.0 31.7 32.7 33.6 34.3 33.6 39.6 38.4 40.5 Table 6: Results for the Multimodal Translation task on the English-French Multi30K Test 2017 data. •LIUMCVC MNMT C •NICT NMTrerank C •DCU-ADAPT MultiMT C OREGONSTATE 2NeuralTranslation C LIUMCVC NMT C CUNI NeuralMonkeyTexutalMT C CUNI NeuralMonkeyMultimodalMT C OREGONSTATE 1NeuralTranslation C SHEF ShefClassInitDec C SHEF ShefClassProj C Baseline (text-only NMT) BLEU ↑ Meteor ↑ TER ↓ 45.9 45.1 44.5 44.1 43.6 43.0 42.9 41.2 37.2 36.8 35.1 65.9 65.6 64.1 63.8 63.4 62.5 62.5 61.6 57.3 57.0 55.8 34.2 34.7 35.2 36.7 37.4 38.2 38.2 37.8 42.4 44.5 45.8 Table 7: Results for the Multimodal Translation task on the English-French Ambiguous COCO dataset. Baseline (target monolingual) CUNI NeuralMonkeyCaptionAndMT C CUNI NeuralMonkeyCaptionAndMT U CMU NeuralEncoderDecoder C CUNI NeuralMonkeyBilingual C BLEU ↑ Meteor ↑ TER ↓ 9.1 4.2 6.5 9.1 2.3 23.4 22.1 20.6 19.8 17.6 91.4 133.6 91.7 63.3 112.6 Table 8: Results for the Multilingual Image Description task on the English-German Multi30K 2017 test data. last year. The main difference is that source language descriptions are no longer observed for images at test time. The training data remains the same and contains images with both source and target language descriptions. The aim is thus to leverage multilingual training data to improve a monolingual task. Table 8 shows the results for the Multilingual image description task. This task attracted fewer submissions than last year, which may be because it was no longer possible to re-use a model designed for Multimodal Translation. The evaluation metrics do not agree on the ranking of the submissions, with major differences in the ranking using either BLEU or TER instead of Meteor. The main result is that none of the submissions outperform the monolingual German baseline according to Meteor. All of the submissions are statistically significantly different compared to the baseline. However, the CMU NeuralEncoderDecoder C submission marginally outperformed the baseline according to TER and equalled its BLEU score. asked to evaluate the semantic relatedness between the source sentence in English and the target sentence in German or French. The image was shown along with the source sentence and the candidate translation and evaluators were told to rely on the image when necessary to obtain a better understanding of the source sentence (e.g. in cases where the text was ambiguous). Note that the reference sentence is not displayed during the evaluation, in order to avoid influencing the assessor. Figure 3 shows an example of the direct assessment interface used in the evaluation. The score of each translation candidate ranges from 0 (meaning that the meaning of the source is not preserved in the target language sentence) to 100 (meaning the meaning of the source is “perfectly” preserved). The human assessment scores are standardized according to each individual assessor’s overall mean and standard deviation score. The overall score of a given system (z) corresponds to the mean standardized score of its translations. 5 The French outputs were evaluated by seven assessors, who conducted a total of 2,521 DAs, resulting in a minimum of 319 and a maximum of 368 direct assessments per system submission, respectively. The German outputs were evaluated by 25 assessors, who conducted a total of 3,485 DAs, resulting in a minimum of 291 and a maximum of 357 direct assessments per system submission, respectively. This is somewhat less than the recommended number of 500, so the results should be considered preliminary. Tables 9 and 10 show the results of the human evaluation for the English to German and the English to French Multimodal Translation task (Multi30K 2017 test data). The systems are ordered by standardized mean DA scores and clustered ac- Human Judgement Results This year, we conducted a human evaluation in addition to the text-similarity metrics to assess the translation quality of the submissions. This evaluation was undertaken for the Task 1 German and French outputs for the Multi30K 2017 test data. This section describes how we collected the human assessments and computed the results. We would like to gratefully thank all assessors. 5.1 Methodology The system outputs were manually evaluated by bilingual Direct Assessment (DA) (Graham et al., 2015) using the Appraise platform (Federmann, 2012). The annotators (mostly researchers) were 5.2 Results Figure 3: Example of the human direct assessment evaluation interface. cording to the Wilcoxon signed-rank test at p-level p ≤ 0.05. Systems within a cluster are considered tied. The Wilcoxon signed-rank scores can be found in Tables 11 and 12 in Appendix A. When comparing automatic and human evaluations, we can observe that they globally agree with each other, as shown in Figures 4 and 5, with German showing better agreement than French. We point out two interesting disagreements: First, in the English→French language pair, CUNI NeuralMonkeyMultimodalMT C and DCUADAPT MultiMT C are significantly better than LIUMCVC MNMT C, despite the fact that the latter system achieves much higher metric scores. Secondly, across both languages, the text-only LIUMCVC NMT C system performs well on metrics but does relatively poorly on human judgements, especially as compared to the multimodal version of the same system. 6 Discussion Visual Features: do they help? Three teams provided text-only counterparts to their multimodal systems for Task 1 (CUNI, LIUMCVC, and OREGONSTATE), which enables us to evaluate the contribution of visual features. For many systems, visual features did not seem to help reliably, at least as measured by metric evaluations: in German, the CUNI and OREGONSTATE text-only systems outperformed the counterparts, while in French, there were small improvements for the CUNI multimodal system. However, the LIUMCVC multimodal system outperformed their text-only system across both languages. The human evaluation results are perhaps more promising: nearly all the highest ranked systems (with the exception of NICT) are multimodal. An intruiging result was the text-only LIUMCVC NMT C, which ranked highly on metrics but poorly in the human evaluation. The LIUMCVC systems were indistinguishable from each other in terms of Meteor scores but the standardized mean direct assessment score showed a significant difference in performance (see Tables 11 and 12): further analysis of the reasons for humans disliking the text-only translations will be necessary. The multimodal Task 1 submissions can be broadly categorised into three groups based on how they use the images: approaches useing double-attention mechanisms, initialising the hidden state of the encoder and/or decoder networks with the global image feature vector, and alternative uses of image features. The doubleattention models calculate context vectors over the source language hidden states and locationpreserving feature vectors over the image; these vectors are used as inputs to the translation decoder (CUNI NeuralMonkeyMultimodalMT). Encoder and/or decoder initialisation involves initialising the recurrent neural network with an affine transformation of a global image feature vector (DCU-ADAPT MultiMT, OREGONSTATE 1NeuralTranslation) or initialising the encoder and decoder with the 1000 dimension softmax probability vector over the object classes in ImageNet object recognition challenge NICT NMTrerank C 54 LIUMCVC NMT C LIUMCVC MNMT C UvA-TiCC IMAGINATION U 52 Meteor 50 48 UvA-TiCC IMAGINATION C CUNI NeuralMonkeyTextualMT U OREGONSTATE 2NeuralTranslation C DCU-ADAPT MultiMT C CUNI NeuralMonkeyMultimodalMT U CUNI NeuralMonkeyTextualMT C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyMultimodalMT C 46 SHEF ShefClassInitDec C 44 42 40 −1 SHEF ShefClassProj C Baseline −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 Human judgments (z) 0.6 0.8 1 Figure 4: System performance on the English→German Multi30K 2017 test data as measured by human evaluation against Meteor scores. The AFRL-OHIOSTATE-MULTIMODAL U system has been ommitted for readability. 74 LIUMCVC MNMT C 72 NICT NMTrerank C LIUMCVC NMT C Meteor 70 DCU-ADAPT MultiMT C OREGONSTATE 2NeuralTranslation C 68 66 CUNI NeuralMonkeyMultimodalMT C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyTextualMT C 64 Baseline 62 60 −1 SHEF ShefClassInitDec C SHEF ShefClassProj C −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 Human judgments (z) 0.6 0.8 1 Figure 5: System performance on the English→French Multi30K 2017 test data as measured by human evaluation against Meteor scores. # Raw z English→German System 1 77.8 0.665 LIUMCVC MNMT C 2 74.1 0.552 UvA-TiCC IMAGINATION U 3 70.3 68.1 68.1 65.1 60.6 59.7 55.9 54.4 54.2 53.3 49.4 46.6 0.437 0.325 0.311 0.196 0.136 0.08 -0.049 -0.091 -0.108 -0.144 -0.266 -0.37 NICT NMTrerank C CUNI NeuralMonkeyTextualMT U DCU-ADAPT MultiMT C LIUMCVC NMT C CUNI NeuralMonkeyMultimodalMT U UvA-TiCC IMAGINATION C CUNI NeuralMonkeyMultimodalMT C OREGONSTATE 2NeuralTranslation C CUNI NeuralMonkeyTextualMT C OREGONSTATE 1NeuralTranslation C SHEF ShefClassProj C SHEF ShefClassInitDec C 15 39.0 36.6 -0.615 -0.674 Baseline (text-only NMT) AFRL-OHIOSTATE MULTIMODAL U Table 9: Results of the human evaluation of the WMT17 English-German Multimodal Translation task (Multi30K 2017 test data). Systems are ordered by standardized mean DA scores (z) and clustered according to Wilcoxon signed-rank test at p-level p ≤ 0.05. Systems within a cluster are considered tied, although systems within a cluster may be statistically significantly different from each other (see Table 11). Systems using unconstrained data are identified with a gray background. (SHEF ShefClassInitDec). The alternative uses of the image features include element-wise multiplication of the target language embeddings with an affine transformation of a global image feature vector (LIUMCVC MNMT), summing the source language word embeddings with affine-transformed 1000 dimension softmax probability vector (SHEF ShefClassProj), using the visual features in a retrieval framework (AFRL-OHIOSTATE MULTIMODAL), and learning visually-grounded encoder representations by learning to predict the global image feature vector from the source language hidden states (UvATiCC IMAGINATION). Overall, the metric and human judgement results in Sections 4 and 5 indicate that there is still a wide scope for exploration of the best way to integrate visual and textual information. In particular, the alternative approaches proposed in the LIUMCVC MNMT and UvA-TiCC IMAGINATION submissions led to strong performance in both the metric and human judgement results, surpassing the more common approaches using initialisation and double attention. Finally, the text-only NICT system ranks highly across both languages. This system uses hierarchical phrase-based MT with a reranking step based on a neural text-only system, since their multimodal system never outperformed the text-only variant in development (Zhang et al., 2017). This is in line with last year’s results and the strong Moses baseline (Specia et al., 2016), and suggests a continuing role for phrase-based MT for small homogeneous datasets. Unconstrained systems The Multi30k dataset is relatively small, so unconstrained systems use more data to complement the image description translations. Three groups submitted systems using external resources: UvA-TiCC, CUNI, and AFRLOHIOSTATE. The unconstrained UvA-TiCC and CUNI submissions always outperformed their respective constrained variants by 2–3 Meteor points and achieved higher standardized mean DA scores. These results suggest that external parallel text corpora (UvA-TiCC and CUNI) and external monolingual image description datasets (UvA-TiCC) can usefully improve the quality of multimodal translation models. However, tuning to the target domain remains important, even for relatively simple image captions. # Raw z English→French System 1 79.4 74.2 74.1 0.446 0.307 0.3 NICT NMTrerank C CUNI NeuralMonkeyMultimodalMT C DCU-ADAPT MultiMT C 4 71.2 65.4 61.9 60.8 60.5 0.22 0.056 -0.041 -0.078 -0.079 LIUMCVC MNMT C OREGONSTATE 2NeuralTranslation C CUNI NeuralMonkeyTextualMT C OREGONSTATE 1NeuralTranslation C LIUMCVC NMT C 9 54.7 54.0 -0.254 -0.282 SHEF ShefClassInitDec C SHEF ShefClassProj C 11 44.1 -0.539 Baseline (text-only NMT) Table 10: Results of the human evaluation of the WMT17 English-French Multimodal Translation task (Multi30K 2017 test data). Systems are ordered by standardized mean DA score (z) and clustered according to Wilcoxon signed-rank test at p-level p ≤ 0.05. Systems within a cluster are considered tied, although systems within a cluster may be statistically significantly different from each other (see Table 12). We ran the best-performing English→German WMT’16 news translation system (Sennrich et al., 2016a) on the English→German Multi30K 2017 test data to gauge the performance of a state-ofthe-art text-only translation system trained on only out-of-domain resources5 . It ranked 10th in terms of Meteor (49.9) and 11th in terms of BLEU (29.0), placing it firmly in the middle of the pack, and below nearly all the text-only submissions trained on the in-domain Multi30K dataset. The effect of OOV words The Multi30k translation training and test data are very similar, with a low OOV rate in the Flickr test set (1.7%). In the 2017 test set, 16% of English test sentences include a OOV word. Human evaluation gave the impression that these often led to errors propagated throughout the whole sentence. Unconstrained systems may perform better by having larger vocabularies, as well as more robust statistics. When we evaluate the English→German systems over only the 161 OOV-containing test sentences, the highest ranked submission by all metrics is the unconstrained UvA-TiCC IMAGINATION submission, with +2.5 Metor and +2.2 BLEU over the second best system (LIUMCVC NMT; 45.6 vs 43.1 Meteor and 24.0 vs 21.8 BLEU). The difference over non-OOV-containing sen5 http://data.statmt.org/rsennrich/ wmt16_systems/en-de/ tences is not nearly as stark, with constrained systems all performing best (both LIUMCVC systems, MNMT and NMT, with 56.6 and 56.3 Meteor, respectively) but unconstrained systems following close behind (UvA-TiCC with 55.4 Meteor, CUNI with 53.4 Meteor). Ambiguous COCO dataset We introduced a new evaluation dataset this year with the aim of testing systems’ ability to use visual features to identify word senses. However, it is unclear whether visual features improve performance on this test set. The text-only NICT NMTrerank system performs competitively, ranking in the top three submissions for both languages. We find mixed results for submissions with text-only and multimodal counterparts (CUNI, LIUMCVC, OREGONSTATE): LIUMCVC’s multimodal system improves over the text-only system for French but not German, while the visual features help for German but not French in the CUNI and OREGONSTATE systems. We plan to perform a further analysis on the extent of translation ambiguity in this dataset. We will also continue to work on other methods for constructing datasets in which textual ambiguity can be disambiguated by visual information. Multilingual Image Description It proved difficult for Task 2 systems to use the English data to improve over the monolingual German baseline. In future iterations of the task, we will consider a lopsided data setting, in which there is much more English data than target language data. This setting is more realistic and will push the use of multilingual data. We also hope to conduct human evaluation to better assess performance because automatic metrics are problematic for this task (Elliott and Keller, 2014; Kilickaya et al., 2017). No. ANR-15-CHR2-0006-01 – Loı̈c Barrault and Fethi Bougares), and by the MultiMT project (EU H2020 ERC Starting Grant No. 678017 – Lucia Specia). Desmond Elliott acknowledges the support of NWO Vici Grant No. 277-89-002 awarded to K. Sima’an, and an Amazon Academic Research Award. We thank Josiah Wang for his help in selecting the Ambiguous COCO dataset. 7 A Significance tests Conclusions We presented the results of the second shared task on multimodal translation and multilingual image description. The shared task attracted submissions from nine groups, who submitted a total of 19 systems across the tasks. The Multimodal Translation task attracted the majority of the submissions. Human judgements for the translation task were collected for the first time this year and ranked systems broadly in line with the automatic metrics. The main findings of the shared task are: (i) There is still scope for novel approaches to integrating visual and linguistic features in multilingual multimodal models, as demonstrated by the winning systems. (ii) External resources have an important role to play in improving the performance of multimodal translation models beyond what can be learned from limited training data. (iii) The differences between text-only and multimodal systems are being obfuscated by the well-known shortcomings of text-similarity metrics. Multimodal systems often seem to be prefered by humans but not rewarded by metrics. Future research on this topic, encompassing both multimodal translation and multilingual image description, should be evaluated using human judgements. In future editions of the task, we will encourage participants to submit the output of single decoder systems to better understand the empirical differences between approaches. We are also considering a Multilingual Multimodal Translation challenge, where the systems can observe two language inputs alongside the image to encourage the development of multi-source multimodal models. Acknowledgements This work was supported by the CHIST-ERA M2CR project (French National Research Agency Tables 11 and 12 show the Wilcoxon signed-rank test used to create the clustering of the systems. UvA-TiCC IMAGINATION U 2.8e-2 3.1e-4 NICT NMTrerank C 4.9e-2 4.8e-5 CUNI NeuralMonkeyTextualMT U - 1.8e-2 1.1e-6 DCU-ADAPT MultiMT C - 1.2e-3 4.5e-8 LIUMCVC NMT C 1.3e-2 4.3e-2 - 6.8e-5 6.6e-10 CUNI NeuralMonkeyMultimodalMT U 1.6e-3 1.3e-2 4.0e-2 - 3.5e-6 6.2e-13 8.1e-5 6.8e-4 4.9e-3 - 2.0e-8 7.9e-18 CUNI NeuralMonkeyMultimodalMT C 9.6e-8 2.4e-6 3.0e-5 5.7e-3 1.7e-2 - 3.2e-12 1.0e-18 OREGONSTATE 2NMT C 1.0e-8 4.9e-7 3.3e-6 1.4e-3 5.7e-3 2.8e-2 - 2.1e-13 2.9e-17 CUNI NeuralMonkeyTextualMT C 4.6e-8 1.3e-6 1.3e-5 2.2e-3 5.6e-3 3.3e-2 - 4.7e-12 3.2e-18 OREGONSTATE 1NMT C 9.8e-9 3.1e-7 2.2e-6 7.5e-4 2.8e-3 1.3e-2 - 5.6e-13 5.2e-26 SHEF ShefClassProj C 1.3e-13 1.6e-11 1.4e-10 2.2e-06 4.5e-06 6.2e-05 6.5e-03 1.9e-02 4.7e-02 - 3.2e-19 1.2e-30 SHEF ShefClassInitDec C 2.2e-17 6.8e-15 5.3e-14 8.5e-09 2.7e-08 3.3e-07 1.5e-04 1.1e-03 3.3e-03 1.2e-02 - 9.1e-24 2.2e-26 3.8e-23 9.8e-23 2.0e-15 1.2e-15 3.5e-14 1.5e-10 2.0e-09 8.3e-08 9.1e-07 4.7e-05 2.0e-03 3.9e-33 5.5e-41 BASELINE LIUMCVC MNMT C Table 11: English → German Wilcoxon signed-rank test at p-level p ≤ 0.05. ‘-’ means that the value is higher than 0.05. BASELINE AFRL-OHIOSTATE MULTIMODAL C NICT NMTrerank C CUNI NMT U DCU-ADAPT MultiMT C LIUMCVC NMT C CUNI MNMT U UvA-TiCC IMAGINATION C CUNI NMMT C OREGONSTATE 2NMT C CUNI NMT C OREGONSTATE 1NMT C SHEF ShefClassProj C SHEF ShefClassInitDec C UvA-TiCC IMAGINATION U LIUMCVC MNMT C UvA-TiCC IMAGINATION C English→German - 4.0e-28 4.7e-25 2.4e-24 4.0e-17 9.7e-18 5.0e-16 1.7e-12 3.6e-11 2.1e-09 2.6e-08 1.3e-06 1.2e-04 1.2e-34 3.1e-43 AFRL-OHIOSTATE MULTIMODAL C CUNI NeuralMonkeyMultimodalMT C - DCU-ADAPT MultiMT C - LIUMCVC MNMT C 1.0e-03 8.8e-03 2.2e-02 - 2.5e-05 2.9e-04 1.3e-03 CUNI NeuralMonkeyTextualMT C 1.7e-02 - 5.5e-08 5.2e-07 6.7e-06 OREGONSTATE 1NeuralTranslation C 2.7e-03 - 1.5e-09 5.0e-08 7.9e-07 LIUMCVC NMT C 1.2e-02 - 1.8e-07 2.9e-06 2.3e-05 SHEF ShefClassInitDec C 5.3e-07 1.7e-04 7.7e-03 2.4e-02 1.8e-02 2.2e-16 2.4e-14 2.2e-12 - 2.0e-06 1.8e-04 7.7e-03 1.8e-02 1.6e-02 6.8e-15 1.1e-13 1.3e-11 SHEF ShefClassProj C NICT NMTrerank C 3.2e-04 1.8e-03 5.9e-14 7.3e-11 5.0e-08 2.3e-07 5.6e-07 5.8e-26 2.3e-24 3.9e-21 Table 12: English → French Wilcoxon signed-rank test at p-level p ≤ 0.05. ‘-’ means that the value is higher than 0.05. 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Noname manuscript No. (will be inserted by the editor) Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ arXiv:1106.1194v2 [cs.NE] 19 Sep 2013 Angelos A. Anastassi Abstract A methodology that can generate the optimal coefficients of a numeri- cal method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a specific system of ordinary differential equations numerically. The case we are examining here concerns an explicit two-stage Runge-Kutta method for the numerical solution of the two-body problem. Following the implementation of the network, the latter is trained to obtain the optimal values for the coefficients of the Runge-Kutta method. The comparison of the new method to others that are well known in the literature proves its efficiency and demonstrates the capability of the network to provide efficient algorithms for specific problems. Keywords Feedforward artificial neural networks, Gradient descent, Backprop- agation, Initial value problems, Ordinary differential equations, Runge-Kutta methods 1 Introduction The literature that involves solving ordinary or partial differential equations with the use of artificial neural networks is quite limited, but has grown significantly in the past decade. To name a few, Dissanayake et al. used a ”universal approximator” (see [1][2]) to transform the numerical problem of solving partial differential equations to an unconstrained minimization problem of an objective function [3]. Meade et al. demonstrated feedforward neural networks that could approximate linear and nonlinear ordinary differential equations [4][5]. Puffer et al. constructed cellular neural networks that were able to approximate the solution of various ⋆ The final publication is available at http://link.springer.com/article/10.1007%2Fs00521013-1476-x A.A. Anastassi Department of Informatics, University of Piraeus, Karaoli & Dimitriou 80, 18534 Piraeus, GREECE E-mail: ang.anastassi@gmail.com 2 Angelos A. Anastassi partial differential equations [6]. Lagaris et al. introduced a method for the solution of initial and boundary value problems, that consists of an invariable part, which satisfies by construction the initial/boundary conditions and an artificial neural network, which is trained to satisfy the differential equation [7]. S. He et al. used feedforward neural networks to solve a special class of linear first-order partial differential equations [8]. The radial basis function (RBF) network architecture [10] has also been used for the solution of differential equations in [9], where Jianyu et al. demonstrated a method for solving linear ordinary differential equations based on multiquadric RBF networks. Ramuhalli et al. proposed an artificial neural network with an embedded finite-element model, for the solution of partial differential equations [11]. Tsoulos et al. demonstrated a hybrid method for the solution of ordinary and partial differential equations, which employed neural networks, based on the use of grammatical evolution, periodically enhanced using a local optimization procedure [12]. In all the aforementioned cases, the neural networks functioned as direct solvers of differential equations. In this work, however, we use the constructed neural network, not as a direct solver, but as a means to generate proper Runge-Kutta coefficients. In this aspect, there has been little relevant published material. For instance in [13], Tsitouras constructs a multilayer feedforward neural network that uses input data associated with an initial value problem and trains the network to produce the solution of this problem as output data, thus generating (by construction of the network) coefficients for a predefined number of Runge-Kutta stages. In this work we construct an artificial neural network that can generate the coefficients of two-stage Runge-Kutta methods. We consider the two-body problem, which is a typical case of an initial value problem where Runge-Kutta methods apply, and therefore the resulting method is specialized in solving it. The comparison shows that the new method is more efficient than the classical methods and thus proves the capability of the constructed neural network to create new Runge-Kutta methods. The structure is as follows: in Sections 2 and 3 we present the basic theory for Runge-Kutta methods and Artificial Neural Networks respectively. In Section 4 we present the implementation of the neural network that applies to a two-stage Runge-Kutta method which solves the two-body problem numerically, while in Section 5 the derivation of the method is provided. In Section 6 we demonstrate the final results along with the comparison of the new method to other well-known methods and finally in Section 7 we reach some conclusions about this work. 2 Runge-Kutta methods We consider a two-stage Runge-Kutta method to solve the first order initial value problem v′ (t) = f (t, v) and v(t0 ) = v0 (1) At each step of the integration interval, we approximate the solution of the initial value problem at tn+1 , where tn+1 = tn + h. For the two-stage Runge-Kutta method, the approximate solution vn+1 is given by v n + 1 = v n + h ( b1 k 1 + b2 k 2 ) (2) Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 3 where k 1 = f ( tn , v n ) (3) k2 = f (tn + c2 h, vn + h a21 k1 ) (4) The coefficients for this set of methods can be presented by the Butcher tableau given below c2 a21 b1 b2 An explicit two-stage Runge-Kutta method can be of second algebraic order at most (see [14]). In order for that to hold, the following conditions must be satisfied b1 + b2 = 1 (5) 1 (6) 2 The extra condition that needs to be satisfied in order for the method to be consistent is a21 = c2 (7) b2 c2 = 3 Artificial Neural Networks An artificial neural network (ANN) is a network of interconnected artificial processing elements (called neurons) that co-operate with each other in order to solve specific problems. ANNs are inspired by the structure and functional aspects of biological nervous systems and therefore present a resemblance. Haykin in [15] defines an artificial neural network as ”a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use”. ANNs, similarly to brains, acquire knowledge through a learning process, which is called training. That knowledge is stored in the form of synaptic weights, whose values express the strength of the connection between two neurons. There are many types of artificial neural networks, depending on the structure and the means of training. An ANN, in its simplest form, consists of a single neuron, in what we call the Perceptron model. The Perceptron is connected with a number of inputs: x1 , x2 , ..., xn . A weight corresponds to each of the neuron’s connections with the inputs and expresses the specific input’s significance to the calculation of the Perceptron’s output. Therefore, there is a number of weights equal to the number of inputs, with wi being the weight that corresponds to input xi , and wi xi being the combined weighted input. The neuron is also connected with a weight, which is not connected with any input and is called bias (symbolized by b). The weighted input data are essentially being mapped to an output value, through the transfer function of the neuron. The transfer function consists of the net input function and the activation function. The first function undertakes the task of combining the various weighted inputs into a scalar value. Typically, the summation function is used for this purpose, thus the net input in that case is described by the following expression 4 Angelos A. Anastassi Fig. 1 An artificial neuron (Perceptron) u= n X wi x i + b (8) i=1 For ease of use, the bias can be treated as any other weight and be represented as w0 , with x0 = 1, so the previous expression becomes u= n X wi x i (9) i=0 The activation function receives the output u of the net input function and produces its own output y (also known as the activation value of the neuron), which is a scalar value as well. The general form of the activation function is y = f (u) (10) What is required of a neural network, such as the Perceptron, is the ability to learn. That means the ability to adjust its weights in order to be able to produce a specific set of output data for a specific set of input data. In the supervised type of learning (which is used to train the Perceptron and other types of neural networks), besides the input data, we provide the network with target data, which are the data that the output should ideally converge to, with the given set of input. The Perceptron uses an estimation of the error as a measure that expresses the divergence between the actual output and the target. In that sense, training is essentially the learning process of adjusting the synaptic weights, in order to minimize the aforementioned error. To achieve this, a gradient descent Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 5 algorithm called delta rule is employed, also known as the Least Mean Square (LMS) method or the Widrow-Hoff rule, developed by Widrow and Hoff in [16]. The training process that uses the delta rule has the following steps: 1. Assign random values to the weights 2. Generate the output data for the set of the training input data 3. Calculate the error E , which is given by a norm of the differences between the target and the output data 4. Adjust the weights according to the following rule ∆wi = −η ∂E ∂u ∂E = −η = ηδxi ∂wi ∂u ∂wi (11) Where η is a small positive constant, called learning step or learning rate, and δ (delta error) is equal to the following expression δ=− ∂E ∂u (12) 5. Repeat from step 2, unless the error E is less than a small predefined constant ǫ or the maximum number of iterations has been reached With each iteration of the learning method, the weights are adjusted in such manner as to reduce the error. Notice that the correction ∆wi of the weights is ∂E , since the desired goal is the minimization of proportional to the negative of ∂w i the error. Strictly speaking, the Perceptron is not technically a network, since it consists of a single neuron. An actual network and the most common form of ANNs is the multilayer Perceptron (MLP), which falls in the general category of feedforward neural networks (FFNNs). Feedforward are the neural networks that contain no feedback connections, i.e. the connections do not form a loop, but are instead all directed towards the output of the network. A multilayer Perceptron consists of various layers, namely the input layer, the output layer, and one or more hidden layers. The hidden layers are not visible externally of the network, hence the ”hidden” property. Each of the layers consists of one or more neurons, except for the input layer, whose input nodes simply function as an entry point for the input data. Each layer is connected to the previous and the next layer, thus providing a pathway for the data to travel throughout the network. When a layer is connected to another, each neuron of the first layer is connected to every neuron of the second layer. In this way, the output of one layer becomes the input for the next one. Therefore, to calculate the output of the MLP, the data (either input data or activation values) are being propagated forward through the neural network. To train the MLP, the backpropagation method is used. Backpropagation is a generalization of the delta rule that is used to train the Perceptron. The method, essentially, functions in the same way for each neuron of the MLP, as simple delta rule does for the Perceptron. The difference lies in the fact that, since the learning procedure requires the calculation of the δ error, the neurons that reside inside the hidden layers must be provided with the δ errors of the neurons of the next layer. For this reason, the δ errors are first calculated for the neurons of the output layer and are propagated backwards (hence the backpropagation term), all across the network. The procedure is the following: 6 Angelos A. Anastassi Fig. 2 A typical multilayer Perceptron 1. Assign random values to the weights 2. Generate the activation values of each neuron, starting from the first hidden layer and continue until the output layer 3. Calculate the error E , which is given by a norm of the differences between the target and the output data 4. Calculate the δ errors of each neuron of the output layer, according to (12) 5. Calculate the δ errors of each neuron of the previous layer, according to δ = f ′ (u) M X w i δi (13) i=1 Where f (u) is the activation function of the neuron, M is the number of the neurons in the next layer, and δi is the δ error of the ith neuron of the next layer 6. Repeat step 5 for the previous layer, until all δ errors for all the hidden layers have been calculated 7. Adjust the weights according to (11) 8. Repeat from step 2, unless the error E is less than a small predefined constant ǫ or the maximum number of iterations has been reached 4 Implementation We introduce a feedforward neural network that can generate the coefficients of two-stage Runge-Kutta methods, specifically designed to apply to the needs of the two-body problem. The two-body problem is described by the following system of equations p x y x′′ = − 3 , y ′′ = − 3 , r = x2 + y 2 (14) r r whose analytical solution is given by x = cos t, y = sin t (15) Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 7 Since a Runge-Kutta method can only solve a system of first order ordinary differential equations, v according to the notation of equation (1), can be given by v = [v1 , v2 , v3 , v4 ] = x, y, x′ , y ′ (16) and h f (t, v) = v′ (t) = v1′ , v2′ , v3′ , v4′ = v3 , v4 , − p v2 v1 , − 3 , r = v12 + v22 r3 r i (17) From now on we will be using x, y, x′ , y ′ , for being more straightforward than v1 , v2 , v3 , v4 . The neural network we have constructed has six inputs: The coordinates of the second body: x and y , the derivatives of the coordinates with respect to the independent variable t: x′ , y ′ , the steplength h and a ”dummy” input with the constant value of 1. The input data consist of a matrix, where each column is a vector, intended to be used by the corresponding input of the network. The matrix satisfies the following equation ′ xn , yn , x′n , yn , h, 1 = [cos tn , sin tn , − sin tn , cos tn , h, 1] (18) With tn+1 = tn + h, where n = 1, 2, ..., N and N is the number of total steps. Similarly, the target matrix satisfies the following equation [xn+1, yn+1 , x′n+1 , yn′ +1 ] = [cos tn+1 , sin tn+1 , − sin tn+1 , cos tn+1 ] (19) The input and the target data are generated through a certain procedure. In this procedure, we set a number of parameters, namely the integration interval [ts , te ], the x0 , y0 , x′0 , y0′ values that correspond to v0 as notated in equation (1), and the constant steplength h. With the use of the integration interval and the steplength, the number of steps N is calculated. For each step, the process generates the input and target data that correspond to the starting and ending point of the integration step respectively. Thus, two matrices are generated by using the theoretical solution of the problem and consist of x, y, x′ , y ′ at the beginning and at the end of each integration step. Apart from the input layer, the network comprises of thirteen additional layers, each consisting of a single neuron. Each layer is connected to one or more other layers, always in a feedforward fashion. Most of the connections have a fixed, unadjustable weight, whose value is equal to one. Practically, that means that the significance of the specific connections does not vary in the progress of the training phase. The connections that do have an adjustable weight are those related with the three coefficients of the Runge-Kutta method, namely a21 , b1 , and b2 . The coefficient c2 is equal to a21 , therefore it does not need to be explicitly calculated through the use of the neural network. The neural network is constructed in such manner as to replicate the actual Runge-Kutta methodology and produce the numerical solution for the two-body problem for each step of the integration. In contrast to the actual methodology, the approximate solution at the end of each step is not used as a starting point for the next step, but instead of this, theoretical values are used as starting points for each step’s calculations. 8 Angelos A. Anastassi Fig. 3 The constructed neural network Most of the layers do not use the standard summation function as the net input function, but a custom one, different in most of the cases. The activation function in every neuron is the identity function, described by the following equation y = f (u) = u (20) Therefore, the transfer function in each case is, in essence, the net input function. Below, there is the list of the inputs and the layers of the network, their input and output connections (with any of the 6 inputs or 13 layers) and in parentheses the expression that they represent: Input 1: – Output connections: layers 1 (k1x′ ), 2 (k1y ′ ), 6 (k2x′ ), 7 (k2y ′ ), 10 (xn+1 ) – Value: xn Input 2: – Output connections: layers 1 (k1x′ ), 2 (k1y ′ ), 6 (k2x′ ), 7 (k2y ′ ), 11 (yn+1 ) – Value: yn Input 3: – Output connections: layers 8 (k2x ), 10 (xn+1 ), 12 (x′n+1 ) Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 9 – Value: x′n = k1x Input 4: ′ – Output connections: layers 9 (k2y ), 11 (yn+1 ), 13 (yn +1 ) ′ – Value: yn = k1y Input 5: ′ – Output connections: layers 3 (h a21 ), 10 (xn+1 ), 11 (yn+1 ), 12 (x′n+1 ), 13 (yn +1 ) – Value: h Input 6: – Output connections: layers 4 (b1 ), 5 (b2 ) – Value: 1 Layer 1: – Input connections: inputs 1 (xn ), 2 (yn ) – Output connections: layers 6 (k2x′ ), 7 (k2y ′ ), 12 (x′n+1 ) – Net input function: fx′ (xn , yn ) = − √ 2xn 2 3 ( xn +yn ) – Output value: k1x′ Layer 2: – Input connections: inputs 1 (xn ), 2 (yn ) ′ – Output connections: layers 6 (k2x′ ), 7 (k2y ′ ), 13 (yn +1 ) – Net input function: fy ′ (xn , yn ) = − √ 2yn 2 3 ( xn +yn ) – Output value: k1y ′ Layer 3: – – – – Input connections: input 5 (h) Output connections:P layers 6 (k2x′ ), 7 (k2y′ ), 8 (k2x ), 9 (k2y ) Net input function: Output value: h a21 Layer 4: – – – – Input connections: input 6 (1) Output connections:P layers 10 (xn+1), 11 (yn+1), 12 (x′n+1), 13 (yn′ +1 ) Net input function: Output value: b1 Layer 5: – – – – Input connections: input 6 (1) Output connections:P layers 10 (xn+1), 11 (yn+1), 12 (x′n+1), 13 (yn′ +1 ) Net input function: Output value: b2 Layer 6: – – – – Input connections: inputs 1 (xn ), 2 (yn ), layers 1 (k1x′ ), 2 (k1y′ ), 3 (h a21 ) Output connections: layer 12 (x′n+1) Net input function: f2x′ = fx′ (xn + h a21 k1x′ , yn + h a21 k1y′ ) Output value: k2x′ 10 Angelos A. Anastassi Layer 7: – – – – Input connections: inputs 1 (xn ), 2 (yn ), layers 1 (k1x′ ), 2 (k1y′ ), 3 (h a21 ) Output connections: layer 13 (yn′ +1 ) Net input function: f2y′ = fy′ (xn + h a21 k1x′ , yn + h a21 k1y′ ) Output value: k2y′ Layer 8: – – – – Input connections: input 3 (x′n = k1x ), layer 3 (h a21 ) Output connections: layer 10 (xn+1) Net input function: f2 = x′n + h a21 k1x = x′n · (1 + h a21 ) Output value: k2x Layer 9: – – – – Input connections: input 4 (yn′ = k1y ), layer 3 (h a21 ) Output connections: layer 11 (yn+1 ) Net input function: f2 = yn′ + h a21 k1y = yn′ · (1 + h a21 ) Output value: k2y Layer 10: – Input connections: inputs 1 (xn ), 3 (x′n = k1x ), 5 (h), layers 4 (b1 ), 5 (b2 ), 8 ( k2 x ) – Output connections: network output – Net input function: f3 = xn + h · (b1 k1x + b2 k2x ) – Output value: xn+1 Layer 11: ′ – Input connections: inputs 2 (yn ), 4 (yn = k1y ), 5 (h), layers 4 (b1 ), 5 (b2 ), 9 ( k2 y ) – Output connections: network output – Net input function: f3 = yn + h · (b1 k1y + b2 k2y ) – Output value: yn+1 Layer 12: – Input connections: inputs 3 (x′n = k1x ), 5 (h), layers 1 (k1x′ ), 4 (b1 ), 5 (b2 ), 6 ( k2 x ′ ) – Output connections: network output – Net input function: f3 = x′n + h · (b1 k1x′ + b2 k2x′ ) – Output value: x′n+1 Layer 13: ′ – Input connections: inputs 4 (yn = k1y ), 5 (h), layers 2 (k1y′ ), 4 (b1 ), 5 (b2 ), 7 ( k2 y ′ ) – Output connections: network output ′ – Net input function: f3 = yn + h · ( b 1 k1 y ′ + b 2 k2 y ′ ) ′ – Output value: yn +1 Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 11 5 Method derivation The derivation of the coefficients for the RK method described in (2), includes certain subroutines: – Data generation – Neural network training – Conversion to fraction During the data generation phase, we generate the input and target data that are going to be used to train the neural network. We have used various integration intervals [ts , te ], from [0, 2π ] up to [0, 200π ] and various steplengths, from h = π4 π down to 512 . As we mentioned before, the input and the target data are described by the equations (18) and (19) respectively. Therefore, after the data generation, we obtain an input matrix of size 6 × N and a target matrix of size 4 × N . Where N is the number of steps and the following equation holds N= te − ts h (21) During the training phase, we use the generated data to train the neural network. The method used to conduct the training is a variation of the typical backpropagation method, called Gradient Descent with Momentum and Adaptive Learning Rate Backpropagation or GDX. The back-propagation algorithm and its numerous variants constitute the most popular learning technique for feedforward neural networks [15]. However, a shortcoming of the original backpropagation algorithm is that it can be easily trapped in local minima. In order to deal with this disadvantage, the addition of a momentum term has been proposed. GDX includes the momentum term in order to be able to escape from local minima, but also has been found to present faster convergence and lower training times compared to other competing methods [17]. Instead of a standard error estimation function, we use a custom one for this purpose, which is shown below \|\|Output − Target\|\|∞ + \|b1 + b2 − 1\| + \|a21 b2 − 1 \| 2 (22) The error to be minimized is the maximum absolute difference between the output and the target data, plus some added expressions to satisfy the algebraic conditions. The terms \|b1 + b2 − 1\| and \|a21 b2 − 12 \| are used to satisfy the conditions (5) and (6) respectively. The coefficient a21 is used interchangeably with c2 , due to (7). As a result of the training phase, the obtained coefficients a21 , b1 and b2 satisfy the conditions to a certain degree of accuracy. Additionally, the Runge-Kutta method constructed with the generated coefficients can produce an output, sufficiently close to the target data. During the last phase, the coefficient a21 is converted into a fraction to simplify the method, with an insignificant loss of accuracy. The rest of the coefficients are calculated with the use of the fractional a21 , in order for the algebraic and consistency conditions, (5), (6) and (7) respectively, to be satisfied. The coefficients of the new method are presented in Table 1. For the current implementation, the number of maximum iterations was set at 10000. That value was selected empirically, as the neural network provided a 12 Angelos A. Anastassi solution after approximately 5000 iterations and effectively completed the training, as it could not improve on the performance any further. 6 Results The best method was provided by using the integration interval [0, 2π ] and the π . Training with other integration intervals and steplengths have steplength h = 128 returned similar results. 2π represents a full oscillation for the two-body problem, which in part explains why training over wider intervals does not yield better results. Apart from the new method, the training has also generated some well known classical methods, some of which are given later in this section. We compare the new method with three classical two-stage explicit RungeKutta methods. The corresponding Butcher tableaus of all methods are shown in Tables 1-4. 11 26 11 26 2 − 11 13 11 Table 1 New method 1 1 1 2 1 2 Table 2 Heun’s method 1 2 1 2 0 1 Table 3 Midpoint method 2 3 2 3 1 4 3 4 Table 4 Third classical method A comparison between the classical methods and the new method is presented in figure 4, which regards the numerical integration of the two-body problem over the interval [0, 1000]. The vertical axis represents the accuracy as expressed by −log10 (maximum absolute error), while the horizontal axis represents the total number of function evaluations that are required for the computation. The latter is provided by the formula F.E. = s · N , where s = 2 and stands for the number of stages of the RK method and N is the number of total steps. We can see that the new method is more efficient among all methods compared. Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks 13 ⋆ Two-body Problem 8 7 New Method Midpoint Classical 3 Heun -log10(error) 6 5 4 3 2 1 2.5E+05 2.5E+06 2.5E+07 2.5E+08 Function Evaluations Fig. 4 Efficiency of the compared methods for the integration of the two-body problem 7 Conclusions We have constructed an artificial neural network that can generate the optimal coefficients of an explicit two-stage Runge-Kutta method. The network is specifically designed to apply to the needs of the two-body problem, and therefore the resulting method is specialized in solving that problem. Following the implementation of the network, the latter is trained to obtain the optimal values for the coefficients of the method. The comparison has shown that the new method developed in this work is more efficient than other well known methods and thus proved the capability of the constructed neural network to create new efficient numerical algorithms. References 1. Cybenko G (1989) Approximations by superpositions of sigmoidal functions. Mathematics of Control, Signals, and Systems 2:303–314 2. Hornik K (1991) Approximation Capabilities of Multilayer Feedforward Networks. Neural Networks 4:251–257 3. Dissanayake MWMG, Phan-Thien N (1994) Neural-network-based approximations for solving partial differential equations. Communications in Numerical Methods in Engineering 10:195–201 4. 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SPATIAL DIFFUSENESS FEATURES FOR DNN-BASED SPEECH RECOGNITION IN NOISY AND REVERBERANT ENVIRONMENTS Andreas Schwarz, Christian Huemmer, Roland Maas, Walter Kellermann Multimedia Communications and Signal Processing Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) Cauerstr. 7, 91058 Erlangen, Germany arXiv:1410.2479v2 [cs.CL] 16 Feb 2015 {schwarz, huemmer, maas, wk}@lnt.de ABSTRACT We propose a spatial diffuseness feature for deep neural network (DNN)-based automatic speech recognition to improve recognition accuracy in reverberant and noisy environments. The feature is computed in real-time from multiple microphone signals without requiring knowledge or estimation of the direction of arrival, and represents the relative amount of diffuse noise in each time and frequency bin. It is shown that using the diffuseness feature as an additional input to a DNN-based acoustic model leads to a reduced word error rate for the REVERB challenge corpus, both compared to logmelspec features extracted from noisy signals, and features enhanced by spectral subtraction. Index Terms— Speech Recognition, Reverberation, Diffuse Noise, Deep Neural Networks 1. INTRODUCTION In automatic speech recognizers (ASR) based on Gaussian Mixture Models and Hidden Markov Models (GMM-HMM), a wide variety of transformations and feature extraction steps is currently being employed with the aim of extracting and normalizing the information contained in the time-domain input signal as efficiently as possible. Recently, with the development of effective training methods for acoustic models based on multiple-layer neural networks, which are often summarized under the term “deep neural networks” (DNN) [1], it has become possible for the acoustic model to learn relationships between features and phonemes to a higher degree than it is possible with manually implemented feature transformation steps. For example, it has been found that simple filterbank features outperform mel-frequency cepstral coefficients (MFCCs) [2, 3], and it is conceivable that, given large amounts of training data and sufficiently complex network structures, time-domain signals may at some point even be directly used as inputs to a neural network. Although the trend in ASR goes towards replacing explicit processing stages by implicit learning, for noise- and reverberation-robust ASR using microphone arrays, spatial information is still predominantly being exploited in a separate speech The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for supporting this work (contract number KE 890/4-2). enhancement preprocessor, e.g., in the form of beamforming [4], multichannel linear prediction [5], blocking matrix-based postfilters [6] or coherence-based postfilters [7]. The single-channel output of the preprocessor is then used to compute features for ASR. In some GMM-HMM-based systems, spatial information is exploited indirectly in uncertainty decoding-based approaches, e.g., in [8], where the feature uncertainty is derived from a noise estimate obtained in a multichannel signal enhancement stage. For DNN-based acoustic models, “noise-aware training” has been proposed [3], where a noise estimate is appended to the noisy feature vector. This has been evaluated for stationary noise estimates [3] and noise-estimates derived from time-frequency masking [9], but may in principle also be used for noise estimates obtained from spatial processing. In [10] and [11], feature vectors from multiple microphones are concatenated to form the input of a DNN-based acoustic model, however, no spatial phase information is exploited. Inspired by the trend towards moving more explicit feature processing steps into the DNN, we propose to exploit spatial information about the diffuseness of the sound field directly by incorporating it into the acoustic model of a DNN-based speech recognizer. The diffuseness estimate is derived from the complex coherence between two omnidirectional microphones and has been used for signal enhancement based on the assumption that late reverberation and noise components can be modeled as diffuse noise [7]. Using the diffuseness as a feature is motivated by the fact that humans exploit similar spatial information for speech recognition in reverberant and noisy environments [12, 13], as it was found that the human auditory system treats spectro-temporal variations in the interaural coherence as “a perceptual surrogate for spectro-temporal variations in the energy of speech signals” [13]. The aim is to learn similar behavior in a DNN-based acoustic model. We first describe the signal model for the estimation of the diffuseness from the instantaneous spatial coherence of a reverberated and noisy speech signal. Then, we show how this estimate is integrated into a feature extraction scheme for ASR, and describe the structure of the DNN-based speech recognizer. Finally, we evaluate the proposed feature on the two-channel task of the REVERB challenge [14], showing that the proposed approach outperforms both noisy multi-condition training and multichannel spectral subtraction-based signal enhancement. 2. BLIND DIFFUSENESS ESTIMATION We consider a reverberated and noisy speech signal recorded by two omnidirectional microphones. The signal xi (t) recorded at the i-th microphone is composed of the desired signal component si (t) and the undesired noise component ni (t) comprising additive noise and late reverberation, i.e., xi (t) = si (t) + ni (t), i = 1, 2. The microphone, desired, and noise signals are represented in the short-time Fourier transform (STFT) domain by the corresponding uppercase letters, i.e., Xi (k, f ), Si (k, f ) and Ni (k, f ), respectively, with the discrete frame index k and continuous frequency f , and the auto- and cross-power spectra Φxi xj (k, f ), Φsi sj (k, f ), Φni nj (k, f ). Note that the continuous frequency f is used here for generality; in practice, f denotes discrete values along the frequency axis. It is assumed that the auto-power spectra of all signal components are identical at both microphones, i.e., Φsi si (k, f ) = Φs (k, f ), Φni ni (k, f ) = Φn (k, f ). The timeand frequency-dependent signal-to-noise ratio (SNR) of the microphone signals can then be defined as SNR(k, f ) = Φs (k, f ) . Φn (k, f ) (1) The aim in the following is to estimate the SNR from the coherence of the mixed sound field Γx (k, f ). This coherence is first estimated as Φ̂x1 x2 (k, f ) , Γ̂x (k, f ) = q Φ̂x1 x1 (k, f )Φ̂x2 x2 (k, f ) where the spectral estimates Φ̂xi xj (k, f ) are obtained by recursive averaging: Φ̂xi xj (k, f ) = λΦ̂xi xj (k−1, f ) + (1−λ)Xi (k, f )Xj∗ (k, f ), (7) with a constant forgetting factor λ between 0 and 1. In [15, 7], it was shown that (3) can be solved for the SNR without requiring knowledge of Γs , using only the assumption that the desired signal is fully coherent, i.e., \|Γs \| = 1. This yields a “blind” estimator for the SNR (or coherent-to-diffuse ratio, CDR) from the mixture coherence Γ̂x (k, f ) which does not require knowledge or estimation of the signal DOA. The estimator is given in (8) at the bottom of this page (the indices k and f are omitted for brevity). The CDR can be transformed into the diffuseness [16] d D̂(k, f ) = [CDR(k, f ) + 1]−1 , The complex spatial coherence functions of the desired signal and noise components are given by Φs1 s2 (k, f ) Φn1 n2 (k, f ) Γs (f ) = , Γn (f ) = , Φs (k, f ) Φn (k, f ) (2) and are assumed to be time-invariant, i.e., dependent only on the spatial characteristics of the signal components. It is furthermore assumed that signal and noise components are orthogonal, such that Φx (k, f ) = Φs (k, f ) + Φn (k, f ). The complex spatial coherence of the mixed sound field can then be written as a function of the SNR and the signal and noise coherence functions: Γx (k, f ) = SNR(k, f )Γs (f ) + Γn (f ) . SNR(k, f ) + 1 (3) The direct sound is now modeled as a plane wave with an unknown direction of arrival (DOA) and therefore unknown time difference of arrival ∆t, while the undesired noise and late reverberation component is modeled as a diffuse (spherically isotropic) sound field. The corresponding spatial coherence functions for the direct and diffuse sound components are then given by Γs (f ) = ej2πf ∆t , (4) d Γn (f ) = Γdiffuse (f ) = sinc(2πf ), c (5) respectively. The direct signal coherence has a magnitude of one with an unknown phase determined by the DOA, while the diffuse noise coherence only depends on the known microphone spacing d. 2 d CDR(k, f) = Γn Re{Γ̂x } − \|Γ̂x \| − (6) (9) which can be thought of as the relative amount of diffuse signal power in the respective time and frequency bin. Since the diffuseness is bounded between 0 and 1, it is more convenient to use as basis for feature computation than the CDR itself. 3. FEATURE EXTRACTION FOR ASR Fig. 1 shows the block diagram of the proposed feature extraction scheme. The microphone signals are first windowed and transformed into the STFT domain. The upper path then corresponds to a classical feature extraction of NMel -dimensional logmelspec (often termed “log-filterbank” or “Log FBank”) features, where the two microphone signals are combined by averaging the spectral powers computed from each microphone, and NMel triangular Mel-scaled weighting filters are applied. The second path shows the extraction of enhanced logmelspec features, where signal enhancement based on the diffuseness estimate is performed by multiplication in the STFT domain with a gain factor G(k, f ), which is computed as described in [7] according to the spectral magnitude subtraction rule. The third path illustrates the computation of the proposed “meldiffuseness” features: the diffuseness D̂(k, f ) is estimated as described in the previous section, and the same NMel triangular Mel weighting filters that are used in the logmelspec feature extraction are applied to create an output vector of the dimensionality NMel . Finally, for comparison, the Mel-weighted magnitude-squared coherence (“melmsc”) is computed as a feature. While the magnitude-squared coherence of a q 2 2 2 Γ2n Re{Γ̂x } − Γ2n \|Γ̂x \| + Γ2n − 2 Γn Re{Γ̂x } + \|Γ̂x \| 2 \|Γ̂x \| − 1 (8) Window X1(k, f ) STFT \|⋅\| 2 logmelspec NSTFT Window avg X2(k, f ) STFT log Mel weighting NMel \|⋅\|2 G(k, f ) NSTFT enhanced logmelspec log Mel weighting NMel Coherence estimation Γˆ x(k, f ) Diffuseness estimation ˆ f) D(k, meldiffuseness Mel weighting NMel melmsc \|⋅\|2 Mel weighting NMel Fig. 1. Feature extraction of logmelspec, enhanced logmelspec, meldiffuseness and melmsc features from 2-channel signals. 1 http://www.lms.lnt.de/files/publications/icassp2015-diffuseness.zip Mel band a) logmelspec 20 10 100 200 300 0 −5 −10 −15 −20 Mel band b) enhanced logmelspec 20 10 100 200 300 0 −5 −10 −15 −20 c) meldiffuseness Mel band mixed sound field is also related to the amount of diffuse noise, this relationship is strongly dependent on the signal DOA and the microphone spacing, therefore the melmsc feature is expected to perform worse than the proposed diffuseness estimate. The interesting question is now how using concatenated logmelspec and meldiffuseness features as input to the neural network compares to using logmelspec features which have been enhanced in the STFT domain. Since the trend in DNN-based acoustic modeling goes towards replacing explicit feature preprocessing and normalization steps by implicit learning, one might consider using the complex spatial coherence directly as feature. Note, however, that the proposed diffuseness feature has two significant advantages over the complex coherence. The complex coherence depends on two additional variables, namely the DOA and the microphone spacing, both of which would need to be sufficiently represented in the training data. Moreover, the diffuseness is a characteristic of the sound field which is independent of the microphone array geometry, and may therefore also be estimated from microphone arrays with other geometries, e.g., spherical arrays [17] or arrays consisting of directional microphones [18], without requiring adaptation of the acoustic model. It is interesting to note that the additional temporal smoothing which is required for the estimation of the coherence (and therefore the diffuseness) has parallels in the human auditory system, where reaction to changes in interaural coherence was found to be more sluggish than reaction to changes in energy [19]. For the results presented in this paper, the time-domain signals (sampled at 16 kHz) are windowed using a 25 ms Hann window with a frame shift of 10 ms and transformed using a 512point DFT, resulting in NSTFT = 257 subbands in the STFT domain. The spatial coherence is estimated using the forgetting factor λ = 0.68. NMel = 24 triangular Mel-scale weighting filters are used, covering a frequency range from 64 to 8000 Hz. MATLAB code for the feature computation is provided online1 . Fig. 2 illustrates the features computed from a noisy and reverberated speech signal taken from the multi-condition training set of the REVERB challenge corpus (LargeRoom2). The coherence-based spectral enhancement visibly reduces the noise floor and the smearing of the speech features over time. The meld- 20 10 100 200 300 1 0.8 0.6 0.4 0.2 0 frame Fig. 2. Features for the reverberated utterance “The statute allows for a great deal of latitude”. iffuseness clearly highlights portions of the signal where noise or reverberation components are dominant. 4. DNN-BASED SPEECH RECOGNITION We employ the Kaldi toolkit [20] as ASR back-end system using the WSJ0 trigram 5k language model of the REVERB challenge and 3551 context-dependent triphone-states in the acoustic model. In a first step, we set up a GMM-HMM baseline system based on Weninger et al. [4] by extracting 13 mean and variance normalized MFCCs (including the zeroth cepstral coefficient), followed by ±4 frame splicing, linear discriminant analysis (LDA), maximum likelihood linear transform (MLLT), and feature-space maximum likelihood linear regression (fMLLR) (see [4, 21] for a detailed description). After conventional maximum likelihood training, discriminative training is performed with the boosted maximum mutual information (bMMI) criterion [4]. The GMM- Table 1. ASR Word Error Rate for the REVERB challenge evaluation and development test sets. Recognizer Feature GMM-HMM MFCC-LDA-MLLT-fMLLR logmelspec+∆+∆∆ enhanced logmelspec+∆+∆∆ DNN-HMM logmelspec+∆+meldiffuseness logmelspec+∆+melmsc Room 1 near far 6.61 7.50 5.74 6.67 6.61 7.12 5.91 6.06 6.17 6.32 SimData Room 2 near far 9.42 16.59 7.65 13.92 7.65 12.18 6.94 10.96 7.04 12.25 HMM system is trained on the clean WSJCAM0 Cambridge Read News REVERB corpus [22]. The alignment of the training data to the HMM states is then extracted from the clean training data and used for the later multi-condition training of the DNN-HMM system. This technique is known to yield better results than a multi-condition state-frame alignment [9, 23]. The hybrid DNN-HMM Kaldi system is based on “Dan’s implementation” [20] using a maxout network with 2-norm nonlinearities/activation functions and 4 hidden layers, each one with an input dimension of 2000 and an output dimension of 400. In accordance with [2, 3], and as described in the previous section, we extract NMel = 24 static logmelspec coefficients, with or without applying coherence-based spectral subtraction enhancement in the STFT domain. Depending on the particular setup in Table 1, also Delta (∆), acceleration (∆∆), melmsc, and/or the proposed meldiffuseness features are derived. Mean and variance normalization and ±5 frame splicing is applied to the entire resulting feature vector. The training is performed on the REVERB multi-condition training set [14], consisting of 7861 noisy and reverberated utterances from the WSJCAM0 corpus, using greedy layer-wise supervised training, preconditioned stochastic gradient descent, “mixing up” [24] as well as final model combination [24]. 5. EVALUATION RESULTS We evaluate the proposed system using the two-channel task of the REVERB challenge [14]. The REVERB evaluation test set consists of ∼5000 reverberated and noisy utterances, partially created by convolution of clean WSJCAM0 utterances with impulse responses and mixing with recorded noise sequences (“SimData”), and partially consisting of multichannel recordings of speakers in a reverberant and noisy room from the MC-WSJAV corpus (“RealData”). For SimData, the reverberation times of the three rooms are approx. 0.25 s, 0.5 s and 0.7 s and the sourcemicrophone spacing is 0.5 m (near) or 2 m (far). For RealData, the reverberation time is approx 0.7 s and the source-microphone distance is 1 m (near) or 2.5 m (far). In both cases, an 8-channel circular microphone array with a diameter of 20 cm was used, of which two microphones with a spacing of d = 8 cm are selected for the two-channel recognition task which is evaluated here. First, we evaluate the word error rate (WER) obtained from the GMM-based recognizer with MFCC features, which is used to obtain the alignment. For the DNN-based recognizer, we compare logmelspec features extracted from the noisy signals, and Evaluation Set Room 3 near far 11.05 20.69 8.65 14.62 8.32 14.57 8.24 12.88 8.17 13.87 Avg 11.98 9.54 9.41 8.50 8.97 RealData Room 1 near far 31.17 30.15 28.45 29.14 28.46 29.07 27.82 26.27 27.34 27.99 Development Set SimData RealData Avg 30.66 28.80 28.77 27.05 27.67 Avg 12.14 9.68 9.13 7.92 8.68 Avg 31.61 24.93 25.29 24.19 24.73 enhanced logmelspec features. In both cases, the feature vector is extended by first- (∆) and second-order (∆∆) derivatives. Then, we evaluate the combination of noisy logmelspec features with spatial meldiffuseness or melmsc features; in this case, only firstorder derivatives (∆) are computed for the logmelspec features, in order to keep the overall dimension of the feature vectors the same (3NMel ). Table 1 shows the WER results for the REVERB challenge evaluation test set, and the average WER for the development test set. As expected, the DNN-based acoustic model achieves a lower WER than the GMM-based model. The diffuseness-based signal enhancement has a negligible effect on WER. This seems to contradict [15], where the same signal enhancement method led to a significantly lower WER, however, there, acoustic models were trained on clean speech. Apparently the effect of the multichannel spectral subtraction for signal enhancement is compensated by noisy multi-condition training. Using the combined noisy logmelspec and diffuseness features as input to the neural network however yields a significantly reduced WER. This confirms that the spatial information extracted from the coherence can be exploited more successfully by the DNN than by speech enhancement using spectral subtraction, even though, in this case, the frequency resolution of the meldiffuseness features is reduced compared to the diffuseness estimate used for spectral subtraction. The melmsc feature also leads to a reduced WER compared to noisy logmelspec features, although the improvement is smaller than with meldiffuseness features. 6. CONCLUSION It has been shown that spatial information extracted from multiple microphones does not necessarily have to be exploited in a signal enhancement front-end, but may be used more effectively as an additional feature input for a DNN-based speech recognizer. 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Multiparameter spectral analysis for aeroelastic instability problems Arion Pons1,* and Stefanie Gutschmidt2 1 Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK. Department of Mechanical Engineering, University of Canterbury, Christchurch 8140, New Zealand. 2 Key words AMS classification multiparameter eigenvalue problems; instability; aeroelasticity. 97M50; 65F15; 15A18; 15A69 This paper presents a novel application of multiparameter spectral theory to the study of structural stability, with particular emphasis on aeroelastic flutter. Methods of multiparameter analysis allow the development of new solution algorithms for aeroelastic flutter problems; most significantly, a direct solver for polynomial problems of arbitrary order and size, something which has not before been achieved. Two major variants of this direct solver are presented, and their computational characteristics are compared. Both are effective for smaller problems arising in reduced-order modelling and preliminary design optimization. Extensions and improvements to this new conceptual framework and solution method are then discussed. 1. Introduction Predicting and controlling aeroelastic instability forms a major part of the discipline of aeroelasticity. Many different physical systems show flutter instability, and many models exist to describe them. In a linear system, or the linearization of a nonlinear system, the onset of flutter or divergence can be formulated in the well-known stability criterion: Im(𝜒) > 0 for stability (1) where 𝜒 are the time-eigenvalues of the system according to the Fourier transform 𝑞(𝑡) = 𝑞̂𝑒 𝜄𝜒𝑡 for the system coordinate 𝑞 [1]. These eigenvalues may be nondimensionalised. Flutter occurs when the system parameters are such that the system is on the stability boundary, Im(𝜒f ) = 0. The flutter point may then be described as a tuple which includes the set of system parameters (particularly, an airspeed parameter) and the modal frequency of instability, 𝜒f . Typically only one or two flutter or divergence points are of industrial relevance. Eq. 1 is not however the only criterion that can be used to characterize instability. This leads us into the study of aeroelastic methods. Apart from the four established methods – the pmethod, classical flutter analysis, the k-method (or U-g method, or V-g method) and the p-k method; all of which are detailed and discussed in a number of reference works [1–3] – recent years have seen a proliferation of new aeroelastic methods. Several authors have refined the existing established methods for particular scenarios or applications [4–6]. The application of concepts from robust control theory have yielded a series of methods, including the 𝜇-method by Lind and Brenner [7,8], the 𝜇-k method by Borglund [9–12], and others [13–15]. The prime advantage of these 𝜇-type methods is that they facilitate the propagation of uncertainty * Corresponding author, adp53@cam.ac.uk, tel. +44 755 366 3296 Multiparameter analysis for instability problems distributions through the system, allowing a worst-case flutter speed estimate to be made in a system with high uncertainty. Other developments have come from other fields: Afolabi [16,17] characterized coupled-mode flutter as a loss of eigenvector orthogonality, using methods from catastrophe theory. Irani and Sazesh [18] used stochastic methods, while Gu et al. [19] devised a genetic algorithm, and a number of authors [20–23] have applied neural networks to the detection of flutter points. It is in the context of these developments that we propose our method of analyzing flutter problems. The central methodological contribution of this paper is the concept that the solution of an aeroelastic system for its flutter points is nothing other than a multiparameter eigenvalue problem. We will show the simple link between the aeroelastic stability problem and multiparameter spectral theory, and how this allows for direct solution of a variety of flutter problems. The purpose of this paper is to detail this link and to explore the mechanisms by which this direct solution may be accomplished. For this purpose we will apply our analysis to simple example problems – however, the method does extend to problems that are significantly more complex; these are considered in Pons and Gutschmidt [24, 25]. 2. Aeroelastic flutter as a multiparameter eigenvalue problem (MEP) Consider a linear finite-dimensional system with eigenvector 𝐱 ∈ ℂ𝑛 and arbitrary continuous dependence on both an eigenvalue parameter 𝜒 ∈ ℂ, and another structural or environmental parameter 𝑝 ∈ ℝ: (2) A(𝜒, 𝑝)𝐱 = 𝟎 𝑛×𝑛 where A ∈ ℂ . Any complex structural parameter can of course be split into two real parameters. The stability problem for this system (with respect to parameter 𝑝) is to find 𝑝 such that an eigenvalue of the problem (𝜒) has zero imaginary part. This point is the ‘stability boundary’: for a system with multiple structural parameters, the stability boundary may be a line or other higher-dimensional surface. We then note that the condition Im(𝜒) = 0 is equivalent to modifying the original definition of the problem such that 𝜒 ∈ ℝ and not 𝜒 ∈ ℂ. Such a manoeuvre does not seem to be immediately useful: under 𝜒 ∈ ℝ, a solution to Eq. 2 only exists on the stability boundary, and nowhere else. In order to develop, for example, iterative methods for flutter point calculation, we need to be able to define some form of solution in the subcritical and supercritical areas (above and below the stability boundary, respectively). There is an easy way of doing this. Following [26], we take the complex conjugate of Eq. 2 as another equation: A(𝜒, 𝑝)𝐱 = 𝟎, ̅ (𝜒, 𝑝)𝐱̅ = 𝟎. A (3) As 𝑝 ∈ ℝ and 𝜒 ∈ ℝ are unaffected by the conjugation, this operation enforces these conditions. This procedure has been utilized before in the analysis of delay differential equations [26], and (in a limited form) in the context of Hopf bifurcation prediction [27], but has never been applied to aeroelastic or other structural stability problems. Equation 3 is nothing other than a multiparameter eigenvalue problem (MEP): an eigenvalue problem in which the eigenvalue point is not simply defined by a scalar and an eigenvector, but by an 𝑛tuple and an eigenvector. A number of methods of analysis have been developed for such problems, and in this paper we will explore some of these. However, as the methods that are available depend strongly on the structure of matrix A, we will first define a system to work with. Page 2 of 20 Multiparameter analysis for instability problems 3. An Example Section model 3.1. Formulation Consider first the simple section model shown in Figure 1. This model has two degrees of freedom: plunge ℎ and twist 𝜃. The governing equations for this model are easy to derive; they are: 𝑚ℎ̈ + 𝑑ℎ ℎ̇ + 𝑘ℎ ℎ − 𝑚𝑥𝜃 𝜃̈ = −𝐿(𝑡), 𝐼𝑃 𝜃̈ + 𝑑𝜃 𝜃̇ + 𝑘𝜃 𝜃 − 𝑚𝑥𝜃 ℎ̈ = 𝑀(𝑡), (4) where 𝑚 and 𝐼𝑃 are the section mass and polar moment of inertia, 𝑘ℎ and 𝑘𝜃 are the section plunge and twist stiffnesses, 𝑑ℎ and 𝑑𝜃 are the section plunge and twist damping coefficients, and 𝑥𝜃 is the section’s static imbalance – defined as the distance along the 𝑥-axis from the pivot point to the centre of mass. Taking the Fourier transform, [ℎ(𝑡), 𝜃(𝑡)] = [ℎ̂, 𝜃̂]𝑒 𝜄𝜒𝑡 , of this model, we obtain: (−𝑚𝜒 2 + 𝜄𝑑ℎ 𝜒 + 𝑘ℎ )ℎ̂ + 𝑚𝑥𝜃 𝜒 2 𝜃̂ = 𝐿(𝜒, ℎ̂, 𝜃̂), 𝑚𝑥𝜃 𝜒 2 ℎ̂ + (−𝐼𝑃 𝜒 2 + 𝜄𝑑𝜃 𝜒 + 𝑘𝜃 )𝜃̂ = 𝑀(𝜒, ℎ̂, 𝜃̂). (5) To model the aerodynamic loads in the frequency domain we use Theodorsen’s unsteady aerodynamic theory [3]: 𝐿 = −𝜒 2 (𝐿ℎ ℎ̂ + 𝐿𝜃 𝜃̂), 𝑀 = 𝜒 2 (𝑀ℎ ℎ̂ + 𝑀𝜃 𝜃̂ ). (6) The aerodynamic coefficients {𝐿ℎ , 𝐿𝜃 , 𝑀ℎ , 𝑀𝜃 } are complex functions of 𝜅 – the reduced frequency; an aerodynamic parameter related to the airspeed (𝑈) by 𝜅 = 𝑏𝜒⁄𝑈. Other structural and environmental parameters involved are the air density (𝜌), the semichord (𝑏) and the distance along the 𝑥-axis from the midchord to the pivot point, as a fraction of the semichord (𝑎). See Pons [24] or Hodges and Pierce [3] for details. We assume that the flow over the airfoil is quasisteady: that is, that Theodorsen’s function takes a value of 1 universally [1,3]. We will deal with general Theodorsen aerodynamics in a later paper, as this requires iterative or approximate multiparameter solution methods which we will not cover here. We also assume without loss of generality a lift-angle of attack coefficient of 𝐶𝐿𝛼 = 2𝜋, as per thin-airfoil theory [28]. It is then customary to nondimensionalise Eq. 5. Further details of this are given in Pons [24]. The final result is a flutter problem of the form 1 1 ((M0 + G0 + G1 + G2 2 ) 𝜒 2 − D0 𝜒 − K 0 ) 𝐱 = 𝟎. 𝜅 𝜅 (7) with dimensionless parameters defined as in Table 1 and the nomenclature, and the matrix coefficients 𝑎 1 1 G0 = [𝑎 (1 + 𝑎2 )], 𝜇 8 (8) 1 −2𝜄 2𝜄(1 − 𝑎) G1 = [ ], 𝜇 −𝜄(1 + 2𝑎) 𝜄𝑎(1 − 2𝑎) Page 3 of 20 Multiparameter analysis for instability problems Table 1: Dimensionless parameter values for the section model Parameter Value mass ratio – 𝜇 20 radius of gyration – 𝑟 0.4899 0.5642 rad/s bending nat. freq. – 𝜔ℎ 1.4105 rad/s torsional nat. freq. – 𝜔𝜃 bending damping – 𝜁ℎ 1.4105 % torsional damping – 𝜁𝜃 2.3508 % −0.1 static imbalance – 𝑟𝜃 pivot point location – 𝑎 −0.2 Figure 1: Diagram of section model G2 = 1 0 [ 𝜇 0 2𝜄𝜁ℎ 𝜔ℎ D0 = [ 0 2 ], 1 + 2𝑎 M0 = [ 0 ], 2 2𝜄𝑟 𝜁𝜃 𝜔𝜃 1 −𝑟𝜃 𝜔2 K0 = [ ℎ 0 −𝑟𝜃 ], 𝑟2 0 2 2 ], 𝑟 𝜔𝜃 (8) Note that this nondimensionalisation is a convenience and not necessary prerequisite for using multiparameter solution methods. However something that will be of great use is to rearrange Eq. 7 into two polynomial forms by defining new eigenvalue parameters: Υ = 𝑈⁄𝑏, 𝜏 = 1⁄𝜅 , and 𝜆 = 1⁄𝜒. These two forms are then ((M0 + G0 )𝜒 2 + G1 Υ𝜒 + G2 Υ 2 − D0 𝜒 − K 0 )𝐱 = 𝟎, (9) which we term the Υ-𝜒 form, and ((M0 + G0 ) + G1 𝜏 + G2 𝜏 2 − D0 𝜆 − K 0 𝜆2 )𝐱 = 𝟎, (10) which we term the 𝜏-𝜆 form. Both are preferable to the 𝑘-𝜒 form, Eq. 7. We should note at this point that, although we have here derived these two forms for a small 2-DOF section model, these forms arise in many other models; often with significantly larger matrices. The solution methods we develop for this form will thus be applicable to a wide variety of problems. 3.2. Undamped system In many aeroelastic systems structural damping is negligible, in which case D0 = 0. While this may seem to be a trivial case of the preceeding systems, the omission of the structural damping term does allow us to change the structure of the system, leading to a faster computation time (as we will show later). In the case of Eq. 10, we have ((M0 + G0 ) + G1 𝜏 + G2 𝜏 2 − K 0 Λ)𝐱 = 𝟎, (11) where Λ = 𝜆2. This system is now linear in Λ, whereas it was previously quadratic in 𝜆. Note that the 𝜏-𝜒 form is the only polynomial form presented which allows us to make one parameter linear in this situation. Page 4 of 20 Multiparameter analysis for instability problems 3.3. Transformation into MEPs The three polynomial systems that we have introduced in this section (Eq. 9, 10 and 11) are not yet fully constrained MEPs, as they only consist of one equation. To constrain the system we employ the method introduced in Section 2 – the addition of an equation representing the conjugate of the initial equation. We obtain: ((M0 + G0 ) + G1 𝜏 + G2 𝜏 2 − K 0 Λ)𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝜏 2 − K ̅0 + G ̅ 0 Λ)𝐱̅ = 𝟎, ((M (12) ((M0 + G0 ) + G1 𝜏 + G2 𝜏 2 − D0 𝜆 − K 0 𝜆2 )𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝜏 2 − D ̅0 + G ̅0𝜆 − K ̅ 0 𝜆2 )𝐱̅ = 𝟎, ((M (13) ((M0 + G0 )𝜒 2 + G1 Υ𝜒 + G2 Υ 2 − D0 𝜒 − K 0 )𝐱 = 𝟎, ̅0 )𝜒 2 + G ̅1 Υ𝜒 + G ̅2 Υ 2 − D ̅0 + G ̅0𝜒 − K ̅ 0 )𝐱̅ = 𝟎. ((M (14) and These systems are all polynomial multiparameter eigenvalue problems [29]. A number of solution methods are known for such systems. 4. Direct solution via linearization 4.1. Linearization Any polynomial MEP can be made into a linear one (consisting of a zeroth-order term and first-order terms in each eigenvalue) via the process of linearization [30]. This process bears resemblance to the linearization of single-parameter polynomial eigenvalue problems; a process which is well-known [31]. Multiparameter linearization is relevant because a number of direct solution methods exist for linear MEPs. Consider first Eq. 12. This equation contains only one quadratic variable (𝜏), as Λ is already linear. To linearize this system, we define a new eigenvector which contains a factor of 𝜏: 𝐪 = [𝐱; 𝜏𝐱]. By expanding the system’s coefficient matrices and using this factor of 𝜏 in the eigenvector to reduce the order of the quadratic term, we obtain a linear problem of double the size: M0 + G0 0 ̅0 ̅ M +G ([ 0 0 ([ 0 −K ]+[ 0 −𝐼𝑛 0 ̅ 0 ] + [−K 0 −𝐼𝑛 0 G 0 ]Λ + [ 1 𝐼𝑛 0 ̅ 0] Λ + [G1 𝐼𝑛 0 G2 ] 𝜏) 𝐪 = 𝟎 0 ̅2 G ̅=𝟎 ] 𝜏) 𝐪 0 (15) The upper rows of Eq. 15 represents Eq. 12 directly, and the lower row represents the identity 𝐼𝑛 (𝜏𝐱) = 𝜏𝐼𝑛 𝐱. Note that other linearizations of similar form are possible. However, irrespective of the exact linearization used, the resulting system will be of the form: (A + BΛ + C𝜏)𝐪 = 𝟎, ̅+B ̅Λ + C̅𝜏)𝐪 (A ̅ = 𝟎. (16) Note that it is not actually necessary to linearize the second (i.e. conjugate) equation of the aeroelastic system, because the linearized conjugate equation will be the conjugate of the linearized first equation. This is a property of this method of linearization. This same linearization process can be applied to any quadratic MEP. Any problem of the form (A + B𝜆 + C𝜏 + D𝜆𝜏 + E𝜆2 + F𝜏 2 )𝐱 = 𝟎, (17) Page 5 of 20 Multiparameter analysis for instability problems can be linearized as A B ([ 0 −𝐼𝑛 0 0 C 0 0 ] + [𝐼𝑛 −𝐼𝑛 0 0 0 +[0 0 𝐼𝑛 0 D E 0 0] 𝜆 0 0 F 𝐱 0] 𝜏) [𝜆𝐱] = 𝟎. 0 𝜏𝐱 (18) We have A = M0 + G0 D = −K 0 (19) B = G1 E=0 C = −D0 F = G2 for Eq. 13, and changing 𝜆 → 𝜒 and 𝜏 → Υ, A = −K 0 D = M0 + G0 (20) B = −D0 E = G1 C=0 F = G2 for Eq. 14. These linearised systems have matrix coefficients triple the size of the original problem. 4.2. Direct solution Consider Eq. 16. Post-multiplying the first equation in this system by C̅𝐲 and premultiplying the second by C𝐱, we have (A + BΛ + C𝜏)𝐱 ⊗ (C̅𝐲) = 0, (21) ̅+B ̅Λ + C̅𝜏)𝐲 = 0. (C𝐱) ⊗ (A These two equations are both equal to zero so we may equate them. After cancelling the terms in 𝜏, the expression can be manipulated into: (22) Δ1 𝐳 = ΛΔ0 𝐳 and an enlarged eigenvector 𝐳 = 𝐱 ⊗ 𝐲 and the operator determinants ̅ Δ0 = B ⊗ C̅ − C ⊗ B ̅ Δ1 = C ⊗ A − A ⊗ C̅ ̅. ̅−B⊗A Δ2 = A ⊗ B (23) Eq. 22 is a generalized eigenvalue problem (GEP), in the single parameter 𝜆. Solvers for the generalized eigenvalue problem are very widely available. If the linear system has square coefficients of size 𝑚 then the operator determinants are of size 𝑚2 . The operator determinants can also be used to define a second GEP in 𝜏. By multiplying the ̅𝐲 and B𝐱 respectively, we can also show that: first and second equations of Eq. 16 by B (24) Δ2 𝐳 = 𝜏Δ0 𝐳. However, it is only necessary to solve one of Eq. 22 or Eq. 24: once one has been solved, then its solutions can be substituted back into the original polynomial system, which yields another smaller GEP. Alternatively, if the eigenvalue is simple then it may be computed more cheaper via Rayleigh quotients in 𝐳 or (decomposing 𝐳 = 𝐱 ⊗ 𝐲), 𝐱 and 𝐲. The system is thus completely solved for its flutter points. This direct solution method is known in mathematical literature as the operator determinant method. Its computational complexity is 𝒪(𝑛6 ), irrespective of whether 𝑛 is the size of the linear coefficients (A, B, etc.) or the polynomial coefficients (M0 , G0 , etc.) [30,32,33]. This large complexity arises from solving the generalized eigenvalue problem, an 𝒪(𝑚3 ) process by the QZ algorithm [34], with operator Page 6 of 20 Multiparameter analysis for instability problems determinants of size 𝑚 = 𝒪(𝑛2 ). The operator determinant method has not previously been used in aeroelasticity, and has only rarely seen engineering application in the study of dynamic model updating [35,36]. 4.3. Singularity One important caveat of the operator determinant approach is that the matrix Δ0 must not be singular. If it is, then the eigenvalues of the original polynomial system (e.g. Eq. 12-14) will not generally coincide with those of the GEPs of the linearized problem (Eq. 22 and 24) [30,37,38]. A linear MEP with singular Δ0 is said to be singular MEP. A large proportion of the linear flutter problems that arise in the study of aircraft aeroelasticity are singular. This is largely because the linearization of polynomial problems tends to generate singular linear problems, even if the original polynomial problem has all its matrix coefficients at full rank – compare Eq. 18. Some smaller linearizations are not necessarily singular: for example Eq. 15. However, here we find that (for our aerodynamic model) the coefficient matrix G2 is not full rank and so the linearized problem is singular anyway. In many circumstances we will be dealing with a singular problem, and for a long time the lack of a working solver for such singular problems has been a major obstacle to the application of multiparameter methods to real-world problems. However, recently a solution has been proposed. Muhič and Plestenjak [29] proved that the eigenvalues of a polynomial system are equivalent to the finite regular eigenvalues of the pair of singular operator determinant GEPs constructed via linearization. The finite regular eigenvalues of a linear two-parameter system are the pairs (𝜆, 𝜇) such that 𝑖 ∈ {1,2} [37]: rank(A𝑖 + B𝑖 𝜆 + C𝑖 𝜇) < max 2 rank(A𝑖 + B𝑖 𝑠 + C𝑖 𝑡), (𝑠,𝑡)∈ℂ (25) that is, the finite regular eigenvalues are the set of points that cause the singular problem to have its maximum rank, even though this is not full rank. On the basis of this proof, Muhič and Plestenjak [29] devised a set of algorithms which would extract the common regular part of the singular matrix pencils (i.e. matrix-valued functions polynomial in a variable 𝜆) Δ1s − 𝜆Δ0s and Δ2s − 𝜆Δ0s . This common regular part is represented by two smaller nonsingular matrix pencils (Δ1 − 𝜆Δ0 and Δ2 − 𝜆Δ0 ), the eigenvalues of which are the finite regular eigenvalues of the singular problem and thus the eigenvalues of the polynomial problem. In practical terms, this can be seen as a compression of the singular operator determinant matrices into smaller full-rank matrices. The operator determinant method, as presented in Section 4.2, can be applied to these compressed pencils. The algorithms involved in the extraction of the common regular part are presented in [29] and published also in MATLAB code [39]. We will not detail them here as they are complex. 5. Direct solution via Quasi-linearization 5.1. Quasi-linearization Hochstenbach et al. [30] recently presented another method of linearization for polynomial MEPs. Instead of increasing the size of the coefficient matrices, this method of linearization increases the number of parameters and equations in the system. To differentiate it from strict linearization, Hochstenbach et al. [30] term this new method quasi-linearization. The process is as follows. Considering again Eq. 12, we define a new eigenvalue parameter 𝛼 = 𝜏 2 . The equations then become Page 7 of 20 Multiparameter analysis for instability problems ((M0 + G0 ) + G1 𝜏 + G2 𝛼 − K 0 Λ)𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝛼 − K ̅0 + G ̅ 0 Λ)𝐱̅ = 𝟎. ((M (26) This is now a linear three-parameter eigenvalue problem, but with only two equations. We need a third equation to constrain the system. We have the relation 𝛼 − 𝜏 2 = 0, but this is nonlinear. However, noticing that we can write this relation as 𝛼 𝜏 (27) det ([ ]) = 0 𝜏 1 we can recast it as a new MEP to add to Eq. 26 (using an arbitrary eigenvector 𝐲): ((M0 + G0 ) + G1 𝜏 + G2 𝛼 − K 0 Λ)𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝛼 − K ̅0 + G ̅ 0 Λ)𝐱̅ = 𝟎, ((M 0 0 0 1 1 0 ([ ]+[ ]𝜏 + [ ] 𝛼) 𝐲 = 𝟎. 0 1 1 0 0 0 (28) In a similar way, we can linearize Eq. 13 with the definitions 𝛼 = 𝜏 2 and 𝛽 = 𝜆2 : ((M0 + G0 ) + G1 𝜏 + G2 𝛼 − D0 𝜆 − K 0 𝛽)𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝛼 − D ̅0 + G ̅ 0𝜆 − K ̅ 0 𝛽)𝐱̅ = 𝟎, ((M 0 0 0 1 1 0 ([ ]+[ ]𝜏 +[ ] 𝛼) 𝐲𝟏 = 𝟎, 0 1 1 0 0 0 0 0 0 1 1 0 ([ ]+[ ]𝜆 + [ ] 𝛽) 𝐲𝟐 = 𝟎, 0 1 1 0 0 0 (29) and Eq. 14 with the definitions 𝛼 = 𝜒 2 , 𝛽 = Υ 2 , 𝛾 = Υ𝜒: ((M0 + G0 )𝛼 + G2 𝛽 + G1 𝛾 − D0 𝜒 − K 0 )𝐱 = 𝟎, ̅0 )𝛼 + G ̅2𝛽 + G ̅1 𝛾 − D ̅0 + G ̅0𝜒 − K ̅ 0 )𝐱 = 𝟎, ((M 1 0 0 1 0 0 ([ ]𝛼 + [ ]𝜒 +[ ]) 𝐲𝟏 = 𝟎, 0 0 1 0 0 1 0 1 0 0 1 0 ([ ]𝛼 + [ ]𝛽 + [ ] 𝛾) 𝐲𝟐 = 𝟎, 0 0 1 0 0 1 (30) These systems can be solved via the operator determinant method, in a more general form than that presented in Section 4.2. 5.2. The general operator determinant method The general form of a nonhomogeneous MEP is 𝑁 𝑊𝑖 (𝜼) = A𝑖0 𝐱 𝑖 + ∑ 𝜂𝑗 A𝑖𝑗 𝐱 𝑖 , 𝑖 = 1, … , 𝑛 (31) 𝑗=1 where 𝜼 is the vector of eigenvalues (𝜂𝑗 being the individual eigenvalues), A𝑖𝑗 are the coefficient matrices, which can be complex and of different sizes for each equation, and 𝐱 𝑖 are the eigenvectors. Eq. 31 can be visualized as a non-square array of matrices: A10 A20 [ ⋮ A𝑁0 A11 A21 ⋮ A𝑁1 A12 A22 ⋮ A𝑁2 ⋯ A1𝑁 ⋯ A2𝑁 ]. ⋱ ⋮ ⋯ A𝑁𝑁 Page 8 of 20 (32) Multiparameter analysis for instability problems In a process analogous to that presented in Section 4.4, we can construct a series of operator determinants for this system. There are 𝑁 + 1 such operator determinants (Δ0 through to Δ𝑁 ), of size 𝑛𝑁 for constant coefficient size 𝑛. They correspond to taking determinants of Eq. 32, with certain columns removed or inserted, and with the normal scalar multiplication operation replaced by a Kronecker product between matrices. Definitions and derivations of these determinants may be found in [40–44], and one software implementation in [39]. In the two-parameter case this analysis collapses into that of Section 4.2. These general operator determinants allow us to compute the eigenvalues of the Eq. 31 via generalized eigenvalue problems. Providing Δ0 is nonsingular, it holds that Δ𝑖 𝐱 = 𝜂𝑖 Δ0 𝐱. (33) In this way we are able to solve the quasi-linearized systems given in Section 5.1 (we will discuss singularity in Section 5.4). However, note that already by using this quasilinearization process we have made the operator determinants smaller than they were with standard linearization. Using Eq. 12, the operator determinants are square and of size 2𝑛2 , and using Eq. 13 or 14 they are of size 4𝑛2 . With standard linearization they are of size 4𝑛2 and 9𝑛2 , respectively. 5.3. Computing operator determinants In the two-parameter case, the computation of the operator determinants is trivial. However in the case of larger systems we must find some other approach. In the general case, the operator determinants may be computed by modifying the Leibniz formula for the determinant [40]: 𝑁 Δ(M) = ∑ sgn(𝐬) ⊗ M𝑖,𝐬𝑖 , 𝐬 ∈ 𝑆𝑁 𝑖=1 (34) where 𝑆𝑁 is the set of permutations of the set {1, … , 𝑁}, sgn(𝐬) is the sign of the permutation vector 𝐬 ∈ 𝑆𝑁 . M is the square matrix array (a modification of Eq. 32) corresponding to the operator determinant desired. The notation ⊗𝑁 𝑖 denotes the repeated application of the Kronecker product. Note that the tensor determinant definition in [40] is slightly erroneous as the factor (−1)sgn(𝜎) in their tensor determinant expressions should be either sgn(𝜎) or (−1)𝑁(𝜎) , where 𝑁(𝜎) is the number of inversions in 𝜎. Alternatively, Muhič and Plestenjak [39] devised an operator determinant Laplace expansion [45] that is able to compute the operator determinants of a system of arbitrary size, via a process of recursion: 𝑁 Δ(M) = ∑(−1)𝑖+1 M𝑖1 ⊗ 𝑚𝑖1 , (35) 𝑖=1 where 𝑚𝑖1 denotes the minor entry corresponding to M𝑖1 . The summation in Eq. 35 follows the first column of M, though, of course, many other summation paths could be used. It should be noted that the use of the Laplace or Leibniz methods for computing the determinant of an ordinary matrix have very high non-polynomial computational complexity costs – 𝒪(𝑛!) and 𝒪(𝑛! 𝑛), respectively [46,47]. 5.4. Singularity One major advantage of the direct solver based on quasi-linearization is that it generates linearized problems that are not always singular – when the coefficient matrices are of full rank, Eq. 28-30 are in general nonsingular. However in our case the coefficient G2 is singular, and so it happens that all these three equations are singular anyway. The theory of the Page 9 of 20 Multiparameter analysis for instability problems compression process noted in Section 4.3 has never been extended to the general multiparameter case, and so we have no rigorous method of solving singular MEPs with 𝑁 > 2. However, we do have a practical method of doing so. Numerical experiments (some of which are detailed in Section 6) indicate that, although the compression algorithm takes only Δ0 , Δ1 and Δ2 as an input, it will successfully compress these three operator determinants for the general multiparameter problem (ignoring all the others). In the case of Eq. 28-30, with these three operator determinants we can solve for all the eigenvalue parameters of the system, irrespective of the order in which we arrange the eigenvalues. We should note that there is no justification for this procedure other than the experimental evidence that the algorithm works – evidence confirmed also by the nonrigourous 𝑁 > 2 compression process utilized in [39]. The common regular part relationship proved by [29] applies only to two-parameter eigenvalue problems arising from linearization, and indeed it is not clear how the concept of the common regular part would extend to an 𝑁 > 2 problem with more than two pencils. That the two-parameter algorithm does work for the first three operator determinants of an 𝑁 > 2 problem implies that the algorithm will generalize – it is a question of working out what this generalization is. This is an interesting area for further research. For the purposes of this paper, however, we will use the compression algorithm for the quasi-linearized Δ0 , Δ1 and Δ2 without proof. 6. Numerical experiments 6.1. Undamped model We are now in a position to compute the flutter points of the models we introduced in Section 3. We have devised two direct solvers to do so: a direct solver with linearization and a direct solver with quasi-linearization. Consider first the undamped section model (Eq. 12), with parameter values as per Table 1. To validate our direct solution methods, we first produce a modal damping plot (Figure 2). Three points of neutral stability may be observed 𝜏 = ±1.00 and 𝜏 = 0. The point 𝜏𝐹 = 1.00 is the only physical flutter point. We can link the modal damping curve for this flutter point with the lower modal frequency path in the Re(𝜒) plot (this requires data not observed directly on Figure 2). We thus can estimate that 𝜒𝐹 = 1.32 rad/s (Λ 𝐹 = 0.57s2 /rad2 ). The point at 𝜏 = −1.00 along the same mode corresponds to a nonphysical flutter event occurring at negative airspeed. Finally, at 𝜏 = 0 both modes have neutral stability (at 𝜒 = 0.548 rad/s and 𝜒 = 1.426 rad/s or Λ = 3.33 s2 /rad2 and Λ = 0.492s2 /rad2 ). This, however, merely represents the fact that the structure is undamped at zero airspeed due to the lack of any structural damping in the model. The divergence point of this system does not appear on this plot, as it occurs at infinite 𝜏 and Λ. However, the computation of divergence points does not require multiparameter methods anyway, as if the frequency 𝜒 is assumed to be zero in the governing equation (e.g. Eq. 9) then the problem becomes a single parameter eigenvalue problem in the airspeed parameter (e.g. Υ). This problem can then be solved with single-parameter solvers. We then compute the flutter points of the system via our two direct solution methods, yielding flutter points identical to those detailed above for the presented accuracy. With the solver based on linearization, the uncompressed operator determinants are of size 16 and the compressed ones of size 4. With the solver based on quasi-linearization the size of the uncompressed determinants can be reduced significantly – down to 8 – while producing compressed determinants of the same size. The quasi-linearization method thus reduces the time required for the compression process, as the finite-regular part extraction algorithm Page 10 of 20 Multiparameter analysis for instability problems works by successively increasing the rank of the system (not all at once). However, for this problem the computation time is too small for any meaningful assessment: we will investigate computation time more fully in Section 6.3. Figure 3 shows the system flutter points superimposed on a contour plot [24]. As can be seen, there is an exact agreement between the direct solutions and the contour plot solutions (the intersections of Re(𝑑) = 0 and Im(𝑑) = 0). Figure 2: Modal damping plot for the undamped section model. Figure 3: Contour plot for the undamped section model, showing identical solutions from the two direct solvers. Page 11 of 20 Multiparameter analysis for instability problems 6.2. Damped model We now simulate the damped system, in both forms (Eq. 13 and 14). Figure 4 shows a contour plot for the 𝜏-𝜆 form of this system (Eq. 16), with direct solutions from both solution methods. There is a single physical flutter point at 𝜏𝐹 = 1.650, 𝜆𝐹 = 0.8336 s/rad, and a nonphysical flutter point at a small negative 𝜏. The divergence point occurs at infinite 𝜏 and 𝜆. As expected, the solutions from both direct solution methods are identical to each other and the solutions from the contour plot. Figure 5 shows a contour plot of the Υ-𝜒 form (Eq. 14) with direct solutions from both solution methods. This physical flutter point can be located at Υ𝐹 = 1.98 Hz, 𝜒𝐹 = 1.20 rad/s and the divergence point at Υ𝐷 = 3.99 Hz. Again, the flutter points computed with direct solvers agree exactly with those seen on the contour plot, and the results from the system as a whole agree with those of the 𝜏-𝜆 form (𝜆𝐹 = 1⁄𝜒𝐹 = 0.83 s/rad and 𝜏𝐹 = Υ𝐹 ⁄𝜆𝐹 = 1.65). 6.3. Computation time Given the high computational complexity of these direct solution methods – 𝒪(𝑛6 ) – we are interested in the maximum system size for which a direct solution is practical. We have already found that for 𝑛 = 2 the computational effort required is tiny, and so at the very least these direct solvers are useful for small reduced-order models as might be used in a preliminary design analysis. This alone is of use, as the directness of these solvers makes them ideal for use in optimization routines or other applications in which a large number of computations must be performed with limited user guidance. To gain a better understanding of the computational complexity characteristics of our methods, we simulate a series of systems of increasing matrix coefficient size. We generate random complex-valued matrices for the polynomial coefficients (M0 , G0 , etc.), of size 𝑛 = 2𝑘 with 𝑘 ∈ {1, 2, … , 5}. For robustness, we average the results for 𝑘 = 1 over 50 random matrices, for 𝑘 = 2 over 10 random matrices, for 𝑘 = 3 over 5 random matrices; and for systems larger than this we generate only one matrix. Figure 6 shows the wall-clock solution time against system size for the direct solver with linearization, for a 64-bit Intel i7-4770 with 3.4 GHz processor and 16 GB RAM, running MATLAB R2014b. Note that the 𝜆 solution time denotes the time required to compute the 𝜆 components of the solution via a series of one-parameter eigenvalue problems. As can be seen, the compression process is the most expensive component of the algorithm, making up a constant fraction of about 65% of the total computation time over the entire range of 𝑛. The GEP solution time is initially completely negligible but becomes more significant as system size increases: by 𝑛 = 32 it makes up 34% of the total computation time. The 𝜆-solution and setup process are never of much significance. We expect from computational complexity theory that the gradient of the GEP-solution time curve in log-log units will be approximately 6.0 (Section 4.2). Fitting a linear curve through this data yields a gradient of 5.46, with an 𝑅 2 of 0.994. However, the GEP solution-time curve is slightly concave, with a maximum gradient of 6.47. The algorithm can then be said to have complexity 𝒪(𝑛6 ) overall. Figure 7 shows the wall-clock solution time against system size for the direct solver with quasi-linearization. This simulation was run on the same Intel i7-4770 platform with MATLAB R2014b. The random coefficient matrices used are identical to those in Figure 6.And although now the quasi-linearized system in no longer singular, we still apply the compression algorithm to capture some of its overhead costs.. As can be seen, these costs are not large but are still more significant than the GEP and 𝜆 solution times for some system sizes. Page 12 of 20 Multiparameter analysis for instability problems Figure 4: Contour plot for the damped section model (𝜏-𝜆 form), showing identical solutions from the two direct solvers. Figure 5: Contour plot for the damped section model (Υ-𝜒 form), showing identical solutions from the two direct solvers. Page 13 of 20 Multiparameter analysis for instability problems Figure 6: Wall-clock solution time against system size for the direct solver with linearization. Figure 7: Wall-clock solution time against system size for the direct solver with quasi-linearization. Page 14 of 20 Multiparameter analysis for instability problems These GEP and 𝜆 solution times are effectively identical to those of the direct solver with linearization, as the algorithm is solving a GEP of exactly the same size, yielding exactly the same eigenvalues. However, the most striking aspect of Figure 7 is the fact that the setup time occupies the majority of the required computation time right up to 𝑛 = 16 (after which it is surpassed by the GEP solution time). The setup process involved using defining the system array (Eq. 32) in a cell array, and computing the necessary operator determinants of this array (Δ0 and Δ2 ) using the modified Leibniz formula (Eq. 34). On investigating these two procedures we find that it is the computation of the Kronecker products within the operator determinant which occupies the greatest time. However, despite this, the actual computational complexity of the algorithm as a whole is lower than that of the direct solver with linearization. GEP solution time is still approximately 𝒪(𝑛6 ), though the concavity that was noted in Figure 6 is still present. Finally, Figure 8 shows a comparison of the total computation times for the two algorithms. As can be seen, below a system coefficient size of approximately 10, linearisation is faster than quasi-linearisation (at 𝑛 = 2, over an order of magnitude faster). Above this coefficient size, quasi-linearisation is faster (at 𝑛 = 32, twice as fast). To keep computation times below 10s, the system size must be below 11; to keep it below 1s, 8, and below 0.1s, 5. As it stands, the operator determinant method is not suitable for use with finite-element models or any other systems with a large number of degrees of freedom. It is, however, useful for the solution of simple reduced order models, e.g. in a preliminary design optimization. Figure 8: Total wall-clock solution time against system size for the two direct solver variants. Page 15 of 20 Multiparameter analysis for instability problems 7. Discussion 7.1. Alternative solution methods We have so far discussed the operator determinant method – a solution method which is direct but computationally expensive. However, two other classes of method are also available. The Sylvester-Arnoldi type methods are a set of closely-related algorithms that are only valid for two-parameter problems, but are fast and can handle large systems [48]. They still use operator determinants to reformulate the system into a set of generalized eigenvalue problems, but instead of solving this set of GEPs directly, the solution procedure is optimized based on the given knowledge that the operator determinants consist of a difference of two Kronecker products. For example, each step of a Krylov subspace procedure solution procedure for the GEP reduces to the solution of a Sylvester equation. Several related algorithms may be devised this way, and in some cases the computational complexity of the solution process can be reduced all the way down to 𝒪(𝑛3 ) [48]. However, this technique shows little potential for generalization to 𝑁-parameter systems, as it relies on being able to reformulate the operator determinant GEP into a simple and well-known matrix equation. As the operator determinant definition becomes more complex, the resulting matrix equation changes and efficient solvers may not be available. There is also no easy extension for singular problems. Subspace methods for one-parameter eigenvalue problems are based around generating a series of linear spaces that eventually approximate one of the system’s eigenspaces (the linear space of eigenvectors corresponding to a given eigenvalue). The Jacobi-Davidson and Rayleigh-Ritz methods are well-known one-parameter subspace methods, which can be generalized to apply to two-parameter systems [26,32,49–51], though this is not without difficulties [49]. These methods do not invoke the operator determinants, and show potential for generalization both to 𝑁-parameter and to polynomial systems. The Jacobi-Davidson is applicable to singular systems and has previously been tested on singular linearized aeroelastic stability problems [25]; but its performance was observed to be poorer than the operator determinant method. However the solution of singular MEPs has only been achieved very recently (2010 [29]), and so it is likely that the next few years will bring significant developments in this area. 7.2. Extensions to the concept The core methodology presented in this paper can be extended to a wide variety of problems. Not only more complex two-parameter flutter models – considered in Pons [24] and Pons and Gutschmidt [25] – but also stability problems from a wide variety of fields, including systems with entirely different eigenvalue definitions. Indeed, any combination of model parameters in a stability model can indeed be treated as multiparameter eigenvalues; and scalar constraints on these parameters can cast as eigenvalue problems by introducing a scalar eigenvector [24]. For example, in aeroelastic model, given the location of a flutter point, we could solve for the sets of parameter values that could generate such a flutter point – allowing us to perform model identification based on flutter point information. This could pave the way for a least-squares approach for overconstrained multiparameter systems. Alternatively, we could introduce a flight altitude / air density / Mach number parameter and compute points on the aeroelastic flutter envelope of the aircraft. The multiparameter formulation provides a versatile way of analysing stability problems in any combination of parameters. Page 16 of 20 Multiparameter analysis for instability problems 8. Conclusion In this paper we have demonstrated and discussed the use of multiparameter solution techniques for the solution of aeroelastic and related stability problems. We have introduced the link between multiparameter spectral theory and stability analysis, and we showed how this link can be used to reformulate stability problems with a complex-valued stability metric and a pertinent environmental parameter into a two-parameter eigenvalue problem. We demonstrated that this allows the direct solution of stability problems that are linear or polynomial in these parameters, and we discussed aspects of the solution process, including the linearization and quasi-linearization of polynomial problems, general N-parameter problems, computational costs and approaches to problem singularity. 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Cluster Synchronization of Coupled Systems with Nonidentical Linear Dynamics Zhongchang Liu∗ and Wing Shing Wong arXiv:1502.07481v2 [cs.SY] 30 Dec 2015 Abstract This paper considers the cluster synchronization problem of generic linear dynamical systems whose system models are distinct in different clusters. These nonidentical linear models render control design and coupling conditions highly correlated if static couplings are used for all individual systems. In this paper, a dynamic coupling structure, which incorporates a global weighting factor and a vanishing auxiliary control variable, is proposed for each agent and is shown to be a feasible solution. Lower bounds on the global and local weighting factors are derived under the condition that every interaction subgraph associated with each cluster admits a directed spanning tree. The spanning tree requirement is further shown to be a necessary condition when the clusters connect acyclicly with each other. Simulations for two applications, cluster heading alignment of nonidentical ships and cluster phase synchronization of nonidentical harmonic oscillators, illustrate essential parts of the derived theoretical results. Index Terms Cluster synchronization; Coupled linear systems; Nonidentical systems; Graph topology I. I NTRODUCTION Understanding the interaction of coupled individual systems continues to receive interest in the engineering research community [1]. Recently, more attention has been paid to cluster synchronization problems which have much wide applications, such as segregation into small subgroups for a robotic team [2] or physical particles [3], predicting opinion dynamics in social networks [4], and cluster phase synchronization of coupled oscillators [5], [6]. In the models reported in most of the literature, the clustering pattern is predefined and fixed; research focuses are on deriving conditions that can enforce cluster synchronization for This work is supported by the Hong Kong RGC Earmarked Grant CUHK 14208314. The authors are with Department of Information Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. ∗ Corresponding author. E-mail: zcliu@ie.cuhk.edu.hk. various system models [7]–[17]. Preliminary studies in [7]–[10] reported algebraic conditions on the interaction graph for coupled agents with simple integrator dynamics. Subsequently, a cluster-spanning tree condition is used to achieve intra-cluster synchronization for firstorder integrators (discrete time [11] or continuous time [12]), while inter-cluster separations are realized by using nonidentical feed-forward input terms. For more complicated system models, e.g., nonlinear systems ([13]–[15]) and generic linear systems ([16], [17]), both control designs and inter-agent coupling conditions are responsible for the occurrence of cluster synchronization. For coupled nonlinear systems, e.g., chaotic oscillators, algebraic and graph topological clustering conditions are derived for either identical models ([13]) or nonidentical models ([14], [15]) under the key assumption that the input matrix of all systems is identical and it can stabilize the system dynamics of all individual agents via linear state feedback (i.e., the so-called QUAD condition). For identical generic linear systems which are partial-state coupled [16], [17], a stabilizing control gain matrix solved from a Ricatti inequality is utilized by all agents, and agents are pinned with some additional agents so that the interaction subgraph of each cluster contains a directed spanning tree. The system models introduced above can describe a rich class of applications for multiagent systems. A common characteristic is that the uncoupled system dynamics of all the agents can be stabilized by linear state feedback attenuated by a unique matrix (i.e., static state feedback). This simplification allows the derivation of coupling conditions to be independent of the control design of any agent, and thus offers scalability to a static coupling strategy. This kind of benefit still exists for nonidentical nonlinear systems which are full-state coupled, as all the system dynamics can be constrained by a common Lipchitz constant (Lipchitz can imply the QUAD condition [18]). However, for the class of partial-state coupled nonidentical linear systems, the stabilizing matrices for distinct linear system models are usually different. Then the coupling conditions under static couplings will be correlated with the control designs of all individual systems. This correlation not only harms the scalability of a coupling strategy but also increases the difficulty in specifying a graph topological condition on the interaction graph. The goal of this paper is to achieve state cluster synchronization for partial-state coupled nonidentical linear systems, where agents with the same uncoupled dynamics are supposed to synchronize together. This is a problem of practical interest, for instance, maintaining different formation clusters for different types of interconnected vehicles, providing different synchronization frequencies for different groups of clocks using coupled nonidentical harmonic oscil- lators, reaching different consensus values for people with different opinion dynamics, and so on. In order to relieve the difficulties in using the conventional static couplings, couplings with a dynamic structure is proposed by introducing a vanishing auxiliary variable which facilitates interactions among agents. With the proposed dynamic couplings, an algebraic necessary and sufficient condition, which is independent of the control design, is derived. This newly derived algebraic condition subsumes those published for integrator systems in [7]–[10] as special cases. Due to the entanglement between nonidentical system matrices and the parameters from the interaction graph, the algebraic condition is not straightforward to check. Thus, a graph topological interpretation of the algebraic condition is provided under the assumption that the interaction subgraph associated with each cluster contain a directed spanning tree. We also derive lower bounds for the local coupling strengths in different clusters, which are independent of the control design due to the dynamic coupling structure. This spanning tree condition is further shown to be a necessary condition when the clusters and the intercluster links form an acyclic structure. This conclusion reveals the indispensability of direct links among agents belonging to the same cluster, and further strengthens the sufficiency statement presented initially in [16]. Another contribution of the proposed dynamic couplings in comparison to those static couplings in [17] is that the lower bound of a global factor which weights the whole interaction graph is also independent of the control design. For this reason, the least exponential convergence rate of cluster synchronization is characterized more explicitly than that in [17]. The derived results in this paper are illustrated by simulation examples for two applications: cluster heading alignment of nonidentical ships and cluster phase synchronization of nonidentical harmonic oscillators. The organization of this paper is as follows: Following this section, the problem formulation is presented in Section II. In Section III, both algebraic and graph topological conditions for cluster synchronization are developed. Simulation examples are provided in Section IV. Concluding remarks and discussions for potential future investigations follow in Section V. II. P ROBLEM S TATEMENT Consider a multi-agent system consisting of L agents, indexed by I = {1, . . . , L}, and S N ≤ L clusters. Let C = {C1 , . . . , CN } be a nontrivial partition of I, that is, N i=1 Ci = I, Ci 6= ∅, and Ci ∩ Cj = ∅, ∀i 6= j. We call each Ci a cluster. Two agents, l and k in I, belong to the same cluster Ci if l ∈ Ci and k ∈ Ci . Agents in the same cluster are described by the same linear dynamic equation: ẋl (t) = Ai xl (t) + Bi ul (t), l ∈ Ci , i = 1, . . . , N (1) where xl (t) ∈ Rn with initial value, xl (0), is the state of agent l and ul (t) ∈ Rmi is the control input; Ai ∈ Rn×n and Bi ∈ Rn×mi are constant system matrices which are distinct for different clusters. A. Interaction graph topology and graph partitions A directed interaction graph G = (V, E, A) is associated with system (1) such that each agent l is regarded as a node, vl ∈ V, and a link from agent k to agent l corresponds to a directed edge (vk , vl ) ∈ E. An agent k is said to be a neighbor of l if and only if (vk , vl ) ∈ E. The adjacency matrix A = [alk ] ∈ RL×L has entries defined by: alk 6= 0 if (vk , vl ) ∈ E, and alk = 0 otherwise. In addition, all = 0 to avoid self-links. Note that alk < 0 means that the influence from agent k to agent l is repulsive, while links with alk > 0 are cooperative. P Define L = [blk ] ∈ RL×L as the Laplacian of G, where bll = Lk=1 alk and blk = −alk for any k 6= l. Corresponding to the partition C = {C1 , . . . , CN }, a subgraph Gi , i = 1, . . . , N , of G contains all the nodes with indexes in Ci , and the edges connecting these nodes. See Fig. 1 for illustration. Without loss of generality, we assume that each cluster Ci , i = 1, . . . , N , consists P of li ≥ 1 agents ( N i=1 li = L), such that C1 = {1, . . . , l1 }, . . ., Ci = {σi + 1, . . . , σi + li }, Pi−1 . . ., CN = {σN + 1, . . . , σN + lN } where σ1 = 0 and σi = j=1 lj , 2 ≤ i ≤ N . Then, the Laplacian L of the graph G can be partitioned  L11 L12  L  21 L22 L= . ..  .. .  LN 1 LN 2 into the following form:  · · · L1N  · · · L2N   , .. ...  .  · · · LN N (2) where each Lii ∈ Rli ×li specifies intra-cluster couplings and each Lij ∈ Rli ×lj with i 6= j, specifies inter-cluster influences from cluster Cj to Ci , i, j = 1, · · · , N . Note that Lii is not the Laplacian of Gi in general. Construct a new graph by collapsing any subgraph of G, Gi , into a single node and define a directed edge from node i to node j if and only if there exists a directed edge in G from a node in Gi to a node in Gj . We say G admits an acyclic partition with respect to C, if G¹1 G1 1 2 01 1 3 4 02 3 4 G¹2 G2 Fig. 1. 2 A graph topology partitioned into two subgraphs. the newly constructed graph does not contain any cyclic components. If the latter holds, by 4 1 2 relabeling the clusters and the nodes in G, we can represent the Laplacian L in a lower -a triangular form 1 a  -1  1 L 0  11   ..  . . L= . . , 3   -1 LN 1 · · · LN N -1 (3) 4 so that each cluster Ci receives no input from clusters Cj if j > i. In Fig. 1, the two subgraphs -1 1 G1 and G2 illustrate an acyclic partition of the whole graph. -1 1 1 c1 B. The cluster synchronization problem 1 5 3 -1 c1 c2 The main task in this paper is to achieve cluster synchronization for the states of systems 2 6 -1 1 4 in (1) via distributed couplings through the control inputs ul (t) which is defined as follows: for l ∈ Ci , i = 1, . . . , N ul (t) = Ki ηl (t) " 1 X η̇l (t) = (Ai + Bi Ki )ηl + c ci c1 c1 2 3 alk (ηk − ηl + xl − xk ) -1 (4a) 4 1 k∈Ci  + X alk (ηk − ηl +1xl −-1xk ) , 1 -1 (4b) k∈C / i 1 n c c 3 where Ki is the control gain matrix to be specified; the7 vector3 ηl (t)8 ∈ R , l ∈ I is an5 6 -1 auxiliary control variable with initial value, ηl (0); c > 0 is the global weighting factor for the whole interaction graph G; each ci > 0 is a local weighting factor used to adjust the intra-cluster coupling strength of cluster Ci . Note that the couplings in (4) takes a dynamic structure. The reasons why conventional static couplings (e.g., those in [13]–[17]) are not c2 preferred will be explained in details in the main part. The cluster synchronization problem is defined below. Definition 1: A linear multi-agent system in (1) with couplings in (4) is said to achieve N -cluster synchronization with respect to the partition C if the following holds: for any xl (0) and ηl (0), l ∈ I, limt→∞ kxl (t) − xk (t)k = 0 ∀k, l ∈ Ci , i = 1, . . . , N , limt→∞ ηl (t) = 0 ∀l ∈ I, and for any set of xl (0), l ∈ I there exists a set of ηl (0), l ∈ I such that lim supt→∞ kxl (t) − xk (t)k > 0 ∀l ∈ Ci , ∀k ∈ Cj , ∀i 6= j. In the definition, all auxiliary variables, ηl (t), l ∈ I are required to decay to zero so as to guarantee that the control effort of every agent is essentially of finite duration. For state separations among distinct clusters, one should not expect them to happen for any set of xl (0)’s and ηl (0)’s; an obvious counterexample is that all system states will stay at zero when xl (0) = ηl (0) = 0 for all l ∈ I. Some assumptions throughout the paper are in order. Assumption 1: Each of the pairs (Ai , Bi ), i = 1, . . . , N is stabilizable. Assumption 2: Each Ai has at least one eigenvalue on the closed right half plane. This assumption excludes trivial scenarios where all system states synchronize to zero. To deal with stable Ai ’s, one may introduce distinct feed-forward terms in ul (t) as studied in [11], [12]. In order to segregate the system states according to the uncoupled system dynamics in (1), an additional mild assumption is made on the system matrices Ai ’s, namely, they can produce distinct trajectories. Rigorously, for any i 6= j, the solutions xi (t) and xj (t) to the linear differential equations ẋi (t) = Ai xi (t) and ẋj (t) = Aj xj (t), respectively satisfy lim supt→∞ kxi (t) − xj (t)k > 0 for almost all initial states xi (0) and xj (0) in the Euclidean space Rn . Assumption 3: Every block Lij of L defined in (2) has zero row sums, i.e., Lij 1lj = 0. This assumption guarantees the invariance of the clustering manifold {x(t) = [xT1 (t), . . . , xTL (t)]T : x1 (t) = · · · = xl1 (t), . . . , xσN +1 (t) = · · · = xL (t)}. It is imposed frequently in the literature to result in cluster synchronization for various multi-agent systems (see [7]–[10], [13], [14], [16], [17]). To fulfill it, one can let positive and negative weights be balanced for all of the links directing from one cluster to any agent in another cluster. The negative weights for inter-cluster links is supposed to provide desynchronizing influences. Note also that with Assumption 3 each Lii is the Laplacian of a subgraph Gi , i = 1, . . . , N . Notation: 1n = [1, 1, . . . , 1]T ∈ Rn . The identity matrix of dimension n is In ∈ Rn×n . The symbol blockdiag{M1 , . . . , MN } represents the block diagonal matrix constructed from the N matrices M1 , . . . , MN . “⊗” stands for the Kronecker product. A symmetric positive (semi-) definite matrix S is represented by S > 0(S ≥ 0). Reλ(A) is the real part of the eigenvalue of a square matrix A, and σ(A) is the spectrum of A. III. C ONDITIONS FOR ACHIEVING C LUSTER S YNCHRONIZATION In this section, we first present a necessary and sufficient algebraic clustering condition that entangles parameters from the Laplacian L and the system matrices Ai ’s. Then, we present some graph topological conditions which offer more intuitive interpretations. The following discussion makes use  cL  1 11  .. Lc =  .  LN 1 of the weighted graph Laplacian  ··· L1N   .. ...  ∈ RL×L , .  · · · cN L N N (5) and the following matrix:  c L̂ ···  1 11  .. ... L̂c =  .  L̂N 1 · · · L̂1N .. . cN L̂N N     ∈ R(L−N )×(L−N ) ,  (6) where each L̂ij , i, j = 1, . . . , N is a block matrix defined as L̂ij = L̃ij − 1li γijT , (7) with γij = [bσi +1,σj +2 , · · · , bσi +1,σj +lj ]T ∈ Rlj −1 ,   b · · · bσi +2,σj +lj  σi +2,σj +2    .. .. . . L̃ij =  .  ∈ R(li −1)×(lj −1) . . .   bσi +li ,σj +2 · · · bσi +li ,σj +lj The two matrices Lc and L̂c have the following relation, whose proof is shown in Appendix I. Lemma 1: Under Assumption 3, each diagonal block Lii in Lc has exactly one zero eigenvalue if and only if the corresponding matrix L̂ii defined in (7) is nonsingular. Moreover, Lc defined in (5) has exactly N zero eigenvalues if and only if the matrix L̂c defined in (6) is nonsingular. A. Algebraic clustering conditions Under Assumption 1, for each i = 1, . . . , N there exists a matrix Pi > 0 satisfying the Riccati equation Pi Ai + ATi Pi − Pi Bi BiT Pi = −I. (8) Choose the control gain matrices as Ki = −BiT Pi , and denote Â = blockdiag{Il1 −1 ⊗ A1 , . . . , IlN −1 ⊗ AN }. Then, we have the following algebraic condition to check the cluster synchronizability. Theorem 1: Under Assumptions 1 to 3, the multi-agent system in (1) with couplings in (4) achieves N -cluster synchronization if and only if the matrix Â − cL̂c ⊗ In is Hurwitz, where L̂c is defined in (6). The proof is given in Appendix II. The matrix Â − cL̂c ⊗ In contains parameters from the interaction graph that entangle intimately with those from the system dynamics. In general, it is not possible to verify the above synchronization condition by simply comparing the eigenvalues of L̂ with those of Ai ’s. However, one can do so for a homogeneous multi-agent system as stated in the following corollary. Corollary 1: Under Assumptions 1 to 3, and with identical system parameters: Ai = A, Bi = B, Ki = K, for all i = 1, . . . , N , a multi-agent system in (1) with couplings in (4) achieves N -cluster synchronization if and only if the following holds: min Reλ(cL̂c ) > max Reλ(A). σ(L̂c ) σ(A) (9) A sketch of the proof for this corollary is given in Appendix III. Remark 1: In words, the algebraic condition (9) states that the weighted graph Laplacian Lc has exactly N zero eigenvalues, and all the nonzero eigenvalues have large enough positive real parts to dominate the unstable system dynamics described by A. This condition implies that related results in [7]–[9] are special cases with A = 0, B = 1 and K = 1. It also includes part of the results in [10], which are obtained for identical double integrators. Note that with identical system parameters, one can use static controllers without involving the auxiliary variables ηl ’s. However, in that case the synchronized state in each cluster depends linearly on the initials states xl (0)’s only. For certain initial state sets, state separations in the limit cannot be guaranteed. B. Graph topological conditions The matrix Â − cL̂c ⊗ In in Theorem 1 can be proven to be Hurwitz for certain graph topologies in conjunction with some lower bounds on the weighting factors. To do so, the following well-known result for subgraphs will be useful. Lemma 2 ([19]): Let Gi be a non-negatively weighted subgraph. Then, the Laplacian Lii of Gi has a simple zero eigenvalue and all the nonzero eigenvalues have positive real parts if and only if Gi contains a directed spanning tree. By Lemma 1, there exists a positive definite matrix Ŵi ∈ R(li −1)×(li −1) such that Ŵi L̂ii + L̂Tii Ŵi > 0, i = 1, . . . , N, (10) if the corresponding subgraph Gi satisfies the conditions in Lemma 2. Denote Ŵ = blockdiag{Ŵ1 , . . . , ŴN }, and let L̂o = L̂c − L̂d , (11) with L̂d = blockdiag{c1 L̂11 , . . . , cN L̂N N }. The following theorem states the main result of this subsection. Theorem 2: Under Assumptions 1 to 3, a multi-agent system in (1) with couplings in (4) achieves N -cluster synchronization exponentially fast with the least rate of 21 [c − λmax (Â + ÂT )], if each subgraph, Gi , contains only cooperative edges and has a directed spanning tree, and the weighting factors satisfy c> max λmax (Ai + ATi ), (12) i∈{1,...,N } and for each i = 1, . . . , N ci ≥ λmax (Ŵ) − λmin (Ŵ L̂o + L̂To Ŵ) λmin (Ŵi L̂ii + L̂Tii Ŵi ) , (13) where each Ŵi satisfies (10). Proof: Following the proof of the sufficiency part of Theorem 1, we need to show that the system ζ̇(t) = (Â − cL̂c ⊗ In )ζ(t) (14) is exponentially stable under the conditions in Theorem 2. First, these conditions guarantee the existence of positive definite matrices, Ŵi ’s, satisfying (10). Hence, (13) can be written as ci λmin (Ŵi L̂ii + L̂Tii Ŵi ) + λmin (Ŵ L̂o + L̂To Ŵ) ≥ λmax (Ŵ) for i = 1, . . . , N . These inequalities together with Weyl’s eigenvalue theorem ([20]) yield the following: λmin (Ŵ L̂c + L̂Tc Ŵ) = λmin (Ŵ L̂d + L̂Td Ŵ + Ŵ L̂o + L̂To Ŵ) ≥ λmin (Ŵ L̂d + L̂Td Ŵ) + λmin (Ŵ L̂o + L̂To Ŵ) ≥ λmax (Ŵ), which further implies that Ŵ L̂c + L̂Tc Ŵ ≥ Ŵ. (15) Now, consider the Lyapunov function candidate V (t) = ζ(t)T (Ŵ ⊗ In )ζ(t) for the system (14). Taking time derivative on both sides of V (t), one gets V̇ (t) = ζ T (t)(Ŵ ⊗ In )(Â − cL̂c ⊗ In ) + (Â − cL̂c ⊗ In )T (Ŵ ⊗ In )ζ(t) = ζ T (t)[(Ŵ ⊗ In )(Â + ÂT ) − c(Ŵ L̂c + L̂Tc Ŵ) ⊗ In ]ζ(t) ≤ ζ T (t)[(Ŵ ⊗ In )(Â + ÂT ) − cŴ ⊗ In ]ζ(t) ≤ −[c − λmax (Â + ÂT )]V (t), where the last inequality follows from (12). This confirms the exponential stability of system (14), and therefore cluster synchronization can be achieved exponentially fast with the least rate of 21 [c − λmax (Â + ÂT )]. We have the following comments on the condition in (12): 1) From the above proof, one can find another lower bound for c as follows: c> λmax ((Ŵ ⊗ In )(Â + ÂT )) λmin (Ŵ L̂c + L̂Tc Ŵ) . (16) This bound is tighter than that in (12) since the inequality in (15) and λmax (Ŵi ) > 0, λmax (Ai + ATi ) ≥ 0 for any i imply that the right-hand side (RHS) of (16) ≤ λmax (Ŵ)λmax (Â + ÂT ) = RHS of (12). However, this tighter bound only guarantees λmax (Ŵ) that V̇ (t) < 0, and does not provide a lower bound on the convergence rate, which could be quite slow. Moreover, the RHS of (16) involves all the ci ’s in L̂c , and no known distributed algorithm is available for the computation. 2) Note that the role of c is more essential in stabilizing the unstable modes of the system matrices, Ai ’s, than in strengthening the connective ability of the interaction graph. A global weighting factor similar to c is utilized in a related paper [17] where the clustering problem for identical linear systems are solved. In that paper, the global factor serves as a parameter in a Ricatti inequality so as to result in a control gain matrix. However, using a larger value for that factor does not necessarily increase the convergence rate. In contrast, the selection of c in this paper is independent of the control design in (8). And the rate of convergence can be improved definitely by using a larger value for c. The following two remarks explain why we prefer the dynamic couplings in (4) than static couplings when dealing with nonidentical linear systems. Remark 2: To achieve state cluster synchronization for a group of generic linear systems, a natural choice of static couplings is the following (slightly modified from couplings of homogeneous linear systems in [16], [17]): for each l ∈ Ci , i = 1, . . . , N   X X ul (t) = Ki ci blk xk (t) + blk xk (t) k∈Ci (17) k∈C / i However, following a similar procedure as in [17], one will need the following condition ci λmin ((Ŵi L̂ii + L̂Tii Ŵi ) ⊗ Pi Bi BiT Pi ) ≥ ρ, (18) for every i = 1, . . . , N , where ρ = λmax ((Ŵ ⊗ In )PBBT P) − λmin (PBBT P(ŴLo ⊗ In ) + (LTo Ŵ ⊗ In )PBBT P). To compute ρ, one needs information on the control design of all agents, i.e., BT P = blockdiag{Il1 −1 ⊗ B1 P1 , . . . , IlN −1 ⊗ BN PN }. This fact renders the selection of local weighting factors, ci ’s, a centralized decision. Moreover, (18) cannot be satisfied by any ci in the nontrivial case that ρ > 0, and Pi Bi BiT Pi is singular for some i. In contrast to (18), the condition (13) specifies explicitly the requirements for ci ’s, and it is independent of the design of control gain matrices. In this sense, the dynamic couplings in (4) are preferable to the static ones in (17). Remark 3: For nonidentical nonlinear systems of the form, ẋl (t) = fi (xl , t), l ∈ Ci , static couplings are used to result in closed-loop systems as follows ([14], [15]):   X X ẋl (t) = fi (xl , t) − Γ ci blk xk (t) + blk xk (t) , k∈Ci k∈C / i where Γ is a constant (usually nonnegative-definite) matrix. It was shown that clustering conditions involve the graph Laplacian only (see [15]) if all individual self-dynamics are constrained by the so-called QUAD condition: for any x, y ∈ Rn , (x − y)T [fl (x) − fl (y) − Γ(x−y)] ≤ −ω(x−y)T (x−y), where ω > 0 is a prescribed positive scalar. For generic linear systems with static couplings in (17), this QUAD condition requires that for any x ∈ Rn , xT (Ai − Γ)x ≤ −ωxT x with Γ = Bi Ki for all i = 1, . . . , N . Given a Γ, for the existence of control gains Ki ’s, one needs all Bi ’s to satisfy Rank(Bi ) = Rank([Bi Γ]). However, this rank condition is too restrictive. For example, for the models in (21), an applicable choice of Γ is I2 , but then Rank(Bi ) < Rank([Bi Γ]) and thus no Ki can be solved from Γ = Bi Ki . In contrast, the dynamic couplings in (4) do not impose such constraints on the system models. Generally, it is not always necessary to let every subgraph contain a directed spanning tree. In fact, agents belonging to a common cluster may not need to have direct connections at all as long as the algebraic condition in Theorem 1 is satisfied. This point is illustrated by a simulation example in the next section. Nevertheless, the spanning tree condition turns out to be necessary under some particular graph topologies as stated by the corollary below. Corollary 2: Let G be an interaction graph with an acyclic partition as in (3), and let the edge weights of every subgraph Gi be nonnegative. Under Assumptions 1 to 3, a multi-agent system (1) with couplings in (4) achieves N -cluster synchronization if and only if every Gi contains a directed spanning tree, and the weighting factors satisfy c · ci > maxσ(Ai ) Reλ(Ai ) minσ(L̂ii ) Reλ(L̂ii ) , ∀i = 1, . . . , N, (19) where each L̂ii is defined in (7). Proof: By Theorem 1, we can examine the stability of Â − cL̂c ⊗ In . Let Ti ∈ R(li −1)×(li −1) , i = 1, . . . , N , be a set of nonsingular matrices such that Ti−1 L̂ii Ti = Ji , where Ji is the Jordan form of L̂ii . Denote T = blockdiag{T1 ⊗ In , . . . , TN ⊗ In }. Then, the 3 4 3 02 c1 4 G¹2 G2 1 G1 2 G1 1 1 2 1 2 -1 1 -5 -1 1 5 -1 -4 1 4 -1 1 1 1 3 4 1 4 G2 (b) (a) Fig. 2. Interaction graph partitioned into two clusters C1 = {1, 2} and C2 = {3, 4} (a) cyclic partition (b) acyclic partition. 1 3 5 -1 1 1 1 3 G2 c2 -1 4 -1 G1 c1 c2 −1 s1 1 1 2 block triangular matrix T (Â − cL̂c ⊗1 In )T has diagonal blocks Ai − c̃i λk (L̂ii )In , where 6 -1 4 -5 5 c̃i = c ·1 ci , k = 1, . . . , li − 1, i = 1, . . . , N . Hence, the matrix Â − cL̂c ⊗ In is Hurwitz if 1 and only if c̃i mink Reλk (L̂ii ) > maxm Reλ1m (Ai )-1for any i. This claim is equivalent to the c1 2 3 -1 1 4 3 the first 1claim of Lemma 2 conclusion of4 this corollary due to sLemma 2, 1, and Assumption 1 2 that requires maxm Reλm (Ai ) ≥ 0. -1 1 G2 c2 reveals the indispensability of direct links among agents in the same cluster This corollary under 1 an acyclicly partitioned interaction graph. Note that such direct communication require7 c3 8 5 ments -1 for intra-cluster agents is not necessary under a nonnegatively weighted interaction graph (see [11], [12], [15] for references). Remark 4: It is worth mentioning for the condition in (19) that one can set ci = 1 for all i, and adjust the global factor c only to result in cluster synchronization. In contrast, without the acyclic partitioning structure, the local weighting factors ci ’s need to satisfy the lower bound conditions in (13). Note that (19) specifies the tightest lower bound for c, while a lower bound reported in [16] for identical linear systems via Lyapunov stability analysis can be quite loose. IV. S IMULATION E XAMPLES In this section, we provide application examples for cluster synchronization of nonidentical linear systems. We also conduct numerical simulations using these models to illustrate the derived theoretical results. A. Example 1: Heading alignment of nonidentical ships Consider a group of four ships with the interaction graph described by Fig. 2(a), where ship 1 and 2 (respectively, ship 3 and 4) are of the same type. The purpose is to synchronize the heading angles for ships of the same type. The steering dynamics of a ship is described by the well-known Nomoto model [21]: ψ̇l (t) = vl (t) 1 κi v̇l (t) = − vl (t) + ul (t) τi τi (20) where ψl is the heading angle (in degree) of a ship l ∈ I, vl (deg/s) is the yaw rate, and ul is the output of the actuator (e.g., the rudder angle). The parameter τi is a time constant, and κi is the actuator gain, both of which are related to the type of a ship. Define for i = 1, 2 the system matrices     0 1 0  , Bi =   , Ai =  κi 0 − τ1i τi (21) and assume that τ1 = 42.21, τ2 = 107.3, κ1 = 0.181, κ2 = 0.185. The solutions to the   22.3 233.2 34 580  and P2 =  , Riccati equations in (8) are given by P1 =  233.2 3915.4 580 16875 which lead to the control gain matrices K1 = −[1 16.79] and K2 = −[1 29.09]. Since maxi=1,2 λmax (Ai + ATi ) = 0.99, we set c = 1 according to (12). The weighted graph Laplacian is given by   0 0 5 −5   −c c   1 1 1 −1 Lc =  ,  −1 1  0 0   0 0 −c2 c2  c1 4   using the definition in (6). So, L̂11 = 1, L̂22 = 1, and for any which yields L̂c =  −1 c2 Ŵ1 > 0 and Ŵ2 > 0, the inequalities in (10) hold. We choose Ŵ1 = Ŵ2 = 1. It follows that λmax (Ŵ) = 1, λmin (Ŵ L̂o + L̂To Ŵ) = −3, and λmin (Ŵi L̂ii + L̂Tii Ŵi ) = 2 for i = 1, 2. Then, we can choose c1 = c2 = 2 so that the inequalities in (13) are satisfied. Simulation result in Fig. 3(a) shows that cluster synchronization is achieved for the heading angles (the velocity vl (t) of every agent will converge to zero as shown in Fig. 3(b)). Now, let c1 = 0 so that agents 1 and 2 in cluster C1 have no direct connection. Cluster synchronization is still achieved as shown in Fig. 3(c). This example illustrates that intracluster connections are not necessary for cluster synchronization under a cyclicly partitioned ψ(t) (deg) 40 20 ψ1 ψ2 ψ3 ψ4 0 −20 −40 0 v1 v2 v3 v4 2 v(t) (deg/s) 60 1 0 −1 −2 100 200 t (sec) 300 0 100 200 t (sec) 300 (a) Under the graph in Fig. 2(a), cluster synchro- (b) Under the graph in Fig. 2(a), the velocity nization is achieved with c1 = c2 = 2. vl (t) of every ship converges to zero. 60 ψ(t) (deg) ψ1 ψ2 ψ3 ψ4 20 0 −20 −40 0 ψ(t) (deg) 50 40 ψ1 ψ2 ψ3 ψ4 0 −50 100 200 t (sec) 300 0 100 200 t (sec) 300 (c) Under the graph in Fig. 2(a), cluster synchro- (d) Under the acyclic partitioning graph in Fig. nization is achieved with c1 = 0, c2 = 2. 2(b), ψl ’s in the first cluster cannot synchronize together. Fig. 3. Evolutions of ψl (t) and vl (t) for the ship heading steering dynamics in (20) connected with graphs in Fig. 2. interaction graph. However, under an acyclic partition as in Fig. 2(b), the first cluster of agents, having no direct connections, cannot achieve state synchronization as shown in Fig. 3(d). B. Example 2: Cluster synchronization of oscillators The studied cluster synchronization problem for nonidentical linear systems may find applications in the coexistence of oscillators with different frequencies. To see this, let us consider two clusters of coupled harmonic oscillators with graph topology in Fig. 4(a), where the first cluster contains a sender s1 and two receivers r1 and r2 , the second cluster contains a sender s2 and two receivers r3 and r4 , and the four receivers are coupled by some directed links. Assume the angular frequencies of the two clusters of oscillators are w1 = 2 rad/s and 1 3 4 1 3 G2 G2 (b) (a) G1 3 s1 1 1 r1 -1 1 4 1 s2 1 r4 1 r3 x1l (t) -5 5 10 0 −10 0 G2 (a) c2 s1 r1 r2 s2 r3 r4 20 r2 c1 4 4 π 2π t (sec) 3π (b) Fig. 4. (a) Interaction graph partitioned into two clusters C1 = {s1 , r1 , r2 } and C2 = {s2 , r3 , r4 }. (b) Evolutions of the first components of the nonidentical harmonic oscillators under the graph in Fig. 4(a). 5 w2 = 2.5 rad/s, respectively. So, the dynamic equation of each oscillator is ẋ1l (t) = x2l (t), ẋ2l (t) = −wi2 x1l (t) + ul (t), l ∈ Ci , i = 1, 2 (22) which corresponds to the following system matrices:     0 1 0  , Bi =   , i = 1, 2. Ai =  −wi2 0 1 The objective is to let the receivers of each cluster follow the state of the sender. By a similar design procedure as in the previous example, we can set K1 = −[0.1231 1.1163], K2 = −[0.0554 1.0539], c = 6, and c1 = c2 = 13. Simulation result in Fig. 4(b) shows the synchronous oscillations of the harmonic oscillators with two distinct angular frequencies. V. C ONCLUSIONS This paper investigates the state cluster synchronization problem for multi-agent systems with nonidentical generic linear dynamics. By using a dynamic structure for coupling strategies, this paper derives both algebraic and graph topological clustering conditions which are independent of the control designs. For future studies, cluster synchronization which can only be achieved for the system outputs is a promising topic, especially for linear systems with parameter uncertainties or for heterogeneous nonlinear systems. For completely heterogenous linear systems, research works following this line are conducted by the authors in [22] and others in [23]. For nonlinear heterogeneous systems, the new theory being established for complete output synchronization problems [24], [25] may be further extended. Another interesting challenge existing in cluster synchronization problems is to discover other graph topologies that meet the algebraic conditions. A PPENDIX I P ROOF OF L EMMA 1  1 0   ∈ Rli ×li for i = Proof: Let S = blockdiag{S1 , . . . , SN }, where Si =  1li −1 Ili −1   1 0 . By direct computation 1, . . . , N . Clearly, Si has the inverse matrix Si−1 =  −1li −1 Ili −1 one can show that   0 γij . Si−1 Lij Sj =  0 L̂ij This implies the first claim when i = j. For the second claim, consider that  0 γ11  0 c L̂ 1 11  . .. −1 . S Lc S =  . .  0 γN 1  0 L̂N 1 ··· 0 γ1N  ··· ... 0 .. . L̂1N .. . ··· 0 γN N     .     ··· 0 cN L̂N N Rearrange the columns and rows of S−1 Lc S by permutation and similarity transformations to get the following block upper-triangular  01×N  ..  .    01×N  0(L−N )×N matrix γ11 .. . ··· .. . γN 1 · · ·  γ1N ..  .   , γN N   L̂c where L̂c is defined in (6). Then, the second claim of this lemma follows immediately. A PPENDIX II P ROOF OF T HEOREM 1 Proof: The closed-loop system equations for (1) using couplings (4) are given as   X X żl = Aci zl − c ci blk Ezk + blk Ezk  , (23) k∈C / i k∈Ci for all l ∈ Ci , i = 1 . . . , N , where zl = [xTl , ηlT ]T and     Ai Bi Ki 0 0 , E =  . Aci =  0 Ai + Bi Ki −In In (24) Let el (t) := zl (t) − zσi +1 (t) for l ∈ Ci and l 6= σi + 1, i = 1, . . . , N . It follows from (23) and Assumption 3 that " ėl (t) = Aci el (t) − c ci X (blk − bσi +1,k )Eek (t) k∈Ci  + X (blk − bσi +1,k )Eek (t) . (25) k∈C / i Define a nonsingular transformation matrix Q as follows:     In 0 I 0  , Q−1 =  n , Q= In In −In In (26) and let εl (t) := [ξlT (t), ζlT (t)]T = Q−1 el (t). Clearly, ξl = xl − xσi +1 and ζl = ηl − ησi +1 − xl + xσi +1 . By (25), one can obtain the following dynamic equations: ξ˙l (t) = (Ai + Bi Ki )ξl (t) + Bi Ki ζl (t), " X (blk − bσi +1,k )ζk (t) ζ̇l (t) = Ai ζl (t) − c ci k∈Ci  + X (blk − bσi +1,k )ζk (t) , k∈C / i for l ∈ Ci and l 6= σi + 1, i = 1, . . . , N . Since Ki stabilizes (Ai , Bi ), the variable εl (t) tends to zero as t → ∞ if and only if ζl (t) tends to zero. Denote ζ(t) = [ζσT1 +2 (t), . . . , ζσT1 +l1 (t), · · · , ζσTN +2 (t), . . . , ζσTN +lN (t)]T , which evolves with the following differential equation ζ̇(t) = Â − cL̂c ⊗ In ζ(t). (27) Clearly, ζ(t) and every εl (t) (hence every el (t)) all converge to zero if and only if Â−cL̂c ⊗In is Hurwitz. That is, we have shown that limt→∞ kxl (t) − xk (t)k = 0 and limt→∞ kηl (t) − ηk (t)k = 0, ∀l, k ∈ Ci , i = 1, . . . , N . Next, we prove that ηl (t) for any l ∈ I vanishes as t → ∞. To this end, for each i = 1, . . . , N , let ηi (t) be the solution of η̇i (t) = (Ai + Bi Ki )ηi (t) with an arbitrary initial P value ηi (0). Since k∈Cj blk = 0 ∀l ∈ I by Assumption 3, we have that η̇i (t) = (Ai + Bi Ki )ηi (t) " = (Ai + Bi Ki )ηi (t) − c ci ( X blk )(ησi +1 − xσi +1 ) k∈Ci  N X X + ( blk )(ησj +1 − xσj +1 ) , j=1,j6=i k∈Cj for any l ∈ Ci . Subtracting the above from (4b) yields η̇l (t) − η̇i (t) = (Ai + Bi Ki )(ηl (t) − ηi (t))   N X X X − c ci blk ζk + blk ζk  . k∈Ci j=1,j6=i k∈Cj The above system is exponentially stable and driven by inputs which all converge to zero exponentially fast. Therefore, for any ηl (0), l ∈ I, we have ηl (t) → ηi (t) → 0 ∀l ∈ Ci , as t → ∞. Lastly, we show that inter-cluster state separations can be achieved for any initial states xl (0)’s by selecting ηl (0)’s properly. Given any set of xl (0), l ∈ I, choose ηl (0), l ∈ I such that xl (0)−ηl (0) = xσi +1 (0)−ησi +1 (0) for all l ∈ Ci , i = 1 . . . , N , and lim supt→∞ keAi t [xl (0)− ηl (0)] − eAj t [xl (0) − ηl (0)]k = 6 0 for any i 6= j. Considering the definition of ζl and the linear differential equation (27), one has xl (t) − ηl (t) = xσi +1 (t) − ησi +1 (t) for all t > 0. This together with (4) lead to the following dynamics ẋl (t) − η̇l (t) = Ai (xl (t) − ηl (t)), ∀l ∈ Ci . It follows that xl (t) = eAi t [xl (0) − ηl (0)] + ηl (t) → eAi t [xl (0) − ηl (0)], ∀l ∈ Ci as t → ∞. Therefore, lim supt→∞ kxl (t) − xk (t)k = 6 0 ∀l ∈ Ci , ∀k ∈ Cj , ∀i 6= j. This completes the proof. A PPENDIX III P ROOF OF C OROLLARY 1 Proof: The proof for the necessity and sufficiency of (9) is straightforward using the results in Lemma 1 and Theorem 1, and thus is omitted for simplicity. We only show that state separations are possible for any initial states xl (0), l ∈ I by using the dynamic couplings even for systems with identical parameters. Constellate the states zl (t) = [xTl (t), ηlT (t)]T of all L agents to form z(t) := [z1T (t), z2T (t), . . . , zLT (t)]T . It follows that ż(t) = (IL ⊗ Ac − cLc ⊗ E)z(t), (28)     A BK 0 0  and E =   . 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ArbiText: Arbitrary-Oriented Text Detection in Unconstrained Scene Daitao Xing1,3 , Zichen Li1,3 , Xin Chen4 , and Yi Fang1,2,3∗ arXiv:1711.11249v1 [cs.CV] 30 Nov 2017 1 NYU Multimedia and Visual Computing Lab 2 NYU Abu Dhabi, UAE 3 New York University, Brooklyn, NY, USA 4 HERE Technologies Abstract Arbitrary-oriented text detection in the wild is a very challenging task, due to the aspect ratio, scale, orientation, and illumination variations. In this paper, we propose a novel method, namely Arbitrary-oriented Text (or ArbText for short) detector, for efficient text detection in unconstrained natural scene images. Specifically, we first adopt the circle anchors rather than the rectangular ones to represent bounding boxes, which is more robust to orientation variations. Subsequently, we incorporate a pyramid pooling module into the Single Shot MultiBox Detector framework, in order to simultaneously explore the local and global visual information, which can therefore generate more confidential detection results. Experiments on established scene-text datasets, such as the ICDAR 2015 and MSRA-TD500 datasets, have demonstrated the superior performance of the proposed method, compared to the state-of-the-art approaches. Figure 1: Text detection results of the proposed ArbiText method. Images in the first row shows examples from ICDAR2015 dataset, the second row shows examples MSRATD500 dataset 1. Introduction Understanding texts in the wild plays an important role in many real-world applications such as PhotoOCR [2], road sign detection in intelligent vehicles [3], license plate detection [17], and assistive technology for the visually impaired [6] [1]. To achieve this goal, the task of accurate arbitraryoriented text detection becomes extremely important. Conventionally, when dealing with horizontal texts under controlled environments, this task can be accomplished through character-based methods such as [9], [10], [18], and [24] considering that individual letters can be easily segmented and distinguished. However, in an unconstrained naturalscene image, text detection becomes rather challenging due ∗ 5 Metrotech Center LC024, Brooklyn, NY, 11201; yfang@nyu.edu; Tel: +1-646-854-8866 to uncontrolled text variations and uncertainties, such as multi-orientation, text distortion, background noise, occlusion, and illumination changes. To address these problems, a lot of recent efforts have been devoted to employing state-of-the-art generic object detectors, such as the Fully Convolutional Network(FCN) [20], Region-based Convolutional Neural Network(R-CNN) [19], and Single Shot Detector(SSD) [14], for the purpose of text detection in the wild. Despite their promising performance in generic object detection, these methods suffered from bridging gaps between the data distributions of texts and generic objects. To enhance the generative abilities of existing deep mod- Email: 1 Figure 2: The Framework of the Proposed Method. Given an input image with a size of 384×384,VGG-16 base network outputs the first feature map from conv4 3 layer. More feature maps with cascading sizes are extracted from extra layers following the first feature map. The first feature map is also used to produce different sub-region representations through the Pyramid Pooling Module. These representation layers are then concatenate feature maps with same size to output the final feature maps. Finally, those maps are fed into a convolution layer to get the final. els, [7] proposed to naturally blend rendered words onto wild images for training data augmentation. The trained model based on such training data is robust to noises and uncontrolled variations. [31] and [13] attempted to integrate region-proposal layers into the deep neural networks, which can generate text-specific proposals (e.g. bounding boxes with larger aspect ratios). In [16], the bounding-box rotation proposals were introduced to make the proposed model more adaptive for unknown orientations of texts in naturalscene images. Nevertheless, the aforementioned scene-text detectors either needed to consider a lot of proposal hypotheses, thus dramatically decreasing the computational efficiency, or utilize insufficient preset bounding-box characteristics to handle severe visual variations of scene-texts in unconstrained natural images. the conventional rectangular ones. The Single Shot Detector (SSD), one of the state-of-art object detectors, is employed, considering its fast detection speed and promising accuracy in generic object detection. Besides the feature maps generated by the original SSD, we additionally incorporate a pyramid pooling module, which can build multiple feature representations on different spatial scales. By merging those different kinds of feature maps, both the local and global information can be preserved, such that texts in unconstrained natural scenes can be more reliably detected. Subsequently, the merged feature maps are fed into a text detection module, consisting of several fully connected convolutional layers, to predict confidential circle anchors. Furthermore, in order to overcome the difficulty of deciding positive points caused by unfixed sizes of circle anchors, we introduce a novel mask loss function by assigning those ambiguous points to a new class. To obtained the final detection results, the Locality-Aware Non-MaximumSuppression (LANMS) scheme [33] is employed. It should be noted that we do not utilize any proposal, which makes the proposed method more computationally efficient. To address the aforementioned drawbacks of existing methods, we propose a novel proposal-free model for arbitrary-oriented text detection in natural images based on the circle anchors and the Single Shot Detector (SDD) framework. More specifically, we adopt circle anchors to represent the bounding boxes, which are more robust to orientation, aspect ratio, and scale variations, compared to In summary, the contributions of our work mainly lie in 2 three-fold: text-line hypothesis. Although these methods can achieve state-of-art results even with scene-text detection in the wild, the requirement for a sophisticated post-processing step of word partitioning and false positive removal can be too time consuming and computationally intensive for realworld applications. A more recent method proposed in [33], however, seeks to make dramatic improvement in efficiency over [29] and [26] by eliminating intermediate steps such as candidate aggregation and word partitioning in the neural network. Nevertheless, the inherent nature of the segmentation approach dense per-pixel processing and prediction - is still a bottleneck that prevents segmentation-based methods from outperforming its competitors. Region-Proposal-based Methods: Although regionproposal based models like R-CNN have already been a state-of-art object detector, it can not be implemented for the purpose of text detection without modifications since its anchor box design is not ideal for the large aspect ratio of words/text-lines. [31] addresses this problem by proposing a novel Region Proposal Network(RPN) called InceptionRPN, which contains a preset of text characteristic prior bounding boxes to generate text-specific proposals and thus filtering out low-quality word regions. However, [31] only performs well on horizontal texts since bounding box characteristics are extremely unpredictable for scene-texts in the wild; multi-oriented and distorted texts can create countless possibilities and variations of bounding-box size, shape, and orientation. To address this challenge, some researchers designed novel region-proposal methods: the rotation proposal method in [16] has the ability to predict the orientation of a text line and thus generate inclined bounding-boxes for oriented texts, while the quadrilateral sliding windows in [15] create a much tighter bounding-box fit around text regions, thus dramatically reduce background noise and interference. On the other hand, some researchers propose methods to modify model architecture like the one proposed in [5], which adds 2D offsets in the standard convolution to enable free form deformation of the sampling grid, and the one proposed in [8], which utilizes direct bounding-box regression originating from a center anchor point in a proposal region. SSD-based Methods. SSD-based method is highly stable and efficient in generating word proposals because SSD is one of the fastest object detector that is also as accurate as slower region-proposal based models like R-CNN. However, SSD possesses similar shortcomings in terms of anchor box design when it comes to scene-text detection. Thus, [13] supplements SSD with ”textbox layers” that can generate bounding-boxes with larger aspect ratios and simultaneously predict text presence and bounding boxes. Unfortunately, this method only works on horizontal texts, • We propose a novel proposal-free method for detecting arbitrary-oriented texts in unconstrained natural scene images, based on the circular anchor representation and the Single Shot Detector framework. The circular anchors are more robust to different aspect ratio, scale, and orientation variations, compared to conventional rectangular ones. • We incorporate a pyramid pooling module into SSD, which can explore both the local and global visual information for robust text detection. • We develop a new mask loss function to overcome the difficulty of deciding positive points caused by unfixed sizes of circle anchors, which can therefore improve the final detection accuracy. 2. Related Works Character-based detection methods have already achieved state-of-art results on horizontal texts in relatively controlled and stable environments. Methods like those proposed in [9], [10], [18], and [24] either detect individual characters by classification of sliding windows or utilize some form of connected-component and regionbased framework such as the Maximally Stable Extremal Regions(MSER) detector. However, some of these methods might not be ideal for detecting scene-texts or multi-oriented texts as more and more environmental variations and uncertainties in terms of text distortion, orientation, occlusion, and noise are introduced. Detecting individual characters in close clusters or ones that blend into the background can also be challenging. Hence, many researchers decide to tackle the problem by approaching the task of text detection as object detection: treating words and/or text-lines as the target object. Region-based Convolutional Neural Network(RCNN)[19], Single Shot Detector(SSD)[14], and segmentation-based Fully Convolutional Network(FCN)[20] are frequently re-purposed for text detection because their superior speed and accuracy are better suited for the time and resource-constraining nature of the task. Expectedly, a majority of the cutting-edge research in scene-text detection, including ours, are based on one of the aforementioned object detection models, which we will analyze below. Segmentation-based Methods: Both [29] and [26] accomplish semantic segmentation of text lines by utilizing Fully Convolutional Network(FCN), which has achieved great performance in pixel-level classification tasks. In [29], for example, pixel-wise text/non-text salient map is first produced via the FCN and subsequently, geometric and character processing is implemented to generate and filter 3 conv10 2 and global layers. However, local information is lost as layer goes deeper and deeper, which results in poor detection precisions especially on texts with complex contextual information. Inspired by [30], we introduced the Pyramid Pooling Module to leverage low-level visual information even in deeper layers. This module fuses feature maps with different pyramid scales. As shown in Fig. 2, the first feature map from the based network is separated on pyramid levels into different sub-regions and output pooled representations after a 1 ∗ 1 convolution layer. Thus, the low-level information of the original image could be preserved in multi-scale level feature maps, which will be further concatenated with ones of the same size to form the final feature map for text detection. By merging these two types of features, both the local and global visual information can be explored. Finally, the text detection layer applies a 3∗3 convolution kernel on the fused feature map to output prediction on the text bounding box. Figure 3: Matching of Circle Anchors The red circle is generated from the ground-truth coordinates, and the blue circle anchor is the one with same size with red circle on the feature point. The blue circle which associates the red one by a vector(cx, cy, area, diameter, angle) where cx and cy are offset between centers of circles. 3.2. Circle Anchors and not scene-texts. Thus in this paper, we attempt to solve the aforementioned limitations of previous detection models by utilizing a proposal-free method based on circular anchors and the SSD framework. Our method is computationally more efficient than both segmentation-based and Region-Proposalbased models because the removal of the region-proposal layer in our network. Our method also improves upon the existing SSD-based method by having the ability to detect both arbitrary-oriented texts and generic objects. As illustrated in Fig. 3, instead of the traditional rectangular anchors, we use circle anchors to represent the bounding box. Specifically, a bounding box can be represented by a 5-dimensional vector (x, y, a, r, θ), where a, r, and θ denotes the area, radius, and rotated angle of a circle anchor. a 1 arcsin( 2 ) 2 (2r ) x2 = r · cos(α + θ), y2 = r · sin(α + θ) α= 3. Method x3 = r · cos(α − θ), y3 = r · sin(α − θ) (1) x1 , y1 = −x3 , −y3 , x4 , y4 = −x2 , −y2 In this section, we will describe the details of our proposed model - ArbiText. We will first introduce the framework and network architecture of our method. Subsequently, we will elaborate on the key components such as the circle anchor representation and the proposed loss function. x1 , x2 , x3 , x4 = x1 + x, x2 + x, x3 + x, x4 + x y1 , y2 , y3 , y4 = y1 + y, y2 + y, y3 + y, y4 + y On a feature map of size (w ∗ w), location, denoted (i, j), associates a circle anchor C0 with (c, ∆x, ∆y, ∆a, ∆r, ∆θ), indicating that a unique circle anchor, represented by (x, y, a, r, θ), is detected with confidence c, where 3.1. Model Framework Our proposed method, in essence, is a multi-scale, proposal-free framework based on the Single Shot Detector. As shown in Fig.2, our model mainly consists of the following four components: 1) the backbone-based network for converting original images into dense feature representations; 2) the feature maps component with cascading map size for detecting multi-scale texts; 3) the Pyramid Pooling Module [30] for extracting sub-region feature representations; and 4) the final text detection layer for circle anchor prediction. We adopt VGG-16 [22] as our base network, and utilize the 6 feature maps at the conv4 3, conv7, con8 2, conv9 2, ra ra + j, y = ∆y · +i w w ra r = exp(∆r) · w ra a = exp(∆a) · w θ = ∆θ x = ∆x · (2) Here, we use the area a and radius r for computational stability. Also, we multiply each value by a factor rwa where ra =1.5. 4 (a) (b) Figure 4: (a) shows the the coordinates of a rectangle. Given the area and diagonal of rectangle, p2 and p3 can be calculated by rotation angles. p1 and p4 are diagonal points of p3 and p1, respectively, which also have negative values. (b) shows the predictable vertical flag, which is 0 if the angles of the bounding box are between −45◦ and 45◦ ; otherwise, it is set to 1. Figure 6: The Selection of Ground Truth. The blue rectangle is the bounding box and the 10 ∗ 10 grid represents a feature map with the same size. Only the feature points in the red eclipse are labeled as positives; the points outside of the yellow eclipse are labeled as negatives. The feature points in the yellow region will be labeled as their own separate class. Figure 5: Score Distribution. (a) shows a bounding box the text. The red and yellow rectangles are possible bounding boxes that have maximum IOU overlap scores with ground truth . (b) shows the eclipse-shape like score function we use in ArbiText. For a SSD-based method, all points on a feature map will potentially be used for minimizing a specific loss function. Each feature point needs to be labeled as either positive or negative. Specifically, in SSD, the points that are labeled as positive are chosen from the regions where the overlap between the default anchor and ground-truth bounding box is larger than 0.5. However, there is no default anchor in our method, but we can still calculate a confidence score for each point. As illustrated in Fig. 5 (a), the feature point on the edge of the bounding box can have a maximum overlap of 0.5(bounding boxes are colored in red and yellow). So the score follows an eclipse distribution (as illustrated in Fig. 5 (b)). The scores at the center of the eclipse have a maximum value of 1.0 and it decreases to 0.5 when the points reach the edge. We use a semi-ellipse as the function to compute the score for each point. As a result, the points outside of the eclipse have a score value 0. Imagine an eclipse score function which has rotation angle θ, the semi-major axes and semi-minor axes have length of w2 and h2 , respectively, where w and h are the width and height of the bounding box. Thus, the score function can be The angle θ is the intersection angle between the long edge of the bounding box and the horizontal axis. Thus, the value of θ ranges from −90◦ to 90◦ . In a deep neural network, each layer has a receptive field that indicates how much contextual information we can utilize. Although the circle anchor representation is invariant to scale variations, [32] has shown that feature maps have limited receptive fields that are much smaller than theoretical ones, especially on high-level layers. As a result, if we do not utilize multi-scale feature maps, the detection scope of the proposed circle anchor representation will be restricted. And considering the size of the extra feature layers, this operation only adds a small amount of computational cost. 3.3. Training Labels Rebuilding and the Loss Function Formulation . 5 4. Experiments represented as: A · (x)2 + B · (x) · (y) + C · (y)2 + F · s2 = F w h a = ;b = 2 2 2 2 2 A = a · sin (θ) + b · cos2 (θ) 4.1. Datasets In order to evaluate the performance of the proposed method, we ran experiments on two benchmark datasets: the ICDAR 2015 dataset and the MSRA-TD500(TD500) dataset. SynthText in the Wild[7] dataset contains more than 800,000 synthetic images created by blending rendered words on wild images. Only samples with width larger than 10 pixels are chosen for training. ICDAR 2015[34] incidental text dataset is from Challenge 4 of ICDAR 2015 Robust Reading Competition that includes 1000 training images and 500 testing images. Since those images are collected by Google Glasses, they suffer from motion blur. The blurry texts have a label of ”###” and are excluded from our experiment. We also included training and testing images from the ICDAR 2013 dataset[12], which helps us in building a more robust text detector. MSRA-TD500(TD500)[4] is a multilingual dataset that includes oriented texts in both Chinese and English. Unlike ICDAR 2015, texts in MSRA-TD500 are annotated at the text-line level and the images were captured more formally, thus texts are much clearer and standardized. There are a total of 500 images, 300 of them were used as training data and 200 were used as testing data. (3) B = −2(a2 − b2 ) · sin(θ) · cos(θ) C = a2 · cos2 (θ) + b2 · sin2 (θ) F = a2 · b2 where x,y are the distances between a feature point and the center of the bounding box respectively,s is score. According to the score function, all points inside the eclipse have a score greater than 0.5. However, the closer the point is to the edge of the bounding box, the large the noise outside of the bounding box will be, which could make the training of networks harder to converge. Thus, only the points with score large than a threshold α will be treated as positives (as shown in Fig. 6, only points inside the red zone are labeled as ”positive”). Points outside of the bounding box will be labeled as ”negative”. For those points with a score between 0.5 and α, we assign them an additional label. Thus, there will be a total of (N + 1) classes(without background). This additional class will only be involved in calculating the classification loss. The feature maps with different sizes can detect texts with different scales. A default box will be labeled as positive if hg 1< < 4. (4) ch 4.2. Implementation Details Base network In our experiment, we uses a pre-trained VGG-16 as our based network. This network is widely used in object detection tasks. All images are resized to 384 ∗ 384 after data augmentation. We extracted five layers with cascading resolution as our feature maps, which are conv4 3, conv7, conv8, conv9, and conv10. We first trained our model on the SynthText dataset for 50,000 iterations with a learning rate of 0.001. Then, we fine-tuned our model using the other datasets with a 0.0005 learning rate. The details of training different datasets are described in later sections. We tested our model on different α values and we discovered that our model achieves optimal performance when α is set to 0.7 for text detection. Locality-Aware NMS In the post-processing stage, bounding boxes with a confidence score greater than 0.5 will be used to produce the final output by NMS merging. The naive NMS has O(n2 ) computational complexity, which is not ideal for real-world applications. We adopt LocalityAware NMS [33] to improve speed of merging bounding boxes. Hard Negative Mining Hard Negative Mining is essential for SSD-based methods because of the imbalance between positive and negative training samples. We adopt the same configuration in SSD[14] by selecting the top 3K where hg is the height of bounding box and ch is the height of cell on feature map. For training, we use the following objective loss function: L(mask, c, l, g) = λ1 λ2 N +1 1 X maski · Lcls (ci )+ Ncls i=0 N 1 X maski · Lloc (l, g)+ Nreg i=1 (5) N 1 X maski · Lcls (vertical) Nreg i=1 where N is the number of object categories, l is the prediction location, and g is the ground-truth location. For each class, maski = 1 if the corresponding point is labeled as ”positive” and belongs to the i-th class. is the loss for vertical bounding box classification that only includes the positive points. We adopt L1 loss for smoothing and Softmax loss as the classification losses. 6 Figure 7: Results of ArtTex on the ICDAR2015 dataset. Blue rectangles are detected text regions by using ArtTex. Table 1: Comparison results of various methods on the ICDAR 2015 Incidental Text dataset negative training samples, where k is number of positive training samples. Thus, we adopt Locality-Aware NMS[33] in our experiment. This algorithm can produce bounding boxes with greater precision in shorter time. Data Augmentation We utilize a data augmentation pipeline that is similar to the one in SSD to make our model more robust against different text variations. The original image is randomly cropped into patches. The crop size is chosen from [0.1, 1] of original image size. Each sample patch will be horizontally flipped with a probability of 0.5. In order to balance samples of different orientations, we also augmented datasets by randomly rotating images by θ degrees. θ is randomly chosen from the following angle set: (-90, -75, -60, -45, -30, -15, 0, 15, 30, 45, 60, 75, 90). Method HUST MCLAB NJU Text StradVision-2 MCLAB FCN[29] CTPN[23] Megcii-Image++ Yao et al.[26] Seglink [21] ArbiText Precision 47.5 72.7 77.5 70.8 51.6 72.4 72.3 73.1 79.2 Recall 34.8 35.8 36.7 43.0 74.2 57.0 58.7 76.8 73.5 F-score 40.2 48.0 49.8 53.6 60.9 63.8 64.8 75.0 75.9 4.3. Detection Results 4.3.1 Detecting Oriented English Text official off-line evaluation scripts. We list the results of our model along with other state-of-art object and text detection methods. The results were obtained from the original papers. The best result of this dataset was obtained by Seglink[21], which achieved a F-measure score of 75.0%. However, our model obtained a score of 75.9%. The improvement comes from the high precision rate we obtained, which outperforms the second First, our model is tested on ICDAR 2015 dataset. The pre-trained model is fine-tuned using both the ICDAR 2013 and the ICDAR 2015 training datasets after 20k iterations. Considering all images in ICDAR 2015 have high resolution, testing images are first resized to 768 ∗ 768. The threshold α is set to 0.7, similar to the one used in the pre-training stage. Performance is evaluated using the 7 Figure 8: Results of ArtTex on the MSRA-TD500 dataset. Blue rectangles are detected text regions by using ArtTex. Table 2: Comparison results of various methods on the MSRA-TD500 dataset highest model by 6.1% Figure.7 shows several detection results taken from the testing dataset of ICDAR 2015. Our proposed method ArbiText can distinguish and localize all kinds of scene text in noisy backgrounds. 4.3.2 Method Kang et al.[11] Yao et al.[25] Yin et al.[27] Yin et al.[28] Zhang et al.[29] Yao et al.[26] Seglink [21] ArbiText Detecting Multi-Lingual Text in Long Lines We further tested our method on the TD500 dataset consists of long text in English and non-Latin scripts. We augmented this dataset by doing the following: 1) Randomly place an image on a canvas of n times of the original image size filled by mean values where n ranges from 1 to 3. 2)We applies random crop according to the overlap strategy described in section 4.3. Thus, we obtained enough images for training. The pre-trained model is fine-tuned for 20K iterations. All images are resized to 384 ∗ 384, which is consistent with the training stage. The experiment has demonstrated that this technique can dramatically increase detection speed without losing much precision. As illustrated in Table 2, ArbiText achieved comparable F-measure scores with other state-of-the-art methods. However, benefiting from lighter network architecture and simplified anchors mechanism, ArbiText has the highest FPS of 12.1. Figure.8 shows ArbiText can detect long lines of text in mixed languages(English and Chinese) without changing any parameters or structures. Precision 71 63 81 71 83 77 86 78 Recall 62 63 63 61 67 75 70 72 F-score 66 60 74 65 74 76 77 75 FPS 0.14 0.71 1.25 0.48 ∼1.61 8.9 12.1 4.4. Limitations As shown in Figure.9.a,b, curved texts can’t be represented by circle anchors. Moreover, Figure.9.c shows our model’s weakness in detecting hand-written texts. 5. Conclusion We have presented ArbiText, a novel, proposal-free object detection method that can be utilized to detect both arbitrary-oriented texts and generic objects simultaneously. Its outstanding performance on different benchmarks demonstrates that ArbiText is accurate, robust, and flexible for real-world applications. In the future, we will extend 8 [13] (a) (b) (c) [14] Figure 9: Failure Cases On MSRA-TD500 The blue rectangles are true positives. The red ones are false negatives and yellow ones are false positives. In (a) and (b), ArbiText fails to detect curved texts. 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arXiv:1609.06782v2 [cs.CV] 22 Feb 2018 1 Deep Learning for Video Classification and Captioning Zuxuan Wu (University of Maryland, College Park), Ting Yao (Microsoft Research Asia), Yanwei Fu (Fudan University), Yu-Gang Jiang (Fudan University) 1.1 Introduction Today’s digital contents are inherently multimedia: text, audio, image, video, and so on. Video, in particular, has become a new way of communication between Internet users with the proliferation of sensor-rich mobile devices. Accelerated by the tremendous increase in Internet bandwidth and storage space, video data has been generated, published, and spread explosively, becoming an indispensable part of today’s big data. This has encouraged the development of advanced techniques for a broad range of video understanding applications including online advertising, video retrieval, video surveillance, etc. A fundamental issue that underlies the success of these technological advances is the understanding of video contents. Recent advances in deep learning in image [Krizhevsky et al. 2012, Russakovsky et al. 2015, Girshick 2015, Long et al. 2015] and speech [Graves et al. 2013, Hinton et al. 2012] domains have motivated techniques to learn robust video feature representations to effectively exploit abundant multimodal clues in video data. In this chapter, we review two lines of research aiming to stimulate the comprehension of videos with deep learning: video classiﬁcation and video captioning. While video classiﬁcation concentrates on automatically labeling video clips based on their semantic contents like human actions or complex events, video captioning 4 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning Benchmarks and challenges Video classiﬁcation Video captioning … Basketball Graduation … A woman is playing Frisbee with a black dog on the grass. Basic deep learning modules Figure 1.1 An overview of the organization of this chapter. attempts to generate a complete and natural sentence, enriching video classiﬁcation’s single label to capture the most informative dynamics in videos. There have been several efforts surveying the literature on video content understanding. Most of the approaches surveyed in these works adopted handcrafted features coupled with typical machine learning pipelines for action recognition and event detection [Aggarwal and Ryoo 2011, Turaga et al. 2008, Poppe 2010, Jiang et al. 2013]. In contrast, this chapter focuses on discussing state-of-the-art deep learning techniques not only for video classiﬁcation but also video captioning. As deep learning for video analysis is an emerging and vibrant ﬁeld, we hope this chapter could help stimulate future research along the line. Figure 1.1 shows the organization of this chapter. To make it self-contained, we ﬁrst introduce the basic modules that are widely adopted in state-of-the-art deep learning pipelines in Section 1.2. After that, we discuss representative works on video classiﬁcation and video captioning in Section 1.3 and Section 1.4, respectively. Finally, in Section 1.5 we provide a review of popular benchmarks and challenges in that are critical for evaluating the technical progress of this vibrant ﬁeld. 1.2 Basic Deep Learning Modules In this section, we brieﬂy review basic deep learning modules that have been widely adopted in the literature for video analysis. 1.2 Basic Deep Learning Modules 5 1.2.1 Convolutional Neural Networks (CNNs) Inspired by the visual perception mechanisms of animals [Hubel and Wiesel 1968] and the McCulloch-Pitts model [McCulloch and Pitts 1943], Fukushima proposed the “neocognitron” in 1980, which is the ﬁrst computational model of using local connectivities between neurons of a hierarchically transformed image [Fukushima 1980]. To obtain the translational invariance, Fukushima applied neurons with the same parameters on patches of the previous layer at different locations; thus this can be considered the predecessor of convolutional neural networks (CNN. Further inspired by this idea, LeCun et al. [1990] designed and trained the modern framework of CNNs LeNet-5, and obtained the state-of-the-art performance on several pattern recognition datasets (e.g., handwritten character recognition). LeNet-5 has multiple layers and is trained with the back-propagation algorithm in an end-toend formulation, that is, classifying visual patterns directly by using raw images. However, limited by the scale of labeled training data and computational power, LeNet-5 and its variants [LeCun et al. 2001] did not perform well on more complex vision tasks until recently. To better train deep networks, Hinton et al. in 2006 made a breakthrough and introduced deep belief networks (DBNs) to greedily train each layer of the network in an unsupervised manner. And since then, researchers have developed more methods to overcome the difﬁculties in training CNN architectures. Particularly, AlexNet, as one of the milestones, was proposed by Krizhevsky et al. in 2012 and was successfully applied to large-scale image classiﬁcation in the well-known ImageNet Challenge. AlexNet contains ﬁve convolutional layers followed by three fully connected (fc) layers [Krizhevsky et al. 2012]. Compared with LeNet-5, two novel components were introduced in AlexNet: 1. ReLUs (Rectiﬁed Linear Units) are utilized to replace the tanh units, which makes the training process several times faster. 2. Dropout is introduced and has proven to be very effective in alleviating overﬁtting. Inspired by AlexNet, several variants, including VGGNet [Simonyan and Zisserman 2015], GoogLeNet [Szegedy et al. 2015a], and ResNet [He et al. 2016b], have been proposed to further improve the performance of CNNs on visual recognition tasks: VGGNet has two versions, VGG16 and VGG19, which contain 16 and 19 layers, respectively [Simonyan and Zisserman 2015]. VGGNet pushed the depth of CNN architecture from 8 layers as in AlexNet to 16–19 layers, which greatly 6 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning improves the discriminative power. In addition, by using very small (3 × 3) convolutional ﬁlters, VGGNet is capable of capturing details in the input images. GoogLeNet is inspired by the Hebbian principle with multi-scale processing and it contains 22 layers [Szegedy et al. 2015a]. A novel CNN architecture commonly referred to as Inception is proposed to increase both the depth and the width of CNN while maintaining an affordable computational cost. There are several extensions upon this work, including BN-Inception-V2 [Szegedy et al. 2015b], Inception-V3 [Szegedy et al. 2015b], and Inception-V4 [Szegedy et al. 2017]. ResNet, as one of the latest deep architectures, has remarkably increased the depth of CNN to 152 layers using deep residual layers with skip connections [He et al. 2016b]. ResNet won the ﬁrst place in the 2015 ImageNet Challenge and has recently been extended to more than 1000 layers on the CIFAR-10 dataset [He et al. 2016a]. From AlexNet, VGGNet, and GoogLeNet to the more recent ResNet, one trend in the evolution of these architectures is to deepen the network. The increased depth allows the network to better approximate the target function, generating better feature representations with higher discriminative power. In addition, various methods and strategies have been proposed from different aspects, including but not limited to Maxout [Goodfellow et al. 2013], DropConnect [Wan et al. 2013], and Batch Normalization [Ioffe and Szegedy 2015], to facilitate the training of deep networks. Please refer to Bengio et al. [2013] and Gu et al. [2016] for a more detailed review. 1.2.2 Recurrent Neural Networks (RNNs) The CNN architectures discussed above are all feed-forward neural networks (FFNNs) whose connections do not form cycles, which makes them insufﬁcient for sequence labeling. To better explore the temporal information of sequential data, recurrent connection structures have been introduced, leading to the emergence of recurrent neural networks (RNNs). Unlike FFNNs, RNNs allow cyclical connections to form cycles, which thus enables a “memory” of previous inputs to persist in the network’s internal state [Graves 2012]. It has been pointed out that a ﬁnite-sized RNN with sigmoid activation functions can simulate a universal Turing machine [Siegelmann and Sontag 1991]. The basic RNN block, at a time step t, accepts an external input vector x (t) ∈ Rn and generates an output vector z(t) ∈ Rm via a sequence of hidden states 1.2 Basic Deep Learning Modules 7 h(t) ∈ Rr : h(t) = σ Wx x (t) + Whh(t−1) + bh (1.1) z(t) = softmax Wz h(t) + bz where Wx ∈ Rr×n, Wh ∈ Rr×r , and Wz ∈ Rm×r are weight matrices and bh and bz are biases. The σ is deﬁned as sigmoid function σ (x) = 1+e1−x and softmax (.) is the softmax function. A problem with RNN is that it is not capable of modeling long-range dependencies and is unable to store information about past inputs for a very long period [Bengio et al. 1994], though one large enough RNN should, in principle, be able to approximate the sequences of arbitrary complexity. Speciﬁcally, two well-known issues—vanishing and exploding gradients, exist in training RNNs: the vanishing gradient problem refers to the exponential shrinking of gradients’ magnitude as they are propagated back through time; and the exploding gradient problem refers to the explosion of long-term components due to the large increase in the norm of the gradient during training sequences with long-term dependencies. To solve these issues, researchers introduced Long short-term memory models. Long short-term memory (LSTM) is an RNN variant that was designed to store and access information in a long time sequence. Unlike standard RNNs, nonlinear multiplicative gates and a memory cell are introduced. These gates, including input, output, and forget gates, govern the information ﬂow into and out of the memory cell. The structure of an LSTM unit is illustrated in Figure 1.2. h(t–1) f (t) Memory cell o(t) i (t) Figure 1.2 × c (t) × Output x(t) Input × h(t) The structure of an LSTM unit. (Modiﬁed from Wu et al. [2016]) 8 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning More speciﬁcally, given a sequence of an external input vector x (t) ∈ Rn, an LSTM maps the input to an output vector z(t) ∈ Rm by computing activations of the units in the network with the following equations recursively from t = 1 to t = T : i(t) = σ (Wxi x (t) + Whi h(t−1) + Wci c(t) + bi ), (1.2) f (t) = σ (Wxf x (t) + Whf h(t) + Wcf c(t) + bf ), c(t) = f (t)c(t−1) + it tanh(Wxc x (t) + Whc h(t−1) + bc ), o(t) = σ (Wxo x (t) + Who h(t−1) + Wco c(t) + bo ), h(t) = o(t) tanh(c(t)), where x (t) , h(t) are the input and hidden vectors with the subscription t denoting the t-th time step, while i(t) , f (t) , c(t) , o(t) are, respectively, the activation vectors of the input gate, forget gate, memory cell, and output gate. Wαβ denotes the weight matrix between α and β. For example, the weight matrix from the input x (t) to the input gate i(t) is Wxi . In Equation 1.2 and Figure 1.2 and at time step t, the input x (t) and the previous states h(t−1) are used as the input of LSTM. The information of the memory cell is updated/controlled from two sources: (1) the previous cell memory unit c(t−1) and (2) the input gate’s activation it . Speciﬁcally, c(t−1) is multiplied by the activation from the forget gate f (t), which learns to forget the information of the previous states. In contrast, the it is combined with the new input signal to consider new information. LSTM also utilizes the output gate o(t) to control the information received by hidden state variable h(t). To sum up, with these explicitly designed memory units and gates, LSTM is able to exploit the long-range temporal memory and avoids the issues of vanishing/exploding gradients. LSTM has recently been popularly used for video analysis, as will be discussed in the following sections. 1.3 Video Classification The sheer volume of video data has motivated approaches to automatically categorizing video contents according to classes such as human activities and complex events. There is a large body of literature focusing on computing effective local feature descriptors (e.g., HoG, HoF, MBH, etc.) from spatio-temporal volumes to account for temporal clues in videos. These features are then quantized into bagof-words or Fisher Vector representations, which are further fed into classiﬁers like 1.3 Video Classiﬁcation 9 support vector machines (SVMs). In contrast to hand crafting features, which is usually time-consuming and requires domain knowledge, there is a recent trend to learn robust feature representations with deep learning from raw video data. In the following, we review two categories of deep learning algorithms for video classiﬁcation, i.e., supervised deep learning and unsupervised feature learning. 1.3.1 Supervised Deep Learning for Classification 1.3.1.1 Image-Based Video Classification The great success of CNN features on image analysis tasks [Girshick et al. 2014, Razavian et al. 2014] has stimulated the utilization of deep features for video classiﬁcation. The general idea is to treat a video clip as a collection of frames, and then for each frame, feature representation could be derived by running a feed-forward pass till a certain fully-connected layer with state-of-the-art deep models pre-trained on ImageNet [Deng et al. 2009], including AlexNet [Krizhevsky et al. 2012], VGGNet [Simonyan and Zisserman 2015], GoogLeNet [Szegedy et al. 2015a], and ResNet [He et al. 2016b], as discussed earlier. Finally, frame-level features are averaged into video-level representations as inputs of standard classiﬁers for recognition, such as the well-known SVMs. Among the works on image-based video classiﬁcation, Zha et al. [2015] systematically studied the performance of image-based video recognition using features from different layers of deep models together with multiple kernels for classiﬁcation. They demonstrated that off-the-shelf CNN features coupled with kernel SVMs can obtain decent recognition performance. Motivated by the advanced feature encoding strategies in images [Sánchez et al. 2013], Xu et al. [2015c] proposed to obtain video-level representation through vector of locally aggregated descriptors (VLAD) encoding [Jégou et al. 2010b], which can attain performance gain over the trivial averaging pooling approach. Most recently, Qiu et al. [2016] devised a novel Fisher Vector encoding with Variational AutoEncoder (FV-VAE) to quantize the local activations of the convolutional layer, which learns powerful visual representations of better generalization. 1.3.1.2 End-to-End CNN Architectures The effectiveness of CNNs on a variety of tasks lies in their capability to learn features from raw data as an end-to-end pipeline targeting a particular task [Szegedy et al. 2015a, Long et al. 2015, Girshick 2015]. Therefore, in contrast to the imagebased classiﬁcation methods, there are many works focusing on applying CNN models to the video domain with an aim to learn hidden spatio-temporal patterns. Ji et al. [2010] introduced the 3D CNN model that operates on stacked video frames, 10 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning extending the traditional 2D CNN designed for images to the spatio-temporal space. The 3D CNN utilizes 3D kernels for convolution to learn motion information between adjacent frames in volumes segmented by human detectors. Karpathy et al. [2014] compared several similar architectures on a large scale video dataset in order to explore how to better extend the original CNN architectures to learn spatiotemporal clues in videos. They found that the performance of the CNN model with a single frame as input achieves similar results to models operating on a stack of frames, and they also suggested that a mixed-resolution architecture consisting of a low-resolution context and a high-resolution stream could speed up training effectively. Recently, Tran et al. [2015] also utilized 3D convolutions with modern deep architectures. However, they adopted full frames as the inputs of 3D CNNs instead of the segmented volumes in Ji et al. [2010]. Though the extension of conventional CNN models by stacking frames makes sense, the performance of such models is worse than that of state-of-the-art handcrafted features [Wang and Schmid 2013]. This may be because the spatio-temporal patterns in videos are too complex to be captured by deep models with insufﬁcient training data. In addition, the training of CNNs with inputs of 3D volumes is usually time-consuming. To effectively handle 3D signals, Sun et al. [2015] introduced factorized spatio-temporal convolutional networks that factorize the original 3D convolution kernel learning as a sequential process of learning 2D spatial kernels in the lower layer. In addition, motivated by the fact that videos can naturally be decomposed into spatial and temporal components, Simonyan and Zisserman [2014] proposed a two-stream approach (see Figure 1.3), which breaks down the learning of video representation into separate feature learning of spatial and temporal clues. More speciﬁcally, the authors ﬁrst adopted a typical spatial CNN to model appearance information with raw RGB frames as inputs. To account for temporal clues among adjacent frames, they explicitly generated multiple-frame dense optical ﬂow, upon which a temporal CNN is trained. The dense optical ﬂow is derived from computing displacement vector ﬁelds between adjacent frames (see Figure 1.4), which represent motions in an explicit way, making the training of the network easier. Finally, at test time, each individual CNN generates a prediction by averaging scores from 25 uniformly sampled frames (optical ﬂow frames) for a video clip, and then the ﬁnal output is produced by the weighted sum of scores from the two streams. The authors reported promising results on two action recognition benchmarks. As the two-stream approach contains many implementation choices that may affect the performance, Ye et al. [2015b] evaluated different options, including dropout ratio and network architecture, and discussed their ﬁndings. Very recently, there have been several extensions of the two-stream approach. Wang et al. utilized the point trajectories from the improved dense trajectories 1.3 Video Classiﬁcation ·· ·· ·· ·· 11 ·· ··· Individual frame Spatial stream CNN Score fusion ·· ·· ·· ·· ·· ··· Stacked optical ﬂow Motion stream CNN Figure 1.3 Two-stream CNN framework. (From Wu et al. [2015c]) Figure 1.4 Examples of optical ﬂow images. (From Simonyan and Zisserman [2014]) [Wang and Schmid 2013] to pool two-stream convolutional feature maps to generate trajectory-pooled deep-convolutional descriptors (TDD) [Wang et al. 2015]. Feichtenhofer et al. [2016] improved the two-stream approach by exploring a better fusion approach to combine spatial and temporal streams. They found that two streams could be fused using convolutional layers rather than averaging classiﬁcation scores to better model the correlations of spatial and temporal streams. Wang et al. [2016b] introduced temporal segment networks, where each segment is used as the input of a two-stream network and the ﬁnal prediction of a video clip is produced by a consensus function combining segment scores. Zhang et al. [2016] proposed to replace the optical ﬂow images with motion vectors with an aim to achieve real-time action recognition. More recently, Wang et al. [2016c] proposed to learn feature representation by modeling an action as a transformation from an initial state (condition) to a new state (effect) with two Siamese CNN networks, operating on RGB frames and optical ﬂow images. Similar to the original two-stream approach, they then fused the classiﬁcation scores from two streams linearly to obtain ﬁnal predictions. They reported better results on two challenging benchmarks 12 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning than Simonyan and Zisserman [2014], possibly because the transformation from precondition to effect could implicitly model the temporal coherence in videos. Zhu et al. [2016] proposed a key volume mining approach that attempts to identify key volumes and perform classiﬁcation at the same time. Bilen et al. [2016] introduced the dynamic image to represent motions with rank pooling in videos, upon which a CNN model is trained for recognition. 1.3.1.3 Modeling Long-Term Temporal Dynamics As discussed earlier, the temporal CNN in the two-stream approach [Simonyan and Zisserman 2014] explicitly captures the motion information among adjacent frames, which, however, only depicts movements within a short time window. In addition, during the training of CNN models, each sweep takes a single frame (or a stacked optical frame image) as the input of the network, failing to take the order of frames into account. This is not sufﬁcient for video analysis, since complicated events/actions in videos usually consist of multiple actions happening over a long time. For instance, a “making pizza” event can be decomposed into several sequential actions, including “making the dough,” “topping,” and “baking.” Therefore, researchers have recently attempted to leverage RNN models to account for the temporal dynamics in videos, among which LSTM is a good ﬁt without suffering from the “vanishing gradient” effect, and has demonstrated its effectiveness in several tasks like image/video captioning [Donahue et al. 2017, Yao et al. 2015a] (to be discussed in detail later) and speech analysis [Graves et al. 2013]. Donahue et al. [2017] trained two two-layer LSTM networks (Figure 1.5) for action recognition with features from the two-stream approach. They also tried to ﬁne-tune the CNN models together with LSTM but did not obtain signiﬁcant performance gain compared with only training the LSTM model. Wu et al. [2015c] fused the outputs of LSTM models with CNN models to jointly model spatio-temporal clues for video classiﬁcation and observed that CNNs and LSTMs are highly complementary. Ng et al. [2015] further trained a 5-layer LSTM model and compared several pooling strategies. Interestingly, the deep LSTM model performs on par with single frame CNN on a large YouTube video dataset called Sports-1M, which may be because the videos in this dataset are uploaded by ordinary users without professional editing and contain cluttered backgrounds and severe camera motion. Veeriah et al. [2015] introduced a differential gating scheme for LSTM to emphasize the change in information gain to remove redundancy in videos. Recently, in a multi-granular spatio-temporal architecture [Li et al. 2016a], LSTMs have been utilized to further model the temporal information of frame, motion, and clip streams. Wu et al. [2016] further employed a CNN operating on spectrograms derived from 1.3 Video Classiﬁcation Figure 1.5 13 LSTM LSTM yt LSTM LSTM yt+1 LSTM LSTM yt+2 LSTM LSTM yt+n Utilizing LSTMs to explore temporal dynamics in videos with CNN features as inputs. soundtracks of videos to complement visual clues captured by CNN and LSTMs, and demonstrated strong results. 1.3.1.4 Incorporating Visual Attention Videos contain many frames. Using all of them is computationally expensive and may degrade the performance of recognizing a class of interest as not all the frames are relevant. This issue has motivated researchers to leverage the attention mechanism to identify the most discriminative spatio-temporal volumes that are directly related to the targeted semantic class. Sharma et al. [2015] proposed the ﬁrst attention LSTM for action recognition with a soft-attention mechanism to attach higher importance to the learned relevant parts in video frames. More recently, Li et al. [2016c] introduced the VideoLSTM, which applied attention in convolutional LSTM models to discover relevant spatio-temporal volumes. In addition to soft-attention, VideoLSTM also employed motion-based attention derived from optical ﬂow images for better action localization. 1.3.2 Unsupervised Video Feature Learning Current remarkable improvements with deep learning heavily rely on a large amount of labeled data. However, scaling up to thousands of video categories presents signiﬁcant challenges due to insurmountable annotation efforts even at video level, not to mention frame-level ﬁne-grained labels. Therefore, the utilization of unsupervised learning, integrating spatial and temporal context information, is a promising way to ﬁnd and represent structures in videos. Taylor et al. [2010] 14 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning proposed a convolutional gated boltzmann Machine to learn to represent optical ﬂow and describe motion. Le et al. [2011] utilized two-layer independent subspace analysis (ISA) models to learn spatio-temporal models for action recognition. More recently, Srivastava et al. [2015] adopted an encoder-decoder LSTM to learn feature representations in an unsupervised way. They ﬁrst mapped an input sequence into a ﬁxed-length representation by an encoder LSTM, which would be further decoded with single or multiple decoder LSTMs to perform different tasks, such as reconstructing the input sequence, or predicting the future sequence. The model was ﬁrst pre-trained on YouTube data without manual labels, and then ﬁne-tuned on standard benchmarks to recognize actions. Pan et al. [2016a] explored both local temporal coherence and holistic graph structure preservation to learn a deep intrinsic video representation in an end-to-end fashion. Ballas et al. [2016] leveraged convolutional maps from different layers of a pre-trained CNN as the input of a gated recurrent unit (GRU)-RNN to learn video representations. Summary The latest developments discussed above have demonstrated the effectiveness of deep learning for video classiﬁcation. However, current deep learning approaches for video classiﬁcation usually resort to popular deep models in image and speech domain. The complicated nature of video data, containing abundant spatial, temporal, and acoustic clues, makes off-the-shelf deep models insufﬁcient for videorelated tasks. This highlights the need for a tailored network to effectively capture spatial and acoustic information, and most importantly to model temporal dynamics. In addition, training CNN/LSTM models requires manual labels that are usually expensive and time-consuming to acquire, and hence one promising direction is to make full utilization of the substantial amounts of unlabeled video data and rich contextual clues to derive better video representations. 1.4 Video Captioning Video captioning is a new problem that has received increasing attention from both computer vision and natural language processing communities. Given an input video, the goal is to automatically generate a complete and natural sentence, which could have a great potential impact, for instance, on robotic vision or on helping visually impaired people. Nevertheless, this task is very challenging, as a description generation model should capture not only the objects, scenes, and activities presented in the video, but also be capable of expressing how these objects/scenes/activities relate to each other in a natural sentence. In this section, we elaborate the problem by surveying the state-of-the-art methods. We classify exist- 1.4 Video Captioning 15 Input video: … Video tagging: baby, boy, chair Figure 1.6 … Video (frame) captioning: Two baby boys are in the chair. Video captioning: A baby boy is biting ﬁnger of another baby boy. Examples of video tagging, image (frame) captioning, and video captioning. The input is a short video, while the output is a text response to this video, in the form of individual words (tags), a natural sentence describing one single image (frame), and dynamic video contents, respectively. ing methods in terms of different strategies for sentence modeling. In particular, we distill a common architecture of combining convolutional and recurrent neural networks for video captioning. As video captioning is an emerging area, we start by introducing the problem in detail. 1.4.1 Problem Introduction Although there has already been extensive research on video tagging [Siersdorfer et al. 2009, Yao et al. 2013] and image captioning [Vinyals et al. 2015, Donahue et al. 2017], video-level captioning has its own characteristics and thus is different from tagging and image/frame-level captioning. A video tag is usually the name of a speciﬁc object, action, or event, which is recognized in the video (e.g., “baby,” “boy,” and “chair” in Figure 1.6). Image (frame) captioning goes beyond tagging by describing an image (frame) with a natural sentence, where the spatial relationships between objects or object and action are further described (e.g., “Two baby boys are in the chair” generated on one single frame of Figure 1.6). Video captioning has been taken as an even more challenging problem, as a description should not only capture the above-mentioned semantic knowledge in the video but also express the spatio-temporal relationships in between and the dynamics in a natural sentence (e.g., “A baby boy is biting ﬁnger of another baby boy” for the video in Figure 1.6). Despite the difﬁculty of the problem, there have been several attempts to address video caption generation [Pan et al. 2016b, Yu et al. 2016, Xu et al. 2016], which are mainly inspired by recent advances in machine translation [Sutskever 16 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning et al. 2014]. The elegant recipes behind this are the promising developments of the CNNs and the RNNs. In general, 2D [Simonyan and Zisserman 2015] and/or 3D CNNs [Tran et al. 2015] are exploited to extract deep visual representations and LSTM [Hochreiter and Schmidhuber 1997] is utilized to generate the sentence word by word. More sophisticated frameworks, additionally integrating internal or external knowledge in the form of high-level semantic attributes or further exploring the relationship between the semantics of sentence and video content, have also been studied for this problem. In the following subsections we present a comprehensive review of video captioning methods through two main categories based on the strategies for sentence generation (Section 1.4.2) and generalizing a common architecture by leveraging sequence learning for video captioning (Section 1.4.3). 1.4.2 Approaches for Video Captioning There are mainly two directions for video captioning: a template-based language model [Kojima et al. 2002, Rohrbach et al. 2013, Rohrbach et al. 2014, Guadarrama et al. 2013, Xu et al. 2015b] and sequence learning models (e.g., RNNs) [Donahue et al. 2017, Pan et al. 2016b, Xu et al. 2016, Yu et al. 2016, Venugopalan et al. 2015a, Yao et al. 2015a, Venugopalan et al. 2015b, Venugopalan et al. 2015b]. The former predeﬁnes the special rule for language grammar and splits the sentence into several parts (e.g., subject, verb, object). With such sentence fragments, many works align each part with detected words from visual content by object recognition and then generate a sentence with language constraints. The latter leverages sequence learning models to directly learn a translatable mapping between video content and sentence. We will review the state-of-the-art research along these two dimensions. 1.4.2.1 Template-based Language Model Most of the approaches in this direction depend greatly on the sentence templates and always generate sentences with syntactical structure. Kojima et al. [2002] is one of the early works that built a concept hierarchy of actions for natural language description of human activities. Tan et al. [2011] proposed using predeﬁned concepts and sentence templates for video event recounting. Rohrbach et al.’s conditional random ﬁeld (CRF) learned to model the relationships between different components of the input video and generate descriptions for videos [Rohrbach et al. 2013]. Furthermore, by incorporating semantic unaries and hand-centric features, Rohrbach et al. [2014] utilized a CRF-based approach to generate coherent video descriptions. In 2013, Guadarrama et al. used semantic hierarchies to choose an appropriate level of the speciﬁcity and accuracy of sentence fragments. Recently, a 1.4 Video Captioning 17 deep joint video-language embedding model in Xu et al. [2015b] was designed for video sentence generation. 1.4.2.2 Sequence Learning Unlike the template-based language model, sequence learning-based methods can learn the probability distribution in the common space of visual content and textual sentence and generate novel sentences with more ﬂexible syntactical structure. Donahue et al. [2017] employed a CRF to predict activity, object, and location present in the video input. These representations were concatenated into an input sequence and then translated to a natural sentence with an LSTM model. Later, Venugopalan et al. [2015b] proposed an end-to-end neural network to generate video descriptions by reading only the sequence of video frames. By mean pooling, the features over all the frames can be represented by one single vector, which is the input of the following LSTM model for sentence generation. Venugopalan et al. [2015a] then extended the framework by inputting both frames and optical ﬂow images into an encoder-decoder LSTM. Inspired by the idea of learning visual-semantic embedding space in search [Pan et al. 2014, Yao et al. 2015b], [Pan et al. 2016b] additionally considered the relevance between sentence semantics and video content as a regularizer in LSTM based architecture. In contrast to mean pooling, Yao et al. [2015a] proposed to utilize the temporal attention mechanism to exploit temporal structure as well as a spatio-temporal convolutional neural network to obtain local action features. Then, the resulting video representations were fed into the text-generating RNN. In addition, similar to the knowledge transfer from image domain to video domain [Yao et al. 2012, 2015c], Liu and Shi [2016] leveraged the learned models on image captioning to generate a caption for each video frame and incorporate the obtained captions, regarded as the attributes of each frame, into a sequence-to-sequence architecture to generate video descriptions. Most recently, with the encouraging performance boost reported on the image captioning task by additionally utilizing high-level image attributes in Yao et al. [2016], Pan et al. [2016c] further leveraged semantic attributes learned from both images and videos with a transfer unit for enhancing video sentence generation. 1.4.3 A Common Architecture for Video Captioning To better summarize the frameworks of video captioning by sequence learning, we illustrate a common architecture as shown in Figure 1.7. Given a video, 2D and/or 3D CNNs are utilized to extract visual features on raw video frames, optical ﬂow images, and video clips. The video-level representations are produced by mean 18 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning … … a small red #end … Mean pooling/ soft-attention LSTM LSTM LSTM … … #start Figure 1.7 LSTM 3D CNN … Optical ﬂow images … 2D CNN … Raw video frames a small street A common architecture for video captioning by sequence learning. The video representations are produced by mean pooling or soft-attention over the visual features of raw video frames/optical ﬂow images/video clips, extracted by 2D/3D CNNs. The sentence is generated word by word in the following LSTM, based on the video representations. pooling or soft attention over these visual features. Then, an LSTM is trained for generating a sentence based on the video-level representations. Technically, suppose we have a video V with Nv sample frames/optical images/clips (uniform sampling) to be described by a textual sentence S, where S = {w1 , w2 , . . . , wNs } consisting of Ns words. Let v ∈ RDv and wt ∈ RDw denote the Dv dimensional visual features of a video V and the Dw -dimensional textual features of the t-th word in sentence S, respectively. As a sentence consists of a sequence of words, a sentence can be represented by a Dw × Ns matrix W ≡ [w1 , w2 , . . . , wNs ], with each word in the sentence as its column vector. Hence, given the video representations v, we aim to estimate the conditional probability of the output word sequence {w1 , w2 , . . . , wNs }, i.e., Pr (w1 , w2 , . . . , wNs \|v). (1.3) Since the model produces one word in the sentence at each time step, it is natural to apply the chain rule to model the joint probability over the sequential words. Thus, the log probability of the sentence is given by the sum of the log probabilities over the words and can be expressed as: log Pr (W\|v) = Ns t=1 log Pr wt v, w1 , . . . , wt−1 . (1.4) 1.5 Benchmarks and Challenges 19 In the model training, we feed the start sign word #start into LSTM, which indicates the start of the sentence generation process. We aim to maximize the log probability of the output video description S given the video representations, the previous words it has seen, and the model parameters θ, which can be formulated as ∗ θ = arg max θ Ns log Pr wt v, w1 , . . . , wt−1; θ . (1.5) t=1 This log probability is calculated and optimized over the whole training dataset using stochastic gradient descent. Note that the end sign word #end is required to terminate the description generation. During inference, we choose the word with maximum probability at each time step and set it as the LSTM input for the next time step until the end sign word is emitted. Summary The introduction of the video captioning problem is relatively new. Recently, this task has sparked signiﬁcant interest and may be regarded as the ultimate goal of video understanding. Video captioning is a complex problem and has been initially forwarded by the fundamental technological advances in recognition that can effectively recognize key objects or scenes from video contents. The developments of RNNs in machine translation have further accelerated the growth of this research direction. The recent results, although encouraging, are still indisputably far from practical use, as the forms of the generated sentences are simple and the vocabulary is still limited. How to generate free-form sentences and support open vocabulary are vital issues for the future of this task. 1.5 Benchmarks and Challenges We now discuss popular benchmarks and challenges for video classiﬁcation (Section 1.5.1) and video captioning (Section 1.5.2). 1.5.1 Classification Research on video classiﬁcation has been stimulated largely by the release of the large and challenging video datasets such as UCF101 [Soomro et al. 2012], HMDB51 [Kuehne et al. 2011], and FCVID [Jiang et al. 2015], and by the open challenges organized by fellow researchers, including the THUMOS challenge [Jiang et al. 2014b], the ActivityNet Large Scale Activity Recognition Challenge [Heilbron et al. 2015], and the TRECVID multimedia event detection (MED) task [Over et al. 2014]. 20 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning Table 1.1 Popular benchmark datasets for video classification, sorted by the year of construction Dataset KTH #Video #Class Released Year Background 600 6 2004 Clean Static 81 9 2005 Clean Static 1,358 25 2007 Dynamic 430 8 2008 Dynamic Hollywood2 1,787 12 2009 Dynamic MCG-WEBV 234,414 15 2009 Dynamic 800 16 2010 Dynamic HMDB51 6,766 51 2011 Dynamic CCV 9,317 20 2011 Dynamic UCF-101 13,320 101 2012 Dynamic THUMOS-2014 18,394 101 2014 Dynamic ≈31,000 20 2014 Dynamic Sports-1M 1,133,158 487 2014 Dynamic ActivityNet 27,901 203 2015 Dynamic EventNet 95,321 500 2015 Dynamic MPII Human Pose 20,943 410 2014 Dynamic FCVID 91,223 239 2015 Dynamic Weizmann Kodak Hollywood Olympic Sports MED-2014 (Dev. set) In the following, we ﬁrst discuss related datasets according to the list shown in Table 1.1, and then summarize the results of existing works. 1.5.1.1 Datasets KTH dataset is one of the earliest benchmarks for human action recognition [Schuldt et al. 2004]. It contains 600 short videos of 6 human actions performed by 25 people in four different scenarios. Weizmann dataset is another very early and simple dataset, consisting of 81 short videos associated with 9 actions performed by 9 actors [Blank et al. 2005]. Kodak Consumer Videos dataset was recorded by around 100 customers of the Eastman Kodak Company [Loui et al. 2007]. The dataset collected 1,358 1.5 Benchmarks and Challenges 21 video clips labeled with 25 concepts (including activities, scenes, and single objects) as a part of the Kodak concept ontology. Hollywood Human Action dataset contains 8 action classes collected from 32 Hollywood movies, totaling 430 video clips [Laptev et al. 2008]. It was further extended to the Hollywood2 [Marszalek et al. 2009] dataset, which is composed of 12 actions from 69 Hollywood movies with 1,707 video clips in total. This Hollywood series is challenging due to cluttered background and severe camera motion throughout the datasets. MCG-WEBV dataset is another large set of YouTube videos that has 234,414 web videos with annotations on several topic-level events like “a conﬂict at Gaza” [Cao et al. 2009]. Olympic Sports includes 800 video clips and 16 action classes [Niebles et al. 2010]. It was ﬁrst introduced in 2010 and, unlike in previous datasets, all the videos were downloaded from the Internet. HMDB51 dataset comprises 6,766 videos annotated into 51 classes [Kuehne et al. 2011]. The videos are from a variety of sources, including movies and YouTube consumer videos. Columbia Consumer Videos (CCV) dataset was constructed in 2011, aiming to stimulate research on Internet consumer video analysis [Jiang et al. 2011]. It contains 9,317 user-generated videos from YouTube, which were annotated into 20 classes, including objects (e.g., “cat” and “dog”), scenes (e.g., “beach” and “playground”), sports events (e.g., “basketball” and “soccer”), and social activities (e.g., “birthday” and “graduation”). UCF-101 & THUMOS-2014 dataset is another popular benchmark for human action recognition in videos, consisting of 13,320 video clips (27 hours in total) with 101 annotated classes such as “diving” and “weight lifting” [Soomro et al. 2012]. More recently, the THUMOS-2014 Action Recognition Challenge [Jiang et al. 2014b] created a benchmark by extending the UCF-101 dataset (used as the training set). Additional videos were collected from the Internet, including 2,500 background videos, 1,000 validation videos, and 1,574 test videos. TRECVID MED dataset was released and annually updated by the task of MED, created by NIST since 2010 [Over et al. 2014]. Each year an extended dataset based on datasets from challenges of previous years is constructed and released for worldwide system comparison. For example, in 2014 the MED 22 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning dataset contained 20 events, such as “birthday party,” “bike trick,” etc. According to NIST, in the development set, there are around 8,000 videos for training and 23,000 videos used as dry-run validation samples (1,200 hours in total). The MED dataset is only available to the participants of the task, and the labels of the ofﬁcial test set (200,000 videos) are not available even to the participants. Sports-1M dataset consists of 1 million YouTube videos in 487 classes, such as “bowling,” “cycling,” “rafting,” etc., and has been available since 2014 [Karpathy et al. 2014]. The video annotations were automatically derived by analyzing online textual contexts of the videos. Therefore the labels of this dataset are not clean, but the authors claim that the quality of annotation is fairly good. ActivityNet dataset is another large-scale video dataset for human activity recognition and understanding and was released in 2015 [Heilbron et al. 2015]. It consists of 27,801 video clips annotated into 203 activity classes, totaling 849 hours of video. Compared with existing datasets, ActivityNet contains more ﬁne-grained action categories (e.g., “drinking beer” and “drinking coffee”). EventNet dataset consists of 500 events and 4,490 event-speciﬁc concepts and was released in 2015 [Ye et al. 2015a]. It includes automatic detection models for its video events and some constituent concepts, with around 95,000 training videos from YouTube. Similarly to Sports-1M, EventNet was labeled by online textual information rather than manually labeled. MPII Human Pose dataset includes around 25,000 images containing over 40,000 people with annotated body joints [Andriluka et al. 2014]. According to an established taxonomy of human activities (410 in total), the collected images (from YouTube videos) were provided with activity labels. Fudan-Columbia Video Dataset (FCVID) dataset contains 91,223 web videos annotated manually into 239 categories [Jiang et al. 2015]. The categories cover a wide range of topics, such as social events (e.g., “tailgate party”), procedural events (e.g., “making cake”), object appearances (e.g., “panda”), and scenes (e.g., “beach”). 1.5.1.2 Challenges To advance the state of the art in video classiﬁcation, several challenges have been introduced with the aim of exploring and evaluating new approaches in realistic settings. We brieﬂy introduce three representative challenges here. 1.5 Benchmarks and Challenges 23 THUMOS Challenge was ﬁrst introduced in 2013 in the computer vision community, aiming to explore and evaluate new approaches for large-scale action recognition of Internet videos [Idrees et al. 2016]. The three editions of the challenge organized in 2013–2015 made THUMOS a common benchmark for action classiﬁcation and detection. TRECVID Multimedia Event Detection (MED) Task aims to detect whether a video clip contains an instance of a speciﬁc event [Awad et al. 2016, Over et al. 2015, Over et al. 2014]. Speciﬁcally, based on the released TRECVID MED dataset each year, each participant is required to provide for each testing video the conﬁdence score of how likely one particular event is to happen in the video. Twenty pre-speciﬁed events are used each year, and this task adopts the metrics of average precision (AP) and inferred AP for event detection. Each event was also complemented with an event kit, i.e., the textual description of the event as well as the potentially useful information about related concepts that are likely contained in the event. ActivityNet Large Scale Activity Recognition Challenge was ﬁrst organized as a workshop in 2016 [Heilbron et al. 2015]. This challenge is based on the ActivityNet dataset [Heilbron et al. 2015], with the aim of recognizing highlevel and goal-oriented activities. By using 203 activity categories, there are two tasks in this challenge: (1) Untrimmed Classiﬁcation Challenge, and (2) Detection Challenge, which is to predict the labels and temporal extents of the activities present in videos. 1.5.1.3 Results of Existing Methods Some of the datasets introduced above have been popularly adopted in the literature. We summarize the results of several recent approaches on UCF-101 and HMDB51 in Table 1.2, where we can see the fast pace of development in this area. Results on video classiﬁcation are mostly measured by the AP (for a single class) and mean AP (for multiple classes), which are not introduced in detail as they are well known. 1.5.2 Captioning A number of datasets have been proposed for video captioning; these commonly contain videos that have each been paired with its corresponding sentences annotated by humans. This section summarizes the existing datasets and the adopted evaluation metrics, followed by quantitative results of representative methods. 24 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning Table 1.2 Comparison of recent video classification methods on UCF-101 and HMDB51 datasets Methods Table 1.3 LRCN [Donahue et al. 2017] 82.9 — LSTM-composite [Srivastava et al. 2015] 84.3 — FST CN [Sun et al. 2015] 88.1 59.1 C3D [Tran et al. 2015] 86.7 — Two-Stream [Simonyan and Zisserman 2014] 88.0 59.4 LSTM [Ng et al. 2015] 88.6 — Image-Based [Zha et al. 2015] 89.6 — Transformation CNN [Wang et al. 2016c] 92.4 63.4 Multi-Stream [Wu et al. 2016] 92.6 — Key Volume Mining [Zhu et al. 2016] 92.7 67.2 Convolutional Two-Stream [Feichtenhofer et al. 2016] 93.5 69.2 Temporal Segment Networks [Wang et al. 2016b] 94.2 69.4 Comparison of video captioning benchmarks Dataset Context MSVD Multi-category AMT workers — TV16-VTT Multi-category Humans 2,000 YouCook Cooking AMT workers 88 — 2,668 TACoS-ML Cooking AMT workers 273 14,105 52,593 Movie DVS 92 48,986 55,905 94 68,337 68,375 653,467 7,180 10,000 200,000 1,856,523 M-VAD MPII-MD MSR-VTT-10K 1.5.2.1 UCF-101 HMDB51 Sentence Source #Videos Movie Script+DVS 20 categories AMT workers #Clips 1,970 — #Sentences 70,028 4,000 #Words 607,339 — 42,457 — 519,933 Datasets Table 1.3 summarizes key statistics and comparisons of popular datasets for video captioning. Figure 1.8 shows a few examples from some of the datasets. Microsoft Research Video Description Corpus (MSVD) contains 1,970 YouTube snippets collected on Amazon Mechanical Turk (AMT) by requesting workers to pick short clips depicting a single activity [Chen and Dolan 2011]. Annotators then label the video clips with single-sentence descrip- 1.5 Benchmarks and Challenges (a) MSVD dataset Sentences: • A dog walks around on its front legs. • The dog is doing a handstand. • A pug is trying for balance walk on two legs. Sentences: • A man lights a match book on ﬁre. • A man playing with ﬁre sticks. • A man lights matches and yells. (b) M-VAD dataset Sentence: • Later he drags someone through a jog. Sentence: • A waiter brings a pastry with a candle. (c) MPII-MD dataset Sentence: • He places his hands around her waist as she • opens her eyes. Sentence: • Someone’s car is stopped by a couple of • uniformed police. (d) MSR-VTT-10K dataset Sentences: • People practising volleyball in the play ground. • A man is hitting a ball and he falls. • A man is playing a football game on green land. Figure 1.8 Sentences: • A cat is hanging out in a bassinet with a baby. • The cat is in the baby bed with the baby. • A cat plays with a child in a crib. Examples from (a) MSVD, (b) M-VAD, (c) MPII-MD, and (d) MSR-VTT-10K datasets. 25 26 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning tions. The original corpus has multi-lingual descriptions, but only the English descriptions are commonly exploited on video captioning tasks. Specifically, there are roughly 40 available English descriptions per video and the standard split of MSVD is 1,200 videos for training, 100 for validation, and 670 for testing, as suggested in Guadarrama et al. [2013]. YouCook dataset consists of 88 in-house cooking videos crawled from YouTube and is roughly uniformly split into six different cooking styles, such as baking and grilling [Das et al. 2013]. All the videos are in a third-person viewpoint and in different kitchen environments. Each video is annotated with multiple human descriptions by AMT. Each annotator in AMT is instructed to describe the video in at least three sentences totaling a minimum of 15 words, resulting in 2,668 sentences for all the videos. TACoS Multi-Level Corpus (TACoS-ML) is mainly built [Rohrbach et al. 2014] based on MPII Cooking Activities dataset 2.0 [Rohrbach et al. 2015c], which records different activities used when cooking. TACoS-ML consists of 185 long videos with text descriptions collected via AMT workers. Each AMT worker annotates a sequence of temporal intervals across the long video, pairing every interval with a single short sentence. There are 14,105 distinct intervals and 52,593 sentences in total. Montreal Video Annotation Dataset (M-VAD) is composed of about 49,000 DVD movie snippets, which are extracted from 92 DVD movies [Torabi et al. 2015]. Each movie clip is accompanied by one single sentence from semiautomatically transcribed descriptive video service (DVS) narrations. The fact that movies always contain a high diversity of visual and textual content, and that there is only one single reference sentence for each movie clip, has made the video captioning task on the M-VAD dataset very challenging. MPII Movie Description Corpus (MPII-MD) is another collection of movie descriptions dataset that is similar to M-VAD [Rohrbach et al. 2015b]. It contains around 68,000 movie snippets from 94 Hollywood movies and each snippet is labeled with one single sentence from movie scripts and DVS. MSR Video to Text (MSR-VTT-10K) is a recent large-scale benchmark for video captioning that contains 10K Web video clips totalling 41.2 hours, covering the most comprehensive 20 categories obtained from a commercial video 1.5 Benchmarks and Challenges 27 search engine, e.g., music, people, gaming, sports, and TV shows [Xu et al. 2016]. Each clip is annotated with about 20 natural sentences by AMT workers. The training/validation/test split is provided by the authors with 6,513 clips for training, 2,990 for validation, and 497 for testing. The TRECVID 2016 Video to Text Description (TV16-VTT) is another recent video captioning dataset that consists of 2,000 videos randomly selected from Twitter Vine videos [Awad et al. 2016]. Each video has a total duration of about 6 seconds and is annotated with 2 sentences by humans. The human annotators are asked to address four facets in the generated sentences: who the video is describing (kinds of persons, animals, things) and what the objects and beings are doing, plus where it is taking place and when. 1.5.2.2 Evaluation Metrics For quantitative evaluation of the video captioning task, three metrics are commonly adopted: BLEU@N [Papineni et al. 2002], METEOR [Banerjee and Lavie 2005], and CIDEr [Vedantam et al. 2015]. Speciﬁcally, BLEU@N is a popular machine translation metric which measures the fraction of N-gram (up to 4-gram) in common between a hypothesis and a reference or set of references. However, as pointed out in Chen et al. [2015], the N-gram matches for a high N (e.g., 4) rarely occur at a sentence level, resulting in poor performance of BLEU@N especially when comparing individual sentences. Hence, a more effective evaluation metric, METEOR, utilized along with BLEU@N , is also widely used in natural language processing (NLP) community. Unlike BLEU@N, METEOR computes unigram precision and recall, extending exact word matches to include similar words based on WordNet synonyms and stemmed tokens. Another important metric for image/video captioning is CIDEr, which measures consensus in image/video captioning by performing a Term Frequency Inverse Document Frequency (TF-IDF) weighting for each N-gram. 1.5.2.3 Results of Existing Methods Most popular methods of video captioning have been evaluated on MSVD [Chen and Dolan 2011], M-VAD [Torabi et al. 2015], MPII-MD [Rohrbach et al. 2015b], and TACoS-ML [Rohrbach et al. 2014] datasets. We summarize the results on these four datasets in Tables 1.4, 1.5, and 1.6. As can be seen, most of the works are very recent, indicating that video captioning is an emerging and fast-developing research topic. Table 1.4 Reported results on the MSVD dataset, where B@N , M, and C are short for BLEU@N , METEOR, and CIDEr-D scores, respectively Methods B@1 B@2 B@3 B@4 M C FGM [Thomason et al. 2014] — — — 13.7 23.9 — LSTM-YT [Venugopalan et al. 2015b] — — — 33.3 29.1 — MM-VDN [Xu et al. 2015a] — — — 37.6 29.0 — S2VT [Venugopalan et al. 2015a] — — — — 29.8 — S2FT [Liu and Shi 2016] — — — — 29.9 — 80.0 64.7 52.6 41.9 29.6 51.7 — — — 42.1 31.4 — 78.8 66.0 55.4 45.3 31.0 — SA [Yao et al. 2015a] Glove+Deep Fusion [Venugopalan et al. 2016] LSTM-E [Pan et al. 2016b] GRU-RCN [Ballas et al. 2016] h-RNN [Yu et al. 2016] — — — 43.3 31.6 68.0 81.5 70.4 60.4 49.9 32.6 65.8 All values are reported as percentages (%). Table 1.5 Reported results on (a) M-VAD and (b) MPII-MD datasets, where M is short for METEOR M-VAD dataset Methods M SA [Yao et al. 2015a] 4.3 Mean Pool [Venugopalan et al. 2015a] 6.1 Visual-Labels [Rohrbach et al. 2015a] 6.4 S2VT [Venugopalan et al. 2015a] 6.7 Glove+Deep Fusion [Venugopalan et al. 2016] 6.8 LSTM-E [Pan et al. 2016b] 6.7 MPII-MD dataset Methods M SMT [Rohrbach et al. 2015b] 5.6 Mean Pool [Venugopalan et al. 2015a] 6.7 Visual-Labels [Rohrbach et al. 2015a] 7.0 S2VT [Venugopalan et al. 2015a] 7.1 Glove+Deep Fusion [Venugopalan et al. 2016] 6.8 LSTM-E [Pan et al. 2016b] 7.3 All values are reported as percentages (%). 1.6 Conclusion Table 1.6 29 Reported results on the TACoS-ML dataset, where B@N , M, and C are short for BLEU@N , METEOR, and CIDEr-D scores, respectively Methods B@1 B@2 B@3 B@4 M C CRF-T [Rohrbach et al. 2013] 56.4 44.7 33.2 25.3 26.0 124.8 CRF-M [Rohrbach et al. 2014] 58.4 46.7 35.2 27.3 27.2 134.7 LRCN [Donahue et al. 2017] 59.3 48.2 37.0 29.2 28.2 153.4 h-RNN [Yu et al. 2016] 60.8 49.6 38.5 30.5 28.7 160.2 All values are reported as percentages (%). 1.6 Conclusion In this chapter, we have reviewed state-of-the-art deep learning techniques on two key topics related to video analysis, video classiﬁcation and video captioning, both of which rely on the modeling of the abundant spatial and temporal information in videos. In contrast to hand crafted features that are costly to design and have limited generalization capability, the essence of deep learning for video classiﬁcation is to derive robust and discriminative feature representations from raw data through exploiting massive videos with an aim to achieve effective and efﬁcient recognition, which could hence serve as a fundamental component in video captioning. Video captioning, on the other hand, focuses on bridging visual understanding and language description by joint modeling. We also provided a review of popular benchmarks and challenges for both video classiﬁcation and captioning tasks. Though extensive efforts have been made in video classiﬁcation and captioning with deep learning, we believe we are just beginning to unleash the power of deep learning in the big video data era. 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DOI: 10.1017 /S0956792515000492. 209 Index 1-bit compressive sensing for sketch similarity, 121 1 Million Song Corpus for audition, 37–38 3D CNN model, 9–10 3DMark Ice Storm Benchmark, 305–306 Abstraction in situations, 166, 168 Accelerometers in multimodal analysis of social interactions, 55–56 Acoustic backgrounds, 41 Acoustic events, 41 Acoustic intelligence, 32–33 Action Control in EventShop platform, 179 Actionable situations, 168 Actions of audition objects, 47 ActivityNet dataset for video classiﬁcation, 22 ActivityNet Large Scale Activity Recognition Challenge, 23 Actuators in eco-systems, 162 AdaPtive HFR vIdeo Streaming (APHIS) for cloud gaming, 309–310 Additive and multiplicative homomorphism, 88–90 Additive quantization in similarity searches, 131 ADMM (alternating-direction method of multipliers), 63–66 Advanced Micro Devices (AMD) for cloud gaming, 289 AES (Advanced Encryption Standard), 98 Affective ratings in emotion and personality type recognition, 240, 243–247 Aggregation in situation recognition, 175– 176 Agreeableness personality dimension, 237 Alerts in EventShop platform, 182 AlexNet, 5, 9 Algebraic homomorphic property in encrypted content, 101 All-in-one software pack for GamingAnywhere, 294–295 Alternating-direction method of multipliers (ADMM), 63–66 AMD (Advanced Micro Devices) for cloud gaming, 289 amtCNN (asymmetric multi-task CNN) model, 149 Animation rendering crowdsourced, 273–275 resource requirements, 256 ANN (approximate nearest neighbors) strategies, 105, 107–111 Annotations eye ﬁxations as, 226–236 user cues, 220 Anti-sparse coding for sketch similarity, 126–127 APHIS (AdaPtive HFR vIdeo Streaming) for cloud gaming, 309–310 Applications for audition, 49 Approximate nearest neighbors (ANN) strategies, 105, 107–111 Approximate search algorithms, 107–108 Architecture for video captioning, 17–18 Arousal dimensions in emotion and 380 Index personality type recognition, 236– 237, 243–247 Arrival times in Poisson processes, 195 Asthma/allergy risk recommendations, 163, 183–184 Asymmetric multi-task CNN (amtCNN) model, 149 Asymmetric sketch similarity schemes, 123 Attention mechanism for LSTM, 13 Audio emotion recognition, 221 multimodal analysis of social interactions, 55–56 multimodal pose estimation, 60 segment grouping, 43–44 Audio processing in encrypted domain audio editing quality enhancement, 99–100 speech/speaker recognition, 95–99 Audition for multimedia computing applications, 49 background, 35–39 Computer Audition ﬁeld, 32–35 conclusion, 49–50 data for, 39–40 generative models of sound, 45 generative structure of audio, 44–45 grouping audio segments, 43–44 nature of audio data, 40–41 NELS, 47–48 overview, 31–32 peculiarities of sound, 41–49 representation and parsing of mixtures, 42–43 structure discovery in audio, 46–47 weak and opportunistic supervision, 48–49 Augmented Lagrangian in ADMM, 64 Augmented-reality cloud gaming, 313 Average query time for similarity searches, 111 B-trees in similarity searches, 107 Bag-of-words representation object recognition, 232–233 similarity searches, 116 Basic linear algebra subprograms (BLAS) for similarity searches, 111 Behavior analysis in user-multimedia interaction, 146 Benchmarks in deep learning, 19–29 Best-bin-ﬁrst strategy in similarity searches, 107 BGN (Boneh-Goh-Nissim) cryptosystem, 99 BGV (Brakerski-Gentry-Vaikuntanathan) scheme, 101 Big-ﬁve marker scale (BFMS) questionnaire for emotion and personality type recognition, 238 Big-ﬁve model for emotion recognition, 237 Binary regularization term in multimodal pose estimation, 62 Biometric data, SPED for, 82–83 Biometric recognition for image processing, 88 BLAS (basic linear algebra subprograms) for similarity searches, 111 BLEU@N metric for video classiﬁcation, 27–29 Blind source separation (BSS) in audition, 36 BOINC platform, 258 Boneh-Goh-Nissim (BGN) cryptosystem, 99 Bottlenecks in Hawkes processes, 208–209 Bounding box annotations in scene recognition, 228–229 Brakerski-Gentry-Vaikuntanathan (BGV) scheme, 101 Branching structure in Hawkes processes, 200–202, 213 Broadcasts in situation-aware applications, 164 BSS (blind source separation) in audition, 36 Building blocks in situation recognition, 171 Business models in fog computing, 261 Index C-ADMM (coupled alternating direction method of multipliers), 64–66 Caller/callee pairs of streams in speech/ speaker recognition, 97 Captioning video, 14–19, 23–29 Capture servers for GamingAnywhere, 296 CASA (computational auditory scene analysis), 36–37 Cascade size in Hawkes processes, 213–215 CBIR (content-based information retrieval), 85 CCV (Columbia Consumer Videos) dataset, 21 CDC (cloud data center) architecture, 80 Cell-probe model and algorithms for similarity searches description, 108 hash functions, 113–118 introduction, 113–114 query mechanisms, 118–120 CF (collaborative ﬁltering) recommender systems, 151–152 SPED for, 82 CG (computationally grounded) factor deﬁnitions for situations, 167 Characterization in situation recognition, 175–176 Chi-Square in similarity searches, 133 CHIL Acoustic Event Detection campaign, 38 Children events in Hawkes processes, 212–213 Chroma features in audition, 37–38 CIDEr metric for video classiﬁcation, 27–29 Classiﬁcation audition, 38 situation recognition, 175–176 Client-server privacy-preserving image retrieval framework, 86 Cloud computing. See Fog computing Cloud data center (CDC) architecture, 80 Cloud gaming, 256 adaptive transmission, 308–310 cloud deployment, 298–302 381 communication, 307–310 conclusion, 314 future paradigm, 310–314 GamingAnywhere, 291–298 hardware decoders, 306 interaction delay, 302–304 introduction, 287–289 multiplayer games, 310–311 research, 289–291 thin client design, 302–306 Cloudlet servers, 259 Clusters of offspring in Hawkes processes, 201–202 CNNs (convolutional neural networks) end-to-end architectures, 9–12 similarity searches, 107 video, 5–6 Collaborative ﬁltering (CF) recommender systems, 151–152 SPED for, 82 Color SIFT (CSIFT) features in object recognition, 234–236 Columbia Consumer Videos (CCV) dataset, 21 Column-wise regularization in C-ADMM, 64 Common architecture for video captioning, 17–18 Common principles of interactions between users and multimedia data, 155 Communications cloud gaming, 307–310 fog computing, 257 Complementary hash functions, 117–118 Complex mathematical operations in encrypted multimedia analysis, 102–103 Complexity encrypted multimedia content, 101 similarity searches, 109–111 Compositional models of sound, 42–43 Compressed-domain distance estimation in similarity searches, 127–128 Compressed speaker recognition (CSR) systems, 97 382 Index Compression in cloud gaming, 307–308 Computational auditory scene analysis (CASA), 36–37 Computational bottlenecks in Hawkes processes, 208–209 Computational complexity in encrypted multimedia content, 101 Computational/conditional security in encrypted multimedia content, 103 Computational cost in similarity searches, 109 Computationally grounded (CG) factor deﬁnitions for situations, 167 Computations in fog computing, 257 Computer Audition. See Audition for multimedia computing Conditional random ﬁeld (CRF) in video captioning, 17–18 Conﬁdentiality of data, SPED for, 83–84 Connectivity for cloud gaming, 311 Conscientiousness personality dimension, 237, 246 Containers in fog computing, 261, 282–284 Content-based information retrieval (CBIR), 85 Content-centric computing, 138, 143 Content delay in CrowdMAC framework, 266–267 “Content Is Dead; Long Live Content!” panel, 142 Content quality in Hawkes processes, 210 Context-aware cloud gaming, 313–314 Controlled minions in fog computing, 257 Conversion operator in EventShop platform, 182 Convolutional neural networks (CNNs) end-to-end architectures, 9–12 similarity searches, 107 video, 5–6 Cooperative component sharing in cloud gaming, 312 Correctness in fog computing, 262 Cosine similarity and indexing similarity searches, 133 sketch similarity, 121–124 Cost CrowdMAC framework, 266–267 crowdsourced animation, 273 “Coulda, Woulda, Shoulda: 20 Years of Multimedia Opportunities” panel, 142 Counter-Strike game, 302 Coupled alternating direction method of multipliers (C-ADMM), 64–66 CRF (conditional random ﬁeld) in video captioning, 17–18 Crowdedness monitors in fog computing, 284 CrowdMAC framework, 263–267 Crowdsensing in SAIS, 268 Crowdsourced animation rendering services, 273–275 Cryptology. See Encrypted domain multimedia content analysis CSIFT features in object recognition, 234–236 CSR (compressed speaker recognition) systems, 97 CubeLendar system, 188 D-CASE (Detection and Classiﬁcation of Acoustic Scenes and Events) challenge, 38 DAGs (direct acyclic graphs) in fog computing, 276–278 Data Box project, 188 Data-centric computing, 137–138, 143 Data collection of social media, 144–145 Data compression in cloud gaming, 307– 308 Data ingestion EventShop platform, 179, 181 situation recognition, 173 Data overhead in encrypted domain multimedia content analysis, 101 Data representation and analysis multimedia analysis, 154 situation recognition, 174, 185–186 Index Data-source Panel in EventShop platform, 178 Data uniﬁcation in situation recognition, 173 Datasets in video classiﬁcation, 19–23 DBNs (deep belief networks) in convolutional neural networks, 5 DCT sign correlation for images, 88 dE-mages, 185 Decoders in cloud gaming, 306 Decomposition in Hawkes processes, 204–205 Deep belief networks (DBNs) in convolutional neural networks, 5 Deep learning benchmarks and challenges, 19–29 conclusion, 29 introduction, 3–4 modules, 4–8 video captioning, 14–19 video classiﬁcation, 8–14, 19–23 Delay bounded admission control algorithm, 266 Delays cloud gaming, 302–304 CrowdMAC framework, 266–267 Demographic information inference from user-generated content, 147 Descriptors in situation deﬁnitions, 167 Detection and Classiﬁcation of Acoustic Scenes and Events (D-CASE) challenge, 38 Device-aware scalable applications for cloud gaming, 291 Difference-of-Gaussian (DoG) transforms in SIFT, 86–87 Digital watermarking, SPED for, 81 Direct acyclic graphs (DAGs) in fog computing, 276–278 Direct memory access (DMA) channels for cloud gaming, 300 Discrete Fourier transformation for probabilistic cryptosystems, 80 383 Distance embeddings in similarity searches, 133 Distance metric learning, intentionoriented, 149–150 Distributed principal component analysis, 275–280 DMA (direct memory access) channels for cloud gaming, 300 Docker containers for fog computing, 283–284 DoG (Difference-of-Gaussian) transforms in SIFT, 86–87 Dropout in AlexNet, 5 E-mages EventShop platform, 178–181 situation recognition, 173 E2LSH techniques in similarity searches, 131 ECGs (electrocardiograms) in emotion and personality type recognition, 238, 241–242 Eco-systems for situation recognition, 162–164 Edge effects in Hawkes processes, 208 EEG (electroencephalogram) devices emotion and personality type recognition, 238, 241–242 user cues, 220–221 Efﬁciency encrypted domain multimedia content analysis, 103 situation recognition, 174 SPED for, 84 Efﬁcient Task Assignment (ETA) algorithm in SAIS, 269–272 EigenBehaviors in situation recognition, 187 Electrocardiograms (ECGs) in emotion and personality type recognition, 238, 241–242 Electroencephalogram (EEG) devices emotion and personality type recognition, 238, 241–242 384 Index Electroencephalogram (EEG) devices (continued) user cues, 220–221 Electronic voting, SPED for, 81 ElGamal cryptosystem, 79 Emerging applications in encrypted domain multimedia content analysis, 102– 103 Emotion and personality type recognition introduction, 236–238 materials and methods, 238–240 personality scores vs. affective ratings, 243–247 physiological feature extraction, 240–243 physiological signals, 247–250 user cues, 221 Encoder-decoder LSTM, 14 Encoding strategies in sketch similarity, 125–126 Encrypted domain multimedia content analysis audio processing, 95–100 conclusion, 104 future research and challenges, 101–104 image processing, 84–90 introduction, 75–78 SPED, 78–84 video processing in encrypted domain, 91–96 End-to-end CNN architectures in imagebased video classiﬁcation, 9–12 Energy usage in CrowdMAC framework, 266, 272 Enhancement layer in cloud gaming, 308 Environmental sound classiﬁcation in audition, 38 Epidemic type aftershock-sequences (ETAS) model, 209 Equivalent counting point processes, 194 ESP game, 220 ETA (Efﬁcient Task Assignment) algorithm in SAIS, 269–272 ETAS (epidemic type aftershock-sequences) model, 209 Ethical issues in situation recognition, 188 Euclidean case in similarity searches, 106–107, 114–115 Evaluation criteria for similarity searches, 109–113 Event-driven servers for GamingAnywhere, 296 EventNet dataset for video classiﬁcation, 22 Events, point processes for. See Point processes for events EventShop platform asthma/allergy risk recommendation, 183–184 heterogeneous data, 180–182 operators, 181–182 overview, 177–178 situation-aware applications, 182–185 system design, 178–180 Exact search algorithms, 107–108 Exhaustive search algorithms, 107–108 Exogenous events in Hawkes processes, 198 Expectation in similarity searches, 128 Expected number of future events in Hawkes processes, 212–213 Expected value in Poisson processes, 194 Exploding gradients in recurrent neural networks, 7 Expressive power concept in situation recognition, 170 Extraversion personality dimension, 237, 244 Eye ﬁxations and movements in object recognition discussion, 232 emotion recognition, 237 ﬁxation-based annotations, 232–236 free-viewing and visual search, 227–232 introduction, 226–227 materials and methods, 227 scene semantics inferences from, 222– 226 user cues, 220 Eysenck’s personality model, 237 F-formation detection in head and body pose estimation, 69–73 Index Face detectors in fog computing, 283–284 Face swapping in video surveillance systems, 92 Facial expressions, differentiating via eye movements in, 224–226 Facial landmark trajectories in emotion and personality type recognition, 242–243 Fairness in cloud gaming, 311 FCGs. See Free-standing conversational groups (FCGs) FCVID (Fudan-Columbia Video Dataset) dataset for video classiﬁcation, 22 Feature extraction in image processing, 86–88 Feed-forward neural networks (FFNNs), 6 Filtering probabilistic cryptosystems, 80 recommender systems, 151–152 situation recognition, 175 SPED for, 82 Fisher Vector encoding with Variational AutoEncoder (FV-VAE), 9 Five-factor model for emotion recognition, 237 Fixed-base annotations for object recognition, 232–236 Flickr videos in audition, 39 user favorite behavior patterns, 148 Fog computing challenges, 260–262 conclusion, 285–286 CrowdMAC framework, 263–267 crowdsourced animation rendering service, 273–275 introduction, 255–258 open-source platforms, 280–285 related work, 258–260 scalable and distributed principal component analysis, 275–280 Smartphone-Augmented Infrastructure Sensing, 268–272 Fraud in fog computing, 262 385 Frechet distances in similarity searches, 133 Free-standing conversational groups (FCGs), 51 conclusion, 73–74 F-formation detection, 69–73 head and body pose estimation, 56–57, 66–69 introduction, 52–55 matrix completion for multimodal pose estimation, 59–66 matrix completion overview, 57–58 multimodal analysis of social interactions, 55–56 SALSA dataset, 58–59 Free-viewing (FV) tasks eye movements, 227–232 scene recognition, 226 Freelance minions in fog computing, 257 Fudan-Columbia Video Dataset (FCVID) dataset for video classiﬁcation, 22 Fully homomorphic encryption (FHE) techniques, 88–90 Function hiding, SPED for, 81 Future actions (FA) in situation deﬁnitions, 166 Future events in Hawkes processes, 212–213 FV-VAE (Fisher Vector encoding with Variational AutoEncoder), 9 G-cluster cloud gaming company, 290 Galvanic skin response (GSR) in emotion and personality type recognition, 238, 241–242 Games as a Service (GaaS), 291 GamingAnywhere community participation, 297–298 environment setup, 294–295 execution, 296–297 introduction, 291–292 research, 297 system architecture, 293–294 target users, 292–293 Gaussian distribution in situation recognition, 185 386 Index Gaussian mixture model (GMM) CSR systems, 97–99 speech recognition, 36 Generative models Hawkes processes, 213–215 sound, 45 Generative structure of audio, 44–45 Geolocation in audition, 39 GIST descriptors in similarity searches, 132 GMM (Gaussian mixture model) CSR systems, 97–99 speech recognition, 36 Goal based (GB) factor in situation deﬁnitions, 166 Google search engine, 151 GoogLeNet, 6, 9 GPUs (graphical processing units) for cloud gaming, 298–302 Gradients in recurrent neural networks, 7 Graph-based approaches for similarity searches, 132–134 Graph-cut approach in F-formation detection, 69–70 Graphical processing units (GPUs) for cloud gaming, 298–302 Graphics compression in cloud gaming, 307–308 Graphs in social graph modeling, 145 Grouping audio segments, 43–44 GSR (galvanic skin response) in emotion and personality type recognition, 238, 241–242 Hamming space and embedding similarity searches, 114–115, 132 sketch similarity, 121–122 Hardware decoders in cloud gaming, 306 Hash functions similarity searches, 113–118 sketch similarity, 123–124 Hawkes processes, 197 branching structure, 200–202 conclusion, 217–218 estimating, 211–212 expected number of future events, 212–213 generative model, 213–215 hands-on tutorial, 215–217 Hawkes model for social media, 209–217 information diffusion, 210–211 intensity function, 198–200 introduction, 192–193 likelihood function, 205–206 maximum likelihood estimation, 207– 209, 211–212 parameter estimates, 205–209 sampling by decomposition, 204– 205 self-exciting processes, 197–198 simulating events, 202–205 thinning algorithm, 202–204 Head and body pose estimates (HBPE) experiments, 66–69 F-formation detection, 69–73 free-standing conversational groups, 53–54 overview, 56–57 SALSA dataset, 59 Heart rate in emotion and personality type recognition, 238, 241–243 Heterogeneous data EventShop platform, 180–182 user-multimedia interaction, 146 Hidden Markov models (HMM) in speech/ speaker recognition, 95–98 Hierarchical structure in audition, 46–47 High-intensity facial expressions in eye movements, 224–226 High-rate quantization theory in similarity searches, 117 Histogram of oriented gradient (HOG) descriptors, 59 HMDB51 dataset for video classiﬁcation, 21, 24 HMM (Hidden Markov models) in speech/ speaker recognition, 95–98 HOG (Histogram of oriented gradient) descriptors, 59 Index Hollywood Human Action dataset for video classiﬁcation, 21 Homomorphic encryption image search and retrieval, 86 SPED, 79 speech/speaker recognition, 99 video processing in encrypted domain, 94 Homophily hypothesis, 154 Honest-but-curious model, 80–81 Hotspots in mobile Internet, 264–268 Hough voting method (HVFF-lin), 69–70 Hough voting method multi-scale extension (HVFF-ms), 69–70 Householder decomposition in sketch similarity, 124 Human actuators in eco-systems, 162 Human behavior in situation recognition, 187 Human-centric computing, 138 Human pose in free-standing conversational groups, 53 Human sensors in eco-systems, 162 HVFF-lin (Hough voting method), 69–70 HVFF-ms (Hough voting method multi-scale extension), 69–70 Hybrid approaches to similarity searches, 131–132 Hybrid compression in cloud gaming, 307–308 Hyper-diamond E8 lattices in similarity searches, 115 Hypervisors for cloud gaming, 299 IARPA Aladdin Please program, 38 Image-based video classiﬁcation, 9 Image collectors in fog computing, 283–284 Image processing in encrypted domain biometric recognition, 88 feature extraction, 86–88 image search and retrieval, 84–86 quality enhancement, 88–90 Image representation learning, intentionoriented, 148–149 387 ImageNet, 9 Imbalance factor (IF) in similarity searches, 116–117 Immigrant events in Hawkes processes, 198, 200, 204, 210 Impact sounds, 45 In-situ sensing in SAIS, 270–271 Independent sample-wise processing in SPED, 79–80 Indexing schemes image search and retrieval, 85 similarity searches, 107–109, 112, 133 Inferring scene semantics from eye movements, 222–226 Inﬂuence-based recommendation, 152 Information diffusion in Hawkes processes, 210–211 Information theoretic/unconditional security in encrypted multimedia content, 103 Infrared detection in multimodal pose estimation, 60 Infrastructure sensing in fog computing, 268–272 Input/output memory management units (IOMMUs) for cloud gaming, 300 Instance-level inferences in audition, 33 Integrity of data encrypted domain multimedia content analysis, 103–104 SPED for, 83 Intensity Hawkes processes, 198–200, 210 Poisson processes, 195 Intention-oriented learning distance metric, 149–150 image representation, 148–149 Inter-arrival times in Poisson processes, 194–196 Interaction delay in cloud gaming, 302–304 Interactions of objects in audition, 47 Interactive scenes, saccades in, 223 Intermediate Query Panel in EventShop platform, 178 388 Index Internal Storage in EventShop platform, 179 Interoperable common principles in social media, 155 Interpolation operator in EventShop platform, 182 Intuitive query and mental model in situation recognition, 174 Inverse transform sampling technique for Hawkes processes, 202–203 IOMMUs (input/output memory management units) for cloud gaming, 300 Job assignment algorithms in SAIS, 269 Johnson-Lindenstrauss Lemma, 107 Joint estimation in multimodal pose estimation, 62 JPEG 2000 images in encrypted domains, 88 K-means in similarity searches, 117, 127– 129 k-NN graphs for similarity searches, 109 KD-trees in similarity searches, 107, 116– 117 Kernel function in Hawkes processes, 198–200, 204–205 Kernel PAC (KPCA) in similarity searches, 133 Kernelized LSH (KLSH) in similarity searches, 133 Keyword recognizers (KWR) in speech/ speaker recognition, 95–97 Keyword searches for images, 85 Kgraph method for similarity searches, 133 KLSH (kernelized LSH) in similarity searches, 133 Kodak Consumer Videos dataset for video classiﬁcation, 20–21 KPCA (kernel PAC) in similarity searches, 133 Kronecker product in multimodal pose estimation, 62 KTH dataset for video classiﬁcation, 20 Kubernetes platform cloud applications, 256 fog computing, 281–283 Kusanagi project, 290 KVM technology in fog computing, 282 KWR (keyword recognizers) in speech/ speaker recognition, 95–97 LabelMe database, 220 Lagrangian in ADMM, 64 Laplacian matrix in multimodal pose estimation, 62 Large margin nearest neighbor (LMNN), 149–150 Last-mile technology, 142 Latent variable analysis (LVA) approach in audition, 37 Lattices in similarity searches, 115 Learned quantizers in similarity searches, 115–116 Leech lattices in similarity searches, 115 Legal issues in cloud gaming, 291 LeNet-5 framework, 5 Likelihood function in Hawkes processes, 205–206 Limb tracking with visual-inertial sensors, 57 Linear ﬁltering in probabilistic cryptosystems, 80 Local maxima in Hawkes processes, 207– 208 Local minima in Hawkes processes, 212 Locality-Sensitive Hashing (LSH) hash functions, 114–115 query-adaptive, 119–120 similarity searches, 105–106 sketch similarity, 122, 126–127 Long short-term memory (LSTM) modeling, 12–13 recurrent neural networks, 7–8 video classiﬁcation, 17–19 Long-term temporal dynamics modeling, 12–13 Look-up operations in similarity searches, 110 Low-intensity facial expressions in eye movements, 224–226 Index Lower the ﬂoor concept in situation recognition, 171 LSH. See Locality-Sensitive Hashing (LSH) LSTM (long short-term memory) modeling, 12–13 recurrent neural networks, 7–8 video classiﬁcation, 17–19 LXC containers for fog computing, 282 M-VAD (Montreal Video Annotation Dataset) for video classiﬁcation, 26 Macroscopic behavior analysis, 146 Magnetoencephalogram (MEG) signals in emotion recognition, 221 Magnitude of inﬂuence in Hawkes processes, 210 Mahout-PCA library for fog computing, 279–280 Malicious model in SPED, 81 MapReduce platform, 279–280 Marked Hawkes processes, 210–211 Marked memory kernel in Hawkes processes, 211 Markov process in Hawkes processes, 204–205 Massively multiplayer online role-playing games (MMORPGs), 310 Matching biometric data, SPED for, 82–83 Matrix completion (MC) free-standing conversational groups, 54 head and body pose estimation, 57–58 multimodal pose estimation, 59–66 Matrix completion for head and body pose estimation (MC-HBPE), 60–61, 66–69 Maximum likelihood estimation in Hawkes processes, 207–209, 211–212 McCulloch-Pitts model for convolutional neural networks, 5 MCG-WEBV dataset for video classiﬁcation, 21 MDA (Module Deployment Algorithm), 282, 284 Mean average precision in similarity searches, 112 389 MediaEval dataset for audition, 39 Medical data, SPED for, 83 MEG (magnetoencephalogram) signals in emotion recognition, 221 Mel-Frequency Cepstral Coefﬁcients (MFCCs) in audition, 37 Memory over time in Hawkes processes, 210 Memory reads in similarity searches, 110–111 Memorylessness property in Poisson processes, 195–196 Mesoscopic behavior analysis, 146 Meta inferences in audition, 33 METEOR metric for video classiﬁcation, 27–29 MFCCs (Mel-Frequency Cepstral Coefﬁcients) in audition, 37 Microscopic behavior analysis, 146 Microsoft Research Video Description Corpus (MSVD) dataset for video classiﬁcation, 24–28 Minions in fog computing, 257–258 MIREX evaluations in audition, 37 Missing and uncertain data in situation recognition, 185–186 Mixtures in audition, 42–43 MMORPGs (massively multiplayer online role-playing games), 310 Mobile Internet, CrowdMAC framework for, 263–267 Mobile ofﬂoading in fog computing, 258– 259 Modeling approach in situation recognition, 171–172 Module Deployment Algorithm (MDA), 282, 284 Montreal Video Annotation Dataset (M-VAD) for video classiﬁcation, 26 MPEG-X standard, 156–157 MPII Human Pose dataset for video classiﬁcation, 22 MPII Movie Description Corpus (MPII-MD) for video classiﬁcation, 26 MSR Video to Text (MSR-VTT-10K) dataset for video classiﬁcation, 26–27 390 Index MSVD (Microsoft Research Video Description Corpus) dataset for video classiﬁcation, 24–28 Multi-probe LSH similarity searches, 118–119 sketch similarity, 122 Multi-task learning approach in head and body pose estimation, 57 Multimedia Commons initiative for audition, 39 Multimedia data social attributes for, 154–155 user interest modeling from, 147–148 Multimedia fog platforms, 257–258 Multimedia-sensed social computing, 156 Multimodal analysis encrypted domain multimedia content analysis, 102 free-standing conversational groups. See Free-standing conversational groups (FCGs) sentiment, 150 social interactions, 55–56 Multimodal pose estimation matrix completion, 59–66 model, 60–63 optimization method, 63–66 Multiplayer games, 310–311 Music signals in audition, 37–38 Natural audio, 33–34 Near neighbors in similarity searches, 114 Need gap vs. semantic gap, 139–141 Neighbors in similarity searches, 106–111, 114 NELL (Never Ending Language Learner) system, 48 NELS (Never-Ending Sound Learner) proposal, 47–48 Neuroticism personality dimension, 237, 244–245 NN-descent algorithm for similarity searches, 133 Non-Euclidean metrics for similarity searches, 132–134 Non-homogeneous Poisson process, 195– 197 Non-interactive scenes, saccades in, 223 Normalized deviation in crowdsourced animation rendering service, 274– 275 NP-hard problem in multimodal pose estimation, 61 Object interactions with saccades, 222–224 Object recognition with eye ﬁxations as implicit annotations, 226–236 Observed situations, 168 Observers in GamingAnywhere, 293 Offspring events in Hawkes processes, 200–201 Olympic Sports dataset for video classiﬁcation, 21 OnLive gaming, 290 Open-source platforms cloud gaming systems, 292 fog computing, 280–285 OpenCV package in EventShop platform, 179 Openness personality dimension, 237, 246 OpenPDS project, 188 OpenStack platform cloud applications, 256 fog computing, 281 Operators Panel in EventShop platform, 178 Opportunistic supervision in audition, 48–49 Optimization techniques cloud gaming, 288 hash functions, 117–118 multimodal pose estimation, 63–66 OTT service for cloud gaming, 310 P2P (peer-to-peer) paradigm CrowdMAC framework, 265 fog computing, 259 Paillier cryptosystem homomorphic encryption, 79 SIFT, 86–87 speech/speaker recognition, 99 Index ParaDrop for fog computing, 259 Parsing of mixtures in audition, 42–43 Pascal animal classes Eye Tracking (PET) database, 226–228 PASCAL Visual Object Classes for user cues, 220 Pass-through GPUs for cloud gaming, 300 Pattern matching in situation recognition, 175–176 PCA (principal component analysis), 275– 280 peer-to-peer (P2P) paradigm CrowdMAC framework, 265 fog computing, 259 Percepts in audition, 43 Personality scores in emotion and personality type recognition vs. affective ratings, 243–247 description, 240 Personality trait recognition, 247–248 Personalization in EventShop platform, 182 Personalized alerts in situation-aware applications, 164 Persuading user action, 187–188 PET (Pascal animal classes Eye Tracking) database, 226–228 Physiological signals in emotion and personality type recognition experiments, 249–250 feature extraction, 240–243 materials and methods, 238–240 overview, 236–238 personality scores vs. affective ratings, 243–247 responses, 238–239 trait recognition, 247–249 Play-as-you-go services in cloud gaming, 298 Point processes for events conclusion, 217–218 deﬁning, 193–194 Hawkes processes. See Hawkes processes introduction, 191–193 Poisson processes, 193–197 Poisson processes deﬁnition, 194–195 391 memorylessness property, 195– 196 non-homogeneous, 196–197 points, 193–194 POSIX platforms for GamingAnywhere, 295–296 Power consumption in cloud gaming, 305–306 Power-law kernel function for Hawkes processes, 215–216 Pre-binarized version vectors in sketch similarity, 126 Principal component analysis (PCA), 275– 280 Privacy encryption for. See Encrypted domain multimedia content analysis fog computing, 262 situation recognition, 174, 188 Probabilistic cryptosystems, 80 Probability density function in Poisson processes, 195 Product quantization in similarity searches, 128–131 Project+take sign approach for sketch similarity, 124–127 Protocols in emotion and personality type recognition, 240 Proximity sensors in multimodal analysis of social interactions, 55–56 Public video surveillance systems, 91 Quality enhancement in image processing in encrypted domain, 88–90 Quality of Experience (QoE) metrics in cloud gaming, 288 Quality of Service (QoS) cloud gaming, 288 fog computing, 262 Quantization in similarity searches, 106, 127–131 Quantization-optimized LSH in sketch similarity, 126–127 Quantizers in similarity searches, 115–116, 128–129 392 Index Queries in similarity searches mechanisms, 118–120 preparation cost, 110 query-adaptive LSH, 119–120 times, 109, 111 Query plan trees in EventShop platform, 180 Query Processing Engine in EventShop platform, 179–180 R-trees in similarity searches, 107 Raise the ceiling concept in situation recognition, 171 Rank-ordered image searches, 85 Rapid prototyping toolkit in situation recognition, 172 Raspberry Pis for fog computing, 283–284 Raw vectors in similarity searches, 112 Real time strategy (RTS) games, 303 Real time streaming protocol (RTSP), 293–294 Real time transport protocol (RTP), 293–294 Recognition problem in situation deﬁnitions, 168 Recommendation in social-sensed multimedia, 151–152 Rectiﬁed Linear Units (ReLUs) in AlexNet, 5 Recurrence Quantiﬁcation Analysis (RQA) in scene recognition, 229–230 Recurrent neural networks (RNNs), 6–8 Redundant computation in fog computing, 278 Region-of-interest rectangles, 225 Registered Queries in EventShop platform, 178 Reliable data collection of social media, 144–145 ReLUs (Rectiﬁed Linear Units) in AlexNet, 5 RenderStorm cloud rendering platform, 273 Representation and parsing of mixtures in audition, 42–43 Request delays in CrowdMAC framework, 266–267 Residual vectors in similarity searches, 129 ResNet, 6 Resource management in fog computing, 261–262 Results Panel in EventShop platform, 178 Revenue in CrowdMAC framework, 266–267 RNNs (recurrent neural networks), 6–8 Round trip time (RTT) jitter in cloud gaming, 308 Rowe, Larry, 142 Rowley eye detectors, 225 RQA (Recurrence Quantiﬁcation Analysis) in scene recognition, 229–230 RSA cryptosystem, 79 RSA-OPRF protocol, 95 RTP (real time transport protocol), 293–294, 303 RTSP (real time streaming protocol), 293– 294 SaaS (Software as a Service) for cloud gaming, 290 Saccades object interactions, 222–224 scene recognition, 229 SAIS (Smartphone-Augmented Infrastructure Sensing), 268–272 SALSA (Synergistic sociAL Scene Analysis) dataset free-standing conversational groups, 58–59 head and body pose estimation, 66–69 SaltStack platform cloud applications, 256 fog computing, 281–282 Sampling by decomposition in Hawkes processes, 204–205 Scalable and distributed principal component analysis, 275–280 Scalable solutions, SPED for, 84 Scalable video coded (SVC) videos, 94–95 Scale invariant feature transform (SIFT) fog computing, 275–276 image processing in encrypted domain, 86–87 object recognition with ﬁxation-based annotations, 234–236 Index similarity searches, 107, 132 Scene understanding from eye movements, 222–226 user cues, 220 Searches images, 84–86 similarity. See Similarity searches SPED, 82 speech/speaker recognition, 97 visual, 226–232 Secure domains for video encoding, 92–93 Secure processing of medical data, SPED for, 83 Secure real time transport protocol (SRTP), 98 Security vs. accuracy, 102–103 encryption. See Encrypted domain multimedia content analysis fog computing, 262 Selectivity in similarity searches, 110, 112 Self-exciting processes Hawkes processes, 197–198 point processes, 192 Semantic gap bridging, 145 vs. need gap, 139–141 Semantic inferences in audition, 33 Semi-controlled minions in fog computing, 257 Semi-honest model for SPED, 80–81 Semi-supervised hierarchical structure in audition, 46 Sensors eco-systems, 162 fog computing, 257 Sequence learning for video captioning, 17 SETI@Home application, 258 SEVIR (Socially Embedded VIsual Representation Learning) approach, 149 Shaderlight cloud rendering platform, 273 Shallow models and approaches audition, 46 393 intention-oriented image representation learning, 148–149 Shannon Information Theory, 138 Shape gain in similarity searches, 115, 128 SIDL (Social-embedding Image Distance Learning) approach, 149–150 SIFT. See Scale invariant feature transform (SIFT) Signal processing in encrypted domain (SPED), 80 applications, 81–83 background, 78–81 beneﬁts, 83–84 introduction, 76–78 Similarity estimation for sketches, 121–123 Similarity searches background, 106–107 cell-probe algorithms, 113–120 conclusion, 134 evaluation criteria, 109–113 hash functions, 113–118 hybrid approaches, 131–132 introduction, 105–106 non-Euclidean metrics and graph-based approaches, 132–134 quantization, 127–131 query mechanisms, 118–120 sketches and binary embeddings, 120– 127 types, 107–108 Simulating events in Hawkes processes, 202–205 Situation awareness in SAIS, 268 Situation deﬁnitions data, 168–169 existing, 165–167 features, 169 overview, 164 proposed, 167–168 recognition problem, 168 Situation recognition using multimodal data asthma risk-based recommendations, 163 challenges and opportunities, 185–188 394 Index Situation recognition using multimodal data (continued) conclusion, 188–189 eco-system, 162–164 EventShop platform. See EventShop platform framework, 170–177 individual behavior, 186–187 introduction, 159–161 missing and uncertain data, 185–186 persuading user action, 187–188 privacy and ethical issues, 188 situation-aware applications, 163–164 situation deﬁned, 164–170 situation estimation, 186 situation evaluation, 174–176 situation responses, 177 workﬂow, 172–176 Sketches and binary embeddings hash function design, 123–124 introduction, 120–121 project+take sign approach, 124–127 similarity estimation, 121–123 similarity search, 120–127 SLA in fog computing, 262 Smartphone-Augmented Infrastructure Sensing (SAIS), 268–272 Social attributes for users and multimedia, 154–155 Social-embedding Image Distance Learning (SIDL) approach, 149–150 Social graph modeling, 145 Social interactions in multimodal analysis, 55–56 Social knowledge on user-multimedia interactions, 143 Social representation of multimedia data, 145 Social-sensed multimedia computing conclusion, 157 demographic information inference from user-generated content, 147 exemplary applications, 150–153 future directions, 153–157 intention-oriented distance metric learning, 149–150 intention-oriented image representation learning, 148–149 introduction, 137–139 MPEG-X, 156–157 multimedia sentiment analysis, 150 overview, 142–144 recent advances, 146–150 reliable data collection, 144–145 semantic gap vs. need gap, 139–141 social attributes for users and multimedia, 154–155 social representation of multimedia data, 145 social-sensed multimedia recommendation, 151–152 social-sensed multimedia search, 151 social-sensed multimedia summarization, 152–153 social-sensed video communication, 153 user interest modeling from multimedia data, 147–148 user-multimedia interaction behavior analysis, 146 user proﬁling and social graph modeling, 145 Socially Embedded VIsual Representation Learning (SEVIR) approach, 149 Sociometric badges in SALSA dataset, 58–59 Software as a Service (SaaS) for cloud gaming, 290 Sound. See Audition for multimedia computing Source separation in audition, 36 Space and time (ST) in situation deﬁnitions, 166 Space complexity in similarity searches, 111–112 Space-Time ARIMA (STARIMA) models, 186 Sparse coding in object recognition, 234 Spatial inferences in audition, 33 Spatial pyramid histogram representation in object recognition, 232–234 Index Spatio-temporal aggregation in situation recognition, 173–174 Spatio-temporal convolutional networks, 10 Spatio-temporal situations, 168 sPCA algorithm, 277–279 SPED. See Signal processing in encrypted domain (SPED) Speech/speaker recognition, 36, 95–99 Sports-1M dataset LSTM, 12 video classiﬁcation, 22 ST (space and time) in situation deﬁnitions, 166 STARIMA (Space-Time ARIMA) models, 186 Stimuli in emotion and personality type recognition, 240 Stop-words in similarity searches, 116 Storage in fog computing, 257 Stream selection in situation recognition, 172–173 Structural questions in audition, 33 Structure discovery in audio, 46–47 Structured quantizers in similarity searches, 115–116 STTPoints EventShop platform, 180–181 situation recognition, 173 Subcritical regime in Hawkes processes, 201 Sum-product of two signals in probabilistic cryptosystems, 80 Super-bits in sketch similarity, 124 Supercritical regime in Hawkes processes, 201 Supervised deep learning for video classiﬁcation, 9–13 Surface information in audition, 47 Surveillance systems, 91 SVC (scalable video coded) videos, 94–95 Synergistic sociAL Scene Analysis (SALSA) dataset free-standing conversational groups, 58–59 head and body pose estimation, 66–69 395 System architecture in GamingAnywhere, 293–294 System design in EventShop platform, 178–180 TACoS Multi-Level Corpus (TACoS-ML) dataset for video classiﬁcation, 26 Tagging video, 14–19 TDD (trajectory-pooled deep-convolutional descriptors), 11 Template-based language model for video captioning, 16–17 Temporal inferences in audition, 33 Temporal segment networks, 11 Temporary engagement in cloud gaming, 311 Thin client design for cloud gaming, 302–306 Thinning algorithm for Hawkes processes, 202–204 Third-party service providers security and privacy concerns, 75 Throughput-driven GPUs for cloud gaming, 299 THUMOS Challenge, 23 Time complexity in similarity searches, 109–110 Trajectory-pooled deep-convolutional descriptors (TDD), 11 Transductive support vector machines (TSVMs), 68 Transferable common principles in social media, 155 Transmission efﬁciency in encrypted multimedia content analysis, 104 TRECVID 2016 Video to Text Description (TV16-VTT) dataset for video classiﬁcation, 27 TRECVID MED dataset for video classiﬁcation, 21–23 Trust-based approaches in recommender systems, 152 TSVM (transductive support vector machines), 68 396 Index TV16-VTT (TRECVID 2016 Video to Text Description) dataset for video classiﬁcation, 27 Tweets dataset for fog computing, 278–280 Two-layer LSTM networks, 12 Two-stream approach for end-to-end CNN architectures, 10–11 UCF-101 & THUMOS-2014 dataset for video classiﬁcation, 21, 24 Unicast alerts in situation-aware applications, 164 Unimodal approaches for wearable devices, 52 Unitary regularization term in multimodal pose estimation, 62 Unsupervised hierarchical structure in audition, 46–47 Unsupervised video feature learning, 13–14 User-centric multimedia computing, 143– 144 User cues conclusion, 250–251 emotion and personality type recognition, 236–250 eye ﬁxations as implicit annotations for object recognition, 226–236 introduction, 219–222 scene semantics inferences from eye movements, 222–226 User-generated content, demographic information inference from, 147 User interest modeling from multimedia data, 147–148 User-multimedia interaction behavior analysis, 146 User proﬁling, 145 Users, social attributes for, 154–155 VA-ﬁles (vector approximation ﬁles) for similarity searches, 109–110 Valence dimensions in emotion and personality type recognition, 236– 237, 243–247 Vanishing gradients in recurrent neural networks, 7 Variable bit rate (VBR) encoding in speech/speaker recognition, 98 Vector approximation ﬁles (VA-ﬁles) for similarity searches, 109–110 Vector ARMA (VARMA) models, 186 Vector identiﬁers in similarity searches, 112 Vector of locally aggregated descriptors (VLAD) encoding, 9 in similarity searches, 132 Vegas over Access Point (VoAP) algorithm in cloud gaming, 309 VGGNet, 5–6, 9 Video convolutional neural networks, 5–6 introduction, 3–4 recurrent neural networks, 6–8 social-sensed communication, 153 Video captioning approaches, 16–17 common architecture, 17–18 overview, 14–15 problem, 15–16 research, 23–29 Video classiﬁcation overview, 8–9 research, 19–23 supervised deep learning, 9–13 unsupervised video feature learning, 13–14 Video in cloud gaming compression, 307 decoders, 305–306 sharing, 311–312 Video processing in encrypted domain introduction, 91 making video data unrecognizable, 91–92 secure domains for video encoding, 92–93 security implementation, 93–96 VideoLSTM, 13 Viola-Jones face detectors, 225 Index Virality in Hawkes processes, 201, 210 Virtualization cloud gaming, 298–302, 312–313 fog computing, 261, 282 Visual data and inertial sensors in head and body pose estimation, 57 Visual Objects Classes (VOC) challenge in scene recognition, 226 Visual search (VS) tasks eye movements, 227–232 scene recognition, 226 Visualization EventShop platform, 182 situation recognition, 173 VLAD (vector of locally aggregated descriptors) encoding, 9 in similarity searches, 132 VoAP (Vegas over Access Point) algorithm in cloud gaming, 309 VOC (Visual Objects Classes) challenge in scene recognition, 226 VOC2012 dataset, 227 Vocal sounds, 45 Voice over Internet protocol (VoIP) trafﬁc, 95–99 Volunteer computing in fog computing, 258 Voronoi diagrams for similarity searches, 109 Voting, SPED for, 81 397 Watermarking, SPED for, 81 Weak and opportunistic supervision in audition, 48–49 Wearable sensing devices, 52 head and body pose estimation, 58 multimodal analysis of social interactions, 55–56 SALSA dataset, 58 Weizmann dataset for video classiﬁcation, 20 Windows platforms for GamingAnywhere, 295–296 Wisdom sources in eco-systems, 162 Wonder Shaper for fog computing, 283 Word-of-mouth diffusion Hawkes processes, 210 point processes, 192 Wrappers in EventShop platform, 181 Xen technology in fog computing, 282 XFINITY Games, 310 Yahoo Flickr Creative Commons 100 Million dataset (YFCC100M) for audition, 39 YouCook dataset for video classiﬁcation, 26 Zero-knowledge watermark detection protocol, 81 ZYNC cloud rendering platform, 273 Editor Biography Shih-Fu Chang Shih-Fu Chang is the Richard Dicker Professor at Columbia University, with appointments in both Electrical Engineering Department and Computer Science Department. His research is focused on multimedia information retrieval, computer vision, machine learning, and signal processing. A primary goal of his work is to develop intelligent systems that can extract rich information from the vast amount of visual data such as those emerging on the Web, collected through pervasive sensing, or available in gigantic archives. His work on content-based visual search in the early 90s—VisualSEEk and VideoQ—set the foundation of this vibrant area. Over the years, he continued to develop innovative solutions for image/video recognition, multimodal analysis, visual content ontology, image authentication, and compact hashing for large-scale indexing. His work has had major impacts in various applications like image/video search engines, online crime prevention, mobile product search, AR/VR, and brain machine interfaces. His scholarly work can be seen in more than 350 peer-reviewed publications, many best-paper awards, more than 30 issued patents, and technologies licensed to seven companies. He was listed as the Most Inﬂuential Scholar in the ﬁeld of Multimedia by Aminer in 2016. For his long-term pioneering contributions, he has been awarded the IEEE Signal Processing Society Technical Achievement Award, ACM Multimedia Special Interest Group Technical Achievement Award, Honorary Doctorate from the University of Amsterdam, the IEEE Kiyo Tomiyasu Award, and IBM Faculty Award. For his contributions to education, he received the Great Teacher Award from the Society of Columbia Graduates. He served as Chair of ACM SIGMM 400 Editor Biography (2013–2017), Chair of Columbia Electrical Engineering Department (2007–2010), the Editor-in-Chief of the IEEE Signal Processing Magazine (2006–2008), and advisor for several international research institutions and companies. In his current capacity as Senior Executive Vice Dean at Columbia Engineering, he plays a key role in the School’s strategic planning, special research initiatives, international collaboration, and faculty development. He is a Fellow of the American Association for the Advancement of Science (AAAS), a Fellow of the IEEE, and a Fellow of the ACM.	1
1 FLiER: Practical Topology Update Detection Using Sparse PMUs arXiv:1409.6644v3 [cs.SY] 22 Jul 2016 C. Ponce D. S. Bindel Abstract—In this paper, we present a Fingerprint Linear Estimation Routine (FLiER) to identify topology changes in power networks using readings from sparsely-deployed phasor measurement units (PMUs). When a power line, load, or generator trips in a network, or when a substation is reconfigured, the event leaves a unique “voltage fingerprint” of bus voltage changes that we can identify using only the portion of the network directly observed by the PMUs. The naive brute-force approach to identify a failed line from such voltage fingerprints, though simple and accurate, is slow. We derive an approximate algorithm based on a local linearization and a novel filtering approach that is faster and only slightly less accurate. We present experimental results using the IEEE 57-bus, IEEE 118-bus, and Polish 19992000 winter peak networks. Index Terms—topology changes, phasor measurement units, voltage fingerprint, approximation, linearization, filtering I. I NTRODUCTION A. Motivation Detection of topology changes is an important network monitoring function, and is a key part of the power grid state estimation pipeline, either as a pre-processing step or as an integrated part of a generalized state estimator. If a topology error processing module fails to detect an error in the model topology, poor and even dangerous control actions may result [1], [2], as unexpected topology changes, such as those due to failed lines, may put stress on the remaining lines and destabilize the network. Thus, it is important to identify topology changes quickly in order to take appropriate control actions. Substations and transmission lines in transmission networks have sensors that directly report failures (or switch open/closed status). However, if a sensor malfunctions, then finding the topology change is again difficult. This can happen due to normal equipment malfunctions, or because a cyber-attacker wishes to mislead network operators. Although failure to The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award Number de-ar0000230. The information, data, or work presented herein was funded in part by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. This research was conducted with Government support under and awarded by DoD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a correctly identify a topology error is less common in a transmission network, the stakes are higher: state estimation based on incorrect topology assumptions can lead to incorrect estimates, causing operators to overlook system instability, and in the worst case, leading to avoidable blackouts. Thus, it is important to have more than one way to monitor network topology. B. Prior work State estimation and topology detection have co-evolved since at least the late 1970s [3]; see [4] for an overview of the state of the art as of 2000. These industry-standard methods use low time-resolution SCADA data about power flows and digital status, together with a reference model, to infer system voltages and currents as well as the current topology. Since the turn of the century, researchers have increasingly proposed PMU-based methods for state estimation, model calibration, fault detection, and wide area monitoring and control [5], [6], [7]. Some PMU-based methods yield state estimates [8], [9], information about oscillations [10], [11], [12], [13], and indicators of faults or contigencies [14], [15] without SCADA data and with little or no reference to a system model during regular operation. These “model-free” methods are attractive because power system models are often not wholly correct. Current PMU deployments do not provide complete observability of most of the power grids, which has lead to a broad literature on optimally placing what PMUs are available [16], [17], [18]. And where PMU data is available, it is much faster than other data sources: within an operating area, SCADA sensors typically report at most every few seconds; and the system data exchange (SDX) model of the North American Electric Reliability Corporation (NERC) provides inter-area topology information only on an hourly basis [19]. According to the IEEE specification, data from a correctly functioning PMU also satisfy tight phase and magnitude error tolerances [20]. Hence, many methods combine PMU data, SCADA data, and model information for state estimation [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33] and for detection and localization of faults [34], [35], [36], [37], [38], [14], [39], [15], [40], [41], [42], [43]. Conventional state estimators use loss functions such as least absolute variation (LAV) or Huber loss [44], [21] for robustness to inconsistent outlier equations caused by bad data or by errors in the model topology. The shape of residuals associated with the outlier equations reflects topology errors, and a quarter century of work on topology estimators exploits this [45], [46], [47], [48], [49], [50]. Robust regression is also used in many hybrid state estimators [21], [22], [25], [31], 2 as deployed PMUs sometimes have errors greater than those nominally allowed by the IEEE standard [?], [?], [?], [?], whether due to poor GPS synchronization (which may cause persistent phase errors), deficiencies in the signal processing algorithms (which may cause oscillations in the PMU output), or other issues. The methods in this paper use differences between PMU predictions, so are insensitive to absolute phase errors; and we filter the signal to reduce sensitivity to deficiencies in signal processing algorithms and to ambient oscillations. Nonetheless, we also use robust regression to defend against other types of sensor errors. Because PMUs measure current and voltage phasors directly, PMU-assisted methods can observe topology changes even in the sparse case when there is not a measurement directly incident on the affected element. This is the basis for several methods for line outage detection; the closest to our work is [34], though other closely related work includes [35], [36], [37]. Our work extends these prior approaches by using PMU signals to identify not only line trips, but also load trips, generator trips, and substation reconfigurations. As with many SCADA topology estimation techniques, we model possible substation reconfigurations using a breaker-level model [51], [52], [53], [44], [54]. In the SCADA literature, one sometimes uses expanded bus-section models only in regions where a substation reconfiguration is suspected [55], but we use a fixed breaker-level model and an extended system with a minimal set of multipliers corresponding to the edges in trees associated with each connected set of bus sections [56], [57]. C. Our work We present here an efficient method to identify topology changes in networks with a (possibly small) number of PMUs. We assume that a complete state estimate is obtained shortly before a topology change, e.g. through conventional SCADA measurements, and we use discrepancies between this state estimate and PMU measurements to identify failures. Our method does not require complete observability from PMU data; it performs well even when there are few PMUs in the network, though having more PMUs does improve the accuracy. Though our approach is similar in spirit to [34], this paper makes three key novel contributions: • We treat not only line outages, but also load trips, generator trips, and substation reconfigurations. This is not new for standard topology estimators, but to our knowledge it is new for PMU-based methods. • We describe a novel subspace-based filtering method to rule out candidate topology changes at low cost. • We take advantage of existing state estimation procedures by linearizing the full AC power flow about a previously estimated state. This increases accuracy compared to the DC approximation used in prior work. II. P ROBLEM F ORMULATION AND F INGERPRINTS Let yik = gik + bik denote the elements of the admittance matrix Y ∈ Cn×n in a bus-branch network model; let Pℓ and Qℓ denote the real and reactive power injections at bus ℓ; and let vℓ = \|vℓ \| exp(θℓ ) denote the voltage phasor at bus ℓ. These quantities are related by the power flow equations H(v; Y ) − s = 0 (1) where X n gℓh Hℓ = \|vℓ \|\|vh \| −bℓh Hn+ℓ h=1 with θℓh = θℓ − θh and s = P1 · · · Pn Q1 bℓh gℓh cos(θℓh ) , sin(θℓh ) (2) T (3) ··· Qn . We note that H is quadratic in v, but linear Y . In a breaker-level model, we use a similar system in which variables are associated with bus sections, and H represents the power flows when all breakers are open. We then write the power flow equations as H(v; Y ) + Cλ − s = 0 C T v = b, (4) where the constraint equations C T v = b have the form cTk v = (ei − ej )T v = vi − vj = bk = 0, i.e. voltage variable j for a “slave” bus section is constrained to be the same as voltage variable i for a “master” bus section. In addition, we include constraints of the form cTk v = eTi v = bk to assign a voltage magnitude at a PV bus or the phase angle at a slack bus. We could trivially eliminate these constraints, but keep them explicit for notational convenience. Our goal is to use the power flow equations to diagnose topology changes such as single line failures, substation reconfigurations, or load or generator trips. We assume the network remains stable and the state shifts from one quasisteady state to another. In practice, of course, there may be oscillations, whether due to ringdown after a topology change or ambient forcing; hence we recommend applying a low-pass filter to extract the mean behavior at each quasi-steady state. Under a topology update, the voltage vector shifts from v to v̂ = v + ∆v. We assume m voltage phasor components, indicated by the rows of E ∈ {0, 1}m×n, are directly observed by PMUs, with appropriate filtering. Assuming the loads and generation vary slowly, modulo high-frequency fluctuations removed by the low-pass filter, we can predict what E∆v should be for each possible contingency. That is, we can match the observed voltage changes E∆v to a list of voltage fingerprints to identify simple topology changes. We note that the same approach used to produce fingerprints for load and generator trips can also be used to identify significant changes in load or generation at a single source. Multiple contingencies can have the same or practically indistinguishable fingerprints. For example, one of two parallel lines with equal admittance may fail, or two lines that are distant from all PMUs but near each other may yield similar fingerprints. But even when a contingency is not identifiable, our method still produces valuable information. When multiple lines have the same effect on the network, our technique can be used to identify a small set of potential lines or breakers to inspect more closely. 3 III. A PPROXIMATE F INGERPRINTS To compute the exact fingerprint for a contingency, we require a nonlinear power flow solve. In a large network with many possible contingencies, this computation becomes expensive. We approximate the changing voltage in each contingency by linearizing the AC power flow equations about the pre-contingency state. As in methods based on the DC approximation, we use the structure of changes to the linearized system to compute voltage change fingerprints for each contingency with a few linear solves. By using information about the current state, we observe better diagnostic accuracy with our AC linearization than with the DC approximation. We consider three different types of contingencies: bus merging or bus splitting due to substation reconfiguration, and line failure. In each case, we assume the pre-contingency state is x = (v, λ) satisfying (4). We denote the post-contingency state by primed variables x′ = (v ′ , λ′ ); we assume in general that the power injections s are the same before and after the contingency. The exact shift in state is ∆x′ = x′ − x, and our approximate fingerprints are based from the approximation δx′ ≈ ∆x to the shift in state. The computation of δx′ for each contingency involves the pre-contingency Jacobian matrix ∂H (v; Y ) C ∂v . A= CT 0 We assume a factorization of A is available, perhaps from a prior state estimate. A. Bus Merging Fingerprints In the case of two bus sections becoming electrically tied due to a breaker closing, we augment C by two additional constraints C ′ to tie together the voltage magnitudes and phase angles of the previously-separate bus sections. That is, the post-contingency state satisfies the augmented system H(v ′ ; Y ) + Cλ′ + C ′ γ − s = 0 C T v′ = b (5) ′T ′ C v = 0. We linearize (5) about the original state x (with γ = 0); because the first two equations are satisfied at this state, we have the approximate system ′ A U δx′ 0 C = − , U = (6) UT 0 γ C ′T v 0 The slack variables γ let the voltage phasor for a “breakaway” group of previously-slaved sections differ from a phasor at the former master section. The two columns of F ∈ {0, 1}n×2 indicate rows of C T that constrain the breakaway voltage magnitudes and the phase angles, respectively. The third equation says no power flows across the open breaker. We linearize (9) about the original state x (with γ = 0); because the first two equations are satisfied at this state, we have the approximate system A U δx′ 0 0 = − , U = . (10) UT 0 γ FTλ F The bordered systems (10) has the same form as (6); and, as before, block Gaussian elimination requires only two solves with A, some dot products, and a 2 × 2 system solve. C. Load/Generator Trip Fingerprints When a load or generator trips offline, that bus becomes a zero-injection PQ nodes. In the case of a PQ load tripping offline, the network itself does not change, so we do not need to augment the matrix A as is done in Equations (6) and (10), but we compute the approximate fingerprint with just A. Rather, it is the power injection vector s that changes. In the case of a generator at a PV bus tripping offline, we need to convert that bus into a PQ bus with zero power injection. We write the augmented system H(v ′ ; Y ) + Cλ′ − s′ = 0 C T v′ + f γ = b T (11) ′ f λ = 0. The slack variable γ lets the voltage magnitude of the bus of interest shift. The vector f is an indicator vector such that the third equation constrains the reactive power injection slack variable to zero. Note that s has also been changed to s′ to represent the real power injection shifting to zero. The resulting system is A u δx′ 0 0 = − , u = . (12) uT 0 γ fT λ f The bordered system (12) has the same form as (10) and (6). In this case, only one solve with A is required. We then solve the system by block elimination to obtain γ = (U T A−1 U )−1 (C ′T v) ′ −1 δx = −A Uγ (7) (8) The formulas (7)–(8) only require two significant linear solves (to evaluate A−1 U ), some dot products, and a 2 × 2 solve. B. Bus Splitting Fingerprints When a bus splits after a breaker opens, the postcontingency state satisfies the augmented system H(v ′ ; Y ) + Cλ′ − s = 0 C T v′ + F γ = b T ′ F λ = 0. (9) D. Line Failure Fingerprints In principle, line failures can be handled in the same way as substation reconfigurations that lead to bus splitting: explicitly represent two nodes on a line that are normally connected (physically corresponding to two sides of a breaker) with a multiplier that forces them to be equal, and compute the fingerprint by an extended system that negates the effect of that multiplier. In practice, we may prefer to avoid the extra variables in this model. The following formulation requires no explicit extra variables in the base model, and can be used with either a breaker-level model or a bus-branch model with no breakers (i.e. C an empty matrix). 4 For line failures, the admittance changes to Y ′ = Y + ∆Y ′ where ∆Y ′ is a rank-one update. The post-contingency state satisfies the system More concretely, we have Uik =   ′ ′ \|vi \|2 P̌i − gii \|vi \|2 P̌i + gii −Q̌i − b′ii \|vi \|2  Q̌k + b′ \|vk \|2 P̌k − g ′ \|vk \|2 P̌k + g ′ \|vk \|2  kk kk kk    P̌i − g ′ \|vi \|2 Q̌i + b′ii \|vi \|2  Q̌i − b′ii \|vi \|2 ii ′ −P̌k + gkk \|vk \|2 Q̌k + b′kk \|vk \|2 Q̌k − b′kk \|vk \|2   1 −1 0 0 0 . VikT = 0 0 \|vi \|−1 0 0 0 \|vk \|−1 H(v ′ ; Y ′ ) + Cλ′ − s = 0 C T v ′ = b, and linearization about x gives ∂H ′ ′ δv H(v; ∆Y ′ ) ∂v (v; Y ) C =− 0 CT 0 δλ′ (13) where H(v; ∆Y ′ ) = H(v; Y ′ ) − H(v; Y ). As we show momentarily, Because H(v, ∆Y ′ ) does not depend on any voltage phasors other than those at nodes i and j, we may write ∂H ∂H ∂H (v; Y ′ ) − (v; Y ) = (v; ∆Y ′ ) = U 0 (V 0 )T . ∂v ∂v ∂v ∂H(v; ∆Y ′ ) T = Eik Dik Eik = U 0 (V 0 )T ∂v e ek ∈ R2n×4 . Eik = i ei ek where U 0 and V 0 each have three columns. That is, the matrix in the system (13) is a rank-three update to A. We can solve such a system by the Sherman-Morrison-Woodbury update formula, also widely known as the Inverse Matrix Modification Lemma [58], [59]. We use the equivalent extended system ′ A U δx r = − , (14) V T −I γ 0 where U= U0 , 0 V = V0 , 0 r= H(v; ∆Y ′ ) . 0 γ = (I + V T A−1 U )−1 (V T r) δx′ = −A−1 (r + U γ). and U 0 = Eik Uik , (16) ′ = P̌ik + gii \|vi \|2 Q̌i ≡ Hi+n = Q̌ik − b′ii \|vi \|2  P̌i  P̌k  0  H(v, ∆Y ′ ) = Eik   Q̌i  = U z, Q̌k  ′ = P̌ki + gkk \|vk \|2 where ′ P̌ik gik ≡ \|vi \|\|vk \| −b′ik Q̌ik b′ik ′ gik cos(θik ) , sin(θik ) and P̌ki , Q̌ki are defined similarly. Let Dik ≡ ∂(P̌i , P̌k , Q̌i , Q̌k ) ∈ R4×4 ; ∂(θi , θk , \|v\|i , \|v\|k ) by the chain rule, we can write Dik = Uik VikT where ∂(P̌i , P̌k , Q̌i , Q̌k ) ∈ R4×3 ∂(θik , log \|v\|i , log \|v\|k ) ∂(θik , log \|v\|i , log \|v\|k ) ∈ R3×4 . ≡ ∂(θi , θk , \|v\|i , \|v\|k ) Uik ≡ VikT (19)   0 z = 1/2 , 1/2 δx′ = −A−1 U (z + γ). (20) IV. F ILTERING In Section III we discussed how to approximate voltage shifts δv ′ associated with several types of contingencies. This approach to predicting voltage changes costs less than a nonlinear power flow solve, but may still be costly for a large network with many contingencies to check. In the current section we show how to rule out contingencies without any solves by computing a cheap lower bound on the discrepancy between the observed voltage changes and the predicted voltage changes under the contingencies. For each contingency, we define the fingerprint score t = kE∆v − Eδv ′ k Q̌k ≡ Hk+n = Q̌ki − b′kk \|vk \|2 , V 0 = Eik Vik . Moreover, we note that (15) The work to evaluate (15)–(16) is three linear solves (for A−1 U ), some dot products, and a small 3 × 3 solve. We will show momentarily how to avoid the solve involving r. We now show that the Jacobian matrix changes by a rank-3 update. For a failed line between nodes i and k, the vector H(v, ∆Y ′ ) has only four nonzero entries: P̌k ≡ Hk (18) so that we may rewrite (16) as We again solve by block elimination: P̌i ≡ Hi where (17) (21) where ∆v is the observed voltage shift and δv ′ is the voltage shift predicted for the contingency. For the contingencies we have described, Eδv ′ has the form Eδv ′ = ĒA−1 U γ (22) where Ē = E 0 simply ignores the multiplier variables λ, and γ is some short vector of slack variables. The expression ĒA−1 does not depend on the contingency, and can be precomputed at the cost of m linear solves (one per observed phasor component). After this computation, the main cost in evaluating (22) is the computation of γ, which involves a contingency-dependent linear system with A as an intermediate step. However, we do not need γ for the filter score τ = min kE∆v − ĒA−1 U µk ≤ t. µ (23) 5 PMUs FLiER FLiER+noise DC Approx DC Approx+noise Algorithm 1 FLiER Compute and store ĒA−1 . For each contingency i, compute τi via (23). Order the contingencies in ascending order by τ . for ℓ = 2, 3, . . . do Compute fingerprint score tℓ Break if tℓ < τℓ+1 end for Return contingencies with computed tℓ tk < τi ≤ ti , without ever computing ti . Exploiting this fact leads to the FLiER method (Algorithm 1)1 . We note that PQ load trips require no filtering, as they involve no change to the system matrix. Filter score computations are embarrassingly parallel and can be spread across processors. Nonetheless, for huge networks with many PMUs and many contingencies, the filter computations might be deemed too expensive for very rapid diagnosis (e.g. in less than a second). However, the concept of a filter subspace can be adapted to these cases. First, one can define a coarse filtering subspaces that is the sum of the filtering subspaces for a set of contingencies. For example, one might define a coarse filtering subspace associated with all possible breaker reconfigurations inside a substation. The coarse subspace filter score provides a lower bound on the filter scores (and hence the fingerprint scores) for all contingencies in the set. Hence, it may not even be necessary to compute individual filter scores for all contingencies considered. Second, one can work with a projected filtering subspace W T (EA−1 U ) where W is a matrix with orthonormal columns. The distance from a projected measurement vector to the projected filtering subspace again gives a lower bound on the full filter score. In addition to reducing the cost of filter score computations, projections can also be used to eliminate faulty or missing PMU measurements from consideration. V. E XPERIMENTS Our standard experimental setup is as follows. For each possible topology change, we compute and pass to FLiER both the full pre-contingency state and the subset of the postcontingency state that would be observed by the PMUs. We test both with no noise and with independent random Gaussian noise with standard deviation 1.7 · 10−3 (≈ 0.1 degrees for phase angles) added to both the initial state estimate and the can be Sparse 68(77) 66(78) 52(74) 49(64) All 78(78) 78(78) 72(77) 66(68) TABLE I IEEE 57- BUS NETWORK ACCURACY COMPARISON FOR 78 LINE FAILURE CONTINGENCIES . W E REPORT COUNTS OF LINE FAILURES CORRECTLY IDENTIFIED AND THOSE SCORED IN THE TOP THREE ( IN PARENTHESES ). In the Euclidean norm, τ is simply the size of the residual in a least squares fit of E∆v to the columns of ĒA−1 U , which can be computed quickly due to the sparsity of U . If U is the augmentation matrix associated with contingency i, we refer to ĒA−1 U as its filtering subspace. Filter score computations are cheap; and if the filter score τi for contingency i exceeds the fingerprint score tk for contingency k, then we know 1 Example Python code of this algorithm https://github.com/cponce512/FLiER Test Suite Single 55(73) 40(65) 17(40) 5(22) found at PMU readings. In [34], 0.1 degrees of Gaussian random noise was applied to phase angles, then smoothed by passing a simulated time-domain signal through a low pass filter; we apply the noise without filtering, so the effect is more drastic. One of the possibilities FLiER checks is that there has been no change; in this case, we use the norm of the fingerprint as both the fingerprint score and the filter score. By including this possibility among those checked, FLiER acts simultaneously as a method for topology change detection and identification. We run tests on the IEEE 57 bus and 118 bus networks, with three different PMU arrangements on each: • Single: Only one PMU is placed in the network, at a lowdegree node (bus 35 in the 57-bus network and 65 in the 118-bus network, providing a total of 3 and 5 bus voltage readings, respectively). This represents a near-worst-case deployment for our method. • Sparse: A few PMUs are placed about the network (on buses 4, 13, and 34 in the 57-bus network and on buses 5, 17, 37, 66, 80, and 100 in the 118-bus network, providing a total of 15 and 40 bus voltage readings, respectively). We consider this a realistic scenario in which sparselydeployed PMUs do not offer full network observability. • All: PMUs are placed on all buses. Any error is due purely to the linear approximation. We did not test changes that cause convergence failure in our power flow solver. We assume such contingencies result in collapse without some control action. A. Accuracy 1) Line Failures: Figure 1 shows the accuracy of FLiER in identifying line failures in the IEEE 57-bus test network. For each PMU deployment, we show the cumulative distribution function of ranks, i.e. the ranks of each simulated contingency in the ordered list produced by FLiER. We show further results in Table I. With PMUs everywhere, the correct answer was chosen in all 78 of 78 cases, even in the presence of noise. The case with three PMUs is also quite robust to noise. In the test with a single unfavorably-placed PMU, FLiER typically ranks the correct line among the top three in the absence of noise; with noise, the accuracy degrades, though not completely. In Figure 2, we repeat the experiment of Figure 1, but with the DC approximation used in [34] rather than the AC linearization used in FLiER. We also present comparisons in Table I. With PMUs everywhere, there is little difference in accuracy. With fewer PMUs, FLiER is more accurate. In the sparse case, the DC approximation without noise behaves similarly to FLiER with noise, while in the single PMU 1.0 0.8 0.8 0.6 0.4 0.2 0.0 Fraction of Tests All Sparse Single 1 2 3 4 5 6 Rank 7 8 Fraction of Tests 1.0 0.6 0.4 1.0 1.0 0.8 0.8 0.6 All Sparse Single Single BF 0.4 0.2 0.0 1 2 3 4 5 6 Rank 7 8 Fig. 1. Cumulative distribution function showing the fraction of line failures where FLiER assigned the correct line at most a given rank (up to 10). Top: Noise-free case. Bottom: Entries with Gaussian noise with σ = 0.0017. “Single BF” is the result from using a brute-force approach with a single PMU and represents the signal-to-noise ratio in that PMU’s information. 1 2 3 4 5 6 Rank 7 8 9 10 0.6 All Sparse Single Single BF 0.4 0.2 0.0 9 10 All Sparse Single 0.2 0.0 9 10 Fraction of Tests Fraction of Tests 6 1 2 3 4 5 6 Rank 7 8 9 10 Fig. 2. CDF of line failures where the DC approximation of [34] assigned the correct line at most a given rank (up to 10). Top: Noise-free case. Bottom: Entries with Gaussian noise with σ = 0.0017. “Single BF” is the result from using a brute-force approach with a single PMU and represents the signal-tonoise ratio in that PMU’s information. deployment the DC results without noise are much worse than those from FLiER even with noise. Figure 3 shows the raw scores computed by FLiER with three PMUs. In this plot, each column represents the fingerprint scores computed for one line failure scenario. The black crosses represent the scores of lines that get past the filter, while the green circles and yellow triangles represent the scores for the correct answer. If there is a green circle, then our algorithm correctly identified the actual line that failed. If there is a yellow triangle, the correct line was not chosen but was among the top three lines selected by the algorithm. In Figure 4, we show one case that FLiER misidentifies. PMUs are deployed on buses marked with blue squares, and lines are colored and thickened according to the FLiER score. The best-scoring line is adjacent to the line that failed. 2) Substation Reconfigurations: Next, we show the accuracy of FLiER as it applies to substation reconfigurations. For these tests, we suppose that every bus in the IEEE 57-bus test network is a ring substation with each bus section on the ring possessing either load, generation, or a branch. We then suppose a substation splits when two of its circuit breakers open. We do not consider cases that isolate a node with a nonzero power injection. Line failures are a subset of this scenario: if the breakers on either side of a section with a branch open, that section becomes a zero-injection leaf bus, which disappears in the quasi-static setting. Figure 5 shows the accuracy of FLiER on substation re- Approximation error 100 10−1 10−2 10−3 10−4 10−5 0 10 20 30 40 50 60 70 80 Line number (sorted) Fig. 3. Test of our algorithm on the IEEE 57-bus network with the sparse PMU deployment. Each column represents one test. Black crosses are fingerprint scores for incorrect lines. Green dots and yellow triangles indicate the scores of the correct line in the case of correct diagnosis or diagnosis in the top three, respectively. configurations with and without noise. With three PMUs and no noise, FLiER is right in 164 of 193 possibilities, and ranks the correct answer among the top three scores in 160 cases. With PMUs everywhere, FLiER is right 181 times, but gets the answer in the top three every single time. With few PMUs, FLiER is more susceptible to noise when diagnosing substation reconfigurations. This is expected, as there are significantly more possibilities to choose from in this case. Also, FLiER sometimes filters out the correct answer in the presence of noise. One possible remedy for this would be to be 7 Contingency Correct % Top 3 % 26 75.2 95.4 Substation of contingency 85.4 96.5 TABLE II A CCURACY OF FL I ER WITH 100 RANDOMLY- PLACED PMU S ON THE P OLISH NETWORK . R ESULTS ARE OUT OF 6283 TESTS . 24 27 1.0 Fig. 4. Line (24, 26) isqthe line removed in this test. Lines are colored and thickened according to t−1 ik . Line (26, 27) was chosen by the algorithm. Fraction of Tests 28 0.8 0.6 0.4 All Sparse Single 0.2 1.0 1 2 3 4 5 6 Rank 7 9 10 1.0 0.4 All Sparse Single 0.2 0.0 1 2 3 4 5 6 Rank 7 8 9 10 0.8 0.6 0.4 All Sparse Single 0.2 1.0 0.0 Fraction of Tests 8 0.6 Fraction of Tests Fraction of Tests 0.0 0.8 0.8 1 2 3 4 5 6 Rank 7 8 9 10 0.6 All Sparse Single Single BF 0.4 0.2 0.0 1 2 3 4 5 6 Rank 7 8 9 10 Fig. 5. Rank CDF for substation reconfigurations without noise (top) and with noise (bottom). “Single BF” is the result from using a brute-force approach with a single PMU and represents the signal-to-noise ratio in that PMU’s information. more lenient with filtering, only throwing a possibility away if τℓ is greater than the kth smallest tℓ , for example. Finally, we demonstrate the effectiveness of using FLiER for substation reconfigurations on a large-scale network by running FLiER on the 400, 220, and 100 kV subset of the Polish network during peak conditions of the 1999-2000 winter, taken from [60]. This is a larger network with 2,383 buses. We placed 100 PMUs randomly around the network, and tested every substation reconfiguration contingency. We summarize the results in Table II. We could likely further improve the accuracy with a thoughtful deployment of PMUs. 3) Including Load and Generator Trips: Here we include load and generator trips in our results. In Figure 6, we show Fig. 6. Rank CDF for generator trips without noise (top) and with noise (bottom). the results of testing for generator trips, in Figure 7, we show the results of testing for load trips, and in Figure 8, we show the results of including load and generator trips in the set of contingencies to test and check for along with substation splits. All tests use the IEEE 57-bus network. In the noisy case for Figure ??, we allow some extra slack in the filtering procedure. In particular, rather than checking if tk < τi , we check if tk < τi − σ, where σ is the standard deviation of the included noise. We did this because the filtering method sometimes filtered out the correct answer for load trips. In a real-world setting one would need to choose that slack value intelligently, but this test shows that slightly less-stringent filtering can improve the method in some cases. 4) PMU Placement: Here we demonstrate the robustness of FLiER to PMU placement. We tested this by sampling three busses from the IEEE 57-bus network uniformly at random, running FLiER for the substation reconfiguration case (as in Figure 5) and recording the fraction of contingencies FLiER identified correctly, and the fraction of contingencies with the solution ranked in the top 3. We performed this test 200 times. We show cumulative distribution functions for the fraction correct and fraction in the top 3 in Figure 9. 1.0 0.8 0.8 0.6 0.4 All Sparse Single 0.2 0.0 1 2 3 4 5 6 Rank 7 8 0.6 0.4 1.0 1.0 0.8 0.8 0.6 All Sparse Single Single BF 0.4 0.2 0.0 1 2 3 4 5 6 Rank 7 8 Fig. 7. Rank CDF for load trips without noise (top) and with noise (bottom). In the noisy case, we allow a filtering slack equal to one noise standard deviation, as otherwise the filter is too stringent. “Single BF” is the result from using a brute-force approach with a single PMU and represents the signalto-noise ratio in that PMU’s information. We do not include zero-injection busses in this test. This figure shows that FLiER tends to be quite robust to PMU placement. In fact, the median fraction correct and median fraction in top 3 are only slightly below those for the noiseless test in Figure 5, where we attempted to place the three PMUs favorably. Furthermore, the standard deviation for fraction correct is only 3.65%, while the standard deviation for fraction in top 3 is 2.67%. 5) Robustness to Bad Data: Here we demonstrate how one can easily modify FLiER for enhanced robustness to bad data and heavy-tailed noise. In Equations (21) and (23), we use the L2 loss function to measure the distance between an estimated fingerprint or subspace and the actual fingerprint. The L2 loss function, while simple and useful, is sensitive to outliers. A single large error component drastically increases the t or τ value, even if all other components have very small error. The result is using the L2 loss can make FLiER sensitive to bad data and heavy-tailed noise. An alternative is to use a loss function that is robust to large outliers. One popular such loss function is the Huber loss [44]: kek2H = Lδ (ei ) m X Lδ (ei ) i=1 1 2 2 ei = δ \|ei \| − 12 δ (24) \|ei \| ≤ δ otherwise (25) This function behaves like the L2 loss for error components near zero, but for error components larger than δ, the loss 1 2 3 4 5 6 Rank 7 8 9 10 0.6 All Sparse Single Single BF 0.4 0.2 0.0 9 10 All Sparse Single 0.2 0.0 9 10 Fraction of Tests Fraction of Tests Fraction of Tests 1.0 1 2 3 4 5 6 Rank 7 8 9 10 Fig. 8. Rank CDF for substation reconfigurations and load/generator trips without noise (top) and with noise (bottom). “Single BF” is the result from using a brute-force approach with a single PMU and represents the signal-tonoise ratio in that PMU’s information. 1.0 Fraction of Samples Fraction of Tests 8 0.8 Correct Top 3 0.6 0.4 0.2 0.0 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Fraction Correct/Fraction Top3 Fig. 9. CDF for fraction of substation reconfiguration contingencies that FLiER gets correct (blue) and in the top 3 (green) with random uniform placement of three PMUs. Noise is not included in this test. grows linearly rather than quadratically. The result is a loss function that is robust to outliers in data. In Figures 10 and 11 we show the usefulness of the Huber loss in certain cases. In Figure 10, we run FLiER on the line failures test (as in the noisy case of Figure 1), but with the PMU reading from bus 4 given a large post-event bias, always returning an after-event angle reading that is 5 degrees too large. As shown in the figure, this single bad reading can cause major problems with FLiER accuracy using L2 loss. With Huber loss, however, that bad data is almost completely ignored, resulting in accuracy essentially identical to that of the noisy case in Figure 1. 9 0.8 0.6 0.4 L2 Loss Huber Loss 0.2 0.0 1 2 3 4 5 6 Rank 7 8 1.0 0.8 0.8 0.6 0.4 All Sparse Single 0.2 0.0 0.0 9 10 Fig. 10. CDF for line failures with a single PMU reading that systematically gives angle readings that are too large by 5 degrees, along with Gaussian random noise with standard deviation 0.0017. PMUs are at locations 4, 13, and 34. Huber scale parameter δ = 1.365 · 0.0017 Fraction of Tests Fraction of lines filtered 1.0 0.2 0.4 0.6 0.8 Fraction of failure scenarios 0.6 0.8 0.6 0.4 All Sparse Single 0.2 0.0 0.0 0.2 0.4 0.6 0.8 Fraction of failure scenarios 0.4 L2 Loss Huber Loss 0.2 0.0 1 2 3 4 5 6 Rank 7 8 9 10 1.0 1.0 Fraction of lines filtered Fraction of Tests 1.0 1.0 Fig. 12. Cumulative distribution function of fraction of lines for which tik need not be computed when a line in the IEEE 57-bus (top) or 118-bus (bottom) network fails uniformly at random. Fig. 11. CDF for substation reconfigurations with Cauchy noise with a scale parameter of 0.001. In Figure 11, we show the utility of the Huber loss in the presence of heavy-tailed noise. Here we test substation reconfigurations with noise given by a Cauchy distribution with scale parameter 0.001. As shown in the figure, the use of the Huber loss significantly improves FLiER’s performance under this type of noise. It is unclear if either of these situations are likely to arise in practice. In the former case (with a bad PMU), it is likely that this bad data stream would be identified in an earlier state estimation stage and already removed from the fingerprint. In the latter case, the IEEE specification [20] requires that the PMU noise profile not have heavy tails, so systematic heavytailed noise is unlikely. B. Filter Effectiveness and Speed The cost of FLiER depends strongly on the effectiveness of the filtering procedure. In Figure 12, we show how often the filter saves us from computing fingerprint scores in experiments on the IEEE 57-bus and 118-bus networks when checking for line failures. For each PMU deployment, we show the cumulative distribution function of the fraction of lines for which fingerprint scores need not be computed for each line failure. The filter performs well even for the sparse PMU deployments; we show a typical case in Figure 13. Approximation error 100 10−1 10−2 10−3 10−4 0 10 20 30 40 50 60 70 80 Line number (sorted) Fig. 13. Example of effective filtering in Algorithm 1. Each column represents a line checked. Blue dots are the lower bounds τik , while red squares are true scores tik . Columns are sorted by τik . In this case, tik only needs to be computed for eight lines. Finally, we demonstrate the importance of the filter by running FLiER on the large Polish network [60] with 100 randomly placed PMUs. Table III shows FLiER run times with and without the filter on ten randomly selected branches. The code is unoptimized Python, so these timings do not indicate of how fast FLiER would run in a performance setting. However, they give a sense of the speedup one expects from filtering. Note also that FLiER correctly identified the failed branch in 9 of 10 cases. In the one case in which it failed, on branch (2346, 2341), tℓ for the correct answer was 6.25 · 10−5 ; this suggests the failure had a negligible impact on the network. We also performed this timing test for a randomly selected 10 Line (1502, 917) (1502, 1482) (557, 556) (2346, 2341) (909, 1155) (644, 629) (591, 737) (559, 542) (378, 336) (101, 94) FLiER (s) 0.27 0.27 0.27 2.95 0.26 0.29 0.29 0.32 0.28 0.26 Solution rank 1 1 1 14 1 1 1 1 1 1 # t’s computed 2 2 4 502 2 7 6 13 6 2 FLiER n.f. (s) 14.14 15.30 14.75 14.40 14.32 14.59 17.27 16.59 16.16 15.29 TABLE III FL I ER RUN TIMES FOR TEN LINE FAILURES WITH AND WITHOUT FILTERING . A BOUT 3000 CONTINGENCIES ARE CONSIDERED . Bus / Split nodes 86/1,2,3 176/1,2 539/7 702/4,5 754/2,3,4,5 994/2,3,4,5 1131/2 1513/4,5,6 1663/1,2,3,4 2164/5 FLiER (s) 4.54 4.70 8.11 4.66 4.97 4.86 5.22 6.65 7.83 4.56 Sol. rank / Sol. bus rank 1/1 3/3 1/1 1/1 1/1 1/1 1/1 4/1 1/1 1/1 # t’s computed 2 4 38 4 7 6 9 23 35 3 FLiER n.f. (s) 823.0 836.9 829.8 820.1 829.8 835.8 850.3 862.3 928.1 875.3 TABLE IV FL I ER RUN TIMES FOR TEN SUBSTATION RECONFIGURATIONS WITH AND WITHOUT FILTERING . N EARLY 7000 CONTINGENCIES ARE CONSIDERED . set of substation reconfigurations, shown in Table IV. Again, the results give a sense of the speedup one expects from filtering in the substation reconfiguration case. VI. C ONCLUSION AND F UTURE W ORK We have presented FLiER, a new algorithm to identify topology changes involving load and generator trips, line outages, and substation reconfigurations using a sparse deployment of PMUs. Our method uses a linearization of the power flow equations together with a novel subspace-based filtering approach to provide fast diagnosis. Unlike prior approaches based on DC approximation, our approach takes advantage of a state estimate obtained shortly before the topology changes, assuming that the network specifications remain unchanged or change in a known way as a result of the failure. 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arXiv:1503.02436v2 [math.GR] 15 Mar 2017 RATIONAL DISCRETE COHOMOLOGY FOR TOTALLY DISCONNECTED LOCALLY COMPACT GROUPS I. CASTELLANO AND TH. WEIGEL Abstract. Rational discrete cohomology and homology for a totally disconnected locally compact group G is introduced and studied. The Hom-⊗ identities associated to the rational discrete bimodule Bi(G) allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretin’s group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension. For a unimodular t.d.l.c. group G of type FP it is possible to define an Euler-Poincaré characteristic χ(G) which is a rational multiple of a Haar measure. This value is calculated explicitly for Chevalley groups defined over a non-discrete, non-archimedean local field K and some other examples. 1. Introduction For a topological group G several cohomology theories have been introduced and studied in the past. In many cases the main motivation was to obtain an interpretation of the low-dimensional cohomology groups in analogy to discrete groups (cf. [3], [33], [34], [35]). The main subject of this paper is rational discrete cohomology for totally disconnected, locally compact (= t.d.l.c.) groups. Discrete cohomology can be defined in a very general frame work. We will call a topological group G to be of type OS, if (1.1) Osgrp(G) = { O ⊆ G \| O an open subgroup of G } is a basis of neighbourhoods of 1 in G, e.g., any profinite group is of type OS, and, by van Dantzig’s theorem (cf. [48]), any t.d.l.c. group is of type OS. For any topological group G of type OS and any commutative ring R the category R[G] dis of discrete left R[G]-modules is an abelian category with enough injectives. This allows to define dExt•R[G] ( , ) as the right derived functors of HomR[G] ( , ), and dH• (R[G], ) as the right derived functors of the fixed point functor G . Even in this general context one has a Hochschild-Lyndon-Serre spectral sequence for short exact sequences of topological groups of type OS (cf. §2.8), and a generalized Eckmann-Shapiro type lemma in the first argument (cf. §2.9). In case that G is a t.d.l.c. group it turns out that Q[G] dis is also an abelian category with enough projectives (cf. Prop. 3.2). Moreover, any projective rational discrete right Q[G]-module is flat with respect to short exact sequences of rational Date: March 16, 2017. 2010 Mathematics Subject Classification. 20J05, 22D05, 57T99, 22E20, 51E42, 17B67. Key words and phrases. Discrete cohomology, totally disconnected locally compact groups, duality groups, discrete actions on simplicial complexes, E-spaces. 1 2 I. CASTELLANO AND TH. WEIGEL discrete left Q[G]-modules (cf. (4.4)). Thus for a discrete left Q[G]-module M one may define rational discrete cohomology with coefficients in M by (1.2) dHk (G, M ) = dExtkG (Q, M ), k ≥ 0, and discrete rational homology with coefficients in M by (1.3) dHk (G, M ) = dTorG k (Q, M ), k≥0 in analogy to discrete groups (cf. §4). The rational discrete cohomological dimension cdQ (G) (cf. (3.11)) reflects structural information on a t.d.l.c. group G, e.g., (1) G is compact if, and only if, cdQ (G) = 0 (cf. Prop. 3.7(a)); (2) the flat rank of G is bounded by cdQ (G) (cf. Prop. 3.10(b)); (3) if G is acting discretely on a tree with compact stabilizers then cdQ (G) ≤ 1 (cf. Prop. 5.4(b)); (4) more generally, if G is acting discretely on a contractable simplical complex Σ with compact stabilizers, then cdQ (G) ≤ dim(Σ) (cf. Prop. 6.6(a)); (5) a nested t.d.l.c. group G satisfies cdQ (G) ≤ 1 (cf. Prop. 5.8). The existence of finitely generated projective rational discrete Q[G]-modules allows to define the FPk -property, k ∈ N0 ∪ {∞}, (cf. §3.6) in analogy to discrete groups (cf. [14, §VIII.5]). A t.d.l.c. group G is compactly generated if, and only if, it is of type FP1 (cf. Thm. 5.3). Using the theory of rough Cayley graphs due to H. Abels (cf. [1]) and Bass-Serre theory, one may also introduce the notion of a compactly presented t.d.l.c. groups (cf. §5.8). Such a group must be of type FP2 . For a t.d.l.c. group G there is not necessarily a group algebra but two natural rational discrete Q[G]-bimodules Bi(G) and Cc (G, Q) and both turn out to be useful in this context. If G is unimodular, they are isomorphic, but not necessarily canonically isomorphic; if G is not unimodular, they are non-isomorphic (cf. Prop. 4.11(b)). The Q[G]-bimodule Cc (G, Q) just consists of the continuous functions with compact support from G to the discrete field Q. The rational discrete left Q[G]-module Bi(G) is defined by (1.4) Bi(G) = lim −→ Q[G/O], O∈CO(G) where CO(G) = { O ∈ Osgrp(G) \| O compact } and the direct limit is taken along a rational multiple of the transfers (cf. (4.15)). It carries naturally also the structure of a discrete right Q[G]-module. The rational discrete Q[G]-bimodule Bi(G) can be used to establish Hom-⊗ identities which are well known for discrete groups (cf. §4.3). This allows to define the notion of a rational t.d.l.c. duality group. In more detail, a t.d.l.c. group G will be said to be a rational duality group of dimension d ≥ 0, if (i) G is of type FP∞ ; (ii) cdQ (G) < ∞; (iii) dHk (G, Bi(G)) = 0 for k 6= d. Note that any discrete virtual duality group in the sense of R. Bieri and B. Eckmann (cf. [9]) of virtual cohomological dimension d ≥ 0 is indeed a rational t.d.l.c. duality group of dimension d ≥ 0, and the same is true for discrete duality groups of dimension d over Q (cf. [8, §9.2]). For these t.d.l.c. groups one has a nontrivial relation between discrete cohomology and discrete homology (cf. Prop. 4.10). RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 3 However, it differs slightly from the usual form. A t.d.l.c. group G satisfying (i) and (ii) will be said to be of type FP. Note that a t.d.l.c. group G satisfying (i), (ii) and (iii) must also satisfy cdQ (G) = d (cf. Prop. 4.7). The rational discrete right Q[G]-module DG = dHd (G, Bi(G)) (1.5) will be called the rational dualizing module. A compact t.d.l.c. group G is a rational t.d.l.c. duality group of dimension 0 with dualizing module isomorphic to the trivial right Q[G]-module Q. If G is a compactly generated t.d.l.c. group satisfying cdQ (G) = 1, then G is a rational t.d.l.c. duality group of dimension 1. Provided that G acts discretely on a locally finite tree T with compact stabilizers one has an isomorphism of rational discrete left Q[G]-modules (1.6) × DG ≃ Hc1 (\|T \|, Q) ⊗ Q(∆) (cf. §6.10), where \|T \| denotes the topological realization1 of T (cf. §A.2), and Hc1 ( , Q) denotes cohomology with compact support with coefficients in the discrete field Q. If Σ is a locally finite C-discrete simplicial G-complex (cf. §6.4), then Hck (\|Σ\|, Q) carries naturally the structure of a rational discrete left Q[G]-module (cf. §A.4). Here Q(∆) denotes the 1-dimensional rational discrete left Q[G]-module which action is given by the modular function2 ∆ : G → Q+ of G (cf. Remark 4.13). For a right Q[G]-module M we denote by × M the associated left Q[G]-module, i.e., for m ∈ M and g ∈ G one has g · m = m · g −1 . Let G be a t.d.l.c. group. The family of compact open subgroups C of G is closed under conjugation and finite intersections. Hence there exists a G-classifying space E C (G) for the family C (cf. [32, §2], [47, Chap. 1, Thm. 6.6]) which is unique up to G-homotopy equivalence. E.g., if Π = π1 (A, Λ, x0 ) is the fundamental group of a graph of profinite groups, then the tree T = T (A, Λ, Ξ, E + ) arising in Bass-Serre theory (cf. [41, §I.5.1, Thm. 12]) is an E C (Π)-space (cf. §6.10). For algebraic groups defined over a non-discrete non-archimedean t.d.l.c. field K a celebrated theorem of A. Borel and J-P. Serre yields the following (cf. §6.11). Theorem A. Let G be a simply-connected semi-simple algebraic group defined over a non-discrete non-archimedian local field K, let (C, S) be the affine building associated to G, and let \|Σ(C, S)\| denote the topological realization of the Bruhat-Tits realization of (C, S). Then G(K) is a rational t.d.l.c. duality group of dimension d = rk(G) = dim(Σ(C, S)), and (1.7) × DG(K) ≃ Hcd (\|Σ(C, S)\|, Q). Moreover, \|Σ(C, S)\| is an E C (G(K))-space. Here rk(G) denotes the rank of G as algebraic group defined over K, i.e., rk(G) coincides with the rank of a maximal split torus of G. As already mentioned in [12] one may think of × DG(K) as a generalized Steinberg module of G(K). Another source of rational t.d.l.c. duality groups arises in the context of topological Kac-Moody groups as introduced by B. Rémy and M. Ronan in [40, §1B]. In principal, these t.d.l.c. groups behave very similarly as the examples described 1Although this topological space is also known as the realization of the tree T (cf. [41, §I.2, p. 14]), we have chosen this name in order to avoid the linguistic ambiguity with the geometric realization of buildings. 2Throughout this paper Haar measures are considered to be left invariant. 4 I. CASTELLANO AND TH. WEIGEL in Theorem B, but there are also quite notable differences. Any split Kac-Moody group G(F) defined over a finite field F has an action on the positive part Ξ+ of the associated twin building. Assuming that the associated Weyl group (W, S) is infinite one defines Ḡ(F) as the permutation closure of G(F) on its action on the chambers of Ξ+ . Then Ḡ(F) acts also on Σ(Ξ+ ), the Davis-Moussang realization of Ξ+ , which is a finite dimensional locally finite simplicial complex. The subgroup Ḡ(F)◦ of Ḡ(F), which is the closure of the image of the type preserving automorphisms of Ξ, is of finite index in Ḡ(F). This group has the following properties. Theorem B. Let F be a finite field, let Ḡ(F) be a topological Kac-Moody group obtained by completing the split Kac-Moody group G(F), and let (W, S) denote its associated Weyl group. (a) Then Σ(Ξ) is a tame C-discrete simplicial Ḡ(F)◦ -complex, and \|Σ(Ξ)\| is an E C (Ḡ(F)◦ )-space. (b) cdQ (Ḡ◦ (F)) = cdQ (Ḡ(F)) = cdQ (W ). (c) For d = cdQ (Ḡ◦ (F)) one has × DḠ(F)◦ ≃ Hcd (\|Σ(Ξ)\|, Q). (d) Ḡ(F)◦ (and hence Ḡ(F)) is a rational t.d.l.c. duality group if, and only if, W is a rational duality group. One may verify easily that whenever W is a hyperbolic Coxeter group, the topological Kac-Moody group Ḡ(F)◦ is indeed a rational t.d.l.c. duality group (cf. Remark 6.12). However, one may also construct examples of topological Kac-Moody groups which are not rational t.d.l.c. duality groups (cf. Remark 6.13). We close our investigations with a short discussion of the Euler-Poincaré characteristic χ(G) one may define for a unimodular t.d.l.c. group G of type FP. The value χ(G) is a rational multiple of a left invariant Haar measure µO , µO (O) = 1, for some compact open subgroup O of G. Indeed, for a discrete group Π of type FP it is straightforward to verify that χ(Π) = χΠ · µ{1} , where χΠ denotes the classical Euler-Poincaré characteristic (cf. [14, §XI.7]). Although we present some explicit calculations in section 7.2, we are still very far away from understanding the generic behaviour of the Euler-Poincaré characteristic. We conjecture that a unimodular compactly generated t.d.l.c. group G satisfying cdQ (G) = 1 should have non-positive Euler-Poincaré characteristic (cf. Question 6). An affirmative answer to this question would resolve the accessibility problem, and thus would yield a complete description of unimodular compactly generated t.d.l.c. groups of rational discrete cohomological dimension 1. Acknowledgements: Sections 2, 3, 4 and 5 are part of the first author’s PhD thesis (cf. [20]). The authors would like to thank M. Bridson for some helpful discussions concerning CAT(0)-spaces. 2. Discrete cohomology 2.1. Topological groups of type OS. Let G be a topological group of type OS. If H is a closed subgroup of G, then { O ∩ H \| O ∈ Osgrp(G) } is a basis of neighbourhood of 1 in H. Hence H is also of type OS. Moreover, if π : G → Ḡ is a continuous, open, surjective homomorphism of topological groups, then { π(O) \| O ∈ Osgrp(G) } is a basis of neighbourhoods of 1 in Ḡ. Thus Ḡ is also of type OS. Finally, if G is homeomorphic to an open subgroup of a topological group Y , then Y is of type OS as well. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 5 Any compact totally-disconnected topological group is profinite and therefore, by definition, of type OS (cf. [42, Prop. 1.1.0]). By van Dantzig’s theorem (cf. [48]), every t.d.l.c. group contains a compact open subgroup. Hence from the previously mentioned properties one conlcudes the following. Fact 2.1. Any totally-disconnected locally-compact group is of type OS. 2.2. Discrete modules of topological groups of type OS. Let R be a commutative ring with 1 ∈ R. For a group G we denote by R[G] its R-group algebra, and by R[G] mod the abelian category of left R[G]-modules. Let G be a topological group of type OS. For M ∈ ob(R[G] mod) the subset (2.1) d(M ) = { m ∈ M \| stabG (m) open in G }, where stabG (m) = { g ∈ G \| g · m = m }, is an R[G]-submodule of M , the largest discrete R[G]-submodule of M . Here we call a left R[G]-module B discrete, if the map · : G × B → B is continuous, where B carries the discrete topology. If φ : B → M is a morphism of left R[G]-modules and B is discrete, one concludes that im(φ) ⊆ d(M ). This has the following consequences (cf. [49, Prop. 2.3.10]). Fact 2.2. Let G be a topological group of type OS, and let R[G] dis denote the full subcategory of R[G] mod the objects of which are discrete left R[G]-modules. Then (a) R[G] dis is an abelian category; (b) d : R[G] mod → R[G] dis is a covariant additive left exact functor, which is the right-adjoint of the forgetful functor f : R[G] dis → R[G] mod; (c) d is mapping injectives to injectives; (d) R[G] dis has enough injectives. The abelian category R[G] mod admits minimal injective envelopes, i.e., for M ∈ ob(R[G] mod) the maximal essential extension iM : M → I(M ) is a minimal injective envelope (cf. [31, §III.11]). Thus, if M ∈ ob(R[G] dis), then d(iM ) : M → d(I(M )) is a minimal injective envelope in R[G] dis. This has the following consequence. Fact 2.3. The abelian category R[G] dis admits minimal injective resolutions. If M and N are left R[G]-modules, then M ⊗ N = M ⊗R N is again a left R[G]-module. Moreover, if G is a topological group of type OS and M and N are discrete left R[G]-modules, then M ⊗ N is discrete as well. 2.3. Closed subgroups. Let H be a closed subgroup of the topological group G of type OS. Then one has a restriction functor (2.2) resG H( ): R[G] dis −→ R[H] dis which is exact. The discrete coinduction functor (2.3) G dcoindG H ( ) = d ◦ coindH ( ) : R[H] dis −→ R[G] dis where coindG H (M ) = HomR[H] (R[G], M ), M ∈ ob(R[H] mod), is the coinduction functor, is left exact. It is the right-adjoint of the restriction functor. In particular, dcoindG H ( ) is mapping injectives to injectives (cf. [49, Prop. 2.3.10]). Let N be a closed normal subgroup of the topological group G of type OS. The canonical projection π : G → Ḡ, where Ḡ = G/N , is surjective, continuous and open. One has an inflation functor (2.4) inf G Ḡ ( ) : R[Ḡ] dis −→ R[G] dis 6 I. CASTELLANO AND TH. WEIGEL which is exact. The inflation functor is left-adjoint to the N -fixed point functor (2.5) N : R[G] dis −→ R[Ḡ] dis which is left exact. In particular, one has the following (cf. [49, Prop. 2.3.10]). Fact 2.4. Let G be a topological group of type OS, and let N be a closed normal subgroup of G. Then the N -fixed point functor N : R[G] dis −→ R[Ḡ] dis, Ḡ = G/N , is mapping injectives to injectives. 2.4. Open subgroups. Let H ⊆ G be an open subgroup P of the topological group G of type OS, and let M ∈ ob(R[H] dis). Then for y = 1≤i≤n ri gi ⊗mi ∈ indG H (M ) T one has 1≤i≤n g stabH (mi ) ⊆ stabG (y). In particular, M is a discrete left R[G]module, and one has an exact induction functor (2.6) indG H( ): R[H] dis −→ R[G] dis. which is the left adjoint of the restriction functor resG H ( ). Hence one has the following (cf. [49, Prop. 2.3.10]). Fact 2.5. Let H ⊆ G be an open subgroup of the topological group G of type OS. Then resG H ( ) is mapping injectives to injectives. 2.5. Discrete cohomology. Let G be a topological group of type OS. For M ∈ ob(R[G] dis) we denote by (2.7) dExtkR[G] (M, ) = Rk HomR[G] dis (M, ) the right derived functors of HomR[G] (M, ) in bifunctors (2.8) dExtkR[G] ( , ) : op R[G] dis R[G] dis. The contra-/covariant × R[G] dis −→ R mod, k ≥ 0, admit long exact sequences in the first and in the second argument. Moreover, dExt0R[G] (M, ) ≃ HomR[G] (M, ), and dExtkR[G] (M, I) = 0 for every injective, discrete left R[G]-module I. We define the k th discrete cohomology group of G with coefficients in R[G] dis by (2.9) dHk (R[G], ) = dExtkR[G] (R, ), k ≥ 0, where R denotes the trivial left R[G]-module. Example 2.6. Let R = Z. (a) If G is a discrete group, G is of type OS and Z[G] dis = Z[G] mod. In particular, dH• (Z[G], ) = H • (G, ) coincides with ordinary group cohomology (cf. [14]). (b) If G is a profinite group, G is of type OS and Z[G] dis coincides with the abelian category of discrete G-modules. Thus dH• (Z[G], ) = H • (G, ) coincides with the Galois cohomology groups discussed in [42]. 2.6. The restriction functor. Let H ⊆ G be a closed subgroup of the topological group G of type OS, and let M ∈ ob(R[G] dis). Let (I • , ∂ • , µG ) be an injective resolution of M in ob(R[G] dis), and let (J • , δ • , µH ) be an injective resolution of resG H (M ) in R[H] dis. By the comparison theorem in homological algebra, one has a RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 7 • • morphism of cochain complexes of left R[H]-modules φ• : resG H (I ) → J such that the diagram (2.10) resG H (M ) µG / resG (I 0 ) H ∂0 / resG (I 1 ) H φ0 resG H (M ) / J0 µH ∂1 / resG (I 2 ) H φ1 δ 0 / J1 ∂2 / ... φ2 δ / J2 1 δ2 / ... commutes. Moreover, φ• is unique up to chain homotopy equivalence with respect to this property. Let φ̃k : (I k )G (2.11) / (I k )H (φk )H / (J k )H . Then φ̃• : ((I • )G , ∂˜• ) → (J • )H , δ̃ • ) is a mapping of cochain complexes of R-modules. Applying the cohomology functor yields a mapping of cohomological functors (2.12) res•G,H (M ) : dH• (R[G], M ) −→ dH• (R[H], resG H (M )). The map res•G,H ( ) will be called the restriction functor in discrete cohomology. 2.7. The inflation functor. Let N be a closed normal subgroup of the topological group of type OS, and let π : G → Ḡ, Ḡ = G/N , denote the canonical projection. Let M ∈ ob(R[Ḡ] dis), and let (I • , ∂ • , µḠ ) be an injective resolution of M in R[Ḡ] dis. Let (J • , δ • , µG ) be an injective resolution of inf G Ḡ (M ) in R[G] dis. By the comparison theorem in homological algebra, one has a mapping of cochain complexes of left • • R[G]-modules φ• : inf G Ḡ (I ) → J which is unique up to chain homotopy equivalence such that the diagram (2.13) inf G Ḡ (M ) inf G (µḠ ) Ḡ / inf G (I 0 ) Ḡ ∂0 φ0 inf G Ḡ (M ) µG / J0 / inf G (I 1 ) Ḡ φ1 δ0 / J1 commutes. Applying first the fixed point functor functor one obtains a functor (2.14) ∂1 / inf G (I 2 ) Ḡ ∂2 / ... φ2 δ1 G / J2 δ2 / ... and then the cohomology inf •Ḡ,G (M ) : dH• (R[Ḡ], M ) −→ dH• (R[G], inf G Ḡ (M )), which will be called the inflation functor in discrete cohomology. 2.8. The Hochschild - Lyndon - Serre spectral sequence. Let N be a closed normal subgroup of the topological group G of type OS, and let π : G → Ḡ, Ḡ = G/N , denote the canonical projection. The functor G : R[G] dis −→ R mod can be decomposed by G = ( Ḡ ) ◦ ( N ). Moreover, N : R[G] dis −→ R[Ḡ] dis is mapping injectives to injectives (cf. Fact 2.4). Thus for M ∈ ob(R[G] dis) the Grothendieck spectral sequence (cf. [49, Thm. 5.8.3]) yields a Hochschild-Lyndon-Serre spectral sequence (2.15) s+t E2s,t (M ) = dHs (R[Ḡ], dHt (R[N ], resG (R[G], M ) N (M ))) =⇒ dH 8 I. CASTELLANO AND TH. WEIGEL of cohomological type concentrated in the first quadrant which is functorial in M . The edge homomorphisms are given by esh : dHs (R[Ḡ], M N ) (2.16) inf sḠ,G (M N ) 0,t etv : E∞ (M ) / E s,0 (M ) 2 / E 0,t (M ) 2 restG,N (M)N s,0 / E∞ (M ), / dHt (R[N ], M )G/N , and one has a 5-term exact sequence (2.17) 0 / dH1 (R[Ḡ], M N ) e1h / dH1 (R[G], M ) e1v / dH1 (R[N ], M )G/N d0,1 2 dH2 (R[G], M ) o e2h dH2 (R[Ḡ], M N ) 2.9. An Eckmann-Shapiro type lemma. Let H be an open subgroup of the topological group G of type OS, and let M ∈ ob(R[G] dis). Let (I • , ∂ • , µG ) be • G • G an injective resolution of M in R[G] dis. Then (resG H (I ), resH (∂ ), resH (µG )) is an G injective resolution of resH (M ) in R[H] dis (cf. Fact 2.5). For B ∈ ob(R[H] dis) one has natural isomorphisms k G k HomR[G] (indG H (B), I ) ≃ HomR[H] (B, resH (I )) (2.18) k ≥ 0, which yield isomorphisms k G dExtkR[G] (indG H (B), M ) ≃ dExtR[H] (B, resH (M )), (2.19) k ≥ 0. In particular, for B = R one obtains natural isomorphisms k εk : dExtkR[G] (indG H (R), ) −→ dH (R[H], ) (2.20) such that (2.21) dExt•R[G] (indG H (R), ) ❚❚❚❚ ❧5 ❧ ❧ ❚❚❚ε❚• ❧❧❧ ❧ ❚❚❚❚ ❧ ❧ ❧ ❚❚❚) ❧ • ❧❧❧ resG,H ( ) • / dExtR[G] (R, ) dH• (R[H], resG H ( )) dExt• R[G] (ε) is a commutative diagram of cohomological functors with values in ε : indG H (R) → R is the canonical map. R[G] dis, where 3. Rational discrete cohomology for t.d.l.c. groups Throughout this section we will assume that G is a t.d.l.c. group. Furthermore, we choose R = Q to be the field of rational numbers. From now on we will omit the appearance of the field Q in the notation, i.e., we put dH• (G, ) = dH• (Q[G], ), dExt•Q[G] ( , ) = dExt•G ( , ), etc. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 9 3.1. Profinite groups. It is well known that a t.d.l.c. group is compact if, and only if, it is profinite (cf. [42, Prop. 1.1.0]). The main goal of this subsection is to establish the following property which will turn out to be essential for our purpose. Proposition 3.1. Let G be a profinite group. Then every discrete left Q[G]-module M is injective and projective in Q[G] dis. Proof. The proposition is a consequence of several facts. Claim 3.1.1. One has dHk (G, M ) = 0 for every M ∈ ob(Q[G] dis) and k ≥ 1. In particular, every finite dimensional discrete left Q[G]-module is projective. Proof of Claim 3.1.1. By [42, §2.2, Prop. 8], one has (3.1) dHk (G, M ) = limU⊳ G dHk (G/U, M U ), −→ ◦ where the projective limit is running over all open normal subgroups U of G. By Maschke’s theorem, one has dHk (G/U, M U ) = 0 for k ≥ 1, and thus, by (3.1), dHk (G, M ) = 0 for all k ≥ 1. Let U be an open subgroup of G. Since dExt1U (Q, ) = 0, HomU (Q, ) : Q[U] dis → Q mod is an exact functor. In particuG lar, Q ∈ ob(Q[U] dis) is projective. Thus, as resG U ( ) is exact, Q[G/U ] ≃ indU (Q) is projective for every open subgroup U of G (cf. [49, Prop. 2.3.10]). By Maschke’s theorem, every finite dimensional discrete left Q[G]-module is a direct summand of Q[G/U ] for some open subgroup U of G. This yields the claim. As a consequence one obtains the following. Claim 3.1.2. Let S ∈ ob(Q[G] dis) be irreducible, and let M ∈ ob(Q[G] dis). Then dExtkG (S, M ) = 0 for all k ≥ 1. Proof of Claim 3.1.2. Note that every irreducible discrete left Q[G]-module S is of finite dimension, and therefore projective (cf. Claim 3.1.1). For a discrete left Q[G]-module M we denote by soc(M ) ⊆ M the socle of M , i.e., soc(M ) is the sum of all irreducible left Q[G]-submodules of M . Moreover, as m ∈ M is contained in the finite dimensional discrete left Q[G]-module Q[G] · m, one has that soc(M ) = 0 if, and only if, M = 0. Claim 3.1.3. Every discrete left Q[G]-module M is injective in ob(Q[G] dis). Proof of Claim 3.1.3. Let (I • , δ • ) be a minimal injective resolution of M in Q[G] dis (cf. Fact 2.3). By Claim 3.1.2, for any irreducible S ∈ ob(Q[G] dis) one has for k ≥ 1 that 0 = dExtkG (S, M ) = HomG (S, I k ) (cf. [6, Cor. 2.5.4(ii)]). In particular, soc(I k ) = 0, and therefore I k = 0. Hence M is injective. Claim 3.1.4. The category Q[G] dis has enough projectives. Proof of Claim 3.1.4. Let M ∈ ob(Q[G] dis). By definition, for m ∈ M the left Q[G]submodule Q[G] · m ⊆ M is finite dimensional and discrete. Thus, by Claim 3.1.1, ` Q[G] · m is projective in Q[G] dis, and therefore, P = m∈M Q[G] · m is projective in Q[G] dis. The canonical map π : P → M is surjective, and this yields the claim. Claim 3.1.5. Every discrete left Q[G]-module is projective in Q[G] dis. 10 I. CASTELLANO AND TH. WEIGEL Proof of Claim 3.1.5. Let M ∈ ob(Q[G] dis). By Claim 3.1.4, there exist a projective discrete left Q[G]-module P and a map π : P → M such that (3.2) 0 / ker(π) /P π /M /0 is a short exact sequence in Q[G] dis. By Claim 3.1.3, ker(π) is injective. Hence (3.2) splits, and M is isomorphic to a direct summand of P , i.e., M is projective. 3.2. Discrete permutation modules. Let G be a t.d.l.c. group. A left G-set Ω is said to be discrete, if stabG (ω) is an open subgroup of G for any ω ∈ Ω. In this case · : G × Ω → Ω is continuous, where Ω is considered to be a discrete topological space and G × Ω carries the product topology. We say that Ω is a discrete left G-set with compact stabilizers, if stabG (ω) is a compact and open subgroup for any ω ∈ Ω. Let Ω be a left G-set. Then Q[Ω] - the free Q-vector space over the set Ω carries canonically the structure of left Q[G]-module. Moreover, Ω is a discrete left G-set if, and only if, Q[Ω] is a discrete left Q[G]-module. For a discrete left G-set Ω, Q[Ω] is also called a discrete left Q[G]-permutation module. Additionally, one has the following property. Proposition 3.2. Let G be a t.d.l.c. group, and let Ω be a discrete left G-set with compact stabilizers. Then Q[Ω] is projective in Q[G] dis. In particular, the abelian category Q[G] dis has enough projectives. F Proof. Let Ω = i∈I Ωi be the decomposition of Ω into G-orbits. Since Q[Ω] = ` i∈I Q[Ωi ] and as the coproduct of projective objects remains projective, it suffices to prove the claim for transitive discrete left G-sets Ω with compact stabilizers, i.e., we may assume that for ω ∈ Ω one has an isomorphism of left G-sets Ω ≃ G/O, and an isomorphism of left Q[G]-modules Q[Ω] ≃ indG O (Q), where O = stabG (ω). The induction functor is left adjoint to the restriction functor which is exact. Since the trivial left Q[O]-module is projective in Q[O] dis (cf. Prop. 3.1), and as indG O ( ) is mapping projectives to projectives (cf. §2.4 and [49, Prop. 2.3.10]), one concludes that Q[Ω] is projective. Let M ∈ ob(Q[G] dis). For each element m in M let Om be a compact open subgroup of stabG (m). Then πm : indG Om (Q) → M given by πm (g ⊗ 1) = g · m is a mapping of left Q[G]-modules. It is straightforward to verify that ` ` (3.3) π = m∈M πm : m∈M indG Om (Q) −→ M ` is F a surjective mapping of discrete left Q[G]-modules, and m∈M indG Om (Q) ≃ Q[ m∈M G/Om ] is projective. It is somehow astonishing that for a t.d.l.c. group G, the category Q[G] dis is a full subcategory of Q[G] mod with enough projectives and enough injectives (cf. Fact 2.2), but the reader may have noticed that this is a direct consequence of Proposition 3.1. Corollary 3.3. Let G be a t.d.l.c. group. A discrete left Q[G]-module M is projective if, and only if, it is a direct summand of a discrete left Q[G]-permutation module Q[Ω] for some discrete left G-set Ω with compact stabilizers. From Proposition 3.2 and Corollary 3.3 one deduces the following properties. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 11 Proposition 3.4. Let G be a t.d.l.c. group and, let H ⊆ G be a closed subgroup of G. Then resG H : Q[G] dis → Q[H] dis is mapping projectives to projectives. Moreover, if H is open in G, then indG H : Q[H] dis → Q[G] dis is mapping projectives to projectives. Proof. By Corollary 3.3, it suffices to prove that resG H (Q[Ω]) is projective for any discrete left Q[G]-permutation with compact stabilizers. Since resG H ( ) commutes with coproducts, we may also assume that Ω is a transitive discrete left G-set with compact stabilizers, i.e., for ω ∈ Ω and O = stabG (ω) one has Ω ≃ G/O.FLet R ⊆ G be a set of representatives of the (H, O)-double cosets in G, i.e., G = r∈R H r O. Then ` r (3.4) resG H (Q[G/O]) ≃ r∈R Q[H/(H ∩ O)]. Since H is closed and O compact and open, H ∩ r O is compact and open in H. Hence, by Proposition 3.2, resG H (Q[G/O]) is a projective discrete left Q[H]-module. Assume that H is open in G. Since indG H ( ) : Q[H] dis → Q[G] dis is the left adjoint G of the exact functor resG ( ) : dis → Q[G] Q[H] dis, indH ( ) is mapping projectives H to projectives (cf. [49, Prop. 2.3.10]). We close this subsection with the following characterization of compact t.d.l.c. groups. Proposition 3.5. Let G be a t.d.l.c. group, and let O be a compact open subgroup of G. Then HomG (Q, Q[G/O]) 6= 0 if, and only if, G is compact. Moreover, if G is compact, then HomG (Q, Q[G/O]) ≃ Q. Proof. If G is not compact, then \|G : O\| = ∞. Hence HomG (Q, Q[G/O]) = 0 (cf. [14, §III.5, Ex. 4]). If G is compact, then \|G : O\| < ∞, and Q[G/O] coincides with the coinduced module coindG O (Q) (cf. [14, §III, Prop. 5.9]). Hence HomG (Q, Q[G/O]) ≃ HomO (Q, Q) ≃ Q. This yields the claim. 3.3. Signed rational discrete permutation modules. For some application it will be useful to consider signed discrete permutation modules. Let (Ω, ¯ , ·) be a signed discrete left G-set, i.e., (Ω, ¯) is a signed set (cf. §A.5), and (Ω, ·) is a left G-set satisfying g · ω = g · ω̄, (3.5) for all g ∈ G and ω ∈ Ω. Then (cf. (A.16)) (3.6) Q[Ω] = Q[Ω]/spanQ { ω + ω̄ \| ω ∈ Ω } is a discreteF left Q[G]-module. For ω ∈ Ω let ω denote its canonical image in Q[Ω]. Let Ω = j∈J Ωr be a decomposition of Ω in G × C2 -orbits, where C2 = Z/2Z and σ = 1 + 2Z ∈ C2 acts by application of ¯ . We say that G acts without inversion on Ωj , if g · ω 6= ω for all g ∈ G and ω ∈ Ωj , otherwise we say that G acts with inversion. In this case, there exists g ∈ G and ω ∈ Ωj such that g ·ω = ω̄. Moreover, the action of G±ω = stabG ({ω, ω̄}) on Q · ω ⊆ Q[Ω] is given by a sign character sgnω : G±ω −→ {±1}. (3.7) Moreover, by choosing representatives of the G × C2 -orbits, one obtains an isomorphism of discrete left Q[G]-modules a a (3.8) Q[Ω] ≃ indG indG G±ω (Q(sgnω )) ⊕ G̟ (Q)). ω∈Rw ̟∈Rwo 12 I. CASTELLANO AND TH. WEIGEL Note that by the ± -construction (cf. §A.5) any rational discrete permutation module Q[Ω] can be considered as the signed rational discrete permutation module Q[Ω± ]. 3.4. The rational discrete cohomological dimension. Let M be a discrete left Q[G]-module for a t.d.l.c. group G. Then M is said to have projective dimension less or equal to n ≤ ∞, if M admits a projective resolution (P• , ∂• , ε) in Q[G] dis satisfying Pk = 0 for k > n, i.e., the sequence (3.9) / Pn 0 ∂n ∂n−1 / Pn−1 / ... ∂2 / P1 ∂1 / P0 ε /M /0 is exact. The projective dimension projQ dim(M ) ∈ N0 ∪ {∞} of M ∈ ob(Q[G] dis) is defined to be the minimum of such n ≤ ∞. As in the discrete case one has the following property. Lemma 3.6. Let G be a t.d.l.c. group, and let M be a discrete left Q[G]-module. Then the following are equivalent. (i) projQ dim(M ) ≤ n; (ii) dExti (M, ) = 0 for all i > n; (iii) dExtn+1 (M, ) = 0; (iv) In any exact sequence (3.10) 0 in /K Q[G] dis / Pn−1 ∂n−1 / ... ∂2 / P1 ∂1 / P0 ε /M /0 with Pi , 0 ≤ i ≤ n − 1, projective, one has that K is projective. Proof. The proof of [14, Chap. VIII, Lemma 2.1] can be transferred verbatim. For a t.d.l.c. group G (3.11) cdQ (G) = projQ dim(Q) will be called the rational discrete cohomological dimension of G, i.e., one has cdQ (G) =projQ dim(Q) (3.12) = min({ n ∈ N0 \| dHi (G, ) = 0 for all i > n } ∪ {∞}) = sup{ n ∈ N0 \| dHn (G, M ) 6= 0 for some M ∈ ob(Q[G] dis) }. Proposition 3.7. Let G be a t.d.l.c. group. (a) One has cdQ (G) = 0 if, and only if, G is compact. (b) If N is a closed normal subgroup of G, then (3.13) cdQ (G) ≤ cdQ (N ) + cdQ (G/N ). (c) If H is a closed subgroup of G, then (3.14) cdQ (H) ≤ cdQ (G). In particular, if H is normal and co-compact, then equality holds in (3.14). Proof. (a) If G is compact, then G is profinite, and thus, by Proposition 3.1, cdQ (G) = projQ dim(Q) = 0. Suppose that G satisfies cdQ (G) = 0, i.e., Q ∈ ob(Q[G] dis) is projective. By van Dantzig’s theorem, there exists a compact open subgroup O of G. By hypothesis, the canonical projection ε : Q[G/O] → Q admits a section. In particular, HomG (Q, Q[G/O]) 6= 0. Hence one concludes that RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 13 \|G : O\| < ∞, and G is compact. (b) is a direct consequence of the HochschildLyndon-Serre spectral sequence (cf. (2.15)) and (3.12). (c) By Proposition 3.4, the restriction of a projection resultion of Q in Q[G] dis is a projective resolution of Q in Q[H] dis showing the inequality (3.14). The final remark is a consequence of (3.13), (3.14) and part (a). Remark 3.8. Let G be a t.d.l.c. group, and let H ⊆ G be such that \|G : H\| < ∞. One concludes from Proposition 3.7(c) that cdQ (H) = cdQ (G). However, the following question will remain unanswered in this paper. Question 1. Suppose that H is a closed co-compact subgroup of the t.d.l.c. group G. Is it true that cdQ (H) = cdQ (G)? Remark 3.9. Proposition 3.7(c) implies that for any discrete group D of a t.d.l.c. group G one has (3.15) cdQ (D) ≤ cdQ (G). 3.5. The flat rank of a t.d.l.c. group. Let G be a t.d.l.c. group, and let H be a closed subgroup of G. Then H is called a flat subgroup of G, if there exists a compact open subgroup O of G which is tidy for all h ∈ H (cf. [5, §2, Def. 1]). For such a subgroup H(1) = { h ∈ H \| s(h) = 1 } is a normal subgroup, and H/H(1) is a free abelian group over some set IH (cf. [51, Cor. 6.15]). Moreover, rk(H) = card(IH ) ∈ N ∪ {∞} is called the rank of the flat subgroup H. The flat rank frk(G) of G is defined by (3.16) frk(G) = sup({ rk(H) \| H a flat subgroup of G } ∪ {0}) ∈ N0 ∪ {∞} (cf. [4, §1.3]). We call the t.d.l.c. group G to be N -compact, if for any compact open subgroup O of G, the open subgroup NG (O) is also compact. One has the following property. Proposition 3.10. Let G be an N -compact t.d.l.c. group. (a) Let H be a flat subgroup of G. Then rk(H) = cdQ (H). (b) frk(G) ≤ cdQ (G). Proof. (a) For any abelian group A which is a free over a set IA one has (3.17) cdQ (A) = cdZ (A) = card(IA ) ∈ N0 ∪ {∞}. Let O be a compact open subgroup such that O is tidy for all h ∈ H. By hypothesis, NG (O) is compact, and therefore C = H ∩ NG (O) is compact. Then, by Proposition 3.7(a) and the Hochschild-Lyndon-Serre spectral sequence (cf. (2.15)), one has natural isomorphisms dHk (H, ) ≃ dHk (H/C, C ) for all k > 0. Hence, by (3.17), cdQ (H) = cdQ (H/C) = rk(H). (b) is a direct consequence of (a). Remark 3.11. (Neretin’s group of spheromorphisms) Let Td+1 be a d+1-regular tree, and let G = Hier◦ (Td+1 ) = Nd+1,d be the group of all almost automorphisms (or spheromorphisms) of Td+1 introduced by Y. Neretin in [37]. It is well known that G is a compactly generated simple (in particular topologically simple) t.d.l.c. group. The group G contains a discrete subgroup D which is isomorphic to the HigmanThompson group Fd+1,d . From this fact one concludes that G contains discrete subgroups isomorphic to Zn for all n ≥ 1. Hence, by Remark 3.9, cdQ (G) = ∞. 14 I. CASTELLANO AND TH. WEIGEL 3.6. Totally disconnected locally compact groups of type FP∞ . Let G be a t.d.l.c. group, and let M be a discrete left Q[G]-module. Then M is said to be finitely generated, if there exist ` compact open subgroups O1 , . . . , On of G and a surjective homomorphism π : 1≤j≤n Q[G/Oj ] → M . A partial projective resolution (3.18) Pn ∂n / Pn−1 ∂n−1 / ... ∂2 / P1 ∂1 ε / P0 /M /0 of M will be called to be of finite type, if Pj is finitely generated for all 0 ≤ j ≤ n. One has the following property. Proposition 3.12. Let G be a t.d.l.c. group, and let M be a discrete left Q[G]module. Then the following are equivalent: (i) There is a partial projective resolution (Pj , ∂j , ε)0≤j≤n of finite type of M in Q[G] dis; (ii) the discrete left Q[G]-module M is finitely generated and for every partial projective resolution of finite type (3.19) Qk in Q[G] dis ðk / Qk−1 ðk−1 / ... ð2 / Q1 ð1 / Q0 ε′ /M /0 with k < n, ker(ðk ) is finitely generated. Proof. The proposition is a direct consequence of generalized form of Schanuel’s lemma (cf. [14, Chap. VIII, Lemma 4.2 and Prop. 4.3]). The discrete left Q[G]-module M will be said to be of type FPn , n ≥ 0, if M satisfies either of the hypothesis (i) or (ii) in Proposition 3.12, i.e., M is of type FP0 if, and only if, M is finitely generated. If M is of type FPn for all n ≥ 0, then M will be said to be of type FP∞ . The t.d.l.c. group G will be called to be of type FPn , n ∈ N ∪ {∞}, if the trivial left Q[G]-module Q is of type FPn . 4. Rational discrete homology for t.d.l.c. groups β α For a profinite group O any short exact sequence 0 → L → M → N → 0 of discrete left Q[O]-modules splits (cf. Prop. 3.1). Hence, if Q ∈ ob(disQ[O] ) denotes the trivial right Q[O]-module, the sequence (4.1) 0 / Q ⊗O L idQ ⊗α / Q ⊗O M idQ ⊗β / Q ⊗O N /0 is exact. Let G be a t.d.l.c. group, let O ⊆ G be a compact open subgroup, and β α let 0 → L → M → N → 0 be a short exact sequence of discrete left Q[G]-modules. Since indG O ( ) : disQ[O] → disQ[G] is an exact functor, and as indG O (Q) ⊗G (4.2) ≃ Q ⊗O resG O( ) : Q[G] dis −→ Q mod are naturally isomorphic additive functors, the sequence (4.3) 0 / indG O (Q) ⊗G L id⊗α / indG O (Q) ⊗G M id⊗β / indG O (Q) ⊗G N /0 : Q[G] dis → Q mod is an exact functor. Moreover, as is exact, i.e., indG O (Q) ⊗G ⊗G commutes with direct limits in the first and in the second argument (cf. [14, §VIII.4, p. 195]), the sequence (4.4) 0 / Q ⊗G L idQ ⊗α / Q ⊗G M idQ ⊗β / Q ⊗G N /0 RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 15 is exact for any projective discrete right Q[G]-module Q (cf. Cor. 3.3), i.e., Q is flat. For a discrete right Q[G]-module C, we denote by dTorG k (C, ) : (4.5) Q[G] dis −→ Q mod the left derived functors of the right exact functor C ⊗G define the rational discrete homology of G by : Q[G] dis → Q mod, and dHk (G, ) = dTorG k (Q, ). (4.6) By definition, one has the following properties. Proposition 4.1. Let G be a t.d.l.c. group, let C be a discrete right Q[G]-module, and let M be a discrete left Q[G]-module. Then (a) dHk (G, M ) = 0 for all k > cdQ (G); (b) if P ∈ ob(Q[G] dis) is projective, one has dTorG k (C, P ) = 0 for all k ≥ 1; (c) if G is compact, one has dTorG (C, M ) = 0 for all k ≥ 1. k Proof. (a) is a direct consequence of the definition of cdQ (G) (cf. (3.11)). (b) Using the previously mentioned arguments one concludes that ⊗G P : disQ[G] → Q mod is an exact functor. This property implies that for the left derived functors LM k of G M P ⊗G M one has that dTork (C, M ) ≃ Lk (C). Since P is projective, Lk = 0 for all k > 1. This yields the claim. (c) is a direct consequence of Proposition 3.1. 4.1. Invariants and coinvariants. Let O be a profinite group, and let M be a discrete left Q[O]-module. The Q-vector space M O = { m ∈ M \| g · m = m for all g ∈ O } ⊆ M (4.7) is called the space of O-invariants of M , and MO = Q ⊗O M, (4.8) is called the space of O-coinvariants of M . By definition, one has a canonical map φM,O : M O −→ MO (4.9) given by φM,O (m) = 1 ⊗O m for m ∈ M . One has the following. Proposition 4.2. Let O be a profinite group, and let M be a discrete left Q[O]module. Then φM,O is an isomorphism. Moreover, if U ⊆ O is an open subgroup, one has commutative diagrams MO (4.10) φM,O ρO,U MU where φM,U MO O / MO φM,O / MO O λU,O / MU MU φM,U / MU X 1 r · m, \|O : U \| r∈R X 1 ρO,U (1 ⊗O n) = 1 ⊗U r−1 · n, \|O : U \| λU,O (m) = (4.11) (4.12) r∈R U m ∈ M , n ∈ N , and R ⊆ O is any set of coset representatives of O/U . 16 I. CASTELLANO AND TH. WEIGEL Proof. The space of O-invariants M O ⊆ M is a discrete left Q[O]-submodule of M . Hence, by Claim 3.1.3, M = M O ⊕ C for some discrete left Q[O]-submodule C of M . Since C is a discrete left Q[O]-module, one has C = limi∈I Ci , where Ci are −→ finite-dimensional discrete left Q[G]-submodules of C. By construction, CiO = 0, and thus, by Maschke’s theorem, (Ci )O = 0. Since Q ⊗O commutes with direct limits, this implies that CO = 0. Hence C is contained in the kernel of the canonical map τM : M → MO . Let φM,O = (τM )\|M O denote the restriction of τM to the Q[O]-submodule M O . Since dH1 (O, ) = 0 (cf. Prop 3.7(a), Prop. 4.1(a)), the long exact sequence for dH• (O, ) implies that the map (M O )O → MO is injective. As τM O : M O → (M O )O is a bijection, the commutativity of the diagram /M MO ◆ ◆◆◆ ◆◆φ◆M,O τM O τM ◆◆◆ ◆◆& / MO (M O )O (4.13) yields that φM,O is injective. As τM is surjective and C ⊆ ker(τM ), one concludes that φM,O is bijective, and C = ker(τM ). The commutativity of the diagrams (4.10) can be verified by a straightforward calculation. 4.2. The rational discrete standard bimodule Bi(G). For a t.d.l.c. group G the set of compact open subgroups CO(G) of G with the inclusion relation ”⊆” is a directed set, i.e., (CO(G), ⊆) is a partially ordered set, and for U, V ∈ CO(G) one has also U ∩ V ∈ CO(G). For U, V ∈ CO(G), V ⊆ U , one has an injective mapping of discrete left Q[U ]-modules X 1 r V, (4.14) η̃U,V : Q −→ Q[U/V ], η̃U,V (1) = \|U : V \| r∈R G U where R ⊂ U is any set of coset representatives of U/V . As indG V = indU ◦ indV , η̃U,V induces an injective mapping X 1 x r V, x ∈ G. (4.15) ηU,V : Q[G/U ] −→ Q[G/V ], ηU,V (x U ) = \|U : V \| r∈R By construction, one has for W ∈ CO(G), W ⊆ V ⊆ U , that ηU,W = ηV,W ◦ ηU,V . For g ∈ G one has an isomorphism of discrete left Q[G]-modules cU,g : Q[G/U ] −→ Q[G/U g ], (4.16) g where U = g −1 cU,g (x U ) = x g U g , x ∈ G, U g. Moreover, for g, h ∈ G and U, V ∈ CO(G), V ⊆ U , one has (4.17) cV,g ◦ ηU,V = ηU g ,V g ◦ cU,g , (4.18) cU g ,h ◦ cU,g = cU,gh . Let Bi[G] = limU∈CO(G) (Q[G/U ], ηU,V ). Then, by definition, Bi[G] is a discrete left −→ Q[G]-module. By (4.17), the assignment (4.19) (xU ) · g = cU,g (x U ) = xg U g , g, x ∈ G, U ∈ CO(G), defines a Q-linear map · : Bi(G) × Q[G] → Bi(G), and, by (4.18), this map defines a right Q[G]-module structure on Bi(G). By (4.19), Bi(G) is a discrete right Q[G]-module making Bi(G) a rational discrete Q[G]-bimodule. Therefore, we RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 17 will call Bi(G) the rational discrete standard bimodule of G. It has the following straightforward properties. Proposition 4.3. Let G be a t.d.l.c. group. (a) Bi(G) is a flat rational discrete right Q[G]-module, and a flat rational discrete left Q[G]-module, i.e., (4.20) G dTorG k (A, Bi(G)) = dTork (Bi(G), B) = 0 for all A ∈ ob(disQ[G] ), B ∈ ob(Q[G] dis) and k ≥ 1. (b) One has ( Q if G is compact, (4.21) HomG (Q, Bi(G)) ≃ 0 if G is not compact. Proof. (a) The direct limit of flat objects is flat (cf. [14, §VIII.4, p. 195]). Thus, by Proposition 4.1(b), Bi(G) is a flat rational discrete left Q[G]-module. A similar argument shows that Bi(G) is also a flat rational discrete right Q[G]-module. (b) is a direct consequence of Proposition 3.5. Let M ∈ ob(Q[G] dis) and U ∈ CO(G). The composition of maps (cf. Prop. 4.2) (4.22) θM,U : Q[U \G] ⊗G M ξM,U / Q ⊗U M φ−1 M,U / MU , where ξM,U (U g ⊗G m) = 1 ⊗U gm, g ∈ G, m ∈ M , is an isomorphism. Moreover, if V ∈ CO(G), V ⊆ U , one has by (4.10) a commutative diagram (4.23) Q[U \G] ⊗G M θM,U / MU νU,V ⊗G idM Q[V \G] ⊗G M / MV θM,V which is natural in M , where νU,V : Q[U \G] → Q[V \G] is given by X 1 V r−1 x, (4.24) νU,V (U x) = \|U : V \| r∈R x ∈ G, and R ⊆ U is a set of coset representatives of U/V , i.e., U = Thus the family of maps (θM,U )U∈CO(G) induces an isomorphism (4.25) θM : Bi(G) ⊗G M −→ M F r∈R rV . which is natural in M . Note that the map θM can be described quite explicitly. For g ∈ G and m ∈ M the subgroup W = stabG (g · m) is open. In F particular, for U ∈ CO(G) the intersection U ∩W is of finite index in U . Let U = r∈R r·(U ∩W ). Then X 1 (4.26) θM (U g ⊗G m) = · r · m. \|U : (U ∩ W )\| r∈R From the commutative diagram (4.23) one concludes the following property. Proposition 4.4. Let G be a t.d.l.c. group. Then one has a natural isomorphism θ : Bi(G) ⊗G −→ idQ[G] dis . 18 I. CASTELLANO AND TH. WEIGEL Remark 4.5. Let G be a t.d.l.c. group. One verifies easily using (4.26) that · (4.27) = θBi(G) : Bi(G) ⊗G Bi(G) → Bi(G) defines the structure of an associative algebra on (Bi(G), ·). However, (Bi(G), ·) contains a unit 1Bi(G) ∈ Bi(G) if, and only if, G is compact. Nevertheless, the algebra (Bi(G), ·) has an augmentation or co-unit, i.e., the canonical map ε : Bi(G) → Q, (4.28) x ∈ G, U ∈ CO(G) ε(xU ) = 1, is a mapping of Q[G]-bimodules satisfying ε(a · b) = ε(a) · ε(b) for all a, b ∈ Bi(G). The algebra Bi(G) comes also equipped with an antipode (4.29) S= × : Bi(G) → Bi(G), S(U g) = g −1 U, g ∈ G, U ∈ CO(G). Again by (4.26) one verifies also that θM : Bi(G) ⊗G M → M defines the structure of a Bi(G)-module for any rational discrete Q[G]-module M . 4.3. The Hom-⊗-identity. Let G be a t.d.l.c. group, and let Q, M ∈ ob(Q[G] dis). Then one has a canonical map (cf. (4.25)) (4.30) ζQ,M : HomG (Q, Bi(G)) ⊗G M −→ HomG (Q, M ), ζQ,M (h ⊗G m)(q) = θM (h(q) ⊗G m), for h ∈ HomG (Q, Bi(G)), q ∈ Q, m ∈ M . Let U ⊆ G be a compact open subgroup of G. From now on we will assume that the canonical embedding ηU : Q[G/U ] −→ Bi(G) (4.31) is given by inclusion. By the Eckmann-Shapiro-type lemma (cf. §2.9), the map (4.32) ˆ : HomG (Q[G/U ], Bi(G)) −→ Bi(G), ĥ = h(U ), h ∈ HomG (Q[G/U ], Bi(G)), is injective satisfying im(ˆ ) = spanQ { x U x \| x ∈ G } ⊆ Bi(G). (4.33) In particular, one has an isomorphism jU : HomG (Q[G/U ], Bi(G)) → Q[U \G] (4.34) induced by ˆ . Hence, by the Eckmann-Shapiro-type lemma (cf. §2.9), (4.22), and the commutativity of the diagram (4.35) HomG (Q[G/U ], Bi(G)) ⊗G M ζQ[G/U ],M / HomG (Q[G/U ], M ) O jU ⊗G idM Q[U \G] ⊗G M / MU φ−1 M,U / MU one concludes that ζQ[G/U],M is an isomorphism for all M ∈ ob(Q[G] dis). The additivity HomG ( , M ) yields the following. Proposition 4.6. Let G be a t.d.l.c. group, let M, P ∈ ob(Q[G] dis), and assume further that P is finitely generated and projective. Then (4.36) ζP,M : HomG (P, Bi(G)) ⊗G M −→ HomG (P, M ) is an isomorphism. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 19 4.4. Dualizing finitely generated projective rational discrete Q[G]-modules. From (4.34) one concludes that for any finitely generated projective rational discrete left Q[G]-module P , HomG (P, Bi(G)) is a finitely generated projective rational discrete right Q[G]-module. In order to obtain a functor ⊛ : Q[G] disop → Q[G] dis we put for M ∈ ob(Q[G] dis) (4.37) M ⊛ = Hom× G (M, Bi(G)) = { ϕ ∈ HomQ (M, Bi(G)) \| ∀g ∈ G ∀m ∈ M : ϕ(g · m) = m · g −1 }. Obviously, × ◦ : HomG (M, Bi(G)) → M ⊛ (cf. (4.29)) yields an isomorphism of Q-vector spaces. Note that for g ∈ G, ϕ ∈ M ⊛ and m ∈ M one has (g · ϕ)(m) = g · ϕ(m). (4.38) By construction, if P is a finitely generated projective rational discrete left Q[G]module, then P ⊛ is a finitely generated projective rational discrete left Q[G]-module as well. Let M ∈ ob(Q[G] dis), let m ∈ M and let ϕ ∈ M ⊛ . Put ϑM (m)(ϕ) = ϕ(m)× . (4.39) Then for g ∈ G one has ϑM (m)(g · ϕ) = ((g · ϕ)(m))× = (g · ϕ(m))× = ϕ(m)× · g −1 = ϑM (m)(ϕ) · g −1 , (4.40) i.e., ϑM (m) ∈ M ⊛⊛ , and thus one has a Q-linear map ϑM : M → M ⊛⊛ . As ϑM (g · m)(ϕ) = ϕ(g · m)× = (ϕ(m) · g −1 )× = g · ϕ(m)× = (g · ϑM (m))(ϕ), (4.41) ϑM : M → M ⊛⊛ is a mapping of left Q[G]-modules. It is straightforward to verify that one obtains a natural transformation ϑ : idQ[G] dis −→ (4.42) ⊛⊛ . By (4.32) and (4.34) one has for U ∈ CO(G) that ϑQ[G/U] (U ) = jU× , where jU× is the map making the diagram (4.43) HomG (Q[G/U ), Bi(G)) × / Q[U \G] × ◦... Q[G/U ]⊛ jU × jU / Bi(G) commute. This yields that ϑQ[G/U] is an isomorphism. Hence the additivity of ϑ and the functors idQ[G] dis and ⊛⊛ imply that ϑP : P → P ⊛⊛ is an isomorphism for every finitely generated projective rational discrete Q[G]-module P . 4.5. T.d.l.c. groups of type FP. A t.d.l.c. group G is said to be of type FP, if (i) cdQ (G) = d < ∞; (ii) G is of type FP∞ (cf. [14, §VIII.6]). By Lemma 3.6 and Proposition 3.12, for such a group G the trivial left Q[G]-module Q possesses a projective resolution (P• , ∂• , ε) which is finitely generated and concentrated in degrees 0 to d. One has the following property. 20 I. CASTELLANO AND TH. WEIGEL Proposition 4.7. Let G be a t.d.l.c. group of type FP. Then (4.44) cdQ (G) = max{ k ≥ 0 \| dHk (G, Bi(G)) 6= 0 }. Proof. Let m = max{ k ≥ 0 \| dHk (G, Bi(G)) 6= 0 }. Then m ≤ d = cdQ (G). By definition, there exists a discrete left Q[G]-module M such that dHd (G, M ) 6= 0. As G is of type FP∞ , the functor dHd (G, ) commutes with direct limits (cf. [14, §VIII.4, Prop. 4.6]). Since M = limi∈I Mi , where Mi are finitely generated Q[G]−→ submodules of M , we may assume also that M is finitely generated, i.e., there P exist finitely many elements m1 , . . . , mn ∈ M such that M = 1≤j≤n Q[G] · mj . Let Oj be a compact open subgroup contained in stabG (mj ). Then one has a ` canonical surjective map 1≤j≤n Q[G/Oj ] → M . Thus, as dHd (G, ) is right ` exact, dHd (G, 1≤j≤n Q[G/Oj ]) 6= 0, and therefore dHd (G, Q[G/Oj ]) 6= 0 for some element j ∈ {1, . . . , d}. Let O = Oj . For any pair of compact open subgroups U, V of G, V ⊆ U , which are contained in O the map ηU,V : Q[G/U ] → Q[G/V ] (cf. (4.15)) is split injective (cf. Prop. 3.1). From the additivity of dHd (G, ) one concludes that dHd (ηU,V ) : dHd (G, Q[G/U ]) → dHd (G, Q[G/V ]) is injective. Hence (4.45) dHd (G, Bi(G)) = limU∈CO (G) dHd (G, Q[G/U ]) 6= 0, −→ O i.e., m ≥ d, and this yields the claim. For a t.d.l.c. group G of type FP with d = cdQ (G) we will call the rational discrete right Q[G]-module DG = dHd (G, Bi(G)) (4.46) the rational dualizing module of G. It has the following fundamental property. Proposition 4.8. Let G be a t.d.l.c. group of type FP with d = cdQ (G). Then there exists a canonical isomorphism υ : dHd (G, ) −→ DG ⊗G (4.47) of covariant additive right exact functors. Proof. Let (P• , ∂•P , εQ ) be a projective resolution of the trivial left Q[G]-module Q, which is finitely generated and concentrated in degrees 0 to d, and let (Q• , ð• , εDG ) be a projective resolution of the discrete right Q[G]-module DG . Let (C• , ∂•C ) be the chain complex of discrete right Q[G]-modules given by (4.48) Ck = HomG (Pd−k , Bi(G)), ∂kC = HomG (∂d−k+1 , Bi(G)) : Ck → Ck−1 , 0 ≤ k ≤ d, (cf. (4.37)). In particular, by construction, H0 (C• , ∂•C ) is canonically isomorphic to DG . Since (C• , ∂• ) is a chain complex of projectives, and as (Q• , ð• , εDG ) is exact, the comparison theorem in homological algebra implies that there exists a mapping of chain complexes χ• : (C• , ∂•C ) → (Q• , ð• ) which is unique up to chain homotopy equivalence. For M ∈ ob(Q[G] dis), χ• induces a chain map (4.49) χ•,M = χ• ⊗G idM : (C• ⊗G M, ∂•C ⊗G idM ) −→ (Q• ⊗G M, ð• ⊗G idM ). The right exactness of (4.50) ⊗G M implies that H0 (χ•,M ) : H0 (C• ⊗G M ) −→ DG ⊗G M RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 21 is an isomorphism. By the Hom-⊗ identity (4.36), one has a natural isomorphism of chain complexes (4.51) C• ⊗G M ≃ HomG (Pd−• , M ). Hence H0 (C• ⊗G M ) ≃ dHd (G, M ), and this yields the claim. Remark 4.9. A straightforward modification of the proof of Proposition 4.8 shows that for a t.d.l.c. group G of type FP one has natural isomorphisms (4.52) χ̄•, : dHd−• (G, ) −→ dTorG • (C• , [[0]]), where dTorG • ( , ) denotes the hyper-homology (cf. [6, §2.7], [14, §VIII.4, Ex. 7]). 4.6. Rational duality groups. A t.d.l.c. group G is said to be a rational duality group of dimension d ≥ 0, if (i) G is of type FP; (ii) dHk (G, Bi(G)) = 0 for all k 6= d. By Proposition 4.7, for such a group one has d = cdQ (G). These groups have the following fundamental property. Proposition 4.10. Let G be t.d.l.c. group which is a rational duality group of dimension d ≥ 0, and let DG denote its rational dualizing module. Then one has natural isomorphisms of (co)homological functors (4.53) dH• (G, ) ≃ dTorG d−• (DG , ), × dH• (G, ) ≃ dExtd−• G ( DG , ). Proof. Let (P• , ∂• , εQ ) be a projective resolution of the trivial left Q[G]-module Q, which is finitely generated and concentrated in degrees 0 to d. Then (4.54) Qk = HomG (Pd−k , Bi(G)), ðk = HomG (∂d−k+1 , Bi(G)), k ∈ {0, . . . , d}, is a chain complex of projective rational discrete right Q[G]-modules concentrated in degrees 0, . . . , d which satisfies ( DG for k = 0, (4.55) Hk (Q• , ð• ) ≃ 0 for k 6= 0, By Proposition 4.6, one has for M ∈ ob(Q[G] dis) isomorphisms (4.56) d−k Hk (Q• ⊗G M, ð• ⊗G idM ) ≃ dTorG k (DG , M ) ≃ dExtG (Q, M ) which are natural in M . This yields the first natural isomorphism in (4.53). Let (P•⊛ [d], ∂•⊛ [d]) be the chain complex (P•⊛ , ∂•⊛ ) concentrated in homological degrees −d, . . . , −0 moved d-places to the left such that it is concentrated in homological degrees 0, . . . , d, i.e., ( × DG for k = 0, ⊛ ⊛ (4.57) Hk (P• [d], ∂• [d]) ≃ 0 for k 6= 0. In particular, (P•⊛ [d], ∂•⊛ [d]) is a projective resolution of the rational discrete left Q[G]-module × DG . Then, by (4.42) and the remarks following it, one has an isomorphism of chain complexes of projective rational discrete left Q[G]-modules (4.58) (P•⊛ [d]⊛ [d], ∂•⊛ [d]⊛ [d]) ≃ (P• , ∂• ). 22 I. CASTELLANO AND TH. WEIGEL By the same arguments as used before, one obtains for M ∈ ob(Q[G] dis) and (4.59) Rk = HomG (P ⊛ [d]d−k , Bi(G)), δk = HomG (∂ ⊛ [d]d−k+1 , Bi(G)), k ∈ {0, . . . , d}, isomorphisms (4.60) d−k × Hk (R• ⊗G M, δ• ⊗G idM ) ≃ dTorG k (Q, M ) ≃ dExtG ( DG , M ) which are natural in M . This yields the claim. 4.7. Locally constant functions with compact support. Let G be a t.d.l.c. group, and let C(G, Q) denote the Q-vector space of continuous functions from G to Q, where Q is considered as a discrete topological space. Then C(G, Q) is a Q[G]-bimodule, where the G-actions are given by (4.61) (g · f )(x) = f (g −1 x), (f · g)(x) = f (x g −1 ), g, x ∈ G, f ∈ C(G, Q). For any compact open set Ω ⊆ G let IΩ : G → Q denote the continuous function given by IΩ (x) = 1 for x ∈ Ω and IΩ (x) = 0 for x ∈ G \ Ω. Then g · IΩ = Ig Ω and IΩ · g = IΩ g . Moreover, (4.62) Cc (G, Q) = spanQ { IgU \| g ∈ G, U ∈ CO(G) } ⊆ C(G, Q), coincides with the set of locally constant functions from G to Q with compact support. It is a rational discrete Q[G]-bisubmodule of C(G, Q). The rational discrete left Q[G]-module Bi(G) is isomorphic to the left Q[G]module Cc (G, Q), but the isomorphism is not canonical. Let O ⊆ G be a fixed compact open subgroup of G, and put COO (G) = { U ∈ CO(G) \| U ⊆ O }. The (O) Q-linear map ψU : Q[G/U ] → Cc (G, Q) given by (4.63) (O) ψU (xU ) = \|O : U \| · IxU is an injective homomorphism of rational discrete left Q[G]-modules. For U, V ∈ (O) (O) (O) COO (G), V ⊆ U , one has ψV ◦ ηU,V = ψU (cf. (4.15)), and thus (ψU )U∈COO (G) induces an isomorphism of left Q[G]-modules (4.64) (O) ψ (O) : Bi(G) → Cc (G, Q), ψ (O) \|Q[G/U] = ψU . This isomorphism has the following properties. Proposition 4.11. Let G be a t.d.l.c. group with modular function ∆ : G → Q+ , i.e., if µ : Bor(G) → R+ 0 ∪ {∞} is a left invarinat Haar measure on G, one has µ(S · g) = ∆(g) · µ(S) for all S ∈ Bor(G). Let O ∈ CO(G). (a) If U is a compact open subgroup of G containing O, then ψ (U ) = \|U : O\| · ψ (O) . (4.65) (b) For all g, h ∈ G and u ∈ Bi(G) one has (4.66) ψ (O) (g · W · h) = ∆(h−1 ) · g · ψ (O) (W ) · h. Proof. (a) Let g ∈ G and W ∈ CO(G). Then, by subsection 4.2, X 1 (4.67) gW = · g · r · (W ∩ O). \|W : W ∩ O\| r∈R Here we omitted the mappings ηW,W ∩O in the notation. Hence (4.68) ψ (O) (gW ) = µ(O) \|O : W ∩ O\| · IgW = · IgW , \|W : W ∩ O\| µ(W ) RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 23 for some left invariant Haar measure µ on G, and (4.69) ψ (U ) (gW ) = µ(U) · IgW . µ(W ) As µ(U) = \|U : O\| · µ(O), this yields the claim. (b) As ψ (O) is a homomorphism of left Q[G]-modules, it suffices to prove the claim F for g = 1 and u = W ∈ CO(G). If W h = h−1 · W · h = s∈S s · (W h ∩ O), one obtains by (a) that ψ (O) (W h) = h · ψ (O) (W h ) X 1 =h· · ψ (O) (s · (W h ∩ O)). h h \|W : W ∩ O\| s∈S h (4.70) = µ(O) \|O : W ∩ O\| · IW h = · IW h . \|W h : W h ∩ O\| µ(W h ) In particular, ψ (O) (W h) = µ(W )/µ(W h ) · ψ (O) (W ) · h. Since (4.71) µ(W h ) = µ(h−1 W h) = µ(W h) = ∆(h) · µ(W ), this completes the proof. L Remark 4.12. With a particular choice of Haar measure µ : Bor(G) → R+ 0 ∪ {∞} on G one can make (Cc (G, Q), ∗µ ) an associative algebra where ∗µ is convolution with respect to µ. However, we have seen that Bi(G) carries an algebra structure which is independent of the choosen Haar measure µ (cf. Remark 4.5). Both rational discrete Q[G]-bimodules Bi(G) and Cc (G, Q) will turn out to be useful. The standard rational Q[G]-bimodule Bi(G) seem to be the canonical choice for studying Hom-⊗ identities or dualizing functors (cf. §4.3, §4.4); while Cc (G, Q) seem to be the right choice for relating the cohomology groups dH• (G, Cc (G, Q)) with the cohomology with compact support of certain topological spaces associated to G (cf. §6.8). Remark 4.13. Let G be a t.d.l.c. group, and let ∆ : G → Q+ denote its modular function. Let Q(∆) denote the rational discrete left Q[G]-module which is isomorphic - as abelian group - to Q and which G-action is given by (4.72) g · q = ∆(g) · q, g ∈ G, q ∈ Q(∆). Let Q(∆)× denote the rational discrete right Q[G]-module associated to Q(∆), i.e., Q(∆)× = Q(∆) and for g ∈ G and q ∈ Q(∆)× one has q · g = g −1 · q. Then, by Proposition 4.11(b), one has a (non-canonical) isomorphism (4.73) Bi(G) ≃G Cc (G) ⊗ Q(∆)× of rational discrete right Q[G]-modules. Considering Q(∆)× as a trivial left Q[G]module, the isomorphism (4.73) can be interpreted as an isomorphism of rational discrete Q[G]-bimodules. 24 I. CASTELLANO AND TH. WEIGEL 4.8. The trace map. For a compact open subgroup O of G let µO denote the left-invariant Haar measure on G satisfying µO (O) = 1, i.e., if U is a compact open subgroup of G containing O, then µO = \|U : O\| · µU . We also denote by h(G) = Q · µO (4.74) the 1-dimensional Q-vector space generated by all Haar measures µO , O ∈ CO(G). One has a Q-linear map tr = ψ (O) ( )(1) · µO : Bi(G) −→ h(G), (4.75) which is independent of the choice of the compact open subgroup O of G, i.e., for all u ∈ Bi(G) and all O, U ∈ CO(G) one has (4.76) tr(u) = ψ (O) (u)(1) · µO = ψ (U ) (u)(1) · µU (cf. (4.65)). In particular, by definition, ( µO (4.77) tr(Og) = 0 if Og = O, if Og = 6 O. In case that G is unimodular one has the following. Proposition 4.14. Let G be a unimodular t.d.l.c. group. Then one has (4.78) tr(g · u) = tr(u · g) for all u ∈ Bi(G) and g ∈ G. In particular, putting Bi(G) = Bi(G)/h g · u − u · g \| u ∈ Bi(G), g ∈ G iQ , the Q-linear map tr induces a Q-linear map tr : Bi(G) −→ h(G). Proof. Since (g · f )(1) = (f · g)(1) = f (g −1 ) for all g ∈ G and f ∈ C(G, Q), one concludes from (4.66) that (4.79) tr(g · u) − tr(u · g) = (ψ (O) (u)(g −1 ) − ψ (O) (u)(g −1 )) · µO = 0. This yields the claim. 4.9. Homomorphism into the bimodule Cc (G, Q). Let G be a t.d.l.c. group. In order to simplify notations we put Cc (G) = Cc (G, Q) (cf. (4.62)). For a rational discrete left Q[G]-module M we put also M ⊙ = HomG (M, Cc (G)), (4.80) and consider M ⊙ as left Q[G]-module, i.e., for g ∈ G and h ∈ M ⊙ one has (g · h)(m) = h(m) · g −1 . (4.81) One has a homomorphism of Q-vector spaces (4.82) ∨ M : HomG (M, Cc (G)) −→ HomQ (M, Q), which is given by evaluation in 1 ∈ G, i.e., for m ∈ M and h ∈ M ⊙ one has (4.83) h∨ M (m) = h(m)(1). The Q-vector space M ∗ = HomQ (M, Q) carries canonically the structure of a left Q[G]-module, i.e., for g ∈ G and f ∈ M ∗ one has (4.84) (g · f )(m) = f (g −1 · m). RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 25 However, M ∗ is not necessarily a discrete Q[G]-module. For m ∈ M , g ∈ G and h ∈ HomG (M, Cc (G)) one concludes from (4.81) and (4.84) that −1 (g · h)∨ )(1) = h(m)(g) M (m) = (g · h)(m) (1) = (h(m) · g −1 −1 −1 (4.85) = g · h(m) (1) = h(g · m)(1) = h∨ · m), M (g i.e., ∨ M is a homomorphism of left Q[G]-modules. This homomorphism has the following property. Proposition 4.15. Let G be a t.d.l.c. group, and let M ∈ ob(Q[G] dis). Then ∨ ⊙ → M ∗ is injective. M: M Proof. Let h ∈ M ⊙ , h 6= 0, i.e., there exists m ∈ M with h(m) 6= 0. In particular, there exists g ∈ G such that h(m)(g) 6= 0. Since (4.86) −1 h(m)(g) = (g −1 · h(m))(1) = h(g −1 · m)(1) = h∨ · m), M (g this implies h∨ M 6= 0. Let α : B → M be a homomorphism of rational discrete left Q[G]-modules. Then for h ∈ M ⊙ and b ∈ B one has (cf. (4.83)) ⊙ α⊙ (h)∨ B (b) = α (h)(b) (1) = h(α(b))(1) ∗ ∨ = h∨ M (α(b)) = α (hM )(b), (4.87) i.e., the diagram M⊙ (4.88) α⊙ / B⊙ ∨ M M∗ ∨ B ∗ α / B∗ is commutative. 4.10. Proper signed discrete left G-sets. Let G be a t.d.l.c. group. A signed discrete left G-set Ω (cf. §3.3) will be said to be proper, if G has finitely many orbits on Ω and stabG (ω) is compact for all ω ∈ Ω. In particular, Q[Ω] is a finitely generated projective rational discrete left Q[G]-module (cf. (3.8)). For ω ∈ Ω define ω ∗ ∈ Q[Ω]∗ by   if ξ = ω, 1 ∗ (4.89) ω (ξ) = −1 if ξ = ω̄, and   0 if ξ 6∈ {ω, ω̄}. In particular, if g ∈ G then (4.90) g · ω ∗ = (g · ω)∗ . For a proper signed discrete left G-set Ω we put also (4.91) Q[Ω∗ ] = spanQ { ω ∗ \| ω ∈ Ω} ⊆ Q[Ω]∗ . In particular, as Ω∗ = { ω ∗ \| ω ∈ Ω } is a proper signed discrete left G-set, Q[Ω∗ ] is a finitely generated projective discrete left Q[G]-module. One has the following property. 26 I. CASTELLANO AND TH. WEIGEL Proposition 4.16. Let G be a t.d.l.c. group, and let Ω be a proper signed discrete ∗ ∨ left G-set. Then im( ∨ Q[Ω] ) = Q[Ω ]. In particular, Q[Ω] induces an isomorphism ∗ ⊙ ∨ ]. −→ Q[Ω : Q[Ω] Ω Proof. In suffices to prove the claim for a proper signed discrete left G-set Ω with one G × C2 -orbit (cf. (3.8)). We distinguish the two cases. Case 1: G acts without inversion on Ω. Let ω ∈ Ω and put O = stabG (ω). By hypothesis, Ω = G · ω ⊔ G · ω̄. Let R ⊆ G be a set of representatives for G/O. For x ∈ R there exists hx ∈ Q[Ω]⊙ = HomG (Q[Ω], Cc (G)) given by hx (ω) = IOx−1 . Moreover, as O Cc (G) = spanQ { IOx−1 \| x ∈ R } (cf. (4.34) and (4.64)), one has Q[Ω]⊙ = spanQ { hx \| x ∈ R }. For ̟ ∈ G · ω there exists a unique element y ∈ R such that ̟ = y · ω. Hence ∨ (4.92) (hx )∨ Q[Ω] (̟) = (hx )Q[Ω] (y · ω) = hx (y · ω)(1) = IyOx−1 (1) = IyO (x) = δx,y . ∨ ∗ Thus (hx )∨ Q[Ω] = (x·ω) . From this fact one concludes that Q[Ω] induces a mapping ∗ ⊙ −→ Q[Ω ], and that Ω is surjective. Hence the claim follows from Ω : Q[Ω] Proposition 4.15. Case 2: G acts with inversion on Ω. Let ω ∈ Ω and σ ∈ G be such that σ·ω = ω̄. Put U = stabG ({ω, ω̄}) and O = stabG (ω). In particular, O is normal in U and σ ∈ U. Let S ⊆ G be a set of representatives for G/U, and put R = S ⊔ Sσ. Then R is a set of representatives for G/O. For x ∈ S there exists kx ∈ Q[Ω]⊙ = HomG (Q[Ω], Cc (G)) given by kx (ω) = IOx−1 − IOσ−1 x−1 . Moreover, as before one has O Cc (G) = spanQ { IOz−1 \| z ∈ R }, and spanQ { kx \| x ∈ S } coincides with the eigenspace of the endomorphism σ◦ ∈ End(O Cc (G)) with respect to the eigenvalue −1. Thus Q[Ω]⊙ = spanQ { kx \| x ∈ S }. By hypothesis, Ω = G · ω. Let Ω+ = S · ω. Then Ω = Ω+ ⊔ Ω− , where − Ω = {̟ ¯ \| ̟ ∈ Ω+ }. Hence for ̟ ∈ Ω+ there exists a unique element y ∈ S such that ̟ = y · ω. This yields ∨ (kx )∨ Q[Ω] (̟) = (kx )Q[Ω] (y · ω) = kx (y · ω)(1) = IyOx−1 (1) − IyOσ−1 x−1 (1). (4.93) = IyOx−1 (1) = δx,y Here we used the fact that IyOσ−1 x−1 (1) = IyU x−1 (1) − IyOx−1 (1) = 0. Hence ∗ (kx )∨ Q[Ω] = (x · ω) and the claim follows by the same argument as in the previous case. One also concludes the following. Proposition 4.17. Let G be a t.d.l.c. group, let Ξ and Ω be proper signed discrete left G-sets, and let α : Q[Ξ] → Q[Ω] be a proper homomorphism of left Q[G]-modules (cf. (A.20)). Then one has a commutative diagram (4.94) Q[Ω]⊙ α⊙ ∨ Ω Q[Ω∗ ] / Q[Ξ]⊙ ∨ Ξ α ∗ / Q[Ξ∗ ], where the vertical maps are isomorphisms. Proof. This is an immediate consequence of Proposition 4.16 and the commutativity of the diagram (4.88). RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 27 5. Discrete actions of t.d.l.c. groups on graphs The notion of graph which will be used throughout the paper coincides with the notion used by J-P. Serre in [41], i.e., a graph Γ = (V(Γ), E(Γ)) will consist of a set of vertices V(Γ), a set of edges E(Γ), an origin mapping o : V(Γ) → E(Γ), a terminus mapping t : V(Γ) → E(Γ) and an edge inversion mapping ¯ : E(Γ) → E(Γ) satisfying ¯ = e, ē ē 6= e (5.1) t(ē) = o(e), o(ē) = t(e), for all e ∈ E(Γ) (cf. [41, §I.2.1]). In particular, E(Γ) is a signed set (cf. §A.5). Such a graph is said to be combinatorial, if the map (t, o) : E(Γ) −→ V(Γ) × V(Γ) (5.2) is injective. Moreover, Γ is said to be locally finite, if stΓ (v) = { e ∈ E(Γ) \| o(e) = v } is finite for every vertex v ∈ V(Γ). 5.1. The exact sequence associated to a graph. For a graph Γ let (5.3) V(Γ) = Q[V(Γ)] denote the free Q-vector space over the set of vertices of Γ, and let E(Γ) = Q[E(Γ)] = Q[E(Γ)]/h e + ē \| e ∈ E(Γ) i, (5.4) (cf. (A.16)), i.e., if e ∈ E[Γ] denotes the image of e ∈ E(Γ) in E(Γ), one has ē = −e. One has a canonical Q-linear mapping ∂ : E(Γ) → V(Γ) given by ∂(e) = t(e) − o(e), (5.5) e ∈ E(Γ), which has the following well known properties (cf. [41, §2.3, Cor. 1]). Fact 5.1. Let Γ = (V(Γ), E(Γ)) be a graph, and let ∂ : E(Γ) → V(Γ) be the map given by (5.5). Then (a) ker(∂) ≃ H1 (\|Γ\|, Q), where \|Γ\| denotes the topological realization of Γ. (b) coker(∂) ≃ Q[V(Γ)/ ∼], where ∼ is the connectedness relation, i.e., Γ is connected, if and only if, coker(∂) ≃ Q. In particular, Γ is a tree if, and only if, ker(∂) = 0 and coker(∂) ≃ Q. Let L(Γ) = ker(∂), i.e., if Γ is a connected graph the sequence (5.6) 0 / L(Γ) / E(Γ) ∂ / V(Γ) ε /Q /0 is exact. 5.2. Rough Cayley graphs. Let G be a t.d.l.c. group, let O be a compact open subgroup of G, and let S ⊆ G \ O be a symmetric subset of G intersecting O trivially, i.e., s ∈ S implies s−1 ∈ S. The rough Cayley graph Γ = Γ(G, S, O) associated with (G, S, O) is the graph given by V(Γ) = G/O and (5.7) E(Γ) = { (gO, gsO) \| g ∈ G, s ∈ S }. The origin mapping is given by the projection on the first coordinate, the terminus mapping is given by the projection on the second coordinate, while the edge inversion mapping permutes the first and second coordinate. The definition we have chosen here follows the approach used in [29, §2]. However, in our setup edges of a graph are directed, and we do not claim Γ to be connected. By construction, G has a discrete left action on Γ = Γ(G, S, O). It is straightforward to verify that Γ is combinatorial. One has the following properties. 28 I. CASTELLANO AND TH. WEIGEL Proposition 5.2. Let G be a t.d.l.c. group, let O be a compact open subgroup of G, let S be a symmetric subset of G \ O, and let Γ = Γ(G, S, O) be the rough Cayley graph associated with (G, S, O). (a) G acts transitively on the set of vertices of Γ; (b) if S is a finite set, then Γ is locally finite; (c) Γ is connected if, and only if, G is generated by O and S, i.e., G = h S, O i. Proof. (a) is obvious. (b) Suppose that S is finite. By (a), it suffices to show that stΓ (O) is a finite set. By definition, stΓ (O) = { (ωO, ωsO \| ω ∈ O, s ∈ S }. Since O is compact open in G, any double coset OsO is the union of finitely many right cosets g1 (s)O,. . . , gα(s) (s)O, i.e., t(stΓ (O)) = { gj (s)O \| s ∈ S, 1 ≤ j ≤ α(s) }. As Γ is combinatorial, the restriction of t to stΓ (O) is injective. This yields the claim. (c) Suppose that Γ is connected, and let g ∈ G. Then there exists a path p = e0 · · · en from O to gO, i.e., o(e0 ) = O, t(en ) = gO and o(ej ) = t(ej−1 ) for 1 ≤ j ≤ n. Let gj ∈ G, sj ∈ S such that ej = (gj O, gj sj O) for 0 ≤ j ≤ n, i.e., g0 = ω0 ∈ O. By induction, one concludes that gj ∈ h S, O i for all 1 ≤ j ≤ n. By construction, there exists ωn ∈ O such that gn sn ωn = g. In particular, g ∈ h S, O i, and thus G = h S, O i. Suppose now that G = h S, O i. By (a), it suffices to show that for any vertex gO ∈ V(Γ), there is a path from O to gO. By hypothesis, there exists elements s1 , . . . , sn ∈ S and ω0 , . . . , ωn ∈ O such that g = ω0 s1 · · · sn ωn . Hence for e0 = (ω0 O, ω0 s1 O) and ek = (ω0 s1 · · · sk−1 ωk−1 O, ω0 s1 · · · sk O) for 1 ≤ k ≤ n one verifies easily that p = e0 · · · en is a path from O to gO. Thus Γ is connected. 5.3. T.d.l.c. groups of type FP1 . It is well known that a discrete group is finitely generated if, and only if, it is of type FP1 (cf. [14, Ex. 1d of §I.2 and Ex. 1 of §VIII]). For t.d.l.c. groups one has the following. Theorem 5.3. Let G be a t.d.l.c. group. Then the following are equivalent. (i) The group G is compactly generated. ε (ii) The discrete Q[G]-module ker(Q[G/O] → Q) is finitely generated for any compact open subgroup O of G, where ε is the augmentation map. ε (iii) There exists a compact open subgroup O of G such that ker(Q[G/O] → Q) is finitely generated. (iv) The group G is of type FP1 . Proof. The implication (ii)⇒(iii) is trivial, and, by Proposition 3.12, (iii) and (iv) are equivalent. Suppose that G is compactly generated, and let O be any compact open subgroup of G. Then there exists a finite symmetric subset S ⊆ G \ O such that G = h S, O i. Let Γ = Γ(G, S, O) be the rough Cayley graph associated with (G, S, O). Then, as Γ is connected (cf. Prop. 5.2), one has a short exact sequence (cf. (5.6)) (5.8) E(Γ) ∂ / Q[G/O] ε /Q / 0. Moreover, as E(Γ) is a rational discrete left Q[G]-module being generated by the finite subset { (O, sO) \| s ∈ S }, ker(ε) is finitely generated. This shows (i)⇒(ii). Let O be a compact open subgroup of G such that K = ker(εO : Q[G/O] → Q) is a finitely generated discrete left Q[G]-module, i.e., there exist finitely many elements u1 , . . . , un ∈ K generating K as Q[G]-module. The set B = { gO −O \| g ∈ G\O } is RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 29 a generating set of K as Q-vector space, i.e., any element uj is a linear combination of finitely many elements of B. Hence there exists a finite subset S ⊂ G \ O such that any element uj is a linear combination of Σ = { sO − O \| s ∈ S }. Adjoining the inverses of elements in S if necessary, we may also assume that S is symmetric, i.e., s ∈ S implies s−1 ∈ S. Note that, by construction, one has P P (5.9) K = 1≤j≤n Q[G] · uj = s∈S Q[G] · (sO − O). Let Γ = Γ(G, S, O) be the rough Cayley graph associated with (G, S, O), and let ∂ : E(Γ) → Q[G/O] denote the canonical map, where we have identified V(Γ) with Q[G/O]. Then im(∂) ⊆ ker(εO ), and sO − O ∈ im(∂) for all s ∈ S (cf. (5.5), (5.7)). In particular, by (5.9), im(∂) = K, and coker(∂) ≃ Q, i.e., Γ is connected (cf. Fact 5.1(b)). Therefore, G = h S, O i (cf. Prop. 5.2(c)), and G is compactly generated. This yields the implication (iii)⇒(i) and completes the proof. 5.4. T.d.l.c. groups acting discretely on a tree. Let G be a group acting discretely on a graph Γ, i.e., all vertex stabilizers Gv = stabG (v), v ∈ V(Γ), and all edge stabilizers Ge = stabG (e), e ∈ E(Γ) are open. For e ∈ E(Γ) the set {e, ē} will be called a geometric edge of Γ. By E g (Γ) we denote the set of all geometric edges of Γ, i.e., E g (Γ) = E(Γ)/C2 , where C2 is the cyclic group of order 2 where the involution is acting by edge inversion. Let G{e} denote the stabilizer of the geometric edge {e, ē}, i.e., one has a sign character sgne : G{e} −→ {±1}. (5.10) By Qe we denote the 1-dimensional rational discrete left Q[G{e} ]-module with action given by (5.10), i.e., for g ∈ G{e} and q ∈ Qe one has g · q = sgne (g) · q. For a tree T one has L(T ) = 0. Hence the exact sequence (5.6) specializes to a short exact sequence. This fact implies the following version of a result of I. Chiswell (cf. [14, p. 179], [23]). Proposition 5.4. Let G be a t.d.l.c. group acting discretely on a tree T . Let RV ⊆ V(T ) be a set of representatives of the G-orbits on V(T ), and let RE ⊆ E(T ) be a set of representatives of the G × C2 -orbits on E(T ). (a) For M ∈ ob(Q[G] dis) one has a long exact sequence (5.11) Q Q k k / / / dHk (G, M ) ... v∈RV dH (Gv , M ) e∈RE dH (G{e} , M ⊗ Qe ) / ... (b) Suppose that Gv is compact for all v ∈ V(T ). Then cdQ (G) ≤ 1. In particular, (5.11) specializes to an exact sequence Q 0 / dH0 (G, M ) / (5.12) 0 v∈RV dH (Gv , M ) 0o dH1 (G, M ) o Q e∈RE dH0 (G{e} , M ⊗ Qe ) Proof. (a) By hypothesis, one has isomorphisms ` V(T )≃ v∈RV indG Gv (Q), (5.13) ` E(T )≃ e∈RE indG G{e} (Qe ), 30 I. CASTELLANO AND TH. WEIGEL and a short exact sequence (5.14) 0 / E(T ) ∂ / V(T ) ε /Q / 0. of rational discrete left Q[G]-modules. Hence (5.11) is a direct consequence of the long exact sequence associated to dExt•G ( , M ) and (5.14), the Eckmann-Shapiro lemma (cf. §2.9), and the isomorphisms dExtkG{e} (Qe , M ) ≃ dHk (G{e} , M ⊗ Qe ) which are natural in M . (b) By hypothesis, Gv is compact open, and Ge is open for all v ∈ V(T ) and e ∈ E(T ). As Ge ⊆ Gt(e) , this implies that Ge and G{e} are also compact and open. Hence (5.14) is a projective resolution of Q in Q[G] dis, and thus cdQ (G) ≤ 1. The exact sequence (5.12) follows from (5.11) and the fact that in this case one has cdQ (Gv ) = cdQ (G{e} ) = 0 (cf. Prop. 3.7(a)). Remark 5.5. If G is a compactly generated t.d.l.c. group, then dH1 (G, Bi(G)) must have dimension 0, 1 or ∞. It is shown in [21] that in `the latter case G is either isomorphic to a free product with amalgamation H U O, where H is an open subgroup of G, O is a compact open subgroup of G and \|O : O ∩ H\| 6= 1, or G is isomorphic to an HNN-extension HNN(H, ϕ) for some open subgroup H, O and U are compact open subgroups of H with at least one of them different from H and ϕ : O → U is an isomorphism, i.e., Stallings’ decomposition theorem holds for compactly generated t.d.l.c. groups. As a consequence a compactly generated t.d.l.c. group G satisfying dim(dH1 (G, Bi(G))) ≥ 2 must have a non-trivial discrete action on some tree T . 5.5. Fundamental groups of graphs of profinite groups. Let Λ be a connected graph. A graph of profinite groups (A, Λ) based on the graph Λ consists of (i) a profinite group Av for every vertex v ∈ V(Λ); (ii) a profinite group Ae for every edge e ∈ E(Λ) satisfying Ae = Aē ; (iii) an open embedding ιe : Ae → At(e) for every edge e ∈ E(Λ). For a graph of groups (A, Λ) one defines the group F (A, Λ) (cf. [41, §I.5.1]) as the group generated by Av , v ∈ V(Λ), and e ∈ E(Λ) subject to the relations (5.15) e−1 = ē and e ιe (a)e−1 = ιē (a) for all e ∈ E(Λ), a ∈ Ae . Let Ξ ⊆ Λ be a maximal subtree of Λ. The fundamental group π1 (A, Λ, Ξ) of the graph of groups (A, Λ) with respect to Ξ is given by (5.16) Π = π1 (A, Λ, Ξ) = F (A, Λ)/h e \| e ∈ E(Ξ) i (cf. [41, §I.5.1]). Choosing an orientation E + ⊂ E(Λ) of Λ one may construct a tree T = T (A, Λ, Ξ, E + ) with a canonical left Π-action (cf. [41, §I.5.1, Thm. 12]). The fundamental group of a graph of profinite groups carries naturally the structure of t.d.l.c. group. Indeed, the set of all open subgroups of vertex stabilizers are a neighborhood basis of the identity element of the topology. Thus from the Main Theorem of Bass-Serre theory one concludes the following. Proposition 5.6. Let (A, Λ) be a graph of profinite groups, let Ξ ⊆ Λ be a maximal subtree of Λ, and let E + ⊂ E(Λ) be an orientation of Λ. Then Π = π1 (A, Λ, Ξ) carries naturally the structure of a t.d.l.c. group, and its action on T = T (A, Λ, Ξ, E + ) is discrete and proper, i.e., vertex stabilizers and edge stabilizers are compact and open. In particular, cdQ (Π) ≤ 1. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 31 Proof. This follows from the Main Theorem of Bass-Serre theory and Proposition 5.4(b). Remark 5.7. The Main Theorem of Bass-Serre theory would yield more information than stated in Proposition 5.6. Indeed, a discrete and proper action of a t.d.l.c. group G on a tree T can be used to define a graph of profinite groups (A, Λ) such that G ≃ π1 (A, Λ, Ξ). 5.6. Nested t.d.l.c. groups. A t.d.l.c. group G is said to be nested, if there exists a countable ascending sequence of compact open subgroups (Ok )k≥0 of G such that S (5.17) G = k≥0 Ok . Such a group can be represented as the fundamental group of the graph of profinite groups (5.18) (A, Λ) : O1 • O1 O2 • O2 O3 • O3 O4 • O4 O5 • O5 ... based on the straight infinite half line Λ, where all injections are the canonical ones, i.e., one has (5.19) G ≃ π1 (A, Λ, Λ). Thus from Proposition 5.6 one concludes the following. Proposition 5.8. Let G be a nested t.d.l.c. group. Then cdQ (G) ≤ 1. S Remark 5.9. Let p be a prime number. Then Qp = k≥0 p−k Zp , the additive group of the p-adic numbers is a nested t.d.l.c. group, but Qp is not compact. Hence cdQ (Qp ) = 1. 5.7. Generalized presentations of a t.d.l.c. groups. Let G be a t.d.l.c. group. A generalized presentation of G is a graph of profinite groups (A, Λ) together with a continuous open surjective homomorphism (5.20) φ : π1 (A, Λ, Ξ) −→ G, such that φ\|Av is injective for all v ∈ V(Λ). Proposition 5.10. Let G be a t.d.l.c. group. (a) G has a generalized presentation ((A, Λ0 ), φ), where Λ0 is a graph with a single vertex. (b) For any generalized presentation ((A, Λ), φ), K = ker(φ) is a discrete free group contained in the quasi-center (5.21) QZ(Π) = { g ∈ Π \| CΠ (g) is open in Π }. of Π = π1 (A, Λ, Ξ). In particular, (5.22) R(φ) = K/[K, K] ⊗Z Q is a rational discrete left Q[G]-module. (c) If ((Ai , Λi ), φi ), i = 1, 2, are two presentations of G, then R(φ1 ) and R(φ2 ) are stably equivalent, i.e., there exist projective rational discrete Q[G]-modules P1 and P2 such that R(φ1 ) ⊕ P1 ≃ R(φ2 ) ⊕ P2 . 32 I. CASTELLANO AND TH. WEIGEL Proof. (a) By van Dantzig’s theorem there exists a compact open subgroup O of G. Choose a subset S of G such that G =< O, S >. Without loss of generality we may assume that S = S −1 , and choose a subset S + ⊆ S such that S = S + ∪ (S + )−1 . Let Λ0 be the graph with a single vertex v with a loop attached for any s ∈ S + , i.e., V(Λ0 ) = {v} and E(Λ0 ) = { es , ēs \| s ∈ S + }. Let (A, Λ0 ) be the graph of profinite groups based on Λ0 defined as follows: - Av = O; - Aes = Aēs = O ∩ s−1 Os for all s ∈ S + ; is Av for all s ∈ S + , where is is left conjugation by - αes : Aes −→ s−1 Os −→ s; - αēs : Aes → Av is the canonical inclusion for all s ∈ S + . Clearly, (A, Λ0 ) is a graph of profinite groups, and for the canonical map (5.23) φ0 : E(Λ0 ) ∪ O → G, φ(es ) = φ(ēs )−1 = s, φ0 (w) = w, w ∈ O, the relations (5.15) are satisfied. The edge set of the maximal subtree Ξ is empty. Hence F (A, Λ0 ) → π1 (A, Λ0 , Ξ) is an isomorphism. This show that there exists a unique group homomorphism φ : π1 (A, Λ0 , Ξ) → G such that φ\|E(Λ0 )∪O = φ0 . By construction, φ is surjective and φ\|Av is injective. Hence ((A, Λ0 ), φ) is a generalized presentation. (b) Let T = T (A, Λ, Ξ, E + ). Then, by construction, any non-trivial element of K acts without inversion of edges and without fixed points on T . Thus, by Stallings’ theorem (cf. [41, §I.3.3, Thm. 4]), K is a free discrete group. As K is normal in Π and discrete, K must be contained in QZ(Π). This implies that R(φ) is a rational discrete left Q[G]-module. (c) Let Πi = π1 (Ai , Λi , Ξi ), and let Ti = T (Ai , Λi , Ξi , Ei+ ) be trees associated to (Ai , Λi ). In particular, Πi is acting without inversion on Ti . Put Ki = ker(φi ). Then for the quotient graphs Γi = Ti //Ki , (5.24) E(Γi ) ∂i / V(Γi ) εi /Q are partial projective resolutions of Q in Q[G] dis. Hence, by Schanuel’s lemma (cf. [14, §VIII.4, Lemma 4.2]), ker(∂1 ) and ker(∂2 ) are stably equivalent. By Fact 5.1(a), one has ker(∂1 ) ≃ R(φ1 ) and ker(∂2 ) ≃ R(φ2 ). This yields the claim. Remark 5.11. Let ((A, Λ), φ) be a presentation of the t.d.l.c. group G. One should think of R(φ) as the relation module of the presentation ((A, Λ), φ). By Proposition 5.10, its stable isomorphism type is an invariant of the t.d.l.c. group G. 5.8. Compactly presented t.d.l.c. groups. A t.d.l.c. group G is said to be compactly presented, if there exists a presentation ((A, Λ), φ), such that (i) Λ is a finite connected graph, and (ii) K = ker(φ) is finitely generated as a normal subgroup of Π = π1 (A, Λ, Ξ). Condition (i) implies that such a group G must be compactly generated. Hence G is of type FP1 (cf. Thm. 5.3). From condition (ii) follows that R(φ) is a finitely generated rational discrete Q[G]-module. Thus, as R(φ) ≃ L(Γ), where T = T (A, Λ, Ξ, E + ) and Γ = T //K (cf. (5.6)), G must be of type FP2 (cf. Prop. 3.12). It has been an open problem for a long time whether discrete groups of type FP2 are indeed finitely presented. Finally, it has been answered negatively by M. Betsvina and N. Brady in [7]. In our context the following question arises. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 33 Question 2. Does there exist a non-discrete t.d.l.c. group G which is of type FP2 , but which is not compactly presented? 6. G-spaces for topological groups of type OS Throughout this section we assume that G is a topological group of type OS, and that F ⊆ Osgrp(G) is a non-empty subset of open subgroups of G satisfying (F1 ) for A ∈ F and g ∈ G one has g A = gAg −1 ∈ F; (F2 ) for A, B ∈ F one has A ∩ B ∈ F; In some cases it will be helpful to assume that the class F satisfies additionally the following condition. (F3 ) for A, B ∈ Osgrp(G), B ⊆ A, \|A : B\| < ∞ and B ∈ F one has A ∈ F. 6.1. F-discrete G-spaces. Let G be a topological group of type OS. We fix a nonempty set of open subgroups F satisfying (F1 ) and (F2 ). A non-trivial topological space X together with a continuous left G-action · : G × X → X will be called a left G-space. Such a space will be said to be F-discrete, if stabG (x) ∈ F for all x ∈ X. A continuous map f : X → Y of left G-spaces which commutes with the G-action is called a mapping of G-spaces. The non-empty G-space X together with an increasing filtration (X n )n≥0 of closed subspaces X n ⊆ X is called a (G, F)CW-complex, if S (C1 ) X = n≥0 X n ; (C2 ) X 0 is an F-discrete subspace of X; (C3 ) for n ≥ 1 there is an F-discrete G-space Λn , G-maps f : S n−1 × Λn → X n−1 and fb: B n × Λn → X n such that (6.1) S n−1 × Λn f / X n−1 B n × Λn fb / Xn is a pushout diagram, where S n−1 denotes the unit sphere and B n the unit ball in euclidean n-space (with trivial G-action); (C4 ) a subspace Y ⊂ X is closed if, and only if, Y ∩ X n is closed for all n ≥ 0. The dimension of X is defined by (6.2) dim(X) = min({ k ∈ N0 \| X k = X } ∪ {∞}). Moreover, X is said to be of type Fn , n ∈ N0 ∪ {∞}, if G has finitely many orbits on Λk for 0 ≤ k ≤ n. 6.2. The F-universal G-CW-complex. Let G be a topological group of type OS, and let F ⊆ Osgrp(G) be a non-empty set of open subgroups satisfying (F1 ) and (F2 ). A left G-CW-complex X said to be F-universal or an E F (G)-space, if (U1 ) X is contractable; (U2 ) for all x ∈ X one has stabG (x) ∈ F; (U3 ) for all A ∈ F the set of A-fixed points X A is non-empty and contractable. It is well known that an F-universal G-CW-complex E F (G) exists and is unique up to G-homotopy. The G-CW-complex E F (G) is universal with respect to all G-CW-complexes with point stabilizers in F, i.e., if X is a G-CW-complex with 34 I. CASTELLANO AND TH. WEIGEL stabG (x) ∈ F for all x ∈ X, then there exists a G-map X → E F (G) which is unique up to G-homotopy (cf. [32, Remark 2.5], [47, Chap. 1, Ex. 6.18.7]). This space can be explicitly constructed as the union of the n-fold joins of the discrete G-space F ΩF = A∈F G/A (cf. [32, §2], [47, Chap. 1, §6]). This notion leads to an obvious notion of dimension, i.e., (6.3) dimF (G) = min{ dim(X) \| X an E F (G)-space } ∈ N0 ∪ {∞} may be considered as the topological F-dimension of G. Although this might be the most natural dimension to be considered, we prefered to introduce and study a combinatorial version of this dimension, the simplicial F-dimension of G. This choice is motivated by the fact, that for the examples we are interested in the E F (G)spaces arise naturally as the topological realization of certain simplicial complexes. 6.3. Simplicial G-complexes. A simplicial complex (cf. §A.1) Σ with a left Gaction will be called a simplicial G-complex. The simplicial G-complex Σ is said to be of type Fn , n ∈ N0 , if G has finitely many orbits on Σq for all q ∈ {0, . . . , n }, and of type F∞ , if it is of type Fn for all n ∈ N0 . For a simplicial G-complex Σ and a q-simplex ω = {x0 , . . . , xq } ∈ Σq we denote by T stabG (ω) = 0≤j≤q stabG (xj ), (6.4) stabG {ω} = { g ∈ G \| g · ω = g }, the pointwise and setwise stabilizer of ω, respectively. In particular, one has stabG (ω) ⊆ stabG {ω}, and \|stabG {ω} : stabG (ω)\| < ∞. The simplicial G-complex Σ will be said to be G-tame3 if stabG (ω) = stabG {ω} for all ω ∈ Σ. For the convenience of the reader some basic properties of simplicial complexes are recollected in §A. 6.4. F-discrete simplical G-complexes. Let G be a topological group of type OS, and let F ⊆ Osgrp(G) be a non-empty set of open subgroups satisfying (F1 ), (F2 ) and (F3 ). A simplicial G-complex Σ will be said to be F-discrete, if stabG (x) ∈ F for all x ∈ Σ0 . Thus, by (F2 ), for an F-discrete simplicial G-complex one has stabG (ω) ∈ F and, by (F3 ), stabG {ω} ∈ F for all ω ∈ Σ. This property has the following consequence. Fact 6.1. Let G be topological group of type OS, and let F ⊆ Osgrp(G) be a nonempty set of open subgroups of G satisfying (F1 ), (F2 ) and (F3 ). Then for every F-discrete simplicial G-complex Σ, its topological realization \|Σ\| (cf. §A.2) is an F-discrete G-CW-complex. Proof. The property of being a G-CW-complex is straightforward. For any point z ∈ \|Σ\|, there exists a unique q-simplex ω ∈ Σ such that z is an interior point of \|Σ(ω)\| (cf. Fact A.3). Hence stabG (ω) ⊆ stabG (z) ⊆ stabG {ω}, and thus stabG (z) ∈ F by (F3 ). 6.5. Contractable F-discrete simplical G-complexes. The F-discrete simplical G-complex Σ will be said to be contractable, if \|Σ\| is contractable. But note that we do not claim that \|Σ\| is G-homotopic to a point (cf. [32, Thm. 2.2]). 3In case that Σ = Σ(Γ) for a graph Γ (cf. Ex. A.2), then Σ is a G-tame simplicial G-complex if, and only if, G acts without inversion of edges on the graph Γ. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 35 Example 6.2. Let G be a topological group of type OS, and let F ⊆ Osgrp(G) be a non-empty set of open subgroups satisfying (F1 ), (F2 ) and (F3 ). Let A ∈ F, and put Ω(A) = Σ[G/A] (cf. Ex. A.1). Then Ω(A) is an F-discrete simplical G-complex. By Fact A.4, \|Ω(A)\| is contractable, i.e., Ω(A) is a contractable F-discrete simplicial G-complex. We define the simplicial F-dimension of G by (6.5) sdimF (G) = min({ n ∈ N0 \| ∃ a contractable F-discrete simplicial G-complex Σ: dim(Σ) = n } ∪ {∞}) (cf. (A.4)). For n ∈ N0 ∪ {∞} we call G to be of type sFn (F), if there exists a contractable F-discrete simplicial G-complex Σ which is of type Fn (cf. §6.3). Moreover, G will be said to be of type sF(F), if there exists a contractable F-discrete simplicial G-complex Σ of type F∞ satisfying dim(Σ) < ∞. 6.6. The Bruhat-Tits property. The F-discrete simplicial G-complex Σ is said to have the Bruhat-Tits property, if for all A ∈ F one has \|Σ\|A 6= ∅. For every point z ∈ \|Σ\|A there exists a unique q-simplex ω(z) ∈ Σ such that z is contained in the interior of \|Σ(ω(z))\| (cf. Fact A.3). Hence A ⊆ stabG (z) ⊆ stabG {ω(z)}. 6.7. Classes of open subgroups for t.d.l.c. groups. Let G be a t.d.l.c. group. Obviously, the class of all open subgroups O = Osgrp(G) and the class C = CO(G) of all compact open subgroups satisfy the conditions (F1 ), (F2 ) and (F3 ). The same is true for (6.6) vN = { O ∈ Osgrp(G) \| ∃ U ⊆ O, U open and nested, \|O : U\| < ∞ }, the class of all virtually nested open subgroups of G (cf. §5.6). Proposition 6.3. Let G be a t.d.l.c. group. Then vN satisfies the conditions (F1 ), (F2 ) and (F3 ). Proof. Obviously, vN satisfies (F1 ) and (F3 ). Let S O ∈ vN, and let U ⊆ O be an open, nested subgroup of finite index, i.e., U = k≥0 Uk , Uk ⊆ Uk+1 , Uk ∈ CO(G). Since \|O : U\| < ∞, we may also assume that U is normal in O. Let V ∈ vN. Then V ∩ U is a nested open subgroup of G. Moreover, as V ∩ U = ker(V → O/U), one has that \|V : V ∩ U\| < ∞. Hence vN satisfies (F2 ) as well. Remark 6.4. The proof of Proposition 5.8 show also that N - the class of all nested open subgroups of a t.d.l.c. group G - satisfies the conditions (F1 ) and (F2 ). Remark 6.5. It is a remarkable fact, that although the Higman-Thompson groups Fn,r , Tn,r , Gn,r are not of finite cohomological dimension, they still satisfy the FP∞ -condition (cf. [15]). The Neretin group Nn,r of spheromorphism of a regular rooted tree Tn,r can be seen as a t.d.l.c. analogue of the Higman-Thompson groups (cf. Remark 3.11). In particular, this group has a factorization (6.7) Nn,r = O · Fn,r , O ∩ Fn,r = {1}, where Fn,r is a discrete subgroup isomorphic to the Richard Thompson group Fn,r and O is a nested open subgroup of Nn,r . Hence the following questions arise. Question 3. Is Nn,r of type sF(N)∞ or even of type sF(C)∞ ? Question 4. Does there exist an E N (Nn,r )-space which is of type F∞ ? 36 I. CASTELLANO AND TH. WEIGEL The finiteness conditions we discussed in subsection §6.5 for C-discrete simplicial G-complexes for a t.d.l.c. group G have the following consequence for its cohomological finiteness conditions. Proposition 6.6. Let G be a t.d.l.c. group. Then one has the following. (a) sdimC (G) ≥ cdQ (G). (b) Let n ∈ N0 ∪ {∞}. If G is of type sFn (C), then G is of type FPn . (c) If G is of type sF(C), then G is of type FP. Proof. Let Σ be a C-discrete simplicial G-complex, let ω = {x0 , . . . , xq } ∈ Σq , and e q (cf. (A.8)). Then stabG {ω} is acting on {±ω}, i.e., one let ω = x0 ∧ · · · ∧ xq ∈ Σ has a continuous group homomorphism (6.8) sgnω : stabG {ω} −→ {±1}. Let Qω denote the 1-dimensional rational discrete left stabG {ω}-module associated to sgnω , i.e., for g ∈ stabG {ω} and z ∈ Qω one has g · z = sgnω (g) · z. Thus if Rq ⊆ Σq is a set of representatives for the G-orbits on Σq , one has an isomorphism ` (6.9) Cq (Σ) ≃ ω∈Rq indG stabG {ω} (Qω ) of rational discrete left Q[G]-modules (cf. §A.3). In particular, Cq (Σ) is a projective rational discrete left Q[G]-module. If Σ is contractable, one has H0 (\|Σ\|, Q) ≃ Q and Hk (\|Σ\|, Q) = 0 for k 6= 0. Hence (C• (Σ), ∂• ) is a projective resolution of the rational discrete left Q[G]-module Q (cf. Fact A.6). In particular, cdQ (G) ≤ dim(Σ). This yields (a). If Σ is contractable and of type Fn , n ∈ N0 ∪ {∞}, then, by (6.9), Ck (Σ) is a finitely generated projective rational discrete left Q[G]-module for 0 ≤ k ≤ n. This yields (b). Part (c) follows by similar arguments. 6.8. Simplicial t.d.l.c. rational duality groups. As in the discrete case it is possible to detect from a group action that a t.d.l.c. group G is indeed a rational duality group. Theorem 6.7. Let G be a t.d.l.c. group, and let Σ be a C-discrete simplicial G-complex such that (D1 ) Σ is contractable; (D2 ) dim(Σ) < ∞ and Σ is of type F∞ ; (D3 ) Σ is locally finite. Then d = cdQ (G) < dim(Σ) and one has (6.10) × DG = Hcd (\|Σ\|, Q) ⊗ Q(∆). In particular, if (D4 ) Hck (\|Σ\|, Q) = 0 for k 6= d, then G is a t.d.l.c. rational duality group of dimension d. Proof. By (D1 ), (C• (Σ), ∂• ) is a projective resolution of Q in Q[G] dis, and by (D2 ), Cq (Σ) is a finitely generated projective rational discrete Q[G]-modules for all q ≥ 0. As Σ is locally finite by (D3 ), ∂q : Cq (Σ) → Cq−1 (Σ), q ≥ 1, has an adjoint map e ∗ ] → Q[Σ e ∗ ], where one has to replace Σ e 0 by Σ e ± (cf. §A.5). Moreover, ∂q∗ : Q[Σ q−1 q 0 (6.11) ∨ eq Σ ∗ eq] : Cq (Σ)⊙ −→ Q[Σ RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 37 is an isomorphism (cf. Prop. 4.16), and one has the commutative diagram (6.12) ∨ eq Σ Cq (Σ)⊙ / Q[Σ e ∗] q Ccq (Σ) ∂∗ q+1 ⊙ ∂q+1 ∨ e Σ Cq+1 (Σ)⊙ q+1 e ∗q+1 ] / Q[Σ ðq Ccq+1 (Σ) (cf. Prop. 4.17). In particular, one obtains canonical isomorphisms (cf. Fact A.7) dHk (G, Cc (G)) ≃ Hck (\|Σ\|, Q)× (6.13) of rational discrete right Q[G]-modules, and non-canonical isomorphisms × (6.14) dHk (G, Bi(G)) ≃ Hck (\|Σ\|, Q) ⊗ Q(∆) of rational discrete left Q[G]-modules (cf. Remark 4.13). In particular, as G is of type FP (cf. Prop. 6.6(c)), one has (6.15) cdQ (G) = max{ k ∈ N0 \| Hck (\|Σ\|, Q) 6= 0 } (cf. Prop. 4.7). Thus for d = cdQ (G) one has an isomorphism (6.16) × DG = Hcd (\|Σ\|, Q) ⊗ Q(∆). This yields the claim. A t.d.l.c. group G admitting a C-discrete simplicial G-complex Σ satisfying (Dj ), 1 ≤ j ≤ 4, will be called a simplicial t.d.l.c. rational duality group. It should be mentioned that the dimension of Σ as simplicial complex does not have to coincide with d = cdQ (G). 6.9. Universal C-discrete simplicial G-complexes. Let Σ be a C-discrete simplical G-complex, and let δ : \|Σ\| × \|Σ\| → R+ 0 be a G-invariant metric on its topological realization. Then (Σ, δ) is called a C-discrete CAT(0) simplicial G-complex, if (E1 ) (\|Σ\|, δ) is a complete CAT(0)-space4, (E2 ) Σ is G-tame. Note that property (E1 ) implies that \|Σ\| is contractable (cf. [2, Prop. 11.7]). Let O be a compact open subgroup of G. By hypothesis, any orbit O · x, x ∈ \|Σ\|, is finite and thus in particular bounded. In particular, O has a fixed point z in \|Σ\| (cf. [2, Thm. 11.23]). Let ω(z) ∈ Σ denote the unique simplex such that z is contained in the interior of ω(z) (cf. Fact A.3). Then, by property (E2 ), O ⊆ stabG (z) = stabG {ω(z)} = stabG (ω(z)), i.e., Σ has the Bruhat-Tits property (cf. §6.6). By property (E2 ), ΣO is a simplicial subcomplex of Σ, and, by (E1 ), (\|ΣO \|, δ) is a complete CAT(0)-space. In particular, \|ΣO \| is contractable. This shows the following. Proposition 6.8. Let G be a t.d.l.c. group, and let Σ be a C-discrete CAT(0) simplical G-complex. Then \|Σ\| is an E C (G)-space. 4For details on CAT(0)-spaces see [13]. 38 I. CASTELLANO AND TH. WEIGEL 6.10. Compactly generated t.d.l.c. groups of rational discrete cohomological dimension 1. Let G be a compactly generated t.d.l.c. group satisfying cdQ (G) = 1. Then G is of type FP (cf. Thm. 5.3). As G is not compact (cf. Prop. 3.7(a)), one has (6.17) dH0 (G, Bi(G)) = HomG (Q, Bi(G)) = 0 (cf. (4.21)). Hence G is a rational t.d.l.c. duality group. Suppose that Π = π1 (A, Λ, Ξ) is the fundamental group of a finite graph of profinite groups (A, Λ), i.e., Λ is a finite connected graph. Moreover, Π acts on the tree T = T (A, Λ, Ξ, E + ) without inversion of edges. L We also assume that Π is not compact. Then cdQ (G) = 1 (cf. Prop. 5.6). Hence the simplicial version of the tree T - which we will denote by the same symbol - has the following properties: (1) T is a locally finite simplicial complex of dimension 1; (2) T is a C-discrete simplicial Π-complex; (3) Π has finitely many orbits on the simplicial complex T , i.e., T is of type F∞ ; (4) T is Π-tame; (5) the standard metric δ : \|T \| × \|T \| → R+ 0 gives (\|T \|, δ) the structure of a complete CAT(0)-space. (6) Hc0 (\|T \|, Q) = 0. Hence (1), (2), (3) and (5) imply that Π is a simplicial t.d.l.c. duality group. Moreover, by Theorem 6.7, one has (6.18) × DΠ ≃ Hc1 (\|T \|, Q) ⊗ Q(∆). From (4) and (5) one concludes that T is a C-discrete CAT(0)-simplicial Π-complex, and thus an E C (Π)-space (cf. Prop. 6.8). 6.11. Algebraic groups defined over a non-discrete, non-archimedean local field. Let K be a non-discrete non-archimedean local field5 with residue field F; in particular, F is finite. Let G be a semi-simple, simply-connected algebraic group defined over K, and let G(K) denote the group of K-rational points. In particular, G(K) carries naturally the structure of a t.d.l.c. group. Suppose that G(K) is not compact. Then G(K) has a type preserving action on an affine building (C, S), the Bruhat-Tits building, where (W, S) is an affine Coxeter group. Moreover, d = \|S\| − 1 coincides with the algebraic K-rank of G. For further details see [17], [18], [46] and [50]. The Tits realization Σ(C, S) of the Bruhat-Tits building (C, S) is a discrete simplicial G(K)-complex of dimension d with the following properties: (1) For ω ∈ Σ(C, S) its stabilizer stabG(K) (ω) is a parahoric subgroup and hence compact, i.e., Σ(C, S) is a C-discrete simplicial G(K)-complex. (2) G(K) has finitely many orbits on Σ(C, S), i.e., Σ(C, S) is of type F∞ . (3) As G(K) acts type preserving, Σ(C, S) is G(K)-tame. (4) Σ(C, S) is locally finite. 5By a local field we understand a non-discrete locally compact field. Such a field comes equipped with a valuation v, and, in case that v is non-archimedean, v is also discrete. Hence such a field is either isomorphic to a finite extension of the p-adic numbers Qp or isomorphic to a Laurent power series ring Fq [t−1 , t]] for some finite field Fq . RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 39 (5) There exists a G(K)-invariant metric δ : \|Σ(C, S)\| × \|Σ(C, S)\| → R+ 0 making (\|Σ(C, S)\|, δ) a complete CAT(0)-space (cf. [2, Thm. 11.16]). In particular, \|Σ(C, S)\| is contractable. (6) By a theorem of A. Borel and J-P. Serre, one has Hck (\|Σ(C, S)\|, Q) = 0 for k 6= d and Hcd (\|Σ(C, S)\|, Q) 6= 0 (cf. [10], [12]). From these properties one concludes the following. Theorem 6.9. Let G be a semi-simple, simply-connected algebraic group defined over the non-discrete non-archimedean local field K. (a) The topological realization \|Σ(C, S)\| of the Tits realization of the affine building (C, S) is an E C (G(K))-space. (b) G(K) is a rational t.d.l.c. duality group of dimension rk(G) = dim(Σ(C, S)). (c) For d = cdQ (G(K)) one has × DG(K) ≃ Hcd (\|Σ(C, S)\|, Q). Proof. (a) is a direct consequence of the properties (1), (3), (5) and Proposition 6.8. (b) and (c) follow from the properties (1), (2), (4), (5), (6) and Theorem 6.7. Note that G(K) is in particular a simplicial t.d.l.c. rational duality group and (6.19) dimC (G(K)) = sdimC (G(K)) = cdQ (G(K)). 6.12. Topological Kac-Moody groups. Let F be a finite field, and let G(F) denote the rational points of an almost split Kac-Moody group defined over F (cf. [38, 12.6.3]) with infinite Weyl group (W, S). Such a group acts naturally on a twin building Ξ (cf. [40, 1.A]). Let Σ(Ξ+ ) denote the Davis-Moussang realization of the positive part of Ξ (cf. [24], [36]). In particular, (1) Σ(Ξ+ ) is a locally finite simplicial complex of finite dimension. Moreover, G(F) is acting on Ξ+ and hence on Σ(Ξ+ ), and thus one has a homomorphism of groups π : G(F) → Aut(Σ(Ξ+ )). As Σ(Ξ+ ) is locally finite, Aut(Σ(Ξ+ )) carries naturally the structure of a t.d.l.c. group. The closure Ḡ(F) of im(π) is called the topological Kac-Moody group associate to G(F) (cf. [40, 1.B and 1.C]). By construction, (2) Σ(Ξ+ ) is a C-discrete Ḡ(F)-simplicial complex. Indeed, if F is of positive characteristic p, then stabḠ(F) (ω) is a virtual pro-p group for all ω ∈ Σ(Ξ+ ) (cf. [40, p. 198, Theorem]). Let G(F)◦ denote the subgroup of G(F) which elements are type preserving on Ξ, and let Ḡ(F)◦ denote the closure of π(G(F)◦ ). Then Ḡ(F)◦ is open and of finite index in Ḡ(F). As G(F)◦ is type preserving, one concludes that (3) Σ(Ξ+ ) is Ḡ(F)◦ -tame. Since G(F) (and thus Ḡ(F)) acts δ-2-transitive on chambers of Ξ, one has that (4) Σ(Ξ+ ) is a C-discrete simplicial Ḡ(F)◦ -complex of type F∞ (cf. §6.3). By a remarkable result of M.W. Davis (cf. [24]) the topological realization \|Σ(Ξ+ )\| admits a Ḡ(F)◦ -invariant CAT(0)-metric d : \|Σ(Ξ+ )\| × \|Σ(Ξ+ )\| → R+ 0 (cf. [2, Thm. 12.66]). Hence Proposition 6.8 implies the following. Theorem 6.10. Let F be a finite field, let G(F) denote the rational points of an almost split Kac-Moody group defined over F with infinite Weyl group, and let Ḡ(F) be the associated topological Kac-Moody group. Then the topological realization \|Σ(Ξ+ )\| of the David-Moussang realization Σ(Ξ+ ) is an E C (Ḡ(F)◦ )-space. 40 I. CASTELLANO AND TH. WEIGEL Note that by Davis’ theorem, (5) Σ(Ξ+ ) is a contractable C-discrete simplicial Ḡ(F)-complex. Hence Theorem 6.7 implies that (6.20) d = cdQ (Ḡ(F)) = cdQ (Ḡ(F)◦ ) ≤ dim(Σ(Ξ+ )) < ∞, Ḡ(F) is of type FP, and since Ḡ(F) is unimodular (cf. [39, p. 811, Thm.(i)]), × (6.21) DḠ(F) ≃ Hcd (\|Σ(Ξ+ )\|, Q). Moreover, by (6.15), (6.22) cdQ (Ḡ(F)) = max{ k ≥ 0 \| Hck (\|Σ(Ξ+ )\|, Q) 6= 0 }. However, Ḡ(F) may or may not be a rational t.d.l.c. duality group (cf. Remark 6.12 and 6.13). The cohomology with compact support of \|Σ(Ξ+ )\| with coefficients in Z was computed in [26]. In more detail, a bT (Ξ+ ), Hck (\|Σ(Ξ+ )\|, Z) = (6.23) H k (K, K S−T ) ⊗ A T ∈S and a similar formula holds for the Davis realization of the Coxeter complex Σ(C) associated to (W, S) (cf. [25]), i.e., a bT (C), Hck (\|Σ(C)\|, Z) = (6.24) H k (K, K S−T ) ⊗ A T ∈S Here S denotes the set of non-trivial spherical subsets of S, and K is a simplicial complex associated to (W, S) build up from the spherical subsets of S. A more bT (Ξ+ ) and A bT (C) yields the following. detailed analysis of the coeffcient modules A Theorem 6.11. Let F be a finite field, let G(F) denote the rational points of an almost split Kac-Moody group defined over F with infinite Weyl group (W, S), and let Ḡ(F) be the associated topological Kac-Moody group. Then (6.25) cdQ (Ḡ(F)◦ ) = cdQ (W ). Moreover, Ḡ(F)◦ (and thus Ḡ(F)) is a rational t.d.l.c. duality group if, and only if, W is a rational discrete duality group. bT (Ξ+ ) and A bT (C) are free Z-modules. Moreover, for T ∈ S one Proof. Note that A bT (C) 6= 0 and A bT (Ξ+ ) 6= 0 (cf. [26, §6, p. 570, Remark and Definition 7.4]). has A As the abelian groups H k (K, K S−T ) are finitely generated, the universal coefficient theorem implies that cdQ (W ) = max{ k ≥ 0 \| ∃ T ∈ S : H k (K, K S−T ) is not torsion } = max{ k ≥ 0 \| Hck (\|Σ(Ξ+ )\|, Q) 6= 0 } (6.26) = cdQ (Ḡ(F)) (cf. Thm. 6.7). Let d = cdQ (W ) = cdQ (Ḡ(F)). Then W is a rational duality group if, and only if, H k (K, K ST ) is a torsion group for all T ∈ S and all k 6= d. By (6.24), this is equivalent to Hck (\|Σ(Ξ+ )\|, Q) = 0 for all k 6= d. Thus Theorem 6.7 yields the claim. Remark 6.12. Let F be a finite field, let G(F) denote the rational points of a split Kac-Moody group defined over F, and let Ḡ(F) be the associated topological KacMoody group. Suppose further that RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 41 (1) the associated generalized Cartan matrix is symmetrizable, and that (2) the associated Weyl group (W, S) is hyperbolic (cf. [28, §6.8]). Condition (2) implies that W is a lattice in the real Lie group O(n, 1), \|S\| = n + 1 (cf. [28, p. 140, and the references therein]). Moreover, as W is the Weyl group of a (split) Kac-Moody Lie algebra with property (1), the Tits representation of the Weyl group W is integral (cf. [19, §16.2]). Hence W is an arithmetic lattice in O(n, 1). Thus by A. Borel’s and J-P. Serre’s theorem (cf. [11]), W is a virtual duality group, and thus, in particular, a rational duality group. Hence Theorem 6.11 implies that Ḡ(F) is a rational discrete t.d.l.c. duality group. Remark 6.13. Consider the Coxeter group (W, S) associated to the Coxeter diagram •✤✷ ☞☞✤ ✷✷✷ ☞ ✷ ☞ ☞☞ ✤ ∞ ✷✷✷ ☞ ✤ ☞ ✷✷ ✤ ☞☞ ✷✷ ☞ ☞ ✷✷ ✤▼ ☞ • (6.27) q ✷ ☞ ▼ ▼∞ ✷✷ ☞☞ ∞q q ▼ ☞ ✷ q ☞q ▼ ✷✷ ▼• ☞☞ •q ` e2 ) W (A1 ). In particular, cdQ (W ) = 2. By Stallings’ decomposiThen W ≃ W (A tion theorem (cf. [44]), H 1 (W, Q[W ]) 6= 0. Thus W is not a rational duality group. Let G(F) denote the rational points of a split Kac-Moody group defined over F with associated Coxeter group is (W, S), and let Ḡ(F) be the associated topological Kac-Moody group. Then Theorem 6.11 implies, that Ḡ(F) is of rational cohomological dimension 2, but that Ḡ(F) is not a rational t.d.l.c. duality group, i.e., dH1 (Ḡ(F), Bi(Ḡ(F)) 6= 0. The t.d.l.c. version of Stalling’s decomposition theorem established in [21] implies that Ḡ(F) can be decomposed non-trivially as a free product with amalgamation in a compact open subgroup (where one factor is a compact open subgroup as well), or as an HNN-extension with amalgamation in a compact open subgroup. It would be interesting to understand how this decomposition is related to the root datum of G(F). We close this section with a question which is motivated by (6.19). Question 5. For which topological Kac-Moody groups Ḡ(F), F a finite field, does dimC (Ḡ(F)) = cdQ (Ḡ(F)) hold (cf. §6.3)? 7. The Euler-Poincaré characteristic of a unimodular t.d.l.c. group of type FP In this subsection we indicate how one can associate to any unimodular t.d.l.c. group G of type FP an Euler-Poincaré characteristic (7.1) χ(G) ∈ h(G) = Q · µO , where O is a compact open subgroup, and µO denotes the left invariant Haar measure on G satisfying µO (O) = 1. If h ∈ Q+ · µO we simply write h > 0; similarly h < 0 shall indicate that h ∈ Q− · µO . In some particular cases discussed in §6.10 and §6.11 it will be possible to calculate the Euler-Poincaré characteristic χ(G) explicitly. It is quite likely, that similar calculation can be done also for the examples decribed in §6.12. However, one of the 42 I. CASTELLANO AND TH. WEIGEL most interesting question from our perspective which was our original motivation to introduce and study this notion we were not able to answer. Question 6. Let G be a compactly generated t.d.l.c. group satisfying cdQ (G) = 1. Does this imply that χ(G) ≤ 0? An affirmative answer of Question 6 would resolve the problem of accessibility for compactly generated t.d.l.c. groups of rational discrete cohomological dimension 1 in analogy to the discrete case (cf. [30]). 7.1. The Hattori-Stallings rank of a finitely generated projective rational discrete Q[G]-module. Let G be a t.d.l.c. group, and let P be a finitely generated projective rational discrete Q[G]-module. The evaluation map (7.2) evP : HomG (P, Bi(G)) ⊗Q P → Bi(G), φ ∈ HomG (P, Bi(G)), evP (φ ⊗ p) = φ(p), p ∈ P, induces a mapping evP : HomG (P, Bi(G)) ⊗G P → Bi(G) such that the diagram (7.3) HomG (P, Bi(G)) ⊗Q P / HomG (P, Bi(G)) ⊗G P evP evP / Bi(G) Bi(G) commutes, where the horizontal maps are the canonical ones and Bi(G) is as defined in §4.8. Then - provided that G is unimodular - one has a map (7.4) ρP : HomG (P, P ) −1 ζP,P / HomG (P, Bi(G)) ⊗G P evP / Bi(G) tr / h(G). (cf. Prop. 4.6, Prop. 4.14). The value rP = ρP (idP ) ∈ h(G) will be called the Hattori-Stallings rank of P . Proposition 7.1. Let G be a unimodular t.d.l.c. group, and let O ⊆ G be a compact open subgroup. Then rQ[G/O] = µO . Proof. Let ηO : Q[G/O] → Bi(G) be the canonical embedding (cf. (4.31)). Then jO (ηO ) = O ∈ O\G (cf. (4.34)). Hence, (7.5) (jO ⊗G idQ[G/O] )(ηO ⊗ O) = O ⊗G O and O φ−1 Q[G/O],O (O ⊗G O) = O ∈ Q[G/O] . (7.6) Under the Eckmann-Shapiro isomorphism α: Q[G/O]O → HomG (Q[G/O], Q[G/O]) the element O is mapped to idQ[G/O] . Hence, by (4.35), (7.7) ζQ[G/O],Q[G/O](ηO ⊗G O) = idQ[G/O] , −1 (idQ[G/O] )) = O, where O denotes the canonical image and evQ[G/O] (ζQ[G/O],Q[G/O] of O in Bi(G). Thus, by the definition of tr : Bi(G) → h(G) (cf. (4.77)), (7.8) and this yields the claim. rQ[G/O] = tr(O) = µO RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 43 Remark 7.2. Let G be a unimodular t.d.l.c. group, and let P ∈ ob(Q[G] dis) be finitely generated and projective. In [22] it is shown, that rP ≥ 0, and rP = 0 if, and only if, P = 0. 7.2. The Euler-Poincaré characteristic. Let G be a unimodular t.d.l.c. group, and let P1 , P2 ∈ ob(Q[G] dis) be finitely generated and projective. Then, by (7.4), one has ρP1 ⊕P2 (idP1 ⊕P2 ) = ρP1 (idP1 ) + ρP2 (idP2 ), i.e., rP1 ⊕P2 = rP1 + rP2 . (7.9) Let (P• , δ• ) be a finite projective resolution of Q in Q[G] dis, i.e., there exists N ≥ 0 such that Pk = 0 for k > N , and Pk is finitely generated for all k ≥ 0. From the identity (7.10) one concludes that the value P (7.10) χ(G) = k≥0 (−1)k · rPk ∈ h(G) is independent from the choice of the projective resolution (P• , δ• ) (cf. [45, Proposition 4.1]) for a similar argument). We will call χ(G) the Euler-Poincaré characteristic of G. 7.3. Examples. 7.3.1. Compact t.d.l.c. groups. Let O be a compact t.d.l.c. group. Then Proposition 7.1 implies that χ(G) = rQ = µO . 7.3.2. Fundamental groups of finite graphs of profinite groups. Let (A, Λ) be a finite graph of profinite groups. Assume further that Π = π1 (A, Λ, x0 ) is unimodular6. Then, by Proposition 7.1, X X µAv − µAe . (7.11) χ(Π) = v∈V(Λ) e∈E + (Λ) In particular, if (A, Λ) is a finite graph of finite groups one obtains X X 1 1 (7.12) χ(Π) = · µ = χΠ · µ{1} , − \|Av \| \|Ae \| + v∈V(Λ) e∈E (Λ) where χΠ denotes the Euler characteristic of the discrete group Π (cf. [41, §II.2.6, Ex. 3]). One has the following: Proposition 7.3. Let (A, Λ) be a finite graph of profinite groups, such that Π = π1 (A, Λ, x0 ) is unimodular and non-compact. Then χ(Π) ≤ 0. Proof. Suppose that one of the open embeddings αe : Ae → At(e) is surjective. Then removing the edge e form Λ, idetifying t(e) with o(e) and taking the induced graph of profinite groups (A′ , Λ′ ) does not change the fundamental group, i.e., one has a topological isomorphism π1 (A, Λ, x0 ) ≃ π1 (A′ , Λ′ , x′0 ). Thus we may assume that the finite graph of profinite groups has the property that none of the injections αe : Ae → At(e) is surjective. Since Π is not compact, Λ cannot be a single vertex. Thus, as \|At(e) : αe (Ae )\| ≥ 2 one has µAe ≥ µAt(e) + µAo(e) for any edge e ∈ E(Λ). Choosing a maximal subtree of Λ then yields the claim. 6For a given finite graph of profinite groups it is possible to decide when Π is unimodular. However, the complexity of this problem will depend on the rank of H 1 (\|Λ\|, Z). E.g., if Λ is a finite tree, then Π will be unimodular. 44 I. CASTELLANO AND TH. WEIGEL 7.3.3. The automorphism group of a locally finite regular tree. Let Td+1 be a (d+1)regular tree, 1 ≤ d < ∞, and let G = Aut(Td+1 )◦ denote the group of automorphism not inverting the orientation of edges. Then, as G is the fundamental group of a finite graph of profinite groups based on a tree with 2 vertices, G is unimodular. One concludes from Bass-Serre theory and §7.3.2 that χ(Aut(Td+1 )◦ ) = (7.13) 1−d · µG e , 1+d where Ge is the stabilizer of an edge. Moreover, χ(Aut(Td+1 )◦ ) < 0. 7.3.4. Chevalley groups over non-discrete non-archimedean local fields. Let X be a simply-connected simple Chevalley group scheme, and let K be a non-archimedean local field with finite residue field F. Put q = \|F\|. Let W be the finite (or spherical) f denote the associated affine Weyl group Weyl group associated to X, and let W f . Let n = associated to W . We also fix a Coxeter generating system ∆ of W f rk(W ) = rk(W ) − 1 = \|∆\| − 1 denote the rank of W as Coxeter group. It is wellknown that X(K) modulo its center is simple. In particular, X(K) is a unimodular t.d.l.c. group. Let Σ be the affine building associated to X(K) (cf. §6.11). The stablizer Iw = stabX(K) (ω) of a simplex ω ∈ Σn of maximal dimension is also called a Iwahori subgroup of X(K). Let Σ(ω) = { ̟ ∈ Σ \| ̟ ⊆ ω } = P(ω) \ {∅}. For ̟ ∈ Σ(ω) let P̟ = stabX(K) (̟) denote the parahoric subgroup associated to ̟, i.e., Pω = Iw. Then X (−1)\|̟\| · µP̟ , (7.14) χ(X(K)) = ̟∈Σ(ω) where \|̟\| is the degree of ̟, i.e., ̟ ∈ Σ\|̟\| (cf. §6.11). Any parahoric subgroup P̟ , ̟ ∈ Σ(ω), corresponds to a unique proper subset ∆(̟) of ∆ of cardinality \|ω\| − \|̟\|, i.e., ∆(ω) = ∅. Moreover, \|P̟ : Iw\| = pW (∆(̟)) (q), (7.15) where pW (∆(̟)) (t) ∈ Z[t] is the Poincaré polynomial associated to the finite Weyl group W (∆(̟)). Thus, by [28, §5.12, Proposition], one obtains that χ(X(K)) = X ̟∈Σ(ω) (7.16) (−1)\|̟\| · µIw \|P̟ : Iw\| = (−1)n+1 · X (−1)\|I\|−1 (−1)n+1 · µIw = · µIw . pW (I) (q) pW f (q) I⊆∆ I6=∆ Q mi ), where mi = di − 1 By R. Bott’s theorem, one has pW f (t) = pW (t)/ 1≤i≤n (1 − t are the exponents of W (cf. [28, §8.9, Theorem]). Thus χ(X(K)) can be rewritten as (7.17) χ(X(K)) = − Y 1 · (q mi − 1) · µIw . pW (q) 1≤i≤n In particular, χ(X(K)) < 0. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 45 Remark 7.4. One may speculate whether in a situation which is controlled by geometry the Euler-Poincaré characteristic should be related to a meromorphic ζfunction evaluated in −1. There are some phenomenon supporting this idea which arise even in a completely different context (cf. [16]). Indeed, using Hecke algebras one may define a formal Dirichlet series ζG,O (s) for certain t.d.l.c. groups G and any compact open subgroup O of such a group G. It turns out that for the example discussed in §7.3.4 one obtains χ(X(K)) = (−1)n+1 · ζX(K),Iw (−1)−1 · µIw (cf. [22]). Nevertheless, further investigations seem necessary in order to shed light on such a possible connection. We close the discussion of the Euler-Poincaré characteristic with a question which is motivated by the examples discussed in §7.3.2 and §7.3.4. Question 7. Suppose that G is a t.d.l.c. group admitting a C-discrete simplicial G-complex Σ such that (1) (2) (3) (4) (5) dim(Σ) = cdQ (G) < ∞; Σ is of type F∞ ; Σ is locally finite; Σ is G-tame; \|Σ\| admits a G-invariant metric δ : \|Σ\| × \|Σ\| → R+ 0 making (\|Σ\|, δ) a complete CAT(0)-space (cf. §6.9). What further condition ensures that χ(G) ≤ 0? The most trivial examples (cf. §7.3.1) show that the conditions (1)-(5) are not sufficient (for G compact Σ can be chosen to consist of 1 point). Appendix A. Simplicial complexes A.1. Simplicial complexes. A simplicial complex Σ is a non-empty set of nontrivial finite subsets of a set X with the property that if B ∈ Σ and F A ⊆ B then A ∈ Σ. Any simplicial complex Σ carries a canonical grading Σ = q≥0 Σq , where (A.1) Σq = { A ∈ Σ \| card(A) = q + 1 }. Example A.1. Let X be a set, and let Σ[X] ⊆ P(X) \ {∅} denote the set of all non-trivial finite subsets of X. Then Σ[X] is a simplicial complex - the simplicial complex generated by X. Example A.2. Let Γ = (V(Γ), E(Γ)) be a combinatorial graph (cf. §5.4). Then Σ(Γ) given by Σ0 (Γ) = V(Γ) and Σ1 (Γ) = { {o(e), t(e)} \| e ∈ E(Γ) } is a simplicial complex - the simplicial complex associated with the graph Γ. In a simplicial complex Σ and n ≥ 0 the subset (A.2) Σ(n) = { A ∈ Σ \| card(A) ≤ n + 1 } is again simplicial subcomplex of Σ - the n-skeleton of Σ. A subset Y ⊆ Σ0 defines a simplicial subcomplex (A.3) Σ(Y ) = { C ∈ Σ \| C ⊆ Y }. For a simplicial complex Σ one defines the dimension by (A.4) dim(Σ) = min({ q ∈ N0 \| Σq+1 = ∅ } ∪ {∞}). 46 I. CASTELLANO AND TH. WEIGEL In particular, dim(Σ(X)) = card(X) − 1. A simplicial complex of dimension 0 is just a set, while simplicial complexes of dimension 1 are unoriented graphs (cf. Ex. A.2). For A ∈ Σq one defines the set of neighbours NΣ (A) of A by NΣ (A) = { B ∈ Σq+1 \| A ⊂ B }. (A.5) Moreover, Σ is called locally finite if for all A ∈ Σq one has card(NΣ (A)) < ∞. A.2. The topological realization of a simplicial complex. With any simplicial complex Σ ⊆ P(X) \ {∅} one can associate a topological space \|Σ\| - the topological realization of Σ - which is a CW-complex. The space \|Σ\| can be constructed as follows: Let V = R[X] denote the free R-vector space over the set X, and for A ∈ Σq , A = { a0 , . . . , aq }, put Pq Pq (A.6) \|A\| = { j=0 tj · aj \| 0 ≤ tj ≤ 1, j=0 tj = 1 } ⊂ R[X]. S Then \|Σ\| = A∈Σ \|A\| ⊂ R[X], and \|Σ\| carries the weak topology with respect to the embeddings iA : \|A\| → \|Σ\|, where \|A\| ⊂ R[A] carries the induced topology (cf. [43, §3.1]). By construction, one has the following property. Fact A.3. Let Σ be a simplical complex, and let z ∈ \|Σ\|. Then there exists a unique simplex A ∈ Σ such that z is an interior point of \|Σ(A)\| ⊆ \|Σ\|. The topological space \|Σ\| has the following well-known property. Fact A.4. Let Σ be a simplicial complex, and let C be a compact subspace of \|Σ\|. Then there exists a finite subset Y ⊆ Σ1 such that C ⊆ \|Σ(Y )\|. Proof. (cf. [27, p. 520, Prop. A.1]). The just mentioned property has the following consequence. Fact A.5. Let X be a non empty set. Then \|Σ[X]\| is contractable. Proof. Put Σ = Σ[X]. As \|Σ\| is a connected CW-complex, one concludes from Whitehead’s theorem (cf. [27,SThm. 4.5]) that it suffices to show that πi (\|Σ\|, x0 ) = 1 for all i ≥ 1. Note that Σ = A∈Px Σ(A), where 0 (A.7) Px0 = { A ⊆ X \| x0 ∈ A, card(A) < ∞ }. By Fact A.4, for any compact subset C of \|Σ\|, there exists a finite set A ∈ Px0 such that C ⊆ \|Σ(A)\|. Hence πi (\|Σ\|, x0 ) = limA∈P πi (\|Σ(A)\|, x0 ). As \|Σ(A)\| coincides −→ x0 with the standard (card(A) − 1)-simplex, one has πi (\|Σ(A)\|, x0 ) = 1. A.3. The chain complex of a simplicial complex. Let Σ ⊆ P(X) \ {∅} be a simplicial complex. For q ≥ 0 put e q = { x0 ∧ · · · ∧ xq \| {x0 , . . . , xq } ∈ Σq } ⊂ Λq+1 (Q[X]), (A.8) Σ where Λ• (Q[X]) denotes the exterior algebra of the free Q-vector space over the set X, and define e q ) ⊆ Λq+1 (Q[X]). (A.9) Cq (Σ) = span (Σ Q Then (C• (Σ), ∂• ), where (A.10) ∂q (x0 ∧ · · · ∧ xq ) = P j 0≤j≤q (−1) x0 ∧ · · · ∧ xj−1 ∧ xj+1 ∧ · · · ∧ xq is a chain complex of Q-vector spaces - the rational chain complex of the simplicial complex Σ (cf. [43, §4.1]). It has the following well-known property (cf. [43, §4.3, Thm. 8]). RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 47 Fact A.6. Let Σ be a simplicial complex. Then one has a canonical isomorphism (A.11) H• (C• (Σ), ∂• ) ≃ H• (\|Σ\|, Q). A.4. Cohomology with compact support. Let Σ ⊆ P(X) \ {∅} be a locally finite simplicial complex. Put X ∗ = { x∗ \| x ∈ X }, and think of X ∗ as a subset of the Q-vector space Cc (X, Q), the functions from X to Q with finite support, i.e., for x, y ∈ X one has x∗ (y) = δx,y . Here δ.,. denotes Kronecker’s function. For q ≥ 0 put (A.12) and define (A.13) b q = { x∗ ∧ · · · ∧ x∗ \| {x0 , . . . , xq } ∈ Σq } ⊂ Λq (Q[X ∗ ]), Σ 0 q b q ) ⊆ Λq (Q[X ∗ ]). Ccq (Σ) = spanQ (Σ By definition, for A ∈ Σq the set (A.14) IΣ (A) = { x ∈ Σ1 \ A \| A ∪ {x} ∈ Σq+1 } is finite. Then (Cc• (Σ), ð• ), where (A.15) ðq (x∗0 ∧ · · · ∧ x∗q ) = P z∈IΣ ({x0 ,...,xq }) z ∧ x∗0 ∧ · · · ∧ x∗q b q , is a cochain complex. It has the following well-known property x∗0 ∧ · · · ∧ x∗q ∈ Σ (cf. [27, p. 242ff]). Fact A.7. Let Σ be a locally finite simplicial complex. Then one has a canonical isomorphism H • (Cc• (Σ), ð• ) ≃ Hc• (\|Σ\|, Q), where Hc• ( , Q) denotes cohomology with compact support with coefficients in Q. A.5. Signed sets. A set X together with a map ¯ : X → X satisfying x̄¯ = x and x̄ 6= x for all x ∈ X will be called a signed set. A map of signed sets φ : X → Y is a map satisfying φ(x̄) = φ(x) for all x ∈ X. Every signed set X defines a Q-vector space (A.16) Q[X] = Q[X]/spanQ { x + x̄ \| x ∈ X }. For a signed set X let X ∗ = { x∗ \| x ∈ X } denote the signed set satisfying x̄∗ = x∗ . Then we may consider Q[X ∗ ] as a Q-subspace of Q[X]∗ = HomQ (Q[X], Q), i.e.,   if y = x, 1 ∗ (A.17) x (y) = −1 if y = x̄,   0 if y 6∈ {x, x̄}. Any map of signed sets φ : X → Y defines a Q-linear map φQ : Q[X] → Q[Y ], but it does not necessarily possess an adjoint map φ∗Q : Q[Y ∗ ] → Q[X ∗ ]. However, if φ is proper, i.e., φ has finite fibres, then there exists a unique map φ∗Q satisfying (A.18) hv ∗ , φQ (u)iY = hφ∗Q (v), uiX , u ∈ Q[X], v ∈ Q[Y ∗ ], where h., .iX and h., .iY denote the evaluation mappings, respectively. Let X + ⊂ X and Y + ⊂ Y be a set of representative for the Z/2Z-orbits on X and Y , respectively. For a map ψ : Q[X] → P Q[Y ] and x ∈ Q[X] - the canonical image of x ∈ X + in Q[X] - one has ψ(x) = y∈Y + λy (x) · y for λy (x) ∈ Q. Put (A.19) supp(ψ, x) = { y ∈ Y + \| λy (x) 6= 0 }. 48 I. CASTELLANO AND TH. WEIGEL We call the map ψ : Q[X] → Q[Y ] proper, if for all y ∈ Y + the set (A.20) csupp(ψ, y) = { x ∈ X + \| y ∈ supp(φ, x) } is finite. 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Generalized Preconditioning and Network Flow Problems Jonah Sherman∗ University of California, Berkeley June 28, 2016(preliminary draft) arXiv:1606.07425v2 [cs.DS] 27 Jun 2016 Abstract We consider approximation algorithms for the problem of finding x of minimal norm kxk satisfying a linear system Ax = b, where the norm k · k is arbitrary and generally non-Euclidean. We show a simple general technique for composing solvers, converting iterative solvers with residual error kAx−bk ≤ t−Ω(1) into solvers with residual error exp(−Ω(t)), at the cost of an increase in kxk, by recursively invoking the solver on the residual problem b̃ = b − Ax. Convergence of the composed solvers depends strongly on a generalization of the classical condition number to general norms, reducing the task of designing algorithms for many such problems to that of designing a generalized preconditioner for A. The new ideas significantly generalize those introduced by the author’s earlier work on maximum flow, making them more widely applicable. As an application of the new technique, we present a nearly-linear time approximation algorithm for uncapacitated minimum-cost flow on undirected graphs. Given an undirected graph with m edges labelled with costs, and n vertices labelled with demands, the algorithm takes −2 m1+o(1) -time and outputs a flow routing the demands with total cost at most (1 + ) times larger than minimal, along with a dual solution proving near-optimality. The generalized preconditioner is obtained by embedding the cost metric into `1 , and then considering a simple hierarchical routing scheme in `1 where demands initially supported on a dense lattice are pulled from a sparser lattice by randomly rounding unaligned coordinates to their aligned neighbors. Analysis of the generalized condition number for the corresponding preconditioner follows that of the classical multigrid algorithm for lattice Laplacian systems. 1 Introduction A fundamental problem in optimization theory is that of finding solutions x of minimum-norm kxk to rectangular linear systems Ax = b. Such solvers have extensive applications, due to the fact that many practical optimization problems reduce to minimum norm problems. When the norm is Euclidean, classical iterative solvers such as steepest descent and conjugate gradient methods produce approximately optimal solutions with residual error kAx − bk exponentially small in iteration count, The rate of exponential convergence depends strongly on the condition number of the square matrix AA∗ . Therefore, the algorithm design process for such problems typically consists entirely of efficiently constructing preconditioners: easily computable left-cancellable operators P, for which the transformed problem PA = Pb is well-conditioned so iterative methods converge rapidly. In a seminal work, Spielman and Teng[17] present a nearly-linear-time algorithm to construct preconditioners with condition number polylogarithmic in problem dimension for a large class of operators A, including Laplacian systems and others arising from discretization of elliptic PDEs. There are many fundamental optimization problems that can be expressed as minimum-norm problems with respect to more general, non-Euclidean norms, including some statistical inference problems and network flow problems on graphs. In particular, the fundamental maximum-flow and uncapacitated minimum-cost flow problems reduce respectively to `∞ and `1 minimum-norm problems in the undirected case. For such problems, there is presently no known black-box iterative solver analagous to those for `2 that converge exponentially with rate independent of problem dimension. Therefore, one is forced to choose between interior point methods to obtain zero or exponentially small error[4], at the cost of iteration count depending strongly on problem dimension, or alternative methods that have error only polynomially small with respect to iterations[15, 9, 14]. ∗ Research supported by NSF Grant CCF-1410022 An important idea implicitly used by classical iterative `2 solvers is residual recursion: reducing the error of an approximate solution x by recursively applying the solver to b̃ = b − Ax. Such recursion is implicit in those solvers due to the fact that in `2 , the map b 7→ xopt is linear, with xopt = (AA∗ )+ b. Therefore, the classical methods are effectively recursing on every iteration, with no distinction between iterations and recursive solves. In general norms, an xopt may not be unique, and there is generally no linear operator mapping b to some xopt . In this paper, we make two primary contibutions. Our first contribution is to extend analysis of residual recursion to arbitrary norms. We present a general and modular framework for efficiently solving minimumnorm problems by composing simple well-known base solvers. The main tool is the composition lemma, describing the approximation parameters of a solver composed via residual recursion of two black-box solvers. The resulting parameters depend on a generalization of the condition number to arbitrary norms; therefore, much like `2 , the algorithm design process is reduced to constructing good generalized preconditioners. Our second contribution is to present, as a practical application of those tools, a nearly-linear time approximation algorithm for the uncapacitated minimum-cost flow problem on undirected graphs. Having established the former framework, the latter algorithm is entirely specified by the construction of a generalized preconditioner, and analyzed by bounding its generalized condition number. Our framework builds upon earlier work[16], where we leveraged ideas introduced by Spielman and Teng to obtain nearly-linear time approximation algorithms for undirected maximum flow. We have since realized that some of the techniques applied to extend `2 -flows to `∞ -flows are in no way specific to flows, and indeed the composition framework presented here is a generalization of those ideas to non-Euclidean minimum-norm problems. Moreover, a significantly simpler max-flow algorithm may be recovered using black-box solvers and general composition as presented here, requiring only the truly flow-specific part of the earlier work: the congestion approximator, now understood to be a specific instance of a more widely-applicable generalized preconditioner. We proceed to discuss those contributions separately in more detail, and outline the corresponding sections of the paper. We also discuss some related work. 1.1 Minimum Norm Problems In section 2, we define the minimum norm problem more precisely, and introduce a useful bicriteria notion of approximation we call (α, β)-solutions. In that notion, α quantifies how much kxk relatively exceeds kxopt k, and β quantifies the residual error kAx − bk, with an appropriate relative scale factor. We may succinctly describe some frequently used algorithms as (α, β)-solvers; we discuss some examples in section 4, including steepest descent and conjugate gradient for `2 , and multiplicative weights for `1 and `∞ . In section 3, we present the composition lemma, which shows that by composing a (α1 , β1 )-solver with a (α2 , β2 )-solver in a black-box manner, we obtain a (α3 , β3 )-solver with different parameter tradeoffs. The composed algorithm’s parameters (α3 , β3 ) depends crucially on a generalization of the classical condition number to rectangular matrices in general norms; Demko[7] provides essentially the exact such generalization needed, so we briefly recall that definition before stating and proving the composition lemma. The proof of the composition lemma is not difficult; it follows quite easily once the right definitions of (α, β)-solutions and a generalized condition number have been established. Nevertheless, the modularity of the composition lemma proves to quite useful. By recursively composing a solver with itself t times, we may decrease β exponentially with t, at the cost of a single fixed increase in α. By composing solvers with different parameters, better parameters may be obtained than any single solver alone. A particularly useful example uses a (M, 0)-solver, where M is uselessly-large on its own, to terminate a chain of t solvers and obtain zero error. As the final step in the chain, we get all of the benefits (zero error), and almost none of the costs, as it contributes M 2−t instead of M to the final solver. By composing the existing solvers discussed in section 4, we obtain solvers for `1 and `∞ problems with residual error exponentially small in t. We state the final composed algorithms parameters. As a result, the algorithm design problem for such problems is reduced to that of designing a generalized preconditioner for A. We briefly discuss such preconditioning in section 5. 1.2 Minimum Cost Flow The second part of this paper applies generalized preconditioning to solve a fundamental problem in network optimization. In the uncapacitated minimum-cost flow problem on undirected graphs, we are given a connected graph G with m edges, each annotated with a cost, and a specified demand at each vertex. The 2 problem is to find an edge flow with vertex divergences equal to the specified demands, of minimal total cost. Our main result is a randomized algorithm that, given such a problem, takes −2 m1+o(1) time and outputs a flow meeting the demands with cost at most (1 + )-times that of optimal. We define various flow-related terms and give a more precise statement of the problem and our result in section 6. We describe and analyze the algorithm in section 7. Having built up our general tools in earlier sections, our task is reduced to the design and analysis of a generalized preconditioner for min-cost flow. We begin by interpreting the edge costs as lengths inducing a metric on the graph, and then apply Bourgain’s smalldistortion embedding[3] into `1 . That is, by paying a small distortion factor in our final condition number, we may focus entirely on designing a preconditioner for min-cost flow in `1 space. The construction of our precondtitioner is based on an extremely simple hierarchical routing scheme, inspired by a combination of the multigrid algorithm for grid Laplacians, and the Barnes-Hut algorithm for n-body simulation[2]. The routing scheme proceeds on a sequence of increasingly dense lattices V0 , V1 , . . . in `1 , where Vt are the lattice points with spacing 2−t , and the input demands are specified on the densest level VT . Starting at level t = T , the routing scheme sequentially eliminates the demand supported on Vt \ Vt−1 by pulling the demand from level t − 1 using a random-rounding based routing: each vertex x ∈ Vt picks a random nearest neighbor in Vt−1 , corresponding to randomly rounding its unaligned coordinates, and then pulls a fractional part of its demand from that neighbor via any shortest path. Afterwards, all demands on Vt \ Vt−1 are met, leaving a reduced problem with demands supported on Vt−1 . The scheme then recurses on the reduced problem. At level 0, the demands are supported on the corners of a hypercube, for which the simple routing scheme of pulling all demand from a uniformly random corner performs well-enough. Having described the routing scheme, we need not actually carry it out. Instead, our preconditioner crudely estimates the cost of that routing. Furthermore, while the routing has been described on an infinite lattice, we’ll observe that if the demands are only initially supported on a small set of n vertices, the support remains small throughout. 1.3 Related Work Spielman and Teng present a nearly-linear time algorithm for preconditioning and solving symmetric diagonally dominant matrices, including those of graph Laplacians[17]. The ideas introduced in that work have led to breakthroughs in various network cut and flow algorithms for the case of undirected graphs. Christiano et. al. apply the solver as a black-box to approximately solve the maximum flow problem in Θ(m4/3 ) time. Madry[13] opens the box and uses the ideas directly to give a family of algorithms for approximating cut-problems, including a mo(1) -approximation in m1+o(1) time. Both the present author[16] and Kelner et. al.[11] combine madry’s ideas with `∞ optimization methods to obtain (1 − )-approxmations to maximum flow in m1+o(1) −O(1) time. While those two algorithms use similar ideas, the actual path followed to achieve the result is rather different. Kelner et. al. construct an oblivious routing scheme that is no(1) -competetive, and then show how to use it to obtain a flow. The algorithm maintains a demand-respecting flow at all times, and minimizes a potential function measuring edge congestion. In contrast, our earlier algorithm short-cuts the need to explicitly construct the routing scheme by using Madry’s construction directly, maintaining a flow that is neither demand nor capacity respecting, aiming to minimize a certain potential function that measures both the congestion and the demand error. For min-cost flow, earlier work considers the more general directed, capacitated case, with integer capacities in {1, . . . , U }. The Θ(nm log log U )-time double-scaling algorithm of Ahuja, Goldberg, Orlin, and Tarjan[1] remained the fastest algorithm for solving min-cost flow for 25 years between its publication and the Laplacian breakthrough. Shortly after that breakthrough, Daitch and Spielman[5] showed how to use the Laplacian solver with interior-point methods to obtain an Θ(m3/2 log2 U )-time algorithm. Lee and Sidford √ further reduce this to Θ(m n logO(1) U ) by a general improvement in interior point methods[12]. We are not aware of prior work 2 Approximate Solutions Let X , Y be finite dimensional vector spaces, where X is also a Banach space, and let A ∈ Lin(X , Y) be fixed throughout this section. We consider the problem of finding a minimal norm pre-image of a specified b in the image of A; that is, finding x ∈ X with Ax = b and kxkX minimal. Let xopt be an exact solution, with Axopt = b, and kxopt kX minimal. There are multiple notions of 3 approximate solutions for this problem. The most immediate is x ∈ X with Ax = b and kxk ≤α kxopt k (1) . We call such x an (α, 0)-solution; we shall be interested in finding (1 + , 0)-solutions for small . A weaker notion of approximation is obtained by further relaxing Ax = b to Ax u b. Quantifying that requires more structure on Y, so let us further assume Y to also be a Banach space. In that case, we say x is an (α, β)-solution if equation 1 holds and kAx − bk ≤β (2) kAkkxopt k The practical utility of such solutions depends on the application. However, the weaker notion has the distinct advantage of being approachable by a larger family of algorithms, such as penalty and dual methods, by avoiding equality constraints. We discuss such existing algorithms in section 4, but mention that they typically yield (1, )-solutions after some number of iterations. For `2 , the iteration dependency on is O(log(1/)), while for some more general norms it is −O(1) . 3 Recursive Composition and Generalized Condition Numbers We now consider how to trade an increase in α for a decrease in β. Residual recursion suggests a natural strategy: after finding an (α, β)-solution, recurse on b̃ = b − Ax. More precisely, we define the composition of two algorithms as follows. Definition 3.1. Let Fi be an (αi , βi )-algorithm for A, for i ∈ {1, 2}. The composition F2 ◦ F1 takes input b, and first runs F1 on b to obtain x. Next, setting b̃ = b − Ax, it runs F2 on input b̃ to obtain x̃. Finally, it outputs x + x̃. Success of composition depends on kb̃kY being small implying b̃ has a small-norm pre-image. The extent to which that is true is quantified by the condition number of A. The condition number in `2 may be defined several ways, resulting in the same quantity. When generalized to arbitrary norms, those definitions differ. We recall two natural definitions, following Demko[7]. Definition 3.2. The non-linear condition number of A : X → Y is kAkX →Y kxkX κ̃X →Y (A) = min : Ax 6= 0 kAxkY The linear condition number is defined by κX →Y (A) = min {kAkX →Y kGkY→X : G ∈ Lin(Y, X ) : AGA = A} Of course, κ̃ ≤ κ. Having defined the condition number, we may now state how composition affects the approximation parameters. Theorem 3.3 (Composition). Let Fi be an (αi , βi /κ̃)-algorithm for A : X → Y, where A has non-linear condition number κ̃ Then, the composition F2 ◦ F1 is an (α1 + α2 β1 , β1 β2 /κ̃-algorithm for the same problem. Before proving lemma 3.3, we state two useful corollaries. The first concerns the result of recursively composing a (α, β/κ̃)-algorithm with itself. Corollary 3.4. Let F be a (α, β/κ̃)-algorithm for β < 1. Let F t be the sequence formed by iterated composition, with F 1 = F , F t+1 = F t ◦ F . Then, F t is a (α/(1 − β), β t /κ̃)-algorithm. Proof. By induction on t. To start, an (α, β/κ̃)-algorithm is trivially a (α/(1−β), β/κ̃)-algorithm. Assuming the claim holds for F t , lemma 3.3 implies F t+1 is a (α + αβ/(1 − β), β t+1 /κ̃)-algorithm. We observe that to obtain (1 + O(), δ)-solution, only the first solver in the chain need be very accurate. Corollary 3.5. Let F be a (1+, /2κ̃-solver and G be a (2, 1/2κ̃)-solver. Then, Gt ◦F is a (1+5, 2−t−1 /κ̃solver. For some problems, there is a very simple (M, 0)-solver known. A final composition with that solver serves to elimate the error. Corollary 3.6. Let F be a (1 + , δ/κ̃)-solver and G be a (M, 0)-solver. Then G ◦ F is a (1 + (1 + δM ), 0)solver. 4 3.1 Proof of lemma 3.3 Since F1 is an (α1 , β1 /κ̃)-algorithm, we have kxk kb̃k kAk ≤ α1 kxopt k β1 kxopt k κ̃ ≤ By the definition of κ̃, kx̃opt k ≤ κ̃ kb̃k ≤ β1 kxopt k kAk Since F2 is an (α2 , β2 /κ̃)-algorithm, kx̃k kAx̃ − b̃k kAk ≤ α2 kx̃opt k ≤ α2 β1 kxkopt ≤ β2 β 1 β2 kx̃opt k ≤ κ̃ κ̃ The conclusion follows from kx + x̃k ≤ kxk + kx̃k. 4 Solvers for `p The recursive composition technique takes existing (α, β)-approximation algorithms and yields new algorithms with different approximation parameters. In this section, we discuss the parameters of existing well-known base solvers. Let A : Rm → Rn be linear and fixed throughout this section. To concisely describe results and ease comparison, for a norm k · kX on Rm and a norm k · kY in Rn , we write (α, β)X →Y to denote an (α, β) solution with respect to A : X → Y. The classical `2 solvers provide a useful goalpost for comparison. Theorem 4.1 ([4]). The steepest-descent algorithm produces a (1, δ)2→2 -solution after O(κ2→2 (A)2 log(1/δ)) simple iterations The conjugate gradient algorithm produces a (1, δ)2→2 -solution after O(κ2→2 (A) log(1/δ)) simple iterations. There are many existing algorithms for p-norm minimization, and composing them yields algorithms with exponentially small error. For this initial manuscript, we simply state the `1 version required for min-cost flow. The general cases consist of describing the many existing algorithms[14] in terms of (α, β)-solvers. Theorem 4.2. There is a (1 + , δ)1→1 -solver with simple iterations totalling O κ̃1→1 (A)2 log(m) −2 ) + log(δ −1 ) Theorem 4.3. There is a (1 + , δ)∞→∞ -solver with simple iterations totalling O κ̃∞→∞ (A)2 log(n) −2 + log(δ −1 ) The preceding theorems follow by combining the multiplicative weights algorithm[15, 9] with the composition lemma. The former algorithm applies in a more general online setting; for our applications, the algorithm yields the “weak” approximation algorithms that shall be composed. A support oracle for a compact-convex set C takes input y and returns some x ∈ C maximizing x · C. Let 4n be the unit simplex in Rn (i.e., the convex hull of the standard basis). The multiplicative weights method finds approximate solutions to the saddle point problem, max min w · z w∈4n z∈C In particular, it obtains an additive -approximation in O(ρ2 −2 log n) iterations with each iteration taking O(n) time plus a call to a support oracle for C. 5 Let Bp be the unit ball in `p . A point w ∈ B1 is represented as w = w+ − w− for (w+ , w− ) ∈ 42n . The base solvers follow by considering the saddle-point problems, max min y ∗ · (Ax − µb) max min y ∗ · (Ax − µb) y ∗ ∈b1 x∈b∞ x∈b1 y ∗ ∈b∞ where µ is scaled via binary-search as needed. The search adds a factor O(log κ̃) to the complexity. This factor may be avoided by using `p -norm for p = log n and p = (log m)/ regularization, respectively, instead. 5 Generalized Preconditioning Let A : X → Y be linear. The minimum-norm problem for A does not intrinsically require Y to be normed; nor does the definition of a (α, 0)-solution. Thus, an algorithm designer seeking to solve a class of minimum norm problems may freely choose how to norm Y. The best choice requires balancing two factors. First, the norm should be sufficiently “simple” that (1, ) solutions for polynomially-small are easy to find. Second, the norm should be chosen so that κ̃(A)X →Y is small. If only the latter constraint existed, the ideal choice would be the optimal A pre-image norm,1 kbkopt(A) = min{kxkX : Ax = b} By construction, κ̃(A)X →opt(A) = 1. Of course, simply evaluating that norm is equivalent to the original problem we aim to solve. It follows that Y should be equipped with the relatively closest norm to opt(A) that can be efficiently minimized. Following the common approach in `2 , when X = `p , we propose to choose a generalized pre-conditioner P : Y → `p injective and consider the norm kPbkp . Again, P should be chosen such that κ̃(PA)p→p is small, and P, P∗ are easy to compute. 6 Graph Problems In this section we discuss some applications of generalized preconditioning to network flow problems. Let G = (V, E) be an undirected graph with n vertices and m edges. While undirected, we assume the edges are oriented arbitrarily. We denote by V ∗ , E ∗ the spaces of real-valued functions f : V → R and f : E → R on vertices and edges, respectively. We denote by V, E the corresponding dual spaces of demands and flows. For S ⊆ V , we write 1S ∈ V ∗ for the indicator function on S. The discrete derivative operator D∗ : V ∗ → E ∗ is defined by (D∗ f )(xy) = f (y) − f (x) for each oriented edge xy. The constant function has zero derivative, D∗ 1V = 0. The (negative) adjoint −D : E → V is the discrete divergence operator ; for a flow  ∈ E and vertex x ∈ V , −(D)(x) is the net quantity being transported away from vertex x by the flow. A single-commodity flow problem is specified by a demand vector b ∈ V, and requires finding a flow  ∈ E satisfying D = b, minimizing some cost function. When the cost-function is a norm on E, the problem is a minimum-norm pre-image problem. Assuming G is connected, b has a pre-image iff the total demand 1V · b is zero. We hereafter assume G to be connected and total demands equal to zero. Having established the generalized preconditioning framework, we may now design fast algorithms for fundamental network flow problems by designing generalized preconditioners for the corresponding minimumnorm problems. As a result, we obtain nearly-linear time algorithms for max-flow and uncapacitated min-cost flow in undirected graphs. The max-flow algorithm was presented previously[16], and involved a combination of problem-specific definitions (e.g. congestion approximators) and subroutines (e.g. gradient-based `∞ minimization). The brief summary in subsection 6.1 shows how the same result may be immediately obtained by combining theorem ?? with only small part of the earlier flow-specific work[16]: the actual construction of the preconditioner. In section 7, we present a nearly-linear time algorithm for uncapacitated min-cost flows, another fundamental network optimization problem. Once again, all that is needed is description and analysis of the preconditioner. 1 We remark this is only defined on the image space of A, rather than the entire codomain. However, we only consider b in that image space 6 6.1 Max-Flow A capacitated graph associates with each edge e, a capacity c(e) > 0. Given capacities, we define the capacity P norm on E ∗ by kf kc∗ = e c(e)\|f (e)\|; the dual congestion norm on E is defined by kkc = maxe \|(e)\| c(e) . ∗ The boundary capacity of a set S ⊆ V is kD 1S kc , which we abbreviate as c(∂S). The single-commodity flow problem with capacity norm is undirected maximum flow problem, and is a fundamental problem in network design and optimization. Following the work of Spielman and Teng for `2 -flows[17], and Madry[13] for cut problems, the author has obtained a nearly-linear time algorithm for maximum flow[16]. We have since realized that only certain parts of the ideas introduced in that paper are actually specific to maximum-flow, and the present work has been obtained through generalizing the other parts. The (previously presented as) flow-specific composition and optimization parts of that paper are entirely subsumed by the earlier sections of this paper. The truly flow-specific part of that work required to complete the algorithm is the generalized preconditioner construction, which we have previously called a congestion-approximator. We briefly describe the main ideas of the construction, and some simple illustrative examples. For the full details, we refer the reader to the third section of [16]. Let F ⊆ 2V be a family of subsets. Such a family, together with a capacity norm on E, induces a cut\|1S ·b\| . That is, for each set in the family, consider the congestion semi-norm on V via kbkc(F) = maxS∈F c(∂S) ratio of the total aggregate demand in that set to the total capacity of all edges entering that set. If the indicator functions of sets in F span V, then it is a norm. Note also that we may write kbkc(F) = kPbk∞ , 1S ·b . where P : Rn → RF is defined by (Pb)S = c(∂S) V If F is taken to be the full power set F = 2 , the max-flow, min-cut theorem is equivalent to the statement that the non-linear condition number κ̃c→c(F) (D∗ ) is exactly one. If F is taken to be the family of singleton sets, the non-linear condition number is the inverse of the combinatorial conductance of G. That is, for highconductance graphs, composition yields a fast algorithm requiring only a simple diagonal preconditioner. This is closely analagous to the situation for `2 , where diagonal preconditioning yields fast algorithms for graphs of large algebraic conductance. A√particularly illustrative example consists of a unit-capacity 2t × 2t 2-D grid. With low (Θ(2−t ) = Θ(1/ n) conductance, the singleton family F performs poorly. A simple but dramatically better family consists of taking F to be all power-of-two size, power-of-two aligned subgrids. The latter family yields linear condition number O(t) = O(log n). Furthermore, computing the aggregate demand in all such sets requires only O(n) time, with each sub-grid aggregate equal to the sum of its four child aggregates. This algorithm is analagous to an `∞ version of multi-grid [?]. Note that the indicator “step” functions 1S are different from the bilinear “pyramid” functions used by multigrid. Theorem 6.1 ([16]). There is an algorithm that given a capacitated graph (G, c) with n vertices and m edges, takes m1+o(1) time and outputs a data structure of size n1+o(1) that efficiently represents a preconditioner P with κ(PD)c→∞ ≤ no(1) . Given the data structure, P and P∗ can be applied in n1+o(1) time. Let C : Rm → E be the diagonal capacity matrix; (Cx)(e) = c(e)x(e). The nearly-linear time algorithm follows by using the preconditioner P in the preceding theorem, applying theorem ?? to the problem of minimizing kxk∞ subject to PDCx = b. A final minor flow-specific part is needed to achieve zero error: the simple (m, 0)-approximation algorithm that consists of routing all flow through a maximum-capacity spanning tree. 7 Uncapacitated Min-Cost Flow In this section, we present a nearly-linear time algorithm for the uncapacitated minimum-cost flow problem on undirected graphs. A cost graph associates P a cost l(e) > 0 with each edge e in G. The cost norm on the space of flows E is defined by kkl = e k(e)kl(e). The undirected uncapacitated minimum-cost flow problem is specified by the graph G, lengths `, and demands b ∈ V, and consists of finding  ∈ E with D = b minimizing kkl . Note that, unlike max-flow, the single-source uncapacitated min-cost flow problem is significantly easier than the distributed source case: the optimal solution is to route along a single-source shortest path away from the source. The problem only becomes interesting with distributed sources and sinks. As a new application of generalized preconditioning, we obtain a nearly-linear time algorithm for approximately solving this problem. 7 Theorem 7.1. There is a randomized algorithm that, given a length-graph G = (V, E, l), takes and outputs a (1 + , 0)-solution to the undirected uncapacitated min-cost flow problem. m1+o(1) 2 time As expected, the bulk of the algorithm lies in constructing a good preconditioner. After introducing some useful definitions and lemmas, we spend the majority of this section constructing and analyzing a preconditioner for this problem. Theorem 7.2. There is a randomized algorithm that, given a length-graph G, takes O(m log2 n + n1+o(1) ) time and outputs a n1+o(1) × n matrix P with κlto∞ (PD) ≤ no(1) . Every column of P has no(1) non-zero entries. Dual to cost is the stretch norm on E ∗ is kf kl∗ = maxe \|f (e)\|/l(e). We say a function φ ∈ V is L-Lipschitz iff kD∗ φkl∗ ≤ L; that is, \|φ(x) − φ(y)\| ≤ Ll(xy) for all edges xy. The dual problem is to maximize φ · b over 1-Lipschitz φ ∈ V. A length function l induces an intrinsic metric d : V × V → R, with distances determined by shortest paths. The preceding definition of Lipschitz is equivalent to \|φ(x) − φ(y)\| ≤ Ld(x, y) for all x, y ∈ V . That is, the Lipschitz constant of φ depends only on the metric, and is otherwise independent of the particular graph or edge lengths inducing that metric. As the dual problem depends only on the induced metric, so must the value of the min-cost flow. Therefore, we may unambiguously write kbkopt(d) = max{φ · b : φ is 1-Lipschitz w.r.t. d} The monotonicity of cost with distance follows immediately. Lemma 7.3. If d(x, y) ≤ d̃(x, y) for all x, y ∈ V, then for all b ∈ V⊥1 , kbkopt(d) ≤ kbkopt(d̃) We recall that in some cases, the identity operator is a good preconditioner; for max-flow, such is the case in constant-degree expanders. For min-cost flow, the analagous case is a metric where distances between any pair of points differ relatively by small factors. Definition 7.4. Let U ⊆ V . We say U is r-separated if d(x, y) ≥ r for all x, y ∈ U with x 6= y. We say U is R-bounded if d(x, y) ≤ R for all x, y ∈ U . Lemma 7.5. Let U ⊂ V be R-bounded and r-separated with respect to d. Let b ∈ V⊥1 be demands supported on U . Then, R r kbk1 ≤ kbkopt(d) ≤ kbk1 2 2 Proof. Note 2r kbk1 is exactly the min-cost of routing b in the metric where all distances are exactly r (consider the star graph with edge lengths r/2, with a leaf for each x ∈ U ). The inequalities follow by monotonicity. To complete the proof of theorem 7.1, assuming theorem 7.2, we also need a simple approximation algorithm that yields a poor approximation, but with zero error. This is easily achieved by the algorithm that routes all flow through a minimum cost (i.e., total length) spanning tree T . Lemma 7.6. Let T be a minimum cost spanning tree with respect to l on G. Consider the algorithm that, given b ∈ V⊥1 , outputs the flow  ∈ E routing b using only the edges in T . Then, kk ≤ nkbkopt(d) Proof. The flow on T is unique, and therefore, optimal for the metric d̃ induced by paths restricted to lie in T . As T is a minimum cost spanning tree, d̃(x, y) ≤ nd(x, y). We now prove theorem 7.1. Let L : Rm → E be the diagonal length matrix (Lx)(e) = `(e)x(e), and let P be the preconditioner from theorem 7.2. Computing P takes O(m log2 n) + n1+o(1) time. The minimum-cost flow problem is equivalent to the problem of minimizing kxk1 subject to PDL−1 x = Pb. As P has n1+o(1) 1+o(1) non-zero entries and κ1→1 (PDL−1 ) ≤ no(1) , the total running time is m 2 . 8 8 Preconditioning Min-Cost Flow A basic concept in metric spaces is that of an embedding: ψ : V → Ṽ , with respective metrics d, d̃. An embedding has distortion L if for some µ > 0, 1≤µ d̃(ψ(x), ψ(y)) ≤L d(x, y) Metric embeddings are quite useful for preconditioning; if d embeds into d̃ with distortion L, then the monotonicity lemma implies the relative costs of flow problems differ by a factor L. It follows that if we can construct a preconditioner P for the min-cost flow problem on d̃, the same P may be used to precondition min-cost flow on d, with condition number at most L-factor worse. Our preconditioner uses embeddings in two ways. The preconditioner itself is based on a certain hierarchical routing scheme. Hierarchical routing schemes based on embeddings into tree-metrics have been studied extensively(see e.g. [8]), and immediately yield preconditioners with polylogarithmic condition number. However, we are unable to efficiently implement those schemes in nearly-linear time. Instead, we apply Bourgain’s O(log n)-distortion embedding[3] into `1 . That is, by paying an O(log n) factor in condition number, we may completely restrict our attention to approximating min-cost flow in `1 space. Theorem 8.1 (Bourgain’s Embedding[3]). Any n-point metric space embeds into `p space with distortion O(log n). If d is the intrinsic metric of a length-graph G = (V, E) with m edges, there is a randomized algorithm that takes O(m log2 n) time and w.h.p. outputs such an embedding, with dimension O(log2 n). The complexity of constructing and evaluating√our preconditioner is exponential in the embedding’s √ dimension. Therefore, we reduce the dimension to Θ( log n), incuring another distortion factor of exp(O( log n)). Theorem 8.2 (Johnson-Lindenstrauss Lemma[10, 6]). Any n points in Euclidean space embed into kdimensional Euclidean space√with distortion nO(1/k) √ , for k ≤ Ω(log n). In particular, k = O(log n) yields O(1) distortion, and k = O( log n) yields exp(O( log n)) distortion. If the original points are in d > k dimensions, there is a randomized algorithm that w.h.p outputs such an embedding in O(nd) time. Our√reduction consists of first embedding the vertices into `2 via theorem 8.1, reducing the dimension to k = O( log √ n) using lemma 8.2, and then using the identity embedding of `2 into `1 , for a total distortion of exp(O( log n)). To estimate routing costs in `1 , we propose an extremely simple hierarchical routing algorithm in `1 , and show it achieves a certain competetive ratio. The routing scheme applies to the exponentially large graph corresponding to a lattice discretization of the k-dimensional cube in `1 ; after describing such a scheme, we show that if the demands in that graph are supported on n vertices, the distributed routing algorithm finds no demand to route in most of the graph, and the same routing algorithm may be carried out in a k − d-tree instead of the entire lattice. On the lattice, the routing algorithm consists of sequentially reducing the support set of the demands from a lattice with spacing 2−t−1 to a lattice with spacing 2−t , until all demand is supported on the corners of a cube containing the original demand points. On each level, the demand on a vertex not appearing in the coarser level is eliminated by pushing its demand to a distribution over the aligned points nearby. 8.1 Lattice Algorithm For t ∈ Z, let Vt = (2−t Z)k be the lattice with spacing 2−t . We design a routing scheme and preconditioner for min-cost flow in `1 , with initial input demands bT supported on a bounded subset of VT . For t = T, T −1, . . ., the routing scheme recursively reduces a min-cost flow problem with demands supported on Vt to a min-cost flow problem with demands supported on the sparser lattice Vt−1 . Given demands bt supported on Vt , the scheme consists of each vertex x ∈ Vt pulling its demand bt (x) uniformly from the closest points to x in Vt−1 along any shortest path, the lengths of which are at most k2−t . For example, a point x = (x1 , . . . , xk ) ∈ V1 with j ≤ k non-integer coordinates and demand b1 (x) pulls b1 (x)2−j units of flow from each of the 2j points in V0 corresponding to rounding those j coordinates in any way. By construction, all points in Vt \ Vt−1 have their demands met exactly, and we are left with a reduced problem residual demands bt−1 supported on Vt−1 . As we show, eventually the demands are supported entirely on the 2k corners of a hypercube forming a 9 bounding box of the original support set, at which point the the simple scheme of distributing the remaining demand uniformly to all corners is a k-factor approximation. We do not actually carry out the routing; rather, we state a crude upper-bound on its cost, and then show that bound itself exceeds the true min-cost by a factor of some κ. For the preconditioner, we shall only require the sequence of reduced demands bt , bt−1 , . . ., and not the flow actually routing them. Theorem 8.3. kbt kopt ≤ t X k2−s kbs k1 ≤ 2k(t + 1)kbt kopt s=0 Assuming theorem 8.3 holds, we take our preconditioner to be a matrix P such that, kPbT k1 = T X k2−t kbt k1 t=0 The proof of theorem 8.3 uses two essential properties of the routing scheme described. The first is that the cost incurred in the reduction from bt to bt−1 is not much larger than the min-cost of routing bt . The second is the reduction does not increase costs. That is, the true min-cost for routing the reduced problem bt−1 is at most that of routing the original demands bt . Lemma 8.4. Let bt−1 be the reduced demands produced when the routing scheme is given bt . Then, 1. The cost incurred by that level of the scheme is at most k2−t kbt k1 ≤ 2kkbt kopt 2. The reduction is non-increasing with respect to min-cost: kbt−1 kopt ≤ kbt kopt Assuming lemma 8.4, we now prove 8.3. We argue the sum is an upper bound on the total cost incurred by the routing scheme, when applied to bt . The left inequality follows immediately, as the true min-cost is no larger. The total cost of the scheme is at most the cost incurred on each level, plus the cost of the final routing of b0 . Lemma 8.4(a) accounts for all terms above s = 0. For the final routing, the demands b0 are supported in {0, 1}k , so the cost of routing all demand to a random corner is at most 12 kkb0 k1 . For the right inequality, we observe t X k2−t kbt k1 ≤ s=0 t X 2kkbs kopt s=0 The inequality follows termwise, using lemma 8.4(a) for s ≥ 1; the s = 0 case follows from lemma 7.5 because V0 is 1-separated. 8.2 Proof of Lemma 8.4 For the first part, each vertex x ∈ Vt routes its demand bt (x) to the closest points in Vt−1 any shortest path. As such paths have length at most k2−t , each point x contributes at most k2−t \|bt (x)\|. For the second part, by scale-invariance it suffices to consider t = 1. Moreover, it suffices to consider the case where b1 consists of a unit demand from between two points x, y ∈ V1 with kx − yk1 = 21 . To see this, we observe any flow routing arbitrary demands b1 consists of a sum of such single-edge flows. As the reduction is linear, if the cost of each single-edge demand-pair is not increased, then the cost their sum is not increased. By symmetry, it suffices to consider x, y with x = (0, z) and y = ( 21 , z). Then, the reduction distributes x’s demand to a distribution over (0, z 0 ) where z 0 ∼ Z 0 ; y’s demand is split over (0, z 0 ) and (1, z 0 ) where z 0 ∼ Z 0 . Therefore, half of the demand at (0, z 0 ) is cancelled, leaving the residual problem of routing 1/2 unit from (0, z 0 ) to (1, z 0 ). That problem has cost 1/2, by routing each fraction directly from (0, z 0 ) to (1, z 0 ). References [1] Ravindra K Ahuja, Andrew V Goldberg, James B Orlin, and Robert E Tarjan. Finding minimum-cost flows by double scaling. Mathematical programming, 53(1-3):243–266, 1992. 10 [2] Josh Barnes and Piet Hut. A hierarchical o (n log n) force-calculation algorithm. nature, 324(6096):446– 449, 1986. [3] J. Bourgain. On lipschitz embedding of finite metric spaces in hilbert space. Israel Journal of Mathematics, 52(1):46–52, 1985. [4] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, New York, NY, USA, 2004. [5] Samuel I Daitch and Daniel A Spielman. Faster approximate lossy generalized flow via interior point algorithms. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 451–460. ACM, 2008. [6] Sanjoy Dasgupta and Anupam Gupta. An elementary proof of a theorem of johnson and lindenstrauss. Random Structures & Algorithms, 22(1):60–65, 2003. [7] Stephen Demko. Condition numbers of rectangular systems and bounds for generalized inverses. Linear Algebra and its Applications, 78:199–206, 1986. [8] Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 448–455. ACM, 2003. [9] Yoav Freund and Robert E Schapire. Adaptive game playing using multiplicative weights. Games and Economic Behavior, 29(1):79–103, 1999. [10] William B Johnson and Joram Lindenstrauss. Extensions of lipschitz mappings into a hilbert space. Contemporary mathematics, 26(189-206):1, 1984. [11] Jonathan A Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 217–226. Society for Industrial and Applied Mathematics, 2014. [12] Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming: Solving linear programs in o (vrank) iterations and faster algorithms for maximum flow. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 424–433. IEEE, 2014. [13] Aleksander Madry. Fast approximation algorithms for cut-based problems in undirected graphs. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 245–254. IEEE, 2010. [14] Yu Nesterov. Smooth minimization of non-smooth functions. Math. Program., 103(1):127–152, May 2005. [15] Serge A. Plotkin, David B. Shmoys, and Eva Tardos. Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research, 20:257–301, 1995. [16] Jonah Sherman. Nearly maximum flows in nearly linear time. In Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on, pages 263–269. IEEE, 2013. [17] Daniel A. Spielman and Shang-Hua Teng. Nearly-linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. CoRR, abs/cs/0607105, 2006. 11	8
Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time ⋆ Seok-Hee Hong1 and Hiroshi Nagamochi2 arXiv:1608.07662v2 [cs.CG] 2 Sep 2016 1 University of Sydney, Australia seokhee.hong@sydney.edu.au 2 Kyoto University, Japan nag@amp.i.kyoto-u.ac.jp Abstract. Thomassen characterized some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph is drawable in straight-lines if and only if it does not contain the configuration [C. Thomassen, Rectilinear drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988]. In this paper, we characterize some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph can be reembedded into a straight-line drawable 1-plane embedding of the same graph if and only if it does not contain the configuration. Re-embedding of a 1-plane embedding preserves the same set of pairs of crossing edges. We give a linear-time algorithm for finding a straight-line drawable 1plane re-embedding or the forbidden configuration. 1 Introduction Since the 1930s, a number of researchers have investigated planar graphs. In particular, a beautiful and classical result, known as Fáry’s Theorem, asserts that every plane graph admits a straight-line drawing [5]. Indeed, a straight-line drawing is the most popular drawing convention in Graph Drawing. More recently, researchers have investigated 1-planar graphs (i.e., graphs that can be embedded in the plane with at most one crossing per edge), introduced by Ringel [13]. Subsequently, the structure of 1-planar graphs has been investigated [4, 12]. In particular, Pach and Toth [12] proved that a 1-planar graph with n vertices has at most 4n − 8 edges, which is a tight upper bound. Unfortunately, testing the 1-planarity of a graph is NP-complete [6, 11], however linear-time algorithms are available for special subclasses of 1-planar graphs [1, 3, 7]. Thomassen [14] proved that every 1-plane graph (i.e., a 1-planar graph embedded with a given 1-plane embedding) admits a straight-line drawing if and only if it does not contain any of two special 1-plane graphs, called the Bconfiguration or W-configuration, see Fig. 1. ⋆ Research supported by ARC Future Fellowship and ARC Discovery Project DP160104148. This is an extended abstract. For a full version with omitted proofs, see [9]. u2 outer face u2 v1 u4 c u1 outer face c s u3 u3 u1 w4c w3c u1 u3 v2 (a) u4 v3 c w2c w1c u2 (c) (b) Fig. 1. (a) B-configuration with three edges u1 u2 , u2 u3 and u3 u4 and one crossing c made by an edge pair {u1 u2 , u3 u4 }, where edge u2 u3 may have a crossing when the configuration is part of a 1-plane embedding; (b) W-configuration with four edges u1 u2 , u2 u3 , v1 v2 and v2 v3 and two crossings c and s made by edge pairs {u1 u2 , v2 v3 } and {u2 u3 , v1 v2 }, where possibly u1 = v1 and u3 = v3 ; (c) Augmenting a crossing c ∈ χ made by edges u1 u3 and u2 u4 with a new cycle Qc = (u1 , w1c , u2 , w2c , u3 , w3c , u4 , w4c ) depicted by gray lines. Recently, Hong et al. [8] gave an alternative constructive proof, with a lineartime testing algorithm and a drawing algorithm. They also showed that some 1-planar graphs need an exponential area with straight-line drawing. We call a 1-plane embedding straight-line drawable (SLD for short) if it admits a straight-line drawing, i.e., it does not contain a B- or W-configuration by Thomassen [14]. In this paper, we investigate a problem of “re-embedding” a given non-SLD 1-plane embedding γ into an SLD 1-plane embedding γ ′ . For a given 1-plane embedding γ of a graph G, we call another 1-plane embedding γ ′ of G a cross-preserving embedding of γ if exactly the same set of edge pairs make the same crossings in γ ′ . More specifically, we first characterize the forbidden configuration of 1-plane embeddings that cannot admit an SLD cross-preserving 1-plane embedding. Based on the characterization, we present a linear-time algorithm that either detects the forbidden configuration in γ or computes an SLD cross-preserving 1-plane embedding γ ′ . Formally, the main problem considered in this paper is defined as follows. Re-embedding a 1-Plane Graph into a Straight-line Drawing Input: A 1-planar graph G and a 1-plane embedding γ of G. Output: Test whether γ admits an SLD cross-preserving 1-plane embedding γ ′ , and construct such an embedding γ ′ if one exists, or report the forbidden configuration. To design a linear-time implementation of our algorithm in this paper, we introduce a rooted-forest representation of non-intersecting cycles and an efficient procedure of flipping subgraphs in a plane graph. Since these data structure and procedure can be easily implemented, it has advantage over the complicated decomposition of biconnected graphs into triconnected components [10] or the SPQR tree [2]. 2 2 Plane Embeddings and Inclusion Forests Let U be a set of n elements, and let S be a family of subsets S ⊆ U . We say that two subsets S, S ′ ⊆ U are intersecting if none of S ∩ S ′ , S − S ′ and S ′ − S is empty. We call S a laminar if no two subsets in S are intersecting. For a laminar S, the inclusion-forest of S is defined to be a forest I = (S, E) of a disjoint union of rooted trees such that (i) the sets in S are regarded as the vertices of I, and (ii) a set S is an ancestor of a set S ′ in I if and only if S ′ ⊆ S. Lemma 1. For a cyclic sequence (u1 , u2 , . . . , uδ ) of δ ≥ 2 elements, define an interval (i, j) to be the set of elements uk with i ≤ k ≤ j if i ≤ j and (i, j) = (i, δ) ∪ (1, j) if i > j. Let S be a set of intervals. A pair of two intersecting intervals in S (when S is not a laminar) or the inclusion-forest of S (when S is a laminar) can be obtained in O(δ + \|S\|) time. Throughout the paper, a graph G = (V, E) stands for a simple undirected graph. The set of vertices and the set of edges of a graph G are denoted by V (G) and E(G), respectively. For a vertex v, let E(v) be the set of edges incident to v, N (v) be the set of neighbors of v, and deg(v) denote the degree \|N (v)\| of v. A simple path with end vertices u and v is called a u, v-path. For a subset X ⊆ V , let G − X denote the graph obtained from G by removing the vertices in X together with the edges in ∪v∈X E(v). A drawing D of a graph G is a geometric representation of the graph in the plane, such that each vertex of G is mapped to a point in the plane, and each edge of G is drawn as a curve. A drawing D of a graph G = (V, E) is called planar if there is no edge crossing. A planar drawing D of a graph G divides the plane into several connected regions, called faces, where a face enclosed by a closed walk of the graph is called an inner face and the face not enclosed by any closed walk is called the outer face. A planar drawing D induces a plane embedding γ of G, which is defined to be a pair (ρ, ϕ) of the rotation system (i.e., the circular ordering of edges for each vertex) ρ, and the outer face ϕ whose facial cycle Cϕ gives the outer boundary of D. Let γ = (ρ, ϕ) be a plane embedding of a graph G = (V, E). We denote by F (γ) the set of faces in γ, and by Cf the facial cycle determined by a face f ∈ F , where we call a subpath of Cf a boundary path of f . For a simple cycle C of G, the plane is divided by C in two regions, one containing only inner faces and the other containing the outer area, where we say that the former is enclosed by C or the interior of C, while the latter is called the exterior of C. We denote by Fin (C) the set of inner faces in the interior of C, by Ein (C) the set of edges in E(Cf ) with f ∈ Fin (C), and by Vin (C) the set of end-vertices of edges in Ein (C). Analogously define Fex (C), Eex (C) and Vex (C) in the exterior of C. Note that E(C) = Ein (C) ∩ Eex (C) and V (C) = Vin (C) ∩ Vex (C). For a subgraph H of G, we define the embedding γ\|H of γ induced by H to be a sub-embedding of γ obtained by removing the vertices/edges not in H, keeping the same rotation system around each of the remaining vertices/crossings and the same outer face. 3 2.1 Inclusion Forests of Inclusive Set of Cycles In this and next subsections, let (G, γ) stand for a plane embedding of γ = (ρ, ϕ) of a biconnected simple graph G = (V, E) with n = \|V \| ≥ 3. Let C be a simple cycle in G. We define the direction of C to be an ordered pair (u, v) with uv ∈ E(C) such that the inner faces in Fin (C) appear on the right hand side when we traverse C in the order that we start u and next visit v. For simplicity, we say that two simple cycles C and C ′ are intersecting if Fin (C) and Fin (C ′ ) are intersecting. Let C be a set of simple cycles in G. We call C inclusive if no two cycles in C are intersecting, i.e., {Fin (C) \| C ∈ C} is a laminar. When C is inclusive, the inclusion-forest of C is defined to be a forest I = (C, E) of a disjoint union of rooted trees such that: (i) the cycles in C are regarded as the vertices of I, and (ii) a cycle C is an ancestor of a cycle C ′ in I if and only if Fin (C ′ ) ⊆ Fin (C). Let I(C) denote the inclusion-forest of C. For a vertex subset X ⊆ V , let C(X) denote the set of cycles C ∈ C such that x ∈ V (C) for some vertex x ∈ X, where we denote C({v}) by C(v) for short. Lemma 2. For (G, γ), let C be a set ofP simple cycles of G. Then any of the following tasks can be executed in O(n + C∈C \|E(C)\|) time. (i) Decision of the directions of all cycles in C; (ii) Detection of a pair of two intersecting cycles in C when C is not inclusive, and construction of the inclusion-forests I(C(v)) for all vertices v ∈ V when C is inclusive; and (iii) Construction of the inclusion-forest I(C) when C is inclusive. 2.2 Flipping Spindles A simple cycle C of G is called a spindle (or a u, v-spindle) of γ if there are two vertices u, v ∈ V (C) such that no vertex in V (C) − {u, v} is adjacent to any vertex in the exterior of C, where we call vertices u and v the junctions of C. Note that each of the two subpaths of C between u and v is a boundary path of some face in F (γ). Given (G, γ), we denote the rotation system around a vertex v ∈ V by ργ (v). For a spindle C in γ, let J(C) denote the set of the two junctions of C. Flipping a u, v-spindle C means to modify the rotation system of vertices in Vin (C) as follows: (i) For each vertex w ∈ Vin (C) − J(C), reverse the cyclic order of ργ (w); and (ii) For each vertex u ∈ J(C), reverse the order of subsequence of ργ (u) that consists of vertices N (u) ∩ Vin (C). Every two distinct spindles C and C ′ in γ are non-intersecting, and they always satisfy one of Ein (C) ∩ Ein (C ′ ) = ∅, Ein (C) ⊆ Ein (C ′ ), and Ein (C ′ ) ⊆ Ein (C). Let C be a set of spindles in γ, which is always inclusive, and let I(C) denote the inclusion-forest of C. 4 When we modify the current embedding γ by flipping each spindle in C, the resulting embedding γC is the same, independent from the ordering of the flipping operation to the spindles, since for two spindles C and C ′ which share a common junction vertex u ∈ J(C) ∩ J(C ′ ), the sets N (u) ∩ Vin (C) and N (u) ∩ Vin (C ′ ) do not intersect, i.e., they are disjoint or one is contained in the other. Define the depth of a vertex v ∈ V in I to be the number of spindles C ∈ C such that v ∈ Vin (C) − J(C), and denote by p(v) the parity of depth of vertex v, i.e., p(v) = 1 if the depth is odd and p(v) = −1 otherwise. For a vertex v ∈ V , let C[v] denote the set of spindles C ∈ C such that v ∈ J(C), and let γC[v] be the embedding obtained from γ by flipping all spindles in C[v]. Let revhσi mean the reverse of a sequence σ. Then we see that ργC (v) = ργC[v] (v) if p(v) = 1; and ργC (v) = revhργC[v] (v)i otherwise. To obtain the embedding γC from the current embedding γ by flipping each spindle in C, it suffices to show how to compute each of p(v) and ργC[v] (v) for all vertices v ∈ V . Lemma 3. Given (G, γ), let C be a set P of spindles of γ. Then any of the following tasks can be executed in O(n + C∈C \|E(C)\|) time. (i) Decision of parity p(v) of all vertices v ∈ V ; and (ii) Computation of ργC[v] (v) for all vertices v ∈ V . 3 Re-embedding 1-plane Graph and Forbidden Configuration A drawing D of a graph G = (V, E) is called a 1-planar drawing if each edge has at most one crossing. A 1-planar drawing D of graph G induces a 1-plane embedding γ of G, which is defined to be a tuple (χ, ρ, ϕ) of the crossing system χ of E, the rotation system ρ of V , and the outer face ϕ of D. The planarization G(G, γ) of a 1-plane embedding γ of graph G is the plane embedding obtained from γ by regarding crossings also as graph vertices, called crossing-vertices. The set of vertices in G(G, γ) is given by V ∪χ. For a notational convenience, we refer to a subgraph/face of G(G, γ) as a subgraph/face in γ. Let γ = (χ, ρ, ϕ) be a 1-plane embedding of graph G. We call another 1-plane embedding γ ′ = (χ′ , ρ′ , ϕ′ ) of graph G a cross-preserving 1-plane embedding of γ when the same set of edge pairs makes crossings, i.e., χ = χ′ . In other words, the planarization G(G, γ ′ ) is another plane embedding of G(G, γ) such that the alternating order of edges incident to each crossing-vertex c ∈ χ is preserved. To eliminate the additional constraint on the rotation system on each crossingvertex c ∈ χ, we introduce “circular instances.” We call an instance (G, γ) of 1-plane embedding circular when for each crossing c ∈ χ, the four end-vertices of the two crossing edges u1 u3 and u2 u4 that create c (where u1 , u2 , u3 and u4 appear in the clockwise order around c) are contained in a cycle Qc = (u1 , w1c , u2 , w2c , u3 , w3c , u4 , w4c ) of eight crossing-free edges for some vertices wic , i = 1, 2, 3, 4 of degree 2, as shown in Fig. 1(c). By definition, c and each wic not necessarily appear along the same facial cycle in the planarization G(G, γ). For example, path (v, w, u) is part of such a cycle Qs for the crossing s in the 5 circular instance in Fig. 2(a), but c and w are not on the same facial cycle in the planarization. A given instance can be easily converted into a circular instance by augmenting the end-vertices of each pair of crossing edges as follows. In the plane graph, G(G, γ), for each crossing-vertex c ∈ χ and its neighbors u1 , u2 , u3 and u4 that appear in the clockwise order around c, we add a new vertex wic , i = 1, 2, 3, 4 and eight new edges ui wic and wic ui+1 , i = 1, 2, 3, 4 (where u5 means u1 ) to form a cycle Qc of length 8 whose interior contains no other vertex than c. Let H be the resulting graph augmented from G, and let Γ be the resulting 1-plane embedding of H augmented from γ. Note that \|V (H)\| ≤ \|V (G)\| + 4\|χ\| holds. We easily see that if γ admits an SLD cross-preserving embedding γ ′ then Γ admits an SLD cross-preserving embedding Γ ′ . This is because a straight-line drawing Dγ ′ of γ ′ can be changed into a straight-line drawing DΓ ′ of some crosspreserving embedding Γ ′ of Γ by placing the newly introduced vertices wic within the region sufficiently close to the position of c. We here see that cycle Qc can be drawn by straight-line segments without intersecting with other straight-line segments in Dγ ′ . Note that the instance (G, γ ′ ) remains circular for any cross-preserving embedding γ ′ of γ. In the rest of paper, let (G, γ) stand for a circular instance (G = (V, E), γ = (χ, ρ, ϕ)) with n ≥ 3 vertices and let G denote its planarization G(G, γ). Fig. 2 shows examples of circular instances (G, γ), where the vertexconnectivity of G is 1. As an important property of a circular instance, the subgraph G(0) with crossing-free edges is a spanning subgraph of G and the four end-vertices of any two crossing edges are contained in the same block of the graph G(0) . The biconnectivity is necessary to detect certain types of cycles by applying Lemma 2. outer face ϕ outer face ϕ u’ u’ w s c’ v s u s’ s’ c u v c’ c z v’ v’ (a) (b) Fig. 2. Circular instances (G, γ) with a cut-vertex u of G, where the crossing edges are depicted by slightly thicker lines: (a) hard B-cycles C = (u, c, v, s) and C ′ = (u′ , c′ , v ′ , s′ ), (b) hard B-cycle C = (u, c, v, s) and a nega-cycle C ′ = (u′ , c′ , v ′ , s′ ) whose reversal is a hard B-cycle, where vertices u, v, u′ , v ′ ∈ V and crossings c, s, c′ , s′ ∈ χ. 6 3.1 Candidate Cycles, B/W Cycle, Posi/Nega Cycle, Hard/Soft Cycle For a circular instance (G, γ), finding a cross-preserving embedding of γ is effectively equivalent to finding another plane embedding of G so that all the current B- and W-configurations are eliminated and no new B- or W-configurations are introduced. To detect the cycles that can be the boundary of a B- or Wconfiguration in changing the plane embedding of G, we categorize cycles containing crossing vertices in G. A candidate posi-cycle (resp., candidate nega-cycle) in G is defined to be a cycle C = (u, c, v) or C = (u, c, v, s) in G with u, v ∈ V and c, s ∈ χ such that the interior (resp., exterior) of C does not contain a crossing-free edge uv ∈ E and any other crossing vertex c′ adjacent to both u and v. outer face ϕ ϕ u e c’ v (a) C C p c s c’ ϕ ϕ u c’ c e c v v (c) C C n (b) C C p u u c’ c s v (d) C C n Fig. 3. Candidate posi- and nega-cycles C = (u, c, v) and C = (u, c, v, s) in G, where white circles represent vertices in V while black ones represent crossings in χ: (a) candidate posi-cycle of length 3, (b) candidate posi-cycle of length 4, (c) candidate nega-cycle of length 3, and (d) candidate nega-cycle of length 4. Fig. 3(a)-(b) and (c)-(d) illustrate candidate posi-cycles and candidate negacycles, respectively. Let C p and C n be the sets of candidate posi-cycles and candidate nega-cycles, respectively. By definition we see that the set C p ∪ C n ∪ {Cf \| f ∈ F (γ)} is inclusive, and hence \|C p ∪ C n ∪ {Cf \| f ∈ F (γ)}\| = O(n). A candidate posi-cycle C with C = (u, c, v) (resp., C = (u, c, v, s)) is called a B-cycle if (a)-(B): the exterior of C contains no vertices in V − {u, v} adjacent to c (resp., contains exactly one vertex in V − {u, v} adjacent to c or s). Note that uv ∈ E when C = (u, c, v) is a B-cycle, as shown in Fig. 4(a). Fig. 4(b) and (d) illustrate the other types of B-cycles. A candidate posi-cycle C = (u, c, v, s) is called a W-cycle if (a)-(W): the exterior of C contains no vertices in V − {u, v} adjacent to c or s. Fig. 4(c) and (e) illustrate W-cycles. Let CW (resp., CB ) be the set of W-cycles (resp., B-cycles) in γ. Clearly a W-cycle (resp., B-cycle) gives rise to a W-configuration (resp., B-configuration). 7 outer face ϕ ϕ u e c (a) C C B-C+ c v (f) C C- v (b) C C B-C+ (c) C C W-C+ c c v (e) C C W C+ ϕ u u c c s v (h) C C n-C- (g) C C- c v s v u s (d) C C B C+ ϕ u ϕ u s c s s ϕ u v ϕ u s c s v ϕ ϕ u v (i) C C n-C- Fig. 4. Illustration of types of cycles C = (u, c, v) and C = (u, c, v, s) in G, where white circles represent vertices in V while black ones represent crossings in χ: (a) B-cycle of length 3, which is always soft, (b) soft B-cycle of length 4, (c) soft W-cycle, (d) hard B-cycle of length 4, (e) hard W-cycle, (f) nega-cycle whose reversal is a hard B-cycle, (g) nega-cycle whose reversal is a hard W-cycle, (h) candidate nega-cycle of length 4 that is not a nega-cycle whose reversal is a hard B-cycle, and (i) candidate nega-cycle of length 4 that is not a nega-cycle whose reversal is a hard W-cycle. Conversely, by choosing a W-configuration (resp., B-configuration) so that the interior is minimal, we obtain a W-cycle (resp., B-cycle). Hence we observe that the current embedding γ admits a straight-line drawing if and only if CW = CB = ∅. A W- or B-cycle C is called hard if (b): length of C is 4, and the interior of C = (u, c, v, s) contains no inner face f whose facial cycle Cf contains both vertices u and v, i.e., some path connects c and s without passing through u or v. On the other hand, a W- or B-cycle C = (u, c, v, s) of length 4 that does not satisfy condition (b) or a B-cycle of length 3 is called soft. We also call a hard B- or W-cycle a posi-cycle. Fig. 4(d) and (e) illustrate a hard B-cycle and a hard W-cycles, respectively, whereas Fig. 4(a) and (b) (resp., (c)) illustrate soft B-cycles (resp., a soft Wcycle). A cycle C = (u, c, v, s) is called a nega-cycle if it becomes a posi-cycle when an inner face in the interior of C is chosen as the outer face. In other words, a nega-cycle is a candidate nega-cycle C = (u, c, v, s) of length 4 that satisfies the following conditions (a’) and (b’), where (a’) (resp., (b’)) is obtained from the above conditions (a)-(B) and (a)-(W) (resp., (b)) by exchanging the roles of “interior” and “exterior”: (a’): the interior of C contains at most one vertex in V − {u, v} adjacent to c or s; and 8 (b’): the exterior of C contains no face f whose facial cycle Cf contains both vertices u and v. Fig. 4(f) and (g) illustrate nega-cycles, whereas Fig. 4(h) and (i) illustrate candidate nega-cycles that are not nega-cycles. Let C + (resp., C − ) denote the set of posi-cycles (resp., nega-cycles) in γ. By definition, it holds that C + ⊆ CW ∪ CB ⊆ C p and C − ⊆ C n . 3.2 Forbidden Cycle Pairs We define a forbidden configuration that characterizes 1-plane embeddings, which cannot be re-embedded into SLD ones. A forbidden cycle pair is defined to be a pair {C, C ′ } of a posi-cycle C = (u, c, v, s) and a posi- or nega-cycle C ′ = (u′ , c′ , v ′ , s′ ) in G with u, v, u′ , v ′ ∈ V and c, s, c′ , s′ ∈ χ to which G has a u, u′ -path P1 and a v, v ′ -path P2 such that: (i) when C ′ ∈ C + , paths P1 and P2 are in the exterior of C and C ′ , i.e., V (P1 ) − {u, u′ }, V (P2 ) − {v, v ′ } ⊆ Vex (C) ∩ Vex (C ′ ), where C and C ′ cannot have any common inner face; and (ii) when C ′ ∈ C − , paths P1 and P2 are in the exterior of C and the interior of C ′ , i.e., V (P1 ) − {u, u′ }, V (P2 ) − {v, v ′ } ⊆ Vex (C) ∩ Vin (C ′ ), where C is enclosed by C ′ . In (i) and (ii), P1 and P2 are not necessary disjoint, and possibly one of them consists of a single vertex, i.e., u = u′ or v = v ′ . The pair of cycles C and C ′ in Fig. 5(a) (resp., Fig. 5(b)) is a forbidden cycle pair, because there is a pair of a u, u′ -path P1 = (u, x, z, y, u′ ) and a v, v ′ -path P2 = (v, x′ , z, y ′ , v ′ ) that satisfy the above conditions (i) (resp., (ii)). Note that the pair of cycles C and C ′ in Fig. 2(a)-(b) is not forbidden cycle pair, because there are no such paths. Our main result of this paper is as follows. Theorem 1. A circular instance (G, γ) admits an SLD cross-preserving embedding if and only if it has no forbidden cycle pair. Finding an SLD cross-preserving embedding of γ or a forbidden cycle pair in G can be computed in linear time. Proof of necessity: The necessity of the theorem follows from the next lemma. For a cycle C = (u, c, v, s) ∈ C + (resp., C − ) with u, v ∈ V and c, s ∈ χ in G, we call a vertex z ∈ V an in-factor of C if the exterior of C ∈ C + (resp., the interior of C ∈ C − ) has a z, u-path Pz,u and a z, v-path Pz,v , i.e., V (Pz,u − {u}) ∪ V (Pz,v − {v}) is in Vex (C) (resp., Vin (C)). Paths Pz,u and Pz,v are not necessarily disjoint. Lemma 4. Given G = G(G, γ), let γ ′ be a cross-preserving embedding of γ. Then: (i) Let z ∈ V be an in-factor of a cycle C ∈ C + ∪ C − in G. Then cycle C is a posi-cycle (resp., a nega-cycle) in G(G, γ ′ ) if and only if z is in the exterior (resp., interior) of C in γ ′ ; 9 outer face ϕ outer face ϕ u’ y u’ u x u x y c’ s’ s z c z s’ s x’ x’ v v’ c’ c y’ v y’ v’ (a) (b) Fig. 5. Illustration of circular instances (G, γ) with a cut-vertex z of G, where the crossing edges are depicted by slightly thicker lines: (a) forbidden cycle pair with hard B-cycles C = (u, c, v, s) and C ′ = (u′ , c′ , v ′ , s′ ) (b) forbidden cycle pair with a hard B-cycle C = (u, c, v, s) and a nega-cycle C ′ = (u′ , c′ , v ′ , s′ ) whose reversal is a hard B-cycle, where vertices u, v, u′ , v ′ ∈ V and crossings c, s, c′ , s′ ∈ χ. (ii) For a forbidden cycle pair {C, C ′ }, one of C and C ′ is a posi-cycle in G(G, γ ′ ) (hence any cross-preserving embedding of γ contains a B- or W-configuration and (G, γ) admits no SLD cross-preserving embedding). Proof of sufficiency: In the rest of paper, we prove the sufficiency of Theorem 1 by designing a linear-time algorithm that constructs an SLD crosspreserving embedding of an instance without a forbidden cycle pair. 4 Biconnected Case In this section, (G, γ) stands for a circular instance such that the vertex-connectivity of the plane graph G is at least 2. In a biconnected graph G, any two posi-cycles C = (u, c, v, s), C ′ = (u′ , c′ , v ′ , s′ ) ∈ C + with u, v, u′ , v ′ ∈ V give a forbidden cycle pair if they do not share an inner face, because there is a pair of u, u′ -path and v, v ′ -path in the exterior of C and C ′ . Analogously any pair of a posi-cycle C and a nega-cycle C ′ such that C ′ encloses C is also a forbidden cycle pair in a biconnected graph G. To detect such a forbidden pair in G in linear time, we first compute the sets Cp , Cn , CW , CB , C + and C − in γ in linear time by using the inclusion-forest from Lemma 2. Lemma 5. Given (G, γ), the following in (i)-(iv) can be computed in O(n) time. (i) (ii) (iii) (iv) The sets Cp , Cn and the inclusion-forest I of Cp ∪ Cn ∪ {Cf \| f ∈ F (γ)}; The sets CW and CB ; The sets C + , C − and the inclusion-forest I ∗ of C + ∪ C − ; and A set {fC \| C ∈ (CW ∪CB )− C + } such that fC is an inner face in the interior of a soft B- or W-cycle C with V (Cf ) ⊇ V (C). 10 Given (G, γ), a face f ∈ F (γ) is called admissible if all posi-cycles enclose f but no nega-cycle encloses f . Let A(γ) denote the set of all admissible faces in F (γ). Lemma 6. Given (G, γ), it holds A(γ) 6= ∅ if and only if no forbidden cycle pair exists in γ. A forbidden cycle pair, if one exists, and A(γ) can be obtained in O(n) time. By the lemma, if (G, γ) has no forbidden cycle pair, i.e., A(γ) 6= ∅, then any new embedding obtained from γ by changing the outer face with a face in A(γ) is a cross-preserving embedding of γ which has no hard B- or W-cycle. 4.1 Eliminating Soft B- and W-cycles Suppose that we are given a circular instance (G, γ) such that G is biconnected and C + = ∅. We now show how to eliminate all soft B- and W-cycles in G in linear time using the inclusion-forest from Lemma 2 and the spindles from Lemma 3. Lemma 7. Given (G, γ) with C + = ∅, there exists an SLD cross-preserving embedding γ ′ = (χ, ρ′ , ϕ′ ) of γ such that V (Cϕ′ ) ⊇ V (Cϕ ) for the facial cycle Cϕ (resp., Cϕ′ ) of the outer face ϕ (resp., ϕ′ ), which can be constructed in O(n) time. Given an instance (G, γ) with a biconnected graph G, we can test whether it has either a forbidden cycle pair or an admissible face by Lemmas 5 and 6. In the former, it cannot have an SLD cross-preserving embedding by Lemma 4. In the latter, we can eliminate all hard B- and W-cycles by choosing an admissible face as a new outer face, and then eliminate all soft B- and W-cycles by a flipping procedure based on Lemma 7. All the above can be done in linear time. To treat the case where the vertex-connectivity of G is 1 in the next section, we now characterize 1-plane embeddings that can have an SLD cross-preserving embedding such that a specified vertex appears along the outer boundary. For a vertex z ∈ V in a graph G, we call a 1-plane embedding γ of G z-exposed if vertex z appears along the outer boundary of γ. We call (G, γ) z-feasible if it admits a z-exposed SLD cross-preserving embedding γ ′ of γ. Lemma 8. Given (G, γ) such that A(γ) 6= ∅, let z be a vertex in V . Then: (i) The following conditions are equivalent: (a) γ admits no z-exposed SLD cross-preserving embedding; (b) A(γ) contains no face f with z ∈ V (Cf ); and (c) G has a posi- or nega-cycle C to which z is an in-factor; (ii) A z-exposed SLD cross-preserving embedding or a posi- or nega-cycle C to which z is an in-factor can be computed in O(n) time. 11 5 One-connected Case In this section, we prove the sufficiency of Theorem 1 by designing a linear-time algorithm claimed in the theorem. Given a circular instance (G, γ), where G may be disconnected, obviously we only need to test each connected component of G separately to find a forbidden cycle pair. Thus we first consider a circular instance (G, γ) such that the vertex-connectivity of G is 1; i.e., G is connected and has some cut-vertices. A block B of G is a maximal biconnected subgraph of G. For a biconnected graph G, we already know how to find a forbidden cycle pair or an SLD crosspreserving embedding from the previous section. For a trivial block B with \|V (B)\| = 2, there is nothing to do. If some block B of G with \|V (B)\| ≥ 3 contains a forbidden cycle pair, then (G, γ) cannot admit any SLD cross-preserving embedding by Lemma 4. We now observe that G may contain a forbidden cycle pair even if no single block of G has a forbidden cycle pair. Lemma 9. For a circular instance (G, γ) such that the vertex-connectivity of G is 1, let B1 and B2 be blocks of G and let P1,2 be a z1 , z2 -path of G with the minimum number of edges, where V (Bi ) ∩ V (P1,2 ) = {zi } for each i = 1, 2. If γ\|Bi has a posi- or nega-cycle Ci to which zi is an in-factor for each i = 1, 2, then {C1 , C2 } is a forbidden cycle pair in G. For a linear-time implementation, we do not apply the lemma for all pairs of blocks in B. A block of G is called a leaf block if it contains only one cut-vertex of G, where we denote the cut-vertex in a leaf block B by vB . Without directly searching for a forbidden cycle pair in G, we use the next lemma to reduce a given embedding by repeatedly removing leaf blocks. Lemma 10. For a circular instance (G, γ) such that the vertex-connectivity of G = G(G, γ) is 1 and a leaf block B of G such that γ\|B is vB -feasible, let H = G−(V (B)−{vB }) be the graph obtained by removing the vertices in V (B)−{vB }. Then (i) The instance (H, γ\|H ) is circular; and ∗ (ii) If (H, γ\|H ) admits an SLD cross-preserving embedding γH , then an SLD ∗ cross-preserving embedding γ of γ can be obtained by placing a vB -exposed ∗ SLD cross-preserving embedding γB of γ\|B within a space next to the cut∗ vertex vB in γH . Given a circular instance (G, γ) such that G = G(G, γ) is connected, an algorithm Algorithm Re-Embed-1-Plane for Theorem 1 is designed by the following three steps. The first step tests whether G has a block B such that γ\|B has a forbidden cycle pair, based on Lemma 8. If one exists, the algorithm outputs a forbidden cycle pair and halts. After the first step, no block has a forbidden cycle pair. In the current circular instance (G, γ), one of the following holds: 12 (i) the number of blocks in G is at least two and there is at most one leaf block B such that γ\|B is not vB -feasible; (ii) G has two leaf blocks B and B ′ such that γ\|B is not vB -feasible and γ\|B ′ is not vB ′ -feasible; and (iii) the number of blocks in G is at most one. In (ii), vB is an in-factor of a cycle C in γ\|B and vB ′ is an in-factor of a cycle C ′ in γ\|B ′ by Lemma 8, and we obtain a forbidden cycle pair {C, C ′ } by Lemma 9. Otherwise if (i) holds, then we can remove all leaf blocks B such that γ\|B is not vB -feasible by Lemma 10. The second step keeps removing all leaf blocks B such that γ\|B is not vB -feasible until (ii) or (iii) holds to the resulting embedding. If (i) occurs, then the algorithm outputs a forbidden cycle pair and halts. When all the blocks of G can be removed successfully, say in an order of B 1 , B 2 , . . . , B m , the third step constructs an embedding with no B- or Wcycles by starting with such an SLD embedding of B m and by adding an SLD embedding of B i to the current embedding in the order of i = m−1, m−2, . . . , 1. By Lemma 10, this results in an SLD cross-preserving embedding of the input instance (G, γ). ∗ Note that we can obtain an SLD cross-preserving embedding γH 1 of γ in the third step when the first and second step did not find any forbidden cycle pair. Thus the algorithm finds either an SLD cross-preserving embedding of γ or a forbidden cycle pair. This proves the sufficiency of Theorem 1. By the time complexity result from Lemma 8, we see that the algorithm can be implemented in linear time. References 1. Auer, C., Bachmaier, C., Brandenburg, F. J. , Gleißner, A., Hanauer, K., Neuwirth D., Reislhuber J.: Outer 1-Planar Graphs, Algorithmica, 74(4), 1293–1320 (2016) 2. 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Action selection in growing state spaces: Control of Network Structure Growth arXiv:1606.07777v2 [cs.SI] 27 Dec 2016 Dominik Thalmeier1 , Vicenç Gómez 2 , and Hilbert J. Kappen 1 1 Donders Institute for Brain, Cognition and Behaviour Radboud University Nijmegen, the Netherlands 2 Department of Information and Communication Technologies Universitat Pompeu Fabra. Barcelona, Spain Abstract The dynamical processes taking place on a network depend on its topology. Influencing the growth process of a network therefore has important implications on such dynamical processes. We formulate the problem of influencing the growth of a network as a stochastic optimal control problem in which a structural cost function penalizes undesired topologies. We approximate this control problem with a restricted class of control problems that can be solved using probabilistic inference methods. To deal with the increasing problem dimensionality, we introduce an adaptive importance sampling method for approximating the optimal control. We illustrate this methodology in the context of formation of information cascades, considering the task of influencing the structure of a growing conversation thread, as in Internet forums. Using a realistic model of growing trees, we show that our approach can yield conversation threads with better structural properties than the ones observed without control. Keywords: control, complex Networks, sampling, conversation threads 1 Introduction Many complex systems can be described as dynamic processes which are characterized by the topology of an underlying network. Examples of such systems are human interaction networks, where the links may represent transmitting opinions Olfati-Saber et al. [2007], Dai and Mesbahi [2011], Centola and Baronchelli [2015], habits Centola [2010], Farajtabar et al. [2014], money Gai and Kapadia [2010], Amini et al. [2016], Giudici and Spelta [2016] or viruses Pastor-Satorras and Vespignani [2001], Eguı́luz and Klemm [2002]. Being able to control, or just influence in some way, the dynamics of such complex networks may lead to important progress, for example, avoiding financial crises, preventing epidemic outbreaks or maximizing information spread in marketing campaigns. The control of the dynamics on networks is a very challenging problem that has attracted significant interest recently Liu et al. [2011], Cornelius et al. [2013], Gao et al. [2014], Yan et al. [2015]. Existing approaches typically consider network controllability as the controllability of 1 the dynamical system induced by the underlying network structure. While it is agreed that network controllability critically depends on the network structure, the problem of how to control the network structure itself while it is evolving remains open. The network structure is determined by the dynamics of addition and deletion of nodes and links over time. In this paper, we address the problem of influencing this dynamics in the framework of stochastic optimal control. The standard way to address these problems is through the Bellman equation and dynamic programming. Dynamic programming is only feasible in small problems and requires approximations when the state and action spaces are large. In the setting of network growth, this problem is more severe, since the state space increases (super-)exponentially with the number of nodes. In order to deal with this curse of dimensionality, we propose to approximate the network growth control problem by a special class of stochastic optimal control problems, known as Kullback-Leibler (KL) control or Linearly-Solvable Markov Decision Problems (LMDPs) Todorov [2009], Kappen et al. [2012]. For this class of problems, one can use efficient adaptive importance sampling methods that scale well in high dimensions. The optimal solution for the KL-control problem tends to be sparse, so that only a few next states become relevant, effectively reducing the branching factor of the original problem. The obtained solution of the KL-control problem is then used to compute the optimal action in the original problem that does not belong to the KL-control class. In the next section we present our proposed general methodology. We then apply it to a realistic problem: influencing the growth process of cascades in online forums, in order to maximize structural network measures that are connected to the quality of an online conversation thread. We conclude the paper with a discussion. 2 Optimal Network Growth as a Control Problem We now formulate the network growth control problem as a stochastic optimal control problem. Let xt ∈ X , with X being the set of all possible network structures, denote the growing structure (state) of the network at time t and let P (x0 \|x, u) describe the network dynamics, where the control variable u ∈ U denotes possible actions we can perform in order to manipulate the network. Let us label the default action, which means not interacting with the system, with u = 0. We denote the corresponding dynamics without control as the uncontrolled process p(x0 \|x) := P (x0 \|x, u = 0). At each time-step t, we incur an arbitrary cost function on the network state r(x, t) which is assigned when the state is reached. The state cost r(x, t) penalizes network structures that are not convenient in the particular context under consideration. For example, if one wants to favour networks with large average clustering coefficient C(x), then r(x, t) = −C(x). Alternatively, one can consider more complex functions, such as the structural virality or Wiener index Mohar and Pisanski [1988], as proposed recently Goel et al. [2015], to maximize the influence in a social network. In general, any measure that can be (efficiently) computed from x fits the presented framework. Our objective is to find the control function u(x, t) : X × N 7→ U which minimizes the total cost over a time horizon T, starting at state x at initial time t = 0 * T + X 0 C (x, t = 0, u(·)) = r (x, 0) + r xt0 , t , (1) t0 =t+1 2 P (x1:T \|x,u(·),t=0) where the expectation is taken with respect to the probability P (x1:T \|x, u(·), t = 0) over paths x1:T in the state space, given state x at time Q t = 0 using the control-function u(·). The probability of a path is given by P (xt+1:T \|x, u(·), t = 0) = T−1 s=t P (xs+1 \|xs , u(xs , s), s). Computing the optimal control can be done by dynamic programming Bertsekas [1995]. We introduce the optimal cost-to-go J(x, t) = min C (x, t, u(·)) , (2) u(·) which is an expectation of the cumulative cost starting at state x and time t and acting optimally thereafter. This can be computed using the Bellman equation J(x, t) = min r(x, t) + J(x0 , t + 1) P (x0 \|x,u,t) . (3) u From J(x, t), the optimal control is obtained by a greedy local optimization: u∗ (x, t) = argminu r(x, t) + J(x0 , t + 1) P (x0 \|x,u,t) . (4) In general, the solution to equation (3) can be computed recursively using dynamic programming Bertsekas [1995] for all possible states. This is however infeasible for controlling network growth, as the computation is of polynomial order in the number of states and the state space of networks increases super-exponentially on the number of nodes. E.g. for directed unweighed networks, there 2 are 2N possible networks with N labelled nodes. 3 Approximating the network growth problem by a Kullback-Leibler control problem In this section we present our main approach, which first computes the optimal cost-to-go on a relaxed problem and then uses it as a proxy for the original optimal cost-to-go. In the next subsection, we introduce the class of KL-control problems that we use as a relaxation. We then illustrate KL-control using a tractable example of tree growth. In subsection 3.3, we explain how can we approximate the KL-control solution using the cross-entropy method. Finally, in subsection 3.4 we show how can we use that result to compute the action selection in the original problem. 3.1 Kullback-Leibler control In order to efficiently compute the optimal cost-to-go, we make the assumption that our controls directly specify the transition probabilities between two subsequent network structures, e.g. P (x0 \|x, u(t)) ≈ u(x0 \|x, t). Further, we define the natural growth process of the network (the uncontrolled dynamics) as a Markov chain with transition probabilities p(x0 \|x). Because our influence on the network dynamics is limited, we add a regularization term to the total cost defined in equation (1) that penalizes deviations from p(x0 \|x). The approximated control cost becomes * T + X λ CKL (x, t, u(·)) =λKL [u (xt+1:T \|x, t) k p (xt+1:T \|x, t)] + r (x, t) + r xt0 , t0 , (5) t0 =t+1 3 u(xt+1:T \|x,t) with the KL-divergence KL [u (xt+1:T \|x, t) k p (xt+1:T \|x, t)] = u (xt+1:T \|x, t) log p (xt+1:T \|x, t) , u(xt+1:T \|x,t) which measures the closeness of the two path distributions, p (xt+1:T \|x, t) and u (xt+1:T \|x, t). The parameter λ thereby regulates the strength of this penalization. λ w.r.t. the control u(x0 \|x, t) With this assumption, the control problem consisting in minimizing CKL belongs to the KL-control class and has a closed form solution Todorov [2009], Kappen et al. [2012]. The probability distribution of an optimal path u∗KL (xt+1:T \|x, t) that minimizes equation (5) is u∗KL (xt+1:T \|x, t) = p (xt+1:T \|x, t) φ(xt+1:T ), hφ(xt+1:T )ip(xt+1:T \|x,t) with φ(xt+1:T ) := exp −λ −1 T X (6) ! 0 r(xt0 , t ) . (7) t0 =t+1 Plugging this into equation (5) and minimizing gives the optimal cost-to-go λ JKL (x, t) = r(x, t) − λ log hφ(xt+1:T )ip(xt+1:T \|x,t) , (8) which can be numerically approximated using paths sampled from the uncontrolled dynamics p(xt+1:T \|x, t). The optimal control corresponding to equation (4) corresponds to a state transition probability distribution that is obtained by marginalization in equation (6). It is expressed in terms of the uncontrolled transition probability p(x0 \|x) and the (exponentiated) optimal cost-to-go: λ (x0 , t + 1) X JKL ∗ 0 ∗ 0 0 uKL (x \|x, t) = uKL xt+1 = x , xt+2:T \|x, t ∝ p(x \|x) exp − . (9) λ x t+2:T This resembles a Boltzmann distribution with temperature λ where the optimal cost-to-go takes the role of an energy. The effect of the temperature becomes clear: for high values of λ, u∗KL (x0 \|x, t) deviates only a little from the uncontrolled dynamics p(x0 \|x), thus the optimal control has a weak influence on the system. In contrast, for low values of λ, the exponential in equation (9) becomes very λ (x0 , t + 1), suppressing the transition pronounced for the state(s) x0 with the smallest cost-to-go JKL 0 probabilities to suboptimal states x . Thus the control has a very strong effect on the process. In λ (x0 , t + 1) is not the limit of λ going to zero, the controlled process becomes deterministic, if JKL 0 degenerate (meaning there is a unique state x which minimizes the optimal cost-to-go). In this case the control is so strong that it overpowers the noise completely. We thus approximate our original (possibly difficult) control problem as a KL-control problem, parametrized by the temperature λ. The approximated optimal cost-to-go J(x0 , t + 1) of equation λ (x0 , t + 1) of (4) is replaced by the corresponding optimal cost-to-go of the KL-control problem JKL equation (9) and used to compute the action selection in the original problem. 3.2 A Tractable Example We now present a tractable example amenable for exact optimal control computation. This example already belongs to the KL-control class, so no approximation is made. The purpose of this analysis it 4 to show how different values of the temperature λ may lead to qualitatively different optimal solutions and other interesting phenomena. Let’s consider a tree that grows at discrete time-steps, starting with the root node at time t = 0. We represent the tree at time t as a vector xt = (x0 , x1 , ..., xt ), where xt indicates the label of the parent of the node attached at time t. At every time-step, either the tree remains the same or a new node is attached to it. The root node has label 1 and the label 0 is specially used to indicate that no node was added at a given time-step (it is also the label of the parent of the root node). The nodes are labelled in increasing order as they arrive to the tree, so that at time-step t, for a tree with k nodes, k ≤ t, xt = 0, 1, . . . , k corresponds to the parent of node k + 1 if a node is added or zero otherwise. Thus, the parent vector at time t = 1 is always x1 = (0, 1). Our example is a finite horizon task of T = 10 time-steps and end-cost only. The end-cost implements two control objectives: it prefers trees of large Wiener index while penalising trees with many nodes (more than five, in this case). The Wiener index is the sum of the lengths of the shortest paths between all nodes in a graph. It is maximal for a chain and minimal for a star. The uncontrolled process is biased to the root: new nodes choose to link the root with probability 3/5 and uniformly otherwise. More precisely ( 3 for j = 1 p(xt+1 = j\|xt ) = 5 2 (10) for (j = 0) or j ∈ {2, . . . , kxt k0 } 5kxt k0 ( −Wiener(xt )δt,T if kxt k0 < 5 r(xt , t) = (11) δt,T otherwise where kxk0 denotes the number of non-zero elements in x and Wiener the (normalized) Wiener index. In this setting, the uncontrolled process p tends to grow trees with more than five nodes with many of them attached to the root node, i.e. with low Wiener index. We want to influence this dynamics so that the target configuration, a chain of five nodes (maximal Wiener index) is more likely to be obtained. Figure 1 (top) shows the state cost r of the final tree that results from choosing the most probable control (MAP solution) as a function of the temperature λ. The exact solution is calculated using dynamic programming Kappen et al. [2012]. We can differentiate three types of solutions, denoted as A, B and C in the figure. For low temperatures (region A) the control aims to fulfil both control objectives: to find a small network with maximal Wiener index. The optimal strategy does not add nodes initially and then builds a tree of maximal Wiener index (see inset of initial controls in left column of the figure). This type of control (to wait while the target is far in the future) is reminiscent of the delayed choice mechanism described previously Kappen [2005]. This initial waiting period makes sense because if the chain of length 5 would be grown immediately, then at time 6 the size of 5 is already reached. If now an additional node attaches, then the final cost would be zero. However if one first waits and then grows the chain, an accidental node insertion before time 6 would not be so disastrous (actually it may help), as one can then just wait until time 7 to start growing the rest of the tree. So delaying the decision when to start growing the tree helps compensating accidental events. For intermediate temperatures (region B), the initial control becomes less extreme, as we observe if we compare the left plots between regions A and B. For λ ≈ 0.07, the solution that builds the tree with maximal Wiener index is no longer optimal, since it deviates too much from the uncontrolled 5 Most Probable solution 0 state cost C -0.5 B A -1 ∗ 1 A u 0.5 0 1 B 10 -2 (λ = 0.01) 0 u∗ 1 10 -1 10 0 λ 1 1 1 1 1 1 2 3 4 5 1 1 1 1 1 1 2 3 4 5 1 0 2 1 6 1 2 2 3 7 1 1 2 4 2 4 3 3 5 8 9 u∗ (λ = 100) 3 1 0 1 1 2 6 1 2 7 3 3 1 8 2 10 2 1 4 9 3 4 2 1 5 10 3 1 C 0.5 0 1 (λ = 0.29) 0.5 0 10 1 1 2 1 2 1 3 2 4 3 2 1 4 4 5 3 3 4 2 4 2 3 3 2 4 2 4 2 5 1 9 5 1 10 2 4 1 1 5 1 9 8 67 8 6 5 5 7 6 1 6 6 7 8 7 3 5 6 7 8 9 10 Figure 1: Example of optimal control of tree growth. (Top): the state cost of the most probable solutions as a function of the temperature λ. In region A, the optimal strategy waits until the last time-steps and then grows a tree with maximal Wiener index. In region B, it builds a star of five nodes. Finally, in region C, it follows the uncontrolled dynamics and builds a star of ten nodes. (Left): for each region, the optimal probabilities u∗ (xt+1 \|xt ) at t = 1 for the two actions which are initially available: no node addition (0) and adding a node to the root (1). In regions A, B the optimal control favours not adding a new node initially. The sequences on the right show how the tree grows. When a new node is added to the tree, it is coloured in red. dynamics. In region B, the control aims to build a network of five nodes or less, but no longer aims to maximize the Wiener index. The control is characterized by an initial waiting period and the subsequent growth of a tree of five nodes, which are in this case all attached to the root node. Finally, for high temperatures (region C, λ > 0.4), the control essentially ignores the cost r and the optimal strategy is to add one node to the root at every time-step, following the uncontrolled process. From these results we conclude that KL-control as a mechanism for controlling network growth can capture complex phenomena such as transitions between qualitatively different optimal solutions and delayed choice effects. 3.3 Sampling from the KL-optimally controlled dynamics In this subsection, we explain how we can sample from the optimally controlled dynamics and thereby λ (x, t) of equation (8). obtain an estimate of the optimal cost-to-go JKL The probability of an optimally controlled path, equation (6), corresponds to the product of the uncontrolled dynamics by the exponentiated state costs. Hence a naive way to obtain samples from the optimal dynamics, would consist in sampling paths from the uncontrolled dynamics p(x0 \|x) and 6 weight them by their exponentiated state costs. Using these samples we can then compute expectations from the optimally controlled dynamics. We use that for any function f (xt+1:T ) we have: + * φ(xt+1:T ) . hf (xt+1:T )iu∗ (xt+1:T \|x,t) = f (xt+1:T ) KL hφ(xt+1:T )ip(xt+1:T \|x,t) p(xt+1:T \|x,t) More precisely, provided a learned model or a simulator of the uncontrolled dynamics p(x0 \|x), we (i) (i) generate M sample paths xt+1:T , i = 1, . . . , M from p(x0 \|x) and compute the weights denominator thereby gives with equation (8) an estimate of the optimal cost-to-go as φ(xt+1:T ) φ̂ . The M hφ(xt+1:T )ip(xt+1:T \|x,t) 1 X (i) ≈ φ̂ := φ xt+1:T . M i=1 This method can be combined with resampling techniques Douc and Cappé [2005], Hol et al. [2006] opt,(i) to obtain unweighted samples xt+1:T from the optimal dynamics (for the numerical methods in this article, we have used structural resampling Douc and Cappé [2005], Hol et al. [2006]). Using such a naive sampling method, however, can be inefficient, specially for low temperatures. (i) φ(x ) t+1:T are more or less equal, for low temperWhile for high temperatures λ basically all weights φ̂ atures only a few samples with very large weights contribute to the approximation, resulting in very poor estimates. This is a standard problem in Monte Carlo sampling and can be addressed using the Cross-Entropy (CE) method De Boer et al. [2005], Kappen and Ruiz [2016], which is an adaptive importance sampling algorithm that incrementally updates a baseline sampling policy or sequence of controls. Here we propose to use the CE method in the discrete formulation and use a parametrized Markov process e uω (x0 \|x, t), with parameters ω, to approximate u∗KL . The CE method in our setting alternates the following steps: 1. In the first step, the optimal control is estimated using M sample paths drawn from a parametrized proposal distribution e uω (x0 \|x, t). 2. In the second step, the parameters ω are updated so that the proposal distribution becomes closer to the optimal probability distribution. As a proposal distribution e uω (x0 \|x, t), we use JeKL (x0 , ω(t)) e uω (x0 \|x, t) ∝ p(x0 \|x) exp − λ ! , (12) which has the same functional form as the optimally controlled transition probabilities in equation (9). The KL-optimal cost-to-go is thereby approximated by a linear sum of time-dependent feature vectors ψkt (x) X JeKL (x, ω(t)) = ωk (t)ψkt (x). (13) k The probability distribution of an optimally controlled path, equation (6), can be written as ! (i) T p xt+1:T \|x, t X (i) (i) (i) 0 ∗ −1 uKL xt+1:T \|x, t ∝ e uω (xt+1:T \|x, t) exp −λ r(xt0 , t ) . (14) (i) e uω (xt+1:T \|x, t) 0 t =t+1 7 This shows that we can draw samples from the proposal distribution and reweight them with the combined weights (i) p(xt+1:T \|x, t) (i) w(i) = φ xt+1:T . (i) e uω (xt+1:T \|x, t) The parameters ωk (t) of the importance sampler are initialized with zeros, which makes the initial proposal distribution equivalent to the uncontrolled dynamics. The procedure requires the gradients of e uω (x0 \|x, t) at each iteration. We describe the details of the CE method in A. We measure the efficiency of an obtained proposal control using the effective sample size (EffSS), which estimates how many effective samples can be drawn from the optimal distribution. Given M samples with weights w(i) , the EffSS is given by 1 PM (i) 2 i=1 w M EffSS = P (15) 2 . M 1 (i) i=1 w M If the weights w(i) are all about the same value, the EffSS is high, indicating that many samples contribute to statistical estimates using the weighted samples. If all weights are equal, the EffSS is equal to the number of samples M. Conversely if the weights w(i) have a large spread, the EffSS is low, indicating that only few independent samples contribute to statistical estimates. In the extreme case, when one weight is much larger then all others, the EffSS approaches 1. 3.4 Action selection using the KL-approximation λ , we need to select an action u ∈ U in the original Once we have an estimate of the cost-to-go JKL control problem, which is not of the KL-control type. We select the optimal action according to D E λ u∗ (x, t) ≈ argminu r(x, t) + JKL (x0 , t + 1) , (16) 0 P (x \|x,u,t) λ (x which requires the computation of JKL t+1 , t + 1) for every reachable state xt+1 . In growing networks, the number of possible next states (the branching factor) increases quickly, and visiting all of them soon becomes infeasible. In this subsection we highlight an important benefit of using the KL-approximation as a relaxation of the original problem: the optimally controlled process tends to discard many irrelevant states, specially for small values of λ. This means that u∗KL (x0 \|x, t) is sparse on x0 (only a few next λ (x0 , t) is very large for the corresponding x0 where states are relevant for the task), since the cost JKL u∗KL (x0 \|x, t) ≈ 0. opt Let xt+1:T denote a trajectory sampled from the optimally controlled process, as described in the previous section. We compute u∗KL (x0 \|x, t) using: û∗KL (x0 \|x, t) = δxopt (t+1),x0 u∗KL (xt+1:T \|x,t) , (17) where xopt (t + 1) is the first element of the trajectory and δxopt (t+1),x0 is the Kronecker delta which is equal one if xopt (t + 1) is equal to x0 , and zero otherwise. We then compute the optimal cost using equation (9): ∗ ûKL (x0 \|x, t) λ 0 JKL (x , t + 1) ∼ − log , (18) p(x0 \|x, t) 8 Figure 2: Task illustration: example of an Internet news forum. News are posted periodically and users can write comments either to the original post or to other user’s comments, forming a cascade of messages. The figure shows an example of conversation thread taken from Slashdot about Google’s AlphaGo. The control task is to influence the structure of the conversation thread (shown as a growing tree in the top-right). where we dropped a term which does not depend on x0 and therefore plays no role in the minimization of equation (16). The KL-approximation can help reducing the branching factor because it needs only λ (x0 , t + 1) only for the x0 where u∗ (x0 \|x, t) > 0 and thus J λ (x0 , t) a few samples to calculate JKL KL KL has a finite value. As mentioned earlier, u∗KL (x0 \|x, t) tends to be more sparse for small values of λ, when the KLcontrol problem is less noisy. In B we provide analytical details of the two extreme conditions, when λ is zero or infinite, respectively. 4 Application to Conversation Threads We have described a framework for controlling growing graphs. We now illustrate this framework in the context of growing information cascades. In particular, we focus on the task of controlling the growth of online conversation threads. These are information cascades that occur, for example, in online forums such as weblogs Leskovec et al. [2007], news aggregators Gómez et al. [2008] or the synthesis of articles of Wikipedia Laniado et al. [2011]. In conversation threads, after an initial post appears, different users react writing comments either to the original post or to comments from other users. 9 Figure 3: Our proposed control mechanism: in addition to the the threaded conversation, we highlight a comment (red node in the growing tree), suggested to be replied by the user. The choice of suggested comment, shown at the bottom of the page, is calculated using the method described in section 3.3. Figure 2 shows an example of a conversation thread, taken from Slashdot (www.slashdot.org). Users see a conversation thread using a similar hierarchical interface. The task we consider is to optimize the structure of the generated conversation thread while it grows. The state is thus defined as a growing tree. We assume an underlying (not observed) population of users that keep adding nodes to this tree. Since we can not control directly what is the node that will receive the next comment, we propose the user interface as a control mechanism to influence indirectly the growth process. This can be done in different ways, for example, manipulating the layout of the comments. In our case, the control signal will be to recommend a comment (by highlighting it) to which the next user can reply. Figure 3 illustrates such a mechanism. The action selection strategy introduced in section 3.4 is used to select the comment to highlight. Our goal is thus to modify the structure of a cascade in certain way while it evolves, by influencing its growth indirectly. It is known that the structure of online threads is strongly related with the complexity of the underlying conversation Gómez et al. [2008], Gonzalez-Bailon et al. [2010]. To fully define our control problem, we need to specify the structural cost function, the uncontrolled dynamics, i.e. the equivalent of equations (10) and (11) for this task, and a model of how an action (highlighting a node) changes the dynamics. Globally, this application differs from the toy example of subsection 3.2 in some important ways: 1. The state-space is larger (threads typically receive more than 10 comments). 2. We choose as state-cost function the Hirsch index (h-index), which makes the control task 10 highly non-trivial. 3. The original problem is not a KL-control problem. We use the action selection method described in section 3.4 to control the growth of the conversation thread. 4.1 Structural Cost Function We propose to optimize the Hirsch index (h-index) as structural measure. In our context, a cascade with h-index h has h comments each of which have received at least h replies. It is a sensible quantity to optimize, since it measures how distributed the comments of users on previous comments are. A high h-index prevents two extreme cases that occur in a rather poor conversation: the case where a small number of posts attract most of the replies, thus there is no interaction, and the case with deep chains, characteristic of a flame war of little interest for the community. Both cases have a low h-index, while a high h-index spreads the conversation over multiple levels of the cascade. The h-index is a function of the degree sequence of all nodes in the tree, where the degree of a node is this case is the number of replies plus one, as there is also a link to the parent (replied comment or post). Therefore we use the degree histogram as features ψkt (x) for the parametrized form of the optimal cost-to-go, equation (13). That is, feature ψkt (x) is the number of nodes with degree k in the tree x at time-step t. We model the problem as a finite horizon task with end-cost. Thus, the state cost is defined as r(x, t) = −δt,T · h(x), where h(x) is the h-index of the tree x. 4.2 Uncontrolled Dynamics for Online Conversation Threads As uncontrolled dynamics, we use a realistic model that determines the probability of a comment to attract the replies of other users at any time, by means of an interplay between the following features: • Popularity α: number of replies that a comment has already received. • Novelty τ : the elapsed time since the comment appeared in the thread. • Root node bias β: characterizes the level of trendiness of the main post. Such a model has proven to be successful in capturing the structural properties and the temporal evolution of discussion threads present in very diverse platforms Gómez et al. [2013]. Notice that these features θ = (α, τ, β) should not be confused with the features ψkt (x) used to encode the costto-go. We represent the conversation thread as a vector of parents xt = (x0 , x1 , ..., xt ). Given the current state of the thread xt , the uncontrolled dynamics attaches a new node t + 1 to an existing node j with probability 1 pθ (xt+1 = j\|xt ) = degj,t α + δj,1 β + τ t+1−j (19) Zt+1 with Zt+1 a normalization constant, degj,t the degree of node j at time t and δj,1 the Kronecker delta function, so parameter β is only nonzero for the root. Given a dataset composed of S threads D := {x(1) , . . . , x(S) } with respective sizes \|x(k) \|, k ∈ {1, . . . S}, the parameter vector θ can be learned by minimizing (k) − log L(D; θ) = − S \|x X X\| k=1 t=2 11 (k) (k) log pθ (xt+1 \|xt ). We learn the parameters using the Slashdot dataset, which consists of S = 9, 820 threads, containing more than 2 · 106 comments among 93, 638 users. In Slashdot, the most relevant feature is the preferential attachment, as detailed in Gómez et al. [2013]. This will have implications in the optimal control solution, as we show later. 4.3 Control interaction The control interaction is done by highlighting a single comment of the conversation. We assume a behavioural model for the user inspired by Craswell et al. [2008], where the user looks at the highlighted comment and decides to reply or not. For simplicity, we assume that the user chooses the highlighted comment with a fixed probability p0 = α/(1 + α) and with probability 1 − p0 she chooses to ignore it. If the highlighting of the comment is ignored, the thread grows according to the uncontrolled process. Therefore, α parametrizes the strength of the influence the controller has on the user. For α → ∞, we can fully control the behaviour and for α = 0, the thread evolves according to the uncontrolled process. A typical control would have a small α as usually the influence of an controlling agent on a social systems is weak. 4.4 Experimental Setup To evaluate the proposed framework we use a simulated environment, without real users. We consider a finite horizon task with T = 50 with the goal to maximize the h-index at end-time, starting from a thread with a single node as initial condition. The state-space consists of 50! ≈ 364 states. The thread grows in discrete time-steps. At each time-step, a new node is added to the thread by a (simulated) user. For that, we first choose which node to highlight (optimal action) as described in section 3.4 using equation (16). We then simulate the user as described in section 4.3, so the highlighted node is selected with probability p0 = α/(1 + α) as the parent of the new node. Otherwise, with probability 1 − p0 , the user ignores the highlighted node and the parent of the new node is chosen according to the Slashdot model, equation (19). This is repeated until the end time. 4.5 Experimental Results We first analyse the performance of the adaptive importance sampling algorithm described in section 3.3 for different fixed values of λ. Figure 4 shows the effective sample size (EffSS), equation (15) as a function of the number of iterations of the CE method. We observe that the EffSS increases to reach a stable value. As expected, large temperature (easier) problems result in higher values of EffSS. We can also see that, even for hard problems with low temperature, the obtained EffSS is significantly larger than zero, which allows us to compute the KL-optimal control. In general, the curves are less smooth for smaller values of λ, because a few qualitatively better samples dominate the EffSS, resulting in higher variance. On the other hand we also observe that the EffSS never reaches 100%. This is expected, as this would mean that our parametrized importance sampler perfectly resembles the optimal control, and this is not possible due to the approximation error introduced by the use of features. We can better understand the learned control by analysing the linear coefficients of the parametrized optimal cost-to-go, equation (13), for this problem. Figure 5 shows the feature weights ωk (t), at different times t = 1, . . . , T, after convergence of the CE method. Feature k corresponds to the number of nodes with degree k in the tree, after a new node arrives. The parent node to which the new node 12 100 λ=1 λ = 1/3 λ = 0.2 λ = 0.1 90 80 EffSS in % 70 60 50 40 30 20 10 0 20 40 60 80 Iteration 100 120 140 Figure 4: Evaluation of the inference step: The Effective sampling size (EffSS) increases after several iterations of the cross-entropy method. As expected, large values of the temperature λ result in higher values of EffSS. We use M = 105 samples to compute the EffSS. The EffSS is measured here in percent of the maximum number of samples M . feature weight forλ=0.2 0.1 5 10 0.05 15 0 Time 20 -0.05 25 30 -0.1 35 -0.15 40 -0.2 45 5 10 15 20 feature degree of parent 25 30 Figure 5: The learned importance sampler: The figure shows the time-dependent parameters of the learned expected cost-to-go for λ = 0.2. Each pixel is the parameter of a feature at a certain time. The features are the degrees of the parent node after the new child attaches. The colour represents the weight of the parameter. Large negative weights (pixels in blue colour) stand for a low cost and thus a desirable state, while large positive weights (red pixels) stand for high cost and thus undesirable states. At all times there is a desirable degree which the parent should have and higher as well as lower degrees are inhibited. This desirable degree is small at early times and becomes larger at later times. 13 has attached is thereby the only node whose degree changes (the degree increases by 1). Thus a high weight for a feature which measures the number of nodes with a certain degree k results in a low probability of attaching to a node with degree k − 1. Conversely low, or large negative weights thus correspond to nodes which have a high probability of becoming the parent of the next node which is added. We observe that there is an intermediate preferred degree (large negative weight, in blue). This is the preferred degree of the parent of the new node, and this preferred degree increases with time, reaching a value of 5 at t = 50. Does this strategy make sense? The maximum h-index of a tree of 50 nodes is 7, and it is achieved if 6 nodes have exactly 7 children and one node has 8. However, achieving such a configuration requires a very precise control. For example, increasing too much the degree of a node, say up to 9, prevents the maximum h-index to be reached, as there are not enough links left, due to the finite horizon. Thus, in this setting, steering for the maximal possible h-index is not optimal. The controller prefers all parents to have a degree of 5 and not less, but also not much more. As having more than five parents with degree at least five will result in an h-index of 5 we conclude that the control seems to aim for a target h-index of 5, while preventing wasting links to higher or lower degree nodes, which would not contribute to achieve that target. The interpretation of why the preferred degree increases with time involves the uncontrolled dynamics. Remember that the most relevant term in equation (19) for the considered dataset corresponds to the preferential attachment, parametrized by α. This term boosts high-degree nodes to get more links. If this happens, most of the links end up attached to a few parents, and this effect can only be suppressed by a strong control. The controller prevents that self-amplifying effect by aiming initially for an overall low degree, preventing a high impact of the preferential attachment. This keeps the process controllable and allows for a more equal distribution of the links. After having evaluated the sampling algorithm, we evaluate the proposed mechanism for actual control of the conversation thread. As described in section 3.4, in our simulated scenario, we highlight the node as the parent which minimizes the computed expected cost-to-go. Figure 6 shows the evolution of the h-index using different control mechanisms. The blue curve shows how the h-index changes under the uncontrolled dynamics. On average, it reaches a maximum of about 3.7 after 50 time steps. In green, we show the evolution of the h-index under a KL-optimal controlled case, for temperature λ = 0.2. As expected, we observe a faster increase, on average, than using the uncontrolled dynamics. The maximum is about 4.7. The red and black curves show the evolution of the h-index using the control mechanism described λ of the KLin subsections 3.4 and 4.3, where we select actions using the expected cost-to-go JKL optimal control with λ = 0.2, for α = 1 and α = 0.5, respectively. In both cases the obtained h-index is even higher than the one obtained with the KL-control relaxation. Therefore, the objective for this task, to increase the h-index, can be achieved through our action selection strategy. As expected, a stronger interaction strength α = 1 leads to higher h-indices than a lower strength α = 0.5. Finally, in Figure 7 we show examples of a real discussion thread from the dataset (Slashdot), a thread generated from the learned model (uncontrolled process) and one resulting from applying our action selection strategy. The latter has higher h-index. 5 Discussion We have addressed the problem of controlling the growth process of a network using stochastic optimal control with the objective to optimize a structural cost that depends on the topology of the 14 6 p 5.5 u max-costtogo α=1 5 max-costtogo α=0.5 4.5 h-index 4 3.5 3 2.5 2 1.5 1 0 5 10 15 20 25 time 30 35 40 45 50 Figure 6: Evaluation of the actual control: uncontrolled dynamics (blue), KL-optimally controlled dynamics (green) action selection based control for α = 1 (red) and α = 0.5 (black). The KLoptimally controlled dynamics, which optimize the sum of the λ-weighted KL-term and the end cost, shifts the final mean value from about 3.7 to about 4.7. The action selection based control, which is aiming to optimize the end cost only, is able to shift the h-index to even higher values then the KL-optimal control. For the controlled dynamics, λ = 0.2 for all three cases. To compute the control in each time-step we sample 1000 trajectories. The statistics where computed using 1000 samples for each of the three cases. growing network. The main difficulty of such a problem is the exploding size of the state space, which grows (super-)exponentially with the number of nodes in the network and renders exact dynamic programming infeasible. We have shown that a convenient way to address this problem is using KL-control, where a regularizer is introduced which penalizes deviations from the natural network growth process. One advantage of this approach is that the optimal control can be solved by sampling. The difficulty of the sampling is controlled by the strength of the regularization, which is parametrized by a temperature parameter λ: for high temperatures the sampling is easy, while for low temperatures, it becomes hard. This is in contrast to standard dynamic programming, whose complexity is directly determined by the number of states and independent of λ. In order to tackle the more challenging low temperature case, we have introduced a featurebased parametrized importance sampler and used adaptive importance sampling for optimizing its parameters. This allows us to sample efficiently in the low temperature regime. For control problems which cannot directly be formulated as KL-control problems, we have proposed to use the solution of a related KL-control problem as a proxy to estimate the effective values of possible next network states. These expected effective values are subsequently used in a greedy strategy for action selection in the original control problem. This action selection mechanism benefits from the sparsity induced by the optimal KL-control solution. We have illustrated the effectiveness of our method on the task of influencing the growth of 15 Slashdot thread Uncontrolled thread Controlled thread Figure 7: Examples of threads. A thread from the data (Slashdot), an uncontrolled thread generated from the model and a controlled thread. The nodes that contribute to the h-index are coloured in yellow. The h-index for the data and the uncontrolled thread is 4 and 6 for the controlled one. conversation cascades. Our control seeks to optimize the structure of the cascade, as it evolves in time, to maximize the h-index at a final time. This task is non-trivial and characterized by a sparse, delayed reward, since the h-index remains constant during most of the time, and therefore a greedy strategy is not possible. Our approach for controlling network growth is inspired in recent approaches to optimal decisionmaking with information-processing constraints Todorov [2009], Tishby and Polani [2011], Kappen et al. [2012], Theodorou and Todorov [2012], Rawlik et al. [2012]. The Cross-Entropy method has been explored previously in the continuous case Kappen and Ruiz [2016]. The continuous formulation of this class of problems has been used in robotics, using parametrized policies Theodorou et al. [2010], Levine and Koltun [2013], Gómez et al. [2014]. In economics, the question of altering social network structure in order to optimize utility has been addressed mainly from a game theoretical point of view, under the name of strategic network formation Jackson and Watts [2002], Bloch and Jackson [2007]. To the best of our knowledge, the problem of network formation has not yet been addressed from a stochastic optimal control perspective. The standard approach to address the problem of controlling a complex, networked system is to directly try to control the dynamics on the network Liu et al. [2011], Cornelius et al. [2013]. This approach considers the classical notion of structural controllability as the capability of being driven from any initial state to any desired final state within finite time. Optimal control in thus referred to the situation where a network can be fully controlled using only one driving signal. This idea is also prevalent in the influence maximization problem in social networks Kempe et al. [2003], Farajtabar et al. [2014, 2015], which consists in finding the subset of driver (most influential) nodes in a network. Since the controllability of the dynamics on the network depends crucially on the topology, several works have considered the idea of changing the network structure is some way that favours structural controllability. For example, the perturbation approach introduced in Wang et al. [2012] looks for the minimum 16 number of links that needs to be added so that the perturbed network can be fully controlled using a single input signal. In Hou et al. [2015], a method to enhance structural controllability of a directed network by changing the direction of a small fraction of links is proposed. More recently, Wang et al. [2016] analyzed node augmentation of directed networks while insisting that the minimum number of drivers remains unchanged. The main difference between our approach and these approaches is that, rather than considering the controllability of the dynamical system on the underlying network, our optimal control task is defined on the structure of the network itself, regardless of the dynamical system defined on it. In some sense, our results complement these approaches. For example, one could use our optimal control approach to shape the growth of the network in a way that the structural controllability, understood as the state cost function, is optimized. Acknowledgements This project is co-financed by the Marie Curie FP7-PEOPLE-2012-COFUND Action, Grant agreement no: 600387, the Marie Curie Initial Training Network NETT, project N. 289146 and the Spanish Ministry of Economy and Competitiveness under the Marı́a de Maeztu Units of Excellence Programme (MDM-2015-0502). A Adaptive Importance Sampling for KL-Optimal Control Computation using the Cross-Entropy method Here we show how the time-dependent weights ωk (t) of the importance sampler are updated such that e uω (x0 \|x, t) becomes closer to the optimal sampling distribution. This corresponds to the second step of the Cross-Entropy method described in subsection 3.3. For clarity in the derivations, we will replace p(x1:T \|x, 0) and u∗KL (x1:T \|x, 0) by p and u∗KL , respectively, in the expectations. The closeness of the two distributions e uω (x0 \|x, t) and u∗KL (x0 \|x, t) can be measured as the cross entropy between the path x1:T probabilities under these two Markov processes: u∗KL (x1:T \|x, 0) ∗ KL [uKL (x1:T \|x, 0) k e uω (x1:T \|x, 0)] = log e uω (x1:T \|x, 0) u∗ KL = − hlog e uω (x1:T \|x, 0)iu∗ + const. =: −D(ω), KL (20) where the constant term hlog u∗KL (x1:T \|x, 0)iu∗ is dropped. KL We minimize equation (20) by gradient descent. At iteration l, the gradient D(ω (l) ) with respect to ωk (t) is given by ∂D(ω (l) ) =− ∂ωk (t) ∂ log e uω(l) (x1:T \|x, 0) ∂ωk (t) u∗ KL 17 where T−1 Y 1 JeKL (xt+1 , ω(t)) e uω(l) (x1:T \|x, 0) = p(x1:T \|x, 0) exp − Z λ t=0 T−1 !+ Y JeKL (xt0 +1 , ω(t0 )) Z= exp − λ 0 t =0 ! p with the normalization constant Z. This leads to !+ T−1 e X ∂D(ω (l) ) ∂ JKL (xt0 +1 , ω(t0 )) log p(x1:T \|x, 0) − =− − log Z ∂ωk (t) ∂ωk (t) λ 0 (21) u∗KL t =0 where we can drop the first term as it is independent of ω(t). The second term can be evaluated using the definition of JeKL , equation (13). Further, plugging in Z we get * T−1 !+ + Y ∂ JeKL (xt0 +1 , ω(t0 )) ∂D(ω (l) ) −1 t =λ ψk (xt+1 ) u∗ + log exp − KL ∂ωk (t) ∂ωk (t) λ 0 t =0 = λ−1 ψkt (xt+1 ) = λ−1 ψkt (xt+1 ) u∗KL u∗KL p T−1 Y ∂ JeKL (xt0 +1 , ω(t0 )) log exp − ∂ωk (t) λ t0 =0 T−1 !+ Y 1 ∂ JeKL (xt0 +1 , ω(t0 )) + exp − Z ∂ωk (t) 0 λ + t =0  u∗KL !+ p p !+  0 e 0 1 JKL (xt +1 , ω(t ))  = λ−1  ψkt (xt+1 ) u∗ − ψkt (xt+1 ) exp − KL Z λ 0 t =0 p = λ−1 ψkt (xt+1 ) u∗ − ψkt (xt+1 ) eu (x \|x,0) KL ω (l) 1:T  D E p(x1:T \|x,0) t (x φ (x ) ψ ) 1:T k t+1   euω (x1:T \|x,0) e uω(l) (x1:T \|x,0) D E = λ−1  − ψkt (xt+1 ) eu (x \|x,0)  ,  p(x1:T \|x,0) ω (l) 1:T φ (x ) 1:T e uω (x1:T \|x,0) T−1 Y e uω(l) (x1:T \|x,0) (22) where we have used the estimates from the importance sampling step and equation (9). The update rule for the parameters becomes (l+1) ωk (l) (t) = ωk (t) + η ∂D(ω (l) ) , ∂ωk (t) for some learning rate η. Algorithm 1 summarizes the CE method applied to this context. 18 (23) Algorithm 1 Cross-Entropy Method for KL-control Require: importance sampler e uω , feature space ψ(·), number of samples M, learning rate η l←0 (l) ωk (t) ← 0, Initialize weights for all k, t, l (i) xt+1:T ← draw M sample trajectories ∼ e uω(l) , i = 1, . . . , M repeat (l) ) compute gradient ∂D(ω ∂ωk (t) using equation (21) (l+1) ωk (l) (l) ) (t) ← ωk (t) + η ∂D(ω ∂ωk (t) for all k, t, l (i) xt+1:T ← draw M samples ∼ e uω(l+1) l ←l+1 until convergence B Analyzing the KL-optimal cost-to-go based action selection We have introduced an action selection framework which is based on an approximation of the optimal λ (x0 , t + 1) of a parametrized family of KL-control cost-to-go J(x0 , t) by the optimal cost-to-go JKL problems which share the same state cost r(x, t). Why is this a good idea? Consider the two extreme cases where the temperature λ, which parametrizes the family of equivalent KL-control problems, is zero or infinite, respectively. Extreme case λ → 0 (zero temperature): The total cost in the KL-control problem becomes equal to the total cost in the original control problem, equation (1), as the KL term vanishes. The KL-optimal control becomes deterministic: λ 0 ( J (x ,t+1) λ (x0 , t + 1) p(x0 \|x) exp − KL λ 1 for x0 = argmin JKL lim u∗KL (x0 \|x, t) = lim = , (24) λ→0 λ→0 Z 0 otherwise where Z is a normalization constant. Thus, for λ → 0, the KL-control problem becomes identical to the original problem if the system is fully controllable, i.e. for every t, x and x̃ there is a ux̃ ∈ U such that p(x0 \|x, t, ux̃ ) = δx̃,x0 . Extreme case λ → ∞ (infinite temperature): For this case, using equation (8) we get ∞ λ JKL (x, t) = lim JKL (x, t) λ→∞ * T X = r(x, t) − lim λ log  exp −λ−1 λ→∞ * = r(x, t) + T X t0 =t+1 t0 =t+1 + r(xt0 , t0 ) . p(xt+1:T \|x,t) 19 !+ r(xt0 , t0 )   p(xt+1:T \|x,t) Using equation (1) and the definition of the uncontrolled dynamics, we can write * T + X ∞ JKL (x, t) = r (x, t) + r xt0 , t0 = C (x, t, 0) . t0 =t+1 (25) P (xt+1:T \|x,0,t) Thus, for λ → ∞, the KL-optimal cost-to-go becomes equal to the total cost in the original control problem under the uncontrolled dynamics (using u = 0). Having this equation (16) can be written as u∗ (x, t) ≈ argminu r(x, t) + C x0 , t + 1, 0 P (x0 \|x,u,t) . (26) In this case, the action selection is equivalent to optimize an expected total cost assuming the system will evolve according to the free dynamics in the future. 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Submitted Spectral Precoding for Out-of-band Power Reduction under Condition Number Constraint in OFDM-Based System Lebing Pan 1 No.50 Research Institute of China Electronic Technology Group Corporation, Shanghai 200311, China. forza@aliyun.com Abstract: Due to the flexibility in spectrum shaping, orthogonal frequency division multiplexing (OFDM) is a promising technique for dynamic spectrum access. However, the out-of-band (OOB) power radiation of OFDM introduces significant interference to the adjacent users. This problem is serious in cognitive radio (CR) networks, which enables the secondary system to access the instantaneous spectrum hole. Existing methods either do not effectively reduce the OOB power leakage or introduce significant biterror-rate (BER) performance deterioration in the receiver. In this paper, a joint spectral precoding (JSP) scheme is developed for OOB power reduction by the matrix operations of orthogonal projection and singular value decomposition (SVD). We also propose an algorithm to design the precoding matrix under receive performance constraint, which is converted to matrix condition number constraint in practice. This method well achieves the desirable spectrum envelope and receive performance by selecting zero-forcing frequencies. Simulation results show that the OOB power decreases significantly by the proposed scheme under condition number constraint. Keyword: Spectral precoding, Out-of-band, Orthogonal frequency division multiplexing (OFDM), Sidelobe suppression, Condition number constraint. 1. Introduction Dynamic spectrum access [1, 2] technology is extensively studied as an effective scheme to achieve high spectral efficiency, which is a crucial step for cognitive radio (CR) networks. Due to the flexible operability over non-continuous bands, orthogonal frequency division multiplexing (OFDM) is considered as a candidate transmission technology for CR system [3]. However, due to the use of rectangular pulse shaping, the power attenuation of its sidelobe is slow by the square of the distance to the main lobe in the 1 Submitted frequency domain. Therefore, the out-of-band (OOB) power radiation or sidelobe leakage of OFDM causes severe interference to the adjacent users. Furthermore, this problem is serious in CR networks, which enables the secondary users to access the instantaneous spectrum hole. Therefore, these secondary users need to ensure that the interference level of the power emission is acceptable for primary users. In practice, typically of the order of 10% guard-band is needed for an OFDM signal in the long term evolution (LTE) system [4]. Therefore, the spectrum efficiency is significantly reduced. The traditional method of sidelobe suppression is based on windowing techniques [5], such as the raised cosine windowing [6], which is applied to the time-domain signal wave. However, this scheme requires an extended guard interval to avoid signal distortion, and the spectrum efficiency is also reduced for large guard intervals. The cancellation carriers (CCs) [7, 8] technique inserts a few carriers at the edge of the spectrum in order to cancel the sidelobe of the data carriers. However, this technique degraded the signalto-noise ratio (SNR) at the receiver. The subcarrier weighting method [9] is based on weighting the individual subcarriers in a way that the sidelobe of the transmission signal are minimized according to an optimization algorithm. However, the bit-error-rate (BER) increases in the receiver, and when the number of subcarriers is large, it is difficult to implement in real-time scenario. The multiple choice sequence method [10] maps the transmitted symbol into multiple equivalent transmit sequences. Therefore, the system throughput is reduced when the size of sequences set grows. Constellation adjustment [11] and constellation expansion [12] are difficult to implement when the order of quadrature amplitude modulation (QAM) is high. Strikingly, the methods [10, 11] require the transmission of side information. The adaptive symbol transition [13] scheme usually provides weak sidelobe suppression in frequency ranges closely neighboring the secondary user occupied band. However, the schemes [7-14] depend heavily on the transmitted data symbols. The precoding technology is widely used in OFDM system to enhance the performance of transmission reliability over the wireless environment, in which there are many precoding methods 2 Submitted proposed for OOB power reduction [15]. There are two main optimization schemes with obtaining large suppression performance. One is to force the frequency response of some frequency points to be zero by orthogonal projection [16-19]. These frequency points are regarded as zero-forcing frequencies. The other method [20, 21] is to minimize the power leakage in an optimization frequency region by adopting the quadratic optimization method, using matrix singular value decomposition (SVD). The precoding scheme in [16] is designed to satisfy the condition that the first N derivatives of the signals are continuous at the edges of symbols. However, this method introduces an error floor in error performance. The sidelobe suppression with orthogonal projection (SSOP) method in [19] adopts one reserved subcarrier for recovering the distorted signal in the receiver. To maintain the BER performance, data cost is introduced in [17] by exploiting the redundant information in the subsequent OFDM symbol. The sidelobe suppression in [21] is based on minimizing the OOB power by selecting some frequencies in an optimized region. The suppression problem is first treated as a matrix Frobenius norm minimization problem, and the optimal orthogonal precoding matrix is designed based on matrix SVD. In [20], an approach is proposed for multiuser cognitive radio system. This method ensures user independence by constructing individual precoder to render selected spectrum nulls. Unlike the methods that focus on minimizing or forcing the sidelobe to zero, the mask compliant precoder in [22] forces the spectrum below the mask by solving an optimization problem. However, the algorithm leads to high complexity. In this paper, a spectral precoding scheme is proposed with matrix orthogonal projection and SVD for OOB reduction in OFDM-based system. The main idea of this scheme is to reduce the OOB power under the receive quality. The condition number of the precoding matrix indicates the BER loss in the receiver. Therefore, we develop an iteration algorithm to design the precoding matrix under matrix condition number constraint. The proposed method has an appropriate balance among suppression performance, spectral efficiency and receive quality. Consequently, it is flexible for practical implementation. 3 Submitted The rest of the paper is organized as follows. In Section 2, we introduce the system model of OOB power reduction by the spectral precoding method. In Section 3, we present the proposed spectral precoding approach. Next, in Section 4, we provide an iteration algorithm to design the precoding matrix according to the desirable spectrum envelope, spectral efficiency and BER performance. Simulations are presented in Section 5 to demonstrate the performance of the proposed method, followed by a summary in Section 6. s d Precoding Add Guard IFFT DAC&UP Conversion Channel d s Decoding Remove Guard FFT Down Conversion &ADC Fig. 1 System diagram of spectrally precoded OFDM. 2. System Model The block diagram of a typical OFDM system using a precoding technique is illustrated in Fig. 1. The number of total carriers used in the transmitter is M . The digital spectral precoding process before inverse fast Fourier transform (IFFT) operation is expressed as s  Pd, (1) where d is the original OFDM symbol of size N 1 , and s is the precoded vector. The size of the precoding matrix P is M  N ( M  N ) and the coding redundancy R  M  N is usually small. The spectral precoding method achieves better suppression performance in zero-padding (ZP) OFDM system than cyclic-prefix (CP) OFDM [19, 21]. What’s more, the ZP scheme has already been proposed as an alternative to the CP in OFDM transmissions [23] and particularly for cognitive radio [24]. The proposed method also can be directly applied to CP systems with degraded performance on sidelobe suppression. 4 Submitted Conventionally, power spectral density (PSD) analysis for multicarrier systems is based on an analog model with a sinc kernel function [25]. The PSD converges to the sinc function with the sampling rate increasing. In addition, the oversampling constraint presented in [26] ensures the desirable power spectral sidelobe envelope property after precoding for DFT-based OFDM. In a general OFDM system, the time-domain signal can be defined by a rectangular function (baseband-equivalent) as 0t T elsewhere 1, g (t )   0, (2) where T is the symbol duration. The frequency domain representation of m-th subcarrier is written as Gm ( )  e jT /2 sin((  m )T / 2) , (  m ) / 2 (3) where m is the center frequency of m-th subcarrier. Therefore, the magnitude envelope in the OOB region (   m ) is expressed by Gm ( )  sin((  m )T / 2) (  m ) / 2 2  .   m (4) Then we define the function Cm ( ) to indicate the magnitude envelope by Cm ( )  1 .   m (5) The expression (5) is obtained from (4) by ignoring a constant factor. This operation does not affect the mathematical analysis for the design of the precoding matrix [19]. The complex computing by (3) is converted to real operation, which reduce the complexity of spectral precoding in the following. Based on (5), using the superposition of M carriers, the target function indicating the PSD at OOB frequency  is defined by 5 Submitted A( )  M 1 s C m0 m m 2 ( ) , (6) where sm is the m-th element in the precoded OFDM vector s . In OFDM system, the power spectrum of its sidelobe decays slowly as ( )2 , where     m is the frequency distance to the mainlobe. From (1) and (6), the PSD target function of the OFDM signal is expressed in the matrix form as P( )    2 1 1   A( )   cT Pd , Ts Ts (7) where c  C0 ( ), C1 ( ),...CM 1 ( ) , (.)T and  denote transpose operation and expectation respectively. T P( ) and C ( ) are not the PSD presentation of the OFDM signal and the frequency response respectively, but indicate their envelope character. The goal of the precoding method is to design P to reduce the emission in OOB region. Simultaneously, the process is irrespective of the value of vector d . 3. The Proposed Joint Spectral Precoding (JSP) Method In this paper, a joint spectral precoding (JSP) method is proposed with two times orthogonal projection and one time SVD. The process is given by three steps: Inner orthogonal projection → SVD → Outside orthogonal projection. As presented in the previous papers [19, 21], in each step, the main idea of orthogonal projection is to force the power of zero-forcing frequency points to be zero, and the SVD operation is to minimize the power in optimized region. However, the precoding matrix derived from the final step does not achieve the goal in each step again. Unlike the methods that focus on minimizing or forcing the sidelobe to zero, we reduce the OOB power under the receive quality constraint by selecting the frequency points in each step. Therefore, the proposed method has an appropriate balance between the suppression performance and the receive quality. In addition, when the number of the reserved carriers is small, such as only one, then the number of zero-forcing frequencies in orthogonal projection method is one. But in the JSP, we set the zero-forcing frequency in inner and outside orthogonal projection to be different and better suppression performance is obtained. 6 Submitted data carriers reserved carriers zero-forcing frequency … optimized region … …  Fig. 2 The spectrum diagram for OOB power reduction. Generally, the subcarriers of OFDM-based system are considered to be continuous. The proposed method also can be used for non-continuous multi-carriers system. The number of employed carriers is M and the index is from 0 to M  1 . Fig. 2 illustrates the frequency region of   m . The R reserved carriers at the upper and lower spectral edges, which are used to achieve sidelobe suppression and maintain the receive quality. Two groups of zero-forcing frequency points ω a and ωb are selected in inner and outside orthogonal projection respectively. The corresponding number is N a and N b ( N a  R , Nb  R ). K frequency points in the optimized region are chosen to reduce the OOB power by SVD operation. We first give the process of JSP in this section, and the detail of how to select the parameters is presented in the next. In the first step, the group of zero-forcing frequency points ω a ( ωa  [a1 ,...ai ,...aN ] , ai  0 or a a  M 1 ) are chosen to achieve i Ca Padˆ  0, (8) Where d̂ is a vector of size M , derived from the original data symbol d N1 by adding zero in the location of the reserved carriers, i.e., dˆ  [0, 0,...dT ,...0, 0]T . Ca is the magnitude response matrix of size N a  M , whose element Ca (i, j )  C j (ai ) computed by (5). j is the index of the carriers. The solution of (8) is equivalent to map d̂ to the nullspace of Ca , In [27], the orthogonal projector Pa mapping vector onto Ca  ( Ca  is the nullspace of Ca ) along Ca is given by 7 Submitted Pa  I M  Ca (CTa Ca )1 CTa , (9) where I M is a unit matrix of size M  M . This projector has a property that the precoded data vector Pa dˆ , in the nullspace of Ca , is closest to d̂ as min Padˆ  dˆ . Padˆ  Ca  (10) Obviously, Pa is non-orthogonal. Therefore, it will cause significant BER performance degradation in the receiver. After the step of the inner orthogonal projection, the precoded data is given by s a  Padˆ . (11) Then in the second step, the magnitude response matrix of K frequency points ω o in the optimized region is written as Cop  Co Pa , where Co is a magnitude response matrix of the K optimized frequency points. The elements in Co is computed by (5). We then use the SVD operation to minimize the power leakage in the optimized region. The problem is determined by Po  arg min Cop P . (12) P By decomposing the matrix Cop into Cop  Uc Σc VcT using SVD, the optimal precoding matrix Po to achieve (12) is derived from Po  [VcR , VcR1 ,...VcM 1 ], (13) where Vci is the i-th column of Vc . The matrix Po of size M  N is composed of the last N columns in Vc . Po is an orthogonal matrix that PoT Po  I N , where I N is a unit matrix of size N  N . The original data d is mapped to Pod , while the length is extended from N to M . The average power of each symbol of M 1 N 1 i 0 i 0 before and after precoding are Ps  i2 and Ps  i2 , where i is the i-th diagonal element in Σ c and Ps 8 Submitted is the average power of each symbol d . The second step is to abandon the largest R singular values in Σ c , then the OOB power is reduced after precoding by Pod . In the third step, the group of zero-forcing frequency points ωb ( ωb  [b1 ,...bi ,...bN ] , bi  0 or b b  M 1 ) are chosen. The orthogonal projection matrix Pb is obtained similar to (9) by i Pb  I M  Cb (CTb Cb )1 CTb , (14) where the matrix Cb of size Nb  M , is the magnitude response matrix of the frequency points b . Finally, the precoding matrix is obtained by P  Pb Po . (15) After precoding, the minimum error problem of decoding in the receiver, is given by ˆ -d , min PPd (16) where P̂ is the decoding matrix. The orthogonal projector Pb map the vector onto Cb  , so rank (Pb )  M  Nb  N , where rank (.) denotes the rank of a matrix. Due to rank (Po )  N and the value of N b is small, the case of that P is full column rank, is easy to be achieved in practice. Then the pseudo- ˆ  I , which is given by Pˆ  (PT P)1 PT . inverse matrix of P is the optimal solution for (16) to achieve PP N This JSP scheme is unlike the methods that minimize the OOB power or forcing the sidelobe to zero. In the third step, we map Pod to the nullspace of Cb rather than the nullspace of Cb Po . If the nullspace of Cb Po is selected, the JSP method turns to the traditional orthogonal projection method, which introduces large deterioration in the receiver. Furthermore, after the operations in the last two steps, the precoding matrix P does not achieve the goal in the first step of mapping d to the nullspace of Ca . In addition, after the operation in the third step, P also does not achieve the goal of minimizing the power leakage in the optimized region in the second step. 9 Submitted As presented in the SVD operation [21] or orthogonal projection method [19], the receive quality and suppression performance have not been well balanced. We also had some tests to examine other combinations that only employing the first two steps or the last two steps in the JSP method. The results indicate that the suppression performance is similar to only using orthogonal projection or SVD operation. The reason is that the better OOB power reduction performance by the JSP method is obtained by two times orthogonal projection. In addition, the BER performance is improved by using reserved carriers in the second step. What’s more, the desirable spectrum envelope also can be achieved by selecting the frequency points or optimized region in the three steps independently. 4. Design of The Precoding Matrix Under Condition Number Constraint In this section, we develop an algorithm to design the precoding matrix, obtaining the desirable spectrum envelope under receive performance constraint. As illustrated in Fig.1, the data after FFT process in the receiver is expressed as s  Qd + n, (17) where Q = HP , H is a complex diagonal matrix with the channel frequency response of M subcarriers and n is complex additive white Gaussian noise (AWGN) vector with zero mean. Q = P for AWGN channel. s is the received signal from noise measurement. The matrix P is full column but PT P  I N . Thus, this non-orthogonal precoding matrix will introduce BER loss in the receiver. If some singular values of P are too small, compared to the other values, low additive noise will result of large errors. Therefore, the condition number  of P is introduced, which is to measure the sensitivity of the solution of linear equations to errors in the data [28].  is given by   Con(P)= max , min (18) 10 Submitted where max and min is the largest and the smallest singular value of P . Con(.) denotes the condition number of a matrix. The value of  indicates the BER loss in the precoding.  [1, ) and larger value of  leads to worse BER performance. Fig. 3 The condition number of transition matrix Q in AWGN channel and Rayleigh fading channel. The transition matrix Q significantly influence the receive quality. We examine the average condition number of Q through different channels in Fig. 3. The results of through Rayleigh channel are averaged over 2000 realizations. Two Rayleigh channel is selected: 3GPP extended pedestrian A (EPA) model [29], whose excess tap delay = (0 30 70 90 110 190 410)ns with the relative power = (0 -1 -2 -3 -8 17.2 -20.8)dB, and a ten-tap Rayleigh block fading channel with exponentially decaying powers set as E (\| hl (i) \|2 )  el /3  9 j 0 e j /3 , j  0,1,...,9 . As illustrated in Fig. 3, the receive performance is better through the AWGN channel than the Rayleigh fading channel. Compared to the transmission without precoding, the condition number of Q linearly increases with condition number constraint  0 in AWGN channel, but decrease in the Rayleigh fading channel when the value of  0 is small. The zero-forcing equalizer is used for AWGN channel and fading channel respectively by 11 Submitted (PT P)1 PT s, d =   H 1 H  (Q Q) Q s, AWGN channel (19) Fading channel where (.) H and d denotes conjugate transpose and the decoded data respectively. In this part, we develop an algorithm to obtain P under a condition number constraint by selecting the frequency points. In order to keep the receive performance decreases slightly, the number of zeroforcing frequency points in the first orthogonal projection is chosen as small as possible. In practice, we select N a  1 for single-side suppression or N a  2 for double-side suppression to keep the spectrum symmetric. If the special case of R  1 is selected, then we fix N a  1 . The envelope of the precoded PSD curve is mainly determined by the outside orthogonal projection. Thus, Nb  R is selected for high suppression performance. The two group zero-forcing frequencies ω a and ωb are arranged in the different OOB region, far from or close to the mainlobe. The one far from the mainlobe is fixed, for the amplitude response is weak by (5). The one closed to the mainlobe is used to maintain the receive quality by adjusting its location. In the SVD operation, in order to effectively abandon the largest R singular values in Σ c , K should larger or equal to R . We select ωo  ωb for simple implementation and K  R . Then the variables for obtaining P are only the group closed to the mainlobe. Thus, we change these frequency points to achieve P  maxCon(P) P s.t. Con(P)   0 , (20) where  0 is the given condition number according to the BER constraint. The signal estimation performance under condition number constraint in a communication system is illustrated in [28, 30]. In addition, as presented in [16, 19], the zero-forcing frequency point close to mainlobe leads to quicker power reduction on the edge of the mainlobe, but the power far from the mainlobe is larger. Inversely, if these points are far from the mainlobe, the power decreases slowly on the edge, while the emission from the mainlobe decreases largely. 12 Submitted Table 1 The correlation between OFDM performance and the parameters in the proposed method. BER Case 0 R Na ― ↓ ― ↓ Case A: Quicker power reduction on the edge of the mainlobe Na=1: Single-side suppression ↑ ↑ Case B: Lower power leakage far Na=2: Double-side suppression from the mainlobe ↓: Negative correlation; ↑: Positive correlation; ―: Weak or no effect. OOB power reduction With summarizing the properties analyzed above, we first give the correlation between the parameters and the performance of OFDM transceiver in Table 1, which is also presented in the simulation section. In the following, we develop an algorithm to obtain the precoding matrix P by selecting the frequencies ωb and ω a , according to the summary in Table 1. The main spectrum envelope is decided in the initialization process step by step as (a.) The value of N a and Case is selected according to the power spectrum envelope property. (b.) R is chosen according to sidelobe suppression performance and spectral efficiency. (c.)  0 is the given condition number according to the BER quality constraint. If Case A is required, we fix ω a far from the mainlobe and adjust ωb . The process to solve (20) is given in the algorithm. 1. Algorithm. 1: Initialization:  0 , R , N a , iteration increment  , ωb  ωb0 , ωa  ωa0 ,   0 , i  0 . 1. i : i  i +1 . i-th iteration. 2. Design Pi by Section 3 and compute the condition number   Con(Pi ) . 3. if (  0 ) ωb :  ωb   , go back to Step 1. else Stop. Output: P  Pi1 . where ωb0 is the initializing frequency points close to the mainlobe, and ω a0 is the fixed frequencies far from the mainlobe. If the number of reserved carriers of R is large, a little adjusting of ωb will lead to 13 Submitted large change to  . Therefore, the iteration increment  also should be small, and the distance between the frequency points in ωb should not be too small. In practice, we select R  4 . If Case B is required, we fix ωb far from the mainlobe and adjust ωa :  ωa   in the Step 3. The distance between the frequency points in ωb also should not be too small or too far from the mainlobe. Otherwise, the matrix Cop may be close to singular or badly scaled. That may lead to that the results may be inaccurate in (13) by SVD in practice. 5. Simulation Results and Discussions In this section, some numerical results are presented to demonstrate the performance of the proposed method. An instance in LTE is selected that the subcarrier spacing is 15 kHz. The number of the subcarriers is N  300 in 5 MHz bandwidth [4]. The frequency axis is normalized to the spacing 2 / T . The index of data subcarriers is from -150 to 150, while the direct current carrier   0 is not employed. To illustrate the OOB power reduction effect, the PSD is obtained by computing the power of DFT coefficients of time-domain OFDM signal over a time span and averaging over thousands of symbols. The frequency-domain oversampling rate is eight and the QPSK modulation is employed. The simulated are mainly ZP-OFDM system, unless noted otherwise. A. Sidelobe Suppression Performance In the first experiment, the OOB reduction performance comparison is presented using different precoding technologies. The number of reserved subcarriers is selected as R  2 . The SSOP [19], the orthogonal projection (OP) [16], the optimal orthogonal precoding (OOP) [21] and CC [8] methods are selected as a comparison. The zero-forcing frequencies are fixed at ω  180 for OP and SSOP method. The optimized region for OOP and CC scheme is also selected nearby the frequencies ω  180 . N a  2 for double-side suppression to keep the spectrum symmetric in the proposed JSP method. The spectral 14 Submitted coding rate is 300/302, ωa  4000 and ωb  180 . The condition number of the precoding matrix is   3.2279 . Fig. 4 PSD of the OFDM signal with different precoding techniques. As the PSD curves illustrated in Fig. 4, all the transmitted signals are ZP-OFDM, besides CP-OFDM signal is presented by the proposed JSP method. It is obviously that the OOB power using the JSP method is lower than other approaches. The OOP/OP/SSOP methods have similar suppression performance. The CC, which is not a spectral precoding method, achieves less OOB power reduction. In addition, the results show the performance degradation of sidelobe suppression in CP systems. Fig. 5 Quicker power reduction on the edge of the mainlobe (Case A) with different condition number constraint. 15 Submitted Fig. 6 Lower power leakage far from the mainlobe (Case B) with different condition number constraint. In Fig. 5, the precoding matrix is designed under a condition number constraint of 0  {1.5,3,5,10, 20} respectively. The PSD curves of Case A is presented with N a  2 . ω a is fixed far from the mainlobe and ωa  4000 , R  2 and   0.25 . Obviously, the suppression performance is improved by relaxing the condition number constraint while the distance between ωb and mainlobe increasing. In Fig. 6, the Case B is presented with N a  2 . We fix ωb far from the mainlobe and ωb  4000 , and the other parameters are same with in Fig. 5. The suppression performance is also improved by relaxing  0 . The OOB power in the region far from the mainlobe is lower than Case A. However, the power on the edge of mainlobe decreases slowly. Fig. 7 Single-side suppression with different number of revered carriers, Na=1. Fig. 8 PSD of Multi-band OFDM signal. In Fig. 7, the Case B is selected, N a  1 and  0  1.5 . The right-side suppression is presented by choosing ω a and ωb in the right side of the mainlobe. The distance between two adjacent frequencies in ω a is five when R  1 . As the symmetry presented in (5), (9) and (14), the difference between adopting the OOB frequency ω and ω to design the precoding matrix is only in the second step of SVD. Therefore, the power on the left side OOB region is also reduced, while the power emission is higher than 16 Submitted on the right side. The suppression performance is improved by increasing the number of revered carriers. Although the simulation is not illustrated, this property is also presented in the case of N a  2 by adjusting R. In cognitive radio system, the secondary users access the spectrum hole dynamically. In addition, carrier aggregation (CA) [31] is a critical technology in LTE system, which enable a user to employ noncontiguous subbands. Therefore, in Fig. 8, the PSD of two non-contiguous subbands occupied by one user or two is presented. The condition number is  0  10 for all the cases. The number of the carriers of each subband is 150. The design of the precoding matrix for two users is independent, while the data transmitted by one user through two subbands is dependent. The results indicate that the proposed scheme is also suitable for multi-users transmitting through non-contiguous subbands. B. BER performance Fig. 9 BER performance with different spectral precoding techniques through AWGN channel. Fig. 10 BER performance with same condition number constraint through AWGN channel, Na=1. A comparison of the BER performance using spectral precoding techniques is shown in Fig.9. The condition number constraint is  0  2 for the JSP. The optimized region or zero-forcing frequencies for OOP, SSOP and OP method is same with the JSP method. As illustrated in Fig. 4, the sidelobe suppression 17 Submitted performance by OOP, SSOP and OP method is similar. The difference is that OOP scheme introduces no quality loss in the receiver and the OP method leads to large error as presented in Fig. 9. The SSOP method improves the receive quality by adopting a reserved subcarrier to decode the distorted signal. The BER quality loss by the JSP method is not notable. In Fig. 10, the condition number constraint is fixed as  0  10 , as well as single-side suppression is selected as N a  1 . The results illustrated that the receive quality is approximate identical when the condition number and N a are fixed, although both of the number of reserved carriers R and the Case are different. This property indicates that if N a has been selected according to single-side suppression or double-side suppression, the BER performance constraint can be converted to the condition number constraint in practice. Fig. 11 BER performance with difference condition number constraint and Na through AWGN channel, R=2. In Fig. 11, the results demonstrate that the BER performance of N a  1 is better than N a  2 under same constraint. This is why we select the value of N a as small as possible to maintain the receive quality. 18 Submitted Fig. 12 BER performance with difference condition number constraint through AWGN channel. Fig. 13 BER performance with difference condition number constraint through 3GPP EPA fading channel. In Fig. 12, the receive quality is presented with N a  2 . The number of revered carriers is R  2 . The results show that the BER performance decreases by relaxing  0 , but the suppression performance is improved shown in Fig. 5-6. For example, the power in the OOB region can be reduced by nearly 40dB when  0  10 , while the SNR loss is less than 1dB at BER=10−7. When the condition number of precoding matrix is small, such as  0  1.5 in Fig. 12, the BER performance is slightly better than that of without precoding. Because the number of the modulated carriers of each symbol is extended from N to M , the receive quality is improved by frequency diversity. However, this property is not notable when the value of  0 is large. In Fig. 13, the parameters are set same with in Fig.12. Comparing to without precoding, the BER performance through the fading channel is similar to AWGN channel, when SNR is small. The receive quality decreases by relaxing  0 . But with SNR increasing, the BER line converges to without precoding. 6. Conclusions In this paper, a joint spectral precoding (JSP) method is proposed to reduce the OOB power emission in OFDM-based system, as well as an iteration algorithm is given to obtain the precoding matrix. We also 19 Submitted convert the BER performance constraint to the condition number constraint of the precoding matrix. As summarized in Table I and presented in the simulations, the proposed method has an appreciate balance between suppression performance and spectral efficiency, under a receive quality constrain. We also can obtain the desirable spectrum envelope by parameters configuration. Therefore, this method is flexible for spectrum shaping in both conventional OFDM systems and non-continuous multi-carriers cognitive radio networks. 7. References 1. Xing, Y.P., Chandramouli, R., Mangold, S., and Shankar, N.S., 'Dynamic Spectrum Access in Open Spectrum Wireless Networks', IEEE Journal on Selected Areas in Communications, 2006, 24, (3), pp. 626-637. 2. Zhao, Q. and Sadler, B.M., 'A Survey of Dynamic Spectrum Access', IEEE Signal Processing Magazine, 2007, 24, (3), pp. 79-89. 3. Mahmoud, H.A., Yucek, T., and Arslan, H., 'Ofdm for Cognitive Radio: Merits and Challenges', IEEE Wireless Communications, 2009, 16, (2), pp. 6-14. 4. Dahlman, E., Parkvall, S., and Skold, J., 4g: Lte/Lte-Advanced for Mobile Broadband, (Academic Press, 2013) 5. Faulkner, M., 'The Effect of Filtering on the Performance of Ofdm Systems', IEEE Transactions on Vehicular Technology, 2000, 49, (5), pp. 1877-1884. 6. Lin, Y.P. and Phoong, S.M., 'Window Designs for Dft-Based Multicarrier Systems', IEEE Transactions on Signal Processing, 2005, 53, (3), pp. 1015-1024. 7. Schmidt, J.F., Costas-Sanz, S., and Lopez-Valcarce, R., 'Choose Your Subcarriers Wisely: Active Interference Cancellation for Cognitive Ofdm', IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2013, 3, (4), pp. 615-625. 8. Brandes, S., Cosovic, I., and Schnell, M., 'Reduction of out-of-Band Radiation in Ofdm Systems by Insertion of Cancellation Carriers', IEEE Communications Letters, 2006, 10, (6), pp. 420-422. 9. Cosovic, I., Brandes, S., and Schnell, M., 'Subcarrier Weighting: A Method for Sidelobe Suppression in Ofdm Systems', IEEE Communications Letters, 2006, 10, (6), pp. 444-446. 10. Cosovic, I. and Mazzoni, T., 'Suppression of Sidelobes in Ofdm Systems by Multiple-Choice Sequences', European Transactions on Telecommunications, 2006, 17, (6), pp. 623-630. 11. Li, D., Dai, X.H., and Zhang, H., 'Sidelobe Suppression in Nc-Ofdm Systems Using Constellation Adjustment', IEEE Communications Letters, 2009, 13, (5), pp. 327-329. 12. Liu, S., Li, Y., Zhang, H., and Liu, Y., 'Constellation Expansion-Based Sidelobe Suppression for Cognitive Radio Systems', Communications, IET, 2013, 7, (18), pp. 2133-2140. 20 Submitted 13. Mahmoud, H.A. and Arslan, H., 'Sidelobe Suppression in Ofdm-Based Spectrum Sharing Systems Using Adaptive Symbol Transition', IEEE Communications Letters, 2008, 12, (2), pp. 133-135. 14. Xu, R., Chen, M., Zhang, J., Wu, B., and Wang, H., 'Spectrum Sidelobe Suppression for Discrete Fourier Transformation-Based Orthogonal Frequency Division Multiplexing Using Adjacent Subcarriers Correlative Coding', IET Communications, 2012, 6, (11), pp. 1374-1381. 15. Huang., X., Zhang., J.A., and Guo, Y.J., 'Out-of-Band Emission Reduction and a Unified Framework for Precoded Ofdm', IEEE Communications Magazine, 2015, 53, (6), pp. 151-159. 16. van de Beek, J. and Berggren, F., 'N-Continuous Ofdm', IEEE Communications Letters, 2009, 13, (1), pp. 1-3. 17. Zheng, Y.M., Zhong, J., Zhao, M.J., and Cai, Y.L., 'A Precoding Scheme for N-Continuous Ofdm', IEEE Communications Letters, 2012, 16, (12), pp. 1937-1940. 18. van de Beek, J., 'Sculpting the Multicarrier Spectrum: A Novel Projection Precoder', IEEE Communications Letters, 2009, 13, (12), pp. 881-883. 19. Zhang, J.A., Huang, X.J., Cantoni, A., and Guo, Y.J., 'Sidelobe Suppression with Orthogonal Projection for Multicarrier Systems', IEEE Transactions on Communications, 2012, 60, (2), pp. 589599. 20. Zhou, X.W., Li, G.Y., and Sun, G.L., 'Multiuser Spectral Precoding for Ofdm-Based Cognitive Radio Systems', IEEE Journal on Selected Areas in Communications, 2013, 31, (3), pp. 345-352. 21. Ma, M., Huang, X.J., Jiao, B.L., and Guo, Y.J., 'Optimal Orthogonal Precoding for Power Leakage Suppression in Dft-Based Systems', IEEE Transactions on Communications, 2011, 59, (3), pp. 844853. 22. Tom, A., Sahin, A., and Arslan, H., 'Mask Compliant Precoder for Ofdm Spectrum Shaping', IEEE Communications Letters, 2013, 17, (3), pp. 447-450. 23. Muquet, B., Wang, Z.D., Giannakis, G.B., de Courville, M., and Duhamel, P., 'Cyclic Prefixing or Zero Padding for Wireless Multicarrier Transmissions?', IEEE Transactions on Communications, 2002, 50, (12), pp. 2136-2148. 24. Lu, H., Nikookar, H., and Chen, H., 'On the Potential of Zp-Ofdm for Cognitive Radio', Proc. WPMC’09, 2009, pp. 7-10. 25. Van Waterschoot, T., Le Nir, V., Duplicy, J., and Moonen, M., 'Analytical Expressions for the Power Spectral Density of Cp-Ofdm and Zp-Ofdm Signals', Signal Processing Letters, IEEE, 2010, 17, (4), pp. 371-374. 26. Wu, T.W. and Chung, C.D., 'Spectrally Precoded Dft-Based Ofdm and Ofdma with Oversampling', IEEE Transactions on Vehicular Technology, 2014, 63, (6), pp. 2769-2783. 27. Meyer, C.D., Matrix Analysis and Applied Linear Algebra, (SIAM, 2000) 28. Tong, J., Guo, Q.H., Tong, S., Xi, j.T., and Yu, Y.G., 'Condition Number-Constrained Matrix Approximation with Applications to Signal Estimation in Communication Systems', IEEE Communications Letters, 2014, 21, (8), pp. 990-993. 29. 3GPP TS 36.141: 'E-Utra Base Station (Bs) Conformance Testing', 2015 21 Submitted 30. Aubry, A., De Maio, A., Pallotta, L., and Farina, A., 'Maximum Likelihood Estimation of a Structured Covariance Matrix with a Condition Number Constraint', Signal Processing, IEEE Transactions on, 2012, 60, (6), pp. 3004-3021. 31. Yuan, G.X., Zhang, X., Wang, W.B., and Yang, Y., 'Carrier Aggregation for Lte-Advanced Mobile Communication Systems', IEEE Communications Magazine, 2010, 48, (2), pp. 88-93. 22	5
arXiv:1604.00656v1 [math.AC] 3 Apr 2016 DEPTH, STANLEY DEPTH AND REGULARITY OF IDEALS ASSOCIATED TO GRAPHS S. A. SEYED FAKHARI Abstract. Let K be a field and S = K[x1 , . . . , xn ] be the polynomial ring in n variables over K. Let G be a graph with n vertices. Assume that I = I(G) is the edge ideal of G and J = J(G) is its cover ideal. We prove that sdepth(J) ≥ n−νo (G) and sdepth(S/J) ≥ n − νo (G) − 1, where νo (G) is the ordered matching number of G. We also prove the inequalities sdepth(J k ) ≥ depth(J k ) and sdepth(S/J k ) ≥ depth(S/J k ), for every integer k ≫ 0, when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg(S/I) ≤ νo (G). 1. Introduction and Preliminaries Let K be a field and let S = K[x1 , . . . , xn ] be the polynomial ring in n variables over K. Let M be a finitely generated Zn -graded S-module. Let u ∈ M be a homogeneous element and Z ⊆ {x1 , . . . , xn }. The K-subspace uK[Z] generated by all elements uv with v ∈ K[Z] is called a Stanley space of dimension \|Z\|, if it is a free K[Z]-module. Here, as usual, \|Z\| denotes the number of elements of Z. A decomposition D of M as a finite direct sum of Stanley spaces is called a Stanley decomposition of M. The minimum dimension of a Stanley space in D is called the Stanley depth of D and is denoted by sdepth(D). The quantity sdepth(M) := max sdepth(D) \| D is a Stanley decomposition of M is called the Stanley depth of M. We say that a Zn -graded S-module M satisfies Stanley’s inequality if depth(M) ≤ sdepth(M). In fact, Stanley [22] conjectured that every Zn -graded S-module satisfies Stanley’s inequality. This conjecture has been recently disproved in [1]. However, it is still interesting to find the classes of Zn -graded S-modules which satisfy Stanley’s inequality. For a reader friendly introduction to Stanley depth, we refer to [18] and for a nice survey on this topic, we refer to [11]. Let G be a graph with vertex set V (G) = x1 , . . . , xn and edge set E(G) (by abusing the notation, we identify the vertices of G with the variables of S). For a vertex xi , the neighbor set of xi is NG (xi ) = {xj \| xi xj ∈ E(G)} and We set NG [xi ] = NG (xi ) ∪ {xi } and call it the closed neighborhood of xi . For every subset 2000 Mathematics Subject Classification. Primary: 13C15, 05E99; Secondary: 13C13. Key words and phrases. Cover ideal, Edge ideal, Ordered matching, Regularity, Stanley depth, Stanley’s inequality. 1 2 S. A. SEYED FAKHARI A ⊂ V (G), the graph G \ A is the graph with vertex set V (G \ A) = V (G) \ A and edge set E(G \ A) = {e ∈ E(G) \| e ∩ A = ∅}. A bipartite graph is one whose vertex set is partitioned into two (not necessarily nonempty) disjoint subsets in such a way that the two end vertices for each edge lie in distinct partitions. A matching in a graph is a set of edges such that no two different edges share a common vertex. A subset W of V (G) is called an independent subset of G if there are no edges among the vertices of W . A subset C of V (G) is called a vertex cover of the graph G if every edge of G is incident to at least one vertex of C. A vertex cover C is called a minimal vertex cover of G if no proper subset of C is a vertex cover of G. Next, we define the notion of ordered matching for a graph. It was introduced in [5] and plays a central role in this paper. Definition 1.1. Let G be a graph, and let M = {{ai , bi } \| 1 ≤ i ≤ r} be a nonempty matching of G. We say that M is an ordered matching of G if the following hold: (1) A := {a1 , . . . , ar } ⊆ V (G) is a set of independent vertices of G; and (2) {ai , bj } ∈ E(G) implies that i ≤ j. The ordered matching number of G, denoted by νo (G), is defined to be νo (G) = max{\|M\| \| M ⊆ E(G) is an ordered matching of G}. The edge ideal I(G) of G is the ideal of S generated by the squarefree monomials xi xj , where {xi , xj } is an edge of G. The Alexander dual of the edge ideal of G in S, i.e., the ideal \ J(G) = I(G)∨ = (xi , xj ), {xi ,xj }∈E(G) is called the cover ideal of G in S. The reason for this name is due to the well-known fact that the generators of J(G) correspond to minimal vertex covers of G. The main goal of This paper is to study the Stanley depth of cover ideals and their power. In Theorem 2.4, we prove that for every graph G, the inequalities sdepth(J(G)) ≥ n−νo (G) and sdepth(S/J(G)) ≥ n−νo (G)−1 hold. In that theorem, we also prove that the same inequalities hold, if one replaces sdepth by depth. Then, in Corollary 2.5, we conclude that for every graph G we have reg(S/I) ≤ νo (G). This inequality was previously proved by Constantinescu and Varbaro [5, Remark 4.8]. However, our proof is more elementary. In Section 3, we consider the Stanley depth of powers of cover ideal of bipartite graphs. Let G be a bipartite graph. In [20, Corollary 3.6], the author proved that the k ∞ sequences {sdepth(J(G)k )}∞ k=1 and {sdepth(S/J(G) )}k=1 are non-increasing. Thus the both sequences are convergent. In Theorem 3.3, we provide lower bounds for the limit value of theses sequences. Indeed, we prove that for every bipartite graph G, we have lim sdepth(J(G)k ) ≥ n − νo (G) and k→∞ lim sdepth(S/J(G)k ) ≥ n − νo (G) − 1. k→∞ Then we conclude in Corollary 3.4 that J(G)k and S/J(G)k satisfy the Stanley’s inequality, for every integer k ≫ 0. Theorem 3.3 also shows that a conjecture of the DEPTH, SDEPTH AND REGULARITY 3 author is true for the powers of cover ideal of bipartite graphs (see Conjecture 3.5 and the paragraph after it). 2. First Power The first main result of this paper is Theorem 2.4, which provides a lower bound for the depth and the Stanley depth of cover ideal of graphs. We first need the following three simple lemmas. The first one shows that the ordered matching number of a graph strictly decreases when we delete the closed neighborhood of a non-isolated vertex. Lemma 2.1. Let G be a graph and x be a non-isolated vertex of G. Then we have νo (G \ NG [x]) ≤ νo (G) − 1. Proof. Assume that νo (G\NG [x]) = t and let M = {{ai , bi } \| 1 ≤ i ≤ t} be an ordered matching of G \ NG [x]. Since x is not isolated, we may choose a vertex y ∈ NG (x). Set at+1 = x and bt+1 = y. Then {a1 , . . . , at+1 } is a set of independent vertices of G, because a1 , . . . , at are vertices of G\NG [x]. By the same reason, at+1 is not adjacent to b1 , . . . , bt . This shows that M ∪{at+1 , bt+1 } is an ordered matching of G and therefore, νo (G) ≥ t + 1. The next Lemma shows that how the cover ideal of a graph G and that of G \ NG [x] are related, when x is an arbitrary vertex of G. Lemma 2.2. Let G ba a graph with vertex set V (G) = {x1 , . . . , xn }. Assume that Q x ∈ V (G) is a vertex of G. Set u = xi ∈NG (x) xi and J ′ = J(G \ NG [x])S. Then J(G) + (x) = uJ ′ + (x). Proof. Let C be a vertex cover of G with x ∈ / C. Then NG (x) ⊆ C and C \ NG (x) is a vertex cover of G \ NG [x]. This shows that J(G) + (x) ⊆ uJ ′ + (x). For the converse inclusion, assume that D is a vertex cover of G \ NG [x]. Then D ∪ NG (x) is a vertex cover of G. This shows that uJ ′ + (x) ⊆ J(G) + (x) and completes the proof. The following lemma provides a combinatorial description for the colon of cover ideals. Lemma 2.3. Let G ba a graph with vertex set V (G) = {x1 , . . . , xn }. Assume that x ∈ V (G) is a vertex of G. Set J ′ = J(G \ x)S. Then (J(G) : x) = J ′ . Proof. If C is a vertex cover of G, then C \ {x} is a vertex cover of G \ x. This shows that (J(G) : x) ⊆ J ′ . On the other hand, if D is a vertex cover of G \ x, then D ∪ {x} is a vertex cover of G. This shows that J ′ ⊆ (J(G) : x). We are now ready to prove the first main result of this paper. As we mentioned in introduction, the second part of this theorem is known by [5, Remark 4.8]. But our argument is completely different and provides a simple proof for it. Theorem 2.4. Let G be a graph and J(G) be its cover ideal. Then 4 S. A. SEYED FAKHARI (i) sdepth(J(G)) ≥ n − νo (G) and sdepth(S/J(G)) ≥ n − νo (G) − 1, (ii) depth(S/J(G)) ≥ n − νo (G) − 1. Proof. We prove (i) and (ii) simultaneously by induction on the number of edges of G. If G has only one edge, then νo (G) = 1 and J(G) is generated by two variables. Then depth(S/J(G)) = n − 2. Also, sdepth(S/J(G)) = n − 2 by [19, Theorem 1.1] and sdepth(J(G)) ≥ n − 1 by [11, Corollary 24] and [13, Lemma 3.6]. Therefore, in these cases, the inequalities in (i) and (ii) are trivial. We now assume that G has at least two edges. Note that, G has at least one nonisolated vertex. Without loss of generality, we may assume that x1 is a non-isolated vertex of G. Let S ′ = K[x2 , . . . , xn ] be the polynomial ring obtained from S by deleting the variable x1 and consider the ideals J ′ = J(G) ∩ S ′ and J ′′ = (J(G) : x1 ). Now J(G) = J ′ S ′ ⊕ x1 J ′′ S and S/J(G) = (S ′ /J ′ S ′ ) ⊕ x1 (S/J ′′ S) (as vector spaces) and therefore by definition of the Stanley depth we have sdepth(J(G)) ≥ min{sdepthS ′ (J ′ S ′ ), sdepthS (J ′′ )}, (1) and (2) sdepth(S/J(G)) ≥ min{sdepthS ′ (S ′ /J ′ S ′ ), sdepthS (S/J ′′ )}. On the other hand, by applying the depth lemma on the exact sequence 0 −→ S/(J(G) : x1 ) −→ S/J(G) −→ S/(J(G), x1 ) −→ 0 we conclude that (3) depth(S/J(G)) ≥ min{depthS ′ (S ′ /J ′ S ′ ), depthS (S/J ′′ )}. Using Lemma 2.3, it follows that J ′′ = J(G \ x1 )S. Hence our induction hypothesis implies that depthS (S/J ′′ ) = depthS ′ (S ′ /J ′′ ) + 1 ≥ n − 1 − νo (G \ x1 ) − 1 + 1 ≥ n − νo (G) − 1. Also, it follows from [13, Lemma 3.6] that sdepthS (S/J ′′ ) = sdepthS ′ (S ′ /J ′′ ) + 1 ≥ n − 1 − νo (G \ x1 ) − 1 + 1 ≥ n − νo (G) − 1, and sdepthS (J ′′ ) = sdepthS ′ (J ′′ ) + 1 ≥ n − 1 − νo (G \ x1 ) + 1 ≥ n − νo (G). On the other hand, it follows from Lemma 2.2 that there exists a monomial u ∈ S ′ such that J ′ S ′ = uJ(G \ NG [x1 ])S ′ . Since uJ(G \ NG [x1 ])S ′ and J(G \ NG [x1 ])S ′ , (up to a shift) are isomorphic as graded S ′ -Modules, we conclude that depthS ′ (J ′ S ′ ) = depthS ′ (J(G \ NG [x1 ])S ′ ). On the other hand, it follows from [7, Theorem 1.1] that sdepthS ′ (J ′ S ′ ) = sdepthS ′ (J(G\NG [x1 ])S ′ ) and sdepthS ′ (S ′ /J ′ S ′ ) = sdepthS ′ (S ′ /J(G\ NG [x1 ])S ′ ). Therefore by [13, Lemma 3.6], Lemma 2.1 and the induction hypothesis we conclude that sdepthS ′ (J ′ S ′ ) = sdepthS ′ (J(G \ NG [x1 ])S ′ ) ≥ n − 1 − νo (G \ NG [x1 ]) ≥ n − νo (G), DEPTH, SDEPTH AND REGULARITY 5 and similarly sdepthS ′ (S ′ /J ′ S ′ ) ≥ n−νo (G) −1 and depthS ′ (S ′ /J ′ S ′ ) ≥ n−νo (G) −1. Now the assertions follow by inequalities (1), (2) and (3). Let M be a finitely generated graded S-Module. The Castelnuovo-Mumford regularity (or simply, regularity) of M, denoted by reg(M), is defined as follows: reg(M) = max{j − i\| TorSi (K, M)j 6= 0}. The regularity of a module is one of the most important homological invariants of it. Computing the regularity of edge ideals or finding bounds for it has been studied by a number of researchers (see for example [8], [10], [14], [15], [23]). An immediate consequence of the second part of theorem 2.4 is the following corollary. Corollary 2.5. For every graph G, we have reg(S/I(G)) ≤ νo (G). Proof. It follows from Theorem 2.4 and the Auslander-Buchsbaum Formula that the projective dimension of J(G) is at most νo (G). Then it follows from Terai’s theorem [12, Theorem 8.1.10] that reg(S/I(G)) ≤ νo (G). Remark 2.6. One can give a direct proof for the above corollary, by applying [16, Corollary 18.7] on the following exact sequence. 0 −→ S/(I(G) : x1 ) −→ S/I(G) −→ S/(I(G), x1 ) −→ 0 However, this proof is essentially the same as given above. In [10], Hà and Van Tuyl proved that the for every graph G, the regularity of S/I(G) is less than or equal to the maximum cardinality of matchings of G. In fact, it follows from their proof (and was explicitly stated in [24]) that the reg(S/I(G)) is at most the minimum cardinality of maximal matchings of G. The following examples show that this bound is not comparable with the bond given in Corollary 2.5. Examples 2.7. (1) Let G = C4 be the 4-cycle-graph. Then one can easily check that νo (G) = 1 and the cardinality of every maximal matching of G is equal to 2. Thus, in this example, νo (G) is strictly less than the minimum cardinality of maximal matchings of G. We also have reg(S/I(G)) = 1 = νo (G). (2) Let G = P4 be the path with 4 vertices. Then one can easily check that νo (G) = 2, while the minimum cardinality of maximal matchings of G is equal to 1. Thus, in this example, the minimum cardinality of maximal matchings of G is strictly less than νo (G). We also have reg(S/I(G)) = 1 is equal to the minimum cardinality of maximal matchings of G. 3. High Powers The aim of this section is to prove that the high powers of cover ideal of bipartite graphs satisfy the Stanley’s inequality. To do this, in Theorem 3.3, we provide a lower bound for the Stanley depth of cover ideal of bipartite graphs. Before that, in Lemma 3.2, we prove that the different powers of cover ideal of a bipartite graphs, can be 6 S. A. SEYED FAKHARI obtained from each other by taking colon with respect to a suitable monomial. To prove Lemma 3.2, we need to remind the definition of symbolic powers. Definition 3.1. Let I be a squarefree monomial ideal in S and suppose that I has the irredundant primary decomposition I = p1 ∩ . . . ∩ pr , where every pi is an ideal of S generated by a subset of the variables of S. Let k be a positive integer. The kth symbolic power of I, denoted by I (k) , is defined to be I (k) = pk1 ∩ . . . ∩ pkr . The proof of the following lemma is based on the fact that the symbolic and the ordinary powers of cover ideal of bipartite graphs coincide. Lemma 3.2. Let G be a bipartite graph Q and assume that V (G) = U ∪ W is a bipartition for the vertex set of G. Set u = xi ∈U xi . Then for every integer k ≥ 1, we have (J(G)k : u) = J(G)k−1 . Proof. It follows from [9, Corollary 2.6] that for every integer k ≥ 1 we have J(G)k = J(G)(k) . On the other hand, for every edge e = {xi , xj } of G, we have \| e ∩ U \|= 1. Thus, ((xi , xj )k : u) = (xi , xj )k−1 , for every integer k ≥ 1. Hence \ (J(G)k : u) = (J(G)(k) : u) = ((xi , xj )k : u) {xi ,xj }∈E(G) = \ (xi , xj )k−1 = J(G)(k−1) = J(G)k−1 . {xi ,xj }∈E(G) As we mentioned in the the first section, the sequences {sdepth(J(G)k )}∞ k=1 and {sdepth(S/J(G)k )}∞ are convergent. In the following theorem, we provide lower k=1 bounds for the limit of theses sequences. Theorem 3.3. Let G be a bipartite graph. Then for every integer k ≥ 1, the inequalities sdepth(J(G)k ) ≥ n − νo (G) and sdepth(S/J(G)k ) ≥ n − νo (G) − 1 hold. Proof. Assume that V (G) = U ∪ W is a bipartition for the vertex set of G. Without loss of generality, we may assume that U = {x1 , . . . , xt } and W = {xt+1 , . . . , xn }, for some integer t with 1 ≤ t ≤ n. Let m be the number of edges of G. We prove the assertions by induction on m + k. First, we can assume that G has no isolated vertex. Because deleting the isolated vertices does not change the cover ideal and the ordered matching number of G. For k = 1, the assertions follow from Theorem 2.4. If m = 1, then G has two vertices and νo (G) = 1. In this case, the first inequality follows from [11, Corollary 24] and the DEPTH, SDEPTH AND REGULARITY 7 second inequality is trivial. Therefore, assume that k, m ≥ 2. Let S1 = K[x2 , . . . , xn ] be the polynomial ring obtained from S by deleting the variable x1 and consider the ideals J1 = J(G)k ∩ S1 and J1′ = (J(G)k : x1 ). Now J(G)k = J1 ⊕ x1 J1′ and S/J(G)k = (S1 /J1 ) ⊕ x1 (S/J1′ ) (as vector spaces) and therefore by definition of the Stanley depth we have (†) sdepth(J(G)k ) ≥ min{sdepthS1 (J1 ), sdepthS (J1′ )}, and (‡) sdepth(S/J(G)k ) ≥ min{sdepthS1 (S1 /J1 ), sdepthS (S/J1′ )}. Notice that J1 = (J(G) ∩ S1 )k . Hence, by Lemma 2.2 we conclude that there exists a monomial u1 ∈ S1 such that J1 = uk1 J(G \ NG [x1 ])k S1 . It follows from [7, Theorem 1.1] that sdepthS1 (J1 ) = sdepthS1 (J(G \ NG [x1 ])k S1 ) and sdepthS1 (S1 /J1 S1 ) = sdepthS1 (S1 /J(G \ NG [x1 ])k S1 ). Therefore, by [13, Lemma 3.6], Lemma 2.1 and the induction hypothesis, we conclude that sdepthS1 (J1 ) = sdepthS1 (J(G \ NG [x1 ])k S1 ) ≥ n − 1 − νo (G \ NG [x1 ]) ≥ n − νo (G), and similarly sdepthS1 (S1 /J1 ) ≥ n−νo (G)−1. Thus, using the inequalities (†) and (‡), it is enough to prove that sdepthS (J1′ ) ≥ n−νo (G) and sdepthS (S/J1′ ) ≥ n−νo (G)−1. For every integer i with 2 ≤ i ≤ t, let Si = K[x1 , . . . , xi−1 , xi+1 , . . . , xn ] be the polynomial ring obtained from S by deleting the variable xi and consider the ideals ′ ′ ∩ Si . : xi ) and Ji = Ji−1 Ji′ = (Ji−1 Claim. For every integer i with 1 ≤ i ≤ t − 1 we have ′ sdepth(Ji′ ) ≥ min{n − νo (G), sdepth(Ji+1 )} and ′ sdepth(S/Ji′ ) ≥ min{n − νo (G) − 1, sdepth(S/Ji+1 )}. Proof of the Claim. For every integer i with 1 ≤ i ≤ t − 1, we have Ji′ = Ji+1 ⊕ ′ ′ ) (as vector spaces) and therefore by and S/Ji′ = (Si+1 /Ji+1 ) ⊕ xi+1 (S/Ji+1 xi+1 Ji+1 definition of the Stanley depth we have (∗) ′ sdepth(Ji′ ) ≥ min{sdepthSi+1 (Ji+1 ), sdepthS (Ji+1 )}, and (∗∗) ′ )}. sdepth(S/Ji′ ) ≥ min{sdepthSi+1 (Si+1 /Ji+1 ), sdepthS (S/Ji+1 Notice that for every integer i with 1 ≤ i ≤ t−1, we have Ji′ = (J(G)k : x1 x2 . . . xi ). Thus Ji+1 = Ji′ ∩ Si+1 = ((J(G)k ∩ Si+1 ) :Si+1 x1 x2 . . . xi ). Hence, it follows from [17, Proposition 2] and [6, Proposition 2.7] (see also [20, Proposition 2.5]) that (∗ ∗ ∗) sdepthSi+1 (Ji+1 ) ≥ sdepthSi+1 (J(G)k ∩ Si+1 ). and (∗ ∗ ∗∗) sdepthSi+1 (Si+1 /Ji+1 ) ≥ sdepthSi+1 (Si+1 /(J(G)k ∩ Si+1 )). 8 S. A. SEYED FAKHARI By Lemma 2.2 we conclude that there exists a monomial ui+1 ∈ Si+1 such that J(G) ∩ Si+1 = ui+1 J(G \ NG [xi+1 ])Si+1 . Therefore J(G)k ∩ Si+1 = uki+1 J(G \ NG [xi+1 ])k Si+1 and it follows from [7, Theorem 1.1] that sdepthSi+1 (J(G)k ∩ Si+1 ) = sdepthSi+1 (J(G \ NG [xi+1 ])k Si+1 ) and sdepthSi+1 (Si+1 /(J(G)k ∩ Si+1 )) = sdepthSi+1 (Si+1 /J(G \ NG [xi+1 ])k Si+1 ). Therefore by [13, Lemma 3.6], Lemma 2.1 and the induction hypothesis we conclude that sdepthSi+1 (J(G)k ∩ Si+1 ) ≥ n − 1 − νo (G \ NG [xi+1 ]) ≥ n − νo (G), and similarly sdepthSi+1 (Si+1 /(J(G)k ∩ Si+1 )) ≥ n − νo (G) − 1. Now the claim follows by inequalities (∗), (∗∗), (∗ ∗ ∗) and (∗ ∗ ∗∗). Now, Jt′ = (J(G)k : x1 x2 . . . xt ) and hence, Lemma 3.2 implies that Jt′ = J(G)k−1 and thus, by induction hypothesis we conclude that sdepth(Jt′ ) ≥ n − νo (G) and sdepth(S/Jt′ ) ≥ n − νo (G) − 1. Therefore, using the claim repeatedly implies that sdepth(J1′ ) ≥ n − νo (G) and sdepth(S/J1′ ) ≥ n − νo (G) − 1. This completes the proof of the theorem. Let I ⊂ S be a monomial ideal. A classical result by Burch [3] states that min depth(S/I k ) ≤ n − ℓ(I), k where ℓ(I) analytic spread of I, that is, the dimension of R(I)/mR(I), where L is the n I = S[It] ⊆ S[t] is the Rees ring of I and m = (x1 , . . . , xn ) is the maxR(I) = ∞ n=0 imal ideal of S. By a theorem of Brodmann [2], depth(S/I k ) is constant for large k. We call this constant value the limit depth of I, and denote it by limk→∞ depth(S/I k ). Brodmann improved the Burch’s inequality by showing that (♯) limk→∞ depth(S/I k ) ≤ n − ℓ(I). Let I ⊂ S be an arbitrary ideal. An element f ∈ S is integral over I, if there exists an equation f k + c1 f k−1 + . . . + ck−1 f + ck = 0 with ci ∈ I i . The set of elements I in S which are integral over I is the integral closure of I. The ideal I is integrally closed, if I = I. It is known that the equality holds, in inequality (♯), if I is a normal ideal. By [9, Corollary 2.6] and [12, Theorem 1.4.6], we know that J(G) is a normal ideal, for every bipartite graph G. Also, it follows from [5, Theorem 2.8] that for every bipartite graph G, we have ℓ(J(G)) = νo (G) + 1. Thus, we conclude that lim depth(S/J(G)k ) = n − 1 − νo (G). k→∞ DEPTH, SDEPTH AND REGULARITY 9 (This equality is explicitly stated in [4, Theorem 4.5].) Therefore, Theorem 3.3 implies the following result Corollary 3.4. Let G be a bipartite graph and J(G) be its edge ideal. Then there exists an integer n0 ≥ 1 such that J(G)k and S/J(G)k satisfy the Stanley’s inequality, for every integer k ≥ n0 . In [21], the author proposed the following conjecture regarding the Stanley depth of integrally closed monomial ideals. Conjecture 3.5. ([21, Conjecture 2.6]) Let I ⊂ S be an integrally closed monomial ideal. Then sdepth(S/I) ≥ n − ℓ(I) and sdepth(I) ≥ n − ℓ(I) + 1. Let G be a bipartite graph. As we mentioned above J(G) is a normal ideal. Thus, every power of J(G) is integrally closed. Therefore, Theorem 3.3 shows that Conjecture 3.5 is true for the powers of cover ideal of bipartite graphs. References [1] A. M. Duval, B. Goeckner, C. J. Klivans, J. L. Martin, A non-partitionable Cohen-Macaulay simplicial complex, preprint. [2] M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 1, 35–39. [3] L. Burch, Codimension and analytic spread, Proc. Cambridge Philos. Soc. 72 (1972), 369–373. [4] A. Constantinescu, M. R. Pournaki, S. A. Seyed Fakhari, N.Terai, S. Yassemi, CohenMacaulayness and limit behavior of depth for powers of cover ideals, Comm. Algebra, 43 (2015), no. 1, 143–157. [5] A. Constantinescu, M. Varbaro, Koszulness, Krull dimension, and other properties of graphrelated algebras, J. Algebraic Combin. 34 (2011), no. 3, 375–400. [6] M. Cimpoeaş, Several inequalities regarding Stanley depth, Romanian Journal of Math. and Computer Science 2, (2012), 28–40. [7] M. Cimpoeaş, Stanley depth of monomial ideals with small number of generators, Central European Journal of Mathematics, 7 (2009), 629–634. [8] H. Dao, C. Huneke, J. Schweig, Bounds on the regularity and projective dimension of ideals associated to graphs, J. Algebraic Combin. 38 (2013), 37–55. [9] I. Gitler, E. Reyes, R. H. Villarreal, Blowup algebras of ideals of vertex covers of bipartite graphs, Contemp. Math. 376 (2005), 273–279. [10] H. T. Hà, A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27 (2008), 215–245. [11] J. Herzog, A survey on Stanley depth. In ”Monomial Ideals, Computations and Applications”, A. Bigatti, P. Giménez, E. Sáenz-de-Cabezón (Eds.), Proceedings of MONICA 2011. Lecture Notes in Math. 2083, Springer (2013). [12] J. Herzog, T. Hibi, Monomial Ideals, Springer-Verlag, 2011. [13] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra 322 (2009), no. 9, 3151–3169. [14] M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30 (2009), 429–445. [15] E. Nevo, Regularity of edge ideals of C4 -free graphs via the topology of the lcm-lattice, J. Combin. Theory Ser. A 118 (2011), 491–501. [16] I. Peeva, Graded syzygies, Algebra and Applications, vol. 14, Springer-Verlag London Ltd., London, 2011. 10 S. A. SEYED FAKHARI [17] D. Popescu, Bounds of Stanley depth, An. St. Univ. Ovidius. Constanta, 19(2),(2011), 187–194. [18] M. R. Pournaki, S. A. Seyed Fakhari, M. Tousi, S. Yassemi, What is . . . Stanley depth? Notices Amer. Math. Soc. 56 (2009), no. 9, 1106–1108. [19] A. Rauf, Stanley decompositions, pretty clean filtrations and reductions modulo regular elements, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 50(98) (2007), no. 4, 347–354. [20] S. A. Seyed Fakhari, Stanley depth and symbolic powers of monomial ideals, Math. Scand., to appear. [21] S. A. Seyed Fakhari, Stanley depth of the integral closure of monomial ideals, Collect. Math. 64 (2013), 351–362. [22] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175–193. [23] A. Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs: vertex decom posability and regularity, Arch. Math. (Basel) 93 (2009), 451–459. [24] R. Woodroofe, Matchings, coverings, and Castelnuovo-Mumford regularity, J. Commut. Algebra 6 (2014), 287–304. S. A. Seyed Fakhari, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran. E-mail address: fakhari@khayam.ut.ac.ir URL: http://math.ipm.ac.ir/∼fakhari/	0
arXiv:1206.5648v3 [cs.PL] 19 Sep 2013 C to O-O Translation: Beyond the Easy Stuff∗ Marco Trudel · Carlo A. Furia · Martin Nordio Bertrand Meyer · Manuel Oriol 19 April 2013 Abstract Can we reuse some of the huge code-base developed in C to take advantage of modern programming language features such as type safety, object-orientation, and contracts? This paper presents a source-to-source translation of C code into Eiffel, a modern object-oriented programming language, and the supporting tool C2Eif. The translation is completely automatic and handles the entire C language (ANSI, as well as many GNU C Compiler extensions, through CIL) as used in practice, including its usage of native system libraries and inlined assembly code. Our experiments show that C2Eif can handle C applications and libraries of significant size (such as vim and libgsl), as well as challenging benchmarks such as the GCC torture tests. The produced Eiffel code is functionally equivalent to the original C code, and takes advantage of some of Eiffel’s features to produce safe and easy-to-debug translations. 1 Introduction Programming languages have significantly evolved since the original design of C in the 1970’s as a “system implementation language” [29] for the Unix operating system. C was a high-level language by the standards of the time, but it is pronouncedly low-level compared with modern programming paradigms, as it lacks advanced features—static type safety, encapsulation, inheritance, and contracts [20], to mention just a few—that can have a major impact on programmer’s productivity and on software quality and maintainability. C still fares as the most popular general-purpose programming language [38], and countless C applications are being actively written and maintained, that take advantage of the language’s conciseness, speed, ubiquitous support, and huge code-base. An automated solution to translate and integrate C code into a modern language would combine the large availability of C programs in disparate domains with the integration in a modern language that facilitates writing safe, robust, and easy-to-maintain applications. The present paper describes the fully automatic translation of C programs into Eiffel, an object-oriented programming language, and the tool C2Eif, which implements ∗ This work was partially supported by ETH grant “Object-oriented reengineering environment”. 1 the translation. While the most common approaches to re-use C code in other host languages are based on “foreign function APIs” (see Section 5 for examples), sourceto-source translation solves a different problem, and has some distinctive benefits: the translated code can take advantage of the high-level nature of the target language and of its safer runtime. Main features of C2Eif. Translating C to a high-level object-oriented language is challenging because it requires adapting to a more abstract memory representation, a tighter type system, and a sophisticated runtime that is not directly accessible. There have been previous attempts to translate C into an object-oriented language (see the review in Section 5). A limitation of the resulting tools is that they hardly handle the trickier or specialized parts of the C language [7], which it is tempting to dismiss as unimportant “corner cases”, but figure prominently in real-world programs; examples include calls to pre-compiled C libraries (e.g., for I/O), inlined assembly, and unrestricted branch instructions including setjmp and longjmp. One of the distinctive features of the present work is that it does not stop at the core features but extends over the often difficult “last mile”: it covers the entire C language as used in practice. The completeness of the translation scheme is attested by the set of example programs to which the translation was successfully applied, as described in Section 4, including major applications such as the vim editor (276 KLOC), major libraries such as libgsl (238 KLOC), and the challenging “torture” tests for the GCC C compiler. C2Eif is available at http://se.inf.ethz.ch/research/c2eif. The webpage includes C2Eif’s sources, pre-compiled binaries, source and binaries of all translated programs of Table 1, and a user guide. Sections 2–3 describe the distinctive features of the translation: it supports the complete C language (including pointer arithmetic, unrestricted branch instructions, and function pointers) with its native system libraries; it complies with ANSI C as well as many GNU C Compiler extensions through the CIL framework [22]; it is fully automatic, and it handles complete applications and libraries of significant size; the generated Eiffel code is functionally equivalent to the original C code (as demonstrated by running thorough test suites), and takes advantage of some advanced features, such as strong typing and contracts, to facilitate debugging some programming mistakes. In our experiments, C2Eif translated completely automatically over 900,000 lines of C code from real-world applications, libraries, and testsuites, producing functionally equivalent1 Eiffel code. Safer code. Translating C code to Eiffel with C2Eif is quite useful to reuse C programs in a modern environment, and is the main motivation behind this work. However, it also implies other valuable side-benefits, which we demonstrate in Section 4. First, the translated code blends reasonably well with hand-written Eiffel code because it is not a mere transliteration from C; it is thus modifiable with Eiffel’s native tools and environments (EiffelStudio and related analysis and verification tools). Second, the 1 As per standard regression testsuites and general usage. 2 translation automatically introduces simple contracts, which help detect recurring mistakes such as out-of-bound array access or null-pointer dereferencing. To demonstrate this, Section 4.4 discusses how we easily discovered a few unknown bugs in widely used C programs (such as libgmp) just by translating them into Eiffel and running standard tests. While the purpose of C2Eif is not to debug C programs, the source of errors is usually more evident when executing programs translated into Eiffel—either because a contract violation occurs, or because the Eiffel program fails sooner, before the effects of the error propagate to unrelated portions of the code. The translated C code also benefits from the tighter Eiffel runtime, so that certain buffer overflow errors are harder to exploit than in native C environments. Why Eiffel? We chose Eiffel as the target language not only out of our familiarity with it, but also because it offers features that complement C’s, such as an emphasis on correctness [44] through the native support of contracts. Another reason for the choice has to do with the size of Eiffel’s community of developers, which is quite small compared to those of other object-oriented languages such as Java or C#. While Java users can avail of countless libraries and frameworks offering high-quality implementations of a wide array of functionalities, Eiffel users often have to implement such functionalities themselves from scratch. An automatic translation tool could therefore bring immediate benefits to reuse the rich C code-base in Eiffel; in fact, translations produced by C2Eif are already being used by the Eiffel community. Finally, Eiffel uncompromisingly epitomizes the object-oriented paradigm; hence translating C into it cannot take the shortcut of merely transliterating similar constructs (as it would have been possible, for example, with C++). The results of the paper are thus applicable to other object-oriented languages.2 An abridged version of this paper was presented at WCRE 2012 [42]; a companion paper presents the tool C2Eif from the user’s point of view [40]. 2 Overview and Architecture C2Eif is a compiler with graphical user interface that translates C programs to Eiffel programs. The translation is a complete Eiffel program which replicates the functionality of the C source program. C2Eif is implemented in Eiffel. High-level view. Figure 1 shows the overall picture of how C2Eif works. C2Eif inputs C projects (applications or libraries) processed with the C Intermediate Language (CIL) framework. CIL [22] is a C front-end that simplifies programs written in ANSI C or using the GNU C Compiler extensions into a restricted subset of C amenable to program transformations; for example, there is only one form of loop in CIL. Using CIL input to C2Eif ensures complete support of the whole set of C statements, without having to deal with each of them explicitly (although it also introduces some limitations, discussed in Section 4.5). C2Eif then translates CIL programs to Eiffel projects 2 As proof of this, the first author has made some progress towards applying the technique described in this paper to automatic translations from C to Java; and has recently been granted startup funding to support this development and turn it into a commercial product. 3 Helper Classes C application or library CIL CIL C file C2Eif Eiffel application or library Eiffel Compiler Binary Figure 1: Usage workflow of C2Eif. consisting of collections of classes that rely on a small set of Eiffel helper classes (described below). Such projects can be compiled with any standard Eiffel compiler. Incremental translation. C2Eif implements a translation T from CIL C to Eiffel as a series T1 , . . . , Tn of successive incremental transformations on the Abstract Syntax Tree. Every transformation Ti targets exactly one language aspect (for example, loops or inlined assembly code) and produces a program in an intermediate language Li which is a mixture of C and Eiffel constructs: the code progressively morphs from C to Eiffel code. The current implementation uses around 45 such transformations (i.e., n = 45). Combining several simple transformations improves the decoupling among different language constructs and facilitates reuse (e.g., to implement a translator of C to Java) and debugging: the intermediate programs are easily understandable by programmers familiar with both C and Eiffel. Helper classes. The core of the translation from C to Eiffel must ensure that Eiffel applications have access to objects with the same capabilities as their counterparts in C; for example, an Eiffel class that translates a C struct has to support field access and every other operation defined on structs. Conversely, C external pre-compiled code may also have to access the Eiffel representations of C constructs; for example, the Eiffel translation of a C program calling printf to print a local string variable str of type char∗ must grant printf access to the Eiffel object that translates str, in conformance with C’s conventions on strings. To meet these requirements, C2Eif includes a limited number of hand-written helper Eiffel classes that bridge the Eiffel and C environments; their names are prefixed by CE for C and Eiffel. Rather than directly replicating or wrapping commonly used external libraries (such as stdio and stdlib), the helper classes target C fundamental language features and in particular types and type constructors. This approach works with any external library, even non-standard ones, and is easier to maintain because involves only a limited number of classes. We now give a concise description of the most important helper classes; Section 3 shows detailed examples of their usage. • CE POINTER [G] represents C pointers of any type through the generic parameter G. It includes features to perform full-fledged pointer arithmetic and to get pointer representations which C can access but the Eiffel’s runtime will not modify (in particular, the garbage collector will not modify pointed addresses nor relocate memory areas). 4 • CE CLASS defines the basic interface of Eiffel classes that correspond to unions and structs. It includes features (members) that return instances of class CE POINTER pointing to a memory representation of the structure that C can access. • CE ARRAY [G] extends CE POINTER and provides consistent array access to both C and Eiffel (according to their respective conventions). It includes contracts that check for out-of-bound access. • CE ROUTINE represents function pointers. It supports calls to Eiffel routines through agents—Eiffel’s construct for function objects (closures or delegates in other languages)—and calls to (and callbacks from) external C functions through raw function pointers. • CE VA LIST supports variadic functions, using the Eiffel class TUPLE (sequences of elements of heterogeneous type) to store a variable number of arguments. It offers an Eiffel interface that extends the standard C’s (declared in stdarg.h), as well as output in a format accessible by external C code. 3 Translating C to Eiffel This section presents the major details of the translation T from C to Eiffel implemented in C2Eif, and illustrates the general rules with a number of small examples. The presentation breaks down T into several components that target different language aspects (for example, TTD maps C type declarations to Eiffel classes); these components mirror the incremental transformations Ti of C2Eif (mentioned in Section 2) but occasionally overlook inessential details for presentation clarity. External functions in Eiffel. Eiffel code translated from C normally includes calls to external C pre-compiled functions, whose actual arguments correspond to objects in the Eiffel runtime. This feature relies on the external Eiffel language construct: Eiffel routines can be declared as external and directly execute C code embedded as Eiffel strings3 or call functions declared in header files. For example, the following Eiffel routine (method) sin twice returns twice the sine of its argument by calling the C library function sin (declared in math.h): sin twice (arg: REAL 32): REAL 32 external C inline use <math.h> alias return 2sin($arg); end Calls using external can exchange arguments between the Eiffel and the C runtimes only for a limited set of primitive type: numeric types (that have the same underlying machine representation in Eiffel and C) and instances of the Eiffel system class POINTER that corresponds to raw untyped C pointers (not germane to Eiffel’s pointer representation, unlike CE POINTER). In the sin twice example, argument arg of numeric type REAL 32 is passed to the C runtime as $arg. Every helper class (described in Section 2) includes an attribute c pointer of type POINTER that offers access to a C-conforming representation usable in external calls. 3 For readability, we will omit quotes in external strings. 5 The mechanism of external calls is used not only in the translations produced by C2Eif but also in the Eiffel standard libraries, wherever interaction with the operating system is required, such as for input/output, file operations, and memory management. Supporting calls to native code is also the only strict requirement to be able to implement, in languages other than Eiffel, this paper’s approach to automatic translation from C. Java, for example, offers similar functionalities through the Java Native Interface (JNI). 3.1 Types and Type Constructors C declarations T v of a variable v of type T become Eiffel declarations v :TTY (T), where TTY is the mapping from C types to Eiffel classes described in this section. Numeric types. C numeric types correspond to Eiffel classes INTEGER (signed integers), NATURAL (unsigned integers), REAL (floating point numbers) with the appropriate bit-size as follows4 : C TYPE T char short int int, long int long long int float double long double E IFFEL CLASS TTY (T ) INTEGER 8 INTEGER 16 INTEGER 32 INTEGER 64 REAL 32 REAL 64 REAL 96 Unsigned variants follow the same size conventions as signed integers but for class NATURAL; for example TTY (unsigned short int) is NATURAL 16. Pointers. Pointer types are translated using class CE POINTER [G] with the generic parameter G instantiated with the pointed type: TTY (T ∗) = CE POINTER [TTY (T)] with the convention that TTY (void) maps to Eiffel class ANY, ancestor to every other class (Object in Java). The definition works recursively for multiple indirections; for example, CE POINTER[CE POINTER[REAL 32]] stands for TTY (float ∗∗). Function pointers. Function pointers are translated to Eiffel using class CE ROUTINE: TTY (T0 (∗) (T1 , ...,Tn )) = CE ROUTINE CE ROUTINE inherits from CE POINTER [ANY], and hence it behaves as a generic pointer, but customizes its behavior using references to agents that wrap the functions pointed to; Section 3.2 describes this mechanism. Arrays. Array types are translated to Eiffel using class CE ARRAY[G] with the generic parameter G instantiated with the array base type: TTY (T [n]) = CE ARRAY[TTY (T)]. 4 We implemented class REAL 96 specifically to support long double on Linux machines. 6 The size parameter n, if present, does not affect the declaration, but initializations of array variables use it (see Section 3.2). Multi-dimensional arrays are defined recursively as arrays of arrays: TTY (T [n1 ][n2 ]...[nm ]) is then CE ARRAY[TTY (T[n2 ]...[nm ])]. Enumerations. For every type enum E defined or used, the translation introduces an Eiffel class E defined by the translation TTD (for type definition): TTD (enum E {v1 = k1 , . . . , vm = km }) = class E feature v1 : INTEGER 32 =k1 ; . . . ; vm : INTEGER 32 =km end Class E has as many attributes as the enum type has values, and each attribute is an integer that receives the corresponding value in the enumeration. Every C variable of type E also becomes an integer variable in Eiffel (that is, TTY (enum E ) = INTEGER 32), and the class E is only used to assign constant values according to the enum naming scheme. Structs and unions. For every compound type struct S defined or used, the translation introduces an Eiffel class S: class S inherit CE CLASS feature TF (T1 v1 ) . . . TF (Tm vm ) end for TTD (struct S {T1 v1 ; . . . ; Tm vm }). Correspondingly, TTY (S) = S; that is, variables of type S become references of class S in Eiffel. The translation TF (T v) of each field v of the struct S introduces an attribute of the appropriate type in class S, and a setter routine set v which also updates the underlying C representation of v: v: TTY (T) assign set v −− declares ‘set v’ as the setter of v set v (a v: TTY (T)) do v := a v ; update memory field (”v”) end Class CE CLASS, from which S inherits, implements update memory field using reflection, so that the underlying C representation is created and updated dynamically only when needed during execution (for example, to pass a struct instance to a native C library), thus avoiding any data duplication overhead whenever possible. Example 1. Consider a C struct car that contains an integer field plate num and a string field brand: typedef struct { unsigned int plate num; char∗ brand; } car; The translation TTD introduces a class CAR as follows: class CAR inherit CE CLASS feature plate num: NATURAL 32 assign set plate num brand: CE POINTER [INTEGER 8] assign set brand set plate num (a plate num: NATURAL 32) do plate num := a plate num; update memory field (”plate num”) end set brand (a brand: CE POINTER [INTEGER 8]) do brand := a brand; update memory field (”brand”) end end 7 The translation of union types follows the same lines as that of structs, with the only difference that classes translating unions generate the underlying C representation in any case upon initialization, even if the union is not passed to the C runtime; calls to update memory field update all attributes of the class to reflect the correct memory value. We found this to be a reasonable compromise between performance and complexity of memory management of union types where, unlike structs, fields share the same memory space. 3.2 Variable Initialization and Usage Initialization. Eiffel variable declarations v : C only allocate memory for a reference to objects of class C, and initialize it to Void (null in Java). The only exceptions are, once again, numeric types: a declaration such as n: INTEGER 64 reserves memory for a 64-bit integer and initializes it to zero. Therefore, every C local variable declaration T v of a variable v of type T may also produce an initialization, consisting of calls to creation procedures of the corresponding helper classes, as specified by the declaration mapping TDE :   (NT) v : TTY (T ) TDE (T v;) = v : TTY (T ); create v.make( n1 , . . . , nm ) (AT)   v : TTY (T ); create v.make (OT) where definition (NT) applies if T is a numeric type; (AT) applies if T is an array type S[n1 ],. . ., [nm ]; and (OT) applies otherwise. The creation procedure make of CE ARRAY takes a sequence of integer values to allocate the right amount of memory for each array dimension; for example int a[2][3] is initialized by create a.make(2, 3). Memory management. Helper classes are regular Eiffel classes; therefore, the Eiffel garbage collector disposes instances when they are no longer referenced (for example, when a local variable gets out of scope). Upon collection, the dispose finalizer routines of the helper classes ensure that the C memory representations are also appropriately deallocated; for example, the finalizer of CE ARRAY frees the array memory area by calling free on the attribute c pointer. To replicate the usage of malloc and free, we offer wrapper routines that emulate the syntax and functionalities of their C homonym functions, but operate on CE POINTER: they get raw C pointers by external calls to C library functions, convert them to CE POINTER, and record the dynamic information about allocated memory size. The latter is used to check that successive usages conform to the declared size (see Section 4.4). Finally, the creation procedure make cast of the helper classes can convert a generic pointer returned by malloc to the proper pointed type, according to the following translation scheme: C CODE T∗ p; p = (T ∗)malloc(sizeof(T)); free(p); T RANSLATED E IFFEL CODE p: CE POINTER[TTY (T)] create p.make cast (malloc (σ(T))) free(p) 8 where σ is an encoding of the size information. Variable usage. The translation of variable usage is straightforward: variable reads in expressions are replicated verbatim, and C assignments (=) become Eiffel assignments (:=); the latter is, for CE ARRAY, CE POINTER, and classes translating C structs and unions, syntactic sugar for calls to setter routines that achieve the desired effect. The only exceptions occur when implicit type conversions in C must become explicit in Eiffel, which may spoil the readability of the translated code but is necessary with strong typing. For example, the C assignment cr = ’s’—assigning character constant ’s’ to variable cr of type char—becomes the Eiffel assignment cr := (’s’).code.to integer 8 that encodes ’s’ using the proper representation. Variable address. Whenever the address &v of a C variable v of type T is taken, v is translated as an array of unit size and type T: TDE (T v) = TDE (T v[1]), and every usage of v is adapted accordingly: &v becomes just v, and occurrences of v in expressions become ∗v. This little hack makes it possible to have Eiffel assignments translate C assignment uniformly; otherwise, usages of v should have different translations according to whether the information about v’s memory location is copied around (with &) or not. Dereferencing and pointer arithmetic. The helper class CE POINTER features a query item which translates dereferencing (∗) of C pointers. Pointer arithmetic is translated verbatim, because class CE POINTER overloads the arithmetic operators to be aliases of proper underlying pointer manipulations, so that an expression such as p + 3 in Eiffel, for references p of type CE POINTER, hides the explicit expression c pointer + 3 ∗ element size. Example 2. Consider an integer variable num, a pointer variable p, and a double pointer variable carsp with target type struct car (defined in Example 1). The following C code fragment declares num, p, and carsp, allocates space for an array of num consecutive cars pointed to by carsp, and makes p point to the array’s third element. int num; car ∗ p; car ∗∗ carsp; ∗carsp = malloc(num ∗ sizeof(car)); p = ∗carsp + 2; C2Eif translates the C fragment as follows, where the implicit C conversion from signed to unsigned int become explicit in Eiffel (calls to routine to natural 32). num: INTEGER 32 p: CE POINTER [CAR] carsp: CE POINTER [CE POINTER [CAR]] carsp.item := create {CE POINTER [CAR]}.make cast (malloc (num.to natural 32 ∗ (create {CAR}.make).structure size. to natural 32)) p := carsp.item + 2 9 2:libffi 1:agent Eiffel C 3:libffi Figure 2: Calls through function pointers. Using function pointers. Class CE ROUTINE, which translates C function pointers, is usable both in the Eiffel and in the C environment (see Figure 2). On the Eiffel side, its instances wrap Eiffel routines using agents—Eiffel’s mechanism for function objects. A private attribute routine references objects of type ROUTINE [ANY, TUPLE], an Eiffel system class that corresponds to agents wrapping routines, with any number of arguments and argument types stored in a tuple. Thus, Eiffel code can use the agent mechanism to create instances of class ROUTINE. For example, if foo denotes a routine of the current class and fp has type CE ROUTINE, create fp.make agent (agent foo) makes fp’s attribute routine point to foo. On the C side, when function pointers are directly created from C pointers (e.g., references to external C functions), CE ROUTINE behaves as a wrapper of raw C function pointers, and dynamically creates invocations to the pointed functions using the library libffi [9]. The Eiffel interface to CE ROUTINE will then translate calls to wrapped functions into either agent invocations or external calls with libffi according to how the class has been instantiated. Assume, for example, that fp is an object of class CE ROUTINE that wraps a procedure with one integer argument. If fp has been created with an Eiffel agent foo as above, calling fp.call ([42]) wraps the call foo (42) (edge 1 in Figure 2); if, instead, fp only maintains a raw C function pointer, the same instruction fp.call ([42]) creates a native C call using libffi (edge 2 in Figure 2). The services of class CE ROUTINE behave as an adapter between the procedural and object-oriented representation of routines: the signatures of C functions must change when they are translated to Eiffel routines, because routines in object-oriented languages include references to a target object as implicit first argument. Calls from external C code to Eiffel routines are therefore intercepted at runtime with libffi callbacks (edge 3 in Figure 2) and dynamically converted to suitable agent invocations. 3.3 Conditionals, Loops, and Functions The translation TCF takes care of constructs to structure the control flow of computations: conditionals, loops, and function definitions and calls. Conditionals and loops. Sequential composition, conditional instructions, and loop instructions are similar in C and Eiffel; therefore, their translation is straightforward.5 5 Eiffel loops have the general form: from init until exit loop B end. init is executed once before the ac- 10 TCF (I1 ; I2 ) TCF (if (c) {TB} else {EB}) = = TCF (while (c) {LB}) = T(I1 ) ; T(I2 ) if T(c) then T(TB) else T(EB) end from until not T(c) loop T(LB) end Note that Eiffel has only one type of loop, which translates the unique type of C loops produced by CIL. Functions. Function declarations and function calls in C translate to routine declarations and routine calls in Eiffel. Given a function foo with arguments a1 , . . . , an of type T1 , . . . , Tn and return type T0 , the translation: TCF T0 foo (T1 a1 , . . . , Tn an ){ B } is defined as follows: ( foo(a1: TTY (T1 ); . . . ; an: TTY (Tn )) do T(B) end if T0 is void foo(a1: TTY (T1 ); . . . ; an: TTY (Tn )) : TTY (T0 ) do T(B) end otherwise Correspondingly, function calls in C become routine calls in Eiffel with the actual arguments also recursively translated: TCF (foo (e1 , . . . , en )) = foo (T(e1 ), . . . , T(en )) Variadic functions. The C language supports variadic functions, also known as varargs functions, which work on a variable number of arguments. The Eiffel translation of variadic function declarations uses the class TUPLE to wrap lists of arguments: the translation of a variadic function var foo with n ≥ 0 required arguments a1 , . . . , an of type T1 , . . . , Tn , optional additional arguments, and return type T0 : TCF T0 var foo (T1 a1 , . . . , Tn an , . . .){ B } is defined as: var foo (args: TUPLE[a1 : TTY (T1 ); . . . ; an : TTY (Tn )]): TTY (T0 ) assuming T0 is not void; otherwise, the return type is omitted, as in the translation of standard functions. Calls to variadic functions with n required arguments may provide m ≥ 0 additional actual arguments en+1 , . . . , en+m after the n required ones e1 , . . . , en . The translation combines all arguments, required and additional, into an (n + m)-TUPLE denoted by square brackets: TCF (var foo (e1 , . . . , en , en+1 , . . . , en+m )) = var foo ([T(e1 ), . . . , T(en ), T(en+1 ), . . . , T(en+m )]) tual loop; the loop body B is executed until the Boolean condition exit becomes true; since the exit condition is tested before each iteration, the body may not be executed. Body B and init may be empty. 11 The correctness of this translation relies on Eiffel’s type conformance rule for instances of the TUPLE class: every (n + m)-TUPLE with types [T1 , . . . , Tn , Tn+1 , . . . , Tn+m ], for n, m ≥ 0, conforms to any shorter n-TUPLE with types [T1 , . . . , Tn ]. Thus, translated calls are type-safe because the assignment of actual to formal arguments is covariant [12]. The bodies of variadic functions can refer to the required arguments by name using the usual syntax. The C type va list and functions va start and va arg in library stdarg provide a means to access the optional arguments: va start(argp, last) initializes a variable argp of type va list to point to the first optional argument after last (the name of the last declared argument). Every successive invocation of va arg(argp, T) returns the argument of type T pointed to by argp, and then moves argp right after it. The Eiffel translation relies on the helper class CE VA LIST which replicates stdarg’s functionality: TDE ( va list argp ) T( va start(argp, last) ) T( va arg(argp, T) ) = = = argp: CE VA LIST create argp.make (args, index) argp.TTY (T ) item where args is the name of the argument TUPLE in the enclosing variadic function and index is the index of the first optional argument. While CE VA LIST provides a uniform interface to both Eiffel and C, the actual arguments of a variadic function may also be accessed in the Eiffel translation using the standard syntax for TUPLEs: in a variadic function with n required arguments, the expression args.ai refers to the ith named argument ai , for 1 ≤ i ≤ n; and args.Tj item(j) refers to the jth argument aj , for any 1 ≤ j ≤ n (required argument) or j > n (optional argument), where Tj is aj ’s type. Example 3. Continuing Examples 1 and 2, consider a variadic function init cars that takes a pointer carsp to an array of cars and num pairs of unsigned integers and strings (passed as optional arguments), and initializes the array with num cars whose plate numbers and brands respectively correspond to the integer and string in each pair. void init cars(car ∗∗carsp, int num, . . .) { va list argp; car ∗ccar; int n; ∗carsp = malloc(num ∗ sizeof(car)); ccar = ∗carsp; va start(argp, num); for(n = num ; n >0; n−−) { ccar →plate num = va arg(argp, unsigned int); ccar →brand = va arg(argp, char∗); ccar++; } } The translation into Eiffel uses the class CE VA LIST and accesses the optional unsigned integer arguments with argp.natural 32 item, and the optional strings with 12 argp.pointer item. The latter returns a generic pointer, which is cast to char ∗ with an explicit cast (line 15). 13 init cars (args: TUPLE [carsp: CE POINTER [CE POINTER [CAR]]; num: INTEGER 32]) local argp: CE VA LIST ccar: CE POINTER [CAR] n: INTEGER 32 do args.carsp.item := create {CE POINTER [CAR]}.make cast ( malloc ( num.to natural 32 ∗ (create {CAR}.make).structure size. to natural 32 ) ) ccar := args.carsp.item create argp.make (args, 3) n := num from until not n >0 loop ccar.item.plate num := argp.natural 32 item 15 ccar.item.brand := (create {CE POINTER [INTEGER 8]}.make cast (argp. pointer item)) ccar := ccar + 1 n := n − 1 end end 3.4 Unstructured Control Flow In addition to constructs for structured programming, C offers control-flow breaking instructions such as jumps. This section discusses their translation to Eiffel. Jumps. Eiffel enforces structured programming, and hence it lacks control-flow breaking instructions such as goto, break, continue, and return. The translation TCF eliminates such instructions along the lines of the global version—using Harel’s terminology [13]—of the structured programming theorem. The body of a function using goto determines a list of instructions s0 , s1 , . . . , sn , where each si is a maximal sequential block of instructions, with no labels after the first instruction or jumps before the last one in the block. TCF translates the body into a single loop over an auxiliary integer variable pc that emulates a program counter:6 6 Eiffel’s inspect/when instructions corresponds to a restricted form of switch/case without fall through. 14 from pc := 0 until pc = −1 loop inspect pc when 0 then T(s0 ) ; upd(pc) when 1 then T(s1 ) ; upd(pc) TCF (hs0 , s1 , . . . , sn i) = ..  .     when n then T(sn ) ; upd(pc)     end    end              Variable pc is initially zero; every iteration of the loop body executes spc for the current value of pc, and then updates pc (with upd(pc)) to determine the next instruction to be executed: blocks ending with jumps modify pc directly, other blocks increment it by one, and exit blocks set it to −1, which makes the overall loop terminate (whenever the original function terminates). This translation supports all control-flow breaking instructions, and in particular continue, break, and return, which are special cases of goto. TCF , however, improves the readability in these special simpler cases by directly using auxiliary Boolean flag variables with the same names as the instruction they replace. The flags are tested as part of the translated exit conditions for the loops where the control-flow breaking instructions appear. Using this alternative translation scheme where the generality of gotos is not required makes for translations with little changes to the code structure, which are usually more readable. The loop while (n > 0){ if (n == 3) break; n−−; }, for example, becomes: from until break or not n >0 loop if n = 3 then break := True end if not break then n := n − 1 end end Long jumps. The C library setjmp provides functions setjmp and longjmp to save an arbitrary return point and jump back to it across function call boundaries. The wrapping mechanism used for external functions (see Section 3.5) does not work to replicate long jumps, because the return values saved by setjmp wrapped as an external function are no longer valid after execution leaves the wrapper. Therefore, C2Eif translates setjmp and longjmp by means of the helper class CE EXCEPTION. As the name suggests, CE EXCEPTION uses Eiffel’s exception propagation mechanism to go back in the call stack to the allocation frame of the function that called setjmp. There, translated goto instructions jump to the specific point saved with setjmp within the function body. Example 4. Consider a C function target that executes setjmp(buf) to save a location using a global variable buf of type jmp buf. In the translated Eiffel code, buf becomes an attribute of type CE EXCEPTION, and the call to setjmp in target becomes a creation of an instance attached to buf: buf := create {CE EXCEPTION}.make (. . .); make’s actual arguments (omitted for simplicity) contain references to the stack frame of the current instance of target, and to the specific instruction where to jump. 15 Later during the execution, another function source called from target performs a longjmp(buf, 3), which diverts execution to the location previously marked by setjmp and returns value 3. In Eiffel, the longjmp becomes a raising of the exception object buf enclosing the return value: buf.raise (3). The flow of what happens next depends on the semantics of Eiffel exceptions, which is significantly different than that of other object-oriented languages such as Java and C#. The raised exception propagates to the recipient target, where it is handled by a rescue clause. Such exception handling blocks are routine-specific in Eiffel, rather than being associated to arbitrary scopes such as in Java’s try /catch blocks. The translation of setjmp also took care of setting up target’s rescue clause: if the raised exception stores a reference to the current instance of routine target, the code in the rescue clause modifies a variable pc local to target to point to the location of the setjmp and, using Eiffel’s retry instruction, executes target’s body again with this new value. Based on the value, the new execution of target’s body jumps to the correct location following a mechanism similar to the previously discussed translation of goto instructions. Finally, if the handled exception references a routine instance other than the current execution of target, the rescue clause propagates the exception through the call stack, so that it can reach the correct stack frame. 3.5 Externals and Encapsulation The translation T uses classes to wrap header and source files, which give a simple modular structure to C programs. Externals. For every included system header header.h, T defines a class HEADER with wrappers for all external functions and variables declared in header.h. The wrappers are routines using the Eiffel external mechanism and performing the necessary conversions between the Eiffel and the C runtimes. In particular, external functions using only numeric types, which are interoperable between C and Eiffel, directly map to wrapper routines; for example, exit in stdlib.h becomes: exit (status: INTEGER 32) external C inline use <stdlib.h> alias exit($status); end When external functions involve types using helper classes in Eiffel, a routine passes the underlying C representation to the external calls; for example, fclose in stdio.h generates: fclose (stream: CE POINTER [ANY]): INTEGER 32 do Result := c fclose (stream.c pointer) end c fclose (stream: POINTER): INTEGER 32 external C inline use <stdio.h> alias return fclose($stream); end In some complex cases—typically, with variadic external functions—the wrapper can only assemble the actual call on the fly at runtime; this is done using CE ROUTINE. For example, printf is wrapped as: 16 printf (args: TUPLE[format: CE POINTER[INTEGER 8]]): INTEGER 32 do Result := (create {CE ROUTINE}.make shared (c printf)).integer 32 item (args) end c printf: POINTER external C inline use <stdio.h> alias return &printf; end The translation can also inline assembly code, using the same mechanisms as external function calls. Globals. For every source file source.c, T defines a class SOURCE that includes translations of all function definitions (as routines) and global variables (as attributes) in source.c. Class SOURCE also inherits from the classes translating the other system header files that source.c includes, to have access to their declarations. For example, if foo.c includes stdio.h, FOO is declared as class FOO inherit STDIO. 3.6 Formatted Output Optimization The library function printf is the standard C output function, which displays values according to format strings passed as argument. In contrast, Eiffel provides the command Io.put string to put plain text strings on standard output, as well as type-specific formatter classes, such as FORMAT INTEGER, to produce string representations of arbitrary types. With the goal of making the translated code as close as possible to standard Eiffel, C2Eif tries to replace calls to printf with equivalent calls to Io.put string and routines of the formatter classes. Whenever printf’s returned value is not used and the format string is a constant literal, C2Eif parses the literal format string and encodes it as equivalent calls to Io.put string and formatters. This process may find mismatches between some format specifiers and the types of the corresponding value arguments, which C2Eif reports as warnings. Another situation where C2Eif can replace calls to printf with calls to put string is when only one argument is passed to the former, which is then interpreted verbatim as a string. Whenever a warning is issued or the usage of printf cannot be rendered using put string, C2Eif falls back to calling a wrapper of the native printf function (described in Section 3.5). The translation also implements similar optimizations for variants of printf such as fprintf. 4 Evaluation and Discussion This section evaluates the translation T and its implementation in C2Eif. The bulk of the evaluation assesses correctness (Section 4.1) and performance (Section 4.2) based on experiments with 14 open-source programs. We also qualitatively discuss other aspects that influence maintainability (Section 4.3), the advantages in terms of safety deriving from switching to the Eiffel runtime (Section 4.4), as well as the current limitations of the C2Eif approach (Section 4.5). 17 4.1 Correct Behavior To experimentally assess the translations produced by C2Eif, we applied it to 14 opensource C programs, including 7 applications, 6 libraries, and one testsuite; most of them are widely-used in Linux and other “nix” distributions. hello world is the only toy application, which is however useful to have a baseline of translating from C to Eiffel with C2Eif. The other applications are: micro httpd 12dec2005, a minimal HTTP server; xeyes 1.0.1, a widget for the X Windows System which shows two googly eyes following the cursor movements; less 382-1, a text terminal pager; wget 1.12, a command-line utility to retrieve content from the web; links 1.00, a simple web browser; vim 7.3, a powerful text editor. The libraries are: libSDL mixer 1.2, an audio playback library; libmongoDB 0.6, a library to access MongoDB databases; libpcre 8.31, a library for regular expressions; libcurl 7.21.2, a URL-based transfer library supporting protocols such as FTP and HTTP; libgmp 5.0.1, for arbitrary-precision arithmetic; libgsl 1.14, a powerful numerical library. The gcc “torture tests” are short but semantically complex pieces of C code, used as regression tests for the GCC compiler. Table 1 shows the results of translating the 14 programs into Eiffel using C2Eif, running on a GNU/Linux box (kernel 2.6.37) with a 2.66 GHz Intel dual-core CPU and 8 GB of RAM, GCC 4.5.1, CIL 1.3.7, EiffelStudio 7.1.8. For each application, library, and testsuite Table 1 reports: (1) the size (in lines of code) of the CIL version of the C code and of the translated Eiffel code; (2) the number of Eiffel classes created; (3) the time (in seconds) spent by C2Eif to perform the source-to-source translation (not including compilation from Eiffel source to binary); (4) the size of the binaries (in MBytes) generated by EiffelStudio.7 We ran extensive trials on the translated programs to verify that they behave as in their original C version, thus validating the correctness of the translation T and its implementation in C2Eif. The trials comprised informal usage, systematic performance tests for some of the applications, and running the standard testsuites available for the three biggest libraries (libcurl, libgmp, and libgsl). The rest of this section describes the translated testsuites and their behavior with respect to correctness; Section 4.2 discusses the quantitative performance results. Library libcurl comes with a client application and a testsuite of 583 tests defined in XML and executed by a Perl script calling the client; libgmp and libgsl respectively include testsuites of 145 and 46 tests, consisting of client C code using the libraries. All tests execute and pass on both the C and the translated Eiffel versions of the libraries, with the same logged output. For libcurl, C2Eif translated the library and the client application. For libgmp and libgsl, it translated the test cases as well as the libraries. The gcc torture testsuite includes 1116 tests; the GCC version we used fails 4 of them; CIL (which depends on GCC) fails another 110 tests among the 1112 that GCC passes; finally, C2Eif (which depends on CIL) passes 989 (nearly 99%) and fails 13 of the 1002 tests passed by CIL. Given the challenging nature of the torture testsuite, this result is strong evidence that C2Eif handles the complete C language used in practice, and produces correct translations. 7 We do not give a binary size for libraries, because EiffelStudio cannot compile them without a client. 18 Table 1: Translation of 14 open-source programs. S IZE (LOCS) CIL E IFFEL hello world micro httpd xeyes less wget links vim libSDL mixer libmongoDB libpcre libcurl libgmp libgsl gcc (torture) T OTAL 8 565 1,463 16,955 46,528 70,980 276,635 7,812 7,966 18,220 37,836 61,442 238,080 147,545 932,035 15 1,934 10,661 22,545 57,702 100,815 395,094 11,553 10,341 24,885 65,070 79,971 344,115 256,246 1,380,947 #E IFFEL CLASSES T IME (S) B INARY SIZE (MB) 1 16 78 75 183 211 663 47 43 38 289 370 978 2,569 5,561 1 1 1 5 25 33 144 3 3 14 18 21 85 79 433 1.3 1.5 1.8 2.6 4.5 13.9 24.2 – – – – – – 1,576 1,626 The 13 torture tests failing after translation to Eiffel target the following unsupported features. One test reads an int from a va list (variadic function list of arguments) which actually stores a struct whose first field is a double; the Eiffel typesystem does not allow this, and inspection suggests that it is probably a copy-paste error rather than a feature. Two tests exercise GCC-specific optimizations, which are immaterial after translation to Eiffel. Six tests target exotic GCC built-in functions, such as builtin frame address; one test performs explicit function alignment; and three rely on special bitfield operations. 4.2 Performance We analyzed the performance of 5 applications, the GCC torture testsuite, and the standard testsuites of 3 libraries (described in Section 4.1) translated to Eiffel using C2Eif. Table 2 shows the result of the performance trials, running on the same system used for the experiments of Section 4.1.8 For each program or testsuite, Table 2 reports the execution time (in seconds), the maximum percentage of CPU, and the maximum amount of RAM (in MBytes) used while running. The table compares the performance of the original C versions (columns “C”) against the Eiffel translations with C2Eif (columns “T”), and, for the simpler examples, against manually written Eiffel implementations (columns “E”) that transliterate the original C implementations using the closest Eiffel constructs (for example, putchar becomes Io.put character) with as little changes as possible to the code structure (we manually wrote these transliterations 8 We compiled all CIL-processed C programs with the GCC options -O2 -s; and all Eiffel programs with disabled void checking, inlining size 100, and stripped binaries. 19 Table 2: Performance comparison for 5 applications and 4 testsuites. hello world micro httpd less wget vim libcurl libgmp libgsl gcc (torture) E XECUTION TIME ( S ) C T E M AX % CPU C T E M AX MB RAM C T E 0 5 36 16 85 199 44 25 0 0 99 – 22 – – – – – 1.3 2.3 – 4.4 – – – – – 0 37 36 16 85 212 728 1501 5 0 46 – – – – – – – 30 99 – 22 – – – – – 30 99 – – – – – – – 5.5 7.8 – 69 – – – – – 5.3 5.6 – – – – – – – ourselves). Maximum CPU and RAM usages are immaterial for the testsuites (torture and libraries), because their execution consists of a large number of separate calls. The rest of this section discusses the performance results together with other qualitative performance assessments. The performance of hello world demonstrates the base overhead, in terms of CPU and memory usage, of the default Eiffel runtime (objects, automatic memory management, and contract checking—which can however be disabled for applications where sheer performance is more important than having additional checks). The test with micro httpd consisted in serving the local download of a 174 MB file (the Eclipse IDE); this test boils down to a very long sequence (approximately 200 million iterations) of inputting a character from file and outputting it to standard output. The translated Eiffel version incurs a significant overhead with respect to the original C version, but is faster than the manually written Eiffel transliteration. This might be due to feature lookups in Eiffel or to the less optimized implementation of Eiffel’s character services. As a side note, we did the same exercise of manually transliterating micro httpd using Java’s standard libraries; this Java translation ran the download example in 170 seconds, using up to 99% of CPU and 150 MB of RAM. The test with wget downloaded the same 174 MB Eclipse package over the SWITCH Swiss network backbone. The bottleneck is the network bandwidth, and hence differences in performance are negligible, except for memory consumption, which is higher in Eiffel due to garbage collection (memory is deallocated only when necessary, thus the maximum memory usage is higher in operational conditions). The test with libcurl consisted in running all 583 tests from the standard testsuite mentioned before. The total runtime is comparable in translated Eiffel and C. The tests with libgmp and libgsl ran their respective standard testsuites. The overall slow-down seems significant, but a closer look shows that the large majority of tests run in comparable time in C and Eiffel: 30% of the libgmp tests take up over 95% of the running time; and 26% of the libgsl tests take up almost 99% of the time. The GCC torture tests incur only a moderate slow-down, concentrated in 3 tests that take 97% of the time. In all these experiments, the tests responsible for the conspicuous 20 slow-down target operations that execute slightly slowlier in the translated Eiffel than in the native C (e.g., accessing a struct field) and repeat it a huge number of times, so that the basic slow-down increases many-fold. These bottlenecks are an issue only in a small fraction of the tests and could be removed manually in the translation. The interactive applications (xeyes, less, links, and vim) run smoothly with good responsiveness, comparable to their original implementations. The test for less and vim consisted in scrolling through large text files one line at a time (by holding “arrow down” in less) or one page at a time (by holding “page down” in vim). Table 2 shows the running times, which are the same in C and Eiffel; the screen refresh rate is also indistinguishable. In all, the performance overhead in switching from C to Eiffel significantly varies with the program type but, even when it is significant, does not preclude the usability of the translated application or library in normal conditions—as opposed to the behavior in a few specific test cases. 4.3 Usability and Maintainability A tool such as C2Eif, which provides automatic translation between languages, is applicable in different contexts within general software maintenance and evolution processes. This section discusses some of these applications and how suitable C2Eif can be for each of them. Reuse in clients. The first, most natural application is using C2Eif to automatically reuse large C code-bases in Eiffel. This is not merely a possibility, but something extremely valuable for Eiffel, whose user community is quite small compared to those of other mainstream languages such as C, Java, or C++. Since we released C2Eif to the public as open-source, we have been receiving several requests from the community to produce Eiffel versions of C libraries whose functionalities are sorely missed in Eiffel, and whose native implementation would require a substantial effort to get to software of quality comparable to the widely tested and used C implementations. This was the case, in particular, of libSDL mixer, libmongoDB, and libpcre, whose automatic translations created using C2Eif were requested from the community and are now being used in Eiffel applications. We are also aware of some Eiffel programmers directly trying to use C2Eif to translate useful C libraries and deploy them in their own software. This form of reuse mainly entails writing Eiffel client code that accesses translated C components. Supporting it requires tools that handle the full C language as used in practice, and that produce translated APIs understandable and usable with a programming style sufficiently close to what is the norm in Eiffel, without requiring in-depth understanding of the C conventions. To give an idea of how C2Eif fares in this respect, consider writing a simple class PRINT SOURCE SEHOME, which uses the API of cURL to retrieve and print the HTML code of the home page at http://se.inf.ethz.ch. Table 3 shows two versions of this client class. The one on the left uses the translation of libcurl automatically generated by C2Eif; the one on the right uses the wrapper of libcurl part of the Eiffel standard libraries included with the EiffelStudio IDE, written by EiffelSoftware programmers. The two 21 solutions are quite similar in structure and style. The version based on C2Eif still requires adhering to a couple of conventions that are a legacy of the translation from C: the arguments passed to routine curl easy setopt must be listed in a tuple, and the routine call itself includes a do nothing which is needed whenever a function that returns a value is used as an instruction (Eiffel enforces separation between functions and procedures). On the other hand, the “native” solution has a slightly more complex control structure, because it has to check that the dynamically linked cURL library is actually accessible at runtime; this is unnecessary with the implementation based on C2Eif, which does not depend on dynamically linked libraries since libcurl is available translated to Eiffel. When a C library undergoes maintenance, the introduced changes have the same impact on the C and on the Eiffel clients of the library. In particular, if the changes to the library do not break client compatibility (that is, the API does not change), one should simply run C2Eif again on the new library version and replace it in the Eiffel projects that depends on it. If the API changes, clients may have to change too, independent of the language they are written in. Evolution of translated libraries. Once a program is translated from C to Eiffel, one may decide it has become part of the Eiffel ecosystem, and hence it will undergo maintenance and evolution as any other piece of Eiffel code. In this scenario, C2Eif class PRINT SOURCE SEHOME class PRINT SOURCE SEHOME inherit LIBCURL CONSTANTS inherit CURL OPT CONSTANTS create make create make feature feature make local easy: P EASY STATIC handle: CE POINTER [ANY] ret: NATURAL 32 do create easy handle := easy.curl easy init easy.curl easy setopt ([handle, Curlopt url, ce string (”se.inf.ethz.ch”)]).do nothing ret := easy.curl easy perform (handle) easy.curl easy cleanup (handle) end end make local easy: CURL EASY EXTERNALS handle: POINTER ret: INTEGER 32 do create easy if easy.is dynamic library exists then handle := easy.init easy.setopt string (handle, Curlopt url, ”se.inf.ethz.ch”) ret := easy.perform (handle) easy.cleanup (handle) else Io.error.put string (”cURL not found.%N”) end end end Table 3: Two Eiffel clients of cURL: using libcurl translated with C2Eif (left) and using the library wrapper provided by EiffelStudio (right). 22 provides an immediately applicable solution to port C code to Eiffel, whereas the ensuing maintenance effort is distributed over an entire lifecycle and devoted to improve the automatic translation (for example, removing the application-dependent performance bottlenecks highlighted in Section 4.2) and completely conforming it to the Eiffel style. Such maintenance of translations is the easier the closer the generated code follows Eiffel conventions and, more generally, the object-oriented paradigm. Providing a convincing empirical evaluation of the readability and maintainability of the code generated by C2Eif (not just from the perspective of writing client applications) is beyond the scope of the present work. Notice, however, that C2Eif already follows numerous Eiffel conventions such as for the names of classes, types, and local variables, which might look verbose to hard-core C programmers but are de rigueur in Eiffel. Follow-up work, which we briefly discuss in Section 6, has targeted the object-oriented reengineering of C2Eif translations. In all, while the translations produced by C2Eif still retain some “C flavor”, we believe they are overall understandable and modifiable in Eiffel with reasonable effort. Two-way propagation of changes. One more maintenance scenario occurs if one wants to be able to independently modify a C program and its translation to Eiffel, while still being able to propagate the changes produced in each to the counterpart. For example, this scenario applies if a C library is being extended with new functionality, while its Eiffel translation produced by C2Eif undergoes refactoring to optimize it to the Eiffel environment. This scenario is the most challenging of those discussed in this section; it poses problems similar to those of merging different development branches of the same project. While merge conflicts are still a bane of collaborative development, modern version control systems (such as Git or Mercurial) have evolved to provide powerful support to ease the process of conflict reconciliation. Thus, they could be very useful also in combination with automatic translators such as C2Eif to be able to integrate changes in C with other changes in Eiffel. 4.4 Safety and Debuggability Besides the obvious advantage of reusing the huge C code-base, translating C code to Eiffel using C2Eif leverages some high-level feature which may improve safety and make debugging easier in some conditions. Uncontrolled format string is a well-known vulnerability [4] of C’s printf library function, which permits malicious clients to access data in the stack by supplying special format strings. Consider for example the C program: int main (int argc, char ∗ argv[]) { char ∗secret = ”This is secret!”; if (argc >1) printf(argv[1]); return 0; } If we call it with: ./example "{Stack: %x%x%x%x%x%x} --> %s", we get the output {stack: 0b7[. . .]469} --> This is secret!, which reveals the local string secret. The safe way to achieve the intended behavior would be the instruction printf (”%s”, argv[1]) instead of printf (argv[1]), so that the input string is interpreted literally. 23 What is the behavior of code vulnerable to uncontrolled format strings, when translated to Eiffel with C2Eif? In simple usages of printf with just one argument as in the example, the translation replaces calls to printf with calls to Eiffel’s Io.put string, which prints strings verbatim without interpreting them; therefore, the translated code is not vulnerable in these cases. The replacement was possible in 65% of all the printf calls in the programs of Table 1. C2Eif translates more complex usages of printf (for example, with more than one argument and no literal format string such as printf (argv[1], argv[2])) into wrapped calls to the external printf function, and hence the vulnerability still exists. However, it is less extensive or more difficult to exploit in Eiffel: primitive types (such as numeric types) are stored on the stack in Eiffel as they are in C, but Eiffel’s bulkier runtime typically stores them farther up the stack, and hence longer and more complex format strings must be supplied to reach the stack data (for instance, a variation of the example with secret requires 386 %x’s in the format string to reach local variables). On the other hand, non-primitive types (such as strings and structs) are wrapped by Eiffel classes in C2Eif, which are stored in the heap, and hence unreachable directly by reaching stack data. In these cases, the vulnerability vanishes in the Eiffel translation. Debugging format strings. C2Eif also parses literal format strings passed to printf and detects type mismatches between format specifiers and actual arguments. This analysis, necessary when moving from C to a language with a stronger type system, helps debug incorrect and potentially unsafe usages of format strings. Indeed, a mismatch detected while running the 145 libgmp tests revealed a real error in the library’s implementation of macro TESTS REPS: char ∗envval, ∗end; /∗ ... ∗/ long repfactor = strtol(envval, &end, 0); if(∗end \|\| repfactor ≤ 0) fprintf (stderr, ”Invalid repfactor: %s.\n”, repfactor); String envval should have been passed to fprintf instead of long repfactor. GCC with standard compilation options does not detect this error, which may produce garbage or even crash the program at runtime. Interestingly, version 5.0.2 of libgmp patches the code in the wrong way, changing the format specifier %s into %ld. This is still incorrect because when envval does not encode a valid “repfactor”, the outcome of the conversion into long is unpredictable. Finally, notice that C2Eif may also report false positives, such as long v = ”Hello!”; printf(”%s”, v) which is acceptable (though probably not commendable) usage. Out-of-bound error detection. C arrays translate to instances of class CE ARRAY (see Section 3.1), which includes contracts that signal out-of-bound accesses to the array content. Therefore, out-of-bound errors are much easier to detect in Eiffel applications using components translated with C2Eif. Simply by translating and running the libgmp testsuite, we found an off-by-one error causing out-of-bound access (our patch has been included in more recent versions of the library); the error does not manifest itself when running the original C version. More generally, contracts help detect the precise location of array access errors. Consider, for example: 24 int ∗ buf = (int ∗) malloc(sizeof (long long int) ∗ 10); buf = buf − 10; buf = buf + 29; ∗buf = ’a’; buf++; ∗buf = ’b’; 1 2 3 4 5 buf is an array that stores 20 elements of type int (which has half the size of long long int). The error is on line 5, when buf points to position 20, out of the array bounds; line 2 is instead OK: buf points to an invalid location, but it is not written to. This program executes without errors in C; the Eiffel translation, instead, stops exactly at line 5 and signals an out-of-bound access to buf. Array bound checking may be disabled, which is necessary in borderline situations where out-of-bound accesses do not crash because they assume a precise memory layout. For example, links and vim use statements of the following form block ∗p = (block ∗)malloc(sizeof(struct block)+ len), with len >0, to allocate struct datatypes of the form struct block { /∗... ∗/char b[1]; }. In this case, p points to a struct with room for 1 + len characters in p→b; the instruction p→b[len]=‘i’ is then executed correctly in C, but the Eiffel translation assumes p→b has the statically declared size 1, hence it stops with an error. Another borderline situation is with multidimensional arrays, such as double a[2][3]. An iteration over a’s six elements with double ∗p = &a[0][0] translated to Eiffel fails to go past the third element, because it sees a[0][0] as the first element of an array of length 3 (followed by another array of the same length). A simple cast double ∗ p = (double∗)a achieves the desired result without fooling the compiler, hence it works without errors also in translated code. These situations are examples of unsafe programming more often than not. More safety in Eiffel. Our experiments found another bug in libgmp, where function gmp sprintf final had three formal input arguments, but was only called with one actual through a function pointer. Inspection suggests it is a copy-paste error of the other function gmp sprintf reps. The Eiffel version found the mismatch when calling the routine and reported a contract violation. Easily finding such bugs demonstrates the positive side-effects of translating widely-used C programs into a tighter, higher-level language. 4.5 Limitations The only significant limitations of the translation T implemented in C2Eif in supporting C programs originate in the introduction of strong typing: programming practices that implicitly rely on a certain memory layout may not work in C applications translated to Eiffel. Section 4.4 mentioned some examples in the context of array manipulation (where, however, the checks on the Eiffel side can be disabled). Another example is a function int trick (int a, int b) that returns its second argument through a pointer to the stack, with the instructions int ∗p = &a; return ∗(p+1). C2Eif’s translation assumes p points to a single integer cell and cannot guarantee that b is stored in the next cell. Another limitation is the fact that C2Eif takes input from CIL, and hence it does not support legacy C such as K&R C. The support can, however, be implemented by di25 rectly extending the pre-processing CIL front-end. Similarly, the GCC torture testsuite highlighted a few exotic GCC features currently unsupported by C2Eif (Section 4.1), which may be handled in future work. Code using unrestricted gotos poses the biggest hurdles to producing readable Eiffel code. This is arguably unavoidable when translating to any programming language that does not have jumps. The translation T does, however, avoid the most complicated general translation scheme with the simpler control-flow breaking instructions continue, break, and return instructions, whose translation is normally much more readable than when using unrestricted gotos (see the example in Section 4). 5 Related Work There are two main approaches to reuse source code written in a “foreign” language (e.g., C) in a different “host” language (e.g., Eiffel): define wrappers for the components written in the foreign language; and translate the foreign source code into functionally equivalent host code. We discuss related work pursuing these approaches in Sections 5.1 and 5.2. In Section 5.3, we review the major solutions to automate object-oriented reengineering, which is a natural follow-up of automatic translation into object-oriented languages. 5.1 Wrapping Foreign Code Wrappers enable the reuse of foreign implementations through the API of bridge libraries. This approach (e.g., [6, 5, 30]) does not modify the foreign code, whose functionality is therefore not altered; moreover, the complete foreign language is supported. On the other hand, the type of data that can be retrieved through the bridging API is often restricted to a simple subset common to the host and foreign language (e.g., primitive types). C2Eif uses wrappers only to translate external functions and assembly code. 5.2 Translating Foreign Code Industrial practices have long encompassed manual migrations of legacy code. Some semi-automated tools exist that help translate code written in legacy programming languages such as old versions of COBOL [37, 21], Fortran-77 [1, 34], and K&R C [46]. Terekhov et al. [37] review how automated language conversion is applied in industry; based on their experience, they conclude that “creating 100% automated conversion tools is neither possible, nor desirable”. Our experience with C2Eif, however, suggests that such a conclusion has only relative validity: in the rich design space of automatic translators, there are scenarios where trading off some performance for complete automation is possible and desirable. Rather than imposing an upfront heavyweight burden on developers in charge of migration, we suggest to start with an automatically translated version which is suboptimal but works out of the box, and then devote the manual programming effort to improving and adapting what is necessary— 26 incrementally with an agile approach which also depends on the specific application domain and requirements. Some translators focus on the adaptation of code into an extension (superset) of the original language. Examples include the migration of legacy code to object-oriented code, such as Cobol to OO-Cobol [25, 32, 45], Ada to Ada95 [35], and C to C++ [15, 47]. Some of such efforts try to go beyond the mere hosting of the original code, and introduce refactorings that take advantage of the object-oriented paradigm. Most of these refactorings are, however, limited to restructuring modules into classes (see the focused discussion in Section 5.3). C2Eif follows a similar approach, but also takes advantage of some advanced features (such as contracts) to improve the reliability of translated code. Ephedra [18] is a tool that translates legacy C to Java. It first translates K&R C to ANSI C; then, it maps data types and type casts; finally, it translate the C source code to Java. Ephedra handles a significant subset of C, but cannot translate frequently used features such as unrestricted gotos, external pre-compiled libraries, and inlined assembly code. A case study evaluating Ephedra [19] involved three small programs: the implementation of fprintf, a monopoly game (about 3 KLOC), and two graph layout algorithms. The study reports that the source programs had to be manually adapted to be processable by Ephedra. In contrast, C2Eif is completely automatic, and works with significantly larger programs. Other tools (proprietary or open-source) to translate C (and C++) to Java or C# include: C2J++ [39], C2J [27], and C++2C# and C++2Java [36]. Table 4 shows a feature comparison among the currently available tools that translate C to an objectoriented language, showing: • The target language. • Whether the tool is completely automatic, that is whether it generates translations that are ready for compilation. • Whether the tool is available for download and usable. In a couple of cases we could only find papers describing the tool but not a version of the implementation working on standard machines. • An (subjective to a certain extent) assessment of the readability of the code produced. In each case, we tried to evaluate if the translated code is sufficiently similar to the C source to be readily understandable by a programmer familiar with the latter. We judged C2J’s readability negatively because the tool puts data into a single global array to support pointer arithmetic. This is quite detrimental to readability and also circumvents type checking in the Java translation. • Whether the tool supports unrestricted calls to external libraries, unrestricted pointer arithmetic, unrestricted gotos, and inlined assembly code. The table demonstrates that C2Eif is arguably the first completely automatic tool that handles the complete C language as used in practice. In previous work, we developed J2Eif, an automatic source-to-source translator from Java to Eiffel [43]; translating between two object-oriented languages does not 27 completely automatic available readability external libraries pointer arithmetic gotos inlined assembly Ephedra Convert2Java C2J++ C2J C++2Java C++2C# C2Eif target language Table 4: Tools translating C to O-O languages. Java Java Java Java Java C# Eiffel no no no no no no yes no no no yes yes yes yes + + + − + + + no no no no no no yes no no no yes no no yes no no no no no no yes no no no no no no yes pose some of the formidable problems of bridging wildly different abstraction levels, which C2Eif had to deal with. TXL [2] is an expressive programming language designed to support source code analysis and transformation. It has been used to implement several language translation frameworks such as from Java to TCL and to Python. We could have used TXL to implement C2Eif; using regular Eiffel, however, allowed us to be independent of third-party closed-source tools, and to retain complete control over the implemented functionalities. Safer C. Many techniques exist aimed at ameliorating the safety of existing C code; for example, detection of format string vulnerability [3], out-of-bound array accesses and other memory errors [28, 24], or type errors [23]. C2Eif has a different scope, as it offers improved safety and debuggability as side-benefits of automatically porting C programs to Eiffel. This shares a little similarity with Ellison and Rosu’s formal executable semantics of C [7], which also finds errors in C program as a “side effect” of a rigorous translation. 5.3 Object-Oriented Reengineering After code written in a procedural language has been migrated to an object-oriented environment, it is natural to reengineer it to conform to the object-oriented design style, taking full advantage of features such as inheritance; this is the goal of object-oriented reengineering. Object-oriented reengineering is beyond the scope of this paper; and we tackled it in follow-up work based on C2Eif which we mention in Section 6. Nonetheless, it is still useful to compare existing approaches to reengineering with C2Eif, solely based on features such as degree of automation and tool support; see our follow-up work [41] for a discussion focused on the object-oriented reengineering techniques. Table 5 summarizes some features of the main approaches to object-oriented reengineering of procedural code: • The source and the target languages (or if it is a generic methodology). 28 full language evaluated methodology methodology C–C++ C–C++ C–C++ C++–C++ Cobol–OOSM Cobol–Java Cobol–Java PL/I–Java C–Eiffel completely automatic Gall [11] Jacobson [14] Livadas [17] Kontogiannis [16] Frakes [10] Fanta [8] Newcomb [26] Mossienko [21] Sneed [33] Sneed [31] C2Eif tool support source–target Table 5: Comparison of approaches to O-O reengineering. no no yes yes yes yes yes yes yes yes no no no no no no yes no yes yes yes – – no ? no no no no no no yes yes yes no 10KL 2KL 120KL 168KL 25KL 200KL 10KL 932KL YES • Whether tool support was developed, that is whether there exists a tool or the paper explicitly mentions the implementation of a tool. A YES in small caps denotes the only currently publicly available tool, namely C2Eif. • Whether the approach is completely automatic, that is does not require any user input other than providing a source procedural program. • Whether the approach supports the full source language (as used in practice) or only a subset thereof. • Whether the approach has been evaluated, that is whether the paper mentions evidence, such as a case study, that the approach was tried on real programs. If available, the table indicates the size of the programs used in the evaluation. Newcomb’s [26] and Sneed’s [33] are the only automatic tools which have been evaluated on programs of significant size. Newcomb’s tool [26], however, produces hierarchical object-oriented state machine models (OOSM); the mapping from OOSM to an object-oriented language is out of the scope of the work. Sneed’s tool [33] translates Cobol to Java; the paper reports that manual corrections of the automatically generated code were needed to get to a correct translation. While these corrections have successively been incorporated as extensions of the tool, the full Cobol language remains unsupported, according to the paper. 6 Conclusions and Future Work This paper presented the complete translation of C applications into Eiffel, and its implementation into the freely available automatic tool C2Eif. C2Eif supports the com29 plete C language, including unrestricted pointer arithmetic and pre-compiled libraries. Experiments in the paper showed that C2Eif correctly translates complete applications and libraries of significant size, and takes advantage of some of Eiffel’s advanced features to produce code of good quality. Future work. Future work will improve the readability and maintainability of the generated code. CIL, in particular, optimizes the code for program analysis, which is sometimes detrimental to readability of the Eiffel code generated by C2Eif. For example, CIL does not preserve comments, which are therefore lost in translation. We will also optimize the helper classes to improve on the few performance bottlenecks mentioned in Section 4.2. A major follow-up to the work described in this paper is the object-oriented reengineering of C code translated to Eiffel. In recent work [41], we have developed an automatic reengineering technique, and implemented it atop the translation produced by C2Eif and described in the present paper. The technique encapsulates functions and type definitions into classes that achieve low coupling and high cohesion, and introduces inheritance and contracts when possible. Other future work still remains in the direction of reengineering, such as in automatically replacing C data structure implementations (e.g., hash tables) with their Eiffel equivalents. 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arXiv:1704.01302v1 [math.ST] 5 Apr 2017 Extremes in Random Graphs Models of Complex Networks Natalia Markovich1 V.A.Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya 65, 117997 Moscow, Russia, Moscow Institute of Physics and Technology State University, Kerchenskaya 1, 117303 Moscow, Russia, (E-mail: nat.markovich@gmail.com) Abstract. Regarding the analysis of Web communication, social and complex networks the fast finding of most influential nodes in a network graph constitutes an important research problem. We use two indices of the influence of those nodes, namely, PageRank and a Max-linear model. We consider the PageRank as an autoregressive process with a random number of random coefficients that depend on ranks of incoming nodes and their out-degrees and assume that the coefficients are independent and distributed with regularly varying tail and with the same tail index. Then it is proved that the tail index and the extremal index are the same for both PageRank and the Max-linear model and the values of these indices are found. The achievements are based on the study of random sequences of a random length and the comparison of the distribution of their maxima and linear combinations. Keywords: Extremal Index, PageRank, Max-Linear Model, Branching Process, Autoregressive Process, Complex Networks. 1 Introduction Regarding the analysis of Web communication, social and complex networks the fast finding of most influential nodes in a network graph constitutes an important research problem. PageRank remains the most popular characteristic of such influence. We aim to find an extremal index of PageRank whose reciprocal value determines the first hitting time, i.e. a minimal time to reach the first influential node by means of a PageRank random walk. The extremal index θ ∈ [0, 1] has many other interpretations and plays a significant role in the theory of extreme values. Particularly, the limit distribution of maxima of stationary random variables (r.v.s) depends on θ. For independent r.v.s θ = 1 holds. θ has a connection to the tail index that shows the heaviness of the tail of a stationary distribution of an underlying process. Google’s PageRank defines the rank R(Xi ) of the Web page Xi as R(Xi ) = c X Xj ∈N (Xi ) R(Xj ) + (1 − c)qi , Dj i = 1, ..., n, (1) where N (Xi ) is the set of pages that link to Xi (in-degree), Dj is the number of outgoing links of page Xj (out-degree), c ∈ (0, 1) is a damping factor, q = (q1 , q2 , ..., qn ) is P a personalization probability vector or user preference such n that qi ≥ 0 and i=1 qi = 1, and n is the total number of pages, [11]. We omit in (1) the term with dangling nodes for simplicity. PageRank of a randomly selected page (a node in the graph) with random inand out-degrees may be considered as a branching process (Cf. [4], [5], [14]) Ri = Ni X (j) Aj Ri + Qi , i = 1, ..., n, (2) j=1 (j) denoting Ri = R(Xi ), Aj =d c/Dj , Qi = (1 − c)qi , [14]. Ri are ranks of descendants of node i, i.e. nodes with incoming links to node i. The r.v. Ni determines an in-degree, i.e. a number of directed edges to the ith node, and a number of nodes in the first generation of descendants belonging to the ith node as a parent, {Qi } is a sequence of i.i.d. r.v.s. Starting from the initial page (node) X0 , a PageRank random walk determines a regenerative process or Harris recurrent process {Xt }, letting it visits pages-followers of the underlying node with probability c and it restarts with probability 1 − c by jumping to a random independent node. A Max-linear model can be considered as an alternative characteristic of the node influence. This model is obtained by a substitution of sums in Google’s definition of PageRank by maxima, i.e. Ri = Ni _ (j) Aj Ri ∨ Qi , i = 1, ..., n, (3) j=1 is proposed in [6]. Formally, (2) can be considered as an autoregressive process with the random number Ni of random coefficients and the independent random term Qi . The extremal index of AR(1) processes with regularly varying stationary distribution and its relation to the tail index were considered in [9]. The extremal index of AR(q), q ≥ 1 processes with q random coefficients was obtained in [10] in a form which is not convenient for calculations. In [7] the results by [9] were extended to multivariate regularly varying distributed random sequences and the extremal and tail indices of sum and maxima of such sequences with l ≥ 1 r.v.s were derived. Our achievements extend and adapt the results by [7] to PageRank and Maxlinear processes. The problem concerns the finding of the extremal index of a random graph that models a real network where incoming nodes of the root node may be linked and, hence, be dependent. Such a random graph is called a Thorny Branching Tree (TBT) since any node may have outbound stubs (teleportations) to arbitrary nodes of the network, [4]. In this respect, such a graph cannot be considered as a pure Galton-Watson branching process where descendants of any node are mutually independent and teleportations are impossible. The paper is organized as follows. In Section 2 we recall necessary results regarding the relation between the tail and extremal indices obtained in [7] for multivariate random sequences which are regularly varying distributed (Theorems 1 and 2). Linear combinations and maxima of the random sequences of a fixed length are considered and it is derived that they have the same tail and extremal indices. In Section 3 we extend Theorem 2 to the case of unequal tail indices assuming r.v.s of a random sequence (Theorem 3). In Section 4 we consider sequences of random lengths and obtain the tail and extremal indices of their linear combinations and maxima (Theorem 4). We further discuss how these results can be applied to PageRank and the Max-linear processes in Section 5. 2 Related Work Let {Rj } be a stationary sequence with distribution function F (x) and maxima Mn = max1≤j≤n Rj . We shall interpret {Rj } as PageRanks of Web pages. Definition 1. A stationary sequence {Rn }n≥1 is said to have extremal index θ ∈ [0, 1] if for each 0 < τ < ∞ there is a sequence of real numbers un = un (τ ) such that lim n(1 − F (un )) = τ and (4) n→∞ lim P {Mn ≤ un } = e−τ θ (5) n→∞ hold ([12], p.53). In [7] the following theorems are proved which we will use to find the extremal (l) (2) (1) and tail indices of PageRank and a Max-linear model. Let Yn , Yn , ..., Yn , n ≥ 1, l ≥ 1 be sequences of r.v.s having stationary distributions with tail indices k1 , ..., kl and extremal indices θ1 , ..., θl , respectively, i.e. P {Yn(i) > x} ∼ c(i) x−ki as x → ∞, (i) where c are some real positive constants. Let us consider the weighted sum Yn (z) = z1 Yn(1) + z2 Yn(2) + ... + zl Yn(l) , z1 , ..., zl > 0 (6) and denote its tail index by k(z) and extremal index by θ(z). Supposing that there is a minimal tail index among k1 , ..., kl , the following theorem states the corresponding k(z) and θ(z). Theorem 1. ([7]) Let k1 < ki , i = 2, ..., l hold. Then Yn (z) has the tail index k(z) = k1 and the extremal index θ(z) = θ1 . (1) (2) (l) In the next theorem it is assumed that sequences Yn , Yn , ..., Yn are mutually independent with equal tail indices k1 = ... = kl = k. We denote (7) Yn∗ (z) = max z1 Yn(1) , z2 Yn(2) , ..., zl Yn(l) . Theorem 2. ([7]) The sequences Yn∗ (z) and Yn (z) have the same tail index k and the same extremal index equal to θ(z) = c(1) z1k c(l) zlk θ + ... + θl . 1 c(1) z1k + ... + c(l) zlk c(1) z1k + ... + c(l) zlk 3 Generalization of Theorem 2 Theorem 3 is a generalization of Theorem 2 to the case of unequal tail indices. (j) Theorem 3. Let {Yn }, n ≥ 1, j = 1, ..., l be mutually independent regularly varying r.v.s with tail indices k1 , ..., kl , respectively. Let km < ki , i = 1, ..., l, i 6= m hold. Then r.v.s Yn∗ (z) and Yn (z) have the same tail index k(z) = km and the same extremal index θ(z) = θm . Proof. First we show that P {Yn∗ (z) > x} ∼ c(z)x−km , Pl where c(z) = i=1 x → ∞, (8) c(i) ziki 1{ki = km }. Similar to [7] and as P {zi Yn(i) > x} ∼ c(i) ziki x−ki (9) holds, we have P {Yn∗ (z) > x} = P {max(z1 Yn(1) , .., zl Yn(l) ) > x} = 1 − P {max(z1 Yn(1) ≤ x} · ... · P {zl Yn(l) ) ≤ x} = l X + l X (−1) ∼ l X c(i) ziki x−ki + l X (−1)k−1 P {zi Yn(i) > x} i=1 k−1 l X P {zi1 Yn(i1 ) > x} · ... · P {zik Yn(ik ) > x} l X c(i1 ) zi1 1 x−ki1 · ... · c(ik ) zik k x−kik i1 <i2 <...<ik ;i1 ,i2 ,...,ik =1 k=2 i=1 ki ki i1 <i2 <...<ik ;i1 ,i2 ,...,ik =1 k=2 −km ∼ c(z)x + o(x−km ), x → ∞. (10) Thus, P {Yn (z) > x} ∼ c(z)x−km follows from Theorem 1. Now we show that Yn∗ (z) and Yn (z) have the same extremal index θ(z) = θm . We use the same notations as in [7] (i) (i) Mn(i) = max{Y1 , Y2 , ..., Yn(i) }, i = 1, .., l; Mn (z) = max{Y1 (z), Y2 (z), ..., Yn (z)}, Mn∗ (z) = max{Y1∗ (z), Y2∗ (z), ..., Yn∗ (z)}, n ≥ 1. By (7) it holds (1) (l) Mn∗ (z) = max{z1 Y1 , ..., z1 Yn(1) , ..., zl Y1 , ..., zl Yn(l) } = max{z1 Mn(1) , ..., zl Mn(l) }. Then we get P {Mn∗ (z)n−1/k ≤ x} = P {z1 Mn(1) n−1/k ≤ x, ..., zl Mn(l) n−1/k ≤ x} (11) Since km is the minimal tail index we have P {zi Mn(i) n−1/km ≤ x} = P {zi Mn(i) n−1/ki ≤ xn1/km −1/ki }. It implies zi Mn(i) n−1/km →P 0, i = 1, ..., l, i 6= m as n → ∞ (12) (i) since limn→∞ P {zi Mn n−1/ki ≤ x} = exp(−c(i) θi ziki x−ki ). By (11) it holds km θm x−km ), P {Mn∗ (z)n−1/km ≤ x} → exp(−c(m) zm n → ∞. Now we have to show that P {Mn∗ (z)n−1/km ≤ x} ∼ P {Mn (z)n−1/km ≤ x}. Let us denote un = xn1/km . Note that the event {Mn∗ (z) ≤ un } follows from {Mn (z) ≤ un }. Then, as in [7], we obtain 0 ≤ P {Mn∗ (z) ≤ un } − P {Mn (z) ≤ un } (13) ∗ ∗ = P {Mn (z) ≤ un } − P {Mn (z) ≤ un , Mn (z) ≤ un } n X = P {Mn∗ (z) ≤ un , Mn (z) > un } ≤ P {Mn∗ (z) ≤ un , Yk (z) > un } k=1 ≤ n X P {Yk∗ (z) ≤ un , Yk (z) > un } = nP {Yn∗ (z) ≤ un , Yn (z) > un } k=1 due to the stationarity of the sequences Yn∗ (z) and Yn (z). Lemma 1 in [7] states that P {Yn∗ (z) ≤ un \|Yn (z) > un } → 0, n → ∞, (14) (j) for i.i.d. regularly varying {Yn } with equal tail index. This can be extended to the case of unequal k1 , ..., kl . Since nP {Yn (z) > un } → c(z)x−km , n → ∞, (15) and (14) hold, it follows lim (P {Mn∗ (z) ≤ un } − P {Mn (z) ≤ un }) = 0. n→∞ 4 Extremal Index of PageRank and the Max-Linear Processes (j) (j) (j) We denote in (2) Ri as Yi (z) and Aj Ri = cRi /Dj , j = 1, ..., Ni as zj Yi . Then we can represent (2) in the form (6) as Yi (z) = Ni X j=1 (j) zj Yi + Qi , i = 1, ..., n, (16) where Ni is a nonnegative integer-valued r.v.. In the context of PageRank zj = c, j = 1, 2, ..., Ni , Qi = z ∗ qi with z ∗ = 1 − c and Ni represents the node in-degree. It is realistic to assume that Ni is a power law distributed r.v. with parameter α > 0, i.e. P {Ni = ℓ} ∼ ℓ−α (17) and Ni is bounded by a total number of nodes in the network. The distribution of Ni is in the domain of attraction of the Fréchet distribution with shape parameter α > 0 and P {Ni > x} = x−α ℓ(x), ∀x > 0, where ℓ(x) is a slowly varying function, since it satisfies a sufficient condition for this property, i.e. the von Mises type condition limn→∞ nP {Ni = n}/P {Ni > n} = α, [1]. Theorem 4 is an extension of Theorems 2 and 3 to maxima and sums of multivariate random sequences of random lengths, that can be applied to PageRank and the Max-linear processes. Let us turn to (16) and denote YN∗n (z) = max(z1 Yn(1) , .., zNn Yn(Nn ) , Qn ), YNn (z) = z1 Yn(1) + .. + zNn Yn(Nn ) + Qn . (j) Theorem 4. Let {Yn }, n ≥ 1, j = 1, ..., Nn and qn = Qn /z ∗ be mutually independent regularly varying i.i.d. r.v.s with tail indices k > 0 and β > 0, respectively, and Nn be regularly varying r.v. with tail index α > 0. Let (1) (N ) Yn ,.., Yn n have extremal indices θ1 , ..., θNn , respectively. Then r.v.s YN∗n (z) and YNn (z) are regularly varying distributed with the same tail index k(z) = min(k, α, β) and the same extremal index θ(z) such that θ(z) = (z ∗ )β , if k ≥ β, ∞ X c(i) θi zik /c(z), if k < β, θ(z) = (18) i=1 where c(z) = P∞ i=1 c(i) zik holds. Proof. We shall show first that P {YN∗n (z) > x} ∼ P {YNn (z) > x} ∼ x− min(k,α,β) . (19) (j) Since r.v.s {Yn }j≥1 are subexponential and i.i.d. it holds P {z1 Yn(1) + .. + z⌊x⌋ Yn(⌊x⌋) > x} ∼ P {max(z1 Yn(1) , .., z⌊x⌋ Yn(⌊x⌋) ) > x} ∼ xP {z1 Yn(1) > x}, x → ∞, (20) (j) [8]. Due to mutual independence of Qn and {Yn } and similar to (10) we get P {YN∗n (z) > x} = P {YN∗n (z) > x, Nn ≤ x} + P {YN∗n (z) > x, Nn > x} ∗ (z) > x} + P {Nn > x} ≤ P {Y⌊x⌋ = 1 − P {max(z1 Yn(1) , .., z⌊x⌋ Yn(⌊x⌋) ) ≤ x}P {Qn ≤ x} + P {Nn > x} ∼ cN x−α + cq (z ∗ )β x−β + c(z)x−k ∼ x− min{k,α,β} , (21) as x → ∞, where cN , cq > 0, c(z) = P∞ i=1 c(i) zik . On the other hand, P {YN∗n (z) > x} ≥ 0 + P {YN∗n (z) > x, Nn > x} ∗ ∗ (z) ≤ x, Nn ≤ x} − 1 (z) > x} + P {Nn > x} + P {Y⌈x⌉ ≥ P {Y⌈x⌉ ∼ x− min{k,α,β} (22) ∗ holds, since P {Y⌈x⌉ (z) ≤ x, Nn ≤ x} → 1 as x → ∞. Due to (21) and (22) we obtain P {YN∗n (z) > x} ∼ x− min{k,α,β} . The same is valid for YNn (z) by substitution of the maximum by the sum due to (20). Hence, (19) follows. Let us prove that YN∗n (z) and YNn (z) have the same extremal index θ(z). Let us denote ∗ (z) = max{YN∗1 (z), YN∗2 (z), ..., YN∗n (z)} MN n (N1 ) (1) = max{z1 Y1 , ..., zN1 Y1 (23) , Q1 , ..., z1 Yn(1) , ..., zNn Yn(Nn ) , Qn } and MNn (z) = max{YN1 (z), YN2 (z), ..., YNn (z)} (1) = max{z1 Y1 (N1 ) + ... + zN1 Y1 + Q1 , ..., z1 Yn(1) + ... + zNn Yn(Nn ) + Qn }. Without loss of generality we may assume that Nn = max{N1 , ..., Nn }. Then (N ) (1) we can complete vectors (z1 Yi , ..., zNi Yi i ), i = 1, 2, ..., n by zeros up to the dimension Nn and separate the vector (Q1 , ..., Qn ). We rewrite (23) as (1) ∗ (z) = max{z1 Y1 , ..., z1 Yn(1) , ..., zNn · 0, ..., zNn · 0, ..., zNn Yn(Nn ) , MN n Q1 , ..., Qn } = max(z1 Mn(1) , z2 Mn(2) , ..., zNn Mn(Nn ) , Mn(Q) ). (Q) Here, Mn = max{Q1 , ..., Qn } relates to the second term in the rhs of (16) corresponding to the user preference term Qi in (2). Following the same arguments as after (11) in Section 3 the statement follows. Really, denoting ∗ k ∗ = min{k, β} and un = xn1/k , x > 0, we get ∗ (z) > un } P {MN n ∗ ∗ (z) > un , Nn ≤ un } = P {MNn (z) > un , Nn > un } + P {MN n ∗ (z) > un } + P {Nn > un }. ≤ P {M⌈u n⌉ On the other hand, ∗ ∗ (z) > un } ≥ P {M⌈u (z) > un , Nn > un } P {MN n n⌉ ∗ ∗ (z) ≤ un , Nn ≤ un } − 1. = P {Nn > un } + P {M⌈un ⌉ (z) > un } + P {M⌈u n⌉ ∗ Note that P {M⌈u (z) ≤ un , Nn ≤ un } − 1 tends to zero as n → ∞. Hence, it n⌉ holds ∗ ∗ (z) > un } + P {Nn > un }, n → ∞. (24) (z) > un } ∼ P {M⌈u P {MN n n⌉ (Q) (i) ∗ If k < β holds, then Mn · n−1/k →P 0 as n → ∞ since P {zi Mn n−1/k ≤ x} → exp(−c(i) θi zik x−k ), i = 1, 2, .... Since P {Nn > un } ∼ u−α n → 0 as n → ∞ holds, then by (24) it follows ∗ ∗ (z)n−1/k ≤ x} = exp{−c(z)θ∗ (z)x−k }, lim P {MN n n→∞ (25) P∞ P∞ where θ∗ (z) = i=1 c(i) θi zik /c(z) and c(z) = i=1 c(i) zik . ∗ (i) (Q) If k ≥ β holds, then zi Mn ·n−1/k →P 0, i = 1, 2, ... follows since P {Mn n−1/β ≤ ∗ β −β x} → exp(−cq (z ) x ) as n → ∞. Thus, we obtain ∗ ∗ (z)n−1/k ≤ x} = exp(−cq (z ∗ )β x−β ). lim P {MN n n→∞ (26) Since {qi } are i.i.d., its extremal index is equal to one. Then by (25) and (26) the extremal index of YN∗n (z) satisfies (18) irrespectively of α. It remains to show that YN∗n (z) and YNn (z) have the same extremal index. Similarly to [7], we have to derive that ∗ ∗ ∗ (z)n−1/k ≤ x}. lim P {MNn (z)n−1/k ≤ x} = lim P {MN n n→∞ n→∞ (27) ∗ Since from the event {MNn (z) ≤ un } it follows {MN (z) ≤ un }, and P {MNn (z) ≤ n ∗ un } ≤ P {MNn (z) ≤ un } holds, we obtain similarly to (13) ∗ (z) ≤ un } − P {MNn (z) ≤ un } 0 ≤ P {MN n ∗ ∗ (z) ≤ un , MNn (z) ≤ un } (z) ≤ un } − P {MN = P {MN n n ∗ = P {MNn (z) ≤ un , MNn (z) > un } ∗ (z) ≤ un , MNn (z) > un , Nn > un } = P {MN n ∗ (z) ≤ un , MNn (z) > un , Nn ≤ un } + P {MN n ∗ (z) ≤ un , M⌊un ⌋ (z) > un , Nn ≤ un } ≤ P {Nn > un } + P {MN n ⌊un ⌋ ≤ P {Nn > un } + X P {Yk∗ (z) ≤ un , Yk (z) > un } k=1 = P {Nn > un } + ⌊un ⌋P {Yk∗ (z) ≤ un , Yk (z) > un } due to the stationarity of {Yk∗ (z)} and {Yk (z)}. (1) (N ) Completing vectors (z1 Yk , ..., zNk Yk k ) by zeroes (28) up to the maximal dimen- sion ⌊un ⌋, we get (1) (⌊un ⌋) P {Yk∗ (z) ≤ un , Yk (z) > un } = P {max(z1 Yk , ..., z⌊un ⌋ Yk (1) z1 Yk (⌊un ⌋) + ... + z⌊un ⌋ Yk , Qk ) ≤ un , + Qk > un } Then (27) follows from (14) and (15) since in (28) P {Yk∗ (z) ≤ un , Yk (z) > un } = P {Yk (z) > un }P {Yk∗ (z) ≤ un \|Yk (z) > un } holds. 5 Application to Indices of Complex Networks Theorem 4 can be applied to PageRank and the Max-linear processes. These processes then have the same tail index and the same extremal index. Theorem 4 is in the agreement with statements in [5] and [14], namely, that the stationary PNi (j) + Qi is regularly varying and distribution of PageRank R =d j=1 Aj Ri its tail index is determined by a most heavy-tailed distributed term in the (j) triple (Ni , Qi , Ai Ri ). This is derived if all terms in the triple are mutually independent. In contrast, Theorem 4 is valid for an arbitrary dependence (j) structure between Nn and {Yn } as well as Nn and Qn , and {Ni } are not necessarily independent. The novelty of Theorem 4 is that the extremal index of both PageRank and the Max-linear processes is the same and it depends on (j) the tail indices in the couple (Qi , Ai Ri ), irrespective of the tail index of Ni . The assumptions of both Theorem 4 and the statements in [5] and [14] do not reflect properly the complicated dependence between node ranks due to the entanglement of links in a real network. For better understanding let us consider the matrix   (N ) (1) (2) 0 0 Q1 z1 Y1 z2 Y1 ...zN1 Y1 1    z1 Y2(1) z2 Y2(2) ...zN1 Y2(N1 ) ...zN2 Y2(N2 ) 0 Q2    ... ... ... ... ...   ... (N ) (N ) (N ) (1) (2) z1 Yn z2 Yn ...zN1 Yn 1 ...zN2 Yn 2 ...zNn Yn n Qn (k, θ1 ) (k, θ2 ) ...(k, θN1 ) ...(k, θN2 ) ......(k, θNn ) ...(β, (z ∗ )β ) corresponding to (16) and completed by zeros up to the maximal dimension, let’s say Nn . Strings of the matrix correspond to generations of descendants of nodes with numbers 1, 2, ..., n. Each column may contain descendants of different nodes having the same extremal index θi , i = 1, 2, ..., Nn . All columns apart of the last one are identically regularly varying distributed with the same tail index k. The columns are mutually independent. In terms of some network, the conditions of Theorem 4 imply that ranks of all nodes with incoming links to a root node (i.e. its followers) are mutually independent, but followers of different nodes may be dependent and, thus, they are combined into clusters. The reciprocal of the extremal index approximates the mean cluster size, [12]. The statement (18) implies that the extremal index of PageRank is equal to θ(z) = (1 − c)β if the user preference dominates (i.e. its distribution tail is heavier than the tail of ranks of followers). If the damping factor c is close to one, then θ(z) is close to zero. The latter means the huge-sized cluster of nodes around a root-node in the presence of rare teleportations. If c is close to zero, then θ(z) is close to one due to the independence of frequent teleportations. If k < β holds, then roughly, the mean size of the cluster is determined by the consolidation of all clusters related to the followers of the underlying root. In practice, the followers of a node may be linked and their ranks can therefore be dependent. The future work will focus on the extremal index of PageRank (j) process when the terms {Yi } in (16) are mutually dependent. References 1. C.W. Anderson. Local limit theorems for the maxima of discrete random variables. 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Issue 3, 42–51, 2013, Moscow state university ISBN 9785211056527 (In russian) 8. C.M. Goldie and C. Klüppelberg. Subexponential Distributions, A Practical Guide to Heavy Tails, eds R. Adler, R. Feldman and M.S. Taqqu, Birkhäuser, Boston, MA, 435–459, 1998 9. L. de Haan, S. Resnick, H. Rootzén and G. de Vries. Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stochastic Processes and Applications, 32, 213–224, 1989. 10. C. Klüppelberg and S. Pergamenchtchikov. Extremal behaviour of models with multivariate random recurrence representation. Stochastic Processes and their Applications, 117, 432-456, 2007. 11. A. N. Langville and C. D. Meyer. Google’s PageRank and Beyond: The Science of Search Engine Rankings, Princeton University Press, 2006. 12. M.R. Leadbetter, G. Lingren and H. Rootzén. Extremes and Related Properties of Random Sequence and Processes, ch.3, Springer, New York, 1983. 13. H. Rootzén. Maxima and exceedances of stationary markov chains. Advances in Applied Probability, 20, 371–390, 1988. 14. Y.V. Volkovich and N. Litvak. Asymptotic analysis for personalized web search. Advances in Applied Probability, 42(2), 577–604, 2010.	10
Item Recommendation with Evolving User Preferences and Experience ∗ Subhabrata Mukherjee† , Hemank Lamba‡ and Gerhard Weikum† Planck Institute for Informatics, ‡ Carnegie Mellon University Email: {smukherjee, weikum}@mpi-inf.mpg.de, hlamba@cs.cmu.edu arXiv:1705.02519v1 [cs.AI] 6 May 2017 † Max Abstract—Current recommender systems exploit user and item similarities by collaborative filtering. Some advanced methods also consider the temporal evolution of item ratings as a global background process. However, all prior methods disregard the individual evolution of a user’s experience level and how this is expressed in the user’s writing in a review community. In this paper, we model the joint evolution of user experience, interest in specific item facets, writing style, and rating behavior. This way we can generate individual recommendations that take into account the user’s maturity level (e.g., recommending art movies rather than blockbusters for a cinematography expert). As only item ratings and review texts are observables, we capture the user’s experience and interests in a latent model learned from her reviews, vocabulary and writing style. We develop a generative HMM-LDA model to trace user evolution, where the Hidden Markov Model (HMM) traces her latent experience progressing over time — with solely user reviews and ratings as observables over time. The facets of a user’s interest are drawn from a Latent Dirichlet Allocation (LDA) model derived from her reviews, as a function of her (again latent) experience level. In experiments with five real-world datasets, we show that our model improves the rating prediction over state-of-the-art baselines, by a substantial margin. We also show, in a use-case study, that our model performs well in the assessment of user experience levels. more than recommendations for new blockbusters. Also, the facets of an item that a user focuses on change with experience. For example, a mature user pays more attention to narrative, light effects, and style rather than actors or special effects. Similar observations hold for ratings of wine, beer, food, etc. Our approach advances state-of-the-art by tapping review texts, modeling their properties as latent factors, using them to explain and predict item ratings as a function of a user’s experience evolving over time. Prior works considering review texts (e.g., [12], [15], [18], [23], [24]) did this only to learn topic similarities in a static, snapshot-oriented manner, without considering time at all. The only prior work [16], considering time, ignores the text of user-contributed reviews in harnessing their experience. However, user experience and their interest in specific item facets at different timepoints can often be observed only indirectly through their ratings, and more vividly through her vocabulary and writing style in reviews. Use-cases: Consider the reviews and ratings by two users on a “Canon DSLR” camera about the facet camera lens. • User 1: My first DSLR. Excellent camera, takes great pictures in I. I NTRODUCTION • User 2: The EF 75-300 mm lens is only good to be used outside. HD, without a doubt it brings honor to its name. [Rating: 5] The 2.2X HD lens can only be used for specific items; filters are Motivation and State-of-the-Art: Collaborative filtering algouseless if ISO, AP,... are correct. The short 18-55mm lens is cheap rithms are at the heart of recommender systems for items like and should have a hood to keep light off lens. [Rating: 3] movies, cameras, restaurants and beer. Most of these methods The second user is clearly more experienced than the first exploit user-user and item-item similarities in addition to the one, and more reserved about the lens quality of that camera history of user-item ratings — similarities being based on model. Future recommendations for the second user should take latent factor models over user and item features [11], and more into consideration the user’s maturity. As a second use-case, recently on explicit links and interactions among users [6][25]. consider the following reviews of Christopher Nolan movies All these data evolve over time leading to bursts in item where the facet of interest is the non-linear narrative style. popularity and other phenomena like anomalies[7]. State-ofthe-art recommender systems capture these temporal aspects by • User 1 on Memento (2001): “Backwards told is thriller noir-art empty ultimately but compelling and intriguing this.” introducing global bias components that reflect the evolution • User 2 on The Dark Knight (2008): “Memento was very compliof the user and community as a whole[10]. A few models also cated. The Dark Knight was flawless. Heath Ledger rocks!” consider changes in the social neighborhood of users[14]. What • User 3 on Inception (2010): “Inception is a triumph of style over is missing in all these approaches, though, is the awareness of substance. It is complex only in a structural way, not in terms of how experience and maturity levels evolve in individual users. plot. It doesn’t unravel in the way Memento does.” Individual experience is crucial in how users appreciate items, and thus react to recommendations. For example, a mature The first user does not appreciate complex narratives, making cinematographer would appreciate tips on art movies much fun of it by writing her review backwards. The second user prefers simpler blockbusters. The third user seems to appreciate ∗ This is an extended version of the paper published in ICDM 2015 [20]. the complex narration style of Inception and, more of, Memento. Refer to [19] for a general (continuous) version of this model with fine-grained temporal evolution of user experience, and resulting language model using We would consider this maturity level of the more experienced Geometric Brownian Motion and Brownian Motion, respectively. User 3 to generate future recommendations to her. Approach: We model the joint evolution of user experience, interests in specific item facets, writing style, and rating behavior in a community. As only item ratings and review texts are directly observed, we capture a user’s experience and interests by a latent model learned from her reviews, and vocabulary. All this is conditioned on time, considering the maturing rate of a user. Intuitively, a user gains experience not only by writing many reviews, but she also needs to continuously improve the quality of her reviews. This varies for different users, as some enter the community being experienced. This allows us to generate individual recommendations that take into account the user’s maturity level and interest in specific facets of items, at different timepoints. We develop a generative HMM-LDA model for a user’s evolution, where the Hidden Markov Model (HMM) traces her latent experience progressing over time, and the Latent Dirichlet Allocation (LDA) model captures her interests in specific item facets as a function of her (again, latent) experience level. The only explicit input to our model is the ratings and review texts upto a certain timepoint; everything else – especially the user’s experience level – is a latent variable. The output is the predicted ratings for the user’s reviews following the given timepoint. In addition, we can derive interpretations of a user’s experience and interests by salient words in the distributional vectors for latent dimensions. Although it is unsurprising to see users writing sophisticated words with more experience, we observe something more interesting. For instance in specialized communities like beeradvocate.com and ratebeer.com, experienced users write more descriptive and fruity words to depict the beer taste (cf. Table V). Table I shows a snapshot of the words used at different experience levels to depict the facets beer taste, movie plot and bad journalism, respectively. We apply our model to 12.7 million ratings from 0.9 million users on 0.5 million items in five different communities on movies, food, beer, and news media, achieving an improvement of 5% to 35% for the mean squared error for rating predictions over several competitive baselines. We also show that users at the same (latent) experience level do indeed exhibit similar vocabulary, and facet interests. Finally, a use-case study in a news community to identify experienced citizen journalists demonstrates that our model captures user maturity fairly well. Contributions: To summarize, this paper introduces the following novel elements: a) This is the first work that considers the progression of user experience as expressed through the text of item reviews, thereby elegantly combining text and time. b) An approach to capture the natural smooth temporal progression in user experience factoring in the maturing rate of the user, as expressed through her writing. c) Offers interpretability by learning the vocabulary usage of users at different levels of experience. d) A large-scale experimental study in five real world datasets from different communities like movies, beer and food; and an interesting use-case study in a news community. Experience Beer Movies News Level 1 Level 2 Level 3 stupid, bizarre storyline, epic realism, visceral, nostalgic bad, stupid biased, unfair opinionated, fallacy, rhetoric bad, shit sweet, bitter caramel finish, coffee roasted TABLE I: Vocabulary at different experience levels. II. OVERVIEW A. Model Dimensions Our approach is based on the intuition that there is a strong coupling between the facet preferences of a user, her experience, writing style in reviews, and rating behavior. All of these factors jointly evolve with time for a given user. We model the user experience progression through discrete stages, so a state-transition model is natural. Once this decision is made, a Markovian model is the simplest, and thus natural choice. This is because the experience level of a user at the current instant t depends on her experience level at the previous instant t-1. As experience levels are latent (not directly observable), a Hidden Markov Model is appropriate. Experience progression of a user depends on the following factors: • Maturing rate of the user which is modeled by her activity in the community. The more engaged a user is in the community, the higher are the chances that she gains experience and advances in writing sophisticated reviews, and develops taste to appreciate specific facets. • Facet preferences of the user in terms of focusing on particular facets of an item (e.g., narrative structure rather than special effects). With increasing maturity, the taste for particular facets becomes more refined. • Writing style of the user, as expressed by the language model at her current level of experience. More sophisticated vocabulary and writing style indicates higher probability of progressing to a more mature level. • Time difference between writing successive reviews. It is unlikely for the user’s experience level to change from that of her last review in a short time span (within a few hours or days). • Experience level difference: Since it is unlikely for a user to directly progress to say level 3 from level 1 without passing through level 2, the model at each instant decides whether the user should stay at current level l, or progress to l+1. In order to learn the facet preferences and language model of a user at different levels of experience, we use Latent Dirichlet Allocation (LDA). In this work, we assume each review to refer to exactly one item. Therefore, the facet distribution of items is expressed in the facet distribution of the review documents. We make the following assumptions for the generative process of writing a review by a user at time t at experience level et : • A user has a distribution over facets, where the facet preferences of the user depend on her experience level et . • A facet has a distribution over words where the words used to describe a facet depend on the user’s vocabulary at Level 1: film will happy people back supposed good wouldnt cant Level 2: storyline believable acting time stay laugh entire start funny Level 3 & 4: narrative cinema resemblance masterpiece crude undeniable admirable renowned seventies unpleasant myth nostalgic Level 5: incisive delirious personages erudite affective dramatis nucleus cinematographic transcendence unerring peerless fevered E1 E1 E2 E3 E4 E5 0 348 504 597 768 720 E1 E2 E3 E4 E5 E1 0 141 126 204 371 E2 146 0 36 129 321 E3 129 39 0 39 202 E4 225 153 42 0 106 E5 430 393 240 125 0 640 E2 384 0 449 551 742 560 480 E3 437 318 0 111 190 400 320 E4 509 352 109 0 138 240 E5 452 164 129 Experience Levels 0 80 0 300 250 200 150 100 160 577 400 350 Experience Levels Experience Levels Level 1: stupid people supposed wouldnt pass bizarre totally cant Level 2:storyline acting time problems evil great times didnt money ended simply falls pretty Level 3: movie plot good young epic rock tale believable acting Level 4: script direction years amount fast primary attractive sense talent multiple demonstrates establish Level 5: realism moments filmmaker visual perfect memorable recommended genius finish details defined talented visceral nostalgia Experience Levels 50 0 (a) Divergence of language model (b) Divergence of facet preference as a function of experience. as a function of experience. Fig. 1: KL Divergence as a function of experience. Hypothesis 2: Facet Preferences Depend on Experience Level. The second hypothesis underlying our work is that users experience level et . Table II shows salient words for two at similar levels of experience have similar facet preferences. facets of Amazon movie reviews at different levels of user In contrast to the LM’s where words are observed, facets are experience, automatically extracted by our latent model. The latent so that validating or falsifying the second hypothesis is facets are latent, but we can interpret them as plot/script not straightforward. We performed a three-step study: and narrative style, respectively. • We use Latent Dirichlet Allocation (LDA) [2] to compute a As a sanity check for our assumption of the coupling between latent facet distribution hfk i of each review. user experience, rating behavior, language and facet prefer- • We run Support Vector Regression (SVR) [4] for each user. ences, we perform experimental studies reported next. The user’s item rating in a review is the response variable, with the facet proportions in the review given by LDA as B. Hypotheses and Initial Studies features. The regression weight wkue is then interpreted as Hypothesis 1: Writing Style Depends on Experience Level. the preference of user ue for facet fk . We expect users at different experience levels to have • Finally, we aggregate these facet preferences for each divergent Language Models (LM’s) — with experienced users experience level e to get the facet preference P corresponding ue ue exp(wk ) having a more sophisticated writing style and vocabulary distribution given by < >. #ue than amateurs. To test this hypothesis, we performed initial Figure 1b shows the KL divergence between the facet prefstudies over two popular communities1 : 1) BeerAdvocate erences of users at different experience levels in BeerAdvocate. (beeradvocate.com) with 1.5 million reviews from 33, 000 We see that the divergence clearly increases with the difference users and 2) Amazon movie reviews (amazon.com) with 8 in user experience levels; this confirms the hypothesis. The million reviews from 760, 000 users. Both of these span a heatmap for Amazon is similar and omitted. period of about 10 years. Note that Figure 1 shows how a change in the experience In BeerAdvocate, a user gets points on the basis of likes level can be detected. This is not meant to predict the experience received for her reviews, ratings from other users, number of level, which is done by the model in Section IV. posts written, diversity and number of beers rated, time in the community, etc. We use this points measure as a proxy for the III. B UILDING B LOCKS OF OUR M ODEL user’s experience. In Amazon, reviews get helpfulness votes Our model, presented in the next section, builds on and from other users. For each user, we aggregate these votes over compares itself against various baseline models as follows. all her reviews and take this as a proxy for her experience. We partition the users into 5 bins, based on the points A. Latent-Factor Recommendation / helpfulness votes received, each representing one of the According to the standard latent factor model (LFM) [9], experience levels. For each bin, we aggregate the review texts the rating assigned by a user u to an item i is given by: of all users in that bin and construct a unigram language model. rec(u, i) = βg + βu + βi + hαu , φi i (1) The heatmap of Figure 1a shows the Kullback-Leibler (KL) where h., .i denotes a scalar product. βg is the average rating divergence between the LM’s of different experience levels, for the BeerAdvocate case. The Amazon reviews lead to a very of all items by all users. βu is the offset of the average rating similar heatmap, which is omitted here. The main observation given by user u from the global rating. Likewise βi is the rating is that the KL divergence is higher — the larger the difference bias for item i. αu and φi are the latent factors associated with is between the experience levels of two users. This confirms our user u and item i, respectively. These latent factors are learned hypothesis about the coupling of experience and user language. using gradient descent by minimizing the mean squared error (M SE) between observed P ratings r(u, i) and predicted ratings 1 Data available at http://snap.stanford.edu/data/ rec(u, i): M SE = \|U1 \| u,i∈U (r(u, i) − rec(u, i))2 TABLE II: Salient words for two facets at five experience levels in movie reviews. B. Experience-based Latent-Factor Recommendation The most relevant baseline for our work is the “user at learned rate” model of [16], which exploits that users at the same experience level have similar rating behavior even if their ratings are temporarily far apart. Experience of each user u for item i is modeled as a latent variable eu,i ∈ {1...E}. Different recommenders are learned for different experience levels. Therefore Equation 1 is parameterized as: receu,i (u, i) = βg (eu,i )+βu (eu,i )+βi (eu,i )+hαu (eu,i ), φi (eu,i )i (2) The parameters are learned using Limited Memory BFGS with the additional constraint that experience levels should be non-decreasing over the reviews written by a user over time. However, this is significantly different from our approach. All of these models work on the basis of only user rating behavior, and ignore the review texts completely. Additionally, the smoothness in the evolution of parameters between experience levels is enforced via L2 regularization, and does not model the natural user maturing rate (via HMM) as in our model. Also note that in the above parametrization, an experience level is estimated for each user-item pair. However, it is rare that a user reviews the same item multiple times. In our approach, we instead trace the evolution of users, and not user-item pairs. C. User-Facet Model In order to find the facets of interest to a user, [22] extends Latent Dirichlet Allocation (LDA) to include authorship information. Each document d is considered to have a distribution over authors. We consider the special case where each document has exactly one author u associated with a Multinomial distribution θu over facets Z with a symmetric Dirichlet prior α. The facets have a Multinomial distribution φz over words W drawn from a vocabulary V with a symmetric Dirichlet prior β. Exact inference is not possible due to the intractable coupling between Θ and Φ. Two ways for approximate inference are MCMC techniques like Collapsed Gibbs Sampling and Variational Inference. The latter is typically much more complex and computationally expensive. In our work, we thus use sampling. Fig. 2: Supervised model for user facets and ratings. Fig. 3: Supervised model for user experience, facets, and ratings. hyper-parameter α for the document-facet Multinomial distribution Θ is parametrized as αd,z = exp(xTd λz ). The model is trained using stochastic EM which alternates between 1) sampling facet assignments from the posterior distribution conditioned on words and features, and 2) optimizing λ given the facet assignments using L-BFGS. Our approach, explained in the next section, follows a similar approach to couple the User-Facet Model and the Latent-Factor Recommendation Model (depicted in Figure 2). IV. J OINT M ODEL : U SER E XPERIENCE , FACET P REFERENCE , W RITING S TYLE We start with a User-Facet Model (UFM) (aka. AuthorTopic Model [22]) based on Latent Dirichlet Allocation (LDA), where users have a distribution over facets and facets D. Supervised User-Facet Model have a distribution over words. This is to determine the The generative process described above is unsupervised and facets of interest to a user. These facet preferences can be does not take the ratings in reviews into account. Supervision interpreted as latent item factors in the traditional Latent-Factor is difficult to build into MCMC sampling where ratings are Recommendation Model (LFM) [9]. However, the LFM is continuous values, as in communities like newstrust.net. supervised as opposed to the UFM. It is not obvious how to For discrete ratings, a review-specific Multinomial rating incorporate supervision into the UFM to predict ratings. The distribution πd,r can be learned as in [13], [21]. Discretizing the user-provided ratings of items can take continuous values (in continuous ratings into buckets bypasses the problem to some some review communities), so we cannot incorporate them into extent, but results in loss of information. Other approaches [12], a UFM with a Multinomial distribution of ratings. We propose [15], [18] overcome this problem by learning the feature an Expectation-Maximization (EM) approach to incorporate supervision, where the latent facets are estimated in an Eweights separately from the user-facet model. An elegant approach using Multinomial-Dirichlet Regression Step using Gibbs Sampling, and Support Vector Regression is proposed in [17] to incorporate arbitrary types of observed (SVR) [4] is used in the M-Step to learn the feature weights continuous or categorical features. Each facet z is associated and predict ratings. Subsequently, we incorporate a layer for with a vector λz whose dimension equals the number of features. experience in the UFM-LFM model, where the experience Assuming xd is the feature vector for document d, the Dirichlet levels are drawn from a Hidden Markov Model (HMM) in the E-Step. The experience level transitions depend on the evolution of the user’s maturing rate, facet preferences, and writing style over time. The entire process is a supervised generative process of generating a review based on the experience level of a user hinged on our HMM-LDA model. A. Generative Process for a Review example, a user at a high level of experience may choose to write on the beer “hoppiness” or “story perplexity” in a movie. The word that she writes depends on the facet chosen and the language model for her current experience level. Thus, she draws a word from the multinomial distribution φetd ,zdi with a symmetric Dirichlet prior δ. For example, if the facet chosen is beer taste or movie plot, an experienced user may choose to use the words “coffee roasted vanilla” and “visceral”, whereas an inexperienced user may use “bitter” and “emotional” resp. Algorithm 1 describes this generative process for the review; Figure 3 depicts it visually in plate notation for graphical models. We use MCMC sampling for inference on this model. Consider a corpus with a set D of review documents denoted by {d1 . . . dD }. For each user, all her documents are ordered by timestamps t when she wrote them, such that tdi < tdj for i < j. Each document d has a sequence of Nd words denoted by d = w1 . . . .wNd . Each word is drawn from a vocabulary V having unique words indexed by {1 . . . V }. Consider a set of U users involved in writing the documents in the corpus, Algorithm 1: Supervised Generative Model for a User’s where ud is the author of document d. Consider an ordered set Experience, Facets, and Ratings of experience levels {e1 , e2 , ...eE } where each ei is from a set for each facet z = 1, ...Z and experience level e = 1, ...E do E, and a set of facets {z1 , z2 , ...zZ } where each zi is from a choose φe,z ∼ Dirichlet(β) set Z of possible facets. Each document d is associated with end a rating r and an item i. for each review d = 1, ...D do At the time td of writing the review d, the user ud has Given user ud and timestamp td experience level etd ∈ E. We assume that her experience level /Current experience level depends on previous level/ transitions follow a distribution Π with a Markovian assumption 1. Conditioned on ud and previous experience etd−1 , choose and certain constraints. This means the experience level of ud etd ∼ πetd−1 at time td depends on her experience level when writing the /User’s facet preferences at current experience level are previous document at time td−1 . influenced by supervision via α – scaled by hyper-parameter ρ controlling influence of supervision/ πei (ej ) denotes the probability of progressing to experience 2. Conditioned on supervised facet preference αud ,etd of ud level ej from experience level ei , with the constraint ej ∈ at experience level etd scaled by ρ, choose {ei , ei + 1}. This means at each instant the user can either stay θud ,etd ∼ Dirichlet(ρ × αud ,etd ) at her current experience level, or move to the next one. for each word i = 1, ...Nd do The experience-level transition probabilities depend on the /Facet is drawn from user’s experience-based facet interests/ rating behavior, facet preferences, and writing style of the user. 3. Conditioned on ud and etd choose a facet The progression also takes into account the 1) maturing rate of zdi ∼ M ultinomial(θud ,etd ) ud modeled by the intensity of her activity in the community, /Word is drawn from chosen facet and user’s and 2) the time gaps between writing consecutive reviews. We vocabulary at her current experience level/ incorporate these aspects in a prior for the user’s transition 4. Conditioned on zdi and etd choose a word wdi ∼ M ultinomial(φetd ,zdi ) rates, γ ud , defined as: end D u d γ ud = + λ(td − td−1 ) /Rating computed via Support Vector Regression with Dud + Davg chosen facet proportions as input features to learn α/ Dud and Davg denote the number of reviews written by ud 5. Choose rd ∼ F (hαud ,etd , φetd ,zd i) and the average number of reviews per user in the community, end respectively. Therefore the first term models the user activity with respect to the community average. The second term reflects the time difference between successive reviews. The B. Supervision for Rating Prediction user experience is unlikely to change from the level when writing the previous review just a few hours or days ago. λ The latent item factors φi in Equation 2 correspond to controls the effect of this time difference, and is set to a very the latent facets Z in Algorithm 1. Assume that we have small value. Note that if the user writes very infrequently, the some estimation of the latent facet distribution φe,z of each second term may go up. But the first term which plays the document after one iteration of MCMC sampling, where e dominating role in this prior will be very small with respect denotes the experience level at which a document is written, to the community average in an active community, bringing and let z denote a latent facet of the document. We also have down the influence of the entire prior. Note that the constructed an estimation of the preference of a user u for facet z at HMM encapsulates all the factors for experience progression experience level e given by θu,e (z). outlined in Section II. For each user u, we compute a supervised regression function At experience level etd , user ud has a Multinomial facet- Fu for the user’s numeric ratings with the – currently estimated preference distribution θud ,etd . From this distribution she draws – experience-based facet distribution φe,z of her reviews as input a facet of interest zdi for the ith word in her document. For features and the ratings as output. e The learned feature weights hαu,e (z)i indicate the user’s Let mei−1 denote the number of transitions from experience i preference for facet z at experience level e. These feature level ei−1 to ei over all users in the community, with the weights are used to modify θu,e to attribute more mass to the constraint ei ∈ {ei−1 , ei−1 + 1}. Note that we allow selffacet for which u has a higher preference at level e. This is transitions for staying at the same experience level. The counts reflected in the next sampling iteration, when we draw a facet z capture the relative difficulty in progressing between different from the user’s facet preference distribution θu,e smoothed by experience levels. For example, it may be easier to progress αu,e , and then draw a word from φe,z . This sampling process to level 2 from level 1 than to level 4 from level 3. is repeated until convergence. The state transition probability depending on the previous In any latent facet model, it is difficult to set the hyper- state, factoring in the user-specific activity rate, is given by: e m i−1 +I(ei−1 =ei )+γ u parameters. Therefore, most prior work assume symmetric i P (ei \|ei−1 , u, e−i ) = meei−1 u +I(e i−1 =ei )+Eγ . Dirichlet priors with heuristically chosen concentration paramwhere I(.) is an indicator function taking the value 1 when eters. Our approach is to learn the concentration parameter α the argument is true, and 0 otherwise. The subscript −i denotes of a general (i.e., asymmetric) Dirichlet prior for Multinomial th the value of a variable excluding the data at the i position. distribution Θ – where we optimize these hyper-parameters to All the counts of transitions exclude transitions to and from learn user ratings for documents at a given experience level. ei , when sampling a value for the current experience level ei during Gibbs sampling. The conditional distribution for the C. Inference experience level transition is given by: We describe the inference algorithm to estimate the distriP (E\|U, Z, W ) ∝ P (E\|U ) × P (Z\|E, U ) × P (W \|Z, E) (4) butions Θ, Φ and Π from observed data. For each user, we Here the first factor models the rate of experience progression factoring in user activity; the second and third factor models the facet-preferences of user, and language model at a specific level of experience respectively. All three factors combined decide whether the user should stay at the current level of experience, or has matured enough to progress to next level. In Gibbs sampling, the conditional distribution for each hidden variable is computed based on the current assignment of other hidden variables. The values for the latent variables are sampled repeatedly from this conditional distribution until convergence. In our problem setting we have two sets of latent Du Y du U Y E Y Z N variables corresponding to E and Z respectively. Y Y P (U, E, Z, W, θ, φ, π; α, δ, γ) = { We perform Collapsed Gibbs Sampling [5] in which we first u=1 e=1 i=1 z=1 j=1 sample a value for the experience level ei of the user for the × P (θu,e ; αu,e ) × P (zi,j \|θu,ei ) P (πe ; γ u ) × P (ei \|πe ) current document i, keeping all facet assignments Z fixed. In {z } \| {z } \| experience transition distribution user experience facet distribution order to do this, we consider two experience levels ei−1 and × P (φe,z ; δ) × P (wi,j \|φei ,zi,j ) } ei−1 + 1. For each of these levels, we go through the current {z } \| document and all the token positions to compute Equation 4 — experience facet language distribution (3) and choose the level having the highest conditional probability. Let n(u, e, d, z, v) denote the count of the word w occurring Thereafter, we sample a new facet for each word wi,j of the in document d written by user u at experience level e belonging document, keeping the currently sampled experience level of to facet z. In the following equation, (.) at any position in a the user for the document fixed. The conditional distributions for Gibbs sampling for the joint distribution indicates summation of the above counts for the update of the latent variables E and Z are given by: respective argument. compute the conditional distribution over the set of hidden variables E and Z for all the words W in a review. The exact computation of this distribution is intractable. We use Collapsed Gibbs Sampling [5] to estimate the conditional distribution for each hidden variable, which is computed over the current assignment for all other hidden variables, and integrating out other parameters of the model. Let U, E, Z and W be the set of all users, experience levels, facets and words in the corpus. In the following, i indexes a document and j indexes a word in it. The joint probability distribution is given by: Exploiting conjugacy of the Multinomial and Dirichlet ui = u, {zi,j = zj }, {wi,j = wj }, e−i ) ∝ distributions, we can integrate out Φ from the above distribution E-Step 1: P (ei = e\|ei−1 ,Y P (ei \|u, ei−1 , e−i ) × P (zj \|ei , u, e−i ) × P (wj \|zj , ei , e−i ) ∝ to obtain the posterior distribution P (Z\|U, E; α) of the latent j variable Z given by: e U Y E Y u=1 e=1 P Q Γ( z αu,e,z ) z Γ(n(u, e, ., z, .) + αu,e,z ) P P Q z αu,e,z ) z Γ(αu,e,z )Γ( z n(u, e, ., z, .) + where Γ denotes the Gamma function. Similarly, by integrating out Θ, P (W \|E, Z; δ) is given by E Y Z Y e=1 z=1 P Q Γ( v δv ) v Γ(n(., e, ., z, v) + δv ) Q P P v Γ(δv )Γ( v n(., e, ., z, v) + v δv ) Y P j zj mei−1 + I(ei−1 = ei ) + γ u i × ei−1 m. + I(ei−1 = ei ) + Eγ u n(u, e, ., zj , .) + αu,e,zj n(., e, ., zj , wj ) + δ P ×P n(u, e, ., zj , .) + zj αu,e,zj wj n(., e, ., zj , wj ) + V δ E-Step 2: P (zj = z\|ud = u, ed = e, wj = w, z−j ) ∝ n(u, e, ., z, .) + αu,e,z n(., e, ., z, w) + δ P P ×P n(u, e, ., z, .) + α z z u,e,z w n(., e, ., z, w) + V δ (5) The proportion of the z th facet in document d with words {wj } written at experience level e is given by: PNd j=1 φe,z (wj ) φe,z (d) = Nd For each user u, we learn a regression model Fu using these facet proportions in each document as features, along with the user and item biases (refer to Equation 2), with the user’s item rating rd as the response variable. Besides the facet distribution of each document, the biases < βg (e), βu (e), βi (e) > also depend on the experience level e. We formulate the function Fu as Support Vector Regression [4], which forms the M -Step in our problem: 1 M-Step: min αu,e T αu,e + C× αu,e 2 Du X (max(0, \|rd − αu,e T < βg (e),βu (e), βi (e), φe,z (d) > \| − ))2 d=1 The total number of parameters learned is [E×Z+E×3]×U . Our solution may generate a mix of positive and negative real numbered weights. In order to ensure that the concentration parameters of the Dirichlet distribution are positive reals, we take exp(αu,e ). The learned α’s are typically very small, whereas the value of n(u, e, ., z, .) in Equation 5 is very large. Therefore we scale the α’s by a hyper-parameter ρ to control the influence of supervision. ρ is tuned using a validation set by varying it from {100 , 101 ...105 }. In the E-Step of the next iteration, we choose θu,e ∼ Dirichlet(ρ × αu,e ). We use the LibLinear2 package for Support Vector Regression. V. E XPERIMENTS Setup: We perform experiments with data from five communities in different domains: BeerAdvocate (beeradvocate.com) and RateBeer (ratebeer.com) for beer reviews, Amazon (amazon.com) for movie reviews, Yelp (yelp.com) for food and restaurant reviews, and NewsTrust (newstrust.net) for reviews of news media. Table III gives the dataset statistics3 . We have a total of 12.7 million reviews from 0.9 million users from all of the five communities combined. The first four communities are used for product reviews, from where we extract the following quintuple for our model < userId, itemId, timestamp, rating, review >. NewsTrust is a special community, which we discuss in Section VI. For all models, we used the three most recent reviews of each user as withheld test data. All experience-based models consider the last experience level reached by each user, and corresponding learned parameters for rating prediction. In all the models, we group light users with less than 50 reviews in training data into a background model, treated as a single user, to avoid modeling from sparse observations. We do not ignore any user. During the test phase for a light user, we take her parameters from the background model. We set Z = 20 for BeerAdvocate, RateBeer and Yelp facets; and Z = 100 for Amazon movies and NewsTrust which have much richer 2 http://www.csie.ntu.edu.tw/ cjlin/liblinear http://www.yelp.com/dataset challenge/ 3 http://snap.stanford.edu/data/, Dataset #Users #Items #Ratings Beer (BeerAdvocate) Beer (RateBeer) Movies (Amazon) Food (Yelp) Media (NewsTrust) 33,387 40,213 759,899 45,981 6,180 66,051 110,419 267,320 11,537 62,108 1,586,259 2,924,127 7,911,684 229,907 134,407 TOTAL 885,660 517,435 12,786,384 TABLE III: Dataset statistics. Models Beer Rate Advocate Beer News Trust AmazonYelp Our model (most recent experience level) f) Our model (past experience level) e) User at learned rate c) Community at learned rate b) Community at uniform rate d) User at uniform rate a) Latent factor model 0.363 0.309 0.373 1.174 1.469 0.375 0.362 0.470 1.200 1.642 0.379 0.383 0.391 0.394 0.409 0.336 0.334 0.347 0.349 0.377 0.575 0.656 0.767 0.744 0.847 1.293 1.203 1.203 1.206 1.248 1.732 1.534 1.526 1.613 1.560 TABLE IV: MSE comparison of our model versus baselines. latent dimensions. For experience levels, we set E = 5 for all. However, for NewsTrust and Yelp datasets our model categorizes users to belong to one of three experience levels. A. Quantitative Comparison Baselines: We consider the following baselines for our work, and use the available code4 for experimentation. a) LFM: A standard latent factor recommendation model [9]. b) Community at uniform rate: Users and products in a community evolve using a single “global clock” [10][27][26], where the different stages of the community evolution appear at uniform time intervals. So the community prefers different products at different times. c) Community at learned rate: This extends (b) by learning the rate at which the community evolves with time, eliminating the uniform rate assumption. d) User at uniform rate: This extends (b) to consider individual users, by modeling the different stages of a user’s progression based on preferences and experience levels evolving over time. The model assumes a uniform rate for experience progression. 4 http://cseweb.ucsd.edu/ jmcauley/code/ 60 50 40 30 20 10 0 BeerAdvocate RateBeer NewsTrust User at learned rate Community at learned rate User at uniform rate Latent factor model Amazon Yelp Community at uniform rate Fig. 4: MSE improvement (%) of our model over baselines. Experience Level 1: drank, bad, maybe, terrible, dull, shit e) User at learned rate: This extends (d) by allowing each user to evolve on a “personal clock”, so that the time to Experience Level 2: bottle, sweet, nice hops, bitter, strong light, head, smooth, good, brew, better, good reach certain experience levels depends on the user [16]. f) Our model with past experience level: In order to determine Expertise Level 3: sweet alcohol, palate down, thin glass, malts, poured thick, pleasant hint, bitterness, copper hard how well our model captures evolution of user experience over time, we consider another baseline where we randomly Experience Level 4: smells sweet, thin bitter, fresh hint, honey end, sticky yellow, slight bit good, faint bitter beer, red brown, sample the experience level reached by users at some timepoint good malty, deep smooth bubbly, damn weak previously in their lifecycle, who may have evolved thereafter. We learn our model parameters from the data up to this time, Experience Level 5: golden head lacing, floral dark fruits, citrus sweet, light spice, hops, caramel finish, acquired taste, and again predict the user’s most recent three item ratings. Note hazy body, lacing chocolate, coffee roasted vanilla, creamy that this baseline considers textual content of user contributed bitterness, copper malts, spicy honey reviews, unlike other baselines that ignore them. Therefore it is better than vanilla content-based methods, with the notion TABLE V: Experience-based facet words for the illustrative of past evolution, and is the strongest baseline for our model. beer facet taste. Discussions: Table IV compares the mean squared error (MSE) 0.6 for rating predictions, generated by our model versus the six baselines. Our model consistently outperforms all baselines, 0.5 reducing the MSE by ca. 5 to 35%. Improvements of our model 0.4 over baselines are statistically significant at p-value < 0.0001. 0.3 Our performance improvement is most prominent for the 0.2 NewsTrust community, which exhibits strong language features, and topic polarities in reviews. The lowest improvement (over 0.1 the best performing baseline in any dataset) is achieved 0 RateBeer BeeeAdvocate NewsTrust Amazon Yelp for Amazon movie reviews. A possible reason is that the Level 1 Level 2 Level 3 Level 4 Level 5 community is very diverse with a very wide range of movies and that review texts heavily mix statements about movie plots Fig. 5: Proportion of reviews at each experience level of users. with the actual review aspects like praising or criticizing certain facets of a movie. The situation is similar for the food and movies (e.g., by genre or production studios) that may better restaurants case. Nevertheless, our model always wins over the distinguish experienced users from amateurs or novices in best baseline from other works, which is typically the “user at terms of their refined taste and writing style. MSE for different experience levels: We observe a weak trend learned rate” model. Evolution effects: We observe in Table IV that our model’s that the MSE decreases with increasing experience level. Users predictions degrade when applied to the users’ past experience at the highest level of experience almost always exhibit the level, compared to their most recent level. This signals that lowest MSE. So we tend to better predict the rating behavior for the model captures user evolution past the previous timepoint. the most mature users than for the remaining user population. Therefore the last (i.e., most recent) experience level attained by This in turn enables generating better recommendations for the a user is most informative for generating new recommendations. “connoisseurs” in the community. Experience progression: Figure 5 shows the proportion of B. Qualitative Analysis reviews written by community members at different experience Salient words for facets and experience levels: We point levels right before advancing to the next level. Here we plot out typical word clusters, with illustrative labels, to show the users with a minimum of 50 reviews, so they are certainly not variation of language for users of different experience levels “amateurs”. A large part of the community progresses from and different facets. Tables II and V show salient words to level 1 to level 2. However, from here only few users move describe the beer facet taste and movie facets plot and narrative to higher levels, leading to a skewed distribution. We observe style, respectively – at different experience levels. Note that the that the majority of the population stays at level 2. facets being latent, their labels are merely our interpretation. User experience distribution: Table VI shows the number of users per experience level in each domain, for users with Other similar examples can be found in Tables I and VII. BeerAdvocate and RateBeer are very focused communities; > 50 reviews. The distribution also follows our intuition of so it is easier for our model to characterize the user experience a highly skewed distribution. Note that almost all users with evolution by vocabulary and writing style in user reviews. We < 50 reviews belong to levels 1 or 2. observe in Table V that users write more descriptive and fruity Language model and facet preference divergence: Figure 6b words to depict the beer taste as they become more experienced. and 6c show the KL divergence for facet-preference and For movies, the wording in reviews is much more diverse language models of users at different experience levels, as and harder to track. Especially for blockbuster movies, which computed by our model. The facet-preference divergence tend to dominate this data, the reviews mix all kinds of aspects. increases with the gap between experience levels, but not as A better approach here could be to focus on specific kinds of smooth and prominent as for the language models. On one hand, Datasets e=1 e=2 e=3 e=4 e=5 Models F1 N DCG BeerAdvocate RateBeer NewsTrust Amazon Yelp 0.05 0.03 - 0.59 0.42 0.72 - 0.19 0.35 0.15 0.13 0.30 0.10 0.18 0.60 0.10 0.68 0.07 0.02 0.25 0.05 0.02 User at learned rate [16] Our model 0.68 0.75 0.90 0.97 TABLE VI: Distribution of users at different experience levels. Level 1: bad god religion iraq responsibility Level 2: national reform live krugman questions clear jon led meaningful lives california powerful safety impacts Level 3: health actions cuts medicare nov news points oil climate major jobs house high vote congressional spending unemployment strong taxes citizens events failure TABLE VII: Salient words for the illustrative NewsTrust topic US Election at different experience levels. this is due to the complexity of latent facets vs. explicit words. On the other hand, this also affirms our notion of grounding the model on language. Baseline model divergence: Figure 6a shows the facetpreference divergence of users at different experience levels computed by the baseline model “user at learned rate” [16]. The contrast between the heatmaps of our model and the baseline is revealing. The increase in divergence with increasing gap between experience levels is very rough in the baseline model, although the trend is obvious. TABLE VIII: Performance on identifying experienced users. based on community engagement, time in the community, other users’ feedback on reviews, profile transparency, and manual validation. We use these member levels to categorize users as experienced or inexperienced. This is treated as the ground truth for assessing the prediction and ranking quality of our model and the baseline “user at learned rate” model [16]. Table VIII shows the F1 scores of these two competitors. We also computed the Normalized Discounted Cumulative Gain (NDCG) [8] for the ranked lists of users generated by the two models. NDCG gives geometrically decreasing weights to predictions at various positions of ranked list: N DCGp = DCGp IDCGp where DCGp = rel1 + Pp reli i=2 log2 i Here, reli is the relevance (0 or 1) of a result at position i. As Table VIII shows, our model clearly outperforms the baseline model on both F1 and N DCG. VII. R ELATED W ORK State-of-the-art recommenders based on collaborative filtering [9][11] exploit user-user and item-item similarities by latent factors. Explicit user-user interactions have been exploited in trust-aware recommendation systems [6][25]. The temporal VI. U SE -C ASE S TUDY aspects leading to bursts in item popularity, bias in ratings, or So far we have focused on traditional item recommendation the evolution of the entire community as a whole is studied for items like beers or movies. Now we switch to a different in [10][27][26]. Other papers have studied temporal issues kind of items - newspapers and news articles - tapping into the for anomaly detection [7], detecting changes in the social NewsTrust online community (newstrust.net). NewsTrust neighborhood [14] and linguistic norms [3]. However, none of features news stories posted and reviewed by members, many these prior work has considered the evolving experience and of whom are professional journalists and content experts. behavior of individual users. The recent work[16], which is one of our baselines, modeled Stories are reviewed based on their objectivity, rationality, and the influence of rating behavior on evolving user experience. general quality of language to present an unbiased and balanced However, it ignores the vocabulary and writing style of users in narrative of an event. The focus is on quality journalism. reviews, and their natural smooth temporal progression. In conIn our framework, each story is an item, which is rated trast, our work considers the review texts for additional insight and reviewed by a user. The facets are the underlying topic into facet preferences and smooth experience progression. distribution of reviews, with topics being Healthcare, Obama Prior work that tapped user review texts focused on other Administration, NSA, etc. The facet preferences can be mapped issues. Sentiment analysis over reviews aimed to learn latent to the (political) polarity of users in the news community. Recommending News Articles: Our first objective is to topics [13], latent aspects and their ratings [12][24], and userrecommend news to readers catering to their facet preferences, user interactions [25]. [15][23] unified various approaches viewpoints, and experience. We apply our joint model to this to generate user-specific ratings of reviews. [18] further task, and compare the predicted ratings with the ones observed leveraged the author writing style. However, all of these for withheld reviews in the NewsTrust community. The mean prior approaches operate in a static, snapshot-oriented manner, squared error (MSE) results are reported in Table IV in without considering time at all. Section V. Table VII shows salient examples of the vocabulary From the modeling perspective, some approaches learn a by users at different experience levels on the topic US Election. document-specific discrete rating [13][21], whereas others Identifying Experienced Users: Our second task is to find learn the facet weights outside the topic model (e.g., [12], experienced members of this community, who have potential [15], [18]). In order to incorporate continuous ratings, [1] for being citizen journalists. In order to find how good our proposed a complex and computationally expensive Variational model predicts the experience level of users, we consider the Inference algorithm, and [17] developed a simpler approach following as ground-truth for user experience. In NewsTrust, using Multinomial-Dirichlet Regression. The latter inspired our users have Member Levels calculated by the NewsTrust staff technique for incorporating supervision. E3 E4 E5 138 123 93 18 135 105 E2 E3 134 0 33 8 65 90 75 112 34 0 21 43 60 E4 89 8 18 0 30 45 30 E5 19 63 48 31 0 E1 120 15 E1 E2 E3 E4 E5 0 180 71 140 125 E2 38 0 175 47 150 175 125 E3 83 149 5 0 88 100 75 E4 E5 86 155 1 E2 E3 0 180 48 203 100 140 180 150 259 E2 0 73 120 90 60 65 83 0 30 E3 160 2217 E2 181 0 1910 E3 601 309 0 1750 1500 1250 1000 750 500 0 Experience Levels 25 0 E2 0 2000 210 50 116 E1 E1 240 E3 86 0 E1 175 E1 0 Experience Levels 200 Experience Levels E2 0 Experience Levels E1 Experience Levels Experience Levels E1 250 0 Experience Levels 0 Experience Levels (a) User at learned rate [16]: Facet preference divergence with experience. E3 E4 E5 0 104 76 141 114 E1 E2 E3 E4 E5 0 192 199 324 161 135 E2 120 105 98 0 78 77 91 90 E3 75 78 0 89 106 75 60 E4 320 E1 132 74 90 0 52 45 109 86 106 52 0 E2 200 0 20 267 167 E3 211 20 0 271 244 160 120 E4 332 284 296 0 129 80 E5 15 240 200 163 0 Experience Levels E2 E3 0 216 234 E1 E2 E3 0 19 34 225 E1 30 E5 E1 54 280 Experience Levels Experience Levels E1 66 45 23 0 200 E1 175 Experience Levels E2 Experience Levels E1 150 E2 204 0 162 125 100 75 E3 156 222 50 0 42 36 E2 20 0 57 18 E3 55 34 0 12 6 0 Experience Levels 30 24 25 40 48 0 Experience Levels 0 Experience Levels (b) Our model: Facet preference divergence with experience. E3 E4 E5 0 193 209 154 201 200 E1 E1 E2 E3 E4 E5 0 571 563 1242 1440 197 0 71 100 162 150 125 E3 E4 213 71 0 77 140 100 75 156 98 76 0 102 50 E5 203 161 138 102 Experience Levels 0 25 0 RateBeer Experience Levels Experience Levels E2 1350 E2 575 0 78 303 390 1050 900 E3 593 105 0 333 362 750 600 E4 1264 326 329 0 213 E1 E2 E3 0 72 96 450 E1 1441 393 339 194 Experience Levels Amazon Movies 0 E2 E3 0 32 210 200 70 60 E2 72 0 50 76 40 30 E3 96 76 150 E1 80 20 0 0 Experience Levels 175 150 125 E2 33 0 132 E3 209 132 0 100 75 50 10 300 E5 E1 90 1200 175 Experience Levels E2 Experience Levels E1 E1 Experience Levels 25 0 0 Yelp NewsTrust (c) Our model: Language model divergence with experience. Fig. 6: Facet preference and language model KL divergence with experience. A general (continuous) version of this work is presented in [19] with fine-grained temporal evolution of user experience, and resulting language model using Geometric Brownian Motion and Brownian Motion, respectively. VIII. C ONCLUSION Current recommender systems do not consider user experience when generating recommendations. In this paper, we have proposed an experience-aware recommendation model that can adapt to the changing preferences and maturity of users in a community. We model the personal evolution of a user in rating items that she will appreciate at her current maturity level. We exploit the coupling between the facet preferences of a user, her experience, writing style in reviews, and rating behavior to capture the user’s temporal evolution. Our model is the first work that considers the progression of user experience as expressed in the text of item reviews. Our experiments – with data from domains like beer, movies, food, and news – demonstrate that our model substantially reduces the mean squared error for predicted ratings, compared to the state-of-the-art baselines. This shows our method can generate better recommendations than those models. 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Learning Invariants using Decision Trees Siddharth Krishna, Christian Puhrsch, and Thomas Wies arXiv:1501.04725v1 [cs.PL] 20 Jan 2015 New York University, USA Abstract. The problem of inferring an inductive invariant for verifying program safety can be formulated in terms of binary classification. This is a standard problem in machine learning: given a sample of good and bad points, one is asked to find a classifier that generalizes from the sample and separates the two sets. Here, the good points are the reachable states of the program, and the bad points are those that reach a safety property violation. Thus, a learned classifier is a candidate invariant. In this paper, we propose a new algorithm that uses decision trees to learn candidate invariants in the form of arbitrary Boolean combinations of numerical inequalities. We have used our algorithm to verify C programs taken from the literature. The algorithm is able to infer safe invariants for a range of challenging benchmarks and compares favorably to other ML-based invariant inference techniques. In particular, it scales well to large sample sets. 1 Introduction Finding inductive invariants is a fundamental problem in program verification. Many static analysis techniques have been proposed to infer invariants automatically. However, it is often difficult to scale those techniques to large programs without compromising on precision at the risk of introducing false alarms. Some techniques, such as abstract interpretation [9], are effective at striking a good balance between scalability and precision by allowing the analysis to be fine-tuned for a specific class of programs and properties. This fine-tuning requires careful engineering of the analysis [11]. Instead of manually adapting the analysis to work well across many similar programs, refinement-based techniques adapt the analysis automatically to the given program and property at hand [8]. A promising approach to achieve this automatic adaptation is to exploit synergies between static analysis and testing [15,17,34]. Particularly interesting is the use of Machine Learning (ML) to infer likely invariants from test data [14,31,32]. In this paper, we present a new algorithm of this type that learns arbitrary Boolean combinations of numerical inequalities. In most ML problems, one is given a small number of sample points labeled by an unknown function. The task is then to learn a classifier that performs well on unseen points, and is thus a good approximation to the underlying function. Binary classification is a specific instance of this problem. Here, the sample data is partitioned into good and bad points and the goal is to learn a predicate that separates the two sets. Invariant inference can be viewed as a binary classification problem [31]. If the purpose of the invariant is to prove a safety property, then the good points are the forward-reachable safe states of the program and the bad points are the backward-reachable unsafe states. These two sets are sampled using program testing. The learned classifier then represents 2 Siddharth Krishna, Christian Puhrsch, and Thomas Wies a candidate invariant, which is proved safe using a static analysis or theorem prover. If the classifier is not a safe invariant, the failed proof yields a spurious counterexample trace and, in turn, new test data to improve the classifier in a refinement loop. Our new algorithm is an instance of this ML-based refinement scheme, where candidate invariants are inferred using a decision tree learner. In this context, a decision tree (DT) is a binary tree in which each inner node is labeled by a function f from points to reals, called a feature, and a real-valued threshold t. Each leaf of the tree is labeled with either “good” or “bad”. Such a tree encodes a predicate on points that takes the form of a Boolean combination of inequalities, f pxq ď t, between features and thresholds. Given sets of features and sample points, a DT learner computes a DT that is consistent with the samples. In our algorithm, we project the program states onto the numerical program variables yielding points in a d-dimensional space. The features describe distances from hyperplanes in this space. The DT learner thus infers candidate invariants in the form of arbitrary finite unions of polyhedra. However, the approach also easily generalizes to features that describe nonlinear functions. Our theoretical contribution is a probabilistic completeness guarantee. More precisely, using the Probably Approximately Correct model for learning [33], we provide a bound on the sample size that ensures that our algorithm successfully learns a safe inductive invariant with a given probability. We have implemented our algorithm for specific classes of features that we automatically derive from the input program. In particular, inspired by the octagon abstract domain [22], we use as features the set of all hyperplane slopes of the form ˘xi ˘ xj , where 1 ď i ă j ď d. We compared our implementation to other invariant generation tools on benchmarks taken from the literature. Our evaluation indicates that our approach works well for a range of benchmarks that are challenging for other tools. Moreover, we observed that DT learners often produce simpler invariants and scale better to large sample sets compared to other ML-based invariant inference techniques such as [14, 30–32]. 2 Overview In this section, we discuss an illustrative example and walk through the steps taken in our algorithm to compute invariants. To this end, consider the program in Fig. 1. Our goal is to find an inductive invariant for the loop on line 4 that is sufficiently strong to prove that the assertion x ‰ 0 on line 12 is always satisfied. Good and Bad States. We restrict ourselves to programs over integer variables x “ px1 , . . . , xd q without procedures. Then a state is a point in Zd that corresponds to some assignment to each of the variables. For simplicity, we assume that our example program has a single control location corresponding to the head of the loop. That is, its states are pairs pv1 , v2 q where v1 is the value of x and v2 the value of y. When our program begins execution, the initial state could be p0, 1q or p0, ´3q, but it cannot be p2, 3q or p0, 0q because of the precondition specified by the assume statement in the program. A good state is defined as any state that the program could conceivably reach when it is started from a state consistent with the precondition. If we start execution at Learning Invariants using Decision Trees 1 2 var x, y: Int ; assume x “ 0 ^ y ‰ 0; 3 4 3 5 6 7 8 9 10 while (y ‰ 0) { if (y ă 0) { x := x ´ 1; y := y + 1; } else { x := x + 1; y := y ´ 1; } } 2 y 4 0 2 4 11 12 assert x ‰ 0; 4 2 0 x 2 4 Fig. 1: Example program. The right side shows some of the good and bad states of the program, in blue and red respectively p0, ´3q, then the states we reach are tp´1, ´2q, p´2, ´1q, p´3, 0qu and thus these are all good states. Similarly, bad states are defined to be the states such that if the program execution was to be started at that point (if we ran the program from the loop head, with those values) then the loop will exit after a finite point and the assertion will fail. For example, p0, 0q is a bad state, as the loop will not run, and we directly go to the assertion and fail it. Similarly, p´2, ´2q is a bad state, as after one iteration of the loop the state becomes p´1, ´1q, and after another iteration we reach p0, 0q, which fails the assertion. The right-hand side of Fig. 1 shows some of the good and bad states of the program. A safe inductive invariant can be expressed in terms of a disjunction of the indicated hyperplanes, which separate the good from the bad states. Our algorithm automatically finds such an invariant. Overview. The high-level overview of our approach is as follows: good and bad states are sampled by running the program on different initial states. Next, a numeric abstract domain that is likely to contain the invariant is chosen manually. We use disjunctions of octagons by default. For each hyperplane (up to translation) in the domain we add a new “feature” to each of the sample points corresponding to the distance to that hyperplane. A Decision Tree (DT) learning algorithm is then used to learn a DT that can separate the good and bad states in the sample, and this tree is converted into a formula that is a candidate invariant. Finally, this candidate invariant is passed to a theorem prover that verifies the correctness of the invariant. We now discuss these steps in detail. Sampling. The first step in our algorithm is to sample good and bad states of this program. We sample the good states by picking states satisfying the precondition, running the program from these states and collecting all states reached. To sample bad states, we look at all points close to good states, run the program from these. If the loop exits within a bounded number of iterations and fails the assert, we mark all states reached as bad states. The sampled good and bad states are shown in Fig. 1. 4 Siddharth Krishna, Christian Puhrsch, and Thomas Wies Features. The next step is to choose a candidate hyperplane set for the inequalities in the invariant. For most of our benchmarks, we used the octagon abstract domain, which consists of all linear inequalities of the form: bi ¨ xi ` bj ¨ xj ď c where 1 ď i ă j ď d, bi , bj P t´1, 0, 1u, and c P Z. We then let H “ tw1 , w2 , . . . u be the set of hyperplane slopes for this domain. Then we transform our sample points (both good and bad) according to these slopes. For each sample point x, we get a new point z given by zi “ x ¨ wi . In our example, the octagon slopes H and some of our transformed good and bad points are fi » fi » 0 » fi 1 0 —0 1ffi , H“– 1 ´1fl 1 1 and 0 1 1 ´1 1 — 1 0 1 1ffi — 1 0ffi ffi — —´1 0ffi ffi „ —  —´1 0 ´1 ´1ffi ffi — ffi — ... . . . ffi 1 0 1 1 ffi — X ¨ HT “ — — 0 0ffi ¨ 0 1 ´1 1 “ — 0 0 0 0ffi . ffi — ffi — — 2 ´2 4 0ffi — 2 ´2ffi fl – fl – ´1 1 ... ´1 1 ´2 ... 0 Learning the DT. After this transformation, we run a Decision Tree learning algorithm on the processed data. A DT (see Fig. 2 for an example) is a concise way to represent a set of rules as a binary tree. Each inner node is labeled by a feature and a threshold. Given a sample point and its features, we evaluate the DT by starting at the root and taking the path given by the rules: if the features is less than or equal to the threshold, we go to the left child, otherwise the right. Leaves specify the output label on that point. Most DT learning algorithms start at an empty tree, and greedily pick the best feature to split on at each node. From the good and bad states listed above, we can easily see that a good feature to split on must be the last one, as all bad states have the last column 0. Indeed, the first split made by the DT is to split on z4 at ´0.5. Since w4 “ p1, 1q, this split corresponds to the linear inequality x ` y ď ´0.5. Now, half the good states are represented in the left child of the root (corresponding to z4 ď ´0.5). The right child contains all the bad states and the other half of the good states. So the algorithm leaves the left child as is and tries to find the best split for the right child. Again, we see the same pattern with z4 , and so the algorithm picks the split z4 ă 1. Now, all bad states fall into the left child, and all good states fall into the right child, and we are done. The computed DT is shown in Fig. 2. Finally, we need to convert the DT back into a formula. To do this, we can follow all paths from the root that lead to good leaves, and take the conjunction of all inequalities on the path, and finally take the disjunction of all such paths. In our example, we get the candidate invariant: px ` y ď ´0.5q _ px ` y ą ´0.5 ^ x ` y ą 0.5q . This can be simplified to px ` y ‰ 0q. Verifying the Candidate Invariant. Our program is then annotated with this invariant and passed to a theorem prover to verify that the invariant is indeed sufficient to prove the program correct. Learning Invariants using Decision Trees 5 FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE Good TRUE TRUE Good Good Good Good FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE Bad TRUE Bad Bad Bad Bad Bad Good Good FALSE FALSE Good Good Good Good Good Good Good Fig. 2: Decision tree learned from the program and sample data in Fig. 1 3 Preliminaries Problem Statement. We think of programs as transition systems P “ pS, R, init, safeq, where S is a set of states, R Ď S ˆ S is a transition relation on the states, init Ď S is a set of initial states, and safe Ď S is the set of safe states that we want our program to remain within. For any set of states X, the set postpXq represents all the successor states with respect to R. More formally, postpXq “ t x1 \| Dx P X. Rpx, x1 q u . Then, we can define the set of reachable states of the program (the good states) to be the least fixed point, good “ lfppλX. init Y postpXqq . Similarly, we can define prepXq “ t x \| Dx1 P X. Rpx, x1 q u and bad “ lfppλX. error Y prepXqq, where error is the complement of safe. Then we see that the program is correct, with respect to the safety property given by safe, if good X bad “ H. Thus, our task is to separate the good states from the bad states. One method to show that these sets are disjoint is to show the existence of a safe inductive invariant. A safe inductive invariant is a set inv that satisfies the following three properties: – init Ď inv, – postpinvq Ď inv, – inv Ď safe. It is easy to see that these conditions imply, in particular, that inv separates the good and bad states: good Ď inv and inv X bad “ H. 6 Siddharth Krishna, Christian Puhrsch, and Thomas Wies Invariant Generation as Binary Classification. The set up for most machine learning problems is as follows. We have an input space X and an output space Y, and one is given a set of samples X Ď X that are labeled by some unknown function f : X Ñ Y. We fix a hypothesis set H Ď Y X , and the aim is to find the hypothesis h P H that most closely approximates f . The samples are often given in terms of feature vectors, and thus a sample x can be thought of as a point in some d-dimensional space. Binary classification is a common instance of this problem, where the labels are restricted to a binary set Y “ t0, 1u. Following [31], we view the problem of computing a safe inductive invariant for a program P “ pS, R, init, safeq as a binary classification problem by defining the input space as the set of all program states X “ S. Sample states x P bad are labeled by 0 and x P good are labeled by 1. Thus, the unknown function f is the characteristic function of the set of good states. The hypothesis to be learned is a safe inductive invariant. Note that if sampling the program shows that some state x is both good and bad, there exists no safe inductive invariant, and the program is shown to be unsafe. Hence, we assume that the set of sample states can be partitioned into good and bad states. Decision Trees. Instead of considering a hypothesis space that can represent arbitrary invariants, we restrict it to a specific abstract domain, namely those invariants that can be represented using decision trees. A Decision Tree (DT) [23] is a binary tree that represents a Boolean function. Each inner node v of T is labeled by a decision of the form xi ď t, where xi is one of the input features and t is a real valued threshold. We denote this inequality by v.cond. We denote the left and right children of an inner node by v.left and v.right respectively. Each leaf v is labeled by an output v.label. To evaluate an input, we trace a path down from the root node T.root of the tree, going left at each inner node if the decision is true, and right otherwise. The output of the tree on this input is the label of the leaf reached by this process. The hypothesis set corresponding to all DTs is thus arbitrary Boolean combinations of linear inequalities of the form xi ď t (axis-aligned hyperplanes). As one can easily see, many DTs can represent the same underlying function. However, the task of finding the smallest (in terms of number of nodes) DT for a particular function can be shown to be NP-complete [19]. Standard algorithms to learn DTs work by greedily selecting at each node the co-ordinate and threshold that separates the remaining training data best [7,27]. This procedure is followed recursively until all leaves have samples labeled by a single class. The criterion for separation is normally a measure such as conditional entropy. Entropy is a commonly used measure of uncertainty in a system. It is a function that is low when the system is homogeneous (in this case, think of when all samples reaching a node have the same label), and high otherwise. Conditional entropy, analogously, measures how homogeneous the samples are after choosing a particular co-ordinate and threshold. More formally, at each node, we look at the samples that reach that node, and define the conditional entropy of splitting feature xi at threshold t as Hpy\|xi : tq “ ppxi ă tqHpy\|xi ă tq ` ppxi ě tqHpy\|xi ě tq, ÿ where Hpy\|xi ă tq “ ´ ppy “ a\|xi ă tq log ppy “ a\|Xi ă tq, aPY Learning Invariants using Decision Trees 7 Algorithm 1: DTInv: Invariant generation algorithm using DT learning def DTInv (P : program): safe inductive invariant for P or fail “ val X, y “ Sampler(P) val H “ Slopes(P) val Z “ X ¨ H T val T “ LearnDT(Z, y) val ϕ “ DTtoForm(T.root) if IsInvariant(P , ϕ) then ϕ else fail def DTtoForm (v: node of a decision tree): formula represented by subtree rooted at v “ if v is a leaf then v.label else pv.cond ^ DTtoForm(v.left) _ v.cond ^ DTtoForm(v.right)q and ppAq is the empirical probability, i.e., the fraction of sample points reaching this node that satisfy the condition A. Hpy\|xi ě tq is defined similarly to Hpy\|xi ă tq. Note that if a particular split perfectly separates good and bad samples, then the conditional entropy is 0. The greedy heuristic is to pick the feature and split that minimize the conditional entropy. There are other measures as well, such as the Gini index, which is used by the DT learner we used in our experiments [25]. 4 Algorithm We now present our DT-learning algorithm. We assume that we have black box procedures for sampling points from the given program, and for getting a set of slopes from a chosen abstract domain. To this end, let Sampler be a procedure that takes a program and returns an n ˆ d matrix X of n sample points, and an n-dimensional vector y corresponding to the label of each sample point (1 for good states, 0 otherwise). Similarly, let Slopes be a procedure that takes a program and returns an m ˆ d matrix H of m hyperplane slopes. We describe the actual procedures used in our experiments in Section 5.1. Our final algorithm is surprisingly simple, and is given in Algorithm 1. We get the sample points and the slopes from the helper functions mentioned above, and then transform the sample points according to the slopes given. We run a standard DT learning algorithm on the transformed sample to obtain a tree that classifies all samples correctly. The tree is then transformed into a formula that is a candidate invariant, by a simple procedure DTtoForm. Finally, the program is annotated with the candidate invariant and verified. This final step is realized by another black box procedure IsInvariant, which checks that the invariant satisfies the three conditions necessary to be a safe inductive invariant. For example, this can be done by encoding init, inv, post and safe as SMT formulas and feeding the three conditions into an SMT solver. To convert the DT into a formula, we note that the set of states that reach a particular leaf is given by the conjunction of all predicates on the path from the root to that leaf. Thus, the set of all states classified as good by the DT is the disjunction of the sets of states that reach all the good leaves. A simple conversion is then to take the disjunction 8 Siddharth Krishna, Christian Puhrsch, and Thomas Wies over all paths to good leaves of the conjunction of all predicates on such paths. The procedure DTtoForm computes this formula recursively by traversing the learned DT. Since the tree was learnt on the transformed sample data, the predicate at each node of the tree will be of the form zi ď c where zi is one of the columns of Z and c is a constant. Since Z “ X ¨ H T , we know that zi pxq “ x ¨ wi for a sample x. Thus, the predicate is equivalent to x ¨ wi ď c, which is a linear inequality in the program variables. Combining this with the conversion procedure above, we see that our algorithm outputs an invariant which is a Boolean combination of linear inequalities over the numerical program variables. Soundness. From the above discussion, we see that the formula ϕ returned by the procedure DTtoForm is of the required format. Moreover, we assume that the procedure IsInvariant is correct. Thus, we can say that our invariant generation procedure DTInv is sound: if it terminates successfully, it returns a safe inductive invariant. Probabilistic Completeness. It is harder to prove that such invariant generation algorithms are complete. We see that the performance of our algorithm depends heavily on the sample set, if the sample is inadequate, it is impossible for the DT learner to learn the underlying invariant. One could augment our algorithm with a refinement loop, which would make the role of the sampler less pronounced, for if the invariant is incorrect, the theorem prover will return a counterexample that could potentially be added to the sample set and we can re-run learning. However, we find in practice that we do not need a refinement loop if our sample set is large enough. We can justify this observation using Valiant’s PAC (probably approximately correct) model [33]. In this model, one can prove that an algorithm that classifies a large enough sample of data correctly has small error on all data, with high probability. It must be noted that one key assumption of this model is that the sample data is drawn from the same distribution as the underlying data, an assumption that is hard to justify in most of its applications, including this one. In practice however, PAC learning algorithms are empirically successful on a variety of applications where the assumption on distribution is not clearly true. Formally, we can give a generalization guarantee for our algorithm using this result of Blumer et al. [6]: Theorem 1. A learning algorithm for a hypothesis class H that outputs a hypothesis h P H will have true error at most with probability at least 1 ´ δ if h is consistent log 13 with a sample of size maxp 4 log 2δ , 8V CpHq q. In the above theorem, a hypothesis is said to be consistent with a sample if it classifies all points in the sample correctly. The quantity V CpHq is a property of the hypothesis class called the Vapnik-Chervonenkis (VC) dimension, and is a measure of the expressiveness of the hypothesis class. As one might imagine, more complex classes lead to a looser bound on the error, as they are more likely to over-fit the sample and less likely to generalize well. Thus, it suffices for us to bound the VC dimension of our hypothesis class, which is all finite Boolean combinations of hyperplanes in d dimensions. The VC dimension of a class H is defined as the cardinality of the largest set of points that H can shatter. A set of points is said to be shattered by a hypothesis class H if for every possible labeling of the points, there exists a hypothesis in H that is consistent with it. Unfortunately, Learning Invariants using Decision Trees 9 DTs in all their generality can shatter points of arbitrarily high cardinality. Given any set of m points, we can construct a DT with m leaves such that each point ends up at a different leaf, and now we can label the leaf to match the labeling given. Since in practice we will not be learning arbitrarily large trees, we can restrict our algorithm a-priori to stop growing the tree when it reaches K nodes, for some fixed K independent of the sample. Now one can use a basic, well-known lemma from [24] combined with Sauer’s Lemma [29] to get that the VC dimension of bounded decision trees is OpKd log Kq. Combining this with Theorem 1, we get the following polynomial bound for probabilistic completeness: Theorem 2. Under the assumptions of the PAC model, the algorithm DTInv returns an invariant that has true error at most with probability at least 1 ´ δ, given that its sample size is Op 1 Kd log K log 1δ q . Complexity. The running time of our algorithm depends on many factors such as the running time of the sampling (which in turn depends on the benchmark being considered), and so is hard to measure precisely. However, the running time of the learning routine for DTs is Opmn logpnqq, where m is the number of hyperplane slopes in H and n is the number of sample points [7, 25]. The learning algorithm therefore scales well to large sets of sample data. Nonlinear invariants. An important property of our algorithm is that it generalizes elegantly to nonlinear invariants as well. For example, if a particular program requires invariants that reason about px mod 2q for some variable x, then we can learn such invariants as follows: given the sampled states X, we add to it a new column that corresponds to a variable x1 , such that x1 “ x mod 2. We then run the rest of our algorithm as before, but in the final invariant, replace all occurrences of x1 by px mod 2q. As is easy to see, this procedure correctly learns the required nonlinear invariant. We have added this feature to our implementation, and show that it works on benchmarks requiring these nonlinear features (see Section 5.2). 5 5.1 Implementation and Evaluation Implementation We implemented our algorithm in Python, using the scikit-learn library’s decision tree classifier [25] as the DT learner LearnDT. This implementation uses the CART algorithm from [7] which learns in a greedy manner as described in Section 3, and uses the Gini index. We implemented a simple and naive Sampler: we considered all states that satisfied the precondition where the value of every variable was in the interval r´L, Ls. For these states, we ran the program with a bound I on the number of iterations of loops, and collected all states reached as good states. To find bad states, we looked at all states that were a margin M away from every good state, ran the program from this state (again with at most I iterations), and if the program failed an assertion, we collected all the states on this path as bad states. The bounds L, I, M were initialized to low values and increased until we had sampled enough states to prove our program correct. 10 Siddharth Krishna, Christian Puhrsch, and Thomas Wies For the Slopes function, we found that for most of the programs in our benchmarks it sufficed to consider slopes in the octagonal abstract domain. This consists of all vectors in t´1, 0, 1ud with at most two non zero elements. In a few cases, we needed additional slopes (see Table 1). For the programs hola15 and hola34, we used a class of slopes of vectors in C d where C is a small set of constants that appear in the program, and their negations. We also developed methods to learn the class of slopes needed by a program by looking at the states sampled. In [31], the authors suggest working in the null space of the good states, viewed as a matrix. This is because if the good states lie in some lower dimensional space, this would automatically suggest equality relationships among them that can be used in the invariant. It also reduces the running time of the learning algorithm. Inspired by this, we propose using Principal Component Analysis [20] on the good states to generate slopes. PCA is a method to find the basis of a set of points so as to maximize the variance of the points with respect to the basis vectors. For example, if all the good points lie along the line 2x ` 3y “ 4, then the first PCA vector will be p2, 3q, and intuition suggests that the invariant will use inequalities of the form 2x ` 3y ă c. Finally, for the IsInvariant routine, we used the program verifier Boogie [4], which allowed us to annotate our programs with the invariants we verified. Boogie uses the SMT solver Z3 [12] as a back-end. 5.2 Evaluation We compared our algorithm with a variety of other invariant inference tools and static analyzers. We mainly focused on ML-based algorithms, but also considered tools based on interpolation and abstract interpretation. Specifically, we considered: – ICE [14]: an ML algorithm based on ICE-learning that uses an SMT solver to learn numerical invariants. – MCMC [30]: an ML algorithm based on Markov Chain Monte Carlo methods. There are two versions of this algorithm, one that uses templates such as octagons for the invariant, and one with all constants in the slopes picked from a fixed bag of constants. The two algorithms have very similar characteristics. We ran both versions 5 times each (as they are randomized) and report the better average result of the two algorithms for each benchmark. – SC [31]: an ML algorithm based on set cover. We only had access to the learning algorithm proposed in [31] and not the sampling procedure. To obtain a meaningful comparison, we combined it with the same sampler that we used in the implementation of our algorithm DTInv. – CPAchecker [5]: a configurable software model checker. We chose the default analysis based on predicate abstraction and interpolation. – UFO [2]: a software model checker that combines abstract interpretation and interpolation (denoted CPA). – InvGen [18]: an inference tool for linear invariants that combines abstract interpretation, constraint solving, and testing. Learning Invariants using Decision Trees 11 For our comparison, we chose a combination of 22 challenging benchmarks from various sources. In particular, we considered a subset of the benchmarks from [13, 14, 18, 31]. We chose the benchmarks at random among those that were hard for at least one other tool to solve. Due to this bias in the selection, our experimental results do not reflect the average performance of the tools that we compare against. Instead, the comparison should be considered as an indication that our approach provides a valuable complementary technique to existing algorithms. We ran our experiments on a machine with a quad-core 3.40GHz CPU and 16GB RAM, running Ubuntu GNU/Linux. For the analysis of each benchmark, we used a memory limit of 8GB and a timeout of 5 minutes. The results of the experiment are summarized in Table 1. Here are the observations from our experiments (we provide more in-depth explanations for these observations in the next section, where we discuss related work in more detail): – Our algorithm DTInv seems to learn complex Boolean invariants as easily as simple conjunctions. – ICE seems to struggle on programs that needed large invariants. For some of these (gopan, popl07), this is because the constraint solver runs out of time/memory, as the constants used in the invariants are also large. For hola19 and prog4, ICE stops because the tool has an inbuilt limit for the complexity of Boolean templates. – However, ICE solves sum1 and trex3 quickly, even though they need many predicates, because the constant terms are small, so this space is searched first by ICE. – Similarly, we notice that MCMC has difficulty finding large invariants, again because the search space is huge. – SC’s learning algorithm is consistently slower than DTInv, due to its higher running time complexity. It also runs out of memory for large sample sizes. – DTInv is able to easily handle benchmarks that CPA, UFO and InvGen struggle on. This is mainly because they are specialized for reasoning about linear invariants, and have issues dealing with invariants that have complicated Boolean structure. We also learned some of the weak points in our current approach: – DTInv is slow in processing fig1 and prog4, both of which are handled by at least one other tool without much effort. However, we note that most of this time is spent in the sampling routine, which is currently a naive implementation. We therefore believe that DT learning could benefit from a combination with a static analysis that provides approximations of the good and bad states to guide the sampler. – We also note that the method of “constant slopes” which we used to handle the non octagonal benchmarks (hola15, hola34) is ad-hoc, and might not work well for larger benchmarks. Beyond octagons. As mentioned in Section 4, we implemented a feature to learn certain nonlinear invariants. We were able to verify some benchmarks that needed reasoning about the modulus of certain variables, as shown in Table 1. Finally, we show one example where we were able to infer a nonlinear invariant (specifically s “ i2 ^ i ď n for square). 12 Siddharth Krishna, Christian Puhrsch, and Thomas Wies Name Vars Type Octagonal: ex23 [14] 4 conj. fig6 [14] 2 conj. fig9 [14] 2 conj. hola10 [13] 4 conj. nested2 [31] 4 conj. nested5 [18] 4 conj. fig1 [14] 2 disj. test1 [31] 4 disj. cegar2 [14] 3 ABC gopan [31] 2 ABC hola18 [13] 3 ABC hola19 [13] 4 ABC popl07 [31] 2 ABC prog4 [31] 3 ABC sum1 [14] 3 ABC trex3 [14] 8 ABC Non octagonal: hola15 [13] 3 conj. Modulus: hola02 [13] 4 conj. hola06 [13] 4 conj. hola22 [13] 4 conj. Non octagonal modulus: hola34 [13] 4 ABC Quadratic: square 3 conj. \|ϕ\| Samp DT SC DTInv ICE MCMC 3 2 2 8 3 4 3 2 5 8 6 7 7 8 6 9 0.10 0.00 0.00 0.03 0.03 2.48 14.61 0.90 0.03 0.03 1.60 0.19 0.03 2.32 0.01 8.44 0.01 0.00 0.01 0.00 0.00 0.02 0.01 0.01 0.01 0.00 0.04 0.01 0.00 0.02 0.01 0.06 4.73 2.61 2.31 2.45 2.39 MO F 7.86 2.64 2.54 MO 3.47 2.72 MO 2.61 MO 0.11 0.00 0.01 0.03 0.03 2.50 14.62 0.91 0.04 0.03 1.64 0.20 0.03 2.34 0.02 8.50 CPA UFO InvGen 8.82 0.01 19.77 1.50 0.30 0.00 1.68 0.13 0.33 0.00 1.73 0.13 49.21 TO 2.03 F 62.02 0.09 1.86 0.12 60.95 31.28* 2.08 0.35 0.38 5.13 1.75 1.64 0.39 TO 1.71 F 4.86 17.30 1.97 0.18 F TO 63.85 58.29 TO 21.93* TO 8.38 F TO F F F TO 110.81 15.20 F 0.13 F F 1.32 29.04* F 0.17 4.51 NA F 0.18 0.02 0.01 0.01 F 0.03 0.03 F 0.04 F F F F F F F F 2 0.52 0.02 0.01 0.54 0.53 4 3 5 0.03 0.03 F 1.79 0.03 MO 0.04 0.00 F 0.06 1.82 0.04 F F F NA NA NA F F F F F F F F F 6 1.14 0.03 3.10 1.17 F NA F F F 3 0.27 0.24 MO 0.51 F NA F F F 0.04 MO 0.13 0.02 Table 1: Results of comparison. The table is divided according to what kinds of invariants the benchmark needed. The “Type” column denotes the Boolean structure of the invariant - conjunctive, disjunctive and arbitrary Boolean combination are denoted as conj., disj. and ABC respectively. The column \|ϕ\| contains the number of predicates in the invariant found by DTInv. The columns ’Samp’ and ’DT’ show the running time in seconds of our sampling and DT learning procedures respectively. We show SC next, as we only compare learning times with SC. Then follows the total time of our tool (DTInv), followed by those for other tools. Each entry of the tool columns shows the running time in seconds if a safe invariant was found, or otherwise one of the following entries. ’NA’: program contains arithmetic operations that are not supported by the tool; ’F’: analysis terminated without finding a safe invariant; ’TO’: timeout; ’MO’: out of memory. The times for MCMC have an asterisk if at least one of the repetitions timed out. In this case the number shown is the average of the other runs. Learning Invariants using Decision Trees 13 We believe our experiments show that Decision Trees are a natural representation for invariants, and that the greedy learning heuristics guide the algorithm to discover simple invariants of complex structures without additional overhead. 6 Related Work and Conclusions Our experimental evaluation compared against other algorithms for invariant generation. We discuss these algorithms in more detail. Sharma et. al. [31] used the greedy set cover algorithm SC to learn invariants in the form of arbitrary Boolean combinations of linear inequalities. Our algorithm based on decision trees is simpler than the set cover algorithm, works better on our benchmarks (which includes most of the benchmarks from [31]), and scales much better to large sample sets of test data. The improved scalability is due to the better complexity of DT learners. The running time of our learning algorithm is Opmn logpnqq where m is the number of features/hyperplane slopes that we consider, and n the number of sample points. On the other hand, the set cover algorithm has a running time of Opmn3 q. This is because the greedy algorithm for set cover takes time Ophn2 q where h is the number of hyperplanes, and [31] considers one hyperplane for every candidate slope and sample point, yielding h “ mn. When invariant generation is viewed as binary classification, then there is a problem in the refinement loop: if the learned invariant is not inductive, it is unclear whether the counterexample model produced by the theorem prover should be considered a “bad” or a “good” state. The ICE-learning framework [14] solves this problem by formulating invariant generation as a more general classification problem that also accounts for implication constraints between points. We note that our algorithm does not fit within this framework, as we do not have a refinement loop that can handle counterexamples in the form of implications. However, we found in our experiments that we did not need any refinement loop as our algorithm was able to infer correct invariants directly after sampling enough data. Nevertheless, considering an ICE version of DT learning is interesting as sampling without a refinement loop becomes difficult for more complex programs. The paper [14] also proposes a concrete algorithm for inferring linear invariants that fits into the ICE-learning framework (referred to as ICE in our evaluation). If we compare the complexity of learning given a fixed sample, our algorithm performs better than [14] both in terms of running time and expressiveness of the invariant. The ICE algorithm of [14] iterates through templates for the invariant. This iteration is done by dovetailing between more complex Boolean structures and increasing the range of the thresholds used. For a fixed template, it formulates the problem of this template being consistent with all given samples as a constraint in quantifier-free linear integer arithmetic. Satisfiability of this constraint is then checked using an SMT solver. We note that the size of the generated constraint is linear in the sample size, and that solving such constraints is NP-complete. In comparison, our learning is sub-quadratic time in the sample size. Also, we do not need to fix templates for the Boolean structure of the invariant or bound the thresholds a priori. Instead, the DT learner automatically infers those parameters from the sample data. 14 Siddharth Krishna, Christian Puhrsch, and Thomas Wies Another ICE-learning algorithm based on randomized search was proposed in [30] (the algorithm MCMC in our evaluation). This algorithm searches over a fixed space of invariants S that is chosen in advance either by bounding the Boolean structure and coefficients of inequalities, or by picking some finite sub-lattice of an abstract domain. Given a sample, it randomly searches using a combination of random walks and hill climbing until it finds a candidate invariant that satisfies all the samples. There is no obvious bound on the time of this search other than the trivial bound of \|S\|. Again, we have the advantage that we do not have to provide templates of the Boolean structure and the thresholds of the hyperplanes. These parameters have to be fixed for the algorithm in [30]. Furthermore, the greedy nature of DT learning is a heuristic to try simpler invariants before more complex ones, and hence the invariants we find for these benchmarks are often much simpler than those found by MCMC. Decision trees have been previously used for inferring likely preconditions of procedures [28]. Although this problem is related to invariant generation, there are considerable technical differences to our algorithm. In particular, the algorithm proposed in [28] only learns formulas that fall into a finite abstract domain (Boolean combinations of a given finite set of predicates), whereas we use decision trees to learn more general formulas in an infinite abstract domain (e.g., unions of octagons). We believe that the main value of our algorithm is its ability to infer invariants with a complex Boolean structure efficiently from test data. Other techniques for inferring such invariants include predicate abstraction [16] as well as abstract interpretation techniques such as disjunctive completion [10]. However, for efficiency reasons, many static analyses are restricted to inferring conjunctive invariants in practice [3, 11]. There exist techniques for recovering loss of precision due to imprecise joins using counterexample-guided refinement [1, 21, 26]. 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Belief Propagation Min-Sum Algorithm for Generalized Min-Cost Network Flow arXiv:1710.07600v1 [stat.ML] 20 Oct 2017 Andrii Riazanov1, Yury Maximov2 and Michael Chertkov3 Abstract— Belief Propagation algorithms are instruments used broadly to solve graphical model optimization and statistical inference problems. In the general case of a loopy Graphical Model, Belief Propagation is a heuristic which is quite successful in practice, even though its empirical success, typically, lacks theoretical guarantees. This paper extends the short list of special cases where correctness and/or convergence of a Belief Propagation algorithm is proven. We generalize formulation of Min-Sum Network Flow problem by relaxing the flow conservation (balance) constraints and then proving that the Belief Propagation algorithm converges to the exact result. I. INTRODUCTION Belief Propagation algorithms were designed to solve optimization and inference problems in graphical models. Since a variety of problems from different fields of science (communication, statistical physics, machine learning, computer vision, signal processing, etc.) can be formulated in the context of graphical model, Belief Propagation algorithms are of great interest for research during the last decade [1], [2]. These algorithms belong to message-passing heuristic, which contains distributive, iterative algorithms with little computation performed per iteration. There are two types of problems in graphical models of the greatest interest: computation of the marginal distribution of a random variable, and finding the assignment which maximizes the likelihood. Sum-product and Min-sum algorithms were designed for solving these two problems under the heuristic of Belief Propagation. Originally, the sum-product algorithm was formulated on trees ([3], [4], [5]), for which this algorithm represents the idea of dynamic programming in the message-passing concept, where variable nodes transmit messages between each other along the edges of the graphical model. However, these algorithms showed surprisingly good performance even when applied to the graphical models of non-tree structure ([6], [7], [8], [9], [10]). Since Belief Propagation algorithms can be naturally *The work was supported by funding from the U.S. Department of Energy’s Office of Electricity as part of the DOE Grid Modernization Initiative. 1 Andrii Riazanov is with the Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA. riazanov@cs.cmu.edu 2 Yury Maximov is with Skolkovo Institute of Science and Technology, Center for Energy Systems, and Los Alamos National Laboratory, Theoretical Division T-4 & CNLS, Los Alamos, NM 87544, USA yury@lanl.gov 3 Michael Chertkov is with Skolkovo Institute of Science and Technology, Center for Energy Systems, and Los Alamos National Laboratory, Theoretical Division T-4, Los Alamos, NM 87544, USA chertkov@lanl.gov implemented and paralleled using the simple idea of message passing, these instruments are widely used for finding approximate solutions to optimization of inference problems. However, despite the good practical performance of this heuristic in many cases, the theoretical grounding of these algorithm remain unexplored (to some extend), so there are no actual proofs that the algorithms give correct (or even approximately correct) answers for the variety of problem statements. That’s why one of important tasks is to explore the scope of the problems, for which these algorithms indeed can be applied, and to justify their practical usage. In [11] the authors proved that the Belief Propagation algorithm give correct answers for Min-Cost Network Flow problem, regardless of the underlying graph being a tree or not. Moreover, the pseudo-polynomial time convergence was proven for these problems, if some additional conditions hold (the uniqueness of the solution and integral input). This work significantly extended the set of problems, for which Belief Propagation algorithms are justified. In this paper we formulate the extension of Min-Cost Network Flow problem, which we address as Generalized Min-Cost Network Flow problem (GMNF). This problem statement is much broader then the original formulation, but we amplify the ideas of [11] to prove that Belief Propagation algorithms also give the correct answers for this generalization of the problem. This extension might find a lot of applications in various fields of study, since GMNF problem is the general problem of linear programming with additional constraints on the cycles of the underlying graphs (more precise, on the coefficient of corresponding vertices), which might be natural for some practical formulations. II. GENERALIZED MIN-COST NETWORK FLOW A. Problem statement Let G = (V, E) be a directed graph, where V is the set of vertices and E is the set of edges, \|V \| = n, \|E\| = m. For any vertex v we denote Ev as the set of edges incident to v, and aev is the coefficient related to this pair (v, e), such that aev > 0 if e is an out-arc with respect to v (e.g. e = (v, w) for some vertex w), and aev < 0 if e is an in-arc with respect to v (e.g. e = (w, v) for some vertex w). For any vertex v and edges e1 , e2 incident to v we ae1 define δ(v, e1 , e2 ) , ev2 . Then we consider the following av property of the graph: Definition 2.1: The graph G is called ratio-balanced if for every non-directed cycle C which consists of vertices {v1 , v2 , v3 , . . . , vk } and edges {e1 , e2 , e3 , . . . , ek } it holds: k Y δ(vi , ei , vi−1 ) = δ(v1 , ek , e1 ) · δ(v2 , e1 , e2 ) · . . . · × i=1 ×δ(vk−1 , ek−2 , ek−1 ) · δ(vk , ek−1 , ek ) = 1. (1) follows: c(C) , c1 + δ(v2 , e1 , e2 )× ! × c2 + δ(v3 , e2 , e3 ) c3 + · · · + δ(vk , ek−1 , ek )ck · · · = = c1 + Here by non-directed cycle we mean that for every pair (vi , vi−1 ) and the pair (v1 , vk ) it holds that either (vi , vi−1 ) ∈ E or (vi−1 , vi ) ∈ E. It is not hard to verify that it suffices for equation (1) to hold only for every simple non-directed cycle of G, since then the equation (1) can be easily deduced to hold for arbitrary non-directed cycle. To check that the given graph is ratio-balances, one then need to check whether (1) holds for any simple cycle. If m is the number of edges, and C is the number of simple cycles of G, one obviously needs at least O(m + C) time to iterate trough all simple cycles. In fact, the optimal algorithm for this task was introduced in [12], which runs for O(m + C) time. Then, to check whether a graph is ratio-balances, one may use this algorithm to iterate through all simple cycles and to check (1) for every one of them. We formulate the Generalized Min-Cost Network Flow problem for ratio-balanced graph G as follows: minimize X ce xe e∈E subject to X aev xe = fv , ∀v ∈ V, (GMNF) e∈Ev 0 ≤ xe ≤ ue , ∀e ∈ E. Here the first set of constrains are balance constraints which must hold for each vertex. The second set of constrains consists of capacity constraints on each edge of G. Coefficients ce and ue , defined for each edge e ∈ E, are called the cost and the capacity of the edge, respectively. Any assignment of x in this problem which satisfies the balance and capacity constraints P is referred as flow. Finally, the objective function g(x) = e∈E ce xe is called the total flow. B. Definitions and properties For the given (GMNF) problem on the graph G and flow x on this graph, the residual network G(x) is defined as follows: G(x) has the same vertex set as G, and for each edge e = (v, w) ∈ E if xe < ue then e is an arc in G(x) with the cost cxe = ce and coefficients (aev )x = aev , (aew )x = aew . Finally, if xe > 0 then there is an arc e′ = (w, v) in G(x) with the cost cxe′ = −ce and coefficients (aev )x = −aev , (aew )x = −aew . It is not hard to see that G(x) is ratiobalanced whenever G is, since only the absolute values of the coefficients aev occur in the definition of this property. Then for each directed cycle C = ({v1 , v2 , v3 , . . . , vk }, {e1 , e2 , e3 , . . . , ek }) we define the cost of this cycle as k X i=2 ci i Y δ(vj , ej−1 , ej ) j=2 It is easy to see that c(C) is properly defined whenever the graph G is ratio-constrained. Then we define σ(x) , min{cx (C)}, where the minimum C is taken over all cycles C in the residual network G(x). Lemma 2.1: If (GMNF) has a unique solution x∗ , then σ(x∗ ) > 0. Proof: We will show that for every directed cycle C from G(x∗ ) we can push the additional flow through the edges of this cycle such that the linear constraints in (GMNF) will still be satisfied, but the total flow in the cycle will change by ε · cx (C) for some ε > 0. Let C = ({v1 , v2 , v3 , . . . , vk }, {e1 , e2 , e3 , . . . , ek }). From the definition of the residual network it follows that we can increase the flow in every edge of the cycle for some positive quantity such that the capacity constraints will still be satisfied. Let’s push additional flow ε > 0 trough e1 . In order to satisfy the balance constraint in v1 , we need to adjust the flow in e2 . We have 4 cases: either one of e1 , e2 can be in G, or their opposites can be in G. If e ∈ C is in G, we will say that it is ’direct’ arc, otherwise it will be ’opposite’. Then there are four cases: 1) e1 , e2 are direct arcs. Then we have the new flow on edge e − 1: y1 = x∗1 + ε. In order to satisfy the balance constraint for the vertex v2 , is must hold aev12 x∗1 + aev22 x∗2 = aev12 y1 + aev22 y2 = aev12 x∗1 + aev12 ε + aev22 y2 ⇒ aev22 (x∗2 − y2 ) = aev12 ε ⇒ y2 = e1 a v2 ∗ x2 − ε e2 . Since both e1 and e2 are direct, it means av2 that e1 = (v1 , v2 ) ∈ E, and e2 = (v2 , v3 ) ∈ E, and thus, by definition, aev12 < 0, and aev22 > 0. Therefore, we have y2 = x∗2 + εδ(v2 , e1 , e2 ). 2) e1 is direct, e2 is opposite. Then again, y1 = x∗1 + ε, but now ’pushing’ the flow through e2 (as the edge in the residual network) means decreasing x2 . The same equalities holds, so ae1 y2 = x∗2 − ε ev22 = x∗2 − εδ(v2 , e1 , e2 ). Since e2 av2 is opposite arc, that means that we should push additional εδ(v2 , e1 , e2 ) through e2 . 3) e1 is opposite, e2 is direct – similar to the case 2). 4) e1 , e2 are opposite arcs – similar to the case 1). So, if we push ε through e1 , we need to push ε2 = εδ(v2 , e1 , e2 ) through e2 to keep the balance in v2 . Then, analogically, to maintain the balance in v3 , we need to push additional ε3 = ε2 δ(v3 , e2 , e3 ) = εδ(v2 , e1 , e2 )δ(v3 , e2 , e3 ) through e3 . Then, consequently adjusting the balance in all the vertexes of C, we will retrieve that to keep the balance Qk in vk , we need to push εk = εk−1 δ(vk , ek−1 , ek ) = ε i=2 δ(vi , ei−1 , ei ) through ek . Now it suffices to show that the balance in v1 is also satisfied. Similarly, we know that if we push εk Q in ek , then we need to push ε1 = εk δ(v1 , ek , e1 ) = ε ki=1 δ(vi , ei−1 , e1 ) = ε (since G is ratio-balanced) through e1 , and that is exactly the amount which we assumed to push at the beginning of this proof. So we indeed push consistent flow through all the edges of C in such a way that the balance constraints in all the vertices is satisfied. We now only need to mention that we can take ε > 0 as small as it is needed to satisfy also all the capacity constraints in the cycle. Now the additional total cost of such adjusting will be k X Proof: l(R) = c1 + δ(v2 , e1 , e2 ) c2 + δ(v3 , e2 , e3 )× ! = × · · · ck−2 + δ(vk−1 , ek−2 , ek−1 )ck−1 · · ·  =  c1 +  + \| cxi εi = cx1 ε + cx2 εδ(v2 , e1 , e2 )+ +cx2 εδ(v2 , e1 , e2 )δ(v3 , e2 , e3 ) + · · · + cxk ε δ(vi , ei−1 , ei ) = × ci i=2 δ(vj , ej−1 , ej ) j=2 We also define the ’reducer’ of the path as follows: t(S) , min j=2,...,k−1 j Y \| m X i=2 p−1 Y j=2 " c′i i Y } # + δ(vj′ , e′j−1 , e′j ) j=2 {z × cp + }   δ(vj′ , e′j−1 , e′j ) ·δ(vp , e′m , ep )× {z k−1 X i=p+1 " } i Y ci " p−1 X i=2 j=p+1 ci i Y j=2  k−1 X i=p+1 " i Y ci # δ(vj , ej−1 , ej )  ≥ # δ(vj , ej−1 , ej )  + δ(vj , ej−1 , ej ) ·  δ(vi , ei−1 , ei ) {z c(C) ≥  c1 + +  (δ(v1′ ,e′m ,e′1 ))−1  i=2 To prove the main result of this paper we will use the following crucial lemma: Lemma 2.2: Let G be any ratio-balanced graph, or a residual network of some ratio-balanced graph (as we already mentioned, the residual network will also be ratio-balanced in this case). Let S = ({v1 , · · · , vk }, {e1, · · · , ek−1 }) ′ be a directed path in G, and C = ({v1′ , v2′ , · · · , vm }, {e′1 , e′2 , · · · , e′m }) be a cycle with v1′ = vp . Let R be ′ , v1′ = the path R = {v1 , v2 , · · · , vp = v1′ , v2′ , . . . , vm vp , vp+1 , . . . , vk }. Then l(R) ≥ l(S) + T c(C), where T = minS t(S) is the minimum of all the reducers among all directed paths S in G. m Y × cp +  δ(vj , ej−1 , ej )  + δ(vj , ej−1 , ej ) · δ(vp , ep−1 , e′1 )× j=2 ! × · · · ck−2 + δ(vk−1 , ek−2 , ek−1 )ck−1 · · · = = c1 + j=2  l(S) , c1 + δ(v2 , e1 , e2 ) c2 + δ(v3 , e2 , e3 )× i Y  +  j=2 # ≥T p−1 Y ∗ k−1 X j=2  Now it is obvious that if cx (C) ≤ 0 for some C, we can change the flow in G such that the total cost will not increase. It means that either x∗ is not an optimal flow, or it is not the unique solution of (GMNF). Next we define the cost of a directed path in G or G(x): Definition 2.2: Let S = ({v1 , · · · , vk }, {e1, · · · , ek−1 }) be a directed path. Then the cost of this path is defined as ci i Y δ(vj , ej−1 , ej ) · δ(vp , ep−1 , e′1 ) × \| i=2 = ε · cx (C) i=2 × c′1 + i=1 k Y p−1 Y " p−1 X e′ e av1′ avp−1 p e′ av1p \| j=p+1 1 e′m v1′ a {z e′ avm p × e avpp } δ(vp ,ep−1 ,ep ) # δ(vj , ej−1 , ej )  + +T c(C) = # " p−1 i Y X δ(vj , ej−1 , ej )  + ci =  c1 +  j=2 i=2 + p Y δ(vj , ej−1 , ej )× j=2  × cp + k−1 X i=p+1 " ci i Y j=p+1 # δ(vj , ej−1 , ej )  + T c(C) = # " i p−1 Y X δ(vj , ej−1 , ej ) + ci = c1 + B. Computation trees One of the important notions used for proving correctness and/or convergence for BP algorithm is the computation tree # " i k−1 ([11], [13], [14], [15]) (unwrapped tree in some sources). The Y X δ(vj , ej−1 , ej ) + T c(C) = ci + idea under this construction is the following: for the fixed j=2 i=p edge e of the graph G, one might want to build a tree of depth N , such that performing N iterations of BP on graph = l(S) + T c(C) G gives the same estimation of flow on e, as the optimal solution of the appropriately defined (GMNF) problem on the computation tree TeN . Since the proof of our result is based on the compuIII. B ELIEF P ROPAGATION ALGORITHM FOR GMNF tation trees approach, in this subsection we describe the construction in details. We will use the same notations for A. Min-Sum algorithm computation tree, as in [11] (section 5). Algorithm 1 represents the Belief Propagation Min-Sum In this paper we consider the computation trees, correalgorithm for (GMNF) from [11]. In the algorithm the func- sponding to edges of G. We say that e ∈ E is the ”root” tions φe (z) and ψv (z) are the variable and factor functions, for N -level computation tree TeN . Each vertex or edge of respectively, defined for v ∈ V, e ∈ E as follows: TeN is a duplicate of some vertex or edge of G. Define the ′ N N ( mapping ΓN e : V (Te ) → V (G) such that if v ∈ V (Te ) is N ′ ce z e if 0 ≤ ze ≤ ue , a duplicate of v ∈ V (G), then Γe (v ) = v. In other words, φe (z) = this function maps each duplicate from V (TeN ) to its inverse +∞ otherwise. ( P in V (G). e 0 if e∈Ev av ze = fv , The easiest way to describe the construction is inductively. ψv (z) = +∞ otherwise. Let e = (v, w) ∈ E(G). Then the tree Te0 consists of two vertices v ′ , w′ , such that Γ0e (v ′ ) = v, Γ0e (w′ ) = w, and an edge e′ = (v ′ , w′ ). We say that v ′ , w′ belong to 0-level Algorithm 1 BP for (GMNF) of Te0 . Note that for any two vertices v ′ , w′ ∈ V (Te0 ) it 0 0 holds that (v ′ , w′ ) ∈ E(Te0 ) ⇔ (Γ0e (v ′ ), Γ0e (w′ )) ∈ E(G), 1: Initialize t = 0, messages me→v (z) = 0, me→w (z) = so the vertices in a tree are connected with an edge if and 0, ∀z ∈ R for each e = (v, w) ∈ E. only if their inverse in the initial graph are connected. This 2: for t = 1, 2, . . . N do property will hold for all trees TeN . Now assume that we 3: For each e = (v, w) ∈ E update messages as defined a tree TeN , such that for any v ′ , w′ ∈ V (TeN ) it follows: ′ N ′ holds that (v ′ , w′ ) ∈ E(TeN ) ⇔ (ΓN e (v ), Γe (w )) ∈ E(G). N N mte→v (z) = φe (z)+ Denote by L(Te ) the set of leafs of Te (vertices which X are connected by edge with exactly one another vertex). For mt−1 zẽ ) , ∀z ∈ R ψw (~z) + + min e→w (~ \|E \| w any u′ ∈ L(TeN ), denote by P (u′ ) the vertex, with which ~ z ∈R ,~ ze =z ẽ∈Ew \e ′ u is connected by edge (so either (u′ , P (u′ )) ∈ E(TeN ) or (P (u′ ), u′ ) ∈ E(TeN )). We now build TeN +1 by extending mte→w (z) = φe (z)+ the three TeN as follows: for every u′ ∈ L(TeN ) let u = X N ′ N ′ mt−1 (~zẽ ) , ∀z ∈ R Γe (u ), and consider the set Bu′ = Su \ {Γe (P (u ))}, ψv (~z) + + min e→v ~ z ∈R\|Ev \| ,~ ze =z where Su is the set of neighbors of u in G. Then for every ẽ∈Ev \e vertex w ∈ Bu′ add vertex w′ to expand V (TeN ) and an edge 4: t := t + 1 (u′ , w′ ) if (u, w) ∈ E or an edge (w′ , u′ ) if (w, u) ∈ E to +1 5: end for expand E(TeN ), and set ΓN (w′ ) = w. Also set the level e ′ 6: For each e = (v, w) ∈ E, set the belief function as of w to be equal (N + 1). N N N So, the tree TeN +1 contains TeN as an induced subtree, and be = φe (z) + me→v (z) + me→w (z) also contains vertices on level N + 1, which are connected 7: Calculate the belief estimate by finding x̂N ∈ to leafs of T N (in fact, it is easy to see that new vertices e e arg min bN e (z) for each e ∈ E. are connected only with leafs from N th level). From the 8: Return x̂N as an estimation of the optimal solution of construction, one may see that for any v ′ , w′ ∈ V (TeN +1 ) it +1 ′ +1 (GMNF). holds that (v ′ , w′ ) ∈ E(TeN +1 ) ⇔ (ΓN (v ), ΓN (w′ )) ∈ e e E(G). In fact, any vertex of TeN +1 with level less then We address the reader to the article [11] for more details, (N + 1) is a local copy of the corresponding vertex from G. intuition and justifications on the Belief Propagation algo- More precisely: let v ′ ∈ V (TeN +1 ) and the level of v ′ is less +1 ′ rithm for general optimization problems, linear programs, or or equal then N . Denote v = ΓN (v ). Then for any vertex e Min-Cost Network Flow in particular. w ∈ V (G) such that either (v, w) ∈ E(G) or (w, v) ∈ E(G), i=2 j=2 there exist exactly one vertex w′ ∈ V (TeN +1 ) such that +1 ΓN (w′ ) = w and v ′ is connected with w′ in the same e way (direction) as v and w are connected in G. Then it is clear that we can extend the mapping Γ on edges by saying +1 ′ +1 ′ +1 ΓN (e = (v ′ , w′ )) = (ΓN (v ), ΓN (w′ )). Now for e e e ′ N +1 every vertex v ∈ V (Te ) and an incident edge ẽ, we +1 ′ can define the coefficient aẽv′ = aev , where v = ΓN (v ), e N +1 and e = Γe (ẽ). We also set the cost and capacity on the computation tree correspondingly to the initial graph, so +1 +1 cẽ = cΓN (ẽ) and uẽ = uΓN (ẽ) e e Now assume there is a (GMNF) problem stated for a graph G. We define the induced (GMNF)N e problem on a computation tree TeN in the following way. Let V 0 (TeN ) ⊂ V (TeN ) be a set of vertices with levels less than N . Then consider the problem: X minimize cẽ xẽ ẽ∈E(TeN ) subject to X aẽv′ xẽ = fv′ , ∀v ′ ∈ V 0 (TeN ), ẽ∈Ev′ ∀ẽ ∈ E(TeN ). (GMNFN e ) N Roughly speaking, (GMNFe ) is just a simple (GMNF) on a computation tree, except that there are no balance constraints for the vertices of N th level. Keeping in mind that the computation tree is locally equivalent to the initial graph, and that Min-Sum algorithm belongs to messagepassing heuristic, which means that the algorithm works locally at each step, one can intuitively guess that BP for (GMNFN e ) works quite similar as BP for the initial (GMNF). This reasoning can be formalized in the following lemma from [11]. Lemma 3.1: Let x̂N e be the value produced by BP for (GMNF) at the end of iteration N for the flow value on edge e ∈ E. Then there exists an optimal solution y ∗ of ∗ ′ N N (GMNFN e ) such that ye′ = x̂e , where e is the root of Te . Though this lemma was proven only for ordinary MinCost Network Flow problem, where \|aev \| = 1 for all v, e, its proof doesn’t rely on these coefficients at any point, which allows us to extend it for any values of these coefficients. 0 ≤ xẽ ≤ uẽ , C. Main results We will now use lemma 3.1 to prove our main result of correctness of BP Min-Sum for (GMNF). The following theorem is the generalization of Theorem 4.1 from [11], and our proof shares the ideas from the original proof. Let n = \|V (G)\|, and denote by x̂N the estimation of flow after N iterations of Algorithm 1. Theorem 3.2: Suppose (GMNF) has a unique solution x∗ . Define L to be the maximum absolute value of the cost of a simple directed path in G(x∗ ), and T as the minimum of the reducersamong all such paths. Then for any N ≥ L + 1 n, x̂N = x∗ . 2σ(x∗ )T Proof: Suppose to the contrary that there exists e0 = L (vα , vβ ) ∈ E and N ≥ + 1 n such that 2σ(x∗ )T ∗ x̂N e0 6= xe0 . By Lemma 3.1, there exist an optimal solution ∗ ∗ N ∗ ∗ y of GMNFN e0 such that ye0 = x̂e0 , and thus ye0 = xe0 . ∗ ∗ Then, without loss of generality, assume ye0 > xe0 . We will show that it it possible to adjust y ∗ in such way, that the flow in GMNFN e0 will decrease, which will contradict to the optimality of y ∗ . Let e′0 = (vα′ , vβ′ ) be the root edge of the computation tree ∗ N Te0 . Since y ∗ is a feasible solution of GMNFN e0 and x is a feasible solution of GMNF: X X e′ fΓ(vα′ ) = aẽvα′ yẽ∗ = av0α′ ye∗′0 + aẽvα′ yẽ∗ ẽ∈Ev′ ẽ∈Ev′ α \e0 α fΓ(vα′ ) = X e′ ∗ 0 aẽΓ(vα′ ) x∗ẽ = aΓ(v ′ ) xe0 + α ẽ∈EΓ(v′ α) X aẽΓ(vα′ ) x∗ẽ ẽ∈Ev′ α \e0 Since the nodes and the edges in the computation tree TeN0 are copies of nodes and vertexes in G, aẽΓ(v′ ) = aẽvα′ . Then α from above equalities it follows that there exists e′1 6= e′0 ′ e incident to vα′ in TeN0 such that av1α′ (x∗Γ(e′ ) − ye∗1 ′ ) > 0. If e′ 1 av1α′ > 0, then e′1 is an in-arc for vα′ , and we say that e′1 has the same orientation, as e′0 . In such case, x∗Γ(e′ ) > ye∗1 ′ . 1 Otherwise, we say that e′1 has the opposite orientation, and x∗Γ(e′ ) < ye∗1 ′ . Using the similar arguments, we will find 1 e′−1 6= e′0 incident to vβ′ satisfying similar condition. Then we can apply the similar reasoning for the other ends of e′1 , e′−1 , using the balance constraints and inequalities on components of x∗ and y ∗ for corresponding vertexes. In the end, we will have a non-directed path starting and ending in leaves of TeN0 : X = {e′−N , e′−N +1 , . . . , e′−1 , e′0 , e′1 , . . . , e′N } such that for −N ≤ i ≤ N one of two cases holds: ∗ ∗ ′ ′ ′ • ye′ > xΓ(e′ ) . Then ei = (v , w ) has the same orientation as e0 . In this case, define Aug(e′ ) = (v ′ , w′ ) and e′ ∈ D. ∗ ∗ ′ ′ ′ • ye′ < xΓ(e′ ) . Then ei = (v , w ) and e0 have opposite i i ′ orientations. Define Aug(e ) = (w′ , v ′ ) and e′ ∈ O. Note that the capacity constraints are similar for corresponding vertices from GMNFN e0 and GMNF, and since , hence, every e′ ∈ X it holds y ∗ is feasible for GMNFN e0 0 ≤ ye∗′ ≤ ue′ = uΓe′ . Then, for any e′ ∈ D, we have x∗Γ(e′ ) < ye∗′ ≤ ue′ , which means that Γ(e′ ) ∈ G(x∗ ) (from the definition of the residual network). Note that in this case e′ = Aug(e′ ), so Γ(Aug(e′ )) ∈ E(G(x∗ )). Next, let’s now e′ = (v ′ , w′ ) ∈ O, and thus x∗Γ(e′ ) > ye∗′ ≥ 0. Again, out of the definition of the residual network, (Γ(w′ ), Γ(v ′ )) ∈ E(G(x∗ )). But since e′ ∈ O, we have Aug(e′ ) = (w′ , v ′ ), thus Γ(Aug(e′ )) ∈ E(G(x∗ )). Therefore, for every edge e′ ∈ X it holds Γ(Aug(x∗ )) ∈ E(G(x∗ )). From the definition of Aug(e′ ) for e′ ∈ X one can see that all Aug(e′ ) have the same direction as e0 . Therefore, W = {Aug(e′−N ), Aug(e′1−N ), . . . , Aug(e′−1 ), Aug(e′0 ), Aug(e′1 ), . . . , Aug(e′N −1 ), Aug(e′N )} is the directed path in TeN0 , and we will call it augmenting path of y ∗ with respect to x∗ . Γ(W ) is also a directed walk on G(x∗ ), which can be decomposed into a simple directed path P and a collection of k simple directed cycles C1 , C2 , . . . , Ck . Since each simple cycle or path has at most n edges and W has 2N + 1 edges, it holds l(W ) ≤ n + kn. On the other Ln + n + 1. Then we obtain hand, l(W ) = 2N + 1 = σ(x∗ )T L . Further we would denote by c∗ (·) and l∗ (·) k > σ(x∗ )T costs of cycles and paths in the residual network G(x∗ ). For each cycle Ci we have c∗ (Ci ) ≥ σ(x∗ ) > 0 by Lemma (2.1), while the cost of P is at least −L. Then using Lemma (2.2) we have: L l∗ (W ) ≥ l∗ (P ) + kT σ(x∗ ) > −L + T σ(x∗ ) = 0 σ(x∗ )T We will now ”extract” flow from W in TeN0 . Let’s redefine the numeration of arcs in W for convenience: W = {w1 , w2 , . . . , w2N +1 }. This edges correspond to X = ′ ′ {w1′ , w2′ , . . . , w2N +1 }, where wi and wi have the same ′ orientation if wi ∈ D, and have the opposite orientation if wi′ ∈ O. Hence extraction of flow from wi means decreasing ∗ ∗ if wi′ ∈ O. if wi′ ∈ D, and increasing yw yw i i Then similarly to the proof of Lemma 2.1, to keep the balance in all the vertexes of W (except the start and the end, since there are no balance constraints for them in GMNFN e0 ), we have to adjust the flow in the following way: ( ∗ yw if wi′ ∈ D ′ − λ, 1 ỹw1′ = (2) ∗ yw if wi′ ∈ O ′ + λ, 1 ( Qi ∗ ′ ′ yw if wi′ ∈ D ′ − λ j=2 δ(vj , wi−1 , wi ), i ỹwi′ = (3) Qi ∗ ′ ′ yw′ + λ j=2 δ(vj , wi−1 , wi ), if wi′ ∈ O i for i = 2, 3, . . . , 2N + 1. Obviously, there exist small enough λ > 0 such that ỹe > x∗Γ(e) ≥ 0 ∀e ∈ D, and ỹe < x∗Γ(e) ≤ uΓ(e) ∀e ∈ O, so the capacity constraints are satisfied for ỹ. (The balance constraints are satisfied by the construction). So, ỹ is a feasible solution of GMNFN e0 . Now we explore how the total cost changes after such transformation of the flow: X cΓ(e′ ) ye∗′ − e′ ∈E(TeN0 ) = X cΓ(e′ ) ỹe′ = e′ ∈E(TeN0 ) X cΓ(e′ ) (ye∗′ − ỹe′ ) = e′ ∈E(TeN0 ) = X cΓ(e′ ) λe′ − e′ ∈D = X e′ ∈D = 2N +1 X i=1 c∗Γ(w ) λ i cΓ(e′ ) λe′ = e′ ∈O c∗Γ(e′ ) λe′ +  X X e′ ∈O i Y j=2 c∗Γ(e′ ) λe′ = X c∗Γ(e′ ) λe′ = e′ ∈W  δ(vj , wj−1 , wj ) = l(W )λ > 0 Here we used that c∗Γ(e′ ) = cΓ(e′ ) for e′ ∈ D and c∗Γ(e′ ) = −cΓ(e′ ) for e′ ∈ O, that is obvious from the construction of W, D, and O. So we found the feasible solution of GMNFN e0 with the total cost less then of y ∗ . It means that y ∗ is not an optimal solution of GMNFN e0 , which leads us to the contradiction. IV. CONCLUSIONS The proved correctness of the Belief Propagation algorithms for General Min-Cost Network Flow problems may serve as justification of applying these algorithms in practice for real problems. The statement of GMNF problem is broad enough, and thus many practical problems may fall under this formulation, which means that these problems may be solved correctly using BP algorithms. The future research is required to determine the speed of convergence of BP for this Generalized Min-Cost Network Flow, since in our paper we have only proved that after the finite iterations of the algorithms the answer will not change and be the correct one. However, since σ(x∗ ) in 3.2 may be arbitrary small for some problems, the number of iterations until the algorithm will give the correct answer may be arbitrary big, and further analysis is needed to reasonably bound the number of steps that should be done. R EFERENCES [1] M. J. Wainwright, M. I. Jordan et al., “Graphical models, exponential families, and variational inference,” Foundations and Trends R in Machine Learning, vol. 1, no. 1–2, pp. 1–305, 2008. [2] K. P. Murphy, Y. Weiss, and M. I. 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Neural Distributed Autoassociative Memories: A Survey V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov Abstract Introduction. Neural network models of autoassociative, distributed memory allow storage and retrieval of many items (vectors) where the number of stored items can exceed the vector dimension (the number of neurons in the network). This opens the possibility of a sublinear time search (in the number of stored items) for approximate nearest neighbors among vectors of high dimension. The purpose of this paper is to review models of autoassociative, distributed memory that can be naturally implemented by neural networks (mainly with local learning rules and iterative dynamics based on information locally available to neurons). Scope. The survey is focused mainly on the networks of Hopfield, Willshaw and Potts, that have connections between pairs of neurons and operate on sparse binary vectors. We discuss not only autoassociative memory, but also the generalization properties of these networks. We also consider neural networks with higher-order connections and networks with a bipartite graph structure for non-binary data with linear constraints. Conclusions. In conclusion we discuss the relations to similarity search, advantages and drawbacks of these techniques, and topics for further research. An interesting and still not completely resolved question is whether neural autoassociative memories can search for approximate nearest neighbors faster than other index structures for similarity search, in particular for the case of very high dimensional vectors. Keywords: distributed associative memory, sparse binary vector, Hopfield network, Willshaw memory, Potts model, nearest neighbor, similarity search. DOI: https://doi.org/10.15407/kvt188.02.005 This front page has been added to assist automated indexing. The paper as published begins on the next page. Please cite as: V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov. Neural distributed autoassociative memories: A survey. Cybernetics and Computer Engineering. 2017. N 2 (188). P. 5–35. Информатика и информационные технологии DOI: https://doi.org/10.15407/kvt188.02.005 УДК 004.22 + 004.93'11 V.I. GRITSENKO1, Corresponding Member of NAS of Ukraine, Director, e-mail: vig@irtc.org.ua D.A. RACHKOVSKIJ 1, Dr (Engineering), Leading Researcher, Dept. of Neural Information Processing Technologies, e-mail: dar@infrm.kiev.ua A.A. FROLOV2, Dr (Biology), Professor, Faculty of Electrical Engineering and Computer Science FEI, e-mail: docfact@gmail.com R. GAYLER3, PhD (Psychology), Independent Researcher, e-mail: r.gayler@gmail.com D. KLEYKO4 graduate student, Department of Computer Science, Electrical and Space Engineering, e-mail: denis.kleyko@ltu.se E. OSIPOV4 PhD (Informatics), Professor, Department of Computer Science, Electrical and Space Engineering, e-mail: evgeny.osipov@ltu.se 1 International Research and Training Center for Information Technologies and Systems of the NAS of Ukraine and of Ministry of Education and Science of Ukraine, ave. Acad. Glushkova, 40, Kiev, 03680, Ukraine 2 Technical University of Ostrava, 17 listopadu 15, 708 33 Ostrava-Poruba, Czech Republic 3 Melbourne, VIC, Australia 4 Lulea University of Technology, 971 87 Lulea, Sweden NEURAL DISTRIBUTED AUTOASSOCIATIVE MEMORIES: A SURVEY Introduction. Neural network models of autoassociative, distributed memory allow storage and retrieval of many items (vectors) where the number of stored items can exceed the vector dimension (the number of neurons in the network). This opens the possibility of a sublinear time search (in the number of stored items) for approximate nearest neighbors among vectors of high dimension. The purpose of this paper is to review models of autoassociative, distributed memory that can be naturally implemented by neural networks (mainly with local learning rules and iterative dynamics based on information locally available to neurons). Scope. The survey is focused mainly on the networks of Hopfield, Willshaw and Potts, that have connections between pairs of neurons and operate on sparse binary vectors. We ã V.I. GRITSENKO, D.A. RACHKOVSKIJ, A.A. FROLOV, R. GAYLER, D. KLEYKO, E. OSIPOV, 2017 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 5 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov discuss not only autoassociative memory, but also the generalization properties of these networks. We also consider neural networks with higher-order connections and networks with a bipartite graph structure for non-binary data with linear constraints. Conclusions. In conclusion we discuss the relations to similarity search, advantages and drawbacks of these techniques, and topics for further research. An interesting and still not completely resolved question is whether neural autoassociative memories can search for approximate nearest neighbors faster than other index structures for similarity search, in particular for the case of very high dimensional vectors. Keywords: distributed associative memory, sparse binary vector, Hopfield network, Willshaw memory, Potts model, nearest neighbor, similarity search. INTRODUCTION In this paper, we review some artificial neural network variants of distributed autoassociative memories (denoted by Neural Associative Memory, NAM) [1– 159]. Varieties of associative memory [93] (or content addressable memory) can be considered as index structures performing some types of similarity search. In autoassociative memory, the output is the word of memory, most similar to the key at the input. We restrict our initial attention to systems where the key and memory words are binary vectors. Therefore, autoassociative memory answers nearest neighbor queries for binary vectors. In distributed memory, different vectors (items to be stored) are stored in shared memory cells. That is, each item to be stored consists of a pattern of activation across (potentially) all the memory cells of the system and each memory cell of the system contributes to the storage and recall of many (potentially all) stored items. Some of types of distributed memory have attractive properties of parallelism, resistance to noise and malfunctions, etc. However, exactly correct answers to the nearest neighbor queries from such memories are not guaranteed, especially when too many vectors are stored in the memory. Neurobiologically plausible variants of distributed memory can be represented as artificial neural networks. These typically perform one-shot memorization of vectors by a local learning rule modifying connection weights and retrieve a memory vector in response to a query vector by an iterative procedure of activity propagation between neurons via their connections. In the first Section, we briefly introduce Hebb's theory of brain functioning based on cell assemblies because it has influenced many models of NAM. Then we introduce a generic scheme of NAMs and their characteristics (discussed in more details in the other sections). The following three Sections discuss the widespread matrix-type NAMs (where each pair of neurons is connected by two symmetric connections) of Hopfield, Willshaw, and Potts that work best with sparse binary vectors. The next Section is devoted to the function of generalization, which differs from the function of autoassociative memory and emerges in some NAMs. The following Section discusses NAMs with higherorder connections (more than two neurons have a connection) and NAMs without connections. Then some recent NAMs with a bipartite graph structure are considered. The last Section provides discussion and concludes the paper. 6 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey CELL ASSEMBLIES AND GENERIC NAM Hebb's paradigm of cell assemblies. According to Hebb [65], nerve cells of the brain are densely interconnected by excitatory connections, forming a neural network. Each neuron determines its membrane potential as the sum of other active neurons' outputs weighted by connection weights. A neuron becomes active if this potential (the input sum) exceeds the threshold value. During network functioning, connection weights between simultaneously active neurons (encoding various items) are increased (the Hebbian learning rule). This results in organization of neurons into cell assemblies — groups of nerve cells most often active together and consequently mutually excited by connection weights between neurons in the assembly. At the same time, the process of increased connection within assemblies leads to mutual segregation of assemblies. When a sufficient part of a cell assembly is activated, the assembly becomes active as a whole because of the strong excitatory connection weights between the cells within the assembly. Cell assemblies may be regarded as memorized representations of items encoded by the distributed patterns of active neurons. The process of assembly activation by a fragment of the memorized item may be interpreted as the process of pattern completion or the associative retrieval of similar stored information when provided with a partial or distorted version of the memorized item. Hebb's theory of brain functioning — interpretation of various mental phenomena in terms of cell assemblies — has turned out to be one of the most profound and generative approaches to brain modeling and has influenced the work of many researchers in the fields of artificial intelligence, cognitive psychology, modeling of neural structures, and neurophysiology (see also reviews in [39, 40, 54, 75, 98, 104, 120, 121, 134]). A generic scheme and characteristics of NAMs. Let us introduce a generic model of the NAM type, inspired by Hebb's paradigm, that will be elaborated in the sections below devoted to specific NAMs. We mainly consider NAMs of the distributed and matrix-type, which are fully connected networks of binary neurons (but see Sections "NAMs with Higher-Order Connections and without Connections", "NAMs with a Bipartite Graph Structure for Nonbinary Data with Constraints" for other NAM types). Each of the neurons (their number is D ) represents a component of the binary vector z . That is, each of the D neurons can be in the state 0 or 1. Each pair of neurons has two mutual connections (one in each direction). The elements of the connection matrix W( D ´ D) represent the weights of all these connections. In the learning mode, the vectors y from the training or memory set (which we call the "base") are "stored" (encoded or memorized) in the matrix W by using some learning rule that changes the values of wij (initially each wij is usually zero). In the retrieval mode, an input binary vector x (probe or key or query vector) is fed to the network by activation of its neurons: z = x . The input sum of the i -th neuron si = å j =1, D wij z j ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 7 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov is calculated. The neuron state is determined as zi (t + 1) = 1 (active) for si (t ) ³ Ti (t ) and zi (t + 1) = 0 (inactive) for si (t ) < Ti (t ) ; Ti is the value of the neuron threshold. For parallel (synchronous) network dynamics, the input sums and the states of all D neurons are calculated (updated) at each step t of iterative retrieval. For sequential (asynchronous) dynamics, zi is calculated for one neuron i , selected randomly. For simplicity, let us consider random selection without replacement, and one step of the asynchronous dynamics to consist of update of the states of all D neurons. The parameters W and T are set so that after a single, or several, steps of dynamics the state of the network (neurons) reaches a stable state (typically, the state vector does not change with t , but cyclic state changes are also considered as "stable"). At the stable state, z is the output of the network. The query vector x is usually a modified version of one of the stored vectors y . In the literature, this might be referred to as a noisy, corrupted, or distorted version of a vector. While the number of stored vectors is not too high, the output z is the stored y closest to x (in terms of dot product simdot (x, y ) º á x, y ñ ). That is, z is the base vector y with the maximum value of á x, y ñ . In this case, NAM returns the (exact) nearest neighbor in terms of simdot . For binary vectors with the same number of unit (i.e. with value equal to 1) components, this is equivalent to the nearest neighbor by the Hamming distance ( dist Ham ). The time complexity (runtime) of one step of the network dynamics is O( D 2 ) . Thus, if a NAM can be constructed that stores a base of N > D vectors so that they can be successfully retrieved from their distorted versions, then the retrieval time via the NAM could be less than the O( DN ) time required for linear search (i.e. the sequential comparison of all base vectors y to x ). Since the memory complexity of this NAM type is O( D 2 ) , as D increases, one can expect an increasing in the size N of the bases that could be stored and retrieved by NAM. Unfortunately, the vector at the NAM output may not be the nearest neighbor of the query vector, and possibly not even a vector of the base. (Note that if one was not concerned with biological plausibility, one can quickly check whether the output vector is in the base set by using a hash table to store all base vectors.) In some NAMs, it is only possible to store many fewer vectors N than D , with high probability of accurate retrieval, especially if the query vectors are quite dissimilar to the base vectors. For NAM analysis, base vectors are typically selected randomly independently from some distribution of binary vectors (e.g., vectors with the probability p of 1-components equal to 1/2, or vector with pD 1-components, 8 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey for some p from interval (0,1)). The assumption of independence simplifies analytical approaches, but is likely unrealistic for real applications of NAMs. The query vectors are typically generated by as modifications of the base vectors. Distortion by deletion randomly changes some of the 1s to 0s (the remaining components are guaranteed to agree). A more complex distortion by noise randomly changes some 1s to 0s, and some 0s to 1s while (exactly or approximately) preserving the total number of 1s. For a random binary vector of dimension D with the probability p of a component to be 1, the Shannon entropy H = Dh( p) , Where h( p) = - p log p - (1 - p)log(1 - p) . For D >> 1 , a random vector with pD of 1s has approximately the same entropy. The entropy of N vectors is NDh( p) . When N vectors are stored in NAM, the entropy per connection is [40, 41]. a = NDh( p) / D 2 = Nh( p) / D . Knowing h( p) , it is easy to determine N for a given a . When too many vectors are stored, NAM becomes overloaded and the probability of accurate retrieval drops (even to 0). The value of a for which a NAM still works reliably depends on the mode of its use (in addition to the NAM design and distributions of base and query vectors). The mode where the undistorted stored base vectors are still stable NAM states (or stable states differ by few components from the intended base vectors), has the largest value of a . (We denote the largest value of a for this mode as "critical", a crit , and the corresponding N as N crit .) For a > a crit the stored vectors become unstable. Note that checking if an input vector is stable does not allow one to extract information from the NAM, since vectors not stored can also be stable. The information (in the Shannon information-theoretic sense) that can be extracted from a NAM is determined by the information efficiency (per connection) E . This quantity is bounded above by some specific a (the entropy per connection that still permits information extraction), which in its turn is bounded above by a crit . A NAM may work in recognition or correction mode. In recognition mode, the NAM distinguishes whether the input (query) vector is from the base or not, yielding extracted information quantified by Erecog . When NAM answers the nearest neighbor queries (correction mode), information quantified by Ecorr is extracted from the NAM by correction (completion) of the distorted query vectors. The more distorted the base vector used as the query at the NAM input, the more information Ecorr is extracted from the NAM (provided that the intended base vector is sufficiently accurately retrieved). However, more distorted input vectors lower the value of a corr at which the NAM is still able to retrieve the correct base vectors, and so lowers Ecorr (which is constrained to be less than a corr ). We refer to these informationISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 9 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov theoretic properties of NAMs as their information characteristics. Let us now consider specific variants of the generic NAM. Hereafter we use the terms "NAM" and "(neural) network" interchangeably. HOPFIELD NAMs Hopfield networks with dense vectors. In the Hopfield NAM, "dense" random binary vectors (with the components from {0,1} with the probability p = 1 / 2 of 1) are used [68]. The learning procedure forms a symmetric matrix W of connection weights with positive and negative elements. The connection matrix is constructed by successively storing each of the base vectors y according to the Hopfield learning rule: wij = wij + ( yi - q)( y j - q) with parameter q = p = 1 / 2 ; wii = 0 . (For brevity, we use the same name for generalization of this rule with q < 1 / 2 , though Hopfield did not propose it, Subsection "Hopfield networks with sparse vectors"). The dynamics in [68] is sequential (in many subsequent studies and implementations it is parallel) with the threshold T = 0 . It was shown [68] that each neuron state update decreases the energy function -(1 / 2) å i , j =1, D zi wij z j , so that a (local) minimum of energy is eventually reached and such a network comes into a stable activation state. As D ® ¥ , various methods of analysis and approximation of experimental (modeling) data obtain a crit » 0.14 [68, 8, 5, 71, 35] which gives N crit » 0.14 D since h(1 / 2) = 1 . Note that similar values of a crit are achieved at rather small finite D . For rigorous proofs of (smaller) a crit see refs in [107]. max for distortion by noise, 0.092 was obtained by the method of As for Ecorr approximate dynamical equations of the mean field [71], and 0.095 by approximating the experiments to D ® ¥ [35]. By the coding theory methods in [112] it was shown that asymptotically (as D ® ¥ ) it is possible to retrieve (with probability approaching 1) exact base vectors with query vectors distorted by noise (so that their dist Ham < D / 2 from the base vectors), for N = D / (2ln D) stored base vectors if non-retrieval of some is permitted. If one requires the exact retrieval of all stored base vectors, the maximum number of vectors which can be stored decreases to N = D / (4ln D) . These values of N were shown to be the lower and upper bounds in [25, 20]. Note that in [47] a crit = 2 was obtained for "optimal" W (obtained by a non-Hebbian learning rule); a pseudoinverse rule (e.g. [125, 140]) gives a crit = 1 . For correlated base vectors, the storage capacity N crit depends on the structure of the correlation. When the base vectors are generated by a onedimensional Markov chain [107], N crit is somewhat higher than it is for 10 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey independent vectors. This and other correlation models were considered in [108]. Hopfield networks with sparse vectors. Hopfield NAMs operating with sparse vectors p < 1 / 2 appeared to have better information characteristics [154] (see also Sections "Willshaw NAMs", "Potts NAMs") than those operating with dense vectors ( p = 1 / 2 ). For example, they attain values N > D . In the usual Hopfield NAM and learning rule (with q = p = 1 / 2 and threshold T = 0 ) the number of active neurons is kept near D / 2 by the balance of negative and positive connections in W . Using the Hopfield rule with q = p < 1 / 2 one can not set T = 0 . This is especially evident for the Hebb learning rule (which we obtain from the Hopfield rule by setting q = 0 ). All connections become non-negative, and T = 0 eventually activates all neurons. Similar behavior is demonstrated by the Hopfield NAM and learning rule with q = p < 1 / 2 . The problem of network activity control (i.e. maintaining some average activity level chosen by the designer) can be solved by applying an appropriate uniform activation bias to all neurons [9, 21]. This is achieved by setting an appropriate positive value of the time-varying threshold T (t ) [21] to ensure, for example, pD < D / 2 active neurons (to match pD in the stored vectors) for parallel dynamics. Note that the Hopfield rule with q = p < 1 / 2 provides better information characteristics than the pure Hebb rule with q = 0 [41, 35]. However, the Hebb rule requires modification of only ( pD)2 connections per vector, whereas the Hopfield rule modifies all connections per vector. As D ® ¥ and p ® 0 ( pD >> 1 , and often p ~ ln D / D ) the theoretical analysis (e.g., [154, 41, 34] and others) gives a crit = (log e) / 2 = 1 / (2ln 2) » 0.72 for the Hopfield rule with q = p , the Hebb rule, and the "optimal" W [47]. In [34] they use a scaled sparseness parameter e = (ln p) -1/2 to investigate max max . For e << 1 they obtained a crit » 0.72 . However convergence of a crit to a crit for e = 0.1 (corresponding to p = 10-8 and to D > 109 ), a crit = 0.43 only. max In [122] it was shown that Erecog = 1 / (4ln 2) » 0.36 (by the impractical exhaustive enumeration procedure of checking that all vectors of the base are stable and all other vectors with the same number of 1-components are not stable). This empirical estimate coincides with the estimate [41]. For retrieval by a single step of dynamics, max Ecorr = 1 / (8ln 2) » 0.18 for distortion by deletion of half the 1-components of a base vector [40, 41, 119, 146]. Let us note again, all these results are obtained for D ® ¥ and p ® 0 . For these conditions, multi-step retrieval (the usual mode of NAM retrieval as explained in Subsection "A generic scheme and characteristics of NAMs") is not ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 11 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov required since NAM reaches a stable state after a single step. In terms of N , since h( p) ® 0 for p ® 0 , it follows that N >> D , that is N crit = a crit D / h ( p ) ® ¥ much faster than D ® ¥ . The same is valid for N corresponding to E . In the experiments [146], for the Hebb rule and multi-step retrieval Ecorr values up to 0.09 were obtained. Detailed studies of the information characteristics of the finite- and infinite-dimensional networks with the Hopfield rule, can be found in [34, 35]. Different degrees of distortion by noise for vectors with pD unit components were used. The dynamic threshold ensured pD active neurons at each step of the parallel dynamics. It was shown [35] that with this choice of threshold the stable states are static (some vector) or cycles of length 2 (two alternating vectors on adjacent steps of the dynamics). (This is the same behavior as for the fixed static threshold and is valid for all networks with symmetric connections.) It has been demonstrated experimentally [35] that even if after the first step of dynamics simdot (y , z ) < simdot (y , x ) (where y is the correct base vector, z is the network state, and x is the distorted input), the correct base vector can sometimes be retrieved by the subsequent steps of the dynamics. Conversely, increasing simdot (y , z ) at the first step of the dynamics does not guarantee correct retrieval [5]. These results apply to both the dense and sparse vector cases. The study [35] used analytical methods developed for the dense Hopfield network and adapted for sparse vectors, including the statistical neurodynamics (SN) [5, 34], the replica method (RM) [8], and the single step method (SS) [80]. All these analytical methods rather poorly predicted the behavior of finite networks for highly sparse vectors, at least for the parallel dynamics studied. (Note that all these methods (SN, RM, SS) provide accurate results for D ® ¥ and p ® 0 , where retrieval by a single step of dynamics is enough.) Empirical experimentation avoids the shortcomings of these analytical methods by directly simulating the behavior of the networks. These simulations allow a corr and information efficiency, Ecorr , to be estimated as a functions of p, D and the level of noise in the input vectors. The value of Ecorr monotonically increases as D increases for a constant p . For p = 0.001 – 0.01, which corresponds to the activity level of neurons in the brain, the maximum value of Ecorr » 0.205 was obtained by approximating experimental results to the case D ® ¥ [35] (higher max than Ecorr » 0.18 for p ® 0 ). In [38] the time of retrieval in the Hopfield network was investigated (using the number of retrieval steps observed in simulation experiments; this number somewhat increases with D ). They conclude that for random vectors with small p , large D , and large N , Hopfield networks may be faster than some other algorithms (e.g., the inverted index) for approximate and exact nearest neighbor 12 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey querying. An increase in the number N crit of stable states corresponding to the stored vectors proportional to ( D / ln D) 2 for p ~ ln D / D is shown asymptotically in [4] (although non-rigorously, see also [34]). Note, this result also follows from N = a D / h( p) by approximating h( p) » - p log p for small enough p ( - ln p >> 1 ). In [67] they give a rigorous analysis of a Hopfield network variant (neurons are divided into parts, see Section "Potts NAMs"), with the Hebb learning rule and p slightly less than ln D / D , for retrieval by a single step of parallel dynamics with fixed T . The lower and upper bounds of N were obtained for which the memory vectors are stable states (with probability approaching 1 as D ® ¥ ), and can also be exactly retrieved from query vectors distorted by noise. The lower and upper bounds of N found in [67] are of the same order as those found in [4]. For this mode of network operation, if we approximate the number of retrieval steps as ln D , we may estimate speed-up as D / (ln D)3 relative to linear search (see Subsection "A generic scheme and characteristics of NAMs"). For both dense and sparse vectors, the NAM capacity N max grows with increasing D . Also, in order to maintain an adequate information content for a sparse vector ( Dh( p) for h( p) << 1 ), it is necessary to have a sufficiently high D . The number of connections grows as D squared (because Hopfield networks are fully connected), which is unattainable even on modern computers at D of millions. Besides, the neurobiologically plausible number of connections per neuron is on the order of 10,000. Therefore, the development of "diluted" networks that perform NAM functions without being fully connected is attractive, e.g. [105, 150, 151, 41, 142]. This partial connectivity can be used to reduce the memory complexity of NAM from quadratic to linear in D [98, 99]. WILLSHAW NAMs Willshaw networks with sparse vectors. NAMs with binary connections from {0,1} are promising since they require only one bit per connection. Such networks were proposed both in heteroassociative [157] and autoassociative versions (e.g. [118, 156, 16, 152, 49, 115, 119, 122, 24, 48, 41, 42, 43, 44, 81]). The learning rule (let's call it the Willshaw rule) becomes: wij = wij Ú ( yi Ù y j ) , where Ú is disjunction, Ù is conjunction. Various strategies for threshold setting can be used, e.g., setting threshold T to ensure pD active neurons, as in Subsection "Hopfield networks with sparse vectors". Note that this NAM can not work with dense vectors, since storing only a small number of dense vectors will set almost all the connection weights to 1. Moreover, for the same reason, the Willshaw networks (unlike the Hopfield networks) cease to work at any constant p and a as D ® ¥ . The number N of random binary vectors able to be stored and retrieved in the Willshaw NAM ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 13 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov grows with decreasing vector density. Even for not very large networks and not very sparse vectors ( p ~ D / D ), N can exceed D (e.g. D = 4096 allowed storage and retrieval of up to N = 7000 vectors distorted by noise in the experiments of [16]). The particular N values reached (for D and p fixed) vary depending on the degree of the query vector distortion and on the desired probability of retrieval. The maximum theoretical a crit = ln 2 » 0.69 is reached as D ® ¥ for p ~ ln D / D [40, 119, 41]. In [122] they obtained max Erecog = ln 2 / 2 » 0.346 (using a computationally expensive exhaustive enumeration procedure). In [42, max 43] the same a crit and Erecog were obtained analytically for the sparseness parameter b = log(1 / p) / ( pD) equal to 1, i.е. for pD somewhat less than log D . (The probability of a connection to be modified after storing N vectors is 1 - (1 - p 2 ) N » 1 - exp( - Np 2 ) = 1 - exp( -a Dp 2 / h( p)) = 1 - exp( -a p log(1 / p) / ( b h( p))) » 1 - exp( -a / b ) <1 (we used p ® 0 ); thus the network can be analyzed for fixed a and b at D ® ¥ .) The same upper bound of E is given as the maximum entropy of W learnt by the Willshaw rule. In [119] the efficiency max Ecorr = ln 2 / 4 » 0.173 was theoretically shown for single-step (as well as multi-step) retrieval and distortion by deletion. For multi-step retrieval in finite Willshaw NAMs (with distortion by deletion) Ecorr up to 0.19 (at D = 20000) was obtained experimentally in [146]. Experiments in [146, 44] show that in the Willshaw NAM (unlike the Hopfield NAM), the values a corr and Ecorr for not too large D are higher than for D ® ¥ (see also [42]). Note that the quality of retrieval in the Willshaw NAM is higher than in the Hopfield NAM; the retrieved vectors more often coincide exactly with the stored vectors of the base. From the detailed analytical and experimental study of the values of a corr in [44] (at various levels of sparsely, parameterized as b , degrees of distortion by noise, and D up to 100000), it was found that Ecorr » 0.13 per connection can be reached in the experiments (for small networks, N = 640, pD = 20, b = 0.25 ). It was also shown that the results of the analytical methods SS [80] and GR [48] are far from the experimental results (in most cases, worse than them). Due to the connections being binary, the efficiency per bit of connection implemented in computer memory is higher than that for the Hopfield network (where Ecorr » 0.205 per connection [41]). A review of NAM studies in [81] concludes that for Willshaw networks having connection matrix W with probability of a nonzero element close to 0 or 1, compression of W improves information characteristics compared with the 14 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey usual uncompressed Willshaw NAM. Such compressed W are obtained when the base vectors have the number of 1s sublogarithmic or superlogarithmic in D . Their comparison of the retrieval time in compressed Willshaw networks and the inverted index has shown the advantage of the inverted index for most parameters. An analytical and experimental comparison of the Willshaw, the GB (Subsection "Willshaw-Potts network"), and the Hopfield networks (with the Hebb rule [4]) for vectors with p of the order of ln D / D and distortion by deletion was carried out in [59]. They investigate single-step retrieval theoretically (asymptotically, for D ® ¥ with probability approaching 1). For all models, the lower bound of N of the order ( D / ln D) 2 is obtained, and for the Willshaw network the matching upper bound is shown. In experiments, the results are worse for a fixed threshold than for a variable threshold. The Hopfield network performed worse, in terms of empirical probability of retrieval versus N , compared with the other NAMs, probably because of the non-optimal Hebb learning rule and non-optimal threshold selection. For the diluted Willshaw networks [24, 6, 41, 99] the optimal pD (providing approximately the same capacity N ) is higher than for the fully connected networks. Willshaw networks in the index structures for nearest neighbor search. In [159] the base of binary sparse vectors is divided into disjoint sets (of the same cardinality) and each is stored in a Willshaw NAM with its own W . When the query vector x is input, simdot (x, Wx) is calculated for the matrices W of all sets, and the vectors of sets with the maximum similarity are used as the nearest neighbor candidates (verified by linear search). Analysis and experiments for bases of random vectors with small random distortions of query vectors showed that up to a certain number of vectors in each network the nearest neighbor is found with a high probability (in experiments, without error) and faster than by linear search only. If this number of vectors per network is exceeded, both the probability of finding an incorrect nearest neighbor and its distance to the correct vector increase. A somewhat lower speedup relative to linear search is shown for real, nonrandom data, versus synthetic, random data. In [60] similar results were obtained analytically (asymptotically for D ® ¥ and error probability approaching zero) and experimentally for bases of sparse and dense random vectors (for the Hopfield rule). POTTS NAMs Potts networks. The NAM from [77] can be considered as a network of neurons that are divided into non-overlapping parts ("columns"), with d neurons in each column and only one active neuron in the state z = d - 1 , for the remaining column neurons z = -1 . That is, the sum of activations over all the neurons is zero in each column. The Hebb rule is used for learning. For the more convenient version of this model with the neurons having the states from {0,1} and single active neuron per column, the Hopfield rule is used. The connection matrix W for the entire network is constructed so that wij = 0 for neurons i and j in the same column (this implies that wii = 0 ). That is, the ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 15 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov network is structured as a multipartite graph. Network dynamics (parallel or sequential) activates the one neuron in each column with the maximum input sum s (one of these neurons is randomly activated for neurons with equal s , but see the GB network below). For the number of columns D , the value of N crit / D of the Potts network was estimated by [77] to be d (d - 1) / 2 times more than 0.14 (i.e., a crit for the Hopfield network with p = 1 / 2 ). However, to approximate the number of connections in the Hopfield network, the Potts network must have D / d columns. Note also that one "Potts vector" contains only ( D / d )log d bits of information [79]. The Potts network with parallel and sequential dynamics, and with singlestep and multi-step retrieval was analytically explored in [109]. For exact retrieval (asymptotically, as D ® ¥ , with probability approaching 1), the upper and lower bounds of N were estimated both for the mode of querying with stable stored vectors and for the correction mode querying with distorted query vectors. In both cases, N = cD / ln D , where the constant c increases quadratically in d , but with different c depending on the degree of distortion and the desired probability of state to be stable or vector to be retrieved. Willshaw-Potts network. For binary connections with the Willshaw learning rule, the Potts network becomes the Willshaw-Potts network [79]. When a vector is stored, a clique (a complete subgraph) is created in the connection graph. As for the Hopfield network, only static stable states or cycles of length 2 were experimentally observed for parallel dynamics. According to [79], the information characteristics of this network are close to the Willshaw network at the same vector sparsity. Since the information content of Willshaw-Potts vectors is low, the N crit is higher than for the Willshaw network. This network was rediscovered as the GB network in [58] with various modifications [3, 59] and hardware implementations (for example, [117] with non-binary connections). The GB network is oriented for exact retrieval of vectors with distortion by deletion (columns without values activate all neurons). The peculiarities of the GB network include: connections of neurons with themselves; the possibility for several neurons in a column (with the maximum input sum) to be in the active state; the contribution from each column to the input sum of a neuron is not more than 1; various options for threshold management; the possibility of working with vectors having all zero components in some columns, etc. A theoretical GB analysis for single-step retrieval, as well as an experimental comparison with the Hopfield and the Willshaw networks for multi-step retrieval is given in [59], see also Subsection "Willshaw networks with sparse vectors". Processing of realistic data. To represent arbitrary binary vectors in the Potts network, they are divided into segments of dimension log d and each segment is encoded by the activation of one neuron of its d -dimensional column [97]. Components of integer vectors can be represented in a similar way. Simulated data are typically generated as independent random samples. This 16 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey ensures that the vectors to be stored are very nearly equally and maximally dissimilar. However, data generated from situations in the world is very unlikely to be so neatly distributed. When working with (unevenly distributed) real data, NAM is used non-optimally (many connections are not modified, others are "oversaturated"). To overcome this in the GB network, a free column neuron is allocated when the number of connections of a neuron (encoding some value) exceeds the threshold [19]. During retrieval, all column neurons that encode a certain value are activated. For better balancing the number of connections, in [64] the number of neurons in the column allocated to represent a vector component is proportional to the frequency of its 1-value in the vectors of the base. During storage, the various neurons representing the component are activated in turn. The authors of [64] also propose an algorithm for finding (with a high probability) all vectors of the base closest to the query vector distorted by deletion; the algorithm often significantly reduces the number of required queries. GENERALIZATION IN NAMs The Hebbian learning in matrix-type distributed memories naturally builds a kind of correlation matrix where the frequencies of joint occurrence of active neurons are accumulated in the updated connection weights. The neural assemblies thus formed in the network may have a complex internal structure reflecting the similarity structure of stored data. This structure can be revealed as stable states of the network — in the general case, different from the stored data vectors. That is, it is possible for vectors retrieved from a network to not be identical to any of the vectors stored in the network (which is generally undesirable). Similarity preserving binary vectors. Similarity of patterns of active neurons (represented as binary vectors) are assumed to reflect the similarity of items (of various complexity and generality) they encode. The similarity value is measured in terms of the number or fraction of common active neurons (or overlap, i.e. normalized dot product of the representing binary vectors). Moreover, the similarity "content" is available as the identities of common active neurons (the IDs of the common 1-components of the representing binary vectors). Note that such data representation schemes by similarity preserving binary vectors have been developed for objects represented by various data types, mainly for (feature) vectors (see survey in [131]), but also for structured data types such as sequences [102, 72, 85,86] and graphs [127, 128, 148, 136, 62, 134]. A significant part of this research is developed in the framework of distributed representations [45, 76, 106, 126, 89], including binary sparse distributed representations [102, 98, 103, 127, 128, 113, 114, 137, 138, 139, 148, 135, 136, 61, 134, 129, 130, 131, 132, 31, 33] and dense distributed representations [75, 76] (see [82, 84, 87, 88, 83] for examples of their applications). Complex internal structure of cell assemblies for graded connections. When binary vectors reflecting similarity of real objects are stored in a NAM by variants of the Hebb (or Hopfield) rule, the weights of connections between the neurons frequently activated together will be greater than the mean value of all ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 17 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov the weights. On the other hand, rare combinations of active neurons will have smaller weights. Thus, neuron assemblies (cell assemblies in terms of Hebb [65]) formed in the network may have a very complicated structure. Hebb and Milner introduced the notions of "cores" and "fringes" ([65] pp. 126–134; for more recent research see [101]) to characterize qualitatively such complex internal structure of assemblies. The notions of core (kernel, nucleus) and fringe (halo) of assemblies have attracted attention to the function of assemblies distinct from the function of associative memory. This different function is not just memorization of individual activity patterns (vectors), but emergence of some generalized internal representations, that were not explicitly presented to the network as vectors for memorization. Some assembly cores may correspond to "prototypes" representing subsets of attributes (encoded by the active neurons represented by the 1-components of the corresponding vectors) often present together in some input vectors. Note that some of these subsets of components/attributes may never be present in any single vector. Stronger cores, corresponding to stable combinations of a small number of typical attributes present in many vectors employed for learning, may correspond to some more abstract or general object category (class). Cores formed by more attributes may represent more specific categories, or object prototypes. However, object tokens (category instances) may also have strong cores if they were often presented to the network for learning. Note that some mechanisms may exist to prevent repeated learning of vectors that are already "familiar" to the network. Also, the rate of weight modification may vary based on the "importance" of the input vector. The representations of real objects have different degrees of similarity with each other. Similarities in various combinations of features often form different hierarchies of similarities that reflect hierarchies of categories of different degrees of generalization (object — class of objects — a more general class, etc.). So, assemblies formed in the network (cores and fringes of different "strength") may have a complex and rich hierarchical structure, with multiple overlapping hierarchies reflecting the structure of different contents and values of similarity implicitly present in the base of vectors used for the unsupervised network learning by the employed variant of the Hebb rule. Thus, many types of the category-based hierarchies (also known as generalization or classification or type-token hierarchies) may naturally emerge in the internal structure of assemblies formed in a single assembly neural network (NAM). Complex internal structure of a neural assembly allows a virtually continuous variety of hierarchical transitions. To reveal various types of categories and prototypes and instances formed in the network, the corresponding assemblies should be activated. To activate only stronger cores, higher values of threshold should be used. Lower threshold values may additionally activate fringes. Research of generalization function in NAMs. Additional stable states that emerge in NAM after memorizing random base vectors and do not coincide with the base vectors are known as false or spurious states or memories, e.g., [69, 155]. In [155] they regarded the emergence of spurious attractors in the Hopfield networks as a side effect of the main function of distributed NAMs, consisting 18 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey not in memorizing individual patterns, but in formation of prototypes. Such an interpretation is close to the earlier work [13, 11, 12] that considered formation of concepts, prototypes, and taxonomic hierarchies as a natural generalization of correlated patterns memorized in a distributed memory. Research of hierarchically correlated patterns and states in Hopfield networks has been initiated by physicists who studied the "ultrametric" organization of ground states in spin-glasses (e.g., [123, 32, 63]; see also [6] and its references). While these earlier works required explicit representation of patterns at various hierarchical levels to be used in the learning rule, more neurobiologically plausible and practical Hebb and Hopfield rules applied to (hierarchically) correlated sparse binary patterns themselves (obtained with some simple correlation model) were considered, e.g., in [153, 66]. [153] theoretically showed "natural" formation of stable cores and fringes as well as traveling through different levels of hierarchies by uniform changing of the threshold. More complex probabilistic neuron dynamics and threshold control expressing neuronal fatigue was modeled in [66]. Dynamics of transitions between stable memory states that models human free recall data and can also be used with hierarchically organized data was considered in [143]. The "neurowindow" approach of [74] may be considered as using multiple thresholds to activate cores or fringes. Revealing the stable states corresponding to emergent assemblies is used for data mining (binary factor analysis) in [36, 37]. Generalization in NAMS with binary connections. The Willshaw learning rule does not form assemblies with the complex internal structure needed for generalization functions, such as emergence of generalization (type-token) hierarchies. The Willshaw learning rule causes the connectivity of an assembly (corresponding to a vector) to become full after a single learning act (vector storage) and not change thereafter. To preserve the capability of forming assemblies with a non-uniform connectivity in NAMs with binary connections, a stochastic analogue of the Hebbian learning rule for binary connections was proposed in [100, 98]: wij = wij Ú ( yi Ù y j Ù xij ) , where xij is a binary random variable equal to 1 with the probability that determines the learning rate. The connectivity value for some set of neurons is determined here by the number of their 1-weight connections. Neurons that have more than some fraction of 1-weight connections with the other neurons of the same assembly may be attributed to the core part of the assembly. In [15] they experimentally studied formation of assemblies with cores and fringes using the above mentioned "stochastic Willshaw" rule ( D = 4096, pD = 120 – 200, about 60 neurons in the core and 60–140 neurons in the fringe). Tests have been performed on retrieving a core by its part; a core and its full fringe by the core and a part of the fringe (the most difficult test); and, a core by a part of its fringe. As expected, experiments with correlated base vectors have shown a substantial decrease of storage capacity compared to random independent vectors. A special learning rule was proposed to increase the stability of fringes. ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 19 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov Formation of prototypes with the stochastic Willshaw rule was also investigated in [7, 23]; a model of paired-associate learning in humans is considered in [141]. Generalization in modular NAMs. A modular structure of neural networks where the Hebb assemblies are formed inside the modules was proposed and developed in [50–56]. The modular assembly neural network is intended for recognition of a limited number of classes. The network is artificially partitioned into several modules (sub-networks) according to the number of classes that the network is required to recognize. Each module network is full-connected, connections are graded. The features extracted from all objects of a certain class are encoded into activation of the patterns of neurons within the corresponding sub-network. After learning, the Hebb assemblies are formed in each module network. In this modular structure, the network acquires the capability to generalize the description of each class within the corresponding module (subnetwork), i.e. separately and completely independently from all other classes. In [56] it was shown that the number of connections in each module can be reduced without loss of the recognition capability. NAMs WITH HIGHER-ORDER CONNECTIONS AND WITHOUT CONNECTIONS Neural networks in the previous sections have connections of order n = 2 (а connection is between two neurons). In this section we consider values of n other than 2, for the NAMs with the structure of the Hopfield network (unlike Section "Hopfield NAMs" where we only considered the case n = 2 ). Neural networks with higher-order connections. In the higher-order (order n > 2 ) generalization of the Hopfield network, n neurons are connected by single connection instead of just two (for example, [124, 17, 46, 1, 70, 94, 26, 95, 96, 30]). For the neuron with states from {–1, + 1} ( p = 1/2), the network dynamics can be defined as zi = sign(å j ... j ¹i wij1 ... jn z j1 ... jn ) . 1 n The analogue of the Hebb learning rule becomes wi1 ...in = 1 D n -1 å m =1, N y im y im2 ... y im . 1 n Other learning rules can also be used. The number of stable states corresponding to the stored random binary vectors (possibly slightly different from them) is estimated in the mentioned papers to be N crit » a crit (n) D n -1 . As in NAMs with connections between pairs of neurons, a crit depends on the specific type of learning rule and network dynamics. a crit does not exceed 2 and decreases with increasing n [94]. For the absence of errors (with a probability approaching 1), the number of stored vectors N » 1 D n -1 / ln D cn 20 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey (for example, [46, 26, 95, 30]). In [95] they obtain cn > 2(2 n - 3)!! So, the exponential in n growth of N is due to the exponential growth of the connection number, and the characteristics per connection deteriorate with increasing n . The generalization of Krotov-Hopfield. For networks with higher-order connections, the network energy in [95] is written as -å m =1, N F (á z, y m ñ) with a smooth function F (u) . For polynomial F (u) and n = 2 , this gives the energy of the usual Hopfield network ([68] and Subsection "Hopfield networks with dense vectors"). For small n , many memory vectors y m have approximately the same values of F (u) and make a comparable contribution to the energy. For n ® ¥ , the main contribution to the energy is given by the memory vector y with the largest á z , yñ . For intermediate n , a large contribution is made by several nearest memory vectors. In [30] it is proved that for F (u) = exp(u) this memory allows one to retrieve N = exp(a D) randomly distorted vectors (within dist Ham < D / 2 from the stored vectors) by a single step of the sequential dynamics, for some 0 < a < ln 2 / 2 , depending on the distortion, with probability converging to 1 for D®¥. In [95] they consider the operation of such a network in the classification mode, where each stored base vector corresponds to one of the categories to be recognized. In particular, to classify the handwritten digits of the MNIST base into 10 classes, in addition to the "visible" neurons to which 28x28 images (with pixel values in [–1, + 1]) are input, there are 10 "classification" neurons. The value of the output is obtained by a non-linearity g ( s ) applied to the input sum s , for example, tanh( s) (instead of the sign( s) function used in the memory mode). The outputs of visible neurons are fixed, and the outputs of classification neurons are determined by a single step of the dynamics. Vectors of N memory states are formed by learning on the training set. The N = 2000 memory vectors minimizing the classification error for the 60,000 images of the MNIST training set were obtained with the stochastic gradient descent algorithm. For a single step of the dynamics this structure is equivalent to a perceptron with one layer of N hidden neurons [95]. The nonlinearity at the output of the visible neurons is f (u ) = F (u) , and that at the output of hidden neurons is g (u ) . The learned memory vectors (with components normalized to [–1, + 1]) are encoded in the weight vectors of the connections between the visible neurons and the hidden neurons. It is shown that when n changes the visualized memory vectors change. For small n , the memorized vectors correspond to the features of the digit images, and for large n they become prototypes of individual digits [95, 96]. Neural networks with first-order connections. In a neural network with connection order n = 1 , each neuron is connected only with itself. They can be ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 21 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov considered as networks without connections, where learning changes the state of the neurons themselves (with "neuron plasticity" [39]). Thus, memory is a single vector of the dimension of the vectors of the base. For binary connections, we get the Bloom filter (see the reviews [22, 149]), which exactly recognizes the absence of an undistorted query vector in the stored database by absence of at least one of its 1-components in the memory vector. If the 1-components of the query vector are a subset of 1-components of the memory vector, the vector is recognized as the base vector, but it is necessary to check this, since there is a false positive probability due to "ghosts" (vectors not from the base, the 1-components of which belong to the memory vector). Ghosts can be considered as analogous to spurious memories (Subsection "Research of generalization function in NAMs"). An analysis of the probability of their appearance under certain restrictions on stored random vectors is given in [144]. In [158], they reduce the probability of false positives. In [57], a Bloom filter version is analyzed which recognizes the absence of distorted query vectors. The autoscaling Bloom filter approach proposed in [92] suggests a generalization of the counting Bloom filter approach based on the mathematics of sparse hyperdimensional computing and allows elastic adjustment of its capacity with probabilistic bounds on false positives and true positives. In [90], the formation of sparse memory vectors (with an additional operation of context-dependent thinning [134]) is considered, and in [91] the probability of correct recognition is estimated. The use of graded connections (the formation of the memory vector is done by addition), including subsequent binarization, and the classification problem for vectors not from the base, are considered in [89, 91]. For real-valued vectors and connections, the recognition of random undistorted vectors is analyzed in [10, 126]. In [73] they allow distortion of vectors. In [126, 73], the analysis of non-random base vectors is given. NAMs WITH A BIPARTITE GRAPH STRUCTURE FOR NONBINARY DATA WITH CONSTRAINTS In some recent papers (e.g., [78, 145, 110, 111]), in order to create NAMs which can store and retrieve (from rather noisy input vectors) the number N of (not always binary) vectors with N near exponential in D , the vectors considered are not arbitrary random but satisfy (linear) constraints. The neural network has the structure of a bipartite graph. One set of neurons (not connected with each other) is used to represent the vectors of the base, neurons of the other set represent constraints. A rectangular matrix of connections between these two sets is learned on the vectors of the base. The connection vector of each constraint neuron represents the vector of that particular constraint. Iterative algorithms with local neuron computations are used for retrieval. Iterative algorithms for learning constraint matrix and vector recovery . In [78, 145], they consider the problem of the exact retrieval (with high probability) of vectors that belong to a subspace of dimension less than D . The graded weights of the bipartite graph connections representing linear constraints are learned from the vectors of the base (which have only non-negative integer components). Iterative algorithms are used for learning. The weights are constrained to be sparse, which is required for analyzing the retrieval algorithm. 22 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey The input (query) vectors x are obtained from the vectors y of the base by additive noise: x = y + e , where e are random sparse vectors with (bipolar) integer components. During retrieval, activity propagates first from the data neurons to the constraint neurons and then in the opposite direction, and so on for multistep retrieval. Non-linear transformations are used in neurons. In a stable state, the data neurons represent a base vector, and all constraint neurons obtain a total weighted zero input from the associated data neurons. In [78] the vectors are divided into intersecting parts. Any part of the vector belongs to a subspace of smaller dimension than the vector dimension of that part. А subset of the constraint neurons corresponds to each part. They are not looking for an orthogonal basis of constraints, but for vectors orthogonal to the corresponding parts of the data vectors from the base: W ( k ) y ( k ) = 0 , where k is the part number. To do this, the objective function is formulated and optimized with a stochastic gradient descent (several times for each part). During retrieval, they first independently correct errors in each part by performing several steps of the network dynamics. The correction is based on the fact that W ( k ) y ( k ) = W ( k ) e( k ) . Then, exploiting intersection of the parts, the parts without errors are used to correct the parts with errors. In [145], y from a subspace of dimension d < D are considered. Training forms a matrix W of D - d non-zero linearly independent vectors orthogonal to the vectors y of the base: Wy = 0 for all y of the base. An iterative algorithm of activity propagation in the network retrieves y . Algorithms [78, 145], described above, are claimed to store the number N of vectors (generated from their respective data models) exponential in D ( O( a D ) , a > 1 ) with the possibility of correcting a number of random errors that is linear in the vector dimension, D . However, to ensure a high probability of retrieval, a graph with a certain structure must be obtained, which is not guaranteed by the learning algorithms used. NAMs based on sparse recovery algorithms. To create autoassociative memory on the basis of a bipartite graph, in [110, 111] they use connection matrices W which allow them to reconstruct a sparse noise vector e which additively distorts the vector y of the base to form the query vector x . Then the required base vector is obtained as y = x - e . The noise vector is calculated using sparse recovery methods (that is, methods that find the solution vector with the least number of non-zero components). These methods require knowledge of the linear constraints matrix W such that Wy = 0 for all vectors y of the base. For some models of vectors (i.e. constraints or generative processes for the base vectors), such W can be obtained in polynomial time from the base of vectors generated by the model. In contrast to [78, 145], finding W is guaranteed with high probability, and adversarial rather than random errors are used as noise. In [110] real-valued vectors are used as the base, satisfying a set of nonsparse linear constraints. The data model, where the vectors of the base are given ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 23 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov by linear combinations of vectors with sub-Gaussian components, allows storing the number of vectors N up to exp( D3/4 ) . The data model with a basis of orthonormal vectors provides N up to exp(d ) , where 1 £ d £ D . Both models allow for accurate recovery from vectors with significant noise. In [111], as in [145], the vectors of the base are from a subspace defined by sparse linear constraints. They consider both real-valued vectors and binary vectors from {-1, +1}D satisfying W models of a certain type (sparse-subGaussian model). Learning is based on solving the dictionary learning problem with a square dictionary [111]. An iterative retrieval algorithm uses the fact that W is an expander graph with good properties [111]. The memory capacity and resistance to distortion is increased relative to [110]. Note the drawback of the methods considered in this section is that bases of real data may not correspond to the data models used. DISCUSSION In addition to being an interesting model of biological memory, neural network autoassociative distributed memories (NAMs) have also been considered as index structures that give promise to speed up nearest neighbor search relative to linear search (and, hopefully, to some other index structures). This mainly concerns sparse binary vectors of high dimension, because the number of such vectors that it is possible to memorize and retrieve from a significantly distorted version may far exceed the dimensionality of the vectors in some matrix-type NAMs, and the ratio N / D may be similar to the speed-up relative to linear search (see the first four Sections). Distributed NAMs have some drawbacks relative to traditional computer science methods for nearest neighbor search. The vector retrieved by a NAM may not be the nearest neighbor of the query vector. This could be tolerable if the output vector is an approximate nearest neighbor from the set of stored vectors. However, in NAMs the output vector may not even be a vector of the base set ("spurious memories"). (Up to a certain number of stored vectors and query vector distortion these problems remain insignificant.) For dense binary vectors, the number of vectors able to be reliably stored and retrieved is (much) smaller than the vector dimension. Also, NAMs are usually analyzed for the average case of random vectors and distortions, whereas real data are not like that, which results in poorer performance. However, available comparisons with the inverted index for sparse binary vectors in the average case do not clearly show the advantage of one or other algorithm in query time (Subsections "Hopfield networks with sparse vectors", "Willshaw networks with sparse vectors") An obvious approach to improve the memory and time complexity of the matrix-type NAMs from quadratic to linear in vector dimension is the use of incompletely connected networks with constant (but rather large) number of connections per neuron. An interesting direction is index structures for similarity search in which NAM modules are used at some stages. The index structure of Subsection "Willshaw networks in the index structures for nearest neighbor search" uses several NAMs to memorize parts of the base, and the similarity of the result of 24 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey single-step retrieval with the query vector is used to select the "best" NAM on which to perform an exact linear search against its stored vectors. Some studies are aimed at more efficient use of NAMs when working with real data. For example, the GB network with binary connections uses different neurons to represent the same component of the source vector, which allows for more balanced use of connections. In Section "Generalization in NAMs" we discussed the use of NAMs for generalization rather than exact retrieval from associative memory. In the NAMs that use versions of the Hebb learning rule, storage of vectors (even random ones) is accompanied by emergence of additional stable states. For correlated vectors, their common 1-components become "tightly" connected and stable states corresponding to them arise. Revealing these stable states can be used for data mining, e.g. for binary factor analysis [36, 37]. Research of complex (possibly hierarchical) structure of stable states (discussed in terms of cores and fringes of neural assemblies) may appear useful both for modeling brain function and for applications. Real data in many cases are not binary sparse vectors of high dimension with which the NAMs considered in the first four Sections work best. So, similarity preserving transformations to that format are required (Subsection "Similarity preserving binary vectors"). However, the obtained vectors (as well as the initial real data) are not random and independent, so the analytical and experimental results available for random, independent vectors usually can not predict NAM characteristics for real data. Using data vectors (often non-binary) that satisfy some linear constraints (instead of random independent vectors) allows bipartite graph based NAM construction with capacity near exponential in the vector dimension (Section "NAMs with a Bipartite Graph Structure for Nonbinary Data with Constraints"). However, again, this requires data from specific vector models (to which real data often do not fit). In NAMs with higher-order connections, connections are not between a pair, but between a larger number of neurons (this number being the order). So, the NAM becomes of tensor-type instead of matrix-type. These NAMs (Section "NAMs with Higher-Order Connections and without Connections") allow storing the number of dense vectors exponential in the order. However, this is achieved by the corresponding increase in the number of connections, and therefore in memory and in query time. The higher-order NAMs are generalized in [95, 96], where, roughly, the sum of polynomial functions of the dot products between all memory vectors and the network state is used as the input sum of a neuron. Such a treatment makes it possible to draw interesting analogies with perceptrons and kernel methods in the classification problem. However, for nearest neighbor search, this seems impose a query time exceeding that of linear search. Overcoming these and other drawbacks and knowledge gaps, and improving NAMs are promising topics for further research. Let's note other directions of research in fast similarity search of binary (non- sparse) vectors. Examples of index structures for exact search are [28] (with a fixed query radius and analysis for worst-case data; however impractical due to the small query radius required for sub-linear query time and moderate ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 25 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov memory costs) and [116] (practical, with variable radius of the query and analysis for random data). Theoretical algorithms for approximate search (providing: sublinear search time, a specified maximum difference of the result from the result of the exact search, and no false negatives) in [2] are modifications of more practical algorithm classes related to Locality Sensitive Hashing and Locality Sensitive Filtering (see [18, 14, 147]). However, the latter allow false negatives (with low probability). Unlike NAMs, these algorithms provide guarantees for the worstcase data, but require a separate index structure for each degree of distortion of the query vector. 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Фролов2, д-р биол.наук, проф., факультет электротехники и информатики, e-mail: docfact@gmail.com Р. Гейлер3 PhD (психология), исследователь, e-mail: r.gayler@gmail.com Д. Клейко4, аспирант, факультет информатики, электрической и космической техники. e-mail: denis.kleyko@ltu.se Е. Осипов4, PhD (информатика), проф., факультет информатики, электрической и космической техники e-mail: evgeny.osipov@ltu.se 1 Международный научно-учебный центр информационных технологий и систем НАН Украины и МОН Украины, пр. Академика Глушкова, 40, г. Киев, 03187, Украина 2 Технический университет Остравы, 17 listopadu 15, 708 33 Острава-Поруба, Чешская Республика 3 Мельбурн, штат Виктория, Австралия 4 Технологический университет Лулео, 971 87 Лулео, Швеция НЕЙРОСЕТЕВАЯ РАСПРЕДЕЛЕННАЯ АВТОАССОЦИАТИВНАЯ ПАМЯТЬ: ОБЗОР В настоящем обзоре рассмотрены модели автоассоциативной распределенной памяти, которые могут быть естественным образом реализованы нейронными сетями. Модели используют для запоминания векторов в основном локальном правиле обучения путем модификации значений весов межнейронных связей, которые существуют между всеми нейронами (полносвязные сети). В распределенной памяти различные векторы запоминают в одних и тех же ячейках памяти, которым в рассматриваемом случае нейронной сети соответствуют одни и те же связи. Обычно исследуют запоминание векторов, случайно выбранных из некоторого распределения. При подаче на вход автоассоциативной памяти искаженных вариантов запомненных в ней векторов осуществляется извлечение (восстановление) ближайшего запомненного вектора. Это реализуется за счет итеративной динамики нейронной сети на основе локально доступной в нейронах информации, полученной по связям от других нейронов сети. Вплоть до определенного количества запомненных в сети векторов и степени их искажения на входе, в результате динамики сеть с симметричными связями приходит в устойчивое состояние, соответствующее запомненному в сети вектору, имеющему наибольшее сходство с входным вектором (сходство обычно измеряют в терминах скалярного произведения). Такие нейросетевые варианты автоассоциативной памяти позволяют запомнить с возможностью восстановления такого количества векторов, которое может превышать размерность векторов (совпадающую с количества нейронов в сети). Для векторов большой размерности это открывает возможность поиска приближенного ближайшего соседа с временной сложностью, сублинейной от количества запомненных в нейронной сети векторов. К недостаткам такой памяти относится то, что восстановленный динамикой сети вектор может не быть ближайшим ко входному или даже может вообще не принадлежать к множеству запомненных векторов и значительно отличаться от любого из них. Исследования различных типов нейросетевой автоассоциативной памяти направлены на выявление диапазонов параметров, при которых указанные недостатки проявляются с малой вероятностью, а достоинства выражены в максимальной степени. ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 33 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov Основное внимание уделено сетям с парными связями типа Hopfield, Willshaw, Potts и работе с бинарными разреженными векторами (векторами с количеством единичных компонентов, малым по сравнению с количеством их нулевых компонентов), т.к. только для таких векторов удается запомнить с возможностью восстановления большое количество векторов. Помимо функции автоассоциативной памяти, для этих сетей также обсуждается функция обобщения. Обсуждаются также неполносвязные сети. Кроме того, рассмотрена автоассоциативная память в нейронных сетях со связями высшего порядка — то есть со связями не между парами, а между большим количеством нейронов. Рассмотрена также автоассоциативная память в нейронных сетях со структурой двудольного графа, где одно множество нейронов представляет запоминаемые векторы, а другое — линейные ограничения, которым они подчиняются. Эти сети выполняют функцию автоассоциативной памяти и для небинарных данных, удовлетворяющих заданной модели ограничений. Обсуждаются отношение рассмотренных в обзоре моделей нейросетевой автоассоциативной распределенной памяти к проблематике поиска по сходству, достоинства и недостатки рассмотренных методов, направления дальнейших исследований. Один из интересных и все еще не полностью разрешенных вопросов заключается в том, может ли нейронная автоассоциативная память искать приближенных ближайших соседей быстрее других индексных структур для поиска по сходству, в частности, для случая векторов очень больших размерностей. Ключевые слова: распределенная ассоциативная память, разреженный бинарный вектор, сеть Хопфилда, память Уиллшоу, модель Поттса, ближайший сосед, поиск по сходству. В.І. Гриценко1, член-кореспондент НАН України, директор, e-mail: vig@irtc.org.ua Д.А. Рачковський1, д-р техн. наук, пров. наук. співроб. відд. нейромережевих технологій оброблення інформації, e-mail: dar@infrm.kiev.ua А.А. Фролов2, д-р біол. наук, проф., факультет електротехніки та інформатики, e-mail: docfact@gmail.com Р. Гейлер3, PhD (психологія), дослідник, e-mail: r.gayler@gmail.com Д. Клейко4, аспірант, факультет інформатики, електричної та космичної техніки. e-mail: denis.kleyko@ltu.se Е. Осипов4, PhD (інформатика), проф., факультет інформатики, електричної та космичної техніки e-mail: evgeny.osipov@ltu.se 1 Міжнародний науково-учбовий центр інформаціоних технологій та систем НАН України та МОН України, пр. Академіка Глушкова, 40, м. Київ, 03187, Україна 2 Технічний університет Острави, 17 listopadu 15, 708 33 Острава-Поруба, Чеська Республіка 3 Мельбурн, штат Вікторія, Австралія 4 Технологічний університет Лулео, 971 87 Лулео, Швеція НЕЙРОМЕРЕЖНА РОЗПОДІЛЕНА АВТОАССОЦІАТИВНА ПАМ’ЯТЬ: ОГЛЯД У цьому огляді розглянуто моделі автоасоціативної розподіленої пам’яті, які можуть бути природним чином реалізовані нейронними мережами. Моделі використовують для запам’ятовування векторів в основному локальному правилі навчання шляхом 34 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey модифікації значень ваг міжнейронних зв’язків, які існують між всіма нейронами (повнозв’язні мережі). У розподіленій пам’яті різні вектори запам’ятовуються в одних і тих самих елементах пам’яті, яким в цьому випадку нейронної мережі відповідають одні і ті ж зв’язки. Зазвичай досліджують запам’ятовування векторів, випадково вибраних з деякого розподілу. Якщо на вхід автоасоціативної пам’яті подаються спотворені варіанти запам’ятованих в ній векторів, здійснюється витяг (відновлення) найближчого раніше запам’ятованого вектора. Це реалізується за рахунок ітераційної динаміки нейронної мережі на основі локально доступної в нейронах інформації, отриманої від інших нейронів мережі. До певної кількості запам’ятованих в мережі векторів і ступеня їх спотворення на вході, в результаті динаміки мережа із симетричними зв’язками приходить в стійкий стан, відповідний запам’ятованому в мережі вектору, який має найбільшу схожість з вхідним вектором (схожість зазвичай вимірюють як скалярний добуток). Такі нейромережні варіанти автоасоціативної пам’яті дозволяють запам’ятати з можливістю відновлення таку кількість векторів, яка може перевищувати розмірність векторів (що збігається з кількістю нейронів в мережі). Для векторів великої розмірності це відкриває можливість пошуку наближеного найближчого сусіда з складністю, сублінейною від кількості запам’ятованих в нейронній мережі векторів. До недоліків такої пам’яті відноситься те, що відновлений динамікою мережі вектор може не бути найближчим до вхідного або навіть може взагалі не належати до множини запам’ятованих векторів і значно відрізнятися від будь-якого з них. Дослідження різних типів нейромережної автоасоціативної пам’яті спрямовано на виявлення діапазонів параметрів, при яких зазначені недоліки проявляються з малою імовірністю, а достоїнства виражені в максимальному ступені. Основну увагу приділено мережам з парними зв’язками типу Hopfield, Willshaw, Potts і роботі з бінарними розрідженими векторами (векторами з кількістю одиничних компонентів, яке є малим у порівнянні з кількістю їх нульових компонентів), так як тільки для таких векторів вдається запам’ятати з можливістю відновлення велику кількість векторів. Крім функції автоасоціативної пам’яті, для цих мереж також обговорюється функція узагальнення. Обговорюються також неповнозв’язкові мережі. Крім того, розглянуто автоасоціативну пам’ять в нейронних мережах зі зв’язками вищого порядку — тобто зі зв’язками не між парами, а між великою кількістю нейронів. Розглянуто також автоасоціативна пам’ять в нейронних мережах зі структурою двудольного графа, де одна множина нейронів надає вектори, які запам’ятовуються, а інша — лінійні обмеження, яким вони підкорюються. Ці мережі виконують функцію автоасоціативної пам’яті також для небінарних даних, які відповідають заданій моделі обмежень. Обговорюються можливості використання розглянутих в огляді моделей нейромережної автоасоціативної розподіленої пам’яті у проблематиці пошуку за схожістю, достоїнства і недоліки розглянутих методів, напрямки подальших досліджень. Один із цікавих і все ще не повністю вирішених питань полягає в тому, чи може нейронна автоасоціативна пам’ять шукати наближених найближчих сусідів швидше інших індексних структур для пошуку за схожістю, зокрема, у випадку векторів дуже великих розмірностей. Ключові слова: розподілена асоціативна пам’ять, розріджений бінарний вектор, мережа Хопфілда, пам’ять Уілшоу, модель Потса, найближчий сусід, пошук за схожістю. ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) View publication stats 35	9
Multi-Rate Control over AWGN Channels via Analog Joint Source–Channel Coding arXiv:1609.07715v3 [cs.SY] 28 Oct 2016 Anatoly Khina, Gustav M. Pettersson, Victoria Kostina and Babak Hassibi Abstract— We consider the problem of controlling an unstable plant over an additive white Gaussian noise (AWGN) channel with a transmit power constraint, where the signaling rate of communication is larger than the sampling rate (for generating observations and applying control inputs) of the underlying plant. Such a situation is quite common since sampling is done at a rate that captures the dynamics of the plant and which is often much lower than the rate that can be communicated. This setting offers the opportunity of improving the system performance by employing multiple channel uses to convey a single message (output plant observation or control input). Common ways of doing so are through either repeating the message, or by quantizing it to a number of bits and then transmitting a channel coded version of the bits whose length is commensurate with the number of channel uses per sampled message. We argue that such “separated source and channel coding” can be suboptimal and propose to perform joint source– channel coding. Since the block length is short we obviate the need to go to the digital domain altogether and instead consider analog joint source–channel coding. For the case where the communication signaling rate is twice the sampling rate, we employ the Archimedean bi-spiral-based Shannon– Kotel’nikov analog maps to show significant improvement in stability margins and linear-quadratic Gaussian (LQG) costs over simple schemes that employ repetition. Index Terms— Networked control, Gaussian channel, joint source–channel coding. I. I NTRODUCTION Networked control systems, especially those for which the links connecting the different components of the system (plant, observer, and controller, say) are noisy, are increasingly finding applications and, as a result have been the subject of intense recent investigations [1]–[3]. In many of these applications the rate at which the output of the plant is sampled and observed, as well as the rate at which control inputs are applied to the plant, is different from the signaling rate with which communication occurs. We shall henceforth A. Khina, V. Kostina and B. Hassibi are with the Dept. of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA (E-mails: {khina, vkostina, hassibi}@caltech.edu). G. M. Pettersson is with the Dept. of Aeronautical and Vehicle Engineering, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden (E-mail: gupet@kth.se). The work of A. Khina was supported in part by a Fulbright fellowship, Rothschild fellowship and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłlodowska-Curie grant agreement No 708932. The work of G. M. Pettersson at Caltech was supported by The Boeing Company under the SURF program. The work of B. Hassibi was supported in part by the National Science Foundation under grants CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from Qualcomm Inc., by NASA’s Jet Propulsion Laboratory through the President and Directors Fund, and by King Abdullah University of Science and Technology. call such systems multi-rate networked control systems. The rate at which the plant is sampled and controlled is often governed by how fast the dynamics of the plant is, whereas the signaling rate of the communication depends on the bandwidth available, the noise levels, etc. As a result, there is no inherent reason why these two rates should be related and, in fact, the communication rate is almost always higher than the sampling rate. This latest fact clearly gives us the opportunity to improve the performance of the system by having the possibility to convey information about each sampled output of the plant, and/or each control signal, through multiple uses of the communication channel. An obvious strategy is to simply repeat the transmitted signal (so-called repetition coding). In analog communication this simply adds a linear factor to the SNR (3 dB for a single repetition); in digital communication over a memoryless packet erasure link, say, it simply reduces the probability of packet loss exponentially in the number of retransmissions. A more sophisticated solution would be to first quantize the analog message (the sampled output or the control signal) and then protect the quantized bits with an error-correcting channel code whose block length is commensurate with the number of channel uses available per sample. A yet more sophisticated solution would be to use a tree code which collectively encodes the quantized bits in a causal fashion over all channel uses [4]–[6]. The latter two solutions implicitly assume what is called the “separation between source and channel coding”, i.e., that quantization of the messages and channel coding of the quantized bits (using either a block code or a tree code) can be done independently of one another. While this is asymptotically true in communication systems (where it is a celebrated result), it is not true for control systems where the overall objective is to minimize a linear-quadratic Gaussian (LQG) cost [7]. To minimize an LQG cost what is needed is joint source–channel coding (JSCC). Unfortunately, in its full generality, this is known to be a notoriously difficult problem and so it has rarely been attempted (especially, in a control context). Nonetheless, this is what we shall attempt in this paper. We assume the communication links are AWGN (additive white Gaussian noise) channels with a certain signal-to-noise ratio (SNR). As we show below, this SNR puts an upper limit on the size of the maximum unstable eigenvalue of the plant that can be stabilized. We further assume that the signaling rate of the communication channel is not much larger than the sampling rate of the plant, say only a factor of 2 to 10 larger. Thus, if one sets aside the (daunting) task of performing coding over multiple messages (a la tree codes) then one is left with constructing a joint source–channel code of relatively short length — something that could very well be feasible. In particular, since both the message and transmitted signals are analog, in this short block regime it is not even clear whether it is necessary to go through a digitization process. Thus, we shall focus on analog JSCC, originally proposed by Shannon [8] and Kotel’nikov [9], which can simply be viewed as an appropriately chosen nonlinear mapping from the analog message to the analog transmitted signal(s). Finally, we should mention that we view this work as a first step and the results as preliminary. Nonetheless, these already indicate that one can obtain substantial gains (in the LQG cost) over simple schemes, such as repetition, by using the ideas mentioned above. The design of more sophisticated JSCC schemes, as well as a comprehensive comparison of different schemes will be deferred to future work. II. P ROBLEM S ETUP We now formulate the control–communication setting that will be treated in this work, depicted also in Fig. 1. We concentrate on the simple case of a scalar full observable state and a scalar AWGN channel. The model and solutions can be extended to more complex cases of vector states and multi-antenna channels. Consider the scalar system with the plant evolution: xt+1 = αxt + wt + ut , (1) where xt is the (scalar) state at time t, wt is an AWGN of power W , α > 1 is a known scalar, and ut is the control signal. Assume further that x0 is Gaussian with power P0 . The measured output is equal to the state corrupted by noise: y t = x t + vt , (2) where vt is an AWGN of power V . In contrast to classical control settings, the observer and the controller are not co-located, and are connected instead via an AWGN channel bi = ai + ni , (3) where bi is the channel output, ai is the channel input subject to a unit power constraint, and ni is an AWGN of power 1/SNR.1 We assume an integer ratio KC between the sample rate of (2) and the signaling rate over the channel (3). That is, KC channel uses (3) are available per one control sample (1), (2). In this work we further assume that the observer knows all past control signals {ui \|i = 1, . . . , t − 1}; for a discussion of the case when such information is not available at the observer, see Section V. 1 This representation is w.l.o.g., as the case of an average power P and C noise power N , can always be transformed to an equivalent channel with average power 1 and √ noise power N/PC , 1/SNR by multiplying both sides of (3) by 1/ PC . vt wt Plant xt xt+1 = αxt + wt + ut yt ut Channel Controller/ Receiver ai Observer/ Transmitter bi ni Fig. 1. Scalar control system with white driving and observation AWGNs and an AWGN communication channel. The dashed line represents the assumption that the past control signals are available at the transmitter. Similarly to the classical LQG control (in which the controller and the observer are co-locatedco-located controller and observer), we wish to minimize the average stage LQG cost after the total number of observed samples T : " # T X 1 2 2 2 Qxt + Rut , J¯T , E F xT +1 + T t=1 for some non-negative constants F , Q and R, by designing appropriate operations at the observer [which also plays the role of the transmitter over (3)] and the controller [which also serves as the receiver of (3)]. The infinite horizon cost is defined as J¯∞ , lim J¯T . T →∞ To that end, we recall next known results from information theory for joint source–channel coding design with low delay. III. L OW-D ELAY J OINT S OURCE –C HANNEL C ODING In this section, we review known results from information theory and communications for transmitting an i.i.d. zeromean Gaussian source st of power PS over an AWGN channel (3). The number of source samples generated per a time instant is not necessarily equal to the channel uses of (3) per the same time. In general, consider the case where KC channel uses of bi are available for every KS source samples of st . The goal of the transmitter is to convey the source st to the receiver with a minimal possible average distortion, where the appropriate distortion measure for our case of interest is the mean square error distortion. To that end, the transmitter applies a mapping E that transforms every KS source samples to KC channel inputs: (a1 , . . . , aKC ) = E (s1 , . . . , sKS ) , such that the input power constraint is satisfied: i 1 h 2 E {E (s1 , . . . , sKS )} ≤ 1. KS The receiver, upon receiving the KC channel outputs of (3) — corresponding to the KC transmitted channel inputs — applies a mapping to these measured outputs to recover estimates ŝt of the source samples: (ŝ1 , . . . , ŝKS ) = D (b1 , . . . , bKC ) . The resulting average distortion of this scheme is D= KS h i 1 X 2 E (st − ŝt ) , KS t=1 and the corresponding (source) signal-to-distortion ratio (SDR) is defined as PS . D Our results here are more easily presented in terms of unbiased errors, as these can be regarded as uncorrelated additive noise in the sequel (when used as part of the developed control scheme). Therefore, we consider the use of (samplewise) correlation-sense unbiased estimators (CUBE), namely, estimators that satisfy SDR , SDRlin = 2SNR, E [st (st − ŝt )] = 0. We note that any estimator ŝB t can be transformed into a CUBE ŝt by multiplying by a suitable constant: E s2 ; ŝt = tB ŝB E st ŝt t for a further discussion of such estimators and their use in communications the reader is referred to [10]. Shannon’s celebrated result [11] states that the minimal achievable distortion, using any transmiter–receiver scheme, is dictated, in the case of a Gaussian source, by2 KS 2 log (1 + SDR) = KS R(D) ≤ KC C = KC 2 log (1 + SNR) where R(D) is the rate–distortion function of the source and C is the channel capacity [11]. Thus, the optimal SDR, commonly referred to as optimum performance theoretically achievable (OPTA) SDR, is given by SDROPTA = (1 + SNR) For finite blocklengths, (4) cannot be exactly attained, except for specific cases in which the source and the distortion measure are probabilistically matched to the channel [12], and strictly tighter outer bounds on the distortion can be derived [13]–[15]. One eminent case where such a matching occurs is that of a Gaussian source and a Gaussian channel with matching number of samples/uses KC = KS . In this case, sending each source sample as is, up to a possible power adjustment, proves optimal and achieves (4) with KC = KS = 1 (and hence also any other positive integer) [16]. Unfortunately, this breaks down when KC 6= KS , and consequently led to the proposal and study of various techniques for low-delay JSCC schemes.3 We next concentrate on the simple case of KS = 1 and KC = 2. That is, the case in which one source sample is conveyed over two channel uses. A naı̈ve approach is to send the source as is over both channel uses, up to a power adjustment. The corresponding unbiased SDR in this case is KC /KS − 1. (4) Shannon’s proof (for a more general case of not necessarily Gaussian source or channel) is based upon the separation principle, according to which the source samples are partitioned into blocks and quantized together, resulting in (approximately) uniform independent bits. These bits are then partitioned again into blocks and encoded together to form the channel inputs. At the receiver first the coded bits are recovered, followed by the reconstruction of the source samples from these bits. However, this compression–coding separation-based technique is optimal only in the limit when the blocklengths KS and KC grow to infinity for a fixed ratio between the two, which implies, in turn, very large delays. 2 The rate–distortion function here is written in terms of the unbiased SDR, in contrast to the more common biased SDR expression log(SDR). a linear improvement rather than an exponential one as in (4). This scheme approaches (4) for very low SNRs, but suffers great losses at high SNRs. We note that the linear factor 2 comes from the fact that the total power available over both channel uses has doubled, and the same performance can be attained by allocating all of the available power to the first channel use and remaining silent during the second channel use. This suggests that better mappings that truly exploit the extra channel use can be constructed. The first to propose an improvement for the 1:2 case were Shannon [8] and Kotel’nikov [9], in the late 1940s. In their works, the source sample is viewed as a point on a singledimensional line, whereas the two channel uses correspond to a two-dimensional space. In these terms, the linear scheme corresponds to mapping the one-dimensional source line to a straight line in the two-dimensional channel space (see Fig. 2), and hence clearly cannot provide any improvement (since AWGN is invariant to rotations). However, by mapping the one-dimensional source line into a twodimensional curve that fills better the space, a great boost in performance can be attained. Specifically, consider the Archimedean bi-spiral, which was considered in several works [17]–[20] (depicted in Fig. 2): ( reg areg s cos(ωs) = creg \|s\| cos(ω\|s\|) sign(s) 1 (s) = c reg a2 (s) = creg s sin(ωs) sign(s) = creg \|s\| sin(ω\|s\|) sign(s) (5) where ω determines the rotation frequency, the factor creg is chosen to satisfy the power constraint, and the sign(s) term is needed to avoid overlap of the curve for positive and 3 The term JSCC is somewhat misleading, as in many of these schemes there is no use of digital components, let alone coding, including the Shannon–Kotel’nikov (SK) maps which are described in detail and used in the sequel. mance (4) for a specific design SNR (4), and improve linearly for higher SNRs. Similar behavior is observed also in Fig. 3 where the optimal ω value varies with the (design) SNR, and mimics closely the quadratic growth in the SDR. Above the design SNR, linear growth is achieved for a particular choice of ω. We further note that the distortion component incurred when a threshold event happens, grows with \|s\|. To avoid stretch this behavior, instead of increasing the magnitude a stretch proportionally to the phase ∠ a as in (6), we increase it slightly faster at a pace that guarantees that the incurred distortion does not grow with \|s\|: ( abounded (s) = cbounded \|s\|λβ cos ω\|s\|λ sign(s) 1 (7) abounded (s) = cbounded \|s\|λβ sin ω\|s\|λ sign(s) 2 Fig. 2. Linear repetition and Archimedean spiral curves. negative values of s (for each of which now corresponds a distinct spiral, and the two meet only at the origin). This spiral allows to effectively improve the resolution w.r.t. small noise values, since the one-dimensional source line is effectively stretched compared to the noise, and hence the noise magnitude shrinks when the source curve is mapped (contracted) back. However, for large noise values, a jump to a different branch — referred to as a threshold effect — may occur, incurring a large distortion. Thus, the value ω needs to be chosen to be as large as possible to allow maximal stretching of the curve for the same given power, while maintaining a low threshold event probability. The SDRs for different values of ω are depicted in Fig. 3a. Another ingredient that is used in conjunction with (5) is stretching s prior to mapping it to a bi-spiral using φλ (s) , sign(s)\|s\|λ : ( stretch astretch (s) = areg \|s\|λ cos ω\|s\|λ sign(s) 1 1 (φλ (s)) = c stretch astretch (s) = areg \|s\|λ sin ω\|s\|λ sign(s) 2 2 (φλ (s)) = c (6) The choice λ = 0.5 promises great boost in performance in the region of high SNRs, as is seen in Fig. 3b. We further note that although the optimal decoder is a minimum mean square error (MMSE) estimator E [s\|b1 , b2 ], in this case, the maximum-likelihood (ML) decoder, p(b1 , b2 \|s), achieves similar performance for moderate and high SNRs. A joint optimization of λ and ω for each SNR, for both ML and MMSE decoding, was carried out in [19] and is depicted in Fig. 3. A desired property of the linear JSCC schemes is their SDR proportional improvement with the channel SNR (“SNR universality”). Such an improvement is not allowed by the separation-based technique, as it fails when the actual SNR is lower than the design SNR, and does not promise any improvement for SNRs above it. This motivated much work in designing JSCC schemes whose performance improves with the SNR, even for the case of large blocklengths [21]– [23]. The schemes in these works achieve optimal perfor- for some β > 1. This has only a slight effect on the resulting SDRs, as is illustrated in Fig. 3. Finally note that in no way do we claim that the spiralbased Shannon–Kotel’nikov (SK) scheme is optimal. Various other techniques exist, most using a hybrid of digital and analog components [24]–[26], which outperform the spiral-based scheme for various parameters. Nevertheless, this scheme is the earliest technique to be considered and gives good performance boosts which suffice for our demonstration. IV. C ONTROL VIA L OW-D ELAY JSCC In this section we construct a Kalman-filter-like solution [27] by employing JSCC schemes. We note that the additional complication here is due to the communication channel (3) and its inherent input power constraint. Denote by x̂rt1 \|t2 the estimate of xt1 at the receiver given {bi \|i = 1, . . . , t2 }, where ‘r’ stands for ‘receiver’, and by x̂tt1 \|t2 the estimate of xt1 given {yi \|i = 1, . . . , t2 }, where ‘t’ stands for ‘transmitter’. Denote their mean square erh further i h i rors (MSEs) by Ptr1 \|t2 , E x̃rt1 \|t2 and Ptt1 \|t2 , E x̃tt1 \|t2 , where x̃tt1 \|t2 , xt1 − x̂tt1 \|t2 and x̃rt1 \|t2 , xt1 − x̂rt1 \|t2 . Then, the scheme works as follows. At time instant t, the controller constructs an estimate x̂rt\|t of xt . It then applies the control signal ut = −Lt x̂rt\|t to the plant, for a pre-determined gain Lt 6= 0. Note that, since both the controller and the observer know the previously applied control signals {uj \|j = 1, . . . , t}, they also know x̂rt\|t and x̂rt+1\|t . Hence, in order to describe xt , the controller aims to convey its best estimate of the state x̂tt\|t . To that end, it can save transmit power by transmitting the error signal (x̂tt\|t − x̂rt\|t−1 ), instead of x̂tt\|t . The controller can then add back x̂rt\|t−1 to the received signal to construct x̂rt\|t . Remark 1: Note that even in the case of a fully observable state, i.e., when V = 0, the state is corrupted by nt when conveyed over the AWGN channel (3) to the controller. The performance of the transmission and the estimation processes applied by the observer and the controller, respectively, determine in turn, the total effective observation noise. The general scheme used throughout this work is detailed below. • Sends the KC channel inputs ai over the channel (3). Controller/Receiver: At time t • Receives the KC channel outputs bi corresponding to time sample t. ˆt = s̄t +neff • Recovers a CUBE of the source signal s̄t : s̄ t , eff where nt ⊥ s̄t is an additive noise of power of (at most) 1/SDR0 . ˆt to construct an estimate of st : • Unnormalizes s̄ q t s̄ r ˆt (10a) − Pt\|t ŝt = Pt\|t−1 q r t = Pt\|t−1 − Pt\|t s̄t + neff (10b) t q t neff . (10c) r − Pt\|t = x̂tt\|t − x̂rt\|t−1 + Pt\|t−1 t • (a) λ = 1. Constructs an estimate x̂rt\|t of xt from all received channel outputs until and including at time t. Since ŝt ⊥ x̂rt\|t−1 , the linear MMSE estimate amounts to4 x̂rt\|t = x̂rt\|t−1 + SDR0 ŝt , 1 + SDR0 (11) with an MSE of 1 r t Pt\|t−1 + SDR0 Pt\|t . 1 + SDR0 Generates the control signal (Lt is given next): r Pt\|t = • (12) ut = −Lt x̂rt\|t , and the receiver prediction of the next system state x̂rt\|t−1 = αx̂rt−1\|t−1 + ut−1 . (b) λ = 0.5. Fig. 3. Performances of the JSCC linear repetition scheme, OPTA bound, and the JSCC SK spiral scheme for optimized λ and ω, for the standard case (β = 1) and distortion-bounded case. The solid lines depict the performance of the standard spiral for various values of ω for two stretch parameters λ = 0.5 and 1, which perform better at high and low SNRs, respectively. Observer/Transmitter: At time t • Generates the desired error signal • • St = α2 R St+1 + Q, St+1 + R (13b) ST = F . (13c) Using (12) and (1), the prediction error at the decoder is given by the following recursion: α2 r t r Pt+1\|t = Pt\|t−1 + SDR0 Pt\|t + W. (14) 1 + SDR0 Scheme 1: st = x̂tt\|t − x̂rt\|t−1 (8a) = x̃rt\|t−1 − x̃tt\|t (8b) r Pt\|t−1 The control (LQG) signal gain Lt is given by (see, e.g., [27]): αSt+1 , (13a) Lt = St+1 + R t Pt\|t of average power − (determined in the sequel). Since the channel input is subject to a unit power constraint, st is normalized: 1 s̄t = q st . (9) r t Pt\|t−1 − Pt\|t Constructs KC channel inputs ai corresponding to s̄t , using a bounded-distortion JSCC scheme of choice of rate ratio 1 : KC with (maximum given any input) average distortion 1/SDR0 for the given channel SNR. The estimates x̂tt\|t can be generated via Kalman filtering (see, e.g., [27]): ỹt = yt − αx̂tt−1\|t−1 − ut−1 , x̂tt\|t = αx̂tt−1\|t−1 + ut−1 + Ktt ỹt (15a) , (15b) where the Kalman filter coefficients are generated via the recursion [27]: Ktt = t Pt\|t−1 t Pt\|t−1 +V , (16a) 4 If the resulting effective noise neff is not an AWGN with power that t does not depend on the channel input, then a better estimator than that in (11) may be constructed. t t = α2 Pt\|t−1 1 − Ktt + W, Pt+1\|t t Pt\|t = Ktt V. (16b) (16c) The recursive relations (14) and (16) lead to the following condition for the stabilizability of the control system. Theorem 1 (Achievable): The scalar control system of Section II is stabilizable using Scheme 1 if α2 < 1 + SDR0 , and its infinite-horizon average stage LQG cost J¯r is upper bounded by Q + α2 − 1 S P t − P̄ t , (17a) J¯r ≤ J¯t + 2 1 + SDR0 − α J¯t = QP̄ t + S P t − P̄ t , (17b) where J¯t is the average stage cost achievable at the observer,5 and P t and P̄ t are the infinite-horizon values of t t Pt\|t−1 and Pt\|t , respectively; S and P t are given as the positive solutions of S 2 − Q + α2 − 1 R S − QR = 0 and Pt 2 − α2 − 1 V + W P t − V W = 0 respectively, and P tV . Pt + V The following theorem is an adaptation of the lower bound in [28] to our setting of interest. Theorem 2 (Lower bound): The scalar control system of Section II is stabilizable only if α2 < 1 + SDROPTA , and the optimal achievable infinite-horizon average stage LQG cost is lower bounded by Q + α2 − 1 S r t ¯ ¯ J ≥J + P t − P̄ t , (18) 1 + SDROPTA − α2 P̄ t = where P t , P̄ t and S are as in Theorem 1, and SDROPTA is given in (4). By comparing (17a) and (18) we see that the possible gap between the two bounds stems from the gap between the bounds on the achievable SDR over the AWGN channel (3). It is interesting to note that in this case, in stark contrast to the classical LQG setting in which the system is stabilizable for any values of α, V and W , low values of SDR render the system unstable. Hence, it provides, among others, the minimal required transmit power for the system to remain stable. The difference from the classical LQG case stems from the additional input power constraint, which effectively couples the power of the observation noise with that of the estimation error, and was previously observed in, e.g., [7], [28]–[30] for the fully-observed setting. Remark 2: In the limit SNR → ∞, we attain also SDR → ∞. In this case, the estimate (11) and the prediction error (14) at the decoder coincide with those at the encoder, (15b) and (16b), respectively, recovering the renowned results of classical (without a communication channel) LQG control. 5 Alternatively, Fig. 4. Optimal average stage LQG cost J¯r of a single representative run for KC = KS = 1, α = 2, and SNRs 2 and 4 which correspond to a stabilizable and an unstabilizable systems. The driving noise and observation noise powers and the LQG penalty coefficients are Q = R = F = W = V = 1. this is the cost J¯r in the limit SNR → ∞. We next discuss the special cases of KC = 1 and KC = 2 channel uses per sample in Sections IV-A and IV-B, respectively. A. Source–Channel Rate Match In this subsection we treat the case of KC = 1, namely, where the sample rate of the control system and the signaling rate of the communication channel match. As we saw in Section III, analog linear transmission of a Gaussian source over an AWGN channel achieves optimal performance (even when infinite delay is allowed), namely, the OPTA SDR (4), and given any input value. Thus, the JSCC scheme that we use in this case is linear transmission — the source is transmitted as is, up to a power adjustment [recall (8) and (9)]: at = s̄t 1 =q st . r t Pt\|t−1 − Pt\|t Since in this case SDR0 = SDROPTA , the upper and lower bounds of Theorems 1 and 2 coincide, establishing the optimum performance in this case. Corollary 1: The scalar control system of Section II with KC = KS = 1 is stabilizable if only if α2 < 1 + SNR, and the optimal achievable infinite-horizon average stage LQG cost satisfies (17a) with equality where SDR0 = SNR. Remark 3: The stabilizability condition and optimum MMSE performance were previously established in [29], [30] for the case of no observation noise V = 0. The optimal averate stage LQG cost is illustrated in Fig. 4, where the normalized in time LQG cost J¯r is evaluated for a system with α = 2 and two SNRs — 2 and 4. SNR = 4 satisfies the stabilizability condition α2 < 1 + SNR, whereas SNR = 2 fails to do so. Unit LQG penalty coefficients Fig. 5. Average stage LQG costs when using the (distortion-bounded) SK Archimedean bi-spiral, repetition and the lower bound of Theorem 2 for α = 3, W = 1, V = 0, Q = 1, R = 0. The vertical dotted lines represent the minimum SNR below which the cost diverges to infinity. Q = R = F = 1 and unit driving noise and observation noise powers W = V = 1 are used. B. Source–Channel Rate Mismatch We now consider the case of KC = 2 channel uses per sample. As we saw in Section III, linear schemes are suboptimal outside the low-SNR region. Instead, by using non-linear maps, e.g., the (modified) Arcimedean spiralbased SK maps (7), better performance can be achieved. We note that the improvement in the SDR of the JSCC scheme is substantial when α2 is of the order of SDR. That is, when the SDR of the linear scheme is close to α2 −1, using an improved scheme with better SDR improves substantially the LQG cost. Unfortunately, the spiral-based SK schemes do not promise any improvement for SNRs below 5dB under maximum-likelihood (ML) decoding. Remark 4: By replacing the ML decoder with an MMSE one, strictly better performance can be achieved over the linear scheme for all SNR values. The effect of the SDR improvement is illustrated in Fig. 5 for a fully-observable (V = 0) system with α = 3 and W = 1, for Q = 1 and R = 0, by comparing the achievable costs and lower bound of Theorems 1 and 2. Remark 5: The resulting effective noise at the output of the JSCC receiver is not necessarily Gaussian, and hence the resulting system states xt , are not necessarily Gaussian either. Nevertheless, for the bounded-distortion scheme (7), this has no effect on the resulting performance. V. D ISCUSSION AND F UTURE R ESEARCH In this paper we considered the simplest case of scalar systems, and KS = 1 and KC = 2. Clearly, an (exponentially) large gain in performance can be achieved for KC > 2. We further note that the results of Theorems 1 and 2 readily extend to systems with vector states xt and vector control signals ut but scalar observed outputs yt . Interestingly, for the case of vector observed, state and control signals, even if the signaling rate of the channel and the sample rate of the observer are equal (rate matched case), conveying several analog observations over a single channel input may be of the essence. This is achieved by a compression JSCC scheme, e.g., by reversing the roles of the source and the channel inputs in the SK spiral-based scheme and similarly promises exponentially growing gains with the SNR and dimension; see [8], [9], [17]–[19], [26], [31]. In this work, we assumed that the observer knows all past control signals. This case can be viewed as a two-sided side-information scenario. Nevertheless, although this is a common situation in practice, there are scenarios in which the observer is oblivious of the control signal applied or has only a noisy measurement of control signal generated by the controller. 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arXiv:1711.09860v1 [math.GR] 27 Nov 2017 TWISTED LAWRENCE-KRAMMER REPRESENTATIONS ANATOLE CASTELLA Abstract. Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected. Here, we investigate an analogue of these representations — introduced by Digne in the spherical cases — for every Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under a group of graph automorphisms, and for the corresponding Artin-Tits group. Under the same assumptions on the parameters as in the small type cases, we first show that these so-called “twisted Lawrence-Krammer representations” are faithful, and we then prove, by computing their formulas when the group of graph automorphisms is of order two or three, their irreducibility in all the spherical and connected cases but one. Introduction Lawrence-Krammer representations (shortened to LK-representations in what follows) are a family of linear representations of Artin-Tits monoids and groups of small type, introduced by Lawrence [17] and Krammer [15] for the braid groups, and intensively studied since then (see the references listed below). Under some assumptions on the defining ring and on the parameters (see condition (⋆) of theorem 6 below), they are known to be faithful for the monoids [16, 1, 8, 12, 22, 14, 6] which ensure their faithfulness for the groups when the type is spherical, irreducible when the type is connected [23, 18, 7, 6], and have proved to be useful in the study of several other properties of Artin-Tits groups [18] and related objects [7, 19]. It is therefore an interesting question to ask if there exists some analogous linear representations for Artin-Tits monoids and groups of non-small type. In [12, end of section 3], Digne defines such objects for Artin-Tits monoids and groups of spherical type Bn , F4 and G2 , using the fact that they appear as the submonoids and subgroups of fixed points of Artin-Tits monoids and groups of small and spherical type (namely of type A2n−1 , E6 and D4 respectively) under the action of a graph automorphism. These new linear representations — that I call “twisted” LK-representations — share the same good combinatorial properties with respect to the associated root systems as in the small type cases, and Digne proves their faithfulness for his choice of parameters in [12, Cor. 3.11]. Date: November 28, 2017. 1 2 ANATOLE CASTELLA The aim of this article is to generalize this construction to every Artin-Tits monoid that appears as the submonoid of fixed elements of an Artin-Tits monoid of small type under a group of graph automorphisms, and to the associated ArtinTits group. Under the same condition (⋆) on the defining ring and on the parameters as in the small type cases, we prove their faithfulness for the monoids, which again ensure their faithfulness for the groups when the type is spherical, and we prove their irreducibility for all the connected and spherical types but one. The paper is organized as follows. We recall the basic needed notions on Coxeter groups, Artin-tits monoids an groups, standard root systems and graph automorphisms in section 1. We recall the definition and the main properties of LK-representations in the small type cases, as stated in [6], in subsection 2.1, and turn to the definition of the twisted LK-representations in subsection 2.2. We prove our faithfulness result in subsection 2.3 (theorem 13). We explicit the formulas of these twisted LK-representations when the group of graph automorphisms is of order two or three in section 3. We use these formulas in section 4 to study the spherical cases. We prove our irreducibility result in subsection 4.1 (theorem 35), and conclude in subsection 4.2 by giving a sufficient condition on the parameters for non-equivalence in the family of twisted LK-representations of a given type. 1. Coxeter matrices and related objects A Coxeter matrix is a matrix Γ = (mi,j )i,j∈I over an arbitrary set I with mi,j = mj,i ∈ N>1 ∪ {∞}, and mi,j = 1 ⇔ i = j for all i, j ∈ I. As usual, we encode the data of Γ by its Coxeter graph, i.e. the graph with vertex set I, an edge between the vertices i and j if mi,j > 3, and a label mi,j on that edge when mi,j > 4. We say that a Coxeter matrix Γ is connected when so is its Coxeter graph. In this paper, we will always assume that I is finite. This condition could be removed at the cost of some refinements in certain statements below (see [4, Ch. 11] for some of them), which are left to the reader. 1.1. Coxeter groups and Artin-Tits monoids and groups. To a Coxeter matrix Γ = (mi,j )i,j∈I , we associate the Coxeter group W = WΓ , the Artin-Tits group B = BΓ and the Artin-Tits monoid B + = BΓ+ , given by the following presentations : W = h si , i ∈ I \| si sj si · · · = sj si sj · · · if mi,j 6= ∞, and s2i = 1 i, \| {z } \| {z } mi,j terms B = h si , i ∈ I \| si sj si · · · = sj si sj · · · if mi,j 6= ∞ i, \| {z } \| {z } mi,j terms B + = mi,j terms mi,j terms h si , i ∈ I \| si sj si · · · = sj si sj · · · if mi,j 6= ∞ i+ . \| {z } \| {z } mi,j terms mi,j terms We denote by ℓ the length function on W and on B + relatively to their generating sets {si , i ∈ I} and {si , i ∈ I} respectively. TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 3 We denote by 4 the left divisibility in the monoid B + , i.e. for g, h ∈ B + , we write h 4 g if there exists h′ ∈ B + such that g = hh′ . This leads to the natural notions of left gcd ’s and right lcm’s in B + . Let J be a subset of I. We denote by ΓJ the submatrix (mi,j )i,j∈J of Γ and by WJ the subgroup hsj , j ∈ Ji of W . We say that J and ΓJ are spherical if WJ is finite, or equivalently if the elements sj , j ∈ J, have a common right multiple in B + (see [3, Thm. 5.6]). In that case, there exists a unique element rJ of maximal length in WJ , and the elements sj (j ∈ J) have a unique right lcm in B + that we denote by ∆J (see [3, Props. 4.1 and 5.7]). 1.2. Standard root system. We refer the reader to [11] for the basic notions on standard root systems. We denote by Φ = ΦΓ = {w(αi ) \| w ∈ W, i ∈ I} the standard root system associated with Γ in the real vector space E = ⊕i∈I Rαi , where the action of W on E is defined, for a generator si and a basis element αj , by si (αj ) = αj +2 cos mπi,j αi . T It is known that Φ = Φ+ ⊔ Φ− , where Φ+ = Φ (⊕i∈I R+ αi ) and Φ− = −Φ+ . As in [6], we will represent Φ+ — and any of its subsets — as a graded graph, where two elements α and β of Φ+ are linked by an edge labeled i if α = si (β), and where the grading is by the depth function on Φ+ defined, for α ∈ Φ+ , by dp(α) = min{ℓ(w) \| w ∈ W such that w(α) ∈ Φ− }. By convention in those graded graphs, we choose to place roots of great depth above roots of small depth, so edges like the following ones will all mean that α = si (β) and dp(β) > dp(α) : sβ sβ sβ i ❏i i✡ ✡ sα , s α , ❏s α . . . In particular in section 3, we will focus on some particular subsets of Φ+ , closure of subsets of Φ+ under the action of some subgroup WJ of W , that we call, after Hée, meshes : Notation For J ⊆ I, we call J-mesh of a subset Ψ of Φ+ the set MJ (Ψ) = T 1. + WJ (Ψ) Φ = {w(α) \| w ∈ WJ , α ∈ Ψ such that w(α) ∈ Φ+ }. 1.3. Graph automorphisms. We call graph automorphism of a Coxeter matrix Γ = (mi,j )i,j∈I every permutation σ of I such that mσ(i),σ(j) = mi,j for all i, j ∈ I, and we denote by Aut(Γ) the group they constitute. Any graph automorphism σ of Γ acts by an automorphism on W (resp. on B and B + ) by permuting the generating set {si \| i ∈ I} (resp. {si \| i ∈ I}). If Σ is a subgroup of Aut(Γ), we denote by W Σ (resp. B Σ , resp. (B + )Σ ) the subgroup of W (resp. subgroup of B, resp. submonoid of B + ) of fixed points under the action of the elements of Σ. It is known that W Σ (resp. (B + )Σ ) is a Coxeter group (resp. Artin-Tits monoid), and that the analogue holds for B Σ when Γ is spherical, or more generally of FC-type (see [13, 21] for the Coxeter case, [20, 9, 10, 5] for the Artin-Tits case). More precisely, if we denote by I Σ the set of spherical orbits of I under Σ, then Σ W (resp. (B + )Σ , and B Σ when Γ is of FC-type) has a Coxeter (resp. ArtinTits) presentation associated with a Coxeter matrix ΓΣ = (mΣ J,K )J,K∈I Σ easily 4 ANATOLE CASTELLA computable from Γ, where the generators are the elements rJ (resp. ∆J ) for J running through I Σ (see for example [9, 10, 5]). Similarly, any graph automorphism σ of Γ acts by a linear automorphism on the vector space E = ⊕i∈I Rαi by permuting the basis (αi )i∈I . This action stabilizes Φ and Φ+ , and the induced action on those sets is w(αi ) 7→ (σ(w))(ασ(i) ). It is easily seen on the definition that the action of Aut(Γ) on Φ+ thus defined respects the depth function on Φ+ . In particular, we can define the depth of an orbit Θ of Φ+ under Σ as the depth of any element α ∈ Θ. 2. Twisted Lawrence-Krammer representations From now on, we fix a Coxeter matrix Γ = (mi,j )i,j∈I of small type, i.e. with mi,j ∈ {2, 3} for all i, j ∈ I with i 6= j. 2.1. Lawrence-Krammer representations - the small type case. Let R be a (unitary) commutative ring and V be a free R-module with basis (eα )α∈Φ+ . We denote by V ⋆ the dual of V and by R× the group of units of R. For f ∈ V ⋆ and e ∈ V , we denote by f ⊗ e the endomorphism of V defined by v 7→ f (v)e. 4 Definition 2 ([6, Def. 7]). For (a, b, c, d) ∈ R and i ∈ I, we denote by ϕi = ϕi,(a,b,c,d) the endomorphism of V given on the basis (eα )α∈Φ+ by  ϕi (eα ) = 0 if α = αi ,     ϕ (e ) = de if α s ✐i , i α α (  β s  ϕi (eβ ) = beα   i in Φ+ . if  ϕi (eα ) = aeα + ceβ αs For a linear form fi ∈ V ⋆ , we define the Lawrence-Krammer map — or the LK-map for short — associated with (a, b, c, d) and fi to be the endomorphism ψi = ψi,(a,b,c,d),fi of V given by ψi = ϕi + fi ⊗ eαi . Proposition 3 ([6, Lem. 9 and Prop. 12]). Assume that d2 − ad − bc = 0 and that the family (fi )i∈I ∈ (V ⋆ )I satisfies the following properties : (i) for i, j ∈ I with i 6= j, fi (eαj ) = 0, (ii) for i, j ∈ I with mi,j = 2, fi ϕj = dfi , (iii) for i, j ∈ I with mi,j = 3, fi ϕj = fj ϕi . Then si 7→ ψi defines a linear representation ψ : B + → L (V ). Moreover if b, c, d and fi (eαi ), i ∈ I, belong to R× , then the image of ψ is included in GL(V ) and hence ψ induces a linear representation ψgp : B → GL(V ). Definition 4 ([6, Def. 11]). We call LK-family (relatively to (a, b, c, d)) any family (fi )i∈I ∈ (V ⋆ )I satisfying conditions (i), (ii) and (iii) of proposition 3 above, and we call LK-representation (relatively to (a, b, c, d) and (fi )i∈I ) the induced representation ψ and, when appropriate, ψgp . Remark 5. When b is invertible in R, an LK-family (fi )i∈I necessarily satisfies fi (eαi ) = fj (eαj ) for every i, j ∈ I with mi,j = 3 (see [6, Prop. 34]). In particular when Γ is connected, we thus get that the elements fi (eαi ), i ∈ I, are all equal. For sake of brevity when this is the case, we will simply denote by f the common value of the fi (eαi ), i ∈ I. TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 5 Moreover when Γ is connected and spherical, one can show that the LK-family (fi )i∈I is entirely determined by this common value f (see [6, Section 3.2]). The first main properties of these representations can be stated as follows : Theorem 6 ([6, Thm. A]). Let ψ : B + → L (V ) be an LK-representation over R associated with parameters (a, b, c, d) ∈ R4 and (fi )i∈I ∈ (V ⋆ )I . Assume that the following condition holds : (⋆) R is an integral domain, fi (eαi ) 6= 0 for all i ∈ I, and there exists a totally ordered integral domain R0 and a ring homomorphism ρ : R → R0 , such that ρ(t) > 0 for t ∈ {a, b, c, d}, and Im(fi ) ⊆ ker(ρ) for all i ∈ I. By extension of the scalars from R to its field of fractions K, consider ψ as acting on the K-vector space VK = K ⊗R V . Then ψ has its image included in GL(VK ), hence induces an LK-representation ψgp : B → GL(VK ), and the following holds. (i) The LK-representation ψ is faithful, and so is ψgr if Γ is spherical. (ii) If Γ is connected, the LK-representations ψ and ψgp are irreducible on VK . (iii) Assume that Γ has at least one edge and that ψ ′ is an LK-representation of B + associated with parameters (a′ , b′ , c′ , d′ ) ∈ R4 and (fi′ )i∈I ∈ (V ⋆ )I that satisfy the conditions of the preamble for the fixed ρ. Then if a′ 6= a, or if d′ 6= d or if fi′ (eαi ) 6= fi (eαi ) for some i ∈ I, the LK-representations ψ and ψ ′ are not equivalent on VK . 2.2. Definition of the twisted Lawrence-Krammer representations. Let us fix a subgroup Σ of Aut(Γ) and an LK-representation ψ : B + → L (V ), g 7→ ψg , of B + associated with an LK-family (fi )i∈I ∈ (V ⋆ )I . Recall that Σ naturally acts on B + , and that the submonoid (B + )Σ of fixed points of B + under Σ is the Artin-Tits monoid BΓ+Σ associated with a certain Coxeter matrix ΓΣ (see subsection 1.3). Moreover, Σ acts on Φ+ and this action induces an action of Σ on V by permutation of the basis (eα )α∈Φ+ , which we denote by Σ → GL(V ), σ 7→ σV . We denote by V Σ the submodule of fixed points of V under the action of Σ. Let us denote by ϕ : B + → L (V ), g 7→ ϕg , the LK-representation of B + associated with the trivial LK-family (i.e. where fi is the zero form for all i ∈ I). Lemma 7. For all (g, σ) ∈ B + × Aut(Γ), we get σV ϕg = ϕσ(g) σV . In particular, for every g ∈ (B + )Σ , ϕg stabilizes V Σ and hence ϕ induces a linear representation ϕΣ : (B + )Σ → L (V Σ ), g 7→ ϕΣ g = ϕg \|V Σ . Proof. The action of Aut(Γ) on Φ+ respects the depth, and this clearly implies that σV (ϕi (eα )) = ϕσ(i) (eσ(α) ) in view of the formulas of definition 2. The result follows by linearity and induction on ℓ(g). Proposition 8. Assume that fi = fσ(i) σV in V ⋆ , for every (i, σ) ∈ I ×Σ. Then for every (g, σ) ∈ B + × Σ, we get σV ψg = ψσ(g) σV . In particular, for every g ∈ (B + )Σ , ψg stabilizes V Σ and hence ψ induces a linear representation ψ Σ : (B + )Σ → L (V Σ ), g 7→ ψgΣ = ψg \|V Σ . Moreover if the images of ψ are invertible, then so are the images of ψ Σ . 6 ANATOLE CASTELLA Proof. We have σV ψi = σV ϕi + σV (fi ⊗ eαi ) = σV ϕi + fi ⊗ eασ(i) and ψσ(i) σV = ϕσ(i) σV + (fσ(i) ⊗ eασ(i) )σV = ϕσ(i) σV + (fσ(i) σV ) ⊗ eασ(i) (thanks to the formulas of [6, Rem. 1]), whence σV ψi = ψσ(i) σV by the previous lemma and assumption on fi and fσ(i) . The first point follows by induction on ℓ(g). Moreover if the images −1 of ψ are invertible, then the equality σV ψg σV−1 = ψσ(g) implies σV ψg−1 σV−1 = ψσ(g) , −1 Σ + Σ and hence ψg stabilizes V for every g ∈ (B ) . This gives the result. Definition 9 (twisted LK-representations). Under the assumption of the previous proposition, we call twisted LK-representation the linear representation ψ Σ : (B + )Σ → L (V Σ ) of the Artin-Tits monoid (B + )Σ = BΓ+Σ , and, when appropriate, Σ the induced representation ψgp : BΓΣ → GL(V Σ ) of the Artin-Tits group BΓΣ . The assumption fi = fσ(i) σV for every (i, σ) ∈ I × Σ is equivalent to fi (eα ) = fσ(i) (eσ(α) ) for every (i, α, σ) ∈ I × Φ+ × Σ. It is not always satisfied : for example if i and σ(i) are not in the same connected component of Γ, then fi (eαi ) and fσ(i) (eασ(i) ) can be chosen to be distinct (see [6, Section 3.1]). I do not know if this assumption is always satisfied when Γ is connected, but we have the following partial result : Proposition 10. Let (fi )i∈I be an LK-family. Assume that b, c, d are invertible in R and that we are in one of the following cases : (i) Γ is spherical and connected (i.e. of type ADE), or (ii) Γ is affine (i.e. of type ÃD̃Ẽ), or (iii) Γ has no triangle and (fi )i∈I is the LK-family of Paris, as in [6, Def. 42]. Then fi = fσ(i) σV for every (i, σ) ∈ I × Aut(Γ). Proof. The condition fi (eα ) = fσ(i) (eσ(α) ) for every (i, α, σ) ∈ I × Φ+ × Aut(Γ), for the three situations (note that the first one is a consequence of the third one), is easy to see by induction on dp(α), using the inductive construction of the fi (eα ), (i, α) ∈ I × Φ+ , and the independence results at the inductive steps, of [6, Sections 3.2, 3.3 and 3.4] respectively, and using the fact that the action of Aut(Γ) on Φ+ respects the depth. 2.3. Twisted faithfulness criterion. The aim of this subsection is to prove that the classical faithfulness criterion of [14], as stated in theorem 6 above, also works in the twisted cases. For g ∈ B + , we set I(g) = {i ∈ I \| si 4 g}. Lemma 11. Let ψ : (B + )Σ → M be a monoid homomorphism where M is left cancellative. If ψ satisfies ψ(g) = ψ(g ′ ) ⇒ I(g) = I(g ′ ) for all g, g ′ ∈ (B + )Σ , then ψ is injective. Proof. Let g, g ′ ∈ (B + )Σ be such that ψ(g) = ψ(g ′ ). We prove by induction on ℓ(g) that g = g ′ . If ℓ(g) = 0, i.e. if g = 1, then I(g) = I(g ′ ) = ∅, hence g ′ = 1 and we are done. If ℓ(g) > 0, fix i ∈ I(g) = I(g ′ ). Since the action of Σ on B + respects the divisibility and since g is fixed by Σ, the orbit J of i under Σ is included in I(g) = I(g ′ ), but then J is spherical and there exist g1 , g1′ ∈ B + such that g = ∆J g1 and g ′ = ∆J g1′ . Since the elements g, g ′ and ∆J are fixed by Σ (recall that ∆J is a generator of (B + )Σ ), so are g1 and g1′ by cancellation in B + . We thus get ψ(∆J )ψ(g1 ) = ψ(∆J )ψ(g1′ ) in M , whence ψ(g1 ) = ψ(g1′ ) by cancellation in M . We therefore get g1 = g1′ by induction and finally g = g ′ . TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 7 + + Notation 12. We denote every Pby Φ /Σ the set of orbits of Φ under Σ and, for + Θ ∈ Φ /Σ, we set eΘ = α∈Θ eα . The family (eΘ )Θ∈Φ+ /Σ is a basis of V Σ . Theorem 13. Assume that fi = fσ(i) σV for every (i, σ) ∈ I × Σ, so that the twisted LK-representation ψ Σ : (B + )Σ → L (V Σ ) is defined (see proposition 8), and assume that condition (⋆) holds : (⋆) R is an integral domain, fi (eαi ) 6= 0 for all i ∈ I, and there exists a totally ordered integral domain R0 and a ring homomorphism ρ : R → R0 , such that ρ(t) > 0 for t ∈ {a, b, c, d}, and Im(fi ) ⊆ ker(ρ) for all i ∈ I. By extension of the scalars from R to its field of fractions K, consider ψ Σ as acting on the K-vector space VKΣ = K⊗R V Σ , so that ψ Σ has its image included in GL(VKΣ ), Σ and hence induces a twisted LK-representation ψgp : BΓΣ → GL(VKΣ ). Then the Σ Σ twisted LK-representation ψ is faithful, and so is ψgp if ΓΣ is spherical. Proof. Under condition (⋆), the elements b, c, d and fi (eαi ), i ∈ I, are non-zero in R, so become units of the field of fractions K of R. By propositions 3 and 8, Im(ψ Σ ) Σ is then included in GL(VKΣ ) — hence is cancellative — and ψgp : BΓΣ → GL(VKΣ ) Σ Σ is defined. Moreover when Γ is spherical, the faithfulness of ψ implies the one of Σ ψgp by [6, Lem. 6]. In order to prove the theorem, it then suffices to see that ψ Σ satisfies the assumption of lemma 11. So let g, g ′ ∈ (B + )Σ be such that ψgΣ = ψgΣ′ and let us show that I(g) = I(g ′ ). We need for that some notations of [6, Section 2.2]. If we denote by V0 the free R0 -module with basis (eα )α∈Φ+ , then the morphism ρ : R → R0 induces a natural monoid homomorphism ρ̃ : L (V ) → L (V0 ), ϕ 7→ ϕ. By (⋆), the homomorphism ρ̃ sends Im(ψ) into L + (V0 ), the submonoid of L (V0 ) composed of the endomorphisms of V0 whose matrix in the basis (eα )α∈Φ+ has non-negative coefficients. Now for h ∈ (B + )Σ and β ∈ Φ+ , let us denote by Rh (β) the support of ψh (eβ ) in the basis (eα )α∈Φ+ , and for Ψ ⊆ Φ+ , let us set S Rh (Ψ) = β∈Ψ Rh (β). Since the coefficients of the matrix of ψh in the basis (eα )α∈Φ+ of V0Pare non-negative, the set Rh (Ψ), when Ψ is finite, is precisely the support of ψh ( β∈Ψ eβ ) in the basis (eα )α∈Φ+ . In order to show that I(g) = I(g ′ ), it suffices to prove, in view of [14, Prop. 2] (see [6, Lem. 20]), that Rg (Φ+ ) = Rg′ (Φ+ ). But since ψg and ψg′ coincide on V Σ , we get in particular that ψg (eΘ ) = ψg′ (eΘ ), S and hence Rg (Θ) = Rg′ (Θ) for every Θ ∈ Φ+ /Σ. And finally since Φ+ = Θ∈Φ+ /Σ Θ, we get Rg (Φ+ ) = S S + Θ∈Φ+ /Σ Rg′ (Θ) = Rg′ (Φ ), whence the result. Θ∈Φ+ /Σ Rg (Θ) = 3. Case of an automorphism of order two or three Let Γ = (mi,j )i,j∈I be a Coxeter matrix of small type and fix Σ = hσi 6 Aut(Γ) of order two or three. Recall that the submonoid (B + )Σ of fixed points of B + under Σ is generated by the elements ∆J , for J running through the spherical orbits of I under Σ. We fix four parameters (a, b, c, d) ∈ R such that d2 − ad − bc = 0 and consider an LK-family (fi )i∈I associated with (a, b, c, d) that satisfy fi = fσ(i) σV for every (i, σ) ∈ I × Σ, so that the associated twisted LK-representation ψ Σ : (B + )Σ → L (V Σ ) is defined. 8 ANATOLE CASTELLA Σ The aim of this section is to compute and study the map ψ∆ for a spherical J Σ Σ Σ orbit J of I under Σ. For sake of brevity, we set ψJ := ψ∆J and ϕΣ J := ϕ∆J for any spherical orbit J. Notation 14. Any spherical orbit J of I under Σ is of one of the following types : (i) type A : J = {i}, (ii) type B : J = {i, j} with mi,j = 2, (iii) type C : J = {i, j, k} with mi,j = mj,k = mk,i = 2, (iv) type D : J = {i, j} with mi,j = 3. Notice that types B and D only occur when Σ is of order two, and type C only occurs when Σ is of order three. We will make a constant use of these types A to D in the following. Notation 15. Let J be a spherical orbit of I under Σ. • We denote by ΘJ the set {αi \| i ∈ J} ; it is an orbit of Φ+ under Σ. • In case D, {αi + αj } is also an orbit of Φ+ under Σ, that we denote by Θ′J . Notation 16. Let J be a spherical orbit of I under Σ. Since we are assuming that fi = fσ(i) σV for every i ∈ I, the linear forms fi , for i ∈ J, coincide on V Σ . Depending on J, we define the following linear forms on V Σ : (i) for type A : fJ = fi \|V Σ , where J = {i}, (ii) for type B : fJ = dfi \|V Σ , for any i ∈ J, (iii) for type C : fJ = d2 fi \|V Σ , for any i ∈ J, (iv) for type D : fJ = fj (bc Id +aϕi )\|V Σ and fJ′ = cfj ϕi \|V Σ , where J = {i, j}. Notice that the two linear forms for case D really only depend on J (not on the choice of i and j in J) since fi \|V Σ = fj \|V Σ by assumption and since fj ϕi = fi ϕj by definition of an LK-family. Lemma 17. If i, j ∈ I with i 6= j, then fj (bc Id +aϕi ) = fj ϕ2i in V ⋆ . Moreover if mi,j = 3, then fj ϕ2i = fi ϕj ϕi in V ⋆ . Proof. By [6, Lem. 11], the endomorphism ϕ2i − aϕi − bc Id has its image included in Reαi , which is included in ker(fj ) if i 6= j since then fj (eαi ) = 0 by definition of an LK-family. So fj (ϕ2i − aϕi − bc Id) is the zero form and this gives the first point. The second point is clear since then fj ϕi = fi ϕj by definition of an LK-family. Proposition 18. Let J be a spherical orbit of I under Σ. Then (i) ψJΣ = ϕΣ J + fJ ⊗ eΘJ if J is of type A, B or C, ′ (ii) ψJΣ = ϕΣ J + fJ ⊗ eΘJ + fJ ⊗ eΘ′J if J is of type D. Proof. Recall that for every i ∈ I, ψi is given by ψi = ϕi + fi ⊗ eαi . The result for type A is trivial, since then ψJΣ = ψi \|V Σ . For type B, we have ψJΣ = ψi ψj \|V Σ . But in that case, we also have ϕi (eαj ) = deαj (by definition of ϕi ) and fi (eαj ) = 0 (by definition of an LK-family), thus we get, thanks to the formulas of [6, Rem. 1], ψi ψj = ϕi ϕj + fi ϕj ⊗ eαi + dfj ⊗ eαj , whence the result since fi ϕj = dfi by definition of an LK-family. Similarly for type C, we have ψJΣ = ψi ψj ψk \|V Σ and since ϕi (eαj ) = deαj , ϕi (eαk ) = ϕj (eαk ) = deαk , and fi (eαj ) = fi (eαk ) = fj (eαk ) = 0, we get, thanks to [6, Rem. 1], ψi ψj ψk = ϕi ϕj ϕk + fi ϕj ϕk ⊗ eαi + dfj ϕk ⊗ eαj + d2 fk ⊗ eαk , TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 9 whence the result since fi ϕj ϕk = dfi ϕk = d2 fi and fj ϕk = dfj by definition of an LK-family. Finally for type D, we have ψJΣ = ψi ψj ψi \|V Σ and ϕi (eαj ) = aeαj + ceαi +αj , ϕi ϕj (eαi ) = bceαj , fi (eαj ) = fj (eαi ) = 0, and fi ϕj (eαi ) = 0 (since fi ϕj = fj ϕi and ϕi (eαi ) = 0), hence we get, thanks to [6, Rem. 1], ψi ψj ψi = ϕi ϕj ϕi + fi ϕj ϕi ⊗ eαi + [bcfi + afj ϕi ] ⊗ eαj + cfj ϕi ⊗ eαi +αj , whence the result by lemma 17 and by the fact that fi \|V Σ = fj \|V Σ . Remark 19. Let J be a spherical orbit of I under Σ and let Θ be an orbit of Φ+ under T Σ. Recall that we denote by MJ (Θ) the J-mesh of Θ, i.e. the set WJ (Θ) Φ+ . Notice that this subset of Φ+ is a union of orbits of Φ+ under Σ. It is clear that all the maps ϕi , for i ∈ J, stabilize the submodule of V generated by the elements eβ for β running through MJ (Θ). So the map ϕΣ J stabilizes the submodule of V Σ generated by the elements eΘ for Θ running through the orbits Σ included in MJ (Θ), and hence the matrix of ϕΣ J in the basis (eΘ )Θ∈Φ+ /Σ of V + is block diagonal, relatively to the partition of Φ /Σ induced by those J-meshes MJ (Θ) for Θ ∈ Φ+ /Σ. In subsections 3.1 to 3.4 below, we explicit the different possible blocks of ϕΣ J, for the four possible types of J and for the different possible J-meshes MJ (Θ) when Θ runs through Φ+ /Σ. For each such J-mesh X = MJ (Θ), we denote by MX the corresponding block in ϕΣ J and by PX its characteristic polynomial. Notation 20. For sake of brevity in what follows, we set dˇ := a − d in R. In ˇ and ddˇ = −bc. particular, the roots of the polynomial X 2 − aX − bc are d and d, 3.1. Type A. We assume here that J = {i}, and hence that ϕΣ J = ϕi \|V Σ . • Configuration A1 : Θ = ΘJ . The corresponding block is MA1 = 0 . i • Configuration A2 : Θ s✐ or Θ The corresponding block is Θ2 • Configuration A3 : MA2 = d . s Θ2 i Θ1 s✐ s✐ i i or Θ or s Θ1 s s i i s s or Θ2 Θ1 The corresponding block is MA3 eΘ1 eΘ2 a b , = c 0 whose characteristic polynomial is ˇ PA3 = (X − d)(X − d). si✐ si✐ si✐. s s i i s i s s s . 10 ANATOLE CASTELLA 3.2. Type B. We assume here that J = {i, j} with mi,j = 2, and hence that ϕΣ J = (ϕi ϕj )\|V Σ . • Configuration B1 : Θ = ΘJ . The corresponding block is MB1 = 0 . si✐ j✐ • Configuration B2 : Θ s✐ s✐ j✐ i j✐ i . or Θ MB2 • Configuration B3 : Θ2 s j✐ i s i✐ j Θ1 s j✐ s i✐ The corresponding block is = d2 . . The corresponding block is MB3 whose characteristic polynomial is eΘ1 eΘ2 ad bd , = cd 0 ˇ PB3 = (X − d2 )(X − dd). Θ3 i • Configuration B4 : Θ2 Θ1 s ❅j ❅s s ❅ j ❅s MB4 whose characteristic polynomial is . The corresponding block is i 3  eΘ2 1 eΘ2 e2Θ a 2ab b = ac bc 0 , c2 0 0 ˇ PB4 = (X − d2 )(X − dd)(X − dˇ2 ). Θ4 s s ❅ j ❅i s ❅s ❅s Θ3 s . The corresponding block is ❅i ❅ j j ❅s ❅s i • Configuration B5 : Θ2 Θ1 MB5  eΘ21 a ac = ac c2 whose characteristic polynomial is eΘ2 eΘ3 eΘ4  2 ab ab 0 bc bc 0 0 0 b 0 , 0 0 ˇ 2 (X − dˇ2 ). PB5 = (X − d2 )(X − dd) Moreover, it is easily checked that the polynomial PB4 already annihilates MB5 . TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 11 3.3. Type C. We assume here that J = {i, j, k} with mi,j = mj,k = mk,i = 2, and hence that ϕΣ J = (ϕi ϕj ϕk )\|V Σ . • Configuration C1 : Θ = ΘJ . The corresponding block is MC1 = 0 . k✐ s j✐i✐ or • Configuration C2 : Θ k✐ k✐ k✐ s s s j✐i✐ j✐i✐ j✐i✐. Θ The corresponding block is MC2 = d3 . • Configuration C3 : Θ2 sk✐ i✐ sk✐ j✐si✐ j✐ Θ1 sk✐ i✐ sk✐ j✐si✐ j✐ i j MC3 k . The corresponding block is eΘ21 eΘ22 ad bd = , cd2 0 whose characteristic polynomial is ˇ PC3 = (X − d3 )(X − d2 d). Θ4 • Configuration C4 : Θ2 Θ1 k✐ i✐ j✐ s s s ✑◗✑j ◗ k ✑ j✑ ◗ k i ✑◗✑◗ ◗ ✑ s i✐✑ s ✐s✑◗s ✐◗s ✐◗s ✐ k✐ ◗ ◗ j j◗✑k k ✑ i ✑ j. Θ3 i◗ ◗✑◗✑ ✑ i ◗✑ s ◗s✑◗s✑ k✐ i✐ j✐ The corresponding block is MC4 eΘ4 eΘ3 eΘ1 eΘ2  a2 d abd abd b2 d acd 0 bcd 0  , = acd bcd 0 0  c2 d 0 0 0  whose characteristic polynomial is ˇ 2 (X − ddˇ2 ). PC4 = (X − d3 )(X − d2 d) ˇ −ddˇ2 ) already Moreover, one can check that the polynomial (X −d3 )(X −d2 d)(X annihilates MC4 . s Θ4 Θ3 • Configuration C5 : Θ2 Θ1 k j❅i s s ❅s k i j❅ ❅ j i s ❅s k❅s ❅ i j ❅s k 12 ANATOLE CASTELLA The corresponding block is MC5  eΘ1 a3 a2 c = ac2 c3 eΘ2 3a2 b 2abc bc2 0 eΘ3 3ab2 b2 c 0 0 whose characteristic polynomial is eΘ4  3 b 0 , 0 0 ˇ − ddˇ2 )(X − dˇ3 ). PC5 = (X − d3 )(X − d2 d)(X s ✏P s ✏sP ✏P PP ✏✏ ✏P ✏P ✏✏ ✏P PP PPPP ✏ ✏ P ✏ ✏ ✏ s s s ✏P s ✏P s ✏sP✏ Ps P ✏ ✏ ✏ P P ✏s✏P ✏s PP P P P P P✏ ✏ ✏ ✏ ✏ P✏ P✏ P✏ P✏ P✏ P✏ P P P P P P ✏ ✏ ✏ ✏ ✏ ✏ P P P P✏ sP✏✏ s ✏✏ s ✏ s ✏ s ✏ P✏ P✏ Ps✏ PsP✏ PsPPs ✏ P ✏ PP PP ✏✏ ✏✏✏✏ PP P ✏ PPPP✏ P✏ ✏ ✏ P s P✏ s ✏ P✏ Ps✏✏ Θ8 Θ5 • Configuration C6 : Θ2 Θ1 Θ6 , Θ7 Θ3 , Θ4 (the labels i, j, k on the edges are omitted). The corresponding block is MC6 eΘ3 2 eΘ2 2 eΘ31 a a2 c  2 a c  2 a c = ac2  2 ac  2 ac c3 a b 0 abc abc 0 0 bc2 0 a b abc 0 abc 0 bc2 0 0 eΘ4 2 a b abc abc 0 bc2 0 0 0 eΘ5 2 ab 0 0 b2 c 0 0 0 0 eΘ6 2 ab 0 b2 c 0 0 0 0 0 eΘ7 2 ab b2 c 0 0 0 0 0 0 whose characteristic polynomial is eΘ8  3 b 0  0  0 , 0  0  0 0 ˇ 3 (X − ddˇ2 )3 (X − dˇ3 ). PC6 = (X − d3 )(X − d2 d) And one can check that the polynomial PC5 already annihilates MC6 . 3.4. Type D. We assume here that J = {i, j} with mi,j = 3, and hence that ϕΣ J = (ϕi ϕj ϕi )\|V Σ . • Configuration D1 : ΘJ and Θ′J . The corresponding block is 0 0 MD1 = . 0 0 • Configuration D2 : Θ si✐ j✐ or Θ s✐ si✐. j✐ i j✐ MD2 Θ3 • Configuration D3 : Θ2 Θ1 s j✐ s i✐ j i s The corresponding block is = d3 . s . The corresponding block is i j s j✐ s i✐ TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 13  eΘ21 eΘ2 eΘ23  ad abd b d = acd bcd 0 , c2 d 0 0 MD3 whose characteristic polynomial is ˇ 3 ). PD3 = (X − d3 )(X 2 + (dd) Θ4 j s Θ3 • Configuration D4 : Θ 2 s ❅i ❅s i j s s . The corresponding block is ❅ j ❅s i Θ1 eΘ4 eΘ3 eΘ2  2eΘ1  a(a + bc) 2a2 b 2ab2 b3  a2 c 2abc b2 c 0  , MD4 =  2  ac bc2 0 0 c3 0 0 0 whose characteristic polynomial is ˇ 3 )(X − dˇ3 ). PD4 = (X − d3 )(X 2 + (dd) Θ6 i s Θ4 • Configuration D5 : Θ2 Θ1  MD5 s s ❅j❅i s ❅s ❅s Θ5 j i j i s s s ❅j ❅ i ❅s ❅s s Θ3 . The corresponding block is i eΘ2 eΘ1 eΘ3 a(a2 + bc) a2 b a2 b  a2 c abc abc   a2 c abc abc =  ac2 0 bc2   ac2 bc2 0 3 c 0 0 eΘ4 ab2 0 b2 c 0 0 0 eΘ5 ab2 b2 c 0 0 0 0 eΘ6  3 whose characteristic polynomial is b 0  0 , 0  0 0 ˇ 3 )2 (X − dˇ3 ). PD5 = (X − d3 )(X 2 + (dd) Moreover, it is easily checked that the polynomial PD4 already annihilates MD5 . 3.5. Consequences on annihilating polynomials. Lemma 21. Let ϕ ∈ L (V ), f, f ′ ∈ V ⋆ and e, e′ ∈ V be such that ϕ(e) = ϕ(e′ ) = 0 and f (e′ ) = f ′ (e) = 0. We set fe = f (e) and fe′ ′ = f ′ (e′ ). Then we get the following identities in L (V ), for all n ∈ N : "n−1 # X (i) (ϕ + f ⊗ e)n = ϕn + f fek ϕn−1−k ⊗ e, k=0 14 ANATOLE CASTELLA ′ ′ n n (ii) (ϕ+f ⊗e+f ⊗e ) = ϕ +f "n−1 X k=0 fek ϕn−1−k # ⊗e+f ′ "n−1 X Proof. By induction on n, using the identities of [6, Rem. 1]. k=0 fe′k′ ϕn−1−k # ⊗e′ . Lemma 22. Let J = {i, j} be an orbit of I under Σ, with mi,j = 3. Then fJ (eΘ′J ) = fJ′ (eΘJ ) = 0 and fJ (eΘJ ) = fJ′ (eΘ′J ). Proof. With the formulas of notation 16 and lemma 17, we get for the first point : fJ (eΘ′J ) = fi ϕj ϕi (eαi +αj ) = fi ϕj (beαj ) = 0, and fJ′ (eΘJ ) = cfi ϕj (eαi + eαj ) = cfj ϕi (eαi ) + cfi ϕj (eαj ) = 0, and for the second point : fJ (eΘJ ) = fi ϕj ϕi (eαi + eαj ) = fi ϕj ϕi (eαj ) = fi (bceαi ) = bcfi (eαi ), and fJ′ (eΘ′J ) = cfi ϕj (eαi +αj ) = cfi (beαi ) = bcfi (eαi ). Proposition 23. Let J be a spherical orbit of I under Σ, and consider P ∈ R[X]. Then there exists a polynomial Q ∈ R[X] (that depends on J and P ) such that : Σ (i) P (ψJΣ ) = P (ϕΣ J ) + fJ Q(ϕJ ) ⊗ eΘJ if J is of type A, B or C, ′ Σ Σ Σ (ii) P (ψJ ) = P (ϕJ ) + fJ Q(ϕΣ J ) ⊗ eΘJ + fJ Q(ϕJ ) ⊗ eΘ′J if J is of type D. Proof. For types A, B and C, the result follows from proposition 18 and lemma 21 (i), which applies since ϕΣ J (eΘJ ) = 0 (cf. configurations A1, B1 and C1). For type D, lemma 21 (ii) applies to the decomposition of ψJΣ given in proposition 18 Σ (ii) since ϕΣ J (eΘJ ) = ϕJ (eΘ′J ) = 0 (cf. configuration D1), and since fJ (eΘ′J ) = ′ fJ (eΘJ ) = 0 by the previous lemma. Hence we get two polynomials Q and Q′ such ′ ′ Σ Σ that P (ψJΣ ) = P (ϕΣ J ) + fJ Q(ϕJ ) ⊗ eΘJ + fJ Q (ϕJ ) ⊗ eΘ′J . But the two polynomials Q and Q′ thus obtained are in fact equal since, again by the previous lemma, fJ (eΘJ ) = fJ′ (eΘ′J ). Pd n Remark 24. More precisely, if P = in proposition 23, then Q = n=0 pn X Pd−1 n q X with q = p and q = p + q f (e d−1 d n−1 n n J ΘJ ) for every 1 6 n 6 d − 1. n=0 n Notation 25. Let J be a spherical orbit of I under Σ. Depending on J, we define a polynomial PJ ∈ R[X] by : ˇ if J is of type A, (i) PJ = (X − d)(X − d) 2 ˇ (ii) PJ = (X − d )(X − dd)(X − dˇ2 ) if J is of type B, 3 2ˇ (iii) PJ = (X − d )(X − d d)(X − ddˇ2 )(X − dˇ3 ) if J is of type C, ˇ 3 )(X − dˇ3 ) if J is of type D. (iv) PJ = (X − d3 )(X 2 + (dd) Lemma 26. Let J be a spherical orbit of I under Σ. Then ′ (i) PJ (ϕΣ J )(eΘ ) = 0 when Θ 6= ΘJ (resp. when Θ 6= ΘJ and Θ 6= ΘJ ) if J is of type A, B or C (resp. D), ′ (ii) PJ (ϕΣ J )(eΘ ) = PJ (0)eΘ when Θ = ΘJ (resp. when Θ = ΘJ or Θ = ΘJ ) if J is of type A, B or C (resp. D). Proof. In view of the results of subsections 3.1 to 3.4 above, the polynomial PJ Σ annihilates the non-zero blocks of ϕΣ J , hence we get that PJ (ϕJ )(eΘ ) = 0 if Θ 6= ′ ΘJ (resp. if Θ 6= ΘJ and Θ 6= ΘJ ) for types A, B and C (resp. D). Moreover Σ since ϕΣ J (eΘJ ) = 0 for all cases, and since ϕJ (eΘ′J ) = 0 for type D, we get that Σ PJ (ϕJ )(eΘJ ) = PJ (0)eΘJ for all cases, and that PJ (ϕΣ J )(eΘ′J ) = PJ (0)eΘ′J for type D. TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 15 Proposition 27. Let J be a spherical orbit of I under Σ. The image of PJ (ψJΣ ) is included in ReΘJ (resp. in ReΘJ ⊕ ReΘ′J ) if J is of type A, B or C (resp. D). As a consequence, the polynomial (X − fJ (eΘJ ))PJ annihilates ψJΣ . Proof. The previous lemma shows that the image of PJ (ϕΣ J ) is included in ReΘJ (resp. in ReΘJ ⊕ ReΘ′J ) for types A, B and C (resp. D). But by proposition 23, Σ PJ (ψJΣ ) is the sum of PJ (ϕΣ whose image is J ) and of some endomorphism of V included in ReΘJ (resp. in ReΘJ ⊕ ReΘ′J ) for types A, B and C (resp. D), whence the first point of the proposition. The second point for types A, B and C (resp. D) follows from proposition 18 (i) (resp. proposition 18 (ii) and lemma 22), and the studies of configurations A1, B1, and C1 (resp. D1), which show that eΘJ (resp. each of eΘJ and eΘ′J ) is an eigenvector of ψJΣ for the eigenvalue fJ (eΘJ ). Remark 28. Let us denote by f the common value of the fi (eαi ) for i ∈ J (recall that we are assuming that fi = fσ(i) σV for all i ∈ I). Then it is easily checked on the definitions that the value of fJ (eΘJ ) is f (resp. df , resp. d2 f , resp. bcf ) if J is of type A (resp. B, resp. C, resp. D). 4. The spherical case We assume here that Γ is of spherical type An (n > 2), Dn (n > 4) or E6 . In all the cases but D4 , we fix Σ = Aut(Γ) (of order two), and when Γ is of type D4 , we take for Σ a subgroup of Aut(Γ) (which is of order six) of order two or three. The Coxeter matrix ΓΣ = (mΣ J,K )J,K∈I Σ — which encodes the presentations of the Coxeter group W Σ , the Artin-Tits monoid (B + )Σ and the Artin-Tits group B Σ — is then of type Bn if Γ = A2n−1 , A2n or Dn+1 with \|Σ\| = 2, of type F4 if Γ = E6 , or of type G2 if Γ = D4 with \|Σ\| = 3 (see for example [9, 5]). We fix an integral domain R and (a, b, c, d) ∈ R4 such that d2 − ad − bc = 0. Let (fi )i∈I ∈ (V ⋆ )I be an LK-family relatively to (a, b, c, d) and consider the associated LK-representation ψ : B + → L (V ) of B + . We will always assume in this section that the elements b, c and d are non-zero in R, so that they become invertible in the field of fractions K of R. By extension of the scalars from R to K, we will consider ψ as acting on the K-vector space VK = V ⊗R K. It then follows from proposition 10 that ψ induces a twisted LKrepresentation ψ Σ : (B + )Σ → L (VKΣ ) of (B + )Σ over VKΣ = K ⊗R V Σ . By [6, Section 3.2], the LK-family (fi )i∈I , seen as an element of (VK⋆ )I , is entirely determined by the common value f of the elements fi (eαi ), i ∈ I. Moreover when f 6= 0, then ψ has its image included in GL(VK ) and hence induces an LK-representation ψgp : B → GL(VK ) of the Artin-Tits group B, and a twisted Σ LK-representation ψgp : B Σ → GL(VKΣ ) of the Artin-Tits group B Σ . In view of the results of the previous section, in order to really understand the Σ maps ψJΣ = ψ∆ , J ∈ I Σ , we need to list the possible configurations of the J-meshes J + MJ (Θ) in Φ , when Θ runs through Φ+ /Σ. This is done in the following lemma. Lemma 29. Fix an orbit J of I under Σ. We list, in the following tables, the number NX of occurrences of configuration X (with the notations of subsections 3.1 to 3.4) among the J-meshes MJ (Θ) in Φ+ , when Θ runs through Φ+ /Σ. (The configurations that do not occur are omitted). 16 ANATOLE CASTELLA • If Γ = A2n−1 , or Dn with \|Σ\| = 2, or E6 , then J can be of type A or B and we get : A2n−1 Dn (\|Σ\| = 2) E6 NA1 1 1 1 if J is of type A NA2 NA3 (n − 1)2 n−1 n2 −6n+10 2n−5 9 7 NB1 1 1 1 if J is of type B NB2 NB3 NB4 (n − 2)2 2n − 4 1 (n−2)(n−3) 0 n−2 6 4 3 • If Γ = A2n , then J can be of type B (if n > 2) or D and we get : NB1 1 if J is of type B NB2 NB3 (n − 1)(n − 2) 2n − 3 NB4 1 if J is of type D ND1 ND2 ND3 1 (n − 1)2 n − 1 • If Γ = D4 with \|Σ\| = 3, then J can be of type A or C and we get : if J is of type A NA1 NA2 NA3 1 1 2 Proof. Left to the reader. if J is of type C NC1 NC2 NC5 1 1 1 4.1. Irreducibility. In this subsection, we exclude the case Γ = A2n , so that there is no orbit of type D in I, and hence no configurations D1 to D5 in Φ+ . Lemma 30. For all Θ ∈ Φ+ /Σ with dp(Θ) > 2, there exists J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ). Proof. Fix α ∈ Θ and i ∈ I such that dp(si (α)) < dp(α), and let us denote by J the orbit of i under Σ. Necessarily, the J-mesh MJ (Θ) is in configuration A3, B3, B4 or C5, and Θ is not of minimal depth in MJ (Θ). In case of a configuration A3, B3, B4 with Θ of maximal depth in MJ (Θ), or C5 with Θ of maximal or second-maximal depth in MJ (Θ), then dp(rJ (Θ)) < dp(Θ) and we are done. In the other cases, then (dp(rJ (Θ)) > dp(Θ) and) a look at the graph of Φ+ in the different possible situations reveals that there exists i′ ∈ I with dp(si′ (α)) < dp(α) for which, if we denote by J ′ its orbit under Σ, then MJ ′ (Θ) is in configuration A3 or B3, so here again we are done, up to changing J for J ′ . In the following proposition, we prove the first part of our irreducibility criterion. Notation 31. If J if an orbit of I under Σ, we consider the polynomial PJ as in notation 25. In view of proposition 23, there exist a polynomial QJ ∈ R[X] such Σ that PJ (ψJΣ ) = PJ (ϕΣ J ) + fJ QJ (ϕJ ) ⊗ eΘJ . + By lemma 26, we get PJ (ψJΣ )(eΘ ) = fJ QJ (ϕΣ J )(eΘ )eΘJ for every Θ ∈ Φ /Σ with ) = 0 for J, K ∈ I Σ with )(e Θ 6= ΘJ . Moreover it can be checked that fJ QJ (ϕΣ ΘK J = 2. mΣ J,K Σ with Proposition 32. Assume that fJ QJ (ϕΣ J )(eΘK ) 6= 0 for every J, K ∈ I Σ Σ Σ Σ mJ,K > 3. If U is an invariant subspace of VK under ψ such that PK (ψK )(U ) 6= {0} for some K ∈ I Σ , then U = VKΣ . TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 17 T Proof. By proposition 27 and since U is invariant, we get PJ (ψJΣ )(U ) ⊆ U ReΘJ Σ )(U ) 6= {0}. Now if J ∈ I Σ is distinct for all J ∈ I Σ , whence eΘK ∈ U since PK (ψK Σ from K, we get that PJ (ψJ )(eΘK ) = fJ QJ (ψJΣ )(eΘK )eΘJ belongs to U , whence Σ eΘJ ∈ U if mΣ J,K > 3 since then, by assumption, fJ QJ (ϕJ )(eΘK ) 6= 0. We thus get that eΘJ ∈ U for all J ∈ I Σ by connectivity of ΓΣ . Now consider Θ ∈ Φ+ /Σ, of depth p > 2, such that every eΘ′ belongs to U whenever dp(Θ′ ) < p. By lemma 30, there exists J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ), so in particular erJ (Θ) ∈ U . Necessarily, the J-mesh MJ (Θ) is in configuration A3, B3, B4 with Θ of maximal depth in MJ (Θ), or C5 with Θ of maximal or second-maximal depth in MJ (Θ), and hence ψJΣ (erJ (Θ) ) is a linear combination of eΘ , erJ (Θ) , eΘJ and possibly some other eΘ′ for Θ′ ⊆ MJ (Θ) with dp(Θ′ ) < p, in which the coefficient of eΘ is c, cd, c2 , c3 or bc2 respectively. Since U is invariant by ψJΣ and since bcd 6= 0 by assumption, we thus get, in all the cases, the element eΘ as a linear combination of elements of U , hence eΘ ∈ U and we conclude by induction that U = VKΣ . The following lemma explicits the condition on fJ QJ (ϕΣ J )(eΘK ) of the previous proposition (recall that f is the common value of the fi (eαi ) for i ∈ I, and that we set dˇ = a − d) : Lemma 33. Consider two orbits J, K ∈ I Σ with mΣ J,K > 3. Then the coefficient Σ fJ QJ (ϕJ )(eΘK ) is equal to : (i) −\|K\|af if J is of type A, (ii) ad2 f (dˇ2 − df ) if J is of type B, (iii) ad5 f (−d3 f 2 + addˇ2 f − dˇ5 ) if J is of type C. Proof. First notice that the assumption mΣ J,K > 3 implies that mj,k = 3 for some (j, k) ∈ J × K (see [9, Section 3] or [5, Section 4.2]). Now it is easy to see that if J is of type A (resp. B, resp. C), then ΘK necessarily appears at the bottom of a configuration A3 (resp. B3 or B4, resp. C5). The result then follows from direct computations, using the expression of QJ deduced from remark 24 and notation 25, and the expressions of the encountered fJ (eΘ ) as multiples of f deduced from [6, Table 1]. Now we turn to the second part of the irreducibility criterion. Proposition 34. Assume that there exists a prime ideal Q of R with bcd 6∈ Q and Im(fi ) ⊆ Q for all i ∈ I. If U is an invariant subspace of VKΣ under ψ Σ such that PJ (ψJΣ )(U ) = {0} for all J ∈ I Σ , then U = {0}. Proof. Recall that the image of PJ (ψJΣ ) is included in KeΘJ . Since U is invariant, we get PJ (ψJΣ )ψgΣ (U ) = {0} for every (J, g) ∈ I Σ × (B + )Σ , and hence, if we denote by L(J,g),Θ the element of R such that PJ (ψJΣ )ψgΣ (eΘ ) = L(J,g),Θ eΘJ , and by L the matrix (L(J,g),Θ )(J,g)∈I Σ ×(B + )Σ ,Θ∈Φ+ /Σ , we get that U is included in ker(L). Now we claim that under the assumptions of the proposition, the matrix L is non-singular, which thus implies that U = {0}. To prove our claim, it suffices to construct some pairs (JΘ , gΘ ) ∈ I Σ ×(B + )Σ for Θ running through Φ+ /Σ, such that the square submatrix L′ = (L(JΘ ,gΘ ),Ω )Θ,Ω∈Φ+ /Σ of L is invertible. We construct the pairs (JΘ , gΘ ) ∈ I Σ × (B + )Σ by induction on dp(Θ) as follows. 18 ANATOLE CASTELLA If dp(Θ) = 1, i.e. if Θ = ΘJ for some J ∈ I Σ , we set JΘ = J and gΘ = 1. Now by induction if dp(Θ) > 2, then we fix some J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ) (which exists by lemma 30) and we set JΘ = JrJ (Θ) and gΘ = grJ (Θ) ∆J . We are going to see that k l m Σ (i) PJΘ (ϕΣ JΘ )ϕgΘ (eΘ ) = ±b c d eΘJΘ for some k, l, m ∈ N, Σ Σ (ii) PJΘ (ϕJΘ )ϕgΘ (eΩ ) = 0 for Ω ∈ Φ+ /Σ with dp(Ω) > dp(Θ) and Ω 6= Θ. Σ Since PJ (ψJΣ )ψgΣ (eΘ ) ≡ PJ (ϕΣ J )ϕg (eΘ ) mod Q, points (i) and (ii) will show that ′ the matrix L is lower triangular modulo Q — if we choose an order on Φ+ /Σ that is consistent with depth — with non-zero diagonal coefficients modulo Q, since bcd 6∈ Q. Hence this will show that the determinant of L′ is non-zero modulo Q (recall that Q is prime), and therefore is non-zero in R, whence the result. So it remains to prove (i) and (ii). Let us prove (i). If Θ = ΘJ for some J ∈ I Σ , then by lemma 26 we get 3 6 PJ (ϕΣ J )(eΘJ ) = PJ (0)eΘJ , with PJ (0) = −bc (resp. (bc) , resp. (bc) ) if J is of type A (resp. B, resp. C), whence the result when dp(Θ) = 1. Now assume that dp(Θ) > 2 and consider the fixed J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ). If MJ (Θ) is in configuration A3, B3, B4 or C5, with Θ of maximal depth in 2 3 MJ (Θ), then ϕΣ J (eΘ ) = λerJ (Θ) with λ = b, bd, b or b respectively, whence the result by induction. The only other possible configuration is C5 with Θ of second maximal depth in MJ (Θ), which occurs when Γ = D4 with \|Σ\| = 3. If we label the vertices of D4 as in [2, Planche IV], we thus get I Σ = {J, K} with J = {1, 3, 4} and K = {2}, Θ = {α1 + α2 + α3 , α1 + α2 + α4 , α2 + α3 + α4 }, gΘ = ∆K ∆J , and Σ 9 7 Σ 3 JΘ = J, whence ϕΣ gΘ (eΘ ) = b ceΘJ and PJΘ (ϕJΘ )ϕgΘ (eΘ ) = b c eΘJΘ . Let us prove (ii). By lemma 26, it suffices to see that ϕΣ gΘ (eΩ ) is a linear combi′ ′ nation of some eΘ with Θ 6= ΘJΘ . This is clear if Θ = ΘJ since then gΘ = 1. So assume that dp(Θ) > 2 and consider the fixed J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ). In view of the results of section 3, ϕΣ J (eΩ ) is a linear combination of some elements eΩ′ where Ω′ is an orbit included in MJ (Ω), which hence satisfies dp(Ω′ ) > dp(Ω)−p, where p = 1, 2 or 3 when J is of type A, B or C respectively. If dp(Ω′ ) > dp(rJ (Θ)) and Ω′ 6= rJ (Θ) for every orbit Ω′ ⊆ MJ (Ω), we get by ′ induction that ϕΣ grJ (Θ) (eΩ′ ) is a linear combination of elements eΘ′ with Θ distinct from ΘJrJ (Θ) = ΘJΘ , whence the result. The only obstructions to this situation are the two following configurations : • MJ (Θ) in configuration B3, MJ (Ω) in configuration B4, and dp(Θ) = dp(Ω) maximal in MJ (Θ) and in MJ (Ω). • MJ (Θ) = MJ (Ω) in configuration C5, Ω of maximal depth, and Θ of second maximal depth, in MJ (Θ). The second point occurs if Γ = D4 with Ω = {α1 + α2 + α3 + α4 } and Θ, J and K 3 Σ as in the proof of (i) above (hence gΘ = ∆K ∆J ), whence ϕΣ gΘ (eΩ ) = b ϕK (eΘK ) = 0 and the result in that case. The first point occurs once if Γ = E6 , for J = {3, 5}, Ω = {α2 + α3 + α4 + α5 } and Θ = {α1 + α3 + α4 + α5 , α3 + α4 + α5 + α6 } with the notations of [2, Planche V]. But the possibilities for (JΘ , gΘ ) are then ({1, 6}, ∆J ∆{2} ∆J ), (J, ∆{1,6} ∆{2} ∆J ) 3 2 or (J, ∆{2} ∆{1,6} ∆J ), whence ϕΣ gΘ (eΩ ) = b d e{α2 } and the result in that case. The first point also occurs if Γ = A2n−1 (n > 4), for J = {n − j, n + j}, Pn+3j Pn−j Pn+j Ω = { i=n−j αi } and Θ = { i=n−3j αi , i=n+j αi } where 1 6 j 6 n−1 3 , with the notations of [2, Planche I]. But in that case we necessarily get gΘ = grK rJ (Θ) ∆K ∆J TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 19 where K = {n − k, n + k} for some j + 1 6 k 6 3j. Hence the K-mesh MK (rJ (Ω)) Σ 2 is in configuration B2 and we get ϕΣ K ϕJ (eΩ ) = (bd) erJ (Ω) , with dp(rJ (Ω)) = dp(Θ) − 2 = dp(rK rJ (Θ)) and rJ (Ω) 6= rK rJ (Θ), whence the result in those cases by induction on rK rJ (Θ). Since there is no configuration B3 in Φ+ for Γ = Dn , the proof is completed. Theorem 35. Assume that condition (⋆) holds : (⋆) R is an integral domain, fi (eαi ) 6= 0 for all i ∈ I, and there exists a totally ordered integral domain R0 and a ring homomorphism ρ : R → R0 , such that ρ(t) > 0 for t ∈ {a, b, c, d}, and Im(fi ) ⊆ ker(ρ) for all i ∈ I. Σ Then the LK-representations ψ Σ and ψgp are irreducible over VKΣ . Proof. The result follows from propositions 32 and 34 : the second one applies with Q = ker(ρ), and the first one applies since under condition (⋆), we get a, d and ˇ < 0 (since ρ(dd) ˇ = ρ(−bc) < 0), so f = fi (eαi ) non-zero in R, f ∈ ker(ρ) and ρ(d) Σ Σ lemma 33 shows that the coefficients fJ QJ (ϕJ )(eΘK ), for mJ,K > 3, are non-zero in R (look at the image of the parenthesized expressions by ρ). Remark 36. This result extends to all the spherical and connected cases but A2n what was done for Γ = E6 in [18, Section 6.1], by a totally different method. The question for Γ = A2n (n > 2) is still open, as far as I know. The strategy used here does not directly apply because of the orbit J of type D in Γ : first, lemma 33 above is not true in this case (look at the orbit Θ′J ), and second, the image of PJ (ψJΣ ) is not of dimension one, but two (at least when f 6= 0). The case Γ = A2 is trivial however, since then I Σ = {I}, VKΣ = KeΘI ⊕ KeΘ′I , and ψIΣ is the homothety of ratio bcf , hence in this case, ψ Σ is not irreducible. 4.2. Non-equivalence. In corollary 39 and theorem 41 below, we study the case of two twisted LKrepresentations ψ Σ and ψ ′Σ of BΓ+Σ ≈ (BΓ+ )Σ induced by two LK-representations ψ and ψ ′ of BΓ+ for a fixed Coxeter matrix Γ. On the contrary in proposition 42 below, we focus on the particular case of the Coxeter type Bn , n > 3, and study the case of three twisted LK-representations ′ ′′ + ψ Σ , ψ ′Σ and ψ ′′Σ of BB induced by three LK-representations ψ, ψ ′ and ψ ′′ of n + + + BΓ , BΓ′ and BΓ′′ respectively, where Γ = A2n−1 , Γ′ = Dn+1 and Γ′′ = A2n . But let us begin with a result on the diagonalizability of the maps ψJΣ . Proposition 37. Let J be an orbit of I under Σ. We set dˇ = a − d. (i) If J is of type A and if d, dˇ and f are pairwise distinct, then ψJΣ is diagonalizable over K, with eigenvalues d, dˇ and f of multiplicity NA2 + NA3 , NA3 and NA1 respectively. ˇ dˇ2 and df are pairwise distinct, then ψ Σ (ii) If J is of type B and if d2 , dd, J ˇ dˇ2 and df of multiplicity is diagonalizable over K, with eigenvalues d2 , dd, NB2 + NB3 + NB4 , NB3 + NB4 , NB4 and NB1 respectively. Proof. Let us prove (i). In view of proposition 27, the map ψJΣ is diagonalizable ˇ But we more precisely claim over K and its eigenvalues can be fJ (eΘJ ) = f , d or d. that for any J-mesh M in configuration A1 (resp. A2, resp. A3), some suitable linear combinations of eΘJ and eΘ , Θ ⊆ M , are eigenvectors of ψJΣ for the value f ˇ whence the result. (resp. d, resp. d and d), 20 ANATOLE CASTELLA The claim is obvious for A1 since eΘJ is clearly an eigenvector of ψJΣ for the value f . For configuration A2, the linear combination eΘ + λeΘJ is suitable if λ satisfies fJ (eΘ ) + λf = dλ, and such a λ exists in K since f 6= d by assumption. Now since ˇ the block of ϕΣ for a J-mesh M in configuration A3 is certainly similar d 6= d, J ˇ i.e. there certainly exists a linear combination to the diagonal matrix Diag(d, d), ẽ of eΘ and erJ (Θ) , where M = Θ ∪ rJ (Θ), that is an eigenvector of ϕΣ J for the ˇ eigenvalue d (resp. d). The linear combination ẽ + λeΘJ is then an eigenvector of ˇ whenever fJ (ẽ) + λf = dλ (resp. dλ), ˇ and such a ψJΣ for the eigenvalue d (resp. d) ˇ λ exists in K since f 6= d (resp. f 6= d) by assumption. The proof of (ii) is similar (recall that there is no configuration B5 in Φ+ ). Remark 38. One can prove an analogue of the previous results for the orbit J of type C in Γ = D4 (when \|Σ\| = 3), but we will not need it in what follows. The ˇ ddˇ2 , eigenvalues in this case would be d3 (of multiplicity NC2 + NC5 = 2), and d2 d, 3 2 ˇ d and d f (of multiplicity NC5 = NC1 = 1). There is also an analogue for the orbit J of type D in Γ = A2n , if we assume ˇ 3 of PJ splits in K[X]. that −ddˇ = bc is a square in K, so that the factor X 2 + (dd) Now let us fix another LK-representation ψ ′ : B + → L (V ) associated with a quadruple (a′ , b′ , c′ , d′ ) ∈ R4 (such that d′2 − a′ d′ − b′ c′ = 0) and an LK-family (fi′ )i∈I relatively to (a′ , b′ , c′ , d′ ). We assume that b′ , c′ and d′ are non-zero in R. As in the preamble of the current section, the LK-family (fi′ )i∈I , seen as an element of (VK⋆ )I , is entirely determined by the common value f ′ of the fi′ (eαi ) for i ∈ I, and the LK-representation ψ ′ induces a twisted LK-representation ψ ′Σ : (B + )Σ → L (VKΣ ) of (B + )Σ . Corollary 39. We set dˇ = a − d and ď′ = a′ − d′ . (i) Assume that Γ 6= A2n , A3 and that d, dˇ and f (resp. d′ , ď′ and f ′ ) are pairwise distinct. Then for any orbit J ∈ I Σ of type A, the maps ψJΣ and ˇ f ) = (d′ , ď′ , f ′ ). ψJ′Σ are similar over K if and only if (d, d, ˇ dˇ2 and df (resp. d′2 , d′ ď′ , (ii) Assume that Γ = A2n , n > 3, and that d2 , dd, ď′ 2 and d′ f ′ ) are pairwise distinct. Then for any orbit J of type B, the ˇ f ) = ±(d′ , ď′ , f ′ ). maps ψJΣ and ψJ′Σ are similar over K if and only if (d, d, ˇ f ) 6= (d′ , ď′ , f ′ ) (resp. if Γ = A2n , n > 3, In particular, if Γ 6= A2n , A3 and (d, d, ′ ′ ′ ˇ and (d, d, f ) 6= ±(d , ď , f )), then the twisted LK-representations ψ Σ and ψ ′Σ are not equivalent over K. Proof. Of course if ψ Σ and ψ ′Σ are equivalent, then the maps ψJΣ and ψJ′Σ are similar for every J ∈ I Σ . But in view of proposition 37 above, the maps ψJΣ and ψJ′Σ are similar if and only if they have the same eigenvalues, with same multiplicity. In case (i), the three multiplicities are NA2 + NA3 > NA3 > NA1 (the second inequality is strict since we avoid the case Γ = A3 ), whence the result. In case (ii), the four multiplicities are NB2 + NB3 + NB4 > NB3 + NB4 > NB4 = NB1 (the first inequality is strict since we avoid the case Γ = A4 ), so ψJΣ and ψJ′Σ are similar if and only if d2 = d′2 , ddˇ = d′ ď′ and {dˇ2 , df } = {ď′ 2 , d′ f ′ }. And these ˇ f ) = ±(d′ , ď′ , f ′ ). three equalities are clearly equivalent to (d, d, Remark 40. The case Γ = A3 (resp. A4 ) can be dealt with as in case (i) (resp. (ii)) of the previous proposition and its proof, but the criterion of similarity of ψJΣ and TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 21 ˇ f } = {ď′ , f ′ } ψJ′Σ for J of type A (resp. B) as then to be relaxed to d = d′ and {d, 2 ′ ′ 2 2 ′2 ′ ′ ′ ˇ ˇ (resp. {d , dd} = {d , d ď } and {d , df } = {ď , d f }). The case Γ = A2 is trivial since then I Σ = {I}, VKΣ = KeΘI ⊕ KeΘ′I , and ψIΣ (resp. ψI′Σ ) is the homothety of ratio bcf (resp. b′ c′ f ′ ). Hence in this case, the twisted LK-representations ψ Σ and ψ ′Σ are equivalent if and only if bcf = b′ c′ f ′ . Theorem 41. Assume that Γ 6= A2 and that condition (⋆) holds for both ψ and ˇ f ) 6= ψ ′ , for the same morphism ρ : R → R0 if Γ = A3 or Γ = A2n . Then if (d, d, ′ ′ ′ Σ ′Σ (d , ď , f ), the twisted LK-representations ψ and ψ are not equivalent over K. Proof. Under condition (⋆), the elements d, dˇ and f are pairwise distinct since ˇ the result their image by ρ are positive, negative and zero respectively (for d, 2 ˇ follows from the identity dd = −bc). Similarly, the elements df , d , ddˇ and dˇ2 are ˇ < 0, ρ(d2 ) > 0 and ρ(dˇ2 ) > 0, pairwise distinct : indeed, we get ρ(df ) = 0, ρ(dd) 2 2 2 2 ˇ ˇ ˇ and d 6= d since d − d = (d − d)a and hence ρ(d2 − dˇ2 ) > 0. The analogue occurs for ψ ′ and hence the results of corollary 39 and remark 40 applies. This gives the result if Γ 6= A2n , A3 . If Γ = A2n , n > 3, then ˇ f ) 6= ±(d′ , ď′ , f ′ ), but we clearly have the condition stated in corollary 39 is (d, d, ′ ′ ′ ′ ˇ (d, d, f ) 6= −(d , ď , f ) since ρ(d) and ρ(d ) are positive, whence the result in that case. Similarly for Γ = A3 (resp. A4 ), the condition of non-equivalence derived ˇ f } 6= {ď′ , f ′ } (resp. {d2 , dd} ˇ = from remark 40 should be d 6= d′ or {d, 6 {d′2 , d′ ď′ } or 2 ′ ′ ′ 2 ˇ f } = {ď′ , f } is equivalent to (d, ˇ f ) = (ď′ , f ′ ) {dˇ , df } 6= {ď′ , d f }). But here {d, ′ ′ since ρ(d) and ρ(d ) are positive and ρ(f ) = ρ(f ) = 0, whence the result for Γ = A3 . Finally for Γ = A4 , since the images by ρ of d2 , dˇ2 , d′2 and ď′ 2 (resp. of ddˇ and d′ ď′ , resp. of df and d′ f ′ ) are positive (resp. negative, resp. zero), the ˇ = {d′2 , d′ ď′ } and {dˇ2 , df } = {ď′ 2 , d′ f ′ } of remark 40 is equivalent condition {d2 , dd} 2 2 ˇ f ) = (d′ , ď′ , f ′ ) since, again, ρ(d) ˇ ˇ to (d , dd, d , df ) = (d′2 , d′ ď′ , ď′ 2 , d′ f ′ ), i.e. (d, d, ′ and ρ(d ) are positive. Let us finally turn to the case of the Coxeter type Bn by considering three twisted ′ ′′ + LK-representations ψ Σ , ψ ′Σ and ψ ′′Σ of BB induced by three LK-representations n + + + ′ ′′ ψ, ψ and ψ of BΓ , BΓ′ and BΓ′′ respectively, where Γ = A2n−1 , Γ′ = Dn+1 and Γ′′ = A2n . ′′ The twisted LK-representation ψ ′′Σ is clearly non-equivalent to the two others ′′ since it is of dimension \|Φ+ Γ′′ /Σ \| = n(n+1) whereas the two others are of dimension ′ + + ′ 2 \|ΦΓ /Σ\| = \|ΦΓ′ /Σ \| = n . The following proposition proves that ψ Σ ans ψ ′Σ are not equivalent, under the usual assumptions on the parameters (a, b, c, d) and (fi )i∈I of ψ, and (a′ , b′ , c′ , d′ ) and (fi′ )i∈I ′ of ψ ′ . We set as above dˇ = a − d and f = fi (eαi ) for any i ∈ I. Proposition 42. Assume that d, dˇ and f (resp. df , d2 , ddˇ and dˇ2 ) are pairwise distinct, and that the analogue holds for ψ ′ , which is the case for example if condition (⋆) holds for both ψ and ψ ′ . Then the twisted LK-representations ψ Σ and ′ ψ ′Σ are not equivalent over K. Proof. The fact that condition (⋆) implies the given conditions on the parameters has been shown in the proof of theorem 42. But then proposition 37 applies and + shows that the images by ψ and ψ ′ of a given standard generator of BB do not n have the same number of eigenvalues, whence the result. 22 ANATOLE CASTELLA References [1] S. Bigelow. Braid groups are linear. J. Amer. Math. Soc. 14 (2001), 471–486. [2] N. Bourbaki. 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THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC BEN HAYES AND ANDREW SALE arXiv:1601.03286v1 [math.GR] 13 Jan 2016 Abstract. Given sofic approximations for countable, discrete groups G, H, we construct a sofic approximation for their wreath product G ≀ H. Sofic groups, introduced by Gromov [10] and developed by Weiss [19], are a large class of groups which can be approximated, in some sense, by finite groups. There are many examples, including all amenable groups, all residually finite groups, and all linear groups (by Malcev’s Theorem). However, because of the weakness of the approximation by finite groups, few permanence properties of soficity are properly understood. Relatively straight-forward examples include closure under direct product and increasing unions, and the soficity of residually sofic groups. More substantial results generally require some amenability assumption. For example, an amalgamated product of two sofic groups is know to be sofic if the amalgamated subgroup is amenable (see [9],[15],[4],[17]). This was extended to encompass the fundamental groups of all graphs of groups with sofic vertex groups and amenable edge groups [3]. In the same paper, it is shown that the graph product of sofic groups is sofic. Also, if H is sofic and is a coamenable subgroup of G, then G is sofic too [8]. We prove a new permanence result, namely that soficity is closed under taking wreath products: Theorem 1. Let G, H be countable, discrete, sofic groups. Then G ≀ H is sofic. We remark that our result is general and requires no amenability or residual finiteness assumptions. The special case of Theorem 1 when G is abelian was proved by Paunescu [15], who used methods of analysis and the notion of sofic equivalence relations developed by Elek and Lippner [5]. While finishing this paper, we learnt that Holt and Rees have dealt with the case when H is residually finite and G sofic [12]. We prove the general result directly, by constructing a sofic approximation for the wreath product, giving a proof that is constructive, quantitative (see Proposition 2.1), and entirely self-contained. The notion of hyperlinearlity gives a class of groups defined in a similar vein to sofic groups, but where they are approximated instead by unitary groups. Our construction extends to the situation where G is hyperlinear and H sofic, showing that G ≀ H is hyperlinear, see [11]. Via their approximations by finite groups, sofic groups have applications to problems of current mathematical interest in a wide area of fields. Sofic groups are relevant to ergodic theory because they are the largest class of groups for which Bernoulli shifts are classified by their base entropy (see [1],[13]) and for which Gottschalk’s surjuncitivity conjecture holds (see e.g. [10],[13]). In the study of group rings they are useful because they are the largest class of groups for which Kaplansky’s direct finiteness conjecture (see [6]) is known. In the field of L2 –invariants, they are the largest class of groups for which the determinant conjecture is known (see [7]), which is necessary to define L2 –torsion (see [14] Conjecture 3.94). They are also the largest class of groups for which an analogue of Lück approximation is known (see [18]). We refer the reader to [14] for applications of L2 –invariants to geometry and group theory. See also [16],[2] for a survey of sofic groups. Date: April 11, 2018. 1 THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 2 Acknowledgments. The first named author would like to thank Jesse Peterson for asking him if wreath products of sofic groups are sofic at the NCGOA Spring Institute in 2012 at Vanderbilt University. 1. Preliminaries We begin with the necessary definitions, as well as a useful lemma to help us identify sofic approximations in wreath products. Definition 1. Let A be a finite set. The normalized Hamming distance, denoted dHamm , on Sym(A) is defined by dHamm (σ, τ ) = 1 \|{a ∈ A : σ(a) 6= τ (a)}\|. \|A\| Definition 2. Let G be a countable discrete group, F a finite subset of G, and ε > 0. Fix a finite set A and a function σ : G → Sym(A). We say that σ is (F, ε)–multiplicative if max dHamm (σ(g)σ(h), σ(gh)) < ε. g,h∈F We say that σ is (F, ε)–free if min dHamm (σ(g), Id) > 1 − ε. g∈F \{1} We say that σ is an (F, ε)–sofic approximation if it is (F, ε)–multiplicative, (F, ε)–free, and furthermore σ(1) = Id . Lastly, we say that G is sofic if for every finite F ⊆ G and ε > 0, there is a finite set A and an (F, ε)–sofic approximation σ : G → Sym(A). Our aim is to use sofic approximations for G and H and build a sofic approximation for G ≀ H. First recall that the wreath product is defined as M G≀H = G⋊H H L where the action of h ∈ H is given via αh ∈ Aut ( H G), defined by αh (gx )x∈H = (gh−1 x )x∈H . A homomorphism π : L G ≀ H → K, for some group K, can be decomposed into a pair of homomorphisms π1 : H G → K, π2 : H → K which satisfy the following equivariance condition: L π2 (h)π1 (g) = π1 (αh (g))π2 (h), for all h ∈ H, g ∈ H G. The following lemma gives an “approximate analogue” to this situation. Lemma 1.1. Let G, H beLcountable, discrete groups. For every finite set F0 ⊆ G ≀ H there are finite sets E1 ⊆ H G, E2 ⊆ H such that the following holds: Let ε > 0 and Ω be a finite set. Suppose σ : G ≀ H → Sym(Ω) is a map such that M • the restriction of σ to G is (E1 , ε/6)–multiplicative, H • the restriction of σ to H is (E2 , ε/6)–multiplicative, • • max g∈E1 ,h∈E2 max g∈E1 ,h∈E2 dHamm σ(g, h), σ(g, 1)σ(1, h) < ε/6, dHamm (σ(1, h)σ(g, 1), σ(αh (g), 1)σ(1, h)) < ε/6. Then σ is (F0 , ε)–multiplicative. THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 3 Proof. After making the right definitions for E1 , E2 , we apply the triangle inequality several times to obtain the result. We require that if (g, h), (ĝ, ĥ) are in F0 , then g, ĝ, αh (ĝ) ∈ E1 , and h, ĥ ∈ E2 . This is true if we define E1 = αh (g) : h ∈ πH (F0 ) ∪ {1}, g ∈ πG (F0 ) , E2 = πH (F0 ). We leave verification that this is sufficient to the reader. To see how this gives an “approximate analogue” of the situation for homomorphisms, notice that the above lemma says that an approximate homomorphismL σ : G≀H → Sym(A) can be thought of as a pair of approximate homomorphisms σ1 : H G → Sym(A), σ2 : H → Sym(A) so that σ2 (h)σ1 (g) ≈ σ1 (αh (g))σ2 (g) L for all g in a large enough finite subset of H G and all h in a large enough finite subset of H. (Here we use ≈ to indicate that both maps agree on a sufficiently large subset of A). It is not hard to see that the above lemma is valid with G≀H replaced with any semidirect product and Sym(A) replaced with any group equipped with a bi-invariant metric. 2. The sofic approximation for wreath products To facilitate our proof, we need to introduce some notation. Let G, H be countable discrete groups and σA : G → Sym(A), σB : H → Sym(B) be two functions (not assumed to be homomorphisms). For h ∈ H, b0 ∈ B, define ! M (h) σA,b0 : G → Sym A B by (h) ab )b∈B , σA,b0 (g) (ab )b∈B = (b where b ab = ( σA (g)(ab ), ab , if b = σB (h)b0 , otherwise. Now suppose that E ⊆ H is finite, and take a subset B0 ⊆ {b ∈ B : σB (h1 )b 6= σB (h2 )b for all h1 , h2 ∈ E with h1 6= h2 .}. (h ) (h ) Note that σA,b1 0 (g1 ) and σA,b2 0 (g2 ) commute if b ∈ B0 , g1 , g2 ∈ G, h1 , h2 ∈ H and h1 6= h2 . Thus it makes sense to define, for b ∈ B0 , ! M M σA,b : G → Sym A E by B Y (x) σA,b (gx ). σA,b (gx )x∈E = x∈E In our applications σB will be a sofic approximation, so we can take B0 to make up the majority of B. Thus σA,b will be defined for “most” b ∈ B. We package all these maps together as a single map ! ! M M σc G → Sym A ⊕B A: E by σc A (g)(a, b) = B ( σA,b (g)(a), b , if b ∈ B0 (a, b), if b ∈ B \ B0 . THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC We extend σc A to define L H 4 L L / E G. Now G by declaring that σc A (g) = Id if g ∈ H G, but g ∈ ! ! M σc A ⊕B B : H → Sym B by σc B (h)(a, b) = a, σB (h)b . Finally, define σ b : G ≀ H → Sym by M ! A B ⊕B ! σ b (g, h) = σc c A (g)σ B (h). Note that σ b is determined by the choice of E, B0 and the two maps σA and σB . Proposition 2.1. Let F ⊆ G ≀ H be finite and ε > 0. Then there are finite sets EA ⊆ G, EB ⊆ H and an ε′ > 0 so that if σA : G → Sym(A) and σB : H → Sym(B) are (EA , ε′ ), (EB , ε′ )–sofic approximations for G, H respectively, then σ b is an (F, ε)–sofic approximation. The remainder of this section is dedicating to proving Proposition 2.1. We will see below that it is possible to compute an explicit upper bound on ε′ . It will depend only on ε and the set F . We remark that σ b(1, 1) = Id by construction. We need to show it is (F, ε)–multiplicative and (F, ε)–free. First we explain how to define the sets E, EA and EB . L Let F ⊆ G ≀ H be finite and ε > 0. Define projections πG : G ≀ H → H G and πH : G ≀ H → H by πG (g, h) = g, πH (g, h) = h. Let E1 , E2 be as in Lemma 1.1 for the finite set F0 = F ∪ {1} ∪ F −1 . As in the proof of Lemma 1.1, we have E1 = αh (g) : h ∈ πH (F0 ), g ∈ πG (F0 ) , E2 = πH (F0 ). Recall that for g = (gx )x∈H ∈ ⊕H G the support of g, denoted Supp(g), is the set of x ∈ H with gx 6= 1. We set [ E = E2 ∪ h Supp(g), g∈E1 h∈E2 EA = gx ∈ G : (gx )x∈H ∈ E1 , EB = E −1 E. Then our collections of finite sets satisfy the following properties, each of which we need later on: EA ⊇ {gx : (gx ) ∈ E1 , x ∈ E}, E E ⊇ ⊇ h Supp(g) for all h ∈ E2 , g ∈ E1 , E2 , EB ⊇ E ∪ E −1 ∪ E −1 E. ε Let σA : G → Sym(A), σB : H → Sym(B) be Choose ε′ so that 0 < ε′ < 48\|E\| 2. (EA , ε′ ), (EB , ε′ )–sofic approximations respectively. Set B01 = {b ∈ B : σB (h1 )b 6= σB (h2 )b for all h1 , h2 ∈ E, h1 6= h2 }, B02 = {b ∈ B : σB (h1 h2 )b = σB (h1 )σB (h2 )b for all h1 , h2 ∈ E}, B0 = B01 ∩ B02 . THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 5 Since σB is a sofic approximation, we can intuitively think of B0 as making up most of the set B. Indeed, Lemma 2.2 below confirms this. Use the sets E ⊂ H, B0 ⊆ B and the maps σA , σB to define the maps σc c b, as constructed at the start of this section. A, σ B, σ We claim that the map σ b is an (F, ε)–sofic approximation. We first make the following preliminary observation. Lemma 2.2. Let κ > 0. If ε′ < Proof. Note that κ 4\|E\|2 \ B01 = then \|B0 \| ≥ (1 − κ)\|B\|. {b ∈ B : σB (h1 )b 6= σB (h2 )b}, h1 ,h2 ∈E h1 6=h2 B02 = \ {b ∈ B : σB (h1 )σB (h2 )b = σB (h1 h2 )b}. h1 ,h2 ∈E h1 6=h2 So \|B \ B01 \| ≤ \|B\| X h1 ,h2 ∈E h1 6=h2 \|{b ∈ B : σB (h1 )b 6= σB (h2 )b}\| . 1− \|B\| We have for all h1 , h2 ∈ E with h1 6= h2 : \|{b ∈ B : σB (h1 )b 6= σB (h2 )b}\| \|B\| = dHamm (σB (h1 ), σB (h2 )) = dHamm (σB (h2 )−1 σB (h1 ), Id). Since dHamm is a invariant under left multiplication, and EB ⊇ E ∪ E −1 we have that −1 ′ dHamm (σB (h2 )−1 , σB (h−1 2 )) = dHamm (Id, σB (h2 )σB (h2 )) < ε . Inserting this into the above two inequalities and using that dHamm is invariant under right multiplication we see that: \|{b ∈ B : σB (h1 )b 6= σB (h2 )b}\| \|B\| = dHamm (σB (h1 ), σB (h2 )) = dHamm (σB (h2 )−1 σB (h1 ), Id) > ′ dHamm (σB (h−1 2 )σB (h1 ), Id) − ε ≥ dHamm (σB (h2−1 h1 ), Id) − 2ε′ > 1 − 3ε′ , where in the last two lines we again use that EB ⊇ E ∪ E −1 ∪ E −1 E. Thus \|B01 \| ≥ (1 − 3 \|E\|2 ε′ ). \|B\| Similarly, (EB , ε′ )–multiplicativity of σB gives X \|B02 \| ≥1− 1 − dHamm σB (h1 h2 ), σB (h1 )σB (h2 ) ≥ 1 − \|E\|2 ε′ . \|B\| h1 ,h2 ∈E This proves the Lemma. ε ε and choose ε′ > 0 as in Lemma 2.2, so ε′ < 48\|E\| Fix κ > 0 with κ < 12 2 . To complete the proof of Proposition 2.1, we break it up into two steps: we first show that this choice of ε′ gives us that σ b is (F, ε)–multiplicative, and then show it is (F, ε)–free. Step 1. We show that σ b is (F, ε)–multiplicative. THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 6 To prove Step 1, we apply Lemma 1.1, verifying below the four necessary conditions. We first check that the restriction to ⊕H G is (E1 , ε/6)–multiplicative. Let g, g ′ ∈ E1 , then ′ ′ dHamm σc c c A (gg ), σ A (g)σ A (g ) M (a, b) : b ∈ B0 , σA,b (g)σA,b (g ′ ) 6= σA,b (gg ′ ) 1 A ⊕ (B \ B ) + ≤ 0 \|A\|\|B\| \|B\| B \|A\|\|B\| \|B\| 1 X ≤κ+ dHamm (σA,b (g)σA,b (g ′ ), σA,b (gg ′ )). \|B\| b∈B0 Q (x) Write g = (gx ), g ′ = (gx′ ). Recall, σA,b (g) = x∈E σA,b (gx ). In the following, we use that the different terms in the product commute, to be precise: for x1 6= x2 ∈ E we have (x ) (x ) [σA,b1 (gx1 ), σA,b2 (gx′ 2 )] = 1. This gives ! Y (x) Y (x) (x) ′ ′ ′ ′ dHamm (σA,b (g)σA,b (g ), σA,b (gg )) = dHamm σA,b (gx )σA,b (gx ), σA,b (gx gx ) . x∈E x∈E Next, using the bi-invariance of the Hamming distance, the triangle inequality, and the (EA , ε′ )–multiplicativity of σA to see that X (x) (x) (x) dHamm (σA,b (g)σA,b (g ′ ), σA,b (gg ′ )) ≤ dHamm σA,b (gx )σA,b (gx′ ), σA,b (gx gx′ ) x∈E = X dHamm (σA (gx )σA (gx′ ), σA (gx gx′ )) x∈E < \|E\| ε′ . Thus we get the required multiplicativity: ′ ′ ′ dHamm (σc c c A (gg ), σ A (g)σ A (g )) < κ + \|E\| ε < ε . 6 The fact that the restriction to H is (E2 , ε/6)–multiplicative is more straight-forward. Indeed, for h, h′ ∈ E2 we have ′ ′ ′ ′ ′ dHamm (σc c c B (hh ), σ B (h)σ B (h )) = dHamm (σB (hh ), σB (h)σB (h )) < ε , where we note that we can use the multiplicative property of σB since E2 ⊆ EB . The third condition of Lemma 1.1 is automatically satisfied by σ b, by construction. We finish this step by verifying the bound on the Hamming distance between σ b (1, h)b σ (g, 1) and σ b (αh (g), 1)b σ (1, h) for h ∈ E2 , g ∈ E1 . Indeed, for such g, h we have dHamm (σc c c c B (h)σ A (g), σ A (αh (g))σ B (h)) ! M 1 ≤ A ⊕ B \ (B0 ∩ σB (h)−1 (B0 )) \|B\| \|A\| \|B\| B 1 (a, b) : b ∈ B0 ∩ σB (h)−1 B0 , σA,b (g)a 6= σA,σB (h)b (αh (g))a \|A\|\|B\| \|B\| 1 (a, b) : b ∈ B0 ∩ σB (h)−1 B0 , σA,b (g)a 6= σA,σB (h)b (αh (g))a . ≤ 2κ + \|B\| \|A\| \|B\| + Since Supp(αh (g)) = h Supp(g), and E contains both Supp(g) and h Supp(g), it follows that for every b ∈ B0 ∩ σB (h)−1 (B0 ) we have Y Y (hx) (x) σA,σB (h)b (gx ). σA,σB (h)b (gh−1 x ) = σA,σB (h)b (αh (g)) = x∈h Supp(g) x∈Supp(g) THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 7 Note that we have used that σA (1) = Id to restrict the number of terms in the product. (x) (hx) We use that for h ∈ E (and hence for h ∈ E2 ) we have that σA,σB (h)b (g) = σA,b (g). Inserting this into the above equation we see that Y (x) σA,b (g) = σA,b (g). σA,σB (h)b (αh (g)) = x∈Supp(g) Returning to the above inequality, we have shown that the remaining sets involved are empty, implying that ε dHamm (σc c c c B (h)σ A (g), σ A (αh (g))σ B (h)) < 2κ < . 6 The hypotheses of Lemma 1.1 have been checked, completing Step 1. Step 2. We show that σ b is (F, ε)–free. Suppose (g, h) ∈ F. If h 6= 1, then Thus (a, b) : σ b(g, h)(a, b) 6= (a, b) ⊇ M ! A B ⊕ b ∈ B : σB (h)b 6= b . dHamm σ b(g, h), Id ≥ dHamm σB (h), Id ≥ 1 − ε′ > 1 − ε. We may therefore assume that h = 1. When b ∈ B0 , the permutation σ b(g, 1) will act by (a, b) 7→ (σA,b (g)a, b), for all a ∈ ⊕B A. We then see that ! ! M (a, b) : σ b (g, 1)(a, b) = (a, b) ⊆ A ⊕ B \ B0 ∪ (a, b) : b ∈ B0 , σA,b (g)a = a . B Assume g = (gx ) 6= 1 and consider the proportion of elements fixed by σ b(g, 1). By the above we have L 1 X a ∈ B A : σA,b (g)a = a 1 (a, b) : σ b(g, 1)(a, b) = (a, b) ≤ κ + . \|B\| \|A\|\|B\| \|B\| \|A\|\|B\| b∈B0 Since g 6= 1, we can find x0 ∈ Supp(g). For every b ∈ B0 , we have that σB (h1 )b 6= σB (h2 )b for every h1 , h2 ∈ Supp(g) with h1 6= h2 . It follows that the (σB (x0 )b)–coordinate of σA,b (g)a is σA (gx0 )aβ , where aβ is the (σB (x0 )b)–coordinate of a. Thus  ( )  M M A ⊕ a ∈ A : σA (gx0 )a = a . a∈ A : σA,b (g)a = a ⊆  B So B\{σB (x0 )b} a∈ L B A : σA,b (g)a = a \|A\|\|B\| ≤ 1 − dHamm (σA (gx0 ), 1). Since gx0 ∈ EA we can use the freeness of σA , yielding 1 \|A\|\|B\| \|B\| \|{(a, b) : σ b (g, 1)(a, b) = (a, b)}\| ≤ κ + \|B0 \| (1 − dHamm (σA (g), 1)) ≤ κ + ε′ . \|B\| Equivalently, dHamm (b σ (g, 1), Id) ≥ 1 − κ − ε′ > 1 − ε. This verifies that σ b is (F, ε)–free, and thus completes the proof of Proposition 2.1, and hence of Theorem 1. THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 8 References [1] L. Bowen. 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Hall-Littlewood-PushTASEP and its KPZ limit Promit Ghosal arXiv:1701.07308v1 [math.PR] 25 Jan 2017 Columbia University Department of Statistics, 1255 Amsterdam Avenue, New York, NY 10027 e-mail: pg2475@columbia.edu Abstract: We study a new model of interactive particle systems which we call the randomly activated cascading exclusion process (RACEP). Particles wake up according to exponential clocks and then take a geometric number of steps. If another particle is encountered during these steps, the first particle goes to sleep at that location and the second is activated and proceeds accordingly. We consider a totally asymmetric version of this model which we refer as Hall-Littlewood-PushTASEP (HL-PushTASEP) on Z≥0 lattice where particles only move right and where initially particles are distributed according to Bernoulli product measure on Z≥0 . We prove KPZ-class limit theorems for the height function fluctuations. Under a particular weak scaling, we also prove convergence to the solution of the KPZ equation. Keywords and phrases: Stochastic six vertex model, KPZ universality, Interacting particle system. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model and main results . . . . . . . . . . . . . . . . . . . 3 Transition Matrix & Laplace Transform . . . . . . . . . . 4 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Weak Scaling Limit . . . . . . . . . . . . . . . . . . . . . . A Fredholm Determinant . . . . . . . . . . . . . . . . . . . . B eigenfunction of the Transition Matrix in HL-PushTASEP C Moment Formula . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 14 21 43 61 63 66 69 1. Introduction We introduce and study a special class of interacting particle systems which we call randomly activated cascading exclusion processes (RACEP). Fix a directed graph G = (V, E) with conductances ce (i.e., non-negative weights) along directed edges e ∈ E and define a random walk measure as the Markov chain on vertices with transition probability from v to v ′ given by cv→v′ normalized by the sum of all conductances out of v. The state space for RACEP is {0, 1}V where 1 denotes a particle and 0 denotes a hole. Each particle is activated according to independent Poisson clocks. Once active, a particle moves randomly to one its adjacent sites. Therein, it chooses an independent random number of steps according to a geometric distribution and then performs an independent random walk (according to the random walk measure just described) of that many steps. However, if along that random walk trajectory, the first particle arrives at a site occupied by a second particle, then, the first particle stays 1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 2 at that site (and goes back to sleep) and the second particle becomes active and proceeds as if its Poisson clock had rung (according to the above rules). The Poisson clock activation processes are supposed to model a much slower process than the random walk cascades, so for simplicity we assume that these random walk cascades occur instantaneously. There is some work necessary to prove well-definedness of this process in general, and we do not pursue that here since we will mainly focus on one concrete case. RACEP is a special type of exclusion process (see, e.g. [Lig05, Lig99]) in which the jump distribution depends on the state around the particle in a rather non-trivial way. Exclusion processes are important models of lattice gases, transport (such as in various ecological or biological contexts), traffic, and queues in series (see for example [SCN10, Cor12] and references therein). The prototypical example of an exclusion process is the asymmetric simple exclusion process (ASEP) which was introduced in the biology literature in 1968 [MGP68] (see also more recent applications such as in [CSN05, GS10]) and two years later in the probability literature [Spi70]. RACEP can also be thought of as a variant of a frog model (see, e.g. [AMP02, AMPR01]) in which particles do not fall asleep once active but continue to move and wake up other particles. A natural interpolation between RACEP and the frog model is to have the probability that a particle goes back to sleep be in (0, 1). These relations to exclusion processes and frog models suggest a number of natural probabilistic questions for RACEP, such as understanding its hydrodynamic and fluctuation limit theorems for various families of infinite graphs with simple conductances. We attack these questions exactly for the special case of RACEP in which the underlying graph is Z≥0 and the random walk is totally asymmetric (in the positive direction). From here on out, we will only focus on this one-dimensional case of RACEP. ASEP and RACEP on Z are siblings in that they both arise as (different) special limits of the stochastic six vertex model [GS92, BCG16] – see Section 2.2. They are, in fact, part of a broader hierarchy of integrable probability particle systems which are solvable due to connections to quantum integrable systems – see [CP16, BP16a, BP16b]. A totally asymmetric version of RACEP also comes up as a marginal of certain continuous time RSK-type dynamics which preserve the class of Hall-Littlewood processes – see [BBW16] wherein they refer to RACEP as the t-pushTASEP due to its similarity with the model of q-PushTASEP [BP16a, CP15]. Here, we are mostly interested in this one dimensional totally asymmetric specialization of RACEP. To avoid the confusion with the time parameter t, we refer RACEP as HallLittlewood-PushTASEP (HL-PushTASEP) throughout the rest of the paper. These systems enjoy many concise and exact formulas from which one can readily perform asymptotics. Such results for ASEP go back to the now seminal papers [TW08, TW11] (and earlier to [Joh00] for TASEP), and results for the stochastic six vertex model were worked out in [BCG16, AB16]. In our present paper, we work out the analogous asymptotics for the HLPushTASEP on Z≥0 using the approach of [BCG16]. (We also provide some additional details in the asymptotics compared to the previous works.) Owing to the recent work [Bor16, BO16] such asymptotics could alternatively be performed via reduction to Schur process asymptotics. We do not pursue that route here and instead opt for the more direct (albeit technically demanding) route. The asymptotic results we show for HL-PushTASEP on Z≥0 (as well as those previously shown for ASEP and the stochastic six vertex model) demonstrate its membership in the Kardar-Parisi-Zhang universality class (see, e.g. the review [Cor12] and references therein). In particular, in Theorem 2.4 we show that for step Bernoulli initial data, HL-PushTASEP imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 3 has fluctuations in its rarefaction fan of order t1/3 and with statistics determined by the GUE Tracy-Widom distribution. At the edge of the rarefaction fan, we also demonstrate the occurrence the Baik-Ben Arous-Péché crossover distributions in [BBAP05] which arises also in ASEP [TW09b], TASEP [BBAP05, BAC11], and the stochastic six vertex model [AB16]. See [Cor14] for further references to asymptotics of other integrable probabilistic systems. The centering and scaling that our result demonstrates agrees with the predictions of KPZ scaling theory from the physics literature – see Section 2.4. The KPZ scaling theory also suggests that under certain weak scalings a broad class of interacting particle systems should converge to the Cole-Hopf solution to the KPZ stochastic PDE (i.e., the KPZ equation). There are, in fact, many different choices of weak scalings under which the KPZ equation remains statistically invariant (see, e.g.[QS15, Section 2.4]). The main motivation for our present investigation into the KPZ equation limit for HL-PushTASEP was our desire to solve the analogous question for the stochastic six vertex model. Let us briefly explain why that question is difficult and what our HL-PushTASEP results suggest regarding it. For ASEP under weak asymmetry scaling, the KPZ equation limit was proved in [BG97] (see also [ACQ11, DT16, CST16]). That work relies upon two main identities. The first is the Gärtner (or microscopic Cole-Hopf) transform which turns ASEP into a discrete stochastic heat equation (SHE). The second is a non-trivial key identity (Proposition 4.8 and Lemma A.1 in [BG97]) which allows one to identify white-noise as the limit of the martingale part of the discrete SHE. For the stochastic six vertex model, one still has an analogous Gärtner transform (in fact, this holds for all higher-spin vertex models in the hierarchy of [CP16] due to the duality shown therein). The discrete time nature of the stochastic six vertex model, however, renders the identification of the white-noise much more complicated and presently it is unclear how to proceed. For higher-spin vertex models with unbounded particle occupation capacity, [CT15] proved convergence to the KPZ equation under certain weak scalings. Even though these models are still discrete time, the scaling was such that the overall particle density goes to zero with the scaling parameter ǫ. Owing to that fact, there was no need for an analog of the key estimate used in the case of ASEP. Our present analysis of HL-PushTASEP also involves a weak scaling under which the local density of particles goes to zero. Consequently, we are able to prove the KPZ limit without the key estimate. This suggests that it may be simpler to derive the KPZ equation directly from the stochastic six vertex model (or other finite-spin integrable stochastic vertex models) when the weak scaling facilitates local density decay to zero. Indeed, there are many different weak scalings which should all lead to the KPZ equation. This can be anticipated, for instance, from analyzing moment or one-point distribution formulas, see e.g. [BO16, Section 12]. We intend on investigating such zero-density KPZ equation limits of the stochastic six vertex model in subsequent work. There is another approach for proving KPZ limit of particle system (when started from their invariant measure) via energy solutions. This approach was introduced by Assing [Ass02] and then significantly developed in [GJ14, GP15]. The invariant measure is Bernoulli product measure for HL-PushTASEP (as well as for the stochastic six vertex model), so it would be interesting to see if these methods apply. Presently, the energy solution approach is only developed for continuous time systems, so an analysis of the stochastic six vertex model results would require developing a discrete time variant. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 4 Outline We introduced above RACEP in an arbitrary infinite directed graph. In Section 2, we describe the dynamics of HL-PushTASEP in one dimension and its connection with stochastic six vertex model. Further, we state our results on asymptotic fluctuation and KPZ limit of HLPushTASEP in Section 2. At the end of this section, we explain the KPZ scaling theory for HL-PushTASEP in brief. In Section 3, we first derive the eigenfunctions of the transition matrix of HL-PushTASEP and then use it to get Fredholm determinant formulas. Section 4 contains the proof of the limiting fluctuation results under the assumption of step Bernoulli initial data. Interestingly, we see a phase transition in the limiting behavior of the fluctuation. In Section 5, we prove the KPZ limit theory for HL-PushTASEP. First, we start with the construction of a discrete SHE. Next, we prove few estimates on the discrete heat kernel and subsequently, show the tightness of the rescaled SHE. At the end, we elaborate on the equivalence of all the limit points by solving the martingale problem for HL-PushTASEP. Acknowledgements Special thanks are due to Ivan Corwin for suggesting the problem and guiding throughout this work. We thank Guillaume Barraquand and Peter Nejjar for several constructive comments on the first draft of this paper. We would also like to thank Li-Cheng Tsai and Rajat Subhra Hazra for helpful discussions. 2. Model and main results We now define the HL-PushTASEP on Z≥0 , describe its connection to the stochastic six vertex model, and then state our main results. Theorems 2.4 and 2.5 give the fluctuations limits for HL-PushTASEP started with step Bernoulli initial data. Theorem 2.9 and 2.10 contain the KPZ equation limits for HL-PushTASEP under weak scaling given in Definition 2.7. 2.1. HL-PushTASEP on Z≥0 We will consider HL-PushTASEP with a finite number of particles supported on Z≥0 . The dynamics of HL-PushTASEP are such that the behavior of particles restricted to the interval [0, L] is Markov. Thus, even if we are interested in HL-PushTASEP with an infinite number of particles, if we only care about events involving the restriction to finite intervals, then it suffices to consider the finite particle number version we define here. The N -particle HL-PushTASEP on Z≥0 has state at time t given by (x1 (t), . . . , xN (t)) where xi (t) ∈ Z≥0 for all t ∈ R+ and x1 (t) < . . . < xN (t). For notational simplicity we add two virtual particles fixed for all time so that x0 = −1 and xN +1 = +∞. For later reference, we denote the state space of N -particle HL-PushTASEP on Z≥0 by X N := ~x = (−1 = x0 < x1 < x2 < . . . < xN < xN +1 = ∞) : ∀i, xi ∈ Z≥0 . We illustrate typical moves in HL-PushTASEP on Z≥0 in Fig 1 and Fig 2. Each of the particles in HL-PushTASEP carries an exponential clock of rate 1. When the clock in a particle rings, it jumps to the right. Unlike the simple exclusion processes, if the particle at xm (t) gets imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit xm−2 (t) xm−1 (t) xm (t) xm+1 (t) 5 xm+2 (t) Fig 1: Assume that particle at xm (t) (colored green) becomes active at time t. It immediately takes a step to the right and then continues to jump according to a geometric distribution with parameter b. Thus, the probability that it takes a jump of exactly size 2 is b(1 − b). With probability b2 it reaches the position of xm+1 (t) (colored red). In that scenario, the green particle occupies the position xm+1 (t) and activates the red particle as shown in Fig. 2. xm−2 (t) xm−1 (t) xm (t) xm+1 (t) xm+2 (t) Fig 2: Once red particle has been activated, it moves by one to the right, and then proceeds according to the same rules which had governed the green particle. For instance, it can jump by j steps with probability (1 − b)bj−1 for 1 ≤ j < xm+2 (t) − xm+1 (t) and with probability bj−1 if j = xm+2 (t) − xm+1 (t). We illustrated here the cases when j equals xm+2 (t) − xm+1 (t) and xm+2 (t) − xm+1 (t) − 1. excited at time t, then it can jump j steps to the right with a geometric probability (1− b)bj−1 for 1 ≤ j < xm+1 (t) − xm (t). It can also knock its neighbor on the right side at xm+1 (t) out from its position with probability bxm+1 (t)−xm (t)−1 . In that case, particle at position xm+1 (t) will also get excited and jumps in the same way independently to others. To be precise, the HL-PushTASEP shares a large extent of resemblance with the particle dynamics in the case of stochastic six vertex model (SSVM) (see, [BCG16, Section 2.2]). Although the dynamics in latter case is governed by a discrete time process, but pushing effect of the particles comes into play in the same way. We embark more upon the connection between these two model later in the following section. One can see similar pushing mechanism in the case of Push-TASEP [BF08], q-PushASEP[CP15] which are also continuous time countable state space Markov processes. But in contrast with the HL-PushTASEP, each of the particles in those cases can only jump by one step when their clocks ring. For HL-PushTASEP on Z≥0 , we are interested in studying the evolution of the empirical particles counts, i.e, for any ν ∈ R, what is the limit shape and fluctuation of the number of particles in an interval [0, ⌊νt⌋] as the time evolves to infinity. One can note that empirical particle counts fits into the stereotype of the height function for the exclusion processes. In what follows, we provide the explicit definitions of the height function and the step Bernoulli initial condition. Definition 2.1. Fix a positive integer N . Define a function Nx (t; .) and ηt (x; .) of ~x = (x1 < x2 < . . . < xN ) ∈ X N via 1 xi = x for some i. Nx (t; ~x) = #{i; xi ≤ x}, ηx (t; ~x) = 0 otherwise. For the rest of the paper, we prefer to use Nx (t) and ηt (x) instead of Nx (t; ~x) and ηt (x; ~x) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 6 with the understanding that those are evaluated at ~x. We call Nx (t) as the height function of HL-PushTASEP and ηt (x) as the occupation variable. Definition 2.2 (Step Bernoulli Initial Data). Fix any positive integers L. In step Bernoulli initial condition with parameter ρ over [0, L], each integer lattice site is occupied by a particle with probability ρ independently of others. Whenever ρ = 1, we refer it as step initial condition. 2.2. Connection with Stochastic Six Vertex Model Six vertex model was first studied in [Lie67] to compute the residual entropy of the ice structure. One can think of six vertex model as a lattice model where the configurations are the assignments of six different types of H2 O molecular structure to the vertices of a square grid so that O atoms are at the vertices of the grid. To each O atoms, there are two H atoms attached so that they are at the angles 90◦ or 180◦ to each other. Later, a stochastic version of the six vertex model was introduced by Gwa and Spohn in [GS92]. Recently, several new discoveries related to the stochastic six vertex model came into light in works including [BCG16, BP16a, BP16b, AB16, Agg16]. Furthermore, there are three equivalent ways of describing stochastic six vertex model: (i) as a ferro-electric asymmetric six vertex model on a long rectangle with specific vertex weights (see, [GS92] or [BCG16, Section 2.2]); (ii) as an interacting particle system subject to the pushing effect and exclusion of mass (see, [BCG16, Section 2.2], [GS92]); or (iii) as a probability measure on directed path ensemble (see, [BP16a, Section 1], [AB16, Section 1.1]). In what follows, we elaborate in details on the second description of stochastic six vertex model. Particle dynamics of stochastic six vertex model is governed by a discrete time Markov chain with local interactions. Fix b1 , b2 ∈ [0, 1]. Consider a particle configuration Yb1 ,b2 (t) = (y1 (t) < y2 (t) < . . .) at time t where 0 < b1 , b2 < 1. Then, at time t + 1, each particle decides with probability b1 whether it will stay in its position, or not. If not, then it moves to the right for j steps with probability (1 − b2 )b2j−1 where j can be any integer in between 1 and the distance of the right neighbor from the particle. Further, the particle at yi (t) can jump to yi+1 (t) with probability byi+1 (t)−yi (t)−1 . In that case, the particle which was initially at yi+1 (t) is pushed towards right by one step and starts to move in the same way. It has been noted in [BCG16, Section 2.2] that the dynamics of the limit Y b (t) := lim Y1−ǫ,b (tǫ−1 ) ǫ→0 (2.1) are those of HL-PushTASEP on Z≥0 . Similarly, they show that the dynamics of Z p,q (t) := lim {Yǫp,ǫq (tǫ−1 ) − tǫ−1 } ǫ→0 are those of ASEP with jump parameters p and q (see also [Agg16]). For any finite particle configuration, it is easy to show that the limiting dynamics in (2.1) indeed matches with HLPushTASEP on Z≥0 . Heuristically, this is attributed to the facts that in an time interval [t, t+ ∆t], (i) the expected number of jumps for each of the particles of the limiting process Y b (t) varies linearly with ∆t and (ii) the probability of more than one jumps decays quadratically with ∆t. In particular, (i) shows the number of jumps of each of the particles follows a Poisson point process with intensity 1 and (ii) shows that the Poisson clocks in all particles are imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 7 independent of each other. In the case of countable particle configurations, one can complete the proof of the convergence of stochastic six vertex model to HL-PushTASEP on Z≥0 using the similar arguments as in [Agg16]. 2.3. Main Results In the following theorems, we state the asymptotic phase transition result for the height function Nνt (t) of HL-PushTASEP on Z≥0 under the step Bernoulli initial data. Let us explain the scalings used in the paper. Assume that initial condition for the HL-PushTASEP dynamics is step Bernoulli with parameter ρ. Thus, it is expected that the macroscopic density of the particles varies between 0 to ρ. In fact, it will be shown that whenever ν ≤ (1 − b)−1 , then macroscopic density around νt will be 0. Furthermore, in the case when (1 − b)−1 < ν < (1 − ρ)−2 (1 − b)−1 , macroscopic density varies as 1 − (ν(1 − b))−1/2 around the position νt. Consequently, one can expect KPZ scaling theory to hold in such scenario. On the contrary, when ν > (1 − ρ)−2 (1 − b)−1 , density profile around the position νt turns out to be flat, thus yielding the gaussian fluctuation. Before stating the theorems, we must define three distributions, namely the GUE Tracy-Widom distribution, square of GOE Tracy-Widom distribution and Gaussian distribution which will arise as limits in three different scenario. Definition 2.3. (1) Consider a piecewise linear curve Γ(1) consists of two linear segments: from ∞e−iπ/3 to 0 and from 0 to ∞eiπ/3 . The distribution function FGU E (x) of the GUE Tracy-widom distributions is defined by FGU E (x) = det(I + KAi )L2 (Γ(1) ) where KAi is the Airy kernel. For any w, w′ ∈ Γ(1) , Airy kernel is expressed as Z e2πi/3 ∞ w3 /3−sw 1 e 1 ′ dv KAi (w, w ) = 3 /3−sv v 2πi e−2πi/3 ∞ e (v − w)(v − w′ ) where the contour of v is piecewise linear from −1 + e−2πi/3 ∞ to −1 to −1 + e2πi/3 ∞. It is important to note that the real part of v − w is always negative. (2) Let δ > 0. Consider a piecewise linear curve Γ(2) which is composed of two linear segments: 2 from −δ + ∞e−iπ/3 to −δ and from −δ to −δ + ∞eiπ/3 . Then, the FGOE is defined by 2 FGOE (xs) = det(I + KGOE,2 )L2 (Γ(2) ) where KGOE,2 is given by 1 KGOE,2 (w, w ) = 2πi ′ Z −2δ+∞e2πi/3 −2δ+∞e−2πi/3 3 v ew /3−sw 1 dv 3 /3−sv ′ v (v − w)(v − w ) w e for aany w, w′ ∈ Γ(2) . The contour of v is piecewise linear from −2δ + ∞e−2πi/3 to −2δ to −2δ + ∞e2πi/3 . (3) For completeness, we also add here a very brief description of the Gaussian distribution. Density of the standard Gaussian distribution function Φ(x) is given by 2 x 1 √ exp − . 2 2π Theorem 2.4. Consider HL-PushTASEP on Z≥0 . Assume initial data is step Bernoulli data. −1 Set α = 1−ρ ρ . Let us also fix a real number ν > (1 − b) . Define the parameters, p p p b1/3 ( ν(1 − b) − 1)2/3 ( ν(1 − b) − 1)2 , σν := , ̺ := b( ν(1 − b) − 1), mν := 1−b (1 − b)1/2 ν 1/6 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 1 ν − , m̃ν := 1 + α α(1 − b) σ̃ν := s 8 ν(1 − b)(1 − ρ)2 − 1 . α2 (1 − b) (a) If ρ > 1 − (ν(1 − b))−1/2 , then lim P t→∞ = FGU E (s). (2.2) 2 = FGOE (s). (2.3) (2.4) mν t − Nνt (t) ≤s σν t1/3 mν t − Nνt (t) ≤s σν t1/3 (b) If ρ = 1 − (ν(1 − b))−1/2 , then lim P t→∞ (c) If ρ < 1 − (ν(1 − b))−1/2 , then lim P t→∞ m̃ν t − Nνt (t) ≤s σ̃ν t1/2 = Φ(s). Let us denote the position of m-th particle at time t by xm (t). To this end, one can note that for any y, t ∈ R+ and m ∈ Z≥0 , the event {Ny (t) ≥ m} is same as the event {xm (t) ≤ y}. Thus, Theorem 2.4 translates to the following results (noted down below) in terms of the particle position xm (t). Theorem 2.5. Continuing with all the notations introduced in Theorem 2.4 above, we have (a) if ρ > 1 − (ν(1 − b))−1/2 , then lim P xtmν (t) ≤ νt + t→∞ ∂mν ∂ν −1 ∂mν ∂ν −1 1/3 ! = FGU E (s), (2.5) σν st1/3 ! 2 = FGOE (s), (2.6) ! (2.7) σν st (b) if ρ = 1 − (ν(1 − b))−1/2 , then lim P xtmν (t) ≤ νt + t→∞ (c) if ρ < 1 − (ν(1 − b))−1/2 , then lim P xtm̃ν (t) ≤ tν + t→∞ ∂ m̃ν ∂ν −1 σ̃ν st 1/2 = Φ(s). Using the theorems above, we derive the following law of large numbers for HL-PushTASEP. Corollary 2.6 (Law of large numbers). Consider HL-PushTASEP on Z≥0 with step initial condition. Denote ν0 := (1 − b)−1 . Then, ( √ ( ν(1−b)−1)2 Nνt (t) when ν > ν0 , (1−b) = lim t→∞ t 0 otherwise. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 9 Interactive particle systems which obey KPZ class conjecture (see, Section 2.4) form the KPZ universality class. Models is the KPZ universality class are in a sense characterized by the KPZ equation which is written as ∂t H(t, x) = √ 1 2 1 δ∂x H(t, x) + κ(∂x H(t, x))2 + Dξ(t, x), 2 2 for δ, κ ∈ R and D > 0 where ξ(., .) denotes the space time white noise. In a seminal work [KPZ86], Karder, Parisi and Zhang introduced the KPZ equation as a continuum model for the randomly growing interface which can be subjected to the local dynamics like smoothing, lateral growth and space-time noise. Most prominent feature of the models in KPZ universality class is that the scaling exponents of the fluctuation, the space and the time follow a ratio 1 : 2 : 3. Further, it is expected that the long time asymptotics of any model in the KPZ universality class coincides with the corresponding statistics of the KPZ equation when initial condition of the model corresponds with the initial condition in KPZ equation. To illustrate, consider ASEP on Z. Initially, if all the sites in Z≤0 are occupied and the rest are left empty, then [TW09b] shows that the height fluctuation in t1/3 scale converges to the Tracy-Widom GUE distribution. Later, it is verified in the work of [ACQ11], [Dot10], [CDR10] and [SS10] that under narrow wedge initial condition the height fluctuation of the KPZ equation in t1/3 scale has the same asymptotic distribution. It is now natural to ask what are the other distributions which arise as the long time fluctuation limits of stochastic models in KPZ universality class. There are only very few instances where this question has been addressed. In the case of ASEP under the step Bernoulli initial condition (sites in Z≤0 are occupied w.p ρ and other places are empty), Tracy and Widom [TW09b] demonstrates that the limiting behavior of the height fluctuation undergoes a phase transition. In particular, whenever ρ is close to 1, scaling exponent of the fluctuations is still t1/3 and limiting distribution is Tracy-Widom GUE. On the contrary, when ρ nears 0, the fluctuation converges to the Gaussian distribution under t1/2 scaling. In between those two regime, there lies a critical point where the limiting distribution of the fluctuation is F12 . Nevertheless, the scaling exponent of the fluctuation at the critical point remains 1/3. It is shown in [CQ13] that the asymptotic limit of the fluctuation (in ASEP) at the critical point corresponds to the Half-Brownian initial data of the KPZ equation. Similar phase transition was in fact first unearthed in the work of Baik, Ben Arous and Péché [BBAP05], where they observed a transition phenomenon (now known as BBP transition) in the limiting law of the largest eigenvalue λ1 of large spiked covariance matrices. In a follow up paper, [Péc06] proved the same for the finite rank perturbation of the GUE matrices. When the perturbation has rank one, then the asymptotic distribution of λ1 at criticality coincides with F12 . In the context of interacting particle systems, Barraquand [Bar15] established the BBP transition in the case of q − T ASEP with few slower particles. More recently, Aggarwal and Borodin [AB16] showed that the same phenomenon also holds in the case of stochastic six vertex model when subjected to generalized Bernoulli initial data. Our next result is on the convergence of an exponential transform of the height function Nx (t) to a mild solution of the stochastic heat equation (SHE). The stochastic heat equation is written as κ√ 1 DZ(t, x)ξ(t, x) ∂t Z(t, x) = δ∂x2 Z(t, x) + 2 δ imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 10 ξ(·, ·) denotes the space time white noise. Let us call Z(t, r) ∈ C([0, ∞), C(R)) a mild solution of the SHE starting from the initial condition Zin if Z tZ Z Pt−s (r − r ′ )Zin (s, r − r ′ )ξ(ds, dr ′ ) (2.8) Ps (r − r ′ )Zin (s, r ′ )dr ′ + Z(t, r) = 0 R √1 2πt R exp(−r 2 /2t) where Pt (r) := is the standard heat kernel. For existence, uniqueness, continuity and positivity of the solutions to (2.8), see [Cor12, Proposition 2.5] and [BG97], [BC95], [Mue91]. Interestingly, the Cole - Hopf solution H(t, x) of the KPZ equation is defined by H(t, x) = κδ log Z(t, x) whenever Z is a mild solution (see, (2.8)) of the SHE. Bertini and Giacomin [BG97] proved that ASEP under weak asymmetric scaling and near equilibrium initial condition converges to the Cole - Hopf solution of the KPZ equation. Later, the same result has been shown for the narrow wedge initial data in [ACQ11]. In both of these works, Gärtner transformation which is a discrete analogue of Cole - Hopf transform plays crucial roles. For weakly asymmetric exclusion process with hopping range m more than 1, similar result has first appeared in the work of Dembo and Tsai [DT16]. But, their proof breaks down when m ≥ 4. In the following discussion, we show that under a particular weak scaling, HL-PushTASEP on Z≥0 started from the step initial condition converges to the KPZ equation. Definition 2.7 (Gärtner Tranform & Weak Noise Scaling). Here, we turn to define explicitly the exponential transformation (or, Gärtner transformation) N νǫ when x + ⌊t/(1 − bνǫ )⌋ ∈ Z≥0 b x+⌊t/(1−bνǫ )⌋ (t)−(1−νǫ )(x+⌊t/(1−b )⌋) exp(−µǫ t) Z(t, x) = 0 o.w (2.9) √ where t ∈ R+ , νǫ = ( 5 − 1)/2 and µǫ = (1 − νǫ ) log b b−1 − 1 . + b−νǫ − 1 1 − bν ǫ (2.10) In the rest of our discussion, we will keep on making an abuse of notation by writing t/(1−bνǫ ) 1/2 instead of ⌊t/(1 − bνǫ )⌋. Let us mention that we use a weak noise scaling b = bǫ := e−λǫ ǫ , −3/2 where ǫ → 0 and λǫ = νǫ . We extend the process Z(t, x), defined for t ∈ R+ and x ∈ Z, to a continuous process in R+ × R by linearly interpolating in x. Let us introduce the scaled process Zǫ (t, x) := Z(ǫ−1 t, ǫ−1 x). (2.11) Furthermore, we endow the space C(R+ ), and C(R+ × R) with the topology of uniform convergence and use ⇒ to denote the weak convergence. Definition 2.8 (Near-Equilibrium Condition). Discrete SHE Zǫ (., .) is defined to have near equilibrium initial condition when the initial data Zǫ (0, x) satisfies the following two conditions: kZǫ (0, x)k2k ≤ Ceǫτ \|x\|, kZǫ (0, x) − Zǫ (0, x′ )k2k v ′ ≤ C ǫ\|x − x′ \| eǫτ (\|x\|+\|x \|) , (2.12) (2.13) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 11 for some v ∈ (0, 1) and τ > 0. Theorem 2.9. Let Z be the unique C(R+ × R)- valued solution of the SHE starting from some initial condition ∆ and let Zǫ (t, x) ∈ C(R+ × R) as defined in (2.11) evolved from some near equilibrium initial condition. If Zǫ (0, .) converges to Z(0, .) whenever ǫ tends to 0, then Zǫ (., .) ⇒ Z(., .) on C(R+ × R) as ǫ → 0. It can be noted that the step initial condition, i.e, xn (0) = n for n ≥ Z+ and xn (0) = −∞ for n ∈ Z− , doesn’t belong to the family of near equilibrium initial conditions in Definition 2.8. Nevertheless, the rescaled height function of HL-PushTASEP started with step initial condition also has KPZ limit. We illustrate this further in the following theorem. Theorem 2.10. Let Z be the unique C(R+ × R)- valued solution of the SHE starting from delta initial measure. Consider Z̃ǫ (t, x) := ǫ−1 (1 − exp(−λǫ νǫ ))Zǫ (t, x) in C(R+ × R) evolving from the step initial condition. Then, Z̃ǫ (., .) ⇒ Z(., .) on C(R+ × R), as ǫ tends to 0. Corollary 2.11. Let Z be as in the Theorem 2.9 so that H(t, x) := log Z(t, x) be the unique Cole-Hopf solution of the KPZ equation starting from the delta initial measure. Then, log Z̃ǫ (., .) ⇒ H(., .) on C(R+ × R) as ǫ → 0. P Proof of Theorem 2.10. To begin with, note that ǫ ζ∈Ξ(0) Z̃ǫ (0, ζ) = 1. Thus, Z̃ǫ (0, .) converges to the delta initial measure as ǫ goes to 0. Below, in Proposition 2.12 which we prove in Section 5, we show the moment estimates for Z̃ǫ at any future time point t. Thus, using Theorem 2.9, one can complete the proof in the same way as in [ACQ11, Section 2]. Proposition 2.12. Fix any T > 0. Consider Z̃(t, x) = ǫ−1 (1 − exp(−λǫ νǫ ))Z(t, x) evolving from the step initial condition. Then, for any t ∈ (0, ǫ−1 T ], ζ1 , ζ2 ∈ Ξ(t) and v ∈ (0, 1/2), we have kZ̃(t, ζ)k ≤ C min{ǫ−1/2 , (ǫt)−1/2 }, ′ ′ v −(1+v)/2 kZ̃(t, ζ) − Z̃(t, ζ )k ≤ (ǫ\|ζ − ζ \|) (ǫt) . (2.14) (2.15) where the constant C depend only on T and v. 2.4. KPZ Scaling Theory Here, we explain how the asymptotic fluctuation results in Theorem 2.4 confirms the KPZ scaling theory (see, [KMHH92], [Spo14]) of the physics literature. To start with, we present the predictions of KPZ scaling theory in the context of exclusion process following [Spo14]. For imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 12 that, one need to assume that spatially ergodic and time stationary measures of the exclusion process on Z are precisely labelled by the density of the particles ρ, where n X 1 η(n). n→∞ 2n + 1 ρ := lim k=−n We denote the stationary measure arising out of the density ρ by νρ . Consider the average steady state current j(ρ) which counts the average number of particles transported from 0 to 1 in unit time where the system is distributed as νρ . Further, denote the integrated covariance P j∈Z Cov(η0 , ηj ) by A(ρ) where the covariance between the occupation variables η0 and ηj is computed under the stationary measure νρ . One expects that on the microscopic scale the rescaled particle density ̺(x, t) given as ̺(x, τ ) = lim P (there is a particle at site ⌊xt⌋ at time ⌊τ t⌋) t→∞ satisfies the conservation equation ∂ ∂ ̺(x, τ ) + j(̺(x, τ )) = 0 ∂τ ∂x when the system is started with the step initial condition, i.e, 1 x≥0 ̺(x, 0) = 0 x < 0. (2.16) (2.17) This result can also be phrased as a hydrodynamic limit theory. For any exclusion process, consider the height function h(., .) : Z × R+ → Z given by  Pj if j > 0  Nt + i=1 ηt (j) h(j, t) := N if j = 0 P t  Nt − −j η (−i) if j < 0. i=1 t Then, using the arguments in [Spo91, Part II, Section 3.3], one can prove that (2.16) implies the following law of large numbers h(νt, t) = φ(ν) t→∞ t lim where the limit shape φ(.) is given by φ(y) = sup {yρ − j(ρ)}. ρ∈[0,1] Furthermore, Let us denote λ(ρ) = −j ′′ (ρ). Under such parametrization, we have the following KPZ class conjecture. KPZ Class Conjecture:([Spo14]) Let y be such that φ is twice differentiable at y with φ′′ 6= 0. Set ρ0 = φ′ (y). If 0 ≤ ρ0 < 1, A(ρ0 ) < ∞ and λ(ρ0 ) 6= 0, then under the step initial condition in (2.17), one has ! tφ(y) − h(⌊yt⌋, t) ≤ s = FGU E (s). lim P t→∞ (− 12 λA2 )1/3 t1/3 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 13 In the case when φ(y) is linear or cusp like, then KPZ conjecture doesn’t hold. It is likely that the scaling of the fluctuations would be different in those cases. Historically, the conjecture on the scaling exponent t1/3 of the fluctuation above dates back to the the work [KPZ86]. To that end, the magnitude of the fluctuation was obtained in [KMHH92] and the limit distribution was first surfaced in [Joh00]. Lemma 2.13. Fix 0 ≤ ρ ≤ 1. Consider the HL-PushTASEP model on Z≥0 . Further, assume that a jet of particles is entering into the region Z≥0 through 0 at rate τ where τ := ρ(1 − ρ)−1 (1 − b)−1 . Then, the product Bernoulli measure νρ will be the invariant measure for the HL-PushTASEP model. In particular, each of the occupation variables {ηt (x)\|x ∈ Z≥0 } follows Ber(ρ) independent of others. Proof. Assume, at time t = 0, each site on Z+ is occupied by at most one particle with probability ρ independently of others. First, we show νρ (ηt (0) = 1) = ρ for any t ≥ 0. Note that influx rate of particles at site x = 0 is equal to τ . Then, rate at which particle leaves site 0 is given by χ1 + χ2 + χ3 χ1 := τ ρ, χ2 := τ b(1 − ρ), and , χ3 := ρ. To begin with, consider the case when initially the site 0 was occupied. Probability of this event is ρ. In such scenario, χ1 captures the rate of out flux of any particle to the right from the site 0 after a particle arrives there. On the contrary, probability that the site 0 was initially unoccupied is 1 − ρ. Thus, if a particle arrives when the site 0 is empty, it moves further one step towards the right with probability b. Henceforth, the value of χ2 synchronizes perfectly with the rate of outflux from the site 0 whenever initially it was unoccupied. Finally, the particle which was initially at the site 0 jumps to the right at rate 1 and this contribution has added up in the total rate out flux through χ3 . Due to the specific choice of τ , we get τ = χ1 + χ2 + χ3 . Consequently, the rate of in flux matches with the rate of out flux of the particles from the site 0. Therefore, probability that the site 0 is occupied doesn’t change over time. More importantly, the rate at which particles will move into the site 1 is also equal to τ . This further implies P(ηt (1) = 1) = ρ for any t ≥ 0. Using induction, one can now prove that the occupation variable ηt (x) for any x ∈ Z+ and t ≥ 0 follows Ber(ρ). In fact, using similar argument, one can show that at any time t, the rate of influx of the particles into any interval [x1 , x2 ] of finite length over Z+ is equal to the rate at which the particles will come out of the interval [x1 , x2 ]. Now, we turn to prove the independence of any two occupation variables ηt (x1 ) and ηt (x2 ) for x1 < x2 ∈ Z+ . Let us consider (ǫx1 , ǫx1 +1 , . . . , ǫx2 ) ∈ {0, 1}x2 −x1 +1 . Further, denote ηt (a) if ǫa = 1, ǫa ηt (a) := 1 − ηt (a) if ǫa = 0. To this end, one can note that E x2 Y a=x1 ! ηtǫa (a) =E x2 Y a=x1 ! η0ǫa (a) for any choice of the variables (ǫx1 , ǫx1 +1 , . . . , ǫx2 ) in the space {0, 1}x2 −x1 +1 . This is again due to the principle of conservation of mass, i.e, the influx rate into the interval [x1 , x2 ] is imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 14 same as the rate of out flux of the particles. Thus, we get   xY 2 −1 X E(ηt (x1 )ηt (x2 )) = E  ηtǫa (a) ηt (x1 )ηt (x2 )  = E ǫx1 +1 ,...,ǫx2 −1 a=x1 +1 X xY 2 −1 η0 (x1 )η0 (x2 ) ǫx1 +1 ,...,ǫx2 −1 a=x1 +1  η0ǫa (a) = E (η0 (x1 )η0 (x2 )) = ρ2 . To that effect, covariance of ηt (x1 ) and ηt (x2 ) becomes 0. As these are Bernoulli random variables, thus the independence is proved. Now, we turn to showing how Theorem 2.4 verifies the KPZ class conjecture in the context of HL-PushTASEP. In Lemma 2.13, it is proved that all the stationary translation invariant measure of HL-PushTASEP are given by product measure νρ where each site follows Ber(ρ) distribution. Further, it has been also shown that under stationary measure νρ steady state current will be ρ(1 − ρ)−1 (1 − b)−1 . To that effect, we have ρ φ(y) = sup ρy − . (2.18) (1 − ρ)(1 − b) ρ∈[0,1] If y(1 − b) > 1, then the function g(ρ) = yρ − ρ(1 − ρ)−1 (1 − b)−1 is maximized when ρ = 1 − (y(1 − b))−1/2 . In this scenario, we get φ(y) = p 2 y(1 − b) − 1 (1 − b) . One can further derive that the equilibrium density ρ corresponding to y is 1 − (y(1 − b))−1/2 . To this end, we have λ(ρ) = −j ′′ (ρ) = −2y 3/2 b(1 − b)1/2 . Further, stationarity of the product Bernoulli measure implies A(ρ) = ρ(1 − ρ) = (y(1 − b))−1/2 (1 − (y(1 − b))−1/2 ). Finally, it can be noted that 2/3 p 1/3 1/3 y(1 − b) − 1 b 1 = σy . = − λ(ρ)A2 (ρ) 2 y 1/6 (1 − b)1/2 Thus, KPZ class conjecture implies that the height function N⌊yt⌋ (t) of HL-PushTASEP started from the step initial condition has a limit shape given by tφ(y) in (2.18) and more importantly, the fluctuation around the limit shape under the scaling of (−2−1 λA2 t)1/3 converges to the Tracy-Widom GUE distribution. As one can see, Theorem 2.4 verifies the conjecture and at the same time, identifies the form of the limit shape and the correct scaling of the fluctuation. 3. Transition Matrix & Laplace Transform For proving the asymptotic results, we need to have concrete knowledge on the distribution of the positions of the particles at any fixed time. In most of the cases of interactive particle systems (see, [TW11] for ASEP, [BCG16] for the stochastic six vertex model), transition imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 15 matrices for the underlying Markov processes play important roles towards that goal. In particular, transition matrices assists in getting suitable Fredholm determinant formulas for the Laplace transforms of some of the important observables like in the case of several quantum integrable interacting particle systems (see, [BC14] for a survey). In the following subsections, we elaborate on the derivation of the transition matrix of the HL-PushTASEP and its use to proclaim the formulas for the Laplace transforms. 3.1. Transition Matrix & Transition Probabilities Primary goal of this section is to compute the transition matrix in the case of finite particle configurations of the HL-PushTASEP. Later in this section, we use the transition matrix to specialize on the transition probabilities of a single particle in a system of finite particle configuration. At first, fix some positive integer N . For any A ⊆ X N define X (N ) P~x (A; t) := Tt (~x → ~y ) ~ x∈A (N ) where Tt (~x → ~y ) denotes the probability of transition to ~y ∈ X N from ~x ∈ X N . In the (N ) following proposition, we derive the eigenfunctions of the transfer matrix Tt . Proposition 3.1. For N > 0, fix N small complex numbers z1 , z2 , . . . , zN such that \|bzi \| < 1 for 1 ≤ i ≤ N and 1 − (2 + b)zj + bzi zj 6= 0 for all 1 ≤ i 6= j ≤ N . Fix a permutation σ ∈ S(N ). Define, Y 1 − (1 + b)zσ(i) + bzσ(i) zσ(j) Aσ = (−1)σ . 1 − (1 + b)zi + bzi zj i<j Then the function Φ(x1 , x2 , . . . , xN ; z1 , z2 , . . . , zN ) = X σ∈S(n) Aσ N Y xi zσ(i) i=1 is an eigenfunction of the transfer matrix T (N ) , i.e X y ∈X N ~ (N ) Tt (~x → ~y )Φ(~y ; z1 , z2 , . . . , zn ) = exp N X i=1 1 − zi −t 1 − bzi ! Φ(~y ; z1 , z2 , . . . , zn ). (3.1) As pointed out in Section 2.2, HL-PushTASEP can be considered as a continuum version of the stochastic six vertex model for which the transfer matrix has been worked in full details in the past. See [Lie67], [Nol92] for the derivation in the case of the stochastic six vertex model under periodic boundary condition and [BCG16, Theorem 3.4] for the derivation on Z. Let us note that one can derive the relation (3.1) by taking appropriate limit (see, (2.1)) of the transfer matrix of stochastic six vertex model in [BCG16, Eq.16]. Although, one has to be careful because the limit might not exist. For completeness, we present a self contained proof of the result in Appendix B. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 16 Proposition 3.2. For N > 0, fix N small complex numbers z1 , z2 , . . . , zN such that \|bzi \| < 1 for 1 ≤ i ≤ N and 1 − (2 + b)zj + bzi zj 6= 0 for all 1 ≤ i 6= j ≤ N . For any permutation σ ∈ S(N ), define Y 1 − (1 + b)zσ(j) + bzσ(i) zσ(j) A′σ = (−1)σ 1 − (1 + b)zj + bzi zj i<j and A′′σ = (−1)σ Y 1 − (1 + b−1 )zσ(i) + b−1 zσ(i) zσ(j) 1 − (1 + b−1 )zi + b−1 zi zj i<j . Let us assume ~x = (x1 , . . . , xN ) and ~y = (y1 , . . . , yN ). Denote the transition matrix for N (N ) particles in time t by Tt . Then, we have (N ) Tt (~x → ~y) = Z X (Cr )N σ∈S(N ) A′σ n Y −xi yi −1 zσ(i) zi exp xi zσ(i) zi−yi −1 exp i=1 1 − zi −t 1 − bzi dz1 . . . dzN (3.2) (3.3) and (N ) Tt (~x → ~y ) = Z X (CR )N σ∈S(N ) A′′σ n Y i=1 1 − zi−1 −t 1 − bzi−1 dz1 . . . dzN where Cr is small positively oriented circle leaving all the singularities outside and CR is large positively oriented circle containing all the singularities inside. Proof. Using the relation (3.1), both the formulas in (3.2) and (3.3) can be proved in the same way as in [BCG16, Theorem 3.6] (see also [TW11, Section II.4.b]). We omit here further details of the proof. Theorem 3.3. Fix an integer N > 0 and a positive real number t. Let us assume that we have N particles which are initially at the position ~y = (y1 , y2 , . . . , yN ) ∈ X N evolve according the dynamics of HL-PushTASEP. Then, for 1 ≤ m ≤ N , we have X X k−1 bκ(S,Z>0 )−mk−k(k−1)/2 P~y (xm = x; t) = (−1)m−1 bm(m−1)/2 m−1 b m≤k≤N \|S\|=k I I Y zj − zi 1 . . . × (2πi)k 1 − (1 + b−1 )zi + b−1 zi zj i,j∈S,i<j Q Y 1 − i∈S zi 1 − zi−1 x−yi −1 dzi zi exp −t ×Q 1 − bzi−1 i∈S (1 − zi ) i∈S where the summation goes over S ⊂ {1, 2, . . . , N } of size k, κ(S, Z>0 ) is the sum of the elements in S, and contours are positively oriented large circles of equal radius which contains all the singularities. Proof. This result is in the same spirit of [BCG16, Theorem 4.9] where they found out explicitly the probability of the similar event in the case of the stochastic six vertex model. They closely followed the techniques used in [TW08, Section 6] in the context of N-particle ASEP. It begins with the realization that P(xm (t) = x; 1, b) is the sum of the probabilities imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 17 (N ) Tt (~y → ~x) over all ~x = (x1 , x2 , . . . , xN ) ∈ X N such that −∞ < x1 < x2 < . . . < xm−1 < xm = x and x < xm+1 < . . . < xN < ∞. Now, one have to make use of the formulas for (N ) Tt (~y → ~x) mentioned in (3.2) and (3.3). Notice that for the sum over all ~x ∈ X N such that −∞ < x1 < x2 < . . . < xm−1 < xm = x contours of integration is Cr and for other part of the sum contours will be CR . For brevity, we are skipping here the details of the proof. 3.2. Moments Formulas In this subsection, we aim to present a series of results on moments of the height function of HL-PushTASEP. One can see [BCG16, Section 4.4] for similar results in the case of the stochastic six vertex model. In Section 4, these formulas turns out to be instrumental for obtaining the asymptotics. For a function f : X N → R, let E~y (f ; t) represent the expectation of f at time t, i.e X (N ) E~y (f ; t) = Tt (~y → ~x). ~ x∈X N Also, let us introduce (a; q)k and (a; q)∞ by defining (a; q)k := k Y (1 − aq j−1 ) and (a; q)k := j=1 ∞ Y (1 − aq j−1 ). j=1 Furthermore, there is a q - analogue of binomials defined as follows (q N −k+1 ; q)k n (1 − q N )(1 − q N −1 ) . . . (1 − q N −k+1 ) = . := (1 − q)(1 − q 2 ) . . . (1 − q k ) (q; q)k k q In both of the above definitions, we assume k is a non-negative integers. Proposition 3.4. Consider HL-PushTASEP on Z≥0 . Fix a positive real number t. Assume that the process has been started from the step Bernoulli initial condition with parameter ρ. Denote the initial data by stepb which belongs to X N . Then for L = 0, 1, 2, . . . and any positive integer x, we have ! L k−2 Y X b−k(k−1)/2+L−k ρk (1 − b1−k+L bi ) Estepb (bLNx ; t) = 1 + (b−L − 1) (b−1 ; b−1 )k I Ik=1 Yi=1 zj − zi 1 × ... k (2πi) 1 − (1 + b−1 )zi + b−1 zi zj 1≤i<j≤k × k Y i=1 (ρ + (1 − zix (b−1 − 1) ρ)b−1 − zi b−1 )(1 1 − zi−1 exp −t − zi ) 1 − bzi−1 dzi (3.4) where the contours of integrations are largely oriented circles with equal radius and contain all the singularities of the integrand. Proof. One can find a similar result in [BCG16, Proposition 4.11] in the case of stochastic six vertex model with step initial data. For the sake of clarity, we present an independent proof in C. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 18 Proposition 3.5. Consider HL-PushTASEP on Z≥0 . Fix a positive number t. As usual stepb ∈ X N denotes the step Bernoulli initial condition with parameter ρ. Then for L = 0, 1, 2, . . . and for any integer x, we have Estepb (b LNx bL(L−1)/2 ; t) = (2πi)L I x I Y L tui dui uA − uB Y 1 + ui exp , ... −1 uA − buB b 1 + ui b ui (1 − αui b−1 ) A<B i=1 where the positively oriented contour for uA includes 0, −b, but does not include bρ(1 − ρ)−1 or {buB }B>A . Note that α := (1 − ρ)ρ−1 . Proof. To prove this result, one needs to substitute zi = (1 + ui )/(1 + ui b−1 ) into the formula (3.4). Consequently, one can notice ui − uj zj − zi = −1 −1 1 − (1 + b )zi + b zi zj ui − buj , dui ρ(b−1 − 1)dzi = −1 −1 (ρ + (1 − ρ)b − zi b )(1 − zi ) ui (1 − αui b−1 ) where α = (1 − ρ)/ρ. Let us mention that a similar result is proved in the case of stochastic six vertex model in [BCG16, Theorem 4.12]. See [BCS14, Theorem 4.20] for a different proof in the case of ASEP. Rest of the calculation after the substitution can be completed in the same way as in [BCG16, Theorem 4.12]. We omit any further details from here. 3.3. Fredholm Determinant Formula In this section, we discuss the representation of the moment formulas of HL-PushTASEP in terms of Fredholm determinant. In the last ten years, a large number of works had been put forward in the literature of KPZ universality class and Fredholm determinant formulas play a crucial role in reaching out such results. For similar exposition in stochastic six vertex model, see [BCG16, Section 4.5]. See [BC14, Section 3.2] and [BCS14, Section 3 and 5] for corresponding formulas in the case of q-Whittaker process and q-TASEP. Moreover, various other instances can be found in the works like [CP16, Theorem 4.2], [BC16, Section 3.3], [BCF14]. Definition and some of the useful properties of the Fredholm determinants are discussed in Appendix A. Definition 3.6. Here, we define the contours DR,d,δ and DR,d,δ;κ . Let us fix positive numbers R, δ, d such that d, δ < 1. Then the contour DR,d,δ is composed of five linear sections: DR,d,δ := (R − i∞, R − id] ∪ [R − id, δ − id] ∪ [δ − id, δ + id] ∪ [δ + id, R + id] ∪ [R + id, R + i∞). Let κ be an integer which is greater than R. Denote two points at which circle of radius κ + δ and centre at 0 crosses (R − i∞, R − id] and (R − i∞, R − id] by ζ and ζ ′ . Let us call that minor arc joining ξκ and ξκ′ to be Iκ . Then DR,δ,d;κ is defined to be a positively oriented closed contour formed by the union DR,d,δ := (ξκ , R − id] ∪ [R − id, δ − id] ∪ [δ − id, δ + id] ∪ [δ + id, R + id] ∪ [R + id, ξκ′ ) ∪ Iκ . See Figure 3 for details. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 19 R + i∞ ζ δ + id R + id κ 1 R − id δ − id ζ′ R − i∞ Fig 3: Plot of the contour DR,d,δ and DR,d,δ;κ . Theorem 3.7. Consider HL-PushTASEP on Z≥0 . Fix a positive number t. Let stepb ∈ WN stands for the step Bernoulli initial data with parameter ρ. Let Cr be a positively oriented circle centered at some negative real number r such that Cr include 0 and −b, but does not contain bα−1 . Furthermore, Cr does not intersect with bCr , i.e, bCr is contained in the interior of Cr . Then, there exist a sufficiently large real number R > 1, and sufficiently small real numbers δ, d < 1 such that for all ζ ∈ C\R≥0 , we have (A) inf ′ w,w ∈Cr κ∈(2R,∞)∩Z s∈DR,d,δ;κ \|bs w − w′ \| > 0, (B) sup w∈Cr κ∈(2R,∞)∩Z s∈DR,d,δ;κ g(w) < ∞. g(bs w) Moreover, for any ζ ∈ C\R≥0 , we have 1 b , Estepb ; t = det I + K ζ (ζbNx ; b)∞ L2 (Cr ) (3.5) (3.6) where the kernel Kζb is given by Kζb (w, w′ ) 1 = 2πi Z DR,d,δ g(w; 1, b, x, t) ds (−ζ)s · · s , s sin(πs) g(b w; 1, b, x, t) b w − w′ (3.7) integration contour DR,d,δ is oriented from bottom to top and x 1 tz 1 g(z; 1, b, x, t) = exp . 1 + zb−1 b(1 − b) (αb−1 z; b)∞ where α = ρ−1 (1 − ρ). imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 20 Proof. We first prove the existence of R, d and δ such that (A) and (B) in (3.5) are satisfied. Note that Cr doesn’t contain 0 on it and it is a bounded closed set in C, thus, compact. Hence, \|w′ w−1 \| has a lower bound whenever w, w′ ∈ Cr . Let us choose R in such a way that bR becomes less than the lower bound of \|w′ w−1 \|. This ensures that the infimum of \|bs w − w′ \| is bounded away from 0 along (ξk , R − id] ∪ [R + id, ξk′ ) ∪ Ik . Further, note that bCr never touches Cr . Thereafter, using boundedness of Cr , we can found δ sufficiently close to 1 such that bs Cr remains disjoint from Cr for all s ∈ [δ − id, δ + id]. However, it implies somewhat much stronger statement. On can say further that bs Cr escapes from Cr for all s ∈ [δ − id, R − id] ∪ [δ + id, R + id]. Hence, condition (A) in (3.5) is satisfied. Furthermore, one can see g(w) 1 + bs−1 w tw(1 − bs ) (αbs−1 w; b)∞ = . exp g(bs w) 1 + b−1 w b(1 − b) (αb−1 w; b)∞ Choice of Cr and R, δ, d implies immediately that condition (B) in (3.5) is also met. Now, we turn into proving (3.6). Techniques for similar determinantal formula is quite well known by now, thanks to the vast literature mentioned above. We will sketch here the basic outline of such technique. For brevity, we skip major portion of details in the proof. We first look into the expansion of 1/(ζbNx ; b)∞ . Using [AAR99, Corollary 10.2.2 a], one can say ∞ X ζ L bLNx 1 = (ζbNx ; b)∞ (b; b)L L=0 where (b; b)L = QL i=1 (1 − b · bi−1 ). Recall that in Lemma 3.5, we have bL(L−1)/2 µL := Estepb (bLNx ; t) = (2πi)L where f (u) := exp tu b I ... I Y L uA − uB Y f (ui ) dui , uA − buB ui A<B 1+u 1 + ub−1 x i=1 1 1 − αub−1 and the integration contours do include 0, −b inside, but, not α−1 b. Moreover, one can notice that f doesn’t have a pole in an open neighborhood of the line joining 0 P and −b. Let us denote any partition λ = (λ1 , λ2 , . . .) of any positive integer k by λ ⊢ k where ∞ i=1 λi = k. Further, denote number of positive λi ’s in λ by l(λ). Thus, using [BCF14, Proposition 5.2], one can conclude that X 1 1 µL = (b; b)L m1 !m2 ! . . . (2πi)l(λ) λ⊢L 1m1 2m2 ... × Z ... C Z C det −1 bλi wi − wj l(λ) l(λ) Y f (wj )f (bwj ) . . . f (bλj −1 wj )dwj i,j=1 j=1 where the contour C also avoids 0, −b and doesn’t intersect with bn C for any n ∈ N. To this end, one can conform the contour C to the the circle Cr without passing any singularity of the integrand. Further, using the form of the function g, it is easy to see g(wj ; 1, b, x, t) = f (wj )f (bwj ) . . . f (bλj −1 wj ). g(bλj wj ; 1, b, x, t) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 21 Consequently, we can write ∞ ∞ X X ζ L µL = (b; b)L L=0 L=0 X λ⊢L 1m1 2m2 ... 1 ζL m1 !m2 ! . . . (2πi)l(λ) × Z Cr ... Z det Cr (3.8) g(wj ; 1, b, x, t) −1 λ i b wi − wj g(bλj wj ; 1, b, x, t) l(λ) l(λ) Y dwj . i,j=1 j=1 At this point, we will make an interchange of two summations above. Let us fix a positive integer k and first sum over all partitions λ which has size k, i.e l(λ) = k. Interchange of summations is justified by the linearity of the integrals. Furthermore, one can write ζ L = ζ λ1 +...+λl(λ) . To that effect, the right side of (3.8) simplifies to !k Z Z ∞ k ∞ X Y X g(wj ; 1, b, x, t) (−ζ)s 1 det . . . dwj . k!(2πi)k C bL wi − wj g(bL wj ; 1, b, x, t) C k=0 i,j=1 j=1 L=1 Now, it remains to show that ∞ X L=1 g(wj ; 1, b, x, t) (−ζ)s 1 = L L b wi − wj g(b wj ; 1, b, x, t) 2πi Z DR,d,δ (−ζ)s g(w; 1, b, x, t) ds · · s . (3.9) s sin(πs) g(b w; 1, b, x, t) b w − w′ Note that the function 1/ sin(πs) decays exponentially as \|s\| → ∞ along the segments (R − i∞, R − id] and [R + id, R + i∞). On the flip side, (bs w − w′ )−1 g(w)/g(bs w) is uniformly bounded on DR,d,δ,κ for all k ≤ R. Thus, integrand in (3.9) goes to zero at at both ends of the line Re(z) = R. Instead of integrating over DR,d,δ , if we choose to integrate over DR,d,δ;κ , then using residue theorem, we obtain Z 1 (−ζ)s g(w; 1, b, x, t) ds · · s (3.10) s 2πi DR,d,δ;κ sin(πs) g(b w; 1, b, x, t) b w − w′ κ X g(w; 1, b, x, t) 1 (−ζ)s · · . Ress=L = sin(πs) g(bs w; 1, b, x, t) bs w − w′ L=1 Moreover, if we let κ grows to infinity, then the integral along the semi-circular arc Iκ tends 0 thanks again to the exponential growth of sin(πs). Thus, integral over DR,d,δ;κ converges to the right side in (3.9) whereas when κ → ∞, the right side in (3.10) recovers the other side of the equality in (3.9). This completes the proof. 4. Asymptotics Main aim of this section is to prove the Theorem 2.4 and Theorem 2.5. Asymptotic analysis that we present here adapts closely to the similar expositions in [BCG16, Section 5]. The crux of the proof of Theorem 2.4 essentially is governed by the the steepest descent analysis of the kernel of the Fredholm determinant in Theorem 3.7. In the following propositions, large time limit of the Fredholm determinant will be provided. We conclude the proof of Theorem 2.4 after Lemma 4.2 modulo the proof of these propositions. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 22 Proposition 4.1. Recall the definition of the contour Cr from Theorem 3.7. Adopt the notations used in Theorem 2.4. 1/3 (a) Assume ρ > 1 − (ν(1 − b))−1/2 . Set x(t) = ⌊νt⌋ and ζt = −b−mν t+sσν t lim det 1 + Kζbt 2 = FGU E (s). t→∞ . Then, we have (4.1) L (Cr ) (b) Let ρ = 1 − (ν(1 − b))−1/2 . For the same choices of x(t) and ζt as above, we have b 2 lim det 1 + Kζt 2 = FGOE (s). t→∞ L (Cr ) ′ 1/3 (c) In the case when ρ < 1 − (ν(1 − b))−1/2 , set x(t) = ⌊ν ′ t⌋ and ζt = −b−mν t+sσ νt one can get lim det 1 + Kζbt 2 = Φ(s). t→∞ L (Cr ) . Then, (4.2) Lemma 4.2. ([BC14, Lemma 4.1.39]) Consider a set of functions {ft }t≥0 mapping R → [0, 1] such that for each t, ft (x) is strictly decreasing in x with a limit of 1 at x = −∞ and 0 at x = ∞ and for each δ > 0, on R\[−δ, δ], ft converges uniformly to 1(x ≤ 0). Define the r-shift of ft as fnr (x) = fn (x − r). Consider a set of random variables Xt indexed by t ∈ R+ such that for each r ∈ R, t→∞ E[ftr (Xt )] −→ p(r) and assume that p(r) is a continuous probability distribution. Then Xt weakly converges in distribution to a random variable X which is distributed according to P(X ≤ r) = p(r). Proof of Theorem 2.4. For part (a) and (b) of Theorem 2.4, we use Lemma 4.2 with the functions 1 , t ∈ R+ . ft (z) := 1/3 z −t (−b ; b)∞ Note that ft (z) is a monotonously decreasing function of the real argument of z, yielding limt→∞ ft (z) = 0 if z > 0 and limt→−∞ ft (Z) = 1 if z ≤ 0. Set Xt = σν−1 t−1/3 (mν t − Nνt (t)). Thus, for any s ∈ R, we can write fts (Xt ) = 1 (ζt bNνt (t) ; b)∞ 1/3 where ζt = b−mν t+sσν t . To this end, part (a) and (b) of Proposition 4.1 shows that the limit of E (fts (Xt )) are indeed probability distribution function. Hence, this proves part (a) and (b) of Theorem 2.4. For part (c), we replace ft by gt which is defined as gt (z) := 1 (−b −t1/2 z ; b)∞ , t ∈ R+ . 1/3 Setting Xt = (σ̃ν )−1 t−1/2 (m̃ν t − Nνt (t)) and ζt = b−mν t+sσ̃ν t , one can note gts (Xt ) = (ζt bNνt (t) ; b)−1 ∞ . Therefore, rest of the proof follows from the part (c) of Proposition 4.1. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 23 Proof of Proposition 4.1 We start with some necessary pre-processing of the kernel Kζbt . Notice that in the statements of all three parts of Proposition 4.1, ζt is substituted with −bβ and x is substituted with ⌊νt⌋ where β is some function of the time variable t and ν > (1 − b)−1 . Under these substitutions, the form of the kernel Kζbt effectively reduces to 1 2i Kζbt (w, w′ ) = Z DR,d,δ g(w; 1, b, νt, t) dh bhβ · · sin(πh) g(bh w; 1, b, νt, t) bh w − w′ where the contour ofthe integration is oriented from the bottom to top and g(z; 1, b, x, t) = x 1 t exp b(1−b) (αb−1 z; b)−1 ∞ . Like in [BCG16], the particular form of the integrand in 1+z/b (3.7) calls for the substitution bh w = v. Unfortunately, the latter substitution is not injective. Thus, to study how this effect the integral, we have to divide the contour DR,d,δ into countably many parts which have one-to-one images under the above substitution. For instance, each of the linear sections Il := [R + i(d + 2π log(b)−1 l), R + i(d + 2π(log(b))−1 (l + 1))] of the contour of h for l ∈ Z\g{−1} maps to the same image. Furthermore, other parts of DR,d,δ , i.e., the union [R − i(d + 2π log(b)−1 ), R − id] ∪ [R − id, δ − id] ∪ [δ − id, δ + id] ∪ [δ + id, R + id] maps to a closed contour DR,d,δ,w composed of four different portions. For completeness, we present here a formal definition of the contour DR,d,δ,w . Definition 4.3. For each w, define a contour DR,d,δ,w as the union T1 ∪ T2 ∪ T3 ∪ T4 . We describe Ti ’s successively. To begin with, T1 is a major arc of a small circle around origin with radius bR \|w\|. To be exact, T1 is the image of the segment [R − i(d + 2π log(b)−1 ), R − id] under the map h 7→ bh w. Similarly, T2 , image of [δ − id, δ + id], is a minor arc of a larger circle of radius bδ \|w\|. Apparently, these two arcs of two different circle are connected by two linear segments T2 and T3 . Precisely, T2 connects bR−id w to bδ−id w and T3 connects bδ+id w to bR+id w. It is evident from the definition of T2 (T3 ) that it is the image of [R − id, δ − id] ([δ + id, R + id]) under the above map. Due to the bottom-to-top orientation of the contour DR,d,δ in Theorem 3.7, DR,d,δ,w will be anti clockwise oriented. See Figure 4 for further details. For notational convenience, let us denote the circle of radius s with center at origin in the complex plane C by Cs (note the difference from Cr ). In fact, CbR \|w\| is the image of each linear segments Il for l ∈ Z\{−1}. Thus, one can write Kζbt (w, w′ ) Z ∞ X 1 = 2i C R k=−∞ k6=−1 1 + 2i Z DR,d,δ,w b (v/w)β \|w\| sin π log(b) (log(v/w)) (v/w)β + 2πik · dv g(w; 1, b, νt, t) · g(v; 1, b, νt, t) v log(b)(v − w′ ) dv g(w; 1, b, νt, t) · · . (4.3) π g(v; 1, b, νt, t) v log(b)(v − w′ ) (log(v/w)) − 2πi sin log(b) At this point, we would make a surgery over the contour DR,d,w,k . Let us connect a minor arc −1 from bR−id w to bR−i(d+2π log(b) ) w in the circle CbR \|w\| (major arc being T1 ) and call it T1′ . We imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit T4 T3 24 wbh R + id T2 R − id T1 R − i(d + 2π log(b)−1 ) Fig 4: Figure on the right side indicates the section of the contour DR,d,δ which under the map h 7→ wbh pertains to the contour T1 ∪ T2 ∪ T3 ∪ T4 shown in the left. use −T1′ to indicate clockwise orientation. Therefore, D̃R,d,w,k := T2 ∪ T3 ∪ T4 ∪ −T1′ denotes a closed contour oriented anti clockwise. Using the properties of the contour integration, it can be said that integral over DR,d,w,k in (4.3) is a sum of the integrals over D̃R,d,w,k and CbR \|w\| . Further, notice that there is no pole of the integrand sitting inside the contour D̃R,d,w,k for any w ∈ Cr (see, Theorem 3.7 for w) thanks to the conditions (A) and (B) in (3.5). Thus, integral over D̃R,d,w,k vanishes. To this effect, one can write Kζbt (w, w′ ) Z ∞ X 1 = 2i C R k=−∞ b (v/w)β \|w\| sin π log(b) (log(v/w)) + 2πik · dv g(w; 1, b, νt, t) · . g(v; 1, b, νt, t) v log(b)(v − w′ ) (4.4) 4.1. Case ρ > 1 − (ν(1 − b))−1/2 , Tracy-Widom Fluctuations (GUE). Fix ν > (1 − b)−1 . Set β = −mν t + sσν t1/3 . Define a function Λ as Λ(z) = z − ν log(1 + z/b) + mν log(z). b(1 − b) (4.5) Thus, one can further simplify (4.4) into Kζbt (w, w′ ) ∞ Z X 1 = 2i log(b) C R k=−∞ b exp(t(Λ(w) − Λ(v)) + t1/3 sσν (log(v) − log(w)))dv π ′ ) sin (log(v/w)) + 2πik (v − w \|w\| log(b) × dv (αb−1 v; b)∞ × . −1 (αb w; b)∞ v (4.6) In a nutshell, the asymptotic behavior of the kernel is governed by the variations of the real part of Λ. In what follows, we exhibit the steepest descent contours, which helps in localizing the main contribution in a neighborhood of a critical point of Re(Λ). Thereafter, using Taylor imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 25 expansion of the arguments of the exponential inside the kernel in (4.6) and appropriate estimates for the kernel in the region away from the critical points, we prove the limit. To this end, notice that νb−1 mν 1 − + . Λ′ (z) = b(1 − b) 1 + z/b z √ ( ν(1−b)−1)2 Plugging mν = , we get 1−b Λ′ (z) = (z − ̺)2 , b2 (1 − b)(1 + ̺/b)̺ p where ̺ = b( ν(1 − b) − 1). This shows that Λ′ has an unique double critical point at ̺. In particular, in a neighborhood of ̺, we have Λ(z) = Λ(̺) + σν ̺ 3 (z − ̺)3 + R((z − ̺)) where (z − ̺)−3 R((z − ̺)) → 0 when z → ̺. thanks to fact that σν ̺ 3 p −1 ν(1 − b) − 1 (1 − b)3/2 ν 1/2 = b2 = Λ′′′ (̺). Now, we would like to deform the contours of v and w suitably so that Re(Λ(w) − λ(v)) < −c for some c > 0 whenever v and w are away from the critical point ̺. In the following lemma, we describe the level curves of Re(Λ(z)) = Re(Λ(̺)) in details. We call a closed contour (let’s say C) star shaped if for any φ ∈ [0, 2π), there exists exactly one point z ∈ C such that Arg(z) = φ. Lemma 4.4. There are exactly three level curves L1 , L2 and L3 of Re(Λ(z)) = Re(Λ(̺)) in the complex plane C satisfying the following properties: (a) both L1 and L2 are simple closed contours meeting with L3 at the point ̺. Further, L1 (L2 ) cuts the negative half of the real line at some point d1 (d2 ) in the interval (−b, 0) ((−∞, −b)), (b) L3 resides in the right half of the complex plane, escapes to infinity as \|z\| → ∞, (c) both L1 and L2 are star shaped, (d) L1 \{̺} is completely contained inside the interior of the region enclosed by L2 . Furthermore, L3 meet the contours L1 and L2 at no other point point except ̺, (e) L1 (L2 ) leaves the x-axis at an angle 5π/6 (π/2) from the point ̺ and meets again at −5π/6 (−π/2). Moreover, upper half (lower half ) of the contour L3 leaves the real line from ̺ at an angle π/6 (−5π/6). Proof. We closely adapt here the proofs of the related results in the context of stochastic six vertex model (see, [BCG16, Section 5.1]). (a) It can be observed in (4.5) that Λ(z) has singularities at 0 and −b on R. Thus, there are at least two level curves L1 and L2 of Re(Λ(z)) = Re(Λ(̺)) which originate from the point ̺ and in the way back to their origin, they cut R− at some points d1 ∈ (−b, 0) and d2 ∈ (−∞, −b) respectively. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 26 (b) Tracing the changes in sign of Re(Λ(z)) − Re(Λ(̺)) in Figure 5, one can infer that outside L2 , there should be a region where Re(Λ(z)) − Re(Λ(̺)) < 0. But, if we take a large circle CM of radius M in C, then for z ∈ CM and Re(z) < 0, we have Re(z) − ν log \|zb−1 (1 + z −1 b)\| + mν log \|z/b\| b(1 − b) Re(z) − (ν − mν ) log \|z\| + ν log(b) − mν log(b) + o(1) < 0 = b(1 − b) Re(Λ(z)) = (4.7) because mν < ν. Consequently, there doesn’t exists any other level curve which contains 0 or −b inside. But, there are another level curve L3 escaping to the infinity in the right half of the complex plane after taking off from ̺. This behavior can also be predicted from (4.7). Let us sketch the necessary arguments behind such predictions. Note that if we increase the value Re(z) to the infinity with a fixed value of Im(z), then the right side of (4.7) goes to infinity as well. This shows that at least in the right half of C, some more solutions (other than those on L1 and L2 ) of the equation Re(Λ(z)) = Re(Λ(̺)) exist. This verifies the second claim. (c) To show that both L1 and L2 are star shaped, we need to prove that any line from origin can cut L1 or L2 exactly at two points. Let us fix a line rz0 in C where z0 is fixed with Re(z0 ) = 1 and r varies over the real line. To this end, we have (z0 ) νz0 b−1 1 ∂ Re(Λ(z)) = Re − + mν . (4.8) ∂r b(1 − b) 1 + rzo b−1 r One can note that solutions of the equation that the right side of (4.8) equals to some fixed value is given by a cubic polynomial. But any cubic polynomial over R can have either one ∂ or three real roots. Next, we show ∂r Re(Λ(rz0 ))) = 0 has exactly three real roots unless Im(z0 ) = 0. It can be readily verified that (i) when r ↓ 0, then right side of (4.8) increases ∂ Re(Λ(rz0 ))) → b−1 (1 − b)−1 and when r = ̺, then to infinity, (ii) when r → ∞, then ∂r ∂ ∂ ∂r Re(Λ(rz0 ))) < 0 unless Im(z0 ) = 0. This shows the equation ∂r Re(Λ(rz0 ))) = 0 has at + least two distinct solutions on R unless Im(z0 ) = 0. Further, co-efficient of r 3 and the constant term in the denominator of (4.8) have same sign. This proves the existence of three real roots out of which two must be positive and last one should be negative. Even those two positive roots are distinct unless Im(z0 ) = 0. Thus, except for the case when Im(z0 ) = 0, none of the roots is an inflection point. We have seen Re(Λ(z)) = Re(Λ(̺)) has at least five roots along the line rz0 (four on L1 & L2 and fifth one on L3 ). As ∂ Re(Λ(rz0 )) vanishes in between any two roots of the same sign, therefore, we know ∂r number of roots of Re(Λ(rz0 )) = Re(Λ(̺)) cannot be greater than five. Otherwise, pairs of consecutive roots with the same sign must be greater than three contradicting the fact that the numerator of the right hand side in (4.8) is a polynomial of degree 3 in r. Thus, the third claim is proved. ∂ (d) Fourth claim again follows from the fact that the equation ∂r Re(Λ(rz0 ))) = 0 has exactly three solutions in r except when Im(z0 ) = 0. If L1 meets L2 at some point ̺′ 6= ̺, ∂ Re(Λ(r̺′ ))) = 0 is supposed to have at most two real solutions. Thus, ̺′ cannot then ∂r be different from ̺. But in the latter case, Re(Λ(r̺)) = (Λ(̺)) is satisfied for three distinct values of r, namely, 1, d1 ̺−1 and d2 ̺−1 . As a matter of fact, we know d1 6= d2 . Consequently, L1 has no other point of intersection with L2 except at ̺. Similarly, one imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 27 can prove L3 doesn’t meet L1 or L2 at any other point besides ̺. Moreover, positions of d1 and d2 over the real line implies L1 \{̺} is completely contained inside the interior of the region bounded by L2 . (e) To begin with, note that Λ(z) in a neighbouhood of the point ̺ behaves as Λ(̺) + σν3 ̺−3 (z − ̺)3 . To that effect, possible choices of the tangents for any of the level curves of the equation Re(Λ(z)) = Re(Λ(̺)) are given by {π/6 ± 2π/3} ∪ {π/6 ± 4π/3} = {±π/6, ±π/2, ±5π/6}. Moreover, we know L1 \{0} is completely contained inside the interior of the region bounded by the contour L2 . Also, L3 stays outside of both the contours L1 and L2 . These together justify the claims in part (e). Definition 4.5. We define two new contours Lw and Lv . Both Lv and Lw consist of a (1) (1) (2) (2) piecewise linear segment (Lv and Lw ) and a curved segment (Lv and Lw ). To begin with, (1) (1) Lw extends linearly from ̺ + ǫe−iπ/3 to ̺ and from there to ̺ + ǫeiπ/3 . Similarly, Lv goes from ̺−̺σν−1 t−1/3 +ǫe−2iπ/3 to ̺−̺σν−1 t−1/3 to ̺−̺σν−1 t−1/3 +ǫe2iπ/3 . Here, ǫ will be chosen (1) (1) in such a way that (i) Lv is completely contained inside L2 and Lw \{̺} does not intersect L2 anywhere else, (ii) \|R((z − ̺))\| is bounded above by c\|z − ̺\|3 for some small constant c. (2) Next, Lw starts from the point ̺ + ǫeiπ/3 and encircles around L2 to join ̺ + ǫe−iπ/3 . On the (2) contrary, Lv starts from the point ̺ − ̺σν−1 t−1/3 + ǫe2iπ/3 and curves around 0 and −b to meet ̺ − ̺σν−1 t−1/3 + ǫe−2iπ/3 . But, it never goes out of the contour L2 . It can be ascertained that both Lv and Lw are bounded away from L2 in a ǫ-neighborhood of ̺ by choosing ǫ appropriately. L3 Lw L2 + + Lv − −b L1 ̺ bα−1 − L3 Fig 5: In the above plot, dashed lines denote the level curves of the equation Re(Λ(z)) = Re(Λ(̺)) whereas solid lines indicate the contour Lv and Lw respectively. Signature of the function Re(Λ(z) − Λ(̺)) has been shown in the respective regions with ± signs. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 28 We deform the contours of v and w to Lv and Lw respectively. Such deformations is possible because Lv sits completely inside Lw whereas previously, contour for v was CbR \|w\| . It can be also noted that we didn’t allow any poles to cross in between in doing the deformations of the contours. Consequently, we have Re(Λ(w)) < Re(Λ(v)) for all w ∈ Lw and for all v ∈ Lv . In what follows, we show how the contribution of the integrals of Kb (ζt ) over the contour Lw dominates the rest in the Fredholm determinant in Proposition 4.1 (4.1). Lemma 4.6. There exists some c, C > 0 and t0 > 0, such that for all t ≥ t0 \|Kζbt (w, w′ )\| ≤ C max{ǫ−1 , ̺−1 σν t1/2 } exp −c(t + \|w\|) + sσ̃ν t1/3 \|w\| ∨ \|w\|−1 log(1 + ǫ) (4.9) (2) where w ∈ Lw , w′ ∈ Lw and ǫ is defined in Definition 4.5. Moreover, the following convergence holds lim det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) = 0 t→∞ w (4.10) Proof. To show (4.9), we need to bound the integrand in (5.13) with suitable integrable functions. In the following discussion, we enlist the upper bounds on the absolute value of each of the components of the integrand. (2) (i) For any two points v and w on the contours Lv and Lw respectively, we know Re(Λ(w)) < Re(Λ(v)). In fact, using the compactness of the respective contours, we can further say that there exists c > 0 such that Re(Λ(w)) − Re(Λ(v)) < −c whenever v ∈ Lv and (1) w ∈ Lw . Furthermore, for any complex number z, we have log \|z\| ≤ \|z\| ∨ \|z\|−1 . To sum up, we get exp(t(Λ(w) − Λ(v)) + t1/3 sσν (log(v) − log(w))) ≤ exp(−tc + sσν t1/3 \|w\| ∨ \|w\|−1 ). (4.11) (ii) For any k ∈ Z\{0}, a crude upper bound on the sine function in the integrand is given by π (log(v/w)) + 2πik \| ≥ C ′ exp(2π\|k\|) (4.12) \| sin log(b) for some constant C ′ > 0. For the case when k = 0, \| sin((π/ log(b)) log(v/w))\| is bounded (2) below by C ′′ minv∈L ,w∈L(2) \| log \|(v/w)\|\|. Recall that Lw stays at least ǫ away from the v w contour L2 . Thus, we have min (2) v∈Lv ,w∈Lw \| log \|(v/w)\|\| ≥ log(1 + ǫ). (4.13) Henceforth, (4.12) and (4.13) implies that X k∈Z 1 sin π log(b) (log(v/w)) + 2πik ≤ ∞ X 2 exp(−2πk). log(1 + ǫ) (4.14) k=0 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 29 (iii) Among the other components, the following bounds can be readily proved. −1 αb (αb−1 v; b)∞ ≤ exp (\|v\| + \|w\|) , (αb−1 w; b)∞ 1−b 1 1 ≤ max{ǫ−1 , ̺−1 σν t1/2 }. ≤ max v∈Lv ,w ′ ∈Lw v − w ′ v−w (4.15) Thus, these completes the proof of (4.9). Next, we show the limit in (4.10). For that, decompose Kζb as Kζbt (w, w′ ) := Kζb,1 (w, w′ ) + Kζbt (w, w′ ), t (4.16) b (2) ′ = Kζbt (w, w′ )1(w ∈ L(1) w ) + Kζt (w, w )1(w ∈ Lw ). Using the definition of the Fredholm determinant in (A.1), one can write det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) (4.17) w Z Z X 1 \| det(Kζbt (wi , wj )) − det(Kζb,1 (wi , wj ))\|dw1 . . . dwn . ≤ ... t n!(2πi)n Lw Lw n≥0 Furthermore, (4.16) implies \| det(Kζbt (wi , wj )) − det(Kζb,1 (wi , wj ))\| ≤ t X τ1 ,...,τn ∈{1,2} (τ1 ,...,τn ) 6=(1,...,1) b,τ \| det(Kζt j (wi , wj ))\| (4.18) b,τ where for any fixed (τ1 , . . . , τn ) ∈ {1, 2}n , (Kζt j (wi , wj ))ni,j=1 denotes a matrix whose j-th b,τ column is given by the j-th column of Kζt j . Using the bound on the kernel in (4.9), we have ! n n X X ǫ−n tn/2 b,τj −1 1/2 . \| det(Kζt (wi , wj ))\| ≤ C \|wi \| ∨ \|wi \| \|wi \| + sσν t exp −tc + c (log(1 + ǫ))n i=1 i=1 Moreover, there are 2n − 1 many terms in the sum in the right side of (4.18). These implies one can further bound the right side of (4.17) by det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) ≤ C exp(−ct + c′ t1/2 ) w (4.19) for some constant C, c, c′ > 0. Notice that the right side of (4.19) converges to 0 as t tends to ∞. This implies that the difference of two Fredholm determinant in (4.10) is exponentially decaying to 0. We have decomposed the kernel Kζbt in (4.16). Apart from that, there is another kind of canonical decomposition of Kζbt based on the division of the integral in (4.6) into two parts. For instance, let us consider (1) (2) Kζbt (w, w′ ) = Kζ1 (w, w′ ) + Kζt (w, w′ ) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 30 where for l = 1, 2 (1) Kζt ∞ Z exp t(Λ(v) − Λ(w)) + sσν t1/3 (log(v) − log(w)) 1 X := (l) π 2i (log(v) − log(w)) + 2πik sin log(b) k=−∞ Lv αb−1 v; b ∞ exp (s (log(v) − log(w))) dv × × × . ′ −1 log(b)(v − w ) (αb w; b)∞ v (2) In what follows, we show that the contribution of the kernel Kζ2 in det(1 + Kζbt )L2 (Lw ) fades way as t grows up. Lemma 4.7. There exist c, C > 0 such that ǫ−1 (2) \|Kζt (w, w′ )\| ≤ exp −c(t + \|w\|) + sσν t1/3 \|w\| ∨ \|w\|−1 log(1 + ǫ) (4.20) (1) holds for all w, w′ ∈ Lw . Thus, one would get (1) lim det(1 + Kζbt )L2 (L(1) ) − det(1 + Kζt )L2 (L(1) ) = 0. t→∞ w w (4.21) Proof. Like in the proof of (4.9), (4.20) can be proved using the bound on each of the com(2) ponents of the integrand of Kζt . Bound for the exponential term in the integrand would be exactly same as what had been derived in (4.11). For the series over inverse sin function, we use the bounds in (4.12) and (4.13) together to get (4.14). We must point out here that mini(2) (1) mum value of \| log \|v/w\|\| for any v ∈ Lv and w′ ∈ Lw is again bounded below by log(1 + ǫ), thus, verifying (4.13) in the present scenario. Bound on the infinite product term in (4.15) will (2) (1) again be the same. Minimum distance between any two points on the contour Lv and Lw is ǫ. Thus, bound on \|1/(v − w′ )\| for the present scenario is only ǫ−1 . These all together establish the inequality in (4.20). Using the expansion of the Fredholm determinant as in (4.17) and bound on difference between two determinants like in (4.18), one can complete the proof of (2) (4.21) given the bound on the kernel Kζt . As the bounds in (4.9) and (4.20) are almost same, thus, it implies that difference of two Fredholm determiant in (4.21) also decays exponentially fast as t escapes to ∞. Now, we make a substitutions of the variables v, w and w′ in the ǫ-neighborhood of ̺. In particular, we transform v 7→ ̺ + t−1/3 ̺σν−1 ṽ, w 7→ ̺ + t−1/3 ̺σν−1 w̃, w′ 7→ ̺ + t−1/3 ̺σν−1 w̃′ . (4.22) (1) To the effect of these substitutions, transformed kernel Kζt looks like (1) Kζt ∞ Z t−1/3 ̺σν−1 X = (1) 2i Lṽ sin k=−∞ exp t(Λ(1 + t−1/3 σν−1 ṽ) − Λ(1 + t−1/3 σν−1 w̃)) + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃)) + 2πik exp sσν t1/3 log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) × log(b)(ṽ − w̃′ ) αb−1 (̺ + t−1/3 ̺σν−1 ṽ); b ∞ dṽ × × . (4.23) −1 −1 −1/3 −1/3 αb (̺ + t ̺σν w̃); b ∞ ̺ + t ̺σν−1 ṽ π log(b) (log(1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit (1) 31 (2) We must point out here that Lṽ , Lṽ have been used to denote the contours for the transformed variable ṽ. Similar change in notations for the contours of w̃ and w̃′ will also be made in the subsequent discussion. Next, we state two lemmas which essentially prove part (a) of Proposition 4.1. (1) Lemma 4.8. For all w̃, w̃′ ∈ Lw̃ , we have (1) \|Kζt (w̃, w̃′ )\| ≤ C exp −c1 \|Re(w̃3 )\| − sσν \|w\| (4.24) for some constants C, c1 , c2 > 0. Moreover, the following convergence Z ∞eiπ/3 exp w̃3 − ṽ3 + sṽ − sw̃ dṽ 3 3 1 (1) lim Kζt (w̃, w̃′ ) = t→∞ 2πi ∞e−iπ/3 (ṽ − w̃′ )(ṽ − w̃) (4.25) holds. (1) (1) Proof. For all w̃ ∈ Lw̃ and ṽ ∈ Lṽ , we have w̃3 ṽ 3 t Λ(̺ + t−13 ̺σν−1 w̃) − Λ(̺ + t−13 ̺σν−1 w̃) = − (4.26) 3 3 + t R(t−1/3 ̺σν−1 w̃) − R(t−1/3 ̺σν−1 ṽ) where t(\|R(t−1/3 ̺σν−1 w̃)\| + \|R(t−1/3 ̺σν−1 ṽ)\|) converges to 0 as t → ∞. One can note that ǫ (1) (1) in Definition 4.5 has been chosen in such a way that for ṽ ∈ Lṽ and w̃ ∈ Lw̃ , we have t(\|R(t−1/3 ̺σν−1 w̃)\| + \|R(t−1/3 ̺σν−1 ṽ)\|) ≤ c \|ṽ\|3 + \|w̃\|3 . (1) (1) for some small constant c > 0. Further, we know Re(w3 − v 3 ) < 0 for w ∈ Lw and v ∈ Lv . That implies real part of the right side in (4.26) is bounded above by −c1 \|Re(ṽ)3 \| + \|Re(w̃)3 \| for some constant c1 > 0. Among the other components of the integrand in (4.23), we have (i) ≤ exp (sσν (\|v\| − \|w\|)) exp sσν t1/3 log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) (4.27) (ii) and, as t → ∞, exp sσν t1/3 log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) → exp(s(ṽ − w̃)),(4.28) t1/3 sin and, π log(b) log(b) sin πt−1/3 ̺σν−1 σ \| log(b)\| ≤ ν 2\|ṽ − w̃\| log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) (4.29) π t→∞ −1/3 −1 −1/3 −1 log(1 + t σν ṽ) − log(1 + t σν w̃) −→ ṽ − w̃, log(b) (4.30) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 32 (iii) t −1/3 ∞ X k=−∞ k6=0 1 sin π log(b) log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) + 2πik t→∞ −→ 0, (4.31) (iv) (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)∞ → 1, (αb−1 (̺ + ̺σν−1 t−1/3 w̃); b)∞ (4.32) whenever t → ∞. Convergence in (4.28) and (4.30) are easy to see. To prove the inequality in (4.27), note that we have Re t1/3 (log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃)) ≤ σν−1 (\|v\| − \|w\|) for all large t. For showing (4.29), we used the fact that \| sin(x)\| ≥ 2x/π for all x ∈ −1 −1 [−π/2, π/2]. For large values of t, log(1 + t−/3σν ṽ ) and log(1 + t−/3σν ṽ ) will be close enough to 0. This validates the bound in (4.29). In the third case, when k 6= 0, sin(. + 2πik) grows exponentially fast as \|k\| increases. This makes the series in (4.31) convergent and hence, when multiplied with t−1/3 , it goes to 0. To see (4.32), notice that αb−1 ̺ < 1 thanks to the condition ρ > 1 − (ν(1 − b))−1/2 . Therefore, (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)m t→∞ −→ 1. (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)m uniformly over m. Moreover, using some simple inequalities like \|(1 + z)\| ≤ exp(\|z\|) for any z ∈ C and (1 + z)−1 ≤ exp(\|z\|) for z ∈ C with \|Re(z)\| < 1, it is also easy to see following crude bound (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)∞ ≤ exp(σν−1 t−1/3 (\|v\| + \|w\|)). (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)∞ Thus, the integrand in (4.23) is uniformly bounded by an integrable function C 1 exp(−c1 (\|Re(ṽ 3 )\| + \|Re(w̃)3 \|) + c2 (\|v\| − \|w\|)). 1 + \|ṽ\| This integrable envelop proves the bound on the kernel in (4.24). Further, using dominated convergence theorem, we get the convergence in (4.25) whereas (4.28)-(4.32) shows the limiting value. Proof of Proposition 4.1(a). Using Lemma 4.6 and 4.7, we have lim det(1 + Kζbt )L2 (Cr ) = lim det(1 + Kζbt )L2 (L(1) ) . t→∞ Furthermore, the kernel (1) (1) Kζt t→∞ w̃ has an integrable upper bound as shown in (4.24) for all w̃, w̃′ ∈ Lw̃ . Thus, using Corollary A.3, one can establish that the limit of the Fredholm determinant det(1 + Kζbt )L2 (L(1) ) is same as the Fredholm determinant of the limiting kernel in (4.25). This w̃ completes the proof. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 33 2 Fluctuations. 4.2. Case ρ = 1 − (ν(1 − b))−1/2 , FGOE In this scenario, the double critical point ̺ of the function Λ becomes equal to bα−1 . Thus, the deformation of the contour of w to Lw is no more viable because in Definition 4.5, Lw is described to pass through ̺ whereas the contour of w must avoid the pole at bα−1 as stated in Theorem 3.7. Furthermore, the result in (4.32) is also not going hold. In what follows, we make a mild change in the definition of contours Lw and Lv to remedy those issues. Definition 4.9. Recall all the notations from Definition 4.5. To avoid the pole at bα−1 , we (1) redefine Lw to go from ̺ − t−1/3 δ̺σν−1 + ǫe−iπ/3 to ̺ − t−1/3 δ̺σν−1 to ̺ − t−1/3 δ̺σν−1 + ǫeiπ/3 . (1) Similarly, we shift the contour Lv to the left in the complex plane by t−1/3 δ̺σν−1 . For instance, (1) in this new settings, Lv extends from ̺ − t−1/3 (1 + δ)̺σν−1 + ǫe−i2π/3 to ̺ − t−1/3 δ̺σν−1 to ̺ − t−1/3 (1 + δ)̺σν−1 + ǫei2π/3 . Here, δ can be any positive real number. Furthermore, ǫ must be chosen is such a way that following conditions, given as (1) (1) both end points ̺ − t−1/3 δ̺σν−1 + ǫe−iπ/3 and ̺ − t−1/3 δ̺σν−1 + ǫeiπ/3 of Lw lie outside the level curve L2 (see, Theorem 4.4 for definition), (1) (1) (2) Re(Λ(w)) < Re(Λ(w)) for all v ∈ Lv and Lw , 3 (3) \|R((z − ̺))\| < c\|z − ̺\| for any z in an ǫ-neighbourhood of ̺, (2) (2) are satisfied. The residual parts, namely Lw and Lv are defined exactly the same way as in Definition 4.5 except the necessary adjustments of their origins and end points. For instance, (2) Lw now starts from ̺ − t−1/3 δ̺σν−1 + ǫe−iπ/3 and ends at ̺ − t−1/3 δ̺σν−1 + ǫeiπ/3 . Thanks to the new definitions above, we can deform of the contour of v and w to Lv and Lw without crossing any singularities. Moreover, one can note that bα−1 lies outside contour Lw . To this end, one can prove the results like Lemma 4.6 and Lemma 4.7 almost in the same way in the present −1context except possibly with a different treatment for the quantity αb−1 v; b ∞ αb−1 w; b ∞ in (4.15). In what follows, we carry out such changes. Lemma 4.10. Assume ρ = 1 − (ν(1 − b))−1/2 . There exist c′ , C ′ > 0 such that whenever max{\|w − ̺\|, \|v − ̺\|} < c′ , then αb−1 v; b ∞ v−̺ < C′ . −1 (αb w; b)∞ w−̺ (4.33) (1) (1) Furthermore, consider the substitutions in (4.22). Denote the contours Lv and Lw after the (1) (1) (1) (1) substitution by Lṽ and Lw̃ respectively. Then, for any ṽ ∈ Lṽ and w̃ ∈ Lw̃ , we have αb−1 (̺ + t−1/3 ̺σν−1 ṽ); b ∞ ṽ = . lim −1 t→∞ αb−1 (̺ + t−1/3 ̺σν w̃); b w̃ ∞ (4.34) Proof. To prove (4.33), let us write αb−1 v; b ∞ =A·B·C (αb−1 w; b)∞ imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit where v−̺ , A := w−̺ Qm j−1 v) j=1 (1 − αb B := Qm , j−1 w) j=1 (1 − αb C := 34 (αbm v; b)∞ . (αbm w; b)∞ Further, m is chosen in such a way that the inequality bm α (̺ + c′ ) < 1/2 is satisfied. To that effect, we have      ∞ ∞ m+j−1 j−1 X X 1 + αb v  b (v − w)  \|C\| = exp  log = exp αbm Re  m+j−1 1 + αb v 1 + ψ(v, w, j) j=1 j=1    ∞ j−1 X b \|v − w\|  2αbm \|v − w\| m  ≤ exp αb . (4.35) ≤ exp 1−b 1 − 21 j=1 In the second equality above, we have used the mean value theorem for the complex variables. Thus, ψ(v, w, j) denotes a complex number in the line joining αbm+j v and αbm+j w. Consequently, one have v−w \|v − w\| \|v − w\| Re ≤ . ≤ 1 + ψ(v, w, j) 1 − \|ψ(v, w, j)\| 1 − bm α (̺ + c′ ) This shows the inequalities in (4.35). Lastly, B involves product of finitely many terms which are bounded above. Hence, this completes the proof of (4.33). Further, one can write α(̺ + t−1/3 b̺σν−1 ṽ); b ∞ αb−1 (̺ + t−1/3 ̺σν−1 ṽ); b ∞ (−t−1/3 ̺σν−1 ṽ) = . (4.36) × (−t−1/3 ̺σν−1 w̃) αb−1 (̺ + t−1/3 ̺σν−1 w̃); b ∞ α(̺ + t−1/3 b̺σν−1 w̃); b ∞ Note that bn α̺ is bounded away from 1 for all non-negative powers of b. Thus, the second term in the product of the right side in (4.36) converges to 1 as t → ∞. This shows the limit in (4.34). Now, we note down two results in the same spirit of Lemma 4.6 and Lemma 4.7. Proofs are exactly same except one has to use the bound in (4.33) instead of (4.15). For brevity, we omit the proofs. Lemma 4.11. There exists some c, C > 0 and t0 > 0, such that for all t ≥ t0 \|Kζbt (w, w′ )\| ≤ C 1 max{ǫ−1 , ̺−1 σν t1/3 } exp −c(t + \|w\|) + sσν t1/3 \|w\| ∨ \|w\|−1 \|w − ̺\| log(1 + ǫ) (2) where w ∈ Lw , w′ ∈ Lw and ǫ is defined in Definition 4.5. Moreover, the following convergence lim det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L(1) = 0 t→∞ w also holds in the present scenario. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 35 Lemma 4.12. There exist c, C > 0 such that (2) \|Kζt (w, w′ )\| ≤ ǫ−1 1 exp −c(t + \|w\|) + sσν t1/3 \|w\| ∨ \|w\|−1 \|w − ̺\| log(1 + ǫ) (1) holds for all w, w′ ∈ Lw . Thus, one would get (1) lim det(1 + Kζbt )L2 (L(1) ) − det(1 + Kζt )L2 (L(1) ) = 0. t→∞ w w Lemma 4.13. Recall all the notations from the previous section and Definition 4.9. For all (1) w̃, w̃′ ∈ Lw̃ , we have 1 (1) \|K̃ζt (w̃, w̃′ )\| ≤ C exp −c1 \|Re(w̃3 )\| − sσν \|w\| δ (4.37) for some positive constants c1 , c2 , C and (1) lim K̃ζt (w̃, w̃′ ) = t→∞ 1 2πi Z −(1+δ)+ei2π/3 exp w̃ 3 3 − ṽ3 3 + s(ṽ − w̃) (ṽ − w̃)(ṽ − w̃′ ) −(1+δ)+e−i2π/3 × ṽdṽ . w̃ (4.38) Proof. To get the bound in (4.37), note that integrand in (4.23) is bounded above by 1 exp(−c1 (\|Re(ṽ 3 )\| + \|Re(w̃)3 \|) + c2 (\|v\| − \|w\|)) C (1 + \|ṽ\|) αb−1 (̺ + t−1/3 ̺σν−1 ṽ); b ∞ . αb−1 (̺ + t−1/3 ̺σν−1 w̃); b ∞ We have illustrated this fact in Lemma 4.8. Thus, we omit further details on it. Let us recall that Lw is shifted to the left by t−1/3 δ̺σν−1 from ̺. To this effect, \|w̃\| & δ and henceforth, the ratio of the infinite products in (4.34) gets bounded above by C ′ \|ṽ\|/δ thanks to (4.33). This accounts for the term 1/δ in the bound on the right side in (4.37). Proof of (4.38) follows from the similar arguments like in Lemma 4.8 except the limit in (4.32) should be replaced by the result (4.34) of Lemma 4.10. To summarize, all these results above prove lim det(1 + Kζbt )L2 (Cr ) = det(1 + Kcrit )L2 (L̄w̃ ) t→∞ where Kcrit (w̃, w̃′ ) := 1 2πi Z −(1+δ)+ei2π/3 −(1+δ)+e−i2π/3 exp w̃ 3 3 − ṽ3 3 + s(ṽ − w̃) (ṽ − w̃)(ṽ − w̃′ ) × ṽdṽ w̃ and L̄w̃ denotes a piecewise linear contour which extends linearly from −δ + ∞e−iπ/3 to −δ and from there again linearly to −δ + ∞eiπ/3 . Hence, the proof of Proposition 4.1 (b) follows. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 36 4.3. Case ρ < 1 − (ν(1 − b))−1/2 , Gaussian Fluctuation In this section, we prove part (c) of Proposition 4.1. Main reason behind a separate analysis is that whenever ρ < 1 − (ν(1 − b))−1/2 , contours like Lv and Lw in Definition 4.5 or even their modifications in Definition 4.9 contain bα−1 inside. To circumvent this illegal inclusion, we adjust the choice of mν in the function Λ (see (4.5)) in such a way that bα−1 turns out to be a new critical point of Λ. To begin with, the form of the kernel Kζbt is given by Kζbt (w, w′ ) ∞ Z X 1 = 2i log(b) C R k=−∞ b exp(t(Λ(w) − Λ(v)) + t1/2 sσ̃ν (log(v) − log(w)))dv π (v − w′ ) sin log(b) (log(v/w)) + 2πik \|w\| × dv (αb−1 v; b)∞ × . −1 (αb w; b)∞ v (4.39) 1/2 and v respectively. Under the these where ζt and bh w are substituted with b−m̃ν t+sσ̃ν t recalibration, the explicit form of the function Λ is given by Λ(z) = z − ν log(1 + zb−1 ) + m̃ν log(z). b(1 − b) (4.40) In what follows, we perform some calculations de rigueur the asymptotic analysis. Plugging 1 1 ν − α(1−b) in (4.40), one can write m̃ν = 1+α Λ′ (z) = (z − bα−1 )(z − ̺′ ) . zb(z + b)(1 − b) where ̺′ := b (ν(1 − ρ)(1 − b) − 1). Under the condition ρ < 1− (ν(1− b))−1/2 , it can be noted that ̺′ > ̺. To this end, we have Λ′′ (bα−1 ) = − (1 − ρ)2 ν(1 − b) − 1 . b2 ρ(1 − b) Further, using Taylor’s expansion, one would get −1 Λ(z) = Λ(bα 1 )− 2 σ̃ν bα−1 2 (z − bα−1 )2 + R̃(z − bα−1 ) (4.41) where \|z − bα−1 \|2 R̃(z − bα−1 ) converges to 0 whenever z → bα−1 . In the following lemma, we illustrate on the properties of the level curves Re(Λ(z)) = Re(Λ(bα−1 )). Lemma 4.14. There exists two simple closed contours L1 and L2 such that for all z ∈ L1 ∪L2 , we have Re(Λ(z)) = Re(Λ(bα−1 )). They further satisfy the following properties. (a) Both the contours originate at the point bα−1 and encircle around 0 to meet again at their origin. For instance, L1 (L2 ) in its way back to bα−1 crosses the real line for the second time at a point d1 ∈ (−b, 0) (d2 ∈ (−∞, −b)). (b) Contour L1 \{bα−1 } is completely contained in the interior of the region bounded by L2 . (c) Contour L1 (L2 ) leaves x axis from the point bα−1 at an angle 3π/4 (π/4) and meet again at angle −3π/4 (−π/4). imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 37 Apart from L1 and L2 , there exists another level curve L3 of the equation Re(Λ(z)) = Re(Λ(bα−1 )) lying completely in the right half of the complex plane. Furthermore, L3 doesn’t intersect L1 or L2 and escapes to infinity whenever Re(z) → ∞. Proof. Proof of the existence of L1 , L2 and L3 are similar to that in Lemma 4.4. Moreover, property (a) and (b) can also be proved in the same way. To see part (c), note that −1 −1 2 −1 )2 in a neighborhood around bα−1 . Λ(z) − Λ(bα−1 ) behaves as −2 (σ̃ν αb ) ν(z − bα −1 −1 Thus, Re Λ(z) − Λ(bα ) = 0 implies Arg(z − bα ) = ± π4 ± 2π 2 . This indicates part (c). We are left to show that L3 doesn’t intersect L1 or L2 . To start with, note that L3 doesn’t pass through bα−1 . This is because there are only four possible choices of the angles (± π4 ± π2 ) −1 that any level curve of the equation Re Λ(z) − Λ(bα ) = 0 can make at bα−1 . Thus, there can be at most two level curves which pass through the point bα−1 . In fact, L3 crosses the real line at a point further right of bα−1 . To see this, notice that Re Λ(z) − Λ(bα−1 ) < 0 if Im(z) = 0 and bα−1 < Re(z) < bα−1 + δ for some δ > 0. Additionally, Re(Λ(z)) increases to infinity with increasing Re(z). Thus, there must exists another point on the real line to the right of bα−1 where once again Re Λ(z) − Λ(bα−1 ) = 0 holds. This shows the claim. Moreover, one can mimick almost the same proof as in Lemma 4.4 to show that L3 doesn’t intersect L1 or L2 at any other point. This completes the proof. We must point out here that L2 is no more star shaped (see Figure 6). This urges for the necessary changes in the definitions of the contours Lv and Lw in the present scenario. In what follows, we enunciate those changes formally. (1) Definition 4.15. We define the linear segment Lv of Lv to extend linearly from −ǫi + Intbα−1 ,2δ to ǫi + Intbα−1 ,2δ crossing the real line at Intbα−1 ,2δ where Intx,τ = x − t−1/3 τ xδ(σ̃ν )−1 for any x ∈ C, τ ∈ R+ . (1) On the flip side, we define the piecewise linear segment Lw to extend linearly from ǫe−iπ/6 + Intbα−1 ,δ to Intbα−1 ,δ and from there to ǫeiπ/6 + Intbα−1 ,δ . At this junction, it is worthwhile to point out that any particular choice of ǫ must satisfies the following properties: (a) ǫe−iπ/6 + Intbα−1 ,δ and ǫeiπ/6 + Intbα−1 ,δ should lie outside the contour L2 , (b) in the ǫ - neighborhood of the point bα−1 , \|R(z − bα−1 )\| in (4.41) must be less than c\|z − bα−1 \|2 for small constant (2) (2) c. In particular, c = 1/4 suitably fits into our analysis. Curved segments Lv and Lw are (2) (1) defined similarly as in Definition 4.5. For instance, Lv starts from the top end point of Lv (1) and encircles around 0 to meet at the bottom end point of Lv . To be precise, it remains (1) (1) (2) always inside of the contour L2 . Unlike Lv , starting from the top end point Lw , Lw circles (1) outside the contour L2 to meet the bottom end point of Lw . In the next lemma, we show that the contribution of the kernel Kζbt in the Fredholm (2) determinant det(1 + Kζbt )L2 (Lw ) over the segment Lw decays down exponentially. (2) Lemma 4.16. For any w ∈ Lw and w′ ∈ Lw , we have \|Kζbt (w, w′ )\| ≤ C max{ǫ−1 , δ−1 t1/2 } exp −tc + sσ̃ν t1/2 \|w\| ∨ \|w\|−1 + α \|w\| (4.42) 1−b imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit − L2 − + α−1 b L1 L3 Lw Lv + −b 38 ̺ − L3 Fig 6: Solid lines are used to denote the contour Lw and Lv whereas dotted lines indicate the level curves of the equation Re(Λ(z) − Λ(bα−1 )) = 0. for some constants c, C > 0. Here, ǫ is same as in Definition 4.15. To that effect, we have lim det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) = 0. t→∞ w (4.43) (2) Proof. To begin with, note that for all w ∈ Lw and all v ∈ Lv there exists a constant c > 0 (2) such that Re (Λ(w) − Λ(v)) < −c. This holds due to the particular way the contours Lw , Lv are chosen and the compactness of each of them. Other components of the integrand in (4.39) are bounded in the following ways (i) exp sσ̃ν t1/2 (log(v) − log(w)) ≤ exp sσ̃ν t1/2 (\|v\| ∨ \|v\|−1 + \|w\| ∨ \|w\|−1 ) ,(4.44) (ii) ∞ X k=−∞ 1 sin π log(b) log(v/w) + 2πik ≤ ∞ X 2 exp(−2πk) exp k=0 π −1 −1 (\|v\| ∨ \|v\| + \|w\| ∨ \|w\| ) , log(b) (4.45) (iii) 1 ≤C v(v − w) minv∈L 1 (2) ′ v ,w ∈Lw \|v − w′ \| ≤ C max{ǫ−1 ∨ δ−1 t1/2 } (4.46) for some large constant C, imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 39 (iv) ∞ m ∞ Y Y Y (αb−1 v; b)∞ k−1 m−1 −1 ≤ (1 + b α\|v\|) \|(1 − b α\|w\|)\| (1 + 2bk−1 α\|w\|) αb−1 w; b)∞ k=1 k=0 k=m+1 α ≤ C exp (\|v\| + 2\|w\|) (4.47) 1−b where for all k ≥ m, one has (1 + 2bk−1 α\|w\|)(1 − bk−1 α\|w\|) ≤ 1. Inequality in (4.44) is a simple consequence of the fact that \| exp(log(v) − log(w))\| ≤ exp Re(v) ∨ Re(v −1 ) + Re(w) ∨ Re(w−1 ) ≤ exp \|v\| ∨ \|v\|−1 + \|w\| ∨ \|w\|−1 for any two complex numbers v, w. To see the bound in the right side of (4.45), note that sin(u + 2πik) is bounded above by 2 exp(\|u\| − 2πk) for all u ∈ C and k ∈ Z. It is easy to comprehend the first inequality in (4.46). For the second inequality, recall that both the end (1) points of Lw are at a distance ǫ away from the point bα−1 for large enough t. Thus, it is (2) possible to construct the contour Lw at a distance of ǫ far away from L2 . As we know Lv is contained in the region enclosed by L2 , thus the minimum value of \|v − w′ \| must be greater (2) (1) than ǫ for all v ∈ Lv and w′ ∈ Lw . On the flip side, if w′ ∈ Lw , then minimum value of \|v − w\| should be greater than δt−1/2 , thus shows the claim in (4.46). First inequality in straightforward. To prove the second inequality, note that the finite product (4.47) Qm is quite m−1 α\|w\|)\|−1 is bounded by a large constant. To bound the other components \|(1 − b k=0 apart from the finite product term, we use a simple inequality (1 + x) ≤ ex which holds for any x ∈ R. These all together imply the inequality in (4.42). Next, we prove the limit in (4.43). Let us write down b (2) ′ Kζbt (w, w′ ) = Kζbt (w, w′ )1(w ∈ L(1) w ) + Kζt (w, w )1(w ∈ Lw ). . Then, using the inequality in (4.18) and the bound Denote Kζbt (w, w′ )1(w ∈ Lw ) by Kζb,res t (1) on Kζbt (w, w′ )1(w ∈ Lw ) from (4.42), we have (2) n det Kζbt (wi , wj ) i,j=1 n (w , w ) − det Kζb,res i j t i,j=1 ≤ C2n max{ǫ−n , (δ−2 t)n/2 } exp(−ct). Further, using series expansion of Fredholm determinant, we get the following bound det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) = det(1 + Kζbt )L2 (Lw ) − det(1 + Kζb,res )L2 (Lw ) t w ≤ C(1 + exp(−2ǫ−1 )) exp(−ct + 2δ−1 t1/2 ) (4.48) where the right side of the inequality above goes to 0 as t → ∞. This completes the proof. Likewise in (4.23), we further represent Kζbt as the sum of two different kernels to identify the leading contribution in the computation of the Fredholm determinant det(1 + Kζbt )L2 (L(1) ) . w In particular, we set (1) (2) Kζbt = Kζt + Kζt imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 40 where for l = 1, 2, (l) Kζt ∞ Z 1 X := (l) 2i Lv sin k=−∞ exp (t(Λ(v) − Λ(w))) π log(b) (log(v) − log(w)) + 2πik (4.49) αb−1 v; b ∞ exp (s (log(v) − log(w))) dv × × . × ′ −1 log(b)(v − w ) (αb w; b)∞ v Following lemma which is in the same spirit of the last Lemma 4.48 proves that one can even (2) ignore the effect of Kζt as t increases to infinity. (1) Lemma 4.17. For any w, w′ ∈ Lw , we have b ′ −1 1/2 \|Kζt (w, w )\| ≤ C(ǫδ) t exp −tc + sσ̃ν t1/2 \|w\| ∨ \|w\|−1 + α \|w\| 1−b (4.50) for some constants c, C > 0. Thus, we claim (1) lim det(1 + Kζbt )L2 (L(1) ) − det(1 + Kζt )L2 (L(1) ) = 0. t→∞ (4.51) w w (2) Proof. Proof of (4.50) hinges upon the fact that Re(Λ(w) − Λ(v)) < −c for all v ∈ Lv and all (1) w ∈ Lw . This again follows from the compactness of both of the contours, continuity of the map Re(Λ(.)) and more importantly, strategic position of the contours around L2 . Bounds on the other component of the integrand in (4.49) can be derived as in (4.44) - (4.47). For instance, bounds in (4.44) and (4.45) work perfectly for the current result. In what follows, we recollect bounds on other two terms and sharpen it for the present scenario. (i) 1 1 ≤C ≤ Cǫ−1 v(v − w) minv∈L(2) ,w′ ∈L(1) \|v − w′ \| v (4.52) w for some large constant C (ii) ∞ m ∞ Y Y Y (αb−1 v; b)∞ k−1 m−1 −1 ≤ (1 + b α\|v\|) \|(1 − b α\|w\|)\| (1 + 2bk−1 α\|w\|) αb−1 w; b)∞ k=1 k=0 k=m+1 α ≤ Cδ−1 t1/2 exp (\|v\| + 2\|w\|) (4.53) 1−b where for all k ≥ m, one has (1 + 2bk−1 α\|w\|)(1 − bk−1 α\|w\|) ≤ 1. One can prove the inequality in (4.52) by recalling that the minimum value of \|v − w′ \| over (2) (1) all v ∈ Lv , w′ ∈ Lw is bounded below by ǫ (see, Definition 4.15). In (4.53), we cannot now bound the finite product term by a large constant uniformly over all choices of t and w. This is because min \|1 − αb−1 w\| = δt−1/2 . (1) w∈Lw imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 41 This causes the factor δ−1 t1/2 to come in the right side of the second inequality. To show (4.51), we use the inequality in (4.18) to conclude n n (1) ≤ C2n (δǫ)−n tn/2 exp(−ct). − det Kζt (wi , wj ) det Kζbt (wi , wj ) i,j=1 i,j=1 (1) whenever w1 , . . . , wn ∈ Lw . To that effect, Fredholm determinant expansion provides the sim(1) ilar bound as in (4.48) on the difference between det(1 + Kζbt )L2 (L(1) ) and det(1 + Kζt )L2 (L(1) ) . w w This ends the proof. Above two lemmas show that in order to find out the limit of the Fredholm determinant (1) (1) in (4.2), it suffices to study Kζt over the segment Lw . Thus, like in (4.22), we make the following substitutions v 7→ bα−1 + t−1/2 b(ασ̃ν )−1 ṽ, w 7→ bα−1 + t−1/2 b(ασ̃ν )−1 w̃, w′ 7→ bα−1 + t−1/2 b(ασ̃ν )−1 w̃′ . (4.54) (1) To this end, one can further write Kζt in the following form - (1) Kζt = t−1/2 b(ασ̃ ν )−1 2i Z 2 2 exp ṽ2 − w̃2 + t R̃(t−1/2 b(ασ̃ν )−1 w̃) − R̃(t−1/2 b(ασ̃ν )−1 w̃) (1) π −1/2 (σ̃ )−1 ṽ) − log(1 + t−1/2 (σ̃ )′ w̃)) + 2πik Lṽ k=−∞ sin (log(1 + t ν ν log(b) ∞ X × × (1) exp sσ̃ν t1/2 (4.55) log(1 + t−1/2 (σ̃ν )−1 ṽ) − log(1 + t−1/2 (σ̃ν )−1 w̃) log(b)(ṽ − w̃′ ) αb−1 (bα−1 + t−1/2 b(ασ̃ν )−1 ṽ); b αb−1 (bα−1 (1) + t−1/2 b(ασ̃ (1) ν )−1 w̃); b ∞ × ∞ dṽ bα−1 + t−1/2 b(ασ̃ −1 ν ) ṽ (1) where Lṽ and Lw̃ denote the segments Lv and Lw respectively after the substitutions made in (4.54). In the next result, we show that in fact the Fredholm determinant of the (1) kernel Kζ1 converges to the normal distribution. Theorem 4.18. For all s ∈ R, we have (1) lim det 1 + Kζt t→∞ (1) L2 (Lw ) = G(s). Proof. To begin with, we enlist below how each of the components of the integrand in (4.55) is bounded above by some integrable function and what are their limits as t → ∞. (i) We first look at the inequalities shown below 2 w̃2 ṽ −1/2 −1 −1/2 −1 − + t R̃(t b(ασ̃ν ) w̃) − R̃(t b(ασ̃ν ) w̃) (4.56) exp 2 2 2 w̃2 ṽ 2 2 − + c\|ṽ\| + c\|w̃\| ≤ C ′ exp −c′1 \|ṽ\|2 + c′2 Re(w̃2 ) ≤ exp Re 2 2 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 42 for some constants C ′ , c′1 > 0 and c′2 . To see the first inequality, recall that \|R̃(z − bα−1 )\| is less than c\|z −bα−1 \|2 in the ǫ-neighborhood of bα−1 (see, Definition 4.15). For showing the second inequality in (4.56), one can argue that Re(ṽ 2 ) + c\|ṽ\|2 ≤ −c′1 \|ṽ\|2 for some c′1 > 0. This is because the complex variable ṽ varies over −2δ + iR. Thus, Re(ṽ 2 ) converges to −∞ as \|ṽ 2 \| → ∞. In fact, we get Re(ṽ 2 )/\|ṽ\|2 → −1 whenever \|ṽ\| goes to ∞. This completes the proof of the inequalities above. Furthermore, as t → ∞, both tR̃(t−1/2 b(ασ̃ν )−1 ṽ) and R̃(t−1/2 b(ασ̃ν )−1 w̃) goes to 0. Thus, we get 2 2 ṽ w̃2 w̃2 ṽ exp − + t R̃(t−1/2 b(ασ̃ν )−1 w̃) − R̃(t−1/2 b(ασ̃ν )−1 w̃) − . → exp 2 2 2 2 (ii) Next, we illustrate on the limit of the infinite sum over k ∈ Z. Note that for any k ∈ Z, we have π −1/2 −1 −1/2 ′ sin (log(1 + t (σ̃ν ) ṽ) − log(1 + t (σ̃ν ) w̃)) + 2πik log(b) ≥ C(1 + t−1/2 \|ṽ\|) exp(2π\|k\|). Thus, it shows the series in (4.55) converges absolutely and uniformly over ṽ. Hence, as t goes to infinity, lim t→∞ ∞ X k=−∞ t−1/2 π(log(b))−1 sin π log(b) (log(1 + t−1/2 (σ̃ν )−1 ṽ) − log(1 + t−1/2 (σ̃ν )′ w̃)) + 2πik = 1 . ṽ − w̃ (iii) Likewise in Lemma 4.10, one can prove that for all large t, there exists constants C ′ such that αb−1 (bα−1 + t−1/2 b(ασ̃ν )−1 ṽ); b ∞ ṽ . ≤ C′ −1 −1 −1/2 −1 w̃ αb (bα + t b(ασ̃ν ) w̃); b ∞ Moreover, we have αb−1 (bα−1 + t−1/2 b(ασ̃ν )−1 ṽ); b αb−1 (bα−1 + t−1/2 b(ασ̃ν )−1 w̃); b ∞ t→∞ −→ ∞ ṽ . w̃ . (iv) Lastly, we can put a very crude bound like exp sσ̃ν t1/2 log(1 + t−1/2 (σ̃ν )−1 ṽ) − log(1 + t−1/2 (σ̃ν )−1 w̃) ≤ C ′′ exp(c(\|ṽ\| + \|w̃\|)) bα−1 bα−1 + t−1/2 b(ασ̃ν )−1 ṽ 1 . 1 + \|ṽ\| for some positive constant C ′′ , c′′ . Notice that even this crude bound doesn’t affect integrability of the product of all bounds together because of the dominating term exp(−c′ \|ṽ\|2 ) in (4.56). Further, we have the following limit bα−1 exp sσ̃ν t1/2 log(1 + t−1/2 (σ̃ν )−1 ṽ) − log(1 + t−1/2 (σ̃ν )−1 w̃) bα−1 + t−1/2 b(ασ̃ν )−1 ṽ t→∞ −→ exp (s(ṽ − w̃)) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 43 for the remaining component of the integrand in (4.55). To this end, using dominated convergence theorem, one can write 2 ṽ2 w̃ Z − + s( w̃ − ṽ) exp 2 2 ṽ 1 (1) · · dṽ lim K (w̃, w̃′ ) = t→∞ ζt 2πi −2δ+iR (ṽ − w̃′ )(ṽ − w̃) w̃ (4.57) (1) for all w̃, w̃′ in the limiting form of the contour Lw̃ . In Appendix A, we prove that Fredholm determinant of the limiting kernel is indeed Gaussian distribution function. Proof of Theorem 2.5. To show (2.2), note that {xmν t ≤ νt + ∂mν sLt1/3 } = {Nνs (t) ≥ mν t}. ∂ν ν −2/3 ) and L, s ∈ R. It is easy to see that νs equals to ν when where νs := ν(1 + sLσν ∂m ∂ν t s = 0. Moreover, using Taylor’s theorem on mνs , one can obtain mνs = mν + sLσν Let us choose L = ∂mν −2/3 Lt + o(t−2/3 ). ∂ν ∂mν −1 . ∂ν To that effect, we have P (Nνs (t) ≥ mν t) = P mνs t − Nνs t ≤ sσν t1/3 + o(t1/3 ) (4.58) Theorem 2.2 implies that the right side in (4.58) converges to FGU E (s). Thus, it completes the proof of (2.5). Similarly, the proofs of (2.6) and (2.7) follow from (2.3) and (2.4) respectively. Proof of Corollary 2.6 Theorem 2.4 implies (t−1 Nνt (t) − mν ) = Op (t−2/3 ) when ν > ν0 . Thus, it proves the first case. Further, note that for any ǫ > 0, there exists δǫ > 0 such that whenever ν ≤ ν0 + δǫ , limit shape mν < ǫ. To that effect, for any ν ≤ ν0 , we have lim sup t→∞ N(ν0 +δǫ )t (t) Nνt (t) ≤ lim ≤ǫ t→∞ t t with high probability. Henceforth, this shows the second claim. 5. Weak Scaling Limit Our goal here is to prove the Theorem 2.9. Following two propositions provide the necessary arguments needed to complete the proof. Proposition 5.1. The collection of processes {Zǫ }ǫ is tight in C(R+ × R). Proposition 5.2. Any limiting point Z of {Zǫ }ǫ solves the SHE. To prove Proposition 5.1-5.2, we first show that the Gärtner transform (recall Definition 2.7) of HL-PushTASEP satisfies a discrete SHE. This step is in harmony with the other works like [BC95, Section 3.2], [DT16, Section 2], [CT15, Section 3]. Let us note that here onwards, constants will change from line to line. Unless specified, they do not depend on anything else. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 44 5.1. Discrete SHE First, we derive a dynamical equation for Z(t, x). Recall that any particle which is at left side of any site x when gets excited can move across x with certain probability. For instance, once the clock of the particle at site y(< x) rings at time t, it jumps across x with probability bx−y−Nx (t)+Ny (t) . For notational convenience, let us introduce the following notation κ(x, t) = x + t . 1 − bν ǫ (5.1) From the definition of Z(t, x) in (2.9), it is clear that contributions of the all the right hops across x in dZ(t, x) during the interval [t, t + dt] is given by (b−1 − 1)Z(t, x)dPtx where {Ptx }−∞<x<∞ denotes a sequence of Poisson point processes with rates κ(x,t) X ηt (l)bκ(x,t)−l−Nκ(x,t) (t)+Nl (t) l=1 and ηt (l) is the occupation variable for the position l (see, Definition 2.1). There is also a contribution coming from the exponential growth term exp(−µǫ t). Collecting all the terms together, we can write down the following   κ(x,t) X (1 − νǫ ) log b − µǫ  Z(t, x)dt dZ(t, x) = (b−1 − 1) ηt (l)bκ(x,t)−l−Nκ(x,t) (t)+Nl (t) − (1 − bνǫ ) l=1   κ(x,t) X + (b−1 − 1)Z(t, x) dPtx − ηt (l)bκ(x,t)−l−Nκ(x,t) (t)+Nl (t) dt . l=1 For the sake of simplicity, denote dM (t, x) = (b −1 − 1) dPtx − X ηt (l)b κ(x,t)−l−Nκ(x,t) (t)+Nl (t) l=1 ! dt . (5.2) Further, let us define Ωx := (b−1 − 1) X ηt (l)bκ(x,t)−l−Nκ(x,t) (t)+Nl (t) . (5.3) l=1 Lemma 5.3. We have Ωx Z(t, x) = (b−1 − 1) X l<x bνǫ (x−l) Z(t, l) − Z(t, x). (5.4) Proof. Using the form of Ωx given in (5.3), we get X Ωx Z(t, x) = (b−1 − 1) ηt (l)bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l exp(−µǫ t). l≤κ(x,t) We will show X X bνǫ (κ(x,t)−l) bNl (t)−l/2 .(5.5) bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l + exp(µǫ t)Ωx Z(x, t) = b−1 l≤κ(x,t) l<κ(x,t) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 45 One can easily P see that above result implies (5.4). For proving (5.5), we have Ωx Z(t, x) = √ √ λǫ ǫ(1 + C ǫ) l≤κ(x,t) ηt (l)bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l by expanding b−1 − 1 in terms of ǫ. Note P λn+1 ǫn ǫ that the value of C is ∞ n=1 (n+1)! . To this effect, we have X bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l + Ωx Z(x, t) exp(µb t) l≤κ(x,t) = X (1 + l≤κ(x,t) = X l≤κ(x,t) = b−1 √ ǫλǫ ηt (l))bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l + Cλǫ ǫ √ Ωx Z(t, x) 1+C ǫ √ [exp(λǫ ǫηt (l)) − Cλǫ ǫηt (l)]bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l Cλǫ ǫ √ Ωx Z(t, x) + 1+C ǫ X bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l . l<κ(x,t) √ √ We wrote down the equality in the third line by identifying (1+ηt (l)λǫ ǫ) as exp(ηt (l)λǫ ǫ)− Cλǫ ǫηt (l). In the last line, we used the fact Nx (t) − ηt (x) = Nx−1 (t) for any x ∈ Z≥0 . This completes the proof. Using the definition of µǫ in (2.10), one can further write down ! ∞ X 1−b dZ(t, x) = (1 − bνǫ )bνǫ k Z(t, x − k) − Z(t, x) dt + Z(t, x)dM (t, x). b(1 − bνǫ ) k=0 Let {Ri }i≥1 be iid random variables such that 1 , supp(Ri ) = Z≥0 − 1 − bν ǫ and P Ri = k − 1 1 − bν ǫ = (1 − bνǫ )bνǫ k for k ∈ Z≥0(5.6) . Note that Ri satisfies E(Ri ) = 0. Under weak noise scaling, one can note that for small ǫ, √ √ −1 Ri + (νǫ λǫ ǫ)−1 follows Geo(λǫ νǫ ǫ). Let us denote a Poisson point process of rate b1−b−1 νǫ associated with Ri ’s by Γt . Its increments over the interval [t1 , t2 ] will be denoted as Γt1 ,t2 . Further, denote Ξ(t1 , t2 ) := Z≥0 − φǫ for t1 ≤ t2 where φǫ := (b−1 − 1)(t2 − t1 ) . (1 − bνǫ )2 (5.7) In this context, define the transition probability   Γt1 ,t2 X Γt1 ,t2 Ri + pǫ (t1 , t2 , ξ) := P  − φǫ = ξ  1 − bν ǫ i=1 for ξ ∈ Ξ(t1 , t2 ). Note that support space of the transition probability pǫ (t1 , t2 , .) is Ξ(t1 , t2 ). Moreover, one can see that {pǫ (t1 , t2 , ξ)}t1 ≤tP 2 ,ξ∈t1 ,t2 form a semi-group. We use the notation [pǫ (t1 , t2 ) ∗ f (t1 )] to denote the convolution ζ∈Ξ(t1 ,t2 ) pǫ (t1 , t2 , ξ − ζ)f (t1 , ζ) of any function imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 46 f with the semi-group. One can further note that pǫ (t, .) satisfy the following differential equation, ! ∞ X (1 − b) d νǫ νǫ k pǫ (t, x) = (1 − b )b pǫ (t, x − k) − pǫ (t, x) (5.8) dt b(1 − bνǫ ) k=0 pǫ (0, x) = δ0 (x). To put this into words, pǫ behaves as a discrete heat kernel. To that effect, we get the following simple form of a discrete SHE Z t Z(t, x) = [pǫ (t1 , t) ∗ Z(t1 )](x) + [pǫ (s, t) ∗ Z(s)dM (s)](x) where 0 ≤ t1 < t. (5.9) t1 Further, utilizing the particle dynamics of HL-PushTASEP, we derive the time derivative of the quadratic variation of the martingale M (t, x) defined in (5.2). Lemma 5.4. We have (b−1 −1)−2 Z(t, x1 )Z(t, x2 )dhM (t, x1 ), M (t, x2 )i (5.10) bνǫ \|x1 −x2 \| (1 − bνǫ )b−1 = Z(t, x1 ∧ x2 )dt Z(t, x1 ∧ x2 ) [pǫ (t, t + dt) ∗ Z(t)](x1 ∧ x2 ) − 1 − bν ǫ b−1 − 1 Proof. We know that dhM (t, x1 ), M (t, x2 )i equals to the rate at which both the Poisson process associated with M (t, x1 ) and M (t, x2 ) acknowledges a jump in the interval [t, t + dt]. Thus, it must be equals to the rate at which any particle to the left of κ(x1 ∧ x2 , t) hops across κ(x1 , t) and κ(x2 , t) at time t. Here, we denote min{x1 , x2 } and max{x1 , x2 } by x1 ∧ x2 and x1 ∨ x2 respectively. To this effect, one can write X ηt (l)bκ(x1 ∨x2 ,t)−l−Nκ(x1 ∨x2 ,t) (t)+Nl (t) dt. dhM (t, x1 ), M (t, x2 )i = (b−1 − 1)2 l≤κ(x1 ∧x2 ,t) Notice that for l ≤ κ(x1 ∧ x2 , t), we have bκ(x1 ∨x2 ,t)−l−Nκ(x1 ∨x2 ,t) (t)+Nl (t) = Z(t, x1 ∧x2 )Z −1 (x1 ∨x2 )bνǫ \|x1 −x2 \| bκ(x1 ∧x2 ,t)−l−Nκ(x1 ∧x2 ,t) (t)+Nl (t) . Therefore, we can say (b−1 − 1)−2 Z(t, x1 )Z(t, x2 )dhM (t, x1 ), M (t, x2 )i X ηt (l)bκ(x1 ,x2 ,l)l−Nκ(x1 ∧x2 ,t) (t)+Nl (t) dt. = Z 2 (t, x1 ∧ x2 )bνǫ \|x1 −x2 \| l≤κ(x1 ∧x2 ,t) One can recognize the following X ηt (l)bκ(x1 ,x2 ,l)l−Nκ(x1 ∧x2 ,t) (t)+Nl (t) = (b−1 − 1)−1 Ωx1 ∧x2 Z(t, x). l≤κ(x1 ∧x2 ,t) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 47 Thus, using Lemma 5.3, we can further say that (b−1 −1)−2 Z(t, x1 )Z(t, x2 )dhM (t, x1 ), M (t, x2 )i equals to ! ∞ νǫ X 1 − b Z(t, x1 ∧ x2 ) dt. (1 − bνǫ )bνǫ k Z(t, x1 ∧ x2 − k) − −1 (1−bνǫ )−1 bνǫ \|x1 −x2 \| Z(t, x1 ∧x2 ) b −1 k=1 Note that one can simplifies the term inside the bracket in the above expression to (1 − bνǫ )b−1 [pǫ (t, t + dt) ∗ Z(t)](x1 ∧ x2 ) − Z(t, x1 ∧ x2 )dt . b−1 − 1 This shows the claim. 5.2. Heat Kernel Estimates & Tightness Now, we aim at proving the tightness of the sequence {Zǫ }ǫ . For that, we need to specialize on certain properties the discrete heat kernel pǫ (see, (5.8) for definition). In the following proposition, we provides the necessary estimates for the heat kernel. Proposition 5.5. Given any T > 0, u ∈ R and v ∈ (0, 1], there exists C depending on T and u such that (i) X ζ∈Ξ(t1 ,t2 ) pǫ (t1 , t2 , ζ) exp(uǫ\|ζ\|) ≤ C, (5.11) (ii) X ζ∈Ξ(t1 ,t2 ) p(t1 , t2 , ζ)\|ζ\|v exp(uǫ\|ζ\|) ≤ Cǫ−v/2 (\|t2 − t1 \|)v/2 , (iii) √ pǫ (t1 , t2 , v) ≤ C ǫ min{1, (\|t2 − t1 \|)−1/2 }, (5.12) (iv) \|pǫ (t1 , t2 , ζ) − pǫ (t1 , t2 , ζ ′ )\| ≤ Cǫ(1+v)/2 \|ζ − ζ ′ \|v min{1, (t2 − t1 )−(1+v)/2 }, (5.13) for all t1 ≤ t2 ∈ (0, ǫ−1 T ] and ζ, ζ ′ ∈ Ξ(t1 , t2 ). Proof. (i) Recall that φǫ (see, (5.7)) is the mean of a random walk at time t of iid Geometric random variable with parameter (1 − bνǫ ) driven by a Poisson point process of rate (1 − bνǫ )−1 (b−1 − 1) over the interval [t1 , t2 ]. We have noted that the support of the transition probability pǫ (t1 , t2 , .) is given by Z≥0 − φǫ . Also define u′ := ǫ1/2 (νǫ λǫ )−1 u. For any ζ ∈ Ξ(t1 , t2 ) and n ∈ N, let us denote n + ζ + φǫ − 1 B(n, ζ) := . ζ + φǫ − 1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 48 For small ǫ, (1 − bνǫ )−1 (b−1 − 1) ≤ νǫ−1 . Hence, Poisson process associated with the semigroup {pǫ (t1 , t2 , .)} will be dominated by a Poisson process of rate νǫ−1 . For notational simplicity, we introduce the following ν −1 (t2 − t1 )n . m(νǫ , t2 − t1 , n) = exp −νǫ−1 (t2 − t1 ) n n P To that effect, we can bound the sum ζ∈Ξ(t1 ,t2 ) pǫ (t1 , t2 , ζ) exp(uǫ\|ζ\|) by ∞ X X n=0 ζ∈Ξ(t1 ,t2 ) m(νǫ , t2 − t1 , n)B(n, ζ)(1 − bνǫ )n bνǫ (ζ+φǫ ) exp(uǫ\|ζ\|) Using a simple fact euǫ\|ζ\| ≤ euǫζ + e−uǫζ and taking the sum over all ζ ∈ Ξ(t1 , t2 ), one can further bound the sum above by (A) + (B) where !n ∞ X (1 − bνǫ ) m(νǫ , t2 − t1 , n) (A) : = exp(−uǫφǫ ) −1 (1 − bνǫ (1−u′ ) ) n=0 1 − bν ǫ −1 −1 = exp νǫ (t2 − t1 ) − νǫ (t2 − t1 ) − uǫφǫ , (5.14) 1 − bνǫ (1−u′ ) n ∞ X (1 − bνǫ ) exp(uǫφǫ ) m(νǫ , t2 − t1 , n) (B) : = (1 − bνǫ (1+u′ ) ) n=0 1 − bν ǫ −1 −1 = exp νǫ (t2 − t1 ) − νǫ (t2 − t1 ) + uǫφǫ . (5.15) 1 − bνǫ (1+u′ ) Now recall b = e−ǫ 1/2 . Henceforth, one can say 1 − bν ǫ −1 1/2 + O(ǫ) ′ = 1 + uνǫ ǫ 1 − bνǫ (1−u ) and 1 − bν ǫ −1 1/2 + O(ǫ). ′ = 1 − uνǫ ǫ 1 − bνǫ (1+u ) Also, we have uǫφǫ = uǫ(1 − bνǫ )−2 (b−1 − 1)(t2 − t1 ) = u(νǫ−2 ǫ1/2 + O(ǫ))(t2 − t1 ). 1/2 1/2 This enforces νǫ−1 1−b1−b (t2 − t1 ) − νǫ−1 (t2 − t1 ) − uǫφǫ and νǫ−1 1−b1−b (t2 − t1 ) − (1−u′ )/2 (1+u′ )/2 −1 −1 νǫ (t2 − t1 ) + uǫφǫ to be bounded by some constant whenever t1 , t2 ∈ (0, ǫ T ]. Thus, one can note right side in the last line of (5.14) and (5.15) are bounded by some constant C which depends P only on u and T . (ii) We can bound ζ∈E(t1 ,t2 ) pǫ (t1 , t2 , ζ)\|ζ\|v exp(uǫ\|ζ\|) by ∞ X n=0 m(νǫ , t2 − t1 , n) X B(n, ζ)(1 − bνǫ )n bνǫ (ζ+φǫ ) \|ζ\|v exp(uǫ\|ζ\|) ≤ (A′ ) + (B ′ ) ζ∈Ξ(t1 ,t2 ) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 49 where (A′ ) := ∞ X n=0 ≤ (B ′ ) := ∞ X n=0 ≤ n=0 X B(n, ζ)(1 − bνǫ )n bνǫ (ζ+φǫ ) \|ζ\|v exp(uǫζ) ζ∈Ξ(t1 ,t2 ) m(νǫ , t2 − t1 , n) n=0 ∞ X ∞ X m(νǫ , t2 − t1 , n) m(νǫ , t2 − t1 , n) v n 1 − bνǫ (1−u′ ) X − φǫ 1 − bν ǫ 1 − bνǫ (1−u′ ) n exp(−uǫφǫ ), (5.16) B(n, ζ)(1 − b1/2 )n bνǫ (ζ+φǫ ) \|ζ\|v exp(−uǫζ) ζ∈Ξ(t1 ,t2 ) m(νǫ , t2 − t1 , n) v n 1 − bνǫ (1+u′ ) − φǫ 1 − bν ǫ 1 − bνǫ (1+u′ ) n exp(uǫφǫ ). (5.17) Notice g(x) = \|x\|v is a concave function for v ∈ (0, 1). We used Jensen’s inequality for concave function in (5.16) - (5.17) of the above computation. To this end, it can be noted that n (1 − bνǫ )n = + (±C + O(ǫ1/2 ))n ′ ′ 1 − bνǫ (1±u ) (1 − bνǫ (1±u ) )2 for some constant C. Also, φǫ = νǫ−1 (t2 −t1 )(1−bνǫ ) ′ (1−bνǫ (1±u ) )2 ± C ′ (t1 − t1 ) + O(ǫ) for some constant C ′ . For any v < 1, using Hölder’s inequality, we can write E (\|aX + b\|v ) ≤ (E((ax + b)2 )v/2 for any random variable X. Thus, we have ∞ X n=0 m(νǫ , t2 − t1 , n) n v 1 − bν ǫ 1 − bνǫ (1±u′ ) n exp(±uǫφǫ ) − φǫ 1 − bνǫ (1−u′ ) 1 − bν ǫ −1 −1 (t2 − t1 ) − νǫ (t2 − t1 ) ± uǫφǫ ≤ exp νǫ 1 − bνǫ (1±u′ ) v/2 −1 νǫ (t2 − t1 )(1 − bνǫ )2 2 + C1 (t2 − t1 ) . (5.18) × (1 − bνǫ (1±u′ ) )4 To this end, we have νǫ−1 (t2 − t1 )(1 − bνǫ )2 + C1 (t2 − t1 )2 . ǫ−1 (t2 − t2 ). ′ (1 − bνǫ (1±u ) )4 Also, one can note that that other part of the product on the right side of (5.18) appeared in (5.14) and (5.15). Thus, it is bounded by some constant which only depends on T . This completes the proof. (iii) Using the inversion formulae of the characteristic function, we can write down the following pǫ (t1 , t2 , ζ) ≤ ∞ X n=0 exp(−νǫ−1 (t2 ν −n (t2 − t1 )n 1 − t1 )) ǫ n! 2πi Z π −π \|φ(s, ǫ)\|n ds imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 50 where φ(s, ǫ) = ∞ X ξ=0 exp is ξ − (1 − bνǫ )−1 Now, using the estimate \|φ(s, ǫ)\| ≤ Z π −π n \|φ(s, ǫ)\| ds ≤ √ √ ǫ ǫ+s2 √ Z [−π,π] (1 − bνǫ )bνǫ ξ . (5.19) (obtained by Taylor series expansion), we get 2 √ 1 + \|s\|/ ǫ !n √ ds ≤ 2 ǫ Z ∞ 0 √ !n 2 ds. 1+s (5.20) Notice that when n = 1, transition probability is just (1 − b)bζ+φǫ which is anyway less that ǫ. But for n > 1, using (5.20) one can further say Z π √ 1 \|φ(s, ǫ)\|n ds ≤ 21+n/2 ǫ n−1 −π which implies " # ∞ √ −1 n X √ ( 2νǫ ) (t2 − t1 )n −1 −1 pǫ (t1 , t2 , ζ) ≤ C ǫ exp(−νǫ (t2 − t1 )) νǫ (t2 − t1 ) + 2 . n!(n − 1) n=2 (5.21) If X ∼ Poisson(τ ), then using Bernstein’s inequality, we have √ P \|X − τ \| ≥ t τ ≤ 2 exp(−Ct2 ) for some constant. This being said, one can deduce that E X1 ≤ C min{1, τ −1/2 }. One can now see second term in (5.21) has an upper bound min{1, (t2 − t1 )−1/2 }. Also, (t2 − t1 ) exp(−2(t2 − t1 )) can be bounded by (t2 − t1 )−1/2 upto some multiplicative constant when ǫ is small enough. Thus, we get √ pǫ (t1 , t2 , ζ) ≤ C ǫ min{1, (t2 − t1 )−1/2 }. (iv) Using uniform v-Holder continuity of the map x 7→ eix for x ∈ R and inversion formulae of the characteristic functions, one can write down Z π ∞ X (t2 − t1 )n 1 \|s(ζ − ζ ′ )\|v \|φ(s, ǫ)\|n ds. exp(−(t2 − t1 )) \|pǫ (t1 , t2 , ζ) − pǫ (t1 , t2 , ζ ′ )\| ≤ n! 2πi −π n=0 (5.22) where φ(., .) is defined in (5.19). Using the estimate of the characteristic function in the last part, we can have the following bound √ !n Z π Z ∞ 2 v n (1+v)/2 v ds (5.23) s \|φ(s, ǫ)\| ds ≤ 2ǫ s 1 + s −π 0 when n > 2. Note that (n−1)(1+s)−n is a density on R+ . It is easy to see that expectation under that density is 1/(n − 2) whenever n > 2. Thus, using Jensen’s inequality, one can imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 51 conclude that right side of (5.23) is bounded above by 21+n/2 ǫ(1+v)/2 ((n − 1)(n − 2)v )−1 when n exceeds 2. For n = 1, 2, ′ νǫ−1 exp(−νǫ−1 (t2 −t1 ))(t2 −t1 )n (1−b)\|bnζ −bnζ \| ≤ C min{1, (t2 −t1 )−(1+v)/2 }ǫ(1+v)/2 \|ζ−ζ ′ \|v for small enough ǫ. Once again, using the property of the confidence band of Poisson distribution, ∞ X n=2 exp(−νǫ−1 (t2 − t1 )) 2n/2 νǫ−n (t2 − t1 )n ≤ C ′ min{1, (t2 − t1 )−(1+v)/2 }, n!(n − 1)(n − 2)v one can bound right side in (5.22) by C\|ζ − ζ ′ \|v ǫ(1+v)/2 min{1, (t2 − t1 )−(1+v)/2 }. This completes the proof. For the notational convenience, we use Z(t, x) instead of Zǫ (t, x) for the next two lemmas. Let us note that one can modify the scaling appropriately to get necessary results for Zǫ (t, x). In a nutshell, the following results portray the key role in showing the tightness. One can also find similar result in the context of ASEP [BC95, Lemma 4.2, 4.3], higher spin exclusion process [CT15, Lemma 4.3]. Recall the decomposition of Z(t, x) in (5.9). We define Zmg (t1 , t2 , x) and Z∇,mg (t1 , t2 , x) as follows Zmg (t1 , t2 , x) : = ′ Z∇,mg (t1 , t2 , x, x ) : = Z t2 t1 Z t2 t1 [pǫ (s, t2 ) ∗ Z(s)dM (s)](x) [pǫ (s, t2 ) ∗ Z(s)dM (s)](x) − Z t2 t1 [pǫ (s, t2 ) ∗ Z(s)dM (s)](x′ ) In the next lemma, we probe into the higher order norms of the processes Zmg (t1 , t2 , x) and Z∇,mg (t1 , t2 , x). Lemma 5.6. For any k ≥ 1 and v ∈ (0, 1], there exists a large constant C which depends only on k such that for all t1 ≤ t2 ∈ N and ζ, ζ ′ ∈ Ξ(t1 , t2 ), kZmg (t1 ,t2 , ζ)k22k ≤ Cǫ 1/2 Z t2 min{1, (t2 − s)−1/2 } (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZ(s)k22k ] (ζ)ds t1 ′ 2 kZ∇,mg (t1 , t2 , ζ, ζ )k2k (1+v)/2 ′ v Z t2 ≤ Cǫ \|ζ − ζ \| min{1, (t2 − s)−(1+v)/2 } t1 + (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZ(s)k22k ] (ζ ′ ) ds (5.24) (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZ(s)k22k ] (ζ) (5.25) where δ0 is the probability mass function of the distribution which is degenerate at 0 and Pǫ denotes probability mass function of the random variables Ri (see, (5.6) for definition). Proof. Fix any t ∈ (t1 , t2 ). Using Burkholder-Davis-Gundy’s inequality, we have \|Zmg (t1 , t, ζ)\|22k ≤ k[Zmg (t1 , ., ζ), Zmg (t1 , ., ζ)]t kk , (5.26) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 52 and \|Z∇,mg (t1 , t, ζ)\|22k ≤ k[Z∇,mg (t1 , ., ζ), Z∇,mg (t1 , ., ζ)]t kk , (5.27) where [., .] denotes the quadratic variation. Let us denote set all of all time points at which particles hop across ζ, ζ ′ in the interval (s1 , s2 ] by S(s1 ,s2 ] (ζ, ζ ′ ). Recall once again the form of the Martingale in (5.2) and the function κ(., .) in (5.1). Define X ηs (y)Z −1 (ξ1 ∨ ξ2 , s)bνǫ κ(ξ1 ∨ξ2 ,s)+Ny (s)−y Q(ξ1 , ξ2 , y, s) = (b−1 − 1)2 y≤κ(ξ1 ∧ξ2 ,s) which controls the contribution coming out of the Martingale M (t, x) in the total variation terms on right side of (5.26) and (5.27). Using interdependence of the Poisson processes {Pl }−∞<l<∞ , one can say Q(ξ1 , ξ2 , y, s) captures the hopping effect of particles at any position y across κ(ξ1 , s) and κ(ξ2 , s) at time s. Thus, we have X X pξǫ1 ,ξ2 (t1 , t, ζ)Z ξ1 ,ξ2 (s)Q(ξ1 , ξ2 , y, s) (5.28) [Zmg (t1 , ., ζ), Zmg (t1 , ., ζ)]t = ξ1 ,ξ2 s∈S(t1 ,t] (ξ1 ,ξ2 ) and [Z∇,mg (t1 , ., ζ, ζ ′ ), Z∇,mg (t1 , ., ζ, ζ ′ )]t = X X ξ1 ,ξ2 s∈S(t1 ,t] (ξ1 ,ξ2 ) 1 ,ξ2 pξ∇,ǫ (t1 , t, ζ, ζ ′ )Z ξ1 ,ξ2 (s)Q(ξ1 , ξ2 , y, s) (5.29) where Z ξ1 ,ξ2 (s) = Z(ξ1 , s)Z(ξ2 , s), pξǫ1 ,ξ2 (s, t2 , ζ) = pǫ (s, t2 , ζ − ξ1 )pǫ (s, t2 , ζ − ξ2 ) and 1 ,ξ2 pξ∇,ǫ (s, t, ζ) = (pǫ (s, t2 , ζ − ξ1 ) − pǫ (s, t2 , ζ ′ − ξ1 ))(pǫ (s, t2 , ζ − ξ2 ) − pǫ (s, t2 , ζ ′ − ξ2 )). For the notational simplicity, for any two point m > n on the lattice Z≥0 and s ∈ R+ h(m, n, s) := m − n − Nn (s) + Nm (s). On simplifying the summands in (5.28) and (5.29), one can get X ηs (y)bh(y,κ(ξ1 ∧ξ2 ,s),s) bνǫ \|ξ1 −ξ2 \| Z(ξ1 , s)Z(ξ, s)Q(ξ1 , ξ2 , y, s) = ǫZ(ξ1 ∧ ξ2 , s)2 y≤κ(ξ1 ∧ξ2 ,s) where h(y, κ(ξ1 ∧ ξ2 , s), s) counts the number of holes in between y and κ(ξ1 ∧ ξ2 , s) at time √ s. Notice that pξǫ1 ,ξ2 (s, t2 , ζ) ≤ C ǫ max{1, (t2 − s)−1/2 }p (s, t , ζ − ξ ∧ ξ2 ) thanks to the √ P ǫ νǫ 2\|ξ1 −ξ2 \| 1 is constant. Thus, we Proposition 5.5. For any fixed ξ1 , one can note ǫ ξ2 ≥ξ1 b bound right side of (5.28) by X X X ηs (y)bh(y,κ(ξ,s),s) . Cǫ min{1, (t2 − s)−1/2 } pǫ (s, t2 , ζ − ξ)Z(ξ, s)2 ξ y≤κ(ξ,s) s∈S(t1 ,t] (ζ) Divide the interval (t1 , t] into disjoint intervals ∩m i=1 (si , si+1 ] where each of them is of length less than or equals to 1. Next, recall that X dhM (s, ξ), M (s, ξ)i = ηs (y)bh(y,κ(ξ,s),s) dt. (5.30) y≤κ(ξ,s) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 53 Using Lemma 5.10, one can bound Z 2 (s, ξ) times the right side of (5.30) by 1 (1 − bνǫ )b−1 Z(s, ξ) [Pǫ ∗ Z(s)](ξ) − Z(s, ξ)dt . 1 − bν ǫ b−1 − 1 √ Thus, recalling that b = e−λǫ ǫ , we get the following bound √ [Pǫ ∗ kZ(s)k22k ](ξ) + kZ(s, ξ)k22k (νǫ λǫ ǫ)−1 + νǫ kZ(s, ξ)k22k 2 P on the k-th norm of Z(ξ, s)2 y≤κ(ξ,s) ηs (y)bh(y,κ(ξ,s),s) . Also note that, √ max Z(s, x) ≤ exp 2 ǫλǫ N(si ,si+1 ] (x) + 4(νǫ )−1 Z(si , x) s∈(si ,si+1 ] and min s∈(si ,si+1 ] √ Z(s, x) ≥ exp −2 ǫλǫ N(si ,si+1 ] (x) − 4(νǫ )−1 max s∈(si ,si+1 ] Z(si , x) where N(si ,si+1] (x) is independent of Z(si , x) and dominated by a Poisson point process of P rate y≤κ(x,si+1 ) ηsi+1 (y)bh(y,κ(x,si+1 ),si+1 ) . Once again, using the same analysis as above one can show that the rate is bounded above by Cǫ−1/2 . Thus, we have √ E exp 2 ǫkλǫ N(si ,si+1 ] (x) + 4k(νǫ )−1 ≤ C ′ where C ′ is a large constant which only depends on k. To this end, we can bound right side of (5.26) by m X √ X min{1, (t2 − s)−1/2 } pǫ (s, t1 , ζ − ξ) C ǫ ξ k=1 inf s∈(si ,si+1 ] [Pǫ ∗ kZ(s)k22k ](ξ) + kZ(s, ξ)k22k . Recall that the length of each of the intervals (siR, si+1 ] is less than Ror equals to 1. Applying the bound inf s∈(si ,si+1 ] kZ(ξ, s)k2k ≤ (si −sk+1 )−1 kZ(ξ, s)k2k ds = kZ(ξ, s)k2k ds and using the semi-group property of pǫ , we get Z t2 kZmg (t1 , t2 , ζ)k22k ≤ Cǫ1/2 min{1, (t2 − s)−1/2 } (δ0 + Pǫ ) ∗ [pǫ (s, t2 ) ∗ kZ(s)k22k ] (ζ) ds t1 where δ0 is the probability mass function which puts all its mass in 0. Note that we also had made of the fact that under convolution operation, pǫ (t1 , t, .) and Pǫ commute with each other. This holds because pǫ is essentially the semi-group of the random walk of the iid random variables coming from the distribution Pǫ . 1 ,ξ2 (t1 , s, ζ, ζ ′ ) in (5.29) by In the case of Z∇,mg , using Proposition 5.13, we can bound pξ∇,ǫ Cǫ(1+v)/2 min{1, (t2 − s)−(1+v)/2 }\|ζ − ζ ′ \|v pǫ (t1 , s, ζ − ξ1 ∧ ξ2 ) + pǫ (t1 , s, ζ ′ − ξ1 ∧ ξ2 ) . Rest of the terms in (5.28) can be controlled exactly in the same fashion as we have done for bounding kZmg (t1 , t2 , ζ)2 k2k . Thus, one can write down kZ∇,mg (t1 , t2 , ζ)k22k Z t2 ≤ Cǫ(1+v)/2 \|ζ − ζ ′ \|v min{1, (t2 − s)−(1+v)/2 } (δ0 + Pǫ ) ∗ [pǫ (s, t2 ) ∗ kZ(s)k22k ] (ζ) t1 + (δ0 + Pǫ ) ∗ [pǫ (s, t2 ) ∗ kZ(s)k22k ] (ζ ′ ) ds. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 54 In the next result, we present a chaos-series type bound for Z(t, ζ). Lemma 5.7. For any k ≥ 1, t > 0 and ζ ∈ Ξ(t), (5.31) kZ(t, ζ)k2 ≤ 2([pǫ (0, t) ∗ kZ(0)k22k ](ζ)) ∞ Z X hǫ (s1 , s2 ) . . . hǫ (sn , sn+1 )Ψ(sn+1 , ζ)ds1 ds2 . . . dsn+1 +2 n=1 ~s∈∆n+1 (t) where hǫ (si , si+1 ) = Cǫ1/2 min{1, (si − s)−1/2 }1si ≥si+1 , Ψn (sn+1 , ζ) = (δ0 + Pǫ )n ∗ [pǫ (0, sn+1 ) ∗ kZ 2 (0)k2k ] (ζ) and ∆n (t) = {~s ∈ (R≥0 )n \|t ≥ s1 ≥ . . . ≥ sn+1 ≥ 0}. Here, Pǫ denotes probability mass function of Ri (see, (5.6) for definition). Note that here C is a finite constant depended only on k. Proof. Recall the decomposition of Z(t, x) in (5.9). Using one simple inequality, \|x + y\|2 ≤ 2(x2 + y 2 ), we have kZǫ (t, ζ)k22k 2 ≤ 2(k[pǫ (0, t) ∗ Zǫ (0)]k2k ) + 2 Z 2 t 0 [pǫ (s, t) ∗ Zǫ (s)dM (s)](ζ)ds . (5.32) 2k We use Jensen’s inequality to bound the first term on the right side of (5.32) by 2([pǫ (0, t) ∗ kZǫ (0)k22k ]). Moreover, one can also bound second term in (5.32) using (5.24). Thus, we get kZǫ (t, ζ)k22k ≤ 2([pǫ (0, t) ∗ kZǫ (0)k22k ]) (5.33) Z t + Cǫ1/2 min{1, (t − s)−1/2 } (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZǫ (s)k22k ] (ζ)ds. 0 To this end, using successive recursion on kZ(s, .)k22k in RHS of (5.33), one can get an asymptotic expansion. Furthermore, semi-group property [pǫ (t1 , s) ∗ pǫ (s, t2 )](.) = pǫ (t1 , t2 , .) and the fact that Pǫ commutes with the semi-group {pǫ } helps in concluding the final step of the inequality. Our next goal is to prove Proposition 5.1 with some refined estimates of the norms of Zǫ (t, x). Below, we illustrate the idea in details. Proof of Proposition 5.1: First, we prove the following moment estimates: kZǫ (t, x)k2k ≤ Ceǫτ \|x\| , kZǫ (t, x) − Zǫ (t, x′ )k2k ′ kZǫ (t, x) − Zǫ (t , x)k2k v ′ ≤ C ǫ\|x − x′ \| eτ ǫ(\|x\|+\|x \|) , α ≤ C ǫ\|t − t′ \| eτ ǫ\|x\| . (5.34) (5.35) (5.36) for some α := v/2 ∧ 1/4, t, t′ ∈ (t1 , ǫ−1 T ], x, x′ ∈ Ξ(t) where T ∈ (0, ∞) is fixed apriori and v is exactly same as in (2.13). Once those are proved, we can make use of Kolmogorov-Chentsov criteria to conclude the tightness of the sequence {Zǫ }ǫ . imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 55 Proof of (5.34): We have kZ(0, x)k2k ≤ eτ ǫ\|x\| from (2.12). One can further bound \|ξ\| by \|x − ξ\| + \|x\|. Thus, we have X pǫ (0, t, x − ξ) exp(2τ ǫ\|x − ξ\|) exp(2τ ǫ\|x\|) ≤ C exp(2τ ǫ\|x\|). [pǫ (0, t) ∗ kZ(0)k22k ](x) ≤ ξ∈Ξ(t) thanks to (5.11). Consequently, first term in RHS of (5.31) is bounded above by C exp(2τ ǫ\|x\|). Using the same arguments, one can also bound [pǫ (0, sn+1 ) ∗ kZ(0)k22k ](x) of the second term in (5.31) by C1 exp(2τ ǫ\|x\|). Furthermore, we know x X ξ=−∞ ∞ √ X √ √ Pǫ (x − ξ) exp(2τ ǫ\|ξ\|) ≤ exp(2τ ǫ\|x\|) ǫ exp(−k ǫ(νǫ − 2 ǫτ )) ≤ C2 exp(2τ ǫ\|x\|) k=0 for some constant C2 when ǫ is small enough. Thus, we can conclude that [(δ0 + Pǫ )n ∗ [pǫ (0, sn+1 ) ∗ Z(s)]](x) is bounded by C1 C2n exp(2τ ǫ\|x\|) for all small ǫ. Interestingly, we can now easily compute left over integral of the second term in the RHS of (5.31). It turns out that n/2 Z n Y (CΓ(1/2))2 ǫt hǫ (si , si+1 )ds1 ds2 . . . dsn+1 ≤ . (5.37) Γ(n/2) ~ s∈∆n+1 (t) i=1 This makes the sum in (5.31) finite and thus helps us to conclude (5.34). Proof of (5.35): We can decompose Z(t, x) − Z(t, x′ ) as X Z(t, x) − Z(t, x′ ) = pǫ (0, t, ξ) Z(t1 , x − ξ) − Z(t1 , x′ − ξ) ξ∈t t + Z X 0 ξ∈Ξ(s,t) pǫ (s, t, x − ξ) − pǫ (s, t, x′ − ξ) Z(s, ξ)dM (s, ξ) (5.38) Notice that we have the bound on pǫ (0, t, ζ) − pǫ (0, t, ζ ′ ) from Lemma 5.5. One can use (2.13) to bound the first term on the RHS of (5.38) as noted down below. k[p(0, t) ∗ Z(0)](x) − [p(0, t) ∗ Z(0)](x′ )k2k X ≤ pǫ (0, t, ξ)kZ(0, x − ξ) − Z(0, x′ − ξ)k2k ξ∈Ξ(0,t) ≤ X ξ∈Ξ(0,t) v pǫ (0, t, ξ) ǫ\|x − x′ \| exp τ ǫ(\|x − ξ\| + \|x′ − ξ\|) v ≤ ǫ\|x − x′ \| exp τ ǫ(\|x\| + \|x′ \|) X pǫ (0, t, ξ) exp(2τ ǫ\|ξ\|) 2ξ∈Ξ(0,t) v ≤ C ǫ\|x − x \| exp τ ǫ(\|x\| + \|x′ \|) . ′ One can identify the term in the last line of (5.38) with Z∇,mg (t1 , t, x, x′ ). It follows from (5.34) that kZ(s, ξ)k2k ≤ C exp(τ ǫ\|ξ\|) for all ξ ∈ Ξ(t1 , s) and 0 ≤ s ≤ ǫ−1 T where C only depends on T . Using the bound \|x − ξ\| + \|x\| for \|ξ\| in exp(τ ǫ\|ξ\|), one can bound [pǫ (s, t) ∗ kZ(s)k22k ](x) above by C exp(2τ ǫ\|x\|). Similarly, C exp(2τ ǫ\|x′ \|) is also valid as an upper bound imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 56 to [pǫ (s, t) ∗ kZ(s)k22k ](x′ ). So, the integrand in (5.24) can be bounded above by min{1, (t2 − s)−(1+v)/2 }1s≤t2 C exp (2τ ǫ(\|x\| + \|x′ \|)) where t2 is equal to t and s ∈ (0, t). Consequently, we can bound the kZ∇,mg (0, t, x, x′ )k2k by C(ǫ\|x − x′ \|)v (ǫt)(1−v)/2 exp (τ ǫ(\|x\| + \|x′ \|)). This proves (5.35) thanks to the fact t ∈ (0, ǫ−1 T ]. Proof of (5.36): Assume t′ < t. We can write down the following: X X Z(t, x) − Z(t′ , x) = p(0, t, ξ)Z(t1 , x − ξ) − p(0, t′ , ξ ′ )Z(0, x − ξ ′ ) ξ ′ ∈Ξ(t′ ) ξ∈Ξ(0,t) + Z t t′ [pǫ (s, t) ∗ Z(s)dM (s)](x). (5.39) Notice that the first term in the last line of (5.39) is exactly Zmg (t′ , t, x). Furthermore, using semigroup property of pǫ , one can say X X X pǫ (t1 , t, ξ)Z(0, x − ξ) = pǫ (0, t′ , ξ ′ )pǫ (t′ , t, ξ − ξ ′ )Z(0, x − ξ) ξ∈Ξ(t) ξ ′ ∈Ξ(t′ ) ξ∈Ξ(0,t) X = pǫ (0, t′ , ξ ′ ) ξ ′ ∈Ξ(0,t′ ) X ζ∈Ξ(t′ ,t) pǫ (t′ , t, ζ)Z(0, x − ξ ′ − ζ). This helps us to analyse the first term in RHS of (5.39) in the following way. X ξ∈Ξ(t) p(0, t, ξ) Z(t1 , x − ξ) − = X ξ ′ ∈Ξ(t′ ) ≤C X ξ ′ ∈Ξ(t′ )  pǫ (0, t′ , ξ ′ )  X p(0, t′ , ξ ′ )Z(0, x − ξ ′ ) 2k ζ∈Ξ(t′ ,t) pǫ (0, t′ , ξ ′ ) ξ ′ ∈Ξ(t′ ) X X pǫ (t′ , t, ζ) Z(0, x − ξ ′ − ζ) − Z(0, x − ξ ′ ) ζ∈Ξ(t′ ,t) ≤ C(ǫ\|t − t′ \|)v/2 exp(2τ ǫ\|x\|). ǫ\|ζ\|v pǫ (t′ , t, ζ) exp τ ǫ(\|ζ\| + 2\|x\| + 2\|ξ ′ \|) 2k   Note that we have used (5.13) in Proposition 5.5 in the last inequality. One can bound kZmg (t′ , t, x)k2k using (5.24). Moreover, integrand in the RHS of (5.24) is bounded above by √ C ǫ1t′ ≤s≤t min{1, (t′ − s)−1/2 } exp(2τ ǫ\|x\|) thanks to (5.34). This enforces kZmg (t′ , t, x)k22k to be bounded by C(ǫ\|t − t′ \|)1/2 exp(2τ ǫ\|x\|). Hence, the claim follows. Proof of Proposition 2.12: First, we show (2.14). In Lemma 5.7, we multiply both side with ǫ−2 (1 − exp(−λǫ νǫ ))2 . Consequently, we get 2 kZ̃(t, ζ)k22k ≤2 [pǫ (0, t) ∗ Z̃(0)](ζ) ∞ Z X +2 h(s1 , s2 ) . . . h(sn , sn+1 )Ψ̃n (sn+1 , ζ)ds1 . . . dsn+1 n=1 ~s∈∆n+1 (t) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 57 where i h Ψ̃n (sn+1 , ζ) := (δ0 + Pǫ )n ∗ ([pǫ (0, sn+1 ) ∗ Z̃(0)])2 (ζ). Note that Z̃(0, .) is now deterministic. Using the relation ǫ bound on pǫ (0, t, .) from (5.12), we further have P ζ∈Ξ(0) Z̃(0) (5.40) = 1 and the upper [pǫ (0, t) ∗ Z̃(0)](x) ≤ C min{ǫ−1/2 , (ǫt)−1/2 }. Similarly, the term [pǫ (0, sn+1 ) ∗ Z̃(0)] inside Ψ̃n (sn+1 , ζ) in (5.40) is also bounded above by min{ǫ−1/2 , (ǫsn+1 )−1/2 }. To that effect, one can bound the rest of the integral as Z −1/2 h(s1 , s2 ) . . . h(sn , sn+1 )(ǫsn+1 ) ~s∈∆n+1 (t) ds1 . . . dsn+1 (CΓ(1/2))2 ǫt ≤ Γ(n/2) n/2 (5.41) in the same way as in (5.37). Thus, for all t ∈ (0, ǫ−1 T ], summing (5.41) over n ∈ Z+ , we get kZ̃(t, x)k22k ≤ C([pǫ (0, t) ∗ Z̃ǫ (0)](ζ))2 + exp((ǫt)1/2 ) ≤ C ′ min{ǫ−1 , (ǫt)−1 } (5.42) where the constants C, C ′ > 0 depend only on T . Thus the claim follows. Now, we turn to show (2.15). To this end, multiplying Z(t, ζ)−Z(t, ζ ′ ) by ǫ−1 (1−exp(−λǫ νǫ )), we get 2  X \|pǫ (0, t, ζ − ξ) − pǫ (0, t, ζ − ξ)\|Z̃(0, ξ) kZ̃(t, ζ) − Z̃(t, ζ ′ )k22k ≤ 2  ξ∈Ξ(0) + ckZ̃∇,mg (0, t, ζ1 , ζ1′ )k22k for some constant c. Using the bound on \|pǫ (0, t, ζ − ξ) − pǫ (0, t, ζ − ξ)\| from (5.13) and the P relation ǫ ζ∈Ξ(0) Z̃(0, ζ) = 1, one can get the following X ξ∈Ξ(0) \|pǫ (0, t, ζ − ξ) − pǫ (0, t, ζ − ξ)\|Z̃(0, ξ) ≤ C(ǫ\|ζ − ζ ′ \|)v min{ǫ− 1+v 2 , (ǫt)− 1+v 2 }. Moreover, substituting 2v in place of v and taking (t1 , t2 ) = (0, t) in (5.25), we get Z h i 1+2v ′ 2 2v kZ̃∇,mg (0, t, ζ, ζ )k2k ≤ǫ 2 \|ζ − ζ\| min{1, (t − s)−(1+2v)/2 } Ψ̃(s, t, ζ) + Ψ̃(s, t, ζ ′ ) ds (5.43) i h where Ψ̃(s, t, ζ) := (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZ̃(s)k22k ] (ζ). One can further bound the integrand on the right side in (5.43) using bound on kZ̃(s, ζ)k22k from (5.42). Henceforth, this implies kZ̃∇,mg (0, t, ζ, ζ ′ )k22k ≤ C(ǫ\|ζ − ζ ′ \|)2v (ǫt)−v+(1/2) . (5.44) But, we know ǫt ≤ T for all t ∈ (0, ǫ−1 T ]. Thus, we can improve the bound on the right side of (5.44) to C ′ (ǫ\|ζ − ζ\|)2v (ǫt)−(1+v) where C ′ depends only on T . imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 58 5.3. Proof of Proposition 5.2: The Martingale Problem Before going into the details of the proof, we present a brief exposure on the martingale problem from [BG97]. Definition 5.8. Let Z be a C([0, ∞) × (−∞, ∞)) valued process such that for any given T > 0, there exists u < ∞ such that sup sup e−u\|x\| E(Z(t, x)) < ∞. t∈[0,T ] x∈R R For such Z and ψ ∈ Cc∞ (R), let hZ(t), ψi := R Z(t, x)ψ(x)dx. We say Z solves the martingale problem with initial condition Z ic ∈ C(R), if Z(0, .) = Z ic (.) in distribution and Z 1 t ∂2ψ t 7→ Nψ (t) := hZ(t), ψi − hZ(0), ψi − Z(τ ), 2 dτ 2 0 ∂x Z t Z 2 (τ ), ψ 2 dτ t 7→ Nψ′ (t) := (Nψ (t))2 − 0 are local martingales for any ψ ∈ Cc∞ (R). It has been stated in [BG97], that for any initial condition Z ic satisfying kZ ic (x)k2k ≤ Cea\|x\| for some a > 0, the solution of the martingale problem stated above coincides with the solution of the SHE with initial condition Z ic . In our case, if we can show any limit point of {Zǫ }ǫ started from Z ic solves the martingale problem, then it follows from [BG97] that {Zǫ }ǫ converges to a unique process which solves SHE with initial condition being Z ic . For solving the martingale problem, we have to have a discrete analogue of hZ(t), ψi for any Zǫ . Note that we can take X hZ(s), ψiǫ = ǫ Z(t, ξ)ψ(ǫξ) ξ∈Z as the discrete analogue for hZǫ (t), ψi. Let us divide (0, ∞) into a number of disjoint subintervals ∪∞ i=1 (si , si+1 ) where 0 = s1 < s2 < . . . and each has length ǫ. Let us assume that sm ≤ t < sm+1 . On the basis of decomposition of Z(t, x) in (5.9), one can say the following:   m−1 X X X  hZ(t), ψiǫ − hZ(0), ψiǫ = ǫ pǫ (si , si+1 , ξ − ζ)ψ(ǫξ) − ψ(ǫζ) Z(si , ζ) i=1 ζ∈Ξ(si ) X +ǫ ζ∈Ξ(si ) +ǫ  Z t ξ∈Ξ(si ,si+1 )+ζ X  ξ∈Ξ(sm ,t)+ζ m−1 X Z si+1 i=1 +ǫ  si X sm ζ∈Ξ(s) X ζ∈Ξ(s)    pǫ (sm , t, ξ − ζ)ψ(ǫξ) − ψ(ǫζ) Z(sm , ζ)  X X ξ∈Ξ(s,si+1 )+ζ ξ∈Ξ(s,t)+ζ  pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) Z(s, ζ)dM (s, ζ)  pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) Z(s, ζ)dM (s, ζ). imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 59 Let us define Rǫ (t) as Rǫ (t) :=ǫ m−1 X X i=1 ζ∈Ξ(si ) X +ǫ ζ∈Ξ(si ) Also, define Nǫ (t) :=ǫ m Z X i=1 +ǫ Z t    si+1 si  X ξ∈Ξ(si ,si+1 )+ζ X ξ∈Ξ(sm ,t)+ζ X ζ∈Ξ(s) X sm ζ∈Ξ(s)     pǫ (si , si+1 , ξ − ζ)ψ(ǫξ) − ψ(ǫζ) Z(s, ζ)ds  pǫ (sm , t, ξ − ζ)ψ(ǫξ) − ψ(ǫζ) Z(sm , ζ). X ξ∈Ξ(s,si+1 ) X  ξ∈Ξ(s,t)+ζ  pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) Z(s, ζ)dM (s, ζ)  pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) Z(s, ζ)dM (s, ζ). Clearly, Nǫ (t) is a local martingale. Let us denote the quadratic variation of Nǫ (t) by N̂ǫ (t). So, for showing that indeed any limit point of {Zǫ }ǫ solves the martingale problem, it is enough to show the following: E hZ(ǫ−1 t), ψiǫ − hZǫ (t), ψi → 0, Z t ∂2ψ −1 E Rǫ (ǫ t) − Zǫ (s), 2 ds → 0, ∂x 0 Z t E N̂ǫ (ǫ−1 t) − hZǫ2 (s), ψ 2 ids → 0. (5.45) (5.46) (5.47) 0 To prove (5.45) - (5.47), we need the following lemma. Proof of (5.45): Here, we have to show that Z Zǫ (t, x)ψ(x)dx hZǫ (t), ψi = R and hZ(ǫ−1 t), ψiǫ are asymptotically same in L1 . Notice that for x such that \|x − ζ\| ≤ 1 where ζ ∈ Ξ(ǫ−1 t), using smoothness of ψ and moment estimates from (5.34)-(5.36), one can say that the sequences ǫ−v Z(ǫ−1 t, ζ)ψ(ǫζ) − Z(ǫ−1 t, x)ψ(ǫx) are L2k bounded for all k ∈ N. This forces hZ(ǫ−1 t), ψiǫ − hZǫ (t), ψi to go to 0 in L1 as ǫ → 0. Proof of (5.46): Recall that subintervals {[si , si+1 )}∞ i=1 form a partition of (0, ∞) such that length of each of the interval is ǫ. Assume here that ǫsm ≤ t < ǫsm+1 . Using the smoothness of ψ, one can get X X pǫ (si ,si+1 , ξ − ζ)ψ(ǫξ) − ψ(ǫζ) = ǫψ ′ (ǫζ) pǫ (si , si+1 , ξ − ζ)(ξ − ζ) ξ∈Ξ(si ,si+1 )+ζ ξ∈Ξ(si ,si+1 ) + X ǫ2 1 2 ξ∈Ξ(si ,si+1 ) ∂2ψ ∂x (ǫζ) + o(ǫ)B(ζ) pǫ (si , si+1 , ξ − ζ)(ξ − ζ)2 2 (5.48) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 60 P where B belongs to L2 . Note that ξ∈Ξ(s,t) pǫ (si , si+1 , ξ − ζ)(ξ − ζ) = 0. This goes back to P the fact that mean under pǫ (si , si+1 , .) is 0. Moreover, ξ∈Ξ(si ,si+1 ) pǫ (si , si+1 , ξ − ζ)(ξ − ζ)2 = −1 1 −1 + O(ǫ))(ν λ )−2 ǫ−1 . Recall from (si+1 − si )σǫ2 = (si+1 − si ) b1−b−1 νǫ (1−bνǫ )2 = (si+1 − si )(νǫ ǫ ǫ −3/2 Definition 2.7 that λǫ = νǫ Rǫ1 (ǫ−1 t) = ǫ . To this effect, we have 2 m X ∂ ψ 1X (si+1 − si )ǫ Z(si , ζ) (ǫζ) + o(ǫ) 2 ∂x2 i=1 (5.49) ζ∈Ξ(si ) tǫ−1 Z tǫ−1 1 ∂2ψ = ǫ hZ(s), Biǫ ds + D. Z(s), 2 ds + o(ǫ) 2 0 ∂x ǫ 0 E RtD 2 Simple change of variable reduces first term in (5.49) to 2−1 0 Z(s), ∂∂xψ2 ds at a cost of Z ǫ an error D. It follows from the moment estimates in (5.34)-(5.36) that D in (5.49) indeed converges to 0 in L1 as ǫ → 0. Further, we know the fact that B is in L2 . Thus, second term also converegs to L1 . Thereafter, one can R t 0 in −1 R tuse DCT along with (5.45) to conclude 1 L distance between 0 hZ(ǫ s), ∂ 2 ψ/∂x2 iǫ ds and 0 hZǫ (s), ∂ 2 ψ/∂x2 ids converges to 0, thus shows the claim. Proof of (5.47): One can write N̂ (ǫ −1 t) = ǫ 2 + ǫ2 m−1 X Z si+1 i=1 t Z si X X ψ ζ1 ,ζ2 (s, si+1 )Z(s, ζ1 )Z(s, ζ2 )dhM (s, ζ1 ), M (s, ζ2 )i ζ1 ,ζ2 ψ ζ1 ,ζ2 (s, t)Z(s, ζ1 )Z(s, ζ2 )dhM (s, ζ1 ), M (s, ζ2 )i sm ζ ,ζ 1 2 where  ψ ζ1 ,ζ2 (s1 , s2 ) =  X ξ∈Ξ(s1 ,s2 ) Recall from (5.10) that  pǫ (s1 , s2 , ξ − ζ1 )ψ(ǫξ)  X ξ∈Ξ(s1 ,s2 )  pǫ (s1 , s2 , ξ − ζ2 )ψ(ǫξ) . (b−1 − 1)−2 Z(t, ζ1 )Z(t, ζ2 )dhM (t, ζ1 ), M (t, ζ2 )i bνǫ \|ζ1 −ζ2 \| (1 − bνǫ )b−1 = Z(t, ζ1 ∧ ζ2 ) [pǫ (t, t + dt) ∗ Z(t)](ζ1 ∧ ζ2 ) − Z(t, ζ1 ∧ ζ2 )dt . 1 − bν ǫ b−1 − 1 Furthermore, one can write ∞ X [pǫ (t, t + dt) ∗ Z(t)](x) − Z(t, x)dt = (1 − bνǫ )bkνǫ (Z(t, x − k) − Z(t, x)) dt. k=0 Using (5.35), we have ∞ ∞ X X √ √ νǫ kνǫ ǫ\|ǫk\|v exp −λǫ νǫ k ǫ + kǫ = O(ǫv/2 ) (1 − b )b (Z(t, x − k) − Z(t, x)) ≤ k=0 k=0 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 61 for any t > 0. Moreover, one can deduce from (5.48) that X ξ∈Ξ(s,si+1 ) pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) = ψ(ǫζ) + ǫ 1 ∂2ψ (ǫζ)(si+1 − s) + o(ǫ2 )B, 2 ∂x2 where B is in L2 (R). Thus, we get N̂ (ǫ−1 t) equal to m−1 X Z si+1 i=1 si ǫ 2 X ψ ζ1 ,ζ2 (s, si+1 )b νǫ \|ζ1 −ζ2 \| (b −1 ζ1 ,ζ2 − 1) bνǫ −1 − 1 2 v/2 Z (s, ζ1 ∧ ζ2 ) + O(ǫ ) ds. 1 − bν ǫ where ψ ζ1 ,ζ2 (s, si+1 ) = ψ(ǫζ1 )ψ(ǫζ2 ) + O(ǫ2 )B1 . Further, the random variable B1 belongs to L2 . Note that νǫ > 0 (see Definition 2.7) is chosen in way so that it satisfies 1 − νǫ = νǫ2 . For this specific choice of νǫ , we have X ζ2 ≥ζ1 bνǫ \|ζ1 −ζ2 \| (b−1 − 1) √ bνǫ −1 − 1 = 1 + O( ǫ). 1 − bν ǫ Rt Thereafter, using continuity of ψ, one can say further N̂ǫ (ǫ−1 t) − 0 hZ 2 (ǫ−1 ), ψ 2 iǫ ds converges Rt Rt to 0 in L1 as ǫ → 0. To this end, (5.45) shows 0 hZ 2 (ǫ−1 s), ψ 2 iǫ ds and 0 hZǫ2 (s), ψ 2 ids both has same L1 limit and hence, completes the proof. Appendix A: Fredholm Determinant We start with the definition of Fredholm determinant. Definition A.1. Let K(x, y) be the meromorphic function of two complex variable, invariably referred as kernel in the literature. Let Γ ⊂ C be a curve. Assume that K has no singularities on Γ × Γ. Then the Fredholm determinant det(I + K) of the kernel K is defined as the sum of the series of the complex integrals Z Z X 1 det(I + K) = (A.1) . . . det (K(zi , zj ))ni,j=1 dz1 , . . . dzn . n!(2πi)n Γ Γ n≥0 Remark 1. In the usual definition of the kernel of Fredholm determinant (see, [Sim05, Chapter 5]), there is no such pre-factor of 1/(2πi)n as in (A.1). Furthermore, to show the absolute convergence of the series in (A.1), one need to have suitable control on supx∈Γ \|K(x, z)\|. To that end, following inequality (see, [Bha97, Exercise 1.1.3]) proves the absolute convergence. Lemma A.2 (Hadamard Inequality). If D is a N × N complex matrix, then v N uX Y u n 2. t Dij \|det(D)\| ≤ i=1 j=1 Corollary A.3. Let Γ be a curve. Kernel K is defined over Γ×Γ. Furthermore, supx∈Γ \|K(x, z)\| ≤ R K(z), where K(z) satisfies Γ K(z)dz < ∞. Then series in (A.1) converges absolutely. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 62 In the next lemma, we show that Fredholm determinant of the limiting kernel in (4.57) is the Gaussian distribution function. Although, we couldn’t find a direct proof that result, we believe that it is quite well known. For the sake of clarity, we present here a short proof. Lemma A.4. Consider a piecewise linear infinite curve Γ which extends linearly from −δ + ∞e−π/6 to −δ and from −δ to −δ + ∞eπ/6 . Let us define the kernel KG for any two points w, w′ ∈ Γ as 2 w2 v Z − + s(v − w) exp 2 2 v 1 dv. KG := 2πi −2δ+R (v − w)(v − w′ ) w Then, det(1 + KG )L2 (Γ) = Z s −∞ 1 2 √ e−z /2 dz. 2π Proof. To begin with, notice that for all v on the line R ∞ −2δ + iR, Re(v − w) < 0 whenever w ∈ Γ. Thus, we can write exp(s(v − w)) = (v − w) s exp(z(v − w))dz. Fix any σ ∈ S(n). To this end, we have Z Z (A.2) . . . K(w1 , wσ(1) )K(w2 , wσ(2) ) . . . K(wn , wσ(n) )dw1 , . . . dwn Γ Γ P 2 vi wi2 n Z Z n exp − + s(v − w ) Y i i i=1 2 2 vi Y 1 Q dwi dv = i n (2πi)n Γn (−2δ+iR)n wi i=1 (vi − wi )(vi − wσ(i) ) i=1 i=1 P 2 vi wi2 n Z Z Z n n n exp Y Y i=1 2 − 2 + zi (vi − wi ) 1 vi Y Q dwi = dv dz i i n (2πi)n Γn (−2δ+iR)n [−s,∞)n wi i=1 (vi − wσ(i) ) i=1 = 1 (2πi)n Z [s,∞)n Z (−2δ+iR)n Z exp Γn P 2 vi n i=1 i=1 wi2 2 n n n + z (v − w ) Y Y i i i i=1 2 vi Y Qn dwi dvi dzi . wi i=1 (vi − wσ(i) ) − i=1 i=1 i=1 Define a piecewise linear curve Γr which extends linearly from −δ + re−iπ/6 to −δ and from −δ to −δ + reiπ/6 . Notice that Γr tends to Γ as r → ∞. Denote the closed contour formed by Γ and the line joining −δ + re−iπ/6 to −δ + reiπ/6 as Γ̃r . As r increases, integrand inside the paranthesis of the last line in (A.2) exponentially decays to zero uniformly for all w ∈ Γ̃r \Γr . Consequently, the integral over Γ̃r will converges to the integral over Γ. Thus, computing the integral over Γn boils down to finding out the residue of the integrand at z1 = . . . = zn = 0 because these are the only poles sitting inside the contour Γ̃nr for all large value of r. Thereafter, 2 vi wi2 ! Z exp Pn n 2 n − + z (v − w ) X i i i i=1 2 2 vi Y vi Qn dwi = exp + zi vi . wi 2 Γn i=1 (vi − wσ(i) ) i=1 i=1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 63 Using the analytic behavior of the function exp(2−1 v 2 + vz), one can argue further ! Y ! Y Z Z n 2 n n n 2 X X vi vi exp + zi vi + zi vi dvi = dvi exp 2 2 (−i∞,i∞)n (−2δ+iR)n i=1 i=1 i=1 i=1 ! n X √ 2 n = ( 2πi) exp − zi . i=1 To this end, multiple integral in (A.2) becomes equal to (1 − Φ(s))n where Φ(.) is the distribution function of standard normal distribution. This implies for all n ≥ 2 Z Z . . . det(K(wi , wj ))ni,j=1 dw1 . . . dwn = 0. Γ Γ Thus, we have det(1 + KG )L2 (Γ) = 1 − (1 − Φ(s)) = Φ(s). Hence, this completes the proof. Appendix B: eigenfunction of the Transition Matrix in HL-PushTASEP Proof of Proposition 3.1: Here, we sketch a proof of the present proposition extending the arguments in [BCG16, Theorem 3.4] in the continuous case. Let us define N functions gi (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN ) as follows X (t,t+dt) yN gi (xi , xi+1 , . . . , xN ; z1 , z2 , . . . , zN ) = Ti ((xi , . . . , xN ) → (yi , . . . , yN )) ziyi . . . zN (yi ,...,yN ) where Tit,t+dt (.) denotes the probability of the transition from the configuration (xi , . . . , xN ) to (yi , . . . , yN ) once the particle at position xi gets excited in the time interval (t, t + dt). For convenience, let us consider gi (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN ) := g (1) (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN ) (2) + gi (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN ). (B.1) Subsequently, we describe two new functions introduced in (B.1) as follows. In the expansion of gi , one can note the contribution of the event that the particle at position xi moves by j-steps x −1 xi+1 xN zi where 1 ≤ j < xi+1 − xi in the time gap (t, t + dt) is 1−bz (zixi − bxi+1−xi −1 zi i+1 )zi+1 . . . . zN i Thus, define (1) gi (xi , . . . , xN ; zi , . . . , zN ) = zi (1 − b) xi x −1 xi+1 xN . . . zN . (zi − bxi+1 −xi −1 zi i+1 )zi+1 1 − bzi Lastly, g(2) takes account of the event that particle at xi jumps to the position xi+1 and consequently, particle which was previously there in xi+1 starts to hop to the right in the same fashion. To this end, one can write (2) x gi (xi , . . . , xN ; zi , . . . , zN ) = bxi+1 −xi −1 zi i+1 gi+1 (xi+1 , . . . , xN ; zi , . . . , zN ). imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 64 Let us define a function g as X g(x1 , . . . , xN ; z1 , . . . , zN ) := (y1 ,...,yN ) yN T (t,t+dt) ((x1 , . . . , xN ) → (y1 , . . . , yN )) ziyi . . . zN where T (t,t+dt) denotes the transfer matrix in the interval (t, t + dt). Naively, g acts as a probability generating function in the infinitesimal time interval (t, t + dt). In what follows, it is easy to see how g and gi ’s are connected g(x1 , . . . , xN ; z1 , . . . , zN ) = N X x i−1 z1x1 . . . zi−1 gi (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN )dt. i=1 If we take N = 2, then we get g(x1 , x2 ; z1 , z2 ) = z x2 +1 (1 − b) x2 −x1 −1 x2 z1 (1 − b) x1 (z1 − bx2 −x1 −1 z1x2 −1 )z2x2 dt + 2 (b z1 + z1x1 )dt. 1 − bz1 1 − bz2 Let us consider S12 g(x1 , x2 ; z1 , z2 )+S21 g(x1 , x2 ; z2 , z1 ) for some functions S12 and S21 of z1 , z2 . One can write z1 (1 − b) z2 (1 − b) + (S12 z1x1 z2x2 + S21 z2x1 z1x2 )dt S12 g(x1 , x2 ; z1 , z2 ) + S21 g(x1 , x2 ; z2 , z1 ) = 1 − bz1 1 − bz2 x2 x2 −x1 −1 + (1 − b)(z1 z2 ) b S12 1 1 z1 z2 − − + S21 dt. 1 − bz2 1 − bz1 1 − bz1 1 − bz2 Thus, on taking S12 = 1 − (1 + b)z1 + bz1 z2 and S12 = −1 + (1 + b)z2 − bz1 z2 , one can note that S12 z1x1 z2x2 + S21 z1x2 z2x1 is an eigenfunction of the transfer matrix T with the eigenvalue z2 (1−b) z1 (1−b) 1−bz1 + 1−bz2 in the interval (t, t+dt). One can also scale S12 and S21 by 1−(1+b)z1 +bz1 z2 without any major consequences. To obtain the eigenfunction and eigenvalues for any N , we use induction. Assume N = n − 1 and X Aσ g(x1 , x2 , . . . , xn−1 ; zσ(1) , zσ(2) , . . . , zσ(n−1) ) (B.2) σ∈S(n−1) = (1 − b) n−1 X i=1 zi 1 − bzi where Aσ := Y (−1)σ 1≤i<j≤n−1 !  X σ∈S(n−1) x  x1 n−1  Aσ zσ(1) . . . zσ(n−1) dt 1 − (1 + b)zσ(i) + bzσ(i) zσ(j) . 1 − (1 + b)zi + bzi zj We need to show that (B.2) holds even when N = n. Consider the following X Aσ g(x1 , x2 , . . . , xn ; zσ(1) , zσ(2) , . . . , zσ(n) ) = (I) + (II) + (III) σ∈S(n) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 65 where  X (I) : = (y1 ,y2 ,...,yn ) y1 =x1 =  n X j=1  T (t,t+dt) ((x1 , . . . , xn ) → (y1 , . . . , yn ))    (1 − b)  X (II) : = (y1 ,y2 ,...,yn ) x1 <y1 <x2 = (1 − b) n X i=1 i6=j  n X j=1 Aσ σ∈S(n) σ(1)=j Y  zi   X x1 x2 xn  Aσ zσ(1) zσ(2) . . . zσ(n) dt,  1 − bzi  X Aσ σ∈S(n) Aσ σ∈S(n) yi   , (B.3) zσ(i)  σ∈S(n) T (t,t+dt) ((x1 , . . . , xn ) → (y1 , . . . , yn ))  X X  x2 −x1 bx2 −x1 −1 zσ(1) zσ(1) − 1 − bzσ(1) 1 − bzσ(1) ! Y  yi  zσ(i) x1 x2 xn dt, zσ(1) zσ(2) . . . zσ(n) and (III) : = X (y1 ,y2 ,...,yn ) y1 =x2 = (1 − b)b + X  T (t,t+dt) ((x1 , . . . , xn ) → (y1 , . . . , yn ))  (y1 ,y2 ,...,yn ) Aσ σ∈S(n) Y  yi  zσ(i) x3 −x2 ! bx3 −x2 −1 zσ(2) zσ(2) x1 x2 xn − dt zσ(1) zσ(2) . . . zσ(n) Aσ 1 − bzσ(2) 1 − bzσ(2) σ∈S(n)   Y y X i  zσ(i) Aσ . T (t,t+dt) ((x1 , . . . , xn ) → (y1 , . . . , yn ))  x2 −x1 −1 y1 =x2 ,y2 =x3 X X σ∈S(n) For any σ ∈ S(n), one can note that " x2 −x1 # # " x2 −x1 zσ(1) zσ′ (1) zσ(2) zσ′ (2) x2 Aσ − (zσ(1) zσ(2) ) = −Aσ′ − (zσ′ (1) zσ′ (2) )x2 1 − bzσ(1) 1 − bzσ(2) 1 − bzσ′ (1) 1 − bzσ′ (2) where σ ′ = (1, 2) ∗ σ. Thus, there will be telescopic cancellations of large number of terms in the sum (II) + (III). To that effect, we would have (II) + (III) = (1 − b) X σ∈S(n) Aσ zσ(1) xn dt z x1 z x2 . . . zσ(n) 1 − bzσ(1) σ(1) σ(2) (B.4) and consequently, adding (B.3) and (B.4) shows (B.2) when N = n. To this end, denoting (0,t) the transfer matrix for exactly m-many clock counts in the interval (0, t) by Tm (.), one can imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 66 deduce X (y1 ,...,yN ) = Z  Tm(0,t) ((x1 , . . . , xN ) → (y1 , . . . , yN ))  (t0 ,...,tm+1 )∈∆(t) 1 = exp(−N t) m! t N X (1 − b)zi i=1 (1 − bzi ) N X (1 − b)zi i=1 (1 − bzi ) !m   X X σ∈S(n)  y1 yN  Aσ zσ(1) . . . zσ(N ) x1 Aσ zσ(1) σ∈S(n)  xN  . . . zσ(N ) m Y k=1 (B.5) exp(−N (tk − tk−1 ))dtk  !m  X xN  x1  Aσ zσ(1) . . . zσ(N ) σ∈S(n) where ∆(t1 , . . . , tm+1 ) = {0 = t0 < t1 < t2 < . . . < tm+1 = t}. Note that exp(−N (tk − tk−1 )) term inside the integral above accounts for the probability that N independent clocks associated with N particles do not ring in the interval (tk−1 , tk ). Once any of the N clock rings, it P (1−b)zi results in a factor of N i=1 (1−bzi ) at the front. Thus, m many clock counts add upto m such factors. Using (B.5), one can further deduce   X X (N ) y1 yN  Aσ zσ(1) . . . zσ(N Tt ((x1 , . . . , xN ) → (y1 , . . . , yN ))  ) σ∈S(N ) (y1 ,...,yn ) = ∞ X X m=0 (y1 ,...,yn ) = ∞ X m=0  Tm(0,t) ((x1 , . . . , xN ) → (y1 , . . . , yN ))  1 exp(−N t) m! X σ∈S(N ) t N X (1 − b)zi i=1 (1 − bzi ) !m   X σ∈S(n)  y1 yN  Aσ zσ(1) . . . zσ(N )  xN  x1 Aσ zσ(1) . . . zσ(N )  ! N X 1 − zi  X xN  x1 = exp − Aσ zσ(1) . . . zσ(N . ) 1 − bzi i=1 σ∈S(n) This completes the proof. Appendix C: Moment Formula Proof of Proposition 3.4: In what follows, we first find out the moment formula with the initially N many particles. First, we claim that under step Bernoulli initial condition the probability of transition of m-th particle to x by time t in HL-PushTASEP is given as X X κ(S,Z>0 )−mk−k(k−1)/2 Sk k − 1 m−1 m(m−1)/2 b ρ Pstepb (xm (t) = x; t) = (−1) b m−1 b m≤k≤N \|S\|=k !Si −Si−1 I I k Y Y zj − zi 1 1 × ... (2πi)k 1 − (1 + b−1 )zi + b−1 zi zj ) 1 − (1 − ρ)zi−1 · · · zk−1 i=1 1≤i<j≤k Q k 1 − zi−1 1 − ki=1 zi Y x−Si −1 zi exp −t dzi , (C.1) × Qk 1 − bzi−1 i=1 (1 − zi ) i=1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 67 where S = {S1 , . . . , Sk } with 1 ≤ S1 < S2 < . . . < Sk ≤ N and κ(S, Z>0 ) counts the sum of the elements in S. One can note that when HL-PushTASEP is started with particles at N ordered places (y1 , . . . , yn ) (call it Y) in Z≥0 , we can have an explicit expression of P~y (xm (t) = x; t) given in Theorem 3.3. To put all the ingredients together, let us just recall the formula of P~y (xm (t) = x; t) given there. For a fixed N ≥ 0, when the initial data is ~y = (y1 , . . . , yN ), then X X κ(S,Z>0 )−mk−k(k−1)/2 k − 1 m−1 m(m−1)/2 b P~y (xm (t) = x; t) = (−1) b m−1 b m≤k≤N \|S\|=k I I Y zj − zi 1 . . . × (2πi)k 1 − (1 + b−1 )zi + b−1 zi zj ) i,j∈S,i<j Q Y 1 − i∈S zi 1 − zi−1 x−yi −1 ×Q exp −t zi dzi , (C.2) 1 − bzi−1 i∈S (1 − zi ) i∈S where κ(S, Z>0 ) is the sum of the element in S. The term which relates initial data with P~y (xm (t) = x; t) in (C.2), is zi−yi for i ∈ S. To obtain the result in case of step initial condition, we have to sum over all ~y by multiplying P~y (xm (t) = x; t) with appropriate probability. Let us introduce k-many new variables t1 , . . . , tk which are defined as t1 := S1 − 1, t2 := S1 − S2 − 1, . . . , tk := Sk − Sk−1 − 1. P To this end, we can write down Si in terms of ti by Si = ij=1 ti + i. So, initially, we would P have the configuration yS1 = S1 + ii , . . . , ySk = Sk + kl=1 il with probability Sk ρ k Y tl + il tl l=1 (1 − ρ)il . One can note that k k Pj Y X Y tj + ij −S ij −Sj + l=1 il zSj j = (1 − ρ) zSj tj j=1 i1 ,...,ik ≥0 j=1 1 · · · zS−1 1 − (1 − ρ)zS−1 j k !tj +1 .(C.3) Q Q Q −S Now, kj=1 zSj j in (C.3) counts for the replacement of i∈S zix−yi −1 in (C.2) with ki=1 zix−Si −1 in (C.1), whereas extra factor in second line of (C.1) comes from the second term at the end of the right side in (C.3). Thus, the claim is proved. Now, we turn to showing the moment formula. Given any L ∈ N, we multiply (C.1) by bmL and sum over all 1 ≤ m ≤ N . Let us note down the following result (see, [AAR99, Theorem 10.2.1]) known as q- binomial Theorem. For any q, t ∈ C such that \|q\| = 6 1 and any positive integer n, we have n n−1 X Y i k(k−1)/2 n (1 + q t) = q tk . (C.4) k q i=0 k=0 Thus, using (C.4), one can write k X m=1 m−1 m(m−1)/−mk+mL (−1) b k−2 Y k−1 (1 − b1−k+L .b). = bL−k m−1 b i=0 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 68 To that effect, we get min(L,N X ) Estepb bLNx (t) ηx (t) = k=1 1 × (2πi)k 1− × Qk I ... Qk i=1 zi i=1 (1 I Y 1≤i<j≤k k Y − zi ) i=1 k−2 Y i=0 (1 − b 1−k+L ! .b) X bκ(S,Z>0 )−k(k−1)/2+L−k ρSk \|S\|=k k Y zj − zi 1 − (1 + b−1 )zi + b−1 zi zj 1 1 − (1 − ρ)zi−1 · · · zk−1 i=1 zix−Si −1 exp −t 1 − zi−1 1 − bzi−1 !Si −Si−1 dzi , (C.5) where S = {S1 , . . . , Sk } with 1 ≤ S1 < S2 < . . . < Sk ≤ N and κ(S, Z>0 ) counts the sum of the elements in S. Letting N → ∞, we sum (C.5) over all possible k ordered subsets 1 ≤ S1 < S2 < . . . < Sk of various sizes. This infinite sum can be simplified in the following way X b k Y S1 +...+Sk Sk ρ zi−Si i=1 1≤S1 <S2 <...<Sk 1 1 − (1 − ρ)zi−1 · · · zk−1 = bk(k+1)/2 ρk !Si −Si−1 = k Y i=1 1 ρ(b/zi )···(b/zk ) 1−(1−ρ)zi−1 ···zk−1 ρ(b/zi )···(b/zk ) − 1−(1−ρ)z −1 −1 i ···zk 1 . zi · · · zk − (1 − ρ) − bk−i+1 ρ In what follows, we use a symmetrization identity from [TW09a, Section III] to conclude that ! L k−2 X Y b−k(k−1)/2+L−k ρk LNx (t) 1−k+L i Estepb b ηx (t) = (1 − b b) (b−1 ; b−1 )k k=1 i=1 ! I I k Y Y zj − zi 1 × zi ... 1− (2πi)k 1 − (1 + b−1 )zi + b−1 zi zj i=1 1≤i<j≤k × k Y 1 − zi−1 zix (b−1 − 1) exp −t (ρ + (1 − ρ)b−1 − zi b−1 )(1 − zi ) 1 − bzi−1 i=1 dzi . 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Image Captioning using Deep Neural Architectures Parth Shah (orcid.org/0000-0002-7880-2228), Vishvajit Bakarola (orcid.org/0000-0002-6602-9194), and Supriya Pati (orcid.org/0000-0001-5528-6104) arXiv:1801.05568v1 [cs.CV] 17 Jan 2018 Chhotubhai Gopalbhai Patel Institute of Technology, Bardoli, India parthpunita@yahoo.in vishvajit.bakrola@utu.ac.in supriya.pati@utu.ac.in Abstract. Automatically creating the description of an image using any natural languages sentence like English is a very challenging task. It requires expertise of both image processing as well as natural language processing. This paper discuss about different available models for image captioning task. We have also discussed about how the advancement in the task of object recognition and machine translation has greatly improved the performance of image captioning model in recent years. In addition to that we have discussed how this model can be implemented. In the end, we have also evaluated the performance of model using standard evaluation matrices. Keywords: Deep Learning, Deep Neural Network, Image Captioning, Object Recognition, Machine Translation, Natural Language Processing, Natural Language Generation 1 Introduction A single image can contain large amount of information in it. Humans have ability to parse this large amount of information by single glance of it. Humans normally communicate though written or spoken language. They can use languages for describing any image. Every individual will generate different caption for same image. If we can achieve same task with machine it will be greatly helpful for variety of tasks. However, generating captions for an image is very challenging task for machine. In order to perform caption generation task by machine, it requires brief understanding of natural language processing and ability to identify and relate objects in an image. Some of the early approaches that tried to solve these challenge are often based on hard-coded features and well defined syntax. This limits the type of sentence that can be generated by any given model. In order to overcome this limitation the main challenge is to make model free of any hard-coded feature or sentence templates. Rule for forming models should be learned from the training data. Another challenge is that there are large number of images available with their associated text available in the ever expanding internet. However, most of them are noisy hence it can not be directly used in image captioning model. Training an image captioning model requires huge dataset with properly available annotated image by multiple persons. In this paper, we have studied collections of different existing natural image captioning models and how they compose new caption for unseen images. We have also presented results of our implementation of these model and compared them. Section 2 of this paper describes Related Work in detail. Show & Tell model in detailed is described in Section 3. Section 4 contains details about implementation environment and dataset. Results and Discussion is provided in detail in Section 5. At the end we provided our concluding remarks in section 6. 2 Related work Creating captioning system that accurately generate captions like human depends on the connection between importance of object in image and how they will be related to other objects in image. Image can be described using more than one sentence but to efficiently train the image captioning model we requires only single sentence that can be provided as a caption. This leads to problem of text summarization in natural language processing. There are mainly two different way to perform the task of image captioning. These two types are basically retrieval based method and generative method. From that most of work is done based on retrieval based method. One of the best model of retrieval based method is Im2Txt model [1]. It was proposed by Vicente Ordonez, Girish Kulkarni and Tamara L Berg. Their system is divided into mainly two part 1) Image matching and 2) Caption generation. First we will provide our input image to model. Matching image will be retrieved from database containing images and its appropriate caption. Once we find matching images we will compare extracted high level objects from original image and matching images. Images will then reranked based on the content matched. Once it is reranked caption of top-n ranked images will be returned. The main limitation of these retrieval based method is that it can only produce captions which are already present in database. It can not generate novel captions. This limitation of retrieval based method is solved in generative models. Using generative models we can create novel sentences. Generative models can be of two types either pipeline based model or end to end model. Pipeline type models uses two separate learning process, one for language modeling and and one for image recognition. They first identify objects in image and provides the result of it to language modeling task. While in end-to-end models we combine both language modeling and image recognition models in single end to end model [2]. Both part of model learn at the same time in end-to-end system. They are typically created by combination of convolutional and recurrent neural networks. Show & Tell model proposed by Vinyals et al. is of generative type end-toend model. Show & Tell model uses recent advancement in image recognition and neural machine translation for image captioning task. It uses combination of Inception-v3 model and LSTM cells [3]. Here Inception-v3 model will provides object recognition capability while LSTM cell provides it language modeling capability [4][5]. 3 Show & Tell Model Recurrent neural networks generally used in neural machine translation [6]. They encodes the variable length inputs into a fixed dimensional vectors. Then it uses these vector representation to decode to the desired output sequence [7][8]. Instead of using text as input to encoder Show & Tell model uses image as input. This image is then converted to word vector and then this word vector is translated to caption using Recurrent neural networks as decoder. To achieve this goal, Show & Tell model is created by hybridizing two different models. It takes input as the image and provides it to Inception-v3 model. At the end of Inception-v3 model single fully connected layer is added. This layer will transform output of Inception-v3 model into word embedding vector. We input this word embedding vector into series of LSTM cell. LSTM cell provides ability to store and retrieve sequential information through time. This helps to generate the sentences with keeping previous words in context. Training of Show & Tell Fig. 1: Architecture of Show & Tell Model model can be divided into two part. First part is of training process where model learns its parameters. While second part is of testing process. In testing process we infer the captions and we compare and evaluate these machine generated caption with human generated captions. 3.1 Training During training phase we provides pair of input image and its appropriate caption to Show & Tell model. Inception-v3 part of model is trained to identify all possible objects in an image. While LSTM part of model is trained to predict every word in the sentence after it has seen image as well as all previous words. For any given caption we add two additional symbols as start word and stop word. Whenever stop word is encountered it stop generating sentence and it marks end of string. Loss function for model is calculated as L(I, S) = − N X log pt (St ) . (1) t=1 where I represent input image and S represent generated caption. N is length of generated sentence. pt and St represent probability and predicted word at the time t respectively. During the process of training we will try to minimize this loss function. 3.2 Inference From various approaches to generate caption a sentence from given image Show & Tell model uses Beam Search to find suitable words to generate caption. If we keep beam size as K, it recursively consider K best word at each output of the word. At each step it will calculate joint probability of word with all previously generated word in sequence. It will keep producing the output until end of sentence marker is predicted. It will select sentence with best probability and outputs it as caption. 4 Implementation For evaluation of image captioning model we have implemented Show & Tell model. Details about dataset,implementation tool and implementation environment is given as follows: 4.1 Datasets For task of image captioning there are several annotated images dataset are available. Most common of them are Pascal VOC dataset and MSCOCO Dataset. In this work MSCOCO image captioning dataset is used. MSCOCO is a dataset developed by Microsoft with the goal of achieving the state-of-the-art in object recognition and captioning task. This dataset contains collection of day-to-day activity with theri related captions. First each object in image is labeled and after that description is added based on objects in an image. MSCOCO dataset contains image of around 91 objects types that can be easily recognizable by even a 4 year old kid. It contains around 2.5 million objects in 328K images. Dataset is created by using crowdsourcing by thousonds of humans [9]. 4.2 Implementation tool and environment For the implementation of this experiment we have used machine with Intel Xeon E3 processor with 12 cores and 32GB RAM running CentOS 7. Tensorflow liberary is used for creating and training deep neural networks. Tensorflow is a deep learning library developed by Google[10]. It provides heterogeneous platform for execution of algorithms i.e. it can be run on low power devices like mobile as well as large scale distributed system containing thousands of GPUs. To define structure of our network tensorflow uses graph definition. Once graph is defined it can be executed on any supported devices. 5 5.1 Results and Discussion Results By the implementation of the Show & Tell model we can able to generate moderately comparable captions with compared to human generated captions. First of all it model will identify all possible objects in image. Fig. 2: Generated word vector from Sample Image As shown in Fig. 2 Inception-v3 model will assign probability of all possible object in image and convert image into word vector. This word vector is provided as input to LSTM cells which will then form sentence from this word vector as shown in Fig. 3 using beam search as described in previous section. Fig. 3: Generated caption from word vector for Sample Image 5.2 Evaluation Matrices To evaluating of any model that generate natural language sentence BLEU (Bilingual Evaluation Understudy) Score is used. It describes how natural sentence is compared to human generated sentence [11]. It is widely used to evaluate performance of Machine translation. Sentences are compared based on modified n-gram precision method for generating BLEU score [12]. Where precision is calculated using following equation: P C∈{Candidates} pn = P P C 0 ∈{Candidates} ngram∈C P Countclip (ngram) ngram0 ∈C 0 Count(ngram0 ) (2) To evaluate our model we have used image from validation dataset of MSCOCO Dataset. Some of captions generated by Show & Tell model is shown as follows: (a) Input image (b) Generated caption Fig. 4: Experiment Result As you can see in Fig. 4, generated sentence is “a woman sitting at a table with a plate of food.”, while actual human generated sentence are “The young woman is seated at the table for lunch, holding a hotdog.”, “a woman is eatting a hotdog at a wooden table.”, “there is a woman holding food at a table.”, “a young woman holding a sandwich at a table.” and “a woman that is sitting down holding a hotdog.”. This result in BLEU score of 63 for this image. (a) Input image (b) Generated caption Fig. 5: Experiment Result Similarly in Fig. 5, generated sentence is “a woman holding a cell phone in her hand.” while actual human generated sentence are “a woman holding a Hello Kitty phone on her hands”, “a woman holds up her phone in front of her face”, “a woman in white shirt holding up a cellphone”, “a woman checking her cell phone with a hello kitty case” and “the asian girl is holding her miss kitty phone”. This result in BLEU score of 77 for this image. While calculating BLEU score of all image in validation dataset we get average score of 65.5. Which shows that our generated sentence are very similar compared to human generated sentence. 6 Conclusion We can conclude from our findings that we can combine recent advancement in Image Labeling and Automatic Machine Translation into an end-to-end hybrid neural network system. This system is capable to autonomously view an image and generate a reasonable description in natural language with better accuracy and naturalness. References 1. V. Ordonez, G. Kulkarni, and T. L. Berg, “Im2text: Describing images using 1 million captioned photographs,” in Advances in Neural Information Processing Systems, pp. 1143–1151, 2011. 2. A. Karpathy and L. Fei-Fei, “Deep visual-semantic alignments for generating image descriptions,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3128–3137, 2015. 3. O. Vinyals, A. Toshev, S. Bengio, and D. Erhan, “Show and tell: Lessons learned from the 2015 mscoco image captioning challenge,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PP, no. 99, pp. 1–1, 2016. 4. C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–9, 2015. 28. 5. C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” arXiv preprint arXiv:1512.00567, 2015. 37. 6. D. Britz, “Introduction to rnns.” WILDML, http://www.wildml.com/, 2016. [Accessed 4-September-2016]. 7. Y. Wu, M. Schuster, Z. Chen, Q. V. Le, M. Norouzi, W. Macherey, M. Krikun, Y. Cao, Q. Gao, K. Macherey, J. Klingner, A. Shah, M. Johnson, X. Liu, L. Kaiser, S. Gouws, Y. Kato, T. Kudo, H. Kazawa, K. Stevens, G. Kurian, N. Patil, W. Wang, C. Young, J. Smith, J. Riesa, A. Rudnick, O. Vinyals, G. Corrado, M. Hughes, and J. Dean, “Google’s neural machine translation system: Bridging the gap between human and machine translation,” CoRR, vol. abs/1609.08144, 2016. 8. I. Sutskever, O. Vinyals, and Q. V. Le, “Sequence to sequence learning with neural networks,” in Advances in neural information processing systems, pp. 3104–3112, 2014. 9. T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick, “Microsoft coco: Common objects in context,” in European Conference on Computer Vision, pp. 740–755, Springer, 2014. 10. M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, et al., “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv preprint arXiv:1603.04467, 2016. 11. K. Papineni, S. Roukos, T. Ward, and W.-J. Zhu, “Bleu: a method for automatic evaluation of machine translation,” in Proceedings of the 40th annual meeting on association for computational linguistics, pp. 311–318, Association for Computational Linguistics, 2002. 12. D. Jurafsky, Speech & language processing. Pearson Education India, 2000.	1
Learning-based Model Predictive Control for Safe Exploration and Reinforcement Learning arXiv:1803.08287v1 [cs.SY] 22 Mar 2018 Torsten Koller, Felix Berkenkamp, Matteo Turchetta and Andreas Krause Abstract— Learning-based methods have been successful in solving complex control tasks without significant prior knowledge about the system. However, these methods typically do not provide any safety guarantees, which prevents their use in safety-critical, real-world applications. In this paper, we present a learning-based model predictive control scheme that provides provable high-probability safety guarantees. To this end, we exploit regularity assumptions on the dynamics in terms of a Gaussian process prior to construct provably accurate confidence intervals on predicted trajectories. Unlike previous approaches, we do not assume that model uncertainties are independent. Based on these predictions, we guarantee that trajectories satisfy safety constraints. Moreover, we use a terminal set constraint to recursively guarantee the existence of safe control actions at every iteration. In our experiments, we show that the resulting algorithm can be used to safely and efficiently explore and learn about dynamic systems. I. I NTRODUCTION In model-based reinforcement learning (RL, [1]), we aim to learn the dynamics of an unknown system from data, and based on the model, derive a policy that optimizes the long-term behavior of the system. Crucial to the success of such methods is the ability to efficiently explore the state space in order to quickly improve our knowledge about the system. While empirically successful, current approaches often use exploratory actions during learning, which lead to unpredictable and possibly unsafe behavior of the system, e.g. in exploration approaches based on the optimism in the face of uncertainty principle [2]. Applying such approaches to the exploration of a real-world safety-critical system, such as an autonomous car, is undesirable. In this paper we introduce S AFE MPC, a safe model predictive control (MPC) scheme that guarantees the existence of feasible return trajectories to a safe region of the state space at every time step with high-probability. These return trajectories are identified through a novel uncertainty propagation method that, in combination with constrained MPC, allows for formal safety guarantees in learning control. Related Work: One area that has considered safety guarantees is robust MPC. There, we iteratively optimize the performance along finite-length trajectories at each time step, based on a known model that incorporates uncertainties and disturbances acting on the system [3]. In a constrained robust This work was supported by SNSF grant 200020 159557, a fellowship within the FITweltweit program of the German Academic Exchange Service (DAAD), and the Max Planck ETH Center for Learning Systems Torsten Koller is with the Department of Computer Science, University of Freiburg, Germany. Email: torsten.koller@merkur.uni-freiburg.de Felix Berkenkamp, Matteo Turchetta and Andreas Krause are with the Learning & Adaptive Systems Group, Department of Computer Science, ETH Zurich, Switzerland. Email: {befelix, matteotu, krausea}@inf.ethz.ch Fig. 1. Propagation of uncertainty over multiple time steps based on a wellcalibrated statistical model of the unknown system. We iteratively compute ellipsoidal over-approximations (purple) of the intractable image (green) of the learned model for uncertain ellipsoidal inputs. MPC setting, we optimize these local trajectories under additional state and control constraints. Safety is typically defined in terms of recursive feasibility and robust constraint satisfaction. In [4], this definition is used to safely control urban traffic flow, while [5] guarantees safety by switching between a standard and a safety mode. However, these methods are conservative since they do not update the model. In contrast, learning-based MPC approaches (LBMPC, [6]) adapt their models online based on observations of the system. This allows the controller to improve over time, given limited prior knowledge of the system. Theoretical safety guarantees in LBMPC are established in [6], which enforces robust feasibility and constraint satisfaction with learned statistical models. However, the model errors are assumed to be deterministically bounded in a polytope and it is not clear how to obtain this outer bound in the first place. MPC based on Gaussian process (GP, [7]) models is proposed in a number of works, e.g. [8], [9]. The difficulty here is that trajectories have complex dependencies on states and unbounded stochastic uncertainties. Safety through probabilistic chance constraints is considered in [10] and [11] based on approximate uncertainty propagation. While often being empirically successful, these approaches do not theoretically guarantee safety of the underlying system. Another area that has considered learning for control is model-based RL. There, we aim to learn global policies based on data-driven modeling techniques, e.g., by explicitly trading-off between finding locally optimal closed-loop policies (exploitation) and learning the behavior of the system globally through exploration [1]. This results in data-efficient learning of policies in unknown systems [12]. In contrast to MPC, where we optimize finite-length trajectories, in RL we typically aim to find an infinite horizon optimal policy. Hence, enforcing hard constraints in RL is challenging. Control-theoretic safety properties such as Lyapunov stability or robust constraint satisfaction are only considered in a few works [13]. In [14], safety is guaranteed by optimizing parametric policies under stability constraints, while [15] guarantees safety in terms of constraint satisfaction through reachability analysis. Our Contribution: We combine ideas from robust control and GP-based RL to design a MPC scheme that recursively guarantees the existence of a safety trajectory that satisfies the constraints of the system. In contrast to previous approaches, we use a novel uncertainty propagation technique that can reliably propagate the confidence intervals of a GP-model forward in time. We use results from statistical learning theory to guarantee that these trajectories contain the system with high probability jointly for all time steps. In combination with a constrained MPC approach with a terminal set constraint, we then prove the safety of the system. We apply the algorithm to safely explore the dynamics of an inverted pendulum simulation. II. P ROBLEM S TATEMENT We consider a nonlinear, discrete-time dynamical system xt+1 = f (xt , ut ) = h(xt , ut ) + g(xt , ut ) , \| {z } \| {z } prior model (1) unknown error where xt ∈ Rnx is the state and ut ∈ Rnu is the control input to the system at time step t ∈ N. We assume that we have access to a twice continuously differentiable prior model h(xt , ut ), which could be based on a first principles physics model. The model error g(xt , ut ) is a priori unknown and we use a statistical model to learn it by collecting observations from the system during operation. In order to provide guarantees, we need reliable estimates of the modelerror. In general, this is impossible for arbitrary functions g. We make the following additional regularity assumptions. assume that the model-error g is of the form g(z) = PWe ∞ nx α × Rn u , a i=0 i k(z, zi ), αi ∈ R, z = (x, u) ∈ R weighted sum of distances between inputs z and representer points zi = (xi , ui ) ∈ Rnx × Rnu as defined through a symmetric, positive definite kernel k. This class of functions is well-behaved in the sense that they form a reproducing kernel Hilbert space (RKHS, [16]) Hk equipped with an inner-product h·, ·ik . The induced norm \|\|g\|\|2k = hg, gik is a measure of the complexity of a function g ∈ Hk . Consequently, the following assumption can be interpreted as a requirement on the smoothness of the model-error g w.r.t the kernel k. Assumption 1 The unknown function g has bounded norm in the RKHS Hk , induced by the continuously differentiable kernel k, i.e. \|\|g\|\|k ≤ Bg . In the case of a multi-dimensional output nx > 1, we follow [17] and redefine g as a single-output function g̃ such that g̃(·, i) = gi (·) and assume that \|\|g̃\|\|k ≤ Bg . We further assume that the system is subject to polytopic state and control constraints X = {x ∈ Rnx \|Hx x ≤ hx , hx ∈ Rmx }, U = {u ∈ R nu u \|Hu u ≤ h , hu ∈ R mu }, (2) (3) which are bounded. For example, in an autonomous driving scenario, the state region could correspond to a highway lane and the control constraints could represent the physical limits on acceleration and steering angle of the car. Lastly, we assume access to a backup controller that guarantees that we remain inside a given safe subset of the state space once we enter it. In the autonomous driving example, this could be a simple linear controller that stabilizes the car in a small region in the center of the lane at slow speeds. Assumption 2 We are given a controller πsafe (·) and a polytopic safe region Xsafe := {x ∈ Rnx \|Hs x ≤ hs } ⊂ X , (4) which is (robust) control positive invariant (RCPI) under πsafe (·). Moreover, the controller satisfies the control constraints inside Xsafe , i.e. πsafe (x) ∈ U ∀x ∈ Xsafe . This assumption allows us to gather initial data from the system inside the safe region even in the presence of significant model errors, since the system remains safe under the controller πsafe . Moreover, we can still guarantee constraint satisfaction asymptotically outside of Xsafe , if we can show that a finite sequence of control inputs eventually steers the system back to the safe set Xsafe . In principle, it is sufficient that we have an arbitrary convex and bounded RCPI set XRCPI ⊂ X . This set can be inner-approximated by the safety polytope Xsafe to arbitrary precision [18]. Given a controller π, ideally we want to enforce the stateand control constraints at every time step, ∀t ∈ N : fπ (xt ) ∈ X , π(xt ) ∈ U, (5) where xt+1 = fπ (xt ) = f (xt , π(xt )) denotes the closedloop system under π. However, due to limited knowledge of the system, it is in general impossible to design a controller that enforces (5) without additional assumptions. Instead, we slightly relax this requirement to safety with high probability. Definition 1 Let π : Rnx → Rnu be a controller for (1) with the corresponding closed-loop system fπ . Let x0 ∈ X and δ ∈ (0, 1]. A system is δ−safe under the controller π iff: Pr [ ∀t ∈ N : fπ (xt ) ∈ X , π(xt ) ∈ U] ≥ 1 − δ. (6) Based on Definition 1, the goal is to design a control scheme that guarantees δ-safety of the system (1). At the same time, we want to improve our model by learning from observations during operation, which increase the performance of the controller over time. III. BACKGROUND In this section, we introduce the necessary background on GPs and set-theoretic properties of ellipsoids that we need to model our system and perform multi-step ahead predictions. A. Gaussian Processes (GPs) We want to learn the unknown model-error g from data using a GP model. A GP(m, k) is a distribution over functions, which is fully specified through a mean function m : Rd → R and a covariance function k : Rd × Rd → R, where d = nx + nu . Given a set of noisy observations yi = f (zi ) + wi , wi ∼ N (0, λ2 ), i = 1, . . . , n, we choose a zero-mean prior on g as m ≡ 0 and regard the differences ỹn = [y1 − h(z1 ), . . . , yn − h(zn )]T between prior model h and observed system response at input locations Z = [z1 , .., zn ]T . The posterior distribution at z is then given as a Gaussian N (µn (z), σn2 (z)) with mean and variance µn (z) = kn (z)T [Kn + λ2 In ]−1 ỹn σn2 (z) T (7) 2 −1 = k(z, z) − kn (z) [Kn + λ In ] kn (z), (8) where [Kn ]ij = k(zi , zj ), [kn (z)]j = k(z, zj ), and In is the n−dimensional identity matrix. In the case of multiple outputs nx > 1, we model each output dimension with an independent GP, GP(mj , kj ), j = 1, .., nx . We then redefine (7) and (8) as µn (·) = (µn,1 (·), .., µn,nx (·)) and σn (·) = (σn,1 (·), .., σn,nx (·)) corresponding to the predictive mean and variance functions of the individual models. Based on Assumption 1, we can use GPs to model the unknown part of the system (1), which provides us with reliable confidence intervals on the model-error g. Lemma 1 [14, Lemma 2]: Assume \|\|g\|\|k ≤ Bg and that measurements are pcorrupted by λ-sub-Gaussian noise. Let βn = Bg + 4λ γn + 1 + ln(1/δ), where γn is the information capacity associated with the kernel k. Then with probability at least 1−δ we have for all i = 1 ≤ i ≤ nx , z ∈ X × U that \|µn−1,i (z) − gi (z)\| ≤ βn · σn−1,i (z). In combination with the prior model h(z), this allows us to construct reliable confidence intervals around the true dynamics of the system (1). The scaling βn depends on the number of data points n that we gather from the system through the information capacity, γn = maxA⊂Z̃,\|A\|=ñ I(g̃A ; g), Z̃ = X × U × I, ñ = n · nx , i.e. the maximum mutual information between a finite set of samples A and the function g. Exact evaluation of γn is in general NPhard, but γn can be greedily approximated and has sublinear dependence on n for many commonly used kernels [19]. The regularity assumption 1 on our model-error and the smoothness assumption on the covariance function k additionally imply that the function g is Lipschitz. Lemma 2 [20, Lemma 2]: Let g have bounded RKHS norm \|\|g\|\|k ≤ Bg induced by a continuously differentiable kernel k. Then g is Lg -Lipschitz continuous. B. Ellipsoids We use ellipsoids to give an outer bound on the uncertainty of our system when making multi-step ahead predictions. Due to appealing geometric properties, ellipsoids are widely used in the robust control community to compute reachable sets [21], [22]. These sets intuitively provide an outer approximation on the next state of a system considering all possible realizations of uncertainties when applying a controller to the system at a given set-valued input. We briefly review some of these properties and refer to [23] for an exhaustive introduction to ellipsoids and to the derivations for the following properties. We use the basic definition of an ellipsoid, E(p, Q) := {x ∈ Rn \|(x − p)T Q−1 (x − p) ≤ 1}, (9) with center p ∈ R and a symmetric positive definite (s.p.d) shape matrix Q ∈ Rn×n . Ellipsoids are invariant under affine subspace transformations such that for A ∈ Rn×r , r ≤ n with full column rank and b ∈ Rr , we have that n A · E(p, Q) + b = E(p + b, AQAT ). (10) The Minkowski sum E(p1 , Q1 ) ⊕ E(p2 , Q2 ), i.e. the pointwise sum between two arbitrary ellipsoids, is in general not an ellipsoids anymore, but we have that E(p1 , Q1 )⊕E(p2 , Q2 ) ⊂ E p1 +p2 , (1+c−1 )Q1 +(1+c)Q2 (11) for all c > 0. Moreover, the minimizer of the trace of the resulting shape matrix is analytically given as c = p T r(Q1 )/T r(Q2 ). A particular problem that we encounter is finding the maximum distance r to the center of an ellipsoid E := E(0, Q) under a special transformation, i.e. r(Q, S) = max \|\|S(x − p)\|\|2 , x∈E(p,Q) (12) where S ∈ Rm×n with full column rank. We show in the appendix that this is a generalized eigenvalue problem of the pair (Q, S T S) and the optimizer is given as the square-root of the largest generalized eigenvalue. IV. S AFE M ODEL P REDICTIVE C ONTROL In this section, we use the assumptions in Sec. II to design a control scheme that fulfills our safety requirements in Definition 1. We construct reliable, multi-step ahead predictions based on our GP model and use MPC to actively optimize over these predicted trajectories under safety constraints. Using Assumption 2, we use a terminal set constraint to theoretically prove the safety of our method. A. Multi-step Ahead Predictions From Lemma 1 and our prior model h(xt , ut ), we directly obtain high-probability confidence intervals on f (xt , ut ) uniformly for all t ∈ N. We extend this to over-approximate the system after a sequence of inputs (ut , ut+1 , ..). The result is a sequence of set-valued confidence regions that contain the true dynamics of the system with high probability. a) One-step ahead predictions: We compute an ellipsoidal confidence region that contains the next state of the system with high probability when applying a control input, given that the current state is contained in an ellipsoid. In order to approximate the system, we linearize our prior model h(xt , ut ) and use the affine transformation property (10) to compute the ellipsoidal next state of the linearized model. Next, we approximate the unknown model-error g(xt , ut ) using the confidence intervals of our GP model. We finally input u ∈ Rnu , by over-approximating the output of the system f (R, u) = {f (x, u)\|x ∈ R} for ellipsoidal inputs R. Here, we choose p as the linearization center of the state and choose ū = u, i.e. z̄ = (p, u). Since the function f˜µ is affine, we can make use of (10) to compute Fig. 2. Decomposition of the over-approximated image of the system (1) under an ellipsoidal input R0 . The exact, unknown image of f (right, green area) is approximated by the linearized model f˜µ (center, top) ˜ which accounts for the confidence interval and the remainder term d, and the linearization errors of the approximation (center, bottom). The resulting ellipsoid R1 is given by the Minkowski sum of the two individual approximations. f˜µ (R, u) = E(h(z̄) + µ(z̄), AQAT ), resulting again in an ellipsoid. This is visualized in Fig. 2 by the upper ellipsoid in the center. To upper-bound the confidence hyper-rectangle on the right hand side of (21), we upper-bound the term kz − z̄k2 by l(R, u) := apply Lipschitz arguments to outer-bound the approximation errors. We sum up these individual approximations, which result in an ellipsoidal approximation of the next state of the system. This is illustrated in Fig. 2. We formally derive the necessary equations in the following paragraphs. The reader may choose to skip the technical details of these approximations, which result in Lemma 3. We first regard the system f in (1) for a single input vector z = (x, u), f (z) = h(z) + g(z). We linearly approximate f around z̄ = (x̄, ū) via f (z) ≈ h(z̄) + Jh (z̄)(z − z̄) + g(z̄) = f˜(z), (13) where Jh (z̄) = [A, B] is the Jacobian of h at z̄. Next, we use the Lagrangian remainder theorem [24] on the linearization of h and apply a continuity argument on our locally constant approximation of g. This results in an upper-bound on the approximation error, L∇h,i \|\|z − z̄\|\|22 + Lg \|\|z − z̄\|\|2 , (14) \|fi (z) − f˜i (z)\| ≤ 2 where fi (z) is the ith component of f , 1 ≤ i ≤ nx , L∇h,i is the Lipschitz constant of the gradient ∇hi , and Lg is the Lipschitz constant of g, which is defined through Lemma 2. The function f˜ depends on the unknown model error g. We approximate g with the statistical GP model, µ(z̄) ≈ g(z̄). From Lemma 1 we have \|gi (z̄) − µn,i (z̄)\| ≤ βn σn,i (z̄), 1 ≤ i ≤ nx , (15) with high probability. We combine (14) and (15) to obtain L∇h,i \|fi (z) − f˜µ(z),i \| ≤ βn σi (z̄) + \|\|z − z̄\|\|22 + Lg \|\|z − z̄\|\|2 , 2 (16) where 1 ≤ i ≤ nx and f˜µ (z) = h(z̄t ) + Jh (z̄t )(z − z¯t ) + µn (z̄). (17) We can interpret (16) as the edges of the confidence hyperrectangle L∇h m̃(z) = f˜µ (z) ± [βn σn−1 (z̄) + \|\|z − z̄\|\|22 + Lg \|\|z − z̄\|\|2 ], 2 (18) where L∇h = [L∇h,1 , .., L∇h,nx ] and we use the shorthand notation a ± b := [a1 ± b1 ] × [anx ± bnx ], a, b ∈ Rnx . We are now ready to compute a confidence region based on an ellipsoidal state R = E(p, Q) ⊂ Rnx and a fixed (19) max z(x)=(x,u), x∈R \|\|z(x) − z̄\|\|2 , (20) which leads to ˜ u) = βn σn−1 (z̄) + L∇h l2 (R, u)/2 + Lg l(R, u). (21) d(R, Due to our choice of z, z̄, we have that \|\|z(x) − z̄\|\|2 = \|\|x − p\|\|2 and we can use (12) to get l(R, u) = r(Q, Inx ), which corresponds to the largest eigenvalue of Q−1 . Using (20), we can now over-approximate the right side of (21) for inputs R by an ellipsoid ˜ u) ⊂ E(0, Q ˜(R, u)), 0 ± d(R, d (22) where we obtain Qd˜(R, u) by over-approximating the hyper˜ u) with the ellipsoid E(0, Q ˜(R, u)) through rectangle d(R, d √ a ± b ⊂ E(a, nx · diag([b1 , .., bnx ])), ∀a, b ∈ Rnx . This is illustrated in Fig. 2 by the lower ellipsoid in the center. Combining the previous results, we can compute the final over-approximation using (11), R+ = m̃(R, u) = f˜µ (R, u) ⊕ E(0, Qd˜(R, u)). (23) Since we carefully incorporated all approximation errors and extended the confidence intervals around our model predictions to set-valued inputs, we get the following generalization of Lemma 1. Lemma 3 Let δ ∈ (0, 1] and choose βn as in Lemma 1. Then, with probability greater than 1 − δ, we have that: ∀x ∈ R : f (x, u) ∈ m̃(R, u), (24) uniformly for all R = E(p, Q) ⊂ X , u ∈ U. Proof: Define m(x, u) = h(x, u)+µn (x, u)±βn σn−1 + n(x, u). From Lemma 1 we have ∀SR ⊂ X , u ∈ U that, with S high probability, x∈R f (x, u)S⊂ x∈R m(x, u). Due to the over-approximations, we have x∈R m(x, u) ⊂ m̃(R, u). Lemma 3 allows us to compute confidence ellipsoid around the next state of the system, given that the current state of the system is given through an ellipsoidal belief. b) Multi-step ahead predictions: We now use the previous results to compute a sequence of ellipsoids that contain a trajectory of the system with high-probability, by iteratively applying the one-step ahead predictions (23). Given an initial ellipsoid R0 ⊂ Rnx and control input ut ∈ U, we iteratively compute confidence ellipsoids as Rt+1 = m̃(Rt , ut ). (25) We can directly apply Lemma 3 to get the following result. Corollary 1 Let δ ∈ (0, 1] and choose βn as in Lemma 1. Choose x0 ∈ R0 ⊂ X . Then the following holds jointly for all t ≥ 0 with probability at least 1 − δ: xt ∈ Rt , where zt = (xt , ut ) ∈ X × U, R0 , R1 , .. is computed as in (25) and xt is the state of the system (1) at time step t. Proof: Since Lemma 3 holds uniformly for all ellipsoids R ⊂ X and u ∈ U, this is a special case that holds uniformly for all control inputs ut , t ∈ N and for all ellipsoids Rt , t ∈ N obtained through (25). Corollary 1 guarantees that, with high probability, the system is always contained in the propagated belief (25). Thus, if we provide safety guarantees for the propagated belief, we obtain high-probability safety guarantees for the trajectories of (1). c) Predictions under state-feedback control laws: When applying multi-step ahead predictions under a sequence of feed-forward inputs ut ∈ X , the individual sets of the corresponding reachability sequence can quickly grow unreasonably large. This is because these open loop input sequences do not account for future control inputs that could correct deviations from the model predictions. Hence, we extend (23) to affine state-feedback control laws of the form πt (xt ) := Kt (xt − pt ) + ut , (26) where Kt ∈ Rnu ×nx is a feedback matrix and ut ∈ Rnu is the open-loop input. The parameter pt is determined through the center of the current ellipsoid Rt = E(pt , Qt ). Given an appropriate choice of Kt , the control law actively contracts the ellipsoids towards their center. Similar to the derivations (13)-(23), we can compute the function m̃ for affine feedback controllers (26) πt and ellipsoids Rt = E(pt , Qt ). The resulting ellipsoid is m̃(Rt , πt ) = E(h(z̄t )+µ(z̄t ), Ht Qt HtT )⊕E(0, Qd˜(Rt , πt )), (27) where z¯t = (pt , ut ) and Ht = At + Bt Kt . The set E(0, Qd˜(Rt , πt )) is obtained similarly to (20) as the ellipsoidal over-approximation of 0 ± [βn σ(z̄) + L∇h l2 (Rt , St ) + Lg l(Rt , St )], 2 (28) with St = [Inx , KtT ] and l(Rt , St ) = maxx∈Rt \|\|St (z(x) − z¯t )\|\|2 . The theoretical results of Lemma 3 and Corollary 1 directly apply to the case of the uncertainty propagation technique (27). B. Safety constraints The derived multi-step ahead prediction technique provides a sequence of ellipsoidal confidence regions around trajectories of the true system f through Corollary 1. We can guarantee that the system is safe by verifying that the computed confidence ellipsoids are contained inside the polytopic constraints (2) and (3). That is, given a sequence of feedback controllers πt , t = 0, .., T − 1 we need to verify Rt+1 ⊂ X , πt (Rt ) ⊂ U, t = 0, .., T − 1, (29) where (R0 , .., RT ) is given through (25). Since our constraints are polytopes, we have that X = Tm x nx x i=1 Xi , Xi = {x ∈ R \|[Hx ]i,· x − hi ≤ 0}, where x [Hx ]i,· is the ith row of H . We can now formulate the state constraints through the condition Rt = E(pt , Qt ) ⊂ X as mx individual constraints Rt ⊂ Xi , i = 1, .., mx , for which an analytical formulation exists [25], q [Hx ]i,· pt + [Hx ]i,· Qt [Hx ]Ti,· ≤ hxi , ∀i ∈ {1, .., mx }. (30) Moreover, we can use the fact that πt is affine in x to obtain πt (Rt ) = E(kt , Kt Qt , KtT ), using (10). The corresponding control constraint πt (Rt ) ⊂ U is then equivalently given by q u [Hu ]i,· ut + [Hu ]i,· Kt Qt KtT [Hu ]T i,· ≤ hi , ∀i ∈ {1, .., mu }. (31) C. The SafeMPC algorithm Based on the previous results, we formulate a MPC scheme that optimizes the long-term performance of our system, while satisfying the safety condition in Definition 1: minimize Jt (R0 , .., RT ) (32a) subject to Rt+1 = m̃(Rt , πt ), t = 0, .., T − 1 (32b) πt (Rt ) ⊂ U, t = 0, .., T − 1 (32d) π0 ,..,πT −1 Rt ⊂ X , t = 1, .., T − 1 RT ⊂ Xsafe , (32c) (32e) where R0 := {xt } is the current state of the system and the intermediate state and control constraints are defined in (30), (31). The terminal set constraint RT ⊂ Xsafe has the same form as (30) and can be formulated accordingly. The objective Jt can be chosen to suit the given control task. Due to the terminal constraint RT ⊂ Xsafe , a solution to (32) provides a sequence of feedback controllers π0 , .., πT that steer the system back to the safe set Xsafe . We cannot directly show that a solution to MPC problem (32) exists at every time step (this property is known as recursive feasibility) without imposing additional assumption, e.g. on the safety controller πsafe . However, we guarantee that such a sequence of feedback controllers exists at every time step through Algorithm 1 as follows: Given a feasible solution Πt = (πt0 , .., πtT −1 ) to (32) at time t, we apply the first feedback control πt0 . In case we do not find a feasible solution to (32) at the next time step, we shift the previous solution in a receding horizon fashion and append πsafe to the sequence to obtain Πt+1 = (πt1 , .., πtT −1 , πsafe ). We repeat this process until a new feasible solution exists that replaces the previous input sequence. We now state the main result of the paper that guarantees the safety of our system under Algorithm 1. Theorem 2 Let π be the controller defined through algorithm 1 and x0 ∈ Xsafe . Then the system (1) is δ−safe under the controller π. Proof: By induction. Base case: If (32) is infeasible, we are δ-safe using the backup controller πsafe of Assumption 2, Algorithm 1 Safe Model Predictive Control (S AFE MPC) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: Input: Safe policy πsafe , dynamics model h, statistical model GP(0, k). Π0 ← {πsafe , .., πsafe } with \|Π0 \| = T for t = 0, 1, .. do Jt ← objective from high-level planner feasible, Π ← solve MPC problem (32) if feasible then Πt ← Π else Πt ← (Πt−1,1:T −1 , πsafe ) xt+1 ← apply ut = Πt,0 (xt ) to the system (1) since x0 ∈ Xsafe . Otherwise the controller returned from (32) is δ-safe as a consequence of Corollary 1 and the terminal set constraint that leads to xt+T ∈ Xsafe . Induction step: let the previous controller πt be δ-safe. At time step t + 1, if (34) is infeasible then Πt leads to a state xt+T ∈ Xsafe , from which the backup-controller is δ-safe by Assumption 2. If (34) is feasible, then the return path is δ-safe by Corollary 1. D. Optimizing long-term behavior In MPC, we aim to optimize the long-term behavior of our system based on finite-length trajectories. In contrast, the S AFE MPC algorithm guarantees the existence of return strategies. This is undesirable in exploration scenarios, where one aims to explore uncertain areas of the state space. In order to optimize the long-term behavior of the system while maintaining the safety properties of our algorithm, we propose to simultaneously plan a performance trajectory perf s0 , .., sH under a sequence of inputs π0perf , .., πH−1 using a performance-model mperf along with the return strategy that we obtain when solving (32). We do not make any assumptions on the performance model and could be given by one of the approximate uncertainty propagation methods proposed in the literature (see, e.g. [10] for an overview). In order to maintain the safety of our system, we enforce that the first r ∈ {1, ., min{T, H}} controls are the same for both trajectories, i.e. we have that πk = πkperf , k = 0, .., r−1. This extended MPC problem is minimize πt ,..,πt+T −1 perf πtperf ,..,πt+H−1 subject to Jt (st , .., st+H ) (32b) − (32e), t = 0, .., T − 1 st+1 = mperf (st , πtperf ), t = 0, .., H − 1 πt = πtperf , t = 0, .., r − 1, (33) where we replace (32) with this problem in Algorithm 1. The safety guarantees of Theorem 2 directly translate to this setting, since we can always fall back to the return strategy. E. Discussion Algorithm 1 theoretically guarantees that the system remains safe, while actively optimizing for performance via the MPC problem (33). This problem can be solved by commonly used, nonlinear programming (NLP) solvers, such as the Interior Point OPTimizer (Ipopt, [26]). Due to the solution of the eigenvalue problem (12) that is required to compute (23), our uncertainty propagation scheme is not analytic. However, we can still obtain exact function values and derivative information by means of algorithmic differentiation, which is at the core of many state-of-the-art optimization software libraries [27]. One way to further reduce the conservatism of the multistep ahead predictions is to linearize the GP mean prediction µn (xt , ut ), which we omitted for clarity. V. E XPERIMENTS In this section, we evaluate the proposed S AFE MPC algorithm to safely explore the dynamics of an inverted pendulum system. The continuous-time dynamics of the pendulum are given by ml2 θ̈ = gml sin(θ) − η θ̇ + u, where m = 0.15kg and l = 0.5m are the mass and length of the pendulum, respectively, η = 0.1Nms/rad is a friction parameter, and g = 9.81m/s2 is the gravitational constant. The state of the system x = (θ, θ̇) consists of the angle θ and angular velocity θ̇ of the pendulum. The system is controlled by a torque u that is applied to the pendulum. The origin of the system corresponds to the pendulum standing upright. The system is underactuated with control constraints U = {u ∈ R\| − 1 ≤ u ≤ 1}. Due to these limits, the pendulum becomes unstable and falls down beyond a certain angle. We do not impose state constraints, X = R2 . However the terminal set constraint (32e) of the MPC problem (32) acts as a stability constraint and prevents the pendulum from falling. As in [14], our prior model h consists of a linearized and discretized approximation to the true system with a lower mass and neglected friction. The safety controller πsafe is a discrete-time, infinite horizon linear quadratic regulator (LQR, [28]) of the approximated system h with cost matrices Q = diag([1, 2]), R = 20. The corresponding safety region XSafe is given by a conservative polytopic innerapproximation of the true region of attraction of πsafe . We use the same mixture of linear and Matérn kernel functions for both output dimensions, albeit with different hyperparameters. We initially train our model with a dataset (Z0 , ỹ0 ) sampled inside the safe set using the backup controller πSafe . That is, we gather n0 = 25 initial samples Z0 = {z10 , .., zn0 0 } with zi0 = (xi , πsafe (xi )), xi ∈ Xsafe , i = 1, .., n and observed next states ỹ0 = {y00 , .., yn0 0 } ⊂ XSafe . The theoretical choice of the scaling parameter βn for the confidence intervals in Lemma 1 can be conservative and we choose a fixed value of βn = 2 instead. We aim to iteratively collect the most informative samples of the system, while preserving its safety. To evaluate the exploration performance, we use the mutual information I(gZn , g) between the collected samples Zn = {z0 , .., zn } ∪ Z0 and the GP prior on the unknown model-error g, which can be computed in closed-form [19]. 1 1 0 0 0 −1 −1 −1 −1.0 −0.5 0.0 0.5 1.0 Angular velocity θ̇ −1.0 −0.5 0.0 0.5 1.0 −1.0 Angular velocity θ̇ 300 200 100 −0.5 0.0 0.5 Iteration Angle θ 1 1.0 Angular velocity θ̇ Fig. 3. Visualization of the samples acquired in the static exploration setting in Sec. V-A for T ∈ {1, 4, 5}. The algorithm plans informative paths to the safe set Xsafe (red polytope in the center). The baseline sample set for T = 1 (left) is dense around origin of the system. For T = 4 (center) we get the optimal trade-off between cautiousness due to a long horizon and limited length of the return trajectory due to a short horizon. The exploration for T = 5 (right) is too cautious, since the propagated uncertainty at the final state is too large. For a first experiment, we assume that the system is static, so that we can reset the system to an arbitrary state xn ∈ R2 in every iteration. In the static case and without terminal set constraints, a provably close-to-optimal exploration strategy is to, at each iteration n, select state-action pair zn+1 with the largest predictive standard deviation [19] X zn+1 = arg max σn,j (z), (34) z∈X ×U 1≤j≤nx where is the predictive variance (8) of the ith GP(0, ki ) at the nth iteration. Inspired by this, at each iteration we collect samples by solving Pnxthe MPC problem (32) with cost function Jn = − i=1 σn,i , where we additionally optimize over the initial state xn ∈ X . Hence, we visit high-uncertainty states, but only allow for stateaction pairs zn that are part of a feasible return trajectory to the safe set XSafe . Since optimizing the initial state is highly non-convex, we solve the problem iteratively with 25 random initializations to obtain a good approximation of the global minimizer. After every iteration, we update the sample set Zn+1 = Zn ∪{zn }, collect an observation (zn , yn ) and update the GP models. We apply this procedure for varying horizon lengths. The resulting sample sets are visualized for varying horizon lengths T ∈ {1, .., 5} with 300 iterations in Fig. 3, while Fig. 4 shows how the mutual information of the sample sets Zi , i = 0, .., n for the different values of T . For short time horizons (T = 1), the algorithm can only slowly explore, since it can only move one step outside of the safe set. This is also reflected in the mutual information gained, which levels off quickly. For a horizon length of T = 4, the algorithm is able to explore a larger part of the state-space, which means that more information is gained. For larger horizons, the predictive uncertainty of the final state is too large to explore effectively, which slows down exploration initially, when we do not have much information about our system. The results suggest that our approach could further benefit from adaptively choosing the horizon during operation, e.g. by employing a variable horizon MPC approach [29], or by carefully monitoring when the mutual information starts to saturate for the current horizon. 2 σn,i (·) I(gZn , g) A. Static Exploration 100 50 T=1 T=2 0 50 100 Iteration T=3 T=4 150 T=5 200 Fig. 4. Mutual information I(gZn , g), n = 1, .., 200 for horizon lengths T ∈ {1, .., 5}. Exploration settings with shorter horizon gather more informative samples at the beginning, but less informative samples in the long run. Longer horizon lengths result in less informative samples at the beginning, due to uncertainties being propagated over long horizons. However, after having gathered some knowledge they quickly outperform the smaller horizon settings. The best trade off is found for T = 4. B. Dynamic Exploration As a second experiment, we collect informative samples during operation; without resetting the system at every iteration. Starting at x0 ∈ Xsafe , we apply the S AFE MPC,Algorithm 1, over 200 iterations. We consider two settings. In the first, we solve the MPC problem (32) with −Jn given by (34), similar to the previous experiments. In the second setting, we additionally plan a performance trajectory as proposed in Sec. IV-D. We define the states of the performance trajectory as Gaussians st = N (mt , St ) ∈ Rnx × Rnx ×nx and the next state is given by the predictive mean and variance of the current state mt and applied action ut . That is, st+1 = N(mt+1 , St+1 ) with mt+1 = µn (mt , ut ), St+1 = Σn (mt , ut ), t = 0, .., H − 1, (35) where Σn (·) = diag(σn2 (·)) and m0 = xn . This simple approximation technique is known as mean-equivalent uncertainty propagation. We define the cost-function −Jt = PH PT 1/2 T t=1 (mt − pt ) Qperf (mt − pt ), which t=0 trace(St ) + maximizes the confidence intervals along the trajectory s1 , .., sH , while penalizing deviation from the safety trajectory. We choose r = 1 in the problem (33), i.e. the first action of the safety trajectory and performance trajectory are the same. As in the static setting, we update our GP models after every iteration. I(gZ200 , g) 150 100 50 0 T=1 T=2 T=3 T=4 T=5 Fig. 5. Comparison of the information gathered from the system after 200 iterations for the standard setting (blue) and the setting where we plan an additional performance trajectory (green). We evaluate both settings for varying T ∈ {1, .., 5} and fixed H = 5 in terms of their mutual information in Fig. 5. 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Generalized eigenvalue problem Starting at (12), we compute the similar problem r2 (Q, S) = max \|\|S(x − p)\|\|22 = x∈E(p,Q) max sT Q−1 s≤1 sT S T Ss, (36) where we used the definition of ellipsoids and the 2-norm under the variable shift s = x−p. Since S has full column rank and S T S is symmetric positive definite, the maximum of (36) is at the boundary, r2 (Q, S) = maxsT Q−1 s=1 sT S T Ss, which is a generalized eigenvalue problem.	3
PCA from noisy, linearly reduced data: the diagonal case arXiv:1611.10333v1 [math.ST] 30 Nov 2016 Edgar Dobriban∗, William Leeb†, and Amit Singer‡ Abstract Suppose we observe data of the form Yi = Di (Si + εi ) ∈ Rp or Yi = Di Si + εi ∈ Rp , i = 1, . . . , n, where Di ∈ Rp×p are known diagonal matrices, εi are noise, and we wish to perform principal component analysis (PCA) on the unobserved signals Si ∈ Rp . The first model arises in missing data problems, where the Di are binary. The second model captures noisy deconvolution problems, where the Di are the Fourier transforms of the convolution kernels. It is often reasonable to assume the Si lie on an unknown low-dimensional linear space; however, because many coordinates can be suppressed by the Di , this low-dimensional structure can be obscured. We introduce diagonally reduced spiked covariance models to capture this setting. We characterize the behavior of the singular vectors and singular values of the data matrix under high-dimensional asymptotics where n, p → ∞ such that p/n → γ > 0. Our results have the most general assumptions to date even without diagonal reduction. Using them, we develop optimal eigenvalue shrinkage methods for covariance matrix estimation and optimal singular value shrinkage methods for data denoising. Finally, we characterize the error rates of the empirical Best Linear Predictor (EBLP) denoisers. We show that, perhaps surprisingly, their optimal tuning depends on whether we denoise in-sample or out-of-sample, but the optimally tuned mean squared error is the same in the two cases. 1 Introduction Principal component analysis (PCA) is a classical statistical method that decomposes a collection of datapoints s1 , . . . , sn ∈ Rp as a linear combination of vectors that account for the most variability (e.g., Jolliffe, 2002; Anderson, 2003). More formally, if s1 , . . . , sn are drawn from a probability distribution with mean zero and covariance matrix ΣS , then the principal components (PCs) of the distribution are the eigenvectors u1 , . . . , up of ΣS . Typically, we approximate the distribution by projecting it onto the PCs with the largest eigenvalues. ∗ Department of Statistics, Stanford University. E-mail: dobriban@stanford.edu. Supported in part by NSF grant DMS-1407813, and by an HHMI International Student Research Fellowship. † Program in Applied and Computational Mathematics, Princeton University. E-mail: wleeb@math.princeton.edu. Supported by the Simons Collaborations on Algorithms and Geometry. ‡ Department of Mathematics, and Program in Applied and Computational Mathematics, Princeton University. Email: amits@math.princeton.edu. Partially supported by Award Number R01GM090200 from the NIGMS, FA955012-1-0317 from AFOSR, Simons Foundation Investigator Award and Simons Collaborations on Algorithms and Geometry, and the Moore Foundation Data-Driven Discovery Investigator Award. 1 A more specific model arising in many applications is the spiked covariance model (Johnstone, 2001). First, the signal Si is a linear combination of r fixed but unobserved orthonormal PCs uk : Si = r X 1/2 `k zik uk , (1) k=1 where r is a fixed parameter (independent of n and p) and zik are iid standardized random variables. Here `k are the eigenvalues, or equivalently the variances along the PCs uk . Second, the observations are Xi = Si + εi , where εi is noise with iid standardized entries. The spiked covariance model has been widely studied in probability and statistics (e.g., Baik et al., 2005; Baik and Silverstein, 2006; Paul, 2007; Benaych-Georges and Nadakuditi, 2012, etc); see also Paul and Aue (2014); Yao et al. (2015). This paper considers the setting when the vectors Si are not only corrupted by noise, but also linearly reduced. This means that for given matrices Di ∈ Rqi ×p , we observe either Yi = Di Xi = Di Si + Di εi . (2) Yi = Di Si + εi . (3) or We think of the reduction matrix Di as either a projection matrix or a linear filter reducing the information that we observe. In general, it will not be possible to reconstruct a vector v from Di v. We refer to model (2) as the reduced-noise model, and to model (3) as the unreduced-noise model. In the reduced-noise model, both the signal and noise are reduced, while in the unreduced-noise case, only the signal is. These models generalize the spiked covariance model, and arise naturally in several settings. For instance: 1. Missing data: For diagonal matrices Di with zeros or ones the reduced-noise model from (2) corresponds to missing data problems widely encountered in statistics (e.g., Schafer, 1997; Little and Rubin, 2014). For random Di independent of other variables, we are under the assumption of missing completely at random (MCAR). 2. Deconvolution and image restoration: In image processing, an image might be corrupted by “blurring”—convolution with a linear filter—followed by noise. After taking the Fourier transform, this can be modeled as a coordinate-wise multiplication by a diagonal matrix Di , followed by adding noise. This corresponds to the unreduced-noise model from (3). For example, the image formation model in cryo-electron microscopy (cryo-EM) under the linear, weak phase approximation leads to such a model (Frank, 1996). A closely related model was recently used by Bhamre et al. (2016), where Si are Fourier transforms of projection images of molecules, and Di are contrast transfer functions. 3. Structural variability in cryo-EM: In cryo-EM, Si is the three-dimensional structure of a molecule, and Di is a tomographic projection of this volume onto a randomly selected plane. Since the molecule typically has only a few degrees of freedom, such as different conformations or states, it is reasonable to model Si to lie on some unknown low-dimensional space (e.g., Katsevich et al., 2015; Andén et al., 2015). This corresponds to the unreduced-noise model from (3): the data Yi = Di Si + εi are tomographic projections Di Si with added noise εi . 2 4. Signal acquisition and compressed sensing: In some signal acquisition tasks such as hyperspectral imaging, due to resource constraints it is convenient to acquire reduced or compressed measurements of signals. The reductions are often taken to be random projections. It is of interest to reconstruct the PCs of the original measurements (e.g., Chang, 2003; Fowler, 2009). This falls under the reduced-noise model from (2). There may certainly be many other applications fitting this framework. In the above examples it is natural to posit that the distribution of the signal vectors Si is of low effective dimensionality. In this paper we will assume that the distribution lies on some unknown linear space of small dimension r, as in equation (1). Given observations of the form (2) or (3), we address several natural statistical questions: 1. Covariance estimation: How to estimate the covariance matrix of the signals Si ? This is both a fundamental statistical problem, and has numerous applications, including classification and denoising. 2. PCA: How should we estimate the principal components of Si ? This question is of special interest due to the importance of PCA for exploratory data analysis and visualization. 3. Denoising: How can we denoise—or predict—the individual signal vectors Si ? This is a central question both in the missing data problems, where it corresponds to imputation, as well as in the image processing problems, where it amounts to noise reduction. In this paper, we develop new methods for a special class of models, where the matrices Di ∈ Rp×p are diagonal. We will call the observations Yi from (2) or (3) diagonally reduced. The missing data and deconvolution problems belong to this class. In the high-dimensional asymptotic regime where p, n both grow to infinity and p/n → γ > 0, we develop methods that provide clear, quantitative answers to all questions posed above, under quite weak assumptions. Related work by Katsevich et al. (2015) and Andén et al. (2015) develops methods for covariance estimation when the Di ’s are projection matrices mapping a 3-D electron density to its integral on a randomly chosen plane. In this data acquisition model for cryo-EM, the authors propose consistent estimators of the covariance of the electron density. However, their observation models are different from our diagonally reduced models. In Bhamre et al. (2016), the questions of covariance estimation and denoising are studied empirically when the Di ’s come from the contrast transfer function of a microscope and the Si ’s are Fourier transforms of clean tomographic projections. Nadakuditi (2014) develops methods for low-rank matrix estimation with missing data. Our results are more general, and also include methods for covariance estimation and denoising individual datapoints, see Sec. 2.3 for more details. Lounici (2014) develops eigenvalue soft thresholding methods for covariance estimation with missing data. In our somewhat more specialized models, we instead find the optimal eigenvalue shrinkers, and they are different from soft thresholding. Cai and Zhang (2016) develop minimax rate-optimal covariance matrix estimators for missing data, focusing on bandable and sparse models. We instead focus on the spiked covariance model. We next give a brief overview of our results. 1.1 Probabilistic Results A lot of work in random matrix theory studies the asymptotic spectral theory of the spiked covariance model and its variants, see e.g., Paul and Aue (2014); Yao et al. (2015). In Sec. 2, we 3 introduce two general diagonally reduced spiked covariance models corresponding to (2) and (3). We characterize the limiting eigenvalues of the data matrix Y (with rows Yi> ), and the limiting angles of its singular vectors with the population singular vectors (of the matrix S with rows Si> ). More specifically, we show in Thm. 2.1 that the eigenvalue distribution of n−1 Y > Y converges to a general Marchenko-Pastur distribution (Marchenko and Pastur, 1967), while the top few eigenvalues have well-defined almost sure limits. This mirrors the behavior known in unreduced spiked models, (e.g., Baik et al., 2005; Baik and Silverstein, 2006; Benaych-Georges and Nadakuditi, 2012); however, our assumptions in Sec. 2 are very general, and in fact lead to the most general results to date even in the unreduced case when Di = Ip (see Cor. 2.3 and Sec. 2.3 for discussion). In the special case where the entries of Di are iid, the limiting spectrum and the angles between the population and empirical singular vectors are given by explicit formulas related to the standard Marchenko-Pastur law, as described in Cor. 2.2. For general Di , we can compute numerically the quantities specified by Thm. 2.1 with the Spectrode method (Dobriban, 2015); see Sec. 2.4. All computational results of this paper are reproducible with software publicly available at github.com/dobriban/diagonally_reduced/. 1.2 Covariance estimation In Sec. 3, we apply the probabilistic results from Sec. 2 to develop methods for covariance estimation from diagonally reduced data, under the additional assumption that the diagonal entries of the reduction matrices Di are iid. The prototypical example is the missing data problem with uniform missingness, where in the reduced-noise model (2) the entries are Bernoulli(δ). Already for unreduced data from the spiked covariance model, the sample covariance matrix is a poor estimator of the population covariance, since neither the empirical PCs nor the empirical spectrum converge to their population counterparts. Though little can be done about correcting the PCs, one can develop optimal shrinkage estimators of the eigenvalues. In the unreduced case, Donoho et al. (2013) consider estimators of the form U η(Λ)U > , where U is the orthogonal matrix of PCs, Λ is the matrix of eigenvalues of the sample covariance, η : R → R is a shrinkage function, and η(Λ) replaces every diagonal element λ of Λ by η(λ). In Sec. 3, we take a similar approach to the problem of covariance estimation in reduced models. We first define an unbiased estimator of the population covariance of the unreduced signals (Eq. 10). Building on the results of Sec. 2, we describe the asymptotic spectral theory of this estimator, and finally derive shrinkers of the spectrum that are asymptotically optimal for certain loss functions. We explicitly derive the optimal shrinkers in the case of operator norm loss (Eq. (15) for reducednoise and Eq. (23) for unreduced-noise) and Frobenius norm loss (Eq. (18) for reduced-noise and Eq. (24) for unreduced-noise). We also derive the asymptotic errors of the optimal shrinkers (Eq. (17) and (19)), and give a recipe for deriving the optimal shrinkers for a broad class of loss functions. 1.3 Denoising In Sec. 4, we consider denoising, the task of predicting the signal vectors Si based on the observations Yi . For this we study the Best Linear Predictor (BLP) well-known from random effects models (e.g., Searle et al., 2009). The general form of a BLP is Ŝi = ΣS (ΣS + Σε )−1 Yi , where Σε is the covariance matrix of the noise. This is also known as a “linear Bayesian” method (Hartigan, 1969). In other areas such as electrical engineering and signal processing, it is known as the “Wiener 4 filter”, “(linear) Minimum Mean Squared Estimator (MMSE)” (Kay, 1993, Ch. 12), “linear Wiener estimator” (Mallat, 2008, p. 538), or “optimal linear filter” (MacKay, 2003, p. 550-551). The BLP is an “oracle” method, because it depends on unknown population parameters. In practice we can use the empirical BLP (EBLP), Σ̂S (Σ̂S + Σ̂ε )−1 Yi , where the unknown parameters are estimated using the data. Due to the inconsistency of PCA in high dimensions, this is suboptimal to the BLP. However, we can find the asymptotically optimal method for estimating the covariance matrix ΣS using the empirical PCs. This estimator holds several surprises. In particular, the optimal EBLP coefficients are different for in-sample and out-of-sample denoising—but the optimal mean squared error ends up identical! See Thms. 4.1, 4.3 for reduced-noise and Sec. 4.4 for unreduced-noise. It also turns out that the formula for in-sample EBLP, applied to all Yi , is identical to optimal singular value shrinkage estimators (Sec. 4.2.2). This analysis involves characterizing random quantities with an intricate dependence structure, such Yi> Di ûk , where ûk is the k-th PC of the sample covariance matrix of Yi . For this we extend significantly the technique introduced by Benaych-Georges and Nadakuditi (2012) to study the angles between uk and ûk . We call this approach the outlier equation method (see Sec. 4). 2 2.1 Probabilistic results Main probabilistic results This section presents a new result in random matrix theory, which will be the key tool for our work on covariance estimation. Recall that we have diagonally reduced observations Yi = Di (Si + εi ) or Yi = Di Si + εi , i = 1, . . . , n, where Si ∈ Rp are unobserved signals and Di ∈ Rp×p are diagonal Pr 1/2 matrices. The signals have the form Si = k=1 `k zik uk . Here uk are deterministic signal directions with kuk k = 1. We will assume that uk are delocalized, so that \|uk \|∞ ≤ C log(p)B /p1/2 for some constants B, C > 0. The scalars zik are standardized independent random variables, specifying the variation in signal strength from sample to sample. For simplicity we assume that the deterministic spike strengths are different and sorted: `1 > `2 > . . . > `r > 0. Finally εi = Γ1/2 αi is sampling noise, where Γ = diag(g12 , . . . , gp2 ) is diagonal and deterministic, and αi = (αi1 , . . . , αip )> has independent standardized entries. The diagonal matrices Di = diag(Di1 , . . . , Dip ) have the form Di = µ + Σ1/2 Ei , (4) where Ei have independent standardized entries. Here µ = diag(µ1 , . . . , µp ) ∈ Rp×p is the deterministic diagonal mean and Σ = diag(σ12 , . . . , σp2 ) ∈ Rp×p is the deterministic diagonal covariance matrix of the entries of the reduction matrices. Let Hp be the uniform distribution on the p scalars 2 gj2 · EDij = gj2 · (µ2j + σj2 ), j = 1, . . . , p, and let Gp be the analogous object for gj2 , j = 1, . . . , p. It turns out that these are the distributions of noise variances relevant for our models. For dealing with model (2), we assume that as p → ∞, Hp converges to a compactly supported limit distribution H: Hp ⇒ H. For dealing with model (3), we assume that Gp converges to a compactly supported limit distribution G: Gp ⇒ G. We will consider the high-dimensional regime where n, p → ∞ such that p/n → γ > 0. In this setup, our answers will depend on the general Marchenko-Pastur distribution (Marchenko and Pastur, 1967). This law describes the behavior of empirical eigenvalue distributions of sample covariance matrices: If N is an n × p matrix with iid standardized entries, and T is a p × p positive 5 Table 1: Definitions Name Definition Defined in Reduced Observation (2), (3) Signal Yi = Di (Si + εi ) or Yi = Di Si + εi Pr 1/2 Si = k=1 `k zik uk Reduction Matrix Di = µ + Σ1/2 Ei (4) (1) µ = diag(µ1 , . . . , µp ), Σ = diag(σ12 , . . . , σp2 ) Noise Noise Variances εi = Γ1/2 αi , where Γ = diag(g12 , . . . , gp2 ) Pp Hp = p−1 j=1 δgj2 (µ2j +σj2 ) , Hp ⇒ H Pp Gp = p−1 j=1 δgj2 , Gp ⇒ G Stieltjes Transform Fγ,H , F γ,H (x) = γFγ,H (x) + (1 − γ)δ0 R dF γ,H (x) R dFγ,H (x) mγ,H (z) = x−z , mγ,H (z) = x−z R mH (z) = dH(x) x−z D-Transform Dγ,H (x) = x · mγ,H (x) · mγ,H (x) Upper Edge b2H = sup supp(Fγ,H ) General MP Law semidefinite matrix with eigenvalue distribution converging to H, then the eigenvalue distribution of the p × p matrix n−1 T 1/2 N > N T 1/2 converges almost surely (a.s.) to the Marchenko-Pastur distribution Fγ,H (see e.g., Bai and Silverstein, 2009, for a reference). When T = Ip is the identity, Fγ,H is known as the standard Marchenko-Pastur distribution, and has density (if γ ∈ (0, 1)): p (g+ − x)(x − g− ) fγ (x) = I(x ∈ [g− , g+ ]) 2πx √ where g± = (1 ± γ)2 . For general H, Fγ,H does not have a closed form, but it can be studied numerically (see e.g., Dobriban, 2015). Closely related to Fγ,H is the distribution F γ,H (x) = γFγ,H (x) + (1 − γ)δ0 . This is the limit of the eigenvalue distribution Rof the n×n matrix n−1 N > T N . We will also need the Stieltjes transform mγ,H of Fγ,H , mγ,H (z) = (x − z)−1 dFγ,H (x), and the Stieltjes transform mγ,H of F γ,H . Based on these, one can define the D-transform of Fγ,H by Dγ,H (x) = x · mγ,H (x) · mγ,H (x). Up to the change of variables x = y 2 , this agrees with the D-transform defined in BenaychGeorges and Nadakuditi (2012). Let b2H be the supremum of the support of Fγ,H , and Dγ,H (b2H ) = limt↓b Dγ,H (t2 ). It is easy to see that this limit is well defined, and is either finite or +∞. Let us denote the support of a distribution H on R by supp(H). Denote the normalized data matrix Ỹ = n−1/2 Y , with the n × p matrix Y having rows Yi> . Our main probability result, proved later in Sec. 5.1, is the following. Theorem 2.1 (Diagonally reduced spiked models). Consider the observation models (2) and (3), under the above assumptions. Suppose that 6 4 4 1. Eαij < C, EEij < C, and E\|zi \|4+φ < C for some φ > 0 and C < ∞. 2. Under model (2), sup supp(Hp ) → sup supp(H). Under model (3), sup supp(Gp ) → sup supp(G). 3. The squared norms kµuk k2 converge to τk > 0. 4. Under model (2), µuk are generic with respect to M = Γ(Σ + µ2 ) in the sense that −1 u> µuk → I(j = k) · τk · mH (z) j µ(M − zIp ) for all z ∈ C+ , where mH is the Stieltjes transform of H. −1 Under model (3), µuk are generic with respect to M = Γ, i.e., u> µuk → I(j = j µ(M − zIp ) k) · τk · mG (z). Then 1. Under model (2), the eigenvalue distribution of Ỹ > Ỹ converges to the general MarchenkoPastur law Fγ,H a.s. In addition, the k-th largest singular value of Ỹ converges, σk (Ỹ ) → tk > 0 a.s., where −1 if `k > 1/[τk Dγ,H (b2H )], Dγ,H ( τk1`k ) (5) t2k = 2 otherwise. bH Moreover, let νj = µuj /kµuj k be the normalized reduced signals and let ûk be the right singular vector of Ỹ corresponding to σk (Ỹ ). Then (νj> ûk )2 → c2jk a.s., where ( mγ,H (t2k ) if j = k and `k > 1/[τk Dγ,H (b2H )], 0 2 2 D γ,H (tk )τk `k cjk = 0 otherwise. 2. Under model (3), the analogous results hold with G replacing H everywhere. Assumption 4 needs explanation. This assumption generalizes the existing conditions for spiked models. In particular, it is easy to see that it holds when the vectors uk are random with independent coordinates. Specifically, let x, y are two independent random vectors with iid zero-mean entries with variance 1/p. Then Ex> µ(M − zIp )−1 µx = p−1 tr µ(M − zIp )−1 µ. Assumption 4 requires that this converges to τ · mH (z), which happens for instance when the vector µ itself has random independent coordinates with variance τ /p, or when it equals a multiple of the identity. However, Assumption 4 is more general, as it does not require any kind of randomness in uk . Thm 2.1 gives the limiting angles of the empirical eigenvectors ûk with the reduced population eigenvectors νj = µuj /kµuj k. These are in general different from the true eigenvectors uj . However, in our main application (Cor. 2.2 and the following sections) they are the same, because µ is a multiple of identity. One can gain some insight into the result in the simpler case where the noise is uncolored, so that Γ = Ip . In that case, before reduction we have a spike strength ` and an average noise level of unity. P After reduction under model (2), we have a spike strength `kµuk2 , and an average noise 2 level p−1 j EDij = p−1 (kµk2 + kσk2 ), where σ = (σ1 , . . . , σp ). For a delocalized u, we expect 2 −1 2 kµuk ≈ p kµk . Therefore, reduction in model (2) typically decreases the signal strength by a factor of kµk2 . kµk2 + kσk2 7 However, the spike strength after reduction—`kµuk2 —depends on the correlation between µ and u. In particular, the reduction can vary among the different PCs. In contrast it is easy to see in a similar way that reduction in model (3) may increase or decrease the signal strength. The key strength of Thm. 2.1 is its generality. Specifically, there is essentially only one previous result on reduced spiked models, appearing in Nadakuditi (2014). However, that only studies iid Bernoulli projections under restrictive conditions on the noise, whereas we allow for (1) a general diagonal covariance structure Σ in the reduction matrices, as well as (2) a general diagonal noise structure Γ, and (3) more general moment conditions. Moreover, even in unreduced spiked models, our results are already the most general results to date (see Sec. 2.3). We think that the generality is important for several reasons: first, for practical reasons it is good to have results that require as few assumptions as possible, especially unverifiable conditions like “randomness” in uk or “orthogonal invariance” of the noise. As a consequence of these general results, existing tools like singular value shrinkage are shown to apply more generally, so this is a direct improvement. Second, from a theoretical perspective it is good to understand the reason for the “spiking” behavior; our results clarify for instance that “orthogonal invariance” of the noise is not needed. 2.1.1 Comments on the proof The broad outline of the proof is inspired by the argument presented in Benaych-Georges and Nadakuditi (2012) for unreduced spiked models. However, there are several new steps. First, the proof in Benaych-Georges and Nadakuditi (2012) concerns only the unreduced case, and the dependence introduced by the random reduction matrices Di is a new challenge. The observations in model (2) are Di Xi = Di Si + Di εi , so the “signal” Di Si and “noise” Di εi are dependent. However, we show that the dependence is asymptotically negligible. For non-diagonal reduction matrices Di , the depencence may be asymptotically non-negligible; this explains why we currently need the diagonal assumption. As a second novelty, the proof involves finding the limits of certain quadratic forms u> k R(z)uj , where R(z) is a specific resolvent matrix with complex argument z. Since the uk are deterministic, the concentration arguments of Benaych-Georges and Nadakuditi (2012) are not available. Instead, we adapt the “deterministic equivalents” approach of Bai et al. (2007). For this we need to take the imaginary part of the complex argument z to zero, which appears to be a new argument in this context. 2.2 Reduced standard spiked models We will later use the following corollary for the reduced standard spiked model where the reduction coefficients and the noise entries are iid random variables. Suppose that the noise variances are equal to unity, so Γ = Ip and thus εi have independent standardized entries. Moreover assume that the reduction matrices Di have iid random diagonal entries Dij with mean µ = EDij and variance σ 2 = Var[Dij ]. Note that previously µ was a matrix, but from now on it will be a scalar, and there will be no possibility for confusion. Let m = µ2 + σ 2 and δ = µ2 /m. In this case it is easy to see that the reduced eigenvectors are the same as the unreduced ones, i.e., νk = uk . Moreover, Assumption 4 from Thm. 2.1 reduces to u> k ul → 0 as n → ∞, if k 6= l. Our answers can be expressed in terms of the well-known characteristics of the standard spiked model. The asymptotic location of the top singular values will depend on the spike forward map 8 (e.g., Baik et al., 2005): λ(`) = λ(`; γ) = (1 + `) 1 + (1 + γ 1/2 )2 γ ` if ` > γ 1/2 , otherwise. The asymptotic angle between singular vectors will depend on the cosine forward map c(`; γ) ≥ 0 given by (e.g., Paul, 2007; Benaych-Georges and Nadakuditi, 2011, etc): ( 1−γ/`2 if ` > γ 1/2 , 2 2 1+γ/` (6) c(`) = c(`; γ) = 0 otherwise. Corollary 2.2 (Reduced standard spiked models). Under observation model (2), in the above setting, the eigenvalue distribution of m−1 Ỹ > Ỹ converges to the standard Marchenko-Pastur law with aspect ratio γ, a.s. Moreover, m−1/2 σk (Ỹ ) → tk > 0 a.s., where t2k = λ(δ`k ) (7) 2 2 Finally, let ûk be the right singular vector of Ỹ corresponding to σk (Ỹ ). Then (u> j ûk ) → cjk a.s., where 2 c (δ`k ) if j = k c2jk = (8) 0 otherwise. 2 Under observation model (3) in the above setting, we have σk (Ỹ )2 → t2k = λ(µ2 `), while (u> j ûk ) → 2 2 2 2 cjk , where cjk = c (µ `k ) if j = k and 0 otherwise. For the proof, see Sec. 5.3. While in the current paper we only use this corollary of Thm. 2.1, the proof in the special case is essentially as involved as in the general case. For this reason, and for potential future applications, we prefer to state Thm. 2.1 as well. In this special case, reduction in model (2) lowers the spike strength from ` to δ`. This result is related to Thm 2.4 of Nadakuditi (2014) on missing data, but we have the following advantages: (1) our result works under a 4-th order moment condition instead of requiring all bounded moments; (2) our result admits arbitrary diagonal reductions, not just missing data (see Sec. 2.3 for details). Finally, when there is no reduction, i.e., when Di = Ip for all i, then it turns out we do not need the delocalization of uk . Indeed that is only needed to show that the diagonal reductions introduce a negligible amount of dependence, but we do not need this when Di = Ip . Hence, we can state the following corollary for unreduced spiked models: Corollary 2.3 (Standard spiked models). Suppose we observe unreduced signals Yi = Si + εi , and we do not assume of the PCs uk . Suppose that the other assumptions of Prthe delocalization 1/2 Cor. 2.2 hold: Si = k=1 `k zik uk , where u> k uj → δkj , while zik , εij are iid standardized with E\|zi \|4+φ < C, Eε4ij < C for some φ > 0 and C < ∞. Then the conclusions of Cor. 2.2 hold. Specifically, the spectrum of n−1 Y > Y converges to a standard MP law, its spikes converge a.s. to λ(`k ) and the squared cosines between population and sample eigenvectors converge a.s. to c2 (`k ). Again, the key strength of this corollary is its generality, specifically that it only has 4-th moment assumptions, not orthogonal invariance. 9 Figure 1: The effect of reduction on sample spikes and correlations between PCs. Formulas from Thm. 2.1 computed with Spectrode (Dobriban, 2015) overlaid with simulations. See Sec. 2.4. 2.3 Related work There is substantial earlier work on unreduced spiked models, and even in this case our result leads to an improvement. We refer to Paul and Aue (2014); Yao et al. (2015) for general overviews of the area. The paper of Benaych-Georges and Nadakuditi (2012) is closely related to our approach, and we essentially follow their novel technique, relying on controlling certain bilinear forms. When the signal direction u is fixed, their results require the distribution of the noise matrix to be bi-unitarily invariant, which essentially reduces to Gaussian distributions. Our model is more general since it only requires a fourth-moment condition on the noise. The technique introduced by Benaych-Georges and Nadakuditi (2011, 2012) was adapted to non-white Gaussian signal-plus noise matrices Xi = `1/2 zi u + Γ1/2 εi in Chapon et al. (2012). They rely on an integration by parts formula for functionals of Gaussian vectors and the Poincaré-Nash inequality. A Poincaré inequality was also assumed in Capitaine (2013). Our result is stronger, since we only require fourth moment conditions. Previous extensions to the setting of missing data are found in Nadakuditi (2014), which describes the OptShrink method for singular value shrinkage matrix denoising. The method is extended to data missing at random, and in particular the limit spectrum of the data matrix with zeroed-out missing data is found (his Thm. 2.4). This is related to Thm. 2.1, but we have the following advantages: (1) our result works under the optimal 4-th order moment condition instead of requiring all moments to be bounded; (2) our result admits arbitrary reductions, not just binary projection matrices, (3) we extend to reduction matrices Di that have non-iid entries, (4) we allow heteroskedastic diagonal noise εi = Γ1/2 αi ; and (5) we also consider the unreduced-noise model from Eq. (3) (whereas Nadakuditi (2014) considers the reduced-noise model from Eq. (2)). 2.4 A numerical study We report the results of a numerical study to gain insight into our theoretical results. We consider model (2), where Xi = `1/2 zi u + εi , where the noise is heteroskedastic, εi ∼ N (0, Γ), and Γ is 10 the diagonal matrix of eigenvalues of a p × p autoregressive (AR) covariance matrix of order 1, with entries Σij = ρ\|i−j\| . We set Yi = Di Xi , where Di have iid Bernoulli(δ) entries. We set the missingness parameter δ to the values 1/3, 2/3 and 1. We choose the AR autocorrelation coefficient ρ = 0.5, and vary the spikes ` from 0 to 3.5. We compute numerically the formulas in Thm. 2.1, using the recent Spectrode method (Dobriban, 2015), see Sec. 5.1.6 for the details. We compare this with a Monte Carlo simulation with n = 200, γ = 1/2, zi , u generated as Gaussian random variables, and the results averaged over 10 Monte Carlo trials. The results—displayed on Fig. 1—allow us to study the effect of reduction on spiked models. In particular, we observe the following phenomena: • The theory and simulations show good agreement. For eigenvalues, the results are very accurate. For the cosine, the results are more variable, and especially so for small δ. • In the left plot of Fig. 1, we see that the empirical spike is an increasing function of the population spike `. Moreover, the location of the phase transition (PT) decreases with δ, i.e., reduction degrades the critical signal strength. • Similarly, in the right plot of Fig. 1, we see that the cosine between population and empirical PCs increases with the population spike `. For a given `, the cosine decreases as δ → 0. It is not hard, but beyond our scope, to formalize the last two observations into theorems. 3 Covariance matrix estimation In this section, we develop methods for covariance estimation in the reduced-noise model Yi = Di Xi + Di εi (Secs. 3.2, 3.3) and in the unreduced-noise model Yi = Di Xi + εi (Sec. 3.4). We also discuss some related work in Sec. 3.5. Finally, we present numerical experiments illustrating the results in Sec. 3.6. We restrict our attention to a special case of the diagonally reduced model we considered in Sec. 2. We assume as in Sec. 2.2 that the entries of the reduction matrices Di are independently and identically distributed. We also suppose that the noise εi is white, with variance 1 on each coordinate; that is, Cov(εi ) = Ip . We will also require that the coordinates of εi and the diagonal entries of Di both have finite eighth moments. Recall that in the setting of Cor. 2.2, we have 2 µ = EDij and also that σ 2 = Var(Dij ). The second moment is m = EDij = µ2 + σ 2 , and δ = µ2 /m. For data missing uniformly at random, m = µ = δ is the probability that each entry is observed. 3.1 The reduced-noise model In the reduced-noise model, we observe n samples of the random vector Y = D(S + ε). It is then easy to see that we have the following formulas relating the covariance matrix of the signal ΣS and the covariance matrix of the observation ΣY : ΣY = µ2 ΣS + σ 2 diag(ΣS ) + mIp 2 ΣS = σ 1 ΣY − diag(ΣY ) − Ip µ2 mµ2 11 (9) Pn These equations make it clear that the sample covariance matrix Σ̂Y = n−1 i=1 Yi Yi> is a biased estimator of the signal covariance matrix ΣS . Based on the second equation, we consider the following debiased estimator of ΣS : Σ̂S = 1 σ2 Σ̂ − diag(Σ̂Y ) − Ip . Y µ2 mµ2 (10) Here we assume for simplicity that µ, σ 2 are known; but these scalar parameters are straightforward to estimate from the observed Di . In the special case of data missing completely at random, i.e., of iid sampling of entries with probability δ, we have µ2 = δ 2 and m = δ, so this formula becomes 1 1 1 (11) Σ̂S = 2 Σ̂Y + − 2 diag(Σ̂Y ) − Ip . δ δ δ If our goal is to estimate ΣX = ΣS + Ip instead of ΣS , the corresponding unbiased estimator is Σ̂X = Σ̂Y /δ 2 + (δ − 1) diag(Σ̂Y )/δ 2 . This recovers the unbiased estimator of ΣX proposed by Lounici (2014). That paper proposes to estimate ΣX by applying the soft-thresholding function ητ (λ) = (λ − τ )+ to the empirical eigenvalues λ of the covariance Σ̂X . Lounici (2014) proves error bounds for this estimator in both operator and Frobenius norm losses, for covariance matrices ΣX of small effective rank ref f (Σ) = tr(Σ)/kΣkop . In contrast, we want to estimate the covariance matrix ΣS of the signal. For this different task, in the spiked covariance model, the function ητ is not optimal, as we will show in Section 3.3. In the next section, we employ the probabilistic results from Sec. 2 to determine the asymptotic spectral theory of Σ̂Y . In particular, we find asymptotic formulas for the eigenvalues, and the angles between its PCs and those of the population covariance ΣS . Next, we show how to use these results in conjunction with the theory of Donoho et al. (2013) to derive optimal non-linearities η of the spectrum of Σ̂Y to estimate ΣS for a variety of loss functions. 3.2 The asymptotic spectral theory of Σ̂S in reduced-noise In this section, we will analyze the asymptotic spectral theory of the debiased estimator Σ̂S ; that is, the limiting eigenvalue distribution, spikes, and limiting angles of its top eigenvectors with those of ΣS . We will rely on Corollary 2.2 from Section 2 and an argument controlling the diagonal terms in the proof in Sec. 5.4.1. √ Corollary 3.1. Let 1 ≤ k ≤ r. Suppose that `k satisfies `k > γ/δ. Then in the limit p, n → ∞ and p/n → γ, the k th largest eigenvalue of Σ̂S , k = 1, . . . , r, converges almost surely to 1 γ 1 (δ`k + 1) 1 + − . (12) δ δ`k δ The distribution of the bottom p − r eigenvalues of Σ̂S converges to a shifted and scaled Marchenko√ √ Pastur distribution (µM P − 1)/δ supported on the interval [(1 − γ)2 − 1, (1 + γ)2 − 1]/δ. If û0k is the k th eigenvector of Σ̂Y and ûk is the k th eigenvector of Σ̂S , then almost surely we have limn→∞ hû0k , ûk i2 = 1 and lim hûk , uk i2 = n→∞ 12 1 − γ/(δ`k )2 . 1 + γ/(δ`k ) Table 2: Optimal covariance shrinkagepin unreduced-noise model. `˜ is the function defined by equation (16), δ = µ2 /m, and sk = 1 − c2k is the asymptotic sine of the angle between the empirical and population PCs. Loss function Eigenvalue Operator ˜ `(δλ+1) δ Squared Frobenius 2 ˜ ˜ `(δλ+1)·c (`(δλ+1)) δ Asymptotic loss References `1 s1 Pr (15), (17) k=1 (1 − c4k )`2k (18), (19) √ If `k ≤ γ/δ, then the top eigenvalue converges to the upper edge of the shifted MP distribution, and the cosine converges to 0. The shifted Marchenko-Pastur distribution arises as the limiting empirical spectral distribution of the eigenvalues corresponding to noise. This is also the case for the available case sample covariance of pure noise (Jurczak and Rohde, 2015). We will discuss the available-case estimator in Secs. 3.5 and 3.6. 3.3 Optimal shrinkage of the spectrum of Σ̂S in reduced-noise Having characterized the asymptotic spectrum of the debiased estimator Σ̂S , we can apply the technique of Donoho et al. (2013) to derive optimal shrinkers of the eigenvalues of Σ̂S to minimize various loss functions. Any of the 26 loss functions found in Donoho et al. (2013) can be adapted to the setting of diagonally reduced data. In Sec. 5.4.2, we carefully check the details of this program. Write the eigendecomposition of ΣS as ΣS = U ΛU > and the eigendecomposition of the debiased estimator Σ̂S as Σ̂S = Û Λ̂Û . For a given function η : R → [0, ∞), define the matrix Σ̂ηS by Σ̂ηS = Û η(Λ̂)Û > where η(Λ̂) is the diagonal matrix that replaces the k th diagonal element λ̂k of Λ̂ with η(λ̂k ). For any value of p, let Lp (A, B) denote a loss function between two p-by-p symmetric matrices A and B. We consider loss functions with two key properties: first, they must be orthogonally invariant; that is, Lp (A, B) = Lp (U AV, U BV ) for any orthogonal matrices U and V . Second, they must decompose over blocks. This means that if A1 , B1 ∈ Rp1 ×p1 and A2 , B2 ∈ Rp2 ×p2 where p = p1 + p2 , then either Lp (A1 ⊕ A2 , B1 ⊕ B2 ) = max{Lp1 (A1 , B1 ), Lp2 (A2 , B2 )}, in which case we say Lp is max-decomposable; or Lp (A1 ⊕ A2 , B1 ⊕ B2 ) = Lp1 (A1 , B1 ) + Lp2 (A2 , B2 ), in which case we say Lp is sum-decomposable. Operator norm loss Lp (A, B) = kA − Bkop is max-decomposable, whereas squared Frobenius norm loss Lp (A, B) = kA − Bk2F is sum-decomposable. Our goal is to find the function η that minimizes the asymptotic loss over certain classes; that is, we seek: η ∗ = arg min L∞ (ΣS , Σ̂ηS ) η where L∞ (ΣS , Σ̂ηS ) is the almost sure limit of Lp (ΣS , Σ̂ηS ) as n, p → ∞ and p/n → γ. We will show from first principles that this limit is well defined. As in Donoho et al. (2013), we will consider only those functions η that collapse the vicinity of the bulk to 0; that is, for which there is an ε > 0 such 13 that η(λ) = 0 whenever λ ≤ (1 + given in Cor. 3.1). In Sec. 5.4.2, we will show: √ γ)2 /δ − 1/δ + ε (this is the value of the upper bulk edge, as L∞ (ΣS , Σ̂ηS ) = L2r M r A2 (`k ), k=1 r M B2 (η(λk ), ck , sk ) (13) k=1 p ` 0 1 − c2 (`) ≥ 0 the , and ck = c(δ`k ), sk = s(δ`k ), where s(`) = 0 0 asymptotic sine of the angle between the empirical and population PCs, η(λ)c2 (δ`) η(λ)c(δ`)s(δ`) B2 (η(λ), c, s) = . η(λ)c(δ`)s(δ`) η(λ)s2 (δ`) where A2 (`) = Since the loss function is either max-decomposable or sum-decomposable, for η to minimize the right side, it is sufficient that it minimize every individual term L2 (A2 (`k ), B2 (η(λk ), ck , sk )). That is, the asymptotically optimal η minimizes the two-dimensional loss: η ∗ = argmin L2 (A2 (`), B2 (η(λ), c, s)). (14) η This dramatically simplifies the problem, as this minimization can often be done explicitly. Deriving the optimal η now depends on the particular choice of loss function. We consider two representative cases where a simple closed formula is easily found: operator norm loss, and squared Frobenius norm loss. The same recipe of explicitly solving the problem (14) can be used for any orthogonally-invariant and max- or sum-decomposable loss function, including those found in Donoho et al. (2013). 3.3.1 Operator norm loss/max-decomposable losses Since operator norm loss Lp (A, B) = kA − Bkop is max-decomposable, equation (13) implies that L∞ (ΣS , Σ̂ηS ) = max L2 (A(`k ), B(η(λk ), ck , sk )). 1≤k≤r Consequently, the asymptotically optimal η is the one that minimizes the two-dimensional loss function L2 (A(`k ), B(η(λk ), ck , sk )) = kA(`k ) − B(η(λk ), ck , sk )kop . Repeating the derivation in Donoho et al. (2013), the optimal η(λk ) sends λk back to its population value, `k . From formula (12) in Cor. 3.1, η ∗ (λ) = ˜ `(δλ + 1) δ where `˜ inverts the spike forward map ` 7→ λ(`; γ) defined in Sec. 2.2, p 2 ˜ = y − 1 − γ + (y − 1 − γ) − 4γ . `(y) 2 (15) (16) Direct computation shows that L2 (A(`k ), B(η ∗ (λk ), ck , sk )) = `k sk ; consequently, the asymptotic loss is given by the formula: ∗ L∞ (ΣS , Σ̂ηs ) = `1 s1 . 14 (17) ˜ Table 3: Optimal covariance p shrinkage in unreduced-noise model. ` is the function defined by 2 equation (16), and sk = 1 − ck is the asymptotic sine of the angle between the empirical and population PCs. 3.3.2 Loss function Eigenvalue Operator ˜ `(µ λ+1) µ2 Squared Frobenius ˜ 2 λ+1)·c2 (`(µ ˜ 2 λ+1)) `(µ µ2 2 Asymptotic loss References `1 s1 Pr (23), (17) k=1 (1 − c4k )`2k (24), (19) Frobenius norm loss/sum-decomposable losses Since the squared Frobenius loss Lp (A, B) = kA − Bk2F is sum-decomposable, equation (13) implies that L∞ (ΣS , Σ̂ηS ) = r X L2 (A(`k ), B(η(λk ), ck , sk )). k=1 Consequently, the asymptotically optimal η is the one that minimizes the two-dimensional loss function L2 (A(`k ), B(η(λk ), ck , sk )) = kA(`k ) − B(η(λk ), ck , sk )k2F . As derived in Donoho et al. ˜ k + 1)/δ, (2013), the value of η(λk ) that minimizes this is `k c2k . We have already seen that `k = `(δλ 2 2 ˜ where ` is the function defined by (16). Consequently, with c (`) = c (`; γ) being the cosine forward map, the formula for η ∗ (λ) is η ∗ (λ) = ˜ ˜ `(δλ + 1) · c2 (`(δλ + 1)) . δ (18) A straightforward computation shows that L2 (A(`k ), B(`k c2k , ck , sk )) = (1 − c4k )`2k , and consequently, the asymptotic loss is given by the formula: ∗ L∞ (ΣS , Σ̂ηs ) = r X (1 − c4k )`2k . (19) k=1 3.4 The unreduced-noise model In the unreduced-noise model Yi = Di si + εi we can develop similar methods for covariance estimation. The formulas relating ΣS to ΣY are ΣY = µ2 ΣS + σ 2 diag(ΣS ) + Ip (20) 2 ΣS = σ 1 1 ΣY − diag(ΣY ) − Ip . µ2 mµ2 m Consequently, the analogous debiased covariance estimator is: Σ̂S = 1 σ2 1 Σ̂Y − diag(Σ̂Y ) − Ip . 2 2 µ mµ m 15 (21) In this section, we will derive optimal shrinkers of the spectrum of Σ̂S in the unreduced-noise model using the same technique as for the reduced-noise model in Sec. 3.3. Our analysis rests on the following result, proved in Sec. 5.4.1: √ Corollary 3.2. Let 1 ≤ k ≤ r. Suppose that `k satisfies `k > γ/µ2 . Then in the limit p, n → ∞ and p/n → γ, the k th largest eigenvalue of Σ̂S converges almost surely to 1 2 γ 1 (µ ` + 1) 1 + (22) − 2. k 2 2 µ µ `k µ The distribution of the bottom p − r eigenvalues of Σ̂S converges to a shifted Marchenko-Pastur √ √ distribution supported on the interval [(1 − γ)2 − 1, (1 + γ)2 − 1]/µ2 . If û0k is the k th eigenvector of Σ̂Y and ûk is the k th eigenvector of Σ̂S , then almost surely we have limn→∞ hû0k , ûk i2 = 1 and lim hûk , uk i2 = n→∞ 1 − γ/(µ2 `k )2 . 1 + γ/(µ2 `k ) √ If `k ≤ γ/µ2 , then the top eigenvalue converges to the upper edge of the shifted MP distribution, and the cosine converges to 0. ˜ 2 λ̂k + Given an empirical eigenvalue λ̂k of Σ̂S , we estimate the population eigenvalue `k by `(µ 2 1)/µ where `˜ is the function given by formula (16). This estimator converges almost surely to the √ true value `k if `k exceeds the threshold γ/µ2 . This also gives us an estimator of the squared ˜ 2 λ̂k + 1)). We can now derive the optimal non-linear functions on the cosine, by the formula c(`(µ spectrum. For operator norm loss, we have η ∗ (λ) = ˜ 2 λ + 1) `(µ , µ2 (23) ∗ which incurs an asymptotic loss of L∞ (ΣS , Σ̂ηs ) = `1 s1 . For squared Frobenius norm loss, the optimal non-linearity is ˜ 2 λ + 1) · c(`(µ ˜ 2 λ + 1)) `(µ 2 µ Pr ∗ and the asymptotic loss is L∞ (ΣS , Σ̂ηs ) = k=1 (1 − c4k )`2k . η ∗ (λ) = 3.5 (24) Alternative linear systems for estimating ΣS The optimal shrinkers derived in Secs. 3.3 and 3.4 for estimating the covariance ΣS from the reduced-noise observations start from the debiased estimators (10) for reduced-noise and (21) for unreduced noise. Another way of viewing these estimators is as the solution to a linear system: for reduced-noise, this system is given by equation (9), and for unreduced-noise by equation (20). Of course, there are other linear systems yielding unbiased estimators whose spectrum we could shrink. The papers Katsevich et al. (2015); Andén et al. (2015); Bhamre et al. (2016) consider such an estimator, which we will briefly discuss here. 16 By definition, for any k = 1, . . . , n, ΣS = ES,ε [Sk Sk> ], where ES,ε denotes the expectation with respect to the random Sk and εk , but not the Dk . We can therefore write in the unreduced-noise model Yk = Dk Sk + εk : Dk ΣDk = ES,ε [(Dk Sk )(Dk Sk )> ] = ES,ε [(Yk − εk )(Yk − εk )> ] = ES,ε [Yk Yk> ] + Ip . (25) If we knew the values of ES,ε [Yk Yk> ] for every k, we could derive an unbiased estimator of ΣS by solving the n equations given by (25). The papers Katsevich et al. (2015); Andén et al. (2015); Bhamre et al. (2016) instead substitute the observed value Yk Yk> for its expected value, and derive an unbiased estimator of ΣS by the minimization problem n Σ̂S = arg min Σ 1X kDk ΣDk> − Yk Yk> − Ip k2F . n k=1 Differentiating in Σ, we see that Σ̂S must satisfy the linear system: n n n k=1 k=1 k=1 1X > 1X > 1X > Dk Dk ΣDk> Dk = (Dk Yk )(Dk> Yk )> − Dk Dk . n n n We now consider another linear system, defined by averaging the n equations (25): n n k=1 k=1 1X 1X Dk ΣS Dk = ES,ε [Yk Yk> ] − Ip . n n Note that the matrices Dk are fixed in this equation. By replacing ES,ε [Yk Yk> ] with the estimate Yk Yk> , we define a new estimator of ΣS as the solution to the equation n n k=1 k=1 1X 1X Dk ΣDk = Yk Yk> − Ip . n n (26) In the case of reduced-noise, we can repeat the same derivation and arrive at an unbiased estimator that satisfies the system n n n k=1 k=1 k=1 1X 1X 1X 2 Dk ΣDk = Yk Yk> − Dk . n n n (27) We will denote the estimator solving (26) (in the unreduced-noise case) and (27) (in the reduced noise case) by Σ̂0s . Taking the expectation of each side of (27), we arrive at the linear system (20), which defines our estimator Σ̂S in the unreduced-noise model. Similarly, taking the expectation of each side of (26), we arrive at the linear system (9), which defines our estimator Σ̂S in the reducednoise model. Consequently, we expect that, in the limit n, p → ∞, Σ̂S and Σ̂0S will be close. In fact, we can show that the relative error of the estimators converges to 0, as stated in the folowing proposition (proved in Sec. 5.4.3): Proposition 3.3. In both the reduced-noise and unreduced-noise models, the relative difference kΣ̂S − Σ̂0S kF /kΣ̂S kF → 0 almost surely as n, p → ∞ and p/n → γ. 17 3.6 Numerical experiments We perform experiments with missing data, where the diagonal entries of the reduction matrices Di are independent Bernoulli(δ) random variables. In this case, µ = m = δ, and σ 2 = δ(1 − δ). The unbiased estimator to which we apply shrinkage is given by the formula (11). In all experiments, both the signal and the noise are drawn from Gaussian distributions. 3.6.1 The errors in estimating the covariance In the first experiment, we illustrate the dependence of the asymptotic errors on the parameters δ and γ. The clean signal vectors Si are drawn from a rank 1 Gaussian. The noise is also Gaussian noise, of unit variance; the ambient dimension p is fixed at p = 1200 in this experiment, while the number of samples n varies with γ. The top two rows of Fig. 2 shows the errors in estimation for Frobenius loss and operator norm loss, as functions of the parameters γ and δ. The error bars cover the empirical mean error, plus/minus two standard deviations over 200 runs of the experiment. Several phenomena are apparent in these plots. First, the empirical mean of the errors is wellapproximated by the asymptotic error formulas (17) and (19), especially as the number of samples grows (corresponding to smaller γ, as n = p/γ = 1200/γ). Second, the errors decay as δ approaches 1, which is expected as larger δ increases the effective signal strength. Third, the errors grow as γ approaches 1; this is also expected, since large γ leads to higher dimensional problems. 4 4.1 Denoising Setup It is often of interest to denoise the observations Yi and predict the signal components Si . We envision a scenario where the data (Yi , Di ), i = 1, . . . , n is already collected, and we construct the denoisers using this dataset. With in-sample denoising we denoise Yi to predict the signal components Si . This makes sense in many applications where we want to use the entire dataset to construct the denoiser. A closely related scenario is out-of-sample denoising, where we want to denoise a new datapoint (Y0 , D0 ). This arises in applications where new samples are made available after an initial preprocessing of Y1 , . . . , Yn is performed, and it is not desired or not feasible to repeat this processing on the augmented dataset (Y0 , D0 ), (Y1 , D1 ), . . . , (Yn , Dn ) for every new data point. While these two settings are very closely related, it turns out, perhaps surprisingly, that the optimal way to construct the denoisers differs substantially between the two. The reason turns out to be closely related to the observation that in high dimensions, the addition of a single datapoint changes the direction of the PCs. We will explain this phenomenon in detail below. We first study the reduced-noise model from (2) in the setting of Cor. 2.2, where the diagonal entries of Di are drawn iid from a distribution with mean µ and variance σ 2 , and the observations are Yi = Di (Si + εi ). is a special type of random effects model, as the “effects” zik of the PrThis1/2 “factors” uk in Si = k=1 `k zik uk are random from sample to sample. As usual in random effects models, the optimal way to predict Si from a mean squared error (MSE) perspective is to use the Best Linear Predictor—or BLP—(e.g., Searle et al., 2009, Sec. 7.4). The BLP of Si is the predictor Ŝi = M Yi that minimizes EkŜi − Si k2 . −1 It is well known that the BLP is ŜiBLP = Cov [Si , Yi ] Cov [Yi , Yi ] Yi . Under the assumptions of Cor. 2.2, we can show (see Sec. 5.5) that the BLP has the same asymptotic MSE properties as 18 500 γ=.9 γ=.6 γ=.3 450 γ=.9 γ=.6 γ=.3 18 16 errors (operator norm loss) errors (Frobenius loss) 400 350 300 250 200 150 100 14 12 10 8 6 4 2 50 0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 δ 0.6 0.8 1 δ 500 δ=.2 δ=.5 δ=.8 450 δ=.2 δ=.5 δ=.8 18 16 errors (operator norm loss) errors (Frobenius loss) 400 350 300 250 200 150 100 14 12 10 8 6 4 2 50 0 0 0.2 0.4 0.6 0.8 1 γ 0.2 0.4 0.6 0.8 1 γ Figure 2: Top row : Estimation error of the covariance as a function of δ. Left: Frobenius shrinker. Right: operator norm shrinker. For each δ, 200 Monte Carlo tests were averaged. The error bars cover two standard deviations; the lines go through the predicted errors. Bottom row : Same plot as a function of γ. 19 Table 4: Denoising in the single-spiked case. BLP: population Best Linear Predictor using u. EBLP: in-sample empirical Best Linear Predictor using û. EBLP-OOS: out-of-sample empirical Best Linear Predictor using û. Here we abbreviate λ = λ(δ`; γ), c2 = c2 (δ`; γ), s2 = 1 − c2 , β = 1 + γ/(δ`). Name Definition Asy MSE Asy Opt η Asy Opt MSE Ref BLP η · uu> Yi (1 − ηµ)2 ` + η 2 m µ` µ2 `+m m` µ2 `+m Thm. 4.1 EBLP η · ûû> Yi ` + η 2 · mλ − 2η · µ`c2 · β µ`c2 µ2 `+m `· EBLP-OOS η · ûû> Yi ` + η 2 · (µ2 `c2 + m) − 2η · µ`c2 2 µ`c µ2 `c2 +m `· µ2 `c2 s2 +m µ2 `c2 +m 2 2 2 µ `c s +m µ2 `c2 +m Thm. 4.1 Prop. 4.3 a denoiser of the following simpler form: Ŝiτ,B = r X τk uk u> k Yi . (28) k=1 The denoisers Ŝiτ,B are indexed by τ = (τ1 , . . . , τr ), and our argument shows that with the choice τk = µ`k /(µ2 `k + m) they are in fact asymptotically equivalent to BLP denoisers. In practice, the true PCs uk are not known, so we use the Empirical BLP (EBLP), where we estimate the unknown parameters uk using the entire dataset. Here we will use the k-th top right singular vector ûk of the n × p matrix Y with rows Yi> as an estimator of uk . In analogy with the simplified form of the denoisers in (28), we will consider EBLPs scaled by η = (η1 , . . . , ηr ) having the form: r X Ŝiη = ηk ûk û> (29) k Yi . k=1 Our goal is to find the optimal scalars ηk , and characterize their MSE. 4.2 In-sample denoising First, we will study in-sample denoising, where the data to be denoised are also used to construct the denoisers. Our main findings for the asymptotic MSE (AMSE) are summarized in Table 4 (in the single-spiked case) and in the following theorem, proved in Sec. 5.6. Theorem 4.1 (In-sample denoising). In the setting of Cor. 2.2 consider in-sample best linear predictors (BLP) of the signals Si based on the observations Yi . Pr 1. The BLP denoisers Ŝiτ,B = k=1 τk · uk u> k Yi based on the population singular vectors have an AMSE limn,p→∞ EkSi − Ŝiτ,B k2 of AM SE B (τ1 , . . . , τr ; `1 , . . . , `r , γ) = r X k=1 20 AM SE B (τr ; `r , γ), where AM SE B (τ ; `, γ) = (1 − τ µ)2 ` + τ 2 m is the AMSE of the BLP Ŝiτ,B = τ · uu> Yi in a single-spiked model with spike strength ` under the assumptions of Cor. 2.2. The asymptotically optimal coefficients are µ`k τk∗ = 2 . µ `k + m Pr 2. The EBLP denoisers Ŝiη = k=1 ηk · ûk û> k Yi , based on the empirical singular vectors have an AMSE limn,p→∞ EkSi − Ŝiη k2 of AM SE E (η1 , . . . , ηr ; `1 , . . . , `r , γ) = r X AM SE E (ηr ; `r , γ), (30) k=1 where AM SE E (η; `, γ) = ` + η 2 · m · λ(δ`; γ) − 2η · µ` · c2 (δ`; γ) · β is the AMSE of EBLP Ŝiη = η · ûû> Yi in a single-spiked model with spike strength ` under the assumptions of Cor. 2.2. Here λ(δ`; γ) is the limit empirical spike, while c2 = c2 (δ`; γ) is the squared cosine, both corresponding to spike strength δ`, defined in Cor. 2.2. The asymptotically optimal coefficients are ηk∗ = µ`k c2k , µ2 `k + m (31) where c2k = c2 (δ`; γ). The basic discovery is that the optimal coefficients for empirical PCs are different from those for population PCs. The coefficients using empirical PCs are reduced by a squared cosine compared to the coefficients using population PCs: ηk∗ = c2k τk∗ . Note that we chose the optimal coefficients to minimize the limiting MSE. However, the limiting MSE, and thus the optimal coefficients, depend on the unknown parameters δ, `k , c2k . To make this a practical method, we can estimate the unknown parameters. The missingness parameter δ can be estimated by plug-in. Based on Cor. 2.2, the estimation of `k and c2k can be done by inverting the spike forward map `k → λ(`k ), see e.g., Bai and Ding (2012); Donoho et al. (2013). It is worth pointing out that the squared error kSi − Ŝi k2 for each individual column Si of BLP and EBLP does not converge in probability or a.s. In fact, its variability is of unit order, and does not decrease as n, p → ∞. However, as shown in Thm 4.1, its expectation—the MSE—does converge. Furthermore, we will show in Sec. 4.2.2 that the average error of EBLP over the entire data matrix converges almost surely (to the AMSE for a single column). As we will show, this is because EBLP, when applied to all columns of the data matrix, is a singular value shrinkage estimator, defined by modifying the singular values of the data matrix while leaving the empirical singular vectors fixed. As a consequence, in the case of missing data the optimal AMSE agrees with that achieved by optimal singular value shrinkage described in Nadakuditi (2014) and Gavish and Donoho (2014). We emphasize, however, that Thm. 4.1 applies to individual data points, not just to the entire matrix. 4.2.1 Comments on the proof Part 2 of Thm. 4.1 is a nontrivial result, because the AMSE is determined by stochastically dependent random quantities such as u> k Di ûk . These are challenging to study, because Di and ûk are dependent random variables. We will analyze these quantities from first principles. 21 While we will explain our method in detail later, briefly, we use the outlier equation approach (see Lemma 5.14), which reduces studying the inner products w> ûk to certain inner products w> r, where r = r(Y ) are vectors that depend on the entire dataset but not directly on the singular vector ûk . As we will see, these inner products are more convenient to study. The basic method was introduced by Benaych-Georges and Nadakuditi (2012), who used it to study the angles between uk and ûk . We extend their approach to other angles, which are more challenging to study. 4.2.2 Singular value shrinkage and the almost sure convergence of the error in the reduced-noise model A well-studied approach to matrix denoising is known as singular value shrinkage. Here, the singular values of the data matrix Y are replaced with shrunken versions, analogous to the eigenvalue shrinkage of covariance matrices studied in Sec. 3. When applied to every Prrow of the data matrix Y , EBLP is a singular value shrinkage algorithm: indeed, since Ŝiη = k=1 ηk · ûk û> k Yi , we can write the entire denoised matrix in the form Ŝ η = r X ηk · ûk û> kY = k=1 r X ηk σk (Y ) · ûk v̂k> . (32) k=1 In other words, the denoised matrix Ŝ η has the same singular vectors as the data matrix Y , where the singular values have been moved from σk (Y ) to ηk σk (Y ) for k = 1, . . . , r (and the remaining ones set to 0). It turns out that for the value of ηk∗ given by equation (31), ηk σk (Y ) is the optimal singular value of the denoised matrix if we seek to minimize the asymptotic Frobenius loss kS − Ŝ η k2F . The proof of this fact follows easily from the analysis of the optimal singular value shrinkers given by Gavish and Donoho (2014). This paper derives optimal shrinkers only when there is no missing data, and the task is to recover a low rank matrix from noisy observations of its entries. The k th singular value of the asymptotically optimal shrunken matrix (with respect to Frobenius loss) 1/2 is `k ck c̃k , where ck is the asymptotic cosine of the angle between the right population singular vector and the right empirical singular vector, and c̃k is the asymptotic cosine of the angle between the left population singular vector and the left empirical singular vector. In the models considered in Gavish and Donoho (2014), these values are ck = c(`k ; γ), where c is the cosine forward map from (6), and c̃k = c̃(`k ; γ) ≥ 0, where c̃2 (`; γ) = (1 − γ/`2 )/(1 + 1/`). As we can see from Cor. 2.2, in our data model the reduced model behaves like the original model, but with spike strengths reduced from `k to δ`k . In particular, the value of c2k is c2k = c2 (δ`; γ). In fact, it is easy to see from the proof of that this is true for the left singular vectors as well; that is, c̃2 = c̃2 (δ`k ; γ). We now quote the result of Gavish and Donoho (2014) that the optimal singular value is equal 1/2 to `k ck c̃k ; since formula (32) shows that this is equal to ηk∗ σk (Y ), the optimal coefficient ηk∗ is: 1/2 ηk∗ = `k c(δ`k ; γ)c̃(δ`k ; γ) c̃(δ`k ; γ) 1/2 = `k c2 (δ`k ; γ) 1/2 σk (Y ) ` σk (Y )c(δ`k ; γ) k Substituting the asymptotic value for σk (Y ) given from (7), a straightforward algebraic manipulation shows that 1/2 1/2 ηk∗ = `k c2 (δ`k ; γ) `k `k c2k = . δ`k + 1 δ`k + 1 22 This agrees with the value for the optimal coefficient ηk for EBLP derived in Thm. 4.1. In particular, the EBLP estimator we derive for the entire data matrix and the optimal singular value shrinkage estimator are identical. Furthermore, from the analysis of Gavish and Donoho (2014), the error of the entire denoised matrix converges almost surely to the expression given in equation (30). We have proved the following theorem: Theorem 4.2. If Y = [Y1 , . . . , Yn ]> is the n-by-p matrix of observations in the reduced-noise model, then the asymptotically optimal singular value shrinkage estimator of the n-by-p signal matrix S = [S1 , . . . , Sn ]> is equal to the EBLP estimator defined in Thm. 4.1. Furthermore, the squared Frobenius error of this estimator converges a.s. to the formula (30). 4.2.3 Comparison with matrix completion algorithms In the case when the reductions matrices are binary, the task of denoising the data matrix Y to approximate S is a matrix completion problem – see, for instance, Candès and Recht (2009), Recht (2011), Keshevan et al. (2009) and Keshevan et al. (2010). A typical model for matrix completion is a low-rank matrix with eigenvectors satisfying an incoherence condition, with order O(n·poly(log n)) entries revealed uniformly at random. In the setting we study in this paper, the number of observed entries of the matrix Y will be O(n2 ), almost an order of magnitude more. When the noise level is very small compared to the magnitude of the entries in the clean matrix (or put differently, when `r 1), then the smaller number of samples is sufficient to recover the low-rank matrix to high accuracy. The references listed above provide several methods with these guarantees. We compare the in-sample EBLP denoiser to the OptSpace method for matrix completion found in Keshevan et al. (2009) and Keshevan et al. (2010). Briefly, their method removes rows/columns with too many observations, truncates the singular values of the data matrix Y , and then cleans up the resulting matrix by an iterative algorithm. In Figure 3 we plot the squared Frobenius errors in reconstructing a rank 1 matrix, for different choices of spike size ` and missingness parameter δ. In this experiment, both the signal and noise are Gaussian, γ = .8 and the dimension p = 400. Each data point plotted is the average error over 100 Monte Carlo runs of the experiment. We observe that the in-sample EBLP outperforms OptSpace when ` is small relative to the noise level. As the size of ` grows, OptSpace’s performance improves, and for small δ and large ` it outperforms EBLP. This is consistent with the guarantees provided for OptSpace; it does well in the low-noise regime, with a small number of samples. 4.3 Out-of-sample denoising We now study out-of-sample denoising in the reduced-noise model, where we denoise new datapoints from the same distribution using a denoiser constructed on an existing dataset. This is typically faster than recomputing the denoiser on the entire dataset. For the oracle BLP denoiser, which assumes knowledge of u, this is the same as in-sample denoising. For the EBLP, however, it turns out that the optimal shrinkage coefficients in this case are different. To analyze this case, let Y0 = D0 X0 be the new sample from the same distribution. We evaluate the limit of the out-of-sample mean squared prediction error EkS0 − Ŝ0η k2 of the EBLP Pr Ŝ0η = k=1 ηk ûk û> k Y0 , where ûk were formed based on Yi , i ≥ 1. 23 δ = .1 1.8 δ = .7 0.8 EBLP error OptSpace error 1.6 EBLP error OptSpace error 0.7 1.4 0.6 average error average error 1.2 1 0.8 0.6 0.4 0.3 0.2 0.4 0.1 0.2 0 20 0.5 40 60 80 100 120 0 20 140 ℓ 40 60 80 100 120 ℓ Figure 3: Comparison of EBLP and OptSpace for completing/denoising the matrix, for different signal strengths. Left; δ = 1/10. Right: δ = 7/10. Errors are measured as the squared Frobenius norm between the estimator and the full, clean, rank 1 matrix. Every experiment was averaged over 100 runs, with γ = .8 and p = 400. Theorem 4.3. In the setting of Cor. 2.2 consider out-of-sample denoising of a new sample Y0 using the empirical BLP Ŝ0η based on the observations Yi , i = 1, . . . , n. Then the limit of the out-of-sample prediction error is r X η,o E (η1 , . . . , ηr ; `1 , . . . , `r , γ) = E ηk ,o (ηr ; `r , γ), k=1 where E η,o (η; `, γ) = ` + η 2 · (m + µ2 `c2 ) − 2η · µ`c2 is the out-of-sample prediction error of EBLP in a single-spiked model with spike ` under the assumptions of Cor. 2.2. The asymptotically optimal shrinkage coefficients are µ`k c2k ηk∗ = . m + µ2 `k c2k See Sec. 5.10 for the proof. The key point is that the optimal shrinkage for out-of-sample prediction is different from both of the shrinkers from in-sample denoising. We also mention that in the special case when Di = Ip for all i, the optimal shrinkage formula matches the one obtained by Singer and Wu (2013), under slightly more restrictive assumptions. It may be counterintuitive that the optimal coefficient changes when a single data point Y0 is added. However, this can be understood because in high-dimensions, a single data point can drastically change the empirical eigenvectors. For an illustration Pn in a simpler setting, consider the sample covariance matrix based on n samples, Σ̂Y,n = n−1 k=1 Yk Yk> and the corresponding 24 140 sample covariance Σ̂Y,n+1 based on the n + 1 samples Y0 , Y1 , . . . , Yn . Their difference is Σ̂Y,n − Σ̂Y,n+1 = n+1 X 1 1 1 1 Y0 Y0> + Yk Yk> = Y0 Y0> + Σ̂Y,n+1 . n n(n + 1) n n k=1 Since the operator norm of Σ̂Y,n+1 converges a.s. to a finite quantity, the term Σ̂Y,n+1 /n is asymptotically negligible. However, the operator norm of Y0 Y0> /n of size kY0 k2 /n, which converges a.s. to δ. Since the spectral distributions of Σ̂Y,n and Σ̂Y,n+1 converge to the same value, the fact that kΣ̂Y,n − Σ̂Y,n+1 kop is of order 1 is due to the fact that the addition of a single data point Y0 completely changes the direction of the empirical PCs of the data. To summarize and better understand our findings, we plot the optimal shrinkage coefficients and MSE for the three scenarios (BLP, in-sample optimal empirical BLP, and out-of-sample optimal empirical BLP) in a single-spiked model with γ = 1/2 and Di = Ip for all i, on Fig. 4. The optimal shrinkage for out-of-sample EBLP is intermediate between the stronger in-sample EBLP and the weaker out-of-sample BLP shrinkage coefficients. However, perhaps unexpectedly the MSE for the two EBLP scenarios agrees exactly! This prompts us to state the following result, proved in Sec. 5.11. Proposition 4.4 (In-sample vs out-of-sample EBLP). The asymptotically optimal shrinkage coefficient for in-sample EBLP is smaller than the asymptotically optimal shrinkage coefficient for the out-of-sample EBLP. However, the asymptotically optimal MSEs are equal in the two cases. This result is interesting, because it shows that same MSE can be achieved out-of-sample as in-sample. Out-of-sample denoising should be harder, because it involves a new datapoint never seen before. The ”hardness” of out-of-sample denoising should be observed in the leading order finite sample (n) correction to the asymptotic MSE. However, the above result shows that this correction vanishes as n, p → ∞. Using the right amount of shrinkage, the same MSE can be achieved asymptotically even out of sample. 4.4 Unreduced-noise We now study the denoising problem under the unreduced-noise model (3), where Yi = Di Si + εi under the assumptions of Cor. 2.2. The analysis is similar to the reduced-noise model (2). The key conclusions are summarized in Table 5. −1 The BLP of Si based on Yi is ŜiBLP = Cov [Si , Yi ] Cov [Yi , Yi ] Yi . Under the conditions of Cor. Pr 2.2, we can show (see Sec. 5.12.1) that this is asymptotically equivalent to Ŝiτ = k=1 τk uk u> k Yi , where τk = µ`k /(µ2 `k + 1). As before, the AMSE of Siτ with arbitrary τ decouples into the AMSEs over the different spikes `k , and those are equal to the AMSEs for the single-spiked model with spikes equal to `k . For a single-spiked model with spike ` we obtain in Sec. 5.12.1 that EkSi − Ŝiτ k2 → ` + τ 2 (`µ2 + 1) − 2τ `µ. The optimal coefficient is τ ∗ = `µ/(`µ2 + 1)—as it should be, based on the above discussion—and it has an AMSE of `/(`µ2 + 1). The advantage of this calculation is that it provides the MSE for any coefficient τ . Next, for the EBLP in the multispiked case, we use ûk as estimators of uk . Since the form of the BLP is the same as before, the EBLP scaled by η = (η1 , . . . , ηr ) have the form in (29). To compute 25 Figure 4: Optimal shrinkage coefficients (left) and MSE (right) for the three scenarios (BLP, insample optimal empirical BLP, and out-of-sample optimal empirical BLP) for γ = 1/2 and δ = 1. Table 5: Unreduced-noise: Denoising in the single-spiked case. BLP: population Best Linear Predictor using u. EBLP: in-sample empirical Best Linear Predictor using û. EBLP-OOS: out-of-sample empirical Best Linear Predictor using û. Here we abbreviate λ = λ(µ2 `; γ), c2 = c2 (µ2 `; γ). Name Definition Asy MSE Asy Opt η Asy Opt MSE BLP η · uu> Yi ` + η 2 · (µ2 ` + 1) − 2η · µ` µ` µ2 `+1 ` µ2 `+1 EBLP η · ûû> Yi ` + η 2 · λ − 2η · µ`c2 · [1 + γ/(µ2 `)] µ`c2 µ2 `+1 `−λ EBLP-OOS η · ûû> Yi ` + η 2 · (µ2 `c2 + 1) − 2η · µ`c2 µ`c2 µ2 `c2 +1 `· µ`c2 µ2 `+1 2 µ2 `c2 s2 +1 µ2 `c2 +1 the AMSE, it is again not hard to see that it decouples into the corresponding single-spiked AMSEs. In the single-spiked case, we find in Sec. 5.12.2 that with λ = λ(µ2 `; γ), c2 = c2 (µ2 `; γ), EkSi − Ŝiη k2 → ` + η 2 · λ − 2η · µ`c2 · [1 + γ/(µ2 `)]. This shows that the optimal coefficient is η ∗ = µ`c2 /(µ2 ` + 1). Finally, for out-of-sample EBLP denoising, it is again not hard to see that the AMSE decouples over the different spikes, and each term equals the AMSE in the single-spiked case. For the AMSE in the single-spiked case, we let (Y0 , D0 ) be a new sample, and find (Sec. 5.12.3) EkS0 − ηûû> Y0 k2 → ` + η 2 (1 + µ2 `c2 ) − 2ηµ`c2 . The optimal coefficient is η ∗ = µ`c2 /(1+µ2 `c2 ), while the optimal MSE is `(1+µ2 `c2 s2 )/(1+µ2 `c2 ). These findings are summarized in Table 5. 26 4.4.1 Singular value shrinkage and the almost sure convergence of the error in the unreduced-noise model As we discussed in Sec. 4.2.2 for the reduced-noise model, in-sample EBLP applied to every column of the data matrix Y = [Y1 , . . . , Yn ]> is equal to the asymptotically optimal singular value shrinkage estimator of the clean matrix S = [S1 , . . . , Sn ]> . The same reasoning applies verbatim to the unreduced-noise model, replacing the almost sure limits of the empirical eigenvalues and angles with their counterparts for unreduced-noise model. From Cor. 2.2, the value of the asymptotic value of the cosine of the angle between the right empirical singular vector and the right population singular vector is c2k is c2k = c2 (µ2 `; γ); and the same proof of this easily shows that the asymptotic cosine of the angle between the left singular vectors is c̃2 = c̃2 (δ`k ; γ). We now quote the formula from Gavish and Donoho (2014), which says that the optimal singular 1/2 value is equal to `k ck c̃k ; since formula (32) shows that this is equal to ηk∗ σk (Y ), the optimal coefficient ηk∗ is: 1/2 ηk∗ = `k c(δ`k ; γ)c̃(δ`k ; γ) c̃(δ`k ; γ) 1/2 = `k c2 (δ`k ; γ) 1/2 σk (Y ) ` σk (Y )c(δ`k ; γ) k Substituting the asymptotic value for σk (Y ) given from Cor. 2.2, it is easy to see that 1/2 1/2 ηk∗ = `k c2 (δ`k ; γ) `k `k c2k = . δ`k + 1 δ`k + 1 This agrees with the value for the optimal coefficient ηk for EBLP derived in Thm. 4.1. In particular, the EBLP estimator we derive for the entire data matrix and the optimal singular value shrinkage estimator are identical. Furthermore, from the analysis of Gavish and Donoho (2014), the error of the entire denoised matrix converges almost surely to the expression given in equation (30). We have proved the following theorem: Theorem 4.5. If Y = [Y1 , . . . , Yn ]> is the n-by-p matrix of observations in the unreduced-noise model, then the asymptotically optimal singular value shrinkage estimator of the n-by-p signal matrix S = [S1 , . . . , Sn ]> is equal to the EBLP estimator for the unreduced-noise model defined in Sec. 4.4. Furthermore, the squared Frobenius error of this estimator converges a.s. to the AMSE in the unreduced-noise model. 4.5 Simulations Next we perform a simulation to check the finite-sample accuracy of our formulas for denoising. In the reduced-noise model with Di = Ip , we consider a single-spiked model with n = 1000, p = 500, and generate iid Gaussian random variables zi , εi , as well as a standardized iid Gaussian random vector u. We vary the spike strength on a grid, and compare our formulas for in-sample theoretical MSE to those obtained by averaging the denoising error in the first sample kŜ1 − S1 k22 over 50 Monte Carlo simulations. The results in Fig. 5 show that the formulas are accurate up to the sampling error. This validates our results from Thm. 4.1. 27 Figure 5: In-sample MSE of denoising schemes: Theoretical results overlaid with Monte Carlo (MC) results. BLP: Best Linear Predictor assuming known population eigenvector u. Emp BLP: Empirical Best Linear Predictor, using the empirical eigenvector û, with the sub-optimal shrinkage coefficient from the BLP. Opt Emp BLP: Empirical Best Linear Predictor using optimal shrinkage coefficient. No missing data (δ = 1). γ = 1/2. Acknowledgements The authors wish to thank Joakim Anden, Tejal Bhamre, Xiuyuan Cheng, David Donoho, and Iain Johnstone for helpful discussions. 5 5.1 Proofs Proof of Thm. 2.1 The proof of Thm. 2.1 spans multiple sections, until Sec. 5.2.1. The proof of the claims under observation models (2) and (3) are very similar. Therefore, we present the proof of the result under model (2), and outline the argument for model (3) in Sec. 5.1.5. Moreover, to illustrate the idea of the proof, we first prove the single-spiked case, i.e., when r = 1. The proof of the multispiked extension is provided in Sec. 5.2. The form Di = µ + Σ1/2 Ei of the reduction matrices implies the following decomposition for the observations Yi : Yi = (µ + Σ1/2 Ei )Xi = µ(`1/2 zi u + εi ) + Σ1/2 Ei Xi = `1/2 zi µu + [µεi + Σ1/2 Ei Xi ]. This suggests a “signal+noise” decomposition for the reduced vectors Yi . Let us denote by ν = µu/ξ 1/2 the normalized reduced signal, where ξ = kµuk2 → τ , and by ε∗i = µεi +Σ1/2 Ei Xi the noise component. The noise has two parts: µεi is due to sampling, while Σ1/2 Ei Xi is due to projection. 28 In matrix form, with the n × p matrix Y having rows Yi> : Y = (ξ`)1/2 Z̃ν > + E ∗ . (33) This suggests that after reduction, the signal strength ` changes to ξ`, while the noise structure changes from εi to ε∗i . This is not obvious, however, because the noise ε∗i is functionally dependent on the signal Xi . Therefore we cannot rely on existing results. Instead, we will analyze the model from first principles, and show that the dependence is asymptotically negligible. For non-diagonal reduction matrices Di , the depencence may be asymptotically non-negligible; this explains why we currently need the diagonal assumption. 5.1.1 Proof outline We will extend the technique of Benaych-Georges and Nadakuditi (2012) to characterize the spiked eigenvalues in the model (33). We denote the normalized vector Z = n−1/2 Z̃, the normalized noise N = n−1/2 E ∗ and the normalized observable matrix Ỹ = n−1/2 Y . Then, our model is Ỹ = (ξ`)1/2 · Zν > + N. (34) We will assume that n, p → ∞ such that p/n → γ > 0. For simplicity of notation, we will first assume that n ≤ p, implying that γ ≥ 1. It is easy to see that everything works when n ≥ p. By Lemma 4.1 of Benaych-Georges and Nadakuditi (2012), the singular values of Ỹ that are not singular values of N are the positive reals t such that the 2-by-2 matrix t · Z > (t2 In − N N > )−1 Z Z > (t2 In − N N > )−1 N ν 0 (ξ`)−1/2 Mn (t) = > > 2 − ν N (t In − N N > )−1 Z t · ν > (t2 Ip − N > N )−1 ν (ξ`)−1/2 0 is not invertible, i.e., det[Mn (t)] = 0. We will find almost sure limits of the entries of Mn (t), to show that it converges to a deterministic matrix M (t). Solving the equation det[M (t)] = 0 will provide an equation for the almost sure limit of the spiked singular values of Ỹ . For this we will prove the following results: Lemma 5.1 (The noise matrix). The noise matrix N has the following properties: 1. The eigenvalue distribution of N > N converges almost surely (a.s.) to the Marchenko-Pastur distribution Fγ,H with aspect ratio γ ≥ 1. 2. The top eigenvalue of N > N converges a.s. to the upper edge b2H of the support of Fγ,H . This is proved in Sec. 5.1.2. For brevity we write b = bH . Compared to Benaych-Georges and Nadakuditi (2012), the key technical innovation here is to show that the contribution of the reduced signal component Σ1/2 Ei Si to E ∗ is negligible. This is accomplished by an ad-hoc bound on the operator norm of the contribution. Since Ỹ is a rank-one perturbation of N , it follows that the eigenvalue distribution of Ỹ > Ỹ also converges to the MP law Fγ,H . This proves the first claim of Thm 2.1. Moreover, since N N > has the same n eigenvalues as the nonzero of N > N , the two R eigenvalues −1 2 > −1 2 −1 facts in Lemma 5.1 imply that when t > b, n tr(t In −N N ) → (t −x) dF γ,H (x) = −m(t2 ). Here F γ,H (x) = γFγ,H (x) + (1 − γ)δ0 and m = mγ,H is the Stieltjes transform of F γ,H . Clearly this convergence is uniform in t. As a special note, when t is a singular value of the random matrix N , we formally define (t2 Ip − N > N )−1 = 0 and (t2 In − N N > )−1 = 0. When t > b, the complement of this event happens a.s. In fact, from Lemma 5.1 it follows that (t2 Ip − N > N )−1 has a.s. bounded operator norm. Next we control the quadratic forms in the matrix Mn . 29 Lemma 5.2 (The quadratic forms). When t > b, the quadratic forms in the matrix Mn (t) have the following properties: 1. Z > (t2 In − N N > )−1 Z − n−1 tr(t2 In − N N > )−1 → 0 a.s. 2. Z > (t2 In − N N > )−1 N ν → 0 a.s. 3. ν > (t2 Ip − N > N )−1 ν → −m(t2 ) a.s., where m = mγ,H is the Stieltjes transform of the Marchenko-Pastur distribution Fγ,H . Moreover the convergence of all three terms is uniform in t > b + c, for any c > 0. This is proved in Sec. 5.1.3. The key technical innovation is the proof of the third part. Most results for controlling quadratic forms x> Ax are concentration bounds for random x. Here x = ν is fixed, and matrix A = (t2 Ip − N > N )−1 is random instead. For this reason we adopt the “deterministic equivalents” technique of Bai et al. (2007) for quantities x> (zIp − N > N )−1 x, with the key novelty that we can take the imaginary part of the complex argument to zero. The latter observation is nontrivial, and mirrors similar techniques used recently in universality proofs in random matrix theory (see e.g., the review by Erdős and Yau, 2012). Lemmas 5.1 and 5.2 will imply that for t > b, the limit of Mn (t) is −t · m(t2 ) −(τ `)−1/2 . M (t) = −(τ `)−1/2 −t · m(t2 ) By the Weyl inequality, σ2 (Ỹ ) ≤ σ2 ((ξ`)1/2 · Zν > ) + σ1 (N ) = σ1 (N ). Since σ1 (N ) → b a.s. by Lemma 5.1, we obtain that σ2 (Ỹ ) → b a.s. Therefore for any ε > 0, a.s. only σ1 (Ỹ ) can be a singular value of Ỹ in (b + ε, ∞) that is not a singular value of N . It is easy to check that D(x) = x · m(x)m(x) is strictly decreasing on (b2 , ∞). Hence, denoting h = limt↓b D(t2 ), for τ ` > h, the equation D(t2 ) = 1/(τ `) has a unique solution t ∈ (b, ∞). By Lemma A.1 of Benaych-Georges and Nadakuditi (2012), we conclude that for τ ` > h, σ1 (Ỹ ) → t a.s., where t solves the equation det[M (t)] = 0, or equivalently, t2 · m(t2 )m(t2 ) = 1 . τ` If τ ` ≤ h, then we note that det[Mn (t)] → det[M (t)] uniformly on t > b + ε. Therefore, if det[Mn (t)] had a root σ1 (Ỹ ) in (b + ε, ∞), det[M (t)] would also need to have a root there, which is a contradiction. Therefore, we conclude σ1 (Ỹ ) ≤ b+ε a.s., for any ε > 0. Since σ1 (Ỹ ) ≥ σ2 (Ỹ ) → b, we conclude that σ1 (Ỹ ) → b a.s., as desired. This finishes the spike convergence claim in Thm. 2.1. Next, we turn to proving the convergence of the angles between the population and sample eigenvectors. Let Ẑ and û be the singular vectors associated with the top singular value σ1 (Ỹ ) of Ỹ . Then, by Lemma 5.1 of Benaych-Georges and Nadakuditi (2012), if σ1 (Ỹ ) is not a singular value of X, then the vector η = (η1 , η2 ) = (u> û, Z > Ẑ) belongs to the kernel of the matrix Mn (σ1 (Ỹ )). By the above discussion, this 2-by-2 matrix is of course singular, so this provides one linear equation for the vector r (with R = (t2 In − N N > )−1 ) tη1 · Z > RZ + η2 [Z > RN ν − (ξ`)−1/2 ] = 0. By the same lemma cited above, it follows that we have the norm identity (with t = σ1 (Ỹ )) t2 η12 · Z > R2 Z + η22 · ν > N > R2 N ν + 2tη1 η2 · Z > R2 N ν = (ξ`)−1 . 30 (35) This follows from taking the norm of the equation tη1 · RZ + η2 · RN ν = (ξ`)−1/2 Ẑ (see Lemma 5.1 in Benaych-Georges and Nadakuditi (2012)). We will find the limits of the quadratic forms below. Lemma 5.3 (More quadratic forms). The quadratic forms in the norm identity have the following properties: 1. Z > (t2 In − N N > )−2 Z − n−1 tr(t2 In − N N > )−2 → 0 a.s. 2. Z > (t2 In − N N > )−2 N ν → 0 a.s. 3. ν > N > (t2 In − N N > )−2 N ν → m(t2 ) + t2 m0 (t2 ) a.s., where m is the Stieltjes transform of the Marchenko-Pastur distribution Fγ,H . The proof is in Sec. 5.1.4. Again, the key novelty is the proof of the third claim. The standard concentration bounds do not apply, because u is non-random. Instead, we use an argument from complex analysis constructing a sequence of functions fn (t) such that their derivatives are fn0 (t) = ν > N > (t2 In − N N > )−2 N ν, and deducing the convergence fn0 (t) from that of fn (t). R of −1 2 > −2 2 Lemma 5.3 implies that n tr(t In − N N ) → (t − x)−2 dF γ,H (x) = m0 (t2 ) for t > b. Solving for η1 in terms of η2 from the first equation, plugging in to the second, and taking the limit as n → ∞, we obtain that η22 → c2 , where m0 (t2 ) 1 2 2 0 2 c2 + m(t ) + t m (t ) = . 2 2 τ `m(t ) τ` Using D(x) = x · m(x)m(x), we find c2 = m(t2 )/[D0 (t2 )τ `], where t solves (5). From the first equation, we then obtain η12 → c1 , where c1 = m(t2 )/[D0 (t2 )τ `], where t is as above. This finishes the proof of Thm. 2.1 in the single-spiked case. The proof of the multispiked case is a relatively simple extension of the previous argument, so we present it in Sec. 5.2. 5.1.2 Proof of Lemma 5.1 Recall that N = n−1/2 E ∗ , where E ∗ has rows ε∗i = µεi + Σ1/2 Ei Xi . Note ε∗ij = µj εij + σj Eij Xij = [µj + σj Eij ]εij + σj `1/2 Eij zi uj . Since εi = Γ1/2 αi , the terms aij = [µj + σj Eij ]gj εij are independent random variables with 2 2 variance gj2 EDij = gj2 (µ2j + σj2 ). Recall that we assumed that the distribution Hp of gj2 EDij converges weakly to the distribution H. Hence the eigenvalue distribution of the matrix A> A, where A = (n−1/2 aij )ij , converges to Marchenko-Pastur distribution Fγ,H (Bai and Silverstein, 2009, Thm. 4.3). Moreover, since 4 4 2 Eαij < ∞ and EDij < ∞, we have Ea4ij < ∞. In addition, by assumption sup gj2 EDij → sup supp(H). > 2 Thus the largest eigenvalue of A A converges a.s. to the upper edge b of the support of Fγ,H , see Bai and Silverstein (1998) and (Bai and Silverstein, 2009, Cor. 6.6). Therefore, since σj are bounded, it is enough to show that the operator norm of the error matrix E ∗∗ with entries n−1/2 Eij zi uj converges to zero a.s. This will ensure that N = A + E ∗∗ has the same two properties as A above, namely its ESD and operator norm converge. Now, denoting by elementwise products kE ∗∗ k = sup a> E ∗∗ c = n−1/2 kak=kck=1 sup kak=kck=1 31 (a z)> E(c u). We have ka zk ≤ kak max \|zi \| = max \|zi \| and kc uk ≤ kck max \|ui \| = max \|ui \|, hence kE ∗∗ k ≤ kEk max \|zi \| max \|ui \|. Since z has iid standardized entries, and C := Ezi4+φ < ∞, we can derive that P r(max \|zi \| ≥ a) ≤ E max \|zi \|4+φ /a4+φ ≤ nC/a4+φ . 0 0 Taking a = n1/2−φ for φ0 small enough, we obtain, max \|zi \| ≤ n1/2−φ a.s. However, since Eij are iid standardized random variables with bounded 4-th moment, kEk → √ 1 + γ a.s. (Bai and Silverstein, 2009). Since kuk∞ ≤ C log(p)B /p1/2 , we obtain kE ∗∗ k ≤ C(1 + 0 √ γ) · log(p)B n1/2−φ p−1/2 → 0 a.s., as required. 5.1.3 Proof of Lemma 5.2 Since N = A + E ∗∗ , and kE ∗∗ k → 0 a.s., it is enough to show the same concentration statements for A instead of N . Indeed, it is easy to see that the error terms are all negligible. Part 1: For Z > (t2 In − AA> )−1 Z, note that Z has iid entries —with mean 0 and variance 1/n—that are independent of A. We will use the following result: Lemma 5.4 (Concentration of quadratic forms, consequence of Lemma B.26 in Bai and i h √ Silverstein k (2009)). Leth x ∈ R be ia random vector with i.i.d. entries and E [x] = 0, for which E ( kxi )2 = 1 √ and supi E ( kxi )4+φ < C for some φ > 0 and C < ∞. Moreover, let Ak be a sequence of random k × k symmetric matrices independent of x, with a.s. uniformly bounded eigenvalues. Then the quadratic forms x> Ak x concentrate around their means: x> Ak x − k −1 tr Ak →a.s. 0. We apply this lemma with x = Z, k = p and Ap = (t2 In − AA> )−1 . To get almost sure convergence, here it is required that zi have finite 4 + φ-th moment. This shows the concentration of Z > (t2 In − AA> )−1 Z. Part 2: To show Z > (t2 In − AA> )−1 Aν concentrates around 0, we note that w = (t2 In − AA> )−1 Aν is a random vector independent of Z, with a.s. bounded norm. Hence, conditional on w: X X 4 2 2 P r(\|Z > w\| ≥ a\|w) ≤ a−4 E\|Z > w\|4 = a−4 [ EZni wi4 + EZni EZnj wi2 wj2 ] i −4 ≤a X 4 EZn1 ( wi2 )2 i6=j −4 −2 =a n EZ14 · kwk42 i For any C we can write P r(\|Z > w\| ≥ a) ≤ P r(\|Z > w\| ≥ a\|kwk ≤ C) + P r(kwk > C). For sufficiently large C, the second term, P r(kwk > C) is summable in n. By the above bound, the first term is summable for any C. Hence, by the Borel-Cantelli lemma, we obtain \|Z > w\| → 0 a.s. This shows the required concentration. Part 2: Finally we need to show that ν > (t2 Ip − A> A)−1 ν concentrates around a definite value. This is probably the most interesting part, because the vector u is not random. Most results for controlling expressions of the above type are designed for random u; however here the matrix A is random instead. For this reason we will adopt a different approach. 32 Under our assumption we have ν > (Γ(Σ + µ2 ) − zIp )−1 ν → mH (z), for z = t2 + iv with v > 0 fixed. Therefore, Thm 1 of Bai et al. (2007) shows that ν > (zIp − A> A)−1 ν → −m(z) a.s., where m(z) is the Stieltjes transform of the Marchenko-Pastur distribution Fγ,H . A close examination of their proofs reveals that their result holds when v → 0 sufficiently slowly, for instance v = n−α for α = 1/10. The reason is that all bounds in the proof have the rate N −k v −l for some small k, l > 0, and hence they converge to 0 for v of the above form. For instance, the very first bounds in the proof of Thm 1 of Bai et al. (2007) are in Eq. (2.2) on page 1543. The first one states a bound of order O(1/N r ). The inequalities leading up to it show that the bound is in fact O(1/(N r v 2r )). Similarly, the second inequality, stated with a bound of order O(1/N r/2 ) is in fact O(1/(N r/2 v r )). These bounds go to zero when v = n−α with small α > 0. In a similar way, the remaining bounds in the theorem have the same property. To get the convergence for real t2 from the convergence for complex z = t2 + iv, we note that \|ν > (zIp − A> A)−1 ν − ν > (t2 Ip − A> A)−1 ν\| = v\|ν > (zIp − A> A)−1 (t2 Ip − A> A)−1 ν\| ≤ ≤ vk(t2 Ip − A> A)−1 k2 · u> u. As discussed above, when t > b, the matrices (t2 Ip − A> A)−1 have a.s. bounded operator norm. Hence, we conclude that if v → 0, then a.s. ν > (zIp − A> A)−1 ν − ν > (t2 Ip − A> A)−1 ν → 0. Finally, m(z) → m(t2 ) by the continuity of the Stieltjes transform for all t2 > 0 (Bai and Silverstein, 2009). We conclude that ν > (t2 Ip − A> A)−1 ν → −m(t2 ) a.s. This finishes the analysis of the last quadratic form. 5.1.4 Proof of Lemma 5.3 As in Lemma 5.2, it is enough to show the same concentration statements for A instead of N . Parts 1 and 2: The proof of Part 1 and 2 are exactly analogous to those in Lemma 5.2. Indeed, the same arguments work despite the change from (t2 Ip − N > N )−1 to (t2 Ip − N > N )−2 , because the only properties we used are its independence from Z, and its a.s. bounded operator norm. These also hold for (t2 Ip − N > N )−2 , so the same proof works. Part 3: We start with the identity ν > N > (t2 In − N N > )−2 N ν = −ν > (t2 Ip − N > N )−1 ν + 2 > 2 t ν (t Ip − N > N )−2 u. Since in Lemma 5.2 we have already established ν > (t2 Ip − N > N )−1 ν → −m(t2 ), we only need to show the convergence of ν > (t2 Ip − N > N )−2 u. For this we will employ the following derivative trick (see e.g., Dobriban and Wager, 2015). We will construct a function with two properties: (1) its derivative is the quantity ν > (t2 Ip − N > N )−2 u that we want, and (2) its limit is convenient to obtain. The following lemma will allow us to get our answer by interchanging the order of limits: Lemma 5.5 (see Lemma 2.14 in Bai and Silverstein (2009)). Let f1 , f2 , . . . be analytic on a domain D in the complex plane, satisfying \|fn (z)\| ≤ M for every n and z in D. Suppose that there is an analytic function f on D such that fn (z) → f (z) for all z ∈ D. Then it also holds that fn0 (z) → f 0 (z) for all z ∈ D. Accordingly, consider the function fp (r) = −ν > (rIp − N > N )−1 ν. Its derivative is fp0 (r) = ν (rIp − N > N )−2 u. Let S := {x + iv : x > b + ε} for a sufficiently small ε > 0, and let us work on > 33 the set of full measure where kN > N k < b+ε/2 eventually, and where fp (r) → m(r). By inspection, fp are analytic functions on S bounded as \|fp \| ≤ 2/ε. Hence, by Lemma 5.5, fp0 (r) → m0 (r). In conclusion, ν > N > (t2 Ip − N > N )−2 N ν → m(t2 ) + t2 m0 (t2 ), finishing the proof. 5.1.5 Proof of Thm. 2.1: Model (3) The proof for model (3) is very similar to that for model (2). Therefore, we only present the outline. Working again in the single-spiked case for simplicity, we have the following decomposition for the observations Yi : Yi = (µ + Σ1/2 Ei )Si + εi = `1/2 zi µu + [µEi Si + εi ]. Denoting ν = µu/ξ 1/2 , where ξ = kµuk2 → τ , and ε∗i = µEi Si + εi , we have in matrix form Y = (ξ`)1/2 Z̃ν > + E ∗ . As in the proof of Lemma 5.1, it is not hard to see that the operator norm kE ∗ − Ekop → 0. Therefore, the spectral properties of Y are equivalent to those of Ỹ = (ξ`)1/2 Z̃ν > + E. However, this is now a spiked model where the signal component is independent of the noise component. It follows immediately from Thm 4.3 of Bai and Silverstein (2009) that the singular value distribution of E = [α1 , . . . , αn ]> Γ1/2 converges to the general Marchenko-Pastur distribution Fγ,G , where G is the limit of the distributions Gp of gj2 , j = 1, . . . , p. Similarly the top singular value of E converges to the upper edge bG of Fγ,G . Moreover, it also follows that the analogues of Lemmas 5.2 and 5.3 hold in our case. Indeed, the same arguments carry through, because the same assumptions hold. This allows the entire argument from Sec. 5.1.1 to carry through, finishing the single-spiked case of Thm. 2.1. The extension to the multispiked case is analogous to that in model (2). 5.1.6 Numerical computation of the quantities from Thm. 2.1 Spectrode computes the Marchenko-Pastur forward map: given an input limit population spectrum H and an aspect ratio γ, it outputs an accurate numerical approximation to the limit empirical spectral distribution (ESD) Fγ,H . Dobriban (2015) established the numerical convergence of the method, and showed in experiments that it is much faster than previous proposals. The method is publicly available at http://github.com/dobriban/eigenedge. The output of Spectrode includes a numerical approximation m̂ to the Stieltjes transform m of the limit ESD, computed over a dense grid xi on the real line. It also includes an approximation b̂2 of the upper edge b2 of the ESD. From this, we compute an approximation of the D-transform as D̂(xi ) = xi · m̂(xi ) · m̂(xi ), where m̂(xi ) = γ · m̂(xi ) + (γ − 1)/xi . Since D is monotone decreasing on (b2 , ∞), we find the smallest grid point xi such that D̂(xi ) ≤ 1/` to approximately compute D−1 (1/`). Finally, the derivative m0 (x) can be expressed as a function m0 (x) = F(m(x)) by differentiating the Marchenko-Pastur fixed-point equation (see e.g., Dobriban, 2015). Therefore, we compute a numerical approximation to D0 (x) = m(x) · m(x) + x[m(x) · m0 (x) + m0 (x) · m(x)] by approximating m̂0 (x) via the same function m̂0 (x) = F(m̂(x)). Similarly we approximate m̂0 . With these steps, we obtain a full numerical implementation of Thm. 2.1. 34 5.2 Proof of Thm. 2.1 - Multispiked extension 1/2 Let us denote by ui = µui /ξi the normalized reduced signals, where ξiP = kµui k2 → τi , and by r ∗ 1/2 εi = µεi +Σ Ei Xi . For the proof we start as in Sec. 5.1.1, obtaining Yi = k=1 (ξk `k )1/2 zik νk +ε∗i . Defining the r × r diagonal matrices L, ∆ with diagonal entries `k , ξk (respectively), and the n × r, p × r matrices Z, V, with columns Zk = n−1/2 (z1k , . . . , znk )> and uk respectively, we have Ỹ = Z(∆L)1/2 U > + E ∗ . The matrix Mn (t) is now 2r × 2r, and has the form t · Z > (t2 In − N N > )−1 Z Z > (t2 In − N N > )−1 N V 0r Mn (t) = − V > N > (t2 In − N N > )−1 Z t · V > (t2 Ip − N > N )−1 V (∆L)−1/2 (∆L)−1/2 . 0r It is easy to see that Lemma 5.1 still holds in this case. To find the limits of the entries of Mn , we need the following additional statement. Lemma 5.6 (Multispiked quadratic forms). The quadratic forms in the multispiked case have the following properties for t > b: 1. Zk> Rα Zj → 0 a.s. for α = 1, 2, if k 6= j. 2. νk> (t2 Ip − N > N )−α νj → 0 a.s. for α = 1, 2, if k 6= j. This lemma is proved in Sec. 5.2.1, using similar techniques as those in Lemma 5.1. Defining the r × r diagonal matrices T with diagonal entries τk , we conclude that for t > b, Mn (t) → M (t) a.s., where now −t · m(t2 )Ir −(T L)−1/2 . M (t) = −(T L)−1/2 −t · m(t2 )Ir As before, by Lemma A.1 of Benaych-Georges and Nadakuditi (2012), we get that for τk `k > 1/D(b2 ), σk (Ỹ ) → tk a.s., where t2k · m(t2k )m(t2k ) = 1/(τk `k ). This finishes the spike convergence proof. To obtain the limit of the angles for ûk for a k such that `k > τk D(b2 ), consider the left singular vectors Ẑk associated to σk (Ỹ ). Define the 2r-vector (∆L)1/2 V > ûk β α= 1 = . β2 (∆L)1/2 Z > Ẑk The vector α belongs to the kernel of Mn (σk (Ỹ )). As argued by Benaych-Georges and Nadakuditi (2012), the fact that the projection of α into the orthogonal complement of M (tk ) tends to zero, implies that αj → 0 for all j ∈ / {k, k + r}. This proves that νj> ûk → 0 for j 6= k, and the analogous claim for the left singular vectors. The linear equation Mn (σk (Ỹ ))α = 0 in the k-th coordinate, where k ≤ r, reads (with t = σk (Ỹ )): X tαk Zk> RZk − αr+k (ξk `k )−1/2 + Mn (σk (Ỹ ))ik αk = 0. i6=k 35 Only the first two terms are non-negligible due to the behavior of Mn , so we obtain tαk Zk> RZk = αr+k (ξk `k )−1/2 + op (1). Moreover taking the norm of the equation Ẑk = R(tZβ1 + N Vβ2 ) (see Lemma 5.1 in Benaych-Georges and Nadakuditi (2012)), we get X X X t2 αi αj Zi> R2 Zj + αk+i αk+j νi> N > R2 N νj + αi αk+j Zi R2 N νj = 1. i,j≤r i,j≤r i,j≤r 2 From Lemma 5.6 and the discussion above, only the terms αk2 Zk> R2 Zk and αr+k νk> N > R2 N νk are non-negligible, so we obtain 2 t2 αk2 Zk> R2 Zk + αr+k νk> N > R2 N νk = 1 + op (1). Combining the two equations above, Zk> R2 Zk > > 2 2 + ν N R N ν αr+k k = 1 + op (1). k ξk `k (Zk> RZk )2 Since this is the same equation as in the single-spiked case, we can take the limit in a completely analogous way. This finishes the proof. 5.2.1 Proof of Lemma 5.6 As in Lemma 5.2, it is enough to show the same concentration statements for A instead of N . Part 1: The convergence Zk> Rα Zj → 0 a.s. for α = 1, 2, if k 6= j, follows directly from the following well-known lemma, cited from Couillet and Debbah (2011): Lemma 5.7 (Proposition 4.1 in Couillet and Debbah (2011)). Let xn ∈ Rn and yn ∈ Rn be independent sequences of random vectors, such that for each n the coordinates of xn and yn are independent random variables. Moreover, suppose that the coordinates of xn are identically distributed with mean 0, variance C/n for some C > 0 and fourth moment of order 1/n2 . Suppose the same conditions hold for yn , where the distribution of the coordinates of yn can be different from those of xn . Let An be a sequence of n × n random matrices such that kAn k is uniformly bounded. Then x> n An yn →a.s. 0. Part 2: To show νk> (t2 Ip − N > N )−α νj → 0 a.s. for α = 1, 2, if k 6= j, the same technique cannot be used, because the vectors uk are deterministic. However, it is straightforward to check that the method of Bai et al. (2007) that we adapted in proving Part 3 of Lemma 5.2 extends to proving νk> (t2 Ip − N > N )−1 νj → 0. Indeed, it is easy to see that all their bounds hold unchanged. In the final step, as a deterministic equivalent for νk> (t2 Ip − N > N )−1 νj → 0, one obtains νk> (t2 Ip − t2 m(t2 )Σ)−1 νj , which tends to 0 by our assumption, showing νk> (t2 Ip − N > N )−1 νj → 0. Then νk> (t2 Ip − N > N )−2 νj → 0 follows from the derivative trick employed in Part 3 of Lemma 5.3. This finishes the proof. 5.3 Proof of Corollary 2.2 This corollary is a special case of our previous results, so the convergence results hold in this case. We only need to check that the limits are given by the formulas provided. Since very similar analysis has been performed by Benaych-Georges and Nadakuditi (2012) and Nadakuditi (2014), we only give part of the proof. 36 Under model (2), since we consider the singular values of the normalized matrix m−1/2 Ỹ instead of Ỹ as in Thm 2.1, it is easy to see that the relevant equation for the limiting singular values in this case is t2 · m0 (t2 )m0R(t2 ) = 1/δ` instead of t2 · m(t2 )m(t2 ) = 1/τ `. Note that m0 (t2 ) = (x − t2 )−1 dF γ,H (x) = γ · m0 (t2 ) + (γ − 1)t−2 . Thus the equation for t2 reads (γt2 m0 (t2 )+γ −1)m0 (t2 ) = (δ`)−1 . However, it is well-known that m0 (t2 ) obeys the equation γt2 m20 + (t2 + γ − 1)m0 + 1 = 0. From these two relations we obtain t2 · m0 (t2 ) = −1 − 1/(δ`). Plugging this back into the second equation and simplifying, we obtain t2 = (δ` + 1)(1 + γ/(δ`)), as required. Finally, under model (3), the proof is analogous. 5.4 5.4.1 Proofs for covariance estimation Proof of Cor. 3.2 From the spectral analysis of Y contained in Corollary 2.2 from Section 2 and the formula (10), the proof of this corollary is immediate from the first part of the following lemma, proved below: Lemma 5.8. We have the following limits (in operator norm) of the diagonals: 1. limn→∞ k diag(Σ̂Y ) − m · Ip k = 0 a.s. 2. limn→∞ k diag(Σ̂S ) − Ip k = 0 a.s. First, note that the second statement is an immediate corollary of the first. In the proof of the first statement, for notational convenience only, we will assume that r = 1; however, an identical proof goes through for any fixed r. We will therefore drop the subscript k for the proof. Decompose Xi into signal and noise, writing Xi = Si + εi , where Si = `1/2 zi u and Cov(ε) = Ip , and Si and εi are independent. So if Si = (si1 , . . . , sip )> , Xi = (Xi1 , . . . , Xip )> , Yi = (Yi1 , . . . , Yip )> , and εi = (εi1 , . . . , εip )> , we can write Yij = Dij Xij = Dij sij + Dij εij and con2 2 2 2 2 sequently Yij2 = Dij sij + 2Dij sij εij + Dij εij . th The (i, i) element of diag(Σ̂Y ) is then n n n n 1X 2 2 1X 1X 2 2 1X 2 2 Yij = Dij sij + 2Dij sij εij + D ε n j=1 n j=1 n j=1 n j=1 ij ij and we will controleach of the three sums on the right side separately. P n 2 2 sij = m`u2i and so Observe that E n1 j=1 Dij X n 1 1 log4B (n) 2 2 Dij sij = (E[d4 ]E[z 4 ]`2 u4i − m2 `u4i ) ≤ c Var n j=1 n n3 where c > 0 is a constant. Chebyshev’s inequality then gives X n 1 p log4B (n) 2 2 2 2 P Dij sij − m · ` · ui ≥ ε for some i ≤ c . n j=1 ε2 n3 Since u2i ≤ C logB (p)/p, this shows that n−1 p/n → γ. Pn j=1 37 2 2 Dij sij converges a.s. to 0 as p, n → ∞ and 2 2 For the sum of terms Dij εij , observe that En−1 Pn j=1 2 2Dij sij εij = 0, and 2 E\|2Dij sij εij \|4 = 16E(d8 )`2 u4i ≤ c log4B (n) n2 √ where c > 0 is a constant and we have used the estimate \|ui \| ≤ C logB (p)/ p. By using Markov’s inequality for the fourth moment (see Petrov (2012)), we then obtain: X n 1 log4B (n) 2 2Dij P sij εij ≥ ε for some i ≤ c 4 2 . n j=1 ε n Pn 2 s ε converges a.s. to 0 as p, n → ∞ and p/n → γ. This proves that n−1 j=1 2Dij Pn ij ij2 2 −1 2 2 2 2 4 8 8 Finally, to deal with n j=1 Dij εij , observe that EDij εij = m, and that E\|Dij εij \| = Ed Eε which is assumed finite. Therefore using again Markov’s inequality for the fourth moment, X n 1 Ed8 Eε8 2 2 P Dij εij − m ≥ ε for some i ≤ cp 4 4 . n j=1 ε n This proves that n−1 proof. 5.4.2 Pn j=1 2 2 Dij εij converges a.s. to 0 as p, n → ∞ and p/n → γ. This finishes the Proof of (13) The following analysis is adapted from that in Donoho et al. (2013). We start with a fact from linear algebra. Lemma 5.9. Suppose A and B are two p-by-p symmetric matrices, with eigenvectors u1 , . . . , up and v1 , . . . , vp . Suppose that each one has eigenvalue 0 with multiplicity p−r for some 1 ≤ r < p/2, with corresponding eigenvectors ur+1 , . . . , up and vr+1 , . . . , vp . Suppose that no eigenvector v1 , . . . , vr lies in the span of u1 , . . . , ur . Then there is an orthogonal matrix W such that W > AW = diag(`1 , . . . , `r ) ⊕ 0(p−r)×(p−r) (36) W > BW = B̃ ⊕ 0(p−2r)×(p−2r) (37) and where B̃ is a 2r-by-2r matrix whose entries are continuous functions of the inner products u> i vj , 1 ≤ i, j ≤ r, on R2r \ {±1}r . Furthermore, if we assume in addition that u> i vj = ci δi=j for some numbers ci ∈ (−1, 1), and p we let si = 1 − c2i , then the matrix B̃ is of the form: diag(η1 c21 , . . . , ηr c2r ) diag(η1 c1 s1 , . . . , ηr cr sr ) diag(η1 c1 s1 , . . . , ηr cr sr ) diag(η1 s21 , . . . , ηr s2r ) Proof. Form vectors ṽ1 , . . . , ṽp−2r , by performing Gram-Schmidt orthogonalization of v1 , . . . , ṽp−2r against the vectors u1 , . . . , ur . If the columns of W are the vectors in this basis, then clearly (36) holds, since Auk = `k uk for k = 1, . . . , r, and Aṽk = 0 for k = 1, . . . , p − 2r. Furthermore, since 38 Bṽk = 0 for k = r + 1, . . . , p − 2r, (37) holds for some 2r-by-2r block B̃. We need only check that 2r the entries of B̃ are continuous functions of u> \ {±1}r . i vj , 1 ≤ i, j ≤ r, on R Let W2r denote the p-by-2r matrix containing the first 2r columns of W , Ur denote the matrix with columns u1 , . . . , ur , Vr denote the matrix with columns v1 , . . . , vr , and Ṽr the matrix with columns ṽ1 , . . . , ṽr . We can therefore write W2r = [Ur Ṽr ]; By the Gram-Schmidt construction, every vector vk , k = 1, . . . , r, can be written as a linear combination of the vectors u1 , . . . , ur , ṽ1 , . . . , ṽk ; in matrix form, these linear combinations can be expressed as: Vr = Ur (Ur> Vr ) + Ṽr (Ṽ > Vr ) Now, we observe that the 2r-by-2r block B̃ is > U V > W2r BW2r = r> diag(η1 , . . . , ηr )[Ur> V Ṽr> V ] Ṽr V We need only show that the entries of Ṽr> Vr are continuous functions of the inner products 1 ≤ i, j ≤ r. Let R = Ṽr> V . Then Rk,l = 0 whenever k > l. When k = l, we have: v u r k−1 u X X 2− (u> v ) (ṽi> vk )2 Rk,k = ṽk> vk = t1 − i k u> i vj , i=1 i=1 and when k < l we have Rk,l = − Pr Pk−1 > > > > i=1 (ui vl )(ui vk ) + i=1 (ṽi vl )(ṽi vk ) Pr . kvk − i=1 (u> i vk )uk k Since for any l = 2, . . . , r, we have R1,l = ṽ1> vl Pr − i=1 (u> v1 )(u> vl ) Pr i > i = kv1 − i=1 (ui v1 )ui k which is obviously a continuous function of the inner products v1> ui on Rr \ {±1}r , an induction argument easily shows all entries of R are continuous in the inner products u> k vl . Finally, if u> v = c δ , the final assertion follows immediately, finishing the proof. j i i=j i If we apply the permutation π2r = (1, r + 1, 2, r + 2, . . . , r, 2r) to L the rows and columns of the r block matrix B̃ from Lemma 5.9, the corresponding matrix becomes i=1 B2 (ηi , ci , si ) where 2 ηc ηcs B2 (η, c, s) = . (38) ηcs ηs2 Applying the same permutation to the top 2r-by-2r block of the matrix A from Lemma 5.9 turns Lr ` 0 this block into: . i=1 A2 (`i ) where A2 (`) = 0 0 Now, let B̃ be given by the formula (38), but where c = ci = c(δ`i ) is the asymptotic inner product between the population eigenvector ui and the empirical eigenvector ûi , and η = ηi = η(λi ) is the almost sure limit of the ith eigenvalue η(λ̂i ) of Σ̂ηS . Since the shrinkers η collapse the vicinity 39 of the bulk to 0, it follows from Cor. 3.1 that the shrunken estimator has rank at most r a.s. Hence, by Lemma 5.9 and Cor. 3.1, if Lp (A, B) is any orthogonally-invariant loss function that decomposes over blocks, we have the almost sure convergence of Lp (ΣS , Σ̂ηS ) to the quantity L∞ (ΣS , Σ̂ηS ) = L2r M r A2 (`k ), k=1 r M B2 (η(λk ), ck , sk ) . k=1 This is the desired result. 5.4.3 Proof of Props. 3.3 We will only provide a proof for the reduced-noise model; the proof for the unreduced-model is 2 2 even simpler. Define the operator Lp acting Pnon p-by-p matrices by Lp (Σ) = µ Σ + σ diag(Σ) 1 and define the operator Lp by Lp (Σ) = n k=1 Dk ΣDk . Also, define define the p-by-p matrices Pn Bp = Σ̂Y − n1 k=1 Dk2 and Bp = Σ̂Y − mIp . Let ∆Bp = Bp − Bp , and ∆Lp = Lp − Lp . The operator ∆Lp taking p-by-p matrices to p-by-p matrices is diagonal. In this proof, when we refer to the (i, j)th diagonal entry of a diagonal operator on Rp×p , we mean the entry that multipliesPthe (i, j)th coordinate of a matrix. When i 6= j, the n 2 −1 th (i, j)th diagonal entry of ∆L diagonal k=1 Di,k Dj,k Σij ; and when i = j, the (i, i) Pnp (Σ) is2 µ −n −1 . entry of ∆Lp is m − n D k=1 i,k A proof nearly identical to the proof of Cor. 3.1 shows that the maximum of all the p2 diagonal entries of ∆Lp converges almost surely to 0; in other words, if Rp×p is equipped with Frobenius norm, then the operator norm of ∆Lp : Rp×p → Rp×p converges to 0 almost surely. Pn The p-by-p matrix ∆Bp is diagonal, with ith diagonal entry equal to m − n−1 k=1 Dk,i . Again, a proof like the proof of Cor. 3.1 shows that the supremum of these elements converges to zero almost surely as n, p → ∞. Also, note that √ the Frobenius norm of the matrix Σ̂S , which is the sum of squares of its eigenvalues, is of size ≈ n. To see this, observe that p p−r 1X 1X 1 1 kΣ̂S k2F = σk (Σ̂S )2 = σk (Σ̂S )2 + n n n n k=1 k=1 p X σk (Σ̂S )2 . k=p−r+1 Pp−r The first term n1 k=1 σk (Σ̂S )2 converges almost surely to the second moment of the MarchenkoPp Pastur law, which is finite since the distribution has finite support. The second term n1 k=p−r+1 σk (Σ̂S )2 converges to 0. Observe that Lp (Σ̂S ) − ∆Lp (Σ̂S ) = Lp (Σ̂S ) = Bp = Bp + ∆Bp = Lp (Σ̂0S ) + ∆Bp or in other words, Σ̂S − Σ̂0S kΣ̂S k = L−1 p (∆Lp Σ̂S ) kΣ̂S k The result follows immediately. 40 − L−1 p (∆Bp ) kΣ̂S k . 5.5 Proof that BLP is asymptotically diagonalized in PC basis (Sec. 4.1) Pr Pr −1 First, it is easy to check that Cov [Si , Yi ] = µ k=1 `k uk u> = [µ2 j=1 `j uj u> j + k , and M = Cov [Yi , Yi ] Pr Pr mIp +E]−1 , where E = σ 2 j=1 `j diag(uj uj ). Therefore the BLP equals ŜiBLP = µ k=1 `k uk u> M Yi . k Moreover, since uj are delocalized, the operator norm kEk → 0, so that it is easy to check that kŜiBLP − Ŝi0 k2 → 0, where r X Ŝi0 = µ `k uk u> k M0 Yi , k=1 Pr > −1 and M0 = [µ . Therefore, it is enough to show that kŜi0 − Ŝi k2 → 0, where j=1 `j uj uj + mIp ] P r > 2 recall that Ŝi = k=1 µ`k /(µ2 `k +m)uk u> k Yi . We can write, with mk = uk (M0 −Ip /(µ `k +m))Yi . 2 kŜi0 − Ŝi k2 = kµ = kµ r X k=1 r X `k uk u> k M0 Yi − `k uk mk k2 = µ2 k=1 r X 2 µ`k /(µ2 `k + m)uk u> k Yi k k=1 r X `k `j mk mj u> k uj . k,j=1 Therefore, to show kŜi0 − Ŝi k2 → 0, it is enough to show mk → 0. For this, using the formula u> (uu> + T )−1 = u> T −1 /(1 + u> T −1 u), we can write  −1  −1   −1  r r r X X X 2 > 2 µ2  = u> µ2  /  uk  u> `j uj u> `j uj u> ` j uj u> 1 + µ uk µ . k j + mIp k j + mIp j + mIp j=1 j6=k j6=k Under the assumptions of Cor. 2.2, we have u> k uj → δkj , hence it is easy to see that the denominator converges a.s. to 1 + µ2 /m. Indeed by using the formula (V V > + mI)−1 = [I − 1/2 1/2 V (V > V + mI)−1 V > ]/m, for V = µ[`1 u1 , . . . , `r ur ] (excluding uk )  −1 r X > > −1 > µ2  uk = u> u> `j uj u> V uk /m. k j + mI k uk /m − uk V (V V + mI) j6=k Now the entries of the r − 1-dimensional vector vk = V > uk are µ`1/2 u> j uk for j 6= k. Therefore, > −1 they converge to zero a.s. Since the operator norm of (V V + mI) is bounded above by 1/m, this shows that the second term converges to zero a.s. This shows that the denominator converges to 1 + µ2 /m a.s. Finally, by using the formula (V V > + mI)−1 = [I − V (V > V + mI)−1 V > ]/m once again, we conclude similarly that  −1 r X > > −1 > µ2  Yi = u> `j uj u> V Yi /m. u> k j + mI k Yi /m − uk V (V V + mI) j6=k h Denoting Mk = µ2 Pr > j6=k `j uj uj + I i−1 u> u> u> k Yi /m k Yi k Mk Yi − = mk = > m 1 + uk Mk uk 1 + µ2 /m · `k , we have > > 1 1 vk (V V + mI)−1 vk /m − − . > 1 + µ2 /m · `k 1 + uk Mk uk 1 + u> k Mk uk 41 Based on our previous calculations, both terms tend to 0, implying mk → 0. Therefore, we have kŜiBLP − Ŝi k2 → 0. By the triangle inequality, this immediately implies that all MSE properties for ŜiBLP also hold for Ŝi . This proves the desired claim. 5.6 5.6.1 Proof of Thm. 4.1 BLP For the BLP, consider first the single-spiked case where the squared prediction error of Ŝiτ = Ŝiτ,B = τ uu> Yi , with Yi = Di (Si + εi ) = Di (`1/2 zi u + εi ) is kSi − Ŝiτ k2 = k`1/2 zi u − τ u> Yi uk2 = (`1/2 zi − τ u> Yi )2 = [`1/2 zi − τ (`1/2 zi u> Di u + u> Di εi )]2 = (1 − τ u> Di u)2 `zi2 + τ 2 (u> Di εi )2 − 2τ (1 − τ u> Di εi )`1/2 zi u> Di εi . Now, zi and εi are independent of Di , hence, conditional on Di we have E[kSi − Ŝiτ k2 \|Di ] = (1 − τ u> Di u)2 ` + τ 2 u> Di2 u. Moreover, Eu> Di u = µ, E(u> Di u)2 = µ2 + σ 2 kuk44 → µ2 —since u is delocalized—and Eu> Di2 u = m, so the overall expectation converges to EkSi − Ŝiτ k2 → (1 − 2τ µ + τ 2 µ2 )` + τ 2 m. This proves that the asymptotic MSE of BLP is AM SE B (τ ; `, γ) = (1 − τ µ)2 ` + τ 2 m. The minimum is achieved for τ ∗ = µ`/(µ2 ` + m), and equals m`/(µ2 ` + m). In the multispiked case, the prediction error is kSi − Ŝiτ k2 = k r X 1/2 2 `k zik uk − τk u> k Yi uk k k=1 = r X 1/2 2 (`k zik − τk u> k Yi ) + k=1 X 1/2 1/2 > > (`k zik − τk u> k Yi )(`j zij − τj uj Yi ) · uk uj k6=j Now, zik , zij are independent if k 6= j, hence we have if k 6= j 1/2 1/2 > > > E(`k zik − τk u> k Yi )(`j zij − τj uj Yi ) = τk τj Euk Yi uj Yi . Using Yi = Di Xi , Xi = have 1/2 k=1 `k zik uk Pr + εi , and the independence properties described above, we > > > > Eu> k Yi uj Yi = Euk Di Xi Xi Di uj = Euk Di ( r X `m um u> m + Ip )Di uj m=1 = = r X m=1 r X > > 2 `m · E(u> k Di um · uj Di um ) + Euk Di uj > 2 `m · [µ2 (u> k um ) · (uj um ) + σ (uk um )> · (uj um )] + mu> k uj . m=1 Above we denoted by c = a b the vector with entries cj = aj bj . Since the vectors uk are delocalized and we assumed u> k ul → 0 for k 6= l, it is easy to see that the entire expression converges to zero. 42 This shows that EkSi − Ŝiτ k2 − E r X 1/2 2 (`k zik − τk u> k Yi ) → 0. k=1 Therefore, the asymptotic MSE decouples over the different PCs. Therefore, we can use our previous results about the asymptotic MSE in the single-spiked model, as soon as we can show that the same formulas for the MSE of specific coordinates given in Sec. 5.6.1 hold in this setting. However, this is easy to see using a calculation similar to the one given above. Indeed, we have X 1/2 1/2 2 > > > 2 E(`k zik − τk u> τk `1/2 k Yi ) = E(`k (1 − τk uk Di uk )zik − m zim uk Di um − τk uk Di εi ) m6=k = `k E(1 − 2 τk u> k Di uk ) + τk2 X 2 2 > 2 `m E(u> k Di um ) + τk Euk Di uk . m6=k 2 2 > 2 2 For k 6= m, the term E(u> um k2 → 0, while the other terms can k Di um ) = µ (uk um ) + σ kuk be evaluated as before, leading to the same formulas. Therefore, the asymptotic limit of the MSE is the same as that in the single-spiked case. This finishes the claims about BLP. 5.6.2 Empirical BLP The squared error of a general EBLP denoiser is kSi − Ŝiη k2 = k r X 1/2 `k zik uk − = 2 ηk ûk û> k Yi k k=1 k=1 r X r X 1/2 2 k`k zik uk − ηk ûk û> k Yi k + k=1 X 1/2 > 1/2 > (`k zik uk − ηk ûk û> k Yi ) (`j zij uj − ηj ûj ûj Yi ) k6=j The following lemma, proved in Sec. 5.6.3, shows that the cross terms vanish: Lemma 5.10 (Vanishing cross term in MSE). The cross terms in the MSE of EBLP denoisers 1/2 > 1/2 > vanish, i.e., for all k 6= j, E(`k zik uk − ηk ûk û> k Yi ) (`j zij uj − ηj ûj ûj Yi ) → 0 Therefore, the limit of the MSE in the multispiked case decouples into the MSEs for the individual spikes: r X 1/2 2 Ek`k zik uk − ηk ûk û> EkSi − Ŝiη k2 − k Yi k → 0. k=1 To evaluate the limiting MSE, we rely on the following lemma (proved in Sec. 5.7), which finds limiting expectations of inner products of the empirical singular vector ûB with the samples Yi and the population singular vector u. Lemma 5.11 (Denoising risk limit). We have the following convergences: 2 2 2 2 1. E(û> k Yi ) → mλk , where λk = t (δ`k ; γ) is defined in Eq. (7). 1/2 > 2 2 2 2. Ezi (û> k Yi )(uk ûk ) → µ`k ck · βk , where ck = c (δ`k ; γ) is defined in Eq. (8), and β =1+ 43 γ . δ`k (39) The key technical innovation is the proof of the second part. The main argument is an extension of the technique introduced by Benaych-Georges and Nadakuditi (2012) for characterizing the limits of the inner products u> j ûk between population and sample eigenvectors. For our proof, we need to extend this technique to characterize limits w> ûk for arbitrary random vectors w. Since this technique relies on an equation for the “outliers” among the eigenvalues of a finite-rank perturbation of a matrix in the terminology commonly used in random matrix theory (see e.g., Tao, 2013), we call this the outlier equation method. Applying the outlier equation method is nontrivial in our case, because the random vectors w to which we need to apply it—e.g., Xi —are dependent with ûk . For this reason, we need to use rank-one perturbation formulas once again to show that the dependence is negligible. We envision that the proof method could have several other applications. Going back to our main argument, Pr based on Lemma 5.11, the limit of the MSE is the following deterministic quantity: AM SE = k=1 `k + ηk2 · mt2k − 2ηk · µ`k c2k · βk . The optimal η minimizing the AMSE has ηk∗ = µ`k c2k · βk /[mt2k ] = µ`k c2k /[m(1 + δ`k )]. This finishes the EBLP analysis. 5.6.3 Proof of Lemma 5.10 1/2 1/2 > > We need to show that E(`k zik uk − ηk ûk û> k Yi ) (`j zij uj − ηj ûj ûj Yi ) → 0. We expand the paran1/2 1/2 theses and note that the first term is E(`k zik uk )> `j zij uj = 0, while the last term is a multiple > > > of Eû> k ûj · ûk Yi · ûj Yi = 0 (because ûk ûj = 0 for k 6= j). Thus it is enough to show the following claim for k 6= j: > Ezij û> k uj · ûk Yi → 0 > For this, we have u> k ûj → 0 a.s., by Cor. 2.2, thus also zij uk ûj → 0, so by convergence reduction 2 (Lemma 5.12) it is enough to show that E(û> Y ) is uniformly bounded. However, as in the proof k i 2 Y ) is uniformly bounded. This finishes of Part 1 of Lemma 5.11 in Sec. 5.7.1, we obtain that E(û> k i the proof of Lemma 5.10. 5.7 Proof of Lemma 5.11 For simplicity of exposition, we first prove the single-spiked case, when r = 1. The extension to the multispiked case is presented in Sec. 5.9. 5.7.1 Part 1 Similar to Lee et al. (2010) in the unreduced case, we use the following exchangeability argument: E(û> Yi )2 = Eû> Yi Yi> û = n−1 n X Eû> Yi Yi> û = n−1 Eû> Y > Y û = Eσ1 (Ỹ )2 . i=1 Now σ1 (Ỹ )2 → m · λ(δ`; γ) a.s. by Cor. 2.2. Moreover, by a bound on the expectation of top eigenvalues of sample covariance matrices such as that in Srivastava and Vershynin (2013), it is easy to see that Eσ1 (Ỹ )2 is uniformly bounded. Hence, Eσ1 (Ỹ )2 → m · λ(δ`; γ), finishing the proof. 44 5.7.2 Part 2 As a preliminary remark for Part 2, denoting t2 = λ(δ`; γ) we note that β = 1 + γ/(δ`) = 1 − m · γm(t2 )/[1 + m · γm(t2 )]. By a simple calculation, this is equivalent to m · m(t2 ) = −1/(γ + δ`). This can be checked as in the proof of Cor. 2.2 in Sec. 5.3. Therefore it is enough to show that −m · γm(t2 ) −m · γm(t2 ) > > 1/2 2 1/2 2 1/2 2 Ezi (û Yi )(u û) → ` c µ + = µ` c + ` c · . 1 + m · γm(t2 ) 1 + m · γm(t2 ) Expanding by using Yi = Di Xi , Xi = `1/2 zi u + εi , we have Ezi (û> Yi )(u> û) = `1/2 Ezi2 (û> Di u)(u> û) + Ezi (û> Di εi )(u> û). Ezi2 (û> Di u)(u> û) > (40) > → µc and Ezi (û Di εi )(u û) → `1/2 · Therefore, it is enough to show that 2 2 c · [−m · γm(t )]/[1 + m · γm(t )]. Since we already know that (u> û)2 → c2 a.s., our first step is to reduce these two claims by “getting rid” of the u> û term. For this and similar arguments, we will rely repeatedly on the following simple lemma: 2 2 Lemma 5.12 (Convergence Reduction). Suppose Wn ,Yn are random variables such that EWn → D for a constant D, EWn2 is uniformly bounded, Yn → C a.s. for some constant C, and Yn is a.s. uniformly bounded. Then EWn Yn → DC. In the special case when C = 0, the statement EWn → D is not needed. Proof. We have EWn Yn = EWn (Yn − C) + CEWn , so it is enough to show EWn (Yn − C) → 0. Therefore we can assume without loss of generality that C = 0. In that case, \|EWn Yn \| ≤ \|EWn2 \|1/2 \|EYn2 \|1/2 ≤ MW \|EYn2 \|1/2 → 0, because EWn2 are uniformly bounded; and because Yn → 0 a.s. and Yn are bounded, so E(Yn )2 → 0 by the dominated convergence theorem. Clearly, once we assumed C = 0 we did not use EWn → D. This finishes the proof. To use this lemma, let us choose the orientation of û such that u> û ≥ 0. Therefore, we know from our main results (e.g., Thm. 2.1), that u> û → c a.s. Moreover, since kûk = 1, it is easy to see that we can apply Lemma 5.12 to both terms in (40). Specifically, for the first term, we apply it with Wn = zi2 (û> Di u) and Yn = u> û; while for the second term we apply it with Wn = zi (û> Di εi ) and Yn = u> û. As mentioned above, this effectively removes the u> û terms, and thus simplifies the analysis considerably. Indeed, by Lemma 5.12 we conclude that Part 2 follows from the following auxiliary convergence results, proved in Sec. 5.8: Lemma 5.13 (Denoising risk auxiliary limits: Part 2). We have the following convergence results: 1. Ezi2 (û> Di u) → µc, where c2 is defined in Eq. (8). 2 −m·γm(t ) 2 2. Ezi (û> Di εi ) → `1/2 · c · 1+m·γm(t is defined in Eq. (7), and m is the Stieltjes 2 ) , where t transform of the standard Marchenko-Pastur law. Based on the above lemma, we completed the proof of Part 2 of Lemma 5.11. We will prove Part 1 of Lemma 5.11 later in Sec. 5.7.1, because it re-uses many of the techniques and results established in the proof of Part 1. This finishes the proof of Lemma 5.11. 45 5.8 Proof of Lemma 5.13 In this lemma, we need to understand the asymptotics of inner products such as u> Di û. Previous results by Benaych-Georges and Nadakuditi (2012) characterized the asymptotics of the cosines u> û. However, these results do not allow us to understand the asymptotics of the inner products u> Di û, due to the dependence of û and Di . Instead, we must go back to first principles, and extend the outlier equation technique introduced by Benaych-Georges and Nadakuditi (2012) to our setting. We will see that the current case is more challenging than the one handled in Benaych-Georges and Nadakuditi (2012). Let us recall the notation from Sec. 5.1.1. According to (34), the normalized data matrix m−1/2 Ỹ = (nm)−1/2 Y can be written as m−1/2 Ỹ = `1/2 · Zu> + N . Note here that u = ν. Let here t = tp be the singular value of m−1/2 Ỹ with left and right singular vectors Ẑ, û, and suppose that t is not a singular value of N . As in Lemmas 4.1 and 5.1 of Benaych-Georges and Nadakuditi (2012), we then have the following outlier equation that provides an equation for the singular vectors of the perturbed matrix m−1/2 Ỹ : 1 Ẑ t · (t2 In − N N > )−1 (t2 In − N N > )−1 N (u> û) · Z (41) = N > (t2 In − N N > )−1 t · (t2 Ip − N > N )−1 (Z > Ẑ) · u `1/2 û The idea is to take inner products of the right hand side with the quantity we want to characterize (e.g., inner product with Di u to understand u> Di û), and then evaluate the limit of the quantities on the left hand side. This extends the technique of Benaych-Georges and Nadakuditi (2012), who took inner products only with the population and sample singular vectors (u and û). In contrast, by using other vectors, such as Di u and Di εi , we are able to extend vastly the reach of their technique. To formalize this, let w be a p-dimensional vector. By taking the inner product of the outlier equation’s last p coordinates with w, we obtain the following scalar equation: (u> û) · w> N > (t2 In − N N > )−1 Z + t(Z > Ẑ) · w> (t2 Ip − N > N )−1 u = 1 `1/2 w> û. (42) Therefore, we can formalize the outlier equation technique as follows: Lemma 5.14 (Outlier equation method). Under the assumptions of Cor. 2.2, suppose that û, Ẑ are chosen so that û> u ≥ 0, Ẑ > Z ≥ 0. Suppose moreover that w = wp is a sequence of random vectors such that the assumptions of Lemma 5.12 for Wn apply to Wn1 = w> N > (t2 In − N N > )−1 Z and Wn2 = w> (t2 Ip − N > N )−1 u. Specifically, suppose that EWn1 → w1∗ and EWn2 → w2∗ . Then the random variables w> û also converge in expectation, namely Ew> û → `1/2 (c(δ`; γ)w1∗ + t · c̃(δ`; γ)w2∗ ). (43) Here c ≥ 0 where c2 is defined in Eq. (8), while c̃(δ`; γ) ≥ 0 is the limit cosine between Z, Ẑ, c̃(`; γ)2 = (1 − γ/`2 )/(1 + 1`) if ` > γ 1/2 and c̃2 = 0 otherwise. As an important special case, suppose that w2∗ = −κm(t2 ), while w1∗ = 0. Then Ew> û → κc. (44) Proof. The first part follows from the discussion before the lemma. For the second part, from Lemma 5.2, it folows that for w = u, we have Ew> (t2 Ip − N > N )−1 u → −m(t2 ), so that in this case the claim holds with κ = 1. For other vectors w, the result follows by linearity. An analogous version of Lemma 5.14 holds with convergence in expectation replaced by convergence a.s. Next, we will use this lemma to prove the two parts of Lemma 5.13. 46 5.8.1 Part 1 Recall the notation R = (t2 In −N N > )−1 and define R̃ = (t2 Ip −N > N )−1 . To show Ezi2 (û> Di u) → µc, we use Lemma 5.14 with the sequence of vectors w = zi2 Di u. To conclude our result by the second part of Lemma 5.14, it is enough to establish the following claims 1. Ezi2 u> Di N > RZ → 0 2. Ezi2 u> Di R̃u → −µ · m(t2 ). Claim 1 : As in the Proof of Lemma 5.1 in Sec. 5.1.2, we can write N = A + E ∗∗ , where E ∗∗ is independent of z and kE ∗∗ k → 0. Therefore, it is easy to see that it is enough to prove the claim with N replaced by A. Let us define R(A) = (t2 In − AA> )−1 . Then, as in the proof of Part 2 of Lemma 5.2 in Sec. 5.1.3, we obtain u> Di A> R(A)Z → 0 a.s. Clearly, these random variables are bounded a.s., since by the proof of Lemma 5.1, R(A) has a.s. uniformly bounded operator norm. Therefore, by the convergence reduction Lemma 5.12 applied to Wn = zi and Yn = u> Di A> R(A)Z—valid since Ezi2 = 1 and Ezi4 is uniformly bounded—we conclude that Ezi2 u> Di A> R(A)Z → 0. As discussed, this implies the desired Claim 1. Claim 2 : As in Claim 1, it is enough to prove the result with N replaced by A. From now on in this claim, we will only work with A, not N ; therefore, we denote for brevity R = R(A), R̃ = R̃(A) = (t2 Ip − A> A)−1 (and no confusion will arise). Also similarly to Claim 1, it will be enough to prove that u> Di R̃u → −µm(t2 ) a.s. For this, note that Di of course depends on the i-th data vector Yi = Di Xi . Therefore, we cannot use the concentration of quadratic forms directly. However, since the only dependence occurs in the i-th sample, we can use a rank-one perturbation formula to separate the i-th sample, and then control the two resulting terms separately. P −1 For this, we define aj = n−1/2 Dj εj to be the rows of A, so that R̃ = (t2 Ip − j aj a> . We j ) P 2 > −1 also define the perturbed matrix R̃i = (t Ip − j6=i aj aj ) . By the matrix inversion formula (M + aa> )−1 = M −1 − M −1 aa> M −1 /(1 + a> M −1 a), we have R̃ = R̃i + R̃i ai a> i R̃i > 1 − ai R̃i ai (45) Therefore, u> Di R̃u = u> Di R̃i u + (u> Di R̃i ai ) · (u> R̃i ai ) . 1 − a> i R̃i ai (46) > > 2 As in the proof of Part P 3 of Lemma 5.2 in Sec. 5.1.3, we obtain u Di R̃i u + u Di u · m(t ) → 0 a.s. However, u> Di u = j u2j Fj , where Fj are iid random variables with mean µ = EDij . Hence, it is easy to see that u> Di u → µ a.s. This shows that the first term in (46) converges to −µ · m(t2 ) a.s. and in expectation. It remains to show that the second term in (46) converges to 0 in expectation. Since 1/(1 − > > a> i R̃i ai ) is uniformly bounded a.s., it is enough to show that E(u Di R̃i ai )·(u R̃i ai ) → 0. However, > > by independence of εi from Di , R̃i , we have E(u Di R̃i ai ) · (u R̃i ai ) = n−1 Eu> Di R̃i Di2 R̃i u = O(n−1 ) a.s., showing the desired claim. This finishes the proof of Claim 2, and hence that of Part 1 of Lemma 5.13. 47 5.8.2 Part 2 For Part 2 of Lemma 5.13, we need to show that Ezi (û> Di εi ) → −`1/2 · cm · γm(t2 )/[1 + m · γm(t2 )]. Using the Outlier Equation method, Lemma 5.14, as in Part 1, with the sequence of vectors w = zi Di εi , it is enough to establish the following claims > 1. Ezi ε> i Di N RZ → −m·γm(t2 ) 1+m·γm(t2 ) 2. Ezi ε> i Di R̃u → 0. As before, we can work with A instead of N , and with R, R̃ being the resolvents of A. The second claim follows immediately, because zi is independent of ε> i Di R̃u, since R̃ does not depend > on zi . Therefore, Ezi ε> D R̃u = Ez Eε D R̃u = 0. i i i i i P 1/2 For the first claim, noting that A> R = R̃A> , and expanding A> Z = , where j zj aj /n −1/2 aj = n Dj εj , we can write X > > > 2 > Ezi ε> Ezi zj ε> i Di A RZ = Ezi εi Di R̃A Z = Ezi εi Di R̃ai + i Di R̃aj j6=i =n −1 Eε> i Di R̃Di εi = Ea> i R̃ai since zi are independent of ε, R̃, and have mean 0. To control this last term, we use a technique similar to that in Sec. 5.8.1. Using the rank one perturbation formula (45), and denoting βi = −1 > a> εi Di R̃i Di εi , we find i R̃i ai = n > a> i R̃ai = ai R̃i ai + 2 βi (a> i R̃i ai ) = . 1 − βi 1 − βi By independence and the concentration of quadratic forms, βi − n−1 m · tr(R̃i ) → 0, while by the Marchenko-Pastur law, n−1 tr(R̃i ) → −γ · m(t2 ). By the convergence reduction lemma, Ea> i R̃ai → −m · γm(t2 )/[1 + m · γm(t2 )] This finishes the proof of Part 2 of Lemma 5.13. Therefore, the proof of Lemma 5.13 is complete. 5.9 Multispiked EBLP MSE We now show that the formulas from Lemma 5.11 are also valid in the multispiked case. Indeed, 1/2 1/2 2 2 > 2 > > Ek`k zik uk − ηk ûk û> k Yi k = `k + ηk E(ûk Yi ) − 2ηk `k Ezik (ûk Yi )(uk ûk ). We have already showed in the proof of Lemma 5.10 that the limit of the second term is the same as in the single-spiked case. It remains to characterize the third term. For this, we follow a similar pattern to that used in the proof of Lemma 5.10 in Sec. 5.7, using the multispiked outlier equation. We sketch the argument below. As in the proof of Thm. 2.1 in Sec. 5.2, the normalized data matrix m−1/2 Ỹ = (nm)−1/2 Y can be written as m−1/2 Ỹ = ZL1/2 U > + N , where Z is the n × r matrix with entries zik , U is the n × r matrix with columns uk , while and L is the r × r diagonal matrix with entries `k . Let tk be the singular value of m−1/2 Ỹ with left and right singular vectors Ẑk , ûk , and suppose that tk is not a singular value of N . The multispiked outlier equation is now: Z · L1/2 · (U > ûk ) tk · (t2k In − N N > )−1 (t2k In − N N > )−1 N Ẑ = k (47) N > (t2k In − N N > )−1 tk · (t2k Ip − N > N )−1 U · L1/2 · (Z > Ẑk ) ûk 48 For a sequence of random vectors w = wp , denoting the r-vectors Wn1 = w> N > (t2k In − N N > )−1 Z and Wn2 = w> (t2k Ip − N > N )−1 u, the analogue of Eq. (42) is obtained by taking the inner product of the last p coordinates of the multispiked equation with w: Wn1 · L1/2 · (U > ûk ) + tk · Wn2 · L1/2 · (Z > Ẑk ) = w> ûk The outlier equation method formalized in Lemma 5.14 also extends in the natural way, by taking the limits of the above equation. Going back to our main argument about EBLP risks, it remains only to characterize the > limit of Ezik (û> k Yi )(uk ûk ). By the convergence reduction lemma, it is enough to show that 1/2 > Ezik (ûk Yi ) → µ`k ck βk , where ck ≥ 0 is the cosine, while βk = 1 + γ/(δ`k ). Examining closely the proof of Lemma 5.13, we see that all claims hold unchanged, and the proofs go through either unchanged or with minimal modifications (similar to the extension of the spike behavior to the multispiked case). We omit the details. 5.10 Proof of Thm. 4.3 Similarly to our previous calculations, the MSE equals Enη,o = EkS0 − Ŝ0η k2 = r X 1/2 2 Ek`k z0k uk − ηk ûk û> k Y0 k k=1 + X 1/2 1/2 > > E(`k z0k uk − ηk ûk û> k Y0 ) (`j z0j uj − ηj ûj ûj Y0 ). k6=j The cross terms vanish because z0k are independent zero-mean random variables, and û> k ûj = 0. To find the limit, we expand the main term as follows: 1/2 1/2 2 2 > 2 > > Ek`k z0k uk − ηk ûk û> k Y0 k = `k + ηk E(ûk Y0 ) − 2ηk `k Ez0k (ûk Y0 )(uk ûk ) Now, because Y0 are independent of ûj , we can take expectation over the randomess in Y0 to see that   r r X X > > 2 2 Eû> µ ` j uj u> `j diag(uj uj ) ûk k Y0 · ûk Y0 = Eûk j + mIp + σ j=1 = m + µ2 r X 2 2 `j E(û> k uj ) + σ j=1 j=1 r X `j Eû> k diag(uj uj )ûk → m + µ2 `k c2k . j=1 On the last line we used the results of Lemma 2.2, as well as the delocalization of uk , and denoted c2k = c2kk the asymptotic cosine of the k-th singular vectors. Moreover ! r X > > 1/2 > > Ez0k (ûk Y0 )(uk ûk ) = Ez0k `m z0m ûk D0 um + ûk D0 ε0 · (u> k ûk ) = m=1 1/2 > `k Eûk D0 uk 1/2 2 · (u> k ûk ) → µ`k ck . Hence, we have the following convergence: 1/2 2 ηk ,o Ek`k z0k uk − ηk ûk û> (`k ) = `k + ηk2 · (m + µ2 `k c2k ) − 2ηk · (µ`k c2k ) k Y0 k → E Pr Therefore, the limit prediction error is E η,o = k=1 E ηk ,o (`k ), finishing the proof of Thm. 4.3. 49 5.11 Proof of Prop. 4.4 First, clearly the optimal shrinkage for in-sample denoising is smaller, because µ`c2 /(µ2 ` + m) ≤ µ`c2 )/(µ2 `c2 + m). To check that the MSE of in-sample and out-of sample EBLP agree, we verify that ` − µ(µ`c2 · β)2 /(mt2 ) = ` − µ(µ`c2 )2 /(µ2 `c2 + m). Clearly this holds whenever c2 = 0. Considering the case c2 > 0, it is enough to show that β 2 /(mt2 ) = 1/(µ2 `c2 + m), or also that (δ` + γ)2 /(δ`)2 = t2 /(δ`c2 + 1). Since t2 = (δ` + 1)[1 + γ/(δ`)], using the formula for c2 , this follows immediately, finishing the proof. 5.12 Proofs for denoising in the unreduced-noise model (Sec. 4.4) The proofs for denoising in the unreduced-noise model from (3) are very similar to those in the reduced-noise model from (2). Therefore, while we give the full outline of the proofs, we skip some of the more technical parts that are essentially identical to those in the proof of Thms. 4.1 and 4.3. 5.12.1 BLP Pr Pr 1/2 Since Si = k=1 `k zik uk and Di has iid entries with mean µ, we have Cov [Si , Yi ] = µ k=1 `k uk u> k. Moreover, Cov [Yi , Yi ] = EYi Yi> = EDi Si Si> Di + Eεi ε> i = EDi r X ` k uk u> k Di + Ip . k=1 Pr Since the entries σ 2 , the a, b-th entry of the first component equals k=1 `k uka ukb · Pr of Di have variance EDia Dib = k=1 `k uka ukb [µ2 + σ 2 I(a = b)]. Therefore, denoting diag(uk uk ) the diagonal matrix with entries {u2ki }, i = 1, . . . , p, Cov [Yi , Yi ] = µ2 r X 2 `k uk u> k + Ip + σ k=1 r X `k diag(uk uk ). k=1 Since uk are delocalized, the operator norm of the last term converges to zero, k diag(uk uk )kop ≤ −1 −1 C 2 log2 B(p)/p → 0. Hence by using the matrix = Pr inversion difference formula A − B A−1 (B − A)B −1 , for A = Cov [Yi , Yi ] and B = µ2 k=1 `k uk u> + I , so that kA − Bk → 0, we see p k that k Cov [Si , Yi ] A−1 Yi − Cov [Si , Yi ] B −1 Yi k → 0. Thus the BLP is asymptotically equivalent to 2 Pr −1 Pr > µ k=1 `k uk u> Yi . As in the reduced-noise model in Sec. 5.5, it is not hard k µ k=1 `k uk uk + Ip Pr 2 to show that this is asymptotically equivalent to Ŝiτ = k=1 τk uk u> k Yi , where τk = µ`k /(µ `k + 1). τ To calculate the MSE of Si for general τ in the single-spiked case, we write as in the proof of Thm 4.1 in Sec. 5.6 EkSi − Ŝiτ k2 = E(`1/2 zi − τ u> Yi )2 = ` + τ 2 E(u> Yi )2 − 2τ `1/2 Ezi u> Yi . But Yi = Di Si + εi and in the single-spiked case Si = `1/2 zi u, so Ezi u> Yi = E`1/2 zi2 u> Di u + Ezi u> εi = `1/2 µ. Moreover, E(u> Yi )2 = E`zi2 (u> Di u)2 + 2`1/2 Ezi u> Di uu> εi + E(u> εi )2 = `E(u> Di u)2 + 1. Now, denoting by u(k) the entries of u X X X X E(u> Di u)2 = E( u(k)2 Dik )( u(l)2 Dil ) = µ2 u(k)2 u(l)2 + (µ2 + σ 2 ) u(k)4 . k l k6=l 50 k P Since u is delocalized, k u(k)4 → 0, so that E(u> Di u)2 → µ2 . In conclusion, we obtain as required EkSi − Ŝiτ k2 → ` + τ 2 (`µ2 + 1) − 2τ `µ. 5.12.2 EBLP To find the MSE of EBLP in the single-spiked case, we have EkSi − Ŝiη k2 = Ek`1/2 zi u − ηûû> Yi k2 = ` + η 2 E(û> Yi )2 − 2η`1/2 Ezi û> u · û> Yi . As in Lemma 5.11, we find that E(û> Yi )2 → t2 (µ2 `; γ), where t2 (µ2 `; γ) is defined in Eq. (7). Also, by the convergence reduction Lemma 5.12, choosing the orientation of û such that û> u ≥ 0, for the last term it is enough to characterize the limit of Ezi û> Yi . By the same argument as in Lemma 5.11, we find that Ezi · û> Yi → µ`1/2 ·c(µ2 `; γ)·[1+γ/(µ2 `)]. Hence, EkSi − Ŝiη k2 → ` + η 2 · t2 (µ2 `; γ) − 2η · µ` · c2 (µ2 `; γ) · [1 + γ/(µ2 `)], as claimed. This also shows that the optimal coefficient is η ∗ = µ` · c2 (µ2 `; γ)/[µ2 ` + 1]. 5.12.3 EBLP Out-of-sample Finally, for out-of-sample EBLP AMSE in the single-spiked case, we let (Y0 , D0 ) be a new sample, and expand Ek`1/2 z0 u − ηûû> Y0 k2 = ` + η 2 E(û> Y0 )2 − 2η`1/2 Ez0 (û> Y0 )(u> û) Now, because Y0 is independent of û, we can take expectation over the randomess in Y0 to see that Eû> Y0 · û> Y0 = Eû> [Ip + µ2 `uu> + σ 2 diag(u 2 > 2 2 > = 1 + µ `E(u û) + σ Eû diag(u u)]û u)û → 1 + µ2 `c2 . On the last line we used that u is delocalized, and the results of Lemma 2.2. Next, Ez0 (û> Y0 )(u> û) = Ez0 `1/2 z0 û> D0 u + û> ε0 · (u> û) = `1/2 Eû> D0 u · (u> û) → `1/2 µc2 . 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1 Multiplicative Updates for Elastic Net Regularized Convolutional NMF Under β -Divergence arXiv:1803.05159v1 [cs.LG] 14 Mar 2018 Pedro J. Villasana T., Stanislaw Gorlow, Member, IEEE and Arvind T. Hariraman Abstract—We generalize the convolutional NMF by taking the β-divergence as the loss function, add a regularizer for sparsity in the form of an elastic net, and provide multiplicative update rules for its factors in closed form. The new update rules embed the β-NMF, the standard convolutional NMF, and sparse coding alias basis pursuit. We demonstrate that the originally published update rules for the convolutional NMF are suboptimal and that their convergence rate depends on the size of the kernel. Index Terms—Convolution, nonnegative matrix factorization, multiplicative update rules, sparsity, elastic net, β-divergence. I. I NTRODUCTION N ONNEGATIVE matrix factorization finds its application in the fields of machine learning and in connection with inverse problems, mostly. It became immensely popular after Lee and Seung derived multiplicative update rules that made the up until then additive steps in the direction of the negative gradient obsolete [1]. In [2], they gave empirical evidence of their convergence to a stationary point, using (a) the squared Euclidean distance, and, (b) the generalized Kullback–Leibler divergence as the contrast function. The factorization’s origins can be traced back to [3], [4]. To better deal with noisy data, the notion of a basis is (most commonly) abandoned in favor of an overcomplete frame or dictionary, and sparsity becomes a desired property of the coefficient matrix. Early works that encourage such a factorization by means of L1 -penalty are [5], [6]. A convolutional variant of the factorization is introduced in [7] based on the Kullback–Leibler divergence. The main idea is to exploit temporal dependencies in the neighborhood of a point in the time-frequency plane. In their original form, the update rules lead to a biased factorization, because the frame of each time translation is updated separately but only the last frame is considered when updating the coefficient matrix. To provide a remedy, multiple coefficient matrices are computed in [8], one for each translation frame, and the final update is by taking the average over the time-aligned matrices. A nonnegative matrix factor deconvolution in 2D based on the Kullback–Leibler divergence is found in [9]. Not only do the authors give a derivation of the update rules, they show a simple way of making the update rules multiplicative. It may be pointed out that the update rule for the coefficient matrix is different from the one in [8]. The same guiding principles are applied to derive the convolutional factorization based on P. Villasana is a graduate student at the KTH Royal Institute of Technology, 114 28 Stockholm, Stockholm County, Sweden and a holder of a scholarship from the Mexican National Council for Science and Technology CONACYT e-mail: pjvt@kth.se. S. Gorlow is with Dolby Sweden. A. Hariraman is enrolled at the KTH Royal Institute of Technology. the squared Euclidean distance in [10]. But in the attempt to give a formal proof for their update rules, the authors largely rederive a biased factorization comparable to [7]. In [11], nonnegative matrix factorization is generalized to a family of α-divergences under the constraints of sparsity and smoothness, while the unconstrained β-divergence is brought into focus in [12]. For both cases multiplicative update rules are given. In this letter, we derive multiplicative update rules for the factorial deconvolution under the β-divergence and an elastic net regularizer. Furthermore, we show that the updates in [7], [8] are suboptimal and that their convergence rate and stability depend on the size of the convolution kernel. II. N ONNEGATIVE M ATRIX FACTORIZATION The nonnegative matrix factorization (NMF) is an umbrella term for a low-rank matrix approximation of the form V ' WH (1) K×I with V ∈ RK×N , W ∈ R>0 , and H ∈ RI×N >0 >0 , where I is the predetermined rank of the factorization. The letters above help distinguish between visible (v) and hidden variables (h) that are put in relation through weights (w). The factorization is usually formulated as a convex minimization problem with a dedicated cost function Cλ according to minimize Cλ (W, H) W, H subject to wki , hin > 0 (2) with Cλ (W, H) = L(V, W H) + λ R(H), (3) where L is a loss function that assesses the error between V and the factorization W H, R is a regularization term, and λ (λ > 0) is a tuning parameter that controls the importance of R. For λ equal to 0, we obtain a non-regularized NMF. A. β-Divergence The loss in (3) can be expressed by means of a contrast or distance function between the elements of V and W H. Due to its robustness with respect to outliers for certain values of the input parameter β, we resort to the β-divergence [13] as a subclass of the Bregman divergence [14], which for the points p and q in a closed convex set is given by Dβ (p, q) = β 1 p + (β − 1) q β − β p q β−1 , β (β − 1) (4) including the prominent special cases DLS (p, q) = Dβ (p, q)\|β=2 = 1 2 (p − q) , 2 (5) 2 for the squared Euclidean distance or the least-squares loss, p DKL (p, q) = lim Dβ (p, q) = p log − p + q, (6) β→1 q for the generalized Kullback–Leibler divergence, and p p (7) DIS (p, q) = lim Dβ (p, q) = − log − 1, β→0 q q for the Itakura–Saito distance. Accordingly, the β-divergence for matrices V and W H can be defined entrywise, as X X def Dβ (V, W H) = Dβ vkn , wki hin . (8) k,n i B. Elastic Net Regularization The factorization in (1) is an inverse problem, which is ill posed. As such, it can benefit from a regularization term. We set in an L1 -regularizer to favor the solution with the fewest hidden variables, i.e. the lowest model order. More precisely, the regularizer that imposes sparsity on H is defined as X def R(H) = kHk1 = hin with hin > 0, (9) i,n which is simply the sum of the entries in H. To overcome the limitations of the L1 -penalty, we add a quadratic term, which is known as elastic net regularization [15]: III. S PARSE β-CNMF Ensuing from the preliminary considerations in Section II, we adopt the formulation of the CNMF from (13) and derive multiplicative update rules for gradient descent, while taking the entrywise β-divergence from (8) as the loss function and imposing sparsity on H as in (10). The result is referred to as the “sparse β-CNMF”, or β-SCNMF for short. A. Problem Statement Under the premise that V is factorizable into {Wm } and H, m = 0, 1, . . . , M − 1, and given the cost function X m Cλ ({Wm }, H) = Dβ V, Wm H−→ +λ R(H), (14) m we seek to find the multiplicative equivalents of the iterative update rules for gradient descent: t ∂ t t+1 t C W , H , (15a) Wm = Wm −κ λ m t ∂Wm t ∂ C λ Wm , Ht , (15b) Ht+1 = Ht − µ t ∂H where (15a) and (15b) alternate at each iteration (t > 0). The step sizes κ and µ are allowed to change at every iteration. 2 def λ R(H) = λ2 kHkF + λ1 kHk1 , (10) P 2 2 where k·kF = i,n hin is the squared Frobenius norm. B. Multiplicative Updates Rules C. Discrete Convolution As can be seen explicitly in (8), the weight wki for the ith hidden variable hin at point (k, n) is applied using the scalar product. Given that hi evolves with n, we can assume that hi is correlated with its past and future states. We can take this into account by extending the dot product to a convolution in our model. Postulating causality and letting the weights have finite support of cardinality M , the convolution writes Computing the partial derivatives of Cλ w.r.t. Wm and H, and by choosing appropriate values for κ and µ, the iterative update rules from (15) become multiplicative: h ◦(β−1) T i◦−1 t+1 t Wm = Wm ◦ Ut Ht−→ m h i T ◦(β−2) ◦ V ◦ Ut Ht−→ (16a) m , hX tT t◦(β−1) Ht+1 = Ht ◦ Wm U←− + 2λ2 Ht m m h i X ◦−1 tT t◦(β−2) m ◦ U m + λ1 1I×N ] ◦ Wm V←− , (16b) ←− def (wki ∗ hi )(n) = M −1 X m wki (m) hi,n−m . (11) m=0 The operation can be converted to a matrix multiplication by lining up the states hin in a truncated Toeplitz matrix:   hi,1 hi,2 hi,3 · · · hi,N −1 hi,N  0 hi,1 hi,2 · · · hi,N −2 hi,N −1    (12)  .. . .. .. . .. .. ..  .  . . . . 0 0 0 ··· hi,N −M hi,N −M +1 In accordance with (1), the convolutional NMF (CNMF) can be formulated as follows to accommodate the structure given in (12), see also [7], [8]: V' M −1 X m , Wm H−→ (13) m=0 m where ·−→ is a columnwise right-shift operation similar to a logical shift in programming languages that shifts all the columns of H by m positions to the right, and fills the vacant positions with zeros. The operation is size-preserving. It can be seen that the CNMF has M times as many weights as the regular NMF, whereas the number of hidden states is equal. for m = 0, 1, . . . , M − 1, with X t Ut = Wm Ht−→ m , m where ◦ denotes the Hadamard, i.e. entrywise, product, ·◦p is equivalent to entrywise exponentiation and ·◦−1 , respectively, m operator is the leftstands for the entrywise inverse. The ·←− shift counterpart of the right-shift operator. The details of the derivation of (16) can be found in the Appendix. C. Interpretation The two update rules given in (16) are of significant value because: • They are multiplicative, and thus, they converge fast and are easy to implement. • Eq. (16a) extends the update rule of the β-NMF for W to a set of M weight matrices that are linked through a convolution operation. It also extends the corresponding update rule of the existing convolutional NMFs with the squared Euclidean distance or the generalized Kullback– Leibler divergence to the family of β-divergences. 3 Eq. (16b) is even more important: Apart from providing a means to control the sparsity of H through λ1 and λ2 , it yields a complete update of the hidden states at every iteration taking all M weight matrices into account. The update rule in (16b) can be viewed as an equivalent to a descent in the direction of the Reynolds gradient, which is to take the average over the partial derivatives under the group of time translations (in m). Note that this is not the same as what was originally proposed by Smaragdis [8]. The average operator reduces the gradient spread as a function of Wm at each step t and makes the descent converge to a single point that is most likely. • D. Uniqueness and Normalization It is understood that the factorization is not unique. This is easily shown by the equivalence V' M −1 X m Wm H−→ m=0 ≡ M −1 X m = Wm B B−1 H−→ M −1 X (17) fm H e m W −→ m=0 m=0 f m = Wm B and H e = B−1 H, for any B that has an with W I×I inverse, B ∈ R in general. Non-negativity still holds for f m and H e if B is a nonnegative diagonal matrix, which in W this case results in a scaling of the columns of Wm and the corresponding rows of H. We can use this to enforce insight the same p-norm on the matrices Wi = wkm i ∈ RK×M : −1 −1 B = diag kW1 k−1 . (18) p , kW2 kp , . . . , kWI kp In Fig. 1 it can be seen that the mean loss is bounded from below by our updates. Furthermore, one can observe that our updates converge to a local minimum after the same number of iterations, whereas the convergence rate of the updates by Smaragdis depends on the number of time translations M in the convolution: the greater M , the slower the rate. It can be added that the average updates produce a steadily decreasing loss, while the biased updates, where H is updated based on WM only, can diverge as the iteration continues. For M = 2, the loss distributions of our and Smaragdis’ average updates seem to have the same mean but a slightly different standard deviation. To test the hypothesis that the two populations are statistically equivalent w.r.t. the mean, we use Welch’s t-test. The shown p-values indicate that already after the first cycle of updates the null hypothesis can be rejected almost surely, i.e. the two update rules converge to different minima. Quite similar results were obtained for other values of K and N . V. C ONCLUSION To the best of our knowledge, with the exception of [9] in the 2D unconstrained case, our letter is the only to provide a complete and correct derivation of the multiplicative updates for the convolutional NMF. In addition, the contrast function is generalized to the family of β-divergences and a penalty is added in the form of an elastic net for sparse modeling. It is shown by simulation that the originally proposed updates for the convolutional NMF are suboptimal and that convergence rate of the latter, and stability alike, are subject to the size of the convolution kernel. A PPENDIX Let ukn = i,m wki (m) hi,n−m and U = ukn ∈ RK×N . Then, for any p ∈ {1, 2, . . . , K}, q ∈ {1, 2, . . . , I}, and r ∈ {0, 1, . . . , M − 1}: P e If all Wm are multiplied by B before (16b), (16b) yields H immediately. In this way, we can put the significances of the hidden variables in relation to each other. IV. S IMULATION In this section, we compare our update rules with the ones by Smaragdis in terms of convergence performance. For this, we generate 100 different V-matrices from M χ2 -distributed Wm -matrices, wki (m) = 2 X ∂ Cλ ({Wm }, H) ∂wpq (r) ! X uβpn vpn uβ−1 ∂ pn = − n ∂wpq (r) β β−1 X β−1 β−2 = upn − vpn upn hq,n−r . n (21) Choosing κ from (15a) as 2 wki (m) p ∼ χ22 wkip (m) ∼ N (0, 1), (19) κ= P p=1 n and a uniformly distributed H-matrix, hin ∼ U(0, 1). wpq (r) uβ−1 pn hq,n−r , (22) leads to the first update rule (20) The Wm -matrices, which represent random spectrograms, are normalized by the Frobenius norm as per (18). The value for β is set to 1 (Kullback–Leibler) and the regularization term is omitted (λ2 = λ1 = 0). The factorizations are simulated with 10 random initializations of {Wm,0 } and H0 (with non-zero entries). The results shown in Fig. 1 are thus computed over ensembles of 1000 losses at each iteration. In our simulation, the number of visible and hidden variables is K = 1000 and I = 10, and the number of time samples (realizations) is N = 100. = t wki (m) β−2 vkn utkn hti,n−m . P tβ−1 t n ukn hi,n−m P t+1 wki (m) n (23) Further, for any p ∈ {1, 2, . . . , I} and q ∈ {1, 2, . . . , N }: ∂ Cλ ({Wm }, H) ∂hpq ! β−1 uβkn vkn ukn ∂ X = − k,n ∂hpq β β−1 + λ2 ∂ ∂ h2 + λ1 hpq . ∂hpq pq ∂hpq (24) 4 M=2 10 3 10 0 10 -4 10 0 10 1 Mean Villasana et al. Smaragdis (biased) Smaragdis (average) 10 1 10 0 Std 10 1 10 1 10 1 10 0 Villasana et al. Smaragdis (biased) Smaragdis (average) 10 -6 10 0 Villasana et al. Smaragdis (biased) Smaragdis (average) 10 0 10 -4 10 0 10 2 M = 16 10 -6 10 0 10 2 1 10 2 Villasana et al. Smaragdis (biased) Smaragdis (average) Std Mean 10 3 10 1 10 2 1 p-value Villasana et al. vs. Smaragdis (average) Significance level p-value Villasana et al. vs. Smaragdis (average) Significance level 0.05 0.05 10 0 10 1 Iteration 10 2 10 0 10 1 Iteration 10 2 Fig. 1. Simulation results including the mean and the standard deviation of the loss distribution and the p-value from Welch’s t-test for the hypothesis that our and Smaragdis’ update rules on average converge to the same minimum as a function of iteration (M is the number of time translations). It is straightforward to show that ∂ ukn = wkp (n − q) ∂hpq (25) by setting n − m = q m = n − q. As a result, plugging in q + m for n in (24) and using (25), we finally obtain ∂ Cλ ({Wm }, H) ∂hpq X β−2 = wkp (m) uβ−1 − v u k,q+m k,q+m k,q+m k,m + 2λ2 hpq + λ1 . (26) Choosing µ from (15b) as hpq β−1 k,m wkp (m) uk,q+m µ= P + 2λ2 hpq + λ1 , leads to the second update rule P t tβ−2 k,m wki (m) vk,n+m uk,n+m t+1 t hin = hin P . t tβ−1 t k,m wki (m) uk,n+m + 2λ2 hin + λ1 (27) (28) R EFERENCES [1] D. D. Lee and H. S. Seung, “Learning the parts of objects by nonnegative matrix factorization,” Nature, vol. 401, pp. 788–791, 1999. [2] ——, “Algorithms for non-negative matrix factorization,” in Advances in Neural Information Processing Systems, 2001, pp. 556–562. [3] P. Paatero and U. Tapper, “Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values,” Environmetrics, vol. 5, no. 2, pp. 111–126, 1994. [4] P. Paatero, “Least squares formulation of robust non-negative factor analysis,” Chemometrics and Intelligent Laboratory Systems, vol. 37, no. 1, pp. 23–35, 1997. [5] P. O. Hoyer, “Non-negative sparse coding,” in Neural Networks for Signal Processing, 2002, pp. 557–565. [6] J. Eggert and E. Korner, “Sparse coding and NMF,” in Neural Networks, vol. 4, 2004, pp. 2529–2533. [7] P. Smaragdis, “Non-negative matrix factor deconvolution; extraction of multiple sound sources from monophonic inputs,” in Independent Component Analysis and Blind Signal Separation, 2004, pp. 494–499. [8] ——, “Convolutive speech bases and their application to supervised speech separation,” IEEE Transactions on Audio, Speech, and Language Processing, vol. 15, no. 1, pp. 1–12, 2007. [9] M. N. Schmidt and M. Mørup, “Nonnegative matrix factor 2-D deconvolution for blind single channel source separation,” in Independent Component Analysis and Blind Signal Separation, 2006, pp. 700–707. [10] W. Wang, A. Cichocki, and J. A. Chambers, “A multiplicative algorithm for convolutive non-negative matrix factorization based on squared Euclidean distance,” IEEE Transactions on Signal Processing, vol. 57, no. 7, pp. 2858–2864, 2009. [11] A. Cichocki, R. Zdunek, and S. Amari, “Csiszár’s divergences for nonnegative matrix factorization: Family of new algorithms,” in Independent Component Analysis and Blind Signal Separation, 2006, pp. 32–39. [12] C. Févotte and J. Idier, “Algorithms for nonnegative matrix factorization with the β-divergence,” Neural Computation, vol. 23, no. 9, pp. 2421– 2456, 2011. [13] A. Basu, I. R. Harris, N. L. Hjort, and M. C. Jones, “Robust and efficient estimation by minimising a density power divergence,” Biometrika, vol. 85, no. 3, pp. 549–559, 1998. [14] L. M. Bregman, “The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming,” USSR Computational Mathematics and Mathematical Physics, vol. 7, no. 3, pp. 200–217, 1967. [15] H. Zou and T. Hastie, “Regularization and variable selection via the elastic net,” Journal of the Royal Statistical Society, vol. 67, no. 2, pp. 301–320, 2005.	8
Accepted as a workshop contribution at ICLR 2015 L EARNING C OMPACT C ONVOLUTIONAL N ETWORKS WITH N ESTED D ROPOUT N EURAL arXiv:1412.7155v4 [cs.CV] 10 Apr 2015 Chelsea Finn, Lisa Anne Hendricks & Trevor Darrell Department of Computer Science UC Berkeley Berkeley, CA 94704, USA {cbfinn, lisa anne, trevor}@eecs.berkeley.edu A BSTRACT Recently, nested dropout was proposed as a method for ordering representation units in autoencoders by their information content, without diminishing reconstruction cost (Rippel et al., 2014). However, it has only been applied to training fully-connected autoencoders in an unsupervised setting. We explore the impact of nested dropout on the convolutional layers in a CNN trained by backpropagation, investigating whether nested dropout can provide a simple and systematic way to determine the optimal representation size with respect to the desired accuracy and desired task and data complexity. 1 M OTIVATION Supervised convolutional neural networks (CNNs) learn representations that are effective on a wide range of tasks. A drawback of current such approaches, however, is that the selection of such architectures is largely optimized by hand, with researchers explicitly searching architecture hyperparameters via cross-validation. Model selection for network architectures has been explored in the context of learning network connectivity dating back to Optimal Brain Damage from LeCun et al. (1989) and has been continued to be explored in the context of learning optimal sparse models. To our knowledge there is no deep visual network capable of increasing its representation capacity based on the complexity of available data or tasks. The recently proposed nested dropout method implicitly accomplishes this, by learning deep representation units in an incremental fashion. We investigate whether such an approach is applicable to visual CNN models, and propose a visual CNN model which can learn to scale its capacity according to the complexity of the data presented to the network during training. In standard dropout, units in the layer are independently dropped out with probability p, namely the output of that unit is set to zero. This is traditionally applied to convolutional and fully-connected layers during training time, and has been shown to act as a regularizer, discouraging over-fitting to the training data (Hinton et al., 2012), though it has also been applied to entire channels of a convolutional layer output by Tompson et al. (2014). Empirically, when training large networks, such as those trained on ImageNet, drop out is necessary to avoid over fitting (Krizhevsky et al., 2012). Nested dropout, on the other hand, randomly draws unit indices from a geometric distribution and drops out all of the units that follow the number drawn, e.g. if a number k is drawn, then the units 0 through k are kept and the remaining units are dropped. When applied to a single layer semi-linear autoencoder, this technique has been proven to enforce an ordering of the units by their information capacity, while not decreasing the flexibility of the representation nor the quality of the resulting solution (Rippel et al., 2014). The primary contributions of this paper are to (1) demonstrate that nested dropout can successfully be applied to convolutional layers trained by back-propagation, (2) propose nested dropout as an advantageous method to learn CNNs that adapt to task and data complexity in a deep learning setting, and (3) provide our implementation in Caffe (Jia et al., 2014), a widely used deep learning framework, upon publication. 1 Accepted as a workshop contribution at ICLR 2015 2 N ESTED D ROPOUT ON CNN S The nested dropout algorithm for a convolutional layer with n channels is as follows: for each sample in a mini-batch, we draw a number k from a geometric distribution and drop out the latter n − k channels of the output of the layer. Nested dropout can also be applied to multiple layers in a network by applying nested dropout to each layer iteratively. First, the number of filters ni for layer i is determined through nested dropout. After fixing the number of filters ni in layer i, nested dropout can then be used to determine the number of filters, ni+1 , for layer i + 1. We trained our CNNs using Stochastic Gradient Descent (SGD) and mini-batches of 100 samples. Because dropped out units are determined by drawing from a geometric distribution, units with a low index are rarely dropped out and thus converge quickly, whereas latter units are frequently dropout out and thus learned very slowly. Filters are incrementally fixed once they have converged, and only the remaining filters are considered when drawing numbers from the geometric distribution. Though incrementing the sweeping index could be done upon filter convergence, we achieved satisfactory results by simply incrementing the unit sweeping index after a set number of iterations. To implement this in the Caffe framework, we added a nested dropout layer that can follow any layer (e.g. convolutional, fully-connected) in the same way as the standard dropout layer. We also added customizations to the solver to support unit sweeping during training. 3 E XPERIMENTS Figure 1: Accuracy as a function of the number of filters in conv1, with and without nested dropout. The oracle consists of 32 separate networks, whereas the nested dropout curve only requires training a single network. The “brain damaged” baseline is trained without nested dropout and only the k best filters are used during test. The nested dropout network achieves maximum accuracy with a compact representation of 23 filters while requiring considerably fewer iterations than a brute force approach. We apply nested dropout to the first convolutional layer of a CNN trained to classify images in the CIFAR-10 dataset, using the default Caffe architecture and training with a fixed learning rate. In Figure 1, we show the test accuracy of a single network trained with nested dropout as a function 2 Accepted as a workshop contribution at ICLR 2015 of the number of conv1 filters. We report the accuracy with k filters by using the partially-trained network after the kth filter has converged, and testing it with the last n − k filters dropped out. We compare to two naive approaches to select the number of filters in conv1. The first approach simply trains 32 separate networks which differ in the number of conv1 filters. The second approach trains a single network with 32 filters, but at test time, a varying number of conv1 filters are used with the remaining dropped out. Note that because there is no inherent ordering to the learned representation without nested dropout, removing a filter severely damages the network resulting in low accuracies. Our experiments in Figure 1 demonstrate that nested dropout efficiently determines the relationship between model capacity and test accuracy. The nested dropout implementation requires 90,000 iterations of training, whereas the brute force approach requires significantly more, up to millions iterations. For example, training 32 separate networks to completion requires 2,880,000 iterations. (a) (b) Figure 2: conv1 filters of networks trained with nested dropout (a) and without (b). Note that the latter filters of (a) carry very little information, yet both networks achieve similar performance, giving an indication for how many units are necessary for this model. In Figure 2, we visualize the filters learned with and without nested dropout. Though the latter filters carry very little information, the test accuracy of the two sets of filters are comparable with 0.787 test accuracy for the network trained with nested dropout and 0.786 test accuracy for the baseline network. Applying nested dropout to the second convolutional layer yielded similar results to our experiments with conv1. After training a network with nested dropout applied to conv1 filters, we determined only 23 conv1 filters are necessary to achieve the maximum classification accuracy for this network. We next train a network in which nested dropout follows conv2 and with a unit sweeping index of 5,000. After learning 25 conv2 filters, the training accuracy converges to 78%. Thus, by using nested dropout we can reduce the total number of parameters in the first two layers by 25% from 64 filters to 48, while maintaining similar accuracy. 4 D ISCUSSION In summary, we have provided a simple method for determining a more compact representation of the convolutional layers. In our experiments, we learned a representation that achieved the same classification accuracy using 23 conv1 filters and 25 conv2 filters rather than the baseline 32 each, within the same optimization framework. A main advantage of our method is that it enables the network to gradually increase network capacity during training. Additionally, we hope that, in the future, ordering parameters may provide insights into optimization of deep convolutional neural networks and how the network architecture impacts performance. 3 Accepted as a workshop contribution at ICLR 2015 ACKNOWLEDGMENTS This work was supported in part by DARPA’s MSEE and SMISC programs, NSF awards IIS1427425, IIS-1212798, and IIS-1116411, Toyota, and the Berkeley Vision and Learning Center. Chelsea Finn was supported by a Berkeley EECS Fellowship and Lisa Anne Hendricks by an NDSEG Fellowship. R EFERENCES Hinton, Geoffrey E., Srivastava, Nitish, Krizhevsky, Alex, Sutskever, Ilya, and Salakhutdinov, Ruslan. Improving neural networks by preventing co-adaptation of feature detectors. CoRR, abs/1207.0580, 2012. URL http://arxiv.org/abs/1207.0580. Jia, Yangqing, Shelhamer, Evan, Donahue, Jeff, Karayev, Sergey, Long, Jonathan, Girshick, Ross, Guadarrama, Sergio, and Darrell, Trevor. Caffe: Convolutional architecture for fast feature embedding. In Proceedings of the ACM International Conference on Multimedia, pp. 675–678. ACM, 2014. Krizhevsky, Alex, Sutskever, Ilya, and Hinton, Geoffrey E. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012. LeCun, Yann, Denker, John S, Solla, Sara A, Howard, Richard E, and Jackel, Lawrence D. Optimal brain damage. In NIPs, volume 2, pp. 598–605, 1989. Rippel, Oren, Gelbart, Michael A, and Adams, Ryan P. Learning ordered representations with nested dropout. arXiv preprint arXiv:1402.0915, 2014. Tompson, Jonathan, Goroshin, Ross, Jain, Arjun, LeCun, Yann, and Bregler, Christoph. Efficient object localization using convolutional networks. CoRR, abs/1411.4280, 2014. URL http: //arxiv.org/abs/1411.4280. 4	9
IDENTITIES IN PLACTIC, HYPOPLACTIC, SYLVESTER, BAXTER, AND RELATED MONOIDS arXiv:1611.04151v2 [math.CO] 5 Mar 2017 ALAN J. CAIN AND ANTÓNIO MALHEIRO Abstract. This paper considers whether non-trivial identities are satisﬁed by certain ‘plactic-like’ monoids that, like the plactic monoid, are closely connected to combinatorics. New results show that the hypoplactic, sylvester, Baxter, stalactic, and taiga monoids satisfy identities, and indeed give shortest identities satisﬁed by these monoids. The existing state of knowledge is discussed for the plactic monoid and left and right patience sorting monoids. 1. Introduction The ubiquitous plactic monoid, whose elements can be viewed as semistandard Young tableaux, and which appears in such diverse contexts as symmetric functions [Mac08], representation theory [Ful97], algebraic combinatorics [Lot02], Kostka–Foulkes polynomials [LS81, LS78], Schubert polynomials [LS85, LS90], and musical theory [Jed11], is one of a family of ‘placticlike’ monoids that are closely connected with combinatorics. These monoids include the hypoplactic monoid [KT97, Nov00], the sylvester monoid [HNT05], the taiga monoid [Pri13], the stalactic monoid [HNT07, Pri13], the Baxter monoid [Gir12], and the left and right patience sorting monoids [Rey07, CMS]. Each of these monoids is obtained by factoring the free monoid A∗ over the infinite ordered alphabet A = {1 < 2 < 3 < . . .} by a congruence that arises from a so-called insertion algorithm that computes a combinatorial object from a word. For instance, for the plactic monoid, the corresponding combinatorial objects are (semistandard) Young tableaux; for the sylvester monoid, they are binary search trees. An identity is a formal equality between two words in the free monoid, and is non-trivial if the two words are distinct. A monoid M satisfies such an identity if the equality in M holds under every substitution of letters in the words by elements of M . For example, any commutative monoid satisfies the non-trivial identity xy = yx. A finitely generated group has polynomial growth if and only if it is virtually nilpotent [Gro81], and a virtually nilpotent group satisfies a non-trivial identity [Mal53, NT63]. Thus it is natural to ask whether every finitely generated semigroup or monoid with polynomial growth satisfies a non-trivial identity. Schneerson [Shn93] provided the first counterexample, but it remains to be seen whether there is a The ﬁrst author was supported by an Investigador FCT fellowship (IF/01622/2013/CP1161/CT0001). For both authors, this work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações), and the project PTDC/MHC-FIL/2583/2014. 1 Symbol Plactic Hypoplactic Sylvester #-sylvester Baxter Stalactic Taiga Left patience sorting Right patience sorting plac hypo sylv sylv# baxt stal taig lPS rPS Identity satisfied ? xyxy = yxyx xyxy = yxxy yxyx = yxxy yxxyxy = yxyxxy xyx = yxx xyx = yxx None None Discussed in Original result Subsec. Subsec. Subsec. Subsec. Subsec. Subsec. Subsec. Subsec. Subsec. 3.7 3.2 3.4 3.4 3.5 3.3 3.3 3.6 3.6 — Present Present Present Present Present Present [CMS] [CMS] paper paper paper paper paper paper ALAN J. CAIN AND ANTÓNIO MALHEIRO Monoid 2 Table 1. Examples of non-trivial identities satisfied by ‘plactic-like’ monoids. The stated identities are always shortest non-trivial identities satisfied by the corresponding monoid, but there may be other identities of the same length. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 3 ‘natural’ finitely generated semigroup with polynomial growth that does not satisfy a non-trivial identity. It is easy to see that the finite-rank analogues of the plactic monoid and the other related monoids discussed above have polynomial growth. This naturally leads to the question of whether these monoids satisfy non-trivial identities, for if any of them failed to do so, it would certainly be a very natural example of a polynomial-growth monoid that does not satisfy a non-trivial identity. Further motivation for this question comes from a result of Jaszuńska & Okniński, who proved that the Chinese monoid [CEK+ 01], which has the same growth type as the plactic monoid [DK94] but which does not arise from such a natural combinatorial object, satisfies Adian’s identity xyyxxyxyyx = xyyxyxxyyx [JO11, Corollary 3.3.4]. (This is the shortest non-trivial identity satisfied by the bicyclic monoid [Adi66, Chapter IV, Theorem 2]). The goal of this paper is to present new results showing that some of these monoids satisfy non-trivial identities, and to survey the state of knowledge for other monoids in this family. New results show that the hypoplactic, sylvester, baxter, stalactic, and taiga monoids satisfy non-trivial identities. A discussion of the situation for the left and right patience sorting monoids and plactic monoids completes the paper. Table 1 summarizes the results. 2. ‘Plactic-like’ monoids In this section, we recall only the definition and essential facts about the various monoids; for further background, see [Lot02, Ch. 5] on the plactic monoid, [Nov00] on the hypoplactic monoid, [HNT05] on the sylvester monoid; [Pri13, § 5] on the taiga monoid; [Pri13] on the stalactic monoid; [Gir12] on the Baxter monoid; and [CMS] on the patience sorting monoids. 2.1. Alphabets and words. For any alphabet X, the free monoid (that is, the set of all words, including the empty word) on the alphabet X is denoted X ∗ . The empty word is denoted ε. For any u ∈ X ∗ , the length of u is denoted \|u\|, and, for any x ∈ X, the number of times the symbol x appears in u is denoted \|u\|x . Throughout the paper, A = {1 < 2 < 3 < . . .} is the set of natural numbers viewed as an infinite ordered alphabet, and An = {1 < 2 < . . . < n} is set of the first n natural numbers viewed as a finite ordered alphabet. 2.2. Combinatorial objects and insertion algorithms. A Young tableau is a finite array of symbols from A, with rows non-decreasing from left to right and columns strictly increasing from top to bottom, with shorter rows below longer ones, and with rows left-justified. An example of a Young tableau is (2.1) 1 1 1 2 5 . 3 3 5 6 6 The following algorithm takes a Young tableau and a symbol from A and yields a new Young tableau: 4 ALAN J. CAIN AND ANTÓNIO MALHEIRO Algorithm 2.1 (Schensted’s algorithm). Input: A Young tableau T and a symbol a ∈ A. (1) If a is greater than or equal to every entry in the topmost row of T , add a as an entry at the rightmost end of T and output the resulting tableau. (2) Otherwise, let z be the leftmost entry in the top row of T that is strictly greater than a. Replace z by a in the topmost row and recursively insert z into the tableau formed by the rows of T below the topmost. (Note that the recursion may end with an insertion into an ‘empty row’ below the existing rows of T .) A quasi-ribbon tableau is a finite array of symbols from A, with rows non-decreasing from left to right and columns strictly increasing from top to bottom, that does not contain any 2 × 2 subarray (that is, of the form ). An example of a quasi-ribbon tableau is: (2.2) 1 1 1 2 3 3 5 5 . 6 6 Notice that the same symbol cannot appear in two different rows of a quasiribbon tableau. There is also an insertion algorithm for quasi-ribon tableau: Algorithm 2.2 ([Nov00, Algorithm 4.4]). Input: A quasi-ribbon tableau T and a symbol a ∈ A. If there is no entry in T that is less than or equal to a, output the tableau obtained by putting a and gluing T by its top-leftmost entry to the bottom of a. Otherwise, let x be the right-most and bottom-most entry of T that is less than or equal to a. Put a new entry a to the right of x and glue the remaining part of T (below and to the right of x) onto the bottom of the new entry a. Output the new tableau. A right strict binary search tree is a labelled rooted binary tree where the label of each node is greater than or equal to the label of every node in its left subtree, and strictly less than every node in its right subtree. An example of a binary search tree is 5 2 (2.3) 1 1 1 6 5 . 6 3 3 The insertion algorithm for right strict binary search trees adds the new symbol as a leaf node in the unique place that maintains the property of being a right strict binary search tree: Algorithm 2.3 ([HNT05, § 3.3]). Input: A right strict binary search tree T and a symbol a ∈ A. If T is empty, create a node and label it a. If T is non-empty, examine the label x of the root node; if a ≥ x, recursively insert a into the right subtree IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 5 of the root node; otherwise recursively insert a into the left subtree of the root note. Output the resulting tree. A left strict binary search tree is a labelled rooted binary tree where the label of each node is strictly greater than the label of every node in its left subtree, and less than or equal to every node in its right subtree; see the left tree shown in (2.4) below for an example. The insertion algorithm for left strict binary search trees is dual to Algorithm 2.3 above: Algorithm 2.4. Input: A left strict binary search tree T and a symbol a ∈ A. If T is empty, create a node and label it a. If T is non-empty, examine the label x of the root node; if a ≤ x, recursively insert a into the left subtree of the root node; otherwise recursively insert a into the right subtree of the root note. Output the resulting tree. The canopy of a (labelled or unlabelled) binary tree T is the word over {0, 1} obtained by traversing the empty subtrees of the nodes of T from left to right, except the first and the last, labelling an empty left subtree by 1 and an empty right subtree by 0. (See (2.4) below for examples of canopies.) A pair of twin binary search trees consist of a left strict binary search tree TL and a right strict binary search tree TR , such that TL and TR contain the same symbols, and the canopies of TL and TR are complementary, in the sense that the i-th symbol of the canopy of TL is 0 (respectively 1) if and only if the i-th symbol of the canopy of TR is 1 (respectively 0). The following is an example of a pair of twin binary search trees, with the complementary canopies 0110101 and 1001010 shown in grey:   3 (2.4) 5  1 6     3 6  11 1 1   1 5   1 1   2 5 1 0 1 0  2 1 , 5 0 1 3 0 1 0 3 0 1 0 6 0 1 0 6      .       A stalactic tableau is a finite array of symbols of A in which columns are top-aligned, and two symbols appear in the same column if and only if they are equal. For example, (2.5) 3 1 2 6 5 3 1 6 5 1 is a stalactic tableau. The insertion algorithm is very straightforward: Algorithm 2.5 ([HNT07, § 3.7]). Input: A stalactic tableau T and a symbol a ∈ A. 6 ALAN J. CAIN AND ANTÓNIO MALHEIRO If a does not appear in T , add a to the left of the top row of T . If a does appear in T , add a to the bottom of the (by definition, unique) column in which a appears. Output the new tableau. A binary search tree with multiplicities is a labelled binary search tree in which each label appears at most once, and where a non-negative integer called the multiplicity is assigned to each node label. An example of a binary search tree is: 52 (2.6) 1 62 2 13 . 32 (The superscripts on the labels in each node denote the multiplicities.) Algorithm 2.6. ([Pri13, Algorithm 3]) Input: A binary search tree with multiplicities T and a symbol a ∈ A. If T is empty, create a node, label it by a, and assign it multiplicity 1. If T is non-empty, examine the label x of the root node; if a < x, recursively insert a into the left subtree of the root node; if a > x, recursively insert a into the right subtree of the root note; if a = x, increment by 1 the multiplicity of the node label x. An lPS tableau is a finite array of symbols of A in which columns are top-aligned, the entries in the top row are non-decreasing from left to right, and the entries in each column are strictly increasing from top to bottom. For example, (2.7) 1 1 1 2 5 3 3 5 6 6 is an lPS tableau. The insertion algorithm is very straightforward: Algorithm 2.7 ([TY11, § 3.2]). Input: An lPS tableau T and a symbol a ∈ A. If a is greater than or equal to every symbol that appears in the top row of T , add a to the right of the top row of T . Otherwise, let C be the leftmost column whose topmost symbol is strictly greater than a. Slide column C down by one space and add a as a new entry on top of C. Output the new tableau. An rPS tableau is a finite array of symbols of A in which columns are topaligned, the entries in the top row are increasing from left to right, and the entries in each column are non-decreasing from top to bottom. For example, (2.8) 1 2 5 6 1 3 5 1 6 3 is an rPS tableau. Again, the insertion algorithm is very straightforward: Algorithm 2.8 ([TY11, § 3.2]). Input: An rPS tableau T and a symbol a ∈ A. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 7 If a is greater than every symbol that appears in the top row of T , add a to the right of the top row of T . Otherwise, let C be the leftmost column whose topmost symbol is greater than or equal to a. Slide column C down by one space and add a as a new entry on top of C. Output the new tableau. 2.3. Monoids from insertion. Let M ∈ {plac, hypo, sylv, sylv# , baxt, stal, taig, lPS, rPS} and u ∈ A∗ . Using the insertion algorithms described above, one can compute from u a combinatorial object PM (u) of the type associated to M in Table 2. If M 6= baxt, one computes PM (u) by starting with the empty combinatorial object and inserting the symbols of u one-by-one using the appropriate insertion algorithm and proceeding through the word u either left-to-right or right-to-left as shown in the table. For M = baxt, one uses a slightly different procedure: Pbaxt (u) is the pair of twin binary search trees Psylv# (u), Psylv (u) . For each M ∈ {plac, hypo, sylv, sylv# , baxt, stal, taig, lPS, rPS}, define the relation ≡M by u ≡M v ⇐⇒ PM (u) = PM (v). In each case, the relation ≡M is a congruence on A∗ , and so the factor monoid M = A∗/≡M can be formed, and is named as in Table 2. The rankn analogue is the factor monoid Mn = A∗n /≡M , where the relation ≡M is naturally restricted to A∗n × A∗n . It follows from the definition of ≡M (for any M ∈ {plac, hypo, sylv, sylv# , baxt, stal, taig, lPS, rPS}) that each element [u]≡M of the factor monoid M can be identified with the combinatorial object PM (u). The evaluation (also called the content) of a word u ∈ A∗ , denoted ev(u), is the infinite tuple of non-negative integers, indexed by A, whose a-th element is \|u\|a ; thus this tuple describes the number of each symbol in A that appears in u. It is immediate from the definition of the monoids above that if u ≡M v, then ev(u) = ev(v), and hence it makes sense to define the evaluation of an element p of one of these monoids to be the evaluation of any word representing it. Note that Mn is a submonoid of M for all n, and that Mm is a submonoid of Mn for all m ≤ n. In each case, M1 is a free monogenic monoid and thus commutative, but that Mn is non-commutative for n ≥ 2 (and thus M is also non-commutative). 3. Identities Subsections 3.2 to 3.5 find shortest non-trivial identities satisfied by hypo, stal, taig, sylv, sylv# , and baxt, in that order. Subsection 3.6 discusses the situation for lPS and rPS, and Subsection 3.7 discusses the current state of knowledge for plac. 3.1. Preliminaries. An identity over an alphabet X is a formal equality u = v, where u and v are words in the free monoid X ∗ , and is non-trivial if u and v are not equal. The elements of X are called variables. A monoid M satisfies the identity u = v if, for every homomorphism φ : X ∗ → M , the equality φ(u) = φ(v) holds in M . It is often useful to think of this in the informal way mentioned in the introduction: M satisfies the identity u = v if equality in M holds under every substitution of variables in the words u Symbol Combinatorial object Algorithm Direction Example Plactic Hypoplactic Sylvester #-sylvester Baxter Stalactic Taiga Left patience sorting Right patience sorting plac hypo sylv sylv# baxt stal taig lPS rPS Young tableau Quasi-ribbon tableau Right strict binary search tree Left strict binary search tree Pair of twin BSTs Stalactic tableau BST with multiplicities lPS tableau rPS tableau 2.1 2.2 2.3 2.4 — 2.5 2.6 2.7 2.8 Pplac (3613151265) is (2.1) Phypo (3613151265) is (2.2) Psylv (3613151265) is (2.3) Psylv (3613151265) is (2.3) Pbaxt (3613151265) is (2.4) Pstal (3613151265) is (2.5) Ptaig (3613151265) is (2.6) PlPS (3613151265) is (2.7) PrPS (3613151265) is (2.8) L-to-R L-to-R R-to-L L-to-R — R-to-L R-to-L L-to-R L-to-R ALAN J. CAIN AND ANTÓNIO MALHEIRO Monoid 8 Table 2. Insertion algorithms used to compute combinatorial objects, and the corresponding monoids. ‘L-to-R’ and ‘R-to-L’ abbreviate ‘left-to-right’ and ‘right-to-left’, respectively. The algorithm used to compute Pbaxt (·) is a special case and is discussed in the main text. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 9 and v by elements of M . Note that if a monoid satisfies an identity u = v, all of its submonoids and all of its homomorphic images satisfy u = v. Lemma 3.1. Let M be a monoid satisfying a non-trivial identity. Then the shortest (in terms of the sums of the lengths of the two sides) non-trivial identity satisfied by M is an identity over an alphabet with at most two variables. Proof. It is sufficient to show that, for any non-trivial identity over an alphabet with at most three variables, there is a strictly shorter identity with at most two variables. So suppose that u = v is a non-trivial identity over X satisfied by M , where \|X\| ≥ 3. Interchanging u and v if necessary, assume \|u\| ≤ \|v\|. Let eM be the identity (that is, the multiplicative neutral element) of M . First consider the case where u is a prefix of v. By non-triviality, u is strictly shorter than v, so that v = uw for some non-empty word w. Let x ∈ X be a variable that appears in w. So x\|u\|x = x\|v\|x is a nontrivial identity. To see that it is satisfied by M , proceed as follows: Let φ′ : {x}∗ → M be a homomorphism. Extend φ′ to a homomorphism φ : X → M by defining φ(z) = eM for all z ∈ X \ {x}. Note that φ′ (x\|u\|x ) = φ(u) and φ′ (x\|v\|x ) = φ(v) by the definition of φ′ , and that φ′ (u) = φ′ (v) since M satisfies u = v. Combining these equaltities shows that φ(x\|u\|x ) = φ(x\|v\|x ). Hence M satisfies the non-trivial identity x\|u\|x = x\|v\|x over the alphabet {x}. Now suppose u is not a prefix of v. Suppose u = u1 · · · u\|u\| and v = v1 · · · v\|v\| . Let i be minimal such that the variables ui and vi are different; suppose these variables are x and y. Let u′ and v ′ be obtained from u and v respectively by deleting all variables other than x and y. By the choice of i, u′ = v ′ is a non-trivial identity. To see that it is satisfied by M , proceed as follows: Let φ′ : {x, y}∗ → M be a homomorphism. Extend φ′ to a homomorphism φ : X ∗ → M by defining φ(z) = eM for all z ∈ X \ {x, y}. Note that φ(u) = φ′ (u′ ) and φ(v) = φ′ (v ′ ) by the definitions of the words u′ and v ′ and the homomorphism φ; note also that φ(u) = φ(v) since M satisfies the identity u = v. Combining these equalities shows that φ′ (u′ ) = φ′ (v ′ ). So M satisfies the non-trivial identity u′ = v ′ over the alphabet {x, y}. Note that in both cases the resulting identity is strictly shorter than the original one. Lemma 3.2. Let M be a monoid that contains a free monogenic submonoid and suppose that M satisfies an identity u = v over an alphabet X. Then \|u\| = \|v\| and \|u\|x = \|v\|x for all x ∈ X. Consequently, if u = v is non-trivial, then \|X\| ≥ 2. Proof. Let a ∈ M generate a free monogenic submonoid of M and let eM be the identity of M . Let x ∈ X. Define φx : X ∗ → M, x 7→ a, z 7→ eM for all z ∈ X \ {x}. Then a\|u\|x = φx (u) = φx (v) = a\|v\|x since M satisfies u = v. Since a generates a free monogenic submonoid, \|u\|x = \|v\|x . Since this holds for all x ∈ X, it follows that \|u\| = \|v\|. 10 ALAN J. CAIN AND ANTÓNIO MALHEIRO The length of an identity u = v with \|u\| = \|v\| is defined to be \|u\| = \|v\|; note that this applies to any identity satisfied by a monoid that contains a free monogenic submonoid by Lemma 3.2. Two identities u = v and u′ = v ′ are equivalent if one can be obtained from the other by possibly renaming variables and swapping the two sides of the equality. For example, xyxy = yxxy and xyyx = yxyx are equivalent, since the second can be obtained from the first by interchanging the variables x and y and swapping the two sides. Lemma 3.3. (1) Up to equivalence, the only length-2 non-trivial identity that can be satisfied by a monoid that contains a free monogenic submonoid is xy = yx; thus any homogeneous monoid satisfying a length-2 identity is commutative. (2) Up to equivalence, the only length-3 non-trivial identities over a twoletter alphabet that can be satisfied by a monoid that contains a free monogenic submonoid are xxy = xyx, xxy = yxx, xyx = yxx. (1) Let u1 u2 = v1 v2 be a non-trivial identity satisfied by some monoid that contains a free monogenic submonoid. Suppose u1 and v1 are the same variable. Then Lemma 3.2, implies that u2 and v2 are also the same variable, contradicting non-triviality. Thus u1 and v1 are different variables. Then Lemma 3.2, implies that u2 is the same variable as v1 and v2 is the same variable as u1 . Thus, up to equivalence, the identity is xy = yx. (2) Let u1 u2 u3 = v1 v2 v3 (where ui , vi ∈ {x, y}) be a non-trivial identity satisfied by some homogeneous monoid. Let I be the set of indices i ∈ {1, 2, 3} where the variables ui and vi are the same. Clearly, non-triviality shows that \|I\| < 3. If \|I\| = 2, then, interchanging x and y if necessary, there is a unique i ∈ {1, 2, 3} where ui is x and vi is y. Then \|u\|x = 1 + \|v\|x , since the words u and v differ only at the i-th symbol; this contradicts Lemma 3.2. If \|I\| = 0, then every symbol of u differs from every symbol of v. Since \|u\| = 3, at least one of \|u\|x and \|u\|y is at least 2. Interchanging x and y if necessary, assume \|u\|x ≥ 2. Then \|v\|x = \|u\|y ≤ 1, which is a contadiction. The only remaining possibility is \|I\| = 1. Interchanging x and y if necessary, assume that ui and vi are both x for exactly one i ∈ 1, 2, 3. For each of the three possibilities, the conditions \|u\|x = \|v\|x and \|u\|y = \|v\|y require the other symbols in u must be x and y, with the corresponding symbols in v being the opposite. Each possibility gives one of the given identities. Proof. Let M ∈ {plac, hypo, sylv, taig, stal, baxt, lPS, rPS}. Then M contains the free monogenic submonoid M1 . (Note that M1 thus satisfies the non-trivial identity xy = yx.) By Lemma 3.2, if M satisfies a non-trivial identity, it must be over an alphabet with at least two variables. The aim is to find shortest non-trivial identities satisfies by each monoid M: Lemmata 3.1 and 3.2 show that it will suffice to consider identities u = v over {x, y} with IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 11 \|u\| = \|v\|. Note that since none of the monoids M is commutative, any identity must have length at least 3. 3.2. Hypoplactic monoid. The authors proved the following result using a quasi-crystal structure for the hypoplactic monoid [CM17, Theorem 9.3]; this subsection presents a direct proof. Proposition 3.4. The hypoplactic monoid satisfies the following non-trivial identities xyxy = xyyx = yxxy = yxyx; xxyx = xyxx. Furthermore, up to equivalence, these are the shortest non-trivial identities satisfied by the hypoplactic monoid. Proof. The first goal is to show that these identities are satisfied by hypo. Let s, t ∈ hypo, and let p, q ∈ A∗ be words representing s, t, respectively. Let B = {a1 < . . . < ak } be the set of symbols in A that appear in at least one of p and q. By Algorithm 2.2, symbols ai+1 and ai are on the same row of Phypo (w) (for any w ∈ B ∗ ) if and only if the word w does not contain a symbol ai somewhere to the right of a symbol ai+1 (since inserting this symbol ai results in the part of the quasi-ribbon tableau that contains ai+1 being glued below the entry ai ). Suppose pqpq contains a symbol ai somewhere to the right of a symbol ai+1 . Then, regardless of whether these symbols both lie in u, both lie in v, or one lies in u and the other lies in v, the word pqqp also contains a symbol ai to the right of a symbol ai+1 . Similar reasoning establishes that if any of the words pqpq, qppq, qpqp, pqqp, ppqp, or pqpp, contains a symbol ai somewhere to the right of a symbol ai+1 , so do all the others. Hence either the symbols ai and ai+1 are on the same row in all of Phypo (pqpq), Phypo (pqqp), Phypo (qppq), Phypo (qpqp), Phypo (ppqp), and Phypo (pqpp), or on different rows in all of them. A quasi-ribbon tableau is clearly determined by its evaluation and by knowledge of whether adjacent different entries are on the same row. Thus, noting that ev(pqpq) = ev(qppq) = ev(qpqp) = ev(pqqp), it follows from the previous paragraph that stst = Phypo (pqpq) = tsst = Phypo (qppq) = tsts = Phypo (qpqp) = stts = Phypo (pqqp). Similarly, noting that ev(ppqp) = ev(pqpp), it follows that ssts = Phypo (ppqp) = Phypo (pqpp) = stss. The next goal is to show that, up to equivalence, these are the only length-4 non-trivial identities satisfied by hypo. Let u = v be a length-4 non-trivial identity over {x, y} satisfied by hypo. By Lemma 3.2, \|u\|x = \|v\|x and \|u\|y = \|v\|y . By Algorithm 2.2, symbols 1 and 2 are on different rows of Phypo (w) (for any w ∈ {1, 2}∗ ) if and only if the word w contain a symbol 1 somewhere to the right of a symbol 2. Suppose u contains a factor xy. Let φ : {x, y}∗ → 12 ALAN J. CAIN AND ANTÓNIO MALHEIRO hypo map x to 2 and y to 1; then Phypo (φ(u)) has 1 and 2 on different rows. The same must hold for Phypo (φ(v)), and thus v must contain a factor xy. The converse is similar, and the reasoning is parallel for factors yx. Thus u contains a factor xy (respectively, yx) if and only if v contains a factor xy (respectively, yx). If both u and v contain only factors xy, then u = x\|u\|x y \|u\|y = x\|v\|x y \|v\|y = v, which contradicts non-triviality. Similarly, it is impossible for u and v to contain only factors yx. So both u and v contain factors xy and yx. Interchanging x and y if necessary, assume \|u\|x = \|v\|x ≥ \|u\|y = \|v\|y . First consider the case where \|u\|x = \|v\|x = 3 and \|u\|y = \|v\|y = 1. Since u and v both contain xy and yx, this implies that u, v ∈ {xxyx, xyxx}. Now consider the case \|u\|x = \|v\|x = 2 and \|u\|y = \|v\|y = 2. Again, since u and v both contain xy and yx, this implies that u, v ∈ {xyxy, xyyx, yxxy, yxyx}. Thus u = v is equivalent to one of the identities in the statement. The final goal is to prove that no length-3 non-trivial identity is satisfied by hypo. Consider the length-3 identities from Lemma 3.3(2): xxy = xyx, xxy = yxx, xyx = yxx. If one puts x = 1 and y = 2 into the first two identities and x = 2 and y = 1 in the second, one sees that hypo satisfies none of them: Phypo (112) = 1 1 2 6= 1 1 = Phypo (121); 2 Phypo (112) = 1 1 2 6= 1 1 = Phypo (211); 2 Phypo (212) = 1 6= 1 2 2 = Phypo (122). 2 2 Thus hypo satisfies no length-3 identity. 3.3. Stalactic and taiga monoids. Lemma 3.5. The stalactic monoid satisfies the non-trivial identity xyx = yxx. Proof. Let x, y ∈ stal and let u, v ∈ A∗ be words representing x, y, respectively. Since the tree Pstal (w) is computed by applying Algorithm 2.5 to each symbol in the word w ∈ A∗ , proceeding right to left, it is clear that the order of the rightmost appearances of each symbol in w determines order of symbols in the top row of the stalactic tableau Pstal (w). Since the order of rightmost appearances of each symbol in the words uvu and vuu are the same, the order of symbols in the top row of Pstal (uvu) and Pstal (vuu) is also the same. Since ev(uvu) = ev(vuu), the length of corresponding columns in Pstal (uvu) and Pstal (vuu) are equal. Hence xyx = Pstal (uvu) = Pstal (vuu) = yxx. Lemma 3.6. The taiga monoid does not satisfy either of the identities xxy = xyx or xxy = yxx, and does not satisfy a non-trivial identity of length 2. Proof. To see that taig does not satisfy either of the identities xxy = xyx or xxy = yxx, note that the rightmost symbol of w ∈ A∗ labels the root node IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 13 of Ptaig (w), but the rightmost symbols of both these identities are different. Thus substituting 1 for x and 2 for y shows that neither identity can be satisfied by taig. Finally, taig does not satisfy a non-trivial identity of length 2 since it is clearly non-commutative. Since the taiga monoid is a homomorphic image of the stalactic monoid [Pri13, § 5], any identity that is satisfied by stal is satisfied by taig. Combining this with Lemmata 3.3, 3.5, and 3.6 gives the following results: Proposition 3.7. The stalactic monoid satisfies the non-trivial identity xyx = yxx. Furthermore, this is the unique shortest non-trivial identity satisfied by the stalactic monoid. Proposition 3.8. The taiga monoid satisfies the non-trivial identity xyx = yxx. Furthermore, this is the unique shortest non-trivial identity satisfied by the taiga monoid. 3.4. Sylvester monoid. By the definition of a binary search tree, if a tree has a node with label a, then any other node with label a is either above it or in its left subtree. This shows that there is a unique path from the root to a leaf node that contains all nodes labelled a. In particular, there is at most one leaf node with label a. Furthermore, there is a unique node a that is both the leftmost node with label a and the node labelled a that is furthest from the root. Similarly, there is a unique node a that is both the rightmost node with label a and the node labelled a that is closest to the root. Lemma 3.9. The sylvester monoid is left-cancellative. Proof. Let u, v ∈ A∗ and a ∈ A. Suppose that Psylv (au) = Psylv (av). That is, the binary search trees Psylv (au) and Psylv (av) are equal. Note further that the last symbol a inserted into both binary search trees is a leaf node. Deleting this (unique) leaf node labelled a from the equal trees Psylv (au) and Psylv (av) leaves equal binary search trees. That is, Psylv (u) and Psylv (v) are equal. Since u and v are arbitrary words and a represents an arbitrary generator, it follows that sylv is left-cancellative. Lemma 3.10. The sylvester monoid does not satisfy an identity equivalent to one of the form uxx = vyx, for any u, v ∈ {x, y}∗ . Proof. Using the Algorithm 2.3, one sees that 2 Psylv (· · · 22) = 2 2 and Psylv (· · · 12) = 1 ; note that these binary search trees differ in the left child of the root node. Thus to see that the identity uxx = vyx cannot be satisfied by sylv, one can substitute 2 for x and 1 for y. Lemma 3.11. Let p, q, r ∈ sylv be such that ev(p) = ev(q) = ev(r). Then pr = qr. 14 ALAN J. CAIN AND ANTÓNIO MALHEIRO Proof. Let u, v, w ∈ A∗ be words representing p, q, r, respectively. The tree Psylv (x) is computed by applying Algorithm 2.3 to each symbol in x, proceeding right to left. That is, Psylv (uw) and Psylv (vw) are obtained by inserting u and v (respectively) into Psylv (w). Since ev(u) = ev(v) = ev(w), every symbol that appears in u or v also appears in w and thus in Psylv (w). Consider how further symbols from u or v are inserted into Psylv (w): • Let a be the smallest symbol that appears in u, v, and w. Then the leftmost node of Psylv (w) must be labelled by a. Thus, during the computation of Psylv (uw) and Psylv (vw), all symbols a in u and v will certainly be inserted into the left subtree of this leftmost node in Psylv (w), and no other symbols are inserted into this left subtree by the minimality of a. • Let c be some symbol other than a that appears in u, v, and w, and let b be the maximum symbol less than c appearing in u, v, and w. Let Nc be the leftmost node in Psylv (w) labelled by c and let Nb be the rightmost node in Psylv (w) labelled by b. Suppose v is the label of the lowest common ancestor Nv of the nodes Nb and Nc . If Nv is neither Nb not Nc , then b ≤ v < c, which implies v = b by the choice of b, which contradicts the fact that Nb is the rightmost node labelled b. Thus the lowest common ancestor of Nb and Nc must be one of Nb or Nc . That is, one of the following must hold: – Nb is above Nc . Then Nc is in the right subtree of Nb , since b < c. Since there is no node that is to the right of Nb and to the left of Nc , it follows that Nc has an empty left subtree. Thus any symbol c in u or v will be inserted into this currently empty left subtree of Nc , and no other symbols will be inserted into this subtree by the maximality of b among the symbols less than c. – Nc is above Nb . Then Nb is in the left subtree of Nc , since b < c. Since there is no node that is to the right of Nb and to the left of Nc , it follows that Nb is the left child of Nc , and that Nb has an empty right subtree. Thus any symbol c in u or v will be inserted into this currently empty right subtree of Nb , and no other symbols will be inserted into this subtree by the maximality of b among the symbols less than c. Combining these cases, one sees that every symbol d from u or v is inserted into a particular previously empty subtree of Psylv (w), dependent only on the value of the symbol d (and not on its position in u or v), and that unequal symbols are inserted into different subtrees. Since ev(u) = ev(v), the same number of symbols d are inserted, for each such symbol d. Hence Psylv (uw) = Psylv (vw) and thus pr = Pstal (uw) = Pstal (vw) = qr. Proposition 3.12. The sylvester monoid satisfies the non-trivial identity xyxy = yxxy. Furthermore, up to equivalence, this is the unique shortest identity satisfied by the sylvester monoid. Proof. Let x, y ∈ sylv. Let p = r = xy and q = yx. Then ev(p) = ev(q) = ev(r), so pr = qr by Lemma 3.11. Thus sylv satisfies xyxy = yxxy. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 15 Since hypo is a homomorphic image of sylv [Pri13, Example 6], and hypo satisfies no length-3 identity by Proposition 3.4, it follows that sylv cannot satisfy a length-3 identity. So let u = v be some length-4 identity satisfied by sylv. Suppose u = u1 u2 u3 u4 and v = v1 v2 v3 v4 , where ui , vi ∈ {x, y}. Since the rightmost symbol in a word w determines the root node of Psylv (w), it follows that u4 and v4 are the same symbol; interchanging x and y if necessary, assume this is y. If u1 and v1 were the same symbol, then the left-cancellativity of sylv (Lemma 3.9) would imply that sylv satisfied the length-3 identity u2 u3 u4 = v2 v3 v4 , which contradicts the previous paragraph. Thus u1 and v1 are different symbols. Interchanging u and v if necessary, assume that u1 is x and v1 is y. The condition \|u\|y = \|v\|y implies that at least one of u2 and u3 is y, which yields the following three possibilities for u: xyyy, xxyy, xyxy. The conditions \|u\|x = \|v\|x and \|u\|y = \|v\|y now imply the following four possibilities for the identity u = v (with the first possibility for u above giving two different identities): xyyy = yxyy, xyyy = yyxy, xxyy = yxxy, xyxy = yxxy. The last of these is the identity already proven to hold in sylv. The second and third cannot hold in sylv by Lemma 3.10. In the first identity, putting x = 1 and y = 2 proves that it is not satisfied by sylv: 2 Psylv (1222) = 2 2 1 2 6= 2 = Psylv (2122) 1 2 Thus xyxy = yxxy is the unique shortest identity satisfied by sylv. Since the sylvester and #-sylvester monoids are anti-isomorphic, the following results follow by dual reasoning: Lemma 3.13. The #-sylvester monoid is right-cancellative. Lemma 3.14. The #-sylvester monoid does not satisfy an identity equivalent to one of the form xxu = xyv, for any u, v ∈ {x, y}∗ . Lemma 3.15. Let p, q, s ∈ sylv# be such that ev(p) = ev(q) = ev(s). Then sp = sq. Proposition 3.16. The #-sylvester monoid satisfies the non-trivial identity yxyx = yxxy. Furthermore, up to equivalence, this is the unique shortest identity satisfied by the #-sylvester monoid. 3.5. Baxter monoid. Lemma 3.17. Let p, q, r, s ∈ baxt be such that ev(p) = ev(q) = ev(r) = ev(s). Then spr = sqr. Proof. Let t, u, v, w ∈ A∗ represent p, q, r, s, respectively. Let t′ = wt and u′ = wu; note that ev(t′ ) = ev(u′ ). By Lemma 3.11 applied to the elements Psylv (t′ ), Psylv (u′ ), Psylv (v), it follows that Psylv (t′ v) = Psylv (u′ v); thus Psylv (wtv) = Psylv (wuv) 16 ALAN J. CAIN AND ANTÓNIO MALHEIRO Let t′′ = tv and u′′ = uv; note that ev(t′′ ) = ev(u′′ ). By Lemma 3.15 applied to the elements Psylv# (t′ ), Psylv# (u′ ), Psylv# (w), it follows that Psylv# (wt′′ ) = Psylv# (wu′′ ); thus Psylv# (wtv) = Psylv# (wuv). Hence spr = Pbaxt (wuv) = Psylv# (wtv), Psylv (wtv) = Psylv# (wuv), Psylv (wuv) = Pbaxt (wuv) = sqp. Proposition 3.18. The Baxter monoid satisfies the identities yxxyxy = yxyxxy and xyxyxy = xyyxxy. Furthermore, up to equivalence, these are the unique shortest non-trivial identities satisfied by the Baxter monoid. Proof. Let x, y ∈ baxt. Let p = r = xy and q = s = yx. Then ev(p) = ev(q) = ev(r) = ev(s). Thus, by Lemma 3.17, yxxyxy = spr = sqr = yxyxxy. The same reasoning with s = xy shows that xyxyxy = xyyxxy. The next step is to show that, up to equivalence, these are the only identities of length 6 satisfied by baxt. So suppose that u = v is an identity of length 6 satisfied by baxt, with u = u1 u2 · · · u6 and v = v1 v2 · · · v6 , where ui , vi ∈ {x, y}. Since the first symbol in a word w determines the root symbol of the left-hand tree in the pair Pbaxt (u), and since the last symbol determines the root symbol of the right-hand tree, it follows that u1 and v1 must be the same variable, and that u6 and v6 must the same variable. By Lemmata 3.10 and 3.14, u2 and v2 must be the same variable, and u5 and v5 must be the same variable. Since sylv and sylv# are both homomorphic images of baxt [Gir12, Proposition 3.7], the identity u = v is also satified by sylv and sylv# . Since sylv is left-cancellative by Lemma 3.9, u3 u4 u5 u6 = v3 v4 v5 v6 is satisfied by sylv and is thus equivalent to xyxy = yxxy. The aim is to characterize u = v up to equivalence, so assume that u3 u4 u5 u6 = v3 v4 v5 v6 actually is the identity xyxy = yxxy. Since sylv# is right-cancellative by Lemma 3.13, u1 u2 u3 u4 = v1 v2 v3 v4 is satisfied by sylv# and is thus equivalent to yxyx = yxxy, which is equivalent (by swapping the two sides) to yxxy = yxyx and (by interchaning x and y) to xyxy = xyyx. Combining these with u3 u4 u5 u6 = v3 v4 v5 v6 from the previous paragraph shows that u = v is (up to equivalence) either yxxyxy = yxyxxy or xyxyxy = xyyxxy. Finally, it is necessary to show that no length-5 identity is satisfied by baxt. Suppose u = v is an identity of length 5 satisfied by the Baxter monoid. Suppose that u = u1 u2 · · · u5 and v = v1 v2 · · · v5 , where ui , vi ∈ {x, y}. As before, considering the root symbols of the left-hand and right-hand trees in Pbaxt (w) shows that u1 and v1 must be the same symbol, and that u5 and v5 must the same symbol. Since sylv and sylv# are both homomorphic images of baxt [Gir12, Proposition 3.7], both of them satisfy the identity u = v. Furthermore, sylv# is right-cancellative and so satisfies the identity u1 u2 u3 u4 = v1 v2 v3 v4 ; while sylv is left-cancellative and so satisfies the identity u2 u3 u4 u5 = v2 v3 v4 v5 . By Proposition 3.12, the unique length-4 identity satisfied by sylv is xyxy = yxxy, so the identity u = v is either xxyxy = xyxxy or yxyxy = yyxxy. Deleting the rightmost symbol y from each of these identities yields xxyx = IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 17 xyxx and yxyx = yyxx, one of which must the identity u1 u2 u3 u4 = v1 v2 v3 v4 satisfied by sylv# . This is a contradiction, since by Proposition 3.16 the only length-4 identity satisfied by sylv# is yxyx = yxxy. 3.6. Left and right patience sorting monoids. The present authors and Silva [CMS, § 3.8] have shown that the elements PlPS (21) and PlPS (1) generate a free submonoid of lPS , and that the elements PrPS (2) and PrPS (31) generate a free submonoid of rPS. Since free monoids of rank at least 2 satisfy no non-trivial identities, it follows that neither lPS nor rPS satisfy a non-trivial identity. Let lPSn = A∗n /≡lPS (where ≡lPS is naturally restricted to A∗n ×A∗n ) be the left patience sorting monoid of rank n. Let rPSn = A∗n /≡rPS (where ≡rPS is naturally restricted to A∗n ×A∗n ) be the right patience sorting monoid of rank n. Clearly lPSn is a submonoid of lPS and rPSn is a submonoid of rPS. Then by the previous paragraph, lPSn for n ≥ 2 satisfies no non-trivial identities, while lPS1 , as a monogenic monoid, is of course commutative and satisfies the identity xy = yx. Similarly, rPSn for n ≥ 3 satisfies no non-trivial identities, while rPS1 , as a monogenic monoid, is of course commutative and satisfies the identity xy = yx. Finally, rPS2 satisfies the identity xyxy = xyyx, and, up to equivalence, this is the shortest identity satisfied by rPS2 [CMS, Proposition 3.29]. 3.7. Plactic monoid. Whether the plactic monoid satisfies a non-trivial identity remains an open question, but some partial results are known. Let placn = A∗n /≡plac (where ≡plac is naturally restricted to A∗n × A∗n ) be the plactic monoid of rank n; clearly placn is a submonoid of plac. First, plac1 is monogenic and thus commutative and satisfies xy = yx. Kubat & Okniński [KO14] have shown that plac2 satisfies Adian’s identity xyyxxyxyyx = xyyxyxxyyx, and that plac3 satisfies the identity pqqpqp = pqpqqp, where p(x, y) and q(x, y) are respectively the left and right side of Adian’s identity (and so the identity pqqpqp = pqpqqp has sixty variables x or y on each side). Furthermore, plac3 does not satisfy Adian’s identity [KO14, p. 111–2]. Conjecture 3.19. For each n ≥ 4, there is a non-trivial identity satisfied by placn . Conjecture 3.20. There is no non-trivial identity satisfied by plac. References [Adi66] S. Adian. ‘Deﬁning relations and algorithmic problems for groups and semigroups’. Trudy Mat. 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Ser. A, 274 (1963), pp. 1–4. J.-B. Priez. ‘A lattice of combinatorial Hopf algebras: Binary trees with multiplicities’. In Formal Power Series and Algebraic Combinatorics, Nancy, 2013. The Association. Discrete Mathematics & Theoretical Computer Science. url: http://www.dmtcs.org/pdfpapers/dmAS0196.pdf. M. Rey. ‘Algebraic constructions on set partitions’. In Formal Power Series and Algebraic Combinatorics, 2007. url: http://www-igm.univ-mlv.fr/~ rey/articles/rey-fpsac07.pdf. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS [Shn93] [TY11] 19 L. Shneerson. ‘Identities in ﬁnitely generated semigroups of polynomial growth’. J. Algebra, 154, no. 1 (1993), pp. 67–85. doi: 10.1006/jabr.1993.1004. H. Thomas & A. Yong. ‘Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm’. Advances in Applied Mathematics, 46, no. 1 (2011), pp. 610–642. doi: 10.1016/j.aam.2009.07.005. Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829–516 Caparica, Portugal E-mail address: a.cain@fct.unl.pt Departamento de Matemática & Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829–516 Caparica, Portugal E-mail address: ajm@fct.unl.pt	4
A NOTE ON NILPOTENT SUBGROUPS OF AUTOMORPHISM GROUPS OF RAAGS arXiv:1803.08284v1 [math.GR] 22 Mar 2018 JAVIER ARAMAYONA & ANTHONY GENEVOIS Abstract. We observe that automorphism groups of right-angled Artin groups contain nilpotent non-abelian subgroups, namely H3 (Z) the threedimensional integer Heisenberg group, provided they admit a certain type of element, called an adjacent transvection. This represents a (minor) extension of a result of Charney-Vogtmann. 1. Introduction Automorphism groups of right-angled Artin groups (RAAGs) are normally studied through their comparison with linear groups and automorphism groups of free groups (and thus, by extension, with mapping class groups). One fact that distinguishes the linear group GL(n, Z) from the automorphism group Aut(Fn ) of the free group Fn , and from the mapping class group Mod(S), is that every solvable subgroup of the latter two is virtually abelian; see [2, 1] and [3], respectively. In sharp contrast, arbitrary automorphism groups of RAAGs may contain a copy of GL(n, Z) with n ≥ 3. Even when this is not the case, Charney-Vogtmann [6] proved that (outer) automorphism groups of RAAGs contain torsion-free nilpotent and non-abelian subgroups, whenever they contain at least two adjacent transvections; see below for a definition. Further examples may be found in [7]. The purpose of this short note is to observe that the presence of a single adjacent transvection suffices for the (full) automorphism group of a centerless RAAG to contain a nilpotent non-abelian subgroup. We remark that Charney-Vogtmann showed [6], if there are no adjacent transvections then every solvable subgroup is virtually abelian. Before we state our result, denote by ≤ the usual partial order in the vertex set V (Γ) of Γ; see Section 2. Also, let H3 (Z) be the three-dimensional integer Heisenberg group, which has presentation H3 (Z) = hA, B, C \| [A, C] = [B, C] = 1, [A, B] = Ci. With this notation, we will observe: Theorem 1.1. Let Γ be a simplicial graph. If there are adjacent vertices a, b ∈ V (Γ) with a ≤ b, such that b is not adjacent to all the vertices of Γ, then H3 (Z) is a subgroup of Aut(AΓ ). As a consequence, for such Γ the group Aut(AΓ ) does not embed in Mod(S) or Aut(Fn ). A further immediate application, surely well-known to experts, is that no finite-index subgroup of such Aut(AΓ ) may act nicely on non-positively curved space: 1 2 JAVIER ARAMAYONA & ANTHONY GENEVOIS Corollary 1.2. If Γ is as in Theorem 1.1 , then no finite-index subgroup of Aut(AΓ ) can act properly by semi-simple isometries on a CAT(0) space. In sharp contrast, Aut(AΓ ) and Out(AΓ ) are sometimes commensurable to a right-angled Artin group [5, 7], and therefore they act nicely on CAT(0) spaces. 2. Definitions In order to make this note as concise and self-contained as possible, we only introduce the objects that we will need in the proof of our results. We refer the reader to the various papers in the bibliography below for a thorough introduction to automorphism groups of right-angled Artin groups. Let Γ be a simplicial graph, and denote by V (Γ) (resp. E(Γ)) its set of vertices (resp. edges). The right-angled Artin group AΓ defined by Γ is the group given by the presentation AΓ = hv ∈ V (Γ) \| [v, w] = 1 ⇐⇒ vw ∈ E(Γ)i . Given a right-angled Artin group AΓ , we consider its automorphism group Aut(AΓ ). This is a finitely presented group with an explicit generating set [11, 12] and an explicit (although in slightly different terms) presentation [9]. Here, we will need a specific type of element of Aut(AΓ ) and Out(AΓ ), called a tranvection. Given vertices v, w ∈ V (Γ), the transvection tvw is the self-map of AΓ defined by tvw : v 7→ vw and tvw (u) = u for every u 6= v. An easy observation is that tvw ∈ Aut(AΓ ) if and only if lk(v) ⊂ st(w); here, lk(·) denotes the link of a vertex in Γ, while st(·) denotes its star, namely the link union the vertex. As usual in the literature, we will write v ≤ w to mean lk(v) ⊂ st(w), noting that the relation ≤ is in fact a partial order. Finally, we say that tvw is an adjacent transvection if v and w are adjacent in Γ. 3. A homomorphic image of the Heisenberg group We now prove Theorem 1.1: Proof of Theorem 1.1. Let a, b be adjacent vertices of Γ, with a ≤ b and which are not adjacent to all the vertices of Γ. We write ca and cb for the automorphisms of AΓ given by conjugation by a and b, respectively. Finally, let t = tab the transvection that sends a to ab and fixes the rest of generators. First, observe that ca and cb commute since a and b are adjacent. Next, we claim that [cb , t] = 1 also. Indeed, if v 6= a, tcb t−1 (v) = tcb (v) = t(bvb−1 ) = bvb−1 = cb (v), and tcb t−1 (a) = tcb (ab−1 ) = t(bab−2 ) = babb−2 = cb (a). Finally, we claim that tca t−1 = cb ca . Indeed, if v 6= a then tca t−1 (v) = tca (v) = t(ava−1 ) = abvb−1 a−1 = cb ca (v), A NOTE ON NILPOTENT SUBGROUPS OF AUTOMORPHISM GROUPS OF RAAGS 3 while tca t−1 (a) = tca (ab−1 ) = t(a2 b−1 a−1 ) = ababb−1 b−1 a−1 = cb ca (a), as desired. In light of the above, and using notation for the presentation of H3 (Z) given in the introduction, the map H3 (Z) → Aut(AΓ ) given by A 7→ ca , B 7→ t, and C 7→ cb is a homomorphism. Now, we want to prove that this is a monomorphism. From the presentation of H3 (Z), it clearly follows that any of its elements can be written as Am B n C p for some m, n, p ∈ Z. Furthermore, Am B n C p = 1 if and only if m = n = p = 0. Indeed, we first deduce from the equality Am B n C p = 1 that m = n = 0 by looking at the image of Am B n C p into the abelianization of H3 (Z); therefore, our equality reduces to C p = 1, and finally p = 0 follows from the torsion-freeness of H3 (Z). Thus, in order to prove that our homomorphism H3 (Z) → Aut(ΓG) is injective, we only have n p to verify that, for every m, n, p ∈ Z, cm a t cb = 1 implies m = n = p = 0. n p So let m, n, p ∈ Z and suppose that cm a t cb = 1. Noticing that n p m n p −p m n m n m n −m cm = abn , a t cb (a) = ca t (b ab ) = ca t (a) = ca (ab ) = a ab a p we first deduce that n = 0. Therefore, cm a cb = 1. This precisely means that m p a b belongs to the center of AΓ . On the other, the center of AΓ corresponds exactly to the subgroup generated by the vertices which are adjacent to all the vertices of Γ. Because we supposed that neither a nor b is adjacent to all the vertices of Γ, it follows that am bp = 1. Therefore, we deduce that am = bp = 1, and finally that m = p = 0 by torsion-freeness of AΓ . 4. Actions on CAT(0) spaces We prove Corollary 1.2: Proof. Let Γ as in Theorem 1.1, so there exist adjacent vertices a, b ∈ V (Γ) with a ≤ b and which are not adjacent to all the vertices of Γ. As above, write ca and cb for the conjugations on a and b, respectively, and t for the transvection tab . Notice that ca and cb are infinite-order automorphisms because a and b do not belong to the center of AΓ . Consider a finite-index subgroup G < Aut(AΓ ); as such, there exists some m m m ≥ 1 such that cm a , cb , t ∈ G. It is immediate to check that, for all n ∈ N, (1) −n mn tn cm = cm a t a cb . Suppose now that G acts properly by semi-simple isometries on some CAT(0) m space X. As cm a and cb commute, the Flat Torus Theorem [4] implies that X contains an isometrically embedded copy of a Euclidean plane, on which m cm a and cb act by translations with quotient a 2-torus. It follows that, as mn must tend a transformation of this plane, the translation length of cm a cb to infinity as n grows; on the other hand, equation (1) implies that this translation length must be equal to that of cm a . This is a contradiction, and the result follows. A possible interpretation of the previous proof is the following. As Theorem 1.1 proves, Aut(ΓG) contains a copy of the three-dimensional integer 4 JAVIER ARAMAYONA & ANTHONY GENEVOIS Heisenberg group H3 (Z) = hA, B, C \| [A, C] = [B, C] = 1, [A, B] = Ci. It is not difficult to notice that, for any k ≥ 1, the subgroup hAk , B k , C k i ≤ H3 (Z) is naturally isomorphic to H3 (Z) itself, so that any finite-index subgroup of Aut(ΓG) has to contain a copy of H3 (Z). Therefore, Corollary 1.2 follows from the fact that H3 (Z) does not act properly by simple-isometries on a CAT(0) space. This is essentially what we have shown in the previous proof. An alternative argument can be found in [10, Corollary 5.1], where it is furthermore proved that, for any proper action of H3 (Z) on a CAT(0) space, C is necessarily parabolic. It is worth noticing that H3 (Z) has a proper parabolic action on the complex hyperbolic plance H2C , which is a proper finite-dimensional CAT(−1) space [10, Corollary 5.1]. References [1] E. Alibegović, Translation lengths in Out(Fn ), Geom. Dedicata 92 (1), pp. 87–93 (2002). [2] M. Bestvina, M. Feighn, M. Handel, Solvable subgroups of Out(Fn ) are virtually Abelian, Geom. Dedicata 104 (1), pp. 71–96 (2004). [3] J. Birman, A. Lubotzky, J. McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (4), pp. 1107–1120 (1983). [4] M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin (1999). [5] R. Charney, M. Farber, Random groups arising as graph products, Alg. Geom. Topol. 12, pp. 979–995 (2012). [6] R. Charney, K. Vogtmann. Subgroups and quotients of automorphism groups of RAAGs. In Low-dimensional and symplectic topology, pp. 9–27, Proc. Sympos. Pure Math., 82, Amer. Math. Soc., Providence, RI (2011). [7] M. Day, Finiteness of outer automorphism groups of random right-angled Artin groups, Algebr. Geom. Topol. 12 (3), pp. 1553–1583 (2012). [8] M. Day, On solvable subgroups of automorphism groups of right-angled Artin groups, Internat. J. Algebra Comput. 21, pp. 61–70 (2011). [9] M. Day. Peak reduction and finite presentations for automorphism groups of right angled Artin groups, Geom. Topol. 13, pp. 817–855 (2009). [10] K. Fujiwara, T. Shioya, S. Yamagata, Parabolic isometries of CAT(0) spaces and CAT(0) dimensions, Algebr. Geom. Topol. 4, pp. 861–892 (2004). [11] M. R. Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. 52 (2), pp. 318–334 (1995). [12] H. Servatius, Automorphisms of graph groups, J. Algebra 126 (1), pp. 34–60 (1989).	4
arXiv:1711.04052v1 [math.AC] 11 Nov 2017 BIG COHEN-MACAULAY MODULES, MORPHISMS OF PERFECT COMPLEXES, AND INTERSECTION THEOREMS IN LOCAL ALGEBRA LUCHEZAR L. AVRAMOV, SRIKANTH B. IYENGAR, AND AMNON NEEMAN Abstract. There is a well known link from the first topic in the title to the third one. In this paper we thread that link through the second topic. The central result is a criterion for the tensor nilpotence of morphisms of perfect complexes over commutative noetherian rings, in terms of a numerical invariant of the complexes known as their level. Applications to local rings include a strengthening of the Improved New Intersection Theorem, short direct proofs of several results equivalent to it, and lower bounds on the ranks of the modules in every finite free complex that admits a structure of differential graded module over the Koszul complex on some system of parameters. 1. Introduction A big Cohen-Macaulay module over a commutative noetherian local ring R is a (not necessarily finitely generated) R-module C such that some system of parameters of R forms a C-regular sequence. In [16] Hochster showed that the existence of such modules implies several fundamental homological properties of finitely generated R-modules. In [17], published in [18], he proved that big Cohen-Macaulay modules exist for algebras over fields, and conjectured their existence in the case of mixed characteristic. This was recently proved by Y. André in [2]; as a major consequence many “Homological Conjectures” in local algebra are now theorems. A perfect R-complex is a bounded complex of finite projective R-modules. Its level with respect to R, introduced in [6] and defined in 2.3, measures the minimal number of mapping cones needed to assemble a quasi-isomorphic complex from bounded complexes of finite projective modules with differentials equal to zero. The main result of this paper, which appears as Theorem 3.3, is the following Tensor Nilpotence Theorem. Let f : G → F be a morphism of perfect complexes over a commutative noetherian ring R. If f factors through a complex whose homology is I-torsion for some ideal I of R with height I ≥ levelR HomR (G, F ), then the induced morphism ⊗nR f : ⊗nR G → ⊗nR F is homotopic to zero for some non-negative integer n. Date: November 15, 2017. 2010 Mathematics Subject Classification. 13D22 (primary); 13D02, 13D09 (secondary). Key words and phrases. big Cohen-Macaulay module, homological conjectures, level, perfect complex, rank, tensor nilpotent morphism. Partly supported by NSF grants DMS-1103176 (LLA) and DMS-1700985 (SBI). 1 2 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN Big Cohen-Macaulay modules play an essential, if discreet role in the proof, as a tool for constructing special morphisms in the derived category of R; see Proposition 3.7. In applications to commutative algebra it is convenient to use another property of morphisms of perfect complexes: f is fiberwise zero if H(k(p) ⊗R f ) = 0 holds for every p in Spec R. Hopkins [21] and Neeman [25] have shown that this is equivalent to tensor nilpotence; this is a key tool for the classification of the thick subcategories of perfect R-complexes. It is easy to see that the level of a complex does not exceed its span, defined in 2.1. Due to these remarks, the Tensor Nilpotence Theorem is equivalent to the Morphic Intersection Theorem. If f is not fiberwise zero and factors through a complex with I-torsion homology for an ideal I of R, then there are inequalities: span F + span G − 1 ≥ levelR HomR (G, F ) ≥ height I + 1 . In Section 4 we use this result to prove directly, and sometimes to generalize and sharpen, several basic theorems in commutative algebra. These include the Improved New Intersection Theorem, the Monomial Theorem and several versions of the Canonical Element Theorem. All of them are equivalent, but we do not know if they imply the Morphic Intersection Theorem; a potentially significant obstruction to that is that the difference span F − levelR F can be arbitrarily big. Another application, in Section 5, yields lower bounds for ranks of certain finite free complexes, related to a conjecture of Buchsbaum and Eisenbud, and Horrocks. In [7] a version of the Morphic Intersection Theorem for certain tensor triangulated categories is proved. This has implications for morphisms of perfect complexes of sheaves and, more generally, of perfect differential sheaves, over schemes. 2. Perfect complexes Throughout this paper R will be a commutative noetherian ring. This section is a recap on the various notions and construction, mainly concerning perfect complexes, needed in this work. Pertinent references include [6, 26]. 2.1. Complexes. In this work, an R-complex (a shorthand for ‘a complex of Rmodules’) is a sequence of homomorphisms of R-modules X := ∂X X ∂n−1 n → Xn−1 −−−→ Xn−2 −→ · · · · · · −→ Xn −−− such that ∂ X ∂ X = 0. We write X ♮ for the graded R-module underlying X. The ith i X suspension of X is the R-complex Σi X with (Σi X)n = Xn−i and ∂nΣ X = (−1)i ∂n−i for each n. The span of X is the number span X := sup{i \| Xi 6= 0} − inf{i \| Xi 6= 0} + 1 Thus span X = −∞ if and only if X = 0, and span X = ∞ if and only if Xi 6= 0 for infinitely many i ≥ 0. The span of X is finite if and only if span X is an natural number. When span X is finite we say that X is bounded. Complexes of R-modules are objects of two distinct categories. In the category of complexes C(R) a morphism f : Y → X of R-complexes is a family (fi : Yi → Xi )i∈Z or R-linear maps satisfying ∂iX fi = fi−1 ∂iY . It is a quasiisomorphism if H(f ), the induced map in homology, is bijective. Complexes that can be linked by a string of quasi-isomorphisms are said to be quasi-isomorphic. TENSOR NILPOTENT MORPHISMS 3 The derived category D(R) is obtained from C(R) by inverting all quasi-isomorphisms. For constructions of the localization functor C(R) → D(R) and of the derived functors ?⊗LR ? and RHomR (¿, ?), see e.g. [13, 31, 24]. When P is a complex of projectives with Pi = 0 for i ≪ 0, the functors P ⊗LR ? and RHomR (P, ?) are represented by P ⊗R ? and HomR (P, ?), respectively. In particular, the localization functor induces for each n a natural isomorphism of abelian groups ∼ HomD(R) (P, Σn X) . (2.1.1) H−n (RHomR (P, X)) = 2.2. Perfect complexes. In C(R), a perfect R-complex is a bounded complex of finitely generated projective R-modules. When P is perfect, the R-complex P ∗ := HomR (P, R) is perfect and the natural biduality map P −→ P ∗∗ = HomR (HomR (P, R), R) is an isomorphism. Moreover for any R-complex X the natural map P ∗ ⊗R X −→ HomR (P, X) is an isomorphism. In the sequel these properties are used without comment. 2.3. Levels. A length l semiprojective filtration of an R-complex P is a sequence of R-subcomplexes of finitely generated projective modules 0 = P (0) ⊆ P (1) ⊆ · · · ⊆ P (l) = P ♮ ♮ such that P (i − 1) is a direct summand of P (i) and the differential of P (i)/P (i−1) is equal to zero, for i = 1, . . . , l. For every R-complex F , we set F is a retract of some R-complex P that R . level F = inf l ∈ N has a semiprojective filtration of length l By [6, 2.4], this number is equal to the level F with respect to R, defined in [6, 2.1]. In particular, levelR F is finite if and only if F is quasi-isomorphic to some perfect complex. When F is quasi-isomorphic to a perfect complex P , one has levelR F ≤ span P . (2.3.1) Indeed, if P := 0 → Pb → · · · → Pa → 0, then consider the filtration by subcomplexes P (n) := P<n+a . The inequality can be strict; see 2.7 below. When R is regular, any R-complex F with H(F ) finitely generated satisfies levelR F ≤ dim R + 1 . (2.3.2) For R-complexes X and Y one has (2.3.3) levelR (Σi X) = levelR X for every integer i, and R level (X ⊕ Y ) = max{levelR X, levelR Y }. These equalities follow easily from the definitions. Lemma 2.4. The following statements hold for every perfect R-complex P . (1) levelR (P ∗ ) = levelR P . (2) For each perfect R-complex Q there are inequalities levelR (P ⊗R Q) ≤ levelR P + levelR Q − 1 . levelR HomR (P, Q) ≤ levelR P + levelR Q − 1 . 4 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN Proof. (1) If P is a retract of P ′ , then levelR P ≤ levelR P ′ and P ∗ is a retract of (P ′ )∗ . Thus, we can assume P itself has a finite semiprojective filtration {P (n)}ln=0 . The inclusions P (l − i) ⊆ P (l − i + 1) ⊆ P define subcomplexes ∗ P ∗ (i) := Ker(P ∗ −→ P (l − i) ) ⊆ Ker(P ∗ −→ P (l − i + 1)) =: P ∗ (i + 1) of finitely generated projective modules. They form a length l semiprojective filtration of P ∗ , as P ∗ (i − 1)♮ is a direct summand of P ∗ (i)♮ and there are isomorphisms ∗ P ∗ (n) ∼ P (l − n + 1) . = P ∗ (n − 1) P (l − n) This gives levelR P ∗ ≤ levelR P , and the reverse inequality follows from P ∼ = P ∗∗ . (2) Assume first that P has a semiprojective filtration {P (n)}ln=0 and Q has a semiprojective filtration {Q(n)}m n=0 . For all h, i, we identity P (h) ⊗R Q(i) with a subcomplex of P ⊗R Q. For each non-negative integer n ≥ 0 form the subcomplex X C(n) := P (j + 1) ⊗R Q(n − j) j>0 of P ⊗R Q. A direct computation yields an isomorphism of R-complexes Q(n − j) C(n) ∼ X P (j + 1) ⊗R . = C(n − 1) P (j) Q(n − j − 1) j>0 l+m−1 {C(n)}n=0 Thus is a semiprojective filtration of P ⊗R Q. The second inequality in (2) follows from the first one, given (1) and the isomorphism HomR (P, Q) ∼ = P ∗ ⊗R Q. Next we verify the first inequality. There is nothing to prove unless the levels of P and Q are finite. Thus we may assume that P is a retract of a complex P ′ with a semiprojective filtration of length l = levelR P and Q is a retract of a complex Q′ with a semiprojective filtration of length m = levelR Q. Then P ⊗R Q is a retract of P ′ ⊗R Q′ , and—by what we have just seen—this complex has a semiprojective filtration of length l + m − 1, as desired. 2.5. Ghost maps. A ghost is a morphism g : X → Y in D(R) such that H(g) = 0; see [11, §8]. Evidently a composition of morphisms one of which is ghost is a ghost. The next result is a version of the “Ghost Lemma”; cf. [11, Theorem 8.3], [29, Lemma 4.11], and [5, Proposition 2.9]. Lemma 2.6. Let F be an R-complex and c an integer with c ≥ levelR F . When g : X → Y is a composition of c ghosts the following morphisms are ghosts F ⊗LR g : F ⊗LR X −→ F ⊗LR Y and RHomR (F, g) : RHomR (F, X) −→ RHomR (F, Y ) Proof. For every R-complex W there is a canonical isomorphism ≃ RHomR (F, R) ⊗LR W −−→ RHomR (F, W ) , so it suffices to prove the first assertion. For that, we may assume that F has a semiprojective filtration {F (n)}ln=0 , where l = levelR F . By hypothesis, g = h ◦ f where f : X → W is a (c − 1)fold composition of ghosts and h : W → Y is a ghost. Tensoring these maps with the exact sequence of R-complexes ι π 0 −→ F (1) −−→ F −−→ G −→ 0 TENSOR NILPOTENT MORPHISMS 5 where G := F/F (1), yields a commutative diagram of graded R-modules H(F (1) ⊗R X) // H(F ⊗R X) H(F (1)⊗f ) // H(G ⊗R X) H(F ⊗f ) H(F (1) ⊗R W ) H(ι⊗W ) // H(F ⊗R W ) H(F (1)⊗h) H(G⊗f ) H(π⊗W ) // H(G ⊗R W ) H(F ⊗h) H(F (1) ⊗R Y ) H(ι⊗Y ) // H(F ⊗R Y ) // H(G ⊗R Y ) where the rows are exact. Since levelR G ≤ l − 1 ≤ c − 1, the induction hypothesis implies G ⊗ f is a ghost; that is to say, H(G ⊗ f ) = 0. The commutativity of the diagram above and the exactness of the middle row implies that Im H(F ⊗ f ) ⊆ Im H(ι ⊗ W ) This entails the inclusion below. Im H(F ⊗ g) = H(F ⊗ h)(Im H(F ⊗ f )) ⊆ H(F ⊗ h)(Im H(ι ⊗ W )) ⊆ Im H(ι ⊗ Y ) H(F (1) ⊗ h)) =0 The second equality comes from the commutativity of the diagram. The last one holds because F (1) is graded-projective and H(h) = 0 imply H(F (1) ⊗ h) = 0. 2.7. Koszul complexes. Let x := x1 , . . . , xn be elements in R. ♮ We write K(x) for the Koszul complex on x. Thus K(x) is the exterior algebra K on a free R-module K(x)1 with basis {e x1 , . . . , x en }, and ∂ is the unique R-linear map that satisfies the Leibniz rule and has ∂(e xi ) = xi for i = 1, . . . , n. In particular, K(x) is a DG (differential graded) algebra, and so its homology H(K(x)) is a graded algebra with H0 (K(x)) = R/(x). This implies (x) H(K(x)) = 0. Evidently K(x) is a perfect R-complex; it is indecomposable when R is local; see [1, 4.7]. As K(x)i is non-zero precisely for 0, . . . , n, from (2.3.1) one gets levelR K(x) ≤ span K(x) = n + 1 . Equality holds if R is local and x is a system of parameters; see Theorem 4.2 below. However, span K(x) − levelR K(x) can be arbitrarily large; see [1, Section 3]. For any Koszul complex K on n elements, there are isomorphisms of R-complexes n M n Σi K ( i ) . K∗ ∼ = = Σ−n K and K ⊗R K ∼ i=0 See [8, Propositions 1.6.10 and 1.6.21]. It thus follows from (2.3.3) that (2.7.1) levelR HomR (K, K) = levelR (K ⊗R K) = levelR K . In particular, the inequalities in Lemma 2.4(2) can be strict. 3. Tensor nilpotent morphisms In this section we prove the Tensor Nilpotence Theorem announced in the introduction. We start by reviewing the properties of interest. 6 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN 3.1. Tensor nilpotence. Let f : Y → X be a morphism in D(R). The morphism f is said to be tensor nilpotent if for some n ∈ N the morphism f ⊗L · · · ⊗L f : Y ⊗LR · · · ⊗LR Y −→ X ⊗LR · · · ⊗LR X \| R {z R } n is equal to zero in D(R); when the R-complexes X, Y are perfect this means that the morphism ⊗n f : ⊗nR Y → ⊗nR X is homotopic to zero. When X is perfect and f : X → Σl X is a morphism with ⊗n f homotopic to zero the n-fold composition Σl f Σ2l f Σnl X −→ Σl X −−−→ Σ2n X −−−−→ · · · −−−→ Σnl X is also homotopic to zero. The converse does not hold, even when R is a field for in that case tensor nilpotent morphisms are zero. 3.2. Fiberwise zero morphisms. A morphism f : Y → X that satisfies k(p) ⊗LR f = 0 in D(k(p)) for every p ∈ Spec R is said to be fiberwise zero. This is equivalent to requiring k ⊗LR f = 0 in D(k) for every homomorphism R → k with k a field. In D(k) a morphism is zero if (and only if) it is a ghost, so the latter condition is equivalent to H(k ⊗LR f ) = 0. In D(k), a morphism is tensor nilpotent exactly when it is zero. Thus if f is tensor nilpotent, it is fiberwise zero. There is a partial converse: If a morphism f : G → F of perfect R-complexes is fiberwise zero, then it is tensor nilpotent. This was proved by Hopkins [21, Theorem 10] and Neeman [25, Theorem 1.1]. The next result is the Tensor Nilpotence Theorem from the Introduction. Recall that an R-module is said to be I-torsion if each one of its elements is annihilated by some power of I. Theorem 3.3. Let R be a commutative noetherian ring and f : G → F a morphism of perfect R-complexes. If for some ideal I of R the following conditions hold (1) f factors through some complex with I-torsion homology, and (2) levelR HomR (G, F ) ≤ height I , then f is fiberwise zero. In particular, f is tensor nilpotent. The proof of the theorem is given after Proposition 3.7. Remark 3.4. Lemma 2.4 shows that the inequality (2) is implied by levelR F + levelR G ≤ height I + 1 ; the converse does not hold; see (2.7.1). On the other hand, condition (2) cannot be weakened: Let (R, m, k) be a local ring and G the Koszul complex on some system of parameters of R and let f : G −→ (G/G6d−1 ) ∼ = Σd R be the canonical surjection with d = dim R. Then G is an m-torsion complex and levelR G = d + 1; see 2.7. Evidently H(k ⊗R f ) 6= 0, so f is not fiberwise zero. In the proof of Theorem 3.3 we exploit the functorial nature of I-torsion. TENSOR NILPOTENT MORPHISMS 7 3.5. Torsion complexes. The derived I-torsion functor assigns to every X in D(R) an R-complex RΓI X; when X is a module it computes its local cohomology: HnI (X) = H−n (RΓI X) holds for each integer n. There is a natural morphism t : RΓI X −→ X in D(R) that has the following universal property: Every morphism Y → X such that H(Y ) is I-torsion factors uniquely through t; see Lipman [24, Section 1]. It is easy to verify that the following conditions are equivalent. (1) H(X) is I-torsion. (2) H(X)p = 0 for each prime ideal p 6⊇ I. (3) The natural morphism t : RΓI X → X is a quasi-isomorphism. When they hold, we say that X is I-torsion. Note a couple of properties: (3.5.1) (3.5.2) If X is I-torsion, then X ⊗LR Y is I-torsion for any R-complex Y . There is a natural isomorphism RΓI (X ⊗L Y ) ∼ = (RΓI X) ⊗L Y . R R Indeed, H(Xp ) ∼ = H(X)p = 0 holds for each p 6⊇ I, giving Xp = 0 in D(R). Thus (X ⊗LR Y )p ∼ = Xp ⊗LR Y ∼ =0 holds in D(R). It yields H(X ⊗LR Y )p ∼ = H((X ⊗LR Y )p ) = 0, as desired. A proof of the isomorphism in (3.5.2) can be found in [24, 3.3.1]. 3.6. Big Cohen-Macaulay modules. Let (R, m, k) be a local ring. A (not necessarily finitely generated) R-module C is big Cohen-Macaulay if every system of parameters for R is a C-regular sequence, in the sense of [8, Definition 1.1.1]. In the literature the name is sometimes used for R-modules C that satisfy the property for some system of parameters for R; however, the m-adic completion of C is then big Cohen-Macaulay in the sense above; see [8, Corollary 8.5.3]. The existence of big Cohen-Macaulay was proved by Hochster [16, 17] in case when R contains a field as a subring, and by André [2] when it does not; for the latter case, see also Heitmann and Ma [15]. In this paper, big Cohen-Macaulay modules are visible only in the next result. Proposition 3.7. Let I be an ideal in R and set c := height I. When C is a big Cohen-Macaulay R-module the following assertions hold. (1) The canonical morphism t : RΓI C → C from the I-torsion complex RΓI C (see 3.5) is a composition of c ghosts. (2) If a morphism g : G → C of R-complexes with levelR G ≤ c factors through some I-torsion complex, then g = 0. Proof. (1) We may assume I = (x), where x = {x1 , . . . , xc } is part of a system of parameters for R; see [8, Theorem A.2]. The morphism t factors as RΓ(x1 ,...,xc ) (C) −→ RΓ(x1 ,...,xc−1 ) (C) −→ · · · −→ RΓ(x1 ) (C) −→ C . Since the sequence x1 , . . . , xc is C-regular, we have Hi (RΓ(x1 ,...,xj ) (C)) = 0 for i 6= −j; see [8, (3.5.6) and (1.6.16)]. Thus every one of the arrows above is a ghost, so that t is a composition of c ghosts, as desired. (2) Suppose g factors as G → X → C with X an I-torsion R-complex. As noted in 3.5, the morphism X → C factors through t, so g factors as g′ g′′ t G −→ X −→ RΓI C − →C. 8 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN In view of the hypothesis levelR G ≤ c and part (1), Lemma 2.6 shows that RHomR (G, t) : RHomR (G, RΓI C) −→ RHomR (G, C) is a ghost. Using brackets to denote cohomology classes, we get [g] = [tg ′′ g ′ ] = H0 (RHomR (G, t))([g ′′ g ′ ]) = 0 . Due to the isomorphism (2.1.1), this means that g is zero in D(R). Lemma 3.8. Let f : G → F be a morphism of perfect R-complexes, where G is finite free with Gi = 0 for i ≪ 0 and F is perfect. Let f ′ : F ∗ ⊗R G → R denote the composed morphism in the next display, where e is the evaluation map: F ∗ ⊗R f e F ∗ ⊗R G −−−−−−→ F ∗ ⊗R F −−→ R . If f factors through some I-torsion complex, then so does f ′ . The morphism f ′ is fiberwise zero if and only if so is f . Proof. For the first assertion, note that if f factors through an I-torsion complex X, then F ∗ ⊗R f factors through F ∗ ⊗R X, and the latter is I-torsion. For the second assertion, let k be field and R → k be a homomorphism of rings. Let (−) and (−)∨ stand for k ⊗R (−) and Homk (−, k), respectively. The goal is to prove that f = 0 is equivalent to f ′ = 0. Since F is perfect, there are canonical isomorphisms ∼ ∨ = F ∗ ⊗ k −−→ HomR (F, k) ∼ = Homk (k ⊗R F, k) = (F ) . Given this, it follows that f ′ can be realized as the composition of morphisms ∨ (F ) ⊗k f ∨ ∨ e (F ) ⊗k G −−−−−−→ (F ) ⊗k F −−→ k . If F is zero, then f = 0 and f ′ = 0 hold. When F is nonzero, it is easy to verify 6 0 is equivalent to f ′ 6= 0, as desired. that f = Proof of Theorem 3.3. Given morphisms of R-complexes G → X → F such that F and G are perfect and X is I-torsion for an ideal I with levelR HomR (G, F ) ≤ height I , we need to prove that f is fiberwise zero. This implies the tensor nilpotence of f , as recalled in 3.1. By Lemma 3.8, the morphism f ′ : F ∗ ⊗R G → R factors through an I-torsion complex, and if f ′ is fiberwise zero, so if f . The isomorphisms of R-complexes (F ∗ ⊗R G)∗ ∼ = G∗ ⊗R F ∼ = HomR (G, F ) and Lemma 2.4 yield levelR (F ∗ ⊗R G) = levelR HomR (G, F ). Thus, replacing f by f ′ , it suffices to prove that if f : G → R is a morphism that factors through an I-torsion complex and satisfies levelR G ≤ height I, then f is fiberwise zero. Fix p in Spec R. When p 6⊇ I we have Xp = 0, by 3.5(2). For p ⊇ I we have levelRp Gp ≤ levelR G ≤ height I ≤ height Ip , where the first inequality follows directly from the definitions; see [6, Proposition 3.7]. It is easy to verify that Xp is Ip -torsion. Thus, localizing at p, we may further assume (R, m, k) is a local ring, and we have to prove that H(k ⊗R f ) = 0 holds. Let C be a big Cohen-Macaulay R-module. It satisfies mC 6= C, so the canonical γ ε f γ map π : R → k factors as R − →C − → k. The composition G − →R− → C is zero in TENSOR NILPOTENT MORPHISMS 9 D(R), by Proposition 3.7. We get πf = εγf = 0, whence H(k ⊗R π) H(k ⊗R f ) = 0. Since H(k ⊗R π) is bijective, this implies H(k ⊗R f ) = 0, as desired. The following consequence of Theorem 3.3 is often helpful. Corollary 3.9. Let (R, m, k) be a local ring, F a perfect R-complex, and G an R-complex of finitely generated free modules. If a morphism of R-complexes f : G → F satisfies the conditions (1) f factors through some m-torsion complex, (2) sup F ♮ − inf G♮ ≤ dim R − 1, and then H(k ⊗R f ) = 0. Proof. An m-torsion complex X satisfies k(p) ⊗LR X = 0 for any p in Spec R \ {m}. Thus a morphism, g, of R-complexes that factors through X is fiberwise zero if and only if k ⊗LR g = 0. This remark will be used in what follows. Condition (2) implies Gn = 0 for n ≪ 0. Let f ′ denote the composition F ∗ ⊗R f e F ∗ ⊗R G −−−−−−→ F ∗ ⊗R F −−→ R , where e is the evaluation map. Since inf (F ∗ ⊗R G)♮ = − sup F ♮ +inf G♮ , Lemma 3.8 shows that it suffices to prove the corollary for morphisms f : G → R. As f factors through some m-torsion complex, so does the composite morphism f G60 ⊆ G −−→ R It is easy to check that if the induced map H(k ⊗R G60 ) → H(k ⊗R R) = k is zero, then so is H(k ⊗R f ). Thus we may assume Gn = 0 for n 6∈ [−d + 1, 0], where d = dim R. This implies levelR G ≤ d, so Theorem 3.3 yields the desideratum. For some applications the next statement, with weaker hypothesis but also weaker conclusion, suffices. The example in Remark 3.4 shows that the result cannot be strengthened to conclude that f is fiberwise zero. Theorem 3.10. Let R be a local ring and f : G → F a morphism of R-complexes. If there exists an ideal I of R such that (1) f factors through an I-torsion complex, and (2) levelR F ≤ height I, then H(C ⊗LR f ) = 0 for every big Cohen-Macaulay module C. Proof. Set c := height I and let t : RΓI C → C be the canonical morphism. It follows from (3.5.1) that C ⊗LR f also factors through an I-torsion R-complex. Then the quasi-isomorphism (3.5.2) and the universal property of the derived I-torsion functor, see 3.5, implies that C ⊗LR f factors as a composition of the morphisms: t⊗L F R C ⊗LR G −→ (RΓI C) ⊗LR F −−−− −→ C ⊗LR F By Proposition 3.7(1) the morphism t is a composition of c ghosts. Thus condition (2) and Lemma 2.6 imply t ⊗LR F is a ghost, and hence so is C ⊗LR f . 10 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN 4. Applications to local algebra In this section we record applications the Tensor Nilpotence Theorem to local algebra. To that end it is expedient to reformulate it as the Morphic Intersection Theorem from the Introduction, restated below. Theorem 4.1. Let R be a commutative noetherian ring and f : G → F a morphism of perfect R-complexes. If f is not fiberwise zero and factors through a complex with I-torsion homology for an ideal I of R, then there are inequalities: span F + span G − 1 ≥ levelR HomR (G, F ) ≥ height I + 1 . Proof. The inequality on the left comes from Lemma 2.4 and (2.3.1). The one on the right is the contrapositive of Theorem 3.3. Here is one consequence. Theorem 4.2. Let R be a local ring and F a complex of finite free R-modules: F := 0 → Fd → Fd−1 → · · · → F0 → 0 For each ideal I such that I · Hi (F ) = 0 for i ≥ 1 and I · z = 0 for some element z in H0 (F ) \ m H0 (F ), where m is the maximal ideal of R, one has d + 1 ≥ span F ≥ levelR F ≥ height I + 1 . Proof. Indeed, the first two inequalities are clear from definitions. As to third one, pick ze ∈ F0 representing z in H0 (F ) and consider the morphism of complexes f : R → F given by r 7→ re z . Since z is not in m H0 (F ), one has H0 (k ⊗R f ) = k ⊗R H0 (f ) 6= 0 for k = R/m. In particular, k⊗R f is nonzero. On the other hand, f factors through the inclusion X ⊆ F , where X is the subcomplex defined by ( Fi i≥1 Xi = Re z + d(F1 ) i = 0 By construction, we have Hi (X) = Hi (F ) for i ≥ 1 and I H0 (X) = 0, so H(X) is I-torsion. The desired inequality now follows from Theorem 4.1 applied to f . The preceding result is a stronger form of the Improved New Intersection Theorem1 of Evans and Griffith [12]; see also [19, §2]. First, the latter is in terms of spans of perfect complexes whereas the one above is in terms of levels with respect to R; second, the hypothesis on the homology of F is weaker. Theorem 4.2 also subsumes prior extensions of the New Intersection Theorem to statements involving levels, namely [6, Theorem 5.1], where it was assumed that I · H0 (F ) = 0 holds, and [1, Theorem 3.1] which requires Hi (F ) to have finite length for i ≥ 1. In the influential paper [18], Hochster identified certain canonical elements in the local cohomology of local rings, conjectured that they are never zero, and proved that statement in the equal characteristic case. He also gave several reformulations that do not involve local cohomology. Detailed discussions of the relations between these statements and the histories of their proofs are presented in [28] and [20]. 1This and the other statements in this section were conjectures prior to the appearance of [2]. TENSOR NILPOTENT MORPHISMS 11 Some of those statements concern properties of morphisms from the Koszul complex on some system of parameters to resolutions of various R-modules. This makes them particularly amenable to approaches from the Morphic Intersection Theorem. In the rest of this section we uncover direct paths to various forms of the Canonical Element Theorem and related results. We first prove a version of [18, 2.3]. The conclusion there is that fd is not zero, but the remarks in [18, 2.2(6)] show that it is equivalent to the following statement. Theorem 4.3. Let (R, m, k) be a local ring, x a system of parameters for R, and K the Koszul complex on x, and F a complex of free R-modules. If f : K → F is a morphism of R-complexes with H0 (k ⊗R f ) 6= 0, then one has Hd (S ⊗R f ) 6= 0 for S = R/(x) and d = dim R . Proof. Recall from 2.7 that ∂(K) lies in (x)K, that K1 has a basis x e1 , . . . , x ed and K ♮ is the exterior algebra on K1 . Thus Kd is a free R-module with basis x = x e1 · · · x ed and Hd (S ⊗R K) = S(1 ⊗ x), so we need to prove f (Kd ) 6⊆ (x)Fd + ∂(Fd+1 ). Arguing by contradiction, we suppose the contrary. This means f (x) = x1 y1 + · · · + xd yd + ∂ F (y) with y1 , . . . , yd ∈ Fd and y ∈ Fd+1 . For i = 1, . . . , d set x∗i := x e1 · · · x ei−1 x ei+1 · · · x ed ; thus {x∗1 , . . . , x∗d } is a basis of the R-module Kd−1 . Define R-linear maps hd−1 : Kd−1 → Fd by hd : Kd → Fd+1 by hd−1 (x∗i ) = (−1)i−1 yi for i = 1, . . . , d . hd (x) = y . Extend them to a degree one map h : K → F with hi = 0 for i 6= d − 1, d. The map g := f − ∂ F h − h∂ K : K → F is easily seen to be a morphism of complexes that is homotopic to f and satisfies gd = 0. This last condition implies that g factors as a composition of morphisms g′ K −−→ F<d ⊆ F . The complex K is m-torsion; see 2.7. Thus Corollary 3.9, applied to g ′ , yields H(k ⊗R g ′ ) = 0. This gives the second equality below: H0 (k ⊗R f ) = H0 (k ⊗R g) = H0 (k ⊗R g ′ ) = 0 . The first one holds because f and g are homotopic, and the last one because g0 = g0′ . The result of the last computation contradicts the hypotheses on H0 (k ⊗R f ). A first specialization is the Canonical Element Theorem. Corollary 4.4. Let I be an ideal in R containing a system of parameters x1 , . . . , xd . With K the Koszul complex on x and F a free resolution of R/I, any morphism f : K → F of R-complexes lifting the surjection R/(x) → R/I has fd (K) 6= 0. As usual, when A is a matrix, Id (A) denotes the ideal of its minors of size d. Corollary 4.5. Let R be a local ring, x a system of parameters for R, and y a finite subset of R with (y) ⊇ (x). If A is a matrix such that Ay = x, then Id (A) 6⊆ (x) for d = dim R. 12 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN Proof. Let K and F be the Koszul complexes on x and y, respectively. The matrix A defines a unique morphism of DG R-algebras f : K → F . Evidently, H0 (k ⊗R f ) is the identity map on k, and hence is not zero. Since fd can be represented by a column matrix whose entries are the various d×d minors of A, the desired statement is a direct consequence of Theorem 4.3. A special case of the preceding result yields the Monomial Theorem. Corollary 4.6. When y1 , . . . , yd is a system of parameters for local ring, one has (y1 · · · yd )n 6∈ (y1n+1 , . . . , ydn+1 ) Proof. Apply Corollary 4.5 to the inclusion  n y1 0  0 y2n  A:= . ..  .. . 0 0 for every integer n ≥ 1. (y1n+1 , . . . , ydn+1 ) ⊆ (y1 , . . . , yd ) and  ··· 0 ··· 0   ..  . .. . . · · · ydn We also deduce from Theorem 4.3 another form of the Canonical Element Theorem. Roberts [27] proposed the statement and proved that it is equivalent to the Canonical Element Theorem; a different proof appears in Huneke and Koh [22]. Recall that for any pair (S, T ) of R-algebras the graded module TorR (S, T ) carries a natural structure of graded-commutative R-algebra, given by the ⋔-product of Cartan and Eilenberg [10, Chapter XI.4]. Lemma 4.7. Let R be a commutative ring, I an ideal of R, and set S := R/I. Let G → S be some R-free resolution, K be the Koszul complex on some generating set of I, and g : K → G a morphisms of R-complexes lifting the identity of S. For every surjective homomorphism ψ : S → T of of commutative rings there is a commutative diagram of strictly graded-commutative S-algebras S ⊗R K V S H1 (S ⊗R K) α V ψ // // V TorR 1 (S, S) S µS TorR 1 (S,ψ) V T TorR 1 (S, T ) // TorR (S, S) TorR (S,ψ) µT // TorR (S, T ) where α1 = H1 (S⊗R g), the map α is defined by the functoriality of exterior algebras, and the maps µ? are defined by the universal property of exterior algebras. V Proof. The equality follows from ∂ K (K) ⊆ IK and K ♮ = R K1 . The resolution G can be chosen to have G61 = K61 ; this makes α1 surjective, and the surjectivity of α follows. The map TorR 1 (S, ψ) is surjective because it can be identified with the natural map I/I 2 → I/IJ, where J = Ker(R → T ); the surjectivity of V R ψ Tor1 (S, ψ) follows. The square commutes by the naturality of ⋔-products. Theorem 4.8. Let (R, m, k) be a local ring, I a parameter ideal, and S := R/I. For each surjective homomorphism S → T the morphism of graded T -algebras V R µT : T TorR 1 (S, T ) −→ Tor (S, T ) has the property that µT ⊗T k is injective. In particular, µk is injective. TENSOR NILPOTENT MORPHISMS 13 Proof. The functoriality of the construction of µ implies that the canonical surjection π : T → k induces a commutative diagram of graded-commutative algebras V T V π TorR 1 (S, T ) µT // TorR (S, T ) TorR 1 (S,π) V k TorR (S,π) TorR 1 (S, k) µk // TorR (S, k) It is easy to verify that π induces a bijective map R R TorR 1 (S, π) ⊗T k : Tor1 (S, T ) ⊗T k → Tor1 (S, k) , so (∧π TorR (S, π)) ⊗T k is an isomorphism. Thus it suffices to show µk is injective. ≃ Let K be the Koszul complex on a minimal generating set of I. Let G − → S and ≃ F − → k be R-free resolutions of S and k, respectively. Lift the identity map of S and the canonical surjection ψ : S → k to morphisms g : K → G and h : G → F , respectively. We have µS α = H(S ⊗R g) and TorR (S, ψ) = H(S ⊗R h). This implies the second equality in the string R S µkd TorR d (S, π)αd = Tord (S, ψ)µd αd = Hd (S ⊗R hg) 6= 0 . The first equality comes from Lemma 4.7, with T = k, and the non-equality from Theorem 4.3, with f = hg. In particular, we get µkd 6= 0. We have an isomorphism ∼ V k d of graded k-algebras, so µk is injective by the next remark. TorR 1 (S, k) = k V Remark 4.9. If Q is a field, d is a non-negative integer, and λ : Q Qd → B is a homomorphism of graded Q-algebras with λd 6= 0, then λ is injective. V Vd Indeed, the graded subspace Q Qd of exterior algebra Q Qd is contained in every every non-zero ideal and has rank one, so λd 6= 0 implies Ker(λ) = 0. 5. Ranks in finite free complexes This section is concerned with DG modules over Koszul complexes on sequences of parameters. Under the additional assumptions that R is a domain and F is a resolution of some R-module, the theorem below was proved in [3, 6.4.1], and earlier for cyclic modules in [9, 1.4]; background is reviewed after the proof. The Canonical Element Theorem, in the form of Theorem 4.3 above, is used in the proof. Theorem 5.1. Let (R, m, k) be a local ring, set d = dim R, and let F be a complex of finite free R-modules with H0 (F ) 6= 0 and Fi = 0 for i < 0. If F admits a structure of DG module over the Koszul complex on some system of parameters of R, then there is an inequality d (5.1.1) rankR (Fn ) ≥ for each n ∈ Z . n Proof. The desired inequality is vacuous when d = 0, so suppose d ≥ 1. Let x be the said system of parameters of R and K the Koszul complex on x. Since F is a DG K-module, each Hi (F ) is an R/(x)-module, and hence of finite length. First we reduce to the case when R is a domain. To that end, let p be a prime ideal of R such that dim(R/p) = d. Evidently, the image of x in R is a system of 14 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN parameters for R/p. By base change, (R/p) ⊗R F is a DG module over (R/p) ⊗R K, the Koszul complex on x with coefficients in R/p, with ∼ R/p ⊗ H0 (F ) 6= 0 . H0 ((R/p) ⊗R F ) = ♮ Moreover, the rank of F ♮ as an R-module equals the rank of (R/p) ⊗R F as an R/p-module. Thus, after base change to R/p we can assume R is a domain. Choose a cycle z ∈ F0 that maps to a minimal generator of the R-module H0 (F ). Since F is a DG K-module, this yields a morphism of DG K-modules f: K →F with f (a) = az . This is, in particular, a morphism of complexes. Since k ⊗R H0 (F ) 6= 0, by the choice of z, Theorem 4.3 applies, and yields that f (Kd ) 6= 0. Since R is a domain, this implies f (Q ⊗R Kd ) is non-zero, where Q is the field of fractions of R. ♮ Set Λ := (Q ⊗R K) and consider the homomorphism of graded Λ-modules V λ := Q ⊗R f ♮ : Λ → Q ⊗R F ♮ . Qd , Remark 4.9 gives the inequality in the display d rankR (Fn ) = rankQ (Q ⊗R Fn ) ≥ rankQ (Λn ) = . n As Λ is isomorphic to Q Both equalities are clear. The inequalities (5.1.1) are related to a major topic of research in commutative algebra. We discuss it for a local ring (R, m, k) and a bounded R-complex F of finite free modules with F<0 = 0, homology of finite length, and H0 (F ) 6= 0. 5.2. Ranks of syzygies. The celebrated and still open Rank Conjecture of Buchsbaum and Eisenbud [9, Proposition 1.4], and Horrocks [14, Problem 24] predicts that (5.1.1) holds whenever F is a resolution of some module ofPfinite length. That conjecture is known for d ≤ 4. Its validity would imply n rankR Fn ≥ 2d . For d = 5 and equicharacteristic R, this was proved in [4, Proposition 1] by using Evans and Griffth’s Syzygy Theorem [12]; in view of [2], it holds for all R. M. Walker [30] used methods from K-theory to prove that P In a breakthrough, d rank F ≥ 2 holds when R contains 12 and is complete intersection (in particR n n ular, regular), and when R is an algebra over some field of positive characteristic. 5.3. Obstructions for DG module structures. Theorem 5.1 provides a series of obstruction for the existence of any DG module structure on F . In particular, it implies that if rankR F < 2d holds with d = dim R, then F supports no DG module structure over K(x) for any system of parameters x. Complexes satisfying the restriction on ranks were recently constructed in [23, 4.1]. These complexes have nonzero homology in degrees 0 and 1, so they are not resolutions of R-modules. 5.4. DG module structures on resolutions. Let F be a minimal resolution of an R-module M of nonzero finite length and x a parameter set for R with xM = 0. When F admits a DG module structure over K(x) the Rank Conjecture holds, by Theorem 5.1. It was conjectured in [9, 1.2′ ] that such a structure exists for all F and x. An obstruction to its existence was found in [3, 1.2], and examples when that obstruction is not zero were produced in [3, 2.4.2]. On the other hand, by [3, 1.8] the obstruction vanishes when x lies in m annR (M ). TENSOR NILPOTENT MORPHISMS 15 It is not known if F supports some DG K(x)-module structure for special choices of x; in particular, for high powers of systems of parameters contained in annR (M ). References [1] H. Altmann, E Grifo, J. Montaño, W. Sanders, T. Vu, Lower bounds on levels of perfect complexes, J. Algebra 491 (2017), 343–356. [2] Y. André, La conjecture du facteur direct, preprint, arXiv:1609.00345. [3] L. L. Avramov, Obstructions to the existence of multiplicative structures on minimal free resolutions, Amer. J. Math. 103 (1987), 1–31. [4] L. L. Avramov, R.-O. Buchweitz, Lower bounds on Betti numbers, Compositio Math. 86 (1993), 147–158. [5] L. L. Avramov, R.-O. Buchweitz, S. 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Manin, Methods of Homological Algebra, Annals of Math. (2) 114 (1981), 323–333. [14] R. Hartshorne, Algebraic vector bundles on projective spaces: a problem list, Topology 18 (1979), 117–128. [15] R. Heitmann, L. Ma, Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic, preprint, arXiv:1703.08281. [16] M. Hochster, Cohen-Macaulay modules, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., 311, Springer, Berlin, 1973; pp. 120–152. [17] M. Hochster, Deep local rings, Aarhus University preprint series, December 1973. [18] M. Hochster, Topics in the Homological Theory of Modules over Commutative Rings, Conf. Board Math. Sci. 24, Amer. Math. Soc., Providence, RI, 1975. [19] M. Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), 503–553. [20] M. Hochster, Homological conjectures and lim Cohen-Macaulay sequences, Homological and Computational Methods in Commutative Algebra (Cortona, 2016), Springer INdAM Series, Springer, Berlin, to appear. [21] M. Hopkins, Global methods in homotopy theory. Homotopy Theory (Durham, 1985), 73–96, London Math. Soc. Lecture Note Ser. 117, Cambridge Univ. Press, Cambridge, 1987. [22] C. Huneke, J. Koh, Some dimension 3 cases of the Canonical Element Conjecture, Proc. Amer. Math. Soc. 98 (1986), 394–398. [23] S. B. Iyengar, M. E. Walker, Examples of finite free complexes of small rank and homology, preprint, arXiv:1706.02156. [24] J. Lipman, Lectures on local cohomology and duality, Local Cohomology and its Applications (Guanajuato, 1999), Lecture Notes Pure Appl. Math. 226, Marcel Dekker, New York, 2002; pp. 39–89. [25] A. Neeman, The chromatic tower for D(R), Topology 31 (1992), 519–532. [26] P. C. Roberts, Homological Invariants of Modules over Commutative Rings, Sém. Math. Sup. 72, Presses Univ. Montréal, Montréal, 1980. [27] P. C. Roberts, The equivalence of two forms of the Canonical Element Conjecture, undated manuscript. [28] P. C. Roberts, The homological conjectures, Progress in Commutative Algebra 1, de Gruyter, Berlin, 2012; pp. 199–230. 16 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN [29] R. Rouquier, Dimensions of triangulated categories, J. K-Theory 1 (2008), 193–256, 257–258. [30] M. E. Walker, Total Betti numbers of modules of finite projective dimension, Annals of Math., to appear; preprint, arXiv:1702.02560. [31] C. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1994. Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A. E-mail address: avramov@math.unl.edu Department of Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A. E-mail address: iyengar@math.utah.edu Centre for Mathematics and its Applications, Mathematical Sciences Institute Australian National University, Canberra, ACT 0200, Australia. E-mail address: Amnon.Neeman@anu.edu.au	0
arXiv:1802.03734v1 [math.OC] 11 Feb 2018 The Use of Presence Data in Modelling Demand for Transportation Jonathan Epperlein1 , Jaroslaw Legierski2,3 , Marcin Luckner3 , Jakub Mareček1∗ and Rahul Nair1 February 13, 2018 Abstract We consider the applicability of the data from operators of cellular systems to modelling demand for transportation. While individual-level data may contain precise paths of movement, stringent privacy rules prohibit their use without consent. Presence data aggregate the individual-level data to information on the numbers of transactions at each base transceiver station (BTS) per each time period. Our work is aimed at demonstrating value of such aggregate data for mobility management while maintaining privacy of users. In particular, given mobile subscriber activity aggregated to short time intervals for a zone, a convex optimisation problem estimates most likely transitions between zones. We demonstrate the method on presence data from Warsaw, Poland, and compare with official demand estimates obtained with classical econometric methods. 1 Introduction Estimating travel demand is a key step of the transportation planning process. Cities typically build a modeling framework and use surveys along with observations to characterize how, where and by what means citizens move. These estimates are then used for management of existing infrastructure or for design of new services. Such an approach is resource-intensive, time consuming, and therefore can only be performed infrequently providing estimates that may be out-of-date. Alternatively, current movement patterns can be estimated based on data from mobile devices. Several works have studied digital trajectories, or “breadcrumbs”, which provide a very rich demand profile. Nevertheless, legislation and privacy regulation ∗ Jakub can be reached at jakub.marecek@ie.ibm.com. 1: IBM Research – Ireland, B3 IBM Campus Damastown, Dublin 15, Ireland. 2: Orange Labs Poland, 02-691 Warsaw, ul. Obrzezna 7, Poland. 3: Warsaw University of Technology, Faculty of Mathematics and Information Science, 00-662 Warsaw, ul. Koszykowa 75, Poland. 1 often restrict the use of such methods. In Europe, in particular, requirements on processing telecommunications transmission data are set by European directives: I Personal data should be protected, as per directive on e-Privacy(2002/58/EC) and directive on personal data protection (95/46/EC). I The consent of the data subject is required for processing any subscriber personal data. I The anonymized data can be processed without consent only if they are aggregated in the first step of processing and de-anonymization of the data is not possible. The anonymized record cannot not be connected with any subscriber in any case. The use and processing of CDR records without the permission of the subscriber is hence not possible in the Europe, and one may need to rely on aggregate data. In this paper, we present methods to estimate trip generation and trip distribution rates from aggregate presence data. In particular, we consider mobile subscriber’s location statistics aggregated in accordance with the above-mentioned regulations. We the present a linear program to be solved for each period of the discretisation of time to obtain the most likely estimate of movements since the previous period. We also present algorithms for obtaining the estimates of movements of likely movements over a number of periods, or consider a distribution of trip durations. Crucially, we show that the linear program can be solved in linear time, which makes the approach applicable in practice. 2 2.1 The Problem Data The data comes in the form of time-stamped averages of event counts for each of a number of geographical zones: (Zone, Timestamp, Event Count), where the zones are defined by the local public transport operator. More specifically: Say there are NZ zones Z1 , . . . , ZNZ , and we have a total of Nt time intervals [t0 ,t1 ], [t1t2 ], . . . , [tNt −1 ,tNt ] for which the event counts are reported; here, the intervals are all of equal length T , i.e. t` − t`−1 = T , but this is not necessary for the applicability of what follows. Thus, the data consists of tuples of the form (Zk ,t` , Ek,` ) for k = 1, . . . , NZ , ` = 1, . . . , Nt , which encode that “In zone Zk during the time interval [t`−1 ,t` ], Ek,` events were observed.” Note that the timestamp of the end of the interval is reported in the dataset, and that there are NZ · Nt such tuples. For the purposes of this paper, we consider the number of events as a measure of how many unique terminals (phones, pagers, etc.) were present in the give zone over the given time interval. 2 2.2 Modelling user movement The given data suggests how many terminals were present at any zone, but, due to privacy restrictions, not which terminals were present where at what time. Hence, the information as to how users move from one zone to another is lost, and the problem we address in this paper is how to estimate this information from the aggregate data that we are given. The problems treated are as follows: Problem 1 (One-Step Transitions). Given aggregate presence data for disjoint geographical zones at two points in time, and a cost function mapping each pair of geographical zones to a real number, estimate for each zone the proportion of users travelling to each other zone, within the time interval between the two points in time. Problem 2 (k-Step Transitions). Given aggregate presence data for disjoint geographical zones at two points t1 ,t2 in time and a cost function mapping each pair of geographical zones to a real number, estimate for each zone the proportion of users travelling to each other zone, within time k(t2 − t1 ). Problem 3 (Duration-L Transitions). Given aggregate presence data for disjoint geographical zones, for two or more points t1 ,t2 , . . . in time, and a cost function mapping each pair of geographical zones to a real number, and a given trip duration L, estimate for each zone the proportion of users travelling to each other zone, with trip duration being L. Problem 4 (Realistic Transitions). Given aggregate presence data for disjoint geographical zones, for two or more points t1 ,t2 , . . . in time, and a cost function mapping each pair of geographical zones to a real number, and a distribution of trip durations, as a histogram approximation of a probability density function fL , estimate for each zone the proportion of users travelling to each other zone. Absent further information, we have to make assumptions about user behaviour, and the main assumption is that the users redistribute themselves in an optimal way – optimal with respect to a cost function, the design of which requires insights into the underlying transportation structure. This is best explained via a trivial-seeming example: Assume first, that there are only two zones, Z1 and Z2 , and that we have the following data: (Z1 ,t1 , 3), (Z2 ,t1 , 1), (Z1 ,t2 , 2), (Z2 ,t2 , 2). Intuitively, it seems obvious to expect that one user went from Zone 1 to Zone 2, whereas all other stayed put. Of course, it is also possible that two users went from Zone 1 to Zone 2 and the sole user in Zone 2 went to Zone 1. However, if we associate a cost to users moving, then the first, intuitive solution, is clearly the optimal one. Assume now, that there is a third zone, Z3 , and that we additionally have the following data: (Z3 ,t1 , 1), (Z3 ,t2 , 1). A natural assumption is now that the user in Zone 3 simply stayed there. But what if Zone 3 is connected to Zones 1 and 2 through rapid transit, whereas the only convenient way of getting from Zone 1 to Zone 2 is through Zone 3, and this journey takes longer than t2 − t1 ? In this case, the cost of travel from Z1 to Z2 would be quite high, and the 3 most reasonable and optimal solution would be to expect one user to move from Z1 to Z3 and one user to move from Z3 to Z2 . This example illustrates the idea of optimal user movement and highlights, how knowledge of the transportation system needs to be incorporated in the design of the cost function. One can, for instance, benefit from the estimates of the travel-times provided by Google Maps [36], use the Euclidean distance, or a discrete metric, in which adjacent pairs of zones have cost 0 and all other pairs have cost 1. Either way, we will solve that problem by casting it as a linear program, which will be formally defined in the next section. Finally, in Problem 4, note that we treat the travel time of all users of the road network as a random variable L. For this random variable L, one assumes there exists probability density function fX , i.e., a non-negative function: Pr[a ≤ L ≤ b] = Z b a fL (x) dx. (1) On the input, we assume a histogram approximation thereof, with the bins of the histogram centered at multiples of the interval at which we sample the input data, i.e., t2 − t1 = tk+1 − tk . For example for the sampling at 1 minute, we would be given a list: Pr[0 ≤ L < 1.5], Pr[1.5 ≤ L < 2.5], . . . (2) We note that such distributional data are regularly obtained by most operators of public transport, either by surveys or by studying traces of individuals using personal seasonal tickets, although the alignment of the bins may not readily match the sampling interval of the presence data. 3 The Linear Program First, we show that Problem 1 can be solved by a linear program. 3.1 Notation and preliminaries Throughout, Z+ denotes the nonnegative integers, n and k are positive integers, 1n denotes an n-dimensional column vector of all 1s, and In is the n-dimensional identity matrix; the dimension n is omitted if it is clear from the context. We denote matrices by capital letters, and their elements by the corresponding lower-case letters, e.g. for a matrix M, mi j denotes the element in the ith row, jth column; M T denotes the transpose of M; tr(M) = ∑i mii denotes the trace of the square matrix M. Note that with this notation, we obtain the vector η of row-sums of M, i.e. η with ηi = ∑ j mi j , as η = M 1 and analogously we have γ T = 1T M for the vector of column-sums. To keep the notation compact, we consider user movement between two consecutive time intervals only; else, we’d have to include an index corresponding to the time stamp 4 in everything, which is unnecessary for the developments. In other words, we consider a data set of the form (Z1 ,t1 , E11 ), (Z2 ,t1 , E21 ), . . . , (3) (ZNz ,t1 , ENz 1 ), (Z1 ,t2 , E12 ), . . . , (ZNz ,t2 , ENz 2 ). Let E` denote the (column) vector of users present in all zones at time t` , i.e. E1 = T E11 E21 · · · ENZ 1 and so on. The number of users moving from Zi to Z j is denoted by xi j and the matrix of flows is X ∈ RNz ×Nz ; this is the quantity we are interested in finding. We denote the costs of moving from Zi to Z j by ci j ∈ R and collect them in the matrix C ∈ RNz ×Nz . 3.2 Cost functions The costs of moving from Zi to Z j is, in some sense, a parameter to the model. We have conducted our experiments with three natural choices of the cost matrix C ∈ RNz ×Nz , but we do not claim these are original. Let us have each zone Zi associated with a polygon, which is defined by a sequence of corner-points in the Euclidean plane. Let us denote ki corner-points associated with Zi x , zy ), (Px , Py ), . . . , (Px , Py ). In the adjacency metric, we consider: by (Pi,1 i,2 i,2 i,1 i,ki i,ki  0     0.1 Ca (Zi , Z j ) :=     1 ifi = j if ∃1 ≤ k1 ≤ ki , 1 ≤ k2 ≤ k j x , Py ) = (Px , Py ) s.t. ∃(Pi,k j,k2 j,k2 i,k1 1 otherwise (4) Considering the centroid of Zi : k centroid(Zi ) := k y i i Pi,x j ∑ j=1 Pi, j ∑ j=1 , ki ki ! (5) The centroidal distance Cc (Zi , Z j ) is the Euclidean distance between centroids centroid(Zi ) and centroid(Z j ) of the two zones. Finally, considering the closest corner-point associated with the other zone: NN(Zi , Z j ) := arg D(A, B) := q min min y y x ,P x (Pi,k i,k )1≤k1 ≤ki (Pj,k ,Pj,k )1≤k2 ≤ki 1 1 2 D(Pi,k1 , Pj,k2 ) 2 (Ax − Bx )2 + (Ay − By )2 . (6) Distance Cd (Zi , Z j ) is then the Euclidean distance between NN(Zi , Z j ) and NN(Z j , Zi ). One can clearly define many other cost matrices, e.g., considering free-flow travel times, and introducing randomisation. 5 3.3 Constraints assuming a constant number of users over time N N z z Ei1 = ∑i=1 Ei2 = N, i.e. that the number N of users in As a first step, assume that ∑i=1 the entire network is the same in both time intervals. In order for a flow X to explain the observations, it must “remove” Ei1 users from, and “place” E j2 users back in each zone Zi (note that the number of users staying in Zi is xii ). The number of users moving from Zi is then given by the row-sum ∑ j xi j , whereas the number of users moving to Zi is given by the column-sum ∑k xki . Of course, flows have to be nonnegative, too. Using the above notations, this translates to X 1 = E1 (7) T X 1 = E2 (8) X ≥ 0. (9) We note here that this set of constraints defines what is known as the (Nz , Nz )-transportation polytope with marginals E1 and E2 . A further constraint can be added: in reality, only integer numbers of users can move, hence we could also require X ∈ ZNz ×Nz . Since we are only approximating anyway, this might or might not be a good idea. Note also Lemma 5, which suggests that for integer marginals, the minimizer is also integral. N N z z The total cost associated with a flow X is ∑i=1 ci j xi j = tr(CT X), and we find the ∑ j=1 flow X minimizing the overall cost – and hence an estimate of the true movement of users between zones – as the solution to the following optimization problem: z = tr(CT X) minimize X ≥0 subject to 3.4 (LP-C) X 1 = E1 T X 1 = E2 . Constraints allowing for time-varying numbers of users We cannot assume that the number of users stays constant: Users are arriving in the covered area and leaving it, and of course devices are being turned off and on. To address that, we introduce a sink/source zone and index it by Nz + 1. For this zone we have ci,Nz +1 = cost of user disappearing from Zi cNz +1,i = cost of user spawning in Zi The number of users in the sink/source zone can only be computed after the data at t2 is available. Then, we let Nz Nz δ12 = ∑ Ei1 − ∑ Ei2 i=1 i=1 and ENz +1,1 = u, ENz +1,2 = u + δ12 , 6 where selecting a parameter u > 0 allows for consideration of users disappearing from Zi and spawning in Z j instead of travelling from Zi to Z j . One might also consider adding another sink/source for this specific purpose to gain more control over the cost: for instance, if ci,Nz +1 and cNz +1, j are too small compared to ci j , the optimization problem will always prefer disappearing/spawning over travel. 4 The Matrix Manipulations Second, one should like go beyond a snapshot of transitions within a single interval between two points in time, at which the presence data are available. We show that one can manipulate the solution of the linear program (LP) of the previous section to derive first the k-step transitions of Problem 2 and subsequently the solutions to Problems 3 and 4 by straightforward matrix manipulation. 4.1 Extrapolating the most recent one-step transitions One possible solution to Problem 2 involves considering the output of the LP as a matrix and raising it to the appropriate power. Let us return to our example, where we had (Z1 ,t1 , 3), (Z2 ,t1 , 1), (Z1 ,t2 , 2), (Z2 ,t2 , 2). We expect the flow of users to be X = 20 11 , in other words two users remain in Z1 , one goes from Z1 to Z2 , and the user already in Z2 remains there. Another interpretation is in terms of proportions: One third of the users in Z1 move to Z2 , whereas two thirds stay put, and so on. Mathematically, that corresponds to normalizing each of X to h 2column i 1 sum up to one (i.e. making X into a row-stochastic matrix), say Xe = 3 3 . 0 1 If this trend were to continue, then we’d have for the number of users in Z1 and Z2 at time t3 : E13 = xe11 E12 + xe21 E22 and E23 = xe12 E12 + xe22 E22 . This can be written compactly, and more generally, as the matrix-vector product E`+1 = XeT E` , and with E1 given, we have E2 = XeT E1 and E3 = XeT E2 = XeT XeT E1 and eventually we get T E`+1 = Xe` E1 . In the context of probability distributions (where the elements Ek` denote probability mass instead of users), this is a well-known and thoroughly researched Markov chain, and all results apply here. The most important one is that, should the trend continue forever, the system will (under mild conditions) settle into a steady state. More specifically, there is a distribution of users f such that XeT f = f and as ` → ∞, we have E` → f . 7 4.2 Extrapolating k recent one-step transitions If instead of just event counts at two time steps, we have event counts at several consecutive time steps, another solution to Problem 2 would be to multiply by the estimated one-step flow matrices in turn. To return to our example, let us extend it to: (Z1 ,t1 , 3), (Z2 ,t1 , 1), (Z1 ,t2 , 2), (Z2 ,t2 , 2), (Z1 ,t3 , 3), (Z2 ,t3 , 1). The flows between times t1 and t2 would be as before, and from t2 to t3 we’d have X2 = 21 01 , or after normalization: Xe1 = 2 3 1 3 0 1 and 1 e X2 = 1 2 0 1 2 . Now the estimate for E4 would be E4 = Xe1T E3 = Xe1T Xe2T E2 = Xe1T Xe2T Xe1T E1 and in general  `/2   XeT XeT E1 if ` even 1 2 E`+1 = (`−1)/2   XeT XeT Xe1T E1 if ` odd. 1 2 Because the product of row-stochastic matrices is again row-stochastic, this new Markov chain may have a stationary distribution appearing every k time steps. The apparent difficulty with a naive implementation of these approaches is runtime, in particular when k is large. As we will show in the next section, there are surprisingly efficient algorithms, though. 4.3 Approximating duration-L transitions In order to solve Problem 3, it suffices to consider a convex combination of 2 solutions to Problem 2 for suitable k. Consider the computation of duration-L transitions, where there exist presence data at times tk ,tk+1 with tk ≤ L ≤ tk+1 . Consider the k-step transition matrix Sk and (k + 1)-step transition matrix Sk+1 computed as a solution to Problem 2. Clearly, S= L − tk tk+1 − L Sk + Sk+1 tk+1 − tk tk+1 − tk (10) is the solution to Problem 3. 4.4 Approximating realistic transitions Finally, in order to solve Problem 4, one has to notice that the solution is a convex combination of H solutions to Problem 2 for H bins of the histogram approximation. 8 Method Run-time Ref. General-purpose interior-point method Hungarian method Augmenting-path algorithm O(n7 ln(1/ε)) O(n4 ) O(n3 ) [10] [17] [15] The proposed algorithm O(n) Theorem 6 Table 1: The complexity of solving the linear program (LP-C) in dimension n × n. We note that it makes it possible to obtain the objective function value at cost O(n), while retrieving the n × n matrix, at which this value is attained requires time O(n2 ). Consider the H bins of the histogram with values: h1 = Pr[0 ≤ L < 1.5] h2 = Pr[1.5 ≤ L < 2.5] ... hH = Pr[max supp L − 1 ≤ L < max supp L], where max supp L is the largest possible realisation of L. It is clear that hi , 1 ≤ i ≤ H can be seen as a probability mass function on a discretisation of L, with ∑H i=1 hi = 1. One can use hi as weights in a convex combination of the transition matrices, which is a sum. For the first bin, we consider a solution S1 of Problem 1 with weight h1 as the first summand. For the second bin, we consider a 2-step transition matrix S2 computed as a solution to Problem 2 with weight h2 as the second summand. Subsequently, we consider k-step transition matrix Sk , 2 < k ≤ H computed as a solution to Problem 2 with weight hk to obtain: H S = ∑ (hi Si ), (11) i=1 which is the solution to Problem 4. 5 Run-Time Analysis 5.1 Linear Programming Next, let us consider whether a faster algorithm for solving the class of linear programming problems (13) are possible. We make use of: Lemma 5 (E.g. [7, Lemma 2.2 and Corollary 2.11]). For the (p, q) transportation polytope defined by γ and η we have: 1. it is nonempty if and only if 1Tq η = 1Tp γ, in other words if the sums of the marginals are equal; 9 Method Run-time Ref. Standard dense matrix multiplication Present-best dense matrix multiplication Fourier-transform-based methods O(n3 ) Standard [18] [26] O(n2.3728639 ) Õ(k + nb) Table 2: The complexity of multiplying two sparse matrices, each in dimension n × n, each with k non-zero coefficients, where the product has b non-zero coefficients. 2. all its vertices are integral, if η ∈ Zq and γ ∈ Z p , i.e. if the marginals are integral. but we stress that the polytope has been studied very extensively over the past 70 years and refer to [29, 30, 33] as standard references. Now we state the main result: Theorem 6. Given η, γ ∈ Zn+ , with 1T η = 1T γ = k, consider z := max tr(X) s.t. X 1 = γ, X T 1 = η. X∈Zn×n + (IP2) Then, z = ∑ni=1 min ηi , γi . This is computable in time linear in n. For the proof, please see the Appendix. 5.2 Sparse Matrix Multiplication In addition to the complexity of the solving of the instance of the linear programming problem, we may need to solve a number of matrix-matrix multiplication problems, where the two matrices are sparse, each in dimension n × n, each with k non-zero coefficients, and where the product has b non-zero coefficients. Using traditional dense linear algebra, one can obtain the result in time O(n3 ), using more sophisticated methods for dense matrices, one can improve this to O(nω ), ω ≈ 2.37. Considering the sparsity of matrices A, B, however, Pagh [26] has shown methods that there are based on conversion to polynomial: n p(x) = n n ∑ ( ∑ Aik s1 (i)xh1 (i) )( ∑ Bk j s2 ( j)xh2 ( j) ) k=1 i=1 j=1 and utilisation of fast Fourier transform for polynomial multiplication, which can compute c0 , ..., cb−1 such that ∑i ci xi = (p(x) mod xb ) + (p(x) ÷ xb ) in time (n2 + nb), with (AB)i j = s1 (i)s2 ( j)c(h1 (i)+h2 ( j)) mod b. Eventually, Pagh [26] shows that the algorithm computes AB exactly in time (N + nb) with high probability. Although these algorithms may not be easy to implement, related algorithms with simpler hashing functions are readily implementable. 10 Table 3: Types of registered events: First two types are generated by a user (“active”), while the following eight are generated by the network (“passive”). I A cell identifier change reported in the access request procedure of any mobile originated (MO) or mobile terminating (MT) transaction I A cell identifier change reported when a cell identifier different from the one stored in the visitor location register (VLR) is used in response to any mobile terminating (MT) transaction I Location update performed by a mobile station I International mobile subscriber identity (IMSI) attached to a mobile station changes I A subscriber is deleted from the visitor location register I The subscriber switches off the mobile station I Mobile station is considered as detached by the network after a long period of inactivity I Cell identifier change is detected during GPRS activity I Cell identifier change is detected during the provision of subscriber information I During long calls, the visitor location register (VLR) may receive request messages to refresh subscriber data. In this way, the VLR notices that the cell identifier arriving in the message differs from the one stored in a database. 6 Computational Experiments For our preliminary experiments, we have used data collected from the Public Land Mobile Network (PLMN) of Orange Polska within the municipal area of Warsaw. For each base transceiver station (BTS) of each separate cellular system (2G/3G), we have received the total numbers of connections from unique terminals per a unit of time. No data other than the aggregate statistics were collected. In particular, the data were collected by an internal transmission system, in which some types of network events are associated with a location. Such events can be subdivided into events generated by the user (“active”) or generated by the network (“passive”). In Table 3, the first two bullet points capture events generated by the user. These events relate to mobile terminating (MT) transactions, i.e., events registered during an active use of a mobile terminal by the user. The following bullet points list event types generated by the network. These events are generated without an active participation of the user and include operations of the so-called visitor location register (VLR), which tracks subscriber roaming within a mobile switching centre (MSC) location area, cell identifier changes, International Mobile Subscriber Identity (IMSI) changes, location updates performed by a mobile station, and the switching of mobile stations. Within the one week of data collected, the most frequent events were location updates, while the cell identifier change was very rare. Figure 1 presents the evolution of the total number of events during the period. The volumes of both active and passive events are correlated with the time of day; the cyclical nature of the evolution of the volumes of events over time is clear. The evolution of the numbers of passive events is more uneven 11 12 104 11 active passive 104 10 10 Number of events 9 8 8 6 7 6 4 5 2 4 0 Jan 09, 08:00 Jan 10, 08:00 Jan 11, 08:00 Jan 12, 08:00 3 08:00:00 09:00:00 10:00:00 Jan 13, 08:00 2017 Figure 1: Total number of events vs time stamps, split by passive (magenta) and active (black) events. The spike in passive events at 3:15am is an artefact of the data acquisition system’s operation. than that of the active events. The local maxima of the numbers of passive events at 3:15 a. m. are related with the operations of the data acquisition and processing system. Further, note that the observed events are not equally distributed in the city. Figure 2 presents statistics of the number of events observed across all BTSs at a given time. The large difference between mean and median, the wide interquartile range, and the very large difference between maximum and mean number of events at all times suggest a large variability in the distribution of events over the BTSs. This disproportion is the result of differing population density, differing ranges of base transceiver stations, and differing telecommunication technologies. For comparison, we have used the estimates presently employed by the public transport operator in Warsaw (ZTM). One should like to point out that the estimates of ZTM imply a rather different distribution of the population across the zones than the density of events within the Orange network. See Figure 3 for the density of events within the Orange network for three representative time periods. In Figure 4, we compare it to the implied presence data within the ZTM estimate: with each data point represents one zone and the two axes correspond to the presence implied by ZTM for the morning peak and the numbers of events within Orange network during the same time period, averaged over the 5 work days. As can be seen, our model starts with rather different data, and any direct comparison of our estimates of trip distributions with the estimates of the public transport operator (ZTM) will hence be imperfect. For further comparison, we have implemented the doubly-constrained gravity model [9]. There, the presence vectors E` across two consecutive time steps are seen as production and attraction vectors within the framework of a travel-demand model, which aims to generate trip distribution rates. An appropriate normalization can give yield matrix X similar: Ai B j Ei1 E j2 xi j = , (12) cαi j 12 Figure 2: The evolution of some statistics of the active (left) and passive (right) events observed. The statistics are computed over all BTSs and all 5 available days (Monday – Friday), e.g. the mean at 9:00 is the mean of the number of events at all BTSs on all 5 days at 9am. The inner (dark blue) band is the interquartile range, whereas the outer (light blue) band extends between the 5th percentile and the maximum event count at that time. (a) 6 a. m. (b) 9:30 a. m. (c) 4:30 p. m. Figure 3: Density of events across zones for different time periods. Color scale uses a power law normalization. where Ai and B j are zone-specific parameters for the origin and destination, respectively, which are learned from the data, and α is a parameter, which we assume to be α = 1.0. An iterative algorithm [9] can then be used to determine the flow, subject to flow conservation constraints. To test the approaches, data were aggregated spatially to 820 zones defined by the public 13 Figure 4: Presence by zone implied by the estimates of ZTM and the density of events in the Orange network. transport operator (ZTM), and temporally to 15-minute units, prior to the application of any methods described in the present paper. Independently, for each cost function, an 820 × 820 matrix has been calculated, based on a description of each zone as a polygon with points defined by their latitude and longitude. Additionally, we have obtained randomised variants of the objective function by adding uniformly-distributed noise U[0, 0.0001) to each element, e.g., Ca (Zi , Z j ) for all i, j. (Note that the introduction of a small amount of noise makes it possible to obtain multiple optimizers with very similar objective function value.) The data have been normalised to the number N = 1, 000, 000 of users, i.e., such that the marginals, and hence the 672, 400 scalar variables sum up to 1, 000, 000. Subsequently, the linear program (LP-C) of Section 3.3 with 672, 400 scalar variables and 1, 640 dense constraints has been formulated and solved for each pair of consecutive time steps between 8 a. m. and 9.30 a. m. on each of the five work days and each of 4 randomisation of the objective function. The sparse, integral solutions of all linear programs have been averaged to reduce the sparsity in our estimate of the 1-step transition matrix S1 . Subsequently, we have obtained i-step transition matrices for i = 3, 4, . . . 7, i.e., corresponding to trips of 30, 45, 60, 75, and 90 minutes of duration, by the matrix-multiplication procedure of Section 4.1. Finally, we have computed a convex combination of the i-step transition matrices as suggested in Section 4.4, using weights detailed in the caption of Figure 5. Notice that we did not have an accurate estimate of the trip-duration distribution, which Figure 5 suggests has a considerable impact. Still, the match seems encouraging, considering the simplicity of the methods applied: we have considered the simplest cost function Ca , we have assumed the constant number N of users throughout in (LP-C), and the crudest method of arriving at the multi-step transition matrices. 14 (a) 1/6, 1/6, 1/6, 1/6, 1/6, 1/6 (b) 0.5, 0.1, 0.1, 0.1, 0.1, 0.1 (c) 1 − 5ε, ε, ε, ε, ε, ε (d) Gravity model Figure 5: Solutions to Problem 4 obtained using (LP-C) and three variants of the vector h1 , h2 , h3 , h4 , h5 , h6 , compared against the estimates of the public transport operator (ZTM). 7 7.1 Related Work The Demand Modelling There is a long history of the use of individual mobile phone subscriber data in transportation modelling. Outside of Europe, the most common study is based on CDR (Call Data Record). For example in [4], the CDR are used to analyse flows between the city and the suburban area of New York. In [19], researchers analyse the CDR from the main mobile operator in Mexico, and one of the papers focuses on users movements analysis. In [37], CDR data including phone calls, SMS messages, and web browsing 15 of customers of one US mobile operator in the San Francisco Bay area are used to determine patterns of urban road use. The Orange company organized two editions of the D4D Challenge (http://www.d4d.orange.com/en/Accueil), during which the anonymous CDR of Abidjan in Cote d’Ivoire data was made available to developers and researchers. Some of the present authors [23, 28] used the data set to optimize public transport. In a related paper [11], the data from the competition is used to estimate the travel demand and network allocation. The access to the CDR data is particularly limited in Europe, though, where privacy-protection regulation make access to the CDR data impossible without their anonymization and user’s consent, in most cases. Within Europe, [2] discusses the use of GPS data, CDR, and cellular data (details of base stations) to determine the trajectory of subscriber mobility. The data for this study was collected by a group of volunteer users in cooperation with one of the French operators, and the study covers the metropolitan area of Paris (Ile-de-France). The use of anonymous mobile data and their use for locating subscribers anonymously is described in [13]. [35] discusses the use of data from Italy’s Vodafone users to compute mobility patterns in Italian cities. In this case, the real-time events (due to voice, data, and SMS communication) observed between A Interface and IU-CS interface in 2G and 3G mobile networks have been used. In [31], the authors describe the usage of aggregated mobile phone data from Rome, including the information about the base station telephone traffic, expressed in Erlang units. These records are combined with the location and trajectory data of callers and data from the two city transportation companies (taxi and public bus company). [24] describe the aggregation procedure used to derive the dataset we use. 7.2 The Algorithms Notice that the linear program (LP-C) has a well-known structure, in that the feasible set is the transportation polytope: Definition 7 (Transportation Polytope). Given γ ∈ R p , η ∈ Rq , the set of matrices X ∈ R p×q , Xi j ≥ 0 satisfying p ∑i=1 Xi j = η j q ∑ j=1 Xi j = γi ∀ j = 1, . . . , q ∀i = 1, . . . , p (13) is called the (p, q)-transportation polytope defined by the marginals γ and η. Related algorithms for solving special-cases of linear programming are employed throughout pattern recognition and content-based content retrieval. The commonly used statistical distances include the Earth mover’s distance (EMD) [32], also known as the transportation cost metric [16], and Mallows distance [25], which are equivalent [20], in the sense of being dual of each other. These are discretisations of the metrics of Wasserstein and Monge-Kantorovich [8], which are also dual of each other. Computing the distances amounts to solving a certain problem over the so-called transportation polytope. Perhaps the best overview is provided by the two-volume work of Rachev and Rüschendorf [29, 30]. 16 In Computer Vision [27, 34, 12], a variety of exact and approximate methods have been considered. Perhaps the best known exact method [27] uses the min-cost flow algorithm, yielding run-time of O(n2 log n) for n bins. Approximations based on gradient flows of certain partial differential equations [12] can be computed faster than earth mover’s distance, empirically, but do not come with any guarantees, i.e., the approximation ratio. Approximations based on the sum of absolute values of weighted wavelet coefficients of the difference of the histograms [34] have been proposed. Such an approximation can be computed in time linear in the number of bins, but does not come with a bound on its quality, i.e., the approximation ratio. Papers on approximations based on space-filling curves [14] have been withdrawn, recently. In Machine Learning [6], Sinkhorn distances, which are a regularised variant of the Earth mover’s distance, have been explored. Although the algorithms based on the Sinkhorn distance are substantially faster than the network simplex for Earth mover’s distance, It has also been realised [22] that one can employ the following representation with O(n) variables for n bins: leading to run-time cubic in the number of bins. In Theoretical Computer Science, [16] have studied the limits of approximability of earth mover’s distance, [1] have proposed logarithmic approximation in sub-linear time, while [3, 5] proposed algorithms approximation algorithms, whose run-time depends on the dimension of the domain space and the quality of the result required, possibly sub-linear in the number of bins. A similar in spirit is a recent method [21], which considers dithering of the domain space, followed by max-flow computations [27]. We stress that our algorithms are much easier to implement and provide the exact solution. Overall, the present-best algorithms [27] had run-time of O(n2 log n) for n bins, compared to O(n) we present in Theorem 6. Across all the references listed, only the value of the objective function, rather than the value of the argument at which it is achieved, is sought. 8 Conclusions Across many transport-engineering applications, from planning of public transport to balancing of a fleet of shared vehicles, one needs an up-to-date model of demand for transportation. Further, one may be interested in higher moments of the demand, rather than just the commonly-used expectation. The use of mobile phone data makes it possible to address both issues. For the first time, we have formalised the problem of extraction of a model of demand for transport from the aggregate data on the use of the base stations. We have shown that the associated optimisation problems have the form of a trace maximisation over the transportation polytope, which allows for much faster algorithms than the usual simplex or interior point methods. 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Step 1: Since Xi j ≥ 0, we certainly have Xi j ≤ ∑n`=1 X` j = η j and Xi j ≤ ∑n`=1 Xi` = γi . Hence in particular Xii ≤ ηi and Xii ≤ γi , and so by choosing the tightest bound at each i, y = ∑ni=1 Xii ≤ ∑ni=1 min ηi , γi follows. Step 2: Before we proceed with Step 2 proper, we make two assumptions and explain that these are without any loss of generality. Assumption 1: η 6= γ. Notice that if η = γ, one lets Xii = ηi = γi for i = 1, . . . , n and Xi j = 0 for all off-diagonalelements Such an X trivially satisfies all constraints and tr(X) = ∑ni=1 ηi = ∑ni=1 min ηi , γi = k, and hence the result holds for this case. We can hence assume η 6= γ without the loss of generality. Assumption 2:The marginals are ordered such that: ηi ≤ γi for i = 1, . . . , m ηi > γi for i = m + 1, . . . , n. If that is not the case, then let P denote the permutation matrix so that Pη and Pγ are ordered.1 Then, if X̄ is feasible for (IP2) with the modified marginals Pη and Pγ, then X = PX̄P is feasible for (IP2) and achieves the identical objective, since tr(PX̄P) = tr(PPX̄) = tr(X̄). Note that we certainly have 1 ≤ m ≤ n − 1, because m = 0 implies ηi > γi for all i, which in turn implies ∑ni=1 ηi > ∑ni=1 γi , which contradicts ∑ni=1 ηi = k = ∑ni=1 γi , hence m ≥ 1; m = n together with Assumption 1 (η 6= γ) leads to an analogous contradiction. The construction of Step 2 fixes all values on the diagonal of X, the first m columns and the last n − m rows: Xii = ηi , Xi j = 0 for j = 1, . . . , m, Xii = γi , Xi j = 0 for i = m + 1, . . . , n, i 6= j j 6= i. e := η − min{η, γ}, γe := γ − min{η, γ}, Denote the remainders of the marginals by η where the min{·} is taken element-wise. 1 P can be constructed easily: let C := {i \| η ≤ γ }. Then m = card(C ) and we obtain P by rearranging i i the identity matrix In so that the columns with indices in C are the first m columns, in any order. 22 γf = η1 γ1 − η1 η2 γ2 − η2 Y =: γe 0 ηm γm − ηm γm+1 0 γn ηfT = 0 0 0 ηm+1 −γm+1 0 ηn − γn =: ηe 0 Figure 6: Illustration of the proof of Theorem 6: After possibly reordering the marginals, the diagonals are filled in, and the hatched regions are filled with zeros. The elements in the rectangular block Y are then used to satisfy the remaining constraints, which are e 0. collected in γe0 and η Let us now consider several properties of this construction; see also Figure 6 for an illustration: P1 tr X = ∑ni=1 min ηi , γi , which matches the bound of Step 1. P2 ∑ni=1 Xi j = η j for j = 1, . . . , m and ∑nj=1 Xi j = γi for i = m + 1, . . . , n, i.e., all constraints but the m first row sums and last n − m column sums are already satisfied. P3 Only Xi j for i = 1, . . . , m and j = m + 1, . . . , n are unassigned. Denote this m × (n − m) matrix by Y . e = 1T γe =: e e , γe ≥ 0, and η ei = γej = 0 for i = 1, . . . , m and j = m + 1, . . . , n. P4 1T η k, η e and γe by η e 0 and γe0 respectively, i.e. η e 0 ∈ Zn−m Denote the remaining elements of η + 23 and γe0 ∈ Zm + and 0 e= 0 , η e η 0 γe γe = . 0 e 0 = 1Tm γe0 = e k, since we only removed zero elements. We still have 1Tn−m η m×(n−m) By property P1 and P3, if we are able to find a matrix Y ∈ Z+ satisfying Y 1n−m = e 0 , we can recover a feasible X that achieves the bound of Step 1. By γe0 , and Y T 1m = η P3 and P4, this amounts to finding a point in the (m, n − m) transportation polytope e 0 . Since by P4 we have that 1Tn−m η e 0 = 1Tm γe0 , it follows defined by the marginals γe0 and η from Lemma 5 that such a Y does exist, which concludes Step 2. 24	8
Interface Reconciliation in Kahn Process Networks using CSP and SAT arXiv:1503.00622v4 [cs.PL] 12 Jul 2015 Pavel Zaichenkov, Olga Tveretina, Alex Shafarenko Compiler Technology and Computer Architecture Group, University of Hertfordshire, United Kingdom {p.zaichenkov,o.tveretina,a.shafarenko}@ctca.eu Abstract. We present a new CSP- and SAT-based approach for coordinating interfaces of distributed stream-connected components provided as closed-source services. The Kahn Process Network (KPN) is taken as a formal model of computation and a Message Definition Language (MDL) is introduced to describe the format of messages communicated between the processes. MDL links input and output interfaces of a node to support flow inheritance and contextualisation. Since interfaces can also be linked by the existence of a data channel between them, the match is generally not only partial but also substantially nonlocal. The KPN communication graph thus becomes a graph of interlocked constraints to be satisfied by specific instances of the variables. We present an algorithm that solves the CSP by iterative approximation while generating an adjunct Boolean SAT problem on the way. We developed a solver in OCaml as well as tools that analyse the source code of KPN vertices to derive MDL terms and automatically modify the code by propagating type definitions back to the vertices after the CSP has been solved. Techniques and approaches are illustrated on a KPN implementing an image processing algorithm as a running example. Keywords: coordination programming, component programming, Kahn Process Networks, interface coordination, constraint satisfaction, satisfiability 1 Introduction The software intensive systems have reached unprecedented scale by every measure: number of lines of code; number of people involved in the development; number of dependencies between software components, and amount of data stored and manipulated [1]. Many of them include heterogeneous elements, which come from variety of different sources: parts of them are written in different languages and tuned for different hardware/software platforms. Furthermore, when the software is developed and modified by dispersed teams, inconsistencies in the design, implementation and usage are unavoidable. This leads to clashes of assumptions about operation cost, resource availability and algorithm processing rate. Last but not least, parts of the system are constantly changing. Many elements need to be replaced without negative effects on performance or behaviour of the rest of the system. One way to attack the software challenge is to suggest a component-based design: a program is designed as a set of components, each represented by an interface that specifies how they can be used in an application, and one or more implementations which define their actual behaviour. When a designer of the application uses a component, they agree to rely only on the interface specification. Similarly, a developer who creates an implementation for a component is unaware of the context where the component will be used. An algorithm that specifies the behaviour depends solely on self-contained input and its result is produced in the form of a message without a specific destination address. The process network, introduced by G. Kahn (KPNs) [2], is a collection of stream-connected algorithmic building blocks, which are fully independent single-threaded processes that lack a global state. The execution of the network generally requires a supervisory coordination program that manages the progress of the blocks and which provides a message-communication infrastructure for the streams. Since all domain-specific computations are performed by the sequential processes inside the blocks, programming is naturally separated into algorithm and concurrency engineering [3]. The coordination language is responsible for component orchestration, namely 1) dynamic load control and adaptivity for a changing environment; 2) access control to shared resources; and 3) communication safety between components. This paper focuses on the last aspect. Component-based design requires an implementation of a single component to be independent from the rest of the network. It raises a number of software engineering issues: components’ interfaces are required to be specific enough so that components are aware of data structures communicated between them and, at the same time, generic enough to facilitate decontextualisation and software reuse. In this paper we present a solution to the interface reconciliation problem for an interface definition language specifically designed for KPNs. We demonstrate a static mechanism (based on solving Constraint Satisfaction Problem (CSP) and SAT) that checks compatibility of component interfaces connected in a network with support of overloading and structural subtyping. This allows one to design completely decontextualised components, so that they may be reused in different contexts without changing the code. This is especially important when the components are provided as a compiled library and its source is either private or unavailable. The components are compatible with a potentially unlimited number of input/output data formats coming from the environment. We also introduce a flow inheritance [4] mechanism: put simply, a message sent from one component to another may also be required to contain additional data which, although not needed by the recipient itself, is nevertheless required by a component that the recipient sends its own messages to (Fig. 1). We propose a Message Definition Language (MDL) that enables components’ generic interfaces as well as subtyping and flow inheritance; we then recast the interface reconciliation problem as a CSP for the interface variables and propose an original solution algorithm that solves it by iterative approximation while generating an adjunct Boolean SAT problem on the way. We designed and implemented a communication protocol1 for components coded in C++ to demonstrate the capabilities of MDL. We developed tools that 1) automatically derive MDL interfaces from the source code; 2) generate a set of constraints given a netlist2 that describes the topology of the network; 3) solve the CSP; and 4) based on the solution of the CSP generate compilable code for every component with some API provided for run-time support. The process is similar to template specialisation in C++, however, in our case, constraints that are produced by a pair of vertices may affect the whole network, and, consequently, a global constraint satisfaction procedure is required. In this paper we provide a formal description of MDL, the CSP definition and the algorithm designed to solve the CSP. Throughout the paper we demonstrate the utility of the proposed approach on a practical example: an image segmentation algorithm based on k-means clustering (Fig. 2). Related work. Linda [5] is the first language designed to separate computation and coordination models. It is based on a simple tuplespace model. One of the disadvantages of the model is that the knowledge about the communication protocol is required while implementing the processes. The problem of separation of concerns has not been solved in Concurrent Collections from Intel [6] (Linda’s successor). Therefore, generic components, which may be reused in multiple contexts without being modified, are not supported in the language. In the programming language Reo [7] components are communicating through hierarchical connectors that coordinate their activities and manipulate message dataflow. Similar to our approach, a constraint satisfaction engine, which finds a solution that specifies a valid interaction between components, is implemented. S. Kemper describes a SAT-based verification of Timed Constraint Automata that is used for coordination of communicating components [8] as well as in Reo. However, the research mostly focuses on the design of reusable interaction protocols and lacks the description of reusable component interfaces. In previous years there were attempts to design efficient programming languages and run-time systems for parallel programming based on KPN [4,9], however, the interface reconciliation problem stemming from nonlocal inheritance in KPNs has not been given enough attention. 2 Motivation Kahn Process Networks is a concurrency model that introduces data streams in the form of sequential channels that connect independent processes into a network. Decontextualisation of processes is an advantage of the model. Since processes do not share any data, a process’s conformity with the context is defined by its interface, which describes the kinds of message that the process 1 2 for the avoidance of doubt we state that the term “protocol” is used here in the sense of ‘convention governing the structure and interpretation of messages’ and not in any state-transition sense a textual representation of a graph A B Fig. 1: Illustration of flow inheritance. A component A can process a value of type X and return a value of type Y as a result. However, an input message contains not only an element of type X, but also an element of type Z. The latter can be processed by a component B. Flow inheritance provides a mechanism for partial message processing in a pipelined fashion. can send and receive. Our goal is to provide a means of interface coordination that supports genericity, i.e. the ability of an interface to function correctly in a variety of contexts. The distributed components are commonly provided as closed-source services. Each service contains multiple processing functions compatible with a variety of contexts. The input interface of a component is specialised based on the message format that the message producer is capable to produce, and, symmetrically, the output interface is specialised based on the consumer’s requirements. For example, one can define a component that contains two functions with type signatures Int -> Int and Int -> String. The functions implement algorithms that compute different values given the same input. The task is to statically choose the algorithm based the consumer’s requirements. This also demonstrates that input and output types in the interfaces of the KPN components are treated in the same manner. The latter makes services fundamentally different from functions. In functional languages, a type signature that corresponds to the interface of the component in the example can be defined by the intersection type Int -> Int ∧ Int -> String, which is unsound due to its ambiguity. In functional languages, the return type of a function depends solely on input argument types. In contrast, the interfaces of the KPN components form context-dependent relations that offer a selection of output types to a consumer. This makes the typing decisions essentially nonlocal and genuinely multiple. The problem being solved can be seen as a type inference problem; however, it cannot be solved using conventional type inference mechanisms based on firstorder unification due to the presence of polymorphic output types and potential cyclic dependencies in the network (the example in Fig. 2 contains a back edge). A common communication pattern in streaming networks is a pipeline, where a message travels along a chain of components that work on its content. The component can accept a subtype of the input type, but part of the message may be bypassed to another component down the pipeline if the message contains the data the further component will need to use (Fig. 1). Two modes of flow inheritance are envisaged, considered next. Flow inheritance for records. The fundamental type of a message in a variety of systems is record, which is a collection of label-value pairs. Each component processes only a specific set of pairs, however the pairs that the component does not require may be bypassed to the output, so they can be processed in the read 1 init 1 2 1 kMeans 1 2 1 1 2 3 segmented image logs/errors Fig. 2: An image segmentation algorithm based on k-means clustering that is implemented as a Kahn Process Network next stages of the pipeline if they are required. For example, a message that represents a geometric shape and has a type {x:float, y:float, radius:float} may be processed in two steps: the first component processes the position of the shape as defined by the pairs x and y, and the second one needs the pair labelled radius. Flow inheritance for variants. OOP extensively uses overloading [10] to improve modularity and reusability of code. Similarly, support of polymorphic components in KPNs facilitates their reuse in different contexts. At the top level we see a component’s interface as a collection of alternative label- record pairs, called variants, where the label corresponds to the particular implementation that can process the message defined by the given record, e.g. (: cart: {x:float, y:float}, polar: {r:float, phi:float} :). Here the colonised parentheses delimit the collection of variants and each variant is associated with a record written as a set of label-value pairs. Any message that does not belong to one of the accepted variants must cause an error unless there exists another component further down in the pipeline that can process it. In this case, the message should be bypassed to the recipient. In streaming networks flow inheritance alleviates the problem and makes configuration of individual interfaces independent from each other. In our work we developed a solution for the interface reconciliation based on the CSP with support of flow inheritance for records and variants. Our mechanism statically detects implementation variants in components that are not required in the context, which is important for applications running in the Cloud where a user is charged proportionally to the amount of resources their application uses. Example. As a running example we use our implementation [11] of an image segmentation algorithm based on k-means clustering [12]. The applications’s KPN graph is shown in Fig. 2. The network represents a pipeline composed of three components: – The component read opens an image file using an input message Mr1 with the file name, and sends it to the first output channel. The component contains 3 functions that overload component’s behaviour (i.e. the input interface of the component is defined by 3 variants): 1) Vr1 loads the colour image in RGB format; 2) Vr2 loads the greyscale image as an intensity one; and 3) Vr3 loads the image as it is stored in the file. – The component init sets initial parameters for the k-means algorithm. The component contains one processing function Vi1 . The input message can either come from the component read or from the environment with an input message Mi1 if it has been opened and preprocessed before. The input message must contain the number of clusters K and the image itself. – The kMeans component represents an iterative implementation (defined as a function Vk1 ) of the k-means algorithm. The result of each iteration is sent to the first output channel, which is circuited back to the input channel of the component itself. This kind of design gives an opportunity to manage system load in the run-time and execute the next iteration only when sufficient resources are available. Once the cluster centres have converged, the algorithm yields the result to the second output channel. Using flow inheritance for variants Mi1 is routed directly to the init component bypassing a component read. Using flow inheritance for records a parameter K that is contained in Mr1 is implicitly bypassed through read to init. The interface reconciliation algorithm is capable of finding out that Vr2 and Vr3 are not used with the provided input, and functions containing the implementations will be excluded from the generated code. 3 Message Definition Language Now we define the Message Definition Language (MDL) that describes component interfaces. Each component has its associated input and output interface terms. A message is a collection of data entities, defined by a corresponding collection of terms that can contain term variables, Boolean variables and Boolean expressions. Each term is either atomic or a collection in its own right. Atomic terms are symbols, which are identifiers used to represent standard C++ types, such as int or string. To account for subtyping (including the kinds that are not present in C++) we include three categories of collections (see Fig. 3): tuples that demand exact match and thus admit no structural subtyping, records that are subtyped covariantly (a larger record is a subtype) and choices that are contravariantly subtyped using set inclusion (a smaller choice is a subtype). The intention of these terms is to represent 1. extensible data records [13,14], where additional named fields can be introduced without breaking the match between the producer and the consumer and where fields can also be inherited from input to output records by lowering the output type, which is always safe; 2. data-record variants, where generally more variants can be accepted by the consumer than the producer is aware of, and where such additional variants can be inherited from the output back to the input of the producer — hence contravariance — again by raising the input type, which is always safe also. Term variables correspond to four categories of terms. However, for the correctness of the algorithm it is important to distinguish variables that represent choices from variables that represent other term categories (due to two kinds of subtyping defined by the seniority relation in Definition 1). We use an up-coerced term variable, e.g. ↑a, to represent a choice term and a down-coerced term variable, e.g. ↓a, to represent any other term, i.e. a symbol, a tuple or a record. Formally, hterm variablei ::= ↑identifier \| ↓identifier We use symbol 8 instead of ↑ or ↓ symbols in the context where the sort is unimportant, e.g. 8a is a term variable that can be either up-coerced or downcoerced. For brevity, term variables are called variables, Boolean variables are called flags and Boolean expressions are called guards. The following grammar specifies the guards: hbool i ::= (hbool i ∧ hbool i) \| (hbool i ∨ hbool i) \| ¬hbool i \| true \| false \| flag MDL terms are built recursively using the constructors: tuple, record, choice and switch, according to the following grammar: htermi ::= hsymbol i \| hterm variablei \| htuplei \| hrecord i \| hchoicei \| hswitchi htuplei ::= (htermi [htermi]∗ ) hrecord i ::= {[hlabel i(hbool i):htermi[,hlabel i(hbool i):htermi]∗ [\|↓identifier ]]} hchoicei ::= (:[hlabel i(hbool i):htermi[,hlabel i(hbool i):htermi]∗ [\|↑identifier ]]:) hlabel i ::= hsymbol i Informally, a tuple is an ordered collection of terms and a record is an extensible, unordered collection of guarded labeled terms, where labels are arbitrary symbols, which are unique within a single record. A choice is a collection of alternative terms. The syntax of choice is the same as that of record except for the delimiters. The difference between records and choices is in subtyping and will become clear below when we define seniority on terms. We use choices to represent polymorphic messages and component interfaces. Records and choices are defined in tail form. The tail is denoted by a variable that represents a term of the same kind as the construct in which it occurs. For example, in the term {l1 (true): t1 , . . . , ln (true): tn \|↓v} the variable ↓v represents the tail of the record, i.e. its members with labels li : li 6= l1 , . . . li 6= ln . A switch is a set of unlabeled (by contrast to a choice) guarded alternatives. hswitchi ::= <hbool i:htermi[, hbool i:htermi]∗ > Exactly one guard must be true for any valid switch. The switch is substitutionally equivalent to the term marked by the true guard: hfalse: t1 , . . . , true: ti , . . . , false: tn i = htrue: ti i = ti . The switch is an auxiliary construct intended for building conditional terms. For example, ha: int, ¬a: stringi represents the symbol int if a = true, and the symbol string otherwise. For each term t, we use V ↑ (t) to denote the set of up-coerced term variables that occur in t, V ↓ (t) to denote the set of the down-coerced ones, and F (t) to denote the set of flags. A term t is called semi-ground if it does not contain variables, i.e. V ↑ (t) ∪ ↓ V (t) = ∅. A term t is called ground if it is semi-ground and does not contain flags, i.e. V ↑ (t) ∪ V ↓ (t) ∪ F (t) = ∅. A term t is well-formed if it is ground and one of the following holds: 1. t is a symbol. 2. n > 0 and t is a tuple (t1 . . . tn ) where all ti , 0 < i ≤ n, are well-formed. 3. n ≥ 0 and t is a record {l1 (b1 ): t1 , . . . , ln (bn ): tn } where for all 0 ≤ i 6= j ≤ n, bi ∧ bj → li 6= lj and all ti for which bi are true are well-formed. 4. n ≥ 0 and t is a choice (:l1 (b1 ): t1 , . . . , ln (bn ): tn :) where for all 0 ≤ i 6= j ≤ n, bi ∧ bj → li 6= lj and all ti for which bi are true are well-formed. 5. n > 0 and t is a switch hb1 : t1 , . . . , bn : tn i where for some 1 ≤ i ≤ n, bi = true and ti is well-formed and where bj = false for all j 6= i. If an element of a record, choice or switch has a guard that is equal to false, then the element can be omitted, e.g. {l1 (b1 ): t1 , l2 (false): t2 , l3 (bn ): t3 } = {l1 (b1 ): t1 , l3 (b3 ): t3 } . If an element of a record or a choice has a guard that is true, the guard can be syntactically omitted, e.g. {l1 (b1 ): t1 , l2 (true): t2 , l3 (bn ): t3 } = {l1 (b1 ): t1 , l2 : t2 , l3 (bn ): t3 } . We define the canonical form of a well-formed collection as a representation that does not include false guards, and we omit true guards anyway. The canonical form of a switch is its (only) term with a true guard, hence any term in canonical form is switch-free. Next we introduce a seniority relation on terms for the purpose of structural subtyping. In the sequel we use nil to denote the empty record { }, which has the meaning of void type in C++ and represents a message without any data. Similarly, we use none to denote the empty choice (: :). Definition 1 (Seniority relation). The seniority relation ⊑ on well-formed terms is defined in canonical form as follows: 1. 2. 3. 4. none ⊑ t if t is a choice. t ⊑ nil if t is any term but a choice. t ⊑ t. t1 ⊑ t2 , if for some k, m > 0 one of the following holds: i i each 1 ≤ i ≤ k; (a) t1 = t11 . . . tk1 , t2 = t12 . . . tk2 and 1 t11⊑ t2 for k 1 1 k (b) t1 = l1 : t1 , . . . , l1 : t1 and t2 = l2 : t2 , . . . , l2m : tm 2 , where k ≥ m and j i for each j ≤ m there is i ≤ k such that l1 = l2 and ti1 ⊑ tj2 ; (c) t1 = (:l11 : t11 , . . . , l1k : tk1 :) and t2 = (:l21 : t12 , . . . , l2m : tm 2 :), where k ≤ m and for each i ≤ k there is j ≤ m such that l1i = l2j and ti1 ⊑ tj2 ; ... nil symbol tuple subtype record choice ... none Fig. 3: Two semilattices representing the seniority relation for terms of different categories. The lower terms are the subtypes of the upper ones. Given the relation t ⊑ t′ , we say that t′ is senior to t and t is junior to t′ . Proposition 1. The seniority relation ⊑ is trivially a partial order, and (T, ⊑) is a pair of upper and lower semilattices (Fig. 3). The seniority relation represents the subtyping relation on terms. If a term t′ describes the input interface of a component, then the component can process any message described by a term t, such that t ⊑ t′ . Although the seniority relation is straightforwardly defined for ground terms, terms that are present in the interfaces of components can contain variables and flags. Finding such ground term values for the variables and such Boolean values for the flags that the seniority relation holds represents a CSP problem, which is formally introduced next. 4 Constraint Satisfaction Problem for KPN In this section we define a Constraint Satisfaction Problem for Kahn Process Networks (CSP-KPN). We regard a KPN network as a directed weakly connected labeled graph G = (V, E, L), where 1. V is a set of vertices. The vertices correspond to individual Kahn processes. 2. E is a set of edges, where each edge e ∈ E is an ordered pair of vertices (v, v ′ ), v, v ′ ∈ V. The edges correspond to channels connecting Kahn processes. 3. A function L : E → Term × Term assigns a label3 to each edge e ∈ E which represents a pair of MDL terms L(e) = t ⊑ t′ called a constraint 4 . It defines the input requirements and the output properties associated with the channel. Given a graph G = (V, E, L) we define the set of constraints as [ C(G) = L(e), e∈E 3 4 we use the same word “label” to refer to the mark on a graph edge and the symbol that labels a term in a record or a choice; however our intention is always clear from the context. in the rest of the paper symbol ⊑ denotes the seniority relation for a pair ground terms; alternatively, if the terms are not ground, ⊑ specifies a constraint. the sets of up-coerced term variables V ↑ (G) and down-coerced term variables as V ↓ (G) [ [ V ↓ (t) ∪ V ↓ (t′ ), V ↑ (t) ∪ V ↑ (t′ ) and V ↓ (G) = V ↑ (G) = t⊑t′ ∈C(G) t⊑t′ ∈C(G) and the set of flags F (G) as F (G) = [ F (t) ∪ F (t′ ). t⊑t′ ∈C(G) Assume a vector of flags f~ = (f1 , . . . , fl ), a vector of term variables 8~v = (8v1 , . . . , 8vm ), a vector of Boolean values ~b = (b1 , . . . , bl ) and a vector of terms ~s = (s1 , . . . , sm ). Then for each term t 1. t[f~/~b] denotes the vector obtained as a result of the simultaneous replacement of fi with bi for each 1 ≤ j ≤ l; 2. t[8~v /~s] denotes the vector obtained as a result of the simultaneous replacement of 8vi with si for each 1 ≤ i ≤ m; 3. t[f~/~b, 8~v /~s] is a shortcut for t[f~/~b][8~v /~s]. Assume a KPN graph G = (V, E, L) such that \|F (G)\| = l, \|V ↑ (G)\| = m, \|V (G)\| = n and for some l, m, n ≥ 0. ↓ Definition 2 (CSP-KPN). We define a CSP for a KPN graph G (CSP-KPN) as follows: for each t ⊑ t′ ∈ C(G) find a vector of Boolean values ~b = (b1 , . . . , bl ), a vectors of ground terms ~t = (t1 , . . . , tm ), ~t′ = (t′1 , . . . , t′n ) such that t[f~/~b, ↑~v /~t, ↓~v /~t′ ] ⊑ t′ [f~/~b, ↑~v /~t, ↓~v /~t′ ], where f~ = (f1 , . . . , fl ), ↑~v = (↑v1 , . . . , ↑vm ), ↓~v = (↓v1 , . . . , ↓vn ). The tuple (~b, ~t, ~t′ ) is called a solution. A CSP-KPN is decidable since the message definition language we introduced can be seen as a term algebra, and decision problems for term algebras are decidable [15]. 5 Adjunct SAT The CSP-KPN solution algorithm presented in the next section is iterative and takes advantage of the order-theoretical structure of the MDL (Proposition 1). Let B0 ⊆ B1 ⊆ · · · ⊆ Bs be sets of Boolean constraints, and ~a↑ and ~a↓ be vectors of semiground terms such that \|~a↑ \| = \|V ↑ (G)\| and \|~a↓ \| = \|V ↓ (G)\|. The vectors ~a↑ and ~a↓ are conditional approximations of the solution. We seek the solution as a fixed point of a series of approximations in the following form: (B0 , ~a↑0 , ~a↓0 ), . . . , (Bs−1 , ~a↑s−1 , ~a↓s−1 ), (Bs , ~a↑s , ~a↓s ), Algorithm 1 CSP-KPN(G) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: c ← \|C(G)\| i←0 B0 ← ∅ ~a↑0 ← (none, . . . , none) ~a↓0 ← (nil, . . . , nil) repeat for 1 ≤ j ≤ c : tj ⊑ t′j ∈ C(G) do (Bi·c+j , ~a↑i·c+j , ~a↓i·c+j ) ← Solve(Bi·c+j−1 , ~a↑i·c+j−1 , ~a↓i·c+j−1 , true, tj , t′j ) end for i←i+1 until (SAT(Bi·c ), ~a↑i·c , ~a↓i·c ) = (SAT(B(i−1)·c ), ~a↑(i−1)·c , ~a↓(i−1)·c ) if Bi·c is unsatisfiable then return Unsat else return (SATSol(Bi·c ), ~a↑i·c [f~/~b], ~a↓i·c [f~/~b]) end if where for every 1 ≤ k ≤ s and a vector of Boolean values ~b that is a solution to SAT(Bk ) (by SAT(Bk ) we mean a set of Boolean vector satisfying Bk ): ~a↑k−1 [f~/~b] ⊑ ~a↑k [f~/~b] and ~a↓k [f~/~b] ⊑ ~a↓k−1 [f~/~b], (1) where the elements of the vectors are compared pairwise. The starting point is B0 = ∅, ~a↑0 = (none, . . . , none), ~a↓0 = (nil, . . . , nil) and the series terminates as soon as SAT(Bs ) = SAT(Bs−1 ), ~a↑s = ~a↑s−1 , ~a↓s = ~a↓s−1 . The adjunct set of Boolean constraints potentially expands at every iteration of the algorithm by inclusion of further logic formulas called assertions into its conjunction as the algorithm processes constraints C(G). Whether the set of Boolean constraints actually expands or not can be determined by checking the satisfiability of SAT(Bk ) 6= SAT(Bk−1 ) for the current iteration k. We argue below that if the original CSP-KPN is satisfiable then so is SAT(Bs ) and that the tuple of vectors (b~s , ~a↑s [f~/b~s ], ~a↓s [f~/b~s ]) is a solution to the former, where b~s is a solution of SAT(Bs ). In other words, the iterations terminate when the conditional approximation limits the term variables, and when the adjunct SAT constrains the flags enough to ensure the satisfaction of all CSP-KPN constraints. In general, the set SAT(Bs ) can have more than one solution. We select one of them, denoted by SATSol(Bs ) in the algorithm. 6 Algorithm In this section we present Algorithm 1 which solves CSP-KPN for a given KPN graph G = (V, E, L). It performs the following steps. The algorithm iterates over the set of constraints C(G) and at each step it builds a closer approximation of the solution. The relation between two consequent approximations satisfies formulas (1). The function Solve() solves the constraint tj ⊑ t′j (see equation (2) in Lemma 1) and updates the vectors ~a↑i·c+j and ~a↓i·c+j with new values. Furthermore, it adds Boolean assertions presented below that ensure 1) satisfaction of the constraint for any ~b ∈ SAT(Bi·c+j ) as provided by Definition 1; and 2) well-formedness of the terms occurring in it. The algorithm terminates if Bi·c ≡ B(i−1)·c , ~a↑i·c = ~a↑(i−1)·c and ~a↓i·c = ~a↓(i−1)·c . Well-formedness assertions for records and choices. Any pair of elements in a well-formed record/choice cannot have equal labels. Therefore, for each record {l1 (b1 ): t1 , . . . , l1 (bn ): tn } and each choice (:l1 (b1 ): t1 , . . . , l1 (bn ): tn :) occurring anywhere in C(G) the following assertion is added to the SAT: ^ ¬(bi ∧ bj ). ∀1≤i,j≤n : li =lj Well-formedness assertions for switches. A well-formed switch term must have exactly one positive guard. Hence, for each switch hb1 : t1 , . . . , bn : tn i occurring anywhere in C(G) the following assertion is added to the SAT: ^ (b1 ∨ · · · ∨ bn ) ∧ ¬(bi ∧ bj ). ∀1≤i,j≤n : i6=j Order assertions. We generate two kinds of order assertions. 1. If a variable 8x is junior to two incommensurable, identically guarded terms 8x ⊑ h. . . : b : t1 . . .i and 8x ⊑ h. . . : b : t2 . . .i, where neither t1 ⊑ t2 nor t2 ⊑ t1 , the assertion ¬b is added to the adjunct SAT. 2. For each c ∈ C(G) of the form hb1 : t1 , . . . , bn : tn i ⊑ hb′1 : t′1 , . . . , b′m : t′m i , the assertion ¬(bi ∧ b′j ) is added to the adjunct SAT if ti 6⊑ t′j . Further details are found in Appendix A. Lemma 1 (Loop invariant). Algorithm 1 finds a series of approximations in the form of (Bk0 , ~a↑k0 , ~a↓k0 ), . . . , (Bks−1 , ~a↑ks−1 , ~a↓ks−1 ), (Bks , ~a↑ks , ~a↓ks ), where ki = i · \|C(G)\|, and such that the following holds for any ~bki ∈ SAT(Bki ). 1. Bki ⊇ Bki−1 ; 2. ~a↑ki−1 [f~/~bki ] ⊑ ~a↑ki [f~/~bki ] and ~a↓ki [f~/~bki ] ⊑ ~a↓ki−1 [f~/~bki ]; 3. ∄ (~a′ , ~a′′ ): (a) ~a↑ki−1 [f~/~bki ] ⊑ ~a′ [f~/~bki ], ~a′ [f~/~bki ] ⊑ ~a↑ki [f~/~bki ]; (b) ~a↓ [f~/~bk ] ⊑ ~a′′ [f~/~bk ], ~a′′ [f~/~bk ] ⊑ ~a↓ [f~/~bk ]. ki i i i ki−1 i given (Bki−1 , ~a↑ki−1 , ~a↓ki−1 ), the Proof. Let c = \|C(G)\|. To construct algorithm iteratively calls Solve() function (see Appendix A). (Bki , ~a↑ki , ~a↓ki ) (Br , ~a↑r , ~a↓r ) ← Solve(Br−1 , ~a↑r−1 , ~a↓r−1 , true, tj , t′j ) where r = ki−1 + j, 1 ≤ j ≤ c and ki = ki−1 + c. For any tj ⊑ t′j , Solve() constructs the Boolean constraints Br , and finds ~a↑r and ~a↓r by solving the equation t[f~/~br−1, ↑~v /~a↑r−1 , ↓~v /~a↓r ] = t′ [f~/~br−1 , ↑~v /~a↑r , ↓~v /~a↓r−1 ], (2) 1. Bki ⊇ Bki−1 by construction: Solve() only adds new Boolean constraints to the existing set. 2. Solve() iteratively constructs the local approximation for each constraint. The series of local approximations converges to the global approximation. 3. Proof by contradiction. Assume that (~a↑r , ~a↓r ) is a solution of (2) and there exists another solution (~a′ , ~a′′ ), such that ~a′ 6= ~a↑r and ~a′′ 6= ~a↓r . Then t[f~/~br−1 , ↑~v /~a↑r−1 , ↓~v /~a′′ ] = t′ [f~/~br−1 , ↑~v /~a′ , ↓~v /~a↓r−1 ]. Two ground terms are equal only if they represent the same term, and, therefore, ~a′ = ~a↑r and ~a′′ = ~a↓r , which contradicts the initial assumption. ⊓ ⊔ Theorem 1 (Termination). CSP-KPN(G) terminates after a finite number of steps for any KPN graph G. Proof. For a given graph G the number of flags, variables and labels for records and choices is bounded. There are two ways to produce new terms: either to add entries with new labels to records and choices, or to substitute subterms for terms. 1. The number of new terms constructed by adding new entries is bounded because the number of labels in a given G is finite. 2. The number of terms constructed by substituting subterms for other terms is bounded because a) the number of variables is finite (the algorithm does not generate new variables); b) after the variables have been instantiated, the category of the term cannot be changed, otherwise, the seniority relation would be violated. It implies that for each ↑v ∈ V ↑ (G) there exists a ground term t̄, such that ↑v ⊑ t̄, and for each ↓v ∈ V ↓ (G) there exists a ground term t, such that t ⊑ ↓v. Providing that \|SAT(Bki )\| ≤ \|SAT(Bki−1 )\|, the algorithm terminates after a finite number of steps. ⊓ ⊔ Theorem 2. Assume a KPN graph G = (V, E, L). The set of constraints C(G) is inconsistent if and only if CSP-KPN(G) returns Unsat. Proof. As the initial approximation the algorithm selects the weakest approximation (∅, (none, . . . , none), (nil, . . . , nil)), it follows from Lemma 1 that the algorithm iterates over all possible approximations in consecutive order starting from (Bk0 , ~a↑k0 , ~a↓k0 ). Therefore, the algorithm cannot skip a solution if one exists. By Theorem 1 the algorithm terminates after a finite number of steps. Hence, it returns Unsat only if and only if the set of constraints C(G) cannot be satisfied. ⊓ ⊔ message message variant variant variant _1_init(vector<vector<double> img); _2_error(string msg); _1_read_color(string fname) { _1_init(...); ... _2_error(...); } _1_read_grayscale(string fname) {...} _1_read_unchanged(string fname) {...} (a) The source code IN 1: (: read_color(c): {fname: string \| $_rc}, read_grayscale(g): {fname: string \| $_rg}, read_unchanged(u): {fname: string \| $_ru} \| $^r :) OUT 1: (: init(or c g u): {img: vector<vector<double>>, \| $_ro1 } \| $^r :) 2: (: error(or c g u): {msg: string \| $_ro2 } :) $_rc <= $_ro1; $_rg <= $_ro1; $_ru <= $_ro1; $_rc <= $_ro2; $_rg <= $_ro2; $_ru <= $_ro2; (b) The interface Fig. 4: The source code and the interface of the component read of the image processing algorithm 7 Communication Protocol In this section we demonstrate interfaces with flow inheritance and code customisation using the example from Section 2. The interfaces are defined as choiceof-records terms. Labels in the choice term of the input interface correspond to function names that can process messages tagged with corresponding labels. Output messages are produced by calling special functions called salvos. The name of a salvo corresponds to one of the labels in the output choice term. The compatibility of two components connected by a channel is defined by the seniority relation. Consider the source code and the interface of the component read in Fig. 4. Integers that have been added as prefixes to functions in the code specify the channels that messages are received from and sent to. In the interfaces we use prefixes $^ and $_ before identifiers to denote up- and down-coerced variables, respectively. A tail variable $^r in the interface enables flow inheritance for choices: variants from the input channel that cannot be processed by the component (i.e. all variants but read_color, read_grayscale or read_unchanged), are absorbed by $^r. Thus, the messages of type Mr1 that contain the name of the image file are processed by the component and the messages of type Mi1 are inherited to the output and forwarded directly to the component init. Flow inheritance for records is realised by down-coerced variables $_rc, $_rg, $_ru, $_ro1, $_ro2, and a set of auxiliary constraints. A record in the input message contains an entry with the label K, which a processing function does not expect. After solving the CSP, the entry is added to the tail variable $_ro1, because the solver deduces that the element with the label K is required by the component init. Furthermore, we use flags c, g and u to exclude the code that is not used in the context. The guards in the output interface employ the joint set of flags from the input variants that can fire salvos specified in the output interface. In the example all three functions can fire init and error salvos; accordingly, the salvos’ guards are c ∨ g ∨ u. The solver deduces that that the variants read_grayscale and read_unchanged cannot receive any messages, and, therefore, their respective processing functions can be excluded from the code. To facilitate decontextualisation we introduce a wrapper for every component called a shell : an auxiliary configuration file that provides facilities for renaming labels in output records and choices and changing the routing of output messages. The source code and the interfaces for the other two components are available in the repository [11]. 8 Implementation We implemented the solver [16] for the CSP-KPN in OCaml. It works on top of the PicoSAT [17] library, the latter used as a subsolver dealing with Boolean assertions. The input for the solver is a set of constraints and the output is in the form of assignments to flags and term variables. We also developed a toolchain in C++ and OCaml that performs the interface reconciliation in five steps: 1. Given a set of C++ sources (the components), augment them with macros acting as placeholders for the code that enables flow-inheritance. 2. Derive the interfaces from the code. 3. Given the interfaces and a netlist that specifies a KPN graph, construct the constraints to be passed on to the CSP-KPN algorithm. 4. Run the solver. 5. Based on the solution, generate header files for every component with macro definitions. In addition, the tool generates the API functions to be called when a component sends or receives a message. Advantages of the presented design are the following: – Interfaces and the code behind them can be generic as long as they are sufficiently configurable. No communication between component designers is necessary to ensure consistency in the design. – Configuration and compilation of every component is separated from the rest of the application. This prevents source code leaks in proprietary software running in the Cloud.5 5 which is otherwise a serious problem. For example, proprietary C++ libraries that use templates cannot be distributed in binary form due to restrictions of the language’s static specialisation mechanism. 9 Conclusion and Future Work We have presented a new static mechanism for coordinating component interfaces based on CSP and SAT that checks compatibility of component interfaces connected in a network with support of overloading and structural subtyping. We developed a fully decoupled Message Definition Language that can be used in the context of KPN for coordinating components written in any programming language. We defined the interface of C++ components to demonstrate the binding between the MDL and message processing functions. Our techniques support genericity, inheritance and structural subtyping, thanks to the order relation defined on MDL terms. On the theory side, we presented the CSP solution algorithm, showed its correctness and identified the termination condition. Although we assume that the algorithm is NP-complete because of the SAT problem, which needs to be solved as a subproblem, the complexity of the algorithm will be evaluated in further research. The next step will be to support multiple flow inheritance in the MDL, to enable combined structures with inheritance (for example, (union ↓a ↓b) represents a record that contains a union of entries associated with records ↓a and ↓b). This would allow one to design components that perform synchronisation and merge multiple messages into one while preserving the inheritance mechanism of a vertex. In the context of Cloud, our results may prove useful to the software-asservice community since we can support much more generic interfaces than are currently available without exposing the source code of proprietary software behind them. Building KPNs the way we do could enable service providers to configure a solution for a network customer based on components that they have at their disposal as well as those provided by other providers and the customer themselves, all solely on the basis of interface definitions and automatic tuning to nonlocal requirements. References 1. Northrop, L., Feiler, P., Gabriel, R.P., Goodenough, J., Linger, R., Longstaff, T., Kazman, R., Klein, M., Schmidtd, D., Sullivan, K., Wallnau, K.: Ultra-LargeScale Systems: The Software Challenge of the Future. Technical report, Software Engineering Institute, Carnegie-Mellon (2006) 2. Kahn, G.: The Soemantics of Simple Language for Parallel Programming. In: IFIP Congress. (1974) 471–475 3. 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Vrba, v., Halvorsen, P., Griwodz, C., Beskow, P., Espeland, H., Johansen, D.: The Nornir run-time system for parallel programs using Kahn process networks on multi-core machines — a flexible alternative to MapReduce. The Journal of Supercomputing 63(1) (2013) 191–217 10. Strachey, C.: Fundamental concepts in programming languages. Higher-order and symbolic computation 13(1-2) (2000) 11–49 11. Zaichenkov, P.: Image Segmentation using K-means Clustering. https://github.com/zayac/joule/tree/kmeans (June 2015) (accessed June 16, 2015). 12. MacQueen, J.: Some methods for classification and analysis of multivariate observations (1967) 13. Gaster, B.R., Jones, M.P.: A Polymorphic Type System for Extensible Records and Variants. Technical report (1996) 14. Leijen, D.: Extensible records with scoped labels (September 2005) 15. Ferrante, J., Rackoff, C.W.: The computational complexity of logical theories. Lecture notes in mathematics. Springer-Verlag (1979) 16. 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Journal on Satisfiability, Boolean Modeling and Computation (JSAT) (2008) A Appendix: Solve function Algorithm 2 Solve(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: B̂i ← AssertWellFormed(AssertWellFormed(Bi , b, t), b, t′ ) if ((t = none or t′ = nil) and (t 6= none or t′ 6= nil)) or (t and t′ are ground, and t = t′ ) then return (B̂i , ~a↑i , ~a↓i ) else if t = ↑v and t′ = ↑v ′ then Bi+1 , ~a↑i+1 ← set a new approximation for ↑v ′ equal to the one of ↑v else if t = ↓v and t′ = ↓v ′ then Bi+1 , ~a↓i+1 ← set a new approximation for ↓v equal to the one of ↓v ′ return (Bi+1 , ~a↑i , ~a↓i+1 ) else if t is nil, symbol, tuple or record, and t′ = ↓v ′ then return Solve(B̂i , ~a↑i , ~a↓i , b, t, t′ [↑~v /~a↑i , ↓~v /~a↓i ]) else if t is a choice and t′ = ↑v ′ then Bi+1 , ~a↑i+1 ← set a new approximation for ↑v ′ as t[↑~v /~a↑i , ↓~v /~a↓i ] when b return (Bi+1 , ~a↑i+1 , ~a↓i ) else if t = ↓v, and t′ is nil, symbol, tuple or record then Bi+1 , ~a↓i+1 ← set a new approximation for ↓v as t′ [↑~v /~a↑i , ↓~v /~a↓i ] when b return (Bi+1 , ~a↓i , ~a↓i+1 ) else if t = ↑v and t′ is a choice then return Solve(B̂i , ~a↑i , ~a↓i , b, t[↑~v /~a↑i , ↓~v /~a↓i ], t′ ) else if t and t′ are tuples then return SolveTupleTuple(B̂i , ~a↑i , ~a↓i , b, t, t′ ) else if t = nil and t′ is a record then return SolveNilRecord(B̂i , ~a↑i , ~a↓i , b, t′ ) else if t and t′ are records then return SolveRecordRecord(B̂i , ~a↑i , ~a↓i , b, t, t′ ) else if t and t′ are choices then return SolveChoiceChoice(B̂i , ~a↑i , ~a↓i , b, t, t′ ) else if t or t′ is a switch then return SolveSwitch(B̂i , ~a↑i , ~a↓i , b, t, t′ ) else return (B̂i ∪ {¬b}, ~a↑i , ~a↓i ) end if Algorithm 3 AssertWellFormed(Bi , b, t) 1: if t is a record {l1 (b1S): t1 , . . . , ln (bn ): tn } or (:l1 (b1 ): t1 , . . . , ln (bn ): tn :) then {b → ¬(bi ∧ bj )} 2: Bi+1 ← Bi ∀1≤i,j≤n : li =lj 3: else if t is a switch hb1 : t1 , . . . , bn : tS n i then 4: Bi+1 ← Bi ∪ {b1 ∨ · · · ∨ bn } {b → ¬(bi ∧ bj )} ∀1≤i,j≤n : i6=j 5: end if 6: return Bi+1 Algorithm 4 SolveTupleTuple(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: Let t be of the form (t1 . . . tn ) g ← t′ [↑~v /~a↑i , ↓~v /~a↓i ] if g = (t′1 . . . t′m ) and n = m then for i : 1 ≤ j ≤ n do Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b, tj , t′j ) end for return (Bi+n , ~a↑i+n , ~a↓i+n ) else return (Bi ∪ {¬b}, ~a↑i+n , ~a↓i+n ) end if Algorithm 5 SolveNilRecord(Bi , ~a↑i , ~a↓i , b, t′ ) 1: g ← t′ [↑~v /~a↑i , ↓~v /~a↓i ] ′ 2: Let g be of the form {l1′ (b′1 ): t′1 , . . . , lm (b′m ): t′m } n V ¬b′j }, ~a↑i , ~a↓i ) 3: return (Bi ∪ {b → j=1 Algorithm 6 SolveRecordRecord(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: g ← t′ [↑~v /~a↑i , ↓~v /~a↓i ] if t = {l1 (b1 ): t1 , . . . , ln (bn ): tn } then for j : 1 ≤ j ≤ m do if ∃k : lk ∈ t, lk = lj′ then Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b → b′j → bk , tk , t′j ) else Bi+j ← Bi+j−1 ∪ {b → ¬b′j } end if end for else if t = {l1 (b1 ): t1 , . . . , ln (bn ): tn \|↓v} then for j : 1 ≤ j ≤ m do if ∃k : lk ∈ t, lk = lj′ then Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b → b′j → bk , tk , t′j ) else Bi+j , ~a↑i+j , ~a↓i+j ← set a new approximation for ↓v as lj′ (b′j ): t′j when b end if end for end if return (Bi+m , ~a↑i+m , ~a↓i+m ) Algorithm 7 SolveChoiceChoice(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: g ← t[↑~v /~a↑i , ↓~v /~a↓i ] if t′ = (:l1 (b1 ): t1 , . . . , ln (bn ): tn :) then for j : 1 ≤ j ≤ m do if ∃k : lk ∈ t, lk′ = lj then Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b → bj → b′k , tj , t′k ) else Bi+j ← Bi+j−1 ∪ {b → ¬b′j } end if end for else if t′ = (:l1 (b1 ): t1 , . . . , ln (bn ): tn \|↑v:) then for j : 1 ≤ j ≤ m do if ∃k : lk ∈ t, lk = lj′ then Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b → bj → b′k , tj , t′k ) else Bi+j , ~a↑i+j , ~a↓i+j ← set a new approximation for ↑v as (:lj (bj ): tj :) when b end if end for end if return (Bi+m , ~a↑i+m , ~a↓i+m ) Algorithm 8 SolveSwitch(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: if t = hb1 : t1 , . . . , bn : tn i then for j : 1 ≤ i ≤ n do Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b, tj , t′ ) end for else if t′ = hb′1 : t′1 , . . . , b′n : t′n i then for i : 1 ≤ j ≤ n do Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b, t, t′j ) end for end if return (Bi+n , ~a↑i+n , ~a↓i+n )	6
1 Isogeometric Analysis Simulation of TESLA Cavities Under Uncertainty arXiv:1711.01828v1 [physics.comp-ph] 6 Nov 2017 Jacopo Corno∗† , Carlo de Falco†‡ , Herbert De Gersem∗ , Sebastian Schöps∗ ∗ Institut für Theorie Elektromagnetischer Felder, TU Darmstadt, Germany † MOX Modeling and Scientific Computing, Politecnico di Milano, Italy ‡ CEN - Centro Europeo di Nanomedicina, Milano, Italy Abstract—In the design of electromagnetic devices the accurate representation of the geometry plays a crucial role in determining the device performance. For accelerator cavities, in particular, controlling the frequencies of the eigenmodes is important in order to guarantee the synchronization between the electromagnetic field and the accelerated particles. The main interest of this work is in the evaluation of eigenmode sensitivities with respect to geometrical changes using Monte Carlo simulations and stochastic collocation. The choice of an IGA approach for the spatial discretization allows for an exact handling of the domains and their deformations, guaranteeing, at the same time, accurate and highly regular solutions. I. I NTRODUCTION The performance of electromagnetic devices such as, for example, energy transducers, magnetrons, waveguides, antennas and linear accelerators is strongly related to the shape of the devices themselves. In particle accelerator cavities, in particular, the acceleration of the particle beam is achieved by exciting specific eigenmodes in the cavity and the resonant frequencies need to be synchronized with the flying particles to guarantee the acceleration. Even small mechanical deformations, either due to manufacturing imperfections or to the electromagnetic pressure (Lorentz detuning) may cause a nonnegligible frequency shift [2], [7]. A correct representation and handling of the domain is then of great importance when implementing a simulation scheme. It has been shown in [7] that the Isogeometric Analysis (IGA) discretization methods introduced in [3], [4] can produce a highly accurate solution, for example for the coupled electromagnetic-mechanic problem modeling Lorenz detuning. The focus of this paper is on quantifying the uncertainty of linear accelerator superconducting cavities in frequency domain, or more precisely the sensitivity of the solution of Maxwell’s eigenvalue problem ∇ × µ−1 ∇ × E = ω 2 εE (1) with respect to (randomly) perturbed domains Ω(y), as in e.g. [1], where E is the electric field, y are geometry parameters, µ, ε and ω are the permeability, permittivity and angular frequency, respectively. The paper is structured as follows, Section 2 and 3 discuss the main ideas behind IGA and Uncertainty Quantification (UQ). In Section 4 and 5 the methods are applied to a benchmark example (pillbox) and the TESLA cavity [6]. The sensitivity of the eigenfrequencies are investigated with respect to uncertain design parameters. Furthermore, non axis-symmetric deformations are taken into consideration. II. I SOGEOMETRIC A NALYSIS In classical discretization methods such as the Finite Element Method (FEM), two steps are usually required: first, the meshing step, i.e., the discretization of the geometry from a Computer Aided Design (CAD) representation, in, typically, hexahedral or tetrahedral elements. Secondly the discretization of the set of equations describing the problem to solve. In [3], Hughes et. al proposed a new approach, called Isogeometric Analysis, were the same basis functions that CAD uses for the description of geometries (e.g. B-Splines and NURBS) are used also as a basis for the solution of the partial differential equations. The IGA paradigm guarantees the Fig. 1: Cylindrical pillbox cavity. c 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. 2 exact description of the computational domain throughout all the analysis, even at the coarsest level of refinement. d-dimensional B-Splines basis functions are defined in the reference space [0, 1]d following a tensor product approach. Along each dimension let p be the degree and Ξ = [ξ0 , ..., ξn+p+1 ] be a knot vector subdividing the unit segment [0, 1]. The Cox-de Boor formula then defines the n + 1 basis functions {Bip }n0 . A B-spline in the physical space is obtained by a mapping 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Fig. 2: B-Spline basis functions of 2nd degree in 1D obtained with knots Ξ = [0, 0, 0, 0.2, 0.4, 0.6, 0.8, 1, 1, 1]. x= n X Bip Pi (2) i=0 where Pi are called control points and acts as degrees of freedom for the curve. Surfaces and volumes are the result of tensorization. As shown in Fig. 2, B-Splines basis functions have higher regularity properties with respect to classical FEM (a B-Spline of degree p can have in general up to p − 1 global derivatives whereas FEM basis are only C 0 across the element boundaries). As a consequence of this, IGA has proven to have a higher accuracy w.r.t. the number of degrees of freedom than classical FEM. The features of IGA are particulary beneficial for quantification of uncertainty related to shape. The CAD representation is not lost in the meshing process, it is possible to easily modify the geometry and for each modified domain there is no need to perform remeshing when deforming the geometry. are a sufficient measure but often one is interested in statistics of outputs. They can be quantified by stochastic moments as expected value and variance Z ∞ E(f ) = f (E(y))ρ(y) dy (5) −∞ Z ∞ 2 f (E(y)) − E(f ) ρ(y) dy var(f ) = (6) −∞ p or the standard deviation std(f0 ) := var(f0 ). However, those integrals can rarely be solved exactly and thus one relies on numerical methods. One way out is the well-known Monte Carlo (MC) sampling method [5]. In this case the set of equations describing the problem is solved M times, for M realizations yi . The results obtained for E are then used to estimate expectations values of the desired quantity of interest by sample averages: E(f ) ≈ III. U NCERTAINTY Q UANTIFICATION (3) The question is then how the input uncertainty E affects the quantity of interest f (E), e.g., a cavity’s eigenfrequency. In many applications classical (local) sensitivities, i.e., Df (E) = ∂ f (E) ∂E (7) i=1 When dealing with mathematical models, e.g. partial differential equations, one has to consider that the input parameters might be affected by uncertainty. Let θ be a random event and y(θ) a vector containing N random input parameters yi (θ) with density ρ(y). Supposing the problem under consideration, i.e., (1), is well defined for any y, then the solution is itself a random variable E = E(y(θ)). M 1 X f (E(yi )). M (4) More sophisticated approaches exploit the regularity of the solution to increase the speed of convergence, e.g., in the generalized Polynomial Chaos (gPC) the mapping (3) from parameter to solution space is approximated by polynomials of degree w [9]. This approximation can be constructed by a basis of distribution dependent orthonormal polynomials {ψp (y)}w 0 and a grid of points yp in the parameter space (called collocation points), i.e., E(y) ≈ w X E(yp )ψp (y). (8) p=0 If the exact solution has a sufficiently smooth dependency w.r.t. its parameters this method converges exponentially. Unfortunately the rate depends heavily on the number of random parameters such that the advantage is lost for large N . 3 −2 10 −3 RelativeSError 10 −4 10 −5 10 GeoPDEsS−SFD CSTSStudioS−SFD CSTSStudioS−SSens.SAnalysis −6 10 0.495 0.5 RadiusS[m] Relative Error in Mean Frequency 100 10-2 10-4 10-6 10-8 10-10 0.505 Closed-form Solution CST Studio (accuracy 1e-6) GeoPDEs (accuracy 1e-6) 0 2 4 6 8 10 Number of Quadrature Nodes (a) Sensitivity of the fundamental frequency f0 in the cylindrical cavity w.r.t. a change in the radius. (b) Convergence of E(f0 ) using gPC, i.e., stochastic Gauss quadrature, for various models. (c) Design parameters of the TESLA cavity half-cell [6]. Fig. 3: Pillbox sensitivities, convergence of stochastic quadrature and TESLA cavity design 3 IV. P ILLBOX C AVITY As a benchmark example we consider the case of a cylindrical pillbox cavity with uncertain radius r (see Fig. 1). We are interested in the resonant frequency of the fundamental (accelerating) mode f0 and its sensitivity w.r.t. the change in radius close to the design value rd = 0.5 m. In this settings one may exactly characterize the frequency and its derivative as Mean Frequency [GHz] 2.5 2 1.5 1 0 f0 (r) = Gc , 2πr df0 (r) Gc =− dr 2πr2 where G ≈ 2.405 is the first zero of the Bessel function of order 0 and c is the speed of light. In Figure 3a we compare implementations of IGA (GeoPDEs [8]) to FEM (CST EM Studio [10]). First, we compute the fundamental frequency for small perturbation of the radius across the design value, using second order IGA and second order FEM with the same level of accuracy. Secondly we use Finite Differences (FD) to estimate the sensitivity. It is clear from the figure that the FEM approximation suffer heavily from the need of remeshing the geometry each time the radius changes, while, even for this naı̈ve approach, IGA obtains a smooth solution, since the parametrization of the cylinder doesn’t change. Finally, the magenta curve in Fig. 3a shows the results obtained by the automated sensitivity analysis tool available in CST: by using more sophisticated methods CST can achieve higher precision but oscillations are still present. To study the global sensitivity of f0 (r) an (artificial) random radius r ∼ U(0.2 m, 0.8 m) is considered. We compare the numerical approach (gPC with FEM and Monopole Modes Dipole Modes Quadrupole Modes 0.5 0 2 4 6 8 10 12 14 16 # Mode Fig. 4: First 16 modes of one-cell TESLA cavity. IGA) to the analytical solution E(f0 ) = 0.2651115... Hz, std(f0 ) = 0.1095555... Hz. Fig. 3b depicts the rapid convergence until the accuracy of the spatial discretization is reached. V. TESLA C AVITY Let us now consider the single cell TESLA cavity geometry whose seven design parameters are described in Figure 3c. Following [11], we apply UQ for design parameters albeit this restricts possible deviations significantly and more realistic cases, such as nonaxissymmetric deformations, bumps/kinks due to welding, mechanical deformation due to Lorentz forces or misalignment of the irisis cannot be represented. We consider the parameters to be uncertain with mean values equal to the TESLA mid-cell design [6] and deviations yi ∼ ( − 0.125 mm, 0.125 mm). MC sampling 4 6 1.5 Standard Deviation [MHz] 5 Standard Deviation [MHz] Monopole Modes Dipole Modes Quadrupole Modes 4 3 2 Monopole Modes Dipole Modes Quadrupole Modes 1 0.5 1 0 0 0 2 4 6 8 10 12 14 16 # Mode (a) Eigenmodes due to uncertain design parameters 0 2 4 6 8 10 12 14 16 # Mode (b) Eigenmodes due to elliptic deformation Fig. 5: Standard deviations for the first 16 modes of the one-cell TESLA Cavity. is applied to estimate mean values and standard deviation of the resonant frequencies for the first 16 modes. With respect to [11], where a 2D solver was used to compute the accelerating frequency by exploiting the cylindrical symmetry of the geometry, GeoPDEs is able to solve the full 3D cavity, thus allowing for the computation of the full spectrum of frequencies, Fig. 4. In Fig. 5a the standard deviation of these modes are depicted. The values are in the MHz range with a maximum deviation in correspondence of the third monopole mode. A second case investigates elliptic deformations of the cavity. For this the domain is deformed in such a way that the cross section is no more a circle but an ellipse, i.e., breaking symmetry. The deviation with respect to the design geometry is chosen to be a uniformly distributed random variable with zero mean and σ = 6.6667 · 10−5 , i.e., Req ∼ U(103.1mm, 103.5mm). The solution of the eigenproblem is computed on a 10×10 grid of collocation points for the first 16 modes as in the previous case. Results (in Fig. 5b) suggest that possible elliptical deformations of the cavity play a slightly weaker role than the design parameters. Nevertheless the higher order modes are, once again, more effected than the fundamental one. Future studies on the full 9-cell cavity should take into account the necessity of including these modes in the analysis. VI. C ONCLUSIONS In this paper the usage of IGA and gPC for the uncertainty quantification of cavities was proposed since those methods exploit the smoothness of the solution in spatial and parameter domain. Academic and realistic geometries underline that this methodology allows to accurately quantify the impact of uncertainties. ACKNOWLEDGMENTS This work is supported by the “Excellence Initiative” of the German Federal and State Governments. the Graduate School CE at TU Darmstadt and the DFG network “UQ for Cavities” (SCHM3127/21). C. de Falco’s work is partially funded by the “Start-up Packages and PhD Program project”, co-funded by Regione Lombardia through “Fondo per lo sviluppo e la coesione 2007-2013” (FAS program). R EFERENCES [1] L. Xiao et al., Modeling imperfection effects on dipole modes in TESLA cavity, IEEE PAC 2007, 2454–2456, 2007. [2] J. Deryckere, T. Roggen, B. Masschaele, and H. De Gersem, Stochastic response surface method for studying microphoning and lorentz detuning of accelerators cavities, ICAP2012, 158160, 2012. [3] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comp. Meth. in App. Mech. and Eng., 194(39), 41354195, 2005. [4] A. Buffa, G. Sangalli, and R. Vzquez, Isogeometric analysis in electromagnetics: B-splines approximation, Comp. Meth. in App. Mech. and Eng., 199, 1143-1152, 2010. [5] Liu, J.S., Monte Carlo Strategies In Scientific Computing, Harvard University, 2002. [6] B. Aune et al., Superconducting tesla cavities. Physical Review Special Topics - Accelerators and Beams, 3(9):092001, 2000. [7] J. Corno, C. de Falco, H. De Gersem and S. Schöps, Isogeometric Simulation of Lorentz Detuning in Superconducting Accelerator Cavities, MOX Report 31/2014. [8] C. de Falco, A. Reali, R. Vázquez, GeoPDEs: A research tool for Isogeometric Analysis of PDEs, Advances in Eng. Soft., 12(42), 1020-1034, 2011. [9] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, 2010. [10] CST EM STUDIO R , Computer Simulation Technology AG, www.cst.com. [11] C. Schmidt, T. Flisgen, J. Heller, U. van Rienen, Comparison of techniques for uncertainty quantification of superconducting radio frequency cavities, ICEAA 2014, 117–120, 2014. [12] J. Gravesen, A. Evgrafov und D.-M. Nguyen, On the sensitivities of multiple eigenvalues, Structural and Multidisciplinary Optimization 44.4, 583-587, 2011.	5
0 Truthful Mechanisms with Implicit Payment Computation MOSHE BABAIOFF, Microsoft Research, Herzeliya, Israel. ROBERT D. KLEINBERG, Computer Science Department, Cornell University, Ithaca, NY, USA. ALEKSANDRS SLIVKINS, Microsoft Research, New York, NY, USA. arXiv:1004.3630v5 [cs.GT] 15 Nov 2015 First version: April 2010 This version: November 2015 It is widely believed that computing payments needed to induce truthful bidding is somehow harder than simply computing the allocation. We show that the opposite is true: creating a randomized truthful mechanism is essentially as easy as a single call to a monotone allocation rule. Our main result is a general procedure to take a monotone allocation rule for a single-parameter domain and transform it (via a black-box reduction) into a randomized mechanism that is truthful in expectation and individually rational for every realization. The mechanism implements the same outcome as the original allocation rule with probability arbitrarily close to 1, and requires evaluating that allocation rule only once. We also provide an extension of this result to multi-parameter domains and cycle-monotone allocation rules, under mild star-convexity and non-negativity hypotheses on the type space and allocation rule, respectively. Because our reduction is simple, versatile, and general, it has many applications to mechanism design problems in which re-evaluating the allocation rule is either burdensome or informationally impossible. Applying our result to the multi-armed bandit problem, we obtain truthful randomized mechanisms whose regret matches the information-theoretic lower bound up to logarithmic factors, even though prior work showed this is impossible for truthful deterministic mechanisms. We also present applications to offline mechanism design, showing that randomization can circumvent a communication complexity lower bound for deterministic payments computation, and that it can also be used to create truthful shortest path auctions that approximate the welfare of the VCG allocation arbitrarily well, while having the same running time complexity as Dijkstra’s algorithm. Categories and Subject Descriptors: J.4 [Social and Behavioral Sciences]: Economics; K.4.4 [Computers and Society]: Electronic Commerce; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems General Terms: theory, algorithms, economics Additional Key Words and Phrases: algorithmic mechanism design, single-parameter mechanisms, multiarmed bandits, regret, multi-parameter mechanisms This is a merged and revised version of the conference papers [Babaioff et al. 2010, 2013] that have appeared in the ACM Conf. on Electronic Commerce (ACM EC) in 2010 and 2013, respectively. This paper contains all results from [Babaioff et al. 2010] and the main result from [Babaioff et al. 2013] (in Section 8). This version is updated to reflect the current status of the follow-up work and open questions. Parts of this research have been done while R. Kleinberg was a Consulting Researcher at Microsoft Research Silicon Valley. He was also supported by NSF Awards CCF-0643934 and AF-0910940, an Alfred P. Sloan Foundation Fellowship, and a Microsoft Research New Faculty Fellowship. 0:2 M. Babaioff et al. 1. INTRODUCTION Algorithmic Mechanism Design studies the problem of implementing the designer’s goal under computational constraints. Multiple hurdles stand in the way for such implementation. Computing the desired outcome might be hard (as in the case of combinatorial auctions) or truthful payments implementing the goal might not exist (as when exactly minimizing the make-span in machine scheduling [Archer and Tardos 2001]). Even when payments that will generate the right incentives do exist, finding such payments might be computationally costly or impossible due to online constraints. It is widely believed that computing payments needed to induce truthful bidding is somehow harder than simply computing the allocation. For example, the formula for payments in a VCG mechanism involves recomputing the allocation with one agent removed in order to determine that agent’s payment; this seemingly increases the required amount of computation by a factor of n + 1, where n is the number of agents. Likewise, for truthful single-parameter mechanisms the formula for payments of a given agent includes integrating the allocation rule over this agent’s bid [Myerson 1981; Archer and Tardos 2001]. In some contexts with incomplete observable information, such as online pay-per-click auctions, computing these “counterfactual allocations” may actually be information-theoretically impossible. This calls into question the mechanism designer’s ability to compute payments that make an allocation rule truthful, even when such payment functions are known to exist. Rigorous lower bounds based on these observations have been established for the communication complexity [Babaioff et al. 2013] and regret [Babaioff et al. 2014; Devanur and Kakade 2009] of truthful deterministic mechanisms. In contrast to these negative results, we show that the opposite is true for randomized single-parameter mechanisms that are truthful-in-expectation: computing the allocation and payments is essentially as easy as a single call to the allocation rule. This allows for positive results that circumvent the lower bounds for deterministic mechanisms cited earlier. 1.1. Single-parameter mechanisms We consider an arbitrary single-parameter domain. The paradigmatic example is an auction that allocates items between agents whose utility is linear in the number of items they receive. The private information of each agent is expressed by a single parameter: her value per item.1 Each agent submits a bid, then the mechanism performs the allocation and charges payments. A mechanism is called “truthful” if each agent maximizes her utility by submitting her true value per item. The allocation rule in a truthful mechanism is called “truthfully implementable”. It is known that an allocation rule is truthfully implementable if and only if it is “monotone”: increasing one agent’s bid while keeping all other bids the same does not decrease this agent’s allocation [Myerson 1981; Archer and Tardos 2001]. A similar property holds for randomized mechanisms and truthfulness-in-expectation. Our contributions. Our main result is a general procedure to take any monotone-inexpectation allocation rule A and transform it into a randomized mechanism that is truthful-in-expectation, implements the same outcome as A with probability arbitrarily close to 1, and requires evaluating that allocation rule only once. (We refer to this 1 In a general single-parameter domain, the allocation rule selects an outcome from some arbitrary collection of feasible outcomes. Each agent has her own type of “good”, and for each agent there is an arbitrary, publicly known mapping from feasible outcomes to a real-valued amount of the corresponding good. The agent’s utility is linear in this amount; the value per unit amount of good is her private information. Truthful Mechanisms with Implicit Payment Computation 0:3 procedure as the generic transformation.) The allocation rule A is accessed only as a function call, so our result applies even if A is an online algorithm. Moreover, for each realization of randomness an agent never loses by participating in the mechanism and bidding truthfully; thus the agents are protected from undesirable random deviations. We make a distinction between randomness in the mechanism and randomness in “nature”: the environment that the mechanism interacts with. Randomness in nature is subject to modeling assumptions and hence is less “reliable”; moreover, agents’ beliefs about nature may be different from the mechanism’s. On the other hand, randomness in the mechanism is fully controlled by the mechanism. Therefore it is desirable to design mechanisms that are truthful in a stronger sense: in expectation over the mechanism’s random seed, for every realization of randomness in nature; we will call such mechanisms ex-post truthful. It is easy to see from [Myerson 1981; Archer and Tardos 2001] that in any ex-post truthful mechanism the allocation rule must satisfy ex-post monotonicity (which is defined similarly to ex-post truthfulness). In the generic transformation described above, if the original allocation rule A is ex-post monotone then the resulting randomized mechanism is ex-post truthful. Similarly, our result extends to Bayesian incentive-compatibility: if A is monotone in expectation with respect to a Bayesian prior over other agents’ bids, then the mechanism is truthful in expectation over this prior. Our generic transformation is particularly useful for mechanism design problems in which re-evaluating the allocation rule is either burdensome or informationtheoretically impossible. 1.2. Bandit mechanisms A leading problem for which only a single call to the allocation rule can be evaluated is the multi-armed bandit (MAB) mechanism design problem [Babaioff et al. 2014; Devanur and Kakade 2009]. In this problem information about the state of the world is dynamically revealed during the allocation; the particular information that is revealed depends on the prior choices of the allocation, and in turn may impact the future choices. Simulating the allocation rule on different inputs may therefore require information that was not observed on the actual run. This “informational obstacle” (insufficient observable information) is a crucial obstacle for deterministic ex-post truthful MAB mechanisms; it is used in [Babaioff et al. 2014] to derive that the appropriate payments cannot be computed unless the allocation rule is very “naı̈ve” (and therefore suboptimal). To put more context, MAB mechanisms are motivated by online pay-per-click ad auctions, and were suggested in [Babaioff et al. 2014; Devanur and Kakade 2009] as a simple model which combines strategic bidding by agents and online learning by the mechanism. Each agent has a single ad that she wants to display to users, and derives utility only if her ad is clicked. The value per click is her private information. The allocation rule proceeds in rounds: in each round the mechanism allocates one ad to be shown to a user and observes whether this ad was clicked. The click probabilities (also known as “click-through rates”, or CTRs) are unknown to the mechanism, and need to be estimated during the run of the allocation rule. All bids are submitted before the allocation starts, and all payments are assigned after it ends. MAB mechanisms are related to MAB algorithms: the allocation rule is essentially an MAB algorithm whose “rewards” are clicks weighted by the corresponding bids. Moreover, welfare of an MAB mechanism is precisely the same as the total reward of its allocation rule.2 Therefore one could directly compare the performance of truthful 2 This is because payments cancel out: the total amount paid by the agents is equal to the total amount received by the mechanism. 0:4 M. Babaioff et al. MAB mechanisms with that of MAB algorithms; both can be quantified using regret: the loss in welfare compared to the benchmark which always picks the best ad. Following [Babaioff et al. 2014; Devanur and Kakade 2009], we focus on the stochastic version of the problem, i.e. we assume that the CTRs do not change over time. Then the “randomness in nature” corresponds to the random clicks, and ex-post truthfulness means truthfulness for every realization of the clicks (but in expectation over the randomness in the mechanism). Note that ex-post truthfulness is a very strong property which holds even if the clicks are chosen by an oblivious adversary. As discussed in [Babaioff et al. 2014; Devanur and Kakade 2009], this property is highly desirable, compared to the weaker notion of “truthfulness in expectation over clicks”, even if the corresponding mechanism has regret guarantees that only apply to the stochastic setting. Our contributions. Applying our generic transformation to the MAB problem we derive that the problem of designing truthful MAB mechanisms reduces to the problem of designing monotone MAB allocation rules. Such a problem has not been previously studied in the rich literature on MAB. Our main result in this direction is a randomized MAB mechanism that is ex-post truthful and has regret O(T 1/2 ) for the stochastic version. This upper bound on regret matches the information-theoretic lower bound for algorithms in the same setting (i.e., the lower bound holds even in the absence of incentive constraints). This stands in contrast to the lower bound of [Babaioff et al. 2014], where it was shown that deterministic ex-post truthful MAB mechanisms must suffer a larger regret of Ω(T 2/3 ). On a technical level, we design a new MAB allocation rule that is ex-post monotone and has regret O(T 1/2 ) for the stochastic setting. (We use it to obtain a randomized ex-post truthful MAB mechanism with the same regret.) Moreover, we show that UCB1 [Auer et al. 2002a] (and a number of similar MAB algorithms) give rise to MAB allocations that are monotone in expectation over clicks, and therefore can be transformed to randomized MAB mechanisms that are truthful in the same sense and have optimal regret. The new ex-post monotone MAB allocation rule is deterministic, which rigorously confirms the intuition from [Babaioff et al. 2014; Devanur and Kakade 2009] that the impossibility results for deterministic MAB mechanisms are caused by the “informational obstacle” (insufficient observable information about clicks) rather than ex-post monotonicity. 1.3. Other contributions Power of randomization. As a by-product of our analysis of MAB mechanisms, we obtain an unconditional separation between the power of randomized vs. deterministic ex-post truthful mechanisms for welfare maximization, in the online setting. (The separation result is unconditional in the sense that it considers exactly the same setting for both classes of mechanisms.) This complements the result of Dobzinski and Dughmi [2009], which gives a separation between these two classes of mechanisms in the offline setting, under a polynomial communication complexity constraint. It is worth noting that the separation in Dobzinski and Dughmi [2009] applies to a rather unnatural problem (two-player multi-unit auctions in which if at least one item is allocated, then all items are allocated and each player receives at least one item) whereas our separation result is for a natural problem: online pay-per-click ad auctions for a single slot, with unknown click-through rates. For the objective of revenue maximization, separations between randomized and deterministic mechanisms have been known for much longer [Thanassoulis 2004; Manelli and Vincent 2006; Dobzinski et al. 2012; Briest et al. 2014] and are in some Truthful Mechanisms with Implicit Payment Computation 0:5 sense less surprising. Randomization allows the mechanism to access a larger set of possible allocations, i.e. the set of all probability distributions over pure allocations, and in some cases this leads to greater revenue, for example by permitting more finegrained price discrimination between agent types. This is not the case for the objective of maximizing welfare (because VCG mechanisms are deterministic and they maximize welfare pointwise while obeying incentive constraints). For welfare maximization, randomized mechanisms are sometimes more powerful than deterministic ones due to other reasons, such as computational power or informational limitations (as in the problem we study). Offline mechanisms. Our main result also has implications for offline mechanism design. Nisan and Ronen, in their seminal paper [Nisan and Ronen 2001] which started the field of algorithmic mechanism design, cite the apparent n-fold computational overhead of computing VCG payments and pose the open question of whether payments can be computed faster than solving n versions of the original problem, e.g. for VCG path auctions. Our result shows that the answer is affirmative, if one adopts the truthful-in-expectation solution concept and tolerates a mechanism that outputs an outcome whose welfare is a (1 + ǫ)-approximation to that of the VCG allocation, for arbitrarily small ǫ > 0. Babaioff et al. [2013] present a social choice function f in an n-player single-parameter domain such that the deterministic communication complexity required for truthfully implementing f exceeds that required for evaluating f by a factor of n. Our result shows that no such lower bound holds when one considers randomized mechanisms, again allowing for a small amount of random error in the allocation. Extension to multi-parameter mechanisms. We extend our generic transformation from single-parameter to multi-parameter mechanisms. It is known that a multiparameter allocation rule is truthfully implementable if and only if it satisfies a property called “cycle-monotonicity”. (This is a rather strong property which specializes to monotonicity in the single-parameter case.) Similar to the single-parameter case, we present a general procedure to take any cycle-monotone allocation rule A and transform it into a randomized mechanism that is truthful-in-expectation, implements the same outcome as A with probability arbitrarily close to 1, and requires evaluating that allocation rule only once. The technical contribution here is that we find a reduction from the multi-parameter setting to the single-parameter case. While much more general that our single-parameter transformation, this result may be more difficult to apply. This is because cycle-monotonicity is known to be a very restrictive property. However, the follow-up work already provides two applications, see Section 2.1 for details. 1.4. Map of the paper This paper makes four high-level contributions: the generic transformation for singleparameter mechanisms (Sections 4 and 5), the two applications to off-line mechanism design (Section 6), the results on MAB mechanisms (Section 7), and an extension to multi-parameter mechanisms (Section 8). Presenting these results requires a significant amount of preliminaries on mechanisms design (Section 3), multi-armed bandits (Section 7.1), and multi-parameter mechanism (Section 8.1). We conclude with open questions (Section 9). A considerable amount of work followed up on the initial conference publication [Babaioff et al. 2010] of this paper. This work is discussed in Section 2.1. 0:6 M. Babaioff et al. 2. RELATED WORK AND FOLLOW-UP WORK The characterization of truthful mechanisms for single-parameter domains, given by Myerson [1981] for single-item auctions and by Archer and Tardos [2001] for a more general class of single-parameter problems, states that a mechanism is truthful if and only if its allocation rule is monotone and its payment rule charges each agent its value for the realized outcome, minus a correction term expressed as an integral over all types lower than the agent’s declared type. Exact computation of this correction term may be intractable, but Archer et al. [2004] developed a clever workaround: one can use random sampling to compute an unbiased estimator of the correction term, at the cost of evaluating the allocation rule once more. Thus, for n agents, the allocation rule must be evaluated n + 1 times: once to determine the actual allocation, and once more per agent to determine that agent’s payment. Our generic transformation relies on a generalization of this random sampling technique, but we show how to avoid recomputing the allocation rule when determining each agent’s payment, by coupling payment generation with the allocation itself. The question of whether computing payments is computationally harder than computing the allocation was raised by Nisan and Ronen [2001] in the context of VCG path auctions. The most significant progress to date was the communication complexity lower bound of Babaioff et al. [2013] mentioned above. Payment computation in online mechanism design is a central issue in the analysis of truthful MAB mechanisms in Babaioff et al. [2014] and Devanur and Kakade [2009]. The main result of [Babaioff et al. 2014] is a characterization of deterministic ex-post truthful mechanisms. It is more restrictive than the Myerson and ArcherTardos characterization. The reason is that computing an agent’s payment requires knowing how many clicks she would have received if she had submitted a lower bid value, which may require the mechanism to hypothetically go back into the past and allocate impressions to a different agent for the purpose of seeing whether a user would have clicked on that agent’s advertisement. Such counterfactual information is typically impossible to obtain in an online setting. Babaioff et al. [2014] focus on welfare maximization. Using the above characterization, they prove that any deterministic ex-post truthful MAB mechanism must incur regret Ω(T 2/3 ), whereas MAB algorithms for the same setting can achieve regret O(T 1/2 ). Devanur and Kakade [2009] consider revenue maximization, and derive a similar Ω(T 2/3 ) lower bound on loss of revenue compared to the VCG payments.3 Dynamic auctions [Athey and Segal 2013; Bergemann and Välimäki 2010; Bergemann and Said 2011] constitute another setting in which information is revealed “dynamically” (over time). However, while in MAB auctions all information from the agents (the bids) is submitted only once and then information is revealed to the mechanism by the environment over time, in dynamic auctions the agents continuously observe private “signals” from the environment and submit “actions” to the mechanism. Accordingly, providing the right incentives becomes much more challenging. On the other hand, existing work has focused on a fully Bayesian setting with known priors on the signals, whereas all of our results do not rely on priors. Finally, several recent papers have explored the theme of reductions in algorithmic mechanism design. Unlike our work which requires mechanisms to be truthful for every realization of the agents’ types, these papers focus on Bayesian settings and adopt Bayesian incentive-compatibility as their solution concept. A reduction converting any allocation rule into a Bayesian incentive-compatible mechanism with approx3 For revenue-maximizing MAB mechanisms, there is no clear comparison with the performance of MAB algorithms. Truthful Mechanisms with Implicit Payment Computation 0:7 imately the same expected social welfare was developed in [Hartline and Lucier 2010; Bei and Huang 2011; Hartline et al. 2011]. Chawla et al. [2012] considered black-box reductions of mechanism design problems to algorithmic problems with the same objective, and demonstrated significant limitations of this approach. The breakthrough results of Cai et al. [2012; 2013a; 2013b] and Daskalakis and Weinberg [2014] circumvented these limitations by instead reducing to algorithmic problems with a modified objective. In particular, reductions from revenue-maximizing mechanisms to welfaremaximizing algorithms are presented in [Cai et al. 2012, 2013a], whereas Cai et al. [2013b] and Daskalakis and Weinberg [2014] present reductions for non-linear objective functions, such as makespan in scheduling.4 2.1. Follow-up work (subsequent to [Babaioff et al. 2010]) Our generic transformation exhibits high variability in payments, and includes an explicit tradeoff between the variability in payments and the loss in performance. Formally, variability can be expressed as variance, maximal absolute value, or (for positive types) maximal rebate. Performance can be expressed as welfare or revenue. Wilkens and Sivan [2012] have proved this tradeoff to be optimal in a certain worstcase sense: our transformation achieves the optimal worst-case variance in payments for any given worst-case loss in performance, where the worst case is over all monotone allocation rules. Their result applies to any single-parameter domain and any of the above notions of variability and performance. Our generic transformation is likely to be very useful in single-parameter settings which exhibit the “informational obstacle” (insufficient observable information) such as the one found for deterministic MAB mechanisms. The follow-up work describes three additional settings. First, Wilkens and Sivan [2012] observe that the same obstacle arises in offline pay-per-click ad auctions with multiple ad slots, where the CTRs have slot-specific multipliers. In conjunction with our generic transformation, an obvious welfare-maximizing allocation rule for that setting results in a truthfulin-expectation mechanism. Second, Shnayder et al. [2012] describe a packet scheduling problem in a network router, where the “informational obstacle” arises due to the potentially missing information about packet arrival times. (As they observe, this information may be missing not only because it is not observed by the router but also because the router simply does not have space to store it.) They design a monotone allocation rule for their setting, and use our generic transformation to convert it to a truthful-in-expectation mechanism. Third, Gatti et al. [2012] consider an extension of MAB mechanisms to multiple ad slots. While they provide truthful mechanisms based on the simple MAB mechanism from [Babaioff et al. 2014; Devanur and Kakade 2009], our generic transformation could give rise to more efficient truthful mechanisms. Wilkens and Sivan [2012] obtain a similar “single-call reduction” (i.e., a reduction from allocation rules to truthful-in-expectation mechanisms which calls the allocation rule only once) for multi-parameter allocation rules that are maximal-indistributional-range (MIDR). MIDR allocation rules [Dobzinski and Dughmi 2009] pick a welfare-maximizing distribution over outcomes from some fixed collection of distributions; they are precisely the allocation rules for which VCG payments produce a truthful mechanism. This result is an independent work with respect to, and a special case of, the multi-parameter reduction in Section 8. The multi-parameter generic transformation in Section 8 has been used in two recent papers. First, Jain et al. [2011] used it to speed up the payment computation for a mechanism that allocates batch jobs in a cloud system. Second, Huang and Kannan 4 All papers discussed in this paragraph, except [Hartline and Lucier 2010], have appeared after the conference publication of this paper [Babaioff et al. 2010]. 0:8 M. Babaioff et al. [2012] used it to compute payments for their privacy-preserving procurement auction for spanning trees, which is based on the well-known “exponential privacy mechanism” from prior work [McSherry and Talwar 2007]. Simplified payment computation. Our generic transformation is most useful if the allocation rule cannot be invoked more than once, as in “bandit mechanisms” or other examples provided in follow-up work. Segal [2010] has observed that any truthful single-parameter mechanism can be implemented in a much simpler way, as long as two calls to the allocation rule are allowed: one computes the allocation, and the other one generates random payments with the correct expectation. In the first call one uses the original bids. For the second call, one selects an agent uniformly at random, and uses the random sampling trick from Archer and Tardos [2001] described above to compute the payment for this agent, and then scales the payment appropriately.5 Also, a simpler generic transformation is possible if one settles for a weaker notion of Bayesian incentive-compatibility [Hartline 2012]. 3. PRELIMINARIES Single-parameter domains. We present the single parameter model for which we apply our procedure. The model is very similar to the model of Archer and Tardos [Archer and Tardos 2001], yet it is slightly more general. We state the model is terms of values and not costs and allow the values to be both positive and negative. We also allow randomization by nature. All these changes are minor and do not change the fundamental characterization, yet are helpful to later derive our results. Let n be the number of agents and let N = [n] be the set of agents. Each agent i ∈ N has some private type consisting of a single parameter xi ∈ Ti that describes the agent, and is known only to i, everything else is public knowledge. We assume that the domain Ti is an open subset of R which is an interval with positive length (possibly starting from −∞ or going up to ∞). Let T = T1 × T2 × ... × Tn denote the domain of types and let t ∈ T denote the vector of true types. There is some set of outcomes O. For single-parameter domains, agents evaluate outcomes in a particular way that we describe next. For each agent i ∈ N there is a function ai : O → R+ specifying the allocation to agent i. The value of an outcome o ∈ O for an agent i ∈ N with type xi is xi · ai (o). The utility that agent i ∈ N derives from outcome o ∈ O when he is charged pi is quasi-linear: ui = xi · ai (o) − pi . For instance, consider the allocation of k identical units of good to agents with additive valuations: agent i has a value of xi per unit. An outcome o specifies how many items each agent receives: ai (o) is the number of items i receives. His valuation for that outcome is his value per-unit times the number of units he receives. A (direct revelation) deterministic mechanism M consists of the pair (A, P), where A : T → O is the allocation rule and P : T → Rn is the payment rule, i.e. the vector of payment functions Pi : T → R for each agent i. Each agent is required to report a type bi ∈ Ti to the mechanism, and bi is called the bid of agent i. We denote the vector of bids by b ∈ T . The mechanism picks an outcome A(b) and charges agent i payment of Pi (b). The allocation for agent i when the bids are b is Ai (b) = ai (A(b)) and he is charged Pi (b). Agent i’s utility when the agents bid b ∈ T and his type is xi ∈ Ti is ui (xi , b) = xi · Ai (b) − Pi (b) 5 However, (1) more work is needed for domains with negative agents’ types, such as VCG shortest path auctions (see Section 6 for more details). In particular, one needs to carefully define the random sampling of the bid for payment computation, using a version of the argument in Section 5.2 to bound the loss in welfare. Truthful Mechanisms with Implicit Payment Computation 0:9 We also consider randomized mechanisms, which are distributions over deterministic mechanisms. For a randomized allocation rule Ai (b) and Pi (b) will denote the expected allocation and payment charged from agent i, when the bids are b. The expectation is taken over the randomness of the mechanism. Sometimes it will be helpful to explicitly consider the deterministic allocation and payment that is generated for specific random seed. in this case we use w to denote the random seed and use Ai (b; w) and Pi (b; w) to denote allocation and payment when the seed is w. There may be some outside randomization that influences the outcome and is not controlled by the mechanism, e.g. randomness in the realization of clicks in sponsored search auction. We call this randomization by nature. With such randomization Ai (b) and Pi (b) also encapsulate expectations over nature’s randomization. Finally, we use the notation Ai (b; w, r) and Pi (b; w, r) to denote the allocation and payment charged from agent i, when the bids are b, the mechanism random seed is w and nature’s random seed is r. Allocation and Mechanism Properties. Let b−i denote the vector of bids of all agents but agent i. We can now write the vector of bids as b = (b−i , bi ). Similar notation will be used for other vectors. We next list two central properties, truthfulness and individual rationality. — Mechanism M is truthful if for every agent i truthful bidding is a dominant strategy: for every agent i, bidding xi always maximizes her utility, regardless of what the other agents bid. Formally, xi · Ai (b−i , xi ) − Pi (b−i , xi ) ≥ xi · Ai (b) − Pi (b) (2) xi · Ai (b−i , xi ) − Pi (b−i , xi ) ≥ 0 (3) holds for every agent i ∈ N , type xi ∈ Ti , bids of others b−i ∈ T−i and bid bi ∈ Ti of agent i. — Mechanism M is individually rational (IR) if an agent never receives negative utility by participating in the mechanism and bidding truthfully. Formally, holds for every agent i ∈ N , type xi ∈ Ti and bids of others b−i ∈ T−i . It will be helpful to establish terminology for the case that the above hold not only in expectation but also for specific realizations. For example, we will say that a mechanism is universally truthful if Equation (2) holds not only in expectation over the mechanism’s randomness, but rather for every realization of that randomness. In general, every property that we define is defined by some inequality, and if the inequality holds for every realization of the mechanism randomness we say that it holds universally, and if it holds for every realization of nature randomness we say that it holds ex-post. When we want to emphasize that the property holds only in expectation over the nature’s randomness we say that it holds stochastically. Note that in an individually rational mechanism an agent is ensured not to incur any loss in expectation. That is rather unsatisfying as for some realizations the agent might suffer a huge loss. It is more desirable to design mechanisms that are universally ex-post individually rational, that is a truthful agent should incur no loss for every bids of the others and every realization of the random events (not only in expectation). If all types are positive, then in addition to individual rationality it is desirable that all agents are charged a non-negative amount; this is known as the no-positivetransfers property. Finally, the welfare of a truthful mechanism is defined to be the P total utility i xi · Ai (t). Characterization. The following characterization of truthful mechanisms, due to Archer and Tardos [Archer and Tardos 2001], is almost identical to the characteriza- 0:10 M. Babaioff et al. tion presented by Myerson [Myerson 1981] for truthful mechanisms in the special case of single item auctions. The crucial property of an allocation that yields truthfulness is monotonicity, defined as follows: Definition 3.1. Allocation rule A is monotone if for every agent i ∈ N , bids b−i ∈ T−i − and two possible bids of i, bi ≥ b− i , we have Ai (b−i , bi ) ≥ Ai (b−i , bi ). Recall that monotonicity of an allocation rule is also defined universally and/or ex-post. We next present the characterization of truthful mechanisms. In the theorem statement, the expression Ai (b−i , u) is interpreted to equal zero when u 6∈ Ti . T HEOREM 3.2. [Myerson 1981; Archer and Tardos 2001] Consider an arbitrary single-parameter domain. An allocation rule A admits a payment rule P such that the mechanism (A, P) is truthful if and only if A is monotone and moreover for each R bi agent i and bid vector b it holds that −∞ Ai (b−i , u) du < ∞. In this case the payment Pi (b) for each agent i must satisfy R bi Pi (b) = Pi0 (b−i ) + bi Ai (b−i , bi ) − −∞ Ai (b−i , u) du, (4) where Pi0 (b−i ) does not depend on bi . A mechanism is called normalized if for each agent i and every bid vector b, zero allocation implies a zero payment: Ai (b) = 0 ⇒ Pi (b) = 0. C OROLLARY 3.3. The truthful mechanism in Theorem 3.2 is normalized if and only if Pi0 (b−i ) ≡ 0, in which case the mechanism is also individually rational and for positive-only types (T ⊂ Rn+ ) it moreover satisfies the no-positive-transfers property. Both Theorem 3.2 and Corollary 3.3 hold in the “ex-post” sense (resp., “universal” sense), if Ai , Pi and Pi0 (b−i ) are interpreted to mean their respective values for a specific random seed of nature (resp., mechanism). In Corollary 3.3, the mechanism is normalized in the same sense as it is truthful. 4. THE GENERIC TRANSFORMATION FOR SINGLE-PARAMETER DOMAINS This section presents a generic procedure which takes any monotone allocation rule for a single-parameter domain and creates a randomized truthful-in-expectation mechanism which attains the same outcome as the original allocation rule with high probability. The resulting mechanism uses the allocation rule as a “black box,” calls it only once, and allocates according to the this call. Henceforth, we will refer to this procedure as the generic transformation. Our main result — the existence of the generic transformation with the desired properties — can be stated informally as follows. T HEOREM 4.1 (I NFORMAL ). Consider an arbitrary single-parameter domain with n agents. Let A be a monotone allocation rule for this domain. Then for each µ ∈ [0, 1] e P) e with the following properties: there exists a truthful mechanism M = (A, — M executes a single call to A(b̃) to compute the allocation, with a pre-processing step to compute the modified bid vector b̃, and a post-processing step to compute the payments. Both pre- and post-processing steps take O(n) time and do not depend on A. e and A(b) — For any bid vector b and any fixed random seed of nature allocations A(b) are identical with probability at least 1 − nµ. — M is universally ex-post individually rational. If all types are positive, then M is ex-post no-positive-transfers, and never pays any agent i more than bi · Ai (x) · ( µ1 − 1). Truthful Mechanisms with Implicit Payment Computation 0:11 Presenting the formal version of this result (Theorem 4.5) requires defining the generic transformation. We begin with an informal description thereof. As evidenced by Equation (4), the payment for agent i is a difference of two terms: the agent’s reported utility (i.e., the product of her bid and her allocation), minus the integral of the allocation assigned to every smaller bid value. We charge the agent for her reported utility, and we give her a random rebate whose expectation equals the required integral. When integrating a function over a finite interval, an unbiased estimator of the integral can be obtained by sampling a uniformly random point of that interval and evaluating the function at the sampled point. This idea was applied, in the context of mechanism design, by Archer et al. [2004]. Below, we show how to generalize the transformation to allow for integrals over unbounded intervals, as required by Equation (4). Using this transformation it is easy to transform any monotone allocation rule into a randomized mechanism that is truthful in expectation and only evaluates the allocation rule n + 1 times: once to determine the actual allocation, and once more per agent to obtain an unbiased estimate of that agent’s payment. Our main innovation is a transformation that uses the same random sampling trick, but only needs to evaluate the allocation rule once during the entire mechanism. (In other words, it does not require additional calls to the allocation rule to compute the payments.) Assume that a parameter µ ∈ (0, 1) is given. For every player, with probability 1 − µ, we leave their bid unchanged; with probability µ, we sample a smaller bid value. The allocation rule is invoked on these bids. An agent is always charged her reported value of the outcome, but if her bid was replaced with a smaller bid value then we refund her an amount equal to an unbiased estimator of the integral in Equation (4), scaled by 1/µ to counterbalance the fact that the refund is only being applied with probability µ. A naı̈ve application of this plan suffers from the following defect: the random resampling of bids modifies the expected allocation vector, so we need to obtain an unbiased estimator of the integral of the modified allocation rule. However, if we change our sampling procedure to obtain such an estimate, then this modifies the allocation rule once again, so we will still be estimating the wrong integral! What we need is a “fixed point” of this process of redefining the sampling procedure. Below, we give a definition of self-resampling procedures that satisfy the requisite fixed point property, and we give two simple constructions of self-resampling procedures. A self-resampling procedure transforms the bid bi of a given agent i bid into two correlated random values (xi , yi ), where xi is the modified bid presented to the original allocation rule, and yi is used (together with the allocation itself) in computing the payment for this agent. More specifically, yi is needed to correctly normalize the unbiased estimator of the integral in Equation (4) for the modified allocation rule, according to Theorem 4.2 below. For agents with positive types we define a simpler self-resampling procedure for which the unbiased estimator does not depend on yi , and therefore, strictly speaking, the procedure only needs to output xi (more details can be found in Section 4.5). However, we explicitly return the yi even for the positive types so as to be consistent with the general definitions and (perhaps more importantly) because we use it to define self-resampling procedures with general support (see Section 4.4). Thus, the formal description of our generic transformation consists of three parts: (1) a method for estimating integrals by evaluating the integrand at a randomly sampled point, (2) the definition and construction of self-resampling procedures, (3) the generic transformation that uses the previous two ingredients to convert any monotone allocation rule into a truthful-in-expectation randomized mechanism. We now specify the details of each of these three parts. 0:12 M. Babaioff et al. 4.1. Estimating integrals via random sampling Let I be a nonempty open interval in R (possibly with infinite endpoints) and let g be R a function defined on I. Let us describe a procedure for estimating the integral g(z) dz by evaluating g at a single randomly sampled point of I. The procedure is I well known; we describe it here for the purpose of giving a self-contained exposition of our algorithm. T HEOREM 4.2. Let F : I → [0, 1] be any strictly increasing function that is differentiable and satisfies inf z∈I F (z) = 0 and supz∈I F (z) = 1. If Y is a random variable with cumulative distribution function F , then Z g(Y ) . g(z) dz = E F ′ (Y ) I P ROOF. Since inf z∈I F (z) = 0 and supz∈I F (z) = 1, it follows that the random variable Y is supported on the entire interval I. Our assumption that F is differentiable implies that Y has a probability density function, namely F ′ (z). Thus, for any funcR ′ tion h, the expectation of h(Y ) is given by I h(z)F (z) dz. Applying this formula to the function h(z) = g(z)/F ′ (z) one obtains the theorem. 4.2. Self-resampling procedures The basic ingredient of our generic transformation is a procedure for taking a bid bi and a random seed wi , and producing two random numbers xi (bi ; wi ), yi (bi ; wi ). The mechanism will use {xi (bi ; wi )}i∈N for determining the allocation and additionally yi (bi ; wi ) for determining the payment it charges agent i. To prove that the mechanism is truthful in expectation we will require the following properties.6 Definition 4.3. Let I be a nonempty interval in R. A self-resampling procedure with support I and resampling probability µ ∈ (0, 1) is a randomized algorithm with input bi ∈ I, random seed wi , and output xi (bi ; wi ), yi (bi ; wi ) ∈ I, that satisfies the following properties: (1) For every fixed wi , xi (bi ; wi ) and yi (bi ; wi ) are non-decreasing functions of bi . (2) With probability 1 − µ, xi (bi ; wi ) = yi (bi ; wi ) = bi . Otherwise xi (bi ; wi ) ≤ yi (bi ; wi ) < bi . (3) The conditional distribution of xi (bi ; wi ), given that yi (bi ; wi ) = b′i < bi , is the same as the unconditional distribution of xi (b′i ; wi ). In other words, Pr[ xi (bi ; wi ) < ai \| yi (bi ; wi ) = b′i ] = Pr[ x(b′i ; wi ) < ai ], ∀ai ≤ b′i < bi . (4) Consider the two-variable function F (ai , bi ) = Pr[yi (bi ; wi ) < ai \| yi (bi ; wi ) < bi ], which we will call the distribution function of the self-resampling procedure. For each bi , the function F (·, bi ) must be differentiable and strictly increasing on the interval I ∩ (−∞, bi ). As it happens, it is easier to construct self-resampling procedures with support R+ , and one such construction that we call the canonical self-resampling procedure (Algorithm 1) forms the basis for our general construction. We defer the discussion of self-resampling procedures with general support until after we have described and analyzed the generic transformation. 6 To keep the notation consistent, we state Definition 4.3 for a given agent i. Strictly speaking, the subscript i is not necessary. Truthful Mechanisms with Implicit Payment Computation 0:13 Algorithm 1: The canonical self-resampling procedure. 1: Input: bid bi ∈ [0, ∞), parameter µ ∈ (0, 1). 2: Output: (xi , yi ) such that 0 ≤ xi ≤ yi ≤ bi . 3: with probability 1 − µ 4: xi ← bi , yi ← bi . 5: else 6: Pick b′i ∈ [0, bi ] uniformly at random. 7: xi ← Recursive(b′i ), yi ← b′i . 8: 9: 10: 11: 12: 13: function Recursive(bi ) with probability 1 − µ return bi . else Pick b′i ∈ [0, bi ] uniformly at random. return Recursive(b′i ). P ROPOSITION 4.4. Algorithm 1 is a self-resampling procedure with support R+ and resampling probability µ. The distribution function for this procedure is F (ai , bi ) = ai /bi . P ROOF. Properties 1 and 2 in Definition 4.3 are immediate from the description of the algorithm. The random seed wi for the algorithm can be defined as a countably infinite sequence of real numbers drawn independently and uniformly at random from [0, 1] interval. Then in order pick a random number in some range [0, r], the algorithm takes the next number in this sequence and multiplies it by r. Property 3 follows from the recursive nature of the sampling procedure: the event yi (bi ; wi ) = b′i < bi implies that the algorithm has followed the “else” branch on Line 5, and has chosen b′i in Line 6. Finally, the distribution function is F (ai , bi ) = ai /bi since conditional on the event yi (bi ; wi ) < bi , the distribution of yi (bi ; wi ) is uniform in the interval [0, bi ]. Property 4 follows trivially. 4.3. The generic transformation Suppose we are given a monotone allocation rule A and for each agent i ∈ N a self-resampling procedure that has resampling probability µ ∈ (0, 1), support Ti , and output values fi = (xi , yi ). Let Fi (ai , bi ) denote the distribution function of the self-resampling procedure for agent i, and let Fi′ (ai , bi ) denote the partial derivative ∂Fi (ai ,bi ) . Our generic transformation combines these ingredients into a randomized ∂ai mechanism M = AllocToMech(A, µ, f ) that works as follows: If A itself is randomized or if there is randomness arising from nature, then we allocate according to A(x; w, r) and we assume that the algorithm’s random seed w and the nature’s random seed r are independent of the random seeds wi used in the resampling step. We are now ready to present our main result: T HEOREM 4.5. Consider an arbitrary single-parameter domain. Let A be a monotone allocation rule. Suppose we are given an ensemble f of self-resampling procedures fi = (xi , yi ) for each agent i, each with resampling probability µ ∈ (0, 1). Then the e P) e = AllocToMech(A, µ, f ) has the following properties. mechanism M = (A, (a) M is truthful, universally ex-post individually rational, 0:14 M. Babaioff et al. Mechanism 2: Generic transformation M = AllocToMech(A, µ, f ) 1: Solicit bid vector b ∈ T . 2: Execute each agent’s self-resampling procedure using an independent random seed wi , to obtain two vectors of modified bids x = (x1 (b1 ; w1 ) , . . . , xn (bn ; wn )), y = (y1 (b1 ; w1 ) , . . . , yn (bn ; wn )). 3: 4: Allocate according to A(x). Each agent i is charged the amount bi · Ai (x) − Ri , where Ri is the rebate ( 1 i (x) · A if yi < bi , ′ Ri = µ Fi (yi ,bi ) 0 otherwise. (5) (b) For n agents and any bid vector b (and any fixed random seed of nature) allocations e and A(b) are identical with probability at least 1 − nµ. A(b) (c) If T = Rn+ (all types are positive), and each fi is the canonical self-resampling procedure, then mechanism M is ex-post no-positive-transfers, and never pays any agent i more than bi · Ai (x) · ( µ1 − 1). Several remarks are in order. — The mechanism never explicitly computes the payment for each agent i (Equation (4)) but rather implicitly creates the correct expected payments through its randomization of the bids. — The mechanism only invokes the original allocation rule A once. This property is very useful when it is impossible to invoke the allocation rule more than once, e.g. for multi-armed bandit allocations. — The mechanism M is randomized even if A is deterministic. It is truthful in expectation over the randomness used by the self-resampling procedures. — If A is ex-post monotone, then M will be ex-post truthful. To see this, fix nature’s random seed r and apply Theorem 4.5 to the allocation rule Ar induced by this r. — If agents’ types are positive then by part (b), the welfare of M is at least 1 − nµ times that of A. Further results on bounding the welfare loss are presented in Section 5. — By definition of the payment rule, the mechanism is universally ex-post normalized. We will not explicitly mention this property in the subsequent applications. Parameter µ controls the trade-off between the loss in welfare and the variance in payments, as quantified by the rebate size Ri . If µ is very small and the mechanism issues rebate(s), then its revenue may be very low and possibly negative. However, this risk may be mitigated if the auction maker runs many independent auctions, as may be the case in practice. Further, the follow-up paper [Wilkens and Sivan 2012] proves that our welfare vs. variance trade-off is optimal. P ROOF OF T HEOREM 4.5. We start with some notation. Aei (b−i , bi ; q) denotes the allocation for agent i given the bid vector b = (b−i , bi ) and the combined random seed q = (w1 , . . . , wn , w, r). When we write Aei (b−i , u) without indicating the dependence on the q, we are referring to the unconditional expectation of Aei (b−i , u; q) over q. Truthful Mechanisms with Implicit Payment Computation 0:15 To prove that M is truthful, we need to prove two things: that the randomized alloe satisfies cation rule Ae is monotone, and that the expected payment rule P R ei (b) = bi Aei (b−i , bi ) − bi Aei (b−i , u) du. (6) P −∞ The monotonicity of randomized allocation rule Ae follows from the monotonicity of A and the monotonicity property 1 in the definition of a self-resampling procedure. To ei satisfies Equation (6), we begin by recalling that the payment charged to prove that P player i is bi Ai (x) − Ri , where the rebate Ri is defined by Equation (5). The expectation of bi Ai (x) is simply bi Aei (b−i , bi ), so to conclude the proof of truthfulness we must show that Rb E[Ri ] = i Aei (b−i , u) du. (7) −∞ Our proof of Equation (7) begins by observing that the conditional distribution of xi , given that yi = u < bi , is the same as the unconditional distribution of xi (u; wi ), by Property 3 of a self-resampling procedure. Combining this with the fact that the random seed wi is independent of {wj : j 6= i}, we find that the conditional distribution of the tuple x = (x−i , xi ), given that yi = u, is the same as the unconditional distribution of the vector x̂ of modified bids that M would input into the allocation rule A if the bid vector were (b−i , u) instead of (b−i , bi ). Taking expectations, this implies that for all e −i , u). u < bi , we have E[Ai (x) \| yi = u] = E[Ai (x̂)] = A(b Now apply Theorem 4.2 with the function g(u) = Aei (b−i , u). Recalling that Fi (·, bi ) is the cumulative distribution function of yi given that yi < bi , we apply the theorem to obtain " # Z bi ei (b−i , yi ) A Ai (x) e Ai (b−i , u) du = E yi < bi = E yi < bi Fi′ (yi , bi ) Fi′ (yi , bi ) −∞ = µ · E[Ri \| yi < bi ], (8) where the second equation follows from the equation derived at the end of the preceding paragraph, averaging over all u < bi . Observing that Ri = 0 unless yi < bi , an event that has probability µ, we see that E[Ri ] = µ · E[Ri \| yi < bi ]. Combined with Equation (8), this establishes Equation (7) and completes the proof that M is truthful. Mechanism M is universally ex-post individually rational because agent i is never charged an amount greater than bi Aei (b; q). Part (b) follows from the union bound: the probability that xi = bi for all i is at least 1 − nµ. For part (c), note that by Proposition 4.4, the canonical self-resampling procedure has distribution function F (ai , bi ) = ai /bi , hence Fi′ (yi , bi ) = 1/bi , for all i, yi , bi . The rebate Ri is equal either i (x) to 0 or to µ1 · F A = bi · Ai (x) · µ1 . We also charge bi · Ai (x) to agent i. The claimed ′ i (yi ,bi ) upper bound on the amount paid to agent i follows by combining these two terms. 4.4. Self-resampling procedures with general support To construct a self-resampling procedure with support in an arbitrary interval I, we can use the following technique. Suppose h : (0, 1] × I → I is a two-variable function such that the partial derivatives ∂h(zi , bi )/∂zi and ∂h(zi , bi )/∂bi are well-defined and strictly positive at every point (zi , bi ) ∈ (0, 1] × I. Suppose furthermore that h(1, bi ) = bi and inf zi ∈(0,1] {h(zi , bi )} = inf(I) for all bi ∈ I. Then we define the h-canonical selfresampling procedure (xhi , yih ) with support I, by specifying that h xi (bi ; wi ) = h(xi (1; wi ), bi ) (9) yih (bi ; wi ) = h(yi (1; wi ), bi ), 0:16 M. Babaioff et al. where (xi , yi ) is the canonical self-resampling procedure as defined in Algorithm 1. P ROPOSITION 4.6. (xhi , yih ) as defined in Equation (9) is a self-resampling procedure with support I and resampling probability µ. The distribution function for (xhi , yih ) is the unique two-variable function F (ai , bi ) such that h(F (ai , bi ), bi ) = ai for all ai , bi ∈ I, ai < bi . (10) P ROOF. Property 1 in Definition 4.3 holds because of the monotonicity of h, Property 2 holds because h(1, bi ) = bi for all bi , and Property 3 holds because the function h is deterministic and monotone. Let Fh (ai , bi ) and F0 (ai , bi ) be the distribution functions for the h-canonical and canonical self-resampling procedures, respectively. Recall that F0 (ai , bi ) = ai /bi by Proposition 4.4. Note that F (ai , bi ) in Equation (10) is unique (and hence well-defined) by the strict monotonicity of h. The claim that Fh (ai , bi ) = F (ai , bi ) easily follows from in Equation (9). By definition of h we have h(yi (1, wi ), bi ) < bi ⇐⇒ yi (1, wi ) < 1. Therefore, letting yi = yi (1, wi ) we have Fh (ai , bi ) , Pr[h(yi , bi ) < ai \| h(yi , bi ) < bi ] = Pr[yi < F (ai , bi ) \| yi < 1] = F0 ( F (ai , bi ) , 1) = F (ai , bi ). Our assumption that h is differentiable and strictly increasing in its first argument now implies that the same property holds for F , which verifies Property 4. 4.5. A simplified generic transformation for positive types We focus on the important special case of positive types, and present Mechanism 3, a simplified version of the generic transformation (Mechanism 2), for this case. Mechanism 3: A simplified generic transformation for positive types. 1: Parameter: resampling probability µ ∈ (0, 1). 2: 3: 4: 5: 6: 7: 8: Collect bid vector b ∈ (0, ∞)n . Independently for each agent i ∈ [n]: Sample: γi uniformly at random from [0, 1] 1/(1−µ) . Set χi = 1 with probability 1 − µ and otherwise χi = γi Construct the vector of modified bids x = (x1 , . . . , xn ), where xi = χi bi . Allocate according to A(x). ( 1 if χi = 1, For each agent i, assign payment bi · Ai (x) · . 1 1 − µ if χi < 1 We prove that Mechanism 3 is equivalent to the generic transformation (Mechanism 2) with a canonical self-resampling procedure (Algorithm 1). P ROPOSITION 4.7. The allocation and payments in Mechanism 3 coincide with those in Mechanism 2 with a canonical self-resampling procedure. Truthful Mechanisms with Implicit Payment Computation 0:17 To prove Proposition 4.7, we provide a non-recursive version of the canonical selfresampling procedure (Algorithm 1), which we call O NE S HOT . We argue that the output of Mechanism 3 is identical to the output of the mechanism obtained by plugging O NE S HOT into Mechanism 2.7 O NE S HOT is also essential for the analysis in Section 5. Algorithm 4: O NE S HOT : a non-recursive version of Algorithm 1. 1: Input: bid bi ∈ [0, ∞), parameter µ ∈ (0, 1). 2: Output: (xi , yi ) such that 0 ≤ xi ≤ yi ≤ bi . 3: with probability 1 − µ 4: xi ← bi , yi ← bi . 5: else 6: Pick γ1 , γ2 ∈ [0, 1] indep., uniformly at random. 1/(1−µ) 1/(1−µ) 1/µ 7: xi ← bi · γ1 , yi ← bi · max{γ1 , γ2 }. P ROPOSITION 4.8. Algorithm 1 and O NE S HOT generate the same output distribution: for any bid bi ∈ [0, ∞), the joint distribution of the pair (xi , yi ) = (xi (bi ; wi ), yi (bi ; wi )) is the same for both procedures. (Here wi denotes the random seed for each agent i.) The proof of Proposition 4.8 can be found in the Appendix. P ROOF OF P ROPOSITION 4.7. By Proposition 4.8, it suffices to compare Mechanism 3 to Mechanism 2 with self-resampling procedure O NE S HOT. To show that the two mechanisms are equivalent, we must show that they yield the same distribution over allocations and the same payments. First we argue about the allocations. In both mechanisms, each bidder’s bid bi is independently transformed into a random xi , and then the allocation rule A is applied to the vector x = (x1 , . . . , xn ). Furthermore, the conditional distribution of xi given bi is the same in both cases: xi = bi with probability 1 − µ, and otherwise xi = bi · γ 1/(1−µ) where γ is uniformly distributed in [0, 1]. Hence, the two mechanisms yield the same distribution over allocations. To see that the payment rules are the same, consider the distribution function of O NE S HOT, as defined in Definition 4.3: F (ai , bi ) = Pr[yi (bi ; wi ) < ai \| yi (bi ; wi ) < bi ]. By Proposition 4.8, Fi (ai , bi ) is also the distribution function for Algorithm 1. By Proposition 4.4 we have F (ai , bi ) = ai /bi , and consequently Fi′ (ai , bi ) , ∂Fi (ai , bi ) 1 = . ∂ai bi In particular, neither allocation nor payments in this mechanism depend on the yi ’s. Suppressing the yi ’s from mechanism M and plugging in Fi′ (yi , bi ) = b1i , we obtain Mechanism 3. This completes the proof of Proposition 4.7. 7 This observation is due to [Shnayder et al. 2012]. 0:18 M. Babaioff et al. 5. IMPROVED BOUNDS ON WELFARE We present improved bounds on the welfare obtained by our generic transformation. We consider two interesting special cases when the agents’ private types are, respectively, always positive and always negative. In the second case, agents are contractors who incur costs and get paid by the mechanism; one such example is a shortest paths mechanism considered in Section 6. We consider the approximation that is achieved by the mechanism as a function of the approximation of the original allocation rule. Recall that our generic transformation creates a mechanism with an allocation that is identical to the original allocation with probability at least 1 − nµ. For positive types this immediately implies a bound on the approximation which degrades with n, the number of agents. (For negative types such bound does not immediately follow since the cost in the low probability event might be prohibitively high.) For both settings, we present a similar bound that does not degrade with n. 5.1. Positive private types Assume that the agents’ types are always positive, more specifically that the type space is T = (0, ∞)n . Recall that for P agents’ types t ∈ T the social welfare of an outcome o is defined to SW(o, t) = i∈N ti ai (o). The optimal social welfare is OPT(t) = maxo∈O SW(o, t), where O is the set of all feasible outcomes. (A mechanism with) an allocation rule A is α-approximate if it holds that α · E[SW(A(t), t)] ≥ OPT(t) for every t. (11) n T HEOREM 5.1. Consider the setting in Theorem 4.5(c), so that T = (0, ∞) and each fi is the canonical self-resampling procedure. If allocation rule A is α-approximate, then µ mechanism AllocToMech(A, µ, f ) is α/(1 − 2−µ )-approximate. P ROOF. Fix a bid vector b, and let o∗ be the corresponding optimal allocation. Recall that our mechanism outputs allocation A(x), where x is the vector of randomly modified bids. As the original allocation rule A is α-approximate, by Equation (11) it holds that α · SW(A(x), x) ≥ OPT(x). We will show that µ bi for each agent i. (12) E[xi ] = 1 − 2−µ Thus when we evaluate o∗ with respect to bids x we get: P α · SW(A(x), x) ≥ OPT(x) ≥ SW(o∗ , x) = i∈N xi ai (o∗ ) P ∗ α · E[SW(A(x), x)] = E i∈N xi ai (o ) X µ µ ∗ = bi ai (o ) = 1 − OPT(b). 1− 2−µ 2−µ i∈N It remains to prove Equation (12). Let us use O NE S HOT to describe the canonical self-resampling procedure. Recall that O NE S HOT generates xi = xi (bi ; wi ) by setting xi = bi with probability 1 − µ, and otherwise sampling γ1 uniformly at random in [0, 1] 1/(1−µ) and outputting xi = bi · γ1 . Hence R1 1/(1−µ) 1 E[xi \| xi < bi ] = 0 bi · γ1 dγ1 = bi · 1+ 1 1 = bi · 1 − 2−µ 1−µ µ . E[xi ] = (1 − µ) · bi + µ · E[xi \| xi < bi ] = bi · 1 − 2−µ Truthful Mechanisms with Implicit Payment Computation 0:19 For arbitrary self-resampling procedures fi with support R+ , Equation (12) can be α replaced by E[xi ] ≥ (1 − µ) bi , which gives a slightly weaker result, namely an 1−µ approximation to the social welfare. 5.2. Negative private types Now assume that the agents’ types are always negative, more specifically that T = (−∞, 0)n . For negative types approximation is defined with respect to the social cost, which is the negation of the social welfare. An algorithm is α-approximate if for every input it outputs an outcome with cost at most α times the optimal cost. We present an approximation bound for an h-canonical self-resampling procedure, for a suitably chosen h. T HEOREM 5.2. Consider the setting in Theorem 4.5. Assume that T = (−∞, 0)n and √ that each fi is the h-canonical self-resampling procedure, where h(zi , bi ) = bi / zi . Suppose µ ∈ (0,12 ). If allocation rule A is α-approximate, then mechanism µ AllocToMech(A, µ, f ) is α 1 + 1−2µ -approximate. The proof of this theorem is almost identical to that of Theorem 5.1, and thus is omitted. The main modification is that Equation (12) is replaced by the following lemma: L EMMA 5.3. In the setting of Theorem 5.2, letting xh be the vector of modified types, it holds that µ E[xhi ] = bi 1 + 1−2µ for all i. P ROOF. Recall that xh is defined by Equation (9). As in the proof of Equation (12), we will use O NE S HOT to describe the canonical self-resampling procedure. It follows that R1 R1 − 1 1 , E[xhi \| xi < bi ] = 0 r bi 1 dγ1 = 0 bi · γ1 2(1−µ) dγ1 = bi · 1− 1 1 = bi · 1 + 1−2µ (1−µ) γ1 E[xhi ] = (1 − µ) · bi + µ · E[xhi \| xhi < bi ] = bi · 1 + 2(1−µ) µ 1−2µ . 6. APPLICATIONS TO OFFLINE MECHANISM DESIGN The VCG mechanism for shortest paths. The seminal paper Nisan and Ronen [2001] has presented the following question: is there a computational overhead in computing payments that will induce agents to be truthful, compared to the computation burden of computing the allocation. One of their examples is the VCG mechanism for the shortest path mechanism design problem, where a naive computation of VCG payments requires additional computation of n shortest path instances. Yet, an explicit payment computation is not the real goal, it is just a means to an end. The real goal is inducing the right incentives. Our procedure shows that without any overhead in computation, if we move to a randomized allocation rule and settle for truthfulness in expectation (and a small loss in performance) one can induce the right incentives. The shortest path mechanism design problem is the following. We are given a graph G = (V, E) and a pair of source-target nodes (vs , vt ). Each agent e controls an edge e ∈ E and has a cost ce > 0 if picked (thus ve = −ce < 0 and Te = (−∞, 0) for every e). That cost is private information, known only to agent e. The mechanism designer’s goal is to pick a path P from node vs to node vt in the graph with minimal total cost, 0:20 M. Babaioff et al. P that is e∈P ce is minimal. Assume that there is no edge that forms a cut between vs and vt . The VCG mechanism is an cost-optimal and truthful mechanism for this problem. It computes a shortest path P with respect to the reported costs and pays to an agent e the difference between the cost of the shortest path that does not contains e and the total cost shortest path excluding the cost of e. A naive implementation of the VCG mechanism requires computing \|P \| + 1 shortest path instances (where \|P \| denotes the number of edges in path P ). VCG is deterministic, truthful and cost-optimal. Let EFF an the cost-optimal allocation rule for the shortest path problem. We can use our general procedure to derive the following result (its proof follows directly from Theorem 4.5 and Theorem 5.2). T HEOREM 6.1. Fix any µ ∈ (0, 21 ). For each agent i, let fi be the h-canonical self√ resampling procedure, where h(zi , bi ) = bi / zi . Let M = AllocToMech(EFF, µ, {fi }) be the mechanism created by applying AllocToMech() to EFF. Then M has the following properties: • It is truthful and universally individually rational. • It only computes one shortest paths instance. µ • It outputs a path with expected length at most 1 + 1−2µ times the length of the shortest path. Recall that parameter µ controls the trade-off between approximation ratio and the rebate size Ri , which for a given random seed is proportional to µ1 . Communication overhead of payment computation. Babaioff et al. [2013] show that there exists a monotone deterministic allocation rule for which the communication required for computing the allocation is factor Ω(n) less than the communication required to computing prices. This implies that inducing the correct incentives deterministically has a large overhead in communication. Assume that instead of requiring explicit computation of payments we are satisfied with inducing the correct incentives using a randomized mechanism. In such case our reduction shows that the deterministic lower bound cannot be extended to randomized mechanisms, if we allow a small error in the allocation. More concretely, consider a single parameter domain with types that are positive, Ti = (0, ∞) (as in [Babaioff et al. 2013]). For all i, use the canonical self-resampling procedure. Consider any monotone allocation rule A. We can apply Theorem 4.5 to obtain a randomized mechanism that is truthful and only executes that allocation rule A once (thus has no communication overhead at all) and has exactly the same allocation with probability at least (1 − µ)n . For any ǫ > 0 we can find µ > 0 such that the error probability is less than ǫ. 7. MULTI-ARMED BANDIT MECHANISMS In this section we apply the main result to multi-armed bandit (MAB) mechanisms: single-parameter mechanisms in which the allocation rule is (essentially) an MAB algorithm parameterized by the bids. As in any single-parameter mechanism, agents submit their bids, then the allocation rule is run, and then the payments are assigned. This application showcases the full power of the main result, since in the MAB setting the allocation rule is only run once, and (in general) cannot be simulated as a computational routine without actually implementing the allocation. Focusing on the stochastic setting, we design truthful MAB mechanisms with the same regret guarantees as the best MAB algorithms such as UCB1 [Auer et al. 2002a]. First, we prove that allocation rules derived from UCB1 and similar MAB algorithms Truthful Mechanisms with Implicit Payment Computation 0:21 are in fact monotone, and hence give rise to truthful MAB mechanisms. Second, we provide a new allocation rule with the same regret guarantees that is ex-post monotone, and hence gives rise to an ex-post truthful MAB mechanism. Third, we use this new allocation rule to obtain an unconditional separation between the power of randomized and deterministic ex-post truthful MAB mechanisms. 7.1. Preliminaries: MAB mechanisms An MAB mechanism [Babaioff et al. 2014; Devanur and Kakade 2009] operates as follows. There are n agents. Each agent i has a private value vi and submits a bid bi . We assume that bi , vi ∈ [0, bmax ], where bmax is known a priori. The allocation consists of T rounds, where T is the time horizon. In each round t the allocation rule chooses one of the agents, call it i = i(t), and observes a click reward π(t) ∈ [0, 1] for this choice; the chosen agent i receives vi π(t) units of utility. Payments are assigned after the last round of the allocation. Note that the social welfare of the mechanism is equal to the PT total value-adjusted click reward: t=1 vi(t) π(t). The special case of 0-1 click rewards corresponds to the scenario in which agents are advertisers in a pay-per-click auction, and choosing agent i in a given round t means showing this agent’s ad. Then the click reward π(t) is the click bit: 1 if the ad has been clicked, and 0 otherwise. Following the web advertising terminology, we will say that in each round, an impression is allocated to one of the agents. Formally, an MAB allocation rule A is an online algorithm parameterized by n, T, bmax and the bids b. In each round it allocates the impression and observes the click reward. Absent truthfulness constraints, the objective is to maximize the reported P welfare: Tt=1 bi(t) π(t). This formulation generalizes MAB algorithms: the latter are precisely MAB allocation rules with all bids set to 1. Given an MAB algorithm Â, there is a natural way to transform it into an MAB allocation rule A. Namely, A runs algorithm Â with modified click rewards: if agent i is chosen in round t then the click reward reported to Â is π̂(t) = (bi /bmax ) π(t). We will say that algorithm Â induces allocation rule A. From now on we will identify an MAB algorithm with the induced allocation rule, e.g. allocation rule UCB1 is induced by algorithm UCB1 [Auer et al. 2002a]. We will focus on the stochastic MAB setting: in all rounds t in which an agent i is chosen, the click reward π(t) is an independent random sample from some fixed distribution on [0, 1] with expectation µi .8 Following the web advertisement terminology, we will call µi the click-through rate (CTR) of agent i. The CTRs are fixed, but no further information about them (such as priors) is revealed to the mechanism. Regret. The performance of an MAB allocation rule is quantified in terms of regret: PT R(T ; b; µ) , T maxi [ bi µi ] − E[ t=1 bi(t) µi(t) ], the difference in expected click rewards between the algorithm and the benchmark: the best agent in hindsight, knowing the µi ’s. We focus on R(T ) , max R(T ; b; µ), where the maximum is taken over all CTR vectors µ and all bid vectors b such that bi ≤ 1 for all i. 9 Regret guarantees from the vast literature on MAB algorithms easily translate to MAB allocation rules. In particular, allocation rule UCB1 has regret √ R(T ) = O( nT log T ) [Auer et al. 2002a], which is nearly matching the information8 The exact shape of this distribution is not essential. E.g. in the advertising example π(t) ∈ {0, 1}. define R(T ) with bmax = 1 merely to simplify the notation. All regret bounds (scaled up by a factor of bmax ) hold for an arbitrary bmax . 9 We 0:22 M. Babaioff et al. √ theoretically optimal regret bound Θ( nT ) [Auer et al. 2002b; Audibert and Bubeck 2010]. The stochastic MAB setting tends to be easier if the best agent is much better than the second-best one. Let us sort the agents so that b1 µ1 ≥ b2 µ2 ≥ . . . ≥ bn µn . The gap δ of the problem instance is defined as (b1 µ1 − b2 µ2 )/bmax . The δ-gap regret Rδ (T ) is defined as the worst-case regret over all problem instances with gap δ. Allocation rule UCB1 achieves Rδ (T ) = O( nδ log T ) [Auer et al. 2002a]; there is a √ lower bound Rδ (T ) = Ω(min( nδ log T, nT )) [Lai and Robbins 1985; Auer et al. 2002b; Kleinberg et al. 2008a]. Click realizations. A click realization is a n × T table ρ in which the (i, t) entry ρi (t) is the click reward (e.g., the click bit) that agent i receives if it is played in round t. Note that in order to fully define the behavior of any algorithm on all bid vectors one may need to specify all entries in the table, whereas only a subset thereof is revealed in any given run. We view ρ as a realization of nature’s random seed. Thus, we can now define ex-post truthfulness and other ex-post properties: informally, ex-post property is a property that holds for every given click realization. For each agent i, round t, bid vector b and click realization ρ, let Ati (b; ρ) denote the probability that MAB allocation rule A allocates the impression at round t to agent i. (If Â is deterministic, the probability Âti (ρ) is trivial: either 0 or 1.) For MAB algorithm Â, define Âti (ρ) similarly. 7.2. Truthfulness and monotonicity Theorem 4.5(c) reduces the problem of designing truthful MAB mechanisms to that of designing monotone MAB allocations. Let us state this reduction explicitly: T HEOREM 7.1. Consider the stochastic MAB mechanism design problem. Let A be a stochastically monotone (resp., ex-post monotone) MAB allocation rule. Applying the transformation in Theorem 4.5(c)10 to A with parameter µ, we obtain a mechanism M such that: (a) M is stochastically truthful (resp., ex-post truthful), ex-post no-positivetransfers, and universally ex-post individually rational. (b) for each click realization, the difference in expected welfare between A and M is at most µnT bmax . Note that the theorem provides two distinct types of guarantees: game-theoretic guarantees in part (a), and performance guarantees in part (b). We show that a very general class of deterministic MAB algorithms induces monotone MAB allocation rules (to which Theorem 7.1 can be applied). Definition 7.2. In a given run of an MAB algorithm, the round-t statistics is a pair of vectors (π, ν), where the i-th component of π (resp., ν) is equal to the total payoff (resp., the number of impressions) of agent i in rounds 1 to t − 1, for each agent i. Vectors π and ν are called p-stats vector and i-stats vector, respectively. Definition 7.3. A deterministic MAB algorithm Â is called well-formed if for each round t and agent i, letting (π, ν) be the round-t statistics, the following properties hold: — [Âti (ρ) is determined by (π, ν)] there is a function χi (π; ν) that depends only on the round-t statistics such that Âti (ρ) = χi (π; ν) for any click realization ρ and all t. — [χ-monotonicity] χi (π; ν) is non-decreasing in πi for any fixed (π−i , ν). 10 Theorem 4.5(c) is stated for the type space T = (0, ∞)n , but it trivially extends to the case T = (0, bmax )n . Truthful Mechanisms with Implicit Payment Computation 0:23 — [χ-IIA] for each round t, any three distinct agents {i, j, l} and any fixed (π−i , ν−i ), changing (πi , νi ) cannot transfer an impression from j to l. The χ-IIA property above is reminiscent of Independence of Irrelevant Alternatives (IIA) property in the Social Choice literature (hence the name). A similar but technically different property is essential in the analysis of deterministic MAB allocation rules in [Babaioff et al. 2014]. Remark 7.4. For a concrete example of a well-formed MAB algorithm, consider (a version of) UCB1.11 The algorithm is very simple: in each round t, it chooses agent p min arg max πi (t)/νi (t) + 8 log(T )/νi (t) . i L EMMA 7.5. In the stochastic MAB mechanism design problem, let A be a MAB allocation rule induced by a well-formed MAB algorithm. Then A is stochastically monotone. P ROOF. We will use an alternative way to define a realization of random click rewards: a stack-realization is a n × T table in which the (i, t) entry is the click bit that agent i receives the t-th time she is played. Clearly a stack-realization and a bid vector uniquely determine the behavior of A. We will show that: A is monotone for each stack-realization. (13) Then A is monotone in expectation over any distribution over stack-realizations, and in particular it is monotone in expectation over the random clicks in the stochastic MAB setting, so the Lemma follows. Let us prove Claim (13). Throughout the proof, fix stack-realization σ, agent i, and bid vector b−i . Consider two bids bi < b+ i . The claim asserts that agent i receives at least as many clicks with bid b+ than with bid bi . i Let us introduce some notation (letting bi be the bid of agent i). Let A(bi , t) be the agent selected by the allocation rule in round t. For each agent j, let νj (bi , t) and πj (bi , t) be, respectively, the total number of impressions and the total click reward of agent j in the first t rounds. Let π̂j (bi , t) = (bj /bmax ) πj (bi , t) be the corresponding total modified click reward. Let ν(bi , t) (resp., π(bi , t) and π̂(bi , t)) be the n-dimensional vector whose j-th component is νj (bi , t) (resp., πj (bi , t) and π̂j (bi , t)) for each agent j. Note that (π̂(bi , t), ν(bi , t)) is the round-t statistics for the MAB algorithm that A is induced by. For each agent j, νj (bi , t) uniquely determines πj (bi , t): Pνj σ(j, s) where νj = νj (bi , t). (14) πj (bi , t) = s=1 Let us overview the forthcoming technical argument. We will show by induction on t that νi (bi , t) ≤ νi (b+ i , t) for all t. For the induction step we only need to worry about the case when the claim holds for a given t with equality. In this case we show that ν−i (bi , t) = ν−i (b+ i , t). This is trivial for n = 2 agents; the general case requires a rather delicate argument that uses the χ-IIA property in Definition 7.3.12 Now let us carry out the proofs in detail. First, denote ν∗ (bi , t) , t − νi (bi , t), and let us show that for any two rounds t, s it holds that + ν∗ (bi , t) = ν∗ (b+ i , s) ⇒ ν−i (bi , t) = ν−i (bi , s). (15) 11 To ensure the χ-IIA property, we use a slightly modified version of UCB1: log T is used instead of log t, and min is used to break ties (instead of an arbitrary rule). This change does not affect regret guarantees. We will denote this version as UCB1 without further notice. 12 Also, we will use the fact that the probabilities χ (π̂, ν) in Definition 7.3 do not depend on the round (given j j and (π̂, ν)). This is the only place in any of the proofs where we invoke this fact. 0:24 M. Babaioff et al. Let us use induction on ν∗ (bi , t). For ν∗ (bi , t) = 0 the statement is trivial. For the induction step, suppose Equation (15) holds whenever ν∗ (bi , t) = ν∗ , and let us sup′ ′ pose ν∗ (bi , t) = ν∗ (b+ i , s) = ν∗ + 1. Let t and s be the latest rounds such that + ′ ′ ′ ν∗ (bi , t ) = ν∗ (bi , s ) = ν∗ . By the induction hypothesis, ν−i (bi , t′ ) = ν−i (b+ i , s ). It re+ ′ ′ mains to prove that A(bi , t + 1) = A(bi , s + 1), i.e. that the allocation rule’s selections in round t′ + 1 given bids (b−i , bi ), and in round s′ + 1 given bids (b−i , b+ i ), are the same.13 By Definition 7.3 these selections are uniquely determined (given the stackrealization) by the bids and the impression counts ν. By the choice of t′ and s′ , neither of the two selections is i, so by the χ-IIA condition in Definition 7.3 the selections are uniquely determined by b−i and ν−i , and hence are the same. This proves Equation (15). Now, to prove Claim (13) it suffices to show that for all t νi (bi , t) ≤ νi (b+ i , t). (16) Let us use induction on t. The claim is trivial for t = 1, since the impression of agent i in round 1 does not depend on (b; σ). For the induction step, assume that the assertion Equation (16) holds for some t, and let us prove it for t + 1. Note that (using the notation from Definition 7.3) νi (bi , t + 1) = νi (bi , t) + χi (π̂(bi , t); ν(bi , t)). Now, νi (bi , t) ≤ νi (b+ i , t) by induction hypothesis. If the inequality is strict then Equation (16) trivially holds for t+1. Now suppose νi (bi , t) = νi (b+ i , t). Then by Equation (15) we have ν(bi , t) = ν(b+ , t). Moreover, by Equation (14) we have π(bi , t) = π(b+ i i , t) and + + therefore π̂−i (bi , t) = π̂−i (bi , t) and π̂i (bi , t) < π̂i (bi , t). Thus, by the χ-monotonicity property in Definition 7.3 we have + χi (π̂(bi , t); ν(bi , t)) ≤ χi (π̂(b+ i , t); ν(bi , t)). This concludes the proof of Equation (16), and that Claim (13). 7.3. Truthfulness and regret In this subsection we focus on the stochastic MAB setting, and consider the trade-off between regret and various notions of truthfulness. Ideally, one would like an MAB mechanism to be truthful in the strongest possible sense (universally ex-post), and have the same regret bounds as optimal MAB algorithms. Let us start with some background. In Babaioff, Sharma and Slivkins [Babaioff et al. 2014] it was proved that any deterministic mechanism that is ex-post truthful and ex-post normalized (under very mild restrictions), and any distribution over such deterministic mechanisms, incurs much higher regret than an optimal MAB algorithm such as UCB1. Namely, the lower bound in [Babaioff √ et al. 2014] states that R(T ) = Ω(n1/3 T 2/3 ), whereas UCB1 has regret R(T ) = O( nT log T ). 14 For δ-gap instances the difference is even more pronounced: the analysis in [Babaioff et al. 2014] provides a polynomial lower bound of Rδ (T ) = Ω(δ T λ ) for some λ > 0, whereas UCB1 achieves logarithmic regret Rδ (T ) = O( nδ log T ). Our first result is that we can use the machinery from Section 7.2 to match the regret of UCB1 for truthful mechanisms. We apply Theorem 7.1 (with µ = T1 ) and Lemma 7.5 to UCB1 to obtain the following corollary: 13 Then ν (b , t) = ν (b+ , s) because in all rounds from t′ + 2 to t (resp., from s′ + 2 to s) agent i is played. −i i −i i 14 Following the literature on regret minimization, we are mainly interested in the asymptotic behavior of R(T ) as a function of T when n is fixed. Truthful Mechanisms with Implicit Payment Computation 0:25 C OROLLARY 7.6. In the stochastic MAB mechanism design problem, there exists a mechanism M such that (a) M is stochastically truthful, ex-post no-positive-transfers, universally ex-post individually rational. √ (b) M has regret R(T ) = O( nT log T ) and δ-gap regret Rδ (T ) = O( nδ log T ). Remark 7.7. The √ regret and δ-gap regret in the above theorem are within small factors (resp., O( log T ) and O(1)) of the best possible for any MAB allocation rule. Remark 7.8. [Babaioff et al. 2014] provides a weaker result which transforms any monotone MAB algorithm such as UCB1 into a truthful and normalized MAB mechanism with matching regret bounds. The guarantees in [Babaioff et al. 2014] are weaker for the following reasons. First, it only applies to 0-1 click rewards, whereas our setting allows for arbitrary click rewards in [0, 1]. Second, the individual rationality guarantee in [Babaioff et al. 2014] is much weaker: an agent may be charged more than her bid (which never happens in our mechanism), and the charge may be huge, as high as bi × (4n)T ; thus, a risk-averse agent may be reluctant to participate. Third, the nopositive-transfers guarantee is weaker: for some realizations of the click rewards the expected payment may be negative. Finally, the payment rule in [Babaioff et al. 2014] requires (as stated) a prohibitively expensive computation. The truthfulness in Corollary 7.6 is only in expectation over the random click rewards. Thus, after seeing a specific realization of the rewards an agent might regret having been truthful. Accordingly, we would like a stronger property: ex-post truthfulness, i.e. truthfulness for every given realization of the rewards. The main result of this section is an ex-post truthful MAB mechanism with optimal regret bounds. Unlike Corollary 7.6, this result requires designing a new MAB allocation rule.15 This allocation rule and its analysis are the main technical contributions. T HEOREM 7.9. In the stochastic MAB mechanism design problem, there is a mechanism M such that (a) M is ex-post truthful, ex-post no-positive-transfers, and universally ex-post individually rational. √ (b) M has regret R(T ) = O( nT log T ) and δ-gap regret Rδ (T ) = O( nδ log T ). The theorem follows from Theorem 7.1 (with µ = T1 ) if there exists an MAB allocation rule that is ex-post monotone and has the claimed regret bounds. Below we provide such allocation rule, called NewCB. L EMMA 7.10. NewCB is ex-post monotone and satisfies the regret bounds in Theorem 7.9(b). Remark 7.11. NewCB is deterministic. While not essential for Theorem 7.9, this fact confirms the intuition from [Babaioff et al. 2014] that the main obstacle for deterministic ex-post truthful MAB mechanisms is insufficient observable information to compute payments rather than ex-post monotonicity of an allocation rule. NewCB maintains a set of active agents; initially all agents are active. For each round t, there is a designated agent i = 1 + (t mod n). If this agent is active, then it is allo15 In particular, the allocation rule induced by UCB1 is not ex-post monotone and thus cannot be used to achieve ex-post truthfulness using the results of Section 7.2. To see that, consider a simple setting with two agents and two rounds, and a click realization in which both agents are not clicked at the first round, but are clicked at the second. With this click realization, an agent might be better off decreasing his bid in order to lose (i.e., not be selected in) the first round, and then win (i.e., be selected in) the second round. 0:26 M. Babaioff et al. cated. Else, an active agent is chosen (according to some fixed ordering on the agents) and allocated. For each agent i, lower and upper confidence bounds (Li , Ui ) on the product bi µi are maintained (recall that µi is the CTR of agent i). After each round, each agent is de-activated if its upper confidence bound is smaller than someone else’s lower confidence bound. The pseudocode is in Algorithm 5. Algorithm 5: NewCB: ex-post monotone MAB allocation rule. 1: Given: n = #agents, T = #rounds, upper bound bmax . 2: Solicit a bid vector b from the agents; b ← b/bmax . 3: Initialize: set of active agents Sact = {all agents}. 4: for all agent i do 5: ci ← 0; ni ← 0 {total click reward and #impressions} 6: {the totals are only over “designated” rounds} 7: Ui ← bi ; Li ← 0 {Upper and Lower Confidence Bounds} 8: {Main Loop} 9: for rounds t = 1, 2, . . . , T do 10: i ← 1 + (t mod n). {The “designated” agent} 11: if i ∈ Sact then 12: Allocate agent i. 13: ni ← ni + 1; ci ← ci + reward. {Update statistics.} 14: {Update confidence bounds.} 15: if Li < Ui then p 16: (L′i , Ui′ ) ← bi (ci /ni ∓ 8 log(T )/ni )). 17: if max(Li , L′i ) < min(Ui , Ui′ ) then 18: (Li , Ui ) ← (max(Li , L′i ), min(Ui , Ui′ )). 19: else i i 20: (Li , Ui ) ← ( Li +U , Li +U ). 2 2 21: else 22: Allocate agent i = min Sact . 23: for all agent i ∈ Sact do 24: if Ui < maxj∈Sact Lj then 25: Remove i from Sact . Fix realization ρ and bid vector b. Let Sact (t, b) be the set of active agents in the beginning of round t. For each agent i, let Li (t, b) and Ui (t, b) be the values of Li and Ui in the end of round t. The goal of the specific update rules for the confidence bounds (lines 15-20) and the statistics (lines 13) is to guarantee the following two properties: — the statistics are kept only for rounds when a designated agent is played. Moreover, for each agent i and round t, and any two bid vectors b and b′ we have Li (t, b)/bi = Li (t, b′ )/b′i ′ if i ∈ Sact (t, b) ∩ Sact (t, b ) then . (17) Ui (t, b)/bi = Ui (t, b′ )/b′i — for any fixed realization ρ and bid vector b, and each agent i: Li ≤ Ui , and from round to round Li is non-decreasing and Ui is non-increasing. In other words, for each round t it holds that Li (t − 1, b) ≤ Li (t, b) ≤ Ui (t, b) ≤ Ui (t − 1, b). (18) Truthful Mechanisms with Implicit Payment Computation 0:27 The ex-post monotonicity follows from these two properties and the de-activation rule (lines 24-25). Ex-post monotonicity. Let L∗ (t, b) , maxi∈Sact (t,b) Li (t, b). Fix agent i and b+ i > bi , + and let b+ = (b−i , b+ ) be the “alternative” bid vector. Let λ = b /b . i i i C LAIM 7.12. We establish the following sequence of claims: (C1) L∗ (t, b) is non-decreasing in t, for any fixed b. (C2) For each round t, L∗ (t, b) ≤ λ L∗ (t, b+ ). (C3) For each round t, Sact (t, b+ ) \ {i} ⊂ Sact (t, b) \ {i}. (C4) In each round t: if i ∈ Sact (t, b) then i ∈ Sact (t, b+ ). P ROOF. Let us prove the parts (C1-C4) one by one. (C1). We use property (18) and the de-activation rule. Throughout the proof, we omit the bid vector b from the notation. Fix round t ≥ 2. Let i ∈ Sact (t − 1) be an agent such that L∗ (t − 1) = Li (t − 1). If i ∈ Sact (t) then L∗ (t − 1) = Li (t − 1) ≤ Li (t) ≤ L∗ (t) Else i is de-activated in round t, so L∗ (t − 1) = Li (t − 1) ≤ Li (t) ≤ Ui (t) < L∗ (t). (C2). Suppose, for the sake of contradiction, that L∗ (t, b) > λ L∗ (t, b+ ). Let j ∈ Sact (t, b) be an agent such that Lj (t, b) = L∗ (t, b). If j ∈ Sact (t, b+ ) then by property (17) we have L∗ (t, b) = Lj (t, b) = λ Lj (t, b∗ ) ≤ λ L∗ (t, b+ ), contradiction. We conclude that j 6∈ Sact (t, b+ ). Thus with bid vector b+ agent j gets disqualified during some round s < t. Thus, Uj (s, b+ ) < L∗ (s, b+ ) ≤ L∗ (t, b+ ), (19) where the second inequality is by Part (C1). Now using property (18) and property (17) (for the right-most inequality), we get that L∗ (t, b) = Lj (t, b) ≤ Uj (t, b) ≤ Uj (s, b) = λ Uj (s, b+ ). Thus, L∗ (t, b) ≤ λ L∗ (t, b+ ) by Equation (19), the desired contradiction. (C3). Use induction on t. The claim trivially holds for t = 1. Assuming the claim holds for some t we prove it holds for t + 1. Fix agent j ∈ Sact (t + 1, b+ ) \ {i}. We need to prove that j ∈ Sact (t + 1, b). Note that j ∈ Sact (t, b+ ), and so j ∈ Sact (t, b) by the induction hypothesis. Therefore L∗ (t, b) ≤ λ L∗ (t, b+ ) ≤ λ Uj (t, b+ ) = Uj (t, b) (by Part (C2)) (by the de-activation rule) (by property (17)) So agent j is not deactivated in round t under bid vector b, i.e. j ∈ Sact (t + 1, b), completing the proof. (C4). Use induction on t. The base case t = 0 holds because initially all agents are active. For the induction step, assume that the statement holds for some round t ≥ 0. Suppose i ∈ Sact (t + 1, b). We need to prove that i ∈ Sact (t + 1, b+ ). Note that i ∈ Sact (t, b), and so i ∈ Sact (t, b+ ) by the induction hypothesis. Therefore L∗ (t, b) ≤ Ui (t, b) (by the de-activation rule) + = λ Ui (t, b ) (by property (17)). 0:28 M. Babaioff et al. If L∗ (t, b) = λ L∗ (t, b+ ), then L∗ (t, b+ ) = L∗ (t, b)/λ ≤ Ui (t, b+ ), and we are done. From here on, assume L∗ (t, b) 6= λ L∗ (t, b+ ). Then L∗ (t, b) < λ L∗ (t, b+ ) by Part (C2). Pick agent j which maximizes Lj (t, b+ ). If j 6= i then j ∈ Sact (t, b) by Part (C3), so L∗ (t, b) ≥ Lj (t, b) = λ Lj (t, b+ ) ∗ + = λ L (t, b ) by property (17)) (by the choice of j), contradicting our assumption. Then j = i, and so L∗ (t, b+ ) = Li (t, b+ ) ≤ Ui (t, b+ ). Ex-post monotonicity follows easily from (C3-C4). C LAIM 7.13. Consider a fixed round t. Suppose agent i is allocated with bid vector b. Then it is also allocated with bid vector b+ . P ROOF. Since i ∈ Sact (t, b), by (C4) we have i ∈ Sact (t, b+ ). If agent i is the designated agent in round t, then as such it is allocated under both bid vectors. If agent i is not the designated agent in round t, then i = min Sact (t, b). By (C3) it holds that Sact (t, b+ ) ⊂ Sact (t, b), which implies that i = min Sact (t, b+ ). So i is allocated under bid vector b+ , too. Regret analysis. The regret analysis is relatively standard, following the ideas in [Auer et al. 2002a]. For simplicity assume that bmax = 1. Fix a bid vector b. For each agent i, let ci (t) and ni (t) be, respectively, the number of clicks and impressions p in all rounds s ≤ t when it is allocated as the designated agent. Let ri (t) = 8 log(T )/ni (t). Then the event \|µi − ci (t)/ni (t)\| ≤ ri (t) for all rounds t (20) holds with probability at least 1 − T −2. 16 In what follows, let us assume that this event holds for all agents i. (The regret accumulated if this event fails is negligible.) Then it easily follows from the specs of NewCB that for each agent i, Li (t, b) ≤ bi µi ≤ Ui (t, b) Ui (t, b) − Li (t, b) ≤ 2 ri (t) Let i∗ ∈ argmaxi bi µi be a best agent. Note that Ui∗ (t, b) ≥ bi∗ µi∗ ≥ bi µi ≥ Li (b, t) for all agents i and rounds t. It follows that i∗ is never de-activated by the algorithm. Consider some agent i with ∆i , bi∗ µi∗ − bi µi > 0. Then ri (t) < ∆i after O(∆−2 i log T ) rounds in which this agent is allocated as the designated agent. After such round t, Ui (b, t) ≤ bi µi + ri (t) < bi µi + ∆i = bi∗ µi∗ ≤ Li∗ (b, t), and therefore agent i is deactivated. It follows that agent i is allocated as the designated agent at most O(∆−2 i log T ) times. Therefore it is de-activated after at most O(k ∆−2 log T ) rounds. This, in turn, implies the claimed regret bound. i 16 This follows from Azuma-Hoeffding inequality via a standard argument, one version of which we provide below. Fix agent i. For each s ∈ N , let Xs be the click bit for the s-th time this agent is allocated as the designated agent, if s ≤ ni (T ), and otherwise define Xs to be an independent 0-1 random variable with expectation µi . Then the random variables Ys = Xs − µs , s ∈ N form a martingale. Applying p AzumaP 8N log(T ) Hoeffding inequality to Y1 , . . . , YN , for any given N , we obtain that the event \| N s=1 Ys \| ≤ holds with probability at least 1 − T −3 . Taking the Union Bounds over all N ≤ T , and noting that ci (t) = Pni (t) −2 . s=1 Xs , it follows that the event (20) holds with probability at least 1 − T Truthful Mechanisms with Implicit Payment Computation 0:29 7.4. The power of randomization A by-product of Theorem 7.9 is a separation between the power of deterministic and randomized mechanisms, in terms of regret for MAB mechanisms that are ex-post truthful and ex-post normalized. The lower bound for deterministic mechanisms is from [Babaioff et al. 2014]. One challenge here is to ensure that the upper and lower bounds talk about exactly the same problem; as stated, Theorem 7.9 and the main lower bound result from [Babaioff et al. 2014] do not. To bypass this problem, we focus on the case of two agents, and use a more general version of the lower bound: Theorem C.1 in the full version of [Babaioff et al. 2014]. Further, to match [Babaioff et al. 2014] we extend the mechanism from Theorem 7.9 to a setting in which bmax is not known a priori. We formulate the separation theorem as follows. Denote R(T, bmax ) , max R(T ; b; µ), where the maximum is taken over all CTR vectors µ and all bid vectors b such that bi ≤ bmax for all i. T HEOREM 7.14. Consider the stochastic MAB mechanism design problem with two agents. Assume bmax is not known a priori to the mechanism. Suppose M is an MAB mechanism that is (i) ex-post truthful and ex-post normalized, and (ii) has regret R(T, bmax ) = Õ(bmax T γ ) for some γ and any bmax . Then: (a) [Babaioff et al. 2014] If M is deterministic then γ ≥ 23 . (b) There exists such randomized M with γ = 12 . P ROOF OF PART ( B ). Let A′ be the ex-post monotone MAB allocation rule in Theorem 7.9, for bmax = 1. Define an MAB allocation rule A as a rule that inputs the bid vector b and passes the modified bid vector b′ = b/(maxi bi ) to A′ . We claim that A is expost monotone, too. Indeed, w.l.o.g. assume b1 > b2 . If b2 increases (to a value ≤ b1 ), then b′2 increases while b′1 stays the same. Thus, the total click reward of agent 2 increases. If b1 increases then b′2 decreases while b′1 stays the same, so the total click reward of agent 2 does not increase, which implies that the total click reward of agent 1 does not decrease. Claim proved. Now part (b) follows from Theorem 7.1 (with µ = T1 ). 8. EXTENSION TO MULTI-PARAMETER DOMAINS Our general transformation from Section 4 can be extended to multi-parameter mechanisms.It is known that a multi-parameter allocation rule is truthfully implementable if and only if it satisfies a property called “cycle-monotonicity”. Similar to the singleparameter case, we present a general procedure to take any cycle-monotone allocation rule A and transform it into a randomized mechanism that is truthful-in-expectation, implements the same outcome as A with probability arbitrarily close to 1, and requires evaluating that allocation rule only once. The technical contribution here is that we find a reduction from the multi-parameter setting to the single-parameter case. This section is self-contained. For more background on multi-parameter mechanisms for a CS-oriented audience, please refer to [Archer and Kleinberg 2008a,b]. An Economics-oriented background for this area can be found in [Ashlagi et al. 2010]. 8.1. Preliminaries: multi-parameter domains Generalized types. In the full generality, multi-parameter mechanisms are defined as follows. There are n agents and a set O of outcomes. Each agent i is characterized by his type xi : O → R, where xi (o) is interpreted as the agent’s valuation for the outcome o ∈ O. For each agent i there is a set of feasible types, denoted Ti . Denote 0:30 M. Babaioff et al. T = T1 × . . . × Tn and call it the type space; call Ti the type space of agent i. The mechanism knows (n, O, T ), but not the actual types xi ; each type xi is known only to the corresponding agent i. Formally, a problem instance, also called a multi-parameter domain, is a tuple (n, O, T ). Using this general notion of types, we define truthful mechanisms in essentially the same way as in Section 3, with minimal syntactic changes. A (direct revelation) mechanism M consists of the pair (A, P), where A : T → O is the allocation rule and P : T → Rn is the payment rule. Both A and P can be randomized. Each agent i reports a type bi ∈ Ti to the mechanism, which is called the bid of this agent. We denote the vector of bids by b = (b1 , . . . , bn ) ∈ T . The mechanism receives the bid vector b ∈ T , selects an outcome A(b), and charges each agent i a payment of Pi (b). The utilities are quasi-linear and agents are risk-neutral: if agent i has type xi ∈ Ti and the bid vector is b ∈ T , then this agent’s utility is ui (xi ; b) = E [xi (A(b)) − Pi (b)] . M (21) For each type xi ∈ Ti of agent i we use a standard notation (b−i , xi ) to denote the bid vector b̂ such that b̂i = xi and b̂j = bj for every agent j 6= i. Special case: dot-product valuations. For intuition, consider dot-product valuations, an important special case where the type x ∈ Ti of each agent i can be decomposed as a dot product x(o) = βx · ai (o), for each outcome o ∈ O, where βx , ai (o) ∈ Rd are some finite-dimensional vectors. Here the term ai (o) is the same for all types x ∈ Ti (and known to the mechanism), whereas βx is the same for all outcomes o ∈ O and is known only to agent i. The term ai (o) is usually called an “allocation” of agent i for outcome o, and βx is called the “private value”. The single-parameter domains defined in Section 3 correspond to the case d = 1. Note that the type x of each agent i is determined by the corresponding private value βx , and his type space Ti is determined by Di = {βx : x ∈ Ti } ⊂ Rd . Because of this, in the literature on dot-product valuations the term “type” often refers to βx . To avoid ambiguity, in this section we will refer to βx as “private value” rather than “type”, and call D1 × . . . × Dn the private value space. Game-theoretic properties. Truthfulness and individual rationality are defined exactly as in Section 3 if expressed in terms of the agents’ utility: — A mechanism is truthful if for every agent i truthful bidding is a dominant strategy: ui (xi ; (b−i , xi )) ≥ ui (xi ; b) ∀xi ∈ Ti , b ∈ T . (22) An allocation rule is called truthfully implementable if it is the allocation rule in some truthful mechanism. — A mechanism is individually rational (IR) if each agent i never receives negative utility by participating in the mechanism and bidding truthfully: ui (xi ; (b−i , xi )) ≥ 0 ∀xi ∈ Ti , b−i ∈ T−i . (23) The right-hand side in (23) represents the maximal guaranteed utility of an “outside option” (i.e., from not participating in the mechanism). For example, our definition of IR is meaningful whenever this utility is 0, which is a typical assumption for most multi-parameter domains studied in the literature. Our assumptions. We make two assumptions on the type space T : — non-negative types: xi (o) ≥ 0 for each agent i, each type xi ∈ Ti , and each outcome o ∈ O. Truthful Mechanisms with Implicit Payment Computation 0:31 — rescalable types: λxi ∈ Ti for each agent i, each type xi ∈ Ti , and any parameter λ ∈ [0, 1]. For dot-product valuations, types are rescalable if and only if it holds that βx ∈ Di ⇒ λβx ∈ Di for each λ ∈ [0, 1]. Thus, assuming rescalable types is equivalent to assuming that the set Di is star-convex at 0. To ensure non-negative types, it suffices to assume that Di ⊂ Rd+ for each agent i, and all allocations are non-negative: ai (o) ∈ Rd+ for all o ∈ O. In particular, for each agent i there exists a zero type: a type xi ∈ Ti such that xi (·) ≡ 0. Let us say that a mechanism is normalized if for each agent i, the expected payment of this agent is 0 whenever she submits the zero type. Truthfulness characterization. We will use the following characterization of truthful mechanisms. A (randomized) allocation rule A is cycle-monotone if the following property holds: for each bid vector b ∈ T , each agent i, each k ≥ 2, and each k-tuple xi,0 , xi,1 , . . . , xi,k ∈ Ti of this agent’s types, we have   k X xi,j (oi,j ) − xi, (j−1) mod k (oi,j ) ≥ 0, where oi,j = A (b−i , xi,j ) ∈ O. E (24) A j=0 T HEOREM 8.1 (R OCHET [1987]). Consider an arbitrary multi-parameter domain (n, O, T ). A (randomized) allocation rule A is truthfully implementable if and only if it is cycle-monotone. Assuming rescalable types, for any cycle-monotone allocation rule A, a mechanism (A, P) is truthful and normalized if and only if Z 1 E [Pi (b)] = E bi (A(b)) − bi (A(b−i , t bi )) dt . (25) A A t=0 Note that this theorem generalizes Theorem 3.2 for single-parameter mechanisms, as applied to single-parameter domains with private value space [0, 1]n . In particular, Equation (25) generalizes the Myerson payment rule for single-parameter mechanisms. 8.2. The multi-parameter transformation Consider allocation rule A, bid vector b ∈ T , and the rescaling vector λ ∈ [0, 1]n . Denote λ ⊗ b = (λ1 b1 , . . . , λn bn ) ∈ T . In other words, λ ⊗ b is the “rescaled” bid vector where the bid of each agent i is λi bi ; this bid vector is well-defined because we assumed the rescalable types property. Note that for each b the subset Tb = {λ ⊗ b : λ ∈ [0, 1]n } ⊂ T forms a single-parameter type space where each agent i has private value λi ∈ [0, 1] and allocation bi (o) for every outcome o. By abuse of notation, let us treat the allocation / payment rules for Tb as functions from the private value space [0, 1]n rather than the type space Tb . Consider an allocation rule Ab (λ) = A(λ ⊗ b) for the single-parameter type space Tb . If the original allocation rule A is truthfully implementable for type space T using payment rule P, then Ab is truthfully implementable for type space Tb using payment rule Pb (λ) = P(λ ⊗ b), because restricting the allocation and payment rules to Tb only limits the set of possible misreports of an agent. Essentially, the idea will be to apply our single-parameter transformation to Ab . 0:32 M. Babaioff et al. Let fµ = (f1 , . . . , fn ) be the n-tuple of canonical self-resampling procedures with resampling probability µ, for some fixed µ ∈ (0, 1). (See Algorithm 1 on page 13.) For each bid vector b ∈ T , let eb ) = AllocToMech(Ab , µ, fµ ) (Aeb , P (26) be the single-parameter mechanism for type space Tb obtained by applying our singleparameter transformation from Section 4 to allocation Ab .17 The transformed multi-parameter mechanism is defined as e e eb (~1 ) for every b ∈ T . A(b), P(b) = Aeb (~1 ), P (27) This completes the description of our multi-parameter transformation. The useful properties of this transformation are captured in the theorem below. T HEOREM 8.2. Consider an arbitrary multi-parameter domain (n, O, T ) with rescalable, non-negative types. Let A be a cycle-monotone allocation rule. Let Mµ = e P) e be the transformed mechanism defined by Equations(26-27), for some parameter (A, µ ∈ (0, 1). Then Mµ has the following properties: (a) Mµ is truthful and normalized. (b) Mµ is universally ex-post individually rational and ex-post no-positive-transfers. Moreover, given a bid vector b, it never pays any agent i more than bi (o)( µ1 − 1), where o = A(b) ∈ O. e (c) For any bid vector b ∈ T (and any fixed random seed of nature) allocations A(b) and A(b) are identical with probability at least 1 − nµ. 2 -approximate. (d) If A is α-approximate (for social welfare) then Ae is α/ 1 − 1−µ P ROOF. Parts (b) and (c) follow immediately from Theorem 4.5, and part (d) follows immediately from Theorem 5.1. Thus, it remains to prove part (a). Note that the single-parameter allocation rule Aeb has the following property: for each agent i the single-parameter bid λi is rescaled by the (randomly chosen) factor χi ∈ [0, 1] which does not depend on the bid, and then Ab is called. Therefore, letting χ = (χ1 , . . . , χn ), it holds that Aeb (λ) = A(χ ⊗ (λ ⊗ b)) for all b ∈ T and λ ∈ [0, 1]n . (28) We claim that Ae is cycle-monotone. Indeed, fix bid vector b ∈ T , agent i, some k ≥ 2, and a k-tuple xi,0 , xi,1 , . . . , xi,k ∈ Ti of this agent’s types. Let us consider a fixed realization of the random vector χ ∈ [0, 1]n . For each type xi,j , note that (by Equations (28) and (27)) we have e i,j , b−i ) = Ae(x ,b ) ( ~1 ) = A (χ ⊗ (b−i , xi,j )) ∈ O. A(x i,j −i Denote this outcome by oi,j (χ). Apply the cycle-monotonicity of A for bid vector χ ⊗ (xi,j , b−i ):   k X xi,j (oi,j (χ)) − xi, (j−1) mod k (oi,j (χ)) ≥ 0. E (29) A j=0 17 Note that the transformed mechanism depends on µ. We do not make this dependence explicit, to simplify the notation. Truthful Mechanisms with Implicit Payment Computation 0:33 e i,j , b−i ), we observe that for this fixed realization of Recalling that oi,j (χ) = A(x χ, Equation (29) is exactly the inequality in the definition of cycle-monotonicity for e Therefore taking expectation over χ, we obtain the desired inequality (24) for A. e A. Claim proved.18 e P), e the payment rule satIt remains to prove that in the transformed mechanism (A, isfies Equation (25). Fix bid vector b and consider the transformed single-parameter eb ) for the single-parameter type space Tb . In the terminology of mechanism (Aeb , P single-parameter domains, each agent i receives an allocation Aeb, i (λ) = bi (Aeb (λ)) whenever the bid vector is λ ∈ [0, 1]n . Since this is a truthful and normalized singleparameter mechanism, it follows that # " Z λi h i e e e E Pb (λ) = E λi Ab, i (λ) − Ab, i (λ−i , u) du , ∀λ ∈ [0, 1]n . 0 Plugging in λ = ~1 and Equation (27), we obtain the desired Equation (25). 9. OPEN QUESTIONS This paper gives rise to a number of open questions. As discussed in Section 2.1, some of these questions have been partially addressed in the follow-up work. Here we present the current status. Variance vs. expectation tradeoff. Randomized mechanisms constructed via our general transformation exhibit an explicit tradeoff between the variance in payments and the loss in expected welfare compared to the optimal allocation rule. Since the variance in payments can be very high, it is desirable to optimize this tradeoff (to complement the expectation-only guarantees). The worst-case optimality result in Wilkens and Sivan [2012], discussed in Section 2.1, does not resolve this question, since it does not rule out a reduction which achieves a better tradeoff for some (but not all) monotone allocation rules. Further, the optimal tradeoff for a given domain could be achieved by a mechanism that cannot be presented as a reduction from some welfare-optimal allocation rule. Our informal conjecture is that the tradeoff in this paper is optimal for any given single-parameter domain with “informational obstacle”, i.e. whenever payment computation for welfare-optimal allocation rule is impossible due to the insufficient observable information. A specific formal conjecture is that our tradeoff is optimal for MAB mechanisms. To take an extreme version, what welfare loss can be achieved if no rebates (i.e., no positive transfers) are allowed? The power of randomization. We have a separation result for randomized vs. deterministic ex-post truthful MAB mechanisms. Can one obtain similar separation results for other single-parameter domains? The positive side for any such hypothetical separation result is provided by our general reduction, so it remains to produce the corresponding negative result for deterministic mechanisms. However, such negative results are not likely to be easy, considering the difficulties faced by [Babaioff et al. 2014; Devanur and Kakade 2009] for MAB mechanisms. One specific target would be the router scheduling problem proposed in [Shnayder et al. 2012]. 18 Note that the proof of cycle-monotonicity of A e did not use any other property of the canonical selfresampling procedures fµ other than Equation (28). The truthfulness properties of fµ are used in the forthcoming argument about payments. 0:34 M. Babaioff et al. MAB allocation rules. This paper opens up the problem of designing monotone MAB allocation rules, which is a new angle in the rich literature on MAB (also see [Slivkins 2011b]). While we have focused on stochastic MAB, many other MAB settings have been studied in the literature, making various assumptions on payoff evolution over time (e.g., [Auer et al. 2002b; Slivkins and Upfal 2008; Hazan and Kale 2009]), dependencies between arms (e.g., [Flaxman et al. 2005; Pandey et al. 2007; Kleinberg et al. 2008b; Srinivas et al. 2010]), and side information available to the algorithm (e.g., [Kleinberg et al. 2008b; Langford and Zhang 2007; Slivkins 2011a]). For most such settings one could meaningfully define the corresponding mechanism design problem; we have reduced this problem to that of designing monotone MAB allocation rules. In particular, for any given MAB setting one could ask whether monotone MAB allocation rules can achieve optimal regret. One appealing target here is the adversarial MAB setting (with oblivious adversary). The ex-post truthful mechanism in [Babaioff et al. 2014] achieves regret 1/3 2/3 Õ(k √ T ) for this setting, whereas the best known MAB algorithms achieve regret O( kT ) [Auer et al. 2002b; Audibert and Bubeck 2010]; it is not clear what is the tight regret bound. More applications. In addition to the applications presented in this paper and the follow-up work, what other domains can our general reduction (and the multiparameter extension thereof) be fruitfully applied to? In particular, one could consider two generalizations of MAB mechanisms: to multiple ads per agent and to multiple ad slots with slot-dependent values-per-click. APPENDIX: O NE S HOT is equivalent to Algorithm 1 (proof of Proposition 4.8) Let us compare the sampling procedures defined by Algorithm 1 and O NE S HOT . To simplify the notation, we will omit the subscript i from the description of the procedures. That is, a self-resampling procedure inputs a scalar bid b and a random seed w, and outputs two numbers (x, y). To prove that O NE S HOT is equivalent to Algorithm 1, we analyze a family of sampling rules that uses bounded-depth recursion to “interpolate” between O NE S HOT and Algorithm 1. Specifically, define BDRk to be the following family of sampling algorithms parameterized by k ∈ N∪{∞}, where k−1 is interpreted as ∞ when k = ∞. Algorithm 6: The sampling algorithm BDRk : Bounded Depth Recursion. 1: Input: bid b ∈ [0, ∞], parameter µ ∈ (0, 1). 2: Output: (x, y) such that 0 ≤ x ≤ y ≤ b. 3: with probability 1 − µ 4: x ← b, y ← b. 5: else 6: if k = 0 7: Pick γ1 , γ2 ∈ [0, 1] indep., uniformly at random. 1/(1−µ) 1/(1−µ) 1/µ 8: x ← b · γ1 , y ← b · max{γ1 , γ2 }. 9: else k>0 10: Pick b′ ∈ [0, b] uniformly at random. 11: (x′ , y ′ ) = BDRk−1 (b′ , µ). 12: x ← x′ , y ← b′ . Truthful Mechanisms with Implicit Payment Computation 0:35 The reader may easily verify that BDRk is equal to O NE S HOT when k = 0 and that it is equal to Algorithm 1 when k = ∞. Furthermore, for any k < k ′ (where k ′ ≤ ∞) there is an obvious coupling of BDRk with BDRk′ such that the two algorithms have probability at most µk+1 of outputting different results: simply let the two executions share the same randomness until the recursion depth equals k. Thus, as k → ∞, the output distribution of BDRk converges, in total variation distance, to that of BDR∞ . We will prove that for every finite k the algorithms BDRk and BDRk+1 have identical output distributions, from which it follows that their output distribution is identical to that of BDR0 and, therefore, that BDR∞ also has the same output distribution as BDR0 , confirming Proposition 4.8. Couple BDRk and BDRk+1 so that they use shared randomness until the two algorithms reach differing points in their control flow. This occurs when the first algorithm is executing a call to BDR0 and the second algorithm is executing a call to BDR1 on the same input (β, µ). (We are denoting the input in this step of the recursive algorithms by (β, µ) rather than (b, µ), to distinguish β from the value of b on which the two algorithms BDRk , BDRk+1 were originally called.) At this point, with probability 1 − µ both algorithms output (x, y) = (β, β). Conditional on this event not taking place, 1/q 1/p 1/p BDR0 outputs (x, y) = (γ1 β, max{γ1 , γ2 }β) where p = 1 − µ, q = µ. Instead BDR 1 ′ computes β = γ3 β where γ3 ∈ [0, 1] is uniformly random, and it outputs (x, y) = (β ′ , β ′ ) 1/p with probability p and otherwise (x, y) = (γ1 β ′ , β ′ ). Lemma A.1 tells us that these two output distributions are the same. L EMMA A.1. Let γ1 , γ2 , γ3 be mutually independent random variables, each uniformly distributed in [0, 1]. Let p, q > 0 be numbers such that p + q = 1. Define random variables x, y, z by: 1/p x = γ1 1/p 1/q y = max{γ1 , γ2 } ( γ3 if γ2 < p z= 1/p γ1 γ3 if γ2 ≥ p Then the pairs (x, y) and (z, γ3 ) are identically distributed. P ROOF. We will show, equivalently, that the pairs (y, x/y) and (γ3 , z/γ3 ) are identically distributed. The distribution of (γ3 , z/γ3) is completely characterized by the following facts which are immediate from the definition of z. (1) γ3 and z/γ3 are independent; (2) γ3 is uniformly distributed in [0, 1]; (3) z/γ3 is equal to 1 with probability p, and conditional on z/γ3 6= 1, the distribution of (z/γ3 )p is uniform on [0, 1). To finish the proof of the lemma, we shall prove the corresponding facts about y and x/y. Let I1 , I2 ⊆ [0, 1] be any pair of intervals (open, closed, or half-open). Let a, b be the endpoints of I1 and c, d the endpoints of I2 . To compute Pr(y ∈ I1 , x/y ∈ I2 ) it suffices 0:36 M. Babaioff et al. to make the following two observations: 1/p 1/q 1/p Pr(y ∈ I1 , x/y = 1) = Pr a ≤ γ1 ≤ b, 0 ≤ γ2 ≤ γ1 q/p = Pr ap ≤ γ1 ≤ bp , 0 ≤ γ2 ≤ γ1 Z bp Z bp = tq/p dt = t1/p−1 dt = p(b − a) ap ap 1/q Pr(y ∈ I1 , x/y ∈ I2 \ {1}) = Pr(a ≤ γ2 1/q ≤ b, cγ2 1/p ≤ γ1 1/q < dγ2 ) p/q p/q = Pr(aq ≤ γ2 ≤ bq , cp γ2 ≤ γ1 ≤ dp γ2 ) Z bq Z bq p p p/q p p = (d − c )t dt = (d − c ) t1/q−1 dt = q(dp − cp )(b − a). aq aq Therefore, Pr(y ∈ I1 , x/y ∈ I2 ) = (b − a) · q(dp − cp ) if 1 6∈ I2 . p p p + q(d − c ) if 1 ∈ I2 From this formula it follows that y and x/y are independent, y is uniformly distributed, Pr(x/y = 1) = p, and the distribution of (x/y)p conditional on x/y 6= 1 is uniform on [0, 1). Acknowledgments We are indebted to Tim Roughgarden for suggesting that for positive types, an improved bound on the social welfare is possible (see Section 5). 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arXiv:1710.05209v2 [cs.LG] 16 Feb 2018 Settling the Sample Complexity for Learning Mixtures of Gaussians Hassan Ashtiani University of Waterloo Shai Ben-David University of Waterloo Nick Harvey University of British Columbia Abbas Mehrabian McGill University Christopher Liaw University of British Columbia Yaniv Plan University of British Columbia Abstract We prove that Θ̃(kd2 /ε2 ) samples are necessary and sufficient for learning a mixture of k Gaussians in Rd , up to error ε in total variation distance. This improves both the known upper bound and lower bound for this problem. For mixtures of axis-aligned Gaussians, we show that Õ(kd/ε2 ) samples suffice, matching a known lower bound. Moreover, these results hold in an agnostic learning setting as well. The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in Rd has an efficient sample compression. 1 Introduction Estimating distributions from observed data is a fundamental task in statistics that has been studied for over a century. This task frequently arises in applied machine learning and it is very common to assume that the distribution can be modeled using a mixture of Gaussians. Popular software packages have implemented heuristics, such as the EM algorithm, for learning a mixture of Gaussians. The theoretical machine learning community also has a rich literature on distribution learning. For example, the recent survey 1 of Diakonikolas (2016) considers learning of structured distributions, and the survey of Kalai, Moitra, and Valiant (2012) focuses on mixtures of Gaussians. This paper develops a general technique for distribution learning, then employs these techniques in the important setting of mixtures of Gaussians. The theoretical model that we adopt is density estimation: given i.i.d. samples from an unknown target distribution, find a distribution that is close to the target distribution in total variation (TV) distance. Our focus is on sample complexity bounds: using as few samples as possible to obtain a good estimate of the target distribution. For background on this model see, e.g., Devroye and Lugosi (2001, Chapter 5) and Diakonikolas (2016). Our new technique for upper bounds on the sample complexity involves a form of sample compression. If it is possible to “encode” members of a class of distributions using a carefully chosen subset of the samples, then this yields an upper bound on the sample complexity of distribution learning for that class. In particular, by constructing compression schemes for mixtures of axis-aligned Gaussians and general Gaussians, we obtain new upper bounds on the sample complexity of learning with respect to these classes, which are optimal up to logarithmic factors. The compression framework can incorporate a notion of robustness, which leads to sample complexity bounds for agnostic learning. Namely, if the target distribution is close to a mixture of Gaussians (in TV distance), our method uses few samples to find a mixture of Gaussians that is close to the target distribution (in TV distance). 1.1 Main results In this section, all learning results refer to the problem of producing a distribution within total variation distance ε from the target distribution. Our first main result is an upper bound for learning mixtures of multivariate Gaussians. This bound is tight up to logarithmic factors. Theorem 1.1. The class of k-mixtures of d-dimensional Gaussians can be learned using e 2 /ε2 ) samples. This result generalizes to the agnostic setting. O(kd Previously, the best upper bounds on the sample complexity of this problem were e O(kd2 /ε4 ), due to Ashtiani, Ben-David, and Mehrabian (2017), and O(k 4 d4 /ε2 ), based on a VC-dimension bound discussed later. For the case of a single Gaussian (i.e., k=1), a sample complexity bound of O(d2 /ε2 ) is well known (see, e.g., Ashtiani et al. (2017, Theorem 13)). Our second main result is a lower bound matching Theorem 1.1 up to logarithmic factors. 2 Theorem 1.2. Any method for learning the class of k-mixtures of d-dimensional Gaus2 2 e sians has sample complexity Ω(kd /ε ). 2 e Previously, the best lower bound on the sample complexity was Ω(kd/ε ) (Suresh, Orlitsky, Acharya, and 2 2 e 2014). Even for a single Gaussian (i.e., k=1), an Ω(d /ε ) lower bound was not known prior to this work. Our third main result is an upper bound for learning mixtures of axis-aligned Gaussians, i.e., Gaussians with diagonal covariance matrix. This bound is tight up to logarithmic factors. Theorem 1.3. The class of k-mixtures of axis-aligned d-dimensional Gaussians can be 2 e learned using O(kd/ε ) samples. This result generalizes to the agnostic setting. 2 e A matching lower bound of Ω(kd/ε ) samples was shown by Suresh et al. (2014). 4 e Previously, the best known upper bounds were O(kd/ε ), due to Ashtiani et al. (2017), 4 2 3 3 2 and O((k d + k d )/ε ), based on a VC-dimension discussed later. Computational efficiency. Although our approach for proving sample complexity upper bounds is algorithmic, our focus is not on computational efficiency. The resulting algorithms are efficient in terms of sample complexity, but their runtime is exponential in the dimension d and the number of mixture components k. The existence of a polynomial time algorithm for density estimation is unknown even for the class of mixtures of axis-aligned Gaussians (Diakonikolas, Kane, and Stewart, 2017a, Question 1.1). Even for the case of a single Gaussian, the published proofs of the O(d2/ε2 ) bound are not algorithmically efficient. Using ideas from our proof of Theorem 1.1, we show in Appendix B that an algorithmically efficient proof for the single Gaussian case can be obtained simply by computing the empirical mean and covariance matrix of O(d2 /ε2 ) samples. 1.2 Related work Distribution learning is a vast topic and many approaches have been considered in the literature. We briefly review approaches that are most relevant to our problem. For parametric families of distributions, a common approach is to use the samples to estimate the parameters of the distribution, possibly in a maximum likelihood sense, or possibly aiming to approximate the true parameters. For the specific case of mixtures of Gaussians, there is a substantial theoretical literature on algorithms that approximate the mixing weights, means and covariances. Kalai et al. (2012) gave a recent survey of this literature. The strictness of this objective cuts both ways. On the one hand, a 3 successful learner uncovers substantial structure of the target distribution. On the other hand, this objective is clearly impossible when the means and covariances are extremely close. Thus, algorithms for parameter estimation of mixtures necessarily require some assumptions on the target parameters. Also, the basic definition of parameter estimation does not immediately extend to an agnostic setting, although there is literature on agnostic parameter estimation, e.g., Lai, Rao, and Vempala (2016). Density estimation has a long history in the statistics literature, where the focus is on the sample complexity question; see, e.g., Devroye (1987); Devroye and Lugosi (2001) for general background. It was first studied in the computational learning theory community under the name PAC learning of distributions by Kearns, Mansour, Ron, Rubinfeld, Schapire, and Sellie (1994), whose focus is on the computational complexity of the learning algorithm. For density estimation there are various possible measures of distance between distributions, the most popular ones being the TV distance and the Kullback-Leibler (KL) divergence. Here we focus on the TV distance since it has several appealing properties, such as being a metric and having a natural probabilistic interpretation. In contrast, KL divergence is not even symmetric and can be unbounded even for intuitively close distributions. For a detailed discussion on why TV is a natural choice, see Devroye and Lugosi (2001, Chapter 5). A popular method for distribution learning in practice is kernel density estimation (see, e.g., Devroye and Lugosi (2001, Chapter 9)). The few rigorously proven sample complexity bounds for this method require certain smoothness assumptions on the class of densities (e.g., Devroye and Lugosi (2001, Theorem 9.5)). The class of Gaussians is not universally Lipschitz and does not satisfy these assumptions, so those results do not apply to the problems we consider. Another elementary method for density estimation is using histogram estimators. Straightforward calculations show that histogram estimators for mixtures of Gaussians would result in a sample complexity that is exponential in the dimension. The same is true for estimators based on piecewise polynomials. The minimum distance estimate (Devroye and Lugosi, 2001, Section 6.8) is another approach for deriving sample complexity upper bounds for distribution learning. This approach is based on uniform convergence theory. In particular, an upper bound for any class of distributions can be achieved by bounding the VC-dimension of an associated set system, called the Yatracos class (see Devroye and Lugosi (2001, page 58) for the definition). For example, Diakonikolas, Kane, and Stewart (2017b) used this approach to bound the sample complexity of learning high-dimensional log-concave distributions. However, for mixtures of Gaussians and axis-aligned Gaussians in Rd , the best known VC-dimension bound (Anthony and Bartlett, 1999, Theorem 8.14) results in loose upper bounds of O(k 4 d4 /ε2 ) and O((k 4 d2 + k 3 d3 )/ε2 ) respectively. 4 Another approach is to first approximate the mixture class using a more manageable class such as piecewise polynomials, and then study the associated Yatracos class, see, e.g., Chan, Diakonikolas, Servedio, and Sun (2014). However, piecewise polynomials do a poor job in approximating d-dimensional Gaussians, resulting in an exponential dependence on d. For density estimation of mixtures of Gaussians, the current best sample complexity 4 e 2 /ε4 ) for general Gaussians and O(kd/ε e upper bounds (in terms of k and d) are O(kd ) for axis-aligned Gaussians, both due to Ashtiani et al. (2017). For the general Gaussian e 2 /ε2 ) and partitions this sample case, their method takes an i.i.d. sample of size O(kd 2 2 e in every possible way into k subsets. Based on those partitions, k O(kd /ε ) “candidate distributions” are generated. The problem is then reduced to learning with respect to that finite class of candidates. Their sample complexity has a suboptimal factor of 1/ε4, of which 1/ε2 arises in their approach for choosing the best candidate, and another factor 1/ε2 is due to the exponent in the number of candidates. Our approach via compression schemes also ultimately reduces the problem to learning with respect to finite classes. However, our compression technique leads to a more refined bound. In the case of mixtures of Gaussians, one factor of 1/ε2 is again incurred due to learning with respect to finite classes. The key is that the number of compressed samples has no additional factor of 1/ε2, so the overall sample complexity bound has only 2 e a O(1/ε ) dependence on ε. As for lower bounds on the sample complexity, much fewer results are known for learning mixtures of Gaussians. The only lower bound of which we are aware is due to 2 e Suresh et al. (2014), who show a bound of Ω(kd/ε ) for learning mixtures of axis-aligned Gaussians (and hence for general Gaussians as well). This bound is tight for the axisaligned case, as we show in Theorem 1.3, but loose in the general case, as we show in Theorem 1.2. 1.3 Our techniques We introduce a novel method for learning distributions via a form of sample compression. Given a class of distributions, suppose there is a method for “compressing” the samples generated by any distribution in the class. Further, suppose there exists a fixed decoder for the class, such that given the compressed set of instances and a sequence of bits, it approximately recovers the original distribution. In this case, if the size of the compressed set and the number of bits is guaranteed to be small, we show that the sample complexity of learning that class is small as well. More precisely, say a class of distributions admits (τ, t, m) compression if there exists a decoder function such that upon generating m i.i.d. samples from any distribution in 5 the class, we are guaranteed, with reasonable probability, to have a subset of size at most τ of that sample, and a sequence of at most t bits, on which the decoder outputs an approximation to the original distribution. Note that τ, t, and m can be functions of ε, the accuracy parameter. This definition is generalized to a stronger notion of robust compression, where the target distribution is to be encoded using samples that are not necessarily generated from the target itself, but are generated from a distribution that is close to the target. We prove that robust compression implies agnostic learning. In particular, if a class admits (τ, t, m) robust compression, then the sample complexity of agnostic learning with respect e + (τ + t)/ε2 ) (Theorem 3.5). to this class is bounded by O(m An attractive property of robust compression is that it enjoys two closure properties. Specifically, if a base class admits robust compression, then the class of k-mixtures of that base class, as well as the class of products of the base class, are robustly compressible (Lemmas 3.6 and 3.7). Consequently, it suffices to provide a robust compression scheme for the class of single Gaussian distributions in order to obtain a compression scheme for classes of mixtures of Gaussians (and therefore, to be able to bound their sample complexity). We prove e e 2 ), O(d)) e that the class of d-dimensional Gaussian distributions admits (O(d), O(d robust e compression (Lemma 4.2). The high level idea is that by generating O(d) samples from a Gaussian, one can get some rough sketch of the geometry of the Gaussian. In particular, the convex hull of the points drawn from a Gaussian enclose an ellipsoid centered at the mean and whose principal axes are the eigenvectors of the covariance matrix. Using ideas from convex geometry and random matrix theory, we show one can in fact encode the center of the ellipsoid and the principal axes using a convex combination of these samples. Then we discretize the coefficients and obtain an approximate encoding. The above results together imply tight (up to logarithmic factors) upper bounds of 2 e e O(kd2 /ε2 ) for mixtures of k Gaussians, and O(kd/ε ) for mixtures of k axis-aligned Gausd sians over R . The robust compression framework we introduce is quite flexible, and can be used to prove sample complexity upper bounds for other distribution classes as well. Lower bound. For proving our lower bound for mixtures of Gaussians, we first prove e 2 /ε2 ) for learning a single Gaussian. Although the approach is quite a lower bound of Ω(d intuitive, the details are intricate and much care is required to make a formal proof. The 2 main step is to construct a large family (of size 2Ω(d ) ) of covariance matrices such that the associated Gaussian distributions are well-separated in terms of their total variation distance while simultaneously ensuring that their Kullback-Leibler divergences are small. Once this is established, we can then apply a generalized version of Fano’s inequality to complete the proof. 6 2 To construct this family of covariance matrices, we sample 2Ω(d ) matrices from the following probabilistic process: start with an identity covariance matrix. Then choose a random subspace of dimension d/9 and slightly increase the eigenvalues corresponding √ to this eigenspace from 1 to roughly 1 + ε/ d. It is easy to bound the KL divergence between the constructed Gaussians. To lower bound the total variation, we show that for every pair of these distributions, there is some subspace for which a vector drawn from one Gaussian will have slightly larger projection than a vector drawn from the other Gaussian. Quantifying this gap will then give us the desired lower bound on the total variation distance. 1.4 Paper outline We set up our formal framework and notations in Section 2. In Section 3, we define compression schemes for distributions, prove their closure properties, and show their connection with density estimation. Theorem 1.1 and Theorem 1.3 are proved in Section 4. Theorem 1.2 is proved in Section 5. All omitted proofs can be found in the appendix. 2 Preliminaries A distribution learning method or density estimation method is an algorithm that takes as input a sequence of i.i.d. samples generated from a distribution g, and outputs (a description of) a distribution ĝ as an estimation for g. We work with continuous distributions in this paper, and so we identify a probability distribution by its probability density function. Let f1 and f2 be two probability distributions defined over the Borel σ-algebra B. The total variation (TV) distance between f1 and f2 is defined by Z 1 TV(f1 , f2 ) := sup (f1 (x) − f2 (x))dx = kf1 − f2 k1 , 2 B∈B B R where kf k1 := Rd \|f (x)\|dx is the L1 norm of f . The Kullback-Leibler (KL) divergence between f1 and f2 is defined by Z f1 (x) dx. KL(f1 k f2 ) := f1 (x) log f2 (x) Rd In the following definitions, F is a class of probability distributions, and g is a distribution (not necessarily in F ). Definition 2.1 (ε-approximation, (ε, C)-approximation). A distribution ĝ is an ε-approximation for g if kĝ − gk1 ≤ ε. A distribution ĝ is an (ε, C)-approximation for g with respect to F if kĝ − gk1 ≤ C · inf kf − gk1 + ε f ∈F 7 Definition 2.2 (PAC-learning distributions, realizable setting). A distribution learning method is called a (realizable) PAC-learner for F with sample complexity mF (ε, δ), if for all distribution g ∈ F and all ε, δ ∈ (0, 1), given ε, δ, and a sample of size mF (ε, δ) generated i.i.d. by that g, with probability at least 1 − δ (over the samples) the method outputs an ε-approximation of g. Definition 2.3 (PAC-learning distributions, agnostic setting). For C > 0, a distribution learning method is called a C-agnostic PAC-learner for F with sample complexity mC F (ε, δ), if for all distributions g and all ε, δ ∈ (0, 1), given ε, δ, and a sample of size mC F (ε, δ) generated i.i.d. from g, with probability at least 1 − δ the method outputs an (ε, C)approximation of g w.r.t. F . We sometimes say a class can be “C-learned in the agnostic setting” to indicate the existence of a C-agnostic PAC-learner for the class. The case C > 1 is sometimes called P wi = 1 } denote the semi -agnostic learning. Let ∆n := { (w1 , . . . , wn ) : wi ≥ 0, n-dimensional simplex. Definition 2.4 (k-mix(F )). Let F be a class of probability distributions. Then the class of k-mixtures of F , written k-mix(F ), is defined as k-mix(F ) := { Pk i=1 wi fi : (w1 , . . . , wk ) ∈ ∆k , f1 , . . . , fk ∈ F } Let d denote the dimension. A Gaussian distribution with mean µ ∈ Rd and covariance matrix Σ ∈ Rd×d is denoted by N (µ, Σ). If Σ is a diagonal matrix, then N (µ, Σ) is called an axis-aligned Gaussian. For a distribution g, we write X ∼ g to mean X is a random variable with distribution g, and we write S ∼ g m to mean that S is an i.i.d. sample of size m generated from g. Definition 2.5. A random variable X is said to be σ-subgaussian if Pr[\|X\| ≥ t] ≤ 2 exp(−t2 /σ 2 ) for any t > 0. Note that if X ∼ N (0, 1) then X is (1984, formula (7.1.13)). √ 2-subgaussian, see, e.g., Abramowitz and Stegun Definition 2.6. Let A, B be symmetric, positive definite matrices of the same size. The log-det divergence of A and B is defined as LD(A, B) := tr(B −1 A − I) − log det(B −1 A). We will use kvk or kvk2 to denote the Euclidean norm p of a vector v, kAk or kAk2 to denote the operator norm of a matrix A, and kAkF := tr(AT A) to denote the Frobenius norm of a matrix A. For x ∈ R, we will write (x)+ := max{0, x}. 8 3 Compression schemes and their connection with learning Let F be a class of distributions over a domain Z. Definition 3.1 (distribution decoder). A distribution decoder for F is a deterministic S∞ S n n function J : ∞ n=0 {0, 1} → F , which takes a finite sequence of elements of Z n=0 Z × and a finite sequence of bits, and outputs a member of F . Definition 3.2 (robust distribution compression schemes). Let τ, t, m : (0, 1) → Z≥0 be functions, and let r ≥ 0. We say F admits (τ, t, m) r-robust compression if there exists a decoder J for F such that for any distribution g ∈ F , and for any distribution q on Z with kg − qk1 ≤ r, the following holds: For any ε ∈ (0, 1), if the sample S is drawn from q m(ε) , then with probability at least 2/3, there exists a sequence L of at most τ (ε) elements of S, and a sequence B of at most t(ε) bits, such that kJ (L, B) − gk1 ≤ ε. Essentially, the definition asserts that with high probability, there should be a (small) subset of S and some (small number of) additional bits, from which g can be approximately reconstructed. We say that the distribution g is “encoded” with L and B, and in general we would like to have a compression scheme of a small size. This compression scheme is called “robust” since it requires g to be approximately reconstructed from a sample generated from q rather than g itself. Remark 3.3. In the definition above we required the probability of existence of L and B to be at least 2/3, but one can boost this probability to 1 − δ by generating a sample of size m(ε) log(1/δ). Next we show that if a class of distributions can be compressed, then it can be learned; thus we build the connection between robust compression and agnostic learning. We will need the following useful result about PAC-learning of finite classes of distributions, which immediately follows from Devroye and Lugosi (2001, Theorem 6.3) and a standard Chernoff bound. It states that a finite class of size M can be 3-learned in the agnostic setting using O(log(M/δ)/ε2 ) samples. Denote by [M] the set {1, 2, ..., M}. Throughout the paper, a/bc always means a/(bc). Theorem 3.4 (Devroye and Lugosi (2001)). There exists a deterministic algorithm that, given candidate distributions f1 , . . . , fM , a parameter ε > 0, and log(3M 2 /δ)/2ε2 i.i.d. samples from an unknown distribution g, outputs an index j ∈ [M] such that kfj − gk1 ≤ 3 min kfi − gk1 + 4ε, i∈[M ] with probability at least 1 − δ/3. 9 The proof of the following theorem appears in Appendix C.1. Theorem 3.5 (compressibility implies learnability). Suppose F admits (τ, t, m) r-robust compression. Let τ ′ (ε) := τ (ε/6) + t(ε/6). Then F can be max{3, 2/r}-learned in the agnostic setting using 1 τ ′ (ε) log(m( ε ) log(1/δ)) + log(1/δ) ′ τ (ε) ε ε 6 e m log + + 2 =O samples. O m 6 δ ε2 6 ε If F admits (τ, t, m) 0-robust compression, then F can be learned in the realizable setting using the same number of samples. We next prove two closure properties of compression schemes. First, Lemma 3.6 below implies that if a class F of distributions can be compressed, then the class of distributions that are formed by taking products of members of F can also be comQ pressed. If p1 , . . . , pd are distributions over domains Z1 , . . . , Zd , then di=1 pi denotes Qd the standard product distribution nQ o over i=1 Zi . For a class F of distributions, define d d := F i=1 pi : p1 , . . . , pd ∈ F . The following lemma is proved in Appendix C.2. Lemma 3.6 (compressing product distributions). If F admits (τ (ε), t(ε), m(ε)) r-robust compression, then F d admits (dτ (ε/d), dt(ε/d), m(ε/d) log(3d)) r-robust compression. Our next lemma implies that if a class F of distributions can be compressed, then the class of distributions that are formed by taking mixtures of members of F can also be compressed. The proof appears in Appendix C.3. Lemma 3.7 (compressing mixtures). If F admits (τ (ε), t(ε), m(ε)) r-robust compression, then k-mix(F ) admits (kτ (ε/3), kt(ε/3) + k log2 (4k/ε)), 48m(ε/3)k log(6k)/ε) r-robust compression. 4 4.1 Upper bound: learning mixtures of Gaussians by compression schemes Warm-up: learning mixtures of axis-aligned Gaussians by compression schemes In this section, we give a simple application of our compression framework to prove an 2 e upper bound of O(kd/ε ) for the sample complexity of learning mixtures of k axis-aligned Gaussians in the realizable setting. In the following section, we generalize these arguments to work for general Gaussians in the agnostic setting. 10 Lemma 4.1. The class of single-dimensional Gaussians admits a (3, O(log(1/ε)), 3) 0robust compression scheme. Proof. Let c < 1 < C be such that PrX∼N (0,1) [c < \|X\| < C] ≥ 0.99. Let N (µ, σ 2 ) be the target distribution. We first show how to encode σ. Let g1 , g2 ∼ N (µ, σ 2). Then g = √12 (g1 − g2 ) ∼ N (0, σ 2 ). So with probability at least 0.99, we have σc < \|g\| < σC. Conditioned on this event, this implies that there is a λ ∈ [−1/c, 1/c] such that λg = σ. We now choose λ̂ ∈ {0, ±ε/2C 2, ±2ε/2C 2 , ±3ε/2C 2 . . . , ±1/c} satisfying \|λ̂ − λ\| ≤ ε/(4C 2), and encode the standard deviation by (g1 , g2 , λ̂). The decoder then √ estimates σ̂ := λ̂(g1 − g2 )/ 2. Note that \|σ̂ − σ\| ≤ \|λ̂ − λ\|\|g\| ≤ σε/(4C) and that the encoding requires two sample points and O(log(C 2 /ε)) = O(log(1/ε)) bits (for encoding λ̂). Now we turn to encoding µ. Let g3 ∼ N (µ, σ 2 ). Then \|g3 − µ\| ≤ Cσ with probability at least 0.99. We will condition on this event, which implies existence of some η ∈ [−C, C] such that g3 +ση = µ. We choose η̂ ∈ {0, ±ε/2, ±2ε/2, ±3ε/2 . . . , ±C} such that \|η̂−η\| ≤ ε/4, and encode the mean by (g3 , η̂). The decoder estimates µ̂ := g3 + σ̂ η̂. Again, note that \|µ̂ − µ\| = \|ση − σ̂η̂\| ≤ \|ση − ση̂\| + \|ση̂ − σ̂ η̂\| ≤ σε/2. Moreover, encoding the mean requires one sample point and O(log(1/ε)) bits. To summarize, the decoder has \|µ̂ − µ\| ≤ σε/2 and \|σ̂ − σ\| ≤ σε/2. Plugging these bounds into Lemma A.4 gives kN (µ, σ 2) − N (µ̂, σ̂ 2 )k1 ≤ ε, as required. To complete the proof of Theorem 1.3 in the realizable setting, we note that Lemma 4.1 combined with Lemma 3.6 implies that the class of axis-aligned Gaussians in Rd admits a (O(d), O(d log(d/ε)), O(log(3d))) 0-robust compression scheme. Then, by Lemma 3.7, the class of mixtures of k axis-aligned Gaussians admit a (O(kd), O(kd log(d/ε) + k log(k/ε)), O(k log(k) log(3d)/ε)) 0-robust compression scheme. Applying Theorem 3.5 implies that 2 e the class of k-mixtures of axis-aligned Gaussians in Rd can be learned using O(kd/ε ) many samples in the realizable setting. 4.2 Agnostic learning mixtures of Gaussians by compression schemes e 2 /ε2) for the sample complexity of learning In this section we prove an upper bound of O(kd 2 e mixtures of k Gaussians in d dimensions, and an upper bound of O(kd/ε ) for the sample complexity of learning mixtures of k axis-aligned Gaussians, both in the agnostic sense. The heart of the proof is to show that Gaussians have robust compression schemes in any dimension. Lemma 4.2. For any positive integer d, the class of d-dimensional Gaussians admits an O(d log(2d)), O(d2 log(2d) log(d/ε)), O(d log(2d)) 2/3-robust compression scheme. 11 Remark 4.3. This lemma can be boosted to give an r-robust compression schemes for any r < 1 at the expense of worse constants hidden in the big Oh, but this will not yield any improvement in the final results. Remark 4.4. In the special case d = 1, there also exists a (4, 1, O(1/ε)) (i.e., constant size) 0.773-robust compression scheme using completely different ideas. The proof appears in Appendix D.4. Remarkably, this compression scheme has constant size, as the value of τ + t is independent of ε (unlike Lemma 4.2). This scheme could be used instead of Lemma 4.2 in the proof of Theorem 1.3, although it would not improve the sample complexity bound asymptotically. Proof of Theorem 1.1. Combining Lemma 4.2 and Lemma 3.7 implies that the class of k-mixtures of d-dimensional Gaussians admits a O(kd log(2d)), O(kd2 log(2d) log(d/ε) + k log(k/ε)), O(dk log k log(2d)/ε) e 2/3-robust compression scheme. Applying Theorem 3.5 with m(ε) = O(dk/ε) and τ ′ (ε) = e 2 k) shows that the sample complexity of learning this class is O(kd e 2 /ε2). This proves O(d Theorem 1.1. Proof of Theorem 1.3. Let G denote the class of 1-dimensional Gaussian distributions. By Lemma 4.2, G admits an (O(1), O(log(1/ε)), O(1)) 2/3-robust compression scheme. By Lemma 3.6, the class G d admits a (O(d), O(d log(d/ε)), O(log(3d))) 2/3-robust compression scheme. Then, by Lemma 3.7, the class k-mix(G d ) admits (O(kd), O(kd log(d/ε) + k log(k/ε)), O(k log(k) log(3d)/ε)) 2/3-robust compression. Applying Theorem 3.5 implies that the class of k-mixtures of axis-aligned Gaussians in Rd can be 3-agnostically learned 2 e using O(kd/ε ) many samples. 4.3 Proof of Lemma 4.2 Let Q denote the target distribution, which satisfies kQ − N (µ, Σ)k1 ≤ 2/3 for some Gaussian N (µ, Σ) which we are to encode. Note that this implies TV(Q, N (µ, Σ)) ≤ 1/3. Remark 4.5. The case of rank-deficient Σ can easily be reduced to the case of full-rank Σ. If the rank of Σ is k < d, any X ∼ N (µ, Σ) lies in some affine subspace S of dimension k. Thus, any X ∼ Q lies in S with probability at least 2/3. With high probability, after seeing 10d samples from Q, at least k + 1 points from S will appear in the sample. We encode S using these samples, and for the rest of the process we work in this affine space, and discard outside points. Hence, we may assume Σ has full rank d. We first prove a lemma that is similar to known results in random matrix theory (see Litvak, Pajor, Rudelson, and Tomczak-Jaegermann, 2005, Corollary 4.1), but is tailored 12 for our purposes. Its proof appears in Appendix D.1. Let S d−1 := and B2d := y ∈ Rd : kyk ≤ 1 . y ∈ Rd : kyk = 1 Lemma 4.6. Let q1 , . . . , qm be i.i.d. samples from a distribution Q where TV(Q, N (0, Id)) ≤ 2/3. Let √ T := { ±qi : kqi k ≤ 4 d }. Then for a large enough constant C > 0, if m ≥ Cd(1 + log d) then 1 d Pr B 6⊆ conv(T ) ≤ 1/6. 20 2 P P Suppose Σ = di=1 vi viT , where the vi vectors are orthogonal. Let Ψ := di=1 vi viT /kvi k. Note that both Σ and Ψ are positive definite, and that Σ = Ψ2 . Moreover, it is easy to P P see that Σ−1 = di=1 vi viT /kvi k4 and Ψ−1 = di=1 vi viT /kvi k3 . The following lemma is proved in Appendix D.2. Lemma 4.7. Let C > 0 be a sufficiently large constant. Given m = 2Cd(1 + log d) samples S from Q, where TV(Q, N (µ, Σ)) ≤ 1/3, with probability at least 2/3, one can encode vectors vb1 , . . . , b vd , µ b ∈ Rd satisfying kΨ−1 (b vj − vj )k ≤ ε/6d2 ∀j ∈ [d], and kΨ−1 (b µ − µ)k ≤ ε, using O(d2 log(2d) log(d/ε)) bits and the points in S. Lemma 4.2 now follows immediately from the following lemma, which is proved in Appendix D.3. P Lemma 4.8. Suppose Σ = Ψ2 = i∈[d] vi viT , where vi are orthogonal and Σ is full rank, and that kΨ−1 (b vj − vj )k ≤ ρ ≤ 1 ∀j ∈ [d], and kΨ−1 (b µ − µ)k ≤ ζ. Then TV(N (µ, X i∈[d] vi viT ), N (b µ, X i∈[d] 13 vbi b viT )) ≤ p 9d3 ρ2 + ζ 2/2. 5 The lower bound for Gaussians and their mixtures e 2 /ε2 ) for learning a single Gaussian, and In this section, we establish a lower bound of Ω(d 2 2 e then lift it to obtain a lower bound of Ω(kd /ε ) for learning mixtures of k Gaussians in d dimensions. Both our lower bounds consider the realizable setting (so they also hold in the agnostic setting). Our lower bound is based on the following lemma, which follows from Fano’s inequality in information theory (see Lemma E.1). Its proof appears in Appendix E.1. Lemma 5.1. Let F be a class of distributions such that for all small enough ε > 0 there exist N densities f1 , . . . , fN ∈ F with KL(fi k fj ) ≤ κ(ε) and TV(fi , fj ) = Ω(ε) ∀i 6= j ∈ [N]. Then any algorithm that learns F to within total variation distance ε with success probability at least 2/3 has sample complexity Ω (log N/(κ(ε) log(1/ε))). Theorem 5.2. Any algorithm that learns a general Gaussian in Rd in the realizable setting within total variation distance ε and with success probability 2/3 has sample complexity d2 Ω ε2 log3 (1/ε) . 2 Proof. Let r = 9 and λ = Θ(εd−1/2 log(1/ε)). Guided by Lemma 5.1, we will build 2Ω(d ) Gaussian distributions of the form fa := N (0, Σa ) where Σa = Id + λUa UaT , where each Ua is a d × d/r matrix with orthonormal columns. To apply Lemma 5.1, we need to give an upper bound on the KL-divergence between any two fa and fb , and a lower bound on their total variation distance. Upper bounding the KL divergence is easy: by Lemma A.1 and since kUaT Ub k2F ≥ 0, λ Ua UaT )(I + λUb UbT ) − I) 1+λ λ λ2 = Tr(λUb UbT − Ua UaT − Ua UaT Ub UbT ) 1+λ 1+λ λ2 λ (d/r) − kU T Ub k2F = λ(d/r) − 1+λ 1+λ a λ2 d ≤ ≤ λ2 d/(2r) = O(ε2 log2 (1/ε)), r + rλ 2 KL(fa k fb ) = Tr(Σ−1 a Σb − I) = Tr((I − as required. Our next goal is to give a lower bound on the total variation distance between fa and fb . For this, we would like the matrices {Ua } to be “spread out,” in the sense that their columns should be nearly orthogonal. This is formalized in Lemma 5.3 below, where we d show if we choose the Ua randomly, we can achieve kUaT Ub k2F ≤ 2r for any a 6= b. Then, 14 if Sa is the subspace spanned by the columns of Ua , then we expect that a Gaussian drawn from N (0, Σa ) should have a slightly larger projection onto Sa then a Gaussian drawn from N (0, Σb). This will then allow us to give a lower bound on the total variation distance between N (0, Σa ) and N (0, Σ b ). √Moreprecisely, in Lemma 5.4 we show that λ d/r d T 2 √ = Ω(ε), completing the proof. kUa Ub kF ≤ 2r implies TV(fa , fb ) = Ω log(r/λ d) We defer the proofs of the following lemmas to Appendix E.2 and Appendix E.3, respectively. 2 Lemma 5.3. Suppose d ≥ r ≥ 9. Then there exists 2Ω(d /r) orthonormal d × d/r matrices d {Ua } such that for any a 6= b we have kUaT Ub k2F ≤ 2r . √ d/r ∈ (0, 1/3). If kUaT Ub k2F ≤ d/(2r), then Lemma 5.4. Suppose that λ ≤ 1 ≤ r, and λ √ λ d/r √ TV(fa , fb ) = Ω log(r/λ . d) Finally, in Appendix E.4 we prove our lower bound for mixtures. Theorem 5.5. Any algorithm that learns a mixture of k general Gaussians in Rd in the realizable setting within total distance ε and with success probability at least 2/3 variation kd2 has sample complexity Ω ε2 log3 (1/ε) . 6 Further discussion A central open problem in distribution learning and density estimation is characterizing the sample complexity of learning a distribution class. An insight from supervised learning theory is that the sample complexity of learning a class (of concepts, functions, or distributions) may be proportional to some kind of intrinsic dimension of the class divided by ε2 , where ε is the error tolerance. For the case of agnostic binary classification, the intrinsic dimension is captured by the VC-dimension of the concept class (see Vapnik and Chervonenkis (1971); Blumer, Ehrenfeucht, Haussler, and Warmuth (1989)). For the case of distribution learning with respect to ‘natural’ parametric classes, we expect this dimension to be equal to the number of parameters. In this paper, we showed that this is indeed the case for the class of Gaussians, axis-aligned Gaussians, and their mixtures in any dimension. In binary classification, the combinatorial notion of Littlestone-Warmuth compression has been shown to be sufficient (Littlestone and Warmuth, 1986) and necessary (Moran and Yehudayoff, 2016) for learning. In this work, we showed that the new but related notion of robust distribution compression is sufficient for distribution learning. Whether the existence of compression schemes is necessary for learning an arbitrary class of distributions remains an intriguing open problem. 15 We would like to mention that while it may first seem that the VC-dimension of the Yatracos set associated with a class of distributions can characterize its sample complexity, it is not hard to come up with examples where this VC-dimension is infinite while the class can be learned with finite samples. Covering numbers do not work, either; for instance the class of Gaussians do not have a bounded covering number in the TV metric, nevertheless it is learnable with finite samples. A concept related to compression is that of core-sets. In a sense, core-sets can be viewed as a special case of compression, where the decoder is required to be the empirical error minimizer. See the work of (Lucic, Faulkner, Krause, and Feldman, 2017) for using core-sets in maximum likelihood estimation. A Standard results Lemma A.1 (Rasmussen and Williams (2006, Equation A.23)). For two full-rank Gaussians N (µ, Σ) and N (µ′ , Σ′ ), their KL divergence is KL(N (µ, Σ) k N (µ′, Σ′ )) = 1 Tr(Σ−1 Σ′ − I) + (µ − µ′ )T Σ−1 (µ − µ′ ) − log det(Σ′ Σ−1 ) . 2 Lemma A.2 (Pinsker’s Inequality (Tsybakov, 2009, Lemma 2.5)). For any two distributions A and B, we have 2 TV(A, B)2 ≤ KL(A k B). Lemma A.3. For two full-rank Gaussians N (µ, Σ) and N (µ′, Σ′ ), their total variation distance is bounded by 2 TV(N (µ, Σ), N (µ′, Σ′ ))2 ≤ KL(N (µ, Σ k N (µ′, Σ′ )) 1 = LD(Σ, Σ′ ) + (µ − µ′ )T Σ−1 (µ − µ′ ) . 2 Proof. Follows from Lemma A.1 and Lemma A.2. Lemma A.4. For any µ, σ, µ b, σ b ∈ R with \|b µ − µ\| ≤ εσ and \|b σ − σ\| ≤ εσ and ε ∈ [0, 2/3] we have kN (µ, σ 2) − N (b µ, σ b2 )k1 ≤ 2ε. Proof. By Lemma A.3, σ b2 4 TV(N (µ, σ 2), N (b µ, σ b2 ))2 ≤ 2 −1−log σ 2 2 ! σ b2 \|µ − µ b\|2 σ b σ b +ε2 . + ≤ −1−log 2 2 σ σ σ σ Since z := σ b/σ ∈ [1−ε, 1+ε] and ε ≤ 2/3, using the inequality x2 −1−log(x2 ) ≤ 3(x−1)2 valid for all \|x − 1\| ≤ 2/3, we find 1 1 TV(N (µ, σ 2), N (b µ, σ b2 ))2 ≤ (3(z − 1)2 + ε2 ) ≤ (4ε2) = ε2 . 4 4 16 And the lemma follows since the L1 distance is twice the TV distance. Fact A.5. Let X and Y be arbitrary random variables on the same space. For any function f , we have TV(f (X), f (Y )) ≤ TV(X, Y ). Proof. This follows from the observation that Pr [f (X) ∈ A] − Pr [f (Y ) ∈ A] = Pr X ∈ f −1 (A) − Pr Y ∈ f −1 (A) ≤ TV(X, Y ), so taking supremum of the left-hand side gives the result. Lemma A.6 (Laurent and Massart (2000, Lemma 1)). Let X have the chi-squared disP tribution with parameter d; that is, X = di=1 Xi2 where the Xi are i.i.d. standard normal. Then, √ Pr[X − d ≥ 2 dt + 2t] ≤ exp(−t) and √ Pr[d − X ≥ 2 dt] ≤ exp(−t). d. The first inequality above implies, in particular, that Pr[X ≥ 16d] ≤ exp(−3) for any Lemma A.7. Let g1 , . . . , gm ∈ Rd be independent samples from N (0, I). For ε ∈ [0, 1], Pr[ Proof. Note that X = 1 m Pm √1 m i=1 gi Pm i=1 gi 2 ≥ (1 + ε)d/m] ≤ exp(−ε2 d/9). 2 has the chi-squared distribution with parameter d. 2 Applying Lemma A.6 with t = ε d/9 shows that Pr[X ≥ (1 + ε)d] ≤ exp(−ε2 d/9). Lemma A.8 (Theorem 3.1.1 in Vershynin (2018)). Let g ∼ N (0, Id). Then (kgk2 − √ is O(1)-subgaussian. Consequently, (kgk2 − d)+ is also O(1)-subgaussian. √ d) Lemma A.9 (Proposition 2.5.2 in Vershynin (2018)). A random variable X is σ-subgaussian if and only if supp≥1 p−1/2 (E\|X\|p)1/p ≤ Cσ for some global constant C > 0. Lemma A.10 (Hoeffding’s Inequality, Proposition 2.6.1 in Vershynin (2018)). Let X1 , . . . , Xn be independent, mean-zero random variables and suppose Xi is σi -subgaussian. Then, for some global constant c > 0 and any t ≥ 0, # " X −ct2 Pr Xi > t ≤ 2 exp P 2 . i σi i 17 Lemma A.11 (Bernstein’s Inequality, Theorem 2.8.1 in Vershynin (2018)). Let g1 , . . . , gn ∼ N (0, 1) and a1 , . . . , an > 0. Then, there is a global constant c > 0 such that for every t ≥ 0, # " n n X X t2 t 2 Pr . ai gi − ai ≥ t ≤ 2 exp −c min Pn 2 , i=1 ai maxi ai i=1 i=1 Theorem A.12 (Gordon’s Theorem, Theorem 5.32 in Vershynin (2012)). Let G be a m×n √ √ matrix with entries independently drawn from N (0, 1). Then Eσmin (G) ≥ m − n. Lemma A.13 (Corollary 5.50 and Remark 5.51 in Vershynin (2012)). Let X1 , . . . , Xm ∼ N (0, Σ), where Σ is d × d, and let 0 < ε < 1 < t. If m ≥ C(t/ε)2 d, then with probability 1 − 2 exp(−t2 d) we have m 1 X Xi XiT − Σ ≤ εkΣk. m i=1 Lemma A.14 (Corollary 4.2.13 in Vershynin (2018)). For any ε ∈ (0, 1), there exists an ε-net for B2d of size (3/ε)d . B Efficient algorithm for learning a single Gaussian by empirical mean/covariance estimation In this section we give a simple algorithm for learning a single high dimensional Gaussian, with sample complexity O(d2/ε2 ) and computational complexity O(d4/ε2 ). Lemma B.1. Let v1 , . . . , vm ∈ Rd be independent samples from N (µ, Σ). Let v̄ = Pm 1 i=1 vi . Then m Pr[(v̄ − µ)T Σ−1 (v̄ − µ) ≥ 2d/m] ≤ exp(−d/9). Proof. Let gi = Σ−1/2 (vi − µ), so that g1 , . . . , gm are independent samples from N (0, I). Then Pr (v̄ − µ)T Σ−1 (v̄ − µ) ≥ (1 + ε)d/m = Pr 1 m Pm i=1 gi ≤ exp(−ε2 d/9), 2 ≥ (1 + ε)d/m by Lemma A.7. We write B A if A−B is a positive semidefinite matrix. Observe that, x−1−log x ≤ (x − 1)2 for any x ∈ [1/2, ∞). 18 Lemma B.2. Let A, B be symmetric, positive definite matrices, satisfying (1 − α)B A (1 + α)B for some α ∈ [0, 1/2]. Then LD(A, B) ≤ dα2. Proof. Let λ1 , . . . , λd be the eigenvalues of B −1 A. By the hypothesis, each λi ∈ [1 − α, 1 + α]. So, LD(A, B) = tr(B −1 A − I) − log det(B −1 A) = d X i=1 (λi − 1) − log d d X X (λi − 1)2 ≤ dα2 . (λi − 1 − log(λi )) ≤ = d Y λi i=1 i=1 i=1 Our result immediately follows from the following theorem. Theorem B.3. Let m = Cd2 /ε2 for a large enough constant C. Let v1 , . . . , vm be i.i.d. P Pm 1 T samples from N (µ, Σ). Let µ̃ = m1 m i=1 vi and Σ̃ = m i=1 vi vi respectively be the empirical mean and the empirical covariance matrix. Then TV(N (µ̃, Σ̃), N (µ, Σ)) ≤ ε with probability at least 1 − 3 exp(−d/9). Proof. We will show that, with probability at least 1 − 3 exp(−d/9), KL(N (µ̃, Σ̃) k N (µ, Σ)) ≤ ε2 , and the theorem follows from Pinsker’s inequality (Lemma A.2). By e −Σ ≤ standard concentration for Gaussian matrices (see Lemma A.13) we have Σ √ ε/ d =: α with probability at least 1 − 2 exp(−d). That is, (1 − α)Σ Σ̃ (1 + α)Σ. Applying Lemma B.2 shows that LD(Σ̃, Σ) ≤ dα2 = ε2 . Next, by Lemma B.1 we have (µ̃ − µ)T Σ−1 (µ̃ − µ) ≤ 2d/m = ε2 /18d with probability at least 1 − exp(−d/9). So KL(N (µ̃, Σ̃) k N (µ, Σ)) = 1 LD(Σ̃, Σ) + (µ̃ − µ)T Σ−1 (µ̃ − µ) ≤ ε2 , 2 with probability at least 1 − 3 exp(−d/9). It is easy to see that, by multiplying the sample size by log(1/δ), one can boost the success probability to 1 − δ, for any δ ∈ (0, 1). C C.1 Omitted proofs from Section 3 Proof of Theorem 3.5 We give the proof for the agnostic case. The proof for the realizable case is similar. Let q be the target distribution that the samples are being generated from. Let α = inf f ∈F kf −qk1 19 be the approximation error of q with respect to F . The goal of the learner is to find a distribution ĥ such that kĥ − qk1 ≤ max{3, 2/r} · α + ε. First, consider the case α ≤ r. In this case, we develop a learner that finds a distribuε tion ĥ such that kĥ−qk1 ≤ 3α+ε. Let g ∈ F be a distribution such that kg −qk1 ≤ α+ 12 (such a g exists by the definition of α). By assumption, F admits (τ, t, m) compression. Let J denote the corresponding decoder. Given ε, the learner first asks for an i.i.d. sample S ∼ q m(ε/6)·log(2/δ) . By the definition of robust compression, we know that with probability at least 1 − δ/2, there exist L ∈ S τ (ε/6) and B ∈ {0, 1}t(ε/6) such that kJ (L, B) − gk ≤ ε/6 (see Remark 3.3). Let h∗ := J (L, B). The learner is of course unaware of L and B. However, given the sample S, it can try all of the possibilities for L and B and create a candidate set of distributions. More concretely, let H = {J (L, B) : L ∈ S τ (ε/6) , B ∈ {0, 1}t(ε/6) }. Note that ′ \|H\| ≤ (m(ε/6) log(2/δ))τ (ε/6) 2t(ε/6) ≤ (m(ε/6) log(2/δ))τ (ε) . Since H is finite, we can use the algorithm of Theorem 3.4 to find a good candidate ĥ from H. In particular, we set the accuracy parameter in Theorem 3.4 to be ε/16 and the confidence parameter to be δ/2. In this case, Theorem 3.4 requires ′ τ (ε) log(m( 6ε ) log( 1δ )) + log( 1δ ) log(6\|H\|2/δ) e ′ (ε)/ε2 ) = O(τ =O 2 2 2(ε/16) ε additional samples, and its output ĥ will be an (ε,3)-approximation of q: kĥ − qk1 ≤ 3kh∗ − qk1 + 4 ε ε ε ≤ 3(kh∗ − gk1 + kg − qk1 ) + ≤ 3(ε/6 + (α + ε/12)) + 16 4 4 ≤ 3α + ε. ′ e Note that the above procedure uses O(m(ε/6) + τ ε(ε) 2 ) samples, and the probability of failure is at most δ (i.e., the probability of either H not containing a good h∗ , or the failure of Theorem 3.4 in choosing a good candidate among H, is bounded by δ/2 + δ/2 = δ). The other case, α > r, is trivial: the learner outputs some distribution b h. Since b h and 2 b q are density functions, we have kh − qk1 ≤ 2 < · α < max{3, 2/r} · α + ε. r C.2 Proof of Lemma 3.6 The following proposition is standard. Proposition C.1 (Lemma 3.3.7 in Reiss (1989)). For i ∈ [d], let pi and qi be probability P distributions over the same domain Z. Then kΠdi=1 pi − Πdi=1 qi k1 ≤ di=1 kpi − qi k1 . 20 Proof of Lemma 3.6. Let G = Πdi=1 gi be an arbitrary element of F d . Let Q be an arbitrary distribution over Z d , subject to kG − Qk1 ≤ r. Let q1 , . . . , qd be the marginal distributions of Q on the d components. First, observe that, since projection onto a coordinate cannot increase the total variation distance (see Fact A.5), we have kqj −gj k1 ≤ r for each j ∈ [d]. We know that F admits (τ, t, m) r-robust compression. Call the corresponding decoder J , and let m0 := m(ε/d) log(3d), and S ∼ Qm0 . The goal is then to encode an εapproximation of G using dτ (ε/d) elements of S and dt(ε/d) bits. Note that each element of S is a d-dimensional vector. For each i ∈ [d], let Si ∈ Z m0 be the set of the ith components of elements of S. By definition of qi , we have Si ∼ qim0 for each i. Thus, for each i ∈ [d], since kqi −gi k ≤ r, with probability at least 1−1/3d there exists a sequence Li of at most τ (ε/d) elements of Si , and a sequence Bi of at most t(ε/d) bits, such that kJ (Li , Bi ) − gi k1 ≤ ε/d. By the union bound, this assertion holds for all i ∈ [d], with probability at least 2/3. We may encode these L1 , . . . , Ld , B1 , . . . , Bd using dτ (ε/d) elements of S and dt(ε/d) bits. Our decoder for F d then extracts L1 , . . . , Ld , B1 , . . . , Bd Q from these elements and bits, and then outputs di=1 J (Li , Bi ) ∈ F d . Finally, ProposiP tion C.1 gives kΠdi=1 J (Li , Bi ) − Gk1 ≤ di=1 kJ (Li , Bi ) − gi k1 ≤ d × ε/d ≤ ε, completing the proof. C.3 Proof of Lemma 3.7 We will need the following standard proposition. Proposition C.2. Let g and g ∗ be probability densities with kg − g ∗k1 = ρ and g ∗ = P w f , with (w1 , . . . , wk ) ∈ ∆k and where each fi is a density. Then we may write i∈[k] P i i g = i∈[k] wi Gi , such that each Gi is a density, and for each i we have kfi − Gi k1 ≤ ρ. Proof. Write ∗ g =g +h= k X wi fi + h = i=1 k X wi (fi + h) (1) i=1 with khk1 = ρ. Note that fi + h is not necessarily a probability density function. Let D denote the set of probability density functions, that is, the set of nonnegative functions with unit L1 norm. Note that this is a convex set. Since projection is a linear operator, by P projecting both sides of (1) onto D we find g = ki=1 wi Gi , where Gi is the L1 projection of fi + h onto D (since g ∈ D, the projection of g onto D is itself). Also, since fi ∈ D and projection onto a convex set does not increases distances, we have kfi − Gi k1 ≤ kfi − (fi + h)k1 = khk1 = ρ, as required. 21 Proof of Lemma 3.7. Let g be the distribution from which we have 48m(ε/3)k log(6k)/ε samples, and suppose g ∗ ∈ k-mix(F ) is the distribution to be compressed, so kg−g ∗ k1 ≤ r. P Thus we have g ∗ = i∈[k] wi fi , with each fi ∈ F , and (w1 , . . . , wk ) ∈ ∆k . By ProposiP tion C.2, we also have g = i∈[k] wi Gi for some distributions G1 , . . . , Gk , such that for each i we have kfi − Gi k1 ≤ r. We view g as a mixture of these k distributions, so the samples from g can be partitioned into k parts, so that samples from the ith part have distribution Gi . We compress each of the parts individually. Moreover, we compress the mixing weights w1 , . . . , wk using bits, as follows. Consider an (ε/3k)-net in ℓ∞ for ∆k , of size (1 + 3k/ε)k . Such a net can be obtained from a mesh of grid-size ε/3k for [0, 1]k , and projecting each of its points onto ∆k . Let (w b1 , . . . , w bk ) be an element in the net that has k(w b1 , . . . , w bk ) − (w1 , . . . , wk )k∞ ≤ ε/3k; then, wi − wbi ≤ ε/3k for all i. Moreover, the particular element (w b1 , . . . , w bk ) of the net k can be encoded using log2 ((1 + 3k/ε) ) ≤ k log2 (4k/ε) bits. For any i ∈ [k], we say component i is negligible if wi ≤ ε/(6k). Since the total number of samples is 48m(ε/3)k log(6k)/ε, by a standard Chernoff bound combined with a union bound over the k components, with probability at least 5/6, for each non-negligible component i, we have at least m(ε/3) log(6k) samples from i. Let i be a non-negligible component. Since F admits (τ, t, m) robust compression and fi ∈ F and kfi − Gi k1 ≤ r, with probability at least 1 − 1/6k there exists τ (ε/3) samples from part i and t(ε/3) bits, from which the decoder can construct a distribution fbi with kfi −fbi k1 ≤ ε/3. Using a union bound over the k components, this is true uniformly over all non-negligible components, with probability at least 5/6. (Note that, for negligible components i, there is no guarantee P b about fbi .) Hence, given the mixing weights w b1 , . . . , w bk , the decoder outputs w bi fi . The total number of instances used to encode is kτ (ε/3). Similarly, the total number of used bits is not more than kt(ε/3) + k log2 (4k/ε). Thus to complete the proof of the P P b lemma, we need only show that k wi fi − w bi fi k1 ≤ ε. Let L ⊆ [k] denote the set of negligible components. We have X (w bi fbi − wi fi ) i∈[k] ≤ 1 ≤ ≤2 X i∈[k] X i∈L X wi (fbi − fi ) wi (fbi − fi ) wi + i∈L X + X (w bi − wi )fbi i∈[k] 1 + 1 X i∈L / wi (ε/3) + i∈L / wi (fbi − fi ) X i∈[k] completing the proof of the lemma. 1 + 1 X i∈[k] \|w bi − wi \| fbi 1 ε/3k × 1 ≤ ε/3 + ε/3 + ε/3 = ε, 22 D D.1 Omitted proofs from Section 4 Proof of Lemma 4.6 First we show the following proposition implies max\|h y, q i\| ≥ q∈S 1 20 1 Bd 20 2 ⊆ conv(T ): ∀y ∈ S d−1 . (2) For, let P := conv(T ). Its polar is P ◦ = y ∈ Rd : \|h y, q i\| ≤ 1 ∀q ∈ T . So (2) implies 1 )B2d . P ◦ ⊆ 20B2d . As polarity reverses containment, we obtain P ⊇ (20B2d )◦ = ( 20 We now bound the probability that (2) fails. For this, let g ∼ N (0, Id) and let we Xy := h y, g i. Notice that Xy ∼ N (0, 1). Since the pdf of Xy is bounded above by 1, √ 1 have Pr \|Xy \| ≤ 10 ≤ 1/5. Moreover, by Lemma A.6, the probability that kgk2 ≥ 4 d is ≤ exp(−3). Hence √ 1 ∨ kgk2 ≥ 4 d ≤ 1/5 + exp(−3) < 0.25. Pr \|Xy \| ≤ 10 √ 1 Now let Yy,i := h y, qi i and let Ey,i be the event {\|Yy,i \| ≤ 10 ∨ kqi k > 4 d}. As TV(Q, N (0, Id)) ≤ 2/3, we have Pr[Ey,i ] ≤ 0.25 + 2/3 < 0.92. Thus   ^ Pr  Ey,i  < (0.92)m i∈[m] √ √ Let N be an (1/80 d)-net of S d−1 with \|N\| ≤ (240 d)d . By a union since √ d bound, m m ≥ Cd(1+log d) for C large enough, with probability at least 1−(240 √ d) (0.92) ≥ 5/6, 1 for all y ∈ N there exists i ∈ [m] such that \|Yy,i \| ≥ 10 and kqi k ≤ 4 d. √ Suppose this event holds.√ Let y ∈ S d−1 , and let y ′ ∈ N satisfy ky − y ′k2 ≤ 1/80 d. 1 . These imply ±qi ∈ T and Let qi be such that kqi k ≤ 4 d and \|Yy′ ,i \| ≥ 10 √ 1 \|Yy,i \| ≥ \|Yy′ ,i \| − kqi k/80 d ≥ 1/10 − 1/20 = , 20 as required. D.2 Proof of Lemma 4.7 Let X1 , . . . , X2m be the samples, and let Yi := √12 Ψ−1 (X2i − X2i−1 ) for i ∈ [m]. Observe that if X2i and X2i−1 were N (µ, Σ), then √12 Ψ−1 (X2i − X2i−1 ) would have been N (0, I). Since X2i and X2i−1 have TV distance at most 1/3 from N (µ, Σ), Yi has TV distance at most 2/3 from N (0, I) (this can be seen, e.g., by the coupling characterization of the TV 23 √ distance). Let I := {i ∈ [m] : kYi k ≤ 4 d}. By Lemma 4.6, with probability ≥ 5/6 we have 1 d B ⊆ conv{±Yi : i ∈ I} C 2 with C = 20. We give the encoding for b vj conditioned on this event. Fix j ∈ [d]. Observe that Ψ−1 vj = vj /kvj k has unit norm, so we can write X Ψ−1 vj /C = θj,i Yi i∈[m] for some vector θj ∈ [−1, 1]m supported on I. Applying Ψ to both sides, we obtain C X θj,i (X2i − X2i−1 ). vj = √ 2 i∈[m] For encoding vbj , consider an (ε/24Cmd3)-net for [−1, 1]m in ℓ∞ distance. The size of the net is (48Cmd3 /ε)m , so any element of the net can be described using O(m log(d/ε)) bits. Let θbj be an element in the net that is closest to θj subject to support(θbj ) ⊆ I, and let P vj := √C2 i∈[m] θbj,i (X2i − X2i−1 ). We have, b m C X (θj,i − θbj,i )Ψ−1 (X2i − X2i−1 )k kΨ−1 (b vj − vj )k = √ k 2 i=1 √ C ≤ √ m(max \|θj,i − θbj,i \|)(max 2kYi k) i i∈I 2 √ √ C ≤ √ m(ε/24Cmd3 )(4 2 d) ≤ ε/6d2, 2 as required. The total number of bits used to encode each vbj is O(m log(d/ε)), giving a total number of O(d2 log(2d) log(d/ε)) bits for encoding them all. We next describe the encoding of µ b. Let Zi := Ψ−1 (Xi − µ) and observe that Zi has a distribution with TV distance at most 2/3 to N (0, I). So, using Lemma A.6, p √ Pr[kZi k ≥ 4 d] ≤ exp(−3) + 1/3 < 1/6, √ which means, with probability at least 5/6, min{kZ1 k, kZ2k} ≤ 4 d. We give an encoding for µ b provided this event occurs. √ P Without loss of generality, assume kZ1 k ≤ 4 d, and suppose Z1 = j∈[d] λj ej , {ei } P P 2 being the standard basis. This implies µ = X1 − j∈[d] λj vj , with λj ≤ 16d2 . Consider √ d an (ε/3d)-net for 4 dB2 of size (36d2/ε)d , and let b λ be the closest element to λ in this P b net. We encode µ b = X1 − j∈[d] λj vbj . Note that this requires O(d log(d/ε)) more bits, which is dominated by the number of bits for encoding the b vj . 24 Finally, observe that, kΨ−1 (b µ − µ)k = k ≤ X (λbj − λj )Ψ−1 (vj − vbj )k j X j bj )Ψ−1 vj k kλbj (Ψ−1 vj − Ψ−1 vbj ) + (λj − λ n o bj \|kΨ−1 vj k ≤ d max \|λbj \|kΨ−1vj − Ψ−1 b vj k + \|λj − λ j √ ε ≤ d · 4 d · 2 + d(ε/3d) ≤ ε, 6d as required. D.3 Proof of Lemma 4.8 b := Let Σ P i vi vbiT . We will show that b b ≤ 9d3 ρ2 LD(Σ, Σ) (3) If this is true, Lemma A.3 gives 1 b + (µ − µ b 2 ≤ 1 LD(Σ, Σ) b)T Σ−1 (µ − µ b) ≤ (9d3 ρ2 + ζ 2), TV(N (µ, Σ), N (b µ, Σ)) 4 4 completing the proof. For showing (3), note that we have b = LD(Ψ−1 ΣΨ−1 , Ψ−1ΣΨ b −1 ) LD(Σ, Σ) X X X = LD( Ψ−1 vi viT Ψ−1 , Ψ−1 vbi b viT Ψ−1 ) = LD(I, Ψ−1 b vi vbiT Ψ−1 ) i i i For the first equality we have used the fact that, if A and B are positive definite and C is invertible, then LD(A, B) = LD(CAC, CBC). Indeed, as we proved in Lemma B.2, LD(A, B) only depends on the spectrum of B −1 A. Observe that (v, λ) is an eigenvector/eigenvalue pair for B −1 A if and only if (C −1 v, λ) is an eigenvector/eigenvalue pair for (CBC)−1 CAC; hence these two matrices have the same spectrum. P Let B := i Ψ−1 b vi vbiT Ψ−1 . We shall show kB − Ik ≤ 3dρ, which implies −3dρI 4 b = LD(I, B) ≤ 9d3 ρ2 . We B − I 4 3dρI and together with Lemma B.2 implies LD(Σ, Σ) have X X kB − Ik = k (Ψ−1 b vi b viT Ψ−1 − Ψ−1 vi viT Ψ−1 )k ≤ kΨ−1 b vi vbiT Ψ−1 − Ψ−1 vi viT Ψ−1 k i = X i i kxi xTi − yi yiT k, vi and yi := Ψ−1 vi (here we have used the fact that Ψ−1 is symmetric). Note with xi := Ψ−1 b that kyi k = kΨ−1 vi k = 1, and that, by the lemma hypothesis, kxi − yi k ≤ ρ. Applying Lemma D.1 below concludes the proof. 25 Lemma D.1. Suppose x, y satisfy kyk = 1 and kx − yk ≤ ε ≤ 1. Then we have kxxT − yy Tk ≤ 3ε. Proof. Suppose x = y + z with kzk ≤ ε. Then, kxxT − yy Tk = kyz T + zy T + zz T k ≤ kyz T k + kzy T k + kzz T k ≤ ε + ε + ε2 ≤ 3ε. D.4 Proof of Remark 4.4 Recall that N (µ, σ 2) denotes a 1-dimensional Gaussian distribution with mean µ and standard deviation σ. We will need a lemma bounding the L1 distance of two Gaussians in terms of their parameters. Any vector (p1 , . . . , pn ) ∈ ∆n induces a discrete probability distribution over [n] defined by Pr(i) := pi . Let x ∨ y := max{x, y}. Lemma D.2. Let (p1 , . . . , p2n+1 ) ∈ ∆2n+1 and (q1 , . . . , q2n+1 ) ∈ ∆2n+1 be discrete probability distributions with ℓ1 distance between them ≤ t. Suppose we have 2n + 1 bins, numbered 1 to 2n + 1. We throw m balls into these bins, where each ball chooses a bin independently according to qi . We pair bin 1 with bin 2, bin 3 with bin 4, . . . , and bin 2n − 1 with bin 2n; so bin 2n + 1 is unpaired. The probability that, for all pairs of bins, at most one them gets a ball, is not more than !m n X max{p2i−1 , p2i } 2n t/2 + p2n+1 + i=1 Proof. Let P1 = {1, 2}, P2 = {3, 4}, ..., Pn = {2n − 1, 2n}, and let A := {A ⊂ [2n] : \|A ∩ Pi \| = 1 ∀i ∈ [n]}. Clearly \|A\| = 2n . For any A ∈ A, let EA be the event that, the first ball does not choose a bin in A, and let FA be the event that, none of the balls choose a bin in A. Then, X X X Pr[EA ] = qi ≤ TV(p, q) + pi ≤ t/2 + pi i∈[2n+1]\A ≤ t/2 + p2n+1 + i∈[2n+1]\A n X i=1 i∈A / max{p2i−1 , p2i }, P and so Pr[FA ] = Pr[EA ]m ≤ (t/2 + p2n+1 + ni=1 (p2i−1 ∨ p2i ))m . Finally, observe that, if for each pair of bins, at most one them gets a ball, then there exists at least one A ∈ A, such that none of the balls chooses a bin in A. The lemma is thus proved by applying the union bound over all events {FA }A∈A . 26 Theorem D.3. The class of all Gaussian distributions over the real line admits (4, 1, O(1/ε)) 0.773-robust compression. Proof. Let q be any distribution (not necessarily a Gaussian) such that there exists a Gaussian g = N (µ, σ 2) with kq − gk1 ≤ r ≤ 0.773. Our goal is to encode g using samples generated from q. Let m = C/ε for a large enough constant C to be determined, and let S ∼ q m be an i.i.d. sample. The goal is to approximately encode µ and σ using only four elements of S and a single bit. We start by defining the decoder J . Our proposed decoder takes as input four points x1 , x2 , y1 , y2 ∈ R, and one bit b ∈ {0, 1}. The decoder then outputs a Gaussian distribution based on the following rule:  x +x \|y −y \|2  N ( 1 2 2 , 1 9 2 ) : if b = 1 J (x1 , x2 , y1 , y2 , b) =  N ( x1 +x2 , \|y − y \|2 ) : if b = 0 1 2 2 Our goal is thus to show that, with probability at least 2/3, there exists x1 , x2 , y1, y2 ∈ S and b ∈ {0, 1} such that kJ (x1 , x2 , y1 , y2, b) − gk1 ≤ ε. Let M = 1/ε and partition the interval [µ − 2σ, µ + 2σ) into 4M subintervals of length εσ. Enumerate these intervals as I1 to I4M , i.e., Ii = [µ − 2σ + (i − 1)(εσ), µ − 2σ + i(εσ)). S Also let I4M +1 = R \ 4M i=1 Ii . We state two claims which will imply the theorem, and will be proved later. Claim 1. With probability at least 5/6, there exist y1 , y2 ∈ S such that at least one of the following two conditions holds: (a) y1 ∈ Ii and y2 ∈ Ii+M for some i ∈ {M + 1, 2M + 2, ..., 2M}. In this case, we let b = 0, and so J (x1 , x2 , y1 , y2 , b) will have standard deviation \|y1 − y2 \|. (b) y1 ∈ Ii and y2 ∈ Ii+3M for some i ∈ [M]. In this case, we let b = 1, and so 2\| J (x1 , x2 , y1 , y2 , b) will have standard deviation \|y1 −y . 3 Also, if both of (a) and (b) happen, we will go with the first rule. Note that if Claim 1 holds, and σ̂ is the standard deviation of J (x1 , x2 , y1 , y2, b), then we will have \|σ̂−σ\| ≤ εσ. Claim 2. With probability at least 5/6, there exist x1 , x2 ∈ S such that x1 ∈ Ii and 2 =: µ̂. x2 ∈ I4M −i+1 for some i ∈ [2M]. If so, J (x1 , x2 , y1 , y2, b) will have mean x1 +x 2 Also note that if Claim 2 holds, then \|µ̂ − µ\| ≤ εσ. Therefore, if both claims hold, Lemma A.4 gives that J (x1 , x2 , y1 , y2, b) = N (µ̂, σ̂ 2 ) is a 2ε-approximation for N (µ, σ 2) = g. In other words, g can be approximately reconstructed, up to error 2ε, using only four data points (i.e., {x1 , x2 , y1 , y2}) from a sample S of size O(1/ε) and a single bit b (the definition of robust compression requires error ≤ ε. For getting this, one just needs to refine the partition by a constant factor, which multiplies M by a constant factor, and as we will see below, this will only multiply m by a constant factor). Note also that the 27 probability of existence of such four points is at least 1 − (1 − 5/6) − (1 − 5/6) ≥ 2/3. Therefore, it remains to prove Claim 1 and Claim 2. We start with Claim 1. View the sets I1 , . . . , I4M , I4M +1 as bins, and consider the R i.i.d. samples as balls landing in these bins according to q. Let pi := Ii g(x)dx and R qi := Ii q(x)dx for i ∈ [4M + 1]. Note that, by triangle’s inequality, the ℓ1 distance between (p1 , . . . , p4M +1 ) and (q1 , . . . , q4M +1 ) is not more than the L1 distance between g and q, which is at most r. We pair the bins as follows: Ii is paired with Ii+M for i ∈ {M + 1, . . . , 2M}, and Ii is paired with Ii+3M for i ∈ [M]. Therefore, by Lemma D.2, the probability that Claim 1 does not hold can be bounded by !m M 2M X X r 22M (pi ∨pi+M )+ (pi ∨pi+3M )+p4M +1 + 2 i=1 i=M +1 m  5 7 M M M 2 2 X X X r pi+ = 22M pi+p4M +1 +  , pi+ 2 3 M 3M +1 i= 2 M +1 i= 2 +1 where we have used the fact that pi are coming from a Gaussian, and thus p1 ≤ · · · ≤ p2M = p2M +1 ≥ · · · ≥ p4M (we have also assumed, for simplicity, that M is even). Let X ∼ N (µ, σ 2) and Φ(A) := Pr[N(0, 1) ∈ A]. Then using known numerical bounds for Φ, we obtain 2.5M X pi + i=1.5M +1 M X i=M/2+1 pi + 3.5M X pi + p4M +1 + r/2 3M +1 = Pr[X ∈ [µ − σ/2, µ + σ/2]] + 2Pr[X ∈ [µ − 3σ/2, µ − σ]] + Pr[X ∈ / [µ − 2σ, µ + 2σ]] + r/2 r r = Φ([−0.5, 0.5])+2Φ([−1.5, −1])+2Φ((−∞, −2])+ < 0.383 + 0.184 + 0.046 + 2 2 = 0.613 + r/2 ≤ 0.9995. Therefore since M = Θ(1/ε), by making m = C/ε for a large enough C, we can make this probability arbitrarily small, completing the proof of Claim 1. Via a similar argument, the probability that Claim 2 does not hold can be bounded by 22M =2 2M X i=1 2M max{pi , p4M −i+1 } + p4M +1 + r/2 !m = 22M 2M X i=1 pi + p4M +1 + r/2 !m (Φ([−1, 1]) + Φ([2, ∞) + r/2)m < 22M (0.5 + 0.023 + r/2)m < 22M (0.91)m < 1/6, for m = C/ε with a large enough C. 28 Remark D.4. By using more bits and adding more scales, one can show that 1-dimensional Gaussians admit (4, b(r), O(1/ε)) r-robust compression for any fixed r < 1 (the number of required bits and the implicit constant in the O will depend on the value of r). E E.1 Omitted proofs from Section 5 Proof of Lemma 5.1 The proof of the following lemma, which is called the ‘generalized Fano’s inequality,’ uses Fano’s inequality in information theory (Cover and Thomas, 2006, Theorem 2.10.1). It was first proved in (Devroye, 1987, page 77). We write here a slightly stronger version, which appears in (Yu, 1997, Lemma 3). Lemma E.1 (generalized Fano’s inequality). Suppose we have M > 1 distributions f1 , . . . , fM with KL(fi k fj ) ≤ β and kfi − fj k1 > α ∀i 6= j ∈ [M]. Consider any density estimation method that gets n i.i.d. samples from some fi , and outputs an estimate fb (the method does not know i). For each i, define ei as follows: assume the method receives samples from fi , and outputs fb. Then ei := Ekfi − fbk1 . Then, we have max ei ≥ α(log M − nβ + log 2)/(2 log M) . i To prove Lemma 5.1, consider a distribution learning method for learning F with sample complexity m(ε), and consider M distributions f1 , . . . , fM satisfying the hypotheses. Suppose we are in the setup of generalized Fano’s inequality: we get samples from some unknown j ∈ [M], and we are to find which fj are the samples coming from. When m(ε) samples are given to the method, with probability ≥ 2/3 it outputs some g within distance ε to fj . Suppose we repeat this procedure for k times, and the method outputs k distributions. If more than half of the times the method’s output was ε-close to some fi , then we output that fi as the answer; otherwise we output f1 . Our error would be 0 with probability Pr [Bin(k, 2/3) > k/2], and at most 2 with the remaining probability. Thus, the expected error can be upper bounded by exp(−Ω(k)) by the Chernoff bound. Thus, generalized Fano’s inequality gives α(log M − (km(ε))κ(ε) + log 2)/(2 log M) ≤ exp(−Ω(k)). Choosing k = Θ(log(1/ε)) and rearranging gives m(ε) = Ω(log M/κ(ε) log(1/ε)), as required. 29 E.2 Proof of Lemma 5.3 We use the probabilistic method. We let the d/r columns of each Ua to be the first d/r columns of a uniformly random orthonormal basis of Rd . To complete the proof, we need 2 only show that for two such random matrices Ua and Ub , with probability 1 − 2−Ω(d /r) d . we have kUaT Ub k2F ≤ 2r d In the following, U = V means U and V have the same distribution. By rotation d invariance, we may assume Ua = [e1 , . . . , ed/r ], so that kUaT Ub k2F = kUd/r k2F , where Ud/r is the d/r × d/r principal submatrix of a uniformly random orthogonal matrix U (alternatively, the columns of Ud/r are the first d/r coordinates of d/r orthonormal vectors in Rd chosen uniformly at random). Hence, it suffices to show that kUd/r k2F ≤ d/(2r) with 2 probability at least 1 − 2−Ω(d /r) . We will do this indirectly by relating Ud/r to a matrix with independent Gaussian entries. To that end, let G be a d × d/r matrix with i.i.d. N (0, 1/d) entries. Let G = UG ΣG VGT be its SVD, where UG ∈ Rd×d/r and ΣG , VG ∈ Rd/r×d/r . Observe that, by rotation invariance of the Gaussian matrix G, the columns of UG are d/r uniformly random orthonormal vectors and hence, the top d/r rows of UG have the same distribution as Ud/r . Moreover, by rotation invariance again, ΣG is independent of UG . Now let Gd/r be the first d/r rows of G. Then, d kGd/r k = (UG )d/r ΣG VG = Ud/r ΣG VG , and, taking Frobenius norms of both sides, d kGd/r kF = kUd/r ΣG VG kF = kUd/r ΣG kF ≥ σmin (ΣG )kUd/r kF , (4) with σmin (ΣG ) being the smallest singular value of ΣG . Since kGd/r k2F is a sum of i.i.d. √ random variables and concentrates sharply around its mean, and Eσmin (ΣG ) ≥ 1 − 1/ r by Gordon’s Theorem (Theorem A.12), this allows us to control kUd/r kF . In particular, for any p ≥ 1, we can bound a suitably translated moment of kUd/r kF . Let (x)+ := max{0, x}. Then, from (4) we get √ √ EG (kGd/r kF − d/r)p+ ≥ EUd/r ,ΣG (σmin (ΣG )kUd/r kF − d/r)p+ √ = EUd/r EΣG (σmin (ΣG )kUd/r kF − d/r)p+ √ ≥ EUd/r (EΣG σmin (ΣG )kUd/r kF − d/r)p+ √ √ ≥ EUd/r ((1 − 1/ r)kUd/r kF − d/r)p+ , where the second inequality is Jensen’s inequality √ inequality is Gordon’s √ and the third Theorem. Lemma A.6 gives that (kG √d/r kF − d/r) is O(1/ d)-subgaussian, and since √ the moments of (1 − 1/ r)kUd/r kF − d/r)+ are bounded by the moments of this random 30 variable, by equivalence of having moment bounds and being subgaussian (Lemma A.9), p √ √ we find that (1 − 1/r)kUd/r kF − d/r is also O(1/ d)-subgaussian. Hence, for any + t > 0, we have h i p √ 2 Pr (1 − 1/r)kUd/r kF − d/r ≤ t ≥ 1 − 2−Ω(t d) . √ √ Choosing t = d/(12 r) and the assumption that r ≥ 9 gives kUd/r k2F ≤ d/(2r) with 2 probability at least 1 − 2−Ω(d /r) , completing the proof. E.3 Proof of Lemma 5.4 Assume that kUaT Ub k2F ≤ d/(2r). Our goal is to show that TV(fa , fb ) = Ω √ λ d/r √ . log(r/λ d) √ Recall that fa = N (0, Σa ) with Σa = Id + λUa UaT . Let g ∼ N (0, Id). Then Σa g ∼ fa . √ √ Thus our goal is to lower bound TV( Σa g, Σb g). Since total variation distance never increases under a mapping (Fact A.5), we need only show that ! √ p p d/r λ √ . TV(Ua UaT Σa g, Ua UaT Σb g) = Ω log(r/λ d) √ P Now let C := Ua UaT Σb and since C is symmetric, its SVD has form C = i∈[d/r] σi wi wiT for orthonormal {wi } (some σi may be zero). Let S denote the column space of Ua ; so w1 , . . . , wd/r ∈ S, and we may assume w1 , . . . , wd/r is an orthonormal basis for S. This √ d P means Ua UaT Σb g = i∈[d/r] σi gi wi , where the gi are the components of g. Note that X σi2 = kCk2F = Tr(CC T ) = Tr(Ua UaT (1 + λUb UbT )Ua UaT ) i∈[d/r] = Tr(Ua UaT ) + λ Tr(Ua UaT Ub UbT Ua UaT ) = = d + λ Tr(UaT Ua UaT Ub UbT Ua ) r d d + λkUaT Ub k2F ≤ (1 + λ/2). r r √ On the other hand, suppose u1 , . . . , ud/r are the columns of Ua . Then, (Ua UaT Σa )2 = √ √ P d √ (1+λ)Ua UaT and hence, Ua UaT Σa g = 1 + λ i∈[d/r] gi ui . That is, Ua UaT Σa g is a spher√ P d ical Gaussian in the subspace S. So, by its rotation invariance, we have 1 + λ i∈[d/r] gi ui = √ P 1 + λ i∈[d/r] gi wi . Hence our goal is to show √ TV( 1 + λ d/r X gi wi , i=1 provided P i∈[d/r] d/r X σi gi wi ) = Ω i=1 σi2 ≤ d(1 + λ/2)/r. 31 √ λ d/r √ log(r/λ d) ! , 2 By reordering the wi , we may assume σ12 ≤ · · · ≤ σd/r . At most half of the σi2 can be 2 twice their average, which means σ12 ≤ · · · ≤ σd/2r ≤ 2(1 + λ/2) ≤ 3. Now we may project the two random vectors onto the subspace generated by the d/(2r) smallest eigenvectors, and this can only decrease √ the total variation distance. Taking the norms of the projected vectors and dividing by d can only decrease the total variation distance; hence our new goal is to show ! √ d/2r d/2r X λ d/r gi 2 X 2 gi2 √ √ , , (5) σi √ ) = Ω TV((1 + λ) d d log(r/λ d) i=1 i=1 provided P i∈[d/2r] σi2 ≤ d(1 + λ/2)/2r and maxi σi ≤ 3. Observe that     d/2r d/2r 2 2 X X √ √ √ gi g √ −E E (1 + λ) σi2 √i  ≥ (1 + λ)( d/2r) − d(1 + λ/2)/2r = λ d/(4r). d d i=1 i=1 Moreover, by Bernstein’s inequality (Lemma A.11), there exists a global constant c > 0 (independent of λ) such that for any t > 0,   d/(2r) d/(2r) X g2 X g2 √ i i 2   √ − E(1 + λ) √ > t ≤ 2 exp −c min{t , t d} , Pr (1 + λ) d d i=1 i=1 and, since σi2 ≤ 3 for all i,   d/2r d/2r 2 2 X X √ g g Pr  σi2 √i − E σi2 √i > t ≤ 2 exp −c min{t2 , t d} . d d i=1 i=1 √ Applying Lemma E.2 gives (5) (note that ζ ≤ O(log(r/λ d)) in the lemma), as required. Lemma E.2. Let X, Y be continuous random variables such that \|EX − EY \| ≥ ∆. Suppose also there exist c, C, β such that for any t > 0 we have max{Pr [\|X − EX\| ≥ t] , Pr [\|Y − EY \| ≥ t]} ≤ C exp(−c min{t2 , βt}). Let ζ := max{1, ∆, log(4C)/cβ, Then, TV(X, Y ) ≥ ∆/8ζ. p p log(4C)/c, log(8C/cβ∆)/cβ, log(4C/c∆)/c}. 32 Proof. We shall show that, for any h > 0 we have TV(X, Y ) ≥ ∆ − C exp(−c min(h2 , βh))∆ − C exp(−ch2 )/ch − 2C exp(−cβh)/cβ , (6) h+∆ and the lemma will follow by choosing h = ζ. Without loss of generality, we may assume EY ≥ EX and by translating, we may assume EX = −∆/2 and EY = ∆/2. Let I = [EX − h, EY + h], I ′ = R \ I and let f (x), g(x) denote the densities of X and Y . Define random variable Z to be \|X − EX\| if \|X − EX\| > h, and 0 otherwise. Then, since I ′ ⊆ (−∞, EX − h) ∪ (EX + h, ∞), we have Z EZ = \|(x − EX)f (x)\|dx (−∞,EX−h)∪(EX+h,∞) Z ≥ (\|x\|f (x) − \|EX\|f (x))dx ′ ZI = \|xf (x)\|dx − (∆/2)Pr [\|X − EX\| > h] ′ ZI \|xf (x)\|dx − C exp(−c min(h2 , βh))∆/2 ≥ I′ On the other hand, Z ∞ Z ∞ EZ = Pr [Z > t] dt = Pr [\|X − EX\| > t] dt 0 h Z ∞ Z ∞ Z 2 2 ≤ C exp(−c min{t , βt})dt ≤ C exp(−ct )dt + h h ∞ C exp(−cβt)dt h ≤ C exp(−ch2 )/2ch + C exp(−cβh)/cβ, where for the last inequality we have used tail bounds for the standard normal distribution (see Abramowitz and Stegun, 1984, formula (7.1.13)). Thus we find Z \|xf (x)\|dx ≤ C exp(−ch2 )/2ch + C exp(−cβh)/cβ + C exp(−c min(h2 , βh))∆/2. I′ A similar calculation gives the same upper bound for Z Z ∆ = x(g(x) − f (x))dx + x(g(x) − f (x))dx ′ I I Z Z ≤ \|x\|\|g(x) − f (x)\|dx + \|x\|(\|g(x)\| + \|f (x)\|)dx I R I′ \|xg(x)\|dx. Finally, observe that I′ ≤ kf (x) − g(x)k1 (h + ∆)/2 + C exp(−ch2 )/ch + 2C exp(−cβh)/cβ + C exp(−c min(h2 , βh))∆, since \|x\| ≤ (h + ∆)/2 for all x ∈ I. Re-arranging and noting total variation distance is half the L1 distance gives (6). 33 E.4 Proof of Theorem 5.5 We begin with a combinatorial lemma. Let dH (·, ·) denote the Hamming distance between two tuples. Lemma E.3. Let T ≥ 2 and k ∈ N. There exists a set of tuples X ⊆ [T ]k such that \|X \| ≥ 2Ω(k log(T )) and dH (x, y) ≥ k/4 for any pair of distinct x, y ∈ X . Proof. Let x and y be strings of length k where each coordinate is drawn from [T ] independently and uniformly at random. Let Xi be the indicator random variable which is P 1 if xi = yi . Then k − dH (x, y) = ki=1 Xi and E[k − dH (x, y)] =p k/T ≤ k/2. Observe that Xi = 1 with probability 1/T and 0 otherwise; hence it is 1/ log(T )-subgaussian. By Hoeffding’s Inequality (Lemma A.10), " k # X Pr Xi ≥ 3k/4 ≤ 2 · exp(−ck log(T )) i=1 for some absolute constant c > 0. Thus, we conclude that there is some set X with \|X \| ≥ 2Ω(k log(T )) and dH (x, y) ≥ k/4 for any pair of distinct x, y ∈ X . We now prove Theorem 5.5, using Lemma 5.1 again. The proof of Theorem 5.2 2 promises a collection of T = 2Ω(d ) matrices Σ1 , . . . , ΣT ≺ 2Id with KL(N (0, Σi k N (0, Σi′ )) ≤ O(ε2 log2 (1/ε)) and TV(N (0, Σi ), N (0, Σi′ )) ≥ Ω(ε) for i 6= i′ . Choose µ1 , . . . , µk ∈ Rd such that kµi − µi′ k2 ≥ C(kd/ε)10 for all i 6= i′ , where C is a large enough constant. 2 By Lemma E.3, there exists a set X ⊆ [T ]k of size 2Ω(k log(T )) = 2Ω(kd ) such that dH (x, y) ≥ k/4 for any distinct x, y ∈ X . We now define a set of mixture distributions as 1 F = fx := (N (µ1, Σx1 ) + . . . + N (µk , Σxk )) : x ∈ X . k We shall show that for any x 6= y, we have KL(fx k fy ) ≤ O(ε2 log2 (1/ε)) and TV(fx , fy ) ≥ Ω(ε). Lemma 5.1 will then conclude the proof. The proof of upper bound for KL divergence simply follows from convexity of KLdivergence (see Cover and Thomas, 2006, Theorem 2.7.2). and the fact that, for each i, KL(N (µi , Σxi k N (µi , Σyi )) = KL(N (0, Σxi k N (0, Σyi )) ≤ O(ε2 log2 (1/ε)). Next we show that TV(fx , fy ) ≥ Ω(ε). Let o n ′ Aj ∈ argmax Prg∼N (µj ,Σxj ) [g ∈ A] − Prg∼N (µj ,Σyj ) [g ∈ A] . A⊆Rd Since Σi ≺ 2Id for all i, we have Prg∼N (µj ,Σxj ) [kg − µj k22 ≥ 2d + O(d log(k/ε))] ≤ ε2 /k 2 , 34 p and a similar bound holds for N (µj , Σyj ). Now define Aj = A′j ∩ B2d (µj , O( d log(k/ε))), where B2d (µ, r) denotes the ℓ2 ball centered at µ with radius r. Then, Prg∼N (µj ,Σxj ) [g ∈ Aj ] − Prg∼N (µj ,Σyj ) [g ∈ Aj ] ≥ TV(N (µj , Σxj ), N (µj , Σyj )) − ε2 /k 2 . Note that the separation of µ1 , . . . , µk implies that A1 , . . . , Ak are disjoint sets and Prg∼N (µi ,Σyi ) [g ∈ Aj ] ≤ ε2 /k 2 for i 6= j. Let A = ∪j Aj . Finally, to lower bound the total variation distance, we have TV(fx , fy ) ≥ Prg∼fx [g ∈ A] − Prg∼fy [g ∈ A] = k X j=1 Prg∼fx [g ∈ Aj ] − Prg∼fy [g ∈ Aj ] k k 1 XX Prg∼N (µi ,Σxi ) [g ∈ Aj ] − Prg∼N (µi ,Σyi ) [g ∈ Aj ] = k j=1 i=1 k i 1 Xh = Prg∼N (µj ,Σxj ) [g ∈ Aj ] − Prg∼N (µj ,Σyj ) [g ∈ Aj ] k j=1 k 1 XX Prg∼N (µi ,Σxi ) [g ∈ Aj ] − Prg∼N (µi ,Σyi ) [g ∈ Aj ] + k j=1 i6=j i 1 Xh ≥ Prg∼N (µj ,Σxj ) [g ∈ Aj ] − Prg∼N (µj ,Σyj ) [g ∈ Aj ] − ε2 k j=1 k k ≥ 1 X TV(N (µj , Σxj ), N (µj , Σyj )) − ε2 /k 2 − ε2 k j=1 1 (k/4)Ω(ε) − 2ε2 ≥ Ω(ε), k where the last inequality is because TV(N (µj , Σxj ), N (µj , Σyj )) ≥ Ω(ε) whenever xj 6= yj which is the case for at least k/4 of the indices j. ≥ References M. Abramowitz and I. A. Stegun, editors. Handbook of mathematical functions with formulas, graphs, and mathematical tables. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York; National Bureau of Standards, Washington, DC, 1984. ISBN 0-471-80007-4. 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Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model∗ Keren Censor-Hillel Seri Khoury Ami Paz May 17, 2017 arXiv:1705.05646v1 [cs.DC] 16 May 2017 Abstract We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires Ω(n2 / log2 n) rounds in the worst case in the CONGEST model, as well as any algorithm for χ-coloring a graph, where χ is the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and nearquadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairsshortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple Ω(n) lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a complexity of Θ(n/ log n). We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question. ∗ Department of Computer Science, Technion, {ckeren,serikhoury,amipaz}@cs.technion.ac.il. Supported in part by ISF grant 1696/14. 1 1 Introduction It is well-known and easily proven that many graph problems are global for distributed computing, in the sense that solving them necessitates communication throughout the network. This implies tight Θ(D) complexities, where D is the diameter of the network, for global problems in the LOCAL model. In this model, a message of unbounded size can be sent over each edge in each round, which allows to learn the entire topology in D rounds. Global problems are widely studied in the CONGEST model, in which the size of each message is restricted to O(log n) bits, where n is the size of the network. The trivial complexity of learning the entire topology in the CONGEST model is O(m), where m is the number of edges of the communication graph, and since m can be as large as Θ(n2 ), one of the most basic questions for a global problem is how fast in terms of n it can be solved in the CONGEST model. Some global problems admit fast O(D)-round solutions in the CONGEST model, such as √ constructing a breadth-first search tree [59]. Some others have complexities of Θ̃(D + n), such as constructing a minimum spanning tree, and various approximation and verification problems [32, 39, 45, 60, 61, 64]. Some problems are yet harder, with complexities that are nearlinear in n [1, 32, 41, 51, 60]. For some problems, no O(n) solutions are known and they are candidates to being even harder that the ones with linear-in-n complexities. A major open question about global graph problems in the CONGEST model is whether natural graph problems for which a super-linear number of rounds is required indeed exist. In this paper, we answer this question in the affirmative. That is, our conceptual contribution is that there exist super-linearly hard problems in the CONGEST model. In fact, the lower bounds that we prove in this paper are as high as quadratic in n, or quadratic up to logarithmic factors, and hold even for networks of a constant diameter. Our lower bounds also imply linear and near-linear lower bounds for the CLIQUE-BROADCAST model. We note that high lower bounds for the CONGEST model may be obtained rather artificially, by forcing large inputs and outputs that must be exchanged. However, we emphasize that all the problems for which we show our lower bounds can be reduced to simple decision problems, where each node needs to output a single bit. All inputs to the nodes, if any, consist of edge weights that can be represented by polylogn bits. Technically, we prove a lower bound of Ω(n2 / log2 n) on the number of rounds required for computing an exact minimum vertex cover, which also extends to computing an exact maximum independent set (Section 3.1). This is in stark contrast to the recent O(log ∆/ log log ∆)-round algorithm of [8] for obtaining a (2 + )-approximation to the minimum vertex cover. Similarly, we give an Ω(n2 / log2 n) lower bound for 3-coloring a 3-colorable graph, which extends also for deciding whether a graph is 3-colorable, and also implies the same hardness for computing the chromatic number χ or computing a χ-coloring (Section 3.2). These lower bounds hold even for randomized algorithms which succeed with high probability.1 An immediate question that arises is whether only NP-hard problems are super-linearly hard in the CONGEST model. In Section 4, we provide a negative answer to such a postulate, by showing two simple problems that admit polynomial-time sequential algorithms, but in the CONGEST model require Ω(n2 ) rounds (identical subgraph detection) or Ω(n2 / log n) rounds (weighted cycle detection). The latter also holds for randomized algorithms, while for the former we show a randomized algorithm that completes in O(D) rounds, providing the strongest possible separation between deterministic and randomized complexities for global problems in the CONGEST model. Finally, we address the intriguing open question of the complexity of computing exact weighted all-pairs-shortest-paths (APSP) in the CONGEST model. While the complexity of the unweighted version of APSP is Θ(n/ log n), as follows from [32, 42], the complexity of weighted 1 We say that an event occurs with high probability (w.h.p) if it occurs with probability c > 0. 1 1 , nc for some constant APSP remains largely open, and only recently the first sub-quadratic algorithm was given in [28]. With the current state-of-the-art, this problem could be considered as a suspect for having a super-linear complexity in the CONGEST model. While we do not pin-down the complexity of weighted APSP in the CONGEST model, we provide a truly linear lower bound of Ω(n) rounds for it, which separates its complexity from that of the unweighted case. Moreover, we argue that it is not a coincidence that we are currently unable to show super-linear lower bound for weighted APSP, by formally proving that the commonly used framework of reducing a 2-party communication problem to a problem in the CONGEST model cannot provide a super-linear lower bound for weighted APSP, regardless of the function and the graph construction used (Section 5). This implies that obtaining any super-linear lower bound for weighted APSP provably requires a new technique. 1.1 The Challenge Many lower bounds for the CONGEST model rely on reductions from 2-party communication problems (see, e.g., [1,17,25,27,32,41,56,57,61,64]). In this setting, two players, Alice and Bob, are given inputs of K bits and need to a single output a bit according to some given function of their inputs. One of the most common problem for reduction is Set Disjointness, in which the players need to decide whether there is an index for which both inputs are 1. That is, if the inputs represent subsets of {0, . . . , K − 1}, the output bit of the players needs to indicate whether their input sets are disjoint. The communication complexity of 2-party Set Disjointness is known to be Θ(K) [49]. In a nutshell, there are roughly two standard frameworks for reducing the 2-party communication problem of computing a function f to a problem P in the CONGEST model. One of these frameworks works as follows. A graph construction is given, which consists of some fixed edges and some edges whose existence depends on the inputs of Alice and Bob. This graph should have the property that a solution to P over it determines the solution to f . Then, given an algorithm ALG for solving P in the CONGEST model, the vertices of the graph are split into two disjoint sets, VA and VB , and Alice simulates ALG over VA while Bob simulates ALG over VB . The only communication required between Alice and Bob in order to carry out this simulation is the content of messages sent in each direction over the edges of the cut C = E(VA , VB ). Therefore, given a graph construction with a cut of size \|C\| and inputs of size K for a function f whose communication complexity on K bits is at least CC(f ), the round complexity of ALG is at least Ω(CC(f )/\|C\| log n). The challenge in obtaining super-linear lower bounds was previously that the cuts in the graph constructions were large compared with the input size K. For example, the graph construction for the lower bound for computing the diameter in [32] has K = Θ(n2 ) and \|C\| = Θ(n), which gives an almost linear lower bound. The graph construction in [32] for the lower bound √ for computing a (3/2 − )-approximation to the diameter has a smaller cut of \|C\| = Θ( n), but this comes at the price of supporting a smaller input size K = Θ(n), which gives a lower bound that is roughly a square-root of n. To overcome this difficulty, we leverage the recent framework of [1], which provides a bitgadget whose power is in allowing a logarithmic-size cut. We manage to provide a graph construction that supports inputs of size K = Θ(n2 ) in order to obtain our lower bounds for minimum vertex cover, maximum independent set and 3-coloring2 . The latter is also inspired by, and is a simplification of, a lower bound construction for the size of proof labelling schemes [33]. Further, for the problems in P that we address, the cut is as small as \|C\| = O(1). For one of the problems, the size of the input is such that it allows us to obtain the highest possible lower bound of Ω(n2 ) rounds. 2 It can also be shown, by simple modifications to our constructions, that these problems require Ω(m) rounds, for graphs with m edges. 2 With respect to the complexity of the weighted APSP problem, we show an embarrassingly simple graph construction that extends a construction of [56], which leads to an Ω(n) lower bound. However, we argue that a new technique must be developed in order to obtain any superlinear lower bound for weighted APSP. Roughly speaking, this is because given a construction with a set S of nodes that touch the cut, Alice and Bob can exchange O(\|S\|n log n) bits which encode the weights of all lightest paths from any node in their set to a node in S. Since the cut has Ω(\|S\|) edges, and the bandwidth is Θ(log n), this cannot give a lower bound of more than Ω(n) rounds. With some additional work, our proof can be carried over to a larger number of players at the price of a small logarithmic factor, as well as to the second Alice-Bob framework used in previous work (e.g. [64]), in which Alice and Bob do not simulate nodes in a fixed partition, but rather in decreasing sets that partially overlap. Thus, determining the complexity of weighted APSP requires new tools, which we leave as a major open problem. 1.2 Additional Related Work Vertex Coloring, Minimum Vertex Cover, and Maximum Independent Set: One of the most central problems in graph theory is vertex coloring, which has been extensively studied in the context of distributed computing (see, e.g., [9–14, 18, 20, 21, 29–31, 37, 53, 55, 62, 65] and references therein). The special case of finding a (∆ + 1)-coloring, where ∆ is the maximum degree of a node in the network, has been the focus of many of these studies, but is a local problem, which can be solved in much less than a sublinear number of rounds. Another classical problem in graph theory is finding a minimum vertex cover (MVC). In distributed computing, the time complexity of approximating MVC has been addressed in several cornerstone studies [5, 6, 8, 14, 34–36, 44, 46–48, 58, 63]. Observe that finding a minimum size vertex cover is equivalent to finding a maximum size independent set. However, these problems are not equivalent in an approximation-preserving way. Distributed approximations for maximum independent set has been studied in [7,15,22,52]. Distance Computations: Distance computation problems have been widely studied in the CONGEST model for both weighted and unweighted networks [1, 32, 38–42, 50, 51, 56, 60]. One of the most fundamental problems of distance computations is computing all pairs shortest paths. For unweighted networks, an upper bound of O(n/ log n) was recently shown by [42], matching the lower bound of [32]. Moreover, the possibility of bypassing this near-linear barrier for any constant approximation factor was ruled out by [56]. For the weighted case, however, we are still very far from understanding the complexity of APSP, as there is still a huge gap 2 5 between the upper and lower bounds. Recently, Elkin [28] showed an O(n 3 · log 3 (n)) upper bound for weighted APSP, while the previously highest lower bound was the near-linear lower bound of [56] (which holds also for any (poly n)-approximation factor in the weighted case). Distance computation problems have also been considered in the CONGESTED-CLIQUE model [16, 38, 40], in which the underlying communication network forms a clique. In this model [16] showed that unweighted APSP, and a (1 + o(1))-approximation for weighted APSP, can be computed in O(n0.158 ) rounds. Subgraph Detection: The problem of finding subgraphs of a certain topology has received a lot of attention in both the sequential and the distributed settings (see, e.g., [2–4, 16, 23–25, 43, 54, 66] and references therein). The problems of finding paths of length 4 or 5 with zero weight are also related to other fundamental problems, notable in our context is APSP [2]. 3 2 2.1 Preliminaries Communication Complexity In a two-party communication complexity problem [49], there is a function f : {0, 1}K × {0, 1}K → {TRUE, FALSE}, and two players, Alice and Bob, who are given two input strings, x, y ∈ {0, 1}K , respectively, that need to compute f (x, y). The communication complexity of a protocol π for computing f , denoted CC(π), is the maximal number of bits Alice and Bob exchange in π, taken over all values of the pair (x, y). The deterministic communication complexity of f , denoted CC(f ), is the minimum over CC(π), taken over all deterministic protocols π that compute f . In a randomized protocol π, Alice and Bob may each use a random bit string. A randomized protocol π computes f if the probability, over all possible bit strings, that π outputs f (x, y) is at least 2/3. The randomized communication complexity of f , CC R (f ), is the minimum over CC(π), taken over all randomized protocols π that compute f . In the Set Disjointness problem (DISJK ), the function f is DISJK (x, y), whose output is FALSE if there is an index i ∈ {0, . . . , K − 1} such that xi = yi = 1, and TRUE otherwise. In the Equality problem (EQK ), the function f is EQK (x, y), whose output is TRUE if x = y, and FALSE otherwise. Both the deterministic and randomized communication complexities of the DISJK problem are known to be Ω(K) [49, Example 3.22]. The deterministic communication complexity of EQK is in Ω(K) [49, Example 1.21], while its randomized communication complexity is in Θ(log K) [49, Example 3.9]. 2.2 Lower Bound Graphs To prove lower bounds on the number of rounds necessary in order to solve a distributed problem in the CONGEST model, we use reductions from two-party communication complexity problems. To formalize them we use the following definition. Definition 1. (Family of Lower Bound Graphs) Fix an integer K, a function f : {0, 1}K × {0, 1}K → {TRUE, FALSE} and a predicate P for graphs. The family of graphs {Gx,y = (V, Ex,y ) \| x, y ∈ {0, 1}K }, is said to be a family of lower bound graphs w.r.t. f and P if the following properties hold: ˙ B a fixed (1) The set of nodes V is the same for all graphs, and we denote by V = VA ∪V partition of it; (2) Only the existence or the weight of edges in VA × VA may depend on x; (3) Only the existence or the weight of edges in VB × VB may depend on y; (4) Gx,y satisfies the predicate P iff f (x, y) = TRUE. We use the following theorem, which is standard in the context of communication complexitybased lower bounds for the CONGEST model (see, e.g. [1, 25, 32, 40]) Its proof is by a standard simulation argument. Theorem 1. Fix a function f : {0, 1}K ×{0, 1}K → {TRUE, FALSE} and a predicate P . If there is a family {Gx,y } of lower bound graphs with C = E(VA , VB ) then any deterministic algorithm for deciding P in the CONGEST model requires Ω(CC(f )/ \|C\| log n) rounds, and any randomized algorithm for deciding P in the CONGEST model requires Ω(CC R (f )/ \|C\| log n) rounds. Proof. Let ALG be a distributed algorithm in the CONGEST model that decides P in T rounds. Given inputs x, y ∈ {0, 1}K to Alice and Bob, respectively, Alice constructs the part of Gx,y 4 for the nodes in VA and Bob does so for the nodes in VB . This can be done by items (1),(2) and (3) in Definition 1, and since {Gx,y } satisfies this definition. Alice and Bob simulate ALG by exchanging the messages that are sent during the algorithm between nodes of VA and nodes of VB in either direction. (The messages within each set of nodes are simulated locally by the corresponding player without any communication). Since item (4) in Definition 1 also holds, we have that Alice and Bob correctly output f (x, y) based on the output of ALG. For each edge in the cut, Alice and Bob exchange O(log n) bits per round. Since there are T rounds and \|C\| edges in the cut, the number of bits exchanged in this protocol for computing f is O(T \|C\| log n). The lower bounds for T now follows directly from the lower bounds for CC(f ) and CC R (f ). In what follows, for each decision problem addressed, we describe a fixed graph construction G = (V, E), which we then generalize to a family of graphs {Gx,y = (V, Ex,y ) \| x, y ∈ {0, 1}K }, which we show to be a family lower bound graphs w.r.t. to some function f and the required predicate P . By Theorem 1 and the known lower bounds for the 2-party communication problem, we deduce a lower bound for any algorithm for deciding P in the CONGEST model. Remark: For our constructions which use the Set Disjointness function as f , we need to exclude the possibilities of all-1 input vectors. This is for the sake of guaranteeing that the graphs are connected, in order to avoid trivial impossibilities. However, this restriction does not change the asymptotic bounds for Set Disjointness, since computing this function while excluding all-1 input vectors can be reduced to computing this function for inputs that are shorter by one bit (by having the last bit be fixed to 0). 3 3.1 Near-Quadratic Lower Bounds for NP-Hard Problems Minimum Vertex Cover The first near-quadratic lower bound we present is for computing a minimum vertex cover, as stated in the following theorem. Theorem 2. Any distributed algorithm in the CONGEST model for computing a minimum vertex cover or for deciding whether there is a vertex cover of a given size M requires Ω(n2 / log2 n) rounds. Finding the minimum size of a vertex cover is equivalent to finding the maximum size of a maximum independent set, because a set of nodes is a vertex cover if and only if its complement is an independent set. Thus, Theorem 3 is a direct corollary of Theorem 2. Theorem 3. Any distributed algorithm in the CONGEST model for computing a maximum independent set or for deciding whether there is an independent set of a given size requires Ω(n2 / log2 n) rounds. Observe that a lower bound for deciding whether there is a vertex cover of some given size M or not implies a lower bound for computing a minimum vertex cover. This is because computing the size of a given subset of nodes can be easily done in O(D) rounds using standard tools. Therefore, to prove Theorem 2 it is sufficient to prove its second part. We do so by describing a family of lower bound graphs with respect to the Set Disjointness function and the predicate P that says that the graph has a vertex cover of size M . We begin with describing the fixed graph construction G = (V, E) and then define the family of lower bound graphs and analyze its relevant properties. The fixed graph construction: Let k be a power of 2. The fixed graph (Figure 1) consists 5 Figure 1: The family of lower bound graphs for deciding the size of a vertex cover, with many edges omitted for clarity. The node ak−1 is connected to all the nodes in TA1 , and a12 is connected to t0A2 and 1 0 to all the nodes in FA2 \ {fA2 }. Examples of edges from b01 and b02 to the bit-gadgets are also given. An additional edge, which is among the edges corresponding to the strings x and y, is {b01 , b12 }, while the edge {a01 , a02 } does not exist. Here, x0,0 = 1 and y0,1 = 0. of four cliques of size k: A1 = {ai1 \| 0 ≤ i ≤ k − 1}, A2 = {ai2 \| 0 ≤ i ≤ k − 1}, B1 = {bi1 \| 0 ≤ i ≤ k − 1} and B2 = {bi2 \| 0 ≤ i ≤ k − 1}. In addition, for each set S ∈ {A1 , A2 , B1 , B2 }, there are two corresponding sets of nodes of size log k, denoted FS = {fSh \| 0 ≤ h ≤ log k − 1} and TS = {thS \| 0 ≤ h ≤ log k − 1}. The latter are called bit-gadgets and their nodes are bit-nodes. The bit-nodes are partitioned into 2 log k 4-cycles: for each h ∈ {0, . . . , log k − 1} and ` ∈ {1, 2}, we connect the 4-cycle (fAh` , thA` , fBh ` , thB` ). Note that there are no edges between pairs of nodes denoted fSh , or between pairs of nodes denoted thS . The nodes of each set S ∈ {A1 , A2 , B1 , B2 } are connected to nodes in the corresponding set of bit-nodes, according to their binary representation, as follows. Let si` be a node in a set S ∈ {A1 , A2 , B1 , B2 }, i.e. s ∈ {a, b}, ` ∈ {1, 2} and i ∈ {0, . . . , k − 1}, and let ihdenote the bit number h in the binary representation of i. For such a node si` define bin(si` ) = fSh \| ih = 0 ∪ h tS \| ih = 1 , and connect si` by an edge to each of the nodes in bin(si` ). The next two claims address the basic properties of vertex covers of G. Claim 1. Any vertex cover of G must contain at least k − 1 nodes from each of the clique A1 , A2 , B1 and B2 , and at least 4 log k bit-nodes. Proof. In order to cover all the edges of each if the cliques on A1 , A2 , B1 and B2 , any vertex cover must contain at least k − 1 nodes of the clique. For each h ∈ {0, . . . , log k − 1} and ` ∈ {1, 2}, in order to cover the edges of the 4-cycle (fAh` , thA` , fBh ` , thB` ), any vertex cover must contain at least two of the cycle nodes. Claim 2. If U ⊆ V is a vertex cover of G of size 4(k − 1) + 4 log k, then there are two indices i, j ∈ {0, . . . , k − 1} such that ai1 , aj2 , bi1 , bj2 are not in U . Proof. By Claim 1, U must contain k − 1 nodes from each clique A1 , A2 , B1 and B2 , and 4 log k 0 0 bit-nodes, so it must not contain one node from each clique. Let ai1 , aj2 , bi1 , bj2 be the nodes in A1 , A2 , B1 , B2 which are not in U , respectively. To cover the edges connecting ai1 to bin(ai1 ), U 0 must contain all the nodes of bin(ai1 ), and similarly, U must contain all the nodes of bin(bi1 ). If i 6= i0 then there is an index h ∈ {0, . . . , log k − 1} such that ih 6= i0h , so one of the edges 6 (fAh1 , thB1 ) or (thA1 , fBh 1 ) is not covered by U . Thus, it must hold that i = i0 . A similar argument shows j = j 0 . Adding edges corresponding to the strings x and y: Given two binary strings x, y ∈ 2 {0, 1}k , we augment the graph G defined above with additional edges, which defines Gx,y . Assume that x and y are indexed by pairs of the form (i, j) ∈ {0, . . . , k − 1}2 . For each such pair (i, j) we add to Gx,y the following edges. If xi,j = 0, then we add an edge between the nodes ai1 and aj2 , and if yi,j = 0 then we add an edge between the nodes bi1 and bj2 . To prove that {Gxy } is a family of lower bound graphs, it remains to prove the next lemma. Lemma 1. The graph Gx,y has a vertex cover of cardinality M = 4(k − 1) + 4 log k iff DISJ(x, y) = FALSE. Proof. For the first implication, assume that DISJ(x, y) = FALSE and let i, j ∈ {0, . . . , k − 1} be such that xi,j = yi,j = 1. Note that in this case ai1 is not connected to aj2 , and bi1 is not connected to bj2 . We define a set U ⊆ V as the union of the two sets of nodes (A1 \ {ai1 }) ∪ (A2 \ {aj2 }) ∪ (B1 \ {bi1 }) ∪ (B2 \ {bj2 }) and bin(ai1 ) ∪ bin(aj2 ) ∪ bin(bi1 ) ∪ bin(bj2 ), and show that U is a vertex cover of Gx,y . First, U covers all the edges inside the cliques A1 , A2 , B1 and B2 , as it contains k − 1 nodes from each clique. These nodes also cover all the edges connecting nodes in A1 to nodes in A2 and all the edges connecting nodes in B1 to nodes in B2 . Furthermore, U covers any edge connecting some node u ∈ (A1 \{ai1 })∪(A2 \{aj2 })∪(B1 \{bi1 })∪(B2 \{bj2 }) with the bit-gadgets. For each node s ∈ ai1 , aj2 , bi1 , bj2 , the nodes bin(s) are in U , so U also cover the edges connecting s to the bit gadget. Finally, U covers all the edges inside the bit gadgets, as from each 4-cycle (fAh` , thA` , fBh ` , thB` ) it contains two non-adjacent nodes: if ih = 0 then fAh1 , fBh 1 ∈ U and otherwise thA1 , thB1 ∈ U , and if jh = 0 then fAh2 , fBh 2 ∈ U and otherwise thA2 , thB2 ∈ U . We thus have that U is a vertex cover of size 4(k − 1) + 4 log k, as needed. For the other implication, let U ⊆ V be a vertex cover of Gx,y of size 4(k − 1) + 4 log k. As the set of edges of G is contained in the set of edges of Gx,y , U is also a cover of G, and by Claim 2 there are indices i, j ∈ {0, . . . , k − 1} such that ai1 , aj2 , bi1 , bj2 are not in U . Since U is a cover, the graph does not contain the edges (ai1 , aj2 ) and (bi1 , bj2 ), so we conclude that xi,j = yi,j = 1, which implies that DISJ(x, y) = FALSE. Having constructed the family of lower bound graphs, we are now ready to prove Theorem 2. Proof of Theorem 2: To complete the proof of Theorem 2, we divide the nodes of G (which are also the nodes of Gx,y ) into two sets. Let VA = A1 ∪ A2 ∪ FA1 ∪ TA1 ∪ FA2 ∪ TA2 and VB = V \ VA . Note that n ∈ Θ(k), and thus K = \|x\| = \|y\| = Θ(n2 ). Furthermore, note that the only edges in the cut E(VA , VB ) are the edges between nodes in {FA1 ∪ TA1 ∪ FA2 ∪ TA2 } and nodes in {FB1 ∪ TB1 ∪ FB2 ∪ TB2 }, which are in total Θ(log n) edges. Since Lemma 1 shows that {Gx,y } is a family of lower bound graphs, we can apply Theorem 1 on the above partition to deduce that because of the lower bound for Set Disjointness, any algorithm in the CONGEST model for deciding whether a given graph has a cover of cardinality M = 4(k − 1) + 4 log k requires at least Ω(K/ log2 (n)) = Ω(n2 / log2 (n)) rounds. 3.2 Graph Coloring Given a graph G, we denote by χ(G) the minimal number of colors in a proper vertex-coloring of G. In this section we consider the problems of coloring a graph in χ colors, computing χ and approximating it. We prove the next theorem. Theorem 4. Any distributed algorithm in the CONGEST model that colors a χ-colorable graph G in χ colors or compute χ(G) requires Ω(n2 / log2 n) rounds. 7 Figure 2: The family of lower bound graphs for coloring, with many edges omitted for clarity. The node Ca1 is connected to all the nodes in FA1 ∪ TA1 and Cb1 is connected to all the nodes in FB1 ∪ TB1 . The node Ca2 is connected to all the nodes in FA2 ∪ TA2 and Cb2 is connected to all the nodes in FB2 ∪ TB2 . Any distributed algorithm in the CONGEST model that decides if χ(G) ≤ c for a given integer c, requires Ω((n − c)2 /(c log n + log2 n)) rounds. We give a detailed lower bound construction for the first part of the theorem, by showing that distinguishing χ ≤ 3 from χ ≥ 4 is hard. Then, we extend our construction to deal with deciding whether χ ≤ c. The fixed graph construction: We describe a family of lower bound graphs, which builds upon the family of graphs defined in Section 3.1. We define G = (V, E) as follows (see Figure 2). There are four sets of size k: A1 = {ai1 \| 0 ≤ i ≤ k − 1}, A2 = {ai2 \| 0 ≤ i ≤ k − 1}, B1 = {bi1 \| 0 ≤ i ≤ k − 1} and B2 = {bi2 \| 0 ≤ i ≤ k − 1}. As opposed to the construction in Section 3.1, the nodes of these sets are not connected to one another. In addition, as in Section 3.1, for each set S ∈ {A1 , A2 , B1 , B2 }, there are two corresponding sets of nodes of size log k, denoted FS = {fSh \| 0 ≤ h ≤ log k − 1} and TS = {thS \| 0 ≤ h ≤ log k − 1}. For each h ∈ {0, . . . , log k − 1} and ` ∈ {1, 2}, the nodes (fAh` , thA` , fBh ` , thB` ) constitute a 4-cycle. Each node si` in a set S ∈ {A1 , A2 , B1 , B2 } is connected by to all nodes in bin(si` ). Up to here, the construction differs from the construction in Section 3.1 only by not having edges inside the sets A1 , A2 , B1 , B2 . We now add the following two gadgets to the graph. 1. We add three nodes Ca0 , Ca1 , Ca2 connected as a triangle, another set of three nodes Cb0 , Cb1 , Cb2 connected as a triangle, and edges connecting Cai to Cbj for each i 6= j ∈ {0, 1, 2}. We connect all the nodes of the form fAh1 , thA1 , h ∈ {0, . . . , log k − 1}, to Ca1 . Similarly, we 8 connect all the nodes fBh 1 , thB1 to Cb1 , the nodes fAh2 , thA2 to Ca2 and the nodes fBh 2 , thB2 to Cb2 . i i 2. For each set S ∈ A , A , B , B , we add two sets of nodes, S̄ = s̄` \| s` ∈ S and S̄¯ = 1 2 1 2 i i s̄¯` \| s` ∈ S . For each ` ∈ {1, 2} and i ∈ {0, . . . , k − 1} we connect a path (si` , s̄i` , s̄¯i` ), and for each ` ∈ {1, 2} and i ∈ {0, . . . , n − 2}, we connect s̄¯i` to s̄i+1 ` . In addition, we connect the gadgets by the edges: ¯i1 ), for each i ∈ {0, . . . , k − 1}; (Ca2 , ā01 ) and (Ca2 , ā ¯k−1 (a) (Ca2 , ai1 ) and (Ca1 , ā 1 ). (b) (Cb2 , bi1 ) and (Cb1 , ¯b̄i1 ), for each i ∈ {0, . . . , k − 1}; (Cb2 , b̄01 ) and (Cb2 , ¯b̄k−1 1 ). ¯i2 ), for each i ∈ {0, . . . , k − 1}; (Ca1 , ā02 ) and (Ca1 , ā ¯k−1 (c) (Ca1 , ai2 ) and (Ca2 , ā 2 ). (d) (Cb1 , bi2 ) and (Cb2 , ¯b̄i2 ), for each i ∈ {0, . . . , k − 1}; (Cb1 , b̄02 ) and (Cb1 , ¯b̄2k−1 ). Assume there is a proper 3-coloring of G. Denote by c0 , c1 and c2 the colors of Ca0 , Ca1 and respectively. By construction, these are also the colors of Cb0 , Cb1 and Cb2 , respectively. For the nodes appearing in Section 3.1, coloring a node by c0 is analogous to not including it in the vertex cover. Ca2 Claim 3. In each set S ∈ {A1 , A2 , B1 , B2 }, at least one node is colored by c0 . Proof. We start by proving the claim for S = A1 . Assume, towards a contradiction, that all nodes of A1 are colored by c1 and c2 . All these nodes are connected to Ca2 , so they must all be colored by c1 . Hence, all the nodes āi1 , i ∈ {0, . . . , k − 1}, are colored by c0 and c2 . The nodes ¯i1 , i ∈ {0, . . . , k − 1}, are connected to Ca1 , so they are colored by c0 and c2 as well. ā ¯k−1 ¯01 , ā11 , ā ¯11 , . . . āk−1 Hence, we have a path (ā01 , ā 1 , ā1 ) with an even number of nodes, starting k−1 0 ¯1 . This path must be colored by alternating c0 and c2 , but both ā01 and in ā1 and ending in ā k−1 ¯1 are connected to Ca2 , so they cannot be colored by c2 , a contradiction. ā A similar proof shows the claim for S = B1 . For S ∈ {A2 , B2 }, we use a similar argument but change the roles of c1 and c2 . Claim 4. For each i ∈ {0, . . . , k − 1}, the node ai1 is colored by c0 iff bi1 is colored by c0 and the node ai2 is colored by c0 iff bi2 is colored by c0 . Proof. Assume ai1 is colored by c0 , so all of its adjacent nodes bin(ai1 ) can only be colored by c1 or c2 . As all of these nodes are connected to C1a , they must be colored by c2 . Similarly, if a node bj1 in B1 is colored by c0 , then the nodes bin(bj1 ), which are also adjacent to C1b , must be colored by c2 . If i 6= j then there must be a bit i such that ih 6= jh , and there must be a pair of neighboring nodes (fAh1 , thB1 ) or (thA1 , fBh 1 ) which are colored by c2 . Thus, the only option is i = j. By Claim 3, there is a node in B1 that is colored by c0 , and so it must be bi1 . An analogous argument shows that if bi1 is colored by c0 , then so does ai1 . For ai2 and bi2 , similar arguments apply, where c1 plays the role of c2 . 2 Adding edges corresponding to the strings x and y: Given two bit strings x, y ∈ {0, 1}k , we augment the graph G described above with additional edges, which defines Gx,y . Assume x and y are indexed by pairs of the form (i, j) ∈ {0, . . . , k − 1}2 . To construct Gx,y , add edges to G by the following rules: if xi,j = 0 then add the edge (ai1 , aj2 ), and if yi,j = 0 then add the edge (bi1 , bj2 ). To prove that {Gx,y } is a family of lower bound graphs, it remains to prove the next lemma. Lemma 2. The graph Gx,y is 3-colorable iff DISJ(x, y) = FALSE. 9 Proof. For the first direction, assume Gx,y is 3-colorable, and denote the colors by c0 , c1 and c2 , as before. By Claim 3, there are nodes ai1 ∈ A1 and aj2 ∈ A2 that are both colored by c0 . Hence, the edge (ai1 , aj2 ) does not exist in Gx,y , implying xi,j = 1. By Claim 4, the nodes bi1 and bj2 are also colored c0 , so yi,j = 1 as well, giving that DISJ(x, y) = FALSE, as needed. For the other direction, assume DISJ(x, y) = FALSE, i.e, there is an index (i, j) ∈ {0, . . . , k − 1}2 such that xi,j = yi,j = 1. Consider the following coloring. 1. Color Cai and Cbi by ci , for i ∈ {0, 1, 2}. 0 0 2. Color the nodes ai1 , bi1 , aj2 and bj2 by c0 . Color the nodes ai1 and bi1 , for i0 6= i, by c1 , and 0 0 the nodes aj2 and bj1 , for j 0 6= j, by c2 . 3. Color the nodes of bin(ai1 ) by c2 , and similarly color the nodes of bin(bi1 ) by c2 . Color k−i the rest of the nodes in this gadget, i.e. bin(ak−i 1 ) and bin(b1 ), by c0 . Similarly, color k−j bin(aj2 ) and bin(bj2 ) by c0 and bin(ak−j 2 ) and bin(b2 ) by c1 . 4. Finally, color the nodes of the forms s̄i` and s̄¯i` as follows. 0 0 0 0 (a) Color āi1 and b̄i1 by c1 , all nodes āi1 and b̄i1 with i0 < i by c0 , and all nodes āi1 and b̄i1 with i0 > i by c2 . 0 0 (b) Similarly, color āi2 and b̄i2 by c2 , all nodes āi2 and b̄i2 with i0 < i by c0 , and all nodes 0 0 āi2 and b̄i2 with i0 > i by c1 . ¯i10 and ¯b̄i10 with i0 < i by c2 , and all nodes ā ¯i10 and ¯b̄i10 with i0 ≥ i by (c) Color all nodes ā c0 . ¯i20 and ¯b̄i20 with i0 < i by c1 , and all nodes ā ¯i20 and ¯b̄i20 with (d) Similarly, color all nodes ā i0 ≥ i by c0 . It is not hard to verify that the suggested coloring is indeed a proper 3-coloring of Gx,y , which completes the proof. Having constructed the family of lower bound graphs, we are now ready to prove Theorem 4. Proof of Theorem 4: To complete the proof of Theorem 4, we divide the nodes of G (which are also the nodes of Gx,y ) into two sets. Let VA = A1 ∪A2 ∪FA1 ∪TA1 ∪FA2 ∪TA2 ∪ Ca0 , Ca1 , Ca2 ∪ Ā1 ∪ Ā¯1 ∪ Ā2 ∪ Ā¯2 , and VB = V \ VA . Note that n ∈ Θ(k). The edges in the cut E(VA , VB ) are the 6 edges connecting Ca0 , Ca1 , Ca2 and Cb0 , Cb1 , Cb2 , and 2 edges for every 4-cycle of the nodes of FA1 ∪ TA1 ∪ FB1 ∪ TB1 and FA2 ∪ TA2 ∪ FB2 ∪ TB2 , for a total of Θ(log n) edges. Since Lemma 2 shows that {Gx,y } is a family of lower bound graphs with respect to DISJK , K = k 2 ∈ Θ(n2 ) and the predicate χ ≤ 3, we can apply Theorem 1 on the above partition to deduce that any algorithm in the CONGEST model for deciding whether a given graph is 3-colorable requires at least Ω(n2 / log2 n) rounds. Any algorithm that computes χ of the input graph, or produces a χ-coloring of it, may be used to deciding whether χ ≤ 3, in at most O(D) additional rounds. Thus, the lower bound applies to these problems as well. Our construction and proof naturally extend to handle c-coloring, for any c ≥ 3. To this end, we add to G (and to Gx,y ) new nodes denoted Cai , i ∈ {3, . . . , c − 1}, and connect them to all of VA , and new nodes denoted Cbi , i ∈ {3, . . . , c − 1}, and connect them to all of VB and also to Ca0 , Ca1 and Ca2 . The nodes Cai are added to Va , and the rest are added to Vb , which increases the cut size by Θ(c) edges. Assume the extended graph is colorable by c colors, and denote by ci the color of the node Cai (these nodes are connected by a clique, so their colors are distinct). The nodes Cbi , i ∈ {2, . . . , c − 1} form a clique, and they are all connected to the nodes Ca0 , Ca1 and Ca2 , so they are colored by the colors {c3 , . . . , cc−1 }, in some order. All the original nodes of VA are 10 connected to Cai , i ∈ {3, . . . , c − 1}, and all the original nodes of VB are connected to Cbi , i ∈ {3, . . . , c − 1}, so the original graph must be colored by 3 colors, which we know is possible iff DISJ(x, y) = FALSE. We added 2c − 6 nodes to the graph, so the inputs strings are of length K = n − 2c + 6. Thus, the new graphs constitute a family of lower bound graphs with respect to EQK and the predicate χ ≤ c, the communication complexity of EQK is in Ω(K 2 ) = Ω((n − c)2 ), the cut size is Θ(c + log n), and Theorem 1 completes the proof. A lower bound for (4/3 − )-approximation: Finally, we extend our construction to give a lower bound for approximate coloring. That is, we show a similar lower bound for computing a (4/3 − ε)-approximation to χ and for finding a coloring in (4/3 − ε)χ colors. Observe that since χ is integral, any (4/3 − )-approximation algorithm must return the exact solution in case χ = 3. Thus, in order to rule out the possibility for an algorithm which is allowed to return a (4/3 − ε)-approximation which is not the exact solution, we need a more general construction. For any integer c, we show a lower bound for distinguishing between the case χ ≤ 3c and χ ≥ 4c. Claim 5. Given an integer c, any distributed algorithm in the CONGEST model that distinguishes a graph G with χ(G) ≤ 3c from a graph with χ(G) ≥ 4c requires Ω(n2 /(c3 log2 n)) rounds. To prove Claim 5 we show a family of lower bound graphs with respect to the DISJK function, where K ∈ Θ(n2 /c2 ), and the predicate χ ≤ 3c (TRUE) or χ ≥ 4c (FALSE). The predicate is not defined for other values of χ. We create a graph Gcx,y , composed of c copies of Gx,y . The i-th copy is denoted Gx,y (i), and its nodes are partitioned into VA (i) and VB (i). We connect all the nodes of VA (i) to all nodes of VA (j), for each i 6= j. Similarly, we connect all the nodes of VB (i) to all the nodes of VB (j). This construction guarantees that each copy is colored by different colors, and hence if DISJ(x, y) = FALSE then χ(Gcx,y ) = 3c and otherwise χ(Gcx,y ) ≥ 3c. Therefore, Gcx,y is a family of lower bound graphs. Proof of Claim 5: Note that n ∈ Θ(kc). Thus, K = \|x\| = \|y\| = n2 /c2 . Furthermore, observe that for each Gx,y (i), there are O(log n) edges in the cut, so in total Gcx,y contains O(c log n) edges in the cut. Since we showed that Gcx,y is a family of lower bound graphs, we can apply Theorem 1 to deduce that because of the lower bound for Set Disjointness, any algorithm in the CONGEST model for distinguishing between χ ≤ 3c and χ ≥ 4c requires at least Ω(n2 /c3 log2 (n)) rounds. For any > 0 and any c it holds that (4/3 − )3c < 4c. Thus, we can choose c to be an arbitrary constant to achieve the following theorem. Theorem 5. For any constant ε > 0, any distributed algorithm in the CONGEST model that computes a (4/3 − ε)-approximation to χ requires Ω(n2 / log2 n) rounds. 4 Quadratic and Near-Quadratic Lower Bounds for Problems in P In this section we support our claim that what makes problems hard for the CONGEST model is not necessarily them being NP-hard problems. First, we address a class of subgraph detection problems, which requires detecting cycles of length 8 and a given weight, and show a nearquadratic lower bound on the number of rounds required for solving it, although its sequential complexity is polynomial. Then, we define a problem which we call the Identical Subgraphs 11 Figure 3: The family of lower bound graphs for detecting weighted cycles, with many edges omitted for clarity. Here, x0,1 = 1 and yk−1,k−1 = 1. Thus, a01 is connected to a12 by an edge of weight k 3 +k·0+1 = k 3 +1, and bk−1 is connected to bk−1 by an edge of weight k 3 −(k(k−1)+k−1) = k 3 −k 2 +1. 1 2 All the dashed edges are of weight 0. Detection problem, in which the goal is to decide whether two given subgraphs are identical. While this last problem is rather artificial, it allows us to obtain a strictly quadratic lower bound for the CONGEST model, with a problem that requires only a single-bit output. 4.1 Weighted Cycle Detection In this section we show a lower bound on the number of rounds needed in order to decide the graph contains a cycle of length 8 and weight W , such that W is a polylog(n)-bit value given as an input. Note that this problem can be solved easily in polynomial time in the sequential n setting by simply checking all of the at most 8 cycles of length 8. Theorem 6. Any distributed algorithm in the CONGEST model that decides if a graph with edge weights w : E → [0, poly(n)] contains a cycle of length 8 and weight W requires Ω(n2 / log2 n) rounds. Similarly to the previous sections, to prove Theorem 6 we describe a family of lower bound graphs with respect to the Set Disjointness function and the predicate P that says that the graph contains a cycle of length 8 and weight W . The fixed graph construction: The fixed graph construction G = (V, E) is defined as follows. The set of nodes contains four sets A1 , A2 , B1 and B2 , each of size k. To simplify our proofs in this section, we assume that k ≥ 3. For each set S ∈ {A1 , A2 , B1 , B2 } there is a node cS , which is connected to each of the nodes in S by an edge of weight 0. In addition there is an edge between cA1 and cB1 of weight 0 and an edge between cA2 and cB2 of weight 0 (see Figure 3). Adding edges corresponding to the strings x and y: Given two binary strings x, y ∈ 2 {0, 1}k , we augment the fixed graph G defined in the previous section with additional edges, which defines Gx,y . Recall that we assume that k ≥ 3. Let x and y be indexed by pairs of the form (i, j) ∈ {0, . . . , k − 1}2 . For each (i, j) ∈ {0, . . . , k − 1}2 , we add to Gx,y the following edges. If xi,j = 1, then we add an edge of weight k 3 + ki + j between the nodes ai1 and aj2 . If yi,j = 1, then we add an edge of weight k 3 − (ki + j) between the nodes bi1 and bj2 . We denote by InputEdges the set of edges {(u, v) \| u ∈ A1 ∧ v ∈ A2 } ∪ {(u, v) \| u ∈ B1 ∧ v ∈ B2 }, and we denote by w(u, v) the weight of the edge (u, v). 12 Observe that the graph does not contain edges of negative weight. Furthermore, the weight of any edge in InputEdges does not exceed k 3 + k 2 − 1, which is the weight of the edge k−1 (ak−1 1 , a2 ), in case xk−1,k−1 = 1. Similarly, the weight of an edge in InputEdges is not less k−1 3 than k − k 2 + 1, which is the weight of the edge (bk−1 1 , b2 ), in case yk−1,k−1 = 1. Using these two simple observations, we deduce the following claim. Claim 6. For any cycle of weight 2k 3 , the number of edges it contains that are in InputEdges is exactly two. Proof. Let C be a cycle of weight 2k 3 , and assume for the sake of contradiction that C does not contain exactly two edges from InputEdges. In case C contains exactly one edge from InputEdges, then the weight of C is at most k 3 + k 2 − 1 < 2k 3 , because all the other edges of C are of weight 0. Otherwise, in case C contains three or more edges from InputEdges, it holds that the weight of C is at least 3k 3 − 3k 2 + 3 > 2k 3 , because all the other edges on C are of non-negative weight. To prove that {Gx,y } satisfies the definition of a family of lower bound graphs, we prove the following lemma. Lemma 3. The graph Gx,y contains a cycle of length 8 and weight W = 2k 3 if and only if DISJ(x, y) = FALSE. Proof. For the first direction, assume that DISJ(x, y) = FALSE and let 0 ≤ i, j ≤ k − 1 be such that xi,j = 1 and yi,j = 1. Consider the cycle (ai1 , cA1 , cB1 , bi1 , bj2 , cB2 , cA2 , aj2 ). It is easy to verify that this is a cycle of length 8 and weight w(aj1 , ai2 ) + w(bi1 , bj2 ) = k 3 + ki + j + k 3 − ki − j = 2k 3 , as needed. For the other direction, assume that the graph contains a cycle C of length 8 and weight 2k 3 . By Claim 6, C contains exactly two edges from InputEdges. Denote these two edges by (u1 , v1 ) and (u2 , v2 ). Since all the other edges in C are of weight 0, the weight of C is w(u1 , v1 )+w(u2 , v2 ). The rest of the proof is by case analysis, as follows. First, it is not possible that (u1 , v1 ), (u2 , v2 ) ∈ {(u, v) \| u ∈ A1 ∧v ∈ A2 }, since in this case w(u1 , v1 )+w(u2 , v2 ) ≥ w(a01 , a02 )+w(a01 , a12 ) = 2k 3 +1. Similarly, it is not possible that (u1 , v1 ), (u2 , v2 ) ∈ {(u, v) \| u ∈ B1 ∧ v ∈ B2 }, since in this case w(u1 , v1 )+w(u2 , v2 ) ≤ w(b01 , b02 )+w(b01 , b12 ) = 2k 3 −1. Finally, suppose without loss of generality that (u1 , v1 ) ∈ {(u, v) \| u ∈ A1 ∧ v ∈ A2 } and (u2 , v2 ) ∈ {(u, v) \| u ∈ B1 ∧ v ∈ B2 }. Denote 0 0 0 0 u1 = ai1 , u2 = aj1 , v1 = bi1 and v2 = bj2 . It holds that w(ai1 , aj2 ) + w(bi1 , bj2 ) = 2k 3 if and only if i = i0 and j = j 0 , which implies that xi,j = 1 and yi,j = 1 and DISJ(x, y) = FALSE. Having constructed the family of lower bound graphs, we are now ready to prove Theorem 6. Proof of Theorem 6: To complete the proof of Theorem 6, we divide the nodes of G (which are also the nodes of Gx,y ) into two sets. Let VA = A1 ∪ A2 ∪ {cA1 , cA2 } and VB = V \ VA . Note that n ∈ Θ(k). Thus, K = \|x\| = \|y\| = Θ(n2 ). Furthermore, note that the only edges in the cut E(VA , VB ) are the edges (cA1 , cB1 ) and (cA2 , cB2 ). Since Lemma 3 shows that {Gx,y } is a family of lower bound graphs, we apply Theorem 1 on the above partition to deduce that because of the lower bound for Set Disjointness, any algorithm in the CONGEST model for deciding whether a given graph contains a cycle of length 8 and weight W = 2k 3 requires at least Ω(K/ log n) = Ω(n2 / log n) rounds. 4.2 Identical Subgraphs Detection In this section we show the strongest possible, quadratic lower bound, for a problem which can be solved in linear time in the sequential setting. Consider the following sequential specification of a graph problem. 13 Definition 2. (The Identical Subgraphs Detection Problem) Given a weighted input graph G = (V, E, w), with an edge-weight function w : E → {0, . . . , W − 1}, W ∈ poly n, such that the set of nodes V is partitioned into two enumerated sets of the same size, VA = {a0 , ..., ak−1 } and VB = {b0 , ..., bk−1 }, the Identical Subgraphs Detection problem is to determine whether the subgraph induced by the set VA is identical to the subgraph induced by the set VB , in the sense that for each 0 ≤ i, j ≤ k − 1 it holds that (ai , aj ) ∈ E if and only if (bi , bj ) ∈ E and w(ai , aj ) = w(bi , bj ) if these edges exist. The identical subgraphs detection problem can be solved easily in linear time in the sequential setting by a single pass over the set of edges. However, as we prove next, it requires a quadratic number of rounds in the CONGEST model, in any deterministic solution (note that this restriction did not apply in the previous sections). For clarity, we emphasize that in the distributed setting, the input to each node in the identical subgraphs detection problem is its enumeration ai or bi , as well as the enumerations of its neighbors and the weights of the respective edges. The outputs of all nodes should be TRUE if the subgraphs are identical, and FALSE otherwise. Theorem 7. Any distributed deterministic algorithm in the CONGEST model for solving the identical subgraphs detection problem requires Ω(n2 ) rounds. To prove Theorem 7 we describe a family of lower bound graphs. The fixed graph construction: The fixed graph G is composed of two k-node cliques on the node sets VA = {a0 , ..., ak−1 } and VB = {b0 , ..., bk−1 }, and one extra edge (a0 , b0 ). Adding edge weights corresponding to the strings x and y: Given two binary strings x k and y, each of size K = 2 log n, we augment the graph G with additional edge weights, which define Gx,y . For simplicity, assume that x and y are vectors of log n-bit numbers each having k 2 entries enumerated as xi,j and yi,j , with i < j, i, j ∈ {0, . . . , k − 1}. For each such i and j we set the weights of w(ai , aj ) = xi,j and w(bi , bj ) = yi,j , and we set w(a0 , b0 ) = 0. Note that {Gx,y } is a family of lower bound graphs with respect to EQK and the predicate P that says that the subgraphs are identical in the aforemention sense. Proof of Theorem 7: Note that n ∈ Θ(k), and thus K = \|x\| = \|y\| = Θ(n2 log n). Furthermore, the only edge in the cut E(VA , VB ) is the edge (a0 , b0 ). Since we showed that {Gx,y } is a family of lower bound graphs, we can apply Theorem 1 on the above partition to deduce that because of the lower bound for EQK , any deterministic algorithm in the CONGEST model for solving the identical subgraphs detection problem requires at least Ω(K/ log n) = Ω(n2 ) rounds. We remark that in a distributed algorithm for the identical subgraphs detection problem running on our family of lower bound graphs, information about essentially all the edges and weights in the subgraphs induced on VA and VB needs to be sent across the edge (a0 , b0 ). This might raise the suspicion that this problem is reducible to learning the entire graph, making the lower bound trivial. To argue that this is far from being the case, we present a randomized algorithm that solves the identical subgraphs detection problem in O(D) rounds and succeeds w.h.p. This has the additional benefit of providing the strongest possible separation between deterministic and randomized complexities for global problems in the CONGEST model, as the former is Ω(n2 ) and the latter is at most O(D). Our starting point is the following randomized algorithm for the EQK problem, presented in [49, Exersise 3.6]. Alice chooses a prime number p among the first K 2 primes uniformly PK−1 ` at random. She treats her input string x as a binary representation of an integer x̄ = `=0 2 x` , and sends p and x̄ (mod p) to Bob. Bob similarly computes ȳ, compares x̄ mod p with ȳ mod p, 14 and returns TRUE if they are equal and false otherwise. The error probability of this protocol is at most 1/K. We present a simple adaptation of this algorithm for the identical subgraph detection problem. Consider the following encoding of a weighted induced subgraph on VA : for each pair i, j of indices, we have dlog W e + 1 bits, indicating the existence of the edge and its weight (recall that W ∈ poly n is an upper bound on the edge weights). This weighted induced subgraph is thus represented by a K ∈ O(n2 log n) bit-string, denoted x = x0 , . . . , xK−1 , and each pair (i, j) has a set Si,j of indices representing the edge (ai , aj ). Note that the bits {x` \| ` ∈ si,j } are known to both ai and aj , and in the algorithm we use the node with smaller index in order to encode these bits. Similarly, a K ∈ O(n2 log n) bit-string, denoted y = y0 , . . . , yK−1 encodes a weighted induced subgraph on VB . The Algorithm. As standard, assume the input graph is connected. The nodes are enumerated as in Definition 2. The algorithm starts with some node, say, a0 , constructing a BFS tree, which completes in O(D) rounds. Then, a0 chooses a prime number p among the first K 2 primes uniformly at random and sends p to all the nodes over the tree, which takes O(D) rounds. P P Each node ai computes the sum j>i `∈Si,j x` 2` mod p, and the nodes then aggregate P P these local sums modulo p up the tree, until a0 computes the sum x̄ mod p = j6=i `∈Si,j x` 2` mod p. A similar procedure is then invoked w.r.t ȳ. Finally, a0 compares x̄ mod p and ȳ mod p, and downcasts over the BFS tree its output, which is TRUE if these values are equal and is FALSE otherwise. If the subgraphs are identical, a0 always returns TRUE, while otherwise their encoding differs in at least one bit, and as in the case of EQK , a0 returns TRUE falsely with probability at most 1/K ∈ O(1/n2 ). Theorem 8. There is a randomized algorithm in the CONGEST model that solves the identical subgraphs detection problem on any connected graph in O(D) rounds. 5 Weighted APSP In this section we use the following, natural extension of Definition 1, in order to address more general 2-party functions, as well as distributed problems that are not decision problems. For a function f : {0, 1}K1 × {0, 1}K2 → {0, 1}L1 × {0, 1}L2 , we define a family of lower bound graphs in a similar way as Definition 1, except that we replace item (4) in the definition with a generalized requirement that says that for Gx,y , the values of the of nodes in VA uniquely determine the left-hand side of f (x, y), and the values of the of nodes in VB determine the right-hand side of f (x, y). Next, we argue that theorem similar to Theorem 1 holds for this case. Theorem 9. Fix a function f : {0, 1}K1 × {0, 1}K2 → {0, 1}L1 × {0, 1}L2 and a graph problem P . If there is a family {Gx,y } of lower bound graphs with C = E(VA , VB ) then any deterministic algorithm for solving P in the CONGEST model requires Ω(CC(f )/ \|C\| log n) rounds, and any randomized algorithm for deciding P in the CONGEST model requires Ω(CC R (f )/ \|C\| log n) rounds. The proof is similar to that of Theorem 1. Notice that the only difference between the theorems, apart from the sizes of the inputs and outputs of f , are with respect to item (4) in the definition of a family of lower bound graphs. However, the essence of this condition remains the same and is all that is required by the proof: the values that a solution to P assigns to nodes in VA determines the output of Alice for f (x, y), and the values that a solution to P assigns to nodes in VB determines the output of Bob for f (x, y). 15 5.1 A Linear Lower Bound for Weighted APSP Nanongkai [56] showed that any algorithm in the CONGEST model for computing a poly(n)approximation for weighted all pairs shortest paths (APSP) requires at least Ω(n/ log n) rounds. In this section we show that a slight modification to this construction yields an Ω(n) lower bound for computing exact weighted APSP. As explained in the introduction, this gives a separation between the complexities of the weighted and unweighted versions of APSP. At a high level, while we use the same simple topology for our lower bound as in [56], the reason that we are able to shave off the extra logarithmic factor is because our construction uses O(log n) bits for encoding the weight of each edge out of many optional weights, while in [56] only a single bit is used per edge for encoding one of only two options for its weight. Theorem 10. Any distributed algorithm in the CONGEST model for computing exact weighted all pairs shortest paths requires at least Ω(n) rounds. The reduction is from the following, perhaps simplest, 2-party communication problem. Alice has an input string x of size K and Bob needs to learn the string of Alice. Any algorithm (possibly randomized) for solving this problem requires at least Ω(K) bits of communication, by a trivial information theoretic argument. Notice that the problem of having Bob learn Alice’s input is not a binary function as addressed in Section 2. Similarly, computing weighted APSP is not a decision problem, but rather a problem whose solution assigns a value to each node (which is its vector of distances from all other nodes). We therefore use the extended Theorem 9 above. The fixed graph construction: The fixed graph construction G = (V, E) is defined as follows. It contains a set of n − 2 nodes, denoted A = {a0 , ..., an−3 }, which are all connected to an additional node a. The node a is connected to the last node b, by an edge of weight 0. Adding edge weights corresponding to the string x: Given the binary string x of size K = (n − 2) log n we augment the graph G with edge weights, which defines Gx , by having each non-overlapping batch of log n bits encode a weight of an edge from A to a. It is straightforward to see that Gx is a family of lower bound graphs for a function f where K2 = L1 = 0, since the weights of the edges determine the right-hand side of the output (while the left-hand side is empty). Proof of Theorem 10: To prove Theorem 10, we let VA = A ∪ {a} and VB = {b}. Note that K = \|x\| = Θ(n log n). Furthermore, note that the only edge in the cut E(VA , VB ) is the edge (a, b). Since we showed that {Gx } is a family of lower bound graphs, we apply Theorem 9 on the above partition to deduce that because K bits are required to be communicated in order for Bob to know Alice’s K-bit input, any algorithm in the CONGEST model for computing weighted APSP requires at least Ω(K/ log n) = Ω(n) rounds. 5.2 The Alice-Bob Framework Cannot Give a Super-Linear Lower Bound for Weighted APSP In this section we argue that a reduction from any 2-party function with a constant partition of the graph into Alice and Bob’s sides is provable incapable of providing a super-linear lower bound for computing weighted all pairs shortest paths in the CONGEST model. A more detailed inspection of our analysis shows a stronger claim: our claim also holds for algorithms for the CONGEST-BROADCAST model, where in each round each node must send the same (log n)-bit message to all of its neighbors. The following theorem states our claim. Theorem 11. Let f : {0, 1}K1 × {0, 1}K2 → {0, 1}L1 × {0, 1}L2 be a function and let Gx,y be a family of lower bound graphs w.r.t. f and the weighted APSP problem. When applying 16 Theorem 9 to f and Gx,y , the lower bound obtained for the number of rounds for computing weighted APSP is at most linear in n. Roughly speaking, we show that given an input graph G = (V, E) and a partition of the set of vertices into two sets V = VA ∪ VB , such that the graph induced by the nodes in VA is simulated by Alice and the graph induced by nodes in VB is simulated by Bob, Alice and Bob can compute weighted all pairs shortest paths by communicating O(n log n) bits of information for each node touching the cut C = (VA , VB ) induced by the partition. This means that for any 2-party function f and any family of lower bound graphs w.r.t. f and weighted APSP according to the extended definition of Section 5.1, since Alice and Bob can compute weighted APSP which determines their output for f by exchanging only O(\|V (C)\|n log n) bits, where V (C) is the set of nodes touching C, the value CC(f ) is at most O(\|V (C)\|n log n). But then the lower bound obtained by Theorem 9 cannot be better than Ω(n), and hence no super-linear lower can be deduced by this framework as is. ˙ B , E) we denote C = E(VA , VB ). Let V (C) denote Formally, given a graph G = (V = VA ∪V the nodes touching the cut C, with CA = V (C) ∩ VA and CB = V (C) ∩ VB . Let GA = (VA , EA ) be the subgraph induced by the nodes in VA and let GB = (VB , EB ) be the subgraph induced by the nodes in VB . For a graph H, we denote the weighted distance between two nodes u, v by wdistH (u, v). ˙ B , E, w) be a graph with an edge-weight function w : E → Lemma 4. Let G = (V = VA ∪V {1, . . . , W }, such that W ∈ poly n. Suppose that GA , CB , C and the values of w on EA and C are given as input to Alice, and that GB , CA , C and the values of w on EB and C are given as input to Bob. Then, Alice can compute the distances in G from all nodes in VA to all nodes in V and Bob can compute the distances from all nodes in VB to all the nodes in V , using O(\|V (C)\| n log n) bits of communication. Proof. We describe a protocol for the required computation, as follows. For each node u ∈ CB , Bob sends to Alice the weighted distances in GB from u to all nodes in VB , that is, Bob sends {wdistGB (u, v) \| u ∈ CB , v ∈ VB } (or ∞ for pairs of nodes not connected in GB ). 0 , w 0 ) with the nodes V 0 = V ∪ C and edges Alice constructs a virtual graph G0A = (VA0 , EA A B A A 0 0 is defined by w 0 (e) = w(e) for each EA = EA ∪ C ∪ (CB × CB ). The edge-weight function wA A 0 (u, v) for u, v ∈ C is defined to be the weighted distance between u and e ∈ EA ∪ C, and wA B v in GB , as received from Bob. Alice then computes the set of all weighted distances in G0A , {wdistG0A (u, v) \| u, v ∈ VA0 }. Alice assigns her output for the weighted distances in G as follows. For two nodes u, v ∈ VA ∪ CB , Alice outputs their weighted distance in G0A , wdistG0A (u, v). For a node u ∈ VA0 and a node v ∈ VB \ CB , Alice outputs min{wdistG0A (u, x) + wdistGB (x, v) \| x ∈ CB }, where wdistG0A is the distance in G0A as computed by Alice, and wdistGB is the distance in GB that was sent by Bob. For Bob to compute his required weighted distances, for each node u ∈ CA , similar information is sent by Alice to Bob, that is, Alice sends to Bob the weighted distances in GA from u to all nodes in VA . Bob constructs the analogous graph G0B and outputs his required distance. The next paragraph formalizes this for completeness, but may be skipped by a convinced reader. 0 , w0 ) Formally, Alice sends {wdistGA (u, v) \| u ∈ CA , v ∈ VA }. Bob constructs G0B = (VB0 , EB B 0 = E ∪C ∪(C ×C ). The edge-weight function w 0 is defined with VB0 = VB ∪CA and edges EB B A A B 0 (e) = w(e) for each e ∈ E ∪ C, and w 0 (u, v) for u, v ∈ C is defined to be the weighted by wB B A B distance between u and v in GA , as received from Alice (or ∞ if they are not connected in GA ). Bob then computes the set of all weighted distances in G0B , {wdistG0B (u, v) \| u, v ∈ VB0 }. Bob assigns his output for the weighted distances in G as follows. For two nodes u, v ∈ VB ∪ CA , Bob outputs their weighted distance in G0B , wdistG0B (u, v). For a node u ∈ VB0 and a node 17 v ∈ VA \ CA , Bob outputs min{wdistG0B (u, x) + wdistGA (x, v) \| x ∈ CA }, where wdistG0B is the distance in G0B as computed by Bob, and wdistGA is the distance in GA that was sent by Alice. Complexity. Bob sends to Alice the distances from all nodes in CB to all node in VB , which takes O(\|CB \| \|VB \| log n) bits, and similarly Alice sends O(\|CA \| \|VA \| log n) bits to Bob, for a total of O(\|V (C)\| n log n) bits. Correctness. By construction, for every edge (u, v) ∈ CB ×CB in G0A with weight wdistG0A (u, v), there is a corresponding shortest path Pu,v of the same weight in GB . Hence, for any path P 0 = (v0 , v1 , . . . , vk ) in G0A between v0 , vk ∈ VA0 , there is a corresponding path Pv0 ,vk of the same weight in G, where P is obtained from P 0 by replacing every two consecutive nodes vi , vi+1 in P ∩ CB by the path Pvi ,vi+1 . Thus, wdistG0A (v0 , vk ) ≥ wdistG (v0 , vk ). On the other hand, for any shortest path P = (v0 , v1 , . . . , vk ) in G connecting v0 , vk ∈ VA0 , there is a corresponding path P 0 of the same weight in G0A , where P 0 is obtained from P by replacing any sub-path (vi , . . . , vj ) of P contained in GB and connecting vi , vj ∈ CB by the edge (vi , vj ) in G0A . Thus, wdistG (v0 , vk ) ≥ wdistG0A (v0 , vk ). Alice thus correctly computes the weighted distances between pairs of nodes in VA0 . It remains to argue about the weighted distances that Alice computes to nodes in VB \ CB . Any lightest path P in G connecting a node u ∈ VA0 and a nodev ∈ VB \CB must cross at least one edge of C and thus must contain a node in CB . Therefore, wdistG (u, v) = min{wdistG (u, x) + wdistG (x, v) \| x ∈ CB }. Recall that we have shown that wdistG0A (u, x) = wdistG (u, x) for any u, x ∈ VA0 . The sub-path of P connecting x and v is a shortest path between these nodes, and is contained in GB , so wdistGB (x, v) = wdistG (x, v). Hence, the distance min{wdistG0A (u, x) + wdistGB (x, v) \| x ∈ CB } returned by Alice is indeed equal to wdistG (u, v). The outputs of Bob are correct by the analogous arguments, completing the proof. Proof of Theorem 11: Let f : {0, 1}K1 ×{0, 1}K2 → {0, 1}L1 ×{0, 1}L2 be a function and let Gx,y be a family of lower bound graphs w.r.t. f and the weighted APSP problem. By Lemma 4, Alice and Bob can compute the weighted distances for any graph in Gx,y by exchanging at most O(\|V (C)\|n log n) bits, which is at most O(\|C\|n log n) bits. Since Gx,y is a family of lower bound graphs w.r.t. f and weighted APSP, condition (4) gives that this number of bits is an upper bound for CC(f ). Therefore, when applying Theorem 9 to f and Gx,y , the lower bound obtained for the number of rounds for computing weighted APSP is Ω(CC(f )/\|C\| log n), which is no higher than a bound of Ω(n). Extending to t players: We argue that generalizing the Alice-Bob framework to a sharedblackboard multi-party setting is still insufficient for providing a super-linear lower bound for weighted APSP. Suppose that we increase the number of players in the above framework to t players, P0 , . . . , Pt−1 , each simulating the nodes in a set Vi in a partition of V in a family of lower bound graphs w.r.t. a t-party function f and weighted APSP. That is, the outputs of nodes in Vi for an algorithm ALG for solving a problem P in the CONGEST model, uniquely determines the output of player Pi in the function f . The function f is now a function from {0, 1}K0 × · · · × {0, 1}Kt−1 to {0, 1}L0 × · · · × {0, 1}Lt−1 . The communication complexity CC(f ) is the total number of bits written on the shared blackboard by all players. Denote by C the set of cut edges, that is, the edge whose endpoints do not belong to the same Vi . Then, if ALG is a R-rounds algorithm, we have that writing O(R\|C\| log n) bits on the shared blackboard suffice for computing f , and so R = Ω(CC(f )/\|C\| log n). We now consider the problem P to be weighted APSP. Let f be a t-party function and let Gx0 ,...,xt−1 be a family of lower bound graphs w.r.t. f and weighted APSP. We first have the players write all the edges in C on the shared blackboard, for a total of O(\|C\| log n) bits. 18 Then, in turn, each player Pi writes the weighted distances from all nodes in Vi to all nodes in V (C) ∩ Vi . This requires no more than O(\|V (C)\|n log n) bits. It is easy to verify that every player Pi can now compute the weighted distances from all nodes in Vi to all nodes in V , in a manner that is similar to that of Lemma 4. This gives an upper bound on CC(f ), which implies that any lower bound obtained by a reduction from f is Ω(CC(f )/\|C\| log n), which is no larger than Ω((\|V (C)\|n+\|C\|) log n/\|C\| log n), which is no larger than Ω(n), since \|V (C)\| ≤ 2\|C\|. Remark 1: Notice that the t-party simulation of the algorithm for the CONGEST model does not require a shared blackboard and can be done in the peer-to-peer multiparty setting as well, since simulating the delivery of a message does not require the message to be known globally. This raises the question of why would one consider a reduction to the CONGEST model from the stronger shared-blackboard model to begin with. Notice that our argument above for t players does not translate to the peer-to-peer multiparty setting, because it assumes that the edges of the cut C can be made global knowledge within writing \|C\| log n bits on the blackboard. However, what our extension above shows is that if there is a lower bound that is to be obtained using a reduction from peer-to-peer t-party computation, it must use a function f that is strictly harder to compute in the peer-to-peer setting compared with the shared-blackboard setting. Remark 2: We suspect that a similar argument can be applied for the framework of non-fixed Alice-Bob partitions (e.g., [64]), but this requires precisely defining this framework which is not addressed in this version. 6 Discussion This work provides the first super-linear lower bounds for the CONGEST model, raising a plethora of open questions. First, we showed for some specific problems, namely, computing a minimum vertex cover, a maximum independent set and a χ-coloring, that they are nearly as hard as possible for the CONGEST model. However, we know that approximate solutions for some of these problems can be obtained much faster, in a polylogarithmic number of rounds or even less. A family of specific open questions is then to characterize the exact trade-off between approximation factors and round complexities for these problems. Another specific open question is the complexity of weighted APSP, which has also been asked in previous work [26,56]. Our proof that the Alice-Bob framework is incapable of providing super-linear lower bounds for this problem may be viewed as providing evidence that weighted APSP can be solved much faster than is currently known. Together with the recent subquadratic algorithm of [28], this brings another angle to the question: can weighted APSP be solved in linear time? Finally, we propose a more general open question which addresses a possible classification of complexities of global problems in the CONGEST model. Some such problems have complexities √ of Θ(D), such as constructing a BFS tree. Others have complexities of Θ̃(D + n), such as finding an MST. Some problems have near-linear complexities, such as unweighted APSP. 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Stein kernels and moment maps Max Fathi∗ arXiv:1804.04699v1 [math.PR] 12 Apr 2018 April 16, 2018 Abstract We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge-Ampère equation. As a consequence, we show how regularity bounds on these maps control the rate of convergence in the classical central limit theorem, and derive new rates in KantorovitchWasserstein distance in the log-concave situation, with explicit polynomial dependence on the dimension. 1 Introduction Stein’s method is a set of techniques introduced by Stein [33, 34] to estimate distances between probability measures. We refer to the survey [30] for an overview. We shall be interested in one particular way of implementing Stein’s method in the Gaussian setting, based on the notion of Stein kernels. Let µ be a probability measure on Rd . A matrix-valued function τµ : Rd −→ Md (R) is said to be a Stein kernel for µ (with respect to the standard Gaussian measure γ on Rd ) if for any smooth test function f taking values in Rd , we have Z Z x · f dν = hτµ , ∇f iHS dν. (1) For applications, it is Rgenerally enough to consider the restricted class of test functions f satisfying (\|f \|2 + k∇f k2HS )dµ < ∞, in which case both integrals in (1) are well-defined as soon as τµ ∈ L2 (µ), provided µ has finite second moments. The motivation behind the definition is that, since the Gaussian measure is the only probability distribution satisfying the integration by parts formula Z Z x · f dγ = div(f )dγ, (2) the Stein kernel τµ coincides with the identity matrix, denoted by Id, if and only if the measure µ is equal to γ. Hence, the Stein kernel can be used to control how far µ is from being a standard Gaussian measure in terms of how much it ∗ CNRS and Institut de Mathématiques max.fathi@math.univ-toulouse.fr 1 de Toulouse, Université de Toulouse, 2 violates the integration by parts formula (2). It appears implicitly in many works on Stein’s method, and has recently been the topic of more direct investigations [1, 27, 28, 21, 8]. The one-dimensional case, where Stein kernels can be explicitly constructed from the density, has been extensively studied [24]. It has applications to central limit theorems [26], concentration inequalities [29, 21, 32] and random matrix theory [7]. A related quantity is the Stein discrepancy Z 2 S(µ) := inf \|τ − Id \|2 dµ τ where the infimum is taken over all possible Stein kernels for µ, since they may not be unique. This quantity has two main interesting properties: it controls the L2 Kantorovitch-Wasserstein distance to the Gaussian [21], and is monotone along the central limit theorem [8]. The aim of this work is to describe how we can construct Stein kernels using a correspondence between centered measures and convex functions, known as the moment measure problem, or moment map problem, which we shall describe in Section 2.1. The main motivation was to give a construction of Stein kernels using optimal transport maps, of which these moment maps can be viewed as a variant. The Stein kernels we shall build have several nice properties that do not seem to be necessarily satisfied by previous constructions. Most notably they shall always takes values that are symmetric, nonnegative matrices. As an application, we shall derive in Section 3 new bounds on the rate of convergence in the multidimensional central limit theorem when the random variables are log-concave, with explicit dependence on the dimension. 2 2.1 Stein kernels and moment maps Moment maps In [9] (revisited in [31], and following earlier works [36, 11, 3, 23]), the following theorem was established: Theorem 2.1 (Cordero-Erausquin and Klartag 2015). Let µ be a centered measure, with finite first moment and that is not supported on a hyperplane. Then there exists a convex function ϕ such that µ is the pushforward of the probability measure with density e−ϕ by the map ∇ϕ. This result can be seen as a variant of the optimal transport problem, where instead of specifying two measures, we fix a target measure, and look for both an original measure and a transport map while imposing the constraint that the map should be the gradient of the potential of the measure. Indeed, here ∇ϕ is also the Brenier map from optimal transport theory [35] sending e−ϕ onto µ. The convex function given by this theorem may well not be smooth, most notably when µ is a combination of Dirac masses. For example, if µ is the uniform 3 measure on {−1, +1}, viewed as a subset of R, the convex function is ϕ(x) = \|x\| on R, which is not smooth at the origin. This will cause some issues later on. We can however assume it satisfies some weak continuity property on the boundary of its support (the notion of essential continuity, which is described in [9]). A smooth version of this theorem, under extra assumptions, was previously obtained by Berman and Berndtson [3], with earlier results due to Wang and Zhu [36] and Donaldson [11]: Theorem 2.2 (Berman and Berndtson 2013). Assume that µ is supported on a compact, open convex set, and that it has a smooth density ρ on its support. Assume moreover that C ≥ ρ ≥ C −1 on the whole support, for some positive constant C. Then the convex function ϕ of Theorem 2.1 is smooth and supported on the whole space Rd . In this result (which is based on Caffarelli’s regularity theory for MongeAmpère PDEs), the convexity of the support plays an essential role. We can reformulate those statements as pertaining to solutions of the PDE e−ϕ = ρ(∇ϕ) det(∇2 ϕ). (3) This PDE is a variant of the Monge-Ampère equation, sometimes called the toric Kähler-Einstein equation. It has been studied in complex geometry, where it is related to the construction of differential structures with specific properties on toric varieties (i.e. quotients of the complex space (C∗ )n ). More recently, it has been studied in [15, 16, 19, 20], where it was used to establish functional inequalities for log-concave measures. A relevant remark to the connection with Stein’s method that we shall describe in the next section is that the standard Gaussian measure is the only fixed point of the map µ → e−ϕ , where ϕ is the moment map of µ. So in some sense the moment map already contains some information on how far the measure is from being Gaussian. In general, unless the dimension is 1, solutions to (3) are not explicit. One particular case where it can be determined is for the uniform measure on the unit d P cube [−1, 1]d , where the moment map is of the form ϕ(x) = 2 log cosh(xi /2) + i=1 C. This can be generalized to uniform measures on centered parallelipipeds by composing this function with the appropriate linear map. 2.2 The connection with Stein kernels For now, assume that µ has a density with respect to the Lebesgue measure which is strictly positive on its support, and is such that the convex function ϕ given by Theorem 2.1 is C 2 . There exists an optimal transport map sending µ onto e−ϕ , which is necessarily ∇ϕ∗ , where ϕ∗ is the Legendre transform of ϕ. ϕ∗ is then also C 2 : since ∇ϕ∗ is the inverse of ∇ϕ (this is a property of the Legendre transform) and Hess ϕ is strictly positive on the whole space, ∇ϕ∗ inherits C 1 regularity from ∇ϕ. 4 Theorem 2.3. If µ has a density ρ with respect to the Lebesgue measure, and the solution ϕ to the PDE (3) is C 2 and supported on the whole space Rd , then Hess ϕ(∇ϕ∗ ) is a Stein kernel for µ. Moreover, the Stein discrepancy satisfies Z 2 S(µ) ≤ \| Hess ϕ − Id \|2HS e−ϕ dx. In particular, if µ is supported on a compact, convex set and has density bounded from above and below by positive constants, this result applies. R The regularity assumptions can be weakened, indeed if \| Hess ϕ − Id \|2HS e−ϕ dx is finite and µ has a continuous density, then the result will still hold. For general 2,1 measures with density and full support, the moment map is only in Wloc in the interior of its support [10], which is not enough to make the proof work. But this is not surprising, since for heavy-tailed random variables the CLT may fail, and this would rule out existence of a Stein kernel belonging to L2 (µ). For background on regularity theory for Monge-Ampère PDEs, we refer to the lecture notes [12]. Remark 2.1. An interesting byproduct of this result is that the Stein kernel obtained in this way takes values that are symmetric and positive matrices. In particular, this explains why the explicit formula for Stein kernels in dimension one defines a nonnegative function. Remark 2.2. The Stein kernel constructed this way seems to be in general different from the one constructed in [8]. Since when the density is supported on a compact, convex set and has density bounded from above and below by positive constants a Poincaré inequality holds, existence of a Stein kernel in that situation was already proven in [8]. It is the particular structure of the kernel we obtain here that makes it interesting, as we will see when obtaining new rates of convergence in the CLT. Proof. Since ϕ is smooth, we have the Stein equation Z Z −ϕ ∇ϕ · f e dx = Tr(∇f )e−ϕ dx. There is no boundary term remaining when integrating by parts because ϕ grows R at least linearly at infinity, since it is convex and e−ϕ dx < ∞ (see for example Lemma 2.1 in [14]). Fix g a smooth function, and take f (x) = g(∇ϕ(x)) in the above equation. We get Z Z ∇ϕ(x) · g(∇ϕ(x))e−ϕ dx = hHess ϕ, ∇g(∇ϕ)ie−ϕ dx. Applying the change of variable y = ∇ϕ∗ (x), which sends µ onto e−ϕ , we obtain Z Z x · g(x)dµ = hHess ϕ(∇ϕ∗ ), ∇gidµ which ensures that Hess ϕ(∇ϕ∗ ) = (Hess ϕ∗ )−1 is indeed a Stein kernel for µ. 5 The bound on the Stein discrepancy is an immediate consequence of the change of variable: since Hess ϕ(∇ϕ∗ ) is a Stein kernel, by definition of the Stein discrepancy we have Z Z 2 ∗ 2 S(µ) ≤ \| Hess ϕ(∇ϕ ) − Id \|HS dµ = \| Hess ϕ − Id \|2HS e−ϕ dx. The well-known Caffarelli contraction theorem [6] states that the Brenier map sending the standard gaussian map onto a uniformly log-concave measure is lipschitz. Klartag [15] proved an analogous estimate for moment maps, which leads to the following bound on Stein kernels in that setting: Corollary 2.4. Assume that µ is uniformly convex, that is it is of the form e−V dx with Hess V ≥ ǫ Id for some ǫ > 0. Then there exists a Stein kernel with values that are positive symmetric matrices, and which is uniformly bounded, that is \|\|τ \|\|op ≤ ǫ−1 . In dimension one, this result was pointed out in [32]. Such pointwise estimates can be used to derive properties of the density and concentration inequalities [29] and isoperimetric inequalities [32]. Proof. The Stein kernel described in this statement is the one built in Theorem 2.3, all that we need to do is to prove the uniform bound on its operator norm. In [15], it was shown that under the uniform convexity assumption, the moment map indeed satisfies the uniform bound \|\| Hess ϕ\|\|op ≤ ǫ−1 , and the conclusion follows. It would also be possible to build a Stein kernel using the construction of [7, 27] and the optimal transport map sending the standard Gaussian measure onto µ. Existence could be proved in the same setting, but there would be two main downsides: we do not have an analogue to Proposition 3.2 below for those maps, so we would only get a useful quantitative estimate in the uniformly convex setting, and due to the particular form of the construction of [7], even in the latter setting the quantitative estimates would get worse. But we would still get existence of a Stein kernel that is bounded for uniformly log-concave measures. 3 Application to rates of convergence in the central limit theorem We now show how the construction of Stein kernels discussed in the previous section leads to new estimates on the rate of convergence in the central limit theorem. The family of distances we shall consider to estimate the distance in the CLT are the Kantorovitch-Wasserstein distances from optimal transport theory, defined as Z Wp (µ, ν) := inf \|\|x − y\|\|p2 π(dx, dy) π 6 where the infimum runs over all couplings of the measures µ and ν. We refer to the textbook [35] for background on optimal transport. The following statement is a result of [21] on how Lp bounds on a Stein kernel control Wasserstein distances to the standard gaussian measure: Proposition 3.1. Let τ be a Stein kernel for the probability measure µ on Rd . Then for any p ≥ 2 we have 1/p  Z X Wp (µ, γ) ≤ Cp d1−2/p  \|τij − δij \|p dµ i,j 1/p R the value of the p-th moment for a one-dimensional with Cp = R \|x\|p dγ standard gaussian. These results mean that if we get estimates on Hess ϕ, averaged out against we can deduce estimates on transport distances. It turns out that when µ is log-concave and compactly supported, such an estimate was already obtained by Klartag [15]: e−ϕ , Proposition 3.2. Let µ be a log-concave probability measure, supported on an open bounded convex set, and with a density bounded from above and below. Then the essentially-continuous convex function ϕ for which µ is the moment measure is C 2 and satisfies for any p ≥ 1 and any θ ∈ S d−1 Z p Z ∗ p p 2p 2 \|hHess ϕ(∇ϕ )θ, θi\| dµ ≤ 8 p (x · θ) dµ . We shall give a proof of Klartag’s estimate in Section 3.1. It will be the same proof as in [15], reformulated in a different language, which may be of interest to some readers. Remark 3.1. The results of [15] assume C ∞ smoothness, relying on a result of [3] to deduce C ∞ smoothness of ϕ. Since we actually only need C 2 smoothness of ϕ, it turns out we only need continuity of the density. Combining our construction of Stein kernels, Klartag’s estimate and basic arguments from Stein’s method, we get the following application to rates of convergence in the CLT: Theorem 3.3. Let µ be an isotropic log-concave probability measure with strictly n P positive and continuous density on its support. Let µn be the law of n−1/2 Xi , i=1 where the Xi are i.i.d. random variables with law µ. Then for any p ≥ 2 we have Wp (µn , γ) ≤ C̃(p) d2−2/p n1/2 with C̃(p) is a constant that depends only on p (and which can be made explicit). In particular, this estimate does not depend on µ. 7 In the case p = 2, the main result of [8] combined with the best currentlyknown estimate on the Poincaré constant of log-concave measures [22] leads to a rate of convergence of the form Cd3/4 n−1/2 , which is better than the one we obtain here. The Kannan-Lovasz-Simonovits conjecture predicts that the Poincaré constant of isotropic log-concave measures is bounded by some universal constant, p independently of the dimension, so we expect a rate of order d/n. Bonis [5] proved a general rate of convergence for measures with finite fourth moment, again when p = 2. In the case p > 2, this result seems to be the first estimate with the sharp dependence on the number of variables in any dimension. In dimension one, Bonis [5] and Bobkov [4] obtained the sharp rate in a far more general setting. In the situation where µ is uniformly log-concave, the uniform estimate on the operator norm of our Stein kernel leads to an improved dependence on the dimension: Theorem 3.4. Let µ = e−V be an isotropic probability measure with strictly positive and continuous density on its support, and assume that Hess V ≥ ǫ Id for some ǫ > 0. Then for any p ≥ 2 we have d3/2−2/p . ǫn1/2 Proof of Theorem 3.3. We first work in the situation where µ has a compact sup2 ∗ −1 port and a density bounded away from zero. Let τ = know is a (∇n ϕ ) , which we n P P Stein kernel for µ. Then as is standard, τn (x) := E n1 τ (Xi ) \| √1n Xi = x Wp (µn , γ) ≤ C(p) k=1 k=1 is a Stein kernel for µn . Applying Jensen’s inequality, we have Z Z p 1X p τij (xi ) − δij dµ⊗n (x1 , .., xn ), \|(τn )ij − δij \| dµn ≤ n and Rosenthal’s inequality for sums of independent centered random variables [13] yields Z Z \|(τn )ij − δij \|p dµn ≤ Kp n−p/2 \|τij − δij \|p dµ. We then have, given an orthonormal basis (θ1 , .., θd ) of Rd , Z X p/2 X Z \|τij − δij \|2 dµ \|τij − δij \|p dµ ≤ i,j Z X p/2 ≤ 2p dp/2 + \|τij \|2 dµ !p ! Z X hτ θi , θi i dµ ≤ 2p dp/2 + i p/2 ≤ C(p) d (p−1) +d ≤ C(p)dp (1 + 8p p2p ). XZ p hτ θi , θi i dµ 8 Hence and therefore Z \|(τn )ij − δij \|p dµn ≤ Kp n−p/2 C(p)dp (1 + 8p p2p ), (4) d2−2/p . n1/2 For the general case, when the support of µ is not necessarily compact, we can take a sequence of compact sets Fℓ that converge to the support of µ, and apply our results to the restriction of µ to Fℓ (renormalized to remain a centered, isotropic probability measure, so that Fℓ has to be modified to take this into account, but this modification will remain convex and compact). The estimate on the Wasserstein distance does not depend on Fℓ , so that we can let ℓ go to infinity and the result remains valid. Wp (µn , γ) ≤ C̃(p) The proof of Theorem 3.4 follows the exact same argument except that we use the improved bound of Corollary 2.4, so we omit it. 3.1 Proof of Proposition 3.2 We shall now give a proof of Proposition 3.2, omiting many computations taken from [18, 16]. While it is not written in the same way as in [15], it is the same proof, and we stress it is not due to us. We describe it in this form so that it is more easily readable for people with a knowledge of Bakry-Emery calculus Proof of Proposition 3.2 . We introduce the Hessian metric on Rd given by the Riemannian metric tensor g = (∇ϕ)−1 . A result of Kolesnikov [18] asserts that when µ is log-concave and satisfies the regularity conditions of Theorem 2.2, then the metric-measure space M = (Rd , g, e−ϕ ) has Ricci curvature bounded from below by 1/2. Moreover, if we consider the Laplacian on M , which is given by the formula Lϕ f = Tr(∇2 f (∇ϕ)−1 ) + ∇ log ρ(∇ϕ) · ∇f then one can check that 2 Lϕ ∂e ϕ = −∂e ϕ; Γ(∂e ϕ) = ∂ee ϕ; where Γ is the squared norm of the gradient with respect to the metric g. These computations can be found in [18, 16]. We can then use tools from Bakry-Émery theory to obtain estimates on eigenfunctions of the Laplacian for spaces with positive curvature to deduce the desired bound. Indeed, if we denote by Pt the semigroup acting on functions induced by Lϕ , we have for any locally-lipschitz function f 1 Pt (f 2 ), Γ(Pt f ) ≤ t 2(e − 1) 9 see Theorem 4.7.2 in [2]. Taking f = ∂e ϕ, since it is an eigenfunction of Lϕ associated to the eigenvalue 1, we have Pt ∂e ϕ = e−t ∂e ϕ. Therefore Γ(Pt f ) = 2 ϕ and for any t > 0 and p ≥ 1 we have e−2t ∂ee p p 1 1 2 p −2pt 2 p (Pt ((∂e ϕ) )) ≤ Pt ((∂e ϕ)2p ). e (∂ee ϕ) ≤ 2(et − 1) 2(et − 1) Hence after integrating, for any t > 0 we have p Z Z e2t 2 p −ϕ (∂ee ϕ) e dx ≤ (∂e ϕ)2p e−ϕ dx. 2(et − 1) Taking t = ln 2, the result then follows from the bound \|\|f \|\|2p,e−ϕ ≤ 2p\|\|f \|\|2,e−ϕ for any eigenfunction of Lϕ associated with the eigenvalue −1, when a logarithmic Sobolev inequality with constant 1/2 holds, see Section 5.3 of [2]. 4 Transporting Stein kernels for other reference measures The abstract setup of Stein’s method can be generalized to cover non-gaussian reference measures. If we wish to compare some measure µ to a reference measure µ0 = e−V dx, say for a smooth function V that is finite everywhere, then µ0 is characterized by the integration by parts formula Z Z ∇V · f dµ0 = Tr(∇f )dµ0 which leads to a definition of Stein kernel as a matrix-valued function such that Z Z ∇V · f dµ = hτ, ∇f idµ. (5) Assume that V is convex, C 2 and that Hess V > 0, and let µV be the pushforward of µ by ∇V . Then for any vector-valued smooth function f , defining g(x) = f (∇V ∗ (x)), we have Z Z ∇V (x) · f (x)dµ = ∇V (x) · g(∇V (x))dµ Z = x · g(x)dµV , 10 so if we take τ̃V,γ to be a Stein kernel for µV with respect to the gaussian measure (assuming for now it exists), we get Z Z ∇V (x) · f (x)dµ = hτ̃V,γ , ∇gidµV Z = hτ̃V,γ (∇V (x)), ∇g(∇V (x))idµ Z = hτ̃V,γ (∇V (x)), (Hess V (x))−1 ∇f (x)idµ Z = hτ̃V,γ (∇V (x))(Hess V (x))−1 , ∇f (x)idµ and therefore τ̃V,γ (∇V (x))(Hess V (x))−1 is a Stein kernel for µ relative to µ0 . To be valid, in addition to the regularity and convexity assumptions on V , this arguments requires that τ̃V,γ exists. It is okay if it is only defined in the sense of distributions (since ∇V is smooth and bijective, composing a distribution with it is possible). Acknowledgments : This work was partly made while the author was in residence at the Institut Henri Poincaré in July 2017, and at the Mathematical Sciences Research Institute in Berkeley for part of the fall 2017 semester. The author benefited from support from the France Berkeley Fund, the ANR Project ANR-17-CE40-0030 - EFI, and ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. I thank Dimitri Shlyakhtenko for the discussion that initiated this work, as well as Dario Cordero-Erausquin, Thomas Courtade, Bo’az Klartag, Michel Ledoux and Emanuel Milman for their advice and comments, and Adrien Saumard for sharing a preliminary version of [32]. References [1] H. Airault, P. Malliavin, and F. Viens. Stokes formula on the Wiener space and n-dimensional Nourdin-Peccati analysis. Journal of Functional Analysis, 258(5):1763–1783, 2010. [2] D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators. 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arXiv:1702.00141v1 [math.ST] 1 Feb 2017 Reliability study of proportional odds family of discrete distributions Pradip Kundu and Asok K. Nanda∗ Department of Mathematics and Statistics, IISER Kolkata Mohanpur 741246, India Abstract The proportional odds model gives a method of generating new family of distributions by adding a parameter, called tilt parameter, to expand an existing family of distributions. The new family of distributions so obtained is known as Marshall-Olkin family of distributions or Marshall-Olkin extended distributions. In this paper, we consider Marshall-Olkin family of distributions in discrete case with fixed tilt parameter. We study different ageing properties, as well as different stochastic orderings of this family of distributions. All the results of this paper are supported by several examples. Keywords: Marshall-Olkin extended distribution, Stochastic ageing, Stochastic orders. 1 Introduction Since the work of Bennett [1] and Pettitt [25] on the proportional odds (PO) model in the survival analysis context, it has been used by many researchers. The assumption of constant hazard ratio is unreasonable in many practical cases as discussed by Bennett [1], Kirmani and Gupta [17] and Rossini and Tsiatis [26]. The PO model has been used by Bennett [1] to demonstrate the effectiveness of a cure, when the mortality rate of a group having some disease approaches that of a (disease-free) control group as time progresses. After Bennet’s [1] work, the PO model has found many practical applications, see, for instance, Collett [4], Dinse and Lagakos [8], Pettitt [25] and Rossini and Tsiatis [26]. Let X and Y be two random variables with distribution functions F (·), G(·), survival functions F̄ (·), Ḡ(·), probability density functions f (·), g(·) and hazard rate functions rX (·) = f (·)/F̄ (·), rY (·) = g(·)/Ḡ(·). Let the odds functions of X and Y be denoted respectively by θX (t) = F̄ (t)/F (t) and θY (t) = Ḡ(t)/G(t). The random variables X and Y are said to satisfy PO model with proportionality constant α if ∗ Corresponding author, e-mail: asok.k.nanda@gmail.com; asok@iiserkol.ac.in 1 θY (t) = αθX (t). For more discussion on PO models one may refer to Kirmani and Gupta [17]. It is observed that, in terms of survival functions, the PO model can be represented as Ḡ(t) = αF̄ (t) , 1 − ᾱF̄ (t) (1.1) where ᾱ = 1 − α. From the above representation we have rY (t) 1 G(t) = , = rX (t) F (t) 1 − ᾱF̄ (t) so that the hazard ratio is increasing (resp. decreasing) for α > 1 (resp. α < 1) and it convergence to 1 as t tends to ∞. This property of hazard functions makes the PO model reasonable in many practical applications. This is in contrast to the proportional hazards model where the ratio of the hazard rates remains constant with time. The model (1.1), with 0 < α < ∞; gives a method of introducing new parameter α to a family of distributions for obtaining more flexible new family of distributions as discussed by Marshall and Olkin [19]. The family of distributions so obtained is known as Marshall-Olkin family of distributions or Marshall-Olkin extended distributions (for details see [19, 20]). For more discussion and applications of Marshall-Olkin family of distributions one can see [3, 5, 6, 13]. The parameter α is called ‘tilt parameter’. This is because the hazard rate of the new family is shifted below or above the hazard rate of the underlying (baseline) distribution for α ≥ 1 and 0 < α ≤ 1 respectively. Thus, Marshall-Olkin family of distributions has implications both in terms of PO model as well as in generating a new family of flexible distributions, and hence it is worth to investigate this family of distributions. Kirmani and Gupta [17] studied some ageing properties of the PO model with fixed tilt parameter α. Nanda and Das [22] studied different ageing classes of Marshall-Olkin family of distributions taking the tilt parameter as random variable. Ghitany and Kotz [14] studied the reliability properties by taking F̄ as the reliability function of the linear failure-rate distribution. Gupta et al. [16] compared the Marshall-Olkin extended distribution and the original distribution with respect to some stochastic orderings. Nanda and Das [23] compared this family of distributions with respect to different stochastic orderings by taking the tilt parameter random. All the studies mentioned above consider the original (baseline) distribution to be continuous. However, not much work is available in the literature for discrete case. In this paper, we study different ageing properties, as well as different stochastic orderings of this family of distributions with discrete baseline distribution and with fixed tilt parameter. 2 Preliminaries Here we discuss the survival function, the hazard (failure) rate function, and the mean residual life of a discrete random variable X with support N = {1, 2, ...}. Let the probability mass 2 function (pmf) of X be given by f (k) = P {X = k}, and the distribution function be F (·) so that the reliability (survival) function F̄ (·) of X becomes F̄ (k) = P {X > k} = ∞ X f (j), k = 1, 2, ..., j=k+1 with F̄ (0) = 1. The failure rate function r(·) (Shaked et al. [28]) is given by r(k) = P {X = k\|X ≥ k} = P {X = k} f (k) = , P {X ≥ k} F̄ (k − 1) and the reversed hazard rate function is given by r̃(k) = f (k)/F (k). Below we give the definitions of different discrete ageing classes. Definition 2.1 A discrete random variable X is said to be (i) ILR (DLR) i.e. increasing (decreasing) in likelihood ratio if f (k) is log-concave (logconvex), i.e. if f (k + 2)f (k) ≤ (≥) f 2 (k + 1), k ∈ N (Dewan and Sudheesh [9]); (ii) IFR (DFR) i.e. increasing (decreasing) failure rate if r(k) is increasing (decreasing) in k ∈ N. This is equivalent to the fact that F̄ (k + 1)/F̄ (k) is decreasing (increasing) in k ∈ N (Salvia and Bollinger [27], Shaked et al. [28], Gupta et al. [15]); (iii) IFRA (DFRA) i.e. increasing (decreasing) in failure rate average if [F̄ (k)]1/k is decreasing (increasing) in k, i.e., [F̄ (k)]1/k ≥ (≤) [F̄ (k + 1)]1/(k+1) , k ∈ N (Esary et al. [10], Shaked et al. [28]); (iv) NBU (NWU) i.e. new better (worse) than used if F̄ (j + k) ≤ (≥) F̄ (j)F̄ (k), j, k ∈ N (Esary et al. [10], Shaked et al. [28]); (v) DRHR (decreasing reversed hazard rate) if r̃(k) is decreasing in k, i.e. if F (k) is logconcave, i.e. if [F (k + 1)]2 ≥ F (k)F (k + 2), k ∈ N (Nanda and Sengupta [24], Li and Xu [18]). (vi) NBAFR (new better than used in failure rate average) if [F̄ (k)]1/k ≤ F̄ (1), k ∈ N (Fagiuoli and Pellerey [11]). 2.1 Proportional odds family of discrete distributions Let X be a discrete random variable with support N = {1, 2, ...} having pmf f (·), distribution function F (·), survival function F̄ (·), hazard rate function rX (·), and reversed hazard rate function r̃X (·). Starting with the survival function F̄ , the survival function of the proportional odds family (also known as Marshall-Olkin family) of discrete distribution is given by Ḡ(k; α) = αF̄ (k) , k = 1, 2, ..., 0 < α < ∞, ᾱ = 1 − α, 1 − ᾱF̄ (k) 3 (2.1) with Ḡ(0; α) = 1. Let the corresponding random variable be denoted by Y . Now the distribution function of Y is given by G(k; α) = 1 − Ḡ(k; α) = F (k) , 1 − ᾱF̄ (k) (2.2) whereas the pmf is given by g(k; α) = Ḡ(k − 1; α) − Ḡ(k; α) = αf (k) . [1 − ᾱF̄ (k − 1)][1 − ᾱF̄ (k)] (2.3) The corresponding hazard rate and the reversed hazard rate functions are given by rX (k) , 1 − ᾱF̄ (k) (2.4) αr̃X (k) . 1 − ᾱF̄ (k − 1) (2.5) rY (k; α) = r̃Y (k; α) = It is to be mentioned here that different properties of (2.1) have been studied by Déniz and Sarabia [7] by taking F as the cdf of Poisson random variable. 3 Stochastic Ageing properties In this section we study how different ageing properties of X are transmitted to the random variable Y . With the following two counterexamples, one with α > 1 and the other with α < 1, we show that if X is ILR, then Y is neither ILR nor DLR. Counterexample 3.1 Consider the random variable X with the mass function given by   0, if k = 1;        0.1, if k = 2; f (k) = 0.25, if k = 3;     0.35, if k = 4;     0.3, if k = 5. Clearly X is ILR. For α = 5, we have the mass function of Y as   0, if k = 1;     1  if k = 2;   46 , 125 g(k; 5) = 1656 , if k = 3;   175   792 , if k = 4;     15 , if k = 5. 22 It is observed that Y is neither ILR nor DLR. 4 Counterexample 3.2 Consider the random variable X with mass function given by   0, if k = 1;        0.3, if k = 2; f (k) = 0.34, if k = 3;     0.26, if k = 4;     0.1, if k = 5. Here X is ILR. For α = 0.2, we have the mass function of Y as   0, if k = 1;     15  if k = 2;   22 , 425 g(k; 0.2) = 1958 , if k = 3;   325    4094 , if k = 4;    1, if k = 5. 46 It is observed that Y is neither ILR nor DLR. ✷ With the following two counterexamples, one for α > 1 and the other for α < 1, we show that if X is DLR, then Y is neither DLR nor ILR. Counterexample 3.3 Consider the random variable X with mass function given by   0.36, if k = 1;     0.26, if k = 2; f (k) =  0.21, if k = 3;     0.17, if k = 4. Here X is DLR. For α = 2, we have the mass function of Y as  9  if k = 1;  41 ,    650 , if k = 2; 2829 g(k; 2) = 700    2691 , if k = 3;   34 if k = 4. 117 , It is observed that Y is neither DLR nor ILR. Counterexample 3.4 Consider the random variable X with mass function given by   0.26, if k = 1;     0.18, if k = 2; f (k) =  0.24, if k = 3;     0.32, if k = 4. 5 Here X is DLR. For α = 0.4, we have the mass function of Y as  65  if k = 1;  139 ,    2250 , if k = 2; 11537 g(k; 0.4) = 1500  if k = 3;  8383 ,    16 if k = 4. 101 , It is observed that Y is neither DLR nor ILR. ✷ The following theorem gives the condition under which IFR/DFR property of X is transmit ted to the random variable Y . The proof follows from the fact that 1/ 1 − ᾱF̄ (k) is increasing (resp. decreasing) in k for α ≥ (resp. ≤)1. Theorem 3.1 If X is IFR (resp. DFR) and α ≥ (resp. ≤) 1, then Y is IFR (resp. DFR). ✷ The following counterexample shows that if α < 1, then the IFR property of X may not be transmitted to the random variable Y . Counterexample 3.5 Consider the random variable X following discrete IFR distribution (cf. Salvia and Bollinger [27]) with f (k) = (k − c)ck−1 /k!, k ∈ N, 0 < c ≤ 1. Here F̄ (k) = ck /k! and rX (k) = 1 − c/k so that X is IFR. Now rY (k; α) = 1 − c/k . 1 − ᾱck /k! It is observed that, for c = 0.8 and α = 0.2, we have rY (2; 0.2) = 0.8064516, rY (3; 0.2) = 0.7870635, and rY (4; 0.2) = 0.8110739. This shows that Y is neither IFR nor DFR. ✷ The following counterexample shows that the DFR property of X may not be transmitted to the random variable Y when α > 1. Counterexample 3.6 Let X follow the Type I discrete Weibull distribution (cf. Nakagawa and Osaki [21]) with pmf given by β β f (k) = q (k−1) − q k , k ∈ N, q ∈ (0, 1), β > 0. β Then the corresponding survival function is given by F̄ (k) = q k , and the hazard rate function is given by rX (k) = 1 − q k β −(k−1)β . Here X is DFR for 0 < β < 1. Note that β β 1 − q k −(k−1) rY (k; α) = . 1 − ᾱq kβ It is observed that, for β = 0.8, α = 5 and q = 0.5, we have rY (7; 5) = 0.2759209, rY (10; 5) = 0.2834942, and rY (13; 5) = 0.2793229. This shows that Y is neither DFR nor IFR. 6 ✷ Below we see that, for α ≥ (resp. ≤) 1, the NBU (resp. NWU) property of X is transmitted to the random variable Y . Theorem 3.2 If X is NBU (resp. NWU) and α ≥ (resp. ≤) 1, then Y is NBU (resp. NWU). Proof: Let X be NBU (resp. NWU). Then Y will be NBU (resp. NWU) if and only if 1 − ᾱF̄ (j + k) (1 − ᾱF̄ (j))(1 − ᾱF̄ (k)) ≥ (resp. ≤) . F̄ (j + k) αF̄ (j)F̄ (k) This is equivalent to the fact that 1 − ᾱ F̄ (j) + F̄ (k) + ᾱF̄ (k)F̄ (j) 1 , ≥ (resp. ≤) αF̄ (j)F̄ (k) F̄ (j + k) which holds if 1 1 ≥ (resp. ≤) . F̄ (j + k) F̄ (j)F̄ (k) The last inequality follows from the fact that, for α ≥ (resp. ≤) 1, 1 − ᾱ F̄ (j) + F̄ (k) + ᾱF̄ (k)F̄ (j) ≤ (resp. ≥) α. Hence the theorem follows. ✷ The following counterexamples show that, for α < (resp. >) 1, the NBU (resp. NWU) property of X may not be transmitted to the random variable Y . Counterexample 3.7 Consider X following the discrete S-distribution (cf. Bracuemond and Gaudoin [2]) with pmf given by f (k) = p(1 − ak ) k−1 Y (1 − p + pai ), k ∈ N, 0 < p ≤ 1, 0 < a < 1. i=1 This gives the survival function as F̄ (k) = rX (k) = p(1 − ak ). Here X is NBU. Now Ḡ(k; α) = α Qk i=1 (1 Qk 1 − ᾱ i=1 Q k − p + pai ), and hazard rate function as (1 − p + pai ) i=1 (1 − p + pai ) . For j = 2, k = 3, p = 0.3, a = 0.6, α = 0.2, we have Ḡ(j + k; α) = 0.075737 and Ḡ(j; α)Ḡ(k; α) = 0.063494. This shows that Y is not NBU. Counterexample 3.8 Let X follow the distribution as given in Counterexample 3.6. Then clearly X is NWU for β ∈ (0, 1). Now, for j = 2, k = 3, α = 5, q = 0.5, we have Ḡ(j + k; α) = 0.3062174 and Ḡ(j; α)Ḡ(k; α) = 0.3657684. This shows that Y is not NWU. 7 ✷ Kirmani and Gupta [17] have observed that if X is IFRA (DFRA), then Y is IFRA (DFRA) for α > (<) 1. Below we show that if X is IFRA, then Y may not be IFRA or DFRA when α < 1. Counterexample 3.9 Let X follow the distribution as given in Counterexample 3.7. Here X is IFRA. Now, for α = 0.2, p = 0.5, and a = 0.6, we have    0.44444, for k = 1;  1/k [Ḡ(k; 0.2)] = 0.438901, for k = 2;    0.457806, for k = 4. This shows that Y is neither IFRA nor DFRA. ✷ Below we show that if X is DFRA, then, for α > 1, Y may not be DFRA or IFRA. Counterexample 3.10 Let X follow the discrete Pareto distribution with survival function c d , k ∈ N, c, d > 0, F̄ (k) = k+d which is DFR and hence DFRA. Now α Ḡ(k; α) = 1 − ᾱ d k+d c d k+d c . For α = 6, d = 2, c = 3, we have [Ḡ(k; 6)]1/k =     0.7164179, for k = 1; 0.658037,    0.68081, for k = 4; for k = 8, which is neither increasing nor decreasing in k, i.e. Y is neither DFRA nor IFRA. ✷ Following theorem shows that, for α ≤ 1, the DRHR property of X is transmitted to the random variable Y . The proof follows from the fact that 1/ 1 − ᾱF̄ (k − 1) is decreasing in k, for α ≤ 1. Theorem 3.3 If X is DRHR, then Y is DRHR for α ≤ 1. ✷ The following counterexample shows that, for α > 1, DRHR property of X may not be transmitted to the random variable Y . Counterexample 3.11 Consider the random   0,     4    25 , 2 F (k) = 5,   2    3,    1, variable X having distribution function given by if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5. 8 Clearly X is DRHR. For α = 4, the distribution   0,     1    22 , 1 G(k; 4) = 7,     31 ,     1, function of Y is given by if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5, which is not DRHR. ✷ The following counterexample shows that, for α < 1, NBAFR property of X may not be transmitted to the random variable Y . Counterexample 3.12 Consider the random  4   5,    8, 13 F̄ (k) = 1   2,    0, variable X with reliability function given by if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4. Here X is NBAFR. For α = 0.4, we have the reliability function of Y as  8   13 , if 1 ≤ k < 2;    16 , if 2 ≤ k < 3; 41 Ḡ(k; 0.4) = 2  if 3 ≤ k < 4;  7,    0, if k ≥ 4. It is clear that Y is not NBAFR. ✷ We summarize the above findings in Table 1. 4 Stochastic Orderings Let X1 and X2 be two discrete random variables with support N = {1, 2, ...} having respective pmf f1 (·), f2 (·), distribution function F1 (·), F2 (·), and survival function F̄1 (·), F̄2 (·). Let the survival function of the Marshall-Olkin family of discrete distributions be given by Ḡi (k; α) = αF̄i (k) , 0 < α < ∞, ᾱ = 1 − α, 1 − ᾱF̄i (k) and let the corresponding random variable be Yi , i = 1, 2. The following theorem shows that the usual stochastic order between X1 and X2 and that of Y1 and Y2 are equivalent. Theorem 4.1 Y1 ≤st Y2 if and only if X1 ≤st X2 . 9 Table 1: Preservation of ageing classes Ageing properties α<1 α>1 ILR Not Preserved Not Preserved DLR Not Preserved Not Preserved IFR Not Preserved Preserved DFR Preserved Not Preserved NBU Not Preserved Preserved NWU Preserved Not Preserved IFRA Not Preserved Preserved DFRA Preserved Not Preserved DRHR Preserved Not Preserved NBAFR Not Preserved Proof: Note that Y1 ≤st Y2 if, and only if αF̄1 (k) 1 − ᾱF̄1 (k) αF̄2 (k) , 1 − ᾱF̄2 (k) ≤ which is equivalent to the fact that F̄1 (k) ≤ F̄2 (k). Hence the theorem follows. ✷ The following theorem gives condition on α, under which hazard rate order between X1 and X2 is transmitted to that between Y1 and Y1 . Theorem 4.2 If X1 ≤hr X2 , then Y1 ≤hr Y2 , provided α ≥ 1. Proof: Since hazard rate order is stronger than usual stochastic order, we have, for α ≥ 1, 1 1 ≥ . 1 − ᾱF̄1 (k) 1 − ᾱF̄2 (k) Now, using the hypothesis we have, from (2.4), rY1 (k; α) = rX2 (k) rX1 (k) ≥ = rY2 (k; α). 1 − ᾱF̄1 (k) 1 − ᾱF̄2 (k) Hence the theorem follows. ✷ The following counterexample shows that the above theorem does not hold if α < 1. Counterexample 4.1 Consider the random variables X1 and X2 with respective reliability function   1,     1, 2 F̄1 (k) = 2   5,    0, if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4, 10 and   1,     5, 8 F̄2 (k) =  11 ,  20    0, if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4. This shows that X1 ≤hr X2 . For α = 0.2, we have the respective reliability function of Y1 and Y2 as and This shows that Y1 hr Y2 .   1,     1, 6 Ḡ1 (k; α) = 2   17 ,    0, if 1 ≤ k < 2;   1,     1, 4 Ḡ2 (k; α) =  11 ,  56    0, if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4, if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4. ✷ The following theorem gives the condition on α such that reversed hazard rate order between X1 and X2 is transmitted to that between Y1 and Y2 . Theorem 4.3 If X1 ≤rhr X2 , then Y1 ≤rhr Y2 , for 0 < α ≤ 1. Proof: Since reversed hazard rate order is stronger than usual stochastic order, we have, for α ≤ 1, 1 1 ≤ . 1 − ᾱF̄1 (k − 1) 1 − ᾱF̄2 (k − 1) Now, using the hypothesis we have, from (2.5), r̃Y1 (k; α) = αr̃X1 (k) αr̃X2 (k) ≤ = r̃Y2 (k; α). ¯ 1 − ᾱF1 (k − 1) 1 − ᾱF¯2 (k − 1) Hence the theorem follows. ✷ That the above theorem does not hold in case of α > 1 is shown in the following counterexample. Counterexample 4.2 Consider the random variables X1 and X2 with respective distribution function   0,     5    24 , 1 F1 (k) = 2,   3    4,    1, 11 if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5, and F2 (k) =   0,     1    6, if 1 ≤ k < 2; if 2 ≤ k < 3; 5 12 , 2 3,         1, if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5. Clearly X1 ≤rhr X2 . For α = 4, we have the distribution functions of Y1 and Y2 respectively as   0, if 1 ≤ k < 2;     5    81 , if 2 ≤ k < 3; 1 G1 (k; 4) = if 3 ≤ k < 4; 5,   3   if 4 ≤ k < 5;  7,    1, if k ≥ 5, and This shows that Y1 rhr Y2 .   0,     1    21 , 5 G2 (k; 4) = 33 ,   1    3,    1, if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5. ✷ Following two counterexamples show that the likelihood ratio order between X1 and X2 is not necessarily transmitted to that between Y1 and Y2 . Counterexample 4.3 Let X1 and X2 have          f1 (k) =         and f2 (k) = the respective probability mass function 0, if k = 1; 0.3, if k = 2; 0.4, if k = 3; 0.2, if k = 4; 0.1, if k = 5,   0, if k = 1;        0.2, if k = 2; 0.3, if k = 3;     0.2, if k = 4;     0.3, if k = 5. 12 Clearly X1 ≤lr X2 . For α = 5, we have the mass functions of Y1 and Y2 respectively as   0, if k = 1;     3    38 , if k = 2; 50 g1 (k; 5) = 209 , if k = 3;     25 if k = 4;  77 ,    5 , if k = 5, 14 and   0,     1    21 , 5 g2 (k; 5) = 42 ,   5    33 ,    15 , 22 This shows that Y1 lr Y2 . if k = 1; if k = 2; if k = 3; if k = 4; if k = 5. ✷ Counterexample 4.4 Take the random variables X1 and X2 having respective mass functions   0, if k = 1;        0.3, if k = 2; f1 (k) = 0.3, if k = 3;     0.2, if k = 4;     0.2, if k = 5, and   0,        0.2, f2 (k) = 0.3,     0.24,     0.26, if k = 1; if k = 2; if k = 3; if k = 4; if k = 5. Clearly X1 ≤lr X2 . For α = 0.2, we have the mass functions of Y1 and Y2 as   0, if k = 1;     15    22 , if k = 2; 75 g1 (k; 0.2) = 374 , if k = 3;   25    357 , if k = 4;    1 , if k = 5, 21 13 and g2 (k; 0.2) = This shows that Y1 lr Y2 .   0,     5    9,         5 18 , 10 99 , 13 198 , if k = 1; if k = 2; if k = 3; if k = 4; if k = 5. ✷ We summarize the above findings in Table 2. Table 2: Preservation of stochastic orderings Stochastic orders between α<1 α>1 baseline distributions 5 Usual stochastic order Preserved Preserved Hazard rate order Not Preserved Preserved Reversed hazard rate order Preserved Not Preserved Likelihood ratio order Not Preserved Not Preserved Conclusion Marshall and Olkin [19] introduced a method of adding a new parameter, called tilt parameter, to a family of distributions for obtaining more flexible new families of distributions. In the literature, some reliability properties of this family of distributions are studied with continuous baseline distributions. However, not much study is done in the literature, to the best of our knowledge, for discrete baseline distributions. This paper discusses various stochastic ageing properties, as well as different stochastic orderings of this family with discrete baseline distributions. Acknowledgements: The support received from IISER Kolkata to carry out this research work is gratefully acknowledged by Pradip Kundu. The financial support from NBHM, Govt. of India (vide Ref. No. 2/48(25)/2014/NBHM(R.P.)/R&D II/1393 dt. Feb. 3, 2015) is duly acknowledged by Asok K. Nanda. 14 References [1] S. Bennett, Analysis of survival data by the proportional odds model, Statistics in Medicine 2 (1983) 273-277. [2] C. Bracquemond, O. Gaudoin, A survey on discrete lifetime distributions, International Journal of Reliability, Quality and Safety Engineering 10(1) (2003) 69-98. [3] C. Caroni, Testing for the Marshall-Olkin extended form of the Weibull distribution, Statistical Papers 51 (2010) 325-336. [4] D. Collett, Modelling survival data in medical research, 2nd edition, Chapman and Hall/CRC, 2004. [5] G.M. Cordeiro, A.J. Lemonte, E.M.M. Ortega, The Marshall-Olkin family of distributions: mathematical properties and new models, Journal of Statistical Theory and Practice 8(2) (2014) 343-366. [6] G.M. Cordeiro, A.J. Lemonte, On the Marshall-Olkin extended Weibull distribution, Statistical Papers 54(2) (2013) 333-353. [7] E.G. Déniz, J.M. Sarabia, A discrete distribution including the Poisson, Communications in Statistics - Theory and Methods 45(8) (2016) 2311-2322. [8] G.E. Dinse, S.W. Lagakos, Regression analysis of tumour prevalence data, Applied Statistics 32(3) (1983) 236-248. [9] I. Dewan, K.K. Sudheesh, Ageing concepts for discrete data- A relook, In: Proceedings of the IEEE International Conference on Quality and Reliability, Bangkok, Thailand, 2011. [10] J.D. Esary, A.W. Marshall, F. Proschan, Shock models and wear processes, The Annals of Probability 1(4) (1973) 627-649. [11] E. Fagiuoli, F. Pellerey, Preservation of certain classes of life distributions under Poisson shock models, Journal of Applied Probability 31 (1994) 458-465. [12] M.E. Ghitany, E.K. Al-Hussaini, R.A. Al-Jarallah, Marshall-Olkin extended Weibull distribution and its application to censored data, Journal of Applied Statistics 32(10) (2005) 1025-1034. [13] M.E. Ghitany, F.A. Al-Awadhi, L.A. Alkhalfan, Marshall-Olkin extended Lomax distribution and its application to censored data, Communications in Statistics - Theory and Methods 36(10) (2007) 1855-1866. 15 [14] M.E. Ghitany, S. Kotz, Reliability properties of extended linear failure rate distributions, Probability in the Engineering and Informational Sciences 21 (2007) 441-450. [15] P.L. Gupta, R.C. Gupta, R.C. Tripathi, On the monotonic properties of discrete failure rates, Journal of Statistical Planning and Inference 65 (1997) 255-268. [16] R.C. Gupta, S. Lvin, C. Peng, Estimating turning points of the failure rate of the extended Weibull distribution, Computational Statistics and Data Analysis 54 (2010) 924-934. [17] S.N.U.A. Kirmani, R.C. Gupta, On the proportional odds model in survival analysis, Annals of the Institute of Statistical Mathematics 53(2) (2001) 203-216. [18] X. Li, M. Xu, Reversed hazard rate order of equilibrium distributions and a related ageing notion, Statistical Papers 49 (2008) 749-767. [19] A.W. Marshall, I. Olkin, A new method of adding a parameter to a family of distributions with applications to the exponential and Weibull families, Biometrika 84 (1997) 641-652. [20] A.W. Marshall, I. Olkin, Life Distributions. Springer, New York, 2007. [21] T. Nakagawa, S. Osaki, The Discrete Weibull Distribution, IEEE Transactions on Reliability, R-24(5) (1975) 300-301. [22] A.K. Nanda, S. Das, Some ageing properties of Marshall-Olkin extended distribution, International Journal of Mathematics and Statistics. 13(1) (2013) 93-107. [23] A.K. Nanda, S. Das, Stochastic orders of the Marshall-Olkin extended distribution, Statistics and Probability Letters 82 (2012) 295-302. [24] A.K. Nanda, D. Sengupta, Discrete life distributions with decreasing reversed hazard, Sankhyā 55 (2005) 164-168. [25] A.N. Pettitt, Proportional odds models for survival data and estimates using ranks, Applied Statistics 33(2) (1984) 169-175. [26] A.J. Rossini, A.A. Tsiatis, A Semiparametric proportional odds regression model for the analysis of current status data, Journal of the American Statistical Association 91(434) (1996) 713-721. [27] A.A. Salvia, R.C. Bollinger, On discrete hazard functions, IEEE Transactions on Reliability 31 (5) (1982) 458-459. [28] M. Shaked, J.G. Shanthikumar, J.B. Valdez-Torres, Discrete hazard rate functions, Computers and Operations Research 22 (4) (1995) 391-402. 16	10

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Lightweight Classification of IoT Malware based on Image Recognition Jiawei Su arXiv:1802.03714v1 [cs.CR] 11 Feb 2018 School of Information Science and Electrical Engineering, Kyushu University, Japan jiawei.su@inf.kyushu-u.ac.jp Daniele Sgandurra ISG Smart Card and IoT Security Centre, Royal Holloway University of London, UK daniele.sgandurra@rhul.ac.uk Danilo Vasconcellos Vargas Faculty of Information Science and Electrical Engineering, Kyushu University, Japan vargas@inf.kyushu-u.ac.jp Yaokai Feng Faculty of Information Science and Electrical Engineering, Kyushu University, Japan fengyk@ait.kyushu-u.ac.jp ABSTRACT The Internet of Things (IoT) is an extension of the traditional Internet, which allows a very large number of smart devices, such as home appliances, network cameras, sensors and controllers to connect to one another to share information and improve user experiences. Current IoT devices are typically micro-computers for domain-specific computations rather than traditional functionspecific embedded devices. Therefore, many existing attacks, targeted at traditional computers connected to the Internet, may also be directed at IoT devices. For example, DDoS attacks have become very common in IoT environments, as these environments currently lack basic security monitoring and protection mechanisms, as shown by the recent Mirai and Brickerbot IoT botnets. In this paper, we propose a novel light-weight approach for detecting DDos malware in IoT environments. We firstly extract one-channel gray-scale images converted from binaries, and then utilize a lightweight convolutional neural network for classifying IoT malware families. The experimental results show that the proposed system can achieve 94.0% accuracy for the classification of goodware and DDoS malware, and 81.8% accuracy for the classification of goodware and two main malware families. CCS CONCEPTS • Network security → IOT network security; • Machine learning → Convolutional neural network; KEYWORDS Internet of things, Malware image, Convolutional neural network, Light-weight detection Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). Conference’17, July 2017, Washington, DC, USA © 2018 Copyright held by the owner/author(s). ACM ISBN 978-x-xxxx-xxxx-x/YY/MM. https://doi.org/10.1145/nnnnnnn.nnnnnnn Sanjiva Prasad Department of Computer Science and Engineering, Indian Institute of Technology Delhi, India Sanjiva.Prasad@cse.iitd.ac.in Kouichi Sakurai Faculty of Information Science and Electrical Engineering, Kyushu University, Japan sakurai@csce.kyushu-u.ac.jp ACM Reference Format: Jiawei Su, Danilo Vasconcellos Vargas, Sanjiva Prasad, Daniele Sgandurra, Yaokai Feng, and Kouichi Sakurai. 2018. Lightweight Classification of IoT Malware based on Image Recognition. In Proceedings of ACM Conference (Conference’17). ACM, New York, NY, USA, 7 pages. https://doi.org/10.1145/ nnnnnnn.nnnnnnn 1 INTRODUCTION Nowadays the notion of the “Internet” has extended from the connection between personal computers to networks composed of a much larger range of devices. Traditional micro devices, such as many kinds of sensors and controllers, are typically only able to perform function-specific tasks based on pre-defined rules. By substituting these function-specific devices with CPU-controlled ones, and by enabling interconnection among them through the Internet, these “things” become “smart” and can now deal with complex tasks. In addition, by enabling Cloud services on these smart-devices, users can easily receive data reported by them and control them. Despite these advantages, smarter devices imply more vulnerabilities, due to the complexity in hardware and software, with more chances for potential adversaries to threaten them. In addition, IoT systems are generally unsecured due to the difficulty of creating unified standards for the various types of IoT hardware and software platforms. Finally, even if smarter compared with traditional sensors, IoT devices still lack sufficient computational resources to be able to use existing PC-based security solutions. In some cases, Cloud services provide a way for developing security protection for IoT devices, e.g. for malware detection [18, 19]. In this paper, we consider a solution to protect local IoT devices from being abused to perform DDoS attacks by being enslaved in botnets of IoT devices, which is currently a common attack. To accomplish this, we first classify IoT DDoS malware samples recently collected in the wild on two major families, namely Mirai and Linux.Gafgyt. We then propose a lightweight solution for detecting and classifying IoT DDoS malware and benign applications locally on the IoT devices by converting the program binaries to gray-scale images, and by feeding these images to a small size convolutional neural network for classification. In this way, resource-constrained Conference’17, July 2017, Washington, Jiawei Su, DC, Danilo USA Vasconcellos Vargas, Sanjiva Prasad, Daniele Sgandurra, Yaokai Feng, and Kouichi Sakurai IoT devices can afford the computation needed for running the proposed detection system locally. Experimental results show that the proposed system can achieve 94.0% accuracy for classifying goodware and DDoS malware, and 81.8% accuracy for the classification of goodware and two main malware families. The main contributions of this research are the following ones: • this is the first classification system tested on real IoT malware samples: previous works have used regular or mobile malware samples instead, due to the difficulty in obtaining IoT malware samples [2, 11, 13]. Specifically there is currently no publicly available IoT malware dataset and the first IoT honeypot for collecting samples of IoT threats was released relatively recently [1]; • the IoT malware classification system can be deployed on real IoT devices. We show in detail the feasibility of using lightweight image classifier for recognizing IoT malware through malware images. Malware image classification has been proposed for classifying regular malware [4]; however, IoT malware is functionally different. For example, many IoT malware may try to kill other malware to guarantee enough computational resource for themselves; • according to the experimental results, we prove that the proposed system can reliably classify goodware and IoT DDoS malware. • to the best of our knowledge, there is currently no reference to describe the time complexity of CNNs. However, the proposed CNN-based approach is empirically considered to be lightweight since it does not need to maintain any training data for classification, differently from other types of classifiers for malware, such as Support Vector Machine and K-nearest neighbours. The computation of CNN for classification is rather simple, and only involves summation and activation. In addition, the proposed system is based on a two layer shallow network which is much more efficient than common deep learning models. The paper is structured as follows. In Sect. 2 we discuss related work. Sect. 3 explains procedures for extracting IoT DDoS malware images and implementing a small size convolutional neural network for classification. In Sect. 4 the detection results in two different scenarios are listed and in Sect. 5 the achievement of this research is summarized and future work is discussed. 2 RELATED WORKS Even if IoT security is an important topic, few defensive solutions exist in the literature [12]. Only recently, the first honeypot specifically for collecting IoT malware has been established by Pa et al. [1]. Their honeypot systems simulated 8 different CPU architectures and are built for observing attacks coming through the Telnet protocol. Initially they collected 43 distinct malware samples which are mostly DDoS attack malware. Their results show that the DDoS attack is the most common security threat in current IoT network environments. These authors kindly shared their observed data set with us which we have used in this research for evaluating our proposal. To the best of our knowledge, while most other works focus on Android malware detection [24, 25], the“Cloudeye” [2] is in practice the only current work specific for IoT malware detection. The system is a signature matching-based malware detection solution. IoT clients are only responsible for preliminary scanning the software locally, and then sending hashed abstracts of suspicious files to Cloud servers for deep analysis, therefore guaranteeing data privacy and low-cost communications. However, in IoT environments the inherent weakness of signature matching-based detection still exists: for example, Cloudeye is not able to deal with new variants of existing samples. Apart from signature matching, machine learning-based malware detection has been proved as effective in various scenarios [3, 14–16, 22, 23]. In IoT environments, also heavy computation machine learning methods are expected to be suitable too because of the availability of Cloud services. In fact, in a possible scenario, the training can be performed on Cloud server, while resourceconstrained IoT devices can receive the trained classifiers from the servers and run the algorithm locally. Note that several machine learning classifiers are heavy at training but efficient during test phase. Classifying malware images has been proven as an effective way for recognizing common PC malware [9, 26]. It is essentially a method for comparing two malware binaries. Nataraj et al. first utilize malware images for classifying regular Internet malware with knearest neighbors [4]. However, the system requires pre-processing of filtering to extract the image texture as features for classification, which might not fit the resource-constrained IoT environments. Similaly, the artificial neural network (ANN) malware classification proposed by Makandar [28] might not be a good candidate to be run on IoT devices to handle due to the heavy computational cost of multiple fully connected layers in ANN for classification. Yue utilized convolutional neural network for malware family classification [5]. In this research, we use malware images for IoT malware classification and show it is a feasible approach. 3 METHODOLOGY In this Section, we describe the methodology of feeding malware images as features, to a small two-layer convolutional neural network for detection. 3.1 Lightweight IoT DDoS Malware Filter For the scenario of detecting IoT DDoS malware detection locally, as previously pointed out, the main difficulty of deploying malware filters lies in the fact that the computational resources available on current IoT devices is limited. A direct solution under such a condition is relying on the security protection services provided by powerful remote servers, such as in Cloud-enabled IoT environments. Cloud servers are usually better protected against node failures, e.g. due to DDoS attacks. Another advantage of using Cloud servers is that a malware databases can be made more comprehensive and can be updated more rapidly than on IoT devices. For these reasons, we propose a two-tier detection architecture, based on a local IoT detection system and a remote, Cloud-based, classification system. In more detail, firstly a lightweight malware classification system is responsible of recognizing suspicious programs and behaviors locally. Note that, at this stage, the main goal is to provide a score on binary suspiciousness. Lightweight Classification of IoT Malware based on Image Recognition Conference’17, July 2017, Washington, DC, USA detecting zero-day attacks. Finally, converting malware binaries to the corresponding images only requires creating the input vectors to the convolutional neural network, i.e. 8-bit vectors, which is a very fast operation. 3.4 Figure 1: The proposed light-weight malware detection scheme. The local detector is located on the client side and captures potentially suspicious programs by relying on cloud backend for final decision • automatic feature extraction: neural network can automatically extract higher level features from the input raw features. That is, the network can learn deep non-linear features that can be hardly discovered and understood by humanbeings. These are sometimes actually counter-intuitive, but indeed effective. Note that many previous works have focused on extracting effective features for malware detection. However, most of them are only effective under specific scenarios, and this might lead to poor scalability. • test-phase efficiency: the training progress of a convolutional neural network requires heavy computation and, for instance, high-end graphic cards are necessary for accelerating training large networks. However, once trained, the network itself is rather lightweight and can be run with tiny computational resources, since only the trained parameters and information of network structure are kept [29, 30]. In contrast, another supervised lightweight classifier, the oneclass support vector machine (OCSVM), though simpler than normal Two-class SVM, still needs to keep a certain amount of training data when running the classification, while a convolutional neural network does not need to keep any. In practice, the training can be handled by the Cloud servers and only the trained network is sent to IoT nodes. On the local IoT side, the convolutional neural network is run to detect malware. In such a case, the system delivers the files or the corresponding abstracts to a remote Cloud server for deeper analysis. In addition, the cloud side can update and distribute new trained detectors to the clients periodically. In the following, we discuss the local malware filter on the client side. We assume that a set of Cloud servers are able to analyze malware samples and retrain the classifiers using standard machine learning algorithms. The proposal system structure is shown in Fig. 1 3.2 IoT DDoS Malware Families According to recent observations and preliminary analysis [1], even if IoT DDoS malware are functionally similar to existing DDoS malware on PC platforms, they contain specific features that are rarely observed on PCs. For example, some samples try to kill other ones of competitive families to get more system resources for themselves, due to the limited computational capability of IoT devices. In addition, IoT malware often target a wide range of devices, such as Internet cameras, DVR and so on. Finally, IoT malware can be also compatible with different processor architectures, ensuring the maximum possible successful infections. 3.3 Malware Image Classification An interesting and novel way of performing malware classification is to analyze their converted binary images. In particular, a malware binary can be reformatted as an 8-bit sequence and then be converted to a gray-scale image which has one channel and pixel values from 0 to 255 [4]. The resulting image can then be fed into machine learning image classifiers for classification. Note that running machine learning classifiers typically needs fewer local storage than signature-matching systems, which is the most common used malware detection method. This is important for storage-constrained IoT devices. In fact, in a matching signatures system, the signature database is typically large in size as it has to contain information for each malware sample and all of its possible variants. In the case of machine learning, little information has to be kept for classification. For example, k-means clustering needs only the information of centroids and radii for classification once trained. Support vector machine merely keeps a small set of training data (i.e., the support vectors) in the test phase. In addition, machine learning methods overcome signature matching on Neural Network for Malware Detection Convolutional neural networks have been proven to have better performance for image recognition than many other kinds of classifiers. A convolutional neural network has two important characteristics that make it fit the scenario of preliminary filtering malware on local IoT devices: 4 EXPERIMENT AND RESULTS In this section we discuss the experimental setup and the results of the classification of the proposed system. 4.1 Preparing the Dataset For these experiments, we have used an IoT DDoS malware dataset newly collected by IoTPOT [1], the first honeypot for collecting IoT threat samples. The malware samples are labelled using VirusTotal [8] with the majority rule. The dataset originally contains 500 malware samples, where most of them are classified into four big families: Linux.Gafgyt.1, Linux.Gafgyt (other variants of Linux.Gafgyt family) , Mirai [10] and Trojan.Linux.Fgt. The rest of the samples belong to relatively rare families such as Tsunami, Hajime, LightAidra. Then we cluster the samples into two categories: Mirai family, which contains Mirai and Trojan.Linux.Fgt1 , and Linux.Gafgyt family which contains Linux.Gafgyt.1 and the other variants. Instead, the benign binary samples (goodware) are collected from 1 Mirai has been shown to have similar features to Trojan.Linux.Fgt [17]. Conference’17, July 2017, Washington, Jiawei Su, DC, Danilo USA Vasconcellos Vargas, Sanjiva Prasad, Daniele Sgandurra, Yaokai Feng, and Kouichi Sakurai convolution layer(kernel=3, stride=1, depth=32) max pooling layer(kernel=2, stride=2) convolution layer(kernel=3, stride = 1, depth=72) max pooling layer(kernel=2, stride=2) fully connected layer(size=256) softmax classifier Table 1: Structure of the Implemented Convolutional Neural Network Ubuntu 16.04.3 system files. The number of samples are balanced for each family by randomly removing the samples that belong to classes that are too large. After the preprocessing phase, we analyzed 365 samples where each class has the same number of samples. Among them, we utilize 45 sample (each class has 15 samples) for testing, and the rest for training. According to the discussion above, the system proposed is only responsible for preliminary detection. That is, the goal is to identify whether a sample is benign or belongs to one of the big malware families: Mirai and Linux.Gafgyt, but there is no need to understand exactly which kind of variant it is. 4.2 Obtaining the Malware Images We then convert each sample of the dataset into the corresponding malware gray-scale image by following the same procedures implemented in [4]. In particular, a malware binary can be reformatted as an 8-bit string sequence, whose decimal encoding represents the value of a one-channel pixel (in the range [0, 255]). Therefore the entire sequence represents a gray-scale image. We rescale the images to the size of 64X64 to be used as input to a convolutional neural network. Some examples of malware and benign-ware images are shown by Fig. 2, 3 and 4. In these images, the structural difference between malware and goodware images can be easily identified. For example, it can be seen that malware images always are more dense. In particular, the majority of the Mirai malware images have a dense central code payload. On the other hand, the images of goodware tend to have larger header parts than malwares. 4.3 Convolutional Neural Network Configuration To have a lightweight detection system, we have implemented a small, two layer shallow convolutional neural network, compared with common image recognition models, such as ImageNet [20] and VGG [7]. The network structure is shown in Table. 1. The network is trained with 5000 iterations with a training batch size of 32 and learning rate 0.0001. 4.4 Results The classification results are shown in Tables 2 and 3 for the cases of two (benign and malicious) and three-class (benign and two malware families: Mirai and gafgyt) classification. The experiments were conducted five times which each time with a completely different training/test data combination (i.e., there are no shared test samples between any two of five test data sets). Figure 2: Images of Goodware PP PPPredict Benign PP True P Benign Gafgyt Mirai 94.67% 6.67% 0% Gafgyt Mirai 2.67% 72.00% 21.33% 2.67% 21.33% 78.67% Table 2: Confusion Matrix for 3-class Classification PPPredict Benign PP True P Benign 94.67% Malicious 6.67% PP Malicious 5.33% 93.33% Table 3: Confusion Matrix for 2-class Classification According to the results of two-class classification, we find the proposed system can predict the existence of maliciousness with about 94.0% accuracy on the average. The accuracy of three-class Lightweight Classification of IoT Malware based on Image Recognition Conference’17, July 2017, Washington, DC, USA Figure 4: Malware Image Examples of the Mirai Family Figure 3: Malware Image Examples of the Linux.Gafgyt Family Class Time consumption in second Goodware 0.0241 Gafgyt 0.0011 Mirai 0.0003 Table 4: Practical results of time complexity for classifying one image of goodware and two malware families. classification is relatively lower. Specifically, there are 6.67% malicious samples that are mis-classified as benign, all of which belong to Gafgyt family, while there is no misclassification of Mirai family to benign. This indicate the Gafgyt has more similar binaries to goodware. On the other side, the rate of misclassification between Mirai and Gafgyt is exactly the same. Generally, the difference between benign and malicious samples is more recognizable than the difference between two malware families. Comparing with misclassification between benign and malicious samples (i.e., two-class classification), the system is more likely to misclassify the samples of two malware families in the case of three-class classification. This indicates the similarity between these two families. Specifically, samples of two families might be obfuscated in similar ways, Conference’17, July 2017, Washington, Jiawei Su, DC, Danilo USA Vasconcellos Vargas, Sanjiva Prasad, Daniele Sgandurra, Yaokai Feng, and Kouichi Sakurai XXX Systems XXX XXX Metrics Accuracy Classifier Num of layers Num of nodes Fully connect layer Preprocess Input dimension Our method ANN with random projection [5] Weighted loss [3] 94.0% CNN 2 104 256 Re-organizing binary 64X64 scalar matrix 99.5% ANN 2 1536 2048 N-gram binary, Random projection 179 thousand binaries 96.9% CNN (VGG-s) 5 1888 4096X2 Re-organizing binary Unknown Table 5: Comparing proposed system with two previous related works. In particular, the number of hidden layers, number of neurons and the number of nodes in fully connect layers, are shown by “Num of Layers”,“Num of nodes”,“Fully connect layer” respectively. It can be seen that the proposed system is more lightweight than references due to the smaller size of network model and lower dimensions of input, as well as simpler preprocessing. or/and share a part of the malicious functions. In fact, the basic botnet functions of different DDoS malware are similar, and mainly include receiving instructions from the control server and spreading the infection. Also consider that Mirai source code has been available online since its beginning, and several new families of IoT botnets include some portions of Mirai code. In addition, IoT malware has to be lightweight and their functions have to be relatively direct and simple. Our accuracy results compete with similar previous works [3, 5]. In specific, Yue [5] also utilized convolutional neural networks and malware images for classifying several PC malware families. However the results are carried out by using much bigger and complex network structures, namely very deep networks (VGG) which contain more than 10 layers while ours only has two layers. Similarly, a very complex preprocess procedure is needed in [5] which involves initial feature selection and random projection while our proposal directly uses raw features for classification. According to the accuracy results, the proposed system can be utilized as a regular malware detector, or a first-stage malware classifier. That is, it can perform a precise classification to identify benign and maliciousness .The exact classification of the malware family can then be performed on a Cloud backend. A comparison of corresponding experimental accuracy and settings is shown by Table 5. The practical time cost of classifying images is depicted in Table 4. In detail, the experiment is conducted with python and tensorflow on a system running Ubuntu 14.04, with a i7-7700 processor, GTX1080Ti graphic card and 16G memory. The code of this research can be found in: https://github.com/Carina02/IotMalwareImage. 5 CONCLUSION AND FUTURE WORK In this paper we have proposed a lightweight malware image classification scheme for locally detecting IoT DDoS malware, and shown its feasibility. The malware filter proposed in this paper is based on convolutional neural networks and can be tuned to be more efficient by using various techniques of reducing network size. For example, removing the neurons and links that are not critical in the network can reduce the number of parameters needed for classification [21]. Such further optimization can make the proposed system implementable on IoT devices with even less computation resources. In addition, new malware image extraction methods can be proposed to obtain more representative features of malware for classification. For improving the detection rate of IoT malware, more complicated cases in practice can be considered such as malware obfuscation. To the best knowledge, there is currently no systematical evaluation on IoT malware obfuscation and several critical questions are yet to be answered, such as whether IoT malware is obfuscated in a similar way to traditional malware, and how limited resources influence obfuscation methods. ACKNOWLEDGMENTS This research was partially supported by Collaboration Hubs for International Program (CHIRP) of SICORP, Japan Science and Technology Agency (JST), and Project of security in the IoT space funding by Department of Science and Technology (DST), India. REFERENCES [1] Pa, Y.M.P., Suzuki, S., Yoshioka, K., Matsumoto, T., Kasama, T. and Rossow, C., 2015. IoTPOT: analysing the rise of IoT compromises. EMU, 9, p.1. 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Secure Adaptive Group Testing Alejandro Cohen ∗ Department ∗ Asaf Cohen ∗ Sidharth Jaggi † Omer Gurewitz∗ of Communication Systems Engineering, Ben-Gurion University, Israel, {alejandr, coasaf, gurewitzs}@bgu.ac.il of Information Engineering, Chinese University of Hong Kong, Hong Kong, jaggi@ie.cuhk.edu.hk arXiv:1801.04735v1 [cs.IT] 15 Jan 2018 † Department Abstract—Group Testing (GT) addresses the problem of identifying a small subset of defective items from a large population, by grouping items into as few test pools as possible. In Adaptive GT (AGT), outcomes of previous tests can influence the makeup of future tests. This scenario has been studied from an information theoretic point of view. Aldridge 2012 showed that in the regime of a few defectives, adaptivity does not help much, as the number of tests required for identification of the set of defectives is essentially the same as for non-adaptive GT. Secure GT considers a scenario where there is an eavesdropper who may observe a fraction δ of the outcomes, and should not be able to infer the status of the items. In the non-adaptive scenario, the number of tests required is 1/(1 − δ) times the number of tests without the secrecy constraint. In this paper, we consider Secure Adaptive GT. Specifically, when an adaptive algorithm has access to a private feedback link of rate Rf , we prove that the number of tests required for both correct reconstruction at the legitimate user, with high probability, and negligible mutual information at the eavesdropper is 1/min{1, 1 − δ + Rf } times the number of tests required with no secrecy constraint. Thus, unlike non-secure GT, where an adaptive algorithm has only a mild impact, under a security constraint it can significantly boost performance. A key insight is that not only the adaptive link should disregard test results and send keys, these keys should be enhanced through a “secret sharing" scheme before usage. I. I NTRODUCTION Group Testing (GT) was introduced in the seminal study by Dorfman to identify syphilis infected draftees while dramatically reducing the number of required assays [1]. Specifically, the objective of GT is to identify a small subset of K unknown defective items within a much larger set of N items, conducting as few measurements T as possible. This problem has been analyzed in various scenarios [2], one of which, Non-Adaptive Group Testing (NGT), is when the entire pooling strategy is decided on beforehand. This scenario has also been formulated as a channel coding problem, e.g., [3]–[6], where each codeword is associated with the pool tests its associated item participates in. Secure GT protects the items’ privacy such that an eavesdropper who may observe only a fraction of the pool-tests, will not be able to infer the status of any of the items (negligible mutual information between the captured pool-tests and the status of the items). [7] addressed the Secure Nonadaptive Group Testing (SNGT) model. In order to confuse the eavesdropper, instead of each item having a single test vector, determining in which pool-tests it should participate, each item has a random vector, chosen from a known set. [7] proved that when the fraction of tests observed by the eavesdropper (Eve) is 0 ≤ δ < 1, the number of tests required for both correct Figure 1: Analogy between channel scheme with feedback to Secure Adaptive Group Testing. reconstruction at the legitimate user and negligible mutual information at Eve’s side is 1/(1 − δ) times the number of tests required with no secrecy constraint. Thus, the solution in [7] relies on information theoretic security, which offers security at the price of rate, or, in the GT case, trades security and the number of tests. In this paper, we consider Adaptive Group Testing (AGT), in which the outcomes of previous tests can influence the construction of future pools. In general, the adaptivity may benefit from both a reduced number of tests (T ) and efficient decoding techniques [8]. However, it has been shown that in many interesting cases, the decrease in T is negligible. Several studies have analyzed AGT as a channel coding problem with feedback (e.g., [9]–[13]). Utilizing the same techniques as Shannon’s seminal study [14] which proved that feedback does not increase channel capacity, the authors in [9] showed that feedback from the lab does not decrease the number of tests required significantly. That is, if T is the minimal number of tests required for reconstruction with negligible error, when N → ∞ the gain due to the adaptivity is marginal. Even with zero-error, the difference in T is only between O(K 2 log N/ log K) and O(K log N ) [2]. Surprisingly, we show that in secure-adaptive group testing, the adaptive feedback link may decrease the number of tests required significantly, which in some cases coincides with the number of pool tests required with no secrecy constraint at all. While from an information theoretic perspective, previous results indeed show that in communication feedback can increase secrecy capacity [15]–[17], herein, previous techniques do not apply directly, as the test transferred are physical entities which cannot be, e.g., one-time-padded, and, as we explain in the sequel, the “encoder” in this case has no knowledge on which K of the N items it needs to protect. Main Contribution We propose a new Secure Adaptive Group Testing (SAGT) algorithm. This algorithm significantly reduces the number of tests required, yet is sufficient for the legitimate user to identify the defective items, and to keep an eavesdropper ignorant regarding any of the items. The model is depicted in Figure 1. In the suggested solution, the set of indices describing the defective items takes the place of a confidential message; the testing matrix represents the design of the pools, where each row corresponds to a separate item. Each such matrix row is associated with a codeword where 1 denotes the pool tests that the corresponding item participates. The decoding algorithm is analogous to a channel decoding process, yet now the adaptive link from the lab (who examines the pools) to the mixer (who mixes the samples and creates the pooled tests) takes the place of a feedback link. The eavesdropper observation is analogous to the output of an erasure channel, such that only part of the tests sent from the legitimate source (the mixer) to the legitimate receiver are observed by the eavesdropper. We assume a private adaptive link, which is not observable to the eavesdropper. We use the feedback link from the lab to the mixer in order to modify the testing matrix, according to the shared information between them, in a way which can be comprehended by both the lab and the mixer, yet confuses the eavesdropper regarding which item participates in each pool test. However, unlike wiretap channels with feedback, we cannot use the data on the link directly, and must use a coding scheme, to increase the number of keys we can generate. Although the keys generated are dependent, any subset which Eve eventually observes is independent and thus protected. Accordingly, this adaptive algorithm decreases the factor 1/(1 − δ) given in SNGT, and one may use a smaller number of tests. In the case that the information rate on the feedback is equal to the rate of the eavesdropper’s observation, this factor is completely rescinded. Thus, we achieve the same sufficiency bound on T as given for the non-secure GT. to denote their realizations, and calligraphic letters to denote the alphabet. Logarithms are in base 2 and hb (·) denotes the binary entropy function. In general, GT is defined by a testing matrix X = {Xj (t)}1≤j≤N,1≤t≤T ∈ {0, 1}N ×T , where each row corresponds to a separate item j ∈ {1, . . . , N }, and each column corresponds to a separate pool test t ∈ {1, . . . , T }. For the j-th item, entry Xj (t) = 1 if item j participates in the t-th pool test and Xj (t) = 0 if is not. Denoting by Aj ∈ {0, 1} an indicator function indicating whether the j-th item belongs to the defective set, the pool test outcome Y (t) ∈ Y T is Y (t) = N _ j=1 Xj (t)Aj = _ Xd (t), d∈K W where is used to denote the boolean OR operation. In AGT, we assume that the makeup of a testing pool can depend on the outcomes of earlier tests, by adaptive feedback link from the lab to the mixer, such that for any test t > 1 and any item j, Xj (t) = Xj (t\|Y (1), . . . , Y (t − 1)). In the secure model, we assume this link is private, and at a limited rate Rf . That is, symbols Ft , t ∈ {1, . . . , T } are sent over an adaptive private feedback link, secretly from the eavesdropper. The feedback alphabets are denoted by {F1 , . . . , FT }. Their cardinalities must satisfy T 1X log(\|Ft \|) ≤ Rf . T t=1 At each time instant t, Ft−1 is computed by the lab and revealed to the mixer. The symbol Ft at time instant t may depend on Y t−1 , the t−1 prior outcomes of the pool tests, the previous feedback symbols, F t−1 , and some randomness. That is, we assume there exists a distribution p(Ft \|Y t−1 , F t−1 ). Hence, in the secure AGT II. P ROBLEM F ORMULATION Xj (t) = Xj (t\|F t ). In SAGT, a legitimate user wishes to identify a small unknown subset K of defective items form a larger set N , while reducing the number of measurements ,T , as much as possible and keeping an eavesdropper, which is able to observe a subset of the tests results ignorant regarding the status of the N items. The adaptive link allows the outcomes from previous tests to influence the makeup of future tests in order to further reduce the total number of measurements. N = \|N \|, K = \|K\| denote the total number of items, and the number of defective items, respectively. The status of the items, defective or not, should be kept secure from the eavesdropper, but detectable by the legitimate user. We assume that in each round the number K of defective items is known apriori. This is a common assumption in the GT literature [18]. Figure 2 gives a graphical representation of the model. Throughout the paper, we use boldface to denote matrices, capital letters to denote random variables, lower case letters Note that in the classic adaptive case, where simply the previous outcome is revealed to the mixer, we have Ft = Yt−1 , hence \|Ft \| = 2 for all t and therefor Rf = 1. In SAGT, we assume an eavesdropper, which observes a noisy vector Z T = {Z(1), . . . , Z(T )}, generated from the outcome vector Y T . We concentrate on the erasure case, where the probability of erasure is 1 − δ, i.i.d. for each test. That is, on average, T δ outcomes are not erased and are available to the eavesdropper via Z T . N Denote by W ∈ W , {1, . . . , K } the index of the subset of defective items. We assume W is uniformly distributed, that is, there is no apriori bias to any specific subset. Denote by Ŵ (Y T , F T ) the index recovered by the legitimate decoder, after observing Y T . We refer to the adaptive procedure of creating the testing matrix, together with the decoder as a SAGT algorithm. We are interested in the asymptotic behavior of a SAGT algorithm. Figure 2: Noiseless secure adaptive group-testing setup. Definition 1. A sequence of SAGT algorithms with parameters N, K,T and Rf is asymptotically reliable and weakly secure if: (1) At the legitimate receiver, observing Y T , lim P (Ŵ (Y T , F T ) 6= W ) = 0. T →∞ (2) At the eavesdropper, observing Z T , lim T →∞ 1 I(W ; Z T ) = 0. T III. R ELATED W ORK Recently works [3]–[5], [7], [9]–[13] adopted an information theoretic perspective on GT, presenting it as a channel coding problem. We briefly review the most relevant results. In [3], the authors mapped the NGT model to an equivalent channel model, where the defective set takes the place of the message, the testing matrix rows are codewords, and the test outcomes are the received signal. They let Ŝ(X T , Y T ) denote the estimate of the defective subset S, which is random due to the randomness in X and Y . Furthermore, let Pe denote the average probability of error, averaged over all subsets S of cardinality K, variables X T and outcomes Y T , i.e., Pe = P r[Ŝ(X T , Y T ) 6= S]. Then, where (S 1 , S 2 ) denote the partition of defective set S into disjoint sets S 1 and S 2 with cardinalities i, the flowing bounds on the total number of tests required in Bernoulli NGT was given by T ≤ T ≤ T , where for some ε > 0 independent of N and K, log N −K i T = (1 + ε) · max , i=1,...,K I(XS 1 ; XS 2 , Y ) log N −K+i i T = max , i=1,...,K I(XS 1 ; XS 2 , Y ) In [9], the authors considered the AGT model as channel coding with feedback, where future inputs to the channel can depend on past outputs. Shannon proved that feedback does not improve the capacity of a single-user channel [14]. However, due to the non-tightness of the bounds on testing in the non-adaptive case, the authors could not show that adaptive GT requires the same number of tests as non-adaptive testing exactly. Alternatively, they showed that it obeys the same lower bound and requires no more tests than the non-adaptive case. Moreover, in [11], using a similar analogy to channel coding N problem, the authors defined the AGT rate R as log2 K /T and introduced the capacity C, that is, if for any ε > 0, there exists a sequence of algorithms with N log2 K ≤C −ε lim N →∞ Ta and success probability approaching one, yet with N limN →∞ log2 K /T > C it approaches zero. In our earlier work [7], we focused on SNGT. We considered a scenario where there is an eavesdropper which is able to observe a subset of the outcomes. We proposed a SNGT algorithm, which keeps the eavesdropper with leakage probability δ, ignorant regarding the items’ status. Specifically, when the fraction of tests observed by Eve is 0 ≤ δ < 1, [7] proved that the number of tests required for both correct reconstruction at the legitimate user and negligible mutual information at Eve’s 1 times the number of tests required with no secrecy side is 1−δ constraint. Then, where I(XS 1 ; XS 2 , Y ) ≥ i/K, according [7, Claim 1], the flowing bounds on the total number of tests required in Bernoulli SNGT was given by T s ≤ Ts ≤ T s , where for some ε > 0 independent of N and K, K N −K 1+ε Ts = max log , 1 − δ i=1,...,K i i 1 N Ts = log . 1−δ K IV. M AIN R ESULTS Under the model definition given in Section II, our main results are the following sufficiency (direct) and necessity (converse) conditions, characterizing the maximal number of tests required to guarantee both reliability and security. The direct proof and leakage are given in Section V. The remaining proofs are deferred to Section VI. A. Direct (Sufficiency) With a private rate limited feedback link, the sufficiency part is given by the following theorem. Theorem 1. Assume a SAGT model with N items, out of which K = O(1) are defective. For any δ < 1 and private adaptive rate limited feedback 0 < Rf < 1, if K N −K 1+ε Tsa ≥ max log , min{1, 1 − δ + Rf } i=1,...,K i i for some ε > 0 independent of N and K, then there exists a sequence of SAGT algorithms which are reliable and weakly secure. That is, as N → ∞, both the average error probability approaches zero and an eavesdropper with leakage probability δ is kept ignorant. It is important to note that if Rf ≥ δ, as the direct proof will show, the information obtained over the adaptive link between the lab and the mixer is powerful enough to obtain security without increasing T . Hence, in this case, the direct bounds of the non-secure and secure adaptive group testing are equal, that is, T = Tsa . Even when Rf < δ, the information obtained over the feedback between the lab and the mixer reduces the upper bound on the number of the tests required in SNGT, thus, T < Tsa ≤ T s . Using an upper bound, such that, log N −K ≤ i i log (N −K)e , the maximization in Theorem 1 can be solved i easily, leading to a simple bound on T . Corollary 1. For SAGT with parameters K << N and T , reliability and secrecy can be maintained with Tsa ≥ 1+ε K log(N − K)e. min{1, 1 − δ + Rf } Remark 1. In this paper, we consider the asymptotic case in T and negligible error. In this case, without a secrecy constrain, it is well known that feedback does not help [9]. Nonetheless, with a secrecy constraint we show that the link is used only for shared randomness. However, for finite T and zero error [2], the mixer may use the previous outcomes to reduce the number of tests T . Thus, for finite T , zero error and a secrecy constraint, is not trivial what should be shared over the link, either pure randomness in order to cope with the secrecy constraint or single previous outcomes in order to try adaptively reduce the number of tests. B. Converse (Necessity) The necessity part is given by the following theorem. Theorem 2. Let T be the minimum number of tests necessary to identify the defective set Sw of cardinality K among population of size N in a SAGT model with private adaptive rate limited feedback 0 ≤ Rf ≤ 1 while keeping eavesdropper, with pooling outcome test leakage probability δ < 1, ignorant regarding the status of the items. Then, 1 N a log . Ts ≥ K min{1, 1 − δ + Rf } The proof is deferred to Section VI. Note that, compared to the lower bound without a security constraints, there is an increase by a multiplicative factor of 1/ min{1, 1 − δ + Rf }. When Rf ≥ δ, the lower bounds of the non-secure and secure adaptive group testing are equal, i.e., T = Tsa . C. Secrecy capacity in SAGT Returning to the analogy in [11] between channel capacity and group testing, one might define by Cs the (asymptotic) N minimal threshold value for log K /T , above which no reliable and secure scheme is possible. Under this definition, where C is the capacity without the security constraint, the result in this paper show that Cs ≥ (min{1, 1 − δ + Rf })C . Clearly, this can be written as, Cs ≥ min{C, C − C(δ) + C(Rf )}, raising the usual interpretation as the difference between the capacity to the legitimate decoder and that to the eavesdropper, yet, with the capacity of the information obtained over the feedback link between the legitimate decoder and encoder [15]–[17]. Note that as the effective number of tests Eve sees is Te = δT , hence her GT capacity is (δ)C. V. C ODE C ONSTRUCTION WITH A DAPTIVE P RIVATE F EEDBACK AND A P ROOF FOR T HEOREM 1 The goal, in general, is to design a proper testing matrix, or, specifically, an algorithm to adaptively update it. Remember that each row describes the tests an item participates in. We thus construct this matrix in batches (of T tests each), each time selecting an appropriate row for each item. In a batch, for each item we generate a bin, containing several sub-bins, with several rows in each sub-bin (see Figure 3). Internal randomness in the mixer, which is not shared with any other party, is used to select a sub-bin for each item, while data received from the adaptive link is used to select the right row from the sub-bin. While this solution is inspired by codes for wiretap channels with rate limited feedback [16], there are several key differences, which not only change the construction, but also require non-trivial processing of the data received from the feedback. Specifically, first, unlike a wiretap channel, herein there are N items, only K of them, unknown to the mixer, actually participate in the output (“transmit”). Thus, bins and sub-bins sizes should be properly normalized. More importantly, the mixer, which acts as an encoder, does not know which K messages it should protect. Thus, the mixer should artificially blow-up the data it receives from the private feedback: from bits (used as keys) intended to protect the K defective items, it generates a larger number of keys, sufficient N items, satisfying the property that any K out of the N which will eventually participate, will still be protected. In other words, the keys received from the feedback cannot be used as is, and an interesting secret sharing-type scheme must be used. Formally, a (batch-processing) SAGT code consists of an N index set W = {1, 2, . . . K }, its w-th item corresponding to the w-th subset K ⊂ {1, . . . , N }; A discrete memoryless source of randomness at the mixer (RX , pRX ); A discrete memoryless source of randomness at the lab (RY , pRY ); A feedback at rate Rf bits per test, resulting in an index I ∈ {1, . . . , 2T Rf } after T uses; We use a single index due to the batch processing (Section V-A describes a test-by-test adaptive algorithm based on the one herein); The mixer, of course, does not know which items are defective, thus it needs to select a row for each item. However, since only the rows of the defective items affect the output Y T , it is beneficial to define an “encoder" GX : W × RX × I → XSw ∈ {0, 1}K×T , Figure 3: Encoding process for a SAGT code. that is, mapping the message, the randomness and input from the adaptive link to the K codewords which are summed to give the tests output Y T . Note that a stochastic encoder and the causally known feedback message I are similar to encoders ensuring information theoretic security, as randomness is required to confuse the eavesdropper about the actual information [16], [19]. A decoder at the legitimate user is a map Ŵ : Y T × I → W. The probability of error is P (Ŵ 6= W ). The probability that an outcome test leaks to the eavesdropper is δ. We assume a memoryless model, i.e., each outcome Y (t) depends only on the corresponding input XSw (t), and the eavesdropper observes Z(t), generated from Y (t) according to p(Y T , Z T \|XSw ) = T Y p(Y (t)\|XSw (t))p(Z(t)\|Y (t)). t=1 Next we provide the detailed construction and analysis. 1) Codebook Generation: Choose integers F and M such that log2 (F ) = T (Rf /K) and log2 (M ) = T (δ − Rf − )/K. For each item we generate bin of M · F independent and identically distributed codewords. Each codeword of size T is generated randomly, where each Xj (t) is chosen according to P (x) ∼ Bernoulli(ln(2)/K). Consequently, T P (X ) = T Y P (xi ). i=1 Then we split each bin to sub-bins of codewords xT (m, f ), 1 ≤ m ≤ M and 1 ≤ f ≤ F (illustrated in Figure 3). 2) Key Generation: We now describe the generation of the shared keys created from the information sent over the adaptive link. This link is of rate Rf . We do not use it to send information about test results, and simply send random bits. Therefore, in a block of length T we receive S = T Rf secret bits. We divide the secret bits to K secret keys, each constituting T Rf /K = SK bits. Our goal is to take these K keys, and use them to create N new keys, with the property that if the original K keys had a random uniform i.i.d. distribution, then any set of K keys out the N new ones will have the same random uniform i.i.d. distribution. This can be done using a generator matrix of an [N, K] MDS code. Such a generator matrix has the property that any K columns are linearly independent. Thus, taking the K original keys, as an SK × K matrix, and multiplying it by the generator matrix GK×N creates a matrix of size SK × N , where each column is used as the new key. Since any subset of K columns of G is invertible, each set of K new keys is simply a 1 : 1 transformation of the K original keys. The importance of this scheme in our context is as follows: for any subset of K new keys (out of N ), if an eavesdropper has no access to the original K keys, he/she is ignorant regarding the new keys. Moreover, it is important to note that unlike protection using a one-time-pad, these keys cannot be XORed with the rows of the testing matrix, as this operation will change the probability of each item participating in a pool-test, deviating from the optimal distribution of the testing matrix. 3) Testing: The mixer receives the T Rf feedback secret bits, divides them to K keys and uses the MDS code to create N new keys. Each new key is of length T Rf /K. Therefore, at each round and for each item j, the mixer selects a subbin using its internal randomness, and a message within it using the key. The result is a codeword xT (m, f ). Therefore, the SAGT matrix contains N randomly selected codewords of length T , one for each item (defective or not). In the first round of tests, the mixer has no available (feedback) key, hence it operates using a larger number of tests for that round as given for the non-adaptive SGT in [7]. Amortized over multiple rounds, this loss is negligible. 4) Decoding at the Legitimate Receiver: The decoder looks for a collection of K codewords XSTŵ , one from each sub-bin, for which Y T is most likely. Namely, P (Y T \|XSTŵ ) > P (Y T \|XSTw ), ∀w 6= ŵ. Then, the legitimate user declares Ŵ (Y T × F ) as the set of bins in which the rows reside. 5) Reliability: Let I(XS 1 ; XS 2 , Y ) denote the mutual information between XS 1 and (XS 2 , Y ), under the i.i.d. distribution with which the codebook was generated and remembering that Y is the output of a Boolean channel. The following lemma is a key step in proving the reliability of the decoding algorithm suggested herein. This Lemma is a direct consequence of [7, Lemma 1], under the enhancement that the index of each sub-bin is set according to the known key sheared between the legitimate decoder and the mixer at each round of the algorithm. Lemma 1. If the number of tests satisfies i log N −K M i , T ≥ (1 + ε) · max i=1,...,K I(XS 1 ; XS 2 , Y ) then, under the codebook above, as N → ∞ the average error probability approaches zero. Applying [7, Claim 1], which lower bounds the mutual information between XS 1 to (XS 2 , Y ) by i/K, to the expression δ−Rf − in Lemma 1, and substituting M = 2T K , a sufficient condition for reliability is 1+ε N −K i T ≥ max log + T (δ − Rf ) i 1≤i≤K K i K with some small . Rearranging terms results in 1+ε N −K T ≥ max log . 1≤i≤K min{1, 1 − (1 + ε)(δ − Rf )}i/K i This complete the reliability part. Note that the bound holds for large K and N , and ε independent of K and N . 6) Information Leakage at the Eavesdropper: We now prove the security constraint is met. Hence, we wish to show that I(W ; Z T )/T → 0, as T → ∞. Denote by CT the random codebook and by XST the set of codewords corresponding to the true defective items. We have, 1 T I(W ; Z T \|CT ) = 1 T F F I(W, RK RK ; Z T \|CT ) + I(RK RK ; Z T \|W CT ) , (1) where RK is the random variable used by the encoder to choose the sub-bins. That is, the encoder has a random variable for each item, RK denotes the union of K such variables, for F is again a union, this time of K the K defective items. RK keys. These are K “shares", out of the N shares generated F uniquely define XST , by the MDS code. Since W, RK , RK continuing from (1), we have: F = T1 I(XST ; Z T \|CT ) − I(RK RK ; Z T \|W, CT ) F = T1 (I(XST ; Z T \|CT ) − H(RK RK \|W, CT ) F +H(RK RK \|Z T , W, CT )) (a) 1 =T (b) F F \|Z T , W, CT ) ) + H(RK RK (XST ; Z T \|CT ) − H(RK RK δ−R − R ≤δ − T1 K T Kf + T Kf + T1 H(RK \|Z T , W, CT ) (c) ≤T , F where T → 0 as T → ∞. (a) is since both RK and RK are independent of W and the codebook. (b) is since both keys are Rf uniform, the first includes K variables of T ( δ− K − K ) bits R each, and the second K shares of T Kf bits each. (c) follows from [7, Section V.B]. In short, given the true K defectives, the codebook and her output Z T , Eve sees a simple MAC channel, at a rate slightly below her capacity. Therefore, she can identify which codeword was selected for each item, hence F identify both RK and RK . A. Test design depends on the outcome of previous pool-test In this solution, the outcome of each separate pool-test, according to the rate of the private adaptive link, can be available at the mixer to influence on the next test. It is important to note that unlike the first solution, we assume that by the adaptive link, the lab and the mixer can agree\know which pool-test, t ∈ T , depends on the symbols sent over the link. This assumption is not trivial, however, in various Figure 4: Encoding process for a SAGT code where the test design depends on the outcome of previous pool-test. scenarios based on GT [2] this assumption holds, e.g., in the channel problem, when the encoder (mixer) and the decoder (lab) schedule the transmission over the main channel and the feedback. In the code construction phase, we generate code as in Section V, yet, with one codeword in each sub-bin (i.e., the feedback will not used to select the rows in the original testing matrix). Hence for each item we have a row-bin with M rowcodewords. Moreover, for each possible column in the original testing matrix, we generate a column-bin with two columncodewords. Then for each j-th item, the mixer randomly select one row for the original testing matrix randomly as in the first setup from the j-th row-bin, yet, per pool test, one column-codeword is selected from his column-bin to define which items will participate in this pool test. This column selected according to the previous outcome feedback sent after each separate pool-test by the legitimate user to the mixer. If the feedback of the previous outcome is not available to the mixer, since the rate of the feedback is limited, the mixer use the column of the original testing matrix. Figure 4 gives a graphical representation of the code. Specifically, in the Q codebook generation phase, using a T distribution P (X T ) = i=1 P (xi ), for each item we generate a row-bin with M independent and identically distributed row-codewords. QN Then in the same way, using a distribution P (X̃ N ) = i=1 P (x̃i ), for each possible column xN (t) we generate a column-bin with two independent and identically distributed column-codewords. In the testing phase, for each item j, the mixer selects uniformly at random one row-codeword xTj (m) from his rowbin. The lab select randomly one index in I ∈ {1, . . . , 2T Rf } and before which pool-test t ∈ T , one bit of the index will sent to the mixer over the feedback to influence on the next test. Note that the lab is available to send at most T Rf bits. Then, in each pool-test t, using the previous feedback outcome Ft , the N mixer selects one column-codeword x̃N t (Ft ) from the x (t) column bin. In the case that the previous feedback outcome is not available at the mixer, the mixer use the original column. The decoder at the legitimate user (lab) know exactly which pool test was depended on the bits sent over the feedback and what was the indexes of the columns selected from the columnbins by the mixer for each pool-tests. Hence, the decoder procedure after T pool-tests, and the analysis of the reliability and the information leakage at the eavesdropper are almost a direct consequence of Section V. Thus, H(W ) ≤ VI. C ONVERSE (N ECESSITY ) In this section, we derive the necessity bound on the required number of tests, in the setup with rate limited feedback between the legitimate user to the lab. Similar to [16], using Fano’s inequality, for Ŵ = ŵ(Y T , F T ), N H(W \|Ŵ ) ≤ 1 + Pe log = T , K where T → 0 as T → ∞ if Pe → 0. Since Ŵ is a function of Y T , F T , H(W \|Y T , F T ) ≤ H(W \|Ŵ ) ≤ T T . I(W ; Z T ) = T γT , 1 T PT i=1 log(\|Fi \|) ≤ Rf and To conclude, the well-known technique of introducing a time sharing random variable is used. Assume Q is independent of XTS w , Y T , Z T and uniform on {1, . . . , T }, this results in Rf + = Rf + T 1X (I(XSiw ; Yi \|Zi ) + δT T i=1 T 1X (I(XSiw ; Yi \|Zi , Q = i) + δT T i=1 = Rf + I(XSQw ; YQ \|ZQ , Q) + δT H(W \|Z T ) + I(W ; Z T ) = Rf + I(XSw ; Y \|Z, Q) + δT T = H(W \|Z ) + T γT I(W ; Y T , F T \|Z T ) + H(W \|Y T , Z T , F T ) + T γT ≤ I(W ; Y T , F T \|Z T ) + T T + T γT (c) I(W ; F T \|Z T ) + I(W ; Y T \|F T , Z T ) + T δT (d) ≤ H(Fi ) + T δT . i=1 (2) = = T X T 1 1X H(W ) ≤ I(XSiw ; Yi \|Zi ) + Rf + δT . T T i=1 H(W ) (b) ; Yi \|Zi ) + We now use the constraint normalize by T . We have, where γT → 0 as T → ∞. Consequently, (a) I(X Siw i=1 1 H(W ) ≤ T Using the secrecy constraint, = T X T T H(F \|Z ) + I(W, XTS w ; Y T \|F T , Z T ) where XSw := XSQw , Y := YQ , Z := ZQ ,. Letting T → ∞, δT → 0 and T → 0, hence 1 H(W ) ≤ T ≤ = + T δT Rf + I(XSw ; Y \|Z, Q) Rf + I(XSw , Q; Y \|Z) Rf + I(XSw ; Y \|Z). where (a) follows from (2), (b) follows from Fano’s inequality and (c) follows by defining δT = T + γT . The following recursive lemma is now required. Hence, Lemma 2 ( [16]). For each t ∈ {1, . . . T }, From the above bound, and since the eavesdropper can obtain information only from the outcomes of the legitimate user, such that, p(y, z\|x) = p(y\|x)p(z\|y), H(F t \|Z t ) + I(W, XtS w ; Y t \|F t , Z t ) t−1 ≤ H(F t−1 \|Z t−1 ) + I(W, Xt−1 \|F t−1 , Z t−1 ) Sw ; Y t−1 + H(Ft \|W, Xt−1 , Z t−1 ) + I(XStw ; Yt \|Zt ). Sw , F To continue, we use Lemma 2 recursively starting from (d): H(W ) ≤ H(F T \|Z T ) + I(W, XTS w ; Y T \|F T , Z T ) + T δT ≤ H(F T −1 \|Z T −1 ) +I(W, XTS w−1 ; Y T −1 \|F T −1 , Z T −1 ) 1 H(W ) ≤ min{I(XSw ; Y ), I(XSw ; Y \|Z) + Rf }. T 1 H(W ) ≤ min{I(XSw ; Y ), I(XSw ; Y ) − I(XSw ; Z) + Rf }. T N Since H(W ) = log K , we have 1 N log ≤ min{I(XSw ; Y ), I(XSw ; Y )−I(XSw ; Z)+Rf }. T K Now let Z̄ denote the random variable corresponding to the tests which are not available to the eavesdropper. Hence, +I(XSTw ; YT \|ZT ) + H(FT ) + T δT H(Z̄) = I(XSw ; Y ) − I(XSw ; Z). ≤ H(F T −2 \|Z T −2 ) +I(W, XTS w−2 ; Y T −2 \|F T −2 , Z T −2 ) +I(XSTw−1 ; YT −1 \|ZT −1 ) + H(FT −1 ) +I(XSTw ; YT \|ZT ) + H(FT ) + T δT ≤ ... T T X X ≤ I(XSiw ; Yi \|Zi ) + H(Fi ) + δT . i=1 i=1 Denote by Ē the set of tests which are not available to Eve and by Ēγ the event {\|Ē\| ≤ T (1 − δ)(1 + γ)} for some γ > 0. We have H(Z̄) = P (Ēγ )H(Z̄\|Ēγ ) + P (Ēγc )H(Z̄\|Ēγc ) ≤ T (1 − δ)(1 + γ) + T P (Ēγc ) ≤ T (1 − δ)(1 + γ) + T 2−T (1−δ)f (γ) , where the last inequality follows from the Chernoff bound for i.i.d. Bernoulli random variables with parameter (1 − δ) and is true for some f (γ) such that f (γ) > 0 for any γ > 0. Thus, we have N log ≤ T (1 − δ)(1 + γ) + T 2−T (1−δ)f (γ) + Rf . 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Aksoylar, “Correction to “boolean compressed sensing and noisy group testing”[mar 12 1880-1901],” Information Theory, IEEE Transactions on, vol. 61, no. 3, pp. 1507– 1507, 2015. [7] A. Cohen, A. Cohen, S. Jaggi, and O. Gurewitz, “Secure group testing,” arXiv preprint arXiv:1607.04849, 2016. [8] J. M. Hughes-Oliver and W. H. Swallow, “A two-stage adaptive grouptesting procedure for estimating small proportions,” Journal of the American Statistical Association, vol. 89, no. 427, pp. 982–993, 1994. [9] M. Aldridge, “Adaptive group testing as channel coding with feedback,” in Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on. IEEE, 2012, pp. 1832–1836. [10] C. Aksoylar and V. Saligrama, “Information-theoretic bounds for adaptive sparse recovery,” in Information Theory (ISIT), 2014 IEEE International Symposium on. IEEE, 2014, pp. 1311–1315. [11] L. Baldassini, O. Johnson, and M. 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Arrhythmia Classification from the Abductive Interpretation of Short Single-Lead ECG Records Tomás Teijeiro*, Constantino A. García, Daniel Castro and Paulo Félix arXiv:1711.03892v1 [cs.AI] 10 Nov 2017 Centro Singular de Investigación en Tecnoloxías da Información (CITIUS), University of Santiago de Compostela, Santiago de Compostela, Spain Abstract In this work we propose a new method for the rhythm classification of short single-lead ECG records, using a set of high-level and clinically meaningful features provided by the abductive interpretation of the records. These features include morphological and rhythm-related features that are used to build two classifiers: one that evaluates the record globally, using aggregated values for each feature; and another one that evaluates the record as a sequence, using a Recurrent Neural Network fed with the individual features for each detected heartbeat. The two classifiers are finally combined using the stacking technique, providing an answer by means of four target classes: Normal sinus rhythm (N), Atrial fibrillation (A), Other anomaly (O) and Noisy (~). The approach has been validated against the 2017 Physionet/CinC Challenge dataset, obtaining a final score of 0.83 and ranking first in the competition. 1. Introduction The potential of Artificial Intelligence and machine learning techniques to improve the early detection of cardiac diseases using low-cost ECG tests is still largely untapped. The 2017 Physionet/Computing in Cardiology challenge defies the scientific community to propose solutions to the automatic detection of Atrial Fibrillation from short single lead ECG signals [1]. The challenge is posed as a classical machine learning problem: A labeled training set is provided, and the proposals are evaluated against a hidden test set of records. However, even if the only metric for the final ranking is the accuracy of the proposed models, a number of additional properties should be considered for the final adoption of each proposal in the clinical practice. Here, we emphasize on the interpretability of the automatic detection of Atrial Fibrillation, a major concern to ensure trust by the care staff [2]. In this sense, our proposal is based on a high-level description of the target signal by means of the same features used by cardiologists in ECG analysis. This description is generated with a pure knowledge-based approach, using an abductive framework for time series interpretation [3] that looks for the set of explanatory hypotheses that best account for the observed evidence. Only after this description has been built, machine learning methods were used to make up for the lack of the expert criteria applied in the labeling of the training set, and to alleviate the effect of possible errors in the interpretation process. 2. Methods The global architecture of the proposal is depicted in Figure 1, and the processing stages are explained in the following subsections. 2.1. Preprocessing The preprocessing stage aims at improving the quality of the data to be interpreted in the following stages, and involves two different tasks: 2.1.1. Data relabeling: The labeling of the training set was performed by a single expert in a single pass, and as a consequence some inconsistencies appear in the classification criteria. Thus, a thorough manual relabeling was carried out, but trying to be conservative and guided by preliminary classification results. We focused on records classified as N but showing what we consider clear anomalies. A total number of 197 out of 8528 records were relabeled. 2.1.2. Lead inversion detection: A number of records in the training set were found to be inverted, probably due to electrode misplacement. Inverted records are more likely to be classified as abnormal due to the presence of infrequent QRS and T wave morphologies, as well as to the greater difficulty to identify P waves. The inverted records were first identified manually, and then a simple logistic regression classifier was trained considering 14 features obtained from the raw signal and a tentative delineation of the P wave, QRS complex and T wave of every heartbeat detected by the gqrs application from the Physionet library [4]. This delineation was performed using the Construe algorithm [3], limiting the interpretation to the conduction level, that is, avoiding the rhythm interpretation. ECG Signal Lead Inversion Detection Global Features Global Classiﬁcation Per-Beat Features Sequence Classiﬁcation Abductive Interpretation Classiﬁcation Stacking Final Class Figure 1. Classification algorithm steps. 2.2. Abductive interpretation The abductive interpretation of the ECG signal is the most significant stage in the proposed approach. Its objective is to characterize the physiological processes underlying the signal behavior, building a description of the observed phenomena in multiple abstraction levels. This responsibility lies with the Construe algorithm, which applies a non-monotonic reasoning scheme to find the set of hypotheses that best explain the observed evidence, by means of a domain-specific knowledge base composed of a set of observables and a set of abstraction grammars. The knowledge base is the same used in [3], that allows to explain the ECG at the conduction and rhythm abstraction levels, thus providing the same features used by cardiologists in ECG analysis. The initial evidence is the set of waves identified in the wave delineation step, that are abstracted by a set of rhythm patterns to describe the full signal as a sequence of cardiac rhythms, including normal sinus rhythms, bradycardias, tachycardias, atrial fibrillation episodes, etc. The non-monotonic nature of the interpretation process allows us to modify the initial set of evidence, by discarding heartbeats that cannot be abstracted by any rhythm pattern, or by looking for missed beats that are predicted by the pattern selected as the best explanatory hypothesis for a signal fragment. This ability to correct the initial evidence is the main strength of our proposal, since it discards many false anomalies generated by the presence of noise and artifacts in the signal. Figure 2 shows an example of a noisy signal in which the gqrs application detects many false positive beats, that are removed or modified in the final interpretation that concludes with a single normal rhythm hypothesis that explains the full fragment. As we can also see in the Figure, the result of the interpretation stage is a sequence of P waves, QRS complexes and T waves observations, as well as a sequence of cardiac rhythms abstracting all those waves. 2.3. Global feature extraction Considering that each ECG record has to be classified globally, providing a single label for the entire signal duration, after the interpretation stage a set of features are calculated trying to summarize the information provided by Construe. A total number of 79 features are calculated, that are comprehensively described in the published software documentation. The feature set is divided into three main groups: • Rhythm features: This includes statistical measures on the RR sequence, such as the limits, median or median absolute deviation; heart rate variability features such as the PNN5, PNN10, PNN50 and PNN100 measures [5]; and information about the rhythm interpretation, such as the median duration of each rhythm hypothesis. • Morphological features: This includes information about the duration, amplitude and frequency spectrum of the observations in the conduction abstraction level, including P and T waves, QRS complexes, PR and QT intervals, and the TP segments. • Signal quality features: Their purpose is to assess the importance of the morphological features showing conduction anomalies, such as wide QRS complexes or long PR intervals. They are based on the sum of the absolute differences of the signal, which we refer to as profile. Some of the profiled areas of the signal are the baseline segments and the P wave area before each heartbeat (taking a constant window of 250 ms). 2.4. Global classification If a precise definition of the expert knowledge leading to the labeling of the training set were available, then the final classification could be directly developed with a basic rule-based system operating on the features extracted from the abductive interpretation stage, and the accuracy of the system would depend mainly on the accuracy of the interpretations. However, the challenge does not publish any guidelines for the classification, specially for the O class. Therefore, an automatic classifier was trained with two objectives: 1) To reveal the criteria leading to the training set labeling; and 2) to make the classification more accurate by learning possible mistakes of the abductive interpretation. The classification method selected for this stage was the Tree Gradient Boosting algorithm, and particularly the XGBoost implementation [6], which showed a high performance and a certain level of interpretability through the importance given to the classification features. The optimization of the hyperparameters was performed using exhaustive grid search and 8-fold cross-validation, leading to the following values: Maximum tree depth: 6, Learning rate: 0.2, Gamma: 1.0, Column subsample by tree: 0.9, Min. child weight: 20, Subsample: 0.8, and Number of boosting rounds: 60. Figure 2. How the abductive interpretation can fix errors in the initial evidence. [Source: First 10 seconds of the A02080 record. Grey: Original gqrs annotations. Blue: QRS observations. Yellow: T wave observations. Green: P wave observation and Normal rhythm hypothesis.] With respect to the first objective, we were able to formalize a number of specific anomalies that lead to classify a record as O. This identification helped to optimize the training set by defining more specific features to be calculated from the interpretation results. Some of the identified anomalies sharing this class were: • Tachycardia (Mean heart rate over 100 bpm). • Bradycardia (Mean heart rate under 50 bpm). • Wide QRS complex (Longer than 110 milliseconds). • Presence of ventricular or fusion beats. • Presence of at least one extrasystole. • Long PR interval (Longer than 210 milliseconds). • Ventricular tachycardia. • Atrial flutter. For some of these anomalies the classification in the training set seems a bit inconsistent, since examples can be found in several classes. For example, there are various records labeled as normal with PR interval longer than 210 milliseconds, as long as examples of records labeled as atrial fibrillation showing an atrial flutter pattern. Regarding the second objective, even after discovering some of the expert criteria distinguishing the target classes a rule-based system was not still competitive against automatic learned models. From our point of view this shows that the XGBoost classifier is able to improve the results of the interpretation alone. 2.5. Per-beat feature extraction Some of the conditions leading to a certain classification may not be present for the entire duration of a record, so the global features are not the best option to characterize episodic events of abnormalities. For example, a normal record with a single ectopic ventricular beat that does not break the rhythm is quite difficult to classify as abnormal by the global classifier. For this reason, some of the features calculated from the abductive interpretation are disaggregated to the individual heartbeat scope, such as the morphology, duration and amplitude of the P wave, the QRS complex and the T wave. Also the RR interval and the RR variation before and after each beat is included, as long as the profile of the P wave area. A sequence classification approach is then used to learn characteristic temporal patterns of each target class. 2.6. Sequence classification In the proposed approach, sequence classification relies on Recurrent Neural Networks (RNNs), a family of neural networks specialized for recognizing sequences of values. Among the different RNN implementations, we focused on Long Short Term Memory networks (LSTMs) [7], since they are capable of remembering information for long periods of time through the use of a cell state. Furthermore, they are able to avoid vanishing and exploding gradients when doing backpropagation through time. The architecture of the neural net is shown in Figure 3. The timedistributed Multilayer Perceptron (MLP) preprocesses the features described in Section 2.5 to transform the data into a space with easier temporal dynamics. The number of hidden units of the MLP was 256, and the dimension of the output space 128. A Rectified Linear Unit (ReLU) was used as activation function. The LSTM_0 layer preprocesses the resulting sequence of transformed features and returns a new sequence, which is subsequently used by the other LSTMs. The LSTM_2 layer just returns the final state of the network, whereas LSTM_1 and LSTM_3 return new transformed sequences. The pooling layers after LSTM_1 and LSTM_3 remove the temporal dimension by computing the temporal mean and maximum of each feature of the sequences, respectively. All the LSTMs used 128 units. Another MLP (with the same configuration of the time-distributed one) joins and transforms the outputs of each LSTM before a Softmax layer, which outputs a probability for each of the 4 classes. L2 -regularization was applied to all layers, using 10−4 as regularization strength. Finally, dropout was also used to improve generalization by preventing feature co-adaptation [8]. The neural network was trained using the categorical cross-entropy as loss function, a batch size of 32, and Adam [9] as optimizer. Furthermore, 15% of all the data was used as validation set to monitor the performance of the neural network. This permitted us to decrease the learning rate when the validation loss got stuck in a plateau and to avoid overfitting by using early stopping. The initial √ learning rate was set to 0.002 and it was decreased by 2 when the validation loss did not improve for at least 3 epochs. Training was ended after 15 epochs without improvement. Temporal mean pooling LSTM_1 Padded sequence of beat features Time-distributed MLP LSTM_2 LSTM_0 MLP Temporal max pooling LSTM_3 Softmax Sequence Class Figure 3. The neural network architecture. 2.7. Classification stacking of interpretability by the use of high-level and meaningful features. The XGBoost classifier based on global features and the RNN classifier based on the per-beat features were combined using the stacking technique. Stacking (also referred to as stacked generalization) involves training a new classification algorithm to combine the predictions of several classifiers [10]. Usually, the stacked model achieves better performance than the individual models due to its ability to discern when each base model performs best and when it performs poorly. Prior to the application of stacking, the predictions of 3 RNNs were averaged to decrease the variance of the RNN classifier arising from the random initialization of the RNN weights and the random split between test and validation set. Averaging similar models also helps in reducing overfitting. Note that this averaging can be seen as a simple bagging method. The probabilities predicted by the XGBoost and the averaged RNNs are then combined through a Linear Discriminant Analysis (LDA) classifier, which acts as stacker. To avoid possible collinearity issues, only 3 probabilities from each model are used. 3. Evaluation To evaluate the performance of the algorithm, we followed the challenge guidelines and metrics. The final score is assigned as the mean F1 measure of the N, A, and O classes. Table 1 shows an example of the results that the proposed method is able to achieve using 8-fold crossvalidation. Note that the stacker usually achieves better scores than the base models and, furthermore, it has lower variance (not shown in the Table). Table 1. Example of stratified 8-fold cross-validation. Method 0 1 2 Fold Number 3 4 5 6 7 XGBoost 0.84 0.84 0.85 0.85 0.82 0.80 0.82 0.82 RNN 0.82 0.81 0.84 0.83 0.86 0.83 0.83 0.83 LDA-stacker 0.85 0.84 0.86 0.86 0.85 0.83 0.84 0.85 4. Mean 0.83 0.83 0.85 Conclusions This work proves that the combination of knowledgebased and learning-based approaches is effective to build classification systems that exploit sophisticated machine learning methods while maintaining a remarkable degree Acknowledgements This work was supported by the Spanish Ministry of Economy and Competitiveness under project TIN201455183-R. Constantino A. García is also supported by the FPU Grant program from the Spanish Ministry of Education (MEC) (Ref. FPU14/02489). References [1] Clifford G, Liu C, Moody B, Lehman L, Silva I, Li Q, Johnson A, Mark R. AF classification from a short single lead ECG recording: The Physionet Computing in Cardiology Challenge 2017. In Computing in Cardiology. 2017; . [2] Caruana R, Lou Y, Gehrke J, Koch P, Sturm M, Elhadad N. Intelligible Models for HealthCare. In 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM Press, 2015; . [3] Teijeiro T, Félix P, Presedo J, Castro D. Heartbeat classification using abstract features from the abductive interpretation of the ECG. IEEE Journal of Biomedical and Health Informatics 2016;. [4] Goldberger A, et al. PhysioBank, PhysioToolkit and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals. Circulation 2000;101:215–220. [5] Mietus JE, Peng CK, Henry I, Goldsmith RL, Goldberger AL. The pNNx files: re-examining a widely used heart rate variability measure. Heart British Cardiac Society oct 2002; 88(4):378–80. ISSN 1468-201X. [6] Chen T, Guestrin C. XGBoost: A Scalable Tree Boosting System. In 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. mar 2016; . [7] Hochreiter S, Schmidhuber J. Long short-term memory. Neural computation 1997;9(8):1735–1780. [8] Srivastava N, Hinton GE, Krizhevsky A, Sutskever I, Salakhutdinov R. Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research 2014;15(1):1929–1958. [9] Kingma D, Ba J. Adam: A method for stochastic optimization. arXiv preprint arXiv14126980 2014;. [10] Wolpert DH. Stacked generalization. Neural networks 1992;5(2):241–259. Address for correspondence: Tomás Teijeiro Campo Rúa de Jenaro de la Fuente Domínguez, S/N, CITIUS Building 15782 Santiago de Compostela, SPAIN tomas.teijeiro@usc.es	2
An experimental exploration of Marsaglia’s xorshift generators, scrambled arXiv:1402.6246v5 [cs.DS] 13 Oct 2016 Sebastiano Vigna, Università degli Studi di Milano, Italy Marsaglia proposed xorshift generators as a class of very fast, good-quality pseudorandom number generators. Subsequent analysis by Panneton and L’Ecuyer has lowered the expectations raised by Marsaglia’s paper, showing several weaknesses of such generators. Nonetheless, many of the weaknesses of xorshift generators fade away if their result is scrambled by a non-linear operation (as originally suggested by Marsaglia). In this paper we explore the space of possible generators obtained by multiplying the result of a xorshift generator by a suitable constant. We sample generators at 100 points of their state space and obtain detailed statistics that lead us to choices of parameters that improve on the current ones. We then explore for the first time the space of high-dimensional xorshift generators, following another suggestion in Marsaglia’s paper, finding choices of parameters providing periods of length 21024 − 1 and 24096 − 1. The resulting generators are of extremely high quality, faster than current similar alternatives, and generate long-period sequences passing strong statistical tests using only eight logical operations, one addition and one multiplication by a constant. Categories and Subject Descriptors: G.3 [PROBABILITY AND STATISTICS]: Random number generation; G.3 [PROBABILITY AND STATISTICS]: Experimental design General Terms: Algorithms, Experimentation, Measurement Additional Key Words and Phrases: Pseudorandom number generators 1. INTRODUCTION xorshift generators are a simple class of pseudorandom number generators introduced by Marsaglia [2003]. In Marsaglia’s view, their main feature is speed: in particular, a xorshift generator with a 64-bit state generates a new 64-bit value using just three 64-bit shifts and three 64-bit xors (i.e., exclusive ors), thus making it possible to generate hundreds of millions of values per second. Subsequent analysis by Brent [2004] showed that the bits generated by xorshift generators are equivalent to certain linear feedback shift registers. Panneton and L’Ecuyer [2005] analyzed in detail the theoretical properties of the generators, and found empirical weaknesses using the TestU01 suite [L’Ecuyer and Simard 2007]. They proposed an increase in the number of shifts, or combination with another generator, to improve quality. In the first part of this paper, as warm-up we explore experimentally the space of xorshift generators with 64 bits of state using statistical test suites. We sample generators at 100 points of their state space, to easily identify spurious failures. Marsaglia proposes some choice of parameters, that, as we will see, and as already reported by Panneton and L’Ecuyer [2005], are not particularly good. We report results that are actually worse than those of Panneton and L’Ecuyer as we use the entire 64-bit output of the generators. While we can suggest some good parameter choices, the result remains poor. Thus, we turn to the idea of scrambling the result of a xorshift generator using a multiplication, as it is typical, for instance, in the construction of practical hash functions due to the resulting avalanching behavior (bits of the result depend on several bits of the input). This method is actually suggested in passing in Marsaglia’s paper. The third edition of the classic “Numerical Recipes” [Press et al. 2007], indeed, proposes this construction for This work is supported the EU-FET grant NADINE (GA 288956). This paper is an extended version of the paper with the same title published in the ACM Transactions on Mathematical Software [Vigna 2016]. Author’s addresses: Sebastiano Vigna, Dipartimento di Informatica, Università degli Studi di Milano, via Comelico 39, 20135 Milano MI, Italy. 2 S. Vigna A0 A1 A2 A3 A4 A5 A6 A7 x x x x x x x x ^= ^= ^= ^= ^= ^= ^= ^= x x x x x x x x << >> << >> << >> >> << a; a; c; c; a; a; b; b; x x x x x x x x C code ^= x >> ^= x << ^= x >> ^= x << ^= x << ^= x >> ^= x << ^= x >> b; b; b; b; c; c; a; a; x x x x x x x x ^= ^= ^= ^= ^= ^= ^= ^= x x x x x x x x << >> << >> >> << << >> c; c; a; a; b; b; c; c; X1 X3 X2 X4 X5 X6 X7 X8 Fig. 1. The eight possible xorshift64 algorithms. The list is actually derived from Panneton and L’Ecuyer [2005], as they correctly remarked that two of the eight algorithms proposed by Marsaglia were redundant, whereas two (A6 and A7 ) were missing. On the right side we report the name of the linear transformation associated to the algorithm as denoted by Panneton and L’Ecuyer [2005]. With our numbering, algorithms A2i and A2i+1 are conjugate by reversal. Note that contiguous shifts in the same direction can be exchanged without affecting the resulting algorithm. We normalized such contiguous shifts so that their letters are lexicographically sorted. a basic, all-purpose generator. From the wealth of data so obtained we derive generators with better statistical properties than those suggested in “Numerical Recipes”. In the last part of the paper, we follow the suggestion about high-dimensional generators contained in Marsaglia’s paper, and compute several choices of parameters that provide full-period xorshift generators with a state of 1024 and 4096 bits. Once again, we propose generators that use a multiplication to scramble the result. At the end of the paper, we apply the same methodology to a number of popular noncryptographic generators, and we discover that our high-dimensional generators are actually faster and of higher or equivalent statistical quality, as assessed by statistical test suites, than the alternatives. The software used to perform the experiments described in this paper is distributed by the author under the GNU General Public License. Moreover, all files generated during the experiments are available from the author. They contain a large amount of data that could be further analyzed (e.g., by studying the distribution of p-values over the seeds). We leave this issue open for further work. 2. AN INTRODUCTION TO xorshift GENERATORS The basic idea of xorshift generators is that their state is modified by applying repeatedly a shift and an exclusive-or (xor) operation. In this paper we consider 64-bit shifts and states made of 2n bits, with n ≥ 6. We usually append n to the name of a family of generators when we need to restrict the discussion to a specific state size. For xorshift64 generators Marsaglia suggests a number of possible combination of shifts, shown in Figure 1. Not all choices of parameters give a full (264 − 1) period: there are 275 suitable choices of a, b and c and eight variants, totaling 2200 generators. In linear-algebra terms, if L is the 64 × 64 matrix on Z/2Z that effects a left shift of one position on a binary row vector (i.e., L is all zeroes except for ones on the principal subdiagonal) and if R is the right-shift matrix (the transpose of L), each left/right shift and xor can be described as a linear multiplication by I + Ls or I + Rs , respectively, where s is the amount of shifting.1 For instance, algorithm A0 of Figure 1 is equivalent to the Z/2Z-linear transformation X1 = I + La I + Rb I + Lc . 1A more detailed study of the linear algebra behind xorshift generators can be found in [Marsaglia 2003; Panneton and L’Ecuyer 2005]. An experimental exploration of Marsaglia’s xorshift generators, scrambled 3 It is useful to associate with a linear transformation M its characteristic polynomial P (x) = det(M − xI). The associated generator has maximum-length period if and only if P (x) is primitive over Z/2Z. This happens if P (x) is irreducible and if x has maximum period in the ring of n polynomial over Z/2Z modulo P (x), that is, if the powers x, x2 , . . . , x2 −1 are distinct modulo P (x). Finally, to check the latter condition is sufficient to check that n x(2 −1)/p 6= 1 mod P (x) n for every prime p dividing 2 − 1 [Lidl and Niederreiter 1994]. The weight of P (x) is the number of terms in P (x), that is, the number of nonzero coefficients. It is considered a good property for generators of this kind that the weight is close to n/2, that is, that the polynomial is neither too sparse nor too dense [Compagner 1991]. Note that the family of algorithms of Figure 1 is intended to generate 64-bit values. This means that the entire output of the algorithm should be used when performing tests. We will see that this has not always been the case in previous literature. 3. SETTING UP THE EXPERIMENTS In this paper we want to explore experimentally the space of a number of xorshift-based generators. Our purpose is to identify variants with full period which have particularly good statistical properties, and test whether claims about good parameters made in the previous literature are confirmed. The basic idea is that of sampling the generators by executing a battery of tests starting with 100 different seeds that are equispaced in the state space. More precisely, if the state is made of n bits we use the seeds 1 + ib2n /100c, 0 ≤ i < 100. The tests produce a number of statistics, and we decided to use as score the number of failed tests. A higher score, thus, means lower quality. Running multiple tests makes it easy to rule out spurious failures, as suggested also by Rukhin et al. [2001] in the context of cryptographic applications.2 We use two tools to perform our tests. The first and most important is TestU01, a test suite developed by L’Ecuyer and Simard [2007] that contains several tests oriented towards the generation of uniform real numbers in [0 . . 1).3 We also perform tests using Dieharder, a suite of tests developed by Brown [2013], both as a sanity check and to compare the power of the two suites. Dieharder contains all original tests from Marsaglia’s Diehard, plus many more additional tests. We refer frequently to the specific type of tests failed: the reader can refer to the TestU01 and Dieharder documentation for more information. We consider a test failed if its p-value is outside of the interval [0.001 . . 0.999]. This is the interval outside which TestU01 reports a test by default. Sometimes a much stricter threshold is used (For instance, L’Ecuyer and Simard [2007] use [10−10 . . 1 − 10−10 ] when applying TestU01 to a variety of generators), and weaker p-values are called suspicious values, but since we are going to repeat the test 100 times we can use relatively weak pvalues: spurious failures will appear rarely, and we can catch borderline cases (e.g., tests failing on 50% of the seeds) that give us useful information. We call systematic a failure that happens for all seeds. For all such failures in our tests, p-values are smaller than 10−15 . Thus, all conclusions drawn in this paper based on system2 We remark that, arguably, a more principled choice would be choosing seeds that are equispaced in the sequence of states traversed by the generator. Unfortunately, this is possible only for generators with “jumpahead” primitives, and we want our methodology to be universal. We checked that all sequences of states used in our tests on generators with 64 bits of state do not overlap. The chance that this happens with more than 128 bits of state is negligible. 3 We use the double-dot notation for intervals introduced by C. A. R. Hoare and Lyle Ramshaw [Graham et al. 1994]. 4 S. Vigna atic failures would not change even if we lowered significantly the failure threshold. More generally, 90% of the p-values of failed tests are actually smaller than 10−6 . We remark that our choice (counting the number of failures) is somewhat rough; for example, we consider the same failure a p-value very close to 0 and a p-value just below 0.001. Indeed, other, more sophisticated methods might be used to aggregate the result of our samples: combining p-values, for instance, or computing a p-value of p-values [Rukhin et al. 2001]. However, our choice is very easy to interpret, and multiple samples partially compensate this problem (spurious failures will appear in few samples). Of course, the number of experiments is very large—in fact, our experiments were carried out using hundreds of cores in parallel and, overall, they add up to more than a century of computational time. Our strategy is to apply a very fast test to all generators and seeds, in the hope of isolating a small group of generators that behave significantly better with respect to these tests. Stronger tests can then be applied to this subset. The same strategy has been followed by Panneton [2004] in the experimental study of xorshift generators contained in his Ph.D. thesis. TestU01 offers three different predefined batteries of tests (SmallCrush, Crush and BigCrush) with increasing computational cost and increased difficulty. Unfortunately, Dieharder does not provide such a segmentation. Note that Dieharder has a concept of “weak” success and a concept of “failure”, depending on the p-value of the test, and we used command-line options to align its behavior with that of TestU01: a p-value outside of the range [0.001 . . 0.999] is a failure. Moreover, we disabled the initial timing tests so that exactly the same stream of 64-bit numbers is fed to the two test suites. In both cases we implemented our own xorshift generator. Some care is needed in this phase, as both TestU01 and Dieharder are inherently 32-bit test suites: since we want to test xorshift as a 64-bit generator, it is important that all bits produced are actually fed into the test. For this reason, we implemented the generation of a uniform real value in [0 . . 1) by dividing the output of the generator by 264 , but we implemented the generation of uniform 32-bit integer values by returning first the lower and then the upper 32 bits of each 64-bit generated value.4 A possible downside of this approach is that we might fail to detect some failure in the high bits (of the 64-bit, full output) due to the interleaving process: however, the fact that in our tests xorshift generators generate many more failures than those reported previously [Panneton and L’Ecuyer 2005] suggests that the approach is well founded. An important consequence of this choice is that some of the bits are actually not used at all. When analyzing pseudorandom real numbers in the unit interval, there is an unavoidable bias towards high bits, as they are more significant. The very lowest bits have lesser importance and will in any case be perturbed by numerical errors. For this reason, it is a good practice to run tests both on a generator and on its reverse5 [Press et al. 2007]. In our case, this is even more necessary, as the lowest eleven bits returned by the generator are not used at all due to the fact that the mantissa of a 64-bit floating-point number is formed by 53 bits only. A recent example shows the importance of testing the reverse generator. Saito and Matsumoto [2014] propose a different way to eliminate linear artifacts: instead of multiplying the output of an underlying xorshift generator (with 128 bits of state and based on 32-bit shifts) by a constant, they add it (in Z/232 Z) with the previous output. Since the sum in Z/232 Z is not linear over Z/2Z, the result should be free of linear artifacts. However, while their generator passes BigCrush, its reverse fails systematically the LinearComp, Ma- 4 If a real value is generated when the upper 32 bits of the last value are available, they are simply discarded. is, on the generator obtained by reversing the order of the 64 bits returned. 5 That An experimental exploration of Marsaglia’s xorshift generators, scrambled 5 trixRank, MaxOft and Permutation test of BigCrush, which highlights a significant weakness in its lower bits. We remark that in this paper we do not pursue the search for equidistribution—the property that all tuples of consecutive values, seen as vectors in the unit cube, are evenly distributed, as done, for instance, by Panneton and L’Ecuyer [2005]. Brent [2010] has already argued in detail that for long-period generators equidistribution is not particularly desirable, as it is a property of the whole sequence produced by the generator, and in the case of a long-period generator only a minuscule fraction of the sequence can be actually used. Moreover, equidistribution is currently impossible to evaluate exactly for long-period nonlinear generators, and in the formulation commonly used in the literature it is known to be biased towards the high bits [L’Ecuyer and Panneton 2005]: for instance, the WELL1024a generator has been designed to be maximally equidistributed [Panneton et al. 2006], and indeed it has measure of equidistribution ∆1 = 0, but the generator obtained by reversing its bits has ∆1 = 366: a quite counterintuitive result, as in general we expect all bits to be equally important. Another problem with equidistribution is that it is intrinsically unstable, unless we restrict its usage to the class of linear generators, only. Indeed, if we take a maximally equidistributed sequence, no matter how long, and we flip the most significant bit of a single element of the sequence, the new sequence will have the worst possible ∆1 . For instance, by flipping the most significant bit of a single chosen value out of the output of WELL1024a we can turn its equidistribution measure to ∆1 = 4143. But for any statistical or practical purpose the two sequences are indistinguishable—we are modifying one bit out of 25 (21024 − 1). However, in general this paradoxical behaviour is not a big issue, because the modified sequence can no longer be emitted by a linear generator. We note that since multiplication by an invertible constant induces a permutation of the space of 64-bit values (and thus of t-tuples of such values), it preserves some of the equidistribution properties of the underlying generator (this is true of any bijective scrambling function); more details will be given in the rest of the paper. 4. RESULTS FOR xorshift64 GENERATORS First of all, all generators fail at all seeds the MatrixRank test from TestU01’s SmallCrush suite.6 A score-rank plot7 of the SmallCrush scores for all generators is shown in Figure 2. The plot associates with abscissa x the number of generators with x or more failures. We observe immediately that there is a wide range of quality among the generators examined. The “bumps” in the plot corresponds to new tests failed systematically. A closer inspection would confirm that there is just a weak correlation between scores of algorithms conjugate by reversal, because of the bias of TestU01 towards high bits. We thus report in Table I reports the best four generators by combined scores (i.e., adding the scores of conjugate generators), which are the only ones failing systematically just the MatrixRank test. The table reports also results for the generator A0 (13, 7, 17) suggested by Marsaglia in his original paper, claiming that it “will provide an excellent period 264 − 1 RNG, [. . . ] but any of the above 2200 choices is likely to do as well”. Clearly, this is not the case: A0 (13, 7, 17)/A1 (13, 7, 17) ranks 655 in the combined SmallCrush ranking and fails systematically several tests. 6 Panneton and L’Ecuyer [2005] reports that half of the generators fail this test, but the authors have chosen to use only 32 of the 64 generated bits as output bits, in practice applying a kind of decimation to the output of the generator. 7 Score-rank plots are the numerosity-based discrete analogous of the complementary cumulative distribution function of scores. They give a much clearer picture than frequency dot plots when the data points are scattered and highly variable. 6 S. Vigna 2500 2000 Rank 1500 1000 500 0 100 1000 SmallCrush score 10000 Fig. 2. Score-rank plot of the distribution of SmallCrush scores for the 2200 possible full-period xorshift64 generators. Table I. Best four xorshift64 generators following SmallCrush. Algorithm A2 (11, 31, 18) A2 (8, 29, 19) A0 (8, 29, 19) A0 (11, 31, 18) A0 (13, 7, 17) Failures 111 155 159 130 276 Conjugate A3 (11, 31, 18) A3 (8, 29, 19) A1 (8, 29, 19) A1 (11, 31, 18) A1 (13, 7, 17) Failures 120 115 112 150 802 Overall 231 270 271 280 1078 W 25 35 35 25 25 Table II. The generators of Table I tested with BigCrush. Algorithm A2 (11, 31, 18) A2 (8, 29, 19) A0 (8, 29, 19) A0 (11, 31, 18) A2 (4, 35, 21) A0 (13, 7, 17) Failures 762 747 749 748 961 1049 Conjugate A3 (11, 31, 18) A3 (8, 29, 19) A1 (8, 29, 19) A1 (11, 31, 18) A3 (4, 35, 21) A1 (13, 7, 17) Failures 750 780 884 926 1444 5454 Overall 1512 1527 1633 1674 2405 6503 Sanity check 1. Is the result of our experiments dependent on our seed choice? To answer this question, we repeated our experiments on xorshift64 generators with SmallCrush on a different set of seeds, namely the integers in the interval [1 . . 100]. Kendall’s τ [Kendall 1938; Kendall 1945] between the two rankings is 0.98, which makes it clear that the dependence on the seed is negligible. In particular, the four best conjugate pairs in Table I are the same with both seeds. To gather more information, we ran the full BigCrush suite and Dieharder on our four best generators, on Marsaglia’s choice and on the best choice from “Numerical Recipes”: the results are given in Tables II and III. Even the four best generators fail now systematically the BirthdaySpacings, MatrixRank and LinearComp tests. The first two generators, however, turn out to perform slightly better than other two. We also notice that BigCrush draws a much thicker line between our four best generators and the other ones, which now fail several more tests. Not surprisingly, Dieharder cannot really separate our four best generators from A2 (4, 35, 21)/A3 (4, 35, 21). An experimental exploration of Marsaglia’s xorshift generators, scrambled 7 Table III. The generators of Table I tested with Dieharder. Algorithm A2 (11, 31, 18) A2 (8, 29, 19) A0 (8, 29, 19) A0 (11, 31, 18) A2 (4, 35, 21) A0 (13, 7, 17) Failures 182 179 176 181 189 183 Conjugate A3 (11, 31, 18) A3 (8, 29, 19) A1 (8, 29, 19) A1 (11, 31, 18) A3 (4, 35, 21) A1 (13, 7, 17) Failures 162 181 182 186 187 1352 Overall 344 360 358 367 376 1535 Equidistribution score (combined) 300 250 200 150 100 50 0 0 500 1000 1500 2000 2500 3000 SmallCrush score (combined) Fig. 3. Scatter plot of the combined SmallCrush score of conjugate xorshift64 generators versus the combined equidistribution score. 4.1. Equidistribution It is interesting to compare the ranking provided by equidistribution properties and that provided by statistical tests. Note that a xorshift64 generator is 1-dimensionally equidistributed, that is, every 64-bit value appears exactly once except for zero. We refer to the already quoted paper by Panneton and L’Ecuyer [2005] for a detailed description of the equidistribution statistics ∆1 , the sum of dimension gaps: a lower value is better. A maximally distributed generator has ∆1 = 0, and we will refer to ∆1 as to the equidistribution score. We computed the equidistribution score for all generators using the implementation of Harase’s algorithm [Harase 2011] contained in the MTToolBox package from Saito [2013]. Similarly to SmallCrush scores, ∆1 has high-bits bias, and a quite strong one [L’Ecuyer and Panneton 2005]. For a fair comparison, we to thus combine the ∆1 score of a generator and of its reverse. Figure 3 shows that there is some correlation (τ = 0.58) between combined SmallCrush scores and combined equidistribution scores. Nonetheless, even if equidistribution is able to detect reliably generators with a very bad SmallCrush score, is not so good at detecting the generators with the best score, as is visible from the quite noisy lower left part of the plot. Indeed, when we restrict our attention to the best 30 generators (by combined SmallCrush scores) Kendall’s τ drops to 0.3. The first two generators by combined equidistribution score, A4 (8, 29, 19) and A6 (8, 29, 19), rank 20 (combined score 361) and 170 (score 596) in the combined SmallCrush test. When analyzed with the more powerful lens of BigCrush, they have combined scores 3441 and 4082, respectively, and fail systematically almost twenty additional tests with respect to the top four generators of Table II. Definitely, choosing among xorshift64 generators by equidistribution score alone is not a good idea. 8 S. Vigna Table IV. The three multipliers used in the rest of the paper. The subscripts recalls the t for which they have good figures of merit. M32 = 2685821657736338717 M8 = 1181783497276652981 M2 = 8372773778140471301 5. AN INTRODUCTION TO xorshift64* GENERATORS Since a xorshift64 generator exhibits evident linearity artifacts, the next obvious step is to perturb its output using a nonlinear (in Z/2Z sense) transformation. A natural candidate is multiplication by a constant, also because such operation is very fast in modern processors. Note that the current state of the generator is multiplied by a constant before returning it, but the state itself is not affected by the multiplication: thus, the period is the same. We call such a generator xorshift. By choosing a constant invertible modulo 264 (i.e., odd), we can guarantee that the generator will output a permutation of the sequence output by the underlying xorshift generator. This approach was noted in passing in Marsaglia’s paper, and it is also proposed in a more systematic way in the third edition of “Numerical Recipes” [Press et al. 2007] to create a very fast, good-quality pseudorandom number generator. However, in the latter case the authors first compute allegedly good triples for xorshift using Diehard (with results markedly different from ours, and in strident contrast with TestU01’s results, as discussed in Section 4) and then choose a multiplier. There is no reason why the best triples for a xorshift64 generator (which are computed empirically) should continue to be such in a xorshift64 generator: and indeed, we will see that this is not the case. We thus repeated the experiments of the previous section on xorshift64* generators. To choose scrambling constants, we followed the heuristic considerations of [Press et al. 2007]. We consider primitive (e.g., full-period) elements of the multiplicative group of Z/264 Z: these elements have no fixed point except for zero, which is a very desirable property for a scrambling function. Moreover, we choose from L’Ecuyer [1999] primitive elements that have good qualities as multiplicative congruential linear generators, as we expect that multiplication by such elements will combine bits in a non-trivial way. We use a standard theoretical measure of quality, the figure of merit, which is a normalized best distance between the hyperplanes of families covering tuples of length t given by successive outputs of the generators (see L’Ecuyer [1999] for details). Since t is an additional parameter, to further understand the dependency on the multiplier we used three different multipliers, shown in Table IV, which have good figures of merit for different t’s. The first multiplier, M32 (the one used in [Press et al. 2007]) and the second, M8 , have been taken from L’Ecuyer [1999]. The third, M2 , was kindly provided by Richard Simard. We remark that many other choices for scrambling the output of a generator are possible, like adding or xoring a fixed word, xoring the output with the output of another generator, or using a bijective function with strong avalanching behavior, such as those used in the construction of high-quality hash functions. The three factors we considered in our choice are: speed, good results in statistical test suites, and preservation of some equidistribution properties (similarly to the approach taken in [L’Ecuyer and Granger-Piché 2003]). For instance, xoring with an additive Weyl generator (another suggestion in Marsaglia’s paper) makes it in general impossible to prove any equidistribution property—not even that all 64bit value except for zero are output by the generator. Multiplication by a constant is a very fast operation in modern processor, and mixing linear operations on Z/2Z with operations in the ring Z/264 Z is a standard technique to avoid visible artifacts from either type of algebraic structure. A drawback is that the lowest bit is, in fact, not scrambled, and thus it is identical to the lowest bit of the underlying xorshift generator.8 8 As remarked by one of the referees, since our multipliers are all equal to 1 modulo 4, this is true also of the second-lowest bit. An experimental exploration of Marsaglia’s xorshift generators, scrambled 9 SmallCrush score (reverse) 1000 100 10 1 1 10 100 SmallCrush score (standard) 1000 Fig. 4. Scatter plot of the SmallCrush score of xorshift64* generators and their reverse. 2500 Standard Reverse Combined 2000 Rank 1500 1000 500 0 0 1 10 100 SmallCrush Score 1000 Fig. 5. Score-rank plot of the distribution of SmallCrush scores for the 2200 possible xorshift64* generators with multiplier M32 . 6. RESULTS FOR xorshift64* GENERATORS The scatter plot in Figure 4 shows that there is essentially no correlation between the scores assigned by SmallCrush to a generator and its reverse (τ = 0.15).9 Another interesting observation on Figure 4 is that the lower right half is essentially empty. So bad generators have a bad reverse, but there are good generators with a very bad reverse. This suggests that the quality of a xorshift64* generator can vary wildly from the low to the high bits. A score-rank plot of the SmallCrush scores for all generators shown in Figure 5 provides us with further interesting information: almost all generators have no systematic failure, but only about half of the reverse generators have no systematic failure. Moreover, the distribution of standard generators degrades smoothly, whereas the distribution of reverse generators sports again the “bump” phenomenon we observed in Figure 2. Since we need to reduce the number of candidates to apply stronger tests, in the case of M32 we decided to restrict our choice to generators with 3 overall failed tests or less, which left us with 152 generators. Similar cutoff points were chosen for M8 and M2 . 9 We report plots only for M32 , as the ones for the other multipliers are visually identical. 10 S. Vigna 1000 Dieharder score (reverse) Crush score (reverse) 10000 1000 100 10 1 1 10 100 1000 10000 Crush score (standard) 100 10 1 1 10 100 1000 Dieharder score (standard) Fig. 6. Scatter plots for Crush (left) and Dieharder (right) scores on xorshift64* generators with multiplier M32 and their reverse, for the 152 best generators. Dieharder score 1000 100 10 10 100 1000 Crush score 10000 Fig. 7. A scatter plot of Crush and Diehard combined scores of the 152 SmallCrush-best xorshift64* generators. The plot is in log-log scale to accommodate some very high values returned by Crush on reverse generators. The lower-left “sweet spot” corner contains generators that never fail systematically (not even reversed) in both test suites. These generators were few enough so that we could apply both Crush and Dieharder. Once again, we examine the correlation between the score of a generator and its reverse by means of the scatter plots in Figure 6, which confirm the high-bits bias, albeit less so in the Dieharder case. In Figure 7 we compare instead the two scores (Crush and Dieharder) available. The most remarkable feature is there are no points in the upper left corner: there is no generator that is considered good by Crush but not by Dieharder. On the contrary, Crush heavily penalizes (in particular because of the score on the reverse generator) a large number of generators. The generators we will select in the end all belong to the small cloud in the lower left corner, where the two test suite agree. The score-rank plot in Figure 8 shows that our strategy pays off: we started with 152 generators with less than three failures, but analyzing them with the more powerful lens provided by Crush we get a much more fine-grained analysis: in particular, only 73 of them give no systematic failure, and they all belong to the “sweet spot” of Figure 7, that is, they do not give any systematic failure in Dieharder, too. An experimental exploration of Marsaglia’s xorshift generators, scrambled 11 160 Standard Reverse Combined 140 120 Rank 100 80 60 40 20 0 100 1000 Crush Score Fig. 8. Score-rank plot of the distribution of Crush scores for the 152 SmallCrush-best xorshift64* generators using multiplier M32 . Finally, we selected for each multiplier the eight generators with the best Crush scores, and applied the BigCrush suite: we obtained several generators failing systematically the MatrixRank test only and shown in Table V (which should be compared with Table II). 6.1. Equidistribution Multiplication by an invertible element just permutes the elements of Z/264 Z leaving zero fixed, so a xorshift64* generator, like the underlying xorshift64 generator, is 1dimensionally equidistributed. 7. HIGH DIMENSION Marsaglia [2003] describes a strategy for xorshift generators in high dimension: the idea is to use always three low-dimensional shifts, but locating them in the context of a larger t × t block matrix of the form   0 0 0 · · · 0 (I + La )(I + Rb )  I 0 0 ··· 0  0    0 I 0 ··· 0  0 M =  0  0 0 I ··· 0  · · · · · · · · · · · · · · ·  ··· 0 0 0 ··· I (I + Rc ) Marsaglia notes that even in this restricted form there are matrices of full period (he provides examples for 32-bit shifts up to 160 bits). However, this route has not been explored for high-dimensional (say, more than 1024 bits of state) generators. The only similar approach is that proposed by Brent [2007] with his xorgens generators, which however uses four shifts. The obvious question is thus: is the additional shift really necessary to pass a strong statistical test such as BigCrush? We are thus going to look for good, full-period generators with 1024 or 4096 bits of state using 64-bit basic shifts.10 10 The reason why the number 4096 is relevant here is that we know the factorization of Fermat’s numbers k 22 + 1 only up to k = 11. When more Fermat numbers will be factorized, it will be possible to design xorshift or xorgens generators with larger state space [Brent 2007]. Note that, however, in practice a period of 21024 − 1 is more than sufficient for any purpose. For example, even if 2100 computers were to generate sequences of 2100 numbers starting from random seeds using a generator with period 21024 , the chances that two sequences overlap would be less than 2−724 . 12 S. Vigna Table V. Results of BigCrush on the best eight xorshift64* generators found by SmallCrush and Crush in sequence. The generators fail systematically only MatrixRank. Algorithm A7 (11, 5, 45) A7 (17, 23, 52) A1 (12, 25, 27) A1 (17, 23, 29) A5 (14, 23, 33) A5 (17, 47, 29) A1 (16, 25, 43) A7 (23, 9, 57) A5 (11, 5, 32) A2 (8, 31, 17) A5 (3, 21, 31) A3 (17, 45, 22) A4 (8, 37, 21) A3 (13, 47, 23) A3 (13, 35, 30) A4 (9, 37, 31) A7 (13, 19, 28) A3 (9, 21, 40) A1 (14, 23, 33) A7 (19, 43, 27) A1 (17, 47, 28) A5 (16, 11, 27) A4 (4, 35, 15) A7 (13, 21, 18) Failures S R M32 226 128 232 130 230 133 229 137 238 132 231 141 238 138 242 134 M8 229 122 229 126 230 141 241 133 239 136 232 144 244 136 243 141 M2 228 128 228 132 234 142 239 137 240 137 234 144 230 149 238 144 + W 354 362 363 366 370 372 376 376 23 25 31 21 32 24 31 19 351 355 371 374 375 376 380 384 13 21 33 27 33 27 27 27 356 360 376 376 377 378 379 382 23 35 29 23 25 25 35 31 The output of such generators will be given by the last 64 bits of the state. It is well known [Brent 2004; Niederreiter 1992] that every bit of state satisfies a linear recurrence (defined by the characteristic polynomial) with full period, so a fortiori the last 64 bits have full period, too. Since we already know that some deficiencies of low-dimensional xorshift generators are well corrected by multiplication by a constant, we will follow the same approach, thus looking for good xorshift* generators of high dimension.11 Note that since multiplication by an integer invertible in Z/264 Z is a permutation of Z/264 Z, a high-dimension xorshift* generator has the same period of the underlying xorshift generator. We cannot in principle claim full period if we look at a single bit of the output of a xorshift* generator; but this property can be easily proved by purely combinatorial means: 11 As in the xorshift64 case, different choices for the shifts are possible. We will not pursue them here. An experimental exploration of Marsaglia’s xorshift generators, scrambled 13 Proposition 7.1. Let x0 , x1 , . . . , x2n −2 be a list of 2t -bit values, t ≤ n, such that every value appears 2n−t times, except for 0, which appears 2n−t − 1 times. Then, for every fixed bit k the associated sequence has period 2n − 1. Proof. Suppose that there is a k and a p \| 2n − 1 such that the k-th bit of x0 , x1 , . . . , x2n −2 has period p (that is, the sequence of bits associated with the k-th bit is made by (2n − 1)/p repetitions of the same sequence of p bits). The k-th bit runs through 2n−1 − 1 zeroes and 2n−1 ones (as there is a missing zero in the output sequence). This means that (2n − 1)/p \| 2n−1 , too, as the same number of ones must appear in every repeating subsequence, and since (2n − 1)/p is odd this implies p = 2n − 1. Corollary 7.2. Every bit of the output of a full-period xorshift* generator has full period. 7.1. Finding good shifts The first step is identifying values of a, b and c for which the generator has maximum period using the primitivity check on the characteristic polynomial. We performed these computations using the algebra package Fermat [Lewis 2013], with the restriction that a + b ≤ 64 and that a is coprime with b (see [Brent 2007] for the rationale behind this choices, which significantly reduce the search space). The resulting sets of values are those shown in Table VI and VIII. For a state of 1024 bits, we obtain 20 possible parameter choices, which we examined in combination with our three multipliers both through BigCrush and through Dieharder. The results, reported in Table VI and VII, are excellent: with the exception of two pathological choices, no test is failed systematically. For a state of 4096 bits (Table VIII and IX) there are 10 possible parameter choices, and no generator fails a test systematically. 7.2. Equidistribution Looking at the shape of the matrix defining high-dimensional xorshift generators it is clear that if the state is made of n bits the last n/64 output values, concatenated, are equal to the current state. This implies that such generators are n/64-dimensionally equidistributed (i.e., every n/64-tuple of consecutive 64-bit values appears exactly once, except for a missing tuple of zeroes), so xorshift1024 generators are 16-dimensionally equidistributed and xorshift4096 generators are 64-dimensionally equidistributed. Since multiplication by a constant just permutes the space of tuples, the same is true of the associated xorshift* generators. 8. JUMPING AHEAD The simple form of a xorshift generator makes it trivial to jump ahead quickly by any number of next-state steps. If v is the current state, we want to compute vM j for some j. But M j is always expressible as a polynomial in M of degree lesser than that of the characteristic polynomial. To find such a polynomial it suffices to compute xj mod P (x), where P (x) is the characteristic polynomial of M . Such a computation can be easily carried k out using standard techniques (quadratures to find x2 mod P (x), etc.), leaving us with a polynomial Q(x) such that Q(M ) = M j . Now, if Q(x) = n X αi xi , i=0 we have vM j = vQ(M ) = n X i=0 αi vM i , S. Vigna 14 S 25 28 37 37 23 28 34 32 44 37 45 42 35 38 34 43 45 39 31 50 M32 Failures R + 31 56 32 60 24 61 26 63 40 63 37 65 34 68 36 68 28 72 36 73 29 74 34 76 43 78 40 78 45 79 40 83 39 84 51 90 890 921 902 952 275 79 223 65 167 265 113 155 227 59 89 281 363 77 233 81 255 69 111 99 W 1, 13, 7 3, 26, 35 40, 11, 31 15, 16, 19 22, 7, 48 9, 14, 41 41, 7, 29 31, 11, 30 2, 11, 61 10, 11, 61 7, 16, 55 16, 23, 30 25, 8, 15 27, 13, 46 31, 10, 27 9, 5, 60 31, 33, 37 10, 9, 63 51, 1, 46 47, 1, 41 a, b, c S 28 29 24 30 29 32 25 33 25 42 32 35 25 39 40 40 39 31 60 67 M8 Failures R + 19 47 22 51 33 57 32 62 33 62 30 62 38 63 32 65 41 66 25 67 35 67 34 69 45 70 32 71 32 72 36 76 39 78 49 80 896 956 907 974 113 89 77 255 223 167 265 363 81 155 65 59 281 275 233 227 79 69 111 99 W 3, 26, 35 27, 13, 46 25, 8, 15 31, 10, 27 9, 5, 60 1, 13, 7 15, 16, 19 2, 11, 61 41, 7, 29 9, 14, 41 22, 7, 48 31, 11, 30 7, 16, 55 31, 33, 37 10, 11, 61 16, 23, 30 40, 11, 31 10, 9, 63 51, 1, 46 47, 1, 41 a, b, c S 29 41 38 36 24 28 36 40 36 33 37 45 36 37 41 44 38 48 31 47 M2 Failures R + 24 53 20 61 24 62 31 67 43 67 42 70 34 70 30 70 34 70 37 70 35 72 27 72 39 75 39 76 37 78 37 81 48 86 48 96 799 830 799 846 89 275 281 233 227 113 255 81 265 167 223 363 65 79 155 59 77 69 111 99 W Table VI. Results of BigCrush on the xorshift1024* generators. The last two generators fail systematically CouponCollector, Gap, HammingIndep, MatrixRank, SumCollector and WeightDistrib. a, b, c 27, 13, 46 31, 33, 37 22, 7, 48 7, 16, 55 9, 14, 41 41, 7, 29 1, 13, 7 10, 11, 61 9, 5, 60 16, 23, 30 3, 26, 35 25, 8, 15 31, 11, 30 40, 11, 31 31, 10, 27 2, 11, 61 15, 16, 19 10, 9, 63 51, 1, 46 47, 1, 41 31, 33, 37 31, 11, 30 16, 23, 30 41, 7, 29 9, 14, 41 10, 9, 63 22, 7, 48 51, 1, 46 27, 13, 46 25, 8, 15 3, 26, 35 2, 11, 61 40, 11, 31 31, 10, 27 47, 1, 41 9, 5, 60 10, 11, 61 15, 16, 19 7, 16, 55 1, 13, 7 a, b, c M32 Failures S R + 57 67 124 65 61 126 74 56 130 71 61 132 74 64 138 74 66 140 66 75 141 78 63 141 63 79 142 80 64 144 81 66 147 79 71 150 74 76 150 82 71 153 74 79 153 81 75 156 75 84 159 72 88 160 94 68 162 87 76 163 79 363 59 265 167 69 223 111 275 281 89 81 77 233 99 227 155 255 65 113 W 25, 8, 15 16, 23, 30 7, 16, 55 3, 26, 35 10, 11, 61 31, 10, 27 31, 33, 37 47, 1, 41 27, 13, 46 31, 11, 30 10, 9, 63 41, 7, 29 2, 11, 61 9, 14, 41 40, 11, 31 15, 16, 19 51, 1, 46 22, 7, 48 1, 13, 7 9, 5, 60 a, b, c M8 Failures S R + 67 56 123 77 54 131 66 66 132 60 75 135 63 74 137 74 69 143 86 58 144 82 62 144 78 69 147 85 62 147 65 86 151 84 68 152 88 65 153 77 80 157 82 78 160 85 76 161 92 74 166 90 82 172 79 95 174 97 89 186 281 59 65 89 155 233 79 99 275 363 69 265 81 167 77 255 111 223 113 227 W 22, 7, 48 15, 16, 19 10, 9, 63 51, 1, 46 1, 13, 7 40, 11, 31 2, 11, 61 31, 11, 30 25, 8, 15 10, 11, 61 47, 1, 41 9, 5, 60 16, 23, 30 27, 13, 46 7, 16, 55 9, 14, 41 41, 7, 29 31, 10, 27 3, 26, 35 31, 33, 37 a, b, c M2 Failures S R + 56 76 132 66 67 133 70 71 141 65 78 143 80 64 144 80 67 147 85 65 150 75 75 150 74 77 151 79 76 155 70 86 156 70 86 156 81 76 157 78 80 158 92 70 162 87 80 167 87 81 168 82 87 169 92 79 171 98 88 186 Table VII. Results of Dieharder on xorshift1024* generators. No test is failed systematically. 223 255 69 111 113 77 81 363 281 155 99 227 59 275 65 167 265 233 89 79 W An experimental exploration of Marsaglia’s xorshift generators, scrambled 15 S. Vigna 16 Algorithm 14, 41, 15 5, 22, 27 30, 29, 39 25, 3, 49 7, 12, 59 19, 34, 19 12, 11, 61 5, 27, 21 23, 26, 29 11, 9, 25 241 45 177 441 103 291 195 187 49 567 W 5, 22, 27 5, 27, 21 25, 3, 49 7, 12, 59 11, 9, 25 12, 11, 61 19, 34, 19 14, 41, 15 30, 29, 39 23, 26, 29 Algorithm M8 Failures R + 35 69 35 71 37 72 39 73 34 74 33 74 35 74 34 77 37 79 43 81 S 34 36 35 34 40 41 39 43 42 38 45 187 441 103 567 195 291 241 177 49 W 11, 9, 25 5, 27, 21 25, 3, 49 19, 34, 19 23, 26, 29 30, 29, 39 12, 11, 61 14, 41, 15 7, 12, 59 5, 22, 27 Algorithm Table VIII. Results of BigCrush on xorshift4096* generators. M32 Failures R + 27 60 30 64 32 65 38 68 25 68 36 70 39 71 41 75 42 78 44 79 S 33 34 33 30 43 34 32 34 36 35 M2 Failures R + 33 63 27 64 34 67 36 75 35 75 37 75 37 77 42 78 44 82 50 88 S 30 37 33 39 40 38 40 36 38 38 W 567 187 441 291 49 177 195 241 103 45 25, 3, 49 12, 11, 61 30, 29, 39 5, 22, 27 11, 9, 25 19, 34, 19 14, 41, 15 7, 12, 59 23, 26, 29 5, 27, 21 Algorithm M32 Failures S R + 70 70 140 58 83 141 67 77 144 62 84 146 73 75 148 85 66 151 83 74 157 73 85 158 73 88 161 98 67 165 441 195 177 45 567 291 241 103 49 187 W 25, 3, 49 14, 41, 15 30, 29, 39 11, 9, 25 12, 11, 61 19, 34, 19 5, 22, 27 23, 26, 29 5, 27, 21 7, 12, 59 Algorithm M8 Failures S R + 67 70 137 72 69 141 70 75 145 73 77 150 75 80 155 89 67 156 93 65 158 72 87 159 75 84 159 90 77 167 441 241 177 567 195 291 45 49 187 103 W 19, 34, 19 5, 22, 27 25, 3, 49 5, 27, 21 11, 9, 25 14, 41, 15 23, 26, 29 12, 11, 61 7, 12, 59 30, 29, 39 Algorithm Table IX. Results of Dieharder on xorshift4096* generators. M2 Failures S R + 75 64 139 67 77 144 77 71 148 77 71 148 81 76 157 79 78 157 74 84 158 74 85 159 84 79 163 78 89 167 291 45 441 187 567 241 49 195 103 177 W An experimental exploration of Marsaglia’s xorshift generators, scrambled 17 18 S. Vigna and now vM i is just the i-th state after the current one. If we known in advance the αi ’s, computing vM j requires just computing the next state for n times, accumulating by xor the i-th state iff αi 6= 0.12 In general, one needs to compute the αi ’s for each desired j, but the practical usage of this technique is that of providing subsequences that are guaranteed to be non-overlapping. We can fix a reasonable jump, for example 2512 for a xorshift1024* generator, and store the αi ’s for such a jump as a bit mask. Operating the jump is now entirely trivial, as it requires at most 1024 state changes. In Figure 12 we show the jump function for the generator of Figure 11. By iterating the jump function, one can access 2512 non-overlapping sequences of length 2512 (except for the last one, which will be of length 2512 − 1). 9. COMPARISON How do our best xorshift* generators score with respect to more complex generators in the literature? We decided to perform a comparison with the popular Mersenne Twister MT19937 [Matsumoto and Nishimura 1998],13 with WELL1024a/WELL19937a, two generators introduced by Panneton et al. [2006] as an improvement over the Mersenne Twister, and with xorgens4096, a very recent 4096-bit generator introduced by Brent [2007] we mentioned in Section 7. All these generators are non-cryptographic and aim at fast, high-quality generation. As usual, 100 tests are performed at 100 equispaced points of the state space. We choose generators from the xorshift* family that perform well on both BigCrush and Dieharder, have a good weight score and enough large parameters (which provide faster state change spreading): more precisely, the xorshift64* generator A1 (12, 25, 27) · M32 (Figure 10), xorshift1024* with parameters 31, 11, 30 and multiplier M8 (Figure 11), and xorshift4096* with parameters 25, 3, 49 and multiplier M2 . 9.1. Quality Table X compares the BigCrush scores of the generators we discussed. The results are quite interesting. A simple 64-bit xorshift* generator has less linear artifacts than MT19937, WELL1024a or WELL19937a and, thus, a significantly better score. High-dimension xorgens4096 and xorshift* generators perform significantly better, in spite of being extremely simple, and have no systematic failure. The 64-bit xorshift* generator suggested by “Numerical Recipes” fails systematically the BirthdaySpacings test, contrarily the one we have selected.14 We do not report the results of Dieharder, as at this level of quality the suite is unable to make any significant distinction among the generators. 9.2. Escaping zeroland We show in Figure 9 the speed at which a few of the generators of Table X “escape from zeroland” [Panneton et al. 2006]: purely linearly recurrent generators with a very large state space need a very long time to get from an initial state with a small number of ones to a state in which the ones are approximately half. The figure shows a measure of escape time given by the ratio of ones in a window of 4 consecutive 64-bit values sliding over the first 100 000 generated values, averaged over all possible seeds with exactly one bit set (see [Panneton et al. 2006] for a detailed description). As it is known, MT19937 needs hundreds of thousands of iterations to start behaving correctly. xorshift4096* and xorgens4096 need a few thousand (but xorgens4096 oscillates 12 Brent’s ranut generator [Brent 1992] contains one of the first applications of this technique. precisely, with its 64-bit version. 14 Note that we report the number of failed tests on our 100 seeds. L’Ecuyer and Simard [L’Ecuyer and Simard 2007] report the number of types of failed tests (e.g., failing two distinct RandomWalk tests counts as one) on a single run, so some care must be taken when comparing the results we report and those reported by them. 13 More An experimental exploration of Marsaglia’s xorshift generators, scrambled 19 Table X. A comparison of generators using BigCrush. Algorithm A1 (12, 25, 27) · M32 A3 (4, 35, 21) · M32 xorshift1024* xorshift4096* xorgens4096 MT19937 WELL1024a WELL19937a S 230 240 33 33 42 258 441 235 Failures R 133 223 32 34 40 258 441 233 W/n + 363 463 65 67 82 516 882 468 Systematic 0.48 0.38 0.35 0.11 0.23 0.34 0.40 0.43 MatrixRank MatrixRank, BirthdaySpacings — — — LinearComp MatrixRank, LinearComp LinearComp xorgens4096 MT19937 WELL1024a WELL19937a xorshift64* xorshift1024* xorshift4096* Average number of ones 0.6 0.5 0.4 0.3 0.2 0.1 0 1 10 100 1000 10000 100000 Samples Fig. 9. Convergence to “half of the bits are ones in average” plot. always around 1/2), WELL19937a and xorshift1024* a few hundreds, whereas WELL1024a just a few dozens, and xorshift64* is almost unaffected. Table XI condenses Figure 9 into the mean and standard deviation of the displayed values. Clearly, the multiplication step helps in reducing the correlation between the number of ones in the state and the number of ones in the output values. Also, the slowness in recovering from states with too many zeroes it directly correlated to the size of the state space—a very good argument against linear generators with too large state spaces. 9.3. Speed Finally, we benchmark the generators of Table X. Our tests were run on an Intel R CoreTM i7-4770 CPU @3.40GHz (Haswell), and the results are shown in Table XII (variance is undetectable, as we generate 1010 values in each test). We also report as a strong baseline results about SFMT19937, the SIMD-Oriented Fast Mersenne Twister [Saito and Matsumoto 2008], a 128-bit version of the Mersenne Twister based on the SSE2 extended instruction set of Intel processors (and thus not usable, in principle, on other processors). We used suitable options to keep the compiler from unrolling loops or extracting loop invariants. 20 S. Vigna Table XI. Mean and standard deviation for the data shown in Figure 9. Algorithm xorshift64* xorgens4096 xorshift1024* WELL1024a xorshift4096* WELL19937a MT19937 Mean 0.5000 0.5000 0.5000 0.4999 0.4992 0.4983 0.2823 Standard deviation 0.0039 0.0031 0.0035 0.0036 0.0110 0.0185 0.1705 Table XII. Time to emit a 64-bit integer on an Intel R CoreTM i7-4770 CPU @3.40GHz (Haswell). Algorithm xorshift64* xorshift1024* xorshift4096* xorgens4096 MT19937 (64-bit version) SFMT19937 WELL1024a WELL19937a Speed (ns/64 bits) 1.58 1.36 1.36 2.06 2.84 1.80 10.31 7.45 The highest speed is achieved by the high-dimensional xorshift* generators. SFMT19937 is a major improvement in speed over MT19937, albeit slightly slower than a high-dimensional xorshift* generator; it fails systematically, moreover, the same tests of MT19937. A xorshift64* generator is actually slower than its high-dimensional counterparts. This is not surprising, as the three shift/xors in a xorshift64* generator form a dependency chain and must be executed in sequence, whereas two of the shifts of a higher-dimension generator are independent and can be internally parallelized by the CPU. WELL1024a and WELL19937a are heavily penalized by their 32-bit structure. #include <stdint.h> uint64_t x; uint64_t next(void) { x ^= x >> 12; // a x ^= x << 25; // b x ^= x >> 27; // c return x * UINT64_C(2685821657736338717); } Fig. 10. The suggested xorshift64* generator in C99 code. The variable x should be initialized to a nonzero seed before calling next(). 10. CONCLUSIONS After our careful experimental analysis, we reach the following conclusions: A xorshift1024* generator is an excellent choice for a general-purpose, highspeed generator. The statistical quality of the generator is very high (it has, actually, An experimental exploration of Marsaglia’s xorshift generators, scrambled 21 #include <stdint.h> uint64_t s[16]; int p; uint64_t next(void) { const uint64_t s0 = s[p]; uint64_t s1 = s[p = (p + 1) & 15]; s1 ^= s1 << 31; // a s[p] = s1 ^ s0 ^ (s1 >> 11) ^ (s0 >> 30); // b,c return s[p] * UINT64_C(1181783497276652981); } Fig. 11. The suggested xorshift1024* generator in C99 code. The array s should be initialized to a nonzero seed before calling next(). #include <stdint.h> #include <string.h> void jump(void) { static const uint64_t JUMP[] = { 0x84242f96eca9c41d, 0xa3c65b8776f96855, 0x4489affce4f31a1e, 0x2ffeeb0a48316f40, 0x3659132bb12fea70, 0xaac17d8efa43cab8, 0x5ee975283d71c93b, 0x691548c86c1bd540, 0x0b5fc64563b3e2a8, 0x047f7684e9fc949d, 0x284600e3f30e38c3 }; 0x5b34a39f070b5837, 0xdc2d9891fe68c022, 0xc4cb815590989b13, 0x7910c41d10a1e6a5, 0xb99181f2d8f685ca, uint64_t t[16] = { 0 }; for(int i = 0; i < sizeof JUMP / sizeof JUMP; i++) for(int b = 0; b < 64; b++) { if (JUMP[i] & 1ULL << b) for(int j = 0; j < 16; j++) t[j] ^= s[(j + p) & 15]; next(); } for(int j = 0; j < 16; j++) s[(j + p) & 15] = t[j]; } Fig. 12. The jump function for the xorshift1024 generator of Figure 11 in C99 code. It is equivalent to 2512 calls to next(). the best results in BigCrush), and its period is so large that the probability of overlapping sequences is practically zero, even in the largest parallel simulation (and strictly nonoverlapping sequences can be easily generated using the jump function). Nonetheless, the state space is reasonably small, so that seeding it with high-quality bits is not too expensive, and recovery from states with a large number of zeroes happens quickly. The generator is also blazingly fast (it is actually the fastest generator we tested). The reasonable state space makes it also easier, in case a large number of generators is used at the same time, to fit their state into the cache. In any case, with respect to other generators, the state is accessed 22 S. Vigna in a more localized way, as read and write operations happen at two consecutive locations, and thus will generate at most one cache miss. In case memory is an issue, or array access is expensive, a very good generalpurpose generator is a xorshift64* generator. While the generator A1 (12, 25, 27)·M32 fails systematically the MatrixRank test, it has less linear artifacts than MT19937, WELL1024a or WELL19937a, which fail systematically even more tests. It is a very good choice if memory footprint is an issue and a very large number of generators is necessary. It can also be used, for instance, to generate the initial state of another generator with a larger state space using a 64-bit seed. We remark that a xorshift64* generator can also actually be faster than a xorshift1024* generator if the underlying language incurs significant costs when accessing an array: for instance, in Java a xorshift64* generator emits a value in 1.62 ns, whereas a xorshift1024* generator needs 2.06 ns. Linear generators with an excessively long period have a number of problems that are not compensated by higher statistical quality. WELL19937a is almost four slower than xorshift1024*, and has a worse performance in BigCrush; moreover, recovery from states with many zeroes, albeit enormously improved with respect to MT19937, is still very slow, and seeding properly the generator requires almost twenty thousands random bits. In the end, it is in general difficult to motivate state spaces larger than 21024 . Similar considerations are made by Press et al. [2007] and L’Ecuyer and Panneton [2005]. Surprisingly simple and fast generators can produce sequences that pass strong statistical tests. The code in Figure 11 is extremely shorter and simpler than that of MT19937, WELL1024a or WELL19937a. Yet, it performs significantly better on BigCrush. It is a tribute to Marsaglia’s cleverness that just eight logical operations, one addition and one multiplication by a constant can produce sequences of such high quality. xorgens generators are similar with this respect, but use several more operations due to the additional shift and to combination with a Weyl generator to hide linear artifacts [Brent 2007]. The t for which the multiplier has a good figure of merit has no detectable effect on the quality of the generator. If our tests, we could not find any significant difference between the behavior of generators based on M32 , M8 or M2 . It could be interesting to experiment with multipliers having very bad figures of merit, or more generally with multipliers chosen using different heuristics. Equidistribution is more useful as a design feature than as an evaluation feature. While designing generators around equidistribution might be a good idea, as it leads in general to good generators, evaluation by equidistribution is a more delicate matter because of high-bits bias, instability issues, and failure to detect the generators having the best scores in statistical suites. TestU01 has significantly more resolution than Dieharder as a test suite. In particular in the high-dimension case, TestU01 is able to provide useful information, whereas Dieharder scores flatten down. However, TestU01 (as any other test suite with high-bits bias) must always be applied to the reverse generator, too. REFERENCES Richard P. Brent. 1992. Uniform Random Number Generators for Supercomputers. In Supercomputing, the competitive advantage: proceedings of the Fifth Australian Supercomputing Conference. 5ASC Organising Committee, Melbourne, 95–104. Richard P. Brent. 2004. Note on Marsaglia’s Xorshift Random Number Generators. Journal of Statistical Software 11, 5 (2004), 1–5. Richard P. Brent. 2007. Some long-period random number generators using shifts and xors. ANZIAM J. 48 (2007), C188–C202. Richard P. Brent. 2010. The myth of equidistribution for high-dimensional simulation. CoRR abs/1005.1320 (2010). An experimental exploration of Marsaglia’s xorshift generators, scrambled 23 Robert G. Brown. 2013. Dieharder: A Random Number Test Suite (Version 3.31). (2013). Retrieved January 8, 2014 from http://www.phy.duke.edu/∼rgb/General/dieharder.php Aaldert Compagner. 1991. The hierarchy of correlations in random binary sequences. Journal of Statistical Physics 63, 5-6 (1991), 883–896. Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. 1994. Concrete Mathematics (second ed.). Addison–Wesley. Shin Harase. 2011. An efficient lattice reduction method for F2 -linear pseudorandom number generators using Mulders and Storjohann algorithm. J. Comput. Appl. Math. 236, 2 (2011), 141–149. Maurice G. Kendall. 1938. A New Measure of Rank Correlation. Biometrika 30, 1/2 (1938), 81–93. Maurice G. Kendall. 1945. The treatment of ties in ranking problems. Biometrika 33, 3 (1945), 239–251. Pierre L’Ecuyer. 1999. Tables of linear congruential generators of different sizes and good lattice structure. Math. Comput 68, 225 (1999), 249–260. Pierre L’Ecuyer and Jacinthe Granger-Piché. 2003. Combined generators with components from different families. Mathematics and Computers in Simulation 62, 3 (2003), 395–404. Pierre L’Ecuyer and François Panneton. 2005. Fast random number generators based on linear recurrences modulo 2: overview and comparison. In Proceedings of the 37th Winter Simulation Conference. Winter Simulation Conference, 110–119. Pierre L’Ecuyer and Richard Simard. 2007. TestU01: A C library for empirical testing of random number generators. ACM Trans. Math. Softw. 33, 4, Article 22 (2007). Robert H. Lewis. 2013. Fermat: A Computer Algebra System for Polynomial and Matrix Computation (Version 5.1). (2013). Retrieved January 8, 2014 from http://home.bway.net/lewis/ Rudolf Lidl and Harald Niederreiter. 1994. Introduction to finite fields and their applications. Cambridge University Press, Cambridge. George Marsaglia. 2003. Xorshift RNGs. Journal of Statistical Software 8, 14 (2003), 1–6. Makoto Matsumoto and Takuji Nishimura. 1998. Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator. ACM Trans. Model. Comput. Simul. 8, 1 (1998), 3–30. Harald Niederreiter. 1992. Random number generation and quasi-Monte Carlo methods. CBMS-NSF regional conference series in Appl. Math., Vol. 63. SIAM. François Panneton. 2004. Construction d’ensembles de points basé sur une récurrence linéaire dans un corps fini de caractéristique 2 pour la simulation Monte Carlo et l’intégration quasi-Monte Carlo. Ph.D. Dissertation. Université de Montréal. François Panneton and Pierre L’Ecuyer. 2005. On the xorshift random number generators. ACM Trans. Model. Comput. 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arXiv:1707.09652v1 [math.GR] 30 Jul 2017 Choosing elements from finite fields Michael Vaughan-Lee November 2012 1 Introduction Graham Higman wrote two immensely important and influential papers on enumerating p-groups in the late 1950s. The papers were entitled Enumerating p-groups I and II, and were published in the Proceedings of the London Mathematical Society in 1960 (see [1] and [2]). In these two papers Higman proved that for any given n, the function f (pn ) enumerating the number of p-groups of order pn is bounded by a polynomial in p, and he formulated his famous PORC conjecture concerning the form of the function f (pn ). He conjectured that for each n there is an integer N (depending on n) such that for p in a fixed residue class modulo N the function f (pn ) is a polynomial in p. For example, for p ≥ 5 the number of groups of order p6 is 3p2 + 39p + 344 + 24 gcd(p − 1, 3) + 11 gcd(p − 1, 4) + 2 gcd(p − 1, 5). (See [3].) So for p ≥ 5, f (p6 ) is one of 8 polynomials in p, with the choice of polynomial depending on the residue class of p modulo 60. The number of groups of order p6 is Polynomial On Residue Classes. As evidence in support of his PORC conjecture Higman proved that, for any given n, the function enumerating the number of p-class 2 groups of order pn is a PORC function of p. He obtained this result as a corollary to a very general theorem about vector spaces acted on by the general linear group. As another corollary to this general theorem, he also proved that for any given n the function enumerating the number of algebras of dimension n over the field of q elements is a PORC function of q. A key step in Higman’s proof of these results is Theorem 2.2.2 from [2]. Theorem 1 (Higman [2]) The number of ways of choosing a finite number of elements from Fqn subject to a finite number of monomial equations and inequalities between them and their conjugates over Fq , considered as a function of q, is PORC. 1 The statement of this theorem probably requires some explanation! Here we are choosing elements x1 , x2 , . . . , xk (say) from the finite field Fqn (where q is a prime power) subject to a finite set of equations and non-equations of the form xn1 1 xn2 2 . . . xnk k = 1 and xn1 1 xn2 2 . . . xnk k 6= 1, where n1 , n2 , . . . , nk are integer polynomials in the Frobenius automorphism x 7→ xq of Fqn . Higman calls these equations and non-equations monomial. For example, suppose we want to choose x1 , x2 ∈ Fqn such that x1 is the root of an irreducible quadratic over Fq and such that x22 is the product of the roots of this quadratic. Then we require that x1 and x2 satisfy 2 −1 x1q −2 = 1, x1q−1 6= 1, xq+1 1 x2 = 1. (1) 2 The equation x1q −1 = 1 guarantees that x1 is the root of a quadratic over Fq , and the non-equation x1q−1 6= 1 guarantees that x1 ∈ / Fq so that the quadratic is irreducible. The other root of the quadratic is then xq1 , so the last equation guarantees that x22 is the product of the roots. To make sure q n −1 q n −1 = 1, x2 = 1. Higman’s that x1 , x2 ∈ Fqn , we also require that x1 theorem is that the function enumerating the number of solutions to (1) in Fqn is a PORC function of q. In this note we give very precise information about the exact form of the PORC functions needed to enumerate the number of solutions to a set of monomial equations. Higman’s proof of his Theorem 2.2.2 involves five pages of homological algebra. A shorter more elementary proof can be found in [4]. The proof in [4] shows that one way to calculate the number of solutions to a set of monomial equations is to write the equations as the rows of a matrix. So we represent the equations 2 −1 x1q n −1 q −2 = 1, xq+1 1 x2 = 1, x1 n −1 = 1, xq2 =1 by the matrix  q2 − 1 0  q+1 −2  .  n  q −1 0  0 qn − 1  For any given value of q the matrix becomes an integer matrix, and it is shown in [4] that the number of solutions is the product of the elementary 2 divisors in the Smith normal form of this integer matrix. To obtain the number of solutions to (1) in Fqn we subtract the number of solutions to the equations n q n −1 −2 x1q−1 = 1, xq+1 = 1, xq2 −1 = 1. 1 x2 = 1, x1 The number of solutions to these equations is just the product of the elementary divisors in the Smith normal form of the matrix   q−1 0  q+1 −2  .  n  q −1 0  0 qn − 1 (In [4] q is assumed to be prime, but the proof is still valid when q is a prime power.) Matrices used in this way to represent a set of monomial equations have entries which are integer polynomials in q. The columns correspond to the unknowns we are solving for, and since there will always be rows (q n − 1, 0, 0, . . . , 0), (0, q n − 1, 0, . . . , 0), . . . , (0, . . . , 0, q n − 1) corresponding to the requirement that the unknowns are elements in Fqn , it follows that the rank of one of these matrices is the number of columns. So the product of the elementary divisors in the Smith normal form of one of these matrices is the greatest common divisor of the k × k minors, where k is the number of columns. These k × k minors are integer polynomials in q, and it is proved in [4] that the greatest common divisor of a set of integer polynomials in q is a PORC function of q. There is some ambiguity about what “the greatest common divisor of a set of polynomials” means here. Suppose we have some integer polynomials f1 (q), f2 (q), . . . , fs (q). For any given value of q these polynomials evaluate to integers, and by “greatest common divisor of the polynomials” we actually mean “greatest common divisor of the values of the polynomials at q”. It is this integer valued function of q which we claim is PORC, and it turns out that we can be quite precise about the form that this PORC function takes. Theorem 2 The greatest common divisor of a set of integer polynomials in q can be expressed in the form df where f is an integer polynomial in q and where r X d=α+ αi gcd(q − ni , mi ) i=1 for some rational numbers α, α1 , α2 , . . . , αr , some integers m1 , m2 , . . . , mr with mi > 1 for all i, and some integers n1 , n2 , . . . , nr with 0 < ni < mi for all i. 3 Corollary 3 The number of ways of choosing a finite number of elements from Fqn subject to a finite number of monomial equations and inequalities between them and their conjugates over Fq can be expressed as a linear combination of terms of the form df , where f and d are as described in Theorem 2. 2 Choosing field elements To make this note self contained, we give a proof here that we can use a matrix to represent a set of monomial equations over a finite field, and that the number of solutions to the equations is the product of the elementary divisors in the Smith normal form of the matrix. So suppose we have a set of monomial equations in unknowns x1 , x2 , . . . , xk , and suppose that we want to find the number of solutions to these equations in the field Fqn . We represent the equations in a matrix A with k columns, with a row (n1 , n2 , . . . , nk ) for each monomial equation xn1 1 xn2 2 . . . xnk k = 1. We also add in rows (q n − 1, 0, 0, . . . , 0), (0, q n − 1, 0, . . . , 0), . . . , (0, 0, . . . , 0, q n − 1) corresponding to the requirement that x1 , x2 , . . . , xk ∈ Fqn . Note that the entries in the matrix A are integer polynomials in q. We now take a particular value for q so that the matrix becomes a matrix with integer entries. Let ω be a primitive element in Fqn , and write xi = ω mi for i = 1, 2, . . . , k , taking the exponents mi as elements in Zqn −1 . Then a row (β1 , β2 , . . . , βk ) in the matrix A corresponds to a relation β1 m1 + β2 m2 + . . . + βk mk = 0 which we require the exponents mi to satisfy. The matrix A can be reduced to Smith normal form over Z by elementary row and column operations. As we apply these operations, the relations encoded in the matrix change. But we show that at each step the number of solutions to the relations stays constant. This is clear for elementary row operations, since an elementary row operation replaces the relations by an equivalent set of relations. So we need to consider the effect of elementary column operations. We can consider the k-tuples (m1 , m2 , . . . , mk ) as elements in the additive group G = Zqn −1 × Zqn −1 × . . . × Zqn −1 . Let A be one of these relation matrices, and let B be the matrix obtained from A after applying an elementary column operation. For each such operation we define an automorphism σ of G with the property that g ∈ G satisfies the relations given by the rows of A if and only if gσ satisfies the relations given by the rows of B. This shows that 4 the number of elements in G satisfying the relations given by A is the same as the number of elements in G satisfying the relations given by B. If the elementary column operation swaps two columns of A then we let σ be the automorphism which swaps the corresponding entries in (m1 , m2 , . . . , mk ), and if the elementary column operation multiplies a column by −1 we let σ be the automorphism which multiplies the corresponding entry in (m1 , m2 , . . . , mk ) by −1. Finally, if the elementary column operation subtracts α times column j from column i, then we let σ be the automorphism which leaves all the entries in (m1 , m2 , . . . , mk ) fixed except for the j-th entry, which it replaces by mj + αmi . The argument above shows that the number of g ∈ G satisfying the original set of relations given by the rows of A is the same as the number of g ∈ G satisfying the relations given by the Smith normal form A. If the elementary divisors in the Smith normal form are d1 , d2 , . . . , dk , then (m1 , m2 , . . . , mk ) is a solution to these equations if and only if d1 m1 = d2 m2 = . . . = dk mk = 0. Provided we can show that di \|q n − 1 for all i, this shows that the number of solutions is d1 d2 . . . dk , as claimed. If A is one of these relation matrices with k columns, then the rows of A are elements in the free Z-module F = Zk . We let R(A) denote the Zsubmodule of F generated by the rows of A. Our claim that di \|q n − 1 for all i amounts to the claim that (q n − 1)F ≤ R(S), where S is the Smith normal form of our initial relation matrix. The Smith normal form is obtained from the initial matrix by a sequence of elementary row and column operations, and we show that (q n − 1)F ≤ R(B) for all the matrices B generated in this sequence. Let A be the starting matrix. Then it contains rows (q n − 1, 0, 0, . . . , 0), (0, q n − 1, 0, . . . , 0), . . . , (0, 0, . . . , 0, q n − 1), so it is clear that (q n −1)F ≤ R(A). Suppose that at some intermediate stage in the reduction of A to Smith normal form we have two matrices B and C, where C is obtained from B by an elementary row operation or an elementary column operation. We assume by induction that (q n − 1)F ≤ R(B), and we show that this implies that (q n − 1)F ≤ R(C). This is clear if C is obtained from B by an elementary row operation, since then R(B) = R(C). So consider the case when C is obtained from B by an elementary column operation. This column operation corresponds to an automorphism σ of F , and if r is a row of B then the corresponding row of C is rσ. So R(C) = 5 R(B)σ, and the fact that (q n −1)F is a characteristic submodule of F implies that (q n − 1)F ≤ R(C). This completes the proof that the number of solutions to the relations given by the rows of the matrix is equal to the product of the elementary divisors in the Smith normal form. As mentioned in the introduction, the product of the elementary divisors in the Smith normal form of an integer matrix with k columns and rank k is the greatest common divisor of the k ×k minors. In the situation we are concerned with, these minors are integer polynomials in q. So the number of solutions to our monomial equations is the greatest common divisor of a set of integer polynomials in q. More precisely, we have a set of integer polynomials in q, and for any given value of q the number of solutions to our monomial equations is the greatest common divisor over Z of the values of these polynomials at q. 3 Proof of Theorem 2 Let f1 (q), f2 (q), . . . , fs (q) be a set of integer polynomials in q. We want to compute the function whose value at q is the greatest common divisor of the integers f1 (q), f2 (q), . . . , fs (q). In this section there is no requirement that q be a prime power, and to make this clear we define a function h : Z → Z by setting h(x) = gcd(f1 (x), f2 (x), . . . , fs (x)) for x ∈ Z. It is the function h we want to compute. As mentioned above, there is some ambiguity about what we mean by “the greatest common divisor of f1 (x), f2 (x), . . . , fs (x)”. We now exploit this ambiguity, and treat x as an indeterminate and treat f1 (x), f2 (x), . . . , fs (x) as elements of the Euclidean domain Q[x]. We can use the Euclidean algorithm to compute the greatest common divisor f (x) of f1 (x), f2 (x), . . . , fs (x) in Q[x] and we can take f (x) to be a primitive polynomial in Z[x]. We then obtain polynomials g1 , g2 , . . . , gs ∈ Q[x] such that f1 g1 + f2 g2 + . . . + fs gs = f. Let m be the least common multiple of the denominators of the coefficients in g1 , g2 , . . . , gs . Then for any given value of x in Z, the greatest common divisor of the integers f1 (x), f2 (x), . . . , fs (x) is df (x) for some d dividing m. Furthermore, as a function of x, the value of d at x depends only on the 6 residue class of x modulo m. We show that we can express d(x) in the form α+ r X αi gcd(x − ni , mi ) i=1 described in the statement of Theorem 2. Furthermore we show that we can take the integers mi to be of the form dmi for some square free divisors di of m with di < m. This shows that h(x) = d(x)f (x) has the form described in Theorem 2. If m = 1 then d = 1 for all x, and we are done. So suppose that m > 1 and let S be the set of prime factors of m. For each subset T ⊂ S let Y dT = p, p∈T and consider the function k : Z → Z defined by X m k(x) = (−1)\|T \| gcd(x, ). dT T ⊂S Clearly the value of k at any given value of x depends only on the residue class of x modulo m. First consider the case when x = m. X Y m 1 k(m) = (−1)\|T \| = m (1 − ) 6= 0. dT p T ⊂S p∈S Next suppose that 1 ≤ x < m. Then there is at least one p ∈ S with the property that the power of p dividing x is less than the power of p dividing m. Pick one such p and let U = S\{p}. Then X m m \|T \| gcd(x, ) − gcd(x, k(x) = (−1) ) = 0, dT pdT T ⊂U since gcd(x, dmT ) = gcd(x, pdmT ) for all T ⊂ U. So if we let c = k(m) then 1 k(x) takes values 0, 0, . . . , 0, 1 as x takes values 1, 2, . . . , m modulo m. It c follows that if 0 < a < m then 1c k(x − a) takes values 0, . . . , 0, 1, 0, . . . , 0 as x takes values 1, 2, . . . , m modulo m (with the 1 in the ath place). So we can express d(x) as a rational linear combination of the functions k(x − a) for 0 ≤ a < m. This implies that we can express d(x) as a rational linear combination of functions of the form gcd(x − ni , mi ) where mi = dmT for some 7 T ⊂ S. If mi = 1 then we can replace gcd(x − ni , mi ) by the constant 1. Also, we can assume 0 ≤ ni < mi . Finally, using the fact that m i −1 X gcd(x − a, mi ) a=0 is a constant function, we can assume that 0 < ni < mi , provided we add a constant term into our expression for d(x). This completes the proof of Theorem 2. Note that the proof shows that we can assume that the denominators of the rational coefficients which appear in the expression for d(x) divide the constant k(m). References [1] G. Higman, Enumerating p-groups. I: Inequalities, Proc. London Math. Soc. (3) 10 (1960), 24–30. [2] G. Higman, Enumerating p-groups. II: Problems whose solution is PORC, Proc. London Math. Soc. (3) 10 (1960), 566–582. [3] M.F. Newman, E.A. O’Brien, and M.R. Vaughan-Lee, Groups and nilpotent Lie rings whose order is the sixth power of a prime, J. Algebra 278 (2004), 383–401. [4] Michael Vaughan-Lee, On Graham Higman’s famous PORC paper, Internat. J. Group Theory 1 (2012), 65–79. 8	4
Deep Layer Aggregation Fisher Yu Dequan Wang Evan Shelhamer Trevor Darrell UC Berkeley arXiv:1707.06484v2 [cs.CV] 4 Jan 2018 Abstract Visual recognition requires rich representations that span levels from low to high, scales from small to large, and resolutions from fine to coarse. Even with the depth of features in a convolutional network, a layer in isolation is not enough: compounding and aggregating these representations improves inference of what and where. Architectural efforts are exploring many dimensions for network backbones, designing deeper or wider architectures, but how to best aggregate layers and blocks across a network deserves further attention. Although skip connections have been incorporated to combine layers, these connections have been “shallow” themselves, and only fuse by simple, one-step operations. We augment standard architectures with deeper aggregation to better fuse information across layers. Our deep layer aggregation structures iteratively and hierarchically merge the feature hierarchy to make networks with better accuracy and fewer parameters. Experiments across architectures and tasks show that deep layer aggregation improves recognition and resolution compared to existing branching and merging schemes. Dense Connections Feature Pyramids + Deep Layer Aggregation Figure 1: Deep layer aggregation unifies semantic and spatial fusion to better capture what and where. Our aggregation architectures encompass and extend densely connected networks and feature pyramid networks with hierarchical and iterative skip connections that deepen the representation and refine resolution. riers, different blocks or modules have been incorporated to balance and temper these quantities, such as bottlenecks for dimensionality reduction [29, 39, 17] or residual, gated, and concatenative connections for feature and gradient propagation [17, 38, 19]. Networks designed according to these schemes have 100+ and even 1000+ layers. Nevertheless, further exploration is needed on how to connect these layers and modules. Layered networks from LeNet [26] through AlexNet [23] to ResNet [17] stack layers and modules in sequence. Layerwise accuracy comparisons [11, 48, 35], transferability analysis [44], and representation visualization [48, 46] show that deeper layers extract more semantic and more global features, but these signs do not prove that the last layer is the ultimate representation for any task. In fact, skip connections have proven effective for classification and regression [19, 4] and more structured tasks [15, 35, 30]. Aggregation, like depth and width, is a critical dimension of architecture. In this work, we investigate how to aggregate layers to better fuse semantic and spatial information for recognition and localization. Extending the “shallow” skip connections of current approaches, our aggregation architectures incor- 1. Introduction Representation learning and transfer learning now permeate computer vision as engines of recognition. The simple fundamentals of compositionality and differentiability give rise to an astonishing variety of deep architectures [23, 39, 37, 16, 47]. The rise of convolutional networks as the backbone of many visual tasks, ready for different purposes with the right task extensions and data [14, 35, 42], has made architecture search a central driver in sustaining progress. The ever-increasing size and scope of networks now directs effort into devising design patterns of modules and connectivity patterns that can be assembled systematically. This has yielded networks that are deeper and wider, but what about more closely connected? More nonlinearity, greater capacity, and larger receptive fields generally improve accuracy but can be problematic for optimization and computation. To overcome these bar1 porate more depth and sharing. We introduce two structures for deep layer aggregation (DLA): iterative deep aggregation (IDA) and hierarchical deep aggregation (HDA). These structures are expressed through an architectural framework, independent of the choice of backbone, for compatibility with current and future networks. IDA focuses on fusing resolutions and scales while HDA focuses on merging features from all modules and channels. IDA follows the base hierarchy to refine resolution and aggregate scale stage-bystage. HDA assembles its own hierarchy of tree-structured connections that cross and merge stages to aggregate different levels of representation. Our schemes can be combined to compound improvements. Our experiments evaluate deep layer aggregation across standard architectures and tasks to extend ResNet [16] and ResNeXt [41] for large-scale image classification, finegrained recognition, semantic segmentation, and boundary detection. Our results show improvements in performance, parameter count, and memory usage over baseline ResNet, ResNeXT, and DenseNet architectures. DLA achieve stateof-the-art results among compact models for classification. Without further architecting, the same networks obtain stateof-the-art results on several fine-grained recognition benchmarks. Recast for structured output by standard techniques, DLA achieves best-in-class accuracy on semantic segmentation of Cityscapes [8] and state-of-the-art boundary detection on PASCAL Boundaries [32]. Deep layer aggregation is a general and effective extension to deep visual architectures. 2. Related Work We review architectures for visual recognition, highlight key architectures for the aggregation of hierarchical features and pyramidal scales, and connect these to our focus on deep aggregation across depths, scales, and resolutions. The accuracy of AlexNet [23] for image classification on ILSVRC [34] signalled the importance of architecture for visual recognition. Deep learning diffused across vision by establishing that networks could serve as backbones, which broadcast improvements not once but with every better architecture, through transfer learning [11, 48] and metaalgorithms for object detection [14] and semantic segmentation [35] that take the base architecture as an argument. In this way GoogLeNet [39] and VGG [39] improved accuracy on a variety of visual problems. Their patterned components prefigured a more systematic approach to architecture. Systematic design has delivered deeper and wider networks such as residual networks (ResNets) [16] and highway networks [38] for depth and ResNeXT [41] and FractalNet [25] for width. While these architectures all contribute their own structural ideas, they incorporated bottlenecks and shortened paths inspired by earlier techniques. Network-innetwork [29] demonstrated channel mixing as a technique to fuse features, control dimensionality, and go deeper. The companion and auxiliary losses of deeply-supervised networks [27] and GoogLeNet [39] showed that it helps to keep learned layers and losses close. For the most part these architectures derive from innovations in connectivity: skipping, gating, branching, and aggregating. Our aggregation architectures are most closely related to leading approaches for fusing feature hierarchies. The key axes of fusion are semantic and spatial. Semantic fusion, or aggregating across channels and depths, improves inference of what. Spatial fusion, or aggregating across resolutions and scales, improves inference of where. Deep layer aggregation can be seen as the union of both forms of fusion. Densely connected networks (DenseNets) [19] are the dominant family of architectures for semantic fusion, designed to better propagate features and losses through skip connections that concatenate all the layers in stages. Our hierarchical deep aggregation shares the same insight on the importance of short paths and re-use, and extends skip connections with trees that cross stages and deeper fusion than concatenation. Densely connected and deeply aggregated networks achieve more accuracy as well as better parameter and memory efficiency. Feature pyramid networks (FPNs) [30] are the dominant family of architectures for spatial fusion, designed to equalize resolution and standardize semantics across the levels of a pyramidal feature hierarchy through top-down and lateral connections. Our iterative deep aggregation likewise raises resolution, but further deepens the representation by nonlinear and progressive fusion. FPN connections are linear and earlier levels are not aggregated more to counter their relative semantic weakness. Pyramidal and deeply aggregated networks are better able to resolve what and where for structured output tasks. 3. Deep Layer Aggregation We define aggregation as the combination of different layers throughout a network. In this work we focus on a family of architectures for the effective aggregation of depths, resolutions, and scales. We call a group of aggregations deep if it is compositional, nonlinear, and the earliest aggregated layer passes through multiple aggregations. As networks can contain many layers and connections, modular design helps counter complexity by grouping and repetition. Layers are grouped into blocks, which are then grouped into stages by their feature resolution. We are concerned with aggregating the blocks and stages. 3.1. Iterative Deep Aggregation Iterative deep aggregation follows the iterated stacking of the backbone architecture. We divide the stacked blocks of the network into stages according to feature resolution. Deeper stages are more semantic but spatially coarser. Skip connections from shallower to deeper stages merge scales Block OUT Existing Proposed Stage Aggregation Node OUT IN OUT IN (a) No aggregation IN (b) Shallow aggregation OUT IN IN (d) Tree-structured aggregation (c) Iterative deep aggregation OUT OUT IN (e) Reentrant aggregation (f) Hierarchical deep aggregation Figure 2: Different approaches to aggregation. (a) composes blocks without aggregation as is the default for classification and regression networks. (b) combines parts of the network with skip connections, as is commonly used for tasks like segmentation and detection, but does so only shallowly by merging earlier parts in a single step each. We propose two deep aggregation architectures: (c) aggregates iteratively by reordering the skip connections of (b) such that the shallowest parts are aggregated the most for further processing and (d) aggregates hierarchically through a tree structure of blocks to better span the feature hierarchy of the network across different depths. (e) and (f) are refinements of (d) that deepen aggregation by routing intermediate aggregations back into the network and improve efficiency by merging successive aggregations at the same depth. Our experiments show the advantages of (c) and (f) for recognition and resolution. and resolutions. However, the skips in existing work, e.g. FCN [35], U-Net [33], and FPN [30], are linear and aggregate the shallowest layers the least, as shown in Figure 2(b). We propose to instead progressively aggregate and deepen the representation with IDA. Aggregation begins at the shallowest, smallest scale and then iteratively merges deeper, larger scales. In this way shallow features are refined as they are propagated through different stages of aggregation. Figure 2(c) shows the structure of IDA. The iterative deep aggregation function I for a series of layers x1 , ..., xn with increasingly deeper and semantic information is formulated as ( x1 if n = 1 I(x1 , ..., xn ) = (1) I(N (x1 , x2 ), ..., xn ) otherwise, where N is the aggregation node. 3.2. Hierarchical Deep Aggregation Hierarchical deep aggregation merges blocks and stages in a tree to preserve and combine feature channels. With HDA shallower and deeper layers are combined to learn richer combinations that span more of the feature hierarchy. While IDA effectively combines stages, it is insufficient for fusing the many blocks of a network, as it is still only sequential. The deep, branching structure of hierarchical aggregation is shown in Figure 2(d). Having established the general structure of HDA we can improve its depth and efficiency. Rather than only routing intermediate aggregations further up the tree, we instead feed the output of an aggregation node back into the backbone as the input to the next sub-tree, as shown in Figure 2(e). This propagates the aggregation of all previous blocks instead of the preceding block alone to better preserve features. For efficiency, we merge aggregation nodes of the same depth (combining the parent and left child), as shown in Figure 2(f). The hierarchical deep aggregation function Tn , with depth n, is formulated as n n Tn (x) = N (Rn−1 (x), Rn−2 (x), ..., R1n (x), Ln1 (x), Ln2 (x)), (2) where N is the aggregation node. R and L are defined as Ln2 (x) = B(Ln1 (x)), Ln1 (x) = B(R1n (x)), ( Tm (x) if m = n − 1 n Rm (x) = n Tm (Rm+1 (x)) otherwise, where B represents a convolutional block. 3.3. Architectural Elements Aggregation Nodes The main function of an aggregation node is to combine and compress their inputs. The nodes learn to select and project important information to maintain Hierarchical Deep Aggregation Iterative Deep Aggregation Aggregation Node Downsample 2x Conv Block OUT IN Figure 3: Deep layer aggregation learns to better extract the full spectrum of semantic and spatial information from a network. Iterative connections join neighboring stages to progressively deepen and spatially refine the representation. Hierarchical connections cross stages with trees that span the spectrum of layers to better propagate features and gradients. the same dimension at their output as a single input. In our architectures IDA nodes are always binary, while HDA nodes have a variable number of arguments depending on the depth of the tree. Although an aggregation node can be based on any block or layer, for simplicity and efficiency we choose a single convolution followed by batch normalization and a nonlinearity. This avoids overhead for aggregation structures. In image classification networks, all the nodes use 1×1 convolution. In semantic segmentation, we add a further level of iterative deep aggregation to interpolate features, and in this case use 3×3 convolution. As residual connections are important for assembling very deep networks, we can also include residual connections in our aggregation nodes. However, it is not immediately clear that they are necessary for aggregation. With HDA, the shortest path from any block to the root is at most the depth of the hierarchy, so diminishing or exploding gradients may not appear along the aggregation paths. In our experiments, we find that residual connection in node can help HDA when the deepest hierarchy has 4 levels or more, while it may hurt for networks with smaller hierarchy. Our base aggregation, i.e. N in Equation 1 and 2, is defined by: different backbones. Our architectures make no requirements of the internal structure of the blocks and stages. The networks we instantiate in our experiments make use of three types of residual blocks [17, 41]. Basic blocks combine stacked convolutions with an identity skip connection. Bottleneck blocks regularize the convolutional stack by reducing dimensionality through a 1×1 convolution. Split blocks diversify features by grouping channels into a number of separate paths (called the cardinality of the split). In this work, we reduce the ratio between the number of output and intermediate channels by half for both bottleneck and split blocks, and the cardinality of our split blocks is 32. Refer to the cited papers for the exact details of these blocks. 4. Applications We now design networks with deep layer aggregation for visual recognition tasks. To study the contribution of the aggregated representation, we focus on linear prediction without further machinery. Our results do without ensembles for recognition and context modeling or dilation for resolution. Aggregation of semantic and spatial information matters for classification and dense prediction alike. 4.1. Classification Networks N (x1 , ..., xn ) = σ(BatchNorm( X Wi xi + b)), (3) i where σ is the non-linear activation, and wi and b are the weights in the convolution. If residual connections are added, the equation becomes N (x1 , ..., xn ) = σ(BatchNorm( X Wi xi + b) + xn ). (4) i Note that the order of arguments for N does matter and should follow Equation 2. Blocks and Stages Deep layer aggregation is a general architecture family in the sense that it is compatible with Our classification networks augment ResNet and ResNeXT with IDA and HDA. These are staged networks, which group blocks by spatial resolution, with residual connections within each block. The end of every stage halves resolution, giving six stages in total, with the first stage maintaining the input resolution while the last stage is 32× downsampled. The final feature maps are collapsed by global average pooling then linearly scored. The classification is predicted as the softmax over the scores. We connect across stages with IDA and within and across stages by HDA. These types of aggregation are easily combined by sharing aggregation nodes. In this case, we only need to change the root node at each hierarchy by combin- Iterative Deep Aggregation 2s OUT Aggregation Node Upsample 2x 2s 4s 4s 8s Stage 2s IN 2s 2s 4s 8s 16s 4s 8s 16s 32s Figure 4: Interpolation by iterative deep aggregation. Stages are fused from shallow to deep to make a progressively deeper and higher resolution decoder. ing Equation 1 and 2. Our stages are downsampled by max pooling with size 2 and stride 2. The earliest stages have their own structure. As in DRN [46], we replace max pooling in stages 1–2 with strided convolution. The stage 1 is composed of a 7×7 convolution followed by a basic block. The stage 2 is only a basic block. For all other stages, we make use of combined IDA and HDA on the backbone blocks and stages. For a direct comparison of layers and parameters in different networks, we build networks with a comparable number of layers as ResNet-34, ResNet-50 and ResNet-101. (The exact depth varies as to keep our novel hierarchical structure intact.) To further illustrate the efficiency of DLA for condensing the representation, we make compact networks with fewer parameters. Table 1 lists our networks and Figure 3 shows a DLA architecture with HDA and IDA. 4.2. Dense Prediction Networks Semantic segmentation, contour detection, and other image-to-image tasks can exploit the aggregation to fuse local and global information. The conversion from classification DLA to fully convolutional DLA is simple and no different than for other architectures. We make use of interpolation and a further augmentation with IDA to reach the necessary output resolution for a task. IDA for interpolation increases both depth and resolution by projection and upsampling as in Figure 4. All the projection and upsampling parameters are learned jointly during the optimization of the network. The upsampling steps are initialized to bilinear interpolation and can then be learned as in [35]. We first project the outputs of stages 3–6 to 32 channels and then interpolate the stages to the same resolution as stage 2. Finally, we iteratively aggregate these stages to learn a deep fusion of low and high level features. While having the same purpose as FCN skip connections [35], hypercolumn features [15], and FPN top-down connections [30], our aggregation differs in approach by going from shallow-todeep to further refine features. Note that we use IDA twice in this case: once to connect stages in the backbone network and again to recover resolution. 5. Results We evaluate our deep layer aggregation networks on a variety of tasks: image classification on ILSVRC, several kinds of fine-grained recognition, and dense prediction for semantic segmentation and contour detection. After establishing our classification architecture, we transfer these networks to the other tasks with little to no modification. DLA improves on or rivals the results of special-purpose networks. 5.1. ImageNet Classification We first train our networks on the ImageNet 2012 training set [34]. Similar to ResNet [16], training is performed by SGD for 120 epochs with momentum 0.9, weight decay 10−4 and batch size 256. We start the training with learning rate 0.1, which is reduced by 10 every 30 epochs. We use scale and aspect ratio augmentation [41], but not color perturbation. For fair comparison, we also train the ResNet models with the same training procedure. This leads to slight improvements over the original results. We evaluate the performance of trained models on the ImageNet 2012 validation set. The images are resized so that the shorter side has 256 pixels. Then central 224×224 crops are extracted from the images and fed into networks to measure prediction accuracy. DLA vs. ResNet compares DLA networks to ResNets with similar numbers of layers and the same convolutional blocks as shown in Figure 5. We find that DLA networks can achieve better performance with fewer parameters. DLA-34 and ResNet-34 both use basic blocks, but DLA-34 has about 30% fewer parameters and ∼ 1 point of improvement in top-1 error rate. We usually expect diminishing returns of performance of deeper networks. However, our results show that compared to ResNet-50 and ResNet-101, DLA networks can still outperform the baselines significantly with fewer parameters. DLA vs. ResNeXt shows that DLA is flexible enough to use different convolutional blocks and still have advantage in accuracy and parameter efficiency as shown in Figure 5. Our models based on the split blocks have much fewer parameters but they still have similar performance with ResNeXt models. For example, DLA-X-102 has nearly the half number of parameters compared to ResNeXt-101, but the error rate difference is only 0.2%. DLA vs. DenseNet compares DLA with the dominant architecture for semantic fusion and feature re-use. DenseNets are composed of dense blocks that aggregate all of their layers by concatenation and transition blocks that reduce dimensionality for tractability. While these networks can Name DLA-34 DLA-48-C DLA-60 DLA-102 DLA-169 DLA-X-48-C DLA-X-60-C DLA-X-60 DLA-X-102 Block Basic Bottleneck Bottleneck Bottleneck Bottleneck Split Split Split Split Stage 1 16 16 16 16 16 16 16 16 16 Stage 2 32 32 32 32 32 32 32 32 32 Stage 3 1-64 1-64 1-128 1-128 2-128 1-64 1-64 1-128 1-128 Stage 4 2-128 2-64 2-256 3-256 3-256 2-64 2-64 2-256 3-256 Stage 5 2-256 2-128 3-512 4-512 5-512 2-128 3-128 3-512 4-512 Stage 6 1-512 1-256 1-1024 1-1024 1-1024 1-256 1-256 1-1024 1-1024 Table 1: Deep layer aggregation networks for classification. Stages 1 and 2 show the number of channels n while further stages show d-n where d is the aggregation depth. Models marked with “-C” are compact and only have ∼1 million parameters. aggressively reduce depth and parameter count by feature reuse, concatenation is a memory-intensive fusion operation. DLA achieves higher accuracy with lower memory usage because the aggregation node fan-in size is log of the total number of convolutional blocks in HDA. Compact models have received a lot of attention due to the limited capabilities of consumer hardware for running convolutional networks. We design parameter constrained DLA networks to study how efficiently DLA can aggregate and re-use features. We compare to SqueezeNet [20], which shares a block design similar to our own. Table 2 shows that DLA is more accurate with the same number of parameters. Furthermore DLA is more computationally efficient by operation count. SqueezNet-A SqueezNet-B DLA-46-C DLA-46-C DLA-X-60-C Top-1 42.5 39.6 36.8 34.0 32.5 Top-5 19.7 17.5 15.0 13.7 12.0 Params 1.2M 1.2M 1.3M 1.1M 1.3M FMAs 1.70B 0.72B 0.58B 0.53B 0.59B Table 2: Comparison with compact models. DLA is more accurate at the same number of parameters while inference takes fewer operations (counted by fused multiply-adds). 5.2. Fine-grained Recognition We use the same training procedure for all of fine-grained experiments. The training is performed by SGD with a minibatch size of 64, while the learning rate starts from 0.01 and is then divided by 10 every 50 epochs, for 110 epochs in total. The other hyperparameters are fixed to their settings for ImageNet classification. In order to mitigate over-fitting, we carry out the following data augmentation: Inception-style Bird Car Plane Food ILSVRC #Class 200 196 102 101 1000 #Train (per class) 5994 (30) 8144 (42) 6667 (67) 75750 (750) 1,281,167 (1281) #Test (per class) 5794 (29) 8041 (41) 3333 (33) 25250 (250) 100,000 (100) Table 3: Statistics for fine-grained recognition datasets. Compared to generic, large-scale classification, these tasks contain more specific classes with fewer training instances. scale and aspect ratio variation [39], AlexNet-style PCA color noise[23], and the photometric distortions of [18]. We evaluate our models on various fine-grained recognition datasets: Bird (CUB) [40], Car [22], Plane [31], and Food [5]. The statistics of these datasets can be found in Table 3, while results are shown in Figure 6. For fair comparison, we follow the experimental setup of [9]: we randomly crop 224×224 in resized 256×256 images for [5] and 448×448 in resized 512×512 for the rest of datasets, while keeping 224×224 input size for original VGGNet. Our results improve or rival the state-of-the-art without further annotations or specific modules for fine-grained recognition. In particular, we establish new state-of-the-arts results on Car, Plane, and Food datasets. Furthermore, our models are competitive while having only several million parameters. However, our results are not better than stateof-the-art on Birds, although note that this dataset has fewer instances per class so further regularization might help. 5.3. Semantic Segmentation We report experiments for urban scene understanding on CamVid [6] and Cityscapes [8]. Cityscapes is a largerscale, more challenging dataset for comparison with other methods while CamVid is more convenient for examining ResNet ResNeXt DenseNet DLA DLA-X 28 Top-1 Error Rate 27 26 25 24 23 22 21 20 20 40 60 # Params (Million) 80 2 4 6 8 10 12 # Multi-Add (Billion) 14 16 Figure 5: Evaluation of DLA on ILSVRC. DLA/DLA-X have ResNet/ResNeXT backbones respectively. DLA achieves the highest accuracies with fewer parameters and fewer computation. Bird Car 90 86 84 .3 85 .2 70 84 .7 81 .6 78 .4 80 75 92 9 91 2.4 .2 95 84 85.085.1 .4 82 .7 80 .5 .7 88 91 Plane 94 93 94 .0 .9 .1 .5 .0 .6 86 .9 84 84 .1 .7 85 .7 88 .7 84 .2 8 81 2.4 .2 81 .6 79 .2 79 .8 85 .5 89 90. 89 .6 0 .7 87 . .7 8 86 8 82 3.2 .1 74 73 .1 .1 VGGNet Food 9 9 91 92.4 2.9 2.6 .6 92 ResNet Baseline DLA VGGNet Compact ResNet Kernel DLA DLA-X-60-C VGGNet ResNet DLA-34 DLA DLA-60 VGGNet DLA-X-60 ResNet DLA DLA-102 Figure 6: Comparison with state-of-the-art methods on fine-grained datasets. Image classification accuracy on Bird [40], Car [22], Plane [31], and Food [5]. Higher is better. P is the number of parameters in each model. For fair comparison, we calculate the number of parameters for 1000-way classification. V- and R- indicate the base model as VGGNet-16 and ResNet-50, respectively. The numbers of Baseline, Compact [13] and Kernel [9] are directly cited from [9]. ablations. We use the standard mean intersection-over-union (IoU) score [12] as the evaluation metric for both datasets. Our networks are trained only on the training set without the usage of validation or other further data. CamVid has 367 training images, 100 validation images, and 233 test images with 11 semantic categories. We start the training with learning rate 0.01 and divide it by 10 after 800 epochs. The results are shown in Table 8. We find that models with downsampling rate 2 consistenly outperforms those downsampling by 8. We also try to augment the data by randomly rotating the images between [-10, 10] degrees and randomly scaling the images between 0.5 and 2. The final results are significantly better than prior methods. Cityscapes has 2, 975 training images, 500 validation images, and 1, 525 test images with 19 semantic categories. Following previous works [49], we adopt the poly learnepoch−1 0.9 ing rate (1 − total with momentum 0.9 and train the epoch ) model for 500 epochs with batch size 16. The starting learning rate is 0.01 and the crop size is chosen to be 864. We also augment the data by randomly rotating within 10 degrees and scaling between 0.5 and 2. The validation results are shown in 9. Surprisingly, DLA-34 performs very well on this dataset and it is as accurate as DLA-102. It should be noted that fine spatial details do not contribute much for this choice of metric. RefineNet [28] is the strongest network in the same class of methods without the computational costs of additional data, dilation, and graphical models. To make a fair comparison, we evaluate in the same multi-scale fashion as that approach with image scales of [0.5, 0.75, 1, 1.25, 1.5] and sum the predictions. DLA improves by 2+ points. 5.4. Boundary Detection Boundary detection is an exacting task of localization. Although as a classification problem it is only a binary task of whether or not a boundary exists, the metrics require precise spatial accuracy. We evaluate on classic BSDS [1] with multiple human boundary annotations and PASCAL boundaries [32] with boundaries defined by instances masks of select semantic classes. The metrics are accuracies at different thresholds, the optimal dataset scale (ODS) and Method DLA-34 8s DLA-34 2s DLA-102 2s Split Val FCN-8s [35] RefineNet-101 [28] DLA-102 DLA-169 Test mIoU 73.4 74.5 74.4 Method SegNet [2] DeepLab-LFOV [7] Dilation8 [45] FSO [24] DLA-34 8s DLA-34 2s DLA-102 2s 65.3 73.6 75.3 75.9 Table 4: Evaluation on Cityscapes to compare strides on validation and to compare against existing methods on test. DLA is the best-in-class among methods in the same setting. 1 Precision Table 5: Evaluation on CamVid. Higher depth and resolution help. DLA is state-of-the-art. Method SE [10] HED [43] DLA-34 2s DLA-102 2s 0.9 0.8 [F=.80] Human [F=.80] DLA-102 [F=.79] DLA-34 [F=.79] HED [F=.77] DLA-34 4s [F=.76] DLA-34 8s 0.7 0.6 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 mIoU 46.4 61.6 65.3 66.1 66.7 68.5 71.0 DSBD [32] M-DSBD [32] DLA-34 2s DLA-102 2s Train ODS OIS AP BSDS 0.541 0.553 0.642 0.648 0.570 0.585 0.668 0.674 0.486 0.518 0.624 0.623 PASCAL 0.643 0.652 0.743 0.754 0.663 0.678 0.757 0.766 0.650 0.674 0.763 0.752 Recall Figure 7: Precision-recall evaluation on BSDS. DLA is the closest to human performance. Method ODS OIS AP SE [10] DeepEdge [3] DeepContour [36] HED [42] CEDN [43]† UberNet [21] (1-Task)† 0.746 0.753 0.756 0.788 0.788 0.791 0.767 0.772 0.773 0.808 0.804 0.809 0.803 0.807 0.797 0.840 0.821 0.849 DLA-34 8s DLA-34 4s DLA-34 2s DLA-102 2s 0.760 0.767 0.794 0.803 0.772 0.778 0.808 0.813 0.739 0.751 0.787 0.781 Table 6: Evaluation on BSDS († indicates outside data). ODS and OIS are state-of-the-art, but AP suffers due to recall. See Figure 7. more lenient optimal image scale (OIS), as well as average precision (AP). Results are shown in for BSDS in Table 6 and the precision-recall plot of Figure 7 and for PASCAL boundaries in Table 7. To address this task we follow the training procedure of HED [42]. In line with other deep learning methods, we Table 7: Evaluation on PASCAL Boundaries. DLA is stateof-the-art. take the consensus of human annotations on BSDS and only supervise our network with boundaries that three or more annotators agree on. Following [43], we give the boundary labels 10 times weight of the others. For inference we simply run the net forward, and do not make use of ensembles or multi-scale testing. Assessing the role of resolution by comparing strides of 8, 4, and 2 we find that high output resolution is critical for accurate boundary detection. We also find that deeper networks does not continue improving the prediction performance on BSDS. On both BSDS and PASCAL boundaries we achieve state-of-the-art ODS and OIS scores. In contrast the AP on BSDS is surprisingly low, so to understand why we plot the precision-recall curve in Figure 7. Our approach has lower recall, but this is explained by the consensus ground truth not covering all of the individual, noisy boundaries. At the same time it is the closest to human performance. On the other hand we achieve state-of-the-art AP on PASCAL boundaries since it has a single, consistent notion of boundaries. When training on BSDS and transferring to PASCAL boundaries the improvement is minor, but training on PASCAL boundaries itself with ∼ 10× the data delivers more than 10% relative improvement over competing methods. 6. Conclusion Aggregation is a decisive aspect of architecture, and as the number of modules multiply their connectivity is made all the more important. 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It is only used when there are more than 100 layers in the classification network. Three types of convolutional blocks are studied in this paper, as shown in Figure 9, since they are widely used in deep learning literature. Because the convolutional blocks will be combined with additional linear projection in the aggregation nodes, we reduce the ratio of bottleneck blocks from 4 to 2. We compare DLA and DLA-X to other networks in Figure 5 in the submitted paper in terms of network parameters and classification accuracy. DLA includes the networks using residual blocks in ResNet and DLA-X includes those using the block in ResNeXt. For fair comparison, we design DLA and DLA-X networks with similar depth and channels with their counterparts. The ResNet models are ResNet-34, ResNet-50, ResNet-101 and ResNet-152. The corresponding DLA model depths are 34, 60, 102, 169. The ResNeXt models are ResNeXt-50 (32x4d), ResNeXt-101 (32x4d), and ResNeXt-101 (64x4d). The corresponding DLA-X model depths are 60 and 102, while the third DLA-X model double the number of 3 × 3 bottleneck channels, similar to ResNeXt-101 (64x4d). B. Semantic Segmentation We report experiments for semantic segmentation on CamVid and Cityscapes. Table 8 shows a breakdown of the accuracies for the categories. We also add data agumention in the CamVid training, as shown in the third group of Table 8. It includes random rotating the images between -10 and 10 degrees and random scaling between 0.5 and 2. We find the results can be further improved by the augmentation. Table 9 shows the breakdown of categories in Cityscapes on the validation set. We also test the models on multiple scales of the images. This testing procedure is used in evaluating the models on the testing images in the previous works. ReLU BN (b) Plain 3-node Conv Concat ReLU BN Conv Concat 3-Node Conv Block Conv Block 2-Node Conv Block Conv Block (a) 3-level hierarchical aggregation (c) Residual 3-node Figure 8: Illustration of aggregation node architectures. n, 1 x 1, n/2 n/2, 3 x 3, n/2 n/2, 1 x 1, n n, 3 x 3, n n, 3 x 3, n (a) Basic (b) Bottleneck n/32, 1 x 1, n n/32, 3 x 3, n/32 n, 1 x 1, n/32 32 Paths 32 Paths 32 Paths n/32, 1 x 1, n n/32, 3 x 3, n/32 n, 1 x 1, n/32 (c) Split . Tree Sky Car Sign Road Pedestrian Fence Pole Sidewalk Bicyclist mean IoU DLA-60 DLA-102 DLA-169 Building DLA-34 8s DLA-60 8s DLA-34 DLA-60 DLA-102 Data Aug Figure 9: Convolutional blocks used in this paper. Our aggregation architecture is as general as stacking layers, so we can use the building blocks of existing networks. The layer labels indicate output channels, kernel size and input channels. (a) and (b) are derived from [16] and (c) from [41]. No 83.2 83.0 83.2 84.4 84.9 77.2 77.0 76.4 77.7 78.0 91.2 91.4 92.5 92.6 92.5 83.6 84.1 84.6 87.1 86.4 48.8 46.9 52.1 51.4 50.8 94.3 94.1 94.4 95.3 94.9 58.6 58.3 61.5 62.2 62.8 32.0 32.8 29.4 32.1 45.4 27.8 26.0 35.1 36.2 35.7 81.1 81.3 82.0 84.5 83.7 55.4 56.8 57.8 64.1 65.8 66.7 66.5 68.1 69.8 71.0 Yes 86.6 86.6 86.9 79.3 78.8 78.9 92.5 92.2 92.5 90.9 90.3 89.9 55.3 57.9 58.5 96.2 96.5 96.5 65.5 66.7 66.1 48.6 49.6 55.4 37.4 38.7 39.0 86.9 87.9 87.7 66.5 66.7 67.7 73.2 73.8 74.4 Road Sidewalk Building Wall Fence Pole Light Sign Vegetation Terrain Sky Person Rider Car Truck Bus Train Motorcycle Bicycle mean IoU Table 8: Semantic segmentation results on the CamVid dataset. DLA-34 8s DLA-34 DLA-102 DLA-169 97.9 98.0 98.0 98.2 83.2 83.5 84.3 84.8 91.9 92.1 92.3 92.5 47.7 51.0 43.2 45.9 57.7 56.8 56.9 60.0 62.4 64.9 67.2 68.0 68.6 69.6 71.6 72.3 77.3 78.5 80.9 81.1 92.2 92.4 92.5 92.7 60.4 62.9 61.4 61.9 94.8 95.1 94.6 95.1 81.1 81.5 82.7 83.4 59.8 59.6 61.5 63.3 94.1 94.5 94.5 95.3 57.5 59.0 60.3 70.9 76.6 78.4 77.7 80.8 54.2 57.8 53.8 48.1 59.7 62.9 62.2 65.4 76.6 76.9 78.5 79.1 73.4 74.5 74.4 75.7 DLA-34 MS DLA-102 MS DLA-169 MS 98.2 84.7 92.5 54.3 59.5 65.9 71.1 79.5 92.7 64.1 95.3 82.6 61.8 94.7 63.3 83.7 64.6 64.2 77.6 98.5 85.0 92.5 47.1 56.7 66.9 74.4 78.6 93.6 71.7 95.1 85.8 67.4 95.3 55.8 63.5 57.8 68.1 76.1 98.3 85.9 92.8 48.3 61.2 69.0 73.4 82.2 92.9 63.1 95.4 84.2 65.1 95.7 76.3 82.9 49.6 68.5 80.2 76.3 76.1 77.1 Table 9: Performance of DLA on the Cityscapes validation set. s8 indicates the input image is downsampled by 8 in the model output. It is 2 by default. Lower downsampling rate usually leads to higher accuracy. “MS” indicates the models are tested on on multiple scales of the input images.	1
MNRAS 000, 1–13 (2017) Preprint 15 November 2017 Compiled using MNRAS LATEX style file v3.0 Uncertainty quantification for radio interferometric imaging: II. MAP estimation Xiaohao Cai1? , Marcelo Pereyra2? and Jason D. McEwen1? 1 Mullard Space Science Laboratory, University College London (UCL), Surrey RH5 6NT, United Kingdom Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom 2 Maxwell arXiv:1711.04819v1 [astro-ph.IM] 13 Nov 2017 Accepted —. Received —; in original form — ABSTRACT Uncertainty quantification is a critical missing component in radio interferometric imaging that will only become increasingly important as the big-data era of radio interferometry emerges. Statistical sampling approaches to perform Bayesian inference, like Markov Chain Monte Carlo (MCMC) sampling, can in principle recover the full posterior distribution of the image, from which uncertainties can then be quantified. However, for massive data sizes, like those anticipated from the Square Kilometre Array (SKA), it will be difficult if not impossible to apply any MCMC technique due to its inherent computational cost. We formulate Bayesian inference problems with sparsity-promoting priors (motivated by compressive sensing), for which we recover maximum a posteriori (MAP) point estimators of radio interferometric images by convex optimisation. Exploiting recent developments in the theory of probability concentration, we quantify uncertainties by post-processing the recovered MAP estimate. Three strategies to quantify uncertainties are developed: (i) highest posterior density credible regions; (ii) local credible intervals (cf. error bars) for individual pixels and superpixels; and (iii) hypothesis testing of image structure. These forms of uncertainty quantification provide rich information for analysing radio interferometric observations in a statistically robust manner. Our MAP-based methods are approximately 105 times faster computationally than state-of-the-art MCMC methods and, in addition, support highly distributed and parallelised algorithmic structures. For the first time, our MAP-based techniques provide a means of quantifying uncertainties for radio interferometric imaging for realistic data volumes and practical use, and scale to the emerging big-data era of radio astronomy. Key words: techniques: image processing – techniques: interferometric – methods: data analysis – methods: numerical – methods: statistical. 1 INTRODUCTION Radio interferometric (RI) telescopes provide observations of the radio emission of the sky with high angular resolution and sensitivity, and provide a wealth of valuable information for astrophysics and cosmology (Ryle & Vonberg 1946; Ryle & Hewish 1960; Thompson et al. 2008). Radio interferometers essentially acquire Fourier measurements of the sky image of interest. Imaging observations made by radio interferometers thus requires solving an ill-posed linear inverse problem (Thompson et al. 2008), which is an important first step in many subsequent scientific analyses. Since the inverse problem is ill-posed (sometimes seriously), uncertainty information (e.g. error estimates) regarding reconstructed images is critical. Nevertheless, uncertainty information is currently lacking in all RI imaging techniques used in practice. In Cai et al. (2017), ? E-mail: x.cai@ucl.ac.uk (XC); m.pereyra@hw.ac.uk (MP); jason.mcewen@ucl.ac.uk (JDM) c 2017 The Authors the first of these companion articles, we propose uncertainty quantification strategies for RI imaging based on state-of-the-art Markov chain Monte Carlo (MCMC) methods that sample the full posterior distribution of the image, with the sparsity-promoting priors that have been shown in practice to be highly effective (e.g. Pratley et al. 2016). Excellent results were achieved and a variety of different uncertainty quantification strategies were presented. However, it is difficult to scale these strategies to big-data due to their high computational overhead. We address this issue in the current article. Over the coming decades radio astronomy will transition into the so-called big-data era. Generally speaking, the new generation of radio telescopes, such as the LOw Frequency ARray (LOFAR1 ), the Extended Very Large Array (EVLA2 ), the Australian 1 2 http://www.lofar.org http://www.aoc.nrao.edu/evla 2 Cai et al. Square Kilometre Array Pathfinder (ASKAP3 ), and the Murchison Widefield Array (MWA4 ), will achieve much higher dynamic range and angular resolution than previous instruments and will acquire very large volumes of data. The Square Kilometer Array (SKA5 ) will provide a considerable step again in dynamic range (six or seven orders of magnitude beyond prior telescopes) and angular resolution, and will acquire massive volumes of data, ushering in the big-data era of radio astronomy. This emerging era of big-data, inevitably, will bring further challenges and so uncertainty quantification will be increasingly important. As discussed in Cai et al. (2017), existing image reconstruction techniques, such as CLEAN-based methods (Högbom 1974; Bhatnagar & Corwnell 2004; Cornwell 2008; Stewart et al. 2011), the maximum entropy method (MEM) (Ables 1974; Gull & Daniell 1978; Cornwell & Evans 1985), and compressed sensing (CS) methods (Wiaux et al. 2009a,b; McEwen & Wiaux 2011; Li et al. 2011a,b; Carrillo et al. 2012, 2014; Wolz et al. 2013; Dabbech et al. 2015; Dabbech et al. 2017; Garsden et al. 2015; Onose et al. 2016, 2017; Pratley et al. 2016; Kartik et al. 2017), do not provide uncertainty information regarding their reconstructed images. The approaches that do provide some form of uncertainty quantification (Sutter et al. 2014; Junklewitz et al. 2016; Greiner et al. 2017) cannot scale to bigdata due to their high computational cost, are typically restricted to Gaussian or log-normal priors, and are not currently used in practice. Please see our first article in this companion series (Cai et al. 2017) for a more thorough review of RI imaging techniques and their properties. The current state of the field thus triggers an urgent need to develop efficient uncertainty quantification methods for RI imaging that scale to big-data. Furthermore, we seek to support the sparsity-promoting priors that have been demonstrated in practice to be highly effective for RI imaging (e.g. Pratley et al. 2016). In Cai et al. (2017) (the first part of this companion series), we proposed uncertainty quantification methods to address the RI imaging problem with sparse priors. In the current article (the second part of this companion series), we present fast uncertainty quantification methods that not only support sparse priors but also scale to big-data. The techniques presented in this article are very different to those presented in Cai et al. (2017) but support the same forms of uncertainty quantification. The uncertainty quantification methods proposed in Cai et al. (2017) are based on two proximal MCMC sampling methods, i.e. the Moreau-Yoshida unadjusted Langevin algorithm (MYULA) (Durmus et al. 2016) and the proximal Metropolis-adjusted Langevin algorithm (Px-MALA) (Pereyra 2015). The main steps of the uncertainty quantification strategies presented in Cai et al. (2017) can be briefly summarised as follows: firstly, the posterior distribution of the image is MCMC sampled; then, uncertainty quantification is performed by using the generated samples to compute local (pixel-wise) credible intervals, highest posterior density (HPD) credible regions, and to perform hypothesis testing of image structure. Two frameworks – analysis and synthesis models – are considered. While excellent results were achieved in Cai et al. (2017), when it comes to big-data, the proposed approach would suffer due to the long computation time required to sample the posterior distribution (as would be the case for any MCMC sampling approach). 3 4 5 http://www.atnf.csiro.au/projects/askap http://www.mwatelescope.org/telescope http://www.skatelescope.org/ In this article we exploit an analytic method to approximate HPD credible regions from maximum a posteriori (MAP) estimators, as derived in Pereyra (2016a), in order to develop very fast methods to perform uncertainty quantification for RI imaging. Our approach supports sparse priors and scales to massive data sizes, i.e. to big-data. We begin by formulating Bayesian MAP estimation for RI imaging as unconstrained convex optimisation problems, for analysis and synthesis forms. These are subsequently solved efficiently by using convex minimisation algorithms (e.g. Combettes & Pesquet 2010). Recent advances in convex optimisation have resulted in techniques that achieve excellent reconstruction fidelity (with convergence guarantees), are flexible, and exhibit relatively low computational costs. They also afford algorithmic structures that can be highly distributed and parallelised (e.g. Carrillo et al. 2014; Onose et al. 2016). Note, specifically, that only one point estimator is computed here for the analysis or synthesis form, in contrast to sampling approaches that seek to explore the full posterior distribution as in Cai et al. (2017), which is very time consuming. MAP estimation is then followed by various strategies to quantify uncertainties. Precisely, first the method of Pereyra (2016a) is used to obtain approximate HPD credible regions for the recovered image. These HPD regions are then used, for the first time, to compute local credible intervals (cf. error bars) that analyse uncertainty spatially and at different scales (pixles or superpixels). Finally, we also use the HPD credible regions to perform hypothesis tests of image structure. We test our proposed approaches on simulated RI observations to demonstrate their effectiveness and compare with the MCMC methods presented in Cai et al. (2017). The remainder of this article is organised as follows. In Section 2 we review the RI imaging inverse problem. In Section 3 we apply convex optimisation algorithms to solve the MAP estimation problem for RI imaging in the context of sparse priors. Uncertainty quantification techniques for RI imaging based on MAP estimation are formulated in Section 4. The performance of the proposed methods is then evaluated numerically in Section 5, where we compare uncertainties quantified by proximal MCMC methods and by MAP estimation. Finally, we conclude in Section 6 with a summary of our main contributions and a discussion of planned extensions. 2 RADIO INTERFEROMETRIC IMAGING In this section the inverse problem related to RI image reconstruction is introduced. We briefly recall the use of proximal MCMC methods to solve this problem (Cai et al. 2017), which we use as a benchmark in the experiments that follow. Finally, an introduction to Bayesian MAP estimation approaches for RI imaging is presented, which may be solved by efficient convex optimisation strategies. 2.1 Radio interferometry Here, we concisely recall the inverse problem of RI imaging (for further details see Cai et al. 2017 and references therein). When the baselines in an array are co-planar and the field of view is narrow, the visibilities, y, can be measured by correlating the signals from pairs of antennas, separated by the baseline components u = (u, v). Let x represent the sky brightness distribution, described in coordinates l = (l, m), and A(l) represent the primary beam of the telescope. The general RI equation for acquiring y can MNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II be represented as (Thompson et al. 2008) Z y(u) = A(l)x(l)e−2πiu·l d2 l. (1) Recovering the sky intensity signal x from the measured visibilities y acquired according to equation (1) then amounts to solving a linear inverse problem (Rau et al. 2009). In the discretised setting, let x ∈ RN represent the sampled intensity signal (the sky brightness distribution). In particular, x can be represented by X x = Ψa = Ψi ai , (2) i where Ψ ∈ CN ×L is a basis or dictionary (e.g., a wavelet basis or an over-complete frame) and vector a = (a1 , · · · , aL )> represents the synthesis coefficients of x under Ψ. In particular, x is said to be sparse if a contains only K non-zero coefficients, K N , or compressible if many coefficients of a are nearly zero. In practice, it is ubiquitous that natural images are sparse or compressible for approriate choices of Ψ. Refer to Cai et al. (2017) for more details about sparse representation. Let y ∈ CM be the M visibilities observed under a linear measurement operator Φ ∈ CM ×N modelling the realistic acquisition of the sky brightness distribution. Then, we have y = Φx + n or y = ΦΨa + n, (3) where n ∈ CM is the instrumental noise. Without loss of generality, we subsequently consider independent and identically distributed (i.i.d.) Gaussian noise. In practice, y is only observed partially or with limited resolution and thus solving (3) for x presents an ill-posed inverse problem. 2.2 Bayesian inference The RI inverse problem (3) can be solved elegantly in the Bayesian statistical framework, which provides tools to estimate x as well as to quantify the uncertainty in the estimated solutions. Let p(y\|x) be the likelihood function of the statistical model associated with (3). In the case of i.i.d. Gaussian noise this reads p(y\|x) ∝ exp(−ky − Φxk22 /2σ 2 ), (4) where σ represents the standard deviation of the noise level. Recovering x directly from y is not possible because the problem is not well posed. Bayesian methods use prior knowledge to address this difficulty. Precisely, they use a prior distribution p(x) to regularise the problem, reduce uncertainty, and improve estimation results. Here we consider both analysis and synthesis formulations because they are both widely used in RI imaging. For analysis models we use Laplace-type priors of the form p(x) ∝ exp(−µkΨ† xk1 ), (5) where Ψ† denotes the adjoint of Ψ, µ > 0 is a regularisation parameter, and k·k1 is the `1 norm. Analogously, for synthesis models we use p(a) ∝ exp(−µkak1 ). (6) Observe that both formulations are equivalent when Ψ is an orthogonal basis. However, for redundant dictionaries the approaches have very different properties. Please see Cai et al. (2017) for more details about this model and other models used in RI imaging. MNRAS 000, 1–13 (2017) 3 The observed and prior information are then combined by using Bayes’ theorem to obtain the posterior distribution, which models our knowledge about x after observing y. For the analysis formulation this is given by p(x\|y) = R RN p(y\|x)p(x) . p(y\|x)p(x)dx (7) Similarly, for the synthesis model the posterior reads p(y\|a)p(a) , p(y\|a)p(a)da p(a\|y) = R (8) RN with p(y\|a) = p(y\|x) for x = Ψa. Refer to Cai et al. (2017) for more detailed discussion about Bayesian inference in the context of RI imaging. 2.3 Proximal MCMC methods To solve the ill-posed inverse problem in (3) with sparsitypromoting priors, which have been shown in practice to be highly effective (Pratley et al. 2016), while also performing uncertainty quantification, two proximal MCMC methods to perform Bayesian inference for RI imaging were developed in the companion article (Cai et al. 2017). These proximal MCMC methods seek to sample the full posterior density p(x\|y) that models our understanding of the image x given data y, in the context of prior information. From the full posterior, summary estimators of x and other quantities of interest can be computed. In particular, in Cai et al. (2017) these methods are used to perform a range of uncertainty quantification analysis for RI images. One of the proximal MCMC methods presented in Cai et al. (2017), MYULA, scales efficiently to high dimensions but suffers from some estimation bias (Durmus et al. 2016). The other, PxMALA, corrects this bias by using a Metropolis-Hastings correction step, at the expense of a higher computational cost and slower convergence (Pereyra 2015). Since Px-MALA can provide results with corrected bias and thus is more accurate, we use it as a benchmark in the subsequent numerical tests presented in this work. Nevertheless, the MCMC methods discussed in Cai et al. (2017) will suffer when scaling to big-data (as will any MCMC method), which motives us to explore alternative faster methods that can scale to big-data. In this article we develop methods for uncertainty quantification based on MAP estimation. We emphasise that while MCMC methods such as Px-MALA are not as efficient as MAP estimation (the main focus in this article), and do not scale to large RI datasets, they are useful for smaller datasets and as a benchmark for the efficient alternative methods that we propose in Section 4. 2.4 Maximum a posteriori (MAP) estimation As discussed in the previous sections, sampling the full posterior p(x\|y) or p(a\|y) by MCMC methods is difficult because of the high dimensionality involved. Instead, Bayesian estimators that summarise p(x\|y) or p(a\|y) are often computed. In particular, one common approach is to compute MAP (maximum-a-posteriori) estimators given by n o xmap = argmin µkΨ† xk1 + ky − Φxk22 /2σ 2 , (9) x for the analysis model, and for the synthesis model by n o xmap = Ψ × argmin µkak1 + ky − ΦΨak22 /2σ 2 . a (10) 4 Cai et al. As we discuss below, a main computational advantage of the MAP estimators (9) and (10) is that they can be computed very efficiently, even in high dimensions, by using convex optimisation algorithms (e.g. Combettes & Pesquet 2010; Green et al. 2015). There is also abundant empirical evidence suggesting that these estimators deliver accurate reconstruction results (see Pereyra 2016b also for a theoretical analysis of MAP estimation). However, since MAP estimation results in a single point estimator, we typical lose uncertainty information that MCMC methods can provide (Cai et al. 2017). On the contrary, however, as we show in this article it is possible to approximately quantify the uncertainties associated with MAP estimators by leveraging recent results in the theory of probability concentration (Pereyra 2016a). Consequently, using the techniques presented later in this article MAP estimation can provide fast methods that scale to big-data and that quantify uncertainties. 2.5 Convex optimisation methods for MAP estimation There are several convex optimisation methods that can be used to solve the MAP estimation problems (9) and (10) efficiently, such as forward-backward splitting, Douglas-Rachford splitting, or alternating direction method of multipliers (ADMM) (see Combettes & Pesquet 2010). In our experiments (9) and (10) are solved by adopting the simple forward-backward algorithm, which we detail below. Forward-backward splitting algorithms solve optimisation problems of the form argmin(f + g)(x), (11) x∈RN by using a splitting of (f + g)(x). We consider the setting where f ∈ / C 1 is proper, convex and lower semi-continuous (l.s.c.) and g ∈ C 1 is l.s.c. convex and βLip -Lipchitz differentiable, i.e., k∇g(ẑ) − ∇g(z̄)k ≤ βLip kẑ − z̄k, ∀(ẑ, z̄) ∈ CN × CN . (12) Precisely, forward-backward algorithms solve (11) by using the iteration x (i+1) = proxλ(i) f (x (i) (i) (i) − λ ∇g(x )), (13) where λ(i) is the step size in a suitable bounded interval (see, e.g., Combettes & Pesquet 2010). The proximity operator of λf is defined as (Moreau 1965) proxλf (z) ≡ argmin f (u) + ku − zk2 /2λ . (14) u∈RN There are several refinements of (13) with better convergence properties. For example, using relaxation leads to the iteration x(i+1) = (1 − β (i) )x(i) + β (i) x̃(i+1) , (15) where x̃(i+1) is computed by (13), β (i) is a sequence of relaxation parameters, λ(i) ∈ (, 2/βLip − ), β (i) ∈ (, 1), and ∈ (0, min{1, 1/βLip }) (Combettes & Wajs 2005); or with λ(i) = 1/βLip , β (i) ∈ (, 3/2 − ), and ∈ (0, 3/4) (Bauschke & Combettes 2011). Furthermore, algorithmic structures that allow computations to be highly distributed and parallelised can also be developed (e.g. Carrillo et al. 2014; Onose et al. 2016) to assist in scaling to big-data. estimation problems for both the analysis setting (9) and synthesis setting (10). For the sake of brevity, henceforth the labels ¯ and ˆ denote symbols related to the analysis and synthesis models, respectively. 3.1 Analysis For the analysis setting (9), set f¯(x) = µkΨ† xk1 and ḡ(x) = ky − Φxk22 /2σ 2 . Then n o argmin f¯(x) + ḡ(x) (16) x can be solved using the forward-backward iteration formula (13), leading to the iterations x(i+1) = proxλ(i) f¯(x(i) − λ(i) ∇ḡ(x(i) )). (17) Assume for now Ψ† Ψ = I, where I is identity matrix (although this assumption is not essential and relaxed later). We have, ∀z̄ ∈ RN , proxλf¯(z̄) = z̄ + Ψ softλµ (Ψ† z̄) − Ψ† z̄ , (18) and ∇ḡ(x) = Φ† (Φx − y)/σ 2 , (19) where softλ (z) = softλ (z1 ), softλ (z2 ), · · · is the pointwise soft-thresholding operator of vector z defined by ( zj (\|zj \| − λ)/\|zj \| if \|zj \| > λ, softλ (zj ) = (20) 0 otherwise, for every component zj . Refer to Cai et al. (2017) for the derivation of (18). Substituting (18) and (19) into (17), the analysis problem (9) can be solved iteratively by v (i+1) = x(i) − λ(i) Φ† (Φx(i) − y)/σ 2 , x(i+1) = v (i+1) +Ψ softλ(i) µ (Ψ† v (i+1) )−Ψ† v (i+1) ) . (21) (22) As initialisation use, e.g., x(0) = Φ† y, i.e. the dirty image. Remark 3.1. In the analysis form (9), if choosing Ψ such that Ψ† Ψ 6= I, i.e. an over-complete frame Ψ, then proxλf¯(z̄) can be computed in an iterative manner: ! 1 1 u(t− 2 ) (t) † (t) u(t+ 2 ) = λite (1 − proxλk·k /λ(t) ) + Ψ u , (23) (t) 1 ite λite 1 u(t+1) = z̄ − Ψu(t+ 2 ) , (24) (t) λite 2 where ∈ (0, 2/βPar ) (βPar is a constant satisfying kΨzk2 ≤ βPar kzk , ∀z ∈ RL ) is a predefined step size and u(t) → proxγ f¯(z̄); refer to Fadili & Starck (2009) and Jacques et al. (2011) and references therein for details. 3.2 Synthesis For the synthesis setting (10), set fˆ(a) = µkak1 and ĝ(a) = ky − ΦΨak22 /2σ 2 . Then n o argmin fˆ(a) + ĝ(a) (25) x 3 SPARSE MAP ESTIMATION FOR RI IMAGING In this section we present the algorithmic details of implementing the forward-backward splitting algorithm to solve the sparse MAP can be solved using the forward-backward iteration formula (13), leading to the iterations a(i+1) = proxλ(i) fˆ(a(i) − λ(i) ∇ĝ(a(i) )). (26) MNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II We have, ∀ẑ = (ẑ1 , · · · , ẑL ) ∈ RL , 5 Observed visibilities in RI imaging: y proxλfˆ(ẑ) = proxλk·k1 (ẑ) = argmin λµkuk1 + ku − ẑk2 /2 u∈RL MAP image estimation: xmap (27) Approximate HPD 0 credible regions: Cα = softλµ (ẑ) and † † 2 ∇ĝ(a) = Ψ Φ (ΦΨa − y)/σ . (28) Finally, substituting (27) and (28) into (26), the synthesis problem (10) can be solved iteratively by a(i+1) = softλ(i) µ a(i) − λ(i) Ψ† Φ† (ΦΨa(i) − y)/σ 2 . (29) Remark 3.2. Note that in both the analysis and synthesis settings various terms can be precomputed. For example, in (21) and (28) the operators Φ† Φ and Ψ† Φ† ΦΨ can be precomputed offline. Similarly, the terms of Φ† y (the so-called dirty map) and Ψ† Φ† y respectively in (21) and (28) can also be precomputed to improve computation efficiency. We summarise the forward-backward splitting algorithms for the analysis and synthesis reconstruction forms in Algorithms 1 and 2. We consider stopping criteria based on a maximum iteration number and when the relative difference between solutions at two consecutive iterations is within some tolerance, i.e., kx(i+1) − x(i) k2 /kx(i) k2 (for Algorithm 1) and kΨa(i+1) − Ψa(i) k2 /kΨa(i) k2 (for Algorithm 2). The iteration is terminated when either of the stopping criteria are reached. Algorithm 1: Forward-backward algorithm for analysis 2 Input: y ∈ RM , x(0) ∈ RN , σ and λ(i) ∈ (0, ∞) Output: x0 3 do 1 8 update v (i+1) = x(i) − λ(i) Φ† (Φx(i) − y)/σ 2 compute u = Ψ† v (i+1) update x(i+1) = v (i+1) + Ψ softλ(i) µ (u) − u) i=i+1 while Stopping criterion is not reached; 9 set x0 = x(i) 4 5 6 7 Algorithm 2: Forward-backward algorithm for synthesis 2 Input: y ∈ RM , a(0) ∈ RL , σ and λ(i) ∈ (0, ∞) Output: a0 3 do 1 7 compute u = a(i) − λ(i) Ψ† Φ† (ΦΨa(i) − y)/σ 2 update a(i+1) = softλ(i) µ (u) i=i+1 while Stopping criterion is not reached; 8 set a0 = a(i) 4 5 6 Approximate local credible intervals: (ξ− , ξ+ ) Hypothesis testing Figure 1. Our proposed uncertainty quantification procedure for RI imaging based on MAP estimation. The light green areas on the right show the types of uncertainty quantification developed. Firstly, an image is reconstructed by MAP estimation using convex optimisation techniques, which scale to big-data. Then, various forms of uncertainty quantification are performed. Global approximate Bayesian credible regions are computed. These are then used to compute local credible intervals (cf. error bars) corresponding to individual pixels and superpixels and to perform hypothesis testing of image structure to test whether a structure is physical or an artefact. 4 BAYESIAN UNCERTAINTY QUANTIFICATION: MAP ESTIMATION The analysis and synthesis reconstruction models address inverse problems which are generally ill-conditioned or ill-posed (especially when the measurements are only observed partially or with limited resolution). Consequently, the corresponding estimators have significant intrinsic uncertainty that is very challenging to analyse and quantify. In Pereyra (2016a) a general methodology was proposed to use MAP estimators to accurately approximate Bayesian credible regions for p(x\|y). These credible regions indicate the regions of the parameter space where most of the posterior probability mass lies. A remarkable property of the approximation is that it only requires knowledge of xmap and therefore it can be computed very efficiently, even in very large-scale problems. The diagram in Figure 1 shows the main components of our proposed uncertainty quantification methodology based on MAP estimation. As is shown, firstly, an image is reconstructed by MAP estimation. MAP estimation can be computed extremely rapidly and is therefore ideal for application to big-data. Then, various forms of uncertainty quantification are performed. Firstly, global approximate Bayesian credible regions are computed. These are then used to compute local credible intervals (cf. error bars) corresponding to individual pixels and superpixels. Finally, again using the global approximate Bayesian credible regions, hypothesis testing of image structure can be performed to test whether a structure is physical or an artefact. For consistency, we adopt the same notation as in the companion article (Cai et al. 2017). 4.1 Approximate highest posterior density (HPD) credible regions The first step in our uncertainty quantification methodology is to compute a credible region for p(x\|y). A posterior credible region with credible level (1 − α)% is a set Cα ∈ RN that satisfies Z p(x ∈ Cα \|y) = p(x\|y)1Cα dx = 1 − α, (30) x∈RN MNRAS 000, 1–13 (2017) 6 Cai et al. where 1Cα is the indicator function for Cα , defined by 1Cα (u) = 1 if u ∈ Cα and 0 otherwise. Many regions satisfy the above property. We focus on the HPD (Highest Posterior Density) region defined by Cα := {x : f (x) + g(x) ≤ γα }, (31) where the threshold γα which defines an isocontour or level-set of the log-posterior is set such that (30) holds, and we recall that p(x\|y) ∝ exp{−f (x) − g(x)}. This region is decisiontheoretically optimal in the sense of minimum volume (Robert 2001). Computing HPD credible regions in (31) is difficult because of the high-dimensional integral in (30). For RI models that are not too high dimensional, Cα can be computed efficiently by using proximal MCMC method as described in Cai et al. 2017. However, this is not possible in big-data settings. Here we use an approximation of Cα proposed recently in Pereyra (2016a) for convex inverse problems solved by MAP estimation. The approximation is given by Cα0 := {x : f (x) + g(x) ≤ γα0 }, (32) where γα0 is an approximation of the HPD threshold γα given by √ γα0 = f (xmap ) + g(xmap ) + τα N + N, (33) p with universal constant τα = 16 log(3/α). Recall that N is the dimension of x and (1 − α)% the credible level considered. After computing xmap by using modern convex optimisation algorithms, γα0 can be calculated straightforwardly using (33), even in very high dimensions. The approximation given in (33) was motivated from recent results in information theory in terms of a probability concentration inequality (refer to Pereyra 2016a for more details). For any α ∈ (4exp(−N/3), 1), the error between γα0 and γα is bounded by the following inequality √ (34) 0 ≤ γα0 − γα ≤ ηα N + N, p p 0 where ηα = 16 log(3/α) + 1/α. Since the error γα − γα grows at most linearly with respect to N when N is large, the credible region Cα0 associated with γα0 is a stable approximation of Cα . Moreover, since γα0 − γα ≥ 0 the approximation is theoretically conservative in the sense that Cα0 overestimates Cα . Precisely, in the analysis formulation, we first compute the reconstructed image xmap by using Algorithm 1, and then obtain an approximate HPD credible region C̄α0,map := {x : f¯(x) + ḡ(x) ≤ γ̄α0 } (35) with √ γ̄α0 = f¯(xmap ) + ḡ(xmap ) + τα N + N. (36) Similarly, in the synthesis setting we compute amap via Algorithm 2, and then construct Ĉα0,map := {Ψa : fˆ(a) + ĝ(a) ≤ γ̂α0 } (37) with √ γ̂α0 = fˆ(amap ) + ĝ(amap ) + τα N + N. γ̄α0 γ̂α0 (38) Note that and define the HPD credible regions implicitly. The HPD credible regions can be used to quantify uncertainties in a variety of manners. In the reminder of this section we describe two such strategies. 4.2 Local credible intervals The first strategy we propose is a novel approach to compute local credible intervals corresponding to pixels and superpixels, as a means for quantifying uncertainty spatially at different scales. This presents a new form of Bayesian uncertainty quantification tailored for image data and is easy to visualise and interpret. The method is based on the HPD credible regions discussed above and is applicable for any method for which HPD credible regions can be computed. Here we promote the MAP-based approach, based on the approximations (36) and (38), and benchmark our results against the MCMC approach Px-MALA, introduced in Cai et al. (2017). Let Ω = ∪i Ωi be a partition of the image domain Ω into subsets or superpixels Ωi such that Ωi ∩ Ωj = ∅, i 6= j. The image domain can be partitioned at different scales, from a single pixel to larger scales involving blocks of several pixels. To index superpixels we define the index operator ζΩi = (ζ1 , · · · , ζN ) ∈ RN on Ωi , which satisfies ( 1, if k ∈ Ωi , (39) ζk = 0, otherwise. To quantify the uncertainty associated with the region Ωi we calculate the points ξ−,Ωi and ξ+,Ωi that saturate the HPD credible region Cα0,map from above and from below at Ωi , given by ξ−,Ωi = min ξ\|f (xi,ξ ) + g(xi,ξ ) ≤ γα0 , ∀ξ ∈ [0, +∞) , (40) ξ ξ+,Ωi = max ξ\|f (xi,ξ ) + g(xi,ξ ) ≤ γα0 , ∀ξ ∈ [0, +∞) , (41) ξ where xi,ξ = x∗ (I − ζΩi ) + ξζΩi represents a point estimator generated by replacing the intensity of x∗ in Ωi by ξ. We recall that γα0 is the threshold or isocontour level defining Cα0,map . We then construct the interval (ξ−,Ωi , ξ+,Ωi ) that represents the range of intensity values ξ of Ωi for which xi,ξ ∈ Cα0,map . Finally, for visualisation, we gather all the lower and upper bounds ξ−,Ωi , ξ+,Ωi , ∀i, into the following two images: X X ξ− = ξ−,Ωi ζΩi , ξ+ = ξ+,Ωi ζΩi . (42) i i We typically consider the difference image (ξ+ − ξ− ) that shows the length of the local credible intervals (cf. error bars). These images can be constructed at different scales to analyse structure of different sizes. In our experiments, as examples, we consider superpixels of sizes 10 × 10, 20 × 20, and 30 × 30 pixels. 4.3 Hypothesis testing of image structure In a manner akin to the companion article Cai et al. (2017), we use knock-out posterior tests to assess specific areas or structures of interest in the reconstructed images. These tests proceed by constructing a surrogate test image x∗,sgt by carefully replacing the structure of interest in an point estimator x∗ (or Ψa∗ ) with background information. If removing the structure has pushed x∗,sgt outside of the HPD credible region (i.e. x∗,sgt ∈ / Cα0,map ), this indicates that the data strongly supports the structure under consideration. Conversely, if x∗,sgt remains inside of the HPD credible region (i.e. x∗,sgt ∈ Cα0,map ), then the likelihood is insensitive to the modification, indicating lack of strong evidence for the scrutinised structure. Algorithmically, a surrogate x∗,sgt for a test area ΩD ⊂ Ω is generated by performing segmentation-inpainting of x∗ , for example by applying a wavelet filter Λ iteratively by using x(m+1),sgt = x∗ 1Ω−ΩD + Λ† softλthd (Λx(m),sgt )1ΩD , (43) MNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II with x(0),sgt = x∗ or x(0),sgt = Ψa∗ for the synthesis formulation (usually 100 iterations are sufficient for convergence). To determine if x∗,sgt ∈ Cα0,map , it suffices to check if f (x∗,sgt ) + g(x∗,sgt ) ≤ γα0 . 5 (44) Table 1. CPU time in minutes for the proximal MCMC method Px-MALA (generating full posterior samples) and MAP-based methods (computing a point estimator), for the analysis and synthesis models and for test images of M31, Cygnus A, W28 and 3C288. MAP estimation is approximately 105 times faster than Px-MALA and can be scaled to big-data. EXPERIMENTAL RESULTS We now investigate the performance of the proposed uncertainty quantification methodology for the three strategies discussed in Section 4. We also report a detailed comparison with the proximal MCMC method Px-MALA, which is one of the MCMC methods introduced in the companion article (Cai et al. 2017) and that can also support sparsity-promoting priors. Px-MALA produces (asymptotically) exact inferences and therefore we use it here as an accurate benchmark for the methods proposed in this article. 5.1 Simulations In a manner akin to Cai et al. (2017), we perform our experiments with the following four RI images: M31 galaxy (size 256 × 256), Cygnus A galaxy (size 256 × 512), W28 supernova remnant (size 256×256), and 3C288 (size 256×256). These images are depicted in Figure 2 (a) and Figure 3 (a). Radio interferometric observations are simulated for these ground truth images in a similar manner as in Cai et al. (2017). The numerical experiments performed in this article for MAP estimation were run on a Macbook laptop with an i7 Intel CPU and memory of 16 GB, running MATLAB R2015b. The Px-MALA algorithm used as a benchmark is significantly more computationally expensive and required a high-performance workstation (see Cai et al. 2017). For further details about the experiment setup and the implementation of Px-MALA please see Cai et al. (2017). Regarding the models used for the experiments, the `1 regularisation parameter µ in the analysis and synthesis models is set to 104 and the dictionary Ψ in the analysis and synthesis models is set to Daubechies 8 wavelets. In Algorithms 1 and 2, we use λ(i) = 0.5, with stopping criteria set by a maximum iteration number of 500 and relative difference between solutions of 10−4 . In formulas (36) and (38), the range of values for α is [0.01, 0.99]. In particular, credible regions and intervals are reported at α = 0.01, corresponding to the 99% credible level. The maximum number of iterations for segmented-inpainting in (43) is set to 200. 5.2 Image reconstruction As the first step in our analysis we perform Bayesian image reconstruction for the four images considered. Precisely, for each image we compute two Bayesian estimators, the MAP estimator computed by convex optimisation and the sample mean estimator computed with Px-MALA. For completeness, we consider both the analysis and the synthesis models (9) and (10). The Bayesian estimators related to the analysis model are shown in Figures 2 and 3. Observe that both estimators produce similar, excellent reconstruction results. For comparison, dirty maps (reconstructed by applying the inverse Fourier transform directly to the visibilities) of the test images are shown in Figure 2 (b) and Figure 3 (b). As expected, the results of the analysis and synthesis models (9) and (10) under an orthogonal basis Ψ are nearly MNRAS 000, 1–13 (2017) 7 CPU time (min) Analysis Synthesis Images Methods M31 (Fig. 2 ) Px-MALA MAP 1307 .03 944 .02 Cygnus A (Fig. 3 ) Px-MALA MAP 2274 .07 1762 .04 W28 (Fig. 3 ) Px-MALA MAP 1122 .06 879 .04 3C288 (Fig. 3 ) Px-MALA MAP 1144 .03 881 .02 undistinguishable6 (see results for M31 in Figure 2; to avoid redundancy the results for the other images are not reported here). For this reason, in the reminder of this article only the results for the analysis model are presented. We emphasise again that MAP estimators computed by convex optimisation are significantly faster to compute than the estimators that require MCMC methods. In particular, in our experiments there is a gain of order 105 in terms of computation time (see Table 1 for the computation time comparisons with Px-MALA). Furthermore, MAP estimation based on convex optimisation supports algorithmic structures that can be highly distributed (e.g. Carrillo et al. 2014; Onose et al. 2016) to further assist in scaling to bigdata. MCMC algorithms cannot typically be distributed to such a high degree. We have not yet considered distributed MAP algorithms here; our MAP-based methods therefore provide additional performance improvements over MCMC beyond the already dramatic improvements shown in Table 1. 5.3 Approximate HPD credible regions We compute the HPD credible regions for the four images considered. Precisely, we use formulas (36) and (38) to approximate the threshold or isocontour value γα0 defining the HPD regions for the analysis and synthesis models (recall that these are highly efficient approximations derived from the MAP estimates xmap and amap ). Figure 4 shows the threshold values obtained for each image and model, for α ∈ [0.01, 0.99]; observe again that the results of the analysis and synthesis models are consistent with each other, as expected. To assess the approximation error involved in using (36) and (38) instead of an MCMC method, we also computed the HPD threshold values using the Px-MALA algorithm which is asymptotically exact (cf. Cai et al. 2017, Figure 6). Recall than Px-MALA is several orders of magnitude more computationally expensive than MAP estimation (see Table 1). This comparison revealed approximation errors of between 1% and 5% over all cases, which is in close agreement with the results reported in Pereyra (2016a). These experiments confirm that the MAP-based approximations (36) and Note that, when Ψ† Ψ = I, as considered here, the analysis and synthesis models are identical. However, when Ψ† Ψ 6= I, they are very different and we expect different reconstructed images. 6 8 Cai et al. (a) ground truth (b) dirty map (c) Px-MALA for analysis model (d) MAP for analysis model (e) Px-MALA for synthesis model (f) MAP for synthesis model Figure 2. Image reconstructions for M31 (size 256 × 256). All images are shown in log10 scale. (a): ground truth; (b): dirty image (reconstructed by inverse Fourier transform); (c) and (d): point estimators for the analysis model (9) computed by Px-MALA and MAP estimation, respectively; (e) and (f): the same as (c) and (d) but for the synthesis model (10). In particular, the point estimators of Px-MALA are the sample mean. Clearly, consistent results between Px-MALA and MAP estimation and between the analysis and synthesis models are obtained. (a) ground truth (b) dirty map (c) Px-MALA for analysis model (d) MAP for analysis model Figure 3. Image reconstructions for Cygnus A (size 256 × 512), W28 (size 256 × 256), and 3C288 (size 256 × 256) radio galaxies (first to third rows). All images are shown in log10 scale. First column: (a) ground truth. Second to forth columns: (b) dirty images; (c) and (d) point estimators for the analysis model (9) computed by Px-MALA and MAP estimation, respectively. Clearly, consistent results between Px-MALA and MAP estimation are obtained. (38) deliver accurate estimates of the HPD credible regions with a dramatically lower computational cost. 5.4 Approximate local credible intervals We use the approximate HPD regions to calculate local credible intervals for image superpixels. Precisely, Figures 5–8 report the length of local credible intervals for the four test images for superpixel grid sizes of 10 × 10, 20 × 20, and 30 × 30 pixels, computed MNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II 1−α (a) M31 1−α (b) Cygnus A 1−α (c) W28 9 1−α (d) 3C288 MAP Px-MALA 0 and γ̂ 0 computed using MAP-based methods, for test images (a) M31, (b) Cygnus A, (c) W28, and (d) Figure 4. HPD credible region isocontour levels γ̄α α 3C288. In particular, MAP-ana (resp. MAP-syn) represents the results by MAP estimation for the analysis (resp. synthesis) model. Note that the red line in plot (d) is overlaid by the blue line and thus may not be visible, due to the high degree of similarity between the two results. In all cases the results of the analysis and synthesis models are in close agreement. (a) point estimators (b) local credible interval length grid size 10 × 10 pixels (c) local credible interval length grid size 20 × 20 pixels (d) local credible interval length grid size 30 × 30 pixels MAP Px-MALA Figure 5. Length of local credible intervals (99% credible level), cf. error bars, computed for M31 for the analysis model (9). First column: (a) point estimators. Second to fourth columns: (b)–(d) local credible intervals at grid sizes of 10 × 10, 20 × 20, and 30 × 30 pixels, respectively. First row gives exact inferences computed with the MCMC method Px-MALA (Cai et al. 2017). Second row gives MAP-based approximate inferences computed by convex optimisation. Clearly, MAP-based approximations provide estimates of the length of local credible intervals (cf. error bars) that are extremely consistent with the ones obtained by Px-MALA, while the MAP estimates can be computed several orders of magnitude more rapidly (Table 1). Moreover, the length of the approximate credible intervals computed by the MAP-based approach are theoretically conservative and can be seen to slightly overestimate the lengths computed by MCMC sampling). (a) point estimators (b) local credible interval length grid size 10 × 10 pixels (c) local credible interval length grid size 20 × 20 pixels Figure 6. Same as Figure 5 but for Cygnus A. MNRAS 000, 1–13 (2017) (d) local credible interval length grid size 30 × 30 pixels Cai et al. MAP Px-MALA 10 (a) point estimators (b) local credible interval length grid size 10 × 10 pixels (c) local credible interval length grid size 20 × 20 pixels (d) local credible interval length grid size 30 × 30 pixels MAP Px-MALA Figure 7. Same as Figure 5 but for W28. (a) point estimators (b) local credible interval length grid size 10 × 10 pixels (c) local credible interval length grid size 20 × 20 pixels (d) local credible interval length grid size 30 × 30 pixels Figure 8. Same as Figure 5 but for 3C288. w.r.t. the analysis model (the results for the synthesis model are very similar). For comparison, Figures 5–8 also show the exact estimates obtained by Px-MALA based on its posterior sample mean. We conclude the main observations as follows. Firstly, the results obtained with both approaches are extremely consistent with each other, indicating that the approximate credible intervals derived from the MAP estimation are very accurate. Secondly, the length of the approximate local credible intervals computed by MAP estimation are theoretically conservative and can be seen to slightly overestimate the lengths computed by MCMC sampling, and so are trustworthy. Thirdly, note that (i) coarser scales have shorter credible intervals than narrower scales, and (ii) superpixels at object boundaries generally have longer credible intervals than superpixels in homogenous regions. These two observations are direct consequences of the fact that there is more uncertainty about high-frequency image components because of the sampling profile associated with the measurement operator Φ, which mainly covers low frequencies (see Cai et al. 2017, Figure 2). 5.5 Hypothesis testing of image structure We conclude our experimental results by demonstrating our methodology for testing structure in reconstructed images. We consider the same images and structures of interest as in Cai et al. (2017), shown in the yellow rectangular areas in the first column of Figure 9. All of these structures are physical (i.e. present in the ground truth images), except for structure 2 in 3C288 which is a reconstruction artefact. Recall that the methodology proceeds as follows. First, we construct a surrogate image x∗,sgt by modifying the MAP estiMNRAS 000, 1–13 (2017) Uncertainty quantification for RI imaging II 11 Table 2. Hypothesis test results for test structures shown in Figure 9 for M31, Cygnus A, W28, and 3C288. Note that γα represents the isocontour defining the HPD credible region at credible level (1 − α), where here α = 0.01, x∗,sgt represents the surrogate generated from point estimator x∗ (in particular, for Px-MALA x∗ is the sample mean of the MCMC samples), and (f + g)(·) represents the objective function; symbols with labels ¯ and ˆ are related to the analysis model (9) and the synthesis model (10), respectively. Symbol 7 indicates that the test area is artificial (and no strong statistical statement can be made as to the area), while 3 indicates that the test area is physical. All values are in units 106 . Clearly, both Px-MALA and MAP estimation give convincing and consistent hypothesis test results. Note that MAP estimation is dramatically more computationally efficient that Px-MALA (Table 1). Images Test areas Ground truth Method (f¯ + ḡ)(x̄∗,sgt ) Isocontour γ̄0.01 (fˆ + ĝ)(Ψ† x̂∗,sgt ) Isocontour γ̂0.01 Hypothesis test M31 (Fig. 9 ) 1 3 Px-MALA MAP 2.44 2.29 2.34 2.25 2.43 2.29 2.34 2.25 3 3 Cygnus A (Fig. 9 ) 1 3 Px-MALA MAP 1.17 1.02 1.26 1.14 1.18 1.02 1.27 1.14 7 7 W28 (Fig. 9 ) 1 3 Px-MALA MAP 3.38 3.47 1.84 1.89 3.37 3.47 1.85 1.89 3 3 1 3 2 7 Px-MALA MAP Px-MALA MAP 3.27 3.11 1.971 1.844 2.02 1.91 2.027 1.912 3.25 3.11 1.954 1.844 2.01 1.91 2.010 1.912 3 3 7 7 3C288 (Fig. 9 ) mator xmap to remove the structure of interest via segmentationinpaiting, computed using formula (43). Each structure is assessed individually. Second, we check if x∗,sgt ∈ / Cα0,map (i.e. if f (x∗,sgt ) + g(x∗,sgt ) > γα0 ) to determine whether there exists strong evidence in favour of the structure considered. Conclusions are generally not highly sensitive to the exact value of α; here we report results for α = 0.01 related to a 99% credible level. The resulting surrogate images are displayed in the second column of Figure 9. The results of these tests are shown in Table 2. For comparison, we also include the results obtained with the reference method Px-MALA (Cai et al. 2017). Again, the two methods produce excellent results that are consistent with each other. From Table 2, we observe that the methods have correctly classified the three main physical structures of M31, W28, and 3C288, and correctly identified the minor structure of 3C288 as a potential reconstruction artefact. Moreover, the methods have found that it is not possible to make a strong statistical statement about the small physical structure in image Cygnus A, which is difficult because it is only a few pixels in size, isolated, and significantly weaker in intensity than the other structures in the image. Before closing this section, we emphasise again that the methods presented in this article deliver a variety of forms of uncertainty quantification with a very low computational cost. While these new forms of uncertainty quantification can also be achieved by using state-of-the-art proximal MCMC methods, such as Px-MALA and MYULA, as presented in the companion article Cai et al. (2017), MCMC techniques cannot scale to massive data sizes. Nevertheless, they are useful for medium-scale problems and provide accurate benchmarks for the highly efficient methods presented herein, which will scale very well to the emerging big-data era of radio astronomy. 6 CONCLUSIONS Uncertainty quantification is an important missing component in RI imaging that will only become increasingly important as the bigdata era of radio interferometry emerges. No existing RI imaging techniques that are used in practice (e.g. CLEAN, MEM or CS approaches) provide uncertainty quantification. In this article, as an MNRAS 000, 1–13 (2017) alternative to MCMC methods, such as Px-MALA and MYULA that were presented in Cai et al. (2017), we present new uncertainty quantification methods based on MAP estimation by convex optimisation. The proposed uncertainty quantification methods exhibit extremely fast computation speeds and allow uncertainty quantification to be performed practically and in a manner that will scale to the emerging big-data era of RI imaging. Our proposed methods, which inherit the advantages of convex optimisation methods, are much more efficient than proximal MCMC methods that explore the entire posterior distribution of the image. Note, however, that the methods proposed here give an approximation of HPD credible regions and, consequently, the additional forms of uncertainty quantification that are built on the approximate HPD credible regions are also approximate. Nevertheless, we show these approximations are very accurate. Moreover, the approximations are conservative so that uncertainties are not underestimated. In contrast, proximal MCMC methods can theoretically provide HPD credible regions and other forms of uncertainty quantification that are more accurate. Therefore, the proposed fast MAP-based methods and the proximal MCMC methods complement each other, rather than being mutually exclusive. We anticipate that when it comes to the big-data era, we will use predominantly fast uncertainty quantification methods such as those based on MAP estimation, and reserve MCMC methods for benchmarking and detailed comparison. A variety of forms of uncertainty quantification for MAP estimation were constructed, including HPD credible regions, local credible intervals (cf. error bars) for individual pixels and superpixels, and tests for image structure. Our methods were evaluated on four test images that are representative in RI imaging. These experiments demonstrated that our MAP-based methods exhibit excellent performance and can reconstruct images with sharp detail. Moreover, they simultaneously underpin highly accurate approximate techniques to quantify uncertainties. In terms of computation time, MAP techniques were found to be approximately 105 times faster than state-of-the-art proximal MCMC methods, even when MAP estimation is run on a standard laptop and proximal MCMC methods on a high-performance workstation. Moreover, they lead to algorithmic structures that can be highly distributed and parallelised. In the near future we will consider alternative approaches to 12 Cai et al. M31 tations and data. It is our hope that uncertainty quantification, e.g. in the form of recovering error bars (Bayesian credible intervals) and hypothesis testing of image structure, will become an important standard component in RI imaging for statistically principled and robust scientific inquiry. For the first time, we propose techniques for the practical quantification of uncertainties in RI imaging. These techniques can be applied not only to observations made by existing telescopes but also to the emerging big-data era of radio astronomy. W28 Cygnus A 1 1 ACKNOWLEDGEMENTS This work is supported by the UK Engineering and Physical Sciences Research Council (EPSRC) by grant EP/M011089/1, and Science and Technology Facilities Council (STFC) ST/M00113X/1. 1 REFERENCES 3C288 2 1 (a) MAP point estimators (b) inpainted surrogate Figure 9. Hypothesis testing for M31, Cygnus A, W28, and 3C288. The five structures depicted in yellow are considered, all of which are physical (i.e. present in the ground truth images), except for structure 2 in 3C288, which is a reconstruction artefact. First column (a): point estimators obtained by MAP estimation for the analysis model (9) (shown in log10 scale). Second column (b): segmented-inpainted surrogate test images with information in the yellow rectangular areas removed and replaced by inpainted background (shown in log10 scale). Hypothesis testing is then performed to test whether the structure considered is physical by checking whether the surrogate test images shown in (b) fall outside of the HPD credible regions. Results of these hypothesis tests are specified in Table 2. Note that for the case shown in the last row the structures within areas 1 and 2 are tested independently. perform hypothesis testing of image structure. For example, substructure can be tested by creating surrogate test images with the substructure in question removed, for example by smoothing the corresponding region. Such an approach would be valuable for improving our understanding of the astrophysical processes involved in radio galaxies, for example in the processes governing radio jets. Most importantly, we plan to apply the uncertainty quantification techniques presented in this article to RI observations acquired by a variety of different telescopes and to make the methods publicly available. 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Soft + Hardwired Attention: An LSTM Framework for Human Trajectory Prediction and Abnormal Event Detection arXiv:1702.05552v1 [cs.CV] 18 Feb 2017 Tharindu Fernandoa,∗, Simon Denmana , Sridha Sridharana , Clinton Fookesa a Image and Video Research Laboratory, SAIVT, Queensland University of Technology, Australia. Abstract As humans we possess an intuitive ability for navigation which we master through years of practice; however existing approaches to model this trait for diverse tasks including monitoring pedestrian flow and detecting abnormal events have been limited by using a variety of hand-crafted features. Recent research in the area of deeplearning has demonstrated the power of learning features directly from the data; and related research in recurrent neural networks has shown exemplary results in sequenceto-sequence problems such as neural machine translation and neural image caption generation. Motivated by these approaches, we propose a novel method to predict the future motion of a pedestrian given a short history of their, and their neighbours, past behaviour. The novelty of the proposed method is the combined attention model which utilises both “soft attention” as well as “hard-wired” attention in order to map the trajectory information from the local neighbourhood to the future positions of the pedestrian of interest. We illustrate how a simple approximation of attention weights (i.e hard-wired) can be merged together with soft attention weights in order to make our model applicable for challenging real world scenarios with hundreds of neighbours. The navigational capability of the proposed method is tested on two challenging publicly available surveillance databases where our model outperforms the currentstate-of-the-art methods. Additionally, we illustrate how the proposed architecture can ∗ Corresponding author Email address: t.warnakulasuriya@qut.edu.au. (Tharindu Fernando ) Preprint submitted to Neural Networks February 21, 2017 be directly applied for the task of abnormal event detection without handcrafting the features. Keywords: human trajectory prediction, social navigation, deep feature learning, attention models. 1. Introduction Understanding and predicting crowd behaviour in complex real world scenarios has a vast number of applications, from designing intelligent security systems to deploying socially-aware robots. Despite significant interest from researchers in domains such as 5 abnormal event detection, traffic flow estimation and behaviour prediction; accurately modelling and predicting crowd behaviour has remained a challenging problem due to its complex nature. As humans we possess an intuitive ability for navigation which we master through years of practice; and as such these complex dynamics cannot be captured with only a 10 handful of hand-crafted features. We believe that directly learning from the trajectories of pedestrians of interest (i.e. pedestrian who’s trajectory we seek to predict) along with their neighbours holds the key to modelling the natural ability for navigation we posses. The approach we present in this paper can be viewed as a data driven approach 15 which learns the relationship between neighbouring trajectories in an unsupervised manner. Our approach is motivated by the recent success of deep learning approaches (Goroshin et al. (2015); Madry et al. (2014); Lai et al. (2014)) in unsupervised feature learning for classification and regression tasks. 1.1. Problem Definition The problem we have addressed can be defined as follows: Assume that each frame in our dataset is first preprocessed such that we have obtained the spatial coordinates of each pedestrian at every time frame. Therefore the trajectory of the ith pedestrian for the time period of 1 to Tobs can be defined as, xi = [x1 , y1 , . . . , xTobs , yTobs ]. 2 (1) Figure 1: A sample surveillance scene (on the left): The trajectory of the pedestrian of interest is shown in green, and has two neighbours (shown in purple) to the left, one in front and none on right. Neighbourhood encoding scheme (on the right): Trajectory information is encoded with LSTM encoders. A soft attention context vector Cts is used to embed the trajectory information from the pedestrian of interest, and a hardwired attention context vector C h is used for neighbouring trajectories. In order to generate Cts we use a soft attention function denoted at in the above figure, and the hardwired weights are denoted by w. The merged context vector is then used to predict the future trajectory for the pedestrian of interest (shown in red). 20 The task we are interested in is predicting the trajectory of the ith pedestrian for the period of Tobs+1 to Tpred , having observed the trajectory of the ith pedestrian from time 1 to Tobs as well as the trajectories all the other pedestrians in the local neighbourhood during that period. This can be considered a sequence to sequence prediction problem where the input sequence captures contextual information corresponding to the spatial 25 location of the pedestrian of interest and their neighbours, and the output sequence contains the predicted future path of the pedestrian of interest. 1.2. Proposed Solution To solve this problem we propose a novel architecture as illustrated in Fig. 1. For encoding and decoding purposes we utilise Long-Short Term Memory networks 30 (LSTM) due to their recent success in sequence to sequence prediction (Bahdanau et al., 2014; Xu et al., 2015; Yoo et al., 2015). We demonstrate the social navigational capability of the proposed method on two challenging publicly available surveillance 3 databases. We demonstrate that our approach is capable of learning the common patterns in human navigation behaviour, and achieves improved predictions for pedestri35 ans paths over the current state-of-the-art methodologies. Furthermore, an application of the proposed method for abnormal human behaviour detection is shown in Section 5. 2. Related Work 2.1. Trajectory Clustering 40 When considering approaches for learning motion patterns through clustering, Giannotti et al. (2007) have proposed the concept of “trajectory patterns”, which represents the descriptions of frequent behaviours in terms of space and time. They have analysed GPS traces of a fleet of 273 trucks comprising a total of 112,203 points. Deviating from discovering common trajectories, Lee et al. (2007) proposed to discover 45 common sub-trajectories using a partition-and-group framework. The framework partitions each trajectory into a set of line segments, and forms clusters by grouping similar line segments based on density. Morris and Trivedi (2009) evaluated different similarity measures and clustering methodologies to uncover their strengths and weaknesses for trajectory clustering. With reference to their findings, the clustering method had 50 little effect on the quality of the results achieved; however selecting the appropriate distance measures with respect to the properties of the trajectories in the dataset had great influence on final performance. 2.2. Human Behaviour Prediction When predicting human behaviour the most common motion models are social 55 force models (Helbing and Molnár, 1995; Koppula and Saxena, 2013; Pellegrini et al., 2010; Yamaguchi et al., 2011; Xu et al., 2012) which generate attractive and repulsive forces between pedestrians. Several variants of such approaches exist. Alahi et al. (2014) represents it as a social affinity feature by learning the pedestrian trajectories with relative positions where as Yi et al. (2015a) observed the behaviour of stationary 60 crowd groups in order to understand crowd behaviour. With the aid of topic models the 4 authors in Wang et al. (2008) were able to learn motion patterns in crowd behaviour without tracking objects. This approach was extended to incorporate spatio-temporal dependencies in Hospedales et al. (2009) and Emonet et al. (2011). Deviating from the above approaches, a mixture model of dynamic pedestrian 65 agents is presented by Zhou et al. (2015), who also consider the temporal ordering of the observations. Yet, this model ignores the interactions among agents, a key factor when predicting behaviour in real world scenarios. The main drawback in all of the above methods is that they utilise hand-crafted features to model human behaviour and interactions. Hand-crafted features may only 70 capture abstract level semantics of the environment and they are heavily dependent on the domain knowledge that we posses. In Alahi et al. (2016) an unsupervised feature learning approach was proposed. The authors have generated multiple LSTMs for each pedestrian in the scene at that particular time frame. They have observed the position of all the pedestrians from time 75 1 to Tobs and predicted all of their positions for the period Tobs+1 to Tpred . They have pooled the hidden states of the immediately preceding time step for the neighbouring pedestrians when generating their positions in the current time step. A more detailed comparison of this model with our proposed model is presented in Sec. 3.3 80 2.3. Attention Models Attention-based mechanisms are motivated by the notion that, instead of decoding based on the encoding of a single element or fixed-length part of the input sequence, one can attend a specific area (or important areas) of the whole input sequence to generate the next output. Importantly, we let the model learn what to attend to based on 85 the input sequence and what it has produced so far. In Bahdanau et al. (2014) have shown that attention-based RNN models are useful for aligning input and output word sequences for neural machine translation. This was followed by the works by Xu et al. (2015) and Yao et al. (2015) for image and video captioning respectively. According to Sharma et al. (2015), attention based models 90 can be broadly categorised into soft attention and hard attention models, based on the 5 method that it uses to learn the attention weights. Soft attention models (Bahdanau et al., 2014; Xu et al., 2015; Yao et al., 2015; Sharma et al., 2015) can be viewed as “supervised” guiding mechanisms which learn the alignment between input and output sequences through backpropagation. Hard attention (Mnih et al., 2014; Williams, 95 1992) is used by Reinforcement Learning to predict an approximate location to focus on. With reference to Sharma et al. (2015), learning hard attention models can become computationally expensive as it requires sampling. Still, soft aligning multiple feature sequences is computationally inefficient as we need to calculate an attention value for each combination of input and output elements. 100 This is feasible in cases such as neural machine translation where we have a 50-word input sequence and generate a 50-word output sequence, but prohibitively expensive in a surveillance setting when a target has hundreds of neighbours, and we have to learn the attention weight values for all possible value combinations for each of the neighbouring trajectories. We tackle this problem via the merging of soft attention and 105 hardwired attention in our framework. 3. Proposed Approach 3.1. LSTM Encoder-Decoder Framework In contrast to past attention models (Bahdanau et al., 2014; Xu et al., 2015) which align a single input sequence with an output sequence, we need to consider multiple 110 feature sequences in the form of trajectory information from the pedestrian of interest and their neighbours when predicting the output sequence. Aligning all features together is not optimal as they have different degrees of influence (i.e. a person walking directly next to the target has greater influence than a person several meters away). Soft attention models are deterministic models which are trained using back-propagation. 115 Therefore, aligning each input trajectory sequence separately via a separate soft attention model is computationally expensive. We show that we can overcome this problem with a set of hardwired weights which we calculate based on the distance between each neighbour and the pedestrian of interest. When considering navigation, as distance is the key factor which determines the 6 yi,t-1 yi,t St-1 St Decoder Cx*t Cst at,1 Encoder h1 h2 at,2 at,3 at,T h3 xi,1 xi,2 xi,3 Pedestrian of interest hT xi,t Ch w1,1 w1,2 w1,T h’1,1 h’1,2 x1,1 x1,2 wj,1 h’1,T x1,t Neighbour 1 wj,2 … wj,T h’j,1 h’j,2 xj,1 xj,2 h’j,T xj,t Neighbour j Figure 2: The proposed Soft + Hardwired Attention model. We utilise the trajectory information from both the pedestrian of interest and the neighbouring trajectories. We embed the trajectory information from the pedestrian of interest with the soft attention context vector Cts , while neighbouring trajectories are embedded with the aid of a hardwired attention context vector C h . In order to generate Cts we use a soft attention function denoted at in the above figure, and the hardwired weights are denoted by w. Then the merged context vector, Ct∗ , is used to predict the future state yi(t) 120 neighbour’s influence, it acts as a good generalisation. The proposed LSTM Encoder-Decoder framework is shown in Fig. 2. Due to its computational complexities, soft attention (denoted at in Fig. 2) is used only when embedding the trajectory information from the pedestrian of interest. We show that by approximating the required attention for the neighbours through hardwired atten- 125 tion weights (wj ), which we calculate based on the distance between the neighbouring pedestrian and the pedestrian of interest, we can generate a good approximation of their influence. The methodology of generating soft and hardwired attentions is outlined in the following subsections. 3.1.1. LSTM Encoder In a general Encoder-Decoder framework, an encoder recieves an input sequence x from which it generates an encoded sequence h. In the context of this paper, the input 7 sequence for the pedestrian i is given in Equation 1 and the encoded sequence is given by, hi = [h1 , . . . , hT ]. (2) The encoding function is an LSTM, which can be denoted by, ht = LSTM(xt , ht−1 ). 130 (3) With the aid of above equation we encode the trajectory information from the pedestrian of interest as well as each trajectory in the local neighbourhood. 3.1.2. LSTM Decoder Before considering how the combine context vector Ct∗ is formulated, the concept of time dependent context vector can be illustrated as follows. For a general case, let st−1 be the decoder hidden state at time t − 1, yt−1 be the decoder output at time t − 1, Ct be the context vector at time t and f be the decoding function. The decoder output at time t is given by, yt = f (st−1 , yt−1 , Ct ), (4) as defined by Bahdanau et al. (2014) such that distinct context vectors are given for each time instant. The context vector depends on the encoded input sequence h = 135 [h1 , . . . , ht ]. In the proposed approach, the given trajectory (i.e x = [x1 , y1 , . . . , xTobs , yTobs ]) for the pedestrian of interest is encoded and used to generate a soft attention context vector, Cts . With the aid of distinct context vectors we are able to focus different degrees of attention towards different parts of the input sequence, when predicting the output sequence. The soft attention context vector Cts can be computed as a weighted sum of hidden states, Cts = T obs X αtj hj . (5) j=1 In Bahdanau et al. (2014), the authors have shown that the weight αtj can be computed by, exp(etj ) αtj = PT , k=1 exp(etk ) 8 (6) etj = a(st−1 , hj ), (7) and the function a is a feed forward neural network for joint training with other components of the system. As an extension to the decoder model proposed in Bahdanau et al. (2014), we have added a set of hardwired weights which we use to generate hardwired attention (C h ). 140 Utilising the hardwired attention model we combine the encoded hidden states of the neighbouring trajectories in the local neighbourhood. Hardwired attention weights are designed to incorporate the notion of distance between the pedestrian of interest and his or her neighbours into the trajectory prediction model. The closer a neighbouring pedestrian, the higher their associated weight, be- 145 cause that pedestrian has a greater influence on the trajectory that we are trying to predict. The simplest representation scheme can be given by, 1 w(n,j) = , dist(n, j) (8) where dist(n, j) is the distance between the nth neighbour and the pedestrian of interest at the j th time instance, and w(n,j) is the generated hardwired attention weight. This idea can be extended to generate the context vector for the hardwired attention 150 model. 0 Let there be N neighbouring trajectories in the local neighbourhood and h(n,j) be the encoded hidden state of the nth neighbour at the j th time instance, then the context vector for the hardwired attention model is defined as, Ch = N T obs X X 0 w(n,j) h(n,j) . (9) n=1 j=1 We then employ a simple concatenation layer to combine the information from individual attentions. Hence the combined context vector can be denoted as, Ct∗ = tanh(Wc [Cts ; C h ]), (10) where Wc is referred to as the set of weights for concatenation. We learn this weight value also through back-propagation. 9 The final prediction can now be computed as, yt = LSTM(st−1 , yt−1 , Ct∗ ), (11) where the decoding function f in Eq. 4 is replaced with a LSTM decoder as we are employing LSTMs for encoding and decoding purposes. 155 3.2. Model Learning The given input trajectories in the training set are clustered based on source and sink positions and we run an outlier detection algorithm for each cluster considering the entire trajectory. For clustering we used DBSCAN (Ester et al. (1996)) as it enables us to cluster the data on the fly without specifying the number of clusters. Hyper 160 parameters of the DBSCAN algorithm were chosen experimentally. In the training phase trajectories are clustered based on the entire trajectory and after clustering we learnt a separate trajectory prediction model for each generated cluster. When modelling the local neighbourhood of the pedestrians of interest, we have encoded the trajectories of those closest 10 neighbours in each direction, namely 165 front, left and right. If there exist more than 10 neighbours in any direction, we have taken the first (closest) 9 trajectories and the mean trajectory of the rest of the neighbours. If a trajectory has less than 10 neighbours, we create dummy trajectories such that we have 10 neighbours, and set the weight of these dummy neighbours to 0. When testing the model, we are concerned with predicting the pedestrian trajectory 170 given the first Tobs locations. To select the appropriate prediction model to use, the mean trajectory for each cluster for the period of 1 to Tobs is generated and in the testing phase, the given trajectories are assigned to the closest cluster centre while considering those mean trajectories as the cluster centroids. 3.3. Comparison to the Social-LSTM model of Alahi et al. (2016) In this section we draw comparisons between the current state-of-the-art technique and the proposed approach. In Alahi et al. (2016), for each neighbouring pedestrian, the hidden state at time t−1 is extracted out and fed as an input to the prediction model of the pedestrian of interest. Let there be N neighbours in the local neighbourhood and 10 0 h(n,t−1) be the hidden state of the nth neighbour at the time instance t − 1. Then the process can be written as, 0 Ht = N X 0 h(n,t−1) , (12) n=1 and the hidden state of the pedestrian of interest at the tth time instance is given by, 0 ht = LSTM(ht−1 , xt−1 , Ht ), 175 (13) where ht−1 refers to the hidden state and xt−1 refers to the position of the pedestrian 0 of interest at the t − 1 time instance. The authors are passing Ht and xt−1 through embedding functions before feeding it to the LSTM model, but in order to draw direct comparisons we are using the above notation. In Alahi et al. (2016), the hidden state of the pedestrian of interest at the tth time instance depends only on his or her previous 180 hidden state, the position in the previous time instance and the pooled hidden state of the immediately preceding time step for the neighbouring pedestrians (see Eq. 13). Comparing to our model, we are considering the entire set of hidden states for the pedestrian of interest as well as the neighbouring pedestrians when predicting the tth output element (see Eq. 5-11). 185 As humans we tend to vary our intentions time to time. For an example consider the problem of navigating in a train station. A person may start walking towards the desired platform and then realise that he hasn’t got a ticket and then make a sudden change and move towards the ticket counter. When applying the LSTM model proposed by Alahi et al. (2016) to such real world scenarios, by observing the immediate preceding 190 hidden state one can generate reactive behaviour to avoid collisions but when doing long term path planning, even though the LSTM is capable of handling long term relationships, the prediction process may go almost “blindly” towards the end of the sequence (Jia et al. (2015)) as we are neglecting vital information about pedestrian’s behaviour under varying contexts. In contrast, the proposed combined attention model 195 considers the entire sequence of hidden states for both the pedestrian of interest and his or her neighbours and then we utilise time dependent weights which enables us to vary their influence in a timely manner. Additionally, we observed that even in unstructured scenes such as train stations, 11 (a) First 2 clusters (b) Next 5 clusters (c) First 5 clusters (d) Next 5 clusters Figure 3: Clustering results for Grand Central (a, b) and Edinburgh Informatics Forum (c, d) Datasets airport terminals and shopping malls where multiple source and sink positions are 200 present, still there exists dominant motion patterns describing the navigation preference of the pedestrians. For instance taking the same train station example, although the main problem that we are trying to solve here is to navigating while avoiding collisions, humans demonstrate different preferences in doing so. One pedestrian may be there to meet passengers and hence is wandering in a free area, while another pedestrian 205 may aim to get in or out of the train station as quickly as possible. Therefore one single LSTM model is not sufficient to capture such ambiguities in navigational patterns. We observed that such distinct preferences in navigation generate unique trajectory patterns which can be easily segmented via the proposed clustering process. Therefore in contrast to Alahi et al. (2016), we are learning a different trajectory prediction model 210 for each trajectory cluster. 4. Experiments We present the experimental results on two publicly available human trajectory datasets: New York Grand Central (GC) (Yi et al. (2015b)) and Edinburgh Informat- 12 ics Forum (EIF) database (Majecka (2009)). The Grand Central dataset consist of 215 around 12,600 trajectories where as the Edinburgh Informatics Forum database contains around 90,000 trajectories. We have conducted 2 experiments. For the first experiment on the Grand Central dataset, after filtering out short and fragmented trajectories 1 , we are left with 8,000 trajectories, and train our model on 5,000 trajectories and evaluate the prediction accuracy on 3,000 trajectories. In the next experiment we con220 sidered 3 days worth trajectories from Edinburgh Informatics Forum database, trained our model on 10,000 trajectories and tested on 5,000 trajectories. Prior to learning trajectory models, we employ clustering to separate the different modes of human motion. This allows us to learn separate models for different behaviours, such as one model for a pedestrian who is buying tickets and another for 225 those who are directly entering or leaving the train station. We believe that these different motion patterns generate unique pedestrian behavioural styles and that are well captured through separate models. This can be achieved via clustering trajectories based on entire trajectory, but this will produce very large number of clusters or large number of outliers due to the wide variation in the different modes of human motion. 230 As a result, in each cluster, we would have very few examples to train our prediction models on. Therefore as a solution to the above stated problem we cluster the trajectories based only on the enter/exit zones. As illustrated in Fig. 3 this approach works reasonably well at separating different modes of human motion. Even with clustering based on entry and exit points, the way that a person moves 235 through the environment and how they are influenced by neighbours will vary considerably. For instance consider the clusters represented in green and blue in Fig. 3 (b). The way that an intersecting trajectory travelling straight up in the scene, from bottom left towards up left, will affect green and blue clusters differently because of their different exit zones. It’s general human nature to try to avoid collisions while keeping the 240 expected heading direction. Therefore entry/exit zones based clustering is sufficient to 1 We consider short trajectories to be those with length less than the time period that we are considering (40 frames) where as fragmented trajectories are trajectories which have discontinuities between 2 consecutive frames due to noise in the tracking process. 13 capture this. 4.1. Quantitative results Similar to Alahi et al. (2016) we report prediction accuracy with the following 3 error metrics. Let n be the number of trajectories in the testing set, xpred be the i,t 245 predicted position for the trajectory i at tth time instance, and xobs i,t be the respective observed positions then, 1. Average displacement error (ADE): TP pred n P ADE = 2 (xpred − xobs i,t ) i,t i=1 t=Tobs +1 n(Tpred − (Tobs + 1)) . (14) . (15) 2. Final displacement error (FDE) : n q P F DE = i=1 obs 2 (xpred i,Tpred − xi,Tpred ) n 3. Average non-linear displacement error (n-ADE): The average displacement error for the non-linear regions of the trajectory, n P n − ADE = TP pred i=1 t=Tobs +1 n P pred 2 I(xpred − xobs i,t ) i,t )(xi,t TP pred i=1 t=Tobs +1 , (16) I(xpred i,t ) where, I(xpred i,t ) =     1 if    0 o.w d2 yi,t dx2i,t 6= 0. (17) In all experiments we have observed the trajectory (and it’s neighbours) for 20 frames and predicted the trajectory for the next 20 frames. Compared to Alahi et al. (2016), which has considered sequences of 20 frames total length, we are considering 250 more lengthy sequences (with a total of 40 frames) as in Baccouche et al. (2011) the authors have shown that LSTM models tend to generate more accurate results with lengthy sequences. 14 In the experimental results, tabulated in Tab 1, we compare our prediction model with the state-of-the-art. As the baseline models we implemented Social Force (SF) 255 model from Yamaguchi et al. (2011) and Social LSTM (S-LSTM) model given in Alahi et al. (2016). For S-LSTM model a local neighbourhood of size 32px was considered and the hyper-parameters were set according to Alahi et al. (2016). In order to make direct comparisons with Alahi et al. (2016), the hidden state dimensions of encoders and decoders of all OUR models were set to be 300 hidden units. 260 For the SF model, preferred speed, destination, and social grouping factors are used to model the agent behaviour. When predicting the destination, a linear support vector machine was trained with the ground truth destination areas detected in the Sec. 3.2. In order to evaluate the strengths of the proposed model, we compare this combined attention model (OURcmb ) and two variations on our proposed approach: 1) OURsft , 265 which ignore the neighbouring trajectories and considers only the soft attention component derived from the trajectory of the person of interest when making predictions; and 2) OURsc which omits the clustering stage such that only a single model (using combined attention weights) is learnt. Metirc Dataset SF S-LSTM OURsc OURsft OURcmb GC 3.364 1.990 1.878 2.041 1.096 EIF 3.124 1.524 1.392 1.685 0.986 GC 5.808 4.519 4.317 5.277 3.011 EIF 3.909 2.510 2.345 3.089 1.311 GC 3.983 1.781 1.701 2.304 0.985 EIF 3.394 2.398 2.098 2.415 0.901 ADE FDE n-ADE Table 1: Quantitative results. In all the methods forecast trajectories are of length 20 frames. The first 2 rows represents the Average displacement error, rows 3 to 4 are for Final displacement error and the final 2 rows are for Average non-linear displacement error. The proposed model outperforms the SF model and S-LSTM model in both datasets. 270 The error reduction is more evident in the GC dataset where there are multiple source and sink positions, different crowd motion patterns are present and motion paths are heavily crowded. Comparing the results of OURsc (proposed approach without clus15 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) Figure 4: Qualitative results: Given (in green), Ground Truth (in Blue), Neighbouring (in purple) and Predicted trajectories from OURcmb model (in red), from S-LSTM model (in yellow), from SF model (in orange). (a) to (l) accurate predictions and (m) to (o) some erroneous predictions tering) against the S-LSTM model we can see that regardless of the clustering process the proposed combined attention architecture is capable of improving the trajectory 275 prediction. For all the measured error metrics OURsc has outperformed S-LSTM and (OURsft ) verifying that it is important to preserve historical data for both the pedestrian of interest as well as the neighbours. Secondly those results show that hard wired 16 weights act as a good approximation of neighbours influence. When comparing results of OURcmb against OURsc it is evident that the cluster280 ing process has partitioned the trajectories based on these different semantics and via utilising separate models for each cluster, we were able to generate more accurate predictions. The combined attention model was capable of learning how the neighbours influence the current trajectory and how this impact varies under different neighbourhood locations. 285 Furthermore, we would like to point out that, because of the separate model learning process we were able to predict the final destination positions with more precision compared to baseline models where they do not consider the environmental configurations of the unstructured scene. While the proposed approach does not explicitly model the environment, a certain amount of environmental information is inherently encoded 290 in the entry and exit locations, which the proposed approach is able to leverage. 4.2. Qualitative results In Fig. 4 we show prediction results of the S-LSTM model, SF model and our combined attention model (OURcmb ) on the GC dataset. It should be noted that our model generates better predictions in heavily crowded areas. As we are learning a sep- 295 arate model for each cluster, the prediction models are able to learn different patterns of influences from neighbouring pedestrians. For instance in the 1st and 3rd column we demonstrate how the model adapts in order to avoid collisions. In the last row of Fig. 4 we show some failure cases. The reason for such deviations from the ground truth were mostly due to sudden changes in destination. Even though these trajectories do 300 not match the ground truth, the proposed method still generates plausible trajectories. For instance, in Fig. 4 (m) and Fig. 4 (o) the model moves side ways to avoid collusion with the neighbours in the left and right directions. Three example scenarios that illustrates the advantage of attending to all the hidden states within that particular context are shown in Fig. 5. The first row shows an exam- 305 ple where the pedestrian of interest exhibits 2 modes of motion (walking and running) within the same trajectory. In the second row we have an example where the previous context of the pedestrian of interest is useful in tthe final prediction. Third row shows 17 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) Figure 5: Example scenarios: Columns (left to right): first observation of the trajectory; half way through; last observation prior to prediction; prediction from the respective models. 1st row shows an example where the pedestrian exhibits 2 modes of motion (walking and running) within the same trajectory. 2nd row shows an example where the previous context of the pedestrian is useful in prediction. 3rd row shows an example where the pedestrian is moving as part of a group. Colours: Given trajectory (in green), Ground Truth (in Blue), Neighbouring (in purple) and Predicted trajectories from OURcmb model (in red), from S-LSTM model (in yellow), from OURsc model (in cyan). In order to preserve visibility without occlusions we have plotted only the closest 2 neighbours in each direction. an example where pedestrian exhibits a group motion. The first three columns show the progression of the motion from the start of the trajectory (first column) to the point 310 directly before the prediction is made (third column). The given trajectory of the person of interest is shown in green and the neighbouring trajectories are shown in purple. In order to preserve visibility without occlusions we have plotted only the closest 2 neighbours in each direction. Finally in the fourth column we have presented the respective predictions from OURcmb model (in red), from S-LSTM model (in yellow), 315 from OURsc model (in cyan). The ground truth observations are shown in blue. When considering the example shown in Fig. 5 (a)-(d) the pedestrian of interest exhibits a running behaviour when entering the scene (Fig. 5 (a)). Then at half way through her trajectory she shifts her behaviour to walking (Fig. 5 (b)). Therefore at 18 the time of the prediction (Fig. 5 (c)), the hidden states of the baseline model will be 320 dominated by the walking behaviour as it is the most recent behaviour of this particular pedestrian. Therefore the predictions generated by the S-LSTM model are erroneous. But as we are attending to all the previous hidden states before predicting the future sequence, the multi model nature of that pedestrian’s motion has been captured by the OURsc and OURcmb models. 325 Another example scenario where the proposed model out performs it’s baseline is shown in Fig. 5 (e)-(h). When deciding upon whether to give way or to cut through the pedestrian group, models OURsc and OURcmb outperforms the S-LSTM model. While attending to the historical data in 5 (f) we can see the behaviour of the pedestrian of interest under a similar context. Therefore the proposed models can anticipate that 330 the preferred behaviour of the pedestrian of interest under such context is giving way to the others. But in the S-LSTM model as it’s attending only to the immediate preceding hidden states and those long range dependencies are not captured. In the example shown in Fig. 5 (i)-(l) we illustrate the importance of considering the entire history of the neighbours. The way that the neighbours affect to a pedestrian dur- 335 ing group motion vastly differs to the impact group motion has on a pedestrian walking alone. For example pedestrians moving as part of a group tend to walk at a similar velocity, keeping small distances between themselves, stopping or turing together; where as pedestrians walking alone try to keep a safe distance between themselves and their neighbours to avoid collisions. These notions can be quickly captured while observing 340 the neighbourhood history. Therefore the predictions generated by both OURsc and OURcmb models outperforms the baseline S-LSTM model which only considered the immediate preceding hidden state of the neighbours when generating the predictions. When comparing the predictions from OURsc model against OURcmb model it is evident that the more spatial context specific predictions are generated by OURcmb 345 as it has been specifically trained on the examples from that particular spatial region. Therefore it anticipates the motion of a particular pedestrian more accurately. Still in all the example scenarios OURsc is shown to be capable of generating acceptable predictions compared to the baseline model, showing that the proposed combined attention mechanism is capable of generating more accurate and realistic trajectories than 19 350 the current state-of-the-art. 5. Abnormal behaviour detection The proposed framework can be directly applied for detecting abnormal pedestrian behaviour. A naive approach would be to predict the trajectory for the period of Tobs+1 to Tpred while observing the same trajectory over this time period and measuring the 355 deviation between the observed and the predicted trajectories. If the deviation is greater than a threshold, then an abnormality can be said to have occurred. However due to the adaptive nature of deep neural networks, abnormal behaviours such as: i) sudden turns and changes in walking directions; and ii) trajectories with abnormal velocities; may not be classified as abnormal events. 360 We observe that the hidden states of the LSTM encoder decoder framework hold vital information which is used to model the walking behaviour of the pedestrian of interest. Hence, if his or her behaviour is abnormal then the hidden state values for that pedestrian should be distinct from those of a normal pedestrian. With that intuition we randomly selected 500 trajectories from the Grand Central 365 dataset and predicted the trajectories for those pedestrians. The trajectories were hand labeled for abnormal behaviour, considering sudden turns and changes in walking direction and abnormal velocities as the set of abnormal behaviours. The dataset consists of 445 normal trajectories and 55 abnormal trajectories. Then we extracted the encoded hidden states (h(t) = [h(1) , . . . , h(Tobs ) ]) for the given trajectory for that pedestrian and 370 the hidden states used for decoding (s(t) = [s(Tobs+1 ) , . . . , s(Tpred ) ]). The resultant hidden states are passed through DBSCAN to detect outliers. With the proposed approach we detected 441 trajectories as being normal and 59 trajectories as abnormal. The resultant detections are given in Table 2. Analysing the classification results, we see that false alarms are mainly due to be- 375 haviours that are erroneously detected as abnormal being uncommon in the database. Cases such as people changing direction to buy tickets, and passengers wandering in the free area are detected as abnormal due to the fact that they are not significantly present in the subset of trajectories selected for this task. 20 Ground Truth Abnormal Normal Abnormal 47 12 Normal 8 433 Total 55 445 Predicted Table 2: Abnormal Event detection with the proposed algorithm: This approach has detected 47 out of 55 ground truth abnormal events We compare this approach to the naive approach given above. It is evident that 380 some abnormal trajectories are misclassified as normal behaviour due to its lack of deviation from the observed trajectory. See Tab. 3 Ground Truth Abnormal Normal Abnormal 29 24 Normal 26 421 Total 55 445 Predicted Table 3: Abnormal event detection with naive approach: This approach has detected only 29 out of 55 ground truth abnormal events Some examples of detected abnormal events are shown in Fig. 6. The first row shows the abnormal behaviour detected due to sudden change of moving direction. Even though, in the examples shown (d) and (e), there isn’t a significant deviation be385 tween the predicted path and the observed path, our abnormal event detection approach has accurately classified the event due to the sudden circular turn in the trajectory in (d) and abnormal velocity in (e). 6. Conclusion In this paper we have proposed a novel neural attention based framework to model 390 pedestrian flow in a surveillance setting. We extend the classical encoder-decoder 21 (b) (a) (c) (e) (d) Figure 6: Abnormal event detections: (a)-(c) abnormal behaviour detected due to sudden change of moving direction. Abnormal behaviour due to sudden circular turn (d) and abnormal velocity in (e) framework in sequence to sequence modelling to incorporate both soft attention as well as hard-wired attention. This has a major positive impact when handling longer trajectories in heavily cluttered neighbourhoods. The hand-crafted hard-wired attention weights approximate the neighbour’s influence and make the application of atten395 tion models pursuable for real world scenarios with large number of neighbours. We tested our proposed model in two challenging publicly available surveillance datasets and demonstrated state-of-the-art performance. Our new neural attention framework exhibited a stronger ability to accurately predict pedestrian motion, even in the presence of multiple source and sink positions and with high crowd densities observed. 400 Furthermore, we have shown how the proposed approach can support abnormal event detection through hidden state clustering. 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DeepConf: Automating Data Center Network Topologies Management with Machine Learning arXiv:1712.03890v1 [cs.NI] 11 Dec 2017 Christopher Streiffer∗ , Huan Chen∗† , Theophilus Benson+ , Asim Kadav‡ Duke University∗ UESTC, China† Brown University+ NEC Labs‡ ABSTRACT In recent years, many techniques have been developed to improve the performance and efficiency of data center networks. While these techniques provide high accuracy, they are often designed using heuristics that leverage domain-specific properties of the workload or hardware. In this vision paper, we argue that many data center networking techniques, e.g., routing, topology augmentation, energy savings, with diverse goals actually share design and architectural similarity. We present a design for developing general intermediate representations of network topologies using deep learning that is amenable to solving classes of data center problems. We develop a framework, DeepConf, that simplifies the processing of configuring and training deep learning agents that use the intermediate representation to learns different tasks. To illustrate the strength of our approach, we configured, implemented, and evaluated a DeepConf-agent that tackles the data center topology augmentation problem. Our initial results are promising — DeepConf performs comparably to the optimal. 1. INTRODUCTION Data center networks (DCN) are a crucial and important part of the Internet’s ecosystem. The performance of these DCNs impact a wide variety of services from web browsing and videos to Internet of Things. The poor performance of these DCNs can result in as much as $4 million in lost revenue [1]. Motivated by the importance of these networks, the networking community has explored techniques for improving and managing the performance of the data center network topology by: (1) designing better routing or traffic engineering algorithms [6, 8, 13, 10], (2) improving performance of a fixed topology by adding a limited number of flexible links [33, 11, 17, 16], and (3) removing corrupted and underutilized links from the topology to save energy and improve performance [18, 14, 35]. Regardless of the approach, these topology-oriented techniques have three things in common: (1) Each is formalized as an optimization problem. (2) Due to the impracticality of scalably solving these optimizations, greedy heuristics are employed to create approximate solutions. (3) Each heuristic is intricately tied to the application patterns and does not generalize across novel patterns. Existing domain-specific heuristics provide suboptimal performance and are often limited to specific scenarios. Thus as a community, we are forced to revisit and redesign these heuristics when the application pattern or network details changes – even a minor change. For example, while c-through [33] and FireFly [17] solve broadly identical problems, they leverage different heuristics to account for low-level differences. In this paper, we articulate our vision for replacing domain-specific rule-based heuristics for topology management with a more general machine learning-based (ML) model that quickly learns optimal solutions to a class of problems while adapting to changes in the application patterns, network dynamics, and low-level network details. Unlike recent attempts that employ ML to learn point solutions, e.g., cluster scheduling [23] or routing [9], in this paper, we present a general framework, called DeepConf, that simplifies the process of designing ML models for a broad range of DCN topology problems and eliminates the challenges associated with efficiently training new models. The key challenges in designing DeepConf are: (1) tackling the dichotomy that exists between deep learning’s requirements for large amounts of supervised data and the unavailability of these required datasets, and (2) designing a general, but highly accurate, deep learning model that efficiently generalizes to learning a broad array of data center problems ranging from topology management and routing to energy savings. The key insight underlying DeepConf is that intermediate features generated from the parallel convolutional layers using network data, e.g., traffic matrix, allows us to generate an intermediate representation of the network’s state that enables learning a broad range of data center problems. Moreover, while labeled production data crucial for machine learning is unavailable, empir- ical studies [21] show that modern data center traffic is higly predictable and thus amenable to offline learning with network simulators and historical traces. DeepConf builds on this insight by using reinforcement learning (RL), an unsupervised technique, that learns through experience and makes no assumptions on how the network works. Rather, they are trained through the use of a reward signal which “guides” them towards an optimal solution and thus do not require real world data, and, instead, they can be trained using simulators. The DeepConf framework provides a predefined RL model with the intermediate representation, a specific design for configuring this model to address different problems, an optimized simulator to enable efficient learning, and an SDN-based platform for capturing network data and reconfiguring the network. In this paper, we make the following contributions: • We present a novel RL-based SDN architecture for developing and training deep ML models for a broad range of DCN tasks. • We design a novel input feature extraction for DCNs for developing different ML models over this intermediate representation of network state. • We implemented a DeepConf-agent tackling the topology augmentation problem and evaluated it on representative topologies [15, 4] and traces [3], showing that our approach performs comparable to optimal. 2. RELATED WORK Our work is motivated by the recent success of applying machine learning and RL algorithms to computer games and robotic planning [25, 31, 32]. The most closely related work [9] applies RL to packet routing. Unlike [9], DeepConf tackles the topology augmentation problem and explores the use of deep networks as function approximators for RL. Existing applications of machine learning to data centers focus on improving cluster scheduling [23] and more recently by Google to optimize Power Usage Effectiveness(PUE) [2]. In this vision paper, we take a different stance and focus on identifying a class of equivalent data center management operations, namely topology management and configuration, that are amenable to a common machine learning approach and design a modular system that enables different agents to interoperate over a network. 3. BACKGROUND This section provides an overview of data center networking challenges and solutions and provides background on our reinforcement learning methods. Ethernet Fabric Optical Circuit Optical Switch Figure 1: One optical switch with k=4 Fat Tree Topology. 3.1 Data Center Networks Data center networks introduce a range of challenges from topology design and routing algorithms to VM placement and energy saving techniques. Data centers support a large variety of workloads and applications with time-varying bandwidth requirements. This variance in bandwidth requirements leads to hotspots at varying locations and at different points in time. To support these arbitrary bandwidth requirements, data center operators can employ non-blocking topologies; however, non-blocking topologies are prohibitively costly. Instead, these operators employ a range of techniques ranging from hybrid architectures [33, 11, 17, 16], traffic engineering algorithms [6, 8, 13, 10], and energy saving techniques [18, 28]. Below, we describe these techniques and illustrate common designs. Augmented Architectures: This class of approaches build on the intuition that at any given point in time, there are only a small number of hotspots (or congested links). Thus, there is no need to build an expensive topology that supports full bisection (eliminating all potential points of congestion). Instead, the existing topology can be augmented with a small number of links which can be added ondemand and moved to the location of the hotspot or congested links. These approaches augment the data center’s ethernet network with a small number of optical [33, 11], wireless [16], or free optics [17]. 1 For example, Figure 1 shows a traditional Fat-Tree topology augmented with an optical switch – as was proposed by Helios [11]. These proposals argue for monitoring the traffic, using an Integer Linear Program (ILP) or heuristic to detect the hotspots and place the flexible links at the location with these hotspots. Unfortunately, moving these flexible links incurs a large switching time during which the links are not operational. These intelligent and efficient algorithms are developed to effectively detect hotspots and efficiently place links. Traffic Engineering: Orthogonal approaches [6, 8, 13, 10] focus on routing. Instead of changing the topologies, these approaches change the mapping of flows to paths within a fixed topology. These proposals, also, argue for monitoring traffic and detecting hotspots. Instead of changing topologies, these techniques move a subset of 1 The number of augmented links is significantly smaller than the number of data center’s permanent links. flows from congested links to un-congested links. Energy Savings Data centers are notorious for their energy usage [14]. To address this, researchers have proposed techniques to improve energy efficiency by detecting periods of low utilization and selectively turning off links [18, 28]. These proposals argue for monitoring traffic, detecting low utilization, and powering-down links in portions of the data center with low utilizations. A key challenge with these techniques is to turn on the powered-down links before demands rise. Taking a step back, these techniques roughly follow the same design and operate in three steps (1) gather network traffic matrix, (2) run an ILP to predict heavy (or low) usage, and (3) perform a specific action on a subset of the network. The actions range from augmenting flexible links, turning off links, or moving traffic. In all situations, the ILP does not scale to a large network and a domain-specific heuristic is often used in its place. 3.2 Reinforcement Learning Reinforcement learning (RL) algorithms learn through experience with a goal towards maximizing rewards. Unlike supervised learning where algorithms train over labels, RL algorithms learn by interacting with an environment such as a game or a network simulator. In traditional RL, an agent interacts with an environment over a number of discrete time steps. Hence, at each time step t, the agent in a world observes a state st in order to select an action at from a possible action set A. The agent is guided by a policy, π, which is a function that maps state st to actions at . The agent receives a reward rt for each action and transitions to the next state st+1 . The goal of the agent is maximizing the total reward. This process continues until the agent reaches a final state or time limit, after which the environment is reset and a new training episode is played. After a number of training episodes, the agent learns to pick actions that maximize the rewards and can learn to handle unexpected states. RL is effective and has been successfully used to model robotics, game bots, etc. The goal of commonly used policy-based RL is to find a policy, π, that maximizes the cumulative reward and converges to a theoretical optimal policy. In deep policy-based methods, a neural network computes a policy distribution π(at \|st ; θ), where θ represents the set of parameters of the function. Deep networks as function approximators is a recent development and other learning methods can be used. We now describe the REINFORCE and actor-critic policy methods which represent different methods to score the policy J(θ). REINFORCE methods use gradient ascent on E[Rt ], P∞ [34] i where Rt = i=0 γ rt+i is the accumulated reward starting from time step t and discounted at each step by γ ∈ (0, 1], the discount factor. The REINFORCE method, which is the Monte-Carlo method, updates θ using the gradient ∇θ log π(at \|st ; θ)Rt , which is an unbiased estimator of ∇θ E[Rt ]. The value function is computed as V π (st ) = E[Rt \|st ] which is the expected return for following the policy π in state st . This method provides actions with high returns but suffers from highvariance of gradient estimates. Asynchronous Advantage Actor Critic (A3C): A3C [24] improves REINFORCE performance by operating asynchronously and by using a deep network to approximate the policy and value faction. A3C uses the actor-critic method which additionally computes a critic function that approximates the value function. A3C, as used by us, uses a network with two convolutional layers followed by a fully connected layer. Each hidden layer is followed by a nonlinearity function (ReLU). A softmax layer which approximates the policy function and a linear layer to output an estimate of the value function V (st ; θ) together constitute the output of this network. Asynchronous gradient descent using multiple agents is used to train the network and this improves the training speed. A central server (similar to a parameter server) coordinates the parallel agents – each agent calculates the gradients and sends the updates to the server after a fixed number of steps, or when a final state is reached. Furthermore, following each update, the central server propagates new weights to the agents to achieve a consensus on the policy values. There is a cost function with each deep network (policy and value). Using two loss functions has found to improve convergence and produce better-regularized models. The policy cost function is given as: fπ (θ) = log π (at \|st ; θ) (Rt − V (st ; θt ))+βH (π (st ; θ)) (3.1) where θt represents the values of the parameters θ at Pk−1 time t, Rt = i=0 γ i rt+i + γ k V (st+k ; θt ) is the estimated discounted reward.H (π (st ; θ)) is used to favor exploration and its strength is controlled by the factor β. The cost function for the estimated value function is: 2 fv (θ) = (Rt − V (st ; θ)) (3.2) Additionally, we augment our A3C model to learn current states apart from accumulating rewards for good configurations using GAE [30]. The deep network, that replaces the transition matrix as the function approximator learns the value of the given state and the policy of the given state. The model uses GAE to compute the value of a given state that not only returns the reward for the model for the given policy decision but also rewards the model for estimating the value of the state. This helps to guide the model to learn the states instead of just maximizing rewards. 4. VISION Training Harness Network Simulator DeepConf Agent DeepConf DeepConf …… Agent Agent DeepConf Abstraction Layer SDN Controller …… Network Topology Figure 2: DeepConf Architecture Our vision is to automate a subset of data center management and operational tasks by leveraging DeepRL. At a high-level, we anticipate the existence of several DeepRL agents, each trained for a specific set of tasks e.g. traffic-engineering, energy-savings, or topologyaugmentations. Each agent will run as an application atop an SDN controller. The use of SDN provides the agents with an interface for gathering their required network state and a mechanism for enforcing their actions. For example, DeepConf should be able to assemble the traffic matrix by polling the different devices within the network, compute a decision for how the optical switch should be configured to best accommodate the current network load, and reconfigure the network. At a high-level, DeepConf’s architecture consists of three components (Figure 2): the network simulator to enable offline training of the DeepRL agents, the DeepConf abstraction layer to facilitate communication between the DeepRL agents and the network, and the DeepRL agents, called DeepConf-agents, which encapsulate data center functionality. Applying learning: The primary challenges in applying machine learning to network problems are (1) the deficiency of training data pertaining to operator and network behavior and (2) the lack of models and loss functions that can accurately model the problem and generalize to unexpected situations. This shortage presents a roadblock for using supervised-based approaches for training ML models. To address this issue, DeepConf uses RL where the model is trained by exploring different network states and environments generated by the surplus of simulators available in the network community. Coupled with the wide availability of network job traces, this allows for DeepConf to learn a highly generalizable policy. DeepConf Abstraction Layer: Today’s SDN controllers expose a primitive interface with low-level information. The DeepConf applications will instead require highlevel models and interfaces. For example, our agents will require interfaces that provide control over paths rather than over flow table entries. While emerging approaches [19] argue for similar interfaces, these approaches do not provide a sufficiently rich set of interfaces for the broad range of agents we expect to support and do not provide composition primitives for safely combining the output from the different agents. Moreover, existing composition operators [27, 20, 12] assume that the different SDN applications (or DeepConf-agent in our case) are generating non-conflicting actions – hence these operators can not tackling conflict actions. SDN Composition approaches [29, 7, 26] that do tackle conflicting actions, require significant rewrite of the SDNApp which we are unable to do because DeepConfagent are rewritten within the DeepRL paradigm. More concretely, we require high-layer SDN abstractions that enable to RLAgents to more easily learn and act of the network. Additionally, we require novel composition operators that can reason about and tackle conflicting actions generated by RLAgents. Domain-specific Simulators: (Low hanging fruit) Existing SDN research leverages a popular emulation platform, Mininet, which fails to scale to large experiments. A key requirement for employing DeepRL is to have efficient and scalable simulators that replays traces and enables learning from these traces. We extend flowbased simulators to model the various dimensions that are required to train our models. To improve efficiency, we explore techniques that partition the simulation and enables reuse of results — in essence, to enable incremental simulations. In addressing our high-level vision and developing solutions to the above challenges, there are several highlevel goals that a production-scale system must address: (1) our techniques must generalize across topologies, traffic matrixes, and a range of operational settings, e.g., link failures; (2) our techniques must be as accurate and efficient as existing state-of-the-art techniques; and (3) our solutions must incur low operational overheads, e.g., minimizing optical switching time or TCAM utilization. 5. DESIGN In this section, we provide a broad description of how to define and structure existing data center network management techniques as RL tasks, then describe the methods for training the resulting DeepConf-agents. 5.1 DeepConf Agents In defining each new DeepConf-agent, there are four main functions that a developer must specify: state space, action space, learning model, and reward function. The action space and reward are both specific to the management task being performed and are, in turn, unique to the agent. Respectively, they express the set of actions an agent can take during each step and the reward for the actions taken. The state space and learning models are more general and can be shared and reused across different DeepConf-agents. This is because of the fundamental similarities shared between the data center management problems, and because the agents are specifically designed for data centers. In defining a DeepConf agent for the topology augmentation problem, we (1) define state-spaces specific to the topology, (2) design a reward function based on application level metrics, and (3) define actions that correlate to activating/de-activating links. State Space: In general, the state space consists of two types of data – each reflecting the state of the network at a given point in time. First, the general network state that all DeepConf-agents require: the network’s traffic matrix (TM) which contains information on the flows which will executed during the last t seconds of the simulation. Second, a DeepConf-agent specific state-space that captures the impact of the actions on the network. For example, for the topology augmentation problem, this would be the network topology – note, the actions change the topology. Whereas for the traffic engineering problem, this would be a mapping of flows to paths – note, the actions change a flow’s routes. Learning Model: Our learning model utilizes a Convolutional Neural Network (CNN) to compute policy decisions. The exact model for a DeepConf-agent depends on the number of state-spaces used as input. In general, the model will have as many CNN-blocks as there are state spaces – one CNN-block for each state space. The output of these blocks are concatenated together and input into two fully connected layers, followed by a softmax output layer. For example, for the topology-augmentation problem, as observed in Figure 4, our DeepConf-agent has two CNN blocks to operate on both the topology and the TM states spaces in parallel. This allows for the lower CNN layers to perform feature extraction on the input spatial data, and for the fully connected layers to assemble these features in a meaningful way. 5.2 Network Training To train a DeepConf-agent, we run the agent against a network simulator. The interaction between the simulator (described in Section 7) and RL agent can be described as follows (Figure 3): (1) The DeepConf agent receives state si from the simulator at training step i. (2) The DeepConf-agent uses the state information to make a policy decision about the network, and returns the selected actions to the simulator. For the DeepConfagent for the topology augmentation problem, called the Augmentation-Agent, the actions are the links to activate and hence a new topology. (3) If the topology (5) Get reward Simulator Flows Network Topology (4) Update route (3) Add links Control Interface (1) Read state DeepConf Agent (2) Decision Figure 3: DeepConf-agent model training. changes, the simulator re-computes the paths for the active flows. (4) The simulator executes the flows for t seconds. (5) The simulator returns the reward ri and state si+1 to the DeepConf-agent, and the process restarts. Initialization During the initial training phase, we force the model to explore the environment by randomizing the selection process — the probability that an action is picked corresponds to the value of the index which represents the action. For instance, with the AugmentationAgent, link i has probability wi of being selected. As the model becomes more familiar with the states and corresponding values, the model will better formulate its policy decision. At this point, the model will associate a higher probability with the links it believes to have a higher reward, which will cause these links to be selected at a higher frequency. This methodology allows for the model to reinforce its decisions about links, while the randomization helps the model avoid local-minima. Learning Optimization: To improve the efficiency of learning, the RL agent maintains a log containing the state, policy decision, and corresponding reward. The RL agent performs experience replay after n simulation steps. During replay, the log is unrolled to compute the policy loss across the specified number of steps using Equation 3.1. The agent is trained using Adam stochastic optimization [22], with an initial learning rate of 10−4 and a learning rate decay factor of 0.95. We found that a smaller learning rate and low decay helped the model better explore the environment and form a more optimal policy. 6. USE CASE: AUGMENTATION AGENT More formally defined, in the topology augmentation problem the data center consists of a fixed hierarchical topology and an optical switch, which connects all the top-of-rack switches. While the optical switch is physically connected to all ToR switches, unfortunately, the optical switch can only support a limited number of active links. Given this limitation, the rules for the topology problem are defined as: • The model must select n links to activate at a given step during the simulation. • The model receives a reward based on the link utilization and the flow duration. 12 DeepConf Optimal Flow Completion Time (s) 10 • All flows are routed using equal-cost multi-path routing (ECMP). State Space: The agent specific state space is the network topology, which is represented by a sparse matrix where entries within the cells correspond to active links within the network. Action Space: The RL agent interacts with the environment by adding links between edge switches. The action space for the model, therefore, corresponds to the different possible link combinations and is represented as an n dimensional vector. The values within the vector correspond to a probability distribution, where wi is equal to the probability of link i being the optimal pick for the given input state s. The model selects the highest n values from this distribution as the links that should be added to the network topology. Reward: The goal of the model can be summarized as: (1) Maximize link utilization and (ii) Minimize the average flow-completion time. With this in mind, we formulate our reward function as: X X bf (6.1) R(Θ, s, i) = tf 6 4 2 0 Figure 4: The CNN model utilized by the RL agent. • The model collects the reward on a per-link basis after t seconds of simulation. 8 FatTree Topology VL2 Figure 5: Median Flow Completion Time. simulator using a large scale map-reduce traces from Facebook [3]. We evaluate two state-of-the-art clos-style data center topologies: K=4 Fat-tree [5] and VL2 [15]. In our analysis, we focus on flow completion time (FCT) a metric which captures the duration between the first and last packet of a flow. We augment both topologies by adding an optical switch with four links. Here we compare DeepConf against the optimal solution derived from a linear program — Note: this optimal solution can not be solved with larger topologies [8]. Learning Results: The training results demonstrate that the RL agent learns to optimize its policy decision to increase the total reward received across each episode. We observed that the loss decreases as training increases, with the largest decrease occurs during the initial training episodes, a result consistent with the learning rate decay factor employed during training. Performance Results: The results, Figure 5, show that DeepConf performs comparable with optimal [33, 11] across representative topologies and workloads. Thus, our system is able to learn a solution that’s close to the optimal across a range of topologies. Takeaway We believe these initial results are promising, and that more work is required in order to understand and improve the performance of DeepConf. f ∈F l∈f Where F represents all active and completed flows during the previous iteration step, l represents the links used by flow f , bf represents the number of bytes transferred during the step time, and tf represents the total duration of the flow. The purpose of this reward function is to reward for high link utilization but penalize for long lasting flows. The design of this function has the effect of guiding the model towards completing large flows within a smaller period of time. 7. EVALUATION In this section, we analyze DeepConf under realistic workload with representative topologies. Experiment Setup We evaluate DeepConf on a trace driven flow-level 8. DISCUSSION We now discuss open questions: Learning to generalize: In order to avoid over-fitting to a specific topology, we train our agent over a large number of simulator configurations. DeepRL agents need to be trained and evaluated on many different platforms to avoid being overtly specific to few networks and correctly handle unexpected scenarios. Solutions that employ machine learning to address network problems using simulators need to be cognizant of these issues when deciding the training data. Learning new reward functions: DeepRL methods need appropriate reward functions to ensure that they optimize for the correct goals. 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arXiv:1507.06792v2 [stat.ME] 31 Mar 2017 Efficient Estimation for Diffusions Sampled at High Frequency Over a Fixed Time Interval Nina Munkholt Jakobsen Michael Sørensen∗ Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Ø Denmark munkholt@math.ku.dk Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Ø Denmark michael@math.ku.dk April 3, 2017 Abstract Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (in-fill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variance-mixtures, where the mixing distribution generally depends on the full sample path of the diffusion process over the observation time interval. Utilising the concept of stable convergence, we also obtain the more easily applicable result that for a suitable data dependent normalisation, the estimators converge in distribution to a standard normal distribution. The theory is illustrated by a simulation study comparing an efficient and a non-efficient estimating function for an ergodic and a non-ergodic model. Key words: Approximate martingale estimating functions, discrete time sampling of diffusions, in-fill asymptotics, normal variance-mixtures, optimal rate, random Fisher information, stable convergence, stochastic differential equation. Running title: Efficient Estimation for High Frequency SDE Data. 1 1 Introduction Diffusions given by stochastic differential equations find application in a number of fields where they are used to describe phenomena which evolve continuously in time. Some examples include agronomy (Pedersen, 2000), biology (Favetto and Samson, 2010), finance (Merton, 1971; Vasicek, 1977; Cox et al., 1985; Larsen and Sørensen, 2007) and neuroscience (Ditlevsen and Lansky, 2006; Picchini et al., 2008; Bibbona et al., 2010). While the models have continuous-time dynamics, data are only observable in discrete time, thus creating a demand for statistical methods to analyse such data. With the exception of some simple cases, the likelihood function is not explicitly known, and a large variety of alternate estimation procedures have been proposed in the literature, see e.g. Sørensen (2004) and Kessler et al. (2012). Parametric methods include the following. Maximum likelihood-type estimation, primarily using Gaussian approximations to the likelihood function, was considered by Prakasa Rao (1983), Florens-Zmirou (1989), Yoshida (1992), Genon-Catalot and Jacod (1993), Kessler (1997), Jacod (2006), Gloter and Sørensen (2009) and Uchida and Yoshida (2013). Analytical expansions of the transition densities were investigated by Aït-Sahalia (2002, 2008) and Li (2013), while approximations to the score function were studied by Bibby and Sørensen (1995), Kessler and Sørensen (1999), Jacobsen (2001, 2002), Uchida (2004), and Sørensen (2010). Simulation-based likelihood methods were developed by Pedersen (1995), Roberts and Stramer (2001), Durham and Gallant (2002), Beskos et al. (2006, 2009), Golightly and Wilkinson (2006, 2008), Bladt and Sørensen (2014), and Bladt et al. (2016). A large part of the parametric estimators proposed in the literature can be treated within the framework of approximate martingale estimating functions, see the review in Sørensen (2012). In this paper, we derive easily verified conditions on such estimating functions that imply rate optimality and efficiency under a high frequency asymptotic scenario, and thus contribute to providing clarity and a systematic approach to this area of statistics. Specifically, the paper concerns parametric estimation for stochastic differential equations of the form dXt = a(Xt ) dt + b(Xt ; θ) dWt , (1.1) where (Wt )t≥0 is a standard Wiener process. The drift and diffusion coefficients a and b are deterministic functions, and θ is the unknown parameter to be estimated. The drift function a needs not be known, but as examples in this paper show, knowledge of a can be used in the construction of estimating functions. For ease of exposition, Xt and θ are both assumed to be one-dimensional. The extension of our results to a multivariate parameter is straightforward, and it is expected that multivariate diffusions can be treated in a similar way. For n ∈ N, we consider observations (Xt0n , Xt1n , . . . , Xtnn ) in the time interval [0, 1], at discrete, equidistant time-points tin = i/n, i = 0, 1, . . . , n. We investigate the high frequency scenario where n → ∞. The choice of the time-interval [0, 1] is not restrictive since results generalise to other compact intervals by suitable rescaling of the drift and diffusion coef2 ficients. The drift coefficient does not depend on any parameter, because parameters that appear only in the drift cannot be estimated consistently in our asymptotic scenario. It was shown by Dohnal (1987) and Gobet (2001) that under the asymptotic scenario con√ sidered here, the model (1.1) is locally asymptotic mixed normal with rate n and random asymptotic Fisher information !2 Z 1 ∂θ b(X s ; θ) I(θ) = 2 ds. (1.2) b(X s ; θ) 0 √ Thus, a consistent estimator θ̂n is rate optimal if n(θ̂n − θ0 ) converges in distribution to a non-degenerate random variable as n → ∞, where θ0 is the true parameter value. The estimator is efficient if the limit may be written on the form I(θ0 )−1/2 Z, where Z is standard normal distributed and independent of I(θ0 ). The concept of local asymptotic mixed normality was introduced by Jeganathan (1982), and is discussed in e.g. Le Cam and Yang (2000, Chapter 6) and Jacod (2010). Estimation for the model (1.1) under the high frequency asymptotic scenario described above was considered by Genon-Catalot and Jacod (1993, 1994). These authors proposed estimators based on a class of contrast functions that were only allowed to depend on the −1/2 n ; θ) and ∆ n ). The estimators observations and the parameter through b2 (Xti−1 (Xtin − Xti−1 n were shown to be rate optimal, and an efficient contrast function was identified. Dohnal (1987) gave estimators for particular cases of the model (1.1). Apart from one instance, these estimators are not of the type investigated by Genon-Catalot and Jacod (1993, 1994), but all apart from one are covered by the theory in the present paper. In this paper, we investigate estimators based on the extensive class of approximate martingale estimating functions Gn (θ) = n X n ; θ) g(∆n , Xtin , Xti−1 i=1 n ; θ) \| with ∆n = 1/n, where the real-valued function g(t, y, x; θ) satisfies that Eθ (g(∆n , Xtin , Xti−1 κ n ) is of order ∆ for some κ ≥ 2. Estimators are obtained as solutions to the estimatXti−1 n ing equation Gn (θ) = 0 and are referred to as Gn -estimators. Exact martingale estimating functions, where Gn (θ) is a martingale, constitute a particular case that is not covered by the theory in Genon-Catalot and Jacod (1993, 1994). An example is the maximum likeli√ hood estimator for the Ornstein-Uhlenbeck process with a(x) = −x and b(x; θ) = θ, for which g(t, y, x; θ) = (y − e−t x)2 − 21 θ(1 − e−2t ). A simpler example of an estimating function for the same Ornstein-Uhlenbeck process that is covered by our theory, but is not of the Genon-Catalot & Jacod-type, is given by g(t, y, x; θ) = (y − (1 − t)x)2 − θt. The class of approximate martingale estimating functions was also studied by Sørensen (2010), who considered high frequency observations in an increasing time interval for a model like (1.1) where also the drift coefficient depends on a parameter. Specifically, the observation times were tin = i∆n with ∆n → 0 and n∆n → ∞. Simple conditions on g for rate optimality and efficiency were found under the infinite horizon high frequency 3 asymptotics. To some extent, the methods of proof in the present paper are similar to those in Sørensen (2010). However, while ergodicity of the diffusion process played a central role in that paper, this property is not needed here. Another important difference is that expansions of a higher order are needed in the present paper, which complicates the proofs considerably. Furthermore, the theory in the current paper requires a more complicated version of the central limit theorem for martingales, and we need the concept of stable convergence in distribution, in order to obtain practically applicable convergence results. First, we establish results on existence and uniqueness of consistent Gn -estimators. We √ show that n(θ̂n − θ0 ) converges in distribution to a normal variance-mixture, which implies rate optimality. The limit distribution may be represented by the product W(θ0 )Z of independent random variables, where Z is standard normal distributed. The random variable W(θ0 ) is generally non-degenerate, and depends on the entire path of the diffusion process over the time-interval [0, 1]. Normal variance-mixtures were also obtained as the asymptotic distributions of the estimators of Genon-Catalot and Jacod (1993). These distributions appear as limit distributions in comparable non-parametric settings as well, e.g. when estimating integrated volatility (Jacod and Protter, 1998; Mykland and Zhang, 2006) or the squared diffusion coefficient (Florens-Zmirou, 1993; Jacod, 2000). Rate optimality is ensured by the condition that ∂y g(0, x, x; θ) = 0 (1.3) for all x in the state space of the diffusion process, and all parameter values θ. Here ∂y g(0, x, x; θ) denotes the first derivative of g(0, y, x; θ) with respect to y evaluated in y = x. The same condition was found in Sørensen (2010) for rate optimality of an estimator of the parameter in the diffusion coefficient, and it is one of the conditions for small ∆-optimality; see Jacobsen (2001, 2002). Due to its dependence on (X s ) s∈[0,1] , the limit distribution is difficult to use for statistical applications, such as constructing confidence intervals and test statistics. Therefore, we b that converges in probability to W(θ0 ). Using the stable convergence construct a statistic W √ n in distribution of n(θ̂n − θ0 ) towards W(θ0 )Z, we derive the more easily applicable result √ b n−1 (θ̂n − θ0 ) converges in distribution to a standard normal distribution. that n W The additional condition that ∂2y g(0, x, x; θ) = Kθ ∂θ b2 (x; θ) b4 (x; θ) (1.4) (Kθ , 0) for all x in the state space, and all parameter values θ, ensures efficiency of Gn estimators. The same condition was obtained by Sørensen (2010) in his infinite horizon scenario for efficiency of estimators of parameters in the diffusion coefficient. It is also identical to a condition given by Jacobsen (2002) for small ∆-optimality. The identity of the conditions implies that examples of approximate martingale estimating functions which are rate optimal and efficient in our asymptotic scenario may be found in Jacobsen (2002) and Sørensen (2010). In particular, estimating functions that are optimal in the sense of Godambe and Heyde (1987) are rate optimal and efficient under weak regularity conditions. 4 The paper is structured as follows: Section 2 presents definitions, notation and terminology used throughout the paper, as well as the main assumptions. Section 3 states and discusses our main results, while Section 4 presents a simulation study illustrating the results. Section 5 contains main lemmas used to prove the main theorem, and proofs of the main theorem and the lemmas. Appendix A consists of auxiliary technical results, some of them with proofs. 2 2.1 Preliminaries Model and Observations Let (Ω, F ) be a measurable space supporting a real-valued random variable U, and an independent standard Wiener process W = (Wt )t≥0 . Let (Ft )t≥0 denote the filtration generated by U and W. Consider the stochastic differential equation dXt = a(Xt ) dt + b(Xt ; θ) dWt , X0 = U , (2.1) for θ ∈ Θ ⊆ R. The state space of the solution is assumed to be an open interval X ⊆ R, and the drift and diffusion coefficients, a : X → R and b : X × Θ → R, are assumed to be known, deterministic functions. Let (Pθ )θ∈Θ be a family of probability measures on (Ω, F ) such that X = (Xt )t≥0 solves (2.1) under Pθ , and let Eθ denote expectation under Pθ . Let tin = i∆n with ∆n = 1/n for i ∈ N0 , n ∈ N. For each n ∈ N, X is assumed to be sampled at times tin , i = 0, 1, . . . , n, yielding the observations (Xt0n , Xt1n , . . . , Xtnn ). Let Gn,i denote the σ-algebra generated by the observations (Xt0n , Xt1n , . . . , Xtin ), with Gn = Gn,n . 2.2 Polynomial Growth In the following, to avoid cumbersome notation, C denotes a generic, strictly positive, realvalued constant. Often, the notation Cu is used to emphasise that the constant depends on u in some unspecified manner, where u may be, e.g., a number or a set of parameter values. Note that, for example, in an expression of the form Cu (1 + \|x\|Cu ), the factor Cu and the exponent Cu need not be equal. Generic constants Cu often depend (implicitly) on the unknown true parameter value θ0 , but never on the sample size n. A function f : [0, 1] × X2 × Θ → R is said to be of polynomial growth in x and y, uniformly for t ∈ [0, 1] and θ in compact, convex sets, if for each compact, convex set K ⊆ Θ there exist constants C K > 0 such that sup t∈[0,1], θ∈K \| f (t, y, x; θ)\| ≤ C K (1 + \|x\|C K + \|y\|C K ) for x, y ∈ X. pol Definition 2.1. C p,q,r ([0, 1] × X2 × Θ) denotes the class of real-valued functions f (t, y, x; θ) which satisfy that 5 j (i) f and the mixed partial derivatives ∂it ∂y ∂kθ f (t, y, x; θ), i = 0, . . . , p, j = 0, . . . , q and k = 0, . . . , r exist and are continuous on [0, 1] × X2 × Θ. (ii) f and the mixed partial derivatives from (i) are of polynomial growth in x and y, uniformly for t ∈ [0, 1] and θ in compact, convex sets. pol pol pol pol Similarly, the classes C p,r ([0, 1] × X × Θ), Cq,r (X2 × Θ), Cq,r (X × Θ) and Cq (X) are defined for functions of the form f (t, x; θ), f (y, x; θ), f (y; θ) and f (y), respectively. Note that in Definition 2.1, differentiability of f with respect to x is never required. For the duration of this paper, R(t, y, x; θ) denotes a generic, real-valued function defined on [0, 1] × X2 × Θ, which is of polynomial growth in x and y uniformly for t ∈ [0, 1] and θ in compact, convex sets. The function R(t, y, x; θ) may depend (implicitly) on θ0 . Functions R(t, x; θ), R(y, x; θ) and R(t, x) are defined correspondingly. The notation Rλ (t, x; θ) indicates that R(t, x; θ) also depends on λ ∈ Θ in an unspecified way. 2.3 Approximate Martingale Estimating Functions Definition 2.2. Let g(t, y, x; θ) be a real-valued function defined on [0, 1]×X2 ×Θ. Suppose the existence of a constant κ ≥ 2, such that for all n ∈ N, i = 1, . . . , n, θ ∈ Θ, n ). n ; θ) \| Xtn (2.2) = ∆κn Rθ (∆n , Xti−1 Eθ g(∆n , Xtin , Xti−1 i−1 Then, the function Gn (θ) = n X n ; θ) g(∆n , Xtin , Xti−1 (2.3) i=1 is called an approximate martingale estimating function. In particular, when (2.2) is satisfied with Rθ (t, x) ≡ 0, (2.3) is referred to as a martingale estimating function. By the Markov property of X, it follows that if Rθ (t, x) ≡ 0, then (Gn,i )1≤i≤n defined by Gn,i (θ) = i X g(∆n , Xtnj , Xtnj−1 ; θ) j=1 is a zero-mean, real-valued (Gn,i )1≤i≤n -martingale under Pθ for each n ∈ N. The score function of the observations (Xt0n , Xt1n , . . . , Xtnn ) is a martingale estimating function under weak regularity conditions, and an approximate martingale estimating function can be viewed as an approximation to the score function. A Gn -estimator θ̂n is essentially obtained as a solution to the estimating equation Gn (θ) = 0. A more precise definition is given in the following Definition 2.3. Here we make the ωdependence explicit by writing Gn (θ, ω) and θ̂n (ω). Definition 2.3. Let Gn (θ, ω) be an approximate martingale estimating function as defined in Definition 2.2. Put Θ∞ = Θ ∪ {∞} and let Dn = {ω ∈ Ω \| Gn (θ, ω) = 0 has at least one solution θ ∈ Θ} . 6 A Gn -estimator θ̂n (ω) is any Gn -measurable function Ω → Θ∞ which satisfies that for Pθ0 almost all ω, θ̂n (ω) ∈ Θ and Gn (θ̂n (ω), ω) = 0 if ω ∈ Dn , while θ̂n (ω) = ∞ if ω < Dn . For any Mn , 0, the estimating functions Gn (θ) and MnGn (θ) yield identical estimators of θ and are therefore referred to as versions of each other. For any given estimating function, it is sufficient that there exists a version of the function which satisfies the assumptions of this paper, in order to draw conclusions about the resulting estimators. In particular, we can multiply by a function of ∆n . 2.4 Assumptions We make the following assumptions about the stochastic differential equation. Assumption 2.4. The parameter set Θ is a non-empty, open subset of R. Under the probability measure Pθ , the continuous, (Ft )t≥0 -adapted Markov process X = (Xt )t≥0 solves a stochastic differential equation of the form (2.1), the coefficients of which satisfy that pol a(y) ∈ C6 (X) pol b(y; θ) ∈ C6,2 (X × Θ) . and The following holds for all θ ∈ Θ. (i) For all y ∈ X, b2 (y; θ) > 0. (ii) There exists a real-valued constant Cθ > 0 such that for all x, y ∈ X, \|a(x) − a(y)\| + \|b(x; θ) − b(y; θ)\| ≤ Cθ \|x − y\| . (iii) U has moments of any order. The global Lipschitz condition, Assumption 2.4.(ii), ensures that a unique solution X exists. The Lipschitz condition and (iii) imply that supt∈[0,1] Eθ (\|Xt \|m ) < ∞ for all m ∈ N. Assumption 2.4 is very similar to the corresponding Condition 2.1 of Sørensen (2010). However, an important difference is that in the current paper, X is not required to be ergodic. Here, law of large numbers-type results are proved by what is, in essence, the convergence of Riemann sums. We make the following assumptions about the estimating function. Assumption 2.5. The function g(t, y, x; θ) satisfies (2.2) for some κ ≥ 2, thus defining an approximate martingale estimating function by (2.3). Moreover, pol g(t, y, x; θ) ∈ C3,8,2 ([0, 1] × X2 × Θ) , and the following holds for all θ ∈ Θ. (i) For all x ∈ X, ∂y g(0, x, x; θ) = 0. 7 (ii) The expansion g(∆, y, x; θ) = g(0, y, x; θ) + ∆g(1) (y, x; θ) + 12 ∆2 g(2) (y, x; θ) + 16 ∆3 g(3) (y, x; θ) + ∆4 R(∆, y, x; θ) (2.4) holds for all ∆ ∈ [0, 1] and x, y ∈ X, where g( j) (y, x; θ) denotes the j0 th partial derivative of g(t, y, x; θ) with respect to t, evaluated in t = 0. Assumption 2.5.(i) was referred to by Sørensen (2010) as Jacobsen’s condition, as it is one of the conditions for small ∆-optimality in the sense of Jacobsen (2001), see Jacobsen (2002). The assumption ensures rate optimality of the estimators in this paper, and of the estimators of the parameters in the diffusion coefficient in Sørensen (2010). The assumptions of polynomial growth and existence and boundedness of all moments serve to simplify the exposition and proofs, and could be relaxed. 2.5 The Infinitesimal Generator pol For λ ∈ Θ, the infinitesimal generator Lλ is defined for all functions f (y) ∈ C2 (X) by Lλ f (y) = a(y)∂y f (y) + 21 b2 (y; λ)∂2y f (y) . pol For f (t, y, x, θ) ∈ C0,2,0,0 ([0, 1] × X2 × Θ), let Lλ f (t, y, x; θ) = a(y)∂y f (t, y, x; θ) + 21 b2 (y; λ)∂2y f (t, y, x; θ) . (2.5) Often, the notation Lλ f (t, y, x; θ) = Lλ ( f (t; θ))(y, x) is used, so e.g. Lλ ( f (0; θ))(x, x) means Lλ f (0, y, x; θ) evaluated in y = x. In this paper the infinitesimal generator is particularly useful because of the following result. Lemma 2.6. Suppose that Assumption 2.4 holds, and that for some k ∈ N0 , pol a(y) ∈ C2k (X) , pol b(y; θ) ∈ C2k,0 (X × Θ) and pol f (y, x; θ) ∈ C2(k+1),0 (X2 × Θ) . Then, for 0 ≤ t ≤ t + ∆ ≤ 1 and λ ∈ Θ, Eλ ( f (Xt+∆ , Xt ; θ) \| Xt ) = k X ∆i i=0 i! Liλ f (Xt , Xt ; θ) + ∆ Z u1 Z uk Z ··· 0 0 0 Eλ Lk+1 f (X , X ; θ) \| X duk+1 · · · du1 t+u t t k+1 λ where, furthermore, Z ∆ Z u1 Z ··· 0 0 0 uk Eλ Lk+1 f (X , X ; θ) \| X duk+1 · · · du1 = ∆k+1 Rλ (∆, Xt ; θ) . t+u t t k+1 λ 8 The expansion of the conditional expectation in powers of ∆ in the first part of the lemma corresponds to Lemma 1 in Florens-Zmirou (1989) and Lemma 4 in Dacunha-Castelle and Florens-Zmirou (1986). It may be proven by induction on k using Itô’s formula, see, e.g., the proof of Sørensen (2012, Lemma 1.10). The characterisation of the remainder term follows by applying Corollary A.5 to Lk+1 λ f , see the proof of Kessler (1997, Lemma 1). For concrete models, Lemma 2.6 is useful for verifying the approximate martingale property (2.2) and for creating approximate martingale estimating functions. In combination with (2.2), the lemma is key to proving the following Lemma 2.7, which reveals two important properties of approximate martingale estimating functions. Lemma 2.7. Suppose that Assumptions 2.4 and 2.5 hold. Then g(0, x, x; θ) = 0 and g(1) (x, x; θ) = −Lθ (g(0, θ))(x, x) for all x ∈ X and θ ∈ Θ. Lemma 2.7 corresponds to Lemma 2.3 of Sørensen (2010), to which we refer for details on the proof. 3 Main Results Section 3.1 presents the main theorem of this paper, which establishes existence, uniqueness and asymptotic distribution results for rate optimal estimators based on approximate martingale estimating functions. In Section 3.2 a condition is given, which ensures that the rate optimal estimators are also efficient, and efficient estimators are discussed. 3.1 Main Theorem The final assumption needed for the main theorem is as follows. Assumption 3.1. The following holds Pθ -almost surely for all θ ∈ Θ. (i) For all λ , θ, 1 Z 0 b2 (X s ; θ) − b2 (X s ; λ) ∂2y g(0, X s , X s ; λ) ds , 0 , (ii) 1 Z 0 ∂θ b2 (X s ; θ)∂2y g(0, X s , X s ; θ) ds , 0 , (iii) 1 Z 0 2 b4 (X s ; θ) ∂2y g(0, X s , X s ; θ) ds , 0 . 9 Assumption 3.1 can be difficult to check in practice because it involves the full sample path of X over the interval [0, 1]. It requires, in particular, that for all θ ∈ Θ, with Pθ -probability one, t 7→ b2 (Xt ; θ) − b2 (Xt ; λ) is not Lebesgue-almost surely zero when λ , θ. As noted by Genon-Catalot and Jacod (1993), this requirement holds true (by the continuity of the function) if, for example, X0 = U is degenerate at x0 , and b2 (x0 ; θ) , b2 (x0 ; λ) for all θ , λ. For an efficient estimating function, Assumption 3.1 reduces to conditions on X with no further conditions on the estimating function, see the next section. Specifically, the conditions involve only the squared diffusion coefficient b2 (x; θ) and its derivative ∂θ b2 . Theorem 3.2. Suppose that Assumptions 2.4, 2.5 and 3.1 hold. Then, (i) there exists a consistent Gn -estimator θ̂n . Choose any compact, convex set K ⊆ Θ with θ0 ∈ int K, where int K denotes the interior of K. Then, the consistent Gn estimator θ̂n is eventually unique in K, in the sense that for any Gn -estimator θ̃n with Pθ0 (θ̃n ∈ K) → 1 as n → ∞, it holds that Pθ0 (θ̂n , θ̃n ) → 0 as n → ∞. (ii) for any consistent Gn -estimator θ̂n , it holds that √ D n(θ̂n − θ0 ) −→ W(θ0 )Z . (3.1) The limit distribution is a normal variance-mixture, where Z is standard normal distributed, and independent of W(θ0 ) given by 1 Z W(θ0 ) = 0 1 4 2 b (X s ; θ0 ) 1 Z 0 2 ∂2y g(0, X s , X s ; θ0 ) !1/2 ds . (3.2) 2 2 1 2 ∂θ b (X s ; θ0 )∂y g(0, X s , X s ; θ0 ) ds (iii) for any consistent Gn -estimator θ̂n , bn = − W  1/2 n  1 X  2  n ; θ̂n ) g (∆n , Xtin , Xti−1  ∆n i=1 n X (3.3) n ; θ̂n ) ∂θ g(∆n , Xtin , Xti−1 i=1 P b n −→ W(θ0 ), and satisfies that W √ D bn−1 (θ̂n − θ0 ) −→ nW N(0, 1) . The proof of Theorem 3.2 is given in Section 5.1. 10 Dohnal (1987) and Gobet (2001) showed local asymptotic mixed normality with rate so Theorem 3.2 establishes rate optimality of Gn -estimators. √ n, Observe that the limit distribution in Theorem 3.2.(ii) generally depends on not only the unknown parameter θ0 , but also on the concrete realisation of the sample path t 7→ Xt over [0, 1], which is only partially observed. Note also that a variance-mixture of normal distributions can be very different from a Gaussian distribution. It can be much more heavytailed and even have no moments. Theorem 3.2.(iii) is therefore important as it yields a standard normal limit distribution, which is more useful in practical applications. 3.2 Efficiency Under the assumptions of Theorem 3.2, the following additional condition ensures efficiency of a consistent Gn -estimator. Assumption 3.3. Suppose that for each θ ∈ Θ, there exists a constant Kθ , 0 such that for all x ∈ X, ∂2y g(0, x, x; θ) = Kθ ∂θ b2 (x; θ) . b4 (x; θ) Dohnal (1987) and Gobet (2001) showed that the local asymptotic mixed normality property holds within the framework considered here with random Fisher information I(θ0 ) given by (1.2). Thus, a Gn -estimator θ̂n is efficient if (3.1) holds with W(θ0 ) = I(θ0 )−1/2 , and the following Corollary 3.4 may easily be verified. Corollary 3.4. Suppose that the assumptions of Theorem 3.2 and Assumption 3.3 hold. Then, any consistent Gn -estimator is also efficient. It follows from Theorem 3.2 and Lemma 5.1 that if Assumption 3.3 holds, and if Gn is normalized such that Kθ = 1, then √ where 1 D bn2 (θ̂n − θ0 ) −→ N(0, 1) , nI n X bn = 1 n ; θ̂n ). I g2 (∆n , Xtin , Xti−1 ∆n i=1 It was noted in Section 2.3 that not necessarily all versions of a particular estimating function satisfy the conditions of this paper, even though they lead to the same estimator. Thus, an estimating function is said to be efficient, if there exists a version which satisfies the conditions of Corollary 3.4. The same goes for rate optimality. Assumption 3.3 is identical to the condition for efficiency of estimators of parameters in the diffusion coefficient in Sørensen (2010), and to one of the conditions for small ∆-optimality given in Jacobsen (2002). 11 Under suitable regularity conditions on the diffusion coefficient b, the function ḡ(t, y, x; θ) = ∂θ b2 (x; θ) 2 2 (y − x) − tb (x; θ) b4 (x; θ) (3.4) yields an example of an efficient estimating function. The approximate martingale property (2.2) can be verified by Lemma 2.6. When adapted to the current framework, the contrast functions investigated by GenonCatalot and Jacod (1993) have the form Un (θ) = n 1X 2 −1/2 n ; θ), ∆ n ) , f b (Xti−1 (Xtin − Xti−1 n n i=1 for functions f (v, w) satisfying certain conditions. For the contrast function identified as efficient by Genon-Catalot and Jacod, f (v, w) = log v + w2 /v. Using that ∆n = 1/n, it is P n ; θ) then seen that their efficient contrast function is of the form Ūn (θ) = ni=1 ū(∆n , Xtin , Xti−1 with ū(t, y, x; θ) = t log b2 (x; θ) + (y − x)2 /b2 (x; θ) and ∂θ ū(t, y, x; θ) = −ḡ(t, y, x; θ). In other words, it corresponds to a version of the efficient approximate martingale estimating function given by (3.4). The same contrast function was considered by Uchida and Yoshida (2013) in the framework of a more general class of stochastic differential equations. A problem of considerable practical interest is how to construct estimating functions that are rate optimal and efficient, i.e. estimating functions satisfying Assumptions 2.5.(i) and 3.3. Being the same as the conditions for small ∆-optimality, the assumptions are, for example, satisfied for martingale estimating functions constructed by Jacobsen (2002). As discussed by Sørensen (2010), the rate optimality and efficiency conditions are also satisfied by Godambe-Heyde optimal approximate martingale estimating functions. Consider martingale estimating functions of the form g(t, y, x; θ) = a(x, t; θ)∗ f (y; θ) − φtθ f (x; θ) , where a and f are two-dimensional, ∗ denotes transposition, and φtθ f (x; θ) = Eθ ( f (Xt ; θ) \| X0 = x). Suppose that f satisfies appropriate (weak) conditions. Let ā be the weight function for which the estimating function is optimal in the sense of Godambe and Heyde (1987), see e.g. Heyde (1997) or Sørensen (2012, Section 1.11). It follows by an argument analogous to the proof of Theorem 4.5 in Sørensen (2010) that the estimating function with g(t, y, x; θ) = tā(x, t; θ)∗ [ f (y; θ) − φtθ f (x; θ)] satisfies Assumptions 2.5.(i) and 3.3, and is thus rate optimal and efficient. As there is a simple formula for ā (see Section 1.11.1 of Sørensen (2012)), this provides a way of constructing a large number of efficient estimating functions. The result also holds if φtθ f (x; θ) and the conditional moments in the formula for ā are suitably approximated by the help of Lemma 2.6. 12 Remark 3.5. Suppose for a moment that the diffusion coefficient of (2.1) has the form b2 (x; θ) = h(x)k(θ) for strictly positive functions h and k, with Assumption 2.4 satisfied. This holds true, e.g., for a number of Pearson diffusions, including the (stationary) Ornstein-Uhlenbeck and square root processes. (See Forman and Sørensen (2008) for more on Pearson diffusions.) Then I(θ0 ) = ∂θ k(θ0 )2 /(2k2 (θ0 )). In this case, under the assump√ tions of Corollary 3.4, an efficient Gn -estimator θ̂n satisfies that n(θ̂n − θ0 ) −→ Y in distribution where Y is normal distributed with mean zero and variance 2k2 (θ0 )/∂θ k(θ0 )2 , i.e. the limit distribution is not a normal variance-mixture depending on (Xt )t∈[0,1] . Note also that when b2 (x; θ) = h(x)k(θ) and Assumption 3.3 holds, then Assumption 3.1 is satisfied when, e.g., ∂θ k(θ) > 0 or ∂θ k(θ) < 0. ◦ 4 Simulation study This section presents a simulation study illustrating the theory in the previous section. An efficient and an inefficient estimating function are compared for two models, an ergodic and a non-ergodic model. For both models the limit distributions of the consistent estimators are non-degenerate normal variance-mixtures. First, consider the stochastic differential equation dXt = −2Xt dt + (θ + Xt2 )−1/2 dWt , X0 = 0, (4.1) where θ ∈ (0, ∞) is an unknown parameter. The solution X is ergodic with invariant probability density proportional to exp −2θx2 − x4 θ + x2 , x ∈ R. The process satisfies Assumption 2.4. We compare the two estimating functions given by n X Gn (θ) = g(∆n , X , X tin n ti−1 ; θ) and Hn (θ) = i=1 n X n ; θ) h(∆n , Xtin , Xti−1 i=1 where g(t, y, x; θ) = (y − (1 − 2t)x)2 − (θ + x2 )−1 t h(t, y, x; θ) = (θ + x2 )10 (y − (1 − 2t)x)2 − (θ + x2 )9 t . Both g and h satisfy Assumptions 2.5 and 3.1. Moreover, g satisfies the condition for efficiency, while h is not efficient. Let WG (θ0 ) and WH (θ0 ) be given by (3.2), that is 1 Z WG (θ0 ) = − Z 1 2 0 1 !−1/2 1 ds and WH (θ0 ) = − (θ0 + X s2 )2 !1/2 2(θ0 + 0 1 Z 0 X s2 )18 ds . (4.2) (θ0 + X s2 )8 ds Numerical calculations and simulations were done in R 3.1.3 (R Core Team, 2014). First, m = 104 trajectories of the process X given by (4.1) were simulated over the time-interval 13 ● ● ● 4 2 0 −2 −4 ● ● ● −2 0 2 4 4 ● ●● ●● 0 2 4 ● ● ●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● 2 0 ● ●● ● ● ● −4 −4 2 0 −2 −2 ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● −4 4 −4 ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● −2 −4 −2 0 2 ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● −4 −2 0 2 4 −4 −2 0 2 4 bG,n (left) and Z bH,n (right) to the N(0, 1) distribution in the Figure 1: QQ-plots comparing Z case of the ergodic model (4.1) for n = 103 (above) and n = 104 (below). [0, 1] with θ0 = 1. These simulations were performed using the Milstein scheme as implemented in the R-package sde (Iacus, 2014) with step size 10−5 . The simulations were subsampled to obtain samples sizes of n = 103 and n = 104 . Let θ̂G,n and θ̂H,n denote estimates of θ obtained by solving the equations Gn (θ) = 0 and Hn (θ) = 0 numerically, on bG,n and W b H,n were calculated by (3.3). the interval [0.01, 1, 99]. Using these estimates, W For n = 103 , θ̂H,n could not be computed for 30 of the m = 104 sample paths. For n = 104 , and for the efficient estimator θ̂G,n there were no problems. Figure 1 shows QQ-plots of bG,n = Z √ b −1 (θ̂G,n − θ0 ) nW G,n and bH,n = Z √ b −1 (θ̂H,n − θ0 ) , nW H,n compared with a standard normal distribution, for n = 103 and n = 104 respectively. These QQ-plots suggest that, as n goes to infinity, the asymptotic distribution in Theorem 3.2.(iii) becomes a good approximation faster in the efficient case than in the inefficient case. 14 0.25 5 0.20 4 0.15 3 0.10 2 0.00 0.05 1 0 1.4 1.6 1.8 2.0 2.2 2.4 2.6 5 10 15 eG Figure 2: Approximation to the densities of WG (θ0 ) (left) and WH (θ0 ) (right) based on W e and WH in the case of the ergodic model (4.1). Inserting θ0 = 1 into (4.2), the integrals in these expressions may be approximated by Riemann sums, using each of the simulated trajectories of X (with sample size n = 104 eG and W e H to for maximal accuracy). This method yields a second set of approximations W the realisations of the random variables WG (θ0 ) and WH (θ0 ), presumed to be more accurate bG,104 and W b H,104 as they utilise the true parameter value. The density function in R than W was used (with default arguments) to compute an approximation to the densities of WG (θ0 ) eG and W eH. and WH (θ0 ), using the approximate realisations W It is seen from Figure 2 that the distribution of WH (θ0 ) is much more spread out than the distribution of WG (θ0 ). This corresponds well to the limit distribution in Theorem 3.2.(ii) being more spread out in the inefficient case than in the efficient case. Along the same lines, √ √ Figure 3 shows similarly computed densities based on n(θ̂G,n − θ0 ) and n(θ̂H,n − θ0 ) for n = 104 , which may be considered approximations to the densities of the normal variancemixture limit distributions in Theorem 3.2.(ii). These plots also illustrate that the limit distribution of the inefficient estimator is more spread out than that of the efficient estimator. Now, consider the stochastic differential equation dXt = 2Xt dt + (θ + Xt2 )−1/2 dWt , X0 = 0. (4.3) For this model, the solution X is not ergodic, but again Assumption 2.4 holds. We compare the two estimating functions given by g(t, y, x; θ) = (y − (1 + 2t)x)2 − (θ + x2 )−1 t h(t, y, x; θ) = (θ + x2 )10 (y − (1 + 2t)x)2 − (θ + x2 )9 t . For both g and f Assumptions 2.5 and 3.1 hold, and g is efficient, while h is not. 15 0.25 0.20 0.15 0.10 0.05 0.00 −20 −10 0 10 20 √ √ Figure 3: Estimated densities of n(θ̂G,n − θ0 ) (solid curve) and n(θ̂H,n − θ0 ) (dashed curve) for n = 104 in the case of the ergodic model (4.1). Simulations were carried out in the same manner as for the ergodic model. In the nonergodic case, an estimator was again found for every sample path when the efficient estimating function given by g was used. For the inefficient estimating function given by h, there was no solution to the estimating equation (in [0.01, 1.99]) in 14% of the samples for n = 104 and in 39 % of the samples for n = 103 . Figure 4 shows QQ-plots of √ √ bG,n = n W b −1 (θ̂G,n − θ0 ) and Z bH,n = n W b −1 (θ̂H,n − θ0 ) compared with a standard normal Z G,n H,n distribution, for n = 103 and n = 104 respectively. These QQ-plots indicate that in the non-ergodic case there is a slightly slower convergence to the asymptotic distribution in Theorem 3.2.(iii) for the efficient estimating function, and a considerably slower convergence for the inefficient estimating function, when compared to the ergodic case. 16 −4 ● ● ● ●● 4 2 ● ● −2 0 2 −4 −2 0 2 4 ● ● ● 4 ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●● 4 2 2 0 0 −2 −2 ● −4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● −4 ● ● −4 ● ● 0 ● −2 −2 0 2 ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● −2 0 2 4 −4 −2 0 2 4 bG,n (left) and Z bH,n (right) to the N(0, 1) distribution in the Figure 4: QQ-plots comparing Z case of the non-ergodic model (4.3) for n = 103 (above) and n = 104 (below). 17 5 Proofs Section 5.1 states three main lemmas needed to prove Theorem 3.2, followed by the proof of the theorem. Section 5.2 contains the proofs of the three lemmas. 5.1 Proof of the Main Theorem In order to prove Theorem 3.2, we use the following lemmas, together with results from Jacod and Sørensen (2012), and Sørensen (2012, Section 1.10). Lemma 5.1. Suppose that Assumptions 2.4 and 2.5 hold. For θ ∈ Θ, let Gn (θ) = n X n ; θ) g(∆n , Xtin , Xti−1 i=1 sq Gn (θ) n 1 X 2 n ; θ) = g (∆n , Xtin , Xti−1 ∆n i=1 and A(θ; θ0 ) = B(θ; θ0 ) = Z 1 2 0 Z 1 2 1 1 0 b2 (X s ; θ0 ) − b2 (X s ; θ) ∂2y g(0, X s , X s ; θ) ds b2 (X s ; θ0 ) − b2 (X s ; θ) ∂2y ∂θ g(0, X s , X s ; θ) ds 1 Z 1 2 C(θ; θ0 ) = − Z 1 2 0 1 0 ∂θ b2 (X s ; θ)∂2y g(0, X s , X s ; θ) ds b4 (X s ; θ0 ) + 1 2 2 2 b2 (X s ; θ0 ) − b2 (X s ; θ) ∂2y g(0, X s , X s ; θ) ds . Then, (i) the mappings θ 7→ A(θ; θ0 ), θ 7→ B(θ; θ0 ) and θ 7→ C(θ; θ0 ) are continuous on Θ (Pθ0 -almost surely) with A(θ0 ; θ0 ) = 0 and ∂θ A(θ; θ0 ) = B(θ; θ0 ). (ii) for all t ∈ [0, 1], [nt] P 1 X n ; θ0 ) \| Xtn Eθ0 g(∆n , Xtin , Xti−1 −→ 0 √ i−1 ∆n i=1 (5.1) [nt] 2 P 1 X n ; θ0 ) \| Xtn Eθ0 g(∆n , Xtin , Xti−1 −→ 0 i−1 ∆n i=1 (5.2) [nt] P 1 X 4 n , Xtn ; θ0 ) \| Xtn E g (∆ , X −→ 0 θ n t 0 i i−1 i−1 ∆2n i=1 (5.3) and [nt] P 1 X n ; θ0 ) \| Xtn Eθ0 g2 (∆n , Xtin , Xti−1 −→ i−1 ∆n i=1 t Z 1 2 0 2 b4 (X s ; θ0 ) ∂2y g(0, X s , X s ; θ0 ) ds . (5.4) 18 (iii) for all compact, convex subsets K ⊆ Θ, P sup \|Gn (θ) − A(θ; θ0 )\| −→ 0 θ∈K P sup \|∂θ Gn (θ) − B(θ; θ0 )\| −→ 0 θ∈K P sq sup Gn (θ) − C(θ; θ0 ) −→ 0 . θ∈K Lemma 5.2. Suppose that Assumptions 2.4 and 2.5 hold. Then, for all t ∈ [0, 1], [nt] P 1 X n ; θ0 )(Wtn − Wtn ) \| Ftn Eθ0 g(∆n , Xtin , Xti−1 −→ 0 . √ i i−1 i−1 ∆n i=1 (5.5) Lemma 5.3. Suppose that Assumptions 2.4 and 2.5 hold, and let Yn,t [nt] 1 X n ; θ0 ) . = √ g(∆n , Xtin , Xti−1 ∆n i=1 Then the sequence of processes (Yn )n∈N given by Yn = (Yn,t )t∈[0,1] converges stably in distribution under Pθ0 to the process Y = (Yt )t∈[0,1] given by Z t b2 (X s ; θ0 )∂2y g(0, X s , X s ; θ0 ) dBs . Yt = √1 2 0 Here B = (Bs ) s≥0 denotes a standard Wiener process, which is defined on a filtered extension (Ω0 , F 0 , (Ft0 )t≥0 , P0θ0 ) of (Ω, F , (Ft )t≥0 , Pθ0 ), and is independent of (U, W). D st We denote stable convergence in distribution under Pθ0 as n → ∞ by −→. Proof of Theorem 3.2. Let a compact, convex subset K ⊆ Θ with θ0 ∈ int K be given. The functions Gn (θ), A(θ, θ0 ), B(θ, θ0 ), and C(θ, θ0 ) were defined in Lemma 5.1. By Lemma 5.1.(i) and (iii), P Gn (θ0 ) −→ 0 and P sup \|∂θ Gn (θ) − B(θ, θ0 )\| −→ 0 θ∈K (5.6) with B(θ0 ; θ0 ) , 0 by Assumption 3.1.(ii), so Gn (θ) satisfies the conditions of Theorem 1.58 in Sørensen (2012). Now, we show (1.161) of Theorem 1.59 in Sørensen (2012). Let ε > 0 be given, and let B̄ε (θ0 ) and Bε (θ0 ), respectively, denote closed and open balls in R with radius ε > 0, centered at θ0 . The compact set K\Bε (θ0 ) does not contain θ0 , and so, by Assumption 3.1.(i), A(θ, θ0 ) , 0 for all θ ∈ K\Bε (θ0 ) with probability one under Pθ0 . 19 Because inf θ∈K\ B̄ε (θ0 ) \|A(θ, θ0 )\| ≥ inf θ∈K\Bε (θ0 ) \|A(θ, θ0 )\| > 0 Pθ0 -almost surely, by the continuity of θ 7→ A(θ, θ0 ), it follows that ! Pθ0 inf \|A(θ, θ0 )\| > 0 = 1 . θ∈K\ B̄ε (θ0 ) Consequently, by Theorem 1.59 in Sørensen (2012), for any Gn -estimator θ̃n , Pθ0 θ̃n ∈ K\ B̄ε (θ0 ) → 0 as n → ∞ . (5.7) for any ε > 0. By Theorem 1.58 in Sørensen (2012), there exists a consistent Gn -estimator θ̂n , which is eventually unique, in the sense that if θ̄n is another consistent Gn -estimator, then Pθ0 θ̂n , θ̄n → 0 as n → ∞ . (5.8) Suppose that θ̃n is any Gn -estimator which satisfies that Pθ0 θ̃n ∈ K → 1 as n → ∞ . Combining (5.7) and (5.9), it follows that Pθ0 θ̃n ∈ B̄ε (θ0 ) → 1 as n → ∞, (5.9) (5.10) so θ̃n is consistent. Using (5.8), Theorem 3.2.(i) follows. To prove Theorem 3.2.(ii), recall that ∆n = 1/n, and observe that by Lemma 5.3, √ D st nGn (θ0 ) −→ S (θ0 ) (5.11) where S (θ0 ) = 1 Z 0 √1 b2 (X s ; θ0 )∂2 y g(0, X s , X s ; θ0 ) dB s , 2 and B = (Bs ) s∈[0,1] is a standard Wiener process, independent of (U, W). As X is then also independent of B, S (θ0 ) is equal in distribution to C(θ0 ; θ0 )1/2 Z, where Z is standard normal distributed and independent of (Xt )t∈[0,1] . Note that by Assumption 3.1.(iii), the distribution of C(θ0 ; θ0 )1/2 Z is non-degenerate. Let θ̂n be a consistent Gn -estimator. By (5.6), (5.11) and properties of stable convergence (e.g. (2.3) in Jacod (1997)), √     nGn (θ0 ) Dst  S (θ0 )    −→   . ∂θ Gn (θ0 ) B(θ0 ; θ0 ) 20 Stable convergence in distribution implies weak convergence, so an application of Theorem 1.60 in Sørensen (2012) yields √ D n(θ̂n − θ0 ) −→ −B(θ0 , θ0 )−1 S (θ0 ) . (5.12) The limit is equal in distribution to W(θ0 )Z, where W(θ0 ) = −B(θ0 , θ0 )−1C(θ0 ; θ0 )1/2 and Z is standard normal distributed and independent of W(θ0 ). This completes the proof of Theorem 3.2.(ii). Finally, Lemma 2.14 in Jacod and Sørensen (2012) is used to write √ √ √ n(θ̂n − θ0 ) = −B(θ0 ; θ0 )−1 nGn (θ0 ) + n\|θ̂n − θ0 \|εn (θ0 ) , where the last term goes to zero in probability under Pθ0 . By the stable continuous mapping theorem, (5.12) holds with stable convergence in distribution as well. Lemma 5.1.(iii) may P b n −→ be used to conclude that W W(θ0 ), so Theorem 3.2.(iii) follows from the stable version of (5.12) by application of standard results for stable convergence. 5.2 Proofs of Main Lemmas This section contains the proofs of Lemmas 5.1, 5.2 and 5.3 in Section 5.1. A number of technical results are utilised in the proofs, these results are summarised in Appendix A, some of them with a proof. pol Proof of Lemma 5.1. First, note that for any f (x; θ) ∈ C0,0 (X×Θ) and any compact, convex subset K ⊆ Θ, there exist constants C K > 0 such that \| f (X s ; θ)\| ≤ C K (1 + \|X s \|C K ) for all s ∈ [0, 1] and θ ∈ int K. With probability one under Pθ0 , for fixed ω, C K (1+\|X s (ω)\|C K ) is a continuous function and therefore Lebesgue-integrable over [0, 1]. Using this method of constructing integrable upper bounds, Lemma 5.1.(i) follows by the usual results for continuity and differentiability of functions given by integrals. In the rest of this proof, Lemma A.3 and (A.7) are repeatedly used without reference. First, inserting θ = θ0 into (A.1), it is seen that [nt] [nt] X 1 X P 3/2 n ; θ0 ) \| Xtn n ; θ0 ) −→ 0 Eθ0 g(∆n , Xtin , Xti−1 = ∆ R(∆n , Xti−1 √ n i−1 ∆n i=1 i=1 [nt] [nt] X 2 1 X P 3 n ; θ0 ) \| Xtn n ; θ0 ) −→ 0 , Eθ0 g(∆n , Xtin , Xti−1 = ∆n R(∆n , Xti−1 i−1 ∆n i=1 i=1 proving (5.1) and (5.2). Furthermore, using (A.1) and (A.3), n X P n ; θ) \| Xtn Eθ0 g(∆n , Xtin , Xti−1 −→ A(θ; θ0 ) i−1 i=1 21 n X P n ; θ) \| Xtn −→ 0 , Eθ0 g2 (∆n , Xtin , Xti−1 i−1 i=1 so it follows from Lemma A.1 that point-wise for θ ∈ Θ, P Gn (θ) − A(θ; θ0 ) −→ 0 . (5.13) Using (A.3) and (A.5), [nt] 1 X n ; θ) \| Xtn Eθ0 g2 (∆n , Xtin , Xti−1 i−1 ∆n i=1 Z t 2 2 P −→ 12 b4 (X s ; θ0 ) + 21 b2 (X s ; θ0 ) − b2 (X s ; θ) ∂2y g(0, X s , X s ; θ) ds 0 and [nt] P 1 X 4 n , Xtn ; θ) \| Xtn E g (∆ , X −→ 0 , θ n t 0 i i−1 i−1 ∆2n i=1 completing the proof of Lemma 5.1.(ii) when θ = θ0 is inserted, and yielding P sq Gn (θ) − C(θ; θ0 ) −→ 0 (5.14) point-wise for θ ∈ Θ by Lemma A.1, when t = 1 is inserted. Also, using (A.2) and (A.4), n X P n ; θ) \| Xtn −→ B(θ; θ0 ) Eθ0 ∂θ g(∆n , Xtin , Xti−1 i−1 i=1 n X P 2 n n ; θ) −→ 0 . Eθ0 ∂θ g(∆n , Xtin , Xti−1 \| Xti−1 i=1 Thus, by Lemma A.1, also P ∂θ Gn (θ) − B(θ; θ0 ) −→ 0 , (5.15) j point-wise for θ ∈ Θ. Finally, recall that ∂y g(0, x, x; θ) = 0 for j = 0, 1. Then, using Lemmas A.7 and A.8, it follows that for each m ∈ N and compact, convex subset K ⊆ Θ, there exist constants Cm,K > 0 such that for all θ, θ0 ∈ K and n ∈ N, Eθ0 \|(Gn (θ) − A(θ; θ0 )) − (Gn (θ0 ) − A(θ0 ; θ0 ))\|2m ≤ Cm,K \|θ − θ0 \|2m Eθ0 \|(∂θ Gn (θ) − B(θ; θ0 )) − (∂θ Gn (θ0 ) − B(θ0 ; θ0 ))\|2m ≤ Cm,K \|θ − θ0 \|2m sq Eθ0 \|(Gn (θ) − C(θ; θ0 )) − sq (Gn (θ0 ) − C(θ ; θ0 ))\| 0 2m ≤ Cm,K \|θ − θ \| 0 2m (5.16) . By Lemma 5.1.(i), the functions θ 7→ Gn (θ) − A(θ; θ0 ), θ 7→ ∂θ Gn (θ) − B(θ; θ0 ) and θ 7→ sq Gn (θ) −C(θ, θ0 ) are continuous on Θ. Thus, using Lemma A.9 together with (5.13), (5.14), (5.15) and (5.16) completes the proof of Lemma 5.1.(iii). 22 Proof of Lemma 5.2. The overall strategy in this proof is to expand the expression on the left-hand side of (5.5) in such a manner that all terms either converge to 0 by Lemma A.3, or are equal to 0 by the martingale properties of stochastic integral terms obtained by use of Itô’s formula. By Assumption 2.5 and Lemma 2.7, the formulae g(0, y, x; θ) = 12 (y − x)2 ∂2y g(0, x, x; θ) + (y − x)3 R(y, x; θ) g(1) (y, x; θ) = g(1) (x, x; θ) + (y − x)R(y, x; θ) may be obtained. Using (2.4) and (5.17), n ; θ0 )(Wtn − Wtn ) \| Ftn Eθ0 g(∆n , Xtin , Xti−1 i i−1 i−1 2 2 1 n n n n n ) \| Ftn = Eθ0 2 (Xti − Xti−1 ) ∂y g(0, Xti−1 , Xti−1 ; θ0 )(Wtin − Wti−1 i−1 3 n n n n n n n + Eθ0 (Xti − Xti−1 ) R(Xti , Xti−1 ; θ0 )(Wti − Wti−1 ) \| Fti−1 n , Xtn ; θ0 )(Wtn − Wtn ) \| Ftn + ∆n Eθ0 g(1) (Xti−1 i i−1 i−1 i−1 n ) \| Ftn n n n n n + ∆n Eθ0 (Xti − Xti−1 )R(Xti , Xti−1 ; θ0 )(Wti − Wti−1 i−1 2 n n n n n + ∆ Eθ0 R(∆n , Xti , Xti−1 ; θ0 )(Wti − Wti−1 ) \| Fti−1 . (5.17) (5.18) Note that n n n n , Xtn ; θ0 )Eθ ∆n g(1) (Xti−1 0 Wti − Wti−1 \| Fti−1 = 0 , i−1 and that by repeated use of the Cauchy-Schwarz inequality, Lemma A.4 and Corollary A.5, C 3 n \| ) n ) R(Xtn , Xtn ; θ0 )(Wtn − Wtn ) \| Ftn ≤ ∆2nC(1 + \|Xti−1 Eθ0 (Xtin − Xti−1 i i i−1 i−1 i−1 C n \| ) n )R(Xtn , Xtn ; θ0 )(Wtn − Wtn ) \| Ftn ≤ ∆2nC(1 + \|Xti−1 ∆n Eθ0 (Xtin − Xti−1 i i i−1 i−1 i−1 C n \| ) n ; θ0 )(Wtn − Wtn ) \| Ftn ≤ ∆5/2 ∆2n Eθ0 R(∆n , Xtin , Xti−1 n C(1 + \|Xti−1 i i−1 i−1 for suitable constants C > 0, with [nt] 1 X m/2 P C n \| ) −→ 0 ∆n C(1 + \|Xti−1 √ ∆n i=1 for m = 4, 5 by Lemma A.3. Now, by (5.18), it only remains to show that [nt] P 1 X 2 2 n , Xtn ; θ0 )Eθ n − Xtn ) (Wtn − Wtn ) \| Ftn ∂y g(0, Xti−1 (X −→ 0 . √ t 0 i i i−1 i−1 i−1 i−1 ∆n i=1 Applying Itô’s formula with the function 2 n ) (w − wtn ) f (y, w) = (y − xti−1 i−1 23 (5.19) n , conditioned on (Xtn , Wtn ) = (xtn , wtn ), it follows that to the process (Xt , Wt )t≥ti−1 i−1 i−1 i−1 i−1 2 n ) (Wtn − Wtn ) (Xtin − Xti−1 i i−1 Z Z tn i n n (X s − Xti−1 )(W s − Wti−1 )a(X s ) ds + =2 n ti−1 n ti−1 Z +2 + Z tin n ti−1 n ti n ti−1 tin n )b(X s ; θ0 ) ds + 2 (X s − Xti−1 Z tin n ti−1 2 n )b (X s ; θ0 ) ds (W s − Wti−1 n )(W s − Wtn )b(X s ; θ0 ) dW s (X s − Xti−1 i−1 (5.20) 2 n ) dW s . (X s − Xti−1 By the martingale property of the Itô integrals in (5.20), 2 n ) (Wtn − Wtn ) \| Ftn Eθ0 (Xtin − Xti−1 i i−1 i−1 Z tn i n )(W s − Wtn )a(X s ) \| Ftn =2 Eθ0 (X s − Xti−1 ds i−1 i−1 n ti−1 + Z +2 tin n ti−1 Z 2 n )b (X s ; θ0 ) \| Ftn Eθ0 (W s − Wti−1 ds i−1 tin n ti−1 n )b(X s ; θ0 ) \| Xtn Eθ0 (X s − Xti−1 ds . i−1 Using the Cauchy-Schwarz inequality, Lemma A.4 and Corollary A.5 again, Z tin n ti−1 C n \| ), n )(W s − Wtn )a(X s ) \| Ftn ds ≤ C∆2n (1 + \|Xti−1 Eθ0 (X s − Xti−1 i−1 i−1 and by Lemma 2.6 n n n ; θ0 ) , n )b(X s ; θ0 ) \| Xtn Eθ0 (X s − Xti−1 = (s − ti−1 )R(s − ti−1 , Xti−1 i−1 so also Z tin n ti−1 C n )b(X s ; θ0 ) \| Xtn n \| ). Eθ0 (X s − Xti−1 ds ≤ C∆2n (1 + \|Xti−1 i−1 Now Z tn [nt] i 1 X 2 n n n )(W s − Wtn )a(X s ) \| Ftn ∂y g(0, Xti−1 , Xti−1 ; θ0 ) Eθ0 (X s − Xti−1 ds √ i−1 i−1 n ∆n i=1 ti−1 Z tn [nt] i 1 X 2 n , Xtn ; θ0 ) n )b(X s ; θ0 ) \| Xtn ∂y g(0, Xti−1 + √ Eθ0 (X s − Xti−1 ds i−1 i−1 n ∆n i=1 ti−1 ≤ ∆3/2 n C [nt] X P C n , Xtn ; θ0 ) (1 + \|Xtn \| ) −→ 0 ∂2y g(0, Xti−1 i−1 i−1 i=1 24 (5.21) by Lemma A.3, so by (5.19) and (5.21), it remains to show that Z tn [nt] i 1 X 2 P 2 n )b (X s ; θ0 ) \| Ftn n n ∂y g(0, Xti−1 , Xti−1 ; θ0 ) Eθ0 (W s − Wti−1 ds −→ 0 . √ i−1 n ∆n i=1 ti−1 Applying Itô’s formula with the function 2 n )b (y; θ0 ) , f (y, w) = (w − wti−1 and making use of the martingale properties of the stochastic integral terms, yields Z tn i 2 n )b (X s ; θ0 ) \| Ftn Eθ0 (W s − Wti−1 ds i−1 n ti−1 = tin Z n ti−1 + + s Z n ti−1 Z 1 2 Z n ) \| Ftn Eθ0 a(Xu )∂y b2 (Xu ; θ0 )(Wu − Wti−1 du ds i−1 tin n ti−1 tin n ti−1 s Z n ti−1 Z s n ti−1 n ) \| Ftn Eθ0 b2 (Xu ; θ0 )∂2y b2 (Xu ; θ0 )(Wu − Wti−1 du ds i−1 n Eθ0 b(Xu ; θ0 )∂y b2 (Xu ; θ0 ) \| Fti−1 du ds . Again, by repeated use of the Cauchy-Schwarz inequality and Corollary A.5, Z tn i 5/2 2 C 2 n \| )(∆ + ∆ n )b (X s ; θ0 ) \| Ftn ds ≤ C(1 + \|Xti−1 Eθ0 (Wtin − Wti−1 n ). n i−1 n ti−1 Now Z tn [nt] i 1 X 2 2 n )b (X s ; θ0 ) \| Ftn n , Xtn ; θ0 ) Eθ0 (W s − Wti−1 ds ∂y g(0, Xti−1 √ i−1 i−1 n ∆n i=1 ti−1 [nt] 3/2 X P C 2 n , Xtn ; θ0 ) C(1 + \|Xtn \| ) −→ 0 , ≤ ∆n + ∆ n ∂2y g(0, Xti−1 i−1 i−1 i=1 thus completing the proof. Proof of Lemma 5.3. The aim of this proof is to establish that the conditions of Theorem IX.7.28 in Jacod and Shiryaev (2003) hold, by which the desired result follows directly. For all t ∈ [0, 1], [ns] [nt] 1 X 1 X n ; θ 0 ) \| Xt n n , Xtn ; θ0 ) \| Xtn sup √ Eθ0 g(∆n , Xtin , Xti−1 g(∆ , X ≤ E √ θ n t 0 i i−1 i−1 i−1 s≤t ∆n i=1 ∆n i=1 and since the right-hand side converges to 0 in probability under Pθ0 by (5.1) of Lemma 5.1, so does the left-hand side, i.e. condition 7.27 of Theorem IX.7.28 holds. From (5.2) and (5.4) of Lemma 5.1, it follows that for all t ∈ [0, 1], [nt] 2 1 X 2 n ; θ0 ) \| Xtn n n n Eθ0 g (∆n , Xtin , Xti−1 − E g(∆ , X , X ; θ ) \| X θ0 n ti ti−1 0 ti−1 i−1 ∆n i=1 25 P Z −→ 1 2 0 t 2 b4 (X s ; θ0 ) ∂2y g(0, X s , X s ; θ0 ) ds , establishing that condition 7.28 of Theorem IX.7.28 is satisfied. Lemma 5.2 implies condition 7.29, while the Lyapunov condition (5.3) of Lemma 5.1 implies the Lindeberg condition 7.30 of Theorem IX.7.28 in Jacod and Shiryaev (2003), from which the desired result now follows. Theorem IX.7.28 contains an additional condition 7.31. This condition has the same form n n , where N = (Nt )t≥0 is any bounded as (5.5), but with Wtin − Wti−1 replaced by Ntin − Nti−1 martingale on (Ω, F , (Ft )t≥0 , Pθ0 ), which is orthogonal to W. However, since (Ft )t≥0 is generated by U and W, it follows from the martingale representation theorem (Jacod and Shiryaev, 2003, Theorem III.4.33) that every martingale on (Ω, F , (Ft )t≥0 , Pθ0 ) may be written as the sum of a constant term and a stochastic integral with respect to W, and therefore cannot be orthogonal to W. A Auxiliary Results This section contains a number of technical results used in the proofs in Section 5.2. Lemma A.1. (Genon-Catalot and Jacod, 1993, Lemma 9) For i, n ∈ N, let Fn,i = Ftin (with Fn,0 = F0 ), and let Fn,i be an Fn,i -measurable, real-valued random variable. If n X P Eθ0 (Fn,i \| Fn,i−1 ) −→ F n X and i=1 P 2 Eθ0 (Fn,i \| Fn,i−1 ) −→ 0 , i=1 for some random variable F, then n X P Fn,i −→ F . i=1 Lemma A.2. Suppose that Assumptions 2.4 and 2.5 hold. Then, for all θ ∈ Θ, (i) n ; θ) \| Xtn Eθ0 g(∆n , Xtin , Xti−1 i−1 2 2 2 n ; θ0 ) − b (Xtn ; θ) ∂ g(0, Xtn , Xtn ; θ) + ∆ R(∆n , Xtn ; θ) , = 21 ∆n b2 (Xti−1 y n i−1 i−1 i−1 i−1 (A.1) (ii) n ; θ) \| Xtn Eθ0 ∂θ g(∆n , Xtin , Xti−1 i−1 2 2 2 1 n n ; θ) ∂ ∂θ g(0, Xtn , Xtn ; θ) = 2 ∆n b (Xti−1 ; θ0 ) − b (Xti−1 y i−1 i−1 2 2 n ; θ)∂ g(0, Xtn , Xtn ; θ) + ∆ R(∆n , Xtn ; θ) , − 21 ∆n ∂θ b2 (Xti−1 y n i−1 i−1 i−1 26 (A.2) (iii) n ; θ) \| Xtn Eθ0 g2 (∆n , Xtin , Xti−1 i−1 2 2 2 2 2 1 n ; θ0 ) − b (Xtn ; θ) n ; θ0 ) + n , Xtn ; θ) (A.3) b (X ∂ = 21 ∆2n b4 (Xti−1 g(0, X t t y 2 i−1 i−1 i−1 i−1 n ; θ) , + ∆3n R(∆n , Xti−1 (iv) 2 n ; θ) , n n ; θ) = ∆2n R(∆n , Xti−1 \| Xti−1 Eθ0 ∂θ g(∆n , Xtin , Xti−1 (A.4) n ; θ) . n ; θ) \| Xtn = ∆4n R(∆n , Xti−1 Eθ0 g4 (∆n , Xtin , Xti−1 i−1 (A.5) (v) Proof of Lemma A.2. The formulae (A.1), (A.2) and (A.3) are implicitly given in the proofs of Sørensen (2010, Lemmas 3.2 & 3.4). To prove the two remaining formulae, note first that using (2.5), Assumption 2.5.(i) and Lemma 2.7, Liθ0 (g4 (0; θ))(x, x) = 0 , Liθ0 (g3 (0, θ)g(1) (θ))(x, x) = 0 , i = 1, 2, 3 i = 1, 2 Lθ0 (g (0, θ)g (θ) )(x, x) = 0 2 (1) 2 Lθ0 (g3 (0, θ)g(2) (θ))(x, x) = 0 Lθ0 (∂θ g(0, θ)2 )(x, x) = 0 . The verification of these formulae may be simplified by using e.g. the Leibniz formula for the n’th derivative of a product to see that partial derivatives are zero when evaluated in y = x. These results, as well as Lemmas 2.6 and 2.7, and (A.8) are used without reference in the following. 2 n ; θ) n \| Xti−1 Eθ0 ∂θ g(∆n , Xtin , Xti−1 2 n ; θ) \| Xtn = Eθ0 ∂θ g(0, Xtin , Xti−1 i−1 (1) n ; θ)∂θ g n ; θ) \| Xtn + 2∆n Eθ0 ∂θ g(0, Xtin , Xti−1 (Xtin , Xti−1 i−1 2 n n n + ∆n Eθ0 R(∆n , Xti , Xti−1 ; θ) \| Xti−1 2 2 2 n , Xtn ; θ) + ∆n Lθ (∂θ g(0, θ) )(Xtn , Xtn ) + ∆ R(∆n , Xtn ; θ) = ∂θ g(0, Xti−1 n 0 i−1 i−1 i−1 i−1 (1) n n n n n + 2∆n ∂θ g(0, Xti−1 , Xti−1 ; θ)∂θ g (Xti−1 , Xti−1 ; θ) + ∆n R(∆n , Xti−1 ; θ) n ; θ) , = ∆2n R(∆n , Xti−1 proving (A.4). Similarly, n ; θ) \| Xtn Eθ0 g4 (∆n , Xtin , Xti−1 i−1 27 n ; θ) \| Xtn = Eθ0 g4 (0, Xtin , Xti−1 i−1 (1) n ; θ)g n ; θ) \| Xtn + 4∆n Eθ0 g3 (0, Xtin , Xti−1 (Xtin , Xti−1 i−1 (1) 2 n ; θ)g n ; θ) \| Xtn + 6∆2n Eθ0 g2 (0, Xtin , Xti−1 (Xtin , Xti−1 i−1 (2) n ; θ)g n ; θ) \| Xtn + 2∆2n Eθ0 g3 (0, Xtin , Xti−1 (Xtin , Xti−1 i−1 (1) 3 n ; θ)g n ; θ) \| Xtn + 4∆3n Eθ0 g(0, Xtin , Xti−1 (Xtin , Xti−1 i−1 (1) (2) n ; θ)g n ; θ)g n ; θ) \| Xtn + 6∆3n Eθ0 g2 (0, Xtin , Xti−1 (Xtin , Xti−1 (Xtin , Xti−1 i−1 (3) n ; θ)g n ; θ) \| Xtn (Xtin , Xti−1 + 23 ∆3n Eθ0 g3 (0, Xtin , Xti−1 i−1 n ; θ) \| Xtn + ∆4n Eθ0 R(∆n , Xtin , Xti−1 i−1 4 4 1 2 2 n , Xtn ; θ) + ∆n Lθ (g (0; θ))(Xtn , Xtn ) + ∆ L (g (0; θ))(Xtn , Xtn ) = g4 (0, Xti−1 0 2 n θ0 i−1 i−1 i−1 i−1 i−1 (1) 3 n , Xtn ; θ) n , Xtn ) + 4∆n g (0, Xtn , Xtn ; θ)g (Xti−1 + 61 ∆3n L3θ0 (g4 (0; θ))(Xti−1 i−1 i−1 i−1 i−1 3 2 3 (1) n , Xtn ) + 2∆ L (g (0; θ)g n , Xt n ) + 4∆2n Lθ0 (g3 (0; θ)g(1) (θ))(Xti−1 (θ))(Xti−1 n θ0 i−1 i−1 2 3 2 (1) (1) n , Xt n ) n , Xtn ; θ) + 6∆ Lθ (g (0; θ)g n , Xtn ; θ)g (θ)2 )(Xti−1 (Xti−1 + 6∆2n g2 (0, Xti−1 n 0 i−1 i−1 i−1 (2) 3 3 (2) n , Xtn ; θ)g n , Xtn ; θ) + 2∆ Lθ (g (0; θ)g n , Xtn ) + 2∆2n g3 (0, Xti−1 (Xti−1 (θ))(Xti−1 n 0 i−1 i−1 i−1 3 (1) n , Xtn ; θ) n , Xtn ; θ)g (Xti−1 + 4∆3n g(0, Xti−1 i−1 i−1 (2) (1) n , Xtn ; θ) n , Xtn ; θ)g n , Xtn ; θ)g (Xti−1 (Xti−1 + 6∆3n g2 (0, Xti−1 i−1 i−1 i−1 (3) n , Xtn ; θ) n , Xtn ; θ)g + 23 ∆3n g3 (0, Xti−1 (Xti−1 i−1 i−1 n ; θ) + ∆4n R(∆n , Xti−1 n ; θ) , = ∆4n R(∆n , Xti−1 which proves (A.5). Lemma A.3. Let x 7→ f (x) be a continuous, real-valued function, and let t ∈ [0, 1] be given. Then Z t [nt] X P n ) −→ ∆n f (Xti−1 f (X s ) ds . 0 i=1 Lemma A.3 follows easily by the convergence of Riemann sums. Lemma A.4. Suppose that Assumption 2.4 holds, and let m ≥ 2. Then, there exists a constant Cm > 0, such that for 0 ≤ t ≤ t + ∆ ≤ 1, Eθ0 \|Xt+∆ − Xt \|m \| Xt ≤ Cm ∆m/2 1 + \|Xt \|m . (A.6) Corollary A.5. Suppose that Assumption 2.4 holds. Let a compact, convex set K ⊆ Θ be given, and suppose that f (y, x; θ) is of polynomial growth in x and y, uniformly for θ in K. Then, there exist constants C K > 0 such that for 0 ≤ t ≤ t + ∆ ≤ 1, Eθ0 (\| f (Xt+∆ , Xt , θ)\| \| Xt ) ≤ C K 1 + \|Xt \|C K 28 for all θ ∈ K. Lemma A.4 and Corollary A.5, correspond to Lemma 6 of Kessler (1997), adapted to the present assumptions. For use in the following, observe that for any θ ∈ Θ, there exist constants Cθ > 0 such that ∆n [nt] X n ) ≤ C θ ∆n Rθ (∆n , Xti−1 [nt] X C n \| θ , 1 + \|Xti−1 i=1 i=1 so it follows from Lemma A.3 that for any deterministic, real-valued sequence (δn )n∈N with δn → 0 as n → ∞, δn ∆n [nt] X P n ) −→ 0 . Rθ (∆n , Xti−1 (A.7) i=1 Note that by Corollary A.5, it holds that under Assumption 2.4, Eθ0 (R (∆, Xt+∆ , Xt ; θ) \| Xt ) = R(∆, Xt ; θ) . (A.8) Lemma A.6. Suppose that Assumption 2.4 holds, and that the function f (t, y, x; θ) satisfies that pol f (t, y, x; θ) ∈ C1,2,1 ([0, 1] × X2 × Θ) with f (0, x, x; θ) = 0 (A.9) for all x ∈ X and θ ∈ Θ. Let m ∈ N be given, and let Dk( · ; θ, θ0 ) = k( · ; θ) − k( · ; θ0 ). Then, there exist constants Cm > 0 such that 2m Eθ0 D f (t − s, Xt , X s ; θ, θ0 ) Z t 2m 2m−1 ≤ Cm (t − s) Eθ0 D f1 (u − s, Xu , X s ; θ, θ0 ) du (A.10) s Z t 2m + Cm (t − s)m−1 Eθ0 D f2 (u − s, Xu , X s ; θ, θ0 ) du s for 0 ≤ s < t ≤ 1 and θ, θ0 ∈ Θ, where f1 and f2 are given by f1 (t, y, x; θ) = ∂t f (t, y, x; θ) + a(y)∂y f (t, y, x; θ) + 12 b2 (y; θ0 )∂2y f (t, y, x; θ) f2 (t, y, x; θ) = b(y; θ0 )∂y f (t, y, x; θ) . Furthermore, for each compact, convex set K ⊆ Θ, there exists a constant Cm,K > 0 such that Eθ0 \|D f j (t − s, Xt , X s ; θ, θ0 )\|2m ≤ Cm,K \|θ − θ0 \|2m for j = 1, 2, 0 ≤ s < t ≤ 1 and all θ, θ0 ∈ K. 29 Proof of Lemma A.6. A simple application of Itô’s formula (when conditioning on X s = x s ) yields that for all θ ∈ Θ, Z t Z t f (t − s, Xt , X s ; θ) = f1 (u − s, Xu , X s ; θ) du + f2 (u − s, Xu , X s ; θ) dWu (A.11) s s under Pθ0 . By Jensen’s inequality, it holds that for any k ∈ N, Z Z t k   t 0 jk k−1 0 j   du Eθ0 D f j (u − s, Xu , X s ; θ, θ ) D f j (u − s, Xu , X s ; θ, θ ) du  ≤ (t − s) Eθ0  s s (A.12) for j = 1, 2, and by the martingale properties of the second term in (A.11), the BurkholderDavis-Gundy inequality may be used to show that Z Z t 2m  m!  t  0 Eθ0  . D f2 (u − s, Xu , X s ; θ, θ ) dWu  ≤ Cm Eθ0 D f2 (u − s, Xu , X s ; θ, θ0 )2 du s s (A.13) Now, (A.11), (A.12) and (A.13) may be combined to show (A.10). The last result of the lemma follows by an application of the mean value theorem. Lemma A.7. Suppose that Assumption 2.4 holds, and let K ⊆ Θ be compact and convex. Assume that f (t, y, x; θ) satisfies (A.9) for all x ∈ X and θ ∈ Θ, and define Fn (θ) = n X n ; θ) . f (∆n , Xtin , Xti−1 i=1 Then, for each m ∈ N, there exists a constant Cm,K > 0, such that Eθ0 Fn (θ) − Fn (θ0 ) 2m ≤ Cm,K \|θ − θ0 \|2m en (θ) = ∆−1 for all θ, θ0 ∈ K and n ∈ N. Define F n F n (θ), and suppose, moreover, that the functions h1 (t, y, x; θ) = ∂t f (t, y, x; θ) + a(y)∂y f (t, y, x; θ) + 21 b2 (y; θ0 )∂2y f (t, y, x; θ) h2 (t, y, x; θ) = b(y; θ0 )∂y f (t, y, x; θ) h j2 (t, y, x; θ) = b(y; θ0 )∂y h j (t, y, x, θ) satisfy (A.9) for j = 1, 2. Then, for each m ∈ N, there exists a constant Cm,K > 0, such that en (θ) − F en (θ0 ) 2m ≤ Cm,K \|θ − θ0 \|2m E θ0 F for all θ, θ0 ∈ K and n ∈ N. 30 Proof of Lemma A.7. For use in the following, define, in addition to h1 , h2 and h j2 , the functions h j1 (t, y, x; θ) = ∂t h j (t, y, x; θ) + a(y)∂y h j (t, y, x; θ) + 21 b2 (y; θ0 )∂2y h j (t, y, x; θ) h j21 (t, y, x; θ) = ∂t h j2 (t, y, x; θ) + a(y)∂y h j2 (t, y, x; θ) + 12 b2 (y; θ0 )∂2y h j2 (t, y, x; θ) h j22 (t, y, x; θ) = b(y; θ0 )∂y h j2 (t, y, x; θ) for j = 1, 2, and, for ease of notation, let 0 n 0 n H n,i j (u; θ, θ ) = Dh j (u − ti−1 , Xu , Xti−1 ; θ, θ ) for j ∈ {1, 2, 11, 12, 21, 22, 121, 122, 221, 222}, where Dk( · ; θ, θ0 ) = k( · ; θ) − k( · ; θ0 ). Recall that ∆n = 1/n. First, by the martingale properties of ∆n n Z X i=1 r 0 n,i 0 n ,tn ] (u)H 1(ti−1 2 (u; θ, θ ) dWu , i the Burkholder-Davis-Gundy inequality is used to establish the existence of a constant Cm > 0 such that    2m  m n Z tn n Z tn  X   2 X  i i n,i n,i  0 0 2 Eθ0  ∆n H2 (u; θ, θ ) dWu  ≤ Cm Eθ0  ∆n H2 (u; θ, θ ) du  . tn tn i=1 i=1 i−1 i−1 Now, using also Ito’s formula, Jensen’s inequality and Lemma A.6,   2m  n  X  0  n ; θ, θ ) Eθ0  ∆n D f (∆n , Xtin , Xti−1  i=1     2m  2m  n Z tn n Z tn  X  X   i i  ≤ Cm Eθ0  ∆n H1n,i (u; θ, θ0 ) du  + Cm Eθ0  ∆n H2n,i (u; θ, θ0 ) dWu  n n i=1 ti−1 i=1 ti−1 Z n   2m  m n n Z tn  ti X   2 X  i n,i n,i  0 0 2 ≤ Cm ∆n Eθ0  H1 (u; θ, θ ) du  + Cm Eθ0  ∆n H2 (u; θ, θ ) du  n n i=1 ti−1 i=1 ti−1      2m  m  Z tn n   1 Z tin X     i 1     n,i n,i    0 0 2 Eθ0  ≤ Cm ∆2m+1 n   ∆n tn H1 (u; θ, θ ) du  + Eθ0  ∆n tn H2 (u; θ, θ ) du  i−1 i−1 i=1  Z Z n n n  t t X n,i  n,i  i i 2m 0 2m 0 2m (A.14) Eθ0 \|H (u; θ, θ )\| du + Eθ0 \|H (u; θ, θ )\| du ≤ Cm ∆n  ≤ Cm,K \|θ n ti−1 i=1 − θ0 \|2m ∆2m n , 1 n ti−1 2 thus   2m  n  X  0 2m −2m 0  ≤ Cm,K \|θ − θ0 \|2m n ; θ, θ ) Eθ0 \|DFn (θ, θ )\| = ∆n Eθ0  ∆n D f (∆n , Xtin , Xti−1  i=1 31 for all θ, θ0 ∈ K and n ∈ N. In the case where also h j and h j2 satisfy (A.9) for all x ∈ X, θ ∈ Θ and j = 1, 2, use Lemma A.6 to write Eθ0 \|H1n,i (u; θ, θ0 )\|2m Z u n,i n 2m−1 ≤ Cm (u − ti−1 ) Eθ0 \|H11 (v; θ, θ0 )\|2m dv n ti−1 + Cm (u − u Z n m−1 ti−1 ) n ti−1 Z n 2m−1 ≤ Cm (u − ti−1 ) u n ti−1 n,i Eθ0 \|H11 (v; θ, θ0 )\|2m dv u Z + Cm (u − n m−1 ti−1 ) + Cm (u − n m−1 ti−1 ) n,i Eθ0 \|H12 (v; θ, θ0 )\|2m dv n ti−1 u Z n ti−1  Z  (v − tn )2m−1 i−1 v n ti−1  Z  n m−1 (v − t ) i−1 Eθ0 v n ti−1 Eθ0 2m n,i H121 (w; θ, θ0 ) 2m n,i H122 (w; θ, θ0 )   dw dv   dw dv n 2m n 3m ≤ Cm,K \|θ − θ0 \|2m (u − ti−1 ) + (u − ti−1 ) , and similarly obtain n 2m n 3m Eθ0 \|H2n,i (u; θ, θ0 )\|2m ≤ Cm,K \|θ − θ0 \|2m (u − ti−1 ) + (u − ti−1 ) . Now, inserting into (A.14),   2m  n  X  0  n ; θ, θ ) Eθ0  ∆n D f (∆n , Xtin , Xti−1  i=1 Z n  X n,i Z tin 2m 0 2m ≤ Cm,K ∆n Eθ0 \|H1 (u; θ, θ )\| du +  i=1 n ti−1 ≤ Cm,K \|θ − θ0 \|2m ∆2m n n Z X tin n ti−1 i=1 tin n ti−1  n,i  0 2m Eθ0 \|H2 (u; θ, θ )\| du n 2m n 3m (u − ti−1 ) + (u − ti−1 ) du 5m ≤ Cm,K \|θ − θ0 \|2m ∆4m , n + ∆n and, ultimately, Eθ0 \|DF̃n (θ, θ0 )\|2m   2m  n  X  0  n , Xtn ; θ, θ ) D f (∆ , X = Eθ0  ∆−1 n t n i i−1  i=1   2m  n  X  0  ∆n  n , Xtn ; θ, θ ) = ∆−4m D f (∆ , X n t n Eθ0  i i−1   i=1 0 2m ≤ Cm,K \|θ − θ \| (1 + ∆n ) ≤ Cm,K \|θ − θ \| . 0 2m pol Lemma A.8. Suppose that Assumption 2.4 is satisfied. Let f ∈ C0,1 (X × Θ). Define Z 1 F(θ) = f (X s ; θ) ds 0 32 and let K ⊆ Θ be compact and convex. Then, for each m ∈ N, there exists a constant Cm,K > 0 such that for all θ, θ0 ∈ K, Eθ0 \|F(θ) − F(θ0 )\|2m ≤ Cm,K \|θ − θ0 \|2m . Lemma A.8 follows from a simple application of the mean value theorem. Lemma A.9. Let K ⊆ Θ be compact. Suppose that Hn = (Hn (θ))θ∈K defines a sequence (Hn )n∈N of continuous, real-valued stochastic processes such that P Hn (θ) −→ 0 point-wise for all θ ∈ K. Furthermore, assume that for some m ∈ N, there exists a constant Cm,K > 0 such that for all θ, θ0 ∈ K and n ∈ N, Eθ0 Hn (θ) − Hn (θ0 ) 2m ≤ Cm,K \|θ − θ0 \|2m . (A.15) Then, P sup \|Hn (θ)\| −→ 0 . θ∈K Proof of Lemma A.9. (Hn (θ))n∈N is tight in R for all θ ∈ K, so, using (A.15), it follows from Kallenberg (2002, Corollary 16.9 & Theorem 16.3) that the sequence of processes (Hn )n∈N is tight in C(K, R), the space of continuous (and bounded) real-valued functions on K, and thus relatively compact in distribution. Also, for all d ∈ N and (θ1 , . . . , θd ) ∈ K d ,     0  Hn (θ1 )  .  D  .   ..  −→  ..  ,     Hn (θd ) 0 D so by Kallenberg (2002, Lemma 16.2), Hn −→ 0 in C(K, R) equipped with the uniform D metric. Finally, by the continuous mapping theorem, supθ∈K \|Hn (θ)\| −→ 0 , and the desired result follows. Acknowledgement We are grateful to the referees for their insightful comments and suggestions that have improved the paper. Nina Munkholt Jakobsen was supported by the Danish Council for Independent Research \| Natural Science through a grant to Susanne Ditlevsen. Michael Sørensen was supported by the Center for Research in Econometric Analysis of Time Series funded by the Danish National Research Foundation. The research is part of the Dynamical Systems Interdisciplinary Network funded by the University of Copenhagen Programme of Excellence. 33 References Aït-Sahalia, Y. (2002). 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Evaluation strategies for monadic computations Tomas Petricek Computer Laboratory University of Cambridge United Kingdom tomas.petricek@cl.cam.ac.uk Monads have become a powerful tool for structuring effectful computations in functional programming, because they make the order of effects explicit. When translating pure code to a monadic version, we need to specify evaluation order explicitly. Two standard translations give call-by-value and call-by-name semantics. The resulting programs have different structure and types, which makes revisiting the choice difficult. In this paper, we translate pure code to monadic using an additional operation malias that abstracts out the evaluation strategy. The malias operation is based on computational comonads; we use a categorical framework to specify the laws that are required to hold about the operation. For any monad, we show implementations of malias that give call-by-value and call-by-name semantics. Although we do not give call-by-need semantics for all monads, we show how to turn certain monads into an extended monad with call-by-need semantics, which partly answers an open question. Moreover, using our unified translation, it is possible to change the evaluation strategy of functional code translated to the monadic form without changing its structure or types. 1 Introduction Purely functional languages use lazy evaluation (also called call-by-need) to allow elegant programming with infinite data structures and to guarantee that a program will not evaluate a diverging term unless it is needed to obtain the final result. However, reasoning about lazy evaluation is difficult thus it is not suitable for programming with effects. An elegant way to embed effectful computations in lazy functional languages, introduced by Moggi [16] and Wadler [26], is to use monads. Monads embed effects in a purely functional setting and explicitly specify the evaluation order of monadic (effectful) operations. Wadler [26] gives two ways of translating pure programs to a corresponding monadic version. One approach leads to a call-by-value semantics, where effects of function arguments are performed before calling a function. However, if an argument has an effect and terminates the program, this may not be appropriate if the function can successfully complete without using the argument. The second approach gives a call-by-name semantics, where effects are performed only if the argument is actually used. However, this approach is also not always suitable, because an effect may be performed repeatedly. Wadler leaves an open question whether there is a translation that would correspond to call-by-need semantics, where effects are performed only when the result is needed, but at most once. The main contribution of this paper is an alternative translation of functional code to a monadic form, parameterized by an operation malias. The translation has the following properties: • A single translation gives monadic code with either call-by-name or call-by-value semantics, depending on the definition of malias (Section 2). When used in languages such as Haskell, it is possible to write code that is parameterized by the evaluation strategy (Section 4.1). J. Chapman and P. B. Levy (Eds.): Fourth Workshop on Mathematically Structured Functional Programming (MSFP 2012). EPTCS 76, 2012, pp. 68–89, doi:10.4204/EPTCS.76.7 Tomas Petricek 69 • The translation can be used to construct monads that provide the call-by-need semantics (Section 4.2), which partly answers the open question posed by Wadler. Furthermore, for some monads, it is possible to use parallel call-by-need semantics, where arguments are evaluated in parallel with the body of a function (Section 4.3). • The malias operation has solid foundations in category theory. It arises from augmenting a monad structure with a computational semi-comonad based on the same functor (Section 3). We use this theory to define laws that should be obeyed by malias implementations (Section 2.2). This paper was inspired by work on joinads [19], which introduced the malias operation for a similar purpose. However, operations with the same type and similar laws appear several times in the literature. We return to joinads in Section 4.4 and review other related work in Section 5. 1.1 Translating to monadic code We first demonstrate the two standard options for translating purely functional code to monadic form. Consider the following two functions that use pureLookupInput to read some configuration property. Assuming the configuration is already loaded in memory, we can write the following pure computation1 : chooseSize :: Int → Int → Int chooseSize new legacy = if new > 0 then new else legacy resultSize :: Int resultSize = chooseSize (pureLookupInput "new_size") (pureLookupInput "legacy_size") The resultSize function reads two different configuration keys and chooses one of them using chooseSize. When using a language with lazy evaluation, the call pureLookupInput "legacy_size" is performed only when the value of "new_size" is less than or equal to zero. To modify the function to actually read configuration from a file as opposed to performing in-memory lookup, we now use lookupInput which returns IO Int instead of the pureLookupInput function. Then we need to modify the two above functions. There are two mechanical ways that give different semantics. Call-by-value. In the first style, we call lookupInput and then apply monadic bind on the resulting computation. This reads both of the configuration values before calling the chooseSize function, and so arguments are fully evaluated before the body of a function as in the call-by-value evaluation strategy: chooseSizecbv :: Int → Int → IO Int chooseSizecbv new legacy = return (if new > 0 then new else legacy) resultSizecbv :: IO Int resultSizecbv = do new ← lookupInput cbv "new_size" legacy ← lookupInput cbv "legacy_size" chooseSize cbv new legacy 1 Examples are written in Haskell and can be found at: http://www.cl.cam.ac.uk/~tp322/papers/malias.html Evaluation strategies for monadic computations 70 In this version of the translation, a function of type A → B is turned into a function A → M B. For example, the chooseSizecbv function takes integers as parameters and returns a computation that returns an integer and may perform some effects. When calling a function in this setting, the arguments may not be fully evaluated (the functional part is still lazy), but the effects associated with obtaining the value of the argument happen before the function call. For example, if the call lookupInput cbv "new_size" read a file and then returned 1024, but the operation lookupInput cbv "legacy_size" caused the program to crash because a specified key was not present in a configuration file, then the entire program would crash. Call-by-name. In the second style, we pass unevaluated computations as arguments to functions. This means we call lookupInput to create an effectful computation that will read the input, but the computation is then passed to chooseSize, which may not need to evaluate it: chooseSizecbn :: IO Int → IO Int → IO Int chooseSizecbn new legacy = do newVal ← new if newVal > 0 then new else legacy resultSizecbn :: IO Int resultSizecbn = chooseSize cbn (lookupInput cbn "new_size") (lookupInput cbn "legacy_size") The translation turns a function of type A → B into a function M A → M B. This means that the chooseSize cbn function takes a computation that performs the I/O effect and reads information from the configuration file, as opposed to taking a value whose effects were already performed. Following the mechanical translation, chooseSizecbn returns a monadic computation that evaluates the first argument and then behaves either as new or as legacy, depending on the obtained value. When the resulting computation is executed, the computation which reads the value of the "new_size" key may be executed repeatedly. First, inside the chooseSizecbn function and then repeatedly when the result of this function is evaluated. In this particular example, we can easily change the code to perform the effect just once, but this is not generally possible for computations obtained by the call-by-name translation. 2 Abstracting evaluation strategy The translations demonstrated in the previous section have two major problems. Firstly, it is not easy to switch between the two – when we introduce effects using monads, we need to decide to use one or the other style and changing between them later on involves rewriting of the program and changing types. Secondly, even in the IO monad, we cannot easily implement a call-by-need strategy that would perform effects only when a value is needed, but at most once. 2.1 Translation using aliasing To solve these problems, we propose an alternative translation. We require a monad m with an additional operation malias that abstracts out the evaluation strategy and has a type m a → m (m a). The term aliasing refers to the fact that some part of effects may be performed once and their results shared in multiple monadic computations. The translation of the previous example using malias looks as follows: Tomas Petricek 71 chooseSize :: IO Int → IO Int → IO Int chooseSize new legacy = do newVal ← new if newVal > 0 then new else legacy resultSize :: IO Int resultSize = do new ← malias (lookupInput "new_size") legacy ← malias (lookupInput "legacy_size") chooseSize new legacy The types of functions and access to function parameters are translated in the same way as in the call-byname translation. The chooseSize function returns a computation IO Int and its parameters also become computations of type IO Int. When using the value of a parameter, the computation is evaluated using monadic bind (e.g. the line newVal ← new in chooseSize). The computations passed as arguments are not the original computations as in the call-by-name translation. The translation inserts a call to malias for every function argument (and also for every letbound value, which can be encoded using lambda abstraction and application). The computation returned by malias has a type m (m a), which makes it possible to perform the effects at two different call sites: • When simulating the call-by-value strategy, all effects are performed when binding the outer monadic computation before a function call. • When simulating the call-by-name strategy, all effects are performed when binding the inner monadic computation, when the value is actually needed. These two strategies can be implemented by two simple definitions of malias. However, by delegating the implementation of malias to the monad, we make it possible to implement more advanced strategies as well. We discuss some of them later in Section 4. We keep the translation informal until Section 2.3 and discuss the malias operation in more detail first. Implementing call-by-name. To implement the call-by-name strategy, the malias operation needs to return the computation specified as an argument inside the monad. In the type m (m a), the outer m will not carry any effects and the inner m will be the same as the original computation: malias :: m a → m (m a) malias m = return m From the monad laws (see Figure 1), we know that applying monadic bind to a computation created from a value using return is equivalent to just passing the value to the rest of the computation. This means that the additional binding in the translation does not have any effect and the resulting program behaves as the call-by-name strategy. A complete proof can be found in Appendix A. Implementing call-by-value. Implementing the call-by-value strategy is similarly simple. In the returned computation of type m (m a), the computation corresponding to the outer m needs to perform all the effects. The computation corresponding to the inner m will be a computation that simply returns the previously computed value without performing any effects: malias :: m a → m (m a) malias m = m >>= (return ◦ return) Evaluation strategies for monadic computations 72 Functor with unit and join: unit :: a → m a map :: m a → (a → b) → m b join :: m (m a) → m a Definition using bind (>>=) and return: return :: a → m a >>= :: m a → (a → m b) → m b Join laws (unit and map laws omitted): Monad laws about bind and return: return a >>= f ≡ f a m >>= return ≡ m (m >>= f ) >>= g ≡ m >>= (λ x → f x >>= g) join ◦ map join = join ◦ join join ◦ map unit = id = join ◦ unit join ◦ map (map f ) = map f ◦ join Figure 1: Two equivalent ways of defining monads with monad laws In Haskell, the second line could be written as liftM return m. The liftM operation represents the functor associated with the monad. This means that binding on the returned computation performs all the effects, obtains a value v and returns a computation return v. When calling a function that takes an argument of type m a, the argument passed to it using this implementation of malias will always be constructed using the return operation. Hence the resulting behaviour is equivalent to the original call-by-value translation. Detailed proof can be found in Appendix A. 2.2 The malias operation laws In order to define a reasonable evaluation strategy, we require the malias operation to obey a number of laws. The laws follow from the theoretical background that is discussed in Section 3, namely from the fact that malias is the cojoin operation of a computational semi-comonad. The laws that relate malias to the monad are easier to write in terms of join, map and unit than using the formulation based on >>= and return. For completeness, the two equivalent definitions of monads with the monad laws are shown in Figure 1. Although we do not show it, one can be easily defined in terms of the other. The required laws for malias are the following: map (map f ) ◦ malias map malias ◦ malias malias ◦ unit join ◦ malias = = = = malias ◦ (map f ) malias ◦ malias unit ◦ unit id (naturality) (associativity) (computationality) (identity) The first two laws follow from the fact that malias is a cojoin operation of a comonad. The naturality law specifies that applying a function to a value inside a computation is the same as applying the function to a value inside an aliased computation. The associativity law specifies that aliasing an aliased computation is the same as aliasing a computation produced by an aliased computation. The computationality law is derived from the fact that the comonad defining malias is a computational comonad with unit as one of the components. The law specifies that aliasing of a pure computation creates a pure computation. Finally, the identity law relates malias with the monadic structure, by requiring that join is a left inverse of malias. Intuitively, it specifies that aliasing a computation of type m a and then joining the result returns the original computation. All four laws hold for the two implementations of malias presented in the previous section. We prove that the laws hold for any monad using the standard monad laws. The proofs can be found in Appendix B. We discuss the intuition behind the laws in Section 2.4 and describe their categorical foundations in Section 3. The next section formally presents the translation algorithm. Tomas Petricek 73 JxKcbn Jλ x.eKcbn Je1 e2 Kcbn Jlet x = e1 in e2 Kcbn = = = = x unit (λ x.JeKcbn ) bind Je1 Kcbn (λ f . f Je2 Kcbn ) (λ x.Je2 Kcbn ) Je1 Kcbn Figure 2: Wadler’s call-by-name translation of λ calculus. JxKcbv Jλ x.eKcbv Je1 e2 Kcbv Jlet x = e1 in e2 Kcbv = = = = unit x unit (λ x.JeKcbv ) bind Je1 Kcbv (λ f . bind Je2 Kcbv (λ x. f x)) bind Je1 Kcbv (λ x.Je2 Kcbv ) Figure 3: Wadler’s call-by-value translation of λ calculus. 2.3 Lambda calculus translation The call-by-name and call-by-value translations given in Section 1.1 were first formally introduced by Wadler [26]. In this section, we present a similar formal definition of our translation based on the malias operation. For our source language, we use a simply-typed λ calculus with let-binding: e ∈ Expr e ::= x \| λ x.e \| e1 e2 \| let x = e1 in e2 τ ∈ Type τ ::= α \| τ1 → τ2 The target language of the translation is identical but for one exception – it adds a type scheme M τ representing monadic computations. The call-by-name and call-by-value translation of the lambda calculus are shown in Figure 2 and Figure 3, respectively. In the translation, we write >>= as bind. The translation of types and typing judgements are omitted for simplicity and can be found in the original paper [26]. Our translation, called call-by-alias, is presented below. The translation has similar structure to Wadler’s call-by-name translation, but it inserts malias operation in the last two cases: Jα Kcba Jτ1 → τ2 Kcba = α = M Jτ1 Kcba → M Jτ2 Kcba JxKcba Jλ x.eKcba Je1 e2 Kcba Jlet x = e1 in e2 Kcba = = = = x unit (λ x.JeKcba ) bind Je1 Kcba (λ f .bind (malias Je2 Kcba ) f ) bind (malias Je1 Kcba ) (λ x.Je2 Kcba ) Jx1 : τ1 , . . . , xn : τn ⊢ e : τ Kcba = x1 : M Jτ1 Kcba , . . . , xn : M Jτn Kcba ⊢ JeKcba : M Jτ Kcba The translation turns user-defined variables of type τ into variables of type M τ . A variable access x is translated to a variable access, which now represents a computation (that may have effects). A lambda expression is turned into a lambda expression wrapped in a pure monadic computation. The two interesting cases are application and let-binding. When translating function application, we bind on the computation representing the function. We want to call the function f with an aliased computation as an argument. This is achieved by passing the translated argument to malias and then applying bind again. The translation of let-binding is similar, but slightly simpler, because it does not need to use bind to obtain a function. Evaluation strategies for monadic computations 74 The definition of J−Kcba includes a well-typedness-preserving translation of typing judgements. The details of the proof can be found in Appendix C. In the translation, let-binding is equivalent to (λ x.e2 ) e1 , but we include it to aid the intuition and to simplify motivating examples in the next section. 2.4 The meaning of malias laws Having provided the translation, we can discuss the intuition behind the malias laws and what they imply about the translated code. The naturality law specifies that malias is a natural transformation. Although we do not give a formal proof, we argue that the law follows from the parametricity of the malias type signature and can be obtained using the method described by Voigtländer [25]. Effect conservation. The meaning of associativity and identity can be informally demonstrated by treating effects as units of information. Given a computation type involving a number of occurrences of m, we say that there are effects (or information) associated with each occurrence of m. The identity law specifies that join ◦ malias does not lose effects – given a computation of type m a, the malias operation constructs a computation of type m (m a). This is done by splitting the effects of the computation between two monadic computations. Requiring that applying join to the new computation returns the original computation means that all the effects of the original computation are preserved, because join combines the effects of the two computations. The associativity law specifies how the effects should be split. When applying malias to an aliased computation of type m (m a), we can apply the operation to the inner or the outer m. Underlining second aliasing, the following two computations should be equal: m (m (m a)) = m (m (m a)). The law forbids implementations that split the effects in an asymmetric way. For instance, if m a represents a step-wise computation that can be executed by a certain number of steps, malias cannot execute the first half of steps and return a computation that evaluates the other half. The proportions associated with individual m values would be 1/4, 1/4 and 1/2 on the left and 1/2, 1/4 and 1/4 on the right-hand side. Semantics-preserving transformations. The computationality and identity (again) laws also specify that certain semantics-preserving transformations on the original source code correspond to equivalent terms in the code translated using the call-by-alias transformation. The source transformations corresponding to computationality and identity, respectively, are the following: let f = λ x.e1 in e2 ≡ e2 [ f ← λ x.e1 ] let x = e in x ≡ e The construction of the lambda function in the first equation is the only place where the translation inserts unit. The computationality law can be applied when a lambda abstraction appears in a position where malias is inserted. Thanks to the first monad law (Figure 1), the value assigned to f in the translation is unit (λ x.e1 ), which is equivalent to the translation of a term after the substitution. In the second equation, the left-hand side is translated as bind (malias e) id, which is equivalent to join (malias e), so the equation directly corresponds to the identity law. Discussion of completeness. The above discussion, together with the theoretical foundations introduced in the next section, supports the claim that our laws are necessary. We do not argue that our set of laws is complete – for instance, we might want to specify that aliasing of an already aliased computation has no effect, which is difficult to express in an equational form. Tomas Petricek 75 However, the definition of completeness, in this context, is elusive. One possible approach that we plan to investigate in future work is to expand the set of semantics-preserving transformations that should hold, regardless of an evaluation strategy. 3 Computational semi-bimonads In this section, we formally describe the structure that underlies a monad having a malias operation as described in the previous section. Since the malias operation corresponds to an operation of a comonad associated with a monad, we first review the definitions of a monad and a comonad. Monads are wellknown structures in functional programming. Comonads are dual structures to monads that are less widespread, but they have also been used in functional programming and semantics (Section 5): Definition 1. A monad over a category C is a triple (T, η , µ ) where T : C → C is a functor, η : IC → T is a natural transformation from the identity functor to T , and µ : T 2 → T is a natural transformation, such that the following associativity and identity conditions hold, for every object A: • µA ◦ T µA = µA ◦ µTA • µA ◦ ηTA = idTA = µA ◦ T ηA Definition 2. A comonad over a category C is a triple (T, ε , δ ) where T : C → C is a functor, ε : T → IC is a natural transformation from T to the identity functor, and δ : T → T 2 is a natural transformation from T to T 2 , such that the following associativity and identity conditions hold, for every object A: • T δA ◦ δA = δTA ◦ δA • εTA ◦ δA = idTA = T εA ◦ δA In functional programming terms, the natural transformation η corresponds to unit :: a → m a and the natural transformation µ corresponds to join :: m (m a) → m a. A comonad is a dual structure to a monad – the natural transformation ε corresponds to an operation counit :: m a → a and δ corresponds to cojoin :: m a → m (m a). An equivalent formulation of comonads in functional programming uses an operation cobind :: m a → (m a → b) → m b, which is dual to >>= of monads. A simple example of a comonad is the product comonad. The type m a stores the value of a and some additional state S, meaning that TA = A × S. The ε (or counit) operation extracts the value A ignoring the additional state. The δ (or cojoin) operation duplicates the state. In functional programming, the product comonad is equivalent to the reader monad TA = S → A. In this paper, we use a special variant of comonads. Computational comonads, introduced by Brookes and Geva [5], have an additional operation γ together with laws specifying its properties: Definition 3. A computational comonad over a category C is a quadruple (T, ε , δ , γ ) where (T, ε , δ ) is a comonad over C and γ : IC → T is a natural transformation such that, for every object A, • εA ◦ γA = idA • δA ◦ γA = γTA ◦ γA . A computational comonad has an additional operation γ which has the same type as the η operation of a monad, that is a → m a. In the work on computational comonads, the transformation γ turns an extensional specification into an intensional specification without additional computational information. Evaluation strategies for monadic computations 76 In our work, we do not need the natural transformation corresponding to counit :: m a → a. We define a computational semi-comonad, which is a computational comonad without the natural transformation ε and without the associated laws. The remaining structure is preserved: Definition 4. A computational semi-comonad over a category C is a triple (T, δ , γ ) where T : C → C is a functor, δ : T → T 2 is a natural transformation from T to T 2 and γ : IC → T is a natural transformation from the identity functor to T , such that the following associativity and computationality conditions hold, for every object A: • T δA ◦ δA = δTA ◦ δA • δA ◦ γA = γTA ◦ γA . Finally, to define a structure that models our monadic computations with the malias operation, we combine the definition of a monad and computational semi-comonad. We require that the two structures share the functor T and that the natural transformation η : IC → T of a monad coincides with the natural transformation γ : IC → T of a computational comonad. Definition 5. A computational semi-bimonad over a category C is a quadruple (T, η , µ , δ ) where (T, η , µ ) is a monad over a category C and (T, δ , η ) is a computational semi-comonad over C , such that the following additional condition holds, for every object A: • µA ◦ δA = idTA The definition of computational semi-bimonad relates the monadic and comonadic parts of the structure using an additional law. Given an object A, the law specifies that taking TA to T 2 A using the natural transformation δA of a comonad and then back to TA using the natural transformation µA is identity. 3.1 Revisiting the laws The laws of computational semi-bimonad as defined in the previous section are exactly the laws of our monad equipped with the malias operation. In this section, we briefly review the laws and present the category theoretic version of all the laws demonstrated in Section 2.2. We require four laws in addition to the standard monad laws (which are omitted in the summary below). A diagrammatic demonstration is shown in Figure 4. For all objects A and B of C and for all f : A → B in C : T 2 f ◦ δA T δA ◦ δA δA ◦ η A µA ◦ δA = = = = δB ◦ T f δTA ◦ δA ηTA ◦ ηA idTA (naturality) (associativity) (computationality) (identity) The naturality law follows from the fact that δ is a natural transformation and so we did not state it explicitly in Definition 5. However, it is one of the laws that are translated to the functional programming interpretation. The associativity law is a law of comonad – the other law in Definition 2 does not apply in our scenario, because we only work with semi-comonad that does not have natural transformation ε (counit). The computationality law is a law of a computational comonad and finally, the identity law is the additional law of computational semi-bimonads. Tomas Petricek TA δA Tf T 2A T2 f 77 // // T B δB T 2B (naturality) TA δA δA T 2A // T 2 A T δA δTA // T 3 A (associativity) A ηA ηA // TA δA TA ηTA // T 2 A (computationality) TA❉ idTA ❉❉ ❉❉ ❉❉ δA ❉"" T 2A // TA ③<< ③ µA ③③ ③③ ③③ (identity) Figure 4: Diagramatic representation of the four additional properties of semi-bimonads 4 Abstracting evaluation strategy in practice In this section, we present several practical uses of the malias operation. We start by showing how to write monadic code that is parameterized over the evaluation strategy and then consider expressing callby-need in this framework. Then we also briefly consider parallel call-by-need and the relation between malias and joinads and the docase notation [19]. 4.1 Parameterization by evaluation strategy One of the motivations of this work is that the standard monadic translations for call-by-name and callby-value produce code with different structure. Section 2.3 gave a translation that can be used with both of the evaluation strategies just by changing the definition of the malias operation. In this section, we make one more step – we show how to write code parameterized by evaluation strategy. We define a monad transformer [13] that takes a monad and turns it into a monad with malias that implements a specific evaluation strategy. Our example can then be implemented using functions that are polymorphic over the monad transformer. We continue using the previous example based on the IO monad, but the transformer can operate on any monad. As a first step, we define a type class named MonadAlias that extends Monad with the malias operation. To keep the code simple, we do not include comments documenting the laws: class Monad m ⇒ MonadAlias m where malias :: m a → m (m a) Next, we define two new types that represent monadic computations using the call-by-name and call-byvalue evaluation strategy. The two types are wrappers that make it possible to implement two different instances of MonadAlias for any underlying monadic computation m a: newtype CbV m a = CbV {runCbV :: m a} newtype CbN m a = CbN {runCbN :: m a} The snippet defines types CbV and CbN that represent two evaluation strategies. Figure 5 shows the implementation of three type classes for these two types. The implementation of the Monad type class is the same for both types, because it simply uses return and >>= operations of the underlying monad. The implementation of MonadTrans wraps a monadic computation m a into a type CbV m a and CbN m a, respectively. Finally, the instances of the MonadAlias type class associate the two implementations of malias (from Section 2.1) with the two data types. Evaluation strategies for monadic computations 78 instance Monad m ⇒ Monad (CbV m) where return v = CbV (return v) (CbV a) >>= f = CbV (a >>= (runCbV ◦ f )) instance Monad m ⇒ Monad (CbN m) where return v = CbN (return v) (CbN a) >>= f = CbN (a >>= (runCbN ◦ f )) instance MonadTrans CbV where lift = CbV instance MonadTrans CbN where lift = CbN instance Monad m ⇒ MonadAlias (CbV m) where malias m = m >>= (return ◦ return) instance Monad m ⇒ MonadAlias (CbN m) where malias m = return m Figure 5: Instances of Monad, MonadTrans and MonadAlias for evaluation strategies Example. Using the previous definitions, we can now rewrite the example from Section 1.1 using generic functions that can be executed using both runCbV and runCbN. Instead of implementing malias for a specific monad such as IO a, we use a monad transformer t that lifts the monadic computation to either CbV IO a or to CbN IO a. This means that all functions will have constraints MonadTrans t, specifying that t is a monad transformer, and MonadAlias (t m), specifying that the computation implements the malias operation. In Haskell, this can be succinctly written using constraint kinds [3], that make it possible to define a single constraint EvalStrategy t m that combines both of the conditions2 : type EvalStrategy t m = (MonadTrans t, MonadAlias (t m)) Despite the use of the type keyword, the identifier EvalStrategy actually has a kind Constraint, which means that it can be used to specify assumptions about types in a function signature. In our example, the constraint EvalStrategy t IO denotes monadic computations based on the IO monad that also provide an implementation of MonadAlias: chooseSize :: EvalStrategy t IO ⇒ t IO Int → t IO Int → t IO Int chooseSize new legacy = do newVal ← new if newVal > 0 then new else legacy resultSize :: EvalStrategy t IO ⇒ t IO Int resultSize = do new ← malias $ lift (lookupInput "new_size") legacy ← malias $ lift (lookupInput "legacy_size") chooseSize new legacy Compared to the previous version of the example, the only significant change is in the type signature of the two functions. Instead of passing computations of type IO a, they now work with computations t IO a 2 Constraint kinds are available in GHC 7.4 and generalize constraint families proposed by Orchard and Schrijvers [18] Tomas Petricek 79 with the constraint EvalStrategy t IO. The result of lookupInput is still IO Int and so it needs to be lifted into a monadic computation t IO Int using the lift function. The return type of the resultSize computation is parameterized over the evaluation strategy t. This means that we can call it in two different ways. Writing runCbN resultSize executes the function using call-by-name and writing runCbV resultSize executes it using call-by-value. However, it is also possible to implement a monad transformer for call-by-need. 4.2 Implementing call-by-need strategy In his paper introducing the call-by-name and call-by-value translations for monads, Wadler noted: “It remains an open question whether there is a translation scheme that corresponds to call-by-need as opposed to call-by-name” [26]. We do not fully answer this question, however we hope to contribute to the answer. In particular, we show how to use the mechanism described so far to turn certain monads into an extended versions of such monads that provide the call-by-need behaviour. In the absence of effects, the call-by-need strategy is equivalent to the call-by-name strategy, with the only difference in that performance characteristics. In the call-by-need (or lazy) strategy, a computation passed as an argument is evaluated at most once and the result is cached afterwards. The caching of results needs to be done in the malias operation. This cannot be done for any monad, but we can define a monad transformer similar to the ones presented in the previous section. In particular, we use Svenningsson’s package [22], which defines a transformer version of the ST monad [11]. As documented in the package description, the transformer should be applied only to monads that yield a single result. Combining lazy evaluation with non-determinism is a more complex topic that has been explored by Fischer et al. [6]. newtype CbL s m a = CbL {unCbL :: STT s m a} Unlike CbV and CbN, the CbL type is not a simple wrapper that contains a computation of type m a. Instead, it contains a computation augmented with some additional state. The state is used for caching the values of evaluated computations. The type STT s m a represents a computation m a with an additional local state “tagged” with a type variable s. The use of a local state instead of e.g. IO means that the monadic computation can be safely evaluated even as part of purely functional code. The tags are used merely to guarantee that state associated with one STT computation does not leak to other parts of the program. Implementing the Monad and MonadTrans instances follows exactly the same pattern as instances of other transformers given in Figure 5. The interesting work is done in the malias function of MonadAlias: instance Monad m ⇒ MonadAlias (CbL s m) where malias (CbL marg) = CbL $ do r ← newSTRef Nothing return (CbL $ do rv ← readSTRef r case rv of Nothing → marg >>= λ v → writeSTRef r (Just v) >> return v Just v → return v) The malias operation takes a computation of type m a and returns a computation m (m a). The monad transformer wraps the underlying monad inside STT, so the type of the computation returned by malias is equivalent to a type STT s m (STT s m a). Evaluation strategies for monadic computations 80 The fact that both outer and inner STT share the same tag s means that they operate on a shared state. The outer computation allocates a new reference and the inner computation can use it to access and store the result computed previously. The allocation is done using the newSTRef function, which creates a reference initialized to Nothing. In the returned (inner) computation, we first read the state using readSTRef . If the value was computed previously, then it is simply returned. If not, the computation evaluates marg, stores the result in a reference cell and then returns the obtained value. Discussion. After implementing runCbL function (which can be done easily using runST), the example in Section 4.1 can be executed using the CbL type. With the call-by-need semantics, the program finally behaves as desired: if the value of the "new_size" key is greater than zero, then it reads it only once, without reading the value of the "legacy_size" key. The value of "legacy_size" key is accessed only if the value of the "new_size" key is less than zero. Showing that the above definition corresponds to call-by-need formally is beyond the scope of this paper. However, the use of STT transformer adds a shared state that keeps the evaluated values, which closely corresponds to Launchbury’s environment-based semantics of lazy evaluation [9]. We do not formally prove that the malias laws hold for the above implementation, but we give an informal argument. The naturality law follows from parametricity. Computations considered in the associativity law are of type m (m (m a)); both sides of the equation create a computation where the two outer m computations allocate a new reference (where the outer points to the inner) and the single innermost m actually triggers the computation. The computationality law holds, because aliasing of a unit computation cannot introduce any effects. Finally, the left-hand side of identity creates a computation that allocates a new reference that is encapsulated in the returned computation and cannot be accessed from elsewhere, so no sharing is added when compared with the right-hand side. 4.3 Parallel call-by-need strategy In this section, we consider yet another evaluation strategy that can be implemented using our scheme. The parallel call-by-need strategy [2] is similar to call-by-need, but it may optimistically start evaluating a computation sooner, in parallel with the main program. When carefully tuned, the evaluation strategy may result in a better performance on multi-core CPUs. We present a simple implementation of the malias operation based on the monad for deterministic parallelism by Marlow et al. [14]. By translating purely functional code to a monadic version using our translation and the Par monad, we get a program that attempts to evaluate arguments of every function call in parallel. In practice, this may introduce too much overhead, but it demonstrate that parallel call-by-need strategy also fits with our general framework. Unlike the previous two sections, we do not define a monad transformer that can embody any monadic computation. For example, performing IO operations in parallel might introduce non-determinism. Instead, we implement malias operation directly for computations of type Par a: instance MonadAlias Par where malias m = spawn m >>= return ◦ get The implementation is surprisingly simple. The function spawn creates a computation Par (IVar a) that starts the work in background and returns a mutable variable (I-structure) that will contain the result when the computation completes. The inner computation that is returned by malias calls a function get :: IVar a → Par a that waits until IVar has been assigned a value and then returns it. Tomas Petricek 81 Using the above implementation of malias, we can now translate purely functional code to a monadic version that uses parallel call-by-need instead of the previous standard evaluation strategies (that do not introduce any parallelism). For example, consider a naive Fibonacci function: fibSeq n \| n 6 1 = n fibSeq n = fibSeq (n − 1) + fibSeq (n − 2) The second case calls the + operator with two computations as arguments. The translated version passes these computations to malias, which starts executing them in parallel. The monadic version of the + operator then waits until both computations complete. If we translate only the second case and manually add a case that calls sequential version of the function for inputs smaller than 30, we get the following: fibPar n \| n < 30 = return (fibSeq n) fibPar n = do n1 ← malias $ fibPar (n − 1) n2 ← malias $ fibPar (n − 2) liftM2 (+) n1 n2 Aside from the first line, the code directly follows the general translation mechanism described earlier. Arguments of a function are turned to monadic computations and passed to malias. The inner computations obtained using monadic bind are then passed to a translated function. Thanks to the manual optimization that calls fibSeq for smaller inputs, the function runs nearly twice as fast on a dual-core machine3 . Parallel programming is one of the first areas where we found the malias operation useful. We first considered it as part of joinads, which are discussed in the next section. Discussion. Showing that the above implementation actually implements parallel call-by-need could be done by relating our implementation of malias to the multi-thread transitions of [2]. Informally, each computation that may be shared is added to the work queue (using spawn) when it occurs on the righthand side of a let binding, or as an argument in function application. The queued work evaluates in parallel with the main program and the get function implements sharing, so the semantics is lazy. As previously, the naturality law holds thanks to parametricity. To consider other laws, we need a formal model that captures the time needed to evaluate computations. We assume that evaluating a computation created by unit takes no time, but all other computations take non-zero time. Moreover, all spawned computations start executing immediately (i.e. the number of equally fast threads is unlimited). In the associativity law, the left-hand side returns a computation that spawns the actual work and then spawns a computation that waits for its completion. The right-hand side returns a computation that schedules a computation, which then schedules the actual work. In both cases, the actual work is started immediately when the outer m computation is evaluated, and so they are equivalent. The computationality law holds, because a unit computation evaluates in no time. Finally, the left-hand side of identity returns a computation that spawns the work and then waits for its completion, which is semantically equivalent to just running the computation. 4.4 Simplifying Joinads Joinads [20, 19] were designed to simplify programming with certain kinds of monadic computations. Many monads, especially from the area of concurrent or parallel programming provide additional oper3 When called with input 37 on a Core 2 Duo CPU, the sequential version runs in 8.9s and the parallel version in 5.1s. Evaluation strategies for monadic computations 82 ations for composing monadic computations. Joinads identify three most common extensions: parallel composition, (non-deterministic) choice and aliasing. Joinads are abstract computations that form a monad and provide the three additional operations mentioned above. The work on joinads also introduces a syntactic extension for Haskell and F# that makes it easier to work with these classes of computations. For example, the following snippet uses the Par monad to implement a function that tests whether a predicate holds for all leafs of a tree in parallel: all :: (a → Bool) → Tree a → Par Bool all p (Leaf v) = return (p v) all p (Node left right) = docase (all p left, all p right) of (False, ?) → return False (?, False) → return False (allL, allR) → return (allL ∧ allR) The docase notation intentionally resembles pattern matching and has similar semantics as well. The first two cases use the special pattern ? to denote that the value of one of the computations does not have to be available in order to continue. When one of the sub-branches returns False, we know that the overall result is False and so we return immediately. Finally, the last clause matches if neither of the two previous are matched. It can only match after both sub-trees are processed. Similarly to do notation, the docase syntax is desugared into uses of the joinad operations. The choice between clauses is translated using the choice operator. If a clause requires the result of multiple computations (such as the last one), the computations are combined using parallel composition. If a computation, passed as an argument, is accessed from multiple clauses, then it should be evaluated only once and the clauses should only access aliased computation. This motivation is similar to the one described in this article. Indeed, joinads use a variant of the malias operation and insert it automatically for all arguments of docase. This is very similar to how the translation presented in this paper uses malias. If aliasing was not done automatically behind the scenes, we would have to write: all p (Leaf v) = return (p v) all p (Node left right) = do l ← malias (all p left) r ← malias (all p right) docase (l, r) of (False, ?) → return False (?, False) → return False (allL, allR) → return (allL ∧ allR) One of the limitations of the original design of joinads is that there is no one-to-one correspondence between the docase notation and what can be expressed directly using the joinad operations. This is partly due to the automatic aliasing of arguments, which inserts malias only at very specific locations. We believe that integrating a call-by-alias translation, described in this article, in a programming language could resolve this situation. It would also separate the two concerns – composition of computations using choice and parallel composition (done by joinads) and automatic aliasing of computations that allows sharing of results as in call-by-need or parallel call-by-need. Tomas Petricek 5 83 Related work Most of the work that directly influenced this work has been discussed throughout the paper. Most importantly, the question of translating pure code to a monadic version with call-by-need semantics was posed by Wadler [26]. To our knowledge, this question has not been answered before, but there is various work that either uses similar structures or considers evaluation strategies from different perspectives. Monads with aliasing. Numerous monads have independently introduced an operation of type m a → m (m a). The eagerly combinator of the Orc monad [10] causes computation to run in parallel, effectively implementing the parallel call-by-need evaluation strategy. The only law that relates eagerly to operations of a monad is too restrictive to allow the call-by-value semantics. A monad for purely functional lazy non-deterministic programming [6] uses a similar combinator share to make monadic (non-deterministic) computations lazy. However, the call-by-value strategy is inefficient and the call-by-name strategy is incorrect, because choosing a different value each time a non-deterministic computation is accessed means that generate and test pattern does not work. The share operation is described together with the laws that should hold. The HNF law is similar to our computationality. However, the Ignore law specifies that the share operation should not be strict (ruling out our call-by-value implementation of malias). A related paper [4] discusses where share needs to be inserted when translating lazy non-deterministic programs to a monadic form. The results may be directly applicable to make our translation more efficient by inserting malias only when required. Abstract computations and comonads. In this paper, we extended monads with one component of a comonadic structure. Although less widespread than monads, comonads are also useful for capturing abstract computations in functional programming. They have been used for dataflow programming [23], array programming [17], environment passing, and more [8]. In general, comonads can be used to describe context-dependent computations [24], where cojoin (natural transformation δ ) duplicates the context. In our work, the corresponding operation malias splits the context (effects) in a particular way between two computations. We only considered basic monadic computations, but it would be interesting to see how malias interacts with other abstract notions of computations, such as applicative functors [15], arrows [7] or additive monads (the MonadPlus type-class). The monad for lazy non-deterministic programming [6], mentioned earlier, implements MonadPlus and may thus provide interesting insights. Evaluation strategies. One of the key results of this paper is a monadic translation from purely functional code to a monadic version that has the call-by-need semantics. We achieve that using the monad transformer [13] for adding state. In the absence of effects, call-by-need is equivalent to call-by-name, but it has been described formally as a version of λ -calculus by Ariola and Felleisen [1]. This allows equational reasoning about computations and it could be used to show that our encoding directly corresponds to call-by-need, similarly to proofs for other strategies in Appendix A. The semantics has been also described using an environment that models caching [9, 21], which closely corresponds to our actual implementation. Considering the two basic evaluation strategies, Wadler [27] shows that call-by-name is dual to callby-value. We find this curious as the two definitions of malias in our work are, in some sense, also dual or symmetric as they associate all effects with the inner or the outer monad of the type m (m a). Furthermore, the duality between call-by-name and call-by-value can be viewed from a logical perspective Evaluation strategies for monadic computations 84 thanks to the Curry-Howard correspondence. We believe that finding a similar logical perspective for our generalized strategy may be an interesting future work. Finally, the work presented in this work unifies monadic call-by-name and call-by-value. In a nonmonadic setting, a similar goal is achieved by the call-by-push-value calculus [12]. The calculus is more fine-grained and strictly separates values and computations. Using these mechanisms, it is possible to encode both call-by-name and call-by-value. It may be interesting to consider whether our computations parameterized over evaluation strategy (Section 4.1) could be encoded in call-by-push-value. 6 Conclusions We presented an alternative translation from purely functional code to monadic form. Previously, this required choosing either the call-by-need or the call-by-value translation and the translated code had different structure and different types in both cases. Our translation abstracts the evaluation strategy into a function malias that can be implemented separately providing the required evaluation strategy. Our translation is not limited to the above two evaluation strategies. Most interestingly, we show that certain monads can be automatically turned into an extended version that supports the call-by-need strategy. This answers part of an interesting open problem posed by Wadler [26]. The approach has other interesting applications – it makes it possible to write code that is parameterized by the evaluation strategy and it allows implementing a parallel call-by-need strategy for certain monads. Finally, we presented the theoretical foundations of our approach using a model described in terms of category theory. We extended the monad structure with an additional operation based on computational comonads, which were previously used to give intensional semantics of computations. In our setting, the operation specifies the evaluation order. The categorical model specifies laws about malias and we proved that the laws hold for call-by-value and call-by-name strategies. Acknowledgements. The author is grateful to Sebastian Fischer, Dominic Orchard and Alan Mycroft for inspiring comments and discussion, and to the latter two for proofreading the paper. We are also grateful to anonymous referees for detailed comments and to Paul Blain Levy for shepherding of the paper. The work has been partly supported by EPSRC and through the Cambridge CHESS scheme. References [1] Zena M. Ariola & Matthias Felleisen (1997): The Call-By-Need Lambda Calculus. Journal of Functional Programming 7, pp. 265–301, doi:10.1017/S0956796897002724. [2] Clem Baker-Finch, David J. King & Phil Trinder (2000): An Operational Semantics for Parallel Lazy Evaluation. In: Proceedings of ICFP 2000, doi:10.1145/351240.351256. [3] Max Bolingbroke (2011): Constraint Kinds for GHC (Unpublished manuscript). Available at http://blog. omega-prime.co.uk/?p=127. [4] B. Braßel, S. Fischer, M. Hanus & F. Reck (2011): Transforming Functional Logic Programs into Monadic Functional Programs. In: Proceedings of WFLP 2010, LNCS 6559, pp. 30–47, doi:10.1.1.157.4578. [5] Stephen Brookes & Shai Geva (1992): Computational Comonads and Intensional Semantics. In: Applications of Categories in Computer Science: Proceedings of the LMS Symposium, 177, pp. 1–44, doi:10.1.1. 45.4952. [6] Sebastian Fischer, Oleg Kiselyov & Chung-chieh Shan (2011): Purely Functional Lazy Non-Deterministic Programming. Journal of Functional Programming 21(4–5), pp. 413–465, doi:10.1145/1631687.1596556. Tomas Petricek 85 [7] John Hughes (1998): Generalising Monads to Arrows. Science of Computer Programming 37, pp. 67–111, doi:10.1016/S0167-6423(99)00023-4. [8] Richard B. Kieburtz (1999): Codata and Comonads in Haskell (Unpublished manuscript). [9] John Launchbury (1993): A Natural Semantics for Lazy Evaluation. In: Proceedings of POPL 1993, pp. 144–154, doi:10.1145/158511.158618. [10] John Launchbury & Trevor Elliott (2010): Concurrent orchestration in Haskell. In: Proceedings of Haskell Symposium, Haskell 2010, pp. 79–90, doi:10.1145/1863523.1863534. [11] John Launchbury & Simon Peyton Jones (1995): State in Haskell. In: Proceedings of the LISP and Symbolic Computation Conference, LISP 1995, doi:10.1007/BF01018827. [12] Paul Blain Levy (2004): Call-By-Push-Value. Springer. 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In: Proceedings of FLOPS 2010, pp. 56–71, doi:10.1007/978-3-642-12251-4_6. [19] Tomas Petricek, Alan Mycroft & Don Syme: Extending Monads with Pattern Matching. In: Proceedings of Haskell Symposium, Haskell 2011, doi:10.1145/2096148.2034677. [20] Tomas Petricek & Don Syme (2011): Joinads: A Retargetable Control-Flow Construct for Reactive, Parallel and Concurrent Programming. In: Proceedings of PADL 2011, doi:10.1007/978-3-642-18378-2_17. [21] S. Purushothaman & Jill Seaman (1992): An Adequate Operational Semantics for Sharing in Lazy Evaluation. In: Proceedings of the 4th European Symposium on Programming, doi:10.1007/3-540-55253-7_ 26. [22] Josef Svenningsson (2011): The STMonadTrans package. Available at http://hackage.haskell.org/ package/STMonadTrans-0.3.1. [23] Tarmo Uustalu & Varmo Vene (2006): The Essence of Dataflow Programming. Lecture Notes in Computer Science 4164, pp. 135–167, doi:10.1007/11894100_5. [24] Tarmo Uustalu & Varmo Vene (2008): Comonadic Notions of Computation. Electron. Notes Theor. Comput. Sci. 203, pp. 263–284, doi:10.1016/j.entcs.2008.05.029. [25] Janis Voigtländer (2009): Free Theorems Involving Type Constructor Classes: Functional Pearl. In: Proceedings of ICFP 2009, ICFP 209, ACM, New York, NY, USA, pp. 173–184, doi:10.1145/1596550.1596577. [26] Philip Wadler (1990): Comprehending Monads. In: Proceedings of Conference on LISP and Functional Programming, LFP 1990, ACM, New York, NY, USA, pp. 61–78, doi:10.1145/91556.91592. [27] Philip Wadler (2003): Call-By-Value Is Dual to Call-By-Name. In: Proceedings of ICFP, ICFP 2003, pp. 189–201, doi:10.1145/944705.944723. Evaluation strategies for monadic computations 86 Je1 e2 Kcba = bind Je1 Kcba (λ f .bind (malias Je2 Kcba ) f )) = bind Je1 Kcba (λ f .bind (unit Je2 Kcba ) f )) = bind Je1 Kcba (λ f . f Je2 Kcba ) = bind Je1 Kcbn (λ f . f Je2 Kcbn ) = Je1 e2 Kcbn (translation) (definition) (left identity) (induction hypothesis) (translation) Jlet x = e1 in e2 Kcba = bind (malias Je1 Kcba ) (λ x.Je2 Kcba ) = bind (unit Je1 Kcba ) (λ x.Je2 Kcba ) = (λ x.Je2 Kcba ) Je1 Kcba = (λ x.Je2 Kcbn ) Je1 Kcbn = Jlet x = e1 in e2 Kcbn (translation) (definition) (left identity) (induction hypothesis) (translation) Figure 6: Proving that using an appropriate malias is equivalent to call-by-name. A Equivalence proofs In this section, we show that our translation presented in Section 2.3 can be used to implement standard call-by-name and call-by-value. We prove that using an appropriate definition of malias from Section 2.1 gives the same semantics as the standard translations described by Wadler [26]. Call-by-name. The translation of types is the same for our translation and the call-by-name translation. In addition, the rules for translating variable access and lambda functions are also the same. This means that we only need to prove that our translation of let-binding and function application are equivalent. When implementing call-by-name using our translation, we use the following definition of malias: malias m = unit m Using this definition and the left identity monad law, we can now show that our call-by-alias translation is equivalent to the translation from Figure 2. The Figure 6 shows the equations for function application and let-binding. Call-by-value. Proving that appropriate definition of malias gives a term that corresponds to the one obtained using standard call-by-value translation is more difficult. In the call-by-value translation, functions are translated to a type τ1 → M τ2 , while our translation produces functions of type M τ1 → M τ2 . As a reminder, the definition of malias that gives the call-by-value behaviour is the following: malias m = bind m (unit ◦ unit) To prove that the two translations are equivalent, we show that the following invariant holds: when using the above definition of malias and our call-by-alias translation, a monadic computation of type M τ that is assigned to a variable x always has a structure unit xv where xv is a variable of type τ . The sketch of the proof is shown in Figure 7. We write e1 ∼ = e2 to mean that the expression e1 in the call-by-alias translation corresponds to an expression e2 in call-by-value translation. This means that expressions of form unit xv translate to values x, variable declarations of xv (in lambda abstraction and let-binding) translate to declarations of x and all function values of type M τ1 → M τ2 become τ1 → M τ2 . Tomas Petricek 87 JxKcba = unit xv ∼ = x = JxKcbv (assumption) (correspondence) (translation) Je1 e2 Kcba = bind Je1 Kcba (λ f .bind (malias Je2 Kcba ) f ) = bind Je1 Kcba (λ f .bind (bind Je2 Kcba (unit ◦ unit)) f ) = bind Je1 Kcba (λ f .bind Je2 Kcba (λ xv .bind (unit (unit xv )) f )) = bind Je1 Kcba (λ f .bind Je2 Kcba (λ xv . f (unit xv ))) ∼ = bind Je1 Kcbv (λ f .bind Je2 Kcbv (λ x. f x)) = Je1 e2 Kcbv (translation) (definition) (associativity) (left identity) (correspondence) (translation) Jlet x = e1 in e2 Kcba = bind (malias Je1 Kcba ) (λ x.Je2 Kcba ) = bind (bind Je1 Kcba (unit ◦ unit)) (λ x.Je2 Kcba ) = bind Je1 Kcba (λ xv .bind (unit (unit xv )) (λ x.Je2 Kcba )) = bind Je1 Kcba (λ xv .(λ x.Je2 Kcba ) (unit xv )) ∼ = bind Je1 Kcbv (λ x.Je2 Kcbv ) = Jlet x = e1 in e2 Kcbv (translation) (definition) (associativity) (left identity) (correspondence) (translation) Figure 7: Proving that using an appropriate malias is equivalent to call-by-value. B Proofs for two implementations In this section, we prove that the two implementations of malias presented in Section 2.1 obey the malias laws. We use the formulation of monads based on a functor with additional operations join and unit. The proof relies on the following laws that hold about join and unit: map (g ◦ f ) unit ◦ f join ◦ map (map f ) join ◦ map join join ◦ unit = = = = = (map g) ◦ (map f ) map f ◦ unit map f ◦ join join ◦ join join ◦ map unit = id (functor) (natural unit) (natural join) (assoc join) (identity) The first law follows from the fact that map corresponds to a functor. The next two laws hold because unit and join are both natural transformations. Finally, the last two laws are additional laws that are required to hold about monads (a precise definition can be found in Section 3). The proofs that the two definitions of malias (implementing call-by-name and call-by-value strategies) are correct can be found in Figure 8 and Figure 9, respectively. The proofs use the above monad laws. We use a definition malias ≡ map unit, which is equivalent to the definition in Appendix A. The figures include proofs for the four additional malias laws as defined in Section 2.2: naturality, associativity, computationality and identity. Evaluation strategies for monadic computations 88 map (map f ) ◦ malias = map (map f ) ◦ unit = unit ◦ map f = malias ◦ map f (definition) (naturality) (definition) map malias ◦ malias = map unit ◦ unit = unit ◦ unit = malias ◦ malias (definition) (natural unit) (definition) malias ◦ unit = unit ◦ unit (definition) join ◦ malias = join ◦ unit = id (definition) (identity) Figure 8: Call-by-name definition of malias obeys the laws map (map f ) ◦ malias = map (map f ) ◦ map unit = map (map f ◦ unit) = map (unit ◦ f ) = map unit ◦ map f = malias ◦ map f (definition) (functor) (natural unit) (functor) (definition) map malias ◦ malias = map (map unit) ◦ (map unit) = map (map unit ◦ unit) = map (unit ◦ unit) = map unit ◦ map unit = malias ◦ malias (definition) (functor) (natural unit) (functor) (definition) malias ◦ unit = map unit ◦ unit = unit ◦ unit (definition) (natural unit) join ◦ malias = join ◦ map unit = id (definition) (identity) Figure 9: Call-by-value definition of malias obeys the laws Tomas Petricek C 89 Typing preservation proof In this section, we show that our translation preserves typing. Given a well-typed term e of type τ , the translated term JeKcba is also well-typed and has a type Jτ Kcba . In the rest of this section, we write J−K for J−Kcba . To show that the property holds, we use induction over the typing rules, using the fact that Jτ K = M τ ′ for some τ ′ . The inductive construction of the typing derivation follows the following rules: JΓ, x : τ ⊢ x : τ K = JΓK, x : M Jτ K ⊢ x : M Jτ K JΓ, x : τ1 ⊢ e : τ2 K JΓ ⊢ λ x.e : τ1 → τ2 K = JΓK, x : M Jτ1 K ⊢ JeK : M Jτ2 K JΓK ⊢ unit (λ x.JeK) : M (M Jτ1 K → M Jτ2 K) JΓ ⊢ e1 : τ1 → τ2 K JΓ ⊢ e2 : τ1 K JΓ ⊢ e1 e2 : τ2 K = JΓK ⊢ Je1 K : M (M Jτ1 K → M Jτ2 K) Γ ⊢ Je2 K : M Jτ1 K Γ ⊢ bind Je1 K (λ f .bind (malias Je2 K) f )) : M Jτ2 K JΓ ⊢ e1 : τ1 K JΓ, x : τ1 ⊢ e2 : τ2 K JΓ ⊢ let x = e1 in e2 : τ2 K = JΓK ⊢ Je1 K : M Jτ1 K JΓK, x : M Jτ1 K ⊢ Je2 K : M Jτ2 K JΓK ⊢ bind (malias Je1 K) (λ x.Je2 K) : M Jτ2 K	6
Relevant change points in high dimensional time series Holger Dette, Josua Gösmann Ruhr-Universität Bochum Fakultät für Mathematik arXiv:1704.04614v2 [math.ST] 3 Nov 2017 44780 Bochum, Germany e-mail: holger.dette@rub.de josua.goesmann@rub.de November 7, 2017 Abstract This paper investigates the problem of detecting relevant change points in the mean vector, say µt = (µ1,t , . . . , µd,t )T of a high dimensional time series (Zt )t∈Z . While the recent literature on testing for change points in this context considers hypotheses for the equality of the means (1) (2) µh and µh before and after the change points in the different components, we are interested in a null hypothesis of the form (1) (2) H0 : \|µh − µh \| ≤ ∆h for all h = 1, . . . , d where ∆1 , . . . , ∆d are given thresholds for which a smaller difference of the means in the hth component is considered to be non-relevant. This formulation of the testing problem is motivated by the fact that in many applications a modification of the statistical analysis might not be necessary, if the differences between the parameters before and after the change points in the individual components are small. This problem is of particular relevance in high dimensional change point analysis, where a small change in only one component can yield a rejection by the classical procedure although all components change only in a non-relevant way. We propose a new test for this problem based on the maximum of squared and integrated CUSUM statistics and investigate its properties as the sample size n and the dimension d both converge to infinity. In particular, using Gaussian approximations for the maximum of a large number of dependent random variables, we show that on certain points of the boundary of the null hypothesis a standardised version of the maximum converges weakly to a Gumbel distribution. This result is used to construct a consistent asymptotic level α test and a multiplier bootstrap procedure is proposed, which improves the finite sample performance of the test. The finite sample properties of the test are investigated by means of a simulation study and we also illustrate the new approach investigating data from hydrology. Keywords: high dimensional time series, change point analysis, CUSUM, relevant changes, precise hypotheses, physical dependence measure AMS Subject Classification: 62M10, 62F05, 62G10 1 1 Introduction In the context of high dimensional time series it is typically unrealistic to assume stationarity. A simple form of non-stationarity, which is motivated by financial time series, where large panels of asset returns routinely display break points, is to assume structural breaks at different times (the change points) in the individual components. One goal of statistical inference is to correctly estimate these “change points” such that the original data can be partitioned into shorter stationary segments. This field is called change point analysis in the statistical literature and since the seminal papers of Page (1954, 1955) numerous authors have worked on the problem of detecting structural breaks or change points in various statistical models [see Aue and Horváth (2013) for a recent review]. There exists in particular an enormous amount of literature on testing for and estimating the location of a change in the mean vector µt = (µ1,t , . . . , µd,t )T = E [Zt ] of a multivariate time series (Zt )nt=1 [see Chu et al. (1996), Horváth et al. (1999), Kirch et al. (2015) among others]. A common feature in these references consists in the fact that the dimension, say d, of the time series is fixed. High dimensional change point problems, where the dimension d increases with sample size have only been recently considered in the literature [see Bai (2010), Zhang et al. (2010), Horváth and Hus̆ková (2012) and Enikeeva and Harchaoui (2014), Jirak (2015a), Cho and Fryzlewicz (2015) and Wang and Samworth (2016) among others]. Some of this work uses information across the coordinates in order to detect smaller changes than could be observed in any individual component series. In the simplest case of one structural break in each component many authors attack the problem of detecting the change point by means of hypothesis testing. For example, Jirak (2015a) investigates the hypothesis of no structural break in a high-dimensional time series by testing the hypotheses H0 : µh,1 = µh,2 = . . . = µh,n for all h = 1, . . . , d, (1.1) where µh,t denotes the h-th component of the mean vector µt of the random variable Zt (t = 1, . . . , n). The alternative is then formulated (in the simplest case of one structural break) as (1) H1 : µh = µh,1 = µh,2 = · · · = µh,2 6= (1.2) µh,kh +1 = µh,kh +2 = · · · = µh,n = (2) µh for at least one h ∈ {1, . . . , d}, where kh ∈ {1, . . . , n} denotes the (unknown) location of the change point in the h-th component. While - even under sparsity assumptions - the detection of small changes in each component is a very challenging problem, a modification of the statistical analysis (such as prediction) might not be necessary if the actual size of change is small. For example, in risk management situations, one is interested in fitting a suitable model for forecasting Value at Risk from data after the last change point [see e.g. Wied (2013)], but in practice, small changes in the parameter are perhaps not very interesting because they do not yield large changes in the Value at Risk. The forecasting quality might only improve slightly, but this benefit could be negatively overcompensated by transaction costs, in particular in high-dimensional portfolios. Moreover, even, if the null hypothesis (1.1) is not rejected, it is difficult to quantify the statistical uncertainty for the subsequent statistical analysis 2 (conducted under the assumption of stationarity), as there is no control about the type II error in this case. The present work is motivated by these observations and proposes a test for the null hypothesis of no relevant change point in a high dimensional context, that is (1) (2) (1.3) (1) (2) (1.4) H0,∆ : \|µh − µh \| ≤ ∆h for all h = 1, . . . , d versus HA,∆ : \|µh − µh \| > ∆h for at least one h ∈ {1, . . . , d}, (1) (2) where µh and µh are the parameters before and after the change point in the h-th component and ∆1 , . . . , ∆d are given thresholds for which a smaller difference of the means in the h-th component is considered as non-relevant. The problem of testing for a relevant difference between (one dimensional) means of two samples has been discussed by numerous authors mainly in the field of biostatistics (see Wellek (2010) for a recent review). In particular testing relevant hypotheses avoids the consistency problem as mentioned in Berkson (1938), that is: Any consistent test will detect any arbitrary small change in the parameters if the sample size is sufficiently large. Dette and Wied (2016) considered relevant hypotheses in the context of change point problems for general parameters, but did not discuss the high dimensional setup, where the dimension increases with sample size. In this case the statistical problems are completely different. The alternative approach requires the specification of the thresholds ∆h > 0, and this has to be carefully discussed and depends on the specific application. We also note that the hypotheses (1.3) contain the classical hypotheses (1.1), which are obtained by simply choosing ∆h = 0 for all h = 1, . . . , d. Nevertheless we argue that from a practical point of view it might be very reasonable to think about this choice more carefully and to define the size of change in which one is really interested. In particular it is often known that ∆h 6= 0 although one is testing “classical hypotheses” of the form (1.1) and (1.2). Moreover, a decision of no relevant structural break at a controlled type I error can be easily achieved by interchanging the null hypothesis and alternative in (1.3), i.e. considering the hypotheses versus e 0,∆ : \|µ(1) − µ(2) \| > ∆h for at least one h ∈ {1, . . . , d} H h h (1) (2) e HA,∆ : \|µ − µ \| ≤ ∆h for all h = 1, . . . , d. h h (1.5) (1.6) In this paper we propose for the first time a test for the hypotheses of a relevant structural break in any of the components of a high dimensional time series. The basic ideas are explained in Section 2 (without going into any technical details), where we propose to calculate for any component the integral of the squared CUSUM process and reject the null hypotheses whenever the maximum of these integrals (calculated with respect to all components) is large. In order to obtain critical values for this test we derive in Section 3 the asymptotic distribution of an appropriately standardized version of the maximum as the sample size and the dimension converge to infinity. We also provide several auxiliary results, which are of own interest, and investigate the case where the maximum is only calculated over a subset of the components. These results are then used in Section 4 to prove that the proposed test yields to a valid statistical inference, i.e. it is a consistent and asymptotic 3 level α test. It turns out that - in contrast to the classical change point problem - the analysis of the test for no relevant structural breaks is substantially harder as the null hypothesis does not correspond to a stationary process (non-relevant changes in the means are allowed). Section 5 is devoted to the investigation of a multiplier block bootstrap procedure. In particular we prove that the quantiles generated by this resampling method also yield to a consistent asymptotic level α test. The finite sample properties of the new test are investigated in Section 6, where we also illustrate our approach analysing a data example from hydrology. Finally some of the technical details are deferred to Section 7. 2 Relevant changes in high dimensions - basic principles In this Section we explain the basic idea of our approach to test for a relevant change in at least one component of the mean vector of a high dimensional time series. For the sake of a transparent representation we try to avoid technical details at this stage and refer to the subsequent sections, where we present the basic assumptions and mathematical details establishing the validity of the proposed method. Throughout this paper we consider an array of real valued random variables {Zj,h }j∈Z,h∈N such that Zj,h = µj,h + Xj,h , (2.1) where µj,h ∈ R for all j ∈ Z, h ∈ N and {Xj,h }j∈Z,h∈N denotes an array of centered and real valued random variables, which implies µj,h = E [Zj,h ] for all j ∈ Z, h ∈ N. It follows from the assumptions made in Section 3 that for each fixed d ∈ N the time series {(Xj,1 , . . . , Xj,d )T }j∈Z (2.2) is stationary. Suppose that Z1 = (Z1,1 , . . . , Z1,d )T , . . . , Zn = (Zn,1 , . . . , Zn,d )T ∈ Rd are d-dimensional observations from the array {Zj,h }j∈Z,h∈N and assume that for each component h ∈ {1, . . . , d} there exists an unknown constant th ∈ (0, 1), such that (1) µh = µj,h , j = 1, . . . , bnth c, (2) (2.3) µh = µj,h , j = bnth c + 1, . . . , n, where bxc = sup{z ∈ Z \| z ≤ x} denotes the larges integer smaller or equal than x. In this case the random variables {Zj,h }h=1,...,d;j=1,...,n also depend on n, i.e. they form a triangular array, but for the sake of readability, we will suppress this dependence in our notation. (1) (2) We define ∆µh = µh −µh as the unknown difference between the means in the h-th component before and after the change point th . Note, that in the case ∆µh 6= 0 the actual location kh = bnth c of the change point depends on the sample size n, which is a common assumption in the literature 4 on change point problems to perform asymptotic inference. It simply ensures that the number of observations before and after the change point is growing proportional to n. For each h with ∆µh = 0 the observable process {Zj,h }j∈Z is stationary and to avoid misunderstandings we set th = 1/2, whenever ∆µh = 0. The reader should notice that in this case the actual value is of no interest in the following discussion. With this notation the hypotheses in (1.3) can be rewritten as versus H0,∆ : \|∆µh \| ≤ ∆h for all h ∈ {1, . . . , d} (2.4) HA,∆ : \|∆µh \| > ∆h for some h ∈ {1, . . . , d}. (2.5) To develop an appropriate test statistic for these hypotheses we rely on the widely used concept of CUSUM-statistics. For each component h ∈ {1, . . . , d} we consider the corresponding CUSUMprocess for the h-th component defined by bnsc bnsc n 1X bnsc X n − bnsc X bnsc Un,h (s) = Zj,h − 2 Zj,h = Zj,h − 2 n n n2 n j=1 j=1 j=1 Under the assumptions stated in Section 3 it is shown in Section 7.1 that Z 1 3 2 E Un,h (s)ds = ∆µ2h + o(1) (th (1 − th ))2 0 as n → ∞ and therefore our considerations will be based on the statistic Z 1 3 2 U2n,h (s)ds, M̂n,h = 2 (t̂h (1 − t̂h )) 0 n X Zj,h . (2.6) j=bnsc+1 (2.7) (2.8) where t̂h denotes an appropriate estimator for the unknown location th of the structural break, that will be precisely defined in Section 3.2. The null hypothesis H0,h : \|∆µh \| ≤ ∆h (2.9) of no relevant change in the h-th component will be rejected for large values of M̂2n,h , and in order to determine a critical value we introduce the normalization √ n (∆) T̂n,h = (2.10) M̂2n,h − ∆2h , τ̂h σ̂h ∆h where σ̂h denotes an estimator for the unknown long-run variance (see Section 3.3 for a precise definition), and the quantity τ̂h is a function of the estimate of the change point defined by q 2 1 + 2t̂h (1 − t̂h ) √ τ̂h = τ (t̂h ) := , (2.11) 5t̂h (1 − t̂h ) which arises due to the integral in equation (2.8). It will be shown in Section 7.3.2 that for a fixed component h ∈ {1, . . . , d} (∆) D T̂n,h =⇒ N (0, 1) 5 as n → ∞, (2.12) D if ∆h = ∆µh , where the symbol =⇒ represents weak convergence of a random variable. Moreover monotonicity arguments show that the test, which rejects the null hypothesis (2.9) of a relevant (∆) change point in the mean of the h-th component, whenever T̂n,h exceeds the (1 − α)-quantile of the standard normal distribution, is a consistent asymptotic level α test. In order to construct a test for the hypotheses H0,∆ versus HA,∆ in (2.4) and (2.5) of a relevant change point in at least one of the components of a high dimensional time series we propose a simultaneous test, which rejects the null hypothesis for large values of the statistic d (∆) max T̂n,h . h=1 Note that a similar approach has been investigated by Jirak (2015a), who considered the “classical” change point problem in high dimension, that is H0,class : \|∆µh \| = 0 for all h ∈ {1, . . . , d} versus (2.13) HA,class : \|∆µh \| > 0 for some h ∈ {1, . . . , d}. (∆) In this case the weak convergence (2.12) does not hold (in fact T̂n,h = oP (1) under H0,class ) and a different statistic has to be considered. As it is well known that the (adjusted) maximum of standard normal distributed random variables converges weakly to a Gumbel distribution if they exhibit an appropriate dependence structure [see for example Berman (1964))], it is reasonable to consider the following d (∆) (2.14) Td,n = ad max T̂n,h − bd h=1 for the high-dimensional change point problem (2.4), where ad and bd denote appropriate sequences, such that the left hand side converges weakly at specific points of the “boundary” of the parameter space corresponding to the null hypothesis. To make these arguments more precise, define the sequences p 2 log d, (2.15) ad = p p bd = ad − log(4π log d)/2ad = 2 log d − log(4π log d)/(2 2 log d), (2.16) and note that the parameter spaces corresponding to the null hypothesis (2.4) and the alternative (2.5) are given by n o H0 = (x, y) ∈ Rd × Rd \|xh − yh \| ≤ ∆h ∀h ∈ {1, . . . , d} and HA = R2d \ H0 , respectively (here x = (x1 , . . . , xd )T , y = (y1 , . . . , yd )T ) denote d-dimensional vectors). Define for k = 0, . . . , d the “(d − k)-dimensional boundary of the hypotheses (2.4) and (2.5)” by n o Ak = (x, y) ∈ R2d \|xh − yh \| ≤ ∆h ∀ h with equality for precisely k components ⊆ H0 . (2.17) For the case d = 2, we illustrate this decomposition of the null hypothesis parameter space in Figure 1. In fact, a large part of this paper is devoted to prove the following statements (under appropriate assumptions - see Theorem 4.1 in Section 4): 6 Figure 1: Decomposition of the parameter space corresponding to the null hypothesis in (2.4) into the sets A0 , A1 and A2 for the case d = 2 . (1) If (µ(1) , µ(2) ) ∈ Ad , the weak convergence D Td,n =⇒ G holds as both n, d → ∞, where G denotes a Gumbel distribution, i.e. P (G ≤ x) = e−e −x . (2.18) (1) (2) (2) If (µ(1) , µ(2) ) ∈ Ak and there exists a constant C, such that \|µh − µh \| ≤ C < ∆h , whenever (1) (2) \|µh − µh \| = 6 ∆h and limd→∞ k/d = c ∈ (0, 1], we have D Td,n =⇒ G + 2 log c as n, d → ∞. (3) If (µ(1) , µ(2) ) ∈ Ak and limd→∞ k/d = 0 we have D Td,n =⇒ −∞ as n, d → ∞. Let g1−α denote the (1 − α) quantile of the Gumbel distribution, then it follows from these considerations that the test which rejects H0,∆ in favour of the alternative hypothesis HA,∆ , whenever Td,n > g1−α , 7 (2.19) is an asymptotic level α test. More precisely, it follows (under appropriate assumptions stated below) that PH0,∆ Td,n > g1−α −→    α α0    0 if (µ(1) , µ(2) ) ∈ Ad if (µ(1) , µ(2) ) ∈ Ak and limd→∞ k/d = c ∈ (0, 1] (2.20) if (µ(1) , µ(2) ) ∈ Ak and limd→∞ k/d = 0 as n, d → ∞, where 0 ≤ α0 ≤ α. Additionally the test is consistent (see Theorem 4.3). In the following sections we will make these arguments more rigorous. Moreover, in order to improve the finite sample approximation of the nominal level we also introduce a multiplier bootstrap procedure and prove its consistency in Section 5. 3 Asymptotic properties In this Section we provide the theoretical background for the test suggested in Section 2. We begin introducing some notations and assumptions. After stating the main assumptions we provide in Section 3.2 the asymptotic theory for an analogue of the statistic Td,n defined in (2.14), where the centering in (2.10) is performed by the ”true” squared differences \|∆µh \|2 and the estimates of the variances σh and the locations of the change points th have been replaced by their ”true” counterparts. In Section 3.3 we introduce estimators for the locations of the structural breaks (which may occur at a different location in each component) and investigate their consistency properties. These are then used to define appropriate variance estimators (note that variances have to be estimated in the case of changes in the mean). Finally, we consider in Section 3.4 the asymptotic distribution of the maximum of the statistics (2.10) where again centering is performed by \|∆µh \|2 instead of ∆2h . These results will then be used in the subsequent Section 4 to investigate the statistical properties of the the test (2.19). Throughout this paper we will use the following notation and symbols. Let x ∧ y and x ∨ y define the minimum and maximum of two real numbers x and y, respectively. For an appropriately integrable random variable Y and q ≥ 1, let kY kq = E [\|Y \|q ]1/q denote the Lq -norm. By the symbols D P =⇒ and −→ we denote weak convergence and convergence in probability, respectively. Moreover, we use the notation xn . yn , whenever the inequality xn ≤ C · yn , holds for some constant C > 0 which does not depend on the sample size and dimension and whose actual value is of no further interest. Due to its frequent appearance, G will always represent a (standard) Gumbel distribution defined by (2.18). In the high dimensional setup the dimension d converges to infinity with n → ∞. Recall the definition of model (2.1) and assume that the time series {Xj,h }j∈Z,h∈N forms a physical system [see e.g. Wu (2005)], that is Xj,h = gh (εj , εj−1 , . . . ) , (3.1) where {εj }j∈Z is a sequence of i.i.d. random variables with values in some measure space S such that for each h ∈ N the function gh : SN → R is measurable. Note that it follows from (3.1) that 8 the time series defined in (2.2) is stationary. Let ε00 be an independent copy of ε0 and define 0 Xj,h = gh (εj , εj−1 , . . . , ε1 , ε00 , ε−1 , . . . ). (3.2) 0 is used to quantify the (temporal) dependence The distance between Xj,h and its counterpart Xj,h of the physical system, and for this purpose we introduce the coefficients 0 ϑj,h,p := kXj,h − Xj,h kp , (3.3) which measure the influence of ε0 on the random variable Xj,h . Let us also define some additional parameters. For h, i ∈ N φh,i (j) = Cov(X0,h , Xj,i ) = Cov(Z0,h , Zj,i ) denotes the covariance function of the h-th and i-th component at lag j Accordingly, the autocovariance function for the h-th component is given by φh (j) = φh,h (j) = Cov(X0,h , Xj,h ) and we obtain the well-known representations X X φh (j) (3.4) γh,i = φh,i (j) and σh2 = j∈Z j∈Z for the long-run covariance and long-run variance, respectively. If we have σh , σi > 0, we can additionally define the long-run correlations by ρh,i = γh,i σh σi (3.5) and it will be crucial for our work, that these quantities become sufficiently small with an increasing temporal distance \|h − i\|. This will be precisely formulated in the following section. 3.1 Assumptions Operating in a high-dimensional framework usually needs stronger assumptions than those for the finite-dimensional case. Mainly, we need uniform dependence and moment conditions among all components to exclude extreme cases and to ensure, that the unknown parameters can be estimated accurately. In the high-dimensional setup considered in this paper the number of parameters th , σh grows together with the dimension d, since even under the null hypothesis of no relevant change points each component exhibits its own variance and change point structure. The precise assumptions made in this paper are the following. Assumption 3.1 (temporal assumptions) Suppose that there exist constants δ ∈ (0, 1) and σ− > 0 such that for some p > 4 the physical dependence coefficients ϑj,h,p and long run variances σh defined in (3.3) and (3.4), respectively, satisfy (T1) suph∈N ϑj,h,p . δ j , There exists constants 0 < σ− ≤ σ+ such that for all h ∈ N 9 (T2) σ− ≤ inf h∈N σh ≤ σ+ . Assumption 3.2 (spatial assumptions) The dimension d increases with the sample size at a polynomial rate, i.e. we assume that for constants C1 and D (S1) d = C1 nD , where the exponent D satisfies (S2) 0 < D < min{p/2 − 2, p/2 − B(p/2 + 1) − 1}, and p refers to the Lp -norm k · kp used to measure the physical dependence in Assumption 3.1. Here B ∈ (0, 1) denotes a constant used to control the bandwidth of an variance estimator, that will be defined in Section 3.4. Further we assume for the long-run correlations in (3.5) (S3) sup\|h−i\|>1 \|ρh,i \| ≤ ρ+ < 1, (S4) \|ρh,i \| ≤ C2 log(\|h − i\| + 2)−2−η , where ρ+ > 0, η > 0 and C2 > 0 denote global constants. Assumption 3.3 (moment assumptions) Suppose, that there exists a positive sequence (Md )d∈N and constants C3 > 1 and C4 > 0, such that (M1) maxdh=1 E [exp(\|X1,h \|/Md )] ≤ C3 (M2) Md ≤ C4 nm with m < 3/8. Assumption 3.4 (location of the change points) There exists a constant t ∈ (0, 1/2), such that for all h ∈ N the unknown locations bnth c of the structural breaks satisfy (C1) t ≤ th ≤ 1 − t. Let us give a brief explanation for the Assumptions 3.1 - 3.4. The temporal Assumptions (T1) and (T2) define the temporal dependence structure and bounds for the long-run variance. Further (T1) implies the existence of the quantities σh , γh,i and ρh,i defined in (3.4) and (3.5). Condition (S3) and (S4) refer to the spatial dependence and are only needed to derive the desired extreme value convergence. Assumption (S1) gives a relation between the number of observations and its dimension, while (S2) is a slightly technical assumption, which enables reasonable estimations of the variance σh and the locations th of the structural breaks. For a proof of the uniform consistency of the estimates for the latter quantity we must further rely on Assumption (C1), which makes the change points identifiable and is a common assumption in the literature. Roughly speaking, it simply ensures to have enough data before and after the change point in each component. Assumptions (S1) and (S2) together with n → ∞ directly imply d → ∞. It is also worth mentioning, that (S1) can be replaced by d . nD , if one additionally supposes that d → ∞. 10 The moment Assumptions (M1) and (M2) are required for a Gaussian approximation and are satisfied, if {Xj,h }j∈Z,h∈{1,...,d} is stationary with respect to the index j and for each h ∈ N the random variable X1,h is sub-Gaussian with parameters vh , Vh > 0, i.e. E [exp(λX1,h )] ≤ Vh exp(λ2 vh ), for all λ ∈ R, where the constants (vh )h=1,...,d and (Vh )h=1,...,d satisfy d max vh < n3/4 and h=1 max Vh ≤ C. h∈N √ for some constant C > 0. Choosing Md = maxdh=1 vh we directly obtain condition (M2). Condition (M1) follows by a straightforward calculation, that is d d h=1 h=1 2 max E [exp(\|X1,h \|/Md )] ≤ 2 max Vh evh /Md ≤ 2Ce < ∞. 3.2 Asymptotic properties - known variances and locations In this section we assume that the locations th of the changes and the long-run variances σh are known. Recalling the approximation (2.7) we define Z 1 1 (3.6) U2 (s)ds M2n,h = (th (1 − th ))2 0 n,h and investigate the asymptotic properties of the maximum of the statistics √ n Tn,h = M2n,h − ∆µ2h , τh σh \|∆µh \| (3.7) where τh = τ (th ) and the function τ is defined by (2.11). Note that Tn,h is the analogue of the (∆) statistic T̂n,h , where the thresholds ∆h , estimates t̂h and σ̂h have been replaced by the unknown quantities ∆µh , th and σh , respectively. Our first result shows that an appropriate standardized version of the maximum of the statistics Tn,h converges weakly to a Gumbel distribution. The proof is complicated and we indicate the main steps in this section deferring the more technical arguments to the appendix - see Section 7. Theorem 3.5 Assume that the Assumptions 3.1 - 3.4 are satisfied and that additionally there exist constants C` , Cu (independent of h and d) such that the inequalities C` ≤ \|∆µh \| ≤ Cu hold for all h = 1, . . . , d . Then d D ad max Tn,h − bd =⇒ G h=1 as d, n → ∞, where the sequences ad and bd are defined in (2.15) and (2.16), respectively. Proof of Theorem 3.5 (main steps). Observing the definition (3.6) and (3.7) we obtain by a straightforward calculation the representation Z 1 √ 3 n (1) (2) 2 2 (3.8) Tn,h = Un,h (s)ds − µh (s, th )ds = Tn,h + Tn,h , 2 (th (1 − th )) τh σh \|∆µh \| 0 11 where µh (s, th ) = (th ∧ s − sth ) (µh,1 − µh,2 ) (1) (3.9) (2) and the statistics Tn,h and Tn,h are defined by (1) Tn,h (2) Tn,h √ Z 1 3 n = (Un,h (s) − µh (s, th ))2 ds, (th (1 − th ))2 τh σh \|∆µh \| 0 √ Z 1 6 n = µh (s, th )(Un,h (s) − µh (s, th ))ds. (th (1 − th ))2 τh σh \|∆µh \| 0 For the following discussion, we introduce the additional notation √ Z 1 6 n (2) T̄n,h = µh (s, th ) (Un,h (s) − E [Un,h (s)]) ds, (th (1 − th ))2 τh σh \|∆µh \| 0 (3.10) (3.11) (3.12) (1) Our first auxiliary result shows that the first term Tn,h in the decomposition (3.8) is asymptotically negligible and is proved in Section 7.2.1. Lemma 3.6 If the assumptions of Theorem 3.5 are satisfied, we have d (1) P ad max Tn,h −→ 0 h=1 as d, n → ∞, where the sequence ad is defined in (2.15). (2) By Lemma 3.6 it suffices now to deal with the term Tn,h . The next step in the proof of Theorem (2) 3.5 consists in a (uniform) approximation of the distribution of the maximum of the statistics T̄n,h by of the distribution of the maximum of (dependent) Gaussian random variables. The proof of the following result is given in Section 7.2.2, where we make use of new developments on Gaussian approximations for maxima of sums of random variables [see Chernozhukov et al. (2013) and Zhang and Cheng (2016). Lemma 3.7 If the Assumptions 3.1 - 3.4 are satisfied, there exists a Gaussian distributed random vector N = (N1 , . . . , Nd )T with mean E [N ] = 0 and covariance matrix (Σh,i )dh,i=1 , such that d d (2) sup P max T̄n,h ≤ x − P max Nh ≤ x = o(1) x∈R h=1 h=1 for d, n → ∞. Moreover, the entries of the matrix Σ are bounded by \|Σh,i \| ≤ \|ρh,i \|, where ρh,i are the long-run correlations defined in (3.5). By Lemma 3.7 it suffices to establish the desired limiting distribution for the maximum of a Gaussian distributed vector. Nowadays, this is a well-understood area of mathematics and we can rely on results of Berman (1964), who originally examined the behavior of the maximum of dependent Gaussian random variables. A straightforward adaption of these arguments shows that the sequence of random variables {Ni }i∈N constructed in Lemma 3.7 satisfies 12 d D ad max Nh − bd =⇒ G (3.13) h=1 as d → ∞, where the sequences ad and bd are defined in (2.15) and (2.16), respectively. Now, combining Lemma 3.7 and (3.13) directly leads to d D (2) ad max T̄n,h − bd =⇒ G. h=1 By the assumption \|∆µh \| ≤ Cu it follows that maxdh=1 (µh (s, th ) − E [Un,h (s)]) = O(n−1 ), which yields d D (2) ad max Tn,h − bd =⇒ G. (3.14) h=1 (1) Due to Tn,h ≥ 0, we obtain the inequalities d d d d (1) (2) (2) ad max Tn,h − bd ≤ ad max Tn,h − bd ≤ ad max Tn,h + ad max Tn,h − bd , h=1 h=1 h=1 h=1 which together with Lemma 3.6 yields the assertion of Theorem 3.5. In the next step we will replace the unknown quantities th and σh in (3.7) by suitable estimators, say t̂h and σ̂h , and obtain the statistics √ n T̂n,h = (3.15) M̂2n,h − ∆µ2h , h ∈ {1, . . . , d} . τ̂h σ̂h \|∆µh \| (∆) We emphasize again that the statistics T̂n,h do not coincide with the statistics T̂n,h in (2.10), which (1) (2) are actually used in the test (2.19) except in the case where ∆µ2h = \|µh − µh \|2 = ∆2h for all h = 1, . . . , d. Thus centering is still performed with respect to the unknown difference of the means before and after the change points. In the following two subsections we give a precise definition of the two estimators and derive an analogue of Theorem 3.5 in the case of estimated change points and variances. 3.3 Estimation of long-run variances and change point locations Determining the relative locations th of the structural breaks and constructing an estimator for the long-run variances σh for all components h ∈ {1, . . . , d} is a rather difficult task in a high dimensional setting. A further difficulty in the problem of testing for relevant structural breaks consists in the fact that even under the null hypothesis there may appear structural breaks in the mean and the corresponding process is not stationary. Therefore in contrast of testing the “classical” hypotheses in (2.13) the construction of a suitable variance estimator is not trivial. A P standard long-run variance estimator in terms of i≤βn φ̂h (i) for an increasing sequence {βn }n∈N and appropriate estimators φ̂h (i) of the auto-covariance from the full sample may not be consistent due to possible changes of the mean. 13 Following Jirak (2015a) we define for each component h = 1, . . . , d the sets Dh,1 := {Zj,h \| 1 ≤ j ≤ bnth c} and Dh,2 := {Zj,h \| bnth c < j ≤ n} , which are the observations before and after the (unknown) change point bnth c, respectively. Since these points are usually unknown, we need to estimate them and for this purpose we propose the common estimator bnsc t̂h := argmax Un,h (s) = argmax X s∈(t,1−t) j=1 s∈(t,1−t) n Zj,h − bnsc X Zj,h . n (3.16) j=1 The following Lemma shows that these estimators are uniformly consistent with respect to all components, where a change point exists. Its proof follows by an adaption of Corollary 3.1 in Jirak (2015a), and is therefore omitted. Lemma 3.8 Let Sd = {1 ≤ h ≤ d \| \|∆µh \| = 0} . (3.17) denote the set of all components h ∈ {1, . . . , d}, where the corresponding time series {Zj,h }j∈Z is stationary, and define µ?d = minc \|∆µh \|. h∈Sd (3.18) Suppose that Assumptions 3.1 - 3.4 are satisfied and assume further that the condition lim sup d,n→∞ log n = 0. µ?d n (3.19) is satisfied. Then for a sufficiently small constant C > 0 it holds that maxc \|t̂h − th \| = oP (n−C ), h∈Sd Moreover, if additional the condition µ?d ≥ C` holds for some constant C` > 0 it follows that maxc \|t̂h − th \| = oP (n−1/2 ). h∈Sd Roughly speaking condition (3.19) guarantees that the decreasing sequence µ?d does not converge too fast to 0 if the dimension of the time series converge to infinity. Otherwise it is not possible to identify the (relative) locations of all change points simultaneously. In view of Lemma 3.8 it is reasonable to estimate the unknown sets Dh,1 and Dh,2 by bh,1 : = Zj,h \| 1 ≤ j ≤ n max{S t̂h , t} , D (3.20) bh,2 : = Zj,h \| n − n max{S(1 − t̂h ), t} < j ≤ n , D 2 and σ̂ 2 be the respectively, where S ∈ (0, 1) denotes a user-specified separation constant. Let σ̂h,1 h,2 bh,1 and D bh,2 , respectively. Then we standard long-run variance estimators based on the samples D 14 2 and σ̂ 2 to estimate the long run variances σ 2 , for example can use any convex combination of σ̂h,1 h,2 h 2 + σ̂ 2 ). To simplify the technical arguments in Section 7 we consider a truncated σ̂h2 = 21 (σ̂h,1 h,2 version, that is 2 2 σ̂h2 = min s2+ , max s2− , ( 12 (σ̂h,1 + σ̂h,2 ) . (3.21) where 0 < s− and s+ are a sufficiently small and large constant, respectively. The following statement is a straightforward implication of the results in Section 3 of Jirak (2015a) and yields the consistency of these estimators (uniformly with respect to the spatial component). Lemma 3.9 Suppose that Assumptions 3.1 - 3.4 are satisfied and additionally that there exists a constant C` > 0 such that the inequality C` ≤ \|∆µh \| hold for all h = 1, . . . , d. Then we have d max \|σ̂h − σh \| = op n−η h=1 for a sufficiently small constant η > 0. 3.4 Weak convergence Equipped with Lemmas 3.8 and 3.9 we are now able to state the main result of this section, which will be proved in Section 7.2.3. Theorem 3.10 If the assumptions of Theorem 3.5 are satisfied, then the statistics T̂n,h defined in (3.15) satisfy d D Td,n = ad max T̂n,h − bd =⇒ G (3.22) h=1 as d, n → ∞, where the sequences ad and bd are defined in (2.15) and (2.16), respectively. (∆) Recall again that the statistics T̂n,h and T̂n,h in (2.10) do not coincide in general. Thus Theorem 3.10 does not show that the test (2.19) is an asymptotic level α test because it does not cover all parameter configurations of the the null hypothesis (2.4). However, if the vector (µ(1) , µ(2) ) is an (∆) element of the set Ad defined in (2.17) we have T̂n,h = T̂n,h and it follows from this result that the probability of rejection converges to α, that is lim P(µ(1) ,µ(2) )∈Ad Td,n > g1−α = α. d,n→∞ We conclude this section with a result, which can be used to control the type I error of the test (2.19) for other values of the vector (µ(1) , µ(2) ). Corollary 3.11 Let {Md }d∈N be an increasing sequence of subsets of {1, . . . , d} (as d, n → ∞). If the assumptions of Theorem 3.5 hold, then   G if limd→∞ \|Md \|/d = 1  D ad max T̂n,h − bd =⇒ G + 2 log c if limd→∞ \|Md \|/d = c for some c ∈ (0, 1),  h∈Md  −∞ if limd→∞ \|Md \|/d = 0. Irrespective of the sequence {Md }d∈N , the bound lim sup P ad max T̂n,h − bd > x ≤ P (G > x) d,n→∞ h∈Md is valid for all x ∈ R. 15 (3.23) 4 Relevant changes in high dimensional time series Recall the problem of testing the hypotheses of a relevant change defined in (2.4) and (2.5). We propose to reject the null hypothesis of no relevant change in any component of the high dimensional mean vector, whenever the inequality (2.19) holds, that is Td,n > g1−α , where the test statistic Td,n is defined in (2.14) and g1−α denotes the (1 − α) quantile of the Gumbel distribution. The following two results make the discussion at the end of Section 2 rigorous and show that the test introduced in (2.19) defines in fact a consistent and asymptotic level α test. Theorem 4.1 Suppose that the Assumptions 3.1 - 3.4 are satisfied, α ∈ (0, 1 − e−1 ] and that there exist constants ∆− , ∆+ such that the thresholds ∆h satisfy the inequalities 0 < ∆− ≤ ∆h ≤ ∆+ < ∞ (4.1) for all h = 1, . . . , d. Then, under the null hypothesis H0,∆ of no relevant change, it follows lim sup P (Td,n > g1−α ) ≤ α. (4.2) d,n→∞ Moreover, let Bd = {h ∈ {1, . . . , d} \| ∆h = \|∆µh \|} and assume further that ∆h − \|∆µh \| ≥ C∆ > 0 for all h ∈ Bdc , for some constant C∆ > 0, then, under H0,∆ , we have   G if limd→∞ \|Bd \|/d = 1,    D Td,n =⇒ G + 2 log c if limd→∞ \|Bd \|/d = c for c ∈ (0, 1),     −∞ if limd→∞ \|Bd \|/d = 0. (4.3) (4.4) Remark 4.2 Condition (4.1) is actually not a very strong restriction since the thresholds ∆h are defined by the user. Nevertheless, the condition is crucial since we use the factor 1/∆h as a (∆) normalisation in the statistics T̂n,h defined in (2.10). Note that under the null hypothesis (2.4) the inequality ∆h ≤ ∆+ is equivalent to \|∆µh \| ≤ Cu , which was one of the assumptions in Theorem 3.10. Consequently the assertion of Theorem 4.1 follows from Theorem 3.10 in the case where \|∆µh \| = ∆h for all h ∈ {1, . . . , d}. However, in the general case the proof of Theorem 4.1 is more complicated and deferred to Section 7.3.1, where we also handle the case \|∆µh \| < ∆h . In the following result we investigate the consistency of the new test. Interestingly it requires less assumptions than Theorem 4.1. Theorem 4.3 Suppose that the Assumptions (T1), (T2) and (C1) hold. Then under the alternative hypothesis HA,∆ of at least one relevant change point we have P Td,n −→ ∞, as d, n → ∞, which in particular gives lim P (Td,n > g1−α ) = 1. d,n→∞ 16 If the test (2.19) rejects the null hypothesis H0,∆ in (2.4) we conclude (at a controlled type I error) that there is at least one component with a relevant change in mean. As there could exist relevant changes in several components, the next step in the statistical inference is the identification of the set Rd = {1 ≤ h ≤ d \| \|∆µh \| > ∆h } , (4.5) of all components with a relevant change. Note that the hypotheses in (2.4) and (2.5) are equivalent to H0,∆ : Rd = ∅ versus HA,∆ : Rd 6= ∅ . In light of Theorem 4.1 and 4.3 a natural estimator for this set is therefore given by n o b d (α) = 1 ≤ h ≤ d \| T̂ (∆) > g1−α /ad + bd . R (4.6) n,h The following theorem provides a consistency result of this estimate. Theorem 4.4 Suppose that the Assumptions 3.1 - 3.4 hold and assume additionally that there exist two constants 0 < C < 1/2, Cu > 0 such that nC min (\|∆µh \| − ∆h ) = ∞ and h∈Rd max \|∆µh \| ≤ Cu . h∈Rd (4.7) Then, the set estimator defined in (4.6) satisfies for α ∈ (0, 1 − e−1 ] b d (α) = 1. lim P Rd ⊂ R (4.8) Moreover we have the following lower bound b d (α) ≥ 1 − α. lim inf P Rd = R (4.9) d,n→∞ d,n→∞ 5 Bootstrap The testing procedure introduced in the previous sections is based on the weak convergence of the maximum of appropriately standardized statistics to a Gumbel distribution, and it is well known that the speed of convergence in limit theorems of this type is rather slow. As a consequence the approximation of the nominal level of the test (2.19) for finite sample sizes may not be accurate. A common way to improve the performance of the test, is to obtain the critical values from an appropriate bootstrap procedure. In the context of testing for relevant change points the construction of an appropriate resampling procedure is not obvious as - in contrast to classical change point problems - the parameter space under the null hypothesis is rather large. In particular it will be necessary to simulate the distribution of the statistic Td,n in case \|∆µh \| = ∆h for all h ∈ {1, . . . , d} such that one can replace the quantile of the Gumbel distribution by the corresponding quantile of the bootstrap statistics. A further problem is to mimic the dependence of the underlying times series, which we will address employing a Gaussian multiplier bootstrap, where observations are block-wise multiplied with independent Gaussian random variables [see Künsch (1989) or Lahiri (2003)]. 17 To handle the problem of potential change points under the null hypothesis (2.4) of no relevant changes, observations from blocks in a neighborhood of estimated change points will not be used in the estimate. Furthermore, components without a change point will be ignored when the bootstrap statistic is constructed. We begin describing this idea in more detail and show in Theorem 5.5, that the bootstrap statistic converges weakly to a Gumbel distribution as well. In the sequel we assume without loss of generalization that n = KL and will split the sample into L blocks of length K, and additionally L ∼ n` and K ∼ n1−` for ` ∈ (0, 1). (5.1) We obtain the following quantities, which control the number of blocks before and after the estimated change point. b − = sup{` ∈ N \| `K + K/2 ≤ t̂h n}, L h + b Lh = inf{` ∈ N \| `K − K/2 ≥ t̂h n}, (5.2) where t̂h denotes the estimator of the location th of the change in the h-th component defined in Section 3.4. The corresponding sample means are given by b− KL 1 1 Xh − Zj,h and Z̄h+ = Z̄h = − b b+ ) KL K(L − L h j=1 h KL X Zj,h , (5.3) b + +1 j=K L h which can be used to define an estimator for the unknown amount of change ∆µh = µh,1 − µh,2 in the h-th component, that is ∆b µh = Z̄h− − Z̄h+ . Moreover, these estimators can also be used to  −   Zj,h − Z̄h Zbj,h = 0   + Zj,h − Z̄h (5.4) define the “mean corrected” sample b− , for j ≤ K L h b− < j ≤ K L b+ , for K L h h + b , for j > K L h . (5.5) Finally, we define blocking variables (that are sums with respect to the different blocks) as Vb`,h (k) = `K X Zbj,h I{j ≤ k} (5.6) j=(`−1)K+1 and introduce the notation `K X Vb`,h = Vb`,h (n) = bj,h . Z j=(`−1)K+1 From the representation (5.5) we directly obtain, that blocks near to the estimator t̂h are ignored, i.e. we have b − + 1, . . . , L b + }. Vb`,h = 0 if ` ∈ {L h h 18 Further denote by Zn = σ(Zj,h \| 1 ≤ j ≤ n, 1 ≤ h ≤ d) the σ-field generated by the sample (Zj,h )1≤j≤n,1≤h≤d and by {ξ` }`∈N a sequence of i.i.d. standard Gaussian random variables, which is independent of Zn . Now we consider a multiplier version of the CUSUM-process from the L blocks, that is (L) Un,h (s) L L `=1 `=1 1X b bnsc X b = ξ` V`,h (bnsc) − 2 ξ` V`,h (n), n n and introduce the bootstrap integral statistics (for each component) √ Z 1 6 n (L) U (s)k(s, t̂ )ds · I \|∆b µh \| > n−1/4 Bn,h = h n,h 2 sbh τ (t̂h )(t̂h (1 − t̂h )) 0 + I \|∆b µh \| ≤ n−1/4 · bd , (5.7) (5.8) where k(x, y) = x ∧ y − xy denotes the covariance kernel of the standard Brownian bridge and L ŝ2h = 1X 2 ξ` · σ̂h L `=1 is the variance estimate from the bootstrap sample. In an analogous manner to the previous sections, we define a normalized maximum of the bootstrap statistics Bn,h by d Bd,n = ad max Bn,h − bd , (5.9) h=1 where the normalizing sequences ad and bd are given by (2.15) and (2.16), respectively. Remark 5.1 Let us give a brief heuristic explanation, why (5.8) and (5.9) define an appropriate bootstrap statistic. Our basic aim is to mimic the distribution of the test statistics Td,n on the “0dimensional boundary” of the null hypothesis H0,∆ , i.e. in case of \|∆µh \| = ∆h for all h = 1, . . . , d. µh (s, th ) Note, that we have = sign(∆µh )k(s, th ) and it was outlined in Section 3, that in this ∆h setting the representation √ Z 1 3 n Tn,h = (Un,h (s) − µh (s, th ))2 ds, (th (1 − th ))2 τh σh \|∆µh \| 0 √ Z 1 6 n + µh (s, th )(Un,h (s) − µh (s, th ))ds (th (1 − th ))2 τh σh \|∆µh \| 0 holds, where by Lemma 3.6 the first summand on the right-hand side is of order oP (1). The component-wise bootstrap statistic Bn,h is supposed to imitate the second summand in this decomposition. However, this approach is only sensible for all components h, that actually contain a change and for this reason we introduce the indicator function I \|∆b µh \| > n−1/4 . To be more precise, we will show in the Appendix that d Bd,n = ad max Bn,h − bd ≈ ad maxc Bn,h − bd h=1 h∈Sd 19 as d, n → ∞, where the set Sd is defined in (3.17). The statistic Bd,n will then be used to generate bootstrap quantiles for the statistic Td,n . In order to prove that this is a valid approach we require the following additional assumptions. Assumption 5.2 (assumptions for the bootstrap) For the constants p, D in Assumption 3.2 and the exponent ` in (5.1) assume that (B1) D < min{(1 − `)(p/2 − 2), `(p/4 − 1)} with ` > 3/4 and p > 8, (B2) limn→∞ log n K(µ∗d )2 = 0, where µ∗d is defined in (3.18). Assumption (B1) is a rather technical condition relating the dimension d ∼ nD , the number of blocks L = n` and the constant p, which was initially introduced in Assumption 3.1. Assumption (B2) is only a restriction for the monotone decreasing sequence µ∗d = minh∈Sdc \|∆µh \|, that is not allowed to decrease arbitrarily fast. We are now ready to state the main results of this section. Our first lemma shows, that we are able to identify the set of stable components Sd correctly. Lemma 5.3 If Assumptions 3.1 - 3.4 and Assumption 5.2 are satisfied, then P ad max Bn,h − bd −→ 0, h∈Sd where the set Sd is the set of all components with no change point defined in (3.17) and the sequences ad and bd are defined in (2.15) and (2.16), respectively. Theorem 5.4 If Assumptions 3.1 - 3.4 and Assumption 5.2 are satisfied, then   G if limd→∞ \|Sdc \|/d = 1,    D ad maxc Bn,h − bd =⇒ G + 2 log c if limd→∞ \|Sdc \|/d = c for c ∈ (0, 1),  h∈Sd    −∞ if limd→∞ \|Sdc \|/d = 0 conditional on Zn in probability, where the sequences ad and bd are defined in (2.15) and (2.16), respectively. Finally, the representation Bd,n = max ad max Bn,h − bd , ad maxc Bn,h − bd , h∈Sd h∈Sd Lemma 5.3 and Theorem 5.4 directly yield the following main result of this section. Theorem 5.5 If Assumptions 3.1 - 3.4 and Assumption 5.2 are satisfied, then   G if limd→∞ \|Sdc \|/d = 1,    D Bd,n =⇒ max{G + 2 log c, 0} if limd→∞ \|Sdc \|/d = c for c ∈ (0, 1),     0 if limd→∞ \|Sdc \|/d = 0 conditional on Zn in probability. 20 (5.10) ∗ Remark 5.6 In the following let g1−α denote the (1 − α)-quantile of the distribution of the bootstrap statistic Bd,n and define the bootstrap test by rejecting the null hypothesis (2.4) in favour of (2.5), whenever ∗ Td,n > g1−α , (5.11) where the statistics Td,n is defined in (2.14). If the alternative hypothesis of at least one relevant P change point holds, Theorem 4.3 shows Td,n −→ ∞, which due to Theorem 5.5 directly yields consistence of the bootstrap test. Under the null hypothesis, we consider different cases. Recall the definition of the sets Bd = {h ∈ {1, . . . , d} \| \|∆µh \| = ∆h }, Sd = {h ∈ {1, . . . , d} \| \|∆µh \| = 0}, where we always have Bd ⊂ Sdc ⊂ {1, . . . , d}. A combination of Theorem 4.1 and Theorem 5.5 now shows, that the rule in (5.11) gives an asymptotic level α test, whenever the limit limd→∞ \|Sdc \|/d exists. 6 Finite sample properties In this section we examine the finite sample properties of the asymptotic and bootstrap test by means of a small simulation study and illustrate its application in an example. 6.1 Simulation study The results of the previous section demonstrate that a test which rejects the null hypothesis in (1.3) for large values of the statistic Td,n defined in (2.14) is consistent and has asymptotic level α, provided that the critical values are either chosen by the asymptotic theory or estimated by the bootstrap procedure introduced in Section 5. It turns out that a bias correction, which does not change the asymptotic properties, yields substantial improvements of the finite sample properties of the asymptotic and bootstrap test. To be precise, recall the decomposition in (3.8), that is (1) (2) Tn,h = Tn,h + Tn,h , (1) (2) where Tn,h and Tn,h are defined in (3.10) and (3.11), respectively. It is shown in Section 3, that the (1) (2) maximum of the terms Tn,h is of order op (1), while the maximum of the terms Tn,h (appropriately (1) standardized) converges weakly to a Gumbel distribution. However, Tn,h is always nonnegative, which may lead to a non negligible bias in applications, in particular when the sample size is small relative to the dimension. To solve this problem for the asymptotic test (2.19), note that we have h i (1) E Tn,h = √ 3σh n(th (1 − th ))2 τh \|∆µh \| Z 0 1 σh (1 + o(1)) k(s, s)ds(1 + o(1)) = √ 2 n(th (1 − th ))2 τh \|∆µh \| 21 and therefore we subtract the term σ̂h √ 2 n(t̂h (1 − t̂h ))2 τ̂h ∆h (∆) from each statistic T̂n,h defined in (2.10). Similarly, a bias correction is also suggested for the bootstrap test in Section 5. Recall that the Bootstrap statistic is already constructed to mimic the (2) distribution of the statistic Tn,h . Consequently we use √ Z 1 2 3 n (L) ds U (s) n,h sbh τ̂h (t̂h (1 − t̂h ))2 ∆h 0 (1) to mimic the distribution of Tn,h and we add it (for each h) to the statistic Bn,h defined in (5.8), (∆) while the statistics T̂n,h in (2.10) are left unchanged. To guarantee a stable long-run variance estimation, we replaced the standard long-run variance estimator used in the theory of Section 3.3, by an estimator using the Bartlett kernel [see Newey and West (1987)], that is (for component h) ( X 1 − \|x\| if \|x\| ≤ 1 j φ̂h (0) + k φ̂h (j) with k(x) = βn 0 if \|x\| > 1 . 1≤\|j\|≤βn In order to have a more conservative test we use the maximum of the two variance estimates based bh,1 and D bh,2 defined in (3.20) as the final variance estimation. on the sets D We will focus on two scenarios with independent innovations in model (2.1), i.e. (I) Xj,h ∼ N (0, 1) i.i.d. (II) Xj,h ∼ Exp(1) i.i.d. and on two models of dependent data, an ARMA-process and a MA-process, defined by (III) Xj,h = 0.2Xj−1,h − 0.3Xj−2,h − 0.4Yj,h + 0.8Yj−1,h , P −3 where Yk,h = εk + 19 i=1 i εk−i,h and εk,h ∼ N (0, 1) i.i.d. (IV) Xj,h = εj,h + 1 10 29 P k −3 εj−k,h , where εk,h ∼ N (0, 1) i.i.d. k=1 All innovations are constructed such that Var(Xj,h ) ≈ 1. Throughout this section we assume that different components are independent and we are interested in testing the relevant hypotheses versus H0,∆ : \|∆µh \| ≤ 1 for all h ∈ {1, . . . , d} (6.1) HA,∆ : \|∆µh \| > 1 for at least one h ∈ {1, . . . , d}, (6.2) that is ∆h = 1 for all h ∈ {1, . . . , d}. Power and level of the tests are simulated for d-dimensional vectors with means  0 if j ≤ bnt c h E[Zj,h ] = , h = 1, . . . , d, µ if j > bnth c 22 where different values of the parameter µ are considered and the time of change is th = 1/2. All results presented in this section are based on 1000 simulation runs and the used significance level is always α = 0.05. Further, the constant S involved in (3.20) was fixed to 0.9. In our first example we investigate the finite sample properties of the asymptotic test (2.19) which uses the quantiles of the Gumbel distribution. In Table 1 we display the simulated type I error of this test at the “0-dimensional boundary” of the null hypothesis (that is ∆h = ∆µh = 1 for all h ∈ {1, . . . , d}) for different values of n and d. It is well known that the approximation of the distribution of the maximum of normally distributed random variables by a Gumbel distribution is not very accurate for small samples and therefore we consider relatively large sample sizes and dimensions in order to illustrate the properties of the asymptotic test. The results reflect the asymptotic properties in Section 3. For the independent cases (I) and (II) the approximation of the nominal level is more precise as for the dependent scenarios (III) and (IV), where the test is more conservative. We also mention that the rejection probabilities are increasing with ∆µh as predicted in Section 2 and 4 (these results are not presented for the sake of brevity). n=2000 model (I) (II) (III) (IV) d=500 8.8% 5.4% 4.9% 9.7% d=1000 11.1 % 5.9 % 4.5 % 13 % n=5000 d=500 6% 5% 1.9% 6.4% d=1000 7.5% 5.5% 2.5% 5.2% f n=10000 d=500 6.4% 4% 2.5% 6.2% d=1000 7.4% 5% 4% 5.9% Table 1: Empirical rejection probabilities of the asymptotic test (2.19) under a specific point at the “0-dimensional boundary” of null hypothesis, that is ∆h = ∆µh = 1 for all h ∈ {1, . . . , d}. \|∆µh \| model (I) (I) (II) (II) (III) (III) (IV) (IV) K 1 4 1 4 5 10 2 4 0.9 0.95 1.0 1.05 1.1 2.2% 5.2% 0.6% 0.9% 0% 0.1% 3.3% 5% 4.2% 9.1% 0.9% 3.2% 0% 1.3% 5.9% 10.3% 8.2% 17.3% 2.5% 9.2% 0.6% 4.6% 10.9% 18.4% 14.7% 29.6% 7.8% 20.2% 3.2% 21.6% 21.5% 32.5% 26.9% 47.8% 17.3% 36.6% 13.7% 59.7% 35.8% 48.7% Table 2: Empirical rejection probabilities of the bootstrap test (5.11) for n = d = 100 at specific points in the alternative and null hypothesis (∆h = 1 for all h ∈ {1, . . . , d}). Different block length K in the multiplier bootstrap are considered. Next we analyse the properties of the bootstrap test (5.11), which was developed in Section 5. Here we focus on relatively small sample sizes, that is n = 100, n = 200 and a relative large 23 Figure 2: Simulated power of the Bootstrap test (5.11) for the hypothesis (6.1). The sample size is n = 100 and the dimension d = 100. All differences are given by ∆µh = µ, where the choice µ = 1 corresponds to a point of the “0-dimensional boundary” Ad of the hull hypothesis. Left panel: Solid line: Model (I), dashed line: Model (II). Right panel: Solid line Model (III), dashed line: Model (IV). dimension compared to the sample sizes, that is d = 50, d = 100. It is well known that the multiplier bootstrap is sensitive with respect to the choice of the block length and this dependence is also observed for the bootstrap test proposed here. Exemplarily we show in Table 2 the simulated rejection probabilities for the different models (I) - (IV), different values of K and ∆µh = µ. Here the values \|∆µh \| ≤ 1 correspond to the null hypothesis. For \|∆µh \| = 1 (for all h ∈ {1, . . . , d}) the results of Section 5 predict that at this point the level of the test should be close to α = 0.05. Note that for the case of independent innovations (model (I) and (II)) the choice K = 1 (which means that no blocks are used) leads to the most reasonable results, given by rejection probabilities on the “0-dimensional boundary” Ad of the null hypothesis of 8.2% and 2.5%, respectively, while larger values of K such as K = 4 yield a too large type I error. On the other hand in the dependent models (III) and (IV), the block length needs to be carefully adapted to the time series structure. For model (III) K = 10 seems to be optimal, while for model (IV) choosing K = 2 gives acceptable results. The larger block length for model (III) might be required due to its autoregressive structure. Moreover, inspecting the results in rows 6 and 8 of Table 2 shows that too small values of K lead to a loss of power, while - similar to the first two models - too large values can cause an uncontrolled type I error. Next we display in Figure 2 the simulated power of the bootstrap test for all four models under consideration (where we use the optimal K from Table 2). Note that the rejection probabilities are increasing with ∆µh as predicted by the asymptotic arguments of the previous sections. In the left 24 panel we show the results for the independent scenarios (I) and (II), which are rather similar. On the other hand an inspection of the right panel shows larger difference in the dependent case. The different dependency structures in model (III) and (IV) yield substantial differences in power of the bootstrap test (5.11). We conclude the discussion of the bootstrap test investigating the sample size n = 200. The corresponding results are presented in Table 3 for the dimensions d = 50 and d = 100. \|∆µh \| model (I) (II) (III) (IV) K 1 1 20 2 0.9 0.95 d = 50 1.0 0.5% 0.3% 0% 0% 2.3% 1% 0.6% 0% 5.1% 4.1% 7.6% 9.5% 1.05 1.1 0.9 14.7% 11% 46.6% 47.56% 34.6% 30.4% 94.8% 100% 0.9% 0% 0% 0% 0.95 d = 100 1.0 1.05 1.1 2.3% 1.2% 0.3% 0% 6.9% 3.7% 6.4% 13.1% 18% 10.6% 51% 100% 40.7% 26.6% 95.3% 100% Table 3: Empirical rejection probabilities of the bootstrap test (5.11) at specific points in the alternative and null hypothesis (∆h = 1 for all h ∈ {1, . . . , d}). The sample sitze is n = 200. 6.2 Data example In this section we illustrate in a short example, how the new test can be used in applications. Our dataset is taken from hydrology and consists of average daily flows (m3 /sec) of the river Chemnitz at Goeritzhain (Germany) in the period 1909-2014. This data set has been recently analysed by Sharipov et al. (2016) using a statistical model from functional data analysis. Following these authors we subdivide the data into n = 105 years with d = 365 days per year. To avoid confusion, the reader should note that the German hydrological year starts on the 1st of November. Equipped with our new methodology, we are now able to test if there is a relevant change in the mean of at least one component. To specify the term ’relevant’, we exemplarily set the thresholds for all components to ∆1 = ∆2 = · · · = ∆365 = 0.63, which is close to 10% of the overall mean of the data under consideration. For a significance level of 5% the Bootstrap test defined in Section 5 rejects the null hypothesis of no relevant change for the given thresholds. Moreover, we can also identify the components, where the individual test (∆) ∗ ∗ statistic leads to a rejection, that is ad (T̂n,h − bd ) > g1−α , where g1−α is the (1 − α) quantile of the bootstrap distribution used in (5.11). For the data under consideration we found four components with a relevant change, given by h = 53, 99, 137 and 252 with corresponding estimators nt̂53 = 56, nt̂99 = 70, nt̂137 = 47 and nt̂252 = 41, respectively. This corresponds to the 23th of December 1965, the 7th of February 1979, the 18th of March 1956 and to the 10th of July 1950, respectively. In Figure 3 we display for these cases the time series before and after the year. For example the panel in the first row shows the average annual flow curves before and after 1950 and the other three 25 Figure 3: Average annual flow curves for the river Chemnitz at Goeritzhain for the periods before (gray) and after (black) the estimated year of change. The four figures correspond to different days of the year, where a change point has been localised: 10th of July (first row), 18 of March(second row) 23th of December (third row) and 7th of February (fourth row). 26 years are interpreted separately. In all cases we observe a large difference between both curves close the estimated component (marked by the vertical dashed line). We finally note that the approach of Sharipov et al. (2016) identifies only one change point, namely the year 1965. In contrast our analysis indicates that there might be additional change points in the years 1950, 1956, 1965 and 1979 corresponding to different parts of the hydrological year. Acknowledgments. This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt A1, C1) of the German Research Foundation (DFG). The authors would also like to thank Martina Stein, who typed parts of this paper with considerable technical expertise and Moritz Jirak for extremely helpful discussions. Moreover we are grateful to Andreas Schuhmann and Svenja Fischer from the Institute of Hydrology in Bochum, who provided hydrological data for Section 6.2. 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(Preprint). 7 Technical details Throughout the proofs we will use that Assumption (C1) directly implies the existence of two constants τ− , τ+ , such that the function τ defined in (2.11) satisfies 0 < τ− ≤ τ (th ) ≤ τ+ < ∞ holds for all h ∈ N. 28 7.1 Proof of assertion (2.7) Straightforward calculations yield E [Un,h (s)] = µh (s, th ) + O(n−1 )∆µh n Var (Un,h (s)) = σh2 (s − s2 ) + o(1), (7.1) uniformly with respect to s ∈ [0, 1], where µh (s, th ) is defined in (3.9). An application of Fubini’s theorem now gives i Z 1 hZ 1 2 E U2n,h (s) ds Un,h (s)ds = E 0 0 Z 1 = E (Un,h (s) − µh (s, th ))2 + 2µh (s, th )E [Un,h (s)] − µ2h (s, th ) ds + o(1) 0 Z 1 σh2 (th (1 − th ))2 = µ2h (s, th )ds + o(1) = ∆µh + + o(1). 6n 3 0 7.2 7.2.1 Details in the proof of Theorem 3.5 Proof of Lemma 3.6 Observing the definition (2.11) and Assumptions (T2) and (C1) it easily follows that there exist constants c and C such that the inequalties c < (th (1 − th ))2 σh < C (7.2) hold for all h ∈ {1, . . . , d}. With these inequalities we obtain √ Z 1 3 n d d (1) (Un,h (s) − µh (s, th ))2 ds 0 ≤ ad max Tn,h = ad max h=1 h=1 (th (1 − th ))2 τh σh \|∆µh \| 0 Z 1 d √ . ad max n (Un,h (s) − µh (s, th ))2 ds h=1 0 √ ad d ≤ √ max sup ( n\|Un,h (s) − µh (s, th )\|)2 . n h=1 s∈[0,1] Using (7.1) and maxdh=1 \|∆µh \| ≤ Cu this is further bounded by 2 √ ad d 2 √ max sup n\|Un,h (s) − E [Un,h (s)] \| + o(1) n h=1 s∈[0,1] √ Introducing the notation ed = 2 2 log 2d it follows that the last term can be bounded by the random variable √ ad d e d 2 4 √ max sup n\|Un,h (s) − E [Un,h (s)] \| − + o(1) 4 n h=1 s∈[0,1] and the claim follows from Theorem 2.5 in (Jirak, 2015a) noting that {Un,h (s) − E [Un,h (s)]}s∈[0,1] corresponds to a CUSUM-process under the classical null hypothesis of no change point, that is (1) (2) µh = µh for all h ∈ {1, . . . , d} (note that there is a typo in the original paper, which has been corrected in the arXiv version). 29 7.2.2 Proof of Lemma 3.7 A straightforward calculation yields n Cov(Un,h (s1 ), Un,i (s2 )) = k(s1 , s2 )γh,i + rn,h,i (s1 , s2 ), where k(s1 , s2 ) = s1 ∧s2 −s1 s2 denotes the covariance kernel of a Brownian bridge and the remainder term satisfies √ lim n sup sup \|rn,h,i (s1 , s2 )\| = 0. n→∞ s1 ,s2 ∈[0,1] h,i∈N An application of Fubini’s theorem shows that sign(∆µ ) sign(∆µ )γ τe(t , t ) i h,i h h i (2) (2) Cov T̄n,h , T̄n,i = + rn,h,i , σh σi τ (th )τ (ti ) (7.3) where the function τe is given by 36 τe(t, t ) = (t(1 − t)t0 (1 − t0 ))2 0 Z 1Z 1 0 k(s1 , t)k(s2 , t0 )k(s1 , s2 )ds1 ds2 (7.4) 0 and the remainder term rn,h,i satisfies lim n→∞ √ n sup \|rn,h,i \| = 0. h,i∈N Moreover, for the function τ defined in (2.11) we obtain the representation s Z 1Z 1 6 τ (t) = k(s1 , t)k(s2 , t)k(s1 , s2 )ds1 ds2 , (t(1 − t))2 0 0 and it follows that τ (th )τ (th ) = τe(th , th ). Therefore we obtain as a special case the estimate (2) Var T̄n,h = 1 + rn,h,h . (7.5) Furthermore the representation bnsc n 1X bnsc X Un,h (s) − E [Un,h (s)] = Xj,h =: U0n,h (s) Xj,h − 2 n n j=1 j=1 yields (2) T̄n,h √ Z 1 6 n = µh (s, th )U0n,h (s)ds. (th (1 − th ))2 τh σh \|∆µh \| 0 and the proof can now be performed in two steps: Step 1: For two constants c, C > 0 it holds that d d (2) e ≤ Cn−c , sup P max T̄n,h ≤ x − P max Nh ≤ x x∈R h=1 h=1 (7.6) e is a d-dimensional centered Gaussian distributed random variable with same where N (2) (2) covariance structure as (T̄n,1 , . . . , T̄n,d )T . 30 Step 2: There exist two constants c, C > 0 such that d d e sup P max Nh ≤ x − P max Nh ≤ x ≤ Cn−c , h=1 x∈R h=1 (7.7) where N is a centered d-dimensional Gaussian random variable with covariance matrix Σ = (Σh,i )i,j=1,...,d satisfying \|Σh,i \| ≤ \|ρh,i \| (7.8) for all h, i ∈ {1, . . . , d}. Step 1: At the end of this proof we derive the following representation n 1 X (2) cn,j,h Xj,h , T̄n,h = √ n (7.9) j=1 where the coefficients cn,j,h are uniformly bounded, that is sup \|cn,j,h \| ≤ c0 < ∞ n,j,h∈N Next we apply the Gaussian approximation in Corollary 2.2 of Zhang and Cheng (2016) to the random variables Yn,j,h = cn,j,h Xj,h . For this purpose we check the assumptions of this result. By Assumption (M1) we obtain a sequence (Md0 )d∈N by Md0 = c0 · Md , which still satisfies n d n d max max E exp(\|Yn,j,h \|/Md0 ) ≤ max max E [exp(\|Xj,h \|/Md )] ≤ C0 . j=1 h=1 j=1 h=1 Moreover, Assumption (M2) yields that Md0 . nm with m < 3/8. This means that for sufficiently small b we have m < (3 − 17b)/8 and by Assumption (S1) it follows that d . exp(nb ). Using the identity (3.1) the triangular array {Yn,j,h \| 1 ≤ j ≤ n, 1 ≤ h ≤ d}n∈N exhibits the following structure Yn,j,h = cn,j,h · gh (εj , εj−1 , . . . ) := g̃n,j,h (εj , εj−1 , . . . ). Define the coefficients n ϑYn,j,h,p = max kg̃n,i,h (εi , . . . ) − g̃n,i,h (εi , , . . . , εi−j+1 , ε0i−j , εi−j−1 , . . . )kp , i=1 where ε0i−j is an independent copy of εi−j , and observe the inequality ∞ X j=u d max ϑYn,j,h,p h=1 ≤ ∞ X d n max sup max \|cn,i,h \|ϑj,h,p . c0 h=1 n∈N i=1 j=u ∞ X δj . δu, j=u which holds uniformly with respect to n. By (7.5) there exist constants c1 and c2 , such that d d (2) (2) 0 < c1 < min Var T̄n,h ≤ max Var T̄n,h < c2 h=1 h=1 31 if n is sufficiently large. Since all requirements are met, Corollary 2.2 of Zhang and Cheng (2016) e having the same covariance matrix as the implies the existence of a Gaussian random variable N (2) (2) T vector (T̄n,1 , . . . , T̄n,d ) and satisfying inequality (7.6). Step 2: We choose the random variable N to be d-dimensional centered Gaussian with covariance matrix given by Σh,i = sign(∆µh ) sign(∆µi )γh,i τe(th , ti ) τe(th , ti ) = sign(∆µh ) sign(∆µi )ρh,i , σh σi τ (th )τ (ti ) τ (th )τ (ti ) e the covariance matrix of the where the function τe is defined in equation (7.4). Next denote with Σ e from Step 1. By (7.3) we have vector N d d h,i=1 h,i=1 e h,i \| = max θd := max \|Σh,i − Σ sign(∆µh ) sign(∆µi )γh,i τe(th , ti ) (2) (2) − Cov(Tn,h , Tn,i ) . n−1/2 σh σi τ (th )τ (ti ) and an application of the Gaussian comparison inequality, in Lemma 3.1 of Chernozhukov et al. (2013) gives d d eh ≤ x . θ1/3 max{1, log(d/θd )}2/3 . n−C . sup P max Nh ≤ x − P max N d x∈R h=1 h=1 The proof of Step 2 is now completed observing the bound \|e τ (th , ti )\| ≤ \|τ (ti )\|\|τ (th )\|, which is a consequence of the (generalised) Cauchy-Schwarz inequality. Proof of the representation (7.9). Recall the definition k(s, t) = s ∧ t − st, then 1 Z 0 n−1 Z (i+1)/n 1X k(s, th )Un,h (s)ds = n i=0 = 1 n i/n n bnsc X bnsc X k(s, th ) Xj,h ds Xj,h − n j=1 n−1 n X Z (i+1)/n X i=0 i/n j=1 k(s, th ) (I{j ≤ i} − i/n) Xj,h ds j=1 ! n n−1 Z 1 X X (i+1)/n k(s, th ) (I{j ≤ i} − i/n) ds Xj,h . = n i/n j=1 i=0 (2) Now observe the representation for T̄n,h in (3.12), where µh (s, t) = 2∆µh k(s, t). Define n−1 Z (i+1)/n cn,j,h X 6∆µh := (th (1 − th ))2 τ (th )σh \|∆µh \| i=0 k(s, t) (I{j ≤ i} − i/n) ds, i/n then the representation (7.9) holds, and the proof is completed observing the inequalities n−1 X Z (i+1)/n 6 12 \|cn,j,h \| ≤ 2ds ≤ c0 := . 4 τ− σ− t i=0 i/n τ− σ− t4 32 7.2.3 Proof of Theorem 3.10 As \|∆µh \| > Cu for all h ∈ {1, . . . , d} we have Sd = ∅ and Sdc = {1, . . . , d}, and Lemma 3.8 implies d max \|t̂h − th \| = op (n−1/2 ) (7.10) h=1 Observing that t̂h ∈ [t, 1 − t] it follows that t2h (1 − th )2 − 1 = op (n−1/2 ) 2 2 t̂h (1 − t̂h ) d max h=1 (7.11) Moreover, one easily verifies that the function t → τ (t) defined in (2.11) is Lipschitz-continuous on the interval [t, 1 − t] and therefore we obtain from (7.10) d max \|τ (th ) − τ (t̂h )\| = op (n−1/2 ). (7.12) h=1 Finally, we note that for a sufficiently small constant C > 0 the estimate d max \|σ̂h τ (t̂h ) − σh τ (t̂h )\| = op (n−C ) (7.13) h=1 holds, which is a direct consequence Lemma 3.9 and assertion (7.12). After these preparations we return to the proof of Theorem 3.10. We recall the definition (2.8) and introduce the notation √ n 0 T̂n,h = M̂2n,h − ∆µ2h τ (th )σh \|∆µh \| We will first show the weak convergence D d 0 ad max T̂n,h − bd =⇒ G (7.14) h=1 For a proof of (7.14) we recall the definition of the statistic Tn,h in (3.7) and obtain from Theorem 3.5 D d ad max Tn,h − bd =⇒ G. h=1 With the representation for M̂2n,h and M2n,h in (2.8) and (3.6), respectively, and the notation q̂h = (th (1 − th ))2 /(t̂h (1 − t̂h ))2 it now follows that √ √ n(q̂h − 1)∆µ2h n d d 2 2 0 Mn,h − ∆µh − bd + = Bn,d ad max T̂n,h − bd = ad max q̂h h=1 h=1 τ (th )σh \|∆µh \| τ (th )σh \|∆µh \| where √ d Bn,d := ad max q̂h Tn,h − bd + (q̂h − h=1 1)∆µ2h n . τ (th )σh \|∆µh \| It is easy to see that this term can be bounded by d d ad max q̂h Tn,h − bd − Rn,d ≤ Bn,d ≤ ad max q̂h Tn,h − bd + Rn,d , h=1 h=1 33 where the remainder satisfies d d h=1 h=1 Rn,d = ad max \|q̂h − 1\| max √ n P ∆µ2h −→ 0, τ (th )σh \|∆µh \| which follows observing the inequalities (7.2), (7.11) and the condition C` ≤ \|∆µh \| ≤ Cu . Thus (7.14) follows, if we can establish D d ad max q̂h Tn,h − bd =⇒ G. (7.15) h=1 For a proof of this result we fix x ∈ R and define ud (x) = x/ad + bd . By d d d P 0 ≤ max q̂h max Tn,h ≤ ud (x) ≤ P 0 ≤ max q̂h Tn,h ≤ ud (x) h=1 h=1 h=1 d d h=1 h=1 ≤ P 0 ≤ min q̂h max Tn,h ≤ ud (x) . (7.16) and d d d d d P min q̂h max Tn,h ≥ 0 = P max q̂h Tn,h ≥ 0 = P max q̂h max Tn,h ≥ 0 , h=1 h=1 h=1 h=1 h=1 we obtain d d d d d P max q̂h max Tn,h ≤ ud (x) ≤ P max q̂h Tn,h ≤ ud (x) ≤ P min q̂h max Tn,h ≤ ud (x) . h=1 h=1 h=1 h=1 h=1 (7.17) From (7.11) and d = C1 nD it follows that d P P d ad bd min q̂h − 1 −→ 0 and ad bd max q̂h − 1 −→ 0, h=1 h=1 which due to Slutsky’s theorem directly implies d d −x −x d d P min q̂h max Tn,h ≤ ud (x) −→ e−e , P max q̂h max Tn,h ≤ ud (x) −→ e−e . h=1 h=1 n→∞ h=1 h=1 n→∞ Thus we have established (7.15) and proved (7.14). To complete the proof of Theorem 3.10 note that the assertion (3.22) is equivalent to d −x P max T̂n,h ≤ ud (x) −→ e−e , n→∞ h=1 (7.18) where we use again ud (x) = x/ad + bd . To prove this statement define d Q− d := min h=1 σ̂h τ̂h σ̂h τ̂h d , Q+ , d := max h=1 σh τ (th ) σh τ (th ) − and consider the set Qd := Q+ d − 1 ∨ Qd − 1 ≤ δd , where the involved sequence is given by δd = (log d)−2 . The inequalities (7.2) and the estimate (7.13) yield d d σ̂h τ̂h P Q+ − 1 > δ ≤ P max σ̂ τ̂ − σ τ (t ) > δ c = o(1). − 1 > δ = P max d d h h h h d d h=1 σh τ (th ) h=1 34 c d By a similar argument for the term Q− d − 1 we obtain P (Qd ) → 0. If maxh=1 T̂n,h ≥ 0 holds, we can conclude that √ n 1 d 1 d 0 2 2 max T̂ = max M̂ − ∆µ n,h n,h h h=1 h=1 τ (th )σh \|∆µh \| Q+ Q+ d d √ n 1 d d 0 ≤ max M̂2n,h − ∆µ2h = T̂n,h = − max T̂n,h . h=1 τ (t̂h )σ̂h \|∆µh \| Qd h=1 Therefore the following inequalities hold d d d + 0 0 P 0 ≤ max T̂n,h ≤ ud (x)Q− ≤ P 0 ≤ max T̂ ≤ u (x) ≤ P 0 ≤ max T̂ ≤ u (x)Q n,h d d n,h d d . (7.19) h=1 h=1 h=1 Observing the identity d d 0 P max T̂n,h ≥ 0 = P ∃h : M̂2n,h ≥ ∆µ2h = P max T̂n,h ≥ 0 h=1 h=1 we can derive from (7.19) d d d + 0 0 ≤ P P max T̂n,h ≤ ud (x)Q− max T̂ ≤ u (x) ≤ P ≤ u (x)Q max T̂ n,h d d n,h d d . h=1 h=1 h=1 (7.20) Hence, we directly obtain d d + + c 0 0 P max T̂n,h ≤ ud (x)Qd ≤ P(Qd ) + P max T̂n,h ≤ ud (x)Qd ∩ Qd h=1 h=1 d 0 ≤ ud (x) + ud (x)δd . ≤ o(1) + P max T̂n,h h=1 Now we fix ε > 0 and note that the inequality ud (x − ε) < ud (x) + ud (x)δd < ud (x + ε) holds if n (or equivalently d) is sufficiently large. The weak convergence (7.14) then yields d d −(x+ε ) 0 0 ≤ lim sup P ≤ u (x + ε) = e−e . lim sup P max T̂n,h ≤ ud (x)Q+ max T̂ d n,h d d,n→∞ h=1 d,n→∞ h=1 Using Bonferroni’s inequality we can proceed similarly for the lower bound of (7.20), i.e. d d − − 0 0 lim inf P max T̂n,h ≤ ud (x)Qd ≥ lim inf P max T̂n,h ≤ ud (x)Qd ∩ Qd d,n→∞ d,n→∞ h=1 h=1 d 0 ≥ lim inf P max T̂n,h ≤ ud (x) − ud (x)δd − P (Qcd ) d,n→∞ h=1 d −(x−ε ) 0 ≥ lim inf P max T̂n,h ≤ ud (x − ε) = e−e . d,n→∞ h=1 The assertion (7.18) then follows by ε → 0, which completes the proof of Theorem 3.10. 7.2.4 Proof of Corollary 3.11 At first we consider the case where md := \|Md \| = c · d + o(d) for some constant c ∈ (0, 1] and note that in this case ad ad max T̂n,h − bd = am max T̂n,h − amd bmd + amd bmd − ad bd . h∈Md amd d h∈Md 35 D Theorem 3.10 yields amd maxh∈Md T̂n,h − amd bmd =⇒ G and furthermore we have ad −→ 1 and amd bmd − ad bd −→ 2 log(c) d→∞ amd d→∞ D A short calculation therefore leads to ad maxh∈Md T̂n,h − bd =⇒ G + 2 log(c). The case md = o(d) can be treated similarly. Finally, statement (3.23) is a consequence of the inequality d max T̂n,h ≤ max T̂n,h . h=1 h∈Md 7.3 7.3.1 Proofs of the results in Section 4 Proof of Theorem 4.1 By Assumption (C1) and the definition of t̂h in (3.16), there exists a global constant C(t) > 1 such that (th (1 − th ))2 ≤ C(t). (t̂h (1 − t̂h ))2 (7.21) Recall the definition of the set Sd in (3.17) , choose a constant ∆2− > ζ > 0 and consider the following decomposition of the set {1, . . . , d} \ Sd p Id := {h ∈ {1, . . . , d} \| (∆h − ζ)/ C(t) ≥ \|∆µh \| > 0}, p (7.22) Ed := {h ∈ {1, . . . , d} \| ∆h ≥ \|∆µh \| > (∆h − ζ)/ C(t)}. Using the representation o n (∆) (∆) (∆) Td,n = max ad max T̂n,h − bd , ad max T̂n,h − bd , ad max T̂n,h − bd . h∈Id h∈Sd h∈Ed the first assertion (4.2) follows from the following three statements P (∆) ad max T̂n,h − bd −→ −∞, h∈Sd P (∆) ad max T̂n,h − bd −→ −∞, h∈Id (∆) lim P ad max T̂n,h − bd ≥ g1−α ≤ α. d,n→∞ h∈Ed (7.23) (7.24) (7.25) (∆) Proof of (7.23): Observing the definition of T̂n,h in (2.10) we obtain the inequality √ ad (∆) max T̂n,h h∈S d − bd √ n n 2 ≤ ad max M̂n,h − ad min ∆2h − ad bd . h∈Sd τ (t̂h )σ̂h ∆h h∈Sd τ (t̂h )σ̂h ∆h The first summand of this expression is further bounded by √ √ n ad max M̂2n,h . ad max n · M2n,h , h∈Sd τ (t̂h )σ̂h ∆h h∈Sd 36 (7.26) (7.27) and arguing as in the proof of Lemma 3.6 yields ad maxh∈Sd √ √ summand of (7.26) we can use n ∆2 τ (t̂h )σ̂h ∆h h P n · M2n,h −→ 0. For the second > 0 and ad bd ∼ log d to obtain √ ad min h∈Sd n ∆2h + ad bd −→ ∞, d,n→∞ τ (t̂h )σ̂h ∆h which yields (7.23). Proof of (7.24): By definition of the set Id , we get ∆2h ≥ C(t)∆µ2h + ζ, which leads to √ √ n n (∆) 2 2 ad max T̂n,h − bd ≤ ad max ζ. M̂n,h − C(t)∆µh − bd − ad min h∈Id h∈Id τ (t̂h )σ̂h ∆h h∈Id τ (t̂h )σ̂h ∆h (7.28) For the second summand of the last expression it holds that √ √ ad n ζ & −→ ∞, ad n min h∈Id τ (t̂h )σ̂h ∆h τ+ s+ ∆+ n→∞ (7.29) From inequality (7.21), we obtain M̂2n,h ≤ C(t)M2n,h , which gives the following bound for the first summand of (7.28) √ √ n n 2 2 ad max M̂n,h − C(t)∆µh − bd . ad max M2n,h − ∆µ2h − bd . (7.30) h∈Id τ (t̂h )σ̂h ∆h h∈Id τ (t̂h )σ̂h ∆h Similar to (3.8) we can use the following decomposition √ n (1) (2) M2n,h − ∆µ2h = Sn,h + Sn,h , τ (t̂h )σ̂h ∆h (1) (2) where the quantities Sn,h and Sn,h are given by (1) Sn,h (2) Sn,h √ Z 1 3 n = 2 (Un,h (s) − µh (s, th ))2 ds, 2 th (1 − th ) τ (t̂h )σ̂h ∆h 0 √ Z 1 6 n = 2 µh (s, th )(Un,h (s) − µh (s, th ))ds, th (1 − th )2 τ (t̂h )σ̂h ∆h 0 respectively. Now we have an upper bound for (7.30) given by (1) (2) ad max Sn,h + ad max Sn,h − bd . h∈Id h∈Id (1) (2) Similar as in the proof of (7.23) one easily shows that ad max Sn,h = op (1). In the case that Sn,h ≥ 0 h∈Id we have (2) Sn,h . (3) Sn,h √ Z 1 6 n := 2 µh (s, th )(Un,h (s) − µh (s, th ))ds, th (1 − th )2 τ (th )σh \|∆µh \| 0 37 which gives n o n o (2) (3) (3) ad max Sn,h − bd . ad max max Sn,h , 0 − bd ≤ max ad max Sn,h − bd , −ad bd . h∈Id h∈Id h∈Id Applying Lemma 3.7 yields (3) lim sup P ad max Sn,h − bd > x ≤ P (G > x) h∈Id d,n→∞ for all x ∈ R and consequently the right hand side of (7.30) is of order Op (1). Now (7.24) follows from (7.28) and (7.29). Proof of (7.25): Observing that dh := ∆2h − ∆µ2h ≥ 0 we obtain √ n (∆) ad max T̂n,h − bd ≤ ad max M̂2n,h − ∆µ2h − bd , (7.31) h∈Ed h∈Ed τ (t̂h )σ̂h ∆h 1 )) ≥ 0, As α ∈ 0, 1 − e−1 the quantile of the Gumbel distribution satisfies g1−α = − log(log( 1−α and we can proceed as follows \|∆µh \| (∆) P ad max T̂n,h − bd > g1−α ≤ P max · ad max T̂n,h − bd > g1−α h∈Ed ∆h h∈Ed h∈Ed ≤ P ad max T̂n,h − bd > g1−α . h∈Ed An application of Corollary 3.11 now yields lim sup P ad max T̂n,h − bd > g1−α ≤ P (G > g1−α ) = α, d,n→∞ h∈Ed which gives assertion (7.25) and completes the proof of assertion (4.2). It remains to show assertion (4.4) under the additional assumption of (4.3). Note that under the latter assumption, we can further decompose the set Ed into Ed = (Ed \ Bd ) ∪ Bd and observe that (4.3) yields p Ed \ Bd = {h ∈ {1, . . . , d} \| ∆h − C∆ ≥ \|∆µh \| > (∆h − ζ)/ C(t)}. Again, we can examine both sets separately. For Ed \ Bd we have (∆) ad max T̂n,h − bd ≤ ad max T̂n,h − bd − ad min h∈Ed \Bd h∈Ed \Bd h∈Ed \Bd √ n . τ (t̂h )σ̂h ∆h (7.32) By definition of Ed \ Bd we obtain that the second summand on the right-hand side of (7.32) tends p (in probability) to −∞. Due to the lower bound minh∈Ed \Bd \|∆µh \| > (∆− − ζ)/ C(t), which holds uniformly in d, we can apply Corollary 3.11 to the first summand of the right-hand side of (7.32), which then gives P (∆) ad max T̂n,h − bd −→ −∞. h∈Ed \Bd On the set Bd = Ed we can directly apply Corollary 3.11, so that we obtain (4.4). 38 7.3.2 Proof of Theorem 4.3 It follows from Theorem 3 of Wu (2005) that n 1 bnsc o X D √ =⇒ {σh Ws }s∈[0,1] , Zj,h − E [Zj,h ] n s∈[0,1] (7.33) j=1 where {Ws }s∈[0,1] denotes the (standard) Brownian motion on the interval [0, 1]. The definition of µh (s, th ) in (3.9), E [Un,h (s)] = µh (s, th )(1 + o(1)) (uniformly with respect to s ∈ [0, 1]) and the continuous mapping theorem yield √ D n(Un,h (s) − µh (s, th )) s∈[0,1] =⇒ {σh Bs }s∈[0,1] , (7.34) where {Bs }s∈[0,1] denotes a (standard) Brownian bridge. Observing (3.8) we get √ Z 1 √ 3 n 2 2 (Un,h (s) − µh (s, th ))2 ds n Mn,h − ∆µh = (th (1 − th ))2 0 √ Z 1 6 n + µh (s, th )(Un,h (s) − µh (s, th ))ds. (th (1 − th ))2 0 Statement (7.34) and the continuous mapping theorem imply Z 1 D √ 6 2 2 n Mn,h − ∆µh =⇒ µh (s, th )σh B(s)ds. (th (1 − th ))2 0 It is well known, that the expression on the right-hand side follows a centered normal distribution and a straightforward calculation shows that its variance is given by ∆µ2h τ 2 (th )σh2 . Replacing σh and th by the estimators σ̂h and t̂h we obtain from Lemmas 3.8 and 3.9 the weak convergence √ n 2 D (7.35) M̂n,h − ∆µ2h =⇒ N (0, ∆µ2h ) , τ (t̂h )σ̂h for each (fixed) h ∈ N provided that \|∆µh \| > 0. After these preparations we are ready to prove the consistency of the test (2.19). If the alternative hypothesis HA,∆ is valid, we can fix k ∈ {1, . . . , d}, such that dk := ∆µ2k − ∆2k > 0. From the definition of the test statistic Td,n in (2.14) we obtain √ √ n n (∆) 2 2 Td,n ≥ ad T̂n,k − bd = ad M̂k − ∆µk + dk − bd , τ (t̂k )σk ∆k τ (t̂k )σk ∆k which gives √ √ g n n 1−α 2 2 P (Td,n M̂k − ∆µk > + bd − dk . ad τ (t̂k )σk ∆k τ (t̂k )σk ∆k √ √ Using bd ∼ log d and τ (t̂ )σn ∆ & n leads to k k k √ g1−α n P + bd − dk −→ −∞. ad τ (t̂k )σk ∆k > g1−α ) ≥ P On the other hand we obtain from (7.35) √ n D \|∆µk \| M̂2k − ∆µ2k =⇒ · N (0, 1) , ∆k τ (t̂k )σk ∆k and the assertion of Theorem 4.3 follows. 39 (7.36) (7.37) 7.3.3 Proof of Theorem 4.4 Due to g1−α /ad + bd > 0 we deduce b d (α)) = P min T̂ (∆) > g1−α /ad + bd P(Rd ⊂ R h∈Rd n,h √ n ≥ P min M̂2n,h − ∆2h > g1−α /ad + bd . h∈Rd τ (t̂h )σ̂h \|∆µh \| Using the notation dh = ∆µ2h − ∆2h we get √ √ n n 2 2 b M̂n,h − ∆µh > g1−α /ad + bd − min dh P(Rd ⊂ Rd (α)) > P min h∈Rd h∈Rd τ (t̂h )σ̂h \|∆µh \| τ (t̂h )σ̂h \|∆µh \| √ n = P min T̂n,h > g1−α /ad + bd − min dh h∈Rd h∈Rd τ (t̂h )σ̂h \|∆µh \| √ n = P ad min T̂n,h + bd ≥ g1−α + 2ad bd − ad min dh h∈Rd h∈Rd τ (t̂h )σ̂h \|∆µh \| By assumption (4.7) we have nC min dh ≥ nC min (\|∆µh \| − ∆h ) ∆− −→ ∞, h∈Rd n→∞ h∈Rd which implies (as 1 . τ (t̂h )σ̂h \|∆µh \| . 1) √ ad bd − ad min dh h∈Rd n −→ −∞. τ (t̂h )σ̂h \|∆µh \| d,n→∞ By arguments similar to those in the proofs of Section 3 one can show that for all x ∈ R lim inf P ad min T̂n,h + bd ≥ x ≥ P (−G ≥ x) , d,n→∞ h∈Rd which yields assertion (4.8). For a proof of (4.9) we apply Bonferroni’s inequality, which gives b d (α) = Rd = P R b d (α) ⊂ Rd , Rd ⊂ R b d (α) ≥ 1 − P R b d (α) * Rd − P Rd * R b d (α) . P R b d (α) = o(1) and Theorem 4.1 By the arguments in the previous paragraph we have P Rd * R gives (∆) b lim sup P Rd (α) * Rd = lim sup P maxc T̂n,h > g1−α /ad + bd ≤ α, d,n→∞ h∈Rd d,n→∞ which finishes the proof. 7.4 Proofs of the results in Section 5 To establish the bootstrap results, we recall the definition of the set Sd in (3.17) and we introduce the set n o b− , K L b+ ) , Ld = ∀ h ∈ Sdc : nth ∈ (K L (7.38) h h which represents the event, that the locations of all change points are identified correctly. We need the following basic properties. 40 Lemma 7.1 If the assumptions of Section 3.1 and Assumption 5.2 hold, then d P (i) nC max \|∆b µh − ∆µh \| −→ 0 if C < 1/2, h=1 (ii) P (Lcd ) . nC for a sufficiently small constant C > 0. Proof. For assertion (i) fix ε > 0 and observe b− d P nC max \|∆b µh − ∆µh \| > ε ≤ P nC max d h=1 h=1 KL 1 Xh Z − µ > ε/2 ∩ L j,h h,1 d b− KL h j=1 d 1 n X h=1 b+ ) K(L − L h b h +1 j=K L + P nC max Zj,h − µh,2 > ε/2 ∩ Ld + o(1). (7.39) The first two summands of the right-hand side of (7.39) exhibit the same structure, so we only treat the first of them. Note that on the event Ld , it holds that b − }. b − } = Xj,h I{j ≤ K L (Zj,h − µh,1 ) I{j ≤ K L h h b − ≤ nt̂h Further there exists a constant 0 < C0 < 1, such that the inequalities C0 n ≤ nt − K ≤ K L h hold. This implies − − b b KL L d h X 1 KX Xh 1 −C Z − µ > ε/2 ∩ L ≤ P X > n ε/2 P nC max j,h h,1 d j,h b− b− h=1 K L KL h j=1 h j=1 h=1 d d k n X X ≤ P max Xj,h > C0 n1−C ε/2 . h=1 k=1 j=1 Since 1 − C > 1/2 we obtain by the Fuk-Nagaev inequality (see Lemma E.3 in Jirak (2015b)) for sufficiently large n d k n X X P max Xj,h > C0 n1−C ε/2 . nD h=1 k=1 j=1 n n(1−C)p = nD+(C−1)p+1 ≤ nD−p/2+1 = o(1), where we also used D ≤ p/2 − 2. Assertion (ii) is shown in the proof of Theorem C.12 in Jirak (2015b). Proof of Lemma 5.3. This is a consequence of Lemma 7.1 (i) and the inequality (for any ε > 0) P ad max Bn,h − bd > ε ≤ P max \|∆b µh \| > n−1/4 . h∈Sd h∈Sd 41 7.4.1 Proof of Theorem 5.4 For the proof we introduce the following more simple version of the bootstrap CUSUM-process (L) {Un,h (s)}s∈[0,1] introduced in (5.7) bLsc L X bLsc X b e (L) (s) := 1 b U ξ V − ξ` V`,h , ` `,h n,h n Ln `=1 `=1 where the truncation is not conducted within the blocks. We make also use of the following extra notation √ Z 1 6 n e (L) (s)k(s, th )ds e = U B n,h σh τ (th )(th (1 − th ))2 0 n,h √ Z 1 6 n ∗ e (L) (s)k(s, t̂h )ds e U Bn,h = σh τ (th )(th (1 − th ))2 0 n,h √ Z 1 6 n (L) ∗ Bn,h = Un,h (s)k(s, t̂h )ds 2 σh τ (th )(th (1 − th )) 0 √ Z 1 6 n (L) (I) Un,h (s)k(s, t̂h )ds. Bn,h = 2 sbh τ (t̂h )(t̂h (1 − t̂h )) 0 The theorem’s claim is a direct consequence of the next five lemmas. We will use the notation P\|Zn (·) = P (·\|Zn ). and frequently apply that the implication P (An ) = o(1) ⇒ P\|Zn (An ) = op (1) holds for all sequences of measurable sets {An }n∈N . Lemma 7.2 The weak convergence D e ad maxc Bn,h − bd =⇒ h∈Sd      G if Sdc = {1, . . . , d}, G + 2 log c if \|Sdc \| = c · d + o(d) for c ∈ (0, 1),     0 if \|Sdc \| = o(d) holds conditionally on Zn . Proof. Without loss of generality assume that the sets Sd and Sdc are given by Sdc = {1, . . . , s} and Sd = {s + 1, . . . , d} with s = \|Sdc \|. Let N = (N1 , . . . , Ns )T denote a centered s-dimensional Gaussian vector with covariance matrix Σ = (Σij )di,j=1 defined by Σi,j = γi,j τe(ti , tj ) , σi σj τ (ti )τ (tj ) where the function τe is defined in (7.4). Our aim is to control the (conditional) Kolmogorov-distance e and maxh∈S c Nh . Since the random variables {ξ` } between maxh∈Sdc B `∈N are independent, we n,h d can directly calculate the conditional covariance Cov\|Zn e , B en,i B = n,h L X 36 Vb`,h Vb`,i β`,h β`,i σh σi τ (ti )τ (th )(th (1 − th ))2 (ti (1 − ti ))2 n `=1 42 with the extra notation Z 1 β`,h = 0 bLsc k(s, th ) I{` ≤ bLsc} − ds. L Let θd denote the distance e , B en,i − Σh,i . Cov\|Zn B n,h θd = max 1≤h,i≤s Using the fact that maxh∈Sdc maxL `=1 \|β`,h \| ≤ 1 a straightforward adaption of Lemma E.8 in Jirak (2015a) gives P L max 1≤h,i≤s 36 1 Xb b −δ V > n . L−2 n−C V β β − γ τ e (t , t ) I Ld `,h `,i `,h `,i h,i h i (th (1 − th ))2 (ti (1 − ti ))2 KL `=1 for sufficiently small constants C, δ > 0, where the set Ld is defined in (7.38). Using the lower 2 τ 2 yields bound σh σi τ (th )τ (ti ) ≥ σ− − P C(δ)C . n−C (7.40) for the set o n C(δ) := θd ILd ≤ n−δ . For a sufficiently small C > 0 we have C C C C C −C = o(1). ≤ nC E P\|Zn (LC P P\|Zn (LC d ∪ C(δ) ) = n P Ld ∪ C(δ) d ∪ C(δ) ) ≥ n Now, we can derive the following upper bound d d e ≤ x − P max Nh ≤ x sup P\|Zn max B n,h h=1 h=1 x∈R d d e ≤ sup P\|Zn max Bn,h ≤ x − P max Nh ≤ x ILd ∩C(δ) + OP (n−C ). h=1 h=1 x∈R (7.41) The identities γh,h = σh2 and τ̃ (th ) = τ 2 (th ) give Σh,h = 1 für alle h ∈ {1, . . . , d}, and by Lemma 3.1 in Chernozhukov et al. (2013) on the set Ld ∩ C(δ) we have for the first summand in (7.41) e sup P\|Zn maxc Bn,h ≤ x − P maxc Nh ≤ x ILd ∩C(δ) (7.42) x∈R h∈Sd h∈Sd 1/3 max{1, log(d/θd }2/3 ILd ∩C(δ) ≤ n−C , (7.43) Combining (7.41) and (7.42) yields for the Kolmogorov distance e sup P\|Zn maxc Bn,h ≤ x − P maxc Nh ≤ x = op (1). (7.44) . θd x∈R h∈Sd h∈Sd 43 The proof now follows observing the bound max1≤h,i≤s \|Σh,i \| ≤ max1≤h,i≤s \|ρh,i \| for the covariance matrix of N , which was derived (see the proof of Lemma 3.7) and using adapted scaling sequences (see proof of Corollary 3.11), which yields   G if Sdc = {1, . . . , d},    D ad maxc Nh − bd =⇒ G + 2 log c if \|Sdc \| = c · d + o(d) for c ∈ (0, 1),  h∈Sd    0 if \|Sdc \| = o(d). Lemma 7.3 Conditionally on Zn it holds that P e ∗ −→ e − ad max B 0. ad maxc B n,h n,h c h∈Sd h∈Sd Proof. The covariance kernel k satisfies for all t, t0 , s ∈ [0, 1] \|k(s, t) − k(s, t0 )\| ≤ 2\|t − t0 \|. Due to (th (1 − th ))2 τ (th ) ≥ t4 τ− we derive the bound Z 1 √ 1 e (L) ∗ e e ad maxc Bn,h − maxc Bn,h . ad n maxc Un,h (s) \|k(s, th ) − k(s, t̂h )\|ds h∈Sd h∈Sd h∈Sd 0 σh √ 1 e (L) Un,h (s) maxc \|t̂h − th \|. . ad n maxc max h∈Sd s∈[0,1] σh h∈Sd Choosing C sufficiently it follows from Corollary 3.3 in Jirak (2015a) that for all ε > 0 C P\|Z n maxc \|th − t̂h \| > ε = op (1). h∈Sd (7.45) Theorem 2.5 and 4.4 from the same reference imply the weak convergence √ 1 e (L) ed D ed n maxc max =⇒ G Un,h (s) − h∈Sd s∈[0,1] σh 4 p conditionally on Zn in probability with ed = 2 log(2d). This yields √ −C 1 e (L) P\|Zn ad nn maxc max Un,h (s) > ε = op (1) h∈Sd s∈[0,1] σh for all ε > 0, which completes the proof of Lemma 7.3. Lemma 7.4 Conditionally on Zn it holds that e ∗ − ad max B ∗ −→ 0. ad maxc B n,h n,h c P h∈Sd h∈Sd Proof. Define PL \|Zn (·) = P\|Zn (· ∩ Ld ) with Ld introduced in (7.38). By Lemma 7.1 the claim follows if we can verify ε ∗ ∗ e∗ e ∗ > ε ≤ PL − maxc B = op (1). (7.46) PL maxc Bn,h n,h \|Zn maxc Bn,h − Bn,h > \|Zn h∈Sd h∈Sd h∈Sd ad ad 44 We have the trivial bound ∗ e∗ . e ∗ ≤ max B ∗ − B maxc Bn,h − maxc B n,h n,h n,h c h∈Sd h∈Sd h∈Sd Assumption (T2) and (C1) imply σh ≥ σ− and τ (th )(th (1 − th ))2 ≥ τ− t4 , which yields √ Z 1 6 n (L) ∗ ∗ e (L) (s) k(s, t̂h )ds e Bn,h − Bn,h = U (s) − U n,h n,h σh τ (th )(th (1 − th ))2 0 Z 1 √ (L) e (L) (s) k(s, t̂h )ds . n U (s) − U . √ n,h 0 n,h (L) e (L) (s) ≤ S (1) + S (2) , n max Un,h (s) − U n,h n,h n,h s∈[0,1] where we use the notation bLsc L X 1 X b (1) Sn,h = max √ ξj Vj,h (bnsc) − ξj Vbj,h (n) , n s∈[0,1] j=1 (2) Sn,h = max s∈[0,1] j=1 bLsc bnsc √ − √ n n L n L X ξj Vbj,h (n) j=1 In the proof of Theorem C.4 from Jirak (2015b) it is shown, that ε (1) = op (1) PL \|Z maxc Sn,h > h∈Sd ad for all ε > 0. The second summand can be bounded by L (2) Sn,h = max s∈[0,1] 1 ≤ √ nL bnsc bLscK 1 X b √ − ξj Vj,h (n) K K nL j=1 L X ξj Vbj,h (n) j=1 and it follows again from the above reference PL \|Zn L ε 1 X b maxc √ ξj Vj,h (n) > h∈Sd ad nL = op (1) j=1 for all ε > 0, which finishes the proof of Lemma 7.4 Lemma 7.5 The weak convergence   G if Sdc = {1, . . . , d},    D (I) ad maxc Bn,h − bd =⇒ G + 2 log c if \|Sdc \| = c · d + o(d) for c ∈ (0, 1),  h∈Sd    0 if \|Sdc \| = o(d) holds conditionally on Zn . 45 (7.47) Proof. Combining Lemmas 7.2, 7.3 and 7.4 already gives assertion (7.47) for the random variables ∗ . We use the additional notation Bn,h Q− d = minc h∈Sd sbh τ̂h (t̂h (1 − t̂h ))2 sbh τ̂h (t̂h (1 − t̂h ))2 + und Q = max d h∈Sdc σh τ (th )(th (1 − th ))2 σh τ (th )(th (1 − th ))2 + −2 and the set Qd = Q− d − 1 ∨ Qd − 1 ≤ δd with δd = (log d) . Let ud (x) = x/ad + bd , then we obtain for fixed ε > 0 and d sufficiently large (I) ∗ c P\|Zn maxc Bn,h ≤ ud (x − ε) − P\|Zn (Qd ) ≤ P\|Zn maxc Bn,h ≤ ud (x) h∈Sd h∈Sd ∗ ≤ ud (x + ε) + P\|Zn (Qcd ) . ≤ P\|Zn maxc Bn,h h∈Sd Combining Proposition 3.5 from Jirak (2015a) with assertion (7.45) one can easily verify that P\|Zn (Qcd ) = op (1) and the remainder of the proof can be done analogously to the proof of Theorem 3.10. Lemma 7.6 Conditionally on Zn it holds that (I) P ad maxc Bn,h − ad maxc Bn,h −→ 0. h∈Sd h∈Sd Proof. For fixed ε > 0 we have (I) (I) P ad maxc Bn,h − maxc Bn,h > ε ≤ P ad maxc Bn,h − Bn,h > ε h∈Sd h∈Sd h∈Sd n o −1/4 −1/4 ≤ P maxc I \|∆b µh \| ≤ n > ε = P minc \|∆b µh \| ≤ n . h∈Sd h∈Sd The proof now follows by −1/4 −1/4 −1/4 P minc \|∆b µh \| ≤ n = P minc \|∆b µh \| ≤ n , minc \|∆µh \| ≤ 2n h∈Sd h∈Sd h∈Sd −1/4 −1/4 + P minc \|∆b µh \| ≤ n , minc \|∆µh \| > 2n h∈Sd h∈Sd ≤ P minc \|∆µh \| ≤ 2n−1/4 + P minc \|∆µh \| − minc \|∆b µh \| > n−1/4 h∈Sd h∈Sd h∈Sd −1/4 ≤ P K minc \|∆µh \| ≤ 2 + P maxc \|∆µh − ∆b µh \| > n = o(1), h∈Sd h∈Sd where we also used that K = n1−` and that Assumption 5.2 implies 1 − ` < 1/4 together with lim K minc \|∆µh \| = ∞ . n,d→∞ h∈Sd 46	10
Annals of Mathematical Sciences and Applications Volume 0, Number 0, 1–9, 0000 arXiv:1711.04451v1 [cs.CV] 13 Nov 2017 Visual Concepts and Compositional Voting Jianyu Wang, Zhishuai Zhang, Cihang Xie, Yuyin Zhou, Vittal Premachandran, Jun Zhu, Lingxi Xie, Alan Yuille It is very attractive to formulate vision in terms of pattern theory [26], where patterns are defined hierarchically by compositions of elementary building blocks. But applying pattern theory to real world images is currently less successful than discriminative methods such as deep networks. Deep networks, however, are black-boxes which are hard to interpret and can easily be fooled by adding occluding objects. It is natural to wonder whether by better understanding deep networks we can extract building blocks which can be used to develop pattern theoretic models. This motivates us to study the internal representations of a deep network using vehicle images from the PASCAL3D+ dataset. We use clustering algorithms to study the population activities of the features and extract a set of visual concepts which we show are visually tight and correspond to semantic parts of vehicles. To analyze this we annotate these vehicles by their semantic parts to create a new dataset, VehicleSemanticParts, and evaluate visual concepts as unsupervised part detectors. We show that visual concepts perform fairly well but are outperformed by supervised discriminative methods such as Support Vector Machines (SVM). We next give a more detailed analysis of visual concepts and how they relate to semantic parts. Following this, we use the visual concepts as building blocks for a simple pattern theoretical model, which we call compositional voting. In this model several visual concepts combine to detect semantic parts. We show that this approach is significantly better than discriminative methods like SVM and deep networks trained specifically for semantic part detection. Finally, we return to studying occlusion by creating an annotated dataset with occlusion, called VehicleOcclusion, and show that compositional voting outperforms even deep networks when the amount of occlusion becomes large. Keywords: Pattern theory, deep networks, visual concepts. 1. Introduction It is a pleasure to write an article for a honoring David Mumford’s enormous contributions to both artificial and biological vision. This article ad1 2 dresses one of David’s main interests, namely the development of grammatical and pattern theoretic models of vision [48][26]. These are generative models which represent objects and images in terms of hierarchical compositions of elementary building blocks. This is arguably the most promising approach to developing models of vision which have the same capabilities as the human visual system and can deal with the exponential complexity of images and visual tasks. Moreover, as boldly conjectured by David [27, 17], they suggest plausible models of biological visual systems and, in particular, the role of top-down processing. But despite the theoretical attractiveness of pattern theoretic methods they have not yet produced visual algorithms capable of competing with alternative discriminative approaches such as deep networks. One reason for this is that, due to the complexity of image patterns, it very hard to specify their underlying building blocks patterns (by contrast, it is much easier to develop grammars for natural languages where the building blocks, or terminal nodes, are words). But deep networks have their own limitations, despite their ability to learn hierarchies of image features in order to perform an impressive range of visual tasks on challenging datasets. Deep networks are black boxes which are hard to diagnose and they have limited ability to adapt to novel data which were not included in the dataset on which they were trained. For example, figure (1) shows how the performance of a deep network degrades when we superimpose a guitar on the image of a monkey. The presence of the guitar causes the network to misinterpret the monkey as being a human while also mistaking the guitar for a bird. The deep network presumably makes these mistakes because it has never seen a monkey with a guitar, or a guitar with a monkey in the jungle (but has seen a bird in the jungle or a person with a musical instrument). In short, the deep network is over-fitting to the typical context of the object. But over-fitting to visual context is problematic for standard machine learning approaches, such as deep networks, which assume that their training and testing data is representative of some underlying, but unknown, distribution [37, 38]. The problem is that images, and hence visual context, can be infinitely variable. A realistic image can be created by selecting from a huge range of objects (e.g., monkeys, guitars, dogs, cats, cars, etc.) and placing them in an enormous number of different scenes (e.g., jungle, beach, office) and in an exponential number of different three-dimensional positions. Moreover, in typical images, objects are usually occluded as shown in figure (1). Occlusion is a fundamental aspect of natural images and David has recognized its importance both by proposing a 2.1 D sketch representation 3 Figure 1: Caption: Adding occluders causes deep network to fail. We refer such examples as adversarial context examples since the failures are caused by misusing context/occluder info. Left Panel: The occluding motorbike turns a monkey into a human. Center Panel: The occluding bicycle turns a monkey into a human and the jungle turns the bicycle handle into a bird. Right Panel: The occluding guitar turns the monkey into a human and the jungle turns the guitar into a bird. [28] and also by using it as the basis for a “dead leaves” model for image statistics [18]. Indeed the number of ways that an object can be occluded is, in itself, exponential. For example, if an object has P different parts then we can occlude any subset of them using any of N different occluders yielding O(P N ) possible occluding patterns. In the face of this exponential complexity the standard machine learning assumptions risk breaking down. We will never have enough data both to train, and to test, black box vision algorithms. As in well known many image datasets have confounds which mean that algorithms trained on them rarely generalize to other datasets [4])İnstead we should be aiming for visual algorithms that can learn objects (and other image structures) from limited amounts of data (large, but not exponentially large). In addition, we should devise strategies where the algorithms can be tested on potentially infinite amounts of training data. One promising strategy is to expand the role of adversarial noise [34][10][45], so that we can create images which stress-test algorithms and deliberately target their weaknesses. This strategy is more reminiscent of game theory than the standard assumptions of machine learning (e.g., decision theory). This motivates a research program which we initiate in this paper by addressing three issues. Firstly, we analyze the internal structure of deep networks in order both to better understand them but, more critically, to extract internal representations which can be used as building blocks for pattern theoretic models. We call these representations visual concepts and extract them by analyzing the population codes of deep networks. We quantify their ability for detecting semantic parts of objects. Secondly, we develop 4 a simple compositional-voting model to detect semantic parts of objects by using these building blocks. These models are similar to those in [1]. Thirdly, we stress test these compositional models, and deep networks for comparison, on datasets where we have artificially introduced occluders. We emphasize that neither algorithm, compositional-voting or the deep network, has been trained on the occluded data and our goal is to stress test them to see how well they perform under these adversarial conditions. In order to perform these studies we have constructed two datasets based on the PASCAL3D+ dataset [43]. Firstly, we annotate the semantic parts of the vehicles in this dataset to create the VehicleSemanticPart dataset. Secondly, we create the Vehicle Occlusion dataset by superimposing cutouts of objects from the PASCAL segmentation dataset [8] into images from the VehicleSemanticPart dataset. We use the VehicleSemanticPart dataset to extract visual concepts by clustering the features of a deep network VGG16 [33], trained on ImageNet, when the deep network is shown objects from VehicleSemanticPart. We evaluate the visual concepts, the compositionalvoting mode;s, and the deep networks for detecting semantic parts image with and without occlusion, i.e. VehicleSemanticPart and Vehicle Occlusion respectively. 2. Related Work David’s work on pattern theory [26] was inspired by Grenander’s seminal work [11] (partly due to an ARL center grant which encouraged collaboration between Brown University and Harvard). David was bold enough to conjecture that the top-down neural connections in the visual cortex were for performing analysis by synthesis [27]. Advantages of this approach include the theoretical ability to deal with large numbers of objects by part sharing [46]. But a problem for these types of models was the difficulty in specifying suitable building blocks which makes learning grammars for vision considerably harder than learning them for natural language [35, 36]. For example, researchers could learn some object models in an unsupervised manner, e.g., see [49], but this only used edges as the building blocks and hence ignored most of the appearance properties of objects. Attempts have been made to obtain building blocks by studying properties of image patches [30] and [5] but these have yet to be developed into models of objects. Though promising, these are pixel-based and are sensitive to spatial warps of images and do not, as yet, capture the hierarchy of abstraction that seems necessary to deal with the complexity of real images. 5 Deep networks, on the other hand, have many desirable properties. such as hierarchies of feature vectors (shared across objects) which capture increasingly abstract, or invariant, image patterns as they ascend the hierarchy from the image to the object level. This partial invariance arises because they are trained in a discriminative way so that decisions made about whether a viewed object is a car or not must be based on hierarchical abstractions. It has been shown, see [47][22][44][32], that the activity patterns of deep network features do encode objects and object parts, mainly be considering the activities of individual neurons in the deep networks. Our approach to study the activation patterns of populations of neuron [40] was inspired by the well-known neuroscience debate about the neural code. On one extreme is the concept of grandmother calls described by Barlow [3] which is contrasted to population encoding as reported by Georgopoulos et al. [9]. Barlow was motivated by the finding that comparatively few neurons are simultaneously active and proposed sparse coding as a general principle. In particular, this suggests a form of extreme sparsity where only one, or a few cells, respond to a specific local stimulus (similar to a matched template). As we will see in section (6), visual concepts tend to have this property which is developed further in later work. There are, of course, big differences between studying population codes in artificial neural networks and studying them for real neurons. In particular, as we will show, we can modify the neural network so that the populations representing a visual concept can be encoded by visual concept neurons, which we call vc-neurons. Hence our approach is consistent with both extreme neuroscience perspectives. We might speculate that the brain uses both forms of neural encoding useful for different purposes: a population, or signal representation, and a sparser symbolic representation. This might be analogous to how words can be represented as binary encodings (e.g., cat, dog, etc.) and by continuous vector space encodings which similarity between vectors captures semantic information [25]. Within this picture, neurons could represent a manifold of possibilities which the vc-neurons would quantize into discrete elements. We illustrate the use of visual concepts to build a compositional-voting model inspired by [1], which is arguably the simplest pattern theoretic method. We stress that voting schemes have frequently been used in computer vision without being thought of as examples of pattern theory methods [12][19][24][29]. To stress test our algorithms we train them on un-occluded images and test them on datasets with occlusion. Occlusion is almost always present in natural images, though less frequently in computer vision datasets, and 6 is fundamental to human perception. David’s work has long emphasized its significance [28], [18] and it is sufficiently important for computer vision researchers to have addressed it using deep networks [39]. It introduces difficulties for tasks such as object detection [16] or segmentation [20]. As has already been shown, part-based models are very useful for detecting partially occluded objects [7] [6][21]. 3. The Datasets In this paper we use three related annotated datasets. The first is the vehicles in the PASCAL3D+ dataset [43] and their keypoint interactions. The second is the VehicleSemanticPart dataset which we created by annotating semantic parts on the vehicles in the PASCAL3D+ dataset to give a richer representation of the objects than the keypoints. The third is the Vehicle Occlusion dataset where we occluded the objects in the first two datasets. The vehicles in the PASCAL3D+ dataset are: (i) cars, (ii) airplanes, (iii) motorbikes, (iv) bicycles, (v) buses, and (vi) trains. These images in the PASCAL3D+ dataset are selected from the Pascal and the ImageNet datasets. In this paper we report results only for those images from ImageNet, but we obtain equivalent results on PASCAL, see arxiv paper [40]. These images were supplemented with keypoint annotations (roughly ten keypoints per object) and also estimated orientations of the objects in terms of the azimuth and the elevation angles. These objects and their keypoints are illustrated in figure (2)(upper left panel). We use this dataset to extract the visual concepts, as described in the following section. In order to do this, we first re-scale the images so that the minimum of the height and width is 224 pixels (keeping the aspect ratio fixed). We also use the keypoint annotations to evaluate the ability of the visual concepts to detect the keypoints. But there are typically only 10 keypoints for each object which means they can only give limited evaluation of the visual concepts. In order to test the visual concepts in more detail we defined semantic parts on the vehicles and annotated them to create a dataset called VehicleSemanticParts. There are thirty nine semantic parts for cars, nineteen for motorbikes, ten for buses, thirty one for aeroplanes, fourteen for bicycles, and twenty for trains. The semantic parts are regions of 100 × 100 pixels and provide a dense coverage of the objects, in the sense that almost every pixel on each object image is contained within a semantic part. The semantic parts have verbal descriptions, which are specified in the webpage 7 Figure 2: The figure illustrates the keypoints on the PASCAl3D+ dataset. For cars, there are keypoints for wheels, headlights, windshield corners and trunk corners. Keypoints are specified on salient parts of other objects. Best seen in color. http://ccvl.jhu.edu/SP.html. The annotators were trained by being provided by: (1) a one-sentence verbal description, and (2) typical exemplars. The semantic parts are illustrated in figure (3) where we show a representative subset of the annotations indexed by “A”, “B”, etc. The car annotations are described in the figure caption and the annotations for the other objects are as follows: (I) Airplane Semantic Parts. A: nose pointing to the right, B: undercarriage, C: jet engine, mainly tube-shape, sometimes with highly turned ellipse/circle, D: body with small windows or text, E: wing or elevator tip pointing to the left, F: vertical stabilizer, with contour from top left to bottom right, G: vertical stabilizer base and body, with contour from top left to bottom right. (II) Bicycle Semantic Parts. A: wheels, B: pedal, C: roughly triangle structure between seat, pedal, and handle center, D: seat, E: roughly T-shape handle center. (III) Bus Semantic Parts. A: wheels, B: headlight, C: license plate, D: window (top) and front body (bottom), E: display screen with text, F: window and side body with vertical frame in the middle, G: side windows and side body. (IV) Motorbike Semantic Parts. A: wheels, B: headlight, C: engine, D: seat. (V) Train Semantic Parts. A: front window and headlight, B: headlight, C: part of front window on the left and side body, D: side windows or doors and side body, E: head body on the right or bottom right and background, F: head body on the top left and background.). In order to study occlusion we create the VehicleOcclusion dataset. This consists of images from the VehicleSemanticPart dataset. Then we add 8 Figure 3: This figure illustrates the semantic parts labelled on the six vehicles in the VehicleSemanticParts dataset (remaining panels). For example, for car semantic parts. A: wheels, B: side body, C: side window, D: side mirror and side window and wind shield, E: wind shield and engine hood, F: oblique line from top left to bottom right, wind shield, G: headlight, H: engine hood and air grille, I: front bumper and ground, J: oblique line from top left to bottom right, engine hood. The parts are weakly viewpoint dependent. See text for the semantic parts of the other vehicles. occlusion by randomly superimposing a few (one, two, or three) occluders, which are objects from PASCAL segmentation dataset [8], onto the images. To prevent confusion, the occluders are not allowed to be vehicles. We also vary the size of the occluders, more specifically the fraction of occluded pixels from all occluders on the target object, is constrained to lie in three ranges 0.2−0.4, 0.4−0.6 and 0.6−0.8. To compute these fractions requires estimating the sizes of the vehicles in the VehicleSemanticPart dataset, which can be done using the 3D object models associated to the images in PASCAL3D+. 4. The Visual Concepts We now analyze the activity patterns within deep networks and show that they give rise to visual concepts which correspond to part/subparts of objects and which can be used as building blocks for compositional models. In section (4.2) we obtain the visual concepts by doing K-means clustering followed by a merging stage. Next in section (4.3), we show an alternative unsupervised method which converges to similar visual concepts, but which can be implemented directly by a modified deep network with additional (unsupervised) nodes/neurons. Section (4.4) gives visualization of the visual concepts. Section (4.5) evaluates them for detecting keypoints and semantic parts. 9 Figure 4: Examples for occluded cars with occlusion level 1,5 and 9 respectively. Level 1 means 2 occluders with fraction 0.2-0.4 of the object area occluded, level 5 means 3 occluders with fraction 0.4-0.6 occluded, and level 9 means 4 occluders with fraction 0.6-0.8 occluded. 4.1. Notation We first describe the notation used in this paper. The image is defined on the image lattice and deep network feature vectors are defined on a hierarchy of lattices. The visual concepts are computed at different layers of the hierarchy, but in this paper we will concentrate on the pool-4 layer, of a VGG-16 [33] trained on ImageNet for classification, and only specify the algorithms for this layer (it is trivial to extend the algorithms to other layers). The groundtruth annotations, i.e. the positions of the semantic parts, are defined on the image lattice. Hence we specify correspondence between points of the image lattice and points on the hierarchical lattices. Visual concepts will never be activated precisely at the location of a semantic part, so we specify a neighborhood of allow for tolerance in spatial location. In the following section, we will use more sophisticated neighborhoods which depend on the visual concepts and the semantic parts. An image I is defined on the image lattice L0 by a set of three-dimensional color vectors {Iq : q ∈ L0 }. Deep network feature vectors are computed on a hierarchical set of lattices Ll , where l indicates the layer with Ll ⊂ Ll−1 for l = 0, 1, .... This paper concentrates on the pool-4 layer L4 . We define correspondence between the image lattice and the pool-4 lattice by the mappings π07→4 (.) from L0 to L4 and π47→0 (.) from L4 to L0 respectively. The function π47→0 (.) gives the exact mapping from L4 to L0 (recall the lattices are defined so that L4 ⊂ L0 ). Conversely, π07→4 (q) denotes the closest position at the L4 layer grid that corresponds to q, i.e., π07→4 (q) = arg minp Dist(q, L0 (p)). The deep network feature vectors at the pool-4 layer are denoted by {fp : p ∈ L4 }. These feature vectors are computed by fp = f (Ip ), where the function f is specified by the deep network and Ip is an image patch, a 10 subregion of the input image I, centered on a point π47→0 (p) on the image lattice L0 . The visual concepts {VCv : v ∈ V} at the pool-4 layer are specified by a set of feature vectors {fv : v ∈ V}. They will be learnt by clustering the pool-4 layer feature vectors {fp } computed from all the images in the dataset. The activation of a visual concept VCv to a feature vector fp at p ∈ L4 is a decreasing function of \|\|fv − fp \|\| (see later for more details). The visual concepts will be learnt in an unsupervised manner from a set T = {In : n = 1, ..., N } of images, which in this paper will be vehicle images from PASCAL3D+ (with tight bounding boxes round the objects and normalized sizes). The semantic parts are given by {SPs ∈ S} and are pre-defined for each vehicle class, see section (3) and the webpage http://ccvl.jhu.edu/ SP.html. These semantic parts are annotated on the image dataset {In : n = 1, ..., N } by specifying sets of pixels on the image lattice corresponding to their centers (each semantic part corresponds to an image patch of size 100 × 100). For a semantic part SPs , we define Ts+ to be the set of points q on the image lattices L0 where they have been labeled. From these labels, we compute a set of points Ts− where the semantic part is not present (constrained so that each point in Ts− is at least γ pixels from every point in Ts+ , where γ is chosen to ensure no overlap). We specify a circular neighborhood N (q) for all points q on the image lattices L0 , given by {p ∈ L0 s.t. \|\|p − q\|\| ≤ γth }. This neighborhood is used when we evaluate if a visual concept responds to a semantic part by allowing some uncertainty in the relative location, i.e. we reward a visual concept if it responds near a semantic part, where nearness is specified by the neighborhood. Hence a visual concept at pixel p ∈ L4 responds to a semantic part s at position q provided \|\|fv − fp \|\| is small and π47→0 (p) ∈ N (q). In this paper the neighborhood radius γth is set to be 56 pixels for the visual concept experiments in this section, but was extended to 120 for our late work on voting, see section (5). For voting, we start with this large neighborhood but then learn more precise neighborhoods which localize the relative positions of visual concept responses to the positions of semantic parts. 4.2. Learning visual concepts by K-means We now describe our first method for extracting visual concepts which uses the K-means algorithm. First we extract pool-4 layer feature vectors {fp } using the deep network VGG-16 [33] from our set of training images T . We normalize them to unit norm, i.e. such that \|fp \| = 1, ∀p. 11 Figure 5: Left Panel: The images I are specified on the image lattice L0 and a deep network extracts features on a hierarchy of lattices. We concentrate on the features {fp } at pool-4 lattice L4 . Projection functions π47→0 () and π07→4 () map pixels from L4 to L0 and vice versa. We study visual concepts VCv at layer L4 and relate them to semantic parts SPs defined on the image lattice. Center Panel: the visual concepts are obtained by clustering the normalized feature vectors {fp }, using either K-means or a mixture of Von Mises-Fisher. Right Panel: extracting visual concepts by a mixture of Von Mises-Fisher is attractive since it incorporates visual concepts within the deep network by adding additional visual concept neurons (vc-neurons). Then we use K-means++ [2], to cluster the feature vectors into a set V of visual concepts. Each visual concept VCv has an index v ∈ {1, 2, . . . , \|V\|} and is specified by its clustering center: fv ∈ R512 (where 512 is the dimension of the feature vector at pool-4 of VGG-net.) In mathematical terms, P the K-means algorithm attempts to minimize a cost function F (V, {fv }) = v,p Vp,v \|\|fp − fv \|\|2 , with respect to the assignment variable V and the cluster centers {fv }. The assignment variables V impose hard assignment so that each feature vector fp is assigned to only one visual concept fv , i.e. for each p, Vp,v∗ = 1 if v∗ = argminv \|\|fp − fv \|\|, and Vp,v = 0 otherwise. The K-means algorithm minimizes the cost function F (V, fv }) by minimizing with respect to V and {fv } alternatively. The Kmeans++ algorithm initializes K-means by taking into account the statistics of the data {fp }. We took a random sample of 100 feature vectors per image as input. The number of clusters K = \|V\| is important in order to get good visual concepts and we found that typically 200 visual concepts were needed for each object. We used two strategies to select a good value for K. The first is to specify a discrete set of values for K (64, 128, 256, 512), determine 12 how detection performance depends on K, and select the value of K with best performance. The second is start with an initial set of clusters followed by a merging stage. We use the Davies-Boundin index as a measure for the σk +σm “goodness” of a cluster k. This is given by DB(k) = maxm6=k \|\|f , where k −fl \|\| fk and fm denote the centers of clusters k and m respectively. σk and σm denote the average distance of all data points in clusters k and m to their P respective cluster centers, i.e., σk = n1k p∈Ck \|\|fp − fk \|\|2 , where Ck = {p : Vp,k = 1} is the set of data points that are assigned to cluster k. The DaviesBoundin index take small values for clusters which have small variance of the P feature vectors assigned to them, i.e. where p:Vp,v =1 \|fp − fv \|2 /nv is small P (with nv = p:Vp,v =1 1), but which are also well separated from other visual clusters, i.e., \|fv − fµ \| is large for all µ 6= v. We initialize the algorithm with K clusters (e.g., 256, or 512) rank them by the Davies-Boundin index, and merge them in a greedy manner until the index is below a threshold, see our longer report [40] for more details. In our experiments we show results with and without cluster merging, and the differences between them are fairly small. 4.3. Learning Visual Concepts by a mixture of von Mises-Fisher distributions We can also learn visual concepts by modifying the deep network to include additional vc neurons which are connected to the neurons (or nodes) in the deep network by soft-max layers, see figure (5). This is an attractive alternative to K-means because it allows visual concepts to be integrated naturally within deep networks, enabling the construction of richer architectures which share both signal and symbolic features (the deep network features and the visual concepts). In this formulation, both the feature vectors {fp } and the visual concepts {fv } are normalized so lie on the unit hypersphere. Theoretical and practical advantages of using features defined on the hypersphere are described in [41]. This can be formalized as unsupervised learning where the data, i.e. the feature vectors {fp }, are generated by a mixture of von Mises-Fisher distributions [13]. Intuitively, this is fairly similar to K-means (since K-means relates to learning a mixture of Gaussians, with fixed isotropic variance, and von-Mises-Fisher relates to an isotropic Gaussian as discussed in the next paragraph). Learning a mixture of von Mises-Fisher can be implemented by a neural network, which is similar to the classic result relating K-means to competitive neural networks [14]. 13 The von-Mises Fisher distribution is of form: P (fp \|fv ) = (1) 1 exp{ηfp · fv }, Z(η) where η is a constant, Z(η) is a normalization factor, and fp and fv are both unit vectors. This requires us to normalize the weights of the visual concepts, so that \|fv \| = 1, ∀v, as well as the feature vectors {fp }. Arguably it is more natural to normalize both the feature vectors and the visual concepts (instead of only normalizing the feature vectors as we did in the last section). There ia a simple relationship between von-Mises Fisher and isotropic Gaussian distributions. The square distance term \|\|fp − fv \|\| becomes equal to 2(1 − fp · fv ) if \|\|fp \|\| = \|\|fv \|\| = 1. Hence an isotropic Gaussian distribution on fp with mean fv reduces to von Mises-Fisher if both vectors are required to lie on the unit sphere. Learning a mixture of von Mises-Fisher can be formulated in terms of neural networks where the cluster centers, i.e. the visual concepts, are specified by visual concept neurons, or vc-neurons, with weights {fv }. This can be shown as follows. Recall that Vp,v is the assignment variable between a feature vector fp and each visual concept fv , and denote the set of assignment by Vp = {Vp,v : v ∈ V} During learning this assignment variable Vp is unknown but we can use the EM algorithm, which involving replacing the Vp,v by a distribution qv (p) for the probability that Vp,v = 1. The EM algorithm can be expressed in terms of minimizing the following free energy function with respect to {qv } and {fv }: (2) X X X F({qv }, {fv }) = − { qv (p) log P (fp , Vp \|{fv }) + qv (p) log qv (p)}, p v v The update rules for the assignments and the visual concepts are respectively: (3) exp{ηfp · fvt } qvt = P t µ exp{ηfp · fµ } fvt+1 = fvt − ηqvt fv + η, where η is a Lagrange multiplier to enforce \|fvt+1 \| = 1. Observe that the update for the assignments are precisely the standard neural network soft-max operation where the {fv } are interpreted as the weights of vc-neurons and the neurons compete so that their activity sums to 1 [14]. After the learning algorithm has converged this softmax activation indicates soft-assignment of a feature vector fp to the vc-neurons. This corresponds to the network shown in figure (5). 14 This shows that learning visual concepts can be naturally integrated into a deep network. This can either be done, as we discussed here, after the weights of a deep network have been already learnt. Or alternatively, the weights of the vc neurons can be learnt at the same times as the weights of the deep network. In the former case, the algorithm simply minimizes a cost function: (4) − X v X exp{ηfp · fv } P log exp{ηfp · fν }, ν exp{ηfp · fν } ν which can be obtained from the EM free energy by solving for the {qv } p ·fv } directly in terms of {fv }, to obtain qv = Pexp{ηf , and substituting this ν exp{ηfp ·fν } back into the free energy. Learning the visual concepts and the deep network weights simultaneously is performed by adding this cost function to the standard penalty function for the deep network. This is similar to the regularization method reported in [23]. Our experiments showed that learning the visual concepts and the deep network weights simultaneously risks collapsing the features vectors and the visual concepts to a trivial solution. 4.4. Visualizing the Visual Concepts We visualize the visual concepts by observing the image patches which are assigned to each cluster by K-means. We observe that the image patches for each visual concept roughly correspond to semantic parts of the object classes with larger parts at higher pooling levels, see figure (6). This paper concentrates on visual concepts at the pool-4 layer because these are most similar in scale to the semantic parts which were annotated. The visual concepts at pool-3 layer are at a smaller scale, which makes it harder to evaluate them using the semantic parts. The visual concepts at the pool-5 layer have two disadvantages. Firstly, they correspond to regions of the image which are larger than the semantic parts. Secondly, their effective receptive field sizes which were significantly smaller than their theoretical receptive field size (determined by using deconvolution networks to find which regions of the image patches affected the visual concept response) which meant that they appeared less tight when we visualized them (because only the central regions of the image patches activated them, so the outlying regions can vary a lot). Our main findings are that the pool-4 layer visual concepts are: (i) very tight visually, in the sense that image patches assigned to the same visual 15 Figure 6: This figure shows example visual concepts on different object categories from the VehicleSemanticPart dataset, for layers pool3, pool4 and pool5. Each row visualizes three visual concepts with four example patches, which are randomly selected from a pool of the closest 100 image patches. These visual concepts are visually and semantically tight. We can easily identify the semantic meaning and parent object class. Figure 7: This figure shows that visual concepts typically give a dense coverage of the different parts of the objects, in this case a car. concept tend to look very similar which is illustrated in figure (8) and (ii) give a dense coverage of the objects as illustrated in figure (7). Demonstrating tightness and coverage of the visual concepts is hard to show in a paper, so we have created a webpage http://ccvl.jhu.edu/cluster.html where readers can see this for themselves by looking at many examples. To illustrate visual tightness we show the average of the images and edge maps of the best 500 patches for a few visual concepts, in figure (8), and compare them to the averages for single filters. These averages show that the clusters are tight. 16 (a) pool4 visual concept (b) pool4 single filter Figure 8: This figure shows four visual concept examples (left four) and four single filter examples (right four). The top row contains example image patches, and the bottom two rows contain average edge maps and intensity maps respectively obtained using top 500 patches.Observe that the means of the visual concepts are sharper than the means of the single filters for both average edge maps and intensity maps, showing visual concepts capture patterns more tightly than the single filters. 4.5. Evaluating visual concepts for detecting keypoints and semantic parts We now evaluate the visual concepts as detectors of keypoints and semantic parts (these are used for evaluation only and not for training). Intuitively, a visual concept VCv is a good detector for a semantic part SPs if the visual concept “fires” near most occurrences of the semantic part, but rarely fires in regions of the image where the semantic part is not present. More formally, a visual concept VCv fires at a position p ∈ L4 provided \|fp − fv \| < T , where T is a threshold (which will be varied to get a precision-recall curve). Recall that for each semantic part s ∈ S we have a set of points Ts+ where the semantic part occurs, and a set Ts− where it does not. A visual concept v has a true positive detection for q ∈ Ts+ if minp∈N (q) \|fp −fv \| < T . Similarly it has a false positive if the same condition holds for s ∈ Ts− . As the threshold T varies we obtain a precision-recall curve and report the average precision. We also consider variants of this approach by seeing if a visual concept is able to detect several semantic parts. For example, a visual concept which fires on semantic part “side-window” is likely to also respond to the semantic parts “front-window” and “back-window”, and hence should be evaluated by its ability to detect all three types of windows. Note that if a visual concept is activated for several semantic parts then it will tend to have fairly bad average precision for each semantic part by itself (because the detection for the other semantic parts will count as false positives). 17 We show results for two versions of visual concept detectors, with and without merging indicated by S-VCM and S-VCK respectively. We also implemented two baseline methods for comparison methods. This first baseline uses the magnitude of a single filter response, and is denoted by S-F. The second baseline is strongly supervised and trains a support vector machines (SVM) taking the feature vectors at the pool-4 layer as inputs, notated by SVM-F. The results we report are based on obtaining visual concepts by K-means clustering (with or without merging), but we obtain similar results if we use the von Mises-Fisher online learning approach, 4.5.1. Strategies for Evaluating Visual Concepts. There are two possible strategies for evaluating visual concepts for detection. The first strategy is to crop the semantic parts to 100×100 image patches, {P1 , P2 , · · · , Pn }, and provide some negative patches {Pn+1 , Pn+2 , · · · , Pn+m } (where the semantic parts are not present). Then we can calculate the response of V Cv by calculating Resi = \|\|fi − fv \|\|. By sorting those m + n candidates by the VC responses, we can get a list Pk1 , Pk2 , · · · , Pkm+n in which k1 , k2 , · · · , km+n is a permutation of 1, 2, · · · , m + n. By varying the threshold from Resk1 to Reskm+n , we can calculate the precision and recall curve. The average precision can be obtained by averaging the precisions at different recalls. This first evaluation strategy is commonly used for detection tasks in computer vision, for example to evaluate edge detection, face detection, or more generally object detection. But it is not suitable for our purpose because different semantic parts of object can be visually similar (e.g., the front, side, and back windows of cars). Hence the false positives of a semantic part detector will often be other parts of the same object, or a closely related object (e.g., car and bus semantic parts can be easily confused). In practice, we will want to detect semantic parts of an object when a large region of the object is present. The first evaluation strategy does not take this into account, unless it is modified so that the set of negative patches are carefully balanced to include large numbers of semantic parts from the same object class. Hence we use a second evaluation strategy, which is to crop the object bounding boxes from a set of images belonging to the same object class to make sure they only have limited backgrounds, and then select image patches which densely sample these bounding boxes. This ensures that the negative image patches are strongly biased towards being other semantic parts of the object (and corresponds to the typical situation where we will want to detect semantic parts in practice). It will, however, also introduce image patches which partly overlap with the semantic parts and hence a visual concept 18 will typically respond multiple times to the same semantic part, which we address by using non-maximum suppression. More precisely, at each position a visual concept VCv will have a response \|\|fp − fv \|\|. Next we apply nonmaximum suppression to eliminate overlapping detection results, i.e. if two activated visual concepts have sufficiently similar responses, then the visual concept with weaker response (i.e. larger \|\|fp − fv \|\|) will be suppressed by the stronger one. Then we proceed as for the first strategy, i.e. threshold the responses, calculate the false positives and false negatives, and obtain the precision recall curve by varying the threshold. (A technical issue for the second strategy is that some semantic parts may be close together, so after non-maximum suppression, some semantic parts may be missed since the activated visual concepts could be suppressed by nearby stronger visual concept responses. Then a low AP will be obtained, even though this visual concept might be very for detecting that semantic part). 4.5.2. Evaluating single Visual Concepts for Keypoint detection. To evaluate how well visual concepts detect keypoints we tested them for vehicles in the PASCAL3D+ dataset and compared them to the two baseline methods, see table (1). Visual concepts and single filters are unsupervised so we evaluate them for each keypoint, by by selecting the visual concept, or single filter, which have best detection performance, as measured by average precision. Not surprisingly, the visual concepts outperformed the single filters but were less successful than the supervised SVM approach. This is not surprising, since visual concepts and single filters are unsupervised. The results are fairly promising for visual concepts since for almost all semantic parts we either found a visual concept that was fairly successful at detecting them. We note that typically several visual concepts were good at detecting each keypoint, recall that there are roughly 200 visual concepts but only approximately 10 keypoints. This suggests that we would get better detection results by combining visual concepts, as we will do later in this paper. 4.5.3. Evaluating single Visual Concepts for Semantic Part detection. The richer annotations on VehicleSemanticPart allows us to get better understanding of visual concepts. We use the same evaluation strategy as for keypoints but report results here only for visual concepts with merging S-VC and for single filters S-F. Our longer report [40] gives results for different variants of visual concepts (e.g., without merging, with different values of K, etc.) but there is no significant difference. Our main findings are: (i) that visual concepts do significantly better than single filters, and (ii) for every semantic part there is a visual concept that detects it reasonably 19 Car Bicycle Motorbike 1 2 3 4 5 6 7 mAP 1 2 3 4 5 mAP 1 2 3 4 mAP S-F .86 .44 .30 .38 .19 .33 .13 .38 .23 .67 .25 .43 .43 .40 .26 .60 .30 .22 .35 S-VCM .92 .51 .27 .41 .36 .46 .18 .45 .32 .78 .30 .55 .57 .50 .35 .75 .43 .25 .45 S-VCK .94 .51 .32 .53 .36 .48 .22 .48 .35 .80 .34 .53 .63 .53 .40 .76 .44 .43 .51 SVM-F .97 .65 .37 .76 .45 .57 .30 .58 .37 .80 .34 .71 .64 .57 .37 .77 .50 .60 .56 Bus Train Aeroplane 1 2 3 4 5 6 mAP 1 2 3 4 5 mAP 1 2 3 4 5 mAP S-F .45 .42 .23 .38 .80 .22 .42 .39 .33 .24 .16 .15 .25 .41 .25 .22 .13 .31 .26 S-VCM .41 .59 .26 .29 .86 .51 .49 .41 .30 .30 .28 .24 .30 .21 .47 .31 .16 .34 .30 S-VCK .41 .51 .26 .33 .86 .52 .48 .42 .32 .30 .28 .25 .32 .31 .47 .31 .20 .35 .33 SVM-F .74 .70 .52 .63 .90 .61 .68 .71 .49 .50 .36 .39 .49 .72 .60 .50 .32 .49 .53 Table 1: AP values for keypoint detection on PASCAL3D+ dataset for six object categories. Visual concepts, with S-VCm or without S-VCK merging, achieves much higher results than the single filter baseline S-F. The SVM method SVM-F does significantly better, but this is not surprising since it is supervised. The keypoint number-name mapping is provided below. Cars – 1: wheel 2: wind shield 3: rear window 4: headlight 5: rear light 6: front 7: side; Bicycle –1: head center 2: wheel 3: handle 4: pedal 5: seat; Motorbike – 1: head center 2: wheel 3: handle 4: seat; Bus – 1: front upper corner 2: front lower corner 3: rear upper corner 4: rear lower corner 5: wheel 6: front center; Train – 1: front upper corner 2: front lower corner 3: front center 4: upper side 5: lower side; Aeroplane – 1: nose 2: upper rudder 3: lower rudder 4: tail 5: elevator and wing tip. well. These results, see Table (2), support our claim that the visual concepts give a dense coverage of each object (since the semantic parts label almost every part of the object). Later in this paper, see table (3) (known scale), we compare the performance of SVMs to visual concepts for detecting semantic parts. This comparison uses a tougher evaluation criterion, based on the interSection over union (IOU) [8], but the relative performance of the two methods is roughly the same as for the keypoints. 4.5.4. What do the other visual concepts do?. The previous experiments have followed the “best single visual concept evaluation”. In other words, for each semantic part (or keypoint), we find the single visual concept which best detects that semantic part. But this ignores three issues: (I) A visual concept may respond to more than one semantic parts (which means that its AP for one semantic part is small because the others are treated as false positives). (II) There are far more visual concepts than semantic parts, so our previous evaluations have not told us what all the visual concepts are doing (e.g., for cars there are roughly 200 visual concepts but only 39 semantic parts, so we are only reporting results for twenty percent of the visual concepts). (III) Several visual concepts may be good for detecting the 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 S-F .86 .87 .84 .68 .70 .83 .83 .82 .72 .18 .22 .49 .33 .25 .22 .18 .37 .18 S-VCM .94 .97 .94 .93 .94 .95 .94 .92 .94 .42 .48 .58 .45 .48 .46 .54 .60 .34 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 mAP S-F .28 .13 .16 .14 .30 .20 .16 .17 .15 .27 .19 .25 .18 .19 .16 .07 .07 .36 S-VCM .38 .19 .26 .37 .42 .32 .26 .40 .29 .40 .18 .41 .53 .31 .57 .36 .26 .53 (a) Car 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mAP S-F .32 .42 .08 .61 .58 .42 .49 .31 .23 .09 .07 .20 .12 .29 .29 .07 .09 .28 S-VCM .26 .21 .07 .84 .61 .44 .63 .42 .34 .15 .11 .44 .23 .59 .65 .08 .15 .37 (b) Aeroplane S-F S-VCM 1 2 3 4 5 6 7 8 9 10 11 12 13 mAP .77 .84 .89 .91 .94 .92 .94 .91 .91 .56 .53 .15 .40 .75 .91 .95 .98 .96 .96 .96 .97 .96 .97 .73 .69 .19 .50 .83 (c) Bicycle 1 2 3 4 5 6 7 8 9 10 11 12 mAP S-F .69 .46 .76 .67 .66 .57 .54 .70 .68 .25 .17 .22 .53 S-VCM .89 .64 .89 .77 .82 .63 .73 .75 .88 .39 .33 .29 .67 (d) Motorbike 1 2 3 4 5 6 7 8 9 mAP S-F .90 .40 .49 .46 .31 .28 .36 .38 .31 .43 S-VCM .93 .64 .69 .59 .42 .48 .39 .32 .27 .53 (e) Bus 1 2 3 4 5 6 7 8 9 10 11 12 13 14 mAP S-F .58 .07 .20 .15 .21 .15 .27 .43 .17 .27 .16 .25 .17 .10 .23 S-VCM .66 .50 .32 .28 .24 .15 .33 .72 .36 .41 .27 .45 .27 .47 .39 (f) Train Table 2: The AP values for semantic part detection on VehicleSemanticPart dataset for cars (top), aeroplanes (middle) and bicycle (bottom). We see that visual concepts S-VCm achieves much higher AP than single filter method S-F. The semantic part names are provided in the supplementary material. same semantic part, so combining them may lead to better semantic part detection. In this section we address issues (I) and (II), which are closely related, and leave the issue of combining visual concepts to the next section. We find that some visual concepts do respond better to more than one semantic part, in the sense that their AP is higher when we evaluate them for detecting 21 a small subset of semantic parts. More generally, we find that almost all the visual concepts respond to a small number of semantic parts (one, two, three, or four) while most of the limited remaining visual concepts appear to respond to frequently occuring “backgrounds”, such as the sky in the airplane images. We proceed as follows. Firstly, for each visual concept we determine which semantic part it best detects, calculate the AP, and plot the histogram as shown in Figure 9 (SingleSP). Secondly, we allow each visual concept to select a small subset – two, three, or four – of semantic parts that it can detect (penalizing it if it fails to detect all of them). We measure how well the visual concept detects this subset using AP and plot the same histogram as before. This generally shows much better performance than before. From Figure 9 (MultipleSP) we see that the histogram of APs gets shifted greatly to the right, when taking into account the fact that one visual concept may correspond to one or more semantic parts. Figure 10 shows the percentage of how many semantic parts are favored by each visual concept. If a visual concept responds well to two, or more, semantic parts, this is often because those parts are visually similar, see Figure 11. For example, the semantic parts for car windows are visually fairly similar. For some object classes, particularly aeroplanes and trains, there remain some visual concepts with low APs even after allowing multiple semantic parts. Our analysis shows that many of these remaining visual concepts are detecting background (e.g., the sky for the aeroplane class, and railway tracks or coaches for the train class), see Figure 12. A few other visual concepts have no obvious interpretation and are probably due to limitations of the clustering algorithm and the CNN features. Our results also imply that there are several visual concepts for each semantic part. In other work, in preparation, we show that these visual concepts typically correspond to different subregions of the semantic parts (with some overlap) and that combining them yields better detectors for all semantic parts with a mean gain of 0.25 AP. 4.6. Summary of Visual Concepts This section showed that visual concepts were represented as internal representations of deep networks and could be found either by K-means or by an alternative method where the visual concepts could be treated as hidden variables attached to the deep network. The visual concepts were visually tight, in the sense that image patches assigned to the same visual concept looked similar. Almost all the visual concepts corresponded to semantic parts 22 1 0.7 0.7 0.7 0.6 0.6 0.6 0.4 Percentage 0.8 0.5 0.5 0.4 0.5 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0~0.2 0 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin (a) Car 0.1 0~0.2 0 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin SingleSP MultipleSP (c) Bus SingleSP MultipleSP 0.9 0.7 0.7 0.6 0.6 0.6 Percentage 0.7 Percentage 0.8 0.4 0.5 0.4 0.5 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0 0~0.2 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin SingleSP MultipleSP 0.9 0.8 0.5 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin 1 1 0.8 0 0~0.2 (b) Aeroplane 1 0.9 SingleSP MultipleSP 0.9 0.8 Percentage Percentage SingleSP MultipleSP 0.9 0.8 0 Percentage 1 1 SingleSP MultipleSP 0.9 0.1 0~0.2 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin (e) Bicycle (d) train 0 0~0.2 0.2~0.4 0.4~0.6 0.6~0.8 0.8~1.0 Average Precision Bin (f) Motorbike Figure 9: The histograms of the AP responses for all visual concepts. For “SingleSP”, we evaluate each visual concept by reporting its AP for its best semantic part (the one it best detects). Some visual concepts have very low APs when evaluated in this manner. For “MultipleSP”, we allow each visual concept to detect a small subset of semantic parts (two, three, or four) and report the AP for the best subset (note, the visual concept is penalized if it does not detect all of them). The APs rise considerably using the MultipleSP evaluation, suggesting that some of the visual concepts detect a subset of semantic parts. The remaining visual concepts with very low APs correspond to the background. 0.6 Car Aeroplane Bus Train Bicycle Motorbike 0.5 Percentage 0.4 0.3 0.2 0.1 0 1 2 3 4 Number of Favored Semantic Parts for Each Visual Concept Figure 10: Distribution of number of semantic parts favored by each visual concept for 6 objects. Most visual concepts like one, two, or three semantic parts. of the objects (one, tow, three, or four), several corresponded to background regions (e.f., sky), but a few visual concepts could not be interpreted. More results and analysis can be found in our longer report [40] and the webpage http://ccvl.jhu.edu/cluster.html. In particular, we observed that visual concepts significantly outper- 23 Figure 11: This figure shows two examples of visual concepts that correspond to multiple semantic parts. The first six columns show example patches from that visual concept, and the last two columns show prototypes from two different corresponding semantic parts. We see that the visual concept in the first row corresponds to “side window” and “front window”, and the visual concept in the second row corresponds to “side body and ground” and “front bumper and ground”. Note that these semantic parts look fairly similar. Figure 12: The first row shows a visual concept corresponding to sky from the aeroplane class, and the second and third rows are visual concepts corresponding to railway track and tree respectively from the train class. formed single filters for detecting semantic parts, but performed worse than supervised methods, such as support vector machines (SVMs), which took the same pool-4 layer features as input. This worse performance was due to several factors. Firstly, supervised approaches have more information so they generally tend to do better on discriminative tasks. Secondly, several visual concepts tended to fire on the same semantic part, though in different locations, suggesting that the visual concepts correspond to subparts of the semantic part. Thirdly, some visual concepts tended to fire on several semantic parts, or in neighboring regions. These findings suggest that visual concepts are suitable as building blocks for pattern theoretic models. We will evaluate this in the next section where we construct a simple compositional-voting model which combines several visual concepts to detect semantic parts. 24 5. Combining Visual Concepts for Detecting Semantic Parts with Occlusion The previous section suggests that visual concepts roughly correspond to object parts/subparts and hence could be used as building blocks for compositional models. To investigate this further we study the precise spatial relationships between the visual concepts and the semantic parts. Previously we only studied whether the visual concepts were active within a fixed size circular window centered on the semantic parts. Our more detailed studies show that visual concepts tend to fire in much more localized positions relative to the centers of the semantic parts. This study enables us to develop a compositional voting algorithm where visual concepts combine together to detect semantic parts. Intuitively the visual concepts provide evidence for subparts of the semantic parts, taking into account their spatial relationships, and we can detect the semantic parts by combining this evidence. The evidence comes both from visual concepts which overlap with the semantic parts, but also from visual concepts which lie outside the semantic part but provide contextual evidence for it. For example, a visual concept which responds to a wheel gives contextual evidence for the semantic part “front window” even though the window is some distance away from the wheel. We must be careful, however, about allowing contextual evidence because this evidence can be unreliable if there is occlusion. For example, if we fail to detect the wheel of the car (i.e. the evidence for the wheel is very low) then this should not prevent us from detecting a car window because the car wheel might be occluded. But conversely, if can detect the wheel then this should give contextual evidence which helps detect the window. A big advantage of our compositional voting approach is that it gives a flexible and adaptive way to deal with context which is particularly helpful when there is occlusion, see figure (13). If the evidence for a visual concept is below a threshold, then the algorithm automatically switches off the visual concept so that it does not supply strong negative evidence. Intuitively “absence of evidence is not evidence of absence”. This allows a flexible and adaptive way to include the evidence from visual concepts if it is supportive, but to automatically switch it off if it is not. We note that humans have this ability to adapt flexibly to novel stimuli, but computer vision and machine learning system rarely do since the current paradigm is to train and test algorithms on data which are presume to be drawn from the same underlying distribution. Our ability to use information flexibly means that our system is in some cases able to detect a semantic part from context even if the part 25 Figure 13: The goal is to detect semantic parts of the object despite occlusion, but without being trained on the occlusion. Center panel: yellow indicates unoccluded part, blue means partially occluded part, while red means fully occluded part. Right panel: Visual concepts (the circles) give evidence for semantic parts. They can be switched off automatically in presence of occlusion (the red circles) so that only the visual concepts that fire (green circles) give positive evidence for the presence of the parts. This robustness to occlusion if characteristic of our compositional model and enables it to outperform fully supervised deep networks. itself is completely occluded and, by analyzing which visual concepts have been switched off, to determine if the semantic part itself is occluded. In later work, this switching off can be extended to deal with the “explaining away” phenomena, highlighted by work in cognitive science [15]. 5.1. Compositional Voting: detecting Semantic Parts with and without Occlusion Our compositional model for detecting semantic parts is illustrated in figure (14). We develop it in the following stages. Firstly, we study the spatial relations between the visual concepts and the semantic parts in greater detail. In the previous section, we only use spatial relations (e.g., is the visual concept activated within a window centered on the semantic part, where the window size and shape was independent of the visual concept or the semantic part). Secondly, we compute the probability distributions of the visual concept responses depending on whether the semantic part is present or nor. This enables us to use the log-likelihood ratio of these two distributions as evidence for the presence of the semantic part. Thirdly, we describe how the evidence from the visual concepts can be combined to vote for a semantic part, taking into account the spatial positions of the parts and allowing visual concepts to switch off their evidence if it falls below threshold (to deal with occlusion). 26 Annotated SP w/ the Best Neighboring VC 1 3 summarization 2 1 42 3 6 7 5 8 Collecting +/- VC Cues for voting SP Prob. Distributions % 𝑇",$ + − 4 quantization 6 + 5 7 loglikelihood % 𝑇",$ − 8 Neighborhood Score Function Figure 14: Illustration of the training phase of one (VCv , SPs ) pair. Only one negative point is shown in each image (marked in a yellow frame in the third column), though the negative set is several times larger than the positive set, see [42] for details. 5.1.1. Modeling the Spatial Relationship between Visual Concepts and Semantic Parts. We now examine the relationship between visual concepts and semantic parts by studying the spatial relationship of each (VCv , SPs ) pair in more detail. Previously we rewarded a visual concept if it responded in a neighborhood N of a semantic part, but this neighborhood was fixed and did not depend on the visual concept or the semantic part. But our observations showed that if a visual concept VCv provides good evidence to locate a semantic part SPs , then the SPs may only appear at a restricted set of positions relative to VCv . For example, if VCv represents the upper part of a wheel, we expect the semantic part (wheel) to appear slightly below the position of VCv . Motivated by this, for each (VCv , SPs ) pair we learn a neighborhood function Nv,s indicating the most likely spatial relationship between VCv and SPs . This is illustrated in figure (14). To learn the neighborhood we measured the spatial statistics for each (VCv , SPs ) pair. For each annotated ground-truth position q ∈ Ts+ we computed the position p? in its L4 neighborhood which best activates VCv , i.e. p? = arg minp∈N(q) kf (Ip ) − fv k, yielding a spatial offset ∆p = p? − q. We normalize these spatial offsets to yield a spatial frequency map Frv,s (∆p) ∈ [0, 1]. The neighborhood Nv,s is obtained by thresholding the spatial frequency map. For more details see [42]. 27 5.1.2. Probability Distributions, Supporting Visual Concepts and Log-likelihood Scores. We use loglikelihood ratio tests to quantify the evidence a visual concept VCv ∈ V can give for a semantic part SPs ∈ S. For each SPs , we use the positive and negative samples Ts+ and Ts− to compute the conditional distributions: d + + Pr min kf (Ip ) − fv k 6 r \| q ∈ Ts , (5) Fv,s (r) = dr p∈Nv,s(q) (6) F− v,s (r) d − = Pr min kf (Ip ) − fv k 6 r \| q ∈ Ts . dr p∈Nv,s(q) We call F+ v,s (r) the target distribution, which specifies the probably activity pattern for VCv if there is a semantic part SPs nearby. If (VCv , SPs ) is a good pair, then the probability F+ v,s (r) will be peaked close to r = 0, (i.e., there will be some feature vectors within Nv,s (q) that cause VCv to activate). The second distribution, F− v,s (r) is the reference distribution which specifies the response of the feature vector if the semantic part is not present. The evidence is provided by the log-likelihood ratio [1]: (7) Λv,s (r) = log F+ v,s (r) + ε F− v,s (r) + ε , where ε is a small constant chosen to prevent numerical instability. In practice, we use a simpler method to first decide which visual concepts are best suited to detect each semantic part. We prefer (VCv , SPs ) pairs for which F+ v,s (r) is peaked at small r. This corresponds to visual concepts which are fairly good detectors for the semantic part and, in particular, those which have high recall (few false negatives). This can be found from our studies of the visual concepts described in section (4). For more details see [42]. For each semantic part we select K visual concepts to vote for it. We report results for K = 45, but good results can be found with fewer. For K = 20 the performance drops by only 3%, and for K = 10 it drops by 7.8%. 5.2. Combining the Evidence by Voting The voting process starts by extracting CNN features on the pool-4 layer, i.e. {fp }) for p ∈ L4 . We compute the log-likelihood evidence Λv,s for each (VCv , SPs ) at each position p. We threshold this evidence and only keep those positions which are unlikely to be false negatives. Then the vote that 28 a visual concepts gives for a semantic part is obtained by a weighted sum of the loglikelihood ratio (the evidence) and the spatial frequency. The final score which is added to the position p + ∆p is computed as: (8) Votev,s (p) = (1 − β) Λv,s (Ip ) + β log Fr(∆p) . U The first term ensures that there is high evidence of VCv firing, and the second term acts as the spatial penalty ensuring that this VCv fires in the right relative position Here we set β = 0.7, and define log Fr(∆p) = −∞ when U Fr(∆p) = 0, and U is a constant. U In order to allow the voting to be robust to occlusion we need this ability to automatically “switch off” some votes if they are inconsistent with other votes. Suppose a semantic part is partially occluded. Some of its visual concepts may still be visible (i.e. unoccluded) and so they will vote correctly for the semantic part. But some of the visual concepts will be responding to the occluders and so their votes will be contaminated. So we switch off votes which fall below a threshold (chosen to be zero) so that failure of a visual concept to give evidence for a semantic part does not prevent the semantic part from being detected – i.e. absence of evidence is not evidence of absence. Hence a visual concept is allowed to support the detection of a semantic part, but not allowed to inhibit it. The final score for detecting a semantic part SPs is the sum of the thresholded votes of its visual concepts: (9) Scores (p) = X max {0, Votev,s (p)}. VCv ∈Vs In practical situations, we will not know the size of the semantic parts in the image and will have to search over different scales. For details of this see [42]. Below we report results if the scale is known or unknown. 5.3. Experiments for Semantic Part Detection with and without Occlusion We evaluate our compositional voting model using the VehicleSemanticParts and the VehicleOcclusion datasets. For these experiments we use a tougher, and more commonly used, evaluation criterion [8], where a detected semantic part is a true-positive if itsIntersection-over-Union (IoU) ratio between the groundtruth region (i.e. the 100 × 100 box centered on the semantic part) 29 Unknown Scale Object S-VCM SVM-VC FR airplane 10.1 18.2 45.3 bicycle 48.0 58.1 75.9 bus 6.8 26.0 58.9 car 18.4 27.4 66.4 motorbike 10.0 18.6 45.6 train 1.7 7.2 40.7 mean 15.8 25.9 55.5 Known Scale VT S-VCM SVM-VC 30.6 18.5 25.9 77.8 61.8 73.8 58.1 27.8 39.6 63.4 28.1 37.9 53.4 34.0 43.8 35.5 13.6 21.1 53.1 30.6 40.4 VT 41.1 81.6 60.3 65.8 58.7 51.4 59.8 Table 3: Detection accuracy (mean AP, %) and scale prediction loss without occlusion. Performance for the deep network FR is not altered by knowing the scale. and the predicted region is greater or equal to 0.5, and duplicate detection is counted as false-positive. In all experiments our algorithm, denoted as VT, is compared to three other approaches. The first is single visual concept detection S-VCm (described in the previous section). The others two are baselines: (I) FasterRCNN, denoted by FR, which trains a Faster-RCNN [31] deep network for each of the six vehicles, where each semantic part of that vehicle is considered to be an “object category”. (II) the Support Vector Machine, denoted by SVM-VC, used in the previous section. Note that the Faster-RCNN is much more complex than the alternatives, since it trains an entire deep network, while the others only use the visual concepts plus limited additional training. We first assume that the target object is not occluded by any irrelevant objects. Results of our algorithm and its competitors are summarized in Table 3. Our voting algorithm achieves comparable detection accuracy to Faster-RCNN, the state-of-the-art object detector, and outperforms it if the scale is known. We observe that the best single visual concept S-VCM performs worse than the support vector machine SVM-VC (consistent with our results on keypoints), but our voting method VT performs much better than the support vector machine. Next, we investigate the case that the target object is partially occluded using the VehicleOcclusion dataset. The results are shown in Table 4. Observe that accuracy drops significantly as the amount of occlusion increases, but our compositional voting method VT outperforms the deep network FR for all levels of occlusion. Finally we illustrate, in figure (15), that our voting method is able to detect semantic parts using context, even when the semantic part itself is 30 2 Occluders 0.2 6 r < 0.4 Object FR VT airplane 26.3 23.2 bicycle 63.8 71.7 bus 36.0 31.3 car 32.9 35.9 motorbike 33.1 44.1 train 17.9 21.7 mean 35.0 38.0 3 Occluders 0.4 6 r < 0.6 FR VT 20.2 19.3 53.8 66.3 27.5 19.3 19.2 23.6 26.5 34.7 10.0 8.4 26.2 28.6 4 Occluders 0.6 6 r < 0.8 FR VT 15.2 15.1 37.4 54.3 18.2 9.5 11.9 13.8 17.8 24.1 7.7 3.7 18.0 20.1 Table 4: Detection accuracy (mean AP, %) when the object is partially occluded and the scale is unknown. Three levels of occlusion are considered. occluded. This is a benefit of our algorithm’s ability to use evidence flexibly and adaptively. 6. Discussion: Visual Concepts and Sparse Encoding Our compositional voting model shows that we can use visual concepts to develop a pattern theoretic models which can outperform deep networks when tested on occluded images. But this is only the starting point. We now briefly describes ongoing work where visual concepts can be used to provide a sparse encoding of objects. The basic idea is to encode an object in terms of those visual concepts which have significant response at each pixel. This gives a much more efficient representation than using the high-dimensional feature vectors (typically 256 or 512). To obtain this encoding we threshold the distance between the feature vectors (at each pixel) and the centers of the visual concepts. As shown in figure (16) we found that a threshold of Tm = 0.7 ensured that most pixels of the objects were encoded on average by a single visual concept. This means that we can approximately encode a feature vector f by specifying the visual concept that is closest to it, i.e. by v̂ = arg minv \|f − fv \| – provided this distance is less than a threshold Tm . More precisely, we can encode a pixel p with feature vectors fp by a binary feature vector bp = (bp1 , ..., bpv , ...) where v indexes the visual concepts and bpv = 1 if \|fp − fv \| < Tm and bpv = 0 otherwise. In our neural network implementation, the bpv correspond to the activity of the vc-neurons (if soft-max is replaced by hard-max). 31 Object: car Actual Case: LOW Occlusion Object: car Actual Case: LOW Occlusion Object: car Actual Case: LOW Occlusion SP: wheel Prediction: LOW Occlusion SP: wheel Prediction: LOW Occlusion SP: wheel Prediction: MOD Occlusion 33 Supporting Visual Concepts 35 Supporting Visual Concepts 22 Supporting Visual Concepts Average Distance: 14.5703 (Low) Average Distance: 19.2344 (Low) Average Distance: 44.4923 (Moderate) Object: car Actual Case: HIGH Occlusion Object: car SP: h-light Prediction: HIGH Occlusion SP: h-light Actual Case: HIGH Occlusion Object: car Prediction: HIGH Occlusion SP: h-light Actual Case: HIGH Occlusion Prediction: LOW Occlusion 25 Supporting Visual Concepts 22 Supporting Visual Concepts 30 Supporting Visual Concepts Average Distance: 73.6239 (High) Average Distance: 75.9826 (High) Average Distance: 22.1090 (Low) Figure 15: This figure shows detection from context: when there are occlusions, the cuesmay come from context which is far away from the part There are several advantages to being able to represent spatial patterns of activity in terms of sparse encoding by visual concepts. Instead of representing patterns as signals f in the high-dimensional feature space we can give them a more concise symbolic encoding in terms of visual concepts, or by thresholding the activity of the vc-neurons. More practically, this leads to a natural similarity measure between patterns, so that two patterns with similar encodings are treated as being the same underlying pattern. This encoding perspective follows the spirit of Barlow’s sparse coding ideas [3] and is also helpful for developing pattern theory models, which are pursuing in current research. it makes it more practical to learning generative models of the spatial structure of image patterns of an object or a semantic part Ss . This is because generating the binary vectors {b} is much easier than generating high-dimensional feature vectors {f }. ItQ can be down, for example, by a simply factorized model P ({bp }\|SPs ) = p P (fp \|SPs ), Q where P (fp \|SPs ) = v P (bv,p \|SPs ), and bv,p = 1 if the v th vc-neuron (or visual concept) is above threshold at position p. It also yields simple distance measures between different image patches which enables one-shot/few-shot 32 Figure 16: This figure shows there is a critical threshold Tm , at roughly 0.7, where most pixels are encoded (i.e. activate) a small number of visual concepts. Below this threshold most pixels have no visual concepts activated. By contrast, above the threshold three of more visual concepts are activated. These results are shown for level four features on the cars (left panel), bikes (center panel) and trains (right panel) in VehicleSemanticPart, and we obtain almost identical plots for the other objects in VehicleSemanticPart. learning, and also to cluster image patches into different classes, such as into different objects and different viewpoints. 7. Conclusion This paper is a first attempt to develop novel visual architectures, inspired by pattern theory, which are more flexible and adaptive than deep networks but which build on their successes. We proceeded by studying the activation patterns of deep networks and extracting visual concepts, which both throws light on the internal representations of deep networks but which can also be used as building blocks for compositional models. We also argued that the complexity of vision, particularly due to occlusions, meant that vision algorithms should be tested on more challenging datasets, such as those where occlusion is randomly added, in order to ensure true generalization. Our studies showed that compositional models outperformed deep networks when both were trained on unoccluded datasets but tested when occlusion was present. 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A para-model agent for dynamical systems arXiv:1202.4707v6 [math.OC] 11 Mar 2018 * 10 years of model-free control methodology * Loı̈c MICHEL Abstract Consider a dynamical system u 7→ x, ẋ = fnl (x, u) where fnl is a nonlinear (convex or nonconvex) function, or a combination of nonlinear functions that can eventually switch. We present, in this preliminary work1 , a generalization of the standard model-free control, that can either control the dynamical system, given an output reference trajectory, or optimize the dynamical system as a derivative-free optimization based ”extremumseeking” procedure. Multiple applications2 are presented and the robustness of the proposed method is studied in simulation. 1 This work is distributed under CC license http://creativecommons.org/licenses/ by-nc-sa/4.0/. Email of the corresponding author : loic.michel54@gmail.com 2 The control of the Epstein frame (described in §3.4) has been experimentally validated and the results have been presented at the French Symposium of Electrical Engineering in Grenoble, Jun. 2016 http://sge2016.sciencesconf.org/. Contents 1 Introduction 1 2 General Principle 2.1 Definition of the closed-loop . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definition of the PMA algorithm . . . . . . . . . . . . . . . . . . . . 2.3 Performances in simulation . . . . . . . . . . . . . . . . . . . . . . . . 1 2 2 4 3 Applications of the Cπ -control 3.1 Motor control in the dq frame . . 3.2 Control of switched non-minimum 3.3 Ballistic and the fire control . . . 3.4 Control of the HIV-1 model . . . 3.5 Control of the Epstein frame . . . . . . . . 6 6 9 16 25 28 4 Derivative-free & ”extremum-seeking” control 4.1 Proposed Cπ -control scheme . . . . . . . . . . . . . . . . . . . . . . . 4.2 Numerical applications . . . . . . . . . . . . . . . . . . . . . . . . . . 35 35 36 5 Concluding remarks 37 2 . . . . . . . . and minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . phase systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction Based on the model-free control methodology, originally proposed by Fliess & Join [1] [2] [3] ten years ago, which is referred to as a self-tuning controller in [4] and which has been widely and successfully applied to many mechanical and electrical processes, the para-model agent (PMA) aims to generalize the model-free control by not only controlling nonlinear system, but also performing an ”extremum-seeking” control. On the one hand, we studied the dynamic performances when controlling generic switched minimum phase, non-minimum phase systems (e.g. [5] [6] [7] [8] [9] [10] [11]) as well as the control of some nonlinear systems taken from applications in physics. On the other hand, we present how the PMA can be used to find the optimum of some classes of nonlinear functions. The proposed para-model agent3 is a simple derivation of the discrete model-free control law [12]. The last progresses result in two contributions: first, the substitution of the computation of the numerical derivatives in the original model-free control approach [3], by an initialization function that makes the controller more robust when stabilizing for example switched processes4 . Then, we propose to extend the properties of the model-free controller to include the extremum seeking control of nonlinear systems without any computation of derivative or gradient. In this case, instead of tracking a working point of nonlinear systems, an appropriate choice of the output reference may stabilize nonlinear systems to their extremum. The paper is structured as follows. Section 2 presents the general principle of the proposed para-model agent. In Section 3, some examples illustrate the control of generic switched linear systems, the control of a three-phase motor, the control of a ballistic fire, the control of a biological system and the control of the measure of magnetic hysteresis in the framework of magnetic materials characterization (this last application is also referred to as the control of nonlinear switched systems). Section 4 presents how the proposed PMA approach can be used as derivative-free optimization / ”extremum-seeking” control. 2 General Principle We consider a nonlinear SISO dynamical system to control: ẋ = fnl (x, u) u 7→ y, y = Cx (1) where fnl is a nonlinear system, the para-model agent is an application (y ∗ , y) 7→ u whose purpose is to control the output y of (1) following an output reference y ∗ . 3 A justification of the proposed name ”para-model” is given in the note of the conclusion. The first steps toward the elaboration of the proposed algorithm were to extend the capabilities of the model-free control regarding the control of switched non-minimum phase systems [13]. 4 1 In simulation, the system (1) is controlled in its ”original formulation” without any modification / linearization. 2.1 Definition of the closed-loop Consider the control scheme depicted in Fig. 1 where Cπ is the proposed PMA controller. Figure 1: Proposed PMA scheme to control or optimize a nonlinear system. 2.2 Definition of the PMA algorithm For any discrete moment tk , k ∈ N∗ , one defines the discrete controller Cπ such that symbolically: Cπ : (y, y ∗ ) 7→ uk = Z 0 t R2 → R i Ki εk−1 d τ uk−1 + Kp (kα e−kβ k − yk−1 ) {z } \| k−1 (2) uik where: y ∗ is the output reference trajectory; Kp and KI are real positive tuning gains; εk−1 = yk∗ − yk−1 is the tracking error; uik = uik−1 + Kp (kα e−kβ k − yk−1 ) is an internal recursive term where kα e−kβ k − yk−1 is the associated exponential tracking error in which kα e−kβ k is an initialization function where kα and kβ are real constants; practically, the integral part is discretized using e.g. Riemann sums. The internal recursion5 on uik is defined such as: uik = uik−1 + Kp (kα e−kβ k − yk−1 ). The set of Cπ -parameters of the controller, that needs to be adjusted by the user, is defined as the set of coefficients {Kp , Ki , kα , kβ }. 5 We refine the definition of the PMA algorithm, for which we aim to optimize the construction; in particular, further investigations concern the study of a direct recursion on uk taking into account the Ki -integration and thus comparing internal recursion (involving uik , uik−1 ) vs external recursion (involving uk , uik−1 )... 2 Practical algorithm A possible algorithmic implementation of the simple definition (2) is given below (for all ii ∈ N∗ ): y_int(ii) = k_alphaexp(-k_betaii); % exp. init. function para_exp_err = y_int(ii-1) - y(ii-1); % exp. tracking error para_stand_err(ii) = y_ref(ii) - y(ii-1); % stand. tracking error para_u(ii) = para_u(ii-1) + Kppara_exp_err; % internal recursion para_G(ii) = Kintpara_stand_err(ii); % def. of the integral part para_tr(ii) = para_tr(ii-1) + h(para_G(ii) + para_G(ii-1))/2; % trapezoidal integration para_u_output = para_u(ii)para_tr(ii); % final product (integrator X internal recursion) where: • ii is the index of the sample in the (optional) vectorized process; • exp is the exponential function; • para exp err is the exponential tracking error; • para stand err is the (standard) output tracking error; • para u is the ”internal” recursion; • para G constitutes the discrete integrator; • para u output is the output of the controller that corresponds to the final product between the discrete integrator and the internal recursive function. and k alpha, k beta, Kp and Kint are real constants. Remark The proposed PMA algorithm could been seen as a ”deformed” integrator Rt since the internal recursion uik multiplies directly the integrator 0 Ki εk−1 d τ . 3 2.3 Performances in simulation 2.3.1 Optimization of the closed-loop Optimizing the performances in simulation means that we want to solve the problem of finding the most appropriate set of Cπ -parameters relating to the minimization of some performances index 6 , that may quantify the dynamical performances of the closed-loop. This problem is thus equivalent to an optimization problem for which any optimization solver can be a priori used. In particular, meta-heuristic solvers or derivative-free optimization solvers are preferred due to the pretty complexity of the closed-loop nonlinear form (in general). We are interested in using the ”Brute Force Optimization” (BFO) solver [14] that is very convenient and efficient to use. Figure 2 illustrates a closed-loop first order system, whose the controlled transient has been BFO-optimized in Fig. 3. Figure 2: Simulation with a set of Cπ -parameters arbitrary fixed to ensure at least asymptotic stability. 2.3.2 Sobol-based sensibility of the controlled transient To investigate the interactions between the Cπ -parameters that influence the minimization of the performances index, we propose a preliminary study of the sensibility of the Cπ -parameters using the Sobol index methodology [15]. Consider a controlled nonlinear system, for which the ISE index is evaluated under strict conditions w.r.t. 6 The classical performance index that are available are IAE, ISE, ITAE, and ITSE. see e.g. (http://www.mathworks.com/matlabcentral/fileexchange/ 18674-learning-pid-tuning-iii--performance-index-optimization/content/html/ optimalpidtuning.html) for a quick review (in the context of PID tuning). 4 Figure 3: Simulation with a set of Cπ -parameters BFO-optimized to minimize a transient performance index. the management of the Cπ -parameters7 , the Sobol-based sensibility is evaluated only during the initialization / transient of the closed-loop. Figure 4 shows that a priori the coefficient kβ does not influence the dynamical performances during the transient. Figure 4: Sobol-based analysis of the ISE index during the closed-loop initialization. 7 case 1 : the value of the evaluated index is bounded to 100 and a tolerance of ±10% is permitted on the Cπ -parameters; case 2 : the value of the evaluated index is not bounded and a tolerance of ±50% is permitted on the Cπ -parameters. The results are very similar between the two cases and the Sobol index for {Kp , Ki , kα , kβ } are respectively {0.3324, 0.3318, 0.3326, 0.0002}. 5 3 3.1 Applications of the Cπ -control Motor control in the dq frame Consider a three-phase motor driven in the dq frame; the motor is supplied by a e voltage source rotating at an ω angular frequency and modeled by a simple RC circuit with an additional voltage source ed that acts as an (internal) disturbance. Figure 5 depicts the proposed model (a single phase is represented) where P (θ) and P i (θ) are respectively the Park and the inverse Park transform. The purpose is to control simultaneously the d and q axis with an a priori unknown disturbance ed considering also that the angular frequency ω is increasing according to the time. Figure 5: Model of the motor in the dq frame including an explicit disturbance ed . The disturbance ed is of the general form:   ed1 = A1 (t) sin(ωd (t)t) ed2 = A2 (t) sin ωd (t)t − ed :=  ed3 = A3 (t) sin ωd (t)t + 2π 3 2π 3 (3) where the amplitude Ai and the angular frequency ωd of each phase i could depend on the time. The control structure is composed of two Cπ controllers: the d axis is ”maintained” close to zero (d∗ denotes the output ref. and dmes , the controlled output) and the q axis tracks a specific reference (q ∗ denotes the output ref. and qmes , the controlled output). The following figures illustrate some cases with different ”behavior” of the disturbance ed . Figure 6 presents the most simple case where A1 = A2 = A3 = Cst and ωd = Cst; in Fig. 7 is considered a time-varying disturbance where A1 = A2 = A3 are increasing according to the time; in Fig. 8, small variations of amplitudes of A1 , A2 and A3 are considered (symbolically, A1 ∼ A2 ∼ A3 ), and finally, Fig. 9 depicts the case where ωd is time-varying only over a short period of time. 6 Figure 6: Control in the dq frame with ”simple” disturbance ed . Figure 7: Control in the dq frame with an increasing amplitude of each component of ed . 7 Figure 8: Control in the dq frame with variation of amplitude of each component of ed . Figure 9: Control in the dq frame with variation of the ωd frequency of ed . 8 3.2 Control of switched non-minimum and minimum phase systems Consider the set Σ of stable linear systems such that Σ = {Σi }, i = 1, 2, · · · n, which are minimum and non-minimum phase systems, and which are considered as unknown in the sense that no explicit model has been identified for control purposes. Assume now that for all systems, there exists an integer p = {1, · · · , 8}, called the switching index, such that during a short time window, we have: ẋ(t) = Ap x(t) + Bp u(t) Σp (u 7−→ y) := (4) y = Cp x(t) where u and y are respectively the input and the output of the system Σp (p is the switching index). The step responses of these p systems are presented Fig. 10. Figure 10: Step responses of each system Σi . Figures 11 12 13 14 present some examples of the application of the Cπ -control under different arbitrary switching sequences that involve both minimum and non-minimum phase systems. The first switching time is t1 , the second is t2 and the third is t3 . Consider now the existence of a delay τ on y that modify (4) such that: ẋ(t) = Ap x(t) + Bp u(t) Σp (u 7−→ y) := (5) y = Cp x(t − τ ) This delay can e.g. simulate the propagation delay inside a sensor network. Figures 15 and 16 present two examples of the application of the Cπ -control under different switching sequences that involve both minimum and non-minimum phase systems. 9 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 11: Switched sequence #1 for t1 = 0.01 s and t2 = 0.05 s. 10 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 12: Switched sequence #2 for t1 = 0.025 s and t2 = 0.072 s. 11 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 13: Switched sequence #3 for t1 = 0.018 s, t2 = 0.035 s and t3 = 0.072 s. 12 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 14: Switched sequence #4 for t1 = 0.35 s, t2 = 0.58 s. 13 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 15: Switched sequence #5 for t1 = 0.015 s, t2 = 0.055 s. An addition of a time-delay occurs at t = 0.06 s. 14 (a) Step responses of each system Σi . (b) Controlled switched sequence. Figure 16: Switched sequence #6 for t1 = 0.025 s, t2 = 0.072 s. An addition of a time-delay occurs at t = 0.06 s. 15 Discussions These results have been obtained with a single specific set of the Cπ -parameters; to improve the tracking performances, one may consider an on-line adjustment of the Cπ -parameters. Although resonances occur at the instants of switches, the stability of the control is preserved (regarding the studied cases) when switching from the different types of systems. In the same manner, a direct tuning of the Cπ -parameters could damp (ideally, could cancel) the resonant effects. The presented simulation results show that the proposed control law is robust to ”strong” model variations and in particular when the model is a switching nonminimum phase or minimum phase system that include eventually time-delay. Moreover, the proposed control law seems to have the same properties than the original model-free control [2] [3] for which its performances have been successfully verified especially in simulation. 3.3 Ballistic and the fire control If one fire a projectile at an initial angle and an initial speed, then general physics allows calculating how far it will travel... but would it be possible to control the initial speed needed in such manner that the projectile reaches a precise distance? That’s what we propose to do using the Cπ -control. 3.3.1 Ballistic simplified model We define first a simple model of the trajectory w(t) = (wx (t), wz (t)) of a projectile of mass m in the usual frame of reference (Oxz) fired with an initial speed magnitude v0 that makes a fire angle θ with the horizontal reference i.e. v(t) = v0 cos(θ)ex + v0 sin(θ)ez . The origin (0, 0) of the frame reference is considered as the initial position of the projectile. Denote a the acceleration vector and v the speed vector of the projectile. From Newton law, considering the action of the gravity g and the air resistance cv2 , we have: m d2 w(t) = ma = mg − cv2 d t2 (6) with the initial conditions: d wx (t) dt d wz (t) dt = v0 cos θ, (0,0) = v0 sin θ (0,0) Considering no air resistance i.e. c = 0, (6) is simplified: 16 (7)      m d wx (t) =0 dt (8)     m d wz (t) = −mg dt whose solution reads:    wx (t) = v0 cos θ t + cte (9)   wz (t) = v0 sin θ t − 1 gt2 + cte 2 From (9), the range (or the target) of the projectile xd at z = 0 can be easily deduced. We have: d= 3.3.2 2v02 cos θ sin θ . g (10) Ballistic-fire control methodology Proposed strategy Consider by hypothesis that a ”virtual” trajectory w∗ of the projectile hits a target x∗d from an initial speed v0∗ and an initial fire angle θ∗ . Consider now the ”true” projectile to fire with a trajectory w. To hit the target x∗d (at z = 0) from a small initial speed vε , one controls the projectile in such manner that the projectile reaches quickly the initial speed v0∗ and the initial fire angle θ∗ required to hit the specified target x∗d according to (10). During such ”launching” phase, that we define as the ”launching” distance ∆x0 for which the trajectory of the projectile is fully controlled, we start from an initial condition that prevents the projectile to reach its target x∗d i.e. : d wx (t) dt d wz (t) dt = vε cos θε , (0,0) = vε sin θε (11) (0,0) where vε < v0∗ and θε ≤ θ∗ are positive resp. initial speed magnitude and fire angle. Figure 17 illustrates simulation examples of a ”virtual” trajectory (subj. to v0∗ and θ∗ ) and a ”true” trajectory (subj. to vε and θ∗ ) that is not controlled; the simulations of the trajectories considering c 6= 0 are presented in Fig. 17(b). The goal is to accelerate the projectile using a specific mechanical device in such manner that the references v0∗ and θ∗ are reached quickly. We consider therefore controlling the trajectory w(t) of the projectile over the launching distance ∆x0 . 17 (a) Case c = 0 (b) Case c = 0 and c 6= 0 Figure 17: Example of comparison of the uncontrolled ”true” projectile (subj. to vε and θ∗ ) and the ”virtual” trajectory (subj. to v0∗ and θ∗ ), considering c = 0 and c 6= 0. We took c = 0.05, θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 18 Controllable ballistic model To simulate a controllable model of the trajectory w of the projectile, we consider adding an external acceleration force in (6) that represents the mechanical action of the specific mechanical device over the distance ∆x0 . We have: d2 w(t) = mg − cv2 + aext (12) d t2 ext where aext (t) = (aext x (t), az (t)) is equivalent to the external force provided by the mechanical device. Since such device acts only over ∆x0 , then we assume that aext = 0 for all x > ∆x0 . m 3.3.3 Implementation of the Cπ -controller A possible control scheme is to consider controlling the trajectory w that must be ”as close as possible” to the reference w∗ over the distance ∆x0 . Therefore, w is physically measured and the external acceleration aext is driven by the Cπ -controller, through the specific mechanical device. We build a closed-loop that creates a feedback between (2) and (12). We have ”symbolically”, for all x ≤ ∆x0 :  Z t  ext ∗   uk = ak = Ki (wk−1 − wk−1 )d τ    0         m uik−1 + Kp (αe−βk − wk−1 ) {z } k−1 \| uik (13) d2 w(t) = mg − cv2 + uk d t2 Determination of the distance ∆x0 ∗ Since we expect that the projectile is fired from ∆x0 with a speed that is very close to v0∗ (and follows, via the Cπ -control, the same trajectory i.e. w ≈ w∗ over ∆x0 ), then, we propose a possible definition of the theoretical launching distance ∆x∗0 , (considered only over the x axis) that corresponds to the solution in wx of: d wx (t) = v0∗ x (14) dt Geometrically, the theoretical launching distance ∆x∗0 is associated to the speed vx that is reached by the projectile (launched with the initial speed vε < v0∗ ) when vx is close to v0∗ x . 19 3.3.4 Numerical simulations Case ∆x0 > ∆x0 ∗ Consider the simulated ”virtual” trajectory, presented in Fig. 17, as the control reference w∗ ; to simplify, we consider θ as constant. Figure 18 presents the case where the fire is controlled considering c = 0 over ∆x0 = 0.11 m. In particular, Fig. 18(a) presents, at the top, the evolution of the controlled trajectory w in comparison with the reference w∗ , and, at the bottom, the calculated speed d wx /d t in comparison with the initial speed v0 cos θ. Figure 18(b) presents the complete ”true” controlled trajectory in comparison with the virtual trajectory. Figure 19 presents the same simulations in the case c 6= 0. Case ∆x0 ∼ ∆x0 ∗ Consider the simulated ”virtual” trajectory, presented in Fig. 17, as the control reference w∗ ; to simplify, we consider θ as constant. Figure 20 presents the case where the fire is controlled considering c = 0 over ∆x∗0 . In particular, Fig. 20(a) presents, at the top, the evolution of the controlled trajectory w in comparison with the reference w∗ , and, at the bottom, the calculated speed d wx /d t in comparison with the initial speed v0 cos θ. Figure 20(b) presents the complete ”true” controlled trajectory in comparison with the virtual trajectory. Figure 21 presents the same simulations in the case c 6= 0. Discussions These results have been obtained using the same set of the Cπ -parameters. The properties of stabilization of the control law, like in the previous case when dealing with switching systems (§3.2) , seem to be preserved and ensure good tracking performances in particular when considering c = 0 and c 6= 0. Further generalizations would allow using multiple and parallel Cπ -controllers in order to control simultaneous physical quantities. In particular, the speed profile v could be controlled simultaneously with the trajectory w... 20 (a) At the top, controlled trajectory w relating to the reference w∗ ; at the bottom, calculated speed d wx /d t relating to the initial speed v0 cos θ. (b) Complete controlled ”true” trajectory in comparison with the virtual trajectory. Figure 18: Example of controlled trajectory considering c = 0 over ∆x0 = 0.11 m. We took θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 21 (a) At the top, controlled trajectory w relating to the reference w∗ ; at the bottom, calculated speed d wx /d t relating to the initial speed v0 cos θ. (b) Complete controlled ”true” trajectory in comparison with the virtual trajectory. Figure 19: Example of controlled trajectory considering c 6= 0 over ∆x0 = 0.11 m. We took c = 0.05, θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 22 (a) At the top, controlled trajectory w relating to the reference w∗ ; at the bottom, calculated speed d wx /d t relating to the initial speed v0 cos θ. (b) Complete controlled ”true” trajectory in comparison with the virtual trajectory. Figure 20: Example of controlled trajectory considering c = 0 over ∆x∗0 . We took θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 23 (a) At the top, controlled trajectory w relating to the reference w∗ ; at the bottom, calculated speed d wx /d t relating to the initial speed v0 cos θ. (b) Complete controlled ”true” trajectory in comparison with the virtual trajectory. Figure 21: Example of controlled trajectory considering c 6= 0 over ∆x∗0 . We took c = 0.05, θε = θ∗ = π/3, v0 = 10 m/s and vε = 9 m/s. 24 3.4 Control of the HIV-1 model The problem is to control the predator-prey like model that describes the evolution of the HIV-1 when subjected to an external ”medical agent”. From a mathematical point of view, we study the possibility of controlling the model (15) for which the purpose is to control the output y (corresponding to the viral load) using the double inputs u1 and u2 in such manner that y converges rapidly to zero 8 [16] [17].    ẋ1 = s − dx1 − (1 − u1 )βx1 x3  ẋ2 = (1 − u1 )βx1 x3 − µx2 (15) ẋ3 = (1 − u2 )kx  2 − cx3   y= 0 0 γ x where (mathematically) : d = 0.02, k = 100, s = 10, β = 2.4.10−5 , µ = 0.24, c = 2.4. γ is a scaling factor that allows normalizing the output. Figure 22 presents the evolution of the output y in open-loop when u1 = u2 = 0 i.e. when no medical drug is considered. Figures 23 and 24 illustrate the control of y considering two different ratios between u1 and u2 . Figure 22: Transient response of y considering u1 = u2 = 0. 8 Since we are trying to control this model only from the mathematical point of view, we do not take into account the constraints that are medically imposed. 25 (a) Output y (b) Input u Figure 23: Controlled output y (viral load) in correspondence with u1 = u2 = u. 26 (a) Output y (b) Input u Figure 24: Controlled output y (viral load) in correspondence with u1 = u2 = 12 u. 27 3.5 Control of the Epstein frame The Epstein frame (see Fig. 25 9 ) aims to characterize a magnetic material by determining its B − H hysteresis curve. The principle is to create a magnetic field H inside the material using a ”magnetizing” current iH . The material gives a response to the field H that physically corresponds to the measurable magnetic induction field B. This B field creates a voltage vB through magnetic induction and the quantities vB and iH is a representation of the magnetic hysteresis curve B − H. To describe experimentally the major B − H hysteresis loop, the material under study has to be magnetized using a current iH that is alternative and of enough magnitude in order to describe the magnetic behavior at saturation. The purpose of the control law implementation is to control iH such that the output voltage of the Epstein frame vB remains ”as close as possible” to a desired reference waveform. Figure 25: An experimental Epstein frame to characterize magnetic materials. 3.5.1 Epstein frame control Proposed Cπ -control scheme Consider the control scheme depicted in Fig. 26 where Cπ is the proposed PMA controller. Kin and Kout are positive real gains. Denote fBH the numerical Jiles-Atherton model that is associated to the magnetic hysteresis B − H and fJA is the complete hysteresis to control. 9 Picture taken from Wikipdia http://upload.wikimedia.org/wikipedia/commons/5/51/ Epstein_frame.jpg. 28 Figure 26: Proposed PMA scheme to control the electrical waveforms measured from a magnetic hysteresis. Jiles-Atherton based hysteresis model The Jiles-Atherton model [18] describes a magnetic hysteresis cycle B − H. It reads: 1 Man − M c d Man dM = + dH 1 + c δk − α(Man − M ) 1 + c d H (16) where c, δk, Man , α are physical coefficients well identified from magnetic hysteresis measurement and we assume that the current iH corresponds to the magnetization H i.e. iH ∝ H and the voltage vB corresponds to the derivative of the magnetic induction field response B = µ0 H + JBH (H) (where JBH describes the B − H hysteresis via (16)). Simplified model of the Epstein frame The Epstein frame admits a complex model based on the Jiles-Atherton model that represents all electric phenomena that occur inside the Epstein frame10 . To simplify the model of the Epstein frame to control, we consider controlling a nonlinear function fJA , which represents a modified Jiles-Atherton model. Denote vH = fJA (iH ) the nonlinear dynamical system that describes the B − H hysteresis as a function of iH , and consider controlling directly the hysteresis cycle in such manner that Cπ controls iH in order to get vH = fJA (iH ) as close as possible to a reference waveform. To define the global fJA hysteresis function that is controlled by Cπ , which includes the scaling coefficients needed by the Cπ corrector, consider iH = Kin u, y = Kout vH and a coefficient Ke such that: y = fJA (u) = Kin Ke u + Kout JBH (Kin u). 10 (17) The Epstein frame is equivalent to a transformer and the ”mutual interactions” between primary and secondary coils must be taken into account in addition to the hysteresis behavior. 29 The small coefficient Ke # 10−4 compensates the very small variations of fBH when B is close to Bsat . Such variations may induce time-delays in the response of the Cπ -controller that induce some distortions of the output signal y. The model (17) could be seen as an ”affine” derivation of the original Jiles-Atherton model. The B − H loops, obtained from the Jiles-Atherton model, are depicted in Fig. 27 considering the frequencies 5 Hz, 50 Hz, 500 Hz, 1 kHz and 10 kHz. Since this hysteresis model does not allow H > Hmax , a limitation is necessary to bound the evolution of H, that may occur eventually during the transient of the dynamic stabilization of the control loop (ex. in Fig. 31). . Figure 27: Simulation of the Jiles-Atherton model for different operating frequencies. 3.5.2 Simulation results Figures 28, 29, 30, 31 and 32 depict the input u and the (rescaled) output y of the control loop according to the time. Given a particular operating frequency, for which a particular H − B hysteresis is studied (see Fig. 27), and assuming that the output reference y ∗ is a sinusoid, whose magnitude corresponds to the theoretical Hmax of the H − B hysteresis, different frequencies are considered (5 Hz, 50 Hz, 500 Hz, 1 kHz and 10 kHz) in order to highlight the behavior of the controlled voltage vb when the frequency changes. In particular, high frequencies (e.g. Figures 31 and 32) introduce an important transient response on y due to the fact that the variations of y ∗ are too fast to get an immediate stabilization to the dynamic working point of the hysteresis. The simulation show that the Cπ -controller gives very interesting 30 dynamic performances over a wide range of dynamic working points relating to the operating frequency. An optimization algorithm (see §2.3) has been used to adjust the parameters of the Cπ -controller in such manner that the shape of the output response y is ”as close to” a sine shape11 . Remarks This hysteresis model is composed of three subsystems (the first magnetization branch, the increasing and decreasing branches) that switch depending on the value of dH/dt. When H << Hmax , the switch between the branches may not be smooth and such ”connection” may induce a small transient on y. An illustration is presented in Fig. 33 at a low frequency in comparison with Fig. 30. The closed-loop has been also tested using a triangular shaped output reference y ∗ . Figure 34 depicts the magnitude of the magnetic field H and the corresponding (rescaled) magnetic induction B during the control loop process according to the time. The frequency of 5 Hz has been considered as an example holding the parameters of the simulation with the sine reference at 5 Hz. Figure 28: Simulated u and y signals according to the time at 5 Hz. 11 Remember that the purpose of the optimization procedure is to minimize the tracking error y − y in such manner that ideally y ≡ y ∗ for the particular sine output reference y ∗ . ∗ 31 Figure 29: Simulated u and y signals according to the time at 50 Hz. Figure 30: Simulated u and y signals according to the time at 500 Hz. 32 Figure 31: Simulated u and y signals according to the time at 1 kHz. Figure 32: Simulated u and y signals according to the time at 10 kHz. 33 Figure 33: Simulated u and y signals according to the time at 500 Hz (H << Hmax ). Figure 34: Simulated H and y signals according to the time at 5 Hz. 34 4 Derivative-free & ”extremum-seeking” control To describe how the PMA could be used as a derivative-free optimization (DFO) algorithm (e.g. [19] [20]) or as an ”extremum-seeking” (ES) control scheme (e.g. [21] [22] [23]), we first define each element of the associated control scheme and then, we derive the operating conditions that would allow to minimize nonlinear functions. We assume that it is possible to derive a control scheme such that the PMA can be used to minimize nonlinear functions. 4.1 Proposed Cπ -control scheme Definition of the closed loop Consider the control scheme depicted in Fig. 35 where Cπ is the proposed PMA ”extremum-seeking” controller. Kin and Kout are s positive real gains. We consider either a static nonlinear function fnl (regarding DFO), which does not have any internal dynamical properties, or a nonlinear SISO dynamical system (1) (regarding ES), to minimize. The function Q is e.g. a basic first order transfer function. Figure 35: Proposed PMA scheme to minimize a nonlinear function fnl . Function to control s • Define the fnl static function to optimize such that: s fnl : Rn → R u0 7→ y or consider a nonlinear SISO dynamical system (1). 35 (18) This function represents the ”nonlinear optimization problem”. Currently, the assumption n ≤ 2 is considered and we denote u0x the input variable for n = 1. • Q is a standard linear transfer function (typically of first order), such that: R→R 0 Q : du dt + γ u0 \|k = uk (19) k where γ is a time-constant, chosen in such manner that the step response of Q s in order to is very fast. As presented in the Fig. 35, Q is associated with fnl s to control. provide some minimal dynamical properties regarding the system fnl Obviously, the Q function is not necessary when fnl is already a dynamical function like (1)12 . A single Cπ -controller13 drives a single input of fnl (eventually through Q), as presented in Fig. 35. 4.2 Numerical applications Since the PMA is designed for nonlinear systems and does not contain any derivatives, it is assumed that the ”extremum-seeking” control is possible considering a specific definition of y ∗ in order to reach and stabilize fnl to its minimum. Let us assume that the following (eventually constrained) minimization problem (described for a single variable): min fnl (x), x∈R (x = u0 identically inside the control scheme) (20) is equivalent to the control scheme described in Fig. 35, for which the output reference y ∗ ”follows” the minimum value of fnl . We denote x = xopt the value that gives the minimum of fnl . Results For each case in 1D, are plotted: the difference between two iterations yk , yk−1 and the error between xopt and the evolution of x through the closed-loop. In these cases, all the parameters of Cπ have been set experimentally to give interesting performances but are not optimal (the choice of the y ∗ function may influence the speed of the convergence). The following numerical cases are studied: 12 Last investigations suggest that the linear transfer function Q may be not necessary even for s a fnl function to control. The properties of the para-model algorithm are currently under study considering nonlinear systems that are ”non-dynamical”. 13 To extend this scheme to multi-input variables (n > 2) of fnl , one may consider the use of a Cπ -controller per input variable. 36 • See Fig. 36 regarding the minimization of a 1D convex function such that: min (x − 30)2 x (21) • See Fig. 37 regarding the minimization of a 1D convex function with a minimum that changes according to the time at an unknown instant t1 such that: t ? 1 min (x − 30)2 → min(x − 40)2 x x (22) • See Fig. 38 regarding the minimization of a 1D non convex function such that: min 10(1.5 cos(x) − x) + exp(x − 5) + 100 subj. to : y ≥ 15x − 60 x,y 5 (23) Concluding remarks We presented how the proposed para-model agent14 , as a model-free and derivativefree based controller, can be used to control nonlinear systems or perform optimization / ”extremum-seeking” control. Further investigations include extensive tests and applications to complex systems as well as a complete study of the stability. Acknowledgement The author is sincerely grateful to Dr. Edouard Thomas for his strong guidance and his valuable comments that improved this paper. The author is also sincerely grateful to Olivier Ghibaudo, Ph.D. student at G2Elab-CNRS (Grenoble) France, for having worked on the numerical version of the hysteresis model. 14 Why ”para-model”? Based on the ”model-free” methodology, for which the model is not correlated to the controller in the sense that the controller does not need an explicit definition of the model to be parametrized and can thus rebuild an ultra-local model from measurements, we propose the prefix ”para” to highlight the fact that the agent is ”in close proximity” to the model of the process that he controls. Although no specific information is needed for the PMA control part, some basic information may be needed to configure properly the PMA and the output reference y ∗ in the case of extremum-seeking control approach. The stability study is an essential part that should formalize the proposed PMA approach in order to justify theoretically the operating conditions of the Cπ -control. 37 Figure 36: min (x − 30)2 (initial condition in red spot). x t ? 1 Figure 37: min fnl2 = (x − 30)2 → min fnl1 = (x − 40)2 (initial condition in red spot). x x 38 Figure 38: min 10(1.5 cos(x) − x) + exp(x − 5) + 100 subj. to : y ≥ 15x − 60 (inix,y tial condition in red spot). References [1] M. Fliess, C. Join, M. Mboup and H. Sira-Ramirez, ”Vers une commande multivariable sans modèle”, Conférence internationale francophone d’automatique (CIFA 2006), France, 2006. (available in french at https://arxiv.org/pdf/math/ 0603155.pdf). [2] M. Fliess and C. 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1 Location-Aided Coordinated Analog Precoding for Uplink Multi-User Millimeter Wave Systems Flavio Maschietti‡ , David Gesbert‡ , Paul de Kerret‡ arXiv:1711.03031v1 [eess.SP] 8 Nov 2017 ‡ Communication Systems Department, EURECOM, Sophia-Antipolis, France Email: {flavio.maschietti, david.gesbert, paul.dekerret}@eurecom.fr Abstract Millimeter wave (mmWave) communication is expected to have an important role in next generation cellular networks, aiming to cope with the bandwidth shortage affecting conventional wireless carriers. Using side-information has been proposed as a potential approach to accelerate beam selection in mmWave massive MIMO (m-MIMO) communications. However, in practice, such information is not error-free, leading to performance degradation. In the multi-user case, a wrong beam choice might result in irreducible inter-user interference at the base station (BS) side. In this paper, we consider locationaided precoder design in a mmWave uplink scenario with multiple users (UEs). Assuming the existence of direct device-to-device (D2D) links, we propose a decentralized coordination mechanism for robust fast beam selection. The algorithm allows for improved treatment of interference at the BS side and in turn leads to greater spectral efficiencies. I. I NTRODUCTION The large bandwidths available at mmWave carrier frequencies are expected to help meet the throughput requirements for future mobile networks [1]. Since smaller wavelength signals are more prone to absorption, mmWave communications require beamforming in order to guarantee appropriate link margins and coverage [2], [3]. To this end, m-MIMO techniques [4] are envisioned as high-gain directional antennas with small form factor can be designed for mmWave usage [5]. However, configuring those massive antennas to operate with large bandwidths entails an additional effort. The high cost and power consumption of the radio components impact on the UEs and small BSs, thus limiting the practical implementation of a fully-digital beamforming architecture [1]. Moreover, the large number of antennas at both ends of the radio links would require unfeasible CSI-training overhead to design the precoders. 2 One step towards simplification consists in replacing the fully-digital architecture with a hybrid analog-digital one [6]–[11]. In mixed analog-digital architectures, a low-dimensional digital processor is concatenated with an RF analog beamformer, implemented through phase shifters. Note that while the latter is sufficient to achieve a good part of the overall beamforming gain – through beam steering towards desired spatial directions – the digital stage is essential when processing multiple streams and users. Interestingly, existing works on hybrid architectures typically ignore multi-user interference issues in the analog domain and cope with them in the digital part. For instance, in [12], a procedure is proposed for the downlink transmission, where the analog stage is intended to find the best beam directions for each UE (regardless of multi-user interference), while the digital one applies the conventional Zero-Forcing (ZF) beamformer on the resulting effective channel. The strength of this approach lies in the fact that it is possible to use the existing beam training algorithms for single-user links – such as [13]–[15] – in the analog stage. Such algorithms have been developed bearing in mind the need for fast link establishment in low-latency applications. Nevertheless, the reduced number of digital chains might not always allow to resolve the residual multi-user interference which remains after the analog beamforming stage. In particular, in a mmWave propagation scenario [2], [3], multiple closely located UEs will likely share some common reflectors, causing an alignment of the main path’s angles of arrival at the BS receiver and preventing it from resolving the interference, even at the digital decoding stage. To solve this problem, a principal idea consists in treating interference before it takes place, i.e. the UE side, as is done for example in [16], [17]. Although showing significant performance advantages over the existing solutions, these works assume perfect CSIR for analog beamforming and single-antenna UEs, which might not be realistic in all mmWave contexts [12]. Rather, we are interested in statistically-driven analog beamforming at the UE TX side. In this paper, we point out that simple analog UE beam selection can be designed so as to enable the analog receive beam on the BS side to discriminate for interference. We propose to do this through the help of low-rate side-information at the UEs. Several works can be found in the mmWave literature, where side-information is exploited to improve performance without burdening overhead. Side-information can be obtained from various sources, such as automotive sensors [18], UHF band [19], GNSS [20], or also past multipath fingerprints measurements [21]. We bring forward the idea that position-based side-information can be exploited in order to develop a coordination mechanism between the UEs, so that the interference at the BS side can 3 be treated efficiently through both the analog and digital parts of the receiver, as opposed to relying on the digital part alone. The main intuition is to use coordination to make sure the selected analog beams at the BS convey the full rank of multi-user channels towards the digital part to preserve invertibility. As in some previous work [22], we are interested in establishing a robust form of coordination which accounts for possible noise in the positioning information made available to the UEs. However, [22] targets a single-user scenario only. In the multi-user scenario, the lack of a realtime communication channel prevents the UEs from exchanging instantaneous CSI. We consider, instead, the existence of a low-rate unidirectional D2D channel, allowing communication of GPS-type data. In particular, we consider a hierarchical set-up where higher ranked UEs receive position information from lower ranked ones. The unidirectional aspect and the limitation to position information exchange help keep the D2D overhead much lower than real-time D2D. Our main contributions read as follows: • We formulate the problem of per-user analog precoding with side position information and recast it as a decentralized beam selection problem. • Our algorithm exploits the hierarchical structure of the information, in order to perform robust (with respect to position data noise) interference mitigation at both analog and digital stages. • Under the proposed method, the UEs coordinate to select beams which, while being suboptimal in terms of average power, help attain the full rank condition needed at the BS for interference suppression. II. S YSTEM M ODEL Consider the single-cell uplink multi-user mmWave scenario in Fig. 1. The BS is equipped with NBS 1 antennas to support K UEs with NUE 1 antennas each. The UEs are assumed to reside in a disk of a given radius rcl , which will be used to control inter-UE average distance. Each UE sends one data stream to the BS. We assume that the BS has NRF = K RF chains available, each one connected to all the NBS antennas, assuming a fully-connected hybrid architecture [1]. The u-th UE precodes the data su ∈ C through the analog precoding vector vu ∈ CNUE ×1 . We assume that the UEs have one RF chain each, i.e. UEs are limited to analog beamforming via phase shifters (constant-magnitude elements) [7]. In addition, E[kvu su k2 ] ≤ 1, assuming normalized power constraints. 4 The reconstructed signal after mixed analog/digital combining at the BS can be expressed as follows – assuming no timing and carrier mismatches: x̂ = K X H H WD WRF Hu vu su + WD WRF n (1) u=1 u NBS ×NUE where H ∈ C is the channel matrix from the u-th UE to the BS and n ∈ CNBS ×1 is the thermal noise vector, with zero mean and covariance matrix σn2 INBS . WRF ∈ CNBS ×NRF is, instead, the analog combining matrix, containing the vectors relative to each of the K RF chains (subject to the same hardware constraints as described above), while WD ∈ CK×NRF denotes the digital combining matrix. The received SINR for the u-th UE at the BS is expressed as follows: H \|wDu WRF Hu vu \|2 γu = P u H 2 w w 2 w6=u \|wD WRF H v \| + σñ (2) where wDu ∈ C1×NRF denotes the row of WD related to the u-th UE (one RF chain for each UE), H n for the filtered thermal noise. and where we used the short-hand notation ñ = WD WRF BS UE 1 rcl UE 2 Fig. 1: Scenario example with L = 3 propagation paths, two reflectors, and K = 2 UEs. The UEs are assumed to reside in a disk of radius rcl , which is relatively common in realistic scenarios, e.g. dense UE distribution in a coffee house. In this illustration, two closely located UEs are sharing some reflectors, and paths reflecting on the top reflector arrive quasi-aligned at the BS while originating from distinct UEs. 5 A. Channel Model Unlike the conventional UHF band propagation environment, the mmWave one does not exhibit rich-scattering [2] and is in fact modeled as a geometric channel with a limited number of dominant propagation paths which survive high attenuation. The UE u is thus subject to the channel matrix Hu ∈ CNBS ×NUE , expressed as the sum of L components or contributions [7]: Hu = NBS NUE L 1/2 X α`u aBS (ϑu` )aHUE (φu` ) (3) `=1 where α`u ∼ CN (0, (σ`u )2 ) denotes the complex gain for the `-th path of the u-th UE. Furthermore, we assume that the variances (σ`u )2 , ` ∈ {1, . . . , L}; u ∈ {1, . . . , K} of the paths are such as P u 2 ` (σ` ) = 1, ∀u ∈ {1, . . . , K}. The variables φu` ∈ [0, 2π) and ϑu` ∈ [0, 2π) are the angles of departure (AoDs) and arrival (AoAs) for each contribution, for a given UE u, where one angle pair corresponds to the LoS direction while other might account for the presence of strong reflectors (e.g. buildings, hills) in the environment. The positions of those points of reflection depend on the position of the considered UE (see Fig. 1). We will denote the reflecting points for the u-th UE with Rui , i ∈ {1, . . . , L − 1} in the rest of the paper. The vectors aUE (φu` ) ∈ CNUE ×1 and aBS (ϑu` ) ∈ CNBS ×1 denote the antenna steering vectors at the u-th UE and the BS for the corresponding AoDs φu` and AoAs ϑu` , respectively. We assume to use ULAs at both sides, so that [23]: h iT 1 −iπ cos(φ) −iπ(NUE −1) cos(φ) aUE (φ) = 1, e ,...,e (NUE )1/2 h iT 1 −iπ cos(ϑ) −iπ(NBS −1) cos(ϑ) aBS (ϑ) = 1, e , . . . , e 1/ (NBS ) 2 (4) (5) B. Codebooks for Analog Beams The most recognized method to implement the analog beamformer is through a network of digitally-controlled phase shifters [24] (refer to [1] for alternative architectures). Thus, the phase of each element of the analog beamformer is limited to fixed quantized values, and therefore, the beamforming vectors need to be selected from a finite set (or codebook). We denote the codebooks used for analog beamforming as: VUE = {v1 , . . . , vMUE }, VBS = {w1 , . . . , wMBS } where VUE is assumed to be shared between all the UEs, to ease the notation. (6) 6 For ULAs, a suitable design for the fixed beamforming vectors in the codebook consists in selecting steering vectors over a discrete grid of angles [12], [14]: vp = aUE (φ̄p ), p ∈ {1, . . . , MUE } (7) wq = aBS (ϑ̄q ), q ∈ {1, . . . , MBS } (8) where the angles φ̄p , ∀p ∈ {1, . . . , MUE } and ϑ̄q , ∀q ∈ {1, . . . , MBS } can be chosen according to different strategies, including regular and non-regular sampling of the [0, π] range [22]. Remark 1. Given the one-to-one correspondence between the beamforming vectors in VBS (resp. VUE ), and the grid angles ϑ̄q , ∀q ∈ {1, . . . , MBS } (resp. φ̄p , ∀p ∈ {1, . . . , MUE }), we will make the abuse of notation q ∈ VBS (resp. p ∈ VUE ) to denote the vector wq ∈ VBS (resp. vp ∈ VUE ). III. I NFORMATION M ODEL In this section, we describe the structure of the channel state- and side-information available at both UEs and BS sides. We start with defining the nature of information in an ideal setting before turning to a realistic (noisy) case. Definition 1. The average beam gain matrix Gu ∈ RMBS ×MUE contains the power level associated with each combined choice of analog beam pair between the BS and the u-th UE after averaging over small scale fading. It is defined as: Guq,p =E αu h 2 wqH Hu vp i (9) where the expectation is carried out over the channel coefficients αu = [α1u , α2u , . . . , αLu ] and with Guq,p denoting the (q, p)-element of Gu . Definition 2. The position matrix Pu ∈ R2×(L+1) contains the two-dimensional location coordinates pun = [punx puny ]T for node n, where n indifferently refers to either the BS, the u-th UE or one of the reflectors Rui , i ∈ {1, . . . , L − 1}. It is defined as follows: h i Pu = puBS puR1 . . . puRL−1 puUE (10) We will denote as P the set containing all the position matrices Pu , ∀u ∈ {1, . . . , K}. As shown in [22], the matrix Gu can be expressed as a function of the matrix Pu . We recall here the deterministic relationship that is found between those two matrices. 7 Lemma 1. We can write the average beam gain matrix relative to the u-th UE as follows: Guq,p (Pu ) L X = (σ`u )2 \|LBS (∆u`,q )\|2 \|LUE (∆u`,p )\|2 (11) `=1 where we remind the reader that (σ`u )2 denotes the variance of the channel coefficients α`u and we have defined: u LUE (∆u`,p ) ei(π/2)∆`,p sin((π/2)NUE ∆u`,p ) = u (NUE )1/2 ei(π/2)NUE ∆`,p sin((π/2)∆u`,p ) LBS (∆u`,q ) ei(π/2)∆`,q sin((π/2)NBS ∆u`,q ) = u (NBS )1/2 ei(π/2)NBS ∆`,q sin((π/2)∆u`,q ) 1 1 (12) u (13) and ∆u`,p = (cos(φ̄p ) − cos(φu` )) (14) ∆u`,q = (cos(ϑu` ) − cos(ϑ̄q )) (15) with the angles φu` , ` ∈ {1, . . . , L} and ϑu` , ` ∈ {1, . . . , L} obtained from the position matrix Pu through simple algebra (refer to [22] for more details). A. Distributed Noisy Information Model In the distributed model, each UE u receives its own estimates of the position matrices Pw , ∀w ∈ {1, . . . , K}. We will use the superscript with parenthesis (u) to denote any information known at the u-th UE. In particular, we denote as P̂w,(u) ∈ R2×(L+1) , ∀w ∈ {1, . . . , K} the local information available at the u-th UE about the position matrix Pw . This information is modeled as follows: P̂w,(u) = Pw + Ew,(u) ∀w ∈ {1, . . . , K} where Ew,(u) denotes the following matrix: h i w,(u) w,(u) w,(u) w,(u) Ew,(u) = eBS eR1 . . . eRL−1 eUE (16) (17) containing the random position errors which the u-th UE made in estimating pw n . Such error comes with an arbitrary, yet known, probability density function few,(u) . n Definition 3. We will denote as P̂ (u) , where: P̂ (u) = {P̂1,(u) , . . . , P̂K,(u) } (18) the overall local information available at the u-th UE containing all the estimated position matrices P̂w,(u) , ∀w ∈ {1, . . . , K}. 8 B. Hierarchical Location-Information Exchange The hierarchical (or nested) model is a sub-case of the distributed model in which the u-th UE has access to the estimates of the UEs u + 1, . . . , K. As we will see in the next section, this information structure enables some coordination for just half of the overhead needed in a conventional two-way exchange mechanism. One consequence in particular is that the u-th UE is able to retrieve the beam decisions carried out at (the less informed) UEs u + 1, . . . , K. C. Additional Information In what follows the number of dominant paths, and their average powers (σ`u )2 , ` ∈ {1, . . . , L}; u ∈ {1, . . . , K} are assumed to be known at each UE, based on prior averaged measurements. Likewise, statistical distributions fenw,(u) , ∀u, w are supposed to be quasi-static and as such are supposed to be available to each UE. In other words, the u-th UE is aware of the amount of error in the position estimates which it and other UEs have to cope with. IV. M ULTI -U SER L OCATION -A IDED H YBRID P RECODING In order to maximize the received SNR γ u defined in (2) for each UE, the mutual optimization of both analog and digital components must be taken into account. A common approach consists in decoupling the design, as the analog beamformer can be optimized in terms of long-term channel statistics, whereas the digital one can be made dependent on instantaneous information [25]. A. Uncoordinated Beam Selection We first review here the approach given in [12], where the authors proposed to design the analog beamformers to maximize the received power for each UE, neglecting multi-user interference. Once the analog beamformers are fixed at both UE and BS sides, the design of the digital beamformer at the BS follows the conventional MU-MIMO approach. In this respect, a common choice is to consider ZF combining. Therefore, the digital beamforming matrix WD is the pseudo-inverse of the effective channel matrix H̃ ∈ CNRF ×K , which is defined as follows [23]: WD = H̃H H̃ −1 H̃H (19) where h i h i H H H H̃ = WRF H1 v1 WRF H2 v2 . . . WRF HK vK , and with WRF = w1 w2 . . . wK . 9 When position and path average power information is available, the beam selection (quun ∈ VBS , pun u ∈ VUE ) at the analog stage of the algorithm proposed in [12] – which we will denote as uncoordinated (un) – can be expressed as follows: u (u) ) = argmax R P̂ , q , p (quun , pun u u , ∀u u (20) qu ∈VBS pu ∈VUE where we have defined the single-user rate Ru as [22]: Guqu ,pu (Pu ) u R (P, qu , pu ) = log2 1 + σn2 (21) Equation (20) can be solved through direct search of the maximum in the matrix Gu , derived from P̂ (u) through (11). While being simple to implement, the information at each UE in this method is treated as perfect, although some Bayesian robustization can be introduced [22]. Another limitation of this approach is that each UE solves its own beam selection problem in a way which is independent of other UEs, thus ignoring the possible impairments in terms of interference. We illustrate this effect in Fig. 2, where we plot the mean rate per UE obtained when the analog precoders are chosen through (20), in case of K = 2 UEs, perfect position information, as a function of rcl . As the inter-UE average distance decreases, the performance of this procedure degrades, since the UEs have much more chance to share common best propagation paths (which results in severe interference at the analog stage at the BS). The action of the ZF is noticeable but not sufficient for small cluster radii. In what follows, we consider different flavors of coordination. B. Naive-Coordinated Beam Selection In order to improve performance, we design the analog precoders according to the following figure of merit, which takes into account the average multi-user interference at the analog stage: R(P, q1:K , p1:K ) = K X u=1 log2 1 + P Guqu ,pu (Pu ) w w 2 w Gqu ,pw (P ) + σn (22) Remark 2. We used here the short-hand q1:K , p1:K to denote the indexes q1 , . . . , qK and p1 , . . . , pK , respectively. The hierarchical model allows the u-th UE to predict the beam selected at UEs u + 1, . . . , K. However, for a full coordination, the u-th UE would also need to know the precoding strategies of the more informed UEs, i.e. UE 1, . . . , u − 1, which involves some guessing [26]. 10 14 Mean rate per user [b/s/Hz] 12 10 8 6 4 2 0 Single−User Hybrid Analog 1 3 5 7 9 11 13 Cluster radius [m] 15 17 19 21 Fig. 2: Mean rate per UE vs Cluster radius. The performance degrades sharply as the inter-UE average distance decreases. As a first approximation, the u-th UE can assume that its estimates are perfect (error-free) and global (shared between all the UEs). Since UEs 1, . . . , u − 1 have in fact different estimates, and since such information is not error-free, we call this approach naive-coordinated (nc). The beam indexes (qunc ∈ VBS , pnc u ∈ VUE ) associated to the u-th UE are then found as follows: (u) o o (q̃1:u−1 , qunc , p̃1:u−1 , pnc ) = argmax R P̂ , q , p (23) qu+1:K ,pu+1:K 1:u 1:u u q1 ,...,qu ∈VBS p1 ,...,pu ∈VUE o Remark 3. We make here an abuse of notation. The subscripts qu+1:K , pou+1:K acknowledge for the known strategies at the u-th UE. Those strategies are fixed parameters of the function R. The same notation will be used in the rest of the paper. Remark 4. The u-th UE will use the precoding vector associated to the index pnc u ∈ VUE to reach the BS and will discard the remaining beam indexes q̃1:u−1 , p̃1:u−1 found for the other UEs. Indeed, those indexes only correspond to guesses realized at the u-th UE which do not necessarily correspond to the true beams used for transmission at UEs 1, . . . , u − 1. We have introduced the notation q̃u to denote such beams. 11 C. Statistically-Coordinated Beam Selection The naive-coordinated approach relies on the correctness of the position estimates available at each UE. As a consequence, its performance is expected to degrade in case of GPS inaccuracies or lost location awareness. As the precision of position estimates decreases, the beam selection is expected to have more confidence in long-term statistics alone, to predict the behavior of the UE which are higher ranked in the information chain. In this case, the position estimates are not exploited, and each UE relies on prior statistics to figure out other UEs’ information. We denote the resulting statistically-coordinated (sc) beam indexes relative to the u-th UE as (qusc ∈ VBS , psc u ∈ VUE ), which read as follows: (q̃1:u−1 , qusc , p̃1:u−1 , psc u) h i o o = argmax EP\|rcl Rqu+1:K ,pu+1:K P, q1:u , p1:u (24) q1 ,...,qu ∈VBS p1 ,...,pu ∈VUE This is a long-term optimization which is updated only if prior statistics change. Thus, (24) represents a simple stochastic optimization problem [27] which can be solved through e.g. approximation of the expectation operator (carried out over prior statistics) with Monte-Carlo iterations. D. Robust-Coordinated Beam Selection The UEs have also access to the statistics of their local position estimates. In the previous approach, each UE used prior statistics to guess the precoding strategies of the more informed UEs. This helps in case the local information available at the u-th UE is not accurate enough to gain more knowledge about the UEs 1, . . . , u−1. In the opposite case, local statistical information can be exploited to supplement prior information. In this approach, the UEs look for beam selection strategies which progressively pass from exploiting local information only – in case of perfect local information – to exploiting statistical information only – in case of poor local information. We denote this approach as robust-coordinated (rc). The beams (qurc ∈ VBS , prc u ∈ VUE ) for the u-th UE are obtained through: h i o o R P, q , p (q̃1:u−1 , qurc , p̃1:u−1 , prc ) = argmax E qu+1:K ,pu+1:K 1:u 1:u u P\|P̂ (u) ,rcl (25) q1 ,...,qu ∈VBS p1 ,...,pu ∈VUE Remark 5. Here, the u-th UE considers its locally-available position estimates as imperfect and globally-shared. Also in this case, an approximate solution can be obtained through Monte-Carlo methods, generating possible matrices P according to the (known) distribution P\|P̂ (u) , rcl . 12 We summarize the proposed robust-coordinated beam selection used at the u-th UE in Algorithm 1. In Step 1, the u-th UE retrieves the processing carried out at less informed UEs u + 1, . . . , K. The K-th UE skips this step. In Step 2, beam selection is performed through (22) (an approximation) and (25). Algorithm 1 frc : Robust-Coordinated Beam Selection (u-th UE) INPUT: P̂ (w) , ∀w ∈ {u, . . . , K}, pdf of (P\|P̂ (w) , rcl ), ∀w ∈ {u, . . . , K} Step 1 1: for w = K : u + 1 do 2: o (qw , pow ) = frc (P̂ (w) . The K-th UE skips this decreasing for loop o , qw+1:K , pow+1:K ) 3: end for Step 2 4: return (qurc , prc u ) ← Evaluate (25) . Refer to Algorithm 2 for implementation details Algorithm 2 Implementation details for Step 2 in Algorithm 1 (u-th UE) o o , pou+1 , . . . , poK INPUT: P̂ (u) , pdf of (P\|P̂ (u) , rcl ), qu+1 , . . . , qK 1: for i = 1 : M do . Approximate expectation over (P\|P̂ (w) , rcl ), ∀w with M Monte-Carlo iterations 2: Generate possible position matrices P through sampling over the distribution (P\|P̂ (w) , rcl ) 3: Compute possible gain matrices Ĝw , ∀w ∈ {1, . . . , K} through (11) using the generated P 4: o o , pou+1 , . . . , poK ) S UM R ATE E VAL(Ĝ1 , . . . , ĜK , qu+1 , . . . , qK . Described in Algorithm 3 5: end for 6: Compute the average sum-rate over the Monte-Carlo iterations for all possible beam pairs 7: (qurc , prc u ) ← Indexes relative to the beams achieving maximum average sum-rate 8: The pair of vectors with indexes (qurc , prc u ) is assigned to the u-th UE V. S IMULATION R ESULTS We evaluate here the performance of the proposed algorithms. We consider L = 3 multipath components. A distance of 100 m is assumed from the UEs’ cluster center and the BS. The radius of the UEs’ cluster is set to rcl = 7 m. Both the BS and UEs are equipped with NUE = NBS = 64 antennas (ULA). The number of elements in the beam codebooks is MUE = MBS = 64, with grid angles spaced according to the inverse cosine function so as to guarantee equal gain losses among adjacent angles [22]. All the plotted rates are the averaged – over 10000 Monte-Carlo runs – rates per UE. 13 Algorithm 3 Function evaluating an approximated average sum-rate (22) (u-th UE) o o 1: function S UM R ATE E VAL(Ĝ1 , . . . , ĜK , qu+1 , . . . , qK , pou+1 , . . . , poK ) 2: o o KnownInds = {qu+1 , . . . , qK , pou+1 , . . . , poK } 3: for w = 1 : u − 1 do 4: . The most informed UE 1 skips this loop P v (qw , . . . , qu , pw , . . . , pu ) = max Ĝw . with given KnownInds qw ,pw /( v6=w Ĝqw ,pv + N0 ) 5: KnownInds = {qw , pw } ∪ KnownInds 6: Discard all the other indexes qw+1 , . . . , qu , pw+1 , . . . , pu . Updated set of indexes to be used in the next iteration 7: end for 8: P return Ĝuqu ,pu /( w6=u Ĝw qu ,pw + N0 ) for all possible qu , pu . with given KnownInds 9: end function A. Location Information Model In the simulations, we adopt a uniform bounded error model for location information [20], [22]. In particular, we assume that all the position estimates lie somewhere inside disks centered in the actual positions pun , n ∈ {UE, BS, Rui }; i ∈ {1, . . . , L − 1}; u ∈ {1, . . . , K}. Let S(r) be the two-dimensional closed ball centered at the origin and of radius r, which is S(r) = {υ ∈ w,(u) R2 : kυk ≤ r}. We model the estimation errors en w,(u) in S(rn w,(u) ), where rn as a random variable uniformly distributed is the maximum positioning error for the node n of the w-th UE as seen from the u-th UE. B. Results and Discussion To evaluate the performance, we start with a simple configuration with K = 2 UEs. 1) Strong LoS: In what follows, we consider a stronger (on average) LoS path with respect to the reflected paths, being indeed the prominent propagation driver in mmWave bands [2], [3]. The reflected paths are assumed to have the same average power. The average power of such (1) (2) paths is assumed to be shared across the UEs, i.e. (σ` )2 , = (σ` )2 , ∀` ∈ {1, . . . , L}. In Fig. 3, we consider the performance of the proposed algorithm as a function of the precision of the information available at the less informed UE. In particular, an error radius of 5 m means w,(2) rn = 5 m, ∀w, n. As for the most informed UE, we consider perfect information in Fig. 3a, w,(1) i.e. rn w,(1) = 0 m, ∀w, n, and 3 m of precision in Fig. 3b, i.e. rn = 3 m, ∀w, n. Fig. 3a and Fig. 3b show that both the uncoordinated and the naive approaches degrade fast as the error radius for the less informed UE increases. This is due to the fact that the UEs build their 14 strategies according to their available position estimates, which become unreliable to perform beam selection. In particular, when the precision is less than 6 m, the statistically-coordinated approach – based on statistical information only – behaves better. The robust-coordinated approach outperforms all the other algorithms, being able to discriminate for interference at the BS side, while taking into account the noise present in position information. 9.5 8 Robust−Coordinated Stat.−Coordinated Naive−Coordinated Uncoordinated 7.5 8.5 Mean rate per UE [b/s/Hz] Mean rate per UE [b/s/Hz] 9 8 7.5 7 6.5 6 Robust−Coordinated Stat.−Coordinated Naive−Coordinated Uncoordinated 7 6.5 6 5.5 5 0 2 4 6 8 10 12 14 16 Pos. information precision for less informed UE [m] 18 (a) Most informed UE with perfect information 20 4.5 0 2 4 6 8 10 12 14 16 Pos. information precision for less informed UE [m] 18 20 (b) Most informed UE with 3 m precision Fig. 3: Mean rate per UE vs Position information precision. Strong LoS. 2) LoS Blockage: It is also interesting to observe how the proposed algorithms behave in case of total line-of-sight blockage, while having one stronger reflected path. Fig. 4 compares the proposed algorithms as a function of the error radius for the less informed UE. The most informed UE is assumed to have access to perfect information. The same considerations outlined for the strong LoS case remain valid. It is possible to observe that the uncoordinated approach performs worse than before (with strong LoS), since the UEs choose to use the beams pointing towards the same stronger reflected path. As a consequence, there is much more chance to arrive at the BS with non-distinguishable AoAs. In such cases, coordination between the UEs is essential to combat multi-user interference. VI. C ONCLUSIONS In mmWave communications, multi-user interference has to be handled in the analog stage as well. In this respect, suitable strategies for multi-user interference minimization can be applied in the beam domain through e.g. exploitation of location-dependent information. 15 9.5 Robust−Coordinated Stat.−Coordinated Naive−Coordinated Uncoordinated Mean rate per UE [b/s/Hz] 9 8.5 8 7.5 7 6.5 6 0 2 4 6 8 10 12 14 16 Pos. information precision for less informed UE [m] 18 20 Fig. 4: Mean rate per UE vs Position information precision. LoS blockage. Dealing with the imperfections in location information is not trivial, due to the decentralized nature of the information, which leads to disagreements between the UEs affecting performance. In this work, we introduced a decentralized robust algorithm which aim to select the best precoder for each UE taking both the noise present in location information and multi-user interference in the analog stage into account. Numerical experiments have shown that good performance can be achieved with the proposed algorithm and have confirmed that coordination is essential to counteract inter-UE interference in mmWave multi-user environments. Exploiting Machine Learning tools [28] for solving the proposed algorithms in a more efficient manner is an interesting and challenging problem which we aim to tackle in future studies. ACKNOWLEDGMENTS F. Maschietti, D. Gesbert and P. de Kerret are supported by the ERC under the European Union’s Horizon 2020 research and innovation program (Agreement no. 670896). 16 R EFERENCES [1] R. W. Heath, N. González-Prelcic, S. Rangan, W. Roh, and A. M. Sayeed, “An overview of signal processing techniques for millimeter wave MIMO systems,” IEEE J. Sel. Topics Signal Process., Apr. 2016. [2] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” IEEE J. Sel. 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C ARMA: Collective Adaptive Resource-sharing Markovian Agents Luca Bortolussi Rocco De Nicola Vashti Galpin Stephen Gilmore Saarland University University of Trieste ISTI - CNR IMT Lucca University of Edinburgh University of Edinburgh Jane Hillston Diego Latella Michele Loreti Mieke Massink University of Edinburgh ISTI - CNR Università di Firenze IMT Lucca ISTI - CNR In this paper we present C ARMA, a language recently defined to support specification and analysis of collective adaptive systems. C ARMA is a stochastic process algebra equipped with linguistic constructs specifically developed for modelling and programming systems that can operate in openended and unpredictable environments. This class of systems is typically composed of a huge number of interacting agents that dynamically adjust and combine their behaviour to achieve specific goals. A C ARMA model, termed a collective, consists of a set of components, each of which exhibits a set of attributes. To model dynamic aggregations, which are sometimes referred to as ensembles, C ARMA provides communication primitives that are based on predicates over the exhibited attributes. These predicates are used to select the participants in a communication. Two communication mechanisms are provided in the C ARMA language: multicast-based and unicast-based. In this paper, we first introduce the basic principles of C ARMA and then we show how our language can be used to support specification with a simple but illustrative example of a socio-technical collective adaptive system. 1 Introduction Collective adaptive systems (CAS) typically consist of very large numbers of components which exhibit autonomic behaviour depending on their properties, objectives and actions. Decision-making in such systems is complicated and interaction between their components may introduce new and sometimes unexpected behaviours. CAS are open, in the sense that components may enter or leave the collective at anytime. Components can be highly heterogeneous (machines, humans, networks, etc.) each operating at different temporal and spatial scales, and having different (potentially conflicting) objectives. We are still far from being able to design and engineer real collective adaptive systems, or even specify the principles by which they should operate. CAS thus provide a significant research challenge in terms of both representation and reasoning about their behaviour. The pervasive yet transparent nature of the applications developed in this paradigm makes it of paramount importance that their behaviour can be thoroughly assessed during their design, prior to deployment, and throughout their lifetime. Indeed their adaptive nature makes modelling essential and models play a central role in driving their adaptation. Moreover, the analysis should encompass both functional and non-functional aspects of behaviour. Thus it is vital that we have available robust modelling techniques which are able to describe such systems and to reason about their behaviour in both qualitative and quantitative terms. To move towards this goal, we consider it important to develop a theoretical foundation for collective adaptive systems that would help in understanding their distincN. Bertrand and M. Tribastone (Eds.): QAPL 2015 EPTCS 194, 2015, pp. 16–31, doi:10.4204/EPTCS.194.2 c L. Bortolussi et al. This work is licensed under the Creative Commons Attribution License. L. Bortolussi et al. 17 tive features. In this paper we present C ARMA, a language designed within the QUANTICOL project1 specifically for the specification and analysis of CAS, with the particular objective of supporting quantitive evaluation and verification. C ARMA builds on a long tradition of stochastic process algebras such as PEPA [13], MTIPP [12], EMPA [2], Stochastic π-Calculus [15], Bio-PEPA [5], MODEST [3] and others [11, 4]. It combines the lessons which have been learned from these languages with those learned from developing languages to model CAS, such as SCEL [8] and PALOMA [9], which feature attribute-based communication and explicit representation of locations. SCEL [8] (Software Component Ensemble Language), is a kernel language that has been designed to support the programming of autonomic computing systems. This language relies on the notions of autonomic components representing the collective members, and autonomic-component ensembles representing collectives. Each component is equipped with an interface, consisting of a collection of attributes, describing different features of components. Attributes are used by components to dynamically organise themselves into ensembles and as a means to select partners for interaction. The stochastic variant of SCEL, called StocS [14], was a first step towards the investigation of the impact of different stochastic semantics for autonomic processes, that relies on stochastic output semantics, probabilistic input semantics and on a probabilistic notion of knowledge. Moreover, SCEL has inspired the development of the core calculus AbC [1] that focuses on a minimal set of primitives that defines attribute-based communication, and investigates their impact. Communication among components takes place in a broadcast fashion, with the characteristic that only components satisfying predicates over specific attributes receive the sent messages, provided that they are willing to do so. PALOMA [9] is a process algebra that takes as starting point a model based on located Markovian agents each of which is parameterised by a location, which can be regarded as an attribute of the agent. The ability of agents to communicate depends on their location, through a perception function. This can be regarded as an example of a more general class of attribute-based communication mechanisms. The communication is based on a multicast, as only agents who enable the appropriate reception action have the ability to receive the message. The scope of communication is thus adjusted according to the perception function. A distinctive contribution of the new language is the rich set of communication primitives that are offered. C ARMA supports both unicast and broadcast communication, and locally synchronous, but globally asynchronous communication. This richness is important to enable the spatially distributed nature of CAS, where agents may have only local awareness of the system, yet the design objectives and adaptation goals are often expressed in terms of global behaviour. Representing these rich patterns of communication in classical process algebras or traditional stochastic process algebras would be difficult, and would require the introduction of additional model components to represent buffers, queues and other communication structures. Another feature of C ARMA is the explicit representation of the environment in which processes interact, allowing rapid testing of a system under different open world scenarios. The environment in C ARMA models can evolve at runtime, due to the feedback from the system, and it further modulates the interaction between components, by shaping rates and interaction probabilities. Furthermore the large scale nature of CAS systems makes it essential to support scalable analysis techniques, thus C ARMA has been designed anticipating both a discrete and a continuous semantics in the style of [16]. The focus of this paper is the presentation of the language and its discrete semantics, which are presented in the F U TS style [7]. The structure of the paper is as follows. Section 2 presents the syntax 1 http://www.quanticol.eu 18 C ARMA: Collective Adaptive Resource-sharing Markovian Agents of the language and explains the organisation of a model in terms of a collective of agents that are considered in the context of an environment. In Section 3 we give a detailed account of the semantics, particularly explaining the role of the environment. The use of C ARMA is illustrated in Section 4 where we describe a model of a simple bike sharing system. Some conclusions are drawn in Section 5. 2 C ARMA syntax A C ARMA system consists of a collective (N) operating in an environment (E ). The collective consists of a set of components. It models the behavioural part of a system and is used to describe a set of interacting agents that cooperate to achieve a set of given tasks. The environment models all those aspects which are intrinsic to the context where the agents under consideration are operating. The environment also mediates agent interactions. We let S YS be the set of C ARMA systems S defined by the following syntax: S ::= N in E where N is a collective and E is an environment. The latter provides the global state of the system and governs the interactions in the collective. We let C OL be the set of collectives N which are generated by the following grammar: NkN N ::= C A collective N is either a component C or the parallel composition of two collectives (N k N). A component C can be either the inactive component, which is denoted by 0, or a term of the form (P, γ), where P is a process and γ is a store. A term (P, γ) models an agent operating in the system under consideration: the process P represents the agent’s behaviour whereas the store γ models its knowledge. A store is a function which maps attribute names to basic values. We let: • ATTR be the set of attribute names a, a0 , a1 ,. . . , b, b0 , b1 ,. . . ; • VAL be the set of basic values v, v0 , v1 ,. . . ; • Γ be the set of stores γ, γ1 , γ 0 , . . . i.e. functions from ATTR to VAL. We let C OMP be the set of components C generated by the following grammar: (P, γ) C ::= 0 We let P ROC be the set of processes P, Q,. . . defined by the following grammar: P, Q ::= \| \| \| \| \| \| − act ::= α ? [π]h→ e iσ → − \| α [π]h e iσ nil kill act.P P+Q P\|Q [π]P A \| \| − α ? [π](→ x )σ → − α [π]( x )σ e ::= a \| this.a \| x \| v \| · · · 4 (A = P) π ::= > \| ⊥ \| e1 ./ e2 \| ¬π \| π ∧ π \| · · · − In C ARMA processes can perform four types of actions: broadcast output (α ? [π]h→ e iσ ), broadcast → − → − → − ? input (α [π]( x )σ ), output (α [π]h e iσ ), and input (α [π]( x )σ ). Where: L. Bortolussi et al. 19 • α is an action type in the set of action type ACT T YPE; • π is an predicate; • x is a variable in the set of variables VAR; −· indicates a sequence of elements; • → • σ is an update, i.e. a function from Γ to Dist(Γ) in the set of updates Σ; where Dist(Γ) is the set of probability distributions over Γ. The admissible communication partners of each of these actions are identified by the predicate π. This is a predicate on attribute names. Note that, in a component (P, γ) the store γ regulates the behaviour of P. Primarily, γ is used to evaluate the predicate associated with an action in order to filter the possible synchronisations involving process P. In addition, γ is also used as one of the parameters for computing the actual rate of actions performed by P. The process P can change γ immediately after the execution of an action. This change is brought about by the update σ . The update is a function that when given a store γ returns a probability distribution over Γ which expresses the possible evolutions of the store after the action execution. − The broadcast output α ? [π]h→ e iσ models the execution of an action α that spreads the values result− ing from the evaluation of expressions → e in the local store γ. This message can be potentially received by any process located at components whose store satisfies predicate π. This predicate may contain references to attribute names that have to be evaluated under the local store. These references are prefixed by the special name this. For instance, if loc is the attribute used to store the position of a component, action − α ? [distance(this.loc, loc) ≤ L]h→ v iσ potentially involves all the components located at a distance that is less than or equal to a given threshold L. The broadcast output is non-blocking. The action is executed even if no process is able to receive the values which are sent. Immediately after the execution of an action, the update σ is used to compute the (possible) effects of the performed action on the store of the hosting component where the output is performed. − To receive a broadcast message, a process executes a broadcast input of the form α ? [π](→ x )σ . This → − action is used to receive a tuple of values v sent with an action α from a component whose store − − satisfies the predicate π[→ v /→ x ]. The transmitted values can be part of the predicate π. For instance, α ? [x > 5](x)σ can be used to receive a value that is greater than 5. The other two kinds of action, namely output and input, are similar. However, differently from broadcasts described above, these actions realise a point-to-point interaction. The output operation is blocking, in contrast with the non-blocking broadcast output. Choice and parallel composition are the usual definitions for process algebras. Processes can be guarded so that [π]P behaves as the process P if the predicate π is satisfied. Finally, process kill is used to destroy a component. We assume that this term always occurs under the scope of an action prefix. C ARMA collectives operate in an environment E . This environment is used to model the intrinsic rules that govern, for instance, the physical context where our system is situated. An environment consists of two elements: a global store γg , that models the overall state of the system, and an evolution rule ρ. The latter is a function which, depending on the global store and the current state of the collective, i.e. the configurations of each component in the collective, returns a tuple of functions ε = hµ p , µr , µu i known as the evaluation context where ACT = ACT T YPE ∪ {α ? \|α ∈ ACT T YPE} and: 20 C ARMA: Collective Adaptive Resource-sharing Markovian Agents • µ p : Γ × ACT → [0, 1], expresses the probability to receive a message; • µr : Γ × Γ × ACT → R≥0 , computes the execution rate of an action; • µu : Γ × ACT → Σ × C OL, determines the updates on the environment (global store and collective) induced by the action execution. These functions regulate system behaviour. Function µ p , which takes as parameters the local stores of the two interacting components, i.e. the sender and the receiver, and the action used to interact, returns the probability to receive a message. Function µr computes the rate of an unicast/broadcast output. This function takes as parameter the local store of the component performing the action and the action on which interaction is based. Note that the environment can disable the execution of a given action. This happens when the function µr (resp. µ p ) returns the value 0. Finally, the function µu is used to update the global store and to install a new collective in the system. The function µu takes as parameters the store of the component performing the action together with the action type and returns a pair (σ , N). Within this pair, σ identifies the update on the global store whereas N is a new collective installed in the system. This function is particularly useful for modelling the arrival of new agents into a system. All of these functions are determined by an evolution rule ρ depending on the global store and the actual state of the components in the system. For instance, the probability to receive a given message may depend on the concentration of components in a given state. Similarly, the actual rate of an action may be a function of the number of components whose store satisfies a given property. 3 C ARMA operational semantics In this section we define the operational semantics of C ARMA specifications. This operational semantics · is defined in three stages. First, we introduce the transition relation −−+· that describes the behaviour of · a single component. Second, this relation is used to define the transition relation −−→· which describes · the behaviour of collectives. Finally, the transition relation 7−−→ will be defined to show how C ARMA systems evolve. All these transition relations are defined in the F U TS style [7]. Using this approach, a transition relation is described using a triple of the form (N, `, N ). The first element of this triple is either a component, or a collective, or a system. The second element is a transition label. The third element is a function associating each component, collective, or system with a non-negative number. A non-zero value represents the rate of the exponential distribution characterising the time needed for the execution of the action represented by `. The zero value is associated with unreachable terms. We use the F U TS style semantics because it makes explicit an underlying Action Labelled Markov Chain, which can be simulated with standard algorithms [10] but is nevertheless more compact than Plotkin-style semantics, as the functional form allows different possible outcomes to be treated within a single rule. A complete description of F U TS and their use can be found in [7]. 3.1 Operational semantics of components We use the transition relation + − ε ⊆ C OMP × L AB × [C OMP → R≥0 ] to define the behaviour of a single component. In this relation [C OMP → R≥0 ] denotes the set of functions from C OMP to R≥0 and L AB is L. Bortolussi et al. 21 the set of transition labels ` which are generated by the following grammar: − ` ::= α ? [π]h→ v i, γ Broadcast output \| \| − α ? [π](→ v ), γ → − α [π]h v i, γ Broadcast input Unicast Output \| − α [π](→ v ), γ − τ[α [π]h→ v i, γ] \| − R[α ? [π](→ v ), γ] Broadcast Input Refusal \| Unicast Input Unicast Synchronization The first four labels are associated with the four C ARMA input-output actions and they contain a reference to the action which is performed (α or α ? ), the store of the component where the action is executed (γ), − and the value which is transmitted or received. The transition label τ[α [π]h→ v i, γ] is the one which → − ? is associated with unicast synchronisation. The final label R[α [π]( v ), γ] denotes the case where a component is not able to receive a broadcast output. This arises at the level of the single component either because the associated message has been lost, or because no process is willing to receive that − message. We will observe later in this section that the use of R[α ? [π](→ v ), γ] labels are crucial to handle appropriately dynamic process operators, namely choice and guard. The transition relation + − ε , as formally defined in Table 1 and Table 2, is parametrised with respect to an evaluation context ε. This is used to compute the actual rate of process actions and to compute the probability to receive messages. The process nil denotes the process that cannot perform any action. The transitions which are induced by this process at the level of components can be derived via rules Nil and Nil-F1. These rules respectively say that the inactive process cannot perform any action, and always refuses any broadcast input. Note that, the fact that a component (nil, γ) does not perform any transition is derived from the fact that any label that is not a broadcast input refusal leads to function 0/ (rule Nil). Indeed, 0/ denotes the 0 constant function. Conversely, Nil-F1 states that (nil, γ) can always perform a transition labelled − R[α ? [π](→ v ), γ] leading to [(nil, γ) 7→ 1], where [C 7→ v] denotes the function mapping the component C to v ∈ R≥0 and all the other components to 0. − The behaviour of a broadcast output (α ? [π1 ]h→ e iσ .P, γ) is described by rules B-Out, B-Out-F1 and − B-Out-F2. Rule B-Out states that a broadcast output α ? [π]h→ e iσ can affect components that satisfy 0 2 π = JπKγ . The action rate is determined by the evaluation context ε = hµ p , µr , µu i and, in particular, by the function µr . This function, given a store γ and the kind of action performed, in this case α ? , returns a value in R≥0 . If this value is greater than 0, it denotes the execution rate of the action. However, the evaluation context can disable the execution of some actions. This happens when µr (γ, α ? ) = 0. The possible next local stores after the execution of an action are determined by the update σ . This takes the store γ and yields a probability distribution p = σ (γ) ∈ Dist(Γ). In rule B-Out, and in the rest of the paper, the following notations are used: • let P ∈ P ROC and p ∈ Dist(Γ), (P, p) is a probability distribution in Dist(C OMP) such that:  P ≡ Q\|kill ∧ C ≡ 0  1 p(γ) C ≡ (P, γ) ∧ P 6≡ Q\|kill (P, p)(C) =  0 otherwise • let c ∈ Dist(C OMP) and r ∈ R≥0 , r · c denotes the function C : C OMP → R≥0 such that: C (C) = r · c(C) 2 We let J·Kγ denote the evaluation function of an expression/predicate with respect to the store γ. 22 C ARMA: Collective Adaptive Resource-sharing Markovian Agents − ` 6= R[α ? [π](→ v ), γ] Nil ` Nil-F1 − R[α ? [π](→ v ),γ] (nil, γ) + − ε 0/ (nil, γ) −−−−−−−−+ε [(nil, γ) 7→ 1] JπKγ = π 0 − − J→ e Kγ = → v p = σ (γ) ε = hµ p , µr , µu i − α ? [π 0 ]h→ v i,γ − (α ? [π]h→ e iσ .P, γ) −−−−−−−+ε µr (γ, α ? ) · (P, p) B-Out → − R[β ? [π2 ]( v ),γ] − − (α ? [π1 ]h→ e iσ .P, γ) −−−−−−−−+ε [(α ? [π1 ]h→ e iσ .P, γ) 7→ 1] JπKγ = π 0 B-Out-F1 − − J→ e Kγ = → v − − ` 6= α ? [π 0 ]h→ v i, γ ` 6= R[β ? [π 0 ](→ v 1 ), γ] B-Out-F2 ` − (α ? [π]h→ e iσ .P, γ) + − ε 0/ − − Jπ2 [→ v /→ x ]Kγ2 = π20 γ1 \|= π20 − − p = σ [→ v /→ x ](γ2 ) ε = hµ p , µr , µu i γ2 \|= π1 → − α ? [π1 ]( v ),γ1 − − − (α ? [π2 ](→ x )σ .P, γ2 ) −−−−−−−+ε µ p (γ1 , γ2 , α ? ) · (P[→ v /→ x ], p) − − Jπ2 [→ v /→ x ]Kγ2 = π20 − (α ? [π2 ](→ x )σ .P, γ2 ) γ1 \|= π20 − R[α ? [π1 ](→ v ),γ1 ] −−−−−−−−−+ε γ2 \|= π1 [(α ? [π − − Jπ2 [→ v /→ x ]Kγ2 = π20 ε = hµ p , µr , µu i → − ? 2 ]( x )σ .P, γ2 ) 7→ 1 − µ p (γ1 , γ2 , α )] (γ1 6\|= π20 or γ2 6\|= π1 ) → − α ? [π1 ]( v ),γ1 − (α ? [π2 ](→ x )σ .P, γ2 ) −−−−−−−+ε 0/ B-In-F1 B-In-F2 − − ` 6= α ? [π1 ](→ v ), γ1 ` 6= R[α ? [π1 ](→ v ), γ1 ] B-In-F3 ` → − (α ? [π ]( x )σ .P, γ ) + − 0/ 2 2 ε α 6= β (α ? [π R[β ? [π → − 1 ]( v ),γ1 ] → − − −−−−−−−−+ε [(α ? [π2 ](→ x )σ .P, γ2 ) 7→ 1] 2 ]( x )σ .P, γ2 ) − B-In-F4 Table 1: Operational semantics of components (Part 1) B-In L. Bortolussi et al. 23 Note that, after the execution of an action a component can be destroyed. This happens when the continuation process after the action prefixing contains the term kill. For instance, by applying rule α ? [π1 ]hvi,γ B-Out we have that: (α ? [π1 ]hviσ .(kill\|Q), γ) −−−−−−+ε [0 7→ r]. Rule B-Out-F1 states that a broadcast output always refuses any broadcast input, while B-Out-F2 − − states that a broadcast output can be only involved in labels of the form α ? [π]h→ v i, γ or R[β ? [π2 ](→ v ), γ]. − Transitions related to a broadcast input are labelled with α ? [π1 ](→ v ), γ1 . There, γ1 is the store of the component executing the output, α is the action performed, π1 is the predicate that identifies the − target components, while → v is the sequence of transmitted values. Rule B-In states that a component − (α ? [π2 ](→ x )σ .P, γ2 ) can perform a transition with this label when its store γ2 satisfies the target predicate, − − i.e. γ2 \|= π1 , and the component executing the action satisfies the predicate π2 [→ v /→ x ]. The evaluation context ε = hµ p , µr , µu i can influence the possibility to perform this action. This transition can be performed with probability µ p (γ1 , γ2 , α ? ). Rule B-In-F1 models the fact that even if a component can potentially receive a broadcast message, the message can get lost according to a given probability regulated by the evaluation context, namely 1 − µ p (γ1 , γ2 , α ? ). Rule B-In-F2 models the fact that if a component is not in the set of possible receivers (γ2 6\|= π1 ) or the sender does not satisfy the expected requirements (γ1 6\|= π20 ) then the component cannot − receive a broadcast message. Finally, rules B-In-F3 and B-In-F4 model the fact that (α ? [π2 ](→ x )σ .P, γ2 ) can only perform a broadcast input on action α and that it always refuses input on any other action type β 6= α, respectively. The behaviour of unicast output and unicast input is defined by the first six rules of Table 2. These rules are similar to the ones already presented for broadcast output and broadcast input. The only difference is that both unicast output (Out-F1) and unicast input (In-F1) always refuse any broadcast input with probability 1. The other rules of Table 2 describe the behaviour of other process operators, namely choice P + Q, parallel composition P\|Q, guard and recursion. The term P + Q identifies a process that can behave either as P or as Q. The rule Plus states that the components that are reachable by (P + Q, γ), via a transition that is not a broadcast input refusal, are the ones that can be reached either by (P, γ) or by (Q, γ). In this rule we use C1 ⊕ C2 to denote the function that maps each term C to C1 (C) + C2 (C), for any C1 , C2 ∈ [C OMP → R≥0 ]. At the same time, process P + Q refuses a broadcast input when both the process P and Q do that. This is modelled by Plus-F1, where, for each C1 : C OMP → R≥0 and C2 : C OMP → R≥0 , C1 + C2 denotes the function that maps each term of the form (P + Q, γ) to C1 ((P, γ)) · C2 ((Q, γ)), while any other component is mapped to 0. Note that, differently from rule Plus, when rule Plus-F1 is applied operator + is not removed after the transition. This models the fact that when a broadcast message is refused the choice is not resolved. In P\|Q the two composed processes interleave for all the transition labels except for broadcast input refusal (Par). For this label the two processes synchronise (Par-F1). This models the fact that a message is lost when both processes refuse to receive it. In the rules the following notations are used: • for each component C and process Q we let: 0 C≡0 C\|Q = (P\|Q, γ) C ≡ (P, γ) Q\|C is symmetrically defined. • for each C : C OMP → R≥0 and process Q, C \|Q (resp. Q\|C ) denotes the function that maps each term of the form C\|Q (resp. Q\|C) to C (C), while the others are mapped to 0; • for each C1 : C OMP → R≥0 and C2 : C OMP → R≥0 , C1 \|C2 denotes the function that maps each term of the form (P\|Q, γ) to C1 ((P, γ)) · C2 ((Q, γ)), while the others are mapped to 0. 24 C ARMA: Collective Adaptive Resource-sharing Markovian Agents − − J→ e Kγ = → v JπKγ = π 0 p = σ (γ) ε = hµ p , µr , µu i − α [π 0 ]h→ v i,γ − (α [π]h→ e iσ .P, γ) −−−−−−+ε µr (γ, α) · (P, p) Out Out-F1 → − R[β ? [π2 ]( v ),γ2 ] − − (α [π1 ]h→ e iσ .P, γ1 ) −−−−−−−−−+ε [(α [π1 ]h→ e iσ .P, γ1 ) 7→ 1] − − J→ e Kγ = → v − − ` 6= α [π 0 ]h→ v i, γ ` 6= R[α ? [π 0 ](→ v ), γ] Out-F2 ` − (α [π]h→ e iσ .P, γ) + − ε 0/ JπKγ = π 0 − − Jπ2 [→ v /→ x ]Kγ2 = π20 γ1 \|= π20 − − p = σ [→ v /→ x ](γ2 ) ε = hµ p , µr , µu i γ2 \|= π1 → − α [π1 ]( v ),γ1 − − − (α [π2 ](→ x )σ .P, γ2 ) −−−−−−−+ε µ p (γ1 , γ2 , α) · (P[→ v /→ x ], p) → − R[β ? [π1 ]( v ),γ1 ] − − (α [π2 ](→ x )σ .P, γ2 ) −−−−−−−−−+ε [(α [π2 ](→ x )σ .P, γ2 ) 7→ 1] − − Jπ2 [→ v /→ x ]Kγ2 = π20 (γ1 6\|= π20 or γ2 6\|= π1 ) → − α [π1 ]( v ),γ1 − (α [π2 ](→ x )σ .P, γ2 ) −−−−−−−+ε 0/ ` In-F2 2 (Q, γ) + − ε C2 2 − ` 6= R[α ? [π 0 ](→ v ), γ] ` (P + Q, γ) + − ε C1 ⊕ C2 − R[α ? [π 0 ](→ v ),γ] In-F1 − − ` 6= α [π1 ](→ v ), γ1 ` 6= R[β ? [π1 ](→ v ), γ1 ] In-F3 ` − (α [π ](→ x )σ .P, γ ) + − 0/ ` (P, γ) + − ε C1 In ε Plus − R[α ? [π 0 ](→ v ),γ] (P, γ) −−−−−−−−+ε C1 (Q, γ) −−−−−−−−+ε C2 Plus-F1 − R[α ? [π 0 ](→ v ),γ] (P + Q, γ) −−−−−−−−+ε C1 + C2 ` (P, γ) + − ε C1 ` (Q, γ) + − ε C2 − ` 6= R[α ? [π](→ v ), γ] ` (P\|Q, γ) + − ε C1 \|Q ⊕ P\|C2 − R[α ? [π](→ v ),γ] (P, γ) −−−−−−−−+ε C1 − R[α ? [π](→ v ),γ] (Q, γ) −−−−−−−−+ε C2 − R[α ? [π](→ v ),γ] 4 A=P Par-F1 ` (P, γ) + −ε C − ` 6= R[α ? [π](→ v ), γ] ` ([π]P, γ) + −ε C γ 6\|= π − ` 6= R[α ? [π](→ v ), γ] ` ([π]P, γ) + − ε 0/ (A, γ) + −ε C γ \|= π Guard ` (P, γ) + −ε C ` (P\|Q, γ) −−−−−−−−+ε C1 \|C2 γ \|= π Par Rec − R[α ? [π](→ v ),γ] (P, γ) −−−−−−−−+ε C − R[α ? [π](→ v ),γ] Guard-F1 ([π]P, γ) −−−−−−−−+ε [π]C γ 6\|= π Guard-F2 − R[α ? [π](→ v ),γ] ([π]P, γ) −−−−−−−−+ε [([π]P, γ) 7→ 1] Table 2: Operational semantics of components (Part 2) Guard-F3 L. Bortolussi et al. 25 − α ? [π](→ v ),γ Zero ` (P, γ) −−−−−−→ε − ` 6= R[α ? [π](→ v ), γ] ` ` (P, γ) → −ε N − α ? [π]h→ v i,γ N1 −−−−−−→ε N1o − α [π]h→ v i,γ N1 −−−−−−→ε N1 Comp − α ? [π](→ v ),γ − α ? [π]h→ v i,γ − α ? [π](→ v ),γ N2 −−−−−−→ε N2 B-In-Sync N1 k N2 −−−−−−→ε N1 k N2 − α ? [π]h→ v i,γ N2 −−−−−−→ε N2o k N2i ) ⊕ (N1i − α [π]h→ v i,γ − α [π]h→ v i,γ Comp-B-In − α ? [π](→ v ),γ N2 −−−−−−→ε (N1o N2 −−−−−−→ε N2 N1 ⊕N2 ⊕N1 +⊕N2 − α ? [π](→ v ),γ N1 −−−−−−→ε N1i N1 k N1 −−−−−−→ε N1 (P, γ) −−−−−−−−+ε N2 − α ? [π](→ v ),γ 0→ − ε 0/ (P, γ) + −ε N − R[α ? [π](→ v ),γ] (P, γ) −−−−−−+ε N1 k − α ? [π](→ v ),γ N2 −−−−−−→ε N2i − α [π](→ v ),γ Out-Sync N1 k N2 −−−−−−→ε N1 k N2 ⊕ N1 k N2 B-Sync N2o ) − α [π](→ v ),γ N1 −−−−−−→ε N1 N2 −−−−−−→ε N2 − α [π](→ v ),γ In-Sync N1 k N2 −−−−−−→ε N1 k N2 ⊕ N1 k N2 − τ[α [π]h→ v i,γ] − α [π]h→ v i,γ − α [π](→ v ),γ − τ[α [π]h→ v i,γ] − α [π]h→ v i,γ − α [π](→ v ),γ N1 −−−−−−−→ε N1s N1 −−−−−−→ε N1o N1 −−−−−−→ε N1i N2 −−−−−−−→ε N2s N2 −−−−−−→ε N2o N2 −−−−−−→ε N2i N1 k − τ[α [π]h→ v i,γ] (N1s kN2 )·⊕N1i N2 −−−−−−−→ε ⊕N i i 1 +⊕N2 ⊕ (N1 kN2s )·⊕N2i ⊕N1i +⊕N2i ⊕ (N1o kN2i ) ⊕N1i +⊕N2i ⊕ (N1i kN2o ) ⊕N1i +⊕N2i Sync Table 3: Operational semantics of collective Rule Rec is standard. The behaviour of ([π]P, γ) is regulated by rules Guard, Guard-F1, Guard-F2 and Guard-F3. The first two rules state that ([π]P, γ) behaves exactly like (P, γ) when γ satisfies predicate π. However, in the first case the guard is removed when a transition is performed. In contrast, the guard still remains active after the transition when a broadcast input is refused. This is similar to what we consider for the rule Plus-F1 and models the fact that broadcast input refusals do not remove dynamic operators. In rule Guard-F1 we let [π]C denote the function that maps each term of the form ([π]P, γ) to C ((P, γ))) and any other term to 0, for each C : C OMP → R≥0 . Rules Guard-F2 and Guard-F3 state that no component can be reached from ([π]P, γ) and all the broadcast messages are refused when γ does not satisfy predicate π. 3.2 Operational semantics of collective The operational semantics of a collective is defined via the transition relation → − ε ⊆ C OL × L AB × [C OL → R≥0 ]. This relation is formally defined in Table 3. We use a straightforward adaptation of the notations introduced in the previous section. Rules Zero, Comp-B-In and Comp describe the behaviour of the single component at the level of collective. Rule Zero is similar to rule Nil of Table 1 and states that inactive component 0 cannot perform any action. Rule Comp-B-In states that the result of a broadcast input of a component at the level of collective is obtained by combining (summing) the transition at the level of components − − labelled α ? [π](→ v ), γ with the one labelled R[α ? [π](→ v ), γ]. This value is then renormalised to obtain a probability distribution. There we use ⊕N to denote ∑N∈C OL N (N). The renormalisation guarantees a reasonable computation of broadcast output synchronisation rates (see comments on rule B-Sync below). 26 C ARMA: Collective Adaptive Resource-sharing Markovian Agents Note that each component can always perform a broadcast input at the level of collective. However, we are not able to observe if the message has been received or not. Moreover, thanks to renormalisation, if − α ? [π](→ v ),γ C −−−−−−→ε N then ⊕N = 1, i.e. N is a probability distribution over C OL. Rule Comp simply states that for the single component C 6= 0 all the transition labels that are not a broadcast input, the relation ` ` → − ε coincides with the relation + − ε. Rules B-In-Sync and B-Sync describe broadcast synchronisation. The former states that two collectives N1 and N2 that operate in parallel synchronise while performing a broadcast input. This models the fact that the input can be potentially received by both of the collectives. In this rule we let N1 k N2 denote the function associating the value N1 (N1 ) · N2 (N2 ) with each term of the form N1 k N2 and 0 with − α ? [π](→ v ),γ all the other terms. We can observe that if N −−−−−−→ε N then, as we have already observed for rule Comp-B-In, ⊕N = 1 and N is in fact a probability distribution over C OL. Rule B-Sync models the synchronisation consequent of a broadcast output performed at the level of a collective. For each N1 : C OL → R≥0 and N2 : C OL → R≥0 , N1 ⊕ N2 denotes the function that maps each term N to N1 (N) + N2 (N). − At the level of collective a transition labelled α ? [π]h→ v i, γ identifies the execution of a broadcast output. When a collective of the form N1 k N2 is considered, the result of these kinds of transitions must be computed (in the F U TS style) by considering: − α ? [π]h→ v i,γ • the broadcast output emitted from N1 , obtained by the transition N1 −−−−−−→ε N1o − α ? [π](→ v ),γ • the broadcast input received by N1 , obtained by the transition N1 −−−−−−→ε N1i − α ? [π]h→ v i,γ • the broadcast output emitted from N2 , obtained by the transition N2 −−−−−−→ε N2o − α ? [π](→ v ),γ • the broadcast input received by N2 , obtained by the transition N2 −−−−−−→ε N2i Note that the first synchronises with the last to obtain N1o k N2i , while the second synchronises with the third to obtain N1i k N2o . The result of such synchronisations are summed to model the race condition between the broadcast outputs performed within N1 and N2 respectively. We have to remark that above N1o (resp. N2o ) is 0/ when N1 (resp. N2 ) is not able to perform any broadcast output. Moreover, the label of a broadcast synchronisation is again a broadcast output. This allows further synchronisations in a derivation. Finally, it is easy to see that the total rate of a broadcast synchronisation is equal to the total rate of broadcast outputs. This means that the number of receivers does not affect the rate of a broadcast that is only determined by the number of senders. Rules Out-Sync, In-Sync and Sync control the unicast synchronisation. Rule Out-Sync states that a collective of the form N1 k N2 performs a unicast output if this is performed either in N1 or in N2 . This is rendered in the operational semantics as an interleaving rule, where for each N : C OL → R≥0 , N k N2 denotes the function associating N (N1 ) with each collective of the form N1 k N2 and 0 with all other collectives. Rule In-Sync is similar to Out-Sync. However, it considers unicast input. Finally, rule Sync regulates the unicast synchronisations and generates transitions with labels of the − − form τ[α [π]h→ v i, γ]. This is the result of a synchronisation between transitions labelled α [π](→ v ), γ, i.e. → − an input, and α [π]h v i, γ, i.e. an output. − v i, γ]), unicast output In rule Sync, Nks , Nko and Nki denote the result of synchronisation (τ[α [π]h→ → − → − (α [π]h v i, γ) and unicast input (α [π]( v ), γ) within Nk (k = 1, 2), respectively. The result of a transition − labelled τ[α [π]h→ v i, γ] is therefore obtained by combining: • the synchronisations in N1 with N2 : N1s k N2 ; L. Bortolussi et al. 27 − α ? [π]h→ v i,γ ρ(γg , N) = ε = hµr , µ p , µu i N −−−−−−→ε N µu (γg , α ? ) = (σ , N 0 ) − α ? [π]h→ v i,γ Sys-B N in (γg , ρ) 7−−−−−−→ N k N 0 in (σ (γg ), ρ) − τ[α [π]h→ v i,γ] ρ(γg , N) = ε = hµr , µ p , µu i N −−−−−−−→ε N µu (γg , α) = (σ , N 0 ) − τ[α [π]h→ v i,γ] Sys N in (γg , ρ) 7−−−−−−−→ N k N 0 in (σ (γg ), ρ) Table 4: Operational Semantics of Systems. • the synchronisations in N2 with N1 : N1 k N2s ; • the output performed by N1 with the input performed by N2 : N1o k N2i ; • the input performed by N1 with the output performed by N2 : N1i k N2o . To guarantee a correct computation of synchronisation rates, the first two addendi are renormalised by considering inputs performed in N2 and N1 respectively. This, on one hand, guarantees that the total rate − − of synchronisation τ[α [π]h→ v i, γ] does not exceed the output capacity, i.e. the total rate of α [π]h→ v i, γ in N1 and N2 . On the other hand, since synchronisation rates are renormalised during the derivation, it also ensures that parallel composition is associative [7]. 3.3 Operational semantics of systems The operational semantics of systems is defined via the transition relation → 7− ⊆ S YS × L AB × [S YS → R≥0 ] that is formally defined in Table 4. Only synchronisations are considered at the level of systems. The first rule is Sys-B. This rule states that a system of the form N in (γg , ρ) can perform a broadcast output when the collective N, under the environment evaluation ε = hµr , µ p , µu i = ρ(γg , N), can evolve − at the level of collective with the label α ? [π]h→ v i, γ to N . After the transition, the global store is updated and a new collective can be created according to function µu . In rule Sys-B the following notations are used. For each collective N2 , N : C OL → R≥0 , S : S YS → R≥0 and p ∈ Dist(Γ) we let N in (p, ρ) denote the function mapping each system N in (γ, ρ) to N (N) · p(γ). The second rule is Sys that is similar to Sys-B and regulates unicast synchronisations. 4 C ARMA at work In this section we will use C ARMA to model a bike sharing system [6, 17]. These systems are a recent, and increasingly popular, form of public transport in urban areas. As a resource-sharing system with large numbers of independent users altering their behaviour due to pricing and other incentives, they are a simple instance of a collective adaptive system, and hence a suitable case study to exemplify the C ARMA language. The idea in a bike sharing system is that bikes are made available in a number of stations that are placed in various areas of a city. Users that plan to use a bike for a short trip can pick up a bike at a suitable origin station and return it to any other station close to their planned destination. One of the major issues in bike sharing systems is the availability and distribution of resources, both in terms of available bikes at the stations and in terms of available empty parking places in the stations, where users will park the bikes after using them. 28 C ARMA: Collective Adaptive Resource-sharing Markovian Agents In our scenario we assume that the city is partitioned in homogeneous zones and that all the stations in the same zone can be equivalently used by any user in that zone. Below, we let {z0 , . . . , zn } be the n zones in the city, each of which contains k parking stations. Each parking station is modelled in C ARMA via a component of the form: ( G\|R , {zone = `, bikes = i, slots = j}) where • zone is the attribute identifying the zone where the parking station is located; • bikes is the attribute used to count the number of available bikes; • slots is the attribute containing the total number of parking slots in the parking station. Processes G and R, which model the procedure to get and return a bike in the parking station, respectively, are defined as follow: 4 G = [bikes > 0] get[zone = this.zone]h•i{bikes ← bikes − 1}.G 4 R = [slots > bikes] ret[zone = this.zone]h•i{bikes ← bikes + 1}.R Process G, when the value of attribute bikes is greater than 0, executes the unicast output with action type get that potentially involves components satisfying the predicate zone = this.zone, i.e. the ones that are located in the same zone3 . When the output is executed the value of the attribute bikes is decreased by one to model the fact that one bike has been retrieved from the parking station. Process R is similar. It executes the unicast output with action type ret that potentially involves components satisfying predicate zone = this.zone. This action can be executed only when there is at least one parking slot available, i.e. when the value of attribute bikes is less than the value of attribute slots. When the output considered above is executed, the value of attribute bikes is increased by one to model the fact that one bike has been returned in the parking station. Users, who can be either bikers or pedestrians, are modelled via components of the form: (Q, {zone = `}) where zone is the attribute indicating where the user is located, while Q models the current state of the user and can be one of the following processes: 4 B = move? [⊥]h•i{zone ← U(z0 , . . . , zn )}.B + stop? [⊥]h•i.W S 4 W S = ret[zone = this.zone](•).P 4 P = go? [⊥]h•i.W S 4 W B = get[zone = this.zone](•).B Process B represents a biker. When a user is in this state (s)he can either move from the current zone to another zone or stop to return the bike to a parking station. These activities are modelled with the 3 Here we use • to denote the unit value. L. Bortolussi et al. 29 10 Min. bikes per Park Max. bikes per Park Avg. bikes per Park Bikes 8 6 4 2 0 0 20 40 60 80 100 Time Figure 1: Simulation of bike scenario. execution of a broadcast output via action types move and stop, respectively. Note that in both of these cases, the predicate used to identify the target of the actions is ⊥, denoting the value false. This means that neither of the two actions actually synchronise with any component (since no component satisfies ⊥). This kind of interaction is used in C ARMA to model spontaneous actions, i.e. actions that render the execution of an activity and that do no require synchronisation. After the broadcast move? the value of attribute zone is updated by randomly selecting the next zone in {z0 , . . . , zn }. With {zone ← U(z0 , . . . , zn )} we denote the update σ such that σ (γ) is the probability distribution giving probability n1 to each store γ[zone ← zi ]. This update models a random movement of the user among the city zones. When process B executes broadcast stop? , it evolves to process W S. This process models a user who is waiting for a parking slot. This process executes an input over ret. This models the fact that the user has found a parking station with an available parking slot in their zone. After the execution of this input process P is executed. The latter component definition models a pedestrian user. The user remains in this state until the spontaneous action go? is performed. After that it evolves to process W B which models a user waiting for a bike. The behaviour of W B is similar to that of W S described above. Using a custom-built prototype simulator, we are able to simulate this modelled scenario. The output on one simulation run is presented in Figure 1. In the graph we show the minimum, average and maximum number of bikes in one zone of the city. We consider a scenario with four zones each containing four parking stations. The total number of users is 150. 5 Conclusions We have presented C ARMA, a new stochastic process algebra for the representation of systems developed in the CAS paradigm. The language offers a rich set of communication primitives, and the use of attributes, captured in a store associated with each component, allows attribute-based communication. For most CAS systems we anticipate that one of the attributes will be the location of the agent and thus it is straightforward to capture systems in which, for example, there is a limited scope of communication, or restriction to only interact with components that are co-located. As demonstrated in the case study presented in Section 4, attributes can also be used to capture the "state" of a component, such as the available number of bikes/slots at a bike station. C ARMA reflects the experience that we have gained through earlier languages such as SCEL [8], its Markovian variants [14] and PALOMA [9]. Compared with SCEL, the representation of knowledge here is more abstract, and not designed for detailed reasoning during the evolution of the model. This 30 C ARMA: Collective Adaptive Resource-sharing Markovian Agents reflects the different objectives of the languages. Whilst SCEL is designed to support the programming of autonomic computing systems, the primary focus of C ARMA is quantitative analysis. In stochastic process algebras such as PEPA, MTIPP and EMPA, data is typically abstracted away, and the influence of data on behaviour is captured only stochastically. When the data is important to differentiate behaviour it must be implicitly encoded in the state of components. In the context of CAS we wish to support attribute-based communication to reflect the flexible and dynamic interactions that occur in such systems. Thus it is not possible to entirely abstract from data. On the other hand, the level of abstraction means that choices within the system will be captured stochastically rather than through the rich policies for reasoning offered by SCEL. We believe that this offers a reasonable compromise between expressiveness and tractability. Another key feature of C ARMA is the inclusion of an explicit environment in which components interact. In PALOMA there was a rudimentary form of environment, termed the perception function but this proved cumbersome to use, and it could not itself be influenced by the behaviour of the components. In C ARMA, in contrast, the environment not only modulates the rates and probabilities related to interactions between components, it can also itself evolve at runtime, due to feedback from the collective. The focus of this paper has been the discrete semantics in the structured operational style of FUTS [7], but in future work we plan to develop differential semantics in the style of [16]. This latter approach will be essential in order to support quantitative analysis of CAS systems of realistic scale, but it may not be possible to encompass the full rich set of language features of C ARMA with such efficient analysis. Further work is needed to investigate this issue, and which language features can be supported for the various forms of quantitative analysis available. Additional work involves the development of an appropriate high-level language for designers of CAS which will be mapped to the process algebra, and hence will enable qualitative and quantitive analysis of CAS during system development by enabling a design workflow and analysis pathway. The intention of this high-level language is not to add to the expressiveness of C ARMA, which we believe to be well-suited to capturing the behaviour of CAS, but rather to ease the task of modelling for users who are unfamiliar with process algebra and similar formal notations. Acknowledgements This work is partially supported by the EU project QUANTICOL, 600708. This research has also been partially funded by the German Research Council (DFG) as part of the Cluster of Excellence on Multimodal Computing and Interaction at Saarland University. References [1] Yehia Abd Alrahman, Rocco De Nicola, Michele Loreti, Francesco Tiezzi & Roberto Vigo (2015): A Calculus for Attribute-based Communication. In: Proceedings of SAC 2015, doi:10.1145/2695664.2695668. To appear. [2] Marco Bernardo & Roberto Gorrieri (1998): A Tutorial on EMPA: A Theory of Concurrent Processes with Nondeterminism, Priorities, Probabilities and Time. Theoretical Computer Science 202(1-2), pp. 1–54, doi:10.1016/S0304-3975(97)00127-8. [3] H.C. Bohnenkamp, P.R. D’Argenio, H. Hermanns & J-P. Katoen (2006): MODEST: A Compositional Modeling Formalism for Hard and Softly Timed Systems. IEEE Trans. Software Eng. 32(10), pp. 812–830, doi:10.1109/TSE.2006.104. L. Bortolussi et al. 31 [4] Luca Bortolussi & Alberto Policriti (2010): Hybrid dynamics of stochastic programs. Theor. Comput. Sci. 411(20), pp. 2052–2077, doi:10.1016/j.tcs.2010.02.008. [5] Federica Ciocchetta & Jane Hillston (2009): Bio-PEPA: A Framework for the Modelling and Analysis of Biological Systems. Theoretical Computer Science 410(33), pp. 3065–3084, doi:10.1016/j.tcs.2009.02.037. [6] Paola De Maio (2009): Bike-sharing: Its History, Impacts, Models of Provision, and Future. Journal of Public Transportation 12(4), pp. 41–56, doi:10.5038/2375-0901.12.4.3. [7] Rocco De Nicola, Diego Latella, Michele Loreti & Mieke Massink (2013): A uniform definition of stochastic process calculi. ACM Comput. Surv. 46(1), p. 5, doi:10.1145/2522968.2522973. [8] Rocco De Nicola, Michele Loreti, Rosario Pugliese & Francesco Tiezzi (2014): A Formal Approach to Autonomic Systems Programming: The SCEL Language. TAAS 9(2), p. 7, doi:10.1145/2619998. 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[13] Jane Hillston (1995): A Compositional Approach to Performance Modelling. CUP. [14] Diego Latella, Michele Loreti, Mieke Massink & Valerio Senni (2014): Stochastically timed predicate-based communication primitives for autonomic computing. In Nathalie Bertrand & Luca Bortolussi, editors: Proceedings Twelfth International Workshop on Quantitative Aspects of Programming Languages and Systems, QAPL 2014, Grenoble, France, 12-13 April 2014., EPTCS 154, pp. 1–16, doi:10.4204/EPTCS.154.1. [15] Corrado Priami (1995): Stochastic π-calculus. The Computer Journal 38(7), pp. 578–589, doi:10.1093/comjnl/38.7.578. [16] Mirco Tribastone, Stephen Gilmore & Jane Hillston (2012): Scalable Differential Analysis of Process Algebra Models. IEEE Transactions on Software Engineering 38(1), pp. 205–219, doi:10.1109/TSE.2010.82. [17] Wikipedia (2013): Bicycle sharing system — Wikipedia, The Free Encyclopedia. Available at http: //en.wikipedia.org/w/index.php?title=Bicycle_sharing_system&oldid=573165089. 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AN INVITATION TO 2D TQFT AND QUANTIZATION OF HITCHIN SPECTRAL CURVES arXiv:1705.05969v1 [math.AG] 17 May 2017 OLIVIA DUMITRESCU AND MOTOHICO MULASE Abstract. This article consists of two parts. In Part 1, we present a formulation of twodimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Necessary backgrounds from Frobenius algebras, topological quantum field theories, and cohomological field theories are reviewed. A result on Frobenius algebra twisted topological recursion is included at the end of Part 1. In Part 2, we explain a geometric theory of quantum curves. The focus is placed on the process of quantization as a passage from families of Hitchin spectral curves to families of opers. To make the presentation simpler, we unfold the story using SL2 (C)-opers and rank 2 Higgs bundles defined on a compact Riemann surface C of genus greater than 1. In this case, quantum curves, opers, and projective structures in C all become the same notion. Background materials on projective coordinate systems, Higgs bundles, opers, and non-Abelian Hodge correspondence are explained. Contents 0. Part 1. 2. 3. 4. 5. 6. Preface: An inspiration from the past 2 1. Topological Quantum Field Theory Frobenius algebras TQFT Cohomological field theory Category of cell graphs 2D TQFT from cell graphs TQFT-valued topological recursion Part 2. Quantization of Higgs Bundles 7. Quantum curves 8. Projective structures, opers, and Higgs bundles 9. Semi-classical limit of SL2 (C)-opers 10. Non-Abelian Hodge correspondence between Hitchin moduli spaces References 5 5 9 13 21 26 29 33 33 36 47 48 50 2010 Mathematics Subject Classification. Primary: 14H15, 14N35, 81T45; Secondary: 14F10, 14J26, 33C05, 33C10, 33C15, 34M60, 53D37. Key words and phrases. Topological quantum field theory; quantum curves; opers; Hitchin moduli spaces; Higgs bundles; Hitchin section; quantization; topological recursion. 1 2 O. DUMITRESCU AND M. MULASE 0. Preface: An inspiration from the past A recent discovery of a cuneiform tablet dated around 350 to 50 B.C.E. suggests that ancient Babylonians must have used geometry of time-momentum space to establish accurate calculations of Jupiter’s orbit [87]. This impressive paper also contains the picture of the tablet which describes the Babylonian’s method of integration. Figure 0.1. The cuneiform tablet used in the analysis of [87]. The orbit of the Jupiter is a graph in coordinate space-time. By considering the graph of momentum of the Jupiter in time-momentum space, Babylonians visualized integral of the momentum by the area underneath the curve, and using a trapezoidal approximation, they actually obtained an estimated value of the integral. This gives a prototype of Newton’s Fundamental Theorem of Calculus. This idea of Babylonians relating geometry of timemomentum space with the analysis of actual orbit of the Jupiter is striking, because it suggests their equal treatment of coordinate space the momentum space. Although it is a stretch, we could imagine the very foundation of symplectic geometry here. Many mathematical cuneiform tablets recording numbers and algebraic calculations have been our source of imagination. The most famous is Plimpton 322 of around 1,800 B.C.E. (see Figure 0.2). It lists 15 Pythagorean numbers in the increasing order of hypotenuse angles from about 45 degrees to 60 degrees [88]. Figure 0.2. Plimton 322 INVITATION TO 2D TQFT AND QUANTIZATION 3 Babylonians seem to have known an algorithm to calculate an approximate value of the square root of any √ number. For example, there is a cuneiform tablet that shows the sexagesimal expansion of 2. Although there have been many speculations for practical purposes of Plimton 322 and mechanisms to come up with the listed numbers, our imagination goes to the surprise of the creator of the tablet. For the pair of numbers (y, z) listed in the second and the third columns, z 2 − y 2 is always a perfect square. Therefore, the square root algorithm terminates in a finite number of steps for these values, and gives an exact answer. They must have found a finiteness in the forest of infinity. This is a strong sentiment that resonates with our mind today. The tablet of Figure 0.1 does not show any geometry. The author of [87] convincingly argues that behind the list of these seemingly meaningless numbers on the tablets, there is a profound geometric investigation of the planetary motion. Our imagination is piqued by the discovery: the interplay between algebra, geometry, and astronomy; and the equal treatment of momentum space and coordinate space. What the feeling of the authors of the tablet would have been, when they were creating it? It must have been a sharp happiness of discovery that mathematics correctly predicts the very nature surrounding us, such as planetary motion. The accuracy of calculations of Babylonians is another astonishment, in addition to the lack of pictures. It is in sharp contrast to Euclid’s Elements, which was written around the same time in Greece. In Elements, we see many beautiful geometric pictures, and proofs. The discovery of axiomatic method culminated in it. This dichotomy, being able to calculate a quantity to a high precision, vs. having a proof of the formula based on a finiteness property, is our very motivation of writing these notes. There are a lot of amazing formulas around us in the direction that we describe. At this moment, we do not have the final understanding of them, yet. In the first part of these lectures, we explore two-dimensional topological quantum field theories formulated in terms of cell graphs. A cell graph of type (g, n) is the 1-skeleton of a cell-decomposition of a compact oriented topological surface of genus g ≥ 0 with n labeled 0-cells, or vertices. Such graphs are called in many different names: ribbon graphs, dessins d’enfants, maps, embedded graphs, etc. We use the terminology “cell graph” indicating the different nature of the graphs on surfaces, which record degeneration of surfaces. Although a cell graph is a 1-dimensional object, the notion of a face is well defined as a 2-cell of the cell decomposition that the cell graph defines on a given surface. The degree of a vertex of a cell graph is the number of incident half-edges. We call a 1-cell an edge of the graph. As explained in our previous lecture notes [30], the number of 1 Cm , where cell graphs of type (0, 1) with the unique vertex of degree 2m is 2m 1 2m Cm = m+1 m is the m-th Catalan number. A generating function ∞ X Cm z = z(x) = m=0 1 x2m+1 of Catalan numbers satisfies an algebraic equation 1 (0.0.1) x=z+ . z The story of [30, Section 2] tells us that enumeration problem of cell graphs of arbitrary type (g, n) is solved by the quantization of the algebraic curve (0.0.1). The key formula 4 O. DUMITRESCU AND M. MULASE for the quantization comes from the Laplace transform of a combinatorial equation [30, Catalan Recursion, Section 2.1]. There, the combinatorial formula is derived by analyzing the effect of edge-contraction operations on cell graphs. The new story we wish to tell in these lectures is that the category of cell graphs carries the information of all two-dimensional topological quantum field theories. In Part 1 of this article, we will present an axiomatic setup of 2D TQFT. The key idea is simple: exact same edge-contraction operations characterize Frobenius algebras. When we contract an edge of a cell graph, two vertices collide. It represents multiplication. When we shrink a loop, since it is a cycle on the topological surface, the process breaks a vertex into two vertices. This is the operation of comultiplication. The topological structure of a cell graph makes these operations dual to one another in the context of Frobenius algebras. We start with defining Frobenius algebras. Topological quantum field theories (TQFT) and cohomological field theories (CohFT) are introduced in the following sections. A category of cell graphs is defined. We explain that this category generates all Frobenius algebras. We then make a connection between cell graphs and 2D TQFT. The theory of topological recursion is a leitmotiv in this article. It is another manifestation of a quantization procedure. Since there are many review articles on topological recursion, we touch the subject only tangentially in this article. Our new result related to this topic is the 2D TQFT-twisted topological recursion, which is explained in the end of Part 1. The cohomology ring of a closed oriented manifold X is a Frobenius algebra. When we consider an even dimensional manifold and only even degree cohomologies (0.0.2) A = H even (X, Q), then we have a commutative Frobenius algebra, which is equivalent to a 2D TQFT. The Gromov-Witten invariants of X of genus 0 determine a quantum deformation of the ring A, known as the big quantum cohomology. This quantum structure then defines a holomorphic object Y , the mirror of X. Thus the holomorphic geometry of Y captures quantum cohomology of X. Gromov-Witten invariants are generalized to arbitrary genera. Here arises a question: Question 0.1. What should be the holomorphic geometry on Y that corresponds to highergenus Gromov-Witten theory of X? If we consider the passage from genus 0 to higher genera Gromov-Witten theory as quantization, then on the side on Y , we need a quantized holomorphic geometry. A good candidate is the D-module theory. The simplest such theory fitting to this context is the notion of quantum curves. The relation between quantum curves and B-model geometry is considered in [2, 19, 17, 27, 28, 30, 49, 50] and others. The key point for this idea to work is when the holomorphic geometry Y corresponding to the Gromov-Witten theory of X is captured by an algebraic or analytic curve. These curves often appear in other areas of mathematics as spectral curves, such as integrable systems, random matrix theory, and Hitchin’s theory of Higgs bundles. A spectral curve naturally comes with a projection to a base curve, making it a covering of the base curve. A quantum curve is a D-module on this base curve, quantizing the spectral curve. Since a CohFT based on the Frobenius algebra H even (X, Q) plays a role similar to Gromov-Witten theory of X, CohFT is a quantization of 2D TQFT. This process consists of two steps of quantizations. The first one is from the classical cohomology ring A to its quantum deformation, which is the mirror geometry. If the mirror is a curve, then the second step should be parallel to the quantization of a spectral curve to a quantum curve. We may be able to understand the reconstruction [36] of CohFT based on a semi-simple INVITATION TO 2D TQFT AND QUANTIZATION 5 Frobenius algebra from the 2D TQFT, that is obtained as the degree 0 part of the cohomology H 0 (Mg,n , Q) of the moduli space of stable curves, analogous to the construction of quantum curves. Part 2 deals with quantum curves. Since the authors have produced a long article [30] explaining the relation between quantum curves and topological recursion, the present article is focused on geometry of the process of quantization of Hitchin spectral curves from a perspective of opers. Instead of providing a general theory of [25, 26, 31], we use SL2 (C) to explain our ideas here. We will show that the classical notion of projective structures on a compact Riemann surface C of genus g ≥ 2 studied by Gunning [51], SL2 (C)-opers, and quantum curves of Hitchin spectral curves, are indeed the exact same notion. A conjecture of Gaiotto [42] relating non-Abelian Hodge correspondence and opers, together with its solution by [26], will be briefly explained in the final section. Acknowledgement. The authors are grateful to the organizers of the Warsaw Advanced School on Topological Quantum Field Theory for providing the opportunity to produce this article. They are in particular indebted to Piotr Sulkowski for constant encouragement and patience. The joint research of authors presented in this article is carried out while they have been staying in the following institutions in the last four years: the American Institute of Mathematics in California, the Banff International Research Station, Institutul de Matematică “Simion Stoilow” al Academiei Romane, Institut Henri Poincaré, Institute for Mathematical Sciences at the National University of Singapore, Kobe University, Leibniz Universität Hannover, Lorentz Center Leiden, Mathematisches Forschungsinstitut Oberwolfach, MaxPlanck-Institut für Mathematik-Bonn, and Osaka City University Advanced Mathematical Institute. Their generous financial support, hospitality, and stimulating research environments are greatly appreciated. The research of O.D. has been supported by GRK 1463 Analysis, Geometry, and String Theory at the Leibniz Universität Hannover, Perimeter Institute for Theoretical Physics in Waterloo, and a grant from MPIM-Bonn. The research of M.M. has been supported by IHÉS, MPIM-Bonn, Simons Foundation, Hong Kong University of Science and Technology, Université Pierre et Marie Curie, and NSF grants DMS-1104734, DMS-1309298, DMS-1619760, DMS-1642515, and NSF-RNMS: Geometric Structures And Representation Varieties (GEAR Network, DMS-1107452, 1107263, 1107367). Part 1. Topological Quantum Field Theory 1. Frobenius algebras Throughout this article, we denote by K a field of characteristic 0. Most of the cases we consider K = Q or K = C. Let A be a finite-dimensional, unital, and associative algebra defined over a field K. A bialgebra comes with an extra set of structures, including a comultiplication. Examples of bialgebras include the group algebra A = K[G] of a finite group G. Its algebra structure is canonically determined by the multiplication rule of G. However, the group algebra has two distinct co-algebra structures. One of which makes K[G] a Frobenius algebra, and the other makes it a Hopf algebra. This difference is reflected in which topological invariants we are dealing with. The Frobenius structure is useful for twodimensional topology, and the Hopf structure is natural for three-dimensional topological invariants. Let us start with defining Frobenius algebras. 6 O. DUMITRESCU AND M. MULASE Definition 1.1 (Frobenius algebras). A finite-dimensional, unital, and associative algebra A over a field K is a Frobenius algebra if there exists a linear map : A −→ K called counit such that for every u ∈ A, (ux) = 0, or (xu) = 0, for all x ∈ A implies that u = 0. Proposition 1.2. If A1 and A2 are Frobenius algebras, then A1 ⊕ A2 and A1 ⊗ A2 are also Frobenius algebras. Let us denote by A the category of Frobenius algebras defined over K. Then C = (A, ⊗, K) is a monoidal category. Proof. Denote by i : Ai −→ K the counit of Ai , i = 1, 2. Then (A1 ⊕ A2 , 1 + 2 ) is a Frobenius algebra. Similarly, 1 · 2 : A1 ⊗ A2 3 u1 ⊗ u2 7−→ 1 (u1 )2 (u2 ) ∈ K makes A1 ⊗ A2 a Frobenius algebra. Example 1.3. The one-dimensional vector space A = K, with the identity map (u) = u, is a simple commutative Frobenius algebra. More generally, by taking the direct sum A = K ⊕n and defining (u1 , . . . , un ) = u1 + · · · + un ∈ K, we construct a semi-simple commutative Frobenius algebra K ⊕n . Example 1.4. The full matrix algebra A = Matn (K) consisting of n × n matrices with (u) = trace(u) is an example of a non-commutative simple Frobenius algebra for n ≥ 2. Example 1.5. As mentioned above, the group algebra A = K[G] of a finite group G is a Frobenius algebra. Here, we define : K[G] −→ K by linearly extending the map whose value for every g ∈ G is given by ( 1 g=1 (g) = 0 g 6= 1. Remark 1.6. If we define (g) = 1 for all g ∈ G, then the group algebra becomes a Hopf algebra. Example 1.7. The cohomology ring H ∗ (M, R) of an oriented compact differential manifold M of dimR M = n with the cup product and : H ∗ (M, R) −→ H n (M, R) = R is a Frobenius algebra. The above example makes us wonder if an analogue of Poincaré pairing exists in a general Frobenius algebra. Indeed, the counterpart is a Frobenius bilinear form defined by (1.0.1) η : A ⊗ A −→ K, η(u, v) = (uv) in terms of the counit : A −→ K. The Frobenius form satisfies the Frobenius associativity (1.0.2) η(uv, w) = η(u, vw), u, v, w ∈ A. The condition for the counit of Definition 1.1 exactly means that the Frobenius form η is non-degenerate. It therefore defines a canonical isomorphism (1.0.3) ∼ λ : A −→ A∗ , hλ(u), vi = η(u, v), u, v ∈ A. INVITATION TO 2D TQFT AND QUANTIZATION 7 The isomorphism introduces a unique comultiplication δ : A −→ A ⊗ A by the following commutative diagram: (1.0.4) δ A λ / A⊗A A∗ m∗ λ⊗λ . / A∗ ⊗ A∗ Here, m∗ : A∗ −→ A∗ ⊗ A∗ is the dual of the multiplication operation m : A ⊗ A −→ A in A. Since we are not assuming the multiplication to be commutative, we define the natural pairing (A∗ ⊗ A∗ ) ⊗ (A ⊗ A) −→ K by observing the order of entities as written. For example, for α, β ∈ A∗ and u, v ∈ A, we calculate (1.0.5) hα ⊗ β, u ⊗ vi = hα, hβ, uivi = hβ, uihα, vi. As the dual of the associative multiplication, the comultiplication δ is coassociative, i.e., it satisfies A8 ⊗ A (1.0.6) δ δ⊗1 & A ⊗8 A ⊗ A. A & δ 1⊗δ A⊗A Note that the identity element 1 ∈ A corresponds to ∈ A∗ by λ, i.e., λ(1) = . This is because η(1, u) = η(u, 1) = (u). We now have the full set of data (A, 1, m, , δ) that defines a bialgebra. The algebra and coalgebra structures satisfy a compatibility condition, as described below. Proposition 1.8. The following diagram commutes: (1.0.7) 1⊗δ A⊗A A⊗ 8 A⊗A /A m δ⊗1 & m⊗1 & / A ⊗ A. 8 δ 1⊗m A⊗A⊗A Remark 1.9. Alternatively, we can define a Frobenius algebra as a bialgebra satisfying (1.0.7), together with a non-degenerate Frobenius form satisfying (1.0.2). Proof. Let he1 , e2 , . . . , er i be a K-basis for A, where r = dimK A. The bilinear form η defines r × r matrices (1.0.8) η = [ηij ], η −1 = [η ij ], ηij := η(ei , ej ). 8 O. DUMITRESCU AND M. MULASE It gives the canonical basis expansion of v ∈ A: X X (1.0.9) v= η(v, ea )η ab eb = η(ea , v)η ba eb . a,b a,b From (1.0.4), we calculate δ(v) = X η(v, ei ej )(η jb eb ) ⊗ (η ia ea ) i,j,a,b = X = X η(vei , ej )(η jb eb ) ⊗ (η ia ea ) i,j,a,b (vei ) ⊗ (η ia ea ) i,a = (m ⊗ 1)(v ⊗ δ(1)), using the pairing convention (1.0.5) and (1.0.10) δ(1) = X η ab ea ⊗ eb . a,b We then have  (1 ⊗ m) ◦ (δ ⊗ 1)(u ⊗ v) = (1 ⊗ m)   X η(u, ei ej )(η jb eb ) ⊗ (η ia ea ) ⊗ v  i,j,a,b = X η(uei , ej )(η jb eb ) ⊗ (η ia ea v) i,j,a,b = X η(uei , ej )(η jb eb ) ⊗ η ia η(ea v, ec )η cd ed i,j,a,b,c,d = X uei ⊗ η ia η(ea , vec )η cd ed i,a,c,d = X (uvec ) ⊗ η cd ed c,d = (m ⊗ 1)(uv ⊗ δ(1)) = δ(uv). Similarly,  (m ⊗ 1) ◦ (1 ⊗ δ)(u ⊗ v) = (m ⊗ 1) u ⊗  X η(v, ei ej )(η jb eb ) ⊗ (η ia ea ) i,j,a,b = X uη(vei , ej )(η jb eb ) ⊗ (η ia ea ) i,j,a,b = X uvei ⊗ η ia ea i,a = (m ⊗ 1)(uv ⊗ δ(1)) = δ(uv). This completes the proof of Proposition 1.8. Remark 1.10. We note that if A is commutative, then the quantities (1.0.11) η ei1 · · · eij , eij+1 · · · en = (ei1 · · · ein ), 1 ≤ j < n, INVITATION TO 2D TQFT AND QUANTIZATION 9 are completely symmetric with respect to permutations of indices. Definition 1.11 (Euler element). The Euler element of a Frobenius algebra A is defined by (1.0.12) e := m ◦ δ(1). In terms of basis, the Euler element is given by X (1.0.13) e= η ab ea eb . a,b The Euler element provides the genus expansion of 2D TQFT, allowing us to calculate higher genus correlation functions from the genus 0 part of the theory. Another application of (1.0.9) is the following formula that relates the multiplication and comultiplication: (1.0.14) (λ(u) ⊗ 1) δ(v) = uv. This is because (λ(u) ⊗ 1) δ(v) = (λ(u) ⊗ 1) X (vη ab ea ) ⊗ eb a,b = X = X η(u, vea )η ab eb a,b η(uv, ea )η ab eb = uv. a,b 2. TQFT In this section, we briefly review TQFT. Although there have been explosive mathematical developments in higher dimensional topological quantum field theories mixing different dimensions during the last decade (see for example, [13, 67] and more recent work inspired by them), we restrict our attention to the two-dimensional speciality in these lectures. From now on, Frobenius algebras we consider are finite-dimensional, unital, associative, and commutative. It has been established (see [1, 16]) that 2D TQFT’s are classified by these types of Frobenius algebras. The axiomatic formulation of conformal and topological quantum field theories was discovered in the 1980s by Atiyah [4] and Segal [89]. A (d−1)-dimensional TQFT is a monoidal functor Z from the monoidal category of (d − 1)-dimensional closed (i.e., compact without boundary) oriented smooth manifolds with oriented d-dimensional cobordism as morphisms, to the monoidal category of finite-dimensional vector spaces defined over a field K. The monoidal structure in the category of (d − 1)-dimensional smooth manifolds is defined by the operation of taking disjoint union, which is a symmetric operation. Disjoint union with the empty set is the identity of this operation. Therefore, we define the monoidal category of K-vector spaces by symmetric tensor products, with the field K serving as the identity operation of tensor product. The functor Z satisfies the non-triviality condition Z(∅) = K, and maps a disjoint union of smooth (d − 1)-dimensional manifolds to a symmetric tensor product of vector spaces. 10 O. DUMITRESCU AND M. MULASE Let us denote by M an oriented closed smooth manifold of dimension d − 1, and by M op the same manifold with the opposite orientation. The functor Z satisfies the duality condition Z(M op ) = Z(M )∗ , where Z(M )∗ is the dual vector space of Z(M ). Suppose we have an oriented bordered d-dimensional smooth manifold N . The boundary ∂N is a smooth manifold of dimension d − 1, and the complement N \ ∂N is an oriented smooth manifold of dimension d. We give the orientation induced from N to its boundary ∂N . The TQFT functor Z then gives an element Z(N ) ∈ Z(∂N ). If N is closed, i.e., ∂N = ∅, then Z(N ) ∈ Z(∅) = K is a number that represents a topological invariant of N . Now consider a bordered oriented smooth manifold N1 with boundary ∂N1 = M1op t M2 , meaning that the two separate boundaries carry different orientations. We choose that M2 is given the induced orientation from N1 , and M1 the opposite orientation. We interpret the situation as N1 giving an oriented cobordism from M1 to M2 . In this case, the functor Z defines an element Z(N1 ) ∈ Z(M1 )∗ ⊗ Z(M2 ), or equivalently, a linear map Z(N1 ) : Z(M1 ) −→ Z(M2 ). Suppose we have another smooth manifold N2 with boundary ∂N2 = M2op t M3 corresponding to a linear map Z(N2 ) : Z(M2 ) −→ Z(M3 ). We can then smoothly glue N1 and N2 along the common boundary component M2 forming a new manifold N = N1 ∪M2 N2 . Clearly ∂N = M1op t M3 , and N gives a cobordism from M1 to M3 . The Atiyah-Segal sewing axiom [4] asserts that (2.0.1) Z(M1 ) k Z(M1 ) Z(N1 ) / Z(M2 ) Z(N2 ) Z(N ) / Z(M3 ) k / Z(M3 ). A d-dimensional TQFT Z defines a topological invariant Z(N ) for each closed d-manifold N . The essence of TQFT is to break N into simpler pieces by cutting along (d − 1)dimensional submanifolds, and represent the invariant Z(N ) from that of simpler pieces. The situation is special for the case of 2D TQFT. First of all, we already know all topological invariants of a closed surface. They are just functions of the genus g of the surface. So there should be no reason for another theory to study them. Since there is only one 1-dimensional connected compact smooth manifold, a 2D TQFT is based on a single vector space (2.0.2) Z(S 1 ) = A, INVITATION TO 2D TQFT AND QUANTIZATION 11 and its tensor products. What else could we gain? The biggest surprise is that a 2D TQFT is further quantized [15, 24, 48, 63]. This process starts with a Frobenius algebra A of (0.0.2). It is quantized to a quantum cohomology, and then further to Gromov-Witten invariants. Construction of spectral curves, and then quantizing them, have a parallelism to this story. This is the story we now explore in this and the next sections, but only to take a snapshot of the converse direction, i.e., from a quantum theory to a classical theory. Let us denote by Σg,m̄,n a connected, bordered, oriented, smooth surface of genus g with m + n boundary circles. This is a surface obtained by removing m + n disjoint open discs from a compact oriented two-dimensional smooth manifold of genus g without boundary. We assume that the boundary of Σg,m̄,n itself is a smooth manifold, hence it is a disjoint union of m + n circles. We give the induced orientation from the surface to n boundary circles, and the opposite orientation to the m boundary circles. The surface Σg,m̄,n gives a cobordism of m circles to n circles. The orientation-preserving diffeomorphism class of such a surface is determined by the genus g and the two number m and n of boundary circles with different orientations. Therefore, the oriented equivalence class of cobordism is also determined by (g, m, n). The TQFT functor Z then assigns to each cobordism Σg,m̄,n a multilinear map def ωg,m̄,n = Z(Σg,m̄,n ) : A⊗m −→ A⊗n , which is completely determined by the label (g, m, n). This is the special situation of the two-dimensionality of TQFT, reflecting the simple topological classification of surfaces. Suppose we have another cobordism Σh,k̄,m of genus h from k circles to m circles, with orientation on the m circles induced from the surface, and the opposite orientation on k circles. We can compose two cobordisms, sewing the m circles of the first cobordism with the m circles of the second. Here, we notice that on each pair of circles, one from the first surface and the other from the second, the orientations are the same, and hence we can put one on top of the other. Therefore, the orientations of the two surfaces are consistent after sewing. This sewing process generates a new surface Σg+h+m−1,k̄,n of genus g + h + m − 1. This is because the m pairs of circles sewed together create m − 1 handles (see Figure 2.1). The sewing axiom of Atiyah-Segal [4] then requires that the functor Z associates the composition of linear maps to the composition of cobordisms. In our situation, the composition of two maps generates a new map ωg,m̄,n ◦ ωh,k̄,m = ωg+h+m−1,k̄,n : A⊗k −→ A⊗n , corresponding to the sewing of cobordisms. Figure 2.1. A string of String Interactions. The physics of a simple model for interacting strings is captured by the sewing axiom of Atiyah-Segal. The sewing procedure can be generalized, allowing partial sewing of the boundary circles of matching orientation. For example, if j ≤ ` and j ≤ m, then gluing only j pairs of circles, 12 O. DUMITRESCU AND M. MULASE we have a composition (2.0.3) ωg,m̄,n ◦ ωh,k̄,` = ωg+h+j−1,k+m−j,n+`−j : A⊗k+m−j −→ A⊗n+`−j . Furthermore, since TQFT does not require cobordism to be given by a connected manifold, we can stack up disjoint union of surfaces and apply partial sewing to create a variety of linear maps from A⊗n to A⊗m . It was Dijkgraaf [16] who noticed the equivalence between 2D TQFTs and commutative Frobenius algebras. We can see the resemblance immediately. On the vector space A, we have operations defined by (2.0.4) 1 = ω0,0̄,1 : K −→ A, m = ω0,2̄,1 : A⊗2 −→ A, = ω0,1̄,0 : A −→ K, δ = ω0,1̄,2 : A −→ A⊗2 , η = ω0,2̄,0 : A⊗2 −→ K. A connected cobordism of m incoming circles and n outgoing circles is classified by the genus g of the surface. It does not depend on the history of how the cobordisms are glued together. Thus the diffeomorphism classes of surfaces after sewing cobordisms tell us relations among the operations of (2.0.4). For example, ω0,1̄,0 ◦ ω0,2̄,1 = ω0,2̄,0 =⇒ ◦ m = η, which is (1.0.1). Associativity of multiplication comes from uniqueness of the topology of Σ0,3̄,1 with one boundary circle carrying the induced orientation and three the opposite orientation. Changing the orientation of each boundary component to its opposite provides coassociativity. In this way A = Z(S 1 ) acquires a bialgebra structure. To assure duality between algebra and coalgebra structures, we need to impose another condition here: nondegeneracy of η = ω0,2̄,0 . Then A becomes a commutative Frobenius algebra. Conversely, if we start with a finite-dimenionsal commutative Frobenius algebra A, then we first construct ω0,m̄,n on the list of (2.0.4). More general maps ωg,m̄,n are constructed by partial sewing (2.0.3). By construction, all these maps are associated with cobordism of circles, and hence determine a 2D TQFT [1]. With the above considerations, we give the following definition. Definition 2.1 (Two-dimensional Topological Quantum Field Theory). Let (A, , λ) be a set of data consisting of a finite-dimensional vector space A over K, a non-trivial linear ∼ map : A −→ K, and an isomorphism λ : A −→ A∗ . A Two-dimensional Topological Quantum Field Theory is a system (A, {ωg,m̄,n }) consisting of linear maps ωg,k̄,` : A⊗k −→ A⊗` , 0 ≤ g, k, `, satisfying the following axioms. • TQFT 1. Symmetry: (2.0.5) ωg,k̄,` : A⊗k −→ A⊗` is symmetric with respect to the Sk -action on the domain. • TQFT 2. Non-triviality: (2.0.6) ω0,1̄,0 = : A −→ K. INVITATION TO 2D TQFT AND QUANTIZATION 13 • TQFT 3. Duality: A⊗m (2.0.7) λ⊗m (A∗ )⊗m ωg,m̄,n (ωg,n̄,m )∗ / A⊗n λ⊗n . / (A∗ )⊗n • TQFT 4. Sewing: (2.0.8) ωg,m̄,n ◦ ωh,k̄,` = ωg+h+j−1,k+m−j,n+`−j : A⊗k+m−j −→ A⊗n+`−j , j ≤ m, j ≤ `. Remark 2.2. As noted above, the vector space A acquires the structure of a commutative Frobenius algebra from these axioms. Conversely, if A is a commutative Frobenius algebra, then ω0,m̄,n of (2.0.4) can be extended to ωg,m̄,n that satisfy the TQFT axioms. 3. Cohomological field theory Recall that for any oriented closed manifold X, its cohomology ring H ∗ (X, K) is a Frobenius algebra defined over K. If we restrict ourselves to even dimensional manifold X and consider only even part of the cohomology, then A = H even (X, K) is a finite-dimensional commutative Frobenius algebra. This Frobenius algebra naturally defines a 2D TQFT. The role of a TQFT in general dimension is to represent a topological invariant of higher dimensional manifolds. We are now seeing that the classical topology of a manifold X is a 2D TQFT. Changing our point of view, we can ask if we start with X, then how do we find the corresponding 2D TQFT? Of course the Frobenius algebra structure automatically determine the unique 2D TQFT as we have seen in Section 2. But then the picture of sewing cobordisms is lost in this algebraic formulation. What is the role of the cobordism of circles in the context of understanding the manifold X? The amazing vision emerged in the early 1990s is that 2D TQFT can be further quantized into Gromov-Witten theory, which produces quantum topological invariants of X that cannot be reduced to classical invariants. This epoch making discovery was one of the decisive moments of the fruitful interaction of string theory and geometry, which is still continuing today. String theory deals with flying strings in a space-time manifold X. The trajectory of a moving string is a curved cylinder embedded in the manifold X, like a duct pipe in the attic. When strings are considered as quantum objects, they can interact one another. For example, Figure 2.1 can be interpreted as a string of string interactions. First, three strings collide in a complicated interaction to produce two strings. The complexity of the interaction is classified as “g = 2” in terms of the genus of the first surface. These two strings created in the first interaction then collide in an even more complicated (i.e., g = 3) interaction to produce four strings in the end. The trajectory of these interactions then forms a bordered surface of genus 6 with 7 boundary components embedded in the space-time X. Models for string theory were originally introduced to understand how the space-time X dictates string interactions. In this process, the importance of Calabi-Yau spaces was recognized, through considerations on the consistency of string theory as a physical theory. By changing the point of view 180◦ , researchers then noticed that a simple model of string interactions could be used to obtain a totally new class of topological invariants of X itself. These new invariants are not necessarily captured by the classical topology, such as (co)homology groups and homotopy groups. We use the terminology of quantum topological invariants for the invariants obtained by string theory considerations. 14 O. DUMITRESCU AND M. MULASE Gromov-Witten theory is a mathematically rigorous interpretation of a model for a quantum string theory. A trajectory of interacting strings in X is the image of a map (3.0.1) f : Σg,m̄,n −→ X from a bordered surface Σg,m̄,n into X. Recall that the fundamental group π1 (X, x0 ) of X is determined by the connectivity of the pointed loop space L(X, x0 ) = C ∞ (S 1 , ∗), (X, x0 ) , which is the moduli space of differentiable maps from S 1 to X that send a reference point ∗ ∈ S 1 to x0 ∈ X. In the same spirit, Gromov-Witten invariants are defined by looking at the classical topological invariants of appropriate moduli spaces of maps from Σg,m̄,n to X. When we have the notion of size of a string, the trajectory Σg,m̄,n has a metric on it. If the ambient manifold X has a geometric structure, such as a symplectic structure, complex structure, or a Kähler structure, then the map f needs to be compatible with the structures of the source and the target. For example, if X is Kähler, then we give a complex structure on Σg,m̄,n that is compatible with the metric it has, and require that f is a holomorphic map. Technical difficulties arise in defining moduli spaces of such maps. Even the moduli spaces are defined, they are often very different objects from the usual differentiable manifolds. We need to extend the notion of manifolds here. Thus identifying reasonable classical topological invariants of these spaces also poses a difficult problem. The simplest scenario is the following: We take X = pt to be just a single point. Of course nobody wants to know the topological structure of a point. We all know it! By exploring the Gromov-Witten theory of a point, we learn the structure of the theory itself. Since we can map anything to a point, a point may not be such a simple object, after all. It is like considering a vacuum in physics. Again, anything can be thrown into a vacuum. A quantum theory of a vacuum is inevitably a rich theory. When X = pt is a point, we consider all incoming and outgoing strings to be infinitesimally small. Thus the surfaces we are considering become closed, and boundary components are just several points identified on them. A metric on a closed surface naturally gives rise to a unique complex structure, making it a compact Riemann surface. And every compact Riemann surface acquires a unique projective algebraic structure, making it a projective algebraic curve. Since we are allowing strings to become infinitesimally small during the interaction, the trajectory may contain a process of an embedded S 1 in the surface shrinking to a point, and then becoming a finite size circle again a moment later. Such a process can be understood in complex algebraic geometry as a nodal singularity of an algebraic curve. Locally, every nodal singularity of a curve is the same as the neighborhood of the origin of a singular algebraic curve {(x, y) ∈ C2 \| xy = 0}. We are talking about an abstract notion of trajectories here, because nothing can move in X = pt. In terms of the map idea (3.0.1), we note that there is a unique map f from any projective algebraic curve to a point, which satisfies any reasonable requirement of maps such as being a morphism of algebraic varieties. Then the moduli space of all maps simply means the moduli space of the source. A stable curve is a projective algebraic curve with only nodal singularities and a finite number of smooth points marked on the curve such that it has only a finite number of algebraic automorphisms fixing each marked point. We recall that the holomorphic automorphism group of P1 , Aut(P1 ) = P SL2 (C), INVITATION TO 2D TQFT AND QUANTIZATION 15 acts on P1 triply transitively. The number of automorphisms fixing up to 2 points is always infinity. Since P SL2 (C) is compact and three-dimensional, if we choose three distinct points on P1 and require the automorphism to fix each of these points, then we have only finitely many choices. Actually in our case, it is unique. For an elliptic curve E, since it is an abelian group, E acts on E transitively. To avoid these translations, we need to choose a point on E. We can naturally identify it as the identity element of the elliptic curve as a group. Automorphisms of an elliptic curve fixing a point then form a finite group. If a compact Riemann surface C has genus g = g(C) ≥ 2, then it is known that the order of the analytic automorphism group is bounded by \|Aut(C)\| ≤ 84(g − 1). This bound comes from hyperbolic geometry. It is easy to see the finiteness. First, we note that universal covering of C is the upper half plane H = {z ∈ C \| Im(z) > 0}, and C is constructed by the quotient ∼ C −→ H/ρ(π1 (C)) through a faithful representation (3.0.2) ρ : π1 (C) −→ P SL2 (R) = Aut(H) of the fundamental group of C into the automorphism group of H. Every holomorphic automorphism of C extends to an automorphism of H. We know that C does not have any non-trivial holomorphic vector field v. If it did, then v would be a differentiable vector field with isolated zeros, and each zero comes with positive index, because locally it is given by z n d/dz. We learn from topology that the sum of the indices of isolated zeros of a vector field on C is equal to χ(C) = 2 − 2g < 0. It is a contradiction. Thus Aut(C) ⊂ P SL2 (C) is a discrete subgroup. Since P SL2 (R) is compact, \|Aut(C)\| is finite. An algebraic curve C is stable if and only if (1) every singularity is nodal, and (2) every irreducible component C 0 of C has a finite number of automorphisms. The second condition means that the total number of smooth marked points and singular points of C that are on C 0 has to be 3 or more if g(C 0 ) = 0, and 1 or more if g(C 0 ) = 1. There is no condition for an irreducible component of genus two or more. For a pair (g, n) of integers g ≥ 0 and n ≥ 1 in the stable range 2g − 2 + n > 0, we denote by Mg,n the moduli space of stable curves of genus g and n smooth marked points. It is a complex orbifold of dimension 3g − 3 + n. Quantum topological invariants of a point X = pt is then realized as classical topological invariants of Mg,n . Alas to this day, we cannot identify the cohomology groups H i (Mg,n , Q) for all values of i, g, n. From the very definition of these moduli spaces, we can construct many concrete cohomology classes on each of Mg,n , called tautological classes. What we do not know is indeed how much more we need to know to determine all of H i (Mg,n , Q). The surprise of Witten’s conjecture [93], proved in [62], is that we can actually explicitly write intersection relations of certain tautological classes for all values of g and n. Among many different proofs available for the Witten conjecture (see for example, [61, 62, 71, 72, 80, 85]), [72, 80] deal with recursive relations among surfaces much in the same spirit of TQFT. These relations are first noticed in [20]. We will discuss these relations in connection to topological recursion later in these lectures. 16 O. DUMITRESCU AND M. MULASE Geometric relations among Mg,n ’s for different values of (g, n) have a simple meaning. Let us denote by Mg,n the moduli space of smooth n-pointed curves. Then the boundary Mg,n \ Mg,n consists of points representing singular curves. Simplest singular stable curve has one nodal singularity, which can be described as collision of two smooth points. Analyzing how these singularities occur via degeneration, we come up with three types of natural morphisms among the moduli spaces Mg,n . They are the forgetful morphisms (3.0.3) π : Mg,n+1 −→ Mg,n which simply erase one of the marked points on a stable curve, and gluing morphisms (3.0.4) gl1 : Mg−1,n+2 −→ Mg,n (3.0.5) gl2 : Mg1 ,n1 +1 × Mg2 ,n2 +1 −→ Mg1 +g2 ,n1 +n2 that construct boundary strata of Mg,n . Under a gluing morphism, we put two smooth points of stable curves together to form a one nodal singularity. The first one gl1 glues two points on the same curve together, and gl2 one each on two curves. The fiber of π at a stable curve (C, p1 , . . . , pn ) ∈ Mg,n is the curve C itself, because another marked point can be placed anywhere on C. Thus π is a universal family of curves parameterized by the base moduli space Mg,n . When we place the extra marked point pn+1 at pi , i = 1, . . . , n, then as the point of Mg,n+1 on the fiber of π, the data represented is a singular curve obtained by joining C itself with a P1 at the location of pi ∈ C, but the two marked points pi , pn+1 are actually placed on the line P1 . Assigning this singular curve to (C, p1 , . . . , pn ) defines a section σi : Mg,n −→ Mg,n+1 , which is a right inverse of π. Geometrically, σi sends (C, p1 , . . . , pn ) to the point pi on the fiber C = π −1 (C, p1 , . . . , pn ). Obviously, π ◦ σi : Mg,n −→ Mg,n is the identity map. Gromov-Witten theory for a Kähler manifold X [15] concerns topological structure of the moduli space Mg,n (X, β) = f : (C, p1 , . . . , pn ) −→ X [f (C)] = β of stable holomorphic maps f from a nodal curve C with n smooth marked points p1 , . . . , pn ∈ C to X such that the homology class of the image f (C) agrees with a prescribed homology class β ∈ H2 (X, Z). Here, stability of a map f is again defined by imposing the finiteness of possible automorphisms. Giving a definition of this moduli space is beyond our scope of this article. We refer to [15]. The moduli space, if defined, should come with natural maps ev i Mg,n (X, β) −−−− → X   φy Mg,n , where the forgetful map φ assigns the stabilization of the source (C, p1 , . . . , pn ) to the map f by forgetting about the map itself, and evi f : (C, p1 , . . . , pn ) −→ X = f (pi ) ∈ X, i = 1, . . . , n, INVITATION TO 2D TQFT AND QUANTIZATION 17 is the value of f at the i-th marked point pi ∈ C. Stabilization means that evey irreducible component of (C, p1 , . . . , pn ) that is not stable is shrunk to a point. If we indeed know the moduli space Mg,n (X, β) and that its cohomology theory behaves as we expect, then we would have ev∗ H ∗ Mg,n (X, β), Q ←−−i−− H ∗ (X, Q)   φ! y H ∗ (Mg,n , Q), where φ! is the Gysin map defined by integration along fiber. If Mg,n (X, β) were a manifold, and the map φ : Mg,n (X, β) −→ Mg,n were a fiber bundle, then Mg,n (X, β) would have been locally a direct product, hence the Gysin map φ! associated with φ would be defined by integrating de Rham cohomology classes of Mg,n (X, β) along fiber of φ. Choose any cohomology classes v1 , . . . vn ∈ H ∗ (X, Q) of X. We then could have defined the GromovWitten invariants by Z X,β GWg,n (v1 , . . . , vn ) := φ! ev1∗ (v1 ) · · · evn∗ (vn ) . Mg,n In general, however, construction of the Gysin map does not work as we hope. This is due to the complicated nature of the moduli space Mg,n (X, β), which often has components of unexpected dimensions. The remedy is to define the virtual fundamental class [Mg,n (X, β)]vir of the expected dimension, and avoid the use of φ! by defining Z X,β (3.0.6) GWg,n (v1 , . . . , vn ) := ev1∗ (v1 ) · · · evn∗ (vn ). [Mg,n (X,β)]vir See [15] for more detail. Now recall that A = H even (X, Q) is a Frobenius algebra. Although Gromov-Witten theory goes through a big black box [Mg,n (X, β)]vir , what we wish is a map Ωg,n : H even (X, Q)⊗n −→ H ∗ (Mg,n , Q) whose integral over Mg,n gives the quantum invariants of X. Then what are the properties that Gromov-Witten invariants should satisfy? Can we list the properties as axioms for the above map Ωg,n so that we can characterize Gromov-Witten invariants? This was one of the motivations of Kontsevich and Manin to introduce CohFT in [63]. As we see below, a 2D TQFT can be obtained as a special case of a CohFT. The amazing relation between TQFT and CohFT, i.e., the reconstruction of CohFT from its restriction to TQFT due to Givental and Teleman [48, 91], plays a key role in many new developments (see for example, [3, 36, 39, 68]), some of which are deeply related with topological recursion. Definition 3.1 (Cohomological Field Theory [63]). Let A be a finite-dimensional, unital, associative, and commutative Frobenius algebra with a basis {e1 , . . . , er }. A Cohomological Field Theory is a system (A, {Ωg,n }) consisting of linear maps (3.0.7) Ωg,n : A⊗n −→ H ∗ (Mg,n , K) defined for (g, n) in the stable range 2g − 2 + n > 0 and satisfying the following axioms: CohFT 0: Ωg,n is Sn -invariant, and Ω0,3 (v1 , v2 , v3 ) = η(v1 v2 , v3 ). CohFT 1: Ωg,n+1 (v1 , . . . , vn , 1) = π ∗ Ωg,n (v1 , . . . , vn ). X gl1∗ Ωg,n (v1 , . . . , vn ) = Ωg−1,n+2 (v1 , . . . , vn , ea , eb )η ab . CohFT 2: a,b 18 O. DUMITRESCU AND M. MULASE CohFT 3: gl2∗ Ωg1 +g2 ,\|I\|+\|J\| (vI , vJ ) = X η ab Ωg1 ,\|I\|+1 (vI , ea ) ⊗ Ωg2 ,\|J\|+1 (vJ , eb ), a,b where I t J = {1, . . . , n} is a disjoint partition of the index set, and the tensor product operation on the right-hand side is performed via the Künneth formula of cohomology rings H ∗ (Mg1 ,n1 +1 × Mg2 ,n2 +1 , K) ∼ = H ∗ (Mg1 ,n1 +1 , K) ⊗ H ∗ (Mg2 ,n2 +1 , K). Remark 3.2. The condition Ω0,3 (v1 , v2 , v3 ) = η(v1 v2 , v3 ) says that the product of the Frobenius algebra is determined by the (0, 3)-value of the Gromov-Witten invariants. Proposition 3.3 (2D TQFT is a CohFT). Every 2D TQFT is a CohFT that takes values in H 0 (Mg,n , K). More precisely, let (A, ωg,m̄,n ) be a 2D TQFT. Then ωg,n = ωg,n̄,0 satisfies the CohFT axioms, by identifying K = H 0 (Mg,n , K). Proof. Since Mg,n is connected, the three types of morphisms (3.0.3), (3.0.4), and (3.0.5) all produce isomorphisms of degree 0 cohomologies. Thus the axioms CohFT 1–3 become (3.0.8) (3.0.9) ωg,n+1 (v1 , . . . , vn , 1) = ωg,n (v1 , . . . , vn ), X ωg,n (v1 , . . . , vn ) = ωg−1,n+2 (v1 , . . . , vn , ea , eb )η ab , a,b (3.0.10) ωg1 +g2 ,\|I\|+\|J\| (vI , vJ ) = X η ab ωg1 ,\|I\|+1 (vI , ea ) · ωg2 ,\|J\|+1 (vJ , eb ). a,b We wish to show that (3.0.8)-(3.0.10) are consequences of the partial sewing axiom (2.0.8) of TQFT under the identification ωg,n = ωg,n̄,0 . Since ω0,0̄,1 = 1, we have ωg,n+1,0 ◦(n+1) ω0,0̄,1 = ωg,n̄,0 = ωg,n : A⊗n −→ K from (2.0.8), which is (3.0.8). Here, ◦(n+1) is the composition taken at the (n + 1)-th slot of the input variables of ωg,n+1,0 . Next, we need to identify ω0,0̄,2 : K −→ A ⊗ A. Since P ω0,0̄,2 = ω0,1̄,2 ◦ ω0,0̄,1 , we see that ω0,0̄,2 (1) = δ(1) = a,b η ab ea ⊗ eb . Denoting by ◦(n+1,n+2) to indicate composition taking at the last two slots of variables, we have (3.0.9) ωg−1,n+2,0 ◦(n+1,n+2) ω0,0̄,2 = ωg,n̄,0 : A⊗n −→ K. If we have two disjoint sets of variables vI and vJ , then we can apply composition of ω0,0̄,2 simultaneously to two different maps. For example, we have ωg1 ,\|I\|+1,0 ⊗ ωg2 ,\|J\|+1,0 ◦ ω0,0̄,2 = ωg1 +g2 ,\|I\|+\|J\|,0 : A⊗(\|I\|+\|J\|) −→ K, which is (3.0.10). Notice that a linear map ωg,m+n : A⊗m ⊗ A⊗n −→ K is equivalent to A⊗m −→ (A∗ )⊗n . Thus we can re-construct a map ωg,m̄,n from ωg,m+n by (3.0.11) A⊗m k A⊗m ωg,m+n / (A∗ )⊗n ωg,m̄,n (λ−1 )⊗n / A⊗n k, / A⊗n INVITATION TO 2D TQFT AND QUANTIZATION 19 ∼ where λ : A −→ A∗ is the isomorphism of (1.0.3). For the case of g = 0, m = 2, and n = 1, (3.0.11) implies that λ−1 ω0,3 (v1 , v2 , · ) = ω0,2̄,1 (v1 , v2 ) = v1 v2 for v1 , v2 ∈ A. Or equivalently, we have (3.0.12) ω0,3 (v1 , v2 , v3 ) = η(v1 v2 , v3 ) = (v1 v2 v3 ). This completes the proof. Conversely, the degree 0 part of the cohomology of a CohFT is a 2D TQFT. Proposition 3.4 (Restriction of CohFT to the degree 0 part of the cohomology ring). Let (A, Ωg,n ) be a CohFT associated with a Frobenius algebra A. The Frobenius algebra A itself defines a unique 2D TQFT (A, ωg,m̄,n ). Denote by r : H ∗ (Mg,n , K) −→ H 0 (Mg,n , K) = K the restriction of the cohomology ring to its degree 0 component, and define (3.0.13) ωg,n = r ◦ Ωg,n : A⊗n −→ K. Then we have the equality of maps (3.0.14) ωg,n = ωg,n̄,0 : A⊗n −→ K for all (g, n) with 2g − 2 + n > 0. In other words, the degree 0 restriction of a CohFT is the 2D TQFT determined by the Frobenius algebra A. Proof. We already know that (A, ωg,m̄,n ) defines a CohFT with values in H 0 (Mg,n , K). We need to show that this CohFT is exactly the degree 0 restriction of the given CohFT we start with. First we extend ωg,n to the unstable range by (3.0.15) ω0,1 = : A −→ K, ω0,2 = η : A⊗2 −→ K. We then note that from (2.0.4) and (3.0.15), we see that (3.0.14) holds for ω0,1 and ω0,2 . The general case of (3.0.14) follows by induction on 3g−3+n, provided that ω0,3 is appropriately defined. This is because (3.0.9) and (3.0.10) are induction formula recursively generating ωg,n from those with smaller values of g and n. Since • Case of (3.0.9): 3g − 3 + n = [3(g − 1) − 3 + (n + 2)] + 1, • Case of (3.0.10): 3(g1 + g2 ) − 3 + \|I\| + \|J\| = [3g1 − 3 + \|I\| + 1] + [3g2 − 3 + \|J\| + 1] + 1, we see that the complexity 3g −3+n is always reduced by 1 in each of the recursive formulas. Finally, we see that since M0,3 is just a point as we have noted above in the discussion of Aut(P1 ), we have ω0,3 = Ω0,3 . Hence (3.0.16) ω0,3 (v1 , v2 , v3 ) = η(v1 v2 , v3 ), which makes (3.0.14) holds for all values of (g, n). This completes the proof. Remark 3.5. The above two propositions show that a 2D TQFT can be defined either just by a commutative Frobenius algebra A, by a system of maps {ωg,m̄,n } satisfying the TQFT axioms, or the degree 0 part of a CohFT. From now on, we use (A, ωg,n ) to denote a 2D TQFT, which is less cumbersome and easier to deal with. Proposition 3.6. The genus 0 values of a 2D TQFT is given by (3.0.17) ω0,n (v1 , . . . , vn ) = (v1 · · · vn ). Proof. This is a direct consequence of CohFT 3 and (1.0.9). 20 O. DUMITRESCU AND M. MULASE One of the original motivations of TQFT [4, 89] is to identify the topological invariant Z(N ) of a closed manifold N . In our current setting, it is defined as (3.0.18) Z(Σg ) := λ−1 (ωg,1 ) for a closed oriented surface Σg of genus g. Here, ωg,1 : A −→ K is an element of A∗ , and ∼ λ : A −→ A∗ is the canonical isomorphism (1.0.3). Proposition 3.7. The topological invariant Z(Σg ) of (3.0.18) is given by Z(Σg ) = (eg ), (3.0.19) where e ∈ A is the Euler element of (1.0.12). Lemma 3.8. We have e := m ◦ δ(1) = λ−1 (ω1,1 ). (3.0.20) Proof. This follows from ω1,1 (v) = X ω0,3 (v, ea , eb )η ab = a,b X η(v, ea eb )η ab = η(v, e) a,b for every v ∈ A. Proof of Proposition 3.7. Since the starting case g = 1 follows from the above Lemma, we prove the formula by induction, which goes as follows: X ωg,1 (v) = ωg−1,3 (v, ea , eb )η ab a,b = X ω0,4 (v, ea , eb , ei )ωg−1,1 (ej )η ab η ij i,j,a,b = X = X η(vea eb , ei )ωg−1,1 (ej )η ab η ij i,j,a,b η(ve, ei )ωg−1,1 (ej )η ij i,j = ωg−1,1 (ve) = ω1,1 (veg−1 ) = η(veg−1 , e) = η(v, eg ). A closed genus g surface is obtained by sewing g genus 1 pieces with one output boundaries to a genus 0 surface with g input boundaries. Since the Euler element is the output of the genus 1 surface with one boundary, we obtain the same result g z }\| { Z(Σg ) = ω0,g (e, . . . , e). Finally we have the following: Theorem 3.9. The value of a 2D TQFT is given by (3.0.21) ωg,n (v1 , . . . , vn ) = (v1 · · · vn eg ). INVITATION TO 2D TQFT AND QUANTIZATION 21 Proof. The argument is the same as the proof of Proposition 3.7: ωg,n (v1 , . . . , vn ) = ω1,n (v1 eg−1 , v2 , . . . , vn ) X = ω0,n+2 (v1 eg−1 , v2 , . . . , vn , ea , eb )η ab a,b = (v1 · · · vn eg ). 4. Category of cell graphs In the original formulation of 2D TQFT, the operations of multiplication and comultiplication are associated with an oriented surface of genus 0 with three boundary circles. In this section, we introduce a category of ribbon graphs, which carries the information of all finite-dimensional Frobenius algebras. To avoid unnecessary confusion, we use the terminology of cell graphs in this article, instead of more common ribbon graphs. Ribbon graphs naturally appear for encoding complex structures of a topological surface (see for example, [62, 75]). Our purpose of using ribbon graphs are for degeneration of stable curves, and we label vertices, instead of faces, of a ribbon graph. Definition 4.1 (Cell graphs). A connected cell graph of topological type (g, n) is the 1-skeleton of a cell-decomposition of a connected closed oriented surface of genus g with n labeled 0-cells. We call a 0-cell a vertex, a 1-cell an edge, and a 2-cell a face, of the cell graph. We denote by Γg,n the set of connected cell graphs of type (g, n). Each edge consists of two half-edges connected at the midpoint of the edge. Remark 4.2. • The dual of a cell graph is a ribbon graph, or Grothendieck’s dessin d’enfant. We note that we label vertices of a cell graph, which corresponds to face labeling of a ribbon graph. Ribbon graphs are also called by different names, such as embedded graphs and maps. • We identify two cell graphs if there is a homeomorphism of the surfaces that brings one cell-decomposition to the other, keeping the labeling of 0-cells. The only possible automorphisms of a cell graph come from cyclic rotations of half-edges at each vertex. Definition 4.3 (Directed cell graph). A directed cell graph is a cell graph for which an arrow is assigned to each edge. An arrow is the same as an ordering of the two half-edges forming an edge. The set of directed cell graphs of type (g, n) is denoted by ~Γg,n . Remark 4.4. A directed cell graph is a quiver. Since our graph is drawn on an oriented surface, a directed cell graph carries more information than its underlying quiver structure. The tail vertex of an arrowed edge is called the source, and the head of the arrow the target, in the quiver language. To label n vertices, we normally use the n-set [n] := {1, 2, . . . , n}. However, it is often easier to use any totally ordered set of n elements for labeling. The main reason we label the vertices of a cell graph is we wish to assign an element of a K-vector space A to each vertex. In this article, we consider the case that a cell graph γ ∈ Γg,n defines a linear map (4.0.1) Γg,n 3 γ : A⊗n −→ K. 22 O. DUMITRESCU AND M. MULASE The set of values of these functions γ can be more general. We discuss some of the general cases in [33]. An effective tool in graph enumeration is edge-contraction operations. Often edge contraction leads to an inductive formula for counting problems of graphs. The same edgecontraction operations acquire algebraic meaning in our consideration. Definition 4.5 (Edge-contraction operations). There are two types of edge-contraction operations applied to cell graphs. ~ = p−→ • ECO 1: Suppose there is a directed edge E i pi in a cell graph γ ∈ Γg,n , ~ in γ, and put connecting the tail vertex pi and the head vertex pj . We contract E the two vertices pi and pj together. We use i for the label of this new vertex, and call it again pi . Then we have a new cell graph γ 0 ∈ Γg,n−1 with one less vertices. In this process, the topology of the surface on which γ is drawn does not change. Thus genus g of the graph stays the same. E p i p j p i Figure 4.1. Edge-contraction operation ECO 1. The edge bounded by two vertices pi and pj is contracted to a single vertex pi . ~ for the edge-contraction operation • We use the notation E ~ : Γg,n 3 γ 7−→ γ 0 ∈ Γg,n−1 . (4.0.2) E ~ in γ ∈ Γg,n at the i-th vertex pi . Since • ECO 2: Suppose there is a directed loop L a loop in the 1-skeleton of a cell decomposition is a topological cycle on the surface, its contraction inevitably changes the topology of the surface. First we look at the half-edges incident to vertex pi . Locally around pi on the surface, the directed loop ~ separates the neighborhood of pi into two pieces. Accordingly, we put the incident L half-edges into two groups. We then break the vertex pi into two vertices, pi1 and pi2 , so that one group of half-edges are incident to pi1 , and the other group to pi2 . ~ upward near at vertex The order of two vertices is determined by placing the loop L pi . Then we name the new vertex on its left by pi1 , and on its right by pi2 . ~ and Let γ 0 denote the possibly disconnected graph obtained by contracting L separating the vertex to two distinct vertices labeled by i1 and i2 . ~ is a loop of handle. We use the • If γ 0 is connected, then it is in Γg−1,n+1 . The loop L ~ to indicate the edge-contraction operation same notation L ~ : Γg,n 3 γ 7−→ γ 0 ∈ Γg−1,n+1 . (4.0.3) L • If γ 0 is disconnected, then write γ 0 = (γ1 , γ2 ) ∈ Γg1 ,\|I\|+1 × Γg2 ,\|J\|+1 , where ( g = g1 + g2 . (4.0.4) I t J = {1, . . . , bi, . . . , n} The edge-contraction operation is again denoted by ~ : Γg,n 3 γ 7−→ (γ1 , γ2 ) ∈ Γg ,\|I\|+1 × Γg ,\|J\|+1 . (4.0.5) L 1 2 INVITATION TO 2D TQFT AND QUANTIZATION 23 p i L p i1 p i2 ~ Figure 4.2. Edge-contraction operation ECO 2. The contracted edge is a loop L ~ of a cell graph. Place the loop so that it is upward near at pi to which L is attached. The vertex pi is then broken into two vertices, pi1 on the left, and pi2 on the right. Half-edges incident to pi are separated into two groups, belonging to two sides of the loop near pi . ~ a separating loop. Here, vertices labeled by I belong to In this case we call L the connected component of genus g1 , and those labeled by J are on the other component of genus g2 . Let (I− , i, I+ ) (reps. (J− , i, J+ )) be the reordering of I t {i} (resp. J t {i}) in the increasing order. Although we give labeling i1 , i2 to the two vertices created by breaking pi , since they belong to distinct graphs, we can simply ~ translates use i for the label of pi1 ∈ γ1 and the same i for pi2 ∈ γ2 . The arrow of L into the information of ordering among the two vertices pi1 and pi2 . Remark 4.6. Let us define m(γ) = 2g − 2 + n for a graph γ ∈ Γg,n . Then every edgecontraction operation reduces m(γ) exactly by 1. Indeed, for ECO 1, we have m(γ 0 ) = 2g − 2 + (n − 1) = m(γ) − 1. The ECO 2 applied to a loop of handle produces m(γ 0 ) = 2(g − 1) − 2 + (n + 1) = m(γ) − 1. For a separating loop, we have +) 2g1 − 2 + \|I\| + 1 2g2 − 2 + \|J\| + 1 2g1 + 2g2 − 4 + \|I\| + \|J\| + 2 = 2g − 2 + n − 1. The motivation for our introduction of directed cell graphs is that we need them when we deal with non-commutative Frobenius algebras. The operation of taking disjoint union is symmetric. Therefore, 2D TQFT inevitably leads to a commutative Frobenius algebra. The advantage of our formalism using directed cell graphs is that we can deal with noncommutative Frobenius algebras and non-symmetric tensor products. For the purpose of presenting the idea of the category of cell graphs as simple as possible, we restrict ourselves to undirected cell graphs in this article. Therefore, we will only recover commutative Frobenius algebras and usual 2D TQFT. A more general theory will be given in [33]. We now introduce the category of cell graphs. The most unusual point we present here is that a morphism between cell graphs is not a cell map. Recall that a cell map f : γ −→ γ 0 from a cell graph γ to another cell graph γ 0 is a topological map between 1-dimensional cell complexes. Thus f sends a vertex of γ to a vertex of γ 0 , and an edge of γ to either an edge or a vertex of γ 0 , keeping the incidence relations. In particular, a cell map is continuous with respect to the topological structure on cell graphs indued from the surface on which they are drawn. 24 O. DUMITRESCU AND M. MULASE Definition 4.7 (Category of cell graphs). The category CG of cell graphs is defined as follows. • The set of objects of CG is the set of all cell graphs: a Γg,n . (4.0.6) Ob(CG) = g≥0,n>0 • A morphism f ∈ Hom(γ, γ 0 ) is a composition of a finite sequence of edge-contraction operations and cell graph automorphisms. In particular, Hom(γ, γ) = Aut(γ). If there is no way to bring γ to γ 0 by consecutive applications of edge-contraction operations and automorphisms, then we define Hom(γ, γ 0 ) = ∅, even though there may be cell maps between them. Remark 4.8. The triple (CG, t, ∅) forms a symmetric monoidal category. Remark 4.9. Automorphisms of a cell graph and ECOs of the first kind are cell maps, but ECO 2 operations are not. When an ECO 2 is involved, a morphism between cell graphs does not have to be a cell map. Even it may not be a continuous map. Example 4.10. A few simple examples of morphisms are given below. Note that vertices are all labeled, and automorphisms are required to keep labeling. (4.0.8) Hom (•, •) = {id}. Hom •, • • = ∅. (4.0.9) E1 E2 Hom(• • •, • •) = {E1 , E2 }. (4.0.7) (4.0.10) (4.0.11) E1 E2 Hom(• • •, •) = {E1 E2 = E2 E1 }. E1 Hom • •, • = {E1 , E2 = σ(E1 )}. E2 (4.0.12) Hom • •, • • = {E1 E2 = E2 E1 }. E1 E2 In (4.0.10), we note that E1 E2 := E1 ◦ E2 is equal to E2 E1 := E2 ◦ E1 , because they both produce the same result • • • → • • → •. The cell graph of the left of (4.0.11) and (4.0.12) has an automorphism σ that interchanges E1 and E2 . Thus as an edge-contraction operation, E2 = E1 ◦ σ = σ(E1 ). Note that there is a 2 : 1 covering cell map for the E1 case of (4.0.11) that sends both edges E1 and E2 on • • to the single loop of • , and E2 the two vertices on the first graph to the single vertex on the second. Since it is not an edge-contraction, this cell map is not a morphism. The morphism of (4.0.12) is not a cell map, since it is not continuous. Let Vect be the category of finite-dimensional K-vector spaces. The triple C = (Vect, ⊗, K) forms a monoidal category. Again for simplicity, we are concerned only with symmetric tensor products in this article, so we consider C a symmetric monoidal category. A Kobject in Vect is a pair (V, : V −→ K) consisting of a vector space V and a linear map INVITATION TO 2D TQFT AND QUANTIZATION 25 : V −→ K. We denote by Vect/K the category of K-objects in Vect. It has the unique final object (K, id : K −→ K). Therefore, C/K = Vect/K, ⊗, (K, id : K −→ K) is again a monoidal category. We denote by Fun(C/K, C/K) (4.0.13) the endofunctor category of the monoidal category C/K, which consists of monoidal functors α : C/K −→ C/K as its objects, and their natural transformations τ as morphisms. Schematically, we have V α(h) W β(h) >K g β(f ) α(f ) f h / β(V ) τ α(V ) <K α(g) τ α(W ) / K. < τ / β(W ) β(g) Here, the triangle on the left shows two objects (V, f : V −→ K) and (W, g : W −→ K) of C/K, and a morphism h between them. The prism shape on the right represents two monoidal endofunctors α and β that assigns V 7−→ α(V ), V 7−→ β(V ) W 7−→ α(W ), W 7−→ β(W ), and a natural transformation τ : α −→ β among them. The final object of Fun(C/K, C/K) is the functor (4.0.14) φ : (V, f : V −→ K) −→ (K, idK : K −→ K) which assigns the final object of the codomain C/K to everything in the domain C/K. With respect to the tensor product and the above functor (4.0.14) as its identity object, the endofunctor category Fun(C/K, C/K) is again a monoidal category. Definition 4.11 (ECO functor, [33]). The ECO functor is a monoidal functor (4.0.15) ω : CG −→ Fun(C/K, C/K) satisfying the following conditions. • The graph • ∈ Γ0,1 consisting of only one vertex and no edge corresponds to the identity functor (4.0.16) ω(•) = id : C/K −→ C/K. • Each graph γ ∈ Γg,n corresponds to a functor (4.0.17) ω(γ) : (V, : V −→ K) 7−→ (V ⊗n , ωV (γ) : V ⊗n −→ K). • Edge-contraction operations correspond to natural transformations. Let us recall the notion of Frobenius object. 26 O. DUMITRESCU AND M. MULASE Definition 4.12 (Frobenius object). Let (C, ⊗, K) be a symmetric monoidal category. A Frobenius object is an object V ∈ Ob(C) together with morphisms m : V ⊗ V −→ V, 1 : K −→ V, δ : V −→ V ⊗ V, : V −→ K, satisfying the following conditions: • (V, m, 1) is a monoid object in C. • (V, δ, ) is a comonoid object in C. We also require the compatibility condition (1.0.7) among morphisms m and δ: id⊗δ V ⊗V V ⊗ 8 V ⊗V /V m δ⊗id & m⊗id δ ' / V ⊗ V. 8 id⊗m V ⊗V ⊗V Since we are considering the monoidal category of K-objects in Vect, there is a priori no notion of 1 in V . The existence of the morphism 1 : K −→ V requires a non-degeneracy condition. The following theorem is proved in [33]. Theorem 4.13 (Generation of Frobenius objects [33]). An object (V, : V −→ K) of C/K is a Frobenius object if ωV (• •) : V ⊗V −→ K defines a non-degenerate symmetric bilinear form on V . 5. 2D TQFT from cell graphs The result of Section 3 tells us that a 2D TQFT can be defined as a system (A, ωg,n ) of linear maps (5.0.1) ωg,n : A⊗n −→ K defined for all values of g ≥ 0 and n ≥ 1, satisfying a set of conditions. The required conditions are the following: First, (A, ωg,n ) is a CohFT for 2g − 2 + n > 0. In addition, we require that ω0,1 = : A −→ K, (5.0.2) ω0,2 = η : A ⊗ A −→ K. In this section we give a different formulation of a 2D TQFT, based on cell graphs and a different set of axioms. Our ultimate goal is to relate 2D TQFT, CohFT, mirror symmetry, topological recursion, and quantum curves. Later in these lectures, we introduce quantum curves. Relations between all these subjects will be discussed elsewhere [33]. Theorem 5.1 (Graph independence [29]). Let (A, : A −→ K) be a Frobenius object under the ECO functor ω of Definition 4.11. Then every connected cell graph γ ∈ Γg,n gives rise to the same map (5.0.3) ωA (γ) : A⊗n 3 v1 ⊗ · · · ⊗ vn 7−→ (v1 · · · vn eg ) ∈ K, where e is the Euler element of (1.0.12). Corollary 5.2 (ECO implies TQFT). Define ωg,n (v1 , . . . , vn ) = ωA (γ)(v1 , . . . , vn ) for every γ ∈ Γg,n . Then {ωg,n } is a 2D TQFT. Proof. Since the value of (5.0.3) is the same as (3.0.21), it is a 2D TQFT. INVITATION TO 2D TQFT AND QUANTIZATION 27 The rest of the section is devoted to proving Theorem 5.1. We first give three examples of graph independence. Lemma 5.3 (Edge-removal lemma). Let γ ∈ Γg,n . • Case 1. There is a disc-bounding loop L in γ. Let γ 0 ∈ Γg,n be the graph obtained by simply removing L from γ. Note that we are not contracting L. • Case 2. The graph γ containts two edges E1 and E2 between two distinct vertices pi and pj that bound a disc. Let γ 0 ∈ Γg,n be the graph obtained by removing E2 . Here again, we are just eliminating E2 . • Case 3. Two loops, L1 and L2 , in γ are attached to the i-th vertex pi . If they are homotopic, then let γ 0 ∈ Γg,n be the graph obtained by removing L2 from γ. In each of the above cases, we have ωA (γ)(v1 , . . . , vn ) = ωA (γ 0 )(v1 , . . . , vn ). (5.0.4) Figure 5.1. Removal of disc-bounding edges. Proof. Case 1. Contracting a disc-bounding loop attached to pi creates (γ0 , γ 0 ) ∈ Γ0,1 ×Γg,n , where γ0 consists of only one vertex and no edges. The natural transformation corresponding to ECO 1 then gives X ωA (γ)(v1 , . . . , vn ) = η(vi , ek e` )η ka η `b ωA (γ0 )(ea )ωA (γ 0 )(v1 , . . . , vi−1 , eb , vi+1 . . . , vn ) a,b,k,` = X η(vi , ek e` )η ka η `b η(1, ea )ωA (γ 0 )(v1 , . . . , vi−1 , eb , vi+1 . . . , vn ) a,b,k,` = X = X η(vi , 1 · e` )η `b ωA (γ 0 )(v1 , . . . , vi−1 , eb , vi+1 . . . , vn ) b,k,` η(vi , e` )η `b ωA (γ 0 )(v1 , . . . , vi−1 , eb , vi+1 . . . , vn ) b,` = ωA (γ 0 )(v1 , . . . , vi−1 , vi , vi+1 . . . , vn ). Case 2. Contracting Edge E1 makes E2 a disc-bounding loop at pi . We can remove it by Case 1. Note that the new vertex is assigned with vi vj . Restoring E1 makes the graph exactly the one obtained by removing E2 from γ. Thus (5.0.4) holds. Case 3. Contracting Loop L1 makes L2 a disc-bounding loop. Hence we can remove it by Case 1. Then restoring L1 creates a graph obtained from γ by removing L2 . Thus (5.0.4) holds. Remark 5.4. The three cases treated above correspond to removing degree 1 and 2 vertices from the dual ribbon graph. 28 O. DUMITRESCU AND M. MULASE Definition 5.5 (Reduced graph). A cell graph is reduced if it does not contain any discbounding loops or disc-bounding bigons. In terms of dual ribbon graphs, the dual of a reduced cell graph has no vertices of degree 1 or 2. We can see from Lemma 5.3, Case 1, that every graph γ ∈ Γ0,1 gives the same map (5.0.5) ωA (γ)(v) = (v). Similarly, Cases 2 and 3 of Lemma 5.3 show that every graph γ ∈ Γ0,2 defines ωA (γ)(v1 , v2 ) = η(v1 , v2 ). This is because we can remove all edges and loops but one that connects the two vertices. Then by the natural transformation corresponding to ECO 1, the value of the assignment ωA (γ) is (v1 v2 ) = η(v1 , v2 ). Proof of Theorem 5.1. We use the induction on m = 2g − 2 + n. The base case is m = −1, or (g, n) = (0, 1), for which the theorem holds by (5.0.5). Assume that (5.0.3) holds for all (g, n) with 2g − 2 + n < m. Now let γ ∈ Γg,n be a cell graph of type (g, n) such that 2n − 2 + n = m. Choose an arbitrary straight edge of γ that connects two distinct vertices, say pi and pj . Then the natural transformation of contracting this edge to γ 0 gives ωA (γ)(v1 , . . . , vn ) = ωA (γ 0 )(v1 , . . . , vi−1 , vi vj , vi+1 . . . , vbj , . . . , vn ) = (v1 . . . vn eg ). If we have chosen an arbitrary loop attached to pi , then its contraction by ECO 2 gives two cases, depending on whether the loop is a loop of handle or a separating loop. For the former case, we have a graph γ 0 , and by appealing to (1.0.9) and (1.0.13), we obtain X ωA (γ)(v1 , . . . , vn ) = η(vi , ek e` )η ka η `b ωA (γ 0 )(v1 , . . . , vi−1 , ea , eb , vi+1 , . . . , vn ) a,b,k,` = X η(vi ek , e` )η ka η `b ωA (γ 0 )(v1 , . . . , vi−1 , ea , eb , vi+1 , . . . , vn ) a,b,k,` = X = X η ka ωA (γ 0 )(v1 , . . . , vi−1 , ea , vi ek , vi+1 , . . . , vn ) a,k η ka (v1 · · · vn eg−1 ea eb ) a,k = (v1 · · · vn eg ). For the case of a separating loop, ECO 2 makes γ → (γ1 , γ2 ), and we have X ωA (γ)(v1 , . . . , vn ) = η(vi , ek e` )η ka η `b ωA (γ1 ) vI− , ea , vI+ ωA (γ2 ) vJ− , eb , vJ+ a,b,k,` ! = X ka `b η(vi , ek e` )η η ea Y g1 vc e ! eb c∈I a,b,k,` Y g2 vd e d∈J ! = X η(vi ek , e` )η ka η `b η Y vc , ea eg1 ! eb c∈I a,b,k,` d∈J ! = X a,k η ka η Y c∈I vc eg1 , ea ! vi ek Y Y d∈J vd eg2 vd eg2 INVITATION TO 2D TQFT AND QUANTIZATION 29 ! = vi Y c∈I vc eg1 Y vd eg2 d∈J g1 +g2 = (v1 · · · vn e ). Therefore, no matter how we apply ECO 1 or ECO 2, we always obtain the same result. This completes the proof. 6. TQFT-valued topological recursion There is a direct relation between a Frobenius algebra and Gromov-Witten theory when A is given by the big quantum cohomology of a target space. Since these Frobenius algebras are usually infinite-dimensional over the ground field, they do not correspond to a 2D TQFT discussed in the previous sections. But for the case that the target space is 0-dimensional, the TQFT indeed captures the whole Gromov-Witten theory. In this section, we present a general framework. Fundamental examples are A = Q, which gives ψ-class intersection numbers on Mg,n , and the center of the group algebra of a finite group A = ZC[G], which produces Gromov-Witten invariants of the classifying space BG. The first example is considered in [29, 30]. The latter case will be discussed elswhere [33]. We wish to solve a graph enumeration problem, where as a graph we consider a cell graph, and each of its vertex is colored by a parameter v ∈ A. We also impose functoriality under the edge-contraction of Definition 4.5 for this coloring. The ECOs reduce the complexity of coloring considerably, because the functoriality makes the coloring process graph independent, as we have shown in the last section. Thus the answer is just the number of graphs times the value (v1 · · · vn eg ) for each topological type (g, n). Here comes the difficulty: there are infinitely many graphs for each topological type, since we allow multiple edges and loops. The standard idea for such counting problem is to appeal to the Laplace transform, which is introduced by Laplace for this particular context of counting an infinite number of objects. Let us denote by (6.0.1) Γg,n (µ1 , . . . , µn ) the set of all cell graphs with labeled vertices of degrees (µ1 , . . . , µn ). Denoting by cd the number of d-cells in a cell-decomposition of a surface of genus g, d = 0, 1, 2, we have n = c0 , Pn µ i=1 i = 2c1 , and 2 − 2g = n − c1 + c2 . Therefore, (6.0.1) is a finite set. Counting processes become easier if we do not have any object with non-trivial automorphism. There are many ways to eliminate automorphism, for example, the minimalistic way of imposing the least possible conditions, or an excessive way to kill automorphisms but for the most objects the conditions are redundant. Actually, it is known that most of the cell graphs counted in (6.0.1) are without any non-trivial automorphisms. Since any possible automorphism induces a cyclic permutation of half-edges incident to each vertex, the easiest way to disallow any automorphism is to assign an outgoing arrow to one of these half-edges, as in [34, 79] (but not as a quiver). An automorphism should preserve the arrowed half-edges, in addition to labeled vertices. We denote by b g,n (µ1 , . . . , µn ) (6.0.2) Γ the set of arrowed cell graphs with labeled vertices of degrees (µ1 , . . . , µn ), and by (6.0.3) b g,n (µ1 , . . . , µn ) Cg,n (µ1 , . . . , µn ) := Γ its cardinality. This number is always a non-negative integer, and C0,1 (2m) = the m-th Catalan number. 2m 1 m+1 m is 30 O. DUMITRESCU AND M. MULASE b g,n (µ1 , . . . , µn ). We use the notation ωA (γ) = ωg,n : A⊗n −→ K of Corollary 5.2 for γ ∈ Γ We are interested in considering the Cg,n (µ1 , . . . , µn )-weighted TQFT Cg,n (µ1 , . . . , µn ) · ωg,n : A⊗n −→ K, (6.0.4) and applying the edge-contraction operations to these maps. We contract the edge of γ that carries the outgoing arrow at the first vertex p1 , then place a new arrow to the half-edge next to the original half-edge with respect to the cyclic ordering induced by the orientation of the surface. If the edge that carries the outgoing arrow at p1 is a loop, then after splitting p1 , we place another arrow to the next half-edge at each of the two newly created vertices, again next to the original loop. From this process we obtain the following counting formula: Proposition 6.1. The 2D TQFT weighted by the number of arrowed cell graphs satisfies the following equation. Cg,n (µ1 , . . . , µn ) · ωg,n (v1 , . . . , vn ) = n X µj Cg,n−1 (µ1 + µj − 2, µ2 , . . . , µ cj , . . . , µn ) · ωg,n−1 (v1 vj , v2 , . . . , vbj , . . . , vn ) j=2 (6.0.5) + X Cg−1,n+1 (α, β, µ2 , . . . , µn ) · ωg−1,n+2 δ(v1 ), v2 , . . . , vn α+β=µ1 −2 + X X α+β=µ1 −2 X η(v1 , ek e` )η ka η `b g1 +g2 =g a,b,k,` ItJ={2,...,n} × Cg1 ,\|I\|+1 (α, µI ) · ωg1 ,\|I\|+1 (ea , vI ) Cg2 ,\|J\|+1 (β, µJ ) · ωg2 ,\|J\|+1 (eb , vJ ) . This is exactly the same formula of [34, 79, 92] multiplied by ωg,n (v1 , . . . , vn ) = (v1 . . . vn eg ). Let us now consider a Frobenius algebra twisted topological recursion. To simplify the notation, we adopt the following way of writing: X (6.0.6) (f ⊗ h) ◦ δ = η(•, ek e` )η ka η `b f (ea )h(eb ), a,b,k,` where f, h : A −→ K are linear functions on A. First let us review the original topological recursion of [38] defined on a spectral curve Σ, which is just a disjoint union of r copies of open discs. Let U be the unit disc centered at 0 of the complex z-line. We choose r sets of functions (xα , yα ), α = 1, . . . , r, defined on U with Taylor expansions (6.0.7) 2 xα (z) = z + ∞ X k aα,k z , k=3 yα (z) = z + ∞ X bα,k z k , k=2 and a meromorphic 1-form (Cauchy kernel) dz dz − + ωα z−a z−b on U , where a, b ∈ U , and ωα is a holomorphic 1-form on U . Since each xα : U −→ C is a 2 : 1 map, we have an involution on U that keeps the same xα -value: (6.0.9) σα : U −→ U, xα σα (z) = xα (z). (6.0.8) ωαa−b (z) = INVITATION TO 2D TQFT AND QUANTIZATION 31 To avoid confusion, we label r copies of U by U1 , . . . , Ur , and consider the functions (xα , yα ) to be defined on Uα . The topological recursion is the following recursive equation r I σ (z)−z ωαα (z1 ) 1 X (6.0.10) Wg,n (z1 , . . . , zn ) = 2πi y σ (z) − y (z) dxα (z) α α α ∂U α α=1   No (0,1)  × Wg−1,n+1 z, σα (z), z2 , . . . , zn + X g1 +g2 =g ItJ={2,...,n}  Wg1 ,\|I\|+1 (z, zI )Wg2 ,\|J\|+1 σα (z), zJ   on symmetric meromorphic n-differential forms Wg,n defined on the disjoint union U1n t · · · t Urn for 2g − 2 + n > 0. Here, zI = (zi )i∈I , and “No (0, 1)” in the summation means the partition g = g1 + g2 and the set partition I t J = {2, . . . , n} do not allow g1 = 0 and I = ∅, or g2 = 0 and J = ∅. The integration is performed with respect to z ∈ ∂U . Note that the differential form in the big bracket [ ] in (6.0.10) is a symmetric quadratic differential in the variable z ∈ Uα . The expression 1/dxα (z) is the contraction operator with respect to ∂z ∂ on Uα . Thus the integrand of the recursion becomes a meromorphic the vector field ∂x α ∂z 1-form on Uα in the z-variable, for which the integration is performed. The multiplication σ (z)−z by ωαα (z1 ) is simply the symmetric tensor product with a 1-form proportional to dz1 . The 1-form W0,1 and the 2-form W0,2 are defined separately: (6.0.11) W0,1 (z) := r X yα (z)dxα (z) α=1 is defined on the disjoint union U1 t · · · t Ur . If z ∈ Uα , then W0,1 (z) = yα (z)dxα (z). In the form of (6.0.10), however, W0,1 does not appear anywhere. Similarly, W0,2 is defined by W0,2 (z1 , z2 ) := dz1 ωαz1 −b (z2 ) (6.0.12) if z1 , z2 ∈ Uα . Here, the constant b ∈ Uα does not play any role. This 2-form explicitly appears in the recursion part (the terms in the big bracket [ ]) of the formula. Definition 6.2. A Frobenius algebra twisted topological recursion for Wg,n = Wg,n (z1 , . . . , zn ) : A⊗n −→ K is the following formula: Wg,n (z1 , . . . , zn ; v1 , . . . , vn ) r I X 1 X = Kα z, σα (z), z1 ; ek , e` , v1 2πi α=1 ∂Uα a,b,k,`  (6.0.13)  × Wg−1,n+1 z, σα (z), z2 , . . . , zn ; ea , eb , v2 , . . . , vn )  No (0,1) + X g1 +g2 =g ItJ={2,...,n}  Wg1 ,\|I\|+1 (z, zI ; ea , vI )Wg2 ,\|J\|+1 σα (z), zJ ; eb , vJ  . Here (6.0.14) ωασα (z)−z (z1 )η(ek e` , v1 )η ka η `b Kα z, σα (z), z1 ; ek , e` , v1 = yα σα (z) − yα (z) dxα (z) 32 O. DUMITRESCU AND M. MULASE is the integration-summation kernel. Symbolically we can write (6.0.13) as r I 1 X Kα z, σα (z), z1 Wg,n = 2πi α=1 ∂Uα   (6.0.15) No (0,1) X   × W ◦ δ + W ⊗ W ◦ δ g−1,n+1 g ,\|I\|+1 g ,\|J\|+1 1 2   g1 +g2 =g ItJ={2,...,n} with the usual integration kernel σ (z)−z Kα z, σα (z), z1 := yα ωαα (z1 ) . σα (z) − yα (z) dxα (z) Theorem 6.3. The topological recursion (6.0.13) uniquely determines the (A⊗n )∗ -valued n-linear differential form Wg,n from the initial data W0,2 . If the initial data is given by W0,2 (z1 , z2 ; v1 , v2 ) = W0,2 (z1 , z2 ) · η(v1 , v2 ) for a 2-form (6.0.12), then there exists a solution {Wg,n } of the topological recursion (6.0.10) and a 2D TQFT {ωg,n } such that (6.0.16) Wg,n (z1 , . . . , zn ; v1 , . . . , vn ) = Wg,n (z1 , . . . , zn ) · ωg,n (v1 , . . . , vn ). Proof. The proof is done by induction on m = 2g − 2 + n with the base case m = 0. We assume that (6.0.16) holds for all (g, n) such that 2g − 2 + n < m, and use the value ωg,n = (v1 · · · vn eg ) given by (3.0.21) for the values of (g, n) in the range of induction hypothesis. Then by (6.0.13) and functoriality under ECO 2, we conclude that (6.0.16) also holds for all (g, n) such that 2g − 2 + n = m. Remark 6.4. In comparison to edge-contraction operations, we note that the multiplication case ECO 1 does not seem to have a counterpart in an explicit way. It is actually included in the terms involving g1 = 0, \|I\| = 1 and g2 = 0, \|J\| = 1 in the partition sum, and (1.0.14) is used to change the comultiplication to multiplication. More precisely, if I = {i}, then it gives a term W0,2 (z, zi ; ea , vi )Wg,n−1 σα (z), z2 , . . . , zbi , . . . , zn ; eb , v2 , . . . , vbi , . . . , vn = W0,2 (z, zi )Wg,n−1 σα (z), z2 , . . . , zbi , . . . , zn · ω0,2 (ea , vi )ωg,n−1 (eb , v2 , . . . , vbi , . . . , vn ) in the partition sum, assuming the induction hypothesis. We also note that X η(ek e` , v1 )η ka η `b ω0,2 (ea , vi )ωg,n−1 (eb , v2 , . . . , vbi , . . . , vn ) a,b,k,` = X = X η `b ω0,2 (v1 e` , vi )ωg,n−1 (eb , v2 , . . . , vbi , . . . , vn ) b,` η `b η(v1 vi , e` )ωg,n−1 (eb , v2 , . . . , vbi , . . . , vn ) b,` = ωg,n−1 (v1 vi , v1 , . . . , vbi , . . . , vn ). By taking the Laplace transform of (6.0.5) using the method of [34, 79], we obtain a D (z , . . . , z ) · ω D solution Wg,n = Wg,n 1 n g,n to the topological recursion, where Wg,n is given by INVITATION TO 2D TQFT AND QUANTIZATION 33 [34, (4.14)] with respect to a global spectral curve of [34, Theorem 4.3], and ωg,n is the TQFT corresponding to the Frobenius algebra A. Part 2. Quantization of Higgs Bundles 7. Quantum curves The cohomology ring H ∗ (X, C) of a Kähler variety X is a Z/2Z-graded Z/2Z-commutative Frobenius algebra over C. The genus 0 Gromov-Witten invariants of X define the big quantum cohomology of X, which is a quantum deformation of the cohomology ring. If the mirror symmetry is established for X, then the information of big quantum cohomology of X is supposed to be encoded in holomorphic geometry of a mirror dual space Y . GromovWitten invariants are generalized to all values of (g, n) with 2g − 2 + n > 0, g ≥ 0, n > 0. The question is: Question 7.1. What should be the holomorphic geometry on Y that captures all genera Gromov-Witten invariants of X through mirror symmetry? Since the transition from g = 0 to all values of g ≥ 0 is indeed a quantization, the holomorphic object on Y that should capture higer genera Gromov-Witten invariants of X is a quantum geometry of Y . A naı̈ve guess may be that it should be a D-module that represents Y as its classical limit. Hence the D-module is not defined on Y . Then where does it live? The idea of quantum curves concerns a rather restricted situation, when the mirror geometry Y is captured by an algebraic, or an analytic, curve. Typical examples are the mirror of toric Calabi-Yau orbifolds of three dimensions. Geometry of the mirror Y of a toric Calabi-Yau 3-fold is encoded in a complex curve known as the mirror curve. Another situation is enumeration problems of various Hurwitz-type coverings of P1 , and also many decorated graphs on surfaces. For these examples, although there are no “space” X, the mirror geometry exists, and is indeed a curve. These mirror curves are special cases of more general notion of spectral curves. Besides mirror symmetry, spectral curves appear in theory of integrable systems, random matrix theory, topological recursion, and Hitchin theory of Higgs bundles. A spectral curve Σ has two common features. The first one is that it is a Lagrangian subvariety of a holomorphic symplectic surface. The other is the existence of a projection π : Σ −→ C to another curve C, called a base curve. The quantum curve is a D-module on the base curve C such that its semi-classical limit is the spectral curve realized in the cotangent bundle Σ ⊂ T ∗ C. From the analogy of 2D TQFT and CohFT, we note that a spectral curve is already a quantized object, since it corresponds to quantum cohomology. The even part H even (X, C), which is a commutative Frobenius algebra, is not the one that corresponds to a spectral curve. In this sense, CohFT is a result of two quantizations: the first one from classical cohomology to a big quantum cohomology through n-point Gromov-Witten invariants of genus 0; and the second quantization is the passage from genus 0 to all genera. Remark 7.2. We note that quantum cohomology itself is a Frobenius algebra, though it is not finite dimensional, because it requires the introduction of Novikov ring. The even degree part forms a commutative Frobenius algebra, yet it does not correspond to a 2D TQFT in the way we presented in Part 1. Now let us turn to the topic of Part 2. The prototype of quantization is a Schrödinger equation. Consider a harmonic oscillator of mass 1, energy E, and the spring constant 34 O. DUMITRESCU AND M. MULASE 1/4 in one dimension. It has a geometric description as an elliptical motion of a constant angular momentum in the phase space, or the cotangent bundle of the real axis. Here, the spectral curve is an ellipse 1 2 x + y2 = E 4 in a real symplectic plane. The quantization of this spectral curve is the quantization of the harmonic oscillator, which is a second order stationary Schrödinger equation in one variable: 1 d2 (7.0.1) −~2 2 + x2 − E ψ(x, ~) = 0. dx 4 The quantization we discuss in Part 2 is in complete parallelism to quantization of harmonic oscillator. As a holomorphic symplectic surface, we use (C2 , dx ∧ dy). A plane quadric 1 2 (7.0.2) x − y2 = 1 4 is an example of a spectral curve. In complex coordinates, we identify y = d/dx, ignoring the imaginary unit. An example of quantization of (7.0.2) is a Schrödinger equation ! 1 1 2 d 2 + 1 − ~ − x ψ(x, ~) = 0, (7.0.3) ~ dx 2 4 which is essentially the same as quantum harmonic oscillator equation (7.0.1), and is known as the Hermite-Weber equation. Its solutions are all well studied. Question 7.3. Why do we care this well-known classical differential equation? A surprising answer [28, 30, 34, 79] to this question is that we find the intersection numbers of Mg,n through the asymptotic expansion! First we apply a gauge transformation ! 2 1 2 1 2 d 1 1 e− 4~ x ~ + 1 − ~ − x2 e 4~ x Z(x, ~) dx 2 4 (7.0.4) 2 d d = ~ 2 + ~x + 1 Z(x, ~) = 0, dx dx 1 2 where Z(x, ~) = e− 4~ x ψ(x, ~). Recall the integer valued function Cg,n (µ1 , . . . , µn ) of (6.0.3), and define their generating functions by X Cg,n (µ1 , . . . , µn ) −µ n (7.0.5) Fg,n (x1 , . . . , xn ) := x1 1 · · · x−µ . n µ1 · · · µn µ1 ,...,µn >0 It is discovered in [34] that the derivatives Wg,n = d1 · · · dn Fg,n satisfy the topological recursion based on the spectral curve (0.0.1), which is the semi-classical limit of (7.0.4). With an appropriate adjustment for (g, n) = (0, 1) and (0, 2), we have the following allorder WKB expansion formula [30, 34, 79]: ! X 1 (7.0.6) Z(x, ~) = exp ~2g−2+n Fg,n (x, . . . , x) . n! g,n We find ([34]) that if we change the coordinate from x to t by t+1 t−1 (7.0.7) x = x(t) = + , t−1 t+1 INVITATION TO 2D TQFT AND QUANTIZATION 35 then Fg,n x(t1 ), x(t2 ), . . . , x(tn ) is a Laurent polynomial for each (g, n) with 2g − 2 + n > 0. The coordinate change (7.0.7) is identified in [28] as a normalization of the singular curve (0.0.1) in the Hirzebruch surface P KP1 ⊕ OP1 by a sequence of blow-ups. The highest degree part of this Laurent polynomial is a homogeneous polynomial of degree 6g − 6 + 3n 2di +1 ! n X Y (−1)n ti highest (7.0.8) Fg,n (t1 , . . . , tn ) = 2g−2+n hτd1 · · · τdn ig,n \|2di − 1\|!! , 2 2 i=1 d1 +···+dn =3g−3+n where the coefficients Z (7.0.9) hτd1 · · · τdn ig,n = Mg,n ψ1d1 · · · ψndn are cotangent class intersection numbers on the moduli space Mg,n . Topological recursion is a mechanism to calculate all Fg,n (x1 , . . . , xn ) from the single equation (7.0.3), or equivalently, (7.0.4). Thus the quantum curve (7.0.3) has the information of all intersection numbers (7.0.9). These are the topics discussed in our previous lectures [30]. Although the following topic is not what we deal with in this article, for the moment let us consider a symplectic surface (C∗ × C∗ , d log x ∧ d log y). As a spectral curve, we use the zero locus of the A-polynomial AK (x, y) of a knot defined in [14]. For a given knot K ⊂ S 3 , the SL2 (C)-character variety (7.0.10) Hom π1 (S 3 \ K), SL2 (C) SL2 (C) of the fundamental group of the knot complement determines an algebraic curve in (C∗ )2 defined by AK (x, y) ∈ Z[x, y]. Here, (C∗ )2 , or to be more precise, (C∗ )2 /(Z/2Z), is the SL2 (C)-character variety of the fundamental group of the torus T 2 = S 1 × S 1 , which is the boundary of the knot complement. Now consider a function f (q, n) in 2 variables (q, n) ∈ C∗ × Z+ , and define operators ( (b xf )(q, n) := q n f (q, n) (7.0.11) (b y f )(q, n) := f (q, n + 1), following [41, 44]. These operators satisfy the commutation relation x b · yb = qb y·x b. The procedure of changing x 7−→ x b and y 7−→ yb is the Weyl quantization. Garoufalidis [44] conjectures that there exists a quantization of the A-polynomial such that (7.0.12) bK (b A x, yb; q)JK (q, n) = 0, where JK (q, n) is the colored Jones polynomial of the knot K indexed by the dimension n of the irreducible representation of SL2 (C). Here, the quantization means that the operator bK (b A x, yb; q) recovers the A-poynomial by the restriction bK (x, y; 1) = AK (x, y). A This relations is the semi-classical limit, which provides the initial condition of the WKB analysis. A geometric definition of a quantum curve that arises as the quantization of a Hitchin spectral curve is developed in [31], based on the work of [26] that solves a conjecture of Gaiotto [42, 43]. The WKB analysis of the quantum curve [27, 28, 30, 57, 58] is performed by applying the topological recursion of [38]. The Hermite-Weber differential equation is an example of this geometric theory. Although there have been many speculations [50] of 36 O. DUMITRESCU AND M. MULASE the applicability of the topological recursion to low-dimensional topology, still there is no counterpart of the Hitchin type geometric theory for the case of the quantization of Apolynomials. The appearance of the modularity in this context [22, 60, 95] is a tantalizing phenomenon, on which there has been a great advancement. In the following sections, we unfold a different story of quantum curves. In geometry, there is a process parallel to the passage from a spectral curve (0.0.1) to a quantum curve (7.0.4). This process is a journey from the moduli space of Hitchin spectral curves to the moduli space of opers [25, 31]. The quantization parameter, the Planck constant ~ of (7.0.4), acquires a geometric meaning in this process. We begin the story with finding a coordinate independent description of global differential equations of order 2 on a compact Riemann surface. 8. Projective structures, opers, and Higgs bundles In [27], the authors have given a definition of partial differential equation version of topological recursion for Hitchin spectral curves. When the spectral curve is a double sheeted covering of the base curve, we have shown that this PDE topological recursion produces a quantum curve of the Hitchin spectral curve through WKB analysis. The mechanism is explained in detail in [28, 30]. WKB analysis is certainly one way to describe quantization. Yet the passage from spectral curves to their quantization is purely geometric. This point of view is adopted in [31], based on our work [26] on a conjecture of Gaiotto [42]. Our statement is that the quantization process is a biholomorphic map from the moduli space of Hitchin spectral curves to the moduli space of opers [6]. In this section, we introduce the notion of opers, and construct the biholomorphic map mentioned above. In this passage, we give a geometric interpretation of the Planck constant ~ as a deformation parameter of vector bundles and connections. For simplicity of presentation, we restrict our attention to SL2 (C)-opers. We refer to [26, 31] for more general cases. To deal with linear differential equation of order higher then or equal to 2 globally on a compact Riemann surface C, we need a projective coordinate system. If ω is a global holomorphic or meromorphic 1-form on C, then (8.0.1) (d + ω)f = 0 is a first order linear differential equation that makes sense globally on C. This is because d + ω is a homomorphism from OC to KC , allowing singularities if necessary. Here, KC is the sheaf of holomorphic 1-forms on C. Of course existence of a non-trivial global solution of (8.0.1) is a different matter, because it is equivalent to ω = −d log f . Suppose we have a second order differential equation 2 d − q(z) f (z) = 0 (8.0.2) dz 2 2 d locally on C with a local coordinate z. Clearly dz 2 is not globally defined, in contrast to the exterior differentiation d in (8.0.1). What is the requirement for (8.0.2) to make sense coordinate free, then? Let us find an answer by imposing " # " # d 2 d 2 2 2 (8.0.3) dz − q(z) f (z) = 0 ⇐⇒ dw − q(w) f (w) = 0. dz dw INVITATION TO 2D TQFT AND QUANTIZATION 37 We wish the shape of the equation to be analogous to (8.0.1). Since ω is a 1-form, we impose that q ∈ H 0 (C, KC⊗2 ) is a global quadratic differential on C. Under a coordinate change w = w(z), q satisfies q(z)dz 2 = q(w)dw2 . How should we think about a solution f ? For (8.0.2) to have a coordinate free meeting, we need to identify the line bundle L on C to which f belongs. Let us denote by e−g(w(z)) the transition function of L with respect to the coordinate patch w = w(z). A function g(w) = g w(z) will be determined later. A solution f satisfies the coordinate condition (8.0.4) e−g w(z) f w(z) = f (z). Then " d dz 2 " d dz 2 2 g w(z) 2 g w(z) 0 = dz · e = dz · e = dz 2 · eg w(z) d −g e dz # − q(z) 2 − q(z) 2 f (z) # 2 w(z) e−g w(z) f w(z) f w(z) − dw2 q(w)f (w) 2 d − gw (w)w0 f w(z) − dw2 q(w)f (w) dz " # 2 d d 0 2 = dz 2 · − 2gw (w)w0 − gw (w)w0 − gw (w)w0 f w(z) − dw2 q(w)f (w) dz dz h i 0 2 0 = dz 2 · fw (w)w0 − 2gw (w)w0 fw (w)w0 − gw (w)w0 − gw (w)w0 f (w) = dz 2 · − dw2 q(w)f (w) dw 2 2 dz + fw (w)w00 dz 2 − 2gw (w)fw (w)dw2 = fww (w) dz 0 2 − gw (w)w0 − gw (w)w0 f (w)dz 2 − dw2 q(w)f (w) # " 2 d = dw2 · − q(w) f (w) + fw (w) w00 − 2gw (w)(w0 )2 dz 2 dw 0 2 − gw (w)w0 − gw (w)w0 f (w)dz 2 , where w0 = dw/dz. Therefore, for (8.0.3) to hold, we need (8.0.5) (8.0.6) w00 − 2gw (w)(w0 )2 = 0 0 2 gw (w)w0 − gw (w)w0 = 0. From (8.0.5) we find (8.0.7) gw (w)w0 = 1 w00 . 2 w0 Substitution of (8.0.7) in (8.0.6) yields 00 0 w (z) 1 w00 (z) 2 (8.0.8) sz (w) := − = 0. w0 (z) 2 w0 (z) 38 O. DUMITRESCU AND M. MULASE We thus encounter the Schwarzian derivative sz (w). Since (8.0.7) is equivalent to 0 1 g(w)0 = log(w0 ) , 2 1 0 we obtain g(w) = 2 log w . Here, the constant of integration is 0 because g(z) = 0 when w = z. Then (8.0.4) becomes r dz − 12 log w0 f (w) ⇐⇒ f (z) = f (z) = e f (w), dw −1 which identifies the line bundle to which f belongs: f (z) ∈ KC 2 . We conclude that the coordinate change w = w(z) should satisfy the vanishing of the Schwarzian derivative sz (w) ≡ 0, and the solution f (z) should be considered as a (multivalued) section of the −1 inverse half-canonical KC 2 . The vanishing of the Schwarzian derivative dictates us to use a complex projective coordinate system of C. A holomorphic connection in a vector bundle E on C is a C-linear map ∇ : E −→ KC ⊗E satisfying the Leibniz condition ∇(f s) = f ∇(s) + df ⊗ s for f ∈ OC and s ∈ E. Since C is a complex curve, every connection on C is automatically flat. Therefore, ∇ gives rise to a holonomy representation (8.0.9) ρ : π1 (C) −→ G of the fundamental group π1 (C) of the curve C into the structure group G of the vector bundle E. A flat connection ∇ is irreducible if the image of the holonomy representation (8.0.9) is Zariski dense in the complex algebraic group G. In our case, since G = SL2 (C), this requirement is equivalent that Im(ρ) contains two non-commuting elements of G. The moduli space MdeR of irreducible holomorphic connections (E, ∇) in a G-bundle E has been constructed (see [90]). Definition 8.1 (SL2 (C)-opers). Consider a point (E, ∇) ∈ MdeR consisting of an irreducible holomorphic SL2 (C)-connection ∇ : E −→ E ⊗ KC acting on a vector bundle E. It is an SL2 (C)-oper if there is a line subbundle F ⊂ E such that the connection induces an OC -module isomorphism (8.0.10) ∼ ∇ : F −→ (E/F ) ⊗ KC . The notion of oper is a generalization of projective structures on a Riemann surface. Every compact Riemann surface C amsits a projective structure subordinating the given complex structure (see [51]). For our purpose of quantization of Hitchin spectral curves associated with holomorphic Higgs bundles, let us assume that g(C) ≥ 2 in what follows. When we allow singularities, we can relax this condition and deal with genus 0 and 1 cases. A complex projective coordinate system is a coordinate neighborhood covering [ C= Uα α with a local coordinate zα of Uα such that for every Uα ∩ Uβ , we have a fractional linear transformation aαβ zβ + bαβ aαβ bαβ (8.0.11) zα = , fαβ := ∈ SL2 (C), cαβ dαβ cαβ zβ + dαβ satisfying a cocycle condition [fαβ ][fβγ ] = [fαγ ]. Here, [fαβ ] is the class of fαβ in the projection (8.0.12) 0 −→ Z/2Z −→ SL2 (C) −→ P SL2 (C) −→ 0, INVITATION TO 2D TQFT AND QUANTIZATION 39 which determines the fractional linear transformation. A choice of ± on each Uα ∩ Uβ is an element of H 1 (C, Z/2Z) = (Z/2Z)2g , indicating that there are 22g choices of a lift. We make this choice once and for all, and consider fαβ an SL2 (C)-valued 1-cocycle. Since 1 dzβ , (cαβ zβ + dαβ )2 a transition function for KC is given by the cocycle (cαβ zβ + dαβ )2 on each Uα ∩ Uβ . A dzα = 1 theta characteristic (or a spin structure) KC2 is the line bundle defined by the 1-cocycle (8.0.13) ξαβ = cαβ zβ + dαβ . 1 Here again, we have 22g choices ±ξαβ for a transition function of KC2 . Since we have already made a choice of the sign for fαβ , we have a consistent choice in (8.0.13), as explained below. 1 Thus we see that the choice of the lift (8.0.12) is determined by KC2 . From (8.0.13), we obtain ∂β2 ξαβ = 0. (8.0.14) This property plays an essential role in our construction of global connections on C. First we show that actually (8.0.14) implies (8.0.11). Proposition 8.2 (A condition for projective coordinate). A coordinate system of C with 1 which the second derivative of the transition function of KC2 vanishes is a projective coordinate system. 1 Proof. The condition (8.0.14) means that the transition function of KC2 is a linear polynomial cαβ zβ + dαβ satisfying the cocycle condition. Therefore, dzα 1 = . dzβ (cαβ zβ + dαβ )2 Solving this differential equation, we obtain (8.0.15) za = mαβ (cαβ zβ + dαβ ) − 1/cαβ cαβ zβ + dαβ with a constant of integration mαβ . In this way we find an element of SL2 (C) on each Uα ∩ Uβ . The cocycle condition makes (8.0.15) exactly (8.0.11). The transition function fαβ defines a rank 2 vector bundle E on C whose structure group is SL2 (C). Since fαβ is a constant element of SL2 (C), the notion of locally constant sections of E makes sense independent of the coordinate chart. Thus defining ∇α = d on Uα for each Uα determines a global connection ∇ in E. Suppose we have a projective coordinate system on C. Let E be the vector bundle we have just constructed, and P(E) its projectivization, i.e., the P1 bundle associated with E. Noticing that Aut(P1 ) = P SL2 (C), the local coordinate system {zα } of (8.0.11) is a global section of P(E). Indeed, this section defines a map (8.0.16) zα : Uα −→ P1 , which induces the projective structure of P1 into Uα via pull back. The map (8.0.16) is not a constant, because its derivative dzα never vanishes on the intersection Uα ∩ Uβ . A 40 O. DUMITRESCU AND M. MULASE global section of P(E) corresponds to a line subbundle F of E, such that E is realized as an extension 0 −→ F −→ E −→ E/F −→ 0. The equality 1 za aαβ bαβ zβ = 1 1 cαβ zβ + dαβ cαβ dαβ −1 z shows that a defines a global section of the vector bundle KC 2 ⊗ E, and that the 1 projectivization image of this section is the global section {zα } of P(E) corresponding to F . Here, the choice of the theta characteristic is consistently made so that the ± ambiguity of (8.0.13) and the one in the lift of the fractional linear transformation to fαβ ∈ SL2 (C) cancel. Since this section is nowhere vanishing, it generates a trivial subbundle −1 −1 OC = KC 2 ⊗ F ⊂ KC 2 ⊗ E. 1 −1 Therefore, F = KC2 . Note that det E = OC , hence E/F = KC 2 , and E is an extension 1 −1 0 −→ KC2 −→ E −→ KC 2 −→ 0, 1 −1 determining an element of Ext1 KC 2 , KC2 ∼ = H 1 (C, KC ) ∼ = C. There is a more straightforward way to obtain (8.0.17). (8.0.17) Theorem 8.3 (Projective coordinate systems and opers). Every projective coordinate sys1 tem (8.0.11) determines an oper (E, ∇) of Definition 8.1 with F = KC2 . Proof. Let σαβ = −(d/dzβ )ξαβ ξαβ gαβ : = (8.0.18) = −1 = −cαβ , where ξαβ is defined by (8.0.13). Since σαβ cαβ zβ + dαβ −cαβ −1 = ξαβ (cαβ zβ + dαβ )−1 1 aαβ bαβ zβ −1 , zα cαβ dαβ 1 which follows from aαβ − cαβ zα = aαβ (cαβ zβ + dαβ ) − cαβ (aαβ zβ + bαβ ) 1 = , (cαβ zβ + dαβ ) (cαβ zβ + dαβ ) we find that fαβ and gαβ define the same SL2 (C)-bundle E. The shape of the matrix gαβ immediately shows (8.0.17). Since the connection ∇ in E is simply d on each Uα with respect to fαβ , the differential operator on Uα with respect to the transition function gαβ is given by 1 zα −1 0 0 (8.0.19) ∇α := d =d− dzα . −1 zα 1 1 0 Since the (2, 1)-component of the connection matrix is dzα which is nowhere vanishing, 1 (8.0.20) 1 ∇ F = KC2 −→ E ⊗ KC −→ (E/F ) ⊗ KC ∼ = KC2 given by this component is an isomorphism. This proves that (E, ∇) is an SL2 (C)-oper. In the final step of the proof to show (E, ∇) is an oper, we need that (8.0.20) is an OC -linear homomorphism. This is because we are considering the difference of connections ∇ and ∇\|F in E. More generally, suppose we have two connections ∇1 and ∇2 in the same vector bundle E. Then the Leibniz condition tells us that ∇1 (f s) − ∇2 (f s) = f ∇1 (s) − f ∇2 (s) INVITATION TO 2D TQFT AND QUANTIZATION 41 for f ∈ OC and s ∈ E. Therefore, ∇1 −∇2 : E −→ E⊗KC is an OC -module homomorphism. Although the extension class of (8.0.17) is parameterized by H 1 (C, KC ) = C, the complex structure of E depends only if σαβ = 0 or not. This is because ξαβ σαβ λ−1 ξαβ λ2 σαβ λ (8.0.21) = , −1 −1 ξαβ ξαβ λ−1 λ hence we can normalize σαβ = 0 or σαβ = 1. The former case gives the trivial extension of 1 −1 two line bundles E = KC2 ⊕ KC 2 . Since σαβ = cαβ = 0, the projective coordinate system (8.0.11) is actually an affine coordinate system. Since we are assuming g(C) > 1, there is no affine structure in C. Therefore, only the latter case can happen. And the latter case gives a non-trivial extension, as we will show later. Suppose we have another projective structure in C subordinating the same complex structure of C. Then we can adjust the ± signs of the lift of (8.0.12) and the square root of (8.0.13) so that we obtain the exact same holomorphic vector bundle E of (8.0.17). Since we are dealing with a different coordinate system, the only change we have is reflected in the connection ∇. Thus two different projective structures give rise to two connections in the same vector bundle E. Hence this difference is an OC -linear homomorphism E −→ E ⊗ KC as noted above. This consideration motivates the following. A Higgs bundle of rank r [52, 53] defined on C is a pair (E, φ) consisting of a holomorphic vector bundle E of rank r on C and an OC -module homomorphism φ : E −→ E ⊗ KC . An SL2 (C)-Higgs bundle is a pair (E, φ) of rank 2 with a fixed isomorphism det E = OC and trφ = 0. It is stable if every line subbundle F ⊂ E that is invariant with respect to φ, i.e., φ : F −→ F ⊗ KC , has a negative degree degF < 0. The moduli spaces of stable Higgs bundles are constructed in [90]. We denote by MDol the moduli space of stable holomorphic SL2 (C)-Higgs bundles on C. It is diffeomorphic to the moduli space MdeR of pairs (E, ∇) consisting of an irreducible holomorphic connection in an SL2 (C)-bundle (see [23, 52, 90]). A particular diffeomorphism (8.0.22) ∼ ν : MDol −→ MdeR is the non-Abelian Hodge correspondence, which is explained in Section 10. The total space of the line bundle KC is the cotangent bundle π : T ∗ C −→ C of the curve C. We denote by η ∈ H 0 (T ∗ C, π ∗ KC ) the tautological section η T ∗C = π; ∗ KC / T ∗C KC π / C, which is a holomorphic 1-form on T ∗ C. Since φ is an End(E)-valued holomorphic 1-form on C, its eigenvalues are 1-forms. The set of eigenvalues is thus a multivalued section of KC , and hence a multivalued section of π : T ∗ C −→ C. The image Σ ⊂ T ∗ C of this multivalued section is the Hitchin spectral curve, which defines a ramified covering of C. The formal definition of Hitchin spectral curve Σ is that it is the divisor of zeros in T ∗ C of the characteristic polynomial det(η − π ∗ φ) ∈ π ∗ KC⊗2 . 42 O. DUMITRESCU AND M. MULASE Hitchin fibration [52] is a holomorphic fibration (8.0.23) µH : MDol 3 (E, φ) 7−→ det(η − π ∗ φ) ∈ B, B := H 0 C, KC⊗2 , that defines an algebraically completely integrable Hamiltonian system in MDol . Hitchin 1 notices in [52] that the choice of a spin structure KC2 that we have made allows us to construct a natural section κ : B ,→ MDol . Define 0 0 0 1 1 0 t (8.0.24) X− := , X+ := X− = , H := [X+ , X− ] = . 1 0 0 0 0 −1 These elements generate the Lie algebra hX+ , X− , Hi ∼ = sl2 (C). Lemma 8.4. Let q ∈ B = H 0 (C, KC⊗2 ) be an arbitrary point of the Hitchin base B, and define a Higgs bundle (E0 , φ(q)) consisting of a vector bundle 1 ⊗H 1 −1 = KC2 ⊕ KC 2 (8.0.25) E0 := KC2 and a Higgs field (8.0.26) φ(q) := X− + qX+ = 0 q . 1 0 Then it is a stable SL2 (C)-Higgs bundle. The Hitchin section is defined by (8.0.27) κ : B 3 q 7−→ (E0 , φ(q)) ∈ MDol , which gives a biholomorphic map between B and κ(B) ⊂ MDol . Proof. We first note that X− : E0 −→ E0 ⊗ KC is a globally defined End0 (E0 )-valued 1-form, since it is essentially the constant map 1 (8.0.28) −1 = 1 : KC2 −→ KC 2 ⊗ KC . Similarly, multiplication by a quadratic differential gives −1 3 1 q : KC 2 −→KC2 = KC2 ⊗ KC . Thus φ(q) : E0 −→ E0 ⊗ KC is globally defined as a Higgs field in E0 . The Higgs pair is stable because no line subbundle of E0 is invariant under φ(q), unless q = 0. And when 1 q = 0, the invariant line subbundle KC2 has degree g − 1, which is positive since g ≥ 2. Remark 8.5. Hitchin sections exist for the moduli space of stable G-Higgs bundles for an arbitrary simple complex algebraic group G. The construction utilizes Kostant’s principal three-dimensional subgroup (TDS) [66]. The use of TDS is crucial in our quantization, as noted in [26, 31]. The image κ(B) is a holomorphic Lagrangian submanifold of a holomorphic symplectic space MDol . The holomorphic symplectic structure of MDol is induced from its open dense subspace T ∗ SU(2, C), where SU(2, C) is the moduli space of rank 2 stable bundles of degree 0 on C. Since the codimension of the complement of T ∗ SU(2, C) in MDol is 2, the natural holomorphic symplectic form on the cotangent bundle automatically extends to MDol . Our first step of constructing the quantization of the Hitchin spectral curve Σ is to define ~-connections on C that are holomorphically depending on ~. We use a one-parameter INVITATION TO 2D TQFT AND QUANTIZATION 43 family E of deformations of vector bundles E~ /E / C × H 1 (C, KC ), C × {~} and a C-linear first-order differential operator ~∇~ : E~ −→ E~ ⊗ KC depending holomorphically on ~ ∈ H 1 (C, KC ) ∼ = C for ~ 6= 0. Here, we identify the Planck constant ~ of the quantization as a geometric parameter 1 − 12 1 1 2 ~ ∈ H (C, KC ) = Ext KC , KC ∼ = C, which determines a unique extension 1 −1 0 −→ KC2 −→ E~ −→ KC 2 −→ 0. (8.0.29) This is exactly the same as (8.0.17). The extension E~ is given by a system of transition functions ξαβ ~σαβ (8.0.30) E~ ←→ −1 0 ξαβ on each Uα ∩Uβ . The cocycle condition for the transition functions translates into a condition −1 σαγ = ξαβ σβγ + σαβ ξβγ . (8.0.31) The application of the exterior differentiation d to the cocycle condition ξαγ = ξαβ ξβγ for 1 KC2 yields dξβγ dξαβ dξαγ dzβ ξβγ + ξαβ dzγ = dzγ . dzγ dzβ dzγ 2 = Noticing ξαβ dzβ dzα , we find that σαβ := − (8.0.32) dξαβ = −∂β ξαβ dzβ solves (8.0.31). The negative sign is chosen to relate (8.0.30) and (8.0.11). By the same reason as before, the complex structure of the vector bundle E~ is isomorphic to E1 if ~ 6= 0, and to E0 of (8.0.25) if ~ = 0. The transition function can also be written as (8.0.33) ξαβ ~σαβ 1 0 0 1 = exp log ξαβ exp −~∂β log ξαβ . −1 ξαβ 0 −1 0 0 Therefore, in the multiplicative sense, the extension class is determined by ∂β log ξαβ . Lemma 8.6. The extension class σαβ of (8.0.32) defines a non-trivial extension (8.0.29). 44 O. DUMITRESCU AND M. MULASE Proof. The long exact sequences of cohomologies H 1 (C, C) / H 1 (C, OC ) / H 2 (C, C) k / H 1 (C, O ∗ ) d log / H 1 (C, KC ) C H 1 (C, C∗ ) 0 / H 1 (C, KC ) ∼ c1 = H 2 (C, Z) / H 2 (C, Z) associated with exact sequences of sheaves 0 0 0 /Z 0 /C / OC / KC /0 0 / C∗ / O ∗ d log / KC C /0 0 = /Z /0 d 0 0 show that the class {σαβ } corresponds to the image of {ξab } via the map d log ∗) − −−−→ H 1 (C, KC ). H 1 (C, OC If d log{ξαβ } = 0 ∈ H 1 (C, KC ), then it comes from a class in H 1 (C, C∗ ), which is the moduli space of line bundles with holomorphic connections, as explained in [8]. It leads to a contradiction 1 0 = c1 KC2 = g − 1 > 0, because g(C) ≥ 2. Remark 8.7. Gukov and Sulkowski [50] defines an intriguing quantizability condition for a spectral curve in terms of the algebraic K-group K2 C(Σ) of the function field of spectral curve Σ. They relate the quantizability and Bloch regulators of [8]. The class {σαβ } of (8.0.32) gives a natural isomorphism H 1 (C, KC ) ∼ = C. We identify the deformation parameter ~ ∈ C with the cohomology class {~σαβ } ∈ H 1 (C, KC ) = C. S We trivialize the line bundle KC⊗2 with respect to a coordinate chart C = α Uα , and write q ∈ H 0 (C, KC⊗ 2) as {(q)α } that satisfies the transition relation (8.0.34) 4 (q)α = (q)β ξαβ . The transition function of the trivial extension E0 is given by (8.0.35) H ξαβ = exp(H log ξαβ ). Since X− : E0 −→ E0 ⊗ KC is a globally defined Higgs filed, its local expressions {X− dzα } with respect to a coordinate system satisfies the transition relation (8.0.36) X− dzα = exp(H log ξαβ )X− dzβ exp(−H log ξαβ ) INVITATION TO 2D TQFT AND QUANTIZATION 45 on every Uα ∩ Uβ . The same relation holds for the Higgs field φ(q) as well: (8.0.37) φα (q)dzα = exp(H log ξαβ )φβ (q)dzβ exp(−H log ξαβ ). Theorem 8.8 (Construction of SL2 (C)-opers). On each Uα ∩Uβ define a transition function ξαβ 1 −~∂β log ξαβ ~ (8.0.38) gαβ := exp(H log ξαβ ) exp − ~∂β log ξαβ X+ = , −1 · ξαβ 1 where ∂β = d dzβ , and ~∂β log ξαβ ∈ H 1 (C, KC ). Then ~ } satisfies the cocycle condition • The collection {gαβ ~ ~ ~ gαβ gβγ = gαγ , (8.0.39) which defines the holomorphic vector bundle bundle E~ of (8.0.29). • The local expression 1 (8.0.40) ∇~α (0) := d − X− dza ~ on Uα for ~ 6= 0 defines a global holomorphic connection in E~ , i.e., −1 1 1 ~ ~ (8.0.41) d − X− dzα = gαβ d − X− dzβ gαβ , ~ ~ if and only if the coordinate is a projective coordinate system. We choose one. • With this particular projective coordinate system, every point (E0 , φ(q)) ∈ κ(B) ⊂ MDol of the Hitchin section (8.0.27) gives rise to a one-parameter family of SL2 (C)~ opers E~ , ∇ (q) ∈ MdeR . In other words, the local expression 1 ∇~α (q) := d − φα (q)dzα ~ on every Uα for ~ 6= 0 determines a global holomorphic connection −1 ~ ~ ∇~β (q) gαβ (8.0.43) ∇~α (q) = gαβ (8.0.42) in E~ satisfying the oper condition. • Deligne’s ~-connection ~∇~ (q) : E~ −→ E~ ⊗ KC (8.0.44) interpolates the Higgs pair and the oper, i.e., at ~ = 0, the family (8.0.44) gives the Higgs pair (E, −φ(q)) ∈ MDol , and at ~ = 1 it gives an SL2 (C)-oper E1 , ∇1 (q) ∈ MdeR . • After a suitable gauge transformation depending on ~, the ~ → ∞ limit of the oper ∇~ (q) exists and is equal to ∇~=1 (0). This point corresponds to the C∗ -fixed point on the Hitchin section. Proof. The cocycle condition of gαβ has been established in (8.0.32) and (8.0.33). Proof of (8.0.41) is a straightforward calculation, using the power series expansion of the adjoint action n (8.0.45) e ~A Be −~A ∞ ∞ }\| { X X 1 nz 1 n ~ (adA )n (B) := ~ [A, [A, [· · · , [A, B] · · · ]]]. = n! n! n=0 It follows that −1 ~ ~ gαβ X− gαβ n=0 46 O. DUMITRESCU AND M. MULASE = exp(H log ξαβ ) exp (−~∂β log ξαβ X+ ) X− exp (~∂β log ξαβ X+ ) exp(−H log ξαβ ) = exp(H log ξαβ )X− exp(−H log ξαβ ) − ~∂β log ξαβ H − ~2 (∂β log ξαβ )2 exp(H log ξαβ )X+ exp(−H log ξαβ ). Note that (8.0.14) is equivalent to −1 −2 ∂β ∂β log ξαβ = ∂β ξαβ ∂β ξαβ = −ξαβ (∂β ξαβ )2 = −(∂β log ξαβ )2 , hence to −1 ~ ~ = ∂β log ξαβ H + ~(∂β log ξαβ )2 exp(H log ξαβ )X+ exp(−H log ξαβ ). ∂β gαβ gαβ Therefore, noticing (8.0.36), (8.0.14) is equivalent to −1 −1 1 1 ~ ~ ~ ~ dzβ = exp(H log ξαβ )X− dzβ exp(−H log ξαβ ) + ∂β gαβ gαβ g X− gαβ ~ αβ ~ 1 = X− dzα . ~ The statement follows from Proposition 8.2. To prove (8.0.43), we need, in addition to (8.0.41), the following relation: −1 ~ ~ . (q)β X+ dzβ gαβ (8.0.46) (q)α X+ dzα = gαβ But (8.0.46) is obvious from (8.0.37) and (8.0.38). 1 The line bundle F required in the definition of SL2 (C)-oper is simply KC2 . The isomorphism (8.0.10) is a consequence of (8.0.28). Finally, the gauge transformation of ∇~ (q) by a bundle automorphism " 1 # −2 H ~ (8.0.47) ~− 2 = 1 ~2 on each coordinate neighborhood Uα gives H H 1 1 q (8.0.48) d − φ(q) 7−→ ~− 2 d − φ(q) ~ 2 = d − X− + 2 X+ . ~ ~ ~ This is because H H ~− 2 X− ~ 2 = ~X− and H H 2 2 2 ~− 2 X+ ~ = ~−2 X+ , which follows from the adjoint formula (8.0.45). Therefore, lim ∇~ (q) ∼ d − X− = ∇~=1 (0), ~→∞ where the symbol ∼ means gauge equivalence. This completes the proof of the theorem. The construction theorem yields the following. Theorem 8.9 (Biholomorphic quantization of Hitchin spectral curves). Let C be a compact Riemann surface of genus g ≥ 2 with a chosen projective coordinate system subordinating its complex structure. We denote by MDol the moduli space of stable holomorphic SL2 (C)Higgs bundles over C, and by MdeR the moduli space of irreducible holomorphic SL2 (C)1 connections on C. For a fixed theta characteristic KC2 , we have a Hitchin section κ(B) ⊂ INVITATION TO 2D TQFT AND QUANTIZATION 47 MDol of (8.0.27). We denote by Op ⊂ MdeR the moduli space of SL2 (C)-opers with the 1 condition that the required line bundle is given by F = KC2 . Then the map γ (8.0.49) MDol ⊃ κ(B) 3 (E0 , φ(q)) 7−→ E~ , ∇~ (q) ∈ Op ⊂ MdeR evaluated at ~ = 1 is a biholomorphic map with respect to the natural complex structures induced from the ambient spaces. The biholomorphic quantization (8.0.49) is also C∗ -equivariant. The λ ∈ C∗ action on the Hitchin section is defined by φ 7−→ λφ. The oper corresponding to (E0 , λφ(q)) ∈ κ(B) is d − λ~ φ(q). Proof. The C∗ -equivariance follows from the same argument of the gauge transformation (8.0.47), (8.0.48). The action φ 7−→ λφ on the Hitchin section induces a weighted action B 3 q 7−→ λ2 q ∈ B through µH . Then we have the gauge equivalence via the gauge transformation H − H 2 λ λ 2 λ λ λ2 q d − φ(q) ∼ d − φ(q) = d − X− + 2 X+ . ~ ~ ~ ~ ~ λ ~ H2 : Remark 8.10. In the construction theorem, our use of a projective coordinate system is essential, through (8.0.13). Only in such a coordinate, our particular definition (8.0.42) makes sense. This is due to the vanishing of the second derivative of ξαβ . And as we have seen above, the projective coordinate system determines the origin ∇1 (0) of the space Op of opers. Other opers are simply translation ∇1 (q) from the origin by q ∈ H 0 (C, KC⊗2 ). 9. Semi-classical limit of SL2 (C)-opers A holomorphic connection on a compact Riemann surface C is automatically flat. Therefore, it defines a D-module over C. Continuing the last section’s conventions, let us fix a projective coordinate system on C, and let (E0 , φ(q)) = κ(q) be a point on the Hitchin section of (8.0.27). It uniquely defines an ~-family of opers E~ , ∇~ (q) . In this section, we establish that the ~-connection ~∇~ (q) defines a family of D-modules on C parametrized by B such that the semi-classical limit of the family agrees with the family of spectral curves over B. To calculate the semi-classical limit, let us trivialize the vector bundle E~ on each simply connected coordinate neighborhood Uα with coordinate zα of the chosen projective coordinate system. A flat section Ψα of E~ over Uα is a solution of ~ψ ~ (9.0.1) ~∇α (q)Ψα := (~d − φα (q)) = 0, ψ α ~ Ψ , the function ψ on U satisfies with an appropriate unknown function ψ. Since Ψα = gαβ α β −1 the transition relation (ψ)α = ξαβ (ψ)β . It means that ψ is actually a local section of the −1 line bundle KC 2 . There are two linearly independent solutions of (9.0.1), because q is a holomorphic function on Uα . Since φ(q) is independent of ~ and takes the form 0 q (9.0.2) φ(q) = , 1 0 48 O. DUMITRESCU AND M. MULASE we see that (9.0.1) is equivalent to the second order equation ~2 ψ 00 − qψ = 0 (9.0.3) −1 for ψ ∈ KC 2 . Since we are using a fixed projective coordinate system, the connection ∇~ (q) takes the same form on each coordinate neighborhood Uα . Therefore, the shape of the differential equation of (9.0.3) as an equation for ψ is again the same on every coordinate neighborhood, as we wished to achieve in (8.0.2). This is exactly what we refer to as the quantum curve of the spectral curve det(η − φ(q)) = 0. It is now obvious to calculate the semi-classical limit of the D-module corresponding to ~∇~ (q). Theorem 9.1 (Semi-classical limit of an oper). Under the same setting of Theorem 8.9, let E(q) denote the D-module E~ , ~∇~ (q) associated with the oper of (8.0.49). Then the semiclassical limit of E(q) is the spectral curve Σ ⊂ T ∗ C of φ(q) defined by the characteristic equation det(η − φ(q)) = 0. The semi-classical limit of (9.0.3) is the limit " # 2 1 1 d (9.0.4) lim e− ~ S0 (zα ) ~2 − q e ~ S0 (za ) = y 2 − q, ~→0 dzα where S0 (zα ) is a holomorphic function on Uα so that dS0 = ydzα gives a local trivialization of T ∗ C over Uα . The computation of semi-classical limit is the same as the calculation of the determinant of the connection ~∇~ (q), after taking conjugation by the scalar diagonal 1 matrix e− ~ S0 (zα ) I2×2 , and then take the limit as ~ goes to0. For every ~ ∈ H 1 (C, KC ), the ~-connection E~ , ~∇~ (q) of (8.0.44) defines a global DC module structure in E~ . Thus we have constructed a universal family EC of DC -modules on a given C with a fixed spin structure and a projective structure: ⊃ E~ , ∇~ (q) (9.0.5) EC o C × B × H 1 (C, KC ) o C × {q} × {~} The universal family SC of spectral curves is defined over C × B. (9.0.6) P (KC ⊕ OC ) × B o C ×B o SC o C ×B o = det η − φ(q) ⊃ 0 C × {q}. The semi-classical limit is thus a map of families (9.0.7) EC / SC / C × B. C × B × H 1 (C, KC ) 10. Non-Abelian Hodge correspondence between Hitchin moduli spaces The biholomorphic map (8.0.49) is defined by fixing a projective structure of the base curve C. Gaiotto [42] conjectured that such a correspondence would be canonically constructed through a scaling limit of non-Abelian Hodge correspondence. The conjecture has INVITATION TO 2D TQFT AND QUANTIZATION 49 been solved in [26] for holomorphic Hitchin moduli spaces MDol and MdeR constructed over an arbitrary complex simple and simply connected Lie group G. In this section, we review the main result of [26] for G = SL2 (C) and compare it with our quantization. We denote by E top the topologically trivial complex vector bundle of rank 2 on a compact Riemann surface C of genus g ≥ 2. The correspondence between stability conditions of holomorphic vector bundles on C and PDEs on differential geometric data is used in Narasimhan-Seshadri [83] to obtain topological structures of the moduli space of stable bundles (see also [5, 82]). Extending this classical case, the stability condition for an SL2 (C)-Higgs bundle (E, φ) translates into a system of PDEs, known as Hitchin’s equations, imposed on a set of geometric data [23, 52, 90]. The data we need are a Hermitian fiber metric h on E top , a unitary connection D in E top with respect to h, and a differentiable sl2 (C)-valued 1-form φ on C. In this section we use D for unitary connections to avoid confusion with holomorphic connections we have been using until the last section. Hitchin’s equations are the following system of nonlinear PDEs. ( FD + [φ, φ† ] = 0 (10.0.1) D0.1 φ = 0. Here, FD denotes the curvature 2-form of D, φ† is the Hermitian conjugate of φ with respect to the metric h, and D0,1 is the Cauchy-Riemann part of D defined by the complex structure of the base curve C. D0,1 determines a complex structure in E top , which we simply denote by E. Then φ becomes a holomorphic Higgs field in E because it satisfies the Cauchy-Riemann equation (10.0.1). The pair (E, φ) constructed in this way from a solution of Hitchin’s equations is a stable Higgs bundle. Conversely [90], a stable Higgs bundle (E, φ) gives rise to a unique harmonic Hermitian metric h and the Chern connection D with respect to h so that the data satisfy Hitchin’s equations. The stability condition for the holomorphic Higgs pair (E, φ) is thus translated into (10.0.1). Define a one-parameter family of connections 1 ζ ∈ C∗ . (10.0.2) D(ζ) := · φ + D + ζ · φ† , ζ Then the flatness of D(ζ) for all ζ is equivalent to (10.0.1). The non-Abelian Hodge correspondence [23, 52, 73, 90] is a diffeomorphic correspondence e ∇) e ∈ MdeR . ν : MDol 3 (E, φ) 7−→ (E, Proving the diffeomorphism of these moduli spaces is far beyond of the scope of this article. Here, we only give the definition of the map ν. We start with the solution (D, φ, h) of Hitchin’s equations corresponding to a stable Higgs bundle (E, φ). It induces a family of e in E top by D(ζ = 1)0,1 . Since D(ζ) is flat connections D(ζ). Define a complex structure E 1,0 e := D(ζ = 1) is automatically a holomorphic connection in E. e Stability of (E, φ) flat, ∇ e ∇) e ∈ MdeR . implies that the resulting holomorphic connection is irreducible, hence (E, Since this correspondence goes through the real unitary connection D, the change of the e is not a holomorphic deformation. complex structure of E to that of E Extending the idea of one-parameter family (10.0.2), Gaiotto conjectures: Conjecture 10.1 (Gaiotto [42]). Let (D, φ, h) be the solution of (10.0.1) corresponding to a sable Higgs bundle (E0 , φ(q)) on the SL2 (C)-Hitchin section (8.0.27). Consider the following two-parameter family of connections 1 (10.0.3) D(ζ, R) := · Rφ + D + ζ · Rφ† , ζ ∈ C∗ , R ∈ R+ . ζ 50 O. DUMITRESCU AND M. MULASE Then the scaling limit (10.0.4) lim R→0,ζ→0 ζ/R=~ D(ζ, R) exists for every ~ ∈ C∗ , and forms an ~-family of SL2 (C)-opers. Remark 10.2. (1) The existence of the limit is non-trivial, because the Hermitian metric h blows up as R → 0. (2) Unlike the case of non-Abelian Hodge correspondence, the Gaiotto limit works only for a point in the Hitchin section. Theorem 10.3 ([26]). Gaiotto’s conjecture holds for an arbitrary simple and simply connected complex algebraic group G. Recall that the representation (3.0.2) gives a realization of C from its universal covering space H as C∼ = H ρ π1 (C) . The representation ρ lifts to SL2 (R) ⊂ SL2 (C), and defines a projective structure in C subordinating its complex structure coming from H. This projective structure is what we call the Fuchsian projective structure. Corollary 10.4 (Gaiotto correspondence and quantization [26]). Under the same setting of Conjecture 10.1, the limit oper of (10.0.4) is given by 1 (10.0.5) lim D(ζ, R) = d − φ(q) = ∇~ (q), ~= 6 0, R→0,ζ→0 ~ ζ/R=~ with respect to the Fuchsian projective coordinate system. The correspondence γ (E0 , φ(q)) 7−→ E~ , ∇~ (q) is biholomorphic, unlike the non-Abelian Hodge correspondence. Proof. The key point is that since E0 is made out of KC , the fiber metric h naturally comes from the metric of C itself. Hitchin’s equations (10.0.1) for q = 0 then become a harmonic equation for the metric of C, and its solution is given by the constant curvature hyperbolic metric. This metric in turn defines the Fuchsian projective structure in C. For more detail, we reefer to [25, 26]. 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Olivia Dumitrescu: Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, U.S.A., and Simion Stoilow Institute of Mathematics, Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania E-mail address: dumit1om@cmich.edu Motohico Mulase: Department of Mathematics, University of California, Davis, CA 95616–8633, U.S.A., and Kavli Institute for Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa, Japan E-mail address: mulase@math.ucdavis.edu	0
Containment for Rule-Based Ontology-Mediated Queries Pablo Barceló Gerald Berger Andreas Pieris Center for Semantic Web Research & DCC, University of Chile Institute of Information Systems TU Wien School of Informatics University of Edinburgh arXiv:1703.07994v3 [cs.DB] 19 Apr 2017 pbarcelo@dcc.uchile.cl gberger@dbai.tuwien.ac.at ABSTRACT Following recent work [24, 26, 27, 40], we focus on the case where the ontology is defined by a set of tuple-generating dependencies (tgds), a.k.a. existential rules or Datalog± rules. Handling such OMQs implies new challenges for classical database tasks. Interestingly, some of these challenges are by now well-studied; most notably (a) query evaluation [8, 24, 25, 27]: given an OMQ Q = (S, Σ, q), a database D over S, and a tuple of constants c̄, does c̄ belong to the evaluation of q over every extension of D that satisfies Σ, or, equivalently, is c̄ a certain answer for Q over D? and (b) relative expressiveness [19, 42, 43]: how does the expressiveness of OMQs compare to the one of other query languages? Surprisingly, despite its prominence, no work to date has carried out an in-depth investigation of containment for OMQs based on tgds and UCQs. Query containment is a fundamental static analysis task that amounts to check if the evaluation of a query is always contained in the evaluation of another query. Several database tasks crucially depend on the ability to check query containment; these include, e.g., query optimization, viewbased query answering, querying incomplete databases, integrity checking, and implication of dependencies: cf. [22, 30, 36, 37, 39, 45]. A particularly important instance of the containment problem is the one defined by the class of CQs. It follows from the seminal work of Chandra and Merlin [29] that CQ containment is polynomially equivalent to CQ evaluation, and thus NP-complete. The NP upper bound is not affected if we consider UCQs [54]. This is seen as a positive result for practical applications that rely on UCQ containment, as the input (the two UCQs) is small. In addition, it shows a stark difference with more expressive relational query languages, e.g., relational algebra (or, equivalently, first-order logic), for which containment is undecidable. The main goal of this work is to understand up to which extend the good computational properties of UCQ containment discussed above can be leveraged to the containment problem for OMQs based on tgds and UCQs (simply called OMQs from now on). In particular, we want to understand which classes of tgds guarantee the decidability of the problem, and, whenever this is the case, how can we obtain complexity bounds that are reasonable for practical purposes. We also want to understand what is the exact relationship between OMQ containment and evaluation for such classes. Let us stress that, apart from the traditional applications of containment mentioned above, it has been recently shown that OMQ containment has applications on other important static analysis tasks for OMQs, namely, distribution over components [15], and UCQ rewritability [16]. Many efforts have been dedicated to identifying restrictions on ontologies expressed as tuple-generating dependencies (tgds), a.k.a. existential rules, that lead to the decidability for the problem of answering ontology-mediated queries (OMQs). This has given rise to three families of formalisms: guarded, non-recursive, and sticky sets of tgds. In this work, we study the containment problem for OMQs expressed in such formalisms, which is a key ingredient for solving static analysis tasks associated with them. Our main contribution is the development of specially tailored techniques for OMQ containment under the classes of tgds stated above. This enables us to obtain sharp complexity bounds for the problems at hand, which in turn allow us to delimitate its practical applicability. We also apply our techniques to pinpoint the complexity of problems associated with two emerging applications of OMQ containment: distribution over components and UCQ rewritability of OMQs. 1. apieris@inf.ed.ac.uk INTRODUCTION Motivation and goals. The novel application of knowledge representation tools for handling incomplete and heterogeneous data is giving rise to a new field, recently coined as knowledge-enriched data management [6]. A crucial problem in this field is ontology-based data access (OBDA) [51], which refers to the utilization of ontologies (i.e., sets of logical sentences) for providing a unified conceptual view of various data sources. Users can then pose their queries solely in the schema provided by the ontology, abstracting away from the specifics of the individual sources. In OBDA, one interprets the ontology Σ and the user query q, which is typically a union of conjunctive queries (UCQ), or, equivalently, the expressions defined by the select-project-join-union operators of relational algebra, as two components of one composite query Q = (S, Σ, q), known as ontology-mediated query (OMQ); S is called the data schema, indicating that Q will be posed on databases over S [19]. Therefore, OBDA is often realized as the problem of answering OMQs. 1 The context. As one might expect, when considered in its full generality, i.e., without any restrictions on the set of tgds, the OMQ containment problem is undecidable. To understand, on the other hand, which restrictions lead to decidability, we recall the two main reasons that render the general containment problem undecidable. These are: The above small witness property allows us to devise a simple non-deterministic algorithm, which makes use of query evaluation as a subroutine for checking non-containment of Q1 in Q2 : guess a database D over S of size at most k, and then check if there is a certain answer for Q1 over D that is not a certain answer for Q2 over D. This algorithm allows us to obtain optimal upper bounds for OMQs based on linear and sticky sets of tgds; however, the exact complexity of OMQs based on non-recursive sets of tgds remains open: Undecidability of query evaluation: OMQ evaluation is, in general, undecidable [12], and it can be reduced to OMQ containment. More precisely, OMQ containment is undecidable whenever query evaluation for at least one of the involved languages (i.e., the language of the left-hand or the right-hand side query) is undecidable. • For OMQs based on linear tgds, the problem is in PSpace, and in ΠP 2 if the arity is fixed. The PSpacehardness is shown by reduction from query evaluation [47], while the ΠP 2 -hardness is inherited from [17]. Undecidability of containment for Datalog: decidability of query evaluation does not ensure decidability of query containment. A prime example is Datalog, or, equivalently, the OMQ language based on full tgds. Datalog containment is undecidable [55], and thus, OMQ containment is undecidable if the involved languages extend Datalog. • For OMQs based on sticky sets of tgds, the problem is in coNExpTime, and in ΠP 2 if the arity of the schema is fixed. The coNExpTime-hardness is shown by exploiting the standard tiling problem for the exponential grid, while the ΠP 2 -hardness is inherited from [17]. • Finally, for OMQs based on non-recursive sets of tgds, containment is in ExpSpace and hard for PNEXP , even for fixed arity. The lower bound is shown by exploiting a recently introduced tiling problem [34]. In view of the above observations, we focus on languages that (a) have a decidable query evaluation, and (b) do not extend Datalog. The main classes of tgds, which give rise to OMQ languages with the desirable properties, can be classified into three main families depending on the underlying syntactic restrictions: (i) guarded tgds [24], which contain inclusion dependencies and linear tgds, (ii) non-recursive sets of tgds [35], and (iii) sticky sets of tgds [27]. While the decidability of containment for the above OMQ languages can be established via translations into query languages with a decidable containment problem, such translations do not lead to optimal complexity upper bounds (details are given below). Therefore, the main goal of our paper is to develop specially tailored decision procedures for the containment problem under the OMQ languages in question, and ideally obtain precise complexity bounds. Our second goal is to exploit such techniques in the study of distribution over components and UCQ rewritability of OMQs. We conclude that in all these cases OMQ containment is harder than evaluation, with one exception: the OMQs based on linear tgds over schemas of unbounded arity. Guarded tgds. The OMQ language based on guarded tgds is not UCQ rewritable, which forces us to develop different tools to study its containment problem. Let us remark that guarded OMQs can be rewritten as guarded Datalog queries (by exploiting the translations devised in [9, 43]), for which containment is decidable in 2ExpTime [20]. But, again, the known rewritings are very large [43], and hence the reduction of containment for guarded OMQs to containment for guarded Datalog does not yield optimal upper bounds. To obtain optimal bounds for the problem in question, we exploit two-way alternating parity automata on trees (2WAPA) [32]. We first show that if Q1 and Q2 are guarded OMQs such that Q1 is not contained in Q2 , then this is witnessed over a class of “tree-like” databases that can be represented as the set of trees accepted by a 2WAPA A. We then build a 2WAPA B with exponentially many states that recognizes those trees accepted by A that represent witnesses to non-containment of Q1 in Q2 . Hence, Q1 is contained in Q2 iff B accepts no tree. Since the emptiness problem for 2WAPA is feasible in exponential time in the number of states [32], we obtain that containment for guarded OMQs is in 2ExpTime. A matching lower bound, even for fixed arity schemas, follows from [16]. Similar ideas based on 2WAPA have been recently used to show that containment for OMQs based on expressive description logics (DLs) is in 2ExpTime [16]. In the DL context, schemas consist only of unary and binary relations. Our automata construction, however, is different from the one in [16] for two reasons: (a) we need to deal with higher arity relations, and (b) even for unary and binary relations, our OMQ language allows to express properties that are not expressible by the DL-based OMQ languages studied in [16]. Our contributions. The complexity of OMQ containment for the languages in question is given in Table 1. Using small fonts, we recall the complexity of OMQ evaluation in order to stress that containment is, in general, harder than evaluation. We divide our contributions as follows: Linear, non-recursive and sticky sets of tgds. The OMQ languages based on linear, non-recursive, and sticky sets of tgds share a useful property: they are UCQ rewritable (implicit in [40]), that is, an OMQ can be rewritten into a UCQ. This property immediately yields decidability for their associated containment problems, since UCQ containment is decidable [54]. However, the obtained complexity bounds are not optimal, since the UCQ rewritings are unavoidably very large [40]. To obtain more precise bounds, we reduce containment to query evaluation, an idea that is often applied in query containment; see, e.g., [29, 31, 54]. Consider a UCQ rewritable OMQ language O. If Q1 and Q2 belong to O, both with data schema S, then we can establish a small witness property, which states that noncontainment of Q1 in Q2 can be witnessed via a database over S whose size is bounded by an integer k ≥ 0, the maximal size of a disjunct in a UCQ rewriting of Q1 . For linear tgds, such an integer k is polynomial, but for non-recursive and sticky sets of tgds it is exponential (implicit in [40]). Combining languages. The above complexity results refer to the containment problem relative to a certain OMQ language O, i.e., both queries fall in O. However, it is natural 2 Linear Sticky Arbitrary Arity Bounded Arity PSpace-c ΠP 2 -c PSpace-c NP-c coNExpTime-c ΠP 2 -c ExpTime-c NP-c NEXP Non-recursive Guarded in ExpSpace and P -hard in ExpSpace and PNEXP -hard NExpTime-c NExpTime-c 2ExpTime-c 2ExpTime-c 2ExpTime-c ExpTime-c Table 1: Complexity of OMQ containment – in small fonts, we recall the complexity of OMQ evaluation. to consider the version of the problem where the involved OMQs fall in different languages. Unsurprisingly, if the lefthand side query is expressed in a UCQ rewritable OMQ language (based on linear, non-recursive or sticky sets of tgds), we can use the algorithm that relies on the small witness property discussed above, which provides optimal upper bounds for almost all the considered cases (the only exception is the containment of sticky in non-recursive OMQs over schemas of unbounded arity). Things are more interesting if the ontology of the left-hand side query is expressed using guarded tgds, while the ontology of the right-hand side query is not guarded. By exploiting automata techniques, we show that containment of guarded in non-recursive OMQs is in 3ExpTime, while containment of guarded in sticky OMQs is in 2ExpTime. We establish matching lower bounds, even over schemas of fixed arity, by refining techniques from [31]. size of the UCQs and the maximum arity of the underlying schema, which are typically very small. For such tasks, a single-exponential time procedure is considered to be acceptable, and it is actually the norm in many cases including database and verification problems; see, e.g., [1, 50, 52]. For OMQs based on sticky, non-recursive and guarded sets of tgds, the containment problem becomes coNExpTimecomplete, PNEXP -hard and 2ExpTime-complete, respectively. This means that we require double-exponential time to solve the problem, which is practically not acceptable. Nevertheless, for sticky sets of tgds, the runtime is doubleexponential only in the maximum arity of the schema, while for guarded sets of tgds is double-exponential only in the size of the UCQs and the maximum arity of the schema. This is good news since, as said above, the size of the UCQs and the arity are typically small, and usually UCQs in OMQs are much smaller than the ontologies. For non-recursive sets of tgds, on the other hand, the runtime is double-exponential, not only in the maximum arity, but also in the number of predicates occurring in the ontology. It is unrealistic to assume that the number of predicates occurring in real-life ontologies is small. This fact, together with the fact that the precise complexity of OMQ containment for non-recursive sets of tgds is still open, suggests that a more careful complexity analysis is needed. This is left as an interesting open problem for future work. Applications. Our techniques and results on containment for guarded OMQs can be applied to other important static analysis tasks, in particular, distribution over components and UCQ rewritability. The notion of distribution over components has been introduced in [3], in the context of declarative networking, and it states that the answer to an OMQ Q can be computed by parallelizing it over the (maximally connected) components of the database. If this is the case, then Q can always be evaluated in a distributed and coordination-free manner. The problem of deciding distribution over components for OMQs has been recently studied in [15]. However, the exact complexity of the problem for guarded OMQs has been left open. By exploiting our results on containment, we can show that it is 2ExpTime-complete. It is well-known that the OMQ language based on guarded tgds is not UCQ rewritable. In view of this fact, it is important to study when a given guarded OMQ Q can be rewritten as a UCQ. This has been studied for OMQs based on central Horn DLs [16, 18]. Interestingly, our automata-based techniques for guarded OMQ containment can be adapted to decide in 2ExpTime whether an OMQ based on guarded tgds over unary and binary relations is UCQ rewritable; a matching lower bound is inherited from [16]. Our result generalizes the result that deciding UCQ rewritability for OMQs based on E LHI, one of the most expressive members of the E L-family of DLs, is 2ExpTime-complete [16]. Organization. Preliminaries are in Section 2. In Section 3 we introduce the OMQ containment problem. Containment for UCQ rewritable OMQs is studied in Section 4, and for guarded OMQs in Section 5. In Section 6 we consider the case where the involved queries fall in different languages. In Section 7 we discuss the applications of our results on guarded OMQ containment and we conclude in Section 8. Proofs and additional details can be found in the appendix. 2. PRELIMINARIES Databases and conjunctive queries. Let C, N, and V be disjoint countably infinite sets of constants, (labeled) nulls and (regular) variables (used in queries and dependencies), respectively. A schema S is a finite set of relation symbols (or predicates) with associated arity. We write R/n to denote that R has arity n. A term is a either a constant, null or variable. An atom over S is an expression of the form R(v̄), where R ∈ S is of arity n > 0 and v̄ is an n-tuple of terms. A fact is an atom whose arguments consist only of constants. An instance over S is a (possibly infinite) set of atoms over S that contain constants and nulls, while a database over S is a finite set of facts over S. We may call an instance and a database over S an S-instance and S-database, respectively. Discussion on Applicability. As shown in Table 1, the containment problem for OMQs based on linear sets of tgds is PSpace-complete, and thus can be solved in singleexponential time. This is not a big practical drawback since the containment problem corresponds to a static analysis task. In fact, the runtime is single exponential only in the 3 Ontology-mediated queries. An ontology-mediated query (OMQ) is a triple (S, Σ, q), where S is a schema, Σ is a set of tgds (the ontology), and q is a (U)CQ over S ∪ sch(Σ) (and possibly other predicates), with sch(Σ) the set of predicates occurring in Σ.1 We call S the data schema. Notice that the set of tgds can introduce predicates not in S, which allows us to enrich the schema of the UCQ q. Moreover, the tgds can modify the content of a predicate R ∈ S, or, in other words, R can appear in the head of a tgd of Σ. We have explicitly included S in the specification of the OMQ to emphasize that it will be evaluated over S-databases, even though Σ and q might use additional relational symbols. The semantics of an OMQ is given in terms of certain answers. The certain answers to a UCQ q(x̄) w.r.t. a database D and a set Σ of tgds is the set of tuples: \ {c̄ ∈ dom(I)\|x̄\| \| c̄ ∈ q(I)}. cert (q, D, Σ) = The active domain of an instance I, denoted dom(I), is the set of all terms occurring in I. A conjunctive query (CQ) over S is a formula of the form: q(x̄) := ∃ȳ R1 (v̄1 ) ∧ · · · ∧ Rm (v̄m ) , (1) where each Ri (v̄i ) (1 ≤ i ≤ m) is an atom without nulls over S, each variable mentioned in the v̄i ’s appears either in x̄ or ȳ, and x̄ are the free variables of q. If x̄ is empty, then q is a Boolean CQ. As usual, the evaluation of CQs is defined in terms of homomorphisms. Let I be an instance and q(x̄) a CQ of the form (1). A homomorphism from q to I is a mapping h, which is the identity on C, from the variables that appear in q to the set of constants and nulls C ∪ N such that Ri (h(v̄i )) ∈ I, for each 1 ≤ i ≤ m. The evaluation of q(x̄) over I, denoted q(I), is the set of all tuples h(x̄) of constants such that h is a homomorphism from q to I. We denote by CQ the class of conjunctive queries. A union of conjunctive queries (UCQ) over S is a formula of the form q(x̄) := q1 (x̄) ∨ · · · ∨ qn (x̄), where each qi (x̄) is a CQ of the form (1). The S evaluation of q(x̄) over I, denoted q(I), is the set of tuples 1≤i≤n qi (I). We denote by UCQ the class of union of conjunctive queries. I⊇D,I\|=Σ Consider an OMQ Q = (S, Σ, q). The evaluation of Q over an S-database D, denoted Q(D), is defined as cert (q, D, Σ). It is well-known that cert (q, D, Σ) = q(chase(D, Σ)); see, e.g., [24]. Thus, Q(D) = q(chase(D, Σ)). Ontology-mediated query languages. We write (C, Q) for the OMQ language that consists of all OMQs of the form (S, Σ, q), where Σ falls in the class C of tgds, i.e., C ⊆ TGD (concrete classes of tgds are discussed below), and the query q falls in Q ∈ {CQ, UCQ}. A problem that is quite important for our work is OMQ evaluation, defined as follows: Tgds and the chase procedure. A tuple-generating dependency (tgd) is a first-order sentence of the form: ∀x̄∀ȳ φ(x̄, ȳ) → ∃z̄ ψ(x̄, z̄) , (2) where φ and ψ are conjunctions of atoms without nulls. For brevity, we write this tgd as φ(x̄, ȳ) → ∃z̄ ψ(x̄, z̄) and use comma instead of ∧ for conjoining atoms. Notice that φ can be empty, in which case the tgd is called fact tgd and is written as ⊤ → ∃z̄ ψ(x̄, z̄). We assume that each variable in x̄ is mentioned in some atom of ψ. We call φ and ψ the body and head of the tgd, respectively. The tgd in (2) is logically equivalent to the expression ∀x̄(qφ (x̄) → qψ (x̄)), where qφ (x̄) and qψ (x̄) are the CQs ∃ȳ φ(x̄, ȳ) and ∃z̄ ψ(x̄, z̄), respectively. Thus, an instance I over S satisfies this tgd iff qφ (I) ⊆ qψ (I). We say that an instance I satisfies a set Σ of tgds, denoted I \|= Σ, if I satisfies every tgd in Σ. We denote by TGD the class of (finite) sets of tgds. The chase is a useful algorithmic tool when reasoning with tgds [24, 35, 47, 49]. We start by defining a single chase step. Let I be an instance over a schema S and τ = φ(x̄, ȳ) → ∃z̄ ψ(x̄, z̄) a tgd over S. We say that τ is applicable w.r.t. I if there exists a tuple (ā, b̄) of terms in I such that φ(ā, b̄) holds in I. In this case, the result of applying τ over I with (ā, b̄) ¯ is the instance J that extends I with every atom in ψ(ā, ⊥), ¯ where ⊥ is the tuple obtained by simultaneously replacing each variable z ∈ z̄ with a fresh distinct null not occurring PROBLEM : INPUT : QUESTION : It is well-known that Eval(TGD, CQ) is undecidable; implicit in [12]. This has led to a flurry of activity for identifying syntactic restrictions on sets of tgds that make the latter problem decidable. Such a restriction defines a subclass C of tgds. The known decidable classes of tgds are classified into three main decidability paradigms, which, in turn, give rise to decidable OMQ languages: Guardedness: A tgd is guarded if its body contains an atom, called guard, that contains all the body-variables. Although the chase under guarded tgds does not necessarily terminate, the problem of deciding whether a tuple of constants is a certain answer to a UCQ w.r.t. a database and a set of guarded tgds is decidable. This follows from the fact that the result of the chase has bounded treewidth (see, e.g., [24]). Let G be the class of (finite) sets of guarded tgds. Then: τ,(ā,b̄) in I. For such a single chase step we write I −−−−→ J. Let us assume now that I is an instance and Σ a finite set of tgds. A chase sequence for I under Σ is a sequence: τ0 ,c̄0 Eval(C, Q) An OMQ Q = (S, Σ, q(x̄)) ∈ (C, Q), an S-database D, and c̄ ∈ dom(D)\|x̄\| . Does c̄ ∈ Q(D)? Proposition 1. [24] Eval(G, CQ) and Eval(G, UCQ) are 2ExpTime-complete, and ExpTime-complete for fixed arity. τ1 ,c̄1 I0 −−−→ I1 −−−→ I2 · · · An important subclass of guarded tgds is the class of linear tgds whose body consists of a single atom. We write L for the class of (finite) sets of linear tgds. of chase steps such that: (1) I0 = I; (2) forSeach i ≥ 0, τi is S a tgd in Σ; and (3) i≥0 Ii \|= Σ. We call i≥0 Ii the result of this chase sequence, which always exists. Although the result of a chase sequence is not necessarily unique (up to isomorphism), each such result is equally useful for our purposes, since it can be homomorphically embedded into every other result. Thus, from now on, we denote by chase(I, Σ) the result of an arbitrary chase sequence for I under Σ. Proposition 2. [25, 47] Eval(L, CQ) and Eval(L, UCQ) are PSpace-complete, and NP-complete for fixed arity. 1 OMQs can be defined for arbitrary first-order theories, not only tgds, and first-order queries, not only UCQs [19]. 4 R(x,y), P(y,z) → ∃w T(x,y,w) T(x,y,z) → ∃w S(y,w) PROBLEM : INPUT : QUESTION : R(x,y), P(y,z) → ∃w T(x,y,w) T(x,y,z) → ∃w S(x,w) Whenever O1 = O2 = O, we refer to the containment problem by simply writing Cont(O). In what follows, we establish some simple but fundamental results, which help to better understand the nature of our problem. We first investigate the relationship between evaluation and containment, which in turn allows us to obtain an initial boundary for the decidability of our problem, i.e., we can obtain a positive result only if the evaluation problem for the involved OMQ languages is decidable (e.g., those introduced in the previous section). We then focus on the OMQ languages introduced in Section 2 and observe that, once we fix the class of tgds, it does not make a difference whether we consider CQs or UCQs. In other words, we show that an OMQ in (C, UCQ), where C ∈ {G, L, NR, S}, can be rewritten as an OMQ in (C, CQ). This fact simplifies our later complexity analysis since for establishing upper (resp., lower) bounds it suffices to focus on CQs (resp., UCQs). × (a) (b) R(x,y), P(y,z) → ∃w T(x,y,w) T(x,y,z) → ∃w S(x,w) Figure 1: Stickiness and Marking. Non-recursiveness: A set Σ of tgds is non-recursive (a.k.a. acyclic [35, 48]), if its predicate graph, the directed graph that encodes how the predicates of sch(Σ) depend on each other, is acyclic. Non-recursiveness ensures the termination of the chase, and thus decidability of OMQ evaluation. Let NR be the class of non-recursive (finite) sets of tgds. Then: 3.1 Evaluation vs. Containment As one might expect, OMQ evaluation and OMQ containment are strongly connected. In fact, as we explain below, the former can be easily reduced to the latter. But let us first introduce some auxiliary notation. Consider a database D and a tuple c̄ = (c1 , . . . , cn ) ∈ dom(D)n , where n ≥ 0. We denote by qD,c̄ (x̄), where x̄ = (xc1 , . . . , xcn ), the CQ obtained from the conjunction of atoms occurring in D after replacing each constant c with the variable xc . Consider now an OMQ Q = (S, Σ, q(x̄)) ∈ (C, CQ), where C is some class of tgds, an S-database D, and a tuple c̄ ∈ dom(D)\|x̄\| . It is not difficult to show that Proposition 3. [48] Eval(NR, CQ) and Eval(NR, UCQ) are NExpTime-complete, even for fixed arity. Stickiness: This condition ensures neither termination nor bounded treewidth of the chase. Instead, the decidability of OMQ evaluation is obtained by exploiting query rewriting techniques (more details on query rewriting are given in Section 4). The goal of stickiness is to capture joins among variables that are not expressible via guarded tgds, but without forcing the chase to terminate. The key property underlying this condition can be described as follows: during the chase, terms that are associated (via a homomorphism) with variables that appear more than once in the body of a tgd (i.e., join variables) are always propagated (or “stick”) to the inferred atoms. This is illustrated in Figure 1(a); the left set of tgds is sticky, while the right set is not. The formal definition is based on an inductive marking procedure that marks the variables that may violate the semantic property of the chase described above [27]. Roughly, during the base step of this procedure, a variable that appears in the body of a tgd τ but not in every head-atom of τ is marked. Then, the marking is inductively propagated from head to body as shown in Figure 1(b). Finally, a finite set of tgds Σ is sticky if no tgd in Σ contains two occurrences of a marked variable. Let S be the class of sticky (finite) sets of tgds. Then: c̄ ∈ Q(D) ⇐⇒ (sch(Σ), ∅, qD,c̄ ) ⊆ (sch(Σ), Σ, q). \| {z } \| {z } Q1 Q2 Let O∅ be the OMQ language that consists of all OMQs of the form (S, ∅, q), i.e., the set of tgds is empty, where q is a CQ. It is clear that Q1 ∈ O∅ and Q2 ∈ (C, CQ). Therefore, for every OMQ language O = (C, CQ), where C is a class of tgds, we immediately get that: Proposition 5. Eval(O) can be reduced in polynomial time into Cont(O∅ , O). We now show that the problem of evaluation is reducible to the complement of containment. Let us say that, for technical reasons which will be made clear in a while, we focus our attention on classes C of tgds that are closed under fact tgd extension, i.e., for every set Σ ∈ C, a set obtained from Σ by adding a (finite) set of fact tgds is still in C. This is not an unnatural assumption since every reasonable class of tgds, such as the ones introduced above, enjoy this property. Consider now an OMQ Q = (S, Σ, q(x̄)) ∈ (C, CQ), where C is some class of tgds, an S-database D, and a tuple c̄ ∈ dom(D)\|x̄\| . It is easy to see that Proposition 4. [27] Eval(S, CQ) and Eval(S, UCQ) are ExpTime-complete, and NP-complete for fixed arity. 3. Cont(O1 , O2 ) Two OMQs Q1 ∈ O1 and Q2 ∈ O2 . Does Q1 ⊆ Q2 ? OMQ CONTAINMENT: THE BASICS The goal of this work is to study in depth the problem of checking whether an OMQ Q1 is contained in an OMQ Q2 , both over the same data schema S, or, equivalently, whether Q1 (D) ⊆ Q2 (D) over every (finite) S-database D. In this case we write Q1 ⊆ Q2 ; we write Q1 ≡ Q2 if Q1 ⊆ Q2 and Q2 ⊆ Q1 . The OMQ containment problem in question is defined as follows; O1 and O2 are OMQ languages (C, Q), where C is a class of tgds (e.g., linear, non-recursive, sticky, etc.), and Q ∈ {CQ, UCQ}: c̄ ∈ Q(D) ⇐⇒ (S, Σ⋆D , qc̄⋆ ) 6⊆ (S, ∅, ∃x P (x)), \| {z } \| {z } Q1 Q2 Σ⋆D where is obtained from Σ by renaming each predicate R in Σ into R⋆ ∈ 6 S and adding the set of fact tgds {⊤ → R⋆ (c1 , . . . , ck ) \| R(c1 , . . . , ck ) ∈ D}, 5 3.3 Plan of Attack qc̄⋆ is obtained from q(c̄) by renaming each predicate R into R⋆ 6∈ S, and the predicate P does not occur in S. Indeed, the above equivalence holds since P 6∈ S implies that Q2 (D) = ∅, for every S-database D. Since C is closed under fact tgd extension, Q1 ∈ (C, CQ), while Q2 ∈ O∅ . We write coCont(O1 , O2 ) for the complement of Cont(O1 , O2 ). Hence, for every OMQ language O = (C, CQ), where C is a class of tgds (closed under fact tgd extension), it holds that: We are now ready to proceed with the complexity analysis of containment for the OMQ languages in question. Our plan of attack can be summarized as follows: • We consider, in Section 4, Cont((C, CQ)), for C ∈ {L, NR, S}. These languages enjoy a crucial property, called UCQ rewritability, which is very useful for our purposes. This property allows us to show the following result: if the containment does not hold, then this is witnessed via a “small” database, which in turn allows us to devise simple guess-and-check algorithms. • We then proceed, in Section 5, with Cont((G, CQ)). This OMQ language does not enjoy UCQ rewritability, and the task of establishing a small witness property as above turned out to be challenging. However, we show the following: if the containment does not hold, then this is witnessed via a “tree-shaped” database, which allows us to devise a decision procedure based on two-way alternating parity automata on finite trees. • In Section 6, we study the case where the OMQ containment problem involves two different languages. If the left-hand side language is UCQ rewritable, then we can devise a guess-and-check algorithm by exploiting the above small witness property. The challenging case is when the left-hand side language is (G, CQ), where again we employ techniques based on tree automata. Proposition 6. Eval(O) can be reduced in polynomial time into coCont(O, O∅ ). By definition, O∅ is contained in every OMQ language (C, CQ), where C is a class of tgds. Therefore, as a corollary of Propositions 5 and 6, we obtain an initial boundary for the decidability of OMQ containment: we can obtain a positive result only if the evaluation problem for the involved OMQ languages is decidable. More formally: Corollary 7. Cont(O1 , O2 ) is undecidable if Eval(O1 ) is undecidable or Eval(O2 ) is undecidable. Can we prove the converse of Corollary 7: Cont(O1 , O2 ) is decidable if both Eval(O1 ) and Eval(O2 ) are decidable? The answer to this question is negative. This is due to the fact that containment of Datalog queries is undecidable [55]. Since Datalog queries can be directly encoded in the OMQ language based on the class F of full tgds, i.e., those without existentially quantified variables, we obtain the following: 4. UCQ REWRITABLE LANGUAGES Proposition 8. [55] Cont((F, CQ)) is undecidable. We now focus on OMQ languages that enjoy the crucial property of UCQ rewritability. This result, combined with the fact that Eval(F) is decidable (since the chase under full tgds always terminates), implies that the converse of Corollary 7 does not hold. Proposition 8 also rules out the OMQ languages that are based on classes of tgds that extend F; e.g., the weak versions of the ones introduced in Section 2, called weakly guarded [24], weakly acyclic [35], and weakly sticky [27] that guarantee the decidability of OMQ evaluation.2 The question that comes up concerns the decidability and complexity of containment for the OMQ languages that are based on the non-weak versions of the above classes, i.e., guarded, non-recursive, and sticky. This will be the subject of the next two sections. Definition 1. (UCQ Rewritability) An OMQ language (C, CQ), where C ⊆ TGD, is UCQ rewritable if, for each OMQ Q = (S, Σ, q(x̄)) ∈ (C, CQ) we can construct a UCQ q ′ (x̄) such that Q(D) = q ′ (D) for every S-database D. We proceed to establish our desired small witness property, based on UCQ rewritability. By the definition of UCQ rewritability, for each language O that is UCQ rewritable, there exists a computable function fO from O to the natural numbers such that the following holds: for every OMQ Q = (S, Σ, q(x̄)) ∈ O, and UCQ rewriting q1 (x̄) ∨ · · · ∨ qn (x̄) of Q, it is the case that max1≤i≤n {\|qi \|} ≤ fO (Q), where \|qi \| denotes the number of atoms occurring in qi . Then: 3.2 From UCQs to CQs Before we proceed with the complexity analysis of containment for the OMQ languages in question, let us state the following useful result: Proposition 10. Consider a UCQ rewritable language O, and two OMQs Q ∈ O and Q′ ∈ (TGD, CQ), both with data schema S. If Q 6⊆ Q′ , then there exists an S-database D, where \|D\| ≤ fO (Q), such that Q(D) 6⊆ Q′ (D). Proposition 9. Given an OMQ Q ∈ (C, UCQ), where C ∈ {G, L, NR, S}, we can construct in polynomial time an OMQ Q′ ∈ (C, CQ) such that Q ≡ Q′ . In Proposition 10 we assume that the left-hand side query falls in a UCQ rewritable language, be we do not pose any restriction on the language of the right-hand side query. Thus, we immediately get a decision procedure for Cont(O1 , O2 ) if O1 is UCQ rewritable and Eval(O2 ) is decidable. Given Q1 = (S, Σ1 , q1 (x̄)) ∈ O1 and Q2 = (S, Σ2 , q2 (x̄)) ∈ O2 : The proof of Proposition 9 relies on the idea of encoding boolean operations (in our case the ‘or’ operator) using a set of atoms; this idea has been used in several other works (see, e.g., [14, 21, 41]). Proposition 9 allows us to focus on OMQs that are based on CQs; in fact, Cont((C1 , CQ), (C2 , CQ)) is C-complete, where C1 , C2 ∈ {G, L, NR, S} and C is a complexity class that is closed under polynomial time reductions, iff Cont((C1 , UCQ), (C2 , UCQ)) is C-complete. 1. Guess an S-database D such that \|D\| ≤ fO1 (Q1 ), and a tuple c̄ ∈ dom(D)\|x̄\| ; and 2 2. Verify that c̄ ∈ Q1 (D) and c̄ 6∈ Q2 (D). The idea of those classes is the same: relax the conditions in the definition of the class, so that only those positions that receive null values during the chase are taken into account. We immediately get that: 6 Proposition 14. It holds that \|sch(Σ)\| f(NR,CQ) (S, Σ, q) ≤ \|q\| · max{\|body (τ )\|} . Theorem 11. Cont(O1 , O2 ) is decidable if O1 is UCQ rewritable and Eval(O2 ) is decidable. τ ∈Σ This general result shows that Cont((C, CQ)) is decidable for every C ∈ {L, NR, S}, but it says nothing about its complexity. This will be the subject of the rest of the section. Proposition 14 implies that non-containment for queries that fall in (NR, CQ) is witnessed via a database of at most exponential size. We show next that this bound is optimal: 4.1 Linearity The problem of computing UCQ rewritings for OMQs in (L, CQ) has been studied in [40], where a resolution-based procedure, called XRewrite, has been proposed. This rewriting algorithm accepts a query Q = (S, Σ, q(x̄)) ∈ (L, CQ) and constructs a UCQ rewriting q ′ (x̄) over S by starting from q and exhaustively applying rewriting steps based on resolution. Let us illustrate this via a simple example: Proposition 15. There are sets of (NR, CQ) OMQs n {Qn 1 = (S, Σ1 , q1 )}n>0 R(x, y) → P (y), n {Qn 2 = (S, Σ2 , q2 )}n>0 , n where \|sch(Σn 1 )\| = \|sch (Σ2 )\| = n + 2, such that for every n n−1 S-database D, if Q1 (D) 6⊆ Qn . 2 (D) then \|D\| ≥ 2 Let us now focus on the complexity of Cont((NR, CQ)). The algorithm underlying Theorem 11, together with the exponential bound provided by Proposition 14, implies that coCont((NR, CQ)) is feasible in non-deterministic exponential time with access to a NExpTime oracle, which immediately implies that Cont((NR, CQ)) is in ExpSpace. Unfortunately, the exact complexity of Cont((NR, CQ)) is still an open problem, and we conjecture that is PNEXP -complete; recall that NExpTime ⊆ PNEXP ⊆ ExpSpace. In what follows, we briefly explain how the PNEXP -hardness is obtained. To this end, we exploit a tiling problem that has been recently introduced in [34]. Roughly speaking, an instance of this tiling problem is a triple (m, T1 , T2 ), where m is an integer in unary representation, and T1 , T2 are standard tiling problems for the exponential grid 2n × 2n . The question is whether, for every initial condition w of length m, T1 has no solution with w or T2 has some solution with w. The initial condition w simply fixes the first m tiles of the first row of the grid. We construct in polynomial time two (NR, CQ) queries Q1 and Q2 such that (m, T1 , T2 ) has a solution iff Q1 ⊆ Q2 . The idea is to force every input database to store an initial condition w of length m, and then encode the problem whether Ti has a solution with w into Qi , for each i ∈ {1, 2}. From the above discussion we get that: Example 1. Assume that S = {P, T }. Consider the set Σ consisting of the linear tgds P (x) → ∃y R(x, y), and T (x) → P (x), and the CQ q(x̄) := ∃y(R(x, y) ∧ P (y)). XRewrite will first resolve the atom P (y) in q using the second tgd, and produce the CQ ∃y(R(x, y) ∧ R(x, z)), which is equivalent to the CQ ∃y R(x, y). Then, ∃y R(x, y) will be resolved using the first tgd, and the CQ P (x) will be obtained, which in turn will be resolved using the third tgd in order to produce the CQ T (x). The UCQ rewriting q ′ (x̄) is P (x) ∨ T (x). It is easy to see that, whenever the input OMQ consists of linear tgds, during the execution of XRewrite it is not possible to obtain a CQ that has more atoms than the original one. This is an immediate consequence of the fact that linear tgds have only one atom in their body. Then: Proposition 12. f(L,CQ) (S, Σ, q) ≤ \|q\|. Having the above result in place, it can be shown that the algorithm underlying Theorem 11 guesses a polynomially sized witness to non-containment, and then calls a C-oracle for solving query evaluation under linear OMQs, where C is PSpace in general, and NP if the arity is fixed; these complexity classes are obtained from Proposition 2. Therefore, coCont((L, CQ)) is in PSpace in general, and in ΣP 2 in case of fixed arity. Regarding the lower bounds, Proposition 5 allows us to inherit the PSpace-hardness of Eval(L, CQ); this holds even for constant-free tgds. Unfortunately, in the case of fixed arity, we can only obtain NP-hardness, while Proposition 6 allows to obtain coNP-hardness. Nevertheless, it is implicit in [17] (see the proof of Theorem 9), where the containment problem for OMQ languages based on description logics is considered, that Cont((L, CQ)) is ΠP 2 -hard, even for tgds of the form P (x) → R(x). Then: Theorem 16. Cont((NR, CQ)) is in ExpSpace, and PNEXP -hard. The lower bound holds even if the arity of the schema is fixed and the tgds are without constants. 4.3 Stickiness We now focus on (S, CQ). As shown in [40], given a query (S, Σ, q), there exists an execution of XRewrite that constructs a UCQ rewriting q1 (x̄) ∨ · · · ∨ qn (x̄) over S with the following property: for each i ∈ {1, . . . , n}, if a variable v occurs in qi in more than one atom, then v already occurs in q. This property has been used in [40] to bound the number of atoms that can appear in a single CQ qi . We write T (q) for the set of terms (constants and variables) occurring in q, C(Σ) for the set of constants occurring in Σ, and ar (S) for the maximum arity over all predicates of S. Theorem 13. Cont((L, CQ)) is PSpace-complete, and ΠP 2 -complete if the arity of the schema is fixed. The lower bounds hold even for tgds without constants. Proposition 17. It holds that 4.2 Non-Recursiveness f(S,CQ) ((S, Σ, q)) ≤ \|S\| · (\|T (q)\| + \|C(Σ)\| + 1)\|ar (S)\| . Although the OMQ language (NR, CQ) is not explicitly considered in [40], where the algorithm XRewrite is defined, the same algorithm can deal with (NR, CQ). By analyzing the UCQ rewritings constructed by XRewrite, whenever the input query falls in (NR, CQ), we can establish the following result; here, body(τ ) denotes the body of the tgd τ : Proposition 17 implies that non-containment for (S, CQ) queries is witnessed via a database of at most exponential size. As for (NR, CQ) queries, we can show that this bound is optimal; here, for a set Σ of tgds, we denote by \|\|Σ\|\| the number of symbols occurring in Σ: 7 words, we study the problem for (G, BCQ), where BCQ denotes the class of Boolean CQs. This does not affect the generality of our proof since it is known that Cont((G, CQ)) can be reduced in polynomial time to Cont((G, BCQ)) [16]. Proposition 18. There exists a set of (S, CQ) OMQs n n n 2 {Q = ({S/n}, Σ , q(x̄))}n>0 , where \|\|Σ \|\| ∈ O(n ), such that for every Q = ({S}, Σ′ , q ′ (x̄)) ∈ (TGD, CQ) and {S}-database D, if Qn (D) 6⊆ Q(D) then \|D\| ≥ 2n−2 . A first glimpse. As already said, (G, CQ) is not UCQ rewritable and, therefore, we cannot employ Proposition 10 in order to establish a small witness property as for the languages considered in Section 4. We have tried to establish a small witness property for (G, CQ) by following a different route, but it turned out to be a difficult task. Nevertheless, we can show a tree witness property, which states that non-containment for (G, CQ) is witnessed via a treelike database. This allows us to devise a procedure based on alternating tree automata. Summing up, the proof for the 2ExpTime membership of (G, CQ) proceeds in three steps: We now study the complexity of Cont((S, CQ)). We first focus on schemas of unbounded arity. Proposition 17 implies that the algorithm underlying Theorem 11 runs in exponential time assuming access to a C-oracle, where C is a complexity class powerful enough for solving Eval(S, CQ) and its complement. But, since Eval(S, CQ) is in ExpTime (see Proposition 4), both Eval(S, CQ) and its complement are in NExpTime, and thus, the oracle call is not really needed. Consequently, coCont((C, CQ)) is in NExpTime. A matching lower bound is obtained by a reduction from the standard tiling problem for the exponential grid 2n × 2n . In fact, the same lower bound has been recently established in [15]; however, our result is stronger as it shows that the problem remains hard even if the right-hand side query is a linear OMQ of a simple form – this is also discussed in Section 6, where containment of queries that fall in different OMQ languages is studied. Regarding schemas of fixed arity, Proposition 17 provides a witness for non-containment of polynomial size, which implies that the algorithm underlying Theorem 11 runs in polynomial time with access to an NPoracle. Therefore, coEval(S, CQ) is in ΣP 2 , while a matching lower bound is implicit in [17]. Then: 1. Establish a tree witness property; 2. Encode the tree-like witnesses as trees that can be accepted by an alternating tree automaton; and 3. Construct an automaton that decides Cont((G, CQ)); in fact, we reduce Cont((G, CQ)) into emptiness for two-way alternating parity automata on finite trees. Each one of the above three steps is discussed in more details in the following three sections. 5.1 Tree Witness Property From the above informal discussion, it is clear that treelike databases are crucial for our analysis. Let us make this notion more precise using guarded tree decompositions. A tree decomposition of a database D is a labeled rooted tree T = (V, E, λ), where λ : V → 2dom(D) , such that: (i) for each atom R(t1 , . . . , tn ) ∈ D, there exists v ∈ V such that λ(v) ⊇ {t1 , . . . , tn }, and (ii) for every term t ∈ dom(D), the set {v ∈ V \| t ∈ λ(v)} induces a connected subtree of T . The tree decomposition T is called [U ]-guarded, where U ⊆ V , if, for every node v ∈ V \ U , there exists an atom R(t1 , . . . , tn ) ∈ D such that λ(v) ⊆ {t1 , . . . , tn }. We write root (T ) for the root node of T , and DT (v), where v ∈ V , for the subset of D induced by λ(v). We are now ready to formalize the notion of the tree-like database: Theorem 19. Cont((S, CQ)) is coNExpTime-complete, even if the set of tgds uses only two constants. In the case of fixed arity, it is ΠP 2 -complete, even for constant-free tgds. Clearly, there exists a double-exponential time algorithm for solving Cont((S, CQ)), which might sound discouraging. However, Proposition 17 implies that the runtime is doubleexponential only in the maximum arity of the data schema. 5. GUARDEDNESS We proceed with the problem of containment for guarded OMQs, and we establish the following result: Theorem 20. Cont((G, CQ)) is 2ExpTime-complete. The lower bound holds even if the arity of the schema is fixed, and the tgds are without constants. Definition 2. An S-database D is a C-tree, where C ⊆ D, if there is a tree decomposition T of D such that: The lower bound is immediately inherited from [16], where it is shown that containment for OMQs based on the description logic E LI is 2ExpTime-hard. Recall that a set of E LI axioms can be equivalently rewritten as a constant-free set of guarded tgds using only unary and binary predicates, which implies the lower bound stated in Theorem 20. However, we cannot immediately inherit the desired upper bound since the DL-based OMQ languages considered in [16] are either weaker than or incomparable to (G, CQ). Nevertheless, the technique developed in [16] was extremely useful for our analysis. Actually, our automata-based procedure exploits a combination of ideas from [16, 44]. The rest of this section is devoted to providing a high-level explanation of this procedure. For the sake of technical clarity, we focus on constant-free tgds and CQs, but all the results can be extended to the general case at the price of more involved definitions and proofs. Moreover, for simplicity, we focus on Boolean CQs. In other 1. DT (root (T )) = C and 2. T is [{root (T )}]-guarded. Roughly, whenever a database D is a C-tree, C is the cyclic part of D, while the rest of D is tree-like. Interestingly, for deciding Cont((G, BCQ)) it suffices to focus on databases that are C-trees and \|dom(C)\| depends only on the left-hand side OMQ. Recall that for a schema S we write ar (S) for the maximum arity over all predicates of S. Then: Proposition 21. Let Qi = (S, Σi , qi ) ∈ (G, BCQ), for i ∈ {1, 2}. The following are equivalent: 1. Q1 ⊆ Q2 . 2. Q1 (D) ⊆ Q2 (D), for every C-tree S-database D such that \|dom(C)\| ≤ (ar (S ∪ sch(Σ1 )) · \|q1 \|). 8 for 2WAPA. As usual, given a 2WAPA A, we denote by L(A) the language of A, i.e., the set of labeled trees it accepts. The emptiness problem is defined as follows: given a 2WAPA A, does L(A) = ∅? Thus, given Q1 , Q2 ∈ (G, BCQ), we need to construct a 2WAPA A such that Q1 ⊆ Q2 iff L(A) = ∅. It is well-known that deciding whether L(A) = ∅ is feasible in exponential time in the number of states, and in polynomial time in the size of the input alphabet [32]. Therefore, we should construct A in double-exponential time, while the number of states must be at most exponential. We first need a way to check consistency of labeled trees. It is not difficult to devise an automaton for this task. The fact that (1) ⇒ (2) holds trivially, while (2) ⇒ (1) is shown by using a variant of the notion of guarded unravelling and compactness. Let us clarify that the above result does not provide a decision procedure for Cont((G, BCQ)), since we have to consider infinitely many databases that are Ctrees with \|dom(C)\| ≤ (ar (S ∪ sch(Σ1 )) · \|q1 \|). 5.2 Encoding Tree-like Databases It is generally known that a database D whose treewidth3 is bounded by an integer k can be encoded into a tree over a finite alphabet of double-exponential size in k that can be accepted by an alternating tree automaton; see, e.g., [13]. Consider an alphabet Γ, and let N∗ be the set of finite sequences of natural numbers, including the empty sequence. A Γ-labeled tree is a pair L = (T, λ), where T ⊆ N∗ is closed under prefixes, and λ : T → Γ is the labeling function. The elements of T identify the nodes of L. It can be shown that D and a tree decomposition T of D with width k can be encoded as a Γ-labeled tree L, where Γ is an alphabet of double-exponential size in k, such that each node of T corresponds to exactly one node of L and vice versa. Consider now a C-tree S-database D, and let T be the tree decomposition that witnesses that D is a C-tree. The width of T is at most k = (\|dom(C)\| + ar (S) − 1), and thus, the treewidth of D is bounded by k. Hence, from the above discussion, D and T can be encoded as a Γ-labeled tree, where Γ is of double-exponential size in k. In general, given an Sdatabase D that is a C-tree due to the tree decomposition T , we show that D and T can be encoded as a ΓS,l -labeled tree, with \|dom(C)\| ≤ l and \|ΓS,l \| being double-exponential in ar (S) and exponential in \|S\| and l. Although every C-tree S-database D can be encoded as a ΓS,l -labeled tree, the other direction does not hold. In other words, it is not true that every ΓS,l -labeled tree encodes a C-tree S-database D and its corresponding tree decomposition. In view of this fact, we need the additional notion of consistency. A ΓS,l -labeled tree is called consistent if it satisfies certain syntactic properties – we do not give these properties here since they are not vital in order to understand the high-level idea of the proof. Now, given a consistent ΓS,l -labeled tree L, we can show that L can be decoded into an S-database JLK that is a C-tree with \|dom(C)\| ≤ l. From the above discussion and Proposition 21, we obtain: Lemma 23. Consider a schema S and an integer l > 0. There is a 2WAPA CS,l that accepts a ΓS,l -labeled tree L iff L is consistent. The number of states of CS,l is logarithmic in the size of ΓS,l . Furthermore, CS,l can be constructed in polynomial time in the size of ΓS,l . Now, the crucial task is, given an OMQ Q ∈ (G, BCQ), to devise an automaton that accepts labeled trees which correspond to databases that make Q true. Lemma 24. Let Q = (S, Σ, q) ∈ (G, BCQ). There is a 2WAPA AQ,l , where l > 0, that accepts a consistent ΓS,l labeled tree L iff Q(JLK) 6= ∅. The number of states of AQ,l is exponential in \|\|Q\|\| and l. Furthermore, AQ,l can be constructed in double-exponential time in \|\|Q\|\| and l. The intuition underlying AQ,l can be described as follows. AQ,l tries to identify all the possible ways the CQ q can be mapped to chase(D, Σ), for any C-tree S-database D such that \|dom(C)\| ≤ l. It then arrives at possible ways how the input tree can satisfy Q. These “possible ways” correspond to squid decompositions, a notion introduced in [24] that indicates which part of the query is mapped to the cyclic part C of D, and which to the tree-like part of D. The automaton exhaustively checks all squid decompositions by traversing the input tree and, at the same time, explores possible ways how to match the single parts of the squid decomposition at hand. The automaton finally accepts if it finds a squid decomposition that can be mapped to chase(D, Σ). Having the above automata in place, we can proceed with our main technical result, which shows that Cont(G, BCQ) can be reduced to the emptiness problem for 2WAPA. But let us first recall some key results about 2WAPA, which are essential for our final construction. It is well-known that languages accepted by 2WAPAs are closed under intersection and complement. Given two 2WAPAs A1 and A2 , we write A1 ∩ A2 for a 2WAPA, which can be constructed in polynomial time, that accepts the language L(A1 ) ∩ L(A2 ). Moreover, for a 2WAPA A, we write A for the 2WAPA, which is also constructible in polynomial time, that accepts the complement of L(A). We can now show the following: Lemma 22. Let Qi = (S, Σi , qi ) ∈ (G, BCQ), for i ∈ {1, 2}. The following are equivalent: 1. Q1 ⊆ Q2 . 2. Q1 (JLK) ⊆ Q2 (JLK), for every consistent ΓS,l -labeled tree L, where l = (ar (S ∪ sch(Σ1 )) · \|q1 \|). 5.3 Constructing Tree Automata Having the above result in place, we can now proceed with our automata-based procedure. We make use of twoway alternating parity automata (2WAPA) that run on finite labeled trees. Two-way alternating automata process the input tree while branching in an alternating fashion to successor states, and thereby moving either down or up the input tree; the detailed definition can be found in [11]. Our goal is to reduce Cont((G, BCQ)) to the emptiness problem Proposition 25. Consider Q1 , Q2 ∈ (G, BCQ). We can construct in double-exponential time a 2WAPA A, which has exponentially many states, such that Q1 ⊆ Q2 ⇐⇒ L(A) = ∅. Proof (sketch). Let Qi = (S, Σi , qi ), for i ∈ {1, 2}, and l = (ar (S ∪ sch(Σ1 )) · \|q1 \|). Then A is defined as (CS,l ∩ AQ1 ,l ) ∩ AQ2 ,l . Since ΓS,l has double-exponential size, Lemmas 23 and 24 imply that A can be constructed in 3 Recall that the treewidth of a database D is the minimum width among all possible tree decompositions T = (V, E, λ) of D, while the width of T is defined as maxv∈V {\|λ(v)\|} − 1. 9 The lower bounds hold even if the arity of the schema is fixed. Moreover, for C = L (resp., C ∈ {NR, S}) it holds even for tgds with one constant (resp., without constants). double-exponential time, while it has exponentially many states. Lemma 22 implies that Q1 ⊆ Q2 iff L(A) = ∅. Proposition 25 implies that Cont((G, BCQ)) is in 2ExpTime, and Theorem 20 follows. Thus, there exists a doubleexponential time algorithm for solving Cont((G, CQ)). Interestingly, the runtime is double-exponential only in the size of the CQs and the maximum arity of the schema. This can be obtained by a providing a more refined complexity analysis of the construction of the 2WAPA A in Proposition 25. 6. Upper bounds. The 2ExpTime membership when C = L is an immediate corollary of Theorem 20. This is not true when C ∈ {NR, S} since the right-hand side query is not guarded. But in this case, since (NR, CQ) and (S, CQ) are UCQ rewritable, one can rewrite the right-hand side query as a UCQ, and then apply the machinery developed in Section 5 for solving Cont((G, CQ)). More precisely, given OMQs Q1 ∈ (G, CQ) and Q2 ∈ (C, CQ), where C ∈ {NR, S}, Q1 ⊆ Q2 iff Q1 ⊆ q, where q is a UCQ rewriting of Q2 . Thus, an immediate decision procedure, which exploits the algorithm XRewrite, is the following: COMBINING LANGUAGES In the previous two sections, we studied the containment problem relative to a language O, i.e., both OMQs fall in O. However, it is natural to consider the version of the problem where the involved OMQs fall in different languages. This is the goal of this section. Our analysis proceeds by considering the two cases where the left-hand side (LHS) query falls in a UCQ rewritable OMQ language, or it is guarded. 1. Let q = XRewrite(Q2 ); 2. For each q ′ ∈ q: if Q1 ⊆ q ′ , then proceed; otherwise, reject; and 3. Accept. 6.1 The LHS Query is UCQ Rewritable The above procedure runs in triple-exponential time. The first step is feasible in double-exponential time [40]. Now, for a single CQ q ′ ∈ q (which is a guarded OMQ with an empty set of tgds) the check whether Q1 ⊆ q ′ can be done by using the machinery developed in Section 5, which reduces our problem to checking whether the language of a 2WAPA A is empty. However, it should not be forgotten that q ′ is of exponential size, and thus, the automaton A has double-exponentially many states. This in turn implies that checking whether L(A) = ∅ is in 3ExpTime, as claimed. Although the above algorithm establishes an optimal upper bound for non-recursive OMQs, a more refined analysis is needed for sticky OMQs. In fact, we need a more refined complexity analysis for the problem Cont((G, CQ), UCQ), that is, to decide whether a guarded OMQ is contained in a UCQ. To this end, we provide an automata construction different from the one employed in Section 5, which allows us to establish a refined complexity upper bound for the problem in question. Consider a (G, CQ) query Q, and a UCQ q = q1 ∨ · · · ∨ qn . As usual, we write \|\|Q\|\| and \|\|qi \|\| for the number of symbols that occur in Q and qi , respectively, and we write var ≥2 (qi ) for the set of variables that appear in more than one atom of qi . By exploiting our new automata-based procedure, we show that the problem of checking if Q ⊆ q is feasible in double-exponential time in (\|\|Q\|\| + max1≤i≤n {\|var ≥2 (qi )\|}), exponential time in max1≤i≤n {\|\|qi \|\|}, and polynomial time in n. This result allows us to show that the above procedure establishes 2ExpTime-membership when the right-hand side OMQ is sticky. But first we need to recall the following key properties of the UCQ rewriting q = XRewrite(Q2 ), constructed during the first step of the algorithm: As an immediate corollary of Theorem 11 we obtain the following result: Cont((C1 , CQ), (C2 , CQ)), for C1 6= C2 , C1 ∈ {L, NR, S} and C2 ∈ {L, NR, S, G}, is decidable. By exploiting the algorithm underlying Theorem 11, we establish optimal upper bounds for all the problems at hand with the only exception of Cont((S, CQ), (NR, CQ)). For the latter, we obtain an ExpSpace upper bound, by providing a similar analysis as for Cont((NR, CQ)), while a NExpTime lower bound is inherited from query evaluation by exploiting Proposition 5. It is rather tedious, and not very interesting from a technical point of view, to go through all the containment problems in question4 and explain in details how the exact upper bounds are obtained; we leave this as an exercise to the interested reader. Regarding the matching lower bounds, in most of the cases they are inherited from query evaluation or its complement by exploiting Propositions 5 and 6, respectively. There are, however, some exceptions: • Cont((S, CQ), (L, CQ)) in the case of unbounded arity, where the problem is coNExpTime-hard, even for sets of tgds that use only two constants. This is shown by a reduction from the standard tiling problem for the exponential grid 2n × 2n . • Cont((L, CQ), (S, CQ)) and Cont((S, CQ), (L, CQ)) in the case of bounded arity, where both problems are ΠP 2 -hard even for constant-free tgds; implicit in [17]. 6.2 The LHS Query is Guarded We proceed with the case where the LHS query is guarded, and we show the following result: 1. q consists of double-exponentially many CQs, Theorem 26. Cont((G, CQ), (C, CQ)) is C-complete:   2ExpTime, C ∈ {L, S}, C =  3ExpTime, C = NR. 2. each CQ of q is of exponential size, and 3. for each q ′ ∈ q, var ≥2 (q ′ ) is a subset of the variables of the original CQ that appears in Q2 . 4 There are eighteen different cases obtained by considering all the possible pairs (O1 , O2 ) of OMQ languages, where O1 6= O2 and O1 is UCQ rewritable, and the two cases whether the arity of the schema is fixed or not. By combining these key properties with the complexity analysis performed above, it is now straightforward to show that Cont((G, CQ), (S, CQ)) is in 2ExpTime. 10 has been left open. Our results on containment for guarded OMQs allow us to close this problem. But first we need to recall a key result that semantically characterizes distribution over components. An OMQ Q with data schema S is unsatisfiable if there is no S-database D such that Q(D) 6= ∅. Moreover, for a CQ q, we write co(q) for its components. The next result has been shown in [15]: Lower Bounds. We establish matching lower bounds by refining techniques from [31], where it is shown that containment of Datalog in UCQ is 2ExpTime-complete, while containment of Datalog in non-recursive Datalog is 3ExpTimecomplete; the lower bounds hold for fixed-arity predicates, and constant-free rules. Interestingly, the LHS query can be transformed into a Datalog query such that each rule has a body-atom that contains all the variables, i.e., is guarded. This is achieved by increasing the arity of some predicates in order to have enough positions for all the bodyvariables. However, for each rule, the number of unguarded variables that we need to guard is constant, and thus, the arity of the schema remains constant. We conclude that Cont((G, CQ), (NR, CQ)) is 3ExpTime-hard. Moreover, containment of guarded OMQs in UCQs is 2ExpTime-hard, which in turn allows us to show, by exploiting the construction underlying Proposition 9, that Cont((G, CQ), (L, CQ)) is 2ExpTime-hard, even if the set of linear tgds uses only one constant, while Cont((G, CQ), (S, CQ)) is 2ExpTime-hard, even for tgds without constants. 7. Proposition 27. Let Q = (S, Σ, q(x̄)) ∈ (G, CQ). The following are equivalent: 1. Q distributes over components. 2. Q is unsatisfiable or there exists q̂(x̄) ∈ co(q) such that (S, Σ, q̂(x̄)) ⊆ Q. Checking unsatisfiability can be easily reduced to containment. Thus, the above result, together with Theorem 20, implies that Dist(G, CQ) is in 2ExpTime, while a matching lower bound is implicit in [15]. Then: Theorem 28. Dist(G, CQ) is 2ExpTime-complete. APPLICATIONS Interestingly, our results on Cont((G, CQ)) can be applied to other important static analysis tasks, in particular, distribution over components and UCQ rewritability. Each one of those tasks is considered in the following two sections. 7.2 Deciding UCQ Rewritability Query rewriting is a well-studied method for evaluating OMQs using standard database technology. The key idea is the following: given an OMQ Q = (S, Σ, q(x̄)), combine Σ and q into a new query qΣ (x̄), the so-called rewriting, which can then be evaluated over D yielding the same answer as Q over D, for every S-database D. For this approach to be realistic, though, it is essential that the rewriting is expressed in a language that can be handled by standard database systems. The typical language that is considered in this setting is first-order (FO) queries [28]. Notice, however, that due to Rossman’s Theorem [53], and the fact that OMQs are closed under homomorphisms, FO and UCQ rewritability coincide. Recall that some OMQ languages are UCQ rewritable, such as the ones based on linear, non-recursive and sticky sets of tgds, while others are not, e.g., guarded OMQs. For those languages O that are not UCQ rewritable, it is important to be able to check whether a query Q ∈ O can be rewritten as a UCQ, in which case we say that it is UCQ rewritable. This gives rise to the following fundamental static analysis task for an OMQ language (C, CQ), where C ⊆ TGD: 7.1 Distribution Over Components The notion of distribution over components has been introduced in [3], and it states that the answer to a query can be computed by parallelizing it over the (maximally connected) components of the input database. But let us first make precise what a component is. A set of atoms A is connected if for all c, d ∈ dom(A), there exists a sequence α1 , . . . , αn of atoms in A such that c ∈ dom(α1 ), d ∈ dom(αn ), and dom(αi ) ∩ dom(αi+1 ) 6= ∅, for each i ∈ {1, . . . , n − 1}. We call B ⊆ A a component of A if (i) B is connected, and (ii) for every α ∈ A \ B, B ∪ {α} is not connected.5 Let co(A) be the set of components of A. We are now ready to introduce the notion of distribution over components. Consider an OMQ Q = (S, Σ, q) ∈ (TGD, CQ). We say that Q distributes over components if Q(D) = Q(D1 ) ∪ · · · ∪ Q(Dn ), where co(D) = {D1 , . . . , Dn }, for every S-database D. In this case, Q(D) can be computed without any communication over a network using a distribution where every computing node is assigned some of the components of the database, and every component is assigned to at least one computing node. In other words, Q can be evaluated in a distributed and coordination-free manner; for more details on coordination-free evaluation see [3, 4, 5]. Therefore, it would be quite beneficial if we can decide whether an OMQ distributes over components, and thus, we obtain the following interesting static analysis task: PROBLEM : INPUT : QUESTION : UCQRew(C, CQ) An OMQ Q ∈ (C, CQ). Is it the case that Q is UCQ rewritable? The above problem has been studied in [15], where tight complexity bounds for (L, CQ) and (S, CQ) have been established. However, its exact complexity for guarded OMQs Bienvenu et al. have recently carried out an in-depth study of the above problem for OMQ languages based on central Horn-DLs [16]. One of their main results is that the above problem for the OMQ language based on E LHI, one of the most expressive members of the E L-family of DLs, is 2ExpTime-complete. Interestingly, by adapting the tree automata techniques developed in Section 5, we can generalize the above result: deciding UCQ rewritability for the OMQ language based on guarded tgds over unary and binary relations is in 2ExpTime. Let G2 be the class of (finite) sets of guarded tgds over unary and binary relations. Then: 5 For technical clarity, the notion of component is defined only for sets of atoms that do not contain 0-ary atoms. Theorem 29. UCQRew(G2 , CQ) is 2ExpTime-complete. PROBLEM : INPUT : QUESTION : Dist(C, CQ) An OMQ Q ∈ (C, CQ). Does Q distributes over components? 11 considerably and, in turn, allows us to obtain our desired semantic characterization of UCQ rewritability: Since the lower bound is inherited from [16], we concentrate on the upper bound. As in Section 5, we can focus on BCQs, i.e., it suffices to show that UCQRew(G2 , BCQ) is in 2ExpTime. Our proof proceeds in two steps: Proposition 30. Let Q = (S, Σ, q) ∈ (G2 , BCQ), where q is connected. The following are equivalent: 1. We semantically characterize UCQ rewritability for queries in (G2 , CQ) in terms of a certain boundedness property for the set of C-trees defined in Section 5. 1. Q is UCQ rewritable. 2. There exist k, m ≥ 0 (which depend only on Q) s.t. Q(D) 6= ∅ =⇒ Q(D≤k ) 6= ∅ or Q(D>0 ) 6= ∅ , 2. We extend the techniques developed in Section 5 and construct in double-exponential time a 2WAPA A that has exponentially many states, such that the aforementioned boundedness property does not hold iff L(A) is infinite. (Such an infinity problem for tree automata has been used to obtain the decidability of the boundedness problem for monadic Datalog [32, 56]). for each C-tree S-database D with \|dom(C)\| ≤ 2 · \|q\| and branching degree at most m. The reduction to the infinity problem. We now proceed with step 2, and we explain how the boundedness property established in item (2) of Proposition 30 can be reduced to the infinity problem for 2WAPAs. As in Section 5, we do not reason with C-tree databases directly, but we deal with their encodings as consistent ΓS,l -labeled trees. In fact, using the same ideas as in Lemma 22, we can show by exploiting Proposition 30 that the following are equivalent: Our 2ExpTime upper bound then follows since the infinity problem for a 2WAPA A, i.e., checking if L(A) is infinite, is feasible in exponential time in the number of states, and in polynomial time in the size of the alphabet. This follows from two known results: (a) The 2WAPA A can be converted into an equivalent non-deterministic tree automata B with a single-exponential blow up in the number of states [57], and (b) solving the infinity problem for non-deterministic tree automata is feasible in polynomial time; cf. [56]. It is worth contrasting our proof with the one in [16] for E LHI, which does not make use of the infinity problem for 2WAPA, but applies a different argument based on pumping. This leads to a finer complexity analysis in terms of the size of the different components of the OMQ, but, in our opinion, makes the proof conceptually harder. (i) Q is UCQ rewritable. (ii) There are k, m ≥ 0 such that Q(JLK) 6= ∅ =⇒ Q(JLK≤k ) 6= ∅ or Q(JLK>0 ) 6= ∅ , for every consistent ΓS,l -labeled tree L with l = 2 · \|q\| and whose branching degree is bounded by m. Let us write Boundedness for the property expressed in item (ii) above, which can be reduced to the problem of checking whether some tree language is finite. Let LQ be the set of all ΓS,l -labeled trees L of branching degree at most m such that: (1) Q(JLK) = ∅ and (2) there is some “extension” L′ of L, with branching degree m, such that Q(JL′ K) 6= ∅ and Q(JL′ K>0 ) = ∅. Notice that L′ can increase the depth but not the branching degree of L. It is not difficult to show that Boundedness holds iff LQ is finite. We then devise in double-exponential time a 2WAPA CQ,l , which has exponentially many states, such that LQ = L(CQ,l ). Therefore, the following holds: The semantic characterization. To establish the semantic characterization from step 1, we need to define the notion of distance from the root for an element u in a C-tree database D. Intuitively, this corresponds to the minimal distance between a node that contains u and the root of a tree decomposition T of D that witnesses the fact that D is a C-tree. We do not consider all such tree decompositions, however, but concentrate on a well-behaved subclass, which we call the lean tree decompositions of the C-tree D; the formal definition can be found in [11], as it does not add much to the explanation we provide here. Due to the fact that we focus on unary and binary relations, such lean tree decompositions ensure the invariance of the notion of distance from the root, by severely limiting the level of redundancy allowed in a tree representation of D. Therefore, it does not matter which lean tree decomposition we choose, since in all of them the distance of an element u from the root will be the same. Let D≤k be the subinstance of D induced by the set of elements whose distance from the root is at most k, and let D>k be the subinstance of D induced by the set of elements whose distance from the root is at least k + 1. Another useful notion is the branching degree of a tree decomposition T , that is, the maximum number of child nodes over all nodes of T . Again, lean tree decompositions ensure the invariance of the branching degree. This allows us to define the branching degree of a C-tree database D as the branching degree of a lean tree decomposition that witnesses the fact that D is a C-tree. It follows from [16] that being able to decide containment for the OMQ language (G2 , BCQ) (as we have done in Section 5) allows us to concentrate on connected CQs when deciding UCQ rewritability. This simplifies technicalities Proposition 31. Consider Q ∈ (G2 , BCQ). We can construct in double-exponential time a 2WAPA A, which has exponentially many states, such that Q is UCQ rewritable ⇐⇒ L(A) is finite. Since checking whether L(A) is infinite is feasible in exponential time in the number of states and in polynomial time in the size of the alphabet, Proposition 31 implies that UCQRew(G2 , CQ) is in 2ExpTime, as needed. 8. CONCLUSIONS We have concentrated on the fundamental problem of containment for OMQ languages based on the main decidable classes of tgds. We have also used our techniques to close problems related to distribution over components and UCQ rewritability. We believe that our techniques for solving containment under guarded OMQs can be extended to frontierguarded OMQs, an interesting extension of guardedness [8]. We are also convinced that our solution to the problem of deciding UCQ rewritability of guarded OMQs over unary and binary relations can be extended to guarded (or even frontier-guarded) OMQs over arbitrary schemas. We are currently investigating these challenging problems. 12 APPENDIX PRELIMINARIES Definition of Non-recursiveness In the main body of the paper, we define non-recursive sets of tgds via the notion of predicate graph. Here, we give an alternative definition, based on the well-known notion of stratification, which is more convenient for the combinatorial analysis that we are going to perform in the proof of Proposition 14. Definition 3. Consider a set Σ of tgds. A stratification of Σ is a partition {Σ1 , . . . , Σn }, where n > 0, of Σ such that, for some function µ : sch(Σ) → {0, . . . , n}, the following hold: 1. For each predicate R ∈ sch(Σ), all the tgds with R in their head belong to Σµ(R) , i.e., they belong to the same set of the partition. 2. If there exists a tgd in Σ such that the predicate R appears in its body, while the predicate P appears in its head, then µ(R) < µ(P ). We say that Σ is stratifiable if it admits a stratification. It is an easy exercise to show that the predicate graph of a set Σ of tgds is acyclic iff Σ is stratifiable. Then: Lemma 32. Σ is non-recursive iff Σ is stratifiable. Definition of Stickiness In the main body of the paper, we provide an intuitive explanation of stickiness. Here, we recall the formal definition of sticky sets of tgds, introduced in [27]. Fix a set Σ of tgds; w.l.o.g., we assume that, for every pair (σ, σ ′ ) ∈ Σ × Σ, σ and σ ′ do not share variables. For notational convenience, given an atom α and a variable x occurring in α, pos(α, x) is the set of positions in α at which x occurs; a position P [i] identifies the i-th attribute of the predicate P . The definition of stickiness hinges on the notion of marked variables in a set of tgds. Definition 4. Consider a tgd σ ∈ Σ, and a variable x occurring in the body of σ. We inductively define when x is marked in Σ as follows: 1. If there exists an atom α in the head of σ such that x does not occur in α, then x is marked in Σ; and 2. Assuming that there exists an atom α in the head of σ such that x occurs in α, if there exists σ ′ ∈ Σ (not necessarily different than σ) and an atom β in the body of σ ′ such that (i) α and β have the same predicate and, (ii) each variable in β that occurs at a position of pos(α, x) is marked in Σ, then x is marked in Σ. We are now ready to recall when a set of tgds is sticky: Definition 5. A set Σ of tgds is sticky if, for each σ ∈ Σ, and for each variable x occurring in the body of σ, the following holds: if x is marked in Σ, then x occurs only once in the body of σ. PROOFS OF SECTION 3 Proof of Proposition 5 Consider an OMQ Q = (S, Σ, q(x̄)) ∈ (C, CQ), where C is a class of tgds, an S-database D, and a tuple c̄ ∈ dom(D)\|x̄\| . We show that: c̄ ∈ Q(D) ⇐⇒ (sch(Σ), ∅, qD,c̄ ) ⊆ (sch(Σ), Σ, q) . \| {z } \| {z } Q1 Q2 (⇒) Assume that Q1 6⊆ Q2 . This implies that there exists a sch(Σ)-database D′ , and a tuple t̄ of constants such that t̄ ∈ qD,c̄ (D′ ) and t̄ 6∈ q(chase(D′ , Σ)). Due to the monotonicity of CQs, t̄ ∈ qD,c̄ (chase(D′ , Σ)). Since, by construction, the instance chase(D′ , Σ) satisfies Σ, we conclude that qD,c̄ 6⊆Σ q.6 By exploiting the well-known characterization of CQ containment in terms of the chase, we get that c̄ 6∈ q(chase(D, Σ)), which is equivalent to c̄ 6∈ Q(D), as needed. (⇐) Conversely, assume that c̄ 6∈ Q(D), or, equivalently, c̄ 6∈ q(chase(D, Σ)). This implies that c̄ 6∈ Q2 (D). Observe that c̄ ∈ qD,c̄ (D) holds trivially, which in turn implies that c̄ ∈ Q1 (D). Therefore, Q1 6⊆ Q2 , and the claim follows. 6 This is the standard notation for the fact that qD,c̄ (I) 6⊆ q(I), for every (possibly infinite) instance I that satisfies Σ. 13 Proof of Proposition 9 The construction underlying Proposition 9 relies on the idea of encoding boolean operations (in our case the ‘or’ operator) using a set of atoms; this idea has been exploited in several other works; see, e.g., [14, 21, 41]. Let Q = (S, Σ, q) ∈ (C, UCQ). Our goal is to construct in polynomial time Q′ = (S, Σ′ , q ′ ) ∈ (C, CQ) such that Q ≡ Q′ . We assume, w.l.o.g., that the predicates of S do not appear in the head of a tgd of Σ; we can copy the content of a relation R/k ∈ S into an auxiliary predicate R⋆ /k, using the tgd R(x1 , . . . , xk ) → R⋆ (x1 , . . . , xk ), while staying inside C, and then rename each predicate P in Σ and q with P ⋆ . The set Σ′ consists of the following tgds: 1. For every R/k ∈ S: R(x1 , . . . , xk ) → R′ (x1 , . . . , xk , 1), True(1). These tgds are annotating the database atoms with the truth constant true, indicating that these are true atoms. 2. Assuming that q = ∃ȳ φ(x̄, ȳ), a tgd: True(t) → ∃x̄∃ȳ∃f φ′∧ (x̄, ȳ, f ), ψ(t, f ), where φ′∧ is the conjunction of atoms in φ, after replacing each atom R(v1 , . . . , vk ) with R′ (v1 , . . . , vk , f ), and ψ is the conjunction of atoms Or(t, t, t), Or(t, f, t), Or(f, t, t), Or(f, f, f ). This tgd generates a “copy” of the atoms in q, while annotating them with a null value that represents the truth constant false, indicating that are not necessarily true atoms. Moreover, the truth table of ‘or’ is generated. 3. Finally, for each tgd φ(x̄, ȳ) → ∃z̄ ψ(x̄, x̄) in Σ, a tgd φ′ (x̄, ȳ, w) → ∃z̄ ψ ′ (x̄, z̄, w), where φ′ and ψ ′ are obtained from φ and ψ, respectively, by replacing each atom R(v1 , . . . , vk ) with R′ (v1 , . . . , vk , w). In fact, this is the actual set of tgds Σ, with the difference that the value at the last position of each atom (which indicates whether it is true or false) is propagated to the inferred atoms. Now, assuming that q = q1 ∨ · · · ∨ qn , the CQ q ′ is defined as follows; let x̄ = x1 . . . xn and ȳ = y1 . . . yn+1 : ^ (qi′ [xi ] ∧ Or(yi , xi , yi+1 )) ∧ True(yn+1 )), ∃x̄∃ȳ (False(y1 ) ∧ 1≤i≤n where x̄ and ȳ are fresh variables not in q, and qi′ [xi ] is obtained from qi by replacing each atom R(v1 , . . . , vk ) with R′ (v1 , . . . , vk , xi ). This completes our construction. It is not difficult to show that Q ≡ Q′ , or, equivalently, for every S-database D, q(chase(D, Σ)) = q ′ (chase(D, Σ′ )). The key observation is that in order to satisfy True(yn+1 ) in the CQ q ′ , at least one of the x̄i ’s must be mapped to 1, which means that at least one qi is satisfied by chase(D, Σ). Finally, it is easy to verify that, for each C ∈ {G, L, NR, S}, Σ ∈ C implies Σ′ ∈ C, and Proposition 9 follows. PROOFS OF SECTION 4 Proof of Proposition 10 W ′ ′ We assume that q(x̄) = n i=1 qi (x̄) is a UCQ rewriting of Q. Since, by hypothesis, Q 6⊆ Q , we conclude that q 6⊆ Q , which ′ in turn implies that there exists i ∈ {1, . . . , n} such that qi 6⊆ Q . Let c(x̄) be a tuple of constants obtained by replacing each variable x in x̄ with the constant c(x), and Dqi the S-database obtained from qi after replacing each variable x in qi with the constant c(x). We show that: Lemma 33. c(x̄) 6∈ Q′ (Dqi ). Proof. Since qi 6⊆ Q′ , there exists an S-database D, and a tuple of constants t̄ such that t̄ ∈ qi (D) and t̄ 6∈ Q′ (D). Clearly, there exists a homomorphism h such that h(qi ) ⊆ D and h(x̄) = t̄. Observe also that ρ(Dqi ) ⊆ D, where ρ = h ◦ c−1 . Towards a contradiction, assume that c(x̄) ∈ Q′ (Dqi ). This implies that there exists a homomorphism γ such that γ(q ′ ) ⊆ chase(Dqi , Σ) and γ(ȳ) = c(x̄), where Q′ = (S, Σ, q ′ (ȳ)). It is not difficult to see that there exists an extension ρ′ of ρ such that ρ′ (chase(Dqi , Σ)) ⊆ chase(D, Σ) and ρ′ (x̄) = t̄. Hence, ρ′ (γ(q ′ )) ⊆ chase(D, Σ), which implies that t̄ ∈ q ′ (chase(D, Σ)); thus, t̄ ∈ Q′ (D). But this contradicts the fact that t̄ 6∈ Q′ (D), and the claim follows. Observe that c(x̄) ∈ q(Dqi ), which immediately implies that c(x̄) ∈ Q(Dqi ). Consequently, by Lemma 33, Q(Dqi ) 6⊆ Q′ (Dqi ). The claim follows since, by construction, Dqi is an S-database such that \|Dqi \| ≤ fO (Q). 14 Algorithm 1: The algorithm XRewrite Input: An OMQ Q = (S, Σ, q(x̄)) ∈ (TGD, CQ) Output: A UCQ q ′ (x̄) such that Q(D) = q ′ (D), for every S-database D i := 0; Qrew := {hq, r, ui}; repeat Qtemp := Qrew ; foreach hq, x, ui ∈ Qtemp , where x ∈ {r, f} do foreach σ ∈ Σ do /* rewriting step foreach S ⊆ body (q) such that σ is applicable to S do i := i + 1; q ′ := γS,σ i (q[S/body (σi )]); if there is no (q ′′ , r, ⋆) ∈ Qrew such that q ′ ≃ q ′′ then Qrew := Qrew ∪ {hq ′ , r, ui}; end end /* factorization step foreach S ⊆ body (q) that is factorizable w.r.t. σ do q ′ := γS (q); if there is no (q ′′ , ⋆, ⋆) ∈ Qrew such that q ′ ≃ q ′′ then Qrew := Qrew ∪ {hq ′ , f, ui}; end end end /* query q is now explored Qrew := (Qrew \ {(q, x, u)}) ∪ {(q, x, e)}; end until Qtemp = Qrew ; Qfin := {q \| hq, r, ei ∈ Qrew , and q contains only predicates of S}; return Qfin / / */ The Algorithm XRewrite In view of the fact that the rewriting algorithm XRewrite is heavily used in our complexity analysis, we would like to recall its definition. This algorithm is based on resolution, and thus, before we proceed further, we need to recall the crucial notion of unification. A set of atoms A = {α1 , . . . , αn }, where n > 2, unifies if there exists a substitution γ, called unifier for A, such that γ(α1 ) = · · · = γ(αn ). A most general unifier (MGU) for A is a unifier for A, denoted as γA , such that for each other unifier γ for A, there exists a substitution γ ′ such that γ = γ ′ ◦ γA . Notice that if a set of atoms unify, then there exists a MGU. Furthermore, the MGU for a set of atoms is unique (modulo variable renaming). The algorihtm proceeds by exhaustively applying two steps: rewriting and factorization, which in turn rely on the technical notions of applicability and factorizability, respectively. We assume, w.l.o.g., that tgds and CQs do not share variables. Given a CQ q, a variable x is called shared in q if x is a free variable of q, or it occurs more than once in q. In what follows, we assume, w.l.o.g., that tgds are in normal form, i.e., they have only one atom in the head, and only one occurrence of an existentially quantified variable [27]. We write π∃ (σ) for the position at which the existentially quantified variable of σ occurs; in case σ does not mention an existentially quantified variable, then π∃ (σ) = ε. (Recall that a position P [i] identifies the i-th attribute of a predicate P .) We are now ready to recall applicability and factorizability; in what follows, we write body (q) for the set of atoms occurring in q, and head (σ) for the head-atom of σ. Definition 6. (Applicability) Consider a CQ q and a tgd σ. Given a set of atoms S ⊆ body(q), we say that σ is applicable to S if the following conditions are satisfied: 1. the set S ∪ {head (σ)} unifies, and 2. for each α ∈ S, if the term at position π in α is either a constant or a shared variable in q, then π 6= π∃ (σ). Roughly, whenever σ is applicable to S, this means that the atoms of S may be generated during the chase procedure by applying σ. Therefore, we are allowed to apply a rewriting step (which is essentially a resolution step) that resolves S using σ, i.e., S is replaced by body(σ), and a new CQ that is closer to the input database is obtained. If we start applying rewriting steps blindly, without checking for applicability, then the soundness of the rewriting procedure is not guaranteed. However, it is possible that the applicability condition is not satisfied, but still we should apply a rewriting step. This may happen due to the presence of redundant atoms in a query. For example, given the CQ q = ∃x∃y∃z (R(x, y) ∧ R(x, z)) and the tgd σ = P (u, v) → ∃w R(w, u) 15 the applicability condition fails since the shared variable x in q occurs at the position π∃ (σ) = R[1]. However, q is essentially the CQ q = ∃x∃y R(x, y), and now the applicability condition is satisfied. From the above informal discussion, we conclude that the applicability condition may prevent the algorithm from being complete since some valid rewriting steps are blocked. Because of this reason, we need the so-called factorization step, which aims at converting some shared variables into nonshared variables, and thus, satisfy the applicability condition. In general, this can be achieved by exhaustively unifying all the atoms that unify in the body of a CQ. However, some of these unifications do not contribute in any way to satisfying the applicability condition, and, as a result, many superfluous CQs are generated. It is thus better to apply a restricted form of factorization that generates a possibly small number of CQs that are vital for the completeness of the rewriting algorithm. This corresponds to the identification of all the atoms in the query whose shared existential variables come from the same atom in the chase, and they can be unified with no loss of information. Summing up, the key idea underlying the notion of factorizability is as follows: in order to apply the factorization step, there must exist a tgd that can be applied to its output.7 Definition 7. (Factorizability) Consider a CQ q and a tgd σ. Given a set of atoms S ⊆ body (q), where \|S\| > 2, we say that S is factorizable w.r.t. σ if the following conditions are satisfied: 1. S unifies, 2. π∃ (σ) 6= ε, and 3. there exists a variable x 6∈ var (body (q) \ S) that occurs in every atom of S only at position π∃ (σ). Having the above key notions in place, we are now ready to recall the algorithm XRewrite, which is depicted in Algorithm 1. As said above, the UCQ rewriting of an OMQ q = (S, Σ, q) is computed by exhaustively applying (i.e., until a fixpoint is reached) the rewriting and the factorization steps. Notice that the CQs that are the result of the factorization step, are nothing else than auxiliary queries which are critical for the completeness of the final rewriting, but are not needed in the final rewriting. Thus, during the iterative procedure, the queries are labeled with r (resp., f) in order to keep track which of them are generated by the rewriting (resp., factorization) step. The CQ that is part of the input OMQ, although is not a result of the rewriting step, is labeled by r since it must be part of the final rewriting. Moreover, once the two crucial steps have been exhaustively applied on a CQ q, it is not necessary to revisit q since this will lead to redundant queries. Hence, the queries are also labeled with e (resp., u) indicating that a query has been already explored (resp., is unexplored). Let us now describe the two main steps of the algorithm. In the sequel, consider a triple (q, x, y), where (x, y) ∈ {r, f} × {e, u} (this is how we indicate that q is labeled by x and y), and a tgd σ ∈ Σ. We assume that q is of the form ∃x̄ ϕ(x̄, ȳ). Rewriting Step. For each S ⊆ body (q) such that σ is applicable to S, the i-th application of the rewriting step generates the query q ′ = γS,σ i (q[S/body (σ i )]), where σ i is the tgd obtained from σ by replacing each variable x with xi , γS,σ i is the MGU for the set S ∪ {head (σ i )} (which is the identity on the variables that appear in the body but not in the head of σ i ), and q[S/body (σ i )] is obtained from q be replacing S with body (σ i ). By considering σ i (instead of σ) we basically rename, using the integer i, the variables of σ. This renaming step is needed in order to avoid undesirable clutters among the variables introduced during different applications of the rewriting step. Finally, if there is no (q ′′ , r, ⋆) ∈ Qrew , i.e., an (explored or unexplored) query that is the result of the rewriting step, such that q ′ and q ′′ are the same (modulo bijective variable renaming), denoted q ′ ≃ q ′′ , then (q ′ , r, u) is added to Qrew . Factorization Step. For each S ⊆ body (q) that is factorizable w.r.t. σ, the factorization step generates the query q ′ = γS (q), where γS is the MGU for S. If there is no (q ′′ , ⋆, ⋆) ∈ Qrew , i.e., a query that is the result of the rewriting or the factorization step, and is explored or unexplored, such that q ′ ≃ q ′′ , then (q ′ , f, u) is added to Qrew . Proof of Proposition 14 We assume, w.l.o.g., that the predicates of S do not appear in the head of a tgd of Σ. Since Σ ∈ NR, by Lemma 32, Σ admits a stratification {Σ1 , . . . , Σn } with stratification function µ : sch(Σ) → {0, . . . , n}. Let us briefly explain how the rewriting tree TQ of the OMQ Q = (S, Σ, q) is defined. TQ is a rooted tree with q being its root. The i-th level of TQ consists of the CQs obtained from the CQs of the (i − 1)-th level by applying rewriting steps (see the algorithm XRewrite for details on the rewriting step) using only tgds from Σn−i+1 . It is easy to verify that the CQs of the i-th level contain only predicates P such that µ(P ) < n − i + 1. It is now clear that the n-th level of TQ (i.e., the leaves of TQ ) consists only of CQs obtained during the execution of XRewrite(Q) that contain only predicates of S. Thus, in order to obtained the desired upper bound, it suffices to show that the number of atoms that occur in a CQ that is a leaf of TQ is at most \|q\| · (maxτ ∈Σ {\|body (τ )\|})\|sch(Σ)\| . To this end, let us focus on one branch B of TQ from the root q to a leaf q ′ . Such a branch can be naturally represented as a k-ary forest FQB , where the root nodes are the atoms of q, and whenever an atom α is resolved during the rewriting step using a tgd τ , the atoms of body (τ ), after applying the appropriate MGU, are the child nodes of α. Therefore, to obtain the desired upper bound, it suffices to show that the number of leaves of FQB is at most \|q\| · (maxτ ∈Σ {\|body (τ )\|})\|sch(Σ)\| . By construction, FQB consists, in general, of \|q\| k-ary rooted trees, where k = maxτ ∈Σ {\|body (τ )\|}, of depth n. Hence, the number of leaves of FQB is at most \|q\| · (maxτ ∈Σ {\|body (τ )\|})n . Since n ≤ \|sch(Σ)\|, the claim follows. 7 Let us clarify that for the purposes of the present work we can rely on the naive approach of exhaustively unifying all the atoms that unify in the body of a CQ. However, we would like to be consistent with [40], where the algorithm XRewrite is proposed, and thus, we stick on the slightly more involved notion of factorizability. 16 Proof of Theorem 16 A proof sketch for the coNExpTimeNP upper bound is given in the main body of the paper. We proceed to establish the PNEXP -hardness. Our proof is by reduction from a tiling problem that has been recently introduced in [34], which in turn relies on the standard Exponential Tiling Problem. Let us first recall the latter problem. An instance of the Exponential Tiling Problem is a tuple (n, m, H, V, s), where n, m are numbers (in unary), H, V are subsets of {1, . . . , m} × {1, . . . , m}, and s is a sequence of numbers of {1, . . . , m}. Such a tuple specifies that we desire a 2n × 2n grid, where each cell is tiled with a tile from {1, . . . , m}. H (resp., V ) is the horizontal (resp., vertical) compatibility relation, while s represents a constraint on the initial part of the first row of the grid. A solution to such an instance of the Exponential Tiling Problem is a function f : {0, . . . , 2n − 1} × {0, . . . , 2n − 1} → {1, . . . , m} such that: 1. f (i, 0) = s[i], for each 0 ≤ i ≤ (\|s\| − 1); 2. (f (i, j), f (i + 1, j)) ∈ H, for each 0 ≤ i ≤ 2n − 2 and 0 ≤ j ≤ 2n − 1; and 3. (f (i, j), f (i, j + 1)) ∈ V , for each 0 ≤ i ≤ 2n − 1 and 0 ≤ j ≤ 2n − 2. We will refer to {0, . . . , 2n − 1} × {0, . . . , 2n − 1} as a grid, with the pairs in it being cells. A cell consists of two coordinates, the column-coordinate (for short col-coordinate) and the row-coordinate, and any function on a grid is a tiling. The Exponential Tiling Problem is defined as follows: given an instance T as above, decide whether T has a solution. It is known that this problem is NExpTime-hard (see, e.g., Section 3.2 of [46]). We are now ready to recall the tiling problem introduced in [34], called Extended Tiling Problem (ETP), which is PNEXP hard. An instance of this problem is a tuple (k, n, m, H1 , V1 , H2 , V2 ), where k, n, m are numbers (in unary), and H1 , V1 , H2 , V2 are subsets of {1, . . . , m} × {1, . . . , m}. The question is as follows: is it the case that for every sequence s, where \|s\| = k, of numbers of {1, . . . , m}, (n, m, H1 , V1 , s) has no solution or (n, m, H2 , V2 , s) has a solution? We give a reduction from the ETP to Cont(NR, CQ). More precisely, given an instance T = (k, n, m, H1 , V1 , H2 , V2 ) of the ETP, our goal is to construct in polynomial time two queries Qi = (S, Σi , qi ) ∈ (NR, CQ), for i ∈ {1, 2}, such that T has a solution iff Q1 ⊆ Q2 . Data Schema S The data schema S consists of: • 0-ary predicates Cij , for each i ∈ {0, . . . , k − 1} and j ∈ {1, . . . , m}; the atom Cij indicates that si = j. The Query Q1 The goal of the query Q1 is twofold: (i) to check that the so-called existence property of the input database, i.e., for every i ∈ {0, . . . , k − 1}, there exists at least one atom of the form Cij , is satisfied, and (ii) to check whether (n, m, H1 , V1 , s), where s is the sequence of tilings encoded in the input database, has a solution. To this end, the query Q1 will mention the following predicates: • 0-ary predicate Ci , indicating that there exists at least one atom of the form Cij in the input database. • 0-ary predicate Existence, indicating that the input database enjoys the existence property. • Unary predicate Tilei , for each i ∈ {1, . . . , m}; the atom Tilei (x) states that x is the tile i. • Binary predicate H; the atom H(x, y) encodes the fact that (x, y) ∈ H1 . • Binary predicate V ; the atom V (x, y) encodes the fact that (x, y) ∈ V1 . • 5-ary predicate Ti , for each i ∈ {1, . . . , n}; the atom Ti (x, x1 , x2 , x3 , x4 ) states that x is a 2i × 2i tiling obtained from the 2i−1 × 2i−1 tilings x1 , . . . , x4 – details on the inductive construction of 2i × 2i tilings from 2i−1 × 2i−1 tilings are given below. • Unary predicate Initiali , for each i ∈ {0, . . . , k − 1}; the atom Initiali (x) states that s[i] = x, i.e., the i-th element of the sequence s is x. • Binary predicate Topji , for each i ∈ {1, . . . , n} and j ∈ {0, . . . , k − 1}; the atom Topji (x, y) states that in the 2i × 2i tiling x the tile at position (j, 0) is y. • 0-ary predicate Tiling, indicating that there exists a 2n × 2n tiling that is compatible with the initial tiling s encoded in the input database. • 0-ary predicate Goal, which is derived whenever the predicates Existence and Tiling are derived. Q1 is defined as the query (S, Σ1 , Goal), where Σ1 consists of the following tgds: 17 X1 X2 X2 Y1 Y1 Y2 X3 X4 X4 Y3 Y3 Y4 X3 X4 X4 Y3 Y3 Y4 Z1 Z2 Z2 W1 W1 W2 X1 X2 Y1 Y2 X3 X4 Y3 Y4 Z1 Z2 W1 W2 Z1 Z2 Z2 W1 W1 W2 Z3 Z4 W3 W4 Z3 Z4 Z4 W3 W3 W4 (a) (b) Figure 2: Inductive construction of tilings. • Checking for the existence property of the input database For each i ∈ {0, . . . , k − 1} and j ∈ {1, . . . , m}: Cij → Ci and the tgd that checks for the existence property C0 , . . . , Ck−1 → Existence • Generate the tiles → ∃x1 . . . ∃xm (Tile1 (x1 ), . . . , Tilem (xm )) • Generate the compatibility relations For each (i, j) ∈ H1 : Tilei (x), Tilej (y) → H(x, y) Tilei (x), Tilej (y) → V (x, y) For each (i, j) ∈ V1 : • Generate the 2n × 2n tilings. The key idea is to inductively construct 2i × 2i tilings from 2i−1 × 2i−1 tilings. It is easy to verify that the grid in Figure 2(a) is a 2i × 2i tiling iff the nine subgrids of it, shown in Figure 2(b), are 2i−1 × 2i−1 tilings. This has been already observed in [33], where Datalog with complex values is studied. First, we construct tilings of size 2 × 2 (the base case of the inductive construction): H(x1 , x2 ), H(x3 , x4 ), V (x1 , x3 ), V (x2 , x4 ) → ∃x T1 (x, x1 , x2 , x3 , x4 ) Then, we inductively construct tilings of larger size until we get tilings of size 2n × 2n . This is done using the following tgds. For each i ∈ {2, . . . , n}: Ti−1 (x1 , x11 , x12 , x21 , x22 ), Ti−1 (x2 , x12 , x13 , x22 , x23 ), Ti−1 (x3 , x13 , x14 , x23 , x24 ) Ti−1 (x4 , x21 , x22 , x31 , x32 ), Ti−1 (x5 , x22 , x23 , x32 , x33 ), Ti−1 (x6 , x23 , x24 , x33 , x34 ), Ti−1 (x7 , x31 , x32 , x41 , x42 ), Ti−1 (x8 , x32 , x33 , x42 , x43 ), Ti−1 (x9 , x33 , x34 , x43 , x44 ) → ∃x Ti (x, x1 , x3 , x7 , x9 ) • Extract from the 2n × 2n tilings the tiles at positions (0, 0), (1, 0), . . . , (k − 1, 0). This is done using the following tgds: → Top01 (x, x1 ), Top11 (x, x2 ) → → .. . ℓ−1 Tℓ (x, x1 , x2 , x3 , x4 ), Top0ℓ−1 (x1 , y0 ), · · · , Top2ℓ−1 −1 (x1 , y2ℓ−1 −1 ) → Top02 (x, y0 ), Top12 (x, y1 ) Top22 (x, y0 ), Top32 (x, y1 ) T1 (x, x1 , x2 , x3 , x4 ) T2 (x, x1 , x2 , x3 , x4 ), Top01 (x1 , y0 ), Top11 (x1 , y1 ) T2 (x, x1 , x2 , x3 , x4 ), Top01 (x2 , y0 ), Top11 (x2 , y1 ) Tℓ (x, x1 , x2 , x3 , x4 ), Top0ℓ−1 (x1 , y0 ), · · · k−2ℓ−1 −1 , Topℓ−1 (x1 , yk−2ℓ−1 −1 ) → ℓ−1 Top0ℓ (x, y0 ), · · · , Topℓ2 ℓ−1 Top2ℓ (x, y0 ), · · · −1 (x, y2ℓ−1 −1 ) , Topk−1 (x, yk−2ℓ−1 −1 ), ℓ where ℓ = ⌈log k⌉. Moreover, for each i ∈ {ℓ + 1, . . . , n}: Ti (x, x1 , x2 , x3 , x4 ), Top0i−1 (x1 , y0 ), · · · , Topk−1 i−1 (x1 , yk−1 ) → 18 Top0i (x, y0 ), . . . , Topk−1 (x, yk−1 ) i • Check whether there exists a 2n × 2n tiling that is compatible with the sequence of tilings s For each i ∈ {0, . . . , k − 1} and j ∈ {1, . . . , m}: Cij , Tilej (x) → Initiali (x) and the tgd Top0n (x, y0 ), Initial0 (y0 ), · · · , Topk−1 (x, yk−1 ), Initialk−1 (yk−1 ) → n Tiling • Finally, we have the output tgd Existence, Tiling → Goal This concludes the construction of Q1 . The Query Q2 The goal of the query Q2 is twofold: (i) to check that the so-called uniqueness property of the input database, i.e., for every i ∈ {0, . . . , k − 1}, there exists at most one atom of the form Cij , is satisfied, and (ii) to check whether (n, m, H2 , V2 , s), where s is the sequence of tilings encoded in the input database, has a solution. The query Q2 mentions the same predicates as Q1 , and is defined as (S, Σ2 , Goal), where Σ2 consists of the following tgds: • Checking the uniqueness property For each i ∈ {0, . . . , k − 1} and j, ℓ ∈ {1, . . . , m} with j < ℓ: Cij , Ciℓ → Goal • The rest of Σ2 encodes the tiling problem (n, m, H2 , V2 , s) in exactly the same way as Σ1 encodes (n, m, H1 , V1 , s). This concludes the construction of Q2 . Proof of Proposition 18 The set Σn consists of the following tgds; for brevity, we write x̄ji for xi , xi+1 , . . . , xj :8 S(x1 , . . . , xn ) n i−1 n Pi (x̄i−1 1 , z, x̄i+1 , z, o), Pi (x̄1 , o, x̄i+1 , z, o) P0 (z, . . . , z , z, o) \| {z } → Pn (x1 , . . . , xn ) n → Pi−1 (x̄i−1 1 , z, x̄i+1 , z, o), 1 ≤ i ≤ n, → Ans(z, o), n while q = Ans(0, 1). It can be verified that, for every {S}-database D, Qn (D) 6= ∅ implies that D ⊇ {S(c1 , . . . , cn−2 , 0, 1) \| (c1 , . . . , cn−2 ) ∈ {0, 1}n−2 }, and thus, \|D\| ≥ 2n−2 . Let Q = ({S}, Σ′ , q ′ ), where Σ′ is a set of tgds and q ′ a Boolean CQ, and D an {S}-database. Clearly, Qn (D) 6⊆ Q(D) iff Qn (D) 6= ∅ and Q(D) = ∅. This implies that \|D\| ≥ 2n−2 , and the claim follows. Proof of Theorem 19 The coNExpTime upper bound, as well as the ΠP 2 -hardness in case of fixed-arity predicates, are discussed in the main body of the paper. Here, we show the coNExpTime-hardness. The proof proceeds in two steps: 1. First, we show that Cont((FNR, CQ), (L, UCQ)) is coNExpTime-hard, where FNR denotes the class of full non-recursive sets of tgds, i.e., non-recursive sets of tgds without existentially quantified variables. 2. Then, we reduce Cont((FNR, CQ), (L, UCQ)) to Cont((S, CQ), (L, UCQ)) by showing that (under some assumptions that are explained below) every query in (FNR, CQ) can be rewritten as an (S, CQ) query. By Proposition 9, we immediately get that Cont((S, CQ), (L, CQ)) is coNExpTime-hard, as needed. Step 1: Cont((FNR, CQ), (L, UCQ)) is coNExpTime-hard We show that Cont((FNR, CQ), (L, UCQ)) is coNExpTime-hard, even if we focus on 0-1 queries, that is, queries Q with following property: for every database D, Q(D) = Q(D01 ), where D01 ⊆ D is the restriction of D on the binary domain {0, 1}, i.e., D01 = {R(c̄) ∈ D \| c̄ ⊆ {0, 1}}. The proof is by reduction from the Exponential Tiling Problem, and is a non-trivial adaptation of the one given in [14] for showing that containment of non-recursive Datalog queries is coNExpTime-hard. Theorem 34. Cont((FNR, CQ), (L, UCQ)) is coNExpTime-hard, even for 0-1 queries. Proof. Given an instance T = (n, m, H, V, s) of the Exponential Tiling Problem, we are going to construct a (FNR, CQ) 0-1 query QT = (S, Σ, q) and a (L, UCQ) 0-1 query Q′T = (S, ΣT , qT ) such that T has a solution iff QT 6⊆ Q′T . 8 A similar construction has been used in [40] for showing a lower bound on the size of a CQ in the UCQ rewriting of a (S, CQ) OMQ. 19 Data Schema S The data schema S consists of: • 2n-ary predicates TiledBy i , for each i ≤ m; the atom TiledBy i (x1 , . . . , xn , y1 , . . . , yn ) indicates that the cell with coordinates ((x1 , . . . , xn ), (y1 , . . . , yn )) ∈ {0, 1}n × {0, 1}n is tiled by tile i. Notice that we use n-bit binary numbers to represent a coordinate; this is the key difference between our construction and the one of [14]. The Query QT The goal of the query QT is to assert whether the input database encodes a candidate tiling, i.e., whether the entire grid is tiled, without taking into account the constraints, that is, the compatibility relations and the constraint on the initial part of the first row. To this end, the query QT will mention the following predicates: • Unary predicate Bit; the atom Bit(x) simply says that x is a bit, i.e., x ∈ {0, 1}. • 2n-ary predicate TiledAboveColi , for each i ≤ n; the atom TiledAboveColi (x̄, ȳ) says that for the row-coordinate ȳ there are tiled cells with coordinates (x̄′ , ȳ) for every col-coordinate x̄′ that agrees with x̄ on the first i − 1 bits. In other words, for the row corresponding to ȳ, every column extending the first i − 1 bits of x̄ is tiled. In particular, TiledAboveCol1 (x̄, ȳ) says that the entire row ȳ is tiled. • 2n-ary predicate TiledAboveRowi , for each i ≤ n; the atom TiledAboveRowi (ȳ) says that for every ȳ ′ that agrees with ȳ on the first i − 1 bits, the row ȳ ′ is fully tiled. • n-ary predicate RowTiled; the atom RowTiled(ȳ) says that the row ȳ is fully tiled. • 0-ary predicate AllTiled, which asserts that the entire grid is tiled. • 0-ary predicate Goal, which is derived whenever the predicate AllTiled is derived. QT is defined as the query (S, Σ, Goal), where Σ consists of the following rules: • Generate Bit atoms → Bit(0) → Bit(1). • RowTiled For each j, k ≤ m: TiledBy j (x1 , . . . , xn−1 , 1, y1 , . . . , yn ), TiledBy k (x1 , . . . , xn−1 , 0, y1 , . . . , yn ), Bit(x1 ), . . . , Bit(xn−1 ), Bit(y1 ), . . . , Bit(yn ), Bit(w) → TiledAboveColn (x1 , . . . , xn−1 , w, y1 , . . . , yn ) For each 2 ≤ i ≤ n: TiledAboveColi (x1 , . . . , xi−1 , 1, xi+1 , . . . , xn , y1 , . . . , yn ), TiledAboveColi (x1 , . . . , xi−1 , 0, x′i+1 , . . . , x′n , y1 , . . . , yn ), Bit(wi ), . . . , Bit(wn ) → TiledAboveColi−1 (x1 , . . . , xi−1 , wi , . . . , wn , y1 , . . . , yn ) A row is fully tiled: TiledAboveCol1 (x1 , . . . , xn , y1 , . . . , yn ) → RowTiled(y1 , . . . , yn ) • AllTiled RowTiled(y1 , . . . , yn−1 , 1), RowTiled(y1 , . . . , yn−1 , 0), Bit(w) → TiledAboveRown (y1 , . . . , yn−1 , w) For each 2 ≤ i ≤ n: TiledAboveRowi (y1 , . . . , yi−1 , 1, yi+1 , . . . , yn ), ′ TiledAboveRowi (y1 , . . . , yi−1 , 0, yi+1 , . . . , yn′ ), Bit(wi ), . . . , Bit(wn ) → TiledAboveRowi−1 (y1 , . . . , yi−1 , wi , . . . , wn ) The entire grid is tiled: TiledAboveRow1 (y1 , . . . , yn ) → AllTiled → 20 AllTiled Goal This concludes the construction of the query QT . The Query Q′T Q′T is defined in such a way that Q′T (D) is non-empty exactly when the input database D encodes an invalid tiling, i.e., when one of the constraints on the tiles is violated. The query Q′T will mention the following intensional predicates: • Unary predicate Bit; as above, Bit(x) says that x is a bit. • 2i-ary predicate LastFirsti , for each 1 ≤ i ≤ n; the atom LastFirsti (x1 , . . . , xi , y1 , . . . , yi ) says that (x1 , . . . , xi ) = (1, . . . , 1) and (y1 , . . . , yi ) = (0, . . . , 0). • 2i-ary predicate Succi , for each 1 ≤ i ≤ n; the atom Succi (x̄, ȳ) says that the i-bit binary number ȳ is the successor of the i-bit binary number x̄. • 0-ary predicate Goal. Q′T is defined as the query (S, Σ′ , q ′ ). The set Σ′ consists of the following linear tgds: • Generate Bit atoms: → Bit(0) → Bit(1). • Generate the successor predicates: → → Succ1 (0, 1) LastFirst1 (1, 0). For each 1 ≤ i ≤ n − 1: Succi (x1 , . . . , xi , y1 , . . . , yi ) Succi (x1 , . . . , xi , y1 , . . . , yi ) LastFirsti (x1 , . . . , xi , y1 , . . . , yi ) LastFirsti (x1 , . . . , xi , y1 , . . . , yi ) → Succi+1 (0, x1 , . . . , xi , 0, y1 , . . . , yi ) → Succi+1 (1, x1 , . . . , xi , 1, y1 , . . . , yi ) → Succi+1 (0, x1 , . . . , xi , 1, y1 , . . . , yi ) → LastFirsti+1 (1, x1 , . . . , xi , 0, y1 , . . . , yi ). The UCQ q ′ consists of the following (Boolean) CQs; for brevity, the existential quantifiers in front of the CQs are omitted: • Tile Consistency For each i 6= j ≤ m: TiledBy i (x1 , . . . , xn , y1 , . . . , yn ), TiledBy j (x1 , . . . , xn , y1 , . . . , yn ), Bit(x1 ), . . . , Bit(xn ), Bit(y1 ), . . . , Bit(yn ) • Tile Compatibility For each (i, j) 6∈ V : Succn (x1 , . . . , xn , y1 , . . . , yn ), TiledBy i (w1 , . . . , wn , x1 , . . . , xn ), TiledBy i (w1 , . . . , wn , y1 , . . . , yn ), Bit(w1 ), . . . , Bit(wn ) For each (i, j) 6∈ H: Succn (x1 , . . . , xn , y1 , . . . , yn ), TiledBy i (x1 , . . . , xn , w1 , . . . , wn ), TiledBy i (y1 , . . . , yn , w1 , . . . , wn ), Bit(w1 ), . . . , Bit(wn ) • Tiling of First Row For each j ≤ n, let fj be the function from {1, . . . , n} into {0, 1} such that fj (1) . . . fj (n) is the number j in binary representation, and let k ∈ {1, . . . , m} other than s[j]; recall that s is a sequence of numbers of {1, . . . , m} that represents a constraint on the initial part of the first row of the grid. Then, we have the CQ: TiledBy k (x1 , . . . , xn , z, . . . , z ), Succ1 (z, o) \| {z } n where, for each i ∈ {1, . . . , n}, xi = z if fj (i) = 0, and xi = o if fj (i) = 1. This concludes the definition of the query Q′T . 21 Step 2: Cont((S, CQ), (L, UCQ)) is coNExpTime-hard Our goal is show that every 0-1 query (S, Σ, q) ∈ (F, CQ) can be equivalently rewritten as a 0-1 query (S, Σ′ , q ′ ), where all the tgds of Σ′ are lossless, i.e., all the body-variables appear also in the head, which in turn implies that Σ′ is sticky. Proposition 35. Consider a 0-1 query Q ∈ (F, CQ). We can construct in polynomial time a 0-1 query Q′ ∈ (S, CQ) such that Q ≡ Q′ . Proof. Let Q = (S, Σ, q), and assume that n is the maximum number of variables occurring in the body of a tgd of Σ. We are going to construct in polynomial time a 0-1 query Q′ = (S, Σ′ , q ′ ) ∈ (S, CQ) such that Q ≡ Q′ . The set Σ′ consists of the following tgds: • Initialization Rules We first transform every database atom of the form R(c̄) into an atom R′ (c̄, 0, . . . , 0, 0, 1). This is done as follows: \| {z } n → Bit(0) → Bit(1) and, for each k-ary predicate R ∈ S, we have the lossless tgd R′ (x1 , . . . , xk , 0, . . . , 0) \| {z } R(x1 , . . . , xk ), Bit(x1 ), . . . , Bit(xk ) → n Notice that we can safely force the variables x1 , . . . , xk to take only values from {0, 1} due to the 0-1 property. • Transformation into Lossless Tgds For each tgd σ ∈ Σ of the form R1 (x̄1 ), . . . , Rk (x̄k ) → R0 (x̄0 ) we have the lossless tgd R1′ (x̄1 , 0, . . . , 0), . . . , Rk′ (x̄k , 0, . . . , 0) \| {z } \| {z } n → R0′ (x̄0 , y1 , . . . , yn ), n where, if {v1 , . . . , vℓ }, for ℓ ∈ {1, . . . , n}, is the set of variables occurring in the body of σ (the order is not relevant), then yi = vi , for each i ∈ {1, . . . , ℓ}, and yj = v1 , for each j ∈ {ℓ + 1, . . . , n}. • Finalization Rules Observe that each atom obtained during the chase due to one of the lossless tgds introduced above is of the form R′ (x̄, ȳ), where ȳ ∈ {0, 1}n . If ȳ 6= (0, . . . , 0), then we need to ensure that eventually the atom R′ (x̄, 0, . . . , 0) \| {z } n ′ will be inferred. This is achieved by adding to Σ the following tgds: For each k-ary predicate R occurring in Σ, and for each 1 ≤ i ≤ n, we have the rule: R′ (x1 , . . . , xk , y1 , . . . , yi−1 , 1, yi+1 , . . . , yn ) → R′ (x1 , . . . , xk , y1 , . . . , yi−1 , 0, yi+1 , . . . , yn ). This concludes the definition of Σ′ . The CQ q ′ is defined analogously. More precisely, assuming that q is of the form (the existential quantifiers are omitted) R1 (x̄1 ), . . . , Rk (x̄k ) ′ the CQ q is defined as R1′ (x̄1 , 0, . . . , 0), . . . , Rk′ (x̄k , 0, . . . , 0). \| {z } \| {z } n n It is easy to verify that Σ′ consists of lossless tgds, and thus, Q′ ∈ (S, CQ). It also not difficult to see that, for every database D over S, Q(D01 ) = Q′ (D01 ); thus, by the 0-1 property, Q(D) = Q′ (D), and the claim follows. By Theorem 34 and Proposition 35, we immediately get that Cont((S, CQ), (L, UCQ)) is coNExpTime-hard, as needed. 22 PROOFS OF SECTION 5 Recall that, for the sake of technical clarity, we focus on constant-free tgds and CQs, but all the results can be extended to the general case at the price of more involved definitions and proofs. Moreover, we assume that tgds have only one atom in the head. This does not affect the generality of our proof since every set of guarded tgds can be transformed in polynomial time into a set of guarded tgds with the above property; see, e.g., [24]. Finally, for convenience of presentation, we also assume that the body of a tgd is non-empty, i.e., the body of a tgd is always an atom and not the symbol ⊤. Proof of Proposition 21 Let us start by recalling the key notion of tree decomposition. Notice that the definition of the tree decomposition that we give here is slightly different than the one in the main body of the paper. The reason is because, for convenience of presentation, we prefer to employ a slightly different notation. Definition 8. Let I be an instance. A tree decomposition of I that omits V , where V ⊆ dom(I), is a pair δ = (T , (Xt )t∈T ), where T = (T, E T ) is a tree and (Xt )t∈T a family of subsets of dom(I) (called the bags of the decomposition) such that: 1. For every v ∈ dom(I) \ V , the set {t ∈ T \| v ∈ Xt } is non-empty and connected. 2. For every atom P (s1 , . . . , sn ) ∈ I, there is a t ∈ T such that {s1 , . . . , sn } ⊆ Xt . The width of a tree decomposition δ = (T , (Xt )t∈T ) omitting V is max{\|Xt \| : t ∈ T } − 1. The tree-width of I is the minimum among the widths of all tree decompositions of I that omit V . We call a tree decomposition omitting ∅ simply tree decomposition of I. For v ∈ T , we denote by Iδ (v) the subinstance of I induced by Xv . Notation. We usually denote the strict partial order among the nodes of a tree T of a tree decomposition δ = (T , (Xt )t∈T ) by ≺. Accordingly, we write v w iff v ≺ w or v = w. For brevity, ε will usually denote the root of a tree decomposition at hand. If ambiguities could possibly arise, we shall use subscripts in these notations. Furthermore, when δ is clear from context, we shall omit it from the expression Iδ (v). Let δ = (T , (Xt )t∈T ) be a tree decomposition of I and V ⊆ T . Recall that δ is [V ]-guarded (or guarded except for V ), if for every node v ∈ T \ V , there is an atom P (s1 , . . . , sn ) ∈ I such that Xv ⊆ {s1 , . . . , sn }. A [∅]-guarded tree decomposition of I is simply called guarded tree decomposition. Also recall the crucial notion of C-tree: Definition 9. An S-instance I is a C-tree, where C ⊆ I, if there is a tree decomposition δ = (T , (Xt )t∈T ) of I such that 1. Iδ (ε) = C, i.e., the subinstance of I induced by Xε equals C. 2. δ is guarded except for {ε}. If δ or C is clear from context, we shall often refer to \|dom(C)\| as the diameter of D and to C as the core of D. Remark. The notion of C-tree defined here refers to both instances and databases, i.e., a C-tree may be a (finite) database or an instance. We often do not explicitly mention whether a C-tree at hand is a database or an instance. However, it will be clear from context whether a C-tree is a database or an instance. We proceed to establish the following technical lemma, which in turn allows us to show Proposition 21. It is an adaption of a result in [7] to the case of guarded tgds. Henceforth, for brevity, given a query Q = (S, Σ, q) ∈ (G, BCQ) and an S-database D, we write D \|= Q for the fact that Q(D) 6= ∅. Lemma 36. Let Q = (S, Σ, q) be an OMQ from (G, BCQ). Let D be an S-database and suppose D \|= Q. Then there is a finite S-instance Iˆ such that Iˆ \|= Q and: 1. Iˆ is a C-tree such that \|dom(C)\| ≤ ar (S ∪ sch(Σ)) · \|q\|. 2. There is a homomorphism from Iˆ to D. Before we proceed with its formal proof, let us explain why Proposition 21 is an easy consequence of Lemma 36. The fact that the first item implies the second is trivial. Conversely, suppose that Q1 6⊆ Q2 , which implies that there exists an ˆ where \|dom(C)\| ≤ ar (S ∪ sch(Σ1 )) · \|q1 \|, S-database D such that D \|= Q1 and D 6\|= Q2 . By Lemma 36, there exists a C-tree I, such that Iˆ \|= Q1 . Moreover, there is a homomorphism from Iˆ to D; hence, since Q2 is closed under homomorphisms, it immediately follows that Iˆ 6\|= Q2 . Consequently, the S-database D̂ obtained from Iˆ after replacing each null z with a distinct constant cz is a C-tree such that Q1 (D̂) 6⊆ Q2 (D̂), and Proposition 21 follows. We now proceed with the proof of Lemma 36 which is our main task in this section. Before that, we introduce some additional auxiliary concepts. 23 The Guarded Chase Forest Given a database D and a set Σ of guarded tgds, the guarded chase forest for D and Σ is a forest (whose edges and nodes are labeled) constructed as follows: 1. For each fact R(ā) in D, add a node labeled with R(ā). 2. For each node v labeled with α ∈ chase(D, Σ) and for every atom β resulting from a one-step application of a rule τ ∈ Σ, if α is the image of the guard in this application of τ , then add a node w labeled with β and introduce an arc from v to w labeled with τ . We can assume that the guarded chase forest is always built inductively according to a fixed, deterministic version of the chase procedure. The non-root nodes are then totally ordered by a relation ≺ that reflects their order of generation. Furthermore, we can extend ≺ to database atoms by picking a lexicographic order among them. Notice that one atom can be the label of multiple nodes. Using the order ≺ we can, however, always refer to the ≺-least node. Guarded Unraveling Let I be an instance over S. We say that X ⊆ dom(I) is guarded in I, if there are a1 , . . . , as ∈ dom(I) such that • X ⊆ {a1 , . . . , as } and • there is an R/s ∈ S such that I \|= R(a1 , . . . , as ). A tuple t̄ is guarded in I if the set containing the elements of t̄ is guarded in I. In the following paragraph, we largely follow the notions introduced in [2, 13]. Fix an S-instance I and some X0 ⊆ dom(I). Let Π be the set of finite sequences of the form X0 X1 · · · Xn , where, for i > 0, Xi is a guarded set in I, and, for i ≥ 0, Xi+1 = Xi ∪ {a} for some a ∈ dom(I) \ Xi , or Xi ⊇ Xi+1 . The sequences from Π can be arranged in a tree by their natural prefix order and each sequence π = X0 X1 · · · Xn identifies a unique node in this tree. In this context, we say that a ∈ dom(I) is represented at π whenever a ∈ Xn . Two sequences π, π ′ are a-equivalent, if a is represented at each node on the unique shortest path between π and π ′ . For a represented at π, we denote by [π]a the a-equivalence class of π. The guarded unraveling around X0 is the instance I ∗ over the elements {[π]a \| a is represented at π}, where I ∗ \|= R([π1 ]a1 , . . . , [πn ]an ) ⇐⇒df I \|= R(a1 , . . . , an ) and ∃π ∈ Π, ∀i ∈ {1, . . . , n} : [π]ai = [πi ]ai , for all R/n ∈ S. Lemma 37. For every S-instance I and any X0 ⊆ dom(I), the guarded unraveling I ∗ around X0 is a C-tree over S, where C is the subinstance of I ∗ induced by the elements {[X0 ]a \| a ∈ X0 }. Proof. Let δ = (T , (Xt )t∈T ), where T is the natural tree that arises from ordering the sequences in Π by their prefixes. For π ∈ T , let Xπ := {[π]a \| a is represented at π}. Let ε denote the root of T . We need to show that δ is an appropriate tree decomposition witnessing that I ∗ is a C-tree. First, note that it is clear that I(ε) = {[X0 ]a \| a ∈ X0 } by construction. Let [π]a ∈ dom(I) and consider the set A := {t ∈ T \| [π]a ∈ Xt }. This set is certainly non-empty. Moreover, for t1 , t2 ∈ A, we know that [t1 ]a = [t2 ]a , hence t1 and t2 are a-connected in T . Suppose I ∗ \|= R([π1 ]a1 , . . . , [πn ]an ) for some R/n ∈ S. Then there is a π ∈ T such that [π]ai = [πi ]ai , for all i = 1, . . . , n. Hence, a1 , . . . , an are all represented at π and so {[π1 ]a1 , . . . , [πn ]an } ⊆ Xπ . It remains to show that δ is guarded except for {ε}. Let π 6= ε and consider the set Xt . Since π is a sequence of length greater than one, its last element Y is a guarded set in I. Hence, there are a1 , . . . , as such that Y ⊆ {a1 , . . . , as } and I \|= R(a1 , . . . , as ) for some R/s ∈ S. Let {a1 , . . . , as } \ Y = {b1 , . . . , bm } and define ρ := π · (Y ∪ {b1 }) · (Y ∪ {b1 , b2 }) · · · (Y ∪ {b1 , . . . , bm }). Then I ∗ \|= R([ρ]a1 , . . . , [ρ]as ), as desired. Notice that this lemma implies that the tree-width of I ∗ is bounded by \|X0 \| + ar (S) − 1. We are now ready to prove Lemma 36: Proof of Lemma 36. Let q = ∃ȳ ϕ(ȳ) and µ a homomorphism mapping ϕ(ȳ) to chase(D, Σ). Let R1 (b̄1 ), . . . , Rk (b̄k ) exhaust all facts from D that are the roots of those ≺-least S facts from µ(ϕ(ȳ)) in the guarded chase forest of D and Σ that have an element from dom(D) as argument. Let Gµ := 1≤i≤k {b̄i } and let I ∗ be the unraveling of D around Gµ , regarding all elements from dom(I ∗ ) as labeled nulls. Henceforth, for every a ∈ Gµ , we denote by λa the element [Gµ ]a . We say that λa represents a. Let C be the substructure of I ∗ induced by the set {λa \| a ∈ Gµ }. Notice that I ∗ is an infinite instance that is a C-tree by Lemma 37. We will show later how to get a finite instance from I ∗ that satisfies our constraints. We proceed to show that I ∗ ∪ Σ logically entails q, denoted I ∗ , Σ \|= q: Lemma 38. I ∗ , Σ \|= q. Proof. We will first construct a universal model J of I ∗ and Σ. Recall that an instance U is a universal model of I and Σ, if it can be homomorphically mapped to every model of I ∪ Σ; in particular, it is well-known and easy to prove that chase(I, Σ) is always a universal model of I and Σ. Before constructing J, we introduce some additional notions. In the following, given a guarded set G = {a1 , . . . , ak } in D, a copy of G in I ∗ is a set Γ = {α1 , . . . , αk } which is guarded in I ∗ such 24 that, for i = 1, . . . , k, we have that αi = [πi ]ai for some sequences πi and D \|= R(ai1 , . . . , aim ) iff I ∗ \|= R(αi1 , . . . , αim ) for all R ∈ S and ij ∈ {1, . . . , k}. Copies of guarded tuples are defined accordingly. Consider the structure chase(D, Σ). Let G be a guarded set in D and D ↾ G denote the subinstance of D induced by G. It is well-known and easy to prove that chase(D ↾ G, Σ) is acyclic (cf., e.g., [23]). Henceforth, we loosely call chase(D ↾ G, Σ) the tree attached to G. The model J is constructed as follows. Let J0 be the instance C. Furthermore, for each guarded set G = {a1 , . . . , ak } in D and each copy Γ = {α1 , . . . , αk } of G in I ∗ , construct a new instance JΓ that is isomorphic to the tree attached to G such that (i) the elements ai of G are renamed to αi in JΓ , (ii) dom(J0 ) ∩ dom(JΓ ) = {α1 , . . . , αk }, and (iii) Γ ∩ Θ = dom(JΓ ) ∩ dom(JΘ ), for every copy Θ of G in I ∗ . The model J is the union of J0 and all the JΓ . If a guarded set X in JΓ arises from renaming elements of a guarded set Y in chase(D ↾ G, Σ), we also say that X is a copy of Y in J. Furthermore, the copies of D that are contained in I ∗ (i.e., in J0 ) are also called copies in J. Observe that J is a model of I ∗ by construction. We show that it is a model of Σ. To this end, we show the following claim. Claim 39. Let t̄ be a guarded tuple in J and let q(x̄) be a guarded conjunctive query9 over S ∪ sch(Σ). Suppose t̄ is a copy of s̄ in J, where s̄ is over dom(chase(D, Σ)) and \|t̄\| = \|s̄\|. Then J \|= q(t̄) iff chase(D, Σ) \|= q(s̄). Proof. Suppose J \|= q(t̄). Let {t̄} = {α1 , . . . , αk } be a copy of {s̄} = {a1 , . . . , ak } in J. Since q(x̄) is guarded, there is a Γ ⊇ ({α1 , . . . , αk } ∩ dom(J0 )) such that JΓ \|= q(t̄). Let G ⊇ {s̄} be the guarded set in D of which Γ is a copy in J0 . It clearly holds that chase(D ↾ G, Σ) \|= q(s̄), whence chase(D, Σ) \|= q(s̄) follows. Suppose that chase(D, Σ) \|= q(s̄). Let t̄ = α1 , . . . , αk and s̄ = a1 , . . . , ak and suppose that αi = [πi ]ai (i = 1, . . . , k). The set {a1 , . . . , ak } is guarded in chase(D, Σ). Hence, there is a guarded G ⊇ {a1 , . . . , ak } ∩ dom(D) in D such that chase(D ↾ G, Σ) \|= q(s̄). We show that there is a Γ ⊇ {α1 , . . . , αk } ∩ dom(I ∗ ) which is a copy of G in I ∗ . Suppose G = {b1 , . . . , bl }. Let π = X0 X1 · · · Xm be such that [π]ai = [πi ]ai for all i = 1, . . . , k. For i = 1, . . . , l, define ρi := π · (Xm ∪ {b1 }) · (Xm ∪ {b1 , b2 }) · · · (Xm ∪ {b1 , . . . , bi }). Then bi is represented at ρi . For i = 1, . . . , l, let βi := [ρi ]bi . We claim that Γ := {β1 , . . . , βl } is a copy of G in I ∗ . Let R/s ∈ S and suppose I ∗ \|= R([ρi1 ]bi1 , . . . , [ρis ]bis ). Then we immediately obtain D \|= R(bi1 , . . . , bis ). Conversely, if D \|= R(bi1 , . . . , bis ), let ρ := ρℓ , where ℓ := max{i1 , . . . , is }. Take any j ∈ {i1 , . . . , is }. It is easy to see that ρ and ρj are bj -equivalent. Hence, [ρj ]bj = [ρ]bj and it follows that I ∗ \|= R([ρi1 ]bi1 , . . . , [ρis ]bis ), as required. It follows that Γ is a copy of G in I ∗ and so there is a structure JΓ contained in J that is isomorphic to chase(D ↾ G, Σ) with b1 , . . . , bl respectively renamed to β1 , . . . , βl . Hence, J \|= q(t̄) as required. Now let σ : ϕ(x̄, ȳ) → ∃z̄ α(x̄, z̄) be a guarded rule from Σ. Suppose that J \|= ∃ȳ ϕ(t̄, ȳ). Since every guarded tuple in J is a copy of some guarded tuple in chase(D, Σ), there is an s̄, of which t̄ is a copy, such that chase(D, Σ) \|= ∃ȳ ϕ(s̄, ȳ). Since chase(D, Σ) is a model of Σ, we know that chase(D, Σ) \|= ∃z̄ α(s̄, z̄). It follows that J \|= ∃z̄ α(s̄, z̄) by the above claim, as required. It remains to show that J is universal: Claim 40. J is universal. Proof. It suffices to show that J can be homomorphically mapped to chase(I ∗ , Σ) via a homomorphism η. We let η0 be the homomorphism that maps every element of J0 to itself. It remains to treat the structures JΓ . Consider a copy Γ = {α1 , . . . , αk } in I ∗ of a set G = {b1 , . . . , bk } which is guarded in D. It suffices to show that JΓ can be mapped to chase(I ∗ , Σ). To this end, it we show how to map chase(D ↾ G, Σ) to chase(I ∗ , Σ). We do so by induction on the number of 0 rule applications of chase(D ↾ G, Σ). For the base case, we map D ↾ G to I ∗ as follows. Let ηG (bi ) := αi , for i = 1, . . . , k. Suppose D ↾ G \|= R(bi1 , . . . , bil ) for some R ∈ S and ij ∈ {1, . . . , k}, where j = 1, . . . , l. Recall that Γ is guarded in I ∗ . 0 is indeed a homomorphism Reviewing the construction of I ∗ , it is easy to see that this holds iff I ∗ \|= R(αi1 , . . . , αil ). Hence, ηG i from D ↾ G to I ∗ . The induction step is obvious—we can easily obtain a homomorphism ηG that maps chase k (D ↾ G, Σ) to i chase(I ∗ , Σ). The desired homomorphism ηG is the union of the ηG (i ≥ 0). We then obtain a homomorphism ηΓ from ηG by appropriately renaming the elements from the domain of the latter as we did in the construction of JΓ —which is nothing else than an isomorphic copy of chase(D ↾ G, Σ). Furthermore, each of these homomorphisms maps each element of Γ to itself. The desired homomorphism η that witnesses that J is universal is the union of η0 and the ηΓ . In order to prove I ∗ , Σ \|= q, it remains to show that there is a homomorphism µ̂ that maps q to J. There are guarded S sets G1 , . . . , Gl in D such that µ can be understood to map q to chase( 1≤i≤l (D ↾ Gi ), Σ). By construction, we know S thatS G1 , . . . , Gl can be chosen in such a way that Gµ ⊆ li=1 Gi . Since Σ is guarded, µ can be understood to map q to 1≤i≤l chase(D ↾ Gi , Σ)—assuming that the labeled nulls occurring in these instances are mutually new. Let Cµ := {{b̄1 }, . . . , {b̄k }}. For every X ∈ Cµ , let ΓX := {λb \| b ∈ X}. Notice that ΓX is a copy of X in I ∗ . By construction, all the facts from q that S are mapped via µ to chase(D, Σ) and which have an element from dom(D) in their image under µ are already mapped to X∈Cµ chase(D ↾ X, Σ). For the other facts, the names of the constants in the databases do not matter.10 Let Θ1 , . . . , Θs be arbitrary copies of the sets {G1 , . . . , Gl } \ Cµ in I ∗ . It follows that we can find our desired match µ̂ in the 9 By a guarded conjunctive query we mean here a CQ that contains an atom that contains all the variables occurring in the CQ as argument. 10 Here, it is of course essential to assume constant-free rules. 25 S S S S union of X∈Cµ JΓX and 1≤i≤s JΘi . Notice that X∈Cµ JΓX is isomorphic to X∈Cµ chase(D ↾ X, Σ) with each b ∈ Gµ represented by λb . Now the database I ∗ has the desired form with C being its core. However, I ∗ is infinite. Since I ∗ , Σ \|= q due to Lemma 38, by compactness, there is a finite B̂ ⊆ I ∗ such that B̂, Σ \|= q. Consider a tree decomposition δ = (T , (Xt )t∈T ) witnessing that I ∗ is a C-tree. There is a maximum ℓ such that B̂ contains all the subinstances induced by the bags of depth less or equal ℓ. Let Iˆ be the instance that actually contains all the subinstances induced by the bags of level up to ℓ. Hence, Iˆ is itself a ˆ Σ \|= q, since B̂ ⊆ I. ˆ C-tree and I, Now there is a natural homomorphism mapping Iˆ to D: we simply specify [π]a 7→ a for all a ∈ dom(D). The instance Iˆ is the one we are looking for. Proof of Lemma 22 One can naturally encode instances of bounded tree-width into trees over a finite alphabet such that the alphabet’s size depends only on the tree-width. Our goal here is to appropriately encode C-trees in order to make them accessible to tree automata techniques. Since the tree-width of a C-tree over S depends only on the size of dom(C) and the maximum arity of S, the alphabet of the encoding will depend on the same. Labeled trees. Let Γ be an alphabet and (N \ {0})∗ be the set of finite sequences of positive integers, including the empty sequence ε.11 Let us recall that a Γ-labeled tree is a pair t = (T, µ), where µ : T → Γ is the labeling function and T ⊆ (N \ {0})∗ is closed under prefixes, i.e., x · i ∈ T implies x ∈ T , for all x ∈ (N ∪ {0})∗ and i ∈ (N ∪ {0}). The elements contained in T identify the nodes of t. For i ∈ N \ {0}, nodes of the form x · i ∈ T are the children of x. A path of length n in T from x to y is a sequence of nodes x = x1 , . . . , xn = y such that xi+1 is a child of xi . A branch is a path from the root to a leaf node. For x ∈ T , we set x · i · −1 := x, for all i ∈ N, and x · 0 := x—notice that ε · −1 is not defined. Encoding. Let l ≥ 0 and fix a schema S. Let US,l be the disjoint union of two sets Cl and TS , respectively containing l and 2 · ar (S) elements. The elements from US,l will be called names. Elements from the set Cl will describe core elements, while those of TS will describe the others. Furthermore, neighboring nodes may describe overlapping pieces of the instance. In particular, if one name is used in neighboring nodes, this means that the name at hand refers to the same element—this is why we use 2w elements for the non-root bags. Let KS,l be the finite schema capturing the following information: • For all a ∈ US,l , there is a unary relation Da ∈ KS,l . • For all a ∈ Cl , there is a unary relation Ca ∈ KS,l . n • For each R ∈ S and every n-tuple ā ∈ US,l , there is a unary relation Rā ∈ KS,l . Let ΓS,l := 2KS,l be an alphabet and suppose that D is a (finite) C-tree over S such that \|dom(C)\| ≤ l. Consider a tree decomposition δ = (T , (Xt )t∈T ) witnessing that D is indeed a C-tree and let ε be the root of T . Fix a function f : dom(D) → US,l such that (i) f ↾ dom(C) is injective and (ii) different elements that occur in neighboring bags of δ are always assigned different names from US,l . Using f , we can encode D and δ into a ΓS,l -labeled tree t = (T̂ , µ) such that each node from T corresponds to exactly one node in T̂ and vice versa. For a node v from T , we denote the corresponding node of T by v̂ in the following and vice versa. In this light, the symbols from KS,l have the following intended meaning: • Da ∈ µ(v̂) means that a is used as a name for some element of the bag Xv . • Ca ∈ µ(v̂) indicates that a is used as name for an element of the bag Xv that also occurs in Xε , i.e., a names an element from the core of D. • Rā ∈ µ(v̂) indicates that R holds in D for the elements named by ā in bag Xv . Under certain assumptions, we can decode a ΓS,l -labeled tree t = (T, µ) into a C-tree whose width is bounded by ar (S) − 1. Let names(v) := {a \| Da ∈ µ(v)}. We say that t is consistent, if it satisfies the following properties: 1. For all nodes v it holds that \|names(v)\| ≤ ar (S), except for the root whose number of names are accordingly bounded by l. Furthermore, names(ε) ⊆ Cl . 2. For all Rā ∈ KS,l and all v ∈ T it holds that Rā ∈ µ(v) implies that {ā} ⊆ names(v). 3. For all a ∈ Cl and all v ∈ T it holds that Da ∈ µ(v) iff Ca ∈ µ(v). 4. If Ca ∈ µ(v), then Ca ∈ µ(w) for all w ∈ T on the unique shortest path between v and the root. 5. For all nodes v 6= ε, there is an Rā ∈ KS,l and a node w such that Rā ∈ µ(w), names(v) ⊆ {ā}, and, for all b ∈ names(v), v and w are b-connected. 11 We specify that 0 is included in N as well. 26 Decoding trees. Suppose now that t is consistent. We show how we can decode t into a database JtK which is a C-tree whose diameter is bounded by l. Let a be a name used in t. We say that two nodes v, w of t are a-equivalent if Da ∈ µ(u) for all nodes u on the unique shortest path between v and w. Clearly, a-equivalence defines an equivalence relation and we let [v]a := {(w, a) \| w is a-equivalent to v} and [v]∗a := {w \| (w, a) ∈ [v]a }. The domain of JtK is the set {[v]a \| v ∈ T, a ∈ µ(v)} and, for R/n ∈ S, we define JtK \|= R([v1 ]a1 , . . . , [vn ]an ) ⇐⇒df there is some v ∈ [v1 ]∗a1 ∩ · · · ∩ [vn ]∗an such that Ra1 ,...,an ∈ µ(v). Lemma 41. Let t be a consistent ΓS,l -labeled tree with root node ε. Then JtK is well-defined and a C-tree over S, where C is the subinstance of JtK induced by the set {[ε]a \| a ∈ names(ε)}. Moreover, \|dom(C)\| is bounded by l. Proof. Let t = (T, µ) be a consistent, ΓS,l -labeled tree. The fact that JtK is well-defined is left to the reader. We are going to construct an appropriate decomposition δ = (T , (Xt )t∈T ) for JtK. The tree T has the same structure as t. Furthermore, for v ∈ T , we set Xv := {[v]a \| a ∈ names(v)}. We need to show that δ is indeed a tree decomposition that satisfies the desired properties. Let [v]a ∈ dom(JtK) and consider two nodes v1 , v2 ∈ T such that [v]a ∈ Xv1 and [v]a ∈ Xv2 . Then v1 , v2 ∈ [v]a and so v1 and v2 are a-connected. Hence, w ∈ [v]a for all w ∈ T which lie on the unique shortest path between v1 and v2 . Since a ∈ names(w) for all such w, it follows that [v]a ∈ Xw , and so [v]a is contained in all bags on the unique path between v1 and v2 . Suppose JtK \|= R([v1 ]a1 , . . . , [vn ]an ). Then there is a v ∈ [v1 ]∗a1 ∩ · · · ∩ [vn ]∗an such that Ra1 ,...,an ∈ µ(v). By consistency, {a1 , . . . , an } ⊆ names(v). Moreover, we know that [vi ]ai = [v]ai , for i = 1, . . . , n. It follows that {[v1 ]a1 , . . . , [vn ]an } ⊆ Xv . Now let v ∈ T \ {ε}. By consistency, there is an Ra1 ,...,an ∈ KS,l and a w ∈ T such that names(v) := {ai1 , . . . , ais } ⊆ {a1 , . . . , an } ⊆ names(w), Ra1 ,...,an ∈ µ(w), and v and w are bij -connected for j = 1, . . . , s. By construction, Xv = {[v]ai1 , . . . , [v]ais } and {[w]a1 , . . . , [w]an } ⊆ Xw . The claim follows now since [v]aij = [w]aij for j = 1, . . . , s. It is immediate that \|dom(C)\| is bounded by l. Notation. Given a consistent ΓS,l -labeled tree t = (T, µ) and a label ρ ∈ µ(T ), in order to ease notation we often regard ρ as a database consisting of the facts {R(ā) \| Rā ∈ ρ}. Furthermore, we let names(ρ) := {a \| Da ∈ ρ}. Proof of Lemma 22. The lemma is an easy consequence of Lemma 41 and the fact that, when encoding a C-tree D over S, together with a tree decomposition witnessing that D is a C-tree, into a consistent ΓS,l -labeled tree t, then JtK and D are isomorphic. Roughly, Lemma 22 states that containment among OMQs from (G, BCQ) can be semantically characterized via the decodings of consistent ΓS,l -labeled trees. This makes the problem of deciding containment amenable to tree automata techniques. Proof of Lemma 23 Before proceeding to the proof of Lemma 23, we first introduce the relevant automata model. Automata Techniques For a set of propositional variables X, we denote by B+ (X) the set of Boolean formulas using variables from X, the connectives ∧, ∨, and the constants true, false. Let us now introduce our automata model. Definition 10. A two-way alternating parity automaton (2WAPA) on trees is a tuple A = (S, Γ, δ, s0 , Ω), where S is a finite set of states, Γ an alphabet (the input alphabet of A), δ : S × Γ → B+ (tran(A)) the transition function, where we set tran(A) := {hαis, [α]s \| s ∈ S, α ∈ {−1, 0, ∗}}, s0 ∈ S the initial state, and Ω : S → N the parity condition that assigns to each s ∈ S a priority Ω(s). Elements from tran(A) are called transitions. Intuitively, a transition of the form h0is means that a copy of the automaton should change to state s and stay at the current node. A transition of the form h−1is means that a copy should be sent to the parent node, which is then required to exist, and proceed in state s, while one of the form h∗is means that a copy of the automaton that assumes state s is sent to some child node. The transition [0]s means the same as h0is, while [−1]s means that a copy of the automaton that assumes state s should be sent to the parent node which is there not required to exist at all. Likewise, [∗]s means that a copy of the automaton assuming state s should be sent to all child nodes. W V Notation. We write ✸s for {hαis \| hαis ∈ tran(A), s ∈ S}, ✷s for {[α]s \| [α]s ∈ tran(A), s ∈ S}, and simply s for h0is. Furthermore, for α ∈ N ∪ {−1, ∗}, we define   if α = −1 and x · α ∈ T , {x · α}, Tα (x) := {x · i \| x · i ∈ T, i ∈ N \ {0}}, if α = ∗,  ∅, otherwise. 27 Definition 11. A run of a 2WAPA A = (S, Γ, δ, s0 , Ω) on a Γ-labeled tree (T, η) is a T × S-labeled tree (Tr , ηr ) such that the following holds: 1. ηr (ε) = (ε, s0 ), 2. if y ∈ Tr , ηr (y) = (x, s), and δ(s, η(x)) = ϕ, then there is an I ⊆ tran(A) such that I \|= ϕ holds and the following conditions are satisfied: • If hαis′ ∈ I then there is a node x′ ∈ Tα (x) and a child node y ′ ∈ Tr of y such that ηr (y ′ ) = (x′ , s′ ). • If [α]s′ ∈ I then for all x′ ∈ Tα (x), there is a child node y ′ ∈ Tr of y such that ηr (y ′ ) = (x′ , s′ ). We say that a run (Tr , ηr ) is accepting on A, if on all infinite paths (ε, s0 ), (x1 , s1 ), (x2 , s2 ), . . . in Tr , the maximum priority among Ω(s0 ), Ω(s1 ), Ω(s2 ), . . . that appears infinitely often is even. A accepts a Γ-labeled tree (T, η), if there is an accepting run on (T, η). We denote by L(A) the set of Γ-labeled trees A accepts, i.e., the language accepted by A. Remark. The automaton model defined above resembles that in [58]. However, we explicitly provide transitions that allow the automaton move to the parent node, while the model defined in [58] provides transitions for moving to some neighboring node, including the parent node. Therefore, the automata in [58] offer transitions of the form s, ✸s, and ✷s with their intended meaning as defined above. Using techniques as employed in [57, 58], for a 2WAPA A, one can show that the problem of deciding whether L(A) = ∅ is feasible in exponential time with respect to the number of states of A and in polynomial time with respect to the size of the input alphabet of A. Proof of Lemma 23. We only give an intuitive explanation for the construction of the desired 2WAPA. To check whether a ΓS,l -labeled tree is consistent, we can check each condition for consistency separately by a dedicated 2WAPA and then take the intersection of all of them. Most of the consistency conditions are easy to check. We give here a more detailed verbal explanation for condition (5). A 2WAPA checking this condition can be constructed as follows. At the beginning of its run, the automaton branches universally to all nodes (except the root) in a state whose intended purpose is to find appropriate guards in the input tree for the names available at the current node. To this end, the automaton has to do a reachability analysis on the input tree and store, using exponentially many states in ar (S), the tuple it seeks to guard. By a guard for the node v here, we mean a node w with an Rā ∈ µ(w) such that (i) {ā} contains all the names present at w and (ii) is b-connected to v for all b ∈ names(v). Notice that such a reachability analysis can be easily performed once we have the means to store the information contained in names(v) in a single state. This is, however, possible since for this task we need somewhat O((ar (S)+l)ar (S) ) states, i.e., polynomially many in the size of ΓS,l . Proof of Lemma 24 We first need to introduce some additional auxiliary notions. Strictly Acyclic Queries Let q be a CQ over a schema S. We denote by free(q) the free variables of q; the same notation is used for first-order formulas in general. We can naturally view q as an instance [q] whose domain is the set of variables of q and contains the body atoms of q as facts. In the following, we will often overload notation and write q for both the query q and the instance [q]. The notions of tree-width, acyclicity, etc. then immediately extend to CQs. Given a tree decomposition δ of q (i.e., of [q]), we say that δ is strict, if some bag of δ contains all variables that are free in q (cf. also [38]). Accordingly, q is called strictly acyclic if it has a guarded tree decomposition that is strict. Strictly acyclic queries have the convenient property to be equivalent to guarded formulas of a special form. Recall that the set of guarded formulas over a schema S is built inductively by including all atomic formulas, relativizing quantifiers by atomic formulas, and closing under Boolean connectives. More precisely, all quantifier occurrences have one of the forms ∀ȳ (α(x̄, ȳ) → ϕ) and ∃ȳ (α(x̄, ȳ) ∧ ϕ), such that the free variables of ϕ are among {x̄, ȳ}. We are interested in the guarded formulas that are build up using conjunction and existential quantification; we restrict ourselves to such formulas in the following. We call a formula from this class strictly guarded, if it is of the form ∃ȳ (α(x̄, ȳ)∧ϕ). We explicitly include the case where ȳ is the empty sequence of variables, i.e., if to formulas of the form α(x̄) ∧ ϕ with free(ϕ) ⊆ {x̄}. Notice that every guarded sentence ϕ (i.e., a formula having no free variables) is strictly guarded, since it is equivalent to ∃y (y = y ∧ ϕ). Furthermore, notice that every usual guarded formula that uses only existential quantifiers and conjunction is equivalent to a conjunction of strictly guarded formulas. The following lemma is proved in [38]. Lemma 42. Every strictly acyclic CQ can be rewritten in polynomial time into an equivalent strictly guarded formula that is built up using conjunction and existential quantification only. The converse holds as well. Squid Decompositions Let q be a BCQ over a schema S having n body atoms. An S-cover of q is a BCQ q + that contains all the atoms from q and may additionally contain 2n other body atoms over S. It is pretty straightforward that, for an S-instance I, it holds that I \|= q iff there is an S-cover q + of q such that I \|= q + . 28 Definition 12. Let I be an instance. For V ⊆ dom(I), we say that I is [V ]-acyclic, if it has a guarded tree decomposition that omits V . Definition 13. Let q be a BCQ over S. A squid decomposition of q is a tuple δ = (q + , µ, H, T, V ), where q + is an S-cover of q, µ : var (q + ) → var (q + ) a mapping, V ⊆ var (µ(q + )), and (H, T ) a partition of the atoms µ(q + ) such that • H is the set of atoms of µ(q + ) induced by V , • T = µ(q + ) \ H and T is [V ]-acyclic. Intuitively, a squid decomposition specifies a way how a BCQ can be mapped to an instance that contains some “cyclic parts”—the set H specifies those atoms that are mapped to such cyclic parts, while A declares those atoms that are mapped to the acyclic parts of the instance at hand. We will make this more precise in Lemma 43 below, where we analyze matches in C-trees. Given a CQ q and a set of variables V ⊆ var (q), the V -reduct of q, denoted q V , is the conjunctive query that arises from q by dropping all the existential quantifiers that bind variables in V . Lemma 43. Let J be a C-tree over S and q a BCQ over S. Let (T , (Xt )t∈T ) be a witnessing tree decomposition of J. It holds that J \|= q iff there is a squid decomposition δ = (q + , µ, H, A, V := {x̄}) of q and a homomorphism η : µ(q + ) → J such that 1. C \|= H is witnessed by η, S 2. ε≺v J(v) \|= AV (η(x̄)) is witnessed by η, and 3. there are strictly guarded formulas ϕ1 , . . . , ϕl such that AV (x̄) ≡ ϕ1 ∧ · · · ∧ ϕl . Proof. For the direction from right to left, consider such a given squid decomposition δ and a homomorphism η as in the hypothesis of the lemma. It is immediate that η ◦ µ is a homomorphism mapping q + to J. Since q + is an S-cover of q, we obtain J \|= q as required. For the other direction, suppose that J \|= q is witnessed by a homomorphism θ. For each v ∈ T \ {ε}, let βv be an atom of J such that J(v) \|= βv and βv contains all domain elements from J(v) as arguments. Notice that the βv exist, since δ is guarded except for {ε}. Since θ maps q to J, for each atom α of q, there is a node vα such that θ(α) ∈ J(vα ). Let W be the set of all these nodes and their closure under greatest lower bounds with respect to , excluding the root node ε of T . Consider the set of atoms Q+ := θ(q) ∪ {βv \| v ∈ W }. Notice that at least half of the nodes of W are of the form vα —hence, \|Q+ \| ≤ 3\|q\|. Let q + be a BCQ constructed as follows. Take the conjunction of q and for each βv (a1 , . . . , an ) (v ∈ W ), add an atom βv (x1 , . . . , xn ), where each xi is a newly chosen variable. Then q + is obviously an S-cover of q. Furthermore, by construction, there is a mapping µ : var (q + ) → var (q + ) and an isomorphism η : µ(q + ) → Q+ such that (η ◦ µ)(q + ) = Q+ . Now let H be the greatest set of atoms of µ(q + ) such that η(H) ⊆ J(ε). Moreover, let V := var (H) and A := µ(q + ) \ H. We claim that δ := (q + , µ, H, A, V ) is a squid decomposition of q that satisfies together with η the points mentioned in the statement of the lemma. To see that δ is a squid decomposition of q, the only nontrivial point to prove is that A is indeed [V ]-acyclic. We will prove this below in the course of establishing the third item. The first two items are immediate by construction. We proveS the third item. Suppose V = {x̄} and consider the V -reduct AV (x̄) of A. By construction, the atoms η(A) are contained in ε≺v J(v). Now the set W together with the order T gives rise to a forest consisting of trees T1 , . . . , Tl whose roots are descendants of ε, i.e., the root of T (recall that ε is not contained S S S in W ). Moreover, we have (i) li=1 Ti = W , (ii) Ti ∩ Tj = ∅, for i 6= j, and (iii) v∈Ti Xv ∩ v∈Tj Xv ⊆ dom(C), for i 6= j. S For v ∈ T , let Q+ (v) := {α ∈ Q+ \| J(v) \|= α} and, for i = 1, . . . , l, let Q+ (Ti ) be the set of atoms v∈Ti Q+ (v). Now it is easy to check using the facts stated before that each Q+ (Ti ) is acyclic and, hence, so is η −1 (Q+ (Ti )). Furthermore, denoting by εi the root of Ti , it holds that dom(Q+ (Ti )) ∩ η(V ) ⊆ dom(Q+ (εi ))—indeed, if a ∈ dom(Q+ (v)) ∩ η(V ) for some v εi , then, since εi ≻ ε and a ∈ Xε , it must be the case that a ∈ dom(Q+ (εi )) by connectivity. It follows that the V -reduct of η −1 (Q+ (Ti )) (viewed as Boolean query), henceforth denoted qT+i , is strictly acyclic and is therefore equivalent to a strictly V guarded formula ϕi . Hence, the query AV (x̄) is equivalent to li=1 ϕi . Moreover, it follows that A itself is [V ]-acyclic—notice Vl that A ≡ ∃x̄ i=1 qT+i and that dom(Q+ (Ti )) ∩ dom(Q+ (Tj )) ⊆ η(V ), for i 6= j. Hence, var (qT+i ) ∩ var (qT+j ) ⊆ V , for i 6= j. The claim now follows since every qT+i is acyclic. Derivation trees Let D be an S-database and Σ a set of guarded rules. Let q0 (x̄) be a strictly acyclic query whose free variables are exactly those from x̄ := x1 , . . . , xn and let ā := a1 , . . . , an be a tuple from dom(D). A derivation tree for (ā, q0 (x̄)) with respect to D and Σ is a finite tree T whose nodes are labeled via a function µ with pairs of the form (b1 , . . . , bk ; q(y1 , . . . , yk )), where b1 , . . . , bk are constants from dom(D) and q(y1 , . . . , yk ) is a strictly acyclic query over S ∪ sch(Σ) having exactly y1 , . . . , yk free, such that the following conditions are satisfied: 1. µ(ε) = (ā, q0 (x̄)), where ε is the root node of T . 29 2. If µ(v) = (c1 , . . . , cm ; q(z1 , . . . , zm )) for some node v, then one of the following conditions holds (let c̄ := c1 , . . . , cm and z̄ := z1 , . . . , zm ): (a) v is a leaf node and q(z̄) ≡ β(z̄), for some atomic formula β(z̄) such that D \|= β(c̄). (b) The node v has a successor labeled by (c̄, b̄; p(z̄, ȳ)) and it holds that Σ \|= ∀z̄, ȳ (p(z̄, ȳ) → q(z̄)). (c) The query q(z̄) is logically equivalent to q1 (zi1,1 , . . . , zi1,k1 ) ∧ · · · ∧ ql (zil,1 , . . . , zil,kl ) and v has l successors v1 , . . . , vl respectively labeled by (ci1,1 , . . . , ci1,k1 ; q1 (zi1,1 , . . . , zi1,ki )), . . . , (cil,1 , . . . , ci1,kl ; ql (zil,1 , . . . , zi1,kl )). Lemma 44. Let α(x1 , . . . , xn ) be an atomic formula. (a1 , . . . , an ; α(x1 , . . . , xn )) with respect to D and Σ. Then D, Σ \|= α(a1 , . . . , an ) iff there is a derivation tree for Proof (sketch). Let ā := a1 , . . . , an and x̄ := x1 , . . . , xn . The direction from right to left is an easy induction on the construction of the derivation tree. We sketch the other direction. Consider the guarded chase forest (F, η) for D and Σ, where η is a function labeling the nodes and edges of F. We construct a derivation tree for (ā, α(x̄)) by induction on the number of chase steps required to derive α(ā) from D and Σ. For the base case, if D \|= α(ā), the claim is obvious since we can apply rule 2.(a). Assume that α(ā) is derived using a rule σ: β0 (x̄, ȳ), β1 , . . . , βk → α(x̄), and a homomorphism µ such that µ(x̄) = ā, where β0 (x̄, ȳ) is the guard of σ. If µ({x̄, ȳ}) ⊆ dom(D), the result immediately follows by the induction hypothesis. Otherwise, the image of β0 (x̄, ȳ) under µ contains some labeled nulls as arguments. Assume that all the β1 , . . . , βk contain nulls as their arguments—for those that do not, the induction hypothesis would yield appropriate derivation trees again. Notice that all the nulls occurring in β1 , . . . , βk appear in µ({ȳ}). By construction of F, there is a node v0 that is an ancestor of the nodes having the atoms µ(β0 ), µ(β1 ), . . . , µ(βk ) as labels and which has a label of the form β0 (ā, b̄) which contains no nulls at all as arguments. There is a corresponding atomic formula γ0 (x̄, z̄) whose image under an appropriate homomorphism equals β0 (ā, b̄). Furthermore, there are atoms γ1 , . . . , γl such that dom({γ1 , . . . , γl }) ⊆ {ā, b̄} and Σ \|= β0 (ā, b̄) ∧ γ1 ∧ · · · ∧ γl → ∃ȳ (β0 (ā, ȳ) ∧ β1 ∧ · · · ∧ βk ). Now regard the γi (i = 1, . . . , l) as atomic formulas with free variables among {x̄, z̄}. The formula p(x̄, z̄) := γ0 (x̄, z̄)∧γ1 ∧· · ·∧γl is then a strictly acyclic query that satisfies Σ \|= ∀x̄, z̄ (p(x̄, z̄) → α(x̄)). An application of rule 2.(b) then requires us to find a derivation tree for (ā, b̄; p(x̄, ȳ)), whence an application of rule 2.(c) reduces this task to finding derivation trees for the atoms γ0 , γ1 , . . . , γl and their corresponding tuples of constants. These trees exist by induction hypothesis and we can simply concatenate them appropriately in order to arrive at a derivation tree for (ā, α(x̄)). Given a guarded formula ϕ(x̄) built up from conjunctions and existential quantification, we define the nesting depth of ϕ(x̄), denoted nd(ϕ(x̄)), inductively: • If ϕ(x̄) is an atomic formula, then nd(ϕ(x̄)) := 0. • If ϕ(x̄) = (ψ1 ∧ ψ2 ), then nd(ϕ(x̄)) := max{nd(ψ1 ), nd(ψ2 )}. • If ϕ(x̄) = ∃ȳ (α(x̄, ȳ) ∧ ψ) and ȳ 6= ∅, then nd(ϕ(x̄)) := nd(ψ) + 1. Lemma 45. Let D be a database, Σ a set of guarded rules, and q(x̄) a strictly acyclic conjunctive query. Then D, Σ \|= q(ā) iff there is a derivation tree for (ā, q(x̄)) with respect to D and Σ. Proof (sketch). We again sketch only the direction from left to right. Let ϕ(x̄) be the strictly guarded formula corresponding to q(x̄). We proceed by induction on the nesting depth of ϕ(x̄). If nd(ϕ(x̄)) = 0, then ϕ(x̄) is quantifier free and thus a conjunction of atoms α0 (x̄) ∧ α1 ∧ · · · ∧ αk , where var (αi ) ⊆ {x̄} for i = 1, . . . , k. An application of rule 2.(c) reduces the problem of building a derivation tree for (ā, ϕ(x̄)) to the problem of building corresponding trees for the αi and their corresponding constants from ā. The existence of these trees is guaranteed by Lemma 44. Now suppose that nd(ϕ(x̄)) = n + 1. Let ϕ(x̄) = ∃ȳ (α(x̄, ȳ) ∧ ψ) and ȳ := y1 , . . . , yk . Assume, without loss of generality, that all the bound variables from ϕ(x̄) are pairwise distinct. In the following, we will describe how to construct a derivation tree for (ā, q(x̄)). If D, Σ \|= q(ā), then there is a homomorphism µ mapping each atom of q(x̄) to chase(D, Σ) such that µ(x̄) = ā. Furthermore, µ maps each atom of q(x̄) to a node of the guarded chase forest F of D and Σ. Let αµ (ā, λ1 , . . . , λk ) denote the atom labeling the node of F where α(x̄, ȳ) is mapped to via µ. Let λi1 , . . . , λil exhaust all elements from λ1 , . . . , λk that are not from dom(D) and b̄ := λj1 , . . . , λjm exhaust those from λ1 , . . . , λk that are from dom(D). Let ϕ′ (x̄, yj1 , . . . , yjm ) be the formula ∃yi1 , . . . , yil (α(x̄, ȳ) ∧ ψ). Clearly, Σ \|= ∀x̄, yj1 , . . . , yjm (ϕ′ (x̄, yj1 , . . . , yjm ) → ϕ(x̄)). Hence, we can create a successor of (ā, ϕ(x̄)) that is labeled by (ā, b̄; ϕ′ (x̄, yj1 , . . . , yjm )). Assume now that none of the λ1 , . . . , λk is from dom(D). Furthermore, assume that k ≥ 1, since otherwise we can just simply apply rule 2.(c) to reduce q(x̄) to a conjunction of queries of the desired form. As in the proof of Lemma 44, there is a node v0 in F whose label β0 (ā, b̄) contains only values from dom(D) as arguments and such that v0 is an ancestor of the node labeling αµ (ā, λ1 , . . . , λk ). Furthermore, all the atoms from 30 µ(q(x̄)) that contain an element from λ1 , . . . , λk as argument are also located in the subtree rooted at v0 . Let p be the query that results from deleting all atoms from q(x̄) which are mapped via µ into the subtree rooted at v0 . Notice that p may be empty and has free variables among x̄. Furthermore p is equivalent to a conjunction p1 ∧ · · · ∧ pl of strictly acyclic queries. Let β0 (x̄, z̄) be the atomic formula whose image under an appropriate homomorphism equals β0 (ā, b̄). A similar line of reasoning as in the proof of Lemma 44 shows that there are atomic formulas β1 , . . . , βm such that var (βi ) ⊆ {x̄, z̄} and ∀x̄, z̄ (β0 (x̄, z̄) ∧ β1 ∧ · · · ∧ βm ∧ p → ϕ(x̄)), whence an application of rule 2.(b) and rule 2.(c) reduces the problem of constructing a derivation tree for (ā, ϕ(x̄)) to that of constructing corresponding trees for β0 , . . . , βm and p. Notice that p is a conjunction of strictly guarded formulas of nesting depth at most n. Hence, the induction hypothesis guarantees the existence of such derivation trees. Having the above results in place, it is easy to show the following statement: Lemma 46. Let D be a C-tree over S and Q = (S, Σ, q) an OMQ where Σ is guarded and q a BCQ. Then D \|= Q iff there is a squid decomposition δ = (q + , µ, H, A, V := {x̄}) of q and a homomorphism η : µ(q + ) → chase(D, Σ) such that: 1. F \|= H is witnessed by η, where F is the subinstance of chase(D, Σ) induced by dom(C). 2. There are strictly acyclic queries q1 , . . . , ql such that (a) AV (x̄) ≡ q1 ∧ · · · ∧ ql and (b) for i = 1, . . . , l and free(qi ) = {x̄i }, there are derivation trees for (η(x̄i ), qi ) with respect to D and Σ. Proof. We can easily prove by induction on the number of chase steps that chase(D, Σ) is an F -tree, where F is the subinstance of chase(D, Σ) induced by dom(C). Now the lemma at hand is immediate by combining this fact with Lemma 43 and Lemma 45. We are now ready to proceed with the proof of Lemma 24: Proof of Lemma 24. Lemma 46 will guide the construction of the 2WAPA we are now going to construct. Suppose Q = (S, Σ, q) is an OMQ from (G, BCQ) and let l ≥ 1. We are going to construct a 2WAPA AQ,l = (S, ΓS,l , δ, s0 , Ω) that accepts a consistent ΓS,l -labeled tree t iff JtK \|= Q. In particular, the number of states of AQ,l will be at most exponential in the size of Q and at most polynomial in l, while the construction of AQ,l will be feasible in 2ExpTime. The state set. Let Λ denote the set of all Boolean acyclic queries over S ∪ sch(Σ) that are of size at most 3\|q\|. Notice that each of these queries is equivalent to a strictly guarded formula. Furthermore, assume that Λ is closed under V -reducts, for V ⊆ var (q), provided that they are strictly acyclic as well, i.e., if p ∈ Λ and V ⊆ var (q), then also pV ∈ Λ provided pV is strictly acyclic. For {ā} ⊆ US,l , let Ŝ(ā) := {p(x̄/ā) \| p ∈ Λ, free(p) = {x̄}, \|ā\| = \|x̄\|} and let Ŝ be the union of all the sets Ŝ(ā). Now the set of states S consists of an initial state, denoted s0 , plus the set Ŝ factorized modulo logical equivalence. We denote by [p] the equivalence class of a query p ∈ Ŝ. Furthermore, for a strictly guarded formula ϕ, we may abuse notation and write [ϕ] for the equivalence class of the strictly acyclic query p ∈ Ŝ that is equivalent to ϕ. Notice that the size of S is exponential in the size of Q, since there are only exponentially many CQs of size at most 3\|q\| that are mutually non-equivalent (cf. [10]). The parity condition. We set Ω(s) := 1, for all s ∈ S. This means that only finite trees are accepted. The transition function. In the following, for each ρ ∈ ΓS,l , we denote by Θ̂(ρ) the set of all pairs that are of the form (α1 ∧ · · · ∧ αn , p1 ∧ · · · ∧ pm ) for which there is a squid decomposition of the form (q + , µ, H, T, {x̄}) and a function θ : {x̄} → names(ρ) such that: • H {x̄} (θ(x̄)) ≡ α1 ∧ · · · ∧ αn , where all the αi are relational ground atoms. • T {x̄} (θ(x̄)) ≡ p1 ∧ · · · ∧ pm , where the pi are strictly acyclic queries. Call two pairs (ϕ1 , ψ1 ) and (ϕ2 , ψ2 ) as above equivalent if ϕ1 ≡ ϕ2 and ψ1 ≡ ψ2 . Let Θ(ρ) be the set of equivalence classes under this relation and denote by [ϕ, ψ] the equivalence of a pair (ϕ, ψ) under this relation. Now we fix for each [p] ∈ S \ {s0 } a strictly guarded formula χ[p] that is equivalent to all queries from [p]. Likewise, we fix a function ϑρ : Θ(ρ) → Θ̂(ρ) such that ϑρ ([ϕ, ψ]) ∈ [ϕ, ψ], i.e., which picks a representative for each equivalence class [ϕ, ψ]. Now let ρ ∈ ΓS,l . Specify δ(·, ρ) as follows: 1. For the initial state s0 , set δ(s0 , ρ) := n m ^ _ ^ { [αi ] ∧ [pi ] \| (α1 ∧ · · · ∧ αn , p1 ∧ · · · ∧ pm ) ∈ ϑρ (Θ(ρ))}. i=1 i=1 Intuitively, the automaton selects a squid decomposition where its components are instantiated by names occurring in the root node of the input tree. The automaton tries to verify the single compartments of the squid decomposition, i.e., it tries to match them to the chase expansion of the input database under Σ. 31 2. Let [p] ∈ S \ {s0 }. We define δ([p], ρ) according to a case distinction: (a) Suppose that p ≡ ⊤. Then δ([p], ρ) := true. (b) Suppose χ[p] = ∃ȳ (α(ā, ȳ) ∧ ϕ), where α(ā, ȳ) is an atomic formula (including equality), free(ϕ) ⊆ {ȳ}, and ā exhausts all names occuring in α. If {ā} 6⊆ names(ρ) then δ([p], ρ) := false. Otherwise, _ _ δ([p], ρ) := {[ϕ(ȳ/b̄)] \| ρ \|= α(ā, b̄), {b̄} ⊆ names(ρ)} ∨ ✸[p] ∨ impl(p, ρ), where impl(p, ρ) := {[p1 ] ∧ · · · ∧ [pn ] \| [p1 ], . . . , [pn ] ∈ S \ {s0 }, {b̄} ⊆ names(ρ), p1 ∧ · · · ∧ pn ≡ q, Σ \|= ∀x̄, ȳ (q(ā/x̄, b̄/ȳ) → p(ā/x̄))}. We provide some intuitive explanation for this second case. (a) If p is the empty query, it can be satisfied at any input node and, hence, the automaton accepts unconditionally on this computation branch. (b) Otherwise, we first inspect the strictly guarded formula χ[p] at hand. If the names occurring in the guard α(ā, ȳ) are not present at the current node, it rejects. Otherwise, it tries to satisfy α(ā, ȳ) with all possible assignments for ȳ at the current node and then proceed in state [ϕ(ȳ/b̄)]. Apart from these possibilities, the automaton can decide to move to any neighboring node (i.e., the parent or a child) while remaining in state [p]. This amounts to an exhaustive search of the input tree that tries to satisfy p in the input tree. Furthermore, the automaton may choose to construct derivation trees for p. There, it uses the information provided by Σ in order to find strictly acyclic queries p1 , . . . , pn that imply p. Consequently, it tries to proceed its search with [p1 ], . . . , [pn ]. We shall now briefly comment on the running time needed to construct AQ,l . The interesting part of the construction concerns the transition function δ, in particular point 2.(b) involving impl(p, ρ). We have seen that in the proofs of Lemma 44 and Lemma 45 that there are double-exponentially many candidates for the query q(ā/x̄, b̄/ȳ) that (possibly) implies p(ā/x̄) under Σ. Furthermore, q(ā/x̄, b̄/ȳ) consists of at most exponentially many atoms. Each check whether such a query q at hand implies p requires at most double-exponential time in the size of p. This follows from the well-known fact that checking query implication under a set of guarded rules is feasible in 2ExpTime with respect to the size of the right-hand side query, and in polynomial time with respect to the size of the left-hand side query (cf. [23]), i.e., the data complexity of query answering under guarded tgds is polynomial time. PROOFS OF SECTION 6 Proof of Theorem 26 A proof sketch is given in the main body of the paper. However, the fact that Cont((G, CQ), (S, CQ)) is in 2ExpTime deserves a formal proof. Recall that to establish the latter result we need a more refined complexity analysis of the problem of deciding whether a guarded OMQ is contained in a UCQ; this is discussed in the main body of the paper. In fact, it suffices to show the following result. As in the previous section, we focus on constant-free tgds and CQs, but all the results can be extended to the general case at the price of more involved definitions and proofs. Moreover, we assume that tgds have only one atom in the head. Recall that we write var ≥2 (q) for the variables of q that appear in more than one atom, and we also write var =1 (q) for the variables of q that appear only in one atom. Then: Proposition 47. Consider Q ∈ (G, BCQ) and a Boolean CQ q. The problem of deciding whether Q ⊆ q is feasible in 1. double-exponential time in (\|\|Q\|\| + \|var ≥2 (q)\|); and 2. exponential time in \|var =1 (q)\|. It is easy to verify that the above result, together with the algorithm devised in the main body of the paper, implies that Cont((G, CQ), (S, CQ)) is in 2ExpTime. The rest of this section is devoted to show the above proposition. Our crucial task is, given a CQ q, to devise an automaton that accepts consistent labeled trees which correspond to databases that make q true. Lemma 48. Let q be a Boolean CQ over S. There is a 2WAPA Aq,l , where l > 0, that accepts a consistent ΓS,l -labeled tree t iff JtK \|= q. The number of states of Aq,l is exponential in \|var ≥2 (q)\| and polynomial in (\|var =1 (q)\| + ar (S) + l). Furthermore, Aq,l can be constructed in exponential time. Proof. We are going to construct Aq,l = (S, ΓS,l , δ, s0 , Ω). Let x1 , . . . , xn be the variables of var =1 (q) and fix a total order x1 ≺ x2 ≺ · · · ≺ xn among them. Define the state set S to be S := {sy,θ \| θ : V → US,l , V ⊆ var ≥2 (q), y ∈ var =1 (q) ∪ {♯}}. Notice that \|S\| = O(\|var =1 (q)\| · (ar (S) + l)\|var ≥2 (q)\| ). We set s0 := s♯,∅ , where ∅ denotes the empty substitution. In the following, we treat q as a set of relational atoms and let X = var ≥2 (q). For ρ ∈ ΓS,l and sy,θ ∈ S, define δ(sy,θ , ρ) as follows: 32 • If y = ♯, distinguish the following cases: 1. If there is an atom α ∈ θ(q) such that var (α) ∩ X 6= ∅ and dom(α) ∩ US,l 6⊆ names(ρ), then δ(s♯,θ , ρ) := false. 2. Otherwise, let δ(s♯,θ , ρ) := (W {s♯,η \| η ⊇ θ, ρ \|= ∃x̄ sx1 ,θ , V • Suppose y = xi , for some i = 1, . . . , n. Let αi,θ denote the   sxi+1 ,θ , δ(sxi ,θ , ρ) := true,  ✸s xi ,θ , (η(q) \ θ(q))} ∨ ✸s♯,θ , if X ∩ var (θ(q)) 6= ∅, otherwise. unique atom α ∈ θ(q) such that xi ∈ var (α). Set if ρ \|= ∃x̄ αi,θ and i < n, if ρ \|= ∃x̄ αi,θ and i = n, otherwise. Set the parity condition Ω to be Ω(s) := 1 for all s ∈ S. Intuitively, the automaton works in two passes. The first pass consists of the runs working on states of the form s♯,θ . In this pass, the automaton tries to find an assignment for the variables in the query that appear in at least two distinct atoms. When a candidate assignment θ is found, the automaton changes to state sx1 ,θ which is the beginning of the second pass. A state of the form sxi ,θ means that the assignment θ can be extended to all variables x ≺ xi and, in this state, the automaton tries to extend θ to cover the variable xi . The automaton accepts if it is able to extend the candidate assignment θ to all x1 , . . . , xn . Having the above result in place, we can now reduce the problem in question to the emptiness problem for 2WAPA. Lemma 49. Consider Q ∈ (G, BCQ) and a Boolean CQ q. We can construct in double-exponential time in \|\|Q\|\| and in exponential time in \|\|q\|\| a 2WAPA A, which has exponentially many states in (\|\|Q\|\| + \|var ≥2 (q)\|) and polynomially many states in \|var =1 (q)\|, such that Q ⊆ q ⇐⇒ L(A) = ∅. ′ Proof. Let Q = (S, Σ, q ) and l = ar (S ∪ sch(Σ)) · \|q ′ \|. Then A is defined as: (CS,l ∩ AQ,l ) ∩ Aq,l . It is an easy task to verify that the claim follows from Lemmas 22, 23, 24, and 48. It is clear that Proposition 47 is an easy consequence of Lemma 49. PROOFS OF SECTION 7 Recall that we focus on unary and binary predicates. Moreover, we consider constant-free tgds and CQs, and we assume that tgds have only one atom in the head. Proof of Proposition 30 Basics. Let D be a C-tree of width two. We say that a tree decomposition δ = (T , (Xt )t∈T ) witnessing that D is a C-tree is lean, if it satisfies the following conditions: • The elements from dom(C) occur only in the root of T and its immediate successors. • If w is a child of v in T , then there are unique c, d ∈ dom(D) such that Xv ∩ Xw = {c} and Xw \ Xv = {d}. The element d is called new at w. • It follows from the previous item that every node v 6= ε in T has a unique new element c ∈ dom(D). We additionally require that c appears in the bag of each child of v. Intuitively, C-trees D that have lean tree decomposition represent the actual tree structure of D. It is fairly straightforward to see that every C-tree has a lean tree decomposition. Recall that the Gaifman graph of D is the graph G(D) = (V, E) with V := dom(D) and (a, b) ∈ E if a and b coexist in some atom of D. Given two nodes a, b from G(D), the distance from a to b in G(D), denoted dG(D) (a, b), is the minimum length of a path between a and b, and ∞ if such a path does not exist. For a, b ∈ dom(D), we denote by dδ (a, b) the minimum distance among two nodes of T that respectively have a and b in their bags. We call dδ (a, b) the distance from a to b in δ. Notice that in a tree decomposition δ witnessing that D is a C-tree, any element a ∈ dom(D), if a appears in the bag of v, then it occurs only at v, at v and its children, or at v and its parent. Since furthermore the bag of the root node is uniquely determined by C, each node in the tree has a uniquely determined set of child nodes whose bags are determined by the structure of D alone. Therefore, the following two lemmas follow immediately. Lemma 50. Let δ = (T , (Xt )t∈T ) be a lean tree decomposition witnessing that D is a C-tree. Then dδ (a, b) ≤ dG(D) (a, b) for all a, b ∈ dom(D). 33 Lemma 51. Let δ and δ ′ be two lean tree decompositions witnessing that D is a C-tree. Then dδ (a, b) = dδ′ (a, b) for all a, b ∈ dom(D). In the following, we denote by D≤k the subinstance of D induced by the set of elements whose distance from any a ∈ dom(C) in any lean tree decomposition δ is bounded by k. The subinstance D>k is defined analogously.The branching degree of a lean tree decomposition is the maximum number of child nodes of any node contained in the tree of δ. Notice that two lean tree decompositions of a C-tree D always have the same branching degree; the argument is similar as for the two lemmas above. Hence, we can simply speak about the branching degree of D. Encodings. Recall that a consistent ΓS,l -labeled tree t = (T, µ) encodes information on an S-database D and an appropriate tree decomposition δ of D. It is clear that JtK has a lean tree decomposition, but it is not guaranteed that this is reflected in δ as well. We call (the consistent) t lean, if the tree decomposition δt = (T := (T, E), (Xv )v∈T ) is, where xEy iff y = x · i for some i ∈ N \ {0} and Xv := {[v]a \| a ∈ names(v)}. The following is easy to prove: Lemma 52. There is a 2WAPA on trees LS,l that accepts a consistent ΓS,l -labeled tree iff it is lean. The number of states of LS,l is bounded logarithmically in the size of ΓS,l and LS,l can be constructed in polynomial time in the size of ΓS,l . Let t = (T, µ) be a labeled tree. The branching degree of a node x ∈ T is the cardinality of {i \| x · i ∈ T, i ∈ N \ {0}}; the branching degree of t is the maximum over all branching degrees of its nodes and ∞ is this maximum does not exist. We also say that t is m-ary if the branching degree of t is bounded by m. A node x ∈ T is a leaf node of t if it has branching degree zero. The depth of T is the maximum length among the lengths of all branches and ∞ if this maximum does not exist. Let us remark that the branching degree of the lean ΓS,l -labeled tree t as defined for labeled trees equals the branching degree of JtK as defined above. Lemma 53. Let Q = (S, Σ, q) be an OMQ from (G2 , BCQ). There is an m ≥ 0 such that the following are equivalent: 1. There is an S-database D such that D \|= Q. 2. There is a C-tree D̂ with \|dom(C)\| ≤ 2\|q\| and branching degree at most m such that D̂ \|= Q. Proof. Let l := 2\|q\| and, let AQ,l be the 2WAPA from Lemma 24. Take the intersection of AQ,l with (i) the 2WAPA CS,l from Lemma 23 and (ii) the 2WAPA from Lemma 52 that checks leanness. Call the resulting automaton B. Then B accepts a ΓS,l -labeled tree t iff t is lean and consistent and JtK \|= Q. We let m be the number of states of B and claim that this is the required bound on the branching degree. First of all, notice that the first item of the lemma trivially implies the second independently from the choice of m. For the other direction, suppose that D \|= Q for some S-database D. Then there is a C-tree B such that dom(C) ≤ 2\|q\| and B \|= Q. Being a C-tree, B has a lean tree decomposition δ and the encoding of B together with δ corresponds to a lean ΓS,l -labeled tree t. It follows that t ∈ L(B). By the results of [44], it follows that there is a t′ ∈ L(B) whose branching degree is bounded by the number of states of B, i.e., by m. The tree t′ is lean and consistent, therefore Jt′ K is a C ′ -tree of branching degree at most m for some C ′ ⊆ JtK such that \|dom(C ′ )\| ≤ 2\|q\|. Furthermore, Jt′ K \|= Q, as required. We are now ready to prove Proposition 30: Proof of Proposition 30. We largely follow [16] here. Choose m as in Lemma 53 above. Suppose first that Q is UCQ rewritable. Let p := p1 ∨ · · · ∨ pn be a corresponding UCQ rewriting. Since the query q is connected, we can assume that p is as well. We choose k > max{\|pi \| : i = 1, . . . , n} and suppose that D \|= Q for some C-tree D. Since p is a UCQ rewriting, D \|= pi for some i = 1, . . . , n. Fix a homomorphism µ witnessing that D \|= pi . We distinguish cases. Suppose first that µ(var (pi )) ∩ dom(C) 6= ∅. Since p is connected, it follows D≤k \|= pi by Lemma 50 and so D≤k \|= p. On the other hand, if µ(var (p)) ∩ dom(C) = ∅, then it is also easy to check that D>0 \|= p. For the other direction, suppose that the second item of the proposition’s statement holds, i.e., there is a k ≥ 0 such that for all C-trees D over S with \|dom(C)\| ≤ 2\|q\| and branching degree at most m it holds that D \|= Q implies D≤k \|= Q or D>0 \|= Q. Let Λ be the set of all C-trees such that \|dom(C)\| ≤ 2\|q\| and that have branching degree at most m such that D \|= Q. We regard Λ as a set W of BCQs and regard it as factorized modulo logical equivalence. It is clear that Λ is finite then and we claim that p := n i=1 pi is a UCQ rewriting of Q. We explicitly include the case where Λ is empty, in which case p is equivalent to the empty disjunction ⊥ and there is no database D at all such that D \|= Q. To see that p is indeed a UCQ rewriting of Q, let D be an S-database such that D \|= p. Then there is an i = 1, . . . , n such that D \|= pi . Furthermore, [pi ] \|= Q and so D \|= Q as well, since Q is closed under homomorphisms. Suppose now D \|= Q. We know that there is a C-tree D̂ with \|dom(C)\| ≤ l := 2\|q\| and branching degree at most m such that D̂ \|= Q and—when we regard D̂ as an instance—there is a homomorphism from D̂ to D. Let D′ ⊆ D̂ be a minimal connected subset of D̂ such that D′ \|= Q. ′ ′ D′ is again a C ′ -tree for some C ′ ⊆ D′ . Therefore D≤k \|= Q or D>0 \|= Q. The latter is impossible by minimality of D′ . ′ ′ ′ Hence, D≤k \|= Q and so there is a (logically equivalent) copy of D≤k contained in Λ. Hence, D≤k \|= p, therefore D′ \|= p, and hence D̂ \|= p. Recall that, when D̂ is regarded as an instance, there is a homomorphism from D̂ to D. Therefore, D \|= p. 34 Proof of Proposition 31 Let Q = (S, Σ, q) be an OMQ from (G2 , BCQ) such that q is connected. We are going to show that the desired 2WAPA A can be constructed in 2ExpTime. Notice that, using similar results as in [16], this gives us a decision procedure for deciding UCQRew(G2 , CQ) also for non-connected queries. Let us first introduce some auxiliary notions. 2WAPAs on m-ary trees. A 2WAPA B on m-ary trees is just defined as a 2WAPA, except that its transitions tran(B) are {hkis, [k]s \| −1 ≤ k ≤ m, s ∈ S}, where S is the state set of B. The notion of run is then defined on m-ary trees only and its definition is modified in the obvious way so as to deal with the transitions hkis, [k]s. Intuitively, for k = 1, . . . , m, a transition hkis means that the automaton should move to the k-th child of the current node (which is then required to exist) and assume state s. Correspondingly, [k]s means that the automaton should move to the k-th child and assume state s provided that this k-th child exists at all. We remark that all 2WAPAs constructed in this paper so far can easily be modified to work on m-ary trees as well and we shall assume in the following that they do so. Furthermore, deciding whether L(B) is feasible in exponential time in the number of states of B and in polynomial time in the size of the input alphabet of B (cf. [57]). Let m be as in Proposition 30. In the following, we shall regard all trees mentioned in the following as m-ary and let l := 2\|q\|. Before proceeding to a proof of Proposition 31, we must make the notion of being an “extension” of a labeled tree more precise. Extensions of trees. Let BQ be a 2WAPA that accepts a ΓS,l -labeled tree t iff (i) t is lean and consistent, (ii) JtK \|= Q, and (iii) JtK>0 6\|= Q. Notice that a 2WAPA A>0 Q that accepts a lean and consistent ΓS,l -labeled tree iff JtK>0 6\|= Q can be easily constructed using the construction in Lemma 24. Hence, BQ can be constructed intersecting several 2WAPAs we have already encountered. Let Π be the set of all tuples of the form (s, s′ ), where s and s′ are states of BQ . We define a new alphabet Λ := 2KS,l ∪Π . Notice that Λ is of double-exponential size in the size of Q. For ρ ∈ Λ, we denote by ρ ↾ ΓS,l the restriction of ρ to ΓS,l , that is, ρ ∩ KS,l . The restriction of a Λ-labeled tree t to ΓS,l , denoted t ↾ ΓS,l , is the tree that arises from t when we restrict the label of each node of t to ΓS,l . We say that a Λ-labeled tree is consistent if (i) its restriction to ΓS,l is consistent and (ii) symbols ρ ∈ Λ such that ρ ∩ Π 6= ∅ appear only in leaf nodes of t. Likewise, we say that a consistent t is lean if t ↾ ΓS,l is. The decoding JtK of t is naturally extended to consistent Λ-labeled trees by setting JtK := Jt ↾ ΓS,l K. The following lemma is a straightforward extension of Lemmas 23 and 52. Lemma 54. There are 2WAPAs CΛ and LΛ that respectively accept a Λ-labeled tree iff it is consistent and lean. Both have logarithmically many states in the size of Λ and can be constructed in polynomial time in the size of Λ. Let t be a lean and consistent Λ-labeled tree. We say that t′ is an extension of t if t′ is a ΓS,l -labeled tree that arises from t by attaching ΓS,l -labeled trees to those leaves of t that contain elements from Π. Furthermore, for such nodes, the labels of the corresponding nodes in t′ are those of t restricted to ΓS,l . Definition 14. Let LQ be the set of all lean and consistent Λ-labeled trees t such that JtK 6\|= Q, yet there is an extension t′ of t such that Jt′ K \|= Q and Jt′ K>0 6\|= Q. Lemma 55. LQ is infinite iff Q is not UCQ rewritable. Proof. Suppose LQ is infinite. Since the trees at hand have bounded branching degree, for every k ≥ 0, there is a t ∈ LQ such that JtK is a C-tree (for some C ⊆ JtK) that contains individuals whose distance from any a ∈ dom(C) is greater than or equal to k and JtK 6\|= Q, yet for some extension t′ of t, we have Jt′ K \|= Q but Jt′ K>0 6\|= Q. Suppose now that Q is UCQ rewritable. Let ℓ be such that for all C ′ -trees D (of the appropriate dimensions), D \|= Q implies D≤ℓ \|= Q or D>0 \|= Q. Choose k > ℓ and t, t′ such that (i) t′ is an extension of t, (ii) t has depth greater than k, and (iii) JtK 6\|= Q but Jt′ K \|= Q and Jt′ K>0 6\|= Q. Since Jt′ K \|= Q, we know that Jt′ K≤ℓ \|= Q or Jt′ K>0 \|= Q. The latter is impossible by assumption, the former contradicts the fact JtK 6\|= Q, since k > ℓ. This proves the direction from left to right. The other direction is immediate. We are now ready to establish Proposition 31: Proof of Proposition 31. We are now going to describe the construction of a 2WAPA A such that L(A) = LQ , which will prove the claim by virtue of Lemma 55. This automaton is the intersection of several ones. First of all, we ensure that all the accepted Λ-trees are lean and consistent (cf. Lemma 54). We additionally intersect the automaton with the complement of AQ,l from Lemma 23 (more precisely, the version of it running on Λ-labeled trees) and another automaton DQ = (S, Λ, δ, s0 , Ω) whose construction we shall describe in more detail here. On a high level, DQ will be constructed so as to accept a lean and consistent Λ-labeled tree if and only if there is an extension t′ of t such that BQ accepts t′ . Let Ŝ be the set of states of BQ , δ̂ its transition function, and Ω̂ its parity function. For σ ∈ ΓS,l , let B̂(σ) be the set of tuples (s, s′ ) ∈ Ŝ × (Ŝ ∪ {true}) such that the following holds: • There is a ΓS,l -labeled tree t = (T, η) such that η(ε) = σ and a run tr = (Tr , ηr ) of BQ on t such that 1. ηr (ε) = (ε, s), i.e., tr starts from s;12 12 Strictly speaking, tr is, of course, not a run since it does not start in the initial state. 35 2. s′ = true and tr is accepting on BQ , or there is a node v ∈ Tr such that ηr (v) = (ε, s′ ). Now the set of states of DQ is the same as of BQ , i.e., S := Ŝ. Accordingly, the initial state of DQ is that of BQ . Furthermore Ω(s) := Ω̂(s), for every s ∈ S. Given s ∈ S and ρ ∈ Λ, we let _ δ(s, ρ) := {s′ \| (s, s′ ) ∈ ρ ∩ B̂(ΓS,l ↾ ρ)} ∨ δ̂(s, ΓS,l ↾ ρ). We are going to give an intuitive explanation of this construction in the following. Roughly, a pair (s, s′ ) ∈ B̂(σ) indicates that there is a ΓS,l -labeled tree t and run of BQ on t such that the root of t is labeled with σ, the run starts in state s, and either BQ accepts t, or it traverses the root again at some point, then being in state s′ . The set B̂(σ) can be computed a priori in 2ExpTime; considering that ΓS,l is of double-exponential size in the size of Q, it follows that the collection {B̂(σ)}σ∈ΓS,l can be computed in 2ExpTime. Now the input tree for DQ comes with labels from Π of the form (s, s′ ) in its leaves. These “types” amount to guesses of possible extensions of the input tree. Utilizing the sets B̂(σ), DQ thus explores the possible ways how the given input tree can be extended to a ΓS,l -labeled tree t′ that is accepted by BQ . 9. REFERENCES [1] S. Abiteboul, R. Hull, and V. Vianu. Foundations of Databases. Addison-Wesley, 1995. [2] A. Amarilli, M. Benedikt, P. Bourhis, and M. Vanden Boom. Query answering with transitive and linear-ordered data. 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HYPERBOLICITY AND NEAR HYPERBOLICITY OF QUADRATIC FORMS OVER FUNCTION FIELDS OF QUADRICS arXiv:1609.07100v2 [math.AC] 9 Oct 2017 STEPHEN SCULLY Abstract. Let p and q be anisotropic quadratic forms over a field F of characteristic 6= 2, let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 , and let k denote the dimension of the anisotropic part of q after scalar extension to the function field F (p) of p. We conjecture that dim(q) must lie within k of a multiple of 2s+1 . This can be viewed as a direct generalization of Hoffmann’s Separation theorem. Among other cases, we prove that the conjecture is true if k < 2s−1 . When k = 0, this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism W (F ) → W (F (p)) has dimension divisible by 2s+1 . 1. Introduction Many of the central problems in the algebraic theory of quadratic forms seem to demand an investigation of the behaviour of quadratic forms under scalar extension to function fields of quadrics. This was already apparent from the early beginnings of the subject, with function fields of Pfister quadrics being used in an essential way to prove the fundamental Arason-Pfister Hauptsatz of 1971 ([AP71]). Arason and Pfister’s work led to the foundational papers of Elman-Lam ([EL72]) and Knebusch ([Kne76],[Kne77]), after which properties of function fields of quadrics have been systematically studied and exploited in many of the major developments in the theory; notable examples include the proof of the Milnor conjectures ([Voe03],[OVV07]) and essentially all advances on Kaplansky’s problem concerning the possible u-invariants of fields ([Mer91],[Izh01],[Vis09]). An important problem in this context is the following: Let p and q be anisotropic quadratic forms of dimension ≥ 2 over a field F of characteristic 6= 2, and let F (p) denote the function field of the projective quadric {p = 0}. Under what circumstances does q become isotropic over F (p)? While this problem seems to be rather intractable in general, the extraction of partial information is already enough for non-trivial applications. As a result, it has been of great interest here to identity general constraints coming from the basic invariants of the forms p and q. A key breakthrough in this direction was made in [Hof95], where Hoffmann determined the constraints coming from the simplest invariants of all, namely, the dimensions of p and q. More precisely, Hoffmann’s “Separation theorem” asserts that if dim(q) ≤ 2s < dim(p) for some integer s, then q remains anisotropic over F (p), and this is optimal, in the sense that isotropy can occur in all other cases. An important generalization of Hoffmann’s theorem taking into account one further invariant of p was later given by Karpenko and Merkurjev in [KM03]. The present article is concerned with the following variant of the above question: To what extent can the form q become isotropic over F (p)? Formally, the extent to which a quadratic form is isotropic is measured by its Witt index, i.e., the maximal dimension of a totally isotropic subspace of the vector space on which it is defined. In analogy with 2010 Mathematics Subject Classification. 11E04, 14E05, 14C15. Key words and phrases. Quadratic forms, function fields of quadrics, hyperbolicity, near hyperbolicity, Witt kernels. 1 2 STEPHEN SCULLY Hoffmann’s result, one can ask the following: What constraints do the dimensions of p and q impose on the Witt index iW (qF (p) ) of q after extension to F (p)? Here the results of [Hof95] and [KM03] only go so far. In particular, they are essentially vacuous in the case where dim(p) ≤ dim(q)/2, whereas the problem itself is non-trivial in all dimensions. A similar criticism applies to the main result of [Scu16], which gives a certain refinement the result of Karpenko and Merkurjev. The purpose of this paper is to propose the following conjectural answer to the above problem: Conjecture 1.1. Suppose that p and q are anisotropic quadratic forms of dimension ≥ 2 over F . Let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 , and let k = dim(q) − 2iW (qF (p) ) (i.e., k is the dimension of the anisotropic part of qF (p) ). Then dim(q) = a2s+1 + ǫ for some non-negative integer a and some integer −k ≤ ǫ ≤ k. Again, it is not hard to see that this is optimal, to the extent that there can be no further gaps in the possible values of dim(q) determined by iW (qF (p) ) and dim(p) alone (see Example 1.5 below). In particular, the statement include’s Hoffmann’s theorem: Example 1.2. Suppose that dim(q) ≤ 2s , so that we have separation of dimensions by a power of 2. The reader will easily confirm that, in this case, it is only possible to express dim(q) in the suggested way if k = dim(q), i.e., if q remains anisotropic over F (p). Conjecture 1.1 is vacuously true if k ≥ 2s − 1, so we are interested here in the case where k ≤ 2s − 2. In other words, beyond the Separation theorem, we are looking at the situation in which iW (qF (p)) is “large” (which explains the title of the article). Loosely speaking, the conjecture asserts that the more isotropic q becomes over F (p), the closer its dimension should be to being divisible by 2s+1 . Our main result is the following: Theorem 1.3. Conjecture 1.1 is true when k < 2s−1 . In particular, the conjecture is true in the extreme case where k = 0, which translates as follows: Corollary 1.4. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . If q becomes hyperbolic over F (p), then dim(q) is divisible by 2s+1 . Put another way, Corollary 1.4 says the following: Let W (F (p)/F ) denote the kernel of the natural scalar extension homomorphism W (F ) → W (F (p)) on Witt groups. If q represents an element of W (F (p)/F ), then dim(q) is divisible by 2s+1 . A few low-dimensional cases aside (see, e.g., [Fit83]), this seems to have been unknown, even conjecturally.1. In fact, one can go rather further here, and show that all higher Witt indices of q, except possibly the last, are divisible by 2s+1 in this case (Theorem 3.4). This can be viewed as a precise generalization of a well-known theorem of Fitzgerald ([Fit81]) on the lowdimensional part of W (F (p)/F ) (see 3.B below). Although the proof of the Arason-Pfister Hauptsatz already drew serious attention to the problem of determining Witt kernels of function fields of quadrics, very little progress has been made over the last 40 years, with Fitzgerald’s theorem being among the few highlights. Our computations perhaps raise some new questions here. For example, must q be divisible by an (s + 1)-fold Pfister form in the situation of Corollary 1.4? Conjecturally, this question should have a positive 1Little seems to have been known beyond the fact that dim(q) ≥ 2s+1 in this case, which is an easy consequence of the Cassels-Pfister subform theorem (see [Kne76, Lem. 4.5]). 3 answer in the case where q has degree s + 1, meaning here that q does not represent an element in I s+2 (F ), the (s + 2)-nd power of the fundamental ideal of even-dimensional forms in W (F ). We provide here some further evidence for this claim, including a proof for the case where dim(p) ≤ 16 (Corollary 5.11). While it seems unlikely that such divisibility would hold in general, we are currently unable to provide any counterexample. Theorem 1.3 is a consequence of results of Vishik ([Vis07]) on a certain descent problem for algebraic cycles over function fields of quadrics. The proof is given in §4 below. Beyond this, we also prove Conjecture 1.1 in a number of additional cases; namely, the conjecture is true if (i) 2s+1 − 2 ≤ dim(p) ≤ 2s+1 , (ii) p is a Pfister neighbour, (iii) k ≤ 7, or (iv) dim(q) ≤ 2s+2 + 2s−1 . In the case of (iii) and (iv), we again prove a stronger result at the level of the splitting pattern of q (Theorem 5.3). In the course of treating case (iii), we show that (a stronger version of) our conjecture is implied by a long-standing conjecture of Kahn on the unramified Witt group of a quadric ([Kah95], see §5.B below). We expect that the statement of Conjecture 1.1 is also true in characteristic 2, even if we allow p and q to be degenerate.2 Unfortunately, the methods of the present article rely on the action of mod-2 Steenrod operations on Chow groups of smooth projective varieties, something which not available in characteristic 2 at the present time. Nevertheless, using entirely different methods, we can show that our main results (and more) extend to the case where char(F ) = 2 and q is quasilinear (i.e., diagonalizable). In view of the different nature of the arguments used, this work has been confined to a separate text ([Scu17]). We conclude this introduction with an example which shows that Conjecture 1.1 cannot be improved without stronger hypotheses or permitting the use of additional invariants: Example 1.5. Let p be an anisotropic quadratic form of dimension ≥ 2 over a field E of characteristic 6= 2. Suppose that p is a neighbour of an (s + 1)-fold Pfister form π for some non-negative integer s. Choose a non-negative integer a, a non-negative integer k < 2s , and let ǫ = k − 2l for some integer 0 ≤ l ≤ k/2. Note that we have ǫ + l ≥ 0. Let X = (X1 , . . . , Xa+ǫ+l ) be a set of a + ǫ + l algebraically independent variables over E, and let F = E(X). Let σ be a codimension-l subform of π, and consider the form q = πF ⊗ hX1 , . . . , Xa−1 i ⊥ Xa σF ⊥ hXa+1 , . . . , Xa+ǫ+l i. Since F/E is a purely transcendental extension, πE and σE are anisotropic. It then follows from [EKM08, Cor. 19.6] that q is anisotropic. Note that q has dimension a2s+1 + ǫ. We claim that (qF (p) )an has dimension k. Since l ≤ k/2 < 2s−1 , σ is a neighbour of π. Let τ be its complementary form of dimension l. Since π is a Pfister form, we then have qF (p) = − Xa τ ⊥ hXa+1 , . . . , Xa+ǫ+l i F (p) in W (F (p)). By Hoffmann’s Separation theorem, τE(p) is anisotropic, and since F (p)/E(p) is purely transcendental, [EKM08, Cor. 19.6] then implies that the right-hand side of the above equation is anisotropic. We therefore have that (qF (p) )an ≃ − Xa τ ⊥ hXa+1 , . . . , Xa+ǫ+l i F (p) . In particular, the dimension of (qF (p) )an is equal to dim(τ ) + ǫ + l = 2l + ǫ = k, as claimed. Since the integers k and ǫ in the statement of Conjecture 1.1 must have the same parity, this shows the optimality of the assertion. 2Of course, if q is degenerate, then one can no longer interpret the integer k as the dimension of the anisotropic part of qF (p) . 4 STEPHEN SCULLY Before proceeding, we make the following conventions: Conventions. All fields considered in this paper have characteristic 6= 2, and all quadratic forms are finite-dimensional and non-degenerate. By a variety, we mean an integral separated scheme of finite-type over a field. The field of 2 elements is denoted by F2 . 2. Some preliminary facts and terminology For the remainder of this text, F will denote an arbitrary field of characteristic 6= 2. We assume basic familiarity with the algebraic theory of quadratic forms, and the reader is referred to [EKM08] for all undefined terminology and notation. 2.A. The Knebusch splitting tower of a quadratic form. Let ϕ be a quadratic form over F . Recall the following construction of Knebusch ([Kne76]): Set F0 = F , ϕ0 = ϕan (the anisotropic the kernel of ϕ), and recursively define Fr = Fr−1 (ϕr−1 ) and ϕr = (ϕFr )an , with the understanding that the process stops at the first non-negative integer h(ϕ) for which ϕh(ϕ) is split (i.e., of dimension at most 1). The integer h(ϕ) is called the height of ϕ. By the splitting pattern of ϕ, we will mean the decreasing sequence of integers comprised of the dimensions of the ϕr . For each 0 ≤ r ≤ h(ϕ), we set jr (ϕ) to be the Witt index of ϕ after extension to the field Fr . If ϕ is not split and r ≥ 1, then the integer jr (ϕ) − jr−1 (ϕ) is called the r-th higher Witt index of ϕ, and is denoted ir (ϕ). By [Kne76, Thm. 5.8], the even-dimensional anisotropic forms of height 1 are precisely Nthose which are similar to a Pfister form, i.e., to a form of the shape hha1 , . . . , am ii := m i=1 h1, −ai i. In view of the inductive nature of Knebusch’s construction, it follows that if ϕ is non-split of even dimension, then ϕh(ϕ)−1 is similar to an n-fold Pfister form for some positive integer n. The latter integer is called the degree of ϕ, denoted deg(ϕ). If dim(ϕ) is odd (resp. if ϕ is hyperbolic), then we set deg(ϕ) = 0 (resp. deg(ϕ) = ∞). By a deep result due to Orlov, Vishik and Voevodsky ([OVV07, Thm. 4.3]), deg(ϕ) then coincides with the supremum of the set of all integers d for which ϕ represents an element in the d-th power I d (F ) of the fundamental ideal ideal in the Witt ring W (F ). 2.B. Stable birational equivalence of quadratic forms. Let ψ and ϕ be a pair of anisotropic quadratic forms over F of dimension ≥ 2. We say that ϕ and ψ are stably birationally equivalent if both ϕF (ψ) and ψF (ϕ) are isotropic. For example, an anisotropic Pfister neighbour is stably birationally equivalent to its ambient Pfister form ([EKM08, Rem. 23.11]). More generally, if ψ is similar to a subform of ϕ having codimension less than i1 (ϕ), then ϕ and ψ are stably birationally equivalent (see [EKM08, Lem. 74.1]). Following the previous example, we say in this case that ψ is a neighbour of ϕ. 2.C. Algebraic cycles on quadrics. Given a variety X over a field K, we will write Ch(X) for its total Chow group modulo 2. The group Ch(X) has the natural structure of an F2 -vector space. If L is a field extension of K, then an element of Ch(XL ) is said to be K-rational if it lies in the image of the natural restriction homomorphism Ch(X) → Ch(XL ). Suppose now that X is a smooth projective quadric of dimension n ≥ 1 defined by the vanishing of a quadratic form ϕ over our fixed field F , and let F be an algebraic closure of F . By [EKM08, Prop. 68.1], an F2 -basis of Ch(XF ) is given by the set {li , hi \| 0 ≤ i ≤ [n/2]}, where li (resp. hi ) is the class of an i-dimensional projective linear subspace (resp. a codimension-i hyperplane section) of XF . The following lemma is 5 a basic consequence of the well-know theorem of Springer asserting that odd-degree field extensions do not affect the Witt index of a quadratic form: Lemma 2.1 (see [EKM08, Cor. 72.6]). Let ϕ be a quadratic form of dimension ≥ 2 over F with associated (smooth) projective quadric X, and let 0 ≤ i ≤ [n/2]. If the element li ∈ Ch(XF ) is F -rational, then iW (ϕ) > i. 2.D. The motivic decomposition type and upper motive of a quadric. Given a field K, we will write Chow (K) for the additive category of Grothendieck-Chow motives over K with integral coefficients (as defined in [EKM08, Ch. XII], for example). If X is a smooth projective variety over K, then we will write M (X) for its motive considered as an object of Chow (K). In the special case where X = Spec(K), we simply write Z instead of M (X) (the dependency on the base field is suppressed from the notation). Given an integer i and an object M of Chow (K), we will write M {i} for the i-th Tate twist of M . In particular, Z{i} will denote the Tate motive with shift i in Chow (K). If L is a field extension of K, we write ML to denote the image of an object M in Chow (K) under the natural scalar extension functor Chow (K) → Chow (L). Suppose now that X is a smooth projective quadric of dimension n ≥ 1 defined by the vanishing of a quadratic form ϕ over our fixed field F , and let Λ(n) = {i \| 0 ≤ i ≤ ` [n/2]} {n − i \| 0 ≤ i ≤ [n/2]}. By a result of Vishik (see [Vis04, §§3,4]), any direct summand of M (X) decomposes (in an essentially unique way) into a finite direct sum of indecomposable objects in Chow (F ). If N is a non-zero direct summand of LM (X), then ∼ there exists a unique non-empty subset Λ(N ) ⊆ Λ(n) such that NF = λ∈Λ(N ) Z{λ} L (loc. cit.). In particular, we have Λ M (X) = Λ(n), i.e., M (XF ) ≃ λ∈Λ(n) Z{λ}. Now, if N1 and N2 are distinct indecomposable direct summands of M (X), then Λ(N1 ) and Λ(N2 ) are easily seen to be disjoint ([Vis04, Lem. 4.2]), and so the complete motivic decomposition of X determines in this way a partition of the set Λ(n). This partition is an important discrete invariant of ϕ known as its motivic decomposition type. For the reader’s convenience, we state here the most significant recent advance in the study of this invariant due to Vishik ([Vis11]). Recall first that the upper motive of X is defined as the unique indecomposable direct summand U (X) of M (X) such that 0 ∈ Λ(U (X)), i.e., such that Z is isomorphic to a direct summand of U (X)F . We then have: Theorem 2.2 (see [Vis11, Thm. 2.1]). Let ϕ be an anisotropic quadratic form of dimension ≥ 2 over F with associated (smooth) projective quadric X. Write dim(ϕ) − i1 (ϕ) = 2r1 − 2r2 + · · · + (−1)t−1 2rt for uniquely determined integers r1 > r2 > · · · > rt−1 > rt +1 ≥ 1, and, for each 1 ≤ c ≤ t, set t c−1 X X i−1 ri −1 (−1) 2 + ǫ(c) (−1)j−1 2rj , Dc = i=1 j=l where ǫ(c) = 1 if c is even and ǫ(c) = 0 if c is odd. Then, for any 1 ≤ c ≤ t, we have Dc ∈ Λ(U (X)), i.e., the Tate motive Z{Dc } is isomorphic to a direct summand of U (X)F . 3. Hyperbolicity of quadratic forms over function fields of quadrics The proof of Theorem 1.3 will be given in the next section. Taking this for granted, we give here the basic applications to the study of Witt kernels of function fields of quadrics. We start by recalling the statement of the Cassels-Pfister subform theorem: 6 STEPHEN SCULLY Proposition 3.1 (see [Kne76, Lem. 4.5]). Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F such that qF (p) is hyperbolic. Then, for any non-zero elements a ∈ D(p) and b ∈ D(q), there exists a quadratic form r over F such that q ≃ r ⊥ abp. In particular, dim(p) ≤ dim(q). Taking degrees into account, the dimension inequality can be refined as follows: Corollary 3.2. Let p and q be anisotropic quadratic forms over F such that qF (p) is hyperbolic, and let n = deg(q). Then dim(p) ≤ 2n . Indeed, this follows from the proposition by taking L to be the penultimate entry of the Knebusch splitting tower of q in the following lemma: Lemma 3.3. Let q and p be anisotropic quadratic forms of dimension ≥ 2 over F such that qF (p) is hyperbolic, and let L be a field extension of F . If qL is not hyperbolic, then pL is anisotropic, and (qL )an becomes hyperbolic over L(p). Proof. It is clear that (qL )an becomes hyperbolic over L(p). If pL is isotropic, then L(p) is a purely transcendental extension of L ([EKM08, Prop. 22.9]) and so qL must already be hyperbolic. 3.A. Main result. The statement of Corollary 3.2 can be reformulated as follows: If qF (p) is hyperbolic, then the last higher Witt index of q is equal to 2m for some m ≥ s. The new insight here is that the remaining higher Witt indices are all divisible by 2s+1 : Theorem 3.4. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . If qF (p) is hyperbolic, then ir (q) is divisible by 2s+1 for every 1 ≤ r < h(q). Proof. Since the statement is insensitive to multiplying p and q by non-zero scalars, we can assume that both forms represent 1. We argue by induction on h(q). If h(q) = 1, there is nothing to prove, so assume otherwise. Then, by Lemma 3.3, pF (q) is anisotropic and q1 becomes hyperbolic over F (q)(p). The induction hypothesis therefore implies that ir (q) = ir−1 (q1 ) is divisible by 2s+1 for all 1 < r < h(q). It remains to show that i1 (q) is also divisible by 2s+1 . We begin by showing that i1 (q) > 2s . Suppose, for the sake of contradiction, that this is not the case. Since both p and q represent 1, it follows from the Cassels-Pfister subform theorem (Proposition 3.1 above) that there exists a quadratic form r over F such that q ≃ r ⊥ p. (3.1) We recall the following observation due to Gentile and Shapiro: Lemma 3.5 ([GS78, Lem. 17]). Let L be a field extension of F . In the above situation, either (1) (qL )an ≃ (rL )an ⊥ (pL )an , or (2) (rL )an ≃ (qL )an ⊥ −(pL )an . Proof. Both (1) and (2) are true if we replace the isometry relation with Witt equivalence. In particular, (2) holds if either qL is hyperbolic or (qL )an ⊥ −(pL )an is anisotropic. If not, then pL is anisotropic and (qL )an becomes hyperbolic over L(p) (Lemma 3.3). At the same time, (qL )an and pL represent a common non-zero element of L, and so pL ⊂ (qL )an by the Cassels-Pfister subform theorem. By Witt cancellation, it then follows that (1) holds. We apply this lemma in the case where L = F (q). Note first that since dim(p) > 2s , r has codimension > 2s in q. Since i1 (q) ≤ 2s , it follows from [Vis04, Cor. 4.9] that rF (q) is anisotropic. As pF (q) is also anisotropic, the lemma therefore tells us that either 7 (1) q1 ≃ rF (q) ⊥ pF (q) , or (2) rF (q) ≃ q1 ⊥ −pF (q) . Now (1) cannot hold, since the right-hand side has dimension equal to dim(q), which is larger than dim(q1 ). On the other hand, (2) cannot hold either; indeed, if (2) holds, then dim(q) − dim(q1 ) 2 dim(r) + dim(p) − dim(r) − dim(p) = 2 = dim(p) > 2s , i1 (q) = which contradicts our assumption. We therefore conclude that i1 (q) > 2s . By Karpenko’s theorem on the possible values of the first higher Witt index ([Kar03]), it follows that dim(q) − i1 (q) is divisible by 2s+1 . Now the k = 0 case of Theorem 1.3 tells us that dim(q) is divisible by 2s+1 , and so the same is therefore true of i1 (q). In particular, we get the following more precise version of Corollary 1.4: Corollary 3.6. Let q and p be anisotropic quadratic forms of dimension ≥ 2 over F , let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 , and let n = deg(q). If qF (p) is hyperbolic, then n ≥ s + 1, and: (1) If n = s + 1, then dim(q) = 2s+1 m for some odd integer m. (2) If n ≥ s + 2, then dim(q) is divisible by 2s+2 . Remark 3.7. We remind the reader that, by [OVV07, Thm. 4.3], n = deg(q) coincides with the largest integer d such that [q] ∈ I d (F ). Proof. The inequality n ≥ s + 1 holds by Corollary 3.2. In view of the fact that h(q)−1 n dim(q) = 2 + X 2ir (q), (3.2) r=1 the remaining statements follow immediately from Theorem 3.4. Now, the following statement is well known (see [Vis04, Lem. 6.2] for a proof): Proposition 3.8. Let q be an anisotropic quadratic form over F which is divisible by an m-fold Pfister form for some integer m ≥ 1. Then ir (q) is divisible by 2m for all 1 ≤ r < h(q). Theorem 3.4 therefore raises the following question: Question 3.9. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . If qF (p) is hyperbolic, does it follow that q is divisible by an (s + 1)-fold Pfister form? While it perhaps seems unlikely that this would hold in general, we are unable to provide a counterexample. At the same time, we do expect that the question has a positive answer in the case where deg(q) takes its smallest possible value of s+1. In fact, the hyperbolicity of qF (p) should imply that p is a Pfister neighbour in this case; by [AP71], this would give that q is isometric to the product of an odd-dimensional form and the ambient Pfister form of p. This expectation goes back to Fitzgerald ([Fit83]), and we provide further evidence in its favour in §5 below. In particular, we show that if deg(q) = s + 1 and qF (p) is hyperbolic, then the upper motive of the quadric {p = 0} is a twisted form of a Rost motive (Corollary 5.11). By a result of Karpenko ([Kar01]), this implies that our claim holds if s ≤ 3. Hypothetically, it should imply for all s ([Vis04, Conj. 4.21]). 8 STEPHEN SCULLY 3.B. Fitzgerald’s theorem revisited. The proof of Corollary 3.6 immediately yields the following lower bound for the dimension of an element of the Witt kernel W (F (p)/F ): Corollary 3.10. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 , and let n = deg(q). If qF (p) is hyperbolic, then n ≥ s + 1 and dim(q) ≥ 2n + (h(q) − 1)2s+2 , with equality holding if and only if ir (q) = 2s+1 for all 1 ≤ r < h(q). In particular, we get a satisfying explanation of Fitzgerald’s theorem on the lowdimensional part of W (F (p)/F ): Corollary 3.11 (Fitzgerald, [Fit81, Thm. 1.6]). Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F such that qF (p) is hyperbolic, and let n = deg(q). If dim(p) > 12 (dim(q) − 2n ), then q is similar to a Pfister form. Proof. To say that q is similar to a Pfister form is equivalent to saying that h(q) = 1 ([Kne76, Thm. 5.8]). Let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . Then, by hypothesis, we have dim(q) < 2n + 2dim(p) ≤ 2n + 2s+2 . By Corollary 3.10, it follows that h(q) = 1, as desired. Remark 3.12. As noted by Fitzgerald (see [Fit81, §2]), the inequality of Corollary 3.11 cannot be weakened in general. Here we can clarify the situation further: Let p and q be as in the statement of Corollary 3.10. If we are in the border case where dim(p) = 1 n n s+2 and dim(p) = 2s+1 . 2 (dim(q)−2 ), then our result shows that h(q) = 2, dim(q) = 2 +2 By a theorem of Vishik ([Vis04, Thm. 6.4]), any degree-n form of height ≥ 2 has dimension ≥ 2n + 2n−1 . We therefore have n − 3 ≤ s ≤ n − 1. All three cases can occur. In fact, q should, in this situation, be the product of an (s+1)-fold Pfister form and either an Albert form (if s = n − 3), a 4-dimensional form of non-trivial discriminant (if s = n − 2) or a form of dimension 3 (if s = n − 1) – see [Kah96, §2.4]. This has been shown to be the case when n ≤ 3 (loc. cit.), and can also be proven for n = 4 using more recent results (though this is absent from the literature). The general case remains out of reach at present. 4. Proof of Theorem 1.3 In this section, we give a concrete geometric reformulation of Conjecture 1.1. Theorem 1.3 then follows as a consequence of the results of [Vis07]. We begin by making the following observation: Lemma 4.1. Let d, k and s be positive integers. Assume that d + k is even and that k < 2s . Then the following are equivalent: (1) d = a2s+1 + ǫ for some non-negative integer a and some integer −k ≤ ǫ ≤ k. s (2) d+k 2 = a2 + µ for some non-negative integer a and some integer 0 ≤ µ ≤ k. (3) The binomial coefficient d+k 2 l ( 2s − 2 is even for every k < l ≤ m, where m = 2s−1 if k ≥ 2s−1 if k < 2s−1 . 9 Proof. The equivalence of (1) and (2) is clear. To prove the equivalence of (2) and (3), we need to recall the basic for oddness of binomial coefficients. Let x be a nonP criterion i be its 2-adic expansion. We write π(x) for the (finite) negative integer, and let ∞ x 2 i=0 i set of all non-negative integers i for which xi is non-zero. We then have: Lemma 4.2 (see [Spr98, Lem. 3.4.2]). Let x and y be non-negative integers. Then the binomial coefficient xy is odd if and only if π(y) ⊂ π(x). Returning to the proof of Lemma 4.1, let us write d+k = a2s + µ 2 for some non-negative integer a and some integer 0 ≤ µ < 2s . If 0 ≤ l < 2s , then Lemma 4.2 implies that d+k µ 2 ≡ (mod 2). l l The equivalence (2) ⇔ (3) therefore amounts to the assertion that µ ≤ k if and only if µl is even for every k < l ≤ m. The “only if” implication here is trivial. Conversely, suppose that µl is even for every k < l ≤ m. Since µµ = 1, it follows that either µ ≤ k or µ > m. If µ > m, then (by the definition of m) we either have (i) µ = 2s − 1, or (ii) k < 2s−1 = m < µ. s By Lemma 4.2, 2 −1 is odd for every 0 ≤ l ≤ 2s−1 , so if (i) holds, then we must have l µ s that k = 2 − 1 = m. If (ii) holds, then 2s−1 is even by hypothesis. On the other hand, s−1 s the inequalities 2 < µ < 2 imply in this case that s − 1 ∈ π(µ). By Lemma 4.2, this µ is odd, giving us a contradiction. We conclude that µ ≤ k, and so the means that 2s−1 equivalence of (2) and (3) is proved. Lemma 4.1 reduces Conjecture 1.1 to the problem of checking the parity of certain binomial coefficients. The point now is that these coefficients can be extracted from the action of the Steenrod algebra on the mod-2 Chow ring of the quadric defined by q. For the remainder of this section, let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let Q be the projective quadric of equation q = 0. We fix an algebraic closure F (resp. F (p)) of F (resp. F (p)). For any j ≥ 0, let S j denote the j-th Steenrod operation of cohomological type on the mod-2 Chow ring of a smooth quasi-projective variety. Lemma 4.3. Fix integers 0 ≤ r ≤ [dim(Q)/2] and 0 ≤ j ≤ r. Then the element S j (lr ) ∈ Ch(QF ) is F -rational if and only if the binomial coefficient dim(q) − r − 1 dim(q) − r − 1 − j is even. Proof. By [EKM08, Cor. 78.5], we have dim(q) − r − 1 j S (lr ) = lr−j . j If S j (lr ) is F -rational, then the coefficient here must be even by Lemma 2.1 (because q is anisotropic). The converse is trivial. If Y is a variety over F , then an element α ∈ Ch(YF ) will be called F (p)-rational if αF (p) ∈ Ch(YF (p) ) is F (p)-rational. We recall the following result of Vishik (extended to odd characteristic in [Fin13]): 10 STEPHEN SCULLY Theorem 4.4 ([Vis07, Cor. 3.5], [Fin13, Thm. 1.1, Prop. 2.1]). Let Y be a smooth quasi-projective variety over F , and let α be an F (p)-rational element of Chm (YF ). If the integral Chow group CH(YF ) is torsion-free, then: (1) S j (α) is F -rational for every j > m − [(dim(p) − 1)/2]. (2) If pF (Y ) is not split, then S j (α) is also F -rational for j = m − [(dim(p) − 1)/2]. Applying Theorem 4.4 to the quadric Q, we get the following: Corollary 4.5. Let r = iW (qF (p) ) − 1, and consider the element lr ∈ Ch(QF ). Then: (1) S j (lr ) is F -rational for all j > dim(Q) − r − [(dim(p) − 1)/2]. (2) If pF (q) is not split, then S j (lr ) is also F -rational for j = dim(Q) − r − [(dim(p) − 1)/2] Proof. Since r = iW (qF (p) ) − 1, the element lr is F (p)-rational. As it is represented by a cycle of dimension r, the assertions follow immediately from Vishik’s theorem. Now, as in the statement of Conjecture 1.1, let k = dim(q) − 2iW (qF (p) ) (i.e., let k be the dimension of the anisotropic part of qF (p) ). Corollary 4.6. The binomial coefficient dim(q)+k 2 l is even for all k < l ≤ [(dim(p) − 1)/2] (and also for l = [(dim(p) − 1)/2] + 1 if pF (q) is not split). Proof. To simplify the notation, let n = [(dim(p) − 1)/2]. Let r = iW (qF (p) ) − 1, and let dim(Q) − r − n ≤ j ≤ r. If j = dim(Q) − r − n, assume additionally that pF (q) is not split. Then, by Corollary 4.5, S j (lr ) is F -rational. Since j ≤ r, Lemma 4.3 then implies that the binomial coefficient dim(q) − r − 1 dim(q) − r − 1 − j is even. It now only remains to observe that dim(q) + k , dim(q) − r − 1 = 2 and that dim(Q) − r − n ≤ j ≤ r if and only if k < dim(q) − r − 1 − j ≤ n + 1. This yields Theorem 1.3, as well as some other cases of Conjecture 1.1: Theorem 4.7. Conjecture 1.1 is true in the following cases: (1) The case where k < 2s−1 . (2) The case where 2s+1 − 2 ≤ dim(p) ≤ 2s+1 . (3) The case where p is a Pfister neighbour. Proof. Let p, q, s and k be as in the statement of Conjecture 1.1, and let ( 2s − 2 if k ≥ 2s−1 m= 2s−1 if k < 2s−1 . 11 By Lemma 4.1, the statement of the conjecture holds for the pair (p, q) if and only if dim(q)+k 2 l is even for every k < l ≤ m. Corollary 4.5 shows that this holds in both cases (1) and (2). Finally, suppose that p is a Pfister neighbour with ambient Pfister form π. Since p and π are stably birationally equivalent (see §2.B), we have iW (qF (π) ) = iW (qF (p) ). To prove that the statement of the conjecture holds in this case, we can therefore replace p by π and assume that p is a Pfister form. In particular, we can assume that dim(p) = 2s+1 . The validity of the conjecture in this case therefore follows from its validity in case (2). Further partial results towards Conjecture 1.1 will be given in the next section, but the case where k ≥ 2s−1 remains open in general. From the above arguments, it is easy to see that we have the following reformulation of the conjecture as a refinement of [Vis07, Thm. 3.1] for quadrics. The details are left to the reader: Proposition 4.8. The following are equivalent: (1) Conjecture 1.1 is true. (2) Let p be an anisotropic quadratic form of dimension ≥ 2 over F , and let s be the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . Let Y be a (possibly isotropic) smooth projective F -quadric. If α ∈ Chm (YF ) is F (p)-rational, then S j (α) is F -rational for all j ≥ m − (2s − 2). 5. Additional results In this section, we provide further justification for Conjecture 1.1 beyond Theorem 1.3. First, we prove that the conjecture holds under a certain assumption on the splitting pattern of q. We then explain how the remaining cases would follow if a related, but rather stronger conjecture of Kahn on the unramified Witt ring of a quadric were true. This approach reveals that, in analogy with Theorem 3.4, Conjecture 1.1 should extend to a statement at the level of the splitting pattern of q. We begin by stating this extension. 5.A. A refinement of the main conjecture. Let p and q be anisotropic quadratic forms of dimension ≥ 2 over F , and let k = dim(q) − 2iW (qF (p) ) (i.e., k is the dimension of the anisotropic part of qF (p)). By [Kne76, §5], k is equal to the dimension of q or one of its higher anisotropic kernels. We can therefore propose the following: Conjecture 5.1. In the above situation, let 0 ≤ l ≤ h(q) be such that dim(ql ) = k. Suppose that k < 2s , where s is the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . Then, for each 0 ≤ r < l, there exist integers ar , br ≥ 1 and −k ≤ ǫr ≤ k such that (1) dim(qr ) = 2s+1 ar + ǫr . (2) With one possible exception, either ir+1 (q) ≤ 12 (k+ǫr ) or ir+1 (q) = br 2s+1 +ǫr . The exception is where dim(qr ) = 2s+1 − k, in which case r = l − 1 and ir+1 (q) = 2s − k. Remark 5.2. If k = 0 (i.e., if qF (p) is hyperbolic), then l = h(q) − 1, and this is just Theorem 3.4. In general, the exceptional case of (2) corresponds to the case of Theorem 3.4 in which deg(q) = s + 1, so that ih(q) (q) is not divisible by 2s+1 (being equal to 2s ). Observe that, for k < 2s , Conjecture 1.1 correponds to assertion (1) of Conjecture 5.1 in the case where r = 0 (of course, Conjecture 1.1 is vacuously true if k ≥ 2s ). We will prove here the following: Theorem 5.3. Conjecture 5.1 (and hence Conjecture 1.1) is true if either 12 STEPHEN SCULLY (1) k ≤ 7. (2) dim(q) ≤ 2s+2 + k. The first case, in particular, implies that both our conjectures hold if dim(p) ≤ 16. The proof of Theorem 5.3 is given in §5.C below. Case (2) is treated using a motivic argument which in fact gives more (Theorem 5.8). Case (1), on the other hand, is treated by relating Conjecture 5.1 to Theorem 3.4 and the aforementioned conjecture of Kahn. We we now explain, this approach more clearly illustrates the philosophy underlying this paper. 5.B. A conjecture of Kahn. Let ϕ be an anisotropic quadratic form of dimension ≥ 2 over F . Recall that the unramified Witt ring of the projective quadric {ϕ = 0}, denoted Wnr (F (ϕ)), is defined as the subring of W (F (ϕ)) consisting of those classes of anisotropic quadratic forms which have no non-trivial (second) residues at the codimension-1 points of the projective quadric {ϕ = 0} (see [Kah95]). The image of the canonical scalar extension map W (F ) → W (F (ϕ)) trivially lies in Wnr (F (ϕ)), but equality need not hold in general (see, e.g., [KRS98, Cor. 10]). Nevertheless, Kahn has made the following strong conjecture concerning the “low-dimensional” part of Wnr (F (ϕ)): Conjecture 5.4 (Kahn, [Kah95, Conj. 1]). Let ϕ be an anisotropic quadratic form over F and let [τ ] ∈ Wnr (F (ϕ)). If dim(τ ) < 12 dim(ϕ), then τ is defined over F , i.e., there exists a quadratic form σ over F such that τ ≃ σF (ϕ) . If true, this has the following application to Conjecture 5.1: Proposition 5.5. Assume in the situation of Conjecture 5.1 that dim(ql−1 ) > 2k. Then, (qF (p) )an is defined over F provided that Conjecture 5.4 holds whenever dim(τ ) ≤ k and dim(ϕ) ≥ dim(ql−1 ). Proof. Let (Fr ) be the Knebusch splitting tower of q. The form ql is evidently unramified at the codimension-1 points of the quadric {ql−1 = 0}. Since dim(ql ) = k and dim(ql−1 ) > 2k, our hypothesis implies that ql ≃ τFl for some form τ over Fl−1 . If l > 1, then τ is unramified at the codimension-1 points of X = {ql−2 = 0}. Indeed, since ql ≃ τFl , Corollary 3.2 implies that [τ ⊥ −ql−1 ] ∈ I n (Fl−1 ), where 2n ≥ dim(ql−1 ) > 2k. Thus, if x ∈ X (1) , and π is a uniformizer for OX,x , we have (using [EKM08, Lem. 19.14]) ∂x,π ([τ ]) = ∂x,π ([τ ⊥ −ql−1 ]) ∈ ∂x,π I n (Fl−1 ) ⊆ I n−1 (Fl−2 (x)). But dim(τ ) = k, and since 2n−1 > k, the Arason-Pfister Hauptsatz ([AP71]) then implies that ∂x,π ([τ ]) = 0. Again, our hypothesis now gives that ql is defined over Fl−2 . Continuing in this way, we see that ql is defined over F , say ql ≃ σFl . We claim that (qF (p) )an ≃ σF (p) . Since dim(σ) = k = dim((qF (p) )an ), it suffices to show that (q ⊥ −σ)F (p) is hyperbolic. But (q ⊥ −σ)Fl (p) is hyperbolic, and so, by [EKM08, Lem. 7.15], it only remains to observe that Fl (p) is a purely transcendental extension of F (p). But if r < l, then the form qr becomes isotropic over Fr (p) by the very definition of k. Hence Fr+1 (p) = Fr (p)(qr ) is a purely transcendental extension of Fr (p), and so the claim follows by an easy induction. If one assumes another related conjecture of Vishik, then one can go a bit further here (see Remark 5.13 (2)). In any case, the philosophy underlying Conjecture 5.1 is now clear: If qF (p) is “nearly hyperbolic”, then its anisotropic part should typically be defined over F , and the necessary conclusions can be made using Theorem 3.4. In more details: Lemma 5.6. Conjecture 5.1 is true in the case where (qF (p) )an is defined over F . 13 Proof. By an easy induction on l, it suffices to show that if l > 0, then dim(q) and i1 (q) have the prescribed form. Assume that (qF (p) )an ≃ σF (p) for some form σ over F , and let η = (q ⊥ −σ)an . Since both q and σ are anisotropic, there exists an integer 0 ≤ λ ≤ dim(σ) = k, and codimension-λ subforms qe ⊂ q and σ e ⊂ σ such that η ≃ qe ⊥ −e σ. In particular, setting ǫ = 2λ − k, we have dim(q) = dim(η) + ǫ. Now, by construction, η becomes hyperbolic over F (p). Thus, by Corollary 3.6, we have dim(η) = 2s+1 a for some a ≥ 1 (and also deg(η) ≥ s+1). We therefore have dim(q) = 2s+1 a+ǫ. As for the assertion regarding i1 (q), we may assume that i1 (q) > (k + ǫ)/2 = λ, so that qe is a neighbour of q in the sense of §2.B above. This implies, in particular, that q and qe are stably birationally equivalent. If η is similar to an (s + 1)-fold Pfister form, then the Cassels-Pfister subform theorem implies that q is a subform of η, whence η ≃ q ⊥ −σ by Witt cancellation. Since dim(σ) = k < 2s , q is then a neighbour of η, and so i1 (q) = 2s − k by [Vis04, Cor. 4.9]. Otherwise, Theorem 3.4 implies that i1 (η) = 2s+1 b for some b ≥ 1. Again, since dim(e σ ) < 2s , it follows that qe is a neighbour of η. As q is a neighbour of qe, we see that q and η are stably birationally equivalent, and so i1 (q) = i1 (η) + dim(q) − dim(η) = 2s+1 b + ǫ by another application of [Vis04, Cor. 4.9]. This proves the lemma. 5.C. Proof of Theorem 5.3. For the remainder of the paper, we fix the following notation: • q and p are anisotropic quadratic forms of dimension ≥ 2 over F . • Q and P denote the (smooth) projective quadrics defined by the vanishing of q and p, respectively. • s is the unique non-negative integer such that 2s < dim(p) ≤ 2s+1 . • k := dim(q) − 2iW (qF (p) ) (i.e., k is the dimension of the anisotropic part of qF (p)). • l is the unique integer in [0, h(q)] such that dim(ql ) = k. • i := iW (qF (p) ). We now proceed with the proof of Theorem 5.3. We begin by showing that Conjecture 5.1 is true provided that 2s+1 − k lies in the splitting pattern of q. To make this more precise, we note the following: Lemma 5.7. Assume, in the situation of Conjecture 5.1, that l ≥ 1 (i.e, that qF (p) is isotropic). Then dim(ql−1 ) = 2N − k for some integer N ≥ s + 1. Proof. Let (Fr ) be the Knebusch splitting pattern of q. Since ql−1 becomes isotropic over Fl−1 (p), and since dim(p) > 2s , Hoffmann’s Separation theorem ([Hof95, Thm. 1]) implies that dim(ql−1 ) > 2s . In particular, dim(ql−1 ) = 2N − m for some integers N ≥ s + 1 and 0 ≤ m < 2N −1 . By definition, we then have i1 (ql−1 ) = 12 (2N − m − k), so that 1 dim(ql−1 ) − i1 (ql−1 ) = (2N − m + k) > 2N −2 (remember that k < 2s ). 2 By Karpenko’s theorem on the possible values of the first Witt index ([Kar03]), it follows that 2N −1 divides m − k. Since both both m and k are strictly less than 2N −1 , this implies that m = k, and so dim(ql−1 ) = 2N − k. Our result is now that Conjecture 5.1 holds in the case where N = s + 1: Theorem 5.8. Assume that l ≥ 1 (i.e., that qF (p) is isotropic). If dim(ql−1 ) = 2s+1 − k, then Conjecture 5.1 is true. Proof. The proof is similar to that of [Scu16, Thm. 4.1], and begins with the following observation regarding the motivic decomposition of the quadric Q: 14 STEPHEN SCULLY Lemma 5.9. In the situation of Theorem 5.8, U (P ){i − 1} is isomorphic to a direct summand of M (Q). Proof. By [Vis04, Thm. 4.15], it suffices to check that for every field extension L of F , we have iW (pL ) > 0 ⇔ iW (qL ) ≥ i. The left to right implication is immediate; indeed, if pL is isotropic, then L(p) is a purely transcendental extension of L, and so iW (qL ) = iW (qL(p) ) ≥ iW (qF (p) ) = i. For the other implication, assume that iW (qL ) ≥ i. If (Fr ) denotes the Knebusch splitting tower of q, then it follows from the very definition of l that the free composite Fl · L is a purely transcendental extension of L (compare the end of the proof of Proposition 5.5). To show that iW (pL ) > 0, it therefore suffices to show that p becomes isotropic over Fl . Note first that, by hypothesis, we have 1 (dim(ql−1 ) − dim(ql )) 2 1 = (2s+1 − k) − (2s+1 − 2k) = 2s . 2 On the other hand, the form ql−1 becomes isotropic over Fl−1 (p). Since dim(p) > 2s , and since Fl = Fl−1 (ql−1 ), the desired assertion thus follows from Izhboldin’s “strong incompressibility” theorem ([Izh00, Thm. 0.2]). dim(ql−1 ) − i1 (ql−1 ) = dim(ql−1 ) − The proof of Theorem 5.8 now proceeds by induction on l, with the case where l = 1 being trivial. Indeed, if l = 1, then dim(q) = 2s+1 − k and i1 (q) = (dim(q) − k)/2 = 2s − k. Assume now that l ≥ 2 (in particular, h(q) ≥ 2). Applying the induction hypothesis to the pair (q1 , pF (q) ), we immediately get that the integers dim(qr ) and ir+1 (q) have the prescribed form for all 1 ≤ r < l. In particular, we have dim(q1 ) = 2s+1 a1 + ǫ1 for some a1 ≥ 1 and −k ≤ ǫ1 ≤ k. Since dim(q) = dim(q1 ) + 2i1 (q), we then have k − ǫ1 1 s . (5.1) i = (dim(q) − k) = 2 a1 + i1 (q) − 2 2 It still remains show that dim(q) and i1 (q) are as claimed. We separate two cases: Case 1. i1 (q) ≤ (k − ǫ1 )/2. In this case, the assertions are clear. Indeed, we have 2s+1 a1 + ǫ1 = dim(q1 ) < dim(q) = dim(q1 ) + 2i1 (q) ≤ dim(q1 ) + (k − ǫ1 ) = 2s+1 a1 + k. Since −k ≤ ǫ1 ≤ k, this shows that dim(q) = 2i+1 a1 + ǫ0 for some −k < ǫ0 ≤ k, and we then have ǫ0 − ǫ1 k + ǫ0 dim(q) − dim(q1 ) = ≤ . i1 (q) = 2 2 2 Case 2. i1 (q) > (k − ǫ1 )/2. In this case, let u denote the smallest non-negative integer such that 2u ≥ i1 (q). By Karpenko’s theorem on the possible values of the first Witt index ([Kar03]), the integer dim(q) − i1 (q) is divisible by 2u . Since dim(q) = dim(q1 ) + 2i1 (q), it follows that 2s+1 a1 + ǫ1 + i1 (q) ≡ 0 (mod 2u ). (5.2) We claim that u > i. Suppose that this is not the case. Then (5.2) implies that i1 (q) = 2u µ − ǫ1 for some integer µ. Now, by hypothesis (and the fact that ǫ1 ≥ −k), we have i1 (q) + ǫ1 > k + ǫ1 k − ǫ1 + ǫ1 = ≥ 0, 2 2 15 and so µ > 0. At the same time, we have 2u µ ≤ 2s . Indeed, since 2u ≥ i1 (q), the inequality 2u µ > 2s would imply that ǫ1 = 2u µ − i1 (q) ≥ 2s , which is not the case (since ǫ1 ≤ k < 2s ). The situation is thus as follows: We have dim(q) − i1 (q) = dim(q1 ) + i1 (q) = 2s+1 a1 + 2u µ, 2u µ 2s . (5.3) 2s+1 a1 where a1 ≥ 1 and 0 < ≤ Now the integer can be written in the form r r r r w 1 2 w−1 2 −2 +···+2 − 2 for some integers r1 > r2 > · · · > rw ≥ s + 1, while 2u µ can be r r w+1 w+2 written as 2 −2 + · · · + (−1)t−1 2rt for unique integers s ≥ rw+1 > rw+2 > · · · > rt−1 > rt + 1 ≥ 1. Equation 5.3 can therefore be re-written as dim(q) − i1 (q) = (2r1 − 2r2 + · · · + 2rw−1 − 2rw ) + 2rw+1 − 2rw+2 + · · · + (−1)t−1 2rt . \| {z } \| {z } 2s+1 a1 2u µ 2u µ 2s Unless = and rw = s + 1, this is precisely the presentation of dim(q) − i1 (q) as an alternating sum of 2-powers appearing in the statement of Vishik’s Theorem 2.2. If 2u µ = 2s and rw = s + 1, then the needed presentation is dim(q) − i1 (q) = (2r1 − 2r2 + · · · + 2rw−1 ) −2s . {z } \| 2s+1 (a1 +1) Either way, see that the Tate motive Z{2s a1 } is isomorphic to a direct summand of U (Q)F . Indeed, this follows by applying Vishik’s Theorem with c = w + 1 in the first case, and with c = w in the second. Let j = i1 (q) − 1 − (k − ǫ1 )/2. Since i1 (q) > (k − ǫ1 )/2, we then have 0 ≤ j < i1 (q). Thus, by [Vis11, Cor. 3.10], U (Q){j} is isomorphic to a direct summand of M (Q). By the preceding discussion, Z{2s a1 + j} is isomorphic to a direct summand of U (Q){j}F . But 2s a1 + j = 2s a1 + i1 (q) − (k − ǫ1 )/2 − 1 = i − 1 by equation (5.1). In view of Lemma 5.9, we thus see that there are two indecomposable direct summands of M (Q) containing the Tate motive Z{i − 1} in their respective decompositions over F , namely, U (Q){j} and U (P ){i − 1}. As result, we must have U (Q){j} ∼ = U (P ){i − 1} (see §2.D above). In particular, we have j = i − 1. Since j < i1 (q), and since i > 0, this implies that i = i1 (q). In other words, it implies that l = 1, which provides us with the needed contradiction to our supposition. We can therefore conclude that u > s, i.e., that i1 (q) > 2s . By (5.2), it follows that i1 (q) = 2s+1 b0 − ǫ1 for some positive integer b0 . Moreover, we then have dim(q) = dim(q1 ) + 2i1 (q) = 2s+1 (a1 + 2b0 ) − ǫ1 . Since −k ≤ ǫ1 ≤ k, this proves the theorem. Let us note that the proof of Theorem 5.8 allows us to extract more information. Recall that a direct summand N in the motive of a smooth projective F -quadric is said to be binary if NF is isomorphic to a direct sum of two Tate motives. We then have: Proposition 5.10. If k < 2s−1 + 2s−2 and dim(ql−1 ) = 2s+1 − k, then the upper motive U (P ) of the quadric P is binary. Proof. By Lemma 5.9, U (P ){i − 1} is isomorphic to a direct summand of M (Q). Since i − 1 = jl (q) − 1, it follows from [Vis11, Thm. 4.13] that N := U (P ){jl−1 (q)} is also isomorphic to a direct summand of M (Q). By an argument identical to that given in [Vis11, Proof of Thm. 7.7], it follows that N , and hence U (P ), is binary provided that ir (q) < il (q) for all r > l. But il (q) = (dim(ql−1 ) − k)/2 = 2s − k > 2s−2 by hypothesis, and 16 STEPHEN SCULLY so it only remains to note that ir (q) ≤ 2s−2 for all r > l. Indeed, if dim(qr−1 ) ≤ 2s−1 , then this is trivial; otherwise, our hypothesis again implies that 2s−1 < dim(qr−1 ) < 2s−1 + 2s−2 and so the claim follows from Hoffmann’s Separation theorem (see [Hof95, Cor. 1]). As conjectured by Vishik (cf. [Vis11, Conj. 4.21]), the upper motive of an anisotropic projective quadric over a field of characteristic 6= 2 should be binary only when its underlying quadratic form is a Pfister neighbour (the converse being a well-known result of Rost ([Ros90, Prop. 4])). Significant evidence for the validity of this assertion has been given in [IV00, §6]. Moreover, in [Kar01], Karpenko showed that Vishik’s conjecture holds for forms of dimension ≤ 16. Applying this to the special case where k = 0 and deg(q) = s+1, we get the following result promised in §3.A above: Corollary 5.11. If qF (p) is hyperbolic and deg(q) = s + 1, then: (1) The upper motive U (P ) of the quadric P is binary. (2) If s ≤ 3 (i.e., dim(p) ≤ 16), there exists an (s + 1)-fold Pfister form π and an odd-dimensional form r such that q ≃ π ⊗ r. Moreover, p is a neighbour of π. Proof. The first statement is the k = 0 case of Proposition 5.10. The second statement then follows from the aforementioned result of Karpenko together with [EKM08, Cor. 23.6]. More precisely, Karpenko showed in [Kar01] that if the upper motive of P is binary, and dim(p) ∈ {3, 5, 9}, then p is a Pfister neighbour. But this implies the same assertion in all dimensions ≤ 16 by [IV00, Thm. 6.1] and [Vis04, Cor. 3.9, Cor. 4.7]. We can now give the proof of Theorem 5.3: Proof of Theorem 5.3. We may assume that l ≥ 1, since otherwise there is nothing to prove. If dim(ql−1 ) = 2s+1 − k, then Conjecture 5.1 holds by Theorem 5.8. Thus, by Lemma 5.7, we can assume that dim(ql−1 ) ≥ 2s+2 − k. Now, if dim(q) ≤ 2s+2 + k, then we obviously have dim(ql−1 ) = 2s+2 − k, and the assertions of Conjecture 5.1 hold trivially. This takes care of case (2) of the theorem. It now remains to treat the case where k ≤ 7. As dim(ql−1 ) ≥ 2s+2 − k, Proposition 5.5 and Lemma 5.6 show that it suffices to know that Kahn’s Conjecture 5.4 holds in the following cases: (i) dim(τ ) ∈ {0, 1}, dim(ϕ) ≥ 7. (ii) dim(τ ) ∈ {2, 3}, dim(ϕ) ≥ 13. (iii) dim(τ ) ∈ {4, 5, 6, 7}, dim(ϕ) ≥ 25. But all these cases of Kahn’s conjecture are already known to be true: for the cases where dim(τ ) ≤ 5, see [Kah95, Thm. 2]; for the case where dim(τ ) = 6, see [Lag99, Thm. principale], and for the case where dim(τ ) = 7, see [IV00, Thm. 3.9]. Corollary 5.12. Conjecture 1.1 holds if dim(q) ≤ 2s+1 + 2s−1 . Proof. If k ≥ 2s−1 , apply 5.3; otherwise, apply Theorem 1.3. Remarks 5.13. (1) Laghribi ([Lag99, Théorème principale]) has shown that if dim(ϕ) > 16, then Conjecture 5.4 also holds for some special classes of 8, 9 and 10-dimensional forms τ . This permits to extend Theorem 5.3 to the case where k ∈ {8, 9, 10} under some additional assumptions on q. We refrain from going into the details here. (2) Conjecturally, the form (qF (p) )an should be defined over F in the situation of Conjecture 5.1 provided that either (i) k < 2s−1 + 2s−2 or (ii) l ≥ 1 and dim(ql−1 ) 6= 2s+1 − k. Indeed, that the second condition should be enough is implicit in our proof of Theorem 5.3. As for the first condition, we may additionally assume in this case that l ≥ 1 and dim(ql−1 ) = 2s+1 − k. By Proposition 5.10, this ensures that the upper motive of P 17 is binary. According to [Vis04, Conj. 4.21], this should in turn imply that p is a Pfister neighbour. By replacing p by its ambient Pfister form, we could then assume that dim(p) = 2s+1 > 2k, and so the claim would follow directly from Conjecture 5.4. Acknowledgements. I would like to thank Alexander Vishik for helpful comments. 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[Vis09] A. Vishik. Fields of u-invariant 2r + 1. Algebra, arithmetic and geometry: in honor of Yu. I. Manin. Vol. II, 661–685, Progr. Math., 270, Birkhäuser Boston, Inc., Boston, MA, 2009. [Vis11] A. Vishik. Excellent connections in the motives of quadrics. Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 1, 183–195. [Voe03] V. Voevodsky. Motivic cohomology with Z/2-coefficients. Publ. Math. Inst. Hautes Études Sci. No. 98 (2003), 59–104. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada E-mail address: stephenjscully@gmail.com	0
Logical Methods in Computer Science Vol. 8 (1:29) 2012, pp. 1–36 www.lmcs-online.org Submitted Published Oct. 4, 2011 Mar. 26, 2012 ON IRRELEVANCE AND ALGORITHMIC EQUALITY IN PREDICATIVE TYPE THEORY ∗ ANDREAS ABEL a AND GABRIEL SCHERER b a Department of Computer Science, Ludwig-Maximilians-University Munich e-mail address: andreas.abel@ifi.lmu.de b Gallium team, INRIA Paris-Rocquencourt e-mail address: gabriel.scherer@gmail.com Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To separate static and dynamic code, several static analyses and type systems have been put forward. We consider Pfenning’s type theory with irrelevant quantification which is compatible with a type-based notion of equality that respects η-laws. We extend Pfenning’s theory to universes and large eliminations and develop its meta-theory. Subject reduction, normalization and consistency are obtained by a Kripke model over the typed equality judgement. Finally, a type-directed equality algorithm is described whose completeness is proven by a second Kripke model. 1. Introduction and Related Work Dependently typed programming languages such as Agda [BDN09], Coq [INR10], and Epigram [MM04] allow the programmer to express in one language programs, their types, rich invariants, and even proofs of these invariants. Besides code executed at run-time, dependently typed programs contain much code needed only to please the type checker, which is at the same time the verifier of the proofs woven into the program. Program extraction takes type-checked terms and discards parts that are irrelevant for execution. Augustsson’s dependently typed functional language Cayenne [Aug99] erases types using a universe-based analysis. Coq’s extraction procedure has been designed by Paulin-Mohring and Werner [PMW93] and Letouzey [Let02] and discards not only types but also proofs. The erasure rests on Coq’s universe-based separation between propositional (Prop) and computational parts (Set/Type). The rigid Prop/Set distinction has the drawback of code duplication: A structure which is sometimes used statically and sometimes dynamically needs to be coded twice, once in Prop and once in Set. An alternative to the fixed Prop/Set-distinction is to let the usage context decide whether a term is a proof or a program. Besides whole-program analyses such as data 1998 ACM Subject Classification: F.4.1. Key words and phrases: dependent types, proof irrelevance, typed algorithm equality, logical relation, universal Kripke model. ∗ Revision and extension of FoSSaCS 2011 conference publication. l LOGICAL METHODS IN COMPUTER SCIENCE c DOI:10.2168/LMCS-8 (1:29) 2012 CC A. Abel and G. Scherer Creative Commons 2 A. ABEL AND G. SCHERER flow, some type-based analyses have been put forward. One of them is Pfenning’s modal type theory of Intensionality, Extensionality, and Proof Irrelevance [Pfe01], later pursued by Reed [Ree03], which introduces functions with irrelevant arguments that play the role of proofs.1 Not only can these arguments be erased during extraction, they can also be disregarded in type conversion tests during type checking. This relieves the user of unnecessary proof burden (proving that two proofs are equal). Furthermore, proofs can not only be discarded during program extraction but directly after type checking, since they will never be looked at again during type checking subsequent definitions. In principle, we have to distinguish “post mortem” program extraction, let us call it external erasure, and proof disposal during type checking, let us call it internal erasure. External erasure deals with closed expressions, programs, whereas internal erasure deals with open expressions that can have free variables. Such free variables might be assumed proofs of (possibly false) equations and block type casts, or (possibly false) proofs of wellfoundedness and prevent recursive functions from unfolding indefinitely. For type checking to not go wrong or loop, those proofs can only be externally erased, thus, the Prop/Set distinction is not for internal erasure. In Pfenning’s type theory, proofs can never block computations even in open expressions (other than computations on proofs), thus, internal erasure is sound. Miquel’s Implicit Calculus of Constructions (ICC) [Miq01a] goes further than Pfenning and considers also parametric arguments as irrelevant. These are arguments which are irrelevant for function execution but relevant during type conversion checking. Such arguments may only be erased in function application but not in the associated type instantiation. Barras and Bernardo [BB08] and Mishra-Linger and Sheard [MLS08] have built decidable type systems on top of ICC, but both have not fully integrated inductive types and types defined by recursion (large eliminations). Barras and Bernardo, as Miquel, have inductive types only in the form of their impredicative encodings, Mishra-Linger [ML08] gives introduction and elimination principles for inductive types by example, but does not show normalization or consistency. While Pfenning’s type theory uses typed equality, ICC and its successors interpret typed expressions as untyped λ-terms up to untyped equality. In our experience, the implicit quantification of ICC, which allows irrelevant function arguments to appear unrestricted in the codomain type of the function, is incompatible with type-directed equality. Examples are given in Section 2.3. Therefore, we have chosen to scale Pfenning’s notion of proof irrelevance up to inductive types, and integrated it into Agda. In this article, we start with the “extensionality and proof irrelevance” fragment of Pfenning’s type theory in Reed’s version [Ree02, Ree03]. We extend it by a hierarchy of predicative universes, yielding Irrelevant Intensional Type Theory IITT (Sec. 2). After specifying a type-directed equality algorithm (Sec. 3), we construct a Kripke model for IITT (Sec. 4). It allows us to prove normalization, subject reduction, and consistency, in one go (Sec. 5). A second Kripke logical relation yields correctness of algorithmic equality and decidability of IITT (Sec. 6). Our models are ready for data types, large eliminations, types with extensionality principles, and internal erasure (Sec. 7). 1 Awodey and Bauer [AB04] give a categorical treatment of proof irrelevance which is very similar to Pfenning and Reed’s. However, they work in the setting of Extensional Type Theory with undecidable type checking, we could not directly use their results for this work. IRRELEVANCE IN TYPE THEORY 3 Contribution and Related Work. We consider the design of our meta-theoretic argument as technical novelty, although it heavily relies on previous works to which we owe our inspiration. Allen [All87] describes a logical relation for Martin-Löf type theory with a countable universe hierarchy. The seminal work of Coquand [Coq91] describes an untyped equality check for the Logical Framework and justifies it by a logical relation for dependent types that establishes subject reduction, normalization, completeness of algorithmic equality, and injectivity of function types in one go. However, his approach cannot be easily extended to a typed algorithmic equality, due to problems with transitivity. Goguen introduces Typed Operational Semantics [Gog94] to construct a Kripke logical relation that simultaneously proves normalization, subject reduction, and confluence for a variant of the Calculus of Inductive Constructions. From his results one can derive an equality check based on reduction to normal form. Goguen also shows how to derive syntactic properties, such as closure of typing and equality under substitution, by a Kripke-logical relation [Gog00]. Harper and Pfenning [HP05] popularize a type-directed equality check for the Logical Framework that scales to extensionality for unit types. They prove completeness of algorithmic equality by a Kripke model on simple types which are obtained by erasure from the dependent types. Erasure is necessary since algorithmic equality cannot be shown transitive before it is proven sound; yet soundness hinges on subject reduction which rests on function type injectivity which in turn is obtained from completeness of algorithmic equality—a vicious cycle. While erasure breaks the cycle, it also prevents types to be defined by recursion on values (so-called large eliminations), a common feature of proof assistants like Agda, Coq, and Epigram. Normalization by evaluation (NbE) has been successfully used to obtain a type-directed equality check based on evaluation in the context of dependent types with large eliminations [ACD07]. In previous work [ACD08], the first author applied NbE to justify a variant of Harper and Pfenning’s algorithmic equality without erasure. However, the meta-theoretic argument is long-winded, and there is an essential gap in the proof of transitivity of the Kripke logical relation. In this work, we explore a novel approach to justify type-directed algorithmic equality for dependent types with predicative universes. First, we show its soundness by a Kripke model built on top of definitional equality. The Kripke logical relation yields normalization, subject reduction, and type constructor injectivity, which also imply logical consistency of IITT. Further, it proves syntactic properties such as closure under substitution, following Goguen’s lead [Gog00]. The semantic proof of such syntactic properties relieves us from the deep lemma dependencies and abundant traps of syntactic meta-theory of dependent types [HP05, AC07]. Soundness of algorithmic equality entails transitivity (which is the stumbling stone), paving the way to show completeness of algorithmic equality by a second Kripke logical relation, much in the spirit of Coquand [Coq91] and Harper and Pfenning [HP05]. This article is a revised and extended version of paper Irrelevance in Type Theory with a Heterogeneous Equality Judgement presented at the conference FoSSaCS 2011 [Abe11]. Unfortunately, the conference version has inherited the above-mentioned gap [ACD08] in the proof of transitivity of the Kripke logical relation. This is fixed in the present article by an auxiliary Kripke model (Section 4). Further, we have dropped the heterogeneous approach to equality in favor of a standard homogeneous one. Heterogeneous equality is not necessary for the style of irrelevance we are embracing here. 4 A. ABEL AND G. SCHERER 2. Irrelevant Intensional Type Theory In this section, we present Irrelevant Intensional Type Theory IITT which features two of Pfenning’s function spaces [Pfe01], the ordinary “extensional” (x : U ) → T and the proof irrelevant (x÷U ) → T . The main idea is that the argument of a (x÷U ) → T function is counted as a proof and can neither be returned nor eliminated on, it can only be passed as argument to another proof irrelevant function or data constructor. Technically, this is realized by annotating variables as relevant, x : U , or irrelevant, x÷U , in the typing context, to confine occurrences of irrelevant variables to irrelevant arguments. Expression and context syntax. We distinguish between relevant (t : u or simply t u) and irrelevant application (t ÷ u). Accordingly, we have relevant (λx : U. T ) and irrelevant abstraction (λx÷U. T ). Our choice of typed abstraction is not fundamental; a bidirectional type-checking algorithm [Coq96] can reconstruct type and relevance annotations at abstractions and applications. Var ∋ x, y, X, Y Sort ∋ s ::= Setk (k ∈ N) Ann ∋ ⋆ ::= ÷ \| : universes annotation: irrelevant, relevant s,s′ Exp ∋ t, u, T, U ::= s \| (x⋆U ) → T sort, (ir)relevant function type ⋆ \| x \| λx⋆U. t \| t u lambda-calculus Cxt ∋ Γ, ∆ ::= ⋄ \| Γ. x⋆T empty, (ir)relevant extension Expressions are considered modulo α-equality, we write t ≡ t′ when we want to stress that t and t′ identical (up to α). Similarly, we consider variables bound in a context to be distinct, and when opening a term binder we will implicitly use α-conversion to add a fresh variable in the context. For technical reasons, namely, to prove transitivity (Lemma 4.3) of the Kripke logical s,s′ relation in Section 4, we explicitly annotate function types (x⋆U ) → T with the sorts s of domain U and s′ of codomain T . We may omit the annotation if it is inessential or determined by the context of discourse. In case T does not mention x, we may write U → T for (x : U ) → T . Sorts. IITT is a pure type system (PTS) with infinite hierarchy of predicative universes Set0 : Set1 : .... The universes are not cumulative. We have the PTS axioms Axiom = {(Seti , Seti+1 ) \| i ∈ N} and the rules Rule = {(Seti , Setj , Setmax(i,j) ) \| i, j ∈ N}. As is customary, we will write the side condition (s, s′ ) ∈ Axiom just as (s, s′ ) and likewise (s1 , s2 , s3 ) ∈ Rule just as (s1 , s2 , s3 ). IITT is a full and functional PTS, which means that for all s1 , s2 there is exactly one s3 such that (s1 , s2 , s3 ). There is no subtyping, so that types—and thus, sorts—are unique up to equality. A proof of sort unicity might relieve us from the sort annotation in function types, however, we obtain sort discrimination too late in our technical development (Lemma 5.10). Substitutions. Substitutions σ are maps from variables to expressions. We require that the domain dom(σ) = {x \| σ(x) 6= x} is finite. We write id for the identity substitution and [u/x] for the singleton substitution σ such that σ(x) := u and σ(y) := y for y 6= x. Substitution extension (σ, u/x) is formally defined as σ ⊎ [u/x]. Capture avoiding parallel substitution of σ in t is written as juxtaposition tσ. IRRELEVANCE IN TYPE THEORY 5 Contexts. Contexts Γ feature two kinds of bindings, relevant (x : U ) and irrelevant (x ÷ U ) ones. The intuition, implemented by the typing rules below, is that only relevant variables are in scope in an expression. Resurrection Γ÷ turns all irrelevant bindings (x ÷ T ) into the corresponding relevant ones (x : T ) [Pfe01]. It is the tool to make irrelevant variables, also called proof variables, available in proofs. The generalization Γ⋆ shall mean Γ÷ if ⋆ = ÷, and just Γ otherwise. We write Γ.∆ for the concatenation of Γ and ∆; herein, we suppose dom(Γ) ∩ dom(∆) = ∅. Primitive judgements of IITT. The following three judgements are mutually inductively defined by the rules given below and in Figure 1. ⊢Γ Γ ⊢t:T Γ ⊢ t = t′ : T Context Γ is well-formed. In context Γ, expression t has type T . In context Γ, t and t′ are equal expressions of type T . Derived judgements. To simplify notation, we introduce the following four abbreviations: Γ Γ Γ Γ ⊢t÷T ⊢ t = t′ ÷ T ⊢T ⊢ T = T′ iff iff iff iff Γ÷ ⊢ t : T, Γ ⊢ t ÷ T and Γ ⊢ t′ ÷ T, Γ ⊢ T : s for some s, Γ ⊢ T = T ′ : s for some s. Γ ⊢ t ⋆ T may mean Γ ⊢ t : T or Γ ⊢ t ÷ T , depending on the value of placeholder ⋆; same for Γ ⊢ t = t′ ⋆ T . We sometimes write Γ ⊢ t, t′ ⋆ T to abbreviate the conjunction of Γ ⊢ t ⋆ T and Γ ⊢ t′ ⋆ T . The notation Γ ⊢ T, T ′ is to be understood similarly. 2.1. Rules. Our rules for well-typed terms Γ ⊢ t : T extend Reed’s rules [Ree02] to PTS style. There are only 6 rules; we shall introduce them one-by-one. Variable rule. Only relevant variables can be extracted from the context. ⊢Γ (x : U ) ∈ Γ Γ ⊢x:U There is no variable rule for irrelevant bindings (x ÷ U ) ∈ Γ, in particular, the judgement x÷U ⊢ x : U is not derivable. This essentially forbids proofs to appear in relevant positions. Abstraction rule. Relevant and irrelevant functions are introduced analogously. Γ. x⋆U ⊢ t : T s,s′ Γ ⊢ (x⋆U ) → T s,s′ Γ ⊢ λx⋆U. t : (x⋆U ) → T To check a relevant function λx : U. t, we introduce a relevant binding x : U into the context and continue checking the function body t. In case of an irrelevant function λx÷U. t, we proceed with an irrelevant binding x ÷ U . This means that an irrelevant function cannot computationally depend on its argument—it is essentially a constant function. In particular, λx÷U. x is never well-typed. s,s′ As a side condition, we also need to check that the introduced function type (x⋆U ) → T is well-sorted; the rule is given below. 6 A. ABEL AND G. SCHERER Application rule. Γ ⊢ t : (x⋆U ) → T Γ ⊢u⋆U ⋆ Γ ⊢ t u : T [u/x] This rule uses our overloaded notations for bindings ⋆, that can be specialized into two different instances for relevant and irrelevant applications. For relevant functions, we get the ordinary dependently-typed application rule: Γ ⊢ t : (x : U ) → T Γ ⊢u:U Γ ⊢ t u : T [u/x] When applying an irrelevant function, we resurrect the context before checking the function argument. Γ ⊢ t : (x÷U ) → T Γ÷ ⊢ u : U Γ ⊢ t ÷ u : T [u/x] This means that irrelevant variables become relevant and can be used in u. The intuition is that the application t ÷ u does not computationally depend on u, thus, u may refer to any variable, even the “forbidden ones”. One may think of u as a proof which may refer to both ordinary and proof variables. For example, let Γ = f : (y÷U ) → U . Then the irrelevant η-expansion λx÷U. f ÷ x is well-typed in Γ, with the following derivation: Γ. x ÷ U ⊢ f : (y÷U ) → U Γ. x : U ⊢ x : U Γ. x ÷ U ⊢ f ÷ x : U Γ ⊢ λx ÷ U. f ÷ x : (x÷U ) → U Observe how the status of x changes for irrelevant to relevant when we check the argument of f . Sorting rules. These are the “Axioms” and the “Rules” of PTSs to form types. ⊢Γ (s, s′ ) Γ ⊢ s : s′ Γ ⊢ U : s1 Γ. x⋆U ⊢ T : s2 s1 ,s2 Γ ⊢ (x⋆U ) → T : s3 (s1 , s2 , s3 ) The rule for irrelevant function type formation follows Reed [Ree02]. Γ ⊢ U : s1 Γ. x÷U ⊢ T : s2 Γ ⊢ U : s1 Γ. x : U ⊢ T : s2 (s1 , s2 , s3 ) s1 ,s2 Γ ⊢ (x÷U ) → T : s3 It states that the codomain of an irrelevant function cannot depend relevantly on the function argument. This fact is crucial for the construction of our semantics in Section 4. Note that it rules out polymorphism in the sense of Barras and Bernado’s Implicit Calculus of Constructions ICC∗ [BB08] and Mishra-Linger and Sheard’s Erasure Pure Type Systems EPTS [MLS08]; the type (X÷Set0 ) → (x : X) → X is ill-formed in IITT, but not in ICC∗ or EPTS. In EPTS, there is the following rule: (s1 , s2 , s3 ) s1 ,s2 Γ ⊢ (x÷U ) → T : s3 It allows the codomain T of an irrelevant function to arbitrarily depend on the function argument x. This is fine in an erasure semantics, but incompatible with our typed semantics in the presence of large eliminations; we will detail the issues in examples 2.3 and 2.8. IRRELEVANCE IN TYPE THEORY 7 Another variant is Pfenning’s rule for irrelevant function type formation [Pfe01]. Γ ⊢ U ÷ s1 Γ. x÷U ⊢ T : s2 (s1 , s2 , s3 ) s1 ,s2 Γ ⊢ (x÷U ) → T : s3 It allows the domain of an irrelevant function to make use of irrelevant variables in scope. It does not give polymorphism, e. g., (X÷Set0 ) → (x : X) → X is still ill-formed. However, (X÷Set0 ) → (x÷X) → X would be well-formed. It is unclear how the equality rule for irrelevant function types would look like—it is not given by Pfenning [Pfe01]. The rule Γ ⊢ U = U ′ ÷ s1 Γ. x÷U ⊢ T = T ′ : s2 (s1 , s2 , s3 ) s1 ,s2 s1 ,s2 Γ ⊢ (x÷U ) → T = (x÷U ′ ) → T ′ : s3 would mean that any two irrelevant function types are equal as long as their codomains are equal—their domains are irrelevant. This is not compatible with our typed semantics and seems a bit problematic in general.2 Type conversion rule. We have typed conversion, thus, strictly speaking, IITT is not a PTS, but a Pure Type System with Judgemental Equality [Ada06]. Γ ⊢t:T Γ ⊢ T = T′ Γ′ ⊢ t : T ′ Equality. Figure 1 recapitulates the typing rules and lists the rules to derive context wellformedness ⊢ Γ and equality Γ ⊢ t = t′ : T . Equality is the least congruence over the βand η-axioms. Since equality is typed we can extend IITT to include an extensional unit type (Section 7). Let us inspect the congruence rule for application: Γ ⊢ t = t′ : (x⋆U ) → T Γ ⊢ u = u′ ⋆ U Γ ⊢ t ⋆ u = t′ ⋆ u′ : T [u/x] In case of relevant functions (⋆ = :) we obtain the usual dependently-typed application rule of equality. Otherwise, we get: Γ ⊢ t = t′ : (x÷U ) → T Γ÷ ⊢ u : U Γ ⊢ t ÷ u = t′ ÷ u′ : T [u/x] Γ ÷ ⊢ u′ : U Note that the arguments u and u′ to the irrelevant functions need to be well-typed but not related to each other. This makes precise the intuition that t and t′ are constant functions. 2.2. Simple properties of IITT. In the following, we prove two basic invariants of derivable IITT-judgements: The context is always well-formed, and judgements remain derivable under well-formed context extensions (weakening). Lemma 2.1 (Context well-formedness). (1) If ⊢ Γ. x : U. Γ′ then Γ ⊢ U . (2) If Γ ⊢ t : T or Γ ⊢ t = t′ : T then ⊢ Γ. Proof. By a simple induction on the derivations. 2This is why Reed [Ree02] differs from Pfenning. 8 A. ABEL AND G. SCHERER ⊢Γ Context well-formedness. ⊢Γ Γ ⊢T ⊢ Γ. x⋆T ⊢⋄ Γ ⊢t:T Typing. ⊢Γ (s, s′ ) Γ ⊢ s : s′ ⊢Γ Γ ⊢ U : s1 Γ. x⋆U ⊢ T : s2 (s1 , s2 , s3 ) s1 ,s2 Γ ⊢ (x⋆U ) → T : s3 (x : U ) ∈ Γ Γ ⊢x:U s,s′ Γ. x⋆U ⊢ t : T Γ ⊢ (x⋆U ) → T s,s′ Γ ⊢ λx⋆U. t : (x⋆U ) → T Γ ⊢ t : (x⋆U ) → T Γ ⊢u⋆U ⋆ Γ ⊢ t u : T [u/x] Γ ⊢t:T Γ ⊢ T = T′ Γ′ ⊢ t : T ′ Γ ⊢ t = t′ : T Equality. Computation (β) and extensionality (η). Γ. x⋆U ⊢ t : T Γ ⊢u⋆U ⋆ Γ ⊢ (λx⋆U. t) u = t[u/x] : T [u/x] s,s′ Γ ⊢ t : (x⋆U ) → T s,s′ Γ ⊢ t = λx⋆U. t ⋆ x : (x⋆U ) → T Equivalence rules. Γ ⊢t:T Γ ⊢t=t:T Compatibility rules. Γ ⊢ t = t′ : T Γ ⊢ t′ = t : T Γ ⊢ U = U ′ : s1 s1 ,s2 Γ ⊢ t1 = t2 : T Γ ⊢ t2 = t3 : T Γ ⊢ t1 = t3 : T Γ. x⋆U ⊢ T = T ′ : s2 s1 ,s2 Γ ⊢ (x⋆U ) → T = (x⋆U ′ ) → T ′ : s3 Γ ⊢ U = U ′ : s1 Γ. x⋆U ⊢ T : s2 (s1 , s2 , s3 ) Γ. x⋆U ⊢ t = t′ : T s1 ,s2 Γ ⊢ λx⋆U. t = λx⋆U ′ . t′ : (x⋆U ) → T Γ ⊢ t = t′ : (x⋆U ) → T Γ ⊢ u = u′ ⋆ U Γ ⊢ t ⋆ u = t′ ⋆ u′ : T [u/x] Conversion rule. Γ ⊢ t = t′ : T Γ ⊢ T = T′ Γ ⊢ t = t′ : T ′ Figure 1: Rules of IITT IRRELEVANCE IN TYPE THEORY 9 It should be noted that we only prove the most basic well-formedness statements here. One would expect that Γ ⊢ t : T or Γ ⊢ t = t′ : T also implies Γ ⊢ T , or that Γ ⊢ t = t′ : T implies Γ ⊢ t : T . This is true—and we will refer to these implications as syntactic validity—but this cannot be proven without treatment of substitution, due to the typing rule for application, which requires substitution in the type, and due to the equality rule for a β-redex, which uses substitution in both term and type. Therefore, syntactic validity is delayed until Section 4 (Corollary 4.17), where substitution will be handled by semantic, rather than syntactic, methods. Weakening. We can weaken a context Γ by adding bindings or making irrelevant bindings relevant. Formally, we have an order on binding annotations, which is the order induced by : ≤ ÷, and we define weakening by monotonic extension. A well-formed context ⊢ ∆ extends a well-formed context ⊢ Γ, written ∆ ≤ Γ, if and only if: ∀x ∈ dom(Γ), (x ⋆1 U ) ∈ Γ =⇒ (x ⋆2 U ) ∈ ∆ with ⋆1 ≤ ⋆2 . Note that this allows to insert new bindings or relax existing ones at any position in Γ, not just at the end. Lemma 2.2 (Weakening). Let ∆ ≤ Γ. (1) If ⊢ Γ.Γ′ and dom(∆) ∩ dom(Γ′ ) = ∅ then ⊢ ∆.Γ′ . (2) If Γ ⊢ t : T then ∆ ⊢ t : T . (3) If Γ ⊢ t = t′ : T ′ then ∆ ⊢ t = t′ : T . Proof. Simultaneously by induction on the derivation. Let us look at some cases: Case ⊢Γ (s, s′ ) Γ ⊢ s = s : s′ By assumption ⊢ ∆, thus ∆ ⊢ s = s : s′ . Case (x : U ) ∈ Γ ⊢ Γ Γ ⊢x=x:U Since ∆ ≤ Γ we have (x : U ) ∈ ∆, thus ∆ ⊢ x = x : U . Case Γ ⊢ U = U ′ : s1 Γ. x⋆U ⊢ T : s2 Γ. x⋆U ⊢ t = t′ : T s1 ,s2 Γ ⊢ λx⋆U. t = λx⋆U ′ . t′ : (x⋆U ) → T W. l. o. g., x 6∈ dom(∆). By (1) and definition of context weakening, ∆ ≤ Γ implies ∆. x⋆U ≤ Γ. x⋆U , so all premises can be appropriately weakened by induction hypothesis. 2.3. Examples. Example 2.3 (Relevance of types). sionality principle. ⊢Γ Γ ⊢ 1 : Seti 3 We can extend IITT by a unit type 1 with exten- ⊢Γ Γ ⊢ () : 1 3Example suggested by a reviewer of this paper. Γ ⊢t:1 Γ ⊢ t′ : 1 Γ ⊢ t = t′ : 1 10 A. ABEL AND G. SCHERER Typed equality allows us to equate all inhabitants of the unit type. As a consequence, the Church numerals over the unit type all coincide, e. g., Γ ⊢ λf : 1 → 1. λx : 1. x = λf : 1 → 1. λx : 1. f x : (1 → 1) → 1 → 1. In systems with untyped equality, like ICC∗ and EPTS, these terms erase to untyped Church-numerals λf λx.x and λf λx. f x and are necessarily distinguished. If we trade the unit type for Bool or any other type with more than one inhabitant, the two terms become different in IITT. This means that in IITT, types are relevant, and we need to reject irrelevant quantification over types like in (X÷Set0 ) → (X → X) → X → X. In IITT, the polymorphic types of Church numerals are (X : Seti ) → (X → X) → X → X. Example 2.4 (Σ-types). IITT can be readily extended by weak Σ-types. Γ ⊢ U : s1 Γ. x⋆U ⊢ T : s2 (s1 , s2 , s3 ) Γ ⊢ (x⋆U ) × T : s3 Γ ⊢ u⋆U Γ ⊢ t : T [u/x] Γ ⊢ (x⋆U ) × T Γ ⊢ (u, t) : (x⋆U ) × T Γ ⊢ p : (x⋆U ) × T Γ. x⋆U. y : T ⊢ v : V Γ ⊢ let (x, y) = p in v : V Γ ⊢u⋆U Γ ⊢ t : T [u/x] Γ. x⋆U. y : T ⊢ v : V Γ ⊢ (x⋆U ) × T Γ ⊢ (let (x, y) = (u, t) in v) = v[u/x][t/y] : V Additional laws for equality could be considered, like commuting conversions, or the identity (let (x, y) = p in (x, y)) = p. The relevant form (x : U ) × T admits a strong version with projections fst and snd and full extensionality p = (fst p, snd p) : (x : U ) × T . However, strong irrelevant Σ-types (x÷U ) × T are problematic because of the first projection: Γ ⊢ p : (x÷U ) × T Γ ⊢ fst p ÷ U With our definition of Γ ⊢ u ÷ U as Γ÷ ⊢ u : U , this rule is misbehaved: it allows us get hold of an irrelevant value in a relevant context. We could define a closed function π1 : (x÷U ) × 1 → U , and composing it with ( , ()) : (x÷U ) → (x÷U ) × 1 would give us an identity function of type (x÷U ) → U which magically makes irrelevant things relevant and IITT inconsistent. In this article, we will not further consider strong Σ-types with irrelevant components; we leave the in-depth investigation to future work. Example 2.5 (Squash type). The squash type \|\|T \|\| was first introduced in the context of NuPRL [CAB+ 86]; it contains exactly one inhabitant iff T is inhabited. Semantically, one obtains \|\|T \|\| from T by equating all of T ’s inhabitants. In IITT, we can define \|\|T \|\| as internalization of the irrelevance modality, as already suggested by Pfenning [Pfe01]. The IRRELEVANCE IN TYPE THEORY 11 first alternative is via the weak irrelevant Σ-type. \|\| \|\| : Seti → Seti \|\|T \|\| := ( ÷T ) × 1 [] [x] : (x÷T ) → \|\|T \|\| := (x, ()) sqelim (T : Seti ) (P : \|\|T \|\| → Setj ) (f : (x÷T ) → P [x]) (t : \|\|T \|\|) : Pt := let (x, ) = t in f ÷ x It is not hard to see that \|\| \|\| is a monad. All canonical inhabitants of \|\|T \|\| are definitionally equal: Γ ⊢ t, t′ ÷ T Γ ⊢ [t] = [t′ ] : \|\|T \|\| This is easily shown by expanding the definition of [ ] and using the congruence rule for pairs with an irrelevant first component. However, we cannot show that all inhabitants of \|\|T \|\| are definitionally equal, because of the missing extensionality principles for weak Σ. Thus, the second alternative is to add the squash type to IITT via the rules: Γ ⊢ T : Seti Γ ⊢ \|\|T \|\| : Seti Γ ⊢t÷T Γ ⊢ [t] : \|\|T \|\| Γ ⊢ t, t′ : \|\|T \|\| Γ ⊢ t = t′ : \|\|T \|\| Γ ⊢ t : \|\|T \|\| Γ. x÷T ⊢ v : V Γ ⊢ let [x] = t in v : V Γ ⊢t÷T Γ. x÷T ⊢ v : V Γ ⊢ (let [x] = [t] in v) = v[t/x] : V Our model (Section 4) is ready to interpret these rules, as well as normalization-by-evaluation inspired models [ACP11]. Example 2.6 (Subset type). The subset type {x : U \| T } is definable from Σ and squash as (x : U ) × \|\|T \|\|. To discuss the next example, we consider a further extension of IITT by Leibniz equality and natural numbers: a ≡ b : Seti for A : Seti and a, b : A refl : a ≡ a for A : Seti and a : A Nat : Seti 0, 1, . . . : Nat +, ∗ : Nat → Nat → Nat. Example 2.7 (Composite). 4 Let the set of composite numbers {4, 6, 8, 9, 10, 12, 14, 15, . . . } be numbers that are the product of two natural numbers ≥ 2. Composite = {n : Nat \| (k : Nat) × (l : Nat) × (n ≡ (k + 2) ∗ (l + 2))} Most composite numbers have several factorizations, and thanks to irrelevance the specific composition is ignored when handling composite numbers. For instance, 12 as product of 3 and 4 is not distinguished from the 12 as product of 2 and 6. (12, [(1, (2, refl))]) = (12, [(0, (4, refl))]) : Composite. 4Example suggested by reviewer. 12 A. ABEL AND G. SCHERER Example 2.8 (Large eliminations). 5 The ICC∗ [BB08] or EPTS [MLS08] irrelevant function type (x ÷ A) → B allows x to appear relevantly in B. This extra power raises some issues with large eliminations. Consider T : Bool → Set0 T true = Bool → Bool T false = Bool t = λF : (b÷Bool) → (T b → T b) → Set0 . λg : (F ÷ false (λx : Bool. x)) → Bool. λa : F ÷ true (λx : Bool → Bool.λy : Bool. x y). g a. The term t is well-typed in ICC∗ + T because the domain type of g and the type of a are βη-equal after erasure (−)∗ of type annotations and irrelevant arguments: (F ÷ false (λx : Bool. x))∗ = F (λxx) =βη F (λxλy. x y) = (F ÷ true (λx : Bool → Bool.λy : Bool. x y))∗ While a Curry view supports this, it is questionable whether identity functions at different types should be viewed as one. It is unclear how a type-directed equality algorithm (see Sec. 3) should proceed here; it needs to recognize that x : Bool is equal to λy : Bool. x y : Bool → Bool. This situation is amplified by a unit type 1 with extensional equality. When we change T true to 1 and the type of a to F ÷ true (λx : 1. ()) then t should still type-check, because λx. () is the identity function on 1. However, η-equality for 1 cannot be checked without types, and a type-directed algorithm would end up checking (successfully) x : Bool for equality with () : 1. This algorithmic equality cannot be transitive, because then any two booleans would be equal. Summarizing, we may conclude that the type of F bears trouble and needs to be rejected. IITT does this because it forbids the irrelevant b in relevant positions such as T b; ICC∗ lacks T altogether. Extensions of ICC∗ should at least make sure that b is never eliminated, such as in T b. Technically, T would have to be put in a separate class of recursive functions, those that actually compute with their argument. We leave the interaction of the three different function types to future research. 3. Algorithmic Equality The algorithm for checking equality in IITT is inspired by Harper and Pfenning [HP05]. Like theirs, it is type-directed, but we are using the full dependent type and not an erasure to simple types (which would anyway not work due to large eliminations). We give the algorithm in form of judgements and rules in direct correspondence to a functional program. Algorithmic equality is meant to be used as part of a type checking algorithm. It is the algorithmic counterpart of the definitional conversion rule; in particular, it will only be called on terms that are already know to be well-typed – in fact, types that are well-sorted. We rely on this precondition in the algorithmic formulation. Algorithmic equality consists of three interleaved judgements. A type equality test checks equality between two types, by inspecting their weak head normal forms. Terms found inside dependent types are reduced and the resulting neutral terms are compared by structural equality. The head variable of such neutrals provides type information that is 5Inspired by discussions with Ulf Norell during the 11th Agda Implementers’ Meeting. IRRELEVANCE IN TYPE THEORY 13 then used to check the (non-normal) arguments using type-directed equality, by reasoning on the (normalized) type structure to perform η-expansions on product types. After enough expansions, a base type is reached, where structural equality is called again, or a sort, at which we use type equality. Informally, the interleaved reductions are the algorithmic counterparts of the β-equality axiom, the type and structural equalities account for the compatibility rules, and typedirected equality corresponds to the η-equality axiom. The remaining equivalence rules are emergent global properties of the algorithm. Weak head reduction. Weak head normal forms (whnfs) are given by the following grammar: Whnf ∋ a, b, f, A, B, F Wne ∋ n, N s,s′ ::= s \| (x⋆U ) → T \| λx⋆U. t \| n ::= x \| n ⋆ u whnf neutral whnf Weak head evaluation t ց a and active application f @⋆ u ց a are functional relations given by the following rules. tցf f @⋆ u ց a t ⋆u ց a aցa t[u/x] ց a (λx⋆U. t) @⋆ u ց a n @⋆ u ց n ⋆ u Instead of writing the propositions t ց a and P [a] we will sometimes simply write P [↓t]. Similarly, we might write P [f @⋆ u] instead of f @⋆ u ց a and P [a]. In rules, it is understood that the evaluation judgement is always an extra premise, never an extra conclusion. Algorithmic equality is given as type equality, structural equality, and type-directed equality, which are mutually recursive. The equality algorithm is only invoked on wellformed expressions of the correct type. Type equality. Type equality ∆ ⊢ A ⇐⇒ A′ , for weak head normal forms, and ∆ ⊢ T ⇐⇒ b ′ T , for arbitrary well-formed types, checks that two given types are equal in their respective contexts. ∆ ⊢ N ←→ b N′ : T ∆ ⊢ ↓T ⇐⇒ ↓T ′ ∆ ⊢ T ⇐⇒ b T′ ∆ ⊢ N ⇐⇒ N ′ ∆ ⊢ U ⇐⇒ b U′ ∆ ⊢ s ⇐⇒ s ∆. x : U ⊢ T ⇐⇒ b T′ s,s′ s,s′ ∆ ⊢ (x⋆U ) → T ⇐⇒ (x⋆U ′ ) → T ′ Note that when invoking structural equality on neutral types N and N ′ , we do not care which type T is returned, since we know by well-formedness that N and N ′ must have the same sort. Structural equality. Structural equality ∆ ⊢ n ←→ n′ : A and ∆ ⊢ n ←→ b n′ : T checks the neutral expressions n and n′ for equality and at the same time infers their type, which is returned as output. ∆ ⊢ n ←→ b n′ : T ∆ ⊢ n ←→ n′ : ↓T (x : T ) ∈ ∆ ∆ ⊢ x ←→ b x:T ∆ ⊢ n ←→ n′ : (x : U ) → T ∆ ⊢ u ⇐⇒ b u′ : U ∆ ⊢ n u ←→ b n′ u′ : T [u/x] ∆ ⊢ n ←→ n′ : (x÷U ) → T ∆ ⊢ n ÷ u ←→ b n′ ÷ u′ : T [u/x] 14 A. ABEL AND G. SCHERER Type-directed equality. Type-directed equality ∆ ⊢ t ⇐⇒ t′ : A and ∆ ⊢ t ⇐⇒ b t′ : T checks ′ terms t and t for equality and proceeds by the structure of the supplied type, to account for η. ∆ ⊢ t ⇐⇒ t′ : ↓T ∆ ⊢ t ⇐⇒ b t′ : T ∆. x⋆U ⊢ t ⋆ x ⇐⇒ b t′ ⋆ x : T ∆ ⊢ t ⇐⇒ t′ : (x⋆U ) → T ∆ ⊢ T ⇐⇒ b T′ ∆ ⊢ ↓t ←→ b ↓t′ : T ∆ ⊢ T ⇐⇒ T ′ : s ∆ ⊢ t ⇐⇒ t′ : N Note that in the but-last rule we do not check that the inferred type T of ↓t equals the ascribed type N . Since algorithmic equality is only invoked for well-typed t, we know that this must always be the case. Skipping this test is a conceptually important improvement over Harper and Pfenning [HP05]. Due to dependent typing, it is not obvious that algorithmic equality is symmetric and transitive. For instance, consider symmetry in case of application: We have to show that ∆ ⊢ n′ u′ ←→ b n u : T [u/x], but using the induction hypothesis we obtain this equality only at type T [u′ /x]. To conclude, we need to convert types, which is only valid if we know that u and u′ are actually equal. Thus, we need soundness of algorithmic equality to show its transitivity. Soundness w. r. t. declarative equality requires subject reduction, which is not trivial, due to its dependency on function type injectivity. In the next section (4), we construct by a Kripke logical relation which gives us subject reduction and soundness of algorithmic equality (Section 5), and, finally, symmetry and transitivity of algorithmic equality. A simple fact about algorithmic equality is that the inferred types are unique up to syntactic equality (where we consider α-convertible expressions as identical). Also, they only depend on the left hand side neutral term n. Lemma 3.1 (Uniqueness of inferred types). (1) If ∆ ⊢ n ←→ n1 : A1 and ∆ ⊢ n ←→ n2 : A2 then A1 ≡ A2 . (2) If ∆ ⊢ n ←→ b n1 : T1 and ∆ ⊢ n ←→ b n2 : T2 then T1 ≡ T2 . Extending structural equality to irrelevance, we let ∆÷ ⊢ n ←→ n : A ∆÷ ⊢ n′ ←→ n′ : A ∆ ⊢ n ←→ n′ ÷ A ′ and analogously for ∆ ⊢ n ←→ b n ÷ T. 4. A Kripke Logical Relation for Soundness In this section, we construct a Kripke logical relation in the spirit of Goguen [Gog00] and Vanderwaart and Crary [VC02] that proves weak head normalization, function type injectivity, and subject reduction plus syntactical properties like substitution in judgements and syntactical validity. As an important consequence, we obtain soundness of algorithmic equality w. r. t. definitional equality. This allows us to establish that algorithmic equality on well-typed terms is a partial equivalence relation. IRRELEVANCE IN TYPE THEORY 15 4.1. An Induction Measure. Following Goguen [Gog94] and previous work [ACD08], we first define a semantic universe hierarchy Ui whose sole purpose is to provide a measure for defining a logical relation and proving some of its properties. The limit Uω corresponds to the proof-theoretic strength or ordinal of IITT. We denote sets of expressions by A, B and functions from expressions to sets of expressions by F. Let Ab = {t \| ↓t ∈ A} denote the closure of A by weak head expansion. The b f @ u ∈ F(u)}. dependent function space is defined as Π A F = {f ∈ Whnf \| ∀u ∈ A. By recursion on i ∈ N we define inductively sets Ui ⊆ Whnf × P(Whnf) as follows [ACD08, Sec. 5.1]: (N, Wne) ∈ Ui (Setj , Seti ) ∈ Axiom (Setj , \|Uj \|) ∈ Ui cj b (T [u/x], F(u)) ∈ U (U, A) ∈ Ubi ∀u ∈ A. (Seti , Setj , Setk ) ∈ Rule ((x⋆U ) → T, Π A F) ∈ Uk Herein, Ubi = {(T, A) \| (↓T, A) ∈ Ui } and \|Uj \| = {A \| (A, A) ∈ Uj for some A}. Only interested in computational strength, we treat relevant and irrelevant function spaces alike— at the level of predicates A, irrelevance is anyhow not observable, only by relations as given later. The induction measure A ∈ Seti shall now mean the minimum height of a derivation of (A, A) ∈ Ui for some A. Note that due to universe stratification, A ∈ Seti is smaller than Seti ∈ Setj . 4.2. A Kripke Logical Relation. Let ∆ ⊢ t :=: t′ ⋆ T stand for the conjunction of the propositions • ∆ ⊢ t ⋆ T and ∆ ⊢ t′ ⋆ T , and • ∆ ⊢ t = t′ ⋆ T . By induction on A ∈ s we define two Kripke relations ∆ ⊢ A s A′ : s ∆ ⊢ a s a′ : A. b and the generalization to ⋆. For better readability, together with their respective closures s the clauses are given in rule form meaning that the conclusion is defined as the conjunction of the premises. ∀ and =⇒ are meta-level quantification and implication, respectively. ∆ ⊢ N :=: N ′ : s ∆ ⊢ N s N′ : s ∆ ⊢ n :=: n′ : N ∆ ⊢ n s n′ : N ⊢∆ (s, s′ ) ∆ ⊢ s s s : s′ b U ′ : s1 ∆ ⊢U s b u′ ⋆ U =⇒ Γ ⊢ T [u/x] s b T ′ [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s ,s 1 2 ∆ ⊢ (x⋆U ) → T :=: (x⋆U ′ ) → T ′ : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T s (x⋆U ′ ) → T ′ : s3 (s1 , s2 , s3 ) b u′ ⋆ U =⇒ Γ ⊢ f ⋆ u s b f ′ ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s s,s′ ∆ ⊢ f :=: f ′ : (x⋆U ) → T s,s′ ∆ ⊢ f s f ′ : (x⋆U ) → T 16 A. ABEL AND G. SCHERER tցa T ցA ∆ ⊢T =A ∆ ⊢t=a:A ∆ ⊢ t′ = a′ : A ∆ ⊢ a s a′ : A ∆ ⊢ t :=: t′ : T b t′ : T ∆ ⊢ts ∆÷ ⊢ a s a : A ∆÷ ⊢ a′ s a′ : A ∆ ⊢ a s a′ ÷ A b t:T ∆÷ ⊢ t s t ′ ց a′ b t′ : T ∆ ÷ ⊢ t′ s b t′ ÷ T ∆ ⊢ts It is immediate that the logical relation contains only well-typed and definitionally equal terms. We will demonstrate that it is also closed under weakening and conversion, symmetric and transitive. Lemma 4.1 (Weakening). (1) If ∆ ⊢ a s a′ : A and Γ ≤ ∆ then there exists a derivation of Γ ⊢ a s a′ : A with the same height. b t′ : T . (2) Analogously for ∆ ⊢ t s Proof. By induction on A ∈ s and T ∈ s, resp. Lemma 4.2 (Type conversion). (1) If Γ ⊢ A s A′ : s then Γ ⊢ a s a′ : A iff Γ ⊢ a s a′ : A′ . b T ′ : s then Γ ⊢ t s b t′ : T iff Γ ⊢ t s b t′ : T ′ . (2) If Γ ⊢ T s Proof. Simultaneously induction in A ∈ s and T ∈ s, resp. We show the “if” direction, the “only if” follows analogously. The interesting case is the one of functions. Case b U ′ : s1 ∆ ⊢U s b u′ ⋆ U =⇒ Γ ⊢ T [u/x] s b T ′ [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T :=: (x⋆U ′ ) → T ′ : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T s (x⋆U ′ ) → T ′ : s3 b u′ ⋆ U =⇒ Γ ⊢ f ⋆ u s b f ′ ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s s,s′ ∆ ⊢ f :=: f ′ : (x⋆U ) → T s,s′ ∆ ⊢ f s f ′ : (x⋆U ) → T s,s′ First, ∆ ⊢ f :=: f ′ : (x⋆U ′ ) → T ′ , holds because of the conversion rule for typing and b u′ ⋆ U ′ and show Γ ⊢ f ⋆ u s b equality. Now assume arbitrary Γ ≤ ∆ and Γ ⊢ u s ′ ⋆ ′ ′ ′ b u ⋆ U , thus, f u : T [u/x]. By induction hypothesis on U ∈ s1 we have Γ ⊢ u s ⋆ ′ ⋆ ′ b Γ ⊢ f u s f u : T [u/x] by assumption. By induction hypothesis on T [u/x] ∈ s2 we b f ′ ⋆ u′ : T ′ [u/x]. obtain Γ ⊢ f ⋆ u s b T : s. Lemma 4.3 (Symmetry and Transitivity). Let ∆ ⊢ T s b t′ : T then ∆ ⊢ t′ s b t : T. (1) If ∆ ⊢ t s b t2 : T and ∆ ⊢ t2 s b t3 : T then ∆ ⊢ t1 s b t3 : T . (2) If ∆ ⊢ t1 s IRRELEVANCE IN TYPE THEORY 17 Proof. We generalize the two statements to whnfs ∆ ⊢ A s A : s and prove all four statements simultaneously by induction in A ∈ s and T ∈ s, resp. Case Let us look at the case for functions. b U : s1 ∆ ⊢U s b u′ ⋆ U =⇒ Γ ⊢ T [u/x] s b T [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T :=: (x⋆U ) → T : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U ) → T s (x⋆U ) → T : s3 Case Symmetry: b u′ ⋆ U =⇒ Γ ⊢ f ⋆ u s b f ′ ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s ′ ∆ ⊢ f :=: f : (x⋆U ) → T ∆ ⊢ f s f ′ : (x⋆U ) → T b u⋆U To show ∆ ⊢ f ′ s f : (x⋆U ) → T , assume arbitrary Γ ≤ ∆ and Γ ⊢ u′ s ′ ⋆ ′ ⋆ ′ b and show Γ ⊢ f u s f u : T [u /x]. By induction hypothesis on U ∈ s2 , with b U : s1 , we have Γ ⊢ u s b u′ ⋆ U , thus, Γ ⊢ f ⋆ u s b f ′ ⋆ u′ : T [u/x] weakened Γ ⊢ U s b u ⋆ U, by assumption. Using symmetry and transitivity on U we obtain Γ ⊢ u s b thus, Γ ⊢ T [u/x] s T [u/x] : s2 . By induction hypothesis on T [u/x] ∈ s2 we apply b f ⋆ u : T [u/x], and since Γ ⊢ T [u/x] s b T [u′ /x] : s2 symmetry to obtain Γ ⊢ f ′ ⋆ u′ s we conclude by type conversion (Lemma 4.2). Case Transitivity: b u′ ⋆ U =⇒ Γ ⊢ f1 ⋆ u s b f2 ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s s,s′ ∆ ⊢ f1 :=: f2 : (x⋆U ) → T s,s′ ∆ ⊢ f1 s f2 : (x⋆U ) → T b u′ ⋆ U =⇒ Γ ⊢ f2 ⋆ u s b f3 ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u s s,s′ ∆ ⊢ f2 :=: f3 : (x⋆U ) → T s,s′ ∆ ⊢ f2 s f3 : (x⋆U ) → T s,s′ s,s′ We wish to prove that ∆ ⊢ f1 s f3 : (x⋆U ) → T . We get ∆ ⊢ f1 :=: f3 : (x⋆U ) → T b u′ ⋆ U , immediately by transitivity of definitional equality. Given Γ ≤ ∆ and Γ ⊢ u s b f3 ⋆ u′ : T [u/x]. we need to show that Γ ⊢ f1 ⋆ u s b u ⋆ U , which b : U is a PER by induction hypothesis, we have Γ ⊢ u s As Γ ⊢ s ⋆ ⋆ ′ b b b f 3 ⋆ u′ : entails f1 u s f2 u : T [u/x]. From Γ ⊢ u s u ⋆ U also have Γ ⊢ f2 ⋆ u s b f3 ⋆ u′ : T [u/x] by transitivity at T [u/x]. T [u/x], which allows to conclude Γ ⊢ f1 ⋆ u s Case Now, we consider function spaces: 18 A. ABEL AND G. SCHERER Case Transitivity: b U2 : s 1 ∆ ⊢ U1 s b u′ ⋆ U1 =⇒ Γ ⊢ T1 [u/x] s b T2 [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U1 ) → T1 :=: (x⋆U2 ) → T2 : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U1 ) → T1 s (x⋆U2 ) → T2 : s3 b U3 : s 1 ∆ ⊢ U2 s b u′ ⋆ U2 =⇒ Γ ⊢ T2 [u/x] s b T3 [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u s s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U2 ) → T2 :=: (x⋆U3 ) → T3 : s3 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U2 ) → T2 s (x⋆U3 ) → T3 : s3 b U3 : s1 By transitivity we have ∆ ⊢ (x⋆U1 ) → T1 :=: (x⋆U3 ) → T3 : s3 and ∆ ⊢ U1 s by induction hypothesis on s1 . Note that this is where the arrow sort annotations are useful. Without them we would b U2 : s1 not know that the sorts in both derivations are equal. We could have ∆ ⊢ U1 s ′ ′ b and ∆ ⊢ U2 s U3 : s1 for apparently unrelated s1 and s1 , and would therefore be unable to use transitivity. b u′ ⋆ U1 , we need to show that Γ ⊢ T1 [u/x] s b T3 [u′ /x] : s3 . Given Γ ≤ ∆ and Γ ⊢ u s b at type U is a PER by induction hypothesis, we have Γ ⊢ u s b u⋆U1 , from which As s b b U2 : s1 – we can deduce Γ ⊢ T1 [u/x] s T2 [u/x] : s2 . By conversion using ∆ ⊢ U1 s ′ b u ⋆ U2 , which implies Γ ⊢ T2 [u/x] s b T3 [u′ /x] : s2 . weakened at Γ – we have Γ ⊢ u s This allows us to conclude by transitivity at type s2 . In the following we show that the variables are in the logical relation, i. e., ∆ ⊢ x s x : ∆(x) for well-formed contexts ∆. As usual, this statement has to be generalized to neutrals n to be proven inductively. b n′ ⋆T . Lemma 4.4 (Into the logical relation). Let T ∈ s. If ∆ ⊢ n :=: n′ ⋆T then ∆ ⊢ n s Proof. By induction on T ∈ s. Case N ∈ s and ∆ ⊢ n :=: n′ ⋆ N . Then ∆ ⊢ n s n′ ⋆ N by cases on ⋆, unfolding definitions. Case s ∈ s′ and ∆ ⊢ N :=: N ′ ⋆ s. Then ∆ ⊢ N s N ′ ⋆ s by cases on ⋆. Case (x⋆U ) → T ∈ s3 and ∆ ⊢ n :=: n′ ⋆0 (x⋆U ) → T . First, the case for ⋆0 = :. We have ∆ ⊢ n :=: n′ : (x⋆U ) → T . Assume arbitrary b u′ ⋆ U , which yields Γ ⊢ u :=: u′ ⋆ U and Γ ⊢ T [u/x] s b T [u/x] : s2 . Γ ≤ ∆ and Γ ⊢ u s ⋆ ′ ⋆ ′ By weakening, Γ ⊢ n u :=: n u : T [u/x], thus, by induction hypothesis, Γ ⊢ n ⋆ u s n′ ⋆ u′ : T [u/x], q.e.d. The case for ⋆0 = ÷ proceeds analogously. IRRELEVANCE IN TYPE THEORY 19 b to substitutions, by 4.3. Validity in the Model. We now extend our logical relation s induction on the destination context. b σ′ : Γ b σ ′ (x) ⋆ U σ ∆ ⊢σs ∆ ⊢ σ(x) s b σ′ : ⋄ b σ ′ : Γ. x⋆U ∆ ⊢σs ∆ ⊢σs b for terms. This relation inherits weakening from s We then define the context ( Γ), type (Γ T = T ′ ) and term (Γ relations, by induction on the length of contexts. Γ ⋄ Γ U Γ. x⋆U Γ T = T′ : s Γ T = T′ Γ t = t′ : T ) validity T =T Γ T Γ (Γ T unless T = s) b b t′ σ ′ : T σ ∆ ⊢ σ s σ ′ : Γ =⇒ ∆ ⊢ tσ s Γ t=t:T ′ Γ t=t :T Γ t:T Because of its asymmetric definition, the logical relation on substitutions may not be a PER in general, but it is for valid contexts. b : Γ is symmetric Lemma 4.5 (Substitution relation is a PER). If Γ, then ∆ ⊢ s and transitive. ∀∆, σ, σ ′ , Proof. By induction on Γ. We demonstrate symmetry for the case b σ′ : Γ b σ ′ (x) ⋆ U σ ∆ ⊢σs ∆ ⊢ σ(x) s Γ. x⋆U . b σ ′ : Γ. x⋆U ∆ ⊢σs b σ : Γ, and by symmetry of s b for terms (Lemma 4.3), By induction hypothesis, ∆ ⊢ σ ′ s ′ b σ(x) ⋆ U σ. We instantiate Γ b U σ ′ : s and conclude ∆ ⊢ σ (x) s U to ∆ ⊢ U σ s b σ(x) ⋆ U σ ′ by conversion (Lemma 4.2). ∆ ⊢ σ ′ (x) s = : T is symmetric and transitive. Lemma 4.6 (Validity is a PER). The relation Γ b for substitutions and conversion with ∆ ⊢ T σ s b Proof. Symmetry requires symmetry of s ′ ′ T σ : s , similar as in Lemma 4.5. We demonstrate transitivity in detail. Given Γ t1 = t2 : T and Γ t2 = t3 : T we show Γ t1 = t3 : T . Clearly, Γ and Γ T or T = s by one of our two assumptions. b σ ′ : Γ and show ∆ ⊢ t1 σ s b t3 σ ′ : T σ. By Lemma 4.5, Assume arbitrary ∆ ⊢ σ s b σ : Γ, thus ∆ ⊢ t1 σ s b t2 σ : T σ. Also, ∆ ⊢ t2 σ s b t3 σ ′ : T σ which entails our goal ∆ ⊢σs b by transitivity of s (Lemma 4.3). s1 ,s2 s′ ,s′ 1 2 Lemma 4.7 (Function type injectivity is valid). If Γ (x⋆U ) → T = (x⋆U ′ ) → T′ ′ ′ ′ ′ ′ T = T : s2 . then s1 = s1 and s2 = s2 and Γ U = U : s1 and Γ. x⋆U ′ ′ 1 ,s2 1 ,s2 b σ ′ : Γ. We have ∆ ⊢ (x⋆U σ) s→ b (x⋆U ′ σ ′ ) s→ Proof. Assume arbitrary ∆ ⊢ σ s Tσ s b U σ : s1 —note that sorts T ′ σ ′ : s3 , thus by definition s1 = s′1 and s2 = s′2 and ∆ ⊢ U ′ σ ′ s b and since ∆, σ, σ ′ are closed and therefore invariant by substitution. By symmetry of s, ′ were arbitrary, we have Γ U = U : s1 . b u′ ⋆ U ′ σ and let ρ = (σ, u/x) and ρ′ = (σ ′ , u′ /x). Further, assume arbitrary ∆ ⊢ u s b ρ′ : Γ. x⋆U ′ . We have Note that w. l. o. g., x 6∈ dom(Γ) and x 6∈ FV(U ′ ) and ∆ ⊢ ρ s b T ′ ρ′ : s2 and since ρ, ρ′ were arbitrary, Γ. x⋆U ′ T = T ′ : s2 . ∆ ⊢ Tρ s 20 A. ABEL AND G. SCHERER b id : Γ. Γ then ⊢ Γ and Γ ⊢ id s Lemma 4.8 (Context satisfiable). If Proof. By induction on Γ. The ⋄ case is immediate. In the Γ. x⋆U case, given Γ Γ U Γ. x⋆U we can use inference b id : Γ Γ. x⋆U ⊢ id s b id(x) ⋆ U id Γ. x⋆U ⊢ id(x) s . b id : Γ. x⋆U Γ. x⋆U ⊢ id s b id : Γ, we obtain the first premise by weakening From the induction hypothesis Γ ⊢ id s b It also yields Γ ⊢ U id : sid for some s by definition of Γ of s. U . Using induction hypothesis, ⊢ Γ, this entails ⊢ Γ. x⋆U . Further, Γ. x⋆U ⊢ x = x ⋆ U , and since trivially b x⋆U , by the Lemma 4.4. This concludes Γ. x⋆U ⊢ x ←→ x⋆U , we can derive Γ. x⋆U ⊢ x s b id(x) ⋆ U id. the second premise Γ. x⋆U ⊢ id(x) s We can now show that every equation valid in the model is derivable in IITT. Theorem 4.9 (Completeness of IITT rules). If Γ t = t′ : T then both Γ ⊢ t : T and ′ ′ Γ ⊢ t : T and Γ ⊢ t = t : T and Γ ⊢ T . b t′ : T , which entails Γ ⊢ t, t′ : T and Γ ⊢ t = Proof. Using Lemma 4.8 we obtain Γ ⊢ t s ′ t : T . Analogously, since our assumption entails Γ T by definition, we get Γ ⊢ T . 4.4. Fundamental theorem. We prove a series of lemmata which constitute parts of the fundamental theorem for the Kripke logical relation. Lemma 4.10 (Resurrection). If b σ ′ : Γ÷ . ∆÷ ⊢ σ ′ s b σ ′ : Γ then ∆÷ ⊢ σ s b σ : Γ÷ and Γ and ∆ ⊢ σ s Proof. By induction on Γ, the interesting case being b σ′ : Γ b σ ′ (x) ⋆ U σ ∆ ⊢σs ∆ ⊢ σ(x) s . b σ ′ : Γ. x⋆U ∆ ⊢σs b σ : (Γ÷ . x : U ). By induction hypothesis ∆÷ ⊢ σ s b σ : Γ÷ , and First, we show ∆÷ ⊢ σ s ÷ b σ(x) : U σ. This immediately entails our goal. by definition, ∆ ⊢ σ(x) s b σ ′ : (Γ÷ . x : U ), observe that Γ U , hence ∆ ⊢ U σ s b For the second goal ∆÷ ⊢ σ ′ s b σ ′ (x) : U σ to U σ ′ U σ ′ : s for some sort s. Thus, we can cast our hypothesis ∆ ⊢ σ ′ (x) s and conclude analogously. Corollary 4.11. If Γ⋆ b σ ′ : Γ then ∆ ⊢ uσ s b uσ ′ ⋆ U σ. u : U and ∆ ⊢ σ s b Proof. In case ⋆ = : it holds by definition, but we need resurrection for ⋆ = ÷. If ∆ ⊢ σ s ′ ÷ ÷ ÷ b σ : Γ , so from Γ σ : Γ, then by resurrection (Lemma 4.10) we have ∆ ⊢ σ s u:U ÷ ÷ ′ ′ ′ b b we deduce ∆ ⊢ uσ s uσ : U σ. Analogously we get ∆ ⊢ uσ s uσ : U σ which we cast b uσ ′ : U σ. to ∆÷ ⊢ uσ ′ s Lemma 4.12 (Validity of β-reduction). Γ. x⋆U t : T Γ⋆ u : U Γ (λx⋆U. t) ⋆ u = t[u/x] : T [u/x] IRRELEVANCE IN TYPE THEORY 21 b ρ′ : Γ Proof. Γ is contained in the first hypothesis Γ u ⋆ U . Then, given ∆ ⊢ ρ s ⋆ ′ ′ b t(ρ , uρ /x) : T (ρ, uρ/x) and also ∆ ⊢ T [u/x]ρ s b we need to show ∆ ⊢ (λx⋆U. t)ρ uρ s ′ T [u/x]ρ : s for some s (the latter to get Γ T [u/x]). Let σ = (ρ, uρ/x) and σ ′ = (ρ′ , uρ′ /x). From the second hypothesis and Cor. 4.11 we get b uρ′ ⋆ U ρ, which gives ∆ ⊢ σ s b σ ′ : Γ. x⋆U . By instantiating the first hypothesis ∆ ⊢ uρ s b tσ ′ : T σ, and also (from the premise Γ. x⋆U T ) ∆ ⊢ T σ = T σ ′ , which we get ∆ ⊢ tσ s gives Γ T [u/x]. b tσ ′ we get the desired ∆ ⊢ (λx⋆U. t)ρ ⋆ uρ s b tσ ′ : T σ, as s b is Finally, from ∆ ⊢ tσ s ⋆ closed by weak head expansion to well-typed ∆ ⊢ (λx⋆U. t)ρ uρ : T σ. Lemma 4.13 (Validity of η). Γ Γ t : (x⋆U ) → T t = λx⋆U. t ⋆ x : (x⋆U ) → T b Proof. Γ and Γ (x⋆U ) → T are direct consequences of our hypothesis. Given ∆ ⊢ ρ s ′ ⋆ ′ b ρ : Γ, we need to show ∆ ⊢ tρ s (λx⋆U. t x)ρ : ((x⋆U ) → T )ρ. W. l. o. g., x is not free in the domain nor range of substitutions ρ and ρ′ , thus with t′ := tρ, t′′ := tρ′ , U ′ := U ρ, U ′′ := b λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ . U ρ′ , T ′ := T ρ and T ′′ := T ρ′ it is sufficient to show ∆ ⊢ t′ s ′ ′ ′ ′ b u′ ⋆ U ′ , we show ∆′ ⊢ t′ ⋆ u s b First, given (∆ , u, u ) such that ∆ ≤ ∆ and ∆ ⊢ u s ′′ ′′ ⋆ ⋆ ′ ′ ′ b (λx⋆U . t x) u : T [u/x]. Our hypothesis Γ t : (x⋆U ) → T entails ∆ ⊢ tρ s tρ : ′ ′′ b ((x⋆U ) → T )ρ, that is to say ∆ ⊢ t s t : (x⋆U ′ ) → T ′ . This logical relation at a b t′′ ⋆ u′ : T ′ [u/x], which function type, when instantiated to (∆′ , u, u′ ), gives us ∆′ ⊢ t′ ⋆ u s weak-head expands to the desired goal. Second, we show ∆ ⊢ t′ :=: λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ . • Γ ⊢ t′ : (x⋆U ′ ) → T ′ is a simple consequence of our hypothesis Γ t : (x⋆U ) → T . • Γ ⊢ λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ has the following proof: Γ t : (x⋆U ) → T =====′′======′′======′′ ∆ ⊢ t : (x⋆U ) → T ∆. x⋆U ′′ ⊢ t′′ : (x⋆U ′′ ) → T ′′ weak (∆. x⋆U ′′ )⋆ ⊢ x : U ′′ var ∆. x⋆U ′′ ⊢ t′′ ⋆ x : T ′′ Γ (x⋆U ) → T ========′′======′′ ∆ ⊢ (x⋆U ) → T ∆ ⊢ λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′′ ) → T ′′ ∆ ⊢ λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ conv • ∆ ⊢ t′ = λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′ ) → T ′ . The η-rule of definitional equality gives us ∆ ⊢ t′′ = λx⋆U ′′ . t′′ ⋆ x : (x⋆U ′′ ) → T ′′ . From Γ (x⋆U ) → T we can convert it to the type (x⋆U ′ ) → T ′ , and then conclude by transitivity using ∆ ⊢ t′ = t′′ : (x⋆U ′ ) → T ′ , which is a direct consequence of Γ t : (x⋆U ) → T . Lemma 4.14 (Validity of function equality). Γ Γ U = U′ Γ. x⋆U t = t′ : T (λx⋆U. t) = (λx⋆U ′ . t′ ) : (x⋆U ) → T Proof. Again Γ and Γ (x⋆U ) → T are simple consequences of our hypotheses. Given ′ b ∆ ⊢ ρ s ρ : Γ (w. l. o. g., x is not free in ρ, ρ′ domain or range), we need to show 22 A. ABEL AND G. SCHERER b (λx⋆U ′ ρ′ . t′ ρ′ ) : ((x⋆U ρ) → T ρ). We will skip the proof of ∆ ⊢ ∆ ⊢ (λx⋆U ρ. tρ) s (λx⋆U ρ. tρ) :=: (λx⋆U ′ ρ′ . t′ ρ′ ) : ((x⋆U ρ) → T ρ), as it is similar to the corresponding part of the η-validity lemma. b u′ ⋆ U ρ, we have to show that Given (∆′ , u, u′ ) such that ∆′ ≤ ∆ and ∆′ ⊢ u s ′ ⋆ ′ ′ ′ ′ ⋆ ′ b (λx⋆U ρ . t ρ ) u : T ρ[u/x]. Let σ = (ρ, u/x) and σ ′ = (ρ′ , u′ /x). ∆ ⊢ (λx⋆U ρ. tρ) u s b u′ ⋆ U ρ, we have ∆′ ⊢ σ s b σ ′ : Γ. x⋆U ρ. Instantiating the second As we supposed ∆′ ⊢ u s ′ ′ ′ hypothesis with ∆ , σ, σ therefore gives us ∆ ⊢ tσ = t′ σ ′ : T σ, which can also be written b t′ ρ′ [u′ /x] : T ρ[u/x], which is weak-head expansible to our goal. ∆′ ⊢ tρ[u/x] s Lemma 4.15 (Validity of irrelevant application). Γ t = t′ : (x÷U ) → T Γ÷ u : U Γ t ÷ u = t′ ÷ u′ : T [u/x] Γ÷ u′ : U b ρ′ : Γ and show ∆ ⊢ tρ ÷ uρ s b t′ ρ′ ÷ u′ ρ′ : T (ρ, uρ/x). Proof. Assume arbitrary ∆ ⊢ ρ s b u′ ρ′ ÷ U ρ, which means ∆÷ ⊢ By the first hypothesis, it is sufficient to show ∆ ⊢ uρ s ÷ ′ ′ ′ ′ b b b uρ s uρ : U ρ and ∆ ⊢ u ρ s u ρ : U ρ. By Resurrection (Lemma 4.10), ∆÷ ⊢ ρ s b uρ : U ρ from the second hypothesis. Analogously, we obtain ρ : Γ÷ , hence ∆÷ ⊢ uρ s ÷ ′ ′ ′ ′ ′ b u ρ : U ρ from the third hypothesis which we can cast to U ρ by virtue of ∆ ⊢ uρ s Γ U which we get from Γ (x÷U ) → T by Lemma 4.7. Theorem 4.16 (Fundamental theorem of logical relations). (1) If ⊢ Γ then Γ. (2) If Γ ⊢ t : T then Γ t : T . (3) If Γ ⊢ t = t′ : T then Γ t = t′ : T . Proof. By induction on the derivation. As a simple corollary we obtain syntactic validity, namely that definitional equality implies well-typedness and well-typedness implies well-formedness of the involved type. This lemma could have been proven purely syntactically, but the syntactic proof requires a sequence of carefully arranged lemmata like context conversion, substitution, functionality, and inversion on types [HP05, AC07]. Our “sledgehammer” semantic argument is built into the Kripke logical relation, in the spirit of Goguen [Gog00]. Corollary 4.17 (Syntactic validity). (1) If Γ ⊢ t : T then Γ ⊢ T . (2) If Γ ⊢ t = t′ : T then Γ ⊢ t : T and Γ ⊢ t′ : T . Proof. By the fundamental theorem, Γ ⊢ t = t′ : T implies Γ Thm. 4.9 implies Γ ⊢ t, t′ : T and Γ ⊢ T . t = t′ : T , which by IRRELEVANCE IN TYPE THEORY 23 5. Meta-theoretic Consequences of the Model Construction In this section, we explicate the results established by the Kripke model. 5.1. Admissibility of Substitution. Goguen [Gog00] observes that admissibility of substitution for the syntactic judgements can be inherited from the Kripke logical relation, which is closed under substitution by its very definition. To show that the judgements of IITT are closed under substitution we introduce relations Γ ⊢ σ : Γ′ for substitution typing and Γ ⊢ σ = σ ′ : Γ′ for substitution equality which are given inductively by the following rules: Γ ⊢ σ : Γ′ ⊢Γ Γ ⊢σ:⋄ Γ′ ⊢ U Γ ⊢ σ(x) ⋆ U σ ′ Γ ⊢ σ : Γ . x⋆U Γ ⊢ σ = σ ′ : Γ′ Γ′ ⊢ U Γ ⊢ σ(x) = σ ′ (x) ⋆ U σ ⊢Γ Γ ⊢ σ = σ′ : ⋄ Γ ⊢ σ = σ ′ : Γ′ . x⋆U Substitution typing and equality are closed under weakening. Semantically, substitutions are explained by environments. We define substitution validity as follows, again in rule form but not inductively: Γ Γ′ b b σ ′ ρ′ : Γ ′ Γ σ=σ: ∀∆ ρ s : Γ. ∆ σρ s Γ σ : Γ′ Γ σ = σ ′ : Γ′ Lemma 5.1 (Fundamental lemma for substitutions). (1) If Γ ⊢ σ : Γ′ then Γ σ : Γ′ . (2) If Γ ⊢ σ = σ ′ : Γ′ then Γ σ = σ ′ : Γ′ . Γ′ ρ′ Proof. We demonstrate 2 by induction on Γ ⊢ σ = σ ′ : Γ′ . Case ⊢Γ Γ ⊢ σ = σ′ : ⋄ b σ ′ ρ′ : ⋄ trivially for any We have Γ by Thm. 4.16 and ⋄ trivially. Also, ∆ ⊢ σρ s ′ b ρ : Γ. ∆ ⊢ρs Case Γ ⊢ σ = σ ′ : Γ′ Γ′ ⊢ U Γ ⊢ σ(x) = σ ′ (x) ⋆ U σ Γ ⊢ σ = σ ′ : Γ′ . x⋆U ′ We have Γ and Γ by induction hypothesis and Γ′ U by Thm. 4.16, thus, ′ b ρ′ : Γ and show ∆ ⊢ σρ s b σ ′ ρ′ : Γ′ . x⋆U . Γ . x⋆U . Now assume arbitrary ∆ ⊢ ρ s b σ ′ ρ′ : Γ′ follows by induction hypothesis. The second subgoal ∆ ⊢ First, ∆ ⊢ σρ s ′ ′ b (σρ)(x) s (σ ρ )(x) ⋆ U σρ is just an instance of the second induction hypothesis. Theorem 5.2 (Substitution and functionality). (1) If Γ ⊢ σ : Γ′ and Γ′ ⊢ t : T then Γ ⊢ tσ : T σ. (2) If Γ ⊢ σ : Γ′ . and Γ′ ⊢ t = t′ : T then Γ ⊢ tσ = t′ σ : T σ. (3) If Γ ⊢ σ = σ ′ : Γ′ . and Γ′ ⊢ t : T then Γ ⊢ tσ = tσ ′ : T σ. (4) If Γ ⊢ σ = σ ′ : Γ′ . and Γ′ ⊢ t = t′ : T then Γ ⊢ tσ = t′ σ ′ : T σ. 24 A. ABEL AND G. SCHERER Proof. We demonstrate 4, the other cases are just variations of the theme. First, from b σ ′ : Γ′ by the fundamental lemma for substitutions Γ ⊢ σ = σ ′ : Γ′ we get Γ ⊢ σ s b id : Γ. Now, by the fundamental (Lemma 5.1), using the identity environment Γ ⊢ id s ′ ′ ′ b t σ : T σ, which entails our goal Γ ⊢ tσ = theorem on Γ ⊢ t = t : T we obtain Γ ⊢ tσ s ′ ′ t σ : T σ by Thm. 4.9. 5.2. Context conversion. Context equality ⊢ Γ = Γ′ is defined inductively by the rules ⊢ Γ = Γ′ Γ ⊢ U = U′ . ⊢⋄=⋄ ⊢ Γ. x⋆U = Γ′ . x⋆U ′ All declarative judgements are closed under context conversion. This fact is easy to prove by induction over derivations, but we get it as just a special case of substitution. Lemma 5.3 (Identity substitution). If ⊢ Γ = Γ′ then Γ ⊢ id = id : Γ′ . Proof. By induction on ⊢ Γ = Γ′ . Case ⊢ Γ = Γ′ Γ ⊢ U = U′ ⊢ Γ. x⋆U = Γ′ . x⋆U ′ By induction hypothesis and weakening, Γ. x⋆U ⊢ id = id : Γ′ . Also, Γ. x⋆U ⊢ x = x ⋆ U and by conversion Γ. x⋆U ⊢ x = x ⋆ U ′ . Together, Γ. x⋆U ⊢ id = id : Γ′ . x⋆U ′ . Theorem 5.4 (Context conversion). Let ⊢ Γ′ = Γ. (1) If Γ ⊢ t : T then Γ′ ⊢ t : T . (2) If Γ ⊢ t = t′ : T then Γ′ ⊢ t = t′ : T . Proof. By Thm. 5.2 with Γ′ ⊢ id = id : Γ. U′ As a consequence, context equality is symmetric and transitive (we can trade Γ ⊢ U = for Γ′ ⊢ U = U ′ ). Thus, context conversion can be applied in the other direction as well. 5.3. Inversion, injectivity, and type unicity. A condition for the decidability of type checking is the ability to invert typing derivations. The proof requires substitution. Lemma 5.5 (Inversion). (1) If Γ ⊢ x : T then (x : U ) ∈ Γ for some U with Γ ⊢ U = T . (2) If Γ ⊢ λx⋆U. t : T then Γ. x⋆U ⊢ t : T ′ for some T ′ with Γ ⊢ (x⋆U ) → T ′ = T . (3) If Γ ⊢ t ⋆ u : T then Γ ⊢ t : (x⋆U ) → T ′ and Γ ⊢ u ⋆ U for some U, T ′ with Γ ⊢ T ′ [u/x] = T . (4) If Γ ⊢ s : T then there is (s, s′ ) ∈ Axiom such that Γ ⊢ s′ = T . s1 ,s2 (5) If Γ ⊢ (x⋆U ) → T ′ : T then Γ ⊢ U : s1 and Γ. x⋆U ⊢ T ′ : s2 , and for some s3 we have Γ ⊢ s3 = T and (s1 , s2 , s3 ) ∈ Rule. Proof. Each by induction on the typing derivation. Remark 5.6. The need for inversion during type checking is the only good reason to have separate typing rules and not simply define typing Γ ⊢ t : T as the diagonal Γ ⊢ t = t : T of equality. While by a logical relation argument we will obtain a suitable inversion result for Γ ⊢ (x⋆U ) → T = (x⋆U ) → T —the famous function type injectivity (Theorem 5.7)— it seems hard to get something similar for application t u. IRRELEVANCE IN TYPE THEORY 25 Injectivity for function types w. r. t. typed equality is known to be tricky. It is connected to subject reduction and required for many meta-theoretic results. We harvest it from our Kripke model. s1 ,s2 s′ ,s′ 1 2 Theorem 5.7 (Function type injectivity). If Γ ⊢ (x⋆U ) → T = (x⋆U ′ ) → T ′ : s3 then ′ ′ ′ ′ s1 = s1 and s2 = s2 and Γ ⊢ U = U : s1 and Γ. x⋆U ⊢ T = T : s2 . b Proof. This follows from Lemma 4.7. Or we can prove it directly as follows: Since Γ ⊢ id s s1 ,s2 s ,s 1 2 b (x⋆U ′ ) → T ′ : s3 which by id : Γ we have by the fundamental theorem Γ ⊢ (x⋆U ) → T s ′ ′ ′ b U : s1 and Γ ⊢ U = U ′ : s1 . Since inversion yields first s1 = s1 and s2 = s2 and Γ ⊢ U s b x⋆U , we also obtain Γ. x⋆U ⊢ T s b T ′ : s2 and conclude Γ. x⋆U ⊢ T = T ′ : s2 . Γ. x⋆U ⊢ x s From the inversion lemma we can prove uniqueness of types, since we are dealing with a functional PTS, and we have function type injectivity. Theorem 5.8 (Type unicity). If Γ ⊢ t : T and Γ ⊢ t : T ′ then Γ ⊢ T = T ′ . Proof. By induction on t, using inversion. 5.4. Normalization and Subject Reduction. An immediate consequence of the model construction is that each term has a weak head normal form and that typing and equality is preserved by weak head normalization. Theorem 5.9 (Normalization and subject reduction). If Γ ⊢ t : T then t ց a and Γ ⊢ t = a : T. b t : T which by definition contains a derivation Proof. By the fundamental theorem, Γ ⊢ t s of Γ ⊢ t = ↓t : T . 5.5. Consistency. Importantly, not every type is inhabited in IITT, thus, it can be used as a logic. A prerequisite is that types can be distinguished, which follows immediately from the construction of the logical relation. Lemma 5.10 (Type constructor discrimination). Neutral types, sorts and function types are mutually unequal. (1) Γ ⊢ N 6= s. (2) Γ ⊢ N 6= (x⋆U ) → T . (3) Γ ⊢ s = s′ implies s ≡ s′ . (4) Γ ⊢ s 6= (x⋆U ) → T . Proof. By the fundamental theorem applied to the identity substitution. For instance, assuming Γ ⊢ N = s : s′ we get Γ ⊢ N s s : s′ but this is a contradiction to the definition of s. From normalization and type constructor discrimination we can show that not every type is inhabited. Theorem 5.11 (Consistency). X : Set0 6 ⊢ t : X. 26 A. ABEL AND G. SCHERER Proof. Let Γ = (X : Set0 ). Assuming Γ ⊢ t : X, we have Γ ⊢ a : X for the whnf a of t. We invert on the typing of a. By Lemma 5.10, X cannot be equal to a function type or sort, thus, a can neither be a λ nor a function type nor a sort, it can only be neutral. The only variable X must be in the head of a, but since X is not of function type, it cannot be applied. Thus, a ≡ X and Γ ⊢ X : X, implying Γ ⊢ X = Set0 by inversion (Lemma 5.5). This is in contradiction to Lemma 5.10! 5.6. Soundness of Algorithmic Equality. Soundness of the equality algorithm is a consequence of subject reduction. Theorem 5.12 (Soundness of algorithmic equality). (1) Let ∆ ⊢ t, t′ : T . If ∆ ⊢ t ⇐⇒ b t′ : T then ∆ ⊢ t = t′ : T . ′ (2) Let ∆ ⊢ n, n : T . If ∆ ⊢ n ←→ b n′ : U then ∆ ⊢ n = n′ : U and ∆ ⊢ U = T . Proof. Generalize the theorem to all six algorithmic equality judgments and prove it by induction on the algorithmic equality derivation. Since we have subject reduction, the proof proceeds mechanically, because each algorithmic rule corresponds, modulo weak head normalization, to a declarative rule. Case ∆ ⊢ T : s and ∆′ ⊢ T ′ : s and ∆ ⊢ ↓T ⇐⇒ ↓T ′ ∆ ⊢ T ⇐⇒ b T′ ′ By induction hypothesis, ∆ ⊢ ↓T = ↓T : s. By subject reduction ∆ ⊢ T = ↓T : s and ∆ ⊢ T ′ = ↓T ′ : s. By transitivity ∆ ⊢ T = T ′ : s. Case ∆ ⊢ T ⇐⇒ b T′ ∆ ⊢ T ⇐⇒ T ′ : s By induction hypothesis, ∆ ⊢ T = T ′ : s. 5.7. Symmetry and Transitivity of Algorithmic Equality. Since algorithmic equality is sound for well-typed terms, it is also symmetric and transitive. Lemma 5.13 (Type and context conversion in algorithmic equality). Let ⊢ ∆ = ∆′ . (1) If ∆ ⊢ A, A′ and ∆ ⊢ A ⇐⇒ A′ then ∆′ ⊢ A ⇐⇒ A′ . (2) If ∆ ⊢ n, n′ : A and ∆ ⊢ n ←→ n′ : A then ∆′ ⊢ n ←→ n′ : A′ for some A′ with ∆ ⊢ A = A′ . (3) If ∆ ⊢ t, t′ : A and ∆ ⊢ t ⇐⇒ t′ : A and ∆ ⊢ A = A′ then ∆′ ⊢ t ⇐⇒ t′ : A′ . Proof. By induction on the derivation of algorithmic equality, where we extend the statements to ←→ b and ⇐⇒ b accordingly. (1) Type equality. Case ∆ ⊢ U ⇐⇒ b U′ ∆. x⋆U ⊢ T ⇐⇒ b T′ ∆ ⊢ (x⋆U ) → T ⇐⇒ (x⋆U ′ ) → T ′ By inversion, ∆ ⊢ U, U ′ and by induction hypothesis, ∆′ ⊢ U ⇐⇒ b U ′ . Again by ′ ′ inversion, ∆. x⋆U ⊢ T and ∆. x⋆U ⊢ T , yet by soundness of algorithmic equality, ∆ ⊢ U = U ′ , hence ∆. x⋆U ⊢ T ′ by context conversion. Further, ⊢ ∆. x⋆U = IRRELEVANCE IN TYPE THEORY 27 ∆′ . x⋆U . Thus, we can apply the other induction hypothesis to obtain ∆′ . x⋆U ⊢ T ⇐⇒ b T ′ , which finally yields ∆′ ⊢ (x⋆U ) → T ⇐⇒ (x⋆U ′ ) → T ′ . (2) Structural equality. Case (x : T ) ∈ ∆ ∆ ⊢ x ←→ b x:T Since ⊢ ∆ = ∆′ , there is a unique (x : T ′ ) ∈ ∆′ with ∆ ⊢ T = T ′ . Hence, ∆′ ⊢ x ←→ b x : T ′. Case Type-directed equality. Case ∆ ⊢ t, t′ : T and ∆ ⊢ T = T ′ and T ցA ∆ ⊢ t ⇐⇒ t′ : A ∆ ⊢ t ⇐⇒ b t′ : T By normalization, T ′ ց A′ , and subject reduction ∆ ⊢ A = T = T ′ = A′ . Since by conversion, ∆ ⊢ t, t′ : A, by induction hypothesis ∆′ ⊢ t ⇐⇒ t′ : A′ . Thus, ∆′ ⊢ t ⇐⇒ b t′ : T ′ . Case ∆ ⊢ (x⋆U ) → T = A′ and ∆. x⋆U ⊢ t ⋆ x ⇐⇒ b t′ ⋆ x : T ∆ ⊢ t ⇐⇒ t′ : (x⋆U ) → T By injectivity A′ ≡ (x⋆U ′ ) → T ′ with ∆ ⊢ U = U ′ and ∆. x⋆U ⊢ T = T ′ . Since ⊢ ∆. x⋆U = ∆′ . x⋆U ′ , by induction hypothesis we have ∆′ . x⋆U ′ ⊢ t ⋆ x ⇐⇒ b t′ ⋆ x : ′ ′ ′ ′ ′ T . We conclude ∆ ⊢ t ⇐⇒ t : (x⋆U ) → T . Lemma 5.14 (Algorithmic equality is transitive). Let ⊢ ∆ = ∆′ . In the following, let the terms submitted to algorithmic equality be well-typed. (1) If ∆ ⊢ n1 ←→ b n2 : T and ∆′ ⊢ n2 ←→ b n3 : T ′ then ∆ ⊢ n1 ←→ b n3 : T and ′ ∆ ⊢T =T . (2) If ∆ ⊢ t1 ⇐⇒ b t2 : T and ∆′ ⊢ t2 ⇐⇒ b t3 : T ′ and ∆ ⊢ T = T ′ then ∆ ⊢ t1 ⇐⇒ b t3 : T . ′ (3) If ∆ ⊢ T1 ⇐⇒ b T2 : s and ∆ ⊢ T2 ⇐⇒ b T3 : s then ∆ ⊢ T1 ⇐⇒ b T3 : s Proof. We extend these statements to ←→ and ⇐⇒ and prove them simultaneously by induction on the first derivation. Case ∆′ ⊢ n2 ←→ b n3 : T ′ ∆ ⊢ n1 ←→ b n2 : T ∆ ⊢ n1 ⇐⇒ n2 : N ∆′ ⊢ n2 ⇐⇒ n3 : N ′ By induction hypothesis ∆ ⊢ n1 ←→ b n3 : T , hence, ∆ ⊢ n1 ⇐⇒ n3 : N . Case ∆ ⊢ N1 ←→ b N2 : T ∆ ⊢ N2 ←→ b N3 : T ′ ∆ ⊢ N1 ⇐⇒ N2 ∆ ⊢ N2 ⇐⇒ N3 Analogously. 28 A. ABEL AND G. SCHERER Case s1 ,s2 ∆ ⊢ n1 ←→ n2 : (x : U ) → T ∆ ⊢ u1 ⇐⇒ b u2 : U ∆ ⊢ n1 u1 ←→ b n2 u2 : T [u1 /x] s′ ,s′ 1 2 ∆′ ⊢ n2 ←→ n3 : (x : U ′ ) → T′ ∆′ ⊢ u2 ⇐⇒ b u3 : U ′ ∆′ ⊢ n2 u2 ←→ b n3 u3 : T ′ [u2 /x] s1 ,s2 s1 ,s2 By induction hypothesis we have ∆ ⊢ n1 ←→ n3 : (x : U ) → T and ∆ ⊢ (x : U ) → s′ ,s′ 1 2 T = (x : U ′ ) → T ′ which gives in particular s1 = s′1 , s2 = s′2 , and ∆ ⊢ U = U ′ : s1 by function type injectivity (Thm. 5.7). By induction hypothesis we can then deduce ∆ ⊢ u1 ⇐⇒ b u3 : U , and therefore conclude ∆ ⊢ n1 u1 ←→ n3 u3 : T [u1 /x]. Case ∆ ⊢ U1 ⇐⇒ b U2 : s 1 s1 ,s2 ∆. x⋆U1 ⊢ T1 ⇐⇒ b T2 : s2 s1 ,s2 ∆ ⊢ (x⋆U1 ) → T1 ⇐⇒ (x⋆U2 ) → T2 : s3 ∆ ⊢ U2 ⇐⇒ b U3 : s 1 ∆. x⋆U2 ⊢ T2 ⇐⇒ b T3 : s2 s1 ,s2 s1 ,s2 ∆ ⊢ (x⋆U2 ) → T2 ⇐⇒ (x⋆U3 ) → T3 : s2 We get ∆ ⊢ U1 ⇐⇒ b U3 : s1 by transitivity. To also get ∆. x⋆U1 ⊢ T1 ⇐⇒ b T3 : s2 we need ⊢ ∆. x⋆U2 = ∆. x⋆U1 , but this stems from ∆ ⊢ U1 ⇐⇒ b U2 : s1 by soundness of algorithmic equality. Theorem 5.15. The algorithmic equality relations are PERs on well-typed expressions. Proof. By Lemma 5.14 and an analogous proof of symmetry. 6. A Kripke Logical Relation for Completeness The only open issues in the meta-theory of IITT are completeness and termination of algorithmic equality. In parts, completeness has been established in the last section already, namely, we have shown injectivity and discrimination for type constructors. What is missing is injectivity and discrimination for neutrals, e. g., if ∆ ⊢ n u = n′ u′ : T ′ then necessarily ∆ ⊢ n = n′ : (x : U ) → T and ∆ ⊢ u = u′ : U , plus ∆ ⊢ T [u/x] = T ′ . In untyped λcalculus, this is an instance of Boehm’s theorem [Bar84]. We follow Coquand [Coq91] and Harper and Pfenning [HP05] and prove it by constructing a second Kripke logical relation, c , for completeness which is very similar to the first one, s, but at base types additionally requires algorithmic equality to hold. After proving the fundamental lemma again, we know that definitionally equal terms are also algorithmically so. As a consequence, equality is decidable in IITT, and so is type checking. IRRELEVANCE IN TYPE THEORY 29 6.1. Another Kripke Logical Relation. Again, by induction on A ∈ s we define two Kripke relations ∆ ⊢ A c A′ : s ∆ ⊢ a c a′ : A. c and the generalization to ⋆. This time, however, together with their respective closures c at base types we will additionally require algorithmic equality to hold, more precisely, the relation ∆ ⊢ t :⇐⇒: t′ : T which stands for the conjunction of the propositions • ∆ ⊢ t : T and ∆ ⊢ t′ : T , and • ∆ ⊢ t ⇐⇒ b t′ : T . Note that by soundness of algorithmic equality, :⇐⇒: implies :=:. Again, we allow ourselves rule notation for the defining clauses of c . ∆ ⊢ n :⇐⇒: n′ : N ∆ ⊢ n c n′ : N ∆ ⊢ N :⇐⇒: N ′ : s ∆ ⊢ N c N′ : s ⊢∆ (s, s′ ) ∆ ⊢ s c s : s′ c U ′ : s1 ∆ ⊢U c c u′ ⋆ U =⇒ Γ ⊢ T [u/x] c c T ′ [u′ /x] : s2 ∀Γ ≤ ∆, Γ ⊢ u c s1 ,s2 s1 ,s2 ′ ′ ∆ ⊢ (x⋆U ) → T :=: (x⋆U ) → T : s3 s ,s s ,s 1 2 1 2 ∆ ⊢ (x⋆U ) → T c (x⋆U ′ ) → T ′ : s3 (s1 , s2 , s3 ) c u′ ⋆ U =⇒ Γ ⊢ f ⋆ u c c f ′ ⋆ u′ : T [u/x] ∀Γ ≤ ∆, Γ ⊢ u c s,s′ ∆ ⊢ f :=: f ′ : (x⋆U ) → T s,s′ ∆ ⊢ f c f ′ : (x⋆U ) → T ∆ ⊢ ↓t c ↓t′ : ↓T ∆ ⊢ t :=: t′ : T c t′ : T ∆ ⊢tc ∆÷ ⊢ a c a : A ∆÷ ⊢ a′ c a′ : A ∆ ⊢ a c a′ ÷ A c t:T ∆÷ ⊢ t c c t′ : T ∆ ÷ ⊢ t′ c c t′ ÷ T ∆ ⊢tc This logical relation contains only well-typed and definitionally equal terms. It is symmetric, transitive, and closed under weakening and type conversion. The proofs are in analogy to those of Section 4, which are relying on the fact that the underlying relation :=: is a Kripke PER and closed under type conversion. The relation :⇐⇒: underlying c has the same properties, thanks to soundness of algorithmic equality. Note that in the definition of ∆ ⊢ f c f ′ : (x⋆U ) → T we did not require f and f ′ to be algorithmically equal. This would hinder the proof of the fundamental theorem for c , since algorithmic equality is not closed under application by definition—it will follow from the fundamental theorem, though. In the next lemma we shall prove that f and f ′ are algorithmically equal if they are related by c . The name Escape Lemma was coined by Jeffrey Sarnat [SS08]. Lemma 6.1 (Escape from the logical relation). Let ∆ ⊢ A c A′ : s (1) ∆ ⊢ A ⇐⇒ A′ . c t′ : A then ∆ ⊢ t ⇐⇒ t′ : A. (2) If ∆ ⊢ t c (3) If ∆ ⊢ n ←→ n′ ⋆ A and ∆ ⊢ n = n′ ⋆ A then ∆ ⊢ n c n′ ⋆ A. 30 A. ABEL AND G. SCHERER c T′ : s Corollary 6.2. Let ∆ ⊢ T c (1) ∆ ⊢ T ⇐⇒ b T ′. c t′ : T then ∆ ⊢ t ⇐⇒ (2) If ∆ ⊢ t c b t′ : T . c n′ ⋆ T . (3) If ∆ ⊢ n ←→ b n′ ⋆ T and ∆ ⊢ n = n′ ⋆ T then ∆ ⊢ n c The corollary is a direct, non-inductive consequence of the lemma, so we can use it in the proof of the lemma, quoted as “IH”. Proof of the lemma. Simultaneously by induction on A :⇐⇒: A′ : s. Case ∆ ⊢ N c N ′ : s. Case 1. ∆ ⊢ N ⇐⇒ N ′ by assumption. Case 2. We have ∆ ⊢ ↓t ←→ ↓t′ : , thus ∆ ⊢ t ⇐⇒ t′ : N . Case 3. First, consider ⋆ = :. If ∆ ⊢ n = n′ : N and ∆ ⊢ n ←→ n′ : N then ∆ ⊢ n ⇐⇒ n′ : N and trivially ∆ ⊢ n c n′ : N . Then, take ⋆ = ÷. Note that if ∆÷ ⊢ n = n : N and ∆÷ ⊢ n ←→ n : N then ∆÷ ⊢ n ⇐⇒ n : N and ∆÷ ⊢ n c n : N . This implies that if ∆ ⊢ n = n′ ÷ N and ∆ ⊢ n ←→ n′ ÷ N then ∆ ⊢ n ⇐⇒ n′ ÷ N and ∆ ⊢ n c n′ ÷ N . Case ∆ ⊢ s c s : s′ . Case 1. Clearly, ∆ ⊢ s ⇐⇒ s. c T ′ : s. Then ∆ ⊢ T ⇐⇒ Case 2. Let ∆ ⊢ T c b T ′ by IH 1, thus ∆ ⊢ T ⇐⇒ T ′ : s ′ Case 3. For ⋆ = : let ∆ ⊢ N ←→ N : s. By inversion, ∆ ⊢ N ←→ b N ′ : T for some T . ′ ′ Then ∆ ⊢ N ⇐⇒ N and ∆ ⊢ N c N : s by definition. Considering ⋆ = ÷, it is sufficient to observe that ∆÷ ⊢ N ←→ N : s implies ∆÷ ⊢ N ⇐⇒ N and ∆÷ ⊢ N c N : s by definition. Case ∆ ⊢ (x⋆U ) → T c (x⋆U ′ ) → T ′ : s3 . Case 1. Similar to 2. c t′ : (x⋆U ) → T . It is sufficient to show ∆. x⋆U ⊢ Case 2. By assumption, ∆ ⊢ t c ⋆ ′ ⋆ c U ′ : s1 , which includes ∆ ⊢ U , we have ∆. x⋆U ⊢ t x ⇐⇒ b t x : T . Since ∆ ⊢ U c c t′ ⋆ x : ↓T via x = x ⋆ U . Since also ∆. x⋆U ⊢ x ←→ b x ⋆ U , we obtain ∆. x⋆U ⊢ t ⋆ x c IH 3, ∆. x⋆U ⊢ x c x ⋆ U . IH 2 then entails our goal. Case 3. First, the case for ⋆ = :. We reuse variable ⋆ for a different irrelevance marker. We have ∆ ⊢ n ←→ n′ : (x⋆U ) → T . Assume arbitrary Γ :⇐⇒: Γ ≤ ∆ and c u′ ⋆ U , which yields Γ ⊢ u = u′ ⋆ U and Γ ⊢ T [u/x] c c T [u′ /x] : Seti . In Γ ⊢uc ′ case ⋆ = : we have to apply IH 2 for Γ ⊢ u ⇐⇒ u : ↓U . Otherwise, we obtain directly Γ ⊢ n ⋆ u ←→ n′ ⋆ u′ : ↓(T [u/x]). By IH 3, Γ ⊢ n ⋆ u c n′ ⋆ u′ : ↓(T [u/x]). The case for ⋆ = ÷ proceeds analogously. b we extend c c to substitutions and define the semantic validity judgements In analogy to s c Γ and Γ c t : T and Γ c t = t′ : T based on c c . Since by the escape lemma, c x : ∆(x), we have Γ ⊢ id c c id : Γ for c Γ. Finally, we reprove the fundamental ∆ ⊢xc theorem: c ). Theorem 6.3 (Fundamental theorem for c c (1) If ⊢ Γ then Γ. (2) If Γ ⊢ t : T then Γ c t : T . (3) If Γ ⊢ t = t′ : T then Γ c t = t′ : T . IRRELEVANCE IN TYPE THEORY 31 6.2. Completeness and Decidability of Algorithmic Equality. Derivations of algorithmic equality can now be obtained by escaping from the logical relation. Theorem 6.4 (Completeness of algorithmic equality). Γ ⊢ t = t′ : T implies Γ ⊢ t ⇐⇒ b t′ : T . c id : Γ, we have Γ ⊢ t c c t′ : T by the fundamental theorem, and Proof. Since Γ ⊢ id c conclude with Lemma 6.1.2. Termination of algorithmic equality is a consequence of completeness. When invoking the algorithmic equality check ∆ ⊢ t ⇐⇒ b t′ : T on two well-typed expressions ∆ ⊢ t, t′ : T we know by completeness that t and t′ are related to themselves, i. e., ∆ ⊢ t ⇐⇒ b t : T and ∆ ⊢ t′ ⇐⇒ b t′ : T . This means that t, t′ , and T are weakly normalizing by the strategy the equality algorithm implements: reduce to weak head normal form and recursively continue with the subterms. Running the equality check on t and t′ performs, if successful, exactly the same reductions, and if it fails, at most the same reductions in t, t′ , and T . Hence, testing equality on well-typed terms always terminates. This argument has been applied in previous work to untyped equality [AC07]. Here, we apply it to typed equality; it is an alternative to Goguen’s technique of proving termination for typed equality from strong normalization [Gog05], which, in our opinion, does not scale to dependently-typed equality. Lemma 6.5 (Termination of algorithmic equality). Let ⊢ ∆. (1) Type equality. (a) Let ∆ ⊢ A, A′ . If D :: ∆ ⊢ A ⇐⇒ A and ∆ ⊢ A′ ⇐⇒ A′ then the query ∆ ⊢ A ⇐⇒ A′ terminates. (b) Let ∆ ⊢ T, T ′ . If D :: ∆ ⊢ T ⇐⇒ b T and ∆ ⊢ T ′ ⇐⇒ b T ′ then the query ∆ ⊢ T ⇐⇒ b ′ T terminates. (2) Structural equality. Let ∆ ⊢ n : T and ∆ ⊢ n′ : T ′ . (a) If D :: ∆ ⊢ n ←→ n : A and ∆ ⊢ n′ ←→ n′ : A′ then the query ∆ ⊢ n ←→ n′ : ? terminates. If successfully, it returns A and we have ∆ ⊢ A = T = T ′ = A. (b) If D :: ∆ ⊢ n ←→ b n : T and ∆ ⊢ n′ ←→ b n′ : T ′ then the query ∆ ⊢ n ←→ b n′ : ? ′ terminates. If successfully, it returns T and we have ∆ ⊢ T = T . (3) Type-directed equality. (a) Let ∆ ⊢ t, t′ : A. If D :: ∆ ⊢ t ⇐⇒ t : A and ∆ ⊢ t′ ⇐⇒ t′ : A then the query ∆ ⊢ t ⇐⇒ t′ : A terminates. (b) Let ∆ ⊢ t, t′ : T . If D :: ∆ ⊢ t ⇐⇒ b t : T and ∆ ⊢ t′ ⇐⇒ b t′ : T then the query ′ ∆ ⊢ t ⇐⇒ b t : T terminates. Proof. Simultaneously by induction on derivation D. (1) Type equality. Case A = A′ = s. The query ∆ ⊢ A ⇐⇒ A′ terminates successfully. Case A = (x⋆U ) → T and A′ = (x⋆U ′ ) → T ′ . First, the query ∆ ⊢ U ⇐⇒ b U ′ runs. By induction hypothesis, it terminates. If it fails, the whole query fails. Otherwise, the query ∆. x⋆U ⊢ T ⇐⇒ b T ′ is run. By induction hypothesis on ∆. x⋆U ⊢ T ⇐⇒ b T ′ ′ and ∆. x⋆U ⊢ T ⇐⇒ b T ′ , the query terminates. Case A = N and A′ = N ′ neutral. By induction hypothesis on ∆ ⊢ N ←→ b N :T and ∆ ⊢ N ′ ←→ b N ′ : T ′ , the query ∆ ⊢ N ←→ b N ′ : ? terminates. Hence, the query ∆ ⊢ N ⇐⇒ N ′ terminates. 32 A. ABEL AND G. SCHERER Case Weak head normal forms A, A′ not covered by previous cases: the query ∆ ⊢ A ⇐⇒ A′ fails immediately, since there is no applicable algorithmic type equality rule. Case The query ∆ ⊢ T ⇐⇒ b T ′ first invokes weak head normalization on T and T ′ . Both terminate since ∆ ⊢ T ⇐⇒ b T , which implies T ց A, and analogously T ′ ց A′ ′ ′ since ∆ ⊢ T ⇐⇒ b T by assumption. Then, the query ∆ ⊢ A ⇐⇒ A′ is run, which terminates by induction hypothesis on ∆ ⊢ A ⇐⇒ A and ∆ ⊢ A′ ⇐⇒ A′ . (2) Structural equality. Case n = n′ = x. The query ∆ ⊢ n ←→ b n′ : ? terminates successfully, returning type ∆(x). Since ⊢ ∆, by inversion (Lemma 5.5) ∆ ⊢ T = T ′ = ∆(x). Case Neutral relevant application for ∆ ⊢ n u : T0 and ∆ ⊢ n′ u′ : T0′ . ∆ ⊢ n ←→ n : (x : U ) → T ∆ ⊢ u ⇐⇒ b u:U ∆ ⊢ n u ←→ b n u : T [u/x] ∆ ⊢ n′ ←→ n′ : (x : U ′ ) → T ′ ∆ ⊢ u′ ⇐⇒ b u′ : U ′ ∆ ⊢ n′ u′ ←→ b n′ u′ : T ′ [u′ /x] The query ∆ ⊢ n u ←→ b n′ u′ : ? first invokes query ∆ ⊢ n ←→ n′ : ?. By induction hypothesis on ∆ ⊢ n ←→ n : (x : U ) → T and ∆ ⊢ n′ ←→ n′ : (x : U ′ ) → T ′ the query terminates. If it fails the whole query fails. Otherwise it returns a type A in weak head normal form, which is identical to (x : U ) → T by uniqueness of inferred types (Lemma 3.1). Further, ∆ ⊢ (x : U ) → T = (x : U ′ ) → T ′ , and by function type injectivity (Thm. 5.7), ∆ ⊢ U = U ′ and ∆. x : U ⊢ T = T ′ . Thus, we can invoke the induction hypothesis on ∆ ⊢ u ⇐⇒ b u : U and ∆ ⊢ u′ ⇐⇒ b u′ : U ′ ′ ′ (cast from ∆ ⊢ u ⇐⇒ b u : U , Lemma 5.13) to infer that the second subquery ∆ ⊢ u ⇐⇒ b u′ : U terminates. If this one is successful, then by soundness of algorithmic equality, ∆ ⊢ u = u′ : U , which implies ∆ ⊢ T [u/x] = T ′ [u′ /x]. Case Neutral irrelevant application with typing ∆ ⊢ n : (x÷U1 ) → T1 ∆ ⊢ u ÷ U1 ÷ ∆ ⊢ n u : T1 [u/x] and algorithmic self-equality ∆ ⊢ n ←→ n : (x÷U ) → T ∆ ⊢ n ÷ u ←→ b n ÷ u : T [u/x] ∆ ⊢ u′ ÷ U1′ ∆ ⊢ n′ : (x÷U1′ ) → T1′ ∆ ⊢ n′ ÷ u′ : T1′ [u′ /x] ∆ ⊢ n′ ←→ n′ : (x÷U ′ ) → T ′ ∆ ⊢ n′ ÷ u′ ←→ b n′ ÷ u′ : T ′ [u′ /x] The query ∆ ⊢ n ÷ u ←→ b n′ ÷ u′ : ? invokes query ∆ ⊢ n ←→ n′ : ?, which terminates by induction hypothesis. If successfully, then ∆ ⊢ (x÷U1 ) → T1 (x÷U ) → T = (x÷U ′ ) → T ′ = (x÷U1′ ) → T1′ . By function type injectivity, ∆ ⊢ U1 = U = U ′ = U1′ and ∆. x÷U ⊢ T1 = T = T ′ = T1′ . By conversion ∆ ⊢ u = u′ ÷ U , thus, ∆ ⊢ T1 [u/x] = T [u/x] = T ′ [u′ /x] = T1′ [u′ /x]. Case In all other cases, the query ∆ ⊢ n ←→ b n′ : ? fails immediately. Case The query ∆ ⊢ n ←→ n : ? spawns subquery ∆ ⊢ n ←→ b n′ : ? which terminates ′ by induction hypothesis on ∆ ⊢ n ←→ b n : T and ∆ ⊢ n ←→ b n′ : T ′ . If successfully, it returns type T , and since T ց A, the original query also terminates, returning A. (3) Type-directed equality. Case Function type ∆ ⊢ t, t′ : (x⋆U ) → T . The query ∆ ⊢ t ⇐⇒ t′ : (x⋆U ) → T spawns subquery ∆. x⋆U ⊢ t ⋆ x ⇐⇒ b t′ ⋆ : T . Since ∆. x⋆U ⊢ t ⋆ x, t′ ⋆ x : T and the IRRELEVANCE IN TYPE THEORY 33 subquery terminates by induction hypothesis on ∆. x⋆U ⊢ t ⋆ x ⇐⇒ b t ⋆ x : T and ′ ⋆ ′ ⋆ ∆. x⋆U ⊢ t x ⇐⇒ b t x : T. Case Sort ∆ ⊢ T, T ′ : s. The query ∆ ⊢ T ⇐⇒ T ′ : s calls ∆ ⊢ T ⇐⇒ b T ′ , which terminates by induction hypothesis on ∆ ⊢ T ⇐⇒ b T and ∆ ⊢ T ′ ⇐⇒ b T ′. Case Neutral type N . t′ ց n ′ ∆ ⊢ n′ ←→ b n′ : T ′ tցn ∆ ⊢ n ←→ b n:T ∆ ⊢ t ⇐⇒ t : N ∆ ⊢ t′ ⇐⇒ t′ : N ′ The query ∆ ⊢ t ⇐⇒ t : N first weak head normalizes t and t′ . By assumption, t ց n and t′ ց n′ , so this terminates. The subquery ∆ ⊢ n ←→ b n′ : ? terminates by induction hypothesis. Thus, the whole query terminates. Case If A is neither a function type, a sort, or a neutral type, the query ∆ ⊢ t ⇐⇒ t′ : A fails immediately. Case The query ∆ ⊢ t ⇐⇒ b t′ : T first weak head normalizes T which terminates since T ց A by assumption. Then it calls ∆ ⊢ t ⇐⇒ t′ : A which terminates by induction hypothesis. Theorem 6.6. If ∆ ⊢ t : T and ∆ ⊢ t′ : T then the query ∆ ⊢ t ⇐⇒ b t′ : T terminates. Proof. From the lemma by completeness of algorithmic equality. Thus we have shown that algorithmic equality is correct, i. e., sound, complete, and terminating. Together, this entails decidability of equality in IITT. Theorem 6.7 (Decidability of IITT). (1) Γ ⊢ t = t′ : T is decidable. (2) Γ ⊢ t : T is decidable. Proof. Decidability of equality follows from soundness (Thm. 5.12), completeness (Thm. 6.4), and termination (Thm. 6.6). Decidability of typing follows from decidability of type conversion, weak head normalization, and function type injectivity, using inversion (Lemma 5.5) on typing derivations. Any reasonable type inference algorithm will do. 7. Extensions Data types and recursion. The semantics of IITT is ready to cope with inductive data types like the natural numbers and the associated recursion principles. Recursion into types, aka known as large elimination, is also accounted for since we have universes and a semantics which does not erase dependencies (unlike Pfenning’s model [Pfe01]). Types with extensionality principles. One purpose of having a typed equality algorithm is to handle η-laws that are not connected to the shape of the expression (like η-contraction for functions) but to the shape of the type only. Typically these are types T with at most one inhabitant, i. e., the empty type, the unit type, singleton types or propositions.6 For such T we have the η-law Γ ⊢ t, t′ : T Γ ⊢ t = t′ : T 6Some care is necessary for the type of Leibniz equality [Abe09, Wer08]. 34 A. ABEL AND G. SCHERER which can only be checked in the presence of type T . Realizing such η-laws gives additional “proof” irrelevance which is not covered by Pfenning’s irrelevant quantification (x÷U ) → T . Internal erasure. Terms u÷U in irrelevant position are only there to please the type checker, they are ignored during equality checking. This can be inferred from the substitution principle: If Γ. x÷U ⊢ T and Γ ⊢ u, u′ ÷ U , then Γ ⊢ T [u/x] = T [u′ /x]; the type T has the same shape regardless of u, u′ . Hence, terms like u serve the sole purpose to prove some proposition and could be replaced by a dummy • immediately after type-checking. Internal erasure can be realized by making Γ ⊢ t ÷ T a judgement (as opposed to just a notation for Γ÷ ⊢ t : T ) and adding the rule Γ ⊢ t÷T . Γ ⊢•÷T The rule states that if there is already a proof t of T , then • is a new proof of T . This preserves provability while erasing the proof terms. Conservativity of this rule can be proven as in joint work of the author with Coquand and Pagano [ACP11]. 8. Conclusions We have extended Pfenning’s notion of irrelevance to a type theory IITT with universes that accommodates types defined by recursion. We have constructed a Kripke model s that shows soundness of IITT, yielding normalization, subject reduction and consistency, plus syntactical properties of the judgements of IITT. A second Kripke logical relation c has proven correctness of algorithmic equality and, thus, decidability of IITT. Integrating irrelevance and data types in dependent type theory does not seem without challenges. We have succeeded to treat Pfenning’s notion of irrelevance, but our proof does not scale directly to parametric function types, a stronger notion of irrelevant function types called implicit quantification by Miquel [Miq01b].7 Two more type theories build on Miquel’s calculus [Miq01a], Barras and Bernardo’s ICC∗ [BB08] and Mishra-Linger and Sheard’s Erasure Pure Type Systems (EPTS) [MLS08], but none has offered a satisfying account of large eliminations yet. Miquel’s model [Miq00] features data types only as impredicative encodings. For irrelevant, parametric, and recursive functions to coexist it seems like three different function types are necessary, e. g., in the style of Pfenning’s irrelevance, extensionality and intensionality. We would like to solve this puzzle in future work, not least to implement high-performance languages with dependent types. Acknowledgments. The first author thanks Bruno Barras, Bruno Bernardo, Thierry Coquand, Dan Doel, Hugo Herbelin, Conor McBride, Ulf Norell, and Jason Reed for discussions on irrelevance in type theory. Work on a previous paper has been carried out while he was invited researcher at PPS, Paris, in the INRIA πr 2 team headed by Pierre-Louis Curien and Hugo Herbelin. The second author acknowledges financial support by the École Normale Superiéure de Paris for his internship at the Ludwig-Maximilians-Universität München from May to September 2011. We thank the two anonymous referees, who suggested changes and examples which significantly improved the presentation, and the patience of the editors waiting for our revisions. 7A function argument is parametric if it is irrelevant for computing the function result while the type of the result may depend on it. In Pfenning’s notion, the argument must also be irrelevant in the type. 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A Method for Reducing the Severity of Epidemics by Allocating Vaccines According to Centrality Krzysztof Drewniak kdrewniak@utexas.edu Joseph Helsing JosephHelsing@my.unt.edu arXiv:1407.7288v1 [cs.SI] 27 Jul 2014 ABSTRACT One long-standing question in epidemiological research is how best to allocate limited amounts of vaccine or similar preventative measures in order to minimize the severity of an epidemic. Much of the literature on the problem of vaccine allocation has focused on influenza epidemics and used mathematical models of epidemic spread to determine the effectiveness of proposed methods. Our work applies computational models of epidemics to the problem of geographically allocating a limited number of vaccines within several Texas counties. We developed a graph-based, stochastic model for epidemics that is based on the SEIR model, and tested vaccine allocation methods based on multiple centrality measures. This approach provides an alternative method for addressing the vaccine allocation problem, which can be combined with more conventional approaches to yield more effective epidemic suppression strategies. We found that allocation methods based on in-degree and inverse betweenness centralities tended to be the most effective at containing epidemics. Categories and Subject Descriptors J.3 [Life and Medical Sciences]: Health; G.2.2 [Discrete Mathematics]: Graph Theory—Graph algorithms General Terms Algorithms, Experimentation, Performance Keywords Computational epidemiology, health informatics, vaccine distribution, centrality measures 1. INTRODUCTION One long-standing question in epidemiological research is how best to allocate limited amounts of vaccine or similar preventative measures in order to minimize the severity of an epidemic. Discovering a way to reduce the severity of epidemics would benefit society by not only preventing loss Armin R Mikler mikler@cs.unt.edu of life, but also reducing the overall disease burden of those epidemics. These burdens, or societal costs, can be seen in reduced economic productivity due to widespread sickness, or increased strain on health infrastructure. Much of the literature on the problem of vaccine allocation has focused on influenza epidemics and has used mathematical models of epidemic spread to determine the effectiveness of proposed methods [12, 13, 17, 22, 24]. Most investigations of vaccine allocation performed their analyses on geographically large scales and focused on determining which subpopulations should be prioritized for vaccination. Since previous research focuses on large-scale, subpopulation-based models, our work explores alternative approaches to the question of vaccine allocation. Much work has been devoted to the development of computational models for disease simulation. Mikler et. al. have developed multiple stochastic models of disease spread in a population, which use the standard SEIR(S) model [15, 19]. Some of these models, such as their Global Stochastic Cellular Automata model [16], can be adapted to the problem of vaccine allocation. Simulation techniques relevant to our work have been developed by Indrakanti [10], who implemented a SEIR-based framework for simulating epidemic spread within a county. Our work applies computational models of epidemics to the problem of geographically allocating a limited number of vaccines within several Texas counties. We developed a graph-based, stochastic model for epidemics that is based on the SEIR model and the work of Mikler et. al. [16]. Various centrality measures have been proposed over the years [18, 2, 3, 7, 4], mainly in the field of social network analysis, which provide means for determining which nodes in a graph are the most important. Our model was then used to investigate various centrality-based vaccine allocation strategies, in which vaccines are allocated to various census blocks within the county in order of their centrality measure score. This approach provides an alternative method for addressing the vaccine allocation problem, which can be combined with more conventional approaches to yield more effective epidemic suppression strategies. In this paper, we present the results of several experiments with centrality-based vaccine allocation strategies. These experiments were performed on graphs constructed from the census data of several Texas counties, and have yielded results that indicate directions for further study. 2. BACKGROUND 2.1 Vaccine Allocation Several authors have previously addressed the problem of vaccine allocation. Medlock and Galvani have investigated the question of which groups should be prioritized for influenza vaccination throughout the United States in the event of a shortage [13]. They developed a model, parameterized with real-world data, for analyzing various vaccination strategies under multiple effectiveness criteria. Medlock and Galvani found that optimal epidemic control could be achieved by targeting adults between the ages of 30 and 39 and children, and that current CDC vaccine allocation recommendations were suboptimal under all considered measures. Tuite et. al. used similar methods to investigate vaccine allocation strategies in Canada during a H1N1 epidemic [22]. They found that targeting high-risk groups, such as people with chronic medical conditions, and then targeting age groups, such as children, that are the most likely to develop complications after influenza infection will cause the greatest reduction in the number of deaths and serious illnesses caused by an influenza epidemic. Similar work was performed by Matrajt and Longini [12]. In their work, Matrajt and Longini attempted to determine whether the optimal vaccine allocation strategy varies with the state of the epidemic. They developed an SIR-based model to analyze the spread of influenza among and between several groups of people in idealized countries, and used criteria similar to those of Medlock and Galvani to ascertain the effectiveness of their methods. The authors determined that the demographic structure of the population in an area significantly affects the optimal allocation strategy. They also found that, at the beginning of an epidemic, those groups that are the most likely to transmit influenza should be targeted immediately, while the groups that are most vulnerable should be targeted after a point before, but near, the peak of the epidemic. The work of Mylius et. al modeled influenza epidemics and found that individuals that are most at risk for complications should be targeted if additional vaccines become available during an epidemic [17]. Mylius et. al. also found that schoolchildren should be prioritized for vaccination at the beginning of an epidemic, as they are heavily involved in disease transmission. Their conclusions bear a close resemblance to those of Matrajt and Longini. The problem of large-scale vaccine allocations were addressed by [12, 13, 17, 22]. They used different models to reach their conclusions, but used similar metrics, such as the reduction in the predicted number of deaths from a disease after the vaccination program, to assess the significance of their results. None of these works addressed the question of vaccine allocation at the level of counties or other similar geographic divisions, nor did they address the vaccine allocation problem from a geographic perspective. Another approach to the vaccine allocation problem, which we extend in our work, can be found in the work of Johnson [11]. Johnson’s approach, unlike those of many other researchers, focuses on vaccine allocation in the small scale. Johnson generated synthetic social network graphs and used various measures of centrality to determine which individuals within those graphs should be vaccinated. Her results showed that the optimal vaccination strategy depends on the structure of the social network it is applied to. Our work adapts Johnson’s methods to the problem of allocating vaccinations within a county. 2.2 Centrality and its Applications The notion of centrality has been used in various fields, most notably in the study of social networks. Multiple centrality measures exist, all of which allow the vertices of a graph to be ranked in order of their importance. Out-Degree and in-degree centralities were first defined by Nieminen [18]. In those measures, the centrality c of a node n in a graph G = (V, E), where V is the set of nodes in the graph, u and v are nodes in V, E is the set of edges, and \|V \| is the number of nodes is dn (1) c= \|V \| − 1 where dn is the out-degree or in-degree of n. Eigenvector centrality, first proposed by Bonacich [2], of a graph G = (V, E) can be calculated by representing G as an adjacency matrix A, where ( 1 if (u, v) ∈ E Auv = (2) 0 if (u, v) 6∈ E Then, the eigenvector centrality x is defined as Ax = λx (3) where λ is the largest eigenvalue of A. Eigenvector centrality is useful because it gives a higher centrality score to a highdegree node that is connected to other high-degree nodes than to a high-degree node connected to low-degree nodes [3]. Another important measure of centrality is betweenness centrality. Betweenness centrality, first described by Freeman [7], is defined as follows, for a node n in a graph G = (V, E), where dst (n) is the number of shortest paths from s ∈ V to t ∈ V that include n, and dst is the number of shortest paths from s to t where the shortest path between two nodes is the path where the sum of the weights of its edges is minimized X dst (n) (4) dst s6=n6=t∈V Betweenness centrality gives higher centrality scores to nodes that are most likely to be involved in the transmission of information throughout a graph [4]. Centrality has been applied to multiple epidemiological questions. Rothenberg et. al. applied various centrality measures to a social network graph generated from CDC data on the spread of HIV in Colorado Springs [20]. Rothenberg analyzed the collected data under multiple centrality measures and found that all of the measures identified all but one of the HIV cases as non-central. Their work also noted several differences in response patterns to the CDC questionnaire between people with high centrality and people with low centrality under all measures, which were closely correlated. Similar work was also performed by Christley et. al. [6]. Christley generated synthetic social networks and performed SIR-based simulations in order to study the applicability of centrality to the problem of identifying individuals at high risk of HIV infection. They found that degree centrality performed no worse than other centrality measures, such as betweenness, but noted that the results might not be valid for larger datasets. Centrality has also played a key role in Johnson’s work, which was discussed previously [11]. Johnson investigated how vaccinating those individuals identified as central in a social network would affect the spread of disease in that network. Johnson found that vaccination strategies based on centrality measures were an effective means of mitigating outbreaks. Related work was performed by Rustam, who applied centrality to the spread of viruses and worms in computer networks [21]. He found that nodes with high centrality scores have a large amount of influence on the spread of a virus, especially when those nodes are rated central by multiple measures. 2.3 Computational Simulation of Epidemics Several methods for simulating epidemics computationally are based on the SEIR model of epidemics [15, 19, 16, 10]. In the SEIR model, the population is divided into four classes: susceptible individuals, who can become infected; exposed or latent individuals, who have been infected but are not capable of spreading the infection; infectious individuals, who can spread the disease to susceptible individuals; and recovered individuals, who can no longer be infected. Mikler et. al. proposed a SEIR-based model that uses a stochastic cellular automaton [16]. The concepts used in their model are applicable to a wide range of epidemic simulations. Agent-based and metapopulation models are the two dominant methods for computational epidemic simulation [1]. In agent-based models, each individual within the population is simulated. Such models, which are often implemented using cellular automata [16, 15, 8], are useful because they provide insights into the progression of the disease. However, they are impractical when large populations need to be simulated due to space required for information about each agent in the simulation. Metapopulation models, on the other hand, break the population into subpopulations and then simulate the interactions between and within the subpopulations [10, 23]. Such models sacrifice some of the precision offered by agent-based modeling in exchange for the ability to simulate large populations. Our work uses a metapopulation-based model, where each census block constitutes a subpopulation. This helps to overcome the excessive computational resources that would be required to simulate hundreds of thousands of individuals. A closely related model that we have adapted for our own work has been developed by Indrakanti [10]. Indrakanti developed a county-level epidemic simulator, which uses census blocks as the base unit of simulation. The model allows for contacts to be generated between any two census blocks, and uses an interaction coefficient to determine the likelihood of contact. The model was used to conduct several experiments which found that epidemics with higher infectivity (likelihood of spread during a contact) reached their peak percentage of infected individuals earlier. It also discovered that the distribution of population between census blocks had a significant effect on the spread of disease. 3. METHODS 3.1 Representing Counties The United States Census Bureau provides geographic data on US counties. They subdivide counties into census blocks at the finest granularity, which are typically bounded by roads, streets, or water features. Being that they may be varying sizes and shapes, rural census blocks are often significantly larger than urban ones. Additionally, the population of a census block may vary greatly as there are many cencus-blocks that contain zero population, i.e. farmland, greenspace, or bodies of water, and others that contain highdesntiy residential structures, i.e. appartment buildings, that may have several hundred people [5]. This geographic data, along with the population of each census block, was obtained for multiple Texas counties, specifically Rockwall, Hays, and Denton County, and was stored in a PostGIS database. The centroid of each census block, which is the average of the points defining the census block, was precomputed and stored within the database. After the census block data was obtained, several methods for representing a county as a graph with census block nodes were investigated. Rockwall County was used for these investigations due to its small size, allowing centrality measures to be computed quickly. All of these representations were based on various methods of constructing a graph G = (V, E) to represent a county, where V , the set of nodes, was the set of populated census blocks in the county. Our first approach to representing counties as a graph, which was ineffective, was a representation where E = {(u, v) \| u, v ∈ V, u 6= v, ST Touches(u, v)}. The ST Touches() function, which determines whether two geographic entities are adjacent, was provided by PostGIS. This representation used undirected edges. Another of our early attempts was an approach where E = {(u, v) \| u, v ∈ V, u 6= v, ST Distance(Centroid(u), v) < r} for various constant values of r, where the ST Distance() function was used to compute the shortest distance between the centroid of u and the block v. The Centroid() function retrieves the centroid of a block from the database. A variation of this approach where the r value for each node u ∈ V was lowered by multiplying it by 1− Population(u) 1000 (5) on the assumption that the population of highly-populated blocks was more likely to interact within the block than that of sparsely-populated blocks was also investigated. All of these approaches were ineffective and gave unrealistic results, such as the identification of a large rural block with a population of 5 as the most central block within Rockwall County. Therefore, they were rejected in favor of a representation based on weighted edges. The results of our investigations led us to use the following method to represent US counties in our experiments. In our final model, the county was represented as a directed graph G = (V, E), where V was the set of populated census blocks. The set of edges E was equal to {(u, v, Wu→v \| u ∈ V, v ∈ V, u 6= v}, where Wu→v was the weight of the edge from u to v. The weights were computed by the following formula, which was adapted from the work of Mikler et. al.[16]: Wu→v = Population(u) × Population(v) 10000 × ST Distance(Centroid(u), v) + 0.00001 (6) This definition assigns higher weights to blocks that are more likely to interact, a fact that is used in our simulator. The multiplication by 1/10000 was included to offset the behavior of ST Distance(), which returns distance as real value less than 1. If the distances were left unscaled, the division could have produced weights outside of the range that can be accurately represented by real values in a computer. The addition of 0.00001 was used to address cases where the centroid of block u is located within block v, which ordinarily results in a distance of 0, from producing an error. Once these weights were computed, the graphs were represented using the NetworkX library [9]. 3.2 Centrality Measures We modified the centrality measures described in section 2.2 to work with our representation. Code from [9] was used as a basis for our work. Out-degree and in-degree centralities were trivially adapted; the out-degree and in-degree of a node were redefined to be the sum of the weights of the outedges or in-edges, respectively, instead of a count of those edges. The implementation of eigenvector centrality in NetworkX already allowed for weighted edges, so it was used without modification. Since that implementation of eigenvector centrality used only the out-edges for its computations, a variation of the measure, eigenvector-in centrality, was also tested where the in-edges were used instead. Betweenness centrality was the most heavily modified measure. Since betweenness centrality relies on shortest paths, the reciprocals of all edge weights were taken before centrality computation, so that the shortest path between nodes s and t is the one that the disease is most likely to spread across. Afterwards, the betweenness centrality scores were computed by a variation of Brandes’s algorithm [4], where Dijkstra’s algorithm was applied to each node in parallel as opposed to sequentially. This modification was found to have no effect on the results of the centrality computation. The centrality measure described above was reported as inverse betweenness centrality. Finally, we defined random centrality, which is not a centrality measure, for use as a baseline. In random centrality, each node is assigned a random real value in [0, 1), which is used as its centrality score. The results of all centrality calculations except random centrality, which was re-computed before each set of simulations, were saved to avoid expensive recomputation when multiple experiments were performed on the same data. 3.3 Graph-Based Stochastic Epidemic Simulation A stochastic epidemic simulator based on the SEIR model and the work of [10] was developed for use with our representation of counties. The simulation began by constructing the graph G = (V, E) as described above, and computing all centrality measures that were to be tested. Each node n ∈ V was labeled with the following attributes: \|Nn \|, the total Figure 1: An illustration of the grouping of census blocks that have been split by the simulator. population of the block, \|Sn \| = \|Nn \|, the number of susceptibles, and \|En \|, \|In \|, \|Rn \| = 0, where \|En \| is the number of latent (exposed) individuals in the block, \|In \| the number of infectious, and \|Rn \| the number of recovered individuals. Throughout the simulation, \|Sn + En + In + Rn \| = \|Nn \|. In addition to these counts, each node was labeled with a mapping ME,n = {0 7→ E0,n , 1 7→ E1,n . . . Tlp 7→ ETlp ,n }, where Tlp is the latent period and Em,n is the number of people in block n with 0 ≤ m ≤ Tlp days remaining until they transition to the next simulation state. In this mapping, Tlp X Em,n = \|En \|. A similar mapping was also produced for m=0 the infectious period, MI,n = {0 7→ I0,n . . . Tip 7→ ITip ,n , where Tip is the infectious period. For this mapping, 0 ≤ m ≤ Tip , similarly to ME,n . These attributes are illustrated in Figure 1. For the purposes of experimentation, 20 simulation sets were performed in parallel, with the random number generator reseeded for each set. Each set consisted of multiple simulations with varying values of available vaccine and centrality measures. The simulations were performed on a server with four six-core Intel Xeon E7540 processors and 256 GB of RAM. At the beginning of each set, a proportion p of the census blocks were chosen randomly for initial infection, which remained constant throughout all simulations in the set. The percentage of the population infected initially was varied between experiments. Vaccines were distributed to the entire population of each chosen census until the total supply of vaccines was exhausted. Census blocks were selected for vaccination in a decreasing order based on their centrality. The centrality measure used to allocate vaccines was varied between experiments so that the measures could be compared for effectiveness. The set of population (blocks) that was initially targeted for infection was held constant as the centrality measure and percent of population that was able to receive vaccination were varied. This allowed us to more accurately determine which centrality measure or measures were most effective under varying conditions. In experiments involving the prevention scenario, vaccines were distributed before the initial infection, while they were dis- Contact rate Transmissibility Mobility Latent period (Tlp ) Infectious period (Tip ) Percentage of blocks to infect (p) 20 0.05 0.99 2 days 3 days 1% Table 1: Parameters kept constant throughout all experiments. These parameters serve to characterize the disease being simulated. Figure 2: An illustration of the infection process used by the simulator. tributed six simulated days after initial infection in the intervention scenario. The value of six days was chosen because at the beginning of day six the initial cases would have transitioned to the recovered state. After the initial infections, which were recorded as day zero and placed those affected into ETE ,n , the simulation was executed until no latent or infectious individuals remained in the population. During each day, after any scheduled vaccine distribution, each block’s ME and MI is updated by setting ME = {i ∈ ME 7→ ME (i + 1), Tlp 7→ 0}, and similarly for MI . After these updates, the census blocks are iterated over. For each block n , i contacts are simulated, where i is In multiplied by the contact rate, 20 contacts/day, which was taken from [10]. For each contact, a random real value in [0, 1) is generated. If this number is greater than or equal to the mobility parameter, which was 0.99, the contact takes place within n, otherwise, the contact takes place within a different census block. The mobility parameter expresses the likelihood that an individual will contact someone who resides in a census block that the individual does not reside in. In the case of non-local contacts, i.e. contacts made between agents in separate blocks, the external block is chosen by the following method. First, the weights (given by Equation 6) of the out-edges of the originating block are normalized by dividing by the out-degree of the block and then sorted in descending order. These values were precomputed once as an optimization. A random target value in [0, 1) is generated, and the list of weights is summed element by element until the sum is greater than or equal to the target value. The node t whose associated weight causes the summation to terminate is chosen as the block for contact. This process is illustrated by Figure 2. Once the block that will be contacted is chosen, a diseasespreading contact occurs if a random real number in [0, 1) is less than the transmissibility parameter, which was 0.05. For non-local contacts, this test is performed before block selection to speed up the simulation significantly. When the contact occurs, one person is, if possible, removed from St , the susceptible population of the target block t, and added to Et , the latent population. The new infection is also added to MEt (Tlp ), which ensures that the newly-infected individual remains in the latent period for Tlp days before becoming infectious. After all the contacts are generated, the SEIR states are updated. For each block n ∈ V , MEn (0) people are removed from En , and added to MIn (Tip ) and In . Similarly, MIn (0) people are transferred from In to Rn . At the conclusion of the updates, the number of people throughout the county in each SEIR state and the percentage of the population in that state are reported. In addition, the number of people who are infected, that is, either latent or infectious, is reported, along with the associated percentage. 3.4 Experiments and Parameters In all of the experiments performed, the parameters listed in Table 1 were used. These parameters were not chosen to reflect an extant disease, and were adapted from the work of Indrakanti [10]. Experiments were performed on Rockwall, Hays, and Denton counties using two epidemic scenarios for each county. In the prevention scenario, vaccines were distributed before any infection took place, and 5 percent the population of the blocks that were selected for infection was initially infected. In the intervention scenario 50 percent of the population of the infected blocks were infected initially, and at the beginning of the 6th day of the simulation vaccines were distributed. These scenarios were intended to simulate a naturally-occurring epidemic of a disease such as influenza or a random, mass-exposure bio-terror attack using perhaps smallpox, respectively. Each combination of county and scenario received its own run of 20 simulation sets. Within each simulation set, the following experiments were performed. The initially infected blocks were held constant for all of these experiments within a set. The amount of vaccine available was varied between 30 percent, 50 percent, 75 percent, and 90 percent of the population of the county. Out-degree, in-degree, eigenvector, eigenvector-in, inverse betweenness, and random centralities were tested. 4. RESULTS In Tables 2 and 3, we present the average peak infected percentage from our experiments. This data was obtained by first finding the maximum percent infected for each simulation. Then, the maximums of each of the 20 simulations for each experiment were arithmetically averaged. The maximum percentage infected at any given time indicates the severity of the epidemic and, consequently, the strain that epidemic places on health resources, such as hospitals. These averages, along with their corresponding standard deviations, are reported in Table 2 for the prevention scenario, and Table 3 for the intervention scenario. These results have been rounded to four decimal places to avoid the appearance Figure 5: 200 most central blocks in Hays County according to multiple centrality measures. Figure 3: Average total number of infected individuals vs. number of vaccines for Denton County intervention scenario experiments, log scale. of false precision. We also obtained the total number of infected individuals for each simulation execution. These totals allow us to determine the severity of infections that do not result in outbreaks. These results were averaged together on a perexperiment basis and plotted on a log scale. The plot for an intervention scenario in Denton County is presented in Figure 3. We also plotted the average percentage of the population that was in each state during every day of the simulation for each experiment to better visualize the progress of the simulated epidemics. Such a plot for a prevention scenario in Hays County at 50 percent vaccination is included as Figure 4. We used maps generated by QuantumGIS, which highlighted approximately the top 10 percent most central blocks according to multiple centrality measures, to ensure that our model was producing realistic results. These maps were also used as an aid in the analysis of our results. A map of Hays County, showing the 200 most central blocks according outdegree, in-degree, and inverse betweenness centralities, is included as Figure 5. This map was created by obtaining the list of the 200 most central census blocks under each centrality measure and marking them with distinct colors. Blocks that were central according to multiple centrality measures received their own colors. 5. DISCUSSION We found that in-degree and inverse betweenness centralities tended to be the most effective at containing epidemics. At low vaccination levels, such as 30 percent and 50 percent, where almost all of the population becomes infected under all vaccination strategies, the peak infected percentage was used to compare the effectiveness of various strategies. Lowering the number of infected people at one time reduces strain on public health infrastructure, allowing the cases that exist to be treated more effectively. At higher vaccination levels, where no significant peak occurred, the state totals, along with the plots of the course of the epidemics, were used to determine effectiveness. The peaks were not used to analyze vaccination effectiveness at the 75 percent and 90 percent intervention experiments, as the peaks were found to reflect the degree to which the disease had spread before the vaccines were distributed, and to have little relation to the effectiveness of allocation strategies. These analyses allowed us to determine which of the tested vaccination strategies was most effective at containing the disease. In all but one of our experiments, at least one of in-degree or inverse betweenness had lower peaks than the control, which was utilizing a random vaccination strategy. Neither measure had a lower peak significantly more times than the other, which shows than inverse betweenness and in-degree centralities are both effective in various situations. In-degree was significantly more effective in the intervention scenario, while inverse betweenness was more effective in the prevention scenario. In-degree centrality allocated vaccines to those blocks that were the most vulnerable, that is, the most likely to be contacted. Because it shielded vulnerable blocks from the disease, in-degree was more effective at containing epidemics that had already begun to spread throughout the county at the time of vaccination. Inverse betweenness centrality, however, allocated vaccines to those blocks most likely to be involved in disease transmission. This allocation % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random No vaccination 59.5177 (0.5823) No vaccination 56.9851 (1.1604) No vaccination 62.3313 (0.4791) Prevention 30% (23501) 31.8124 (0.3003) 29.2634 (0.3717) 32.5062 (0.3980) 29.9352 (0.4541) 29.1990 (0.2862) 29.7944 (0.4934) 30% (47132) 32.6743 (0.3588) 26.1614 (0.5565) 28.1819 (0.2369) 27.7457 (0.3377) 28.0340 (0.4022) 29.3217 (0.3645) 30% (198784) 34.9600 (0.1829) 32.2106 (0.1637) 34.4785 (0.1946) 32.1597 (0.1371) 32.3480 (0.1617) 32.3165 (0.2183) Scenario: Rockwall 50% (39168) 12.9924 (0.2802) 9.6473 (0.2685) 13.9375 (0.3912) 12.5988 (0.3685) 10.5134 (0.2083) 10.5293 (0.2976) Hays County 50% (78553) 13.8937 (0.1090) 9.3640 (0.1403) 11.1222 (0.1744) 10.8385 (0.1924) 9.9216 (0.1894) 10.5042 (0.2072) Denton County 50% (331307) 14.4588 (0.0759) 13.1199 (0.0911) 15.3266 (0.0855) 13.8856 (0.1084) 12.8719 (0.1070) 13.0035 (0.0921) County 75% (58752) 0.0207 (0.0079) 0.0181 (0.0086) 1.4525 (0.7278) 1.1630 (0.6695) 0.0220 (0.0106) 0.0184 (0.0135) 90% (70503) 0.0073 (0.0030) 0.0051 (0.0051) 0.0079 (0.0038) 0.0077 (0.0037) 0.0068 (0.0030) 0.0060 (0.0081) 75% (117830) 0.0208 (0.0066) 0.0172 (0.0071) 0.0183 (0.0058) 0.0190 (0.0061) 0.0175 (0.0057) 0.0244 (0.0115) 90% (141396) 0.0053 (0.0029) 0.0059 (0.0036) 0.0052 (0.0023) 0.0050 (0.0025) 0.0062 (0.0025) 0.0095 (0.0094) 75% (496960) 0.0196 (0.0028) 0.0178 (0.0040) 0.0374 (0.0359) 0.0212 (0.0039) 0.0190 (0.0024) 0.0186 (0.0086) 90% (596353) 0.0059 (0.0014) 0.0057 (0.0020) 0.0058 (0.0014) 0.0058 (0.0014) 0.0057 (0.0016) 0.0061 (0.0038) Table 2: For the prevention scenario, average peak infected percentage with standard deviation in parentheses. Lowest values in boldface. Because there were no vaccines to be distributed, there is only one value for no vaccination. % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random % pop (# vaccines) Out-Degree Inverse Betweenness Eigenvector Eigenvector-In In-Degree Random No vaccination 59.3418 (1.0607) No vaccination 58.0324 (1.8390) No vaccination 62.8987 (0.5339) Intervention Scenario: Rockwall County 30% (23501) 50% (39168) 75% (58752) 32.0758 (0.5265) 13.5754 (0.4445) 1.7544 (0.5848) 29.4420 (0.5808) 10.1374 (0.3663) 1.8280 (0.6144) 32.8470 (0.7541) 15.2428 (0.4611) 2.0908 (0.3910) 30.7960 (0.7225) 13.8876 (0.4673) 1.8606 (0.4803) 29.3326 (0.7046) 10.6855 (0.4030) 1.7150 (0.5744) 30.5434 (0.7600) 11.9360 (0.3728) 1.7970 (0.6138) Hays County 30% (47132) 50% (78553) 75% (117830) 32.3683 (0.7623) 13.4622 (0.3566) 1.8578 (0.6219) 27.5758 (0.9813) 10.2484 (0.4134) 1.9364 (0.6338) 28.9617 (0.8530) 12.1266 (0.4515) 1.8978 (0.6402) 28.4758 (0.7967) 11.9360 (0.4113) 1.8689 (0.6336) 28.2601 (0.9398) 9.8329 (0.4272) 1.8171 (0.6087) 29.0436 (1.0581) 11.3709 (0.6273) 1.9226 (0.6457) Denton County 30% (198784) 50% (331307) 75% (496960) 34.8273 (0.4330) 14.5756 (0.1599) 1.7676 (0.3085) 32.8160 (0.2959) 13.5322 (0.1470) 1.8390 (0.3245) 34.6035 (0.2898) 15.4285 (0.1743) 1.7687 (0.3033) 32.3856 (0.2001) 14.1328 (0.1577) 1.7384 (0.3112) 32.2367 (0.2579) 13.0264 (0.1090) 1.7361 (0.3060) 33.0462 (0.4129) 13.6822 (0.1478) 1.8266 (0.3223) 90% (70503) 1.6049 (0.5464) 1.6352 (0.5571) 1.6432 (0.5579) 1.6393 (0.5607) 1.6143 (0.5672) 1.6320 (0.5323) 90% (141396) 1.7342 (0.5743) 1.7479 (0.5707) 1.7367 (0.5627) 1.7339 (0.5832) 1.7230 (0.5756) 1.7459 (0.5652) 90% (596353) 1.6589 (0.2921) 1.6654 (0.3033) 1.6604 (0.2939) 1.6528 (0.2914) 1.6519 (0.2921) 1.6691 (0.3046) Table 3: For the intervention scenario, average peak infected percentage with standard deviation in parentheses. Lowest values in boldface. Because there were no vaccines to be distributed, there is only one value for no vaccination. Figure 4: The average percentage of the population that was latent, infectious, or infected (either latent or infectious) v. time for Hays County prevention scenario (50 percent vaccination). Points with a value of 0 were omitted from the graph. strategy was more effective in the prevention scenario because it blocked off likely transmission paths, which would have allowed the epidemic to spread quickly. Both of these targeting methods appear to be effective at reducing the severity of epidemics at the county level, though they are effective in different scenarios. This conclusion agrees with that of Matrajt and Longini, who found that vaccinating the vulnerable and those likely to transmit the disease was an optimal strategy at the national level [12]. Out-degree centrality was consistently the least effective method of vaccination, with peaks higher than those of random vaccination in most of the 30 percent and 50 percent vaccination experiments. This resulted from out-degree centrality’s tendency to allocate vaccines to high-population areas that are likely to spread the disease once it reaches them, regardless of their probability of infection. This result is supported by Figure 5, which shows out-degree targeting mainly small, urban blocks. Figure 5 also shows that out-degree centrality selected different blocks than the effective measures, which had many blocks in common. At 75 percent vaccination, at least one of in-degree and inverse betweenness was more effective at halting disease spread than random vaccination in all experiments. Similar to the 30 percent and 50 percent levels of vaccination, both measures were the most effective about half the time. In-degree centrality tended to fall slightly faster than inverse betweenness at the beginning of the simulation, though inverse betweenness was more effective at stopping disease spread near the end of the simulation, a results that agrees with those of Matrajt and Longini [12]. Effectively no disease spread occurred under all measures, at 90 percent vaccination. As at previous vaccination levels, in-degree and inverse betweenness were each most effective in half of the experiments. The differences in effectiveness in the prevention and intervention scenarios were much less significant than at lower vaccination levels. The lack of significant disease spread at 90 percent was a consequence of herd immunity, the primary factor influencing disease spread at high vaccination levels [14]. Due to the initial unvaccinated period, epidemics in the intervention scenario at 90 percent vaccination tended to last longer than those in the prevention scenario. Eigenvector-based measures, especially eigenvector-in centrality, which targeted vulnerable blocks that were likely to spread the disease to other vulnerable blocks, showed an interesting trend: their effectiveness increased with county size and the prevalence of urban areas. In fact, in the prevention scenario at 30 percent vaccination in Denton County, eigenvector-in centrality was the most effective, followed by inverse betweenness. In Rockwall County, which is small and sparsely populated, these measures were the worst performers. At 75 percent vaccination in Rockwall, eigenvector and eigenvector-in centralities produced epidemic curves similar to those observed at lower vaccination levels, which did not occur for the other measures. In contrast, eigenvector-based measures, especially eigenvector-in centrality, were more effective in Hays and Denton counties, which are larger and more heavily urbanized. In several experiments, eigenvectorin centrality outperformed random vaccination, though they were much less reliable than in-degree and inverse betweenness. Eigenvector-in centrality’s effectiveness increased in urbanized counties because eigenvector-based measures favor central blocks that are connected to other central blocks. Clusters of vulnerable blocks, which eigenvector-in centrality selects, were prevalent in counties such as Hays and Denton than in Rockwall, leading to the increased effectiveness of that centrality measure. However, as seen in Figure 3, eigenvector-based measures were still generally ineffective compared to other options. In-degree centrality, at 30 percent and 50 percent vaccination, caused the longest delay in the peak of the epidemic. This result arose because contacts within in-degree central blocks, which were more likely to occur than those within non-central ones, did not cause disease spread because those blocks were vaccinated. This resulted in the disease taking longer to spread because it was forced into less probable paths. In previous experiments with our simulator, we received unusual data, which we do not represent here. Specifically, random vaccination was reported as the most effective method for both the intervention and prevention scenarios in Denton County. These results were most likely a result of the stochastic nature of our simulator. It is possible, however, that this result was caused by in-degree centrality’s preference for the large number of highly-populated blocks in Denton County, which left large areas of the county unvaccinated. Some of the limitations of our work are our assumptions that the population is homogenous within each block, that contact rates between census blocks are approximated by Equation 6, and that the individual contact rate is constant. Another limitation is that we are only testing one isolated county at a time. These assumptions do not reflect reality, but are common approximations that have been used in computational models of epidemics such as the Global Stochastic Cellular Automata model [16]. Future work will primarily focus on addressing the limitations described above. Methods for simulating non-homogenous populations [19, 10] will be incorporated into our model. Alternative centrality measures and more accurate weighting formulas will be investigated. We will also attempt to determine what methods would produce results that surpass random vaccination in cases that are currently outliers. 6. REFERENCES [1] M. Ajelli, B. Gonçalves, D. Balcan, V. Colizza, H. Hu, J. J. Ramasco, S. Merler, and A. Vespignani. Comparing large-scale computational approaches to epidemic modeling: agent-based versus structured metapopulation models. BMC infectious diseases, 10:190, Jan. 2010. [2] P. Bonacich. Factoring and weighting approaches to status scores and clique identification. 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Adaptive Bernstein–von Mises theorems in Gaussian white noise arXiv:1407.3397v3 [math.ST] 19 Dec 2016 Kolyan Ray∗ Abstract We investigate Bernstein–von Mises theorems for adaptive nonparametric Bayesian procedures in the canonical Gaussian white noise model. We consider both a Hilbert space and multiscale setting with applications in L2 and L∞ respectively. This provides a theoretical justification for plug-in procedures, for example the use of certain credible sets for sufficiently smooth linear functionals. We use this general approach to construct optimal frequentist confidence sets based on the posterior distribution. We also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved. AMS 2000 subject classifications: Primary 62G20; secondary 62G15, 62G08. Keywords and phrases: Bayesian inference, posterior asymptotics, adaptation, credible set, confidence set. Contents 1 Introduction 2 2 Statistical setting 2.1 Function spaces and the white noise model . . . . . . . . . . . . . . . . . . . 2.2 Weak Bernstein–von Mises phenomena . . . . . . . . . . . . . . . . . . . . . . 2.3 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 7 8 3 Bernstein–von Mises results 3.1 Empirical and hierarchical Bayes in `2 . . . . . . . . . . . . . . . . . . . . . . 3.2 Slab and spike prior in L∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 11 4 Applications 4.1 Adaptive credible sets in `2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Adaptive credible bands in L∞ . . . . . . . . . . . . . . . . . . . . . . . . . . 13 13 15 5 Posterior independence of the credible sets 16 ∗ Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands. E-mail: k.m.ray@math.leidenuniv.nl Most of this work was completed during the author’s PhD at the University of Cambridge. This work was supported by UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 and the European Research Council under ERC Grant Agreement 320637. 1 6 Simulation example 19 7 Proofs 7.1 Proofs of weak BvM results in `2 (Theorems 3.1 and 3.2) 7.2 Proof of weak BvM result in L∞ (Theorem 3.5) . . . . . . 7.3 Credible sets . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Posterior independence of the credible sets . . . . . . . . . 7.5 Remaining proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 23 25 27 30 38 8 Technical facts and results 8.1 Results for `2 -setting . . . . . . . . . 8.2 Results for L∞ -setting . . . . . . . . 8.3 Wavelets . . . . . . . . . . . . . . . . 8.4 Weak convergence . . . . . . . . . . 8.5 Results on empirical and hierarchical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 39 41 43 44 45 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bayes procedures . . . . . . . . . . Introduction A key aspect of statistical inference is uncertainty quantification and the Bayesian approach to this problem is to use the posterior distribution to generate a credible set, that is a region of prescribed posterior probability (often 95%). This can be considered an advantage of the Bayesian approach since Bayesian credible sets can be computed by simulation. In particular, the Bayesian generates a number of posterior draws and then keeps a prescribed fraction of the draws, discarding the remainder which are considered “extreme” in some sense. From a frequentist perspective, key questions are whether such a method has a theoretical justification and what is an effective rule for determining which draws to discard. A natural approach is to characterize such draws using a geometric notion, in particular by considering a minimal ball in some metric. In finite dimensions, the Euclidean distance has a clear interpretation as the natural measure of size. However in infinite dimensions such a notion is less clear-cut: the L2 metric is the natural generalization of the Euclidean norm, but lacks a clear visual interpretation, while L∞ can be easily visualized but is more difficult to treat mathematically. From the Bayesian perspective of simulating credible sets, the practitioner ultimately seeks a practical and effective rule for sorting through posterior draws and such geometric interpretations can be viewed as somewhat artificial impositions. The aim of this article is therefore to study possible geometric choices of credible sets that behave well from a frequentist asymptotic perspective. (n) Consider data Y (n) arising from some probability distribution Pf , f ∈ F. We place a prior distribution Π on F and study the behaviour of the posterior distribution Π(· \| Y (n) ) (n) under the frequentist assumption Y (n) ∼ Pf0 for some non-random true f0 ∈ F as the data size or quality n → ∞. From such a viewpoint, the theoretical justification for posterior based inference using any (Borel) credible set in finite dimensions is provided by the Bernstein–von Mises (BvM) theorem (see [33, 50]). This deep result establishes mild conditions on the prior under which the posterior is approximately a normal distribution centered at an efficient estimator of the true parameter. It thus provides a powerful tool to study the asymptotic 2 behaviour of Bayesian procedures and justifies the use of Bayesian simulations for uncertainty quantification. A BvM in infinite-dimensions fails to hold in even very simple cases. Freedman [20] showed that in the basic conjugate `2 sequence space setting with both Gaussian priors and data, the BvM does not hold for `2 -balls centered at the posterior mean – see also the related contributions [17, 28, 34]. The resulting message is that despite their intuitive interpretation, credible sets based on posterior draws using an `2 -based selection procedure do not behave as in classical parametric models. Recently, Castillo and Nickl [11, 12] have established fully infinite-dimensional BvMs by considering weaker topologies than the classical √ Lp spaces. Their focus lies on considering spaces which admit 1/ n-consistent estimators and where Gaussian limits are possible, unlike Lp -type loss. Credible regions selected using these different geometries are shown to behave well, generating asymptotically exact frequentist confidence sets. In this paper, we explore this approach in practice via both theoretical results for adaptive priors, as well as by numerical simulations. We consider an empirical Bayes, a hierarchical Bayes and a multiscale Bayes approach. Before going into more abstract detail, it is useful to consider an example from [12] to numerically illustrate this approach in practice. Suppose that we observe Y1 , ..., Yn i.i.d. observations from an unknown density f0 on [0, 1]. We take a simple histogram prior Π, L f =2 L −1 2X hk 1ILk , ILk = (k2−L , (k + 1)2−L ], k ≥ 0, (1.1) k=0 L where the hk are drawn from a D(1, ..., 1)-Dirichlet distribution on the unit simplex in R2 . Here we ignore adaptation issues and select L = Ln based on the smoothness of the true function. Letting {ψlk : l ≥ 0, k = 0, ..., 2l − 1} denote the standard Haar wavelets and wl = l1/2+ for > 0 small, consider the multiscale credible ball (1.2) Cn = f : max wl−1 \|hf − fˆn , ψlk i\| ≤ Rn n−1/2 , k,l≤Ln where fˆn denotes the posterior mean and Rn = R(Y1 , ..., Yn ) is chosen such that Π(Cn \| Y1 , ..., Yn ) = 0.95. By Proposition 1 of [12], Pf0 (f0 ∈ Cn ) → 0.95 as n → ∞, whereas no such result is available for the L∞ -credible ball. Due to the conjugacy of the Dirichlet distribution with multinomial sampling, the posterior distribution can be computed straightforwardly and Rn can be easily obtained by simulation. For convenience we take f0 to be a Laplace distribution with location parameter 1/2 and scale parameter 5 that is truncated to [0, 1], that is f0 (x) ∝ e−5\|x−1/2\| 1[0,1] (x) with f0 ∈ H2s ([0, 1)) for s < 3/2. In Figure 1, we plotted the true density (solid black) and the posterior mean (red) in the cases n = 1000, 2000, 5000, 10000. We generated 100,000 posterior draws and plotted the 95% closest to the posterior mean in the M(w) sense (grey) to simulate Cn . We also used the posterior draws to generate a 95% credible band in L∞ by estimating Qn satisfying Π(f : \|\|f − fˆn \|\|∞ ≤ Qn \| Y ) = 0.95 and then plotting fˆn ± Qn (dashed black). We see that the L∞ diameter of Cn is strictly greater than that of the L∞ -credible band, with this difference particularly marked at the peak of the density. However, the diameter of Cn is spatially heterogeneous and has greatest width at the peak, whilst having smaller width around points where the true density is more regular. In all cases, Cn contains the true f0 , whereas the L∞ confidence band has more difficulty capturing the peak. 3 Figure 1: Credible sets based on the Dirichlet prior with the true density function (solid black), the posterior mean (red), a 95% credible band in L∞ (dashed black) and the set Cn given in (1.2) (grey). We have n = 1000, 2000, 5000 and 10000 respectively. The main message of this numerical example is that simulating the credible set Cn , which uses a slightly different geometry, yields a set that does not look particularly strange in practice and in fact resembles an L∞ credible band. Both approaches are methodologically similar, the only difference being the rule for discarding posterior draws. From a theoretical point of view, the difference between the two sets is far more significant, with Cn yielding exact coverage statements at the expense of unbounded L∞ diameter. It is however possible to improve upon the naive implementation of such sets to also obtain the optimal L∞ diameter (see Proposition 1 of [12] and related results below). Modifying the geometry in such a way to obtain an exact coverage statement therefore comes at little additional cost from a practitioner’s perspective. Nonparametric priors typically contain tuning or hyper parameters, and it is a key challenge to study procedures that select these parameters automatically in a data-driven manner. This avoids the need to make unreasonably strong prior assumptions about the unknown parameter of interest, since incorrect calibration of the prior can lead to suboptimal performance (see e.g. [30]). It therefore makes sense to use an automatic procedure, unless a practitioner is particularly confident that their prior correctly captures the fine details of the unknown parameter, such as its level of smoothness or regularity. Adaptive procedures are widely used in practice, with hyper parameters commonly selected using a hyperprior or an empirical Bayes method. In the case of Gaussian white noise, a number of Bayesian procedures have been shown to be rate adaptive over common smoothness classes (see for example [27, 29, 42]). Most such frequentist analyses restrict attention to obtaining contraction rates and do not 4 study coverage properties of credible sets. The focus of this paper is therefore to investigate nonparametric BvMs for adaptive priors, with the goal of studying the coverage properties of credible sets. In the case of Gaussian white noise, there has been recent work [30, 34] circumventing the need for a BvM by explicitly studying the coverage properties of certain specific credible sets. Of particular relevance is a nice recent paper by Szabó et al. [48], where the authors use an empirical Bayes approach combined with scaling up the radius of `2 -balls to obtain adaptive confidence sets under a so-called polished tail condition. Their approach relies on explicit prior computations and provides an alternative to the more abstract point of view taken here. One of our principal goals is exact coverage statements and this seems more difficult to obtain using such an explicit approach. Since adaptive confidence sets do not exist in full generality, we also require self-similarity conditions on the true parameter to exclude certain “difficult” functions [24],[26],[6]. In particular, we shall consider the procedure of [48] in Section 3.1 and obtain exact coverage statements under the self-similarity condition introduced there. We note other work dealing with BvM results in the nonparametric setting. Leahu [34] has studied the impact of prior smoothness on the existence of BvM theorems in the conjugate Gaussian sequence space model. Bickel and Kleijn [3], Castillo [8], Rivoirard and Rousseau [45] and Castillo and Rousseau [13] provide sufficient conditions for BvMs for semiparametric functionals. For the case of finite-dimensional posteriors with increasing dimension, see Ghosal [22] and Bontemps [4] for the case of regression or Boucheron and Gassiat [5] for discrete probability distributions. Much of the approach taken here can equally be applied to other statistical settings such as sparsity and inverse problems [43], but we restrict to the nonparametric regime for ease of exposition. Since our focus lies on BvM results and coverage statements and this changes little conceptually, we omit such generalizations to maintain mathematical clarity. 2 2.1 Statistical setting Function spaces and the white noise model We use the usual notation Lp = Lp ([0, 1]) for p-times Lebesgue integrable functions and denote by `p the usual sequence spaces. We consider the canonical white noise model, which is equivalent to the fixed design Gaussian regression model with known variance. For f ∈ L2 = L2 ([0, 1]), consider observing the trajectory (n) dYt 1 = f (t)dt + √ dBt , n t ∈ [0, 1], (2.1) where dB is a standard white noise. By considering the action of an orthonormal basis {eλ }λ∈Λ on (2.1), it is statistically equivalent to consider the Gaussian sequence space model (n) Yλ 1 ≡ Yλ = fλ + √ Zλ , n λ ∈ Λ, (2.2) where the (Zλ )λ∈Λ are i.i.d. standard normal random variables and the unknown parameter of interest f = (fλ )λ∈Λ is assumed to be in `2 . We denote by Pf0 or P0 the law of Y arising from (2.2) under the true function f0 . In the following, Λ will represent either a Fourier-type basis or a wavelet basis. In the `2 -setting, (2.2) can be interpreted purely in sequence form with Λ = N and we do not need to associate to it a time index t ∈ [0, 1]. 5 In L∞ we consider a multiscale approach so that Λ = {(j, k) : j ≥ 0, k = 0, ..., 2j − 1}. In particular, we consider an S-regular (S ≥ 0) wavelet basis of L2 ([0, 1]), {ψlk : l ≥ J0 − 1, k = 0, ..., 2l − 1}, with J0 ∈ N. For notational simplicity, denote the scaling function φ by the first wavelet ψ(J0 −1)0 . We consider either periodized wavelets or boundary corrected wavelets (see e.g. [38] for more details). Moreover, in certain applications we require in addition that the wavelets satisfy a localization property sup J0 −1 2X J0 /2 \|φJ0 k (x)\| ≤ c(φ)2 < ∞, sup x∈[0,1] k=0 j −1 2X \|ψjk (x)\| ≤ c(ψ)2j/2 < ∞, (2.3) x∈[0,1] k=0 j ≥ J0 (see Section 8.3 for more discussion). The sequence model (2.2) corresponds to estimating the wavelet coefficients flk = hf, ψlk i, for all (l, k) ∈ Λ, since any function f ∈ L2 generates such a wavelet sequence. Conversely, any such sequence P(flk ) generates the wavelet series of a function (or distribution if the sequence is not in `2 ) (l,k) flk ψlk . For s, δ ≥ 0, define the Sobolev spaces at the logarithmic level: ( ) ∞ X s,δ 2 k 2s (log k)−2δ \|fk \|2 < ∞ . H s,δ ≡ H2 := f ∈ `2 : \|\|f \|\|s,2,δ := k=1 From this we recover the usual definition of the Sobolev spaces H s ≡ H2s = H2s,0 and by duality we define for s > 0, H2−s := (H2s )∗ . By standard Hilbert space duality arguments, we can consider `2 as a subspace of H2−s and can similarly define the logarithmic spaces for s < 0 and δ ≥ 0 using the above series definition. In the `2 -setting we shall classify smoothness via the Sobolev hyper rectangles for β ≥ 0: ) ( Q(β, R) = f ∈ `2 : sup k 2β+1 fk2 ≤ R . k≥1 In the L∞ ([0, 1])-setting we consider multiscale spaces: for a monotone increasing sequence w = (wl )l≥1 with wl ≥ 1, define ( ) 1 M = M(w) = x = (xlk ) : \|\|x\|\|M(w) := sup max \|xlk \| < ∞ l≥0 wl k (for further references to multiscale statistics see [12]). A separable closed subspace is obtained by considering the restriction 1 M0 = M0 (w) = x ∈ M(w) : lim max \|xlk \| = 0 , l→∞ wl k that is those (weighted) sequences in M(w) that converge to 0. Note that M contains the space `2 , since \|\|x\|\|M ≤ \|\|x\|\|`2 as wl ≥ 1. In this setting, we consider norm-balls in the Besov β spaces B∞∞ ([0, 1]), H(β, R) = {f = (flk )(l,k)∈Λ : \|flk \| ≤ R2−l(β+1/2) , ∀(l, k) ∈ Λ}. β We recall that B∞∞ ([0, 1]) = C β ([0, 1]), the classical Hölder (-Zygmund in the case β ∈ N) spaces. For more details on these embeddings and identifications see [38]. 6 Whether an `2 -white noise defines a tight random element of M0 (w) depends √ on the weighting sequence (wl ). Recall that we call a sequence (wl )l≥1 admissible if wl / l % ∞ as l → ∞ [12]. Let Z = {Zλ = hZ, eλ i : λ ∈ Λ}, where Zλ ∼ N (0, 1) i.i.d., denote the Gaussian white noise in (2.2). We have from [11, 12] that for δ > 1/2 and (wl ) an admissible sequence, −1/2,δ Z defines a tight Gaussian Borel random variable on H2 and M0 (w) respectively, which we denote Z. In view of this tightness, we can consider (2.1) as a Gaussian shift model: 1 Y(n) = f + √ Z, n √ −1/2,δ where the above inequality is in the H2 - or M0 (w)-sense. Since n(Y(n) − f ) = Z in −1/2,δ H2 or M0 (w), it immediately follows that Y(n) is an efficient estimator of f in either norm. Among the two classes {H2s,δ }s∈R,δ≥0 and {M0 (w)}w of spaces considered, one can show that s = −1/2, δ > 1/2 and admissibility of w determine the minimal spaces where the law of the `2 -white noise Z is tight (see [11, 12] for further discussion). We therefore focus attention on these spaces since they provide the threshold for which a weak convergence approach can −1/2,δ work. For convenience, we denote H ≡ H(δ) ≡ H2 . We further denote the law of Z in H or M0 (w) by N as appropriate. 2.2 Weak Bernstein–von Mises phenomena Due to the continuous embeddings `2 ⊂ H and `2 ⊂ M0 (w), any Borel probability measure on `2 yields a tight Borel probability measure on H and M0 (w). Consider a prior Π on `2 and let Πn = Π(· \| Y (n) ) denote the posterior distribution based on data (2.2). For S a vector space and z ∈ S, consider the map τz : S → S given by √ τz : f 7→ n(f − z). Let Πn ◦τY−1 (n) denote the image measure of the posterior distribution (considered as a measure on H or M0 (w)) under the map τY(n) . Thus for any Borel set B arising from these topologies, √ Πn ◦ τY−1 n(f − Y(n) ) ∈ B \| Y (n) ), (n) (B) = Π( √ so that we can more intuitively write Πn ◦ τY−1 n(f − Y(n) ) \| Y (n) ), where L(f \| Y (n) ) (n) = L( denotes the law of f under the posterior. For convenience, we metrize the weak convergence of probability measures via the bounded Lipschitz metric (defined in Section 8.4). Recalling that we denote by N the law of the white noise Z in (2.2) as an element of S, we define the notion of nonparametric BvM. Definition 1. Consider data generated from (2.2) under a fixed function f0 and denote by P0 the distribution of Y (n) . Let βS be the bounded Lipschitz metric for weak convergence of probability measures on S. We say that a prior Π satisfies a weak Bernstein-von Mises phenomenon in S if, as n → ∞, √ E0 βS (Πn ◦ τY−1 n(f − Y(n) ) \| Y (n) ), N ) → 0. (n) , N ) = E0 βS (L( Here S is taken to be one of H(δ) for δ > 1/2, H −s for s > 1/2 or M0 (w) for (wl )l≥1 an admissible sequence. 7 The weak BvM says that the (scaled and centered) posterior distribution asymptotically looks like an infinite-dimensional Gaussian distribution in some ‘weak’ sense, quantified via the bounded Lipschitz metric (8.9). Weak convergence in S implies that these two probability measures are approximately equal on certain classes of sets, whose boundaries behave smoothly with respect to the measure N (see Sections 1.1 and 4.1 of [11]). 2.3 Self-similarity The study of adaptive BvM results naturally leads to the topic of adaptive frequentist confidence sets. It is known that confidence sets with radius of optimal order over a class of submodels nested by regularity that also possess honest coverage do not exist in full generality (see [26, 40] for recent references). We therefore require additional assumptions on the parameters to be estimated and so consider self-similar functions, whose regularity is similar at both small and large scales. Such conditions have been considered in Giné and Nickl [24], Hoffmann and Nickl [26] and Bull [6] and ensure that we remove those functions whose norms (measuring smoothness) are difficult to estimate and which statistically look smoother than they actually are. We firstly consider the `2 -type self-similarly assumption found in Szabó et al. [48]. Definition 2. Fix an integer N0 ≥ 2 and parameters ρ > 1, ε ∈ (0, 1). We say that a function f ∈ Q(β, R) is self-similar if dρN e X fk2 ≥ εRN −2β for all N ≥ N0 . k=N We denote the class of self-similar elements of Q(β, R) by QSS (β, R, ε). This condition says that each block (fN , ..., fdρN e ) of consecutive components contains at least a fixed fraction (in the `2 -sense) of the size of a “typical” element of Q(β, R), so that the signal looks similar at all frequency levels (see [48, 39, 40] for further discussion). The parameters N0 and ρ affect the results of this article through the sample size at which the asymptotic results take effect, that is n → ∞ implicitly implies statements of the form “for n ≥ n0 large enough”, where n0 depends on N0 and ρ. For this reason, the impact of N0 and ρ is not explicitly mentioned below and one may simply treat these constants as fixed (e.g. N0 = 2 and ρ = 2). The lower bound in Definition 2 can be slightly weakened to permit for example logarithmic deviations from N −2β . However since this results in additional technicality whilst adding little extra insight, we do not pursue such a generalization here. It is possible to consider a weaker self-similarity condition using a strictly frequentist approach [40], though this has not been explored in the Bayesian setting and it is unclear whether our P approach extends in such a way. Let Kj (f ) = k hf, φjk iφjk denote the wavelet projection at resolution level j. In L∞ we consider Condition 3 of Giné and Nickl [24], which can only be slightly relaxed [6]. Definition 3. Fix a positive integer j0 . We say that a function f ∈ H(β, R) is self-similar if there exists a constant ε > 0 such that \|\|Kj (f ) − f \|\|∞ ≥ ε2−jβ for all j ≥ j0 . We denote the class of self-similar elements of H(β, R) by HSS (β, R, ε). 8 In particular, since f ∈ H(β, R), we have that \|\|Kj (f ) − f \|\|∞ 2−jβ for all j ≥ j0 . What we really require is that there is at least one significant coefficient at the level log2 ((n/ log n)1/(2β+1) ) that the posterior distribution can detect. However, this level depends also on unknown constants in practice (see proof of Proposition 4.5) and so we require a statement for all (sufficiently large) resolution levels as in Definition 3. See Giné and Nickl [24] and also Bull [6] for further discussion about this condition. 3 3.1 Bernstein–von Mises results Empirical and hierarchical Bayes in `2 We continue the frequentist analysis of the adaptive priors studied in [29, 48, 47] in `2 . For α > 0 define the product prior on the `2 -coordinates by the product measure Πα = ∞ O N (0, k −2α−1 ), k=1 so that the coordinates are independent. A draw from this distribution will be Πα -almost 0 surely in all Sobolev spaces H2α for α0 < α. The posterior distribution corresponding to Πα is given by ∞ O n 1 Πα (· \| Y ) = N Yk , 2α+1 . (3.1) k 2α+1 + n k +n k=1 Hβ If f0 ∈ and α = β, it has been shown [2, 7, 30] that the posterior contracts at the minimax rate of convergence, while if α 6= β, then strictly suboptimal rates are achieved. Since the true smoothness β is generally unknown, two data-driven procedures have been considered in [29]. The empirical Bayes procedure consists of selecting the smoothness parameter by using a likelihood-based approach. Namely, we consider the estimate α̂n = argmax `n (α), (3.2) α∈[0,an ] where an → ∞ is any sequence such that an = o(log n) as n → ∞ and ∞ n 1X n2 2 log 1 + 2α+1 − 2α+1 `n (α) = − Y 2 k k +n k k=1 is the marginal log-likelihood for α in the joint model (f, Y ) in the Bayesian setting (relative to the infinite product measure ⊗∞ k=1 N (0, 1)). The quantity an is needed to uniformly control the finite dimensional projections of the empirical Bayes procedure to establish a parametric BvM (Theorem 7.2). The posterior distribution is defined via the plug-in procedure Πα̂n (· \| Y ) = Πα (· \| Y ) \|α=α̂n . If there exist multiple maxima to (3.2), then any of them can be selected. A fully Bayesian approach is to put a hyperprior on the parameter α. This yields the hierarchical prior distribution Z ∞ ΠH = λ(α)Πα dα, 0 where λ is a positive Lebesgue density on (0, ∞) satisfying the following assumption (Assumption 2.4 of [29]). 9 Condition 1. Assume that for every c1 > 0, there exists c2 ≥ 0, c3 ∈ R, with c3 > 1 if c2 = 0 and c4 > 0 such that for α ≥ c1 , −c3 c−1 exp (−c2 α) ≤ λ(α) ≤ c4 α−c3 exp (−c2 α) . 4 α The exponential, gamma and inverse gamma distributions satisfy Condition 1 for example. Knapik et al. [29] showed that both these procedures contract to the true parameter adaptively at the (almost) minimax rate, uniformly over Sobolev balls, and a similar result holds for Sobolev hyper rectangles. Both procedures satisfy weak BvMs in the sense of Definition 1. Theorem 3.1. Consider the empirical Bayes procedure described above. For every β, R > 0 and s > 1/2, we have sup E0 βH −s (Πα̂n ◦ τY−1 , N ) → 0 f0 ∈Q(β,R) as n → ∞. Moreover, for δ > 2 we have the (slightly) stronger convergence sup f0 ∈QSS (β,R,ε) E0 βH(δ) (Πα̂n ◦ τY−1 , N ) → 0 as n → ∞. Theorem 3.2. Consider the hierarchical Bayes procedure described above, where the prior density λ satisfies Condition 1. For every β, R > 0 and s > 1/2, we have sup f0 ∈Q(β,R) −1 E0 βH −s (ΠH n ◦ τY , N ) → 0 as n → ∞. Moreover, for δ > 2 we have the (slightly) stronger convergence sup f0 ∈QSS (β,R,ε) −1 E0 βH(δ) (ΠH n ◦ τY , N ) → 0 as n → ∞. The requirement of self-similarity for a weak BvM in H(δ) could conceivably be relaxed, but such an assumption is natural since it is anyway needed for the construction of adaptive confidence sets in Section 4.1. It is not clear whether this is a fundamental limit or a technical artefact of the proof. The condition δ > 2 is also required for technical reasons. Whilst minimax optimality is clearly desirable from a theoretical frequentist perspective, it may be too stringent a goal in our context. Using a purely Bayesian point of view, we derive an analogous result to Doob’s almost sure consistency result. Specifically, a weak BvM holds in H(δ) for prior draws, almost surely under both the empirical Bayes and hierarchical priors. For this, it is sufficient to show that prior draws are self-similar almost surely. Proposition 3.3. The parameter f0 is self-similar in the sense of Definition 2, Πα -almostsurely for any α > 0. Consequently, Πα̂n and ΠH satisfy a weak BvM in H(δ) for δ > 2, Πα -a.s., α > 0, and ΠH -a.s. respectively. In particular, f satisfies Definition 2 with smoothness α and parameters ρ > 1 and ε = ε(α, ρ, R) > 0 sufficiently small and random N0 sufficiently large, Πα -almost surely. As a simple corollary to Theorems 3.1 and 3.2, we have that the rescaled posteriors merge weakly 10 (with respect to weak convergence on H(δ)) in the sense of Diaconis and Freedman [19]. By Proposition 2.1 of [41], we immediately have that the unscaled posteriors merge weakly with respect to the `2 -topology since they are both consistent [29]. However, in the case of bounded Lipschitz functions (rather than the full case of continuous and bounded functions), we can improve this result to obtain a rate of convergence. Corollary 3.4. For every β, R > 0, s > 1/2 and δ > 2, we have sup f0 ∈Q(β,R) −1 −1 E0 βH −s (ΠH n ◦ τY , Πα̂n ◦ τY ) → 0 sup f0 ∈QSS (β,R,ε) −1 −1 E0 βH(δ) (ΠH n ◦ τY , Πα̂n ◦ τY ) → 0 as n → ∞. In particular, for S = H −s or H(δ) as above, Z L H . sup u d(Πn − Πα̂n ) = oP0 √ n u:\|\|u\|\|BL ≤L S 3.2 Slab and spike prior in L∞ Consider the slab and spike prior, whose frequentist contraction rate has been analyzed in Castillo and van der Vaart [15], Hoffmann et al. [27] and Castillo et al. [14]. The assumptions in [27] ensure that prior draws are very sparse and only very few coefficients are fitted. We therefore modify the prior slightly so that the prior automatically fits the first few coefficients of the signal without any thresholding. This ensures that the posterior will have a rough approximation of the signal before fitting wavelet coefficients more sparsely at higher resolution levels. This makes sense from a practical point of view by preventing overly sparse models and is in fact necessary from a theoretical perspective (see Proposition 3.7). Let Jn = blog n/ log 2c be such that n/2 < 2Jn ≤ n and define some strictly increasing sequence j0 = j0 (n) → ∞ such that j0 (n) < Jn . For the low resolutions j ≤ j0 (n) we fit a simple product prior where we draw the fjk ’s independent from a bounded density g that is strictly positive on R. For the middle resolution levels j0 (n) < j ≤ Jn , the fjk ’s are drawn independently from the mixture Πj (dx) = (1 − wj,n )δ0 (dx) + wj,n g(x)dx, n−K ≤ wj,n ≤ 2−j(1+θ) , for some K > 0 and θ > 1/2. All coefficients at levels j > Jn are set to 0. Since this is a product prior, one can sample from the posterior by sampling from each component separately (using either an MCMC scheme or explicit expressions depending on the choice of density g). We have a weak BvM in the multiscale space M0 (w), where the rate at which the admissible sequence (wl ) diverges depends on the how many coefficients we automatically √ fit in the prior via the sequence j0 (n). Recall that a sequence (wl )l≥1 is admissible if wl / l % ∞. Theorem 3.5. Consider the slab and spike prior defined above with lower threshold given by the strictly increasing sequence j0 (n) → ∞. The posterior distribution satisfies a weak BvM in M0 (w) in the sense of Definition 1, that is for every β, R > 0, sup f0 ∈H(β,R) E0 βM0 (w) (Πn ◦ τY−1 , N ) → 0 √ as n → ∞, for any admissible sequence (wl ) satisfying wj0 (n) / log n % ∞. 11 √ Note that in the limiting case wl = l, we recover j0 (n) ' log n, so that the prior automatically fits the same fixed fraction of the full 2Jn ' n coefficients. Since we consider only admissible sequences, the fraction of coefficients that the prior fits automatically is asymptotically vanishing. An alternative way to consider this result √ is in reverse: based on a desired rate in practice, we prescribe an admissible sequence wl = lul , where ul is some divergent sequence, and then pick j0 (n) appropriately. Taking j0 (n) to grow more slowly than any power of log n means (wl ) must grow faster than any power of l, resulting in a greater than logarithmic down-weighting of the wavelet coefficients in M(w). It may therefore be more appropriate to take j0 (n) a power of log n, which yields the following specific case. Corollary 3.6. Consider the slab and spike prior defined above with lower threshold j0 (n) ' 1 (log n) 2+1 for some > 0. Then it satisfies a weak BvM in M0 (w) in the sense of Definition 1, that is for every β, R > 0, sup f0 ∈H(β,R) E0 βM0 (w) (Πn ◦ τY−1 , N ) → 0 as n → ∞ for the admissible sequence wl = l1/2+ ul , where ul is any (arbitrarily slowly) diverging sequence. While the requirement to fit the first few coefficients of the prior is mild and of practical use in nonparametrics, it is naturally of interest to study the behaviour of the posterior distribution with full thresholding, that is when j0 (n) ≡ 0, which we denote by Π0 . In general √ however, the full posterior contracts to the truth at a rate strictly slower than 1/ n in M(w), √ so that a n-rescaling of the posterior cannot converge weakly to a limit. This holds even for self-similar functions. Proposition 3.7. Let (wl ) be any admissible sequence. Then for any β, R > 0, there exists ε = ε(β, R, ψ) > 0 and f0 ∈ HSS (β, R, ε) such that along some subsequence (nm ), E0 Π0 (\|\|f − Y\|\|M(w) ≥ Mnm nm −1/2 \| Y (nm ) ) → 1 for all Mn → ∞ sufficiently slowly. Consequently, for such an f0 , a weak BvM in M0 (w) in the sense of Definition 1 cannot hold. It is particularly relevant that Proposition 3.7 applies to self-similar parameters since a major application of the weak BvM is the construction of adaptive credible regions with √ good frequentist properties under self-similarity (see Proposition 4.5). On the level of a nrescaling as in Definition 1, the rescaled posterior distribution asymptotically puts vanishingly small probability mass on any given M(w)-ball infinitely often. p This occurs because the posterior selects non-zero coordinates by thresholding at the level log n/n rather than √ the √ required 1/ n (Lemma 1 of [27]). The weighting sequence (wl ) regularizes the extra log n factor at high frequencies, but does not do so at low frequencies. This√is the reason that the weighting sequence (wl ) depends explicitly on the thresholding factor log n in Theorem 3.5. It seems that using such an adaptive scheme on low frequencies of the signal causes the weak BvM to fail. This prior closely resembles the frequentist practice of wavelet thresholding, where such a phenomenon has also been observed. For example, Giné and Nickl [23] require similar (though stronger) assumptions on the number of coefficients that need to be fitted automatically to obtain a central limit theorem for the distribution function of the hard thesholding wavelet estimator in density estimation (Theorem 8 of [23]). 12 4 4.1 Applications Adaptive credible sets in `2 We propose credible sets from the hierarchical or empirical Bayes procedures, which we show are adaptive frequentist confidence sets for self-similar parameters. We consider the natural Bayesian approach of using the quantiles of the posterior distribution to obtain a credible set of prescribed posterior probability. By considering sets whose geometry is amenable to the space H(δ), the weak BvM P implies that such credible sets are asymptotically confidence sets. −1 −2δ f 2 . For a given significance level 0 < γ < 1, Recall that \|\|f \|\|2H(δ) = ∞ k=1 k (log k) k consider the credible set √ Cn = f : \|\|f − Y\|\|H(δ) ≤ Rn / n , (4.1) where Rn = Rn (Y, γ) is chosen such that Πα̂n (Cn \|Y ) = 1 − γ or ΠH (Cn \|Y ) = 1 − γ. Since the empirical and hierarchical Bayes procedures both satisfy a weak BvM in H(δ), we have from Theorem 1 of [11] that in both cases Pf0 (f0 ∈ Cn ) → 1 − γ and Rn = OP0 (1) as n → ∞, so that Cn is asymptotically an exact frequentist confidence set (of unbounded `2 -diameter). We control the diameter of the set using either the estimator α̂n or the posterior median as a smoothness estimate, and then use the standard frequentist approach of undersmoothing. In the first case, consider n o p √ C̃n = f : \|\|f − Y\|\|H(δ) ≤ Rn / n, \|\|f − fˆn \|\|H α̂n −n ≤ C log n , (4.2) where fˆn is the posterior mean, Rn is chosen as in Cn , n (chosen possibly data dependently) satisfies r1 /(log n) ≤ n ≤ (r2 / log n) ∧ (α̂n /2) for some 0 < r1 ≤ r2 ≤ ∞ and C > 1/r1 . 0 The undersmoothing by n is necessary since the posterior assigns probability 1 to H α for α0 < α̂n , while probability 0 to H α̂n itself. Geometrically, C̃n is the intersection of two `2 -ellipsoids, Cn and an H α̂n −n -norm ball. For a typical element f in C̃n , the size of the low frequency coordinates of f are determined by Cn , while the smoothness condition in C̃n acts to regularize the elements of Cn (which are typically not in `2 ) by shrinking the higher frequencies. Proposition 4.1. Let 0 < β1 ≤ β2 < ∞, R ≥ 1 and ε > 0. Then the confidence set C̃n given in (4.2) satisfies sup f0 ∈QSS (β,R,ε) β∈[β1 ,β2 ] Pf0 (f0 ∈ C̃n ) − (1 − γ) → 0 as n → ∞. For every β ∈ [β1 , β2 ], uniformly over f0 ∈ QSS (β, R, ε), 0 1/(4β+2) Πα̂n (C̃n \| Y ) = 1 − γ + OP0 n−C n for some C 0 > 0 independent of β, R, ε, N0 , ρ, while the `2 -diameter satisfies for δ > 2, \|C̃n \|2 = OP0 n−β/(2β+1) (log n)(2δβ+1/2)/(2β+1) . 13 The logarithmic correction in the definition of H(δ) that is required for a weak BvM causes the (log n)2δβ/(2β+1) penalty (which is O((log n)2δ ) uniformly over β ≥ 0); this is the price required for using a plug-in approach in H(δ). The remaining (log n)1/(4β+2) factor arises due to the second constraint in C̃n , where the H α̂n -radius must be taken sufficiently large to ensure C̃n has sufficient posterior probability. While the second constraint in (4.2) reduces the credibility below 1 − γ, Proposition 4.1 shows that this credibility loss is very small. The Bayesian approach takes care of this automatically since the posterior concentrates on a much more regular set than `2 . This is corroborated empirically by numerical evidence (see Figure 3), which shows that the credibility of the set C̃n rapidly approaches 1 − γ as n increases. Remark 4.2. A naive interpretation of Cn yields a credible set that is far too large, having unbounded `2 -diameter, with the additional constraint in C̃n needed to regularize the set. In actual fact the posterior does this regularization automatically with Cn being “almost optimal”. Proposition 4.1 could be rewritten for Cn with exact credibility Πα̂n (Cn \| Y ) = 1 − γ and `2 diameter satisfying 2δβ+1/2 0 1/(4β+2) − β Πα̂n (f ∈ Cn : \|\|f − fˆn \|\|2 ≤ Cn 2β+1 (log n) 2β+1 \| Y ) = 1 − γ + OP0 n−C n , for some C, C 0 > 0. In view of this, the sets Cn and C̃n are essentially the same from the point of view of the posterior, with Cn having exact credibility for finite n and correct `2 -diameter asymptotically and C̃n having the reverse. In particular, the finite time credibility “gap” for either having too large radius in Cn or smaller than 1 − γ credibility for C̃n is of the same size. Moreover, the above statement holds without the need for a self-similarity assumption, which is possible since the confidence set does not strictly have optimal diameter. The same notion also holds for Cn arising from the hierarchical Bayes procedure. Remark 4.3. By Lemma 8.2 the empirical Bayes posterior mean fˆn satisfies \|\|fˆn − Y\|\|H(δ) = √ oP0 (1/ n) and so is an efficient estimator of f0 in H(δ). Consequently one can substitute Y with fˆn in the definitions of Cn and C̃n . Replacing the estimate α̂n with the median αnM of the marginal posterior distribution λn (·\|Y ) yields a fully Bayesian analogue. To obtain the necessary undersmoothing over a target range [β1 , β2 ], we consider the shifted estimator β̂n = αnM − (C + 1)/ log n, where C = C(R, β2 , ε, ρ) = maxβ1 ≤β≤β2 C(R, β, ε, ρ) is the constant appearing in Lemma 8.7 (which can be explicitly computed). Consider n o p √ C̃n0 = f : \|\|f − Y\|\|H ≤ Rn / n, \|\|f − fˆn \|\|H β̂n ≤ Mn log n , (4.3) where fˆn is the posterior mean, Mn → ∞ grows more slowly than any polynomial and Rn is chosen as in Cn . Taking Cn arising from the hierarchical Bayesian procedure ΠH , C̃n0 is a “fully Bayesian” object. We have an analogue of Proposition 4.1. Proposition 4.4. Let 0 < β1 ≤ β2 < ∞, R ≥ 1 and ε > 0. Then the confidence set C̃n0 given in (4.3) satisfies sup f0 ∈QSS (β,R,ε) β∈[β1 ,β2 ] Pf0 (f0 ∈ C̃n0 ) − (1 − γ) → 0 14 as n → ∞. For every β ∈ [β1 , β2 ], uniformly over f0 ∈ QSS (β, R, ε), ΠH (C̃n0 \| Y ) = 1 − γ + oP0 (1) , while the `2 -diameter satisfies for δ > 2, \|C̃n0 \|2 = OP0 n−β/(2β+1) (log n)(2δβ+1/2)/(2β+1) . 4.2 Adaptive credible bands in L∞ We provide a fully Bayesian construction of adaptive credible bands using the slab and spike prior. The posterior median f˜ = (f˜n,lk )(l,k)∈Λ (defined coordinate-wise) takes the form of a thresholding estimator (c.f. [1]), which we use to identify significant coefficients. This has the advantage of both simplicity and interpretability and also provides a natural Bayesian approach for this coefficient selection. Such an approach was used by Kueh [31] to construct an asymptotically honest (i.e. uniform in the parameter space) adaptive frequentist confidence set on the sphere using needlets. In that article, the coefficients are selected based on the empirical wavelet coefficients with the thresholds selected conservatively using Bernstein’s inequality. In contrast, we use a Bayesian approach to automatically select the thresholding quantile constants that then yields exact coverage statements. Let √ Dn = {f : \|\|f − Y\|\|M(w) ≤ Rn / n}, (4.4) where Rn = Rn (Y, γ) is chosen such that Π(Dn \| Y ) = 1 − γ. We then define the data driven width of our confidence band r Jn 2l −1 X log n X vn σn,γ = σn,γ (Y ) = sup 1{f˜lk 6=0} \|ψlk (x)\|, (4.5) n x∈[0,1] l=0 k=0 where (vn ) is any (possibly data-driven) sequence such that vn → ∞. Under a local selfsimilarity type condition as in Kueh [31], one could possibly remove the supremum in (4.5) to obtain a spatially adaptive procedure. However, we restrict attention to more global self-similarity conditions here for simplicity. Since we consider wavelets satisfying (2.3), we have r r l −1 Jn 2X Jn X p log n log n X σn,γ ≤ vn sup \|ψlk (x)\| ≤ C(ψ)vn 2l/2 ≤ C 0 vn log n < ∞ a.s., n x∈[0,1] n l=0 k=0 l=0 for all n and γ ∈ (0, 1). Let πmed denote the projection onto the non-zero coordinates of the posterior median and in a slight abuse of notation set l πmed (Y )(x) = Jn 2X −1 X Ylk 1{f˜lk 6=0} ψlk (x), l=0 k=0 where we recall Ylk = R1 ψlk (t)dY (t). Consider the set √ Dn = {f : \|\|f − Y\|\|M(w) ≤ Rn / n, \|\|f − πmed (Y )\|\|∞ ≤ σn,γ (Y )}, 0 (4.6) where Rn is as in (4.4). This involves a two-stage procedure: we firstly calculate the required M(w)-radius Rn and then use the posterior median to select the coefficients deemed significant. 15 Proposition 4.5. Let 0 < β1 ≤ β2 < ∞, R ≥ 1 and ε > 0. Consider the slab and spike prior defined√ above with threshold j0 (n) → ∞ and let (wl ) be any admissible sequence that satisfies wj0 (n) / log n % ∞. Then the confidence set Dn given in (4.6), using the choice (wl ) and σn,γ (Y ) defined in (4.5) for vn → ∞, satisfies sup f0 ∈HSS (β,R,ε) β∈[β1 ,β2 ] Pf0 (f0 ∈ Dn ) − (1 − γ) → 0 as n → ∞. For every β ∈ [β1 , β2 ], uniformly over f0 ∈ HSS (β, R, ε), Π(Dn \| Y ) = 1 − γ + oP0 (1), while the L∞ -diameter satisfies \|Dn \|∞ = OP0 (n/ log n)−β/(2β+1) vn . Under self-similarity, Dn has radius equal to the minimax rate in L∞ up to some factor vn that can be taken to diverge arbitrarily slowly, again mirroring a frequentist undersmoothing penalty. The choice of the posterior median is for simplicity and can be replaced by any other suitable thresholding procedure, for example directly using the posterior mixing probabilities between the atom at zero and the continuous density component. One could also consider other alternatives to σn,γ that simultaneously control the L∞ norm of the credible set whilst also preserving coverage and credibility. A similar construction β̂n -ball, to the credible sets in Section 4.1 could also be pursued by intersecting Dn with a B∞1 where β̂n is a suitable estimate of the smoothness. Alternatively, in view of Remark 4.2, one can also show that ! ! wjn (β) −β/(2β+1) Π f ∈ Dn : \|\|f − Tn \|\|∞ ≤ p n (log n)(β+1)/(2β+1) Y = 1 − γ + oP0 (1) , jn (β) where Tn is an efficient estimator of f0 in M that is alsoprate-optimal in L∞ (e.g. (8.5) or (8.6)) and 2jn ∼ (n log n)1/(2β+1) . The factor wjn (β) / jn (β) can be made to diverge arbitrarily slowly by the prior choice of j0 (n). 5 Posterior independence of the credible sets −1/2,δ As shown above, the spaces H(δ) = H2 and M(w) yield credible sets with good frequentist properties. However, given the different geometries proposed, it is of interest to compare them to more classical credible sets. Consider the `2 -ball studied in [20, 48] (though without the blow-up factor of the latter) Cn`2 = {f : \|\|f − fˆn \|\|2 ≤ Q̃n (α̂n , γ)}, (5.1) where Q̃n (α̂n , γ) is selected such that Πα̂n (Cn`2 \| Y ) = 1 − γ. Since the posterior variance of Πα (· \| Y ) is independent of the data, the radius Q̃n (α̂n , γ) depends only on the data through α̂n . By Theorem 1 of [20] we have Q̃n (α, γ) = Qn n−α/(2α+1) , where Qn → Q > 0. Numerical examples of C̃n and Cn`2 are displayed in Section 6. Given the similarity of C̃n and Cn`2 in Figures 2 and 4, a natural question (voiced for example in [10, 39, 49]) is to what 16 extent these sets actually differ, both in theory and practice. From a purely geometric point of view these sets can be considered as infinite-dimensional ellipsoids with differing orientations. From a Bayesian perspective, an intriguing question is to what degree the decision rules on which these credible sets are based differ with respect to the posterior. For simplicity, we centre C̃n at the posterior mean fˆn , which we can do by Remark 4.3. Theorem 5.1. Suppose an in (3.2) satisfies in addition an ≤ log n/(6 log n log n). Then the (1 − γ)-H(δ)-credible ball C̃n defined in (4.2) and the (1 − γ)-`2 -credible ball Cn`2 defined in (5.1) are asymptotically independent under the empirical Bayes posterior, that is as n → ∞, Πα̂n (C̃n ∩ Cn`2 \| Y ) = Πα̂n (C̃n \| Y )Πα̂n (Cn`2 \| Y ) + oP0 (1) = (1 − γ)2 + oP0 (1) uniformly over f0 ∈ Q(β, R). The first equality above also holds with C̃n replaced by Cn or Cn`2 replaced by the blown-up `2 -credible ball studied in [48]. Moreover, the above statement also holds for the hierarchical Bayes posterior with C̃n replaced by the (1 − γ)-H(δ)-credible ball C̃n0 given in (4.3) and Cn`2 replaced by the corresponding hierarchical Bayes `2 -credible set. Theorem 5.1 says that the Bayesian decision rules leading to the construction of C̃n and Cn`2 are fundamentally unrelated - one contains asymptotically no information about the other. Although we can conclude that C̃n and (blown-up) Cn`2 are frequentist confidence sets with similar properties, they express completely different aspects of the posterior. Note that this is not simply an artefact of the prior choice since the equivalent prior credible sets are not independent under the prior despite its product structure. An alternative interpretation is to consider Bayesian tests based on the credible regions, which have optimal frequentist properties. In this context, the two tests screen different and unrelated features. While both of these approaches are valid, both for the frequentist and the Bayesian, Theorem 5.1 says that neither of these constructions can be reduced to the other. The H(δ)-credible sets are principally determined by the low frequencies (k ≤ kn , where kn → ∞ in the proof), whereas the `2 -credible sets are driven by the high frequencies (k > kn ). The product structure of the posterior asymptotically decouples these two regimes yielding the independence statement. In particular the H(δ)-norm down-weights the higher order frequencies enough that one is dealing with a close to finite-dimensional model. Such a result is unlikely to hold for arbitrary priors, unless there is some degree of posterior independence between the frequency ranges driving the different credible sets (though less independence than a full product posterior is necessary). The numerical simulations in Figure 3 corroborate Theorem 5.1 very closely, indicating that this result provides a good finite sample approximation to the posterior behaviour. The posterior draws plotted in Section 6 are approximately drawn from the posterior distribution conditioned to the respective credible sets. Corollary 5.2 quantifies how close these draws are in terms of the total variation distance \|\| · \|\|T V . `2 Cn n Corollary 5.2. Let ΠC̃ α̂n (· \| Y ), Πα̂n (· \| Y ) denote the posterior distribution conditioned to the sets C̃n , Cn`2 respectively. Then as n → ∞, `2 Cn n \|\|ΠC̃ α̂n (· \| Y ) − Πα̂n (· \| Y )\|\|T V = γ + oP0 (1). Proof. Each conditional distribution consists of the posterior distribution restricted to the relevant credible set and normalized by the same factor (1 − γ). The two distributions are 17 therefore identical on their intersection and so the total variation distance equals ! 1 Πα̂n (C̃n ∩ (Cn`2 )c \| Y ) Πα̂n (C̃nc ∩ Cn`2 \| Y ) 1 2γ(1 − γ) + oP0 (1) = γ + oP0 (1). + = 2 2 1−γ Πα̂n (C̃n \| Y ) Πα̂n (Cn`2 \| Y ) Turning to the L∞ -setting, P for mathematical convenience let us consider the slightly stronger Besov norm kf kB∞1 = l 2l/2 maxk \|hf, ψlk i\| as in Hoffmann et al. [27]. This norm 0 0 0 is closely related to the k · k∞ -norm via the Besov space embbedings B∞1 ⊂ L∞ ⊂ B∞∞ (Chapter 4.3 of [25]). Define ∞ DnL = {f : \|\|f − Tn \|\|B∞1 ≤ Qn (γ)}, 0 (5.2) (2) where Tn = Tn is given by (8.6) and is an efficient estimator of f0 in M that is also rate∞ optimal in L∞ and Qn (γ) is selected such that Π(DnL \| Y ) = 1 − γ. The choice of Tn is not essential, but it is convenient to select an estimator that can simultaneously act as the ∞ centering for both Dn and DnL . We take the density g in Π to be Gaussian to simplify certain computations. Analogous results to those in H(δ) then hold. Theorem 5.3. Consider the slab and spike prior Π with lower threshold j0 (n) → ∞ and let 2 g be the density of √ the Gaussian distribution N (0, τ ). Let (wl ) be any admissible sequence satisfying wj0 (n) / log n % ∞ as n → ∞. Then the (1 − γ)-M(w)-credible ball Dn defined in ∞ (4.6) and the (1 − γ)-L∞ -credible ball DnL defined in (5.2) are asymptotically independent under the posterior, that is as n → ∞, ∞ Π(Dn ∩ DnL ∞ \| Y ) = Π(Dn \| Y )Π(DnL \| Y ) + oP0 (1) = (1 − γ)2 + oP0 (1) uniformly over f0 ∈ H(β, R). In particular the choice of j0 (n) in Corollary 3.6 satisfies the conditions of Theorem 5.3 √ since then wj0 (n) ' un log n, where un can be made to diverge arbitrarily slowly. L∞ Corollary 5.4. Consider the same conditions as in Theorem 5.3 and let ΠDn (· \| Y ), ΠDn (· \| ∞ Y ) denote the posterior distribution conditioned to the sets Dn , DnL respectively. Then as n → ∞, L∞ \|\|ΠDn (· \| Y ) − ΠDn (· \| Y )\|\|T V = γ + oP0 (1). Heuristics for an extension to density estimation The proofs of Theorems 5.1 and 5.3 presented here rely on the independence of the coordinates in the Gaussian white noise model. This model can be viewed as an idealized version of other more concrete statistical models, being mathematically more tractable. In view of the extension of the nonparametric BvM to density estimation in [12], let us briefly discuss a heuristic of what we might expect in this setting. Suppose we observe Y1 , ..., Yn i.i.d. observations from an unknown density f0 on [0, 1]. Assume that f0 is uniformly bounded away from 0 and that f0 ∈ C β ([0, 1]), where 1/2 < β ≤ 1. Consider the simple histogram prior Π specified in (1.1), where we again ignore adaptation issues and select L = Ln → ∞ based on the smoothness of f0 . Such a prior has been shown 18 to contract optimally in L∞ by Castillo [9] and to satisfy a weak BvM in M0 by Castillo and Nickl [12]. Let (ψlk ) denote the Haar wavelet basis on [0, 1] and M the related multiscale space for a suitable admissible sequence (wl ). Further let πA denote the projection onto the elements of the Haar wavelet basis with resolution level contained in A. One can show that for suitable sequences jn,1 , jn,2 → ∞ satisfying 2jn,1 2jn,2 , √ Π(f : \|\|f − Tn \|\|M ≤ Rn / n, \|\|f − Tn \|\|∞ ≤ Q̄n \| Y ) √ = Π(f : kπ≤jn,1 (f − Tn )kM ≤ (Rn + o(1))/ n, kπ≥jn,2 (f − Tn )k∞ ≤ Q̄n + δn \|Y ) + oP0 (1), √ where Tn is a suitable centering, Rn / n and Q̄n are the (1 − γ)-quantiles of the respective credible sets and δn is selected small enough to only change the credibility of the latter set by oP0 (1). Using the conjugacy of the Dirichlet distribution with multinomial sampling, the posterior distribution for (hk ) is D(N1 + 1, ..., NL + 1), where Nk = \|{Yi : Yi ∈ ILk }\|. Observe that the law of flk = hf, ψlk i under the posterior depends principally on the observations falling within supp(ψlk ) = [k2−l , (k + 1)2−l ]. Unlike the Gaussian white noise model, there is dependence across the posterior wavelet coefficients due to the dependence within the Dirichlet distribution and the constraint that the number of observations sums to n. The k · kM -norm in the above display is the weighted maximum of {flk : l ≤ jn,1 }. If jn,1 does not grow too fast, this consists of relatively few “large sample” averages. Heuristically, this term behaves like a central order statistic, being driven by the average sample behaviour. On the contrary, the k·k∞ -term is determined by the largest coefficients at each resolution level l ≥ jn,2 . Since 2jn,1 2jn,2 , these can be seen to behave more like extreme order statistics, being the maximum of many almost independent “small samples” (at least relative to the frequencies l ≤ jn,1 ). Even though order statistics depend, by definition, on all observations, central and extreme order statistics asymptotically depend on the observations in orthogonal ways and become stochastically independent (c.f. Chapter 21 of [50]). One might therefore hope that the two norms in the previous display are asymptotically independent in the sense of Theorems 5.1 and 5.3. An alternative way to understand why the wavelet coefficients at a given resolution level may be considered “almost independent” under the posterior is via Poissonization. It is wellknown that density estimation is asymptotically equivalent to Poisson intensity estimation [37, 44], where one observes a Poisson process with intensity measure nf0 . Equivalently, the Poisson experiment corresponds to observing a Poisson random variable N with expectation n and then independently of N observing Y1 , ..., YN i.i.d. with density f0 . In this framework, the dependence induced by the number of observations summing to n is removed, meaning that the variables (N1 , ..., NL ) defined above are fully independent. The remaining dependence is due to the Dirichlet distribution and becomes negligible as the number of bins Ln → ∞. Since this equivalence is asymptotic in nature, one should expect such a heuristic to manifest itself also asymptotically. 6 Simulation example We now apply our approach in a numerical example. Following on from the example of the −1/2,δ M(w)-based credible set (1.2), we now consider the space H2 . Consider the Fourier sine 19 basis ek (x) = √ 2 sin(kπx), k = 1, 2, ..., and define the true function f0,k = hf0 , ek i2 = k −3/2 sin(k) so that the true smoothness is β = 1. We consider realisations of the data (2.2) at levels n = 500 and 2000 and use the empirical Bayes posterior distribution. We plotted the true f0 (black), the posterior mean (red) and an approximation to the credible sets (grey). To simulate the `2 credible balls Cn`2 given in (5.1), we sampled 2000 curves from the posterior distribution and kept the 95% closest in the `2 sense to the posterior mean and plotted them (grey). We performed the same approach to obtain the full H(δ)-credible set Cn given in (4.1) and then plotted the full adaptive confidence set C̃n given in (4.2) with C = 1 and n = 1/ log n. We also present the approximate credibility of C̃n by considering the fraction of the simulated √ curves from the ˆ posterior that satisfy the extra constraint of C̃n that \|\|f − fn \|\|H α̂n −n ≤ log n. This is given in Figure 2. While the true `2 and H(δ) credible balls are unbounded in L∞ , the posterior draws can be shown to be bounded in L∞ explaining the boundedness of the plots. Sampling from the posterior (and thereby implicitly intersecting the sets Cn`2 and C̃n with the posterior support) seems the natural approach for the Bayesian. Indeed those elements that constitute the “roughest” or least regular elements of the credible sets are not seen by the posterior, that is they have little or no posterior mass (see Lemma 8.3). The posterior contains significantly more information than merely the `2 or H(δ) norm of the parameter of interest, as can be seen by it assigning mass 1 to a strict subset of `2 . For further discussion on plotting such credible sets see [10, 36, 49]. For a given set of 2000 posterior draws, we also computed the credibility of C̃n at a chosen significance level and the credibility of the posterior draws falling in both C̃n and Cn`2 . This latter quantity has value (1 − γ)2 + oP0 (1) by Theorem 5.1. We repeated this 20 times and the average values are presented in Figure 3. The posterior distribution appears to have some difficulty visually capturing the resulting function at its peak. In fact the credible sets do “cover the true function”, but do so in an `2 rather than an L∞ -sense. Indeed, any `2 -type confidence ball will be unresponsive to highly localized pointwise features since they occur on a set of small Lebesgue measure (as in this case). Similar reasoning also explains the performance of the posterior mean at this point. The posterior mean estimates the Fourier coefficients of f0 and hence estimates the true function in an `2 -sense via its Fourier series. In Section 5 it was shown that the two approaches behave very differently theoretically and the numerical results in Figure 3 match this theory very closely. It appears that the two methods do indeed use different rejection criteria in practice resulting in different selection outcomes. The visual similarity between the `2 and H(δ)-credible balls in Figure 2 is therefore a result of the posterior draws themselves looking similar, rather than the methods performing identically. We note that already by n = 500, C̃n has the correct credibility so that the high frequency smoothness constraint is satisfied with posterior probability virtually equal to 1 (c.f. Proposition 4.1). C̃n is therefore an actual credible set for reasonable (finite) sample sizes rather than a purely asymptotic credible set. The posterior distribution already strongly regularizes the high frequencies so that the posterior draws are very regular with high probability. This can be quantitatively seen by the rapidly decaying variance term of the posterior distribution (3.1). This is indeed the case in the simulation, where the credibility gap is negligible, thereby 20 Figure 2: Empirical Bayes credible sets for the Fourier sine basis with the true curve (black) and the empirical Bayes posterior mean (red). The left panels contain the `2 credible ball Cn`2 given in (5.1) and the right panels contain the set C̃n given in (4.2). From top to bottom, n = 500, 2000 and α̂n = 1.29, 1.01, with the right-hand side each having credibility 95%. demonstrating that most of the posterior draws already satisfy the smoothness constraint in C̃n . We repeat the same simulation using the same true function f0,k = k −3/2 sin(k), but with basis equal to the singular value decomposition (SVD) of the Volterra operator (c.f. [30]): √ ek (x) = 2 cos((k − 1/2)πx), k = 1, 2, ... and plot this in Figure 4 for n = 1000.and plot this in Figure 4 for n = 1000. Unlike Figure 2, the resulting function has no “spike” and so both credible sets have no trouble visually capturing the true function (though one should remember that these are `2 rather than L∞ type credible sets). We now illustrate √ the multiscale approach using the slab and spike prior with lower threshold j0 (n) = log n, plotting the true function (solid black) and posterior mean (red) at levels n = 200, 500. We have used Haar wavelets, set g to be N (0, 1/2) and have taken prior weights wj,n = min(n−1 , 2−5.5j ), corresponding to K = 1 and θ = 5. For n = 200 and 500, we have fitted one scaling function plus 28 − 1 = 255 and 29 − 1 = 511 wavelet coefficients respectively (i.e. 2Jn +1 − 1). We again sampled 2000 curves from the posterior distribution and plotted the 95% closest to the posterior mean in the M(w) sense (grey) to simulate Dn in (4.4). We also used the posterior draws to generate a 95% credible band in L∞ by estimating ∞ Q̄n (0.05) and then plotting DnL in (5.2) (dashed black). Finally we computed local 95% credible intervals at every point x ∈ [0, 1] and joined these to form a credible band (dashed blue). This is given in Figure 5. 21 Chosen significance Credibility of C̃n Credibility of C̃n ∩ Cn`2 Expected credibility of C̃n ∩ Cn`2 0.95 0.9500 0.9020 0.9025 Chosen significance Credibility of C̃n Credibility of C̃n ∩ Cn`2 Expected credibility of C̃n ∩ Cn`2 0.95 0.9500 0.9025 0.9025 n=500 0.90 0.85 0.8999 0.8499 0.8102 0.7220 0.8100 0.7225 n=2000 0.90 0.85 0.9000 0.8500 0.8095 0.7226 0.8100 0.7225 0.80 0.8000 0.6406 0.6400 0.80 0.8000 0.6409 0.6400 Figure 3: Table showing the average credibility of C̃n , the average credibility of the posterior draws falling in both sets and the expected value of the latter (from Theorem 5.1). Figure 4: Empirical Bayes credible sets for the Volterra SVD basis with the true curve (black) and the empirical Bayes posterior mean (red) for n = 1000 and α̂n = 1.07. The left and right panels contain the `2 credible ball Cn`2 given in (5.1) and C̃n (credibility 95%) given in (4.2) respectively. We see from Figure 5 that each posterior draw consists of a rough approximation of the signal via frequencies j ≤ j0 (n) with a few “spikes” from the high frequencies; the rather unusual shape is a reflection of the prior choice. It is worth noting that the posterior draws are bounded in L∞ since the posterior contracts rate optimally to the truth in L∞ [27]. We see that the L∞ diameter of Dn is strictly greater than that of the L∞ -credible bands, though this only manifests itself in a few places. The size of the L∞ -bands is driven by the size of the spikes, which are few in a number but occur in every posterior draw, resulting in seemingly very wide credible bands. On the contrary, the local credible intervals ignore the spikes since less than 5% of the draws have a spike at any given point, resulting in much tighter bands. The dashed blue lines in effect correspond to the 95% L∞ -band from a prior fitting exclusively the low frequencies j ≤ j0 (n), which is a non-adaptive prior modelling analytic smoothness. This dramatically oversmoothes the truth resulting in far too narrow credible bands and is highly dangerous since it is known that oversmoothing the truth can yield zero coverage [30, 34]. 22 Figure 5: Slab and spike credible sets with the true curve (black), posterior mean (red), a 95% credible band in L∞ (dashed black), pointwise 95% credible intervals (dashed blue) and the set Dn given in (4.4) (grey). We have n = 200, 500 respectively. 7 Proofs In what follows denote by πj the projection onto either Vj = span{ek : 1 ≤ k ≤ j} or Vj = span{ψlk : 0 ≤ l ≤ j, k = 0, ..., 2l −1} depending on whether we are considering a Fouriertype basis or a wavelet basis. Similarly define π>j to be the projection onto span{ek : k > j} or Vj = span{ψlk : l > j, k = 0, ..., 2l − 1}. 7.1 Proofs of weak BvM results in `2 (Theorems 3.1 and 3.2) √ To prove a weak BvM we need to show that the posterior contracts at rate 1/ n to the truth in the relevant space and that the finite-dimensional projections of the rescaled posterior converge weakly to those of the normal law N (see Theorem 8 of [11] for more discussion). The latter condition is implied by a classical parametric BvM in total variation. Recall that α̂n is the maximum marginal likelihood estimator defined in (3.2). Theorem 7.1. For every β, R > 0 and Mn → ∞, we have sup f0 ∈Q(β,R) E0 Πα̂n (f : \|\|f − f0 \|\|S ≥ Mn Ln n−1/2 \|Y ) → 0, where S = H(δ) or H −s for s > 1/2. If S = H(δ) then Ln = (log n)3/2 (log log n)1/2 ; if in addition f0 ∈ QSS (β, R, ε), then the rate improves to Ln = 1 for δ ≥ 2. If S = H −s for s > 1/2, then Ln = 1. Proof. This contraction result is proved in the same manner as Theorem 2 in [29], with suitable modifications for the different norms used. In the case S = H(δ), self-similarity is needed to obtain a sharp upper bound on the behaviour of α̂n , which is required to bound the posterior bias. Theorem 7.2. The finite dimensional projections of the empirical Bayes procedure satisfy a parametric BvM, that is for every finite dimensional subspace V ⊂ `2 , sup f0 ∈Q(β,R) E0 \|\|Πα̂n (·\|Y ) ◦ TY−1 − NV (0, I)\|\|T V → 0, where πV denotes the projection onto V and Tz : f 7→ 23 √ nπV (f − z). Proof. Without loss of generality, let V = span{ek : 1 ≤ k ≤ J}. Using Pinsker’s inequality and that α̂n ∈ [0, an ] by the choice (3.2), \|\|Πα̂n (· \| Y ) ◦ TY−1 − N (0, IJ )\|\|2T V ≤ sup \|\|Πα (· \| Y ) ◦ TY−1 − N (0, IJ )\|\|2T V α∈[0,an ] ≤ sup KL(Πα (· \| Y ) ◦ TY−1 , N (0, IJ )), α∈[0,an ] where KL denotes the Kullback-Leibler divergence. Using the exact formula for the KullbackLeibler divergence between two Gaussian measures on RJ , KL(Πα (· \| Y ) ◦ TY−1 , N (0, IJ )) " J # J J X X k 4α+2 Yk2 1 X n n −J = +n + log 2 k 2α+1 + n (k 2α+1 + n)2 k 2α+1 + n ≤ . 1 2n k=1 J X k=1 2α+2 J n k=1 k 2α+1 + k 4α+2 Yk2 + k=1 J 4α+3 max Y 2 . n 1≤k≤J k Since E0 max1≤k≤J Yk2 = O(1) for fixed J and α ≤ an = o(log n) by the choice of an , the result follows. Proof of Theorem 3.1. Fix η > 0, let S denote H −s or H(δ) as appropriate and set Π̃α̂n = Πα̂n ◦ τY−1 . By the triangle inequality, βS (Π̃α̂n , N ) ≤ βS (Π̃α̂n , Π̃α̂n ◦ πj−1 ) + βS (Π̃α̂n ◦ πj−1 , N ◦ πj−1 ) + βS (N ◦ πj−1 , N ), for some j > 0. Using the contraction result of Theorem 7.1 and following the argument of Theorem 8 of [11], we deduce that the E0 -expectation of the first term is smaller that η/3 for sufficiently large j, uniformly over the relevant function class (in the case of H(δ) the result holds for all δ > 2 - we recall from the proof of that theorem that if the required contraction is established in H(δ 0 ), then the required tightness argument holds in H(δ) for any δ > δ 0 ). A similar result holds for the third term. For the middle term, note that the total variation distance dominates the bounded Lipschitz metric. For fixed j, we thus have that for n large enough, E0 βS (Π̃α̂n ◦ πj−1 , N ◦ πj−1 ) ≤ E0 \|\|Πα̂n (·\|Y ) ◦ TY−1 − NV (0, I)\|\|T V ≤ η/3, using Theorem 7.2 with V = Vj . A similar situation holds true for the hierarchical Bayesian prior. Theorem 7.3. Suppose that the prior density λ satisfies Condition 1. Then for every β, R > 0 and Mn → ∞, we have sup E0 ΠH f : \|\|f − f0 \|\|S ≥ Mn Ln n−1/2 \|Y → 0, f0 ∈Q(β,R) where S = H(δ) or H −s for s > 1/2. If S = H(δ) then Ln = (log n)3/2 (log log n)1/2 ; if in addition f0 ∈ QSS (β, R, ε), then the rate improves to Ln = 1 for δ ≥ 2. If S = H −s for s > 1/2, then Ln = 1. 24 Proof. This result is proved in the same manner as Theorem 3 in [29], with suitable modifications arising as in the proof of Theorem 7.1. Theorem 7.4. The finite dimensional projections of the hierarchical Bayesian procedure satisfy a parametric BvM, that is for every finite dimensional subspace V ⊂ `2 , sup f0 ∈Q(β,R) E0 \|\|ΠH (·\|Y ) ◦ TY−1 − NV (0, I)\|\|T V → 0, where πV denotes the projection onto V and Tz : f 7→ √ nπV (f − z). Proof. Again let V = span{ek : 1 ≤ k ≤ J}. Using Fubini’s theorem and that the total variation distance is bounded by 1, \|\|ΠH (· \| Y ) ◦ TY−1 − N (0, IJ )\|\|T V Z ∞ Z 1 λ(α \| Y )dΠα (· \| Y ) ◦ TY−1 (x)dα − dN (0, IJ )(x) = 2 RJ 0 Z Z 1 ∞ ≤ λ(α \| Y ) dΠα (· \| Y ) ◦ TY−1 (x)dα − dN (0, IJ )(x) dα 2 0 RJ Z ∞ = λ(α \| Y )\|\|Πα (· \| Y ) ◦ TY−1 − N (0, IJ )\|\|T V dα 0 Z αn −1 λ(α \| Y )dα ≤ sup \|\|Πα (· \| Y ) ◦ TY − N (0, IJ )\|\|T V 0<≤α≤αn 0 Z ∞ + λ(α \| Y )dα, αn where αn is defined in Section 8.5. The first term is oP0 (1) by the same argument as in the proof of Theorem 7.2 and the second term is oP0 (1) by the proof of Theorem 3 of [29]. Since the total variation distance is bounded, convergence in P0 -probability is equivalent to convergence in L1 (P0 ). Proof of Theorem 3.2. The proof is exactly the same as that of Theorem 3.1, using Theorems 7.3 and 7.4 instead of Theorems 7.1 and 7.2. 7.2 Proof of weak BvM result in L∞ (Theorem 3.5) Following Theorem 3.1 of [27], define the sets o n p Jn (γ) = (j, k) ∈ Λ : \|f0,jk \| > γ log n/n for γ > 0. In what follows, we denote by S the support of the prior draw, that is the set of non-zero coefficients of f = (fjk )(j,k)∈Λ drawn from the prior. We require the following contraction result. Theorem 7.5. Consider the slab and spike prior defined in Section 3.2 with lower threshold given by the strictly increasing sequence j0 (n) → ∞. Then for every 0 < βmin ≤ βmax , R > 0 and Mn → ∞, we have sup f0 ∈H(β,R) E0 Π(f : \|\|f − f0 \|\|M(w) ≥ Mn n−1/2 \| Y ) → 0 uniformly over β ∈ [βmin , βmax ], where (wl ) is any admissible sequence satisfying wj0 (n) ≥ √ c log n for some c > 0. 25 Proof of Theorem 7.5. Fix η > 0. Consider the event An = {S c ∩ Jn (γ) = ∅} ∩ {S ∩ Jnc (γ) = ∅} ∩ { max (j,k)∈Jn (γ) \|f0,jk − fjk \| ≤ γ p (log n)/n}. (7.1) By Theorem 3.1 of [27], there exist constants 0 < γ < γ < ∞ (independent of β and R) such that sup E0 Π(Acn \| Y ) . n−B , (7.2) f0 ∈∪β∈[βmin ,βmax ] H(β,R) for some B = B(βmin , βmax , R) > 0 (this follows since the probabilities of the complements of each of the events constituting An satisfy the above bound individually). We then have the following decomposition for some D = D(η) > 0 large enough to be specified later, E0 Π \|\|f − f0 \|\|M ≥ Mn n−1/2 \| Y ≤ E0 Π {\|\|f − f0 \|\|M ≥ Mn n−1/2 } ∩ {\|\|πj0 (f − f0 )\|\|M ≤ Dn−1/2 } ∩ An \| Y (7.3) −1/2 −1/2 + E0 Π {\|\|f − f0 \|\|M ≥ Mn n } ∩ {\|\|πj0 (f − f0 )\|\|M > Dn } ∩ An \| Y + E0 Π(Acn \| Y ). Note that the first term on the right-hand side of (7.3) is bounded by E0 Π({\|\|π>j0 (f − f0 )\|\|M ≥ (Mn − D)n−1/2 } ∩ An \| Y ). Combining this with (7.2), we can upper bound the right hand side of (7.3) by E0 Π({\|\|π>j0 (f − f0 )\|\|M ≥ M̃n n−1/2 } ∩ An \| Y ) + E0 Π(\|\|πj0 (f − f0 )\|\|M > Dn−1/2 \| Y ) + o(1), (7.4) where M̃n = Mn −D → ∞ as n → ∞. We bound the two remaining terms in (7.4) separately. For the first term in (7.4), we can proceed as in the proof of Theorem 3.1 of [27]. By the definition of the Hölder ball H(β, R), there exists Jn (β) such that 2Jn (β) ≤ k(n/ log n)1/(2β+1) for some constant k > 0 such that Jn (γ) ⊂ {(j, k) : j ≤ Jn (β), k = 0, ..., 2j − 1} and sup sup wl−1 max \|f0,lk \| ≤ f0 ∈H(β,R) l>Jn (β) k R2−Jn (β)(β+1/2) 1 p ≤ C(β, R) √ . n Jn (β) Consider now the frequencies j0 < l ≤ Jn (β). On the event An , we have that r log n γ 1 1 1 max \|flk − f0,lk \| ≤ γ ≤ √ , sup wj 0 n c n j0 <l≤Jn (β) wl k √ since wj0 (n) ≥ c log n by hypothesis. We thus have that on the event An , \|\|π>j0 (f − f0 )\|\|M = O(n−1/2 ) for any f0 ∈ H(β, R), which proves that the first term in (7.4) is 0 for n sufficiently large. Consider now the second term in (7.4). We shall use the approach of [9] using the moment generating function to control the low frequency terms. coordinates we Qj0 Recall that on these Π have the simple product prior Π(dx1 , ..., dxj0 ) = k=1 g(xi )dxi . Let E (· \| Y ) denote the 26 expectation with respect to the posterior measure. Following Lemma 1 of [9], we have the subgaussian bound √ 2 E0 EΠ (et n(flk −Ylk ) \| Y ) ≤ Cet /2 for some some C > 0. Using this and proceeding as in the proof of Theorem 4 of [12] yields ! √ √ nE0 EΠ \|\|πj0 (f − Y )\|\|M \| Y = E0 EΠ sup l−1/2 max n\|flk − Ylk \| \| Y ≤ C, j≤j0 k for some C > 0. By Markov’s inequality and then the triangle inequality, the second term in (7.4) is then bounded by √ √ C n n E0 EΠ \|\|πj0 (f − f0 )\|\|M \| Y ≤ E0 EΠ \|\|πj0 (Y − f0 )\|\|M \| Y + D D D (7.5) C E0 \|\|Z\|\|M + . ≤ D D By Proposition 2 of [12] and the fact that (wl ) is an admissible sequence, the first term in (7.5) is also bounded by C 0 /D for some C 0 > 0. Taking D = D(η) > 0 sufficiently large, (7.5) can be then made smaller than η/2. Proof of Theorem 3.5. Fix η > 0 and denote Π̃n = Πn ◦ τY−1 . By the triangle inequality, uniformly over the relevant class of functions, βM0 (Π̃n , N ) ≤ βM0 (Π̃n , Π̃n ◦ πj−1 ) + βM0 (Π̃n ◦ πj−1 , N ◦ πj−1 ) + βM0 (N ◦ πj−1 , N ), √ for fixed j > 0. Since we have a 1/ n-contraction rate in M for the posterior from Theorem 7.5, we can make the E0 -expectation of the first term smaller than η/3 by taking j sufficiently large, again using the arguments of Theorem 8 of [11]. We recall from the proof of that theorem that if the required contraction is established in M(w) for an admissible sequence (wl ), then the required tightness argument holds in M0 (w) for any admissible (wl ) such that wl /wl % ∞. A similar result holds for the third term. For the middle term, note that j0 (n) ≥ j for n large enough. For such n, the projected prior onto the first j coordinates is a simple product prior which satisfies the usual conditions of the parametric BvM, namely it has a density that is positive and continuous at the true (projected) parameter (see Chapter 10 of [50] for more details). Since the total variation distance dominates the bounded Lipschitz metric, this completes the proof. 7.3 Credible sets `2 confidence sets Proof of Proposition 4.1. By Lemma 8.1 and the definition of C̃n , we have √ sup \|P0 (f0 ∈ C̃n ) − P0 (\|\|f0 − Y\|\|H ≤ Rn / n)\| → 0 f0 ∈QSS (β,R,ε) β∈[β1 ,β2 ] as n → ∞, so that it is sufficient to show that the second probability in the above display tends to 1 − γ, uniformly over the relevant self-similar Sobolev balls. This follows directly by Theorem 1 of [11] (an examination of that proof shows that the convergence holds uniformly 27 over the parameter space as long as the weak BvM itself holds uniformly, as is the case here), thereby establishing the required coverage statement. Since n ≥ r1 / log n and C > 1/r1 in the definition (4.2) of C̃n , applying Lemma 8.3 (with η = 1) yields the inequality p 1 Πα̂n \|\|f − fˆn \|\|H α̂n −n ≥ C log n Y . exp −C 0 (log n)n 4α̂n +2 , where C 0 > 0 does not depend on α̂n . Since by Lemma 8.6 we have α̂n ≤ β for large enough n with P0 -probability tending to 1, the right-hand side is bounded by a multiple of exp(−C 00 (log n)n1/(4β+2) ) with the same probability. The completes the credibility statement. Let f1 , f2 ∈ C̃n and set g = f1 − f2 . Picking Jn ∼ [n/(log n)2δ−1 ]1/(1+2α̂n −2n ) yields \|\|g\|\|22 = ∞ X 2 \|gk \| = k=1 Jn X kk −1 2δ−2δ (log k) \|gk \| + k=1 2δ 2 \|\|g\|\|2H(δ) ∞ X k 2(α̂n −n )−2(α̂n −n ) \|gk \|2 k=Jn +1 −2(α̂n −n ) Jn \|\|g\|\|2H α̂n −n ≤ Jn (log Jn ) + = OP0 Jn (log Jn )2δ n−1 + Jn−2(α̂n −n ) (log n) 4δ(α̂n −n )+1 2(α̂ −n ) − 1+2α̂n −2 n n (log n) 1+2α̂n −2n , = OP0 n where the constants do not depend on g. Since \|α̂n − β\| = OP0 (1/ log n) by Lemma 8.6 and n = O(1/ log n) by assumption, some straightforward computations yield that \|\|g\|\|22 = OP0 (n−2β/(2β+1) (log n)(4δβ+1)/(2β+1) ) as n → ∞. Proof of Proposition 4.4. The proof follows in the same way as that of Proposition 4.1, using Lemma 8.7 and an analogue of Lemma 8.1. The only difference is for the credibility statement, where we no longer have an exponential inequality like Lemma 8.3. However, arguing as in [29] with the H β̂n -norm instead of the `2 -norm, one can show that under self-similarity the √ posterior contracts about the posterior mean at rate M̃n log n for any M̃n → ∞ (see also Theorem 1.1 of [49]). It then follows that the second constraint in (4.3) is satisfied with credibility 1 − oP0 (1). L∞ confidence bands Proof of Proposition 4.5. By Lemma 8.4, it suffices to prove all the results on the event Bn defined in (8.1). We firstly establish the diameter of the confidence set. Recall that πmed denotes the projection onto the non-zero coordinates of the posterior median and for a set of coordinates E, let πE denote the projection onto span(E). Taking f1 , f2 ∈ Dn and setting 2Jn (β) ' (n/ log n)1/(2β+1) , we have on Bn , \|\|f1 − f2 \|\|∞ ≤ \|\|f1 − πmed (Y )\|\|∞ + \|\|f2 − πmed (Y )\|\|∞ r Jn (β) 2l −1 X X log n ≤ 2 sup vn \|ψlk (x)\| n x∈[0,1] l=0 k=0 r r Jn (β) log n X l/2 2Jn (β) log n 0 ≤ C(ψ)vn 2 ≤ C vn = OP0 n n l=0 28 log n n β 2β+1 ! vn . We now establish asymptotic coverage. Split f0 = πJn (γ) (f0 ) + πJnc (γ) (f0 ). Since πJnc (γ) ◦ πmed (Y ) = 0 on Bn , we can write \|\|f0 − πmed (Y )\|\|∞ ≤ \|\|πmed (f0 − Y )\|\|∞ + \|\|(id − πmed ) ◦ πJn (γ) (f0 )\|\|∞ + \|\|πJnc (γ) (f0 )\|\|∞ , (7.6) where id denotes the identity operator. For the third term in (7.6), note that since f0 ∈ H(β, R), \|\|πJnc (γ) (f0 )\|\|∞ ≤ ∞ X 2l/2 l=0 max k:(l,k)∈Jnc (γ) Jn (β) ≤ X r 2 l/2 l=0 γ \|hf0 , ψlk i\| (7.7) β X log n 2β+1 log n −lβ 2 ≤ C(β, R) + . n n l>Jn (β) For the second term in (7.6), we note that any indices remaining satisfy (l, k) ∈ Jnc (γ 0 ) and so by the same reasoning as above, this term is also O((log n/n)β/(2β+1) ). By the proof of Proposition 3 of [26], we have that for f0 ∈ HSS (β, R, ε), sup \|hf0 , ψlk i\| ≥ d(b, R, β, ψ)2−j(β+1/2) . (l,k):l≥j ˜ Let J˜n (β) be such that 2 (n/ log n)1/(2β+1) ≤ 2Jn (β) ≤ (n/ log n)1/(2β+1) , where = (b, R, β, ψ) > 0 is small enough so that d/β+1/2 > γ 0 . Using this yields r r log n d(b, R, β, ψ) log n sup \|hf0 , ψlk i\| ≥ > γ0 . β+1/2 n n (l,k):l≥J˜n (β) We therefore have that on the event Bn , there exists (l0 , k 0 ) with l0 ≥ J˜n (β) such that f˜l0 k0 6= 0 and a non-zero coefficient therefore appears in the definition (4.5) of σn,γ . We can thus lower bound s r β log n 2J˜n (β) log n log n 2β+1 0 σn,γ ≥ vn sup \|ψl0 k0 (x)\| ≥ c(ψ)vn = c vn . (7.8) n x∈[0,1] n n Now, since vn → ∞ as n → ∞, we have from (7.7) and the remark after it that for sufficiently large n (depending on β and R), the last two terms in (7.6) satisfy \|\|(id − πmed ) ◦ πJn (γ) (f0 )\|\|∞ + \|\|πJnc (γ) (f0 )\|\|∞ ≤ C log n n β 2β+1 ≤ σn,γ /2. For the first term in (7.6) we recall that on Bn , the posterior median only picks up coefficients (l, k) with l ≤ Jn (β) ≤ Jn . Therefore on this event, X \|\|πmed (f0 − Y )\|\|∞ ≤ sup \|f0,lk − Ylk \|\|ψlk (x)\| x∈[0,1] (l,k):f˜lk 6=0 r ≤ C(ψ) log n n X (l,k):l≤Jn (β) 29 2 l/2 ≤C 0 log n n 1 2β+1 . Using the lower bound (7.8), we deduce that on Bn , \|\|πmed (f0 − Y )\|\|∞ ≤ σn,γ (Y )/2 for n large enough, uniformly over f0 ∈ HSS (β, R). Combining all of the above yields that Bn ⊂ {\|\|f0 − πmed (Y )\|\|∞ ≤ σn,γ }. We therefore conclude that √ P0 (f0 ∈ Dn ) = P0 ({\|\|f0 − Y\|\|M(w) ≤ Rn / n} ∩ {\|\|f0 − πmed (Y )\|\|∞ ≤ σn,γ } ∩ Bn ) + o(1) √ = P0 ({\|\|f0 − Y\|\|M(w) ≤ Rn / n} ∩ Bn ) + o(1) = 1 − γ + o(1), √ where we have used that P0 (Bn ) → 1 and that P0 (\|\|f0 − Y\|\|M(w) ≤ Rn / n) → 1 − γ by Theorem 5 of [12]. Noting finally that both of these probabilities converge uniformly over the relevant self-similar Sobolev balls, the coverage statement also holds uniformly as required. For the credibility statement it suffices to show that the second constraint in (4.6) is satisfied with posterior probability tending to 1. Again using that \|\|πmed (f0 − Y )\|\|∞ ≤ σn,γ (Y )/2 on Bn as well as (7.8), we have that uniformly over f0 ∈ H(β, R), E0 Π(f : \|\|f − πmed (Y )\|\|∞ ≥ σn,γ \| Y ) ≤ E0 Π(f : \|\|f − f0 \|\|∞ ≥ σn,γ /2 \| Y ) + E0 Π(f : \|\|f0 − πmed (Y )\|\|∞ ≥ σn,γ /2 \| Y ) ≤ E0 Π(f : \|\|f − f0 \|\|∞ ≥ c0 vn ((log n)/n)β/(2β+1) /2 \| Y ) + P0 (Bnc ) → 0 since the posterior contracts at rate (log n/n)β/(2β+1) by Theorem 3.1 of [27]. 7.4 Posterior independence of the credible sets Proof of Theorem 5.1. We first consider the fixed-regularity prior Πα with α ∈ [0, an ], re(α) placing the sets C̃n and Cn`2 respectively by the (1 − γ)-H(δ)-credible ball Cn and the (α,`2 ) (1 − γ)-`2 -credible ball Cn for Πα (· \| Y ) (i.e. (4.1) and (5.1) for Πα (· \| Y ) rather than Πα̂n (· \| Y )). By the definition of the posterior distribution (3.1), we can write a posterior draw f ∼ Πα (· \| Y ) as ∞ X 1 √ f − fˆn,α = ζ e , 2α+1 + n k k k k=1 where ζk ∼ N (0, 1) are independent and fˆn,α is the posterior mean. Let kn → ∞ be some sequence satisfying kn = o(n1/(4α+2) ). We shall prove the result by showing that the H(δ)credible ball is determined by the frequencies k ≤ kn , while the `2 -credible ball is determined by the frequencies k ≥ kn . We therefore decompose both credible balls according to the threshold kn . By Lemma 1 of [32], which can be adapted to the case D = ∞, and some elementary computations, we have the following exponential inequalities for any x ≥ 0:   r ∞ 2 X ζk C(δ) x x  P ≥ log kn + + ≤ e−x , (7.9) kn kn k(log k)2δ (log kn )2δ k=kn +1 P kn X ! p ζk2 ≥ kn + 2 kn x + 2x ≤ e−x , (7.10)  1 √ 1 − 23/2 n− 4α+2 x  ≤ e−x , 4α (7.11) k=1  P ∞ X k=kn +1 ζk2 k 2α+1 + 2α n ≤ n− 2α+1 30 where C(δ) < ∞ for δ > 1/2. Moreover, note that P kn X k=1 ζk2 x ≤ n (k 2α+1 + n)k(log k)2δ−1 ! p ≤ P ζ12 /2 ≤ x ≤ x/π, (7.12) using that ζ1 is standard normal. Define the event   ( ) kn ∞  X X ζk2 3C(δ)  2 Ãn,α = ≤ ∩ ζk ≤ 5kn  k(log k)2δ (log kn )2δ−1  k=kn +1 k=1   ∞  X  2α ζk2 1 − 2α+1 n ∩ ≥   k 2α+1 + n 8an k=kn +1 (k ) n X ζk2 1 ∩ ≥ . δ−1/2 (k 2α+1 + n)k(log k)2δ−1 n(log k ) n k=1 1/(2an +1) in (7.11) and Setting x = kn in the inequalities (7.9) and (7.10), x = 2−9 a−2 n n x = (log kn )−(δ−1/2) in (7.12) yields sup Πα (Ãcn,α \| Y ) ≤ 2e−kn + e−2 −9 a−2 n1/(2an +1) n √ + (log kn )−(2δ−1)/4 / π → 0 α∈[0,an ] as n → ∞, since an ≤ log n/(6 log log n) by assumption. We have that on Ãn,α , ∞ X 2α ζk2 1 − 2α+1 5kn n ≤ \|\|f − fˆn,α \|\|22 ≤ + , 8an n k 2α+1 + n k=kn +1 k n X ζk2 1 3C(δ) ˆn,α \|\|2 ≤ \|\|f − f ≤ + . H(δ) 2α+1 (k + n)k(log k)2δ n(log kn )2δ−1 n(log kn )δ−1/2 k=1 √ (α) Recall that Cn has radius equal to Rn / n, where Rn (Y, γ) →P0 R(γ) > 0 by Theorem (α,` ) 1 of [11]. Similarly, Cn 2 has radius Qn n−α/(2α+1) , where Qn → Q > 0 by Theorem 1 of [20]. Using these facts, the above bounds and the definition of kn , the probability 31 (α) (α,`2 ) Πα (Cn ∩ Cn \| Y ) equals o n 2α R2 \|\|f − fˆn \|\|2H(δ) ≤ n , \|\|f − fˆn \|\|22 ≤ Q2n n− 2α+1 ∩ Ãn Y + o(1) n X kn ζk2 1 2 −(2δ−1) = Πα ≤ R + O (log k ) , n n n (k 2α+1 + n)k(log k)2δ k=1 ∞ X 2α 1 ζk2 − 2α+1 − 2α+1 2 ∩ Ãn Y + o(1) n ≤ Qn + O kn n k 2α+1 + n Πα k=kn +1 1 ∞ ζk2 ζk2 Rn2 + o(1) X Q2n + o(n− 4α+2 ) = Πα ≤ , ≤ Y + o(1) 2α n k 2α+1 + n (k 2α+1 + n)k(log k)2δ n 2α+1 k=1 k=kn +1 ! kn X ζk2 Rn2 + o(1) = Πα ≤ Y n (k 2α+1 + n)k(log k)2δ k=1   1 ∞ − 4α+2 2 2 X ζk Q + o(n )  ≤ n Y + o(1), × Πα  2α 2α+1 k +n n 2α+1 X kn k=kn +1 where in the last line we have used the independence of the coordinates under the posterior. Using again the inequalities (7.9)-(7.12), the final line equals Πα \|\|f − fˆn,α \|\|2H(δ) ≤ (Rn2 + o(1))/n Y 2α 1 × Πα \|\|f − fˆn,α \|\|22 ≤ Q2n + o(n− 4α+2 ) n− 2α+1 Y + o(1) (2) =: Π(1) α,n × Πα,n + o(1). Since supα∈[0,an ] Πα (Ãcn,α \| Y ) → 0, the previous display holds uniformly over α ∈ [0, an ] so that we have shown sup (2) Πα (Cn(α) ∩ Cn(α,`2 ) \| Y ) − Π(1) α,n × Πα,n = o(1). (7.13) α∈[0,an ] For the full empirical Bayes posterior, note that the second constraint in (4.2) is satisfied with posterior probability 1 − oP0 (1) uniformly over f0 ∈ Q(β, R) by the proof of Proposition 4.1, so that it suffices to prove the theorem with Cn in (4.1) instead of C̃n . Since an = o(log n), we can take kn → ∞ such that kn = o(n1/(4an +2) ), from which also kn = o(n1/(4α+2) ) for all α ∈ [0, an ], the interval over which α̂n ranges. By (7.13), it therefore remains to check that (1) (2) Πα̂n ,n , Πα̂n ,n →P0 1 − γ as n → ∞. For this we require a finer understanding of the posterior behaviour of the norms in (1) the second to last display. Consider firstly Πα̂n ,n . By Theorem 3.1 and Lemma 8.2, Πα̂n satisfies a weak BvM in H(δ) with centering fˆn = fˆn,α̂ instead of Y. By Theorem 1 of [11], n Rn →P0 Φ̃−1 (1 − γ) > 0, where Φ̃ is defined via Φ̃(t) = N (kZkH(δ) ≤ t) and we recall N is (1) the law of the white noise Z as an element of H(δ). Note that Πα̂n ,n equals N (kZkH(δ) √ ˆ ≤ Rn + o(1)) + O sup \|Πα̂n ( n\|\|f − fn,α̂n \|\|H(δ) ≤ t\|Y ) − N (kZkH(δ) ≤ t)\| . t≥0 32 Since Φ̃ is strictly monotone and continuous, the first term equals Φ̃(Rn + o(1)) = (1 − γ) + oP0 (1). Using that H(δ)-norm balls form a uniformity class for N (see the proof of Theorem 1 of [11]), the second term is oP0 (1) as required. (2) We now turn our attention to Πα̂n ,n . By Theorem 1 of [29], supf0 ∈Q(β,R) P0 (α̂n ∈ [αn , αn ]) → 1, where αn , αn are defined in Section 8.5. By Lemma 1(i) of [29], αn ≥ β − C/ log n ≥ β/2 for n large enough. Since also α̂n ≤ an by the choice (3.2), supf0 ∈Q(β,R) P0 (α̂n ∈ [β/2, an ]) → 1 and we may therefore restrict α̂n to this interval. By Theorem 1 of Freedman [20], we have that for f ∼ Πα (·\|Y ), p kf − fˆn,α k22 = Cn,α + Dn,α Zn,α , R∞ R∞ 4α+1 2α where Cn,α n 2α+1 → Cα = 0 (1 + u2α+1 )−1 du, Dn,α n 2α+1 → Dα = 2 0 (1 + u2α+1 )−2 du, Zn,α has mean 0, variance 1 and Zn,α →d N (0, 1) as n → ∞. We wish to make these three convergence statements uniform over α ∈ [β/2, an ]. The first two statements essentially follow from [20] upon keeping careful track of the remainder terms. Let gα (u) = 1/(1 + u2α+1 ). 2α P∞ 2α+1 + n) and n 2α+1 /(k 2α+1 + n) = h g (kh ), where First note that Cn,α = n α n k=1 1/(k 2α hn = n−1/(2α+1) . Using these yields that for any L > 0, \|Cn,α n 2α+1 − Cα \| is bounded above by L/hn X Z hn gα (khn ) − gα (u)du + 0 k=1 ∞ X L Z ∞ hn gα (khn ) + gα (u)du. L k=L/h+1 The second and third terms are easily bounded by L−2α /(2α). The first term is just the error RL when approximating 0 gα with its (right) Riemann sum with L/h points, which is bounded 1 1 by kgα0 kL∞ [0,L] L2 /(2L/hn ). One can show that kgα0 kL∞ [0,∞) = α1− 2α+1 (α+1)1+ 2α+1 /(2α+1). Substituting this into the bound for the Riemann sum and optimizing the three terms over L gives that the previous display is bounded by a multiple of 1 α 1 1 (2α)2− 2α+1 (2α + 2)1+ 2α+1 2α + 1 2α ! 2α+1 2α hn2α+1 . It can be checked that over the range 0 < β/2 ≤ α ≤ an → ∞, the above display without 2α the hn2α+1 term is maximized at an for n large enough, whereupon it can be bounded by 2α − 2α a constant multiple of an . The continuous function α 7→ hn2α+1 = n (2α+1)2 has a single minimum on [0, ∞) occurring at α = 1/2, is strictly decreasing on [0, 1/2], strictly increasing on [1/2, ∞) and attains its maximal value of 1 at α = 0, ∞. Since we consider only the region α ≥ β/2, it follows that the maximum will occur at an for n large enough, depending only on β. We have therefore shown that the previous display is bounded above by log n − 2an 2 (2a +1) n C(β)an n ≤ C(β) exp log an − → 0, 4an where the inequality holds for n large enough (depending on an ) and the convergence to zero 2α follows since an ≤ log n/(6 log n log n) by assumption. In conclusion, supα∈[β/2,an ] \|Cn,α n 2α+1 − 4α+1 Cα \| → 0. Identical computations yield that supα∈[β/2,an ] \|Dn,α n 2α+1 − Dα \| → 0. 33 It remains only to show the uniformity of the convergence in distribution, which is based on the central limit theorem (Theorem 1 of [20]). Under the posterior, −1/2 Zn,α = Dn,α ∞ X k=1 1 k 2α+1 + n 2 (ζn,k − 1) =: ∞ X Xn,k , k=1 where ζn,k ∼ N (0, 1) are independent and −1/2 2 Xn,k = Dn,α (ζn,k − 1)/(k 2α+1 + n) are also independent. Recalling the exact definition of Dn,α from Theorem 1 of [20], Dn,α = P Pn1/(2α+1) −2 1 −2+ 1 r 2α+1 + n)−2 ≥ 1 2α+1 . Letting λ = E\|ζ 2 2 ∞ n = 2n r k=1 (k k=1 n,k − 1\| , 2 ∞ X  −3/2 E\|Xn,k \|3 ≤ Dn,α λ3  n1/(2α+1) X k=1 k=1 1  1 + n3 X k −6α−3  k>n1/(2α+1) 1 1 ≤ 2(2n2− 2α+1 )3/2 λ3 n−3+ 2α+1 = 25/2 λ3 n− 4α+2 . Let Fn,α and Φ denote the cdfs of Zn,α and the standard normal distribution respectively. By the Berry-Esseen theorem for infinite arrays (Theorem 3.2 of [21] with summability matrix pn,k ≡ 1), there exists a universal constant C0 such that kFn,α − Φk∞ ≤ C0 ∞ X 1 E\|Xn,k \|3 ≤ 25/2 C0 λ3 n− 4α+2 , k=1 which implies supα∈[β/2,an ] kFn,α − Φk∞ → 0 since an = o(log n). We have shown that p 2α+1/2 2α kf − fˆn,α k22 = (Cα + o(1))n− 2α+1 + ( Dα + o(1))n− 2α+1 Zn,α , where all o(1) terms and the convergence in distribution are uniform over α ∈ [β/2, an ]. We (2) consequently see that the remainder term in the posterior probability Πα,n only changes this (2) probability by o(1) for any α ∈ [β/2, an ] and so Πα̂n ,n = (1 − γ) + oP0 (1). (2) Proof of Theorem 5.3. Throughout we will write Tn instead of Tn for convenience, where  ˆ if j ≤ j0 (n),  fn,jk (2) Tn,jk = Yjk 1{f˜jk 6=0} if j0 (n) < j ≤ blog n/ log 2c.   0 if blog n/ log 2c < j, and fˆn denotes the posterior mean of the slab and spike procedure. Since the second constraint in (4.6) is satisfied with posterior probability 1 − oP0 (1) uniformly over f0 ∈ H(β, R) by the proof of Proposition 4.5, it suffices to prove the theorem p with Dn in (4.4) instead of Dn . Let f0 ∈ H(β, R) and jn → ∞ satisfy jn ≤ j0 (n), wjn ≤ j0 (n) and wj2n 2jn = o(2j0 (n) ). Similarly to Theorem 5.1, we shall decompose both credible balls according to the threshold jn . For this we must understand the typical sizes of the projections of f − Tn in both norms under the posterior. 34 Consider firstly k · kM . For the frequencies l > j0 (n), \|\|π>j0 (n) (f −Tn )\|\|M ≤ \|\|π>j0 (n) (f −f0 )\|\|M +\|\|π>j0 (n) (f0 −Y)\|\|M +\|\|π>j0 (n) (Y−Tn )\|\|M . (7.14) p The third term is OP0 (wj−1 (log n)/n) by the proof of Lemma 8.5. For the first term, on (n) 0 the event An defined in (7.1), max wl−1 max \|flk − f0,lk \| ≤ l>j0 (n) k max wl−1 max \|flk − f0,lk \| + max wl−1 max \|f0,lk \| k k l>Jn (β) p p −1 −1 ≤ wj0 (n) (log n)/n + wJn (β) (log n)/n p = O(wj−1 (log n)/n). (n) 0 j0 (n)<l≤Jn (β) For the second term in (7.14), E0 \|\|π>j0 (n) (f0 − Y)\|\|M(w) p j0 (n) √ E0 \|\|π>j0 (n) (Z)\|\|M(√l) = O ≤ wj0 (n) n ! p j0 (n) √ wj0 (n) n using that E0 \|\|Z\|\|M(√l) is finite by Proposition 2 of [12]. Combining these yields Π(f : \|\|π>j0 (n) (f − Tn )\|\|M = O(wj−1 0 (n) p (log n)/n) \| Y ) = 1 − oP0 (1) since j0 (n) . log n. (2) Recall that by definition Tn,lk equals the posterior mean for l ≤ j0 (n) (see (8.6)), so that τ2 (2) flk − Tn,lk \|Ylk ∼ N 0, , 0 ≤ l ≤ j0 (n), 1 + nτ 2 under the posterior. For √ (ζlk ) i.i.d. standard Gaussians, we have the well-known bound E max0≤k<2l \|ζlk \| ≤ C l for some universal constant C. Applying the Borell-SudakovTsirelson inequality [35] to the maximum at level l yields that for M > 0 large enough, p Π(f :k(π>jn − π>j0 (n) )(f − Tn )kM ≥ M jn wj−1 n−1/2 \|Y ) n √ 1 M jn τ √ =P max ζlk ≥ max √ wj n n jn <l≤j0 (n) wl 0≤k<2l 1 + nτ 2 ! √ √ j0 (n) X M jn wl 1 + nτ 2 √ ≤ P max \|ζlk \| − E max \|ζlk \| > − E max \|ζlk \| wjn nτ 0≤k<2l 0≤k<2l 0≤k<2l l=jn +1  !2  √ j0 (n) X w j 0 l √n − C l ≤ Ce−c jn → 0. ≤2 exp −c M wj n l l=jn +1 Combining this with the above inequalities gives p Π(f : \|\|π>jn (f − Tn )\|\|M = O(wj−1 (log n)/n) \| Y ) = 1 − oP0 (1). 0 (n) (7.15) We now show that the k · kM -norm of the remaining frequencies j ≤ jn is of strictly larger size with high probability. For any un → 0, √ Π(f : \|\|πjn (f − Tn )\|\|M ≤ w0 un n−1/2 \| Y ) ≤ Π(f : \|ζ00 \| ≤ cun \| Y ) ≤ 2ceun / 2π = o(1). (7.16) 35 √ In particular, taking un log n/wj0 (n) gives the result. Turn now to k · kB∞1 0 . For f ∈ Dn , \|\|πjn (f − Tn )\|\|B∞1 = 0 jn X 2l/2 max \|flk − Tn,lk \| ≤ k l=0 jn X l=0 Rn 2l/2 wl √ = OP0 n wjn 2jn /2 √ n ! . (7.17) Note that Π(f : kπ>jn (f − Tn )kB∞1 ≤ c2 0 j0 (n)/2 p j0 (n)/n \| Y ) ≤ P max 0≤k<2j0 (n) ζj0 (n)k p ≥ c j0 (n) . 0 (7.18) Using again the Borell-Sudakov-Tsirelson inequality as above gives that the right-hand side 00 is O(2−c j0 (n) ) for c > 0 small enough. Combining (7.15)-(7.18), we have shown that for some √ C1 , ..., C4 > 0 and un → 0 such that un log n/wj0 (n) , √ n log n √ , Ān := \|\|π>jn (f − Tn )\|\|M ≤ C1 wj0 (n) n \|\|πjn (f − Tn )\|\|B∞1 0 un kπjn (f − Tn )kM ≥ C2 √ , n wj 2jn /2 , ≤ C3 n√ n kπ>jn (f − Tn )kB∞1 0 p j0 (n)2j0 (n)/2 o √ ≥ C4 n satisfies Π(Ān \|Y ) = 1 − oP0 (1). √ Write δn := wjn 2jn /2 / n. Note that by the previous display and the assumptions on the p √ ∞ L j (n) 0 growth of jn → ∞, the radius Q̄n (γ) of the credible set Dn satisfies Q̄n & j0 (n)2 / n δn with P0 -probability tending to one. Using the independence of the different coordinates ∞ under the posterior, the probability Π(Dn ∩ DnL \| Y ) equals √ Π \|\|f − Tn \|\|M ≤ Rn / n, \|\|f − Tn \|\|∞ ≤ Q̄n ∩ Ān \| Y + oP0 (1) n p √ = Π \|\|πjn (f − Tn )\|\|M ≤ (Rn + O(wj−1 log n))/ n, (n) 0 o ≤ Q̄n + O(δn ) ∩ Ān \| Y + oP0 (1) \|\|π>jn (f − Tn )\|\|B∞1 0 √ = Π f : \|\|πjn (f − Tn )\|\|M ≤ (Rn + o(1))/ n \| Y × Π f : \|\|π>jn (f − Tn )\|\|B∞1 ≤ Q̄n + O(δn ) \| Y + oP0 (1). 0 Again using that Π(Ān \|Y ) = 1 − oP0 (1), the final line equals √ Π f : \|\|f − Tn \|\|M ≤ (Rn + o(1))/ n × Π f : \|\|f − Tn \|\|B∞1 ≤ Q̄n + O(δn ) \| Y + oP0 (1). 0 (7.19) We now check that the above product has asymptotically the correct posterior probability. By the same argument as in Theorem 5.1 (replacing H(δ) by M), the first probabilityP equals (1 − γ) + oP0 (1). Setting Ml = 2l/2 max0≤k<2l \|flk − Tn,lk \|, we have kf − Tn kB∞1 = l Ml , 0 where the (Ml ) are independent due to the product structure of the posterior. We can therefore write the posterior density of kf − Tn kB∞1 as hn ∗ Gn , where Mj0 (n) has density hn 0 P and l6=j0 (n) Ml has probability distribution Gn , both supported on [0, ∞). 36 Using standard extreme value theory, we can establish the limiting distribution of Mj0 (n) . For Zi ∼ N (0, 1) independent, we have a−1 m (max1≤i≤m \|Zi \| − bm ) converges in distribution to the standard Gumbel distribution (i.e. distribution function F (x) = exp(−e−x )), where p 1 log(4π log 2m) 1 √ am = √ , bm = 2 log 2m − +O log m 2 log 2m 2 2 log 2m 2 (bm is the solution to 2m2 = πb2m ebm ). This follows from Theorem 10.5.2(c) and Example 10.5.3 of [18] with only minor modifications due to the absolute values within the maximum (intuitively it is the same as the maximum of 2m standard Gaussians). Moreover, by Pólya’s Theorem (p. 265 of [16]) the convergence of the distribution functions is uniform: sup \|(Φ(am x + bm ) − Φ(−am x − bm ))m − exp(−e−x )\| → 0, (7.20) x∈R where we recall \|Zi \| has distribution function Φ(x) − Φ(−x). Recall that Mj0 (n) is the sum of i.i.d. (rescaled) Gaussians under the posterior. Using that Q̄n is the (1 − γ)-posterior quantile for kf − Tn kB∞1 0 , \|Π(f :\|\|f − Tn \|\|B∞1 ≤ Q̄n + O(δn ) \| Y ) − (1 − γ)\| 0 X ≤ Π Q̄n − cδn ≤ Ml ≤ Q̄n + cδn Y l Z Q̄n +cδn Z ∞ hn (x − y)dGn (y)dx = Q̄n −cδn Z 0 ∞ Z Q̄n −y+cδn hn (z)dGn (y) = 0 Q̄n −y−cδn ≤ sup P(Mj0 (n) ∈ [t − cδn , t + cδn ]). t≥0 Using the limiting distribution of Mj0 (n) , (7.20) and that the maximum of the standard Gumbel density function is e−1 , the last probability equals ! 2j0 (n)/2 τ max \|ζj0 (n)k \| ∈ [t − cδn , t + cδn ] P √ 1 + nτ 2 0≤k<2j0 (n) ! √ 2 1 + nτ = P a−1 max \|ζ \| − b2j0 (n) ∈ a−1 [t − b2j0 (n) − cδn , t − b2j0 (n) + cδn ] 2j0 (n) 0≤k<2j0 (n) j0 (n)k 2j0 (n) 2j0 (n)/2 τ ! √ 1 + nτ 2 −1 ≤ P Gumbel(0, 1) ∈ a2j0 (n) j (n)/2 [t − b2j0 (n) − cδn , t − b2j0 (n) + cδn ] + o(1) 20 τ r p jn −j0 (n) nj0 (n) 0 −1 2 + o(1) → 0, ≤ c0 e−1 δ + o(1) = c e w j (n)2 n j 0 n 2j0 (n) by the choice of jn . In conclusion, we have shown that the second probability in (7.19) equals (1 − γ) + oP0 (1). This completes the proof. 37 7.5 Remaining proofs Proof of Proposition 3.3. Fix ρ > 1, let ε = ε(α, ρ, R) < (1 − ρ−2α )/(2αR) be sufficiently PdρN e small so that ε ∈ (0, 1) and consider the events Aα,N = { k=N fk2 < εRN −2α }. By a simple PdρN e integral comparison we have that k=N k −2α−1 ≥ (2α)−1 N −2α (1 − ρ−2α ), so that under the conditional prior,   dρN e X Πα (Aα,N ) = P  k −2α−1 gk2 < εRN −2α  k=N   1 k −2α−1 (gk2 − 1) < εRN −2α − ≤ P N −2α (1 − ρ−2α ) 2α k=N   dρN e X k −2α−1 (gk2 − 1) < −ε0 N −2α  , ≤ P dρN e X k=N where the gk ’s are i.i.d. standard normal random variables and ε0 > 0 (by the choice of ε). By (4.2) of Lemma 1 of [32] we have the exponential inequality    1/2 dρN e dρN e X X √   k −4α−2  P k −2α−1 (gk2 − 1) ≤ −2  x ≤ e−x . k=N k=N PdρN e For N ≥ 2, again by an integral comparison we have that k=N k −4α−2 ≤ C(α)N −4α−1 . Using this and letting x = M N , the exponential inequality becomes   dρN e X √ k −2α−1 (gk2 − 1) ≤ −C 0 (α) M N −2α  ≤ e−M N . P k=N √ Taking M sufficiently small so that C 0 (α) M < ε0 , we obtain that Πα (Aα,N ) ≤ e−M N . Since this sequence is summable in N , the result follows from the first Borel-Cantelli Lemma. √ Proof of Proposition 3.7. Under the law P0 , nE0 \|\|Y − f0 \|\|M(w) = E0 \|\|Z\|\|M(w) < ∞ by Proposition 2 of [12]. By the triangle inequality it therefore suffices to show the conclusion of Proposition 3.7 with Y replaced by f0 . Rewrite the multiscale indices Λ = {(l, k) : l ≥ 0, k = 0, ..., 2l − 1} in increasing lexicographic order, so that Λ = {(lm , km ) : m ∈ N}, where lm = i, if 2i ≤ m < 2i+1 , i = 0, 1, 2, ..., km = m − 2i , if 2i ≤ m < 2i+1 , i = 0, 1, 2, ... Consider a strictly increasing subsequence (nm )m≥1 of N such that (log nm )/wl2m → ∞ as m → ∞ (such a subsequence can be constructed for any admissible (wl ) since wl % ∞). Define a function f0 ∈ `2 via its wavelet coefficients p hf0 , ψlm km i = r log nm /nm , 38 where r ≤ γ for γ the value given in the proof of Theorem 7.5. Since r 2 lm (β+1/2) β+1/2 \|hf0 , ψlm km i\| ≤ rm log nm , nm we can ensure f0 is in any given Hölder ball H(β, R) by letting r be sufficiently small and taking the subsequence nm to grow fast enough. Consider now a further subsequence, removing terms corresponding to one index per resolution level, say (l, kl ) (i.e. removing terms with indices m = 2l + kl , l = 0, 1, 2, ..., from the above subsequence), and set 0 \|hf0 , ψlkl i\| = R2−l(β+1/2) . Using the Besov space embedding L∞ ⊂ B∞∞ , \|\|Kj (f ) − f \|\|∞ ≥ C(ψ) max 2l/2 max \|hf0 , ψlk i\| l>j k ≥ C(ψ)2(j+1)/2 \|hf0 , ψ(j+1)kj+1 i\| = C(ψ)R2−β 2−jβ = ε(β, R, ψ)2−jβ , thereby establishing that f0 ∈ HSS (β, R, ε). Let An denote the event defined in (7.1). We have that on Anm , the posterior distribution 0 Π (· \| Y (nm ) ) assigns the (lm , km ) coordinate to the Dirac mass component of the distribution. Consequently, by the choice of (nm ), E0 Π0 ( \|\|f − f0 \|\|M ≤ Mnm nm −1/2 \| Y (nm ) ) = E0 Π0 ({\|\|f − f0 \|\|M ≤ Mnm nm −1/2 } ∩ Anm \| Y (nm ) ) + o(1) ≤ E0 Π0 ({\|flm km − f0,lm km \| ≤ Mnm wlm nm −1/2 } ∩ Anm \| Y (nm ) ) + o(1) p = E0 Π0 ({r log nm /nm ≤ Mnm wlm n−1/2 } ∩ Anm \| Y (nm ) ) + o(1) m p ≤ E0 Π0 (r log nm /wlm ≤ Mnm \| Y (nm ) ) + o(1) = o(1) √ for any sequence Mn such that Mnm = o(wl−1 log nm ) as m → ∞. m 8 8.1 Technical facts and results Results for `2 -setting The following two lemmas describe the behaviour of the posterior mean of the empirical Bayes procedure. The first says that the posterior mean is a consistent estimator of f0 in a sequence of Sobolev norms with data driven exponent. In particular, we are interested in the case n → 0 when the Sobolev exponent tends to the true smoothness β. Note that we require n strictly positive since the posterior mean is itself not an element of H α̂n . The −1/2,δ second says that fˆn is an efficient estimator of f0 in H2 . Both proofs are similar to that of Theorem 2 of [29] and are thus omitted. Lemma 8.1. Let fˆn denote the posterior mean of the empirical Bayes procedure and let n > 0. Then for every β, R > 0 and Mn → ∞, we have sup P0 \|\|fˆn − f0 \|\|H α̂n −n ≥ Mn → 0 f0 ∈QSS (β,R,ε) as n → ∞. 39 Lemma 8.2. Let fˆn denote the posterior mean of the empirical Bayes procedure. Then for f0 ∈ QSS (β, R, ε), δ > 1 and as n → ∞, √ \|\|fˆn − Y\|\|H(δ) = oP0 (1/ n). We have an exponential inequality which measures posterior spread in a variety of Sobolev norms. Since for fixed α, the posterior only depends on the data through the posterior mean fˆn,α , the following probabilities are independent of the observed data Y . Lemma 8.3. Let fˆn,α denote the posterior mean of Πα (· \| Y ). Then for any 0 ≤ s < α and any η > 0, 2(α−s) 1 − 2α+1 2 ˆ Πα f : \|\|f − fn,α \|\|H s ≥ (1 + η) 1 + n Y 2(α − s) η 1 1/4 1/(4α+2) √ . ≤ e exp − 1+ n 2(α − s) 24 Proof. For f ∼ Πα (· \| Y ) we can use the explicit form of the posterior mean in (3.1) to write \|\|f − fˆn,α \|\|2H s = ∞ X k=1 k 2s ζ 2, +n k k 2α+1 where the ζk ∼ N (0, 1) are independent. Letting tn = n1/(2α+1) and using standard tail bounds, ∞ h i X EΠ \|\|f − fˆn,α \|\|2H s \| Y = k 2s 1 X 2s X −2(α−s)−1 ≤ k + k k 2α+1 + n n k=1 k≤tn k>tn 2(α−s) 1 ≤ 1+ n− 2α+1 . 2(α − s) The posterior variance of \|\|f − fˆn,α \|\|2H s is given by ν2 = 2 ∞ X k=1 4(α−s)+1 2 X 4s X −4(α−s)−2 k 4s ≤ 2 k + k ≤ 3n− 2α+1 . 2 + n) n (k 2α+1 k≤tn k>tn Combining the above with the exponential inequality for χ2 -squared random variables found in Proposition 6 of [46], we have ! ∞ √ X k 2s 1/4 −x/ 8 2 e e ≥P (ζ − 1) ≥ νx k 2α+1 + n k k=1 √ − 4(α−s)+1 2(α−s) 1 − 2α+1 2 ˆ 4α+2 ≥ Πα f : \|\|f − fn,α \|\|H s ≥ 1 + n + 3n x Y . 2(α − s) √ Taking x = (η/ 3)[1 + 1/(2α − 2s)]n1/(4α+2) gives the desired result. 40 8.2 Results for L∞ -setting To prove Proposition 4.5 we need to understand the behaviour of the posterior median under the law P0 . Lemma 8.4. Let f˜ = f˜n denote the posterior median (defined coordinate-wise) of the slab and spike prior. Then the event Bn = {f˜lk = 0 ∀(l, k) ∈ Jnc (γ)} ∩ {f˜lk 6= 0 ∀(l, k) ∈ Jn (γ 0 )} √ ∩ { n\|Ylk − f0,lk \| ≤ (8l log 2 + a log n)1/2 ∀l ≤ Jn , ∀k = 0, ..., 2l − 1} (8.1) satisfies inf f0 ∈H(β,R) P0 (Bn ) → 1 as n → ∞, for some constants 0 < γ < γ 0 < ∞ and a > 0. Proof. We show that the P0 -probability of each of these events individually tends to 1. For the first event {f˜lk = 0 ∀(l, k) ∈ Jnc (γ)} ⊇ {Π(flk = 0 \| Y ) ≥ 1/2 ∀(l, k) ∈ Jnc (γ)} ⊇ {Π(flk = 0 ∀(l, k) ∈ Jnc (γ)) ≥ 1/2} = {Π(S ∩ Jnc (γ) = ∅) ≥ 1/2}. By Lemma 1 of [27] the P0 -probability of this last event tends to 1 for some γ > 0 as n → ∞. Consider the third event, √ Ωn = { n\|Ylk − f0,lk \| ≤ (8l log 2 + a log n)1/2 ∀l ≤ Jn , ∀k = 0, ..., 2l − 1}, which by (41) of [27] (or the Borell-Sudakov-Tsireslon inequality [35]) satisfies P0 (Ωcn ) → 0. We shall lastly show that Ωn ⊂ {f˜lk 6= 0 ∀(l, k) ∈ Jn (γ 0 )}, (8.2) which then completes the proof. Consider firstly the case f0,lk ∈ Jn (γ 0 ) with f0,lk > 0. Write Π(flk ≤ 0 \| Y ) = Π(flk = 0 \| Y ) + Π(flk < 0 \| Y ). (8.3) By the proof of Lemma 1 of [27], we have that on the event Ωn and for sufficiently large 0 2 γ 0 , the first posterior probability in (8.3) is bounded above by a multiple of nK+1/2−(γ ) /8 . Again on the event Ωn , we use (42) of [27] to bound the second term via Π(flk < 0 \| Y ) = ≤ wjn R∞ −n (x−Ylk )2 2 g(x)dx −∞ e 2 −n (x−Y ) lk 2 R0 g(x)dx + (1 − wj,n ) −∞ e R −√nYlk − 1 v2 \|\|g\|\|∞ −∞ e 2 dv √ √ = C nΦ̄( nYlk ), 1/2 a(π/n) wjn (8.4) where Φ̄ = 1 − Φ with Φ the distribution function of a standard normal variable. On Ωn , we have for l ≤ Jn , s r r 2Jn log 2 + 12 log n log n log n 0 Ylk = (Ylk − f0,lk ) + f0,lk ≥ − +γ ≥δ n n n 41 for some δ = δ(γ 0 ) > 0 that can be made arbitrarily large by taking γ 0 large enough. Thus applying the standard tail bounds for Φ̄ we have that the right-hand side of (8.4) is bounded above by a multiple of √ 1 1 2 p √ n n2−2δ − 12 δ 2 log n nΦ̄(δ log n) ≤ √ e = C(δ) √ . δ 2π log n log n Combining the above results, we have that for sufficiently large γ 0 (and hence δ), (8.3) is bounded above by a constant times n−B for some B > 0, uniformly over the positive coefficients in Jn (γ 0 ). In particular, the posterior median satisfies f˜lk > 0 for all (l, k) ∈ Jn (γ 0 ) with flk > 0 and n large enough. The case f0,lk < 0 is dealt with similarly, thereby proving (8.2). A simultaneous estimator in M(w) and L∞ It may be of interest to obtain an efficient estimator of f0 in M(w) that is also an element of L∞ , unlike Y. Letting f˜ = f˜n and fˆn denote the posterior median and mean of the slab and spike procedure respectively, define the estimators  if j ≤ j0 (n),  Yjk (1) Tn,jk = Yjk 1{f˜jk 6=0} if j0 (n) < j ≤ blog n/ log 2c. (8.5)   0 if blog n/ log 2c < j, (2) Tn,jk  ˆ  fn,jk = Yjk 1{f˜jk 6=0}   0 if j ≤ j0 (n), if j0 (n) < j ≤ blog n/ log 2c. (8.6) if blog n/ log 2c < j. Lemma 8.5. Consider the slab and spike prior Π with lower threshold j0 (n) → ∞ satisfying j0 (n) = o(log n) and let (wl ) be any admissible sequence satisfying wj0 (n) = o(nv ) for any (i) v > 0. Then the estimators Tn , i = 0, 1, defined in (8.5) and (8.6) satisfy for some M 0 > 0 and any Mn → ∞, √ sup P0 (\|\|Tn(i) − f0 \|\|M(w) ≥ Mn / n) → 0, f0 ∈H(β,R) sup f0 ∈H(β,R) P0 (\|\|Tn(i) − f0 \|\|∞ ≥ M 0 (log n/n)β/(2β+1) ) → 0 (i) as n → ∞. Moreover, \|\|Tn − Y\|\|M(w) = oP0 (n−1/2 ), uniformly over f0 ∈ H(β, R). (i) Proof. Since Y is an efficient estimator of f0 in M, if suffices to establish \|\|Tn − Y\|\|M = (i) oP0 (n−1/2 ) to show that Tn is also an efficient estimator of f0 in M. Consider firstly j > j0 (n), where the estimators coincide, and let Jn (β) be as in the proof of Theorem 7.5. On the event Bn defined in (8.1) and following the proof of Theorem 7.5, we have \|\|π>j0 (n) (Tn(i) − Y)\|\|M ≤ max j0 (n)<l≤Jn (β) wl−1 max \|Ylk 1{f˜lk =0} \| + max wl−1 max \|f0,lk \| k wl−1 l≥Jn (β) k max max (\|Ylk − f0,lk \| + \|f0,lk \|) + o(n−1/2 ) k:(l,k)∈Jnc (γ) p ≤ Cwj−1 (log n)/n + o(n−1/2 ) = o(n−1/2 ) (n) 0 ≤ j0 (n)<l≤Jn (β) 42 by the choice of j0 (n). (1) For j ≤ j0 (n) and i = 1, we trivially have \|\|πj0 (n) (Tn − Y)\|\|M = 0. Consider now i = 2. Arguing as in Theorem 2 of [12] and using the conditions on j0 (n), one obtains the √ uniform bound E0 EΠ [\|\| nπj0 (n) (f − Y)\|\|1+ M(w) \| Y ] ≤ C() for > 0 small enough. From the weak convergence of Π(· \| Y ) ◦ τY−1 towards N and a uniform integrability argument (via the √ moment bound), it follows as in Theorem 10 of [11] that nπj0 (n) (EΠ (f \|Y ) − Y) → EN = 0 in M0 (w) in probability, which implies the result. For L∞ we have on Bn , Jn (β) 2l/2 max \|Ylk − f0,lk \|1{(l,k)∈Jn (γ)} + \|f0,lk \|1{(l,k)∈Jnc (γ)} X \|\|π>j0 (n) (Tn(i) − f0 )\|\|∞ . k l=j0 (n)+1 ∞ X 2l/2 max \|f0,lk \| + k l=Jn (β)+1 r Jn (β)/2 .2 log n + R2−Jn (β)β . n max \|Ylk − f0,lk \| . 2 j0 (n)/2 log n n β 2β+1 . Now j0 (n) \|\|πj0 (n) (Tn(1) − f0 )\|\|∞ . X r 2 l/2 k l=0 (1) log n . n log n n β 2β+1 , (2) thereby proving the second statement for Tn . For Tn , using the convergence of the posterior mean to Y in M0 for j ≤ j0 (n) shown above, j0 (n) \|\|πj0 (n) (Tn(2) − Tn(1) )\|\|∞ . X 2l/2 max \|fˆn,lk − Ylk \| k l=0   j0 (n) = oP0  X l=0 8.3 l/2 2 w √ l  = oP0 n 2j0 (n)/2 wj0 (n) √ n ! = oP0 log n n β 2β+1 ! . Wavelets Let us briefly recall the notion of periodized and boundary corrected wavelets and discuss condition (2.3). Let φ, ψ denote a scaling and corresponding wavelet function on R satisfying X X sup \|φ(x − k)\| < ∞, sup \|ψ(x − k)\| < ∞. (8.7) x∈R k∈Z x∈R k∈Z Examples include Meyer wavelets (see Section 2 in [38] for other choices). Consider firstly the periodic case. As usual define the dilated and translated wavelet at resolution level j and scale position k/2j by φjk (x) = 2j/2 φ(2j x − k), ψjk (x) = 2j/2 ψ(2j x − k) for j, k ∈ Z. Periodize the wavelet functions via X X per φper φjk (x + m), ψjk (x) = ψjk (x + m), x ∈ [0, 1] jk (x) = m∈Z m∈Z 43 per J0 − for j = 0, 1, ... and k = 0, ..., 2j − 1. Then the wavelet system {φper J0 k , ψjm : k = 0, ..., 2 1, m = 0, ..., 2j − 1, j = J0 , J0 + 1, ...} forms an orthonormal wavelet basis of L2 ((0, 1]) and satisfies (2.3) due to (8.7). In the case of R, an orthonormal basis of Vj = span{(φjk )k }, j ≥ J0 , can be obtained by taking 2j−J0 dilations of the orthonormal basis (φJ0 k )k of a basic resolution space VJ0 . In the case of boundary corrected wavelets, the analogous orthonormal basis of the basic resolution space VJ0 , J0 ∈ N, contains 2J0 elements and consists of 3 components. At resolution level J0 , t J0 /2 φlef t (2J0 x), a basis consists of 3 components. Firstly, N left edge functions φlef J0 k (x) = 2 k t k = 0, ..., N − 1, where φlef is a modification of φ that remains bounded and has compact k J0 /2 φright (2J0 x), k = 0, ..., N − 1, support. Secondly, N right edge functions φright J0 k (x) = 2 k with the same properties. Thirdly, 2J0 − 2N interior functions, that are the usual translates of dilations of φ defined on R, that is φJ0 k for k = N, ..., 2J0 − N − 1, which we note are all J0 supported in the interior of [0, 1]. Writing for convenience {φbc J0 k : k = 0, ..., 2 − 1} instead lef t right of {φJ0 k , φJ0 k0 , φJ0 m : k = 0, ..., N − 1, k 0 = 0, ..., N − 1, m = N, ..., 2J0 − N − 1}, we have the first part of (2.3) J0 −1 2X \|φbc J0 k (x)\| ≤2 J0 /2 ≤2 J0 /2 k=0 N t max \|\|φlef k \|\|∞ 0≤k<N C(N, φ) + 2 J0 /2 +2 J0 /2 N \|\|∞ max \|\|φright k 0≤k<N + 2J0X −N −1 2J0 /2 \|φ(2J0 x − k)\| k=N 0 C (φ), (8.8) where we have used that N is fixed and that the original wavelet function on R satisfies (8.7). Starting at resolution level J0 with the usual dilated wavelets on R, ψJ0 k (x) = 2J0 /2 ψ(2J0 x− k), 2J0 ≥ N , it is possible to construct corresponding boundary wavelet functions t 0 J0 − N − 1}. , ψJright {ψJlef 0 , ψJ0 m : k = 0, ..., N − 1, k = 0, ..., N − 1, m = N, ..., 2 0k 0k For j ≥ J0 , we can then define the dilates of the boundary wavelets in the usual way: right (2j−J0 x). (x) = 2(j−J0 )/2 ψJright ψjk 0k t j−J0 lef t (2 x), (x) = 2(j−J0 )/2 ψJlef ψjk 0k lef t right This yields the required wavelets at resolution level j, namely {ψjk , ψjk , ψjm : k = 0 0 j 0, ..., N − 1, k = 0, ..., N − 1, m = N, ..., 2 − N − 1}, which for convenience we write as bc : k = 0, ..., 2j − 1}. Arguing as in (8.8) and again using (8.7) gives the second part of {ψjk (2.3). 8.4 Weak convergence For µ and ν probability measures on a metric space (S, d), define the bounded Lipschitz metric by Z βS (µ, ν) = sup u(s)(dµ(s) − dν(s)) , (8.9) u:\|\|u\|\|BL ≤1 S \|\|u\|\|BL = sup \|u(s)\| + s∈S \|u(s) − u(t)\| . d(s, t) s,t∈S:s6=t sup βS metrizes the weak convergence of probability distributions, that is random variables Xn →d X converge in distribution in (S, d) if and only if βS (L(Xn ), L(X)) → 0, where −1/2,δ L(X) denotes the law of X. In particular, we shall consider the choices S = H(δ) = H2 or S = H −s for s > 1/2 in `2 and S = M0 (w) for {wl }l≥1 an admissible sequence in L∞ . 44 8.5 Results on empirical and hierarchical Bayes procedures Let us recall some definitions and results from [29, 48] that appear in proofs elsewhere. Define hn : (0, ∞) → [0, ∞) to be hn (α) = ∞ 2 log k X n2 k 2α+1 f0,k 1 + 2α n1/(2α+1) log n k=1 (k 2α+1 + n)2 (8.10) and for 0 < l < L define the bounds αn = inf{α > 0 : hn (α) > l} ∧ p log n, αn = inf{α > 0 : hn (α) > L(log n)2 }. The behaviour of the empirical Bayes estimator α̂n defined in (3.2) is contained in Lemma 3.11 of [48], which is summarized below for convenience. Lemma 8.6 (Szabó et al.). Fix βmax > 0. For any 0 < β ≤ βmax and R ≥ 1, there exist constants K1 and K2 such that P0 (β − K1 / log n ≤ α̂n ≤ β + K2 / log n) → 1 uniformly over f0 ∈ QSS (β, R, ε). 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11 Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design H. Boyer, F. Miranville, D. Bigot, S. Guichard, I. Ingar, A. P. Jean, A. H. Fakra, D. Calogine and T. Soubdhan University of La Reunion, Physics and Mathematical Engineering for Energy and Environment Laboratory, LARGE - GéoSciences and Energy Lab., University of Antilles et de la Guyanne, France 1. Introduction The aim of this paper is to briefly recall heat transfer modes and explain their integration within a software dedicated to building simulation (CODYRUN). Detailed elements of the validation of this software are presented and two applications are finally discussed. One concerns the modeling of a flat plate air collector and the second focuses on the modeling of Trombe solar walls. In each case, detailed modeling of heat transfer allows precise understanding of thermal and energetic behavior of the studied structures. Recent decades have seen a proliferation of tools for building thermal simulation. These applications cover a wide spectrum from very simplified steady state models to dynamic simulation ones, including computational fluid dynamics modules (Clarke, 2001). These tools are widely available in design offices and engineering firms. They are often used for the design of HVAC systems and still subject to detailed research, particularly with respect to the integration of new fields (specific insulation materials, lighting, pollutants transport, etc.). 2. General overview of heat transfer and airflow modeling in CODYRUN software 2.1 Thermal modeling This part is detailed in reference (Boyer, 1996). With the conventional assumptions of isothermal air volume zones, unidirectional heat conduction and linearized exchange coefficients, nodal analysis integrating the different heat transfer modes (conduction, convection and radiation) achieve to establish a model for each constitutive thermal zone of the building (one zone being a room of a group of rooms with same thermal behaviour). For heat conduction in walls, it results from electrical analogy that the nodal method leads to the setting up of an electrical network as shown in Fig. 1, the number of nodes depending on the number of layers and spatial discretization scheme : Tsi T1 T2 T3 Tse Fig. 1. Example of associated electrical network associated to wall conduction 228 Evaporation, Condensation and Heat Transfer The physical model of a room (or group of rooms) is then obtained by combining the thermal models of each of the walls, windows, air volume, which constitute what CODYRUN calls a zone. To fix ideas, the equations are of the type encountered below: C si C se dTsi dt dTse dt C ai = hci (Tai − Tsi ) + hri (Trm − Tsi ) + K (Tse − Tsi ) + ϕswi = hce (Tae − Tse ) + hre (Tsky − Tse ) + K (Tsi − Tse ) + dTai dt Nw ∑ hci S j (Tai − Tsi ( j ) + c m (Tae − Tai ) = ϕswe (1) (2) (3) j =1 0 = Nw ∑ hri A j (Tsi ( j ) − Trm ) (4) j =1 Equations of type (1) and (2) correspond to energy balance of nodes inside and outside surfaces. Nw denote the number of walls of the room, and correspond to the following figure: ϕlwi ϕswi ϕ ci ϕ lwe ϕswe ϕce Fig. 2. Wall boundary conditions ϕ being heat flux density, lw, sw, and c indices refers to long wave radiation, short wave radiation and convective exchanges. Indices e and i are for exterior and indoor. Equation (3) comes from the thermo-convective balance of the dry-bulb inside air temperature Tai, taking into account an air flow m through outside (Tae) to inside. (4) represents the radiative equilibrium for the averaged radiative node temperature, Trm. The generic approach application implies a building grid-construction focus. Following the selective model application logic developed (Boyer, 1999), an iterative coupling process is implemented to manage multi-zone projects. From the first software edition, some new models were implemented. They are in relation with the radiosity method, radiative zone coupling (short wavelength, though window glass) or the diffuse-light reduction by close solar masks (Lauret, 2001). Another example of CODYRUN evolution capacity could be illustrated by a specific study using nodal reduction algorithms (Berthomieu, 2003). To resolve it, a new implementation was done: the system was modified into a canonic form (in state space). Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 229 From a user utilization point of view, this thermal module exit is mainly linked to the discretizated elements temperatures, the surfacic heat flows, zone specific comfort indices, energy needs, etc. 2.2 Airflow modeling Thermal and airflow modules are coupled to each other thanks to an iterative process, the coupling variables being the air mass flows. It is useful to implement the inter zones flows [i , j ] then qualifies the mass transfer from the zone . The terms m with the help of a matrix m i to the zone j, as it is depicted . Fig. 3. Interzonal airflow rates matrix Then, a pressure model for calculating previous airflow rates is set and takes into account wind effect and thermal buoyancy. To date, the components are integrated ventilation vents, small openings governed by the equation of crack flow, as : = Cd S (ΔP)n m (5) S is aperture surface, Cd and n (typically 0.5- 0.67) depends on airflow regime and type of openings and ΔP the pressure drop. Therefore, the following scheme can be applied for encountered cases. Outdoor Indoor Indoor Indoor Tai (j) Tai (k) Tae Tai (i) Wind z z Pz(j) Pz(i) Fig. 4. Exterior and interior small openings Taking into account the inside air volume incompressibility hypothesis, the mass weight balance should be nil. Thus, in the established system, called pressure system, the unknowns are the reference pressure for each of the zones. Mechanical ventilation is simply taken into account through its known airflow values. In this way, ventilation inlets are not taken into detailed consideration which could have been the case owing to their flowpressure characteristics. 230 Evaporation, Condensation and Heat Transfer ⎧ i=N i,1) + m vmc ( 1 ) = 0 ⎪ ∑ m( ⎪ i =0,i ≠1 ⎪ i=N ⎪ i,2) + m vmc ( 2 ) = 0 ⎪ ∑ m( ⎪ i = 0,i ≠ 2 ⎨ ⎪ ... ⎪ ⎪ ⎪i = N − 1 ⎪ i,N) + m vmc ( N ) = 0 m( ⎪ ∑ ⎩ i =0 (6) m vmc ( k ) is the flow extracted from zone k (by vents) and N the total number of zones. After the setting up of this non linear system, its resolution is performed using a variant of the Newton-Raphson (under relaxed method). This airflow model has recently been supplemented by a CO2 propagation model in buildings (Calogine, 2010). 2.3 Humidity transfers For decoupling reasons, zones dry temperatures, as well as inter zone airflows, have been previously calculated. At this calculation step, for a given building, each zone's specific humidity evolution depends on the matricial equation : Ch .r = A h .r + Bh (7) According to Duforestel’s model, an improvement of the hydric model (Lucas, 2003) was given by the use of a hygroscopic buffer. If we consider N as the number of zones, a linear system of dimension 2N is obtained and a finite difference scheme is used for numerical solution. 3. Some elements of CODYRUN validation 3.1 Inter model comparison A large number of comparisons on specific cases were conducted with various softwares, whether CODYBA, TRNSYS for thermal part, with AIRNET, BUS, for airflow calculation or even CONTAM, ECOTECT and ESP. To verify the correct numerical implementation of models, we compared the simulation results of our software with those from other computer codes (Soubdhan, 1999). The benchmark for inter-comparison software is the BESTEST procedure (Judkoff, 1983), (Judkoff, 1995). This procedure has been developed within the framework of Annex 21 Task 12 of the Solar Heating and Cooling Program (SHC) by the International Energy Agency (IEA). This tests the software simulation of the thermal behavior of building by simulating various buildings whose complexity grows gradually and comparing the results with other simulation codes (ESP DOE2, SERI-RES, CLIM2000, TRNSYS, ...). The group's collective experience has shown that when a program showed major differences with the so-called reference programs, the underlying reason was a bug or a faulty algorithm. Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 231 3.1.1 IEA BESTEST cases The procedure consists of a series of buildings carefully modeled, ranging progressively from stripped, in the worst case, to the more realistic. The results of numerical simulations, such as energy consumed over the year, annual minimum and maximum temperatures, peak power demand and some hourly data obtained from the referral programs, are a tolerance interval in which the software test should be. This procedure was developed using a number of numerical simulation programs for thermal building modeling, considered as the state of the art in this field in the US and Europe. After correction of certain parts of CODYRUN, an example output from this confrontation is the following concerns and energy needs for heating over a year, for which the results of our application are clearly appropriate. Fig. 5. Exemple of BESTEST result The procedure requires more than a hundred simulations and cases treated with CODYRUN showed results compatible with most referral programs, except for a few. We were thus allowed to reveal certain errors, such as the algorithm for calculating the transmittance of diffuse radiation associated with double glazing and another in the distribution of the incident radiation by the windows inside the envelope. As in the reference codes, two points are to remember: the majority of errors came from a poor implementation of the code and all the efficiency of the procedure BESTEST is rechecked, errors belonging to modules that have been used for years. Concerning airflow validation, IEA multizone airflow test cases describe several theoretical and comparative test cases for the airflow including multizone cases. It was developed within the International Energy Agency (IEA) programs: Solar Heating and Cooling (SHC) Task 34 and Energy Conservation in Building and Community Systems (ECBCS) Annex 43. The tests are designed for testing the capability of building energy simulation programs to predict the ventilation characteristics including wind effect, buoyancy and mechanical fan. CODYRUN passed all the test cases. 232 Evaporation, Condensation and Heat Transfer 3.1.2 TN51 airflow case In (Orme, 1999), a test case composed of a building with three storeys is described and first published in (Tuomala, 1995). All boundary conditions are imposed (wind, external temperature) and indoor conditions are constant, i.e. in steady state. Fig. 6. TN 51 Test case Next figure shows solution of 3 references codes, i.e. COMIS, CONTAM93 and BREEZE. CODYRUN’s results were added on the figure extracted from the paper (Orme, 1999). Fig. 7. Intermodel airflow comparison Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 233 As it can be seen, in terms of numerical results, CODYRUN gives nearly the same values as the other codes, little differences being linked to numerical aspects as algorithms or convergence criteria used. 3.2 Experimental confrontation As part of measurement campaigns on dedicated cells and real homes, a set of elements of models have been to face action (and for some improved), mostly concerning thermal and humidity aspects. Not limited to, some of these aspects are presented in the articles (Lauret, 2001) (Lucas, 2001) and (Mara, 2001), (Mara, 2002). 4. Two applications of the software 4.1 Air solar collector An air solar collector is used to heat air from solar irradiation. More precisely, it allows to convert solar energy (electromagnetic form) into heat (Brownian motion). This type of system can be used to heat buildings or to preheat air for drying systems. Air solar collector is constituted by an absorber encapsulated into an isolated set-up (Fig. 8). To avoid heat loss by the incoming flux side, a glass is located above the absorber, letting a gap between them. This glass allowing to catch long wavelength emitted by the set-up when its temperature rises-up. Air temperature improvement is mostly done by air convection at the absorber interface. This heat is then carried by air in the building or in the drying system. Fig. 8. Air solar collector principle, adapted from (Sfeir, 1984) The enclosure is not strictly speaking a building, but the generic nature of the software allows to study this device. This is a zone (within the captation area), a glazed windows, exterior walls and an inner wall of absorber on which it is possible to indicate that solar radiation is incident. It should be noted that this is part of a class simulation exercise allowing students to learn collector physics basis. 234 Evaporation, Condensation and Heat Transfer 4.1.1 Collector modeling and description of the study It is assumed that before initial moment, there is no solar irradiation and the collector is at thermal equilibrium, i.e. outside air temperature is the set-up temperature. At t=0, an irradiation flux Dh is applied without interactions with the outside air temperature. In a simplified approach, a theoretical study is easily obtained. Some equations below are used to understand the functioning of the collector. In steady state, the incoming irradiation flux reaching the absorber is fully converted and transmitted to the air (assumed to uniform temperature): ρasC asV dTai as (Tai - Tae ) = τ 0 Dh - K pS(Tai - Tae ) - mC dt (8) Where Kp is the thermal conductance (W.m-2.K-1) of each wall through which losses can pass, m dot is a constant air mass flow through the collector, Tai is the inside air temperature, Tae is the outside air temperature, and ρas (1.2 kg.m-3) and Cas (1000 J.kg-1.K-1) are respectively the air thermal conductivity and density. By knowing initial conditions (at t=0, Tai=Tae), the solution of equation (8) appears: Tai ( t ) = Tae + τ 0 Dh K (1− e − t K / ρ as C as V ) (9) The power delivered by the collector is given by as (Tai - Tae ) Pu = mC (10) The studied collector is horizontal, the irradiated face has a surface of 1m², and an air gap thickness of 0.1m. The meteorological file is constituted of a constant outside air temperature (25°C) and a constant direct horizontal solar irradiation of 500W (between 6 am and 7 pm). It is assumed (and further verified) that the steady state is reached at the end of the day. Concerning the building description file, the collector is composed by 1 zone, 3 boundaries (1 for the upper face, 1 for the lower face, and 1 for all lateral faces), and 4 components (glass, lateral walls, lower wall and absorber). Walls are made - from inside to outside - of wood (1cm), polystyrene (5cm) and wood (1cm). Wood walls have a short wave absorptivity of α = 0.3. The glass type is a single layer ( τ 0 = 0.85), and the absorber is a metallic black sheet of 2mm (α = 0.95). The solar irradiation absorption model of the collector is a specific one. This has to be chosen in CODYRUN, because traditionally in buildings the solar irradiation doesn’t impact directly an internal wall (absorber), but the floor. Even if this solar air heater is considered as a very simple one, the following figure gives some details about heat exchanges in the captation area. Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 235 Fig. 9. Solar irradiation absorption process. 4.1.2 Simulation results in the case of forced convection in air gaps An air flow of 1 m3.h-1 is set. After running simulations, the following graph is obtained for two identical days : Fig. 10. Evolution of air temperature at the output of the solar collector with forced convection (1m3.h-1). 236 Evaporation, Condensation and Heat Transfer Simulation results are consistent with simplified study, but not exactly identical. In fact, CODYRUN model is more precise than those use in eq. 8: thermal inertia of walls are taken into account, superficial exchanges are detailed, exchange coefficients are depending on inclination, glass transmittance is depending on solar incidence, … With a low air flow rate of 1 m3.h-1, the collector output power is about 22W and its efficiency about 4.5%. A solution to improve efficiency is to increase air flow rate (see Fig. 11), in link with the fact that conductive thermal losses are decreasing when internal temperature is decreasing. Increase of flow rate leads to a lower air output temperature (and also to change convection exchange coefficients). In some cases like some drying systems where a minimal air output temperature should be needed, this can be problematic. In these cases, CODYRUN can help to choose the best compromise between air output temperature and collector power or efficiency. Fig. 11. Evolution of collector outputs regarding air flow rate. Many other simulations could be conducted, in particular in the case of natural convection. Some of these cases could be applied to air pre-heating in case of passive building design, as for next application. 4.2 Trombe wall modeling 4.2.1 Trombe-Michel wall presentation A Trombe-Michel wall (often called Trombe wall), is a passive heating system invented and patented (in 1881) by Edward Morse. It takes its name from two French people who popularized it in 1964, one engineer and one architect, respectively Félix Trombe and Jacques Michel. A Trombe-Michel wall allows to store solar energy and distribute heat following simple physical principles. This system is composed of a vertical wall, submitted to the sunshine, which absorb and store the radiative energy as thermal energy by inertia. To improve its absorptivity, the wall can be paint in a dark color. Stored energy rises-up the wall Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 237 temperature, inducing some heat transfer such as conduction (into the wall from outdoor to indoor side), radiation and convection (both with in/outdoor). To avoid outdoor losses, a glazing, which is not transmissive to ultraviolet and far infrared, is located few centimeters in front of the wall (outdoor side). So the glazing reduces outside losses by convection and traps wall's radiation due to its temperature (far infrared emission) into the building. But it also slightly reduces the solar incoming flux, because some of it is reflected or absorbed. Nota: Considering the glazing, the sunshine insulation (etc.), it is not easy to choose the optimal set-up configuration; this kind of situations are typical cases where CODYRUN simulations allow to find out. Heat transfer through the wall is done by conduction. According to the wall size and its composition, a thermal delay appears between heat absorption and emission. This characteristic allows to configure specifically the system in function of the needs. Considering the thermal delay, conductive transfer-mode is convenient to heat by night, but not by day. To palliate this lack, a faster transfer mode is brought by a variant of the Trombe-Michel wall, called recycling wall (Fig. 12). The recycling wall is obtained by holes addition at the top and the bottom of the wall. This variant allows to get a natural air flow, transferring energy by sensitive enthalpy between the system side of the wall (also called Trombe side) and the room. Indeed, as long as the glass is far enough of the wall, the temperature difference between them induces some air-convective movement, initiating the global air-circulation, so allowing heat distribution. airstream absorber convection window radiation Fig. 12. Trombe-Michell recycling wall set-up (Sfeir, 1981) Several variants of the Trombe-Michel wall can be used. These one usually evolve in function of the climatic need and can, for example, be defined by: the holes size and presence (or not), the characteristics of the storage wall (e.g. thickness, density, presence of fluid circulation or phase change materials, etc. ) or the glazing type (e.g. simple or double, treated surface, noble gas, etc.). Some technical indications about these considerations are given by Mazria (Mazria, 1975). Though two levels experimental design, (Zalewski, 1997) shows that external glazing emissivity, window type and wall absorptivity mainly characterize the system. Trombe-Michel system (and its variants) inspired numerous publications about temperate climate and proved their efficiency as passive heating system. By extension, there is also a lot of studies about similar system, as solar chimney (Ong, 2003), (Awbi, 2003). 238 Evaporation, Condensation and Heat Transfer There are several values usually used to describe Trombe-Michel wall functioning: • Transmitted energy through the wall by conduction, • Transmitted energy through the holes by sensitive enthalpy, • The system efficiency, defined as the rate between the energy received into the room over the total energy received by the North wall (South hemisphere) in the meantime. • The FGS (solar benefits fraction), defined as the rate between the benefits from the Trombe-Michel system and the energy needed by the room without it, for a room temperature of 22°C. Nota: Even if the FGS refers to active heating systems, it allows to compare easily passive heating systems between them and to other installations. 4.2.2 Presentation of Trombe-Michel wall simulations under CODYRUN This numerical study is resolved by software simulations. The case study case presented here takes place in Antananarivo, Madagascar (Indian Ocean). The aim of the simulations conducted was to help this developing country to face low temperatures in classrooms during winter season. The word 'zone', used previously, is understood as CODYRUN vocable, correspond usually to one room. Into the actual description, isothermal air assumption is made into the capitation area. This consideration is taken as first approximation and can be possibly modified later to improve the model (e.g. replaced by linear gradient hypothesis, etc.). According to the bibliography, one of the main issues mentioned is about the convection coefficient into the Trombe side of the wall. Because of the system successively laminar and turbulent, convection correlation such as - Nu = f(Gr) – cannot be used. In this case, (Zalewski, 1997) says that the laminar coefficient evolve between 2 and 2.3 W.m-2.K-1 and the turbulent one evolve between 2.25 and 3.75 W.m-2.K-1. So, in a first approach, we approximate the temperature-evolving value of the convection coefficient as the average of the turbulent and laminar values (2.9 W.m-2.K-1 ). Considering thermo-circulation, the CODYRUN's airflow module allows to calculate air flows through wall's holes. This consideration and the case study description, the two areas (the Trombe system and the room) are studied coupled, which is a right physical representation of the reality. Results validity are certified by the fact that each module of CODYRUN, and their combinations, has been validated by comparison with experimental results, reference codes and BESTEST procedures. This multi-model code structure (Boyer, 1996) can be efficiently exploited by studies such as this one, where it is necessary to choose the convection coefficients by area or by wall, to mesh slightly the walls to calculate precisely the conductive fluxes, to choose a solar gains repartition model or equivalent sky temperature, etc. CODYRUN also creates numerous output simulation variables, in addition to the one related to the dry temperature area, such as sensible power outputs from mass exchange through wall holes, conductive fluxes through the wall, enthalpic zone balance and the PMV (Predictive Mean Vote, according to Gagge) of interior zone to quantify comfort conditions. It is also possible to explore the conductive flux through the Trombe-Michel wall glass. Simulation results are presented with meteorological data of Antananarivo coming from TRNSYS 16.0 software and its TMY2 documentation data file. According to this one, daily Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 239 sunshine insulation annual average is about 5.5 kWh.m-2. The one on the North and East walls are respectively 2.23 and 2.67 kWh.m-2. Modified in function of the current albedo (0.3), the previous values become 3 and 3.46 kWh.m-2. The simulated building represents a residence unit based on a geometric cell description (9 m² parallelepiped square base ground). Moreover, materials, thermo-physics characteristics and wall type are typical from Madagascar. The room is supposed to be constantly occupied by two persons and have a usual profile for intern lighting. This area is supposed to exchange air with the outside at a rate of 1 volume per hour. Finally, the system area is of 2 m2 and is located on the North wall (South hemisphere). 4.2.3 Simulation Passive heating oriented-simulations are made the first day of July (Julian calender days 181 and following). In a first time, the objective is to explain the set-up functioning. So, some hourly based analyses will be conducted. Finally, the last experiment will compare the same Trombe-Michel wall with and without the recycling variant. 4.2.3.1 Inverse thermo-circulation evidence By night, over some system specific conditions (open holes) and in function of previous day insulations (e.g. low insulation), the flow evolution can sometime be the symmetric of the one by day (Fig. 13). In this configuration, the system functioning is reversed, cooling the room instead of heating it. Between points 19 and 32 (i.e. 7 pm to 8 am next day), the Trombe-area temperature is lower than the room temperature, inducing the inverse thermocirculation. This physical process can be illustrated by the negative part of the sensitive aeraulic benefits for the inside area. Even if the flux values by night are low, it is important to note the following point : the curves are obtained by assuming a sky temperature equal to the outside air temperature. This assumption reduces the impact of the system/sky radiation transfer, so involves an error. This error could be reduced by taking a sky temperature value more realistic, so by considering the night-sky wide wavelength emission and the high glass emissivity (0.9). A better approximation can be done by a sky temperature model such as Tsky=Tae– K, where K=6. Fig. 13. Dry temperatures and sensible power balance through holes. 4.2.3.2 Night hole closing system Hole closing system strategy deserve a specific comment. This one can be done nicely from flow measurement, but a simpler and passive way to do it also exists. This one consists in a 240 Evaporation, Condensation and Heat Transfer plastic foil, only attached on one side and occulting the holes preventing from inverse thermo-circulation. Into CODYRUN, the closure system is designed by the “little opening” component added to a hourly profile (i.e. 24 opening percentages), where the hole is closed by night and opened by day. However, the thermo-circulation (by system inertia) continues during the two first hours of the evening, so the profile includes this pattern too. Because of the small gap with the previous curves (Fig. 13), the results are firstly surprising. An explanation can be done by the small thermal gradient during the considered night. According to Mazria (Mazria, 1975) and Bansal (Bansal, 1994), recycling variant are only useful into cold climates (winter averages: -1 °C to -7 °C). So, simulation results show the minor impact of the hole control in this study case. In other words, it seems that the mass flow have a very low sensibility to the opening state. This hypothesis is rejoined by another author (Zalewski, 1997) who have substantively the same conclusion for a North France located case. 4.2.3.3 Global Trombe wall functioning To obtain the full power delivered by the wall, airflow and conductive heat transfer have to be summed. By night, it appears a constantly positive heat flux inducing a rise-up of the room temperature (Fig. 14). In the morning (e.g. 9am first day), the global flux becomes negative. A full curve interpretation is approached by airflow and conductive heat flux analyze through the wall (right part of Fig. 14). Early morning, the global heat flux inversion can be explained by highly negative conductive heat flux resulting certainly from the heat flux night inversion (around 9 to 14, so a delay of 4 hours). Drawn curves of the incoming heat flux through the wall confirm this hypothesis. As foreseeable (without simulation tools), the curves show the alternate functioning of the conductive and sensible airflow input power (the wall characteristics setting the shifting period). By night, the conductive heat flux through the wall rises-up the room temperature, whereas by day this action is done by the heat flux associated to air displacement, physically expressed by the natural air movement which is a priori synchronous with the sunlight distribution profile. The previous report allows to ask about the holes necessity. In fact, even if there is no heat transfer improvement, the main interest comes from the heat distribution and its timeevolving modulation into the building. Fig. 14. Global and specific fluxes induced by the Trombe wall into the room. To increase the room input power (i.e. the global heat flux), it could be simple to rise-up the wall surface and/or to add some reflectors to raise the incoming sunlight intensity. Heat Transfer in Buildings: Application to Solar Air Collector and Trombe Wall Design 241 However, in every cases the system have to be simulated for summer conditions. In order to reduce the external loss (illustrated by negative heat fluxes), a double glazing could be used, even if it reduces the incoming sunlight intensity (transmission loss). For reference, double glazing can be easily made from simple glazing. Another solution consists in isolating the indoor side of the wall. 4.2.3.4 Trombe-Michel wall and recycling variant comparison The comparison, in the meantime, of the wall's holes presence (or not), expresses a gap between the global heat fluxes. The values corresponding to the recycling variant are shifted to the left compared to the classic Trombe-Michel wall (Fig. 15). The input power delivered with the holes is slightly superior, which confirms the hypothesis of low impact on section 4.2.3.2. A piece of explanation (checkable by simulation) can be given by the thermal gradient between the room and the Trombe system, the decrease of this one reducing the conductive transfer through the glazing. Predicted Mean Votes (PMV) curves (of the right of Fig. 15) express the comfort sensation which evolve between -3 and +3. By comparison, it can be affirmed that the holes improve the comfort sensation. This improvement is related to the previous comment (about the superior input power amount which led to a rise-up of the dry temperature), and to the higher indoor side wall temperatures. Fig. 15. Input power and PMV comparisons 4.2.4 Trombe wall simulation exercise conclusion This example of CODYRUN use allows to find some interesting results about TrombeMichel wall configurations for the specific location of Antananarivo. Results such as the interest of the recycling variant (which is almost not influenced by hole closing procedure), the time-repartition (and impact) of the airflow and conductive heat flow on the room temperature, (etc.) have been found. Considering the case study, much more informations could be deduced from other well designed simulations. The considerable amount of knowledge found was only deduced from simulations, so only following a numerical approach. From this consideration and the results precision, a good estimation about CODYRUN quality and level of development can be done. 5. Conclusion This paper describes the models, validation elements and two applications of a general software simulation of building envelopes, CODYRUN. Many studies and applications have been conducted using this code and illustrate the benefits of this specific development. 242 Evaporation, Condensation and Heat Transfer Conducted and sustained for over fifteen years, the specific developments made around CODYRUN led our team to be owner of a building simulation environment and develop connex themes such as validation or application to large scale in building architectural requirements. This evolving environment has enabled us to drive developments in methodological areas (Mara, 2001) (modal reduction, sensitivity analysis, coupling with genetic algorithms (Lauret, 2005), neural networks, meta models,...) or related to technological aspects (solar masks, integration of split-system, taking into account the radiant barriers products, ...). This approach allows a detailed understanding of the physical phenomena involved, the use of this application in teaching and capitalization within a work tool of a research team. Accompanied by the growth of computer capabilities, the subject of high environmental quality in buildings, our field becomes more complex by the integration of aspects other than those originally associated with the thermal aspects (environmental quality, including lighting, pollutants, acoustics, ...). Simultaneously, it is also essential to achieve a better efficiency in the transfer of knowledge and operational tools. 6. References Awbi H. (2003),Ventilation of buildings, Spon Press, ISBN 0415270-55-3. Balcomb J.D. & al. (1997), Passive solar heating of buildings, in Solar Architecture, Proceedings of the aspen Energy Forum, Ann Arbor Science Publishers Inc., Aspen, Colorado, 1977. Bansal N.K., Hauser G., Minke G. (1994). Passive Building design. A handbook of natural climatic control, Elsevier, ISBN 0-444-81745-X, Amsterdam (Netherlands) Berthomieu T., Boyer H. 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arXiv:1507.06739v5 [math.ST] 30 Nov 2016 Selective inference with a randomized response Xiaoying Tian, Jonathan Taylor∗ Department of Statistics Stanford University Sequoia Hall Stanford, CA 94305, USA e-mail: xtian@stanford.edu jonathan.taylor@stanford.edu Abstract: Inspired by sample splitting and the reusable holdout introduced in the field of differential privacy, we consider selective inference with a randomized response. We discuss two major advantages of using a randomized response for model selection. First, the selectively valid tests are more powerful after randomized selection. Second, it allows consistent estimation and weak convergence of selective inference procedures. Under independent sampling, we prove a selective (or privatized) central limit theorem that transfers procedures valid under asymptotic normality without selection to their corresponding selective counterparts. This allows selective inference in nonparametric settings. Finally, we propose a framework of inference after combining multiple randomized selection procedures. We focus on the classical asymptotic setting, leaving the interesting high-dimensional asymptotic questions for future work. AMS 2000 subject classifications: Primary 62M40; secondary 62J05. Keywords and phrases: selective inference, nonparametric, differential privacy. 1. Introduction Tukey (1980) promoted the use of exploratory data analysis to examine the data and possibly formulate hypotheses for further investigation. Nowadays, many statistical learning methods allow us to perform these exploratory data analyses, based on which we can posit a model on the data generating distribution. Since this model is not given a priori, classical statistical inference will not provide valid tests that control the Type-I errors. Selective inference seeks to address this problem, see Lee et al. (2013a), Lockhart et al. (2014), Lee & Taylor (2014), Fithian et al. (2014). Loosely speaking, there are two stages in selective inference. The first is the selection stage that explores the data and formulates a plausible model for the data distribution. Then we enter the inference stage that seeks to provide valid inference under the selected model which is proposed after inspecting the data. Inference under different models have been studied, notably the Gaussian families Lee et al. (2013a), Tian et al. (2015), Lee & Taylor (2014) as well as other exponential families Fithian et al. (2014). In this work, we consider selective inference in a general setting that include nonparametric settings. In addition, we introduced the use of randomized response in ∗ Supported in part by NSF grant DMS 1208857 and AFOSR grant 113039. 1 Tian and Taylor/Selective inference with a randomized response 2 model selection. A most common example of randomized model selection is probably the practice of data splitting. Assuming independent sampling, we can divide the data into two subsets, using the first for model selection and the second subset for inference. Though not emphasized, this split is often random. Hence, data splitting can be thought of as a special case of randomized model selection. To motivate the use of randomized selection and introduce the inference problem that ensues, we consider the following example. 1.1. A first example Publication bias, (also called the “file drawer effect” by Rosenthal (1979)) is a bias introduced to scientific literature by failure to report negative or non-confirmatory results. We formulate the problem in the simple example below. Example 1 (File drawer problem). Let n X̄n = 1X Xi,n n i=1 be the sample mean of a sample of n iid draws from Fn in a standard triangular array. We set µn = EFn [X1,n ] and assume EFn [(X1,n − µn )2 ] = 1. Suppose that we are interested in discovering positive effects and would only report the sample mean if it survives the file drawer effect, i.e. n1/2 X̄n > 2. (1) Then what is the “correct” p-value to report for an observation X̄n,obs that exceeds the threshold? If we have Gaussian family, namely Fn = N (µn , 1), then the distribution of X̄n surviving the file drawer effect (1) is a truncated Gaussian distribution. We also call this distribution the selective distribution. Formally, its survival function is 1 1/2 P (t) = P X̄n > t\|n X̄n > 2 , X̄n ∼ N µn , n 1/2 1 − Φ n (t − µn ) = 1 − Φ(2 − n1/2 µn ) where Φ is the CDF of an N (0, 1) random variable. Therefore, we get a pivotal quantity 1 − Φ n1/2 (X̄n,obs − µn ) ∼ Unif(0, 1), P (X̄n,obs ) = 1 − Φ(2 − n1/2 µn ) (2) 1 1/2 n X̄n,obs > 2, Xn,obs ∼ N µn , n The pivotal quantity in (2) allows us to construct p-values or confidence intervals for Gaussian families. When the distributions Fn ’s are not normal distributions, central limit theorem states that the sample mean X̄n is asymptotically normal when Fn Tian and Taylor/Selective inference with a randomized response 3 has second moments. Thus a natural question is whether the pivotal quantity in (2) is asymptotically Unif(0, 1) when Xi,n does not come from a normal distribution? The following lemma provides a negative answer to this question in the case when Fn is a translated Bernoulli distribution that has a negative mean. Essentially when the selection event n1/2 X̄n > 2 becomes a rare event with vanishing probability, the pivotal quantity in (2) no longer converges to Unif(0, 1). We defer the proof of the lemma to the appendix. Lemma 1. If Xi,n takes values in {−1.5, 0.5}, with P (Xi,n = −1.5) = P (Xi,n = 0.5) = 0.5. Thus µn = −0.5. Then the pivot in (2) does not converge to Unif(0, 1) P (X̄n ) 6→ Unif(0, 1), for the X̄n ’s surviving the file drawer effect (1). Randomized selection circumvents this problem. In the following, we propose a randomized version of the “file drawer problem”. Example 2 (File drawer problem, randomized ). We assume the same setup of a triangular array of observations Xi,n as in Example 1. But instead of reporting X̄n when it survives the file drawer effect (1), we independently draw ω ∼ G, and only report X̄n if n1/2 X̄n + ω > 2. (3) Note that the selection event is different from that in (1) in that we randomize the sample mean before checking whether it passes the threshold. In this case, if Fn = N (µn , 1), the survival function of X̄n is 1 1/2 ×G P (t) = P X̄n > t\|n X̄n + ω > 2 , (X̄n , ω) ∼ N µn , n = P Z > n1/2 (t − µn )\|Z + ω > 2 − n1/2 µn , (Z, ω) ∼ N (0, 1) × G. (4) To compute the exact form of P (t), we have to compute the convolution of N (0, 1) and G which has explicit forms for many distributions G. Moreover, when G is Logistic or Laplace distribution, we have P (X̄n,obs ) → Unif(0, 1), as long as Fn has centered exponential moments in a fixed neighbourhood of 0. The convergence is in fact uniform for −∞ < µn < ∞. For details, see Lemma 10 in Section 5.2. The only difference between these two examples is the randomization in selection. After selection, we need to consider the conditional distribution for inference, which conditions on the selection event. If we denote by F∗n the distribution used for selective inference, we have in Example 1, 1{n1/2 X̄n >2} dF∗n (X̄n ) = . dFn PFn (n1/2 X̄n > 2) (5) Tian and Taylor/Selective inference with a randomized response 4 We also call the ratio between F∗n and Fn the selective likelihood ratio. In this case, the selective likelihood ratio is simply a restriction to the X̄n ’s that survives the file drawer effect. We observe that √ √ nX̄n = nµn + Z, Z ∼ N (0, 1), which leads to three scenarios for selection. • µn > δ > 0, for some δ > 0. √ In this case, the dominant term for selection is nµn , and since we have a big positive effect, we would always report the sample mean X̄n when n is big. This corresponds to the selection event having probability tending to 1 and the selective likelihood ratio goes to 1 as well. In this case, there is very little selection bias, and the original law is a good approximation to the selective distribution for valid inference. • µn < −δ < 0, for some δ > 0. √ In this case, the dominant term is also nµn , but in the negative direction. As n → ∞, the selection probability vanishes and the selective likelihood becomes degenerate. We almost never report the sample mean in this scenario, but in the rare event where we do, by no means can we use the original distribution for inference. • −δ < n1/2 µn < δ, for some δ > 0. This corresponds to local alternatives. In this case, the selective likelihood neither converges to 1 or becomes degenerate. Rather, it becomes an indicator function of a half interval. Proper adjustment is needed for valid inference in this case. It is in the second scenario that pivotal quantity (2) will not converge to Unif(0, 1). Different distributions will have different behaviors in the tail. Since the conditioning event n1/2 X̄n > 2 becomes a large-deviations event, we cannot expect it to behave like the normal distribution in the tail. On the other hand, in Example 2, if we denote by F̃∗n the law for selective inference, we have Ḡ 2 − n1/2 (X̄n − µn ) − n1/2 µn dF̃∗n Ḡ(2 − n1/2 X̄n ) (X̄n ) = = dFn EFn (Ḡ(2 − n1/2 X̄n )) EFn Ḡ 2 − n1/2 (X̄n − µn ) − n1/2 µn (6) R∞ where Ḡ(t) = t G(du) is the survival function of G. When µn < −δ < 0 for some δ > 0, and G is the Laplace or Logistic distribution so that Ḡ has an exponential tail, the dominant term exp(n1/2 µn ) in both the numerator and the denominator will cancel out, making the selective likelihood ratio properly behaved in this difficult scenario. It turns out that this selective likelihood ratio is fundamental to formalizing asymptotic properties of selective inference procedures. Its behavior determines not only the asymptotic convergence of the pivotal quantities like in (4), but also whether consistent estimation of the population parameters is possible with large samples. Again in the negative mean scenario where µn < −δ < 0, the sample mean X̄n surviving the non-randomized “file drawer effect” cannot be a consistent estimator for the underlying means µn because it will always be positive. But if X̄n is reported as in Tian and Taylor/Selective inference with a randomized response 5 Example 2, it will be consistent for µn even if µn is negative and bounded away from 0. For detailed discussion, see Section 3. In general, the behavior of the selective likelihood ratio can be used to study the asymptotic properties of selective inference procedures. We study consistent estimation and weak convergence for selective inference procedures in Section 3 and Section 5 respectively. We are especially inspired by the field of differential privacy (c.f. Dwork et al. (2014) and references therein) to study the use of randomization in selective inference. Privatized algorithms purposely randomize reports from queries to a database in order to allow valid interactive data analysis. To our understanding, our results are the first results related to weak convergence in privatized algorithms, as most guarantees provided in the differentially private literature are consistency guarantees. Some other asymptotic results in selective inference have also been considered in Tibshirani et al. (2015), Tian & Taylor (2015), though these have a slightly different flavor in that they marginalize over choices of models. We conclude this section with some more examples. 1.2. Linear regression Consider the linear regression framework with response y ∈ Rn , and feature matrix X ∈ Rn×p , with X fixed. We make a homoscedasticity assumption that Cov [y\|X] = σ 2 I, with σ 2 considered known. Of interest is µ = E(y\|X), a functional of F = F(X) the conditional law of y given X. When F is a Gaussian distribution, exact selective tests have been proposed for different selection procedures Tibshirani (1996), Taylor et al. (2014), Tian et al. (2015). Removing the Gaussian distribution on F, Tian & Taylor (2015) showed that the same tests are asymptotically valid under some conditions. Randomized selection in this setting is a natural extension of these works. Fithian et al. (2014) proposed to use a subset of data for model selection, which yields a significant increase in power. In this work, we study general randomized selection procedures. Consider the following example. Due to the sparsity of the solution of LASSO Tibshirani (1996) 1 β̂λ (y) = argmin ky − Xβk22 + λ · kβk1 , β∈Rp 2 a small subset of variables can be chosen for which we want to report p-values or confidence intervals. This problem has been studied in Lee et al. (2013a). However, instead of using the original response y to select the variables, we can independently draw ω ∼ Q and choose the variables using y ∗ = y + ω. Specifically, we choose subset E by solving 1 β̂λ (y, ω) = argmin ky ∗ − Xβk22 + λ · kβk1 , p 2 β∈R y ∗ = y + ω, (7) Tian and Taylor/Selective inference with a randomized response 6 and take E = supp(β̂λ (y, ω)). In Section 4.2.2, we discuss how to carry out inference after this selection procedure, with much increased power. We also discuss the reason behind this increase in Section 4.2. 1.3. Nonparametric selective inference All the previous works on selective inference assume a parametric model like the Gaussian family or the exponential family. In this work, we allow selective inference in a non-parametric setting. Consider the following examples. Suppose in a classification problem, we observe independent samples, iid (xi , yi ) ∼ F, (xi , yi ) ∈ Rp × {0, 1}. with fixed p. This problem is non-parametric if we do not assume any parametric structure for F and are simply interested in some population parameters of the distribution F. In Section 5, we developed asymptotic theory to construct an asymptotically valid test for the population parameters of interest. More details can be found in Section 5.4.1. Also consider a multi-group problem where a response x is measured on p treatment groups. A special case is the two-sample problem where there are two groups. It is of interest to form a confidence interval for the effect size in the “best” treatment group. This arises often in medical experiments where multiple treatments are performed and we are interested to discover whether one of the treatment has a positive effect. The fact we have chosen to report the “best” treatment effect exposes us to selection bias and multiple testing issues Benjamini & Hochberg (1995), and therefore calls for adjustment after selection. Benjamini & Stark (1996) have considered the parametric setting iid where xj ∼ N (µj , σ 2 ) for each group. Suppose for robustness, it is of interest to report the median effect size instead of the mean (assuming responses are not symmetric). Then without any assumptions on the distribution of the measurements, this also becomes a nonparametric problem. But we can apply the theory in Section 5 to cope with this problem, for details, see Section 5.3. 1.4. Outline of the paper There are three main advantages of applying randomization for selective inference, • Consistent estimation under the selective distribution • Increase in power for selective tests • Weak convergence of selective inference procedures In the following sections, Section 2 gives the setup of selective inference and introduced selective likelihood ratio, which is the key for studying consistent estimation and weak convergence of selective inference procedures. Section 4 focuses on linear regression models with different randomization schemes, demonstrating the increase in power. Section 5 proposes an asymptotic test for the nonparametric settings. Theorem 9 proves that the central limit theorem holds under the selective distribution with mild conditions. Applications to the two examples in Section 1.3 are discussed. This is a Tian and Taylor/Selective inference with a randomized response 7 result for fixed dimension p. Finally, Section 6 discusses the possibility of extending our work to the setting, when multiple selection procedures are performed on different randomizations of the original data. One application is selective inference after cross validation for the square-root LASSO Belloni et al. (2011). 2. Selective Likelihood Ratio We first review some key concepts of selective inference. Our data D lies in some measurable space (D, F), with unknown sampling distribution D ∼ F. Selective inference seeks a reasonable probability model M – a subset of the probability measures on (D, F), and carry out inference in M . Central to our discussion is a selection algorithm, a set-valued map b:D→Q Q (8) where Q is loosely defined as being made up of “potentially interesting statistical questions”. For instance, in the linear regression setting, D = Rn , our data D = y and we have a fixed feature matrix X ∈ Rn×p . The unknown sampling distribution is F = L(y\|X), the conditional law of y given X. b might be all linear regression modA reasonable candidate for the range of Q els indexed by subsets of {1, . . . , p} with known or unknown variance. For any selected subset of variables E, we carry out selective inference within the model M = {N (XE βE , σ 2 I), βE ∈ R\|E\| }. Since we use the data to choose the model M , it is only fair to consider the conditional distribution for inference, b D\|M ∈ Q(D), D ∼ F. Therefore, we seek to control the selective Type-I error: b ≤α PM,H0 (reject H0 \| M ∈ Q) (9) b and H0 ⊂ M is the where M is the selected family of distributions in the range of Q null hypothesis. Selective intervals for parametric models M can then be constructed by inverting such selective hypothesis tests, though only the one-parameter case has really been considered to date. 2.1. Randomized selection Randomized selection is a natural extension of the framework above. We enlarge our probability space to include some element of randomization. Specifically, let H denote an auxiliary probability space and Q is a probability measure on H. A randomized selection algorithm is then simply b∗ : D × H → Q. Q Tian and Taylor/Selective inference with a randomized response 8 Note the randomization is completely under the control of the data analyst and hence Q will be fully known. This is an extension of the non-randomized selective inference framework in the sense that we can take Q to be the Dirac measure at 0. Many choices b∗ are natural extensions of Q, b which we will see in many examples. of Q Randomized selective inference is simply based on the law F∗ , which we also call the selective distribution, b∗ (D, ω), (D, ω) ∼ F × Q. D\|M ∈ Q (10) Note that although randomization is incorporated into selection, inference is still carried out using the original data D, after adjusting for the selection bias by considering the conditional distribution F∗ . Similar to the selective inference we defined above, we seek to control the selective Type-I error, b∗ ) ≤ α. PF∗ (reject H0 ) = PM,H0 (reject H0 \|M ∈ Q (11) Moreover, we also want to achieve good estimation, which makes EF∗ ((θ̂(y) − θ(F))2 ) (12) small. b∗ . But In Sections 3 to 5, we will discuss concrete examples of D, D, F and Q before that we first introduce the selective likelihood ratio, which is a crucial quantity in studying the selective distribution F∗ . 2.2. Selective likelihood ratio Selective likelihood ratio provides a way of connecting the original distribution F and its selective counterpart F∗ . It is easy to see from (10) that the selective distribution is simply a restriction of the (D, ω)’s such that model M will be selected. Thus F∗ is absolutely continuous with respect to F, and the selective likelihood ratio is W(M ; D) dF∗ (D) = = `F (D) ∀ F ∈ M, dF EF (W(M ; D)) n o b∗ (D, ω) . W(M ; D) = Q ω : M ∈ Q (13) The numerator in `F (D) is the restriction of (D, ω), integrated over the randomizations ω, and the denominator is simply a normalizing constant. One implication of the selective likelihood ratio is that for distributions F in parametric families, their selective counterparts may have the same parametric structure. 2.2.1. Exponential families One commonly used parametric family is the exponential family. Assume that F = Fθ is an exponential family with natural parameter space Θ and D = Rn and the data Tian and Taylor/Selective inference with a randomized response 9 D = y. Its density with respect to the reference measure dF0 is, dFθ (y) = exp{θT T (y) − ψ(θ)}, dF0 θ ∈ Θ. (14) Through the relationship in (13) we conclude, for any randomization scheme, the law F∗M,θ is another exponential family. Formally, Lemma 2. If Fθ belongs to the exponential family in (14), then for any randomized b∗ , the selective distribution is also an exponential family, selection procedure Q dF∗M,θ (y) ∝ W(M ; y) exp{θT T (y) − ψ(θ)}, dF0 θ ∈ Θ. with the same sufficient statistic T (y) and natural parameters θ. Furthermore, to test H0j : θj = 0, we consider the following law, Tj (y) \| T−j (y), y ∼ F∗M,θ . (15) The first claim of the lemma is quite straight-forward using the relationship in (13). The second claim is a Lehmann–Scheffe (c.f. Chapter 4.4 in Lehmann (1986)) construction which was proposed in Fithian et al. (2014), to construct tests for one of the natural parameters treating the others as nuisance parameters. For detailed construction of such tests in the linear regression setting, see Section 4. 3. Consistent Estimation After Model Selection In this section, we leave the parametric setup and consider general models M . In particular, we study the consistency of estimators under the selective distribution for arbitrary models. We first introduce the framework of asymptotic analysis under the selective model. Then we state conditions for consistent estimation in Lemma 3 and conclude with examples. For any model M , which is a collection of distributions, we define its corresponding selective model, which is the collection of corresponding selective distributions, ∗ ∗ ∗ dF (D) = `F (D), F ∈ M , (16) M = F : dF b∗ }. Selecwhere `F (D) is the selective likelihood ratio for the selection event {M ∈ Q ∗ tive inference is carried out under the selective model M . In order to make meaningful asymptotic statements, we consider a sequence of ranb∗n )n≥1 and models (Mn )n≥1 with each Mn in the domized selection procedures (Q ∗ b range of Qn . Often, we are interested in some population parameter θn , which can be thought be as a functional of the distribution Fn ∈ Mn , θn : Mn → R. Tian and Taylor/Selective inference with a randomized response 10 b∗ , which already incorporates the It is worth pointing out that Mn is selected by Q n statistical questions we are interested in. In this sense, Mn is chosen a posteriori. The selected model Mn∗ does not change our target of inference, it merely changes the distribution under which such inference should be carried out. In other words, if θn is the mean parameter, we are interested in the underlying mean of Fn , not F∗n . We might have a good estimator θ̂n : D → R for θn (Fn ) under Fn , namely i h EFn (θ̂n − θn (Fn ))2 → 0. θ̂n is a consistent estimator if our model Mn is given a priori. But as we use data select Mn , what really cares about is its performance under the selective distribution F∗n . Will this estimator still be consistent under the selective distribution F∗n ? Formally, we say an estimator θ̂n is uniformly consistent in Lp for θn (Fn ) under the sequence (Mn )n≥1 if lim sup sup kθ̂n − θn (Fn )kLp (Fn ) → 0. n Fn ∈Mn Similarly, we say that θ̂n is uniformly consistent in probability for the functional θn (Fn ) under the sequence (Mn )n≥1 if for every > 0 there exists δ() > 0 such that for all δ ≥ δ() lim sup sup Fn (\|θ̂n − θn (Fn )\| > δ) ≤ . n Fn ∈Mn The following lemma states the conditions for consistency of θ̂n under the sequence of corresponding selective models (Mn∗ )n≥1 , b∗n , Mn )n≥1 of randomized selection procedures Lemma 3. Consider a sequence (Q and models. Suppose the selective likelihood ratios satisfies, for some p > 1, lim sup sup k`Fn kLp (Fn ) < C. n (17) Fn ∈Mn Then for any sequence of estimators θ̂n uniformly consistent for θn (Fn ) in Lα , it is also uniformly consistent for θn (Fn ) in Lγ under (Mn∗ )n≥1 , γ ≤ α/q, p1 + 1q = 1. Further, if θ̂n is uniformly consistent for θn in probability, then θ̂n is uniformly consistent for θn in probability under the sequence (Mn∗ )n≥1 . Proof. Let ∆n = θ̂n − θn (Fn ). To prove the first assertion note that for any F∗n ∈ Mn∗ Z k∆n kLγ (F∗n ) = \|∆n \|γ `Fn (y)Fn (dy) Dn ≤ k\|∆n \|γ kLq (Fn ) k`Fn (y)kLp (Fn ) = k∆n kγLγq (Fn ) k`Fn (y)kLp (Fn ) ≤ k∆n kγLα (Fn ) k`Fn (y)kLp (Fn ) ≤ Ck∆n kγLα (Fn ) Tian and Taylor/Selective inference with a randomized response 11 For any δ > 0, F∗n (\|∆n \| Z > δ) = Dn 1{\|∆n \| > δ}`Fn (y)Fn (dy) 1/q ≤ [Fn (\|∆n \| > δ)] k`Fn kLp (F) 1/q ≤ C [Fn (\|∆n \| > δ)] . We illustrate the application of Lemma 3 through our “file drawer effect” examples in Section 1.1. 3.1. Revisit the “file drawer problem” First we note that in Example 1 and 2, we observe data Dn = (X1,n , . . . , Xn,n ), with Xi,n ∼ Fn . The randomized selection in Example 2 can be realized as ( √ report p-values for X̄n , if nX̄n + ω > 2, ∗ b Q (Dn , ω) = √ do nothing, if nX̄n + ω ≤ 2, where we independently draw ω ∼ G. By law of large numbers, we easily see that if we always report X̄n , it will be an unbiased estimator for µn . However, since we only observe the sample means surviving the file drawer effect. Will X̄n still be consistent for µn ? In the most difficult scenario discussed in Section 1.1, where µn < −δ < 0 for some δ > 0, X̄n cannot be a consistent estimator for µn in Example 1. This is easy to see as Example 1 will only report positive sample means. A remarkable feature of randomized selection is that consistent estimation of the population parameters is possible even when the selection event has vanishing probabilities. In fact, the following lemma states that when G is a Logistic distribution, X̄n is consistent for µn after the randomized file drawer effect in Example 2. Lemma 4. Suppose as in Example 2, we observe a triangular array with Xi,n ∼ Fn . Fn has mean µn = µ < 0. If we draw ω ∼ Logistic(κ), where κ is the scale of the Logistic distribution. Then the sample means X̄n surviving the “randomized” file drawer effect are consistent for µ, √ p X̄n → µ, conditional on nX̄n + ω > 2. if Fn has moment generating function in a neighbourhood of 0. Namely, ∃a > 0, such that EFn [exp (a \|Xi,n − µn \|)] ≤ C. Before we prove the lemma, we want to point out that although the selection √ procedure in Example 2 is different from that in Example 1 because of randomization, nµn is still the dominant term in selection. Note that √ √ √ nX̄n + ω = nµn + n(X̄n − µn ) + ω. Tian and Taylor/Selective inference with a randomized response 12 √ Since both n(X̄n − µn ) and ω are Op (1) random variables, the dominant term √ nµn → −∞, would ensure that the selection event has vanishing probabilities in Example 2 as well. Thus it is particularly impressive that Example 2 gives consistent estimation where Example 1 cannot. The proof of Lemma 4 is deferred to the appendix. We also verified this theory of consistent estimation through simulations. Figure 1 shows the empirical distributions of the sample mean X̄n after the file drawer effect in Example 1 or the “randomized” file drawer effect in Example 2. They are marked with “blue” colors or “red” colors respectively. We set the true underlying mean to be µn = µ = −1 and mark it with the dotted vertical line in Figure 1. The upper panel Figure 1a is simulated with n = 100 and the lower panel Figure 1b is simulated with n = 250. We notice that in both simulations, the sample mean in Example 1 concentrates around the thresholding boundary, which is positive. Thus, these sample means can not be possibly for the underlying mean µ = −1. However, the existence of randomization allows us to report negative sample means. As a result, the sample mean in Example 2 will be consistent for µ = −1. We see that as we increase sample size n, the sample means concentrates closer to µ = −1. 4. Inference in linear regression models In the linear regression setting, we assume a fixed feature matrix X ∈ Rn×p , and observe the response vector D = y ∈ Rn . We assume the noises are normally distributed. There are two ways to parametrize a linear model, and both belong to some exponential family. Now we introduce the selected model, n o Msel (E) = N (XE βE , σ 2 I) : βE ∈ R\|E\| , E ⊂ {1, . . . , p} (18) with σ 2 known or unknown or the saturated model, Msat = N (µ, σ 2 I) : µ ∈ Rn (19) with known variance. Now we consider some randomized selection procedures and inference after selection. 4.1. Data splitting and data carving In the introduction, we introduced data splitting Cox (1975) as a special case of randomized selective inference. In Fithian et al. (2014), the term data carving was introduced to demonstrate that data splitting is inadmissible. In data splitting (and data carving) inference makes most sense in the selected model Msel (E), hence we should b as returning a subset E of variables selected. think of Q Let us formalize this notion in our notation. Let Q be some measure on assignments b a selection algorithm defined on datasets of any of n data points into groups and Q size. The distribution Q determines a randomized selective inference procedure with b∗ , an algorithm applied to subsets of the original data set. In this selection algorithm Q 13 Tian and Taylor/Selective inference with a randomized response X̄ after file drawer effect 90 80 4.0 70 3.5 60 3.0 50 2.5 40 2.0 30 1.5 20 1.0 10 0.5 0 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 X̄ after randomized file drawer effect 4.5 0.0 0.2 0.4 0.0 −1.2 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.2 0.4 (a) n = 100 X̄ after file drawer effect 120 X̄ after randomized file drawer effect 7 6 100 5 80 4 60 3 40 2 20 0 -1.2 1 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0 −1.2 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 (b) n = 250 Fig 1: Empirical distributions of sample means X̄n in Example 1 and Example 2, with original or randomized file drawer effect. For the randomization, we draw ω ∼ Logistic(κ), with κ = 0.5. Tian and Taylor/Selective inference with a randomized response 14 case, it is easy to see that D W(E; y) = W(Msel (E); y) ∝ X ω qω · 1{Msel (E)∈Q(y b 1 (y,ω))} where qω is the mass assigned to assignment ω by Q. Multiple assignments or splits considered in Meinshausen et al. (2009), Meinshausen & Bühlmann (2010) can be formalized in a similar fashion. We can construct UMPU tests for βE in the selected model Msel (E) by using Lemma 2, (also see Fithian et al. (2014)). We note that in Fithian et al. (2014) the authors conditioned unnecessarily on the split ω, and we would expect that aggregating over splits would yield a more powerful procedure. However, there are two disadvantages with this randomization scheme. First, it is computationally difficult to aggregate over all random splits. Second, it seems difficult to consider the saturated model Msat for inference, which is more robust to model misspecifications. To overcome those difficulties, we introduce other randomization schemes below. 4.2. Additive noise and more powerful tests Our second randomization scheme in linear regression involves additive noise. Specifically, we draw ω ∼ Q and use the randomized response y ∗ (y, ω) = y + ω for selection In this case, we can consider both the selected model Msel,E and the saturated model Msat . Per Lemma 2, we can perform valid inference for βE in Msel,E or linear functionals of µ in Msat . One major advantage of using a randomized response y ∗ for selective inference is that these procedures yield much more powerful tests, at a small cost of on the quality of the selected models. In other words, small amount of randomization is cause a small loss in the model selection stage, but we gain much more power in the inference stage. The reason for increased power can be explained by a notion called leftover Fisher Information first introduced in Fithian et al. (2014). Since selective inference is essentially inference under the selective distribution F∗n , the Fisher Information under F∗n would determine how efficient the selective tests are. In the saturated model with ∗ Gaussian noise Msat , y−µ σ 2 is the score statistic and its variance under Fn is exactly the leftover Fisher Information (a similar relationship holds in the selected model Msel,E ). Lemma 5 gives a lower bound on this leftover Fisher Information when the randomization noise Q = N (0, γ 2 I). Lemma 5. For either Msat or Msel (E), if we use Gaussian randomization noise Q = b ∗ ) = Q(y b + ω), then the leftover Fisher N (0, γ 2 ), and the selection is based on Q(y information is bounded below by (1 − τ )I(θ), τ = σ 2 /(σ 2 + γ 2 ), and I(θ) is the non-selective Fisher information for θ in Msat or Msel (E). The parameters θ depend on which of the two models we are considering. 15 Tian and Taylor/Selective inference with a randomized response b ∗ Proof. In the saturated model Msat , the score statistic is V = y−µ σ 2 . Since Q(y ) is measurable with respect to y ∗ , h i b ∗ ) ≥ Var [V \| y ∗ ] = 1 Var [y \| y ∗ ] . Var V \| Q(y σ4 Since y and y ∗ = y + ω are both normal distributions with covariance matrices, Cov [y, y ∗ ] = σ 2 I, Var [y ∗ ] = (γ 2 + σ 2 )I, we have the leftover Fisher Information h i b ∗ ) ≥ 1 Var [y \| y ∗ ] Var V \| Q(y σ4 4 1 σ 1 = 4 (σ 2 I − 2 I) = 2 (1 − τ )I = (1 − τ )I(µ). 2 σ γ +σ σ In the selected model Msel,E , the score statistic is V = T XE (y−XE βE ) . σ2 Similarly, h i T b ∗ ) ≥ 1 Var XE Var V \| Q(y y \| y∗ 4 σ 1 σ4 1 2 T T T = 4 σ (XE XE ) − 2 (X XE ) = 2 (1 − τ )XE XE = (1 − τ )I(βE ). σ γ + σ2 E σ When there is no randomization γ = 0, we potentially have no leftover Fisher information. This corresponds to a very rare selection event. However after randomization, even with very extreme selection, there is always leftover Fisher information, which makes the selective tests more powerful. Consider the following examples. 4.2.1. Revisit the “file drawer problem” In Example 1 and Example 2, if we assume Fn = N (µ, 1), they are a special case of the linear regression model, with the feature matrix X = 1, the all ones vector. In this case, nX̄n is the score statistic, and its variance under the selective distribution is the Fisher information. Lemma 5 states that the leftover Fisher information is lower bounded by n(1 − τ ) if we draw randomize using Gaussian variables, Q = N (0, γ 2 ), τ = 1/(1 + γ 2 ). Moreover, the increase in leftover Fisher information with randomization is not specific to Gaussian randomizations. For example, in Figure 1 when we use Logistic randomization, we also observe that under the selective distribution with randomization, X̄n has a much bigger variance than without randomization. As discussed above, this variance multiplied by n2 is exactly the leftover Fisher information, which explains why selective procedures after randomization will have better performances than without. We investigate the relationship between the leftover Fisher information and the length of confidence intervals constructed by inverting the pivot in (4). Specifically, 16 Tian and Taylor/Selective inference with a randomized response 1.0 1.0 0.5 0.5 0.0 0.0 0.5 0.5 1.0 1.0 1.5 2.0 0.2 1.5 Gaussian randomization Nominal 0.1 0.0 0.1 Observed X̄ 0.2 0.3 Logistic randomization Nominal 0.4 (a) Gaussian added noise 2.0 0.2 0.1 0.0 0.1 Observed X̄ 0.2 0.3 0.4 (b) logistic added noise Fig 2: Selective confidence intervals for different added noise in Example 2, after observing a reported sample mean, we want to report confidence intervals for the underlying mean µ. Figure 2 demonstrates the selective intervals (solid lines) after (3) with ω being either Gaussian or Logistic noises. The sample size n = 100. Unlike the nominal confidence intervals (dashed lines), the selective intervals are valid with 90% coverage for the underlying mean. Since Lemma 3 gives a lower bound of (1 − τ )I(µ), we would intuitively expect the selective confidence intervals to be 1/(1 − τ ) the length of the nominal intervals. This is verified in Figure 2a, when we observe really negative sample means. (The sample means can be negative because we added randomization.) On the other hand, for Logistic randomization√in Figure 2b, the intervals are slightly wider than the nominal intervals around the 2/ n, but narrow to roughly the nominal size on both sides of the truncation point. This indicates that added logistic noise might preserve more information than Gaussian additive noise. Both additive noises improve significantly over a non-randomization scheme (c.f. Figure 3 in Fithian et al. (2014)). Of course, the increase in power and shortening of selective confidence intervals does not come without a price. Because we select with a randomized response, we are likely to select a worse model. But the trade-off between model quality and power is highly in favor of randomization. See the following example. 4.2.2. Linear regression with added noise Back to the general setup of linear regression models, we select a model by solving LASSO with the randomized response y ∗ = y + ω and return the active set E of the solution (as in (7)). Then per Lemma 2, we can construct valid selective tests in both Msat and Msel (E). For instance, in Msel (E), we can construct tests for the hypothesis H0j : βj = 0, j ∈ E based on the law, η T y AE (y + ω) ≤ bE , PE\j y, (y, ω) ∼ N (XE βE , σ 2 I) × Q, βj = 0, (20) † T where η = (XE ) ej , ej is the j-th column of the identity matrix, PE\j is the projection matrix onto the column space of E but orthogonal to η, and AE , bE are the appropriate Tian and Taylor/Selective inference with a randomized response 0.5 Type II error 0.4 17 Data splitting Data carving Additive noise 0.3 0.2 0.1 0.0 1.0 0.9 0.8 0.7 Probability of screening 0.6 0.5 Fig 3: Comparison of inference in additive noise randomization vs. data carving. matrix and vector corresponding to LASSO selection. This is a UMPU test due to the Lehmann–Scheffe construction (Fithian et al. 2014) and controls the selective Type-I error (11). Although, we cannot compute the explicit forms of (20), the selection events in (20) are polyhedrons and thus a hit-and-run or Hamiltonian Monte Carlo algorithm Pakman & Paninski (2012) can be used for sampling. Figure 3 compares inference in the additive Gaussian noise scheme to the data carving procedure proposed in Fithian et al. (2014) as well as data splitting. In Msel (E), the probability of screening (i.e. selecting E including all the nonzero β’s) is a surrogate for the quality of the model. As additive noise uses a different randomization scheme than data splitting and data carving, we vary the amount of randomization used in each scheme and match on the probability of screening. Thus Figure 3 is like an ROC curve for the trade-off between model quality and power of tests. The x-axis goes in the direction of increased randomization, with the left most point corresponding to no randomization at all. We see even with a small randomization that barely affects model selection, we can substantially lower the Type-II error from 0.2 to less than 0.05. The trade-off is highly in favor of (small) randomization. We see in Figure 3 that additive noise lowers the Type-II error by almost half than data carving for the same screening probability and they both clearly dominate data splitting. For the concrete setup of the simulation, see Chapter 7 of Fithian et al. (2014). 5. Weak convergence and selective inference for statistical functionals In the nonparametric setting, we assume a triangular array of data, Dn = (d1,n , . . . , dn,n ), iid and di,n ∼ Fn . When Fn = F, it is the special case of independent sampling. We are interested in some functional of the distribution µn = µ(Fn ). Associated with µn is our statistic T which is a linearizable statistic (Chung et al. 2013). iid Definition 6 (Linearizable statistic). Suppose di,n ∼ Fn , we call T a linearizable Tian and Taylor/Selective inference with a randomized response 18 statistic for µn = µ(Fn ) if for any sample size n, n T (Dn ) = 1X ξi,n + R, n i=1 p E [ξi,n ] = µn ∈ R , ξi,n = ξ(di,n ), Cov [ξi,n ] = Σn ∈ R (21) p×p . 1 where ξ a function of the data and R is bounded with probability 1, R = op (n− 2 ) under F. We use the slight abuse of notations to denote ξi,n as iid random variables as well. Throughout this section, we assume the dimension p is fixed. We are interested in establishing a pivotal quantity for Tn = T (Dn ) like (4) in Example 2 where Tn is the sample mean after the randomized “file drawer effect”. It turns out we have an exact pivotal quantity if Tn is normally distributed. To lighten notation, we suppress the script n in the following lemma, which is a finite sample result valid for any n. We prove the lemma in Section 7. Lemma 7. If the statistic T is normally distributed from N (µ, Σ n ) and the model M b∗ (T, ω), where ω ∼ Q. Then for any contrast η, is selected by randomized selection Q b∗ , we have which could depend on the outcome of selection Q R∞ Q(t; Vη ) · exp −n(t − η T µ)2 /2ση2 dt F∗ ηT T T P (T ; η µ, Σ) = R ∞ ∼ Unif(0, 1) (22) Q(t; Vη ) · exp(−n(t − η T µ)2 /2ση2 ) dt −∞ where ση2 = η T Ση, Q(t, Vη ) = Q Vη = I− 1 Σηη T ση2 T b∗ t · Ση/ση2 + Vη , ω ω:M ∈Q . Remark 8. In selected models Msel,E , the selection is often made not only based on (T, ω), but also other statistic of the data, which we call the null statistic N . Thus b∗ (T, N ), ω }. To make notation the selection event should be expressed as {M ∈ Q simpler, we exclude such possibilities. But a slightly modified pivot where we replace Q(t; Vη ) with Q(t; Vη , N ) in (22) and integrate over N , is still Unif(0, 1) distributed. Note that Lemma 7 provides a valid pivotal quantity for any randomized selection b∗ and any randomization noise Q provided that T is normally distributed. procedure Q In fact, Lemma 7 does not require T to be a linearizable statistic. In some sense, the lemma is a reformulation (after rescaling) of the selective tests constructed in the linear regression model with additive noises (see Section 4.2.2). For example, in the selected model Msel,E , to test the hypothesis H0j : βj = 0, j ∈ E, we consider the law (20). After introducing the null statistic N = PE⊥ y, the pivot in (22) is in fact the CDF transform of this law, taking T = PE y, Σ = nσ 2 PE , and the selection event b∗ ((T, N ), ω)} to be the affine selection event defined in (20). With simple {M ∈ Q −2 calculation, it is easy to see Vη = (PE − kηk η T η)y = PE\η y, which we condition on in both (22) and (20). Tian and Taylor/Selective inference with a randomized response 19 Of course the pivot in (22) is very difficult to compute explicitly, and we need to use sampling schemes like in (20). But in a nutshell, P (T ; η T µ, Σ) is simply a CDF transform of the law Σ T ∗ b η T \| Vη , M ∈ Q (T, ω), (T, ω) ∼ N µ, × Q. (23) n After introducing the null statistic, Lemma 7 is agnostic to the selected model Msel,E , where µ = XE βE or the saturated model Msat , where the parameter is simply µ. The nuances between the two models in terms of sampling is that the saturated model condition on N (treating it as part of Vη ), but selected model integrate over N . Lemma 7 is written with T implicitly being the approximate average of n i.i.d variables, hence the distribution N (µ, Σ n ). Linearizable statistics are of particular interest as they converge to N (µ, Σ ) due to central limit theorem. In the following, we seek to esn tablish conditions under which the pivot P (T ; µ, Σ) will be asymptotically Unif(0, 1). 5.1. Selective central limit theorem In other work on asymptotics of selective inference Tian & Taylor (2015), Tibshirani et al. (2015), the setup considered is usually the saturated model Msat . These works b∗ . In considered asymptotics of selective inference marginalized over the range of Q contrast, we consider the convergence for any particular selected model Mn , under b∗n }. Specifically, we allow weak the conditional law of the selection event {Mn ∈ Q convergence of the pivot in (22) in the sequence of selected models (Mn )n≥1 . As explained above, selected models integrate over the null statistics while saturated models condition on those, thus the selective tests should have more power provided that the selected model is believable. In the saturated model, our result provides a finer measure of convergence than in Tian & Taylor (2015). On the other hand, Tian & Taylor (2015) allows high-dimensional setting in some cases while we consider fixed dimension p. Similar to the asymptotic setting in Section 3, we consider the convergence of P (Tn ; η T µn , Σn ) under a sequence of models (Mn )n≥1 selected by a sequence of b∗n )n≥1 . (Tn )n≥1 is a sequence of linearizable statistics defined selection procedures (Q in Definition 6, with asymptotic mean µn and asymptotic covariance matrix Σnn . It turns out that in this setting, the selective likelihood ratio `Fn again plays an important role in the convergence of the pivot. Recall that with randomized selection b∗ (Tn , ω), the selective likelihood is Q W(Tn ; Mn ) , EFn [W(Tn ; Mn )] n o b∗ (Tn , ω) W(Tn ; Mn ) = Q ω : Mn ∈ Q n `Fn (Tn ; Mn ) = (24) It will √ be convenient to rewrite the likelihood ratio in terms of the normalized vector Zn = n(Tn − µn ) `¯Fn (Zn ) = `F (n−1/2 Zn + µn ). (25) as well as the pivot (22) P̄Fn (Zn ) = P (n−1/2 Zn + µn ; ηnT µn , Σn ). (26) 20 Tian and Taylor/Selective inference with a randomized response Our approach is basically a comparison of how the pivot will behave under Fn and its Gaussian counterpart Φn = N (µ(Fn ), Σ(Fn )). Specifically, it is a modification of the proof of Theorem 1.1 of Chatterjee (2005), modified to allow for the fact the derivatives of the pivot and the likelihood are not required to be uniformly bounded. Given a norm Ω on Rp , define n o r/k λΩ ∂ k f (s) exp(−rΩ(s)) : 1 ≤ k ≤ r , (27) r (f ) = sup s∈Rp where ∂ k denotes the k-fold differentiation with respect to the p-dimensional vector s, k · k denotes element wise maximum. Now we state our selective central limit theorem, which we prove in Section 7. Theorem 9 (Selective central limit theorem). Suppose the statistics Tn = T (Dn ) are linearizable statistics according to Definition 6. We also assume the norms Ω : Rp → R are such that for each f ∈ {P̄n , `¯Fn , `¯Φn }, it satisfies sup λΩ 3 (f ) ≤ C1 . (28) Fn ∈Mn Moreover, assume ξi,n has uniformly bounded moment generating function in some neighbourhood of 0. Namely, ∃a > 0, such that sup sup EFn (exp(akξi,n − µ(Fn )k1 )) ≤ C2 . (29) n≥1 Fn ∈Mn Furthermore, we assume lim sup n1/2 · n b∗n ] b∗n ] − P(Φ ×Q) [Mn ∈ Q P(Fn ×Q) [Mn ∈ Q n ≤ C3 . b∗ ] P(Φ ×Q) [Mn ∈ Q n (30) n Then, for any g with uniformly bounded derivatives up to third order EF∗n g P (Tn ) − Z 0 1 g(x)dx ≤ n−1/2 K(g, C1 , C2 , C3 , p), n ≥ n0 (31) where K depends only on the bounds on the derivatives of g, the constants C1 , C2 , C3 and the dimension p. Thus the convergence is uniform in (Mn )n≥1 for models satisfying (28), (29) and (30). Theorem 9 provides a finite sample bound on the convergence of the pivot P (Tn ). Since we allow g to be functions with uniformly bounded derivatives up to the third order, (31) implies convergence of P (Tn ) to Unif(0, 1) under F∗n . In the following examples, we show how to verify conditions (28), (29) and (30). 5.2. Revisit the “file drawer problem” In Examples 1 and 2, we considered only reporting an interval or a p-value about µn when n1/2 X̄n > 2 or n1/2 X̄n + ω > 2. This is an example where we do not really Tian and Taylor/Selective inference with a randomized response 21 select a model, but rather select only a proportion of the data to report. The selective distribution simply refers to the law of the reported sample means, which pass the threshold. The data we observe is Dn = (X1,n,...,Xn,n ) with the linearizable statistic Tn simply being the sample mean X̄n . Example 1 corresponds to the degenerate randomization of adding 0 to X̄n . Work of Tian & Taylor (2015) show that in order for the corresponding pivot to converge weakly we can take, for ∆ < 0 fixed n o 3 Mn = F : EF [X̄n ] > n−1/2 ∆, EF [Xi,n ]<∞ . (32) That is, X̄n will satisfy a selective CLT when the population mean is not too negative. On the other hand, in Example 2, the pivot in (22) is of the form, R∞ √ 2 Ḡ(2 − nt)e−n(t−µn ) /2 dz X̄n , (33) P (X̄n ) = R ∞ √ Ḡ(2 − nt)e−n(t−µn )2 /2 dz −∞ and likelihood `Fn (X̄n ) is defined in (6). When G is the Logistic noise, then condition (28) and (30) can be verified. Formally, we have the following lemma whose proof we defer to the appendix, Lemma 10. If G = Logistic(κ), with κ being the scale parameter, then if centered Xi,n ’s have moment generating functions in the neighbourhood of zero, then the pivot P (X̄n ) is asymptotically Unif(0, 1). In other words, with Logistic randomization noise, we can take the sequence of models to be Mn = Fn : EFn exp a \|X1,n − µn \| < ∞ , for some a > 0. (34) Requiring exponential moments is stricter than the third moment condition in (32), but we would have a stronger conclusion, namely weak convergence uniformly over all µn ’s. 5.3. Two-sample median problem In the two-sample median problem, we have two treatment groups from which we take iid iid measurements, x1i ∼ F1 and x2i ∼ F2 ; for simplicity of notation, we assume we observe n samples from each group, and drop n in the subscript. We will report the bigger median from this group in the non-randomized setting. Exact formulation of randomized selection will be discussed below. Suppose our underlying distribution is F = F1 ×F2 . Let µ = (µ1 , µ2 ) is the population median of the two groups, and T = (T1 , T2 ) is the sample median. The well-known result by Bahadur (1966) states that the sample median is a linearizable statistic for the median when the CDF of the distribution F has positive density f , and f 0 is bounded iid in a neighbourhood of the population median m. Formally, if xi ∼ F , then the sample median n 1 X 1{xi > m} − 1/2 + Rn , (35) T (x1 , . . . , xn ) = m + n i=1 F 0 (m) Tian and Taylor/Selective inference with a randomized response 22 with R = O(n−3/4 log n) with probability 1. b∗ reports Our (randomized) selection algorithm Q ( P (T ; µ1 , Σ), if T1 > T2 + n−1/2 ω P (T ; µ2 , Σ), if T1 ≤ T2 + n−1/2 ω, where ω ∼ Q and Σ = diag( 41 f1 (µ1 )−2 , 14 f2 (µ2 )−2 ) is a diagonal matrix. f1 , f2 are the densities of F1 and F2 . Without loss of generality, we suppose M1 is selected, i.e. the first group is the “best” group. We choose the randomization noise Q to be a Logistic(κ) with mean 0 and κ is the scale, and let Gκ be the CDF. The resulting pivot for µ1 is R∞ √ √ Gκ ( nt − nT2 ) · exp(−n(t − µ1 )2 /2σ12 ) dt 1 T1 , σ12 = . P (T ; µ1 , Σ) = R ∞ √ √ 2 2 4f1 (µ1 )2 G ( nt − nT2 ) · exp(−n(t − µ1 ) /2σ1 ) dt −∞ κ This pivot strikes a similarity with √ the pivot in (33) for Example 2 with the truncation threshold 2 being replaced by nT2 and plugging in the appropriate means and variances of the medians. A result similar to Lemma 10 can be established, which ensures convergence of the pivot uniformly for any underlying medians (µ1 , µ2 ). In order to construct the above pivot, we need knowledge of the variance σ12 . Without selection, there are natural estimates of this variance. One may ask, how will inference be affected if we plug this estimate into our pivot? We revisit this question in Section 5.5. 5.4. Affine selection events In this section, we discuss the special case of affine selection events (regions). This combined with the asymptotic result in Theorem 9 applies to more general settings. In particular, it allows us to approximate non-affine regions. For a concrete example, see Section 5.4.1. We drop the subscript n where possible to simplify notations. Suppose for our model b∗ } can be deM , the selection is based on (T, ω), and the selection event {M ∈ Q scribed as √ { nAM T + ω ∈ KM }, where the affine matrix AM ∈ Rd×p and KM is a region in Rd . Many examples of nonrandomized selective inference can be expressed in this way (c.f. Lee et al. (2013b), Taylor et al. (2014), Lee & Taylor (2014), Fithian et al. (2015)). In this section, we provide conditions under which Theorem √9 can be applied. We again normalize T to be Z = n(T − µ), then the selection event can be rewritten as {AM (Z + ∆) + ω ∈ KM }, (36) √ where nµ = ∆, Z converges to N (0, Σ). Suppose ω ∼ Q, which has distribution function G. Then we introduce some conditions on the selection region KM and the added noise distribution G, Tian and Taylor/Selective inference with a randomized response 23 Lower bound: We assume there is some norm h, such that Z G(dw) ≥ C − exp − inf h(w) , ∀θ ∈ Rd . w∈KM −θ KM −θ Smoothness: Suppose G has density g, we assume the first 3 derivatives of g are integrable, Z k∂ j g(w)kdw ≤ Cj , j = 0, 1, 2, 3 Rd where the norm on the left-hand side is the maximum element-wise of the partial derivatives. The above two conditions essentially require G to be differentiable and have heavier tails than (or equal to) exponential tails. In fact we prove that the lower bound and smoothness conditions ensure that (28) are satisfied under the local alternatives introduced below. Definition 11 (Local alternatives). For the sequence of selected model (Mn )n≥1 , we define the local alternatives of radius of B to be the set all sequences (µn )n≥1 , such that √ dh (0, KMn − AMn ∆) ≤ B, ∆ = nµn where dh (·, ·) is the distance induced by the norm h. The notion of local alternatives is natural in the asymptotic setting as we expect even a small effect size will be more prominent when we collect more and more data. Formally, we have the following lemma, whose proof is deferred to the appendix. Lemma 12. Suppose G, KM satisfy the lower bound and smoothness conditions, then condition (28) are satisfied under the local alternatives. Now, we are left to verify conditions (29) and (30). Condition (29) is essentially a moment condition on the centered statistics ξi,n − µn , which we have to assume. Condition (30) can be verified using the well known results in multivariate CLT (see Gotze (1991)). To be rigorous, we state the following lemma, which we also prove in the appendix. Lemma 13. If Fn is such that the centered statistics ξi,n −µn have finite third moments, then under the local alternatives, condition (30) is satisfied. To summarize, Lemma 12 and Lemma 13 state that if G has integrable derivatives and exponential tails, then the pivot in (22) converges to Unif(0, 1) uniformly for F∗n so long as Fn ’s are such that ξi,n − µn have exponential moments in a neighbourhood of 0. Unlike the sample mean and sample median examples, the pivot is difficult to compute explicitly in this case. However, as we discuss in the beginning of Section 5, the pivot is essentially the CDF transform of the conditional law (23), which we can sample from. As discussed above, we can just take ω to be from a Logistic distribution. Now we apply the above theory to logistic regression. Tian and Taylor/Selective inference with a randomized response 24 5.4.1. Example: randomized logistic lasso iid Suppose we observe independent samples, di = (yi , xi ) ∼ F, where yi ’s are binary observations and xi ∈ Rp . The ordinary logistic regression solves the following problem, β̄ = argminβ∈Rp `(β) " n # X = argminβ∈Rp − yi log π(xi β) + (1 − yi ) log(1 − π(xi β)) , (37) i=1 where π(x) = exp(x)/(1 + exp(x)). This is a nonparametric setting as we do not assume any parametric structure for F. The randomized logistic lasso adds an `1 penalty, a randomization term and a small quadratic term, 1 1 β̂ = argminβ∈Rp √ `(β) + ω T β + kΛβk1 + √ kβk22 , n 2 n (38) iid where ωj ∼ Logistic(κ) is the perturbation to the gradient and Λ is a diagonal matrix which introduces (possibly) unequal feature weights, κ controls the amount of randomization added. The addition of the quadratic term ensures that (38) is strictly convex, thus has a unique solution. A similar formulation for linear regression has been proposed in Meinshausen & Bühlmann (2010). Selective inference in this setting has not been considered before. Without the Gaussian assumptions Lee et al. (2013a) does not apply. The parametric setting of this problem has been discussed in Fithian et al. (2014), but computation of the selective tests are mostly infeasible for general X. Finally, the asymptotic result by Tian & Taylor (2015) does not apply here as the framework require exactly affine selection regions, which is not the case in this setting. Suppose the solution to (38) has nonzero entry set E, then our target of inference ∗ βE , the unique population minimizer which satisfies T ∗ EF [XE (y − π(XE βE ))] = 0. (39) ∗ )) with independently samNote that a parametric model yi \|xi ∼ Bernoulli(π(xi,E βE ∗ pled xi ’s will have βE satisfying (39). But we by no means assume such an underlying ∗ distribution. Rather, for any well-behaved distribution F, βE can be thought of of a statistical functional of the underlying distribution F, depending on the outcome of selection E. Selective inference in this setting is carried out conditioned on (E, sE ), the active set and its signs. We first introduce the following notations, exp(XE βE ) , WE (βE ) = diag(πE (βE )(1 − πE (βE ))), 1 + exp(XE βE ) 1 T 1 T QE (βE ) = XE WE (βE )XE , CE (βE ) = X−E WE (βE )XE , n n DE (βE ) = CE (βE )Q−1 E (βE ) πE (βE ) = 25 Tian and Taylor/Selective inference with a randomized response where X is the feature matrix, and XE , X−E is the columns corresponding to the active set and inactive set respectively. By law of large numbers, we have p def ∗ ∗ QE (βE ) → EF QE (βE ) = Q, p def p def ∗ ∗ CE (βE ) → EF CE (βE ) = C, ∗ DE (βE ) → CQ−1 = D. (40) Now we introduce our linearizable statistics and show that the conditioning event (E, sE ) can be expressed as affine regions of these statistics. Lemma 14. Suppose E is the active set of the solution of (38), and we denote " n # X β̄E = argmin − yi log π(xi,E βE ) + (1 − yi ) log(1 − π(xi,E βE )) βE ∈RE i=1 as the unpenalized MLE restricted to the selected variables E. ∗ The following statistic T is linearizable with asymptotic mean (βE , ρ) and variance Σ/n, β̄E + R, T = 1 T n X−E y − πE (β̄E ) ∗ )) . Morewhere R = op (n−1/2 ) is a small residual, and ρ = E xTi,−E (yi − π(xi,E βE over, √ the selection event {Ê, zÊ = (E, sE )} can be characterized as the affine region { nAM T + BM ω ≤ bM }, where       −SE 0 SE Q−1 0 −SE Q−1 ΛE sE I−E  , BM =  D −I−E  , bM = λ−E − DΛE sE  , AM =  0 0 −I−E −D I−E λ−E + DΛE sE where I−E denotes the identity matrix of n − \|E\| dimensions and ΛE , Λ−E denote the active block and the inactive block of Λ respectively and λ is the diagonal elements of Λ, SE = diag(sE ). The proof of this lemma is also deferred to the appendix. Thus using Lemma 12 and Lemma 13, we can conclude under local alternatives, the pivot (22) converges to Unif(0, 1). To test H0j : βj∗ = 0, we take η = ej , and sample ∗ √ Σ βE η T T \| Vη , nAM T + BM ω ≤ bM , (T, ω) ∼ N , × G, ρ n ∗ where ρ = E xTi,−E (yi − π(xi,E βE )) . Since ρ is the nuisance parameters for testing H0j , j ∈ E, the conditional law above will not depend on its value. A hit-and-run algorithm for sampling this law can be implemented. Moreover, recent development by Tian et al. (2016), Harris et al. (2016) propose more general and efficient sampling schemes for this law. For details, see for example Chapter 3.2 Tian et al. (2016) where the sampling scheme for this very example is considered and simulation results are provided. In Lemma 14, we assume the covariance matrix Σ is known. In applications, we can bootstrap it. But is it valid to plug in the bootstrap estimate of Σ? Tian and Taylor/Selective inference with a randomized response 26 5.5. Plugging in variance estimates In Section 5.3 we derived quantities that were asymptotically pivotal for the best median, up to an unknown variance. In the sample median case, by (35), the variance of the sample median is approximately [4nf (m)2 ]−1 , where f (m) is the PDF evaluated at the median m. A simple consistent estimator for f (m) is to take 1/2 ± √1n quantiles an and bn , then 2 f (m) ≈ √ (41) n(bn − an ) is consistent for f (m) based on which we get a consistent estimator for σ12 . More generally, computing the pivot (22) requires knowledge of Σ. In practice, we usually do not have prior knowledge of the variance Σ and need a consistent estimate for Σ. We might use a bootstrap or jackknife estimator. When p is fixed, the bootstrap estimator is consistent and thus we get a consistent estimator Σ̂. Lemma 3 states that under moment conditions on the likelihood, Σ̂ will be consistent for Σ under F∗n as well, justifying the plug-in estimator of Σ. Figure 4 is some simulation results for the two-sample medians problem. In each case, we take the sample size for each treatment group to be 500, and generate the noise from a skewed distribution N (0, 1) + 0.5Exp(1). We standardize it such that the noise has median 0 under the null hypothesis. We use additive logistic noise with scale κ = 0.8 for randomization. The better group is decided using the randomized sample median, and selective inference is carried out. In Figure 4a, the pivot with plugin variance estimate σ b in (41) is plotted under both the null hypothesis H0 : µbetter = 0 and the HA : µbetter > √1n . The pivot has reasonable power even for identifying local alternatives. The pivot is almost exactly Unif(0, 1) under the null hypothesis with the sample size n = 500. In fact, it is very close at a relatively small sample size n = 50 justifying the application of asymptotics in the nonparametric setting. Figure 4b further illustrates the difference in the unselective v.s. selective distribution and its convergence to its theoretical limit. We see that there is a clear shift in selective distribution that calls for adjustment for the selection. For sample size n = 500, the empirical selective distribution converges to our theoretical distribution. 6. Multiple Randomizations of the Data Most of the examples above focus on a single randomization ω on the data, which we use for model selection. We naturally want to extend it to multiple randomizations, and multiple randomized selections, which will collectively suggest a model for inference. In this section, we allow multiple randomizations in a possibly sequential fashion and discuss how inference can be carried out. 27 Tian and Taylor/Selective inference with a randomized response 1.0 ECDF for the pvalues, two-sample median, n=500 null alternative Better median histogram, n=500 selected pdf median, unselected median, selected 6 5 0.8 4 0.6 3 0.4 2 0.2 0.0 0.0 1 0.2 0.4 0.6 0.8 0 0.3 1.0 0.2 0.1 0.0 0.1 0.2 median values 0.3 0.4 0.5 (a) Null and alternative pivot for the “bet- (b) Selective v.s. unselective distribution, ter” median and theoretical PDF Fig 4: Asymptotic distribution of the median for the selected group 6.1. Selective inference after cross-validation Consider the case where we first choose a regularization parameter by cross-validation, and then fit the square-root LASSO problem Belloni et al. (2011) at this parameter, β̂λ (y; X) = argmin ky − Xβk2 + λkβ\|1 , (42) β where λ is picked from a fixed grid Λ = [λ1 , . . . , λk ]. The discussion below is not specific to selection by square-root LASSO. The model selected by cross-validated square-root LASSO involves two steps of selection. We denote by yCV the response for selecting the randomization parameter, and yselect the response vector for fitting the square-root LASSO at the selected regularization parameter λ. Both vectors are randomized version of the original vector y. Inference after cross validation requires combining two steps of randomized selection. Consider the following procedure. First, we randomize y to get the vector yCV and yselect ∼ N (y, σ12 I) yinter \|y, X yCV \|yinter , y, X yselect \|yinter , y, X ∼ 2 ∼ N (yinter , σ2,CV I) (43) 2 N (yinter , σ2,select I). Note the intermediate vector yinter is introduced convenience of sampling. The above is just one of the plausible randomization schemes. After having randomized, we select λ with K-fold cross-validation using yCV : λ̂ = λ̂(yCV , X) = argminλ∈Λ CVK (yCV , X, λ) (44) where CVK (y, X, λ) is the usual K-fold cross-validation score with coefficients estimated by the square-root LASSO. Alternatively, one could compute the cross-validation score using the OLS estimators of the selected variables. Note that we have left implicit Tian and Taylor/Selective inference with a randomized response 28 the randomization that splits observations into groups. That is λ̂ in (44) above is a function of (yCV , X, ω) where ω is a random partition of {1, . . . , n} into K groups. When we sample yCV below, we redraw ω each time. The subset of variables and signs is selected using the square-root LASSO with response yselect : n o Ê(yCV , yselect , X) = j : β̂λ̂(yCV ,X),j 6= 0 , (45) zÊ (yCV , yselect , X) = sign(β̂λ̂(yCV ,X) ). After seeing the selected variables Ê, we perform inference in the selected model Msel (Ê). Since Msel (Ê), we will still have an exponential family after selection. Per Lemma 2, we sample from the following law, L XjT y λ̂(yCV , X) = λ, (Ê, zÊ ) = (E, zE ), PE\j y . The additional conditioning on the signs are for computational reasons. In fact, recent development in Harris et al. (2016) proposes sampling schemes that overcome these difficulties, so that we do not need to condition on this additional information. To sample from the above law, we use a Gibbs-type sampler, which iterate over y, yinter , yCV and yselect , conditional on the other three and the selection event. It includes the following steps. Sampling yCV Using the conditional independence of yCV and yselect given yinter , we have n o 2 L yCV yinter , yselect , y, λ̂(yCV , X) = λ = N (yinter , σ2,CV I) λ̂(yCV , X) = λ . This is the computational bottleneck, as we do not have good description for the selection event for cross validation. A brute-force sampling scheme will be computationally expensive, as we need to refit the model over a grid of λ’s. Thus, we do not update yCV too often. Sampling yselect The conditional independence of yselect and yCV given yinter implies, L yselect yinter , yCV , y, (Ê, zÊ )(yCV , yselect , X) = (E, zE ), λ̂(yCV , X) = λ n o 2 = N (yinter , σ2,select I) \| (Êλ , zÊ,λ )(yselect , X) = (E, zE ) Tian et al. (2015) has given an explicit description of the selection event {Êλ , zÊ,λ = (E, zE )}. Thus hit-and-run sampling provides a tractable sampling scheme. Sampling yinter This is a simple step. Because the selection event is based on yCV and yselect , we have L yinter y, yselect , yCV  !−1  1 y + σ21 yCV + σ2 1 yselect σ12 1 1 1 2,CV 2,select . , + 2 + 2 =N  1 1 1 σ12 σ2,CV σ2,select + + 2 2 σ σ σ2 1 2,CV 2,select 29 Tian and Taylor/Selective inference with a randomized response Sampling y This is also simple with our randomization scheme. Note that y is conditionally independent of yselect and yCV given yinter , 1 1 1 σ 2 XE βE + σ12 yinter , 1 1 2 σ + 2 2 σ σ1 L y yinter , yCV , yselect = L y yinter = N 1 + 2 σ1 −1 ! Since we condition on PE\j y, we essentially take y and project out the update on the space orthogonal to that of Xj . A chain that iterates through the above four steps will give us samples from the desired distribution for inference. 6.2. Collaborative selective inference One of the motivations of the reusable holdout described in Dwork et al. (2014) is that it allows a data analyst to repeatedly query a database yet still be able to approximately estimate expectations even after asking many questions about the data. Another version of this model may be that several groups wish to model the same data and then, as a consortium, decide on a final model and be able to approximately estimate expectations in this final model. We might call this collaborative selective inference. Formally, suppose each of L groups has its own preferred method of model selecbl )1≤l≤L . We assume there is a central “data” tion, encoded as selection procedures (Q bank that decides what “data” each group is allowed to see. We express this is as a sequence of randomization schemes (yl∗ )1≤l≤L . Formally, this is equivalent to enlarging the probability space to D × B with measure F × B and fixing a function y ∗ (y, ω) = (y1∗ (y, ω), . . . , yl∗ (y, ω)). It may be desirable to choose the law of y ∗ \|y so that the coordinates are conditionally independent given y, though it is not necessary. bl (y ∗ ) ∈ σ(y ∗ ) and convene Now suppose that the L groups choose models M̂l∗ = Q l l to discuss what the best model is M . For every choice of L models (M1 , . . . , ML ) and final model M , the following selective distribution can be used for valid selective inference b ∗ dF∗ B(ω : ∩L l=1 Ql (yl (y, ω)) = Ml ) (y) = . (46) b ∗ = Ml ) dF (F × B)(∩L Q l=1 l When the yl∗ ’s are conditionally independent given y then it is clear that b ∗ B(∩L l=1 Ql (y (y, ω)) = Ml ) = L Y l=1 bl (y ∗ (y, ω)) = Ml ). B(Q l It is possible that the consortium has beforehand decided on an algorithm that will choose a best model automatically, determined by some function S(M1 , . . . , ML ). In this case, one should use the selective distribution dF∗ B(ω : S(M1∗ (y, ω), . . . , ML∗ (y, ω)) = M ) (y) = dF (F × B)(S(M1∗ , . . . , ML∗ ) = M ) (47) Tian and Taylor/Selective inference with a randomized response 30 When the models in question are parametric, perhaps Gaussian distributions, and the randomization is additive Gaussian noise the central data bank can explicitly lower bound the leftover information by ∗ Var(y\|y1∗ , . . . , yL ). This quantity is expressible in terms of the marginal variance of y and the central data bank’s noise generating distribution for y ∗ (y, ω) = (y + ω1 , . . . , y + ωL ). By maintaining a lower bound on the above quantity, the central data bank can maintain a minimum prescribed information in the data for final estimation and/or inference. In a sequential setting, where valid inference is desired at each step, maintaining a lower bound may involve releasing noisier and noisier versions of y. Sampling under this scheme seems quite difficult, and we leave it as an area of interesting future research. 7. Proof 7.1. Proof of Theorem 9 To prove Theorem 9, we first prove the following lemma, which might be of independent interest. Lemma 15.√Suppose Tn is a linearizable statistic for µn = µ(Fn ) as defined in (21). Let Zn = n(Tn − µn ) ∈ Rp and a function f : Rp → R with finite λΩ 3 (f ) for some norm Ω on Rp . Moreover, if Fn has finite centered exponential moments in a neighbourhood of zero. Then 1 −2 \|EFn [f (Zn )] − EΦn [f (Zn )] \| ≤ C(p)λΩ , 3 (f )n n ≥ n0 for some n0 ≥ 1, where C(p) is a constant only dependent on the dimension. Lemma 15 can be seen as an extension of the result by Chatterjee (2005) in the sense that the author in Chatterjee (2005) established result for the case Ω = 0. The proof is also an adaptation of the technique in Chatterjee (2005). Pn Proof. Without loss of generality, we assume Tn = n1 i=1 ξi,n and the residual is 0. First, we define the normalizing operator. For any S ∈ Rn×p , n 1 X N (S) = √ S[i] ∈ Rp , n i=1 where S[i] is the i-th row of S. We also define for any n and 0 ≤ k ≤ n,  0   0  ξ1,n − µ(Fn ), ξ1,n − µ(Fn ), ..     ..     ., .,  0   0  ξk−1,n − µ(Fn ), ξ  − µ(F ), n  k−1,n  0   −     Sn,k =  ξk,n − µ(Fn ),  Sn,k =  0  ξk+1,n − µ(Fn ), ξk+1,n − µ(Fn ),         .. ..     ., ., ξn,n − µ(Fn ) ξn,n − µ(Fn ) Tian and Taylor/Selective inference with a randomized response iid 31 iid 0 where ξi,n ∼ Fn with mean µ(Fn ) and variance Σ(Fn ) and ξi,n ∼ N (µ(Fn ), Σ(Fn )). − − Let Fn,k . Fn,k denote the distribution of Sn,k and Sn,k ’s respectively. Note Fn,k and − Fn,k are determined by Fn . For simplicity of notation we only distinguish the two − distributions by Fn,k and Fn,k , avoiding verbose notations of S, e.g. EFn,k [S] = EFn,k [Sn,k ]. It is then easy to see Zn = N (Sn,0 ). Now by telescoping: EFn [f ] − EΦn [f ] = EFn,0 [f ◦ N (S)] − EFn,n [f ◦ N (S)] n X ≤ EFn,i−1 [f ◦ N (S)] − EF − [f ◦ N (S)] + EF − [f ◦ N (S)] − EFn,i [f ◦ N (S)] . n,i n,i i=1 Let ∂i be the derivative with respect to the i-th row S[i]. Using Taylor’s expansion − at Sn,k , we have EFn,i [f ◦ N ] − EF − [f ◦ N ] n,i i 1 h 1 T = √ EF − [∂i f ◦ N (S)] 0 + Tr EF − ∂i2 f ◦ N (S) · Σ(Fn ) + Rn,i n,i n n n,i where the precise form of the Taylor remainder Rn,i depends on realizing the laws Fn,i − and Fn,i on the same probability space. In order to not introduce new notation, we have avoided explicitly writing out this construction, directing readers to Chatterjee (2005) for details. Nevertheless, 1 Ω − 23 0 0 3 √ \|Rn,i \| ≤ c1 (p)[λ3 (f ) · n ]EF − exp Ω(N (S)) + Ω(ξi,n ) kξi,n k1 n,i n 0 where ξi,n are centered version of ξi,n and c1 is some dimension dependent constant. Let C(Ω) be the constant s.t Ω(·) ≤ C(Ω)k · k1 , C(Ω) only depends on the dimension p. Thus, using the independence of the ξi,n ’s, 1 0 0 3 EF − exp Ω(N (S)) + √ Ω(ξi,n ) kξi,n k1 n,i n !# " 0 C(Ω)kξi,n k1 0 3 √ . ≤EF − [exp (C(Ω)kN (S)k1 )] · EFn kξi,n k1 exp n,i n Now we bound these two expectations. By the exponential moment condition (29) and Lemma 17, it is easy to conclude the first term is bounded by lim sup EFn [exp (C(Ω)kZn k1 )] ≤ c2 (p). n 0 The second expectation is bounded by γ, an upper bound on the third moment of ξi,n , " 0 lim sup EFn kξi,n k31 exp n 0 C(Ω)kξi,n k1 √ n !# ≤ γ, 32 Tian and Taylor/Selective inference with a randomized response Thus it is not hard to see 3 −2 \|Rn,i \| ≤ c1 (p)c2 (p)γλΩ . 3 (f )n Notice the first and second order terms in EFn,i [f ◦ N ] − EF − [f ◦ N ] cancel with n,i those in EFn,i−1 [f ◦ N ] − EF − [f ◦ N ], and therefore we have, n,i EFn [f ] − EΦn [f ] ≤ n X (Rn,i + R̃n,i ) i=1 where R̃n,i is the remainder of EFn,i−1 [f ◦ N ] − EF − [f ◦ N ]. With a similar argument n,i 3 −2 , \|R̃n,i \| ≤ c1 (p)c2 (p)γλΩ 3 (f )n and summing over n terms, we have the conclusion of the lemma. Now we prove the main theorem, Theorem 9. Proof. First, notice that per Lemma 7, we have EΦ∗n [g ◦ P (Tn )] = the selective likelihood ratio, it is easy to see, R1 0 g(x)dx. Using EF∗n [g(P (Tn ))] = EFn [g(P (Tn ))`Fn (Tn )] The same equation holds for Φn = Φ(Fn ), thus we have EF∗n [g(P (Tn ))] − EΦ∗n [g(P (Tn ))] ≤ \|EFn [g(P (Tn ))`Φn (Tn )] − EΦn [g(P (Tn ))`Φn (Tn )]\| + (48) \|EFn [g(P (Tn ))`Fn (Tn )] − EFn [g(P (Tn ))`Φn (Tn )]\| We need to bound both terms. Recall the notation P n and `¯Fn for the normalized statistic Zn . If we let f = g(P̄n ) · `¯Φn , then per Lemma 15, we have \|EFn [g(P̄n ) · `¯Φn ] − EΦn [g(P̄n ) · `¯Φn ]\| ≤ 2C(p) · C1 n−1/2 , where we use the bound in condition (28). Now we replace `Φn with `Fn in the second term. With some algebra, we can bound it by h i b∗ (Tn , ω) PFn ×Q M ∈ Q h i , EFn [g(P (Tn ))`Fn (Tn )] 1 − b∗ (Tn , ω) PΦn ×Q M ∈ Q which in turn is bounded by C(g) b∗ ] − P(Φ ×Q) [M ∈ Q b∗ ] P(Fn ×Q) [M ∈ Q n ≤ C(g)C3 n−1/2 , b∗ ] P(Φ ×Q) [M ∈ Q n per condition (30) and C(g) is a bound on g. Tian and Taylor/Selective inference with a randomized response 33 7.2. Proof of Lemma 7 Proof. Let φµ, n1 Σ denote the density for N (µ, n1 Σ) and T = 1 T ση2 Ση·(η T )+Vη , we see ∗ that Q(η T T, Vη ) is W(M ; T ) in (13). Thus under the selective law F , the distribution of T has density proportional to φµ, n1 Σ (T ) · Q(η T T, Vη ). Since η T T ⊥ Vη under F, we can factorize φµ, n1 Σ (T ) into the product of densities of η T T and Vη . Thus conditioning on Vη , the density of η T T is proportional to n(η T T − η T µ)2 exp − · Q(η T T, Vη ). 2ση2 Therefore, the pivot in (22) is the survival function of η T T under F∗ and is distributed as Unif(0, 1). Moreover, we note the distribution does not depend on the conditioned value of Vη , thus P (T ; η T µ, Σ) in (22) is Unif(0, 1). References Bahadur, R. R. (1966), ‘A note on quantiles in large samples’, The Annals of Mathematical Statistics 37(3), 577–580. Belloni, A., Chernozhukov, V. & Wang, L. 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(2009), ‘p-values for high-dimensional regression’, Journal of the American Statistical Association 104(488), 1671–1681. URL: http://amstat.tandfonline.com/doi/abs/10.1198/jasa.2009.tm08647 Pakman, A. & Paninski, L. (2012), ‘Exact hamiltonian monte carlo for truncated multivariate gaussians’, arXiv:1208.4118 [stat] . arXiv: 1208.4118. URL: http://arxiv.org/abs/1208.4118 Rosenthal, R. (1979), ‘The file drawer problem and tolerance for null results.’, Psychological bulletin 86(3), 638. Taylor, J., Lockhart, R., Tibshirani, R. J. & Tibshirani, R. (2014), ‘Post-selection adaptive inference for least angle regression and the lasso’, arXiv:1401.3889 [stat] . URL: http://arxiv.org/abs/1401.3889 Tian, X., Bi, N. & Taylor, J. (2016), ‘Magic: a general, powerful and tractable method for selective inference’, arXiv preprint arXiv:1607.02630 . Tian, X., Loftus, J. R. & Taylor, J. E. (2015), ‘Selective inference with unknown variance via the square-root lasso’, arXiv preprint arXiv:1504.08031 . Tian, X. & Taylor, J. (2015), ‘Asymptotics of selective inference’, arXiv preprint arXiv:1501.03588 . Tibshirani, R. (1996), ‘Regression shrinkage and selection via the lasso’, Journal of the Royal Statistical Society: Series B 58(1), 267–288. Tibshirani, R. J., Rinaldo, A., Tibshirani, R. & Wasserman, L. (2015), ‘Uniform asymptotic inference and the bootstrap after model selection’, arXiv preprint arXiv:1506.06266 . Tukey, J. W. (1980), ‘We need both exploratory and confirmatory’, The American Statistician 34(1), 23–25. Tian and Taylor/Selective inference with a randomized response 35 Appendix A: Proof of Lemma 1 √ Proof. First we normalize the sample mean as Z = n(X̄n + 0.5) and rewrite the pivot as 1√ 1 − Φ(Z) n, Pe(Z) = √ , Z >2+ 2 1 − Φ(2 + 21 n) and Φ is the CDF of the standard normal distribution. As n → √ ∞, we can use Mills ratio to approximate the normal tail. Specifically, denote bn = 12 n + 2, 1 − Φ(Z) bn 1 ≈ exp (bn + Z)(bn − Z) 1 − Φ(bn ) Z 2 (49) bn 1 = exp − (bn − Z)2 exp [−bn (Z − bn )] , Z > bn . Z 2 We study the behavior of −bn (Z − bn ) for Z > bn . By studying its distribution, we p will also see that Z − bn → 0, for Z > bn , thus the term 1 bn p exp − (bn − Z)2 → 1, as n → ∞. Rn = Z 2 Now we study the distribution of bn (Z − bn ) conditioning on Z > bn . Since X̄n is a translation of a binomial distribution divided by n, we can rewrite Z in terms of a Binomial distribution, which will be useful for calculating the conditional distribution of bn (Z − bn ). Specifically, Z= 2Sn − n √ , n 1 Sn ∼ Bin(n, ). 2 Thus for t ≥ 0, 2Sn − n √ − bn P (bn (Z − bn ) > t) = P bn n √ √ 3 t n =P Sn > n + n + = 4 2bn √ √ t n 1 4 n− n− 2bn X i=0 n 1 · n i 2 To study the conditional distribution P (bn (Z − bn ) > t \| Z > bn ), we essentially need to study the ratio of two partial sums of binomial coefficients. Note for any n, m ∈ Z+ , n > m we have n m m−1 = n n − m+1 m Noticing that k n−k+1 ≤ m n−m+1 , for any k ≤ m, thus Pm−1 n i Pi=0 ≤ m n i=0 i m . n−m+1 36 Tian and Taylor/Selective inference with a randomized response Now let m = 14 n − √ n, and use the above inequality j = Pm−j Pi=0 m n i n i=0 i ≤ m n−m+1 √ t n 2bn times, we have j . Therefore we have, P (bn (Z − bn ) > t \| Z > bn ) = ≤ P (bn (Z − bn ) > t) P (bn (Z − bn ) > 0) √ 1+ t√4 − n n → exp[− log(3)t] √ 3 n + n + 1 4 1 4n (50) p We can draw two conclusions from (50). First, conditional on Z > bn , Z − bn → 0, which implies the first term in the pivot approximation (49) Rn → 1. Moreover, (50) shows that the overshoot bn (Z − bn ) is not Exp(1) distributed in the limit. In fact, we can conclude its limit (if existed) is strictly stochastically dominated by an Exp(1). Thus, exp [−bn (Z − bn )] 6→ Unif(0, 1), and hence the pivot does not converge to Unif(0, 1). Appendix B: Proof of Lemma 14 Proof. We first prove that T is in fact a linearizable statistic. Since β̄E is the restricted MLE, we see that 1 T XE y − πE (β̄E ) n 1 T ∗ ∗ [y − πE (βE )] + Q(βE − β̄E ) + R1 , = XE n 0= ∗ ∗ − β̄E ) + R̃1 , where R̃1 = op (n−1/2 ) is the residual ) − Q)(βE where R1 = (QE (βE ∗ ∗ ) from its from the Taylor’s expansion at βE . R1 = op (n−1/2 ) since deviations QE (βE −1/2 ∗ asymptotic mean should be Op (n ) and β̄E − βE = op (1). Thus, we can deduce β̄E = 1 −1 T ∗ ∗ Q XE [y − πE (βE )] + βE + Q−1 R1 . n Similarly, 1 T 1 T 1 ∗ T ∗ X−E y − πE (XE β̄E ) = X−E [y−πE (βE )]− DXE [y−πE (βE )]+op (n−1/2 ) n n n Thus we can conclude that T is a linearizable statistic with T ∗ QXi,E (yi − π(xi,E βE )) ξi,n = . T ∗ ∗ Xi,−E (yi − π(xi,E βE )) − DXi,E (yi − π(xi,E βE )) Tian and Taylor/Selective inference with a randomized response 37 Now we rewrite the selection event in terms of (T, ω). Using the KKT conditions of (38), √ β̂E T X (y − πE (β̂E )) = n(ω + Λz) + 0 (51) sE β̂E ≥ 0, ku−E k∞ < 1, , sE = sign(β̂E ) and u−E is the subgradient for the inactive sE u−E variables. Using a Taylor expansion on the β̂E as well, we see that 1 1 T 0 Q X [y − πE (β̂E )] = + (β̄E − β̂E ) + op (n−1/2 ). T C X [y − π ( β̄ )] n n E E −E where z = Plugging in the equalities in the KKT conditions, we will have, 1 β̂E = β̄E − √ Q−1 (ωE + ΛE zE ) + op (n−1/2 ) n 1 T 1 T X [y − πE (β̂E )] = X−E [y − πE (β̄E )] + C(β̄E − β̂E ) + op (n−1/2 ) n −E n 1 1 T [y − πE (β̄E )] + √ D(ωE + ΛE zE ) + op (n−1/2 ) = X−E n n Using the inequalities in the KKT conditions, we have the selection event is {AM T + BM ω ≤ bM } with AM , BM and bM defined in the lemma. Appendix C: Proofs related to Logistic noise Throughout the article, logistic noise has played an important role in all the examples. The following lemma on the tail behavior of the logistic distribution is crucial to all the proofs with added logistic noise. Let G be the CDF of Logistic(κ), with κ being the scale parameter. g is the PDF of G. G(w) = eκw , 1 + eκw g(w) = κe−κ\|w\| 1 + e−κ\|w\| 2 . Lemma 16. The following lower bounds hold, Ḡ(κw) ≥ 1 −κ\|w\| e , 2 g(w) ≥ 1 −κ\|w\| e . 4 (52) For k = 0, 1, 2, 3, . . . : ∂k ek e−κ\|w\| . g(w) ≤ κk+1 C ∂wk ek ’s are universal constants. where C (53) Tian and Taylor/Selective inference with a randomized response 38 Proof. We can write ( g(w) = κe−κw h0 (e−κw ) w > 0 κeκw h0 (eκw ) w≤0 where h0 (x) = (1 + x)−2 . For j ≥ 1, define hj (x) = x · h0j−1 (x). By induction, I claim that for each j, hj is rational such that the polynomial in the numerator is of order 2 less than the denominator, and the denominator polynomial is bounded below by 1. Hence, hj ’s are bounded on the interval [0, 1]. Now, it is not hard to see that ( Pk −(−κ)k+1 j=0 cj,k hj (e−κw )e−κw w > 0 ∂k g(w) = P k ∂wk κk+1 j=0 cj,k hj (eκw )eκw w≤0 for universal cj,k ’s and k = 0, 1, 2, . . . . Now we state the following lemmas which are foundations of the proofs of various lemmas in the article. Lemma 17. Assume Tn is a decomposable statistic and ξi,n has mean 0, variance σ 2 , and centered exponential moments in a neighbourhood of zero, i.e satisfies condition √ (29). Denote Zn = n(Tn − µn ), then 2 2 κ σ E [exp (κZn )] → exp , for κ > 0. 2 √ Lemma 18. In Example 2, if we normalize the sample mean Zn = n(X̄n − µn ), we can rewrite the selective likelihood ratio and the pivot as √ Ḡ(2 − Zn − nµn ) , `¯Fn (Zn ) = √ EFn Ḡ(2 − Zn − nµn ) and R∞ P (Zn ) = n RZ ∞ −∞ √ nµn ) exp(−t2 /2) dt . √ Ḡ(2 − t − nµn ) exp(−t2 /2) dt Ḡ(2 − t − Then for any Fn with finite centered exponential moment in a neighbourhood of zero, we have for k = 0, 1, 2, 3 ∂k ¯ `F (Z) ≤ C1 exp[κ\|Z\|], ∂Z k n ∂k P (Z) ≤ C1 . ∂Z k (54) for some C1 only depending on κ. Proof of Lemma 17. Pn Proof. Without loss of generality, we assume Tn = n1 i=1 ξi,n . Since Fn has centered exponential moments in a neighbourhood of zero, it is each to see n κ E [exp (κZn )] = M √ n Tian and Taylor/Selective inference with a randomized response 39 exists as long as √κn < a. M (·) is the moment generating function of ξi,n − µn , M (t) = EFn [exp(t(ξi,n − µ))]. Therefore, √1 t= n κ log [M (κt)] lim n log M √ = lim n→∞ t→0 t2 n κ2 M 00 (κt) κ2 σ 2 = lim = . t→0 2M (κt) 2 To derive the equality, we used M 00 (0) = Var [ξi,n ] = σ 2 and M 0 (0) = 0, M (0) = 1. Proof of Lemma 18. Proof. Noticing the lower bound in (52), we have i 1 h i √ √ 1 h √ E Ḡ(2 − Zn − nµn ) ≥ E e(−κ\|2−Zn − nµn \|) ≥ E e(−κ\|2−Zn \|−κ n\|µn \|) 2 2 On the other hand, using the upper bounds in (53), we have for k = 1, 2, 3, √ ek−1 eκ\|2−Z\| ek−1 e−κ\|2−Z− nµn \| κk C ∂k ¯ κk C h i ≤ 2 ` (Z) ≤ 2 √ F n ∂Z k E e(−κ\|2−Z\|) E e(−κ\|2−Z\|−κ n\|µn \|) Since x−1 is convex on the positive axis, it is hard to see h i 1 ≤ E e(κ\|2−Z\|) ≤ e2κ E eκZ + e−κZ . (−κ\|2−Z\|) E e Thus using Lemma 17, we know E e±κZ → exp(κ2 /2). Thus, we conclude sup n ∂k ¯ `F (Z) ≤ C1 exp[κ \|Z\|], ∂Z k n k = 1, 2, 3. (55) √ To verify the above inequality for k = 0. Notice that for µn ≥ 0, Ḡ(2 − Z − nµn ) ≤ Ḡ(2 − Z). Thus the denominator of `¯Fn (Z) is bounded below using the argument above. For µ < 0, √ √ √ Ḡ(2 − Z − nµn ) ≤ exp(−κ(2 − Z − nµn ) ≤ exp(−κ n \|µn \| + κ \|2 − Z\|). √ The term exp(−κ n \|µn \|) cancels with the one in the denominator, thus (55) holds for k = 0 as well. Analogously, similar bounds can be derived for the derivatives of P (Z) as well, thus we have the conclusion of the lemma. C.1. Proof of Lemma 4 and Lemma 10 . The proof of Lemma 4 is a simple application of Lemma 17 and Lemma 18. Tian and Taylor/Selective inference with a randomized response 40 Proof. By law of large numbers, we know that X̄n is consistent for µ unselectively. Thus, using the result by Lemma 3, we only need to verify that the selective likelihood is integrable in Lq . For simplicity, we take q = 2. First notice from (54) that the selective likelihood ratio is bounded by a multiple of exp[κ\|Z\|]. Then by Lemma 17, lim sup EFn `Fn (X̄)2 ≤ 2C12 exp(2κ2 ). n The proof of Lemma 10 uses results in Lemma 18 and Lemma 15 Proof. It follows simply from (54) that condition (28) are satisfied with the norm function Ω simply being the absolute value function. Therefore, we only need to verify (30). Note for µn = µ < 0 √ √ PFn ×Q [ nXn + ω > 2] − PΦ×Q [ nXn + ω > 2] √ PΦ×Q [ nXn + ω > 2] √ √ EFn Ḡ(2 − Z − nµ) − EΦ Ḡ(2 − Z − nµ) = √ EΦ Ḡ(2 − Z − nµ) √ √ √ exp(−κ nµ)EFn Ḡ(2 − Z − nµ) − EΦ Ḡ(2 − Z − nµ) ≤2 EΦ [exp(−κ\|2 − Z\|)] √ √ ≤2EΦ [exp(2κ + 2\|Z\|)] · C exp(−κ nµ) exp(κ nµ)n−1/2 ≤2C exp(κ2 )n−1/2 , √ Ω The second to last inequality uses Lemma 15 √ and the fact that λ3 (Ḡ) ≤ exp(κ nµ). For µn = µ ≥ 0, the denominator PΦ×Q [ nXn + ω > 2] is bounded below, and Ḡ has bounded derivatives. Therefore, a simple application of the Berry-Esseen Theorem will suffice. Appendix D: Proofs related to affine selection regions D.1. Proof of Lemma 12 The quantity that appears in both the pivot and the selective likelihood ratio is Z Q(z; ∆) = P(A(z + ∆) + ω ∈ K) = G(dw), K−A(z+∆) where ω ∼ G. The associated selective likelihood in terms of z is `F (z; ∆) = R Q(z; ∆) . Q(t; ∆)F(dt) Rq We first rewrite the pivot in terms of U . R∞ Q(t; Lz, ∆) exp(−t2 /2ση2 ) dt ηT z P (z; ∆) = R ∞ , Q(t; Lz, ∆) exp(−t2 /2ση2 ) dt −∞ (56) Tian and Taylor/Selective inference with a randomized response 41 where we use the slight abuse of notation for the one dimensional function Q(t; Lz, ∆) 1 Σηη T ση2 1 Q(t; Lu, ∆) = P(t · 2 AΣη + ALu + A∆ + w ∈ K) ση Z = G(dw). L=I− K−t· σ12 AΣη−ALu−A∆ η We first establish a lower bound on EF [Q(Z; ∆)] = PFn ×Q [A(Z + ∆) + ω ∈ K] under the local alternatives. Lemma 19. If we assume the lower bound condition, then under the local alternatives with radius B, i.e. dh (0, K − A∆) ≤ B, we have Z Q(u; ∆)F(du) ≥ C − C(Φ, h) · e−B , Rp where C(Φ, h) is a constant only depending on the normal distribution Φ = N (0, Σ) and the norm h in the local alternatives condition. Proof. We first see that the lower bound condition gives the following lower bound. Z − Q(u; ∆) = G(dw) ≥ C exp − inf h(w) . (57) w∈K−A(u+∆) K−A(u+∆) Consider Z Z Q(u; ∆)F(du) ≥ C − exp − inf h(w) F(du) w∈K−A(∆+u) Rp Rp Z = C− exp − inf h(w − Au) F(du) w∈K−A∆ Rp Z ≥ C− exp − inf h(w) + h(−Au) F(du) w∈K−A∆ Rp Z = C − · exp − inf h(w) e−h(Au) F(du) w∈K−A∆ Rp Z − =C e−h(Au) F(du) · e−B . Rp Finally, since the exp(−h(Au)) has uniformly bounded derivatives up to the third order, we have Z Z exp(−h(Au))F(du) → exp(−h(Au))Φ(du) Rp Rp as Z → N (0, Σ) in distribution. Let C(Φ, h) = have the conclusion of the lemma. R Rp exp(−h(Au))Φ(du), and we will Tian and Taylor/Selective inference with a randomized response 42 The following lemmas establish the bounds on the derivatives for the likelihood function `F and the pivot P (u; ∆). Lemma 12 is easily obtained using Lemma 20 and Lemma 21 below. Lemma 20. Suppose the smoothness and the lower bound conditions are satisfied, then for local alternatives with radius B, ∂α `F (z; ∆) ≤ C ∗ (B, Φ, h), ∂z α Φ = N (0, Σ). (58) Proof. The smoothness condition implies the following upper bound. For a multi-index α, we have Z ∂α ∂α Q(z; ∆) = α g(w)dw ∂z α ∂z K−A(∆+z) Z ∂α g(w − Az)dw. = α K−A∆ ∂z Therefore, from the smoothness condition, ∂α def Q(z; ∆) ≤ C(A)Cα = Cα+ (A). ∂z α (59) This combined with Lemma 19 gives the conclusion of the lemma. Next, we derive the exponential bounds on the derivatives of the pivot P (z; ∆) with respect to z. Lemma 21. Assuming the conditions of Lemma 12, for a multi-index α up to the order of 3, ∂α P (z; ∆) ≤ Cα∗ · eα·Lip(h)kALzk2 , ∂z α where the norm on the left is the element-wise maximum and C is independent of (z, ∆) and Lip(h) is the Lipschitz constant of h with respect to `2 norm. Proof. To get a lower bound on the denominator, note (57) 1 − Q(t; Lz, ∆) ≥ C exp − inf h(w − t · 2 AΣη) w∈K−ALz−A∆ ση ≥ C − exp − inf h(w) − \|t\|h(AΣη/ση2 ) . w∈K−ALz−A∆ Therefore, the denominator will be lower bounded by Z ∞ 1 √ Q(t; Lz, ∆) exp(−t2 /2ση2 )dt 2π −∞ − 2 2 2 ≥C exp h(AΣη/ση ) ση /2 · exp − inf h(w) w∈K−ALz−A∆ ≥C − exp h(AΣη/ση2 )2 ση2 /2 e−B · e−h(ALz) . 43 Tian and Taylor/Selective inference with a randomized response On the other hand, the upper bound (59) ensures, Z ∂α 1 √ Q(t; Lz, ∆) exp(−t2 /2ση2 )dt ≤ Cα+ (A). 2π R ∂z α Note the derivatives of the pivot will be a polynomial in terms of the form, R∞ ∂ Q(t; Lz, ∆) exp(−t2 /2ση2 )dt ηT z α R∞ Q(t; Lz, ∆) exp(−t2 /2ση2 )dt −∞ and therefore, it is easy to get the conclusion of the lemma. D.2. Proof of Lemma 13 Using Lemma 19 and the following lemma, we can easily prove Lemma 13. √ Lemma 22. Let Zn = n(Tn − µn ) ∈ Rp , and Fn has finite third moments γ. Moreover, suppose the randomization noise ω ∈ Q, a probability measure on Rd . Then for any sequence of sets (Un )n≥1 , Un ⊆ Rp × Rd , we have 1 \|PFn ×Q [(Zn , ω) ∈ Un ] − PΦn ×Q [(Zn , ω) ∈ Un ]\| ≤ C3 γn− 2 , where Φn = N (µ(Fn ), Σ(Fn )) and C3 is a constant depending only on p. Proof of Lemma 22 uses the well known results of Berry-Esseen Theorem. A multivariate extension can be found in Gotze (1991). Proof. For each ω, we denote Un (ω) = {Z ∈ Rp : (Z, ω) ∈ Un } ⊆ Rp . Thus the difference in the two probabilities is \|PFn ×Q [(Z, ω) ∈ Un ] − PΦn ×Q [(Z, ω) ∈ Un ]\| ≤EQ [\|PFn [Z ∈ Un (ω)] − PΦ [Z ∈ Un (ω)]\|] ≤ sup \|Fn (U ) − Φn (U )\| < C3 γn−1/2 , U ∈Rp where C3 only depends on the dimension p. The last inequality is a direct application of equation (1.5) in Gotze (1991).	10
arXiv:1204.2394v5 [math.AC] 26 Feb 2013 ON INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS OF LOCAL COHOMOLOGY MODULES MAJID RAHRO ZARGAR AND HOSSEIN ZAKERI Abstract. Let (R, m) be a commutative Noetherian local ring and let M be an Rmodule which is a relative Cohen-Macaulay with respect to a proper ideal a of R and set n := ht M a. We prove that injdimM < ∞ if and only if injdimHn a (M ) < ∞ and that injdimHn a (M ) = injdimM − n. We also prove that if R has a dualizing complex n and Gid R M < ∞, then Gid R Hn a (M ) < ∞ and Gid R Ha (M ) = Gid R M − n. More- over if R and M are Cohen-Macaulay, then it is proved that Gid R M < ∞ whenever Gid R Hn a (M ) < ∞. Next, for a finitely generated R-module M of dimension d, it is proved d d that if KM c is Cohen-Macaulay and Gid R Hm (M ) < ∞, then Gid R Hm (M ) = depth R−d. The above results have consequences which improve some known results and provide characterizations of Gorenstein rings. 1. introduction Throughout this paper, R is a commutative Noetherian ring, a is a proper ideal of R and M is an R-module. For a prime ideal p of R, the residue class field Rp /pRp is denoted by k(p). For each non-negative integer i, let Hia (M ) denotes the i-th local cohomology module of M with respect to a; see [1] for its definition and basic results. Also, we use injdimR M to denote the usual injective dimension of M . The notion of Gorenstein injective module was introduced by E.E. Enochs and O.M.G. Jenda in [4]. The class of Gorenstein injective modules is greater than the class of injective modules; but they are same classes whenever R is a regular local ring. The Gorenstein injective dimension of M , which is denoted by Gid R M , is defined in terms of resolutions of Gorenstein injective modules. This notion has been used in [3, 15, 21] and has led to some interesting results. Notice that Gid R M ≤ injdimR M and the equality holds if injdimR M < ∞. The principal aim of this paper is to study the injective (resp. Gorenstein injective) dimension of certain R-modules in terms of injective (resp. Gorenstein injective) dimension of its local cohomology modules at support a. In this paper we will use the concept of relative Cohen-Macaulay modules which has been studied in [7] under the title of cohomologically complete intersections. The organization of this paper is as follows. In section 2, among other things, we prove, in 2.5, that if M is relative Cohen-Macaulay with respect to a, then injdimM and injdimHht M a (M ) are a simultaneously finite and there is an equality injdimHaht M a (M ) = injdimM − ht M a. Then, 2000 Mathematics Subject Classification. 13D05, 13D45. Key words and phrases. Injective dimension, Gorenstein injective dimension, Local cohomology, Gorenstein ring, Relative Cohen-Macaulay module. 1 2 M.R. ZARGAR AND H. ZAKERI as a corollary, we obtain a characterization of Gorenstein rings. Next, in 2.10, for all n ≥ 0 and any p ∈ Supp (M ), we establish a comparison between the Bass numbers of HnpRp (Mp ) dim (R/p) (M ) whenever (R, m) is a homomorphic image of a Gorenstein ring and and Hn+ m M is finitely generated. In section 3, we first prove some basic properties about Gorenstein injective dimension of a module. In particular, Proposition 3.6 indicates that Gorenstein injective dimension is a refinement of the injective dimension. As a main result, in Theorem 3.8 we establish a Gorenstein injective version of 2.5. Indeed, it is proved that if, in addition to the hypothesis of 2.5, R has a dualizing complex, then Gid R M < ∞ implies Gid R Hna (M ) < ∞ and the converse holds whenever R and M are Cohen-Macaulay. This theorem has consequences which recover some interesting results that have currently been appeared in the literature. As a first corollary of 3.8, we deduce that Gid R Hnm (M ) = Gid R M − n, wherever M is a Cohen-Macaulay module over the Cohen-Macaulay local ring (R, m) and dim M = n. This corollary improves the main result [15, Theorem 3.10](see the explanation which is offered before 3.9). As a second corollary, we obtain a characterization of Gorenstein local rings which recovers [21, Theorem 2.6]. As a main result, it has been shown in [15, Theorem 3.10] that if R and M are Cohen-Macaulay with dim M = d and Gid R Hdm (M ) < ∞, then Gid R Hdm (M ) = dim R − d. In 3.12, we will use the canonical module of a module to improve the above result without Cohen-Macaulay assumption on R and M . This result provides some characterizations of Gorenstein local rings. 2. local cohomology and injective dimension The starting point of this section is the next proposition, which plays essential role in the proof of Theorems 2.5 and 3.8. Proposition 2.1. Let n be a non-negative integer and let N be an a-torsion R-module. Suppose that Hia (M ) = 0 for all i 6= n. Then Ext iR (N, Hna (M )) ∼ = Ext i+n R (N, M ) for all i ≥ 0. Proof. First we notice that Hom R (N, M ) = Hom R (N, Γa (M )). Hence, in view of [14, Theorem 10.47], we have the Grothendieck third quadrant spectral sequence with E2p,q = Ext pR (N, Hqa (M )) =⇒ Ext p+q R (N, M ). p Now, since Hqa (M ) = 0 for all q 6= n, E2p,q = 0 for all q 6= n. Therefore, this spectral sequence collapses in the column q = n; and hence one gets, for all i ≥ 0, the isomorphism Ext iR (N, Hna (M )) ∼ = Ext i+n R (N, M ), as required. The following corollary, which is an immediate consequence of 2.1, determines the Bass numbers µi (p, Hna (M )) := vdim k(p) Ext iRp (k(p), HnaRp (Mp )) of the local cohomology module Hna (M ). INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 3 Corollary 2.2. Let n and M be as in 2.1. Then, for all p ∈ V(a), µi (p, Hna (M )) = µi+n (p, M ) for each i ≥ 0. Definition 2.3. We say that a finitely generated R-module M is relative Cohen Macaulay with respect to a if there is precisely one non-vanishing local cohomology module of M with respect to a. Clearly this is the case if and only if grade (a, M ) = cd (a, M ), where cd (a, M ) is the largest integer i for which Hia (M ) 6= 0 and grade (a, M ) is the least integer i such that Hia (M ) 6= 0. Observe that the above definition provides a generalization of the concept of CohenMacaulay modules. Also, notice that the notion of relative Cohen-Macaulay modules is connected with the notion of cohomologically complete intersection ideals which has been studied in [7] and has led to some interesting results. Furthermore, such modules have been studied in [8] over certain rings. Remark 2.4. Let M be a relative Cohen-Macaulay module with respect to a and let cd (a, M ) = n. Then, in view of [1, Theorems 6.1.4, 4.2.1, 4.3.2], it is easy to see that Supp Hna (M ) = Supp (M/aM ) and ht M a = grade (a, M ), where ht M a = inf{ dim Rp Mp \| p ∈ Supp (M/aM ) }. The following theorem, which is one of the main results of this section, provides a comparison between the injective dimensions of a relative Cohen-Macaulay module and its non-zero local cohomology module. Here we adopt the convention that the injective dimension of the zero module is to be taken as −∞. Theorem 2.5. Let (R, m) be local and let n be a non-negative integer such that Hia (M ) = 0 for all i 6= n. (i) If injdimM < ∞, then injdimHna (M ) < ∞. (ii) The converse holds whenever M is finitely generated. Furthermore, if M is non-zero finitely generated, then injdimHna (M ) = injdimM − n. Proof. (i). Let s := injdimM < ∞. We may assume that Hna (M ) 6= 0; and hence s − n ≥ 0. Therefore, in view of 2.2, µi+(s−n) (p, Hna (M )) = 0 for all p ∈ Spec (R) and for all i > 0; so that injdimHna (M ) ≤ s − n. (ii). Suppose that M is finitely generated. We first notice that Hna (M ) = 0 if and only if M = aM ; and this is the case if and only if M = 0. Therefore we may assume that Hna (M ) 6= 0. Suppose that t := injdimHna (M ) < ∞. Then there exists a prime ideal q of R such that µt (q, Hna (M )) 6= 0. Hence, by 2.2, µt+n (q, M ) 6= 0. Next we show that µt+n+i (p, M ) = 0 for all p ∈ Spec (R) and for all i > 0. Assume the contrary. Then there exists a prime ideal p of R such that µt+n+j (p, M ) 6= 0 for some j > 0. Let r := dim R/p. Then, by [11, §18, Lemma 4], we have µt+n+j+r (m, M ) 6= 0. Hence, by 2.2, µj+t+r (m, Hna (M )) 6= 0 which is a contradiction in view of the choice of t. Therefore, injdimM ≤ t + n. The final assertion is a consequence of (i) and (ii). 4 M.R. ZARGAR AND H. ZAKERI Next, we provide an example to show that if R is non-local, then Theorem 2.5(ii) is no longer true. Also, in 3.11, we present two examples which show that 2.5(ii) and 2.5(i), respectively, are no longer true without the finiteness and the relative Cohen-Macaulayness assumptions on M . Example 2.6. Suppose that R is a non-local Artinian ring with injdimR = ∞. Let L Max (R) = {m1 , ..., mn }. Then, in view of [17, Exercise 8.49], we have R = m∈Max (R) Γm (R). Now, since the injective dimension of R is infinite, there exists a maximal ideal mt of R L such that the injective dimension of Γmt (R) is infinite. Set M := ER (R/ms ) Γmt (R), where ms ∈ Max (R) with ms 6= mt . Then M is a finitely generated R-module with infinite injective dimension and Hims (M ) = 0 for all i 6= 0; but Γms (M ) is injective. It is well-known that if (R, m, k) is a d-dimensional local ring, then R is Gorenstein if and only if R is Cohen-Macaulay and Hd (R) ∼ = ER (k). The following corollary, which recovers m this fact, is an immediate consequence of 2.5. Corollary 2.7. Let (R, m) be local and let R be relative Cohen-Macaulay with respect to a. Then R is Gorenstein if and only if injdimHaht R a (R) is finite. In particular, if x = x1 , ..., xn is an R-sequence for some non-negative integer n, then R is Gorenstein if and only if injdimHn(x) (R) is finite. The following proposition, which is needed in the proof of 3.8, provides an explicit minimal injective resolution for the non-zero local cohomology module of a relative Cohen-Macaulay module. Proposition 2.8. Suppose that M is relative Cohen-Macaulay with respect to a and that n = cd (a, M ). Then 0 −→ Hna (M ) −→ M µn (p, M )E(R/p) −→ p∈V(a) is a minimal injective resolution for M µn+1 (p, M )E(R/p) −→ · · · p∈V(a) Hna (M ). Furthermore, Ass R Hna (M ) = {p ∈ V(a)\| µn (p, M ) 6= 0}. Proof. Let d−1 d0 dn−1 dn dn+1 0 −→ M −→ E0 (M ) −→ · · · −→ En−1 (M ) −→ En (M ) −→ En+1 (M ) −→ · · · be a minimal injective resolution for M . If there exists a prime ideal p in V(a) with µn−1 (p, M ) 6= 0, then depth Rp Mp ≤ n − 1. On the other hand, since p ∈ Supp (M/aM ), 2.4 implies that HnaRp (Mp ) 6= 0. Therefore n = grade Rp (aRp , Mp ) ≤ depth Rp Mp ≤ n − 1 which is a contradiction. It follows that Γa (E n−1 (M )) = 0; and hence we obtain the minimal injective resolution 0 −→ Hna (M ) −→ Γa (En (M )) −→ Γa (En+1 (M )) −→ · · · for Hna (M ). Now, we may use this resolution to complete the proof. INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 5 The following elementary lemma, which is needed in the proof of the next theorem, can be proved by using a minimal free resolution for M and the concept of localization. Lemma 2.9. Let (R, m, k) be local and let M be finitely generated. Then, for any prime R ideal p of R, vdim k(p) Tor i p (k(p), Mp ) ≤ vdim k Tor R i (k, M ) for all i ≥ 0. The next theorem provides a comparison of Bass numbers of certain local cohomology modules. Theorem 2.10. Suppose that (R, m, k) is a local ring which is a homomorphic image of a Gorenstein local ring and that M is finitely generated. Let n, m be non-negative integers. Then µm (p, Hn (M )) ≤ µm (m, Hn+dim R/p (M )) for all p ∈ Spec (R). p m Proof. Let (R′ , m′ ) be a Gorenstein local ring of dimension n′ for which there exists a surjective ring homomorphism f : R′ → R. Let p be a prime ideal of R and set p′ = f −1 (p). Now Rp′ ′ is a Gorenstein local ring and dim R′ /p′ = dim R/p. Since R′ is Gorenstein, we have dim Rp′ ′ = dim R′ − dim R′ /p′ . In view of [1, Exercise 11.3.1] there is, for each j ∈ Z, j an Rp -isomorphism Ext jR′ (Mp , Rp′ ′ ) ∼ = (Ext R′ (M, R′ ))p . Also, by the Local Duality Thep′ n′ −n−t orem [1, Theorem 11.2.6], we have HnpRp (Mp ) ∼ (Mp , Rp′ ′ ), ERp (k(p)) = Hom Rp Ext R ′ p′ n′ −n−t ∼ as Rp -modules, where t := dim R/p, and Hn+t (M, R′ ), ER (k)). It m (M ) = Hom R (Ext R′ therefore follows that n n′ −n−t m ∼ Ext m (Mp , Rp′ ′ ), ERp (k(p)))) Rp (k(p), HpRp (Mp )) = Ext Rp (k(p), Hom Rp (Ext R′ ′ p ′ p n −n−t ∼ Hom R (Tor R (Mp , R′ ′ )), ER (k(p))). = m (k(p), Ext ′ p Rp ′ p p and n+t Ext m R (k, Hm (M )) n′ −n−t ∼ (M, R′ ), ER (k))) = Ext m R (k, Hom R (Ext R′ ′ n −n−t R ∼ Hom R (Tor (k, Ext ′ (M, R′ )), ER (k)). = m R Since by 2.9 R ′ ′ n −n−t n −n−t vdim k(p) (Tor mp (k(p), Ext R (Mp , Rp′ ′ ))) ≤ vdim k (Tor R (M, R′ ))), one ′ m (k, Ext R′ p′ may use the above isomorphisms to complete the proof. It is known as Bass’s conjecture that if a local ring admits a finitely generated module of finite injective dimension, then it is a Cohen-Macaulay ring. For the proof of this fact the reader is referred to [12] and [13]. In the next corollary we shall use this fact and the concept of a generalized Cohen-Macaulay module. Recall that, over a local ring (R, m), a finitely generated module of positive dimension is a generalized Cohen-Macaulay module if Him (M ) is finitely generated for all 0 ≤ i < dim M . Corollary 2.11. Let the situation be as in 2.10. Then the following statements hold. dim R/p (M ) for all prime ideals p of R and for (i) injdimRp HnpRp (Mp ) ≤ injdimR Hn+ m any n ≥ 0. (ii) If M is generalized Cohen-Macaulay with dimension d such that Hdm (M ) is injective, then Mp is Gorenstein, in the sense of [18], for all p ∈ Supp (M ) \ {m}. 6 M.R. ZARGAR AND H. ZAKERI Proof. (i) is clear by 2.10. (ii) Let p ∈ Supp (M ) \ {m}. By [1, Exercise 9.5.7], Mp is Cohen-Macaulay and dim Mp + dim R/p = dim M . Hence, in view of (i) and 2.5, we have injdimMp = dim Mp . Therefore, by [18, Theorem 3.11], [2, Theorem 3.1.17] and Bass’s conjecture, Mp is Gorenstein. 3. local cohomology and gorenstein injective dimension We first recall some definitions that we will use in this section. Definition 3.1. Following [4], an R-module M is said to be Gorenstein injective if there exists a Hom (Inj, −) exact exact sequence · · · → E1 → E0 → E 0 → E 1 → · · · of injective R-modules such that M = Ker (E 0 → E 1 ). We say that an exact sequence 0 → M → G0 → G1 → G2 → · · · of R-modules and R-homomorphisms is a Gorenstein injective resolution for M , if each Gi is Gorenstein injective. We say that Gid R M ≤ n if and only if M has a Gorenstein injective resolution of length n. If there is no shorter resolution, we set Gid R M = n. Dually, an R-module M is said to be Gorenstein projective if there is a Hom (−, P roj) exact exact sequence · · · → P1 → P0 → P 0 → P 1 → · · · of projective R-modules such that M = Ker (P 0 → P 1 ). Similarly, one can define the Gorenstein projective dimension, Gpd R M , of M . Definition 3.2. For a local ring R admitting the dualizing complex DR , we denote by KM the canonical module of an R-module M , which is defined to be KM = Hd−n (RHom R (M, DR )), where d = dim R and n = dim M . Note that if R is Cohen-Macaulay, then KR coincides with the classical definition of the canonical module of R which is denoted by ωR . Definition 3.3. Let R be a Cohen-Macaulay local ring of Krull dimension d which admits a canonical module ωR . Following [4], let J0 (R) be the class of R-modules M which satisfies the following conditions. (i) Ext iR (ωR , M ) = 0 , for all i > 0. (ii) Tor R i (ωR , Hom R (ωR , M )) = 0, for all i > 0. (iii) The natural map ωR ⊗R Hom R (ωR , M ) → M is an isomorphism. This class of R-modules is called the Bass class. Definition 3.4. Following [19], let a and b be ideals of R. We set W (a, b) = { p ∈ Spec (R) \| an ⊆ p + b for some integer n > 0}. For an R-module M , Γa,b (M ) denotes a submodule of M consisting of all elements of M with support in W (a, b), that is, Γa,b (M ) = { x ∈ M \| Supp (Rx) ⊆ W (a, b)}. INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 7 The following lemma has been proved in [10, Lemma 4.2] and is of assistance in the proof of Proposition 3.6. Lemma 3.5. Let (R, m, k) be local. Then Ext iR (ER (k), M ) ∼ = Ext iR̂ (ER̂ (k), M ⊗R R̂) = Ext iR (E(k), M ⊗R R̂) ∼ for all i ≥ 0. Let (R, m, k) be local and let M be a non-zero non-injective R-module of finite Gorenstein injective dimension. It was shown in [5, Corollary 4.4] that if Ext iR (E, M ) = 0 for all i ≥ 0 and all indecomposable injective R-modules E 6= ER (k), then Gid R M = sup{ i \| Ext iR (ER (k), M ) 6= 0}. Our next proposition, which is concerned with this result, indicates that Gorenstein injective dimension is a refinement of the injective dimension. However we will use 3.6 and 3.7 to prove the main theorem 3.8. Proposition 3.6. Let (R, m, k) be local and let M be non-zero with Gid R M < ∞. If either M is finitely generated or Artinian, then Gid R M = sup{ i \| Ext iR (ER (k), M ) 6= 0}. Proof. First assume that M is finitely generated. Then, by [3, Theorem 3.24], Gid R M = c. Now, since R b is complete and M c is finitely generated as an R-module, b Gid Rb M the proof of [6, Proposition 2.2] in conjunction with [5, Corollary 4.4 ] implies that c = sup{ i \| Ext i (E b (R/ b m), c) 6= 0}. b M Gid Rb M b R R Therefore we can use 3.5 to complete the proof. Next, we consider the case where M is Artinian. By [15, Lemma 3.6 ] and [1, Exercise b m))) b 8.2.4], Gid R (M ) = Gid b (M ) and, by [3, Theorem 4.25], Gfd b (Hom b (M, E b (R/ = R R R R Gid Rb (M ), where, for an R-module X, Gfd R (X), denotes the Gorenstein flat dimension of b m)) b b is finitely generated as an R-module, X. Therefore, since Hom Rb (M, ERb (R/ in view of [3, Theorem 4.24] and [3. Theorem 1.10] we have the first equality in the next display b m))) b Gfd Rb (Hom Rb (M, ERb (R/ b m)), b 6= 0} b R) = sup{ i\| Ext iRb (Hom Rb (M, ERb (R/ = sup{ i\| Ext iR (ER (k), M ) 6= 0}. The last equality follows from [10, Theorem 4.3], because ER (k) and M are Artinian. Lemma 3.7. Let (R, m) be a Cohen-Macaulay local ring and let M be finitely generated. Suppose that x ∈ m is both R-regular and M -regular. Then the following statements are equivalent. (i) Gid R M < ∞. (ii) Gid R/xR M/xM < ∞. Furthermore, Gid R/xR M/xM = Gid R M − 1. Proof. In view of [3, Theorem 3.24], we can assume that R is complete; and hence it admits a canonical module ωR . 8 M.R. ZARGAR AND H. ZAKERI (i)⇒(ii). It follows from [3, Proposition 3.9] that Gid R M/xM < ∞. Therefore we can use [4, Proposition 10.4.22], [11, p.140,lemma 2] and [2, Theorem 3.3.5], to see that Gid R/xR M/xM < ∞. (ii)⇒(i). By [4, Proposition 10.4.23], M/xM ∈ J0 (R/xR). Since ωR/xR ∼ = ωR /xωR , in i R view of [11, p.140, lemma 2] we have Ext R (ωR , M/xM ) = Tor i (ωR , Hom R (ωR , M/xM )) = 0 for all i > 0 and ωR /xωR ⊗R/xR Hom R (ωR , M/xM ) ∼ = M/xM . Now, using the exact x sequence Ext iR (ωR , M ) −→ Ext iR (ωR , M ) −→ Ext iR (ωR , M/xM ) and Nakayama’s lemma, we deduce that Ext iR (ωR , M ) = 0 for all i > 0. Thus we have the exact sequence (3.1) x 0 −→ Hom R (ωR , M ) −→ Hom R (ωR , M ) −→ Hom R (ωR , M/xM ) −→ 0. Now, we may use (3.1) and similar arguments as above to see that Tor iR (ωR , Hom R (ωR , M )) = ∼ M. 0 for all i > 0. Also, in view of [2, Lemma 3.3.2], we can see that ωR ⊗R Hom R (ωR , M ) = Therefore by [4, Proposition 10.4.23], Gid R M < ∞. The final assertion is an immediate consequence of [3, Theorem 3.24]. Theorem 3.8, which is a Gorenstein injective version of 2.5, is one of the main results of this section. As we will see, this theorem has consequences which recover some interesting results that have currently been appeared in the literature. Here we adopt the convention that the Gorenstein injective dimension of the zero module is to be taken as −∞. Theorem 3.8. Suppose that the local ring (R, m) has a dualizing complex and let n be a non-negative integer such that Hia (M ) = 0 for all i 6= n. (i) If Gid R M < ∞, then Gid R Hna (M ) < ∞. (ii) The converse holds whenever R and M are Cohen-Macaulay. Furthermore, if M is non-zero finitely generated with finite Gorenstein injective dimension, then Gid R Hna (M ) = Gid R M − n. Proof. (i) Notice that if Hna (M ) = 0, then there is nothing to prove. So, we may assume that Hna (M ) 6= 0. Hence, by [20, Lemma 1.1], we have n ≤ d, where d = Gid R M . Let d0 d−1 dn−1 d1 dd−1 dn+1 dn 0 −→ M −→ G0 −→ G1 −→ · · · −→ Gn−1 −→ Gn −→ Gn+1 −→ · · · −→ Gd−1 −→ Gd −→ 0 be a Gorenstein injective resolution for M . By applying the functor Γa (−) on this exact sequence, we obtain the complex Γa (d−1 ) Γa (d0 ) Γa (d1 ) 0 −→ Γa (M ) −→ Γa (G0 ) −→ Γa (G1 ) −→ · · · −→ Γa (Gn−1 ) Γa (dn−1 ) −→ Γa (dn ) Γa (Gn ) −→ Γa (Gn+1 ) Γa (dn+1 ) −→ · · · −→ Γa (Gd−1 ) Γa (dd−1 ) −→ Γa (Gd ) −→ 0 in which, by [15, Theorem 3.2], Γa (Gi ) is Gorenstein injective for all 0 ≤ i ≤ d. If n = 0 the result is clear. So suppose that n > 0. Now, since by [20, Lemma 1.1] each Gi is Γa -acyclic for all i, we may use [1, Exercise 4.1.2 ] in conjunction with our assumption on local cohomology modules of M to obtain the following two exact sequences Γa (d−1 ) Γa (d0 ) 0 −→ Γa (M ) −→ Γa (G0 ) −→ · · · −→ Γa (Gn−1 ) and Γa (dn−1 ) −→ Γa (Gn ) −→ Γa (Gn ) −→ 0 Im Γa (dn−1 ) INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 9 0 −→ Im Γa (dn ) = Ker Γa (dn+1 ) ֒→ Γa (Gn+1 ) −→ · · · −→ Γa (Gd−1 ) −→ Γa (Gd ) −→ 0. But, by assumption, Γa (M ) = 0. Therefore, by using the first above exact sequence and n ) [4, Theorem 10.1.4], we see that ImΓΓa (G is Gorenstein injective. Notice that Hna (M ) = n−1 ) a (d n ker Γa(d ) Im Γ (dn−1 ) . Therefore, patching the second above long exact sequence together with the a exact sequence 0 −→ Hna (M ) −→ Γa (Gn ) Γa (Gn ) −→ −→ 0, Im Γa (dn−1 ) ker Γa (dn ) yields the following long exact sequence 0 −→ Hna (M ) −→ Γa (Gn ) −→ Γa (Gn+1 ) −→ · · · −→ Γa (Gd−1 ) −→ Γa (Gd ) −→ 0. Im Γa (dn−1 ) Hence, Gid R Hna (M ) ≤ Gid R M − n. (ii) Suppose that R and M are Cohen-Macaulay. Since H0m (E(R/m)) = E(R/m) and, for any non-maximal prime ideal p of R, H0m (E(R/p)) = 0, we may apply 2.8 to see that Him (Hna (M )) = Hn+i m (M ) for all i ≥ 0. Therefore, we can use the Cohen-Macaulayness of M to deduce that Him (Hna (M )) =  0 if i 6= dim M/aM Hd (M ) if i = dim M/aM, m where d = dim M . Hence, by using part(i) for Hna (M ), we have Gid R Hdm (M ) < ∞. Now, we proceed by induction on d to show that Gid R M is finite. The case d = 0 is clear. Let d > 0 and assume that the result has been proved for d − 1. Suppose that x ∈ m is both R-regular and M -regular. Then one can use the induced exact sequence d d 0 −→ Hd−1 m (M/xM ) −→ Hm (M ) −→ Hm (M ) −→ 0 and [3, Proposition 3.9] to see that Gid R (Hd−1 m (M/xM )) is finite. Hence, by inductive hypothesis, Gid R M/xM is finite. Therefore, since, in view of [9, Corollary 6.2], R admits a canonical module, one can use the same argument as in the proof of 3.7(i)⇒(ii) to deduce that Gid R/xR M/xM < ∞. Therefore Gid R M is finite by 3.7. Now the result follows by induction. For the final assertion, let M be non-zero finitely generated with Gid R M = s < ∞. Then, by part(i), we have Gid R Hna (M ) ≤ s − n. If Gid R Hna (M ) < s − n, Then, in view s−n of [3, Theorem 3.6], we deduce that Ext R (E(k), Hna (M )) = 0. Hence, by Proposition 2.1, Ext sR (E(k), M ) = 0 which is a contradiction by 3.6. Therefore, Gid R Hna (M ) = Gid R M − n. Let (R, m) be a local ring. As a main theorem, it was proved in [15, Theorem 3.10] that if R and M are Cohen-Macaulay with dim M = n and Gid R Hnm (M ) < ∞, then Gid R Hnm (M ) = dim R − n. Notice that if Gid R Hnm (M ) < ∞, then, in view of [15, Lemma 3.6], 3.8(ii) and [3, Theorem 3.24], we have depth R = Gid R M . Therefore, the next corollary, which is established without the assumption that Gid R Hnm (M ) < ∞, recovers [15, Theorem 3.10]. Another improvement of the above result will be given in 3.12. 10 M.R. ZARGAR AND H. ZAKERI Corollary 3.9. Let (R, m) be a Cohen-Macaulay local ring and let M be Cohen-Macaulay of dimension n. Then Gid R Hnm (M ) = Gid R M − n. b is a Cohen-Macaulay R-module b of dimension n. By Proof. First we notice that M ⊗R R n b using [15, Lemma 3.6] and [1, Exercise 8.2.4], we have Gid R H (M ) = Gid b Hnb (M ⊗R R). m R m c are simultaneously finite. Therefore Also, in view of [3, Theorem 3.24], Gid R M and Gid Rb M we may assume that R is complete; and hence it has a dualizing complex. Now, one can use 3.8 to obtain the assertion. In [21, Theorem 2.6] a characterization of a complete Gorenstein local ring R, in terms of Gorenstein injectivity of the top local cohomology module of R, is given. The next corollary together with 2.7 recover that characterization. Corollary 3.10. Let (R, m) be a Cohen-Macaulay local ring which has a dualizing complex. Then the following conditions are equivalent. (i) R is Gorenstein. (ii) Gid R Hna (R) < ∞ for any ideal a of R such that R is relative Cohen-Macaulay with respect to a and that ht R a = n. (iii) Gid R Hna (R) < ∞ for some ideal a of R such that R is relative Cohen-Macaulay with respect to a and that ht R a = n. Proof. The implication (i)⇒(ii) follows from 2.7 and the implication (ii)⇒(iii) is clear. The implication (iii)⇒(i) follows from 3.8(ii) and [3, Proposition 3.11]. Concerning the above corollary, we notice that if Hna (R) is Artinian, then it is not needed to impose the hypothesis that R has a dualizing complex. Therefore [21, Theorem 2.6] follows from 3.10 without the completeness assumption on R. Next, as promised before, we provide examples to show that if M is not finitely generated or M is not relative Cohen-Macaulay, then 2.5(ii) and 2.5(i), respectively, are no longer true. Examples 3.11. (i). Let (R, m) be a Gorenstein local ring with dim R ≥ 2 such that Rp is not regular for some non-maximal prime ideal p of R (for example, one can take R= K[[X,Y,Z]] (X 2 ) and p = (x, y)R, where K is a field). Then one can use [3, Theorem 3.14] to see that there exists a non-zero Gorenstein injective Rp -module Mp which is neither injective nor finitely generated. Hence, by [4, Proposition 10.1.2 ], injdimRp Mp = ∞; so that injdimR Mp = ∞. Now, we notice that, for all x ∈ m − p, injdimR Mx = ∞ because (Mx )pRx ∼ = Mp . It is easy to check that Him (Mx ) = 0 for all i and that Mx is not finitely generated as an R-module. Set N = Mx ⊕ ER (k). Then injdimN = ∞, but Γm (N ) is injective. This example shows that, in 2.5(ii), the finiteness assumption on M is required. (ii). Let R = k[[x, y]]/(xy), where k is a field. Then R is a 1-dimensional complete Gorenstein local ring. Let m be the maximal ideal of R and let J = (y)R. In view of [1, Theorem 8.2.1] and [21, Corollary 2.10], H1J (R) is a non-zero Gorenstein injective R-module. Note that ΓJ (R) 6= 0. Now, we show that H1J (R) is not injective. If H1J (R) were injective, then Hom R (H1J (R), ER (R/m)) = Rn for some positive integer n. Therefore, by using [19, INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 11 Theorem 5.11] and [21, Lemma 3.1], we get an R-isomorphism ψ : Rn = Hom R (H1J (R), ER (R/m)) → Γm,J (R) = J. Now, ψ(xRn ) = xJ = 0 which is a contradiction. Hence, by [4, Proposition 10.1.2], we have injdimH1J (R) = ∞. Therefore, by using the exact sequences 0 −→ ΓJ (R) −→ R −→ R/ΓJ (R) −→ 0 and 0 −→ R/ΓJ (R) −→ Ry+(xy) −→ H1J (R) −→ 0, we achieve injdimΓJ (R) = ∞. As a mentioned just above 3.9, the next theorem is an improvement of [15, Theorem 3.10]. Indeed, we will use the canonical module of a module to prove the above result without assuming that R and M are Cohen-Macaulay. Notice that if M is Cohen-Macaulay, then KM is Cohen-Macaulay. But the converse does not hold in general; see for example [16, Lemma 1.9] and [16, Theorem 1.14]. Theorem 3.12. Assume that (R, m) is local, and M is non-zero finitely generated of dimension d. Then the following statements hold. (i) Gid R Hda (M ) = Gpd Rb ΓmR,a b R b (KM c). d d (ii) If KM c is Cohen-Macaulay and Gid R Hm (M ) < ∞, then Gid R Hm (M ) = depth R−d. Proof. (i) By [1, Theorem 7.1.6], Hda (M ) is Artinian. Therefore, by use of [1, Theorem 4.3.2] and [19, Theorem 5.11], we have Hda (M ) = 0 if and only if ΓmR,a b R b (KM c) = 0. Hence, we may assume that Hda (M ) 6= 0. Now, by [15, Lemma 3.6], Gid R Hda (M ) = Gid Rb Hda (M ) and, by [3, Theorem 4.25], Gid Rb Hda (M ) = Gfd Rb Hom Rb (Hda (M ), ER (k)). Therefore, since b Hom b (Hd (M ), ER (k)) is finitely generated as an R-module, one can use [3, Theorem 4.24] R a and [19, Theorem 5.11] to establish the result. d (ii) First notice that ΓmR,m b R b (KM b (KM c) = KM c. Hence, by part(i), Gid R Hm (M ) = Gpd R c). d b− Therefore, in view of [3, Proposition 2.16] and [3, Theorem 1.25], Gid R H (M ) = depth R m depth KM c. Now, one can use [16, Lemma 1.9(c)] to complete the proof. The following corollary is a generalization of the main result [21, Theorem 2.6]. Corollary 3.13. Assume that (R, m) is local with dim R = d and that KRb is CohenMacaulay. Then the following statements are equivalent. (i) R is Gorenstein. (ii) injdimR Hdm (R) < ∞. (iii) Gid R Hdm (R) < ∞. Proof. The implication (i)⇒(ii) follows from 2.5 while the implication (ii)⇒(iii) is clear. (iii)⇒(i). By 3.12(ii), we have Gid R Hdm (R) = depth R − dim R; and hence R is CohenMacaulay. Now one can use 3.9 to obtain the assertion. 12 M.R. ZARGAR AND H. ZAKERI Corollary 3.14. Let (R, m) be local with dim R = d ≤ 2. Then the following statements are equivalent. (i) R is Gorenstein. (ii) Gid R Hdm (R) < ∞. (iii) Hda (M ) is Gorenstein injective for all finitely generated R–modules M and for all ideals a of R. Proof. Let M be a non-zero finitely generated R-module. Then, by [1, Theorem 7.1.6], Hda (M ) and Hdm (R) are Artinian. Therefore, in view of [15, Lemma 3.5], we may assume that R is complete. Since, by [16, Lemma 1.9], KR is Cohen-Macaulay, (i)⇔(ii) follows immediately from 3.13. The implication (iii)⇒(i) is clear and the implication (i)⇒(iii) follows from [21, Corollary 2.10]. Acknowledgements. The authors would like to thank Alberto Fernandez Boix and the referee for careful reading of manuscript and helpful comments. References [1] M.P. Brodmann and R.Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998. [2] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993. [3] L.W. Christensen, H-B. Foxby, and H. Holm, Beyond totally reflexive modules and back, In: Noetherian and Non-Noetherian Perspectives, edited by M. Fontana, S-E. Kabbaj, B. Olberding and I. Swanson, Springer Science+Business Media, LLC, New York, 2011, 101–143. [4] E.E. Enochs and O.M.G. Jenda, Relative homological algebra, de Gruyter, Berlin, 2000. [5] E.E. Enochs and O.M.G. Jenda, Gorenstein injective dimension and Tor-depth of modules, Arch. Math, 72 (1999) 107–117. [6] R. Fossum, H-B. Foxby, P. Griffith and I. Reiten, Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, Publ. Math. I.H.E.S, 45 (1975) 193–215. [7] M. Hellus and P. Schenzel, On cohomologically complete intersections, Journal of Algebra, 320 (2008) 3733–3748. [8] M. Jahangiri, and A. Rahimi, Relative Cohen-Macaulayness and relative unmixedness of bigraded modules, to appear in Commutative Algebra. [9] T. Kawasaki, On Macaulayfication of Noetherian schemes, Trans. Amer. Math. Soc., 352 (2000) 2517– 2552. [10] B. Kubik, M.J. Leamerb and S. Wagstaff, Homology of artinian and Matlis reflexive modules, I, J. Pure Appl. Algebra, 215 (2011) 2486–2503. [11] H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1992. [12] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S. 42 (1973) 47–119. [13] P. Roberts, Le theoreme d’intersection, C. R. Acad. Sci. Paris Ser. I, 304 (1987) 177–180. [14] J.J. Rotman, An introduction to homological algebra, Second ed., Springer, New York, 2009. [15] R. Sazeedeh, Gorenstein injective of the section functor, Forum Mathematicum, 22 (2010) 1117–1127. [16] P. Schenzel, On the use of local cohomoloy in algebra and geometry, in: Lectures at the Summmer school of Commutative Algebra and Algebraic Geometry, Birkhä, Basel, Ballaterra, 1996. [17] R.Y. Sharp, Steps in commutative algebra, Second Edition, Cambridge University Press, Cambridge, 2000. INJECTIVE AND GORENSTEIN INJECTIVE DIMENSIONS 13 [18] R.Y. Sharp, Gorenstein Modules, Mathematische Zeitschrift, 115 (1970) 117–139. [19] R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra, 213 (2009) 582–600. [20] S. Yassemi, A generalization of a theorem of Bass, Comm. Algebra, 35 (2007) 249–251. [21] T. Yoshizawa, On Gorenstein injective of top local cohomology modules, Proc. Amer. Math. Soc, 140 (2012) 1897–1907. M.R. Zargar and H. Zakeri, Faculty of mathematical sciences and computer, Kharazmi University, 599 Taleghani Avenue, Tehran 15618, Iran E-mail address: zargar9077@gmail.com E-mail address: zakeri@tmu.ac.ir M.R. Zargar, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395–5746, Tehran, Iran.	0
Distributed Compression of Graphical Data Payam Delgosha and Venkat Anantharam arXiv:1802.07446v1 [cs.IT] 21 Feb 2018 Department of Electrical Engineering and Computer Sciences University of California, Berkeley {pdelgosha, ananth} @ berkeley.edu February 22, 2018 Abstract In contrast to time series, graphical data is data indexed by the nodes and edges of a graph. Modern applications such as the internet, social networks, genomics and proteomics generate graphical data, often at large scale. The large scale argues for the need to compress such data for storage and subsequent processing. Since this data might have several components available in different locations, it is also important to study distributed compression of graphical data. In this paper, we derive a rate region for this problem which is a counterpart of the Slepian–Wolf Theorem. We characterize the rate region when the statistical description of the distributed graphical data is one of two types – a marked sparse Erdős–Rényi ensemble or a marked configuration model. Our results are in terms of a generalization of the notion of entropy introduced by Bordenave and Caputo in the study of local weak limits of sparse graphs. 1 Introduction Nowadays, storing combinatorically structured data is of great importance in many applications such as the internet, social networks and biological data. For instance, a social network could be presented as a graph where each node models an individual and each edge stands for a friendship. Also, vertices and edges can carry marks, e.g. the mark of a vertex represents its type, and the mark of an edge represents its shared information. Due to the sheer amount of such data, compressing it has drawn attention, see e.g. [CS12], [Abb16], [DA17]. As the data is not always available in one location, it is important to also consider distributed compression of graphical data. Traditionally, distributed lossless compression is modeled using two (or more) correlated stationary and ergodic processes representing the components of the data at the individual locations. In this case, the rate region is given by the Slepian–Wolf Theorem [CT12]. We adopt an analogous framework, namely that two correlated marked random graphs on the same vertex set are presented to two encoders which then individually compress their data such that a third party can recover both realizations from the two compressed representations, with a vanishing probability of error in the asymptotic limit of data size. We characterize the compression rate region for two scenarios, namely, a marked sparse Erdős–Rényi ensemble and a marked configuration model. We employ the framework of local weak convergence, also called the objective method, as a counterpart for marked graphs of the notion of stochastic processes [BS01, AS04, AL07]. Our characterization is best understood in terms of a generalization of a measure of entropy introduced by Bordenave and Caputo, which we call the BC entropy [BC14]. It turns out that the BC entropy captures the per–vertex growth rate of the Shannon entropy for the ensembles 1 we study in this paper. This motivates it as a natural measure governing the asymptotic compression bounds. The paper is organized as follows. In Section 2 we introduce the notation and formally state the problem. Sections 3 and 4 give a brief introduction to the objective method and the BC entropy, mostly specialized for the examples we study. Finally, in Section 5, we characterize the rate region for the scenarios we present in Section 2. 2 Notations and Problem Statement The set of real numbers is denoted by R. For an integer n, [n] denotes the set {1, 2, . . . , n}. For a probability distribution P , H(P ) denotes its Shannon entropy. Also, for a random variable X, we denote by H(X) its P Shannon entropy. For a positive integer N and a sequence of positive integers {ai }1≤i≤k such that ai ≤ N , we define N N! := . {ai }1≤i≤k a1 ! . . . ak !(N − a1 − · · · − ak )! For sequences of reals an and bn we write an = O(bn ) if, for some constant C ≥ 0, we have \|an \| ≤ C\|bn \| for n large enough. Furthermore, we write an = o(bn ) if an /bn → 0 as n → ∞. We denote by 1 [A] the indicator of the event A. For a probability distribution P , X ∼ P denotes that the random variable X has law P . Throughout the paper logarithms are to the natural base. A marked graph with edge mark set Ξ and vertex mark set Θ is a graph where each edge carries a mark in Ξ and each vertex carries a mark in Θ. We assume that all graphs are simple unless otherwise stated. Also, we assume that all edge and vertex mark sets are finite. For two vertices v and w in a graph G, v ∼G w denotes that v and w are adjacent in G. Let G be a marked graph on a finite vertex set with edges and vertices carrying marks in the sets Ξ and Θ, respectively. We denote the edge mark count vector of G by m ~ G = {mG (x)}x∈Ξ , where mG (x) is the number of edges in G carrying mark x. Furthermore, we denote the vertex mark count vector of G by ~uG = {uG (θ)}θ∈Θ , where uG (θ) denotes the number of vertices in G with mark θ. Additionally, − → for a graph G on the vertex set [n], we denote the degree sequence of G by dgG = {dgG (1), . . . , dgG (n)} where dgG (i) denotes the degree of vertex i. For a degree sequence d~ = (d(1), . . . , d(n)) and an integer k, we define ~ := \|{1 ≤ i ≤ n : d(i) = k}. ck (d) (1) Also, for two degree sequences d~ = (d(1), . . . , d(n)) and d~′ = (d′ (1), . . . , d′ (n)), and two integers k and l, we define ~ d~′ ) := \|{1 ≤ i ≤ n : d(i) = k, d′ (i) = l}\|. ck,l (d, (2) (n) Given a degree sequence d~ = (d(1), . . . , d(n)), we let G ~ denote the set of simple unmarked graphs G d on the vertex set [n] such that dgG (i) = d(i) for 1 ≤ i ≤ n. Throughout this paper, we assume that Ξ1 and Ξ2 are two fixed and finite sets of edge marks. Moreover, Θ1 and Θ2 are two fixed and finite vertex mark sets. For i ∈ {1, 2} and an integer n, let (n) Gi be the set of marked graphs on the vertex set [n] with edge and vertex mark sets Ξi and Θi , (n) (n) respectively. For two graphs G1 ∈ G1 and G2 ∈ G2 , G1 ⊕ G2 denotes the superposition of G1 and G2 which is a marked graph defined as follows: a vertex 1 ≤ v ≤ n in G1 ⊕ G2 carries the mark (θ1 , θ2 ) where θi is the mark of v in Gi . Furthermore, we place an edge in G1 ⊕ G2 between vertices v and w if there is an edge between them in at least one of G1 of G2 , and mark this edge (x1 , x2 ), where, for 1 ≤ i ≤ 2, xi is the mark of the edge (v, w) in Gi if it exists and ◦i otherwise. Here ◦1 and ◦2 are auxiliary marks not present in Ξ1 ∪ Ξ2 . Note that G1 ⊕ G2 is a marked graph with edge and vertex 2 mark sets Ξ1,2 := (Ξ1 ∪ {◦1 }) × (Ξ2 ∪ {◦2 }) \ {(◦1 , ◦2 )} and Θ1,2 := Θ1 × Θ2 , respectively. We use the terminology jointly marked graph to refer to a marked graph with edge and vertex makr sets Ξ1,2 and (n) Θ1,2 respectively. With this, let G1,2 denote the set of jointly marked graphs on the vertex set [n]. Moreover, for i ∈ {1, 2}, we say that a graph is in the i–th domain if edge and vertex marks come from Ξi and Θi , respectively. For a jointly marked graph G1,2 and 1 ≤ i ≤ 2, the i–th marginal of G1,2 , denoted by Gi , is the marked graph in the i–th domain obtained by projecting all vertex and edge marks onto Ξi and Θi , respectively, followed by removing edges with mark ◦i . Note that any jointly marked graph G1,2 is uniquely determined by its marginals G1 and G2 , because G1,2 = G1 ⊕ G2 . Given an edge mark count m ~ = {m(x)}x∈Ξ1,2 , for x1 ∈ Ξ1 ∪ {◦1 }, with an abuse of notation we define X m(x1 ) := (x′1 ,x′2 )∈Ξ1,2 m((x′1 , x′2 )). (3) : x′1 =x1 In a similar fashion, we define m(x2 ) for x2 ∈ Ξ2 ∪ {◦2 }. Likewise, given a vertex mark count vector ~u = {u(θ)}θ∈Θ1,2 , and for θ1 ∈ Θ1 and θ2 ∈ Θ2 , we define u(θ1 ) = X u((θ1 , θ2′ )) u(θ2 ) = X u((θ1′ , θ2 )). (4) θ1′ ∈Θ1 θ2′ ∈Θ2 (n) (n) Assume that we have a sequence of random graphs G1,2 ∈ G1,2 , drawn according to some ensemble distribution. Additionally, assume that there are two encoders who want to compress realizations of such jointly marked graphs in a distributed fashion. Namely, the i–th encoder, 1 ≤ i ≤ 2, has only (n) (n) access to the i–th marginal Gi . We assume that the encoders know the distribution of G1,2 . (n) (n) (n) (n) Definition 1. An hn, L1 , L2 i code is a tuple of functions (f1 , f2 , g (n) ) for each n such that (n) fi (n) : Gi and (n) → [Li ] (n) (n) i ∈ {1, 2}, (n) g (n) : [L1 ] × [L2 ] → G1,2 . (n) (n) The probability of error for this code corresponding to the ensemble of G1,2 , which is denoted by Pe , is defined as (n) (n) (n) (n) (n) Pe(n) := P g (n) (f1 (G1 ), f2 (G2 )) 6= G1,2 . Now we define our achievability criterion. Definition 2. A rate tuple (α1 , R1 , α2 , R2 ) ∈ R4 is said to be achievable for the above scenario if there (n) (n) is a sequence of hn, L1 , L2 i codes such that (n) lim sup n→∞ log Li − (αi n log n + Ri n) ≤0 n i ∈ {1, 2}, (5) (n) and also Pe → 0. The rate region R ∈ R4 is defined as follows: for fixed α1 and α2 , if there are (m) (m) sequences R1 and R2 with limit points R1 and R2 in R, respectively, such that for each m, the (m) (m) rate tuple (α1 , R1 , α2 , R2 ) is achievable, then we include (α1 , R1 , α2 , R2 ) in the set R. In this paper, we characterize the above rate region for the following two sequences of ensembles: 3 The Erdős–Rényi ensemble: Assume that nonnegative real numbers ~p = {px }x∈Ξ1,2 together with a probability distribution ~ q = {qθ }θ∈Θ1,2 are given such that for all x1 ∈ Ξ1 ∪ {◦1 } and x2 ∈ Ξ2 ∪ {◦2 }, we have X X p(x′1 ,x′2 ) > 0. (6) p(x′1 ,x′2 ) > 0 and (x′ ,x′ )∈Ξ1,2 1 2 x′2 =x2 (x′ ,x′ )∈Ξ1,2 1 2 x′1 =x1 (n) For an integer n large enough, we define the probability distribution G(n; p~, ~q) on G1,2 as follows: for each pair of vertices 1 ≤ i < j ≤ n, the edge (i, j) is present Pin the graph and has mark x ∈ Ξ1,2 with probability px /n, and is not present with probability 1 − x∈Ξ1,2 px /n. Furthermore, each vertex in the graph is given a mark θ ∈ Θ1,2 with probability qθ . The choice of edge and vertex marks is done independently. The configuration model ensemble: Assume that a fixed integer ∆ > 0 and a probability distribution ~r = {rk }∆ k=0 supported on the set {0, . . . , ∆} are given, such that r0 < 1. Moreover, assume that probability distributions ~γ = {γx }x∈Ξ1,2 and ~q = {qθ }θ∈Θ1,2 on the sets Ξ1,2 and Θ1,2 , respectively, are given. We assume that for all x1 ∈ Ξ1 ∪ {◦1 } and x2 ∈ Ξ2 ∪ {◦2 }, we have X X γ(x′1 ,x′2 ) > 0. (7) γ(x′1 ,x′2 ) > 0 and (x′1 ,x′2 )∈Ξ1,2 x′2 =x2 (x′1 ,x′2 )∈Ξ1,2 x′1 =x1 Furthermore, let d~(n) = {d(n) (1), . . . , d(n) (n)} be P a sequence of degree sequences such Pnthat for all n n and 1 ≤ i ≤ n, we have d(n) (i) ≤ ∆ and also i=1 d(n) (i) is even. Let mn := ( i=1 d(n) (i))/2. Additionally, if for 0 ≤ k ≤ ∆, ck (d~(n) ) denotes the number of 1 ≤ i ≤ n such that d(n) (i) = k, we assume that for some constant K > 0, ∆ X \|ck (d~(n) ) − nrk \| ≤ Kn1/2 . (8) k=0 (n) Now, we define the law G(n; d~(n) , ~γ , ~ q ) on G1,2 for n large enough as follows. First, we pick an unmarked graph on the vertex set [n] uniformly at random among the set of graphs G with maximum degree ∆ − → such that for each 0 ≤ k ≤ ∆, ck (dgG ) = ck (d~(n) ).1 Then, we assign i.i.d. marks with law ~γ on the edges and i.i.d. marks with law ~q on the vertices. As we will discuss in Section 3 below, the sequence of Erdős–Rényi ensembles defined above converges in the local weak sense to a marked Poisson Galton Watson tree. Moreover, the configuration model ensemble converges in the same sense to a marked Galton Watson process with degree distribution ~r. We will characterize the achievability rate regions in Section 5 in terms of these limiting objects for the above two sequences of ensembles. This will turn out to be best understood in terms of a measure of entropy discussed in Section 4 below. 3 The framework of Local Weak Convergence In this section, we discuss the framework of local weak convergence mainly in the context of the Erdős– Rényi and configuration model ensembles discussed in Section 2. For a general discussion, the reader is referred to [BS01, AS04, AL07]. 1 The fact that each degree is bounded to ∆, r < 1 and the sum of degrees is even implies that d ~(n) is a graphic 0 sequence for n large enough. This is, for instance, a consequence of Theorem 4.5 in [BC14]. 4 Let Ξ and Θ be finite mark sets. A marked graph G with edge and vertex mark sets Ξ and Θ respectively together with a distinguished vertex o, is called a rooted marked graph and is denoted by (G, o). For a rooted marked graph (G, o) and integer h ≥ 1, (G, o)h denotes the h neighborhood of o, i.e. the subgraph consisting of vertices with distance no more than h from o. Note that (G, o)h is connected by definition. Two connected rooted marked graphs (G1 , o1 ) and (G2 , o2 ) are said to be isomorphic if there is a vertex bijection between the two graphs that maps o1 to o2 , preserves adjacencies and also preserves vertex and edge marks. With this, we denote the isomorphism class corresponding to a rooted marked graph (G, o) by [G, o]. We simply use [G, o]h as a shorthand for [(G, o)h ]. Let G∗ (Ξ, Θ) denote the set of isomorphism classes [G, o] of connected rooted marked graphs on a countable vertex set with edge and vertex marks coming from the sets Ξ and Θ, respectively. It can be shown that G∗ (Ξ, Θ) can be turned into a separable and complete metric space [AL07]. For a probability distribution µ on G∗ (Ξ, Θ), let deg(µ) denote the expected degree at the root in µ. For a finite marked graph G and a vertex v in G, let G(v) denote the connected component of v. With this, if v is a vertex chosen uniformly at random in G, we define U (G) be the law of [G(v), v], which is a probability distribution on G∗ (Ξ, Θ). (n) Let G1,2 be a random jointly marked graph with law G(n; p~, ~q) and let vn be a vertex chosen uniformly at random in the set [n]. A simple Poisson approximation implies that Dx (vn ), the number of edges adjacent to vn with mark x ∈ Ξ1,2 , converges in distribution to a Poisson random variables with mean px as n goes to infinity. Moreover, {Dx (vn )}x∈Ξ1,2 are asymptotically mutually independent. A similar argument can be repeated for any other vertex in the neighborhood of vn . Also, it can be (n) shown that the probability of having cycles converges to zero. In fact, the structure of (G1,2 , vn )h converges in distribution to a rooted marked Poisson Galton Watson tree with depth h. ER More precisely, let (T1,2 , o) be a rooted jointly marked tree defined as follows. First, the mark of the root is chosen from distribution ~ q . Then, for x ∈ Ξ1,2 , we independently generate Dx with law Poisson(px ). We then add Dx many edges with mark x to the root o. For each offspring, we repeat the same procedure independently, i.e. choose its mark and edges with each mark from the corresponding ER Poisson distribution. Recursively repeating this, we get a connected jointly marked tree T1,2 rooted at ER o, which has possibly countably infinitely many vertices. Let µ1,2 denote the law of the isomorphism ER class [T1,2 , o]. Note that µER 1,2 is a probability distribution on G∗ (Ξ1,2 , Θ1,2 ). The above discussion (n) ER implies that [G1,2 , vn ]h converges in distribution to [T1,2 , o]h . In fact, even a stronger statement can (n) be proved. More precisely, if we consider the sequence of random graphs G1,2 independently on a (n) joint probability space, U (G1,2 ) converges weakly to µER 1,2 with probability one. With this, we say (n) that, almost surely, µER 1,2 is the local weak limit of the sequence G1,2 , where the term “local” stands for looking at a fixed depth neighborhood of a typical node. ER With the above construction, for 1 ≤ i ≤ 2, let TiER be the i–th marginal of T1,2 . Moreover, let ER ER ER µi be the law of [Ti (o), o]. Therefore, µi is a probability distribution on G∗ (Ξi , Θi ). Similarly, (n) one can see that, almost surely, µER is the local weak limit of the sequence Gi . i CM A similar picture also holds for the configuration model. More precisely, let (T1,2 , o) be a rooted jointly marked random tree constructed as follows. First, we generate the degree of the root with law ~r. Then, for each offspring w of o, we independently generate the offspring count of w with law r~′ = {rk′ }∆−1 k=0 defined as (k + 1)rk+1 , 0 ≤ k ≤ ∆ − 1, rk′ = E [X] where X has law ~r. We continue this process recursively, i.e. for each vertex other than the root, we independently generate its offspring count with law r′ . The distribution ~r′ is called the sized biased distribution, and takes into account the fact that each node other than the root has an extra edge on 5 top of it, and hence its degree should be biased in order to get the correct degree distribution ~r. Then, CM for each vertex and edge existing in the graph T1,2 , we generate marks independently with laws ~q CM CM and ~γ , respectively. Let µ1,2 be the law of [T1,2 , o]. Moreover, for 1 ≤ i ≤ 2, let µCM be the law of i (n) CM (n) CM ~ [T (o), o]. It can be shown that if G has law G(n; d , ~γ , ~q) then, almost surely, µ is the local i 1,2 1,2 (n) G1,2 , (n) Gi , µCM i for 1 ≤ i ≤ 2. weak limit of and is the local weak limit of Given Ξ and Θ, not all probability distributions on G∗ (Ξ, Θ) can appear as the local weak limit of a sequence. In fact, the condition that all vertices have the same chance of being chosen as the root for a finite graph manifests itself as a certain stationarity condition at the limit called unimodularity [AL07]. 4 The BC entropy In this section, we discuss a notion of entropy for probability distributions on a space of rooted marked graphs. This is a marked version of the entropy defined by Bordenave and Caputo in [BC14], which was defined for probability distributions on the space of rooted (unmarked) graphs. To distinguish it from the Shannon entropy, we call this notion of entropy the BC entropy. Let Ξ and Θ be finite mark sets and let µ be a probability distribution on G∗ (Ξ, Θ). Moreover, let m ~ (n) and ~u(n) be sequences of edge and vertex mark counts, respectively, such that for all x ∈ Ξ, (n) 2m (x)/n converges to the expected number of edges with mark x connected to the root in µ, and for all θ ∈ Θ, u(n) (θ)/n converges to the probability of the mark of the root in µ being θ. Let (n) Gm (µ, ǫ) be the set of graphs G on the vertex set [n] with edge and vertex marks in Ξ and Θ, ~ (n) ,~ u(n) respectively, such that m ~G =m ~ (n) , ~uG = ~u(n) , and U (G) is in the ball around µ with radius ǫ with respect to the Lévy–Prokhorov distance [Bil13]. P Definition 3. If an := x∈Ξ m(n) (x), define (n) Σ(µ, ǫ) := lim sup (µ, ǫ)\| − an log n log \|Gm ~ (n) ,~ u(n) n n→∞ Σ(µ, ǫ) := lim inf (n) (µ, ǫ)\| log \|Gm ~ (n) ,~ u(n) − an log n n n→∞ Note that both Σ(µ, ǫ) and Σ(µ, ǫ) decrease as ǫ decreases. Therefore, we may define the upper and lower BC entropies as Σ(µ) := limǫ↓0 Σ(µ, ǫ) and Σ(µ) := limǫ↓0 Σ(µ, ǫ). If Σ(µ) = Σ(µ), we denote the common value by Σ(µ) and call it the BC entropy of µ. Using similar techniques as in the proof of Theorem 1.2 in [BC14], one can show that Σ(µ) and ~ (n) and ~u(n) , and Σ(µ) = Σ(µ) for all µ Σ(µ) do not depend on the specific choice of the sequences m with positive expected degree of the root. Now, we connect the asymptotic behavior of the entropy of the ensembles defined in Section 2 to (n) ER ER the P BC entropy of their local weak limits. Assume that G1,2 has law G(n; p~, ~q). Let d1,2 := deg(µ1,2 ) = x∈Ξ1,2 px . Moreover, we use the following notational conventions for xi ∈ Ξi and θi ∈ Θi , 1 ≤ i ≤ 2. px1 := X p(x1 ,x′2 ) px2 := x′2 ∈Ξ2 ∪{◦2 } qθ1 := X X p(x′1 ,x2 ) x′1 ∈Ξ1 ∪{◦1 } qθ2 := q(θ1 ,θ2′ ) X θ1′ ∈Θ1 θ2′ ∈Θ2 6 q(θ1′ ,θ2 ) (9) P q , it can be easily verified For 1 ≤ i ≤ 2, let dER := deg(µER i i ) = xi ∈Ξi pxi . If Q = (Q1 , Q2 ) has law ~ that with s(x) defined as x2 − x2 log x for x > 0 and zero for x = 0, we have   ER X d 1,2 (n) n log n + n H(Q) + s(px ) + o(n) H(G1,2 ) = 2 x∈Ξ1,2 ! X dER (n) 1 s(px1 ) + o(n) H(G1 ) = n log n + n H(Q1 ) + 2 x1 ∈Ξ1 ! X dER (n) 2 n log n + n H(Q2 ) + H(G2 ) = s(px2 ) + o(n) 2 x ∈Ξ 2 (10a) (10b) (10c) 2 Using a generalization of Theorem 1.3 in [BC14], it can be seen that the coefficient of n in the above ER ER 3 equations are Σ(µER 1,2 ), Σ(µ1 ) and Σ(µ2 ), respectively. (n) Similarly, for the configuration model, let G be distributed according to G(n; d~(n) , ~γ , ~q) and let 1,2 X be a random variable with law ~r. Moreover, let Γk = (Γk1 , Γk2 ), 1 ≤ k ≤ ∆, be an i.i.d. sequence distributed according to ~γ . With this, let X1 := X X 1 Γk1 6= ◦1 k=1 X2 := X X 1 Γk2 6= ◦2 . (11) k=1 Note that for 1 ≤ i ≤ 2, Xi is basically the distribution of the degree of the root in µCM . If dCM 1,2 := i CM CM CM deg(µ1,2 ) and, for 1 ≤ i ≤ 2, di := deg(µi ), it can be seen that (see Appendix A for the details) dCM 1,2 n log n + n − s(dCM 1,2 ) + H(X) − E [log X!] 2 dCM 1,2 H(Γ) + o(n) + H(Q) + 2 CM d (n) H(G1 ) = 1 n log n + n − s(dCM 1 ) + H(X1 ) − E [log X1 !] 2 dCM + H(Q1 ) + 1 H(Γ1 \|Γ1 6= ◦1 ) + o(n) 2 dCM (n) H(G2 ) = 2 n log n + n − s(dCM 2 ) + H(X2 ) − E [log X2 !] 2 dCM + H(Q2 ) + 2 H(Γ2 \|Γ2 6= ◦2 ) + o(n) 2 (n) H(G1,2 ) = (12a) (12b) (12c) CM CM Also, it can be seen that the coefficients of n in the above equations are Σ(µCM 1,2 ), Σ(µ1 ) and Σ(µ2 ), respectively. CM If µ1,2 is any of the two distributions µER 1,2 or µ1,2 , and µ1 and µ2 are its marginals, we define the conditional BC entropies as Σ(µ2 \|µ1 ) := Σ(µ1,2 ) − Σ(µ1 ) and Σ(µ1 \|µ2 ) := Σ(µ1,2 ) − Σ(µ2 ). 5 Main Results Now, we are ready to state our main result, which is to characterize the rate region in Definition 2. In the following, for pairs of reals (α, R) and (α′ , R′ ), we write (α, R) ≻ (α′ , R′ ) if either α > α′ , or α = α′ and R > R′ . We also write (α, R) (α′ , R′ ) if either (α, R) ≻ (α′ , R′ ) or (α, R) = (α′ , R′ ). 7 CM Theorem 1. Assume µ1,2 is either of the two distributions µER 1,2 or µ1,2 defined in Section 4. Then, if R is the rate region for the sequence of ensembles corresponding to µ1,2 defined in Section 2, a rate tuple (α1 , R1 , α2 , R2 ) ∈ R if and only if (α1 , R1 ) ((d1,2 − d2 )/2, Σ(µ1 \|µ2 )) (13a) (α2 , R2 ) ((d1,2 − d1 )/2, Σ(µ2 \|µ1 )) (13b) (α1 + α2 , R1 + R2 ) (d1,2 /2, Σ(µ1,2 )) (13c) where d1,2 = deg(µ1,2 ), d1 = deg(µ1 ) and d2 = deg(µ2 ). We prove the achievability for the Erdős–Rényi case and the configuration model in Sections 5.1 and 5.2, respectively. Afterwards, we prove the converse for the two cases in Sections 5.3 and 5.4, respectively. Before this, we state the following general lemma used in the proofs, whose proof is straightforward using Stirling’s approximation. Lemma 1. Assume that a positive integer k and sequences of integers an and bn1 , . . . , bnk are given. Pk 1. If an /n → a > 0 and for each 1 ≤ i ≤ k, bni /n → bi ≥ 0 where a = i=1 bi , we have ! bi an 1 . = aH log lim n→∞ n a 1≤i≤k {bni }1≤i≤k 2. If an / n 2 → 1 and bni /n → bi ≥ 0, we have an {bn i }1≤i≤k log lim n→∞ where s(x) is defined to be 5.1 x 2 − x 2 − P k n i=1 bi n log n = k X s(2bi ), i=1 log x for x > 0 and 0 if x = 0. Proof of Achievability for the Erdős–Rényi case Here we show that a rate tuple (α1 , R1 , α2 , R2 ) is achievable for the Erdős–Rényi ensemble if it satisfies the following ER ER ER (α1 , R1 ) ≻ ((dER 1,2 − d2 )/2, Σ(µ1 \|µ2 )) (14a) ER ER ER ((dER 1,2 − d1 )/2, Σ(µ2 \|µ1 )) ER (dER 1,2 /2, Σ(µ1,2 )) (14b) (α2 , R2 ) ≻ (α1 + α2 , R1 + R2 ) ≻ (14c) Note that if a rate tuple (α′1 , R1′ , α′2 , R2′ ) satisfies the weak inequalities (13a)–(13c) then, for any ǫ > 0, (α′1 , R1′ + ǫ, α′2 , R2′ + ǫ) satisfies the above strict inequalities. As we show below, this implies that (α′1 , R1′ + ǫ, α′2 , R2′ + ǫ) is achievable. Hence, after sending ǫ → 0, we get (α′1 , R1′ , α′2 , R2′ ) ∈ R. We show that any (α1 , R1 , α2 , R2 ) satisfying (14a)–(14c) is achievable by employing a random (n) binning method. More precisely, for i ∈ {1, 2}, we set Li = ⌊exp(αi n log n + Ri n)⌋ and for each (n) (n) (n) Gi ∈ Gi , we assign fi (Gi ) uniformly at random in the set [Li ] and independent of everything else. To describe our decoding scheme, we first need to define some notation. Let M(n) denote the set of edge count vectors m ~ = {m(x)}x∈Ξ1,2 such that X \|m(x) − npx /2\| ≤ n2/3 . x∈Ξ1,2 8 Moreover, let U (n) denote the set of vertex mark count vectors ~u = {u(θ)}θ∈Θ1,2 such that X \|u(θ) − nqθ \| ≤ n2/3 . θ∈Θ1,2 (n) (n) (n) Furthermore, we define Gp~,~q to be the set of graphs H1,2 ∈ G1,2 such that m ~ H (n) ∈ M(n) and 1,2 (n) (n) (n) (n) ~uH (n) ∈ U (n) . Upon receiving (i, j) ∈ [L1 ] × [L2 ], we form the set of graphs H1,2 ∈ Gp~,~q such that 1,2 (n) (n) (n) (n) (n) (n) (n) f1 (H1 ) = i and f2 (H2 ) = j, where H1 and H2 are the marginals of H1,2 . If this set has only one element, we output this element as the decoded graph; otherwise, we report an error. (n) In what follows, assume that G1,2 is a random graph with law G(n; p~, ~q). We consider the following four error events corresponding to the above scheme (n) := {G1,2 ∈ / Gp~,~q }, (n) := {∃H1,2 6= G1,2 : fi (Hi ) = fi (Gi ), i ∈ {1, 2}}, (n) := {∃H2 (n) := {∃H1 E1 E2 E3 E4 (n) (n) (n) (n) (n) (n) 6= G2 (n) 6= G1 (n) (n) : G1 ⊕ H2 (n) : H1 (n) (n) (n) (n) (n) ∈ Gp~,~q , f2 (H2 ) = f2 (G2 )}, (n) ∈ Gp~,~q , f1 (H1 ) = f1 (G1 )}. ⊕ G2 (n) (n) (n) (n) (n) (n) (n) (n) (n) (n) (n) Note that outside the above four events, the decoder successfully decodes the input graph G1,2 . (n) P(E1 ) Using Chebyshev’s inequality, for some κ > 0 we have as n goes to infinity. Moreover, using the union bound, we have ≤ κn−1/3 , which converges to zero (n) (n) ≤ P E2 \|Gp~,~q \| (n) (n) L1 L2 . (15) (n) (n) ~ H (n) and ~uH (n) are in the sets M(n) and Note that for each graph H1,2 ∈ Gp~,~q , the mark count vectors m 1,2 1,2 U (n) , respectively. Additionally, we have \|M(n) \| ≤ (2n2/3 )\|Ξ1,2 \| and \|U (n) \| ≤ (2n2/3 )\|Θ1,2 \| . Therefore, (n) \|Gp~,~q \| ≤ (2n2/3 )(\|Ξ1,2 \|+\|Θ1,2 \|) max A1 (m, ~ ~u), (16) (n) m∈M ~ u ~ ∈U (n) where A1 (m, ~ ~u) := n {u(θ)}θ∈Θ1,2 ! n 2 {m(x)}x∈Ξ1,2 ! . Now, let m ~ (n) and ~u(n) be sequences in M(n) and U (n) , respectively. Then, for all x ∈ Ξ1,2 and θ ∈ Θ1,2 , we have m(n) (x)/n → px /2 and u(n) (θ)/n → qθ . Thereby, using Lemma 1, we have P log A1 (m ~ (n) , ~u(n) ) − ( x∈Ξ1,2 m(n) (x)) log n lim n→∞ n X = H(~q) + s(px ) = Σ(µER 1,2 ). x∈Ξ1,2 Substituting this into (16) and using the fact that \|m(n) (x) − npx /2\| ≤ n2/3 , we have (n) lim n→∞ log \|Gp~,~q \| − n dER 1,2 2 n 9 log n = Σ(µER 1,2 ). (17) Substituting this into (15), we have lim sup 1 (n) log P E2 n (n) ≤ lim sup log \|Gp~,~q \| − n dER 1,2 2 log n − nΣ(µER 1,2 ) n dER n( 1,2 2 − α1 − α2 ) log n + n(Σ(µER 1,2 ) − R1 − R2 ) n (n) (n) n(α1 + α2 ) log n + n(R1 + R2 ) − log L1 L2 . + lim sup n + lim sup The first term is nonpositive due to (17), the second term is strictly negative due to the assumption (n) (n) (14c), and the third term is nonpositive due to our choice of L1 and L2 . Consequently, the RHS is (n) strictly negative, which implies that P(E2 ) → 0. (n) (n) (n) (n) (n) (n) Now, we show that P(E3 \ E1 ) vanishes. In order to do so, for H1 ∈ G1 , define S2 (H1 ) := (n) (n) (n) (n) (n) {H2 ∈ G2 : H1 ⊕ H2 ∈ Gp~,~q }. Using the union bound, we have (n) (n) ≤ P E3 \ E1 ≤ X (n) (n) (n) P(G1,2 = H1,2 ) (n) (n) L2 (n) H1,2 ∈Gp ~,~ q 1 (n) L2 max (n) (n) H1,2 ∈Gp ~,~ q (n) \|S2 (H1 )\| (18) (n) (n) \|S2 (H1 )\|. It can be shown that (See Appendix B) max (n) lim sup (n) (n) H1,2 ∈Gp ~,~ q (n) ER dER 1,2 − d1 log n 2 n n→∞ where H1 (n) log \|S2 (H1 )\| − n (19) ER ≤ Σ(µER 2 \|µ1 ), (n) is the first marginal of H1,2 . Substituting in (18), we get n 1 (n) (n) lim sup log P E3 \ E1 ≤ lim sup n ER dER 1,2 −d1 2 (n) ER log n + nΣ(µER 2 \|µ1 ) − log L2 n ER dER 1,2 −d1 2 ER − α2 ) log n + n(Σ(µER 2 \|µ1 ) − R2 ) n (n) nα2 log n + nR2 − log L2 . + lim sup n ≤ lim sup n( (20) Note that the first term is strictly negative due to the assumption (14b), while the second term (n) (n) (n) is nonpositive due to our way of choosing L2 . This means that P(E3 \ E1 ) goes to zero as n (n) (n) goes to infinity. Similarly, P(E4 \ E1 ) converges to zero. This means that there exists a sequence of deterministic codebooks with vanishing probability of error, which completes the proof of achievability. 5.2 Proof of Achievability for the Configuration model Our achievability proof for this case is very similar in nature to that for the Erdős–Rényi case, with the modifications discussed below. 10 ~ = ck (d~(n) ) Let D(n) be the set of degree sequences d~ with entries bounded by ∆ such that ck (d) (n) for all 0 ≤ k ≤ ∆. Moreover, redefine M to be the set of mark count vectors m ~ such that P P P 2/3 , where recall that mn = ( ni=1 d(n) (i))/2. We x∈Ξ1,2 \|m(x) − mn γx \| ≤ n x∈Ξ1,2 m(x) = mn and (n) use the same as in the previous section, i.e. the set of vertex mark count vectors ~u P definition for U such that θ∈Θ1,2 \|u(θ) − nqθ \| ≤ n2/3 . In what follows, let X be a random variable with law ~r, X1 and X2 defined as in (11) and Γ = (Γ1 , Γ2 ) a random variable with law ~γ . − → (n) (n) We define W (n) to be the set of graphs H1,2 ∈ G1,2 such that: (i) dgH (n) ∈ D(n) , (ii) m ~ H (n) ∈ M(n) , 1,2 1,2 (iii) ~uH (n) ∈ U (n) , (iv) for all 0 ≤ l ≤ k ≤ ∆, recalling the notation in (2), we have 1,2 − → − → \|ck,l (dgH (n) , dgH (n) ) − nP (X = k, X1 = l) \| ≤ n2/3 , 1,2 1 (21) and (v), for all 0 ≤ l ≤ k ≤ ∆, we have − → − → \|ck,l (dgH (n) , dgH (n) ) − nP (X = k, X2 = l) \| ≤ n2/3 . 1,2 2 (22) We employ a similar random binning framework as in Section 5.1. For decoding, upon receiving (n) (n) (n) (n) (n) a pair (i, j), we form the set of graphs H1,2 ∈ W (n) such that f1 (H1 ) = i and f2 (H2 ) = j. If this set has only one element, we output it as the source graph; otherwise, we output an indication of (n) error. In order to prove the achievability, we consider the four error events Ei , 1 ≤ i ≤ 4, exactly as (n) those in the previous section, with Gp~,~q being replaced with W (n) . (n) (n) It can be shown that if G ∼ G(n; d~(n) , ~γ , ~q), the probability of G ∈ W (n) goes to one as n goes 1,2 1,2 (n) to infinity (see Lemma 4 in Appendix A). Therefore, P(E1 ) goes to zero. (n) To show that P(E2 ) vanishes, similar to the analysis in Section 5.1, we find an asymptotic upper bound for log \|W (n) \|. By only considering the conditions (i), (ii) and (iii) in the definition of W (n) , we have n (n) log \|W (n) \| ≤ log + log \|Gd~(n) \| {ck (d~(n) )}∆ k=0 mn 2/3 \|Ξ1,2 \| + log (2n ) max (23) (n) {m(x)}x∈Ξ1,2 m∈M ~ n . + log (2n2/3 )\|Θ1,2 \| max (n) {u(θ)} u ~ ∈U θ∈Θ1,2 By assumption, we have r0 < 1, hence dCM 1,2 > 0. The condition (8) together with Lemma 3 then implies that (n) d log \|G ~(n) \| − n dCM 1,2 2 log n = −s(dCM 1,2 ) − E [log X!] . n Using this together with Lemma 1 for the other terms in (23), we have lim n→∞ dCM log \|W (n) \| − n 1,2 2 log n ≤ −s(dCM lim sup 1,2 ) + H(X) n n→∞ dCM 1,2 + H(Γ) + H(Q) − E [log X!] = Σ(µCM 1,2 ), 2 where Γ and Q are random variables with law ~γ and ~q, respectively. 11 (24) (n) (n) (n) (n) Now, in order to show that P(E3 \E1 ) vanishes, we prove a counterpart for (19). For H1 ∈ G1 , (n) (n) (n) (n) (n) (n) we define S2 (H1 ) to be the set of graphs H2 ∈ G2 such that H1 ⊕ H2 ∈ W (n) . Then, it can be shown that (see Appendix C) (n) max (n) lim sup H1,2 ∈W (n) (n) log \|S2 (H1 )\| − n n n→∞ (n) Then, similar to (20), this shows that P(E3 This completes the proof of achievability. 5.3 CM dCM 1,2 − d1 log n 2 (25) CM ≤ Σ(µCM 2 \|µ1 ). (n) (n) \ E1 ) vanishes. The proof for P(E4 (n) \ E1 ) is similar. Proof of the Converse for the Erdős–Rényi case In this section, we show that every rate tuple (α1 , R1 , α2 , R2 ) ∈ R for the Erdős–Rényi scenario must satisfy the conditions (13a)–(13c). By definition, for a rate tuple (α1 , R1 , α2 , R2 ) ∈ R, there exist (m) (m) (m) (m) sequences R1 and R2 such that for each m, (α1 , R1 , α2 , R2 ) is achievable, and besides we have (m) (m) (m) (m) R1 → R1 and R2 → R2 . If we show that (α1 , R1 , α2 , R2 ) satisfies (13a)–(13c) for each m, it is easy to see that (α1 , R1 , α2 , R2 ) must also satisfy the same inequalities. Therefore, it suffices to show that any achievable rate tuple satisfies (13a)–(13c). For this, take an achievable rate tuple (α1 , R1 , α2 , R2 ) together with a corresponding sequence of (n) (n) (n) (n) hn, L1 , L2 i codes (f1 , f2 , g (n) ). By definition, we have (n) lim sup − (αi n log n + Ri n) ≤0 n log Li n→∞ i ∈ {1, 2}, (26) (n) (n) and also the error probability Pe goes to zero as n goes to infinity. Now, we define the set A(n) ⊆ G1,2 as (n) (n) (n) (n) (n) (n) (n) (n) A(n) := Gp~,~q ∩ {H1,2 ∈ G1,2 : g (n) (f1 (H1 ), f2 (H2 )) = H1,2 }, (27) (n) where Gp~,~q was defined in Section 5.1. In fact, A(n) is the set of “typical” graphs with respect to the (n) (n) Erdős–Rényi model that are successfully decoded by the code (f1 , f2 , g (n) ). In the following, let (n) (n) q ) be distributed according to the Erdős–Rényi model. Moreover, let PER be the law G1,2 ∼ G (n) (n; p~, ~ (n) (n) (n) (n) (n) (n) (n) of G1,2 , i.e. for H1,2 ∈ G1,2 , PER (H1,2 ) := P(G1,2 = H1,2 ). With this, we define a random variable (n) G̃1,2 (n) whose distribution is the conditional distribution of G1,2 , conditioned on lying in A(n) , i.e. ( (n) (n) (n) PER (H1,2 )/πn H1,2 ∈ A(n) (n) (n) (28) P G̃1,2 = H1,2 = 0 o.t.w. (n) (n) (n) where πn := P G1,2 ∈ A(n) is the normalizing factor. Additionally, let P̃ER be the law of G̃1,2 . If, (n) for i ∈ {1, 2}, M̃i (n) (n) denotes fi (G̃i ), we have (n) (n) log L1 + log L2 (n) (n) (n) (n) ≥ H(M̃1 ) + H(M̃2 ) ≥ H(M̃1 , M̃2 ) (29) (n) = H(G̃1,2 ), (n) where the last equality follows from the fact that, by definition, G̃1,2 takes values among the graphs (n) that are successfully decoded, and hence is uniquely identified given M̃1 12 (n) and M̃2 . (n) (n) (n) Now, we find a lower bound for H(G̃1,2 ). For doing so, note that for H1,2 ∈ G1,2 and n large enough, we have   ! P X X p n x∈Ξ1,2 px x (n) (n) − log PER (H1,2 ) = − mH (n) (x) log − − mH (n) (x) log 1 − 1,2 1,2 n n 2 x∈Ξ1,2 x∈Ξ1,2 X uH (n) (θ) log qθ . − θ∈Θ1,2 1,2 (n) Gp~,~q , (n) H1,2 (n) Gp~,~q On the other hand, due to the definition of if ∈ then, for all x ∈ Ξ1,2 have px px n − n2/3 ≤ mH (n) (x) ≤ n + n2/3 , and 1,2 2 2 nqθ − n2/3 ≤ uH (n) (θ) ≤ nqθ + n2/3 . (30) and θ ∈ Θ1,2 , we 1,2 Substituting these in (30) and using the inequality log(1 − x) ≤ −x which holds for x ∈ (0, 1), for n large enough, we have  P X px X px n x∈Ξ1,2 px (n) (n) n − n2/3 (log n − log px ) +  − log PER (H1,2 ) ≥ n + n2/3  − 2 2 n 2 x∈Ξ1,2 x∈Ξ1,2 X (nqθ − n2/3 ) log qθ . − θ∈Θ1,2 Using P x∈Ξ1,2 px = dER 1,2 and simplifying the above, we realize that there exists a constant c > 0 that (n) (n) does not depend on n or H1,2 , such that for all H1,2 ∈ A(n) , we have (n) (n) − log PER (H1,2 ) ≥ n X px X px X dER 1,2 qθ log qθ − cn2/3 log n log n − n log px + n −n 2 2 2 x∈Ξ1,2 x∈Ξ1,2 θ∈Θ1,2 dER 1,2 2/3 =n log n + nΣ(µER log n. 1,2 ) − cn 2 Now, if (n) G̃1,2 (31) is the random variable defined in (28), we have (n) H(G̃1,2 ) = − (n) X (n) (n) (n) P̃ER (H1,2 ) log P̃ER (H1,2 ) (n) H1,2 ∈A(n) = log πn − 1 πn (n) X (n) (n) (n) PER (H1,2 ) log PER (H1,2 ). (n) H1,2 ∈A(n) (n) Note that since the probability of error of the above code vanishes, i.e. Pe (n) (n) → 0, and P G1,2 ∈ Gp~,~q → (n) (n) 1, we have πn → 1 as n → ∞. On the other hand, with probability one, we have G̃1,2 ∈ Gp~,~q . Also, P (n) (n) by the definition of πn , we have H (n) ∈A(n) PER (H1,2 ) = πn . Thereby, employing the bound (31), we 1,2 have (n) H(G̃1,2 ) − n lim inf n→∞ n dER 1,2 2 13 log n ≥ Σ(µER 1,2 ). (32) Now, using the assumption (26) together with the bound (29), we have (n) 0 ≥ lim sup n→∞ (n) ≥ lim inf (n) log L1 + log L2 − (α1 + α2 )n log n − n(R1 + R2 ) n H(G̃1,2 ) − n n→∞ dER 1,2 2 log n − nΣ(µER n 1,2 ) + lim inf n→∞ n dER 1,2 2 log n + nΣ(µER 1,2 ) − (α1 + α2 )n log n − n(R1 + R2 ) . n (33) The first term is nonnegative due to (32). Consequently, ER d − α − α log n + n(Σ(µER n 1,2 1 2 1,2 ) − R1 − R2 ) 2 . 0 ≥ lim inf n→∞ n (34) ER Note that this is impossible unless α1 + α2 ≥ dER 1,2 /2. Furthermore, if α1 + α2 = d1,2 , it must be the ER ER case that R1 + R2 ≥ Σ(µ1,2 ). But this is precisely (13c) for µ1,2 = µ1,2 . Now, we turn to showing (13a). We have (n) log L1 (n) (n) (n) ≥ H(M̃1 ) ≥ H(M̃1 \|M̃2 ) (n) (n) (n) (n) (n) (n) = H(G̃1 , M̃1 \|M̃2 ) − H(G̃1 \|M̃1 , M̃2 ) (a) (n) (n) = H(G̃1 \|M̃2 ) (b) (n) (35) (n) ≥ H(G̃1 \|G̃2 ) (n) (n) = H(G̃1,2 ) − H(G̃2 ). (n) (n) where (a) uses the facts that M̃1 (n) M̃2 we Now, we (n) (n) can unambiguously determine G̃1,2 and (n) find an upper bound for H(G̃2 ). Note hence (n) and also, since G̃1,2 ∈ A(n) , given M̃1 is a function of G̃1 (n) G̃1 . Also, (b) uses data processing inequality. (n) that since G̃1,2 ∈ A(n) with probability one, we have (n) (n) H(G̃2 ) ≤ log \|A2 \|, where (n) A2 (n) (n) (n) := {H2 (n) (n) ∈ G2 (n) (n) : H1 and (n) ⊕ H2 (n) (36) (n) ∈ A(n) for some H1 (n) ∈ G1 }. (n) But for H1,2 := H1 ⊕ H2 ∈ A2 , since A2 ⊆ Gp~,~q , by definition we have that for all x ∈ Ξ1,2 and all θ ∈ Θ1,2 , X X \|uH (n) (θ) − nqθ \| ≤ n2/3 . \|mH (n) (x) − npx /2\| ≤ n2/3 and x∈Ξ1,2 1,2 1,2 θ∈Θ1,2 P Moreover, for x2 ∈ Ξ2 and θ2 ∈ Θ2 , we have mH (n) (x2 ) = x1 ∈Ξ1 ∪{◦1 } mH (n) ((x1 , x2 )) and uH (n) (θ2 ) = 1,2 2 2 P θ1 ∈Θ1 mH (n) ((θ1 , θ2 )). Using this in the above and using triangle inequality, we realize that for 1,2 (n) H2 (n) (n) (n) (n) ∈ A2 , we have m ~ H (n) ∈ M2 and ~uH (n) ∈ U2 , where M2 is the set of edge mark count 2 2 P (n) vectors m ~ such that x2 ∈Ξ2 \|m(x2 ) − npx2 /2\| ≤ n2/3 , and U2 is the set of vertex mark count vectors P ~u such that θ2 ∈Θ2 \|u(θ2 ) − nqθ2 \| ≤ n2/3 . Consequently, (n) \|A2 \| ≤ (2n2/3 )(\|Ξ2 \|+\|Θ2 \|) max (n) m∈M ~ 2 n 2 {m(x2 )}x2 ∈Ξ2 14 max (n) u ~ ∈U2 n . {u(θ2 )}θ2 ∈Θ2 (n) Using Lemma 1 and the definition of M2 (n) log \|A2 \| − n lim sup n n→∞ dER 2 2 (n) and U2 log n above, with Q = (Q1 , Q2 ) ∼ ~q, we get ≤ H(Q2 ) + X s(px2 ) = Σ(µER 2 ). x2 ∈Ξ2 Substituting into (36), we get (n) log H(G̃2 ) − n lim sup n n→∞ dER 2 2 log n ≤ Σ(µER 2 ). Using this together with (32) and substituting into (35) we get (n) lim inf log L1 − n ER dER 1,2 −d2 2 n n→∞ log n ER ER ER ≥ Σ(µER 1,2 ) − Σ(µ2 ) = Σ(µ1 \|µ2 ). Using a similar method as in (33) and (34), this implies (13a). The proof of (13b) is similar. This completes the proof of the converse for the Erdős–Rényi case. 5.4 Proof of the Converse for the configuration model The proof of the converse for the configuration model is similar to that for the Erdős–Rényi model presented in the previous section. Take an achievable rate tuple (α1 , R1 , α2 , R2 ) together with a (n) (n) (n) (n) sequence of hn, L1 , L2 i codes (f1 , f2 , g (n) ) achieving this rate tuple. Moreover, redefine the set A(n) to be (n) (n) (n) (n) (n) (n) (n) A(n) := W (n) ∩ {H1,2 ∈ G1,2 : g (n) (f1 (H1 ), f2 (H2 )) = H1,2 }, (37) (n) where the set W (n) was defined in Section 5.2. Now, let G1,2 ∼ G(n; d~(n) , ~γ , ~q) be distributed according (n) to the configuration model ensemble, and let G̃1,2 ∈ A(n) have the distribution obtained from that of (n) G1,2 (n) A (n) by conditioning on it lying in the set A(n) . Note that the normalizing constant πn := P(G1,2 ∈ (n) (n) ) goes to 1 as n → ∞ since P(G1,2 ∈ W (n) ) → 1 and the error probability of the code, Pe , (n) (n) (n) (n) vanishes. Moreover, let PCM and P̃CM be the laws of G1,2 and G̃1,2 , respectively. In the following, we show that dCM (n) H(G̃1,2 ) − n 1,2 2 log n ≥ Σ(µCM (38) lim inf 1,2 ), n→∞ n and (n) dCM H(G̃2 ) − n 22 log n lim sup ≤ Σ(µCM (39) 2 ). n n→∞ The rest of the proof is then identical to that of the previous section, so we only focus on proving the above two statements. − → (n) (n) For (38), note that for H1,2 ∈ G1,2 such that dgH (n) ∈ D(n) , where D(n) was defined in Section 5.2, 1,2 we have (n) (n) − log PCM (H1,2 ) = log n {ck (d~(n) )}∆ k=0 (n) d + log \|G ~(n) \| − 15 X x∈Ξ1,2 mH (n) (x) log γx − 1,2 X θ∈Θ1,2 uH (n) (θ) log qθ . 1,2 (n) Now, if H1,2 ∈ W (n) , using the definition of W (n) we realize that there exists a constant c > 0 such that X X n (n) (n) (n) +log \|G \|− − log PCM (H1,2 ) ≥ log m γ log γ − nqθ log qθ −cn2/3 =: Kn . n x x d~(n) {ck (d~(n) )}∆ k=0 x∈Ξ1,2 θ∈Θ1,2 (n) (n) Note that the right hand side is a constant independent of H1,2 and is denoted by Kn . Since G̃1,2 falls in (n) W (n) with probability one, this means that H(G̃1,2 ) ≥ log πn + Kn . But πn → 1 as n → ∞. Therefore, using the assumption (8) together with (24) from Section 5.2 and also the fact that mn /n → dCM 1,2 /2, we realize that (n) H(G̃1,2 ) − n lim inf n→∞ n dCM 1,2 2 log n ≥ H(X) − s(dCM 1,2 ) − E [log X!] + dCM 1,2 H(Γ) + H(Q), 2 where X ∼ ~r, Γ ∼ ~γ and Q ∼ ~ q . Note that the right hand side is precisely Σ(µCM 1,2 ), hence we have proved (38). (n) (n) (n) (n) (n) In order to show (39), note that H(G̃2 ) ≤ log \|A2 \| where A2 consists of graphs H2 ∈ G2 (n) (n) (n) (n) (n) (n) such that for some H1 ∈ G1 , we have H1 ⊕ H2 ∈ A(n) . Since A(n) ⊆ W (n) , for all H2 ∈ A2 we have X X \|mH (n) (x2 ) − mn γx2 \| ≤ n2/3 and (40) \|uH (n) (θ2 ) − nqθ2 \| ≤ n2/3 . 2 x2 ∈Ξ2 2 θ2 ∈Θ2 − → (n) (n) On the other hand, the condition (22) implies that dgH (n) ∈ D2 where D2 denotes the set of degree 2 sequences d~ of size n with elements bounded by ∆ such that ~ − nP (X2 = k) \| ≤ (∆ + 1)n2/3 \|ck (d) ∀0 ≤ k ≤ ∆, where X2 is the random variable defined in (11). Consequently, we have P (n) log \|A2 \| ≤ (n) log \|D2 \| + (n) max log \|G ~ \| d (n) ~ d∈D + 2 max (n) H2 (n) log ∈A2 x2 ∈Ξ2 (41) mH (n) (x2 ) 2 {mH (n) (x2 )}x2 ∈Ξ2 2 n + max log . (n) (n) {uH (n) (θ2 )}θ2 ∈Θ2 H2 ∈A2 (42) 2 (n) Note that (41) implies that \|D2 \| ≤ (2(∆ + 1)n2/3 )∆+1 maxd∈D (n) ~ 2 implies 1 (n) lim sup log \|D2 \| ≤ H(X2 ). n n→∞ n . ~ ∆ {ck (d)} k=0 Therefore, Lemma 1 (43) > 0. Hence, using Lemma 3 in On the other hand, the assumptions r0 < 1 and (7) imply that dCM 2 Appendix A we have lim sup (n) d maxd∈D (n) log \|G ~ ~ \|−n 2 (n) log n n n→∞ Moreover, if H2 dCM 2 2 ≤ −s(dCM 2 ) − E [log X2 !] . (n) is a sequence in A2 , from (40), for all x2 ∈ Ξ2 , we have lim P n→∞ mH (n) (x2 ) 2 ′ x′2 ∈Ξ2 mH2(n) (x2 ) =P γx 2 x′2 ∈Ξ2 16 γx′2 = P (Γ2 = x2 \|Γ2 6= ◦2 ) , (44) where Γ = (Γ1 , Γ2 ) has law ~γ . Additionally, we have 1 X dCM mH (n) (x2 ) = 2 . n→∞ n 2 2 lim x2 ∈Ξ2 Thereby, from Lemma 1, we have 1 lim sup max log n→∞ n H2(n) ∈A2(n) P x2 ∈Ξ2 mH (n) (x2 ) 2 {mH (n) (x2 )}x2 ∈Ξ2 2 ≤ dCM 2 H(Γ2 \|Γ2 6= ◦2 ). 2 (45) Finally, as we have uH (n) (θ2 )/n → qθ2 for all θ2 ∈ Θ2 , another usage of Lemma 1 implies that 2 lim sup n→∞ 1 n max log ≤ H(Q2 ), n H2(n) ∈A2(n) {uH (n) (θ2 )}θ2 ∈Θ2 (46) 2 where Q = (Q1 , Q2 ) has law ~ q . Now, combining (43), (44), (45) and (46) and substituting into (42), (n) (n) and also using the bound H(G̃2 ) ≤ log \|A2 \|, we realize that (n) lim sup n→∞ H(G̃2 ) − n n dCM 2 2 log n ≤ H(X2 ) − s(dCM 2 ) − E [log X2 !] + dCM 2 H(Γ2 \|Γ2 6= ◦2 ) + H(Q2 ). 2 But the right hand side is precisely Σ(µCM 2 ). This completes the proof of (39). As was mentioned before, the rest of the proof is identical to that in the previous section. 6 Conclusion We gave a counterpart of the Slepian–Wolf Theorem for graphical data, employing the framework of local weak convergence. We derived the rate region for two graph ensembles, namely an Erdős–Rényi model and a configuration model. A Asymptotic behavior of the entropy of the configuration model Here, we prove (12a)–(12c). Before this, we set some notation and state some general lemmas. In what follows, we employ the definitions of the sets W (n) and D(n) from Section 5.2. Moreover, Γ = (Γ1 , Γ2 ) and Q = (Q1 , Q2 ) be random variables with laws ~γ and ~q, respectively. Let β1 := P(Γ1 6= ◦1 ) and Γ̃1 (n) be a random variable on Ξ1 with the law of Γ1 conditioned on Γ1 6= ◦1 . Let F1,2 be a simple unmarked (n) (n) graph chosen uniformly at random from the set ∪ ~ (n) G . By definition, G ∼ G(n; d~(n) , ~γ , ~q) is d∈D obtained from (n) F1,2 d~ 1,2 by adding independent edge and vertex marks according to the laws of ~γ and ~q (n) (n) respectively. Let F1 be obtained from F1,2 by independently removing each edge with probability (n) (n) 1 − β1 . Then, G1 can be thought of as obtained from F1 by adding independent vertex and edge (n) (n) (n) (n) marks with the laws of Q1 and Γ̃1 , respectively. Hence, we may consider G1,2 , F1,2 , G1 and F1 on a joint probability space. It is straightforward to see the following. 17 Lemma 2. Assume X is an integer valued random variable taking value in {0, . . . , ∆} and 0 ≤ ǫ ≤ 1. Let {Yi }i≥1 be a sequence of i.i.d. Bernoulli random variables with P (Yi = 1) = ǫ. Define the random PX variable X1 to be i=1 Yi and X2 := X − X1 . Then, we have X H(X1 , X2 ) = H(X) + E [X] H(Y1 ) − E log . X1 The following lemma which is a direct consequence of Theorem 4.5 and Corollary 4.6 in [BC14], is useful in the asymptotic analysis of the count of the graphs with a given degree sequence. Lemma 3. Given an integer ∆, assume that Y is an integer random variable bounded by ∆ such that (n) d := E [Y ] > 0. Moreover, assume that for each n, ~a(n) = (1), . . . , a(n) (n)) is a degree sequence P(a n (n) (i) is even and, for 0 ≤ k ≤ ∆, of length n with entries bounded by ∆ such that bn := i=1 a (n) ck (~a )/n → P(Y = k). Then, we have (n) log \|G~a(n) \| − n→∞ n lim where s(d) := d 2 − d 2 bn 2 log n = −s(d) − E [log Y !] , log d. (n) (n) / W (n) ) ≤ κn−1/3 for some constant κ > 0. q ), we have P(G1,2 ∈ Lemma 4. If G1,2 ∼ G(n; d~(n) , ~γ , ~ Proof. For this, we note the following. Condition (i) in the definition of W (n) always holds for a (n) realization G1,2 . Chebyshev’s inequality implies that conditions (ii) and (iii) hold with probability at least 1 − κ1 n−1/3 , for some κ1 > 0. To show (iv), fix k and l and for 1 ≤ i ≤ n, let Yi be the indicator Pn − → − → of dgG(n) (i) = k and dgG(n) (i) = l. With Y := i=1 Yi , we have ck,l (dgG(n) , dgG(n) ) = Y . Note that 1,2 1 (n) 1,2 1 (n) an edge of G1,2 exists in G1 if its mark is not of the form (◦1 , x2 ), which happens with probability β1 . Therefore, i h h i dg (n) (i) (n) F1,2 E Yi \|F1,2 = 1 dgF (n) (i) = k β1l (1 − β1 )k−l . 1,2 l Consequently, i h k l (n) (n) ~ β (1 − β1 )k−l . E Y \|F1,2 = ck (d ) l 1 Since this is a constant, it is also equal to E [Y ]. Now, if sk,l := P (X = k, X1 = l), we have sk,l = rk kl β1l (1 − β1 )k−l . Hence, the assumption (8) implies that k l β (1 − β1 )k−l . (47) \|E [Y ] − nsk,l \| ≤ n1/2 l 1 (n) Furthermore, since edge marks are chosen independently, conditioned on F1,2 , if i and j are nonadjacent (n) (n) vertices in F1,2 , then Yi are Yj are independent, conditioned on F1,2 . As a result, if I denotes the set 18 (n) of (i, j) with 1 ≤ i 6= j ≤ n such that i and j are not adjacent in F1,2 , we have n i h i X h (n) (n) E Yi2 \|F1,2 + E Y 2 \|F1,2 = i=1 ≤n+ X (i,j)∈I / X 1≤i6=j≤n i h (n) E Yi Yj \|F1,2 i h i h X (n) (n) E Yi Yj \|F1,2 E Yi Yj \|F1,2 + (i,j)∈I ≤ n + 2mn + X i h (n) E Yi Yj \|F1,2 X i i h h (n) (n) E Yi \|F1,2 E Yj \|F1,2 (i,j)∈I (a) = n + 2mn + (i,j)∈I X ≤ n + 2mn + 1≤i6=j≤n i i h h (n) (n) E Yi \|F1,2 E Yj \|F1,2 i2 h (n) = n + 2mn + E Y \|F1,2 , (n) where (a) uses the fact that conditioned on F1,2 , the random variables Yi and Yj are independent for 1/2 (i, j) ∈ I.h From (8), and κ2 := ∆(∆ + 1)/4 is a constant. As we saw i we have \|mn − nd1,2 /2\| ≤ κ2 n (n) above, E Y \|F1,2 = E [Y ]. Therefore, we have Var Y ≤ κ3 n for some κ3 > 0. This together with (47) and Chebyshev’s inequality implies that the condition (iv) holds with probability at least 1 − κ4 n−1/3 , for some κ4 > 0. Similarly, the same statement holds for the condition (v). Now we show (12a)–(12c). In the following, with X ∼ ~r and X1 and X2 defined as in (11), let (n) B1,2 be the set of pairs of degree sequences d~ and ~δ with n elements bounded by ∆ such that for all ~ ~δ) − nP(X1 = k, X ′ = l)\| ≤ n2/3 , where X ′ := X − X1 . Moreover, let B (n) be 0 ≤ k, l ≤ ∆, \|ck,l (d, 1 1 1 ~ be the set of degree ~ ~δ) ∈ B (n) . For d~ ∈ B (n) , let B (n) (d) the set of d~ such that for some ~δ, we have (d, 1 1,2 2\|1 ~ ~δ) ∈ B (n) . sequences ~δ such that (d, 1,2 (n) (n) Now, we show (12a). Since G1,2 is formed by adding independent vertex and edge marks to F1,2 , we have (n) (n) H(G1,2 ) = log \|D(n) \| + log \|Gd~(n) \| + mn H(Γ) + nH(Q). ∆K 1/2 From (8), we have \|mn − ndCM . Moreover, we have E [X] > 0. Consequently, using 1,2 /2\| ≤ 2 n 1 (n) Lemma 3 and the fact that n log \|D \| → H(X), we get (12a). (n) We now turn to showing (12b). Since the expected number of the edges in F1 is ndCM 1 /2, we have dCM (n) (n) H(G1 ) = H(F1 ) + n 1 H(Γ1 \|Γ1 6= ◦1 ) + nH(Q1 ). (48) 2 (n) (n) With this, we focus on H(F1 ). With En being the indicator of G1,2 ∈ / W (n) , we have (n) (n) H(F1 ) = H(F1 \|En = 0)P(En = 0) (n) +H(F1 \|En = 1)P(En = 1). From Lemma 4, we have (n) (n) H(F1 \|En = 1)P(En = 1) ≤ (H(F1,2 ) + mn H(β1 ))κn−1/3 , 19 (49) (n) where H(β1 ) denotes the binary entropy of β1 . Note that as we discussed above, H(F1 ) = O(n log n). Thereby, the RHS of the above is o(n). On the other hand, by the definition of W (n) , if E = 0, − → (n) (n) (n) (n) dgF (n) ∈ B1 . Therefore, H(F1 \|E = 0) ≤ log B1 + maxd∈B \|. The assumption r0 < 0 (n) log \|G ~ d~ 1 1 together with (7) imply that dCM > 0. Hence, using Lemma 3 together with (49), 1 (n) H(F1 ) − n lim sup n (n) Now, let F̃1 note that dCM 1 2 log n ≤ −s(dCM 1 ) + H(X1 ) − E [log X1 !] . (50) (n) (n) be the unmarked graph consisting of the edges removed from F1,2 to obtain F1 (n) (n) (n) (n) and (n) H(F1 ) = H(F1 , F̃1 ) − H(F̃1 \|F2 ) (51) (n) (n) (n) = H(F1,2 ) + mn H(β1 ) − H(F̃1 \|F1 ) − → → (n) − Note that, conditioned on E = 0, we have dgF̃ (n) ∈ B2\|1 (dgF (n) ). Moreover, the assumption (7) 1 1 CM together with r0 < 0 imply that dCM > 0. Hence, using a similar method in proving (50), we 1,2 − d1 have (n) (n) H(F̃1 \|F1 ) − n lim sup n CM dCM 1,2 −d1 2 log n CM ≤ −s(dCM 1,2 − d1 ) +H(X1′ \|X1 ) − E [log X1′ !] . (n) Using this together with the asymptotic of H(F1,2 ) which was derived above in showing (12a) and substituting into (51), followed by a simplification using Lemma 2, we get (n) H(F1 ) − n lim inf n dCM 1 2 log n (52) ≥ −s(dCM 1 ) + H(X1 ) − E [log X1 !] . This together with (50) and (48) completes the proof of (12b). The proof of (12c) is similar. (n) (n) Bounding \|S2 (H1 )\| for the Erdős–Rényi case B (n) (n) (n) Note that for G1,2 ∈ G1,2 and H2 (n) (n) ∈ G2 , if G1 (n) ⊕ H2 (n) ∈ Gp~,~q , we have m ~ G(n) ⊕H (n) ∈ M(n) and 1 2 ~uG(n) ⊕H (n) ∈ U (n) . On the other hand, for fixed m ~ ∈ M(n) and ~u ∈ U (n) , if for all x1 ∈ Ξ1 we have 1 2 (n) m(x1 ) = mG(n) (x1 ) and for all θ1 ∈ Θ1 we have u(θ1 ) = uG(n) (θ1 ), then the number of H2 1 1 m ~ G(n) ⊕H (n) = m ~ and ~uG(n) ⊕H (n) = ~u is at most 1 2 A2 (m, ~ ~u) := 1 Y x1 ∈Ξ1 such that 2 m(x1 ) {m(x1 , x2 )}x2 ∈Ξ2 ∪{◦2 } !! × !  P Y − x1 ∈Ξ1 m(x1 ) × {m(◦1 , x2 )}x2 ∈Ξ2 θ ∈Θ n 2 1 1 ! u(θ1 ) , {u(θ1 , θ2 )}θ2 ∈Θ2 where we have used the notational conventions in (3) and (4). Consequently, we have max (n) (n) G1,2 ∈Gp ~,~ q (n) (n) \|S2 (G1 )\| ≤ \|M(n) \|\|U (n) \| max A2 (m, ~ ~u) (n) m∈M ~ u ~ ∈U (n) ≤ (2n2/3 )(\|Ξ1,2 \|+\|Θ1,2 \|) max A2 (m, ~ ~u). (n) m∈M ~ u ~ ∈U (n) 20 (53) Now, if m ~ (n) and ~u(n) are sequences in M(n) and U (n) , respectively, for all x ∈ Ξ1,2 we have (n) m (x)/n → px /2. Furthermore, for all x1 ∈ Ξ1 and θ1 ∈ Θ1 , we have m(n) (x1 )/n → px1 /2 and ~ (n) and ~u(n) , with Q = (Q1 , Q2 ) u(n) (θ1 )/n → qθ1 . As a result, using Lemma 1, for any such sequences m having law ~ q we have P log A2 (m ~ (n) , ~u(n) ) − ( x2 ∈Ξ2 m(n) (◦1 , x2 )) log n lim n→∞ n ! X X px p(x1 ,x2 ) 1 = s(p◦1 ,x2 ) + H 2 px 1 x2 ∈Ξ2 ∪{◦2 } x2 ∈Ξ2 x1 ∈Ξ1 ! X qθ1 ,θ2 qθ1 H + qθ1 θ2 ∈Θ2 θ1 ∈Θ1 X X = H(Q2 \|Q1 ) + s(px ) − s(px1 ) x∈Ξ1,2 = x1 ∈Ξ1 ER Σ(µER 2 \|µ1 ), where the second inequality follows by rearranging the terms and using the definition of s(.). Using the fact that \|m(n) (◦1 , x2 ) − np◦1 ,x2 /2\| ≤ n2/3 , log An (m ~ (n) , ~u(n) ) − n lim n→∞ n ER dER 1,2 −d1 2 log n ER = Σ(µER 2 \|µ1 ). This together with (53) implies (19). C (n) (n) Bounding \|S2 (H1 )\| for the configuration model (n) (n) (n) Here, we find an upper bound for maxH (n) \|S2 (H1 )\| and use it to show (25). Fix G1,2 ∈ W (n) and 1,2 (n) assume H2 (n) (n) (n) (n) (n) (n) ∈ S2 (G1 ). With H1,2 := G1 ⊕ H2 , let H̃2 (n) be the subgraph of H1,2 consisting of (n) edges not present in G1 . Employing the notation of Appendix A, by the definition of the set W (n) , − → → (n) (n) − (n) we have dgH̃ (n) ∈ B2\|1 (dgG(n) ). Therefore, we can think of H1,2 as constructed from G1 by adding a 2 1 − → (n) graph to G1 with degree sequence dgH̃ (n) , markings its edges, adding second domain marks to edges 2 (n) in G1 , and also adding second domain marks to vertices. Consequently, max (n) H1,2 ∈W (n) (n) (n) log \|S2 (H1 )\| ≤ + max log (n) m∈M ~ + max log u∈U (n) ~ max (n) G1,2 ∈W (n) → (n) − log \|B2\|1 (dgG(n) )\| + 1 mn − x1 ∈Ξ1 m(x1 ) {m(◦1 , x2 )}x2 ∈Ξ2 u(θ1 ) {u(θ1 , θ2 )}θ2 ∈Θ2 ! Y θ1 ∈Θ1 log \|G~δ \| G 1 ! P (n) max → (n) (n) − G1,2 ∈W (n) ,~ δ∈B2\|1 (dg (n) ) Y x1 ∈Ξ1 m(x1 ) {m(x1 , x2 )}x2 ∈Ξ2 ! (n) The definition of B2\|1 implies that → 1 (n) − max log \|B2\|1 (dgH (n) )\| = H(X − X1 \|X1 ). (n) n→∞ n H 1 (n) 1,2 ∈W lim 21 (54) CM Note that the assumption (7) together with r0 < 1 implies that dCM > 0. Therefore, Lemma 3 1,2 − d1 in Appendix A implies that lim sup max n→∞ (n) G1,2 ∈W (n) (n) δ log \|G~ \| − n CM dCM 1,2 −d1 2 log n n CM ≤ −s(dCM 1,2 − d1 ) − E [log(X − X1 )!] . − → ~ δ∈B(dg (n) ) G 1 Furthermore, a usage of Lemma 1 implies that the third and the fourth terms in (54) divided by n dCM converge to 1,2 γ and Q has law ~q. 2 H(Γ2 \|Γ1 ) and H(Q), respectively, where Γ = (Γ1 , Γ2 ) has law ~ Putting these together, we have (n) (n) lim maxG(n) ∈W (n) log \|S2 (G1 )\| − n 1,2 n→∞ CM dCM 1,2 −d1 2 n log n CM = −s(dCM 1,2 − d1 ) + H(X − X1 \|X1 ) −E [log(X − X1 )!] + dCM 1,2 H(Γ2 \|Γ1 ) + H(Q2 \|Q1 ). 2 CM Using Lemma 2 and rearranging, this is precisely equal to Σ(µCM 2 \|µ1 ), which completes the proof of (25). Acknowledgments The authors acknowledge support from the NSF grants ECCS-1343398, CNS- 1527846, CCF-1618145, the NSF Science & Technology Center grant CCF- 0939370 (Science of Information), and the William and Flora Hewlett Foundation supported Center for Long Term Cybersecurity at Berkeley. References [Abb16] Emmanuel Abbe. Graph compression: The effect of clusters. In Communication, Control, and Computing (Allerton), 2016 54th Annual Allerton Conference on, pages 1–8. IEEE, 2016. [AL07] David Aldous and Russell Lyons. Processes on unimodular random networks. Electron. J. Probab, 12(54):1454–1508, 2007. [AS04] David Aldous and J Michael Steele. The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on discrete structures, pages 1–72. Springer, 2004. [BC14] Charles Bordenave and Pietro Caputo. Large deviations of empirical neighborhood distribution in sparse random graphs. Probability Theory and Related Fields, pages 1–73, 2014. [Bil13] Patrick Billingsley. Convergence of probability measures. John Wiley & Sons, 2013. [BS01] Itai Benjamini and Oded Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6:no. 23, 13 pp. (electronic), 2001. [CS12] Yongwook Choi and Wojciech Szpankowski. Compression of graphical structures: Fundamental limits, algorithms, and experiments. IEEE Transactions on Information Theory, 58(2):620–638, 2012. 22 [CT12] Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012. [DA17] Payam Delgosha and Venkat Anantharam. Universal lossless compression of graphical data. In Information Theory (ISIT), 2017 IEEE International Symposium on, pages 1578–1582. IEEE, 2017. 23	7
REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS arXiv:1308.5285v1 [math.AC] 24 Aug 2013 KUEI-NUAN LIN AND CLAUDIA POLINI A BSTRACT. In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M ) onto it, where M is the direct sum of powers of m. We compute a Gröbner basis of the ideal defining R(M ). It turns out that the normal domain R(M ) is a Koszul algebra and from this we deduce that in many instances R(I) is a Koszul algebra as well. 1. I NTRODUCTION In this paper we investigate the Rees algebra R(I) = R[It] as well as the special fiber ring F(I) = R(I) ⊗ k of an ideal I in a standard graded algebra R over a field k. These objects are important to commutative algebraists because they encode the asymptotic behavior of the ideal I and to algebraic geometers because their projective schemes define the blowup and the special fiber of the blowup of the scheme Spec(R) along V (I). One of the central problems in the theory of Rees rings is to describe R(I) and F(I) in terms of generators and relations (see for instance [24, 29, 10, 28, 26, 17, 18, 15, 11, 6, 13, 14]). This is a challenging quest which is open for most classes of ideals, even three generated ideals in a polynomial ring in two variables (see for instance [2, 1, 22]). The goal is to find an ideal A in a polynomial ring S = R[T1 , . . . , Ts ] so that R(I) = S/A. If the ideal I is generated by forms of the same degree, then these forms define rational maps between projective spaces and the special fiber ring and the Rees ring describe the image and the graph of such rational maps, respectively. By computing the defining equations of these algebras, one is able to exhibit the implicit equations of the graph and of the variety parametrized by the map. This classical and difficult problem in elimination theory has also been studied in applied mathematics, most notably in modeling theory, where it is known as the implicitization problem (see for instance [3, 4, 5, 9]). If the ideal I is not generated by forms of the same degree, one can consider the truncation of I past its generator degree. In this paper we treat truncations of complete intersection ideals in a polynomial ring. More precisely, let R = k[x1 , . . . , xn ] be a polynomial ring over a field k with homogeneous maximal ideal m, let f1 , . . . , fr be a homogeneous regular sequence in R AMS 2010 Mathematics Subject Classification. Primary 13A30; Secondary 13H15, 13B22, 13C14, 13C15, 13C40. The first author was partially supported by an AWM-NSF mentor travel grant. The second author was partially supported by NSF grant DMS-1202685 and NSA grant H98230-12-1-0242. Keywords: Elimination theory, Gröbner basis, Koszul algebras, Rees algebra, Special fiber ring. 1 2 K. N. LIN AND C. POLINI of degrees d1 ≥ . . . ≥ dr , let d ≥ d1 be an integer, and write ai = d − di . The truncation I = (f1 , . . . , fr )≥d of the complete intersection (f1 , . . . , fr ) in degree d is the R-ideal generated by P the forms in (f1 , . . . , fr ) of degree at least d. In other words, I = (f1 , . . . , fr ) ∩ md = ri=1 mai fi . The Cohen-Macaulayness of the Rees algebra of such ideals was previously studied in [12], where the authors show that R(I) is Cohen-Macaulay for all d > D and they give a sharp estimate for D. However, the defining equations of R(I) were unknown. In this paper we describe them explicitly for all d when r = 2 and for d ≥ d1 + d2 when r ≥ 3. Furthermore we prove that the Rees ring and the special fiber ring are Koszul algebras for d ≥ d1 + d2 and for d ≥ d1 + d2 − 1 if r = 2. To determine the defining equations of R(I) we map another Rees ring R(M ) onto R(I), where M is the module ma1 ⊕ . . . ⊕ mar . Our aim then becomes to find the defining ideal of R(M ) and the kernel Q, 0 → Q −→ R(M ) −→ R(I) → 0 . The problem of computing the implicit equation of R(M ) is interesting in its own right and it was previously addressed in [23], where the relation type of R(M ) was computed. We solve it in Section 2. It turns out that R(M ) and F(M ) are normal domains whose defining ideals have a Gröbner basis of quadrics; hence, they are Koszul algebras. For r = 2, the kernel Q is a height one prime ideal of the normal domain R(M ); therefore it is a divisorial ideal of R(M ). Our goal is then reduced to explicitly describing ideals that represent the elements in the divisor class group of R(M ). The approach is very much inspired by [21] and [20]. For r = 2 and d ≥ d1 + d2 − 1 or r ≥ 3 and d ≥ d1 + d2 , the ideal I has a linear presentation and Q turns out to be a linear ideal in the T ′ s. Using this we prove that the defining ideal of R(I) has a quadratic Gröbner basis and hence R(I) is a Koszul algebra as well. 2. T HE B LOWUP ALGEBRAS OF DIRECT SUMS OF POWERS OF THE MAXIMAL IDEAL Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k with homogeneous maximal ideal m. Let 0 ≤ a1 ≤ . . . ≤ ar be integers. Write a = a1 , . . . , ar . In this section, we will describe explicitly the Rees algebra and the special fiber ring of the module M = Ma = ma1 ⊕ · · · ⊕ mar in terms of generators and relations and we will prove that they are Koszul normal domains. We will end the section with a study of the divisor class group of the blowup algebras of M . Definition 2.1. Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k. Let a be a positive integer, write Ja and Ja′ for the two sets of multi-indices in (N ∪ {0})n−1 and Nn−1 , respectively, that are defined as follows Ja = {j = (jn−1 , . . . , j1 ) \| 0 ≤ j1 ≤ . . . ≤ jn−1 ≤ a} , Ja′ = {j = (jn−1 , . . . , j1 ) \| 1 ≤ j1 ≤ . . . ≤ jn−1 ≤ a} . REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 3 Write xa,j and xa,j,s for the monomials x a,j = n Y j −ji−1 xi i i=1 with j ∈ Ja , j0 = 0, jn = a and xa,j,s = xs a,j x x1 with j ∈ Ja′ , 1 ≤ s ≤ n . The Rees algebra R(M ) of M is the subalgebra R(M ) = R[{xal ,j tl \| 1 ≤ l ≤ r, j ∈ Jal }] ⊂ R[t1 , . . . , tr ] of the polynomial ring R[t1 , . . . , tr ], while the special fiber ring F(M ) is the subalgebra of R(M ) F(M ) = k[{xal ,j tl \| 1 ≤ l ≤ r, j ∈ Jal ] ⊂ R(M ) . To find a presentation of these algebras we consider the polynomial rings T = Ta = k[{Tl,j \| 1 ≤ l ≤ r, j ∈ Jal }] S = Sa = R ⊗k Ta = Ta [x1 , . . . , xn ] in the new variables Tl,j and the epimorphisms of algebras (1) φ : S → R(M ) ψ : T → F(M ) defined by φ(Tl,j ) = ψ(Tl,j ) = xal ,j tl . Notice that ψ is the restriction of φ to T . We can assume that the ai ’s are all positive because if ai = 0 for 1 ≤ i ≤ s with s ≤ r then the Rees algebra R(M ) is isomorphic to a polynomial ring over the Rees algebra of mas+1 ⊕ · · · ⊕ mar R(⊕rl=1 mal ) ∼ = R(⊕rl=s+1 mal )[t1 , · · · , ts ] , and likewise for the special fiber ring. Furthermore, we can treat simultaneously the special fiber ring and the Rees algebra since R(⊕rl=1 mal ) ∼ = F(m ⊕ (⊕rl=1 mal )) and F(M ) = R(M )/mR(M ) . Definition 2.2. Let τ be the lexicographic order on a set of multi-indices in (N ∪ {0})n , i.e. p > q if the first nonzero entry of p− q is positive. Let 1 ≤ a1 ≤ . . . ≤ ar be integers. Write a = a1 , . . . , ar . Order the set of multi-indices {(l, j) \| 1 ≤ l ≤ r, j ∈ Ja′l } by τ . Write Tl,j,s for the variable Tl,jn−1 ,...,js ,js−1 −1,...,j1 −1 . P +n−2 (1) Let Ba be the n × ( rl=1 aln−1 ) matrix whose entry in the s-row and the (l, j)-column is the variable Tl,j,s with 1 ≤ s ≤ n, 1 ≤ l ≤ r, and j ∈ Ja′l . 4 K. N. LIN AND C. POLINI (2) Let Ca be the n × (1 + Pr l=1 al +n−2 ) n−1  matrix x1  .. Ca =  . xn   Ba  . Example 2.3. For instance if n = 3, r = 2 and a = 1, 2 then Ba is the 3 × 4 matrix   T1,1,1 T2,1,1 T2,2,1 T2,2,2 Ba =  T1,1,0 T2,1,0 T2,2,0 T2,2,1  . T1,0,0 T2,0,0 T2,1,0 T2,1,1 Write L for the kernel of the epimorphism φ defined in (1). Notice that φ(Tl,j,s ) = xa,j,s tl = xs a,j x1 x tl hence one can easily deduce the inclusion I2 (Ca ) ⊂ L . Our goal is to show that the above inclusion is an equality. In order to establish this claim it suffices to show that I2 (Ca ) is a prime ideal of dimension at most n + r = dim R(M ) (see for instance [27, 2.2]). Theorem 2.4. Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k with homogeneous maximal ideal m. Let 1 ≤ a1 ≤ . . . ≤ ar be integers and let M = ma1 ⊕ · · · ⊕ mar . The Rees algebra and the special fiber ring of M are Koszul normal domains. Furthermore, R(M ) = S/I2 (Ca ) F(M ) = T /I2 (Ba ) , where a, Ca , and Ba are as in Definition 2.2. An important step in the proof of Theorem 2.4 is to show that the set of 2 by 2 minors of Ca forms a Gröbner basis for I2 (Ca ). In the next lemma we will show much more. Indeed, the set of 2×2 minors of any submatrix Da of Ca forms a Gröbner basis for I2 (Da ). Notice that τ induces an ordering of the variables of T . With respect to this ordering we consider the reverse lexicographic order on the monomials in the ring T , which we also call τ . Remark 2.5. Let a′ = 1, a1 , . . . , ar and denote with Ta′ the polynomial ring associated with the sequence a′ . Notice that Ta′ /I2 (Ba′ ) ∼ = Sa /I2 (Ca ) and the matrix Ca is equal (after changing the name of the variables) to the matrix Ba′ . Lemma 2.6. Adopt assumption 2.2 and let Da be any submatrix of Ca with the same number of rows. The set of 2 × 2 minors of Da forms a Gröbner basis for I2 (Da ) with respect to τ . We use a general strategy to compute the Gröbner bases that we outline below. REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 5 Strategy 2.7. Let R be a polynomial ring over a field with a fixed monomial order ” > ”. Let D be a collection of 2 × 2 minors of a matrix with entries in R. The Buchberger criterion (see for instance [8, 15.8]) states that a generating set D of an ideal is a Gröbner basis for the ideal if the S-polynomials (or S-pairs) of any two elements of D reduces to zero module D. The following strategy will be used to show that the remainders of the S-pairs of the elements of D reduces to zero modulo D (see for instance [7, Section 2.9, Definition 1 and Theorem 3]). Let ad − bc and hg − ef be two elements of D. To compute the S-polynomial S(ad − bc, hg − ef ) we can assume, for instance, that h = a and ad > bc and ag > ef because otherwise the leading terms of the two polynomials are relatively prime and therefore S(ad − bc, ag − ef ) reduces to zero modulo D (see for instance [7, Section 2.9, Proposition 4]). We consider the matrix   d b 0 M = c a f . G e g We obtain the following equality by computing the determinant of M in two ways d (2) a f e g −g d b c a =b c f G g −f d b . G e Notice that the left hand side of equation (2) is our S-polynomial S(ag − ef, ad − bc). If by a suitable choice of G ∈ R, the polynomials cg − f G and de − bG are in D and each term on the right hand side has order strictly less than adg, then the S-pair S(ag − ef, ad − bc) has remainder zero modulo D and in particular it reduces to zero modulo D (see for instance [7, Section 2.9, Lemma 2]). The choice of the polynomial G and the matrix M depends on the polynomials we start with as we will explain later. Proof of Lemma 2.6 Because of Remark 2.5 it is enough to prove the statement for Da any submatrix of Ba′ with the same number of rows. Let D denote the set of 2 × 2 minors of Da . We use the strategy described above to show that the S-polynomials of any two elements of D reduces to zero module D. To simplify notation, we will use (l, j, s) to represent the entry Tl,j,s in the matrix Da . The leading terms of two elements of D h1 = (l1 , i, s) (l2 , j, s) (l1 , i, t) (l2 , j, t) and h2 = (l3 , p, u) (l4 , q, u) , (l3 , p, v) (l4 , q, v) where 1 ≤ s < t ≤ n, (l2 , j, s) > (l1 , i, s), 1 ≤ u < v ≤ n, and (l4 , q, u) > (l3 , p, u) are (l1 , i, s)(l2 , j, t) and (l3 , p, u)(l4 , q, v). Notice that Tl,j,s > Tl,j,t if s < t. These will be relatively prime unless (l1 , i, s) = (l3 , p, u), or (l1 , i, s) = (l4 , q, v), or (l2 , j, t) = (l3 , p, u), or (l2 , j, t) = (l4 , q, v). Since (l1 , i, s) = (l4 , q, v) and (l2 , j, t) = (l3 , p, u) are symmetric, it suffices to consider three cases. Case 1: Set (l1 , i, s) = a (l2 , j, s) = f (l1 , i, s) = a (l3 , p, s) = c h1 = and h2 = , (l1 , i, t) = e (l2 , j, t) = g (l1 , i, u) = b (l3 , p, u) = d 6 K. N. LIN AND C. POLINI where we can assume s < u ≤ t and l1 ≤ l2 ≤ l3 . We use equation (2) and the matrix M of Strategy 2.7 with G = (l3 , p, t). The S-polynomial S(h1 , h2 ) reduces to zero modulo D because b and e are smaller than any other entries of the two matrices defining h1 and h2 . Case 2: Set h1 = (l1 , i, s) = a (l2 , j, s) = e (l1 , i, t) = f (l2 , j, t) = g and h2 = (l3 , p, u) = d (l1 , i, u) = c (l3 , p, s) = b (l1 , i, s) = a , where u < s < t and l3 ≤ l1 ≤ l2 . We use equation (2) and the matrix M of Strategy 2.7 with G = (l2 , j, u). The S-polynomial S(h1 , h2 ) reduces to zero modulo D because b and f are smaller than any other entries of the two matrices defining h1 and h2 . Case 3: Set h1 = (l1 , i, s) = d (l2 , j, s) = b (l1 , i, t) = c (l2 , j, t) = a and h2 = (l3 , p, u) = g (l2 , j, u) = e (l3 , p, t) = f (l2 , j, t) = a , where we can assume u ≤ s < t and l1 ≤ l3 ≤ l2 . We use equation (2) and the matrix M of Strategy 2.7 with G = (l1 , i, u). The S-polynomial S(h1 , h2 ) reduces to zero modulo D because c and f are smaller than any other entries of the two matrices defining h1 and h2 . Corollary 2.8. Adopt assumptions 2.2 and let Da be any submatrix of Ca with the same number of rows. The ideal I2 (Da ) is prime in Sa . Proof. Write S = Sa . Notice that by Lemma 2.6 the variable u ∈ S appearing in the first column and the last row of the matrix Da does not divide any element in the generating set of inτ I2 (Da ). Hence u is regular on S/I2 (Da ). After localizing at u the ideal I2 (Da )u is isomorphic to an ideal generated by variables, which is a prime ideal in the ring Su . Thus I2 (Da ) is a prime ideal in the ring S. Corollary 2.9. Adopt assumptions 2.2. The initial ideal inτ (I2 (Ba ) is generated by the monomials Tl1 ,i,s Tl2 ,j,t with (l1 , i) <τ (l2 , j) and s < t. Proof. The assertion follows from Lemma 2.6. Proof of Theorem 2.4 We first show that for any sequence of r positive integers a = a1 , . . . , ar , the Rees algebra of M is defined by the ideal of minors I2 (Ca ), R(M ) = S/I2 (Ca ). Let L be the kernel of the epimorphism φ defined in (1). Recall that I2 (Ca ) ⊂ L, where the first ideal is prime, according to Corollary 2.8, and the second ideal has dimension n + r, according to [27, 2.2]. Hence to show that equality holds it will be enough to prove that the dimension of I2 (Ca ) is at most n + r. By Remark 2.5, Ta′ /I2 (Ba′ ) ∼ = Sa /I2 (Ca ). Hence it will be enough to prove that the dimension of I2 (Ba′ ) is at most n + r which is equivalent to show that the dimension of I2 (Ba ) REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 7 is at most n + r − 1. As ht I2 (Ba ) = ht inτ (I2 (Ba )), we can compute the dimension of inτ (I2 (Ba ). Let U be the set of r + n − 1 variables U = {{Tl,0,··· ,0 }1≤l≤r , Tr,ar ,··· ,ar , Tr,ar ,...,ar ,0, . . . , Tr,ar ,0,··· ,0 } . Consider the prime p ∈ Spec(T ) generated by all the variables of T that are not in U . We claim that p is a minimal prime over inτ (I2 (Ba ). It is clear that the image of inτ (I2 (Ba )) in the quotient ring T /p is zero. To show the claim, consider the prime ideal p′ ⊂ p obtained by deleting one variable Tl,j , with j ∈ Jal , from the generating set of p. If l < r, then Tl,j Tr,0 ∈ inτ (I2 (Ba ) \ p′ . Hence the image of inτ (I2 (Ba )) in the quotient ring T /p′ is not zero. If l = r, then 0 < ju ≤ . . . ≤ ji < ar for some u, i with 1 ≤ u ≤ i ≤ n − 1. Thus we can assume that Tl,j = Tr,ar ,...,ar ,ji ,...,ju ,0,...,0 6∈ p′ . Therefore the element Tl,j appears at least in two columns of the matrix Ba , namely the columns corresponding to the sequences r, ar , . . . , ar , ji , . . . , ju , 1, . . . , 1 and r, ar , . . . , ar , ji + 1, . . . , ju + 1, 1, . . . , 1. 2 ∈ in (I (B ) \ p′ . Again the image of in (I (B )) in the quotient ring T /p′ is not zero. Hence Tl,j τ 2 a τ 2 a Hence dim T /I2 (Ba ) = dim T /inτ (I2 (Ba )) = dim T /p = \|U \| = r + n − 1 as claimed, where the second equality follows as I2 (Ba ) is a prime ideal (see for instance [19]). From the above follows that for any sequence of r positive integers a = a1 , . . . , ar the special fiber ring of M is defined by the ideal of minors I2 (Ba ), indeed F(M ) = k ⊗R R(M ) = k ⊗R S/I2 (Ca ) = T /I2 (Ba ). For any sequence a the ideals I2 (Ca ) and I2 (Ba ) have a Gröbner basis of quadrics according to Lemma 2.6, hence both the Rees algebra and the special fiber ring of M are Koszul domains. Normality follows because F(M ) is a direct summand of R(M ) which in turn is a direct summand of R[t1 , . . . , tr ]. The latter claim can be easily seen once we consider R(M ) as a Nr+1 −graded R[t1 , . . . , tr ]-algebra. In the rest of this section we study the divisor class group of the normal domain A = Ta /I2 (Ba ) for any sequence a. Definition 2.10. Let K be the A-ideal generated by all the variables appearing in the first row of Ba , i.e. all the variables Tl,j with 1 ≤ l ≤ r and j ∈ Ja′l . Theorem 2.11. The divisor class group Cl(A) is cyclic generated by K. Proof. To compute the divisor class group we use Nagata’s Theorem: If W ⊂ A is a multiplicatively closed set, then there is an exact sequence of Abelian groups 0 −→ U −→ Cl(A) −→ Cl(AW ) −→ 0 , 8 K. N. LIN AND C. POLINI where U is the subgroup of Cl(A) generated by {[p] \| p a height one prime ideal with p ∩ W 6= ∅} . i i \| i ∈ Z}. Notice Tr,ar 6∈ I2 (Ba ). Hence = Tr,a We use the above theorem with W = {Tr,a r r ,··· ,ar Tr,ar is regular on T /I2 (Ba ). After localizing at Tr,ar the ideal I2 (Ba )Tr,ar is isomorphic to an ideal generated by variables, and the ring ATr,ar is a polynomial ring, hence factorial. Thus Cl(AW ) = 0 and Cl(A) = U . Now we will show that U is cyclic generated by K. Notice that [K] ∈ U because K is a prime ideal of height one containing Tr,ar . Clearly, Tr,ar ∈ K. To show that A/K is a domain of dimension dim A − 1, let R′ = k[x2 , . . . , xn ] be the polynomial ring over k in n − 1 variables, let m′ be its homogeneous maximal ideal, and let M ′ = m′a1 ⊕ · · · ⊕ m′ar . The claim follows by ∼ F(M ′ ) and F(M ′ ) is a domain of dimension n − 1 + r − 1 = dim A − 1. Theorem 2.4 as A/K = Let P be the A-ideal generated by all the variables Tr,j with j ∈ Jar . Clearly, P is prime. Indeed, ∼ F(Ma′ ) with a′ = if r = 1, then A/P = k; if r > 1, then, according to Theorem 2.4, A/P = a1 , . . . , ar−1 . Furthermore, if r > 1, P has height one as dim A/P = dim F(Ma′ ) = dim A − 1. Next we show that every prime ideal p in A containing Tr,ar contains either K or P . Assume that K 6⊂ p, then there exists a variable Tl,i 6= Tr,ar with i ∈ Ja′l that is not in p. Recall that the set of multi indices Jar is ordered by τ . For all j ∈ Jar we show that Tr,j ∈ p by descending induction on Jar . The base case is trivial since Tr,ar ∈ p. Assume Tr,j 6= Tr,ar . Then there exists a multi-index s ∈ Jar with s > j such that the equality Tr,j Tl,i = Tr,s Tl,i,t holds in A for some integer t ≥ 2. Notice that Tl,i,t is well defined because i ∈ Ja′l . By induction Tr,s ∈ p, thus Tr,j ∈ p since p is prime and Tl,i 6∈ p. Hence P ⊂ p. If r = 1 the above inclusion implies that ht p > 1 and hence U is cyclic generated by K. If r > 1, then U is generated by P and K. We conclude by showing that [P ] = ar [K], or equivalently, P = (Tr,ar ) :A K ar . Since both ideals have height one and P is prime it is enough to prove the inclusion P ⊂ (Tr,ar ) :A K ar . The latter follows from the equation (3) Tr,j K ar −j1 ∈ (Tr,ar ) . We prove equation (3) using descending induction on j1 with 0 ≤ j1 ≤ ar . The base case is trivial since j1 = ar implies Tr,j = Tr,ar . If j1 < ar , then the variable Tr,j appears in a row s of Ba with s > 1. Thus for any element λ ∈ K there exists β ∈ S such that the equality Tr,j λ = Tr,jn−1 ,··· ,js+1 ,js +1,··· ,j1 +1 β holds in A. Now the claim follows by induction. According to Theorem 2.11 the classes of the divisorial ideals K (δ) and P (δ) , the δ-th symbolic power of K and P respectively, constitute Cl(A). In the next theorem we exhibit a monomial REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 9 generating set for K (δ) for δ ≥ 1. As in [20] and [25] we identify K (δ) with a graded piece of A. We put a new grading on the ring T , Deg(Tl,j ) = j1 . Notice that I2 (Ba ) is an homogeneous ideal with respect to this grading. Thus Deg induces a grading on A. Let A≥δ be the ideal generated by all monomials m in A with Deg(m) ≥ δ. Theorem 2.12. The δ-th symbolic power of K, K (δ) , equals the monomial ideal A≥δ . Proof. One proceeds as in [21, 1.5]. 3. T HE B LOWUP ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS In this section our goal is to compute explicitly the defining equations of the blowup algebras of truncations of complete intersections. Assumptions 3.1. Let R = k[x1 , . . . , xn ] be a polynomial ring over a field k with homogeneous maximal ideal m. Let r be an integer with 1 ≤ r ≤ n. Let f1 , . . . , fr be a homogeneous regular sequence in R of degree d1 ≥ . . . ≥ dr . Let d ≥ d1 be an integer, write ai = d − di . Let I be the truncation of the complete intersection (f1 , . . . , fr ) in degree d, i.e. the R-ideal generated by {mai fi \| 1 ≤ i ≤ r} , I = (f1 , . . . , fr )≥d = (ma1 f1 , . . . , mar fr ). Theorem 3.2. Adopt assumptions 3.1 and write M = ma1 ⊕ . . . ⊕ mar . There is a short exact sequence 0 → Q → R(M ) → R(I) → 0 where Q is a prime ideal of height r − 1 in the normal domain R(M ). Proof. The natural map ζ M −→ (u1 , . . . , ur ) 7→ I = ma1 f1 + . . . + mar fr → 0 u1 f1 + . . . + ur fr induces a surjection on the level of Rees algebras Ψ : R(M ) −→ R(I) and Ker Ψ is a prime of height r − 1 since R(M ) and R(I) are domains and dim R(M ) = n + r = dim R(I) + r − 1. 10 K. N. LIN AND C. POLINI Definition 3.3. Adopt assumptions 3.1. Let a be the sequence a1 , . . . , ar and let S be the polynomial ring Sa . Consider the algebra epimorphism χ obtained by the composition of the two algebra epimorphisms φ : S −→ R(M ) Ψ : R(M ) −→ R(I) and with φ as in (1) and let A be the S-ideal defined by the short exact sequence χ 0 → A −→ S −→ R(I) → 0 . Let τ be the order on Tl,j defined in Section 2. Notice that, through the algebra epimorphism χ, the order τ induces an order on the set C = xal ,j fl \| 1 ≤ l ≤ r, j ∈ Jal of generators of I. Let ϕ be the presentation matrix of I over R with respect to C. In the following remark we give a resolution of I when r = 2. In particular, we describe explicitly ϕ. Remark 3.4. To compute a resolution of I = (f1 , f2 )≥d we first truncate the Koszul complex of f1 , f2 in degree d. (1) If d2 ≤ d ≤ d1 + d2 we obtain the short exact sequence: ρ ζ 0 −→ R(−d1 − d2 + d) −→ M −→ I −→ 0 , −f2 where ζ is the natural surjection described in Theorem 3.2 and ρ = . Set ϕ′′ to f1 −f2 be the map that rewrites in terms of the k-basis B of M ordered with the order f1 induced by τ . The Eagon-Northcott complex gives us an R-resolution F• of M , while the complex G• that is trivial everywhere except in degree 0 gives us an R-resolution of R(−d1 − d2 + d). The map ρ can be trivially lifted to a morphism of complexes ρ• : G• −→ F• that is trivial in positive degree and is ϕ′′ in degree zero. (2) If d ≥ d1 + d2 + 1 we obtain the short exact sequence: ρ ζ 0 −→ md−d1 −d2 −→ M −→ I −→ 0 , −f2 where ζ is the natural surjection described in Theorem 3.2 and ρ = . Write E f1 for the k-bases of md−d1 −d2 ordered with the order induced by τ . Set ϕ′′ to be the map −f2 that rewrites E in terms of the k-basis B of M ordered with the order induced f1 by τ . From the Eagon-Northcott complex we obtain R-resolutions F• and G• of M and md−d1 −d2 , respectively. The map ρ can be lifted to a morphism of complexes ρ• : G• −→ F• with ϕ′′ in degree zero. REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 11 In both cases the mapping cone C(ρ• ) is a non-minimal free resolution of I. In particular, the presentation matrix of I with respect to C is ϕ = [ϕ′ , ϕ′′ ] where ϕ′ is the matrix presenting M with respect to B. In the following remark we describe explicitly ϕ for any r when d ≥ d1 + d2 . Remark 3.5. Adopt assumptions 3.1. To compute the presentation matrix for I = (f1 , . . . , fr )≥d V we need to truncate the Koszul complex K• (f1 , . . . , fr ) = Re1 ⊕ . . . ⊕ Rer . If d ≥ d1 + d2 then d ≥ di + dj for any 1 ≤ i < j ≤ r and we have: M . . . −→ m d−di −dj ei ∧ ej −→ M = 1≤i<j≤r r M mai −→ I −→ 0 . i=1 md−di −dj , Write B and Ei,j for the k-bases of M and of respectively, ordered with the order induced ′′ by τ . Set ϕ to be the map that rewrites Ei,j (−fj ei + fi ej ) in terms of B for all 1 ≤ i < j ≤ r. The presentation matrix of I with respect to C is ϕ = [ϕ′ , ϕ′′ ] where ϕ′ is the matrix presenting M with respect to B. Definition 3.6. Adopt assumptions 3.1. Let h1 , . . . , ht ∈ S be the homogeneous polynomials obtained by the matrix multiplication [T ]ϕ′′ , where ϕ′′ is the matrix described in Remark 3.4 and 3.5. Think of S as a naturally bigraded ring with deg xi = (1, 0) and deg Tj = (0, 1). If r = 2 and (δ, 1) with δ = d2 −a1 = d1 −a2 = d1 +d2 −d ≥ d1 ≤ d ≤ d1 +d2 , then t = 1 and h1 has bidegree X σi,j + n − 1 0. For any r, if d ≥ d1 + d2 then t = with σi,j = d − di − dj ≥ 0 and the n−1 1≤i<j≤r hk ’s have bidegree (0, 1). Proposition 3.7. Adopt assumptions 3.1, 3.6 with d ≥ d1 +d2 then (h1 , . . . , ht )R(M ) is a non-zero prime ideal of height ≥ r − 1. Proof. Let 1 ≤ i < j ≤ r. Write X λk xdi ,k fi = and fj = X αk xdj ,k k∈Jdj k∈Jdi Let σi,j = d − di − dj ≥ 0. As in Definition 2.1, denote the elements of the basis Ei,j of mσi,j with xσi,j ,t,s , t ∈ Jσ′ i,j . Since we have a one to one correspondence between the hk ’s and the elements of Ei,j , we write hk as hσi,j ,t,s . Let H be the S-ideal generated by {hσi,j ,t,s \| 1 ≤ i < j ≤ r, t ∈ Jσ′ i,j , 1 ≤ s ≤ n}. We obtain X X λk xai ,k+t,s αk xaj ,k+t,s , and xσi,j ,t,s fj = xσi,j ,t,s fi = k∈Jdi k∈Jdj hence hσi,j ,t,s = X k∈Jdi λk Ti,k+t,s − X k∈Jdj αk Tj,k+t,s . 12 K. N. LIN AND C. POLINI Let Hj be the S-ideal generated by {hσ1,l ,t,s \| 2 ≤ l ≤ j, t ∈ Jσ′ 1,l , 1 ≤ s ≤ n}. We show by induction on j with 2 ≤ j ≤ r that the ideal Hj R(M ) is prime of height ≥ j − 1. Let j = 2. Notice that for each t ∈ Jσ′ 1,2 , the column [hσ1,2 ,t,1 , . . . , hσ1,2 ,t,n ]tr is a linear combination of columns of P +n−2 +n−2 Ca . Write Ea for the n × (1 + rq=1 aqn−1 ) matrix obtained by Ca by substituting σ1,2n−1 columns with [hσ1,2 ,t,1 , . . . , hσ1,2 ,t,n ]tr , t ∈ Jσ′ 1,2 . The two S-ideals I2 (Ca ) + H2 and I2 (Ea ) + H2 are equal. The ideal H2 R(M ) is prime according to Corollary 2.8 as S/(I2 (Ea ) + H2 ) ∼ = T ′ /I2 (Da ) for some polynomial ring T ′ and Da a suitable submatrix of Ca with n rows. Now degree considerations show that hσ1,2 ,t,s 6∈ I2 (Ca ), hence H2 R(M ) 6= 0. Thus the ideal H2 R(M ) is prime of height at least one. Let 2 ≤ l ≤ j and assume by induction that Hj−1 R(M ) is prime of height ≥ j − 2, we show Hj R(M ) is prime of height ≥ j − 1. For each t ∈ Jσ′ 1,l , the column [hσ1,l ,t,1 , . . . , hσ1,l ,t,n ]tr P +n−2 is a linear combination of columns of Ca . Write Ea for the n × (1 + rq=1 aqn−1 ) matrix Pj σ1,l +n−2 tr columns with [hσ1,l ,t,1 , . . . , hσ1,l ,t,n ] , 2 ≤ l ≤ j obtained by Ca by substituting l=2 n−1 ′ and t ∈ Jσ1,l . The two S-ideals I2 (Ca ) + Hl and I2 (Ea ) + Hl are equal. The ideal Hj R(M ) is prime according to Corollary 2.8 as S/(Ij (Ea ) + Hj ) ∼ = T ′ /I2 (Da ) for some polynomial ring T ′ and Da a suitable submatrix of Ca with n rows. Notice that for each t ∈ Jσ′ 1,j , the column [hσ1,j ,t,1 , . . . , hσ1,j ,t,n ]tr is a linear combination of a subset of the columns {(1, t) \| t ∈ Ja′1 } and {(j, t) \| t ∈ Ja′j } of Ca , while the column [hσ1,l ,t,1 , . . . , hσ1,l ,t,n ]tr is a linear combination of a subset of the columns {(1, t) \| t ∈ Ja′1 } and {(l, t) \| t ∈ Ja′l } of Ca with 2 ≤ l ≤ j − 1. Hence degree considerations show that hσ1,j ,t,s 6∈ (Hj−1 , I2 (Ca )), hence Hj R(M ) 6= 0. Thus the ideal Hj R(M ) is prime of height ≥ j − 1. Using the same argument one can show that HR(M ) is a prime ideal and its height is at least r − 1 as HR(M ) ⊃ Hr R(M ) . Remark 3.8. If d ≥ d1 +d2 −1, then the ideal I = (f1 , . . . , fr )≥d has a linear resolution. The Rees algebra of linearly presented ideals of height 2 has been described explicitly in terms of generators and relations in [18] under the additional assumption that I is perfect. However, if r < n, the truncations of codimension r complete intersections are never perfect. For large d, the Rees algebra R(I) is Cohen-Macaulay as shown in [12]. But the defining equations of R(I) were unknown, we give them explicitly in Theorem 3.9. If r = 2 we prove that for d ≥ d1 + d2 − 1 the Rees ring R(I) is a Koszul domain (see Corollary 3.13). If r ≥ 3, we prove that R(I) is a Koszul domain for d ≥ d1 + d2 (see Corollary 3.13). In addition in [16] we study the depth and regularity of the blowup algebras of I. REES ALGEBRAS OF TRUNCATIONS OF COMPLETE INTERSECTIONS 13 Theorem 3.9. Adopt assumptions 3.1 and 3.6 with d ≥ d1 + d2 then R(I) = R(Ma )/(h1 , . . . , ht ) = Sa /(I2 (Ca ), h1 , . . . , ht ) . Proof. Adopt the notation of the proof of Proposition 3.7. Let H be the S-ideal generated by {hσi,j ,t,s \| 1 ≤ i < j ≤ r, t ∈ Jσ′ , 1 ≤ s ≤ n}. According to Remark 3.5, we have H ⊂ A. Hence HR(M ) ⊂ AR(M ) where the first ideal is a non-zero prime ideal of height ≥ r − 1 by Proposition 3.7 and the second one has height r − 1. Assume r = 2. In the following theorem we express the Rees algebra of I as defined by a divisor on the Rees algebra of the module M = ma1 ⊕ ma2 that we computed explicitly in the previous section. Indeed, for r = 2 the prime ideal Q ∈ Spec(R(M )) of Theorem 3.2 gives rise to an element of the divisor class Cl(R(M )). This group has been studied explicitly in Theorem 2.11: it is cyclic generated by the prime ideal L, where L be the R(M )-ideal generated by all the variables appearing in the first row of Ca . In the next theorem we identify for which s the ideal L(s) is isomorphic to Q. Definition 3.10. Let L be the R(M )-ideal generated by all the variables appearing in the first row of Ca . We will use the convention that the (symbolic) power of any element or ideal with nonpositive exponent is one or the unit ideal, respectively. Theorem 3.11. Adopt assumptions 3.1, 3.6, and 3.10 with r = 2 then xδ1 AR(M ) = (h1 , . . . , ht )L(δ) . In particular, (1) If d1 ≤ d ≤ d1 + d2 − 1 then the R(M )-ideals xδ1 AR(M ) and h1 L(δ) are equal and the bigraded R(M )-modules A and L(δ) (0, −1) are isomorphic. (2) If d ≥ d1 + d2 then R(I) = Sa /(I2 (Ca ), h1 , . . . , ht ) . Proof. Notice that the first statement follows from (1) and (2) and (2) has been proven in Theorem 3.9. Thus it will be enough to prove (1). Write h = h1 . One proceeds as in [21, 1.11]. For clarity we rewrite part of the proof here since there are some minor differences and in [21, 1.11] the integer δ was assumed to be ≥ 2. Degree considerations show that x1 is not in I2 (Ca ). The ideal I2 (Ca ) is prime, so xδ1 is also not in I2 (Ca ). The second assertion in (1) follows from the first as xδ1 has bi-degree (δ, 0) and h1 has bi-degree (δ, 1). Write to mean image in R(M ). We prove the equality xδ1 AR(M ) = hL(δ) by showing that A = (h̄/x̄δ1 )L(δ) , where the fraction is taken in the quotient field Q of R(M ). 14 K. N. LIN AND C. POLINI Notice that (x̄i1 ) :Q L(i) = (x̄1 , . . . , x̄n )i . This follows as in the proof of claim (1.12) in [21, 1.11]. Furthermore h̄ ∈ mδ = (x̄δ1 ) :Q L(δ) . Thus, h̄L(δ) ⊆ x̄δ1 R(M ). Define D to be the ideal (h̄/x̄δ1 )L(δ) of R(M ). At this point, we see that the ideal D is either zero or divisorial. To show that D is not zero and to establish the equality A = D, it suffices to prove that A ⊆ D, because A is a height one prime ideal of R(M ). This is the only part where the argument differs from [21, 1.11]. Notice that h̄ ∈ D as x̄1 ∈ L. For every w ∈ m, one has Iw = (f1 , f2 )w . Therefore, R[It]w = R[(f1 , f2 )t]w and we obtain (h̄)w = Aw . It follows that h̄ 6= 0 and Aw ⊆ Dw . The rest of the proof follows as in [21, 1.11]. Corollary 3.12. Adopt assumptions 3.1, 3.6, and 2.10. (a) If d ≥ d1 + d2 then F(I) = F(Ma )/(h1 , . . . , ht ) = Ta /(I2 (Ba ), h1 , . . . , ht ) . (b) If r = 2 and d1 ≤ d ≤ d1 + d2 − 1 then F(I) = Ta /K with K∼ = K (δ) (−1). Proof. The proof follows from Theorem 3.9 and Theorem 3.11 and the fact that F(I) = k ⊗ R(I). Also the same argument as in [21, 4.2] shows the isomorphism K ∼ = K (δ) (−1). Corollary 3.13. Adopt assumptions 3.1. If d ≥ d1 + d2 or if r = 2 and d = d1 + d2 − 1, then R(I) and F(I) are Koszul algebras. Proof. We will prove the statement for the Rees ring. The same proof works for the special fiber ring using Corollary 3.12. If d ≥ d1 + d2 , then R(I) = S/(I2 (Ca ), h1 , . . . , ht ) according to Theorem 3.9, and according to the proof of Corollary 2.8 the latter ring is isomorphic to T ′ /I2 (Da ) for some polynomial ring T ′ and Da a suitable submatrix of Ca with n rows. But this ring is Koszul as it has a Gröbner of quadrics by Lemma 2.6. 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1 Anti-jamming Communications Using Spectrum Waterfall: A Deep Reinforcement Learning Approach arXiv:1710.04830v1 [cs.IT] 13 Oct 2017 Xin Liu, Yuhua Xu, Member, IEEE, Luliang Jia, Student Member, IEEE, Qihui Wu, Senior Member, IEEE, and Alagan Anpalagan, Senior Member, IEEE Abstract—This letter investigates the problem of anti-jamming communications in dynamic and unknown environment through on-line learning. Different from existing studies which need to know (estimate) the jamming patterns and parameters, we use the spectrum waterfall, i.e., the raw spectrum environment, directly. Firstly, to cope with the challenge of infinite state of raw spectrum information, a deep anti-jamming Q-network is constructed. Then, a deep anti-jamming reinforcement learning algorithm is proposed to obtain the optimal anti-jamming strategies. Finally, simulation results validate the the proposed approach. The proposed approach is relying only on the local observed information and does not need to estimate the jamming patterns and parameters, which implies that it can be widely used various anti-jamming scenarios. Index Terms—Anti-jamming, Deep Q-Network, Deep Reinforcement Learning I. I NTRODUCTION Anti-jamming is always an active research topic, as wireless transmissions are naturally vulnerable to jamming attacks. The mainstream anti-jamming techniques includes Frequency Hopping Spread Spectrum (FHSS) and Direct-Sequence Spread Spectrum (DSSS) [1]. Recently, to address the interactions between the legitimate users and the jammers, game theory has been widely applied in the literature [2]–[7]. In methodology, these approaches need to know the jamming strategies, which implies that the legitimate users are required to estimate the jamming patterns and parameters from the observed environment. However, with the rapid development of artificial intelligence and universal software radio peripheral (USRP) [8], the jammers can easily create dynamic and intelligent jamming attacks. As a consequence, there are two limitations with regard to estimation-based anti-jamming communications: i) there may be information loss for unknown jamming patterns, and ii) if the intelligent jammer switches its strategies dynamically and rapidly, it is not possible to track and react it in real time. Thus, it is challenging and interesting to investigate antiX. Liu is with the College of Information Science and Engineering, Guilin University of Technology, Guilin 541004, China. (email:leo nanjing@126.com). Y. Xu and L. Jia are with the College of Communication Engineering, PLA Army Engineering University, Nanjing 210007, China. (e-mail: yuhuaenator@gmail.com;jiallts@163.com). Qihui Wu is with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China (e-mail: wuqihui2014@sina.com). Alagan Anpalagan is with the Department of Electrical and Computer Engineering, Ryerson University, Toronto, Canada (alagan@ee.ryerson.ca). jamming communication approaches in dynamic and unknown environment. To overcome the above limitations, a promising way is to design new anti-jamming approaches that utilize the raw environmental information, which is known as spectrum waterfall [9], without estimating jamming patterns and parameters. These kind of anti-jamming approaches would avoid information loss and adapt to the dynamic environment, as can be expected. In addition, online learning is an effective way to solve the decision problems in dynamic environment. The widely used technique is Q-learning [10], which has been used in anti-jamming problems [2], [3]. Unfortunately, Q-learning is not able to deal with the raw environmental information directly because of the infinite state of the environment. Motivated by the deep reinforcement learning technique for learning successful control policies from raw video data in [11], we investigate the anti-jamming problem in unknown and dynamic environment. First, the raw spectrum information is defined as the state of the environment to avoid losing the jammer information as much as possible. Then, a deep anti-jamming Q-network (DAQN) is constructed to realize the direct processing of raw spectrum information. Finally, a deep anti-jamming reinforcement learning algorithm (DARLA) is proposed. Simulation results show that the proposed DARAL achieves the best anti-jamming strategies in various scenarios. The main contributions are summarized as follows. • • Based on the deep reinforcement learning technique, a smart anti-jamming communication scheme is proposed. In particular, the raw spectrum information is defined as a state, which describes the detail features of jammer more accurately. The proposed algorithm is relying only on the locally observed information and does not need to estimate the jamming patterns and parameters the jammer in advance, i.e., it is model-free, which can be widely used in various anti-jamming scenarios. Note that the most related work is [12], which also adopted deep reinforcement learning to investigate the anti-jamming problems. The main differences in this work are as follows: i) the environment state is presented by extracting features of signal-to-interference-plus-noise ratio (SINR) and primary user occupancy in [12], while it is presented by the raw spectrum information in this work, and ii) it requires the jammer to have the same channel-slot transmission structure 2 known as spectrum waterfall [9], as shown in Fig. 2. Taking the swept jamming as example, we can accurately determine the frequency range and intensity (color) of jamming at the next moment by looking at the thermal chart, which also means we can determine the anti-jamming strategy accordingly. In the unknown and dynamic environment, we do not consider estimation-based anti-jamming strategies. Instead, define St as the environment state, and then consider a dynamic decision problem in which the agent (anti-jamming user) interacts with an environment through a sequence of observations of environment (St ), actions (at ) and rewards (rt ). Specifically, an action a can be a combination decisions of frequency, power, coding schemes, spread spectrum, and other kinds of anti-jamming decisions, e.g., a = (f, p) represents the combination actions of frequency (f ) and power (p). The rewards associated with the actions and environment is defined as: R(a) − λδ β(a, S) ≥ βth (a) r(a, S) = , (2) 0 β(a, S) < βth (a) Fig. 1. System model. Fig. 2. Thermodynamic chart of various jamming pattern. with the users in [12]. On the contrary, this requirement does not hold in our work, which makes the proposed approach more general. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION We consider the transmission of one user (a transmitterreceiver pair) against one or several jammers, as shown in Fig. 1. The agent, which is disposed at the receiving end, sends anti-jamming strategies to the transmitter through a reliable control link. Jammers may adopt fixed, random, or possibly intelligent jamming patterns. However, we do not analyze the specific jamming models, but obtain the optimal anti-jamming strategies based on the raw spectrum information. While the receiver receives the desired signal, the agent continuously senses the whole communication bands and stores the sensed values. Denote the spectrum vector as Pt = {pt,1 , pt,2 , · · · , pt,N }, where pt,n is the power of frequency n at time t and N is the number of sampling points in frequency space. In order to sufficiently use history spectrum information, a two-dimensional matrix, which describes timefrequency features of spectrum environment, is expressed as:   pt−1,1 Pt−1  Pt−2   pt−2,1    St =  .  =  .  ..   ..  Pt−M pt−M,1 pt−1,2 pt−2,2 .. . ··· ··· .. . pt−1,N pt−2,N .. . pt−M,2 ··· pt−M,N     . (1)  It is noted that St contains all the spectrum information until time t, as M tends to infinity. However, the difficulty of the decision optimization problem is significantly increased with the increase of M . Therefore, M can take an appropriate value, which would be determined by the time-varying characteristics of the spectrum environment. To illustrate the rationality of using St as the basis of antijamming decision-making, we give the thermodynamic charts of the St matrix of several common jamming patterns, also where R(a) is the bit rate when the action a is selected, λ is the cost when action changes, δ is an indication of action change (δ = 1 if at 6= at−1 ; δ = 0 if at = at−1 ), β(a, S) is the received signal to interference plus noise ratio (SINR) in state S with action a. βth (a) is the required SINR threshold for successful transmission. Note that R(a) and βth (a) are modeled as a function of action a, the reason is as follows: the bit rate and SINR requirements change for different anti-jamming strategies, such as forward error correction and spread spectrum schemes. Then, the goal of the agent is to select anti-jamming actions in a fashion that maximizes cumulative future reward Rt = P∞ i i=0 γ rt+i+1 , where γ is the discount factor. One way of achieving this goal is to compute the following optimal actionvalue (also known as Q) function [10]: Q∗ (S, a) = max E {Rt \| St = S, at = a, π}, π (3) where the anti-jamming policy π = P (a \| S) refers to a probability distribution over the actions. Based on the Bellman equation, n o ∗ ′ ′ Q∗ (S, a) = E r + γ max Q (S , a ) \| S, a . (4) ′ a III. A NTI - JAMMING C OMMUNICATION S CHEME The traditional Q-learning is unable to cope with the antijamming problem described in section II, as the state space size of St is almost infinite. In order to solve this problem, a deep anti-jamming Q-network (DAQN) is constructed to address the interactive decision-making problem with raw spectrum information input, which contains decision network and update network, as shown in Fig. 3 and Fig. 4 respectively. We use a deep convolutional neural network (CNN) to approximate the optimal action-value as shown in decision network, where the input state St is represented by a thermal chart of M ×N pixels. After the processing of two convolutional layers and two fully connected layers, the output is the estimated Q function, where K is the size of action space. At last, the 3 Fig. 3. Decision network of the DAQN. Algorithm 1: Deep Anti-jamming Reinforcement Learning Algorithm (DARLA) Initialize : Set D = ∅, ǫ = 1, Set θ with random weights, Sense initial environment S1 . For t = 1, T do With probability ǫ, select a random action at otherwise, select at = arg max Q(St , a; θ) a Fig. 4. Updating network of the DAQN. decision layer outputs the corresponding action based on the estimated Q function. However, reinforcement learning is known to be unstable or even to diverge when a nonlinear function approximator [11], such as the neural network is used to represent the Q function. The main reason is correlation during the learning process. The idea of experience replay is adopted to address these instabilities as shown in the update network. To perform experience replay, we store the agent’s experiences et = (St , at , rt , St+1 ) at each time-step t in data set Dt = (e1 , · · · et ). When the experience pool is big enough, we construct target values r + γ max Q(S′ , a′ ) by randomly choosing elements in a a′ uniform distribution (S, a, r, S′ ) ∼ U (D), which reduces the correlation during sequential observation. The Q-learning update at iteration i uses the following loss function: i h 2 Li (θi ) = E(S,a,r,S′ )∼U(D) (yi − Q(S, a; θi )) , (5) where θi is the parameter of Q-network at iteration i and yi = r + γ max Q(S′ , a′ ; θi−1 ) is target value computed by a′ Q-network parameter θi−1 with greedy strategy. By assuming that yi is the expected output of CNN with network weight θi when the input is S, we calculate the difference between real output Q(S, a; θi ) and target value yi to determine the update of network parameters. Differentiating the loss function with respect to the weights, we arrive at the following gradient: ∇θi Li (θi ) = E(S,a,r,S′ ) [(yi − Q(S, a; θi )) ∇θi Q(S, a; θi )] . (6) According to the gradient descent algorithm, the network weight θi is updated according to (6). Although there are two Execute action at and compute rt and observe St+1 Store transitions (St , a, r, St+1 ) in D If Sizeof (D) > N (Enough amount of transitions) Sample random minibatch of transistions (S, a, r, S′ ) from D Compute yi = r + γ max Q(S′ , a′ ; θ) a′ Compute gradient based on Eq.(6) and Update θ End If Calculate ǫ = max(0.1, ǫ − ∆ǫ) End For CNN networks with different weights, as shown in Fig. 4, the actual implementation requires only one CNN network, as the computing of target values and the updating of network weights are in different stages. The algorithm for anti-jamming communication based on deep reinforcement learning is presented in Algorithm 1. IV. N UMERICAL R ESULTS AND D ISCUSSIONS In the simulation setting, the user and the jammer combat with each other in a frequency band of 20MHz, where the frequency resolution of spectrum sensing is 100kHz. The user performs a full band sensing every 1ms and retains the spectrum data within the 200ms. Hence, the size of matrix St is 200 × 200. The bandwidth of user signal is 4MHz, and the center frequency is allowed to change in each 10ms with the step of 2MHz, which means K = 9. Both signal and jamming are raised cosine waveform with roll-off factor α = 0.3, in which jamming power is 30dBm and signal power is 0dBm. The demodulation threshold βth at all frequency is set to be 10dB, and the cost of action change λ is set to be 0.2R(a). Four kinds of jamming patterns are given for simulation: i) Sweep jamming (sweep speed is 1GHz/s); ii) Comb jamming (three fixed frequency signals at 2MHz, 10MHz, and 18MHz); 4 1 0.9 Normalized throughput 0.8 0.7 0.6 0.5 0.4 0.3 Sweep Jamming Comb Jamming Random Jamming 0.2 0.1 0 0 0.5 1 Iteration 1.5 2 4 x 10 Fig. 5. Normalized throughput under different jamming patterns. and (c) respectively, and the converging states are given in Fig. 6(d), (e), and (f) respectively. These states that contain time-frequency information can clearly reflect the past actions of user and jamming. Taking sweep jamming as an example, at the beginning of the learning procedure, the user adopts randomized action as it is unfamiliar with environment (the locations of rectangular blocks are randomly distributed), and after convergence, the frequency is properly changed before the jamming arrives (the rectangular blocks are distributed according to the slashes). With regard to the intelligent jamming, since the probability distribution of user actions is the basis for jammer to release jamming, the best strategy for user is that the probability of each action is almost identical. The simulation results in Fig. 7 show the probabilities of each action being selected during the learning procedure, which is consistent with our analysis. V. C ONCLUSION Fig. 6. Environmental states at initial and convergent stages under different jamming patterns. 0.25 Probability 0.2 0.15 In this letter, we investigated the anti-jamming problem in unknown and dynamic environment. Aiming at employing the waterfall spectrum information directly, we constructed a deep anti-jamming Q-network to handle the complex interactive decision-making problem with infinite number of states. Then, a deep anti-jamming reinforcement learning algorithm was proposed. Using the proposed learning algorithm, the user is able to learn the best anti-jamming strategy by constantly trying various actions and sensing the spectrum environment. Simulation results in various scenarios are presented to validate the proposed anti-jamming communication approach. Future work on designing multi-user deep anti-jamming reinforcement learning algorithms is ongoing. R EFERENCES 0.1 0.05 0 10 8 200 6 150 4 100 2 Action index 50 0 0 Iteration/100 Fig. 7. Probability of user actions during learning. iii) Random jamming (frequency is randomly changed every 20ms with the step of 4MHz); iv) Intelligent jamming (the jammer continuously observes the probability that the user signal appears at each frequency point, and chooses the largest one as jamming channel). For all Jamming patterns, the instantaneous bandwidth of the jamming is set to be 4MHz. The normalized average throughput of legitimate user under different jamming patterns is given in Fig. 5. It is shown that the anti-jamming ability of user has been improved significantly with the proposed DARLA learning. Especially in the case of comb jamming, the normalized throughput is close to one after convergence, which indicates that the jamming is almost completely avoided. Environmental states at initial stages under sweep, comb and random jamming patterns are given in Fig. 6(a), (b), [1] L. 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arXiv:1706.03568v1 [cs.DS] 12 Jun 2017 Monitoring of Domain-Related Problems in Distributed Data Streams∗ Pascal Bemmann Felix Biermeier Jan Bürmann Arne Kemper Till Knollmann Steffen Knorr Nils Kothe Alexander Mäcker Manuel Malatyali Friedhelm Meyer auf der Heide Sören Riechers Johannes Schaefer Jannik Sundermeier Heinz Nixdorf Institute & Computer Science Department Paderborn University, Germany {pbemmann, felixbm, jbuerman, kempera, tillk, stknorr, nkothe, amaecker, malatya, fmadh, soerenri, jschaef, janniksu} @mail.uni-paderborn.de Abstract Consider a network in which n distributed nodes are connected to a single server. Each node continuously observes a data stream consisting of one value per discrete time step. The server has to continuously monitor a given parameter defined over all information available at the distributed nodes. That is, in any time step t, it has to compute an output based on all values currently observed across all streams. To do so, nodes can send messages to the server and the server can broadcast messages to the nodes. The objective is the minimisation of communication while allowing the server to compute the desired output. We consider monitoring problems related to the domain Dt defined to be the set of values observed by at least one node at time t. We provide randomised algorithms for monitoring Dt , (approximations of) the size \|Dt \| and the frequencies of all members of Dt . Besides worst-case bounds, we also obtain improved results when inputs are parameterised according to the similarity of observations between consecutive time steps. This parameterisation allows to exclude inputs with rapid and heavy changes, which usually lead to the worst-case bounds but might be rather artificial in certain scenarios. ∗ This work was partially supported by the German Research Foundation (DFG) within the Priority Program “Algorithms for Big Data” (SPP 1736) and by the Federal Ministry of Education and Research (BMBF) as part of the poject “Resilience by Spontaneous Volunteers Networks for Coping with Emergencies and Disaster” (RESIBES), (grant no 13N13955 to 13N13957). 1 1 Introduction Consider a system consisting of a huge amount of nodes such as a distributed sensor network. Each node continuously observes its environment and measures information such as temperature, pollution or similar parameters. Given such a system, we are interested in aggregating information and continuously monitoring properties describing the current status of the system at a central server. To keep the server’s information up to date, the server and the nodes can communicate with each other. In sensor networks, however, the amount of such communication is particularly crucial, as communication translates to energy consumption, which determines the overall lifetime of the network due to limited battery capacities. Therefore, algorithms aim at minimizing the communication required for monitoring the respective parameter at the server. One very basic parameter is the domain of the system defined to be the values currently observed across all nodes. We consider different notions related to the domain and propose algorithms for monitoring the domain itself, (approximations of) its size and (approximations of) the frequencies of values comprising the domain, respectively. Each of these parameters can provide useful information, e.g. the information about the (approximated) frequency of each value allows to approximate very precisely the histogram of the observed values, and this allows to determine (approximations of) several functions of the input, e.g. heavy hitters, quantiles, top-k, frequency moments or threshold problems. 1.1 Model and Problems We consider the continuous distributed monitoring setting, introduced by Cormode, Muthukrishnan, and Yi in [1], in which there are n distributed nodes, each uniquely identified by an identifier (ID) from the set {1, . . . , n}, connected to a single server. Each node observes a stream of values over time and at any discrete time step t node i observes one value vit ∈ {1, . . . , ∆}. The server is asked to, at any point t in time, compute an output f (t) which depends on the ′ values vit (for t′ ≤ t, and i = 1, . . . , n) observed across all distributed streams up to the current time step t. The exact definition of f (·) depends on the concrete problems under consideration, which are defined in the section below. For the solution of these problems, we are usually interested in approximation algorithms. An ε-approximation of f (t) is an output f˜(t) of the server such that (1 − ε)f (t) ≤ f˜(t) ≤ (1 + ε)f (t). We call an algorithm that, for each time step, provides an ε-approximation with probability at least 1 − δ, an (ε, δ)approximation algorithm. To be able to compute the output, the nodes and the server can communicate with each other by exchanging single cast messages or by broadcast messages sent by the server and received by all nodes. Both types of communication are instantaneous and have unit cost per message. That is, sending a single message to one specific node incurs cost of one and so does one broadcast message. Each message has a size of O(log ∆ + log n + log log 1δ ) bits and will usually, besides a constant number of control bits, consist of a value from {1, . . . , ∆}, a node ID and an identifier to distinguish between messages of different instances of an algorithm applied in parallel (as done when using standard probability amplification techniques). Having a broadcast channel is an extension to [1], which was originally proposed in [2] and afterwards applied in [7, 8]. For ease of presentation, we assume that not only the server can send 2 broadcast messages, but also the nodes. This changes the communication cost only by a factor of at most two, as a broadcast by a node can always be implemented by a single cast message followed by a broadcast of the server. Between any two time steps we allow a communication protocol to take place, which may use polylogarithmic O(logc n) rounds, for some constant c. The optimisation goal is the minimisation of the communication cost, given by the number of exchanged messages, required to monitor the considered problem. 1.1.1 Monitoring of Domain-Related Functions. In this paper, we consider the monitoring of different problems related to the domain of the network. The domain at time t is defined as Dt := {v ∈ {1, . . . , ∆} \| ∃i with vit = v}, the set of values observed by at least one node at time t. We study the following three problems related to the domain: • Domain Monitoring. At any point in time, the server needs to know the domain of the system as well as a representative node for each value of the domain. Formally, monitor Dt = {v1 , . . . , v\|Dt \| } ⊆ {1, . . . , ∆}, at any point t in time. Also, maintain a sequence Rt = (j1 , . . . , j∆ ) of nodes such that for all observed values v ∈ Dt a representative i is determined with jv = i and vit = v. For each value v ∈ / Dt which is not observed, no representative is given and jv = nil. • Frequency Monitoring. For each v ∈ Dt monitor the frequency \|Ntv \| of nodes in Ntv := {i ∈ {1, . . . , n} \| vit = v} that observed v at t, i.e. the number of nodes currently observing v. • Count Distinct Monitoring. Monitor \|Dt \|, i.e. the number of distinct values observed at time t. We provide an exact algorithm for the Domain Monitoring Problem and (ε, δ)approximations for the Frequency and Count Distinct Monitoring Problem. 1.2 Our Contribution P For the Domain Monitoring Problem, an algorithm which uses Θ( t∈T \|Dt \|) messages on expectation for T time steps is given in Section 2. This is asymptotically optimal in the worst-case in which Dt ∩ Dt+1 = ∅ holds for all t ∈ T . We also provide an algorithm and an analysis based on the minimum possible number R∗ of changes of representatives for a given input. It exploits situations where Dt ∩ Dt+1 6= ∅ and uses O(log n · R∗ ) messages on expectation. For an (ε,δ)-approximation of the Frequency Monitoring Problem for T time P steps, we first provide an algorithm using Θ( t∈T \|Dt \| ε12 log \|Dδt \| ) messages on expectation in Section 3. We then improve this bound for instances in which observations between consecutive steps have a certain similarity. That is, for inputs fulfilling the property that for all v ∈ {1, . . . , ∆} and some σ ≤ 1/2, the number of nodes observing v does not change by a factor larger than σ between consecutive time steps, we provide an algorithm that uses an expected amount of O(\|D1 \|(max(δ, σ)T + 1) ε12 log \|Dδ1 \| ) messages. In Section 4, we provide an algorithm using Θ(T · ε12 log δ1 ) messages on expectation for the Count Distinct Monitoring Problem for T time steps. For instances which exhibit a certain similarity an algorithm is presented which monitors the problem using log(n)·R∗ 1 Θ (1 + T · max{2σ, δ}) \|Dt \|·ε2 log δ messages on expectation. 3 1.3 Related Work The basis of the model considered in this paper is the continuous monitoring model as introduced by Cormode, Muthukrishnan and Yi in [1]. In this model, there is a set of n distributed nodes each observing a stream given by a multiset of items in each time step. The nodes can communicate with a central server, which in turn has the task to continuously, at any time t, compute a function f defined over all data observed across all streams up to time t. The goal is to design protocols aiming at the minimisation of the number of bits communicated between the nodes and the server. In [1], the monitoring of several functions is studied in their (approximate) threshold variants, in which the server has to output 1 if f ≥ τ and 0 if f ≤ (1 − ε)τ given τ and ε. Precisely, algorithms P , for p m where mi denotes the frequency of for the frequency moments Fp = i i item i for p = 0, 1, 2 are given. F1 represents the simple sum of all items received so far and F0 the number of distinct items received so far. Since the introduction of the model, monitoring of several functions has been studied such as the monitoring of frequencies and ranks by Huang, Yi and Zhang in [5]. The frequency of an item i is defined to be the number of occurrences of i across all streams up to the current time. The rank of an item i is the number of items smaller than i observed in the streams. Frequency moments for any p > 2 are considered by Woodruff and Zhang in [9]. A variant of the Count Distinct Monitoring Problem is considered by Gibbons and Tirthapura in [4]. The authors study a model in which each of two nodes receives a stream of items and at the end of the streams a server is asked to compute F0 based on both streams. A main technical ingredient is the use of so called public coins, which, once initialized at the nodes, provide a way to let different nodes observe identical outcomes of random experiments without further communication. We will adopt this technique in Section 4. Note that the previously mentioned problems are all defined over the items received so far, which is in contrast to the definition of monitoring problems which we are going to consider and which are all defined only based on the current time step. This fact has the implication that in our problems the monitored functions are no longer monotone, which makes its monitoring more complicated. Concerning monitoring problems in which the function tracked by the server only depends on the current time step, there is also some previous work to mention. In [6], Lam, Liu and Ting study a setting in which the server needs to know, at any time, the order type of the values currently observed. That is, the server needs to know which node observes the largest value, second largerst value and so on at time t. In [10], Yi and Zhang consider a system only consisting of one node connected to the server. The node continuously observes a d-dimensional vector of integers from {1, . . . , ∆}. The goal is to keep the server informed about this vector up to some additive error per component. In [3], Davis, Edmonds and Impagliazzo consider the following resource allocation problem: n nodes observe streams of required shares of a given resource. The server has to assign, to each node, in each time step, a share of the resource that is as least as large as the required share. The objective is then given by the minimization of communication necessary for adapting the assignment of the resource over time. 4 2 The Domain Monitoring Problem We start by presenting an algorithm to solve the Domain Monitoring Problem for a single time step. We analyse the communication cost using standard worstcase analysis and show tight bounds. By applying the algorithm for each time step, we then obtain tight bounds for monitoring the domain for any T time steps. The basic idea of the protocol as given in Algorithm 1 is quite simple: Applied at a time t with a value v ∈ {1, . . . , ∆}, the server gets informed whether v ∈ Dt holds or not. To do so, each node i with vit = v essentially draws a value from a geometric distribution and then those nodes having drawn the largest such value send broadcast messages. By this, one can show that on expectation only a constant number of messages is sent. Furthermore, if applied with v = nil, the server can decide whether v ′ ∈ Dt for all v ′ ∈ {1, . . . , ∆} at once with Θ(\|Dt \|) messages on expectation. To this end, for each v ′ ∈ {1, . . . , ∆} independently, the nodes i with vit = v ′ drawing the largest value from the geometric distribution send broadcast messages. In the presentation of Algorithm 1, we assume that vit = v is always true if v = nil. Also, in order to apply it to a subset of nodes, we assume that each node maintains a value statusi ∈ {0, 1} and only nodes i take part in the protocol for which statusi = status holds. Algorithm 1 ConstantResponse(v, status) [for fixed time t] 1. Each node i for which statusi = status and (v 6= nil ⇒ vit = v) hold, draws a value ĥi from a geometric distribution with success probability p := 1/2. 2. Let hi = min{log n, ĥi }. 3. Node i broadcasts its value in round log n−hi unless a node i′ with vit = vit′ has broadcasted before. We have the following lemma, which bounds the expected communication cost of Algorithm 1 and has already appeared in a similar way in [8] (Lemma III.1). Lemma 2.1. Applied for a fixed time t, ConstantResponse(v, 1) uses Θ(1) messages on expectation if v 6= nil and Θ(\|Dt \|) otherwise. Proof. First consider the case where v 6= nil. Regarding the expected communication of ConstantResponse(v, 1) we introduce some notation. Let Xi be a {0, 1}-random variable P indicating whether the node i ∈ Ntv sends a message to the server, and X := Xi . According to the algorithm a sensor i sends a message if and only if its height hi matches the round specified for that height and no other sensor i′ has sent its value beforehand. We obtain Pr [Xi = 1] = Pr [∃r ∈ {1, . . . , log n} : hi = r ∧ ∀i′ ∈ Ntv \ {i} : hi′ ≤ r] nv −1 log Xn 1 1 1− r . ≤ 2r 2 r=1 We know that E[Xi ] = Pr [Xi = 1] and thus v E[X] ≤ n · log Xn r=1 1 2r 5 nv −1 1 1− r . 2 nv −1 has only one extreme point and Observing that f (r) = nv · 21r 1 − 21r f (r) ≤ 2 for all r ∈ [0, log(n)], we use the integral test for convergence to obtain nv −1 nv −1 Z log n 1 1 1 v E[X] ≤ n · 1− r 1− r ≤n dr + 2 2 2r 2 0 r=1 " nv #log n 1 1 1 1− r +2≤ + 2 < 4. ≤ ln (2) 2 ln (2) v log Xn 1 2r 0 For the case v = nil we can apply the same argumentation independently for each value v ∈ Dt . This concludes the proof of the lemma. In order to solve the domain monitoring problem for T time steps, the server proceeds as follows: In each step t the server calls ConstantResponse(nil, 1) to identify all values belonging to Dt as well as a valid sequence Rt . By the previous lemma we then have an overall communication cost Pof Θ(\|Dt \|) for each time step t. For monitoring T time steps, the cost is Θ( t∈T \|Dt \|). This is asymptotically optimal in the worst-case P since on instances where Dt ∩Dt+1 = ∅ for all t, any algorithm has cost Ω( t∈T \|Dt \|). Theorem 2.2. Using ConstantResponse(v, P 1), the Domain Monitoring Problem for T time steps can be solved using Θ( t∈T \|Dt \|) messages on expectation. A Parameterised Analysis Despite the optimality of the result, the strategy of computing a new solution from scratch in each time step seems unwise and the analysis does not seem to capture the essence of the problem properly. It often might be the case that there are some similarities between values observed in consecutive time steps and particularly, that Dt ∩ Dt+1 6= ∅. In this case, there might be the chance to keep a representative for several consecutive time steps, which should be exploited. Due to these observations we next define a parameter describing this behavior and provide a parameterised analysis. To this end, we consider the number of component-wise differences in the sequences of nodes Rt−1 and Rt and call this difference the number of changes of representatives in time step t. Let R∗ denote the minimum possible number of changes of representatives (over all considered time steps T ). The formal description of our algorithm is given in Algorithm 2. Roughly speaking, the algorithm defines, for each value v, phases, where a phase is defined as a maximal time interval during which there exists one node observing value v throughout the entire interval. Whenever a node being a representative for v changes its observation, it informs the server so that a new representative can be chosen (from those observing v throughout the entire phase, which is indicated by statusi = 1). If no new representative is found this way, the server tries to find a new representative among those observing v and for which statusi = 0 and ends the current phase. Additionally, if a node observes a value v at time t for which v ∈ / Dt , a new representative is determined among these nodes. Note that this requires each node to store Dt at any time t and hence a storage of O(∆). 6 Algorithm 2 DomainMonitoring (Node i) 1. Define statusi := 1. 2. If at some time t, vit 6= vit−1 , then 2.1. If vit ∈ / Dt−1 , set statusi = 0 and apply ConstantResponse(vit , 0). t 2.2. If vi ∈ Dt−1 , set statusi = 0. Additionally inform server in case i ∈ Rt−1 . 2.3. If server starts a new phase for v = vit , set statusi = 1. (Server) [Initialisation] Call ConstantResponse(nil, 1) to define D0 and for each v ∈ D0 choose a representative uniformly at random from all nodes which have sent v. [Maintaining Dt and Rt at time t] Start with Dt = Dt−1 and Rt = Rt−1 and apply the following rules: • [Current Phase, (try to) find new representative] If informed by representative of a value v ∈ Dt−1 , 1) Call ConstantResponse(v, 1). 2) If node(s) respond(s), choose new representative among the responding sensors uniformly at random. 3) Else call ConstantResponse(v, 0). End current phase for v and, if there is no response, delete v from Dt and the respective representative from Rt . • [If ConstantResponse(v, 0) leads to received message(s), start new phase] Start a new phase for value v if message from an application of ConstantResponse(v, 0) (by Step 3) initialised by the server or initialised in Step 2.1. by a node) is received. Add or replace respective representative in Rt by choosing a node uniformly at random from those responding to ConstantResponse(v, 0). Theorem 2.3. DomainMonitoring as described in Algorithm 2 solves the Domain Monitoring Problem using O(log n·R∗ ) messages on expectation, where R∗ denotes the minimum possible number of changes of representatives. S Proof. We consider each value v ∈ t Dt separately. Let Nt1 ,t2 := {i \| vit = v ∀t1 ≤ t ≤ t2 } denote the set of nodes that observe the value v at each point in time t with t1 ≤ t ≤ t2 . Consider a fixed phase for v and let t1 and t2 be the points in time where the phase starts and ends, respectively. A phase only ends in Step 3), hence there was no response from ConstantResponse(v, 1), for v we can associate a cost of which implies Ntv1 ,t2 = ∅. Thus, to each phase S at least one to R∗ and this holds for each v ∈ t Dt . Therefore, R∗ is at least the overall number of phases of all values. Next we analyze the expected cost of Algorithm 2 during the considered phase for v. Let w.l.o.g. Nt1 := Nt1 ,t1 = {1, 2, . . . , k}. With respect to the fixed phase, only nodes in Nt1 can communicate and the communication is bounded by the number of changes of the representative for v during the phase. Let t′i be the first time after t1 at which node i does not observe v. Let the nodes be sorted such that i < j implies t′i ≥ t′j . Let a1 , . . . , am be the nodes Algorithm 2 chooses as representatives in the considered phase. We want to show that E[m] = O(log k). To this end, partition the set of time steps t′i into groups Gi . Intuitively, Gi represents the time steps in which the nodes 7 continuously observe value v since time t1 and the size of the initial set of nodes that observed v is halved i times. Formally, Gi contains all time steps tℓi−1 +1 , . . . , tℓi (where ℓ−1 := 0 for convenience) such that ℓi is the largest integer fulfilling \|Nt1 ,t′ℓ \| ∈ (k/2i+1 , k/2i ]. i Let Si be the numberPof changes of representatives in time steps belonging log k to Gi . We have E[m] = i=0 E[Si ]. Consider a fixed Si . Let Ej be the event that the j-th representative chosen in time steps belonging to Gi is the first one k ⌋ . Observe that as soon as this happens, the with an index in 1, . . . , ⌊ 2i+1 respective representative will be the last one chosen in a time step belonging to group Gi . Now, since the algorithm chooses a new representative uniformly at random from the index set 1, . . . , ⌊ 2ki ⌋ , the probability that it chooses a representative k from 1, . . . , ⌊ 2i+1 ⌋ is at least 1/2 except for the first representative of v, where it might be slightly smaller due to rounding errors. Ej occurs only if the first j−2 j − 1 representatives were each not chosen from this set, i.e. Pr [Ej ] ≤ 12 . P P P j = O(1). Hence, E[Si ] = j E[Si \|Ej ] · Pr[Ej ] ≤ j j · ( 12 )j−2 = j 2j−2 3 The Frequency Monitoring Problem In this section we design and analyse an algorithm for the Frequency Monitoring Problem, i.e. to output (an approximation) of the number of nodes currently observing value v. We start by considering a single time step and present an algorithm which solves the subproblem to output the number of nodes that observe v within a constant multiplicative error bound. Afterwards, and based on this subproblem, a simple sampling algorithm is presented which solves the Frequency Monitoring Problem for a single time step up to a given (multiplicative) error bound and with demanded error probability. While in the previous section we used the algorithm ConstantResponse with the goal to obtain a representative for a measured value, in this section we will use the same algorithm to estimate the number of nodes that measure a certain value v. Observe that the expected maximal height of the geometric experiment increases with a growing number of nodes observing v. We exploit this fact and use it to estimate the number of nodes with value v, while still expecting constant communication cost only. For a given a time step t and a value v ∈ Dt , we define an algorithm ConstantFactorApproximation as follows: We apply ConstantResponse(v, 1) with statusi = 1 for all nodes i. If the server receives the first response in communication round r ≤ log n, the algorithm outputs ñvconst = 2r as the estimation for \|Ntv \|. We show that we compute a constant factor approximation with constant probability. Then we amplify this probability using multiple executions of the algorithm and taking the median (of the executions) as a final result. Lemma 3.1. The algorithm ConstantFactorApproximation estimates the number \|Ntv \| of nodes observing the value v at time t up to a factor of 8, i.e. ñvconst ∈ [\|Ntv \|/8, \|Ntv \| · 8] with constant probability. Proof. Let nv be the number of nodes currently observing value v, i.e. nv := \|Ntv \|. Recall that the probability for a single node to draw height h is Pr[hi = h] = 21h , if h < log n, and Pr[hi = h] = 22h , if h = log n. Hence, Pr[hi ≥ h] = 1 for all h ∈ {1, . . . , log n}. 2h−1 8 We estimate the probability of the algorithm to fail, by analysing the cases that ñvconst is larger than log nv + 3 or smaller than log nv − 3. We start with the first case and by applying a union bound we obtain: Pr[∃i : hi > log nv + 3] ≤ Pr[∃i : hi ≥ ⌈log nv ⌉ + 3] ⌈log nv ⌉+2 1 1 v =n · ≤ . 2 4 For the latter case we bound the probability that each node has drawn a height strictly smaller than log nv − 3 by Y Pr[hi < ⌈log nv ⌉ − 3] Pr[∀i : hi < log nv − 3] ≤ i = 1− 1 2⌈log nv ⌉−4 nv ≤ 8 1− v n nv ≤ 1 . e8 Thus, the probability that we compute an 8-approximation is bounded by v n v hi Pr ≤ 2 ≤ 8n = 1 − Pr[∃i : hi > log nv + 3] + Pr[∀i : hi < log nv − 3] 8 1 1 > 0.7 + ≥1− 4 e8 We apply an amplification technique to boost the success probability to arbitrary 1 − δ ′ using Θ(log δ1′ ) parallel executions of the ConstantFactorApproximation algorithm and choose the median of the intermediate results as the final output. Corollary 3.2. Applying Θ log δ1′ independent, parallel instances of ConstantFactorApproximation, we obtain a constant factor approximation of \|Ntv \| with success probability at least 1 − δ ′ using Θ log δ1′ messages on expectation. 1 Proof. Choose d = 45 2 ln δ ′ to be the number of copies of the algorithm and return the median of the intermediate results. Let Ij be the indicator variable for the event that the j-th experiment does not result in an 8-approximation. By Lemma 3.1 the failure probability can be upper bounded by a constant, i.e. Pr [Ij ] ≤ 0.3. Hence, using a Chernoff bound, the probability that at least half of the experiments do meet the required approximation factor of 8 is     d d X X 2 1 · 0.3 · d Ij ≥ 1 + Pr  Ij ≥ d ≤ Pr  2 3 j=1 j=1 2 1 · 3 ·0.3·d ≤ e −( 3 ) 2 2 2 45 = e− 45 ·d = e− 45 · 2 ln 1 δ′ = δ′ . Observe that if at least half of the intermediate results are within the demanded error bound, so is the median. Thus, the algorithm produces an 8-approximation of \|Ntv \| with success-probability of at least 1 − δ ′ , concluding the proof. 9 To obtain an (ε, δ)-approximation, in Algorithm 3 we first apply the ConstantFactorApproximation algorithm to obtain a rough estimate of \|Ntv \|. It is used to compute a probability p, which is broadcasted to the nodes, so that every node observing value v sends a message with probability p. Since the ConstantFactorApproximation result ñvconst in the denominator of p is close to \|Ntv \|, the number of messages sent on expectation is independent of \|Ntv \|. The estimated number of nodes observing v is then given by the number of responding nodes n̄v divided by p, which, on expectation, results in \|Ntv \|. Algorithm 3 EpsilonFactorApprox(v ∈ Dt , ε, δ) (Node i) 1. Receive p from the server. 2. Send a response message with probability p. [for fixed time t] (Server) 1. Set δ ′ := δ3 2. Call ConstantFactorApproximation(v, δ ′ ) to obtain ñvconst . 24 1 3. Broadcast p = min 1, ε2 ñv · ln δ′ . const v 4. Receive n̄ messages. 5. Compute and output estimated number of nodes in Ntv as ñv = n̄v /p. Lemma 3.3. The algorithm EpsilonFactorApprox as given in Algorithm 3 provides an (ε,δ)-approximation of \|Ntv \|. Proof. The algorithm obtains a constant factor approximation ñvconst with probability 1 − δ ′ . The expected number of messages is E [n̄v ] = nv · p. We start by estimating the conditional probability that more than (1+ε) nv p responses are sent under the condition that ñvconst ≤ 8nv and p < 1. In this case we have 3 1 1 24 · ln ′ ≥ 2 v · ln ′ , p= 2 v ε ñconst δ ε n δ hence using a Chernoff bound it follows p1 := Pr [n̄v ≥ (1 + ε)nv p \|ñvconst ≤ 8nv ∧ p < 1 ] ≤ e− ε2 3 nv · ε23nv ·ln 1 δ′ = δ′. Likewise the probability that less than (1 − ε) nv p messages are sent under the condition that ñvconst ≤ 8nv and p < 1 is p2 := Pr [n̄v ≤ (1 − ε)nv p \|ñvconst ≤ 8nv ∧ p < 1 ] ≤ e− ε2 2 nv · ε23nv ·ln 1 δ′ 3 1 ≤ e− 2 ln δ′ < δ ′ . Next consider the case that ñvconst > 8nv and p < 1 holds. Using nv v v v v v Pr [ñconst > 8n ] ≤ Pr ñconst > 8n ∨ ñconst < ≤ δ′ 8 and pi · Pr [ñvconst ≤ 8nv ] ≤ pi for i ∈ {1, 2}, Pr [(1 − ε)nv p < n̄v < (1 + ε)nv p \|p < 1 ] ≥ 1 − (Pr [ñvconst > 8nv ] + (p1 + p2 )) ≥ 1 − 3δ ′ = 1 − δ. 10 For the last case p = 1, we have Pr [(1 − ε)nv p < n̄v < (1 + ε)nv p \|p ≥ 1 ] = 1, by using n̄v = nv . Now, Pr [(1 − ε)nv p < n̄v < (1 + ε)nv p] ≥ 1 − δ directly follows. Lemma 3.4. Algorithm EpsilonFactorApprox as given in Algorithm 3 uses Θ( ε12 log δ1 ) messages on expectation. Proof. Recall that each of the nv nodes sends a message with probability p, leading to nv · p messages on expectation. First assume that the constant factor approximation was successful, i.e. n81 ≤ ñvconst ≤ 8n1 . If p < 1, we have 24 1 24 · 8 1 1 1 . nv · p = nv 2 v · ln ′ ≤ · ln = Θ log ε ñconst δ ε2 δ′ ε2 δ If p = 1, by definition ε2 ñ24 · ln δ1′ ≥ 1, hence ñvconst = O ε12 · log δ1′ . Thus, v const nv p ≤ 8ñvconst p = O ε12 · log δ1′ . For the case constant factor approximation was not successful, that the 1 1 v note that Pr ñvconst < 8·2 holds analogously to the calculation in ≤ e2i+3 in 1 v Lemma 3.1. Also, for ñvconst ≥ 8·2 n and p < 1, we have i 24 1 1 1 . nv p ≤ 8 · 2i · ñvconst · 2 v · ln = 2i · Θ 2 log ε ñconst δ ε δ Similarly, for p = 1, we have nv p ≤ 8 · 2i · ñvconst = 2i · Θ ε12 log δ1 as in this case, ñvconst = O ε12 · log δ1′ . Hence, we can conclude 1 v 1 1 v v · Pr ñconst ≥ n E [n̄ ] ≤ Θ 2 log ε δ 8 ∞ X 1 1 1 v 1 i+1 v v · 2 · Θ Pr n ≤ ñ < n log + const 8 · 2i+1 8 · 2i ε2 δ i=0 ! ! ∞ ∞ X X i+3 2i+1 1 1 ≤Θ ≤ Θ 2 log 2i+1−2 1+ 1+ 2i+3 ε δ e i=0 i=0 ! ∞ X 1 1 1 1 ≤ Θ 2 log 2−i = Θ 2 log 1+ . ε δ ε δ i=0 1 1 log ε2 δ Theorem 3.5. There exists an algorithm that provides an (ε,δ)-approximation for the Frequency Monitoring Problem for T time steps with an expected number P \|Dt \| 1 messages. of Θ t∈T \|Dt \| ε2 log δ Proof. In every time step t we first identify Dt by applying ConstantResponse using Θ (\|Dt \|) messages on expectation. On every value v ∈ Dt we then perform algorithm EpsilonFactorApprox(v,ε, \|Dδt \| ), resulting in an amount of Θ \|Dt \| ε12 log \|Dδt \| messages on expectation for a single time step, while 0 \|δ achieving a probability (using a union bound) of 1 − \|D \|D0 \| = 1 − δ that in one time step the estimations for every v are ε-approximations. Applied for each of the T time steps, we obtain a bound as claimed. 11 A Parameterised Analysis Applying EpsilonFactorApprox in every time step is a good solution in worst case scenarios. But if we assume that the change in the set of nodes observing a value is small in comparison to the size of the set, we can do better. We extend the EpsilonFactorApprox such that in settings where from one time step to another only a small fraction σ of nodes change the value they measure, the amount of communication can be reduced, while the quality guarantees remain intact. We define σ such that ∀t : σ ≥ v v \|Nt−1 \ Ntv \| + \|Ntv \ Nt−1 \| . v \|Nt \| Note that this also implies that Dt = Dt−1 holds for all time steps t, i.e. the set of measured values stays the same over time. The extension is designed so that compared to EpsilonFactorApprox, also in settings with many changes the solution quality and message complexity asymptotically does not increase. The idea is the following: For a fixed value v, in a first time step EpsilonFactorApprox is executed (defining a probability p in Step 3 of Algorithm 3). In every following time step, up to 1/δ consecutive time steps, nodes that start or stop measuring a value v send a message to the server with the same probability p, while nodes that do not observe a change in their value remain silent. In every time step t, the server uses the accumulated messages from the first time step and all messages from nodes that started measuring v in time steps 2 . . . t, while subtracting all messages from nodes that stopped measuring v in the time steps 2 . . . t. This accumulated message count is then used similarly as in EpsilonFactorApprox to estimate the total number of nodes observing v in the current time step. The algorithm starts again if a) 1/δ time steps are over, so that the probability of a good estimation remains good enough, or b) the sum of estimated nodes to start/stop measuring value v is too large. The latter is done to ensure that the message probability p remains fitting to the number of nodes, ensuring a small amount of communication, while guaranteeing an (ε, δ)-approximation. − Let n+ t , nt be the number of nodes that start measuring v in time step t or − v v v v that stop measuring it, respectively, i.e. n+ t = \|Nt \ Nt−1 \|, nt = \|Nt−1 \ Nt \|, + − and n̄t and n̄t the number of them that sent a message to the server in time − step t. In the following we call nodes contributing to n+ t and nt entering and leaving, respectively. Lemma 3.6. For any v ∈ D1 , the algorithm ContinuousEpsilonApprox provides an (ε,δ)-approximation of \|Ntv \|. Proof. By the same arguments as in Lemma 3.3, we obtain an (ε,δ ′ )-approximation of n1 . In any further time step we compute our estimate over the sum of all received messages (n̄1 , arrivals and departures). If too many nodes change their measured value, we redo a complete estimation of the nodes in Ntv . Recall that ñt is the random variable giving the estimated number of nodes − n̄+ n̄− by the algorithm, and ñ+ t = p , ñt = p are the random variables giving the estimated arrivals and departures in that time step. We look at Ptany time step − t > 1 where the restart criteria are not met: Since ñt = ñ1 + i=2 ñ+ i − ñi and the linearity of expectation, for any time t ≥ 1 we can use a Chernoff bound as in Lemma 3.3 to show that the estimation is an (ε, δ ′ )-approximation. 12 Algorithm 4 ContinuousEpsilonApprox(v, ε, δ) (Node i) 1. If t = 1, take part in EpsilonFactorApprox called in Step 2 by the server. 2. If t > 1, broadcast a message with probability p if vit−1 = v ∧ vit 6= v or vit−1 6= v ∧ vit = v. (Server) 1. Set δ ′ := δ 2 . 2. Set t := 1 and run EpsilonFactorApprox(v, ε/3, δ) to obtain n̄1 , p. 3. Output ñ1 = n̄p1 . 4. Repeat at the beginning of every new time step t > 1: 4.1. Receive messages from nodes changing the observed value to obtain − n̄+ t and n̄t . P Pt t + − 4.2. Break if t ≥ 1/δ or /p ≥ n̄1 /2. i=1 n̄i + i=1 n̄i Pt P t − /p. 4.3. Output ñt = n̄1 + i=1 n̄+ i − i=1 n̄i 5. Go to Step 2. Using a union bound on the fail probability of up to 1/δ time steps, we get a 1 − δ1 · δ ′ = 1 − δ probability of having a correct estimation in any time step. 1 1 1 Lemma 3.7. For a fixed value v and T ′ = min{ 2σ , δ }, σ ≤ 2 , time steps, 1 1 ContinuousEpsilonApprox uses Θ ε2 log δ messages on expectation. Proof. The message complexity depends on the initial size \|N1v \| and on the number of nodes leaving and entering N v in those time steps, which is bounded by σ. If EpsilonFactorApprox obtained a correct probability p in Step 1, i.e. p = Θ( n11 ), the expected number of messages (in case p < 1) is    ′  T T′ X X 1 1 −  = E n̄1 +  p=Θ E n̄+ n̄t p = Θ i + n̄i n n 1 1 t=1 i=2   T′ X − n+ p ≤ (n1 + T ′ σn1 ) p = n1 + i + ni i=2 1 1 = n1 (1 + T ′ σ) · 24 · 2 v ln ′ ε ñconst δ 1 1 1 2 σ · 1/ε log . =Θ 1 + min , 2σ δ δ Considering the case where EpsilonFactorApprox estimated wrong, the message complexity could increase greatly if the probability p is too large for the actual number of nodes (i.e. an underestimation leads to high message complexity). But the probability to misestimate by some constant factor (which would increase the message complexity by that factor) decreases exponentially in this factor (as shown in Lemma 3.4 for EpsilonFactorApprox), leaving 13 the expected number of messages to be Θ 1 1 Θ ε2 log δ . 1 + min 1 1 2σ , δ σ · 1 ε2 · log 1δ = Theorem 3.8. There exists an (ε,δ)-approximation algorithm for the Frequency Monitoring Problem for T consecutive time steps which uses an amount of Θ \|D1 \| (1 + T · max{2σ, δ}) ε12 log \|Dδ1 \| messages on expectation, if σ ≤ 1/2. Proof. The algorithm works by first applying ConstantResponse(nil,1) to obtain D1 and then applying ContinuousEpsilonApprox(v, ε, δ/\|D1 \|) for every v ∈ D1 . By Lemma 3.6 we know that in every time step and for all v ∈ D1 , the frequency of v is approximated up to a factor of ε with probability 1−δ/\|D1\|. 1 1 We divide the T time steps into intervals of size T ′ = min{ 2σ , δ } and perform ContinuousEpsilonApprox on each of them for every value v ∈ D1 . There are ⌈ TT′ ⌉ ≤ 1+T ·max{2σ, δ} such intervals. For each of those, by Lemma 3.7 we \|D1 \| 1 1 2 messages on expectation for each v ∈ need Θ 1 + min{ 2σ , δ }σ · 1/ε log δ D1 . This yields a complexity of Θ \|D1 \| (1 + T · max{2σ, δ}) ε12 log \|Dδ1 \| due to 1 1 1 , δ }σ ≤ 2σ · σ = Θ(1). Using a union bound over the fail probability for min{ 2σ 1 \|δ every v ∈ D1 , a success probability of at least 1 − \|D \|D1 \| = 1 − δ follows. By Theorem 3.5, trivially repeating the single step algorithm EpsilonFac \|D1 \| 1 messages on expectation for T (betorApprox needs Θ T \|D1 \| ε2 log δ cause the number of nodes in Ntv for any v ∈ D1 is at least N1v /2 in every time step of that interval). Hence, the number of messages sent when using ContinuousEpsilonApprox is reduced in the order of max{2σ, δ}. 4 The Count Distinct Monitoring Problem In this section we present an (ε,δ)-approximation algorithm for the Count Distinct Monitoring Problem. The basic approach is similar to the one presented in the previous section for monitoring the frequency of each value. That is, we first estimate \|Dt \| up to a (small) constant factor and then use the result to define a protocol for obtaining an (ε, δ)-approximation. If we could assume that, at any fixed time t, each value was observed by at most one node, it would be possible to solve this problem with expected communication cost of O( ε12 log δ1 ) (per time step t and per value v ∈ Dt ) using the same approach as in the previous section. Since this assumption is generally not true, we aim at simulating such behaviour that for each value in the domain only one random experiment is applied. We apply the concept of public coins, which allows nodes measuring the same value to observe identical outcomes of their random experiments. To this end, nodes have access to a shared random string R of fully independent and unbiased bits. This can be achieved by letting all nodes use the same pseudorandom number generator with a common starting seed, adding a constant number of messages to the bounds proven below. We assume that the server sends a new seed in each phase by only loosing at most a constant factor in the amount of communication used. However, we can drop this assumption by checking whether there are nodes that changed their value such that only in 14 rounds in which there are changes new public randomness is needed. The formal description of the algorithm for a constant factor and an ε-approximation are given in Algorithm 5 and Algorithm 6, respectively. We consider the access of the public coin to behave as follows: Initialised with a seed, a node accesses the sequence of random bits R bitwise, i.e. after reading the j’th bit, the node next accesses bit j + 1. Observe the crucial fact that as long as each node accesses the exact same number of bits, each node observes the exact same random bits simultaneously. Algorithm 5 essentially works as follows: In a first step, each node draws a number from a geometrical distribution using the public coin. By this, all nodes observing the same value v obtain the same height hv . In the second step we apply the strategy as in the previous section to reduce communication if lots of nodes observe the same value: Each node i draws a number gi from a geometrical distribution without using the public coin. Afterwards, all nodes with the largest height gi among those with the largest height hv broadcast their height hv . Algorithm 5 ConstantFactorApproximation [for fixed time t] (Node i, observes value v = vi ) 1. Draw a random number hv as follows: Consider the next ∆ · log n random bits b1 , . . . , b∆·log n from R. Let h be the maximal number of bits bv·log n+1 , . . . , bv·log n+1+h that equal 0. Define hv := min{h, log n}. 2. Let gi′ be a random value drawn from a geometric distribution with successprobability p = 1/2 and define gi = min(gi′ , log n) (without accessing public coins). 3. Broadcast drawn height hv in round r = log2 n − (hv − 1) · log n − gi unless a node i′ has broadcasted before. (Server) 1. Receive a broadcast message containing height h in round r. 2. Output dˆt = 2h . Note that only (at most n) nodes that observe value v with hv = maxv′ hv′ may send a message in Algorithm 5. Now, all nodes observing the same value observe the same outcome of their random experiments determining hv . Hence, by a similar reasoning as in Lemma 3.1, one execution of the algorithm uses O(1) messages on expectation. Using the algorithm given in Algorithm 5 and applying the same idea as in the previous section, we obtain an (ε, δ)-approximation as given in Algorithm 6: Each node tosses a coin with a success probability depending on the constant factor approximation (for which we have a result analogous to Corollary 3.2). Again, all nodes use the public coin so that all nodes observing the same value obtain the same outcome of this coin flip. Afterwards, those nodes which have observed a success apply the same strategy as in the previous section, that is, they draw a random value from a geometric distribution, and the nodes having the largest height send a broadcast. 15 Algorithm 6 EpsilonFactorApprox [for fixed time t] (Node i) 1/δ 1. Flip a coin with success probability p = 2−q = c log , q ∈ N as follows: ε2 dˆt Consider the next ∆ · q random bits b1 , . . . b∆·q . The experiment is successful if and only if all random bits bv·q+1 , . . . , bv·q+q equal 0. The node deactivates (and does not take part in Steps 2. and 3.) if the experiment was not successful. 2. Draw a random value h′i from a geometric distribution and define hi = min(h′i , log n) (without accessing public coins). 3. Node i broadcasts its value in round log n−hi unless a node i′ with vit = vit′ has broadcasted before. (Server) 1. Let St be the set of received values. 2. Output d˜t := \|St \|/p Using arguments analogous to Lemmas 3.3 and 3.4 and applying EpsilonFactorApprox for T time steps, we obtain the following theorem. Theorem 4.1. There exists an (ε, δ)-approximation algorithm for the Count Distinct Monitoring Problem for T time steps using O(T · ε12 log δ1 ) messages on expectation. A Parameterised Analysis In this section we consider the problem for multiple time steps and parameterise the analysis with respect to instances in which the domain does not change arbitrarily between consecutive time steps. Recall that for monitoring the frequency from a time step t − 1 to the current time step t, all nodes that left and all nodes that entered toss a coin to estimate the number of changes. However, to identify that a node observes a value which was not observed in the previous time step, the domain has to be determined exactly. We apply the following idea instead: For each value v ∈ {1, . . . , ∆} we flip a (public) coin. We denote the set of values with a successful coin flip as the sample. Afterwards, the algorithm only proceeds on the values of the sample, i.e. in cases in which a node observes a value with a successful coin flip and no node observed this value in previous time steps, this value contributes to the estimate d˜+ t at time t. Regarding the (sample) of nodes that leave the set of observed values, the DomainMonitoring algorithm is applied to identify which (sampled) values are not observed any longer (and thus contribute to d˜− t ). Analogous to Lemma 3.6, we have the following lemma. Lemma 4.2. ContinuousEpsilonApprox achieves an (ε,δ)-approximation of \|Dt \| in any time step t. For the number of messages, we argue based on the previous section. However, in addition the DomainMonitoring algorithm is applied. Observe that the size of the domain changes by at most n/2, and consider the case that this number of nodes observed the same value v. The expected cost (where the expectation is taken w.r.t. whether v is within the sample) is O(log n · R∗ · p) = n·R∗ 1 O log \|Dt \|ε2 log δ . Similar to Theorem 3.8, we then obtain the following theorem. 16 Algorithm 7 ContinuousEpsilonApprox(ε, δ) 1. Compute δ ′ = 2 δ 2 2. Broadcast a new seed value for the public coin. 3. Compute an (ε, δ ′ )-approximation d˜1 of \|D1 \| using Algorithm 6. Furthermore, obtain the success-probability p. 4. Repeat for each time step t > 1: 4.1. Each node i applies Algorithm 2 if the observed value vi is in the sample set. Let dˆ− t be the number of values (in sample set) which number of nodes that join the sample. left the domain and dˆ+ t the P Pt ˆ− t ˜ ˜ 4.2. Server computes dt = d1 + i=2 dˆ+ i /p − i=2 di /p. Pt ˜+ Pt ˜− 4.3. Break if t = 1/δ or d + d /p exceeds d˜1 /2. i=2 i i=2 i 5. Set t = 1 and go to Step 2. Theorem 4.3. ContinuousEpsilonApprox provides and (ε, δ)-approximation for the Count Distinct Monitoring Problem for T time steps using an log(n)·R∗ 1 amount of Θ (1 + T · max{2σ, δ}) \|Dt \|·ε2 log δ messages on expectation, if σ ≤ 1/2. References [1] Graham Cormode, S. Muthukrishnan, and Ke Yi. Algorithms for distributed functional monitoring. In Proceedings of the 19th Annual ACMSIAM Symposium on Discrete Algorithms (SODA ’08), pages 1076–1085. SIAM, 2008. [2] Graham Cormode, S. Muthukrishnan, and Ke Yi. Algorithms for Distributed Functional Monitoring. ACM Transactions on Algorithms, 7(2):21:1–21:20, 2011. [3] Sashka Davis, Jeff Edmonds, and Russell Impagliazzo. Online Algorithms to Minimize Resource Reallocations and Network Communication. In Proceedings of the 9th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th International Conference on Randomization and Computation (APPROX’06/RANDOM ’06), volume 4110 of Lecture Notes in Computer Science, pages 104–115. Springer, 2006. [4] Phillip B. Gibbons and Srikanta Tirthapura. Estimating simple functions on the union of data streams. In Proceedings of the 13th annual ACM Symposium on Parallel Algorithms and Architectures (SPAA ’01), pages 281–291. ACM, 2001. [5] Zengfeng Huang, Ke Yi, and Qin Zhang. Randomized algorithms for tracking distributed count, frequencies, and ranks. In Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS ’12), pages 295–306. ACM, 2012. 17 [6] Tak Wah Lam, Chi-Man Liu, and Hing-Fung Ting. Online Tracking of the Dominance Relationship of Distributed Multi-dimensional Data. In Proceedings of the 8th International Workshop on Approximation and Online Algorithms (WAOA ’10), volume 6534 of Lecture Notes in Computer Science, pages 178–189. Springer, 2010. [7] Alexander Mäcker, Manuel Malatyali, and Friedhelm Meyer auf der Heide. Online Top-k-Position Monitoring of Distributed Data Streams. In Proceedings of the 2015 IEEE International Parallel and Distributed Processing Symposium (IPDPS ’15), pages 357–364. IEEE, 2015. [8] Alexander Mäcker, Manuel Malatyali, and Friedhelm Meyer auf der Heide. On Competitive Algorithms for Approximations of Top-k-Position Monitoring of Distributed Streams. In Proceedings of the 2016 IEEE International Parallel and Distributed Processing Symposium (IPDPS ’16), pages 700–709. IEEE, 2016. [9] David P. Woodruff and Qin Zhang. Tight bounds for distributed functional monitoring. In Proceedings of the 44th Symposium on Theory of Computing (STOC ’12), pages 941–960. ACM, 2012. [10] Ke Yi and Qin Zhang. Multidimensional online tracking. ACM Transactions on Algorithms, 8(2):12, 2012. 18	8
Inverse of a Special Matrix and Application Thuan Nguyen arXiv:1708.07795v1 [cs.DM] 25 Aug 2017 School of Electrical and Computer Engineering, Oregon State University, Corvallis, OR, 97331 Email: nguyeth9@oregonstate.edu Abstract—The matrix inversion is an interesting topic in algebra mathematics. However, to determine an inverse matrix from a given matrix is required many computation tools and time resource if the size of matrix is huge. In this paper, we have shown an inverse closed form for an interesting matrix which has much applications in communication system. Base on this inverse closed form, the channel capacity closed form of a communication system can be determined via the error rate parameter α. Keywords: Inverse matrix, convex optimization, channel capacity. I. M ATRIX C ONSTRUCTION In Wireless communication system or Free Space Optical communication system, due to the shadow effect or the turbulent of environment, the channel conditions can be flipped from “good” to “bad” or “bad” to “good” state such as Markov model after the transmission time σ [1] [2]. For simple intuition, in “bad” channel, a signal will be transmitted incorrectly and in “good” channel, the signal is received perfectly. Suppose a system has total n channels, the “good” channel is noted as “1” and “bad” channel is “0”, respectively, the transmission time between transmitter and receiver is σ, the probability the channel is flipped after the transmission time σ is α. We note that if the system using a binary code such as On-Off Keying in Free Space Optical communication, then the flipped probability α is equivalent to the error rate. Consider a simple case for n = 2, suppose that at the beginning, both channel is “good” channel, the probability of system has both of channels are “good” after transmission time σ, for example, is (1−α)2 . Let call Aij is the probability of system from the state has i − 1 “good” channels and n − i + 1 “bad” channels transfers to state has j − 1 “good” and n − j + 1 “bad” channels. Obviously that 1 ≤ i ≤ n + 1 and 1 ≤ j ≤ n + 1. For example, the transition matrix A2 and A3 for n = 2 and n = 3 are constructed respectively as follows: (1 − α)2  A2 = α(1 − α) α2   (1 − α)3  (1 − α)2 α A3 =  (1 − α)α2 α3 2α(1 − α) (1 − α)2 + α2 2α(1 − α) 3(1 − α)2 α 2(1 − α)α2 + (1 − α)3 2(1 − α)2 α + α3 3α2 (1 − α)  α2 α(1 − α) . (1 − α)2 3α2 (1 − α) 2(1 − α)2 α + α3 2(1 − α)α2 + (1 − α)3 3(1 − α)2 α given by Proposition 2. Moreover, this matrices are obviously central symmetric matrix. Proposition 1. For n channels system, the transition matrix An has size (n + 1) × (n + 1) and all entries An ij in row i column j will be established by An ij = s=max(i−j,0) n+1−i s i−1 j−i+2s n−(j−i+2s) α (1 − α) Proof. From the definition, Anij is the probability from state has i − 1 “good” channels or i − 1 bit “1” transfer to state has j − 1 “good” channels or j − 1 bit “1”. Therefore, suppose s is the number channels in i − 1 “good” channels that is flipped to “bad” channels after the transmission time σ and 0 ≤ s ≤ i − 1. Thus, to maintain j − 1 “good” channels after the time σ, the number of “bad” channels in n + 1 − i “bad” channels must be flipped to “good” channels is: (j − 1) − ((i − 1) − s) = j − i + s Therefore, the total number of channels are flipped their state after transmission time σ is: s + (j − i + s) = j − i + 2s and the total number of channels that preserves their state after transmission time σ is n−(j −i+2s). However, 0 ≤ s ≤ i−1. Similarly, the number of “bad” channels in n + 1 − i “bad” channels must be flipped to “good” channels should be in 0 ≤ j − i + s ≤ n + 1 − i. Hence: ( max s = min(n + 1 − j; i − 1) min s = max(0; i − j) Therefore, An ij can be determined by below form: An ij = j − i + s s=min(n+1−j,i−1) X s=max(i−j,0) n+1−i s i−1 j−i+2s n−(j−i+2s) α (1 − α) Proposition 2. All the entries of inverse matrix A−1 n given in Proposition 1 can be determined via original transition matrix  A n for ∀ α 6= 1/2. α3  (1 − α)α2  . (1 − α)2 α (1 − α)3 These transition matrices are obviously size (n+1)×(n+1) since the number of “good” channels can achieve n+1 discrete values from 0, 1, . . . , n. Moreover, these class matrices have several interesting properties: (1) all entries in matrix An can be determined by Proposition 1; (2) the inverse of matrix An is s=min(n+1−j,i−1) j − i + s X An −1 ij = (−1)i+j An (1 − 2α)n ij Due to the pages limitation, we will show the detailed proof at the end of this paper. To illustrate our result, an example of the inverse matrix A2 are shown as follows: A−1 2  (1 − α)2 1  = −α(1 − α) (1 − 2α)2 α2 −2α(1 − α) (1 − α)2 + α2 −2α(1 − α)  α2 −α(1 − α) . (1 − α)2 Next, base on the existence of inverse matrix closed form, we will show that a capacity closed form for a discrete memory-less channel can be established. We note that in [3], the authors said that haven’t closed form for channel capacity problem. However, with our approach, the closed form can be established for a wide range of channel with error rate α is small. II. O PTIMIZE SYSTEM CAPACITY A discrete memoryless channel is characterized by a channel matrix A ∈ Rm×n with m and n representing the numbers of distinct input (transmitted) symbols xi , i = 1, 2, . . . , m, and output (received) symbols yj , j = 1, 2, . . . , n, respectively. The matrix entry Aij represents the conditional probability that given a symbol xi is transmitted, the symbol xj is received. Let p = (p1 , p2 , . . . , pm )T be the input probability mass vector, where pi denotes the probability of transmitting symbol xi , then the probability mass vector of output symbols q = (q1 , q2 , . . . , qn )T = AT p, where qi denotes the probability of receiving symbol yi . For simplicity, we only consider the case n = m such that the number of transmitted input patterns is equal the number of received input patterns. The mutual information between input and output symbolsis: I(X; Y ) = H(Y ) − H(Y \|X), where H(Y ) = H(Y \|X) = j=n X qj j=1 m X n X − log qj pi Aij log Aij . i=1 j=1 Thus, the mutual information function can be written as: I(X; Y ) = − j=n X (AT p)j log (AT p)j + m X n X pi Aij log Aij , i=1 j=1 j=1 where (AT p)j denotes the jth component of the vector q = (AT p). The capacity C of a discrete memoryless channel associated with a channel matrix A puts a theoretical maximum rate that information can be transmitted over the channel [3]. It is defined as: C = max I(X; Y ). p (1) Therefore, finding the channel capacity is to find an optimal input probability mass vector p such that the mutual information between the input and output symbols is maximized. For a given channel matrix A, I(X; Y ) is a concave function in p [3]. Therefore, maximizing I(X; Y ) is equivalent to minimizing −I(X; Y ), and the capacity problem can be cast as the following convex problem: Minimize: m X n n X X pi Aij log Aij (AT p)j log (AT p)j − i=1 j=1 j=1 Subject to: ( pi 0 1T p = 1 Optimal numerical values of p∗ can be found efficiently using various algorithms such as gradient methods [4] [5]. However, in this paper, we try to figure out the closed form for optimal distribution p via KKT condition. The KKT conditions state that for the following canonical optimization problem: Problem Miminize: f (x) Subject to: gi (x) ≤ 0, i = 1, 2, . . . n, hj (x) = 0, j = 1, 2, . . . , m, construct the Lagrangian function: L(x, λ, ν) = f (x) + n X i=1 λi gi (x) + m X νj hj (x), (2) j=1 then for i = 1, 2, . . . , n, j = 1, 2, . . . , m, the optimal point x∗ must satisfy:   gi (x∗ ) ≤ 0,    ∗   hj (x ) = 0, dL(x,λ,ν) (3) \|x=x∗ ,λ=λ∗ ,ν=ν ∗ = 0, dx   ∗ ∗  λi xi = 0,    λ∗ ≥ 0. i Our transition matrix that is already established in previous part can represent as a channel matrix. In the optical transmission, for example, the transmission bits are denoted by the different levels of energy, for example, in On-Off Keying code bit “1” and “0” is represented by high and low power level. This energy is received by a photo diode and converse directly to the voltage for example. However, these photo diode work base on the aggregate property when collecting all the incident energy, that said, if two channels transmit a bit “1” then the photo diode will receive the same energy “2” even though this energy comes from a different pair of channels. Therefore, the received signal is completely dependent to the number of bits “1” in transmission side. Hence, in receiver side, the photo diode will recognize n + 1 states 0, 1, 2, . . . , n. From this property, the transition matrix A is the previous section is exactly the system channel matrix. The channel capacity of system, therefore, is determined as an optimization problem in (1). Next, we will show that the above optimization problem can be solved efficiently by KKT condition. We note that our method can establish the closed form for general channel matrix and then the results are applied to special matrix An . First, we try to optimize directly with input distribution p, however, the KKT condition for input distribution is too complicated to construct the first derivation. On the other hand, base on the existence of inverse channel matrix, the output variable is more suitable to work with KKT condition since. Due to 0 ≤ qj ≤ 1, the Lagrange function from (2) with output variable q is: L(qj , λj , νj ) = I(X, Y ) + j=n X j=n X qj λj + ν( qj − 1) j=1 j=1 Using KKT conditions, at optimal point qj∗ , λ∗j , ν ∗ :  ∗ qj ≥ 0   Pj=n ∗     j=1 qj = 1   dI(X, Y ) ν ∗ − λ∗j − =0 dqj∗    λ∗ ≥ 0     j∗ ∗ λj qj = 0 P Because 0 ≤ pi ≤ 1, i = 1, . . . , (n) and ni=1 pi = 1, so Pi=n always exist pi > 0. From qj = i=1 pi Aij with ∀ Aij > 0, we can see clearly that qj > 0 with ∀qj or qj∗ > 0 with ∀qj∗ . Therefore with fifth condition, λ∗j = 0 with ∀λ∗j . Then, we have simplified KKT conditions: Pj=n ∗  j=1 qj = 1 dI(X, Y ) =0 ν ∗ − dqj∗ The derivations are determined by: j=n i=n dI(X, Y ) X −1 X Aij log Aij − (1 + log qj ) Aji = dqj j=1 i=1 Let call: i=n X A−1 ji j=n X Aij log Aij = Kj From the second KKT simplified condition, we can compute ∀ qj∗ : ∗ qj∗ = 2Kj −ν −1 And finally: ∗ Due to the channel matrix is a closed form of α, the optimal input vector p and output vector q also is a function of α. However, we note that since the KKT condition works directly to the output variable q, the optimal input p can be invalid pi > 1 or pi < 0. In next step, our simulations shown that for n ≤ 10 and α ≤ 0.2, both output and input vector are valid. That said, our approach will be worked with a good system where the error probability α is small. In case of the invalid optimal input vector, the upper bound of channel capacity, of course, will be established. III. C ONCLUSION In this paper, our contributions are twofold: (1) establish an inverse closed form for a class of channel matrix based on the error probability α; (2) figure out the closed form for channel matrix with small error rate α and determine the upper bound system capacity for a high error rate channel. R EFERENCES [1] Jeff McDougall and Scott Miller. Sensitivity of wireless network simulations to a two-state markov model channel approximation. In Global Telecommunications Conference, 2003. GLOBECOM’03. IEEE, volume 2, pages 697–701. IEEE, 2003. [2] Hong Shen Wang and Nader Moayeri. Finite-state markov channel-a useful model for radio communication channels. IEEE transactions on vehicular technology, 44(1):163–171, 1995. [3] Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012. [4] Michael Grant, Stephen Boyd, and Yinyu Ye. Cvx: Matlab software for disciplined convex programming, 2008. [5] Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. j=1 i=1 A PPENDIX Next, using derivation of I(X,Y) at qj = qj∗ and last KKT condition: ν ∗ = Kj − (1 + log qj ∗ ) Hence: qj∗ = 2Kj −ν ∗ −1 Next, using first KKT simplified condition, we have the sum of all output states is 1. j=n X 2Kj −ν ∗ −1 =1 j=1 2 ν∗ = j=n X 2 Kj −1 j=1 Therefore, ν ∗ can be figured out by: ∗ ∗ pT = q T A−1 ij ν = log j=n X j=1 2Kj −1 Proof for Proposition 2. Proof. To simplify our notation, the “good” and “bad” channel are represented by bit “1” and “0”, respectively. Next, we will use the definition to show that: An An −1 = I If matrix A∗n is constructed by A∗nij = (−1)i+j An ij , then we need to show that: An A∗n = B = (1 − 2α)n I Firstly, we note that the An ij and An ∗ij is only different by sign of the first index (−1)i+j . Therefore, Bij which is computed by product of row i in matrix An ij and column j in matrix An ∗ij , can be computed by: Bij = k=n+1 X k=1 An ik A∗nkj Note that the An ik is the probability from state i “good” channels (with i − 1 bit “1” and n − i + 1 bit “0”) to medium state has k “good” channels (with k − 1 bit “1” and n − k + 1 bit “0”). Moreover, if the sign is ignored, then An ∗kj also is the probability going from state k to state j, too. However, n the state k includes C k−1 sub-states which have a same number of “good” and “bad” channels. For example with n = 2, state k = 2 includes two sub-states that contains one “good” and one “bad” channels are “10” an “01”. Therefore, the total number of sub-states while k runs from 1 to n is P k=n+1 n C k−1 = 2n sub-states. Let compute Bij by divided k=1 into two subsets: Compute Bij for i = j: This means that Bii is the sum of the probability from state i − 1 bit “1” go to states has k − 1 bit “1” then come back to state has i − 1 bit “1”. In 2n sub-states, we can divide back to n + 1 categories by the number of different position between i and k. • If all the bit in i and k are the same, then the probability is: ! ! C n n (1 − α)2n (1 − α)n (1 − α)n = C 0 0 • If all the bit in i and k different at only one position, then the probability is: ! n (1 − α)2(n−1) (1 − α)2 C 1 • If all the bit in i and k different at only two positions, then the probability is: ! n C (1 − α)2(n−2) (1 − α)2×2 2 • If all the bit in i and k different at all positions, then the probability is: ! C n (1 − α)2n n Therefore, Bii can be determined by the probability of all n + 1 categories such as: ! n 2t C α (1−α)2n−2t = ((1 − α)2 − α2 )n = (1−2α)n Bii = t t=0 t=n X A∗nkj into two subsets: Compute Bij for i6= j: Let divide k + j is odd and A∗nkj < 0 or k + j is even and A∗nkj > 0, Pk=n respectively. Therefore, Bij = k=1 An ik A∗nkj also is distributed into positive or negative subsets. Next, we will show that the positive subset in Bij is equal the negative subset then Bij = 0 for i 6= j. Indeed, suppose that state i with i − 1 bit “1” go to state k1 and then to back to state j with j −1 bit “1” and Bik1 is positive value. Next, we will show that existence a state k2 such that Bik2 is negative value and Bik1 = −Bik2 . Let call s is the number of positions where state i and j have a same bit. Obviously that s ≤ n − 1 due to i 6= j. For example if n = 4 and i = 1111 and j = 0001, we have s = 1 because i and j share a same bit “1” in the positions fourth. Suppose that an arbitrary state k1 are picked, we will show how to chose the state k2 with Bik1 = −Bik2 . Consider two follows cases: • If (n − s) is odd. k2 is constructed by maintain s position of k1 where i and j have same bit and flip bit in the n − s rest positions. • If (n − s) is even. k2 is constructed by maintain s + 1 position of k1 where s position are i and j have a same bit and one position where i and j have a different bit, next n − s − 1 rest positions will be flipped. Note that since s ≤ n − 1 then we are able to flip n − s − 1 rest positions. We obviously can see that k1 and k2 satisfied the probability condition \|Bik1 \| = \|Bik2 \| due to the number of flipped bit between i and k1 equals the number of flipped bit between k2 and j and the number of flipped bit between j and k1 equals the number of flipped bit between k2 and i. Next, we will prove that k1 and k2 make Bik1 and Bik2 in different subsets. Indeed, call number of bit “1” in k1 is b1 , number of bit “1” in k2 is b2 , number of bit “1” in s bit same of i and j is bs , respectively. Therefore, the number of bit “1” of k1 in (n − s) rest positions is (k1 − ks ), the number of bit “1” of k2 in (n − s) rest positions is (k2 − ks ). • If (n − s) is odd. Since all bit in (n − s) rest positions of k1 is flipped to create k2 , then total number of bit “1” in n − s bit of k1 and k2 is (k1 − ks + k2 − ks = n − s) is odd. So, (k1 + k2 ) should be an odd number. That said (k1 − k2 ) is odd or (k1 + j) − (k2 + j) is odd. Therefore, Bik1 and Bik2 bring the contradict sign. • If (n − s) is even. Because, we fix one more position to create k2 , then number of flipped bit (n−s−1) is odd number. If one more bit is fixed in k1 is “0”, we have a same result with case (n − s) is odd. If fixed bit is “1”, similarly in first case (k1 − ks − 1) + (k2 − ks − 1) = n − s − 1 is odd number, therefore (k1 + k2) is odd number. That said (k1 − k2 ) is odd or (k1 + j) − (k2 + j) is odd. Therefore, Bik1 and Bik2 bring the contradict sign. Therefore, the state k2 always can be created from a random state k1 and Bik1 and Bik2 bring a contradict sign. That said for i 6= j, Bij = 0. Therefore: B = (1 − 2α)n I The Proposition 2, therefore, are proven.	7
AN INTRODUCTION TO PRESENTATIONS OF MONOID ACTS: QUOTIENTS AND SUBACTS arXiv:1709.08916v1 [math.GR] 26 Sep 2017 CRAIG MILLER AND NIK RUŠKUC Abstract. The purpose of this paper is to introduce the theory of presentations of monoids acts. We aim to construct ‘nice’ general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an M -act A and a subact B of A, on the one hand we construct presentations for B and the Rees quotient A/B using a presentation for A, and on the other hand we derive a presentation for A from presentations for B and A/B. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a subact B has finite complement in an M -act A. We show that if M is a finitely generated monoid and B is finitely presented, then A is finitely presented. We also show that if M belongs to a wide class of monoids, including all finitely presented monoids, then the converse also holds. 1. Introduction The concept of presentations is significant within many areas of algebra. Finite presentability of acts was first studied by P. Normak in 1977 [11], and is a fundamental finiteness condition for the theory of monoid acts (see [9]). The related notion of coherency for monoids was introduced by V. Gould in 1992 [6], and has since been intensively studied by several authors (see [7], [8]). Finite presentability of acts also plays a key role in the monoid properties of being right Noetherian [11] and being completely right pure [5]. However, there has not yet been developed a systematic theory of presentations for acts over monoids. This paper is concerned with introducing such a theory through considering presentations for two of the most basic constructions: quotients and subacts. A follow-on article will deal with various product constructions of acts. The paper is structured as follows. In Section 2, we collect some basic definitions and facts about acts. In Section 3, we introduce the notions of presentations and finite presentability for a monoid act, provide various examples of act presentations, and record several results which will be of vital importance in the rest of the paper. In the remainder of the paper, we study presentations for various constructions. Typically we first obtain a general presentation for a construction and then derive corollaries regarding finite presentability. Section 4 is concerned 2010 Mathematics Subject Classification. 20M30, 20M05. 1 with presentations of Rees quotients. Before moving to presentations of subacts in general in Section 6, we first discuss presentations of unions of subacts in Section 5. Section 6 splits into two parts; in the first part we construct a general presentation for a subact, and in the second part we study a specific case where the subact has finite complement. 2. Preliminaries The theory of monoid acts is essentially the theory of representations of monoids by transformations. A monoid is a semigroup with an identity. One of the most common and universal ways of defining monoids is by means of presentations, and we briefly review the basics here. Let Z be an alphabet. We denote by Z ∗ the monoid of all words in Z. A monoid presentation is a pair hZ \| P iMon, where P ⊆ Z ∗ × Z ∗ . A monoid M is said to be defined by the monoid presentation hZ \| P iMon if M∼ = Z ∗ /ρ, where ρ is the smallest congruence on Z ∗ containing P . Thus we can identify M with Z ∗ /ρ, so that the elements of M are the ρ-classes of words from Z ∗ . To put it differently, each word w ∈ Z ∗ represents an element of M. Let u, v ∈ Z ∗ . We say that v is obtained from u by an application of a relation from P if u = pqr and v = pq ′ r, where p, r ∈ Z ∗ and (q, q ′) ∈ P . We say that u = v is a consequence of P if either u and v are identical words or if there exists a sequence u = w1 , w2 , . . . , wk = v where each wi+1 is obtained from wi by an application of a relation from R. We have the following basic fact: Lemma 2.1. Let M be a monoid defined by a presentation hZ \| P iMon, and let u, v ∈ Z ∗ . Then u = v holds in M if and only u = v is a consequence of R. Let M be a monoid with identity 1. An M-act is a non-empty set A together with a map A × M → A, (a, m) 7→ am such that a(mn) = (am)n and a1 = a for all a ∈ A and m, n ∈ M. For instance, M itself is an M-act via right multiplication. A subset B of an M-act A is a subact of A if bm ∈ B for all b ∈ B, m ∈ M. Note that the right ideals of M are precisely the subacts of the M-act M. A subset U of an M-act A is a generating set for A if for any a ∈ A, there exist u ∈ U, m ∈ M such that a = um. We write A = hUi if U is a generating set for A. An M-act A is said to be finitely generated (resp. cyclic) if it has a finite (resp. one-element) generating set. Note that a right ideal of M can be generated by a set as an M-act or as a semigroup. We introduce the convention that ‘generate’ will always mean ‘generate as an M-act’. For M-acts A and B, a map θ : A → B is an M-homomorphism if (am)θ = (aθ)m for all a ∈ A, m ∈ M. If θ is also bijective, then it is an M-isomorphism, and we write A ∼ = B. 2 An equivalence relation ρ on A is an (M-act) congruence on A if (a, b) ∈ ρ implies (am, bm) ∈ ρ for all a, b ∈ A and m ∈ M. For a congruence ρ on an M-act A, the quotient set A/ρ = {[a] : a ∈ A} becomes an M-act by defining [a]m = [am]. Given an M-act A and a subact B of A, we define the Rees congruence ρB on A by aρB b ⇐⇒ a = b or a, b ∈ B for all a, b ∈ A. We denote the quotient act A/ρB by A/B and call it the Rees quotient of A by B. We usually identify the ρB -class {a} ∈ A/B with a for each a ∈ A \ B, and denote the ρ-class B ∈ A/B by 0. For an M-act A and X ⊆ A × A, we denote by hXicg the smallest congruence on A containing X. A congruence ρ on an M-act A is finitely generated if there exists a finite subset X ⊆ A × A such that ρ = hXicg . Let A be an M-act and let X ⊆ A × A. We introduce the notation X = X ∪ {(u, v) : (v, u) ∈ X}, which will be used throughout the paper. For a, b ∈ A, an X-sequence connecting a and b is any sequence a = p1 m1 , q1 m1 = p2 m2 , q2 m2 = p3 m3 , . . . , qk mk = b, where (pi , qi ) ∈ X and mi ∈ M for 1 ≤ i ≤ k. We now provide the following useful lemma (see [9, Section 1.4] for a proof): Lemma 2.2. Let M be a monoid, let A be an M-act, let X ⊆ A × A and let a, b ∈ A. Then (a, b) ∈ hXicg if and only if either a = b or there exists an X-sequence connecting a and b. A generating set U for an M-act A is a basis of A if for any a ∈ A, there exist unique u ∈ U and m ∈ M such that a = um. An M-act A is said to be free if it has a basis. We have the following structure theorem for free acts. Proposition 2.3. [9, Theorem 1.5.13]. An M-act A is free if and only if it is isomorphic to a disjoint union of M-acts all of which are M-isomorphic to M. This leads to the following explicit construction of a free act. Construction 2.4. Let M be a monoid. Let X be a non-empty set and consider the set X × M. With the operation (x, m)n = (x, mn) for all (x, m) ∈ X × M and n ∈ M, the set X × M is a free M-act with basis X. We denote this M-act by FX,M , although we will usually just write FX . We will also usually write x · m for (x, m) and x for (x, 1). 3 Proposition 2.5. [9, Theorem 1.5.15]. Let A be an M-act and let F be a free M-act with basis X. If φ is any map from X to A, then there exists a unique M-homomorphism θ : F → A such that θ\|X = φ. Further, if Xφ is a generating set for A, then θ is surjective. Corollary 2.6. For any M-act A, there exists a free M-act F such that A is a homomorphic image of F . 3. Presentations of monoid acts Let M be a monoid. An (M-act) presentation is a pair hX \| Ri, where X is a non-empty set and R ⊆ FX × FX is a relation on the free M-act FX . An element x of X is called a generator, while an element (u, v) of R is called a (defining) relation, and is usually written as u = v. An M-act A is said to be defined by the presentation hX \| Ri if A is M-isomorphic to the quotient act FX /ρ, where ρ = hRicg is the smallest congruence on FX containing R. Let A be an M-act and θ : A → FX /ρ an M-isomorphism, where ρ = hRicg . We say an element w ∈ FX represents an element a ∈ A if aθ = [w]ρ . In the context of presentations, we write w1 ≡ w2 if w1 and w2 are equal in FX , and w1 = w2 if they represent the same element of A. Remark 3.1. Let A be an M-act and let X be any generating set for A. By Proposition 2.5, there exists a surjective M-homomorphism θ : FX → A, so we have that A ∼ = FX /ker θ by the First Isomorphism Theorem. Therefore, A is defined by the presentation hX \| Ri where R is any relation which generates ker θ. Hence, every M-act can be defined by a presentation. Definition 3.2. Let hX \| Ri be a presentation and let w1 , w2 ∈ FX . We say that the relation w1 = w2 is a consequence of R if w1 ≡ w2 or there is an R-sequence connecting w1 and w2 . We say that w2 is obtained from w1 by an application of a relation from R if there exists an R-sequence with only two distinct terms connecting w1 and w2 . The next lemma follows immediately from Lemma 2.2. Lemma 3.3. Let hX \| Ri be a presentation, let A be the M-act defined by hX \| Ri, and let w1 , w2 ∈ FX . Then w1 = w2 in A if and only if w1 = w2 is a consequence of R. Let M be a monoid, let A be an M-act generated by a set Y , and let φ : X → Y be a surjective map. Let θ : FX → A be the unique M-homomorphism extending φ, and let R be a subset of FX × FX . We say that A satisfies R (with respect to φ) if for each (u, v) ∈ R, we have uθ = vθ; that is, R ⊆ ker θ. Note that the M-act defined by a presentation hX \| Ri satisfies R. From the definition of an act defined by a presentation and Lemma 2.2, we have: 4 Proposition 3.4. Let M be a monoid, let A be an M-act generated by a set X, and let R ⊆ FX × FX . Then hX \| Ri is a presentation for A if and only if the following conditions hold: (1) A satisfies R; (2) if w1 , w2 ∈ FX such that A satisfies w1 = w2 , then w1 = w2 is a consequence of R. The next fact follows from Proposition 2.5 and the Third Isomorphism Theorem for acts. Proposition 3.5. Let A be an M-act defined by a presentation hX \| Ri, let B be an M-act, and let φ : X → B be a map onto a generating set for B. If the generators Xφ of B satisfy all the relations from R, then there exists a surjective M-homomorphism ψ : A → B. Definition 3.6. A finite presentation is a presentation hX \| Ri where X and R are finite. An M-act A is finitely presented if it can be defined by a finite presentation. Note that a right ideal of a monoid M may be finitely presented as an M-act or as a semigroup. When we say that a right ideal is ‘finitely presented’, we will always mean as an M-act. Example 3.7. (1) The free M-act FX is defined by the finite presentation hX \| i. In particular, if X is finite, then FX is finitely presented. (2) For any monoid M, the M-act M is finitely presented, since M is a free M-act with basis {1}. The following results are specialisations of well-known facts from general algebra. They essentially reflect the fact that congruence-generation is an algebraic closure operator. See, for instance, Section 1.5 and Theorem 2.5.5 in [3] for more details. Proposition 3.8. Let M be a monoid, let A be an M-act defined by a finite presentation hX \| Ri, and let Y be another finite generating set for A. Then A can be defined by a finite presentation in terms of the generators Y . Proposition 3.9. Let M be a monoid, and let A be a finitely presented M-act with a presentation hX \| Si where X is finite and S is infinite. Then there exists a finite subset S ′ ⊆ S such that A is defined by the finite presentation hX \| S ′i. Corollary 3.10. [11, Theorem 2]. Let M be a monoid and let A be a cyclic M-act. Then A is finitely presented if and only if A is isomorphic to a quotient act of M by a finitely generated right congruence on M. Let M be a monoid with a generating set S and let A be an M-act with a generating set X. It can be easily proved, using Proposition 3.4, that the following are all presentations for A: (1) hA \| a · m = am (a ∈ A, m ∈ M)i; 5 (2) hX \| x · m = y · n (x, y ∈ X, m, n ∈ M, xm = yn)i; (3) hA \| a · s = as (a ∈ A, s ∈ S)i. Given the presentation in (3), we immediately have the following: Lemma 3.11. If M is a finitely generated monoid and A is a finite M-act, then A is finitely presented. If M is a non-finitely generated monoid, however, then finite M-acts are not necessarily finitely presented, as the following example demonstrates. Example 3.12. Let M = X ∗ be a free monoid with X infinite and consider the trivial M-act A = {0}. Now A is defined by the presentation h0 \| 0 · x = 0 (x ∈ X)i. If A were finitely presented, then it could be defined by a finite presentation P = h0 \| 0 · x = 0 (x ∈ X0 )i, where X0 is a finite subset of X. But for x 6∈ X0 , the relation 0 · x = 0 is clearly not a consequence of the relations of P , so A is not finitely presented. Remark 3.13. One may be tempted to think that the trivial M-act being finitely presented is equivalent to M being finitely generated. However, the trivial M-act is in fact finitely presented for a much larger class of monoids M. For example, if M is a monoid with a left zero z, it can be easily proved that the trivial M-act {0} is defined by the finite presentation h0 \| 0 = 0 · zi. The following lemma provides a necessary and sufficient condition for the trivial act to be finitely presented. Lemma 3.14. Let M be a monoid. Then the trivial M-act {0} is finitely presented if and only if there exists a finitely presented M-act A which contains a zero. Proof. The direct implication is obvious. For the converse, let A be an M-act with a zero 0, and suppose that A is defined by a finite presentation hX \| Ri where 0 ∈ X. We define a finite set R′ = {0 · m = 0 · n : (x · m, y · n) ∈ R for some x, y ∈ X}, and claim that {0} is defined by the presentation h0 \| R′i. We need to show that for any m ∈ M, the relation 0 · m = 0 is a consequence of R′ . Let m ∈ M. Since 0 · m = 0 holds in A, it is a consequence of R, so we have an R-sequence connecting 0 · m and 0. Now, replacing every x ∈ X appearing in this R-sequence with 0, we obtain an R′ -sequence connecting 0 · m and 0, so 0 · m = 0 is a consequence of R′ . Tietze transformations (for acts) provide a method for yielding a new presentation for a monoid act from a known presentation. Given a presentation hX \| Ri for an M-act A, the elementary Tietze transformations are: 6 (T1) adding new relations ui = vi , i ∈ I, to hX \| Ri, providing that each ui = vi is a consequence of R; (T2) deleting relations ui = vi , i ∈ I, from R, providing that each ui = vi is a consequence of R \ {ui = vi : i ∈ I}; (T3) adding new generating elements yi , i ∈ I, and new relations yi = wi , i ∈ I, to hX \| Ri, for any wi ∈ FX ; (T4) if hX \| Ri has relations xi = wi , i ∈ I, where xi ∈ X and wi ∈ FX ′ where X ′ = X \ {xi : i ∈ I}, then deleting each xi from X, deleting each xi = wi from R, and replacing all remaining appearances of xi with wi . Proposition 3.15. Two presentations define the same M-act if and only if one can be obtained from the other by a finite number of applications of elementary Tietze transformations. Remark 3.16. The proof of Proposition 3.15 is essentially the same as the proof for its analogue in group theory or semigroup theory. To see an idea of the proof, one may consult [13, Theorem 2.5]. Corollary 3.17. Let M be a monoid, and let A be an M-act defined by a presentation hX \| Si where X is finite and S is infinite. Then A is finitely presented if and only if there exists a finite subset S ′ ⊆ S such that every relation from S is a consequence of S ′ . Proof. Suppose that A is finitely presented. By Proposition 3.9, there exists a finite subset S ′ ⊆ S such that A is defined by a finite presentation hX \| S ′ i. Therefore, since every relation from S holds in A, it must be a consequence of S ′ . Conversely, suppose that there exists a finite subset S ′ ⊆ S such that every relation from S is a consequence of S ′ . Using Tietze transformations, we can delete every relation from S \ S ′ . By Proposition 3.15, we have that A is defined by the finite presentation hX \| S ′i. 4. Rees quotients Let M be a monoid, let A be an M-act and let B be a subact of A. Recall that the Rees quotient A/B is the quotient act resulting from the Rees congruence ρB on A given by aρB b ⇐⇒ a = b or a, b ∈ B for all a, b ∈ A. We shall identify the ρB -class {a} ∈ A/B with a for each a ∈ A\B, and denote the ρ-class B ∈ A/B by 0. The purpose of this section is, on the one hand, to construct a presentation for A/B using a presentation for A and a generating set for B, and on the other hand, to derive a presentation for A using presentations for B and A/B. These general presentations will give rise to corollaries pertaining to finite presentability. Let X be any generating set for A. We now give a presentation for A/B in terms of the generators (X \ B) ∪ {0}. 7 Theorem 4.1. Let M be a monoid. Let A be an M-act defined by a presentation hX \| Ri, let B be a subact of A generated by Y , and let h0 \| Si be a presentation for the trivial M-act {0}. For each y ∈ Y , choose wy ∈ FX such that y = wy holds in A, and let R′ = R ∪ {y = wy : y ∈ Y }. We now define the sets R1 = {(u, v) ∈ R : u represents an element of A \ B}, R2 = {u = 0 : u ∈ FX\B ∩ L(X, B), (u, v) ∈ R′ for some v ∈ FX∪Y }, where L(X, B) denotes the set of elements of FX which represent elements of B. Then A/B is defined by the presentation hX \ B, 0 \| R1, R2 , Si. Proof. It is clear that A/B satisfies R1 , R2 and S. Let X ′ = X \ B, and let w1 , w2 ∈ FX ′ ∪{0} such that w1 = w2 holds in A/B. By Proposition 3.4, we just need to show that w1 = w2 is a consequence of R1 , R2 and S. If w1 represents an element of A \ B, then w1 = w2 is a consequence of R1 . Suppose w1 represents 0 in A/B. We claim that w1 = 0 is a consequence of R1 ∪ R2 ∪ S. If w1 ∈ F0 , then w1 = 0 is a consequence of S. Now assume that w1 ∈ FX ′ . By Proposition 3.15, we have that A is defined by the presentation hX, Y \| R′i. Choose w1′ ∈ FY such that w1 = w1′ holds in A. Then w1 = w1′ is a consequence of R′ , so there exists an R′ -sequence w1 = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = w1′ , where (pi , qi ) ∈ R′ and mi ∈ M for 1 ≤ i ≤ k. Note that for i ∈ {1, . . . , k}, if pi represents an element of A \ B, then pi+1 ∈ FX ′ . Therefore, since p1 ∈ FX ′ and pk represents an element of B (pk = qk in A and qk ∈ FY ), we may choose i minimal such that pi ∈ FX ′ and pi represents an element of B. We then have that w1 = pi mi is a consequence of R1 , and we obtain 0 · mi from pi mi by an application of a relation from R2 . Now, since 0 · mi = 0 is a consequence of S, we deduce that w1 = 0 is a consequence of R1 , R2 and S. This proves the claim. Exactly the same argument proves that w2 = 0 is a consequence of R1 ∪ R2 ∪ S, and hence so is w1 = 0 = w2 , as required. Corollary 4.2. Let M be a monoid, let A be a finitely presented M-act and let B be a finitely generated subact of A. Then A/B is finitely presented if and only if the trivial M-act is finitely presented (which includes all finitely generated monoids). Proof. If A/B is finitely presented, then it follows from Lemma 3.14 that the trivial M-act is finitely presented, since A/B contains a zero. The converse follows immediately from Theorem 4.1. We now turn to our second aim in this section: assembling a presentation for A from those for a subact and the Rees quotient. So, let M be a monoid, let A be an M-act and let B be a subact of A. Let X be a generating set for B and let Y be a generating set for A/B, and let Y ′ = Y \ {0}. Note that if A \ B is a subact of A, the element 0 must belong to Y , and so Y = Y ′ ∪ {0}. If A \ B is not a 8 subact of A/B, then there exist elements w ∈ FY ′ that represent 0 in A/B, and the set Y need not contain 0. We shall now give a presentation for A in terms of the generators X ∪ Y ′ . Theorem 4.3. Let M be a monoid, let A be an M-act and let B be a subact of A. Let hX \| Ri and hY \| Si be presentations for B and A/B respectively, and let Y ′ = Y \ {0}. If A \ B is not a subact of A/B, for each w ∈ FY ′ that represents 0 in A/B choose αw ∈ FX such that w = αw in A, and also fix one of them and denote it by z. We now define the sets S1 = {(u, v) ∈ S : u represents an element of A \ B}; S2 = {u = αu : (u, v) ∈ S for some v ∈ FY , u ∈ FY ′ , u = 0 in A/B} ∪ {z = αz }. Then A is defined by the presentation hX, Y ′ \| R, S1, S2 i. Proof. We first claim that if an element w ∈ FY ′ represents an element of B in A, then there exists w ′ ∈ FX such that w = w ′ is a consequence of relations from S1 and S2 . Indeed, we have that w = z holds in A/B, so w = z is a consequence of S; that is, there exists an S-sequence w = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = z, where (pi , qi ) ∈ S and mi ∈ M for 1 ≤ i ≤ k. If all (pi , qi ) ∈ S1 , then w = αz is a consequence of S1 and z = αz . Otherwise, we take (pi , qi ) ∈ S \ S1 with i minimal, so w = pi mi is a consequence of S1 , and we obtain αpi mi from pi mi by an application of a relation from S2 . We shall now show that A is defined by the presentation hX, Y ′ \| R, S1 , S2 i. It is clear that A satisfies R, S1 and S2 . Let w1 , w2 ∈ FX∪Y ′ be such that w1 = w2 in A. If w1 represents an element of A \ B, then w1 = w2 is a consequence of S1 . Now suppose that w1 represents an element of B. Using the above claim, if necessary, we have w1′ , w2′ ∈ FX such that w1 = w1′ and w2 = w2′ are consequences of S1 and S2 (if wi ∈ FX , simply let wi ≡ wi′ ). But then w1′ = w2′ holds in B, so it is a consequence of R. Hence, we have that w1 = w2 is a consequence of R, S1 and S2 . Corollary 4.4. Let M be a monoid, let A be an M-act and let B be a subact of A. If B and A/B are finitely presented, then A is finitely presented. 5. Unions In this section we consider presentations for unions of acts. A union of acts can be of one of two types: disjoint or amalgamated. An amalgamated union of Macts is a union of a family of M-acts intersecting pairwise in a common subact. We only consider the union of two acts, although the results of this section can easily be generalised to any finite number of acts. Throughout the section we aim to prove our results in the general setting where C = A ∪ B is an M-act with 9 A and B subacts, and A ∩ B is potentially non-empty. Each of those results will typically have an immediate corollary for disjoint unions, which we state separately immediately after. The main purpose of the section is to explore under what conditions we have C = A∪B is finitely generated (resp. finitely presented) if and only if A and B are finitely generated (resp. finitely presented), and to provide interesting examples to demonstrate that this does not occur in general. We begin by considering finite generation. Lemma 5.1. Let C = A ∪ B be an M-act with A and B subacts of C. If A and B are finitely generated, then C is finitely generated. Proof. If A = hXi and B = hY i, then C = hX ∪ Y i. In the following example, we show that the converse to Lemma 5.1 does not hold in general by constructing a finitely generated monoid M and right ideals A and B of M such that C = A ∪ B is finitely generated (in fact, finitely presented) but neither A nor B are finitely generated. Example 5.2. Let M = {a, b}∗ . Let X = {ai b : i ≥ 0} and Y = {bi a : i ≥ 0}, and let A and B be the right ideals generated by X and Y respectively. It is clear that A and B are not finitely generated. We have that C = A ∪ B is generated by the set {a, b} and is free with respect to this generating set, so C is finitely presented. Lemma 5.3. Let C = A ∪ B be an M-act with A and B subacts of C, and suppose that A ∩ B is either empty or finitely generated. If C is finitely generated, then A and B are finitely generated. Proof. If A ∩ B = ∅, let U = ∅; otherwise, let A ∩ B = hUi where U is finite. Suppose that C = hZi. Let Y = Z \ B and let X = Y ∪ U. Let a ∈ A. If a ∈ A \ B, then a = ym for some y ∈ Y and m ∈ M. If a ∈ A ∩ B, then a = um for some u ∈ U and m ∈ M. Therefore, we have that A = hXi. Hence, if Z is finite, A is finitely generated, and by symmetry so is B. Corollary 5.4. Let A and B be disjoint M-acts. Then A ∪ B is finitely generated if and only if A and B are finitely generated. We now turn our attention to finite presentability. We begin by giving a general presentation for C = A ∪ B, and we then immediately derive a corollary that gives a sufficient condition for C to be finitely presented. Theorem 5.5. Let C = A ∪ B be an M-act with A and B subacts of C. Let A and B have presentations hX \| Ri and hY \| Si respectively. If A ∩ B 6= ∅, let U be a generating set for A ∩ B; otherwise, let U = ∅. For each u ∈ U, choose ρX (u) ∈ FX and ρY (u) ∈ FY which both represent u in C, and define a set T = {ρX (u) = ρY (u) : u ∈ U}. 10 Then C is defined by the presentation hX, Y \| R, S, T i. Proof. Let w1 , w2 ∈ FX∪Y such that w1 = w2 in C. If w1 , w2 ∈ FX , then w1 = w2 is a consequence of R. If w1 , w2 ∈ FY , then w1 = w2 is a consequence of S. Suppose now that w1 ∈ FX and w2 ∈ FY . Let c = um, with u ∈ U and m ∈ M, be the element of A ∩ B that both w1 and w2 represent. Since w1 = ρX (u)m holds in A, it is a consequence of R, and likewise w2 = ρY (u)m is a consequence of S. We also obtain ρY (u)m from ρX (u)m by an application of a relation from T . Therefore, we have that w1 = w2 is a consequence of R, S and T . Corollary 5.6. Let C = A ∪ B be an M-act with A and B subacts of C, and suppose that A ∩ B is either empty or finitely generated. If A and B are finitely presented, then C is finitely presented. The converse to Corollary 5.6 does not hold in general. Recall that in Example 5.2 we showed that there exists a monoid M and M-acts A and B such that C = A ∪ B is finitely presented but neither A nor B are finitely generated. We now present a more striking example: Example 5.7. There exist a monoid M and finitely generated right ideals A and B of M such that A ∩ B is finitely generated and C = A ∪ B is finitely presented but neither A nor B are finitely presented. Let M be the monoid defined by the presentation ha, b, s, t \| abi a = aba, bai b = bab, sa = a, tb = b (i ≥ 2)iMon . We have a complete rewriting system RM on X = {a, b, s, t} consisting of the rules abi a → aba, bai b → bab, sa → a, tb → b (i ≥ 2), and this yields the following set of normal forms for M: X ∗ \ X ∗ ({abi a, bai b : i ≥ 2} ∪ {sa, tb})X ∗ ; that is, the set of all words in X which do not contain as a subword the left-hand side of one of the rewriting rules. For more information on rewriting systems, one may consult [2] for instance. Let A and B be the right ideals of M generated by {a, t} and {b, s} respectively. From the monoid presentation for M, we see that A is defined by the infinite presentation ha, t \| a · bi a = a · ba (i ≥ 2)i. If A were finitely presented, then it could be defined by a presentation P = ha, t \| a · bi a = a · ba (2 ≤ i ≤ k)i. But if i > k, then the relation a · bi a = a · ba cannot be a consequence of the relations of P , since there do not exist m, n ∈ M such that bi a = mn in M and (a · m, a · ba) ∈ P ; in other words, no relation of P can be applied to a · bi a. 11 Therefore, A is not finitely presented. Similarly, we have that B is not finitely presented. We also that A ∩ B = ha, bi. It is clear from the monoid presentation for M that C = A ∪ B is generated by the set {s, t} and is free with respect to this generating set, so hence C is finitely presented. We now turn to consider conditions for when C = A ∪ B being finitely presented implies that the components A and B are both finitely presented. Theorem 5.8. Let C = A ∪ B be an M-act with A and B subacts of C, and suppose that A ∩ B is either empty or finitely presented. If C is finitely presented, then both A and B are finitely presented. Proof. It clearly suffices to show that A is finitely presented. If A ∩ B = ∅, let U = ∅; otherwise, let A ∩ B be defined by a finite presentation hU \| Si. Suppose C is defined by a finite presentation hZ \| Ri. Let Y = Z \ B and X = Y ∪ U. As in the proof of Lemma 5.3, we have that A = hXi. Also, let Y ′ = Z ∩ B. For each w ∈ FZ such that w represents an element of A∩B, choose ρU (w) ∈ FU which represents the same element of A ∩ B. Also, for each u ∈ U, choose wu ∈ FZ such that wu represents u. We now define the following sets: R1 = {(u, v) ∈ R : u, v ∈ FY }; R2 = {u = wu : u ∈ U, wu ∈ FY }; R3 = {u = ρU (u) : u ∈ FY , (u, v) ∈ R for some v ∈ FY ′ }. Note that the set R2 may be empty (if wu ∈ FY ′ for every u ∈ U); however, this does not affect the argument that follows. We make the following claim: Claim. If an element w ∈ FY represents an element of A ∩ B, then there exists w ′ ∈ FU such that w = w ′ is a consequence of relations from R1 , R2 and R3 . Proof. Let w ∈ FY represent an element c ∈ A ∩ B. Now c = um for some u ∈ U and m ∈ M. Since w = wu m holds in C, it is a consequence of R, so there exists an R-sequence w = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = wu m, where (pi , qi ) ∈ R and mi ∈ M for 1 ≤ i ≤ k. If each (pi , qi ) ∈ R1 , then w = wu m is a consequence of R1 , and we obtain u · m from wu m by an application of a relation from R2 . Otherwise, there exists i minimal such that pi ∈ FY and qi ∈ FY ′ , so w = pi mi is a consequence of R1 , and we obtain ρU (pi )mi from pi mi by an application of a relation from R3 . Returning to the proof of Theorem 5.8, we shall show that A is defined by the finite presentation hX \| R1 , R2 , R3 , Si. Let w1 , w2 ∈ FX such that w1 = w2 holds in A. If w1 represents an element of A \ B, then w1 = w2 is a consequence of R1 . Suppose w1 represents an element of A ∩ B. Using the claim above, if necessary, we have w1′ , w2′ ∈ FU such that w1 = w1′ and w2 = w2′ are consequences of R1 , R2 and R3 (if wi ∈ FU , simply let 12 wi ≡ wi′ ). Since w1′ = w2′ holds in A ∩ B, we have that w1′ = w2′ is a consequence of S. Therefore, w1 = w2 is a consequence of R1 , R2 , R3 and S. Corollary 5.9. Let A and B be disjoint M-acts. Then A ∪ B is finitely presented if and only if A and B are finitely presented. We now investigate how finite presentability of C = A∪B affects the intersection A ∩ B. Proposition 5.10. Let C = A ∪ B be an M-act with A and B subacts of C, and suppose that A ∩ B is non-empty. If C is finitely presented and A and B are finitely generated, then A ∩ B is finitely generated. Proof. Let A and B be generated by finite sets X and Y respectively. Since C is finitely presented, it can be defined by a finite presentation hX, Y \| Ri. For any w ∈ FX , let w denote the element of A which w represents, and define U = {u : u ∈ FX , (u, v) ∈ R for some v ∈ FY } ⊆ A ∩ B. Let c ∈ A ∩ B. Choose w1 ∈ FX and w2 ∈ FY which both represent the element c. Since w1 = w2 holds in C, it is a consequence of R, so there exists an R-sequence w1 = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = w2 , where (pi , qi ) ∈ R and mi ∈ M for 1 ≤ i ≤ k. Now, there exists i ∈ {1, . . . , k} such that pi ∈ FX and qi ∈ FY , and we have that c = pi mi , so c ∈ hUi. Hence, we have that A ∩ B = hUi, so A ∩ B is finitely generated. Corollary 5.11. Let C = A ∪ B be an M-act with A and B finitely presented subacts of C and A ∩ B non-empty. Then C is finitely presented if and only if A ∩ B is finitely generated. Remark 5.12. In the categorical sense, the disjoint union of acts is a coproduct and the amalgamated union of acts is a pushout. In the category of groups, the coproduct is called the free product and the pushout is called the free product with amalgamation. Notice the similarity between Corollary 5.11 and a well-known result, due to G. Baumslag, which states that for two finitely presented groups G1 and G2 such that H = G1 ∩ G2 is a group, the amalgamated free product G = G1 ∗H G2 is finitely presented if and only if H is finitely generated; see [1, Chapter 6] for more details. Given the results concerning finite presentability in this section, the following two questions arise: Do there exist monoids M and M-acts A, B and C with C = A ∪ B, such that: (1) A and B are finitely presented but C is not finitely presented? (2) A and B are finitely presented, while A ∩ B is finitely generated but not finitely presented? 13 In the following, we exhibit examples which provide positive answers to both of the above two questions. Example 5.13. There exists a monoid M and finitely presented right ideals A and B of M such that C = A ∪ B is not finitely presented. Let M be the monoid defined by the presentation ha, b, c \| aci a = bci−1 b (i ≥ 2)iMon . We have a complete rewriting system RM on X = {a, b, s, t} consisting of the rules aci a → bci−1 b (i ≥ 2), and this yields the following set of normal forms for M: X ∗ \ (X ∗ {aci a : i ≥ 2}X ∗ ). Let A and B be the right ideals of M generated by {a} and {b} respectively. We have that A and B are free M-acts and hence finitely presented. Let X = {aci a : i ≥ 2}. It is clear from the monoid presentation for M that X generates A∩B, and that this is a minimal generating set for A ∩ B, so A ∩ B is not finitely generated. It now follows from Corollary 5.11 that C = A ∪ B is not finitely presented. Example 5.14. There exists a monoid M and finitely presented right ideals A and B of M such that A ∩ B is finitely generated but not finitely presented. Let M be the monoid defined by the presentation ha, b, c \| a2 = a, cab = ab, abi a = aba (i ≥ 2)iMon. We have a complete rewriting system RM on X = {a, b, c} consisting of the rules a2 → a, cab → ab, abi a → aba (i ≥ 2), and this yields the following set of normal forms for M: X ∗ \ X ∗ ({abi a : i ≥ 2} ∪ {a2 , cab})X ∗ . Let A and B be the right ideals of M generated by {a} and {c} respectively. From the monoid presentation for M, we see that B is a free M-act (and hence finitely presented), that A is defined by the infinite presentation ha \| a · a = a, a · bi a = a · ba (i ≥ 2)i, and that A ∩ B is defined by the infinite presentation hy \| y · bi a = y · a (i ∈ N)i, where y represents ab. We claim that A is also defined by the finite presentation ha \| a · a = ai. Indeed, for any i ≥ 2, we have a · bi a = (a · a)bi a ≡ a · abi a ≡ a · aba ≡ (a · a)ba ≡ a · ba. It can be shown that A ∩ B is not finitely presented using a similar argument to the one in Example 5.7. 14 Note that since A and B are finitely presented and A ∩ B is finitely generated, Corollary 5.11 implies that C = A ∪ B is finitely presented. In fact, it is easy to see that C is defined by the finite presentation ha, c \| a · a = a, a · b = c · abi. 6. Subacts In this section we consider presentations for subacts of monoid acts. In the first part of the section we construct a general (infinite) presentation for a subact of a monoid act. From this presentation we obtain a method for finding ‘nicer’ presentations in special situations. We note that for general monoids M, finitely generated subacts of finitely presented M-acts are not necessarily finitely presented. In the second part of the section, we shall consider a particular case where we have a subact B with finite complement in an M-act A; we say that B is large in A and A is a small extension of B. This was motivated by the analogous concept of ‘large subsemigroups’ within semigroup theory; see [12] for more details. In particular, it is shown there that various finiteness properties, including finite generation and finite presentability, are inherited by both large subsemigroups and small extensions of semigroups. Given these results, it is natural to ask whether similar results hold in the setting of monoid acts. We shall show that, for finitely generated monoids M, finite generation is inherited by both large subacts and small extensions, and finite presentability is also inherited by small extensions. Somewhat surprisingly, though, there exist finitely generated monoids M for which large subacts of finitely presented M-acts are not necessarily finitely presented. We shall show, however, that there is a large class of monoids for which finite presentability is inherited by large subacts. Let M be a monoid, let A be an M-act defined by a presentation hX \| Ri, and let B be a subact of A generated by a set Y . We seek a presentation for B in terms of the generators Y . For each y ∈ Y , we choose wy ∈ FX which represents y, and let ψ : FY → FX be the unique M-homomorphism extending y 7→ wy . We call ψ the representation map. For an element w ∈ FX which represents an element of B, we have w = x · m for some x ∈ X and m ∈ M, and xm = yn for some y ∈ Y, n ∈ M, so w = (yψ)n holds in B. Therefore, we have a map φ : L(X, B) → FY , where L(X, B) denotes the set of all elements of FX which represent elements of B, satisfying (wφ)ψ = w in A for all w ∈ L(X, B). We call φ a rewriting map. Note that the existence of φ follows from the Axiom of Choice. We now state our first result of this section, giving a presentation for B, which has analogues within group and semigroup theory; see [10, Theorem 2.6] and [4, Theorem 2.1] for more details. 15 Theorem 6.1. Let M be a monoid. Let A be an M-act defined by a presentation hX \| Ri and let B be a subact of A generated by Y . For each y ∈ Y , we choose wy ∈ FX which represents y. Let ψ be the representation map and let φ be a rewriting map, and define the following sets of relations: R1 = {y = wy φ : y ∈ Y }; R2 = {(wm)φ = (wφ)m : w ∈ L(X, B), m ∈ M}; R3 = {(um)φ = (vm)φ : (u, v) ∈ R, m ∈ M, um ∈ L(X, B)}. Then B is defined by the presentation hY \| R1 , R2 , R3 i. Proof. We first show that B satisfies R1 , R2 and R3 . This amounts to showing that uψ = vψ holds in A for each u = v in R1 , R2 and R3 . For each y ∈ Y , we have yψ ≡ wy = (wy φ)ψ holds in A, since wy ∈ L(X, B). For any w ∈ L(X, B) and m ∈ M, we have ((wm)φ)ψ = wm = ((wφ)ψ)m ≡ ((wφ)m)ψ holds in A. Finally, for any (u, v) ∈ R, m ∈ M such that um ∈ L(X, B), we have ((um)φ)ψ = um = vm = ((vm)φ)ψ holds in A. We now claim that for any w ∈ FY , we have that w = (wψ)φ is a consequence of R1 and R2 . Indeed, we have w ≡ y · m for some y ∈ Y and m ∈ M, so wψ ≡ wy m. We obtain (wy φ)m from w by an application of the relation y = wy φ, and since wy ∈ L(X, B), we have that (wψ)φ = (wy φ)m is a relation from R2 . Now let w1 , w2 ∈ FY be such that w1 = w2 holds in B. Since w1 ψ = w2 ψ holds in A, it is a consequence of R, so we have an R-sequence w1 ψ = p1 m1 , q1 m1 = p2 m2 , . . . , qk mk = w2 , where (pi , qi ) ∈ R and mi ∈ M for 1 ≤ i ≤ k. For each i ∈ {1, . . . , k}, we have pi mi ∈ L(X, B), so (w1 ψ)φ = (w2 ψ)φ is a consequence of the relations (pi mi )φ = (qi mi )φ of R3 . Finally, since w1 = (w1 ψ)φ and (w2 ψ)φ = w2 are consequences of R1 and R2 , we conclude that w1 = w2 is a consequence of R1 , R2 and R3 . Remark 6.2. The presentation from Theorem 6.1 has the disadvantage that it always has infinitely many relations if M is an infinite monoid, and neither the rewriting map nor the set L(X, B) have been defined constructively. However, the result does give a method for finding ‘nice’ presentations for subacts in certain cases. Given an M-act A defined by a presentation hX \| Ri and a subact B of A, this method consists of the following: (1) finding a generating set Y for B; (2) finding a rewriting map φ : L(X, B) → FY ; (3) finding a set S ⊆ FY × FY of relations which hold in B and imply the relations of the presentation P given in Theorem 6.1. 16 Using Tietze transformations, we can add S to P (S must be a consequence of the relations of P since these are defining relations for B) and then remove the remaining relations (since they are consequences of S). Hence, by Proposition 3.15, we have that B is defined by the presentation hY \| Si. For the remainder of this section we shall be considering large subacts. Recall that a subact B of an M-act is said to be large in A, and A is said to be a small extension of B, if the set A \ B is finite. We shall investigate how similar an Mact A and a large subact B of A are with regard to finite generation and finitely presentability. We begin by considering finite generation. Lemma 6.3. Let M be a monoid, let A be an M-act and let B be a large subact of A. If B is finitely generated, then A is finitely generated. Proof. If B is generated by a set X, then A is generated by X ∪ (A \ B). In the following, we show that the converse to Lemma 6.3 does not hold for monoids in general, but it does however hold for all groups and all finitely generated monoids. Example 6.4. Let M = X ∗ with X infinite. Let I = X + , so I is a large subact of the cyclic M-act M. Clearly X is a minimal generating set for I, so I is not finitely generated. Lemma 6.5. Let M be a group, let A be an M-act and let B be a large subact of A. If A is finitely generated, then B is finitely generated. Proof. Since A is the disjoint union of its subacts B and A \ B, it follows from Corollary 5.4 that B is finitely generated if A is finitely generated. The following result provides a generating set for a large subact. Proposition 6.6. Let M be a monoid generated by a set Z, let A be an M-act generated by a set X, and let B be a large subact of A. Define a set S = {am ∈ B : a ∈ A \ B, m ∈ Z}, and let Y = (X ∩ B) ∪ S. Then B is generated by the set Y . Proof. Since Y ⊆ B and B is a subact of A, we have hY i ⊆ B. Let b ∈ B. If b = xm for some x ∈ X ∩ B, m ∈ M, then b ∈ hY i. Otherwise, b = xm1 . . . mk for some x ∈ X \ B, mi ∈ Z. Let s be minimal such that xm1 . . . ms ∈ B, and let a = xm1 . . . ms−1 . Then a ∈ A \ B and ams ∈ B, so ams ∈ S. Therefore, we have b = (ams )ms+1 . . . mk ∈ hSi ⊆ hY i. Hence, we have that B = hY i. Corollary 6.7. Let M be a finitely generated monoid, let A be an M-act and let B be a large subact of A. If A is finitely generated, then B is finitely generated. 17 We have shown that, for groups and finitely generated monoids M, finite generation is inherited by both large subacts and small extensions. We now turn our attention to finite presentability. For a monoid M, there are two questions relating to large subacts that arise: (1) Is every small extension of every finitely presented M-act finitely presented? (2) Is every large subact of every finitely presented M-act finitely presented? We first show that the property every small extension of every finitely presented M-act is finitely presented, is equivalent to another monoid property, and from this we immediately derive as a corollary that finitely generated monoids M admit a positive answer to question (1). Proposition 6.8. The following are equivalent for a monoid M: (1) every finite M-act is finitely presented; (2) every small extension of every finitely presented M-act is finitely presented. Proof. (1) ⇒ (2). Let A be a small extension of a finitely presented M-act B. We have that A/B is finite and hence finitely presented by assumption. Since B and A/B are finitely presented, it follows that A is finitely presented by Corollary 4.4. (2) ⇒ (1). Let A be a finite M-act. Choose a finitely presented M-act B disjoint from A. We have that A ∪ B is a small extension of B, so it is finitely presented by assumption. Hence, by Corollary 5.9, we have that A is finitely presented. Corollary 6.9. Let M be a finitely generated monoid. Then every small extension of every finitely presented M-act is finitely presented. We now turn to consider which monoids M give a positive answer to question (2); that is, every large subact of every finitely presented M-act is finitely presented. We first present an example which reveals that there exist finitely generated monoids which do not possess this property, and then we show that there exists a large class of monoids for which the property holds. Example 6.10. Let M be the monoid defined by the presentation ha, b \| abi a = aba (i ≥ 2)iMon . Let I = M \ {1, a}, so I is a large subact of the finitely presented M-act M. Now I = hb, a2 , abi is a disjoint union of hbi, ha2 i and habi. Clearly hbi and ha2 i are free and hence finitely presented. Letting y = ab, we have that hyi is defined by the presentation hy \| y · bi a = y · a (i ∈ N)i. We saw in Example 5.14 that hyi is not finitely presented. It hence follows from Corollary 5.9 that I is not finitely presented. Lemma 6.11. Let M be a group, let A be an M-act and let B be a large subact of A. If A is finitely presented, then B is finitely presented. 18 Proof. Since A is the disjoint union of its subacts B and A \ B, it follows from Corollary 5.9 that B is finitely presented if A is finitely presented. Definition 6.12. A monoid M is right coherent if every finitely generated subact of every finitely presented M-act is finitely presented. Examples of right coherent monoids include groups, Clifford monoids, semilattices, the bicyclic monoid, free commutative monoids and free monoids [8]. Since for any finitely generated monoid M, a large subact of a finitely generated M-act is finitely generated, we have the following result: Lemma 6.13. Let M be a finitely generated right coherent monoid, let A be an M-act and let B be a large subact of A. If A is finitely presented, then B is finitely presented. Before stating our final result of this section, we first introduce a technical definition. Let M be a monoid with a presentation hZ \| P iMon, and let A be an M-act with a presentation hX \| Ri. For a word w in Z ∗ , let w denote the element of M which w represents. We say an element x · w ∈ FX,Z ∗ represents an element a ∈ A if x · w ∈ FX,M represents a ∈ A. Theorem 6.14. Let M be a finitely presented monoid, let A be an M-act and let B be a large subact of A. If A is finitely presented, then B is finitely presented. Proof. We shall prove this result using the method based on Theorem 6.1 and outlined in Remark 6.2. Let M be defined by the presentation hZ \| P iMon, where Z and P are finite. Suppose A is defined by the finite presentation hX \| Ri. We define the finite set S = {am ∈ B : a ∈ A \ B, m ∈ Z}, and let Y = (X ∩ B) ∪ S. We have that B = hY i by Proposition 6.6. Let W denote the set of elements of FX,Z ∗ which represent elements of B. We define a map θ : W → FY,M as follows. For u ∈ W , we have u = x · w for some x ∈ X and w ∈ Z ∗ . If x ∈ B, let uθ = x · w. Suppose x ∈ A \ B. Now w = m1 . . . mk where mi ∈ Z. Let s be minimal such that x · m1 . . . ms represents an element of B, say b, and let uθ = b · ms+1 . . . mk . Let L(X, B) denote the set of elements of FX,M which represent elements of B. For each m ∈ M, choose an element wm ∈ Z ∗ which represents m. We now have a well-defined rewriting map φ : L(X, B) → FY , x · m 7→ (x · wm )θ. 19 Note that for any x ∈ X ∩ B, we have x ≡ xφ. Now, for each y ∈ S, we have y = ay my for some ay ∈ A \ B and my ∈ Z. Choose uy ∈ FX which represents ay in A, so (uy my )φ = y holds in B. We now define the following sets of relations: S1 = {uφ = vφ : (u, v) ∈ R, u ∈ L(X, B)}; S2 = {b · w = c · z : b, c ∈ S, w and z are suffixes of p and q respectively for some (p, q) ∈ P, b · w = c · z holds in B}. Since R, S and P are finite, we have that S1 and S2 are finite. We now make the following claim: Claim. Let x ∈ X and w, w ′ ∈ Z ∗ such that w = w ′ holds in M and x·w represents an element of B. Then (x · w)θ = (x · w ′)θ is a consequence of S2 . Proof. If x ∈ B, then (x · w)θ ≡ x · w ≡ x · w ′ ≡ (x · w ′ )θ. Suppose now that x ∈ A \ B. Since w = w ′ is a consequence of P , it is clearly sufficient to consider the case where w ′ is obtained from w by a single application of a relation from P , so let w = pqr and w ′ = pq ′ r where p, r ∈ Z ∗ and (q, q ′) ∈ P . There are three cases. Case 1: x · p represents an element of B. Since q = q ′ in M, we have (x · w)θ ≡ ((x · p)θ)qr ≡ ((x · p)θ)q ′ r ≡ (x · w ′ )θ. Case 2: x · pq represents an element of A \ B. Now r = m1 . . . mk where mi ∈ Z. Let s be minimal such that x · pqm1 . . . ms represents an element of B, say b. Since x · pq and x · pq ′ represent the same element of A, we have (x · w)θ ≡ b · ms+1 . . . mk ≡ (x · w ′ )θ. Case 3: x · p represents an element of A \ B and x · pq represents an element of B. Now q = m1 . . . mk and q ′ = n1 . . . nl where mi , ni ∈ Z. Let s be minimal such that x · pm1 . . . ms represents an element of B, say b, and let t be minimal such that x · pn1 . . . nt represents an element of B, say c. We have that (x · w)θ ≡ (b · ms+1 . . . mk )r = (c · nt+1 . . . nl )r ≡ (x · w ′ )θ, using an application of a relation from S2 . Returning to the proof of Theorem 6.14, we shall show that B is defined by the finite presentation hY \| S1 , S2 i. We need to show that the relations R1 , R2 and R3 of the presentation for B given in Theorem 6.1 are consequences of S1 and S2 . That is, we show that for any y ∈ S, w ∈ L(X, B) and m ∈ M, and (u, v) ∈ R, n ∈ M such that un ∈ L(X, B), the relations y = (uy my )φ, (wm)φ = (wφ)m and (un)φ = (vn)φ are consequences of S1 and S2 . 20 Let y ∈ S. We have that uy = x · m for some x ∈ X \ B and m ∈ M. Since wm my = wmmy holds in M, we have that y ≡ (x · wm my )θ = (x · wmmy )θ ≡ (uy my )φ is a consequence of S2 by the above claim. Now let u = x · n ∈ L(X, B) and m ∈ M. We have that (um)φ ≡ (x · wnm )θ and (uφ)m ≡ (x · wn wm )θ. Since wnm = wn wm holds in M, we have that (um)φ = (uφ)m is a consequence of S2 by the above claim. Finally, let (u, v) ∈ R and n ∈ M such that un ∈ L(X, B). Suppose first that u ∈ L(X, B). We have that (un)φ = (uφ)n and (vn)φ = (vφ)n are consequences of S2 , and we obtain (vφ)n from (uφ)n by an application of a relation from S1 . Therefore, (un)φ = (vn)φ is a consequence of S1 and S2 . Suppose now that u represents an element of A \ B. We have that u = x · m and v = x′ · m′ for some x, x′ ∈ X and m, m′ ∈ M. Now (un)φ = (x · wm wn )θ and (vn)φ = (x′ · wm′ wn )θ are consequences of S2 by the above claim. We have that wn = m1 . . . mk where mi ∈ Z. Let s be minimal such that x · wm m1 . . . ms represents an element of B, say b. Since x · wm and x′ · wm′ represent the same element of A, we have (x · wm wn )θ ≡ b · ms+1 . . . mk ≡ (x′ · wm′ wn )θ. Therefore, we have that (un)φ = (vn)φ is a consequence of S2 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] G. Baumslag. Topics in Combinatorial Group Theory. Birkhauser, 1993. R. Book and F. Otto. String rewriting systems. Springer-Verlag, 1993. S. Burris and H. Sankappanavar. A Course in Universal Algebra. Springer-Verlag, 1981. C. Campbell, E. Robertson, N. Ruskuc and R. Thomas. Reidermeister-Schreier type rewriting for semigroups. Semigroup Forum, 51:47-62, 1995. V. Gould. Completely right pure monoids. Proc. Royal Irish Acad., 87A:73-82, 1987. V. Gould. Coherent monoids. J. Australian Math. Soc., 53:166-182, 1992. V. Gould, M. Hartmann and N. Ruskuc. Free monoids are coherent. Proc. Edinburgh Math. Soc., 60:127-131, 2017. V. Gould and M. Hartmann. Coherency, free inverse monoids and related free algebras. Math. Proc. Cambridge Phil. Soc., to appear. M. Kilp, U. Knauer and A. Mikhalev. Monoids, Acts and Categories. Walter de Gruyter, 2000. W. Magnus, A. Karrass and D. Solitar. Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Courier Corporation, 2004. P. Normak. On Noetherian and finitely presented acts (in Russian). Tartu Ul. Toimetised, 431:37-46, 1977. N. Ruskuc. On large subsemigroups and finiteness conditions of semigroups. Proc. London Math. Soc., 76:383-405, 1998. N. Ruskuc. Semigroup Presentations. PhD Thesis, University of St Andrews, 1995. 21 School of Mathematics and Statistics, University of St Andrews, St Andrews, Scotland, UK E-mail address: {cm30,nik.ruskuc}@st-andrews.ac.uk 22	4
On the Behaviours Produced by Instruction Sequences under Execution arXiv:1106.6196v2 [cs.PL] 11 Jun 2012 J.A. Bergstra and C.A. Middelburg Informatics Institute, Faculty of Science, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands J.A.Bergstra@uva.nl,C.A.Middelburg@uva.nl Abstract. We study several aspects of the behaviours produced by instruction sequences under execution in the setting of the algebraic theory of processes known as ACP. We use ACP to describe the behaviours produced by instruction sequences under execution and to describe two protocols implementing these behaviours in the case where the processing of instructions takes place remotely. We also show that all finite-state behaviours considered in ACP can be produced by instruction sequences under execution. Keywords: instruction sequence, remote instruction processing, instruction sequence producible process 1 Introduction The concept of an instruction sequence is a very primitive concept in computing. It has always played a central role in computing because of the fact that execution of instruction sequences underlies virtually all past and current generations of computers. It happens that, given a precise definition of an appropriate notion of an instruction sequence, many issues in computer science can be clearly explained in terms of instruction sequences. A simple yet interesting example is that a program can simply be defined as a text that denotes an instruction sequence. Such a definition corresponds to an empirical perspective found among practitioners. In theoretical computer science, the meaning of programs usually plays a prominent part in the explanation of many issues concerning programs. Moreover, what is taken for the meaning of programs is mathematical by nature. On the other hand, it is customary that practitioners do not fall back on the mathematical meaning of programs in case explanation of issues concerning programs is needed. They phrase their explanations from an empirical perspective. An empirical perspective that we consider appealing is the perspective that a program is in essence an instruction sequence and an instruction sequence under execution produces a behaviour that is controlled by its execution environment in the sense that each step performed actuates the processing of an instruction by the execution environment and a reply returned at completion of the processing determines how the behaviour proceeds. This paper concerns the behaviours produced by instruction sequences under execution as such and two issues relating to the behaviours produced by instruction sequences under execution, namely the issue of implementing these behaviours in the case where the processing of instructions takes place remotely and the issue of the extent to which the behaviours considered in process algebra can be produced by instruction sequences under execution. Remote instruction processing means that a stream of instructions to be processed arises at one place and the processing of that stream of instructions is handled at another place. This phenomenon is increasingly encountered. It is found if loading the instruction sequence to be executed as a whole is impracticable. For instance, the storage capacity of the execution unit is too small or the execution unit is too far away. Remote instruction processing requires special attention because the transmission time of the messages involved in remote instruction processing makes it hard to keep the execution unit busy without intermission. In the literature on computer architecture, hardly anything can be found that contributes to a sound understanding of the phenomenon of remote instruction processing. As a first step towards such an understanding, we give rigorous descriptions of two protocols for remote instruction processing at a level of abstraction that captures the underlying essence of the protocols. One protocol is very simple, but makes no effort keep the execution unit busy without intermission. The other protocol is more complex and is directed towards keeping the execution unit busy without intermission. It is reminiscent of an instruction pre-fetching mechanism as found in pipelined processors (see e.g. [26]), but its range of application is not restricted to pipelined instruction processing. The work presented in this paper belongs to a line of research which started with an attempt to approach the semantics of programming languages from the perspective mentioned above. The first published paper on this approach is [7]. That paper is superseded by [8] with regard to the groundwork for the approach: program algebra, an algebraic theory of single-pass instruction sequences, and basic thread algebra, an algebraic theory of mathematical objects that represent in a direct way the behaviours produced by instruction sequences under execution.1 The main advantages of the approach are that it does not require a lot of mathematical background and that it is more appealing to practitioners than the main approaches to programming language semantics: the operational approach, the denotational approach and the axiomatic approach. For an overview of these approaches, see e.g. [32]. The work presented in this paper is based on the instruction sequences considered in program algebra and the representation of the behaviours produced by instruction sequences under execution considered in basic thread algebra. It is rather awkward to describe and analyse the behaviours of this kind using al1 In [8], basic thread algebra is introduced under the name basic polarized process algebra. 2 gebraic theories of processes such as ACP [3,6], CCS [27,31] and CSP [23,29]. However, the objects considered in basic thread algebra can be viewed as representations of processes as considered in process algebra. This allows for the protocols for remote instruction processing to be described using ACP or rather ACPτ , an extension of ACP which supports abstraction from internal actions. Process algebra is an area of the study of concurrency which is considered relevant to computer science, as is witnesses by the extent of the work on algebraic theories of processes such as ACP, CCS and CSP in theoretical computer science. This strongly hints that there are programmed systems whose behaviours can be taken for processes as considered in process algebra. Therefore, it is interesting to know to which extent the behaviours considered in process algebra can be produced by programs under execution, starting from the perception of a program as an instruction sequence. In this paper, we will show that, by apposite choice of instructions, all finite-state processes can be produced by instruction sequences (provided that the cluster fair abstraction rule, see e.g. Section 5.6 of [24], is valid). The instruction sequences considered in program algebra are single-pass instruction sequences, i.e. finite or infinite sequences of instructions of which each instruction is executed at most once and can be dropped after it has been executed or jumped over. Program algebra does not provide a notation for programs that is intended for actual programming: programs written in an assembly language are finite instruction sequences for which single-pass execution is usually not possible. We will also show that all finite-state processes can as well be produced by programs written in a program notation which is close to existing assembly languages. Instruction sequences under execution may make use of services provided by their execution environment such as counters, stacks and Turing tapes. The use operators added to basic thread algebra in e.g. [12] can be used to describe the behaviours produced by instruction sequences under execution that make use of services. Interesting is that instruction sequences under execution that make use of services may produce infinite-state processes. On that account, we will make precise what processes are produced by instruction sequences under execution that make use of services provided by their execution environment. As a continuation of the work on a new approach to programming language semantics mentioned above, the notion of an instruction sequence was subjected to systematic and precise analysis using the groundwork laid earlier. This led among other things to expressiveness results about the instruction sequences considered and variations of the instruction sequences considered (see e.g. [12,18,20,21,36]). Instruction sequences are under discussion for many years in diverse work on computer architecture, as witnessed by e.g. [4,22,25,30,33,34,35,39,41], but the notion of an instruction sequence has never been subjected to any precise analysis before. As another continuation of the work on a new approach to programming language semantics mentioned above, selected issues relating to well-known subjects from the theory of computation and the area of computer architecture were rigorously investigated thinking in terms of instruction sequences 3 (see e.g. [14,15,16,17,19]). The subjects from the theory of computation, namely the halting problem and non-uniform computational complexity, are usually investigated thinking in terms of a common model of computation such as Turing machines and Boolean circuits (see e.g. [1,28,38]). The subjects from the area of computer architecture, namely instruction sequence performance, instruction set architectures and remote instruction processing, are usually not investigated in a rigorous way at all. The general aim of the work in both continuations mentioned is to bring instruction sequences as a theme in computer science better into the picture. The work presented in this paper forms a part of the last mentioned continuation. This paper is organized as follows. The body of the paper consists of three parts. The first part (Sections 2–7) concerns the behaviours produced by instruction sequences under execution as such and includes surveys of program algebra, basic thread algebra and the algebraic theory of processes known as ACP. The second part (Sections 8–10) concerns the issue of implementing these behaviours in the case where the processing of instructions takes place remotely and includes rigorous descriptions of two protocols for remote instruction processing. The third part (Sections 11–14) concerns the issue of the extent to which the behaviours considered in process algebra can be produced by instruction sequences under execution and includes the result that, by apposite choice of instructions, all finite-state processes can be produced by instruction sequences. This paper consolidates material from [11,13,14]. 2 Program Algebra In this section, we review PGA (ProGram Algebra). The starting-point of program algebra is the perception of a program as a single-pass instruction sequence. The concepts underlying the primitives of program algebra are common in programming, but the particular form of the primitives is not common. The predominant concern in the design of program algebra has been to achieve simple syntax and semantics, while maintaining the expressive power of arbitrary finite control. In PGA, it is assumed that a fixed but arbitrary set A of basic instructions has been given. The intuition is that the execution of a basic instruction may modify a state and produces a reply at its completion. The possible replies are the Boolean values T and F. PGA has the following primitive instructions: – – – – – for each a ∈ A, a plain basic instruction a; for each a ∈ A, a positive test instruction +a; for each a ∈ A, a negative test instruction −a; for each l ∈ N, a forward jump instruction #l; a termination instruction !. We write I for the set of all primitive instructions of PGA. On execution of an instruction sequence, these primitive instructions have the following effects: 4 – the effect of a positive test instruction +a is that basic instruction a is executed and execution proceeds with the next primitive instruction if T is produced and otherwise the next primitive instruction is skipped and execution proceeds with the primitive instruction following the skipped one — if there is no primitive instructions to proceed with, inaction occurs; – the effect of a negative test instruction −a is the same as the effect of +a, but with the role of the value produced reversed; – the effect of a plain basic instruction a is the same as the effect of +a, but execution always proceeds as if T is produced; – the effect of a forward jump instruction #l is that execution proceeds with the l-th next instruction of the program concerned — if l equals 0 or there is no primitive instructions to proceed with, inaction occurs; – the effect of the termination instruction ! is that execution terminates. PGA has the following constants and operators: – for each u ∈ I, an instruction constant u ; – the binary concatenation operator ; ; – the unary repetition operator ω . We assume that there is a countably infinite set of variables which includes x, y, z. Terms are built as usual. We use infix notation for concatenation and postfix notation for repetition. A closed PGA term is considered to denote a non-empty, finite or eventually periodic infinite sequence of primitive instructions.2 The instruction sequence denoted by a closed term of the form t ; t′ is the instruction sequence denoted by t concatenated with the instruction sequence denoted by t′ . The instruction sequence denoted by a closed term of the form tω is the instruction sequence denoted by t concatenated infinitely many times with itself. Some simple examples of closed PGA terms are a;b;c, a ; (b ; c)ω . +a ; #2 ; #3 ; b ; ! , On execution of the instruction sequence denoted by the first term, the basic instructions a, b and c are executed in that order and after that inaction occurs. On execution of the instruction sequence denoted by the second term, the basic instruction a is executed first, if the execution of a produces the reply T, the basic instruction b is executed next and after that execution terminates, and if the execution of a produces the reply F, inaction occurs. On execution of the instruction sequence denoted by the third term, the basic instruction a is executed first, and after that the basic instructions b and c are executed in that order repeatedly forever. Closed PGA terms are considered equal if they represent the same instruction sequence. The axioms for instruction sequence equivalence are given in Table 1. In this table, n stands for an arbitrary positive natural number. The term tn , 2 An eventually periodic infinite sequence is an infinite sequence with only finitely many distinct suffixes. 5 Table 1. Axioms of PGA (x ; y) ; z = x ; (y ; z) (xn )ω = xω xω ; y = xω (x ; y)ω = x ; (y ; x)ω PGA1 PGA2 PGA3 PGA4 where t is a PGA term, is defined by induction on n as follows: t1 = t and tn+1 = t ; tn . The unfolding equation xω = x ; xω is derived as follows: xω = (x ; x)ω by PGA2 = x ; (x ; x)ω by PGA4 = x ; xω by PGA2 . Each closed PGA term is derivably equal to a term in canonical form, i.e. a ω term of the form t or t ; t′ , where t and t′ are closed PGA terms in which the repetition operator does not occur. For example: (a ; b)ω ; c ; ! = a ; (b ; a)ω , +a ; (#4 ; b ; (−c ; #5 ; !)ω )ω = +a ; #4 ; b ; (−c ; #5 ; !)ω . The initial models of PGA are considered its standard models. Henceforth, we restrict ourselves to the initial model I PGA of PGA in which: – the domain is the set of all non-empty, finite and eventually periodic infinite sequences over the set I of primitive instructions; – the operation associated with ; is concatenation; – the operation associated with ω is the operation ω defined as follows: • if F is a finite sequence over I, then F ω is the unique eventually periodic infinite sequence F ′ such that F concatenated n times with itself is a proper prefix of F ′ for each n ∈ N; • if F is an eventually periodic infinite sequence over I, then F ω is F . In the sequel, we use the term instruction sequence for the elements of the domain of I PGA , and we denote the interpretations of the constants and operators of PGA in I PGA by the constants and operators themselves. I PGA is loosely called the initial model of PGA because all initial models of PGA are isomorphic, i.e. there exist bijective homomorphism between them (see e.g. [37,40]). 3 Basic Thread Algebra In this section, we review BTA (Basic Thread Algebra). BTA is an algebraic theory of mathematical objects that represent in a direct way the behaviours produced by instruction sequences under execution. The objects concerned are called threads. In BTA, it is assumed that a fixed but arbitrary set A of basic actions, with tau ∈ / A, has been given. Besides, tau is a special basic action. We write Atau for 6 Table 2. Axiom of BTA x E tau D y = x E tau D x T1 A ∪ {tau}. A thread performs basic actions in a sequential fashion. Upon each basic action performed, a reply from an execution environment determines how it proceeds. The possible replies are the Boolean values T and F. Performing tau, which is considered performing an internal action, always leads to the reply T. Although BTA is one-sorted, we make this sort explicit. The reason for this is that we will extend BTA with an additional sort in Section 13. BTA has one sort: the sort T of threads. To build terms of sort T, it has the following constants and operators: – the inaction constant D : T; – the termination constant S : T; – for each a ∈ Atau , the binary postconditional composition operator E a D : T × T → T. We assume that there are infinitely many variables of sort T, including x, y, z. Terms of sort T are built as usual. We use infix notation for the postconditional composition operators. We introduce basic action prefixing as an abbreviation: a ◦ t, where a ∈ Atau and t is a term of sort T, abbreviates t E a D t. The thread denoted by a closed term of the form t E a D t′ will first perform a, and then proceed as the thread denoted by t if the reply from the execution environment is T and proceed as the thread denoted by t′ if the reply from the execution environment is F. The threads denoted by D and S will become inactive and terminate, respectively. Some simple examples of closed BTA terms are a ◦ (S E b D D) , (b ◦ S) E a D D . The first term denotes the thread that first performs basic action a, next performs basic action b, if the reply from the execution environment on performing b is T, after that terminates, and if the reply from the execution environment on performing b is F, after that becomes inactive. The second term denotes the thread that first performs basic action a, if the reply from the execution environment on performing a is T, next performs the basic action b and after that terminates, and if the reply from the execution environment on performing a is F, next becomes inactive. BTA has only one axiom. This axiom is given in Table 2. Using the abbreviation introduced above, axiom T1 can be written as follows: x E tau D y = tau ◦ x. Notice that each closed BTA term denotes a thread that will become inactive or terminate after it has performed finitely many actions. Infinite threads can be described by guarded recursion. A guarded recursive specification over BTA is a set of recursion equations E = {X = tX \| X ∈ V }, where V is a set of variables of sort T and each tX is a BTA term of the form D, S or t E a D t′ with t and t′ that contain only variables 7 Table 3. RDP, RSP and AIP hX\|Ei = htX \|Ei if X = tX ∈ E RDP E ⇒ X = hX\|Ei if X ∈ V(E) RSP V n≥0 πn (x) = πn (y) ⇒ x = y π0 (x) = D πn+1 (S) = S πn+1 (D) = D πn+1 (x E a D y) = πn (x) E a D πn (y) AIP P0 P1 P2 P3 from V . We write V(E) for the set of all variables that occur in E. We are only interested in models of BTA in which guarded recursive specifications have unique solutions, such as the projective limit model of BTA presented in [5]. A simple example of a guarded recursive specification is the one consisting of following two equations: x = x EaD y , y = y EbD S . The x-component of the solution of this guarded recursive specification is the thread that first performs basic action a repeatedly until the reply from the execution environment on performing a is F, next performs basic action b repeatedly until the reply from the execution environment on performing b is F, and after that terminates. For each guarded recursive specification E and each X ∈ V(E), we introduce a constant hX\|Ei of sort T standing for the X-component of the unique solution of E. We write htX \|Ei for tX with, for all Y ∈ V(E), all occurrences of Y in tX replaced by hY \|Ei. The axioms for the constants for the components of the solutions of guarded recursive specifications are RDP (Recursive Definition Principle) and RSP (Recursive Specification Principle), which are given in Table 3. RDP and RSP are actually axiom schemas in which X stands for an arbitrary variable, tX stands for an arbitrary BTA term, and E stands for an arbitrary guarded recursive specification over BTA. Side conditions are added to restrict what X, tX and E stand for. The equations hX\|Ei = htX \|Ei for a fixed E express that the constants hX\|Ei make up a solution of E. The conditional equations E ⇒ X = hX\|Ei express that this solution is the only one. RDP and RSP are means to prove closed terms that denote the same infinite thread equal. We introduce AIP (Approximation Induction Principle) as an additional means to prove closed terms that denote the same infinite thread equal. AIP is based on the view that two threads are identical if their approximations up to any finite depth are identical. The approximation up to depth n of a thread is obtained by cutting it off after it has performed n actions. AIP is also given in Table 3. Here, approximation up to depth n is phrased in terms of the unary projection operator πn : T → T. The axioms for the projection operators are axioms P0–P3 in Table 3. P1–P3 are actually axiom schemas in which a stands for arbitrary basic action and n stands for an arbitrary natural number. We write BTA+REC for BTA extended with the constants for the components of the solutions of guarded recursive specifications, the projection operators and the axioms RDP, RSP, AIP and P0–P3. 8 The minimal models of BTA+REC are considered its standard models.3 Recall that a model of an algebraic theory is minimal iff all elements of the domains associated with the sorts of the theory can be denoted by closed terms. Henceforth, we restrict ourselves to the minimal models of BTA+REC. We assume that a minimal model MBTA+REC of BTA+REC has been given. In the sequel, we use the term thread for the elements of the domain of MBTA+REC , and we denote the interpretations of constants and operators in MBTA+REC by the constants and operators themselves. Let T be a thread. Then the set of states or residual threads of T , written Res(T ), is inductively defined as follows: – T ∈ Res(T ); – if T ′ E a D T ′′ ∈ Res(T ), then T ′ ∈ Res(T ) and T ′′ ∈ Res(T ). Let T be a thread and let A′ ⊆ Atau . Then T is regular over A′ if the following conditions are satisfied: – Res(T ) is finite; – for all T ′ , T ′′ ∈ Res(T ) and a ∈ Atau , T ′ E a D T ′′ ∈ Res(T ) implies a ∈ A′ . We say that T is regular if T is regular over Atau . For example, the x-component of the solution of the guarded recursive specification consisting of the following two equations: x=a◦y , y = (c ◦ y) E b D (x E d D S) has five states and is regular over any A′ ⊆ Atau for which {a, b, c, d} ⊆ A′ . We will make use of the fact that being a regular thread coincides with being a component of the solution of a finite guarded recursive specification in which the right-hand sides of the recursion equations are of a restricted form. A linear recursive specification over BTA is a guarded recursive specification E = {X = tX \| X ∈ V } over BTA, where each tX is a term of the form D, S or Y E a D Z with Y, Z ∈ V . Proposition 1. Let T be a thread and let A′ ⊆ Atau . Then T is regular over A′ iff there exists a finite linear recursive specification E over BTA in which only basic actions from A′ occur such that T is a component of the solution of E. Proof. The implication from left to right is proved as follows. Because T is regular, Res(T ) is finite. Hence, there are finitely many threads T1 , . . . , Tn , with T = T1 , such that Res(T ) = {T1 , . . . , Tn }. Now T is the x1 -component of the solution of the linear recursive specification consisting of the following equations:  if Ti = S S xi = D for all i ∈ [1, n] . if Ti = D  xj E a D xk if Ti = Tj E a D Tk 3 A minimal model of an algebraic theory is a model of which no proper subalgebra is a model as well. 9 Table 4. Defining equations for thread extraction operation \|a\| = a ◦ D \|a ; F \| = a ◦ \|F \| \|+a\| = a ◦ D \|+a ; F \| = \|F \| E a D \|#2 ; F \| \|−a\| = a ◦ D \|−a ; F \| = \|#2 ; F \| E a D \|F \| \|#l\| = D \|#0 ; F \| = D \|#1 ; F \| = \|F \| \|#l + 2 ; u\| = D \|#l + 2 ; u ; F \| = \|#l + 1 ; F \| \|!\| = S \|! ; F \| = S Because T is regular over A′ , only basic actions from A′ occur in the linear recursive specification constructed in this way. The implication from right to left is proved as follows. Thread T is a component of the unique solution of a finite linear specification in which only basic actions from A′ occur. This means that there are finitely many threads T1 , . . . , Tn , with T = T1 , such that for every i ∈ [1, n], Ti = S, Ti = D or Ti = Tj E a D Tk for some j, k ∈ [1, n] and a ∈ A′ . Consequently, T ′ ∈ Res(T ) iff T ′ = Ti for some i ∈ [1, n] and moreover T ′ E a D T ′′ ∈ Res(T ) only if a ∈ A′ . Hence, Res(T ) is finite and T is regular over A′ . ⊓ ⊔ Remark 1. A structural operational semantics of BTA+REC and a bisimulation equivalence based on it can be found in e.g. [10]. The quotient algebra of the algebra of closed terms of BTA+REC by this bisimulation equivalence is one of the minimal models of BTA+REC. 4 Thread Extraction In this short section, we use BTA+REC to make mathematically precise which threads are produced by instruction sequences under execution. For that purpose, A is taken such that A ⊇ A is satisfied. The thread extraction operation \| \| assigns a thread to each instruction sequence. The thread extraction operation is defined by the equations given in Table 4 (for a ∈ A, l ∈ N, and u ∈ I) and the rule that \|#l ; F \| = D if #l is the beginning of an infinite jump chain. This rule is formalized in e.g. [12]. Let F be an instruction sequence and T be a thread. Then we say that F produces T if \|F \| = T . For example, a;b;c +a ; #2 ; #3 ; b ; ! +a ; −b ; c ; ! +a ; #2 ; (b ; #2 ; c ; #2)ω produces produces produces produces a◦b◦c◦D, (b ◦ S) E a D D , (S E b D (c ◦ S)) E a D (c ◦ S) , D E a D (b ◦ D) . In the case of instruction sequences that are not finite, the produced threads can be described as components of the solution of a guarded recursive specification. For example, the infinite instruction sequence (a ; +b)ω 10 produces the x-component of the solution of the guarded recursive specification consisting of following two equations: x=a◦y , y = x EbD y and the infinite instruction sequence a ; (+b ; #2 ; #3 ; c ; #4 ; −d ; ! ; a)ω produces the x-component of the solution of the guarded recursive specification consisting of following two equations: x=a◦y , 5 y = (c ◦ y) E b D (x E d D S) . Algebra of Communicating Processes In this section, we review ACPτ (Algebra of Communicating Processes with abstraction). This algebraic theory of processes will among other things be used to make precise what processes are produced by the threads denoted by closed terms of BTA+REC. For a comprehensive overview of ACPτ , the reader is referred to [3,24]. In ACPτ , it is assumed that a fixed but arbitrary set A of atomic actions, with τ, δ ∈ / A, and a fixed but arbitrary commutative and associative function \| : A ∪ {τ } × A ∪ {τ } → A ∪ {δ}, with τ \| e = δ for all e ∈ A ∪ {τ }, have been given. The function \| is regarded to give the result of synchronously performing any two atomic actions for which this is possible, and to give δ otherwise. In ACPτ , τ is a special atomic action, called the silent step. The act of performing the silent step is considered unobservable. Because it would otherwise be observable, the silent step is considered an atomic action that cannot be performed synchronously with other atomic actions. We write Aτ for A ∪ {τ }. ACPτ has the following constants and operators: – – – – – – – – – – for each e ∈ A, the atomic action constant e ; the silent step constant τ ; the inaction constant δ ; the binary alternative composition operator + ; the binary sequential composition operator · ; the binary parallel composition operator k ; the binary left merge operator ⌊⌊ ; the binary communication merge operator \| ; for each H ⊆ A, the unary encapsulation operator ∂H ; for each I ⊆ A, the unary abstraction operator τI . We assume that there are infinitely many variables, including x, y, z. Terms are built as usual. We use infix notation for the binary operators. The precedence conventions used with respect to the operators of ACPτ are as follows: + binds weaker than all others, · binds stronger than all others, and the remaining operators bind equally strong. Let t and t′ be closed ACPτ terms, e ∈ A, and H, I ⊆ A. Intuitively, the constants and operators to build ACPτ terms can be explained as follows: 11 – the process denoted by e first performs atomic action e and next terminates successfully; – the process denoted by τ performs an unobservable atomic action and next terminates successfully; – the process denoted by δ can neither perform an atomic action nor terminate successfully; – the process denoted by t + t′ behaves either as the process denoted by t or as the process denoted by t′ , but not both; – the process denoted by t · t′ first behaves as the process denoted by t and on successful termination of that process it next behaves as the process denoted by t′ ; – the process denoted by t k t′ behaves as the process that proceeds with the processes denoted by t and t′ in parallel; – the process denoted by t ⌊⌊ t′ behaves the same as the process denoted by t k t′ , except that it starts with performing an atomic action of the process denoted by t; – the process denoted by t \| t′ behaves the same as the process denoted by t k t′ , except that it starts with performing an atomic action of the process denoted by t and an atomic action of the process denoted by t′ synchronously; – the process denoted by ∂H (t) behaves the same as the process denoted by t, except that atomic actions from H are blocked; – the process denoted by τI (t) behaves the same as the process denoted by t, except that atomic actions from I are turned into unobservable atomic actions. The operators ⌊⌊ and \| are of an auxiliary nature. They are needed to axiomatize ACPτ . The axioms of ACPτ are given in Table 5. CM2–CM3, CM5–CM7, C1–C4, D1–D4 and TI1–TI4 are actually axiom schemas in which a, b and c stand for arbitrary constants of ACPτ , and H and I stand for arbitrary subsets of A. ACPτ is extended with guarded recursion like BTA. A recursive specification over ACPτ is a set of recursion equations E = {X = tX \| X ∈ V }, where V is a set of variables and each tX is an ACPτ term containing only variables from V . We write V(E) for the set of all variables that occur in E. Let t be an ACPτ term without occurrences of abstraction operators containing a variable X. Then an occurrence of X in t is guarded if t has a subterm of the form e · t′ where e ∈ A and t′ is a term containing this occurrence of X. Let E be a recursive specification over ACPτ . Then E is a guarded recursive specification if, in each equation X = tX ∈ E: (i) abstraction operators do not occur in tX and (ii) all occurrences of variables in tX are guarded or tX can be rewritten to such a term using the axioms of ACPτ in either direction and/or the equations in E except the equation X = tX from left to right. We are only interested models of ACPτ in which guarded recursive specifications have unique solutions, such as the models of ACPτ presented in [3]. For each guarded recursive specification E and each X ∈ V(E), we introduce a constant hX\|Ei standing for the X-component of the unique solution of E. We 12 Table 5. Axioms of ACPτ x·τ =x B1 x · (τ · (y + z) + y) = x · (y + z) B2 x+y = y+x (x + y) + z = x + (y + z) x+x = x (x + y) · z = x · z + y · z (x · y) · z = x · (y · z) x+δ = x δ·x=δ A1 A2 A3 A4 A5 A6 A7 x k y = x ⌊⌊ y + y ⌊⌊ x + x \| y a ⌊⌊ x = a · x a · x ⌊⌊ y = a · (x k y) (x + y) ⌊⌊ z = x ⌊⌊ z + y ⌊⌊ z a · x \| b = (a \| b) · x a \| b · x = (a \| b) · x a · x \| b · y = (a \| b) · (x k y) (x + y) \| z = x \| z + y \| z x \| (y + z) = x \| y + x \| z CM1 CM2 CM3 CM4 CM5 CM6 CM7 CM8 CM9 ∂H (a) = a if a ∈ /H ∂H (a) = δ if a ∈ H ∂H (x + y) = ∂H (x) + ∂H (y) ∂H (x · y) = ∂H (x) · ∂H (y) D1 D2 D3 D4 τI (a) = a if a ∈ /I τI (a) = τ if a ∈ I τI (x + y) = τI (x) + τI (y) τI (x · y) = τI (x) · τI (y) TI1 TI2 TI3 TI4 a\|b=b\|a (a \| b) \| c = a \| (b \| c) δ\|a=δ τ \|a=δ C1 C2 C3 C4 Table 6. RDP, RSP and AIP hX\|Ei = htX \|Ei if X = tX ∈ E RDP E ⇒ X = hX\|Ei if X ∈ V(E) RSP V n≥0 πn (x) = πn (y) ⇒ x = y AIP π0 (a) = δ πn+1 (a) = a π0 (a · x) = δ πn+1 (a · x) = a · πn (x) πn (x + y) = πn (x) + πn (y) πn (τ ) = τ πn (τ · x) = τ · πn (x) PR1 PR2 PR3 PR4 PR5 PR6 PR7 write htX \|Ei for tX with, for all Y ∈ V(E), all occurrences of Y in tX replaced by hY \|Ei. The axioms for the constants for the components of the solutions of guarded recursive specifications are RDP and RSP, which are given in Table 6. RDP and RSP are actually axiom schemas in which X stands for an arbitrary variable, tX stands for an arbitrary ACPτ term, and E stands for an arbitrary guarded recursive specification over ACPτ . Side conditions are added to restrict what X, tX and E stand for. Closed terms of ACPτ extended with constants for the components of the solutions of guarded recursive specifications that denote the same process cannot always be proved equal by means of the axioms of ACPτ together with RDP and RSP. We introduce AIP to remedy this. AIP is based on the view that two processes are identical if their approximations up to any finite depth are identical. The approximation up to depth n of a process behaves the same as that process, except that it cannot perform any further atomic action after n atomic actions have been performed. AIP is given in Table 6. Here, approximation up to depth 13 n is phrased in terms of a unary projection operator πn . The axioms for the projection operators are axioms PR1–PR7 in Table 6. PR1–PR7 are actually axiom schemas in which a stands for arbitrary constants of ACPτ different from τ and n stands for an arbitrary natural number. We write ACPτ +REC for ACPτ extended with the constants for the components of the solutions of guarded recursive specifications, the projection operators, and the axioms RDP, RSP, AIP and PR1–PR7. The minimal models of ACPτ +REC are considered its standard models. Henceforth, we restrict ourselves to the minimal models of ACPτ +REC. We assume that a fixed but arbitrary minimal model MACPτ +REC of ACPτ +REC has been given. From Section 12, we will sometimes assume that CFAR (Cluster Fair Abstraction Rule) is valid in MACPτ +REC . CFAR says that a cluster of silent steps that has exits can be eliminated if all exits are reachable from everywhere in the cluster. A precise formulation of CFAR can be found in [24]. We use the term process for the elements from the domain of MACPτ +REC , and we denote the interpretations of constants and operators in MACPτ +REC by the constants and operators themselves. Let P be a process. Then the set of states or subprocesses of P , written Sub(P ), is inductively defined as follows: – P ∈ Sub(P ); – if e · P ′ ∈ Sub(P ), then P ′ ∈ Sub(P ); – if e · P ′ + P ′′ ∈ Sub(P ), then P ′ ∈ Sub(P ). Let P be a process and let A′ ⊆ Aτ . Then P is regular over A′ if the following conditions are satisfied: – Sub(P ) is finite; – for all P ′ ∈ Sub(P ) and e ∈ Aτ , e · P ′ ∈ Sub(P ) implies e ∈ A′ ; – for all P ′ , P ′′ ∈ Sub(P ) and e ∈ Aτ , e · P ′ + P ′′ ∈ Sub(P ) implies e ∈ A′ . We say that P is regular if P is regular over Aτ . We will make use of the fact that being a regular process over A coincides with being a component of the solution of a finite guarded recursive specification in which the right-hand sides of the recursion equations are linear terms. Linearity of terms is inductively defined as follows: – – – – δ is linear; if e ∈ Aτ , then e is linear; if e ∈ Aτ and X is a variable, then e · X is linear; if t and t′ are linear, then t + t′ is linear. A linear recursive specification over ACPτ is a guarded recursive specification E = {X = tX \| X ∈ V } over ACPτ , where each tX is linear. Proposition 2. Let P be a process and let A′ ⊆ A. Then P is regular over A′ iff there exists a finite linear recursive specification E over ACPτ in which only atomic actions from A′ occur such that P is a component of the solution of E. 14 Proof. The proof follows the same line as the proof of Proposition 1. ⊓ ⊔ Remark 2. Proposition 2 is concerned with processes that are regular over A. We can also prove that being a regular process over Aτ coincides with being a component of the solution of a finite linear recursive specification over ACPτ if we assume that the cluster fair abstraction rule [24] holds in the model MACPτ +REC . However, we do not need this more general result. P , in } and ti1 , . . . , tin are ACPτ We will write i∈S ti , where S = {i1 , . . .P terms, for ti1 + . . . + tin . The convention is that i∈S ti stands for δ if S = ∅. We will often write X for hX\|Ei if E is clear from the context. It should be borne in mind that, in such cases, we use X as a constant. 6 Program-Service Interaction Instructions Recall that, in PGA, it is assumed that a fixed but arbitrary set A of basic instructions has been given. In the sequel, we will make use a version of PGA in which the following additional assumptions relating to A are made: – a fixed but arbitrary finite set F of foci has been given; – a fixed but arbitrary finite set M of methods has been given; – A = {f.m \| f ∈ F , m ∈ M}. Each focus plays the role of a name of some service provided by an execution environment that can be requested to process a command. Each method plays the role of a command proper. Executing a basic instruction of the form f.m is taken as making a request to the service named f to process command m. A basic instruction of the form f.m is called a program-service interaction instruction. Recall that, in BTA, it is assumed that a fixed but arbitrary set A of basic actions has been given. In the sequel, we will make use of a version of BTA in which A = A. A basic action of the form f.m is called a thread-service interaction action. The intuition concerning program-service interaction instructions given above will be made fully precise in Section 7, using ACP. 7 Process Extraction In this section, we use ACPτ +REC to make mathematically precise which processes are produced by threads. For that purpose, A and \| are taken such that the following conditions are satisfied:4 A ⊇ {sf (d) \| f ∈ F , d ∈ M ∪ B} ∪ {rf (d) \| f ∈ F , d ∈ M ∪ B} ∪ {stop, i} 4 As usual, we will write B for the set {T, F}. 15 Table 7. Defining equations for process extraction operation \|S\|c = stop \|D\|c = δ \|T E tau D T ′ \|c = i · i · \|T \|c \|T E f.m D T ′ \|c = sf (m) · (rf (T) · \|T \|c + rf (F) · \|T ′ \|c ) and for all f ∈ F , d ∈ M ∪ B, and e ∈ A: sf (d) \| rf (d) = i , sf (d) \| e = δ if e 6= rf (d) , e \| rf (d) = δ if e 6= sf (d) , stop \| e = δ i\|e= δ . if e 6= stop , Actions of the forms sf (d) and rf (d) are send and receive actions, respectively, stop is an explicit termination action, and i is a concrete internal action. The process extraction operation \| \| assigns a process to each thread. The process extraction operation \| \| is defined by \|T \| = τ{stop} (\|T \|c ), where \| \|c is defined by the equations given in Table 7 (for f ∈ F and m ∈ M). Let P be a process, T be a thread, and F be an instruction sequence. Then we say that T produces P if τ · τI (\|T \|) = τ · P for some I ⊆ A, and we say that F produces P if \|F \| produces P . Notice that two atomic actions are involved in performing a basic action of the form f.m: one for sending a request to process command m to the service named f and another for receiving a reply from that service upon completion of the processing. Notice also that, for each thread T , \|T \|c is a process that in the event of termination performs a special termination action just before termination. Abstraction from this termination action yields the process denoted by \|T \|. The process extraction operation preserves the axioms of BTA+REC. Before we make this fully precise, we have a closer look at the axioms of BTA+REC. A proper axiom is an equation or a conditional equation. In Table 3, we do not find proper axioms. Instead of proper axioms, we find axiom schemas without side conditions and axiom schemas with side conditions. The axioms of BTA+REC are obtained by replacing each axiom schema by all its instances. Henceforth, we write α∗ , where α is a valuation of variables in MBTA+REC , for the unique homomorphic extension of α to terms of BTA+REC. Moreover, we identify t1 = t2 and ∅ ⇒ t1 = t2 . Proposition 3. Let E ⇒ t1 = t2 be an axiom of BTA+REC, and let α be a valuation of variables in MBTA+REC . Then \|α∗ (t1 )\| = \|α∗ (t2 )\| if \|α∗ (t′1 )\| = \|α∗ (t′2 )\| for all t′1 = t′2 ∈ E. Proof. The proof is trivial for the axiom of BTA and the axioms RDP and RSP. Using the equation \|πn (T )\|c = π2n (\|T \|c ), the proof is also trivial for the axioms AIP and P0–P3. This equation is easily proved by induction on n and case distinction on the structure of T in both the basis step and the inductive step. ⊓ ⊔ 16 Remark 3. Proposition 3 would go through if no abstraction of the above-mentioned special termination action was made. Notice further that ACPτ without the silent step constant and the abstraction operator, better known as ACP, would suffice if no abstraction of the special termination action was made. 8 A Simple Protocol for Remote Instruction Processing In this section and the next section, we consider two protocols for remote instruction processing. The simple protocol described in this section is presumably the most straightforward protocol for remote instruction processing that can be achieved. Therefore, we consider it a suitable starting-point for the design of more advanced protocols for remote instruction processing – such as the one described in the next section. Before this simple protocol is described, an extension of ACP is introduced to simplify the description of the protocols. The following extension of ACP from [2] will be used: the non-branching conditional operator :→ over B. The expression b :→ p, is to be read as if b then p else δ. The additional axioms for the non-branching conditional operator are T :→ x = x and F :→ x = δ . In the sequel, we will use expressions whose evaluation yields Boolean values instead of the constants T and F. Because the evaluation of the expressions concerned are not dependent on the processes denoted by the terms in which they occur, we will identify each such expression with the constant for the Boolean value that its evaluation yields. Further justification of this can be found in [9, Section 9]. The protocols concern systems whose main components are an instruction stream generator and an instruction stream execution unit. The instruction stream generator generates different instruction streams for different threads. This is accomplished by starting it in different states. The general idea of the protocols is that: – the instruction stream generator generating an instruction stream for a thread T E a D T ′ sends a to the instruction stream execution unit; – on receipt of a, the instruction stream execution unit gets the execution of a done and sends the reply produced to the instruction stream generator; – on receipt of the reply, the instruction stream generator proceeds with generating an instruction stream for T if the reply is T and for T ′ otherwise. In the case where the thread is S or D, the instruction stream generator sends a special instruction (stop or dead) and the instruction stream execution unit does not send back a reply. In this section, we consider a very simple protocol for remote instruction processing that makes no effort to keep the execution unit busy without intermission. 17 In the protocols, the generation of an instruction stream start from the thread produced by an instruction sequence under execution instead of the instruction sequence itself. It follows immediately from the definition of the thread extraction operation that the threads produced by instruction sequences under execution are regular threads. Therefore, we restrict ourselves to regular threads. We write I for the set A∪{stop, dead}. Elements from I will loosely be called instructions. The restriction of the domain of MBTA+REC to the regular threads will be denoted by RT . The functions act , thrt , and thrf defined below give, for each thread T different from S and D, the basic action that T will perform first, the thread with which it will proceed if the reply from the execution environment is T, and the thread with which it will proceed if the reply from the execution environment is F, respectively. The functions act :RT → I, thrt :RT → RT , and thrf :RT → RT are defined as follows: thrf (S) = D , act (S) = stop , thrt (S) = D , thrf (D) = D , thrt (D) = D , act (D) = dead , act (T E a D T ′ ) = a , thrt (T E a D T ′ ) = T , thrf (T E a D T ′ ) = T ′ . The function nxt 0 defined below is used by the instruction stream generator to distinguish when it starts with handling the instruction to be executed next between the different instructions that it may be. The function nxt 0 :I ×RT → B is defined as follows: T if act (T ) = a nxt 0 (a, T ) = F if act (T ) 6= a . For the purpose of describing the simple protocol outlined above in ACPτ , A and \| are taken such that, in addition to the conditions mentioned at the beginning of Section 7, the following conditions are satisfied: A ⊇ {si (d) \| i ∈ {1, 2}, d ∈ I} ∪ {ri (d) \| i ∈ {1, 2}, d ∈ I} ∪ {si (r) \| i ∈ {3, 4}, r ∈ B} ∪ {ri (r) \| i ∈ {3, 4}, r ∈ B} ∪ {j} and for all i ∈ {1, 2}, j ∈ {3, 4}, d ∈ I, r ∈ B, and e ∈ A: si (d) \| ri (d) = j , si (d) \| e = δ if e 6= ri (d) , e \| ri (d) = δ if e 6= si (d) , sj (r) \| rj (r) = j , sj (r) \| e = δ if e 6= rj (r) , e \| rj (r) = δ if e 6= sj (r) , j\|e =δ . Notice that the set B is the set of replies. Let T ∈ RT . Then the process representing the simple protocol for remote instruction processing with regard to thread T is described by 0 ∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 ) , 18 0 where the process ISGT is recursively specified by the following equation: X 0 ISGT = nxt 0 (f.m, T ) :→ f.m∈A 0 0 s1 (f.m) · (r4 (T) · ISGthrt (T ) + r4 (F) · ISGthrf (T ) ) + nxt 0 (stop, T ) :→ s1 (stop) + nxt 0 (dead, T ) :→ s1 (dead) , the process IMTC 0 is recursively specified by the following equation: X IMTC 0 = r1 (d) · s2 (d) · IMTC 0 , d∈I the process RTC 0 is recursively specified by the following equation: X RTC 0 = r3 (r) · s4 (r) · RTC 0 , r∈B the process ISEU 0 is recursively specified by the following equation: X ISEU 0 = r2 (f.m) · sf (m) · (rf (T) · s3 (T) + rf (F) · s3 (F)) · ISEU 0 f.m∈A + r2 (stop) + r2 (dead) · δ and H = {si (d) \| i ∈ {1, 2}, d ∈ I} ∪ {ri (d) \| i ∈ {1, 2}, d ∈ I} ∪ {si (r) \| i ∈ {3, 4}, r ∈ B} ∪ {ri (r) \| i ∈ {3, 4}, r ∈ B} . 0 ISGT is the instruction stream generator for thread T , IMTC 0 is the transmission channel for messages containing instructions, RTC 0 is the transmission channel for replies, and ISEU 0 is the instruction stream execution unit. If we abstract from all communications via the transmission channels, then 0 the process denoted by ∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 ) and the process \|T \| are equal modulo an initial silent step. 0 Theorem 1. For each T ∈ RT , τ · τ{j} (∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 )) denotes the process τ · \|T \|. Proof. Let T ∈ RT . Moreover, let E be a finite linear recursive specification over ACPτ with X ∈ V(E) such that \|T \| is the X-component of the solution of E in MACPτ +REC . By Proposition 2 and the definition of the process extraction operation, it is sufficient to prove that 0 τ · τ{j} (∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 )) = τ · hX\|Ei . By AIP, it is sufficient to prove that for all n ≥ 0: 0 πn (τ · τ{j} (∂H (ISGT k IMTC 0 k RTC 0 k ISEU 0 ))) = πn (τ · hX\|Ei) . This is easily proved by induction on n and in the inductive step by case distinction on the structure of T , using the axioms of ACPτ and RDP and in addition the fact that \|T ′ \| ∈ Sub(\|T \|) for all T ′ ∈ Res(T ) and the fact that there exists an bijection between Sub(\|T \|) and V(E). ⊓ ⊔ 19 9 A More Complex Protocol In this section, we consider a more complex protocol for remote instruction processing that makes an effort to keep the execution unit busy without intermission. The specifics of the more complex protocol considered here are that: – the instruction stream generator may run ahead of the instruction stream execution unit by not waiting for the receipt of the replies resulting from the execution of instructions that it has sent earlier; – to ensure that the instruction stream execution unit can handle the runahead, each instruction sent by the instruction stream generator is accompanied with the sequence of replies after which the instruction must be executed; – to correct for replies that have not yet reached the instruction stream generator, each instruction sent is also accompanied with the number of replies received since the last sending of an instruction. This protocol is reminiscent of an instruction pre-fetching mechanism as found in pipelined processors (see e.g. [26]), but its range of application is not restricted to pipelined instruction processing. We write B≤n , where n ∈ N, for the set {u ∈ B∗ \| len(u) ≤ n}.5 It is assumed that a natural number ℓ has been given. The number ℓ is taken for the maximal number of steps that the instruction stream generator may run ahead of the instruction stream execution unit. Whether the execution unit can be kept busy without intermission with the given ℓ depends on the actual execution times of instructions and the actual transmission times over the transmission channels involved. If the execution unit can be kept busy without intermission with the given ℓ, then it is useless to increase ℓ. The set IM of instruction messages is defined as follows: IM = [0, ℓ] × B≤ℓ × I . In an instruction message (n, u, a) ∈ IM: – n is the number of replies that are acknowledged by the message; – u is the sequence of replies after which the instruction that is part of the message must be executed; – a is the instruction that is part of the message. The instruction stream generator sends instruction messages via an instruction message transmission channel to the instruction stream execution unit. We refer to a succession of transmitted instruction messages as an instruction stream. An instruction stream is dynamic by nature, in contradistinction with an instruction sequence. 5 As usual, we write D∗ for the set of all finite sequences with elements from set D and len(σ) for the length of finite sequence σ. Moreover, we write ǫ for the empty sequence, d for the sequence having d as sole element, σσ ′ for the concatenation of finite sequences σ and σ ′ , and tl(σ) for the tail of finite sequence σ. 20 The set SISG of instruction stream generator states is defined as follows: SISG = [0, ℓ] × P(B≤ℓ+1 × RT ) . In an instruction stream generator state (n, R) ∈ SISG : – n is the number of replies that has been received by the instruction stream generator since the last acknowledgement of received replies; – in each (u, T ) ∈ R, u is the sequence of replies after which the thread T must be performed. The functions updpm and updcr defined below are used to model the updates of the instruction stream generator state on producing a message and consuming a reply, respectively. The function updpm : (B≤ℓ × RT ) × SISG → SISG is defined as follows: updpm((u, T ), (n, R)) = (0, (R \ {(u, T )}) ∪ {(uT, thrt(T )), (uF, thrf (T ))}) if act (T ) ∈ A (0, (R \ {(u, T )})) if act (T ) ∈ /A. The function updcr : B × SISG → SISG is defined as follows: updcr (r, (n, R)) = (n + 1, {(u, T ) \| (ru, T ) ∈ R}) . The function sel defined below is used to model the selection of the sequence of replies and the instruction that will be part of the next message produced by the instruction stream generator. The function sel : P(B≤ℓ × RT ) → P(B≤ℓ × RT ) is defined as follows: sel (R) = {(u, T ) ∈ R \| ∀(v, T ′ ) ∈ R • len(u) ≤ len(v)} . Notice that (u, T ) ∈ sel (R) and (v, T ′ ) ∈ R only if len(u) ≤ len(v). By that breadth-first run-ahead is enforced. The performance of the protocol would change considerably if breadth-first run-ahead was not enforced. The set SISEU of instruction stream execution unit states is defined as follows: SISEU = [0, ℓ] × P(B≤ℓ × I) . In an instruction stream execution unit state (n, S) ∈ SISEU : – n is the number of replies for which the instruction stream execution unit still has to receive an acknowledgement; – in each (u, a) ∈ S, u is the sequence of replies after which the instruction a must be executed. The functions updcm and updpr defined below are used to model the updates of the instruction stream execution unit state on consuming a message and producing a reply, respectively. The function updcm : IM × SISEU → SISEU is defined as follows: . . updcm((k, u, a), (n, S)) = (n − k, S ∪ {(tl n−k (u), a)}) .6 21 The function updpr : B × SISEU → SISEU is defined as follows: updpr (r, (n, S)) = (n + 1, {(u, a) \| (ru, a) ∈ S}) . The function nxt defined below is used by the instruction stream execution unit to distinguish when it starts with handling the instruction to be executed next between the different instructions that it may be. The function nxt : I × P(B≤ℓ × I) → B is defined as follows: T if (ǫ, a) ∈ S nxt(a, S) = F if (ǫ, a) ∈ /S. The instruction stream execution unit sends replies via a reply transmission channel to the instruction stream generator. We refer to a succession of transmitted replies as a reply stream. For the purpose of describing the transmission protocol in ACPτ , A and \| are taken such that, in addition to the conditions mentioned at the beginning of Section 7, the following conditions are satisfied: A ⊇ {si (d) \| i ∈ {1, 2}, d ∈ IM} ∪ {ri (d) \| i ∈ {1, 2}, d ∈ IM} ∪ {si (r) \| i ∈ {3, 4}, r ∈ B} ∪ {ri (r) \| i ∈ {3, 4}, r ∈ B} ∪ {j} and for all i ∈ {1, 2}, j ∈ {3, 4}, d ∈ IM, r ∈ B, and e ∈ A: si (d) \| ri (d) = j , si (d) \| e = δ if e 6= ri (d) , e \| ri (d) = δ if e 6= si (d) , sj (r) \| rj (r) = j , sj (r) \| e = δ if e 6= rj (r) , e \| rj (r) = δ if e 6= sj (r) , j\|e =δ . Let T ∈ RT . Then the process representing the more complex protocol for remote instruction processing with regard to thread T is described by ∂H (ISGT k IMTC k RTC k ISEU ) , where the process ISGT is recursively specified by the following equations: ISGT ′ ISG(n,R) ′ = ISG(0,{(ǫ,T )}) , X ′ s1 ((n, u, act (T ))) · ISGupdpm((u,T = ),(n,R)) (u,T )∈sel(R) + X ′ r4 (r) · ISGupdcr (r,(n,R)) r∈B (for every (n, R) ∈ SISG with R 6= ∅) , ′ ISG(n,∅) =j (for every (n, ∅) ∈ SISG ) , 6 . . . As usual, we write i − j for the monus of i and j, i.e. i −j = i−j if i ≥ j and i −j =0 otherwise. As usual, tl n (u) is defined by induction on n as follows: tl 0 (u) = u and tl n+1 (u) = tl(tl n (u)). 22 the process IMTC is recursively specified by the following equation: X IMTC = r1 (d) · s2 (d) · IMTC , d∈IM the process RTC is recursively specified by the following equation: X RTC = r3 (r) · s4 (r) · RTC , r∈B the process ISEU is recursively specified by the following equations: ISEU ′ ISEU(n,S) ′ = ISEU (0,∅) , X ′ = r2 (d) · ISEU updcm(d,(n,S)) d∈IM X ′′ + nxt(f.m, S) :→ sf (m) · ISEU (f,(n,S)) f.m∈A + nxt(stop, S) :→ j + nxt(dead, S) :→ δ (for every (n, S) ∈ SISEU ) , X ′′ ′ ISEU(f,(n,S)) = rf (r) · s3 (r) · ISEU updpr(r,(n,S)) r∈B X ′′ + r2 (d) · ISEU (f,updcm(d,(n,S))) d∈IM (for every (f, (n, S)) ∈ F × SISEU ) , and H = {si (d) \| i ∈ {1, 2}, d ∈ IM} ∪ {ri (d) \| i ∈ {1, 2}, d ∈ IM} ∪ {si (r) \| i ∈ {3, 4}, r ∈ B} ∪ {ri (r) \| i ∈ {3, 4}, r ∈ B} . ISGT is the instruction stream generator for thread T , IMTC is the transmission channel for instruction messages, RTC is the transmission channel for replies, and ISEU is the instruction stream execution unit. The protocol described above has been designed such that, for each T ∈ RT , τ · τ{j} (∂H (ISGT k IMTC k RTC k ISEU )) denotes the process τ · \|T \|. We refrain from presenting a proof of the claim that the protocol satisfies this because this paper is first and foremost a conceptual paper and the proof is straightforward but tedious. The transmission channels IMTC and RTC can keep one instruction message and one reply, respectively. The protocol has been designed in such a way that the protocol will also work properly if these channels are replaced by channels with larger capacity and even by channels with unbounded capacity. Suppose that the transmission times over the transmission channels are small compared with the execution times of instructions. Even then the protocol described in Section 8 will always have to idle for a short time after the execution of an instruction, whereas after an initial phase the protocol described above will never have to idle after the execution of an instruction if the instruction stream generator may run a few steps ahead of the instruction stream execution unit. 23 10 Adaptations of the Protocol In this section, we discuss some conceivable adaptations of the protocol described in Section 9. While we were thinking through the details of that protocol, various variations suggested themselves. The variations discussed below are among the most salient ones. We think they deserve mention. However, their discussion is not in depth. The reason for this is that these variations have not yet been investigated thoroughly. Consider the case where, for each instruction, it is known what the probability is with which its execution leads to the reply T. This might give reason to adapt the protocol described in Section 9. Suppose that the instruction stream generator states do not only keep the sequences of replies after which threads must be performed, but also the sequences of instructions involved in producing those sequences of replies. Then the probability with which the sequences of replies will happen can be calculated and several conceivable adaptations of the protocol to this probabilistic knowledge are possible by mere changes in the selection of the sequence of replies and the instruction that will be part of the next instruction message produced by the instruction stream generator. Among those adaptations are: – restricting the instruction messages that are produced ahead to the ones where the sequence of replies after which the instruction must be executed will happen with a probability ≥ 0.50, but sticking to breadth-first runahead; – restricting the instruction messages that are produced ahead to the ones where the sequence of replies after which the instruction must be executed will happen with a probability ≥ 0.95, but not sticking to breadth-first runahead. At first sight, these adaptations are reminiscent of combinations of an instruction pre-fetching mechanism and a branch prediction mechanism as found in pipelined processors (see e.g. [26]). However, usually branch prediction mechanisms make use of statistics based on recently processed instructions instead of probabilistic knowledge of the kind used in the protocols sketched above. Regular threads can be represented in such a way that it is effectively decidable whether the two threads with which a thread may proceed after performing its first action are identical. Consider the case where threads are represented in the instruction stream generator states in such a way. Then the protocol can be adapted such that no duplication of instruction messages takes place in the cases where the two threads with which a thread possibly proceeds after performing its first action are identical. This can be accomplished by using sequences of elements from B ∪ {∗}, instead of sequences of elements from B, in instruction messages, instruction stream generator states, and instruction stream execution unit states. The occurrence of ∗ at position i in a sequence indicates that the ith reply may be either T or F. The impact of this change on the updates of instruction stream generator states and instruction stream execution unit states is minor. This adaptation is reminiscent of an instruction pre-fetch mechanism 24 as found in pipelined processors that prevents instruction pre-fetches that are superfluous due to identity of branches. 11 Alternative Choice Instructions Process algebra is an area of the study of concurrency which is considered relevant to computer science, as is witnesses by the extent of the work on algebraic theories of processes such as ACP, CCS and CSP in theoretical computer science. This strongly hints that there are programmed systems whose behaviours can be taken for processes as considered in process algebra. Therefore, it is interesting to know to which extent the behaviours considered in process algebra can be produced by programs under execution, starting from the perception of a program as an instruction sequence. In coming sections, we will establish results concerning the processes as considered in ACP that can be produced by instruction sequences under execution. For the purpose of producing processes as considered in ACP, we need a version of PGA with special basic instructions to deal with the non-deterministic choice between alternatives that stems from the alternative composition of processes. Recall that, in PGA, it is assumed that a fixed but arbitrary set A of basic instructions has been given. In the coming sections, we will make use a version of PGA in which the following additional assumptions relating to A are made: – – – – a fixed but arbitrary finite set F of foci has been given; a fixed but arbitrary finite set M of methods has been given; a fixed but arbitrary set AA of atomic actions, with t ∈ / AA, has been given; A = {f.m \| f ∈ F , m ∈ M} ∪ {ac(e1 , e2 ) \| e1 , e2 ∈ AA ∪ {t}}. On execution of a basic instruction ac(e1 , e2 ), first a non-deterministic choice between the atomic actions e1 and e2 is made and then the chosen atomic action is performed. The reply T is produced if e1 is performed and the reply F is produced if e2 is performed. Basic instructions of this kind are material to produce all regular processes by means of instruction sequences. A basic instruction of the form ac(e1 , e2 ) is called an alternative choice instruction. Henceforth, we will write PGAac for the version of PGA with alternative choice instructions. The intuition concerning alternative choice instructions given above will be made fully precise at the end of this section, using ACPτ . It will not be made fully precise using an extension of BTA because it is considered a basic property of threads that they are deterministic behaviours. Recall that we make use of a version of BTA in which A = A. A basic action of the form ac(e1 , e2 ) is called an alternative choice action. Henceforth, we will write BTAac for the version of BTA with alternative choice actions. For the purpose of making precise what processes are produced by the threads denoted by closed terms of BTAac +REC, A and \| are taken such that, in addition to the conditions mentioned at the beginning of Section 7, the following conditions are satisfied: A ⊇ AA ∪ {t} 25 Table 8. Additional defining equation for process extraction operation \|T E ac(e, e′ ) D T ′ \|c = e · \|T \|c + e′ · \|T ′ \|c and for all e, e′ ∈ A: e′ \| e = δ if e′ ∈ AA ∪ {t} . The process extraction operation for BTAac has as defining equations the equations given in Table 7 and in addition the equation given in Table 8. Proposition 3 goes through for BTAac . 12 Instruction Sequence Producible Processes It follows immediately from the definitions of the thread extraction and process extraction operations that the instruction sequences considered in PGA produce regular processes. The question is whether all regular processes are producible by these instruction sequences. In this section, we show that all regular processes can be produced by the instruction sequences with alternative choice instructions. We will make use of the fact that all regular threads over A can be produced by the single-pass instruction sequences considered in PGA. Proposition 4. For each thread T that is regular over A, there exists a PGA instruction sequence F such that F produces T , i.e. \|F \| = T . Proof. By Proposition 1, T is a component of the solution of some finite linear recursive specification E over BTA. There occur finitely many variables X0 , . . . , Xn in E. Assume that T is the X0 -component of the solution of E. Let F be the PGA instruction sequence (F0 ; . . . ; Fn )ω , where Fi is defined as follows (0 ≤ i ≤ n):  !;!;!   #0 ; #0 ; #0     +a ; #3·(j−i)−1 ; #3·(k−i)−2 Fi = +a ; #3·(j−i)−1 ; #3·(n+1−(i−k))−2    +a ; #3·(n+1−(i−j))−1 ; #3·(k−i)−2   +a ; #3·(n+1−(i−j))−1 ; #3·(n+1−(i−k))−2 if if if if if if Xi Xi Xi Xi Xi Xi = = = = = = S∈E D∈E Xj E a D Xk Xj E a D Xk Xj E a D Xk Xj E a D Xk ∈ ∈ ∈ ∈ E E E E ∧i ∧i ∧i ∧i < < ≥ ≥ j j j j ∧i ∧i ∧i ∧i < ≥ < ≥ k k k k. Then F is a PGA instruction sequence such that the interpretation of \|F \| = T . ⊓ ⊔ All regular processes over AA can be produced by the instruction sequences considered in PGAac . Theorem 2. Assume that CFAR is valid in MACPτ +REC . Then, for each process P that is regular over AA, there exists an instruction sequence F in which only basic instructions of the form ac(e, t) occur such that F produces P , i.e. τ · τ{t} (\|\|F \|\|) = τ · P . 26 Proof. By Propositions 1, 2 and 4, it is sufficient to show that, for each finite linear recursive specification E over ACPτ in which only atomic actions from AA occur, there exists a finite linear recursive specification E ′ over BTAac in which only basic actions of the form ac(e, t) occur such that τ ·hX\|Ei = τ ·τ{t} (\|hX\|E ′ i\|) for all X ∈ V(E). Take the finite linear recursive specification E over ACPτ that consists of the recursion equations Xi = ei1 · Xi1 + . . . + eiki · Xiki + e′i1 + . . . + e′ili , where ei1 , . . . , eiki , e′i1 , . . . , e′ili ∈ AA, for i ∈ {1, . . . n}. Then construct the finite linear recursive specification E ′ over BTAac that consists of the recursion equations Xi = Xi1 E ac(ei1 , t) D (. . . (Xiki E ac(eiki , t) D (S E ac(e′i1 , t) D (. . . (S E ac(e′ili , t) D Xi ) . . .))) . . .) for i ∈ {1, . . . n}; and the finite linear recursive specification E ′′ over ACPτ that consists of the recursion equations Zi1 = e′i1 + t · Zi2 , Zi2 = e′i2 + t · Zi3 , .. . Zili = e′ili + t · Xi , Xi = ei1 · Xi1 + t · Yi2 , Yi2 = ei2 · Xi2 + t · Yi3 , .. . Yiki = eiki · Xiki + t · Zi1 , where Yi2 , . . . , Yiki , Zi1 , . . . , Zili are fresh variables, for i ∈ {1, . . . n}. It follows immediately from the definition of the process extraction operation that \|hX\|E ′ i\| = hX\|E ′′ i for all X ∈ V(E). Moreover, it follows from CFAR that τ ·hX\|Ei = τ ·τ{t} (hX\|E ′′ i) for all X ∈ V(E). Hence, τ ·hX\|Ei = τ ·τ{t} (\|hX\|E ′ i\|) for all X ∈ V(E). ⊓ ⊔ For example, assuming that CFAR is valid, the instruction sequence (+ac(r3 (T), t) ; #4 ; +ac(r3 (F), t) ; #5 ; #7; +ac(s4 (T), t) ; #5 ; #9 ; +ac(s4 (F), t) ; #2 ; #9)ω produces the reply transmission channel process RTC of which a guarded recursive specification is given in Section 9. Remark 4. Theorem 2 with “τ · τ{t} (\|\|F \|\|) = τ · P ” replaced by “\|\|F \|\| = P ” can be established if PGA is extended with multiple-reply test instructions, see [11]. In that case, the assumption that CFAR is valid is superfluous. 13 Services and Use Operators An instruction sequence under execution may make use of services. That is, certain instructions may be executed for the purpose of having the behaviour 27 produced by the instruction sequence affected by a service that takes those instructions as commands to be processed. Likewise, a thread may perform certain actions for the purpose of having itself affected by a service that takes those actions as commands to be processed. The processing of an action may involve a change of state of the service and at completion of the processing of the action the service returns a reply value to the thread. The reply value determines how the thread proceeds. The use operators can be used in combination with the thread extraction operation from Section 4 to describe the behaviour produced by instruction sequences that make use of services. In this section, we first review the use operators, which are concerned with threads making such use of services, and then extend the process extraction operation to the use operators. A service H consists of – – – – a set S of states; an effect function eff : M × S → S; a yield function yld : M × S → B ∪ {B}; an initial state s0 ∈ S; satisfying the following condition: ∀m ∈ M, s ∈ S • (yld (m, s) = B ⇒ ∀m′ ∈ M • yld (m′ , eff (m, s)) = B) . The set S contains the states in which the service may be, and the functions eff and yld give, for each method m and state s, the state and reply, respectively, that result from processing m in state s. By the condition imposed on services, once the service has returned B as reply, it keeps returning B as reply. Let H = (S, eff , yld , s0 ) be a service and let m ∈ M. Then the derived service ∂ H, is the service (S, eff , yld , eff (m, s0 )); and of H after processing m, written ∂m the reply of H after processing m, written H(m), is yld (m, s0 ). When a thread makes a request to the service to process m: – if H(m) 6= B, then the request is accepted, the reply is H(m), and the service ∂ proceeds as ∂m H; – if H(m) = B, then the request is rejected and the service proceeds as a service that rejects any request. We introduce the sort S of services. However, we will not introduce constants and operators to build terms of this sort. The sort S, standing for the set of all services, is considered a parameter of the extension of BTA being presented. Moreover, we introduce, for each f ∈ F , the binary use operator /f : T × S → T. The axioms for these operators are given in Table 9. Intuitively, T /f H is the thread that results from processing all actions performed by thread T that are of the form f.m by service H. When a basic action of the form f.m performed by thread T is processed by service H, it is turned into the basic action tau and postconditional composition is removed in favour of basic action prefixing on the basis of the reply value produced. We add the use operators to PGAac as well. We will only use the extension in combination with the thread extraction operation \| \| and define \|F /f H\| = 28 Table 9. Axioms for use operators S /f H = S D /f H = D (x E tau D y) /f H = (x /f H) E tau D (y /f H) (x E g.m D y) /f H = (x /f H) E g.m D (y /f H) if f 6= g ∂ H) if H(m) = T (x E f.m D y) /f H = tau ◦ (x /f ∂m ∂ (x E f.m D y) /f H = tau ◦ (y /f ∂m H) if H(m) = F (x E f.m D y) /f H = tau ◦ D if H(m) = B (x E ac(e1 , e2 ) D y) /f H = (x /f H) E ac(e1 , e2 ) D (y /f H) πn (x /f H) = πn (πn (x) /f H) U1 U2 U3 U4 U5 U6 U7 U8 U9 \|F \| /f H. Hence, \|F /f H\| denotes the thread produced by F if F makes use of H. If H is a service such as an unbounded counter, an unbounded stack or a Turing tape, then a non-regular thread may be produced. In order to extend the process extraction operation to the use operators, we need an extension of ACPτ with action renaming operators ρh , where h:Aτ → Aτ such that h(τ ) = τ . The axioms for action renaming are given in [24]. Intuitively, ρh (P ) behaves as P with each atomic action replaced according to h. We write ρe′ 7→e′′ for the renaming operator ρh with h defined by h(e′ ) = e′′ and h(e) = e if e 6= e′ . For the purpose of extending the process extraction operation to the use operators, A and \| are taken such that, in addition to the conditions mentioned at the beginning of Section 7, with everywhere B replaced by B ∪ {B}, and the conditions mentioned at the end of Section 11, the following conditions are satisfied: A ⊇ {sserv (r) \| r ∈ B ∪ {B}} ∪ {rserv (m) \| m ∈ M} ∪ {stop∗ } and for all e ∈ A, m ∈ M, and r ∈ B ∪ {B}: sserv (r) \| e = δ , e \| rserv (m) = δ , stop \| stop = stop∗ , stop∗ \| e = δ . We also need to define a set Af ⊆ A and a function hf : Aτ → Aτ for each f ∈ F: Af = {sf (d) \| d ∈ M ∪ B ∪ {B}} ∪ {rf (d) \| d ∈ M ∪ B ∪ {B}} ; for all e ∈ Aτ , m ∈ M and r ∈ B ∪ {B}: hf (sserv (r)) = sf (r) , hf (rserv (m)) = rf (m) , V V hf (e) = e if r′ ∈N e 6= sserv (r′ ) ∧ m′ ∈M e 6= rserv (m′ ) . To extend the process extraction operation to the use operators, the defining equation concerning the postconditional composition operators has to be adapted and a new defining equation concerning the use operators has to be 29 Table 10. Adapted and additional defining equations for process extraction operation \|T E f.m D T ′ \|c = sf (m) · (rf (T) · \|T \|c + rf (F) · \|T ′ \|c + rf (B) · δ) \|T /f H\|c = ρstop∗ 7→stop (∂{stop} (∂Af (\|T \|c k ρhf (\|H\|c )))) added. These two equations are given in Table 10, where \|H\|c is the XH component of the solution of X {XH ′ = rserv (m) · sserv (H ′ (m)) · X ∂ H ′ + stop \| H ′ ∈ ∆(H)} , ∂m m∈M where ∆(H) is inductively defined as follows: – H ∈ ∆(H); – if m ∈ M and H ′ ∈ ∆(H), then ∂ ′ ∂m H ∈ ∆(H). The extended process extraction operation preserves the axioms for the use operators. Owing to the presence of axiom schemas with semantic side conditions in Table 9, the axioms for the use operators include proper axioms, which are all of the form t1 = t2 , and axioms that have a semantic side condition, which are all of the form t1 = t2 if H(m) = r. By that, the precise formulation of the preservation result is somewhat complicated. Proposition 5. 1. Let t1 = t2 be a proper axiom for the use operators, and let α be a valuation of variables in MBTA+REC . Then \|α∗ (t1 )\| = \|α∗ (t2 )\|. 2. Let t1 = t2 if H(m) = r be an axiom with semantic side condition for the use operators, and let α be a valuation of variables in MBTA+REC . Then \|α∗ (t1 )\| = \|α∗ (t2 )\| if H(m) = r. Proof. The proof is straightforward. We sketch the proof for axiom U5. By the definition of the process extraction operation, it is sufficient to show that ∂ H)\|c if H(m) = T. In outline, this goes \|(T E f.m D T ′ ) /f H\|c = \|tau ◦ (T /f ∂m as follows: \|(T E f.m D T ′ ) /f H\|c = ρstop∗ 7→stop (∂{stop} (∂Af (sf (m) · (rf (T) · \|T \|c + rf (F) · \|T ′ \|c + rf (B) · δ) k ρhf (\|H\|c )))) ∂ H\|c )))) = i · i · ρstop∗ 7→stop (∂{stop} (∂Af (\|T \|c k ρhf (\| ∂m ∂ c = \|tau ◦ (T /f ∂m H)\| . In the first and third step, we apply defining equations of \| \|c . In the second step, we apply axioms of ACPτ +REC with action renaming, and use that H(m) = T. ⊓ ⊔ Remark 5. Let F be a PGAac instruction sequence and H be a service. Then \|\|F /f H\|\| is the process produced by F if F makes use of H. Instruction sequences that make use of services such as unbounded counters, unbounded stacks or Turing tapes are interesting because they may produce non-regular processes. 30 14 PGLD Programs and the Use of Boolean Registers In this section, we show that all regular processes can also be produced by programs written in a program notation which is close to existing assembly languages, and even by programs in which no atomic action occurs more than once in an alternative choice instruction. The latter result requires programs that make use of Boolean registers. A hierarchy of program notations rooted in PGA is introduced in [8]. One program notation that belongs to this hierarchy is PGLD, a very simple program notation which is close to existing assembly languages. It has absolute jump instructions and no explicit termination instruction. In PGLD, like in PGA, it is assumed that there is a fixed but arbitrary finite set of basic instructions A. The primitive instructions of PGLD differ from the primitive instructions of PGA as follows: for each l ∈ N, there is an absolute jump instruction ##l instead of a forward jump instruction #l. PGLD programs have the form u1 ; . . . ; uk , where u1 , . . . , uk are primitive instructions of PGLD. The effects of all instructions in common with PGA are as in PGA with one difference: if there is no next instruction to be executed, termination occurs. The effect of an absolute jump instruction ##l is that execution proceeds with the l-th instruction of the program concerned. If ##l is itself the l-th instruction, then inaction occurs. If l equals 0 or l is greater than the length of the program, then termination occurs. We define the meaning of PGLD programs by means of a function pgld2pga from the set of all PGLD programs to the set of all closed PGA terms. This function is defined by pgld2pga(u1 ; . . . ; uk ) = (φ1 (u1 ) ; . . . ; φk (uk ) ; ! ; !)ω , where the auxiliary functions φj from the set of all primitive instructions of PGLD to the set of all primitive instructions of PGA are defined as follows (1 ≤ j ≤ k): φj (##l) = #l − j if φj (##l) = #k + 2 − (j − l) if φj (##l) = ! if φj (u) =u if j≤l≤k, 0<l<j, l =0∨l >k , u is not a jump instruction . PGLD is as expressive as PGA. Before we make this fully precise, we introduce a useful notation. Let α is a valuation of variables in I PGA , and let α∗ be the unique homomorphic extension of α to terms of PGA. Then α∗ (t) is independent of α if t is a closed term, i.e. α∗ (t) is uniquely determined by I PGA . Therefore, we write tI PGA for α∗ (t) if t is a closed term. Proposition 6. For each closed PGA term t, there exists a PGLD program p such that \|tI PGA \| = \|pgld2pga(p)I PGA \|. 31 Proof. In [8], a number of functions (called embeddings in that paper) are defined, whose composition gives, for each closed PGA term t, a PGLD program p such that \|tI PGA \| = \|pgld2pga(p)I PGA \|. ⊓ ⊔ Let p be a PGLD program and P be a process. Then we say that p produces P if \|pgld2pga(p)I PGA \| produces P . Below, we will write PGLDac for the version of PGLD in which the additional assumptions relating to A mentioned in Section 11 are made. As a corollary of Theorem 2 and Proposition 6, we have that all regular processes over AA can be produced by PGLDac programs. Corollary 1. Assume that CFAR is valid in MACPτ +REC . Then, for each process P that is regular over AA, there exists a PGLDac program p such that p produces P . We switch to the use of Boolean registers now. First, we describe services that make up Boolean registers. A Boolean register service accepts the following methods: – a set to true method set:T; – a set to false method set:F; – a get method get. We write MBR for the set {set:T, set:F, get}. It is assumed that MBR ⊆ M. The methods accepted by Boolean register services can be explained as follows: – set:T : the contents of the Boolean register becomes T and the reply is T; – set:F : the contents of the Boolean register becomes F and the reply is F; – get : nothing changes and the reply is the contents of the Boolean register. Let s ∈ B ∪ {B}. Then the Boolean register service with initial state s, written BR s , is the service (B ∪ {B}, eff , eff , s), where the function eff is defined as follows (b ∈ B): eff (set:T, b) = T , eff (set:F, b) = F , eff (get, b) = b , eff (m, b) = B if m 6∈ MBR , eff (m, B) = B . Notice that the effect and yield functions of a Boolean register service are the same. Let p be a PGLD program and P be a process. Then we say that p produces P using Boolean registers if (. . . (\|pgld2pga(p)I PGA \| /br:1 BR F ) . . . /br:k BR F ) produces P for some k ∈ N+ . We have that PGLDac programs in which no atomic action from AA occurs more than once in an alternative choice instruction can produce all regular processes over AA using Boolean registers. 32 Theorem 3. Assume that CFAR is valid in MACPτ +REC . Then, for each process P that is regular over AA, there exists a PGLDac program p in which each atomic action from AA occurs no more than once in an alternative choice instruction such that p produces P using Boolean registers. Proof. By the proof of Theorem 2 given in Section 12, it is sufficient to show that, for each thread T that is regular over A, there exist a PGLD program p in which each basic action from A occurs no more than once and a k ∈ N+ such that (. . . (\|pgld2pga(p)I PGA \| /br:1 BR F ) . . . /br:k BR F ) = T . Let T be a thread that is regular over A. We may assume that T is produced by a PGLD program p′ of the following form: +a1 ; ##(3 · k1 + 1) ; ##(3 · k1′ + 1) ; .. . +an ; ##(3 · kn + 1) ; ##(3 · kn′ + 1) ; ##0 ; ##0 ; ##0 ; ##(3 · n + 4) , where, for each i ∈ [1, n], ki , ki′ ∈ [0, n − 1] (cf. the proof of Proposition 2 from [36]). It is easy to see that the PGLD program p that we are looking for can be obtained by transforming p′ : by making use of n Boolean registers, p can distinguish between different occurrences of the same basic instruction in p′ , and in that way simulate p′ . ⊓ ⊔ 15 Conclusions Using the algebraic theory of processes known as ACP, we have described two protocols to deal with the phenomenon that, on execution of an instruction sequence, a stream of instructions to be processed arises at one place and the processing of that stream of instructions is handled at another place. The more complex protocol is directed towards keeping the execution unit busy. In this way, we have brought the phenomenon better into the picture and have ascribed a sense to the term instruction stream which makes clear that an instruction stream is dynamic by nature, in contradistinction with an instruction sequence. We have also discussed some conceivable adaptations of the more complex protocol. The description of the protocols start from the behaviours produced by instruction sequences under execution. By that we abstract from the instruction sequences which produce those behaviours. How instruction streams can be generated efficiently from instruction sequences is a matter that obviously requires investigations at a less abstract level. The investigations in question are an option for future work. We believe that the more complex protocol described in this paper provides a setting in which basic techniques aimed at increasing processor performance, such as pre-fetching and branch prediction, can be studied at a more abstract level than usual (cf. [26]). In particular, we think that the protocol can serve 33 as a starting-point for the development of a model with which trade-offs encountered in the design of processor architectures can be clarified. We consider investigations into this matter an interesting option for future work. The fact that process algebra is an area of the study of concurrency which is considered relevant to computer science, strongly hints that there are programmed systems whose behaviours are taken for processes as considered in process algebra. In that light, we have investigated the connections between programs and the processes that they produce, starting from the perception of a program as an instruction sequence. We have shown that, by apposite choice of basic instructions, all regular processes can be produced by means of instruction sequences as considered in PGA. We have also made precise what processes are produced by instruction sequences under execution that make use of services. 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On the Origin of Deep Learning On the Origin of Deep Learning Haohan Wang haohanw@cs.cmu.edu Bhiksha Raj bhiksha@cs.cmu.edu arXiv:1702.07800v4 [cs.LG] 3 Mar 2017 Language Technologies Institute School of Computer Science Carnegie Mellon University Abstract This paper is a review of the evolutionary history of deep learning models. It covers from the genesis of neural networks when associationism modeling of the brain is studied, to the models that dominate the last decade of research in deep learning like convolutional neural networks, deep belief networks, and recurrent neural networks. In addition to a review of these models, this paper primarily focuses on the precedents of the models above, examining how the initial ideas are assembled to construct the early models and how these preliminary models are developed into their current forms. Many of these evolutionary paths last more than half a century and have a diversity of directions. For example, CNN is built on prior knowledge of biological vision system; DBN is evolved from a trade-off of modeling power and computation complexity of graphical models and many nowadays models are neural counterparts of ancient linear models. This paper reviews these evolutionary paths and offers a concise thought flow of how these models are developed, and aims to provide a thorough background for deep learning. More importantly, along with the path, this paper summarizes the gist behind these milestones and proposes many directions to guide the future research of deep learning. 1 Wang and Raj 1. Introduction Deep learning has dramatically improved the state-of-the-art in many different artificial intelligent tasks like object detection, speech recognition, machine translation (LeCun et al., 2015). Its deep architecture nature grants deep learning the possibility of solving many more complicated AI tasks (Bengio, 2009). As a result, researchers are extending deep learning to a variety of different modern domains and tasks in additional to traditional tasks like object detection, face recognition, or language models, for example, Osako et al. (2015) uses the recurrent neural network to denoise speech signals, Gupta et al. (2015) uses stacked autoencoders to discover clustering patterns of gene expressions. Gatys et al. (2015) uses a neural model to generate images with different styles. Wang et al. (2016) uses deep learning to allow sentiment analysis from multiple modalities simultaneously, etc. This period is the era to witness the blooming of deep learning research. However, to fundamentally push the deep learning research frontier forward, one needs to thoroughly understand what has been attempted in the history and why current models exist in present forms. This paper summarizes the evolutionary history of several different deep learning models and explains the main ideas behind these models and their relationship to the ancestors. To understand the past work is not trivial as deep learning has evolved over a long time of history, as showed in Table 1. Therefore, this paper aims to offer the readers a walk-through of the major milestones of deep learning research. We will cover the milestones as showed in Table 1, as well as many additional works. We will split the story into different sections for the clearness of presentation. This paper starts the discussion from research on the human brain modeling. Although the success of deep learning nowadays is not necessarily due to its resemblance of the human brain (more due to its deep architecture), the ambition to build a system that simulate brain indeed thrust the initial development of neural networks. Therefore, the next section begins with connectionism and naturally leads to the age when shallow neural network matures. With the maturity of neural networks, this paper continues to briefly discuss the necessity of extending shallow neural networks into deeper ones, as well as the promises deep neural networks make and the challenges deep architecture introduces. With the establishment of the deep neural network, this paper diverges into three different popular deep learning topics. Specifically, in Section 4, this paper elaborates how Deep Belief Nets and its construction component Restricted Boltzmann Machine evolve as a trade-off of modeling power and computation loads. In Section 5, this paper focuses on the development history of Convolutional Neural Network, featured with the prominent steps along the ladder of ImageNet competition. In Section 6, this paper discusses the development of Recurrent Neural Networks, its successors like LSTM, attention models and the successes they achieved. While this paper primarily discusses deep learning models, optimization of deep architecture is an inevitable topic in this society. Section 7 is devoted to a brief summary of optimization techniques, including advanced gradient method, Dropout, Batch Normalization, etc. This paper could be read as a complementary of (Schmidhuber, 2015). Schmidhuber’s paper is aimed to assign credit to all those who contributed to the present state of the art, so his paper focuses on every single incremental work along the path, therefore cannot elab2 On the Origin of Deep Learning Year 300 BC 1873 1943 1949 1958 1974 1980 1982 1985 1986 1990 1997 2006 2009 2012 Table 1: Major milestones that will be covered in this paper Contributer Contribution introduced Associationism, started the history of human’s Aristotle attempt to understand brain. introduced Neural Groupings as the earliest models of Alexander Bain neural network, inspired Hebbian Learning Rule. introduced MCP Model, which is considered as the McCulloch & Pitts ancestor of Artificial Neural Model. considered as the father of neural networks, introduced Donald Hebb Hebbian Learning Rule, which lays the foundation of modern neural network. introduced the first perceptron, which highly resembles Frank Rosenblatt modern perceptron. Paul Werbos introduced Backpropagation Teuvo Kohonen introduced Self Organizing Map introduced Neocogitron, which inspired Convolutional Kunihiko Fukushima Neural Network John Hopfield introduced Hopfield Network Hilton & Sejnowski introduced Boltzmann Machine introduced Harmonium, which is later known as Restricted Paul Smolensky Boltzmann Machine Michael I. Jordan defined and introduced Recurrent Neural Network introduced LeNet, showed the possibility of deep neural Yann LeCun networks in practice Schuster & Paliwal introduced Bidirectional Recurrent Neural Network Hochreiter & introduced LSTM, solved the problem of vanishing Schmidhuber gradient in recurrent neural networks introduced Deep Belief Networks, also introduced Geoffrey Hinton layer-wise pretraining technique, opened current deep learning era. Salakhutdinov & introduced Deep Boltzmann Machines Hinton introduced Dropout, an efficient way of training neural Geoffrey Hinton networks 3 Wang and Raj orate well enough on each of them. On the other hand, our paper is aimed at providing the background for readers to understand how these models are developed. Therefore, we emphasize on the milestones and elaborate those ideas to help build associations between these ideas. In addition to the paths of classical deep learning models in (Schmidhuber, 2015), we also discuss those recent deep learning work that builds from classical linear models. Another article that readers could read as a complementary is (Anderson and Rosenfeld, 2000) where the authors conducted extensive interviews with well-known scientific leaders in the 90s on the topic of the neural networks’ history. 4 On the Origin of Deep Learning 2. From Aristotle to Modern Artificial Neural Networks The study of deep learning and artificial neural networks originates from our ambition to build a computer system simulating the human brain. To build such a system requires understandings of the functionality of our cognitive system. Therefore, this paper traces all the way back to the origins of attempts to understand the brain and starts the discussion of Aristotle’s Associationism around 300 B.C. 2.1 Associationism “When, therefore, we accomplish an act of reminiscence, we pass through a certain series of precursive movements, until we arrive at a movement on which the one we are in quest of is habitually consequent. Hence, too, it is that we hunt through the mental train, excogitating from the present or some other, and from similar or contrary or coadjacent. Through this process reminiscence takes place. For the movements are, in these cases, sometimes at the same time, sometimes parts of the same whole, so that the subsequent movement is already more than half accomplished.” This remarkable paragraph of Aristotle is seen as the starting point of Associationism (Burnham, 1888). Associationism is a theory states that mind is a set of conceptual elements that are organized as associations between these elements. Inspired by Plato, Aristotle examined the processes of remembrance and recall and brought up with four laws of association (Boeree, 2000). • Contiguity: Things or events with spatial or temporal proximity tend to be associated in the mind. • Frequency: The number of occurrences of two events is proportional to the strength of association between these two events. • Similarity: Thought of one event tends to trigger the thought of a similar event. • Contrast: Thought of one event tends to trigger the thought of an opposite event. Back then, Aristotle described the implementation of these laws in our mind as common sense. For example, the feel, the smell, or the taste of an apple should naturally lead to the concept of an apple, as common sense. Nowadays, it is surprising to see that these laws proposed more than 2000 years ago still serve as the fundamental assumptions of machine learning methods. For example, samples that are near each other (under a defined distance) are clustered into one group; explanatory variables that frequently occur with response variables draw more attention from the model; similar/dissimilar data are usually represented with more similar/dissimilar embeddings in latent space. Contemporaneously, similar laws were also proposed by Zeno of Citium, Epicurus and St Augustine of Hippo. The theory of associationism was later strengthened with a variety of philosophers or psychologists. Thomas Hobbes (1588-1679) stated that the complex experiences were the association of simple experiences, which were associations of sensations. He also believed that association exists by means of coherence and frequency as its strength 5 Wang and Raj Figure 1: Illustration of neural groupings in (Bain, 1873) factor. Meanwhile, John Locke (1632-1704) introduced the concept of “association of ideas”. He still separated the concept of ideas of sensation and ideas of reflection and he stated that complex ideas could be derived from a combination of these two simple ideas. David Hume (1711-1776) later reduced Aristotle’s four laws into three: resemblance (similarity), contiguity, and cause and effect. He believed that whatever coherence the world seemed to have was a matter of these three laws. Dugald Stewart (1753-1828) extended these three laws with several other principles, among an obvious one: accidental coincidence in the sounds of words. Thomas Reid (1710-1796) believed that no original quality of mind was required to explain the spontaneous recurrence of thinking, rather than habits. James Mill (1773-1836) emphasized on the law of frequency as the key to learning, which is very similar to later stages of research. David Hartley (1705-1757), as a physician, was remarkably regarded as the one that made associationism popular (Hartley, 2013). In addition to existing laws, he proposed his argument that memory could be conceived as smaller scale vibrations in the same regions of the brain as the original sensory experience. These vibrations can link up to represent complex ideas and therefore act as a material basis for the stream of consciousness. This idea potentially inspired Hebbian learning rule, which will be discussed later in this paper to lay the foundation of neural networks. 2.2 Bain and Neural Groupings Besides David Hartley, Alexander Bain (1818-1903) also contributed to the fundamental ideas of Hebbian Learning Rule (Wilkes and Wade, 1997). In this book, Bain (1873) related the processes of associative memory to the distribution of activity of neural groupings (a term that he used to denote neural networks back then). He proposed a constructive mode of storage capable of assembling what was required, in contrast to alternative traditional mode of storage with prestored memories. To further illustrate his ideas, Bain first described the computational flexibility that allows a neural grouping to function when multiple associations are to be stored. With a few hypothesis, Bain managed to describe a structure that highly resembled the neural 6 On the Origin of Deep Learning networks of today: an individual cell is summarizing the stimulation from other selected linked cells within a grouping, as showed in Figure 1. The joint stimulation from a and c triggers X, stimulation from b and c triggers Y and stimulation from a and c triggers Z. In his original illustration, a, b, c stand for simulations, X and Y are outcomes of cells. With the establishment of how this associative structure of neural grouping can function as memory, Bain proceeded to describe the construction of these structures. He followed the directions of associationism and stated that relevant impressions of neural groupings must be made in temporal contiguity for a period, either on one occasion or repeated occasions. Further, Bain described the computational properties of neural grouping: connections are strengthened or weakened through experience via changes of intervening cell-substance. Therefore, the induction of these circuits would be selected comparatively strong or weak. As we will see in the following section, Hebb’s postulate highly resembles Bain’s description, although nowadays we usually label this postulate as Hebb’s, rather than Bain’s, according to (Wilkes and Wade, 1997). This omission of Bain’s contribution may also be due to Bain’s lack of confidence in his own theory: Eventually, Bain was not convinced by himself and doubted about the practical values of neural groupings. 2.3 Hebbian Learning Rule Hebbian Learning Rule is named after Donald O. Hebb (1904-1985) since it was introduced in his work The Organization of Behavior (Hebb, 1949). Hebb is also seen as the father of Neural Networks because of this work (Didier and Bigand, 2011). In 1949, Hebb stated the famous rule: “Cells that fire together, wire together”, which emphasized on the activation behavior of co-fired cells. More specifically, in his book, he stated that: “When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that As efficiency, as one of the cells firing B, is increased.” This archaic paragraph can be re-written into modern machine learning languages as the following: ∆wi = ηxi y (1) where ∆wi stands for the change of synaptic weights (wi ) of Neuron i, of which the input signal is xi . y denotes the postsynaptic response and η denotes learning rate. In other words, Hebbian Learning Rule states that the connection between two units should be strengthened as the frequency of co-occurrences of these two units increase. Although Hebbian Learning Rule is seen as laying the foundation of neural networks, seen today, its drawbacks are obvious: as co-occurrences appear more, the weights of connections keep increasing and the weights of a dominant signal will increase exponentially. This is known as the unstableness of Hebbian Learning Rule (Principe et al., 1999). Fortunately, these problems did not influence Hebb’s identity as the father of neural networks. 7 Wang and Raj 2.4 Oja’s Rule and Principal Component Analyzer Erkki Oja extended Hebbian Learning Rule to avoid the unstableness property and he also showed that a neuron, following this updating rule, is approximating the behavior of a Principal Component Analyzer (PCA) (Oja, 1982). Long story short, Oja introduced a normalization term to rescue Hebbian Learning rule, and further he showed that his learning rule is simply an online update of Principal Component Analyzer. We present the details of this argument in the following paragraphs. Starting from Equation 1 and following the same notation, Oja showed: wit+1 = wit + ηxi y where t denotes the iteration. A straightforward way to avoid the exploding of weights is to apply normalization at the end of each iteration, yielding: wt + ηxi y wit+1 = Pn i 1 ( i=1 (wit + ηxi y)2 ) 2 where n denotes the number of neurons. The above equation can be further expanded into the following form: Pn t w i yx w j yxj wj i + ) + O(η 2 ) wit+1 = i + η( Z Z Z3 P 1 where Z = ( ni wi2 ) 2 . Further, two more assumptions are introduced: 1) η is small. P 1 Therefore O(η 2 ) is approximately 0. 2) Weights are normalized, therefore Z = ( ni wi2 ) 2 = 1. When these two assumptions were introduced back to the previous equation, Oja’s rule was proposed as following: wit+1 = wit + ηy(xi − ywit ) (2) Oja took a step further to show that a neuron that was updated with this rule was effectively performing Principal Component Analysis on the data. To show this, Oja first re-wrote Equation 2 as the following forms with two additional assumptions (Oja, 1982): d t w = Cwit − ((wit )T Cwit )wit d(t) i where C is the covariance matrix of input X. Then he proceeded to show this property with many conclusions from his another work (Oja and Karhunen, 1985) and linked back to PCA with the fact that components from PCA are eigenvectors and the first component is the eigenvector corresponding to largest eigenvalues of the covariance matrix. Intuitively, we could interpret this property with a simpler explanation: the eigenvectors of C are the solution when we maximize the rule updating function. Since wit are the eigenvectors of the covariance matrix of X, we can get that wit are the PCA. Oja’s learning rule concludes our story of learning rules of the early-stage neural network. Now we proceed to visit the ideas on neural models. 8 On the Origin of Deep Learning 2.5 MCP Neural Model While Donald Hebb is seen as the father of neural networks, the first model of neuron could trace back to six years ahead of the publication of Hebbian Learning Rule, when a neurophysiologist Warren McCulloch and a mathematician Walter Pitts speculated the inner workings of neurons and modeled a primitive neural network by electrical circuits based on their findings (McCulloch and Pitts, 1943). Their model, known as MCP neural model, was a linear step function upon weighted linearly interpolated data that could be described as: ( P 1, i wi xi ≥ θ AND zj = 0, ∀j y= 0, otherwise where y stands for output, xi stands for input of signals, wi stands for the corresponding weights and zj stands for the inhibitory input. θ stands for the threshold. The function is designed in a way that the activity of any inhibitory input completely prevents excitation of the neuron at any time. Despite the resemblance between MCP Neural Model and modern perceptron, they are still different distinctly in many different aspects: • MCP Neural Model is initially built as electrical circuits. Later we will see that the study of neural networks has borrowed many ideas from the field of electrical circuits. • The weights of MCP Neural Model wi are fixed, in contrast to the adjustable weights in modern perceptron. All the weights must be assigned with manual calculation. • The idea of inhibitory input is quite unconventional even seen today. It might be an idea worth further study in modern deep learning research. 2.6 Perceptron With the success of MCP Neural Model, Frank Rosenblatt further substantialized Hebbian Learning Rule with the introduction of perceptrons (Rosenblatt, 1958). While theorists like Hebb were focusing on the biological system in the natural environment, Rosenblatt constructed the electronic device named Perceptron that was showed with the ability to learn in accordance with associationism. Rosenblatt (1958) introduced the perceptron with the context of the vision system, as showed in Figure 2(a). He introduced the rules of the organization of a perceptron as following: • Stimuli impact on a retina of the sensory units, which respond in a manner that the pulse amplitude or frequency is proportional to the stimulus intensity. • Impulses are transmitted to Projection Area (AI ). This projection area is optional. • Impulses are then transmitted to Association Area through random connections. If the sum of impulse intensities is equal to or greater than the threshold (θ) of this unit, then this unit fires. 9 Wang and Raj (a) Illustration of organization of a perceptron in (Rosenblatt, 1958) (b) A typical perceptron in modern machine learning literature Figure 2: Perceptrons: (a) A new figure of the illustration of organization of perceptron as in (Rosenblatt, 1958). (b) A typical perceptron nowadays, when AI (Projection Area) is omitted. • Response units work in the same fashion as those intermediate units. Figure 2(a) illustrates his explanation of perceptron. From left to right, the four units are sensory unit, projection unit, association unit and response unit respectively. Projection unit receives the information from sensory unit and passes onto association unit. This unit is often omitted in other description of similar models. With the omission of projection unit, the structure resembles the structure of nowadays perceptron in a neural network (as showed in Figure 2(b)): sensory units collect data, association units linearly adds these data with different weights and apply non-linear transform onto the thresholded sum, then pass the results to response units. One distinction between the early stage neuron models and modern perceptrons is the introduction of non-linear activation functions (we use sigmoid function as an example in Figure 2(b)). This originates from the argument that linear threshold function should be softened to simulate biological neural networks (Bose et al., 1996) as well as from the consideration of the feasibility of computation to replace step function with a continuous one (Mitchell et al., 1997). After Rosenblatt’s introduction of Perceptron, Widrow et al. (1960) introduced a followup model called ADALINE. However, the difference between Rosenblatt’s Perceptron and ADALINE is mainly on the algorithm aspect. As the primary focus of this paper is neural network models, we skip the discussion of ADALINE. 2.7 Perceptron’s Linear Representation Power A perceptron is fundamentally a linear function of input signals; therefore it is limited to represent linear decision boundaries like the logical operations like NOT, AND or OR, but not XOR when a more sophisticated decision boundary is required. This limitation was highlighted by Minski and Papert (1969), when they attacked the limitations of perceptions by emphasizing that perceptrons cannot solve functions like XOR or NXOR. As a result, very little research was done in this area until about the 1980s. 10 On the Origin of Deep Learning (a) (b) (c) (d) Figure 3: The linear representation power of preceptron To show a more concrete example, we introduce a linear preceptron with only two inputs x1 and x2 , therefore, the decision boundary w1 x1 + w2 x2 forms a line in a two-dimensional space. The choice of threshold magnitude shifts the line horizontally and the sign of the function picks one side of the line as the halfspace the function represents. The halfspace is showed in Figure 3 (a). In Figure 3 (b)-(d), we present two nodes a and b to denote to input, as well as the node to denote the situation when both of them trigger and a node to denote the situation when neither of them triggers. Figure 3 (b) and Figure 3 (c) show clearly that a linear perceptron can be used to describe AND and OR operation of these two inputs. However, in Figure 3 (d), when we are interested in XOR operation, the operation can no longer be described by a single linear decision boundary. In the next section, we will show that the representation ability is greatly enlarged when we put perceptrons together to make a neural network. However, when we keep stacking one neural network upon the other to make a deep learning model, the representation power will not necessarily increase. 11 Wang and Raj 3. From Modern Neural Network to the Era of Deep Learning In this section, we will introduce some important properties of neural networks. These properties partially explain the popularity neural network gains these days and also motivate the necessity of exploring deeper architecture. To be specific, we will discuss a set of universal approximation properties, in which each property has its condition. Then, we will show that although a shallow neural network is an universal approximator, deeper architecture can significantly reduce the requirement of resources while retaining the representation power. At last, we will also show some interesting properties discovered in the 1990s about backpropagation, which may inspire some related research today. 3.1 Universal Approximation Property The step from perceptrons to basic neural networks is only placing the perceptrons together. By placing the perceptrons side by side, we get a single one-layer neural network and by stacking one one-layer neural network upon the other, we get a multi-layer neural network, which is often known as multi-layer perceptrons (MLP) (Kawaguchi, 2000). One remarkable property of neural networks, widely known as universal approximation property, roughly describes that an MLP can represent any functions. Here we discussed this property in three different aspects: • Boolean Approximation: an MLP of one hidden layer1 can represent any boolean function exactly. • Continuous Approximation: an MLP of one hidden layer can approximate any bounded continuous function with arbitrary accuracy. • Arbitrary Approximation: an MLP of two hidden layers can approximate any function with arbitrary accuracy. We will discuss these three properties in detail in the following paragraphs. To suit different readers’ interest, we will first offer an intuitive explanation of these properties and then offer the proofs. 3.1.1 Representation of any Boolean Functions This approximation property is very straightforward. In the previous section we have shown that every linear preceptron can perform either AND or OR. According to De Morgan’s laws, every propositional formula can be converted into an equivalent Conjunctive Normal Form, which is an OR of multiple AND functions. Therefore, we simply rewrite the target Boolean function into an OR of multiple AND operations. Then we design the network in such a way: the input layer performs all AND operations, and the hidden layer is simply an OR operation. The formal proof is not very different from this intuitive explanation, we skip it for simplicity. 1. Through this paper, we will follow the most widely accepted naming convention that calls a two-layer neural network as one hidden layer neural network. 12 On the Origin of Deep Learning (a) (b) (c) (d) Figure 4: Example of Universal Approximation of any Bounded Continuous Functions 3.1.2 Approximation of any Bounded Continuous Functions Continuing from the linear representation power of perceptron discussed previously, if we want to represent a more complex function, showed in Figure 4 (a), we can use a set of linear perceptrons, each of them describing a halfspace. One of these perceptrons is shown in Figure 4 (b), we will need five of these perceptrons. With these perceptrons, we can bound the target function out, as showed in Figure 4 (c). The numbers showed in Figure 4 (c) represent the number of subspaces described by perceptrons that fall into the corresponding region. As we can see, with an appropriate selection of the threshold (e.g. θ = 5 in Figure 4 (c)), we can bound the target function out. Therefore, we can describe any bounded continuous function with only one hidden layer; even it is a shape as complicated as Figure 4 (d). This property was first shown in (Cybenko, 1989) and (Hornik et al., 1989). To be specific, Cybenko (1989) showed that, if we have a function in the following form: X f (x) = ωi σ(wiT x + θ) (3) i f (x) is dense in the subspace of where it is in. In other words, for an arbitrary function g(x) in the same subspace as f (x), we have \|f (x) − g(x)\| < where > 0. In Equation 3, σ denotes the activation function (a squashing function back then), wi denotes the weights for the input layer and ωi denotes the weights for the hidden layer. This conclusion was drawn with a proof by contradiction: With Hahn-Banach Theorem and Riesz Representation Theorem, the fact that the closure of f (x) is not all the subspace where f (x) is in contradicts the assumption that σ is an activation (squashing) function. Till today, this property has drawn thousands of citations. Unfortunately, many of the later works cite this property inappropriately (Castro et al., 2000) because Equation 3 is not the widely accepted form of a one-hidden-layer neural network because it does not deliver a thresholded/squashed output, but a linear output instead. Ten years later after this property was shown, Castro et al. (2000) concluded this story by showing that when the final output is squashed, this universal approximation property still holds. Note that, this property was shown with the context that activation functions are squashing functions. By definition, a squashing function σ : R → [0, 1] is a non-decreasing function 13 Wang and Raj Figure 5: Threshold is not necessary with a large number of linear perceptrons. with the properties limx→∞ σ(x) = 1 and limx→−∞ σ(x) = 0. Many activation functions of recent deep learning research do not fall into this category. 3.1.3 Approximation of Arbitrary Functions Before we move on to explain this property, we need first to show a major property regarding combining linear perceptrons into neural networks. Figure 5 shows that as the number of linear perceptrons increases to bound the target function, the area outside the polygon with the sum close to the threshold shrinks. Following this trend, we can use a large number of perceptrons to bound a circle, and this can be achieved even without knowing the threshold because the area close to the threshold shrinks to nothing. What left outside the circle is, in fact, the area that sums to N2 , where N is the number of perceptrons used. Therefore, a neural network with one hidden layer can represent a circle with arbitrary diameter. Further, we introduce another hidden layer that is used to combine the outputs of many different circles. This newly added hidden layer is only used to perform OR operation. Figure 6 shows an example that when the extra hidden layer is used to merge the circles from the previous layer, the neural network can be used to approximate any function. The target function is not necessarily continuous. However, each circle requires a large number of neurons, consequently, the entire function requires even more. This property was showed in (Lapedes and Farber, 1988) and (Cybenko, 1988) respectively. Looking back at this property today, it is not arduous to build the connections between this property to Fourier series approximation, which, in informal words, states that every function curve can be decomposed into the sum of many simpler curves. With this linkage, to show this universal approximation property is to show that any one-hiddenlayer neural network can represent one simple surface, then the second hidden layer sums up these simple surfaces to approximate an arbitrary function. As we know, one hidden layer neural network simply performs a thresholded sum operation, therefore, the only step left is to show that the first hidden layer can represent a simple surface. To understand the “simple surface”, with linkage to Fourier transform, one can imagine one cycle of the sinusoid for the one-dimensional case or a “bump” of a plane in the two-dimensional case. 14 On the Origin of Deep Learning Figure 6: How a neural network can be used to approximate a leaf shaped function. For one dimension, to create a simple surface, we only need two sigmoid functions appropriately placed, for example, as following: f1 (x) = f2 (x) = h 1+ e−(x+t1 ) h 1 + ex−t2 Then, with f1 (x) + f2 (x), we create a simple surface with height 2h from t1 ≤ x ≤ t2 . This could be easily generalized to n-dimensional case, where we need 2n sigmoid functions (neurons) for each simple surface. Then for each simple surface that contributes to the final function, one neuron is added onto the second hidden layer. Therefore, despite the number of neurons need, one will never need a third hidden layer to approximate any function. Similarly to how Gibbs phenomenon affects Fourier series approximation, this approximation cannot guarantee an exact representation. The universal approximation properties showed a great potential of shallow neural networks at the price of exponentially many neurons at these layers. One followed-up question is that how to reduce the number of required neurons while maintaining the representation power. This question motivates people to proceed to deeper neural networks despite that shallow neural networks already have infinite modeling power. Another issue worth attention is that, although neural networks can approximate any functions, it is not trivial to find the set of parameters to explain the data. In the next two sections, we will discuss these two questions respectively. 15 Wang and Raj 3.2 The Necessity of Depth The universal approximation properties of shallow neural networks come at a price of exponentially many neurons and therefore are not realistic. The question about how to maintain this expressive power of the network while reducing the number of computation units has been asked for years. Intuitively, Bengio and Delalleau (2011) suggested that it is nature to pursue deeper networks because 1) human neural system is a deep architecture (as we will see examples in Section 5 about human visual cortex.) and 2) humans tend to represent concepts at one level of abstraction as the composition of concepts at lower levels. Nowadays, the solution is to build deeper architectures, which comes from a conclusion that states the representation power of a k layer neural network with polynomial many neurons need to be expressed with exponentially many neurons if a k − 1 layer structured is used. However, theoretically, this conclusion is still being completed. This conclusion could trace back to three decades ago when Yao (1985) showed the limitations of shallow circuits functions. Hastad (1986) later showed this property with parity circuits: “there are functions computable in polynomial size and depth k but requires exponential size when depth is restricted to k − 1”. He showed this property mainly by the application of DeMorgan’s law, which states that any AND or ORs can be rewritten as OR of ANDs and vice versa. Therefore, he simplified a circuit where ANDs and ORs appear one after the other by rewriting one layer of ANDs into ORs and therefore merge this operation to its neighboring layer of ORs. By repeating this procedure, he was able to represent the same function with fewer layers, but more computations. Moving from circuits to neural networks, Delalleau and Bengio (2011) compared deep and shallow sum-product neural networks. They showed that a function√ that could be expressed with O(n) neurons on a network of depth k required at least O(2 n ) and O((n − 1)k ) neurons on a two-layer neural network. Further, Bianchini and Scarselli (2014) extended this study to a general neural network with many major activation functions including tanh and sigmoid. They derived the conclusion with the concept of Betti numbers, and used this number to describe the representation power of neural networks. They showed that for a shallow network, the representation power can only grow polynomially with respect to the number of neurons, but for deep architecture, the representation can grow exponentially with respect to the number of neurons. They also related their conclusion to VC-dimension of neural networks, which is O(p2 ) for tanh (Bartlett and Maass, 2003) where p is the number of parameters. Recently, Eldan and Shamir (2015) presented a more thorough proof to show that depth of a neural network is exponentially more valuable than the width of a neural network, for a standard MLP with any popular activation functions. Their conclusion is drawn with only a few weak assumptions that constrain the activation functions to be mildly increasing, measurable, and able to allow shallow neural networks to approximate any univariate Lipschitz function. Finally, we have a well-grounded theory to support the fact that deeper network is preferred over shallow ones. However, in reality, many problems will arise if we keep increasing the layers. Among them, the increased difficulty of learning proper parameters is probably the most prominent one. Immediately in the next section, we will discuss the main drive of searching parameters for a neural network: Backpropagation. 16 On the Origin of Deep Learning 3.3 Backpropagation and Its Properties Before we proceed, we need to clarify that the name backpropagation, originally, is not referring to an algorithm that is used to learn the parameters of a neural network, instead, it stands for a technique that can help efficiently compute the gradient of parameters when gradient descent algorithm is applied to learn parameters (Hecht-Nielsen, 1989). However, nowadays it is widely recognized as the term to refer gradient descent algorithm with such a technique. Compared to a standard gradient descent, which updates all the parameters with respect to error, backpropagation first propagates the error term at output layer back to the layer at which parameters need to be updated, then uses standard gradient descent to update parameters with respect to the propagated error. Intuitively, the derivation of backpropagation is about organizing the terms when the gradient is expressed with the chain rule. The derivation is neat but skipped in this paper due to the extensive resources available (Werbos, 1990; Mitchell et al., 1997; LeCun et al., 2015). Instead, we will discuss two interesting and seemingly contradictory properties of backpropagation. 3.3.1 Backpropagation Finds Global Optimal for Linear Separable Data Gori and Tesi (1992) studied on the problem of local minima in backpropagation. Interestingly, when the society believes that neural networks or deep learning approaches are believed to suffer from local optimal, they proposed an architecture where global optimal is guaranteed. Only a few weak assumptions of the network are needed to reach global optimal, including • Pyramidal Architecture: upper layers have fewer neurons • Weight matrices are full row rank • The number of input neurons cannot smaller than the classes/patterns of data. However, their approaches may not be relevant anymore as they require the data to be linearly separable, under which condition that many other models can be applied. 3.3.2 Backpropagation Fails for Linear Separable Data On the other hand, Brady et al. (1989) studied the situations when backpropagation fails on linearly separable data sets. He showed that there could be situations when the data is linearly separable, but a neural network learned with backpropagation cannot find that boundary. He also showed examples when this situation occurs. His illustrative examples only hold when the misclassified data samples are significantly less than correctly classified data samples, in other words, the misclassified data samples might be just outliers. Therefore, this interesting property, when viewed today, is arguably a desirable property of backpropagation as we typically expect a machine learning model to neglect outliers. Therefore, this finding has not attracted many attentions. However, no matter whether the data is an outlier or not, neural network should be able to overfit training data given sufficient training iterations and a legitimate learning algorithm, especially considering that Brady et al. (1989) showed that an inferior algorithm 17 Wang and Raj was able to overfit the data. Therefore, this phenomenon should have played a critical role in the research of improving the optimization techniques. Recently, the studying of cost surfaces of neural networks have indicated the existence of saddle points (Choromanska et al., 2015; Dauphin et al., 2014; Pascanu et al., 2014), which may explain the findings of Brady et al back in the late 80s. Backpropagation enables the optimization of deep neural networks. However, there is still a long way to go before we can optimize it well. Later in Section 7, we will briefly discuss more techniques related to the optimization of neural networks. 18 On the Origin of Deep Learning 4. The Network as Memory and Deep Belief Nets Figure 7: Trade off of representation power and computation complexity of several models, that guides the development of better models With the background of how modern neural network is set up, we proceed to visit the each prominent branch of current deep learning family. Our first stop is the branch that leads to the popular Restricted Boltzmann Machines and Deep Belief Nets, and it starts as a model to understand the data unsupervisedly. Figure 7 summarizes the model that will be covered in this Section. The horizontal axis stands for the computation complexity of these models while the vertical axis stands for the representation power. The six milestones that will be focused in this section are placed in the figure. 4.1 Self Organizing Map The discussion starts with Self Organizing Map (SOM) invented by Kohonen (1990). SOM is a powerful technique that is primarily used in reducing the dimension of data, usually to one or two dimensions (Germano, 1999). While reducing the dimensionality, SOM also retains the topological similarity of data points. It can also be seen as a tool for clustering while imposing the topology on clustered representation. Figure 8 is an illustration of Self Organizing Map of two dimension hidden neurons. Therefore, it learns a two dimension representation of data. The upper shaded nodes denote the units of SOM that are used to 19 Wang and Raj Figure 8: Illustration of Self-Organizing Map represent data while the lower circles denote the data. There is no connection between the nodes in SOM 2 . The position of each node is fixed. The representation should not be viewed as only a numerical value. Instead, the position of it also matters. This property is different from some widely-accepted representation criterion. For example, we compare the case when one-hot vector and one-dimension SOM are used to denote colors: To denote green out of a set: C = {green, red, purple}, one-hot representation can use any vector of (1, 0, 0), (0, 1, 0) or (0, 0, 1) as long as we specify the bit for green correspondingly. However, for a one-dimensional SOM, only two vectors are possible: (1, 0, 0) or (0, 0, 1). This is because that, since SOM aims to represent the data while retaining the similarity; and red and purple are much more similar than green and red or green and purple, green should not be represented in a way that it splits red and purple. One should notice that, this example is only used to demonstrate that the position of each unit in SOM matters. In practice, the values of SOM unit are not restricted to integers. The learned SOM is usually a good tool for visualizing data. For example, if we conduct a survey on the happiness level and richness level of each country and feed the data into a two-dimensional SOM. Then the trained units should represent the happiest and richest country at one corner and represent the opposite country at the furthest corner. The rest two corners represent the richest, yet unhappiest and the poorest but happiest countries. The rest countries are positioned accordingly. The advantage of SOM is that it allows one 2. In some other literature, (Bullinaria, 2004) as an example, one may notice that there are connections in the illustrations of models. However, those connections are only used to represent the neighborhood relationship of nodes, and there is no information flowing via those connections. In this paper, as we will show many other models that rely on a clear illustration of information flow, we decide to save the connections to denote that. 20 On the Origin of Deep Learning to easily tell how a country is ranked among the world with a simple glance of the learned units (Guthikonda, 2005). 4.1.1 Learning Algorithm With an understanding of the representation power of SOM, now we proceed to its parameter learning algorithm. The classic algorithm is heuristic and intuitive, as shown below: Here we use a two-dimensional SOM as example, and i, j are indexes of units; w is weight Initialize weights of all units, wi,j ∀ i, j for t ≤ N do Pick vk randomly Select Best Matching Unit (BMU) as p, q := arg mini,j \|\|wij − vk \|\|22 Select the nodes of interest as the neighbors of BMU. I = {wi,j \|dist(wi,j , wp,q ) < r(t)} Update weights: wi,j = wi,j + P (i, j, p, q)l(t)\|\|wij − vk \|\|22 , ∀i, j ∈ I end for of the unit; v denotes data vector; k is the index of data; t denotes the current iteration; N constrains the maximum number of steps allowed; P (·) denotes the penalty considering the distance between unit p, q and unit i, j; l is learning rate; r denotes a radius used to select neighbor nodes. Both l and r typically decrease as t increases. \|\| · \|\|22 denotes Euclidean distance and dist(·) denotes the distance on the position of units. This algorithm explains how SOM can be used to learn a representation and how the similarities are retained as it always selects a subset of units that are similar with the data sampled and adjust the weights of units to match the data sampled. However, this algorithm relies on a careful selection of the radius of neighbor selection and a good initialization of weights. Otherwise, although the learned weights will have a local property of topological similarity, it loses this property globally: sometimes, two similar clusters of similar events are separated by another dissimilar cluster of similar events. In simpler words, units of green may actually separate units of red and units of purple if the network is not appropriately trained. (Germano, 1999). 4.2 Hopfield Network Hopfield Network is historically described as a form of recurrent3 neural network, first introduced in (Hopfield, 1982). “Recurrent” in this context refers to the fact that the weights connecting the neurons are bidirectional. Hopfield Network is widely recognized because of its content-addressable memory property. This content-addressable memory property is a simulation of the spin glass theory. Therefore, we start the discussion from spin glass. 3. The term “recurrent” is very confusing nowadays because of the popularity recurrent neural network (RNN) gains. 21 Wang and Raj Figure 9: Illustration of Hopfield Network. It is a fully connected network of six binary thresholding neural units. Every unit is connected with data, therefore these units are denoted as unshaded nodes. 4.2.1 Spin Glass The spin glass is physics term that is used to describe a magnetic phenomenon. Many works have been done for a detailed study of related theory (Edwards and Anderson, 1975; Mézard et al., 1990), so in this paper, we only describe this it intuitively. When a group of dipoles is placed together in any space. Each dipole is forced to align itself with the field generated by these dipoles at its location. However, by aligning itself, it changes the field at other locations, leading other dipoles to flip, causing the field in the original location to change. Eventually, these changes will converge to a stable state. To describe the stable state, we first define the total field at location j as X sk sj = oj + ct d2jk k ct where oj is an external field, is a constant that depends on temperature t, sk is the polarity of the kth dipole and djk is the distance from location j to location k. Therefore, the total potential energy of the system is: X X sk PE = sj oj + ct sj (4) d2jk j k The magnetic system will evolve until this potential energy is minimum. 4.2.2 Hopfield Network Hopfield Network is a fully connected neural network with binary thresholding neural units. The values of these units are either 0 or 14 . These units are fully connected with bidirectional weights. 4. Some other literature may use -1 and 1 to denote the values of these units. While the choice of values does not affect the idea of Hopfiled Network, it changes the formulation of energy function. In this paper, we only discuss in the context of 0 and 1 as values. 22 On the Origin of Deep Learning With this setting, the energy of a Hopfield Network is defined as: X X E=− si bi − si sj wi,j i (5) i,j where s is the state of a unit, b denotes the bias; w denotes the bidirectional weights and i, j are indexes of units. This energy function closely connects to the potential energy function of spin glass, as showed in Equation 4. Hopfield Network is typically applied to memorize the state of data. The weights of a network are designed or learned to make sure that the energy is minimized given the state of interest. Therefore, when another state presented to the network, while the weights are fixed, Hopfield Network can search for the states that minimize the energy and recover the state in memory. For example, in a face completion task, when some image of faces are presented to Hopfield Network (in a way that each unit of the network corresponds to each pixel of one image, and images are presented one after the other), the network can calculate the weights to minimize the energy given these faces. Later, if one image is corrupted or distorted and presented to this network again, the network is able to recover the original image by searching a configuration of states to minimize the energy starting from corrupted input presented. The term “energy” may remind people of physics. To explain how Hopfield Network works in a physics scenario will be clearer: nature uses Hopfield Network to memorize the equilibrium position of a pendulum because, in an equilibrium position, the pendulum has the lowest gravitational potential energy. Therefore, whenever a pendulum is placed, it will converge back to the equilibrium position. 4.2.3 Learning and Inference Learning of the weights of a Hopfield Network is straightforward (Gurney, 1997). The weights can be calculated as: X wi,j = (2si − 1)(2sj − 1) i,j the notations are the same as Equation 5. This learning procedure is simple, but still worth mentioning as it is an essential step of a Hopfield Network when it is applied to solve practical problems. However, we find that many online tutorials omit this step, and to make it worse, they refer the inference of states as learning/training. To remove the confusion, in this paper, similar to how terms are used in standard machine learning society, we refer the calculation of weights of a model (either from closed-form solution, or numerical solution) as “parameter learning” or “training”. We refer the process of applying an existing model with weights known onto solving a real-world problem as “inference”5 or “testing” (to decode a hidden state of data, e.g. to predict a label). The inference of Hopfield Network is also intuitive. For a state of data, the network tests that if inverting the state of one unit, whether the energy will decrease. If so, the 5. “inference” is conventionally used in such a way in machine learning society, although some statisticians may disagree with this usage. 23 Wang and Raj network will invert the state and proceed to test the next unit. This procedure is called Asynchronous update and this procedure is obviously subject to the sequential order of selection of units. A counterpart is known as Synchronous update when the network first tests for all the units and then inverts all the unit-to-invert simultaneously. Both of these methods may lead to a local optimal. Synchronous update may even result in an increasing of energy and may converge to an oscillation or loop of states. 4.2.4 Capacity One distinct disadvantage of Hopfield Network is that it cannot keep the memory very efficient because a network of N units can only store memory up to 0.15N 2 bits. While a network with N units has N 2 edges. In addition, after storing M memories (M instances of data), each connection has an integer value in range [−M, M ]. Thus, the number of bits required to store N units are N 2 log(2M + 1) (Hopfield, 1982). Therefore, we can safely draw the conclusion that although Hopfield Network is a remarkable idea that enables the network to memorize data, it is extremely inefficient in practice. As follow-ups of the invention of Hopfield Network, many works are attempted to study and increase the capacity of original Hopfield Network (Storkey, 1997; Liou and Yuan, 1999; Liou and Lin, 2006). Despite these attempts made, Hopfield Network still gradually fades out of the society. It is replaced by other models that are inspired by it. Immediately following this section, we will discuss the popular Boltzmann Machine and Restricted Boltzmann Machine and study how these models are upgraded from the initial ideas of Hopfield Network and evolve to replace it. 4.3 Boltzmann Machine Boltzmann Machine, invented by Ackley et al. (1985), is a stochastic with-hidden-unit version Hopfield Network. It got its name from Boltzmann Distribution. 4.3.1 Boltzmann Distribution Boltzmann Distribution is named after Ludwig Boltzmann and investigated extensively by (Willard, 1902). It is originally used to describe the probability distribution of particles in a system over various possible states as following: Es F (s) ∝ e− kT where s stands for the state and Es is the corresponding energy. k and T are Boltzmann’s constant and thermodynamic temperature respectively. Naturally, the ratio of two distribution is only characterized by the difference of energies, as following: r= Es2 −Es1 F (s1 ) = e kT F (s2 ) which is known as Boltzmann factor. 24 On the Origin of Deep Learning Figure 10: Illustration of Boltzmann Machine. With the introduction of hidden units (shaded nodes), the model conceptually splits into two parts: visible units and hidden units. The red dashed line is used to highlight the conceptual separation. With how the distribution is specified by the energy, the probability is defined as the term of each state divided by a normalizer, as following: Esi e− kT psi = P − Esj kT je 4.3.2 Boltzmann Machine As we mentioned previously, Boltzmann Machine is a stochastic with-hidden-unit version Hopfield Network. Figure 10 introduces how the idea of hidden units is introduced that turns a Hopfield Network into a Boltzmann Machine. In a Boltzmann Machine, only visible units are connected with data and hidden units are used to assist visible units to describe the distribution of data. Therefore, the model conceptually splits into the visible part and hidden part, while it still maintains a fully connected network among these units. “Stochastic” is introduced for Boltzmann Machine to be improved from Hopfield Network regarding leaping out of the local optimum or oscillation of states. Inspired by physics, a method to transfer state regardless current energy is introduced: Set a state to State 1 (which means the state is on) regardless of the current state with the following probability: 1 p= ∆E 1 + e− T where ∆E stands for the difference of energies when the state is on and off, i.e. ∆E = Es=1 − Es=0 . T stands for the temperature. The idea of T is inspired by a physics process that the higher the temperature is, the more likely the state will transfer6 . In addition, the probability of higher energy state transferring to lower energy state will be always greater than the reverse process7 . This idea is highly related to a very popular optimization 6. Molecules move faster when more kinetic energy is provided, which could be achieved by heating. 7. This corresponds to Zeroth Law of Thermodynamics. 25 Wang and Raj algorithm called Simulated Annealing (Khachaturyan et al., 1979; Aarts and Korst, 1988) back then, but Simulated Annealing is hardly relevant to nowadays deep learning society. Regardless of the historical importance that the term T introduces, within this section, we will assume T = 1 as a constant, for the sake of simplification. 4.3.3 Energy of Boltzmann Machine The energy function of Boltzmann Machine is defined the same as how Equation 5 is defined for Hopfield Network, except that now visible units and hidden units are noted separately, as following: X X X X X E(v, h) = − vi bi − hk bk − vi vj wij − vi hk wik − hk hl wk,l i k i,j i,k k,l where v stands for visible units, h stands for hidden units. This equation also connects back to Equation 4, except that Boltzmann Machine splits the energy function according to hidden units and visible units. Based on this energy function, the probability of a joint configuration over both visible unit the hidden unit can be defined as following: p(v, h) = P e−E(v,h) −E(m,n) m,n e The probability of visible/hidden units can be achieved by marginalizing this joint probability. For example, by marginalizing out hidden units, we can get the probability distribution of visible units: P −E(v,h) e p(v) = P h −E(m,n) m,n e which could be used to sample visible units, i.e. generating data. When Boltzmann Machine is trained to its stable state, which is called thermal equilibrium, the distribution of these probabilities p(v, h) will remain constant because the distribution of energy will be a constant. However, the probability for each visible unit or hidden unit may vary and the energy may not be at their minimum. This is related to how thermal equilibrium is defined, where the only constant factor is the distribution of each part of the system. Thermal equilibrium can be a hard concept to understand. One can imagine that pouring a cup of hot water into a bottle and then pouring a cup of cold water onto the hot water. At start, the bottle feels hot at bottom and feels cold at top and gradually the bottle feels mild as the cold water and hot water mix and heat is transferred. However, the temperature of the bottle becomes mild stably (corresponding to the distribution of p(v, h)) does not necessarily mean that the molecules cease to move (corresponding to each p(v, h)). 4.3.4 Parameter Learning The common way to train the Boltzmann machine is to determine the parameters that maximize the likelihood of the observed data. Gradient descent on the log of the likelihood 26 On the Origin of Deep Learning function is usually performed to determine the parameters. For simplicity, the following derivation is based on a single observation. First, we have the log likelihood function of visible units as X X l(v; w) = log p(v; w) = log e−Ev,h − log e−Em,n m,n h where the second term on RHS is the normalizer. Now we take the derivative of log likelihood function w.r.t w, and simplify it, we have: X ∂l(v; w) ∂E(m, n) ∂E(v, h) X p(m, n) =− p(h\|v) + ∂w ∂w ∂w m,n h ∂E(v, h) ∂E(m, n) + Ep(m,n) ∂w ∂w where E denotes expectation. Thus the gradient of the likelihood function is composed of two parts. The first part is expected gradient of the energy function with respect to the conditional distribution p(h\|v). The second part is expected gradient of the energy function with respect to the joint distribution over all variable states. However, calculating these expectations is generally infeasible for any realistically-sized model, as it involves summing over a huge number of possible states/configurations. The general approach for solving this problem is to use Markov Chain Monte Carlo (MCMC) to approximate these sums: = − Ep(h\|v) ∂l(v; w) = − < si , sj >p(hdata \|vdata ) + < si , sj >p(hmodel \|vmodel ) (6) ∂w where < · > denotes expectation. Equation 6 is the difference between the expectation value of product of states while the data is fed into visible states and the expectation of product of states while no data is fed. The first term is calculated by taking the average value of the energy function gradient when the visible and hidden units are being driven by observed data samples. In practice, this first term is generally straightforward to calculate. Calculating the second term is generally more complicated and involves running a set of Markov chains until they reach the current models equilibrium distribution, then taking the average energy function gradient based on those samples. However, this sampling procedure could be very computationally complicated, which motivates the topic in next section, the Restricted Boltzmann Machine. 4.4 Restricted Boltzmann Machine Restricted Boltzmann Machine (RBM), originally known as Harmonium when invented by Smolensky (1986), is a version of Boltzmann Machine with a restriction that there is no connections either between visible units or between hidden units. Figure 11 is an illustration of how Restricted Boltzmann Machine is achieved based on Boltzmann Machine (Figure 10): the connections between hidden units, as well as the connections between visible units are removed and the model becomes a bipartite graph. With this restriction introduced, the energy function of RBM is much simpler: X X X E(v, h) = − vi bi − hk bk − vi hk wik (7) i k 27 i,k Wang and Raj Figure 11: Illustration of Restricted Boltzmann Machine. With the restriction that there is no connections between hidden units (shaded nodes) and no connections between visible units (unshaded nodes), the Boltzmann Machine turns into a Restricted Boltzmann Machine. The model now is a a bipartite graph. 4.4.1 Contrastive Divergence RBM can still be trained in the same way as how Boltzmann Machine is trained. Since the energy function of RBM is much simpler, the sampling method used to infer the second term in Equation 6 becomes easier. Despite this relative simplicity, this learning procedure still requires a large amount of sampling steps to approximate the model distribution. To emphasize the difficulties of such a sampling mechanism, as well as to simplify followup introduction, we re-write Equation 6 with a different set of notations, as following: ∂l(v; w) = − < si , sj >p0 + < si , sj >p∞ ∂w (8) here we use p0 to denote data distribution and p∞ to denote model distribution. Other notations remain unchanged. Therefore, the difficulty of mentioned methods to learn the parameters is that it requires potentially “infinitely” many sampling steps to approximate the model distribution. Hinton (2002) overcame this issue magically, with the introduction of a method named Contrastive Divergence. Empirically, he found that one does not have to perform “infinitely” many sampling steps to converge to the model distribution, a finite k steps of sampling is enough. Therefore, Equation 8 is effectively re-written into: ∂l(v; w) = − < si , sj >p0 + < si , sj >pk ∂w Remarkably, Hinton (2002) showed that k = 1 is sufficient for the learning algorithm to work well in practice. Carreira-Perpinan and Hinton (2005) attempted to justify Contrastive Divergence in theory, but their derivation led to a negative conclusion that Contrastive Divergence is a 28 On the Origin of Deep Learning Figure 12: Illustration of Deep Belief Networks. Deep Belief Networks is not just stacking RBM together. The bottom layers (layers except the top one) do not have the bi-directional connections, but only connections top down. biased algorithm, and a finite k cannot represent the model distribution. However, their empirical results suggested that finite k can approximate the model distribution well enough, resulting a small enough bias. In addition, the algorithm works well in practice, which strengthened the idea of Contrastive Divergence. With the reasonable modeling power and a fast approximation algorithm, RBM quickly draws great attention and becomes one of the most fundamental building blocks of deep neural networks. In the following two sections, we will introduce two distinguished deep neural networks that are built based on RBM/Boltzmann Machine, namely Deep Belief Nets and Deep Boltzmann Machine. 4.5 Deep Belief Nets Deep Belief Networks is introduced by Hinton et al. (2006)8 , when he showed that RBMs can be stacked and trained in a greedy manner. Figure 12 shows the structure of a three-layer Deep Belief Networks. Different from stacking RBM, DBN only allows bi-directional connections (RBM-type connections) on the top one layer while the following bottom layers only have top-down connections. Probably a better way to understand DBN is to think it as multi-layer generative models. Despite the fact that DBN is generally described as a stacked RBM, it is quite different from putting one RBM on the top of the other. It is probably more appropriate to think DBN as a one-layer RBM with extended layers specially devoted to generating patterns of data. Therefore, the model only needs to sample for the thermal equilibrium at the topmost layer and then pass the visible states top down to generate the data. 8. This paper is generally seen as the opening of nowadays Deep Learning era, as it first introduces the possibility of training a deep neural network by layerwise training 29 Wang and Raj 4.5.1 Parameter Learning Parameter learning of a Deep Belief Network falls into two steps: the first step is layer-wise pre-training and the second step is fine-tuning. Layerwise Pre-training The success of Deep Belief Network is largely due to the introduction of the layer-wised pretraining. The idea is simple, but the reason why it works still attracts researchers. The pre-training is simply to first train the network component by component bottom up: treating the first two layers as an RBM and train it, then treat the second layer and third layer as another RBM and train for the parameters. Such an idea turns out to offer a critical support of the success of the later finetuning process. Several explanations have been attempted to explain the mechanism of pre-training: • Intuitively, pre-training is a clever way of initialization. It puts the parameter values in the appropriate range for further fine-tuning. • Bengio et al. (2007) suggested that unsupervised pre-training initializes the model to a point in parameter space which leads to a more effective optimization process, that the optimization can find a lower minimum of the empirical cost function. • Erhan et al. (2010) empirically argued for a regularization explanation, that unsupervised pretraining guides the learning towards basins of attraction of minima that support better generalization from the training data set. In addition to Deep Belief Networks, this pretraining mechanism also inspires the pretraining for many other classical models, including the autoencoders (Poultney et al., 2006; Bengio et al., 2007), Deep Boltzmann Machines (Salakhutdinov and Hinton, 2009) and some models inspired by these classical models like (Yu et al., 2010). After the pre-training is performed, fine-tuning is carried out to further optimize the network to search for the parameters that lead to a lower minimum. For Deep Belief Networks, there are two different fine tuning strategies dependent on the goals of the network. Fine Tuning for Generative Model Fine-tuning for a generative model is achieved with a contrastive version of wake-sleep algorithm (Hinton et al., 1995). This algorithm is intriguing for the reason that it is designed to interpret how the brain works. Scientists have found that sleeping is a critical process of brain function and it seems to be an inverse version of how we learn when we are awake. The wake-sleep algorithm also has two steps. In wake phase, we propagate information bottom up to adjust top-down weights for reconstructing the layer below. Sleep phase is the inverse of wake phase. We propagate the information top down to adjust bottom-up weights for reconstructing the layer above. The contrastive version of this wake-sleep algorithm is that we add one Contrastive Divergence phase between wake phase and sleep phase. The wake phase only goes up to the visible layer of the top RBM, then we sample the top RBM with Contrastive Divergence, then a sleep phase starts from the visible layer of top RBM. Fine Tuning for Discriminative Model The strategy for fine tuning a DBN as a discriminative model is to simply apply standard backpropagation to pre-trained model 30 On the Origin of Deep Learning Figure 13: Illustration of Deep Boltzmann Machine. Deep Boltzmann Machine is more like stacking RBM together. Connections between every two layers are bidirectional. since we have labels of data. However, pre-training is still necessary in spite of the generally good performance of backpropagation. 4.6 Deep Boltzmann Machine The last milestone we introduce in the family of deep generative model is Deep Boltzmann Machine introduced by Salakhutdinov and Hinton (2009). Figure 13 shows a three layer Deep Boltzmann Machine (DBM). The distinction between DBM and DBN mentioned in the previous section is that DBM allows bidirectional connections in the bottom layers. Therefore, DBM represents the idea of stacking RBMs in a much better way than DBN, although it might be clearer if DBM is named as Deep Restricted Boltzmann Machine. Due to the nature of DBM, its energy function is defined as an extension of the energy function of an RBM (Equation 7), as showed in the following: E(v, h) = − X i vi bi − N X X hn,k bn,k − n=1 k X i,k vi wik hk − N −1 X X hn,k wn,k,l hn+1,l n=1 k,l for a DBM with N hidden layers. This similarity of energy function grants the possibility of training DBM with constrative divergence. However, pre-training is typically necessary. 4.6.1 Deep Boltzmann Machine (DBM) v.s. Deep Belief Networks (DBN) As their acronyms suggest, Deep Boltzmann Machine and Deep Belief Networks have many similarities, especially from the first glance. Both of them are deep neural networks originates from the idea of Restricted Boltzmann Machine. (The name “Deep Belief Network” 31 Wang and Raj seems to indicate that it also partially originates from Bayesian Network (Krieg, 2001).) Both of them also rely on layerwise pre-training for a success of parameter learning. However, the fundamental differences between these two models are dramatic, introduced by how the connections are made between bottom layers (un-directed/bi-directed v.s. directed). The bidirectional structure of DBM grants the possibility of DBM to learn a more complex pattern of data. It also grants the possibility for the approximate inference procedure to incorporate top-down feedback in addition to an initial bottom-up pass, allowing Deep Boltzmann Machines to better propagate uncertainty about ambiguous inputs. 4.7 Deep Generative Models: Now and the Future Deep Boltzmann Machine is the last milestone we discuss in the history of generative models, but there are still much work after DBM and even more to be done in the future. Lake et al. (2015) introduces a Bayesian Program Learning framework that can simulate human learning abilities with large scale visual concepts. In addition to its performance on one-shot learning classification task, their model passes the visual Turing Test in terms of generating handwritten characters from the worlds alphabets. In other words, the generative performance of their model is indistinguishable from human’s behavior. Being not a deep neural model itself, their model outperforms several concurrent deep neural networks. Deep neural counterpart of the Bayesian Program Learning framework can be surely expected with even better performance. Conditional image generation (given part of the image) is also another interesting topic recently. The problem is usually solved by Pixel Networks (Pixel CNN (van den Oord et al., 2016) and Pixel RNN (Oord et al., 2016)). However, given a part of the image seems to simplify the generation task. Another contribution to generative models is Generative Adversarial Network (Goodfellow et al., 2014), however, GAN is still too young to be discussed in this paper. 32 On the Origin of Deep Learning 5. Convolutional Neural Networks and Vision Problems In this section, we will start to discuss a different family of models: the Convolutional Neural Network (CNN) family. Distinct from the family in the previous section, Convolutional Neural Network family mainly evolves from the knowledge of human visual cortex. Therefore, in this section, we will first introduce one of the most important reasons that account for the success of convolutional neural networks in vision problems: its bionic design to replicate human vision system. The nowadays convolutional neural networks probably originate more from the such a design rather than from the early-stage ancestors. With these background set-up, we will then briefly introduce the successful models that make themselves famous through the ImageNet Challenge (Deng et al., 2009). At last, we will present some known problems of the vision task that may guide the future research directions in vision tasks. 5.1 Visual Cortex Convolutional Neural Network is widely known as being inspired by visual cortex, however, except that some publications discuss this inspiration briefly (Poggio and Serre, 2013; Cox and Dean, 2014), few resources present this inspiration thoroughly. In this section, we focus on the discussion about basics on visual cortex (Hubel and Wiesel, 1959), which lays the ground for further study in Convolutional Neural Networks. The visual cortex of the brain, located in the occipital lobe which is located at the back of the skull, is a part of the cerebral cortex that plays an important role in processing visual information. Visual information coming from the eye, goes through a series of brain structures and reaches the visual cortex. The parts of the visual cortex that receive the sensory inputs is known as the primary visual cortex, also known as area V1. Visual information is further managed by extrastriate areas, including visual areas two (V2) and four (V4). There are also other visual areas (V3, V5, and V6), but in this paper, we primarily focus on the visual areas that are related to object recognition, which is known as ventral stream and consists of areas V1, V2, V4 and inferior temporal gyrus, which is one of the higher levels of the ventral stream of visual processing, associated with the representation of complex object features, such as global shape, like face perception (Haxby et al., 2000). Figure 14 is an illustration of the ventral stream of the visual cortex. It shows the information process procedure from the retina which receives the image information and passes all the way to inferior temporal gyrus. For each component: • Retina converts the light energy that comes from the rays bouncing off of an object into chemical energy. This chemical energy is then converted into action potentials that are transferred onto primary visual cortex. (In fact, there are several other brain structures involved between retina and V1, but we omit these structures for simplicity9 .) 9. We deliberately discuss the components that have connections with established technologies in convolutional neural network, one who is interested in developing more powerful models is encouraged to investigate other components. 33 Wang and Raj Figure 14: A brief illustration of ventral stream of the visual cortex in human vision system. It consists of primary visual cortex (V1), visual areas (V2 and V4) and inferior temporal gyrus. • Primary visual cortex (V1) mainly fulfills the task of edge detection, where an edge is an area with strongest local contrast in the visual signals. • V2, also known as secondary visual cortex, is the first region within the visual association area. It receives strong feedforward connections from V1 and sends strong connections to later areas. In V2, cells are tuned to extract mainly simple properties of the visual signals such as orientation, spatial frequency, and colour, and a few more complex properties. • V4 fulfills the functions including detecting object features of intermediate complexity, like simple geometric shapes, in addition to orientation, spatial frequency, and color. V4 is also shown with strong attentional modulation (Moran and Desimone, 1985). V4 also receives direct input from V1. • Inferior temporal gyrus (TI) is responsible for identifying the object based on the color and form of the object and comparing that processed information to stored memories of objects to identify that object (Kolb et al., 2014). In other words, IT performs the semantic level tasks, like face recognition. Many of the descriptions of functions about visual cortex should revive a recollection of convolutional neural networks for the readers that have been exposed to some relevant technical literature. Later in this section, we will discuss more details about convolutional neural networks, which will help build explicit connections. Even for readers that barely 34 On the Origin of Deep Learning have knowledge in convolutional neural networks, this hierarchical structure of visual cortex should immediately ring a bell about neural networks. Besides convolutional neural networks, visual cortex has been inspiring the works in computer vision for a long time. For example, Li (1998) built a neural model inspired by the primary visual cortex (V1). In another granularity, Serre et al. (2005) introduced a system with feature detections inspired from the visual cortex. De Ladurantaye et al. (2012) published a book describing the models of information processing in the visual cortex. Poggio and Serre (2013) conducted a more comprehensive survey on the relevant topic, but they didn’t focus on any particular subject in detail in their survey. In this section, we discuss the connections between visual cortex and convolutional neural networks in details. We will begin with Neocogitron, which borrows some ideas from visual cortex and later inspires convolutional neural network. 5.2 Neocogitron and Visual Cortex Neocogitron, proposed by Fukushima (1980), is generally seen as the model that inspires Convolutional Neural Networks on the computation side. It is a neural network that consists of two different kinds of layers (S-layer as feature extractor and C-layer as structured connections to organize the extracted features.) S-layer consists of a number of S-cells that are inspired by the cell in primary visual cortex. It serves as a feature extractor. Each S-cell can be ideally trained to be responsive to a particular feature presented in its receptive field. Generally, local features such as edges in particular orientations are extracted in lower layers while global features are extracted in higher layers. This structure highly resembles how human conceive objects. C-layer resembles complex cell in the higher pathway of visual cortex. It is mainly introduced for shift invariant property of features extracted by S-layer. 5.2.1 Parameter Learning During parameter learning process, only the parameters of S-layer are updated. Neocogitron can also be trained unsupervisedly, for a good feature extractor out of S-layers. The training process for S-layer is very similar to Hebbian Learning rule, which strengthens the connections between S-layer and C-layer for whichever S-cell shows the strongest response. This training mechanism also introduces the problem Hebbian Learning rule introduces, that the strength of connections will saturate (since it keeps increasing). The solution was also introduced by Fukushima (1980), which was introduced with the name “inhibitory cell”. It performed the function as a normalization to avoid the problem. 5.3 Convolutional Neural Network and Visual Cortex Now we proceed from Neocogitron to Convolutional Neural Network. First, we will introduce the building components: convolutional layer and subsampling layer. Then we assemble these components to present Convolutional Neural Network, using LeNet as an example. 35 Wang and Raj Figure 15: A simple illustration of two dimension convolution operation. 5.3.1 Convolution Operation Convolution operation is strictly just a mathematical operation, which should be treated equally with other operations like addition or multiplication and should not be discussed particularly in a machine learning literature. However, we still discuss it here for completeness and for the readers who may not be familiar with it. Convolution is a mathematical operation on two functions (e.g. f and g) and produces a third function h, which is an integral that expresses the amount of overlap of one function (f ) as it is shifted over the other function (g). It is described formally as the following: Z ∞ f (τ )g(t − τ )dτ h(t) = −∞ and denoted as h = f ? g. Convolutional neural network typically works with two-dimensional convolution operation, which could be summarized in Figure 15. As showed in Figure 15, the leftmost matrix is the input matrix. The middle one is usually called a kernel matrix. Convolution is applied to these matrices and the result is showed as the rightmost matrix. The convolution process is an element-wise product followed by a sum, as showed in the example. When the left upper 3×3 matrix is convoluted with the kernel, the result is 29. Then we slide the target 3 × 3 matrix one column right, convoluted with the kernel and get the result 12. We keep sliding and record the results as a matrix. Because the kernel is 3 × 3, every target matrix is 3 × 3, thus, every 3 × 3 matrix is convoluted to one digit and the whole 5 × 5 matrix is shrunk into 3 × 3 matrix. (Because 5 − (3 − 1) = 3. The first 3 means the size of the kernel matrix. ) One should realize that convolution is locally shift invariant, which means that for many different combinations of how the nine numbers in the upper 3 × 3 matrix are placed, the convoluted result will be 29. This invariant property plays a critical role in vision problem because that in an ideal case, the recognition result should not be changed due to shift or rotation of features. This critical property is used to be solved elegantly by Lowe (1999); Bay et al. (2006), but convolutional neural network brought the performance up to a new level. 5.3.2 Connection between CNN and Visual Cortex With the ideas about two dimension convolution, we further discuss how convolution is a useful operation that can simulate the tasks performed by visual cortex. 36 On the Origin of Deep Learning (a) Identity kernel (b) Edge detection kernel (c) Blur kernel (d) Sharpen kernel (e) Lighten kernel (f) Darken kernel (g) Random kernel 1 (h) Random kernel 2 Figure 16: Convolutional kernels example. Different kernels applied to the same image will result in differently processed images. Note that there is a 91 divisor applied to these kernels. The convolution operation is usually known as kernels. By different choices of kernels, different operations of the images could be achieved. Operations are typically including identity, edge detection, blur, sharpening etc. By introducing random matrices as convolution operator, some interesting properties might be discovered. Figure 16 is an illustration of some example kernels that are applied to the same figure. One can see that different kernels can be applied to fulfill different tasks. Random kernels can also be applied to transform the image into some interesting outcomes. Figure 16 (b) shows that edge detection, which is one of the central tasks of primary visual cortex, can be fulfilled by a clever choice of kernels. Furthermore, clever selection of kernels can lead us to a success replication of visual cortex. As a result, learning a meaningful convolutional kernel (i.e. parameter learning) is one of the central tasks in convolutional neural networks when applied to vision tasks. This also explains that why 37 Wang and Raj Figure 17: An illustration of LeNet, where Conv stands for convolutional layer and Sampling stands for SubSampling Layer. many well-trained popular models can usually perform well in other tasks with only limited fine-tuning process: the kernels have been well trained and can be universally applicable. With the understanding of the essential role convolution operation plays in vision tasks, we proceed to investigate some major milestones along the way. 5.4 The Pioneer of Convolutional Neural Networks: LeNet This section is devoted to a model that is widely recognized as the first convolutional neural network: LeNet, invented by Le Cun et al. (1990) (further made popular with (LeCun et al., 1998a)). It is inspired from the Neocogitron. In this section, we will introduce convolutional neural network via introducing LeNet. Figure 17 shows an illustration of the architecture of LeNet. It consists of two pairs of Convolutional Layer and Subsampling Layer and is further connected with fully connected layer and an RBF layer for classification. 5.4.1 Convolutional Layer A convolutional layer is primarily a layer that performs convolution operation. As we have discussed previously, a clever selection of convolution kernel can effectively simulate the task of visual cortex. Convolutional layer introduces another operation after convolution to assist the simulation to be more successful: the non-linearity transform. Considering a ReLU (Nair and Hinton, 2010) non-linearity transform, which is defined as following: f (x) = max(0, x) which is a transform that removes the negative part of the input, resulting in a clearer contrast of meaningful features as opposed to other side product the kernel produces. Therefore, this non-linearity grants the convolution more power in extracting useful features and allows it to simulate the functions of visual cortex more closely. 38 On the Origin of Deep Learning 5.4.2 Subsampling Layer Subsampling Layer performs a simpler task. It only samples one input out every region it looks into. Some different strategies of sampling can be considered, like max-pooling (taking the maximum value of the input), average-pooling (taking the averaged value of input) or even probabilistic pooling (taking a random one.) (Lee et al., 2009). Sampling turns the input representations into smaller and more manageable embeddings. More importantly, sampling makes the network invariant to small transformations, distortions, and translations in the input image. A small distortion in the input will not change the outcome of pooling since we take the maximum/average value in a local neighborhood. 5.4.3 LeNet With the two most important components introduced, we can stack them together to assemble a convolutional neural network. Following the recipe of Figure 17, we will end up with the famous LeNet. LeNet is known as its ability to classify digits and can handle a variety of different problems of digits including variances in position and scale, rotation and squeezing of digits, and even different stroke width of the digit. Meanwhile, with the introduction of LeNet, LeCun et al. (1998b) also introduces the MNIST database, which later becomes the standard benchmark in digit recognition field. 5.5 Milestones in ImageNet Challenge With the success of LeNet, Convolutional Neural Network has been shown with great potential in solving vision tasks. These potentials have attracted a large number of researchers aiming to solve vision task regarding object recognition in CIFAR classification (Krizhevsky and Hinton, 2009) and ImageNet challenge (Russakovsky et al., 2015). Along with this path, several superstar milestones have attracted great attentions and has been applied to other fields with good performance. In this section, we will briefly discuss these models. 5.5.1 AlexNet While LeNet is the one that starts the era of convolutional neural networks, AlexNet, invented by Krizhevsky et al. (2012), is the one that starts the era of CNN used for ImageNet classification. AlexNet is the first evidence that CNN can perform well on this historically difficult ImageNet dataset and it performs so well that leads the society into a competition of developing CNNs. The success of AlexNet is not only due to this unique design of architecture but also due to the clever mechanism of training. To avoid the computationally expensive training process, AlexNet has been split into two streams and trained on two GPUs. It also used data augmentation techniques that consist of image translations, horizontal reflections, and patch extractions. The recipe of AlexNet is shown in Figure 18. However, rarely any lessons can be learned from the architecture of AlexNet despite its remarkable performance. Even more unfortunately, the fact that this particular architecture of AlexNet does not have a wellgrounded theoretical support pushes many researchers to blindly burn computing resources 39 Wang and Raj Figure 18: An illustration of AlexNet to test for a new architecture. Many models have been introduced during this period, but only a few may be worth mentioning in the future. 5.5.2 VGG In the blind competition of exploring different architectures, Simonyan and Zisserman (2014) showed that simplicity is a promising direction with a model named VGG. Although VGG is deeper (19 layer) than other models around that time, the architecture is extremely simplified: all the layers are 3 × 3 convolutional layer with a 2 × 2 pooling layer. This simple usage of convolutional layer simulates a larger filter while keeping the benefits of smaller filter sizes, because the combination of two 3×3 convolutional layers has an effective receptive field of a 5 × 5 convolutional layer, but with fewer parameters. The spatial size of the input volumes at each layer will decrease as a result of the convolutional and pooling layers, but the depth of the volumes increases because of the increased number of filters (in VGG, the number of filters doubles after each pooling layer). This behavior reinforces the idea of VGG to shrink spatial dimensions, but grow depth. VGG is not the winner of the ImageNet competition of that year (The winner is GoogLeNet invented by Szegedy et al. (2015)). GoogLeNet introduced several important concepts like Inception module and the concept later used by R-CNN (Girshick et al., 2014; Girshick, 2015; Ren et al., 2015), but the arbitrary/creative design of architecture barely contribute more than what VGG does to the society, especially considering that Residual Net, following the path of VGG, won the ImageNet challenge in an unprecedented level. 5.5.3 Residual Net Residual Net (ResNet) is a 152 layer network, which was ten times deeper than what was usually seen during the time when it was invented by He et al. (2015). Following the path VGG introduces, ResNet explores deeper structure with simple layer. However, naively 40 On the Origin of Deep Learning Figure 19: An illustration of Residual Block of ResNet increasing the number of layers will only result in worse results, for both training cases and testing cases (He et al., 2015). The breakthrough ResNet introduces, which allows ResNet to be substantially deeper than previous networks, is called Residual Block. The idea behind a Residual Block is that some input of a certain layer (denoted as x) can be passed to the component two layers later either following the traditional path which involves convolutional layers and ReLU transform succession (we denote the result as f (x)), or going through an express way that directly passes x there. As a result, the input to the component two layers later is f (x) + x instead of what is typically seen as f (x). The idea of Residual Block is illustrated in Figure 19. In a complementary work, He et al. (2016) validated that residual blocks are essential for propagating information smoothly, therefore simplifies the optimization. They also extended the ResNet to a 1000-layer version with success on CIFAR data set. Another interesting perspective of ResNet is provided by (Veit et al., 2016). They showed that ResNet behave behaves like ensemble of shallow networks: the express way introduced allows ResNet to perform as a collection of independent networks, each network is significantly shallower than the integrated ResNet itself. This also explains why gradient can be passed through the ultra-deep architecture without being vanished. (We will talk more about vanishing gradient problem when we discuss recurrent neural network in the next section.) Another work, which is not directly relevant to ResNet, but may help to understand it, is conducted by Hariharan et al. (2015). They showed that features from lower layers are informative in addition to what can be summarized from the final layer. ResNet is still not completely vacant from clever designs. The number of layers in the whole network and the number of layers that Residual Block allows identity to bypass are still choices that require experimental validations. Nonetheless, to some extent, ResNet has shown that critical reasoning can help the development of CNN better than blind 41 Wang and Raj experimental trails. In addition, the idea of Residual Block has been found in the actual visual cortex (In the ventral stream of the visual cortex, V4 can directly accept signals from primary visual cortex), although ResNet is not designed according to this in the first place. With the introduction of these state-of-the-art neural models that are successful in these challenges, Canziani et al. (2016) conducted a comprehensive experimental study comparing these models. Upon comparison, they showed that there is still room for improvement on fully connected layers that show strong inefficiencies for smaller batches of images. 5.6 Challenges and Chances for Fundamental Vision Problems ResNet is not the end of the story. New models and techniques appear every day to push the limit of CNNs further. For example, Zhang et al. (2016b) took a step further and put Residual Block inside Residual Block. Zagoruyko and Komodakis (2016) attempted to decrease the depth of network by increasing the width. However, incremental works of this kind are not in the scope of this paper. We would like to end the story of Convolutional Neural Networks with some of the current challenges of fundamental vision problems that may not able to be solved naively by investigation of machine learning techniques. 5.6.1 Network Property and Vision Blindness Spot Convolutional Neural Networks have reached to an unprecedented accuracy in object detection. However, it may still be far from industry reliable application due to some intriguing properties found by Szegedy et al. (2013). Szegedy et al. (2013) showed that they could force a deep learning model to misclassify an image simply by adding perturbations to that image. More importantly, these perturbations may not even be observed by naked human eyes. In other words, two objects that look almost the same to human, may be recognized as different objects by a well-trained neural network (for example, AlexNet). They have also shown that this property is more likely to be a modeling problem, in contrast to problems raised by insufficient training. On the other hand, Nguyen et al. (2015) showed that they could generate patterns that convey almost no information to human, being recognized as some objects by neural networks with high confidence (sometimes more than 99%). Since neural networks are typically forced to make a prediction, it is not surprising to see a network classify a meaningless patter into something, however, this high confidence may indicate that the fundamental differences between how neural networks and human learn to know this world. Figure 20 shows some examples from the aforementioned two works. With construction, we can show that the neural networks may misclassify an object, which should be easily recognized by the human, to something unusual. On the other hand, a neural network may also classify some weird patterns, which are not believed to be objects by the human, to something we are familiar with. Both of these properties may restrict the usage of deep learning to real world applications when a reliable prediction is necessary. Even without these examples, one may also realize that the reliable prediction of neural networks could be an issue due to the fundamental property of a matrix: the existence of null space. As long as the perturbation happens within the null space of a matrix, one may be able to alter an image dramatically while the neural network still makes the 42 On the Origin of Deep Learning (a) (b) (c) (d) (e) (f) (g) (h) Figure 20: Illustrations of some mistakes of neural networks. (a)-(d) (from (Szegedy et al., 2013)) are adversarial images that are generated based on original images. The differences between these and the original ones are un-observable by naked eye, but the neural network can successfully classify original ones but fail adversarial ones. (e)-(h) (from (Nguyen et al., 2015)) are patterns that are generated. A neural network classify them into (e) school bus, (f) guitar, (g) peacock and (h) Pekinese respectively. misclassification with high confidence. Null space works like a blind spot to a matrix and changes within null space are never sensible to the corresponding matrix. This blind spot should not discourage the promising future of neural networks. On the contrary, it makes the convolutional neural network resemble the human vision system in a deeper level. In the human vision system, blind spots (Gregory and Cavanagh, 2011) also exist (Wandell, 1995). Interesting work might be seen about linking the flaws of human vision system to the defects of neural networks and helping to overcome these defects in the future. 5.6.2 Human Labeling Preference At the very last, we present some of the misclassified images of ResNet on ImageNet Challenge. Hopefully, some of these examples could inspire some new methodologies invented for the fundamental vision problem. Figure 21 shows some misclassified images of ResNet when applied to ImageNet Challenge. These labels, provided by human effort, are very unexpected even to many other humans. Therefore, the 3.6% error rate of ResNet (a general human usually predicts with error rate 5%-10%) is probably hitting the limit since the labeling preference of an annotator is harder to predict than the actual labels. For example, Figure 21 (a),(b),(h) are labeled as a tiny part of the image, while there are more important contents expressed by the image. On the other hand, Figure 21 (d) (e) are annotated as the background of the image while that image is obviously centering on other object. 43 Wang and Raj (a) flute (b) guinea pig (c) wig (d) seashore (e) alp (f) screwdriver (g) comic book (h) sunglass Figure 21: Some failed images of ImageNet classification by ResNet and the primary label associated with the image. To further improve the performance ResNet reached, one direction might be to modeling the annotators’ labeling preference. One assumption could be that annotators prefer to label an image to make it distinguishable. Some established work to modeling human factors (Wilson et al., 2015) could be helpful. However, the more important question is that whether it is worth optimizing the model to increase the testing results on ImageNet dataset, since remaining misclassifications may not be a result of the incompetency of the model, but problems of annotations. The introduction of other data sets, like COCO (Lin et al., 2014), Flickr (Plummer et al., 2015), and VisualGenome (Krishna et al., 2016) may open a new era of vision problems with more competitive challenges. However, the fundamental problems and experiences that this section introduces should never be forgotten. 44 On the Origin of Deep Learning 6. Time Series Data and Recurrent Networks In this section, we will start to discuss a new family of deep learning models that have attracted many attentions, especially for the tasks on time series data, or sequential data. The Recurrent Neural Network (RNN) is a class of neural network whose connections of units form a directed cycle; this nature grants its ability to work with temporal data. It has also been discussed in literature like (Grossberg, 2013) and (Lipton et al., 2015). In this paper, we will continue to offer complementary views to other surveys with an emphasis on the evolutionary history of the milestone models and aim to provide insights into the future direction of coming models. 6.1 Recurrent Neural Network: Jordan Network and Elman Network As we have discussed previously, Hopfield Network is widely recognized as a recurrent neural network, although its formalization is distinctly different from how recurrent neural network is defined nowadays. Therefore, despite that other literature tend to begin the discussion of RNN with Hopfield Network, we will not treat it as a member of RNN family to avoid unnecessary confusion. The modern definition of “recurrent” is initially introduced by Jordan (1986) as: If a network has one or more cycles, that is, if it is possible to follow a path from a unit back to itself, then the network is referred to as recurrent. A nonrecurrent network has no cycles. His model in (Jordan, 1986) is later referred to as Jordan Network. For a simple neural network with one hidden layer, with input denoted as X, weights of hidden layer denoted as wh and weights of output layer denoted as wy , weights of recurrent computation denoted as wr , hidden representation denoted as h and output denoted as y, Jordan Network can be formulated as ht = σ(Wh X + Wr y t−1 ) y = σ(Wy ht ) A few years later, another RNN was introduced by Elman (1990), when he formalized the recurrent structure slightly differently. Later, his network is known as Elman Network. Elman network is formalized as following: ht = σ(Wh X + Wr ht−1 ) y = σ(Wy ht ) The only difference is that whether the information of previous time step is provided by previous output or previous hidden layer. This difference is further illustrated in Figure 22. The difference is illustrated to respect the historical contribution of these works. One may notice that there is no fundamental difference between these two structures since yt = Wy ht , therefore, the only difference lies in the choice of Wr . (Originally, Elman only introduces his network with Wr = I, but more general cases could be derived from there.) Nevertheless, the step from Jordan Network to Elman Network is still remarkable as it introduces the possibility of passing information from hidden layers, which significantly improve the flexibility of structure design in later work. 45 Wang and Raj (a) Structure of Jordan Network (b) Structure of Elman Network Figure 22: The difference of recurrent structure from Jordan Network and Elman Network. 6.1.1 Backpropagation through Time The recurrent structure makes traditional backpropagation infeasible because of that with the recurrent structure, there is not an end point where the backpropagation can stop. Intuitively, one solution is to unfold the recurrent structure and expand it as a feedforward neural network with certain time steps and then apply traditional backpropagation onto this unfolded neural network. This solution is known as Backpropagation through Time (BPTT), independently invented by several researchers including (Robinson and Fallside, 1987; Werbos, 1988; Mozer, 1989) However, as recurrent neural network usually has a more complex cost surface, naive backpropagation may not work well. Later in this paper, we will see that the recurrent structure introduces some critical problems, for example, the vanishing gradient problem, which makes optimization for RNN a great challenge in the society. 6.2 Bidirectional Recurrent Neural Network If we unfold an RNN, then we can get the structure of a feedforward neural network with infinite depth. Therefore, we can build a conceptual connection between RNN and feedforward network with infinite layers. Then since through the neural network history, bidirectional neural networks have been playing important roles (like Hopfield Network, RBM, DBM), a follow-up question is that what recurrent structures that correspond to the infinite layer of bidirectional models are. The answer is Bidirectional Recurrent Neural Network. 46 On the Origin of Deep Learning Figure 23: The unfolded structured of BRNN. The temporal order is from left to right. Hidden layer 1 is unfolded in the standard way of an RNN. Hidden layer 2 is unfolded to simulate the reverse connection. Bidirectional Recurrent Neural Network (BRNN) was invented by Schuster and Paliwal (1997) with the goal to introduce a structure that was unfolded to be a bidirectional neural network. Therefore, when it is applied to time series data, not only the information can be passed following the natural temporal sequences, but the further information can also reversely provide knowledge to previous time steps. Figure 23 shows the unfolded structure of a BRNN. Hidden layer 1 is unfolded in the standard way of an RNN. Hidden layer 2 is unfolded to simulate the reverse connection. Transparency (in Figure 23) is applied to emphasize that unfolding an RNN is only a concept that is used for illustration purpose. The actual model handles data from different time steps with the same single model. BRNN is formulated as following: ht1 = σ(Wh1 X + Wr1 ht−1 1 ) ht2 = σ(Wh2 X + Wr2 ht+1 2 ) y = σ(Wy1 ht1 + Wy2 ht2 ) where the subscript 1 and 2 denote the variables associated with hidden layer 1 and 2 respectively. With the introduction of “recurrent” connections back from the future, Backpropagation through Time is no longer directly feasible. The solution is to treat this model as a combination of two RNNs: a standard one and a reverse one, then apply BPTT onto each of them. Weights are updated simultaneously once two gradients are computed. 6.3 Long Short-Term Memory Another breakthrough in RNN family was introduced in the same year as BRNN. Hochreiter and Schmidhuber (1997) introduced a new neuron for RNN family, named Long Short-Term Memory (LSTM). When it was invented, the term “LSTM” is used to refer the algorithm 47 Wang and Raj that is designed to overcome vanishing gradient problem, with the help of a special designed memory cell. Nowadays, “LSTM” is widely used to denote any recurrent network that with that memory cell, which is nowadays referred as an LSTM cell. LSTM was introduced to overcome the problem that RNNs cannot long term dependencies (Bengio et al., 1994). To overcome this issue, it requires the specially designed memory cell, as illustrated in Figure 24 (a). LSTM consists of several critical components. • states: values that are used to offer the information for output. ? input data: it is denoted as x. ? hidden state: values of previous hidden layer. This is the same as traditional RNN. It is denoted as h. ? input state: values that are (linear) combination of hidden state and input of current time step. It is denoted as i, and we have: it = σ(Wix xt + Wih ht−1 ) (9) ? internal state: Values that serve as “memory”. It is denoted as m • gates: values that are used to decide the information flow of states. ? input gate: it decides whether input state enters internal state. It is denoted as g, and we have: g t = σ(Wgi it ) (10) ? forget gate: it decides whether internal state forgets the previous internal state. It is denoted as f , and we have: f t = σ(Wf i it ) (11) ? output gate: it decides whether internal state passes its value to output and hidden state of next time step. It is denoted as o and we have: ot = σ(Woi it ) (12) Finally, considering how gates decide the information flow of states, we have the last two equations to complete the formulation of LSTM: mt =g t it + f t mt−1 ht =ot mt (13) (14) where denotes element-wise product. Figure 24 describes the details about how LSTM cell works. Figure 24 (b) shows that how the input state is constructed, as described in Equation 9. Figure 24 (c) shows how 48 On the Origin of Deep Learning (a) LSTM “memory” cell (b) Input data and previous hidden state form into input state (c) Calculating input gate and forget gate (d) Calculating output gate (e) Update internal state (f) Output and update hidden state Figure 24: The LSTM cell and its detailed functions. 49 Wang and Raj input gate and forget gate are computed, as described in Equation 10 and Equation 11. Figure 24 (d) shows how output gate is computed, as described in Equation 12. Figure 24 (e) shows how internal state is updated, as described in Equation 13. Figure 24 (f) shows how output and hidden state are updated, as described in Equation 14. All the weights are parameters that need to be learned during training. Therefore, theoretically, LSTM can learn to memorize long time dependency if necessary and can learn to forget the past when necessary, making itself a powerful model. With this important theoretical guarantee, many works have been attempted to improve LSTM. For example, Gers and Schmidhuber (2000) added a peephole connection that allows the gate to use information from the internal state. Cho et al. (2014) introduced the Gated Recurrent Unit, known as GRU, which simplified LSTM by merging internal state and hidden state into one state, and merging forget gate and input gate into a simple update gate. Integrating LSTM cell into bidirectional RNN is also an intuitive follow-up to look into (Graves et al., 2013). Interestingly, despite the novel LSTM variants proposed now and then, Greff et al. (2015) conducted a large-scale experiment investigating the performance of LSTMs and got the conclusion that none of the variants can improve upon the standard LSTM architecture significantly. Probably, the improvement of LSTM is in another direction rather than updating the structure inside a cell. Attention models seem to be a direction to go. 6.4 Attention Models Attention Models are loosely based on a bionic design to simulate the behavior of human vision attention mechanism: when humans look at an image, we do not scan it bit by bit or stare at the whole image, but we focus on some major part of it and gradually build the context after capturing the gist. Attention mechanisms were first discussed by Larochelle and Hinton (2010) and Denil et al. (2012). The attention models mostly refer to the models that were introduced in (Bahdanau et al., 2014) for machine translation and soon applied to many different domains like (Chorowski et al., 2015) for speech recognition and (Xu et al., 2015) for image caption generation. Attention models are mostly used for sequence output prediction. Instead of seeing the whole sequential data and make one single prediction (for example, language model), the model needs to make a sequential prediction for the sequential input for tasks like machine translation or image caption generation. Therefore, the attention model is mostly used to answer the question on where to pay attention to based on previously predicted labels or hidden states. The output sequence may not have to be linked one-to-one to the input sequence, and the input data may not even be a sequence. Therefore, usually, an encoder-decoder framework (Cho et al., 2015) is necessary. The encoder is used to encode the data into representations and decoder is used to make sequential predictions. Attention mechanism is used to locate a region of the representation for predicting the label in current time step. Figure 25 shows a basic attention model under encoder-decoder network structure. The representation encoder encodes is all accessible to attention model, and attention model only selects some regions to pass onto the LSTM cell for further usage of prediction making. 50 On the Origin of Deep Learning Figure 25: The unfolded structured of an attention model. Transparency is used to show that unfolding is only conceptual. The representation encoder learns are all available to the decoder across all time steps. Attention module only selects some to pass onto LSTM cell for prediction. Therefore, all the magic of attention models is about how this attention module in Figure 25 helps to localize the informative representations. To formalize how it works, we use r to denote the encoded representation (there is a total of M regions of representation), use h to denote hidden states of LSTM cell. Then, the attention module can generate the unscaled weights for ith region of the encoded representation as: βit = f (ht−1 , r, {αjt−1 }M j=1 ) where αjt−1 is the attention weights computed at the previous time step, and can be computed at current time step as a simple softmax function: exp(βit ) αit = PM t j exp(βj ) Therefore, we can further use the weights α to reweight the representation r for prediction. There are two ways for the representation to be reweighted: • Soft attention: The result is a simple weighted sum of the context vectors such that: t r = M X αjt cj j • Hard attention: The model is forced to make a hard decision by only localizing one region: sampling one region out following multinoulli distribution. 51 Wang and Raj (a) Deep input architecture (b) Deep recurrent architecture (c) Deep output architecture Figure 26: Three different formulations of deep recurrent neural network. One problem about hard attention is that sampling out of multinoulli distribution is not differentiable. Therefore, the gradient based method can be hardly applied. Variational methods (Ba et al., 2014) or policy gradient based method (Sutton et al., 1999) can be considered. 6.5 Deep RNNs and the future of RNNs In this very last section of the evolutionary path of RNN family, we will visit some ideas that have not been fully explored. 6.5.1 Deep Recurrent Neural Network Although recurrent neural network suffers many issues that deep neural network has because of the recurrent connections, current RNNs are still not deep models regarding representation learning compared to models in other families. Pascanu et al. (2013a) formalizes the idea of constructing deep RNNs by extending current RNNs. Figure 26 shows three different directions to construct a deep recurrent neural network by increasing the layers of the input component (Figure 26 (a)), recurrent component (Figure 26 (b)) and output component (Figure 26 (c)) respectively. 52 On the Origin of Deep Learning 6.5.2 The Future of RNNs RNNs have been improved in a variety of different ways, like assembling the pieces together with Conditional Random Field (Yang et al., 2016), and together with CNN components (Ma and Hovy, 2016). In addition, convolutional operation can be directly built into LSTM, resulting ConvLSTM (Xingjian et al., 2015), and then this ConvLSTM can be also connected with a variety of different components (De Brabandere et al., 2016; Kalchbrenner et al., 2016). One of the most fundamental problems of training RNNs is the vanishing/exploding gradient problem, introduced in detail in (Bengio et al., 1994). The problem basically states that for traditional activation functions, the gradient is bounded. When gradients are computed by backpropagation following chain rule, the error signal decreases exponentially within the time steps the BPTT can trace back, so the long-term dependency is lost. LSTM and ReLU are known to be good solutions for vanishing/exploding gradient problem. However, these solutions introduce ways to bypass this problem with clever design, instead of solving it fundamentally. While these methods work well practically, the fundamental problem for a general RNN is still to be solved. Pascanu et al. (2013b) attempted some solutions, but there are still more to be done. 53 Wang and Raj 7. Optimization of Neural Networks The primary focus of this paper is deep learning models. However, optimization is an inevitable topic in the development history of deep learning models. In this section, we will briefly revisit the major topics of optimization of neural networks. During our introduction of the models, some algorithms have been discussed along with the models. Here, we will only discuss the remaining methods that have not been mentioned previously. 7.1 Gradient Methods Despite the fact that neural networks have been developed for over fifty years, the optimization of neural networks still heavily rely on gradient descent methods within the algorithm of backpropagation. This paper does not intend to introduce the classical backpropagation, gradient descent method and its stochastic version and batch version and simple techniques like momentum method, but starts right after these topics. Therefore, the discussion of following gradient methods starts from the vanilla gradient descent as following: θt+1 = θt − η5tθ where 5θ is the gradient of the parameter θ, η is a hyperparameter, usually known as learning rate. 7.1.1 Rprop Rprop was introduced by Riedmiller and Braun (1993). It is a unique method even studied back today as it does not fully utilize the information of gradient, but only considers the sign of it. In other words, it updates the parameters following: θt+1 = θt − ηI(5tθ > 0) + ηI(5tθ < 0) where I(·) stands for an indicator function. This unique formalization allows the gradient method to overcome some cost curvatures that may not be easily solved with today’s dominant methods. This two-decade-old method may be worth some further study these days. 7.1.2 AdaGrad AdaGrad was introduced by Duchi et al. (2011). It follows the idea of introducing an adaptive learning rate mechanism that assigns higher learning rate to the parameters that have been updated more mildly and assigns lower learning rate to the parameters that have been updated dramatically. The measure of the degree of the update applied is the `2 norm of historical gradients, S t = \|\|51θ , 52θ , ... 5tθ \|\|22 , therefore we have the update rule as following: θt+1 = θt − St where is small term to avoid η divided by zero. 54 η 5t + θ On the Origin of Deep Learning AdaGrad has been showed with great improvement of robustness upon traditional gradient method (Dean et al., 2012). However, the problem is that as `2 norm accumulates, the fraction of η over `2 norm decays to a substantial small term. 7.1.3 AdaDelta AdaDelta is an extension of AdaGrad that aims to reducing the decaying rate of learning rate, proposed in (Zeiler, 2012). Instead of accumulating the gradients of each time step as in AdaGrad, AdaDelta re-weights previously accumulation before adding current term onto previously accumulated result, resulting in: (S t )2 = β(S t−1 )2 + (1 − β)(5tθ )2 where β is the weight for re-weighting. Then the update rule is the same as AdaGrad: θt+1 = θt − St η 5t + θ which is almost the same as another famous gradient variant named RMSprop10 . 7.1.4 Adam Adam stands for Adaptive Moment Estimation, proposed in (Kingma and Ba, 2014). Adam is like a combination momentum method and AdaGrad method, but each component are re-weighted at time step t. Formally, at time step t, we have: ∆tθ =α∆t−1 + (1 − α)5tθ θ (S t )2 =β(S t−1 )2 + (1 − β)(5tθ )2 η θt+1 =θt − t ∆t S + θ All these modern gradient variants have been published with a promising claim that is helpful to improve the convergence rate of previous methods. Empirically, these methods seem to be indeed helpful, however, in many cases, a good choice of these methods seems only to benefit to a limited extent. 7.2 Dropout Dropout was introduced in (Hinton et al., 2012; Srivastava et al., 2014). The technique soon got influential, not only because of its good performance but also because of its simplicity of implementation. The idea is very simple: randomly dropping out some of the units while training. More formally: on each training case, each hidden unit is randomly omitted from the network with a probability of p. As suggested by Hinton et al. (2012), Dropout can be seen as an efficient way to perform model averaging across a large number of different neural networks, where overfitting can be avoided with much less cost of computation. 10. It seems this method never gets published, the resources all trace back to Hinton’s slides at http://www.cs.toronto.edu/t̃ijmen/csc321/slides/lecture slides lec6.pdf 55 Wang and Raj Because of the actual performance it introduces, Dropout soon became very popular upon its introduction, a lot of work has attempted to understand its mechanism in different perspectives, including (Baldi and Sadowski, 2013; Cho, 2013; Ma et al., 2016). It has also been applied to train other models, like SVM (Chen et al., 2014). 7.3 Batch Normalization and Layer Normalization Batch Normalization, introduced by Ioffe and Szegedy (2015), is another breakthrough of optimization of deep neural networks. They addressed the problem they named as internal covariate shift. Intuitively, the problem can be understood as the following two steps: 1) a learned function is barely useful if its input changes (In statistics, the input of a function is sometimes denoted as covariates). 2) each layer is a function and the changes of parameters of below layers change the input of current layer. This change could be dramatic as it may shift the distribution of inputs. Ioffe and Szegedy (2015) proposed the Batch Normalization to solve this issue, formally following the steps: n 1X µB = xi n i=1 n 1X 2 σB = (xi − µB )2 n i=1 x i − µB x̂i = σB + yi =σL x̂i + µL where µB and σB denote the mean and variance of that batch. µL and σL two parameters learned by the algorithm to rescale and shift the output. xi and yi are inputs and outputs of that function respectively. These steps are performed for every batch during training. Batch Normalization turned out to work very well in training empirically and soon became popularly. As a follow-up, Ba et al. (2016) proposes the technique Layer Normalization, where they “transpose” batch normalization into layer normalization by computing the mean and variance used for normalization from all of the summed inputs to the neurons in a layer on a single training case. Therefore, this technique has a nature advantage of being applicable to recurrent neural network straightforwardly. However, it seems that this “transposed batch normalization” cannot be implemented as simple as Batch Normalization. Therefore, it has not become as influential as Batch Normalization is. 7.4 Optimization for “Optimal” Model Architecture In the very last section of optimization techniques for neural networks, we revisit some old methods that have been attempted with the aim to learn the “optimal” model architecture. Many of these methods are known as constructive network approaches. Most of these methods have been proposed decades ago and did not raise enough impact back then. Nowadays, with more powerful computation resources, people start to consider these methods again. 56 On the Origin of Deep Learning Two remarks need to be made before we proceed: 1) Obviously, most of these methods can trace back to counterparts in non-parametric machine learning field, but because most of these methods did not perform enough to raise an impact, focusing a discussion on the evolutionary path may mislead readers. Instead, we will only list these methods for the readers who seek for inspiration. 2) Many of these methods are not exclusively optimization techniques because these methods are usually proposed with a particularly designed architecture. Technically speaking, these methods should be distributed to previous sections according to the models associated. However, because these methods can barely inspire modern modeling research, but may have a chance to inspire modern optimization research, we list these methods in this section. 7.4.1 Cascade-Correlation Learning One of the earliest and most important works on this topic was proposed by Fahlman and Lebiere (1989). They introduced a model, as well as its corresponding algorithm named Cascade-Correlation Learning. The idea is that the algorithm starts with a minimum network and builds up towards a bigger network. Whenever another hidden unit is added, the parameters of previous hidden units are fixed, and the algorithm only searches for an optimal parameter for the newly-added hidden unit. Interestingly, the unique architecture of Cascade-Correlation Learning grants the network to grow deeper and wider at the same time because every newly added hidden unit takes the data together with outputs of previously added units as input. Two important questions of this algorithm are 1) when to fix the parameters of current hidden units and proceed to add and tune a newly added one 2) when to terminate the entire algorithm. These two questions are answered in a similar manner: the algorithm adds a new hidden unit when there are no significant changes in existing architecture and terminates when the overall performance is satisfying. This training process may introduce problems of overfitting, which might account for the fact that this method is seen much in modern deep learning research. 7.4.2 Tiling Algorithm Mézard and Nadal (1989) presented the idea of Tiling Algorithm, which learns the parameters, the number of layers, as well as the number of hidden units in each layer simultaneously for feedforward neural network on Boolean functions. Later this algorithm was extended to multiple class version by Parekh et al. (1997). The algorithm works in such a way that on every layer, it tries to build a layer of hidden units that can cluster the data into different clusters where there is only one label in one cluster. The algorithm keeps increasing the number of hidden units until such a clustering pattern can be achieved and proceed to add another layer. Mézard and Nadal (1989) also offered a proof of theoretical guarantees for Tiling Algorithm. Basically, the theorem says that Tiling Algorithm can greedily improve the performance of a neural network. 57 Wang and Raj 7.4.3 Upstart Algorithm Frean (1990) proposed the Upstart Algorithm. Long story short, this algorithm is simply a neural network version of the standard decision tree (Safavian and Landgrebe, 1990) where each tree node is replaced with a linear perceptron. Therefore, the tree is seen as a neural network because it uses the core component of neural networks as a tree node. As a result, standard way of building a tree is advertised as building a neural network automatically. Similarly, Bengio et al. (2005) proposed a boosting algorithm where they replace the weak classifier as neurons. 7.4.4 Evolutionary Algorithm Evolutionary Algorithm is a family of algorithms uses mechanisms inspired by biological evolution to search in a parameter space for the optimal solution. Some prominent examples in this family are genetic algorithm (Mitchell, 1998), which simulates natural selection and ant colony optimization algorithm (Colorni et al., 1991), which simulates the cooperation of an ant colony to explore surroundings. Yao (1999) offered an extensive survey of the usage of evolution algorithm upon the optimization of neural networks, in which Yao introduced several encoding schemes that can enable the neural network architecture to be learned with evolutionary algorithms. The encoding schemes basically transfer the network architecture into vectors, so that a standard algorithm can take it as input and optimize it. So far, we discussed some representative algorithms that are aimed to learn the network architecture automatically. Most of these algorithms eventually fade out of modern deep learning research, we conjecture two main reasons for this outcome: 1) Most of these algorithms tend to overfit the data. 2) Most of these algorithms are following a greedy search paradigm, which will be unlikely to find the optimal architecture. However, with the rapid development of machine learning methods and computation resources in the last decade, we hope these constructive network methods we listed here can still inspire the readers for substantial contributions to modern deep learning research. 58 On the Origin of Deep Learning 8. Conclusion In this paper, we have revisited the evolutionary path of the nowadays deep learning models. We revisited the paths for three major families of deep learning models: the deep generative model family, convolutional neural network family, and recurrent neural network family as well as some topics for optimization techniques. This paper could serve two goals: 1) First, it documents the major milestones in the science history that have impacted the current development of deep learning. These milestones are not limited to the development in computer science fields. 2) More importantly, by revisiting the evolutionary path of the major milestone, this paper should be able to suggest the readers that how these remarkable works are developed among thousands of other contemporaneous publications. Here we briefly summarize three directions that many of these milestones pursue: • Occam’s razor: While it seems that part of the society tends to favor more complex models by layering up one architecture onto another and hoping backpropagation can find the optimal parameters, history says that masterminds tend to think simple: Dropout is widely recognized not only because of its performance, but more because of its simplicity in implementation and intuitive (tentative) reasoning. From Hopfield Network to Restricted Boltzmann Machine, models are simplified along the iterations until when RBM is ready to be piled-up. • Be ambitious: If a model is proposed with substantially more parameters than contemporaneous ones, it must solve a problem that no others can solve nicely to be remarkable. LSTM is much more complex than traditional RNN, but it bypasses the vanishing gradient problem nicely. Deep Belief Network is famous not due to the fact the they are the first one to come up with the idea of putting one RBM onto another, but due to that they come up an algorithm that allow deep architectures to be trained effectively. • Widely read: Many models are inspired by domain knowledge outside of machine learning or statistics field. Human visual cortex has greatly inspired the development of convolutional neural networks. Even the recent popular Residual Networks can find corresponding mechanism in human visual cortex. Generative Adversarial Network can also find some connection with game theory, which was developed fifty years ago. We hope these directions can help some readers to impact more on current society. More directions should also be able to be summarized through our revisit of these milestones by readers. Acknowledgements Thanks to the demo from http://beej.us/blog/data/convolution-image-processing/ for a quick generation of examples in Figure 16. 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An evolutionary approach to the identification of Cellular Automata based on partial observations Witold Bołt†∗ , Jan M. Baetens∗ and Bernard De Baets∗ † Systems arXiv:1508.05752v1 [cs.NE] 24 Aug 2015 ∗ KERMIT, Research Institute, Polish Academy of Sciences, Warsaw, Poland Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Ghent, Belgium Abstract—In this paper we consider the identification problem of Cellular Automata (CAs). The problem is defined and solved in the context of partial observations with time gaps of unknown length, i.e. pre-recorded, partial configurations of the system at certain, unknown time steps. A solution method based on a modified variant of a Genetic Algorithm (GA) is proposed and illustrated with brief experimental results. I. I NTRODUCTION CAs present an attractive and effective modelling technique for a variety of problems. In order to use CAs in a practical modelling task, one needs to understand the underlying rules, relevant to the given phenomenon, and translate them into a CA local rule. Additionally, the state space, tessellation and neighborhood structure need to be pinned down beforehand. This narrows the application area for CAs, since there are problems for which it is hard to manually design a proper local rule. In some cases only the initial and final states of the system are known (e.g. [1]–[3]). Such problems motivate the research on automated CA identification. Various methods have been used, including genetic algorithms (GAs) [4]–[7], genetic programming [8]–[10], gene expression programming [11], ant intelligence [12], machine learning [13], as well as direct search/construction approaches [14]–[17]. Existing methods can be divided into two main groups. Firstly, methods for solving specific, global problems. An example of such a problem is majority classification in which one only knows the initial condition and the desired outcome. Secondly, methods that exploit the entire time series of configurations, where it is assumed that all configurations are known. Only limited research efforts have been devoted to problems involving identification based on partial information [4]. The main goal of the research presented in this paper is to develop methods capable of automated CA identification in case of partial information. The paper is organized as follows. In Section II we start with introducing basic definitions and presenting some well-known facts on CAs. Section III holds the formal definition of the CA identification problem, while in Section IV we reformulate this problem as an optimization task. In Section V the evolutionary algorithm for solving the identification problem is presented. The paper is concluded by Section VI which presents initial results of computational experiments. An introduction to the methods presented in this paper, and a simpler formulation of the discussed algorithm can be found in [18]. 978-1-4799-7492-4/15/$31.00 c 2015 IEEE II. P RELIMINARIES We start by defining a CA. In this paper we will concentrate on 1D, deterministic CAs with a symmetric neighborhood. Let r ∈ N and fA : {0, 1}2 r+1 → {0, 1} be any function, then for N > 0 we define the N –cell global CA rule AN : {0, 1}N → {0, 1}N as: AN (. . . , si , . . .) = (. . . , fA (si−r , . . . , si+r ), . . .), (1) using periodic boundary conditions, i.e. for any i ∈ Z it holds that si+N = si . The function fA used in this definition will be referred to as a local rule, and the integer r will be referred to as the radius of the neighborhood. Any local rule can be uniquely defined by a lookup table (LUT) that lists all of the possible arguments together with the corresponding function values. It is assumed that the arguments are listed in a lexicographic order. The general form of such a LUT in the case of radius r = 1 is shown in Table I. TABLE I. LUT OF LOCAL RULE R = (l8 , l7 , l6 , l5 , l4 , l3 , l2 , l1 )2 111 110 101 100 011 010 001 000 l8 l7 l6 l5 l4 l3 l2 l1 The LUT can be used to enumerate local rules, as the coefficients li can be treated as P8digits in the binary representation of an integer R, i.e. R = i=1 li 2i−1 . Clearly this extends to higher values of the radius. Due to the fact that the ordering of arguments in the LUT is fixed, only the second row needs to be stored, such that a LUT may be represented as a binary vector. The length of such a vector is 22 r+1 . With {0, 1}∗ we will denote the S set of all binary sequences of finite length, i.e. {0, 1}∗ = N >0 {0, 1}N . The function A : {0, 1}∗ → {0, 1}∗ , satisfying A(X) = AN (X) if and only if X ∈ {0, 1}N , where each of the global rules AN is defined with the same local rule fA , will be referred to as a generalized global rule of a CA. Such functions will be frequently used throughout this paper, therefore we will simply refer to them as global rules or rules. In this paper a CA will be identified in terms of its global rule, and by referring to a CA we therefore always refer to its global rule in this generalized sense. Note that rule A is uniquely defined by a given local rule fA , but the opposite is not true. For a given rule A we may find different local rules defining it. Fact 1 highlights the relationship between different local rules defining the same CA. Fact 1. Two local rules f : {0, 1}2 r+1 → {0, 1} and g : {0, 1}2 u+1 → {0, 1}, u ≤ r, define the same CA if and only if it holds: f (s1 , . . . , s2 r+1 ) = g(sr−u+1 , . . . , sr+u+1 ), 2 r+1 for any (s1 , . . . , s2 r+1 ) ∈ {0, 1} (2) . Example 1. Let g : {0, 1} → {0, 1} be defined by g(s) = s and f : {0, 1}3 → {0, 1} be defined by f (s1 , s2 , s3 ) = s2 s3 + s2 (1 − s3 ). We can see that for any s1 , s3 ∈ {0, 1} it holds that f (s1 , s2 , s3 ) = g(s2 ) = s2 , and thus f and g define the same CA rule, which happens to be the identity rule. For a given neighborhood radius r, Ar denotes the set of all CAs that can be expressed with the use of a local rule with a neighborhood of radius r. CAs belonging to A1 are referred to as Elementary CAs (ECAs), and form the most commonly studied class of 2–state CAs [19]. Fig. 1. Space-time diagram of ECA 150 Two important properties of the sets Ar are underlined in Fact 2. 2 r+1 Fact 2. For any r ≥ 0, Ar ⊂ Ar+1 and \|Ar \| = 22 . Let A be a CA, X ∈ {0, 1}M for some M and T > 0. The finite sequence of vectors given by: (X, A(X), A2 (X)), . . . , AT −1 (X)), where At denotes the t–th application of the rule A, will be referred to as the space-time diagram containing T time steps. Each of the elements of a space-time diagram will be referred to as a configuration of the CA, while the first element will be referred to as the initial configuration. If t = 0, 1, . . . , T − 1 and m = 1, . . . , M , then At (X)[m] refers to the state of the m–th cell in the t–th element of the space-time diagram. Example 2. We consider an ECA defined by local rule 150. The LUT of ECA 150 is shown in Table II. TABLE II. LUT OF ECA 150 111 110 101 100 011 010 001 000 1 0 0 1 0 1 1 0 Figure 1 depicts a space-time diagram of ECA 150, starting from a random initial configuration. Following a common convention, the space-time diagram is visualized as a bitmap in which every row corresponds to a configuration at specific time step. The first row in the image is the initial configuration. State one is drawn as a black pixel, while white pixel corresponds to state zero. III. P ROBLEM STATEMENT belonging to the set {0, 1}. Additionally, let the first row I[1] ∈ {0, 1}M . Such an array I will be referred to as an observation. If an observation I does not contain the symbol ?, we refer to it as spatially complete. The first row I[1] is assumed to represent the initial configuration of a CA, and row I[n] for n > 1 represents the configuration at time step τn . It is assumed that τn < τn+1 . Let I be an observation. The number C(I) = #{I[n, m] 6= ?} will be referred to as the number of completely observed states. In our case, for any observation I it holds that C(I) > 0. For each observation I, we define the set com(I) that contains all of the spatially complete observations I 0 , satisfying I 0 [n, m] = I[n, m] for all n, m such that I[n, m] 6= ?. Example 3. Let observation I be given by: 0 I= 0 1 Then the set com(I) is   0 1 0 0 com(I) =  1 1 1 ? 1 0 1 . ? given by: 0 0 1 0 1 , 0 0 1 , 0 1 1 1  0 1 0 0 1 0  0 1 1 , 0 1 1 . 1 1 0 1 1 1  As can be easily counted, C(I) = 7. In this section we define the identification problem. The formulation presented below is based on the concept of an observation of a space-time diagram, which is assumed to be incomplete, i.e. it contains only partial information on the states of the CA. We will say that a CA A fits the observation I if and only if there exists an I 0 ∈ com(I) and a sequence of natural numbers (τn ) such that τn < τn+1 and for any n ∈ {1, 2, . . . , N − 1} it holds: Aτn (I 0 [1]) = I 0 [n + 1]. (3) Formally, we assume that the states of a system, which is assumed to be an unknown CA, were observed at certain, unknown time steps. Let I be an N × M array containing symbols belonging to the set {0, 1, ?}, where the symbols 0 and 1 denote valid states, while ? denotes an unknown state Proposition 3. Rule A fits the observation I if and only if there exist an I 0 ∈ com(I) and a sequence of natural numbers (tn ) such that tn ≤ tn+1 and for any n ∈ {1, 2, . . . , N − 1} it holds: Atn (I 0 [n]) = I 0 [n + 1]. (4) The sequence (τn ) in the definition of fitting, corresponds to the time steps in the CA evolution (which are assigned to the rows of the observation), while the sequence (tn ) in Proposition 3 refers to the number of missing time frames between two P consecutive rows in the observed diagram. Obn viously τn = i=1 ti . In practice, it is useful to be able to use more than one observation for the identification. Therefore, we will consider observation sets I containing a finite number of observations. For simplicity, we assume that the elements of I are numbered, i.e. I = {I1 , . . . , I\|I\| }. We will say that rule A fits the observation set I, if it fits all of the observations in the set. Note that for the sake of simplicity we will write C(I) to express the number of observed P states in all of the observations belonging to I, i.e. C(I) = I∈I C(I). Additionally, we will write M (I) to denote the total number of columns in all of P the observations belonging to I, i.e. M (I) = I∈I MI where MI is the number of columns of observation I. For a non-empty observation set I, the set R(I) will denote all CA rules that fit the observation set I. The identification problem is defined as finding the elements of the set R(I) based on I. In practice, our goal will be limited to finding at least one of the elements of R(I) ∩ Ar for some r > 0. The problem can also be seen from the machine learning perspective in which the observation set is a training set, from which we try to learn and build a set of rules based on this knowledge. The following fact will be used in the design of the identification algorithm, to simplify calculations. Informally, it could be expressed by understanding the observation set I as a set of conditions that the rule needs to meet. Having fewer conditions, it becomes more likely to find solutions meeting those conditions. Fact 4. Let I be an observation set, and let I 0 ⊂ I. Then R(I) ⊂ R(I 0 ). Since we consider only finite observation sets, we know that for every observation set I there exists a T > 0 such that, if a solution exists, and (tIn ) is the time gap sequence of observation I ∈ I, then 1 ≤ tIn ≤ T , for every n. In the construction of the solution algorithm, we will assume that an upper-bound for T is known. IV. CA I DENTIFICATION AS AN OPTIMIZATION PROBLEM The identification problem, defined in Section III, can be formulated as an optimization problem, which in turn enables the use of evolutionary search methods. We start with an auxiliary definition. Let a, b ∈ {0, 1, ?}M be some vectors. We define the distance between a and b as: X dist(a, b) = \|ai − bi \|. (5) ai ,bi ∈{0,1} We assume that if there is no i such that ai 6= ? and bi 6= ? then dist(a, b) = 0. Therefore dist(a, b) = 0 6⇒ a = b. Assume that I is a set of observations of some unknown CA belonging to Ar , i.e. R(I) ∪ Ar 6= ∅. Let A be a CA, and for every I ∈ I, let (τnI ) be a strictly increasing sequence of natural numbers. As a start, we define the error measure EI (A, (τiI )), which measures how well a given CA A fits the observation set I, assuming that τiI is the time step of the i–th row in observation I. The measure EI is defined as: EI (A, (τiI )) = I −1 X NX I dist(Aτn (I[1]), I[n + 1]), (6) I∈I n=1 where NI is the number of rows of observation I ∈ I. The following fact is an direct consequence of the definition of the identification problem. Fact 5. A ∈ R(I) if and only if there exists a sequence (τiI ) such that EI (A, (τiI )) = 0. Note that in the case when I = {I} we will write EI instead of E{I} . Let (ti ) be a sequence of natural numbers, and let A be a A CA rule. Observation I¯(t defined as: i) ( I[n, m], if I[n, m] 6= ?, A I¯(t [n, m] = A i) Atn−1 (I¯(t [n − 1])[m], if I[n, m] = ?, i) will be referred to as the A–completion of I with time gaps A (ti ). Note that any observation I satisfies I[1] = I¯(t [1] for i) any A, (ti ). Fact 6. I¯A ∈ com(I). (ti ) Example 4. Assume that CA A is ECA 150 with LUT given by Table II and local rule f150 . Let (ti )2i=1 = (1, 2). We consider A the observation I defined in Example 3 and compute I¯(t . i) 0 I= 0 1 1 ? 1 0 1 ? 0 1 0 A 0 1 1 I¯(t = ) i 1 1 0 A The calculation is as follows. Firstly we compute I¯(t [2, 2]. i) Since t1 = 1 we simply apply the rule to the first row of A I, i.e. I¯(t [2, 2] = f150 (I[1, 1], I[1, 2], I[1, 3]) = 1. Since i) A t2 = 2, to find I¯(t [3, 3], we first need to compute one i) additional configuration by evaluating the rule on configuration A A I¯(t [2]. It is easy to check that A(I¯(t [2]) = (0, 0, 0), and thus i) i) A I¯(ti ) [3, 3] = f150 (0, 0, 0) = 0. Based on Proposition 3, we define an alternative error eI (A, (tI )) that will turn out to be more useful in measure E i the construction of the solution algorithm. Assuming that (tIi ) eI is a sequence of natural numbers representing time gaps, E is defined as: I −1 I X NX A A eI (A, (tIi )) = ¯ E dist Atn I¯(t I ) [n] , I(tI ) [n + 1] . i i I∈I n=1 (7) A eI without using the Since I¯(t ∈ com(I), we can express E i) function dist as: eI (A, (tIi )) = E I −1 X NX I A ¯A \|Atn I¯(t I ) [n] − I(tI ) [n + 1]\|. i I∈I n=1 i (8) Example 5. We refer again to observation I, CA A and (ti ) used in Example 4 and we compute the error measuresPEI and eI . Let us start with EI . Following the fact that τn = n ti , E i=1 we get (τi ) = (1, 3). The error measure EI can be computed easily by evolving A, starting from the initial configuration I[1] and comparing the results with the values in I, for entries not occupied by ?. Starting from the top: A(I[1]) = (1, 1, 1). Since τ1 = 1 we compare the outcome with the second row of I. As we see, I[2, 1] = 0 6= 1 has an incorrect value, I[2, 2] =? so it does not contribute to the error and I[2, 3] = 1 which is a correct value. Since τ2 = 3 we should further evolve A three times, starting from A(I[1]), but since A((1, 1, 1)) = (1, 1, 1), we can simply compare I[3] with (1, 1, 1) and see that no errors occur. Summing up, the total error is: EI (A, (τi )) = 1. T NI −1 possibilities, which holds a substantial computational burden. Due to this, even in the case of partial observations, we follow the approach described above and treat the time steps independently. The only difference that we introduce is that if for given n, few different candidate values for tIn lead to the same, minimal value of the pairwise error, one of those candidates is being selected randomly. Such an approach, is a stochastic overestimation of the error, i.e. the calculated value will never be lower than the actual error. Additionally, if a given CA is a solution to the problem, recalculating the approximate error measure multiple times increases the probability of finding the exact value, which is found by taking the minimum of all of the obtained results. Such an approach turned out to be sufficient in the discussed context. V. E VOLUTIONARY ALGORITHM eI , by taking pairs of rows Similarly, we find the value of E A A I¯(t [n] and I[n+1] and comparing the results of Atn (I¯(t [n]) i) i) and I[n + 1]. The error in the first pair of rows is the same as in the case of EI . For the second pair the initial condition is A I¯(t [2] = (0, 1, 1), and since A(0, 1, 1) = (0, 0, 0) and since i) A((0, 0, 0)) = (0, 0, 0), we do not further evaluate A. We compare (0, 0, 0) with I[3], which yields 2 incorrect values. eI (A, (ti )) = 3. Summing up, the total error is E Having stated the identification problem as an optimization problem in this section, we describe its solution using an evolutionary algorithm based on the classical GA. In order to follow the GA approach, we need to define the individuals’ representation, the population structure, a fitness function for ranking the individuals, but also the selection procedure for reproduction, and finally the cross-over and mutation operators. Formally, also halting conditions need to be formulated. eI is expressed by the The relation between EI and E following proposition. A. Representation of individuals and population structure Proposition 7. Let A be a CA rule and I an observation set. There exists a strictly increasing sequence (τiI ) of natural numbers, such that EI (A, (τiI )) = 0 if and only if there exists eI (A, (tI )) = 0. a sequence (tIi ) of natural numbers such that E i As a consequence of Proposition 7, the identification problem can be expressed mathematically as the minimization of e Note that this is only possible due to the assumption that E. observation set I contains partial space-time diagrams of some unknown CA. In a more general setting, where the observations could have a more complex origin, such a simplification is not possible. As mentioned earlier, we consider the case where the upper bound for the time gaps is known. Using this knowledge, we eI independently of the selection of define the error measure E (tIi ) as: eI (A) = min E eI (A, (tIi )). E (9) (tIi ) 1≤tIi ≤T Note that the minimum in (9) is always defined, since there is a finite number of possibilities for the choice of tIn . Additionally, note that for a spatially complete observation I, the choice of tIn is independent of the choice of tIm for any n 6= m, and for observations I and J, the choice of (tIi ) is independent from the choice of (tJi ). Consequently, to find the eI in the case of a spatially complete observation value of E P set, we need to examine at most I∈I T (NI − 1) sequences of time gap lengths. (tIi ) In the general case, the choices of the values of are dependent on each other, and thus in order to find the exact value of the error measure we need to examine all of the Here, the individuals that make up the population are CAs, encoded through the LUT of their local rules, which is possible since the LUT of any CA A ∈ Ar can be represented as a bitstring of length 22 r+1 . We assume that the population consists of CA belonging to Ar , for some r > 0. We consider populations of P > 0 individuals. By P i we denote the population of the i–th generation of the GA. The population P 1 is the initial population, and is constructed by randomly selecting P bit-strings. Populations P i for i > 1 are the outcomes of applying the genetic operators, according to the rules described in the remainder of this section. B. Fitness function The fitness function is directly related to the error measure eI defined by (9). Although Proposition 7 states that the E error measures given by (6) and (9) can be used interchangeably, preliminary experiments showed that the later results in efficient and convergent algorithm, while suboptimal results were obtained using the measure given by (6). This follows from the fact that the error in row n is affected by errors appearing in rows 2, . . . , n − 1. As we know from the research on dynamical properties of CAs, small initial perturbations can strongly affect the final system state [20]. For that reason, it eI with a GA as compared to EI . is easier to optimize E 2 r+1 be a LUT of some local rule which Let L ∈ {0, 1}2 defines a CA A. Then fitI (L) denotes the fitness of A, and is defined as: eI (A). fitI (L) = C(I) − M (I) − E (10) The fitness function takes integer values from 0 up to C(I) − M (I), i.e. there are finitely many possible values of the fitness function. The goal of the GA is to maximize fitness, and a CA with a maximal fitness value is a solution of the identification problem. From the above, it is clear that if C(I) − M (I) is close to zero, solving the problem is infeasible, since the number of possible values is very small and the population is not able to gradually increase its fitness. Additionally, if C(I) = M (I), then the problem is trivial because every CA is a solution. The fitness defined by (10) has proven to work effectively, but the computing time needed for its evolution becomes unacceptable if the observation set is large. Therefore, during the evolution, to estimate the value of fitI we use fitI 0 for some non-empty subset I 0 ⊂ I. We start by randomly selecting elements for the subset I 0 . Subsequently, but before evolving a new population we replace one of the elements in the subset I 0 with a randomly selected observation from I. Due to Fact 4 we are sure that such an approach does not result in reducing the solution set. C. Selection operator Having defined the fitness function, we can define the selection operator, which is responsible for selecting the parent individuals that will be used to produce the next generation. We use a random selection method where the selection probability of a given individual is proportional to its fitness. Individuals are selected with replacement, i.e. individuals might be selected multiple times for reproduction. G. Halting conditions The algorithm evolves by generating populations according to the procedure described above until a maximum, predefined number of populations Λ was evolved or, if a CA that fits the observation set was discovered. As mentioned in Subsection V-B during the evolution, the fitness fitI is approximated by fitI 0 for some I 0 ( I, which is effective for selection, but can not be used in the halting condition since fitI 0 (A) = C(I 0 ) − M (I 0 ) does not imply fitI (A) = C(I) − M (I). Therefore, for the individual A with the highest value fitI 0 (A), we additionally calculate fitI (A) and base the halting condition on it, i.e. the algorithm stops as soon an element is found. VI. R ESULTS OF EXPERIMENTS By means of our experiments we verified to what extent the partiality of observations affects the efficiency of the GA in terms of the number of GA iterations required to find a solution. We concentrated on two ECAs: 150 and 180, with LUTs given in Table II and III, respectively. TABLE III. LUT OF ECA 180 111 110 101 100 011 010 001 000 1 0 1 0 1 0 1 0 D. Cross-over operator To produce offspring, we select two parents according to the procedure described in Subsection V-C. A uniform crossover operator is used, i.e. if L1 , L2 denote parents, the outcome of the cross-over operator is a vector Lc with values that are randomly selected from L1 and L2 , i.e. P(Lc [i] = L1 [i]) = P(Lc [i] = L2 [i]) = 0.5. E. Mutation operator Finally, the offspring individual is mutated. A simple bitflip mutation is being used, i.e. for every position of the vector a decision is made whether or not the value should be flipped, with pf being the probability of flipping the value. The expected number of flipped positions in the population is P pf 22 r+1 . F. Elite survival After evolving a new population, the elite survival procedure is applied. Our experiments proved that such an approach is required to reach convergence. The procedure is implemented by a deterministic selection of PE P fittest individuals from the previous population used to replace randomly selected individuals in the newly evolved one. Including this elite survival process can dramatically increase the performance of the algorithm, though there are cases where such an approach causes the population to progress towards a local optimum. To overcome this, we apply a simple, adaptive procedure that deactivates elite survival in cases when the maximum fitness value of the population remained constant for more than Noff generations. The elite survival procedure is again switched on after a predefined number of Non generations, or if the maximum fitness improved. In this experiment, the GA evolution is based on observation sets IA (k) for k = {0, 1, . . . , 150} and ECA A ∈ {150, 180}. The integer k will be referred to as the problem number. The observation set IA (0) is a set of Ω > 0 observations obtained from Ω different, random initial conditions common for both A, by selecting subsequent configurations of ECA A generating time gaps of random length from 1 to T . The set IA (k) for k > 0 is built from observations belonging to IA (k − 1) by modifying them in such a way that π = 2000 randomly selected, completely observed entries are replaced by “?”. In other words, by increasing k the effect of spatial partiality is increased. As a result of such 150 a procedure we obtained a series of observation sets IA (k) k=0 , for which it holds C(IA (k)) − C(IA (k + 1)) = π. The identification algorithm was then executed for each of the obtained observation sets. Given that the family of ECAs contains only 256 members, the identification problem would be relatively easy to tackle, so we set the radius r = 2, i.e. the population contains local rules with radius r = 2 represented as bit-strings of length 32. Without this modification the algorithm is able to find a solution in a few iterations, by examining the entire search space. In order to account for the stochastic nature of the GA, the experiment is repeated L > 0 times for each r, k. The values of of the GA parameters used in our experiment setup are shown in Table IV. The results vary significantly depending on the rule in question, which is not surprising since the dynamics of ECAs 150 and 180 is different. The normalized Maximum Lyapunov Exponent (nMLE) [21]–[23] of the former is the highest among param r pf P PE T C Ω π s Λ L value 2 0.01 512 32 10 69 64 2000 8 5000 20 PARAMETERS USED IN THE EXPERIMENT description rule radius probability of flipping 1-bit in mutation number of individuals in population elite size bound for the time gap length number of rows / columns in each observation number of observations number of cells being removed from each observation set number of samples for fitness approximation maximal number of the GA populations number of repetitions of the GA per rule no. of successful GA executions TABLE IV. 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 problem number k no. of successful GA executions (a) ECA 150 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 problem number k (b) ECA 180 Fig. 3. Fig. 2. Number of successful GA executions Space-time diagram of ECA 180 all of the ECAs, and thereby this CA’s behavior may be considered complex. In contrast, the nMLE of ECA 180 is only approximately 0.48, which hints that, in some sense, the behavior of this ECA is simpler than the one displayed by ECA 150. The differences in the overall dynamical complexity of these two CAs can be acknowledged by examining their spacetime diagrams, which are depicted in Fig. 1 and Fig. 2. To understand the performance of the GA, we first checked for which k the algorithm was able to find a solution (Fig. 3). When comparing the plot for ECA 150 with the one for ECA 180, it is clear that the identification problem turned out to be much more challenging for ECA 150. Indeed, for this ECA, the algorithm was effective only if the number removed observation elements was smaller than 50 π, whereas it mostly failed when more spatial partiality was added. Besides, even for k close to 0, not all of the GA executions were successful. In contrast, identifying ECA 180 was always possible for k < 120, but for k > 120 we see a sudden drop in the performance. Note that in both cases, for k = 150 a solution was easily found, since for this setting the problem is trivial, i.e. almost all CAs can be considered a solution. The above results suggest that, depending on the dynamical characteristics of the CA in question, the maximum allowable number of missing elements in the observations differs. Further research is undertaken to better understand the link between the identifiability and dynamics of CAs. Figure 4 depict the minimum, average and maximum number of GA iterations among the runs resulting in a solution for ECA 150 and ECA 180, respectively. In the case of ECA 180, we see that the efforts needed for finding a solution grows as k increases, up to the point where it becomes impossible. Furthermore, we see that in most cases the difference between maximal and minimal values is relatively low. In the case of ECA 150, the results are much less stable. The differences between maximal and minimal values are substantial, and the efforts needed to find the solution do not steadily grows with the growing spatial partiality. The only similarity between the two CAs seems to be in the fact that there exists some critical k beyond which the problem becomes impossible to solve. S UMMARY In this paper we introduced the identification problem of CAs in the context of partial observations. An evolutionary algorithm for tackling the problem was presented, and its performance was verified for the two ECAs. The initial experiments suggest that the difficulty of the identification problem is somehow linked to the dynamical complexity of the CAs. The problem and solution algorithm presented in this paper, should be considered as one of the first steps in identifying CAs from data originating from real-world phenomenon observations. Unavoidably, such observations will be somehow incomplete in the sense that it is impossible to continuously track the involved processes. R EFERENCES [1] S. Al-Kheder, J. Wang, and J. Shan, “Cellular automata urban growth model calibration with genetic algorithms,” in Proc. Urban Remote Sensing Joint Event, 2007. IEEE, 2007, pp. 1–5. [2] E. Sapin, L. Bull, and A. Adamatzky, “Genetic approaches to search for computing patterns in cellular automata,” IEEE Comput. Intell. 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The Cross-Quantilogram: Measuring Quantile arXiv:1402.1937v2 [math.ST] 21 Jan 2018 Dependence and Testing Directional Predictability between Time Series∗ Heejoon Han† Oliver Linton‡ Tatsushi Oka§ Yoon-Jae Whang¶ March 14, 2016 Abstract This paper proposes the cross-quantilogram to measure the quantile dependence between two time series. We apply it to test the hypothesis that one time series has no directional predictability to another time series. We establish the asymptotic distribution of the cross-quantilogram and the corresponding test statistic. The limiting distributions depend on nuisance parameters. To construct consistent confidence intervals we employ a stationary bootstrap procedure; we establish consistency of this bootstrap. Also, we consider a self-normalized approach, which yields an asymptotically pivotal statistic under the null hypothesis of no predictability. We provide simulation studies and two empirical applications. First, we use the cross-quantilogram to detect predictability from stock variance to excess stock return. Compared to existing tools used in the literature of stock return predictability, our method provides a more complete relationship between a predictor and stock return. Second, we investigate the systemic risk of individual financial institutions, such as JP Morgan Chase, Morgan Stanley and AIG. Keywords: Quantile, Correlogram, Dependence, Predictability, Systemic risk. ∗ We thank a Co-Editor, Jianqing Fan, an Associate Editor and three anonymous referees for constructive comments. Han’s work was supported by the National Research Foundation of Korea (NRF2013S1A5A8021502). Linton’s work was supported by Cambridge INET and the ERC. Oka’s work was supported by Singapore Academic Research Fund (FY2013-FRC2-003). Whang’s work was supported by the SNU Creative Leading Researcher Grant. † Department of Economics, Sungkyunkwan University, Seoul, Republic of Korea. ‡ Faculty of Economics, University of Cambridge, Cambridge, UK. § Department of Economics, National University of Singapore, Singapore. ¶ Department of Economics, Seoul National University, Seoul, Republic of Korea. 1 1 Introduction Linton and Whang (2007) introduced the quantilogram to measure predictability in different parts of the distribution of a stationary time series based on the correlogram of “quantile hits”. They applied it to test the hypothesis that a given time series has no directional predictability. More specifically, their null hypothesis was that the past information set of the stationary time series {yt } does not improve the prediction about whether yt will be above or below the unconditional quantile. The test is based on comparing the quantilogram to a pointwise confidence band. This contribution fits into a long literature of testing predictability using signs or rank statistics, including the papers of Cowles and Jones (1937), Dufour et al. (1998), and Christoffersen and Diebold (2002). The quantilogram has several advantages compared to other test statistics for directional predictability. It is conceptually appealing and simple to interpret. Since the method is based on quantile hits it does not require moment conditions like the ordinary correlogram and statistics like the variance ratio that are derived from it, Mikosch and Starica (2000), and so it works well for heavy tailed series. Many financial time series have heavy tails, see, e.g., Mandelbrot (1963), Fama (1965), Rachev and Mittnik (2000), Embrechts et al. (1997), Ibragimov et al. (2009), and Ibragimov (2009), and so this is an important consideration in practice. Additionally, this type of method allows researchers to consider very long lags in comparison with regression type methods, such as Engle and Manganelli (2004). There have been a number of recent works either extending or applying this methodology. Davis and Mikosch (2009) have introduced the extremogram, which is essentially the quantilogram for extreme quantiles, and Davis et al. (2012) has provided the inference methods based on bootstrap and permutation for the extremogram. See also Davis et al. (2013). Li (2008, 2012) has introduced a Fourier domain version of the quantilogram while 1 Hong (2000) has used a Fourier domain approach for test statistics based on distributions. Further development in the Fourier domain approach has been made by Hagemann (2013) and Dette et al. (2015). See also Li (2014) and Kley et al. (2016). The quantilogram has recently been applied to stock returns and exchange rates, Laurini et al. (2008) and Chang and Shie (2011). Our paper addresses three outstanding issues with regard to the quantilogram. First, the construction of confidence intervals that are valid under general dependence structures. Linton and Whang (2007) derived the limiting distribution of the sample quantilogram under the null hypothesis that the quantilogram itself is zero, in fact under a special case of that where the process has a type of conditional heteroskedasticity structure. Even in that very special case, the limiting distribution depends on model specific quantities. They derived a bound on the asymptotic variance that allows one to test the null hypothesis of the absence of predictability (or rather the special case of this that they work with). Even when this model structure is appropriate, the bounds can be quite large especially when one looks into the tails of the distribution. The quantilogram is also useful in cases where the null hypothesis of no predictability is not thought to be true - one can be interested in measuring the degree of predictability of a series across different quantiles. We provide a more complete solution to the issue of inference for the quantilogram. Specifically, we derive the asymptotic distribution of the quantilogram under general weak dependence conditions, specifically strong mixing. The limiting distribution is quite complicated and depends on the long run variance of the quantile hits. To conduct inference we propose the stationary bootstrap method of Politis and Romano (1994) and prove that it provides asymptotically valid confidence intervals. We investigate the finite sample performance of this procedure and show that it works well. We also provide R code that carries out the computations efficiently.1 We also define a self-normalized version of the statistic for testing the null hypothesis that the quantilogram is zero, following Lobato (2001). This statistic has an asymptotically pivotal distribution, 1 This can be found at http://www.oliverlinton.me.uk/research/software. 2 under the null hypothesis, whose critical values have been tabulated so that there is no need for long run variance estimation or even bootstrap. Second, we develop our methodology inside a multivariate setting and explicitly consider the cross-quantilogram. Linton and Whang (2007) briefly mentioned such a multivariate version of the quantilogram but they provided neither theoretical results nor empirical results. In fact, the cross-correlogram is a vitally important measure of dependence between time series: Campbell, Lo, and MacKinlay (1997), for example, use the cross autocorrelation function to describe lead lag relations between large stocks and small stocks. We apply the cross-quantilogram to the study of stock return predictability; our method provides a more complete picture of the predictability structure. We also apply the cross-quantilogram to the question of systemic risk. Our theoretical results described in the previous paragraph are all derived for the multivariate case. Third, we explicitly allow the cross-quantilogram to be based on conditional (or regression) quantiles (Koenker and Basset, 1978). Using conditional quantiles rather than unconditional quantiles, we measure directional dependence between two time-series after parsimoniously controlling for the information at the time of prediction.2 Moreover, we derive the asymptotic distribution of the cross-quantilogram that are valid uniformly over a range of quantiles. The remainder of the paper is as follows: Section 2 introduces the cross-quantilogram and Section 3 discusses its asymptotic properties. For consistent confidence intervals and hypothesis tests, we define the bootstrap procedure and introduce the self normalized test statistic. Section 4 considers the partial cross-quantilogram and gives a full treatment of its behavior in large samples. In Section 5 we report results of some Monte Carlo simulations to 2 Our analysis includes the cross-quantilogram based on unconditional quantiles as a special case. In this case, the cross-quantilogram is shown to be a functional of the empirical copula introduced by Ruschendorf (1976) and Deheuvels (1979) as some nonparametric measures of dependence, such as Spearman’s rho and Kendall’s tau. In this special case, the asymptotic results for the empirical copula, which are found in Stute (1984), Fermanian et al. (2004) and Segers (2012) among others, can apply for the cross-quantilogram. Generally, however, the cross-quantilogram here differs from the empirical copula process and needs different treatment for analyzing its properties. 3 evaluate the finite sample properties of our procedures. In Section 6 we give two applications: we investigate stock return predictability and system risk using our methodology. Appendix contains all the proofs. We use the following notation: The norm k · k denotes the Euclidean norm, i.e., kzk = Pd for z = (z1 , . . . , zd )⊤ ∈ Rd and the norm k · kp indicates the Lp norm of a d × 1 P random vector z, given by kzkp = ( dj=1 E\|zj \|p )1/p for p > 0. Let 1[·] be the indicator ( 2 1/2 j=1 zj ) function taking the value one when its argument is true, and zero otherwise. We use R, Z and N to denote the set of all real numbers, all integers and all positive integers, respectively. Let Z+ = N ∪ {0}. 2 The Cross-Quantilogram Let {(yt , xt ) : t ∈ Z} be a strictly stationary time series with yt = (y1t , y2t )⊤ ∈ R2 and (1) (d ) xt = (x1t , x2t ) ∈ Rd1 × Rd2 , where xit = [xit , . . . , xit i ]⊤ ∈ Rdi with di ∈ N for i = 1, 2. We use Fyi \|xi (·\|xit ) to denote the conditional distribution function of the series yit given xit with density function fyi \|xi (·\|xit ), and the corresponding conditional quantile function is defined as qi,t (τi ) = inf{v : Fyi \|xi (v\|xit ) ≥ τi } for τi ∈ (0, 1), for i = 1, 2. Let T be the range of quantiles we are interested in evaluating the directional predictability. For simplicity, we assume that T is a Cartesian product of two closed intervals in (0, 1), that is T ≡ T1 × T2 , where Ti = [τ i , τ i ] for some 0 < τ i < τ i < 1.3 We consider a measure of serial dependence between two events {y1t ≤ q1,t (τ1 )} and {y2,t−k ≤ q2,t−k (τ2 )} for an arbitrary pair of τ = (τ1 , τ2 )⊤ ∈ T and for an integer k. In the literature, {1[yit ≤ qi,t (·)]} is called the quantile-hit or quantile-exceedance process for i = 1, 2. The cross-quantilogram is defined as the cross-correlation of the quantile-hit processes 3 E [ψτ1 (y1t − q1,t (τ1 ))ψτ2 (y2,t−k − q2,t−k (τ2 ))] ρτ (k) = q q , 2 2 E ψτ1 (y1t − q1,t (τ1 )) E ψτ2 (y2,t−k − q2,t−k (τ2 )) (1) It is straightforward to extend the results to a more general case, e.g. the case for which T is the union of a finite number of disjoint closed subsets of (0, 1)2 . 4 for k = 0, ±1, ±2, . . . , where ψa (u) ≡ 1[u < 0] − a. The cross-quantilogram captures serial dependence between the two series at different conditional quantile levels. In the special case of a single time series, the cross-quantilogram becomes the quantilogram proposed by Linton and Whang (2007). Note that it is well-defined even for processes {(y1t , y2t )}t∈N with infinite moments. Like the quantilogram, the cross-quantilogram is invariant to any strictly monotonic transformation applied to both series, such as the logarithmic transformation.4 To construct the sample analogue of the cross-quantilogram based on observations {(yt , xt )}Tt=1 , we first estimate conditional quantile functions. In this paper, we consider the linear quantile regression model proposed by Koenker and Bassett (1978) for simplicity and let qi,t (τi ) = x⊤ it βi (τi ) with a di × 1 vector of unknown parameters βi (τi ) for i = 1, 2. To estimate the parameters β(τ ) ≡ [β1 (τ1 )⊤ , β2 (τ2 )⊤ ]⊤ , we separately solve the following minimization problems: β̂i (τi ) = arg min βi ∈Rdi T X t=1 ̺τi yit − x⊤ it βi , where ̺a (u) ≡ u(a − 1[u < 0]). Let β̂(τ ) ≡ [β̂1 (τ1 )⊤ , β̂2 (τ2 )⊤ ]⊤ and q̂i,t (τi ) = x⊤ it β̂i (τi ) for i = 1,2. The sample cross-quantilogram is defined by PT ψτ1 (y1t − q̂1,t (τ1 ))ψτ2 (y2,t−k − q̂2,t−k (τ2 )) qP , T 2 (y − q̂ (τ )) 2 (y ψ ψ − q̂ (τ )) 1,t 1 2,t−k 2 t=k+1 τ1 1t t=k+1 τ2 2,t−k ρ̂τ (k) = qP T t=k+1 (2) for k = 0, ±1, ±2, . . . . Given a set of conditional quantiles, the cross-quantilogram considers dependence in terms of the direction of deviation from conditional quantiles and thus measures the directional predictability from one series to another. This can be a useful del h When one is interested in measuring serial dependence events {q1,t (τ1 ) ≤ y1t ≤ q1,t (τ1 )} l two l h between h l h and {q2,t−k (τ2 ) ≤ y2,t−k ≤ q2,t−k (τ2 )} for arbitrary τ1 , τ1 and τ2 , τ2 , one can use an alternative version of the cross-quantilogram that is defined by replacing ψτi (yit − qi,t (τi )) in (1) with ψ[τ l ,τ h ] (yit − qi,t ( τil , τih )) = 1[qi,t (τil ) < yit < qi,t τih ] − τih − τil . i i 4 For example, if τ1 = [0.9, 1.0] and τ2 = [0.4, 0.6] , the alternative version measures dependence between an event that y1t is in a high range and an event that y2,t−k is in a mid-range. In some cases, such an alternative version could be easier to interpret and therefore be useful. The inference procedure provided in this paper is also valid for the alternative version of the cross-quantilogram. See the working paper version of this paper for an empirical application using the alternative version. 5 scriptive device. By construction, ρ̂τ (k) ∈ [−1, 1] with ρ̂τ (k) = 0 corresponding to the case of no directional predictability. The form of the statistic generalizes to the l dimensional multivariate case and the (i, j)th entry of the corresponding cross-correlation matrices Γτ̄ (k) is given by applying (2) for a pair of variables (yit , xit ) and (yjt−k , xjt−k ) and a pair of con⊤ ditional quantiles (q̂i,t (τi ), q̂j,t−k (τj ))) for τ̄ = (τ1 , . . . , τl ) . The cross-correlation matrices ⊤ possess the usual symmetry property Γτ̄ (k) = Γτ̄ (−k) when τ1 = · · · = τd . Suppose that τ ∈ T and p are given. One may be interested in testing the null hypothesis H0 : ρτ (1) = · · · = ρτ (p) = 0 against the alternative hypothesis that ρτ (k) 6= 0 for some k ∈ {1, . . . , p}. This is a test for the directional predictability of events up to p lags {y2,t−k ≤ q2,t−k (τ2 ) : k = 1, . . . , p} for {y1t ≤ q1,t (τ1 )}. For this hypothesis, we can use the P (p) Box-Pierce type statistic Q̂τ = T pk=1 ρ̂2τ (k). In practice, we recommend to use the BoxP (p) Ljung version Q̌τ ≡ T (T + 2) pk=1 ρ̂2τ (k)/(T − k) which had small sample improvements in our simulations. On the other hand, one may be interested in testing a stronger null hypothesis, i.e. the absence of directional predictability over a set of quantiles: H0 : ρτ (1) = · · · = ρτ (p) = 0, ∀τ ∈ T , against the alternative hypothesis that ρτ (k) 6= 0 for some (k, τ ) ∈ {1, . . . , p} × T with p fixed. In this case, we can use the sup-version test statistic sup Q̂τ(p) τ ∈T = sup T τ ∈T p X ρ̂2τ (k). k=1 (p) Note that the portmanteau test statistic Q̂τ for a specific quantile is a special case of the sup-version test statistic. 3 Asymptotic Properties We next present the asymptotic properties of the sample cross-quantilogram and related test statistics. Since these quantities contain non-smooth functions, we employ techniques widely used in the literature on quantile regression, see Koenker and Bassett (1978) and Pollard 6 (1991) among others. Define yt,k = (y1t , y2,t−k )⊤ , xt,k = (x1t , x2,t−k ), qt,k (τ ) = [q1,t (τ1 ), q2,t−k (τ2 )]⊤ and q̂t,k (τ ) = [q̂1,t (τ1 ), q̂2,t−k (τ2 )]⊤ and let {yt,k ≤ qt,k (τ )} = {y1t ≤ q1 (τ1 \|x1t ), y2,t−k ≤ q2 (τ2 \|x2t−k )} and (k) Fy\|x (·\|xt,k ) = P (yt,k ≤ ·\|xt,k ) for t = k + 1, . . . , T and for some finite integer k > 0. We (k) use ∇G(k) (τ ) to denote ∂/∂vE[Fy\|x (vt,k \|xt,k )] evaluated at vt,k = qt,k (τ ), where vt,k = ⊤ ⊤ di [x⊤ (i = 1, 2). Let d0 = 1 + d1 + d2 . 1t v1 , x2,t−k v2 ] for vi ∈ R Assumption A1. {(yt , xt )}t∈Z is strictly stationary and strong mixing with coefficients {αj }j∈Z+ that P 2s−2 ν/(2s+ν) satisfy ∞ αj < ∞ for some integer s ≥ 3 and ν ∈ (0, 1). For each j=0 (j + 1) (j) (1) (d ) i = 1, 2, E\|xit \|2s+ν < ∞ for all j = 1, . . . , di , given xit = [xit , . . . , xit i ]⊤ . A2. The conditional distribution function Fyi \|xi (·\|xit ) has continuous densities fyi \|xi (·\|xit ), which is uniformly bounded away from 0 and ∞ at qi,t (τi ) uniformly over τi ∈ Ti , for i = 1, 2 and for all t ∈ Z. A3. For any ǫ > 0 there exists a ν(ǫ) such that supτi ∈Ti sups:\|s\|≤ν(ǫ) \|fyi \|xi (qi,t (τi )\|xit ) − fyi \|xi (qi,t (τi ) + s\|xit )\| < ǫ for i = 1, 2 and for all t ∈ Z. (k) A4. For every k ∈ {1, . . . , p}, the conditional joint distribution Fy\|x (·\|xt,k ) has the condi(k) tional density fy\|x (·\|xt,k ), which is bounded uniformly in the neighborhood of quantiles of interest, and also has a bounded, continuous first derivative for each argument uniformly in the neighborhood of quantiles of interest and thus ∇G(k) (τ ) exists over τ ∈ T . A5. For each i = 1, 2, there exist positive definite matrices Mi and Di (τi ) such that PT P ⊤ −1 (a) plimT →∞ T −1 Tt=1 xit x⊤ it = Mi and (b) plimT →∞ T t=1 fyi \|xi (qi,t (τi )\|xit )xit xit = Di (τi ) uniformly in τi ∈ Ti . Assumption A1 imposes the mixing rate used in Andrews and Pollard (1994) and a moment condition on regressors, while allowing for the dependent variables to be processes 7 with infinite moments. For a strong mixing process, ρτ (k) → 0 as k → ∞ for all τ ∈ (0, 1). Assumption A2 ensures that the conditional quantile function given xit is uniquely defined while allowing for dynamic misspecification, or P (yit ≤ qi,t (τi )\|Fit ) 6= τi given some information set Fit containing all “relevant” information available at t for i = 1, 2. In the absence of dynamic misspecification, which is assumed in Hong et al. (2009) under their null hypothesis, the analysis becomes substantially simple because each hit-process {ψτi (y − qi,t (τi ))} is a sequence of iid Bernoulli random variables. As Corradi and Swanson (2006) discuss, however, results under correct dynamic specification crucially rely on an appropriate choice of the information set; specification search for the information set based on pre-testing may have a nontrivial impact on inference. Thus, Assumption A2 is appropriate for the purpose of testing directional predictability given a particular information set xit . Assumption A3 implies that the densities are smooth in some neighborhood of the quantiles of interest. Assumption A4 ensures that the joint distribution of (x1t , x2t−k ) is continuously differentiable. Assumption A5 is standard in the quantile regression literature. To describe the asymptotic behavior of the cross-quantilogram, we define a set of d0 dimensional mean-zero Gaussian process {Bk (τ ) : τ ∈ [0, 1]2 }pk=1 with covariance-matrix function for k, k ′ ∈ {1, . . . , p} and for τ, τ ′ ∈ T , given by ⊤ k′ ′ ′ Ξkk′ (τ, τ ) ≡ E[Bk (τ )B (τ )] = ∞ X l=−∞ ⊤ cov ξl,k (τ ), ξ0,k′ (τ ′ ) , ⊤ ⊤ where ξt,k (τ ) = (1[yt,k ≤ qt,k (τ )], x⊤ 1t 1[y1t ≤ q1,t (τ1 )], x2t 1[y2t ≤ q2,t (τ2 )]) for t ∈ Z. Define ⊤ ⊤ ⊤ B(p) (τ ) = [B1 (τ ) , . . . , Bp (τ ) ] as the d0 p-dimensional zero-mean Gaussian process with the covariance-matrix function denoted by Ξ(p) (τ, τ ′ ) for τ, τ ′ ∈ T . We use ℓ∞ (T ) to denote the space of all bounded functions on T equipped with the uniform topology and (ℓ∞ (T ))p to denote the p-product space of ℓ∞ (T ) equipped with the product topology. Let the notation “⇒” denote the weak convergence due to Hoffman-Jorgensen in order to handle the measurability issues, although outer probabilities and expectations are not used explicitly in this 8 paper for notational simplicity. See Chapter 1 of van der Vaart and Wellner (1996) for a comprehensive treatment of weak convergence in non-separable metric spaces. The next theorem establishes the asymptotic properties of the cross-quantilogram. Theorem 1 Suppose that Assumptions A1-A5 hold for some finite integer p > 0. Then, in the sense of weak convergence of the stochastic process in (ℓ∞ (T ))p we have: √ (p) T ρ̂τ(p) − ρτ(p) ⇒ Λτ(p) B(p) (τ ), (p) ⊤ ⊤ (3) ⊤ where ρ̂τ ≡ [ρ̂τ (1), . . . , ρ̂τ (p)] and Λτ = diag(λτ 1 , . . . , λτ p ) with λτ,k = p 1 τ1 (1 − τ1 )τ2 (1 − τ2 )    1 −∇G(k) (τ )[D1−1 (τ1 ), D2−1 (τ2 )]⊤   . (4) Under the null hypothesis that ρτ (1) = · · · = ρτ (p) = 0 for every τ ∈ T , it follows that sup Q̂τ(p) ⇒ sup kΛτ(p) B(p) (τ )k2 , τ ∈T (5) τ ∈T by the continuous mapping theorem. 3.1 3.1.1 Inference Methods The Stationary Bootstrap The asymptotic null distribution presented in Theorem 1 depends on nuisance parameters. We suggest to estimate the critical values by the stationary bootstrap of Politis and Romano (1994). The stationary bootstrap is a block bootstrap method with blocks of random lengths. The stationary bootstrap resample is strictly stationary conditional on the original sample. Let {Li }i∈N denote a sequence of iid random block lengths having the geometric distribution with a scalar parameter γ ≡ γT ∈ (0, 1): P ∗(Li = l) = γ(1 − γ)l−1 for each positive 9 integer l, where P ∗ denotes the conditional probability given the original sample. We assume that the parameter γ satisfies the following growth condition: √ Assumption A6. T ν/2(2s+ν)(s−1) γ + ( T γ)−1 → 0 as T → ∞, where s and ν are defined in Assumption A1. We need the condition that γ = o(T −ν/2(2s+ν)(s−1) ) for the purpose of establishing uniform convergence over the subset T of [0, 1]2 , given the moment conditions on regressors under Assumption A1. This condition can be relaxed when regressors are uniformly bounded because γ = o(1) when s = ∞. Let {Ki }i∈N be a sequence of iid random variables, which have the discrete uniform distribution on {k + 1, . . . , T } and are independent of both the original data and {Li }i∈N . Ki +Li −1 representing the blocks of length Li starting with the We set BKi ,Li = {(yt,k , xt,k )}t=K i Ki -th pair of observations. The stationary bootstrap procedure generates the bootstrap ∗ samples {(yt,k , x∗t,k )}Tt=k+1 by taking the first (T − k) observations from a sequence of the resampled blocks {BKi ,Li }i∈N . In this notation, when t > T , (yt,k , xt,k ) is set to be (yjk , xjk ), where j = k + (t mod (T − k)) and (yk,k , xk,k ) = (yt,k , xt,k ), where mod denotes the modulo operator.5 Using the stationary bootstrap resample, we estimate the parameter β(τ ) by solving the minimization problem: β̂1∗ (τ1 ) = arg min β1 ∈Rd1 T X t=k+1 ∗ ∗ ̺τ1 (y1t − x∗⊤ 1t β1 ) and β̂2 (τ2 ) = arg min β2 ∈Rd2 T −k X t=1 ∗ ̺τ2 (y2t − x∗⊤ 2t β2 ). ∗ Then the conditional quantile function given the stationary bootstrap resample, qi,t (τi ) ≡ ∗ ∗⊤ ∗ ∗ ∗⊤ ∗⊤ ⊤ x∗⊤ it βi (τi ), is estimated by q̂i,t (τi ) ≡ xit β̂i (τi ) for each i = 1, 2. Define β̂ (τ ) = [β̂1 (τ1 ), β̂2 (τ2 )] ∗ ∗ ∗ ∗ (τ2 )]⊤ . We construct β̂ ∗ (τ ) (τ1 ), q2,t−k (τ2 )]⊤ and q∗t,k (τ ) = [q1,t and let q̂∗t,k (τ ) = [q̂1,t (τ1 ), q̂2,t−k by using (T − k) bootstrap observations, while β̂(τ ) is based on T observations, but the 5 For any positive integers a and b, the modulo operation a mod b is equal to the remainder, on division of a by b. 10 difference of sample sizes is asymptotically negligible given the finite lag order k. The cross-quantilogram based on the stationary bootstrap resample is defined as follows: ρ̂∗τ (k) PT ∗ ∗ ∗ ∗ ψτ1 (y1t − q̂1,t (τ1 ))ψτ2 (y2,t−k − q̂2,t−k (τ2 )) qP . T ∗ ∗ ∗ ∗ 2 2 t=k+1 ψτ1 (y1t − q̂1,t (τ1 )) t=k+1 ψτ2 (y2,t−k − q̂2,t−k (τ2 )) = qP T t=k+1 We consider the stationary bootstrap to construct a confidence interval for each statistic of p cross-quantilograms {ρ̂τ (1), . . . , ρ̂τ (p)} for a finite positive integer p and subsequently construct a confidence interval for the omnibus test based on the p statistics. To maintain the original dependence structure, we use (T −p) pairs of observations {[(yt,1 , xt,1 ), . . . , (yt,p , xt,p )]}Tt=p+1 to resample the blocks of random lengths. (p)∗ Given a vector cross-quantilogram ρ̂τ , we define the omnibus test based on the sta(p)∗ tionary bootstrap resample as Q̂τ (p)∗ = T (ρ̂τ (p) ⊤ (p)∗ − ρ̂τ ) (ρ̂τ (p) − ρ̂τ ). The following theorem shows the validity of the stationary bootstrap procedure for the cross-quantilogram. We use the concept of weak convergence in probability conditional on the original sample, which is denoted by “⇒∗ ”, see van der Vaart and Wellner (1996, p. 181). Theorem 2 Suppose that Assumption A1-A6 hold. Then, in the sense of weak convergence conditional on the sample we have: √ (p)∗ (p) (p) ⇒∗ Λτ B(p) (τ ) (a) T ρ̂τ − ρ̂τ in probability; (b) Under the null hypothesis that ρτ (1) = · · · = ρτ (p) = 0 for every τ ∈ T , (p)∗ (p) sup P sup Q̂τ ≤ z − P sup Q̂τ ≤ z →p 0. ∗ z∈R τ ∈T τ ∈T In practice, repeating the stationary bootstrap procedure B times, we obtain B sets (p)∗ ⊤ of cross-quantilograms and {ρ̂τ,b = [ρ̂∗τ,b (1), . . . , ρ̂∗τ,b (p)] }B b=1 and B sets of omnibus tests (p)∗ (p)∗ (p)∗ (p) ⊤ (p)∗ (p) {Q̂τ,b }B b=1 with Q̂τ,b = T (ρ̂τ,b − ρ̂τ ) (ρ̂τ,b − ρ̂τ ). For testing jointly the null of no directional predictability, a critical value, c∗Q,α , corresponding to a significance level α is 11 (p)∗ given by the (1 − α)100% percentile of B test statistics {supα∈T Q̂α,b }B b=1 , that is, c∗Q,α (p)∗ = inf c : P sup Q̂τ,b ≤ c ≥ 1 − α . ∗ τ ∈T For the individual cross-quantilogram, we pick up percentiles (c∗1k,α , c∗2k,α ) of the bootstrap √ √ ∗ ∗ T (ρ̂∗τ,b (k) − ρ̂τ (k)) ≤ c∗2k,α ) = distribution of { T (ρ̂∗τ,b (k) − ρ̂τ (k))}B b=1 such that P (c1k,α ≤ 1 − α, in order to obtain a 100(1 − α)% confidence interval for ρτ (k) given by [ρ̂τ (k) + T −1/2 c∗1k,α, ρ̂τ (k) + T −1/2 c∗2k,α ]. In the following theorem, we provide a power analysis of the omnibus test statistic (p) supτ ∈T Q̂τ when we use a critical value c∗Q,α . We consider fixed and local alternatives. The fixed alternative hypothesis against the null of no directional predictability is H1 : ρτ (k) 6= 0 for some (τ, k) ∈ T × {1, . . . , p}, (6) and the local alternative hypothesis is given by √ H1T : ρτ (k) = ζ/ T for some (τ, k) ∈ T × {1, . . . , p}, (7) where ζ is a finite non-zero constant. Thus, under the local alternative, there exists a p × 1 (p) vector ζτ (p) (p) (p) such that ρτ = T −1/2 ζτ with ζτ having at least one non-zero element for some τ ∈T. We consider the asymptotic power of a test for the directional predictability over a range of quantiles with multiple lags in the following theorem; however, the results can be applied to test for a specific quantile or a specific lag order. The following theorem shows that the √ cross-quantilogram process has non-trivial local power against the T -local alternatives. Theorem 3 Suppose that Assumptions A1-A6 hold. Then: (a) Under the fixed alternative in (6), (p) ∗ lim P sup Q̂τ > cQ,α → 1. T →∞ τ ∈T 12 (b) Under the local alternative in (7) (p) (p) (p) 2 (p) ∗ lim P sup Q̂τ > cQ,α = P sup kΛτ B (τ ) + ζτ k ≥ cQ,α , T →∞ τ ∈T τ ∈T (p) where cQ,α = inf{c : P (supτ ∈T kΛτ B(p) (τ )k2 ≤ c)) ≥ 1 − α}. 3.1.2 The Self-Normalized Cross-Quantilogram We use recursive estimates to construct a self-normalized cross-quantilogram. The selfnormalized approach was proposed by Lobato (2001) and was recently extended by Shao (2010) to a class of asymptotically linear test statistics.6 The self-normalized approach has a tight link with the fixed-b asymptotic framework proposed by Kiefer et al. (2000).7 The self-normalized statistic has an asymptotically pivotal distribution whose critical values have been tabulated so that there is no need for long run variance estimation or even bootstrap. As discussed in section 2.1 of Shao (2010), the self-normalized and the fixed-b approach have better size properties, compared with the standard approach involving a consistent asymptotic variance estimator, while it may be asymptotically less powerful under local alternatives (see Lobato (2001) and Sun et al. (2008) for instance). Given a subsample {(yt , xt )}st=1 , we can estimate sample quantile functions by solving minimization problems β̂i,s (τi ) = arg min βi ∈Rdi s X t=1 ̺τi yit − x⊤ it βi , for i = 1, 2. Let q̂i,t,s (τi ) = x⊤ it β̂i,s (τi ). We consider the minimum subsample size s larger than [T ω], where ω ∈ (0, 1) is an arbitrary small positive constant. The trimming parameter, ω, 6 Kuan and Lee (2006) apply the approach to a class of specification tests, the so-called M tests, which are based on the moment conditions involving unknown parameters. Chen and Qu (2015) propose a procedure for improving the power of the M test, by dividing the original sample into subsamples before applying the self-normalization procedure. 7 The fixed-b asymptotic has been further studied by Bunzel et al. (2001), Kiefer and Vogelsang (2002, 2005), Sun et al. (2008), Kim and Sun (2011) and Sun and Kim (2012) among others. 13 is necessary to guarantee that the quantiles estimators based on subsamples have standard asymptotic properties and plays a different role to that of smoothing parameters in long-run variance estimators. Our simulation study suggests that the performance of the test is not sensitive to the trimming parameter. A key ingredient of the self-normalized statistic is an estimate of cross-correlation based on subsamples: Ps ψτ1 (y1t − q̂1,t,s (τ1 ))ψτ2 (y2,t−k − q̂2,t−k,s (τ2 )) qP , s 2 2 t=k+1 ψτ1 (y1t − q̂1,t,s (τ1 )) t=k+1 ψτ2 (y2,t−k − q̂2,t−k,s (τ2 )) ρ̂τ,s (k) = qP s t=k+1 (p) ⊤ for [T ω] ≤ s ≤ T . For a finite integer p > 0, let ρ̂τ,s = [ρ̂τ,s (1), . . . , ρ̂τ,s (p)] . We construct an outer product of the cross-quantilogram using the subsample V̂τ,p = T −2 T X s=[T ω] (p) s2 ρ̂τ,s − ρ̂τ(p) (p) ρ̂τ,s − ρ̂τ(p) ⊤ . We can obtain the asymptotically pivotal distribution using V̂τ,p as the asymptotically random normalization. For testing the null of no directional predictability, we define the selfnormalized omnibus test statistic ⊤ −1 (p) Ŝτ(p) = T ρ̂τ(p) V̂τ,p ρ̂τ . (p) The following theorem shows that Ŝτ is asymptotically pivotal. To distinguish the process used in the following theorem from the one used in the previous section, let {B̄(p) (·)} denote a p-dimensional, standard Brownian motion on (ℓ([0, 1]))p equipped with the uniform topology. Theorem 4 Suppose that Assumptions A1-A5 hold. Then, for each τ ∈ T , Ŝτ(p) →d B̄(p) (1) where V̄(p) = R1 ω ⊤ V̄(p) −1 B̄(p) (1), ⊤ {B̄(p) (r) − r B̄(p) (1)}{B̄(p) (r) − r B̄(p) (1)} dr. 14 The joint test based on finite multiple quantiles can be constructed in a similar manner, while the extension of the self-normalized approach to a range of quantiles is not obvious. The asymptotic null distribution in the above theorem can be simulated and a critical value, cS,α , corresponding to a significance level α is tabulated by using the (1 − α)100% percentile of the simulated distribution.8 In the theorem below, we consider a power function of the (p) self-normalized omnibus test statistic, P (Ŝτ > cS,α ). For a fixed τ ∈ T , we consider a fixed alternative H1 : ρτ (k) 6= 0 for some k ∈ {1, . . . , p}, (8) √ H1T : ρτ (k) = ζ/ T for some k ∈ {1, . . . , p}, (9) and a local alternative (p) where ζ is a finite non-zero scalar. This implies that there exists a p-dimensional vector ζτ (p) (p) such that ρτ = T −1/2 ζτ (p) with ζτ having at least one non-zero element. Theorem 5 (a) Suppose that the fixed alternative in (8) and Assumptions A1-A5 hold. Then, lim P Ŝτ(p) > cS,α → 1. T →∞ (b) Suppose that the local alternative in (9) is true and Assumptions A1-A5 hold. Then, (p) lim P Ŝτ > cS,τ = P B̄(p) (1) + (Λτ(p) ∆τ(p) )−1 ζτ(p) T →∞ (p) (p) (p) ⊤ ⊤ V (p) −1 where ∆τ is a d0 p × d0 p matrix with ∆τ (∆τ ) ≡ Ξ(p) (τ, τ ). 8 We provide the simulated critical values in our R package. 15 (p) (p) (p) −1 (p) ≥ cS,α , B̄ (1) + (Λτ ∆τ ) ζτ 4 The Partial Cross-Quantilogram We define the partial cross-quantilogram, which measures the relationship between two events {y1t ≤ q1,t (τ1 )} and {y2,t−k ≤ q2,t−k (τ2 )}, while controlling for intermediate events between t and t − k as well as whether some state variables exceed a given quantile. Let zt ≡ [ψτ3 (y3t − ⊤ q3,t (τ3 )), . . . , ψτl (ylt − ql,t (τl ))] be an (l − 2) × 1 vector for l ≥ 3, where qi,t (τi ) = x⊤ it βi (τi ) for τi and a di × 1 vector xit (i = 3, . . . , l), and zt may include the quantile-hit processes based on some of the lagged predicted variables {y1,t−1 , . . . , y1,t−k }, the intermediate predictors {y2,t−1 , . . . , y1,t−k−1} and some state variables that may reflect some historical events up to t.9 ⊤ For simplicity, we present the results for a single set of quantiles τ̄ = (τ1 , . . . , τl ) and a single lag k, although the results can be extended to the case of a range of quantiles and multiple lags in an obvious way. To ease the notational burden in the rest of this section, we consider the case for which a lag k = 0 without loss of generality and suppress the ⊤ ⊤ dependence on k. Let ȳt = [y1t , . . . , ylt ]⊤ and x̄t = [x⊤ 1t , . . . , xlt ] . We introduce the correlation matrix of the hit processes and its inverse matrix i h ⊤ and Pτ̄ = Rτ̄−1 , Rτ̄ = E ht (τ̄ )ht (τ̄ ) where an l × 1 vector of the hit process is denoted by ht (τ̄ ) = [ψτ1 (y1t − q1,t (τ1 )), . . . , ψτl (ylt − ⊤ ql,t (τl ))] . For i, j ∈ {1, . . . , l}, let rτ̄ ,ij and pτ̄ ,ij be the (i, j) element of Rτ̄ and Pτ̄ , re√ spectively. Notice that the cross-quantilogram is rτ̄ ,12 / rτ̄ ,11 rτ̄ ,22 , and the partial crossquantilogram is defined as ρτ̄ \|z = − √ pτ̄ ,12 . pτ̄ ,11 pτ̄ ,22 9 In principle, the intermediate predictors and state variables do not need to be transformed into quantile hits. As emphasized earlier, however, one of the main advantages of considering qauntile hits is its applicability to more general time series, being robust to the existence of moments. If needed, it is straightforward to extend the results here to the case of the original variables in zt with additional moment conditions. We thank an anonymous referee for pointing this out. 16 The partial cross-correlation also has a form ρτ̄ \|z = δ s τ1 (1 − τ1 ) , τ2 (1 − τ2 ) where δ is a scalar parameter defined in the following regression: ψτ1 (y1t − q1,t (τ1 )) = δψτ2 (y2t − q2,t (τ2 )) + γ ⊤ zt + ut , with a (l − 2) × 1 vector γ and an error term ut . Thus, testing the null hypothesis of ρτ̄ \|z = 0 can be viewed as testing predictability between two quantile hits with respect to information z̄ as in Granger causality test based on the regression form (Granger, 1969). By choosing relevant variables z̄, one can use ρτ̄ \|z for the purpose of testing Granger causality (Pierce and Haugh, 1977). See also Hong et al. (2009) for testing Granger causality in tail distribution. To obtain the sample analogue of the partial cross-quantilogram, we first construct a vector of hit processes, ĥt (τ̄ ), by replacing the population conditional quantiles in ht (τ̄ ) by the sample analogues {q̂1,t (τ1 ), . . . , q̂l,t (τl )}. Then, we obtain the estimator for the correlation matrix and its inverse as R̂τ̄ = T 1X ⊤ ĥt (τ̄ )ĥt (τ̄ ) and P̂τ̄ = R̂τ̄−1 , T t=1 which leads to the sample analogue of the partial cross-quantilogram ρ̂τ̄ \|z = − p p̂τ̄ ,12 , p̂τ̄ ,11 p̂τ̄ ,22 (10) where p̂τ̄ ,ij denotes the (i, j) element of P̂τ̄ for i, j ∈ {1, . . . , l}. In Theorem 6 below, we show that ρ̂τ̄ \|z asymptotically follows a normal distribution, while the asymptotic variance depends on nuisance parameters as in the previous section. To address the issue of the nuisance parameters, we may employ the stationary bootstrap or the 17 self-normalization technique. For the bootstrap, we can use pairs of variables {(ȳt , x̄t )}Tt=1 to generate the stationary bootstrap resample {(ȳt∗ , x̄∗t )}Tt=1 and then obtain the stationary bootstrap version of the partial cross-quantilogram, denoted by ρ̂∗τ̄ \|z , using the formula in (10). When we use the self-normalized test statistics, we estimate the partial cross-quantilogram ρτ̄ ,s\|z based on the subsample up to s, recursively and then use V̂τ̄ \|z = T −2 T X s=[T ω] s2 ρ̂τ̄ ,s\|z − ρ̂τ̄ ,T \|z 2 , to normalize the cross-quantilogram, thereby obtaining the asymptotically pivotal statistics. To obtain the asymptotic results, we impose the following conditions on the conditional distribution function Fyi \|xi (·\|xit ) and its density function fyi \|xi (·\|xit ) of each pair of additional variables (yit , xit ) for i = 1, . . . , l and on the pairwise joint distribution Fij (v1 , v2 \|xit , xjt ) ≡ P (yit ≤ v1 , yjt ≤ v2 \|xit , xjt ) for (v1 , v2 ) ∈ R2 . Assumption A7. (a) {(ȳt , x̄t )}t∈Z is a strictly stationary and strong mixing sequence satisfying the condition in Assumption A1; (b) The conditions in Assumption A2 and A3 hold for the Fyi \|xi (·\|xit ) and fyi \|xi (·\|xit ) at the relevant quantile for t = 1, . . . , T , for i = 1, . . . , l; (c) Fij (·\|xit , xjt ) satisfies the condition in Assumption A4 and there exists a vector ⊤ ∇r Gij ≡ ∂/∂br E[Fij (x⊤ it b1 , xjt b2 \|xit , xjt )] evaluated at (b1 , b2 ) = (βi (τi ), βi (τj )) for (r, i, j) ∈ {1, 2}×{1, . . . , l}2 ; (d) There exist positive definite matrices Mi and Di (τi ) as in Assumption A5 for i = 1, . . . , l. Assumption A7(a) requires the same weak dependence property as in Assumption A1. Assumptions A7(b)-(c) ensure the smoothness of the marginal conditional distribution, marginal density function and the joint distribution of each pair (yit , yjt ) given (xit , xjt ) for 1 ≤ i, j ≤ l. Assumption A7(d) is used to derive a Bahadur representation of q̂it (τi ) for i = 1, . . . , l. We now state the asymptotic properties of the partial cross-quantilogram and the related 18 inference methods. Theorem 6 (a) Suppose that Assumption A7 holds. Then, √ T (ρ̂τ̄ \|z − ρτ̄ \|z ) →d N(0, στ̄2\|z ), for each τ̄ ∈ [0, 1]l , where στ̄2\|z = ξτ̄ t = − X 1≤i,j≤l i6=j and λτ̄ i = P P∞ l=−∞ cov(ξτ̄ l , ξτ̄ 0 ) with pτ̄ ,1i pτ̄ ,2j ψτi (yit − qi,t (τi ))ψτj (yjt − qj,t (τj )) + 1≤j≤l j6=i l X i=1 −1 λ⊤ τ̄ i Di (τi ) xit ψτi (yit − qi,t (τi )), (pτ̄ ,1i pτ̄ ,2j + pτ̄ ,2i pτ̄ ,1j ) ∇1 Gij . (b) Suppose that Assumption A6 and A7 hold. Then, sup P ∗ ρ̂∗τ̄ \|z ≤ s − P ρ̂τ̄ \|z ≤ s →p 0, s∈R for each τ̄ ∈ [0, 1]l . (c) Suppose that Assumption A7 holds. Then, under the null hypothesis that ρτ̄ \|z = 0, we have √ T ρ̂τ̄ \|z 1/2 V̂τ̄ \|z for each τ̄ ∈ [0, 1]l . →d nR 1 ω B(1) {B(1) − rB(r)}2dr o1/2 , We can show that the partial cross-quantilogram has non-trivial local power against a √ sequence of T -local alternatives, applying the similar arguments used in Theorem 3 and Theorem 5, and thus we omit the details. 19 5 Monte Carlo Simulation We investigate the finite sample performance of our test statistics. We adopt the following simple VAR model with covariates and consider two data generating processes for the error terms. y1t = 0.1 + 0.3y1,t−1 + 0.2y2,t−1 + 0.3z1t + u1t y2t = 0.1 + 0.2y2,t−1 + 0.3z2t + u2t , where zit ∼ iid χ2 (3)/3 for i = 1, 2. DGP1: (u1t , u2t )⊤ ∼ iid N (0, I2 ) where I2 is a 2 ×2 identity matrix. We let (u1t , u2t , z1t , z2t ) be mutually independent. DGP2:       u1t   σ1t 0   ε1t    =  ε2t 0 1 u2t 2 2 where (ε1t , ε2t )⊤ ∼ iid N (0, I2) and σ1t = 0.1+0.2u21,t−1+0.2σ1,t−1 +u22,t−1 . We let (ε1t , ε2t , z1t , z2t ) be mutually independent. The sample cross-quantilogram defined in (2) adopts conditional quantiles q̂it (τi ) = ⊤ ⊤ ⊤ x⊤ by quantile regression of the above it β̂i (τi ). We first estimate β(τ ) ≡ [β1 (τ1 ) , β2 (τ2 ) ] VAR model, where x1t = (1, y1,t−1, y2,t−1 , z1t )⊤ and x2t = (1, y2,t−1 , z2t )⊤ and then obtain the sample cross-quantilogram using q̂it (τi ) = x⊤ it β̂i (τi ). Under DGP1, there is no predictability from the event {y2,t−k ≤ q2,t−k (τ2 )} to the event {y1t ≤ q1t (τ1 )} for all quantiles τ1 and τ2 , because Pr [y1t ≤ q1t (τ1 ) \| y2,t−k , x2,t−k ] = Pr [u1t ≤ Φ−1 (τ1 )] = τ1 for all t ≥ 1 and τ1 ∈ (0, 1), where Φ denotes the standard normal cdf. Under DGP2, (u1t ) is defined as the GARCH-X process, where its conditional variance is the GARCH(1,1) process with an exogenous covariate. The GARCH-X process is 20 commonly used for modeling volatility of economic or financial time series in the literature, see Han (2015) and references therein. Under DGP2, there exists predictability from 2 {y2,t−k ≤ q2,t−k (τ2 )} to {y1t ≤ q1t (τ1 )} through σ1t for all quantiles (τ1 ,τ2 ) ∈ (0, 1)2 , except the case τ1 = 0.5 because the conditional distribution of u1t given x1t is symmetric around 0.10 5.1 Results Based on the Bootstrap Procedure We first examine the finite-sample performance of the Box-Ljung test statistics based on the stationary bootstrap procedure. To save space, only the results for the case where τ1 = τ2 are reported here because the results for the cases where τ1 6= τ2 are similar. The Box-Ljung (p) test statistics Q̂τ are based on ρ̂τ (k) for τi = 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9 or 0.95 and k = 1, 2, . . . , 5. Tables 1 and 2 report empirical rejection frequencies of the Box-Ljung test statistics based on the bootstrap critical values at the 5% level. The sample sizes considered are T =500, 1,000 and 2,000. The number of simulation repetitions is 1,000. The bootstrap critical values are based on 1,000 bootstrapped replicates. The tuning parameter γ is set to be 0.01.11 In general, our simulation results in Tables 1-3 show that the test has reasonably good size and power performance in finite samples. Table 1 reports the simulation results for the DGP1, which show the size performance. The rejection frequencies are close to 0.05 in mid quantiles, while the test tends to slightly under-reject in low and high quantiles. Table 2 reports the simulation results for the DGP2, which show the power performance. Except for the median, the rejection frequencies approach one as the sample size increases, which shows that our test is consistent. As expected, the rejection frequencies are close to 10 To see this, note that the conditional distribution of u1t given x1t has median zero because Pr(u1t ≤ 0 \| x1t ) = Pr(σ1t ε1t ≤ 0 \| x1t ) = Pr(ε1t ≤ 0 \| x1t ) = Pr(ε1t ≤ 0) = 0.5 and likewise Pr(u1t ≥ 0 \| x1t ) = 0.5. Therefore, letting Ft = (y2,t−k , x2,t−k ), Pr (y1t < q1,t (0.5) \| Ft ) = Pr (u1t < 0 \| Ft ) = Pr (ε1t < 0 \| Ft ) = 0.5. This implies that there is no predictability from {y2,t−k ≤ q2,t−k (τ2 )} to {y1t ≤ q1,t (τ1 )} at τ1 = 0.5 under DGP2. 11 Recall that 1/γ indicates the average block length. We tried different values for γ including one chosen by the data dependent rule suggested by Politis and White (2004) and the results are still similar particularly for a large sample. The details of the data dependent rule is explained in Section 6. 21 0.05 at the median because there is no predictability at the median under the DGP2 (see Footnote 10 for an explanation). Next, we examine the finite-sample performance of the sup-version of the Box-Ljung test (p) statistic supτ ∈T Q̂τ over a range of quantiles.12 The simulation results in Table 3 show that (p) the sup-version test statistic supτ ∈T Q̂τ also has reasonably good finite sample performance, though it tends to under-reject under DGP1. For DGP2, the rejection frequencies approach one as the sample size increases. 5.2 Results for the Self-Normalized Statistics (p) We also examine the performance of the self-normalized version of Q̂τ under the same setup as above. We fix the trimming constant ω to be 0.1.13 The number of repetitions is 3,000. The empirical sizes of the test are reported in Table 4, where the underlying process is the VAR model with DGP1. The test generally under-rejects under the null hypothesis (DGP1), while at the extreme quantiles (τ = 0.05 or 0.95) the test slightly over-rejects in the small sample (T = 500). This finding is not very surprising because the self-normalized statistic is based on subsamples and at the extreme quantiles there are effectively not enough observations to compute the test statistic accurately. Using the GARCH-X process of DGP2, we obtain empirical powers and present the results in Table 5. With a one-period lag (p = 1), the self-normalized quantilogram at τ1 , τ2 ∈ {0.1, 0.2, 0.8, 0.9} rejects the null by about 23.0-30.0%, 64.3-68.3% and 91.7-94.0% for sample sizes 500, 1,000 and 2,000, respectively. In general, the rejection frequencies increase as the sample size increases, decline as the maximum number of lags p increases, and are not sensitive to the choice of the trimming value. Our results suggest that the selfnormalized statistics may have lower power in finite samples compared with the test statistics based on the stationary bootstrap procedure, see Lobato (2001) for a similar finding. 12 Due to computational burden, we compute the Box-Ljung test statistic as a maximum over nine quantile levels τi = 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9 and 0.95. 13 We also considered 0.03 and 0.05 for ω and the results are similar to those for ω = 0.1. 22 6 Empirical Studies 6.1 Stock Return Predictability We apply the cross-quantilogram to detect directional predictability from an economic state variable to stock returns. The issue of stock return predictability has been very important and extensively investigated in the literature; see Lettau and Ludvigson (2010) for an extensive review. A large literature has considered predictability of the mean of stock return. The typical mean return forecast examines whether the mean of an economic state variable is helpful in predicting the mean of stock return (mean-to-mean relationship). Recently, Cenesizoglu and Timmermann (2008) considered whether the mean of an economic state variable is helpful in predicting different quantiles of stock returns representing left tail, right tail or shoulders of the return distribution. The cross-quantilogram adds one more dimension to analyze predictability compared with the linear quantile regression, and so it provides a more complete picture on the relationship between a predictor and stock returns. Moreover, we can consider very large lags in the framework of the quantilogram. We use daily data from 3 Jan. 1996 to 29 Dec. 2006 with sample size 2,717.14 Stock returns are measured by the log price difference of the S&P 500 index and we employ stock variance as the predictor. The stock variance is treated as an estimate of equity risk in the literature. The risk-return relationship is an important issue in the finance literature; see Lettau and Ludvigson (2010) for an extensive review. The cross-quantilogram can provide a more complete relationship from risk to return, which cannot be examined using existing methods. To measure stock variance, we use the realized variance given by the sum of squared 5-minute returns.15 The autoregressive coefficient for stock variance is estimated to be 0.68 and the unit root hypothesis is clearly rejected. The sample mean and median of stock returns are 0.0003 and 0.0005, respectively. 14 The working paper version of this paper provides the results using the monthly data previously analyzed in Goyal and Welch (2008). 15 The realized variance is obtained from ‘Oxford-Man Institute’s realised library’. 23 (p) In Figures 1-3, we provide the sup-type test statistic supτ ∈T Q̂τ , the cross-quantilogram (p) ρ̂τ (k) and the portmanteau test Q̂τ (we use the Box-Ljung versions throughout) to detect directional predictability from stock variance, representing risk, to stock return. In each graph, we show the 95% bootstrap confidence intervals for no predictability based on 1,000 bootstrapped replicates. The tuning parameter 1/γ is chosen by adapting the rule suggested by Politis and White (2004) (and later corrected in Patton et al. (2009)).16 Since it is for univariate data, we apply it separately to each time series and define γ as the average value. (p) We first examine the sup-version Box-Ljung test statistic supτ ∈T Q̂τ and the results are provided in Figure 1. We consider low and high ranges of quantiles. For the low range, we set T = [0.1, 0.3] and τi = 0.1 + 0.02k for k = 0, 1, · · · , 10. For the high range, we set T = [0.7, 0.9] and τi = 0.7 + 0.02k for k = 0, 1, · · · , 10. In each range, there are eleven different values of τi and we let τ1 = τ2 in calculating ρ̂τ (k) for simplicity. Figure 1 clearly shows that there exists predictability from stock variance to stock return in each range. (p) Next we investigate the cross-quantilogram ρ̂τ (k) and the portmanteau test Q̂τ for different quantile points in Figures 2(a)-3(b). For the quantiles of stock return q1 (τ1 ), we consider τ1 = 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9 and 0.95. For the quantiles of stock variance q2 (τ2 ), we consider τ2 = 0.1 and 0.9. Figures 2(a) and 2(b) are for the case when the stock variance is in the low quantile, i.e. τ2 = 0.1. The cross-quantilograms ρ̂τ (k) for τ1 = 0.05, 0.1, 0.2 and 0.3 are negative and significant for many lags. For example, in case of τ1 = 0.05, it means that when risk is very low, it is less likely to have a large negative loss. On the other hand, the cross-quantilograms for τ1 = 0.7, 0.8, 0.9 and 0.95 is positive and significant for many lags. For example, in case of τ1 = 0.95, it means that when risk is very low, it is less likely to have a large positive gain. However, the cross-quantilogram for τ1 = 0.5 is mostly insignificant, which means that risk is not helpful in predicting whether stock return is located below or above its median. Figure 2(b) shows that the Box-Ljung test statistics are mostly significant except for τ1 = 0.5. 16 Specifically, 1/γ̂ = (2Ĝ2 /D̂SB )1/3 T 1/3 where D̂SB = 2ĝ 2 (0). The definitions of ĝ and Ĝ are given on page 58 of Politis and White (2004). 24 Figures 3(a) and 3(b) are for the case when stock variance is in the high quantile, i.e. τ2 = 0.9. Compared to the previous case of τ2 = 0.1, the cross-quantilograms have similar trends but much larger absolute values. For τ1 = 0.05, the cross-quantilogram ρ̂τ (1) is −0.193, which implies that when risk is higher than its 0.9 quantile, there is an increased likelihood of having a very large negative loss in the next day. For τ1 = 0.95, the cross-quantilogram ρ̂τ (1) is 0.188, which implies that when risk is high (higher than its 0.9 quantile), there is an increased likelihood of having a very large positive gain in the next day. The crossquantilogram for τ1 = 0.5 is mostly insignificant and the Box-Ljung test statistics in Figure 3(b) are mostly significant except for τ1 = 0.5. The results in Figures 1-3 show that stock variance is helpful in predicting stock return and detailed features depend on each quantile of stock variance and stock return. When stock variance is in high quantile, the absolute value of the cross-quantilogram is higher and the cross-quantilogram is significantly different from zero for larger lags. Our results exhibit a more complete relationship between risk and return and additionally show how the relationship changes for different lags. 6.2 Systemic Risk The Great Recession of 2007-2009 has motivated researchers to better understand systemic risk—the risk that the intermediation capacity of the entire financial system can be impaired, with potentially adverse consequences for the supply of credit to the real economy. One approach to measure systemic risk is measuring co-dependence in the tails of equity returns of an individual financial institution and the financial system.17 Prominent examples include the work of Adrian and Brunnermeier (2011), Brownlees and Engle (2012) and White et al. (2012). Since the cross-quantilogram measures quantile dependence between time series, we apply it to measure systemic risk. 17 Bisias et al. (2012) categorize the current approaches to measuring systemic risk along the following lines: 1) tail measures, 2) contingent claims analysis, 3) network models, and 4) dynamic stochastic macroeconomic models. 25 We use the daily CRSP market value weighted index return as the market index return as in Brownlees and Engle (2012). We consider returns on JP Morgan Chase (JPM), Morgan Stanley (MS) and AIG as individual financial institutions. As in Brownlees and Engle (2012), JPM, MS and AIG belong to the Depositories group, the Broker-Dealers group and the Insurance group, respectively. We investigate the cross-quantilogram ρ̂τ (k) between an individual institution’s stock return and the market index return for k = 60 and τ1 = τ2 = 0.05. In each graph, we show the 95% bootstrap confidence intervals for no quantile dependence based on 1,000 bootstrapped replicates. The sample period is from 24 Feb. 1993 to 31 Dec. 2014 with sample size 5,505.18 The data including the financial crisis from 2007 and 2009 might not be suitable to be viewed as a strictly stationary sequence and hence may not fit into our theoretical framework.19 Nevertheless, we provide the empirical results because it would be practically interesting to consider a sample period that includes the recent crisis and post-crisis.20 In Figure 4, each graph in the left column shows the cross-quantilogram from each individual institution to the market. The cross-quantilograms are positive and generally significant for large lags. The cross-quantilogram from JPM to the market reaches its peak (0.146) at k = 12 and declines steadily afterwards. This means that it takes about two weeks for the systemic risk from JPM to reach its peak once JPM is in distress. From MS to the market, the cross-quantilogram reaches its peak (0.127) at k = 2. From AIG to the market, the cross-quantilogram reaches its peak (0.127) at k = 17. When AIG is in distress, the systemic risk from AIG takes a longer time (about three weeks) to reach its peak. When an individual financial institution is in distress, each institution makes an influence on the market in a different way. Each graph in the right column of Figure 4 shows the cross-quantilograms from the 18 The stock return series of Morgan Stanley are available from 24 Feb. 1993. The stock return series of individual financial institutions are obtained from Yahoo Finance. 19 A rigorous treatment of nonstationary time series in our context is a challenging issue and will be reported in a future work. 20 The results for the sample period from 24 Feb. 1993 to 29 Dec. 2006 are also available from the authors upon request. 26 market to an individual institution. The cross-quantilogram for this case is a measure of an individual institution’s exposure to system wide distress and therefore it is similar to the stress tests performed by individual institutions. From the market to each institutions, the cross-quantilogram at k = 1 is relatively low for JPM (0.062) and MS (0.073) while it is higher for AIG (0.104). Overall, when the market is in distress, each institution is influenced by its impact in a different way. But the cross-quantilogram reaches its peak at k = 2 for all cases. The cross-quantilograms at k = 2 are 0.135, 0.131 and 0.139 for JPM, MS and AIG, respectively. As shown in Figure 4, the cross-quantilogram is a measure for either an institution’s systemic risk or an institution’s exposure to system wide distress. Compared to existing methods, one important feature of the cross-quantilogram is that it provides in a simple manner how such a measure changes as the lag k increases. For example, White et al. (2012) adopt an additional impulse response function within the multivariate and multiquantile framework to consider tail dependence for a large k. Moreover, another feature of the cross-quantilogram is that it does not require any modeling. For example, the approach by Brownlees and Engle (2012) is based on the standard multivariate GARCH model and it requires the modeling of the entire multivariate distribution. Next, we apply the partial cross-quantilogram to examine the systemic risk after controlling for an economic state variable. Following Adrian and Brunnermeier (2011) and Bedljkovic (2010), we adopt the VIX index as the economic state variable. Since the VIX index itself is highly persistent and can be modeled as an integrated process, we instead use the VIX index change, the first difference of the VIX index level, as the state variable. For the quantile of the state variable, i.e. τ3 in (10), we let τ3 = 0.95 because a low quantile of a stock return is generally associated with a rapid increase of the VIX index. Figure 5 shows that the partial cross-quantilograms are still significant in some cases even if their values are generally lower than the values of the cross-quantilograms in Figure 4. This indicates that there still remains systemic risk from an individual institution after 27 controlling for an economic state variable. These significant partial cross-quantilograms will be of interest for the management of the systemic risk of an individual financial institution. 7 Conclusion We have established the limiting properties of the cross-quantilogram in the case of a finite number of lags. Hong (1996) established the properties of the Box-Pierce statistic in the case that p = pn → ∞ : after a location and scale adjustment the statistic is asymptotically normal, see also Hong et. al. (2009) for a related work. No doubt our results can be extended to accommodate this case, although in practice the desirability of such a test is questionable, and the chi-squared type limit in our theory may provide better critical values for even quite long lags. The cross-quantilogram is easy to compute and the bootstrap confidence intervals appear to represent modest enlargements of the Bartlett intervals in the series that we examined. The statistic shows the cross dependence structure of the time series in a granular fashion that is more informative than the usual methods. 28 Appendix In appendix, we use C, C1 , C2 , . . . to denote generic positive constants without further clarification. Appendix A. Asymptotic Results of Cross-Quantilogram Lemma A1 Let {zt }t∈Z be a strict stationary, strong mixing sequence of Rd -valuedPrandom variables for some integer d ≥ 1 with strong mixing coefficients {αj }j∈Z+ satisfying ∞ j=0 (j + ν/(2s+ν) 1)2s−2 αj ∞. Then, for some integer s ≥ 2 and ν ∈ (0, 1). Suppose that E[z1 ] = 0 and kz1 k2s+ν < E T X t=1 zt 2s 1−s kz1 k2s ≤ T s C kz1 k2s 2s+ν . 2+ν + T Proof. See Supplemental Material. We define the process indexed by τ ∈ T : T o 1 X n (k) 1[yt,k ≤ qt,k (τ )] − E[Fy\|x (qt,k (τ )\|xt,k )] . Vt,k (τ ) := √ T t=k+1 Also, define a di × 1 vector of random variables indexed by τi ∈ Ti for each i = 1, 2: T 1 X Wi,T (τi ) := √ xit ψτi (yit − qi,t (τi )) . T t=1 The below lemma shows the stochastic equicontinuity of the processes defined above, using a similar argument in Bai (1996). Proposition A1 Suppose Assumption A1-A5 hold. Let k ∈ {1, . . . , p} and P2 define metrics ′ ′ ′ ′ ′ ρi (τi , τi ) = \|τi − τi \| for τi , τi ∈ Ti (i = 1, 2) and a metric ρ(τ, τ ) = i=1 ρi (τi , τi ) for ′ τ, τ ∈ T . Then, (a) VT,k (τ ) is stochastically equicontinuous on (T , ρ); (b) Wi,T (τi ) is stochastically equicontinuous on (Ti , ρi ) for each i = 1, 2. Proof. See Supplemental Material. Because of the importance of the result, we present the central limit theorem for strong mixing sequence in the lemma below. The proof can be found in Corollary 5.1 of Hall and Heyde (1980) or Rio (1997, 2013) among others. A.1 Lemma A2 Suppose that the strict stationary sequence {zt }t∈Z satisfies the strong mixing P ς/(2+ς) condition with E[z1 ] = 0 and E\|z1 \|2+ς < ∞ for some ς ∈ (0, ∞), while ∞ < j=1 αj PT −1/2 2 2 2 2 ∞. Then, limT →∞ E[(T ∈ [0, ∞). If σ > 0, then t=1 zt ) ] = σ for some σ PT −1 −1/2 d σ T t=1 zt → N(0, 1). Define a d0 × 1 vector BT,k (τ ) = [VT,k (τ ), W1,T (τ1 )⊤ , W2,T (τ2 )⊤ ]⊤ for τ ∈ T and k = 1, . . . , p. The following proposition shows the weak convergence of the process {BT,k (τ ) : τ ∈ T }pk=1. Proposition A2 Suppose Assumptions A1-A5 hold. Then, [BT,1 (·), . . . , BT,p (·)]⊤ ⇒ [B1 (·), . . . , Bp (·)]⊤ . Proof. Proposition A1 shows that [BT,1 (·), . . . , BT,p (·)]⊤ is stochastic equicontinuous. Thus, it remains to establish convergence of the finite dimensional distributions. By the CramerWold device, it suffices to show J X j=1 θj p X k=1 d (j) 2 κ⊤ , → N 0, σθ,κ k BT,k τ for any {θj ∈ R}Jj=1 , {κk ∈ Rd }pk=1 , {τ (j) ∈ [0, 1]2 }Jj=1, and J ≥ 1, where 2 σθ,κ = J X J X j=1 j ′ =1 θj θj ′ p p X X ′ (j) , τ (j ) )κk′ . κ⊤ k Ξk,k ′ (τ (A-1) k=1 k ′ =1 The original time-series is a stationary sequence satisfying the strong mixing condition in Assumption A1 and a measurable transformation involving lagged variables satisfies the same mixing condition if the lag order is finite. Hence, the central limit theorem for strong-mixing sequences in Lemma A2 shows that the convergence in distribution to the normal law with the finite variance. Therefore, we establish the weak convergence. Let v = (v1 , v2 ) ∈ Rd1 × Rd2 and vt,k = (v1,t , v2,t−k )⊤ ∈ R2 with vi,t = x⊤ it vi for i = 1, 2 and for t = 1, . . . , T . Define T o 1 X n (k) −1/2 −1/2 1[yt,k ≤ qt,k (τ ) + T vt,k ] − E[Fy\|x (qt,k (τ ) + T vt,k \|xt,k )] , VT,k (τ, v) := √ T t=k+1 and T 1 X Wi,T (τi , vi ) := √ xit 1[yit ≤ qi,t (τi ) + T −1/2 vi,t ] − Fyi \|xi (qi,t (τi ) + T −1/2 vi,t \|xit ) . T t=1 Proposition A3 Suppose Assumption A1-A5 hold. Then, A.2 (a) supτ ∈T supv∈VM \|VT,k (τ, v) − VT,k (τ )\| = op (1) for every M > 0; (b) supτi ∈Ti supvi ∈Vi,M kWi,T (τi , vi ) − Wi,T (τi )k = op (1) for every M > 0 and i = 1, 2, where VM = V1,M × V2,M with Vi,M = {vi ∈ Rdi : kvi k ≤ M} for i = 1, 2. Proof. See Supplemental Material. Proposition A4 Suppose Assumption A1-A5 hold. Then, for i = 1, 2 √ T {β̂i (τi ) − βi (τi )} = −Di−1 (τi )Wi,T (τi ) + op (1), uniformly in τ ∈ Ti . Proof. See Supplemental Material. The below lemma shows that the limiting behavior of the cross-quantilogram process reflects the contributions of estimation errors due to the estimation of the conditional quantile function. Proposition A5 Suppose that Assumption A1-A5 hold. Then, for each k ∈ {1, . . . , p}, √ √ VT,k (τ ) + ∇G(k) (τ )⊤ T {β̂(τ ) − β(τ )} p + op (1), T {ρ̂τ (k) − ρτ (k)} = τ1 (1 − τ1 )τ2 (1 − τ2 ) uniformly in τ ∈ T . P Proof. Let γ̂τ,k = T −1 Tt=k+1 ψτ1 (y1t − q̂1,t (τ1 ))ψτ2 (y2,t−k − q̂2,t−k (τ2 )) and γτ,k = E[ψτ1 (y1t − q1,t (τ1 ))ψτ2 (y2,t−k − q2,t−k (τ2 ))]. Using a similar argument in Lemma 2.1 of Arcones (1998), P we can show supτi ∈Ti \|T −1/2 Tt=1 ψτi (yit − q̂i,t (τi ))\| = op (1) for i = 1, 2, because xit includes a constant term. It follows that, uniformly in τ ∈ T , T −1 T X t=1 ψτ2i (yit − q̂i,t (τi )) = τi (1 − τi ) + op (1), for i = 1, 2, (A-2) and √ T (γ̂τ,k − γτ,k ) = T −1/2 T X (k) 1[yt,k ≤ q̂t,k (τ )] − E Fy\|x (qt,k (τ )\|xt,k ) + op (1). t=k+1 Define VM = {v ≡ (v1 , v2 ) ∈ Rd1 × Rd2 : maxi=1,2 kvi k ≤ M} for some M > 0 and let ⊤ ⊤ vt,k = (x⊤ 1t v1 , x2,t−k v2 ) . Then, Proposition A3 implies T −1/2 T X (k) 1[yt,k ≤ qt,k (τ ) + T −1/2 vt,k ] − E Fy\|x (qt,k (τ )\|xt,k ) t=k+1 = VT,k (τ ) + √ (k) (k) T E Fy\|x (qt,k (τ ) + T −1/2 vt,k \|xt,k ) − Fy\|x (qt,k (τ )\|xt,k ) + op (1), A.3 uniformly in (τ, v) ∈ T × VM for any M > 0. Also, the mean-value theorem implies √ (k) (k) T E[Fy\|x (qt,k (τ ) + T −1/2 vt,k \|xt,k ) − Fy\|x (qt,k (τ )\|xt,k )] = ∇G(k) (τ )⊤ v + o(1) uniformly in (τ, v) ∈ T × VM . Thus, for any M > 0, sup (τ,v)∈T ×VM \|RT (τ, v)\| = op (1), (A-3) where RT (τ, v) :=T −1/2 T X (k) 1[yt,k ≤ qt,k (τ ) + T −1/2 vt,k ] − E Fy\|x (qt,k (τ )\|xt,k ) t=k+1 − VT,k (τ ) + ∇G(k) (τ )⊤ v . Let ǫ be an arbitrary positive constant. Proposition A2 √ and A4 imply that there exists a constant M > 0 such that P (supτ ∈T kβ̂(τ ) − β(τ )k > M/ T ) < ǫ for a sufficiently large T . It follows that there exists an M > 0 such that √ sup P sup RT (τ, T {β̂(τ ) − β(τ )}) > ǫ < ǫ + P \|RT (τ, v)\| > ǫ , τ ∈T (τ,v)∈T ×VM for a sufficiently large T . Thus, (A-3) yields √ √ T (γ̂τ,k − γτ,k ) = VT,k (τ ) + ∇G(k) (τ )⊤ T {β̂(τ ) − β(τ )} + op (1), uniformly in τ ∈ T . This together with (A-2) yields the desired result. Proof of Theorem 1. For each i = 1, 2, Proposition A4 yields an asymptotic linear √ −1 approximation, T {β̂i (τi ) − βi (τ √i )} = −Di (τi )Wi,T (τi )⊤+ op (1) uniformly in τi ∈ Ti , which with Proposition A5 shows that T {ρ̂τ (k) − ρτ (k)} = λτ,k BT,k (τ )+op (1) uniformly in τ ∈ T . For a finite p > 0, we have √ (p) T ρ̂τ(p) − ρτ(p) = Λτ(p) BT (τ ) + op (1), (A-4) uniformly in τ ∈ T . The desired result is obtained from Proposition A2 with the continuous mapping theorem. Appendix B. Stationary Bootstrap A positive integer valued, possibly infinite random variable µ is said to be a stopping time with respect to a filtration {Fn , n ≥ 1} if {µ = n} ∈ Fn , ∀n ∈ N. Given random block lengths {Li }i∈N under the stationary bootstrap, define N = inf{i ∈ N : L1 + · · · + Li ≥ n}. Then, N is a stopping time with respect to {σ(L1 , . . . , Li ) : 1 ≤ i ≤ n}. In the following lemma, we present a moment inequality using ideas found in the literature on the sopped random walk process. See Gut (2009) for a comprehensive treatment. Lemma B1 Let {zt }t∈Z be a strict stationary, strong mixing sequence of Rd -valued random A.4 variables for some integer d ≥ 1 with strong mixing coefficients {αj }j∈Z+ satisfying 2s−2 ν/(2s+ν) αj P∞ j=0 (j + 1) for some integer s ≥ 2 and ν ∈ (0, 1). Suppose that kz1 k2s+ν < ∞ and a stationary bootstrap resample, {zt∗ }Tt=1 , from {ztP }Tt=1 satisfies Assumption A6 with the sample Pk+l−1 ∗ zt∗ . Then, size T > 0. Define Sk,l = t=k zt and Sk,l = k+l−1 t=k E where S̃k,l = ∗ S1,T Pk+l−1 t=k −E ∗ ∗ S1,T 2s ∞ n X s πl E S̃1,l ≤ C (T γ) 2s + E S̃1,T l=1 2s o , (zt − Ezt ) for k, l ∈ N. Proof. See Supplemental Material. Lemma B2 Suppose that the same conditions assumed in Lemma B1 hold. Then, 2s s−1 ∗ ∗ ≤ T s C kz1 k2s kz1 k2s E S1,T − E ∗ S1,T 2+ν + γ 2s+ν , for a sufficiently large T . Proof. See Supplemental Material. We now turn to the asymptotic results of cross-quantilogram based on the stationary bootstrap. Define V∗T,k (τ ) and W∗i,T (τi ) T 1 X ∗ √ 1[yt,k ≤ q∗t,k (τ )] − 1[yt,k ≤ qt,k (τ )] := T t=k+1 T 1 X ∗ ∗ √ := xit ψτi yit∗ − qi,t (τi ) − xit ψτi (yit − qi,t (τi )) T t=k+1 for each i = 1, 2. The lemma below shows the stochastic equicontinuity of the processes, V∗T,k (·) and W∗i,T (·), unconditional on the original sample. Proposition B1 Suppose Assumption A1-A6 hold. Let k ∈ {1, . . . , p} and define metrics ρi (·, ·) for i = 1, 2 and a metric ρ(·, ·) as in Proposition A1. Then, (a) V∗T,k (τ ) is stochastically equicontinuous on (T , ρ); (b) W∗i,T (τi ) is stochastically equicontinuous on (Ti , ρi ) for each i = 1, 2. Proof. See Supplemental Material. Let B∗T,k (τ ) = [V∗T,k (τ ), W∗1,T (τ1 )⊤ , W∗2,T (τ2 )⊤ ]⊤ for (k, τ ) ∈ {1, . . . , p} × T and define (p)∗ BT,k (τ ) := [B∗T,1 (τ ), . . . , B∗T,p (τ )]⊤ . As a norm that introduces the topology of (ℓ∞ (T ))pd0 , we use supτ ∈T k · k defined on (ℓ∞ (T ))pd0 , so that supτ ∈T kf (τ )k for any f ∈ (ℓ∞ (T ))pd0 . A.5 Let BL1 be the set of all Lipschitz continuous, real-valued functions on (ℓ∞ (T ))pd0 with a Lipschitz constant bounded by 1. We prove the following proposition by modifying the argument used in Theorem 2 of Galvao et. al. (2014), where the approach of van der Vaart and Wellner (1996, Theorem 2.9.6) is extended for the dependent process but their setup differs from the one here. Proposition B2 Suppose Assumptions A1-A6 hold. Then, (p)∗ sup E ∗ h(BT ) − E h(B(p) ) →p 0. h∈BL1 Proof. Let δ > 0. Given the compact set T in [0, 1]2 , there exists a finite partition (j) (j) {T (j) }Jj=1 such that max1≤j≤J supτ ′ ,τ ′′ ∈T (j) kτ ′′ −τ ′ k ≤ δ. Pick up τ (j) ≡ (τ1 , τ2 )⊤ ∈ T (j) for j = 1, . . . , J and let Πδ be a map from T to {τ (j) }Jj=1 so that Πδ (τ ) = τ (j) if τ ∈ T (j) . Define (p)∗ (p)∗ (p)∗ BT ◦ Πδ and B(p) ◦ Πδ as the stochastic processes on T , given by BT ◦ Πδ (τ ) = BT (Πδ (τ )) and B(p) ◦ Πδ (τ ) = B(p) (Πδ (τ )) for τ ∈ T . It follows from the triangle inequality that, for any h ∈ BL1 , (p)∗ (p)∗ (p)∗ E ∗ h(BT ) − E h(B(p) ) ≤ E ∗ h(BT ) − E ∗ h(BT ◦ Πδ ) (p)∗ + E ∗ h(BT ◦ Πδ ) − E h(B(p) ◦ Πδ ) + E h(B(p) ◦ Πδ ) − E h(B(p) ) . (A-5) (A-6) (A-7) It suffices to show that (A-5) - (A-7) are op (1) uniformly in h ∈ BL1 . We first consider (A-5). We have (p)∗ (p)∗ (p)∗ (p)∗ ∗ ∗ E sup E h(BT ) − E h(BT ◦ Πδ ) ≤ E sup h(BT ) − h(BT ◦ Πδ ) . h∈BL1 h∈BL1 (p)∗ (p)∗ ∗ Let IT,δ,ǫ := 1[supτ ∈T kBT (τ ) − BT ◦ Πδ (τ )k > ǫ] for ǫ > 0. Proposition B1 implies that (p)∗ (p)∗ ∗ ] < ǫ for every ǫ > 0. Also suph∈BL1 \|h(BT ) − h(BT ◦ Πδ )\| ≤ 2 limδ↓0 limT →∞ E[IT,δ,ǫ because the range of a function h is [−1, 1]. It follows that (p)∗ (p)∗ ∗ lim lim E sup h(BT ) − h(BT ◦ Πδ ) · IT,δ,ǫ ≤ 2ǫ. δ↓0 T →∞ (p)∗ h∈BL1 (p)∗ (p)∗ (p)∗ Since suph∈BL1 \|h(BT ) − h(BT ◦ Πδ )\| ≤ supτ ∈T kBT (τ ) − BT ◦ Πδ (τ )k, we have (p)∗ (p)∗ ∗ E sup h(BT ) − h(BT ◦ Πδ ) · (1 − IT,δ,ǫ) ≤ ǫ. h∈BL1 (p)∗ (p)∗ Thus, limδ↓0 limT →∞ E[suph∈BL1 \|h(BT )−h(BT ◦Πδ )\|] ≤ 3ǫ. An application of the Markov inequality yields that (A-5) is op (1) uniformly in h ∈ BL1 . A.6 (p)∗ Next we shall show that suph∈BL1 \|E ∗ [h(BT ◦ Πδ )] − E[h(B(p) ◦ Πδ )]\| →p 0 for any (p)∗ δ > 0. It suffices to show that {BT (τ (j) )}Jj=1 →d {B(p) (τ (j) )}Jj=1 conditional on the original sample, sequence. To this end, we use the Cramer-Wold device Pp every PJfor almost ⊤ ∗ (j) for some {θj ∈ R}Jj=1 and {κk ∈ Rd }pk=1. Let and consider k=1 κk BT,k τ j=1 θj P P P P (j) ∗ (j) ), where ξt,k (·) is defined in ) and vt = Jj=1 θj pk=1 κ⊤ vt∗ = Jj=1 θj pk=1 κ⊤ k ξt,k (τ k ξt,k (τ ∗ Section 3 and ξt,k (·) is its bootstrap counterpart. Then, we can write J X θj j=1 p X ∗ κ⊤ k BT,k k=1 τ (j) =T −1/2 T X (vt∗ − vt ). t=k+1 As discussed in Proposition A2, {vt }t∈N is a stationary time-series satisfying Assumption 1. As shown in p. 1237 of Kunsch (1989), the moment and strong-mixing assumption imposed on the original time series implies the condition imposed on the forth joint cumulant in (8) of Politis and Romano (1994). Hence, Theorems 1 and 2 of Politis and Romano (1994) 2 2 imply that the bootstrap estimate of the variance converges to σθ,κ in probability, where σθ,κ is defined in (A-1), and that we obtained the distribution convergence conditional on the original sample. Finally, consider (A-7). The process B(p) is uniformly continuous on T , which withthe dominated convergence theorem yields that limδ↓0 suph∈BL1 E h(B(p) ◦ Πδ ) − E h(B(p) ) = 0. Hence, we obtain the desired conclusion. ∗ ∗ ∗ ∗ For v = (v1 , v2 ) ∈ Rd1 × Rd2 , let vt,k = (v1,t , v2,t−k )⊤ with vi,t = x∗⊤ it vi for i = 1, 2. Define V∗T,k (τ, v) := T −1/2 T X ∗ ∗ 1[yt,k ≤ q∗t,k (τ ) + T −1/2 vt,k ] − 1[yt,k ≤ qt,k (τ ) + T −1/2 vt,k ] , t=k+1 and W∗i,T (τi , vi ) := T −1/2 T X ∗ ∗ ∗ xit ψτi yit∗ − qi,t (τi ) − T −1/2 vi,t − xit ψτi yit − qi,t (τi ) − T −1/2 vi,t . t=k+1 Proposition B3 Suppose Assumption A1-A6 hold. Then, (a) supτ ∈T supv∈VM \|V∗T,k (τ, v) − V∗T,k (τ )\| = op (1) for every M > 0; (b) supτi ∈Ti supvi ∈Vi,M kW∗i,T (τi , vi ) − W∗i,T (τi )k = op (1) for every M > 0 and i = 1, 2, where VM = V1,M × V2,M with Vi,M = {vi ∈ Rdi : kvi k ≤ M} for i = 1, 2. Proof. See Supplemental Material. Proposition B4 Suppose Assumption A1-A6 hold. Then, for i = 1, 2, √ T {β̂i∗ (τi ) − βi (τi )} = 1 −Di−1 (τi ) √ T A.7 T X t=k+1 ∗ x∗it ψτi (yit∗ − qi,t (τi )) + op (1), uniformly in τi ∈ Ti . Proof. A similar argument used in Proposition A4 completes the proof and thus the details are omitted. Proposition B5 Suppose that Assumption A1-A6 hold. Then, for each k ∈ {1, . . . , p}, √ √ V∗T,k (τ ) + ∇G(k) (τ )⊤ T {β̂ ∗ (τ ) − β̂(τ )} ∗ p + op (1), T {ρ̂τ (k) − ρ̂τ (k)} = τ1 (1 − τ1 )τ2 (1 − τ2 ) uniformly in τ ∈ T . P ∗ ∗ ∗ ∗ ∗ Proof. Let γ̂τ,k = T −1 Tt=k+1 ψτ1 (y1t − q̂1,t (τ1 ))ψτ2 (y2,t−k − q̂2,t−k (τ2 )). Using a similar arguP ment used to show Lemma 2.1 of Arcones (1998), we can show supτi ∈Ti \|T −1/2 Tt=1 ψτi (yit∗ − ∗ q̂i,t (τi ))\| = op (1) for i = 1, 2. It follows that T −1 T X t=k+1 ψτ2i (yit∗ − q̂it∗ (τi )) = τi (1 − τi ) + op (1), for i = 1, 2, and √ T ∗ γ̂τ,k − γ̂τ,k = T −1/2 T X ∗ 1[yt,k ≤ q̂∗t,k (τ )] − 1[yt,k ≤ q̂t,k (τ )] + op (1), t=k+1 uniformly in τi ∈ Ti and τ ∈ T , respectively. As in Proposition A5, we can show T √ (k) 1 X √ 1[yt,k ≤ q̂t,k (τ )]−E Fy\|x (qt,k (τ )xt,k ) = VT,k (τ )+∇G(k) (τ )⊤ T {β̂(τ )−β(τ )}+oP (1), T t=k+1 uniformly in τ ∈ T . A similar argument used in Proposition A5 together with Proposition B3 and A3 yields that, uniformly in τ ∈ T , T h i 1 X ∗ (k) √ 1[yt,k ≤ q̂∗t,k (τ )] − E Fy\|x (qt,k (τ )\|xt,k ) T t=k+1 = VT,k (τ )∗ + VT,k (τ ) √ + ∇G(k) (τ )⊤ T {β̂ ∗ (τ ) − β(τ )} + op (1). √ √ ∗ It follows that T (γ̂τ,k − γ̂τ,k ) = V∗T,k (τ ) + ∇G(k) (τ )⊤ T {β̂ ∗ (τ ) − β̂(τ )} + op (1) uniformly in τ ∈ T . Thus, we obtained the desired result. √ (p)∗ ∗(p) (p) Proof of Theorem 2. (a) Define the processes ĜT (τ ) := T (ρ̂τ − ρ̂τ ) and G(p) (τ ) := (p) Λτ B(p) (τ ) for τ ∈ T and for an integer p > 0. Let g BL1 denote the set of all Lipschitz ∞ p continuous, real-valued functions on (ℓ (T )) with a Lipschitz constant bounded by 1. It A.8 suffices to show that (p)∗ sup E ∗ h(GT ) − E h(G(p) ) →p 0. g1 h∈BL (p)∗ (p) (p)∗ Let GT (τ ) := Λτ BT (τ ). We can write (p)∗ (p)∗ (p)∗ sup E ∗ h(ĜT ) − E h(G(p) ) ≤ sup E ∗ h(ĜT ) − E ∗ h(GT ) g1 h∈BL g1 h∈BL (p)∗ + sup E ∗ h(GT ) − E h(G(p) ) . g1 h∈BL √ Propositions A4 and B4 imply that T {β̂i∗ (τi )−β̂i (τi )} = −Di−1 (τi )W∗1,T (τi )+op(1) uniformly in τi ∈ Ti for each i = 1, 2. It follows from Proposition B3 that √ ∗ T {ρ̂∗τ (k) − ρ̂τ (k)} = λ⊤ τ,k BT,k (τ ) + op (1), uniformly in τ ∈ T , where λτ,k is defined in (4). This leads to (p)∗ (p)∗ sup ĜT (τ ) − GT (τ ) = op (1). τ ∈T (p)∗ (p)∗ (p)∗ (p)∗ This implies that suph∈BL g1 \|h(ĜT )−h(GT )\| = op (1), because suph∈BL g1 \|h(ĜT )−h(GT )\| ≤ (p)∗ (p)∗ supτ ∈T kĜT (τ ) − GT (τ )k. It follows from the dominated convergence theorem that (p)∗ (p)∗ ∗ ∗ limT →∞ E suph∈BL g1 \|E [h(ĜT )] − E [h(GT )]\| = 0. An application of the Markov inequal(p)∗ (p)∗ ∗ ∗ ity shows that suph∈BL g1 \|E [h(ĜT )] − E [h(GT )]\| = op (1). (p) Under Assumption A4 and A5, Λτ is bounded uniformly in τ ∈ T , we have (p)∗ (p)∗ sup E ∗ h(GT ) − E h(G(p) ) ≤ C1 sup E ∗ g(BT ) − E g(B(p) ) , g∈BL1 g1 h∈BL where the right-hand side is negligible in probability from Proposition B2. Hence, we obtain the desired result. (b) From the continuous mapping theorem, the result in (a) of this theorem yields the desired result. See Theorem 10.8 of Kosorok (2007) for a general argument. Proof of Theorem 3. As shown in (A-4), under both fixed and local alternatives, √ (p) T ρ̂τ(p) − ρτ(p) = Λτ(p) BT (τ ) + op (1) (p) (p) uniformly in τ ∈ T , and it follows from Theorem 1 that Λτ BT (τ ) = OP (1) uniformly in τ ∈T. (p) (a) Under the fixed alternative, there is some τ ∈ T such that ρτ is some non-zero √ (p) √ (p) (p) (p) constant and then T ρ̂τ = Λτ BT (τ ) + T ρτ + op (1) uniformly in τ ∈ T . This im(p) (p) plies that, under the fixed alternative, supτ ∈T Q̂τ = T supτ ∈T kρτ k2 (1 + op (1)). Thus, (p) supτ ∈T Q̂τ →p ∞ under the fixed alternative, whereas the critical value c∗Q,τ is bounded in A.9 (p) probability from Theorem 2. Therefore, limT →∞ P (supτ ∈T Q̂τ > c∗Q,τ ) = 1. Therefore, our test is shown to be consistent under the fixed alternative. √ (p) (p) (p) (b) Under the local alternative, we can write ρτ = ζτ / T , where ζτ is a p-dimensional constant vector, at least one of elements is non-zero. Thus, we have (p) Q̂τ(p) = kΛτ(p) BT (τ ) + ζτ(p) k2 + oP (1), uniformly in τ ∈ T . From Theorem 1 and the continuous mapping theorem, sup Q̂τ(p) ⇒ sup kΛτ(p) B(p) (τ ) + ζτ(p)k2 . τ ∈T τ ∈T (p)∗ Also, Theorem 2 implies supτ ∈T Q̂τ desired result follows. (p) ⇒∗ supτ ∈T kΛτ B(p) (τ )k2 in probability. Thus, the Appendix C. Self-Normalized Cross-Quantilogram Lemma C1 Let {zt }t∈Z be a strict stationary, strong mixing sequence of Rd -valuedPrandom variables for some integer d ≥ 1 with strong mixing coefficients {αj }j∈Z+ satisfying ∞ j=0 (j + ν/(2s+ν) 1)2s−2 αj ∞. Then, for some integer s ≥ 2 and ν ∈ (0, 1). Suppose that E[z1 ] = 0 and kz1 k2s+ν < E sup r∈[0,1] [T r] X t=1 zt 2s ≤ CT s z1 2s 2+ν + T 1−s z1 2s+ν . Proof. The desired result follows from Theorem 6.3 and Annexes C of Rio (2013) as in Lemma A1. We define the process indexed by r ∈ [0, 1] [T r] o 1 X n (k) 1[yt,k ≤ qt,k (τ )] − E[Fy\|x (qt,k (τ )\|xt,k )] , V̄T,k,τ (r) := √ T t=k+1 and [T r] 1 X W̄i,T,τi (r) := √ xit {1[yit ≤ qi,t (τi )] − τi } , T t=1 for each i = 1, 2. The following proposition shows the stochastic equicontinuity of the processes defined above. Proposition C1 Suppose Assumption A1-A5 hold. Let k ∈ {1, . . . , p} and define metrics ρ̄(r, r ′ ) = \|r ′ − r\| for r, r ′ ∈ [0, 1]. Then, (a) V̄T,k,τ (r) is stochastically equicontinuous on ([0, 1], ρ̄). A.10 (b) W̄i,T,τi (r) is stochastically equicontinuous on ([0, 1], ρ̄) for each i = 1, 2. Proof. See Supplemental Material. Define a d0 × 1 vector B̄T,k,τ (r) = [V̄T,k,τ (r), W̄1,T,τ1 (r)⊤ , W̄2,T,τ2 (r)⊤ ]⊤ . for r ∈ [0, 1] and k = 1, . . . , p. The following proposition shows the weak convergence of the process {B̄T,k,τ (r) : r ∈ [0, 1]}pk=1 . Proposition C2 Suppose Assumptions A1-A5 hold. Then, B̄T,1,τ (·), . . . , B̄T,p,τ (·) ⊤ ⊤ ⇒ B̄1,τ (·), . . . , B̄p,τ (·) . ⊤ Proof. Proposition C1 establishes the stochastic equicontinuity of B̄T,1,τ (·), . . . , B̄T,p,τ (·) and it suffices to show convergence of the finite dimensional distributions. Since the finite dimensional convergences can be shown by a similar argument used in Proposition A2, we omit the details. For v = (v1 , v2 ) ∈ Rd1 × Rd2 , we define [T r] h io 1 X n (k) 1[yt,k ≤ qt,k (τ ) + T −1/2 vt,k ] − E Fy\|x qt,k (τ ) + T −1/2 vt,k \|xt,k , V̄T,k,τ (r, v) := √ T t=k+1 and [T r] 1 X W̄i,T,τi (r, vi ) := √ xit 1[yit ≤ qi,t (τi ) + T −1/2 vi,t ] − Fyi \|xi qi,t (τi ) + T −1/2 vi,t \|xit . T t=k+1 Proposition C3 Suppose Assumption A1-A5 hold. Then, (a) supω≤r≤1 supv∈VM \|V̄T,k,τ (r, v) − V̄T,k,τ (r)\| = op (1) for every M > 0; (b) supω≤r≤1 supvi ∈Vi,M kW̄i,T,τi (r, vi ) − W̄i,T,τi (r)k = op (1) for every M > 0 and for i = 1, 2, where VM = V1,M × V2,M with Vi,M = {vi ∈ Rdi : kvi k ≤ M} for i = 1, 2. Proof. See Supplemental Material. Proposition C4 Suppose Assumption A1-A5 hold. Then, for i = 1, 2 and for each τi ∈ Ti , √ T {β̂i,[T r] (τi ) − βi (τi )} = −Di−1 (τi )r −1 W̄i,T,τi (r) + op (1), uniformly in r ∈ [ω, 1]. Proof. The proof follows the line of Proposition A4 with Proposition C3(b). Hence, we omit the details. A.11 Proposition C5 Suppose Assumption A1-A5 hold. Then, for each (k, τ ) ∈ {1, . . . , p} × T , √ √ r −1 V̄T,k,τ (r) + ∇G(k) (τ )⊤ T {β̂[T r] (τ ) − β(τ )} p T ρ̂τ,[T r] (k) − ρτ (k) = + op (1), τ1 (1 − τ1 )τ2 (1 − τ2 ) ⊤ ⊤ ⊤ uniformly in r ∈ [ω, 1], where β̂[T r] = (β̂1,[T r] , β̂2,[T r] ) . Proof. A similar argument used in Proposition A5 with Proposition C3(a) yields the desired result and thus we omit the detail. Proof of Theorem 4. Proposition C4 and C5 imply that, for each (k, τ ) ∈ {1, . . . , p} × T , √ T ρ̂τ,[T r] (k) − ρτ (k) = r −1 λ⊤ τ,k B̄T,k,τ (r) + op (1), uniformly in r ∈ [ω, 1]. It follows that in r ∈ [ω, 1]. This implies √ (p) (p) (p) (p) T (ρ̂τ,[T r] − ρτ ) = r −1 Λτ B̄T,τ (r) + op (1) uniformly n o [T r] (p) (p) (p) (p) √ ρ̂τ,[T r] − ρ̂τ,T = Λτ(p) B̄T,τ (r) − r B̄T,τ (1) + op (1), T (p) (p) (p) uniformly in r ∈ [ω, 1]. From Proposition C2, {Λτ (B̄T,τ (r) − r B̄T,τ (1)) : r ∈ [ω, 1]} weakly (p) (p) (p) converges to {Λτ (B̄τ (r) − r B̄τ (1)) : r ∈ [ω, 1]}, which is equivalent in distribution to (p) a p × 1 vector of the Brownian bridge process {∆τ (B̄(p) (r) − r B̄(p) (1)) : r ∈ [ω, 1]} with (p) (p) ∆τ (∆τ )⊤ ≡ Ξ(p) (τ, τ ),and thus it follows from the continuous mapping theorem that √ (p) T ρ̂τ,T , V̂τ,p →d ∆τ(p) B̄(p) (1), ∆τ(p) V̄(p)(∆τ(p) )⊤ . (p) Thus, we obtain Ŝτ →d B̄(p) (1)⊤ (V̄(p) )−1 B̄(p) (1). This completes the proof. Proof of Theorem 5. Under both fixed and local alternative, the argument used in Theorem 4 gives √ (p) (p) T ρ̂τ,[T r] − ρτ(p) = r −1 Λτ(p) B̄T,τ (r) + op (1), (p) (p) (p) (p) ⊤ thereby yielding V̂τ,p ⇒ (Λτ ∆τ )V̄(p) (Λτ ∆τ ) . √ (p) √ (p) (p) (p) (a) Under the fixed alternative, we have T ρ̂τ,T = Λτ B̄T,τ (1) + T ρτ + op (1), where the right-hand side diverges in probability as T → ∞. Since the critical value we use is finite in probability from Theorem 4, we obtain result. √ (p)the desired (p) (p) (p) (b) Under the local alternative, T ρ̂τ,T = Λτ B̄T,τ (1) + ξτ + op (1). It follows that Ŝτ(p) →d B̄(p) (1) + (Λτ(p) ∆τ(p) )−1 ξτ(p) ⊤ V̄(p) This completes the proof. A.12 −1 (p) B̄ (1) + (Λτ(p) ∆τ(p) )−1 ξτ(p) . Appendix D. Partial Cross-Quantilogram For 1 ≤ i, j ≤ l, let 1ij = 1[yit ≤ qi,t (τi ), yjt ≤ qj,t (τj )] and define VT,ij T T 1 X 1 X =√ (1ij − E [1ij ]) and Wi,T = √ xit ψτi (yit − qi,t (τi )) . T t=1 T t=1 Proof of Theorem 6. We first consider (a). The correlation matrix Rτ̄ is symmetric and R̂τ̄−1 − Rτ̄−1 = −R̂τ̄−1 (R̂τ̄ − Rτ̄ )Rτ̄−1 . It follows that vec(P̂τ̄ − Pτ̄ ) = −Pτ̄ ⊗ P̂τ̄ vec(R̂τ̄ − Rτ̄ ), which implies l l X X √ √ pτ̄ ,1i p̂τ̄ ,2j T (r̂τ̄ ,ij − rτ̄ ,ij ). T (p̂τ̄ ,12 − pτ̄ ,12 ) = − i=1 j=1 Following the line of proof of Theorem 1, we can show P̂τ̄ = Pτ̄ + op (1) and also have √ T (r̂τ̄ ,ii − rτ̄ ,ii ) = op (1) for i = 1, . . . , l, from argument in Lemma 2.1 of Arcones (1998). Thus, we have X √ √ pτ̄ ,1i pτ̄ ,2j T (r̂τ̄ ,ij − rτ̄ ,ij ) + op (1). T (p̂τ̄ ,12 − pτ̄ ,12 ) = − 1≤i,j≤l i6=j Proposition A5 implies √ √ T (r̂τ̄ ,ij − rτ̄ ,ij ) = VT,ij + ∇1 G⊤ ij T {β̂i (τi ) − βi (τi )} √ + ∇2 G ⊤ ij T {β̂j (τj ) − βj (τj )} + op (1), for 1 ≤ i, j ≤ l with i 6= j. Since VT,ij = VT,ji and ∇2 Gij = ∇1 Gji for 1 ≤ i, j ≤ l, √ T (p̂τ̄ ,12 − pτ̄ ,12 ) = − X 1≤i,j≤l i6=j pτ̄ ,1i pτ̄ ,2j VT,ij − l X i=1 √ λ⊤ τ̄ i T {β̂i (τi ) − βi (τi )} + op (1), where λτ̄ i is defined in Theorem 6. Proposition A4 implies √ T (p̂τ̄ ,12 − pτ̄ ,12 ) = − X pτ̄ ,1i pτ̄ ,2j VT,ij + l X −1 λ⊤ τ̄ i Di (τi ) Wi,T + op (1). i=1 1≤i,j≤l i6=j The asymptotic normality can be established by using the central limit theorem for mixing random vectors. The proofs of (b) and (c) are similar to those of Theorems 2 and 4, respectively, and thus we omit the details. A.13 Appendix E. Tables and Figures Table 1. (size) Empirical rejection frequency of the Box-Ljung test statistic the bootstrap procedure (VAR with DGP1 and the nominal level 5%) Quantiles (τ1 = τ2 ) T p 0.05 0.10 0.20 0.30 0.50 0.70 0.80 0.90 500 1 0.051 0.025 0.037 0.045 0.040 0.043 0.043 0.033 2 0.017 0.032 0.043 0.072 0.068 0.060 0.057 0.036 3 0.011 0.022 0.051 0.073 0.066 0.055 0.050 0.032 4 0.007 0.022 0.047 0.062 0.059 0.057 0.046 0.026 5 0.009 0.025 0.035 0.052 0.051 0.052 0.054 0.027 1000 1 0.033 0.030 0.037 0.048 0.047 0.039 0.037 0.052 2 0.018 0.037 0.045 0.051 0.043 0.046 0.052 0.041 3 0.011 0.031 0.049 0.056 0.044 0.054 0.045 0.028 4 0.013 0.027 0.049 0.053 0.041 0.055 0.041 0.022 5 0.007 0.022 0.044 0.040 0.044 0.040 0.036 0.021 2000 1 0.038 0.034 0.040 0.034 0.034 0.048 0.050 0.034 2 0.028 0.025 0.043 0.035 0.045 0.051 0.050 0.035 3 0.023 0.033 0.031 0.045 0.050 0.045 0.042 0.029 4 0.017 0.023 0.042 0.052 0.038 0.036 0.038 0.025 5 0.009 0.025 0.038 0.038 0.035 0.035 0.034 0.019 (p) Q̂τ based on 0.95 0.047 0.012 0.010 0.008 0.006 0.042 0.015 0.006 0.008 0.006 0.054 0.024 0.018 0.016 0.014 Notes: The first and second columns report the sample size T and the number of lags p for the Box(p) Ljung test statistics Q̂τ , respectively. The rest of columns show empirical rejection frequencies based on bootstrap critical values at the 5% significance level. The tuning parameter γ is set to be 0.01. A.14 (p) Table 2. (power) Empirical rejection frequency of the Box-Ljung test statistic Q̂τ based on the bootstrap procedure (VAR with DGP2 (GARCH-X process)) Quantiles (τ1 = τ2 ) T p 0.05 0.10 0.20 0.30 0.50 0.70 0.80 0.90 0.95 500 1 0.361 0.701 0.722 0.383 0.042 0.383 0.713 0.684 0.362 2 0.303 0.610 0.584 0.257 0.063 0.231 0.589 0.589 0.300 3 0.270 0.541 0.491 0.202 0.053 0.174 0.467 0.515 0.246 4 0.230 0.451 0.403 0.172 0.058 0.126 0.378 0.447 0.208 5 0.203 0.393 0.344 0.134 0.060 0.115 0.314 0.386 0.177 1000 1 0.751 0.948 0.942 0.638 0.048 0.619 0.951 0.952 0.760 2 0.708 0.916 0.912 0.425 0.046 0.431 0.908 0.932 0.712 3 0.651 0.877 0.845 0.322 0.052 0.315 0.849 0.897 0.651 4 0.589 0.838 0.784 0.255 0.048 0.250 0.778 0.854 0.596 5 0.537 0.801 0.716 0.203 0.042 0.190 0.714 0.809 0.563 2000 1 0.969 0.999 0.999 0.905 0.044 0.923 0.999 0.998 0.974 2 0.965 1.000 0.999 0.808 0.053 0.817 0.999 1.000 0.979 3 0.959 1.000 0.997 0.688 0.053 0.673 0.998 1.000 0.967 4 0.944 1.000 0.990 0.585 0.047 0.573 0.994 0.999 0.957 5 0.930 1.000 0.982 0.510 0.037 0.485 0.987 0.997 0.938 Notes: Same as Table 1. A.15 Table 3. Empirical Rejection Frequencies of the sup-version of the Box-Ljung test statistic (p) supτ ∈T Q̂τ based on the bootstrap procedure (VAR with DGP1/DGP2 and the nominal level 5%) T p DGP1 (size) DGP2 (power) 500 1 0.004 0.624 2 0.007 0.460 3 0.008 0.356 4 0.008 0.265 5 0.009 0.221 1000 1 0.004 0.976 2 0.011 0.946 3 0.006 0.895 4 0.003 0.825 5 0.007 0.765 2000 1 0.012 1.000 2 0.015 1.000 3 0.020 1.000 4 0.020 1.000 5 0.017 0.999 Notes: The first and second columns report the sample size T and the number of lags p for the sup(p) version of the Box-Ljung test statistic supτ ∈T Q̂τ , respectively. The sup-version test statistic is the Box-Ljung test statistic maximized over nine quantiles τi = 0.05, 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9and 0.95. The third and fourth columns show empirical rejection frequencies based on bootstrap critical values at the 5% significance level. The tuning parameter γ is set to be 0.01. A.16 Table 4. (size) Empirical Rejection Frequencies of the Self-Normalized Statistics (VAR with DGP1 and the nominal level: 5%) Quantiles (τ1 = τ2 ) T p 0.05 0.10 0.20 0.30 0.50 0.70 0.80 0.90 0.95 500 1 0.043 0.000 0.000 0.007 0.003 0.013 0.007 0.000 0.047 2 0.090 0.010 0.007 0.003 0.003 0.003 0.000 0.003 0.127 3 0.130 0.007 0.000 0.007 0.003 0.000 0.003 0.000 0.143 4 0.150 0.007 0.000 0.000 0.000 0.000 0.000 0.000 0.167 5 0.187 0.003 0.000 0.000 0.000 0.000 0.000 0.000 0.177 1000 1 0.010 0.013 0.010 0.013 0.020 0.003 0.007 0.003 0.007 2 0.023 0.007 0.000 0.007 0.000 0.003 0.003 0.007 0.037 3 0.040 0.003 0.010 0.000 0.007 0.003 0.007 0.000 0.047 4 0.043 0.000 0.007 0.000 0.007 0.003 0.003 0.000 0.047 5 0.047 0.000 0.007 0.000 0.000 0.000 0.003 0.000 0.053 2000 1 0.013 0.030 0.017 0.017 0.033 0.013 0.020 0.017 0.027 2 0.007 0.000 0.007 0.007 0.027 0.010 0.027 0.017 0.020 3 0.017 0.000 0.003 0.003 0.013 0.010 0.003 0.003 0.013 4 0.013 0.000 0.003 0.000 0.010 0.007 0.003 0.000 0.013 5 0.010 0.003 0.003 0.000 0.007 0.003 0.000 0.000 0.017 Notes: The first and second columns report the sample size T and the number of lags p for the (p) test statistics Q̂τ , respectively. The rest of columns show empirical rejection frequencies given simulated critical values at 5% significance level. The trimming value ω is set to be 0.1. Table 5. (power) Empirical Rejection Frequencies of the Self-Normalized (VAR with DGP2: GARCH-X process) Quantiles (τ1 = τ2 ) T p 0.05 0.10 0.20 0.30 0.50 0.70 0.80 0.90 500 1 0.067 0.230 0.297 0.077 0.007 0.150 0.300 0.253 2 0.030 0.070 0.113 0.033 0.000 0.037 0.113 0.077 3 0.047 0.010 0.043 0.010 0.000 0.017 0.023 0.020 4 0.063 0.007 0.023 0.000 0.000 0.010 0.013 0.003 5 0.120 0.003 0.007 0.003 0.000 0.003 0.003 0.000 1000 1 0.347 0.643 0.683 0.313 0.010 0.323 0.673 0.663 2 0.153 0.523 0.527 0.177 0.020 0.180 0.543 0.463 3 0.063 0.300 0.347 0.090 0.010 0.097 0.377 0.283 4 0.033 0.210 0.223 0.050 0.000 0.037 0.243 0.153 5 0.047 0.097 0.133 0.030 0.000 0.023 0.127 0.097 2000 1 0.757 0.917 0.923 0.663 0.030 0.693 0.940 0.920 2 0.577 0.873 0.917 0.513 0.013 0.540 0.883 0.863 3 0.427 0.787 0.860 0.400 0.007 0.397 0.800 0.810 4 0.270 0.680 0.807 0.323 0.017 0.297 0.740 0.680 5 0.197 0.567 0.700 0.223 0.003 0.213 0.680 0.590 Notes: Same as Table 4. A.17 Statistics 0.95 0.050 0.010 0.023 0.050 0.080 0.317 0.157 0.063 0.017 0.020 0.707 0.577 0.390 0.250 0.163 Sup−version for low range Sup−version for high range 1400 1800 1600 1200 1400 Portmanteau Portmanteau 1000 800 600 1200 1000 800 600 400 400 200 0 200 0 10 20 30 Lag 40 50 60 0 0 10 (p) 20 30 Lag 40 50 60 Figure 1. Sup-version Box-Ljung test statistic supτ ∈T Q̂τ for each lag p to detect directional predictability from stock variance to stock return. For the low range, we set T = [0.1, 0.3]and τi = 0.1 + 0.02kfor k = 0, 1, . . . , 10. We let τ1 = τ 2 for ρ̂τ (k).For the high range, we set T = [0.7, 0.9] and τi = 0.7 + 0.02k for k = 0, 1, . . . , 10. The dashed lines are the 95% bootstrap confidence intervals centred at the null hypothesis. A.18 τ = 0.05 τ = 0.1 Quantilogram 1 Quantilogram 1 0.2 0.2 0.2 0 0 0 −0.2 0 20 40 τ = 0.3 60 −0.2 0 1 20 40 τ = 0.5 60 −0.2 0.2 0 0 0 0 20 40 τ = 0.8 60 −0.2 0 20 40 τ = 0.9 60 −0.2 0.2 0 0 0 20 40 60 −0.2 0 20 40 Lag 60 20 40 τ = 0.95 60 20 60 1 0.2 0 0 1 0.2 −0.2 20 40 τ = 0.7 1 0.2 −0.2 0 1 0.2 1 Quantilogram τ = 0.2 1 60 −0.2 0 40 Lag Lag Figure 2(a). The sample cross-quantilogram ρ̂τ (k) for τ2 = 0.1 to detect directional predictability from stock variance to stock return. Bar graphs describe sample cross-quantilograms and lines are the 95% bootstrap confidence intervals centred at zero. τ = 0.05 τ = 0.1 Portmanteau 1 500 0 0 20 40 τ = 0.3 60 Portmanteau 1 1000 1000 500 500 0 0 1 Portmanteau τ = 0.2 1 20 40 τ = 0.5 60 60 500 0 0 20 40 60 0 20 40 τ = 0.95 1 60 0 20 60 1 200 50 20 40 τ1 = 0.8 20 40 τ = 0.7 400 100 0 0 1 150 500 0 0 60 0 0 20 40 τ = 0.9 1 60 0 1000 1000 500 500 0 0 Lag 20 40 Lag (p) 60 0 40 Lag Figure 2(b). Box-Ljung test statistic Q̂τ for each lag p and quantile τ using ρ̂τ (k) with τ2 = 0.1. The dashed lines are the 95% bootstrap confidence intervals centred at zero. A.19 τ = 0.05 τ = 0.1 Quantilogram 1 Quantilogram 1 0.2 0.2 0.2 0 0 0 −0.2 0 20 40 τ = 0.3 60 −0.2 0 1 20 40 τ = 0.5 60 −0.2 0.2 0 0 0 0 20 40 τ = 0.8 60 −0.2 0 20 40 τ = 0.9 60 −0.2 0.2 0 0 0 20 40 60 −0.2 0 20 40 Lag 60 20 40 τ = 0.95 60 20 60 1 0.2 0 0 1 0.2 −0.2 20 40 τ = 0.7 1 0.2 −0.2 0 1 0.2 1 Quantilogram τ = 0.2 1 60 −0.2 0 40 Lag Lag Figure 3(a). The sample cross-quantilogram ρ̂τ (k) with τ2 = 0.9 to detect directional predictability from stock variance to stock return. Same as Figure 1(a). τ = 0.05 τ = 0.1 Portmanteau 1 1000 1000 500 500 0 20 40 τ = 0.3 60 0 0 Portmanteau 1 Portmanteau 1 1500 1000 0 τ = 0.2 1 2000 20 40 τ = 0.5 60 60 0 0 20 40 τ = 0.9 1 60 0 1000 2000 2000 500 1000 1000 0 20 40 0 20 40 τ = 0.95 1 60 0 20 60 500 50 0 60 1000 100 20 40 τ1 = 0.8 20 40 τ = 0.7 1 150 0 0 1 500 0 0 60 0 0 Lag 20 40 Lag (p) Figure 3(b). Box-Ljung test statistic Q̂τ Same as Figure 1(b). 60 0 40 Lag for each lag p and quantile τ using ρ̂τ (k) with τ2 = 0.9. A.20 Market to JPM 0.2 0.15 0.15 Quantilogram Quantilogram JPM to Market 0.2 0.1 0.05 0 −0.05 −0.1 0.1 0.05 0 −0.05 0 20 40 −0.1 60 0 20 Lag 0.15 0.15 0.1 0.05 0 −0.05 0.1 0.05 0 −0.05 0 20 40 −0.1 60 0 20 Lag AIG to Market 60 Market to AIG 0.2 0.15 0.15 Quantilogram Quantilogram 40 Lag 0.2 0.1 0.05 0 −0.05 −0.1 60 Market to MS 0.2 Quantilogram Quantilogram MS to Market 0.2 −0.1 40 Lag 0.1 0.05 0 −0.05 0 20 40 60 −0.1 Lag 0 20 40 60 Lag Figure 4. The sample cross-quantilogram ρ̂τ (k). Bar graphs describe sample cross-quantilograms and lines are the 95% bootstrap confidence intervals centred at zero. A.21 Market to JPM 0.2 0.15 0.15 Partial Quantilogram Partial Quantilogram JPM to Market 0.2 0.1 0.05 0 −0.05 −0.1 0 20 40 0.1 0.05 0 −0.05 −0.1 60 0 20 Lag 0.15 0.15 0.1 0.05 0 −0.05 0 20 40 0.1 0.05 0 −0.05 −0.1 60 0 20 Lag 0.15 0.15 0.1 0.05 0 −0.05 20 60 Market to AIG 0.2 Partial Quantilogram Partial Quantilogram AIG to Market 0 40 Lag 0.2 −0.1 60 Market to MS 0.2 Partial Quantilogram Partial Quantilogram MS to Market 0.2 −0.1 40 Lag 40 60 0.1 0.05 0 −0.05 −0.1 Lag 0 20 40 60 Lag Figure 5. The sample partial cross-quantilogram ρ̂τ̄ \|z (k). Bar graphs describe sample partial cross-quantilograms and lines are the 95% bootstrap confidence intervals centred at zero. A.22 References Adrian, T. and M.K. Brunnermeier (2011) CoVaR, Tech. rep., Federal Reserve Bank of New York, Staff Reports. Andrews, D.W.K. and D. 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arXiv:1301.6890v1 [math.AC] 29 Jan 2013 TIGHT CLOSURE WITH RESPECT TO A MULTIPLICATIVELY CLOSED SUBSET OF AN F -PURE LOCAL RING RODNEY Y. SHARP Abstract. Let R be a (commutative Noetherian) local ring of prime characteristic that is F -pure. This paper studies a certain finite set I of radical ideals of R that is naturally defined by the injective envelope E of the simple R-module. This set I contains 0 and R, and is closed under taking primary components. For a multiplicatively closed subset S of R, the concept of tight closure with respect to S, or S-tight closure, is discussed, together with associated concepts of S-test element and S-test ideal. It is shown that an ideal a of R belongs to I if and only if it is the S ′ -test ideal of R for some multiplicatively closed subset S ′ of R. When R is complete, I is also ‘closed under taking test ideals’, in the following sense: for each proper ideal c in I, it turns out that R/c is again F -pure, and if g and h are the unique ideals of R that contain c and are such that g/c is the (tight closure) test ideal of R/c and h/c is the big test ideal of R/c, then both g and h belong to I. The paper ends with several examples. 0. Introduction This paper is concerned with a (commutative Noetherian) local ring R having maximal ideal m and prime characteristic p; the Frobenius homomorphism f : R −→ R, for which f (r) = r p for all r ∈ R, will play a central rôle. Let us (temporarily) use Rf to denote the Abelian group R considered as an (R, R)-bimodule with r1 · r · r2 = r1 rr2p for all r, r1 , r2 ∈ R. The ring R is said to be F -pure if and only if, for every R-module M, the map αM : M −→ Rf ⊗R M for which αM (m) = 1 ⊗ m, for all m ∈ M, is injective. This property can be reformulated in terms of certain left modules over the skew polynomial ring R[x, f ] associated to R and f in the indeterminate x over R, which we refer to as the Frobenius skew polynomial ring over R. Recall that R[x, f ] is, as a left R-module, freely generated by (xi )i∈N0 (we use N0 (respectively N) to denote the set (respectively positive) integers), and so consists of all polynomials Pnof non-negative i i=0 ri x , where n ∈ N0 and r0 , . . . , rn ∈ R; however, its multiplication is subject to the rule xr = f (r)x = r p x for all Lr∞∈ R. Note that R[x, f ] can benconsidered as a positively-graded ring R[x, f ] = n=0 R[x, f ]n , with R[x, f ]n = Rx for n ∈ N0 . If we endow Rxn with its natural structure as an (R, R)-bimodule (inherited from its being a graded component of R[x, f ]), then Rxn is isomorphic (as (R, R)-bimodule) Date: June 22, 2017. 2010 Mathematics Subject Classification. Primary 13A35, 16S36, 13E05, 13E10, 13H05; Secondary 13J10. Key words and phrases. Commutative Noetherian ring, local ring, prime characteristic, Frobenius homomorphism, tight closure, test element, big test element, test ideal, big test ideal, excellent ring, Frobenius skew polynomial ring, F -pure ring, complete local ring. 1 2 RODNEY Y. SHARP to R viewed as a left R-module in the natural way and as a right R-module via f n , the nth iterate of the Frobenius ring homomorphism. In particular, Rx ∼ = Rf as (R, R)-bimodules. A left R[x, f ]-module G is said to be x-torsion-free precisely when xg = 0, where g ∈ G, implies that g = 0. We can say that R is F -pure if and L only if,n for every R-module M, the graded left R[x, f ]-module R[x, f ] ⊗R M = n∈N0 Rx ⊗R M is x-torsion-free. We shall also use the following alternative characterization of F -purity. 0.1. Theorem ([9, Theorem 3.2]). The local ring (R, m) is F -pure if and only if the R-module structure on ER (R/m), the injective envelope of the simple R-module, can be extended to an x-torsion-free left R[x, f ]-module structure. In the paper [7], some useful properties of x-torsion-free left R[x, f ]-modules were developed, and it is appropriate to recall some of them at this point. L The graded two-sided ideals of R[x, f ] are just the subsets of the form n∈N0 an xn , where (an )n∈N0 is an ascending sequence of ideals of R. (Of course, such a sequence a0 ⊆ a1 ⊆ · · · ⊆ an ⊆ · · · is eventually stationary.) Let H be a left R[x, f ]-module. An R[x, f ]-submodule of H is said to be a special annihilator submodule of H if it has the form annH (A) := {h ∈ H : θh = 0 for all θ ∈ A} for some graded two-sided ideal A of R[x, f ]. We shall use A(H) to denote the set of special annihilator submodules of H. The graded annihilator gr-annR[x,f ] H of H is defined to be the largestL graded twosided ideal of R[x, f ] that annihilates H. Note that, if gr-annR[x,f ] H = n∈N0 an xn , then a0 = (0 :R H), which we shall sometimes refer to as the R-annihilator of H. We shall use I(H) (or IR (H) when it is desirable to specify the ring R) to denote the set of R-annihilators of R[x, f ]-submodules of H. 0.2. The Basic Correspondence ([7, §1, §3]). Let G be an x-torsion-free left module over R[x, f ]. (i) By [7, Lemma 1.9], the members of I(G) are all radical ideals of R; they are referred to as the G-special R-ideals;L in fact, the graded annihilator of an R[x, f ]-submodule L of G is equal to n∈N0 axn , where a = (0 :R L). (ii) By [7, Proposition 1.11], there is an order-reversing bijection, Θ : A(G) −→ I(G) given by Θ : N 7−→ gr-annR[x,f ] N ∩ R = (0 :R N). The inverse bijection, Θ−1 : I(G) −→ A(G), also order-reversing, is given by Θ−1 : b 7−→ annG (bR[x, f ]). (iii) By [7, Theorem 3.6 and Corollary 3.7], the set of G-special R-ideals is precisely the set of all finite intersections of prime G-special R-ideals (provided one includes the empty intersection, R, which corresponds under the bijection of part (ii) to the zero special annihilator submodule of G). In symbols, I(G) = {p1 ∩ · · · ∩ pt : t ∈ N0 and p1 , . . . , pt ∈ I(G) ∩ Spec(R)} . (iv) By [7, Corollary 3.11], when G is Artinian as an R-module, the sets I(G) and A(G) are both finite. S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 3 Thus, if (R, m) is F -pure and we endow ER (R/m) with a structure as x-torsionfree left R[x, f ]-module (this is possible, by Theorem 0.1), then the resulting set I(ER (R/m)) is finite. In fact, rather more can be said. 0.3. Theorem ([9, Corollary 4.11]). Suppose that (R, m) is F -pure and local. Then the left R[x, f ]-module R[x, f ] ⊗R ER (R/m) is x-torsion-free, and, furthermore, the set I(R[x, f ] ⊗R ER (R/m)) of (R[x, f ] ⊗R ER (R/m))-special R-ideals, is finite. In fact, for any x-torsion-free left R[x, f ]-module structure on ER (R/m) that extends its R-module structure (and such exist, by Theorem 0.1), we have I(R[x, f ] ⊗R ER (R/m)) ⊆ I(ER (R/m)), and the latter set is finite. It turns out that, in the situation of Theorem 0.3, all the minimal prime ideals of R belong to I(R[x, f ] ⊗R ER (R/m)), and the smallest ideal of positive height in I(R[x, f ] ⊗R ER (R/m)) is the big test ideal of R, that is, the ideal generated by all big test elements for R. (A reminder about big test elements is included in the next section.) In short, to each F -pure local ring (R, m) of characteristic p, there is naturally associated a finite set of radical ideals I(R[x, f ] ⊗R ER (R/m)) of R, and the smallest member of positive height in this set has significance for tight closure theory. The purpose of this paper is to address the following question: what can be said about the other members of I(R[x, f ] ⊗R ER (R/m))? We shall show that, for each multiplicatively closed subset S of R, one can define reasonable concepts of S-tight closure, S-test element and S-test ideal; it turns out that an ideal b of R is a member of I(R[x, f ] ⊗R ER (R/m)) if and only if it is the S-test ideal of R for some choice of multiplicatively closed subset S of R. We shall also show that, when R is complete, I(R[x, f ] ⊗R ER (R/m)) is ‘closed under taking (big) test ideals’, in the following sense: it turns out that, for each proper ideal c ∈ I(R[x, f ] ⊗R ER (R/m)), the ring R/c is again F -pure, and the set I(R[x, f ] ⊗R ER (R/m)) has among its membership the unique ideals g and h of R such that g, h ⊇ c and g/c = τe(R/c), the big test ideal of R/c, and h/c = τ (R/c), the test ideal of R/c. I am grateful to Mordechai Katzman for helpful discussions about the material in this paper. 1. Internal S-tight closure 1.1. Notation. From this point onwards in the paper, R will denote a commutative Noetherian ring of prime characteristic p. We shall only assume that R is local when this is explicitly stated; then, the notation ‘(R, m)’ will denote that m is the maximal ideal of R. As in tight closure theory, we use R◦ to denote the complement in R of the union of the minimal prime ideals of R. The Frobenius homomorphism on R will always be denoted by f . We shall use Φ (or ΦR when it is desirable to specify which ring is being considered) to denote the functor R[x, f ] ⊗R • from the category of R-modules (and all R-homomorphisms) to the category of all N0 -graded left R[x, f ]-modules (and all homogeneous R[x, f ]-homomorphisms). For an R-module M, we shall identify Φ(M) 4 RODNEY Y. SHARP L n with n∈N0 Rx ⊗R M, and (sometimes) identify its 0th component R ⊗R M with M, in the obvious ways. For n ∈ Z, we shall denote the nth component of a Z-graded module L by Ln . Throughout the paper, S will denote a multiplicatively closed subset of R. (We require that each multiplicatively closed subset of R contains 1.) Also throughout the paper, H will denote a left R[x, f ]-module and G will denote an x-torsion-free left R[x, f ]-module. 1.2. Definitions. We define the internal S-tight closure of zero in H, denoted ∆S (H) (or ∆SR (H)), to be the R[x, f ]-submodule of H given by ∆S (H) = {h ∈ H : there exists s ∈ S with sxn h = 0 for all n ≫ 0} . Note that if the left R[x, f ]-module H is Z-graded, then ∆S (H) is a graded submodule. Let M be an R-module, and consider Φ(M), as in 1.1. Now ∆S (Φ(M)) is a graded R[x, f ]-submodule of Φ(M); we refer to the 0th component of ∆S (Φ(M)) as the Stight closure of 0 in M, or the tight closure with respect to S of 0 in M, and denote ∗,S it by 0∗,S when it is clear what M is). M (or by 0 ∗,S Thus 0M is the set of all elements m of M for which there exists s ∈ S such that, ∗,R◦ is the usual for all n ≫ 0, we have sxn ⊗ m = 0 in Rxn ⊗R M. In particular, 0M tight closure of 0 in M. (See M. Hochster and C. Huneke [2, §8].) Now let N be an R-submodule of M. The inverse image of 0∗,S M/N under the natural epimorphism M −→ M/N is defined to be the S-tight closure of N in M, and is ∗,S ∗,S denoted by NM or N ∗,S . Thus NM is the set of all elements m of M for which there exists s ∈ S such that, for all n ≫ 0, the element sxn ⊗ m of Rxn ⊗R M belongs to ∗,R◦ the image of Rxn ⊗R N in Rxn ⊗R M. Note that NM is the usual tight closure of N in M. (See Hochster–Huneke [2, §8].) I am grateful to the referee for pointing out that the S-tight closure of N in M is the tight closure of N in M with respect to C in the sense of Hochster–Huneke [2, Definition (10.1)], where C is {sR : s ∈ S}, the family of principal ideals generated by elements of S, directed by reverse inclusion ⊇. The S-tight closure of a in R is referred to simply as the S-tight closure of a and is denoted by a∗,S . The fact that there is a homogeneous R[x, f ]-isomorphism M n R/a[p ] R[x, f ] ⊗R (R/a) ∼ = n∈N0 n (where the right-hand side has the left R[x, f ]-module structure for which x(r+a[p ] ) = n+1 r p + a[p ] for all r ∈ R) enables one to conclude that n n a∗,S = r ∈ R : there exists s ∈ S with sr p ∈ a[p ] for all n ≫ 0 . ◦ Thus a∗,R is the usual tight closure a∗ of a. (See Hochster–Huneke [2, §3].) 1.3. Examples. In the situation of Definition 1.2, let S, T be multiplicatively closed subsets of R with S ⊆ T . Then clearly ∆S (H) ⊆ ∆T (H). (i) Note that {1} is a multiplicatively closed subset of R, and ∆{1} (H) = {h ∈ H : xn h = 0 for all n ≫ 0} = {h ∈ H : there exists n ∈ N0 with xn h = 0}, S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 5 the x-torsion submodule Γx (H) of H. Thus Γx (H) ⊆ ∆S (H) and therefore (0 :R ∆S (H)) ⊆ (0 :R Γx (H)). (ii) Let M be an R-module and let h, n ∈ N0 . Endow Rxn and Rxh with their natural structures as (R, R)-bimodules (inherited from their being graded components of R[x, f ]). Then there is an isomorphism of (left) R-modules φ : ∼ = Rxn+h ⊗R M −→ Rxn ⊗R (Rxh ⊗R M) for which φ(rxn+h ⊗m) = rxn ⊗(xh ⊗m) for all r ∈ R and m ∈ M. One can use isomorphisms like that described in the above paragraph to see that ∗,S ∗,S ∆S (R[x, f ] ⊗R M) = 0∗,S M ⊕ 0Rx⊗R M ⊕ · · · ⊕ 0Rxn ⊗R M ⊕ · · · . 1.4. Notation. We define MS (G), for the x-torsion-free left R[x, f ]-module G, to be the set of minimal members (with respect to inclusion) of the set {p ∈ Spec(R) ∩ I(G) : p ∩ S 6= ∅} of prime G-special R-ideals that meet S. T When MS (G) is finite, we shall set b := p∈MS (G) p, although we shall write bS,G for b when it is desirable to indicate S and G; in that case, it follows from 0.2(iii) that b is the smallest member of I(G) that meets S (and, in particular, b is contained in every other member of I(G) that meets S). (In the special case where MS (G) = ∅, we interpret b as R, the intersection of the empty family of prime ideals of R.) 1.5. Proposition. Consider the x-torsion-free left R[x, f ]-module G. The set MS (G) (see 1.4) is finite if and only if (0 :R ∆S (G)) ∩ S 6= ∅. When these conditions are satisfied, and b denotes the intersection of the prime ideals in the finite set MS (G), then ∆S (G) = annG (bR[x, f ]) and (0 :R ∆S (G)) = b. Proof. (⇐) Set c := (0 :R ∆S (G)) and assume that there exists s ∈ c ∩ S. Since G is x-torsion-free and ∆S (G) is an R[x, f ]-submodule of G, it follows from 0.2(i) that c is radical and gr-annR[x,f ] ∆S (G) = cR[x, f ]. Now annG (cR[x, f ]) ⊆ annG ((sR)R[x, f ]) ⊆ ∆S (G) ⊆ annG (cR[x, f ]), and so annG (cR[x, f ]) = ∆S (G). Note that c ∈ I(G). If c = R, then ∆S (G) = 0, so that MS (G) is empty because a p ∈ I(G) ∩ Spec(R) with p ∩ S 6= ∅ must satisfy annG (pR[x, f ]) ⊆ ∆S (G) = 0, and this leads to a contradiction to 0.2(ii). We therefore suppose that c 6= R. Let c = p1 ∩ · · · ∩ pt be the minimal primary decomposition of the (radical) ideal c. By [7, Theorem 3.6 and Corollary 3.7], the prime ideals p1 , . . . , pt all belong to I(G); they all meet S. Since Spec(R) satisfies the descending chain condition, each member of {p′ ∈ Spec(R) ∩ I(G) : p′ ∩ S 6= ∅} contains a member of MS (G). In particular, each of p1 , . . . , pt contains a member of MS (G). Now let p ∈ MS (G). Since p meets S, we must have that annG (pR[x, f ]) ⊆ ∆S (G) = annG (cR[x, f ]). It now follows from the inclusion-reversing bijective correspondence of 0.2(ii) that p ⊇ c, so that p contains one of p1 , . . . , pt . We can therefore conclude that p1 , . . . , pt are precisely the minimal members of {p ∈ I(G) ∩ Spec(R) : p ∩ S 6= ∅}. 6 RODNEY Y. SHARP Therefore MS (G) is finite. (⇒) Assume that MS (G) is finite. Then b is the smallest member of I(G) that meets S. Consequently, annG (bR[x, f ]) ⊆ ∆S (G). Let g ∈ ∆S (G), so that there L exist s′ ∈ S and n0 ∈ N0 such that s′ xn g = 0 for all n ≥ n0 . Thus g ∈ annG ( n≥n0 Rs′ xn ) =: J, a special annihilator submodule of G. Let b′ be the Gspecial R-ideal that corresponds toL this special annihilator submodule (in the bijective correspondence of 0.2(ii)). Since n≥n0 Rs′ xn ⊆ gr-annR[x,f ] J = b′ R[x, f ], we have s′ ∈ b′ , so that b′ ∩ S 6= ∅. Therefore b′ ⊇ b, and g ∈ J = annG (b′ R[x, f ]) ⊆ annG (bR[x, f ]). Therefore ∆S (G) ⊆ annG (bR[x, f ]). We conclude that ∆S (G) = annG (bR[x, f ]), and that (0 :R ∆S (G)) = (0 :R annG (bR[x, f ])) = b. All the claims in the statement of the proposition have now been proved. 1.6. Definition. An S-test element for R is an element s ∈ S such that, for every R-module M and every j ∈ N0 , the element sxj annihilates 1 ⊗ m ∈ (Φ(M))0 for every m ∈ 0∗,S M . The ideal of R generated by all the S-test elements for R is called the S-test ideal of R, and denoted by τ S (R). One of the aims of this paper is to show that S-test elements for R exist when R is F -pure and local. This is a suitable point at which to remind the reader of some of the classical concepts related to tight closure test elements. 1.7. Reminder. Recall that a test element for modules for R is an element c ∈ R◦ such that, for every finitely generated R-module M and every j ∈ N0 , the element cxj annihilates 1 ⊗ m ∈ (Φ(M))0 for every m ∈ 0∗M . The phrase ‘for modules’ is inserted because Hochster and Huneke have also considered a concept of a test element for ideals for R, which is defined to be an element c ∈ R◦ such that, for every cyclic Rmodule M and every j ∈ N0 , the element cxj annihilates 1 ⊗ m ∈ (Φ(M))0 for every m ∈ 0∗M . When R is reduced and excellent, the concepts of test element for modules and test element for ideals for R coincide: see [2, Discussion (8.6) and Proposition (8.15)]. T Hochster and Huneke define the test ideal τ (R) of R to be M (0 :R 0∗M ), where the intersection is taken over all finitely generated R-modules M. In [2, Proposition (8.23)(b)] they show that, if R has a test element for modules, then τ (R) is the ideal generated by the test elements for modules, and τ (R) ∩ R◦ is the set of test elements for modules. 1.8. Reminder. Recall also that a big test element for R is an element c ∈ R◦ such that, for every R-module M and every j ∈ N0 , the element cxj annihilates 1 ⊗ m ∈ (Φ(M))0 for every m ∈ 0∗M . We let τe(R) denote the ideal generated by all big test elements for R, and call this the big test ideal of R. Note that an R◦ -test element for R, in the sense of Definition 1.6, is just a big test element for R. Recall that an injective cogenerator of R is an injective R-module E with the property that, for every R-module M and every non-zero element m ∈ M, there exists an R-homomorphism φ : M −→ E such that φ(m) 6= 0. As R is Noetherian, L m∈Max(R) ER (R/m), where Max(R) denotes the set of maximal ideals of R, is one injective cogenerator of R. S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 7 L 1.9. Reminders. Let E := m∈Max(R) ER (R/m), an injective cogenerator of R. (i) Recall from Hochster–Huneke [2, Definition (8.19)] that the finitistic tight S ∗ closure of 0 in E, denoted by 0∗fg E , is defined to be M 0M , where the union is taken over all finitely generated R-submodules M of E. It was shown in [2, Proposition (8.23)(d)] that τ (R) = (0 :R 0∗fg E ). ∗fg ∗ (ii) It is conjectured that (0 :R 0E ) = (0 :R 0E ). This conjecture is known to be true (a) if R is an excellent Gorenstein local ring (see K. E. Smith [12, p. 48]); (b) if R is the localization of a finitely generated N0 -graded algebra over an F -finite field K (of characteristic p and having K as its component of degree 0) at its unique homogeneous maximal ideal (see G. Lyubeznik and K. E. Smith [5, Corollary 3.4]); (c) if R is a Cohen–Macaulay local ring which is Gorenstein on its punctured spectrum (see Lyubeznik–Smith [6, Theorem 8.8]); or (d) if (R, m) is local and an isolated singularity (see Lyubeznik–Smith [6, Theorem 8.12]). (iii) It was shown in [10, Theorem 3.3] that if R has a big test element, then the ◦ big test ideal τe(R) of R is equal to (0 :R ∆R (Φ(E))), and the set of big test ◦ elements for R is (0 :R ∆R (Φ(E))) ∩ R◦ . 2. Existence of S-test elements in F -pure local rings Theorem 0.3 was proved by means of an ‘embedding theorem’ [9, Theorem 4.10]. Similar embedding theorems were established in [8, Theorem 3.5] and [10, Theorem 3.2]. In this paper, the ideas underlying those theorems are going to be pursued further, and so we begin with three remarks and a lemma that can be viewed as addenda to [10, §2]. The notation in this section is as described in 1.1. e denote the graded companion of H described in [10, Example 2.1. Remark. Let H S (H), so that (0 : ∆S (H)) e = ∆^ e = (0 :R ∆S (H)). 2.1]. It is easy to check that ∆S (H) R 2.2. Remark. Let (H (λ) )λ∈Λ be a non-empty family of Z-graded left R[x, f ]-modules, Q and consider the graded product ′λ∈Λ H (λ) of the H (λ) , described in [8, Lemma 2.1] and [10, Example 2.2]. It is routine to check that, if there ideal d0 of R such Q′ is an (λ) H = d0 . that (0 :R ∆S (H (λ) )) = d0 for all λ ∈ Λ, then 0 :R ∆S λ∈Λ L 2.3. Remark. Assume that the left R[x, f ]-module H = n∈Z Hn is Z-graded; let t ∈ Z. Denote by H(t) the result of application of the tth shift functor to H; this is described in [10, Example 2.3]. It is clear that ∆S (H(t)) = ∆S (H)(t), so that (0 :R ∆S (H(t))) = (0 :R ∆S (H)). L 2.4. Lemma. Let b, h ∈ N and let W := n≥b Wn be a graded left R[x, f ]-module. Let (gi )i∈I be a family of arbitrary elements of Wb . Consider the h-place extension exten(W ; (gi )i∈I ; h) of W by (gi )i∈I , defined in [9, Definition 4.5] and [10, §2]. Then (0 :R ∆S (exten(W ; (gi )i∈I ; h))) = (0 :R ∆S (W )). Proof. This can be proved by making obvious modifications to the proof, presented in ◦ ◦ [10, Proposition 2.4(i)], that (0 :R ∆R (exten(W ; (gi )i∈I ; h))) = (0 :R ∆R (W )). 8 RODNEY Y. SHARP The above remarks and lemma are helpful for use in the proof of some of the claims in the following Embedding Theorem. 2.5. Embedding Theorem. (See [10, Theorem 3.2].) Let E be an injective Lcogenerator of R. Assume that there exists an N0 -graded left R[x, f ]-module H = n∈N0 Hn such that H0 is R-isomorphic to E. Let M be an R-module. Then there is a family L(n) n∈N0 of N0 -graded left R[x, f ]modules, where L(n) is an n-place extension of the −nth shift of a graded product of copies of H (for each n ∈ N0 ), for which there exists a homogeneous R[x, f ]monomorphism Y′ M L(n) =: K. (Rxi ⊗R M) −→ ν : Φ(M) = i∈N0 n∈N0 Consequently, (0 :R ∆S (H)) = (0 :R ∆S (K)) ⊆ (0 :R ∆S (Φ(M))). Furthermore, if H is x-torsion-free, then so too is Φ(M), and I(Φ(M)) ⊆ I(H) and (0 :R ∆S (Φ(M))) ∈ I(H). Proof. The existence of K and ν with the stated properties were proved in [10, Theorem 3.2]. The existence of the R[x, f ]-monomorphism ν shows that (0 :R ∆S (K)) ⊆ (0 :R ∆S (Φ(M))), while Remarks 2.2 and 2.3 and Lemma 2.4 show that (0 :R ∆S (K)) = (0 :R ∆S (H)). Now suppose H is x-torsion-free. It follows from [8, Lemmas 2.3 and 2.8] that K is x-torsion-free and that I(K) = I(H). The existence of the R[x, f ]-monomorphism ν shows that Φ(M) is x-torsion-free and R[x, f ]-isomorphic to an R[x, f ]-submodule of K; therefore I(Φ(M)) ⊆ I(K) = I(H). Finally, (0 :R ∆S (Φ(M))) ∈ I(Φ(M)). 2.6. Theorem. Suppose that the local ring (R, m) is F -pure. Then R has an S-test element. In more detail, set E := ER (R/m). Recall that bS,Φ(E) denotes the intersection of all the minimal members of the set {p ∈ Spec(R) ∩ I(Φ(E)) : p ∩ S 6= ∅} (see 1.4). Then S ∩bS,Φ(E) is (non-empty and) equal to the set of S-test elements for R. Furthermore, bS,Φ(E) = (0 :R ∆S (Φ(E))). Proof. By Theorem 0.3, the set I(Φ(E)) of Φ(E)-special R-ideals is finite. Consequently, MS (Φ(E)) is finite, and Proposition 1.5 shows that (0 :R ∆S (Φ(E)))∩S 6= ∅. We use the Embedding Theorem 2.5 with Φ(E) playing the rôle of H. We conclude that, for every R-module M, we have (0 :R ∆S (Φ(E))) ⊆ (0 :R ∆S (Φ(M))). Thus (0 :R ∆S (Φ(E))) is precisely the set of elements of R that annihilate ∆S (Φ(M)) for every R-module M. Since ∆S (Φ(M)) is an R[x, f ]-submodule of Φ(M) (for each R-module M), it follows that (0 :R ∆S (Φ(E))) ∩ S (which is non-empty) is the set of S-test elements for R. It also follows from Proposition 1.5 that ∆S (Φ(E)) = annΦ(E) (bS,Φ(E) R[x, f ]) and (0 :R ∆S (Φ(E))) = bS,Φ(E) . S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 9 In our applications of these ideas, we shall frequently take S to be the complement in R of the union of finitely many prime ideals. In that case, a little more can be said. 2.7. S Lemma. Let A be a commutative ring and let p1 , . . . , pn ∈ Spec(A). Set T := A \ ni=1 pi , and let a be an ideal of A such that a ∩ T 6= ∅. Then a can be generated by elements of a ∩ T . Proof. Let a′ be the ideal of A generated by a ∩ T . Then a ⊆ (a ∩ T ) ∪ (a ∩ (A \ T )) ⊆ a′ ∪ p1 ∪ · · · ∪ pn . Since a ∩ T 6= ∅, we have, for every i ∈ {1, . . . , n}, that a 6⊆ pi . Therefore, by the Prime Avoidance Theorem (in the form given in [4, Theorem 81]), we must have a ⊆ a′ . Sn 2.8. Corollary. Suppose that the local ring (R, m) is F -pure, and S = R \ i=1 pi , where p1 , . . . , pn ∈ Spec(R). It was shown in Theorem 2.6 that R has an S-test element. Set E := ER (R/m). The S-test ideal of R, that is, the ideal of R generated by all S-test elements for R, is bS,Φ(E) , the smallest member of I(Φ(E)) that meets S. In symbols, τ S (R) = bS,Φ(E) . In particular, the big test ideal τe(R) is the smallest member of I(Φ(E)) of positive height. (We interpret the height of the improper ideal R as ∞.) Proof. We saw in Theorem 2.6 that the set S ∩ bS,Φ(E) is non-empty and equal to the set of S-test elements for R. By Lemma 2.7, the ideal bS,Φ(E) can be generated by elements of S ∩ bS,Φ(E) . For the final claim, take S = R◦ and note that a proper ideal of R has positive height if and only if it meets R◦ . We shall actually use variations of the Embedding Theorem 2.5 in Proposition 2.10 below. 2.9. Definitions. Suppose that (R, m) is local and F -pure; set E := ER (R/m). Let M be an R-module. S (i) We define the finitistic S-tight closure 0∗fg,S of 0 in M to be N 0∗,S M N , where the union is taken over all finitely generated submodulesTN of M. (ii) We define the finitistic S-test ideal τ fg,S (R) of R to be L (0 :R 0∗,S L ), where the intersection is taken over all finitely generated R-modules L. 2.10. Proposition. Suppose that (R, m) is local and F -pure; set E := ER (R/m). (i) For every R-module M, we have I(Φ(M)) ⊆ I(Φ(E)) and (0 :R ∆S (Φ(E))) ⊆ (0 :R ∆S (Φ(M))) ∈ I(Φ(E)). ∗fg,S (ii) The ideal (0 :R 0E ) annihilates (0 :R 0∗,S L ) for every finitely generated Rmodule L. ∗fg,S (iii) We have τ fg,S (R) = (0 :R 0E ), and this ideal belongs to I(Φ(E)). ∗,S (iv) For every R-module M, we have (0 :R 0∗,S E ) ⊆ (0 :R 0M ). (v) We have bS,Φ(E) = (0 :R 0∗,S E ). Consequently, when S is the complement in R of the union of finitely many prime ideals, then the S-test ideal τ S (R) of R is equal to (0 :R 0∗,S E ). 10 RODNEY Y. SHARP Proof. (i) This follows from the Embedding T Theorem 2.5, with H taken as Φ(E). (ii) By Krull’s Intersection Theorem, n∈N mn L = 0. We can therefore express the zero submodule of L as the intersection of a countable family (Qi )i∈N of irreducible submodules of finite colength. (A submodule of L is said to be irreducible if it is proper and cannot be expressed as the intersection of two strictly larger submodules.) Note that ER (L/Qi ) = E for all i ∈ N. Q The R-monomorphism Λ0 : L −→ i∈N L/Qi for which λ0 (g) = (g + Qi )i∈N for all g ∈ L can be extended to a homogeneous R[x, f ]-homomorphism ! Y′ M Y′ M Rxn ⊗R (L/Qi ) Φ(L/Qi ) = Rxn ⊗R L −→ λ : Φ(L) = n∈N0 i∈N i∈N n∈N0 whose restriction to the nth component of Φ(L), for n ∈ N0 , satisfies λ(rxn ⊗ g) = (rxn ⊗ (g + Qi ))i∈N for all r ∈ R and g ∈ L. It is clear that the R-monomorphism Q ∗,S λ0 (that is, the component of degree 0 of λ) maps 0∗,S L into i∈N 0L/Qi . But L/Qi is R-isomorphic to a finitely generated submodule of E, and so 0∗,S L/Qi is annihilated by ∗fg,S ∗fg,S (0 :R 0E ) (for all i ∈ N). It follows that 0∗,S ). LT is annihilated by (0 :R 0E ∗fg,S ∗,S (iii) By part (ii), we have (0 :R 0E ) ⊆ L (0 :R 0L ), where the intersection is ∗fg,S taken over all finitely generated R-modules L. Thus (0 :R 0E ) ⊆ τ fg,S (R). On the S other hand, by definition, τ fg,S (R) annihilates N 0∗,S N , where the union is taken over ∗fg,S all finitely generated submodules N of E; therefore τ fg,S (R) ⊆ (0 :R 0E ). For each finitely generated R-module L, ∗,S ∗,S ∆S (Φ(L)) = 0∗,S L ⊕ 0Rx⊗R L ⊕ · · · ⊕ 0Rxn ⊗R L ⊕ · · · , T by Example 1.3(ii), so that (0 :R ∆S (Φ(L))) = n∈N0 (0 :R 0∗,S Rxn ⊗R L ). Note that Rxn ⊗R L is a finitely generated R-module, for each n ∈ N0 . It therefore follows that T T ∗,S S L (0 :R ∆ (Φ(L))) = L (0 :R 0L ), where in both cases the intersection is taken over generated R-modules L. T all finitely fg,S S Therefore τ (R) is equal to the above ideal L (0 :R ∆ (Φ(L))), and the latter is in I(Φ(E)) because each (0 :R ∆S (Φ(L))) is (by part (i)) and I(Φ(E)) is closed under taking arbitrary intersections (by [7, Corollary 1.12]). (iv) By [10, Lemma 3.1], there is a family of graded left R[x, f ]-modules H (λ) λ∈Λ , with each H (λ) equal to Φ(E), and a homogeneous R[x, f ]-homomorphism Y′ H (λ) µ : Φ(M) −→ λ∈Λ such that its component µ0 of degree 0 is a monomorphism. Since µn (sxn ⊗ m) = sxn µ0 (m) for all m ∈ M, s ∈ S and n ∈ N0 , it follows that the R-monomorphism ∗,S ∗,S µ0 maps 0∗,S M into a direct product of copies of 0E . Therefore (0 :R 0E ) annihilates 0∗,S M . Note that this is true for each R-module M. (v) It now follows from part (iv) and the fact (see Example 1.3(ii)) that ∗,S ∗,S ∆S (Φ(E)) = 0∗,S E ⊕ 0Rx⊗R E ⊕ · · · ⊕ 0Rxn ⊗R E ⊕ · · · S S S,Φ(E) that (0 :R 0∗,S , by Theorem 2.6. E ) = (0 :R ∆ (Φ(E))). But (0 :R ∆ (Φ(E))) = b S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 11 When S is the complement in R of the union of finitely many prime ideals, it follows from Corollary 2.8 that τ S (R) = bS,Φ(E) . 2.11. Remarks. Suppose that (R, m) is local and F -pure; set E := ER (R/m). (i) In the special case in which S is taken to be R◦ , Proposition 2.10(v) reduces ◦ to the (probably well-known) result that the big test ideal τe(R) = τ R (R) of R is equal to (0 :R 0∗E ). (ii) In the special case in which S is taken to be R◦ , the first part of Proposition 2.10(iii) reduces to (a special case of) a result of Hochster–Huneke [2, Proposition (8.23)(d)] about the test ideal τ (R): ◦ τ (R) = τ fg,R (R) = (0 :R 0∗fg,R ) = (0 :R 0∗fg E E ). ◦ We have seen that, over an F -pure local ring (R, m), the set I(Φ(E)) (where E := ER (R/m)) of radical ideals includes the test ideal τ (R), the big test ideal ◦ ◦ τe(R) = bR ,Φ(E) = τ R (R) of R and the S-test ideal, for each multiplicatively closed subset S of R which is the complement in R of the union of finitely many prime ideals. It is natural to ask whether every member of the finite set I(Φ(E)) occurs as the S ′ -test ideal for some multiplicatively closed subset S ′ of R. In Theorem 2.12 below, we shall answer this question in the affirmative. 2.12. Theorem. Suppose that the local ring (R, m) is F -pure, and set E := ER (R/m). Let a ∈ I(Φ(E)). Then there exists a multiplicatively closed subset S of R such that a is the S-test ideal of R. Moreover, S can be taken to be the complement in R of the union of finitely many prime ideals. Proof. If a = R, then we can take S = {1} or S = R \ m. We therefore assume henceforth in this proof that a is proper. Let p1 , . . . , pt be the (distinct) associated prime ideals of a; recall from [7, Theorem 3.6 and Corollary 3.7] that they all belong to I(Φ(E)). Let T be the set of all prime members of I(Φ(E)) which neither contain, nor are contained in, any of p1 , . . . , pt . Let q1 , . . . , qu be the maximal members of the set of prime ideals in I(Φ(E)) that are properly contained in one of p1 , . . . , pt , and let U := {q1 , . . . , qu }. (Observe that T and/or U could S be empty; for example, both are empty if a = 0.) Set S := R \ q∈T ∪U q. Our aim is to show that a is the S-test ideal τ S (R) of R. It follows from Corollary 2.8 that τ S (R) = bS,Φ(E), the intersection of the minimal members of the set of prime ideals in I(Φ(E)) that meet S. Let p ∈ I(Φ(E)) ∩ Spec(R). Then, since I(Φ(E)) and therefore T and U are finite, p ∩ S = ∅ if and only if p is contained in some q ∈ T ∪ U. Let i ∈ {1, . . . , t}. Then pi meets S, or else pi ⊆ qj for some j ∈ {1, . . . , u}, and as qj is properly contained in one of p1 , . . . , pt , this would lead to a contradiction to the minimality of the primary decomposition a = p1 ∩ · · · ∩ pt . Furthermore, pi must be a minimal member of the set J := {p ∈ I(Φ(E)) ∩ Spec(R) : p ∩ S 6= ∅} , for otherwise there would exist p ∈ J with p ⊂ pi , so that p would be contained in one of q1 , . . . , qu and therefore disjoint from S. This shows that p1 , . . . , pt are all associated primes of bS,Φ(E) . To complete the proof, it is enough for us to show that there is no other associated prime of this ideal. So suppose that p ∈ ass bS,Φ(E) \ {p1 , . . . , pt } and seek a contradiction. Then p must contain, or be contained in, pi for some i ∈ {1, . . . , t} (or else it would be in T and 12 RODNEY Y. SHARP disjoint from S); if p ⊂ pi , then p would be contained in qj for some j ∈ {1, . . . , t} and so would be disjoint from S; if p ⊃ pi , then p could not be a primary component of the radical ideal bS,Φ(E) . Thus each possibility leads to a contradiction. Therefore ass bS,Φ(E) = {p1 , . . . , pt } and bS,Φ(E) = a. 3. The complete case In this section we shall concentrate on the case where (R, m) is local, F -pure and complete. 3.1. Theorem. Suppose (R, m) is local, F -pure and complete. Set E := ER (R/m). Let c ∈ I(Φ(E)) with c 6= R. In the light of Theorem 2.12, letSp1 , . . . , pw be prime ideals of R for which the multiplicatively closed subset S = R \ w i=1 pi of R satisfies c = τ S (R). Set J := ∆S (Φ(E)), a graded left R[x, f ]-module. ∗,S ∗,S (i) We have J = 0∗,S E ⊕ 0Rx⊗R E ⊕ · · · ⊕ 0Rxn ⊗R E ⊕ · · · . (ii) When we regard J as a graded left (R/c)[x, f ]-module in the natural way, it is x-torsion-free and has IR/c (J) = {g/c : g ∈ I(Φ(E)) : g ⊇ c} . (iii) The 0th component J0 of J is (0 :E c); as R/c-module, this is isomorphic to ER/c ((R/c)/(m/c)). (iv) The ring R/c is F -pure. (v) We have I(ΦR/c (J0 )) ⊆ IR/c (J), so that d : d is an ideal of R with d ⊇ c and d/c ∈ I(ΦR/c (J0 )) ⊆ I(ΦR (E)). Proof. Set R := R/c. (i) It follows from Example 1.3(ii) that ∗,S ∗,S ∆S (Φ(E)) = 0∗,S E ⊕ 0Rx⊗R E ⊕ · · · ⊕ 0Rxn ⊗R E ⊕ · · · . (ii) Since gr-annR[x,f ] ∆S (Φ(E)) = cR[x, f ], we see that c annihilates ∆S (Φ(E)), and so the latter inherits a structure as an x-torsion-free left R[x, f ]-module. As the R[x, f ]-submodules of J are exactly the R[x, f ]-submodules of Φ(E) contained in J, the claim about IR (J) is clear. (iii) Note that c = bS,Φ(E) = τ S (R), and that, by Proposition 2.10(v), this is the ∗,S R-annihilator of 0∗,S E . Since R is complete, we can conclude that 0E = (0 :E c), by Matlis duality (see, for example, [11, p. 154]). (iv),(v) Let N be an R-module. Use the Embedding Theorem 2.5 over the ring R (with J playing the rôle of H) to deduce that ΦR (N) is x-torsion-free and I(ΦR (N)) ⊆ IR (J). These are true for each R-module N, and, in particular, for J0 . It follows that R is F -pure. The final claim follows from the description of IR (J) given in part (ii). 3.2. Corollary. Suppose that (R, m) is local, F -pure and complete. Denote ER (R/m) by E. Let c ∈ I(Φ(E)) with c 6= R. Denote R/c by R, and note that R is F pure, by Theorem 3.1(iv). Let T be a multiplicatively closed subset of R which is the complement in R of the union of finitely many prime ideals. (i) If h denotes the unique ideal of R that contains c and is such that h/c = τ fg,T (R), the finitistic T -test ideal of R, then h ∈ I(Φ(E)). S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 13 (ii) In particular, if h′ denotes the unique ideal of R that contains c and is such that h′ /c = τ (R), the test ideal of R, then h′ ∈ I(Φ(E)). (iii) If g denotes the unique ideal of R that contains c and is such that g/c = τ T (R), the T -test ideal of R, then g ∈ I(Φ(E)). (iv) In particular, if g′ denotes the unique ideal of R that contains c and is such that g′ /c = τe(R), the big test ideal of R, then g′ ∈ I(Φ(E)). Proof. Use the notation of Theorem 3.1. Note that, as R-module, J0 is the injective envelope of the simple R-module. (i) By Proposition 2.10(iii), we have τ fg,T (R) ∈ I(ΦR (J0 )). The result therefore follows from Theorem 3.1(v). ◦ (ii) This is a special case of part (i): take T = R . (iii) By Proposition 2.10(v), we have τ T (R) ∈ I(ΦR (J0 )). The result therefore follows from Theorem 3.1(v). ◦ (iv) This is a special case of part (iii): take T = R . The remainder of the paper is devoted to the provision of some examples of the above ideas. 3.3. Example. Let K be an algebraically closed field of characteristic p, and assume that p ≥ 5 and that p ≡ 1 (mod 3). Let R′ = K[[X, Y, Z]], where X, Y, Z are independent indeterminates, and a = (X 3 + Y 3 + Z 3 ) ∈ Spec(R′ ). By Huneke [3, Examples 4.7 and 4.8], R := R′ /a is F -pure, and the test ideal τ (R) = m. Because R is Gorenstein and excellent, the test ideal τ (R) is equal to the big test ideal τe(R), by 1.9(ii)(a). This means that m must be the smallest ideal in I(Φ(E)) of positive height, so that I(Φ(E)) ∩ Spec(R) = {0, m} . 3.4. Reminder. Suppose that (R, m) is local and F -pure, and set E = ER (R/m). In the case where R is an (F -pure) homomorphic image of an F -finite regular local ring, Janet Cowden Vassilev showed in [13, §3] that there exists a strictly ascending chain 0 = τ0 ⊂ τ1 ⊂ · · · ⊂ τt ⊂ τt+1 = R of radical ideals of R such that, for each i = 0, . . . , t, the reduced local ring R/τi is F -pure and its test ideal is exactly τi+1 /τi . If R is complete, all of τ0 , τ1 , . . . , τt and all their associated primes belong to I(Φ(E)) (by Corollary 3.2(ii) and [7, Theorem 3.6 and Corollary 3.7]). 3.5. Lemma. Assume that (R, m) is local, F -pure and complete. Set E = ER (R/m). (i) There is a strictly ascending chain 0 = τ0 ⊂ τ1 ⊂ · · · ⊂ τt ⊂ τt+1 = R of radical ideals of R such that, for each i = 0, . . . , t, the reduced local ring R/τi is F -pure and its test ideal is τi+1 /τi . We call this the test ideal chain of R. All of τ0 = 0, τ1 , · · · , τt , and all their associated primes, belong to I(Φ(E)). (ii) There is a strictly ascending chain 0 = τe0 ⊂ τe1 ⊂ · · · ⊂ τew ⊂ τew+1 = R of radical ideals in I(Φ(E)) such that, for each i = 0, . . . , w, the reduced local ring R/e τi is F -pure and its big test ideal is τei+1 /e τi . We call this the big test ideal chain of R. All of τe0 = 0, τe1 , · · · , τew , and all their associated primes, belong to I(Φ(E)). Note. We have not assumed that R is F -finite in part (i). If the conjecture mentioned in 1.9(ii) turns out to be true, then the big test ideal chain and the test ideal chain of R would coincide. 14 RODNEY Y. SHARP Proof. (i) We know from Theorem 0.1 that E can be given a structure as an x-torsionfree left R[x, f ]-module (that extends its R-module structure). It therefore follows from work of the present author in [8, Corollary 3.8] that, in this complete case, the test ideal chain of R exists. By Corollary 3.2(ii), all the terms in the test ideal chain of R belong to I(Φ(E)), and all the associated prime ideals of these ideals also belong to I(Φ(E)), by [7, Theorem 3.6 and Corollary 3.7]. (ii) Let c ∈ I(Φ(E)) with c 6= R. By Theorem 3.1(iv), the ring R/c is F -pure. By Corollary 3.2(iv), if g′ denotes the unique ideal of R that contains c and is such that g′ /c = τe(R/c), then g′ ∈ I(Φ(E)). One can therefore construct τe1 (R), τe2 (R), . . . successively until some τet+1 (R) = R, when the process stops. Use Corollary 3.2(iv) and [7, Theorem 3.6 and Corollary 3.7] again to complete the proof. We can now use some of Vassilev’s computations in [13, §3] to give some examples. 3.6. Examples. Let K be an algebraically closed field of characteristic p. In these examples, X, Y, Z, W denote independent indeterminates over K, and x, y, z, w denote the natural images of X, Y, Z, W (respectively) in R′ /a = R for appropriate choices of R′ and a proper ideal a of R′ . (i) As in Vassilev [13, Example 3.12(1)], take R′ = K[[X, Y, Z]], a = (XY, XZ, Y Z) and R = R′ /a. For this R, the test ideal chain is 0 ⊂ m ⊂ R. Since R is an isolated singularity, we have τ (R) = τe(R), by 1.9(ii)(d). Therefore m is the smallest ideal of positive height in I(Φ(E)). Thus in this case, I(Φ(E)) ∩ Spec(R) = {(x, y), (x, z), (y, z), m} . (ii) As in Vassilev [13, Example 3.12(2)], take R′ = K[[X, Y, Z, W ]], a = (XY Z, XY W, XZW, Y ZW ) and R = R′ /a. For this R, the test ideal chain is 0 ⊂ (xy, xz, xw, yz, yw, zw) ⊂ m ⊂ R. It therefore follows from Lemma 3.5(i) that { (x, y), (x, z), (x, w), (y, z),(y, w), (z, w), (x, y, z), (x, y, w), (x, z, w), (y, z, w), m} ⊆ I(Φ(E)) ∩ Spec(R). (iii) As in Vassilev [13, Example 3.12(3)], take R′ = K[[X, Y, Z]], a = (XY, Y Z), and R = R′ /a. For this R, the test ideal chain is 0 ⊂ m ⊂ R. Because R is an isolated singularity, τ (R) = τe(R), by 1.9(ii)(d). This means that m must be the smallest ideal in I(Φ(E)) of positive height, so that I(Φ(E)) ∩Spec(R) = {(x, z), (y), m} . (iv) As in Vassilev [13, Example 3.12(4)], take R′ = K[[X, Y, Z, W ]], a = (XY, ZW ) and R = R′ /a. For this R, the test ideal chain is 0 ⊂ (xy, xz, xw, yz, yw, zw) ⊂ m ⊂ R, so that {(x, z), (x, w), (y, z), (y, w), (x, y, z), (x, y, w), (x, z, w), (y, z, w), m} ⊆ I(Φ(E)) ∩ Spec(R) by Lemma 3.5(i). S-TIGHT CLOSURE OVER AN F -PURE LOCAL RING 15 References [1] R. Fedder, ‘F -purity and rational singularity’, Transactions Amer. Math. Soc. 278 (1983) 461–480. [2] M. Hochster and C. Huneke, ‘Tight closure, invariant theory and the Briançon-Skoda Theorem’, J. Amer. Math. Soc. 3 (1990) 31–116. [3] C. Huneke, ‘Tight closure, parameter ideals and geometry’, Six lectures on commutative algebra, Eds. J. Elias, J. M. Giral, R. M. Miró-Roig and S. Zarzuela, Progress in Mathematics 166 (Birkhäuser, Basel, 1998), pp. 187–239. [4] I. Kaplansky, Commutative rings (Allyn and Bacon, Boston, 1970). [5] G. Lyubeznik and K. E. Smith, ‘Strong and weak F -regularity are equivalent for graded rings’, American J. Math. 121 (1999) 1279–1290. [6] G. Lyubeznik and K. E. Smith, ‘On the commutation of the test ideal with localization and completion’, Transactions Amer. Math. Soc. 353 (2001) 3149–3180. [7] R. Y. Sharp, ‘Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure’, Transactions Amer. Math. Soc. 359 (2007) 4237–4258. [8] R. Y. Sharp, ‘Graded annihilators and tight closure test ideals’, J. Algebra 322 (2009) 3410– 3426. [9] R. Y. Sharp, ‘An excellent F -pure ring of prime characteristic has a big tight closure test element’, Transactions Amer. Math. Soc. 362 (2010) 5455–5481. [10] R. Y. Sharp, ‘Big tight closure test elements for some non-reduced excellent rings’, J. Algebra 349 (2012) 284-316. [11] D. W. Sharpe and P. Vámos, Injective modules, Cambridge Tracts in Mathematics and Mathematical Physics 62 (Cambridge University Press, Cambridge, 1972). [12] K. E. Smith, ‘Tight closure of parameter ideals’, Inventiones math. 115 (1994) 41–60. [13] J. C. Vassilev, ‘Test ideals in quotients of F -finite regular local rings’, Transactions Amer. Math. Soc. 350 (1998) 4041–4051. School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom E-mail address: R.Y.Sharp@sheffield.ac.uk	0
Power Beacon-Assisted Millimeter Wave Ad Hoc Networks Xiaohui Zhou, Jing Guo, Salman Durrani, and Marco Di Renzo arXiv:1703.06611v2 [cs.IT] 6 Sep 2017 Abstract Deployment of low cost power beacons (PBs) is a promising solution for dedicated wireless power transfer (WPT) in future wireless networks. In this paper, we present a tractable model for PB-assisted millimeter wave (mmWave) wireless ad hoc networks, where each transmitter (TX) harvests energy from all PBs and then uses the harvested energy to transmit information to its desired receiver. Our model accounts for realistic aspects of WPT and mmWave transmissions, such as power circuit activation threshold, allowed maximum harvested power, maximum transmit power, beamforming and blockage. Using stochastic geometry, we obtain the Laplace transform of the aggregate received power at the TX to calculate the power coverage probability. We approximate and discretize the transmit power of each TX into a finite number of discrete power levels in log scale to compute the channel and total coverage probability. We compare our analytical predictions to simulations and observe good accuracy. The proposed model allows insights into effect of system parameters, such as transmit power of PBs, PB density, main lobe beam-width and power circuit activation threshold on the overall coverage probability. The results confirm that it is feasible and safe to power TXs in a mmWave ad hoc network using PBs. Index Terms Wireless communications, wireless power transfer, millimeter wave transmission, power beacon, stochastic geometry. X. Zhou, J. Guo and S. Durrani are with the Research School of Engineering, College of Engineering and Computer Science, The Australian National University, Canberra, ACT 2601, Australia (Emails: {xiaohui.zhou, jing.guo, salman.durrani}@anu.edu.au). M. Di Renzo is with the Laboratoire des Signaux et Systèmes, Centre National de la Recherche Scientifique, CentraleSupélec, University Paris Sud, Université Paris-Saclay, 91192 Gif-sur-Yvette, France (Email: marco.direnzo@lss.supelec.fr). 1 I. I NTRODUCTION Wireless power transfer (WPT) can prolong the lifetime of low-power devices in the network and is currently in the spotlight as a key enabling technology in future wireless communication networks [1]–[3]. Compared to energy harvesting from ambient energy sources, e.g., solar, wind or ambient radio frequency (RF) sources, which may change rapidly with time, location and weather conditions, WPT has a significant advantage of being always available and controllable [1]. There are currently two main approaches to WPT: (i) simultaneous information and power transfer (SWIPT) and (ii) power beacon (PB) based approach. While SWIPT, which proposes to extract the information and power from the same signal, has been the subject of intense research in the academic community [1], [4], [5], industry has preferred to adopt the PB approach. In this approach, low cost PBs, which do not require backhaul links, are deployed to provide dedicated power transfer in wireless networks. For example, the Cota Tile is a PB designed to wirelessly charge devices like smartphones in a home environment and was showcased at the 2017 Consumer Electronics Show (CES) [6]. There are two key challenges in the application of PBs to wider networks. The first challenge is the lack of tractable models for analysis and design of such networks. Although simulations can be used in this regard, exhaustive simulation of every possible scenario of interest will be extremely time-consuming and onerous. Hence, it is important to explore tractable models for PB-assisted communications in wireless networks. The second challenge is the use of practical models for WPT, which capture realistic aspects of WPT. For instance, WPT receivers (RXs) can only harvest power if the incident received power is greater than the power circuit activation threshold (typically around −20 dBm [1]). Similarly, WPT transmitters (TXs) have to adhere to maximum transmit power constraints due to safety considerations. Hence, it is important to adopt a realistic and practical model for WPT. A. Related Work Microwave (below 6 GHz) systems: Recently, the investigation of PBs has drawn attention in the literature from different aspects. For point-to-point or point-to-multipoint communication systems, the resource allocation for PB-assisted system was considered in [7], [8], where the authors mainly aimed at finding 2 the optimum time ratio for power transfer (PT) and information transmission (IT). In [9], the authors studied the PB-assisted network in the context of physical layer security, where an energy constrained source is powered by a dedicated PB. For large scale networks, some papers have characterized the performance of PB-assisted communications using stochastic geometry, which is a powerful mathematical tool to provide tractable analysis by incorporating the randomness of users. Specifically, the feasibility of PB deployment in a cellular network, under the outage constraint at the base station, was investigated in [10], where cellular users are charged by PBs for uplink transmission. By considering that the secondary TX is charged by the primary user in a cognitive network, the authors derived the spatial throughput for the secondary network in [11]. Adaptively directional PBs were proposed for sensor network in [12] and the authors found the optimal charging radius for different sensing tasks. In [13], three WPT schemes were proposed to select the PB for charging in a device-to-device-aided cognitive cellular network. The authors in [14] formulated the total outage probability in a PB-assisted ad hoc network by including the energy harvesting sensitivity into the analysis. Note that all the aforementioned works considered the conventional microwave frequency band, i.e., below 6 GHz. MmWave systems: Millimeter wave (mmWave) communication, which aims to use the spectrum band typically around 30 GHz, is emerging as a key technology for the fifth generation systems [15]. Considerable advancements have already been made in the understanding, modelling and analysis of mmWave communication using stochastic geometry [16]–[19]. From the prior work, we can summarize two distinctive features of mmWave communication: (i) owing to the smaller wavelength, mmWave allows a large number of antenna arrays with directional beamforming to be equipped at the TX and RX; (ii) since the mmWave propagation is susceptible to blockage, it causes the large difference for path-loss and fading characteristics between line of sight (LOS) and non light of sight (NLOS) environment. MmWave communication can be beneficial for WPT since both technologies inherently operate over short distances and the narrow beams in mmWave communication can focus the transmit power. Very recently, some papers have used stochastic geometry to analyse mmWave SWIPT networks [20], [21]. The statistics of the aggregate received power from PBs in a mmWave ad hoc network were studied in 3 our preliminary work in [22]. To the best of our knowledge, the study of a PB-assisted mmWave network using stochastic geometry, taking into account realistic and practical WPT and mmWave characteristics such as building blockages, beamforming, power circuit activation threshold, maximum harvested power and maximum transmit power, is not available in the literature. B. Our Approach and Contributions In this paper, we consider a PB-assisted wireless ad hoc network under mmWave transmission where TXs adopt the harvest-then-transmit protocol, i.e., they harvest energy from the aggregate RF signal transmitted by PBs and then use the harvested energy to transmit the information to their desired RXs. Both the PT and IT phases are carried out using antenna beamforming under the mmWave channel environment, which is subjected to building blockages. Using tools from stochastic geometry, we develop a tractable analytical framework to investigate the power coverage probability, the channel coverage probability and the total coverage probability at a reference RX taking a mmWave three-state propagation model and multislope bounded path-loss model into account. In the proposed framework, the power coverage probability is efficiently and accurately computed by numerical inversion using the closed-form expression for the Laplace transform of the aggregate received power1 at the typical TX. The novel contributions of this paper are summarized as follows: • We adopt a realistic model of wirelessly powered TXs by taking into consideration (i) the power circuit activation threshold, which accounts for the minimum aggregate received power required to activate the energy harvesting circuit, (ii) the allowed maximum harvested power, which accounts for the saturation of the energy harvesting circuit and (iii) the maximum transmit power, which accounts for the safety regulation and the electrical rating of the antenna circuit. • For tractable analysis of the channel coverage probability and the total coverage probability, we propose to discretize the transmit power of each TX into a finite number of discrete power levels in the log scale. Using this approximation, we derive the channel coverage probability and the total 1 In this paper, we use the Laplace transform of a random variable to denote the Laplace transform of the distribution of a random variable for brevity. 4 coverage probability at the typical RX. Comparison with simulation results shows that the model, with only 10 discrete levels for the transmit power of TXs, has good accuracy in the range of 5%-10%. • Based on our proposed model, we investigate the impact of varying important system parameters (e.g., transmit power of PB, PB density, allowed maximum harvested power, directional beamforming parameters etc.) on the network performance. These trends are summarized in Table V. • We investigate the feasibility of using PBs to power up TXs while providing an acceptable performance for IT towards RXs in mmWave ad hoc network. Our results show that under practical setups, for PB transmit power of 50 dBm and TXs with a maximum transmit power between 20 − 40 dBm, which are practical and safe for human exposure, the total coverage probability is around 90%. C. Notation and Paper Organization The following notation is used in this paper. Pr(·) indicates the probability measure and E[·] denotes the expectation operator. j is the imaginary number and Re[·] denotes the real part of a complex numR∞ R∞ ber. Γ(x) = 0 tx−1 exp(−t)dt is the complete gamma function and Γ(a, x) = x ta−1 exp(−t)dt is R 1 tb−1 (1−t)c−b−1 Γ(c) dt is the the upper incomplete gamma function, respectively. 2 F1 (a, b; c; z) = Γ(b)Γ(c−b) (1−tz)a 0 Gaussian hypergeometric function. fX (x) and FX (x) denotes the probability density function (PDF) and the cumulative distribution function (CDF) of a random variable X. LX (s) = E[exp(−sX)] denotes the Laplace transform of a random variable X. A list of the main mathematical symbols employed in this paper is given in Table I. The rest of the paper is organized as follows: Section II describes the system model and assumptions. Section III focuses on the PT phase of the system and derives the power coverage probability. Section IV details the IT phase, which covers the analysis of transmit power statistics and channel coverage probability. Section V summaries the total coverage probability. Section VI presents the results and the effect of the system parameters on the network performance. Finally, Section VII concludes the paper. 5 TABLE I: Summary of Main Symbols Used in the Paper. Symbol Definition Symbol Definition φp PB PPP Pp PB transmit power φt TX PPP Pt TX transmit power φn t nth level TX PPP Ptn nth level TX transmit power λp Density of PB PPP kn Portion of TXs at the nth level λt Density of TX PPP N Number of battery levels λn t Density of nth level TX PPP w Step size of each battery level d0 Length of desired TX-RX link σ2 Noise power rmin Radius of the LOS region η Power conversion efficiency rmax Exclusion radius of the OUT region ρ Time switching parameter αL LOS link path-loss exponent γPT Power circuit activation threshold αN NLOS link path-loss exponent P1max Allowed maximum harvested power at active TX gL LOS link channel fading P2max Maximum transmit power of active TX gN NLOS link channel fading γTR SINR threshold m Nakagami-m fading parameter PP cov Power coverage probability min Gmax p , Gp , θp PB beamforming parameters PC cov Channel coverage probability min Gmax t , Gt , θt TX beamforming parameters Pcov Total coverage probability min Gmax r , Gr , θr RX beamforming parameters II. S YSTEM M ODEL We consider a two-dimensional mmWave wireless ad hoc network, where TXs are first wirelessly charged by PBs and then they transmit information to RXs. The locations of PBs are modeled as a homogeneous Poisson point process (PPP) φp in R2 with constant node density λp . TXs are assumed to be randomly independently deployed and their locations are modeled as a homogeneous PPP φt with node density λt . For each TX, it has a desired RX located at a distance d0 in a random direction. Throughout the paper, we use Xi to denote both the random location as well as the ith TX itself, Yi to denote both the location and the corresponding ith RX and Zi to denote both the location and the ith PB, respectively. Note that we assume the indoor-to-outdoor penetration loss is high. Therefore, all the PBs, TXs and RXs can be regarded as outdoor devices. 6 A. Power Transfer and Information Transmission Model We assume that each PB has access to a dedicated power supply (e.g., a battery or power grid) and transmits with a constant power Pp . Time is divided into slots and let T denote one time slot. Each TX adopts the harvest-then-transmit protocol to perform PT and IT. Specifically, each time slot T is divided into two parts with ratio ρ ∈ (0, 1): in the first ρT seconds TX harvests energy from the RF signal transmitted by PBs and stores the energy in an ideal (infinite capacity) battery2 . In the remaining (1 − ρ)T seconds, TXs use all the harvested energy to transmit information to their desired RXs. Hence, there is no interference between the PT and IT stages. We make the following assumptions for realistic modelling of PT: • Different from previous works [10], [21], [24], where energy harvesting activation threshold is not considered and the devices can harvest power from any amount of incident power, we assume that the TX can scavenge energy if and only if the instantaneous aggregate received power from all PBs is greater than a power circuit activation threshold γPT . If this condition is met, then the TX is called an active TX. Otherwise, the TX will be inactive and will not scavenge any energy from the PBs. • Once the energy harvesting circuit is activated, the harvested power at the active TX is assumed to be linearly proportional to the aggregate received power with power conversion efficiency η. Due to the saturation of the energy harvesting circuit, the harvested power at the active TX cannot exceed a maximum level denoted as P1max [25]. In addition, the active TX cannot transmit information with a power greater than P2max because of the safety regulation and the electrical rating of the antenna circuit [26]. B. MmWave Blockage Model Under outdoor mmWave transmissions, each link between the PB and the TX (i.e., PB-TX link) or between the TX and the RX (i.e., TX-RX link) is susceptible to building blockages due to their high diffraction and penetration characteristics [16]. In this work, we adopt the state-of-the-art three-state blockage model as in [17], [27], where each PB-TX or TX-RX link can be in one of the following three 2 In this work, we do not consider the impact of battery imperfections [23]. 7 states: (i) the link is in LOS state if no blockage exists, (ii) the link is in NLOS state if blockage exists and (iii) the link is in outage (OUT) state if the link is too weak to be established. Given that the PB-TX or TX-RX link has a length of r, the probabilities pLOS (·), pNLOS (·) and pOUT (·) of it being in LOS, NLOS and OUT states, respectively, are pOUT (r) = u(r − rmax ); pNLOS (r) = u(r − rmin ) − u(r − rmax ); (1) pLOS (r) = 1 − u(r − rmin ), where u(·) denotes the unit step function, rmin is the radius of the LOS region and rmax is the exclusion radius of the OUT region3 , as illustrated in Fig. 1. The values of rmin and rmax depend on the propagation scenario and the mmWave carrier frequency [17]. Moreover, the communication link between TX and its desired RX is assumed to be always in LOS state. C. MmWave Channel Model It has been shown by the measurements that mmWave links experience different channel conditions under LOS, NLOS and OUT states [30]. Thus, we consider the following path-loss plus block fading channel model. For the path-loss, we adopt and modify a multi-slope path-loss model [31] and define the path-loss of PB-TX or TX-RX link with a propagation distance     1,        r−αL ,  l(r) =    βr−αN ,        ∞, of r as follows 06r<1 1 6 r < rmin , (2) rmin 6 r < rmax rmax 6 r where the first condition is added to avoid the singularity as r → 0, αL denotes the path-loss exponent for the link in LOS state, αN denotes the path-loss exponent for the link in NLOS state (2 6 αL 6 αN ), 3 Note that the two-state blockage model in [28], [29], which does not consider the OUT region, can be considered as a special case of the three-state blockage model with rmax = ∞. 8 the path-loss of the link in OUT state is assumed to be infinite [17] and the continuity in the multi-slope αN −αL path-loss model is maintained by introducing the constant β , rmin [31]. As for the fading, the link under LOS state is assumed to experience Nakagami-m fading, while the link under NLOS state is assumed to experience Rayleigh fading4 . Furthermore, both the LOS and the NLOS links experience additive white Gaussian noise (AWGN) with variance σ 2 . However, under the PT phase, the AWGN power is too small to be harvested by TXs. Hence, we ignore it in the PT phase. D. Beamforming Model To compensate the large path-loss in mmWave band, directional beamforming is necessary for devices [33]. In this work, we consider that mmWave antenna arrays perform directional beamforming at all PBs, TXs and RXs. Similar to [16], [17], the actual antenna array pattern can be approximated by a sectorized gain pattern which is given by Ga (θ) =     Gmax \|θ\| ≤ a ,    Gmin a , θa 2 , (3) otherwise is the main lobe antenna gain, where subscript a = p for PB, a = t for TX and a = r for RX, Gmax a is the side lobe antenna gain, θ ∈ [−π, π) is the angle off the boresight direction and θa is the main Gmin a lobe beam-width. Note that, as shown in Section VI-C, this model can be easily related to specific array geometries, such as an N element uniform planar or linear or circular array [18]. The main beam at the PBs are assumed to be randomly and independently oriented with respect to each other and uniformly distributed in [−π, π). Given a sufficient density of the PBs, this simple strategy ensures that the aggregate received power from PBs at different locations in the network is roughly on the same order and avoids the need for channel estimation and accurate beam alignment. In addition, it has been shown in [33] that the random directional beamforming can perform reasonably well given that more than one users need to be served. 4 We do not consider shadowing but it can be included using the composite fading model in [32]. 9 PB-TX gain Gij Fig. 1: Illustration of mmWave blockage model. TX-RX gain Dij k Gain Gk Probability pk Gain Dk Probability qk 1 max Gmax p Gt θp θt 4π 2 max Gmax t Gr θt θr 4π 2 2 min Gmax p Gt θp (2π−θt ) 4π 2 min Gmax t Gr θt (2π−θr ) 4π 2 3 max Gmin p Gt (2π−θp )θt 4π 2 max Gmin t Gr (2π−θt )θr 4π 2 4 min Gmin p Gt (2π−θp )(2π−θt ) 4π 2 min Gmin t Gr (2π−θt )(2π−θr ) 4π 2 TABLE II: Probability Mass Function of Gij and Dij . Let Gij be the effective antenna gain on the link from the ith PB to the jth TX. Under sectorization, Gij is a discrete random variable with probability pk = Pr(Gij = Gk ) and k ∈ {1, 2, 3, 4}, where its distribution is summarized in Table II. With regards to TX and RX, we assume that each TX points its main lobe towards its desired RX max and the directly. Therefore, the effective antenna gain of the desired TX-RX link is D0 = Gmax t Gr orientation of the beam of the interfering TX is uniformly distributed in [−π, π). Let Dij (i 6= j) be the effective antenna gain on the link from the ith TX to the jth RX. Similar to Gij , Dij is a discrete random variable with probability qk = Pr(Dij = Dk ), where its distribution is given in Table II. E. Metrics In this paper, we are interested in the PB-assisted mmWave wireless ad hoc network in terms of the total coverage probability for RXs (i.e., the probability that a RX can successfully receive the information from its TX after the TX successfully harvests energy from PBs). Based on the system model described above, the success of this event has to satisfy two requirements, which are: • The corresponding TX is in power coverage. Due to the random network topology and the fading channels, the aggregate received power from all PBs is a random variable. If the aggregate received power at a TX is greater than the power circuit activation threshold, the energy harvesting circuit is active and this TX can successfully harvest energy from PBs. As a result, the TX is under power coverage and IT then takes place. 10 • The RX is in channel coverage. The instantaneous transmit power for each active TX depends on its random received power. RX can receive the information from its desired TX (i.e., in channel coverage) if the signal-to-interference-plus-noise ratio (SINR) at the RX is above a certain threshold. By leveraging the Laplace transform of the aggregate received power at a typical TX and the interference at a typical RX, we compute the power coverage probability and channel coverage probability in the following sections. In the subsequent analysis, we condition on having a reference RX Y0 at the origin (0, 0) and its associated TX X0 located at a distance d0 away at (d0 , 0). According to Slivnyak’s theorem, the conditional distribution is the same as the original one for the rest of the network [34]. III. P OWER T RANSFER In this section, we focus on the PT phase of the system. We analyze the aggregate received power at a reference TX from all PBs and find the power coverage probability at the corresponding RX. Since the power harvested from the noise is negligible, the instantaneous aggregate received power at the typical TX X0 from all the PBs can be expressed as PPT = Pp X Gi0 gi0 l(ri ), (4) Zi ∈φp where Pp is the PB transmit power, Gi0 is the effective antenna gain between Zi and X0 , gi0 is the fading power gain between the ith PB Zi and the typical TX X0 , which follows the gamma distribution (under Nakagami-m fading assumption) if the PB-TX link is in LOS state and exponential distribution (under Rayleigh fading distribution) if the PB-TX link is in NLOS state. l(ri ) is the path-loss function given in (2) and ri = \|Zi − X0 \| is the Euclidean length of the PB-TX link between Zi and X0 . Using (4), the power coverage probability is defined as follows. Definition 1: The power coverage probability is the probability that the aggregate received power at the typical TX is higher than the power circuit activation threshold γPT . It can be expressed as PPcov (γPT ) = Pr(PPT > γPT ). (5) Remark 1: Analytically characterizing the power coverage probability in (5) is a challenging open problem in the literature. Generally, it is not possible to obtain a closed-form power coverage probability 11 because of the randomness in the antenna gain, mmWave channels and locations of PBs. The closed-form expression only exists under the unbounded path-loss model with α = 4 and Rayleigh fading for all links, which is shown to be Lévy distribution [34]. To overcome this problem, some works [20], [28], [35] employed the Gamma scaling method. This approach involves introducing a dummy Gamma random variable with parameter N 0 to reformulate the original problem. However, the approach can sometimes lead to large errors with finite N 0 value. Other works adopted the Gil-Pelaez inversion theorem [36] . This approach involves one fold integration and is only suitable for the random variable with a simple Laplace transform. If the Laplace transform is even moderately complicated, this method is not very efficient even if the Laplace transform is in closed-form. In this work, we adopt a numerical inversion method, which is easy to compute, if the Laplace transform of a random variable is in closed-form, and provides controllable error estimation. Following [37], [38], the CDF of the aggregate received power PPT is given as 1 FPPT (x) = 2πj = 1 2πj Z a+j∞ a−j∞ Z a+j∞ a−j∞ LFPPT (s) exp(sx)ds (6a) LPPT (s) exp(sx)ds. s (6b) where (6a) is obtained according to the Bromwich integral [39] and (6b) follows from probability theory that LPPT (s) = sLFPPT (s). Using the trapezoidal rule and the Euler summation, the above integral can be transformed into a finite sum. Therefore, we can express the power coverage probability as B C+b 2−B exp( A2 ) X B X (−1)c LPPT (s) P Pcov (γPT ) =1 − Re , γPT b D s c c=0 b=0 where Re[·] is the real part operator, s = A+j2πc , 2γPT (7) LPPT (s) is the Laplace transform of PPT , Dc = 2 (if c = 0) and Dc = 1 (if c = 1, 2, ..., C + b). A, B and C are positive parameters used to control the estimation accuracy. From (7), the key parameter in order to obtain the power coverage probability is LPPT (s). By the definition of Laplace transform of a random variable, we express LPPT (s) in closed-form in the following theorem. Theorem 1: Following the system model in Section II, the Laplace transform of the aggregate received 12 power at the typical TX from all the PBs in a mmWave ad hoc network is 4 Y −αL 2 LPPT (s) = exp πλp rmin pk mm (m+srmin Pp Gk )−m−1 + πλp pk (sPp Gk )δL (Ξ1 (1) − Ξ1 (rmin )) k=1 + πλp pk sPp Gk β (Ξ2 (rmin ) − Ξ2 (rmax )) + πλp pk (sPp Gk β)δN (Ξ3 (rmin ) − Ξ3 (rmax )) , 2 + αN (8) where mm (r−αL sPp Gk )−δL −m αL Γ(1 + m) mrαL Ξ1 (r) = , (9) 2 F1 1 + m,m + δL ;1 + m + δL ;− (2 + mαL )Γ(m) sPp Gk r2 Ξ2 (r) = α , (10) r N + sPp Gk β (r−αN sPp Gk β)−δN −1 rαN β −1 αN Ξ3 (r) = , sPp Gk β(2 + αN ) −2(r +sPp Gk β)2 F1 1, δN +1; 2+δN ; − rαN + sPp Gk β sPp Gk (11) and Γ(·) is the complete gamma function, function, δL , 2 αL and δN , 2 F1 (·, ·; ·; ·) is the Gaussian (or ordinary) hypergeometric 2 . αN Proof: See Appendix A. By substituting (8) into (7), we can compute the power coverage probability. As shown in Theorem 1, the Laplace transform of PPT is in closed-form; hence, PPcov (γPT ) is just a summation over a finite number of terms. Following the selection guideline of parameters A, B and C in [38], we can achieve a stable numerical result by carefully choosing them. Before ending this section, we validate the analysis for the power coverage probability. Fig. 2 plots the power coverage probability versus power circuit activation threshold. The simulation results are generated by averaging over 108 Monte Carlo simulation runs. We set A = 24, B = 20 and C = 30 in order to achieve an estimation error of 10−10 . The other system parameters follow Table IV. From the figure, we can see that the analytical results match perfectly with the simulation results, which demonstrates the accuracy of the proposed approach. Fig. 2 also shows that the power coverage probability increases with the density of PBs, because the aggregate received power at TX increases as the PB density increases. IV. I NFORMATION T RANSMISSION In this section we focus on the IT phase between the TX and RX. We assume that the TX uses all the harvested energy in the IT phase. As indicated in Section II-A, the transmit power of an active TX is a 13 Power coverage probability 1 =100/km2 p 0.8 =50/km2 p =10/km2 p 0.6 Simulation 0.4 0.2 0 -20 -10 0 10 Power circuit activation threshold 20 PT 30 (dBm) Fig. 2: Power coverage probability versus power circuit activation threshold γPT for different PB densities. Other system parameters follow Table IV. random variable which depends on its harvested power. Hence, we first evaluate the transmit power for an active TX. Then, we calculate the channel coverage probability at the reference RX. Note that the derived channel coverage probability is in fact a conditional probability, which is conditioned on the reference TX-RX link being active. A. Transmit Power and Locations of Active TX Using the PT assumptions in Section II-A, the instantaneous transmit power for each active TX is    ρ  η 1−ρ PPT , min(η −1 P1max , 1−ρ P max ) > PPT > γPT ηρ 2 Pt = , (12)   ρ 1−ρ  min( P max , P max ), P > min(η −1 P max , P max ) 1−ρ 1 2 PT 1 ηρ 2 where 0 6 η 6 1 is the power conversion efficiency. Note that the first condition in (12) comes from the fact that the received power at an active TX must be greater than γPT . For the second condition in (12), P1max is the maximum harvested power at an active TX when the energy harvesting circuit is saturated and P2max is the maximum transmit power for an active TX. Thus, the second condition caps the transmit power by the allowed maximum harvested power constraint or the maximum transmit power constraint. The following remarks discuss the modelling challenges and proposed solution for characterizing Pt . Remark 2: To the best of our knowledge, the closed-form expression for the PDF of Pt is very difficult to obtain. This is because Pt and PPT are correlated and the closed-form CDF of PPT is not available 14 according to Section III. In the literature, some papers [21], [40] have proposed to use the average harvested power as the transmit power for each TX. However, this does not always lead to accurate results. Hence, inspired from the approach in [41], we propose to discretize Pt in (12) into a finite number of levels. We show that this approximation allows tractable computation of the channel coverage probability. The accuracy of this approximation depends on the number of levels. Our results in Section VI-A show that if we discretize the power level in the log scale, a reasonable level of accuracy is reached with as little as 10 levels. Remark 3: From (12), we can see that Pt depends on PPT . Hence, the motivation for discretizing Pt in the log scale comes from looking into two important measures of PPT , the skewness and the kurtosis. The skewness and the kurtosis describe the shape of the probability distribution of PPT . As presented in [22], the distribution of the aggregate received power is skewed to the right with a heavy tail, because both the skewness and the kurtosis of PPT are much greater than 0 for most cases. Therefore, most of the TXs will be at the lowest power level if we discretize Pt in linear scale. Hence, we discretize the power level in the log scale. This improves the accuracy of the approximation. Let N + 1 and w denote the total number of levels and the step size of each level, respectively. P2max )−γPT min(η −1 P1max , 1−ρ ηρ dBm. We further define kn as the portion of TXs They are related by w = N whose Pt is at the nth level, i.e., kn = Pr ((nw + γPT ) dBm 6 PPT < ((n + 1) w + γPT ) dBm) for n = {0, 1, 2, ..., N − 1} and kN = Pr(PPT > min(η −1 P1max , 1−ρ P max )). Combining with the power coverage ηρ 2 probability derived in Section III, we can express kn as     PPcov ((nw + γPT )dBm) − PPcov (((n + 1) w + γPT ) dBm) , kn =    PP (min(η −1 P max , 1−ρ P max )), cov 1 ηρ 2 n = {0, 1, 2, ..., N − 1} . (13) n=N The above expression allows us to determine the portion of TXs whose Pt is at the nth level. The transmit power for the active TX at the nth level is nw+γPT −30 ρ n 10 Pt = η 10 W. 1−ρ (14) The next step is to decide how to model the locations of the TXs whose Pt is at the nth level. This is discussed in the remark below. 15 Remark 4: In general, the location and the transmit power of an active TX are correlated, i.e., a TX has higher chance to be activated and transmits with a larger power, if its location is closer to a PB. However, it is not easy to identify and fit a spatial point process with local clustering to model the location of active TXs [14], [42]. In this paper, for analytical tractability, we assume that the location and the transmit power of an active TX are independent, i.e., a TX in φt can have a transmit power of Ptn with probability kn independently of other TXs. Therefore, using the thinning theorem, we interpret the active TX at the nth level as an independent homogeneous PPP with node density λnt = kn λt , denoted as φnt . The accuracy of this approximation will be validated in Section VI-A. B. Channel Coverage Probability Given that the desired TX is active, the instantaneous SINR at the reference RX, Y0 , is given as SIN R = P PX0 D0 h0 l(d0 ) , 2 Xi ∈φactive PXi Di0 hi0 l(Xi ) + σ (15) where h0 and hi0 denote the fading power gains on the reference link and the ith interference link respectively, D0 and Di0 denote the beamforming antenna gain at the RX from its reference TX and the ith interfering TX respectively and σ 2 is the AWGN power. PX0 and PXi are the transmit power for the reference TX and the active TX Xi , respectively. Using (15), the power coverage probability is defined as follows. Definition 2: The channel coverage probability is the probability that the SINR at the reference RX is above a threshold γTR and can be expressed as PC cov (γTR ) = Pr(SIN R > γTR ). (16) Remark 5: It is possible to employ the numerical inversion method in Section III to find the channel coverage probability. In doing so, the Laplace transform of the term IX +σ 2 PX0 D0 h0 l(d0 ) is required. This Laplace tranform cannot be expressed in closed-form because of the random variables PX0 and h0 in the denominator. Although it is still computable, it leads to greater computation complexity. Consequently, we employ the reference link power gain (RLPG) based method in [38] to efficiently find the channel coverage 16 probability. The basic principle of this approach is to first find the conditional outage probability in terms of the CDF of the reference links fading power gain and then remove the conditioning on the fading power gains and locations of the interferers, respectively. In order to apply this method, the reference TX-RX link is assumed to undergo Nakagami-m fading with integer m. The result for the conditional channel coverage probability is presented in the following proposition. Proposition 1: Following the system model in Section II, the conditional channel coverage probability at the reference RX in a mmWave ad hoc network is PC cov (γTR ) = where IX = PN P n=0 Xi ∈φn t N m−1 X X (−s)l dl kn 2 (s) , L I +σ X l P (γ l! ds P ) PT cov n=0 l=0 Ptn Di0 hi0 l(Xi ) and s = (17) mγTR . Ptn D0 l(d0 ) Proof: See Appendix B. (17) needs the Laplace transform of the interference plus noise. Using stochastic geometry, we can derive it and the result is shown in the following corollary. Corollary 1: Following the system model in Section II and the discretization assumption in Section IV-A, the Laplace transform of the aggregate interference plus noise at the reference RX in a mmWave ad hoc network is LIX +σ2 (s) = N Y 4 Y −αL n 2 exp πλnt rmin qk mm (m+srmin Pt Dk )−m−1 + πλnt qk (sPtn Dk )δL (Ξ01 (1) − Ξ01 (rmin )) n=0k=1 +πλnt qk sPtn Dk β (Ξ02 (rmin ) − Ξ02 (rmax )) + πλnt qk (sPtn Dk β)δN (Ξ03 (rmin ) − Ξ03 (rmax )) exp(−sσ 2 ), 2 + αN (18) where mm (r−αL sPtn Dk )−δL −m αL Γ(1 + m) mrαL = , (19) 2 F1 1 + m,m + δL ;1 + m + δL ;− (2 + mαL )Γ(m) sPtn Dk r2 Ξ02 (r) = α , (20) r N + sPtn Dk β (r−αN sPtn Dk β)−δN −1 n rαN β −1 0 αN n Ξ3 (r) = sPt Dk β(2 + αN ) −2(r +sPt Dk β)2 F1 1, δN +1; 2+δN ; − n . rαN + sPtn Dk β sPt Dk Ξ01 (r) (21) Proof: Following the definition of Laplace transform, we have LIX +σ2 (s) =EIX [exp(−s(IX + σ 2 ))] = EIX [exp(−sIX )] exp(−sσ 2 ) = LIX (s) exp(−sσ 2 ), (22) 17 where the Laplace transform of the aggregate interference can be expressed as    N X X LIX (s) =EIX [exp(−sIX )] = EDi0 ,hi0 ,φnt exp −s Ptn Di0 hi0 l(Xi ) n=0 Xi ∈φn t = N Y n=0    EDi0 ,hi0 ,φnt exp −s X Ptn Di0 hi0 l(Xi ) . (23) Xi ∈φn t Then, following the same steps as the proof of Laplace transform of aggregate received power in Appendix A, we can find the expectation in (23) and arrive at the result in (18). The Laplace transform shown in Corollary 1 is in closed-form. Substituting (18) into (17), we can easily compute the conditional channel coverage probability. Note that (17) requires higher order derivatives of the Laplace transform of the interference plus noise dl 2 (s), L dsl IX +σ which can be yielded in closed-form using chain rules and changing variables. For brevity, the details are omitted here. V. T OTAL C OVERAGE P ROBABILITY As discussed in Section II-E, the event that the information can be successfully delivered to RX has two requirements, i.e., satisfying power coverage and channel coverage. Based on our definition, the total coverage probability is Pcov (γPT , γTR ) = Pr(TX is in power coverage & RX is in channel coverage) = Pr(TX is in power coverage) Pr(RX is in channel coverage \| TX is in power coverage) Combining our analysis presented in Section III and IV, we have Pcov (γPT , γTR ) =PPcov (γPT )PC cov (γTR ) N m−1 X X (−s)l dl = LI +σ2 (s)kn , l! dsl X n=0 l=0 where s = mγTR , Ptn D0 l(d0 ) (24) LIX +σ2 (s) is given in Corollary 1, kn is presented in (13), which is determined by the power coverage probability. The key metrics are summarized in Table III. VI. R ESULTS In this section, we first validate the proposed model and then discuss the design insights provided by the model. Unless stated otherwise, the values of the parameters summarized in Table IV are used. The 18 TABLE III: Summary of the Analytical Model for PB-assisted mmWave Ad Hoc Networks. Performance metrics General form Key factor(s) Power coverage probability (7) LPPT (s) in (8) Channel coverage probability (17) PP cov (γPT ) in (7) & LIX +σ 2 (s) in (18) Total coverage probability (24) PP cov (γPT ) in (7) & LIX +σ 2 (s) in (18) TABLE IV: Parameter Values. Parameter Value Parameter Value Parameter Value Parameter Value λp 50 /km2 m 5 αL 2 ρ 0.5 λt 100 /km2 min Gmax p , Gp , θp [20 dB, −10 dB, 30o ] αN 4 η 0.5 d0 20 m min Gmax t , Gt , θt [10 dB, −10 dB, 45o ] Pp 40 dBm γPT -20 dBm rmin 100 m min Gmax r , Gr , θr [10 dB, −10 dB, 45o ] P1max 20 dBm γTR 30 dBm rmax 200 m σ2 -30 dBm P2max 30 dBm N 10 chosen values are consistent with the literature in mmWave and WPT [1], [16], [17]. Note that the values of rmin and rmax correspond to 28 GHz mmWave carrier frequency [16]. We mainly focus on illustrating the results for total coverage probability and channel coverage probability. As for the power coverage probability, it will be explained within the text. Table V summarizes the impact of varying the important system parameters5 , i.e., SINR threshold γTR , PB density λp , TX density λt , PB transmit power Pp , radius of the LOS region rmin , power circuit activation threshold γPT , the beam-width of the main lobe of TX θt , RX’s main lobe beam-width θr , allowed maximum harvested power at active TX P1max , time switching parameter ρ and TX maximum transmit power P2max on the three network performance metrics. In Table V, ↑, ↓ and - denote increase, decrease and unrelated, respectively. ↑↓ represents that the performance metric first increases then decreases with the system parameter. Please note that the trends in Table V originate from the analysis of the numerical results, which is presented in detail in the following subsections. 5 Note that the trends reported in Table V remain the same for a two-state blockage model. 19 TABLE V: Effect of Important System Parameters. Parameter Power coverage probability Channel coverage probability Total coverage probability Increasing γTR - ↓ ↓ Increasing λp ↑ ↓↑ ↑ Increasing λt - ↓ ↓ Increasing Pp ↑ ↑ ↑ Increasing rmin ↑ ↑ ↑ Increasing γPT ↓ ↑ ↓ Increasing θt and θr ↑ ↓ ↓ Increasing P1max - ↑↓ ↑↓ Increasing ρ - ↑ ↑ Increasing P2max - ↓ ↓ A. Model Validation In this section, we validate the proposed model for the channel coverage probability and the total coverage probability. Fig. 3 plots the channel coverage probability and the total coverage probability for a reference RX against SINR threshold for different densities of PBs and TXs. The analytical results are obtained using Proposition 1 and (24) with 10 discrete levels for Pt . The simulation results are generated by averaging over 108 Monte Carlo simulation runs and do not assume any discretization of power levels. From the figure, we can see that our analytical results provide a good approximation to the simulation. The small gap between them comes from two reasons: (i) discretization of the power levels, as discussed in Remark 3, and (ii) ignorance of the correlation between the location and the transmit power of active TX, as discussed in Remark 4. From Fig. 3, we can see that the gap between the simulation and the analytical results is smaller, when γTR is higher. At γTR = 30 dBm, which is a typical SINR threshold, the relative errors between the proposed model and the simulation results for both channel coverage probability and total coverage probability are between 5% to 10%. This validates the use of 10 discrete levels for Pt , which provides good accuracy. Insights: Comparing the four cases for the different PB and TX densities, Fig. 3 shows that: (i) The channel coverage probability decreases while the total coverage probability increases as PB density 1 1 0.8 0.8 Coverage probability Coverage probability 20 0.6 0.4 0.2 0 10 Channel Total Simulation 20 30 SINR threshold 40 TR 50 0.6 0.4 0.2 0 10 60 (dBm) 0.8 0.8 Coverage probability Coverage probability 1 0.6 0.4 Channel Total Simulation 20 30 SINR threshold 40 TR 50 (dBm) (c) λp = 50 /km2 , λt = 250 /km2 . 30 40 TR 50 60 (dBm) (b) λp = 10 /km2 , λt = 100 /km2 . 1 0 10 20 SINR threshold (a) λp = 50 /km2 , λt = 500 /km2 . 0.2 Channel Total Simulation 60 0.6 0.4 0.2 0 10 Channel Total Simulation 20 30 SINR threshold 40 TR 50 60 (dBm) (d) λp = 10 /km2 , λt = 50 /km2 . Fig. 3: Channel coverage probability and total coverage probability versus SINR threshold γTR . The PB density is 50 and 10 per km2 and the TX density is 500, 100, 250 and 50 per km2 . increases. As the PB density increases, the aggregate received power at TX increases as well as the number of active TXs. Therefore, interfering power received by the RX is higher and the channel coverage probability decreases. However, the total coverage probability increases because the power coverage probability increases with the PB density. (ii) When the PB density is low, the TXs are very likely to be inactive and the total coverage probability is dominated by the power coverage probability. When the PB density is high, the TXs are very likely to be active. Hence, the interference is strong and the channel coverage probability dominates the total coverage probability. (iii) For the same PB density, both the total coverage probability and the channel coverage probability are higher, when the TX density is lower. This is because more interfering TXs exist if TX density increases. 21 B. Effect of PB Transmit Power Fig. 4(a) illustrates the effect of PB transmit power Pp on the total coverage probability and channel coverage probability, with different radius of the LOS region rmin = 50m, 100m. The simulation results are also plotted in the figure, which are averaged over 108 Monte Carlo simulation runs. The accuracy is between 3% to 8%, which again validates the proposed model. Hence, in the subsequent figures in the paper we only show the analytical results and discuss the insights. Fig. 4(b) plots the total coverage probability against the transmit power of PB. We also plot an asymptotic result when Pp approaches infinity. This result is obtained as follows. As Pp approaches infinity, if one or more PBs fall into the LOS or NLOS region of a TX, this TX will be active and transmit with a ρ P1max , P2max ). Hence, the asymptotic power coverage probability is equivalent to power of Pt = min( 1−ρ the probability that at least one PB falls into the LOS or NLOS region of the TX, which is given by 2 lim PPcov = 1 −exp −πλp rmax . Pp →∞ (25) The asymptotic conditional channel coverage probability and the asymptotic total coverage probability can be found by (17) and (24) respectively with the portion of TXs at the nth level as     0, n = {0, 1, 2, ..., N − 1} . lim kn = Pp →∞    PPcov , n = N (26) From the figure, we can see that the analytical and asymptotic results converge as Pp gets large, which validates the derivation of the asymptotic results. In addition, in Fig. 4(b), we have marked the safe RF exposure region with a PB transmit power less than 51 dBm, equivalently power density smaller than 10 W/m2 at 1 m from the PB [26]. We will discuss in detail later in the feasibility study in Section VI-E. Insights: Fig. 4(a) shows that: (i) The channel coverage probability first slightly decreases and then increases with the increase of Pp . This can be explained as follows. At first, both the transmit power of the desired TX and the number of interfering TX increase with Pp . The interplay of this two factors results in the slightly decreasing trend for the channel coverage probability. As Pp further increases, the increase in the number of interfering TX is negligible, while the transmit power of the desired TX continues to 22 1 Total coverage probability 1 Coverage probability Channel coverage probability 0.8 Total coverage probability 0.6 rmin=50m 0.4 rmin=100m Simulation (P ccov) 0.2 Simulation (P cov) 0 30 35 40 45 Asymptotic total coverage probability=0.9953 0.8 0.6 0.4 0.2 50 PB transmit power Pp (dBm) (a) Channel coverage and total coverage with different rmin . Analytical Asymptotic 0 30 Safe level of RF exposure 40 50 Unsafe level of RF exposure 60 70 80 PB transmit power Pp (dBm) (b) Total coverage and asymptotic total coverage. Fig. 4: Coverage probabilities versus PB transmit power Pp . increase, which leads to the increase of the channel coverage probability. (ii) The total coverage probability increases as PB transmit power Pp increases. When Pp is small, the desired TX might not receive enough power to activate the IT process. So the total coverage probability is small and is limited by the power coverage probability. When Pp is large, the channel coverage probability becomes the dominant factor in determining the total coverage probability. Hence, eventually the channel coverage probability and total coverage probability curves merge. (iii) The total coverage probability increase with rmin , because more PBs falls into the LOS region and the path-loss is less severe, which improves the power coverage probability. The benefit of increasing the radius of the LOS region is less significant for the channel coverage probability. C. Effect of Directional Beamforming at PB, TX and RX Fig. 5 plots the total coverage probability and channel coverage probability against the power circuit activation threshold of TX for different beamforming parameters at TX and RX, i.e., [20 dB, −10 dB, 30o ] and [10 dB, −10 dB, 45o ]. Insights: Fig. 5 shows that, for both sets of beamforming parameters, as the power circuit activation threshold γPT increases, the channel coverage probability is always increasing, while the total coverage probability stays roughly the same at first and then decreases. This can be explained as follows. When γPT 23 1 Channel coverage probability 0.9 0.8 Total coverage probability 0.7 o TX, RX [20, -10, 30 ] TX, RX [10, -10, 45 o] 0.6 -30 Total coverage probability Coverage probability 1 Np =4, 9, 16 0.9 1 0.8 0.95 0.7 0.9 0.6 0.85 4 6 8 N =4 10 p Np=9 0.5 Np=16 0.4 -25 -20 -15 Power circuit activation threshold -10 PT (dBm) 5 10 15 20 Number of TX and RX antenna elements Nt , N r Fig. 5: Channel coverage probability and total coverage probability Fig. 6: Total coverage probability versus the numbers of antenna versus power circuit activation threshold γPT with different TX and elements at TX and RX Nt and Nr with different numbers of RX beamforming parameters. antenna elements at PB. increases, the power coverage probability decreases. The reduction in the number of active TXs improves the channel coverage probability. With regards to the total coverage probability, its trend is determined by the interplay of channel coverage probability and power coverage probability. At first, the drop in power coverage is relatively small as shown in Fig. 2; so the total coverage probability is almost unchanged. After a certain point, the power coverage probability drops a lot, which mainly governs the total coverage probability. Hence, the total coverage probability decreases later on. Comparing the curves for the different beamforming parameters, we can see that TX and RX with [20 dB, −10 dB, 30o ] gives a higher total coverage probability in the low power circuit activation threshold region. This is because a narrower main lobe beam-width gives a larger main lobe gain and makes less interfering TXs fall into its main lobe which results in higher channel coverage probability. However, the total coverage probability is limited by the power coverage probability when γPT is large. Impact of number of antenna elements: The beamforming model adopted in this paper can be related to any specific array geometry by substituting the appropriate values for the three beamforming parameters. For instance, a uniform planar square array with half-wavelength antenna element spacing can be used min at the PBs, TXs and RXs. The values for the main lobe antenna gain Gmax a , side lobe antenna gain Ga and main lobe beamwidth θa depend on the number of the antenna elements Na and can be calculated 24 by using the equations below [18]: √ Gmax = Na , Gmin a a √ √ √ Na − 2π3 Na sin( 2√N3 a ) 3 √ √ , θa = √ , = √ 3 3 Na Na − 2π sin( 2√Na ) (27) where subscript a = p for PB, a = t for TX and a = r for RX. Fig. 6 plots the total coverage probability versus the numbers of antenna elements at the TX and RX Nt and Nr with different PB antenna element number Np . The figure shows that the total coverage probability increases with the numbers of antenna elements at the TX and RX, which agrees with our previous findings. However, under our considered system parameters, the total coverage probability stays roughly the same after having more than about 15 TX and RX antenna elements, as the side lobe antenna gain and the main lobe beamwidth stay almost constant with further increase in the number of antenna elements. In addition, the number of antenna elements at the PB does not significantly impact the total coverage probability. D. Effect of Allowed Maximum Harvested Power at TX Fig. 7 plots the total coverage probability and channel coverage probability against the allowed maximum harvested power of TX P1max for different time switching ratios 0.2, 0.5 and 0.8. Note that both the time switching ratio and the allowed maximum harvested power do not affect the power coverage probability. Insights: Fig. 7 shows that the channel coverage probability and the total coverage probability both first increase with P1max , then decrease. The rise of the channel coverage probability is because the possible transmit power of the desired TX increases with its allowed maximum harvested power. However, as P1max further increases, the accumulated harvested energy during the PT phase is higher and the transmit power of other active TX also goes up. As a result, the interfering power received at the RX is higher and the channel coverage probability decreases. The channel coverage probability will converge to a constant value as P1max increases even further, because the maximum transmit power of active TX has limited the channel performance. Comparing the curves for different ρ, we can see that for a given maximum harvested power of TX P1max , increasing ρ improves the coverage probabilities. When ρ is higher, more energy is captured during 25 1 1 Coverage probability Coverage probability Pp =50 dBm 0.8 =0.2 =0.5 =0.8 0.6 0.4 0.2 Channel coverage probability Total coverage probability 0 -20 -10 0 10 20 30 0.9 0.8 Pp =30 dBm 0.7 0.6 Channel coverage probability Total coverage probability 0.5 20 Maximum harvested power Pmax (dBm) 1 25 30 35 40 Maximum TX transmit power Pmax (dBm) 2 Fig. 7: Channel coverage probability and total coverage probability Fig. 8: Channel coverage probability and total coverage probability versus allowed maximum harvested power P1max with different time versus maximum TX transmit power P2max for different PB transmit switching ratios. power with the allowed maximum harvested power of TX being 50 dBm. the PT phase. Therefore, the transmit power of active TX is now limited by the maximum transmit power P2max . As a result, the channel coverage probability and total coverage probability converge and do not vary much with the changes in the allowed maximum harvested power. E. Feasibility of PB-assisted mmWave Wireless Ad hoc Networks Finally, we investigate the feasibility of PB-assisted mmWave wireless ad hoc network. Fig. 8 is a plot of the total coverage probability and channel coverage probability versus maximum TX transmit power P2max with varied PB transmit power, 50 dBm and 30 dBm. To better highlight the impact of the maximum transmit power at TX, we have set P1max equal to 50 dBm which is much higher than P2max . From the figure, we can see that the channel coverage probability and total coverage probability do not change much with the considered maximum TX transmit power, which means that the probability mass function (PMF) of the transmit power for the desired TX remains almost the same. Note that the power coverage probability is independent of the maximum transmit power of TX. Insight: From Fig. 8, the total coverage probability and the channel coverage probability are around 90% if Pp is 50 dBm. If PB transmits with a constant power of 50 dBm, the power density at a distant of 1 m from the PB is 7.95 W/m2 . This power density is smaller than 10 W/m2 , which is the permissible 26 safety level of human exposure to RF electromagnetic fields based on IEEE Standard. Under this safety regulation, the maximum permissible PB transmit power would be 51 dBm. We have marked this value in Fig. 4(b). From Fig. 4(b), we can see that the total coverage probability with a PB transmit power less than 51 dBm can be up to 93.4% of the maximum system performance, as given by the asymptotic analysis in Section VI-B, based on our considered system parameters. The results in Fig. 4(b) and Fig. 8 show that PB-assisted mmWave ad hoc networks are feasible under practical network setup. VII. C ONCLUSION In this paper, we have presented an approximate yet accurate model for PB-assisted mmWave wireless ad hoc networks, where TXs harvest energy from all PBs and then use the harvested energy to transmit information to their corresponding RXs. We first obtained the Laplace transform of the aggregate received power at the TX to compute the power coverage probability. Then, the channel coverage probability and total coverage probability were formulated based on discretizing the transmit power of TXs into a finite number of levels. The simulation results confirmed the accuracy of the proposed model. The results have shown that the total coverage probability improves by increasing the transmit power of PB, narrowing the main lobe beam-width and decreasing the maximum harvested power at the TX. Our results also showed that PB-assisted mmWave ad hoc network is feasible under realistic setup conditions. Future work can consider extensions to other MAC protocols such as carrier-sense multiple access (CSMA) [43], [44] and optimal allocation of the transmit power of an active TX. 27 A PPENDIX A P ROOF OF T HEOREM 1 Following the definition of Laplace transform, the Laplace transform of the aggregate received power can be expressed as    LPPT (s) = EPPT [exp(−sPPT )] = Eφp ,Gi0 ,gi0 exp −sPp X Gi0 gi0 l(ri ) Zi ∈φp " !# = Eφp ,Gi0 ,gi0 exp −sPp X " Eφp ,Gi0 ,gi0 exp −sPp Gi0 gi0 l(ri ) 06ri <1 !# X Gi0 gi0 l(ri ) 16ri <rmin " !# × Eφp ,Gi0 ,gi0 exp −sPp X Gi0 gi0 l(ri ) rmin 6ri <rmax Z πZ 1 EGi0 ,gi0 [1 − exp(−sPp Gi0 gi0 )]λp rdrdθ = exp − −π 0 {z } \| A1 Z π Z rmin −αL EGi0 ,gi0 [1−exp(−sPp Gi0 gi0 r )]λp rdrdθ × exp − −π 1 \| {z } A2 Z π Z rmax −αN EGi0 ,gi0 [1−exp(−sPp Gi0 gi0 βr )]λp rdrdθ . × exp − −π rmin {z } \| (28) A3 The first term A1 is evaluated as follows A1 = exp (−πλp (1 − EGi0 ,gi0 [exp(−sPp Gi0 gi0 )])) Z ∞ = exp −πλp 1 − EGi0 exp(−sPp Gi0 g)fgL (g)dg 0 = exp −πλp + πλp mm EGi0 (m + sPp Gi0 )−m ! 4 X = exp −πλp + πλp mm (m + sPp Gk )−m pk , (29) k=1 where we use the fact that the link in LOS state experiences Nakagami-m fading with fgL (g) = mm g m−1 exp(−mg) . Γ(m) The second term A2 is evaluated as follows −αL 2 A2 = exp πλp EGi0 ,gi0 [1 − exp(−sPp Gi0 gi0 )] − πλp rmin EGi0 ,gi0 1 − exp(−srmin Pp Gi0 gi0 ) h i δL δL − πλp EGi0 ,gi0 (sPp Gi0 ) gi0 γ(1 − δL , sPp gi0 Gi0 ) h i δL −αL +πλp EGi0 ,gi0 (sPp Gi0 )δL gi0 γ(1 − δL , sPp gi0 Gi0 rmin ) (30a) 28 m = exp πλp −πλp m 4 X −m (m + sPp Gk ) 2 pk −πλp rmin k=1 + πλp 4 X (sPp Gk )δL k=1 4 X m + 4 X −αL 2 πλp rmin mm (m + srmin Pp Gk )−m pk k=1 −δL−m m m (sPp Gk ) αL Γ(1+m) pk 2 F1 1 + m,m + δL ;1 + m + δL ;− (2 + mαL )Γ(m) sPp Gk −αL mm (rmin sPp Gk )−δL−m αL Γ(1+m) (2 + mαL )Γ(m) k=1 αL rmin m pk , × 2 F1 1 + m,m + δL ;1 + m + δL ;− sPp Gk − πλp (sPp Gk )δL (30b) where (30a) follows from changing variables and integration by parts and (30b) is obtained after taking the expectation over gL then Gi0 . Similarly, the third term A3 can be worked out by taking the expectation over gN , which has a PDF as fgN (h) = exp(−g). The details are omitted for sake of brevity. Finally, the Laplace transform in Theorem 1 is obtained by substituting A1 , A2 and A3 into (28). A PPENDIX B P ROOF OF P ROPOSITION 1 By substituting (15) into (16), we can express the conditional channel coverage probability as ! PX0 D0 h0 l(d0 ) C Pcov (γTR ) = Pr P > γTR 2 Xi ∈φactive PXi Di0 hi0 l(Xi ) + σ ! PX0 D0 h0 l(d0 ) ≈ Pr PN P (31a) > γTR n 2 P D i0 hi0 l(Xi ) + σ t n=0 Xi ∈φn t γTR (IX + σ 2 ) γTR (IX + σ 2 ) = Pr h0 > = EPX0 ,IX 1 − Fh0 , (31b) PX0 D0 l(d0 ) PX0 D0 l(d0 ) P P where approximation in (31a) comes from our power level discretization, IX = N Ptn Di0 hi0 l(Xi ) n=0 Xi ∈φn t and Fh0 (·) is the CDF of the fading power gain on the reference TX-RX link. Since the desired link is assumed to experience Nakagami-m fading with integer m, the CDF of h0 has a nice form, which is P 1 l Fh0 (h) = 1 − m−1 l=0 l! (mh) exp(−mh). Hence, we can re-write (31b) as "m−1 l # 2 2 X1 γ (I + σ ) γ (I + σ ) TR X TR X PC m exp −m cov (γTR ) =EPX0 ,IX l! 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SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 arXiv:1512.02289v2 [math.RA] 5 Aug 2016 ANTHONY BAK AND ALEXEI STEPANOV Abstract. In 2012 the second author obtained a description of the lattice of subgroups of a Chevalley group G(Φ, A), containing the elementary subgroup E(Φ, K) over a subring K ⊆ A provided Φ = Bn , Cn or F4 , n > 2, and 2 is invertible in K. It turns out that this lattice is a disjoint union of “sandwiches” parameterized by subrings R such that K ⊆ R ⊆ A. In the current article a similar result is proved for Φ = Cn , n > 3, and 2 = 0 in K. In this setting one has to introduce more sandwiches, namely, sandwiches which are parameterized by form rings (R, Λ) such that K ⊆ Λ ⊆ R ⊆ A. The result we get generalizes Ya. N. Nuzhin’s theorem of 2013 concerning the root systems Φ = Bn , Cn , n > 3, where the same description of the subgroup lattice is obtained, but under the condition that A is an algebraic extension of a field K. Introduction Throughout this paper K, R and A will denote commutative rings. Let G = GP (Φ, ) denote a Chevalley–Demazure group scheme with a reduced irreducible root system Φ and weight lattice P . If the weight lattice P is not important, we leave it out of the notation. Denote by E(A) = EP (Φ, A) the elementary subgroup of G(A), i. e. the subgroup generated by all elementary root unipotent elements xα (t), α ∈ Φ, t ∈ A. Let K be a subring of A. We study the lattice L = L E(K), G(A) of subgroups of G(A), containing E(K). The standard description of L is called a sandwich classification theorem. It states that for each H ∈ L there exists a unique subring R between K and A suchthat H lies between E(R) and its normalizer NA (R) in G(A). The lattice L E(R), NA (R) of all subgroups of G(A) that lie between E(R) and NA (R) is called a standard sandwich. Thus, the sandwich classification theorem holds iff L is the disjoint union of all standard sandwiches. In [17] the second author proved the sandwich classification theorem provided that Φ is doubly laced and 2 is invertible in K. In this article we consider the symplectic case of rank n > 3 with 2 = 0 in A, in particular, we always assume that Φ = Cn . In this situation we show that the sandwich classification theorem, as formulated above, does not hold. Let R be a subring of A, containing K. Recall [5] that an additive subgroup Λ of R is called a (symplectic) form parameter if it contains 2R and is closed under multiplication by ξ 2 for all ξ ∈ R. If 2 = 0, the set R2 = {ξ 2 \| ξ ∈ R} is a subring of R, and a form parameter Λ in R is just an R2 -submodule of R. Let Ep2n (R, Λ) denote the subgroup of Sp2n (R) generated by all root unipotents xα (µ) and xβ (λ), where α is a short root and µ ∈ R and β is a long root and λ ∈ Λ. Suppose now that Λ ⊇ K. Then clearly Ep2n (R, Λ) > Ep2n (K). But one can check that if Λ 6= R ′ ′ then Ep2n (R, Λ) is not contained in any standard sandwich L Ep2n (R ), NA (R ) such that K ⊆ R′ . This shows that L cannot be a union of standard sandwiches, unless we enlarge our standard sandwiches or introduce additional ones. It turns out that the latter approach is the correct one. It is analogous to that followed by the first author in [3] and by E. Abe and The work of the second named author under this publication is supported by Russian Science Foundation, grant N.14-11-00297. 1 2 ANTHONY BAK AND ALEXEI STEPANOV K.Suzuki in [2, 1] in order to provide enough sandwiches to classify subgroups of Sp2n (R), which are normalized by Ep2n (R). If A and K are field and A is algebraic over K, then the sandwich classification theorem was obtained by Ya. N. Nuzhin in [14, 15]. In [15] he considered the case of doubly laced root systems in characteristic 2 and G2 in characteristics 2 and 3. In all these cases sandwiches were parameterized by pairs; each pair consists of a subring and an additive subgroup, satisfying certain properties. Let F be a field of characteristic 2. Then there are injections Sp2n (F ) ֌ SO2n+1 (F ) ֌ Sp2n (F ), which turn to isomorphisms if F is perfect, see [16] Theorem 28, Example (a) after this theorem, and Remark before Theorem 29. Note that in this case the Chevalley groups GP (Cn , F ) and GP (Bn , F ) do not depend on the weight lattice P ([16], Exercise after ′ Corollary 5 to Theorem 4 ). Therefore, the lattice L E(Φ, K), G(Φ, F ) embeds to the lattice L Ep2n (F2 ), Sp2n (F ) for Φ = Bn , Cn and a subring K of F , and its description follows from the main result of the current article. Actually, the injections above are constructed in [16] on the level of Steinberg groups. Since over a field K2 (Φ, F ) is generated by Steinberg symbols, they induce the injections of Chevalley groups. Over a ring A there is no appropriate description of K2 available, therefore we can not use the above arguments. Moreover, in the ring case the group GP (Φ, A) depends on P . By these reasons, the adjoint group of type Cn and groups of type Bn over rings will be considered in a subsequent article. We state now our main result. Following [5], we call a pair (R, Λ) consisting of a ring R and a form parameter Λ in R a form ring. Theorem 1. Let A denote a (commutative) ring such 2 = 0 in A. Let K be a subring of A. If H is a subgroup of Sp2n (A) containing Ep2n (K), n > 3, then there is a unique form ring (R, Λ) such that K ⊆ Λ ⊆ R ⊆ A and Ep2n (R, Λ) 6 H 6 NA (R, Λ), where NA (R, Λ) is the normalizer of Ep2n (R, Λ) in Sp2n (A). Let L(R, Λ) = L Ep2n (R, Λ), NA (R, Λ) denote the lattice of all subgroups H of Sp2n (A) such that Ep2n (R, Λ) 6 H 6 NA (R, Λ). From now on, L(R, Λ) is what we shall mean by a standard sandwich. In view of Theorem 1 it is natural to study the lattice structure of L(R, Λ). By definition Ep2n (R, Λ) is normal in NA (R, Λ). Therefore the lattice structure of L(R, Λ) is the same as that of the quotient group NA (R, Λ)/ Ep2n (R, Λ). The lattice L(R, Λ) contains an important subgroup Sp2n (R, Λ), which is called a Bak symplectic group, whose definition will be recalled in Section 1. A group will be called quasi-nilpotent if it is a direct limit of nilpotent subgroups, i. e. the group has a directed system of nilpotent subgroups whose colimit is the group itself. In the following result we use the notion of Bass–Serre dimension of a ring introduced by the first author in [6, § 4]. Recall that the Krull (or combinatorial) dimension of a topological space is the supremum of the lengths of proper chains of nonempty closed irreducible subsets. Bass–Serre dimension d = BS-dim R of a ring R is the smallest integer such that the maximal spectrum Max R is a union of a finite number of irreducible Noetherian subspaces of Krull dimension not greater than d. If there is no such integer, then BS-dim R = ∞. Theorem 2. Let A denote a (commutative) ring and let R be a subring of A and (R, Λ) a form ring. Suppose n > 2. 1. Sp2n (R, Λ) = Sp2n (R) ∩ NA (R, Λ). 2. Sp2n (R, Λ) is normal in NA (R, Λ). SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 3 3. NA (R, Λ)/ Sp2n (R, Λ) is abelian. 4. Sp2n (R, Λ)/ Ep2n (R, Λ) is quasi-nilpotent. 5. Let R0 denotes the subring of R, generated by all elements ξ 2 such that ξ ∈ R. If Bass–Serre dimension of R0 is finite, then Sp2n (R, Λ)/ Ep2n (R, Λ) is nilpotent. In particular, the sandwich quotient group NA (R, Λ)/ Ep2n (R, Λ) is quasi-nilpotent by abelian or nilpotent by abelian if BS-dim R0 < ∞. Although Theorem 1 is proven under the assumptions 2 = 0 and n > 3, we do not invoke these assumptions until the proof of the theorem in Section 6. In particular, Theorem 2 is proven without these assumptions. Of course, we always assume that n > 2. Notation. Let H be a group. For two elements x, y ∈ H we write [x, y] = xyx−1 y −1 for their commutator and xy = y −1xy for the y-conjugate of x. For subgroups X, Y 6 H we let X Y denote the normal closure of X in the subgroup generated by X and Y , while [X, Y ] stands for the mixed commutator group generated by X and Y . By definition it is the group generated by all commutators [x, y] such that x ∈ X and y ∈ Y . The commutator subgroup [X, X] of X will be also denoted by D(X) and we set D k (X) = D k−1 (X), D k−1(X) . Recall that a group X is called perfect if D(X) = X. The identity matrix is denoted by e as well as the identity element of a Chevalley group. We denote by eij the matrix with 1 in position (ij) and zeroes elsewhere. The entries of a matrix g are denoted by gij . For the entries of the inverse matrix we use abbreviation (g −1)ij = gij′ . The transpose of the matrix g is denoted by g T, thus (g T )ij = gji . 1. The symplectic group The symplectic group Sp2n (R), its elementary root unipotent elements, and its elementary subgroup Ep2n (R) will be recalled below. The groups Sp2n (R, Λ) and their elementary subgroups Ep2n (R, Λ) will be defined in Section 2. Since the groups Sp2n (R, Λ) are not in general algebraic (in fact, Sp2n (R, Λ) is algebraic iff Λ = R or Λ = {0}), it is convenient to work with the standard matrix representation of Sp2n (R). On the other hand, we want to use the notions of parabolic subgroup, unipotent radical, etc. from the theory of algebraic groups. Thus, to simplify the exposition, we define below these notions directly in terms of the matrix representation we use. Following Bourbaki [9] we view the root system Cn and its set of fundamental roots Π in the following way. Let Vn denote n-dimensional euclidean space with the orthonormal basis ε1 , . . . , εn . Let Cn = {±εi ± εj \| 1 6 i < j 6 n} ∪ {±2εk \| 1 6 k 6 n}, Π = {αi = εi − εi+1 , where i = 1, . . . , n − 1, αn = 2εn }. The elements ±εi ± εj are called short roots and the elements ±2εk long roots. From the perspective of algebraic groups, we want the standard Borel subgroup of Sp2n (R) to be the group of all upper triangular matrices in Sp2n (R). Accordingly, we take the following matrix description of Sp2n (R). Let J denote the n × n-matrix with 1 in each antidiagonal 0 J position and zeroes elsewhere. Let F = −J 0 . Then, the group Sp2n (R) is the subgroup of GL2n (R) consisting of all matrices which preserve the bilinear form whose matrix is F . In other words, Sp2n (R) = {g ∈ GL2n (R) \| g T F g = F }. Let I = (1, . . . , n, −n, . . . , −1) denote the linearly ordered set whose linear ordering is obtained by reading from left to right. We enumerate the rows and columns of the matrices 4 ANTHONY BAK AND ALEXEI STEPANOV of GL2n (R) by indexes from I. Thus the position of a coordinate of a matrix in GL2n (R) is denoted by a pair (i, j) ∈ I × I. In the language of algebraic groups the set of all diagonal matrices in Sp2n (R) is a maximal torus. Let e denote the 2n × 2n identity matrix. The following matrices are elementary root unipotent elements of Sp2n (R) with respect to the torus above: xεi −εj (ξ) = Tij (ξ) = T−j,−i (−ξ) = e + ξeij − ξe−j,−i, xεi +εj (ξ) = Ti,−j (ξ) = Tj,−i (ξ) = e + ξei,−j + ξej,−i , x−εi −εj (ξ) = T−i,j (ξ) = T−j,i (ξ) = e + ξe−i,j + ξe−j,i , x2εk (ξ) = Tk,−k (ξ) = e + ξek,−k , x−2εk (ξ) = T−k,k (ξ) = e + ξe−k,k , where ξ ∈ R, 1 6 i, j, k 6 n, i 6= j. Note that the subscripts (i, j), (i, j), (i, −j), (j, −i), (−j, −i), (k, −k), and (−k, k) on the T above all belong to the set C̃n = I × I \ {(k, k) \| k ∈ I} of nondiagonal positions of a matrix from Sp2n (R) and exhaust C̃n . There is a surjective map p : C̃n → Cn defined by the following rule. p(i, j) = p(−j, −i) = εi − εj ; p(i, −j) = p(j, −i) = εi + εj ; p(−i, j) = p(−j, i) = −εi − εj ; p(k, −k) = 2εk ; p(−k, k) = −2εk ; where ξ ∈ R, 1 6 i, j, k 6 n, i 6= j. With this notation the correspondence between elementary symplectic transvections Tij (ξ) and root elements xα (ξ) looks as follows. Tij (ξ) = xp(i,j) (− sign(ij)ξ) for all i 6= j ∈ I. Note that p maps symmetric (with respect to the antidiagonal) positions to the same root. Therefore, for (ij) ∈ I and ξ ∈ R we have Ti,j (ξ) = T−j,−i (− sign(ij)ξ). The root subgroup scheme Xα is defined by Xα (R) = {xα (ξ) \| ξ ∈ R}. The scheme Xα is naturally isomorphic to Ga , i. e. xα (ξ)xα (µ) = xα (ξ + µ) for all ξ, µ ∈ R. The following commutator formulas are well known in matrix language, cf. [8, § 3]. They are special cases of the Chevalley commutator formula in the algebraic group theory. c = 2π/3; [xα (λ), xβ (µ)] = xα+β (±λµ), if α + β ∈ Φ, αβ c = π/2; [xα (λ), xβ (µ)] = xα+β (±2λµ), if α + β ∈ Φ, αβ c = 3π/4; [xα (λ), xβ (µ)] = xα+β (±λµ)xα+2β (±λµ2 ), if α + β, α + 2β ∈ Φ, αβ [xα (λ), xβ (µ)] = e, if α + β ∈ / Φ ∪ {0} In our proofs we make frequent use of the parabolic subgroup P1 of Sp2n . In the matrix language above it is defined as follows: SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 5 P1 (R) = {g ∈ Sp2n R \| gi1 = g−1−i = 0 ∀i 6= 1} −1 The definition above of Sp2n (R) shows that g11 = g−1,−1 for any matrix g ∈ P1 (R). The unipotent radical U1 of P1 is the subgroup generated by all root subgroups T1i such that i 6= 1. The Levi subgroup L1 (R) of P1 (R) consists of all g ∈ P1 (R) such that g1i = g−i,−1 = 0 for all i 6= 1. As a group scheme it is isomorphic to Gm × Sp2n−2 . 2. Bak symplectic groups The Bak symplectic group Sp2n (R, Λ) is the particular case of the Bak general unitary group, where the involution is trivial and the symmetry λ = −1. The main references for the definition and the structure of the general unitary group is the book [5] and the paper [8] by N. Vavilov and the first author. In this section we recall definitions and simple properties to be used in the sequel. Let R be a commutative ring. An additive subgroup Λ of R is called a symplectic form parameter in R, if it contains 2R and is closed under multiplication by squares, i. e. µ2 λ ∈ Λ for all µ ∈ R and λ ∈ Λ. Define Sp2n (R, Λ) as the subgroup of Sp2n (R) consisting of all matrices preserving the quadratic form which takes values in R/Λ and is defined by the matrix ( 00 J0 ). Let ∗ denote the involution on the matrix ring Mn (R) given by the formula a∗ = JaT J. Note that this is the reflection of a matrix with respect to the antidiagonal. Define Mn (R, Λ) = {a ∈ Mn (R) \| a = a∗ , an−k+1 k ∈ Λ for all k = 1, . . . , n}. Write a matrix of degree 2n in the block form ( ac db ), where a, b, c, d ∈ Mn (R). It follows from [8, Lemma 2.2] and its proof that under the notation above we have the following formula. a b ∗ ∗ ∗ ∗ Lemma 2.1. Sp2n (R, Λ) = \| a d − c b = e and c a, d b ∈ Mn (R, Λ) . c d It is easy to check that Mn (R, Λ) is a form parameter in the ring Mn (R) with involution ∗, corresponding to the symmetry λ = −1. The minimal form parameter in Mn (R) with the same involution and symmetry is Mn (R, 2R); it is denoted by Minn (R). Let M̄n (R, Λ) denote the additive group Mn (R, Λ)/ Minn (R). Lemma 2.2. The group M̄n (R, Λ) has a natural structure of a left Mn (R)-module under the operation a ◦ b̄ = aba∗ mod Minn (R), where a ∈ Mn (R), b̄ ∈ M̄n (R, Λ), and b is a preimage of b̄ in Mn (R, Λ). By abuse of notation for a ∈ Mn (R) and b ∈ Mn (R, Λ) we shall write a ◦ b instead of a ◦ (b + Minn (R)). It is easy to check that an elementary root unipotent element xα (ξ) ∈ Sp2n (R) belongs to Sp2n (R, Λ) if and only if α is a short root or ξ ∈ Λ. Denote by Ep2n (R, Λ) the group generated by all such elements: Ep2n (R, Λ) = hxα (ξ) \| α ∈ Cnshort & ξ ∈ R ∨ α ∈ Cnlong & ξ ∈ Λi. 3. Subgroups generated by elementary root unipotents In this section we assume that n > 3. Let H be a subgroup of Sp2n (A), containing Ep2n (K). The following lemma shows that we can uncouple a short root element from a long one inside H. 6 ANTHONY BAK AND ALEXEI STEPANOV Lemma 3.1. Let α, β ∈ Cn be a short and a long root, respectively, such that α + β is not a root. Let g = xα (µ)xβ (λ), where λ, µ ∈ A. If Ep2n (K)g 6 H (e. g. g ∈ H), then each factor of g belongs to H. Proof. Since n > 3, there exists a short root γ such that γ + α is a short root and γ + β is not a root. Then, Xβ (A) commutes with Xα (A) and Xγ (A), hence [g, xγ (1)] = xα+γ (±µ) ∈ H. Conjugating this element by an appropriate element from the Weyl group over K and taking the inverse if necessary, we get xα (µ) ∈ H. It follows that Ep2n (K)xβ (λ) 6 H. Now, take a short root δ such that β+δ is a root. Then, [xδ (1), xβ (λ)] = xδ+β (±λ)x2δ+β (±λ) ∈ H. Notice that the root δ + β is short whereas 2δ + β is long. As in the first paragraph of the proof one concludes that xδ+β (±λ) ∈ H, hence x2δ+β (±λ) ∈ H. Again, using the action of the Weyl group and taking inverse if necessary, one shows that xβ (λ) ∈ H. Put Pα (H) = {t ∈ A \| xα (t) ∈ H}. Since the Weyl group acts transitively on the set of roots of the same length, it is easy to see that Pα (H) = Pβ (H) if \|α\| = \|β\|. Let R = RH = Pα (H) for any short root α, and let Λ = ΛH = Pβ (H) for any long root β. Lemma 3.2. With the above notation (R, Λ) is a form ring and K ⊆ Λ ⊆ R ⊆ A. Proof. Clearly, Pα (H) is an additive subgroup of A. Since n > 3, there are two short roots α, α′ such that α + α′ also is short. The commutator formula [xα (λ), xα′ (µ)] = xα+α′ (±λµ) shows that R is a ring. Now, let α, α′ be short roots which are orthogonal in Vn and such that β = α + α′ ∈ Φ is a long root. Then [xα (µ), xα′ (1)] = xβ (±2µ). If µ ∈ R, then this element belongs to H. This proves that 2R ⊆ Λ. Finally, we show that Λ is closed under multiplication by squares in R. Let g = [xβ (λ), x−α (µ)] = xα′ (±λµ)xβ−2α (±λµ2 ) If λ ∈ Λ and µ = 1, then g ∈ H, and Lemma 3.1 shows that xα′ (±λ) ∈ H. Therefore, Λ ⊆ R. On the other hand, if µ ∈ R and λ ∈ Λ, then g also lies in H, and by Lemma 3.1 we have xβ−2α (±λµ2 ) ∈ H. This shows that Λ is stable under multiplication by squares of elements of R. If H is a subgroup of Sp2n (A) containing Ep2n (K), then (RH , ΛH ) is called the form ring associated with H. 4. The normalizer Let (R, Λ) be a form subring of a ring A such that 1 ∈ Λ. In this section we develop properties of the normalizer NA (R, Λ) of the group Ep2n (R, Λ) in Sp2n (A) and prove Theorem 2. Here we do not assume that n > 3. By a result of Bak and Vavilov [7, Theorem 1.1] we know that Ep2n (R, Λ) is normal in Sp2n (R, Λ), thus Sp2n (R, Λ) 6 NA (R, Λ). First, we show that the quotient NA (R, Λ)/ Sp2n (R, Λ) is abelian. Lemma 4.1. Let g ∈ Sp2n (A). If Ep2n (R, Λ)g 6 GL2n (R) then gij gkl ∈ R for all i, j, k, l ∈ I. P Proof. To begin we express a matrix unit ejk as a linear combination ξm a(m) for some (m) ξm ∈ R and a ∈ Ep2n (R, Λ). Note that the set of such linear combinations is closed under multiplication. Suppose j 6= k and let i 6= ±j, ±k. Then ejk = (Tji (1) − e)(Tik (1) − e) and ejj = ejk ekj . It follows that if Ep2n (R, Λ)g 6 GL2n (R), then g −1 ejk g ∈ Mn (R). Thus SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 7 gij′ gkl ∈ R for all i, j, k, l ∈ I (recall that gij′ = (g −1 )ij ). The conclusion of the lemma follows since the entries of g −1 coincide with the entries of g up to sign and a permutation. The next proposition describes the normalizer in the case A = R. Proposition 4.2. NR (R, Λ) = Sp2n (R, Λ). Proof. Let g = ( ac db ) ∈ NR (R, Λ). Since g ∈ Sp2n (R), then d∗ a − b∗ c = e. We have to prove that c∗ a, d∗ b ∈ Mn (R, Λ). Since h = ( 0e Je ) ∈ Ep2n (R, Λ), we have e − aJc∗ aJa∗ −1 ∈ Ep2n (R, Λ) f = ghg = −cJc∗ e + cJa∗ It follows that the matrices −(cJc∗ )∗ (e − aJc∗ ) and (e + cJa∗ )∗ (aJa∗ ) belong to Mn (R, Λ). Since cJc∗ and aJa∗ are automatically in Mn (R, Λ), we have that cJc∗ aJc∗ , aJc∗ aJa∗ ∈ Mn (R, Λ). Modulo Minn (R) we can write (cJ)◦(c∗a), (aJ)◦(c∗ a) ∈ M̄n (R, Λ). By Lemma 2.2 M̄n (R, Λ) is an Mn (R) module, therefore c∗ a + Minn (R) = J(d∗ a − b∗ c)J ◦ (c∗ a) = (Jd∗ ) ◦ (aJ) ◦ (c∗ a) − (Jb∗ ) ◦ (cJ) ◦ (c∗ a) ∈ M̄n (R, Λ). The proof that d∗ b ∈ Mn (R, Λ) is essentially the same. Corollary 4.3. [NA (R, Λ), NA (R, Λ)] 6 Sp2n (R, Λ). Proof. If g, h ∈ NA (R, Λ) and f = [g, h], then by Lemma 4.1 for all indexes p, q we have fpq = n X i,j,k=1 ′ ′ gpi hij gjk hkq = n X ′ (gpi gjk )(h′ij hkq ) ∈ R i,j,k=1 Therefore, f ∈ Sp2n (R) ∩ NA (R, Λ) = NR (R, Λ) = Sp2n (R, Λ). The first 3 items of Theorem 2 are already proved. Our next goal is to show that the main theorem of [12] implies that Sp2n (R, Λ)/ Ep2n (R, Λ) is nilpotent, provided that R0 has finite Bass–Serre dimension. This will imply the rest of Theorem 2. Recall that R0 denotes the subring of R generated by all elements ξ 2 such that ξ ∈ R. Lemma 4.4. If R0 is semilocal and 1 ∈ Λ, then Sp2n (R, Λ) = Ep2n (R, Λ). Proof. Since R is integral over R0 , it is a direct limit of R0 -subalgebras R′ ⊆ R such that R′ is module finite and integral over R0 . By the first theorem of Cohen–Seidenberg, each R′ is semilocal. Thus by [4, Lemma 4] Sp2n (R′ , Λ ∩ R′ ) = Ep2n (R′ , Λ ∩ R′ ) for each R′ . Since Sp2n and Ep2n commute with direct limits, it follows that Sp2n (R, Λ) = Ep2n (R, Λ). Lemma 4.5. If R is a finitely generated Z-algebra then so is R0 . Proof. Since 2ξ = (ξ + 1)2 − ξ 2 − 1, we have 2R ⊆ Q R0 . Let S be a finite set of generators 2 for R as a Z-algebra. We set S0 = {s \| s ∈ S} ∪ {2 s∈S ′ s \| S ′ ⊆ S} and denote by R1 the Z-subalgebra generated by S0 . Clearly, 2R ⊆ R1 . The map ξ 7→ ξ 2 is a ring epimorphism R/2R → R0 /2R, therefore R0 /2R is generated by the images of generators of R. It follows that R0 /2R = R1 /2R, hence R0 = R1 is finitely generated. 8 ANTHONY BAK AND ALEXEI STEPANOV Proof of Theorem 2. Since NR (R, Λ) = NA (R, Λ) ∩ Sp2n (R), the first statement follows from Proposition 4.2, the second is a particular case of [7, Theorem 1.1], and the third one coincides with Corollary 4.3. Recall that Sp02n (R, Λ) = ∩ϕ Ker ϕ, where ϕ ranges over all group homomorphisms Sp2n (R, Λ) → Sp2n (R̃, Λ̃′ )/ Ep2n (R̃, Λ̃) induced by form ring morphisms (R, Λ) → (R̃, Λ̃) such that R̃0 is semilocal. By Lemma 4.4 Sp02n (R, Λ) = Sp2n (R, Λ). Since R is integral over R0 , it is a direct limit of R0 -subalgebras R′ ⊆ R such that R′ is module finite over R0 . By [12][Theorem 3.10] the group Sp02n (R′ , Λ ∩ R′ )/ Ep2n (R′ , Λ ∩ R′ ) is nilpotent, provided that R0 has finite Bass–Serre dimension. Since Sp2n and Ep2n commute with direct limits, this proves (5). Any commutative ring is a direct limit of finitely generated Z-algebras. Let R = inj lim R(i) , where i ranges over some index set I and each R(i) is a finitely generated Z-algebra. Then (i) R0 = inj lim R0 and Sp2n (R, Λ ∩ R)/ Ep2n (R, Λ) = inj lim Sp(R(i) , Λ ∩ R(i) )/ Ep2n (R(i) , Λ ∩ (i) R(i) ). By Lemma 4.5 each R0 is a finitely generated Z-algebra. Therefore, this ring has finite Krull dimension and hence finite Bass–Serre dimension. By (5) each group Sp(R(i) , Λ ∩ R(i) )/ Ep2n (R(i) , Λ ∩ R(i) ) is nilpotent which implies item (4). The following statement is crucial for the proof of our main theorem. For Noetherian rings it is almost immediate consequence of Theorem 2. Lemma 4.6. Let g ∈ Sp2n (A), n > 3. If Ep2n (R, Λ)g 6 NA (R, Λ), then g ∈ NA (R, Λ). Moreover, Ep2n (R, Λ) is a characteristic subgroup of NA (R, Λ). Proof. Let θ be either an automorphism of NA (R, Λ) or an automorphism of Sp2n (A) such that Ep2n (R, Λ)θ 6 NA (R, Λ) (we denote by hθ the image of an element h ∈ NA (R) under the action of θ). Since n > 3, the Chevalley commutator formula (see section 1) implies that the group Ep2n (R, Λ) is perfect. By Corollary 4.3 we have Ep2n (R, Λ)θ = [Ep2n (R, Λ), Ep2n (R, Λ)]θ 6 [NA (R, Λ), NA (R, Λ)] 6 Sp2n (R, Λ). We shall prove that hθ ∈ Ep2n (R, Λ) for any h ∈ Ep2n (R, Λ). Write h as a product of elementary root unipotents xα1 (s1 ) · · · xαm (sm ). Let R′ denote the Z-subalgebra of R generated by all si ’s, and let Λ′ denote the form parameter of R′ generated by those sj for which αj is a long root. Clearly h ∈ Ep2n (R′ , Λ′) and Ep2n (R′ , Λ′ ) is a finitely generated group. Let R′′ denote the R′ -algebra generated by all entries of the matrices y θ , where y ranges over all generators of Ep2n (R′ , Λ′ ). Let Λ′′ = Λ ∩ R′′ . The inclusion Ep2n (R, Λ)θ 6 Sp2n (R, Λ) shows that R′′ ⊆ R. Note that Ep2n (R′ , Λ′ )θ 6 Sp2n (R′′ , Λ′′ ) by the choice of R′′ . Since R′′ is a finitely generated R′ -algebra, it is a finitely generated Z-algebra. By Lemma 4.5 R0′′ is a finitely generated Z-algebra. Therefore, it has finite Krull dimension and hence finite Bass–Serre dimension. Thus by Theorem 2(5), the kth commutator subgroup D k Sp2n (R′′ , Λ′′ ) equals to Ep2n (R′′ , Λ′′ ) for some positive integer k. Now, since Ep2n (R′ , Λ′) is perfect, it is equal to D k Ep2n (R′ , Λ′ ). It follows that Ep2n (R′ , Λ′ )θ = D k Ep2n (R′ , Λ′ )θ 6 D k Sp2n (R′′ , Λ′′) = Ep2n (R′′ , Λ′′ ). In particular, hθ ∈ Ep2n (R′′ , Λ′′ ) 6 Ep2n (R, Λ). Thus, Ep2n (R, Λ) is invariant under θ. If θ is an automorphism of NA (R) this means that Ep2n (R, Λ) is a characteristic subgroup of NA (R). If θ is an inner automorphism defined by g ∈ G(A), then the statement we proved is the first assertion of the lemma. SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 9 The following straightforward corollary shows that the normalizers of all subgroups of the sandwich L Ep2n (R, Λ), NA (R, Λ) lie in that sandwich. Corollary 4.7. For any H 6 NA (R, Λ) containing Ep2n (R, Λ) its normalizer is contained in NA (R, Λ). In particular, the group NA (R, Λ) is self normalizing. 5. Inside a parabolic subgroup Let H be a subgroup of Sp2n (A), normalized by Ep2n (K). Denote by (R, Λ) the form ring, associated with H. In the proof of the following lemma we keep Q using the ordering on the index set I, defined in section 1. For example, the product −1 j=2 means that j = 2, . . . , n, −n, . . . , −1. We assume that the order of factors agrees with the ordering on I. Lemma 5.1. If g ∈ U1 (R) and Ep2n (K)g ∈ H, then g ∈ Ep2n (R, Λ). Q Proof. Let g = −1 j=2 T1j (µj ). We have to prove that µj ∈ R for any j 6= −1 and µ−1 ∈ Λ. Let h be the smallest element from I such that µj 6= 0. We proceed by going down induction on h. If h = −1, then g consists of a single factor and there is nothing to prove. If h = −2, then the result follows from Lemma 3.1. Q Now, let h 6 −2. Denote by i the successor of h in I. Then, [Th,i (−1), g] = T1,i (µh ) T1,j (ξh ). j>i By induction hypothesis µh ∈ R. The element T1h (−µh )g satisfies the conditions of the lemma. Again by induction hypothesis it belongs to Ep2n (R, Λ). Thus, g ∈ Ep2n (R, Λ). It is known that in a Chevalley group the unipotent radicals of two opposite standard parabolic subgroups span the elementary group (see e. g. [18, Lemma 2.1]). The next lemma shows that this holds for the parabolic subgroup P1 of Sp2n (R, Λ) as well. Lemma 5.2. The set {T1i (µ), Ti1 (µ), T±1 ∓1 (λ) \| i 6= ±1, µ ∈ R, λ ∈ Λ} generates the elementary group Ep2n (R, Λ). Proof. If i 6= ±j, ±1 then Tij (µ) = [Ti1 (µ), T1j (1)]. On the other hand, for λ ∈ Λ we have T−ii (λ) = [T−11 (µ), T1i (1)]T−i1 (−λ). Lemma 5.3. Let H be a subgroup of Sp2n (A), containing Ep2n (K) and let (R, Λ) be a form ring, associated with H. Suppose that g commutes with a long root subgroup Xγ (K) and Ep2n (K)g ∈ H. Then g ∈ NA (R, Λ). Proof. Without loss of generality we may assume that γ = 2ε1 is the maximal root. Then g belongs to the standard parabolic subgroup P1 , corresponding to the simple root α1 = ε1 −ε2 , ′ ′ in other words, gi1 = g−1i = g−11 = 0 for all i 6= ±1. Moreover, g11 = g−1−1 = g11 = g−1−1 = 2 µ, where µ = 1. Then g = ab for some b ∈ U1 (A) and a ∈ L1 (A) (a and b are not necessarily in H). For any d ∈ U1 (K) the element dg belongs to H ∩ U1 (A). By Lemma 5.1, dg ∈ Ep2n (R, Λ). If d = T1i (1), then the above inclusion implies that µgij ∈ R for all i 6= 1 and all j ∈ I. It follows that T1i (1)a and Ti1 (1)a belong to Ep2n (R, Λ). On the other hand, a commutes with root subgroups X±γ (K). By the previous lemma we conclude that a normalizes Ep2n (R, Λ). Now, we have Ep2n (K)b 6 Ep2n (R, Λ)g ∈ H and by Lemma 5.1 b ∈ Ep2n (R, Λ). Thus, g ∈ NA (R, Λ) as required. 10 ANTHONY BAK AND ALEXEI STEPANOV 6. Proof of Theorem 1 In this section we assume that 2 = 0 in K. Let G be a Chevalley group with a not simply laced root system, e. g. G = Sp2n . Recall that in this case a short root unipotent element h is called a small unipotent element, see [11]. The terminology reflects the fact that the conjugacy class of h is small. In our settings a small unipotent element is conjugate to Tij (µ), where i 6= ±j and µ ∈ R. The following lemma was obtained over a field by Golubchik and Mikhalev in [10], Gordeev in [11] and Nesterov and Stepanov in [13]. Lemma 6.1. Let Φ = Bn , Cn , F4 and let R be a ring such that 2 = 0. Let α be a long root g and g ∈ G(Φ, R). If h ∈ G(R) is a small unipotent element, then Xα (R)h commutes with Xα (R). g Proof. The identity with constants [Xα (R)h , Xα (R)] = {1} is inherited by subrings and quotient rings. Any commutative ring with 2 = 0 is a quotient of a polynomial ring over F2 which is a subring of a field of characteristic 2. The next lemma is the last ingredient for the proof of Theorem 1. It follows from the normal structure of the general unitary group GU2n (R, Λ) obtained by Bak and Vavilov at the middle of 1990-s. Since this result has not been published yet, we give a prove of a very simple special case of it. Lemma 6.2. A normal subgroup N of Ep2n (R, Λ), containing a root element Tij (1), coincides with Ep2n (R, Λ). Proof. First, suppose that i 6= ±j. Take k 6= ±i, ±j. Then, [Tij (1), Tjk (ξ)] = Tik (µ) ∈ N. Since the Weyl group acts transitively on the set of all short roots, we have Tlm (µ) ∈ N for all l 6= ±m and µ ∈ R. Further, Tl,−m (−λ)[Tlm (1), Tm,−m (λ)] = Tl,−l (λ) ∈ N for all λ ∈ Λ and l ∈ I. Now, let i = m, j = −m and l 6= ±m. Put λ = 1 in the latter commutator identity. Then, Tl,−m (1)Tl,−l (1) ∈ N. By transitivity of the Weyl group we know that Tl,−l (1) ∈ N, therefore Tl,−m (1) ∈ N. By the first paragraph of the proof, N contains all the generators of Ep2n (R, Λ). Now we are ready to prove Theorem 1. The idea of the proof is the same as for the main result of [17]. Proof of Theorem 1. Let (R, Λ) be the form subring associated with H. Put h = T12 (1) and x = T1,−1 (λ), where λ ∈ Λ. Take two arbitrary elements a, b from Ep2n (R, Λ) and consider agb the element c = xh ∈ H. By Lemma 6.1 this element commutes with a long root subgroup and by Lemma 5.3 c ∈ NA (R). Rewrite c in the form b c = g −1 (a−1 h−1 a)g(bxb−1 )g −1(a−1 ha)g Since b ∈ Ep2n (R, Λ), the element bcb−1 is in NA (R). Fix a and let b and λ vary. The subgroup generated by bxb−1 is normal in Ep2n (R, Λ). By Lemma 6.2 it must coincide with −1 −1 −1 Ep2n (R, Λ). Thus, Ep2n (R, Λ)g (a h a)g ∈ NA (R), and by Lemma 4.6 (a−1 h−1 a)g ∈ NA (R). Again, elements of the form a−1 h−1 a, as a ranges over Ep2n (R, Λ), generate a normal subgroup in Ep2n (R, Λ). The minimal normal subgroup of Ep2n (R, Λ) containing h must be equal to Ep2n (R, Λ) by Lemma 6.2. Therefore, Ep2n (R, Λ)g ∈ NA (R). By Lemma 4.6 one has g ∈ NA (R), which completes the proof. SUBRING SUBGROUPS IN SYMPLECTIC GROUPS IN CHARACTERISTIC 2 11 References [1] E. Abe, Normal subgroups of Chevalley groups over commutative rings, Contemp. Math. 83 (1989), 1–17. [2] E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tohoku Math. J. 28 (1976), no. 2, 185–198. [3] A. Bak, The stable structure of quadratic modules, Ph.D. thesis, Columbia Univ., Columbia, USA, 1969. , Odd dimension surgery groups of odd torsion groups vanish, Topology 14 (1975), [4] no. 4, 367–374. , K-theory of forms, Ann. of Math. Stud. 98, Princeton Univ. Press, Princeton [5] N.J., 1981. , Nonabelian K-theory: The nilpotent class of K1 and general stability, K-Theory [6] 4 (1991), 363–397. [7] A. Bak and N. A. Vavilov, Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35–47. [8] , Structure of hyperbolic unitary groups I: Elementary subgroups, Algebra Colloq 7 (2000), no. 2, 159–196. [9] N. Bourbaki, Elements of mathematics. Lie groups and Lie algebras. Chapters 4-6, Springer-Verlag, Berlin-Heidelberg-New York, 2008. [10] I. Z. Golubchik and A. V. Mikhalev, Generalized group identities is classical groups, J. Soviet Math. 27 (1984), no. 4, 2902–2918. [11] N. L. Gordeev, Freedom in conjugacy classes of simple algebraic groups and identities with constants, St.Petersburg Math. J. 9 (1998), no. 4, 709–723. [12] R. Hazrat, Dimension theory and nonstable K1 of quadratic modules, K-Theory 27 (2002), no. 4, 293–328. [13] V. V. Nesterov and A. V. Stepanov, Identity with constant in a Chevalley group of type F4 , St.Petersburg Math. J. 21 (2010), no. 5, 819–823. [14] Ya. N. Nuzhin, Groups contained between groups of Lie type over different fields, Algebra Logic 22 (1983), 378–389. , Intermediate subgroups in the Chevalley groups of type Bl , Cl , F4 , and G2 over [15] the nonperfect fields of characteristic 2 and 3, Sib. Math. J. 54 (2013), no. 1, 119–123. [16] R. G. Steinberg, Lectures on Chevalley groups, Dept. of Mathematics, Yale University, 1968. [17] A. V. Stepanov, Subring subgroups in Chevalley groups with doubly laced root systems, J. Algebra 362 (2012), 12–29. [18] A. V. Stepanov, Elementary calculus in Chevalley groups over rings, J. Prime Research in Math. 9 (2013), 79–95. Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany E-mail address: bak@mathematik.uni-bielefeld.de St.Petersburg State University and St.Petersburg Electrotechnical University E-mail address: stepanov239@gmail.com	4
Joint Modelling of Location, Scale and Skewness Parameters of the Skew Laplace Normal Distribution Fatma Zehra Doğru1* and Olcay Arslan2 1 Giresun University, Faculty of Economics and Administrative Sciences, Department of Econometrics, 28100 Giresun/Turkey fatma.dogru@giresun.edu.tr 2 Ankara University, Faculty of Science, Department of Statistics, 06100 Ankara/Turkey oarslan@ankara.edu.tr Abstract In this article, we propose joint location, scale and skewness models of the skew Laplace normal (SLN) distribution as an alternative model for joint modelling location, scale and skewness models of the skewt-normal (STN) distribution when the data set contains both asymmetric and heavy-tailed observations. We obtain the maximum likelihood (ML) estimators for the parameters of the joint location, scale and skewness models of the SLN distribution using the expectation-maximization (EM) algorithm. The performance of the proposed model is demonstrated by a simulation study and a real data example. Keywords: EM algorithm, joint location, scale and skewness models, ML, SLN, SN. 1. Introduction There are many remarkable and tractable methods for modeling the mean. In practice, modelling the dispersion will be of direct interest in its own right, to identify the sources of variability in the observations (Smyth and Verbyla (1999)). In recent years, joint mean and dispersion models have been used for modeling heteroscedastic data sets. For instance, Park (1966) proposed a log linear model for the variance parameter and described the Gaussian model using a two stage process to estimate the parameters. Harvey (1976) examined the maximum likelihood (ML) estimation of the location and scale effects and also proposed a likelihood ratio test for heteroscedasticity. Aitkin (1987) proposed the modelling of variance heterogeneity in normal regression analysis. Verbyla (1993) estimated the parameters of the normal regression model under the log linear dependence of the variances on explanatory variables using the restricted ML. Engel and Huele (1996) represented an extension of the response surface approach to Taguchi type experiments for robust design by accommodating generalized linear modeling. Taylor and Verbyla (2004) introduced joint modelling of location and scale parameters of the t distribution. Lin and Wang (2009) proposed a robust approach for joint modelling of mean and scale parameters for longitudinal data. Lin and Wang (2011) studied Bayesian inference for joint modelling of location and scale parameters of the t distribution for longitudinal data. Wu and Li (2012) explored the variable selection for joint mean and dispersion models of the inverse Gaussian distribution. Li and Wu (2014) proposed joint modelling of location and scale parameters of the skew normal (SN) (Azzalini (1985, 1986)) distribution. Zhao and Zhang (2015) proposed variable selection of varying dispersion student-t regression models. Recently, Li et al. (2017) proposed variable selection in joint location, scale and skewness models of the SN distribution and Wu et al. (2017) explored variable selection in joint location, scale and skewness models of the STN distribution. The skew exponential power distribution was proposed by Azzalini (1986) to deal with both skewness and heavy-tailedness, simultaneously. Its properties and inferential aspects were studied by DiCiccio and Monti (2004). Gómez et al. (2007) studied the skew Laplace normal (SLN) distribution that is a special case of the skew exponential power distribution. This distribution has wider range of skewness and also more applicable than the SN distribution. In literature, skewness and heavy-tailedness are modelled by using STN distribution for joint location, scale and skewness models. However, the 1 STN distribution has an extra parameter that is the degrees of freedom parameter. Since this parameter should be estimated along with the other parameters, it may be computationally more exhaustive in practice. Therefore, in this paper, we propose to model joint location, scale and skewness models of the SLN distribution as an alternative model for the joint location, scale and skewness models of the STN distribution to model both skewness and heavy-tailedness in the data. The rest of the paper is designed as follows. In Section 2, we give some properties of the SLN distribution. In Section 3, we introduce joint location, scale and skewness models of the SLN distribution. In Section 4, we give the ML estimation of the proposed joint location, scale and skewness model using the EM algorithm. In Section 5, we provide a simulation study to show the performance of the proposed model. In Section 6, modeling applicability of the proposed model is illustrated by using a real data set. The paper is finalized with a conclusion section. 2. Skew Laplace normal distribution Let 𝑌 be a SLN distributed random variable (𝑌 ∼ 𝑆𝐿𝑁(𝜇, 𝜎 2 , 𝜆)) with the location parameter 𝜇 ∈ ℝ, scale parameter 𝜎 2 ∈ (0, ∞) and the skewness parameter 𝜆 ∈ ℝ. The probability density function (pdf) of 𝑌 is given as 𝑓(𝑦) = 2𝑓𝐿 (𝑦; 𝜇, 𝜎)Φ (𝜆 𝑦−𝜇 ), 𝜎 (1) where 𝑓𝐿 (𝑦; 𝜇, 𝜎) represents the pdf of Laplace distribution with 𝑓𝐿 (𝑦; 𝜇, 𝜎) = 1 −\|𝑦−𝜇\| 𝑒 𝜎 2𝜎 and Φ is the cumulative distribution function of the standard normal distribution. Figure 1 displays the plots of the pdf of the SLN distribution for 𝜇 = 0, 𝜎 = 1 and different values of 𝜆. Figure 1. Examples of the SLN pdf for 𝜇 = 0, 𝜎 = 1 and different skewness parameter values of 𝜆. Let the random variables 𝑍 ∼ 𝑆𝑁(0,1, 𝜆) and 𝑉 with the pdf 𝑓𝑉 (𝑣) = 𝑣 −3 exp(−(2𝑣 2 )−1 ), 𝑣 > 0 be two independent random variables. Then, the random variable 𝑌 ∼ 𝑆𝐿𝑁(𝜇, 𝜎 2 , 𝜆) has the following scale mixture form 2 𝑌 =𝜇+𝜎 𝑍 . 𝑉 (2) Further, using the stochastic representation of the SN (Azzalini (1986, p. 201) and Henze (1986, Theorem 1)) distributed random variable 𝑍, the stochastic representation of the random variable 𝑌 is obtained as 𝑌 = 𝜇 +𝜎( 𝜆\|𝑍1 \| √𝑉 2 (𝑉 2 + 𝜆2 ) + 𝑍2 √𝑉 2 + 𝜆2 (3) ), where 𝑍1 ∼ 𝑁(0,1) and 𝑍2 ∼ 𝑁(0,1) are independent random variables. This stochastic representation will give the following hierarchical representation of the SLN distribution. Let 𝑈 = √𝑉 −2 (𝑉 2 + 𝜆2 )\|𝑍1 \|. Then, 𝜎𝜆𝑢 𝜎2 , ), 𝑣 2 + 𝜆2 𝑣 2 + 𝜆2 𝑣 2 + 𝜆2 𝑈\|𝑣 ∼ 𝑇𝑁 ((0, ) ; (0, ∞)) , 𝑣2 𝑉 ∼ 𝑓𝑉 (𝑣) = 𝑣 −3 exp(−(2𝑣 2 )−1 ), 𝑌\|𝑢, 𝑣 ∼ 𝑁 (𝜇 + (4) where 𝑇𝑁(∙) shows the truncated normal distribution. The hierarchical representation will allow us to carry on the parameter estimation using the EM algorithm. Using this hierarchical representation the joint pdf of 𝑌, 𝑈 and 𝑉 can be written as 2 1 −2 1 𝑣 2 (𝑦 − 𝜇)2 𝜆(𝑦 − 𝜇) 2 )−1 ) 𝑓(𝑦, 𝑢, 𝑣) = 𝑣 exp(−(2𝑣 exp {− ( + − (𝑢 ) )}. 𝜋𝜎 2 𝜎2 𝜎 (5) Next we will turn our attention to the conditional distribution of 𝑈 given 𝑌 and 𝑉. Taking the integral of (5) over 𝑈, we obtain the joint pdf of 𝑌 and 𝑉 as 2 1⁄2 −2 𝑣 2𝑠2 𝑓(𝑦, 𝑣) = ( 2 ) 𝑣 exp (−(2𝑣 2 )−1 − ) Φ(𝜆𝑠) , 𝜋𝜎 2 (6) where 𝑠 = (𝑦 − 𝜇)⁄𝜎. Then, dividing (5) by (6) yields the following conditional density function of 𝑈 given the others 𝑓(𝑢\|𝑦, 𝑣) = 1 √2𝜋 exp {− (𝑢 − 𝜆𝑠)2 } Φ(𝜆𝑠). 2 (7) It is clear from the density given in (7) that 𝑈 and 𝑉 are conditionally independent. Therefore, the distribution of 𝑈\|𝑌 = 𝑦 is 𝑈\|𝑌 = 𝑦 ∼ 𝑇𝑁((𝜆𝑠, 1); (0, ∞)). (8) Further, after dividing (6) by (1), we get the following conditional density function of 𝑉 given 𝑌 2 𝑣 2 𝑠 2 \|𝑦 − 𝜇\| 𝑓(𝑣\|𝑦) = √ 𝑣 −2 exp {−(2𝑣 2 )−1 − + }. 𝜋 2 𝜎 3 (9) Now, we are ready to give the following proposition. The proof of this proposition can be easily done using the conditional pdfs given above. Proposition 1. Using the hierarchical representation given in (4), we have the following conditional expectations 𝜎 , \|𝑦 − 𝜇\| Φ(𝜆𝑠) 𝐸(𝑈\|𝑦) = 𝜆𝑠 + , 𝜙(𝜆𝑠) 𝐸(𝑈 2 \|𝑦) = 1 + 𝜆𝑠𝐸(𝑈\|𝑦) . (10) 𝐸(𝑉 2 \|𝑦) = (11) (12) Note that these conditional expectations will be used in the EM algorithm given in Section 4. 3. Joint location, scale and skewness models of the SLN distribution In this study, we consider the following joint location, scale and skewness models of the SLN distribution 𝑦𝑖 ∼ 𝑆𝐿𝑁(𝜇𝑖 , 𝜎𝑖2 , 𝜆𝑖 ), 𝑖 = 1,2, … , 𝑛 𝜇𝑖 = 𝒙𝑇𝑖 𝜷 , log 𝜎𝑖2 = 𝒛𝑇𝑖 𝜸 , 𝑇 { 𝜆𝑖 = 𝒘 𝑖 𝜶 , (13) 𝑇 𝑇 where 𝑦𝑖 is the 𝑖𝑡ℎ observed response, 𝒙𝑖 = (𝑥𝑖1 , … , 𝑥𝑖𝑝 ) , 𝒛𝑖 = (𝑧𝑖1 , … , 𝑧𝑖𝑞 ) and 𝒘𝑖 = (𝑤𝑖1 , … , 𝑤𝑖𝑟 )𝑇 𝑇 are observed covariates corresponding to 𝑦𝑖 , 𝜷 = (𝛽1 , … , 𝛽𝑝 ) is a 𝑝 × 1 vector of unknown parameters 𝑇 in the location model, and 𝜸 = (𝛾1 , … , 𝛾𝑞 ) is a 𝑞 × 1 vector of unknown parameters in the scale model and 𝜶 = (𝛼1 , … , 𝛼𝑟 )𝑇 is a 𝑟 × 1 vector of unknown parameters in the skewness model. These covariate vectors 𝒙𝑖 , 𝒛𝑖 and 𝒘𝑖 are not needed to be identical. 4. ML estimation of joint location, scale and skewness models of the SLN distribution Let (𝑦𝑖 , 𝒙𝑖 , 𝒛𝑖 ), 𝑖 = 1,2, … , 𝑛, be a random sample from model given in (13). Let 𝜽 = (𝜷, 𝜸, 𝜶). Then, the log-likelihood function of 𝜽 based on the observed data is written as 𝑛 𝑛 𝑛 1 \|𝑦𝑖 − 𝒙𝑇𝑖 𝜷\| ℓ(𝜽) = − ∑ 𝒛𝑇𝑖 𝜸 − ∑ + ∑ log Φ(𝜅𝑖 ), 𝑇 2 𝑒 𝒛𝑖 𝜸⁄2 𝑖=1 where 𝜅𝑖 = 𝒘𝑇𝑖 𝜶 (𝑦𝑖 −𝒙𝑇 𝑖 𝜷) 𝑇 ⁄ 𝑒 𝒛𝑖 𝜸 2 𝑖=1 (14) 𝑖=1 . The ML estimator of 𝜽 can be found by maximizing the equation (14). We can see that the direct maximization of this function does not seem very tractable, so numerical algorithms may be needed to approximate the possible maximizer of this function. Since, the SLN distribution has a scale mixture form, the EM algorithm (Dempster et al. (1977)) can be implemented to obtain the ML estimator for 𝜽. To simplify the steps of the EM algorithm, we will use the stochastic representation of the SLN distribution given in (3). 4 Let 𝑉 and 𝑈 be the latent variables. Using the hierarchical representation given in (4), or the model (13) we get the following hierarchical representation 𝑇 𝑌𝑖 \|𝑢𝑖 , 𝑣𝑖 ∼ 𝑁 (𝒙𝑇𝑖 𝜷 + 𝑒 𝒛𝑖 𝜸⁄2 (𝒘𝑇𝑖 𝜶)𝑢𝑖 2 𝑣𝑖2 + (𝒘𝑇𝑖 𝜶) 𝑇 , 𝑒 𝒛𝑖 𝜸 2) , 𝑣𝑖2 + (𝒘𝑇𝑖 𝜶) 2 𝑣𝑖2 + (𝒘𝑇𝑖 𝜶) 𝑈𝑖 \|𝑣𝑖 = 1 ∼ 𝑇𝑁 ((0, ) ; (0, ∞)) , 𝑣𝑖2 −1 (15) 𝑣𝑖 ∼ 𝑓(𝑣𝑖 ) = 𝑣𝑖−3 exp (−(2𝑣𝑖2 ) ). Let 𝒖 = (𝑢1 , … , 𝑢𝑛 ) and 𝒗 = (𝑣1 , … , 𝑣𝑛 ) be the missing data and (𝒚, 𝒖, 𝒗) be the complete data, where 𝒚 = (𝑦1 , … , 𝑦𝑛 ). Then, using the hierarchical representation given in (15), the complete data loglikelihood function of 𝜽 can be written as 𝑛 2 1 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 2 −1 ℓ𝑐 (𝛉; 𝒚, 𝒖, 𝒗) = ∑ {− log 𝜋 − 𝒛𝑇𝑖 𝜸 − 2 log 𝑣𝑖 − (2𝑣𝑖2 ) − ( 𝑣𝑖 𝑇 2 2 𝑒 𝒛𝑖 𝜸 𝑖=1 +𝑢𝑖2 −2 2 𝒘𝑇𝑖 𝜶 𝑇 𝑒 𝒛𝑖 𝜸⁄2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝑢𝑖 + (𝒘𝑇𝑖 𝜶) 𝑇 𝑒 𝒛𝑖 𝜸 2 (16) (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) )}. To obtain the ML estimator of 𝜽, we have to maximize (16). However, the estimators obtained from this maximization will be dependent on the latent variables. Thus, to handle this latency problem, we have to take the conditional expectation of the complete data log-likelihood function given the observed data 𝑦𝑖 𝑛 1 −1 𝐸(ℓ𝑐 (𝛉; 𝒚, 𝒖, 𝒗)\|𝑦𝑖 ) = ∑ {− log 𝜋 − 𝒛𝑇𝑖 𝜸 − 2𝐸(log 𝑉𝑖 \|𝑦𝑖 ) − 𝐸 ((2𝑉𝑖2 ) \| 𝑦𝑖 ) 2 𝑖=1 2 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) − ( 𝐸(𝑉𝑖2 \|𝑦𝑖 ) + 𝐸(𝑈𝑖2 \|𝑦𝑖 ) 𝑇𝜸 𝒛 2 𝑒 𝑖 −2 𝒘𝑇𝑖 𝜶 𝑇 𝑒 𝒛𝑖 𝜸⁄2 2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝐸(𝑈𝑖 \|𝑦𝑖 ) + (𝒘𝑇𝑖 𝜶) 𝑇 𝑒 𝒛𝑖 𝜸 2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) )}. (17) The conditional expectations 𝐸(𝑉𝑖2 \|𝑦𝑖 ), 𝐸(𝑈𝑖 \|𝑦𝑖 ) and 𝐸(𝑈𝑖2 \|𝑦𝑖 ) in (17) can be calculated using the conditional expectations given in (10)-(12). Note that since the other conditional expectations are not related to the parameters, we do not calculate them. Let 𝑇 𝑣̂𝑖 = 𝐸(𝑉𝑖2 \|𝑦𝑖 ) = 𝑒 𝒛𝑖 𝜸̂⁄2 \|𝑦𝑖 − ̂\| 𝒙𝑇𝑖 𝜷 (18) , Φ(𝜅̂ 𝑖 ) , 𝜙(𝜅̂ 𝑖 ) = 𝐸(𝑈𝑖2 \|𝑦) = 1 + 𝜅̂ 𝑖 𝑢̂1𝑖 , 𝑢̂1𝑖 = 𝐸(𝑈𝑖 \|𝑦𝑖 ) = 𝜅̂ 𝑖 + (19) 𝑢̂2𝑖 (20) ̂ where, 𝜅̂ 𝑖 = 𝒘𝑇𝑖 𝜶 ̂ (𝑦𝑖 −𝒙𝑇 𝑖 𝜷) 𝑇 ̂⁄ 𝑒 𝒛𝑖 𝜸 2 . Then, using these conditional expectations in (17) we get the following objective function to be maximized with respect to 𝜽 5 𝑛 2 1 1 (𝑦 − 𝒙𝑇 𝜷) ̂ ) = ∑ {− log 𝜋 − 𝒛𝑇𝑖 𝜸 − ( 𝑖 𝑇 𝑖 𝑄(𝜽; 𝜽 𝑣̂𝑖 + 𝑢̂2𝑖 2 2 𝑒 𝒛𝑖 𝜸 𝑖=1 −2 2 𝒘𝑇𝑖 𝜶 𝑇 𝑒 𝒛𝑖 𝜸⁄2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝑢̂1𝑖 + (𝒘𝑇𝑖 𝜶) 𝑇 𝑒 𝒛𝑖 𝜸 2 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) )}. (21) Now, the steps of the EM algorithm will be as follows: EM algorithm: 1. Take initial value for 𝜽(0) = (𝜷(0) , 𝜸(0) , 𝜶(0) ). 2. E-Step: Given the observed data and the current parameter values, find the conditional expectation of the complete data log-likelihood function given in (16). This corresponds to calculating the conditional expectations given in (18)-(20). This step will be carried on as follows. Compute the following conditional expectations for the 𝑘 = 0,1,2, … iteration 𝑣̂𝑖 (𝑘) = ̂ (𝑘) ) 𝐸(𝑉𝑖2 \|𝑦𝑖 , 𝜽 𝑇 (𝑘) ⁄ 2 = 𝑒 𝒛𝑖 𝜸̂ ̂ (𝑘) \| \|𝑦𝑖 − 𝒙𝑇𝑖 𝜷 (22) , (𝑘) (𝑘) 𝑢̂1𝑖 ̂ (𝑘) = 𝐸(𝑈𝑖 \|𝑦𝑖 , 𝜽 )= (𝑘) 𝜅̂ 𝑖 (𝑘) ̂ (𝑘) ) = 1 𝑢̂2𝑖 = 𝐸(𝑈 2 \|𝑦, 𝜽 (𝑘) where, 𝜅̂ 𝑖 ̂ (𝑘) = 𝒘𝑇𝑖 𝜶 + Φ (𝜅̂ 𝑖 ) , (𝑘) 𝜙 (𝜅̂ 𝑖 ) (𝑘) (𝑘) + 𝜅̂ 𝑖 𝑢̂1𝑖 , (23) (24) ̂ (𝑘) ) (𝑦𝑖 −𝒙𝑇 𝑖𝜷 𝑇 ̂(𝑘) ⁄2 𝑒 𝒛𝑖 𝜸 . ̂ ) and maximize it with respect to 𝜽 to obtain 3. M-Step: Use these conditional expectations in 𝑄(𝜽; 𝜽 new estimates. This maximization step yields the following formulation to update the new estimates. The (𝑘 + 1)𝑡ℎ parameter estimates can be computed using −1 ̂ (𝑘+1) = 𝜽 ̂ (𝑘) + (−𝐻(𝜽 ̂ (𝑘) )) 𝜽 where 𝐺(𝜽) = ̂) 𝜕𝑄(𝜽;𝜽 and 𝐻(𝜽) = 𝜕𝜽 ̂ (𝑘) ), 𝐺(𝜽 ̂) 𝜕2 𝑄(𝜽;𝜽 𝜕𝜽𝜕𝜽𝑇 (25) . 4. Repeat E and M steps until the convergence is satisfied. Remark. For the detail expressions of 𝐺(𝜽) and 𝐻(𝜽) see Appendix. 5. Simulation study In this section, we give a simulation study to show the performance of the proposed location, scale and skewness models of the SLN distribution in terms of mean squared error (MSE). The MSE is given with the following formula 𝑁 ̂ (𝜃̂) = 𝑀𝑆𝐸 1 2 ∑(𝜃̂𝑗 − 𝜃) , 𝑁 𝑗=1 6 where 𝜃 is the true parameter value, 𝜃̂𝑗 is the estimate of 𝜃 for the 𝑗𝑡ℎ simulated data and 𝜃̅ = 1 𝑁 ∑ 𝜃̂ . 𝑁 𝑗=1 𝑗 All simulation studies are conducted as 𝑁 = 1000 times. We set the sample sizes as 50, 100, 150 and 200. Note that the simulation study and real data example are performed using MATLAB R2015b. For all numerical calculations, the convergence rule is taken as 10−6. The data are generated from the following location, scale and skewness models of the SLN distribution 𝑦𝑖 ∼ 𝑆𝐿𝑁(𝜇𝑖 , 𝜎𝑖2 , 𝜆), 𝑖 = 1,2, … , 𝑛 𝜇𝑖 = 𝒙𝑇𝑖 𝜷 , log 𝜎𝑖2 = 𝒛𝑇𝑖 𝜸 , 𝑇 { 𝜆𝑖 = 𝒘𝑖 𝜶. Here, all covariate vectors 𝒙𝑖 , 𝒛𝑖 and 𝒘𝑖 are independently generated from uniform distribution 𝑈(−1,1). To carry out the simulation study, we take the following two cases for true parameter values: Case I: 𝛽0 = (0, −1, −1)𝑇 , 𝛾0 = (0, −1, −1)𝑇 and 𝛼0 = (0, −1, −1)𝑇 , Case II: 𝛽0 = (0,1,1)𝑇 , 𝛾0 = (0,1,1)𝑇 and 𝛼0 = (0,1,1)𝑇 , Case III: 𝛽0 = (1,1,0,0,1)𝑇 , 𝛾0 = (0.7,0.7,0,0,0.7)𝑇 and 𝛼0 = (0.5,0.5,0,0,0.5)𝑇 . The simulation results are summarized in Tables 1, 2 and 3. These tables include the mean of the estimators and the values of MSE. From these tables, we observe the followings. The proposed EM algorithm is working accurately for estimating the parameters. When the sample sizes increase, the values of MSE decrease. Table 1. Mean of the estimators and the values of MSE for the different sample sizes for the Case I. Model Location Model Scale Model Skewness Model 𝑛 𝛽0 𝛽1 𝛽2 𝛾0 𝛾1 𝛾2 𝛼0 𝛼1 𝛼2 50 Mean MSE 0.0010 0.0433 -1.0068 0.0624 -0.9950 0.0643 -0.0655 0.0905 -1.0595 0.3300 -1.1024 0.3445 -0.0527 0.5398 -1.5780 1.7814 -1.6140 2.0557 100 Mean 0.0020 -0.9983 -0.9987 -0.0384 -1.0576 -1.0281 0.0035 -1.1981 -1.2066 MSE 0.0161 0.0243 0.0256 0.0369 0.1376 0.1272 0.0600 0.2234 0.2339 150 Mean -0.0002 -0.9987 -1.0077 -0.0278 -1.0264 -1.0118 -0.0034 -1.1213 -1.1157 MSE 0.0088 0.0143 0.0150 0.0228 0.0829 0.0843 0.0296 0.1113 0.1141 200 Mean -0.0006 -0.9965 -0.9987 -0.0223 -1.0088 -1.0060 -0.0016 -1.0852 -1.0754 MSE 0.0061 0.0109 0.0117 0.0161 0.0594 0.0629 0.0190 0.0727 0.0652 Table 2. Mean of the estimators and the values of MSE for the different sample sizes for the Case II. Model Location Model Scale Model Skewness Model 𝑛 𝛽0 𝛽1 𝛽2 𝛾0 𝛾1 𝛾2 𝛼0 𝛼1 𝛼2 50 Mean 0.0059 1.0046 1.0036 -0.0794 1.0957 1.0562 -0.0130 1.5993 1.5928 MSE 0.0417 0.0623 0.0661 0.0935 0.3421 0.3332 0.4643 1.9956 1.7924 100 Mean 0.0035 1.0040 1.0095 -0.0369 1.0380 1.0357 0.0045 1.2105 1.2117 7 150 MSE 0.0142 0.0247 0.0265 0.0346 0.1282 0.1252 0.0635 0.2636 0.2621 Mean -0.0020 0.9956 1.0085 -0.0209 1.0281 1.0193 -0.0009 1.1210 1.1148 200 MSE 0.0088 0.0158 0.0138 0.0224 0.0796 0.0783 0.0285 0.1035 0.1075 Mean 0.0018 1.0034 1.0036 -0.0239 1.0200 1.0130 -0.0013 1.0717 1.0802 MSE 0.0067 0.0115 0.0105 0.0162 0.0600 0.0580 0.0203 0.0672 0.0705 Table 3. Mean of the estimators and the values of MSE for the different sample sizes for the Case III. Model Location Model Scale Model Skewness Model 𝑛 𝛽0 𝛽1 𝛽2 𝛽3 𝛽4 𝛾0 𝛾1 𝛾2 𝛾3 𝛾4 𝛼0 𝛼1 𝛼2 𝛼3 𝛼4 50 Mean 1.0035 1.0018 0.0117 -0.0151 0.9971 0.5812 0.8583 -0.0148 -0.0303 0.8151 1.3635 1.2163 0.0499 -0.0369 1.3361 100 MSE 0.1734 0.2093 0.2129 0.2219 0.2102 0.1404 0.5997 0.6155 0.5936 0.5718 3.2188 3.1849 1.8253 1.6726 3.4589 Mean 0.9631 1.0057 -0.0181 -0.0006 1.0053 0.6667 0.7411 -0.0024 -0.0136 0.7538 0.7983 0.7769 0.0156 -0.0245 0.7471 150 MSE 0.0900 0.0709 0.0806 0.0784 0.0767 0.0598 0.1412 0.1715 0.1621 0.1561 0.4941 0.4341 0.1861 0.2151 0.3766 Mean 0.9596 0.9966 0.0045 0.0037 0.9958 0.6913 0.7145 0.0170 -0.0168 0.7044 0.6819 0.6210 -0.0059 -0.0103 0.6317 200 MSE 0.0508 0.0415 0.0473 0.0447 0.0372 0.0387 0.0955 0.0963 0.0951 0.0920 0.1866 0.1381 0.0807 0.0764 0.1347 Mean 0.9808 1.0012 0.0091 -0.0095 1.0081 0.6792 0.7226 -0.0147 -0.0110 0.7309 0.6067 0.5894 0.0033 -0.0041 0.5827 MSE 0.0356 0.0296 0.0294 0.0301 0.0287 0.0240 0.0685 0.0612 0.0665 0.0621 0.0975 0.0831 0.0420 0.0499 0.0608 6. Real data example The Martin Marietta data set includes the relationship of the excess rate of returns of the Marietta Company and an index for the excess rate of return for the New York Exchange (CRSP). These rate of returns for the company and the CRSP index were determined monthly over a period of five years. This data set used by Butler et al. (1990) for modelling a simple linear regression with Gaussian errors. Azzalini and Capitanio (2003) analyzed this data set for modelling the linear regression model when the errors have the skew t distribution. Also, Taylor and Verbyla (2004) examined this data set for joint modelling of location and scale parameters of the t distribution. We display the scatter plot of the data set and the histogram of the Martin Marietta excess returns. Since the skewness coefficient of Martin Marietta excess returns is 2.9537 and also according to the Figure 2 (b), we can say that it will be more suitable to model this data set with a joint location, scale and skewness models of a skew distribution. Figure 2. (a) Scatter plot of the data set. (b) Histogram of the Martin Marietta excess returns. In this article, we analyze this data set to illustrate the applicability of the joint location, scale and skewness models of SLN distribution over the joint location, scale and skewness models of the STN distribution. For the comparison of the models, we use the values of the Akaike information criterion 8 (AIC) (Akaike (1973)), the Bayesian information criterion (BIC) (Schwarz (1978)), and the efficient determination criterion (EDC) (Bai et al. (1989)). These criteria have the following form ̂) + 𝑚𝑐𝑛 , −2ℓ(𝛉 where ℓ(∙) represents the maximized log-likelihood, 𝑚 is the number of free parameters to be estimated in the model and 𝑐𝑛 is the penalty term. Here, we take 𝑐𝑛 = 2 for AIC, 𝑐𝑛 = log(𝑛) for BIC and 𝑐𝑛 = 0.2√𝑛 for EDC. We give the estimation results in Table 4 for all models. This table contains the estimates, bootstrap standard errors (BSEs) (Efron and Tibshirani (1993)) of estimates based on 500 random samples, the log-likelihood, and the values of AIC, BIC, and EDC. Note that we take the heteroscedastic t model results given in Taylor and Verbyla (2004) as initial values for the parameters of location and scale models. Also, we set 𝛼0 = 𝛼1 = 0 as initial values for the parameters of skewness model and the degrees of freedom parameter 3.75. In Figure 3, we show the scatter plot of the data set with the fitted regression lines obtained from the joint location, scale and skewness models of the SLN distribution and the joint location, scale and skewness models of the STN distribution. We observe that the joint location, scale and skewness models of the SLN distribution has better fit than the location, scale and skewness models of the STN distribution according to the information criteria and also Figure 3. Table 4. Estimation results for Martin Marietta data set. Location model Scale model Skewness model Information Criteria 𝛽0 𝛽1 𝛾0 𝛾1 𝛼0 𝛼1 ̂) ℓ(𝛉 AIC BIC EDC Skew t Normal Estimate BSE 0.0289 -0.0349 1.1387 0.4888 0.4418 -5.7905 15.8670 25.1552 1.6808 0.5614 33.9531 19.3303 75.9986 -139.9872 -125.3268 -118.3268 9 Skew Laplace Normal Estimate BSE 0.0227 -0.0267 0.3683 0.7344 0.3234 -5.8282 11.8023 17.2765 1.0923 0.4040 9.0474 13.3093 76.9986 -139.9971 -127.4311 -121.4311 Figure 3. The scatterplot of the data set with the fitted regression lines obtained from joint location, scale and skewness models of the SN and SLN distributions. 7. Conclusions In this study, we have proposed the joint location, scale and skewness models of the SLN distribution as an alternative to the joint location, scale and skewness models of the STN distribution. We have obtained the ML estimates via the EM algorithm. We have provided a simulation study to show the estimation performance of the proposed model. We have observed from simulation results that the parameters can be accurately estimated. We have given a real data application to test the applicability of the proposed model and also to compare with the joint, location, scale and skewness models of the STN distribution. We have seen from real data example results that the joint location, scale and skewness models of the SLN distribution gives better fit than the joint, location, scale and skewness models of the STN distribution. 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Appendix Using the objective function given in (21), we obtain the score function 𝐺(𝜽) = ̂) 𝑇 𝜕𝑄(𝜽; 𝜽 = (𝐺1𝑇 (𝜷), 𝐺2𝑇 (𝜸), 𝐺3𝑇 (𝜶)) , 𝜕𝜽 where 𝑛 𝐺1 (𝜷) = ∑ 𝑖=1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒙𝑖 𝑇 𝑒 𝒛𝑖 𝜸 𝑛 𝑛 𝑖=1 𝑖=1 𝑛 (𝑣̂𝑖 + 2 (𝒘𝑇𝑖 𝜶) ) − ∑ 𝑖=1 2 (𝒘𝑇𝑖 𝜶)𝒙𝑖 𝑢̂1𝑖 𝑇 𝑒 𝒛𝑖 𝜸⁄2 , 𝑛 1 1 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒛𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒛𝑖 𝑢̂1𝑖 2 𝑇 𝐺2 (𝜸) = − ∑ 𝒛𝑖 + ∑ (𝑣 ̂ + 𝜶) ) − ∑ , (𝒘 𝑖 𝑖 𝑇 𝑇 2 2 2 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸⁄2 𝑖=1 11 𝑛 𝐺3 (𝜶) = ∑ 𝑛 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒘𝑖 𝑢̂1𝑖 −∑ 𝑇 𝑒 𝒛𝑖 𝜸⁄2 𝑖=1 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒘𝑖 𝑇 𝑒 𝒛𝑖 𝜸 𝑖=1 , and observed Fisher information matrix ̂ ) 𝜕 2 𝑄(𝜽; 𝜽 ̂ ) 𝜕 2 𝑄(𝜽; 𝜽 ̂) 𝜕 2 𝑄(𝜽; 𝜽 𝜕𝜷𝜕𝜷𝑇 𝜕𝜷𝜕𝜸𝑇 𝜕𝜷𝜕𝜶𝑇 2 2 2 2 ̂) ̂ ) 𝜕 𝑄(𝜽; 𝜽 ̂ ) 𝜕 𝑄(𝜽; 𝜽 ̂) 𝜕 𝑄(𝜽; 𝜽 𝜕 𝑄(𝜽; 𝜽 𝐻(𝜽) = = , 𝑇 𝑇 𝑇 𝑇 𝜕𝜽𝜕𝜽 𝜕𝜸𝜕𝜷 𝜕𝜸𝜕𝜸 𝜕𝜸𝜕𝜶 ̂ ) 𝜕 2 𝑄(𝜽; 𝜽 ̂ ) 𝜕 2 𝑄(𝜽; 𝜽 ̂) 𝜕 2 𝑄(𝜽; 𝜽 𝑇 𝑇 𝑇 [ 𝜕𝜶𝜕𝜷 𝜕𝜶𝜕𝜸 𝜕𝜶𝜕𝜶 ] where 𝑛 𝑛 𝑖=1 𝑛 𝑖=1 2 ̂) 𝜕 2 𝑄(𝜽; 𝜽 𝒙𝑖 𝒙𝑇𝑖 (𝒘𝑇𝑖 𝜶) = − ∑ 𝑇 𝑣̂𝑖 − ∑ 𝒙𝑖 𝒙𝑇𝑖 , 𝑇 𝜕𝜷𝜕𝜷𝑇 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸 𝑛 ̂) 𝜕 𝑄(𝜽; 𝜽 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒙𝑖 𝒛𝑇𝑖 (𝒘𝑇𝑖 𝜶)𝒙𝑖 𝒛𝑇𝑖 𝑢̂1𝑖 2 𝑇 = − ∑ (𝑣 ̂ + 𝜶) ) + ∑ , (𝒘 𝑖 𝑖 𝑇 𝑇 𝜕𝜷𝜕𝜸𝑇 2 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸⁄2 2 𝑖=1 𝑛 𝑛 𝑖=1 𝑛 𝑖=1 𝑖=1 ̂) 𝜕 2 𝑄(𝜽; 𝜽 𝒙𝑖 𝒘𝑇𝑖 𝑢̂1𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒙𝑖 𝒘𝑇𝑖 = − ∑ + 2 ∑ , 𝑇 𝑇 𝜕𝜷𝜕𝜶𝑇 𝑒 𝒛𝑖 𝜸⁄2 𝑒 𝒛𝑖 𝜸 𝑛 ̂) 𝜕 𝑄(𝜽; 𝜽 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒛𝑖 𝒙𝑇𝑖 (𝒘𝑇𝑖 𝜶)𝒛𝑖 𝒙𝑇𝑖 𝑢̂1𝑖 2 𝑇 = −∑ (𝑣̂𝑖 + (𝒘𝑖 𝜶) ) + ∑ , 𝑇 𝑇 𝜕𝜸𝜕𝜷𝑇 2 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸⁄2 2 𝑖=1 𝑛 𝑖=1 𝑛 2 ̂) 𝜕 𝑄(𝜽; 𝜽 1 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒛𝑖 𝒛𝑇𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒛𝑖 𝒛𝑇𝑖 𝑢̂1𝑖 2 𝑇 = − ∑ (𝑣 ̂ + 𝜶) ) + ∑ , (𝒘 𝑖 𝑖 𝑇 𝑇 𝜕𝜸𝜕𝜸𝑇 2 4 𝑒 𝒛𝑖 𝜸 𝑒 𝒛𝑖 𝜸⁄2 2 𝑖=1 𝑛 𝑛 𝑖=1 2 ̂) 𝜕 2 𝑄(𝜽; 𝜽 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒛𝑖 𝒘𝑇𝑖 𝑢̂1𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒛𝑖 𝒘𝑇𝑖 = − ∑ + ∑ , 𝑇 𝑇 𝜕𝜸𝜕𝜶𝑇 2 𝑒 𝒛𝑖 𝜸⁄2 𝑒 𝒛𝑖 𝜸 𝑖=1 𝑛 𝑖=1 𝑛 ̂) 𝜕 𝑄(𝜽; 𝜽 𝒘𝑖 𝒙𝑇𝑖 𝑢̂1𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒘𝑖 𝒙𝑇𝑖 = −∑ + 2∑ , 𝑇 𝑇 𝜕𝜶𝜕𝜷𝑇 𝑒 𝒛𝑖 𝜸⁄2 𝑒 𝒛𝑖 𝜸 2 𝑖=1 𝑛 𝑖=1 𝑛 2 ̂) 𝜕 𝑄(𝜽; 𝜽 1 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷)𝒘𝑖 𝒛𝑇𝑖 𝑢̂1𝑖 (𝒘𝑇𝑖 𝜶)(𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒘𝑖 𝒛𝑇𝑖 =− ∑ +∑ , 𝑇 𝑇 𝜕𝜶𝜕𝜸𝑇 2 𝑒 𝒛𝑖 𝜸⁄2 𝑒 𝒛𝑖 𝜸 2 𝑖=1 𝑛 𝑖=1 2 ̂) 𝜕 2 𝑄(𝜽; 𝜽 (𝑦𝑖 − 𝒙𝑇𝑖 𝜷) 𝒘𝑖 𝒘𝑇𝑖 = − ∑ . 𝑇 𝜕𝜶𝜕𝜶𝑇 𝑒 𝒛𝑖 𝜸 𝑖=1 12	10
1 Investigations of a Robotic Testbed with Viscoelastic Liquid Cooled Actuators arXiv:1711.01649v2 [cs.SY] 8 Mar 2018 Donghyun Kim, Junhyeok Ahn, Orion Campbell, Nicholas Paine, and Luis Sentis, Abstract—We design, build, and thoroughly test a new type of actuator dubbed viscoelastic liquid cooled actuator (VLCA) for robotic applications. VLCAs excel in the following five critical axes of performance: energy efficiency, torque density, impact resistence, joint position and force controllability. We first study the design objectives and choices of the VLCA to enhance the performance on the needed criteria. We follow by an investigation on viscoelastic materials in terms of their damping, viscous and hysteresis properties as well as parameters related to the longterm performance. As part of the actuator design, we configure a disturbance observer to provide high-fidelity force control to enable a wide range of impedance control capabilities. We proceed to design a robotic system capable to lift payloads of 32.5 kg, which is three times larger than its own weight. In addition, we experiment with Cartesian trajectory control up to 2 Hz with a vertical range of motion of 32 cm while carrying a payload of 10 kg. Finally, we perform experiments on impedance control and mechanical robustness by studying the response of the robotics testbed to hammering impacts and external force interactions. Index Terms—Viscoelastic liquid cooled actuator, Torque feedback control, Impedance control. I. I NTRODUCTION ERIES elastic actuators (SEAs) [1] have been extensively used in robotics [2], [3] due to their impact resistance and high-fidelity torque controllability. One drawback of SEAs is the difficulty that arises when using a joint position controller due to the presence of the elastic element in the drivetrain. To remedy this problem the addition of dampers has been previously considered [4]–[6]. However, incorporating mechanical dampers makes actuators bulky and increases their mechanical complexity. One way to avoid this complexity is to employ elastomers instead of metal springs. Using a viscoelastic material instead of combined spring-damper systems enables compactness [7] and simplified drivetrains [8]. However, it is difficult to achieve high bandwidth torque control due to the nonlinear behavior of elastomers. To address this difficulty, [9] models the force-displacement curve of elastomer using a “standard linear model.” The estimated elastomer force is employed in a closed-loop force controller. Unfortunately, the hysteresis in the urethane elastomer destabilized the system at frequencies above 2 Hz. In contrast our controllers achieve a bandwidth of 70 Hz. The study on [10] accomplishes reasonably good torque control performance, but the range of torques is small to ensure that the elastomer operates in the linear region; our design and control methods described here achieve more than an order of magnitude higher range of torques with high fidelity tracking. To sufficiently address the nonlinear behavior of elastomers, which severely reduce force control performance, we empiri- S cally analyze various viscoelastic materials with a custom-built elastomer testbed. We measure each material’s linearity, creep, compression set, and damping under preloaded conditions, which is a study under-documented in the academic literature. To achieve stable and accurate force control, we study various feedback control schemes. In a previous work, we showed that the active passivity obtained from motor velocity feedback [11] and model-based control such as disturbance observer (DOB) [12] play an essential role in achieving high-fidelity force feedback control. Here, we analyze the phase margins of various feedback controllers and empirically show their operation in the new actuators. We verify the stability and accuracy of our controllers by studying impedance control and impact tests. To test our new actuator, we have designed a two degree-offreedom (DOF), robotic testbed, shown in Fig. 5. It integrates two of our new actuators, one in the ankle, and another in the knee, while restricting motions to the sagittal plane. With the foot bolted to the floor for initial tests, weight plates can be loaded on the hip joint to serve as an end-effector payload. We test operational space control to show stable and accurate operational space impedance behaviors. We perform dynamic motions with high payloads to showcase another important aspect of our system, which is its cooling system aimed at significantly increasing the power of the robot. The torque density of electric motors is often limited by sustainable core temperature. For this reason, the maximum continuous torque achieved by these motors can be significantly enhanced using an effective cooling system. Our previous study [13] analyzed the improvements on achievable power based on thermal data of electric motors and proposed metrics for design of cooling systems. Based on the metrics from that study, we chose a 120 W Maxon EC-max 40, which is expected to exert 3.59 times larger continuous torque when using the proposed liquid cooling system. We demonstrate the effectiveness of liquid cooling by exerting 860N continous force during 5 min and 4500N peak force during 0.5s while keeping the core temperatures below 115◦C, which is much smaller than the maximum, 155◦C. We accurately track fast motions of 2 Hz while carrying a 10 kg payload for endurance tests. In addition we perform heavy lift tests with a payload of 32.5 kg keeping the motor temperatures under 80◦C. The main contribution of this paper is the introduction of a new viscoelastic liquid cooled actuator and a thorough study of its performance and its use on a multidof testbed. We demonstrate that the use of liquid cooling and the elastomer significantly improve joint position controllability and power density over traditional SEAs. More concretely, we 1) design 2 a new actuator, dubbed the VLCA, 2) extensively study viscoelastic materials, 3) extensively analyze torque feedback controllers for VLCAs, and 4) examine the performance in a multidof prototype. II. BACKGROUND Existing actuators can be characterized using four criteria: power source (electric or hydraulic), cooling type (air or liquid), elasticity of the drivetrain (rigid or elastic), and drivetrain type (direct, harmonic drive, ball screw, etc.) [14], [15]. One of the most powerful and common solutions is the combination of hydraulic, liquid-cooling, rigid and direct drive actuation. This achieves high power-to-weight and torque-to-weight ratios, joint position controllability, and shock tolerance. Existing robots that use this type of actuators include Atlas, Spot, Big Dog, and Wildcat of Boston Dynamics, BLEEX of Berkeley [16], and HyQ of IIT [17]. However, hydraulics are less energy efficient primarily because they require more energy transformations [18]. Typically, a gasoline engine or electric motor spins a pump, which compresses hydraulic fluid, which is modulated by a hydraulic servo valve, which finally causes a hydraulic piston to apply a force. Each stage in this process incurs some efficiency loss, and the total losses can be very significant. The combination of electric, air-cooled, rigid, and harmonic drive actuators are other widely used actuation types. Some robots utilizing these actuator types include Asimo of Honda, HRP2,3,4 of AIST [19], HUBO of KAIST [20], REEM-C of PAL Robotics, JOHNNIE and LOLA of Tech. Univ. of Munich [21], [22], CHIMP of CMU [23], Robosimian of NASA JPL [24], and more. These actuators have precise position control and high torque density. For example, LOLA’s theoretical knee peak torque-density (129N m/kg) is comparable to ours (107N m/kg), although they did not validate their number experimentally and their max speed is roughly 2/3 of our max speed [22]. Compared to us, low shock tolerance, low fidelity force sensing, and low efficiency gearboxes are common drawbacks of these type of actuators. According to Harmonic Drive AGs catalog, the efficiency of harmonic drives may be as poor as 25% and only increases above 80% when optimal combinations of input shaft speed, ambient temperature, gear ratio, and lubrication are present. Conversely, the efficiency of our VLCA is consistently above 80% due to the use of a ball screw mechanism. [25] used liquid cooling for electric, rigid, harmonic drive actuators to enhance continuous power-to-weight ratio. The robots using this type of actuation include SCHAFT and Jaxon [26]. These actuators share the advantages and disadvantages of electric, rigid, harmonic drive actuators, but have a significant increase of the continuous power output and torque density. One of our studies [13], indicates a 2x increase in sustained power output by retrofitting an electric motor with liquid cooling. Other published results indicate a 6x increase in torque density through liquid cooling [14], [27], though such performance required custom-designing a motor specifically for liquid cooling. In our case we use an off-theshelf electric motor. In contrast with our design, these actuators do not employ viscoelastic materials reducing their mechanical robustness and high quality force sensing and control. Although the increased power density achieved via liquid cooling amplifies an electric actuator’s power, the rigid drivetrain is still vulnerable to external impacts. To increase impact tolerance, many robots (e.g. Walkman and COMAN of IIT [28], Valkyrie of NASA [29], MABEL and MARLO in UMich [30], [31], and StarlETH of ETH [32]) adopt electric, air-cooled, elastic, harmonic drive actuators. This type of actuation provides high quality force sensing, force control, impact resistance, and energy efficiency. However, precise joint position control is difficult because of the elasticity in the drivetrain and the coupled effect of force feedback control and realtime latencies [33]. Low efficiency originating from the harmonic drives is another drawback. As an alternative to harmonic drives, ball screws are great drives for mechanical power transmission. SAFFiR, THOR, and ESCHER of Virginia Tech [34]–[36], M2V2 of IHMC [37], Spring Flamingo of MIT [38], Hume of UT Austin [11], and the X1 Mina exoskeleton of NASA [39] use electric, air-cooled, elastic, ball-screw drives. These actuators show energy efficiency, good power and force density, low noise force sensing, high fidelity force controllability, and low backlash. Compared to these actuators our design significantly reduces the bulk of the actuator and increases its joint position controllability. There are some other actuators that have special features such as the electric actuators used in MIT’s cheetah [40], which allow for shock resistance through a transparent but backlash-prone drivetrain. However, the lack of passive damping limits the joint position controllability of these type of actuators compared to us. III. VISCOELASTIC MATERIAL CHARACTERIZATION The primary driver for using elastomers instead of metal springs is to benefit from their intrinsic damping properties. However, the mechanical properties of viscoelastic materials can be difficult to predict, thus making the design of an actuator based on these materials a challenging endeavor. The most challenging aspect of incorporating elastomers into the structural path of an actuator is in estimating or modeling their complex mechanical properties. Elastomers possess both hysteresis and strain-dependent stress, which result in nonlinear force displacement characteristics. Additionally, elastomers also exhibit time-varying stress-relaxation effects when exposed to a constant load. The result of this effect is a gradual reduction of restoration forces when operating under a load. A third challenge when using elastomers in compression is compression set. This phenomenon occurs when elastomers are subjected to compressive loads over long periods of time. An elastomer that has been compressed will exhibit a shorter free-length than an uncompressed elastomer. Compression set is a common failure mode for o-rings, and in our application, it could lead to actuator backlash if not accounted for properly. To address these various engineering challenges we designed experiments to empirically measure the following four properties of our viscoelastic springs: 1) force versus displacement, 2) stress relaxation, 3) compression set, and 4) 3 E-stop Load cell Displacement sensor Elastic material EtherCAT-based embedded control system Belt drive BLDC motor 2500 2000 1500 Force (N) 1000 Ball screw drive (a) Viscoelastic material Testbed 0 -500 Spring steel Viton 75A Polyurethane 80A Polyurethane 90A EPDM 80A Reinforced Silicon 70A Buna- N 90A Silicone 90A -1000 Reinforced silicone 70A Viton 75A Buna-N 90A Polyurethane 80A Polyurethane 90A Spring steel 500 EPDM 80A -1500 -2000 0 1 2 3 4 5 -2500 6 -5 Compression set (%) -4 -3 -2 -1 0 1 2 3 Dispacement (m) 5 10 -4 (c) Force vs Displacement curve Phase (deg) Force (N) Magnitude (dB) (b) Complession set 4 time (sec) (d) Stess relaxation Increasing system bandwidth Spring steel Viton 75A Buna-N 90A Polyurethane 90A Frequency (Hz) (e) Dynamic response of four elastomer Fig. 1. Viscoelastic material test. (a) The elastomer testbed is designed and constructed to study various material properties of candidate viscoelastic materials. (b) We measured each elastomers free length both before and after they were placed in the preloaded testbed. (c) A strong correlation between material hardness and the materials stiffness can be observed. An exception to this correlation is the fabric reinforced silicone which we hypothesize had increased stiffness due to the inelastic nature of its reinforcing fabric. Nonlinear effects such as hysteresis can also be observed in this plot. (d) We command a rapid change in material displacements and then measured the materials force change versus time for 300 seconds. Note that the test of reinforced silicone 70A is omitted due to its excessive stiffness. (d) Although the bandwidths of the four responses are different, their damping ratios (signal peak value) are relatively constant, which implies different damping. frequency response, which will be used to characterize each material’s effective viscous damping. We built a viscoelastic material testbed, depicted in Fig. 1(a), to measure each of these properties. We selected and tested the seven candidate materials that are listed in Table I. The dimension of the tested materials are fairly regular, with 46mm diameter and 27mm thickness. A. Compression set Compression set is the reduction in length of an elastomer after prolonged compression. The drawback of using materials with compression set in compliant actuation is that the materials must be installed with larger amounts of preload forces to avoid the material sliding out of place during usage. To measure this property, we measured each elastomers free length both before and after the elastomer was placed in the preloaded testbed. The result of our compression set experiments are summarized in Table I. B. Force versus displacement In the design of compliant actuation, it is essential to know how much a spring will compress given an applied force. This displacement determines the required sensitivity of a spring-deflection sensor and also affects mechanical aspects of the actuator such as usable actuator range of motion and clearance to other components due to Poisson ratio expansion. In this experiment, we identify the force versus displacement curves for the various elastomer springs. Experimental data for all eight springs as shown in Fig 1(b). Note that there is a disagreement between our empirical measurements and the analytic model relating stiffness to hardness, i.e. the Gent’s relation shown in [41]. This mismatch arises because in our experiments the materials are preloaded whereas the analytical models assume unloaded materials. 4 Materials Compression Linearity set (%) (R-square) Linear stiffness (N/mm) Spring steel 0 0.996 860.8 Polyurethane 90A 2 0.992 8109 Preloaded elastic modulus (N/mm) Material damping (N s/m) Creep (%) Material Cost ($) 0 0 - 112.5 16000 15.3 19.40 Reinforced silicone 70A 2.7 0.978 57570 798.7 242000 - 29.08 Buna-N 90A 2.8 0.975 11270 156.4 29000 25 51.47 Viton 75A 4 0.963 2430 33.7 9000 30.14 105.62 Polyurethane 80A 4.5 0.993 2266 31.4 4000 16.8 19.40 EPDM 80A 6.48 0.939 6499 90.2 16000 23.4 35.28 Silicone 90A - 0.983 12460 172.9 37000 10.7 29.41 TABLE I S UMMARY OF VISCOELASTIC MATERIALS C. Stress relaxation E. Selection of Polyurethane 90A Stress-relaxation is an undesirable property in compliant actuators for two reasons. First, the time-varying force degrades the quality of the compliant material as a force sensor. When a material with significant stress-relaxation properties is used, the only way to accurately estimate actuator force based on deflection data is to model the effect and then pass deflection data through this model to obtain a force estimate. This model introduces complexity and more room for error. The second reason stress-relaxation can be problematic is that it can lead to the loss of contact forces in compression-based spring structures. The experiment for stress relaxation is conducted as follow: 1) enforce a desired displacement to a material, 2) record the force data over time from the load cell, 3) subtract the initially measured force from all of the force data. Empirically measured stress-relaxation properties for each of the materials are shown in Fig. 1 (c), which represents force offsets as time goes under the same displacement enforced. Note that each material shows different initial force due to the different stiffness and each initial force data is subtracted in the plot. A variety of other experiments were conducted to strengthen our analysis and are summarized in Table I. Based on these results, Polyurethane 90A appears to be a strong candidate for viscoelastic actuators based on its high linearity (0.992), low compression set (2%), low creep (15%), and reasonably high damping (16000 N s/m). It is also the cheapest of the materials and comes in the largest variety of hardnesses and sizes. D. Dynamic response In regards to compliant actuation, the primary benefit of using an elastomer spring is its viscous properties, which can characterize the dynamic response of an actuator in series with such a component. To perform this experiment, we generate motor current to track an exponential chirp signal, testing frequencies between 0.001Hz and 200Hz. Given the inputoutput relation of the system, we can fit a second order transfer function to the experimental data to obtain an estimate of the system’s viscous properties. However, this measure also includes the viscoelastic testbed’s ballscrew drive train friction (Fig. 1(a)). To quantify the elastomer spring damping independently of the damping of the testbed drive train, the latter (8000 N s/m) was first characterized using a metal spring, and then subtracted from subsequent tests of the elastomer springs to obtain estimates for the viscous properties of the elastomer materials. Fig. 1(d) shows the frequency response results for current input and force output of three different springs, while controlling the damping ratio. The elastomers have higher stiffness than the metal spring, hence their natural frequencies are higher. IV. V ISCOELASTIC L IQUID C OOLED ACTUATION The design objectives of the VLCA are 1) power density, 2) efficiency, 3) impact tolerance, 4) joint position controllability, and 5) force controllability. Compactness of actuators is also one of the critical design parameters, which encourage us to use elastomers instead of metal springs and mechanical dampers. Our previous work [13] shows a significant improvement in motor current, torque, output power and system efficiency for liquid cooled commercial off-the-shelf (COTS) electric motors and studied several Maxon motors for comparison. As an extension of this previous work, in this new study we studied COTS motors and their thermal behavior models and selected the Maxon EC-max 40 brushless 120 W (Fig. 2(e)), with a custom housing designed for the liquid cooling system (Fig. 2(h)). The limit of continuous current increases by a factor of 3.59 when liquid convection is used for cooling the motor. Therefore, a continuous motor torque of 0.701 N · m is theoretically achievable. Energetically, this actuator is designed to achieve 366 W continuous power and 1098W short-term power output with an 85% ball screw efficiency (Fig. 2(b)) since short-term power is generally three time larger than continuous power. With the total actuator mass of 1.692 kg, this translates into a continuous power of 216W/kg and a short-term power of 650W/kg. The liquid pump, radiator, and reservoir are products of Swiftech which weight approximately 1kg. By combining convection liquid cooling, high power brushless DC (BLDC) motors, and a high-efficiency ball screw, we aim to surpass existing electric actuation technologies with COTS motors in terms of power density. In terms of controls, a common problem with conventional SEAs is their lack of physical damping at their mechanical output. As a result, active damping must be provided from torque produced by the motor [42]. However, the presence of 5 (e) BLDC Motor (a) Timing belt transmission (b) Ball screw drive (c) Load cell (d) Actuator output (f) Quadrature encoder (g) Temperature sensor (h) Liquid cooling jacket Opposite side Motor part Rubber part Load part (i) Tube connector (j) Polyuretane elastomer (k) Compliance deflection sensor (l) Mechanical ground pivot (m) Quadrature encoder (deflection) ground Fig. 2. Viscoelastic Liquid Cooled Actuator. The labels are explanatory. In addition, the actuator contains five sensors: a load cell, a quadrature encoder for the electric motor, a temperature sensor, and two elastomer deflection sensors. One of the elastomer deflection sensors is absolute and the other one is a quadrature encoder. The quadrature encoder gives high quality velocity data of the elastomer deflection. signal latency and derivative signal filtering limit the amount by which this active damping can be increased, resulting in SEA driven robots achieving only relatively low output impedances [33] and thus operating with limited joint position control accuracy and bandwidth. Our VLCA design incorporates damping directly into the compliant element itself, reducing the requirements placed on active damping efforts from the controller. The incorporation of passive damping aims to increase the output impedance while retaining compliance properties, resulting in higher joint position control bandwidth. The material properties we took into consideration will be introduced in Section III. The retention of a compliant element in the VLCA drive enables the measurement of actuator forces based on deflection. The inclusion of a load cell (Fig. 2(c)) on the actuators output serves as a redundant force sensor and is used to calibrate the force displacement characteristics of the viscoelastic element. Mechanical power is transmitted when the motor turns a ball nut via a low-loss timing belt and pulley (Fig. 2 (a)), which causes a ball screw to apply a force to the actuator’s output (Fig. 2(d)). The rigid assembly consisting of the motor, ball screw, and ball nut connects in series to a compliant viscoelastic element (Fig. 2(j)), which connects to the mechanical ground of the actuator (Fig. 2(k)). When the actuator applies a force, the reaction force compresses the viscoelastic element. The viscoelastic element enables the actuator to be more shock tolerant than rigid actuators yet also maintain high output impedance due to the inherent damping in the elastomer. V. ACTUATOR F ORCE F EEDBACK C ONTROL To demonstrate various impedance behaviors in operational space, robots must have a stable force controller. Stable and accurate operational space control (OSC) is not trivial to achieve because of the bandwidth interference between outer position feedback control (OSC) and inner torque feedback control [11]. Since stable torque control is a critical component for a successful OSC implementation, we extensively study various force feedback controls. Jm (kg m2 ) bm (N m s) mr (kg) br (N s/m) kr (N/m) 3.8e−5 2.0e−4 1.3 2.0e4 5.5e6 TABLE II ACTUATOR PARAMETERS The first step in this analysis is to identify the actuator dynamics. The transfer functions of the reaction force sensed in the series elastic actuators (elastomer deflection) are well explained in [43]. When the actuator output is fixed, the transfer function from the motor current input to the elastomer deflection is given by Px = xr ηkτ Nm = , (1) 2 + m )s2 + (b N 2 + b )s + k im (Jm Nm r m m r r where η, kτ , Nm , and im are the ball screw efficiency, the torque constant of a motor, the speed reduction ratio of the motor to the ball screw, and the current input for the motor, respectively. The equations follow the nomenclature in Fig. 3(a). We can find η, kτ , and Nm in data sheets, which are 0.9, 0.0448 N · m/A, and 3316 respectively. The gear ratio of the drivetrain is computed by dividing the speed reduction of pulleys (2.111) with lead length of the ball screw (0.004m) using the equation 2π × 2.111/0.004. However, we need to experimentally identify kr , br , Jm , and bm . We infer kr by dividing the force measurement from the load cell by the elastomer deflection. The other parameters are estimated by comparing the frequency response of the model and experimental data. The frequency response test is done with the ankle actuator while prohibiting joint movement with a load and an offset force command. The results are presented in Fig. 3 with solid gray lines. Note that the dotted gray lines are the estimated response from the transfer function (measured elastomer force/ input motor force) using the parameters of Table. II. The estimated response and experimental result match closely with one another, implying that the parameters we found are close to the actual values. We also study the frequency response for different load masses to understand how the dynamics changes as the joint moves. When 10kg is attached to the end of link, the reflected 6 (a) VCLA -149 o 27.5 Hz Fig. 3. Frequency response of VLCA. Gray solid lines are experimental data and the other lines are estimated response with the model using empirically parameters. (b) Open-loop PDm PIDm PDf PDm + DOB 60 Magnitude (dB) Experiment result 40 20 0 -20 -40 0 Phase margin -45 47.6 -90 41.0 1) Proportional (P) + Derivative (Df ) using velocity signal obtained by a low-pass derivative filtered elastomer deflection 2) Proportional (P) + Derivative (Dm ) using motor velocity signal measured by a quadrature encoder connected to a motor axis The second controller (PDm ) has benefits over the first one (PDf ) with respect to sensor signal quality. The velocity of motor is directly measured by a quadrature encoder rather than low-pass filtered elastomer deflection data, which is relatively noisy and lagged. In addition, Fig. 4 shows that the phase margin of the second controller (47.6) is larger than the first one (17.1). To remove the force tracking error at low frequencies, we consider two options: augmenting the controller either with integral control or with a DOB on the PDm controller. To compare the two controllers, we analyzed the phase margins of all the mentioned controllers. First, we chose to focus on the location where the sensor data returns in order to address the time delay of digital controllers (Fig. 4 (a) and (c)). Next, we have to compute the open-loop transfer function for each closed loop system. For example, the PDf controller’s closed loop transfer function is kr Px kp (Fr − e−T s Fk ) + Fr − kd,f Qd e−T s Fk , N (2) where Fk , Fr , T , and Qd are the measured force from a elastomer deflection, a reference force, a time delay, a low pass derivative filter, respectively.For convenience, we use N instead of the multiplication of three terms, ηkτ Nm . When 34.0 -225 10 (c) 42.8 17.7 -135 -180 mass to the actuator varies from 1500kg to 2500kg because the length of the effective moment arm changes depending on joint position. In Fig. 3(b), the bode plots are presented and the response is not significantly different than the fixed output case. Therefore, we design and analyze the feedback controller based on the fixed output dynamics. For the force feedback controller, we first compare two options, which we have used in our previous studies [11], [12]: Fk = kr -3dB Phase (deg) Inf. mass 2500 kg 2000 kg 1500 kg 0 10 1 10 2 Frequency (Hz) VCLA kr s Fig. 4. Stability analysis of controllers. Phase margins of each controllers and open-loop system are presented. gathering the term with e−T s of Eq. (2), we obtain Fk kr Px (Kp + 1)/N = . Fr 1 + e−T s kr Px (Kp + Kd,f Qd )/N (3) Then, the open-loop transfer function of the closed system with the time delay is open PPD = kr Px (Kp + Kd,f Qd )/N. f (4) We can apply the same method for the PIDm and PDm +DOB controllers. The transfer function of PIDm , which is presented in Fig. 4(c), is 1 kr Px (Fr − e−T s Fk )(Kp + Ki ) + Fr Fk = N s (5) Fk −T s − Kd,m e sNm . kr Then it becomes Fk kr Px (Kp + Ki /s + 1)/N = . Fr 1 + e−T s Px (kr (Kp + Ki /s) + Kd sNm )/N (6) When we apply a DOB instead of integral control, we need the inverse of the plant. In our case, the plant of the DOB is PDm , which is similar to Eq. (6) except that Ki and e−T s are omitted: kr Px (Kp + 1) PPDm (= Pc ) = . (7) N + Px (kr Kp + Kd,m sNm ) 7 Fig. 5. Robotic testbed. Our testbed consists of two VLCAs at the ankle and the knee. The foot of the testbed is fixed on the ground. The linkages are designed to vary the maximum peak torques and velocities depending on postures. As the joint positions change, the ratios between ball screw velocities (L̇0,1 ) and joint velocities (q̇0,1 ) also change because of effective lengths of moment arms vary. The linkages are designed to exert more torque when the robot crouches, which is the posture that the gravitational loads on the joints are large. The formulation of PDm including the DOB, which is shown in Fig. 4(c), is Fk = kr Px (Kp + 1)(Fd − e−T s Pc−1 Qτ d Fk ) , (8) (N + e−T s Px (kr Kp + Kd,m sNm )) (1 − Qτ d ) the hip, and the foot is fixed on the ground. With this testbed, we intended to demonstrate coordinated position control with two VLCAs, the viability of liquid cooling on an articulated platform, cartesian position control of a weighted end effector, and verification of a linkage design. The two joints each have a different linkage structure that was carefully designed so that the moment arm accommodates the expected torques and joint velocities as the robot posture changes (Fig. 5). For example, each joint can exert a peak torque of approximately 270 N m and the maximum joint velocity ranges between 7.5 rad/s and 20+ rad/s depending on the mechanical advantage of the linkage along the configurations. The joints can exert a maximum continuous torque of 91 N m at the point of highest mechanical advantage. This posture dependent ratio of torque and velocity is a unique benefit of prismatic actuators. Given cartesian motion trajectories, which are 2nd order Bspline or sinusoidal functions, the centralized controller computes the torque commands with operational space position and velocity, which are updated by the sensed joint position and velocity. The OSC formulation that we use is −1 τ = AJhip (ẍdes + Kp e + Kd ė − J˙hip q̇) + b + g, (11) where A, b, and g represent inertia, coriolis, and gravity joint torque, respectively. ẍdes , e, and ė are desired trajectory acceleration, position and velocity error, respectively. q̇ ∈ R2 is the joint velocity of the robot and τ is the joint torque. Jhip is a jacobian of the hip, which is a 2 × 2 square matrix and assumed to be full-rank. where Qτ d is a second order low-pass filter. Then the transfer function is VII. R ESULTS Fk kr Px (Kp + 1) . = We first conducted various single actuator tests to show Fd N (1 − Qτ d ) + e−T s (N Qτ d + Px (kr Kp + Kd,m sNm )) (9) basic performance such as torque and joint position controllability, continuous and peak torque, and impact resistance. SubThe open-loop transfer function is sequently, we focused on the performance of OSC using the N Qτ d + Px (kr Kp + Kd,m sNm ) open robotic testbed integrated with DOB based torque controllers PPD = (10) m +DOB N (1 − Qτ d ) to demonstrate actuator efficiency and high power motions. open open open open The bode plots of PPD , P , P , and P PDm PIDm PDm +DOB f are presented in Fig. 4(b). The gains (Kp , Kd,m , Ki ) are the A. Single Actuator Tests same as the values that we use in the experiments presented in Fig. 6(a) shows the experimental results of our frequency Section VII-A, which are 4, 15, and 300, respectively. The PDf response testing as well as the estimated response based on controller uses Kd Nm /kr for Kd,f to normalize the derivative the transfer functions. We compare three types of controllers: gain. The cutoff frequency of the DOB is set to 15Hz because PD , PID , and PD + DOB. As we predicted in the m m m this is where the PDm +DOB shows a magnitude trend similar analysis of Section V, the PD + DOB controller shows less m to the integral controller (PIDm ). The results imply that the phase drop and overshoot than PID . The integral control m PDm +DOB controller is more stable than PIDm with respect feedback gain used in the experiment is 300 and the cutoff to phase margin and maximum phase lag. This analysis is also frequency of the DOB’s Q filter is 60Hz, which shows τd experimentally verified in Section VII-A. similar error to the PID controller (Fig. 6(b)). Another test m VI. ROBOTIC T ESTBED We built a robotic testbed shown in Fig. 5. To demonstrate dynamic motion, we implemented an operational space controller (OSC) incorporating the multi-body dynamics of the robot. We designed and built a robotic testbed (Fig. 5) consisting of two VLCAs - one for the ankle (q0 ) and one for the knee (q1 ). The design constrains motion to the sagittal plane, the robot carries 10kg, 23kg, or 32.5kg of weight at presented in Fig. 6(c) also supports the stability and accuracy of torque control. In the test, we command a ramp in joint torque from 1 to 25N m in 0.1s. The sensed torque (blue solid line) almost overlaps the commanded torque (red dashed line). Fig. 6(d) is the result of a joint position control test designed to show that VLCAs have better joint position controllability than SEAs using springs. In the experiment, we use a joint encoder for position control and a motor quadrature encoder for velocity feedback. To compare the VLCAs performance 8 20 10 1.8 2 2.2 2.4 2.6 2.8 -1.8 JPos (rad) Phase (deg) (c) Torque fast response Open (estimated) (estimated) (estimated) -2 -2.4 1.6 1.8 2 2.2 2.4 2.6 70 1 60 0 50 40 -2 -3 2.8 30 0 1 2 3 time (sec) 20 (f) peak force (d) Position fast response Actuator force (N) 80 1000 150 500 100 0 -500 time (sec) 50 actuator force core temperature (w/ liquid) core temperature (w/o liquid) 0 50 100 150 200 250 0 350 300 Motor temperature (oC) Error (dB) (a) Frequency responses of different controllers Frequency (Hz) 90 2 time (sec) Reference 100 -1 (sim.) metal spring (sim.) elastomer (exp.) command (exp.) sensed -2.2 Frequency (Hz) 110 3 0 1.6 Open x 103 4 command sensed Actuator force (N) Magnitude (dB) 5 Motor temperature (oC) Torque (Nm) 30 time (sec) (e) Continous force and core temperature (b) Error magintude and chirp test trajectories Fig. 6. Torque Feedback Control Test. (a) Experimental data and estimated response based on the transfer functions are presented. Estimated response of PD controller is identical to the PD+DOB since DOB theoretically does not change the transfer function. The plot show PD+DOB shows better performance in terms of less overshoot and smaller phase drop near to the natural frequency. (b) We choose integral controller feedback gain that shows similar accuracy of PD+DOB’s. The left is error magnitude of three controllers. PD controller has larger error than the other two controller in the low frequency region. The right is torque trajectories in the time domain. Load cell (solid holding) Load cell (w/ elastomer) Rubber deflection (solid holding) Rubber deflection (w/ elastomer) 1000 Actuator force (N) with that of spring-based SEAs, we present simulation results for a spring-based SEA on the same plot as the experiment result for the VLCA. The green dashed line is the simulated step response of our actuator and the yellow dotted line is the result of the simulation model using the same parameters except the spring stiffness and damping. The spring stiffness was selected to be 11% of the elastomer’s, based on the results of our tests in Section III, and the damping for the spring case was set to 8000 N s/m which only includes the drivetrain friction. The results show a notable improvement in joint position control when using an elastomer instead of a steel spring. Fig. 6(e) shows the continous force and the motor core temperature trend with and without liquid cooling. The observed continous force is 860N and the motor core temperature settles at 115◦C with liquid cooling. Fig. 6(f) is the the result of shortterm torque test. In the experiment, we fix the output of the actuator and command a 31A current for 0.5s. The observed force measured by a loadcell (Fig. 2(c)) is 4500N, which is a little smaller than the theoretically expected value, 5900N. Considering that the estimated core temperature surpassed 107◦C (< 155◦C limit), we expect that the theoretical value is reasonable. Thus, we conclude that the maximum force density of our actuator is larger than 2700N/kg and potentially 3500N/kg. Fig. 7 shows loadcell and elastomer force data from the impact tests. In the tests, we hit the loadcell connected to the ball screw (Fig. 2(c)) with a hammer falling from a constant height while fixing the actuator in two different places to 500 0 95% interval -500 -1000 -2 0 2 4 6 8 10 12 14 time (ms) Fig. 7. Impact test. 83 trials are plotted and estimated with gaussian process. We can see the deflections of the elastomer, which imply that the elastic element absorbes the external impact force. compare the rigid actuator to viscoelastic actuator response. In the rigid scenario, outer case of ballnut, a blue part in Fig. 2, is fixed to exclude the elastomer from the external impact force path. In the second case, we fixed the ground pin of the actuator, which is depicted by a gray part in Fig. 2(l), to see how the elastomers react to the impact. The impact experiment is challenging because the number of data points we can obtain is very small with a 1ms update rate. To overcome the lack of data points, we estimate the mean and variance of 83 trials by gaussian process regression. The results presented in Fig. 7 imply that there is no significant difference in the forces measured by the loadcell in both cases, which is predictable because the elastic element is placed behind the drivetrain. However, the elastomer does Ankle command sensed 20 0 power supply 14 16 18 20 22 24 14 16 18 20 22 24 40 Wm Wb 0.2 Hip position (m) joint electric motor 50 30 0 -0.1 14 16 18 20 22 24 1 Wk Average efficiency 1.5 0.1 Wk / W m Compliant in horizontal direction servo drive -20 Knee Stiff in vertical direction Torque (Nm) 9 1 0.5 0.8 14 16 (a) Impedance control 22 0 0.6 0.4 -20 12 14 16 18 40 12 14 16 18 Hip position (m) 0.1 10 12 10 12 14 16 18 14 16 18 1 time (sec) (b) Operational space impact test 0.9 0.1 0.85 0 0.8 -0.1 2 command 2.5 sensed 0.9 0.75 0.7 0.8 0.65 0.7 0.6 0.6 1.5 time (sec) 1.5 2 2.5 3 3.5 5 (sec) 3 1 0.5 0.3 0.2 0 -0.2 0.5 1 1.5 2 2.5 3 3.5 0 -0.1 0.8 1 1 time (sec) 0.2 1.5 0.5 -0.4 30 10 1 0 24 Wk / W b Ankle 20 -10 10 Knee Torque (Nm) Hitting down 18 time (sec) 2 2.5 -0.05 0 0.05 (c) Fast up and down (1.7 Hz) Fig. 8. Operational Space Impedance Control Test. (a) The robot demonstrates different impedance: stiff in the vertical direction and compliant in the horizontal direction. The high tracking performance of force feedback control results in the overlapped commanded and sensed torques. (b) To show the stability, we hit the weight with a hammer while operating the impedance control. Even under the impact, force control show stable and accurate tracking. (c) The robot demonstrates a 1.7Hz up and down motion while carrying 10kg weight at the hip, and shows a position error of less than 2.5cm. play a significant role in absorbing energy from the impact which is evident from large elastomer deflection in the second case. Thus, the presence of the elastic element mitigates the propagation of an impulse to the link where the actuator grounds. B. Operational Space Impedance Control Fig. 8 shows our OSC experimental tests (Section VI) carrying a 10kg weight. In the first test presented in Fig. 8(a), the commanded behavior is to be compliant in the horizontal direction (x) and to be stiff in the vertical direction (y). When pushing the hip with a sponge in the x direction, the robot smoothly moves back to comply with the push, but it strongly resists the given vertical disturbance to maintain the Fig. 9. Efficiency analysis of the ankle actuator. Efficiencies of mechanical system using electrical power has 3 steps from a power supply to robot joints. The graph shows the ratio of the mechanical power of the ankle joint and the motor power and the ratio of the joint power and power supply’s input power. commanded height. To show the stability of our controller, we also test the response to impacts by hitting the weight with a hammer (Fig. 8(b)). Even when there are sudden disturbances, the torque controllers rapidly respond to maintain good torque tracking performance as shown in Fig. 6(d). Fig. 8(c) shows the tracking performance of our system while following a fast vertical hip trajectory. While traveling 0.3m with 1.7Hz frequency, the hip position errors are bounded by 0.025m. This result demonstrates that our system is capable of stable and accurate OSC, which is challenging because of the bandwidth conflict induced by its cascaded structure. C. Efficiency Analysis Fig. 9 explains the power flow from the power supply to the robot joint. Input current (Ib ) and voltage (Vb ) are measured in the micro-controllers and the product of those two yields the input power from the power supply. θ̇m is measured by the quadrature encoder connected to the motor’s axis (Fig. 2(f)) and τm is computed from kτ im with im measured in the micro-controller. Joint velocity is low-pass derivative filtered joint positions measured at the absolute joint encoders. The torque (τk ) is computed from projecting the load cell data across the linkage’s effective moment arm. In this test, the robot lifts a 23kg load using five different durations to observe efficiency over a range of different speeds and torques. The results are presented in Fig. 9 with the description of three different power measures. The sensed torque data measured by a load cell is noisy; therefore, we compute the average of the drivetrain efficiency for a clearer comparison. The averages are the integrations of efficiency divided by the time durations. Here we only integrate efficiency while the mechanical power is positive, to prevent confounding our results by incorporating the work done by gravity. 7 7.5 8 200 100 0 -100 0 7 joint torque 7.5 mechanical power 8 400 200 0 -200 -400 -600 200 100 0 6.5 7 7.5 Mechanical power (W) Ankle 7 200 -200 6.5 Knee 8 5 4 6.5 Joint torque (Nm) 7.5 8 (a) 2Hz up and down motion Ankle 0 45 40 -1000 35 -2000 -3000 0 0.5 1 1.5 2 2.5 actuator force 3 3.5 4 motor temperature 30 4.5 80 Knee -1000 60 -2000 -3000 40 -4000 0 Power (W) Ankle Knee 6.5 6 1000 Actuator force (N) command joint encoder motor encoder 2.6 2.4 2.2 2 1.8 0.5 1 1.5 2 2.5 3 3.5 4 Motor temperature (oC) Joint position (rad) 10 4.5 total ankle knee 1500 1000 500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time (sec) (b) Heavy weight lift Fig. 10. High power motion experiment. (a) Joint position data from joint encoder and motor encoder are shown. In this experiment, the maximum observed torque of the ankle joint is 250 N m and the maximum observed mechanical power of the knee joint is 310W. (b) The robot lifts by 0.3m a 32.5 kg load during 0.4s. There is still a safety margin with respect to the limits equal to 5900N and 155◦C. The experimental results show that the drivetrain efficiency is approximately 0.89, which means that we lose only a small amount of power in the drivetrain and most of the torque from the motor is delivered to the joint. This high efficiency indicates only minor drivetrain friction, which is beneficial for dynamics-based motion controllers. D. High Power Motion Experiment To demonstrate high power motions such as fast vertical trajectories and heavy payload lifts, we use the motor position control mode, which uses the quadrature encoders attached directly to the motor for feedback. Fig. 10(a) presents the results of a test comprised of 2Hz vertical motion with 0.32 m of travel while carrying a load of 10 kg at the hip. With respect to mechanical power, the knee joint repeatedly exerts 305W, which is close to the predicted constant power (360W). Although the limited range of motion makes it hard to demonstrate continuous mechanical power, these results convincingly support our claim of enhanced continuous power enabled through liquid cooling. Fig. 10(b) presents another test in which the robot lifts a 32.5kg weight. We can see that the robot operates in the safe region (≤ 5900N and ≤ 155◦C) while demonstrating high power motion. 6(e). As we can see, when turning off liquid cooling the temperature rises quickly above safety limits whereas when turning on the cooling we can sustain large payload torques for long periods of time. The use of elastomers versus steel springs has demonstrated a clear improvement on joint position performance as shown in Fig. 6(d). This capability is important to achieve a large range of output joint or Cartesian space impedances. In the future we will explore further reducing the size of our viscoelastic liquid cooled actuators. Maintaining the current compact design structure we can still reduce another significant percentage the bulk of the actuator by exploring new types of bearings, ballnut sizes and piston bearings at the front end of the actuator. We will also explore using different material for the liquid cooling actuator jacket. The current polyoxymethylene material is easily breakable and develops cracks due to the vibrations and impacts of this kind of robotic applications. In the future we will switch to sealed metal chambers for instance. Further in the future we will consider designing our own motor stators and rotors for improved performance. We expect this kind of actuators to make their way into full humanoid robots and high performance exoskeleton devices and we look forward to participate in such interesting future studies. ACKNOWLEDGMENT VIII. C ONCLUDING R EMARKS Overall our main contribution has been on the design and extensive testing of a new viscoelastic liquid cooled actuator for robotics. One of the tests addressed is impedance control in the operational space instead of joint impedance control. It is often the case that humanoid robots require impedance control in the operational space. For instance, controlling the operational space impedance can enable improved locomotion behaviors such as running. Our controllers demonstrate that we can control the impedance in the Cartesian operational space as a potential functionality for future robotic systems. The use of liquid cooling has allowed to sustain high output torque for prolonged times as shown in the experiments of Fig. The authors would like to thank the members of the Human Centered Robotics Laboratory at The University of Texas at Austin for their help and support. This work was supported by the Office of Naval Research, ONR Grant [grant #N000141512507] and NASA Johnson Space Center, NSF/NASA NRI Grant [grant #NNX12AM03G]. R EFERENCES [1] G. A. Pratt and M. M. Williamson, “Series elastic actuators,” in Intelligent Robots and Systems 95. ’Human Robot Interaction and Cooperative Robots’, Proceedings. 1995 IEEE/RSJ International Conference on, 1995, pp. 399–406. [2] B. Henze, M. A. Roa, and C. Ott, “Passivity-based whole-body balancing for torque-controlled humanoid robots in multi-contact scenarios,” The International Journal of Robotics Research, p. 0278364916653815, Jul. 2016. 11 [3] N. Paine, J. S. Mehling, and J. 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1 Surprise Search for Evolutionary Divergence arXiv:1706.02556v1 [cs.NE] 8 Jun 2017 Daniele Gravina, Antonios Liapis, and Georgios N. Yannakakis Abstract—Inspired by the notion of surprise for unconventional discovery we introduce a general search algorithm we name surprise search as a new method of evolutionary divergent search. Surprise search is grounded in the divergent search paradigm and is fabricated within the principles of evolutionary search. The algorithm mimics the self-surprise cognitive process and equips evolutionary search with the ability to seek for solutions that deviate from the algorithm’s expected behaviour. The predictive model of expected solutions is based on historical trails of where the search has been and local information about the search space. Surprise search is tested extensively in a robot maze navigation task: experiments are held in four authored deceptive mazes and in 60 generated mazes and compared against objective-based evolutionary search and novelty search. The key findings of this study reveal that surprise search is advantageous compared to the other two search processes. In particular, it outperforms objective search and it is as efficient as novelty search in all tasks examined. Most importantly, surprise search is faster, on average, and more robust in solving the navigation problem compared to any other algorithm examined. Finally, our analysis reveals that surprise search explores the behavioural space more extensively and yields higher population diversity compared to novelty search. What distinguishes surprise search from other forms of divergent search, such as the search for novelty, is its ability to diverge not from earlier and seen solutions but rather from predicted and unseen points in the domain considered. Index Terms—Surprise search, novelty search, divergent search, deception, fitness-based evolution, maze navigation, NEAT. I. I NTRODUCTION VER the last 50 years, evolutionary computation (EC) has shown vast potential in numerical and behavioural optimization. The most common approach to optimization in artificial evolution is via an objective function, which rewards solutions based on their ‘goodness’ [1], i.e. how close they are to an optimal behaviour (if such a behaviour is known beforehand) or how much they improve a performance metric. The objective function (or fitness function) encapsulates the principle of evolutionary pressure for fitting (adapting) within the environment. Despite the success of such approaches in a multitude of tasks [1], [2], they are challenged in deceptive fitness landscapes [3] where the global optimum is neighboured by low-quality solutions. In such cases, the local search of an objective-based evolutionary algorithm can guide search away from a global optimum and towards local optima. As a general principle, more deceptive problems challenge the design of a corresponding objective function; this paper follows [4] and views deception as the intuitive definition of problem hardness. Many algorithms have been proposed to tackle the problem O All authors are with the Institute of Digital Games, University of Malta, Msida 2080, Malta (e-mail: daniele.gravina@um.edu.mt; antonios.liapis@um.edu.mt; georgios.yannakakis@um.edu.mt) of deception, primarily revolving around diversity preservation [5]–[7], which deters premature convergence while still rewarding proximity to the objective and divergent search [8] which abandons objectives in favour of rewarding diversity in the population. There are, however, problems which lack an easily defined objective — or a gradient to reaching it. For instance, openended evolution studies within artificial life [9] do not have a goal state and instead prioritize e.g. survival [10], [11]. In evolutionary art, music or design, a large body of research in computational creativity and generative systems [12]–[14] focuses on the creative capacity of search rather than on the objectives. In [13], computational creativity is considered on two core properties of a produced solution: value and novelty [13]. Value is the degree to which a solution is of high quality, whereas novelty is the degree to which a solution (or output) is dissimilar to existing examples. While objective-based EC can be seen as a metaphor of searching for value, a divergent EC strategy as novelty search [8], [15] can be considered as a metaphor of searching for novelty. An effective use of both in EC can lead to highly novel and valuable at the same time outcomes [16], thus realizing quality diversity [17]. However, according to [18], novelty and value are not sufficient for the discovery of highly creative and unconventional solutions to problems. While novelty can be considered as a static property, surprise considers the temporal properties of the discovery, an important dimension to assess the creativity of the generated solution [18], [19]. Further studies in general intelligence and decision making [20] support the importance of unexpectedness for problem-solving. Driven by the notion of computational surprise for the purpose of creatively traversing the search space towards unexpected or serendipitous solutions, this paper proposes surprise search for the purposes of divergent evolutionary search. The hypothesis is that searching for unexpected — not merely unseen — solutions is beneficial to EC as it complements our search capacities with highly efficient and robust algorithms beyond the search for objectives or mere novelty. Surprise search is built upon the novelty search [8] paradigm that rewards individuals which differ from other solutions in the same population and a historical archive. Surprise is assumed to arise from a violation of expectations [21]: as such, it is different than novelty which rewards deviation from past and current behaviours. A computational, quantifiable model of surprise must build expectations based on trends in past behaviours, and predict future behaviours from which it must diverge from. In order to create expected behaviours, the algorithm maintains a lineage of where evolutionary search has been. These groups of evolutionary lineages require the right level of locality in the behavioural space — surprise can be inclusive of all behaviours (globally) or merely consider part 2 of all possible behaviours (locally). Any deviation from these stepping stones of search would elicit surprise; alternatively, they can be viewed as serendipitous discovery if the deviation leads to a surprisingly good point in the behavioural space. The findings of this paper suggest that surprise constitutes a powerful drive for computational discovery as it incorporates predictions of an expected behaviour that it attempts to deviate from; these predictions may be based on behavioural relationships in the solution space as well as historical trends derived from the algorithm’s sampling of the domain. This paper builds upon and extends significantly our earlier work which introduced the notion of surprise search [22] and compared its performance against novelty search and objective-based search [23] in the maze navigation testbed of [8]. The current paper extends the preliminary study of [23] by introducing two new maze navigation problems of increased complexity, and analysing the impact of several parameters to the behaviour of surprise search. This paper also includes an extensive comparison between novelty search and surprise search both in the behavioural and the genotypic space. Finally, to further examine how surprise search performs across a wide range of problems, we test how it scales in sixty randomly generated mazes of varying degrees of complexity. The key findings of the paper suggest that surprise search is as efficient as novelty search and both algorithms, unsurprisingly, outperform fitness-based search. Furthermore, surprise search appears to be the most robust algorithm in the four testbed tasks and to be the most successful and fastest algorithm in the 60 randomly generated mazes. While both novelty and surprise search converge to the solution significantly faster than fitnessbased search, surprise search solves the navigation problem faster, on average, and more often than novelty search. The experiments of this paper validate our hypothesis that surprise can be beneficial as a divergent search approach and provide evidence for its supremacy over novelty search in the tasks examined. II. D ECEPTION , D IVERGENT S EARCH AND Q UALITY D IVERSITY This section motivates surprise search by providing a brief overview of the challenges faced by fitness-based approaches when handling deceptive problems and how divergent search has been used to address such challenges. The section concludes with a discussion on the relationship between surprise search and quality diversity algorithms. A. Deception in Evolutionary Computation The term deception in the context of EC was introduced by [3] to describe instances where highly-fit building blocks, when recombined, may guide search away from the global optimum. Since that first mention, the notion of deception (including deceptive problems and deceptive search spaces) has been refined and expanded to describe several problems that challenge evolutionary search for a solution. [4] argues that “the only challenging problems are deceptive”. EC-hardness is often attributed to deception, as well as sampling error [24] and a rugged fitness landscape [25]. In combinatorial optimisation problems, the fitness landscape can affect optimisation when performing local search. Such a search process assumes that there is a high correlation between the fitness of neighbouring points in the search space, and that genes in the chromosome are independent of each other. The latter assumption refers to epistasis [26] which is a factor of GA-hardness: when epistasis is high (i.e. where too many genes are dependent on other genes in the chromosome), the algorithm searches for a unique optimal combination of genes but no substantial fitness improvements are noticed [26]. As noted, epistasis is evaluated from the perspective of the fitness function and thus is susceptible to deception; [27] argue that deceptive functions can not have low epistasis, although fitness functions with high epistasis are not necessarily deceptive. Such approaches are often based on the concepts of correlation, i.e. the degree to which an individual’s fitness score is well correlated to its neighbours’ in the search space, or epistasis, i.e. the degree of interaction among genes’ effects. As noted by [8], most of the factors of EC-hardness originate from the fitness function itself; however, poorly designed genetic operators and poorly chosen evolutionary parameters can exacerbate the problem. B. Divergent Search By definition, the biggest issue of a deceptive problem is premature convergence to local optima, while the global optimum is difficult to reach as the deceptive fitness landscape lead the search away from it. Several approaches have been proposed to counter this behaviour, surveyed by [15]. For instance, speciation [28] and niching [29] are popular diversity maintenance techniques, which enforce local competition among similar solutions. Similarity can be measured on the genotypical level [5], on the fitness scores [6], or on the age of the individuals [7]. Multiple promising directions are favoured by this localised competition, making premature convergence less likely. Alternatives such as coevolution [30] can lead to an arms race that finds a better gradient for search, but can suffer when individuals’ performance is poor or one individual vastly outperforms others [31]. Finally, techniques from multiobjective optimisation can, at least in theory, explore the search space more effectively by evaluating individuals in more than one measures of quality [32] and thus avoid local optima in one fitness function by attempting to improve other objectives; however, multi-objective optimisation can not guarantee to circumvent deception [33]. Divergent search methods differ from previous approaches as they explicitly ignore the objective of the problem they are trying to solve. While the approaches described above provide control mechanisms, modifiers or alternate objectives which complement the gradient search towards a better solution, divergent algorithms such as novelty search [8] motivates exploration of the search space by rewarding individuals that are phenotypically (or behaviourally) different without considering whether they are objectively “better” than others. Novelty search is neither random walk nor exhaustive search, however, as it gives higher rewards to solutions that are different from others in the current and past populations 3 by maintaining a memory of the areas of the search space it has explored via a novelty archive. The archive contains previously found novel individuals, and the highest-scoring individuals in terms of novelty are continuously added to it as evolution carries on. The distance measure which assesses “difference” is based on a behaviour characterization, which is problem-dependent: for a maze navigation task, distance may be calculated on the agents’ final positions or directions [8], [17], for robot locomotion it may be on the position of a robot’s centre of mass [8], for evolutionary art it may be on properties of the images such as brightness and symmetry [34]. C. Quality Diversity A recent trend in evolutionary computation is inspired by a species’ tendency to face a strong competition for survival within its own niche [17]. EC algorithms of this type seek for the discovery of both quality and diversity at the same time, following the traditional approach within computational creativity of seeking outcomes characterized by both quality (or value) and novelty [13]. Such evolutionary algorithms have been named quality diversity algorithms [17] and aim to find a maximally diverse population of highly performing individuals. Examples of such algorithms include novelty search with local competition [35] and MAP-Elites [36] as well as algorithms that constrain the feasible space of solutions — thereby forcing high quality solutions — while searching for divergence such as constrained novelty search [16], [37]. Surprise search can complement the search for diversity and replace other divergent search algorithms commonly used for quality diversity. It can, for instance, be used in combination with local competition instead of novelty search as in [35]. Towards that goal, a recent study employed surprise search for game weapon generation in a constrained fashion [38]. The constrained surprise search algorithm rewarded the generation of surprising weapons — thereby maintaining diversity — with guaranteed high quality imposed by constraints on weapon balance and effectiveness. III. T HE N OTION OF S URPRISE S EARCH This section examines the notion of surprise as a driver of evolutionary search. To this end, we first describe the concept of surprise, we then highlight the differences between surprise and novelty and, finally, we frame surprise search in the context of divergent search. A. What is Surprise? The study of surprise has been central in neuroscience, psychology, cognitive science, and to a lesser degree in computational creativity and computational search. In psychology and emotive modelling studies, surprise defines one of Ekman’s six basic emotions [39]. Within cognitive science, surprise has been defined as a temporal-based cognitive process of the unexpected [21], [40], a violation of a belief [41], a reaction to a mismatch [21], or a response to novelty [14]. In computational creativity, surprise has been attributed to the creative output of a computational process [14], [18]. While variant types and taxonomies of surprise have been suggested in the literature — such as aggressive versus passive surprise [18] — we can safely derive a definition of surprise that is general across all disciplines that study surprise as a phenomenon. For the purposes of this paper we define surprise as the deviation from the expected and we use the notions surprise and unexpectedness interchangeably due to their highly interwoven nature. B. Novelty vs. Surprise Novelty and surprise are different notions by definition as it is possible for a solution to be both novel and/or expected to variant degrees. Following the core principles of Lehman and Stanley [8] and Grace et al. [18], novelty is defined as the degree to which a solution is different from prior solutions to a particular problem. On the other hand, surprise is the degree to which a solution is different from the expected solution to a particular problem. Expectations are naturally based on inference from past experiences; analogously surprise is built on the temporal model of past behaviours. To exemplify the difference between the notions of novelty and surprise, [22] uses a card memory game where cards are revealed, one at a time, to the player who has to predict which card will be revealed next. The novelty of game outcome (i.e. next card) is the highest if all past revealed cards are different. The surprise value of the game outcome in that case is low as the player has grown to expect a new, unseen, card every time. On the other hand, if seen cards are revealed after a while then the novelty of the game outcome decreases, but surprise increases as the game deviates from the expected behaviour which calls for a new card every time. Surprise is a temporal notion as expectations are by nature temporal. Prior information is required to predict what is expected; hence a prediction of the expected [19] is a necessary component for modelling surprise computationally. By that logic, surprise can be viewed as a temporal novelty process or as novelty on the prediction (rather than the behavioural) space. Surprise search maintains a prediction (a gradient) of where novelty has been on the prediction space, which is derived from the behavioural space. In that sense surprise resembles a time derivative of novelty. C. Novelty Search vs. Surprise Search According to Grace et al. [18], novelty and value (i.e. objective in the context of EC) are not sufficient for the discovery of unconventional solutions to problems (or creative outputs) as novelty does not cater for the temporal aspects of discovery. Novelty search rewards divergence from current behaviours (i.e. other individuals of the population) and prior behaviours (i.e. an archive of previously novel solutions) [8]; in this way it provides the necessary stepping stones toward achieving an objective (i.e. value). Surprise, on the other hand, complements the search for novelty as it rewards divergence from the expected behaviour. In other words while novelty search attempts to deviate from previously seen solutions, surprise search attempts to deviate from solutions that are expected to be seen in the future. 4 As discussed above, surprise search must reward an individual’s deviation from the expected behaviour. This goal can be decomposed into two tasks: prediction and deviation. At the highest descriptive level, surprise search uses local information from past generations to predict behaviour(s) of the population in the current generation; observing the behaviours of each individual in the actual population, it rewards individuals that deviate from predicted behaviour(s): this is summarized in Fig. 1. The following sections will describe the models of prediction and deviation, and the parameters which can affect their behaviour. Fig. 1: High-level overview of the surprise search algorithm when evaluating an individual i in a population at generation t. The h previous generations are considered, with respect to k behavioural characteristics per generation, to predict the expected k behaviours of generation t. The surprise score of individual i is the deviation of the behaviour of i from a subset of these k expected behaviours. Highly relevant to this study is the work on computational models of surprise for artificial agents [42], which however does not consider using such a model of surprise for search. Other aspects of unexpectedness such as intrinsic motivation [43] and artificial curiosity [44] have also been modelled. The concepts of novelty within reinforcement learning research are also interlinked to the idea of surprise search [43], [45]. Artificial curiosity and intrinsic motivation differ from surprise search as the latter is based on evolutionary divergent search and motivated by open-ended evolution, similarly to novelty search. Specifically, surprise search does not keep a persistent world model as [44] does; instead it focuses on the current trajectory of search using the latest points of the search space it has explored (and ordering them temporally). Additionally, it rewards deviations from expected behaviours agnostically rather than based on how those deviations improve a world model. This allows surprise search to backtrack and re-visit areas of the search space it has already visited, which is discouraged in both novelty search and curiosity. Inspired by the above arguments and findings in computational creativity, we view surprise for computational search as the degree to which expectations about a solution are violated through observation [18]. Our hypothesis is that if modelled appropriately, surprise may enhance divergent search and complement or even surpass the performance of traditional forms of divergent search such as novelty. The main findings of this paper validate our hypothesis. IV. T HE S URPRISE S EARCH A LGORITHM This section discusses the principles of designing a surprise search algorithm for any task or search space. To realise surprise as a search mechanism, an individual should be rewarded when it deviates from the expected behaviour, i.e. the evaluation of a population in evolutionary search is adapted. This means that surprise search can be applied to any EC method, such as NEAT [28] in the case study of this paper. A. Model of Prediction As shown in Fig. 1, the predictive model uses local information from previous generations to estimate (in a quantitative way) the expected behaviour(s) in the current population. Formally, predicted behaviours (p) are created via eq. (1), where m is the predictive model that uses a degree of local (or global) behavioural information (expressed by k) from h previous generations. Each parameter (m, h, k) may influence the scope and impact of predictions, and are problem-specific both from a theoretical (as they can affect performance of surprise search) and a practical (as certain domains may limit the possible choice of parameters) perspective. p = m(h, k) (1) How much history of prior behaviours (h) should surprise search consider?: In order to predict behaviours in the current population, the predictive model must consider previous generations. In order to estimate behaviours of a population at generation t, the predictive model must find trends in the populations of generations t − 1, t − 2, · · · , t − h. The minimum number of generations to consider in order to observe an evolutionary trend, therefore, is h = 2 (the two prior generations to the one being evaluated). However, behaviours that have performed well in the past could also be included in a surprise archive, similar to the novelty archive of novelty search [8], and subsequently used to make predictions of interesting future behaviours. Such a surprise archive would serve as a more persistent history (h > 2) but considering only the interesting historical behaviours rather than all past behaviours. How local (k) are the behaviours surprise search needs to consider to make a prediction?: Surprise search can consider behavioural trends of the entire population when creating a prediction (global information). In that case, k = 1 and all behaviours are aggregated into a meaningful average metric for each prior generation. The current generation’s expected behaviours are similarly expressed as a single (average) metric; deviation of individuals in the actual population is derived from that single metric. At the other extreme, surprise search can consider each individual in the population and derive an estimated behaviour based on the behaviours of its ancestors in the genotypic sense (parents, grandparents etc.) or behavioural sense (past individuals with the closest behaviour). In this case k = P where P is the size of the population, and the number of predictions to deviate from will similarly be P . Therefore, 5 the parameter k determines the level of prediction locality which can vary from 1 to P ; intermediate values of k split prior populations into a number of population groups using problem-specific criteria and clustering methods. What predictive model (m) should surprise search use?: Any predictive modelling approach can be used to predict a future behaviour, such as a simple linear regression of a number of points in the behavioural space, non-linear extrapolations, or machine learned models. Again, we consider the predictive model, m, to be problem-dependent and contingent on the h and k parameters. B. Model of Deviation To put pressure on unexpected behaviours, we need an estimate of the deviation of an observed behaviour from the expected behaviour (if k = 1) or behaviours. Following the principles of novelty search [8], this estimate is derived from the behaviour space as the average distance to the n-nearest expected behaviours (prediction points). The surprise score s for an individual i in the population is calculated as: n s(i) = 1X ds (i, pi,j ) n j=0 (2) where ds is the domain-dependent measure of behavioural difference between an individual and its expected behaviour, pi,j is the j-closest prediction point (expected behaviour) to individual i and n is the number of prediction points considered; n is a problem-dependent parameter determined empirically (n≤k). C. Important notes The evolutionary dynamics of surprise search are similar to those achieved via novelty search. A temporal window of where the search has been is maintained by looking at the prior behaviours (expressed by h and k), which are used to make a prediction of the expected behaviours. However, surprise search works on a different search space (the prediction space) which is orthogonal to the behavioural space used by novelty, as it deviates from the expected and not from the actual behaviours. Therefore, a new form of divergent search emerges, which looks at previous behaviours only in an implicit way in order to derive the prediction space. A concern could be if surprise search is merely a version of random walk, especially considering that it deviates from predictions which can differ from actual behaviours. Several comparative experiments in section VII show that surprise search is different and far more effective compared to various random benchmark algorithms. The source code of the surprise search algorithm is publicly available here1 . V. M AZE NAVIGATION T EST B ED Inspired by the work of Lehman and Stanley for testing novelty search [8], we use a maze navigation problem as a testbed for surprise search, as it particularly suits the definition 1 http://www.autogamedesign.eu/mazesurprisesearch (a) Neural Network (b) Sensors Fig. 2: Robot controller for the maze navigation task. Fig. 2a shows the network’s inputs and outputs. Fig. 2b shows the layout of the sensors: the six black arrows are rangefinder sensors, and the four blue pie-slice sensors act as a compass towards the goal. of deceptive problem. The maze navigation task is made of a closed two-dimensional maze, which contains a start position and a goal position: the navigation task has a deceptive landscape—which is directly visible to a human observer— due to the several local optima present in the search space of the problem. Cul-de-sacs added in the shortest path to the goal makes the problem more deceptive, as EC must visiting positions with lower fitness scores before reaching the goal, making the problem harder and more deceptive. In this case, navigation is performed by virtual robot controllers with sensors and mechanisms for controlling their direction and speed: the mapping between the two is provided is via an artificial neural network (ANN) evolved via neuroevolution of augmenting topologies [28]. Starting from the mazes introduced in [8], we designed two additional mazes of enhanced complexity and deceptiveness. This section briefly describes the maze navigation problem, the four mazes adopted, and the parameters for the experiment of Section VII. A. The Maze Navigation Task The maze navigation task consists of finding the path from a starting point to a goal in a two-dimensional maze, in a fixed number of simulation steps. The problem becomes harder when mazes include dead-ends and the goal is far away from the starting point. As in [8], the robot has six range sensors to measure its distance from the closest obstacle, plus four range radars that fire if the goal is in their arc (see Fig. 2b). Therefore, the robot’s ANN receives 10 inputs from the sensors and it controls two actuators, i.e. whether to turn or change the speed (see Fig. 2a). Evolving a controller able to successfully navigate a maze is a challenging problem, as EC needs to evolve a complex mapping between the input (sensors) and the output (movement) in an unknown environment. Even if it can be considered a toy problem, it is an interesting testbed as it stands for a general deceptive search space. Two properties have made this environment a canonical test for divergent search (e.g. [8], [17]): the ease of manually designing deceptive mazes and the low computational burden, which enables researchers to run multiple comparative tests among 6 (a) Medium (b) Hard (c) Very hard (d) Extremely hard Fig. 3: The maze testbeds that appear in [8] (Fig. 3a and 3b) and new mazes introduced in [46] and this paper (Fig. 3c and 3d respectively). The filled circle is the robot’s starting position and the empty circle is the goal. The maze size is 300×150 units for medium and 200×200 for the other mazes. algorithms. Furthermore, the generality of the findings can be tested with automatically generated mazes, as in Section VIII. B. Mazes This paper tests the performance of surprise search on four mazes (see Fig. 3), two of which (medium and hard) have been used in [8]. The medium maze (see Fig. 3a) is somewhat challenging as an algorithm should evolve a robot that avoids dead-ends placed alongside the path to the goal. The hard maze (see Fig. 3b) is more deceptive, due to the dead-end at the leftmost part of the maze; an algorithm must search in less promising (less fit) areas of the maze to find the global optimum. For these two mazes we follow the experimental parameters set in [8] and consider a robot successful if it manages to reach the goal within a radius of five units at the end of an evaluation of 400 simulation steps. Beyond the two mazes of [8], two additional mazes (very hard and extremely hard) were designed to test an algorithm’s performance in even more deceptive environments. The very hard maze (see Fig. 3c) is a modification of the hard maze introduced in [46] with more dead ends and winding passages. The extremely hard maze is a new maze (see Fig. 3d) that features a longer and more complex path from start to goal, thereby increasing the deceptive nature of the problem. If we define a maze’s complexity as the shortest path between the start and the goal, complexity increases substantially from medium (240 units), to hard (360 units), to very hard (442 units) and finally to the extremely hard maze (552 units); note that the last three mazes are of equal size. The high problem complexity of the very hard and the extremely hard mazes led us to empirically increase the number of simulation steps for the evaluation of a robot to 500 and 1000 simulation steps, respectively. By increasing the simulation time in the more deceptive mazes we manage to achieve reasonable performances for at least one algorithm examined which allows for a better analysis and comparison. VI. A LGORITHM PARAMETERS FOR M AZE NAVIGATION This section provides details about the general and specific parameters for all the algorithms compared. We primarily test the performance of three algorithms: objective search, novelty search and surprise search, and include three baseline algorithms for comparative purposes. All algorithms use NEAT to evolve a robot controller with the same parameters as in [8], where the maze navigation task and the mazes of Fig. 3 were introduced. Evolution is carried on a population of 250 individuals for a maximum of 300 generations in the medium and hard maze for a fair comparison to results obtained in [8]. However, the number of generations is increased to 1000 for the more deceptive mazes (very hard and extremely hard) to allow us to analyse the algorithms’ behaviour over a longer evolutionary period. The NEAT algorithm uses speciation and recombination, as described in [28]. The specific parameters of all compared algorithms are detailed below. A. Objective search Objective search uses the agent’s proximity to the goal as a measure of its fitness. Following [8], proximity is measured as the Euclidean distance between the goal and the position of the robot at the end of the simulation. This distance does not account for the maze’s topology and walls, and can be deceptive in the presence of dead-ends. B. Novelty Search Novelty search uses the same novelty metric and parameter values as presented in [8]. In particular, the novelty metric is the average distance of the robot from the nearest neighbouring robots among those in the current population and in a novelty archive. Distance in this case is the Euclidean distance between two robot positions at the end of the simulation; this rewards robots ending in positions that no other robot has explored yet. The parameter for the novelty archive (e.g. the initial novelty threshold for inserting individuals to the archive is 6) is as given in [8]. Sensitivity Analysis: While in [8] novelty is calculated as the average distance from the 15 nearest neighbours, the introduction of new mazes in this paper mandates that the n parameter of novelty search is tested empirically. For that purpose we vary n from 5 to 30 in increments of 5 across all mazes and select the n values that yield the highest number of maze solutions (successes) in 50 independent runs of 300 generations for the medium and hard maze, and 1000 generations for the other mazes. If there is more than one n value that yields the highest number of successes then the lowest average evaluations to solve the maze is taken into account as a selection criterion. Figure 5a shows the results obtained by this analysis across all mazes. The best results are indeed obtained with 15 nearest neighbours for the medium and hard maze, as in [8] (49 and 48 successes, respectively). In the very hard maze there is no difference between 10 and 15 in terms of successes (39) but n = 15 yields less evaluations, while in the extremely hard maze n = 10 yields less evaluations and more successes (24) than any other value tested. In summary, reported results in Section VII use n = 15 for the medium, hard and very hard maze, and n = 10 for the extremely hard maze. C. Surprise search Surprise search uses the surprise metric of eq. (2) to reward unexpected behaviours. As with the other algorithms compared, behaviour in the maze navigation domain is expressed Normalized Evaluations 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 5 10 15 20 25 30 n (a) Generation t − 2 (b) Generation t − 1 Normalized Evaluations 7 0.8 0.6 0.4 0.2 20 40 60 80 100 120 140 160 180 200 220 240 1.0 0.8 0.6 0.4 0.2 20 40 60 80 100 120 140 160 180 200 220 240 k (c) Generation t Fig. 4: The key phases of the surprise search algorithm as applied to the maze navigation domain. Surprise search uses a history of two generations (h = 2) and 10 behavioural clusters (k = 10) in this example. Cluster centroids and prediction points are depicted as empty red (light gray in grayscale) and solid blue (dark gray in grayscale) circles, respectively. as the position of the robot at the end of a simulation. The behavioural difference ds in eq. (2) is the Euclidean distance between the robots’ final position and a considered prediction point, p. Following the general formulation of surprise in Section IV-A, the prediction points are a function of a model m that considers k local behaviours of h prior generations. In this comparative study we use the simplest possible prediction model (m) which is a one-step linear regression of two points (h = 2) in the behavioural space. Thus, only the two previous generations are considered when creating prediction points to deviate from in the current generation (see Fig. 4). In the first two generations the algorithm performs mere random search due to a lack of prediction points. The locality (k) of behaviours is determined by the number of behavioural clusters in the population that is obtained by running k-means on the final robot positions. The surprise search algorithm applies k-means clustering at each generation by seeding the initial configuration of the k centroids with the centroids obtained in the previous generation; this seeding process is omitted only in the first generation due to the lack of earlier clusters. This way the algorithm is able to pair centroids in subsequent generations and track their behavioural history. Using the k pairs of centroids of the last two generations, the algorithm creates k prediction points for the current generation through a simple linear projection. Surprise search rewards the behaviour that obtains a surprising outcome given the temporal sequence of the final points of the robot across generations. Surprise search is thus orthogonal to objective and novelty search as it rewards robots that visit areas outside the predicted space(s), without any explicit knowledge of the final goal. It should be noted that for high values of k the k-means algorithm might end up not assigning any data point to a particular cluster; the chance of this happening increases with k and the sparseness of data (in particular in datasets containing outliers) [47]. Further, the seeding initialization procedure we follow for k-means in this domain aims to behaviourally connect centroids across generations so as to enable us to predict and deviate from the expected behaviour in the next generation. The adopted initialization procedure (i.e. inheriting from centroids of the previous generation) does not guarantee (a) Novelty search parameters (b) Surprise search parameters Fig. 5: Sensitivity Analysis: selecting n for novelty search (Fig. 5b), k and n for surprise search (Fig. 5a). Figures depict the average number of evaluations (normalized by the total number of evaluations) obtained out of 50 runs . Error bars represent the 95% confidence interval. that all seeded centroids will be allocated a robot position during the assignment step of k-means, as robot positions (data points) might change drastically from one generation to the next. In the case of surprise search, when an empty cluster appears in the current generation (i.e. in positions where a cluster existed in a past generation but not currently), then its prediction is not updated (i.e. moved) until a final robot position gets close to the empty cluster’s centroid. Predicted centroids that have not been recently updated (due to empty clusters) are still considered when calculating the surprise score, and indirectly act as an archive of earlier predictions. However, this archive is not persistent as the number of ‘archived’ prediction points can increase or decrease during the course of evolution, and depends on k. Sensitivity Analysis: To choose appropriate parameters for k (information locality) and n (number of prediction points) in the prediction and deviation models respectively, a sensitivity analysis is conducted for all mazes. We obtain k empirically by varying its value between 20 and P in increments of 20 for each maze. We also test all k for n = 1 and n = 2 in this paper. As in the sensitivity analysis for novelty search we select the k and n values that yield the highest number of successes in 50 independent runs. If there is more than one k, n combination that yields the highest number of successes we select the combination that solves the maze in the fewest average evaluations. Figure 5b shows the average number of evaluations for all k values tested, for n = 1 and n = 2. It is clear that higher k values result to less evaluations on average. Moreover, it seems that n = 2 leads to better performance in the two more deceptive mazes. Based on the above selection criteria, we pick k = 200 and n = 1 for the medium maze, which gives the highest number of successes (50) and the lowest number of evaluations (16, 364 evaluations on average). For the hard maze we select k = 100 and n = 1, as it is the most robust (49 success) and fastest (23, 214 evaluations on average) among tested values. Following the same procedure k = 200 and n = 2 in the very hard maze, and k = 220 and n = 2 in the extremely hard maze (see Fig. 5b). 300 280 260 240 220 200 1800 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 3 Average Maximum Fitness In this paper we started our investigations with the simplest possible prediction model (m), which is a linear regression, and the shortest possible time window of two generations for the history parameter (h). The impact of the history parameter and the prediction model on the algorithm’s performance is not examined empirically in this paper and remains open to future investigations. We get back to this discussion in Section IX. Average Maximum Fitness 8 The robot maze navigation problem is used to compare the performance of surprise, novelty and objective search. To test the algorithms’ performance, we follow the approach proposed in [48] and compare their efficiency and robustness in all four test bed mazes. We finally analyse some typical examples on both the behavioural and the genotypical space of the generated solutions. All results reported are obtained from 100 independent evolutionary runs; reported significance and corresponding p values are obtained via two-tailed MannWhitney U-test, with a significance level of 5%. A. Efficiency Efficiency is defined as the maximum fitness over time, where fitness is calculated as 300 − d(i); d(i) the Euclidean distance between the final position of robot i and the goal, as in [8]. Figure 6 shows the average maximum fitness across evaluations for each approach for the four mazes. In the medium maze, we can observe that both surprise and novelty search converge after approximately 35,000 evaluations. Even if novelty seems to yield a higher average maximum fitness values than surprise search, the difference is insignificant. Novelty search, on average, obtains a final maximum fitness of 295.84 (σ = 1.47), while surprise search obtains a fitness of 295.93 (σ = 0.72); p > 0.05. By looking at the 95% confidence intervals, it seems that novelty search yields higher average maximum fitness between 240 220 200 1800 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 3 Average Maximum Fitness 300 280 260 240 220 200 1800 25 50 75 100 125 150 175 200 225 250 Evaluations (x103 ) (c) Very hard maze (b) Hard maze Average Maximum Fitness (a) Medium maze D. Other baseline algorithms VII. S URPRISE S EARCH IN AUTHORED D ECEPTIVE M AZES 260 Evaluations (x10 ) Evaluations (x10 ) Three more baseline algorithms are included for comparative purposes. Random search is a baseline proposed in [8] that uses a uniformly-distributed random value as the fitness function of an individual. The other two baselines are variants of surprise search that test the impact of the predictive model. Surprise search (random), SSr , selects k random prediction points (pi,j in eq. 2) within the maze following a uniform distribution, and tests how surprise search would perform with a highly inaccurate predictive model. Surprise search (no prediction), SSnp , uses the current generation’s actual clusters as its prediction points (pi,j in eq. 2), thereby, omitting the prediction phase of the surprise search algorithm. SSnp uses real data (cluster centroids) from the current generation rather than predicted data regarding the current generation, and tests how the algorithm performs divergent search from real data. Note that SSnp is reminiscent of novelty search, except that it uses deviation from cluster centroids (not points) and does not use a novelty archive. The same parameter values (k and n) are used for these variants of surprise search. 300 280 300 280 260 240 220 200 1800 25 50 75 100 125 150 175 200 225 250 Evaluations (x103 ) (d) Extremely hard maze Fig. 6: Efficiency (average maximum fitness) comparison for the four mazes in Fig. 3. The graphs depict the evolution of fitness over the number of evaluations. Values are averaged across 100 runs of each algorithm and the error bars represent the 95% confidence interval of the average. 7500 and 25,000 evaluations. This difference is due to the predictions that surprise search tries to deviate from. Early during evolution, two consecutive generations may have robots far from each other, lead to distant and erratic predictions. Eventually, we have a convergence and the predictions become more consistent, allowing surprise search to solve the maze. Both objective search and SSnp seem fairly efficient to solve the maze; however, they are not able to find the goal in all the runs. The random baselines, instead, perform poorly and show very little improvement as evolution progresses. The baselines’ performance proves that surprise search is different from a random walk and that the prediction model positively affects the performance of the algorithm. In a more deceptive test, the hard maze, we can see from Fig. 6b that novelty and surprise perform much better than all other algorithms; differences in efficiency between novelty and surprise search are not significant. Surprise search and novelty search find the goal in 99 and 93 out of 100 runs respectively, SSnp finds the solution in 61 runs, while for the rest of the baselines the success rate is far below. It is interesting to note that objective search reaches a high fitness score around 260 at the very beginning of the evolutionary process, and then it stops to improve. This is due to the dead-end at the upper right corner of the maze (Fig. 3b), which prevents the algorithm from discovering the global optimum. In order to discover the global optimum, in fact, the algorithm needs to explore the least fit areas of the search space, such as the bottom-right corner of the maze. In the very hard maze, objective search never finds the 100 80 80 Successes 100 60 40 20 60 40 20 00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 3 00 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 3 Evaluations (x10 ) Evaluations (x10 ) (a) Medium maze (b) Hard maze 100 100 80 80 Successes Successes solution in 100 runs; Fig. 6c shows that this algorithm is not able to reach the goal because, as in the hard maze, it reaches the left-most dead end and is unable to bypass that local optimum. Unsurprisingly, the random and SSR baselines also perform poorly. On the other hand, novelty search finds the solution in 85 out of 100 runs, while surprise search finds a solution in 99 runs. Interestingly, SSnp finds 88 solutions out of 100 runs, similar to novelty search. This can be explained by looking at how SSnp is implemented. Its behaviour is quite similar to novelty search as it merely uses the local behaviours of the current generation; the key difference is that these behaviours are clustered in SSnp . In such a complex maze surprise search seems to handle maze deceptiveness in a better way; it obtains a final maximum fitness of 295.77 (σ = 3.59) which is higher (but not significantly) than that of novelty search (292.15; σ = 9.85); p > 0.05. From the confidence intervals of Fig. 7c, it appears that surprise search is performing better or significantly better than novelty search after 50, 000 evaluations until the end of the run. In the most deceptive (extremely hard) maze, objective search, random and SSr do not find any solution in 100 runs, performing poorly in terms of efficiency. This is not surprising as all of these algorithms perform consistently poorly in all but the simplest mazes. Novelty search and surprise search find the solution 48 and 67 times, respectively, while the SSnp obtains 39 solutions. As can be seen in Fig. 6d, surprise search yields a higher maximum fitness after 150,000 evaluations, with a final maximum fitness of 287.68 (σ = 12.16). Another way of estimating efficiency is the effort it takes an algorithm to find a solution. In this case, surprise clearly manages to be more advantageous. In the medium maze surprise search manages to find the goal, on average, in 16, 084 evaluations (σ = 11, 588) which is faster than novelty (19, 814; σ = 15, 441) and significantly faster than objective search (48, 186; σ = 23, 590) and SSnp (26, 452; σ = 21, 249). We observe the same comparative advantage in the hard maze as surprise search solves the problem in 23, 566 evaluations on average (σ = 15, 925) whereas novelty search, SSnp , and objective search solve it in 28, 493 (σ = 19, 939), 47, 550 (σ = 25, 524) and 73, 643 (σ = 7, 542) evaluations, respectively. Most importantly surprise search is significantly faster (p < 0.05) than novelty search in the more deceptive problems: on average surprise search finds the solution in 76, 261 evaluations (σ = 52, 385) in the very hard maze and in 154, 794 evaluations (σ = 84, 733) in the extremely hard maze, whereas novelty search requires 115, 600 evaluations (σ = 81, 091) and 178, 045 evaluations (σ = 86, 410), respectively. Furthermore surprise search is significantly faster (p < 0.01) than SSnp , which requires 117, 560 (σ = 74, 430) and 200, 190 (σ = 75, 569) evaluations in the very hard and the extremely hard maze, respectively. The findings from the above experiments indicate that, in terms of maximum fitness obtained, surprise search is comparable to novelty search and far more efficient than objective search in deceptive domains. We can further argue that the deviation from the predictions (which are neither random nor omitted) is beneficial for surprise search as indicated by the performances of SSr and SSnp . The performance of this Successes 9 60 40 40 20 20 00 60 25 50 75 100 125 150 175 200 225 250 Evaluations (x103 ) (c) Very hard maze 00 25 50 75 100 125 150 175 200 225 250 Evaluations (x103 ) (d) Extremely hard maze Fig. 7: Robustness comparison for the four mazes in Fig. 3. The graphs depict the evolution of algorithm successes in solving the maze problem over the number of evaluations. baseline appears to be similar to novelty search, especially in harder mazes; this is not surprising as SSnp is conceptually similar to novelty search, as noted in Section VI-D. It is also clear that, on average, surprise search finds the solution faster than any other algorithm in all mazes. B. Robustness Robustness is defined as the number of successes obtained by the algorithm across time (i.e. evaluations). In Figure 7 we compare the robustness of each approach across the four mazes, collected from 100 runs. In the medium maze (Fig. 7a), surprise search is more successful than novelty search in the first 20,000 evaluations; moreover, surprise search finds, on average, the 100 solutions in fewer evaluations compared to the other approaches. As noticed in the previous section, in the first 20,000 evaluations novelty search has a comparable or higher efficiency in Fig. 6a; this points to the fact that while some individuals in surprise search manage to reach the goal, others do not get as close to it as in novelty search. On the other hand, objective search fails to find the goal in 29 runs, because of the several dead-ends present in this maze. The control algorithm SSnp finds the goal 93 times out of 100, but it’s slower compared to novelty and surprise search. Few solutions are found by the baseline random search and SSr , and they are significantly slower than the other approaches. Fig. 7b shows that, in the hard maze, novelty search attains more successes than surprise search in the first 10,000 evaluations but the opposite is true for the remainder of the evolutionary progress. As in the previous maze, this behaviour is not reflected in the efficiency graph (Fig. 6b): this 10 can be explained by how surprise search evolves individuals, as they change their distance to the goal more abruptly, while novelty search evolves behaviours in smooth incremental steps. On the other hand, SSnp finds fewer solutions in this maze, 62 out of 100. Finally, the deceptive properties of this maze are exemplified by the poor performance of objective search and the two random baselines. The capacity of surprise search is more evident in the very hard maze (see Fig. 7c) where the difference in terms of robustness becomes even larger between surprise and novelty search. While in the first 50,000 evaluations novelty and surprise search attain a comparable number of successes, the performance of surprise search is boosted for the remainder of the evolutionary run. Ultimately, surprise search solves the very hard maze in 99 out of 100 times in just 160,000 evaluations whereas novelty search manages to obtain 85 solutions by the end of the 250,000 evaluations. With 88 solutions out of 100, SSnp performs similarly to novelty search in this maze but is generally slower compared to surprise search. Objective search and the two random baselines, as expected, do not succeed in solving the maze. Similarly, in the extremely hard maze (see Fig. 7d) the benefits of surprise search over the other algorithms are quite apparent. While surprise and novelty obtain a similar number of successes in the first 100,000 evaluations, surprise search obtains more successes in the remaining evaluations of the run. At the end of 250,000 evaluations in the most deceptive map examined, surprise search finds solutions in 67 runs versus 48 runs of novelty search. SSnp finds 39 solutions and it is generally slower than novelty and surprise search. As in the very hard maze, the remaining algorithms fail to find a single solution to this maze. C. Analysis As an additional comparison between surprise and novelty search, we study the behavioural and genotypical characteristics of these two approaches. The behavioural space is presented in a number of typical runs collected from the four mazes, while the genotypical space is inspected through the metrics computed from the final ANNs evolved by these two algorithms. Objective search and the other baselines are not further analysed in this section to emphasise on comparisons between surprise and novelty search. 1) Behavioural Space: Typical Examples: Table I shows pairs of typical surprise and novelty search runs for each of the four mazes; in all examples illustrated the maze is solved at different number of evaluations as indicated at the captions of the images. The typical runs are shown as heatmaps which represent the aggregated distribution of the robots’ final positions throughout all evaluations. Moreover, we report the entropy (H) of those positions as a measure of the populations’ spatial diversity in the maze. Surprise search seems to explore more uniformly the space, as revealed by the final positions depicted in the heatmaps. The corresponding H values further support this claim, especially in the more deceptive mazes. 2) Genotypic Space: Table II contains a set of metrics that characterize the final ANNs evolved by surprise and novelty search obtained from all four mazes, which quantify aspects of genomic complexity and genomic diversity. For genomic complexity we consider the number of connections and the number of hidden nodes of the final ANNs evolved, while genomic diversity is measured as the average pairwise distance of the final ANNs evolved. This distance is computed with the compatibility metric, a linear combination of disjoint and excess genes and weight difference, as defined in [28]. As noted in [8], novelty search tends to evolve simpler networks in terms of connections when compared to objective search. Surprise search, on the other hand, seems to generate significantly more densely connected ANNs than novelty search (based on the number of connections). It also evolves slightly larger ANNs than novelty search (based on the number of hidden nodes). Most importantly, surprise search yields population diversity — as expressed by the compatibility metric [8] — that is significantly higher than novelty search. This difference seems to be mostly due to the disjoint factor, which counts the number of mismatching genes between two genomes, depending on whether their genes are within the innovation numbers of the other genome [28]. This suggests that ANNs evolved with surprise search are more diverse in terms of evolutionary history. In the more deceptive mazes, differences in genomic complexity and diversity become significantly larger. In the very hard maze the average number of connections for surprise search grows to 101.94 (σ = 52.45) while novelty search evolves ANNs with 42.03 connections (σ = 14.53), on average; the number of hidden nodes used by surprise search is significantly larger (6.94; σ = 3.62) compared to novelty search. Moreover the diversity metric (compatibility) is around three times that of novelty search. A similar trend can be noticed in the extremely hard maze, where again surprise search evolves denser, larger and more diverse ANNs. As mentioned earlier, handling more complex and larger ANNs has a direct impact on the computational cost of surprise search since it takes more time to simulate new networks across generations. It should be noted that creating larger networks does not imply that this behaviour is beneficial, it is however an indication that surprise search operates differently to novelty search. VIII. S URPRISE S EARCH IN G ENERATED M AZES In the previous sections we showed the power of surprise search in four selected instances of deceptive problems. While surprise search outperforms novelty and objective search both in terms of efficiency and robustness in four human-designed mazes, an important concern is whether these results are general enough across a broader set of problems. In order to assess how surprise search generalises in any maze navigation task, we follow the methodology presented in [49] and test the performance of surprise, novelty and objective search as well as the baselines across numerous mazes generated through an automated process. Moreover, the parameters of k and n which were fine-tuned for the problem at hand in each maze of Section VII are now kept the same, enabling us to observe if a particular parameter setup for surprise search can perform well in unseen problems of varying complexity. 11 TABLE I: Behavioural Space. Typical successful runs solved after a number of evaluations (E) across the four mazes examined. Heatmaps illustrate the aggregated numbers of final robot positions across all evaluations. Note that white space in the maze indicates that no robot visited that position. The entropy (H ∈ [0, 1]) of visited positions is also reported and is calculated as P follows: H = (1/logC) i {(vi /V )log(vi /V )}; where vi is the number of robot visits in a position i, V is the total number of visits and C is the total number of discretized positions (cells) considered in the maze. Medium Maze (E = 25, 000) Novelty Surprise H = 0.63 H = 0.67 Hard Maze (E = 25, 000) Novelty Surprise H = 0.61 H = 0.67 Very Hard Maze (E = 75, 000) Novelty Surprise H = 0.63 H = 0.69 Extremely Hard Maze (E = 75, 000) Novelty Surprise H = 0.64 H = 0.68 TABLE II: Genotypic Space. Metrics of genomic complexity and diversity of the final ANNs evolved using NEAT, averaged across successful runs. Values in parentheses denote standard deviations. Maze Medium Hard Very Hard Extremely Hard Algorithm Surprise Novelty Surprise Novelty Surprise Novelty Surprise Novelty Genomic Complexity Connections Hidden Nodes 33.76 (15.08) 2.46 (1.53) 29.08 (6.10) 2.2 (1.0) 52.34 (28.57) 3.84 (2.66) 32.55 (9.84) 2.48 (1.29) 101.94 (52.45) 6.94 (3.62) 42.03 (14.53) 3.27 (1.90) 158.16 (89.65) 10.63 (5.85) 41.79 (11.77) 3.25 (1.53) Compatibility 42.52 (19.80) 32.55 (7.90) 73.24 (36.82) 39.35 (12.05) 160.31 (67.01) 56.53 (19.66) 260.52 (113.08) 54.05 (14.44) Genomic Diversity Disjoint Weight Difference 27.40 (16.27) 1.22 (0.25) 24.97 (7.05) 1.09 (0.26) 51.60 (27.76) 1.25 (0.24) 31.06 (11.32) 1.19 (0.28) 121.56 (57.11) 1.26 (0.22) 46.65 (19.25) 1.16 (0.27) 207.97 (107.2) 1.31 (0.23) 45.30 (14.55) 1.15 (0.24) Excess 11.44 (11.45) 4.28 (3.41) 17.86 (20.99) 4.71 (4.66) 34.96 (37.34) 6.38 (7.71) 48.62 (54.20) 5.28 (4.93) A. Experiment Description To compare the capabilities in navigation policies of surprise, novelty and objective search in increasingly complex maze problems, we test their performance against 60 randomly generated mazes. These mazes are created by a recursive division algorithm [50], which starts from an empty maze and divides it into two areas by adding a vertical or a horizontal wall with a randomly located hole in it. This process is repeated until no areas can be further subdivided, because doing so would make the maze untraversable or because a maximum number of subdivisions is reached. In this experiment, the starting position and the ending position of the maze have been fixed in the lower left and upper right corner respectively, while the generated mazes have a number of subdivisions chosen randomly between 2 and 6. These values have been chosen empirically to avoid generating mazes that are too easy (solvable by all three methods in few generations) or impossible to solve (because of too many subdivisions). Examples of the mazes generated are shown in Fig. 8. The parameters of surprise search and novelty search are fixed based on well-performing setups with mazes of Section VII: surprise search uses k = 200 and n = 2 (used in the very hard maze) and novelty uses n = 15 (used in medium, hard and very hard mazes). Each generated maze was tested 50 times for each of three methods, measuring the number of successes (i.e. once the agent reaches the goal) in each maze. The number of simulation timesteps is set to 200 and the number of generations to 600. 2 subdivisions 3 subdivisions 4 subdivisions 5 subdivisions 6 subdivisions Fig. 8: Maze generator: Sample generated mazes (200x200 units) created via recursive division, showing the starting location (black circle) and the goal location (white circle). B. Results As a first analysis on the results obtained on the 60 mazes, we focus on which of the evolutionary approaches finds strictly more successes. Table III shows that surprise search has more successes than novelty search in 40% of mazes, while novelty achieves more successes than surprise search in 8% of the generated mazes. Comparing the results of these two approaches against objective search, surprise search outperforms objective search in more mazes (56%) than novelty search (40%). If we look at the baselines, SSnp reaches comparable performance to novelty search, but surprise search remains the most successful algorithm, as it outperforms SSnp in 45% of the considered mazes. Finally, SSr and random search evidently perform poorly compared to the other approaches. An important question to put to the test is how novelty search and surprise search perform with respect to maze deceptiveness. Intuitively, we can say that the deceptiveness (or difficulty) of the maze can be determined by the number of successes obtained by objective search: the more deceptive 12 Fig. 9: Linear regression: relation between the failures of objective search and successes of surprise and novelty search. Fig. 10: Robustness: algorithm successes in solving all the generated mazes over the number of evaluations for each considered method. the maze, the more often objective search would fail to find a solution. Figure 9 shows the number of successes obtained by novelty and surprise search against the number of failures obtained by objective search. Unsurprisingly, surprise search and novelty search find mazes where objective search failed more difficult as well, since their successes are highly correlated with the failures of objective search (adjusted R2 > 0.85 for each method, p < 0.001). Looking at the trends of the linear regression lines, surprise search constantly achieves a greater number of successes, based on the intercept values of the linear regression models which are significantly different according to an ANCOVA test (p < 0.05). Based on the angle of the linear regression line, it also seems that surprise search scales better for more deceptive mazes. As a final analysis, we report the robustness obtained by aggregating all the runs of the 60 generated mazes for each approach, i.e. a total of 3000 runs. From Fig. 10 we can observe that surprise search is faster, on average, than novelty and objective search in reaching the goal from 112, 500 evaluations onward (p < 0.05). Surprise, novelty and objective search require on average 71, 865 (σ = 63, 007), 76, 225 (σ = 64, 961) and 84, 398 (σ = 65, 081) evaluations for each success, respectively. As Fig. 10 shows, some maze problems are easy to solve as all three methods fare similarly in the first 20, 000 evaluations but as the problems become more complex, surprise and novelty search become faster than objective search and eventually surprise search surpasses novelty search in terms of successes. Furthermore, surprise search shows a significant improvement compared to its baseline variants. Respectively, SSnp , SSr and random search obtain 75, 531 (σ = 63, 820), 118, 668 (σ = 53934) and 123, 558 (σ = 50, 568) evaluations; all results are significantly different from the performance of surprise search (p < 0.05). IX. D ISCUSSION This paper identified the notion of surprise, i.e. deviation from expectations, as an alternative measure of divergence to the notion of novelty and presented a general framework for incorporating surprise in evolutionary search in Section IV. In order to highlight the differences between surprise search and other divergent search techniques (such as novelty search) or baselines (such as random search or search with inaccurate predictions), an experiment in the robot maze navigation testbed was carried out and comparisons between algorithms were made on several dimensions. The key findings of these experiments suggest that surprise search yields comparable efficiency to novelty search and it outperforms objective search. Moreover it finds solutions faster and more often than any other algorithm considered. In a broader range of mazes, generated via recursive subdivision, surprise search was also shown to be more robust and generalise well, as it had more successes than novelty search in 40% of generated mazes. The comparative advantages of surprise search over novelty search are inherent to the way the algorithm searches, attempting to deviate from predicted unseen behaviours instead of prior seen behaviours. Compared to novelty search, surprise search may also deviate from expected behaviours that exist in areas that have been visited in the past by the algorithm. The novelty archive operates in a similar fashion; however it contains positions (instead of prediction points) and these positions are always considered for the calculation of the novelty score. In surprise search, instead, the prediction points are derived from clusters that characterise areas in the behavioural space. The findings in the maze navigation experiments show a clear difference between novelty and surprise, both behaviourally and genotypically. Surprise search has greater exploratory capabilities, which are more obvious in later generations. Surprise search also creates genotypically diverse populations, with larger and denser ANNs. It is likely this combination of diverse populations, larger and denser networks and a higher spatial diversity that gives surprise search its advantage over novelty search. Finally, the comparative analysis of surprise search against random search suggests that surprise search is not random search. Clearly it outperforms random search in efficiency and robustness. Furthermore, the poor performance of the two surprise search variants — employing random predictions and omitting predictions — suggests that the prediction of expected behaviour is beneficial for divergent search. By now we have enough evidence for the benefits of surprise search and enough findings suggesting that surprise search is a different and more robust algorithm compared to novelty search in this domain. Furthermore, through our analysis, we have identified qualitative characteristics of the algorithm that gave us critical insights on the way the algorithm operates. However, we still lack empirical evidence on the reasons the algorithm manages to perform that well compared to other divergent search algorithms. An intuition by the anonymous reviewers of a previous paper about the comparative benefits of surprise search is that the algorithm allows search to revisit areas in the behavioural space. Such a behaviour, in contrast, is penalised in novelty search. This difference in how the two algorithms operate leads to the assumption that surprise search is more willing to revisit points in the behaviour space — in a form of backtracking or cyclical manner. As a result of this shifting selection pressure in the behavioural space a different strategy is adopted every time a particular area is revisited as each time the ANN controller is different and potentially larger. Such an algorithmic behaviour appears to be beneficial for search and might explain why ANNs get significantly larger in surprise search. 13 TABLE III: Successes: Percentage of generated mazes for which the algorithm in the row has a strictly greater number of successes than the algorithm in the column. The last row and the last column are respectively the average of each column and the average of each row. Objective Novelty Surprise SSnp SSr Random Total Objective 40 56 51 1 0 29.6 Novelty 15 40 35 1 0 19 Surprise 5 8 11 1 0 5 More importantly than a performance comparison between algorithms, however, is the introduction of surprise as a drive for divergent search. As will be discussed in Section X, the general framework of surprise search can be used with other behaviour characterizations and in other domains. Moreover, while some properties such as the model of prediction are inherent to surprise search, properties such as the clustering of behaviours as well as findings regarding the ability of surprise search to backtrack can be re-used in other divergent search algorithms; examples include a variant of novelty search which considers neighboring cluster centroids rather than neighboring individuals in the behavioural space, or a novelty archive which is pruned (and reduces in size) over time to allow backtracking. We can only hypothesise that the way surprise search operates may result in increased evolvability [51], which is an individual’s capacity to generate future phenotypic variation, or alternatively, the potential for further evolution. Naturally, all these hypotheses need to be tested empirically in future studies as outlined in the next section. SSnp 6 20 45 1 0 16.5 SSr 71 75 78 78 20 62.8 Generation t − 2 Random 78 81 85 85 36 71.8 Total 35 44.8 60.8 52 8 4 - Generation t − 1 Generation t Fig. 11: Behaviour Characterization: The key phases of the surprise search algorithm on the maze navigation task, characterizing behaviour via 200 samples of robot positions over time. Surprise search uses a history of two generations (h = 2) and 10 clusters (k = 10) in this example (for the sake of visualization, only one cluster is shown). Robot trails are depicted as green lines. Cluster centroids in generations t − 2 and t−1 as well as their predictions are depicted, respectively, as red, dark red and blue lines. X. E XTENSIONS AND F UTURE W ORK While this study already offers evidence for the advantages of surprise as a form of divergent search, further work on several directions needs to be performed. Behaviour Characterization: The surprise search algorithm currently characterises a behaviour merely as a point (i.e. the final robot position on the maze after the simulation time elapses). While such a decision was made in order to compare our findings against the initial results of [8], we currently have no evidence suggesting that surprise search would be able to generalise well in behaviours that are characterised by higher dimensions (e.g. a robot trail). To envision the behaviour of surprise search in a high dimensional space, Fig. 11 shows a possible implementation for surprise search with robot trails, sampled over time. Following the implementation described in Section VI-C, we show the three key phases of the algorithm: the robots’ trails are clustered at generation t − 2 via kmeans (see Fig. 11a), then at generation t − 1 we seed the clustering algorithm with the ones computed in the previous generation and we find the new trajectories’ centroids (Fig. 11b). Finally the prediction at generation t is computed via linear interpolation of each point on the computed (centroid) trajectories at generations t−2 and t−1 (Fig. 11c). Preliminary experiments have shown that predictions of robot trails do not affect the performance of surprise search compared to results in this paper; future studies, however, should investigate Ht−2 at generation t−2 Ht−1 at generation t−1 Ht at generation t Fig. 12: Deviation: Surprise search using heatmaps, at generation t. The first two heatmaps are computed in the last two generations by using the final robot positions, Ht−2 and Ht−1 . Using linear interpolation, the difference Ht−1 − Ht−2 is computed and applied to Ht−1 to derive the predicted current population’s Ht . The surprise score penalizes a robot if its position (green point) is on a high concentration cell on the predicted heatmap Ht . the impact of higher dimensional behaviours across several domains, especially on how they affect the predictive model of the algorithm. Deviation: The current algorithm allows for any degree and type of deviation from the expected behaviour. Inspired by novelty search, this paper only investigated a linear deviation from expectations — i.e. the further a behaviour is from the prediction the better. There exist, however, several ways of computing deviation in a non-linear or probabilistic fashion, e.g. as per [18]. In a maze navigation environment for instance, 14 we can alternatively consider a non-distance-based deviation by using heatmaps of the chosen behaviour characterization. Figure 12 shows the key phases of this implementation: in generation t − 2 and t − 1 we compute the heatmaps Ht−2 and Ht−1 which map the final positions of all robots (k = 1), and we use a linear interpolation to compute the predicted heatmap at generation t. The surprise score is then computed by mapping the individual’s position on the predicted heatmap: 1 − Ht (x, y), where x, y is the final position of the robot mapped onto the heatmap. Complexity and generality: To a degree, the experiment with 60 generated mazes tests how generalizable surprise search is without explicit parameter tuning. Since results from that experiment indicated that surprise search scales better to more deceptive problems, we need to further test the algorithm’s potential within the maze navigation domain through more deceptive and complex environments. The capacity of surprise search will then need to be tested in other domains such as robot locomotion or procedural content generation. As an example, the potential of surprise search has been explored for the generation of unexpected weapons in a first person shooter game [38]. In that work, the considered behaviour characterization is a weapon tester agent’s death location: as we have described above, we can employ a heatmap as a probability distribution of the population’s behaviour and predict the next generation’s heatmap by means of linear interpolation of each singular cell. Surprise search has shown its capacity to generate feasible and diverse content, thereby achieving quality diversity. Surprise search has also been successfully implemented to evolve soft robot morphologies [52] constrained by a fixed lattice. The goal of this experiment is to create robots able to travel as far as possible from a starting position, within a number of simulation steps. In this task, surprise search was shown to be as efficient as novelty search; moreover, it evolved more diverse morphologies. When computing behavioural distance for this problem, the surprise score is computed by predicting an entire robot trace in all simulation steps, based on past behaviours (traces). This further supports the claim that surprise is unaffected by the dimensionality of the behaviour characterization. The experiments presented in this paper, in [38] and [52] already demonstrate the generalizability of surprise search across three rather diverse domains: maze navigation, game content generation and robot design. Model of expected behaviour: When it comes to designing a model of expected behaviour there are two key aspects that need to be considered: how much prior information the model requires and how is that information used to make a prediction. In this paper the prediction of behaviour is based on the simplest form of 1-step predictions via a linear regression. This simple predictive model shows the capacity of surprise search given its performance advantages over novelty search. However we can envision that better results can be achieved if machine learned or non-linear predicted models are built on more prior information (h > 2). A possible way of considering more extensive history is to apply linear interpolation of past centroids over time. It is thus possible to compute a 3-dimensional line over the three dimensions considered (x, y, t) and compute the next predicted position over the line by taking the interpolated position at generation t. Linear regression can be easily replaced by a quadratic or cubic regression or even an artificial neural network model or a support vector machine. Prediction locality of surprise search: The algorithm presented in this paper allows for various degrees of prediction locality. We define prediction locality as the amount of local information considered by surprise search to make a prediction. This is expressed by k which is left as a variable for the algorithm designer. Prediction locality can be derived from the behavioural space (as in this paper) but also on the genotypic space. Future investigations should investigate the effect of locality for surprise search. Experiments in this paper (see Fig. 5b) already showcase that algorithm performance is not sensitive with respect to k (as long as k is sufficiently high for the problem at hand). Clustering: Surprise search, in the form presented here, requires some form of behavioural clustering. While k-means was investigated in the experiments of this paper for its simplicity and popularity, any clustering algorithm is applicable. Comparative studies between approaches need to be investigated, including different ways of dealing with (or taking advantage of) empty clusters. XI. C ONCLUSIONS In this paper, we argue that surprise is a concept that can be exploited for evolutionary divergent search, we provide a general definition of the algorithm that follows the principles of searching for surprise and we test the idea in a maze navigation task. Results show that surprise search has clear advantages over other forms of evolutionary divergent search, i. e. novelty search, and outperforms traditional fitness search in deceptive problems. In particular, surprise search has shown a comparable efficiency to novelty search and, most importantly, to be more successful and faster in finding the goal. Moreover, a detailed analysis of the behaviours and the genomes evolved by surprise search has revealed a more diverse population and a higher exploratory capacity. 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Journal of Automated Reasoning manuscript No. (will be inserted by the editor) Mechanized semantics for the Clight subset of the C language arXiv:0901.3619v1 [cs.PL] 23 Jan 2009 Sandrine Blazy · Xavier Leroy the date of receipt and acceptance should be inserted later Abstract This article presents the formal semantics of a large subset of the C language called Clight. Clight includes pointer arithmetic, struct and union types, C loops and structured switch statements. Clight is the source language of the CompCert verified compiler. The formal semantics of Clight is a big-step operational semantics that observes both terminating and diverging executions and produces traces of input/output events. The formal semantics of Clight is mechanized using the Coq proof assistant. In addition to the semantics of Clight, this article describes its integration in the CompCert verified compiler and several ways by which the semantics was validated. Keywords The C programming language · Operational semantics · Mechanized semantics · Formal proof · The Coq proof assistant 1 Introduction Formal semantics of programming languages—that is, the mathematical specification of legal programs and their behaviors—play an important role in several areas of computer science. For advanced programmers and compiler writers, formal semantics provide a more precise alternative to the informal English descriptions that usually pass as language standards. In the context of formal methods such as static analysis, model checking and program proof, formal semantics are required to validate the abstract interpretations and program logics (e.g. axiomatic semantics) used to analyze and reason about programs. The verification of programming tools such as compilers, type-checkers, static analyzers and program verifiers is another area where formal semantics for the languages involved is a prerequisite. While This work was supported by Agence Nationale de la Recherche, grant number ANR-05-SSIA-0019. S. Blazy ENSIIE, 1 square de la Résistance, 91025 Evry cedex, France E-mail: Sandrine.Blazy@ensiie.fr X. Leroy INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France E-mail: Xavier.Leroy@inria.fr 2 formal semantics for realistic languages can be defined on paper using ordinary mathematics [31, 16, 7], machine assistance such as the use of proof assistants greatly facilitates their definition and uses. For high-level programming languages such as Java and functional languages, there exists a sizeable body of mechanized formalizations and verifications of operational semantics, axiomatic semantics, and programming tools such as compilers and bytecode verifiers. Despite being more popular for writing systems software and embedded software, lower-level languages such as C have attracted less interest: several formal semantics for various subsets of C have been published, but only a few have been mechanized. The present article reports on the definition of the formal semantics of a large subset of the C language called Clight. Clight features most of the types and operators of C, including pointer arithmetic, pointers to functions, and struct and union types, as well as all C control structures except goto. The semantics of Clight is mechanized using the Coq proof assistant [10, 4]. It is presented as a big-step operational semantics that observes both terminating and diverging executions and produces traces of input/output events. The Clight subset of C and its semantics are presented in sections 2 and 3, respectively. The work presented in this paper is part of an ongoing project called CompCert that develops a realistic compiler for the C language and formally verifies that it preserves the semantics of the programs being compiled. A previous paper [6] reports on the development and proof of semantic preservation in Coq of the front-end of this compiler: a translator from Clight to Cminor, a low-level, imperative intermediate language. The formal verification of the back-end of this compiler, which generates moderately optimized PowerPC assembly code from Cminor is described in [28]. Section 4 describes the integration of the Clight language and its semantics within the CompCert compiler and its verification. Formal semantics for realistic programming languages are large and complicated. This raises the question of validating these semantics: how can we make sure that they correctly capture the expected behaviors? In section 5, we argue that the correctness proof of the CompCert compiler provides an indirect but original way to validate the semantics of Clight, and discuss other approaches to the validation problem that we considered. We finish this article by a discussion of related work in section 6, followed by future work and conclusions in section 7. Availability The Coq development underlying this article can be consulted on-line at http://compcert.inria.fr . Notations [x, y[ denotes the semi-open interval of integers {n ∈ Z \| x ≤ n < y}. For functions returning “option” types, ⌊x⌋ (read: “some x”) corresponds to success with return value x, and 0/ (read: “none”) corresponds to failure. In grammars, a∗ denotes 0, 1 or several occurrences of syntactic category a, and a? denotes an optional occurrence of syntactic category a. 2 Abstract syntax of Clight Clight is structured into expressions, statements and functions. In the Coq formalization, the abstract syntax is presented as inductive data types, therefore achieving a deep embedding of Clight into Coq. 3 Signedness: Integer sizes: Float sizes: signedness ::= Signed \| Unsigned intsize ::= I8 \| I16 \| I32 floatsize ::= F32 \| F64 Types: τ ::= int(intsize,signedness) \| float(floatsize) \| void \| array(τ ,n) \| pointer(τ ) \| function(τ ∗ , τ ) \| struct(id, ϕ ) \| union(id, ϕ ) \| comp pointer(id) Field lists: ϕ ::= (id, τ )∗ Fig. 1 Abstract syntax of Clight types. 2.1 Types The abstract syntax of Clight types is given in figure 1. Supported types include arithmetic types (integers and floats in various sizes and signedness), array types, pointer types (including pointers to functions), function types, as well as struct and union types. Named types are omitted: we assume that typedef definitions have been expanded away during parsing and type-checking. The integral types fully specify the bit size of integers and floats, unlike the C types int, long, etc, whose sizes are left largely unspecified in the C standard. Typically, the parser maps int and long to size I32, float to size F32, and double to size F64. Currently, 64-bit integers and extended-precision floats are not supported. Array types carry the number n of elements of the array, as a compile-time constant. Arrays with unknown sizes (τ [] in C) are replaced by pointer types in function parameter lists. Their only other use in C is within extern declarations of arrays, which are not supported in Clight. Functions types specify the number and types of the function arguments and the type of the function result. Variadic functions and unprototyped functions (in the style of Ritchie’s pre-standard C) are not supported. In C, struct and union types are named and compared by name. This enables the definition of recursive struct types such as struct s1 { int n; struct * s1 next;}. Recursion within such types must go through a pointer type. For instance, the following is not allowed in C: struct s2 { int n; struct s2 next;}. To obviate the need to carry around a typing environment mapping struct and union names to their definitions, Clight struct and union types are structural: they carry a local identifier id and the list ϕ of their fields (names and types). Bit-fields are not supported. These types are compared by structure, like all other Clight types. In structural type systems, recursive types are traditionally represented with a fixpoint operator µα .τ , where α names the type µα .τ within τ . We adapt this idea to Clight: within a struct or union type, the type comp pointer(id) stands for a pointer type to the nearest enclosing struct or union type named id. For example, the structure s1 defined previously in C is expressed by struct(s1, (n, int(I32, signed))(next, comp pointer(s1))) 4 Expressions: a ::= id \|n \|f \| sizeof(τ ) \| op1 a \| a1 op2 a2 \| a \| a.id \| &a \| (τ )a \| a1 ? a2 : a3 Unary operators: op1 ::= - \| ~ \| ! Binary operators: op2 ::= + \| - \| \| / \| % \| << \| >> \| & \| \| \| ^ \| < \| <= \| > \| >= \| == \| != variable identifier integer constant float constant size of a type unary arithmetic operation binary arithmetic operation pointer dereferencing field access taking the address of type cast conditional expressions arithmetic operators bitwise operators relational operators Fig. 2 Abstract syntax of Clight expressions Incorrect structures such as s2 above cannot be expressed at all, since comp_pointer let us refer to a pointer to an enclosing struct or union, but not to the struct or union directly. Clight does not support any of the type qualifiers of C (const, volatile, restrict). These qualifiers are simply erased during parsing. The following operations over types are defined: sizeof(τ ) returns the storage size, in bytes, of type τ , and field offset(id, ϕ ) returns the byte offset of the field named id in a struct whose field list is ϕ , or 0/ if id does not appear in ϕ . The Coq development gives concrete definitions for these functions, compatible with the PowerPC ABI [48, chap. 3]. Typically, struct fields are laid out consecutively and padding is inserted so that each field is naturally aligned. Here are the only properties that a Clight producer or user needs to rely on: – Sizes are positive: sizeof(τ ) > 0 for all types τ . – Field offsets are within the range of allowed byte offsets for their enclosing struct: if field offset(id, ϕ ) = ⌊δ ⌋ and τ is the type associated with id in ϕ , then [δ , δ + sizeof(τ )[ ⊆ [0, sizeof(struct id′ ϕ )[. – Different fields correspond to disjoint byte ranges: if field offset(idi , ϕ ) = ⌊δi ⌋ and τi is the type associated with id i in ϕ and id 1 6= id 2 , then [δ1 , δ1 + sizeof(τ1 )[ ∩ [δ2 , δ2 + sizeof(τ2 )[ = 0. / – When a struct is a prefix of another struct, fields shared between the two struct have the same offsets: if field offset(id, ϕ ) = ⌊δ ⌋, then field offset(id, ϕ .ϕ ′) = ⌊δ ⌋ for all additional fields ϕ ′ . 2.2 Expressions The syntax of expressions is given in figure 2. All expressions and their sub-expressions are annotated by their static types. In the Coq formalization, expressions a are therefore pairs (b, τ ) of a type τ and a term b of an inductive datatype determining the kind and 5 Statements: Switch cases: s ::= skip \| a1 = a2 \| a1 = a2 (a∗ ) \| a(a∗ ) \| s1 ;s2 \| if(a) s1 else s2 \| switch(a) sw \| while(a) s \| do s while(a) \| for(s1 ,a2 ,s3 ) s \| break \| continue \| return a? sw ::= default : s \| case n : s;sw empty statement assignment function call procedure call sequence conditional multi-way branch “while” loop “do” loop “for” loop exit from the current loop next iteration of the current loop return from current function default case labeled case Fig. 3 Abstract syntax of Clight statements. arguments of the expression. In this paper, we omit the type annotations over expressions, but write type(a) for the type annotating the expression a. The types carried by expressions are necessary to determine the semantics of type-dependent operators such as overloaded arithmetic operators. The following expressions can occur in left-value position: id, a, and a. id. Within expressions, only side-effect free operators of C are supported, but not assignment operators (=, +=, ++, etc) nor function calls. In Clight, assignments and function calls are presented as statements and cannot occur within expressions. As a consequence, all Clight expressions always terminate and are pure: their evaluation performs no side effects. The first motivation for this design decision is to ensure determinism of evaluation. The C standard leaves evaluation order within expressions partially unspecified. If expressions can contain side-effects, different evaluation orders can lead to different results. As demonstrated by Norrish [36], capturing exactly the amount of nondeterminism permitted by the C standard complicates a formal semantics. It is of course possible to commit on a particular evaluation order in a formal semantics for C. (Most C compiler choose a fixed evaluation order, typically right-to-left.) This is the approach we followed in an earlier version of this work [6]. Deterministic side-effects within expressions can be accommodated relatively easily with some styles of semantics (such as the big-step operational semantics of [6]), but complicate or even prevent other forms of semantics. In particular, it is much easier to define axiomatic semantics such as Hoare logic and separation logic if expressions are terminating and pure: in this case, syntactic expressions can safely be used as part of the logical assertions of the logic. Likewise, abstract interpretations and other forms of static analysis are much simplified if expressions are pure. Most static analysis and program verification tools for C actually start by pulling assignments and function calls out of expressions, and only then perform analyses over pure expressions [9, 13, 42, 8, 1, 17]. 6 dcl ::= (τ id)∗ Variable declarations: name and type F ::= τ id(dcl1 ) { dcl2 ; s } (dcl1 = parameters, dcl2 = local variables) Internal function definitions: External function declarations: Fe ::= extern τ id(dcl) Functions: Fd ::= F \| Fe internal or external P ::= dcl;Fd∗ ;main = id Programs: global variables, functions, entry point Fig. 4 Abstract syntax of Clight functions and programs. Some forms of C expressions are omitted in the abstract syntax but can be expressed as syntactic sugar: array access: a1 [a2 ] indirect field access: a->id sequential “and”: a1 && a2 sequential “or”: a1 \|\| a2 ≡ ≡ ≡ ≡ (a1 + a2 ) (a.id) a1 ? (a2 ? 1 : 0) : 0 a1 ? 1 : (a2 ? 1 : 0) 2.3 Statements Figure 3 defines the syntax of Clight statements. All structured control statements of C (conditional, loops, Java-style switch, break, continue and return) are supported, but not unstructured statements such as goto and unstructured switch like the infamous “Duff’s device” [12]. As previously mentioned, assignment a1 = a2 of an r-value a2 to an l-value a1 , as well as function calls, are treated as statements. For function calls, the result can either be assigned to an l-value or discarded. Blocks are omitted because block-scoped variables are not supported in Clight: variables are declared either with global scope at the level of programs, or with function scope at the beginning of functions. The for loop is written for(s1 , a2 , s3 ) s, where s1 is executed once at the beginning of the loop, a2 is the loop condition, s3 is executed at the end of each iteration, and s is the loop body. In C, s1 and s3 are expressions, which are evaluated for their side effects. In Clight, since expressions are pure, we use statements instead. (However, the semantics requires that these statements terminate normally, but not by e.g. break.) A switch statement consists in an expression and a list of cases. A case is a statement labeled by an integer constant (case n) or by the keyword default. Contrary to C, the default case is mandatory in a Clight switch statement and must occur last. 2.4 Functions and programs A Clight program is composed of a list of declarations for global variables (name and type), a list of functions (see figure 4) and an identifier naming the entry point of the program (the main function in C). The Coq formalization supports a rudimentary form of initialization for global variables, where an initializer is a sequence of integer or floating-point constants; we omit this feature in this article. Functions come in two flavors: internal or external. An internal function, written τ id(dcl1 ) { dcl2 ; s }, is defined within the language. τ is the return type, id the name of 7 the function, dcl1 its parameters (names and types), dcl2 its local variables, and s its body. External functions extern τ id(dcl) are merely declared, but not implemented. They are intended to model “system calls”, whose result is provided by the operating system instead of being computed by a piece of Clight code. 3 Formal semantics for Clight We now formalize the dynamic semantics of Clight, using natural semantics, also known as big-step operational semantics. The natural semantics observe the final result of program execution (divergence or termination), as well as a trace of the invocations of external functions performed by the program. The latter represents the input/output behavior of the program. Owing to the restriction that expressions are pure (section 2.2), the dynamic semantics is deterministic. The static semantics of Clight (that is, its typing rules) has not been formally specified yet. The dynamic semantics is defined without assuming that the program is well-typed, and in particular without assuming that the type annotations over expressions are consistent. If they are inconsistent, the dynamic semantics can be undefined (the program goes wrong), or be defined but differ from what the C standard prescribes. 3.1 Evaluation judgements The semantics is defined by the 10 judgements (predicates) listed below. They use semantic quantities such as values, environments, etc, that are summarized in figure 5 and explained later. G, E ⊢ a, M ⇐ ℓ (evaluation of expressions in l-value position) G, E ⊢ a, M ⇒ v (evaluation of expressions in r-value position) G, E ⊢ a∗ , M ⇒ v∗ (evaluation of lists of expressions) t G, E ⊢ s, M ⇒ out, M ′ (execution of statements, terminating case) t G, E ⊢ sw, M ⇒ out, M ′ (execution of the cases of a switch, terminating case) t G ⊢ Fd(v∗ ), M ⇒ v, M ′ (evaluation of function invocations, terminating case) T G, E ⊢ s, M ⇒ ∞ (execution of statements, diverging case) T G, E ⊢ sw, M ⇒ ∞ (execution of the cases of a switch, diverging case) T G ⊢ Fd(v∗ ), M ⇒ ∞ ⊢ P⇒B (evaluation of function invocations, diverging case) (execution of whole programs) Each judgement relates a syntactic element to the result of executing this syntactic element. For an expression in l-value position, the result is a location ℓ: a pair of a block identifier b and a byte offset δ within this block. For an expression in r-value position and for a function application, the result is a value v: the discriminated union of 32-bit integers, 64-bit floating-point numbers, locations (representing the value of pointers), and the special value undef representing the contents of uninitialized memory. Clight does not support assignment between struct or union, nor passing a struct or union by value to a function; therefore, struct and union values need not be represented. Following Norrish [36] and Huisman and Jacobs [21], the result associated with the execution of a statement s is an outcome out indicating how the execution terminated: either normally by running to completion or prematurely via a break, continue or return statement. 8 Block references: b ∈Z Memory locations: ℓ ::= (b, δ ) byte offset δ (a 32-bit integer) within block b Values: v ::= int(n) \| float( f ) \| ptr(ℓ) \| undef integer value (n is a 32-bit integer) floating-point value ( f is a 64-bit float) pointer value undefined value Statement outcomes: out ::= Normal \| Continue \| Break \| Return \| Return(v) Global environments: G ::= (id 7→ b) ×(b 7→ Fd) map from global variables to block references and map from function references to function definitions Local environments: E ::= id 7→ b map from local variables to block references Memory states: M ::= b 7→ (lo,hi, δ 7→ v) map from block references to bounds and contents Memory quantities: κ ::= int8signed \| int8unsigned \| int16signed \| int16unsigned \| int32 \| float32 \| float64 vν ::= int(n) \| float( f ) I/O values: continue with next statement go to the next iteration of the current loop exit from the current loop function exit function exit, returning the value v I/O events: ν ::= id( vν ∗ 7→ vν ) name of external function, argument values, result value Traces: t ::= ε \| ν .t T ::= ε \| ν .T finite traces (inductive) finite or infinite traces (coinductive) Program behaviors: B ::= terminates(t,n) \| diverges(T ) termination with trace t and exit code n divergence with trace T Operations over memory states: alloc(M,lo,hi) = (M′ ,b) Allocate a fresh block of bounds [lo,hi[. free(M,b) = M′ Free (invalidate) the block b. load(κ ,M,b,n) = ⌊v⌋ Read one or several consecutive bytes (as determined by κ ) at block b, offset n in memory state M. If successful return the contents of these bytes as value v. store(κ ,M,b,n,v) = ⌊M′ ⌋ Store the value v into one or several consecutive bytes (as determined by κ ) at offset n in block b of memory state M. If successful, return an updated memory state M′ . Operations over global environments: funct(G,b) = ⌊b⌋ Return the function definition Fd corresponding to the block b, if any. symbol(G,id) = ⌊b⌋ Return the block b corresponding to the global variable or function name id. globalenv(P) = G Construct the global environment G associated with the program P. initmem(P) = M Construct the initial memory state M for executing the program P. Fig. 5 Semantic elements: values, environments, memory states, statement outcomes, etc Most judgements are parameterized by a global environment G, a local environment E, and an initial memory state M. Local environments map function-scoped variables to references of memory blocks containing the values of these variables. (This indirection through memory is needed to allow the & operator to take the address of a variable.) These blocks are allocated at function entry and freed at function return (see rule 32 in figure 10). Likewise, the global environment G associates block references to program-global variables and functions. It also records the definitions of functions. The memory model used in our semantics is detailed in [29]. Memory states M are modeled as a collection of blocks separated by construction and identified by integers b. Each block has lower and upper bounds lo, hi, fixed at allocation time, and associates values 9 Expressions in l-value position: E(id) = b or (id ∈ / Dom(E) and symbol(G,id) = ⌊b⌋) G,E ⊢ a,M ⇒ ptr(ℓ) (1) G,E ⊢ id,M ⇐ (b,0) G,E ⊢ a,M ⇐ (b, δ ) (2) G,E ⊢ a,M ⇐ ℓ type(a) = struct(id′ , ϕ ) field offset(id, ϕ ) = ⌊δ ′ ⌋ G,E ⊢ a.id,M ⇐ (b, δ + δ ) (3) ′ G,E ⊢ a,M ⇐ ℓ type(a) = union(id′ , ϕ ) (4) G,E ⊢ a.id,M ⇐ ℓ Expressions in r-value position: G,E ⊢ n,M ⇒ int(n) (5) G,E ⊢ f ,M ⇒ float( f ) (6) G,E ⊢ sizeof(τ ),M ⇒ int(sizeof(τ )) (7) G,E ⊢ a,M ⇐ ℓ G,E ⊢ &a,M ⇒ ptr(ℓ) G,E ⊢ a1 ,M ⇒ v1 (8) G,E ⊢ a,M ⇒ v G,E ⊢ a1 ,M ⇒ v1 (9) loadval(type(a),M′ ,ℓ) = ⌊v⌋ G,E ⊢ a,M ⇐ ℓ eval unop(op1 ,v1 ,type(a1 )) = ⌊v⌋ (10) G,E ⊢ op1 a1 ,M ⇒ v G,E ⊢ a2 ,M1 ⇒ v2 eval binop(op2 ,v1 ,type(a1 ),v2 ,type(a2 )) = ⌊v⌋ (11) G,E ⊢ a1 op2 a2 ,M ⇒ v G,E ⊢ a1 ,M ⇒ v1 is true(v1 ,type(a1 )) G,E ⊢ a2 ,M ⇒ v2 (12) G,E ⊢ a1 ? a2 : a3 ,M ⇒ v2 G,E ⊢ a1 ,M ⇒ v1 is false(v1 ,type(a1 )) G,E ⊢ a3 ,M ⇒ v3 (13) G,E ⊢ a1 ? a2 : a3 ,M ⇒ v3 G,E ⊢ a,M ⇒ v1 cast(v1 ,type(a), τ ) = ⌊v⌋ G,E ⊢ (τ )a,M ⇒ v (14) Fig. 6 Natural semantics for Clight expressions to byte offsets δ ∈ [lo, hi[. The basic operations over memory states are alloc, free, load and store, as summarized in figure 5. Since Clight expressions are pure, the memory state is not modified during expression evaluation. It is modified, however, during the execution of statements and function calls. The corresponding judgements therefore return an updated memory state M ′ . They also produce a trace t of the external functions (system calls) invoked during execution. Each such invocation is described by an input/output event ν recording the name of the external function invoked, the arguments provided by the program, and the result value provided by the operating system. In addition to terminating behaviors, the semantics also characterizes divergence during the execution of a statement or of a function call. The treatment of divergence follows the coinductive natural approach of Leroy and Grall [30]. The result of a diverging execution is the trace T (possibly infinite) of input/output events performed. In the Coq specification, the judgements of the dynamic semantics are encoded as mutually inductive predicates (for terminating executions) and mutually coinductive predicates (for diverging executions). Each defining case of each predicate corresponds exactly to an inference rule in the conventional, on-paper presentation of natural semantics. We show most of the inference rules in figures 6 to 12, and explain them in the remainder of this section. 10 Access modes: µ ::= By value(κ ) \| By reference \| By nothing access by value access by reference no access Associating access modes to Clight types: A (int(I8,Signed)) A (int(I8,Unsigned)) A (int(I16,Signed)) A (int(I16,Unsigned)) A (int(I32, )) A (pointer( )) = = = = = = By By By By By By value(int8signed) A (array( , )) value(int8unsigned) A (function( , )) value(int16signed) A (struct)( , )) value(int16unsigned) A (union)( , )) value(int32) A (void) value(int32) = = = = = By By By By By reference reference nothing nothing nothing Accessing or updating a value of type τ at location (b, δ ) in memory state M: loadval(τ ,M,(b, δ )) loadval(τ ,M,(b, δ )) loadval(τ ,M,(b, δ )) storeval(τ ,M,(b, δ ),v) storeval(τ ,M,(b, δ ),v) = = = = = load(κ ,M,b, δ ) ⌊(b, δ )⌋ 0/ store(κ ,M,b, δ ,v) 0/ if A (τ ) = By if A (τ ) = By if A (τ ) = By if A (τ ) = By otherwise value(κ ) reference nothing value(κ ) Fig. 7 Memory accesses. 3.2 Evaluation of expressions Expressions in l-value position The first four rules of figure 6 illustrate the evaluation of an expression in l-value position. A variable id evaluates to the location (b, 0), where b is the block associated with id in the local environment E or the global environment G (rule 1). If an expression a evaluates (as an r-value) to a pointer value ptr(ℓ), then the location of the dereferencing expression a is ℓ (rule 2). For field accesses a. id, the location ℓ = (b, δ ) of a is computed. If a has union type, this location is returned unchanged. (All fields of a union share the same position.) If a has struct type, the offset of field id is computed using the field_offset function, then added to δ . From memory locations to values The evaluation of an l-value expression a in r-value position depends on the type of a (rule 8). If a has scalar type, its value is loaded from memory at the location of a. If a has array type, its value is equal to its location. Finally, some types cannot be used in r-value position: this includes void in C and struct and union types in Clight (because of the restriction that structs and unions cannot be passed by value). To capture these three cases, figure 7 defines the function A that maps Clight types to access modes, which can be one of: “by value”, with a memory quantity κ (an access loads a quantity κ from the address of the l-value); “by reference” (an access simply returns the address of the l-value); or “by nothing” (no access is allowed). The loadval and storeval functions, also defined in figure 7, exploit address modes to implement the correct semantics for conversion of l-value to r-value (loadval) and assignment to an l-value (storeval). Expressions in r-value position Rules 5 to 14 of figure 6 illustrate the evaluation of an expression in r-value position. Rule 8 evaluates an l-value expression in an r-value context. The expression is evaluated to its location ℓ. From this location, a value is deduced using 11 ε G,E ⊢ skip,M ⇒ Normal,M (15) ε G,E ⊢ break,M ⇒ Break,M (16) ε G,E ⊢ continue,M ⇒ Continue,M (17) ε ⇒ Return,M (18) G,E ⊢ (return 0),M / G,E ⊢ a,M ⇒ v (19) ε G,E ⊢ (return ⌊a⌋),M ⇒ Return(v),M G,E ⊢ a1 ,M ⇐ ℓ storeval(type(a1 ),M,ℓ,v) = ⌊M′ ⌋ G,E ⊢ a2 ,M ⇒ v ε (20) G,E ⊢ (a1 = a2 ),M ⇒ Normal,M′ t t 2 out,M2 G,E ⊢ s2 ,M1 ⇒ 1 Normal,M1 G,E ⊢ s1 ,M ⇒ (21) t .t 1 2 G,E ⊢ (s1 ;s2 ),M ⇒ out,M2 t G,E ⊢ s1 ,M ⇒ out,M′ out 6= Normal (22) t G,E ⊢ (s1 ;s2 ),M ⇒ out,M′ Fig. 8 Natural semantics for Clight statements (other than loops and switch statements) the loadval function described above. By rule 9, &a evaluates to the pointer value ptr(ℓ) as soon as the l-value a evaluates to the location ℓ. Rules 10 and 11 describe the evaluation of unary and binary operations. Taking binary operations as an example, the two argument expressions are evaluated and their values v1 , v2 are combined using the the eval_binop function, which takes as additional arguments the types τ1 and τ2 of the arguments, in order to resolve overloaded and type-dependent operators. To give the general flavor of eval_binop, here are the cases corresponding to binary addition: τ1 τ2 v1 int( ) int( ) int(n1 ) float( ) float( ) float( f1 ) ptr(τ ) int( ) ptr(b, δ ) int( ) ptr(τ ) int(n) otherwise v2 int(n2 ) float( f2 ) int(n) ptr(b, δ ) eval binop(+, v1 , τ1 , v2 , τ2 ) ⌊int(n1 + n2 )⌋ ⌊float( f1 + f2 )⌋ ⌊ptr(b, δ + n × sizeof(τ ))⌋ ⌊ptr(b, δ + n × sizeof(τ ))⌋ 0/ The definition above rejects mixed arithmetic such as “int + float” because the parser that generates Clight abstract syntax (described in section 4.1) never produces this: it inserts explicit casts from integers to floats in this case. However, it would be easy to add cases dealing with mixed arithmetic. Likewise, the definition above adds two single precision floats using double-precision addition, in violation of the ISO C standard. Again, it would be easy to recognize this case and perform a single-precision addition. Rules 12 and 13 define the evaluation of conditional expressions a1 ? a2 : a3 . The predicates is_true and is_false determine the truth value of the value of a1 , depending on its type. At a float type, float(0.0) is false and any other float value is true. At an int or ptr type, int(0) is false and int(n) (n 6= 0) and ptr(ℓ) values are true. (The null pointer is represented as int(0).) All other combinations of values and types are neither true nor false, causing the semantics to go wrong. Rule 14 evaluates a cast expression (τ )a. The expression a is evaluated, and its value is converted from its natural type type(a) to the expected type τ using the partial function cast. This function performs appropriate conversions, truncations and sign-extensions between integers and floats. We take a lax interpretation of casts involving pointer types: if the 12 Outcome updates (at the end of a loop execution): loop Break ❀ Normal loop loop Return ❀ Return Return(v) ❀ Return(v) while loops: G,E ⊢ a,M ⇒ v is false(v,type(a)) ε G,E ⊢ a,M ⇒ v is true(v,type(a)) t G,E ⊢ s,M ⇒ out,M′ (23) G,E ⊢ (while(a) s),M ⇒ Normal,M loop out ❀ out ′ (24) t G,E ⊢ (while(a) s),M ⇒ out′ ,M′ G,E ⊢ a,M ⇒ v is true(v,type(a)) t1 t2 G,E ⊢ s,M ⇒ (Normal \| Continue),M1 G,E ⊢ (while(a) s),M1 ⇒ out′ ,M2 (25) t .t 1 2 G,E ⊢ (while(a) s),M ⇒ out′ ,M2 for loops: s1 6= skip t 1 G,E ⊢ s1 ,M ⇒ Normal,M1 t 2 G,E ⊢ (for(skip,a2 ,s3 ) s),M1 ⇒ out,M2 (26) t .t 1 2 G,E ⊢ (for(s1 ,a2 ,s3 ) s),M ⇒ out,M2 G,E ⊢ a2 ,M ⇒ v is false(v,type(a2 )) ε G,E ⊢ (for(skip,a2 ,s3 ) s),M ⇒ Normal,M G,E ⊢ a2 ,M ⇒ v is true(v,type(a2 )) t 1 out1 ,M1 G,E ⊢ s,M ⇒ (27) loop out1 ❀ out (28) t G,E ⊢ (for(skip,a2 ,s3 ) s),M ⇒ out,M1 G,E ⊢ a2 ,M ⇒ v is true(v,type(a2 )) t1 t2 G,E ⊢ s,M ⇒ (Normal \| Continue),M1 G,E ⊢ s3 ,M1 ⇒ Normal,M2 t 3 out,M3 G,E ⊢ (for(skip,a2 ,s3 ) s),M2 ⇒ (29) t .t .t 2 3 out,M3 G,E ⊢ (for(skip,a2 ,s3 ) ),M 1 ⇒ Fig. 9 Natural semantics for Clight loops source and destination types are both either pointer types or 32-bit int types, any pointer or integer value can be converted between these types without change of representation. However, the cast function fails when converting between pointer types and float or small integer types, for example. 3.3 Statements and function invocations, terminating case The rules in figure 8 define the execution of a statement that is neither a loop nor a switch statement. The execution of a skip statement yields the Normal outcome and the empty trace (rule 15). Similarly, the execution of a break (resp. continue) statement yields the Break (resp. Continue) outcome and the empty trace (rules 16 and 17). Rules 18–19 describe the execution of a return statement. The execution of a return statement evaluates the argument of the return, if any, and yields a Return outcome and the empty trace. Rule 20 executes an assignment statement. An assignment statement a1 = a2 evaluates the l-value a1 to a location ℓ and the r-value a2 to a value v, then stores v at ℓ using the storeval function of figure 7, producing the final memory state M ′ . We assume that the types of a1 and a2 are identical, therefore no implicit cast is performed during assignment, unlike in C. (The Clight parser described in section 4.1 inserts an explicit cast on the r-value 13 Function calls: G,E ⊢ a f un ,M ⇒ ptr(b,0) G,E ⊢ aargs ,M ⇒ vargs t G ⊢ Fd(vargs ),M ⇒ vres ,M′ funct(G,b) = ⌊Fd⌋ type of fundef(Fd) = type(a f un ) (30) t G,E ⊢ a f un (aargs ),M ⇒ vres ,M′ G,E ⊢ a,M ⇐ ℓ G,E ⊢ a f un ,M ⇒ ptr(b,0) G,E ⊢ aargs ,M ⇒ vargs t funct(G,b) = ⌊Fd⌋ type of fundef(Fd) = type(a f un ) G ⊢ Fd(vargs ),M ⇒ vres ,M1 storeval(type(a),M1 ,ptr(ℓ),vres ) = ⌊M2 ⌋ (31) t G,E ⊢ a = a f un (aargs ),M ⇒ vres ,M2 Compatibility between values, outcomes and return types: Normal,void # undef Return, void # undef Return(v), τ # v when τ 6= void Function invocations: F = τ id(dcl1 ) { dcl2 ; s } alloc vars(M,dcl 1 + dcl2 ,E) = (M1 ,b∗ ) bind params(E,M1 ,dcl1 ,vargs ) = M2 t G,E ⊢ s,M2 ⇒ out,M3 out, τ # vres (32) t G ⊢ F(vargs ),M ⇒ vres ,free(M3 ,b∗ ) Fe = extern τ id(dcl) ν = id( vargs ,vres ) ν (33) G ⊢ Fe(vargs ),M ⇒ vres ,M Fig. 10 Natural semantics for function calls a2 when necessary.) Note that storeval fails if a1 has a struct or union type: assignments between composite data types are not supported in Clight. The execution of a sequence of two statements starts with the execution of the first statement, thus yielding an outcome that determines whether the second statement must be executed or not (rules 21 and 22). The resulting trace is the concatenation of both traces originating from both statement executions. The rules in figure 9 define the execution of while and for loops. (The rules describing the execution of dowhile loops resemble the rules for while loops and are omitted in this paper.) Once the condition of a while loop is evaluated to a value v, if v is false, the execution of the loop terminates normally, with an empty trace (rules 23 and 27). If v is true, the loop body s is executed, thus yielding an outcome out (rules 24, 25, 28 and 29). If out is Normal or Continue, the whole loop is re-executed in the memory state modified by the first execution of the body. In s, the execution of a continue statement interrupts the current execution of the loop body and triggers the next iteration of s. If out is Break, the loop terminates normally; if out is Return, the loop terminates prematurely with the same outcome (rules 24 loop and 28). The ❀ relation models this evolution of outcomes after the premature end of the execution of a loop body. Rules 26–29 describe the execution of a for(s1 , a2 , s3 ) s loop. Rule 26 executes the initial statement s1 of a for loop, which must terminate normally. Then, the loop with an empty initial statement is executed in a way similar to that of a while loop (rules 27–29). If the body s terminates normally or by performing a continue, the statement s3 is executed before re-executing the for loop. As in the case of s1 , it must be the case that s3 terminates normally. 14 t T G,E ⊢ s1 ,M ⇒ ∞ T G,E ⊢ s2 ,M1 ⇒ ∞ G,E ⊢ s1 ,M ⇒ Normal,M1 (34) T (35) t.T G,E ⊢ s1 ;s2 ,M ⇒ ∞ G,E ⊢ s1 ;s2 ,M ⇒ ∞ G,E ⊢ a,M ⇒ v T G,E ⊢ s,M ⇒ ∞ is true(v,type(a)) (36) T G,E ⊢ (while(a) s),M ⇒ ∞ G,E ⊢ a,M ⇒ v is true(v,type(a)) t G,E ⊢ s,M ⇒ (Normal \| Continue),M1 T G,E ⊢ (while(a) s),M1 ⇒ ∞ (37) t.T G,E ⊢ (while(a) s),M ⇒ ∞ G,E ⊢ a f un ,M ⇒ ptr(b,0) funct(G,b) = ⌊Fd⌋ G,E ⊢ aargs ,M ⇒ vargs type of fundef(Fd) = type(a f un ) T G ⊢ Fd(vargs ),M ⇒ ∞ (38) T G,E ⊢ a f un (aargs ),M ⇒ ∞ F = τ id(dcl1 ) { dcl2 ; s } bind params(E,M1 ,dcl1 ,vargs ) = M2 alloc vars(M,dcl1 + dcl2 ,E) = (M1 ,b∗ ) T G,E ⊢ s,M2 ⇒ ∞ (39) T G ⊢ F(vargs ),M ⇒ ∞ Fig. 11 Natural semantics for divergence (selected rules) We omit the rules for switch(a) sw statements, which are standard. Based on the integer value of a, the appropriate case of sw is selected, and the corresponding suffix of sw is executed like a sequence, therefore implementing the “fall-through” behavior of switch cases. A Break outcome for one of the cases terminates the switch normally. The rules of figure 10 define the execution of a call statement a f un (aargs ) or a = a f un (aargs ). The expression a f un is evaluated to a function pointer ptr(b, 0), and the reference b is resolved to the corresponding function definition Fd using the global environment G. This function definition is then invoked on the values of the arguments aargs vres as per the judgment G ⊢ Fd(vargs ), M ⇒ t, M ′ . If needed, the returned value vres is then stored in the location of the l-value a (rules 30 and 31). The invocation of an internal Clight function F (rule 32) allocates the memory required for storing the formal parameters and the local variables of F, using the alloc_vars function. This function allocates one block for each variable id : τ , with lower bound 0 and upper bound sizeof(τ ), using the alloc primitive of the memory model. These blocks initially contain undef values. Then, the bind_params function iterates the storeval function in order to initialize formal parameters to the values of the corresponding arguments. The body of F is then executed, thus yielding an outcome (fourth premise). The return value of F is computed from this outcome and from the return type of F (fifth premise): for a function returning void, the body must terminate by Normal or Return and the return value is undef; for other functions, the body must terminate by Return(v) and the return value is v. Finally, the memory blocks b∗ that were allocated for the parameters and local variables are freed before returning to the caller. A call to an external function Fe simply generates an input/output event recorded in the trace resulting from that call (rule 33). 15 G = globalenv(P) M = initmem(P) t symbol(G,main(P)) = ⌊b⌋ funct(G,b) = ⌊ f ⌋ G ⊢ f (nil),M ⇒ int(n),M′ (40) ⊢ P ⇒ terminates(t,n) G = globalenv(P) symbol(G,main(P)) = ⌊b⌋ M = initmem(P) funct(G,b) = ⌊ f ⌋ T G ⊢ f (nil),M ⇒ ∞ (41) ⊢ P ⇒ diverges(T ) Fig. 12 Observable behaviors of programs 3.4 Statements and function invocations, diverging case Figure 11 shows some of the rules that model divergence of statements and function invocations. As denoted by the double horizontal bars, these rules are to be interpreted coinductively, as greatest fixpoints, instead of the standard inductive interpretation (smallest fixpoints) used for the other rules in this paper. In other words, just like terminating executions correspond to finite derivation trees, diverging executions correspond to infinite derivation trees [30]. A sequence s1 ; s2 diverges either if s1 diverges, or if s1 terminates normally and s2 diverges (rules 34 and 35). Likewise, a loop diverges either if its body diverges, or if it terminates normally or by continue and the next iteration of the loop diverges (rules 36 and 37). A third case of divergence corresponds to an invocation of a function whose body diverges (rules 38 and 39). 3.5 Program executions Figure 12 defines the execution of a program P and the determination of its observable behavior. A global environment and a memory state are computed for P, where each global variable is mapped to a fresh memory block. Then, the main function of P is resolved and applied to the empty list of arguments. If this function invocation terminates with trace t and result value int(n), the observed behavior of P is terminates(t, n) (rule 40). If the function invocation diverges with a possibly infinite trace T , the observed behavior is diverges(T ) (rule 41). 4 Using Clight in the CompCert compiler In this section, we informally discuss how Clight is used in the CompCert verified compiler [27, 6, 28]. 4.1 Producing Clight abstract syntax Going from C concrete syntax to Clight abstract syntax is not as obvious as it may sound. After an unsuccessful attempt at developing a parser, type-checker and simplifier from scratch, we elected to reuse the CIL library of Necula et al. [33]. CIL is written in OCaml and provides the following facilities: 16 1. A parser for ISO C99 (plus GCC and Microsoft extensions), producing a parse tree that is still partially ambiguous. 2. A type-checker and elaborator, producing a precise, type-annotated abstract syntax tree. 3. A simplifier that replaces many delicate constructs of C by simpler constructs. For instance, function calls and assignments are pulled out of expressions and lifted to the statement level. Also, block-scoped variables are lifted to function scope or global scope. 4. A toolkit for static analyses and transformations performed over the simplified abstract syntax tree. While conceptually distinct, (2) and (3) are actually performed in a single pass, avoiding the creation of the non-simplified abstract syntax tree. Thomas Moniot and the authors developed (in OCaml) a simple translator that produces Clight abstract syntax from the output of CIL. Much information produced by CIL is simply erased, such as type attributes and qualifiers. struct and union types are converted from the original named representation to the structural representation used by Clight. String literals are turned into global, initialized arrays of characters. Finally, constructs of C that are unsupported in Clight are detected and meaningful diagnostics are produced. The simplification pass of CIL sometimes goes too far for our needs. In particular, the original CIL transforms all C loops into while(1) { ... } loops, sometimes inserting goto statements to implement the semantics of continue. Such CIL-inserted goto statements are problematic in Clight. We therefore patched CIL to remove this simplification of C loops and natively support while, do and for loops CIL is an impressive but rather complex piece of code, and it has not been formally verified. One can legitimately wonder whether we can trust CIL and our hand-written translator to preserve the semantics of C programs. Indeed, two bugs in this part of CompCert were found during testing: one that we introduced when adding native support for for loops; another that is present in the unmodified CIL version 1.3.6, but was corrected since then. We see two ways to address this concern. First, we developed a pretty-printer that displays Clight abstract syntax tree in readable, C concrete syntax. This printer makes it possible to conduct manual reviews of the transformations performed by CIL. Moreover, experiment shows that re-parsing and re-transforming the simplified C syntax printed from the Clight abstract syntax tree reaches a fixed point in one iteration most of the time. This does not prove anything but nonetheless instills some confidence in the approach. A more radical way to establish trust in the CIL-based Clight producer would be to formally verify some of the simplifications performed. A prime candidate is the simplification of expressions, which transforms C expressions into equivalent pairs of a statement (performing all side effects of the expression) and a pure expression (computing the final value). Based on initial experiments on a simple “while” language, the Coq verification of this simplification appears difficult but feasible. We leave this line of work for future work. 4.2 Compiling Clight The CompCert C compiler is structured in two parts: a front-end compiler translates Clight to an intermediate language called Cminor, without performing any optimizations; a backend compiler generates PowerPC assembly code from the Cminor intermediate representation, performing good register allocation and a few optimizations. Both parts are composed of multiple passes. Each pass is proved to preserve semantics: if the input program P has observable behavior B, and the pass translates P to P′ without reporting a compile-time error, then the output program P′ has the same observable behavior B. The proofs of semantic 17 preservation are conducted with the Coq proof assistant. To facilitate the proof, the compiler passes are written directly in the specification language of Coq, as pure, recursive functions. Executable Caml code for the compiler is then generated automatically from the functional specifications by Coq’s extraction facility. The back-end part of CompCert is described in great detail in [28]. We now give an overview of the front-end, starting with a high-level overview of Cminor, its target intermediate language. (Refer to [28, section 4] for detailed specifications of Cminor.) Cminor is a low-level imperative language, structured like Clight into expressions, statements, and functions. A first difference with Clight is that arithmetic operators are not overloaded and their behavior is independent of the static types of their operands: distinct operators are provided for integer arithmetic and floating-point arithmetic. Conversions between integers and floats are explicit. Likewise, address computations are explicit in Cminor, as well as individual load and store operations. For instance, the C expression a[x] where a is a pointer to int is expressed as load(int32, a +i x i 4), making explicit the memory quantity being addressed (int32) as well as the address computation. At the level of statements, Cminor has only 5 control structures: if-then-else conditionals, infinite loops, block-exit, early return, and goto with labeled statements. The exit n statement terminates the (n + 1) enclosing block statements. Within Cminor functions, local variables can only hold scalar values (integers, pointers, floats) and they do not reside in memory. This makes it easy to allocate them to registers later in the back-end, but also prohibits taking a pointer to a local variable like the C operator & does. Instead, each Cminor function declares the size of a stack-allocated block, allocated in memory at function entry and automatically freed at function return. The expression addrstack(n) returns a pointer within that block at constant offset n. The Cminor producer can use this block to store local arrays as well as local scalar variables whose addresses need to be taken. To translate from Clight to Cminor, the front-end of CompCert C therefore performs the following transformations: 1. Resolution of operator overloading and materialization of all type-dependent behaviors. Based on the types that annotate Clight expressions, the appropriate flavors (integer or float) of arithmetic operators are chosen; conversions between ints and floats, truncations and sign-extensions are introduced to reflect casts; address computations are generated based on the types of array elements and pointer targets; and appropriate memory chunks are selected for every memory access. 2. Translation of while, do and for loops into infinite loops with blocks and early exits. The break and continue statements are translated as appropriate exit constructs. 3. Placement of Clight variables, either as Cminor local variables (for local scalar variables whose address is never taken), sub-areas of the Cminor stack block for the current function (for local non-scalar variables or local scalar variables whose address is taken), or globally allocated memory areas (for global variables). In the first version of the front-end, developed by Zaynah Dargaye and the authors and published in [6], the three transformations above were performed in a single pass, resulting in a large and rather complex proof of semantic preservation. To make the proofs more manageable, we split the front-end in two passes: the first performs transformations (1) and (2) above, and the second performs transformation (3). A new intermediate language called C#minor was introduced to connect the two passes. C#minor is similar to Cminor, except that it supports a & operator to take the address of a local variable. Accordingly, 18 the semantics of C#minor, like that of Clight, allocates one memory block for each local variable at function entrance, while the semantics of Cminor allocates only one block. To account for this difference in allocation patterns, the proof of semantic preservation for transformation (3) exploits the technique of memory injections formalized in [29, section 5.4]. It also involves nontrivial reasoning about separation between memory blocks and between sub-areas of a block. The proof requires about 2200 lines of Coq, plus 800 lines for the formalization of memory injections. The proof of transformations (1) and (2) is more routine: since the memory states match exactly between the original Clight and the generated C#minor, no clever reasoning over memory states, blocks and pointers is required. The Coq proof remains relatively large (2300 lines), but mostly because many cases need to be considered, especially when resolving overloaded operators. 5 Validating the Clight semantics Developing a formal semantics for a real-world programming language is no small task; but making sure that the semantics captures the intended behaviors of programs (as described, for example, by ISO standards) is even more difficult. The smallest mistake or omission in the rules of the semantics can render it incomplete or downright incorrect. Below, we list a number of approaches that we considered to validate a formal semantics such as that of Clight. Many of these approaches were prototyped but not carried to completion, and should be considered as work in progress. 5.1 Manual reviews The standard way to build confidence in a formal specification is to have it reviewed by domain experts. The size of the semantics for Clight makes this approach tedious but not downright impossible: about 800 lines of Coq for the core semantics, plus 1000 lines of Coq for dependencies such as the formalizations of machine integers, floating-point numbers, and the memory model. The fact that the semantics is written in a formal language such as Coq instead of ordinary mathematics is a mixed blessing. On the one hand, the typechecking performed by Coq guarantees the absence of type errors and undefined predicates in the specification, while such trivial errors are common in hand-written semantics. On the other hand, domain experts might not be familiar with the formal language used and could prefer more conventional presentations as e.g. inference rules. (We have not yet found any C language expert who is comfortable with Coq, while several of them are fluent with inference rules.) Manual transliteration of Coq specifications into LATEX inference rules (as we did in this paper) is always possible but can introduce or (worse) mask errors. Better approaches include automatic generation of LATEX from formal specifications, like Isabelle/HOL and Ott do [35, 43]. 5.2 Proving properties of the semantics The primary use of formal semantics is to prove properties of programs and meta-properties of the semantics. Such proofs, especially when conducted on machine, are effective at revealing errors in the semantics. For example, in the case of strongly-typed languages, type 19 soundness proofs (showing that well-typed programs do not go wrong) are often used for this purpose. In the case of Clight, a type soundness proof is not very informative, since the type system of C is coarse and unsound to begin with: the best we could hope for is a subject reduction property, but the progress property does not hold. Less ambitious sanity checks include “common sense” properties such as those of the field_offset function mentioned at end of section 2.1, as well as determinism of evaluation, which we obtained as a corollary of the verification of the CompCert compiler [28, sections 2.1 and 13.3]. 5.3 Verified translations Extending the previous approach to proving properties involving two formal semantics instead of one, we found that proving semantics preservation for a translation from one language to another is effective at exposing errors not only in the translation algorithm, but also in the semantics of the two languages involved. If the translation “looks right” to compiler experts and the semantics of the target language has already been debugged, such a proof of semantic preservation therefore generates confidence in the semantics of the source language. In the case of CompCert, the semantics of the Cminor intermediate language is smaller (300 lines) and much simpler than that of Clight; subsequent intermediate languages in the back-end such as RTL are even simpler, culminating in the semantics of the PPC assembly language, which is a large but conceptually trivial transition function [28]. The existence of semantic-preserving translations between these languages therefore constitutes an indirect validation of their semantics. Semantic preservation proofs and type soundness proofs detect different kinds of errors in semantics. For a trivial example, assume that the Clight semantics erroneously interprets the + operator at type int as integer subtraction. This error would not invalidate an hypothetical type soundness proof, but would show up immediately in the proof of semantic preservation for the CompCert front-end, assuming of course that we did not commit the same error in the translations nor in the semantics of Cminor. On the other hand, a type soundness proof can reveal that an evaluation rule is missing (this shows up as failures of the progress property). A semantic preservation proof can point out a missing rule in the semantics of the target language but not in the semantics of the source language, since it takes as hypothesis that the source program does not go wrong. 5.4 Testing executable semantics Just like programs, formal specifications can be tested against test suites that exemplifies expected behaviors. An impressive example of this approach is the HOL specification of the TCP/IP protocol by Sewell et al. [5], which was extensively validated against network traces generated by actual implementations of the protocol. In the case of formal semantics, testing requires that the semantics is executable: there must exist an effective way to determine the result of a given program in a given initial environment. The Coq proof assistant does not provide efficient ways to execute a specification written using inductive predicates such as our semantics for Clight. (But see [11] for ongoing work in this direction.) As discussed in [3], the eauto tactic of Coq, which performs Prolog-style resolution, can sometimes be used as the poor man’s logic interpreter to execute inductive predicates. However, the Clight semantics is too large and not syntax-directed enough to render this approach effective. 20 On the other hand, Coq provides excellent facilities for executing specifications written as recursive functions: an interpreter is built in the Coq type-checker to perform conversion tests; Coq 8.0 introduced a bytecode compiler to a virtual machine, speeding up the evaluation of Coq terms by one order of magnitude [15]; finally, the extraction facility of Coq can also be used to generate executable Caml code. The recommended approach to execute a Coq specification by inductive predicates, therefore, is to define a reference interpreter as a Coq function, prove its equivalence with the inductive specification, and evaluate applications of the function. Since Coq demands that all recursive functions terminate, these interpretation functions are often parameterized by a nonnegative integer counter n bounding the depth of the evaluation. Taking the execution of Clight statements as an example, the corresponding interpretation function is of the shape exec stmt(W, n, G, E, M, s) = Bottom(t) \| Result(t, out, M ′ ) \| Error where n is the maximal recursion depth, G, E, M are the initial state, and s the statement to execute. The result of execution is either Error, meaning that execution goes wrong, or Result(t, out, M ′ ), meaning that execution terminates with trace t, outcome out and final memory state M ′ , or Bottom(t), meaning that the maximal recursion depth was exceeded after producing the partial trace t. To handle the non-determinism introduced by input/output operations, exec_stmt is parameterized over a world W : a partial function that determines the result of an input/output operation as a function of its arguments and the input/output operation previously performed [28, section 13.1]. The following two properties characterize the correctness of the exec_stmt function with respect to the inductive specification of the semantics: t G, E ⊢ s, M ⇒ out, M ′ ∧ W \|= t ⇔ ∃n, exec stmt(W, n, G, E, M, s) = Result(t, out, M) T G, E ⊢ s, M ⇒ ∞ ∧ W \|= T ⇔ ∀n, ∃t, exec stmt(W, n, G, E, M, s) = Bottom(t) ∧ t is a prefix of T Here, W \|= t means that the trace t is consistent with the world W , in the sense of [28, section 13.1]. See [30] for detailed proofs of these properties in the simpler case of call-by-value λ calculus without traces. The proof of the second property requires classical reasoning with the axiom of excluded middle. We are currently implementing the approach outlined above, although it is not finished at the time of this writing. Given the availability of the CompCert verified compiler, one may wonder what is gained by using a reference Clight interpreter to run tests, instead of just compiling them with CompCert and executing the generated PowerPC assembly code. We believe that nothing is gained for test programs with well-defined semantics. However, the reference interpreter enables us to check that programs with undefined semantics do go wrong, while the CompCert compiler can (and often does) turn them into correct PowerPC code. 5.5 Equivalence with alternate semantics Yet another way to validate a formal semantics is to write several alternate semantics for the same language, using different styles of semantics, and prove logical implications between them. In the case of the Cminor intermediate language and with the help of Andrew Appel, we developed three semantics: (1) a big-step operational semantics in the style of the Clight semantics described in the present paper [27]; (2) a small-step, continuation-based semantics 21 [2, 28]; (3) an axiomatic semantics based on separation logic [2]. Semantics (1) and (3) were proved correct against semantics (2). Likewise, for Clight and with the help of Keiko Nakata, we prototyped (but did not complete yet) three alternate semantics to the big-step operational semantics presented in this paper: (1) a small-step, continuation-based semantics; (2) the reference interpreter outlined above; (3) an axiomatic semantics. Proving the correctness of a semantics with respect to another is an effective way to find mistakes in both. For instance, the correctness of an axiomatic semantics against a big-step operational semantics without traces can be stated as follows: if {P}s{Q} is a valid Hoare triple, then for all initial states G, E, M satisfying the precondition P, either the statement s diverges (G, E ⊢ s, M ⇒ ∞) or it terminates (G, E ⊢ s, M ⇒ out, M ′ ) and the outcome out and the final state G, E, M ′ satisfy postcondition Q. The proof of this property exercises all cases of the big-step operational semantics and is effective at pointing out mistakes and omissions in the latter. Extending this approach to traces raises delicate issues that we have not solved yet. First, the axiomatic semantics must be extended with ways for the postconditions Q to assert properties of the traces generated by the execution of the statement s. A possible source of inspiration is the recent work by Hoare and O’Hearn [20]. Second, in the case of a loop such as {P} while(a) s {Q}, we must not only show that the loop either terminates or diverges without going wrong, as in the earlier proof, but also prove the existence of the corresponding traces of events. In the diverging case, this runs into technical problems with Coq’s guardedness restrictions on coinductive definitions and proofs. In the examples given above, the various semantics were written by the same team and share some elements, such as the memory model and the semantics of Clight expressions. Mistakes in the shared parts will obviously not show up during the equivalence proofs. Relating two independently-written semantics would provide a more convincing validation. In our case, an obvious candidate for comparison with Clight is the Cholera semantics of Norrish [36]. There are notable differences between our semantics and Cholera, discussed in section 6, but we believe that our semantics is a refinement of the Cholera model. A practical issue with formalizing this intuition is that Cholera is formalized in HOL while our semantics is formalized in Coq. 6 Related work Mechanized semantics for C The work closest to ours is Norrish’s Cholera project [36], which formalizes the static and dynamic semantics of a large subset of C using the HOL proof assistant. Unlike Clight, Cholera supports side effects within expressions and accounts for the partially specified evaluation order of C. For this purpose, the semantics of expressions is given in small-step style as a non-deterministic reduction relation, while the semantics of statements is given in big-step style. Norrish used this semantics to characterize precisely the amount of non-determinism allowed by the C standard [37]. Also, the memory model underlying Cholera is more abstract than that of Clight, leaving unspecified a number of behaviors that Clight specifies. Tews et al [46, 47] developed a denotational semantics for a subset of the C++ language. The semantics is presented as a shallow embedding in the PVS prover. Expressions and statements are modeled as state transformers: functions from initial states to final states plus value (for expressions) or outcome (for statements). The subset of C++ handled is close to our Clight, with a few differences: side effects within expressions are allowed (and treated using a fixed evaluation order); the behavior of arithmetic operations in case of overflow is 22 not specified; the goto statement is not handled, but the state transformer approach could be extended to do so [45]. Using the Coq proof assistant, Giménez and Ledinot [14] define a denotational semantics for a subset of C appropriate as target language for the compilation of the Lustre synchronous dataflow language. Owing to the particular shape of Lustre programs, the subset of C does not contain general loops nor recursive functions, but only counted for loops. Pointer arithmetic is not supported. As part of the Verisoft project [40], the semantics of a subset of C called C0 has been formalized using Isabelle/HOL, as well as the correctness of a compiler from C0 to DLX assembly language [26, 44, 41]. C0 is a type-safe subset of C, close to Pascal, and significantly smaller than Clight: for instance, there is no pointer arithmetic, nor break and continue statements. A big-step semantics and a small-step semantics have been defined for C0, the latter enabling reasoning about non-terminating executions. Paper and pencil semantics for C Papaspyrou [39] develops a monadic denotational semantics for most of ISO C. Non-determinism in expression evaluation is modeled precisely. The semantics was validated by testing with the help of a reference interpreter written in Haskell. Nepomniaschy et al. [34] define a big-step semantics for a subset of C similar to Pascal: it supports limited uses of goto statements, but not pointer arithmetic. Abstract state machines have been used to give semantics for C [16] and for C# [7]. The latter formalization is arguably the most complete (in terms of the number of language features handled) formal semantics for an imperative language. Other examples of mechanized semantics Proof assistants were used to mechanize semantics for languages that are higher-level than C. Representative examples include [23, 25] for Standard ML, [38] for a subset of OCaml, and [24] for a subset of Java. Other Java-related mechanized verifications are surveyed in [18]. Many of these semantics were validated by conducting type soundness proofs. Subsets of C Many uses of C in embedded or critical applications mandate strict coding guidelines restricting programmers to a “safer” subset of C [19]. A well-known example is MISRA C [32]. MISRA C and Clight share some restrictions (such as structured switch statements with default cases at the end), but otherwise differ significantly. For instance, MISRA C prohibits recursive functions, but permits all uses of goto. More generally, the restrictions of MISRA C and related guidelines are driven by software engineering considerations and the desire for tool-assisted checking, while the restrictions of Clight stem from the desire to keep its formal semantics manageable. Several tools for static analysis and deductive verification of C programs use simplified subsets of C as intermediate representations. We already discussed the CIL intermediate representation [33]. Other examples include the Frama-C intermediate representation [8], which extends CIL’s with logical assertions, and the Newspeak representation [22]. CIL is richer than Clight and accurately represents all of ISO C plus some extensions. Newspeak is lower-level than Clight and targeted more towards static analysis than towards compilation. 7 Conclusions and future work In this article, we have formally defined the Clight subset of the C programming language and its dynamic semantics. While there is no general agreement on the formal semantics of 23 the C language, we believe that Clight is a reasonable proposal that works well in the context of the formal verification of a compiler. We hope that, in the future, Clight might be useful in other contexts such as static analyzers and program provers and their formal verification. Several extensions of Clight can be considered. One direction, discussed in [29], is to relax the memory model so as to model byte- and bit-level accesses to in-memory data representations, as is commonly done in systems programming. Another direction is to add support for some of the C constructs currently missing, in particular the goto statement. The main issue here is to formalize the dynamic semantics of goto in a way that lends itself well to proofs. Natural semantics based on statement outcomes can be extended with support for goto by following the approach proposed by Tews [45], but at the cost of nearly doubling the size of the semantics. Support for goto statements is much easier to add to transition semantics based on continuations, as the Cminor semantics exemplifies [28, section 4]. However, such transition semantics do not lend themselves easily to proving transformations of loops such as those performed by the front-end of CompCert (transformation 2 in section 4.2). 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CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS arXiv:1011.2083v4 [math.GR] 10 Mar 2016 MANOJ K. YADAV Abstract. In 1904, Issai Schur proved the following result. If G is an arbitrary group such that G/ Z(G) is finite, where Z(G) denotes the center of the group G, then the commutator subgroup of G is finite. A partial converse of this result was proved by B. H. Neumann in 1951. He proved that if G is a finitely generated group with finite commutator subgroup, then G/ Z(G) is finite. In this short note, we exhibit few arguments of Neumann, which provide further generalizations of converse of the above mentioned result of Schur. We classify all finite groups G such that \|G/ Z(G)\| = \|γ2 (G)\|d , where d denotes the number of elements in a minimal generating set for G/ Z(G). Some problems and questions are posed in the sequel. 1. Introduction In 1951, Neumann [18, Theorem 5.3] proved the following result: If the index of Z(G) in G is finite, then γ2 (G) is finite, where Z(G) and γ2 (G) denote the center and the commutator subgroup of G respectively. He mentioned [19, End of page 237] that this result can be obtained from an implicit idea of Schur [23], and his proof also used Schur’s basic idea. However there is no mention of this fact in [18] in which Schur’s paper is also cited. In this note, this result will be termed as ‘the Schur’s theorem’. Neumann also provided a partial converse of the Schur’s theorem [18, Corollary 5.41] as follows: If G is finitely generated by k elements and γ2 (G) is finite, then G/ Z(G) is finite, and bounded by \|G/ Z(G)\| ≤ \|γ2 (G)\|k . Our first motivation of writing this note is to exhibit an idea of Neumann[18, page 179] which proves much more than what is said above on converse of the Schur’s theorem. We quote the text here (with a minor modification in the notations): “Let G be generated by g1 , g2 , . . . , gk . Then Z(G) = ∩κ=k κ=1 CG (gκ ); for, an element of G lies in the centre if and only if it (is permutable) commutes with all the generators of G. If G is an FC-group (group whose all conjugacy classes are of finite length), then \|G : CG (gκ )\| is finite for 1 ≤ κ ≤ k, and Z(G), as intersection of a finite set of subgroups of finite index, also has finite index. The index of the intersection of two subgroups does not exceed the product of the indices of the subgroups: hence in this case one obtains an upper bound for the index of the centre, namely \|G : Z(G)\| ≤ Πκ=k κ=1 \|G : CG (gκ )\|.” Just a soft staring at the quoted text for a moment or two suggests the following. The conclusion does not require the group G to be FC-group. It only requires the finiteness of the conjugacy classes of the generating elements. If a generator of the group G is contained in Z(G), one really does not need to count it. Thus the argument works perfactly well even if G is 2010 Mathematics Subject Classification. Primary 20F24, 20E45. Key words and phrases. commutator subgroup, Schur’s theorem, class-preserving automorphism. 1 CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 2 generated by infinite number of elements, all but finite of them lie in the center of G. Thus the following result holds true. Theorem A. Let G be an arbitrary group such that G/ Z(G) is finitely generated by x1 Z(G), x2 Z(G), . . . , xt Z(G) and the conjugacy class of xi in G is of finite length for 1 ≤ i ≤ t. Then G G/ Z(G) is finite. Moreover \|G/ Z(G)\| ≤ Πti=1 \|xG i \| and γ2 (G) is finite, where xi denotes the conjugacy class of xi in G. Neumann’s result [18, Corollary 5.41] was reproduced by Hilton [12, Theorem 1]. It seems that Hilton was not aware of Neumann’s result. This lead two more publications [21] and [24] dedicated to proving special cases of Theorem A. Converse of the Schur’s theorem is not true in general as shown by infinite extraspecial pgroups, where p is an odd prime. It is interesting to know that example of such a 2-group also exists, which is mentioned on page 238 (second para of Section 3) of [19]. It is a central product of infinite copies of quaternion groups of order 8 amalgamated at the center of order 2. Our second motivation of writing this note is to provide a modification of an innocent looking result of Neumann [20, Lemma 2], which allows us to say little more on converse of the Schur’s theorem. A modified version of this lemma is the following. Lemma 1.1. Let G be an arbitrary group having a normal abelian subgroup A such that the index of CG (A) in G is finite and G/A is finitely generated by g1 A, g2 A, . . . , gr A, where \|giG \| < ∞ for 1 ≤ i ≤ r. Then G/Z(G) is finite. This lemma helps proving the first three statements of the following result. Theorem B. For an arbitrary group G, G/ Z(G) is finite if any one of the following holds true: (i) Z2 (G)/ Z(Z2 (G)) is finitely generated and γ2 (G) is finite. (ii) G/ Z(Z2 (G)) is finitely generated and G/(Z2 (G)γ2 (G)) is finite. (iii) γ2 (G) is finite and Z2 (G) ≤ γ2 (G). (iv) γ2 (G) is finite and G/ Z(G) is purely non-abelian. Our final motivation is to provide a classification of all groups G upto isoclinism (see Section 3 for the definition) such that \|G/ Z(G)\| = \|γ2 (G)\|d is finite, where d denotes the number of elements in a minimal generating set for G/ Z(G), discuss example in various situations and pose some problems. We conclude this section with fixing some notations. For an arbitrary group G, by Z(G), Z2 (G) and γ2 (G) we denote the center, the second center and the commutator subgroup of G respectively. For x ∈ G, [x, G] denotes the set {[x, g] \| g ∈ G}. Notice that \|[x, G]\| = \|xG \|, where xG denotes the conjugacy class of x in G. If [x, G] ⊆ Z(G), then [x, G] becomes a subgroup of G. For a subgroup H of G, CG (H) denotes the centralizer of H in G and for an element x ∈ G, CG (x) denotes the centralizer of x in G. 2. Proofs We start with the proof of Lemma 1.1, which is essentialy same as the one given by Neumann. Proof of Lemma 1.1. Let G/A be generated by g1 A, g2 A, . . . , gr A for some gi ∈ G, where 1 ≤ i ≤ r < ∞. Let X := {g1 , g2 , . . . , gr } and A be generated by a set Y . Then G = hX ∪ Y i and Z(G) = CG (X) ∩ CG (Y ). Notice that CG (A) = CG (Y ). Since CG (A) is of finite index, CG (Y ) is also of finite index in G. Also, since \|giG \| < ∞ for 1 ≤ i ≤ r, CG (X) is of finite index in G. Hence the index of Z(G) in G is finite and the proof is complete. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 3 Proof of Theorem A can be made quite precise by using Lemma 1.1. Proof of Theorem A. Taking A = Z(G) in Lemma 1.1, it follows that G/ Z(G) is finite. Moreover, \|G/ Z(G)\| = \|G/ ∩ti=1 CG (xi )\| ≤ Πti=1 \|G : CG (xi )\| = Πti=1 \|[xi , G]\| = Πti=1 \|xG i \|. That γ2 (G) is finite now follows from the Schur’s theorem. For the proof of Theorem B we need the following result of Hall [9] and the subsequent proposition. Theorem 2.1. If G is an arbitrary group such that γ2 (G) is finite, then G/ Z2 (G) is finite. Explicit bounds on the order of G/ Z2 (G) were first given by Macdonald [14, Theorem 6.2] and later on improved by Podoski and Szegedy [22] by showing that if \|γ2 (G)/(γ2 (G) ∩ Z(G))\| = n, then \|G/ Z2 (G)\| ≤ nc log2 n with c = 2. Proposition 2.2. Let G be an arbitrary group such that γ2 (G) is finite and G/ Z(G) is infinite. Then G/ Z(G) has an infinite abelian group as a direct factor. Proof. Since γ2 (G) is finite, by Theorem 2.1 G/ Z2 (G) is finite. Thus Z2 (G)/ Z(G) is infinite. Again using the finiteness of γ2 (G), it follows that the exponent of Z2 (G)/ Z(G) is finite. Hence by [6, Theorem 17.2] Z2 (G)/ Z(G) is a direct sum of cyclic groups. Let G/ Z2 (G) be generated by x1 Z2 (G), . . . , xr Z2 (G) and H := hx1 , . . . , xr i. Then it follows that modulo Z(G), H ∩ Z2 (G) is finite. Thus we can write Z2 (G)/ Z(G) = hy1 Z(G)i × · · · × hys Z(G)i × hys+1 Z(G)i × · · · , such that (H ∩ Z2 (G)) Z(G)/ Z(G) ≤ hy1 Z(G)i × · · · × hys Z(G)i. It now follows that the infinite abelian group hys+1 Z(G)i × · · · is a direct factor of G/ Z(G), and the proof is complete. We are now ready to prove Theorem B. Proof of Theorem B. Since γ2 (G) is finite, it follows from Theorem 2.1 that G/ Z2 (G) is finite. Now using the fact that Z2 (G)/ Z(Z2 (G)) is finitely generated, it follows that G/ Z(Z2 (G)) is finitely generated. Take Z(Z2 (G)) = A. Then notice that A is a normal abelian subgroup of G such that the index of CG (A) in G is finite, since Z2 (G) ≤ CG (A). Hence by Lemma 1.1, G/ Z(G) is finite, which proves (i). Again take Z(Z2 (G)) = A and notice that Z2 (G)γ2 (G) ≤ CG (A). (ii) now directly follows from Lemma 1.1. If Z2 (G) ≤ γ2 (G), then Z2 (G) is abelian. Thus (iii) follows from (i). Finally, (iv) follows from Proposition 2.2. This completes the proof of the theorem. We conclude this section with an extension of Theorem A in terms of conjugacy classpreserving automorphisms of given group G. An automorphism α of an arbitrary group G is called (conjugacy) class-preserving if α(g) ∈ g G for all g ∈ G. We denote the group of all class-preserving automorphisms of G by Autc (G). Notice that Inn(G), the group of all inner automorphisms of G, is a normal subgroup of Autc (G) and Autc (G) acts trivially on the center of G. A detailed survey on class-preserving automorphisms of finite p-groups can be found in [25]. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 4 Let G be the group as in the statement of Theorem A. Then G is generated by x1 , x2 , . . . , xt along with Z(G). Since Autc (G) acts trivially on the center of G, it follows that (2.3) \| Autc (G)\| ≤ Πti=1 \|xG i \| as there are only \|xG i \| choices for the image of each xi under any class-preserving automorphism. G Since \|xi \| is finite for each xi , 1 ≤ i ≤ t, it follows that \| Autc (G)\| ≤ Πti=1 \|xG i \| is finite. We have proved the following result of which Theorem A is a corollary, because \|G/ Z(G)\| = \| Inn(G)\| ≤ \| Autc (G)\|. Theorem 2.4. Let G be an arbitrary group such that G/ Z(G) is finitely generated by x1 Z(G), x2 Z(G), . . . , xt Z(G) and the conjugacy class of xi in G is of finite length for 1 ≤ i ≤ t. Then Autc (G) is finite. Moreover \| Autc (G)\| ≤ Πti=1 \|xG i \| and γ2 (G) is finite. Proof of Theorem A is also reproduced using IA-automorphisms (automorphisms of a group that induce identity on the abelianization) in [7, Theorem 2.3]. Proof goes on the same way as in the case of class-preserving automorphisms. 3. Groups with maximal central quotient We start with the following concept due to Hall [8]. For a group X, the commutator map aX : X/ Z(X) × X/ Z(X) → γ2 (X) given by aX (x1 Z(X), x2 Z(X)) = [x1 , x2 ] is well defined. Two groups K and H are said to be isoclinic if there exists an isomorphism φ of the factor group K̄ = K/ Z(K) onto H̄ = H/ Z(H), and an isomorphism θ of the subgroup γ2 (K) onto γ2 (H) such that the following diagram is commutative a K̄ × K̄ −−−G−→ γ2 (K)     φ×φy yθ a H̄ × H̄ −−−H−→ γ2 (H). The resulting pair (φ, θ) is called an isoclinism of K onto H. Notice that isoclinism is an equivalence relation among groups. The following proposition (also see Macdonald’s result [14, Lemma 2.1]) is important for the rest of this section. Proposition 3.1. Let G be a group such that G/ Z(G) is finite. Then there exists a finite group H isoclinic to the group G such that Z(H) ≤ γ2 (H). Moreover if G is a p-group, then H is also a p-group. Proof. Let G be the given group. Then by Schur’s theorem γ2 (G) is finite. Now it follows from a result of Hall [8] that there exists a group H which is isoclinic to G and Z(H) ≤ γ2 (H). Since \|γ2 (G)\| = \|γ2 (H)\| is finite, Z(H) is finite. Hence, by the definition of isoclinism, H is finite. Now suppose that G is a p-groups. Then it follows that H/ Z(H) as well as γ2 (H) are p-groups. Since Z(H) ≤ γ2 (H), this implies that H is a p-group. For an arbitrary group G with finite G/ Z(G), we have (3.2) \|G/ Z(G)\| ≤ \|γ2 (G)\|d , where d = d(G/ Z(G)). For simplicity we say that a group G has Property A if G/ Z(G) is finite and equality holds in (3.2) for G. We are now going to classify, upto isoclinism, all groups G having Property A. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 5 Let G be an arbitrary group having Property A. Then by Proposition 3.1 there exists a finite group H isoclinic to G and, by the definition of isoclinism, H has Property A. Thus for classifying all groups G, upto isoclinism, having Property A, it is sufficient to classify all finite group with this property. Let us first consider non-nilpotent finite groups. For such group we prove the following result in [5] Theorem 3.3. There is no non-nilpotent group G having Property A. So we only need to consider finite nilpotent groups. Since a finite nilpotent group is a direct product of it’s Sylow p-subgroups, it is sufficient to classify finite p-groups admitting Property A. Obviously, all abelian groups admit Property A. Perhaps the simplest examples of non-abelian groups having Property A are finite extraspecial p-groups. The class of 2-generated finite capable nilpotent groups with cyclic commutator subgroup also admits Property A. A group G is said ∼ H/ Z(H). Isaacs [13, Theorem 2] proved: to be capable if there exists a group H such that G = Let G be finite and capable, and suppose that γ2 (G) is cyclic and that all elements of order 4 in γ2 (G) are central in G. Then \|G/ Z(G)\| ≤ \|γ2 (G)\|2 , and equality holds if G is nilpotent. So G admits Property A, if G is a nilpotent group as in this statement and G is 2-generated. A complete classification of 2-generated finite capable p-groups of class 2 is given in [15]. Motivated by finite extraspecial p-groups, a more general class of groups G admitting Property A can be constructed as follows. For any positive integer m, let G1 , G2 , . . . , Gm be 2-generated finite p-groups such that γ2 (Gi ) = Z(Gi ) ∼ = X (say) is cyclic of order q for 1 ≤ i ≤ m, where q is some power of p. Consider the central product (3.4) Y = G1 ∗X G2 ∗X · · · ∗X Gm of G1 , G2 , . . . , Gm amalgamated at X (isomorphic to cyclic commutator subgroups γ2 (Gi ), 1 ≤ i ≤ m). Then \|Y \| = q 2m+1 and \|Y / Z(Y )\| = q 2m = \|γ2 (Y )\|d(Y ) , where d(Y ) = 2m is the number of elements in any minimal generating set for Y . Thus Y has Property A. Notice that in all of the above examples, the commutator subgroup is cyclic. Infinite groups having Property A can be easily obtained by taking a direct product of an infinite abelian group with any finite group having Property A. We now proceed to showing that any finite p-group G of class 2 having Property A is isoclinic to a group Y defined in (3.4). Let x ∈ Z2 (G) for a group G. Then, notice that [x, G] is a central subgroup of G. We have the following simple but useful result. Lemma 3.5. Let G be an arbitrary group such that Z2 (G)/ Z(G) is finitely generated by x1 Z(G), x2 Z(G), . . . , xt Z(G) such that exp([xi , G]) is finite for 1 ≤ i ≤ t. Then \| Z2 (G)/ Z(G)\| = t Y exp([xi , G]). i=1 Proof. By the given hypothesis exp([xi , G]) is finite for all i such that 1 ≤ i ≤ t. Suppose that exp([xi , G]) = ni . Since [xi , G] ⊆ Z(G), it follows that [xni i , G] = [xi , G]ni = 1. Thus xni i ∈ Z(G) and no smaller power of xi than ni can lie in Z(G), which implies that the order of xi Z(G) is Qt ni . Since Z2 (G)/ Z(G) is abelian, we have \| Z2 (G)/ Z(G)\| = i=1 exp([xi , G]). CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 6 Let Φ(X) denote the Frattini subgroup of a group X. The following result provides some structural information of p-groups of class 2 admitting Property A. Proposition 3.6. Let H be a finite p-group of class 2 having Property A and Z(H) = γ2 (H). Then (i) γ2 (H) is cyclic; (ii) H/ Z(H) is homocyclic; (iii) [x, H] = γ2 (H) for all x ∈ H − Φ(H); (iv) H is minimally generated by even number of elements. Proof. Let H be the group given in the statement, which is minimally generated by d elements x1 , x2 . . . , xd (say). Since Z(H) = γ2 (H), it follows that H/ Z(H) is minimally generated by x1 Z(H), x2 Z(H), . . . , xd Z(H). Thus by the identity \|H/ Z(H)\| = \|γ2 (H)\|d , it follows that order of xi Z(H) is equal to \|γ2 (H)\| for all 1 ≤ i ≤ d. Since the exponent of H/ Z(H) is equal to the exponent of γ2 (H), we have that γ2 (H) is cyclic and H/ Z(H) is homocyclic. Now by Qt Lemma 3.5, \|γ2 (H)\|d = \|H/ Z(H)\| = i=1 exp([xi , H]). Since [xi , H] ⊆ γ2 (H), this implies that [xi , H] = γ2 (H) for each i such that 1 ≤ i ≤ d. Let x be an arbitrary element in H − Φ(H). Then the set {x} can always be extended to a minimal generating set of H. Thus it follows that [x, H] = γ2 (H) for all x ∈ H − Φ(H). This proves first three assertions. For the proof of (iv), we consider the group H̄ = H/Φ(γ2 (H)). Notice that both H as well as H̄ are minimallay generated by d elements. Since [x, H] = γ2 (H) for all x ∈ H − Φ(H), it follows that for no x ∈ H − Φ(H), x̄ ∈ Z(H̄), where x̄ = xΦ(γ2 (H)) ∈ H̄. Thus it follows that Z(H̄) ≤ Φ(H̄). Also, since γ2 (H) is cyclic, γ2 (H̄) is cyclic of order p. Thus it follows that H̄ is isoclinic to a finite extraspecial p-group, and therefore it is minimally generated by even number of elements. Hence H is also minimally generated by even number of elements. This completes the proof of the proposition. Using the definition of isoclinism, we have Corollary 3.7. Let G be a finite p-group of class 2 admitting Property A. Then γ2 (G) is cyclic and G/ Z(G) is homocyclic. We need the following important result. Theorem 3.8 ([3], Theorem 2.1). Let G be a finite p-group of nilpotency class 2 with cyclic center. Then G is a central product either of two generator subgroups with cyclic center or two generator subgroups with cyclic center and a cyclic subgroup. Theorem 3.9. Let G be a finite p-group of class 2 having Property A. Then G is isoclinic to the group Y , defined in (3.4), for suitable positive integers m and n. Proof. Let G be a group as in the statement. Then by Proposition 3.1 there exists a finite p-group H isoclinic to G such that Z(H) = γ2 (H). Obviously H also satisfies \|H/ Z(H)\| = \|γ2 (H)\|d , where d denotes the number of elements in any minimal generating set of G/ Z(G). Then by Proposition 3.6, γ2 (H) = Z(H) is cyclic of order q = pn (say) for some positive integer n, and H/ Z(H) is homocyclic of exponent q and is of order q 2m for some positive integer m. Since Z(H) = γ2 (H) is cyclic, it follows from Theorem 3.8 that H is a central product of 2-generated CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 7 groups H1 , H2 , . . . , Hm . It is easy to see that γ2 (Hi ) = Z(Hi ) for 1 ≤ i ≤ m and \|γ2 (H)\| = q. This completes the proof of the theorem. We would like to remark that Theorem 3.9 is also obtained in [26, Theorem 11.2] as a consequence on study of class-preserving automorphisms of finite p-group. But we have presented here a direct proof. Now we classify finite p-groups of nilpoency class larger than 2. Consider the metacylic groups E D r+t r r+s t (3.10) K := x, y \| xp = 1, y p = xp , [x, y] = xp , where 1 ≤ t < r and 0 ≤ s ≤ t (t ≥ 2 if p = 2) are non-negative integers. Notice that the nilpotency class of K is at least 3. Since K is generated by 2 elements, it follows from (2.3) that \| Autc (K)\| ≤ \|γ2 (K)\|2 = p2r . It is not so difficult to see that \| Inn(K)\| = \|K/ Z(K)\| = p2r . Since Inn(K) ≤ Autc (K), it follows that \| Autc (K)\| = \| Inn(K)\| = \|γ2 (K)\|2 = \|γ2 (K)\|d(K) (That Autc (G) = Inn(G), is, in fact, true for all finite metacylic p-groups). Thus K admits Property A. Furthermore, if H is any 2-generator group isoclinic to K, then it follows that H admits Property A. For a finite p-groups having Property A, there always exists a p-group H isoclinic G such that \|H/ Z(H)\| = \|γ2 (H)\|d , where d = d(H). The following theorem now classifies, upto isoclinism, all finite p-groups G of nilpotency class larger than 2 having Property A. Theorem 3.11 (Theorem 11.3, [26]). Let G be a finite p-group of nilpotency class at least 3. Then the following hold true. (i) If \|G/ Z(G)\| = \|γ2 (G)\|d , where d = d(G), then d(G) = 2; (ii) If \|γ2 (G)/γ3 (G)\| > 2, then \|G/ Z(G)\| = \|γ2 (G)\|d if and only if G is a 2-generator group with cyclic commutator subgroup. Furthermore, G is isoclinic to the group K defined in (3.10) for suitable parameters; (iii) If \|γ2 (G)/γ3 (G)\| = 2, then \|G/ Z(G)\| = \|γ2 (G)\|d if and only if G is a 2-generator 2-group of nilpotency class 3 with elementary abelian γ2 (G) of order 4. It is clear that the groups G occuring in Theorem 3.11(iii) are isoclinic to certain groups of order 32. Using Magma (or GAP), one can easily show that such groups of order 32 are SmallGroup(32,k) for k = 6, 7, 8 in the small group library. We conclude this section with providing some different type of bounds on the central quotient of a given group. If \|γ2 (G) Z(G)/ Z(G)\| = n is finite for a group G, then it follows from [22, Theorem 1] that \|G/ Z2 (G)\| ≤ n2 log2 n . Using this and Lemma 3.5 we can also provide an upper bound on the size of G/ Z(G) in terms of n, the rank of Z2 (G)/ Z(G) and exponents of certain sets of commutators (here these sets are really subgroups of G) of coset representatives of generators of Z2 (G)/ Z(G) with the elements of G. This is given in the following theorem. Theorem 3.12. Let G be an arbitrary group. Let \|γ2 (G) Z(G)/ Z(G)\| = n is finite and Z2 (G)/ Z(G) is finitely generated by x1 Z(G), x2 Z(G), . . . , xt Z(G) such that exp([xi , G]) is finite for 1 ≤ i ≤ t. Then Yt exp([xi , G]). \|G/ Z(G)\| ≤ n2 log2 n i=1 4. Problems and Examples Theorem B provides some conditions on a group G under which G/ Z(G) becomes finite. It is interesting to solve CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 8 Problem 1. Let G be an arbitrary group. Provide a set C of optimal conditions on G such that G/ Z(G) is finite if and only if all conditions in C hold true. As we know that there is no finite non-nilpotent group G admitting Property A. Since Inn(G) ≤ Autc (G), it is interesting to consider Problem 2. Classify all non-nilpotent finite group G such that \| Autc (G)\| = \|γ2 (G)\|d , where d = d(G). A much stronger result than Theorem 3.3 is known in the case when the Frattini subgroup of G is trivial. This is given in the following theorem of Herzog, Kaplan and Lev [10, Theorem A] (the same result is also proved independently by Halasi and Podoski in [11, Theorem 1.1]). Theorem 4.1. Let G be any non-abelian group with trivial Frattini subgroup. Then \|G/ Z(G)\| < \|γ2 (G)\|2 . The following result with the assertion similar to the preceding theorem is due to Isaacs [13]. Theorem 4.2. If G is a capable finite group with cyclic γ2 (G) and all elements of order 4 in γ2 (G) are central in G, then \|G : Z(G)\| ≤ \|γ2 (G)\|2 . Moreover, equality holds if G is nilpotent. So, there do exist nilpotent groups with comparatively small central quotient. A natural problem is the following. Problem 3. Classify all finite p-groups G such that \|G : Z(G)\| ≤ \|γ2 (G)\|2 . Let G be a finite nilpotent group of class 2 minimally generated by d elements. Then it Qd follows from Lemma 3.5 that \|G/ Z(G)\| ≤ i=1 exp([xi , G]), which in turn implies (4.3) \|G/ Z(G)\| ≤ \|exp(γ2 (G))\|d . Problem 4. Classify all finite p-groups G of nilpotency class 2 for which equality holds in (4.3). Now we discuss some examples of infinite groups with finite central quotient. The most obvious example is the infinite cyclic group. Other obvious examples are the groups G = H × Z, where H is any finite group and Z is the infinite cyclic group. Non-obvious examples include finitely generated F C-groups, in which conjugacy class sizes are bounded, and certain Cernikov groups. We provide explicit examples in each case. Let Fn be the free group on n symbols and p be a prime integer. Then the factor group Fn /(γ2 (Fn )p γ3 (Fn )) is the required group of the first type, where γ2 (Fn )p = hup \| u ∈ γ2 (Fn )i. Now let H = Z(p∞ ) × A be the direct product of quasi-cyclic (Prüfer) group Z(p∞ ) and the cyclic group A = hai of order p, where p is a prime integer. Now consider the group G = H ⋊ B, the semidirect product of H and the cyclic group B = hbi of order p with the action by xb = x for all x ∈ Z(p∞ ) and ab = ac, where c is the unique element of order p in Z(p∞ ). This group is suggested by V. Romankov and Rahul D. Kitture through ResearchGate, and is a Cernikov group. The following problem was suggested by R. Baer in [1]. Problem 5. Let A be an abelian group and Q be a group. Obtain necessary and sufficient conditions on A and G so that there exists a group Q with A ∼ = Z(G) and Q ∼ = G/A. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 9 This problem was solved by Baer himself for an arbitrary abelian group A and finitely generated abelian group G. Moskalenko [16] solved this problem for an arbitrary abelian group A and a periodic abelian group G. He [17] also solved this problem for arbitrary abelian group A and a non-periodic abelian group G such that the rank of G/t(G) is 1, where t(G) denotes the tortion subgroup of G. If this rank is more than 1, then he solved the problem when A is a torsion abelian group. For a given group Q, the existence of a group G such that Q ∼ = G/ Z(G) has been studied extensively under the theme ‘Capable groups’. However, to the best of our knowledge, Problem 5 has been poorly studied in full generality. Let us restate a special case of this problem in a little different setup. A pair of groups (A, Q), where A is an arbitray abelian group and Q is an arbitrary group, is said to be a capable pair if there exists a group G such that A ∼ = Z(G) and Q ∼ = G/ Z(G). So, in our situation, the following problem is very interesting. Problem 6. Classify capable pairs (A, Q), where A is an infinite abelian group and Q is a finite group. Finally let us get back to the situation when G is a group with finite γ2 (G) but infinite G/ Z(G). The well known examples of such type are infinite extraspecial p-groups. Other class of examples can be obtained by taking a central product (amalgamated at their centers) of infinitely many copies of a 2-generated finite p-group of class 2 such that γ2 (H) = Z(H) is cyclic of order q, where q is some power of p. Notice that both of these classes consist of groups of nilpotency class 2. Now if we take G = X × H, where X is an arbitrary group with finite γ2 (X) and H is with finite γ2 (H) and infinite H/ Z(H), then γ2 (G) is finite but G/ Z(G) is infinite. So we can construct nilpotent groups of arbitrary class and even non-nilpotent group with infinite central quotient and finite commutator subgroup. A non-nilpotent group G is said to be purely non-nilpotent if it does not have any non-trivial nilpotent subgroup as a direct factor. With the help of Rahul D. Kitture, we have also been able to construct purely non-nilpotent groups G such that γ2 (G) is finite but G/ Z(G) is infinite. Let H be an infinite extra-special p-group. Then we can always find a field Fq , where q is some power of a prime, containing all pth roots of unity. Now let K be the special linear group sl(p, Fq ), which is a non-nilpotent group having a central subgroup of order p. Now consider the group G which is a central product of H and K amalgamated at Z(H). Then G is a purely non-nilpotent group with the required conditions. It will be interesting to see more examples of this type which do not occur as a central product of such infinite groups of nilpotency class 2 with non-nilpotent groups. By Proposition 2.2 we know that for an arbitrary group G with finite γ2 (G) but infinite G/ Z(G), the group G/ Z(G) has an infinite abelian group as a direct factor. Further structural information is highly welcome. Problem 7. Provide structural information of the group G such that γ2 (G) is finite but G/ Z(G) is infinite? References [1] R. Baer, Groups with preassigned central and central quotient group, Trans. Amer. Math. Soc. 44 (1938), 387-412. [2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. CENTRAL QUOTIENT VERSUS COMMUTATOR SUBGROUP OF GROUPS 10 [3] J. M. Brady, R. A. Bryce, and J. Cossey, On certain abelian-by-nilpotent varieties, Bull. Austral. Math. Soc. 1 (1969), 403 - 416. [4] Y. Cheng, On finite p-groups with cyclic commutator subgroup, Arch. Math. 39 (1982), 295-298. [5] S. Dolfi, E. Pacifici and M. K. Yadav, Bounds for the index of the centre in non-nilpotent groups, preprint. [6] L. Fuchs, Infinite abelian groups, Vol. I Pure and Applied Mathematics, Vol 36 Academic Press, Yew York - London, 1970. [7] D. K. Gumber and H. Kalra, On the converse of a theorem of Schur, Arch. Math. (Basel) 101 (2013), 17-20. [8] P. Hall, The classification of prime power groups, Journal für die reine und angewandte Mathematik 182 (1940), 130-141. [9] P. Hall, Finite-by-nilpotent groups, Proc. Cambridge Phil. Soc. 52 (1956), 611-616. [10] M. Herzog, G. Kaplan and A. Lev, The size of the commutator subgroup of finite groups, J. Algebra 320 (2008), 980-986. [11] Z. Halasi and K. Podoski, Bounds in groups with trivial Frattini subgroups, J. Algebra 319 (2008), 893-896. [12] P. Hilton, On a theorem of Schur, Int. J. Math. Math. 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Soc. 21 (Series A) (1976), 467-472. [21] P. Niroomand, The converse of Schur’s theorem, Arch. Math. 94 (2010), 401-403. [22] K. Podoski and B. Szegedy, Bounds for the index of the centre in capable groups, Proc. Amer. Math. Soc. 133 (2005), 3441-3445. [23] I. Schur, Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, Journal für die reine und angewandte Mathematik 127 (1904), 20-50. [24] B. Sury, A generalization of a converse of Schur’s theorem, Arch. Math. 95 (2010), 317-318. [25] M. K. Yadav, Class-preserving automorphisms of finite p-groups: a survey, Groups St Andrews - 2009 (Bath), LMS Lecture Note Ser. 388 (2011), 569-579. [26] M. K. Yadav, Class-preserving automorphisms of finite p-groups II, Israel J. Math. 209 (2015), 355-396. DOI: 10.1007/s11856-015-1222-4 School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad - 211 019, INDIA E-mail address: myadav@hri.res.in	4
arXiv:1709.00410v2 [cs.NE] 10 Jan 2018 Visual art inspired by the collective feeding behavior of sand–bubbler crabs Hendrik Richter HTWK Leipzig University of Applied Sciences Faculty of Electrical Engineering and Information Technology Postfach 301166, D–04251 Leipzig, Germany. Email: hendrik.richter@htwk-leipzig.de. January 11, 2018 Abstract Sand–bubblers are crabs of the genera Dotilla and Scopimera which are known to produce remarkable patterns and structures at tropical beaches. From these pattern–making abilities, we may draw inspiration for digital visual art. A simple mathematical model is proposed and an algorithm is designed that may create such sand–bubbler patterns artificially. In addition, design parameters to modify the patterns are identified and analyzed by computational aesthetic measures. Finally, an extension of the algorithm is discussed that may enable controlling and guiding generative evolution of the art–making process. 1 Introduction Among the various forms of inspiration from biology for digital visual art, swarms and other types of collective animal behavior are considered to be particularly promising of showing algorithmic ideas and templates that may have the potential to create non–trivial patterns of aesthetic value. Examples are ant– and ant–colony–inspired visual art [12, 25], but our imagination has also been stimulated by the pattern– making of flies [1] and other swarms, flocks, or colonies [15, 22]. In this paper, the collective behavior of another family of animals provides a source of inspiration: crabs of the genera Dotilla and Scopimera, commonly called sand–bubblers. Sand–bubbler crabs are known for their burrow–orientated feeding behavior that produces radial patterns of regular shapes and designs in the wet sand of tropical beaches [2, 3, 7]. The patterns consist of tiny balls that are placed in curves or spirals, straight or bent lines, which finally form overall structures, thus producing astonishing works of natural art [4]. The creation of structures starts at receding tide. The patterns grow in complexity and scale as time goes by until the returning tide cleans them off, in fact resetting the art–making process. This pattern–making behavior, its visual results, and the potentials of thus inspired algorithms producing digital visual art are our topics. We approach these topics by discussing three interconnected main themes in this paper. The first is to describe an aspect of animal behavior observed in nature (the feeding pattern of a colony of sand–bubbler crabs) by economic mathematical means in order to make it accessible for an algorithmic process producing two–dimensional visual art. This goes along with identifying design parameters that allow the patterns to vary within the restriction that they should generally resemble structures as found in nature. The second theme is to study the image–producing process by computational aesthetic measures. Here, the focus is not on numerically evaluating an overall artistic value. There are some strong arguments that such an evaluation is rather elusive and may not really be achievable free from ambiguity, see also the discussion in Sec. 3. Instead, the focus is put on how some aesthetic measures, namely Benford’s law measure, Ross, 1 Figure 1: Feeding patterns of sand–bubbler crabs, observed by the author at Tanjung Beach, Langkawi, Malaysia, in March 2017. Ralph and Zong’s bell–curve measure and a fractal dimension measure, scale to the design parameters. A third theme is to use these measures to guide and control the art–making process. This is done by updating the design parameters while the algorithmic process is running. By employing the relationships between aesthetic measures and design parameters for updating the parameters, the measures can be changed in an intended way. Hence, this theme studies the possibilities of including generative development, which takes up some ideas from evolving and evolutionary art. The paper is structured as follows. Sec. 2 reviews the behavior of sand–bubbler crabs, particularly the burrow–orientated feeding process. Inspired by these behavioral patterns, Sec. 3 presents an algorithm to create visual art, studies computational aesthetic measures and generative development, and finally shows examples of art pieces thus created. Concluding remarks and an outlook end the paper. 2 Behavior of sand–bubbler crabs Sand–bubbler crabs are tiny crustacean decapods of the genera Dotilla and Scopimera in the family Dotillidae that dwell on tropical sandy beaches across the Indo–Pacific region. They permanently inhabit intertidal environments as they adapted to the interplay between high and low tide. Sand–bubblers are surface deposit feeders. Thus, their habitat must be periodically covered by water for their food supply to restore. They are also air–breathers. Thus, the habital zone needs to be periodically uncovered from water for giving them time for feeding. Observing these periodic changes, the crabs dig burrows in the sand where they seek shelter and hide during high tide. In low tide (and daylight) they swarm out in search for food. They pick up sand grains with their pincers and eat the microscopic organic particles that are left over by the receding tide and coat the grains. After they have cleaned the sand grains of eatable components, they form them into small balls, also called pseudo–faecal pellets, and throw them behind. Supposedly, they do so in order to avoid searching the same sand twice. As high tide comes back in, they retreat to their burrows, seal them to trap air, and wait for the next tidal cycle. By this process of feeding, and particularly by molding sand into pellets and distributing them, sand– bubbler crabs create intricate patterns in the sand that remain there until the next high tide washes them away, see Fig. 1 that shows such feeding patterns as observed by the author at Tanjung Beach, Langkawi, Malaysia in March 2017. The patterns in the sand start from a center point, have a radial orientation and comprise of small sand balls along a tiny path. The patterns vary in complexity depending on how many crabs there are contributing, on how long the last high tide is away, and on several other factors discussed below. The patterns in the sand are starting point and inspiration for an algorithm producing two–dimensional visual art. Before presenting such an algorithm in Sec. 3, it may be interesting to explore the feeding process and the pattern–making a bit more deeply. A main feature of the lifestyle of sand–bubblers is their burrow–orientated feeding behavior [2, 3, 5, 7]. 2 This means that the entrance of the burrow is the center of the feeding range and the feeding process is highly structured. Basically, such a structured and stereotype behavior can be found generally among the 14 species of Dotilla and 17 species of Scopimera currently recognized, but there is variety in the details of the feeding. This, in turn, offers to vary the art–making algorithm in order to achieve different graphic effects. The feeding starts from the burrow entrance, where the crab sets out to progress sideways along a straight line radiating from the burrow. While moving along, it sorts sand from organic–rich fine particles which are scraped off and subsequently ingested. The crab only feeds on the upper millimeter or two of the sand. The residual larger nutrient–deficient sand grains are molded into ball–like globules, but not digested, which motivates calling them pseudo–faecal pellets. When a pellet has reached a certain diameter, which scales to the size of the crab, it is moved along the body and pushed away behind the animal. After moving along the straight line radiating from the burrow and ejecting pellets for a certain distance, the crab steps aside, turns around and moves back toward the burrow entrance while continuing with feeding and pellet ejecting. When reaching the entrance, its turns around again, and works the next line (see also Fig. 2 for a schematic description). The pellets are placed in such a way that a straight path to the burrow entrance always remains clear. This is essential for the crab to retreat to the burrow in case of external disturbance or danger. Consequently, the feeding progresses radially line by line while the lines (which are also called trenches in the biological literature) rotate around the burrow entrance just like the hands of a clock. In fieldwork it has been observed that the rotation may be clockwise or anti–clockwise with approximately the same frequency [17, 20]. By such a rotation of the feeding trench, an angular sector of a circle in the sand is excavated, cleared of eatable components, and amassed with pellets. The angular sector of uninterrupted feeding may vary; an average of 60◦ has been reported, but also that occasionally a full circle has been completed [2, 13, 17], see also the right–hand side of Fig. 1. Furthermore, it has been reported that there is a fairly close relationship between the maximum length of a trench and the size of the crab. Such a relationship is plausible as the maximum length together with the crab size scales to the maximum time needed for the crab to seek shelter in the burrow, if an external disturbance requires such an escape. For one species of sand–bubblers, Dotilla wichmanni, even a measurement of the maximum length has been recorded with the body length of eight to nine crabs [17]. Furthermore, another stereotype of pattern has been identified as sometimes the distribution of pellets along the trench follows a rhythmic pattern where the crab regularly suspends pellet ejecting and “concentric rings” emerge, as shown for Dotilla clepsydrodactylus [20]. In view of the structured burrow–orientated feeding behavior of sand–bubbler crabs as described above, it seems not surprising that regular patterns in the sand emerge. Ideally, a well–ordered collection of full circular areas with the burrow entrance in the center should appear that are filled with radial strings of equally distanced pellets. However, looking at patterns from nature (see, for instance, Fig. 1) reveals them as irregular in several respects. There are reasons for these irregularities. A first reason is that for a colony of sand–bubblers the coordinates of the burrow entrances are randomly distributed across the beach where they are dwelling. There is almost always a certain minimum distance between two burrow entrances and the number of burrows for a given square of beach follows a Poisson distribution, as shown for the species Dotilla fenestrata [11]. A second reason is that the orientation of the first trench (and hence the overall orientation of the angular sector) is not different from a random distribution [17, 20]. There is no indication that the orientation is related to the slope of the sand surface, or any compass direction or the sun’s position. Another reason, for instance reported for the species Scopimera inflata [7], is that a crab may move a short distance away from the entrance of the burrow before starting to feed, which results in a gap between burrow and trench. The width of the gap appears to be randomly distributed around the crab’s size. A further reason is that for a given burrow (and therefore a crab of given size) the trenches have not all the same length, even if the feeding is undisturbed [11]. There is a weak correlation between trench length and feeding time, with the trenches growing longer the longer the crab feeds, but generally also trench length must be regarded as to follow a random distribution. Finally, a crab may be externally disturbed at any time, for instance by intruders such as other crabs or human observers, or eating enemies such as shorebirds. 3 y j=36 s Cs j = 4 j=2 Cs J B J C B J^ BMB j = 1 J J Cs s rk J Cs M J Cs Pi j =5P θ1 PP D P JC u x D T (xi , yi ) > j = 6 DD Figure 2: Algorithmically generating a pellet location with the burrow coordinates (xi , yi )T , the trench angle θ1 of the (lower branch of the) first trench and the radial coordinate rk ; see also the first term of the right–hand side of Eq. (1). The arrows indicate the feeding direction of a sand–bubbler crab. The outbound trench j = 3 depicts a radial string of six equally distanced pellets, while for j = 1 there is one pellet; for the other trenches the pellets are not shown. As sand–bubblers are extremely perceptive to movement, in such a case the crab rapidly escapes to the burrow. It may reemerge a little later and resume eating. The feeding may or may not continue along the trench last used before escape. It has been convincingly argued by field experiments that sand–bubblers (for instance the species Dotilla wichmanni) must possess remarkable orientation abilities that allow them to resume the direction along which they were feeding, even in the absence of visual clues such as a partly finished trench [17]. Whether or not a crab uses these abilities in a given situation, however, also appears to follow a random distribution. Lastly, a crab may interrupt its feeding even if there is no evident external trigger, for instance to rest or to restore its water supply, which is done in the burrow. To sum it up: The exact position of each pellet follows from the structure of the burrow–oriented feeding behavior, but is subject to a substantial degree of random. If we reframe the organized and sequential placement of a sufficiently large number of pellets as a dynamic process, the pattern generation could be interpreted as a random walk substantially biased by the feeding structure. This interplay between determinism and random may be one of the reasons why the patterns in the sand appear non–trivial and of some aesthetic value. Finally, we may note that apart from feeding, there are at least two further aspects of the sand–bubbler’s lifestyle that might be interesting as inspiration for visual art, burrow–digging and interaction with other crabs. These aspects could be addressed in future work. 3 3.1 Design of art–making algorithms Generating patterns from pellet distribution The feeding behavior of sand–bubbler crabs as described in Sec. 2 is now used to design templates for algorithmically generating two–dimensional visual art. With respect to the regular and irregular aspects of the sand patterns, the art–making algorithm is also comprised of deterministic and random elements. Describing the algorithm starts with the notion that every pellet has a location (xijk , yijk )T in a two– dimensional plane. The index (ijk) refers to the k–th pellet (k = 1, 2, 3, . . . , Kij ) belonging to the j–th trench (j = 1, 2, 3, . . . , Ji ) of the i–th burrow (i = 1, 2, 3, . . . , I). The pellet location is algorithmically specified by 2 ) N (µijk , σijk xijk xi + rk · cos (θj ) = + , (1) 2 ) yijk yi + rk · sin (θj ) N (µijk , σijk where (xi , yi )T are the coordinates of the i–th burrow, θj is the trench angle belonging to the j–th trench, and rk is the radial coordinate of the k–th pellet. For each burrow, the maximum number of pellets Kij remains to be specified for a given i and j; the same applies to the maximum number of trenches Ji . Fig. 2 unpacks the description and shows an example with one burrow and six trenches (three outbound and three returning). 4 Algorithm 1 Generate patterns from pellet distribution Set maximum number of burrows I for i=1 to I do Generate burrow coordinates (xi , yi )T Set maximum number of trenches Ji for j=1 to Ji do Generate trench angle θj Set maximum number of pellets Kij for k=1 to Kij do Generate pellet radial coordinate rk 2 Set mean µijk and variance σijk Calculate pellet location (xijk , yijk )T by Eq. (1) Return location and store for subsequent coloring and visualization end for end for end for The first term of the right–hand side of Eq. (1) can be seen as to represent the deterministic aspect of the pellet ejecting that characterizes the burrow–orientated feeding behavior of sand–bubblers. The second term represents the random aspect with shifting the position by a realization of a random variable 2 . Accordingly, every pellet location (x , y )T has normally distributed with mean µijk and variance σijk ijk ijk three deterministic parameters: burrow coordinates (xi , yi )T , trench angle θj and radial coordinate rk ; 2 of the normal distribution from which the and two stochastic parameters: mean µijk and variance σijk random shift in position is realized. Eq. (1) not only specifies the location of a pellet, it can also be used to generate patterns consisting of pellets, trenches and burrows, see Algorithm 1. Therefore, for each burrow coordinates (xi , yi )T , one sequence of trench angles (θ1 , θ2 , . . . , θJi ) and Ji sequences of pellet radial coordinates (r1 , r2 , . . . , rKij ) are defined. A major guideline for defining these parameter sequences is the intention to replicate characteristic features of the sand–bubbler pattern as discussed in Sec. 2. These patterns can be labeled according to the graphic effects they convey and subsequently taken as templates to create two–dimensional visual art works. In the numerical experiments reported in Sec. 3.4 the following templates are used: (i) random trench length (RTL), (ii) growing trench length (GTL), (iii) concentric rings (CCR), and (iv) burrow–to–trench gaps (BTG). Each template varies the art–making algorithm and replicates a specific pattern found in nature. For random trench length (RTL), the maximal number of trenches Ji is a random integer uniformly distributed on [1, Jmax ]. The first and the last trench angle (θ1 and θJi ) are chosen as realizations of a uniform random variable distributed on [0, 2π]; the remaining trench angles are calculated from the number Ji to be equidistant between θ1 and θJi . Each trench has a length `j that is a realization of a random variable normally distributed on [µT , σT2 ]. From the length `j , the pellet radial coordinates are calculated to be equidistant with distance dj . Growing trench length (GTL) is similar to RTL (and uses the same general design parameters), but the trench length `j is calculated by a normal distribution where the mean µT grows with trench j by a growth rate ∆µT . Thus, the first trench has mean µT , while the Ji –th trench has mean µT + (Ji − 1)∆µT . The variance σT2 is constant. Concentric rings (CCR) again have trenches with constant mean and variance but do not always have equidistant pellet radial coordinates as there are λ gaps of width gλ where no pellets are placed. Finally, burrow–to–trench gaps (BTG) start with a gap of width g0 in pellet placement. Tab. 1 summarizes the design parameters. Algorithm 1 specifies the location of the pellets and hence the structure of the patterns. In nature, the patterns are monochromatic as the pellets all have the color of the sand they are built from, see Fig. 1. The artistic interpretation of the patterns suggests using different colors for making them more visually appealing and also conveying additional information. As the algorithm imposes a chronological order on pellet placement by a forward counting of burrows, trenches and finally the pellets themselves, a possible 5 Table 1: Design parameters and default values for generating patterns from pellet distribution, see Algorithm 1. The parameters of RTL are general and also apply to the other templates. Template Design parameter Symbol Default value RTL Number of burrows I 3 T (and Burrow coordinates (xi , yi ) N (0, 7) general) Maximum number of trenches Jmax 50 Pellet distance dj 0.25 Mean random shifting µijk −1, 0, 1 2 Variance random shifting σijk 0.3, 0.8 Mean trench length µT 25 Variance trench length σT2 1 GTL Grow rate ∆µT 2 CCR Number of gaps λ 3 Gap width gλ 4 BTG Burrow gap width g0 8 design is to use a colorization that reflects this order. Alternatively, each burrow can be associated with a specific color and the order of the pellet placement can be characterized by shades or brightness of this color. In addition, the burrows using the same template (see Tab. 1) may take colors that are similar. 3.2 Analysis by computational aesthetic measures Next, the templates and their design parameters are analyzed by computational aesthetic measures. This analysis aims at studying two questions. (i) To what extend do the measures vary over the templates and parameters? This goes along with assessing how sensitive these relationships are. (ii) Which templates and/or parameters are most suitable to guide and control the art–making process toward desired values of the aesthetic measures? Three computational aesthetic measures are studied, Benford’s law measure (BFL), Ross, Ralph and Zong’s bell curve measure (RRZ) and a fractal dimension measure (FRD), which were chosen because a recent study reported them to be weakly correlated [6]. All measures are calculated for the images with 512 × 512 pixels. The Benford’s law measure BFL [6, 21] is known for scaling to the naturalness of an image. It is calculated by taking the distribution of the luminosity of an image and comparing it to the Benford’s law distribution: BFL = 1 − dtotal /dmax (2) P9 where dtotal = i=1 (HI (i) − HB (i)), dmax = 1.398 is the maximum possible value of dtotal , HB = (0.301, 0.176, 0.125, is the Benford’s law distribution and HI is the normalized sorted 9–bin histogram of the luminosity of the image. The luminosity of the image is calculated for each pixel of the image by taking the weighted sum of the red (R), green (G) and blue (B) values: lum = 0.2126 · R + 0.7152 · G + 0.0722 · B. Ross, Ralph and Zong’s bell–curve measure RRZ [6, 23] is a measure of color gradient normality. It compares the color gradients of the image to a normal (bell-curve or Gaussian) distribution. Therefore, the color gradient of the colors red, green and blue are calculated for each pixel, summarized and normalized by a detection threshold, see [6, 23] for details. Hence, we obtain a distribution with mean µ and variance σ 2 . The distribution is discretized into 100 bins to obtain a discrete color gradient distribution pi . For µ and σ 2 calculated from the observed color gradient distribution, an expected discrete Gaussian distribution qi is computed. Finally, the Kullback–Leibler divergence between these two distributions gives the bell–curve measures X RRZ = pi log (pi /qi ). (3) i 6 Figure 3: Aesthetic measures as a function of design parameters: number of burrows I, maximum number of trenches Jmax and pellet distance dj . RTL (blue), GTL (magenta), CCR (red), BTG (green); solid lines, 2 = 0.3, dotted lines, σ 2 = 0.8. σijk ijk Note that in deviation from [6, 23], the Kullback–Leibler divergence in Eq. (3) is not amplified by the factor 1000. The third aesthetic measure studied is the fractal dimension FRD [6, 24] of the image. Following empirical studies and an argument by Spehar et al. [24], a fractal image of dimension 1.3 ≤ d ≤ 1.5 is most preferred by human evaluation, compared to images that have a fractal dimension outside this range. Thus, a fractal dimension d ≈ 1.35 is considered to be most desirable and the fractal aesthetic measure FRD = max (0, 1 − \|1.35 − d\|) (4) can be defined, where d is the fractal dimension of the image calculated by box–counting. Fig. 3 shows the measures for the templates RTL, GTL, CCR and BTG for selected design parameters 2 . The results are means over 100 images generated and two values of the variance of the random shifting σijk by Algorithm 1 and the design parameters listed in Tab. 1. The design parameter number of burrows I is varied over 2 ≤ I ≤ 10, the maximum number of trenches Jmax over 20 ≤ Jmax ≤ 100 and the pellet distance dj over 0.05 ≤ dj ≤ 1. The default values of these parameters all lie within the range studied. 7 Furthermore, as the aesthetic measures are calculated from the RGB values of the pixels, the colors of the image have a profound effect on the values of the measures. As discussed in Sec. 3.1, the colorization of the pellets is also a subject of algorithmic design. Thus, to counter the effect of a given colorization and for not singularizing particular colors, the results are calculated as means over 70 hue values randomly selected. From the results in Fig. 3, we see that the different templates (RTL, GTL, CCR, BTG) generally produce characteristic curves over the design parameters for the aesthetic measures BFL and RRZ. Also, the two values of random shifting (which can be interpreted as noise levels) are clearly distinct. The general trend is that BFL and RRZ decrease with increasing number of burrows I and increasing maximum number of trenches Jmax , while there is an increase with increasing pellet distance dj . An interesting exception is the template CCR which for varying pellet distance dj behaves differently and crosses the curves of the other templates. The results for the aesthetic measures FRD are slightly different. For FRD, the templates RTL and BTG (blue and green lines) have almost the same results and for all templates, different noise levels have only negligible effect. A possible explanation is that the difference between RTL and BTG is only that in the images there is a circle free from pellets around the burrow entrance for BTG. The overall structure of the pellet placement is the same, which is recognized by the calculation of the fractal dimension accounting mainly for density of geometrical objects but less for their layout. The same may apply for varying the noise level. Based on these results assessing how sensitive the relationships between design parameters and aesthetic measures are, some conclusions can be drawn about possible ways to guide and control the art– making algorithm. Basically, there are two options. One is to switch to a particular template but keep the values of design parameters, another is to keep the template but change the values of design parameters. (In fact, it would also be thinkable to change both template and parameter value at the same time, but this may appear rather contrived.) For instance, to increase the aesthetic measure BFL, employing the template BTG is suitable, while RRZ is largest for RTL (but not very clearly) and FRD for GTL. Another possibility to raise BFL is to decrease the maximum number of trenches Jmax . The same is also useful to increase RRZ, particularly for the template CCR, but not for FRD. There are more subtle schemes to manipulate the measures, but it is also clear that the three aesthetic measures studied here react differently, yet even contradictory to these manipulations. With this in mind, next section discusses possible designs of such manipulations and their effects on the art–making process. 3.3 Guiding and controlling generative evolution Main features of generative and evolving digital art are that the creating process develops over time, is guided and controlled by the algorithmic design, and thus gains functional autonomy to produce complete art works [10]. A frequently used design for guidance and control is to implement a feedback, for instance via evaluating the works by using a measure of their aesthetic value [18, 22]. Such ideas are, for instance, implemented by employing evolutionary algorithms for finding “optimal” values of the aesthetic measures, which has shown to create interesting works of art [1, 6, 14, 22]. Algorithm 1 given in Sec. 3.1 does not really, in itself, evoke such a perspective of autonomously guiding and controlling generative evolution. The algorithm is suitable for producing visual art works. It also depends on selecting templates and design parameters (see Tab. 1), which enables modifying the works by changing the selection. Thus, promoting generative evolution essentially requires measuring the aesthetic value of the algorithmically generated patterns and designing a feedback. Of course, it is possible to evaluate the pattern by interaction with humans. However, there are some issues with interactive evaluation [8, 9, 22]. The first is that humans evaluating art are usually much slower in doing so than computers algorithmically generating the works, which creates the proverbial “fitness bottleneck”. A second reason is that human evaluation may change over time, for instance caused by fatigue, but also by boredom, which may bias the results towards superficial novelty rather than overall quality. A third reason is that human evaluation of art is always conditioned by personal and cultural “taste”. For this reason, it is no real help to distribute the evaluation to a larger group of human evaluators. In such a case, either the tastes of different subgroups become visible (as for instance shown for evaluating 8 Algorithm 2 Guide and control pattern–making Set up look–up table with template and design parameters that increase computational aesthetic measures Calculate expected aesthetic measures for the look–up table Select random template and take default design parameter while Termination criterion not met do Generate burrow by Algorithm 1 Calculate aesthetic measures if Measure < Expectation then Select randomly increasing template or design parameter else Keep template and parameter end if end while the beauty of abstract paintings [19]), or generally the averaging effect of asking a large group of people about their aesthetic choices is reproduced, but arguably “the unique kind of vision expected from artists” [8] is not achieved. An alternative are computational aesthetic measures [6, 8, 9, 19, 21], which enable an automated rating without human interference or supervision. Sec. 3.2 discussed such measures for evaluating sand–bubbler patterns. These aesthetic measures of two–dimensional visual art are frequently calculated by analyzing the (spatial) distribution of colored pixels (or groups of pixels). In other words, computational aesthetic measures use properties such as pixel–wise color information to deduce an overall rating of an image. However, there is a significant conceptual gap between measuring computational image properties and aesthetic beauty. Apart from the measures considered in Sec. 3.2, there is a huge number of measures that try to bridge this gap, see e.g. [9, 6, 16, 19] for an overview. Some of them are strongly correlated, while others address different aspects of the color distribution. Moreover, recent studies [6, 19] suggested that it seems rather difficult to establish stable correlations between beauty ratings by humans and image properties, which casts some doubts on whether an evaluation of the aesthetic value of visual art based on computational aesthetic measures is really objective, meaningful and feasible. These studies have also shown that computational aesthetic measures account more for certain visual effects (or “visual styles” [6]) than for aesthetic value in general. For instance, Benford’s law measure (BFL) assigns high values for images that have a grainy texture, while Ross, Ralph and Zong’s bell curve measure (RRZ) likes distinct color progression and the fractal dimension measure (FRD) values low colorfulness highly; see den Heijer & Eiben [6] for a discussion and comparison of 7 computational aesthetic measures. In the following, we study an alternative approach for guiding and controlling generative evolution that tries to address the desire to promote certain visual effects, see Algorithm 2. It utilizes the relationships between design parameters and aesthetic measures discussed in Sec. 3.2. Therefore, a look–up–table is set up that lists templates and design parameters that increase aesthetic measures. After generating the pattern for each single burrow, an aesthetic measure is calculated and compared to an “expected value” recorded before, see Fig. 3. If the value is below the expectation, a measure–increasing template or design parameter is activated. If not, the generation process continues unaltered. Which template or parameter is selected is due to chance. These steps are repeated until a termination criterion is met. In this implementation, the pattern generation ends if a maximum number of burrows are placed in the image. (In the rare case that no measure–increasing template or parameter is available, the setting is also kept.) An interesting feature of such an algorithmic design is that it immediately allows to control and intervene while the image is being created. In other words, the control does not wait until the image is completed, but uses feedback for intervening during the process creating the art works. Fig. 4 shows the aesthetic measures BFL, RRZ and FRD for Algorithm 2. The experimental setup is the same as for the templates, see Fig. 3. Again, the 9 Figure 4: Aesthetic measures for guiding and controlling pattern–making. The measures for the templates (RTL, GLT, CCR, BTG) with the default values of the design parameters (see Tab. 1) are compared to the measures obtained by Algorithm 2. GCB means using the measure BFL to guide and control, GCR uses RRZ, GCF uses FRD. results are averages over 100 images and 70 random hue values. The aesthetic measures for the templates in Tab. 1 are compared to using the measures BFL (GCB ), RRZ (GCR ) and FRD (GCF ) for guiding and controlling the art–making process. Apart from the results of the measures that are obtained by using the same measures to guide and control, we also record the results of the other measures to examine cross–effects (GCR and GCF for BFL, GCB and GCF for RRZ and GCB and GCR for FRD). The results in Fig. 4 indicate that using templates and design parameters is mostly suitable to manipulate the aesthetic measures in an intended way. For BFL we obtain that the measure for GCB is increased as compared to the templates RTL, GTL and CCR, while the images do contain a mix of all templates. For FRD, we even find an increase in GCF compared to all templates. However, for RRZ we get a rather strong decrease in GCR . A possible explanation is that the measure RRZ accounts for color gradient normality, while the effect of Algorithm 2 is basically in changing the selection of templates and design parameters but only indirectly in colorization. Hence, the measure RRZ is poorly sensitive to such a changed selection. These speculations are supported by looking at cross–effects as we see that also the measures that are not used for guidance and control show similar results. This additionally allows to conjecture that for images such as the feeding patterns the aesthetic measures are more correlated than initially assumed. Next, we look at the visual results produced by both algorithms1 . 3.4 Examples of art works Fig. 5 shows a gallery of nine art works generated by Algorithm 1 with the parameter values given in Tab. 1. Looking at these images, we can see that they all display structures that have similarity to natural sand– bubbler patterns. However, colorization of the actual choices of the random–dependent design parameters also offers diversity. In addition, another six images generated by Algorithm 2 are shown in Fig. 6. Although there is a substantial degree of similarity between these images and the images generated by Algorithm 1, it is worth observing some subtle differences. For instance, it is noticeable that typically the images produced by Algorithm 2 are more widespread and grainier, see for instance Fig. 6 upper left and lower middle, which may be attributed to the effect of promoting the visual effects associated with the measure BFL. However, at least in the opinion of the author, there was no real success in provoking a distinct color progression or a particularly low colorfulness, as supposedly associated with RRZ and FRD. More analytic work is needed to clarify the relationships between aesthetic measures and visual effects for a given image–producing algorithm. 1 See https://feit-msr.htwk-leipzig.de/sandbubblerart/ for further images and videos. 10 Figure 5: Gallery with examples of sand–bubbler inspired art works generated by Algorithm 1. 4 Concluding remarks and outlook An algorithm to create visual art has been presented that draws inspiration from the collective feeding behavior of sand–bubbler crabs. Thus, the paper exemplifies another instance of an art–making process based on mathematical models of animal behavior observed in nature. Apart from the algorithmic design and its artistic output, we mainly studied the algorithmic process, particularly how computational aesthetic measures scale to the design parameters and how these measures can be used for controlling and guiding generative evolution. The interest in analyzing the algorithmic process of digital art–making is from both a computational and an artistic point of view. The rationale for the latter is that art–theoretically and following an argument of Philip Galanter [8, 10], in generative art the artistic work in itself is not seen as important as the artistic process. In a reminiscence to the art movement of “truth to material”, generative art may focus on “truth to process”. A possible interpretation states that it is far less interesting to just ask if a human observer likes a particular piece of artificial art. What is much more interesting is to analyze the creation procedure of generative art and study the interplay between algorithmic design (and design parameters) and the works thus created. Not only is the question of whether or not an artificially created image is more or less aesthetic from a human point of view merely opening up an inexhaustible topic 11 Figure 6: Gallery with examples of sand–bubbler inspired art works generated by Algorithm 2. of debate and might be not answerable, this question may actually not be very useful for advancing our understanding of what constitutes artistic beauty and how it can be algorithmically (re–)created. In this spirit the galleries of art works shown in Figs. 5 and 6 could be rather seen as accompanying the analytic results, and not the other way around. The analytic and visual results given here only use a small subset of the design space opened up by Algorithm 1 and 2. Future work could further explore this design space. For instance, the images only contain pellets of a given size and color. 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ICA based on the data asymmetry arXiv:1701.09160v1 [math.ST] 31 Jan 2017 P. Spurek, J. Tabor Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Cracow, Poland P. Rola Department of Mathematics of the Cracow University of Economics, Rakowicka 27, 31-510 Cracow, Poland M. Ociepka Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Cracow, Poland Abstract Independent Component Analysis (ICA) - one of the basic tools in data analysis - aims to find a coordinate system in which the components of the data are independent. Most of existing methods are based on the minimization of the function of fourth-order moment (kurtosis). Skewness (third-order moment) has received much less attention. In this paper we present a competitive approach to ICA based on the Split Gaussian distribution, which is well adapted to asymmetric data. Consequently, we obtain a method which works better than the classical approaches, especially in the case when the underlying density is not symmetric, which is a typical situation in the color distribution in images. Keywords: ICA, Split Normal distribution, skewness. 1. Introduction Independent component analysis (ICA) is one of the most popular methods of data analysis and preprocessing. Historically, Herault and Jutten [1] Email addresses: przemyslaw.spurek@ii.uj.edu.pl,jacek.tabor@ii.uj.edu.pl (P. Spurek, J. Tabor), przemyslaw.rola@outlook.com (P. Rola) Preprint submitted to Pattern Recognition February 1, 2017 seem to be the first (around 1983) to have addressed the problem of ICA to separate mixtures of independent signals. In signal processing ICA is a computational method for separating a multivariate signal into additive subcomponents and has been applied in magnetic resonance [2], MRI [3, 4], EEG analysis [5, 6, 7], fault detection [8], financial time series [9] and seismic recordings [10]. Moreover, it is hard to overestimate the role of ICA in pattern recognition and image analysis; its applications include face recognition [11, 12], facial action recognition [13], image filtering [14], texture segmentation [15], object recognition [16, 17], image modeling [18], embedding graphs in pattern-spaces [19, 20], multi-label learning [21] and feature extraction [22]. The calculation of ICA is discussed in several papers [23, 24, 25, 26, 27, 28, 29], where the problem is given various names, in particular it is also called “source separation problem”. ICA is similar in many aspects to principal component analysis (PCA). In PCA we look for an orthonormal change of basis so that the components are not linearly dependent (uncorrelated). ICA can be described as a search for the optimal basis (coordinate system) in which the components are independent. Let us now, for the readers convenience, describe how the ICA works. Data is represented by the random vector x and the components as the random vector s. The aim is to transform the observed data x into maximally independent components s with respect to some measure of independence. Typically we use a linear static transformation W , called the transformation matrix, combined with the formula s = W x. Most ICA methods are based on the maximization of non-Gaussianity. This follows from the fact that one of the theoretical foundations of ICA is given by the dual view at the Central Limit Theorem [30], which states that the distribution of the sum (average or linear combination) of N independent random variables approaches Gaussian as N → ∞. Obviously if all source variables are Gaussian, ICA will not work. The classical measure of non-Gaussianity is kurtosis (the forth central moment), which can be both positive or negative. Random variables that have a negative kurtosis are called subgaussian, and those with the positive one are called supergaussian. Supergaussian random variables have typically a “spiky” pdf with heavy tails, i.e. the pdf is relatively large at zero and at large values of the variable, while being small for intermediate values (ex. the Laplace distribution). Typically non-Gaussianity is measured by the absolute value of kurtosis (the square of kurtosis can also be used). Thus many methods of finding independent components are based on fitting a 2 Figure 1: Comparison of images separation by our method (ICASG ), with FastICA and ProDenICA. density with similar kurtosis as the data, and consequently are very sensitive to the existence of outliers. Moreover, typically data sets are bounded, and therefore the credible estimation of tails is not easy. Another problem with these methods, is that they usually assume that the underlying density is symmetric, which is rarely the case. In our work we introduce and explore a new approach ICASG , based on the asymmetry of the data, which can be measured by the third central moment (skewness). Any symmetric data, in particular gaussian, has skewness equal to zero. Negative values of skewness indicate the data skewed to the left and the positive ones indicate the data skewed to the right1 . Consequently, 1 By skewed to the left, we mean that the left tail is long relative to the right tail. Similarly, skewed to the right means that the long tail is on the right-hand side. 3 skewness is a natural measure of non-Gaussianity. In our approach, instead of approximating the data by product of densities with heavy tails, we approximate it by a product of asymmetric densities (so called Split Gaussians). Contrary to classical approaches which consider third or fourth central moment, our algorithm is based on second moments. This is a consequence of the fact that Split Gaussian distributions arise from merging two opposite halves of normal distributions in their common mode (for more information see Section 4). Therefore we use only second order moments to describe skewness in dataset, and therefore we obtain an effective ICA method which is resistant to outliers. Figure 2: MLE estimation for image histograms with respect to Logistic and Split Gaussian distributions. The results of classical ICA and ICASG in the case of image separation (for more detail comparison we refer to Section 6) is presented in Fig. 1. In the experiment we mixed two images (see Fig. 1 a) by adding and subtracting them (see Fig. 1 b). Our approach gives essentially better results than the classical FastICA approach, compare Fig. 1 c) to Fig. 1 d) and Fig. 1 f) to Fig. 1 g). In the case of classical ICA we can see artifacts in background, which means that the method does not separate signal properly. On the other hand, ProDenICA and ICASG almost perfectly recovered images, compare Fig. 1 c) to Fig. 1 e) and Fig. 1 f) to Fig. 1 h). In general, ICASG in most cases gives better results than other ICA methods, see Section 6, while its numerical complexity lies below the methods 4 Figure 3: Comparison between our approach and classical ICA in the case of resistance on outliers. which obtain comparable results, that is ProDenICA and PearsonICA. This is caused in particular by the fact that asymmetry is more common than heavy tails in real data sets – we performed the symmetry test by using R package lawstat [31] with 5 percent confidence ratio, and it occurred that all image datasets we used in our paper have asymmetric densities. We also verified it in the case of density estimation of our images. We found optimal parameters of Logistic and Split Gaussian distributions and compared the values of MLE function in Fig. 2. As we see, in most cases Split Gaussian distribution fits the data better than the Logistic one. Summarizing the results obtained in the paper, our method works better than classical approaches for asymmetric data, and is more resistant to outliers (see Example 1.1). Example 1.1. We consider the data with heavy tails (a sample from the Logistic distribution) and skewed ones (a sample from the Split Normal distribution). We added to the data outliers uniformly generated from rectangle [min(X1 ) − sd(X1 ), max(X1 ) + sd(X1 )] × [min(X2 ) − sd(X2 ), max(X2 ) + sd(X2 )], where sd(Xi ) is a standard deviation of the i-th coordinate of X. In Fig. 3 we present how the absolute value of the Tucker’s congruence coefficient (the similarity measure of extracted factors, see Section 6) is changing when we add the outliers. As we see, ICASG is more stable and deals better with outliers in the 5 Figure 4: Logistic, Split Normal and Classical Gaussian distribution fitted to data with heavy tails and skew one. data, which follows from the fact that classical ICA typically depends on the moments of order four, while our approach uses moments of order two. This paper is arranged as follows. In the second section, we discuss related works. In the third, the theoretical background of our approach to ICA is presented. We introduce a cost function which uses the General Split Gaussian distribution and show that it is enough to minimize it respectively to only two parameters: vector m ∈ Rd and d×d matrix W . We also calculate the gradient of the cost function, which is necessary for the efficient use in the minimization procedure. The last section describes numerical experiments. The effects of our algorithm are illustrated on simulated and real datasets. 2. Related works Various ICA methods were discussed in [23, 24, 25, 26, 27, 28]. Herault and Jutten seem to be the first who introduced the ICA around 1983. They proposed an iterative real-time algorithm based on a neuro-mimetic architecture, which nevertheless, can show the lack of convergence in a number of cases [32]. It is worth mentioning that in their framework, higher-order statistics were not introduced explicitly. Giannakis et al. [33] addressed the 6 issue of identifiability of ICA in 1987 using third-order cumulants. However, the resulting algorithm required an exhaustive search. Lacoume and Ruiz [34] sketched a mathematical approach to the problem using higher-order statistics, which can be interpreted as a measure of fitting independent components. Cardoso [35, 36] focused on the algebraic properties of the fourth-order cumulants (kurtosis) what is still a popular approach [37]. Unfortunately kurtosis has some drawbacks in practice, when its value has to be estimated from a measured sample. The main problem is that kurtosis can be very sensitive to the outliers. Its value may depend on only a few observations in the tails of the distribution. In high-dimensional problems, where separation process contains PCA (for dimension reduction), whitening (for scale normalization), and standard ICA this effect is called a small sample size problem [38, 39]. This is caused by the fact that for the high-dimensional data sets ICA algorithms tend to extract the independent features simply by the projections that isolate single or very few samples (outliers). To address the difficulty random pursuit and locality pursuit methods were applied [39]. Another commonly used solution is to use skewness [40, 41, 42, 43] instead of kurtosis. Unfortunately, skewness has received much less attention than kurtosis, and consequently methods based on skewness are usually not well theoretically justified. One of the most popular ICA method dedicated to the skew data is PearsonICA [44, 45], which minimizes mutual information using a Pearson [46] system-based parametric model. The model covers a wide class of source distributions including skewed distributions. The Pearson system is defined by the differential equation f 0 (x) = (a1 x − a0 )f (x) , b0 + b1 x + b2 x 2 where a0 , a1 , b0 , b1 and b2 are the parameters of the distribution. The parameters of the Pearson system can be estimated using the method of moments. Therefore such algorithms have strong limitations connected with the optimization procedure. The main problems are number of parameters which have to be fitted and numerical efficiency of the minimization procedure. An important measure of fitting independent components is given by negentropy [47]. FastICA [48], one of the most popular implementations of ICA, uses this approach. Negentropy is based on the information-theoretic 7 quantity of (differential) entropy. This concept leads to the mutual information which is the natural information-theoretic measure of the independence of random variables. Consequently, one can use it as the criterion for finding the ICA transformation [28, 49]. It can be shown that minimization of the mutual information is roughly equivalent to maximization of negentropy and it is easier to estimate since we do not need additional parameters. ProDenICA [50, 51] is based not on a single nonlinear function, but on an entire function space of candidate nonlinearities. In particular, the method works with the functions in a reproducing kernel Hilbert space, and make use of the “kernel trick” to search over this space efficiently. The use of a function space makes it possible to adapt to a variety of sources and thus makes ProDenICA algorithms more robust to varying source distributions. A somewhat similar approach to ICA is based on the maximum likelihood estimation [27]. It is closely connected to the infomax principle since the likelihood is proportional to the negative of mutual information. In recent publications, the maximum likelihood estimation is one of the mot popular [24, 52, 53, 54, 55, 56, 57] approaches to ICA. Maximum likelihood approach needs the source pdf. In the classical ICA it is common to use the superGaussian logistic density or other heavy tails distributions. In this paper we present ICASG , a method which joins the positive aspects of classical ICASG approaches with recent ones like ProDenICA or Pearson ICA. First of all we use a General Split Gaussian distribution, which uses second order moments to describe skewness in dataset, and therefore is relatively robust to noise or outliers. The GSG distribution can be fitted by minimizing a simple function, which depends on only two parameters m ∈ Rd , W ∈ M(Rd ), see Theorem 5.1. Moreover we calculate its gradient, and therefore we can use numerically efficient gradient type algorithms, see Theorem 5.2. 3. Theoretical justification Let us describe the idea2 behind ICA [30]. Suppose that we have a random vector X in Rd which is generated by the model with the density F . Then it is well-known that components of X are independent iff there exist onedimensional densities f1 , . . . , fd ∈ DR , where by DR we denote the set of 2 In fact it is one of the possible approaches, as there are many explanations which lead to similar formula. 8 Figure 5: Logistic and General Split Normal distributions fitted to data with heavy tails and skew ones. densities on R, such that F (x) = f1 (x1 ) · . . . · fd (xd ), for x = (x1 , . . . , xd ) ∈ Rd . Now suppose that the components of X are not independent, but that we know (or suspect) that there is a basis A (we put W = A−1 ) such that in that base the components of X become independent. This may be formulated in the form F (x) = det(W ) · f1 (ω1T (x − m)) · . . . · fd (ωdT (x − m)) for x ∈ Rd , 9 (3.1) where ωiT (x − m) is the i-th coefficient of x − m (the basis is centered in m) in the basis A (ωi denotes the i-th column of W ). Observe, that for a fixed family of one-dimensional densities F ⊂ DR , the set of all densities given by (3.1) for fi ∈ F, forms an affine invariant set of densities. Thus, if we want to find such a basis that components become independent, we need to search for a matrix W and one-dimensional densities such that the approximation F (x) ≈ det(W ) · f1 (ω1T (x − m)) · . . . · fd (ωdT (x − m)), for x ∈ Rd , is optimal. However, before proceeding to practical implementations, we need to precise: 1. how to measure the above approximation, 2. how to deal with data X, since we do not have the density, 3. how to work with the family of all possible densities. The answer to the first point is simple and is given by the Kullback-Leibler divergence, which is defined to be the integral: Z ∞ p(x) dx, DKL (P kQ) = p(x) log q(x) −∞ where p and q denote the densities of P and Q. This can be written as DKL (P kQ) = h(P ) − M LE(P, Q), where h is the classical Shannon entropy. Thus to minimize the KullbackLeibler divergence, we can equivalently maximize the MLE. This is helpful, since for a discrete data X we have nice estimator of the LE (likelihood estimation): 1 X LE(X, Q) = ln(q(x)). \|X\| x∈X Thus we arrive at the following problem. Problem [reduced]. Let X be a data set. Find an unmixing matrix W , center m, and densities f1 , . . . , fd ∈ DR so that the value LE(X, f1 , . . . , fd , m, W ) = P 1 ln(f1 (ω1T (x − m)) . . . fd (ωdT (x − m))) + ln(det(W )) = \|X\| 1 \|X\| x∈X d P P ln(fi (ωiT (x − m))) + ln(det(W )) i=1 x∈X 10 is maximized. However, there is still a problem with the last point, as the search over the space of all densities DR is not feasible. Thus, we naturally have to reduce our search to a subclass of all densities F (which should be parametrized by a finite amount of parameters). Problem [final]. Let X ⊂ Rd be a data set and F ⊂ DR be a set of densities. Find an unmixing matrix W , center m, and densities f1 , . . . , fd ∈ F so that the value d 1 XX ln(fi (ωiT (x − m))) + ln(det(W )) \|X\| i=1 x∈X is maximized. It may seem that the most natural choice is Gaussian densities. However, this is not the case as Gaussian densities are affine invariant, and therefore do not “prefer” any fixed choice of coordinates3 . In other words we have to choose a family of densities which is distant from Gaussian ones. In the classical ICA approach it is common to use the super-Gaussian logistic distribution: f (x; µ, s) = e x−µ s s 1+e 1 2 x−µ sech . = x−µ 2 4s 2s s The main difference between the gaussian and super-gaussian is the existence of the heavy tails. This can be also viewed as the difference in the fourth moments. However, such a choice leads to some negative consequences, namely the model is very sensitive to outliers. Moreover, if the data is not-symmetric, the approximation could not give the expected results, as the model consists only of symmetric densities. The idea behind this paper was to choose the model of densities which wouldn’t have the two above disadvantages. So, instead of choosing the family which differs from the Gaussians by the size of tail (fourth moment), we chose a family which would allow estimation of asymmetric densities – Split Gaussian distribution [58]. 3 In fact one can observe that the choice of gaussian densities leads to PCA, if we restrict to the case of orthonormal bases 11 Figure 6: Level sets of the General Split Normal distribution with different parameters. Example 3.1. In Fig. 4 and Fig. 5 we present a comparison between the Logistic and the Split Normal distribution in 1d and 2d respectively. In experiments we use the classical skew dataset Lymphoma [59, 60] and the classical heavy tails dataset Australian athletes [61]. In the case of heavy tails both methods work nice, since real dataset represent heavy tails which are not symmetric and the skew model is able to detect it. On the other hand, in the case of skew data Split Normal gives essentially better results. 4. Split Gaussian distribution In this section we present our density model. A natural direction for extending the normal distribution is the introduction of some skewness, and several proposals have indeed emerged, both in the univariate and multivariate case, see [62, 63, 64]. One of the most popular approaches is the Split Normal (SN) distribution, or the Split Gaussian (SG) distribution [58]. In our paper we use a generalization of this model, which we call the General Split Normal (GSN) distribution. 12 We start from the one-dimensional case. After that we present a possible generalization of this definition to the multidimensional setting, which corresponds with the formula (3.1). Contrary to the Split Gaussian distribution, we skip the assumption of the orthogonality of coordinates (often called principal components), and obtain an ICA model. 4.1. One-dimensional case The density of the one-dimensional Split Gaussian distribution is given by the formula for x ≤ m, c · exp[− 2σ1 2 (x − m)2 ], 2 2 SN (x; m, σ , τ ) = 1 2 c · exp[− 2τ 2 σ2 (x − m) ], for x > m, q where c = π2 σ −1 (1 + τ )−1 . As we see the split normal distribution arises from merging two opposite halves of two probability density functions of normal distributions in their common mode. In general the use of the Split Gaussian distribution (even in 1D) allows to fit data with better precision (from the likelihood function point of view). In 1982 John [65] showed that the likelihood function can be expressed in an intensive form, in which the scale parameters σ and τ are a function of the location parameter m (see Theorem 3.1 proved by [64]). Thanks to this theorem we can maximize the likelihood function numerically with respect to a single parameter m only. The rest of parameters are explicitly given by simple formulas. 4.2. Multidimensional Split Gaussian distribution A natural generalization of the univariate split normal distribution to the multivariate settings was presented by [64]. Roughly speaking, authors assume that a vector x ∈ Rd follows the multivariate Split Normal distribution, if its principal components are orthogonal and follow the one-dimensional Split Normal distribution. Definition 4.1 (Definition 2.2. [64]). A density of the multivariate Split Normal distribution is given by SNd (x; m, Σ, τ ) = d Y SN (ωjT (x − m); 0, σj2 , τj2 ), j=1 13 where ωj is the eigenvector corresponding to the j-th largest eigenvalue in the spectral decomposition of Σ = W AW T and m = [m1 , . . . , md ]T , A = diag(σ12 , . . . , σd2 ) and τ = [τ12 , . . . , τd2 ]. One can easily observe that the principal components ωjT x are independent. For this generalization a similar theorem, like in the one-dimensional case, is valid. We can extract the maximum likelihood estimation by maximizing the function with respect to two parameters m ∈ Rd and W ∈ Md (R) where columns of W are orthonormal vectors (Md (R) denotes the set of d-dimensional square matrices). We may use this theorem for numerical maximization of the likelihood function w.r.t. m and W . Unfortunately, the optimization process on Stiefel manifold (the set of orthogonal matrices) studied by [66] is numerically ineffective and requires additional tools. This problem can be omitted by using Eulerian angles described by [67]. In the two-dimensional case, W is explicitly parametrized as π π cos(θ) sin(θ) W = , − <θ≤ . − sin(θ) cos(θ) 2 2 In such a case we can straightforwardly apply standard numerical optimization algorithm. Both of these solutions can be applied. Nevertheless, unnatural assumption of the orthogonality of principal components causes two negative effects: the optimization process is time consuming and the model with the restriction that the coordinates are orthogonal can not accommodate data as good as the general one. Therefore, in this article we use more flexible model – the General Split Normal [68] distribution: Definition 4.2. A density of the multivariate General Split Normal distribution is given by 2 2 GSNd (x; m, W, σ , τ ) = det(W ) d Y SN (ωjT (x − m); 0, σj2 , τj2 ), j=1 where ωj is the j-th column of non-singular matrix W , m = (m1 , . . . , md )T , σ = (σ1 , . . . , σd ) and τ = (τ1 , . . . , τd ). 14 Figure 7: Comparison between fitting Gaussian, Split Gaussian and General Split distribution on [59, 60]. Observe that, contrary to Split Gaussian, General Split Gaussian does not have orthogonal basis. Our model is a natural generalization of the multivariate Split Normal distribution proposed in [64] (see Definition 4.1) and is given in the form formulated by (3.1) for the set of Split Gaussian densities. Clearly every Split Normal distribution is a General Split Normal distribution. The above generalization is flexible and allows to fit data with greater precision, see Fig. 7. The level sets of the GSN distribution with different parameters are presented in Fig. 6. We skip the constraints of orthogonality of the principal components. Consequently, we can apply the standard optimization procedure directly. In the next section we discuss how to fit data in our model. 5. Maximum likelihood estimation In the previous section we introduced the GSN distribution. Now we show how to use the likelihood estimation in our setting. As it was mentioned, we have to maximize the likelihood function with respect to four parameters. In the case of the General Split Normal distribution (contrary to the classical Gaussian one) we do not have explicit formulas and consequently we heave to solve the optimization problem. In the first subsection, we reduce our problem to the simpler one by introducing the function l. Minimization of l is equivalent to maximization of the likelihood function. In the second subsection we present how to minimize our function by using the gradient method. 15 5.1. Optimization problem The density of the GSN distribution depends on four parameters m ∈ Rd , W ∈ M(Rd ), σ ∈ Rd , τ ∈ Rd . We can find them by minimizing the simpler function, which depends on only m ∈ Rd and W ∈ M(Rd ). Other parameters are given by explicit formulas. Theorem 5.1. Let x1 , . . . , xn be given. Then the likelihood maximized w.r.t. σ and τ is −3n/2 dn/2 d Y 1 2n gj (m, W ) , (5.1) L̂(X; m, W ) = 2 πe \|det(W )\| 3 j=1 where 1/3 1/3 gj (m, W ) = s1j + s2j , s1j = P [ωjT (xi − m)]2 , Ij = {i = 1, . . . , n : ωjT (xi − m) ≤ 0}, i∈Ij s2j = P i∈Ijc [ωjT (xi − m)]2 , Ijc = {i = 1, . . . , n : ωjT (xi − m) > 0}, and the maximum likelihood estimators of σj2 and τj are σ̂j2 (m, W ) = 1 2/3 s g (m, W ), n 1j j τ̂j (m, W ) = s2j s1j 1/3 . Proof. See Appendix 8. Thanks to the above theorem, instead of looking for the maximum of the likelihood function, it is enough to obtain the maximum of the simpler function (5.1) which depends on two parameters m ∈ Rd and W ∈ M(Rd ) l(X; m, W ) = d Y 1 2 \|det(W )\| 3 gj (m, W ), (5.2) j=1 where ωj stands for the j-th column of matrix W . Consequently, maximization of (5.1) is equivalent to minimization of (5.2), see the following corollary. Corollary 5.1. Let X ⊂ Rd , m ∈ Rd , W ∈ M(Rd ) be given, then argmax L̂(X; m, W ) = argmin l(X; m, W ). m,W m,W 16 Figure 8: Results of image separation with the uses of various ICA algorithms. 5.2. Gradient One of the possible methods of optimization is the gradient method. Since the minimum of l is equal to the minimum of ln(l), in this subsection we 17 calculate the gradient of ln(l). Before we prove suitable Theorem 5.2, we recall the following lemma. Lemma 5.1. Let A = (aij )1≤i,j≤d be a differentiable map from real numbers to d × d matrices then ∂det(A) = adjT (A)ij , (5.3) ∂aij where adj(A) stands for the adjugate of A, i.e. the transpose of the cofactor matrix. Proof. By the Laplace expansion detA = d P (−1)i+j aij Mij where Mij is the j=1 minor of the entry in the i-th row and j-th column. Hence ∂detA = (−1)i+j Mij = adjT (A)ij . ∂aij Now we are ready to calculate gradient of our cost function. Theorem 5.2. Let X ⊂ Rd , m = (m1 , . . . , md )T ∈ Rd , W = (ωij )1≤i,j≤d non T ) ∂ ln l(X;m,W ) singular be given. Then ∇m ln l(X; m, W ) = ∂ ln l(X;m,W , . . . , , ∂m1 ∂md where d P T P T P ∂ ln l(X;m,W ) 1 1 −1 2ωj (xi − m)ωjk + 2 2ωj (xi − m)ωjk . = 1 1 2 ∂mk 3 +s 3 j=1 s1j 2j 3 3s1j i∈Ij Moreover, ∇W ln l(X; m, W ) = ∂ ln l(X;m,W ) ∂ωpk −2 + 31 s2p3 P i∈Ipc = − 23 (ω −1 )Tpk + h 3 3s2j i∈Ijc ∂ ln l(X;m,W ) ∂ωpk 1 1 1 3 +s 3 s1p 2p i 2 1 −3 s 3 1p , where 1≤p,k≤d P 2ωpT (xi − m)(xik − mk )+ i∈Ip T 2ωp (xi − m)(xik − mk ) , and s1j = P [ωjT (xi − m)]2 , Ij = {i = 1, . . . , n : ωjT (xi − m) ≤ 0}, i∈Ij s2j = P i∈Ijc [ωjT (xi − m)]2 , Ijc = {i = 1, . . . , n : ωjT (xi − m) > 0}. 18 Proof. See Appendix 9. Thanks to the above theorem we can use gradient descent, a first-order optimization algorithm. To find a local minimum of the cost function ln(l) using gradient descent, one takes steps proportional to the negative of the gradient of the function at the current point. If instead one takes steps proportional to the positive of the gradient, one approaches a local maximum of that function, see Algorithm 1. Algorithm 1 : Input data set X Initial conditions initialization of mean vector m = mean(X) initialization of matrix W = cov(X) Gradient algorithm obtain new values of m and V by applying gradient method for function log(l) (see formula 5.1): (m, W ) = argmin log(l(X; m̄, W̄ )), m̄,W̄ where ∇m ln l(X; m, W ) ∇W ln l(X; m, W ) are given by Theorem 5.2 calculate σ ∈ Rd and τ ∈ Rd by using Theorem 5.1 Return value return optimal ICA basis (m, W ). At the end of this section we present comparison of computational efficiency between ICASG and various ICA methods, see Fig. 9. In our experiment we consider the classical image separation problem, where we mixed two images by adding and subtracting them. We use ten pairs of images. Each pair was scaled to different sizes. In Fig. 9 we present mean value of computation time. FastICA, Infomax and JADE are the most effective but 19 Figure 9: Comparison of computational efficiency between ICASG and various ICA methods. do not solve the problem of image separation sufficiently well, see Tab. 1. On the other hand, the ProDenICA which gives comparable result to ICASG , is much slower. 6. Experiments and analysis To compare our method to classical ones we use Tucker’s congruence coefficient [69] (uncentered correlation) defined by Pd Cr(s, s̄) = qP d i=1 si s̄i qP d 2 i=1 si . 2 i=1 s̄i Its values range between −1 and +1. It can be used to study the similarity of extracted factors across different samples. Generally, a congruence coefficient of 0.9 indicates a high degree of factor similarity, while a coefficient of 0.95 20 or higher indicates that the factors are virtually identical. In the case of ICA methods multiplying by the scalar any of the sources do not change results. Therefore the sign of congruence coefficient is not important and we can compare absolute value of Tucker’s congruence. We evaluate our method in the context of images, sound, hyperspectral unmixing and EEG data. For comparison we use R package ica [70], PearsonICA [71], ProDenICA [72], tsBSS [73]. The most popular method used in practice is FastICA [48, 74] algorithm, which uses negentropy. In this context we can use three different functions to estimate neg-entropy: logcosh, exp and kurtosis. We also compare our method with algorithm using Information-Maximization (Infomax) approach [49]. Similarly to FastICA we consider three possible nonlinear functions: hyperbolic tangent, logistic and extended Infomax. We also consider algorithm which uses Joint Approximate Diagonalization of Eigenmatrices (JADE) proposed by Cardoso and Souloumiac’s [75, 75, 74]. One of the most popular ICA methods dedicated for skew data is PearsonICA [44, 45], which minimizes mutual information using a Pearson [46] system-based parametric model. Another model we consider is ProDenICA [50, 51], which is based not on a single nonlinear function, but on an entire function space of candidate nonlinearities. In particular, the method works with the functions in a reproducing kernel Hilbert space, and make use of the kernel trick to search over this space efficiently. We also compare our method with FixNA [76], method for blind source separation problem. 6.1. Separation of images One of the most popular application of ICA is the separation of images. In our experiments we use four images from the USC-SIPI Image Database of size 256 × 256 pixels (4.1.01, 4.1.06, 4.1.02, 4.1.03) and eight of size 512 × 512 pixels (4.2.04, 4.2.02, boat.512, elaine.512, 5.2.10, 5.2.08, 5.3.01, 4.2.03). We also use 8 images from the Berkeley Segmentation Dataset of size 482 × 321 with indexes (#119082, #42049, #43074, #38092, #157055, #220075, #295087, #167062). We make random pairs of above images and use them 1 1 as a source signal, combined by the mixing matrix A = . From 1 −1 practical point of view, we simply obtain two new images by adding and dividing sources pictures. Our goal is to reconstruct original images by using only the knowledge about mixed ones. The visualization of this process we 21 ICASG 4.1.01 4.1.02 4.1.06 4.1.03 4.2.04 5.2.10 4.2.02 5.2.08 boat.512 5.3.01 elaine.512 4.2.03 119082 157055 42049 220075 43074 295087 38092 167062 -0.9818 0.992 -0.9609 0.5664 -0.5034 0.2893 0.2305 0.5717 0.3593 0.4316 0.5874 -0.0226 0.9987 0.389 -0.7493 0.4359 -0.7371 -0.3997 -0.5949 0.3255 FastICA logcosh exp 0.5481 -0.5457 0.6696 0.6644 -0.4297 -0.4297 0.2062 0.2062 0.0506 0.0528 -0.0719 -0.0749 -0.0376 0.0203 0.1037 -0.0625 0.0351 0.0314 0.0078 0.0138 0.32 0.32 -0.3196 -0.3196 0.5736 0.5736 -0.3619 -0.3619 0.3009 0.3028 -0.5087 -0.5154 0.0344 0.0323 -0.048 -0.0458 0.0555 0.0564 -0.0025 -0.0041 kurtosis -0.5485 0.6707 -0.4296 0.2057 -0.0499 0.0709 -0.0017 -0.0097 -0.056 -0.0262 -0.32 0.3201 0.5731 -0.3618 -0.299 0.503 0.0429 -0.0566 0.031 0.0425 Infomax JADE PearsonICA tanh tangent logistic 0.548 -0.5484 -0.548 -0.5492 -0.5308 0.6695 0.6705 0.6695 0.6726 0.6696 -0.4297 -0.4297 -0.4296 -0.4296 -0.4297 0.2061 0.206 0.2058 0.2058 -0.2062 0.0505 -0.0512 0.0508 0.0397 0.3123 -0.0717 0.0727 -0.0722 -0.057 -0.4275 0.0377 0.0265 0.0061 -0.0093 -0.1228 -0.1039 -0.0773 -0.0285 0.0086 -0.2913 0.0343 -0.0449 0.0298 0.0356 -0.1046 0.0091 -0.008 0.0164 0.007 0.1061 0.32 -0.32 0.32 0.32 -0.32 -0.3196 0.3199 -0.3196 -0.3202 -0.3195 0.5737 0.5733 0.5735 0.5735 -0.032 -0.3619 -0.3619 -0.3619 -0.3619 0.0046 -0.3005 -0.3031 -0.3007 -0.2898 0.2596 0.5074 0.5168 0.5081 0.4789 0.4838 0.0348 0.0404 0.0342 0.0324 0.0891 -0.0484 -0.0541 -0.0478 -0.0459 -0.1035 -0.0553 0.041 0.0557 0.0375 0.0535 0.0021 0.0241 -0.0029 0.0306 0.0011 ProDenICA FixNA -0.0013 -0.0981 0.4297 0.207 -0.3164 0.4334 0.1282 -0.3091 -0.0461 0.0486 0.0287 -0.048 0.5744 0.3619 0.0421 -0.0645 0.3925 0.4015 0.4036 0.7404 0.5503 -0.6761 0.0148 0.0127 0.1461 -0.1979 0.1235 -0.2931 0.3175 -0.5303 0.2282 -0.2554 0.3695 -0.2446 0.142 -0.1839 0.2458 -0.2406 0.2614 -0.5495 Table 1: The Tucker’s congruence coefficient measure between original images and results of different ICA algorithms. present in Fig. 8. The results of this experiment are presented in Tab. 1 where we exhibit Tucker’s congruence coefficients. In the case of the Tucker’s congruence coefficient measure almost in all situation we obtain better results. The ICASG method essentially better recovers original signals. In Fig. 8 we can sow that ICASG almost perfectly recovers source signal. 6.2. Cocktail-party problem In this subsection we compare our method with classical ones in the case of cocktail-party problem. Imagine that you are in a room where two people are speaking simultaneously. You have two microphones, which you hold in different locations. The microphones give you two recorded time signals, which we could interpret as mixed signal x. Each of these recorded signals is a weighted sum of the speech signals emitted by the two speakers, which we denote by s. The cocktail-party problem is to estimate the two original speech signals. In our experiments we use signal obtained by mixing synthetic sources4 4 We use signals from http://research.ics.aalto.fi/ica/cocktail/cocktail_en. cgi. 22 ICASG source source source source source source source source source source source source source source source source 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 0.1597 0.7739 0.1388 0.9435 0.1985 0.8453 0.232 0.7679 0.1728 0.9424 0.15 0.7417 0.1036 0.908 0.1166 0.8212 logcosh 0.1097 0.7705 0.0899 0.9075 0.079 0.8887 0.0989 0.7798 0.0989 0.9245 0.0404 0.7129 0.0839 0.9016 0.1153 0.8136 FastICA exp 0.1096 0.7713 0.0899 0.9076 0.0791 0.8882 0.0989 0.7799 0.099 0.9243 0.0404 0.7134 0.084 0.9015 0.1156 0.8116 kurtosis 0.1101 0.7672 0.0908 0.898 0.079 0.8889 0.099 0.7793 0.0988 0.9256 0.0402 0.707 0.0839 0.9019 0.1145 0.8195 tanh 0.1097 0.7705 0.0899 0.9074 0.079 0.8887 0.0989 0.7798 0.0989 0.9246 0.0404 0.7132 0.0839 0.9019 0.1152 0.8141 Infomax tangent 0.11 0.7685 0.0899 0.907 0.079 0.8892 0.0989 0.7798 0.0989 0.925 0.0404 0.7125 0.0839 0.9019 0.1148 0.8174 logistic 0.1097 0.7705 0.0899 0.9074 0.079 0.8887 0.0989 0.7798 0.0989 0.9246 0.0404 0.7129 0.0839 0.9017 0.1153 0.8138 JADE PearsonICA ProDenICA FixNA 0.1101 0.7704 0.0908 0.9075 0.079 0.8898 0.099 0.7801 0.0989 0.9245 0.0402 0.7124 0.0839 0.9014 0.1155 0.8165 0.1097 0.7704 0.0899 0.9075 0.0789 0.8898 0.0989 0.7801 0.0989 0.9245 0.0404 0.7124 0.084 0.9014 0.1149 0.8165 0.1412 0.9998 0.0984 0.9989 0.0843 0.8459 0.1153 0.9344 0.0963 0.9729 0.0567 0.9998 0.093 0.9999 0.1427 0.9996 0.109 0.7751 0.0907 0.8988 0.0791 0.8882 0.0989 0.7796 0.0987 0.9273 0.0402 0.7099 0.0836 0.9056 0.1147 0.8176 Table 2: The Tucker’s congruence coefficient measure between original sound and results of different ICA algorithms in the case of cocktail-party problem. 1 1 (similar as before we use mixing matrix A = ). Comparison between 1 −1 methods we present in Tab. 2. In the case of cocktail-party problem our method recovers sources signal better then classical methods. 6.3. Hyperspectral Unmixing Independent component analysis has been recently applied into hyperspectral unmixing as a result of its low computation time and its ability to perform without prior information. However, when applying ICA for hyperspectral unmixing, the independence assumption in the ICA model conflicts with the abundance sum-to-one constraint and the abundance nonnegative constraint in the linear mixture model, which affects the hyperspectral unmixing accuracy. Nevertheless, ICA was recently applied in this area [77, 78]. In this subsection we apply simple example which shows that our method can by used for spectral data. Urban data [79, 80, 81] is one of the most widely used hyperspectral datasets used in the hyperspectral unmixing study. Each image has 307 × 307 pixels, each of which corresponds to a 2 × 2 m area. In this image, there are 210 wavelengths ranging from 400 nm to 2500 nm, resulting in a spectral resolution of 10 nm. After the channels 1–4, 76, 87, 101–111, 136–153 and 198–210 are removed (due to dense water vapor and atmospheric effects), there remain 162 channels (this is a common preprocess for hyperspectral unmixing analyses). There is ground truth [79, 80, 81], which contains 4 channels: #1 Asphalt, #2 Grass, #3 Tree and #4 Roof. 23 ICASG #1 Asphalt #2 Grass #3 Tree #4 Roof 0.6774 -0.7784 0.7267 0.6666 logcosh 0.2859 -0.2746 0.2338 -0.4256 FastICA exp 0.2864 -0.2605 0.2717 0.4279 kurtosis -0.2595 -0.2798 -0.2547 0.4167 Infomax PearsonICA ProDenICA tanh tangent logistic -0.2972 -0.2954 -0.2972 0.20978 0.4928 -0.2814 -0.2816 -0.2814 -0.2412 -0.4323 0.2441 0.2354 0.2442 0.2482 -0.5961 -0.4244 0.4301 -0.4244 0.4193 -0.6128 Table 3: The Tucker’s congruence coefficient measure between reference layers and results of different ICA algorithms in the case of the urban data set. Figure 10: Congruence distance between layers obtain by different ICA algorithms and the closest reference channel. A highly mixed area is cut from the original data set in this experiment (similar example was showed in [77]), with the size of 200 × 150 pixels. In our experiment we apply various ICA methods and report the Tucker’s congruence coefficient measure between each layer and the closest reference channel, see Fig. 10. ICASG and ProDenICA give layers which contain more information then the other approaches. Distance between four best channels to the reference ones we present in Tab. 3. 24 6.4. EEG At the end of this section we present how our method works in the case of EEG signals. In this context, ICA is applied to many different task like eye movements, blinks, muscle, heart and line noise e.t.c.. In this experiment we concentrate on eye movement and blink artifacts. Our goal here is to demonstrate that our method is capable of finding artifacts in real EEG data. However, we emphasize that it does not provide a complete solution to any of these practical problems. Such a solution usually entails a significant amount of domain-specific knowledge and engineering. Nevertheless, from these preliminary results with EEG data, we believe that the method presented in this paper provides a reasonable solution for signal separation, which is simple and effective enough to be easily customized for a broad range of practical problems. For EEG analysis, the rows of the input matrix x are the EEG signals recorded at different electrodes, the rows of the output data matrix s = W x are time courses of activation of the ICA components, and the columns of the inverse matrix, W , give the projection strengths of the respective components onto the scalp sensors. One EEG data set used in the analysis was collected from 40 scalp electrodes (see Fig. 11 a)). The second and the third are located very near to eye and can be understood as a base (we can use them for removing eye blinking artifacts). In Fig. 11 b) we present signals obtained by ICASG . The scale of this figure is large but we can find the data which have spikes exactly in the same place as the two base signals (see Fig. 11 c)). After removing selected signal and going back to the original situation we obtain signal (see Fig. 11 d)) without eye blinking artifacts (compare Fig. 11 a) with Fig. 11 d)). 7. Conclusion In our work we introduce and explore a new approach to ICA which is based on the asymmetry of the data. Roughly speaking in our approach, instead of approximating the data by product of densities with heavy tails, we approximate it by a product of asymmetric densities – the Split Gaussian distribution. Contrary to classical approaches which consider third or fourth central moment, our algorithm in practice is based on second moments. This is a consequence of the fact that Split Gaussian distributions arise from merging two opposite halves of normal distributions in their common mode. 25 Figure 11: Results of ICASG in the case of EEG data. Therefore we use only second order moments to describe skewness in dataset, and therefore we obtain an effective ICA method which is resistant to outliers. We verified our approach on images, sound and EEG data. In the case of source signal reconstructing our approach gives essentially better results (better recover original signals). The main reason is such that kurtosis is very sensitive to the outliers and that the asymmetry of the data is more popular than heavy tails in real data sets. 8. Appendix A Proof of Theorem 5.1. Let X = {x1 , . . . , xn }. We write zi = W (xi − m), zij = ωjT (xi − m), 26 for observation i, where i = 1, . . . , n and coordinates j = 1, . . . , d. Let us consider the likelihood function, i.e. n Q L(X; m, W, σ, τ ) = GSNd (xi ; m, W, σ, τ ) i=1 n Q d Q SN (ωjT (xi − m); 0, σj2 , τj2 ) i=1 j=1 n Q −n Q h d n Q d = c1 \|det(W )\| σj (1 + τj ) exp − = \|det(W )\| j=1 where c1 = q d 2 π i=1 j=1 1 2 z (1{zij ≤0} 2σj2 ij i + τj−2 1{zij >0} ) , . Now we take the log-likelihood function, i.e. ln(L(X; m, W, σ, τ )) n Q −n P i d n P d h σj (1 + τj ) = ln c1 \|det(W )\| + − 2σ1 2 zij2 (1{zij ≤0} + τj−2 1{zij >0} ) j j=1 i=1 j=1 n Q −n d d P −2 P 2 σ −2 P 2 σj zij = ln c1 \|det(W )\| σj (1 + τj ) zij + τj2 − 12 j j=1 j=1 i∈Ijc i∈Ij n Q d d −n P 1 = ln c1 \|det(W )\| σj (1 + τj ) s1j + τ12 s2j . − 2σ 2 j=1 j=1 j j We fix m, W and maximize the log-likelihood function over τ and σ. In such a case we have to solve the following system of equations ∂ ln(L(X;m,W,σ,τ )) ∂σj = − σnj + σj−3 (s1j + τj−2 s2j ) = 0, ∂ ln(L(X;m,W,σ,τ )) ∂τj n = − 1+τ + j s2j τj3 σj2 = 0, for j = 1, . . . , d. By simple calculations we obtain the expressions for the estimators 1/3 s2j 2 1 2/3 σ̂j (m, W ) = n s1j gj (m, W ), τ̂j (m, W ) = . s1j Substituting it into the log-likelihood function, we get dn Q d 2 3 −n dn 2 1 n √ 2 g (m, W ) e− 2 L̂(m, W ) = π \|det(W )\| · n j j=1 dn − 3n2 d 2 Q 1 = 2n . g (m, W ) 2 j πe \|det(W )\| 3 j=1 27 9. Appendix B Proof of Theorem 5.2. Let us start with the partial derivative of ln(l) with respect to m. We have 1 1 d d d 3 +s 3 ) P P P ∂(s1j ∂ ln(gj (m,W )) ∂ ln l(X;m,W ) 2j 1 1 ∂s1j 1 ∂s2j 1 = = . 1 1 2 ∂m + 2 ∂m 1 1 ∂mk ∂mk ∂mk k k 3 +s 3 j=1 s1j 2j j=1 Now, we need ∂s1j ∂mk = P i∈Ij ∂s1j ∂mk ∂s2j , ∂mk and ∂[ωjT (xi −m)]2 ∂mk = 3 +s 3 j=1 s1j 2j 3 3s1j 3 3s2j therefore 2ωjT (xi − m) P i∈Ij ∂ωjT (xi −m) ∂mk = P −2ωjT (xi − m)ωjk . i∈Ij Analogously we get ∂s2j ∂mk = P i∈Ijc −2ωjT (xi − m)ωjk . Hence ∂ ln l ∂mk = d P −1 1 1 3 +s 3 j=1 s1j 2j 1 2 P 3 3s1j i∈Ij 2ωjT (xi − m)ωjk + 1 2 P 3 3s2j i∈Ijc 2ωjT (xi − m)ωjk . Now we calculate the partial derivative of ln l(X; m, W ) with respect to the matrix W . We have ∂ ln l(X;m,W ) ∂ωpk 2 = ∂ ln \|det(W )\|− 3 ∂ωpk + d P j=1 ∂ ln(gj (m,W )) . ∂ωpk To calculate the derivative of the determinant we use Jacobi’s formula (see Lemma 5.1). Hence 2 5 2 ∂ ln(det(W )− 3 ) ) 2 3 = det(W ) det(W )− 3 ∂det(W = − 23 det(W )−1 adjT (W )pk − ∂ωpk 3 ∂ωpk 1 det(W )(W −1 )Tpk = − 23 (ω −1 )Tpk , = − 32 det(W ) where (ω −1 )Tpk is the element in the p-th row and k-th column of the matrix (W −1 )T . Now we calculate 1 1 3 +s 3 ) ∂(s1j ∂ ln(gj (m,W )) 2j 1 1 1 ∂s1j 1 ∂s2j = 1 1 ∂ωpk = 1 1 + 2 ∂ωpk , 2 ∂ω ∂ωpk pk 3 +s 3 s1j 2j 3 +s 3 s1j 2j 28 3 3s1j 3 3s2j where ∂s1j ∂ωpk ( = 0, P P i∈Ij ∂[ωjT (xi −m)]2 ∂ωpk = P 2ωjT (xi − m) i∈Ij ∂ωjT (xi −m) ∂ωpk = if j = 6 p 2ωpT (xi − m)(xik − mk ), if j = p i∈Ip and xik is the k-th element of the vector xi . 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Infinitesimal topological generators and quasi non-archimedean topological groups arXiv:1505.00415v2 [math.GR] 14 Mar 2016 Tsachik Gelander∗and François Le Maître† Abstract We show that connected separable locally compact groups are infinitesimally finitely generated, meaning that there is an integer n such that every neighborhood of the identity contains n elements generating a dense subgroup. We generalize a theorem of Schreier and Ulam by showing that any separable connected compact group is infinitesimally 2-generated. Inspired by a result of Kechris, we introduce the notion of a quasi non-archimedean group. We observe that full groups are quasi non-archimedean, and that every continuous homomorphism from an infinitesimally finitely generated group into a quasi non-archimedean group is trivial. We prove that a locally compact group is quasi non-archimedean if and only if it is totally disconnected, and provide various examples which show that the picture is much richer for Polish groups. In particular, we get an example of a Polish group which is infinitesimally 1-generated but totally disconnected, strengthening Stevens’ negative answer to Problem 160 from the Scottish book. 1 Introduction One of the simplest invariants one can come up with for a topological group G is its topological rank t(G), that is, the minimum number of elements needed to generate a dense subgroup of G. For this invariant to have a chance to be ﬁnite, one needs to assume that G is separable since every ﬁnitely generated group is countable. The oldest result in this topic is probably due to Kronecker in 1884 [Kro31], and says that an n-tuple (a1 , ..., an ) of real numbers projects down to a topological generator of Tn = Rn /Zn if and only if (1, a1 , ..., an ) is Q-linearly independent. Since such n-tuples always exist, the topological rank of the compact connected abelian group Tn is equal to one. Using this result, it is then an amusing exercise to show that t(Rn ) = n+1. This means that as a topological group, Rn remembers its vector space dimension. In particular, we see that the topological rank sometimes contains useful information (another somehow similar instance of this phenomenon was recently discovered by the second-named author for full groups, see [LM14]). In order to deal with general connected Lie groups, it is useful to introduce the following deﬁnition. ∗ Research supported by the ISF, Moked grant 2095/15. Research supported by the Interuniversity Attraction Pole DYGEST and Projet ANR-14-CE25-0004 GAMME. † 1 Definition 1.1. The infinitesimal rank of a topological group G is the minimum n ∈ N ∪ {+∞} such that every neighborhood of the identity in G contains n elements g1 , ..., gn which generate a dense subgroup of G. We denote it by tI (G). One can easily show that tI (Rn ) = t(Rn ) = n + 1. This is relevant for the study of the inﬁnitesimal rank of real Lie groups because if we know that the Lie algebra of a connected Lie group G is generated (as a Lie algebra) by n elements, then using the fact fact that tI (R) = 2 we can deduce that tI (G) 6 2n (see the well-known Lemma 4.2). For various classes of connected Lie groups one can say more. In particular this is the situation for compact connected Lie groups, in which case a uniform result is useful in view of the important role of compact groups in the structure theory of locally compact groups. Auerbach showed that for every compact connected Lie group G, one has tI (G) 6 2 [Aue34]. Moreover, he could show that the set of pairs of topological generators of G has full measure, which then led Schreier and Ulam to the following general result, for which we will include a short proof. Theorem 1.2 (Schreier-Ulam, [SU35]). Let G be a connected compact metrisable group. Then almost every pair of elements of G generates a dense subgroup of G. In particular, tI (G) = t(G) = 2 for any non-abelian such G. Note that there is a vast area between metrizable compact groups and separable ones; R for instance (T1 ) is compact separable, but not metrizable. Conversely, being separable is a minimal assumption for a group to have ﬁnite topological rank. Going back to the n-torus, Kronecker’s result admits a far-reaching generalization due to Halmos and Samelson, which settles the abelian case. Theorem 1.3 (Halmos-Samelson, [HS42, Corollary]). The topological rank of every connected separable compact abelian group is equal to one. Our ﬁrst result is a generalization of Schreier and Ulam’s Theorem to the separable compact case. One of the diﬃculties is that we cannot use their Haar measure argument anymore. Theorem 1.4 (see Theorem 3.1). Let G be a separable compact connected group. If G is non-abelian, then tI (G) = 2, and if G is abelian, then tI (G) = 1. The fact that the topological rank t(G) = 2 was proved by Hoﬀmann and Morris [HM90, Theorem. 4.13]. Note that it follows that tI (G) = t(G) for separable compact connected group. It is natural to ask if the following stronger statement is true: Question 1.5. Let G be a separable compact connected group. Is the set of pairs in G which topologically generate G necessarily of full measure in G × G? Say that a topological group is infinitesimally finitely generated if it has ﬁnite inﬁnitesimal rank. Using the previous result, and the solution to Hilbert’s ﬁfth problem, we establish the following result, which was noted by Schreier and Ulam in the abelian case (see the last paragraph of [SU35]). Theorem 1.6 (see Theorem. 4.1). Let G be a separable locally compact group. Then G is infinitesimally finitely generated if and only if G is connected. 2 Let us now leave the realm of locally compact groups for a moment and discuss the class of Polish groups, i.e. separable topological groups whose topology admits a compatible metric. These groups abound in analysis, for instance the unitary group of a separable Hilbert space or the group of measure-preserving transformations of a standard probability space are Polish groups. Moreover, they form a robust class of groups, e.g. every countable product of Polish groups is Polish (see [Gao09] for other properties of this ﬂavour). Recall that a locally compact group is Polish if and only if it is second-countable. It is not hard to show that Theorem 1.6 fails for Polish groups: for instance RN is connected but not topologically ﬁnitely generated, in particular it is not inﬁnitesimally ﬁnitely generated. The question of the converse is more interesting, even for the following weaker property. Definition 1.7. A topological group G is infinitesimally generated if every neighbourhood of the identity generates G. Clearly every connected group is inﬁnitesimally generated, and every inﬁnitesimally ﬁnitely generated group is inﬁnitesimally generated. Moreover it follows from van Dantzig’s theorem that every inﬁnitesimally generated locally compact group is connected. Question 1.8 (Mazur’s Problem 160 [Mau81]). Must an inﬁnitesimally generated Polish group be connected? In [Ste86], Stevens exhibited the ﬁrst examples of inﬁnitesimally generated Polish group which are totally disconnected. We show that her examples actually have inﬁnitesimal rank 2 and then provide the following stronger negative answer to Question 1.8. Theorem 1.9 (see Theorem. 5.15). There exists a Polish group of infinitesimal rank 1 which is totally disconnected. Let us now introduce the quasi non-archimedean property which is a strong negation of being inﬁnitesimally ﬁnitely generated. Definition 1.10. A topological group is quasi non-archimedean if for every neighborhood of the identity U in G and every n ∈ N, there exists a neighborhood of the identity V such that for every g1 , ..., gn ∈ V , the group generated by g1 , ..., gn is contained in U. Remark. If we switch the quantiﬁers and ask for a V which works for every n ∈ N, it is not hard to see that the deﬁnition then becomes that of a non-archimedean topological group (i.e. admitting a basis of neighborhoods of the identity made of open subgroups). Our inspiration for the above deﬁnition comes from the following result of Kechris: every continuous homomorphism from an inﬁnitesimally ﬁnitely generated group into a full group is trivial (see the paragraph just before Section (E) of Chapter 4 in [Kec10]). We upgrade this by showing that every full group is quasi non-archimedean, and that any continuous homomorphism from an inﬁnitesimally ﬁnitely generated group into a quasi non-archimedean group is trivial (see Proposition 5.5). For locally compact groups, we obtain the following characterisation. Theorem 1.11 (see Theorem. 5.8). Let G be a separable locally compact group. Then G is quasi non-archimedean if and only if G is totally disconnected. 3 Note that full groups are connected and at the same time quasi non-archimedean Polish groups. Moreover, we show that every quasi non-archimedean Polish groups embeds into a connected quasi non-archimedean Polish group (see Proposition 5.4.). We also provide examples of totally disconnected Polish groups which are quasi nonarchimedean, but not non-archimedean (see Corollary. 5.12). On the other hand Theorem 1.9 ensures us that there are totally disconnected Polish groups which are not quasi nonarchimedean. The paper is organised as follows. In Section 2, we prove some basic results on topological generators. In Section 3, we show that separable connected compact groups are inﬁnitesimally 2-generated. Section 4 is devoted to the proof that every connected separable locally compact group is inﬁnitesimally ﬁnitely generated. In Section 5 we introduce quasi non-archimedean groups and study their basic properties. We also give numerous examples, and show that a separable locally compact group is totally disconnected if and only if it is quasi non-archimedean. Finally, a Polish group into which no non-discrete locally compact group can embed is built in Section 6, where we also ask three questions raised by this work. Acknowledgements. We warmly thank Pierre-Emmanuel Caprace for coming up with the terminology “inﬁnitesimally generated”, as well as for useful conversations around this topic. We also thank Yves de Cornulier for his helpful remarks on a ﬁrst version of the paper. 2 Basic results about topological generators We collect some results which will serve us in the preceding sections. Proposition 2.1. Let G be a connected locally compact group. Suppose that K is a profinite normal subgroup of G such that G/K is infinitesimally finitely generated. Then tI (G) = tI (G/K). Moreover, if ḡ1 , . . . , ḡk topologically generate the group G/K, and g1 , . . . , gk are arbitrary respective lifts in G, then g1 , . . . , gk topologically generate G. The proof of Proposition 2.1 will rely on the following two lemmas: Lemma 2.2. Let H be a connected locally compact group and f : H ։ L a finite covering map. Let {l1 , . . . lk } be a topological generating set for L and pick hi ∈ f −1 (li ), i = 1, . . . , k arbitrarily. Then h1 , . . . , hk topologically generate H. Proof. Set F = hh1 , . . . , hk i. Note that f , being a ﬁnite cover, is a closed map, and hence f (F ) is closed in L. Since l1 , . . . , lk ∈ f (F ) we have f (F ) = L. Thus Ker f · F = H. Hence by the Baire category theorem F has a non-empty interior. Since H is connected, this implies that F = H. Lemma 2.3. Let G be a (connected) locally compact group admitting a pro-finite normal subgroup K ⊳G such that L = G/K is a Lie group. Then G is an inverse limit G = lim Lα ←− of finite (central) extensions Lα of L. Although the Lemma holds without the assumption that G is connected, since it signiﬁcantly simplify the proof while being suﬃcient for our needs, we will prove it only under the connectedness assumption. 4 Proof. Given k ∈ K, the image of the orbit map G → K, g 7→ gkg −1 is at the same time connected, since G is connected and the map is continuous, and totally disconnected as the image lies in K. It follows that it is constant and k is central. Since k is arbitrary we deduce that K is central in G. In particular every subgroup of K is normal in G. Let Kα be a net of open subgroups in K with trivial intersection. Then K = lim K/Kα ←− and G = lim G/Kα . Set Lα = G/Kα and note that as Kα is open in K, it is of ﬁnite ←− index there. Thus the map Lα → L is a ﬁnite covering. Proof of Proposition 2.1. Let G and K be as in Proposition 2.1. By Lemma 2.3, G = lim Lα is an inverse limit of ﬁnite covers Lα of L = G/K. Let ḡ1 , . . . , ḡk be topological ←− generators of L, let g1 , . . . , gk be arbitrary lifts in G and denote by giα the projection of gi in Lα , for every i, α. By Lemma 2.2, hgiα : i = 1, . . . , ki is dense in Lα . That is, the group hgi : i = 1, . . . , ki projects densely to all Lα . This implies that it is dense in G. 3 Connected compact separable groups are infinitesimally 2-generated Our aim in this section is to prove the following result. Recall that t(G) is the minimal number of topological generators of G, while tI (G) is the minimal n ∈ N such that every neighborhood of the identity contains n elements which topologically generate G. Theorem 3.1. Let G be a separable compact connected group. Then tI (G) = t(G) = 2 if G is nonabelian and tI (G) = t(G) = 1 if G is abelian. 3.1 The metrizable case The case where G is metrizable was proved by Schreier and Ulam [SU35]. Recall that a compact group is metrizable if and only if it is ﬁrst countable. In that case one can show that almost every pair of elements in G (or a single element if G is abelian) topologically generates G. Let us give a short argument for Theorem 1.2 in that case. First recall that by the Peter–Weyl theorem, G is an inverse limit of compact Lie groups and, being ﬁrst countable, the limit is over a countable net. Since a countable intersection of full measured sets is of full measure, it is enough to prove the analog statement for compact connected Lie groups. Note also that, in complete generality, a subgroup H ≤ G is dense if and only if • H ∩ G′ is dense in G′ , where G′ is the commutator subgroup in G, and • HG′ is dense in G. A connected compact Lie group G is reductive, hence an almost direct product of its commutator G′ with its centre Z. Moreover G′ is connected and a ﬁnite cover of G/Z which is a semisimple group of adjoint type. In view of Lemma 2.2, we deduce: Lemma 3.2. Let G be a connected compact Lie group and H ≤ G a subgroup. Then H is dense in G if and only if both HG′ and HZ(G) are dense in G. Therefore, for a pair (x, y) ∈ G2 to generate a dense subgroup, it is suﬃcient if • the projections of x, y to G/Z generate a dense subgroup in G/Z, and 5 • the projection of x to G/G′ generates a dense subgroup in G/G′ . It is easy to check that both conditions are satisﬁed with Haar probability 1 (cf. [Gel08, Lem. 1.4 and Lem. 1.10]). 3.2 Proof of Theorem 3.1 in the general case Let us ﬁrst deal with abelian groups. By the Halmos–Samelson theorem, whenever G is connected compact separable abelian, one has t(G) = 1. From their result, we deduce the following consequence. Lemma 3.3. Let G be a compact connected separable abelian group. Then tI (G) = 1. Proof. By the Halmos–Samelson theorem, we may and do pick g ∈ G such that hgi is dense in G. Let U be a neighborhood of the identity in G, and ﬁx n ∈ Z \ {0} such that g n ∈ U. Then since hg n i has ﬁnite index in hgi, its closure hg n i has ﬁnite index in hgi = G. Since G is connected, we must have hg n i = G. Now suppose that G is any connected compact group. By the Peter–Weil theorem G = lim Gα is an inverse limit of compact connected Lie groups Gα . ←− Observe that a surjective map f : G1 ։ G2 between groups always satisﬁes f (G′1 ) = G′2 and f (Z(G1 )) ⊂ Z(G2 ), while for reductive Lie groups we also have f (Z(G1 )) = Z(G2 ). Since every connected compact Lie group is reductive, hence the product of its centre and its commutator, we deduce that the same hold for general compact connected groups, i.e. Z(G) = lim Z(Gα ), G′ = lim G′α and G = Z(G)G′ . ←− ←− Moreover, since the G′α are semisimple, and in particular perfect, G′ is also perfect, i.e. G′ = G′′ . It follows that if Γ ≤ G′ is dense, then Γ′ is dense in G′ . Hence we have: Claim. Suppose that a, b ∈ G′ topologically generate G′ , and h ∈ Z(G) is an element whose image mod G′ topologically generates G/G′ . Then ah and b topologically generate G. Suppose from now on that G is separable. Then every quotient of G is also separable. Now G/G′ is connected and abelian, so we have tI (G/G′ ) = 1 by Lemma 3.3. Since Z(G) surjects onto G/G′ , every identity neighbourhood in G contains a central element h whose image in G/G′ generate a dense subgroup. Thus we are left to show that tI (G′ ) = 2. The centre of G′ is totally disconnected since it can be written as Z(G′ ) = lim Z(G′α ), and every G′α has ﬁnite center. In view of Proposition 2.1 we may ←− thus suppose that G′ is center-free. In order to simplify notations, let us suppose below that G itself is center-free. Note that: Lemma 3.4. A center-free connected compact group is a direct product of simple Lie groups. Q Thus, G is of the form G = α∈I Sα with Sα being connected adjoint simple Lie group. Q Lemma 3.5. The group G = α∈I Sα is separable (if and) only if Card(I) ≤ 2ℵ0 . 6 Let us explain the ‘only if’ side. The other direction will follow once we show that Card(I) ≤ 2ℵ0 implies that G has a two generated dense subgroup. Suppose by way of contradiction that Card(I) > 2ℵ0 . Since there are only countably many isomorphism types of (compact adjoint) simple Lie groups, we deduce that there is some compact simple Lie group S and a cardinality κ > 2ℵ0 such that G admits a factor isomorphic to S κ . However by the cardinals version of the pigeon hole principal, if D ⊂ S κ is a countable subset, there must be two factors S1 , S2 of S κ such that the projection of D to S1 × S2 lies in the diagonal. In particular, D cannot be dense, conﬁrming the desired contradiction. Thus, we may suppose below that Card(I) ≤ 2ℵ0 . Definition 3.6. Let S1 , S2 be two groups, F a set and fi : F → Si , i = 1, 2 two maps. We shall say that the maps f1 and f2 are isomorphically related if there is an isomorphism φ : S1 → S2 such that the following diagram is commutative F f1 − → S1 ց f2 ↓ φ S2 Q Q Lemma 3.7. Let ki=1 Si (k ≥ 2) be a product of simple groups and let R < ki=1 Si be a proper subgroup that projects onto every Si . Then there are two factors Si , Sj , i 6= j such that the restrictions of the quotient maps R → Si and R → Sj are isomorphically related. Q Proof. For a subset J ⊂ {1, . . . , k} let us denote SJ = i∈J Si and RJ = ProjSJ (R). Let J ⊂ {1, . . . , k} be a minimal subset such that RJ is a proper subgroup of SJ . By our assumption J exists and satisﬁes \|J\| > 1. We claim that \|J\| = 2. To see this, we may reorder the indices so that J = {1, . . . , j} and suppose by way of contradiction that j ≥ 3. This, together with the minimality of J, implies that for every g ∈ S1 and every 1 < i ≤ j there is an element in RJ whose ﬁrst coordinate is g and whose i’th coordinate is 1. However, multiplying commutators of elements as above, forcing that at each coordinate 1 < i ≤ j at least one of the elements we use is trivial, and using the fact that S1 is perfect, we deduce that S1 ≤ RJ . In the same way we get that Si ≤ RJ for all i ∈ J contradicting the assumption that RJ is proper in SJ . Thus J = {1, 2}. Moreover, as Si is simple and normal, and RJ projects onto Si , while RJ is proper in SJ , it follows that RJ ∩ Si is trivial, for i = 1, 2. Therefore, the restriction of the projection from RJ to Si is an isomorphism, for i = 1, 2, hence the maps R → S1 and R → S2 are isomorphically related. Assembling together isomorphic factors of G, we may decompose G as a direct product Q G = n∈N Gn where diﬀerent n’s correspond to non-isomorphic simple Lie factors Sn , and each Gn is of the form Gn = SnIn with \|In \| ≤ 2ℵ0 . For x ∈ G we shall denote by xαn , α ∈ In its corresponding coordinates. Two elements x, y ∈ G generate a dense subgroup if and only if for any ﬁnite Q set of indices F = {(ni , αi )} the projections of x and y generate a dense subgroup in F Snαii . In view of Lemma 3.7 we have: Corollary 3.8. Two elements x, y ∈ G generate a dense subgroup in G if and only if • hxαn , ynαi is dense in Snα for every n ∈ N and every α ∈ In , and • For every n, m ∈ N and α ∈ In , β ∈ Im the (projection) maps {x, y} → Snα , {x, y} → β Sm are not isomorphically related. 7 Thus, in order to prove the theorem, we should explain how to pick (xαn , ynα) arbitrarily close to the identity in Snα , for every n ∈ N, α ∈ In , which generate a dense subgroup of Sn , such that for every pair (n, α) 6= (m, β) there is no isomorphism Sn → Sm taking β (xαn , ynα ) to (xβm , ym ). Since for n 6= m there is no isomorphism between Sn and Sm we should only consider the case n = m. Let S = Sn . Recall that there is an identity neighbourhood (a Zassenhaus neighbourhood) Ω ⊂ S on which the logarithm is well deﬁned, and for x, y ∈ Ω the group hx, yi is dense in S if log(x) and log(y) generate the Lie algebra Lie(S), (see [Kur49]). Thus we may pick some open set U1 × U2 ⊂ Ω2 such that for all (x, y) ∈ U1 × U2 the group hx, yi is dense in S (cf. [GŻuk02]). Furthermore, we may pick U1 , U2 arbitrarily close to the identity. Since Out(S) is ﬁnite and S is compact, we may also suppose, by taking U1 and U2 suﬃciently small, that for (x, y) ∈ U1 × U2 , if f ∈ Aut(S) satisﬁes (f (x), f (y)) ∈ U1 × U2 , then f is inner. Finally, since the orbit of each (x, y) ∈ U1 × U2 under the action of S on S × S by conjugation, is dim(S) dimensional, while dim(U1 × U2 ) = 2 dim(S), we can pick a section S transversal to the orbits foliation, inside U1 × U2 . Clearly the cardinality of that section is 2ℵ0 , hence we may imbed In inside this section. This imbedding yields a choice of (xα , y α) ∈ S ⊂ U1 × U2 for every α ∈ In , and we have that each pair (xα , y α ) generates a dense subgroup in S, and no two pairs are isomorphically related. This completes the proof of Theorem 3.1. 4 Local generators and locally compact connected groups Recall that a topological group G is infinitesimally finitely generated if tI (G) < +∞, that is, if there exists n ∈ N such that every neighborhood of the identity contains n topological generators for G. Similarly, we say that G is topologically finitely generated if t(G) < ∞, that is, if G admits a dense ﬁnitely generated subgroup. Last but not least, a topological group is infinitesimally generated if it is generated by every neighborhood of the identity. Note that if a group is inﬁnitesimally ﬁnitely generated, then it is also topologically ﬁnitely generated and inﬁnitesimally generated. Our main goal in this section is to prove the following result. Theorem 4.1. Let G be a separable connected locally compact group. Then G is infinitesimally finitely generated. As already noted, the separability condition is necessary for a group to be topologically ﬁnitely generated. The connectedness assumption is also necessary (for local generation) since otherwise G/G◦ , the group of connected components, is non-trivial and by the van Dantzig’s theorem admits a base of identity neighbourhood consisting of open compact subgroups. In particular, if O ≤ G/G◦ is a proper open subgroup, its pre-image in G cannot contain a topological generating set. Let us start by dealing with the nicest connected locally compact groups: Lie groups. For these, one can use the Lie algebra to produce topological generators. The following lemma is well known, but we include a proof for the reader’s convenience. Lemma 4.2 (Folklore). Let G be a connected Lie group, and let g be its Lie algebra. Suppose that g is generated as a Lie algebra by X1 , ..., Xn . Then every neighborhood of 8 the identity in G contains 2n elements g1 , ..., g2n which generate a dense subgroup in G. In particular, G is infinitesimally finitely generated. Proof. Let V be a neighborhood of the identity in G. Let U be a small enough neighborhood of 0 in g such that exp : U → G is a homeomorphism onto its image, and exp(U) ⊆ V . Fix 2n elements Y1 , ..., Y2n of U such that for every i ∈ {1, ..., n}, {Y2i , Y2i+1 } generates a dense subgroup of RXi . For all i ∈ {1, ..., 2n}, let gi = exp(Yi ), we will show that these elements topologically generate G. Let H be the closed subgroup generated by the set {gi }2n i=1 . Note that for all i ∈ {1, ..., n}, the group H contains exp(RXi ) since the restriction of the exponential map to RXi is a continuous group homomorphism and RXi is topologically generated by Y2i and Y2i+1 which are mapped to g2i ∈ H and g2i+1 ∈ H. Furthermore, H is a Lie group by Cartan’s theorem, and since H contains every exp(RXi ) the Lie algebra h of H contains every Xi , so h = g. Because G is connected, we get that G = H. We deduce from the above lemma that for any connected Lie group, tI (G) 6 2 dim(G). Better bounds on tI (G) can be deduced from the analysis in [BG03, BGSS06, Gel08]. Let now G be a general connected locally compact group. Recall the celebrated Gleason-Yamabe theorem (cf. [Kap71, Page 137]): Theorem 4.3. (Gleason–Yamabe) Let G be a connected locally compact group. Then there is a compact normal subgroup K ⊳ G such that G/K is a Lie group. Since G is connected, G/K is a connected Lie group and hence tI (G/K) is ﬁnite. However, K may not be connected. Example 4.4. (1) (The solenoid) For every n, let Tn be a copy of the circle group {z ∈ C : \|z\| = 1}, and whenever m divides n let fn,m : Tn → Tm be the n/m sheeted cover fn,m (z) = z n/m . Let T = lim Tn be the inverse limit group. Then T is connected, abelian ←− and locally compact, but admits no connected co-Lie subgroups. (2) Similarly, as SL2 (R) is homotopic to a circle, we can deﬁne, for every n ∈ N, Gn as the n sheeted cover of SL2 (R). Then whenever m divides n there is a canonical covering morphisms ψn,m : Gn → Gm , and we may let G be the inverse limit G = lim Gn . Then ←− G is a connected locally compact group which admits no nontrivial connected compact normal subgroups. In order to prove Theorem 4.1, we need one last elementary lemma. Lemma 4.5. Let G be a topological group and N a normal subgroup. Then tI (G) ≤ tI (N) + tI (G/N). In particular if G/N and N are infinitesimally finitely generated then so is G. Proof of Theorem 4.1. Let G be a connected separable locally compact group. Let K ⊳ G be a compact normal subgroup such that G/K is a Lie group (see Theorem 4.3), and let K ◦ be its identity connected component. Then K ◦ is characteristic in K and therefore normal in G. Being a closed subgroup of a locally compact separable group, K ◦ is separable [CI77]. Let H = G/K ◦ and K t = K/K ◦ . By the isomorphism theorem, G/K ∼ = H/K t . t Note that K is a pro-ﬁnite group, hence by Proposition 2.1 H is inﬁnitesimally ﬁnitely generated. By Theorem 3.1, K ◦ is inﬁnitesimally ﬁnitely generated, hence, by Lemma 4.5, G is inﬁnitesimally ﬁnitely generated. 9 5 Quasi non-archimedean groups A topological group is non-archimedean if it has a basis of neighborhoods of the identity made of open subgroups. Equivalently, every neighborhood of the identity V contains a smaller neighborhood of the identity U such that the group generated by U is contained in V , which is the same as requiring that for every n ∈ N and g1 , ..., gn ∈ U, the group generated by g1 , ..., gn is contained in V . The deﬁnition that follows is obtained by switching two quantiﬁers in the above condition. Definition 5.1. Say a topological group G is quasi non-archimedean if for all n ∈ N and all neighborhood of the identity V ⊆ G, there exists a neighborhood of the identity U ⊆ V such that for all g1 , ..., gn ∈ U, the group generated by g1 , ..., gn is contained in V . Clearly every non-archimedean group is also quasi non-archimedean. Let us give right away the motivating example for this deﬁnition. We ﬁx a standard probability space (X, µ), that is, a probability space which is isomorphic to the interval [0, 1] with its Borel σ-algebra and the Lebesgue-measure. A Borel bijection T of X is called a non-singular automorphism if for all measurable A ⊆ X, one has µ(A) = 0 if and only if µ(T −1(A)) = 0. The group of all these automorphisms is denoted by Aut∗ (X, µ), two such automorphisms being identiﬁed if they coincide on a full measure set. We then deﬁne the uniform metric du on Aut∗ (X, µ) by: for all T, U ∈ Aut∗ (X, µ), du (T, U) = µ({x ∈ X : T (x) 6= U(x)}). This is a complete metric, though far from being separable (e.g. the group S1 acts freely on itself, yielding an uncountable discrete subgroup of (Aut∗ (X, µ), du )). But among closed subgroups of (Aut∗ (X, µ), du ), full groups are separable. Full groups are invariants of orbit equivalence attached to nonsingular actions of countable groups on (X, µ): given a non-singular action of a countable group Γ on (X, µ), its full group [RΓ ] is the group of all T ∈ Aut∗ (X, µ) such that for every x ∈ X, T (x) ∈ Γ · x. Since every subgroup of a quasi non-archimedean group is quasi non-archimedean for the induced topology, the following result implies that full groups are quasi nonarchimedean. Theorem 5.2 (Kechris). Aut∗ (X, µ) is quasi non-archimedean for the uniform metric. Proof. Deﬁne the support of T ∈ Aut∗ (X, µ) to be the set of all x ∈ X such that T (x) 6= x. Note that du (idX , T ) is precisely the measure of the support of T . Let ǫ > 0 and n ∈ N, and consider the open ball U := Bdu (idX , ǫ). Suppose that g1 , ..., gn belong to V := Bdu (idX , ǫ/n), and let A be the reunion of their supports. By assumption, A has measure less than ǫ. Then the group generated by g1 , ..., gn is contained in the group of elements supported in A, which is itself a subset of U = Bdu (idX , ǫ). The following proposition shows that the class of quasi non-archimedean groups satisﬁes basically the same closure properties as the class of non-archimedean groups. Proposition 5.3. The class of quasi non-archimedean groups is closed under taking subgroups (with the induced topology), products and quotients. 10 The next proposition highlights the main diﬀerence between non-archimedean and quasi non-archimedean groups. From now on, we will restrict ourselves to the narrower but well-behaved class of Polish groups, that is, separable groups whose topology admits a compatible complete metric, e.g. full groups for the uniform topology, the group Aut∗ (X, µ) endowed with the weak topology, or the unitary group of a separable Hilbert space endowed with the strong operator topology. Let us point out that a locally compact group is Polish if and only if it is second-countable (see [Kec95, Thm. 5.3]). Recall that if G is a Polish group and (X, µ) is a standard (non-atomic) probability space, then the group L0 (X, µ, G) of measurable maps from X to G is a Polish group for the topology of convergence in measure, two such maps being identiﬁed if they coincide on a full measure set. A basis of neighborhoods of the identity for this topology is given by the sets Ũǫ = {f ∈ L0 (X, µ, G) : µ({x ∈ X : f (x) 6∈ U}) < ǫ}, where U is an open neighborhood of the identity in G and ǫ > 0. The Polish group L0 (X, µ, G) enjoys the two following nice properties (see e.g. [Kec10, Chap. 19]) . • G embeds into L0 (X, µ, G) via constant maps. • L0 (X, µ, G) is connected, in fact contractible. Proposition 5.4. Let G be a quasi non-archimedean Polish group. Then L0 (X, µ, G) is quasi non-archimedean. In particular any quasi non-archimedean Polish group embeds in a connected quasi non-archimedean Polish group. Proof. Let Ũǫ = {f : µ({x ∈ X : f (x) 6∈ U}) < ǫ} be a basic neighborhood of the identity in L0 (X, µ, G). Let n ∈ N and V be a corresponding neighborhood of the identity witnessing that G is quasi non-archimedean. Consider the following open neighborhood of the identity in L0 (X, µ, G): Ṽǫ/n = {f : µ({x ∈ X : f (x) 6∈ V }) < ǫ/n}. Then if we let f1 , ..., fn ∈ Ṽǫ,n , the reunion of the sets {x ∈ X : fi (x) 6∈ V } has measure less than ǫ. By the deﬁnition of V and U the group generated by f1 , ..., fn is a subset of Ũǫ . Remark. Since non-archimedean groups are totally disconnected, the above proposition implies there are a lot more quasi non-archimedean groups than the non-archimedean ones. The following proposition is inspired by Section (D) of Chapter 4 in [Kec10], where it is shown that any continuous homomorphism from an inﬁnitesimally ﬁnitely generated group into a full group is trivial. Proposition 5.5. Any continuous homomorphism from an infinitesimally finitely generated group into a quasi non-archimedean group is trivial. Proof. Let ϕ : G → H be such a morphism, let V be any neighborhood of the identity in H, and let n = tI (G). Then there is a neighborhood of the identity U in H such that any subgroup of H generated by n elements of U is contained in V . Since ϕ is continuous, ϕ−1 (U) is a neighborhood of the identity in G, and because tI (G) = n we may ﬁnd g1 , ..., gn ∈ ϕ−1 (U) which generate a dense subgroup in G. Then the closure of ϕ(G) coincides with the closure of the group generated by ϕ(g1 ), ..., ϕ(gn ) ∈ U, which by assumption is contained in V . So ϕ(G) is contained in the closure of any neighborhood of the identity in H, and since H is Hausdorﬀ this means that ϕ is trivial. 11 Corollary 5.6. The only topological group which is both infinitesimally finitely generated and quasi non-archimedean is the trivial group. Corollary 5.7. Every continuous homomorphism from a connected separable locally compact group into (Aut∗ (X, µ), du ) is trivial. Proof. Since every connected separable locally compact group is inﬁnitesimally ﬁnitely generated by Theorem 4.1 and Aut∗ (X, µ) is quasi non-archimedean by Theorem 5.2, the previous proposition readily applies. As a consequence of the previous proposition and Theorem 1.6, we have the following interesting characterizations of connectedness and total disconnectedness for locally compact separable groups. Theorem 5.8. Let G be a locally compact separable group. Then the following hold: (1) G is connected if and only if G is infinitesimally finitely generated. (2) G is totally disconnected if and only if G is quasi non-archimedean. Proof. If G is not connected but inﬁnitesimally ﬁnitely generated, then G/G0 must also be inﬁnitesimally ﬁnitely generated, which is impossible by van Dantzig’s theorem. The converse is provided by Theorem 1.6. If G is totally disconnected, then G is non-archimedean by van Dantzig’s theorem. But this implies that G is quasi non-archimedean. For the converse, suppose G is quasi nonarchimedean. Then G0 also is, but then by (1) and Proposition 5.5 it must be trivial. Remark. As was pointed out by Caprace and Cornulier, one can prove (2) more directly. Indeed, if G0 is non-trivial then it admits a non-trivial one-parameter subgroup which is in particular inﬁnitesimally 2-generated, contradicting Proposition 5.5. This actually gives a proof that a locally compact group is quasi non-archimedean if and only if it is totally disconnected, regardless of its separability. Let us now give an example of a totally disconnected Polish group which is quasi non-archimedean, but not non-archimedean. This class of examples was introduced by Tsankov [Tsa06, Sec. 5], using work of Solecki [Sol99]. We denote by S∞ the group of all permutations of the integers, equipped with its Polish topology of pointwise convergence. Recall that every non-archimedean Polish group arises as a closed subgroup of S∞ (see [BK96, Thm. 1.5.1]). Here, the groups that we will consider are subgroups of S∞ , but equipped with a Polish topology which reﬁnes the topology of pointwise convergence. Definition 5.9. A lower semi-continuous submeasure on N is a function λ : P(N) → [0, +∞] such that the following hold: • λ(∅) = 0; • for all n ∈ N, we have 0 < λ({n}) < +∞; • for all A ⊆ B ⊆ N, we have λ(A) 6 λ(B); • (subadditivity) for all A, B ⊆ N, we have λ(A ∪ B) 6 λ(A) + λ(B); • (lower semi-continuity) for every increasing sequence (Ak )k∈N of subsets of N, we S have λ( k∈N Ak ) = limk∈N λ(Ak ). 12 We associate to every lower semi-continuous submeasure λ on N a subgroup of S∞ , denoted by Sλ , deﬁned by Sλ = {σ ∈ S∞ : λ(supp σ \ {0, ..., n}) → 0 [n → +∞]}, where supp σ = {n ∈ N : σ(n) 6= n}. Note that since µ({0, ..., n}) < +∞ for every n ∈ N, the support of every σ ∈ Sλ has ﬁnite measure. Also, if λ is actually a measure, then Sλ = {σ ∈ S∞ : λ(supp σ) < +∞}; furthermore if λ is a probability measure then Sλ = S∞ . The group Sλ is equipped with a natural left-invariant metric dλ analogous to the uniform metric on Aut∗ (X, µ) deﬁned by dλ (σ, σ ′ ) = λ({n ∈ N : σ(n) 6= σ ′ (n)}). Note that the condition λ(supp σ \ {0, ..., n}) → 0 ensures that the countable group of permutations of ﬁnite support is dense in Sλ which is thus separable. It is a theorem of Tsankov that Sλ is actually a Polish group. The following result is a straightforward adaptation of Theorem 5.2, replacing du by dλ . Proposition 5.10. Let λ be a lower semi-continuous submeasure on N. Then Sλ is quasi non-archimedean. It is easily checked that the topology of Sλ reﬁnes the topology induced by S∞ , so that Sλ is always totally disconnected, and that the open subgroups of Sλ separate points from the identity (in particular, Sλ is not locally generated). The following example shows that it can furthermore fail to be non-archimedean. Note that this is just a particular case of a more general phenomenon: one can actually characterize when the topology fails to be zero-dimensional1 (see [Tsa06, Thm. 5.3]; the example below is taken from [Mal15, Cor. 4]). Example 5.11. Consider the measure λ on N deﬁned by X1 . λ(A) = n n∈A Then Sλ is not a non-archimedean group. Indeed, if we ﬁx ǫ > 0 and N ∈ N, we can ﬁnd a ﬁnite family (Ai )N i=1 of disjoint subsets of N such that for every i ∈ {1, ..., N}, ǫ < λ(Ai ) < ǫ. For all i ∈ {1, ..., N}, let σi be a permutation whose support is equal 2 Q to Ai . Then σ = N i=1 σi is at distance at least Nǫ/2 from the identity, so that the ball of radius ǫ around the identity generates a group which contains elements arbitrarily far away from the identity. Corollary 5.12. There exists a totally disconnected Polish group which is quasi nonarchimedean, but not non-archimedean. Let us now give examples of totally disconnected Polish group which are inﬁnitesimally ﬁnitely generated. To do this, we will upgrade a result of Stevens [Ste86] who showed the existence of totally disconnected inﬁnitesimally generated Polish groups: we will show that her examples are actually inﬁnitesimally ﬁnitely generated. This will be a consequence of the following general statement, which also implies the well-known fact that R has a dense Gδ of pairs of topological generators. 1 A topology is zero-dimensional if it has a basis made of clopen sets. 13 Theorem 5.13. Let G be an abelian Polish group which contains the group Z[1/2] of dyadic rationals as a dense subgroup, and assume furthermore that in the topology induced by G on Z[1/2], we have 21n → 0 [n → +∞]. Then for every g0 ∈ Z[1/2], the set of h ∈ G such that hg, hi generate a dense subgroup of G is dense. In particular there is a dense Gδ set of couples of topological generators of G in G2 and so G is infinitesimally finitely generated with infinitesimal rank at most 2. Proof. Let g0 ∈ Z[1/2]. In order for a couple (g0 , h) ∈ G2 to generate a dense subgroup of G, it suﬃces for the closed subgroup they generate to contain 21n for every n ∈ N, for 2 the group Z[1/2] is dense in G. T So the set T := {h ∈ G : hg0 , hi = G} may be written as a countable intersection T = n∈N Tn , where 1 Tn := h ∈ G : n ∈ hg0 , hi . 2 Since G is Polish the set Tn is Gδ , so we only need to show that Tn is dense. To this end, ﬁx ǫ > 0, n ∈ N, and let h0 ∈ Z[1/2]. We want to ﬁnd h ∈ Tn such that d(h, h0 ) < ǫ, where d is a ﬁxed compatible metric on G. Write g0 = 2km1 and h0 = 2km2 , where k1 , k2 ∈ Z and m ∈ N. We will ﬁnd β ∈ N such β 1 that for all N ∈ N, if h = h0 + 2m+N , then hg0 , hi contains 2N+m , so that in particular it 1 1 if and only if we can contains 2n as soon as N + m > n. The group hg0 , hi contains 2N+m 1 ﬁnd u, v ∈ Z such that ug0 + vh = 2m+N . This condition can be rewritten as 2N k1 u + (2N k2 + β)v = 1. So we want to ﬁnd β ∈ N such that for all N ∈ N we have that 2N k1 and 2N k2 + β are relatively prime. Let us furthermore ask that β is odd, so that we only have to make sure that every odd prime divisor of k1 does not divide 2N k2 + β. Let p1 , ..., pk list the odd primes which divide both k1 and k2 , while pk+1, ..., pl are the odd primes which divide k1 but not k2 . Then it is easily checked that β = (2 + p1 · · · pk )pk+1 · · · pl works: for all i 6 k we have that β is invertible modulo pi and pi divides k2 so that 2N k2 + β is not divisible by pi , while for k < i 6 l, β is null modulo pi while 2N k2 is invertible so that 2N k2 + β is not divisible by pi . β 1 But then, since 2m+N tends to zero as N tends to +∞, we also have 2m+N → 0 β [N → +∞]. Then, as explained before, the group generated by g := g0 and h := h0 + 2m+N 1 contains 2m+N , so that (g, h) ∈ Tn , while 0 = d(g, g0) < ǫ and d(h, h0 ) < ǫ if N was chosen large enough. So every Tn is a dense subset of G, which ends the proof since this furthermore shows that the set of couples generating a dense subgroup of G is dense in G2 and this set has to be a Gδ . Let us now apply the previous theorem and describe Steven’s examples of Polish groups which are inﬁnitesimally ﬁnitely generated but totally disconnected. These groups arise as a Polishable subgroups of the real line, constructed by taking a completion of the dyadic rationals with respect to a well chosen norm which makes 2−n have much bigger norm than usually. We ﬁx a biinﬁnite sequence of positive real number (ri )i∈Z such that ri → 0[i → +∞] and for all i ∈ Z, we have ri+1 6 ri 6 2ri+1 . Then one can deﬁne the following group 14 norm2 k·k on the ring Z[1/2] of dyadic rationals: for every x ∈ Z[1/2], ( n ) n X X kxk := inf \|ai \| ri : x = ai 2−i , ai ∈ Z, n ∈ N . i=−n i=−n It is easy to check that this deﬁnes a group norm on Z[1/2] which reﬁnes the usual norm. Using the fact that ri 6 2ri+1 , one can easily show that for all x ∈ Z[1/2], ( n ) n X X kxk = inf \|ai \| ri : x = ai 2−i , ai ∈ {−1, 0, 1}, n ∈ N . i=−n i=−n In particular, we see that for all n ∈ N, we have k2−n k = rn so that 2−n → 0 as n → +∞. Let Z[1/2] k·k denote the completion of Z[1/2] with respect to this norm. Since this norm reﬁnes the usual norm, Z[1/2] k·k is a subgroup of R. Stevens explicitely described the k·k k·k elements of R belonging to Z[1/2] and showed that the group Z[1/2] is inﬁnitesimally generated [Ste86, Thm. 2.1 (ii)], and we see that Theorem 5.13 strengthens this because k·k it implies that Z[1/2] is inﬁnitesimally 2-generated. To obtain totally disconnected examples, we need another result of Stevens stating that the following are equivalent (see [Ste86, Thm. 2.2]): P (i) i∈N ri = +∞, (ii) k·k is not equivalent to \|·\| when restricted to Z[1/2], (iii) Q ∩ Z[1/2] (iv) Z[1/2] k·k k·k = Z[1/2], is totally disconnected, Remark. Note that every subgroup G of R which is not equal to R has to be totally disconnected for the induced topology, since its complement is dense in R so that the sets of the form ]r, +∞[ for r ∈ R \ G are clopen in G. In particular, if G is a proper subgroup of R equipped with a topology which reﬁnes the usual topology of R then G is totally disconnected. So conditions (ii) and condition (iii) clearly imply condition (iv). P So suppose further that i∈N ri = +∞ (e.g. take ri = 1 if i 6 0 and ri = 1i otherwise). k·k Then we see that Z[1/2] is a totally disconnected Polish group which has inﬁnitesimal rank at most 2. Moreover since this group is a subgroup of the real line endowed with a ﬁner topology (see [Ste86, Thm. 2.1]) it cannot be monothetic, so its inﬁnitesimal rank is actually equal to 2. Corollary 5.14. There exists a totally disconnected Polish group which has infinitesimal rank 2, in particular there is a totally disconnected Polish group which is not quasi nonarchimedean. Remark. Note that one can see directly that Stevens’ groups are not quasi non-archimedean, even for n = 1. Indeed, if U is a neighborhood of 0 not containing 1, then if V is another neighborhood of 0 there is some N ∈ N such that 1/2N ∈ V , but the group generated by 1/2N contains 1 hence it is not a subset of U. 2 A norm on an abelian group is a function \|·\| : G → [0, +∞) such that for any x, y ∈ G, \|x + y\| 6 \|x\| + \|y\|, and \|x\| = \|−x\|. 15 We know that while the group of the reals has inﬁnitesimal rank 2, its quotient S1 = R/Z has inﬁnitesimal rank 1. The same is true of Stevens’ examples, which is going to yield the following result. Theorem 5.15. There exists a totally disconnected Polish group which has infinitesimal rank 1. Proof. Let G be a totally disconnected Polish group obtained by Stevens’ construction from a sequence ; then G is a proper subgroup of R containing Z[1/2] as a dense subgroup. Observe that Z is a discrete subgroup of G and we may thus form the Polish group G̃ := G/Z. The group G̃ is a proper dense subgroup of S1 = R/Z, so S1 \ G̃ is thus dense in S1 . Let A = p([0, 1/2[) where p : R → R/Z is the usual projection, then for all g ∈ S1 \ G̃ the set (g + A) ∩ G̃ is clopen in G̃. Moreover since S1 \ G̃ is dense in S1 the family of sets ((g + A) ∩ G̃)g∈S1 \G̃ separates points in G̃, so G̃ is totally disconnected. Furthermore, we have by Theorem 5.13 that there is a dense Gδ of h ∈ G such that the group generated by 1 and h is dense in G. Since 1 ∈ Z and G̃ = G/Z we conclude that there is a dense Gδ of h ∈ G̃ which generate a dense subgroup in G̃, in particular G̃ has inﬁnitesimal rank 1. We don’t know an example of a totally disconnected Polish group which is inﬁnitesimally generated and quasi non-archimedean. Moreover, we want to stress out that all the examples we know of Polish groups which are quasi non-archimedean actually fail the property even for n = 1, so it would be very interesting to have examples having a “non-QNA rank” greater than 1. 6 Further remarks and questions Let us point out how one can easily build Polish groups into which no non-discrete locally compact group can embed. Lemma 6.1. Let Γ be a countable discrete group without elements of finite order. Then every monothetic subgroup of L0 (X, µ, Γ) is infinite discrete. In particular, no nontrivial compact group embeds into L0 (X, µ, Γ). Proof. Given 1 6= f ∈ L0 (X, µ, Γ), ﬁnd A ⊆ X non-null and γ ∈ Γ \ {1} such that f↾A is constant equal to γ. By assumption, for all n ∈ Z \ {0}, the support of f n contains A, and so hf i is discrete. Theorem 6.2. Let Γ be a countable discrete group without elements of finite order, and let G be a separable locally compact group. Then every continuous morphism G → L0 (X, µ, Γ) factors through a discrete group. Proof. Let G0 be the connected component of the identity. Because L0 (X, µ, Γ) is quasi non-archimedean, π factors through G/G0 by Proposition 5.5 and Theorem 1.6. Then by van Dantzig’s theorem and the previous lemma, the kernel of the later map contains an open subgroup of G/G0 , hence it factors through a discrete group. Question 6.3. Is there a Polish group without any locally compact closed subgroup? 16 The group L0 (X, µ, G) was originally introduced to show that every Polish group embeds into a connected group, and we saw that being quasi non-archimedean is somehow opposite to being connected. Because L0 (X, µ, G) can be quasi non-archimedean, one may ask whether every Polish group embeds into a connected not quasi non-archimedean group. The isometry group of the Urysohn space answers this question — it is universal for Polish groups, connected (see [Mel06, Mel10] for stronger versions of these results as well as background on the Urysohn space) and cannot be quasi non-archimedean by universality. However, the same question can be asked replacing not quasi non-archimedean by inﬁnitesimally ﬁnitely generated. It seems to be open wether the isometry group of the Urysohn space is inﬁnitesimally ﬁnitely generated (it is topologically 2-generated by a result of Solecki, see [Sol05]). Question 6.4. Is the isometry group of the Urysohn space inﬁnitesimally 2-generated? Let us end this paper by mentioning a question related to ample generics. A Polish group G has ample generics if the diagonal conjugacy action of G onto Gn has a comeager orbit for every n ∈ N (see [KR07]). It has been recently discovered that there exists Polish groups with ample generics which are not non-archimedean (see[KLM15] and [Mal15]). These examples arise either as full groups or as groups of the form Sλ , which are quasi non-archimedean groups by Theorem 5.2 and Proposition 5.10. This motivates the following question. Question 6.5. Is there a Polish group which has ample generics, but which is not quasi non-archimedean? References [Aue34] H. Auerbach. Sur les groupes linéaires bornés (iii). Studia Mathematica, 5(1):43– 49, 1934. [BG03] E. Breuillard and T. Gelander. On dense free subgroups of Lie groups. J. Algebra, 261(2):448–467, 2003. [BGSS06] Emmanuel Breuillard, Tsachik Gelander, Juan Souto, and Peter Storm. Dense embeddings of surface groups. Geom. Topol., 10:1373–1389, 2006. [BK96] Howard Becker and Alexander S. Kechris. The descriptive set theory of Polish group actions, volume 232 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1996. [CI77] W. W. Comfort and G. L. Itzkowitz. Density character in topological groups. Math. Ann., 226(3):223–227, 1977. 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Proceedings of Machine Learning Research vol 65:1–58, 2017 Towards Instance Optimal Bounds for Best Arm Identification arXiv:1608.06031v2 [cs.LG] 24 May 2017 Lijie Chen Jian Li Mingda Qiao CHENLJ 13@ MAILS . TSINGHUA . EDU . CN LIJIAN 83@ MAIL . TSINGHUA . EDU . CN QMD 14@ MAILS . TSINGHUA . EDU . CN Institute for Interdisciplinary Information Sciences (IIIS), Tsinghua University, Beijing, China. Abstract In the classical best arm identification (Best-1-Arm) problem, we are given n stochastic bandit arms, each associated with a reward distribution with an unknown mean. Upon each play of an arm, we can get a reward sampled i.i.d. from its reward distribution. We would like to identify the arm with the largest mean with probability at least 1 − δ, using as few samples as possible. The problem has a long history and understanding its sample complexity has attracted significant attention since the last decade. However, the optimal sample complexity of the problem is still unknown. Recently, Chen and Li (2016) made an interesting conjecture, called gap-entropy conjecture, concerning the instance optimal sample complexity of Best-1-Arm. Given a Best-1-Arm instance I (i.e., a set of arms), let µ[i] denote the ith largest mean and ∆[i] = µ[1] − µ[i] denote the corPn −2 responding gap. H(I) = i=2 ∆[i] denotes the complexity of the instance. The gap-entropy conjecture states that for any instance I, Ω H(I) · ln δ −1 + Ent(I) is an instance lower bound, where Ent(I) is an entropy-like term determined by the gaps, and there is a δ-correct algorithm for −1 . We note Best-1-Arm with sample complexity O H(I) · ln δ −1 + Ent(I) + ∆−2 ln ln ∆ [2] [2] −2 −1 that Θ ∆[2] ln ln ∆[2] is necessary and sufficient to solve the two-arm instance with the best and second best arms. If the conjecture is true, we would have a complete understanding of the instance-wise sample complexity of Best-1-Arm (up to constant factors). In this paper, we make significant progress towards a complete resolution of the gap-entropy conjecture. For the upper bound, we provide a highly nontrivial algorithm which requires −1 −1 O H(I) · ln δ −1 + Ent(I) + ∆−2 ln ln ∆ polylog(n, δ ) [2] [2] samples in expectation for any instance I. For the lower bound, we show that for any Gaussian Best-1-Arm instance with gaps of the form 2−k , any δ-correct monotone algorithm requires at least Ω H(I) · ln δ −1 + Ent(I) samples in expectation. Here, a monotone algorithm is one which uses no more samples (in expectation) on I ′ than on I, if I ′ is a sub-instance of I obtained by removing some sub-optimal arms. Keywords: best arm identification, instance optimality, gap-entropy 1. Introduction The stochastic multi-armed bandit is one of the most popular and well-studied models for capturing the exploration-exploitation tradeoffs in many application domains. There is a huge body c 2017 L. Chen, J. Li & M. Qiao. C HEN L I Q IAO of literature on numerous bandit models from several fields including stochastic control, statistics, operation research, machine learning and theoretical computer science. The basic stochastic multi-armed bandit model consists of n stochastic arms with unknown distributions. One can adaptively take samples from the arms and make decision depending on the objective. Popular objectives include maximizing the cumulative sum of rewards, or minimizing the cumulative regret (see e.g., Cesa-Bianchi and Lugosi (2006); Bubeck et al. (2012)). In this paper, we study another classical multi-armed bandit model, called pure exploration model, where the decision-maker first performs a pure-exploration phase by sampling from the arms, and then identifies an optimal (or nearly optimal) arm, which serves as the exploitation phase. The model is motivated by many application domains such as medical trials Robbins (1985); Audibert and Bubeck (2010), communication network Audibert and Bubeck (2010), online advertisement Chen et al. (2014), crowdsourcing Zhou et al. (2014); Cao et al. (2015). The best arm identification problem (Best-1-Arm) is the most basic pure exploration problem in stochastic multiarmed bandits. The problem has a long history (first formulated in Bechhofer (1954)) and has attracted significant attention since the last decade Audibert and Bubeck (2010); Even-Dar et al. (2006); Mannor and Tsitsiklis (2004); Jamieson et al. (2014); Karnin et al. (2013); Chen and Li (2015); Carpentier and Locatelli (2016); Garivier and Kaufmann (2016). Now, we formally define the problem and set up some notations. Definition 1.1 Best-1-Arm: We are given a set of n arms {A1 , . . . , An }. Arm Ai has a reward distribution Di with an unknown mean µi ∈ [0, 1]. We assume that all reward distributions are Gaussian distributions with unit variance. Upon each play of Ai , we get a reward sampled i.i.d. from Di . Our goal is to identify the arm with the largest mean using as few samples as possible. We assume here that the largest mean is strictly larger than the second largest (i.e., µ[1] > µ[2] ) to ensure the uniqueness of the solution, where µ[i] denotes the ith largest mean. Remark 1.2 Some previous algorithms for Best-1-Arm take a sequence (instead of a set) of n arms as input. In this case, we may simply assume that the algorithm randomly permutes the sequence at the beginning. Thus the algorithm will have the same behaviour on two different orderings of the same set of arms. Remark 1.3 For the upper bound, everything proved in this paper also holds if the distributions are 1-sub-Gaussian, which is a standard assumption in the bandit literature. On the lower bound side, we need to assume that the distributions are from some family parametrized by the means and satisfy certain properties. See Remark D.4. Otherwise, it is possible to distinguish two distributions using 1 sample even if their means are very close. We cannot hope for a nontrivial lower bound in such generality. The Best-1-Arm problem for Gaussian arms was first formulated in Bechhofer (1954). Most early works on Best-1-Arm did not analyze the sample complexity of the algorithms (they proved their algorithms are δ-correct though). The early advances are summarized in the monograph Bechhofer et al. (1968). For the past two decades, significant research efforts have been devoted to understanding the optimal sample complexity of the Best-1-Arm problem. On the lower bound Mannor and Tsitsiklis Pn side,−2 (2004) proved that any δ-correct algorithm for Best-1-Arm takes Ω( i=2 ∆[i] ln δ−1 ) samples in expectation. In fact, their result is an instance-wise lower bound (see Definition 1.6). Kaufmann et al. 2 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION P −1 (2015) also provided an Ω( ni=2 ∆−2 [i] ln δ ) lower bound for Best-1-Arm, which improved the constant factor in Mannor and Tsitsiklis (2004). Garivier and Kaufmann (2016) focused on the asymptotic sample complexity of Best-1-Arm as the confidence level δ approaches zero (treating the gaps as fixed), and obtained a complete resolution of this case (even for the leading constant).1 Chen and Li (2015) showed that for each n there exists a Best-1-Arm instance with n arms that Pn −2 require Ω ln ln n samples, which further refines the lower bound. ∆ i=2 [i] The algorithms for Best-1-Arm have also been significantly improved in the last two decades Even-Dar et al. (2002); Gabillon et al. (2012); Kalyanakrishnan et al. (2012); Karnin et al. (2013); Jamieson et al. (2014); Chen and Li (2015); Garivier and Kaufmann (2016). Karnin et al. (2013) obtained an upper bound of Xn −1 −1 . O + ln δ ln ln ∆ ∆−2 [i] [i] i=2 The same upper bound was obtained by Jamieson et al. (2014) using a UCB-type algorithm called lil’UCB. Recently, the upper bound was improved to Xn −1 −1 −2 −1 + ln ln ∆ , n) + ln δ ln ln min(∆ O ∆−2 ∆ [2] [2] [i] [i] i=2 by Chen and Li (2015). There is still a gap between the best known upper and lower bound. To understand the sample complexity of Best-1-Arm, it is important to study a special case, which we term as SIGN-ξ. The problem can be viewed as a special case of Best-1-Arm where there are only two arms, and we know the mean of one arm. SIGN-ξ will play a very important role in our lower bound proof. Definition 1.4 SIGN-ξ: ξ is a fixed constant. We are given a single arm with unknown mean µ 6= ξ. The goal is to decide whether µ > ξ or µ < ξ. Here, the gap of the problem is defined to be ∆ = \|µ − ξ\|. Again, we assume that the distribution of the arm is a Gaussian distribution with unit variance. In this paper, we are interested in algorithms (either for Best-1-Arm or for SIGN-ξ) that can identify the correct answer with probability at least 1 − δ. This is often called the fixed confidence setting in the bandit literature. Definition 1.5 For any δ ∈ (0, 1), we say that an algorithm A for Best-1-Arm (or SIGN-ξ) is δcorrect, if on any Best-1-Arm (or SIGN-ξ) instance, A returns the correct answer with probability at least 1 − δ. 1.1. Almost Instance-wise Optimality Conjecture It is easy to see that no function f (n, δ) (only depending on n and δ) can serve as an upper bound of the sample complexity of Best-1-Arm (with n arms and confidence level 1 − δ). Instead, the sample complexity depends on the gaps. Intuitively, the smaller the gaps are, the harder the instance is (i.e., more samples are required). Since the gaps completely determine an instance (for Gaussian arms with unit variance, up to shifting), we use ∆[i] ’s as the parameters to measure the sample complexity. 1. In contrast, our work focus on the situation that both δ and all gaps are variables that tend to zero. In fact, if we let the gaps (i.e., ∆[i] ’s) tend to 0 while maintaining δ fixed, their lower bound is not tight. 3 C HEN L I Q IAO Now, we formally define the notion of instance-wise lower bounds and instance optimality.For algorithm A and instance I, we use TA (I) to denote the expected number of samples taken by A on instance I. Definition 1.6 (Instance-wise Lower Bound) For a Best-1-Arm instance I and a confidence level δ, we define the instance-wise lower bound of I as L(I, δ) := inf TA (I). A:A is δ-correct for Best-1-Arm We say a Best-1-Arm algorithm A is instance optimal, if it is δ-correct, and for every instance I, TA (I) = O(L(I, δ)). Now, we consider the Best-1-Arm problem from the perspective of instance optimality. Unfortunately, even for the two-arm case, no instance optimal algorithm may exist. In fact, Farrell (1964) showed that for any δ-correct algorithm A for SIGN-ξ, we must have lim inf ∆→0 TA (I) = Ω(1). ∆−2 ln ln ∆−1 This implies that any δ-correct algorithm requires ∆−2 ln ln ∆−1 samples in the worst case. Hence, the upper bound of ∆−2 ln ln ∆−1 for SIGN-ξ is generally not improvable. However, for a particular SIGN-ξ instance I∆ with gap ∆, there is an δ-correct algorithm that only needs O(∆−2 ln δ−1 ) samples for this instance, implying L(I∆ , δ) = Θ(∆−2 ln δ−1 ). See Chen and Li (2015) for details. Despite the above fact, Chen and Li (2016) conjectured that the two-arm case is the only obstruction toward an instance optimal algorithm. Moreover, based on some evidence from the previous work Chen and Li (2015), they provided an explicit formula and conjecture that L(I, δ) can be expressed by the formula. Interestingly, the formula involves an entropy term (similar entropy terms also appear in Afshani et al. (2009) for completely different problems). In order to state Chen and Li’s conjecture formally, we define the entropy term first. Definition 1.7 Given a Best-1-Arm instance I and k ∈ N, let X Gk = {i ∈ [2, n] \| 2−(k+1) < ∆[i] ≤ 2−k }, Hk = ∆−2 [i] , i∈Gk and pk = H k / X j Hj . We can view {pk } as a discrete probability distribution. We define the following quantity as the gap entropy of instance I: X 2 Ent(I) = pk ln p−1 k . k∈N:Gk 6=∅ Remark 1.8 We choose to partition the arms based on the powers of 2. There is nothing special about the constant 2, and replacing it by any other constant only changes Ent(I) by a constant factor. Conjecture 1.9 (Gap-Entropy Conjecture (Chen and Li, 2016)) There is an algorithm for Best1-Arm with sample complexity −1 ln ln ∆ O L(I, δ) + ∆−2 [2] , [2] 2. Note that it is exactly the Shannon entropy for the distribution defined by {pk }. 4 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION for any instance I and δ < 0.01. And we say such an algorithm is almost instance-wise optimal for Best-1-Arm. Moreover, Xn −1 L(I, δ) = Θ . · ln δ + Ent(I) ∆−2 [i] i=2 Remark 1.10 As we mentioned before, the term ∆−2 ln ln ∆−1 is sufficient and necessary for distinguishing the best and the second best arm, even though it is not an instance-optimal bound. The gap entropy conjecture states that modulo this additive term, we can obtain an instance optimal algorithm. Hence, the resolution of the conjecture would provide a complete understanding of the sample complexity of Best-1-Arm (up to constant factors). All the previous bounds for Best-1-Arm agree with Conjecture 1.9, i.e., existing upper (lower) bounds are no smaller (larger) the conjectured bound. See Chen and Li (2016) for details. 1.2. Our Results In this paper, we make significant progress toward the resolution of the gap-entropy conjecture. On the upper bound side, we provide an algorithm that almost matches the conjecture. Theorem 1.11 There is a δ-correct algorithm for Best-1-Arm with expected sample complexity Xn −1 −2 −1 −1 O · polylog(n, δ ) . ln ln ∆ · ln δ + Ent(I) + ∆ ∆−2 [2] [2] i=2 [i] P −1 + Ent(I) in Conjecture 1.9. For the Our algorithm matches the main term ni=2 ∆−2 [i] · ln δ additive term (which is typically small), we lose a polylog(n, δ−1 ) factor. In particular, for those instances where the additive term is polylog(n, δ−1 ) times smaller than the main term, our algorithm is optimal. On the lower bound side, despite that we are not able to completely solve the lower bound, we do obtain a rather strong bound. We need to introduce some notations first. We say an instance is discrete, if the gaps of all the sub-optimal arms are of the form 2−k for some positive integer k. We say an instance I ′ is a sub-instance of an instance I, if I ′ can be obtained by deleting some sub-optimal arms from I. Formally, we have the following theorem. Theorem 1.12 For any discrete instance I, confidence level δ < 0.01, and any δ-correct algorithm A for Best-1-Arm, there exists a sub-instance I ′ of I such that Xn −1 TA (I ′ ) ≥ c · , · ln δ + Ent(I) ∆−2 [i] i=2 where c is a universal constant. We say an algorithm is monotone, if TA (I ′ ) ≤ TA (I) for every I ′ and I such that I ′ is a subinstance of I. Then we immediately have the following corollary. Corollary 1.13 For any discrete instance I, and confidence level δ < 0.01, for any monotone δ-correct algorithm A for Best-1-Arm, we have that Xn −1 TA (I) ≥ c · , · ln δ + Ent(I) ∆−2 [i] i=2 where c is a universal constant. We remark that all previous algorithms for Best-1-Arm have monotone sample complexity bounds. The above corollary also implies P that if an algorithm has amonotone sample complexn −2 −1 + Ent(I) on all discrete instances. ity bound, then the bound must be Ω i=2 ∆[i] · ln δ 5 C HEN L I Q IAO 2. Related Work SIGN-ξ and A/B testing. In the A/B testing problem, we are asked to decide which arm between the two given arms has the larger mean. A/B testing is in fact equivalent to the SIGN-ξ problem. It is easy to reduce SIGN-ξ to A/B testing by constructing a fictitious arm with mean ξ. For the other direction, given an instance of A/B testing, we may define an arm as the difference between the two given arms and the problem reduces to SIGN-ξ where ξ = 0. In particular, our refined lower bound for SIGN-ξ stated in Lemma 4.1 also holds for A/B testing. Kaufmann et al. (2015); Garivier and Kaufmann (2016) studied the limiting behavior of the sample complexity of A/B testing as the confidence level δ approaches to zero. In contrast, we focus on the case that both δ and the gap ∆ tend to zero, so that the complexity term due to not knowing the gap in advance will not be dominated by the ln δ−1 term. Best-k-Arm. The Best-k-Arm problem, in which we are required to identify the k arms with the k largest means, is a natural extension of Best-1-Arm. Best-k-Arm has been extensively studied in the past few years Kalyanakrishnan and Stone (2010); Gabillon et al. (2011, 2012); Kalyanakrishnan et al. (2012); Bubeck et al. (2013); Kaufmann and Kalyanakrishnan (2013); Zhou et al. (2014); Kaufmann et al. (2015); Chen et al. (2017), and most results for Best-k-Arm are generalizations of those for Best-1Arm. As in the case of Best-1-Arm, the sample complexity bounds of Best-k-Arm depend on the gap parameters of the arms, yet the gap of an arm is typically defined as the distance from its mean to either µ[k+1] or µ[k] (depending on whether the arm is among the best k arms or not) in the context of Best-k-Arm problem. The Combinatorial Pure Exploration problem, which further generalizes the cardinality constraint in Best-k-Arm (i.e., to choose exactly k arms) to general combinatorial constraints, was also studied Chen et al. (2014, 2016); Gabillon et al. (2016). PAC learning. The sample complexity of Best-1-Arm and Best-k-Arm in the probably approximately correct (PAC) setting has also been well studied in the past two decades. For Best-1-Arm, the tight worst-case sample complexity bound was obtained by Even-Dar et al. (2002); Mannor and Tsitsiklis (2004); Even-Dar et al. (2006). Kalyanakrishnan and Stone (2010); Kalyanakrishnan et al. (2012); Zhou et al. (2014); Cao et al. (2015) also studied the worst case sample complexity of Best-k-Arm in the PAC setting. 3. Preliminaries Throughout the paper, I denotes an instance of Best-1-Arm (i.e., I is a set of arms). The arm with the largest mean in I is called the optimal arm, while all other arms are sub-optimal. We assume that every instance has a unique optimal arm. Ai denotes the arm in I with the i-th largest mean, unless stated otherwise. The mean of an arm A is denoted by µA , and we use µ[i] as a shorthand notation for µAi (i.e., the i-th largest mean in an instance). Define ∆A = µ[1] − µA as the gap of arm A, and let ∆[i] = ∆Ai denote the gap of arm Ai . We assume that ∆[2] > 0 to ensure the optimal arm is unique. We partition the sub-optimal arms into different groups based on their gaps. For each k ∈ N, group Gk is defined as Ai : ∆[i] ∈ 2−(k+1) , 2−k . For brevity, let G≥k and G≤k denoted S∞ Sk −2 i=k Gi and i=1 Gi respectively. The complexity of arm Ai is defined as ∆[i] , while the comPn plexity of instance I is denoted by H(I) = i=2 ∆−2 [i] (or simply H, if the instance is clear from P −2 the context). Moreover, Hk = A∈Gk ∆A denotes the total complexity of the arms in group Gk . 6 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION (Hk )∞ k=1 naturally defines a probability distribution on N, where the probability of k is given by pk = Hk /H. The gap-entropy of the instance I is then denoted by X Ent(I) = pk ln p−1 k . k Here and in the following, we adopt the convention that 0 ln 0−1 = 0. 4. A Sketch of the Lower Bound 4.1. A Comparison with Previous Lower Bound Techniques We briefly discuss the novelty of our new lower bound technique, and argue why the previous techniques are not sufficient to obtain our result. To obtain a lower bound on the sample complexity of Best-1-Arm, all the previous work Mannor and Tsitsiklis (2004); Chen et al. (2014); Kaufmann et al. (2015); Garivier and Kaufmann (2016) are based on creating two similar instances with different answers, and then applying the change of distribution method (originally developed in Kaufmann et al. (2015)) to argue that a certain number of samples are necessary to distinguish such two instances. The idea was further refined by Garivier and Kaufmann (2016). They formulated a max-min game between the algorithm and some instances (with different answers than the given instance) created by an adversary. The value of the game at equilibrium would be a lower bound of the samples one requires to distinguish the current instance and several worst adversary instances. However, we notice that even in the two-arm case, one cannot prove the Ω(∆−2 ln ln ∆−1 ) lower bound by considering only one max-min game to distinguish the current instance from other instance. Roughly speaking, the ln ln ∆−1 factor is due to not knowing the actual gap ∆, and any lower bound that can bring out the ln ln ∆−1 factor should reflect the union bound paid for the uncertainty of the instance. In fact, for the Best-1-Arm problem with n arms, the gap entropy Ent(I) term exists for a similar reason (not knowing the gaps). Hence, any lower bound proof for Best1-Arm that can bring out the Ent(I) term necessarily has to consider the uncertainty of current instance as well (in fact, the random permutation of all arms is the kind of uncertainty we need for the new lower bound). In our actual lower bound proof, we first obtain a very tight understanding of the SIGN-ξ problem (Lemma 4.1).3 Then, we provide an elegant reduction from SIGN-ξ to Best-1-Arm, by embedding the SIGN-ξ problem to a collection of Best-1-Arm instances. 4.2. Proof of Theorem 1.12 Following the approach in Chen and Li (2015), we establish the lower bound by a reduction from SIGN-ξ to discrete Best-1-Arm instances, together with a more refined lower bound for SIGNξ stated in the following lemma. Lemma 4.1 Suppose δ ∈ (0, 0.04), m ∈ N and A is a δ-correct algorithm for SIGN-ξ. P is a probability distribution on {2−1 , 2−2 , . . . , 2−m } defined by P (2−k ) = pk . Ent(P ) denotes the Shannon entropy of distribution P . Let TA (µ) denote the expected number of samples taken by A when it runs on an arm with distribution N (µ, 1) and ξ = 0. Define αk = TA (2−k )/4k . Then, m X pk αk = Ω(Ent(P ) + ln δ−1 ). k=1 3. Farrell’s lower bound Farrell (1964) is not sufficient for our purpose. 7 C HEN L I Q IAO It is well known that to distinguish the normal distribution N (2−k , 1) from N (−2−k , 1), Ω(4k ) samples are required. Thus, αk = TA (2−k )/4k denotes the ratio between the expected number of samples taken by A and the corresponding lower bound, which measures the “loss” due to not knowing the gap in advance. Then Lemma 4.1 can be interpreted as follows: when the gap is drawn from a distribution P , the expected loss is lower bounded by the sum of the entropy of P and ln δ−1 . We defer the proof of Lemma 4.1 to Appendix D. Now we prove Theorem 1.12 by applying Lemma 4.1 and an elegant reduction from SIGN-ξ to Best-1-Arm. Proof [Proof of Theorem 1.12] Let c0 be the hidden constant in the big-Ω in Lemma 4.1, i.e., m X pk αk ≥ c0 · (Ent(P ) + ln δ−1 ). k=1 We claim that Theorem 1.12 holds for constant c = 0.25c0 . Suppose towards a contradiction that A is a δ-correct (for some δ < 0.01) algorithm for Best-1Arm and I = {A1 , A2 , . . . , An } is a discrete instance, while for all sub-instance I ′ of I, TA (I ′ ) < c · H(I)(Ent(I) + ln δ−1 ). Recall that H(I) and Ent(I) denote the complexity and entropy of instance I, respectively. Construct a distribution of SIGN-ξ instances. Let nk be the number of arms in I with gap 2−k , and m be the greatest integer such that nm > 0. Since I is discrete, the complexity of instance I is given by m X H(I) = 4k nk . k=1 −1 −2 −m }. Moreover, Let pk = 4k nk /H(I). Then (pk )m k=1 defines a distribution P on {2 , 2 , . . . , 2 the Shannon entropy of distribution P is exactly the entropy of instance I, i.e., Ent(P ) = Ent(I). Our goal is to construct an algorithm for SIGN-ξ that violates Lemma 4.1 on distribution P . A family of sub-instances of I. Let U = {k ∈ [m] : nk > 0} be the set of “types” of arms that are present in I. We consider the following family of instances obtained from I. For S ⊆ U , define IS as the instance obtained from I by removing exactly one arm of gap 2−k for each k ∈ S. Note that IS is a sub-instance of I. Let S denote U \ S, the complement of set S relative to U . For S ⊆ U and k ∈ S, let τkS denote the expected number of samples taken on all the nk arms with gap 2−k when A runs on IS . Define αSk = 4−k τkS /nk . We note that 4k αSk is the expected number of samples taken on every arm with gap 2−k in instance IS .4 We have the following inequality: XX XX X 4k nk αSk = τkS ≤ TA (IS ) < c · 2\|U \| H(I)(Ent(I) + ln δ−1 ). (1) S⊆U k∈S S⊆U k∈S S⊆U The second step holds because the lefthand side only counts part of the samples taken by A. The last step follows from our assumption and the fact that IS is a sub-instance of I. 4. Recall that a Best-1-Arm algorithm is defined on a set of arms, so the arms with identical means in the instance cannot be distinguished by A. See Remark 1.2 for details. 8 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Construct algorithm Anew from A. Now we define an algorithm Anew for SIGN-ξ with ξ = 0. Given an arm A, we first choose a set S ⊆ U uniformly at random from all subsets of U . Recall that µ[1] denotes the mean of the optimal arm in I. Anew runs the following four algorithms A1 through A4 in parallel: 1. Algorithm A1 simulates A on IS ∪ {µ[1] + A}. 2. Algorithm A2 simulates A on IS ∪ {µ[1] + A}. 3. Algorithm A3 simulates A on IS ∪ {µ[1] − A}. 4. Algorithm A4 simulates A on IS ∪ {µ[1] − A}. More precisely, when one of the four algorithms requires a new sample from µ[1] + A (or µ[1] − A), we draw a sample x from arm A, feed µ[1] + x to A1 and A2 , and then feed µ[1] − x to A3 and A4 . Note that the samples taken by the four algorithms are the same up to negation and shifting. Anew terminates as soon as one of the four algorithms terminates. If one of A1 and A2 identifies µ[1] + A as the optimal arm, or one of A3 and A4 identifies an arm other than µ[1] − A as the optimal arm, Anew outputs “µA > 0”; otherwise it outputs “µA < 0”. Clearly, Anew is correct if all of A1 through A4 are correct, which happens with probability at least 1 − 4δ. Note that since 4δ < 0.04, the condition of Lemma 4.1 is satisfied. Upper bound the sample complexity of Anew . The crucial observation is that when µA = −2−k and k ∈ S, A1 effectively simulates the execution of A on IS\{k} . In fact, since all arms are Gaussian distributions with unit variance, the arm µ[1] + A is the same as an arm with gap 2−k in the original Best-1-Arm instance. Recall that the number of samples taken on each of the arms with S\{k} . Therefore, the expected number of samples taken on A gap 2−k in instance IS\{k} is 4k αk S\{k} is upper bounded by 4k αk .5 Likewise, when µA = −2−k and k ∈ S, A2 is equivalent to the execution of A on IS\{k} , and thus the expected number of samples on A is less than or equal to S\{k} 4k αk . Analogous claims hold for the case µA = +2−k and algorithms A3 and A4 as well. It remains to compute the expected loss of Anew on distribution P and derive a contradiction to Lemma 4.1. It follows from a simple calculation that   m X X X X 1 S\{k}  S\{k} pk αk ≤ αk pk · \|U \|  + αk 2 k=1 k∈U S⊆U :k∈S S⊆U :k∈S = = ≤ 1 2\|U \|−1 1 2\|U \|−1 X X S\{k} pk αk k∈U S⊆U :k∈S X X 4k nk · αS H(I) k S⊆U k∈S 2\|U \| · c · (Ent(I) + ln δ−1 ) < c0 (Ent(P ) + ln(4δ)−1 ). 2\|U \|−1 5. Recall that if A1 terminates after taking T samples from µ[1] + A, the number of samples taken by Anew on A is also T (rather than 4T ). 9 C HEN L I Q IAO The first step follows from our discussion on algorithm Anew . The third step renames the variables and rearranges the summation. The last line applies (1). This leads to a contradiction to Lemma 4.1 and thus finishes the proof. 5. Warmup: Best-1-Arm with Known Complexity To illustrate the idea of our algorithm for Best-1-Arm, we consider the following Pn simplified yet still non-trivial version of Best-1-Arm: the complexity of the instance, H(I) = i=2 ∆−2 [i] , is given, yet the means of the arms are still unknown. 5.1. Building Blocks We introduce some subroutines that are used throughout our algorithm. Uniform sampling. The first building block is a uniform sampling procedure, Unif-Sampl(S, ε, δ), which takes 2ε−2 ln(2/δ) samples from each arm in set S. Let µ̂A be the empirical mean of arm A (i.e., the average of all sampled values from A). It obtains an ε-approximation of the mean of each arm with probability 1 − δ. The following fact directly follows by the Chernoff bound. Fact 5.1 Unif-Sampl(S, ε, δ) takes O(\|S\|ε−2 ln δ−1 ) samples. For each arm A ∈ S, we have Pr [\|µ̂A − µA \| ≤ ε] ≥ 1 − δ. We say that a call to procedure Unif-Sampl(S, ε, δ) returns correctly, if \|µ̂A − µA \| ≤ ε holds for every arm A ∈ S. Fact 5.1 implies that when \|S\| = 1, the probability of returning correctly is at least 1 − δ. Median elimination. Even-Dar et al. (2002) introduced the Median Elimination algorithm for the PAC version of Best-1-Arm. Med-Elim(S, ε, δ) returns an arm in S with mean at most ε away from the largest mean. Let µ[1] (S) denote the largest mean among all arms in S. The performance guarantees of Med-Elim is formally stated in the next fact. Fact 5.2 Med-Elim(S, ε, δ) takes O(\|S\|ε−2 ln δ−1 ) samples. Let A be the arm returned by Med-Elim. Then Pr[µA ≥ µ[1] (S) − ε] ≥ 1 − δ. We say that Med-Elim(S, ε, δ) returns correctly, if it holds that µA ≥ µ[1] (S) − ε. Fraction test. Procedure Frac-Test(S, clow , chigh , θ low , θ high , δ) decides whether a sufficiently large fraction (compared to thresholds θ low and θ high ) of arms in S have small means (compared to thresholds clow and chigh ). The procedure randomly samples a certain number of arms from S and estimates their means using Unif-Sampl. Then it compares the fraction of arms with small means to the thresholds and returns an answer accordingly. The detailed implementation of Frac-Test is relegated to Appendix A, where we also prove the following fact. Fact 5.3 Frac-Test(S, clow , chigh , θ low , θ high , δ) takes O (ε−2 ln δ−1 ) · (∆−2 ln ∆−1 ) samples, where ε = chigh − clow and ∆ = θ high − θ low . With probability 1 − δ, the following two claims hold simultaneously: 10 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION • If Frac-Test returns True, \|{A ∈ S : µA < chigh }\| > θ low \|S\|. • If Frac-Test returns False, \|{A ∈ S : µA < clow }\| < θ high \|S\|. We say that a call to procedure Frac-Test returns correctly, if both the two claims above hold; otherwise the call fails. Elimination. Finally, procedure Elimination(S, dlow , dhigh , δ) eliminates the arms with means smaller than threshold dlow from S. More precisely, the procedure guarantees that at most a 0.1 fraction of arms in the result have means smaller than dlow . On the other hand, for each arm with mean greater than dhigh , with high probability it is not eliminated. We postpone the pseudocode of procedure Elimination and the proof of the following fact to Appendix A. Fact 5.4 Elimination(S, dlow , dhigh , δ) takes O(\|S\|ε−2 ln δ−1 ) samples in expectation, where ε = dhigh − dlow . Let S ′ denote the set returned by Elimination(S, dlow , dhigh , δ). Then with probability at least 1 − δ/2, \|{A ∈ S ′ : µA < dlow }\| ≤ 0.1\|S ′ \|. Moreover, for each arm A ∈ S with µA ≥ dhigh , we have Pr A ∈ S ′ ≥ 1 − δ/2. We say that a call to Elimination returns correctly if both \|{A ∈ S ′ : µA < dlow }\| ≤ 0.1\|S ′ \| and A1 (S) ∈ S ′ hold; otherwise the call fails. Here A1 (S) denotes the arm with the largest mean in set S. Fact 5.4 directly implies that procedure Elimination returns correctly with probability at least 1 − δ. 5.2. Algorithm Now we present our algorithm for the special case that the complexity of the instance is known in advance. The Known-Complexity algorithm takes as its input a Best-1-Arm instance I, the complexity H of the instance, as well as a confidence level δ. The algorithm proceeds in rounds, and maintains a sequence {Sr } of arm sets, each of which denotes the set of arms that are still considered as candidate answers at the beginning of round r. Roughly speaking, the algorithm eliminates the arms with Ω(εr ) gaps at the r-th round, if they constitute a large fraction of the remaining arms. Here εr = 2−r is the accuracy parameter that we use in round r. To this end, Known-Complexity first calls procedures Med-Elim and Unif-Sampl to obtain µ̂ar , which is an estimation of the largest mean among all arms in Sr up to an O(εr ) error. After that, Frac-Test is called to determine whether a large proportion of arms in Sr have Ω(εr ) gaps. If so, Frac-Test returns True, and then Known-Complexity calls the Elimination procedure with carefully chosen parameters to remove suboptimal arms from Sr . The following two lemmas imply that there algorithm for Best-1-Arm that matches is a δ-correct −1 −2 the instance-wise lower bound up to an O ∆[2] ln ln ∆[2] additive term.6 6. Lemma 5.6 only bounds the number of samples conditioning on an event that happens with probability 1 − δ, so the algorithm may take arbitrarily many samples when the event does not occur. However, Known-Complexity can be transformed to a δ-correct algorithm with the same (unconditional) sample complexity bound, using the “parallel simulation” technique in the proof of Theorem 1.11 in Appendix C. 11 C HEN L I Q IAO Algorithm 1: Known-Complexity(I, H, δ) Input: Instance I with complexity H and risk δ. Output: The best arm. S1 ← I; Ĥ ← 4096H; for r = 1 to ∞ do if \|Sr \| = 1 then return the only arm in Sr ; ; εr ← 2−r ; δr ← δ/(10r 2 ); ar ← Med-Elim(Sr , 0.125εr , 0.01); µ̂ar ← Unif-Sampl({ar }, 0.125εr , δr ); if Frac-Test(S r , µ̂ar −1.75εr , µ̂ar − 1.125εr , 0.3, 0.5, δr ) then δr′ ← \|Sr \|ε−2 r /Ĥ δ; Sr+1 ← Elimination(Sr , µ̂ar − 0.75εr , µ̂ar − 0.625εr , δr′ ); else Sr+1 ← Sr ; end Lemma 5.5 For any Best-1-Arm instance I and δ ∈ (0, 0.01), Known-Complexity(I, H(I), δ) returns the optimal arm in I with probability at least 1 − δ. Lemma 5.6 For any Best-1-Arm instance I and δ ∈ (0, 0.01), conditioning on an event that happens with probability 1 − δ, Known-Complexity(I, H(I), δ) takes −1 ln ln ∆ O H(I) · (ln δ−1 + Ent(I)) + ∆−2 [2] [2] samples in expectation. 5.3. Observations We state a few key observations on Known-Complexity, which will be used throughout the analysis. The proofs are exactly identical to those of Observations A.3 through A.5 in Appendix A. The following observation bounds the value of µ̂ar at round r, assuming the correctness of Unif-Sampl and Med-Elim. Observation 5.7 If Unif-Sampl returns correctly at round r, µ̂ar ≤ µ[1] (Sr ) + 0.125εr . Here µ[1] (Sr ) denotes the largest mean of arms in Sr . If both Unif-Sampl and Med-Elim return correctly, µ̂ar ≥ µ[1] (Sr ) − 0.25εr . The following two observations bound the thresholds used in Frac-Test and Elimination by applying Observation 5.7. = µ̂ar − 1.75εr and chigh = µ̂ar − 1.125εr denote the Observation 5.8 At round r, let clow r r two thresholds used in Frac-Test. If Unif-Sampl returns correctly, chigh ≤ µ[1] (Sr ) − εr . If both r low Med-Elim and Unif-Sampl return correctly, cr ≥ µ[1] (Sr ) − 2εr . Observation 5.9 Let dlow = µ̂ar − 0.75εr and dhigh = µ̂ar − 0.625εr denote the two thresholds r r used in Elimination. If Unif-Sampl returns correctly, dhigh ≤ µ[1] (Sr ) − 0.5εr . If both Med-Elim r and Unif-Sampl return correctly, dlow ≥ µ (S ) − ε . r [1] r r 12 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION 5.4. Correctness We define E as the event that all calls to procedures Unif-Sampl, Frac-Test, and Elimination return correctly. We will prove in the following that Known-Complexity returns the correct answer with probability 1 conditioning on E, and Pr [E] ≥ 1 − δ. Note that Lemma 5.5 directly follows from these two claims. Event E implies correctness. It suffices to show that conditioning on E, Known-Complexity never removes the best arm, and the algorithm eventually terminates. Suppose that A1 ∈ Sr . Observation 5.9 guarantees that at round r, the upper threshold used by Elimination is smaller than or equal to µ[1] (Sr ) − 0.5εr < µ[1] . By Fact 5.4, the correctness of Elimination guarantees that A1 ∈ Sr+1 . It remains to prove that Known-Complexity terminates conditioning on E. Define rmax := maxGr 6=∅ r. Suppose r ∗ is the smallest integer greater than rmax such that Med-Elim returns correctly at round r ∗ .7 By Observation 5.9, the lower threshold in Elimination is greater than or equal to µ[1] − εr∗ . The correctness of Elimination implies that \|Sr∗ +1 \|−1 = \|Sr∗ +1 ∩G≤rmax \| ≤ \|Sr∗ +1 ∩G<r∗ \| = \|{A ∈ Sr∗ +1 : µA < µ[1] −εr∗ }\| < 0.1\|Sr∗ +1 \|. It follows that \|Sr∗ +1 \| = 1. Therefore, the algorithm terminates either before or at round r ∗ + 1. E happens with high probability. We first note that at round r, the probability that either Unif-Sampl or Frac-Test fails (i.e., returns incorrectly) is at most 2δr . By a union bound, the probability that at least one call to Unif-Sampl or Frac-Test returns incorrectly is upper bounded by ∞ X r=1 ∞ X δ < δ/2. 2δr = 5r 2 r=1 It remains to bound the probability that Elimination fails at some round, yet procedures Unif-Sampl and Frac-Test are always correct. Define P (r, Sr ) as the probability that, given the value of Sr at the beginning of round r, at least one call to Elimination returns incorrectly in round r or later, yet Unif-Sampl and Frac-Test always return correctly. We prove by induction that for any Sr that contains the optimal arm A1 , P (r, Sr ) ≤ δ Ĥ where M (r, Sr ) := \|Sr ∩ G≤r−2 \| and C(r, Sr ) := 128C(r, Sr ) + 16M (r, Sr )ε−2 , r ∞ X \|Sr ∩ Gi \| i+1 X ε−2 j + j=r i=r−1 rmax X+1 (2) ε−2 i . i=r The details of the induction are postponed to Appendix E. 7. Med-Elim returns correctly with probability at least 0.99 in each round, so r ∗ is well-defined with probability 1. 13 C HEN L I Q IAO Observe that M (1, I) = 0 and C(1, I) = ∞ X \|Sr ∩ Gi \| ∞ X rmax X+1 \|Sr ∩ Gi \|4i + 4rmax ∞ 16 X ≤ 3 X i=0 A∈Sr ∩Gi Pr [E] ≥ 1 − P (1, S1 ) − 4i i=1 i=0  Therefore we conclude that 4j + j=1 i=0 16 ≤ 3 i+1 X !  −2  ∆−2 ≤ A + ∆[2] 32 H(I). 3 δ 2 δ δ 128C(1, I) + 16M (1, I)ε−2 − 1 2 Ĥ δ 32H δ ≥ 1 − 128 · · − ≥ 1 − δ, 4096H 3 2 which completes the proof of correctness. Here the first step applies a union bound. The second step follows from inequality (2), and the third step plugs in C(1, I) ≤ 32H(I)/3 and Ĥ = 4096H. ≥1− 5.5. Sample Complexity As in the proof of Lemma 5.5, we define E as the event that all calls to procedures Unif-Sampl, Frac-Test, and Elimination return correctly. We prove that Known-Complexity takes −1 ln ln ∆ O H(I)(ln δ−1 + Ent(I)) + ∆−2 [2] [2] samples in expectation conditioning on E. Samples taken by Unif-Sampl and By Facts 5.1 and 5.3, procedures Unif-Sampl Frac-Test. −1 −2 −1 −2 and Frac-Test take O εr ln δr = O εr (ln δ + ln r) samples in total at round r. In the proof of correctness, we showed that conditioning on E, the algorithm does not terminate before or at round k (for k ≥ rmax + 1) implies that Med-Elim fails between round rmax + 1 and round k − 1, which happens with probability at most 0.01k−rmax −1 . Thus for k ≥ rmax + 1, the expected number of samples taken by Unif-Sampl and Frac-Test at round k is upper bounded by −1 O 0.01k−rmax −1 · ε−2 (ln δ + ln k) . k Summing over all k = 1, 2, . . . yields the following upper bound: rX max k=1 −1 ε−2 + ln k) + k (ln δ ∞ X −1 0.01k−rmax −1 · ε−2 + ln k) k (ln δ k=rmax +1 −1 −1 . ln δ + ln ln ∆ =O 4rmax (ln δ−1 + ln rmax ) = O ∆−2 [2] [2] Here the first step holds since the first summation is dominated by the last term (k = rmax ), while the second one is dominated by the firstj term (k = k rmax + 1). The second step follows from the −1 observation that rmax = maxGr 6=∅ r = log2 ∆[2] . 14 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Samples taken by Med-Elim and Elimination. By Facts 5.2 and 5.4, Med-Elim and Elimination (if called) take H −2 −2 ′ −2 −1 O(\|Sr \|εr ) + O(\|Sr \|εr ln(1/δr )) = O \|Sr \|εr ln δ + ln \|Sr \|ε−2 r samples in total at round r. We upper bound the number of samples by a charging argument. For each round i, define ri as the largest integer r such that \|G≥r \| ≥ 0.5\|Si \|.8 Then we define   0, j < ri , Ti,j = H −2  ln δ−1 + ln , j ≥ ri εi \|Gj \|εi−2 as the number of samples that each arm in G Pj is charged at round i. We prove in Appendix E that for any i, j \|Gj \|Ti,j is an upper bound on the number of samples taken by Med-Elim and Elimination at the i-th round. Moreover, the expected number of samples that each arm in group Gj is charged is upper bounded by !! X H E[Ti,j ] = O ε−2 . ln δ−1 + ln j \|Gj \|ε−2 j i Note that Hk = P A∈Gk  O X i,j −2 ∆−2 A = Θ(\|Gk \|εk ). Therefore, Med-Elim and Elimination take   ! H  −2 \|G \|ε j j j   X H  Hj ln δ−1 + ln = O Hj j = O H(I) ln δ−1 + Ent(I) \|Gj \|E[Ti,j ] = O  X \|Gj \|ε−2 j ln δ−1 + ln samples in expectation conditioning on E. In total, algorithm Known-Complexity takes −1 −1 + O H(I) ln δ−1 + Ent(I) ln δ + ln ln ∆ O ∆−2 [2] [2] −1 ln ln ∆ =O H(I) ln δ−1 + Ent(I) + ∆−2 [2] [2] samples in expectation conditioning on E. This proves Lemma 5.6. 5.6. Discussion In the Known-Complexity algorithm, knowing the complexity H in advance is crucial to the efficient allocation of confidence levels (δr′ ’s) to different calls of Elimination. When H is unknown, 8. Note that \|G≥0 \| = n − 1 ≥ 0.5\|Si \| and \|G≥r \| = 0 < 0.5\|Si \| for sufficiently large r, so ri is well-defined. 15 C HEN L I Q IAO our approach is to run an elimination procedure similar to Known-Complexity with a guess of H. The major difficulty is that when our guess is much smaller than the actual complexity, the total confidence that we allocate will eventually exceed the total confidence δ. Thus, we cannot assume in our analysis that all calls to the Elimination procedure are correct. We present our ComplexityGuessing algorithm for the Best-1-Arm problem in Appendix A. References Peyman Afshani, Jérémy Barbay, and Timothy M Chan. Instance-optimal geometric algorithms. In Foundations of Computer Science, 2009. FOCS’09. 50th Annual IEEE Symposium on, pages 129–138. IEEE, 2009. Jean-Yves Audibert and Sébastien Bubeck. Best arm identification in multi-armed bandits. In COLT-23th Conference on Learning Theory-2010, pages 13–p, 2010. Robert E Bechhofer. A single-sample multiple decision procedure for ranking means of normal populations with known variances. The Annals of Mathematical Statistics, pages 16–39, 1954. Robert Eric Bechhofer, Jack Kiefer, and Milton Sobel. 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Optimal pac multiple arm identification with applications to crowdsourcing. In Proceedings of the 31st International Conference on Machine Learning (ICML-14), pages 217–225, 2014. 18 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Organization of the Appendix The appendix contains the proofs of our main results. In Section A, we present our algorithm for Best-1-Arm along with a few useful observations. In Section B and Section C, we prove the correctness and the sample complexity of our algorithm, thus proving Theorem 1.11. We present the complete proof of Theorem 1.12 in Section D. Finally, Section E contains the complete proofs of Lemma 5.5 and Lemma 5.6. Appendix A. Upper Bound A.1. Building Blocks We start by presenting the missing implementation and performance guarantees of our subroutines Frac-Test and Elimination. Fraction test. Recall that on input (S, clow , chigh , θ low , θ high , δ), procedure Frac-Test decides whether a sufficiently large fraction (with respect to θ low and θ high ) of arms in S have means smaller than the thresholds clow and chigh . The pseudocode of Frac-Test is shown below. Algorithm 2: Frac-Test(S, clow , chigh , θ low , θ high , δ) Input: An arm set S, thresholds clow , chigh , θ low , θ high , and confidence level δ. ε ← chigh − clow ; ∆ ← θ high − θ low ; m ← (∆/6)−2 ln(2/δ); cnt ← 0; for i = 1 to m do Pick A ∈ S uniformly at random; µ̂A ← Unif-Sampl({A}, ε/2, ∆/6); if µ̂A < (clow + chigh )/2 then cnt ← cnt + 1; end if cnt/m > (θ low + θ high )/2 then return True; else return False; Now we prove Fact 5.3. Fact 5.3 (restated) Frac-Test(S, clow , chigh , θ low , θ high , δ) takes O((ε−2 ln δ−1 ) · (∆−2 ln ∆−1 )) samples, where ε = chigh − clow and ∆ = θ high − θ low . With probability 1 − δ, the following two claims hold simultaneously: • If Frac-Test returns True, \|{A ∈ S : µA < chigh }\| > θ low \|S\|. • If Frac-Test returns False, \|{A ∈ S : µA < clow }\| < θ high \|S\|. Proof The first claim directly follows from Fact 5.1 and m · O(ε−2 ln ∆−1 ) = O((ε−2 ln δ−1 ) · (∆−2 ln ∆−1 )). It remains to prove the contrapositive of the second claim: \|{A ∈ S : µA < clow }\| ≥ θ high \|S\| implies Frac-Test returns True, and \|{A ∈ S : µA < chigh }\| ≤ θ low \|S\| implies Frac-Test returns False. 19 C HEN L I Q IAO Suppose \|{A ∈ S : µA < clow }\| ≥ θ high \|S\|. Then in each iteration of the for-loop, it holds that µA < clow with probability at least θ high . Conditioning on µA < clow , by Fact 5.1 we have µ̂A ≤ µA + ε/2 < clow + ε/2 = (clow + chigh )/2 with probability at least 1 − ∆/6. Thus, the expected increment of counter cnt is lower bounded by θ high (1 − ∆/6) ≥ θ high − ∆/6. Thus, cnt/m is the mean of m i.i.d. Bernoulli random variables with means greater than or equal to θ high − ∆/6. By the Chernoff bound, it holds with probability 1 − δ/2 that cnt/m ≥ θ high − ∆/6 − ∆/6 > (θ low + θ high )/2. An analogous argument proves cnt/m < (θ low + θ high )/2 with probability 1 − δ/2, given \|{A ∈ S : µA < chigh }\| ≤ θ low \|S\|. This completes the proof. Elimination. We implement procedure Elimination by repeatedly calling Frac-Test to determine whether a large fraction of the remaining arms have means smaller than the thresholds. If so, we uniformly sample the arms, and eliminate those with low empirical means. Algorithm 3: Elimination(S, dlow , dhigh , δ) Input: An arm set S, thresholds dlow , dhigh , and confidence level δ. Output: Arm set after the elimination. S1 ← S; dmid ← (dlow + dhigh )/2; for r = 1 to +∞ do δr ← δ/(10 · 2r ); if Frac-Test(Sr , dlow , dmid , 0.05, 0.1, δr ) then high − dmid )/2, δ ); µ̂ ← Unif-Sampl(S r , (d r Sr+1 ← A ∈ Sr : µ̂A > (dmid + dhigh )/2 ; else return Sr ; end We prove Fact 5.4 in the following. Fact 5.4 (restated) Elimination(S, dlow , dhigh , δ) takes O(\|S\|ε−2 ln δ−1 ) samples in expectation, where ε = dhigh − dlow . Let S ′ be the set returned by Elimination(S, dlow , dhigh , δ). Then we have Pr[\|{A ∈ S ′ : µA < dlow }\| ≤ 0.1\|S ′ \|] ≥ 1 − δ/2. Moreover, for each arm A ∈ S with µA ≥ dhigh , we have Pr[A ∈ S ′ ] ≥ 1 − δ/2. 20 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Proof Let ε = dhigh − dlow . To bound the number of samples taken by Elimination, we note that the number of samples taken in the r-th iteration is dominated by that taken by Unif-Sampl, O(\|Sr \|ε−2 ln δr−1 ). It suffices to show that \|Sr \| decays exponentially (in expectation); a direct summation over all r proves the sample complexity bound. We fix a particular round r. Suppose Frac-Test returns correctly (which happens with probability at least 1 − δr ) and the algorithm does not terminate at round r. Then by Fact 5.3, it holds that \|{A ∈ Sr : µA < dmid }\| > 0.05\|Sr \|. For each A ∈ Sr with µA < dmid , it holds with probability 1 − δr that µ̂A < µA + (dhigh − dmid )/2 < dmid + (dhigh − dmid )/2 = (dmid + dhigh )/2. Note that δr = δ/(10 · 2r ) ≤ 0.1. Thus, at most a 0.1 fraction of arms in {A ∈ Sr : µA < dmid } would remain in Sr+1 in expectation. It follows that conditioning on the correctness of Frac-Test at round r, the expectation of \|Sr+1 \| is upper bounded by 0.05\|Sr \| · δr + 0.95\|Sr \| ≤ 0.05\|Sr \|/10 + 0.95\|Sr \| = 0.955\|Sr \|. Moreover, even if Frac-Test returns incorrectly, which happens with probability at most 0.1, we still have \|Sr+1 \| ≤ \|Sr \|. Therefore, E[\|Sr+1 \|] ≤ 0.9 · 0.955E[\|Sr \|] + 0.1E[\|Sr \|] < 0.96E[\|Sr \|]. A simple induction yields E[\|Sr \|] ≤ 0.96r−1 \|S\|. Then the sample complexity of Elimination is upper bounded by ! ∞ ∞ X X r−1 −1 −2 −1 −2 0.96 (ln δ + r) E[\|Sr \|]ε ln δr = O \|S\|ε r=1 r=1 = O \|S\|ε−2 ln δ−1 . Then we proceed to the proof of the second claim. Let E denote the event that all calls to procedure Frac-Test returns correctly. By Fact 5.3 and a union bound, ∞ X Pr [EA ] ≥ 1 − δr ≥ 1 − δ/2. r=1 Conditioning on event E, if the algorithm terminates and returns Sr at round r, Fact 5.3 implies that \|{A ∈ Sr : µA < dlow }\| < 0.1\|Sr \|. This proves the second claim. Finally, fix an arm A ∈ S with µA > dhigh . Define EA as the event that every call to Frac-Test returns correctly in the algorithm, and \|µ̂A − µA \| < (dhigh − dmid )/2 in every round. By Facts 5.1 and 5.3, ∞ X Pr [EA ] ≥ 1 − 2δr ≥ 1 − δ/2. r=1 Then in each round r, it holds conditioning on EA that µ̂A ≥ µA − (dhigh − dmid )/2 > dhigh − (dhigh − dmid )/2 = (dmid + dhigh )/2. Thus, with probability 1 − δ/2, A is never removed from Sr . 21 C HEN L I Q IAO A.2. Overview As shown in Section 5, we can solve Best-1-Arm using −1 ln ln ∆ O H · (Ent + ln δ−1 ) + ∆−2 [2] [2] P samples, if we know in advance the complexity of the instance, i.e., H = ni=2 ∆−2 [i] . The value of H is essential for allocating appropriate confidence levels to different calls of Elimination and achieving the near-optimal sample complexity. When H is unknown, our strategy is to guess its value. The major difficulty with our approach is that when our guess, Ĥ, is much smaller than the actual complexity H, the total confidence that we allocate will exceed the total confidence δ. To prevent this from happening, we maintain the total confidence that we have allocated so far, and terminate the algorithm as soon as the sum exceeds δ.9 After that, we try a guess that is a hundred times larger. As we will see later, the most challenging part of the analysis is to ensure that our algorithm does not return an incorrect answer when Ĥ is too small. We also keep track of the number of samples that have been taken so far. Roughly speaking, when the number exceeds 100Ĥ , we also terminate the algorithm and try the next guess of Ĥ. This simplifies the analysis by ensuring that the number of samples we take for each guess grows exponentially, and thus it suffices to bound the number of samples taken on the last guess. A.3. Algorithm Algorithm Entropy-Elimination takes an instance of Best-1-Arm, a confidence δ and a guess of complexity Ĥt = 100t . It either returns an optimal arm (i.e., “accept” Ĥt ) or reports an error indicating that the given Ĥt is much smaller than the actual complexity (i.e., “reject” Ĥt ). Throughout the algorithm, we maintain Sr , Hr and Tr for each round r. Sr denotes the collection of arms that are still under consideration at the beginning of round r. We say that an arm is removed (or eliminated) at round r, if it is in Sr \ Sr+1 . Roughly speaking, Hr is an estimate of the total complexity of arms in group G1 , G2 , . . . , Gr . When this quantity exceeds our guess Ĥt , Entropy-Elimination directly rejects (i.e., returns an error). Tr is an upper bound on the number of samples taken by Med-Elim and Elimination10 before round r. As mentioned before, we also terminate the algorithm when Tr exceeds 100Ĥt . Intuitively, this prevents Entropy-Elimination from taking too many samples on small guesses of H, which gives rise to an inferior sample complexity. In each round of Entropy-Elimination, we first call Med-Elim to obtain a near-optimal arm ar . Then we use Unif-Sampl to estimate the mean of ar , denoted by µ̂ar . After that, we call Frac-Test with appropriate parameters to find out whether a considerable fraction of arms in Sr have gaps larger than εr . If so, we call procedure Elimination and update the value of Hr+1 accordingly. Note that we set the thresholds {θr } of Frac-Test such that the intervals [θr−1 , θr ] are disjoint. In particular, this property is essential for proving Lemma B.6 in the analysis of the correctness of the algorithm. Our algorithm for Best-1-Arm guesses the complexity of the instance and invokes EntropyElimination to check whether the guess is reasonable. If Entropy-Elimination reports an error, 9. For ease of analysis, we actually use δ 2 instead of δ in the algorithm. 10. As we will see later, the analysis of the sample complexity of Med-Elim and Elimination are different from the other two procedures. 22 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Algorithm 4: Entropy-Elimination(I, δ, Ĥt ) Input: Instance I, confidence δ and a guess of complexity Ĥt = 100t . Output: The best arm, or an error indicating the guess is wrong. S1 ← I; H1 ← 0; T1 ← 0; θ0 ← 0.3; c ← log4 100; for r = 1 to ∞ do if \|Sr \| = 1 then return the only arm in Sr ; εr ← 2−r ; δr ← δ/(50r 2 t2 ); 2 δr′ ← (4\|Sr \|ε−2 r /Ĥ)δ ; −1 −2 δ/Ĥ ln \|S \|ε ; Tr+1 ← Tr + \|Sr \|ε−2 r r t r if (Hr + 4\|Sr \|ε−2 r ≥ Ĥt ) or (Tr+1 ≥ 100Ĥt ) then return error; ar ← Med-Elim(Sr , 0.125εr , 0.01); µ̂ar ← Unif-Sampl({ar }, 0.125εr , δr ); θr ← θr−1 + (ct − r)−2 /10; if Frac-Test(Sr , µ̂ar − 1.75εr , µ̂ar − 1.125εr , δr , θr−1 , θr ) then Hr+1 ← Hr + 4\|Sr \|ε−2 r ; Sr+1 ← Elimination(Sr , µ̂ar − 0.75εr , µ̂ar − 0.625εr , δr′ ); else Sr+1 ← Sr ; Hr+1 ← Hr ; end 23 C HEN L I Q IAO we try a guess that is a hundred times larger. Otherwise, we return the arm chosen by EntropyElimination. Algorithm 5: Complexity-Guessing Input: Instance I and confidence δ. Output: The best arm. for t = 1 to ∞ do Ĥt ← 100t ; Call Entropy-Elimination(I, δ, Ĥt ); if Entropy-Elimination does not return an error then return the arm returned by Entropy-Elimination; end A.4. Observations We start with a few simple observations on Entropy-Elimination that will be used throughout the analysis. We first note that Entropy-Elimination lasts O(t) rounds on guess Ĥt , and our definition of θr ensures that all θr are in [0.3, 0.5]. Observation A.1 The for-loop in Entropy-Elimination(I, δ, Ĥt ) is executed at most ct times, where c = log4 100. Proof When r ≥ ct − 1, ct−1 Hr + 4\|Sr \|ε−2 = Ĥt . r ≥4·4 Thus Entropy-Elimination rejects at the if-statement. Observation A.2 For all t ≥ 1 and 1 ≤ r ≤ ct − 1, 0.3 ≤ θr−1 ≤ θr ≤ 0.5. Proof Clearly θr ≥ θ0 = 0.3. Moreover, r ∞ X 1 X −2 −2 (ct − k) /10 ≤ 0.3 + θr = θ0 + k ≤ 0.5. 10 k=1 k=1 The following observation bounds the value of µ̂ar at round r, conditioning on the correctness of Unif-Sampl and Med-Elim. Observation A.3 If Unif-Sampl returns correctly at round r, µ̂ar ≤ µ[1] (Sr ) + 0.125εr . Here µ[1] (Sr ) denotes the largest mean of arms in Sr . If both Unif-Sampl and Med-Elim return correctly, µ̂ar ≥ µ[1] (Sr ) − 0.25εr . 24 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Proof By definition, µar ≤ µ[1] (Sr ). When Unif-Sampl({ar }, 0.125εr , δr ) returns correctly, it holds that µ̂ar ≤ µar + 0.125εr ≤ µ[1] + 0.125εr . When both Med-Elim and Unif-Sampl are correct, µar ≥ µ[1] (Sr ) − 0.125εr , and thus µ̂ar ≥ µar − 0.125εr ≥ µ[1] (Sr ) − 0.25εr . The following two observations bound the thresholds used in Frac-Test and Elimination by applying Observation A.3. = µ̂ar − 1.75εr and chigh = µ̂ar − 1.125εr denote the Observation A.4 At round r, let clow r r two thresholds used in Frac-Test. If Unif-Sampl returns correctly, chigh ≤ µ[1] (Sr ) − εr . If both r low Med-Elim and Unif-Sampl return correctly, cr ≥ µ[1] (Sr ) − 2εr . Proof Observation A.3 implies that when Unif-Sampl is correct, chigh ≤ µ[1] (Sr ) + 0.125εr − 1.125εr = µ[1] (Sr ) − εr r and when both Med-Elim and Unif-Sampl return correctly, clow ≥ µ[1] (Sr ) − 0.25εr − 1.75εr = µ[1] (Sr ) − 2εr . r Observation A.5 Let dlow = µ̂ar − 0.75εr and dhigh = µ̂ar − 0.625εr denote the two thresholds r r used in Elimination. If Unif-Sampl returns correctly, dhigh ≤ µ[1] (Sr ) − 0.5εr . If both Med-Elim r low and Unif-Sampl return correctly, dr ≥ µ[1] (Sr ) − εr . Proof By the same argument, we have dhigh ≤ µ[1] (Sr ) + 0.125εr − 0.625εr = µ[1] (Sr ) − 0.5εr r when Unif-Sampl returns correctly, and dlow ≥ µ[1] (Sr ) − 0.25εr − 0.75εr = µ[1] (Sr ) − εr r when both Med-Elim and Unif-Sampl are correct. 25 C HEN L I Q IAO Appendix B. Analysis of Correctness B.1. Overview We start with a high-level overview of the proof of our algorithm’s correctness. We first define a good event on which we condition in the rest of the analysis. Let E1 be the event that in a particular run of Complexity-Guessing, all calls of procedure Unif-Sampl and Frac-Test return correctly. Recall that δr , the confidence of Unif-Sampl and Frac-Test, is set to be δ/(50r 2 t2 ) in the r-th round of iteration t. By a union bound, Pr[E1 ] ≥ 1 − 2 ∞ X ∞ X δ/(50t2 r 2 ) = 1 − 2δ(π 2 /6)2 /50 ≥ 1 − δ/3. t=1 r=1 The δ-correctness of our algorithm is guaranteed by the following two lemmas. The first lemma states that Entropy-Elimination accepts a guess Ĥt and returns correctly with high probability when Ĥt is sufficiently large. The second lemma guarantees that Entropy-Elimination rejects a guess Ĥt when Ĥt is significantly smaller than H, the actual complexity. More precisely, we define the following two thresholds: tmax = ⌊log100 H⌋ − 2 and t′max = log100 H(Ent + ln δ−1 )δ−1 + 2. The precise statements of the two lemmas are shown below. Lemma B.1 With probability 1 − δ/3 conditioning on event E1 , Complexity-Guessing halts before or at iteration t′max and it never returns a sub-optimal arm between iteration tmax + 1 and t′max . Lemma B.2 With probability 1 − δ/3 conditioning on event E1 , Complexity-Guessing never returns a sub-optimal arm in the first tmax iterations. Lemma B.1 and Lemma B.2 directly imply the following theorem. Theorem B.3 Complexity-Guessing is a δ-correct algorithm for Best-1-Arm. Proof Recall that Pr[E1 ] ≥ 1 − δ/3. It follows directly from Lemma B.1 and Lemma B.2 that with probability 1 − δ, Entropy-Elimination accepts at least one of Ĥ1 , Ĥ2 , . . . , Ĥt′max . Moreover, when Entropy-Elimination accepts, it returns the optimal arm. Therefore, Complexity-Guessing is δ-correct. B.2. Useful Lemmas To analyze our algorithm, it is essential to bound the probability that a specific guess Ĥt gets rejected by Entropy-Elimination. We hope that this probability is high when Ĥt is small (compared to the true complexity H), while it is reasonably low when Ĥt is large enough. It turns out to be useful to consider the following procedure P obtained from Entropy-Elimination by removing the if-statement that checks whether Hr + 4\|Sr \|ε−2 r ≥ Ĥt and Tr+1 ≥ 100Ĥt . In other 26 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION words, the modified procedure P never rejects, regardless the value of Ĥt . Note that r, the number of rounds, may exceed ct in P, which leads to invalid values of θr . In this case, we simply assume that the thresholds used in Frac-Test are 0.3 and 0.5 respectively, and the following analysis still works. Define random variable H∞ and T∞ to be the final estimation of the complexity and the number of samples at the end of P. More precisely, if P terminates at round r ∗ , then H∞ and T∞ are defined as Hr∗ and Tr∗ , respectively. Note that there is a natural mapping from an execution of P to an execution of EntropyElimination. In particular, if both H∞ < Ĥt and T∞ < 100Ĥt hold in an execution of procedure P, then Entropy-Elimination accepts in the corresponding run. Therefore, we may upper bound the probability of rejection by establishing upper bounds of H∞ and T∞ . The following two lemmas bound the expectation of H∞ and T∞ conditioning on the event that Elimination always returns correctly. Lemma B.4 E[H∞ \|all Elimination return correctly] ≤ 256H. Lemma B.5 Suppose Ĥt ≥ H. E[T∞ \|all Elimination return correctly] ≤ 16(H(Ent + ln δ−1 + ln(Ĥt /H))). Note that it is crucial for the two lemmas above that all Elimination are correct. The following lemma gives an upper bound on the probability that some call of Elimination returns incorrectly. Lemmas B.4 through B.6 together can be used to upper bound the probability of rejecting a guess Ĥ. In the statement of Lemma B.6, we abuse the notation a little bit by assuming A1 ∈ G∞ and ∆−2 [1] = +∞. Lemma B.6 Suppose that s ∈ {2, 3, . . . , n} and r ∗ ∈ N ∪ {∞} satisfy As−1 ∈ Gr∗ . When Entropy-Elimination runs on parameter Ĥt < ∆−2 [s−1] , the probability that there exists a call of procedure Elimination that returns incorrectly before round r ∗ is upper bounded by ! n X −2 ∆[i] δ2 /Ĥt . 3000s i=s The proofs of the three lemmas above are shown below. Proof [Proof of Lemma B.4] In the following analysis, we always implicitly condition on the event that all Elimination return correctly. Define H(r, S) as the expectation of H∞ −Hr at the beginning of the r-th kround of Entropy-Elimination, when the current set of arms is Sr = S. Let rmax denote j log2 ∆−1 [2] . Define C(r, S) = ∞ X \|S ∩ Gi \| i+1 X ε−2 j + j=r i=r−1 rmax X+1 ε−2 i i=r and M (r, S) = \|S ∩ G≤r−2 \|. We prove by induction on r that H(r, S) ≤ 128C(r, S) + 16M (r, S)ε−2 r . (3) We start with the base case at round rmax + 2. Recall that clow and dlow denote the lower r r threshold of Frac-Test and Elimination in round r respectively. For all r ≥ rmax + 2, if Med-Elim 27 C HEN L I Q IAO returns correctly at round r (which happens with probability 0.99), according to Observation A.4 and Observation A.5, we have dlow ≥ clow ≥ µ[1] − 2εr ≥ µ[1] − 2−(rmax +1) ≥ µ[2] . r r Since Frac-Test returns correctly (contioning on E1 ) and \|{A ∈ Sr : µA ≤ clow r }\| ≥ \|{A ∈ Sr : µA ≤ µ[2] }\| = \|Sr \| − 1 ≥ 0.5\|Sr \| ≥ θr \|Sr \| (the last step applies Observation A.2), Frac-Test must return True and Elimination will be called. Since we assume that all calls of Elimination return correctly, we have \|Sr+1 \| − 1 = \|{A ∈ Sr+1 : µA ≤ µ[2] }\| ≤ \|{A ∈ Sr+1 : µA ≤ dlow r }\| ≤ 0.1\|Sr+1 \|, which guarantees that Sr+1 only contains the optimal arm and the algorithm will return correctly in the next round. Let r0 denote the first round after round rmax + 2 (inclusive) in which Med-Elim returns correctly. Then according to the discussion above, we have Pr[r0 = r] ≤ 0.01r−rmax −2 for all r ≥ rmax + 2. Thus it follows from a direct summation on possible values of r0 that ∞ X H(rmax + 2, S) ≤ ≤ ≤ r=rmax +2 ∞ X Pr[r0 = r] · 4\|S\|ε−2 r r−rmax −2 4\|S\|ε−2 r 0.01 r=rmax +2 8\|S\|ε−2 rmax +2 ≤ 16M (rmax + 2, S)ε−2 rmax +2 , which proves the base case. Before proving the induction step, we note the following fact: for r = 1, 2, . . . , rmax + 1, C(r, S) − C(r + 1, S) = = ∞ X i=r−1 ∞ X \|S ∩ Gi \| i+1 X ε−2 j − j=r −2 \|S ∩ Gi \|ε−2 r + εr ∞ X i=r \|S ∩ Gi \| i+1 X −2 ε−2 j + εr j=r+1 (4) i=r−1 = (\|S ∩ G≥r−1 \| + 1)ε−2 r . Suppose inequality (3) holds for r + 1. Consider the following three cases of the execution of Entropy-Elimination in round r. Let Ncur = \|S ∩ Gr−1 \| and Nbig = \|S ∩ G≥r \|. For brevity, let Nsma denote M (r, S) in the following. We have Nsma + Ncur + Nbig = \|S\| − 1. Note that Sr+1 is the set of arms that survive round r. Case 1: Med-Elim returns correctly and Frac-Test returns True. 28 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION According to the induction hypothesis, the expectation of H∞ − Hr in this case can be bounded by: H(r + 1, Sr+1 ) + 4\|S\|ε−2 r −2 ≤128C(r + 1, Sr+1 ) + 16M (r + 1, Sr+1 )ε−2 r+1 + 4\|S\|εr −2 ≤128C(r + 1, S) + 16[(Nsma + Ncur )/10] · (4ε−2 r ) + 4\|S\|εr −2 =128[C(r, S) − (Ncur + Nbig + 1)ε−2 r ] + (6.4Nsma + 6.4Ncur + 4\|S\|)εr =128C(r, S) + (10.4Nsma − 117.6Ncur − 124Nbig − 124)ε−2 r ≤128C(r, S) + 10.4Nsma ε−2 r . Here the third line follows from the fact that Sr+1 ⊆ S and C(r+1, S) is monotone in S. Moreover, the correctness of the Elimination procedure implies that M (r + 1, Sr+1 ) ≤ (Nsma + Ncur )/10. The fourth line applies identity (4). Case 2: Med-Elim returns correctly and Frac-Test returns False. Since Frac-Test is always correct (conditioning on E1 ) and it returns False, Fact 5.3, Observation A.2 and Observation A.4 together imply Nsma ≤ θr \|S\| ≤ \|S\|/2. Thus Nsma ≤ \|S\| − Nsma = Ncur + Nbig + 1. As Elimination is not called in this round, the expectation of H∞ − Hr in this case can be bounded by H(r + 1, S) ≤128C(r + 1, S) + 16M (r + 1, S)ε−2 r+1 −2 ≤128[C(r, S) − (Ncur + Nbig + 1)ε−2 r ] + 64(Nsma + Ncur )εr =128C(r, S) + (64Nsma − 64Ncur − 128Nbig − 128)ε−2 r ≤ 128C(r, S). Here the last step follows from 64Nsma −64Ncur −128Nbig −128 ≤ 64(Nsma −Ncur −Nbig −1) ≤ 0. Case 3: Med-Elim returns incorrectly. In this case, the worst scenario happens when we add 4\|S\|ε−2 r to the complexity Hr , but no arms are eliminated. Then the expectation of H∞ − Hr in this case can be bounded by H(r + 1, S) + 4\|S\|ε−2 r −2 ≤128C(r + 1, S) + 16M (r + 1, S)ε−2 r+1 + 4\|S\|εr −2 ≤128[C(r, S) − (Ncur + Nbig + 1)ε−2 r ] + [64(Nsma + Ncur ) + 4\|S\|]εr −2 =128C(r, S) + (68Nsma − 60Ncur − 124Nbig − 124)ε−2 r ≤ 128C(r, S) + 68Nsma εr . Recall that Case 3 happens with probability at most 0.01. Thus we have: H(r, S) ≤ 0.01 128C(r, S) + 68M (r, S)ε−2 + 0.99 128C(r, S) + 10.4M (r, S)ε−2 r r ≤ 128C(r, S) + 16M (r, S)ε−2 r . 29 C HEN L I Q IAO The induction is completed. Note that (3) directly implies our bound: E [H∞ \|all Elimination return correctly] =H(1, S) ≤ 128C(1, S) + 16M (1, S)   i+1 ∞ X X 4j  \|S ∩ Gi \| ·  =128 j=0 i=0 ≤256 ∞ X 4i+1 \|S ∩ Gi \| i=0 ≤256 ∞ X X ∆−2 A = 256H. i=0 A∈S∩Gi Then we prove Lemma B.5, which is restated below. Lemma B.5. (restated) Suppose Ĥt ≥ H. E[T∞ \|all Elimination return correctly] ≤ 16(H(Ent + ln δ−1 + ln(Ĥt /H))). Proof [Proof of Lemma B.5] Recall that T∞ is the sum of ! −1 −2 H \|S \|ε Ĥ r r t ln = \|Sr \|ε−2 \|Sr \|ε−2 δ + ln δ−1 + ln r r ln H \|Sr \|εr−2 Ĥt (5) for all round r. T∞ serves as an upper bound on the expected number of samples taken by Med-Elim and Elimination (up to a constant factor). Before the technical proof, we discuss the intuition of our analysis. In order to bound T∞ , we attribute each term in (5) to a specific subset of arms. For simplicity, r we assume for now that this term is just \|Sr \|ε−2 r = 4 \|Sr \|. Roughly speaking, we “charge” a cost of r −2 εr = 4 to each arm in group G≥r . We expect that \|G≥r \| is at least a constant times \|Sr \|, so that the number of samples (i.e., 4r \|Sr \|) can be covered by the total charges. Then the analysis reduces to calculating the total cost that each arm is charged. Fix an arm A ∈ Gr′ for some r ′ . As described ′ above, A is charged 4r in round r (1 ≤ r ≤ r ′ ), and thus the total charge is bounded by 4r , which is the actual complexity of A. Now we start the formal proof. Consider the execution of procedure P on Ĥt . We define a collection of random variables {Ti,j : i, j ≥ 1}, where Ti,j corresponds to the cost we charge each arm in Gj at round i. For each i, let ri denote the largest integer such that \|G≥ri \| ≥ 0.5\|Si \|. Note that such an ri always exists, as \|G≥1 \| = \|S1 \| ≥ 0.5\|Si \| and \|G≥r \| = 0 for sufficiently large r. We define Ti,j as  0, j < ri , Ti,j = −2 Ĥ H −1 t εi ln \|G \|ε−2 + ln δ + ln H , j ≥ ri . j i Note that this slightly differs from the proof idea described above: Ti,j might be positive when i > j (i.e., we may not always charge G≥i in round i). In fact, the charging argument described 30 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION in the proof idea works only if, ideally, all calls of Med-Elim are correct. Since actually some Med-Elim may return incorrectly, we have to slightly modify the charging method. Nevertheless, we will show that this difference only incurs a reasonably small cost in expectation. We first claim that X T∞ ≤ 2 \|Gj \| · Ti,j . (6) i,j In other words, the total costh we charge is indeed an upper bound on Ti∞ . Note that the contribution −1 + ln(Ĥ /H) , while its contribution to the ln(H/(\|Sr \|ε−2 of round i to T∞ is \|Si \|ε−2 t r )) + ln δ i right-hand side of (6) is X X −2 −1 )) + ln δ + ln( Ĥ /H) ln(H/(\|G \|ε 2 \|Gj \| · Ti,j = 2 \|Gj \| · ε−2 t j i i j j h i −2 −1 ln(H/(\|S \|ε )) + ln δ + ln( Ĥ /H) ≥ 2\|G≥ri \| · ε−2 r t r i h i −2 −1 ln(H/(\|S \|ε )) + ln δ + ln( Ĥ /H) . ≥ \|Si \|ε−2 r t r i Then identity (6) directly follows from a summation on i. Then we bound the expectation of each Ti,j . When i ≤ j, we have the trivial bound ! Ĥt H −1 −2 . + ln δ + ln ln E[Ti,j ] ≤ εi H \|Gj \|ε−2 i When i > j, we bound the probability that Ti,j > 0. By definition, Ti,j > 0 if and only if ri ≤ j, where ri is the largest integer that satisfies \|G≥ri \| ≥ 0.5\|Si \|. It follows that Ti,j > 0 only if \|G≥j+1 \| < 0.5\|Si \|. Observe that in order to have \|G≥j+1 \| < 0.5\|Si \|, Med-Elim must return incorrectly between round j + 1 and round i − 1. In fact, suppose towards a contradiction that Med-Elim is correct in round k ∈ [j + 1, i − 1]. Then we have \|G≥j+1 \| ≥ \|G≥k \| ≥ \|Sk+1 ∩ G≥k \| > 0.5\|Sk+1 \| ≥ 0.5\|Si \|, a contradiction. Here the third step is due to the fact that when Elimination returns correctly at round k, the fraction of arms in Sk+1 with gap greater than 2−k is less than 0.1. Note that for each specific round, the probability that Med-Elim returns incorrectly is at most 0.01. Thus, the probability that Ti,j > 0 for i > j is upper bounded by 0.01i−j−1 . Therefore, ! Ĥt H −1 i−j−1 −2 + ln δ + ln . E[Ti,j ] ≤ 0.01 εi ln H \|Gj \|εi−2 It remains to sum up the upper bounds of E[Ti,j ] to yield our bound of E[T∞ ]. X X X \|Gj \| · E[Ti,j ] = 2 \|Gj \| · E[Ti,j ] + 2 \|Gj \| · E[Ti,j ]. E[T∞ ] ≤ 2 i,j i≤j 31 i>j C HEN L I Q IAO Here the first part can be bounded by 2 X i≤j j XX Ĥt H + ln δ−1 + ln \|Gj \| · E[Ti,j ] ≤ 2 \|Gj \| · 4 ln i \|Gj \|4 H j i=1 ! X Ĥt H + ln δ−1 + ln ≤4 \|Gj \| · 4j ln j \|Gj \|4 H j ! X H Ĥ t ≤4 Hj ln + Hj ln δ−1 + Hj ln Hj /4 H j ! Ĥt . ≤ 8H Ent + ln δ−1 + ln H i ! The second part can be bounded similarly. 2 X i>j ∞ X X H Ĥt \|Gj \| · E[Ti,j ] ≤ 2 0.01 \|Gj \| · 4 ln + ln δ−1 + ln i \|Gj \|4 H j i=j+1 ! X H Ĥt ≤4 \|Gj \| · 4j ln + ln δ−1 + ln j \|Gj \|4 H j ! X Ĥ H t + Hj ln δ−1 + Hj ln ≤4 Hj ln Hj /4 H j ! Ĥt . ≤ 8H Ent + ln δ−1 + ln H i−j−1 i ! In fact, the crucial observation for both the two inequalities above is that the summation decreases exponentially as i becomes farther away from j. The lemma directly follows. Finally, we prove Lemma B.6, which is restated below. Recall that we abuse the notation a little bit by assuming A1 ∈ G∞ and ∆−2 [1] = +∞. Lemma B.6. (restated) Suppose that s ∈ {2, 3, . . . , n} and r ∗ ∈ N ∪ {∞} satisfy As−1 ∈ Gr∗ . When Entropy-Elimination runs on parameter Ĥt < ∆−2 [s−1] , the probability that there exists a call of procedure Elimination that returns incorrectly before round r ∗ is upper bounded by ! n X δ2 /Ĥt . ∆−2 3000s [i] i=s Proof [Proof of Lemma B.6] Recall that As−1 ∈ Gr∗ . Suppose As ∈ Gr′ . Suppose that we are at the beginning of round r of Entropy-Elimination and the subset of arms that have not been removed is Sr = S. Moreover, we assume that the optimal arm, A1 , is still in Sr . Let P (r, S) denote the probability that some call of procedure Elimination returns incorrectly in round r, r + 1, . . . , r ∗ − 1. 32 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION As in the proof of Lemma B.4, we bound P (r, S) by induction using the potential function method. Define ′ +2 rX i+1 r′ X X −2 ε−2 εj + (s − 1) \|S ∩ Gi \| C(r, S) = j j=r j=r i=r−1 and M (r, S) = \|S ∩ G≤r−2 \|. Then it holds that for 1 ≤ r ≤ r ′ + 1, ′ C(r, S) − C(r + 1, S) = r X −2 −2 \|S ∩ Gi \|ε−2 r + (s − 1)εr ≥ (\|S ∩ G≥r−1 \| + 1)εr . i=r−1 We prove by induction that 2 P (r, S) ≤ 128C(r, S) + 16M (r, S)ε−2 δ /Ĥ. r (7) We first prove the base case at round r ′ + 2. If r ′ + 2 ≥ r ∗ , the bound holds trivially. Otherwise, we consider the ratio α = \|Sr′ +2 ∩ {As , As+1 , . . . , An }\|/\|Sr′ +2 \|, which is the fraction of arms at round r ′ + 2 that are strictly worse than As−1 . Let r0 be the first round after r ′ + 2 (inclusive) in which Med-Elim returns correctly. If Frac-Test returns False in round r0 , according to Fact 5.3 and the correctness of Frac-Test conditioning on event E1 , we have α ≤ θr0 . Consequently, in each of the following rounds (say, round r > r0 ), Frac-Test always returns False since α ≤ θr0 ≤ θr−1 , and Elimination will never be called before round r ∗ . Note that it is crucial that the threshold interval of Frac-Test in diffrent rounds are disjoint. For the other case, suppose Frac-Test returns True and we call Elimination in round r0 . Then after that, assuming Elimination returns correctly, the fraction of arms worse than As−1 will be smaller than 0.1. It also follows that Elimination will never be called after round r0 . Therefore, Elimination is called at most once between round r ′ + 2 and r ∗ − 1, and it can only be called at round r0 . Note that ′ for r ≥ r ′ + 2, Pr[r0 = r] ≤ 0.01r−r −2 . A direct summation on all possible values of r0 yields ′ P (r + 2, S) ≤ ∗ −1 rX Pr[r0 = r] · δr′ r=r ′ +2 = ∗ −1 rX ′ 2 0.01r−r −2 · 4\|S\|ε−2 r δ /Ĥ r=r ′ +2 ∞ X 2 ≤ 4\|S\|ε−2 δ / Ĥ 0.01k 4k ′ r +2 k=0 ≤ 2 5\|S\|ε−2 r ′ +2 δ /Ĥ. ′ ′ ′ Note that C(r ′ +2, S) = (s−1)ε−2 r ′ +2 , M (r +2, S) = \|S ∩G≤r \| and \|S\| ≤ \|S ∩G≤r \|+(s−1). Thus 2 P (r ′ + 2, S) ≤ 5(\|S ∩ G≤r′ \| + s − 1)ε−2 r ′ +2 δ /Ĥ which proves the base case. 2 ≤ 128C(r ′ + 2, S) + 16M (r ′ + 2, S)ε−2 ′ r +2 δ /Ĥ, 33 C HEN L I Q IAO Then we proceed to the induction step. Again, we consider whether Med-Elim returns correctly and whether Frac-Test returns True. Let Ncur = \|S ∩ Gr−1 \| and Nbig = \|S ∩ G≥r \|. Again, we denote M (r, S) by Nsma for brevity. Note that Sr+1 is the set of arms that survive round r. Case 1: Med-Elim returns correctly and Frac-Test returns True. In this case, Elimination is called with confidence level δr′ . Then the conditional probability that some Elimination returns incorrectly in this case is bounded by P (r + 1, Sr+1 ) + δr′ 2 −2 ≤ 128C(r + 1, Sr+1 ) + 16M (r + 1, Sr+1 )ε−2 δ /Ĥ r+1 + 4\|S\|εr −2 ≤ 128C(r + 1, S) + 64(Nsma + Ncur )ε−2 δ2 /Ĥ r /10 + 4\|S\|εr 2 −2 = 128C(r, S) − 128(Ncur + Nbig + s − 1)ε−2 δ /Ĥ r + (6.4Nsma + 6.4Ncur + 4\|S\|)εr ≤[128C(r, S) + 10.4M (r, S)εr−2 ]δ2 /Ĥ. Case 2: Med-Elim returns correctly and Frac-Test returns False. Since Frac-Test returns False, according to Fact 5.3 and Observation A.4, we have Nsma ≤ \|S\| − Nsma = Ncur + Nbig + 1. Then the conditional probability in this case is bounded by 2 P (r + 1, S) ≤ [128C(r + 1, S) + 16M (r + 1, S)ε−2 r+1 ]δ /Ĥ −2 2 ≤ [128C(r, S) − 128(Ncur + Nbig + s − 1)ε−2 r + 64(Nsma + Ncur )εr ]δ /Ĥ 2 ≤ [128C(r, S) + (64Nsma − 64Ncur − 128Nbig − 128(s − 1))ε−2 r ]δ /Ĥ 2 ≤ 128C(r, S)ε−2 r δ /Ĥ. Here the last step follows from 64Nsma − 64Ncur − 128Nbig − 128(s − 1) ≤ 64(Nsma − Ncur − Nbig − 1) ≤ 0. Case 3: Med-Elim returns incorrectly. 2 In this case, the worst scenario is that we call Elimination with confidence δr′ ≤ 4\|S\|ε−2 r δ /Ĥ, yet no arms are removed. So the conditional probability in this case is bounded by 2 P (r + 1, S) + 4\|S\|ε−2 r δ /Ĥ 2 −2 ≤ 128C(r + 1, S) + 16M (r + 1, S)ε−2 δ /Ĥ r+1 + 4\|S\|εr −2 −2 2 ≤[128C(r, S) − 128(Ncur + Nbig + s − 1)ε−2 r + 64(Nsma + Ncur )εr + 4(Nsma + Ncur + Nbig + 1)εr ]δ /Ĥ 2 ≤[128C(r, S) + (68Nsma − 60Ncur − 124Nbig − 124)ε−2 r ]δ /Ĥ ≤ 128C(r, S) + 68M (r, S)ε−2 δ2 /Ĥ. r Recall that Case 3 happens with probability at most 0.01. Thus we have: 2 2 P (r, S) ≤ 0.01 128C(r, S) + 68M (r, S)ε−2 δ /Ĥ + 0.99 128C(r, S) + 10.4M (r, S)ε−2 δ /Ĥ r r ≤ 128C(r, S) + 16M (r, S)ε−2 δ2 /Ĥ. r 34 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION The induction is completed. It follows from (7) that   ′ +2 rX i+1 r′ X X  δ2 /Ĥ ε−2 ε−2 \|Gi \| P (1, S) ≤ 128  j j + (s − 1) r′ ≤ 128 (4/3) " X i+1 \|Gi \|4 i=0 n X ≤ 128 (16/3) ≤ 3000s j=1 j=1 i=0 " + (4/3)(s − 1)4 r′ ∆−2 [i] + (64/3)(s − 1)4 i=s n X i=s r ′ +2 ∆−2 [i] ! # # δ2 /Ĥ δ2 /Ĥ δ2 /Ĥ. B.3. Proof of Lemma B.1 Recall that tmax = ⌊log100 H⌋ − 2 and t′max = ⌈log100 [H(Ent + ln δ−1 )δ−1 ]⌉ + 2. We restate and prove Lemma B.1 in the following. Lemma B.1. (restated) With probability 1 − δ/3 conditioning on event E1 , Complexity-Guessing halts before or at iteration t′max and it never returns a sub-optimal arm between iteration tmax + 1 and t′max . The high-level idea of the proof is to construct three other “good events” E2 , E3 and E4 . We show that each event happens with high probability conditioning on E1 . Moreover, events E1 through E4 together imply the desired event. Proof Recall that tmax = ⌊log100 H⌋ − 2 and t′max = ⌈log100 [H(Ent + ln δ−1 )δ−1 ]⌉ + 2. Let E2 denote the following event: for all t such that t ≥ tmax + 1 and Ĥt < 1003 H, Entropy-Elimination either rejects or outputs the optimal arm. Since Ĥtmax +1 = 100tmax +1 ≥ H/10000, there are at most log100 [1003 H/(H/10000)] + 1 = 6 different values of such Ĥt . For each Ĥt , the probability of returning a sub-optimal arm is bounded by the probability that the optimal arm is deleted, which is in turn upper bounded by δ2 as a corollary of Lemma B.9 proved in the following section. Thus, by a union bound, Pr[E2 \|E1 ] ≥ 1 − 6δ2 . Let E3 be the event that for all Ĥt such that t ≤ t′max and Ĥt ≥ 1003 H (or equivalently, ⌈log100 H⌉ + 3 ≤ t ≤ t′max ), Entropy-Elimination never returns an incorrect answer. In fact, in order for Entropy-Elimination to return incorrectly, some call of Elimination must be wrong. Thus we may apply Lemma B.6 to bound the probability of E3 . Specifically, we apply Lemma B.6 with 35 C HEN L I Q IAO s = 2. Then we have t′max Pr[E3 \|E1 ] ≥ 1 − ≥1− ≥1− 3000s X t=⌈log100 H⌉+3 ∞ X P n −2 i=s ∆[i] Ĥt δ2 6000H 2 δ 100t t=⌈log100 H⌉+3 ∞ X 6000 2 δ ≥ 100k k=3 1 − δ2 /100. Here the third step is due to the simple fact that 100⌈log 100 H⌉ ≥ H. Finally, let E4 denote the event that when Entropy-Elimination runs on Ĥt′max , no Elimination is wrong and the algorithm finally accepts. In order to bound the probability of the last event, we simply apply Markov inequality based on Lemma B.4 and Lemma B.5. Let E0 be the event that no Elimination is wrong when Entropy-Elimination runs on Ĥt′max . Then we have Pr[E4 \|E1 ] ≥ Pr[E0 \|E1 ] − E[H∞ \|E0 ] Ĥt′max − E[T∞ \|E0 ] 100Ĥt′max h i 16H Ent + ln δ−1 + ln(Ĥt′max /H) 256H − + ln δ−1 )δ−2 1003 H(Ent + ln δ−1 )δ−2 256 2 16 Ent + 3 ln δ−1 + ln(1002 (Ent + ln δ−1 )) 2 2 δ δ − ≥1−δ − 1002 1003 (Ent + ln δ−1 ) ≥ 1 − 2δ2 . ≥ 1 − δ2 − 1002 H(Ent Note that conditioning on events E1 through E4 , Entropy-Elimination never outputs an incorrect answer between iteration tmax + 1 and t′max . Moreover, our algorithm terminates before or at iteration t′max . The lemma directly follows from a union bound and the observation that for all δ ∈ (0, 0.01), 6δ2 + δ2 /100 + 2δ2 ≤ δ/3. Remark B.7 The last part of the proof implies a more general fact: for fixed Ĥt , EntropyElimination accepts with probability at least 1 − δ2 − 256H Ĥt − 16H(Ent + ln δ−1 + ln(Ĥt /H)) 100Ĥt . B.4. Mis-deletion of Arms We prove Lemma B.2 in the following. Again, our analysis in this subsection conditions on event E1 , which guarantees that all calls of Frac-Test and Unif-Sampl in Entropy-Elimination are correct. The high-level idea of the proof is to show that a large proportion of arms will not be accidentally removed before they have contributed a considerable amount to the total complexity. Formally, we define the mis-deletion of arms as follows. 36 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Definition B.8 An arm A ∈ Gr is mis-deleted in a particular run of Entropy-Elimination, if A is deleted before or at round r − 1. In particular, the optimal arm is mis-deleted if it is deleted in any round. The following lemma bounds the probability that a certain collection of arms are all mis-deleted. Lemma B.9 For a fixed collection of k arms, the probability that all of them are mis-deleted is at most δ2k . Proof Let S = {A1 , A2 , . . . , Ak } be a fixed set of k arms. (Here we temporarily drop the convention that Ai denotes the arm with the i-th largest mean.) For each Ai , let Eibad denote the event that Ai is mis-deleted, and let ri denote the group that contains Ai (i.e., Ai ∈ Gri ). By definition, µAi ≥ µ[1] − εri . We start by proving the following fact: suppose Elimination is called with confidence level δr′ in round r. Then the probability that all arms in S are mis-deleted in round r simultaneously is bounded by δ′ kr . We assume that r < ri for all i = 1, 2, . . . , k. Otherwise, if r ≥ ri for some i, then Ai cannot be mis-deleted in round r, since the definition of mis-deletion requires that r < ri . To analyze the behaviour of Elimination, we recall that each run of Elimination consists of several stages. (Here we use the term “stage” for an iteration of Elimination, while the term for Entropy-Elimination is “round”.) In each stage, procedure Unif-Sampl is called at line 6 to estimate the means of the arms that have not been eliminated. Let ribad denote the stage in which Ai gets deleted. Recall that dhigh is the upper threshold used in Elimination in round r. According to Observar tion A.5, dhigh ≤ µ[1] (Sr ) − 0.5εr = µ[1] (Sr ) − 2−(r+1) ≤ µ[1] − εri ≤ µAi . r Here the third step follows from our assumption that r < ri . In order for Elimination to eliminate an arm Ai with mean greater than dhigh in stage ribad , the Unif-Sampl subroutine must return an incorrect estimation for Ai (i.e., \|µ̂Ai − µAi \| > (dhigh − dmid )/2), which happens with probability bad . Since the samples taken on different arms are independent, the events that at most δr′ / 10 · 2ri Unif-Sampl returns incorrect estimates for different arms are also independent, and it follows that Qk ′ bad r bad . the probability that each arm Ai is removed at stage ri is bounded by i=1 δr / 10 · 2 i Therefore, the probability that all the k arms in S are mis-deleted in Elimination is upper bounded by ∞ X ∞ X r1bad =1 r2bad =1 = k Y i=1 ≤ k Y ··· ∞ Y k X rkbad =1 i=1 bad δr′ /(10 · 2ri )  ∞ X bad   δr′ / 10 · 2ri  ribad =1 k δr′ = δ′ r . i=1 Then we start with the proof of the lemma. Suppose that we are at the beginning of round r. m arms among S are still in Sr , while the sum of confidence levels allocated in the previous rounds 37 C HEN L I Q IAO Pr−1 ′ is δ′ (i.e., δ′ = i=1 δi ). Let P (r, δ′ , m) denote the probability that all the m remaining arms are mis-deleted in the future. We prove by induction that P (r, δ′ , m) ≤ (δ2 − δ′ )m . (8) Recall that the number of rounds that Entropy-Elimination lasts is bounded by ct according to Observation A.1. Thus when r = ⌈ct⌉ + 1, we have P (r, δ′ , m) = 0. Observe that δ′ never exceeds δ2 according to the behaviour of Entropy-Elimination. Therefore (8) holds for the base case. Now we proceed to the induction step. If Elimination is not called in round r, by induction hypothesis we have P (r, δ′ , m) ≤ P (r + 1, δ′ , m) ≤ (δ2 − δ′ )m , which proves inequality (8). If, on the other hand, Elimination is called with confidence δr′ . We observe that by the claim we proved above, the probability that exactly j arms among the m arms m ′j are mis-deleted is at most δ r . Thus by a simple summation, j ′ P (r, δ , m) ≤ m X m j=0 j j δ′ r ·P (r +1, δ′ +δr′ , m−j) ≤ m X m j=0 j j δ′ r (δ2 −δ′ −δr′ )m−j = (δ2 −δ′ )m , which completes the induction step. Finally, the lemma directly follows from (8) by plugging in r = 1, δ′ = 0 and m = k. Remark B.10 Let Eibad denote the event that Ai is mis-deleted. Note that although the events {Eibad } are not independent, we can still obtain an exponential bound (i.e., δ2k ) on the probability that k such events happen simultaneously. We call such events quasi-independent to reflect this property. Formally, a collection of n events {Ei }ni=1 are δ-quasi-independent, if for all 1 ≤ k ≤ n and 1 ≤ a1 < a2 < · · · < ak ≤ n, we have Pr[Ea1 ∩ Ea2 ∩ · · · ∩ Eak ] ≤ δk . Then the collection of events {Eibad } are δ2 -quasi-independent. The following lemma proves a generalized Chernoff bound for quasi-independent events. Lemma B.11 Suppose v1 , v2 , . . . , vn > 0. {Yi }ni=1 is a collection of random variables, where the support of Yi is {0, vi }. Moreover, the collection of events {YP i = vi } are δ-quasi-independent. Let (S1 , S2 , . . . , Sm ) be a partition of {1, 2, . . . , n} such that j∈Si vj ≤ 1 for all i. Define P 1 Pm δ Pn Xi = j∈Si Yj . Let X = m i=1 Xi and p = m i=1 vi . Then for all q ∈ (p, 1), Pr[X ≥ q] ≤ e−mD(q\|\|p) , where D(x\|\|y) = x ln(x/y) + (1 − x) ln[(1 − x)/(1 − y)] is the relative entropy function. 38 T OWARDS I NSTANCE O PTIMAL B OUNDS Proof Let pi = δ P j∈Si vj . Then p = 1 m Pm i=1 pi . FOR B EST A RM I DENTIFICATION For t > 0, we have Pr[X ≥ q] = Pr[etmX ≥ etmq ] ≤ E[etmX ] . etmq To bound E[etmX ], we consider a collection of independent random P variables Ỹ1 , Ỹ2 , . . . , Ỹn defined by Pr[Ỹi = vi ] = δ and Pr[Ỹi = 0] = 1 − δ. Define X̃i = j∈Si Ỹj for i = 1, 2, . . . , m, 1 Pm tmX can be written as and X̃ = m i=1 X̃i . Note that each term in the Taylor expansion of e Ql α i=1 Ynl , where l ≥ 0, (n1 , n2 , . . . , nl ) ∈ {1, 2, . . . , n}l , and α = tl /(l!) > 0. The correspondQ ing term in etmX̃ is then α li=1 Ỹnl . Let U = \|{ni : i ∈ {1, 2, . . . , l}}\| denote the set of distinct numbers among n1 , n2 , . . . , nl . We have " l # # " l l l Y Y Y Y vn l = E vnl ≤ δ\|U \| Ỹnl . Ynl = Pr[Yi = vi for all i ∈ U ] · E i=1 i=1 i=1 i=1 Summing over all terms in the expansion yields m i h i Y h E etX̃i . E etmX ≤ E etmX̃ = i=1 Here the last step holds since {X̃i } are independent. Note that since X̃i ∈ [0, 1], it follows from Jensen’s inequality that i i h h E etX̃i ≤ E et X̃i + 1 − X̃i = pi et + 1 − pi . Then tmX E e Pm ≤ m Y (pi et + 1 − pi ) ≤ (pet + 1 − p)m . i=1 1 Recall that p = m i=1 pi . Here the last step follows from Jensen’s inequality and the concavity t of ln(e x + 1 − x) for t > 0. By setting t = ln q(1−p) p(1−q) , we have Pr[X ≥ q] ≤ t m pe + 1 − p E[etmX ] ≤ = e−mD(q\|\|p) . etmq etq The following lemma states that if a collection of arms with a considerable amount of total complexity are not mis-deleted, Entropy-Elimination rejects Ĥ. Lemma B.12 S is a set of sub-optimal arms with complexity H(S) > Ĥ. Let r ∗ = maxA∈S log2 ∆−1 A . If in a particular run of Entropy-Elimination, no arm in S is mis-deleted and there exists an arm A∗ outside S with µA∗ ≥ maxA∈S µA such that A∗ is not deleted in the first r ∗ − 1 rounds, then Ĥ is rejected in that run. 39 C HEN L I Q IAO Proof Suppose S = {A1 , A2 , . . . , Ak } and Ai ∈ Gri . Without loss of generality, µA1 ≤ µA2 ≤ · · · ≤ µAk . By definition of r ∗ , we have r ∗ = max1≤i≤k ri = rk . According to our assumption, both Ak and A∗ are not deleted in the first r ∗ − 1 rounds. Thus Entropy-Elimination does not accept in the first r ∗ rounds. Suppose for contradiction that Ĥ is not rejected by Entropy-Elimination in a particular run. Define R = {r ∈ [1, r ∗ − 1] : Elimination is called in round r}. Let N1 = {i ∈ [k] : ∃r ∈ R, r ≥ ri } and N2 = [k] \ N1 . For each i ∈ N1 , since Ai is not mis-deleted, Ai ∈ Sri . Define ri′ = min{r ∈ R : r ≥ ri } as the first round after ri (inclusive) in which Elimination is called. . It follows that Ai ∈ Sri′ . At round ri′ of Entropy-Elimination, Hri′ +1 is set to Hri′ + 4\|Sri′ \|ε−2 r′ i . It follows that Hr∗ is at least the total cost = ε−2 Therefore we can “charge” Ai a cost of 4ε−2 ri′ +1 ri′ P . that arms in N1 are charged, i∈N1 ε−2 ri′ +1 For each i ∈ N2 , we have Ai ∈ Sri and Sri = Sr∗ . Thus it holds that \|Sr∗ \| ≥ \|N2 \|. When the if-statement in Entropy-Elimination is checked in round r ∗ , we have Hr ∗ + 4\|Sr∗ \|ε−2 r∗ ≥ X i∈N1 ε−2 ri′ +1 + N2 ε−2 r ∗ +1 ≥ k X i=1 ε−2 ri +1 ≥ k X ∆−2 Ai = H(S) > Ĥ. i=1 Here the second step follows from ri′ ≥ ri and r ∗ ≥ ri , while the third step follows from ∆Ai ≥ 2−(ri +1) . Therefore Entropy-Elimination rejects in round r ∗ , a contradiction. B.5. Proof of Lemma B.2 Lemma B.2 is restated below. Recall that tmax = ⌊log100 H⌋ − 2. Lemma B.2. (restated) With probability 1 − δ/3 conditioning on event E1 , Complexity-Guessing never returns a sub-optimal arm in the first tmax iterations. The high-level idea of the proof is simple. For each Ĥt , we identify a collection of near-optimal “crucial arms”. By Lemma B.9, the probability that all “crucial arms” are mis-deleted is small, thus we may assume that at least one crucial arm survives. This crucial arm serves as A∗ in Lemma B.12. Then according to Lemma B.12, in order for Entropy-Elimination to accept Ĥt , it must mis-delete a collection of “non-crucial” arms with a significant fraction of complexity. The probability of this event can also be bounded by using the generalized Chernoff bound proved in Lemma B.11. The major technical difficulty is the choice of “crucial arms”. We deal the following three cases separately: (1) Ĥt is greater than ∆−2 [2] , the complexity of the arm with the second largest mean; (2) −2 −2 Ĥt is between ∆−2 [s] and ∆[s−1] for some 3 ≤ s ≤ n; and (3) Ĥt is smaller than ∆[n] . We bound the probability that the lemma is violated in each case, and sum them up using a union bound. Proof [Proof of Lemma B.2] Case 1: ∆−2 [2] ≤ Ĥt ≤ Ĥtmax . We first deal with the case that Ĥt is relatively large. We partition the sequence of sub-optimal arms A2 , A3 , . . . , An into contiguous B1 , B2 , . . . , Bm such that the total complexity in each P blocks −2 −2 −2 block Bi , denoted by H(Bi ) = A∈Bi ∆A , is between ∆[2] and 3∆[2] . To construct such a partition, we append arms to the current block one by one from A2 to An . When the complexity 40 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION of the current block exceeds ∆−2 [2] , we start with another block. Clearly, the complexity of each . Note that the last block may have a complexity less than resulting block is upper bounded by 2∆−2 [2] . In that case, we simply merge it into the second last block. As a result, the total complexity ∆−2 [2] i i h h −2 −2 −2 , 3m∆ . It follows that H ∈ m∆ , 3∆ of every block is in ∆−2 [2] . [2] [2] [2] ≤ Ĥt < For brevity, let B≤i denote B1 ∪B2 ∪· · ·∪Bi and B<i = B≤i−1 . Since H(B1 ) = ∆−2 [2] H = H(B≤m ), therehexists a unique integer i k ∈ [2, m] that satisfies H(B<k ) ≤ Ĥt < H(B≤k ). −2 −2 Then we have Ĥt ∈ (k − 1)∆[2] , 3k∆[2] . Since B≤k contains at least k arms, it follows from Lemma B.9 that with probability 1 − δ2k , at least one arm in B≤k is not mis-deleted. Recall that by Lemma B.12, Entropy-Elimination accepts Ĥt only if either of the following two events happens: (a) no arm in B≤k survives, which happens with probability δ2k ; (b) a collection of arms among B>k with total complexity of at least H(B>k ) − Ĥ are mis-deleted. −2 For i = 2, 3, . . . , n, define vi = ∆−2 [i] /(3∆[2] ) and Yi = vi · I[Ai is mis-deleted]. For i = 1, 2, . . . , m, Xi is defined as Xi = X Aj ∈Bi Yj = X ∆−2 A −2 · I[A is mis-deleted]. 3∆ [2] A∈B i In other words, Xi is the total complexity of the arms in block Bi that are mis-deleted, divided by a −2 constant 3∆−2 [2] . Recall that H(Bi ) ≤ 3∆[2] , so Xi is between 0 and 1. Let X= n m X 1 1 X ∆−2 Xi = −2 [i] · I[Ai is mis-deleted] m 3m∆ [2] i=2 i=1 denote the mean of these random variables. Since the events of mis-deletion of arms are δ2 -quasiindependent, we may apply Lemma B.11. Note that −2 n n Hδ2 δ2 X δ2 X ∆[i] 2 = p= vi = −2 −2 ≤ δ . m m 3∆ 3m∆ [2] [2] i=2 i=2 Here the last step applies H ≤ 3m∆−2 [2] . On the other hand, conditioning on event (b) (i.e., a collection of arms with total complexity H(B>k ) − Ĥ are mis-deleted), we have X= n X 1 ∆−2 [i] · I[Ai is mis-deleted] 3m∆−2 [2] i=2 −2 (m − k)∆−2 H(B>k ) − Ĥ [2] − 3k∆[2] ≥ ≥ 3m∆−2 3m∆−2 [2] [2] ≥ m − 4k m − 12m/10000 1 ≥ ≥ . 3m 3m 6 −2 Here the third step follows from H(B>k ) ≥ (m − k)∆−2 [2] and Ĥ ≤ 3k∆[2] . The last line holds since −2 k∆−2 [2] ≤ Ĥ ≤ Ĥtmax ≤ H/10000 ≤ 3m∆[2] /10000, 41 C HEN L I Q IAO which implies k ≤ 3m/10000. According to Lemma B.11, we have Pr[X ≥ 1/6] ≤ exp −mD 1/6\|\|δ2 1 5 m 5m ln = exp − ln 2 − 6 6δ 6 6(1 − δ2 ) ≤ (6δ2 )m/6 · (6/5)5m/6 ≤ δm/6 . Recall that D(x\|\|y) stands for the relative entropy function. The last step follows from 6δ ·(6/5)5 ≤ 1. Therefore, i h (9) Pr Entropy-Elimination accepts Ĥt ≤ δ2k + δm/6 . i h It remains to apply a union bound to (9) for all values of Ĥt in ∆−2 , Ĥ t max . Recall that k ≥ 2, [2] and the ratio between different guesses Ĥt is at least 100. It follows that the values of k are distinct for different values of Ĥt , and thus the sum of the first term, δ2k , can be bounded by ∞ X k=2 δ2k = δ4 ≤ 2δ4 . 1 − δ2 For the second term, we note that the number of guesses Ĥt between ∆−2 [2] and Ĥtmax is at most m l H −2 − 1 ≤ log100 (3m) − 1. tmax − log100 ∆−2 [2] + 1 ≤ log100 H − 2 − log 100 ∆[2] + 1 = log100 ∆−2 [2] In particular, if m < 1002 /3, no Ĥt will fall into [∆−2 [2] , tmax ]. Thus we focus on the nontrivial case m ≥ 1002 /3. Then the sum of the second term δm/6 can be bounded by 2 /18 δm/6 · (log100 (3m) − 1) ≤ δ100 , since δm/6 · (log100 (3m) − 1) decreases on [1002 /3, +∞) for δ ∈ (0, 0.01). Finally, we have i h 2 , H ] ≤ 2δ4 + δ100 /18 ≤ 3δ4 . Pr Entropy-Elimination accepts Ĥt for some Ĥt ∈ [∆−2 t max [2] −2 Case 2: ∆−2 [s] ≤ Ĥ < ∆[s−1] for some 3 ≤ s ≤ n. In this case, Ĥ is between the complexity of As−1 and As . Our goal is to prove an upper bound of δΩ(s) on the probability of returning a sub-optimal arm for each specific s. Summing Pover all s yields a bound on the total probability. Our analysis depends on the ratio between Ĥ and ni=s ∆−2 [i] , the complexity of arms that are worse than As . Intuitively, when Ĥ is greater than the sum (Case 2-1), the contribution of the arms worse than As to the complexity is negligible. Thus we have to rely on the fact that the s − 1 arms with the largest means will not be mis-deleted simultaneously with high probability. On the other hand, when Ĥ is significantly smaller than the sum (Case 2-2), we may apply the same analysis as in Case 1. Finally, if the value of Ĥ is between the two cases (Case 2-3), it suffices to prove a relatively loose bound, since the number of possible values is small. 42 T OWARDS I NSTANCE O PTIMAL B OUNDS Case 2-1: Ĥ > 300000s FOR B EST A RM I DENTIFICATION Pn −2 i=s ∆[i] . In this case, our guess Ĥ is significantly larger than the total complexity of As , As+1 , . . . , An , yet Ĥ is smaller than the complexity of any one among the remaining arms. Thus intuitively, in order to reject Ĥ, Entropy-Elimination should not mis-delete all the first s − 1 arms. More formally, we have the following fact: in order for Entropy-Elimination to return a sub-optimal arm, it must delete A1 along with at least s − 3 arms among A2 , A3 , . . . , As−1 before round r ∗ , where r ∗ r ∗ +1 ≥ ∆−2 is the group that contains As−1 . In fact, since 4ε−2 r∗ = 4 [s−1] ≥ Ĥt , Entropy-Elimination ∗ terminates before or at round r . If A1 is not deleted before round r ∗ , Entropy-Elimination can only return A1 as the optimal arm, which is correct. If less than s−3 arms among A2 , A3 , . . . , As−1 are deleted before round r ∗ , for example Ai and Aj are not deleted (2 ≤ i < j ≤ s − 1), then both of them are contained in Sr∗ . It follows that Entropy-Elimination does not return before round r ∗ . We first bound the probability that A1 is deleted before round r ∗ . In order for this to happen, some Elimination must return incorrectly. By Lemma B.6, the probability of this event is upper bounded by ! n X δ2 /Ĥt . ∆−2 3000s [i] i=s In fact, we have a more general fact: the probability that a fixed set of k arms among {A2 , A3 , . . . , As−1 } together with A1 are deleted before round r ∗ is bounded by ! ! n n X X δ2(k+1) /Ĥt . ∆−2 δ2 /Ĥt · δ2k = 3000s ∆−2 3000s [i] [i] i=s i=s The proof follows from combining the two inductions in the proof of Lemma B.6 and Lemma B.9, and we omit it here. Since {A2 , A3 , . . . , As−1 } contains s − 2 subsets of size s − 3, the probability that Entropy-Elimination returns an incorrect answer on a particular guess Ĥt is at most ! n X −2 ∆[i] δ2(s−2) /Ĥt . (s − 2) · 3000s i=s It remains to apply a union bound on all Ĥt that fall into this case. Recall that Ĥt > 300000s Pn −2 i=s ∆[i] and Ĥt grows exponentially in t at a rate of 100. Thus the total probability is upper bounded by P n −2 ∞ ∞ 2(s−2) δ2(s−2) (s − 2) X 3000s ∆ X i=s [i] 1 2(s−2) δ (s − 2) δ (s − 2). = = P n −2 k+1 100 99 100k · 300000s i=s ∆[i] k=0 k=0 P Case 2-2: Ĥ < ni=s ∆−2 [i] /(78s). In this case, we apply the technique in the proof of Case 1. We partition the sequence As , As+1 , . . . , An i h −2 −2 into m consecutive blocks B1 , B2 , . . . , Bm such that H(Bi ) ∈ ∆[s] , 3∆[s] . Let B≤i denote Pn −2 B1 ∪ B2 ∪ · · · ∪ Bi . Since H(B1 ) = ∆−2 i=s ∆[i] /(78s) < H(B≤m ), there [s] ≤ Ĥ < exists a unique integer i k ∈ [2, m] such that H(B<k ) ≤ Ĥ < H(B≤k ). It follows that Ĥ ∈ h −2 −2 (k − 1)∆[s] , 3k∆[s] . By Lemma B.12, in order for Entropy-Elimination to accept Ĥ, one of the following two events happens: (a) Entropy-Elimination mis-deletes all arms in B≤k ∪ {A1 , A2 , . . . , As−1 }; (b) the total 43 C HEN L I Q IAO complexity of mis-deleted arms among B>k is greater than H(B>k ) − Ĥ. Since B≤k contains at least k arms, by Lemma B.9, the probability of event (a) is bounded by δ2(s+k−1) . Again, we bound the probability of event (b) using the generalized Chernoff bound in Lemma B.11. −2 For each i = s, s + 1, . . . , n, define vi = ∆−2 [i] /(3∆[s] ) and Yi = vi · I[Ai is mis-deleted]. Define random variables {Xi : i ∈ {1, 2, . . . , m}} as Xi = X Yj = Aj ∈Bi 1 X −2 ∆A · I[A is mis-deleted]. 3∆−2 [s] A∈B i Since H(Bi ) ≤ 3∆−2 [s] , Xi is between 0 and 1. Let X= n m X 1 X 1 ∆−2 Xi = [i] · I[Ai is mis-deleted] −2 m 3m∆ [s] i=s i=1 denote the mean of these random variables. Since the events {Yi = vi } are δ2 -quasi-independent, we may apply Lemma B.11. We have n p= H(B≤m )δ2 δ2 X vi = ≤ δ2 . −2 m 3m∆ [s] i=s Here the last step applies H(B≤m ) ≤ 3m∆−2 [s] . On the other hand, conditioning on event (b) (i.e., a collection of arms in B>k with total complexity H(B>k ) − Ĥ are mis-deleted), we have n X 1 · I[Ai is mis-deleted] ∆−2 X= −2 3m∆[s] i=s [i] −2 (m − k)∆−2 H(B>k ) − Ĥ [s] − 3k∆[s] ≥ ≥ 3m∆−2 3m∆−2 [s] [s] ≥ m − 4m/(26s) 1 m − 4k ≥ ≥ . 3m 3m 6 . The last line holds and Ĥ ≤ 3k∆−2 Here the third step follows from H(B>k ) ≥ (m − k)∆−2 [s] [s] since −2 k∆−2 [s] ≤ Ĥ ≤ H(B≤m )/(78s) ≤ m∆[s] /(26s), which implies k ≤ m/(26s). By Lemma B.11, we have Pr[X ≥ 1/6] ≤ δm/6 , and thus the probability that Entropy-Elimination return an incorrect answer on Ĥt is bounded by δ2(s+k−1) + δm/6 . It remains to apply a union bound on all valus of Ĥt that fall into this case. Since k ≥ 2 and the values of k are distinct, the sum of the first term is bounded by ∞ X k=2 δ2(s+k−1) = δ2s+2 ≤ 2δ2s+2 . 1 − δ2 44 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION For the second term, note that the number of different values of Ĥt between ∆−2 [s] and H(B≤m )/(78s) is bounded by i h log100 H(B≤m )/(78s)/∆−2 [s] + 1 ≤ log 100 [m/(26s)] + 1. Pn −2 i=s ∆[i] /(78s) In particular, if m < 26s, no Ĥt will fall into this case. So in the following we focus on the nontrivial case that m ≥ 26s. Since The sum of the second term is at most δm/6 (log100 [m/(26s)] + 1) ≤ δ13s/3 ≤ δ2s+2 . Here the first step follows from the fact that δm/6 (log100 [m/(26s)] + 1) decreases on [26s, +∞) for all δ ∈ (0, 0.01) and s ≥ 3. The second step follows from s ≤ 3. Therefore, the total probability that Entropy-Elimination returns incorrectly in this sub-case is bounded by 2δ2s+2 + δ2s+2 ≤ 3δ2s+2 . Pn P −2 Case 2-3: Ĥ ∈ [ ni=s ∆−2 i=s ∆[i] ]. [i] /(78s), 300000s In this case, we simply bound the probability of returning an incorrect answer by the probability that at least s − 2 arms in {A1 , A2 , . . . , As−1 } are mis-deleted, which is in turn bounded by (s − 1)δ2(s−2) according to Lemma B.9. As in the argument of Case 2-1, suppose that two arms Ai and Aj (1 ≤ i < j ≤ s − 1) are not mis-deleted. Let r ∗ be the group that contain As−1 . Then both Ai r ∗ +1 ≥ ∆−2 and Aj are contained in Sr∗ . However, as 4ε−2 r∗ = 4 [s−1] ≥ Ĥt , Entropy-Elimination will ∗ reject in round r , which implies that Entropy-Elimination will never return a sub-optimal arm. Note that at most log100 300000s + 1 ≤ 2 log100 s + 5 = log10 s + 5 1/(78s) different values of Ĥ fall into this case. Therefore, the total probability is bounded by δ2(s−2) (log10 s + 5)(s − 1). Combining Case 2-1 through Case 2-3 yields the following bound: the probability that Entropy−2 Elimination outputs an incorrect answer for some 3 ≤ s ≤ n and Ĥ ∈ [∆−2 [s] , ∆[s−1] ) is at most n X 1 2(s−2) 2s+2 2(s−2) δ (s − 2) + 3δ +δ (log10 s + 5)(s − 1) 99 s=3 n X 6 2(s−2) s − 2 + 3δ + (log10 s + 5)(s − 1) δ = 99 ≤ s=3 ∞ X δ2(s−2) (log10 s + 6)(s − 1) s=3 ∞ X 2 ≤δ 0.012(s−3) (log10 s + 6)(s − 1) ≤ 20δ2 . s=3 Case 3: Ĥt < ∆−2 [n] . 45 = C HEN L I Q IAO Finally, we turn to the case that Ĥ is smaller than ∆−2 [n] . In this case, Complexity-Guessing always rejects. Suppose An ∈ Gr∗ . Then in the first r ∗ − 1 rounds of Entropy-Elimination, Frac-Test always returns False. Thus no elimination is done before round r ∗ . Since Ĥt < ∆−2 [n] ≤ ∗ 4ε−2 r ∗ , Entropy-Elimination directly rejects when checking the if-statement at round r . Case 1 through Case 3 together directly imply the lemma, as 3δ4 + 20δ2 < δ/3 for all δ ∈ (0, 0.01). Appendix C. Analysis of Sample Complexity Recall that E1 is the event that all calls of Frac-Test and Unif-Sampl in Entropy-Elimination return correctly. We bound the sample complexity of our algorithm using the following two lemmas. Lemma C.1 Conditioning on E1 , the expected number of samples taken by Med-Elim and Elimination in Complexity-Guessing is O(H(Ent + ln δ−1 )). Lemma C.2 Conditioning on E1 , the expected number of samples taken by Unif-Sampl and Frac-Test in Complexity-Guessing is −1 −1 O(∆−2 [2] ln ln ∆[2] polylog(n, δ )). The two lemmas above directly imply the following theorem. Theorem C.3 Conditioning on E1 , the expected sample complexity of Complexity-Guessing is −1 −1 polylog(n, δ ) . ln ln ∆ O H(ln δ−1 + Ent) + ∆−2 [2] [2] Theorems B.3 and C.3 together imply that Complexity-Guessing is a δ-correct algorithm for Best-1-Arm, and its expected sample complexity is −1 −1 polylog(n, δ ) ln ln ∆ O H(ln δ−1 + Ent) + ∆−2 [2] [2] conditioning on an event which happens with probability at least 1 − δ. However, to prove Theorem 1.11, we need a δ-correct algorithm with the desired sample complexity in expectation (not conditioning on another event). In the following, we prove Theorem 1.11 using a parallel simulation trick developed in Chen and Li (2015). Proof [Proof of Theorem 1.11] Given an instance I of Best-1-Arm and a confidence level δ, we define a collection of algorithms {Ak : k ∈ N}, where Ak simulates Complexity-Guessing on instance I and confidence level δk = δ/2k . Then we construct the following algorithm A: • A runs in iterations. In iteration t, for each k such that 2k−1 divides t, A simulates Ak until Ak requires a sample from some arm A. A draws a sample from A, feeds it to Ak , and continue simulating Ak until it requires another sample. After that, A temporarily suspends Ak . • When some algorithm Ak terminates, A also terminates and returns the same answer. 46 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION We first note that if all algorithms in {Ak } are correct, A eventually returns the correct answer. Recall that Ak is a δ/2k -correct for Best-1-Arm. Thus by a simple union bound, A is P∞ algorithm k correct with probability 1 − k=1 δ/2 = 1 − δ, thus proving that A is δ-correct. It remains to bound the sample complexity of A. According to Theorem C.3, there exist constants C and m, along with a collection of events {Ek }, such that for each k, Pr[Ek ] ≥ 1 − δk , and the expected number of samples taken by Ak conditioning on Ek is at most i h −1 m m −1 (ln n + ln δ ) ln ln ∆ C H · (ln δk−1 + Ent) + ∆−2 k [2] [2] i h −1 m −1 m (ln n + (k ln δ ) ) ln ln ∆ ≤C H · (k ln δ + Ent) + ∆−2 [2] [2] ≤km · T (I). i h −1 m m −1 (ln n + ln δ ) , the desired samln ln ∆ Here T (I) denotes C H · (ln δ−1 + Ent) + ∆−2 [2] [2] ple complexity. The first step follows from the fact that ln δk−1 = ln δ−1 + k ≤ k ln δ−1 for δ < 0.01. Since different algorithms in {Ak } take independent samples, the events {Ek } are independent. Define random variable σ as the minimum number such that event Eσ happens. Then it follows that Pr[σ = k] ≤ Pr[E 1 ∩ E 2 ∩ · · · ∩ E k−1 ] ≤ k−1 Y δi ≤ 0.01k−1 . i=1 Let Tk denote the number of samples taken by Ak if it is allowed to run indefinitely (i.e., A does not terminate). Conditioning on σ = k, we have E[Tk ] ≤ km · T (I). Moreover, A terminates before or at iteration 2k−1 km · T (I). It follows that the number of samples taken by A is bounded by ∞ ∞ X X 2−(i−1) ≤ 2k km · T (I). ⌊2k−1 km · T (I)/2i−1 ⌋ ≤ 2k−1 km · T (I) i=1 i=1 Thus the expected sample complexity of A is bounded by ∞ X Pr[σ = k] · 2k km · T (I) k=1 ≤ ∞ X 0.01k−1 · 2k km · T (I) k=1 ≤100T (I) ∞ X 0.02k km = O(T (I)). k=1 Therefore, A is a δ-correct algorithm for Best-1-Arm with expected sample complexity of −1 −1 O(H · (ln δ−1 + Ent) + ∆−2 [2] ln ln ∆[2] polylog(n, δ )). We conclude the section with the proofs of Lemmas C.1 and C.2. 47 C HEN L I Q IAO Proof [Proof of Lemma C.1] Suppose that Complexity-Guessing terminates after iteration t0 . According to Entropy-Elimination, for each 1 ≤ t ≤ t0 , the algorithm takes O(Ĥt ) = O(100t ) samples in Med-Elim and Elimination when it runs on Ĥt . As 100t grows exponentially in t, it suffices to bound the expectation of the last term, namely 100t0 . Let t∗ = ⌈log100 H(Ent + ln δ−1 ) + 3⌉. We first show that when t ≥ t∗ , Entropy-Elimination accepts Ĥt with constant probability. According to Remark B.7, the probability that EntropyElimination rejects Ĥt is upper bounded by 256H Ĥt + H(Ent + ln δ−1 + ln(Ĥt /H)) 100Ĥt ≤ 256H Ĥt /20 ≤ 1/200. + 3 100 H 100Ĥt The first step follows from the following two observations. First, as Ĥt ≥ Ĥt∗ ≥ 1003 H(Ent + ln δ−1 ), we have H(Ent + ln δ−1 ) ≤ 100−3 Ĥt . Second, since Ĥt /H ≥ 1003 and x ≥ 100 ln x 1 holds for all x ≥ 106 , we have H ln(Ĥt /H) ≤ H · (Ĥt /H) = Ĥt /100. 100 Therefore, the probability that t0 equals t∗ + k is bounded by 200−k for all k ≥ 1. It follows from a simple summation on all possible t0 that ∞ X 100t Pr[t0 = t] E 100t0 = ≤ t=1 t∗ X t 100 · 1 + t=1 ∞ X ∗ +k 100t · 200−k k=1 t∗ = O H(Ent + ln δ−1 ) . = O 100 Proof [Proof of Lemma C.2] When Entropy-Elimination runs on guess Ĥt , Unif-Sampl takes −1 O(ε−2 r ln δr ) samples in the r-th round, while the number of samples taken by Frac-Test is −1 −2 −1 −3 ln(θr − θr−1 )−1 = O ε−2 O ε−2 . r ln δr (θr − θr−1 ) r ln δr (θr − θr−1 ) As the second term dominates the first, we focus on the complexity of Frac-Test in the following analysis. Recall that εr = 2−r , δr = δ/(50r 2 t2 ) ≥ δ2 /(r 2 t2 ) and θr − θr−1 = (ct − r)−2 /10. For each t, suppose r ranges from 1 to rmax , then the complexity at iteration t is bounded by rX max −1 −3 ε−2 r ln δr (θr − θr−1 ) r=1 rX max ≤2 4r (ln δ−1 + ln r + ln t)[(ct − r)−2 /10]−3 r=1 rX max ≤2000 4r (ln δ−1 + ln r + ln t)(ct − r)6 r=1 rmax =O(4 (ln δ−1 + ln t)(ct − rmax )6 ) 48 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION The last step follows from the observation that the last term dominates the summation, and the fact ln rmax = O(ln t) due to Observation A.1. Let random variable t0 denote the last t in the execution of Complexity-Guessing. As in the proof of Lemma C.1, we define t∗ = ⌈log100 H(Ent + ln δ−1 ) + 3⌉. We have also shown that Pr[t ≥ t0 + k] ≤ 200−k for all k ≥ 1. Thus, the expected complexity incurred after iteration t∗ can be bounded by the complexity at iteration t∗ . When t < log100 ∆−2 [2] , it follows from rmax ≤ ct − 1 that the complexity is O(4ct (ln δ−1 + ln t)) = O(100t (ln δ−1 + ln t)). Summing over t = 1, 2, . . . , log100 ∆−2 [2] yields log100 ∆−2 [2] X −1 + ln ln ∆−1 100t (ln δ−1 + ln t) = O(∆−2 [2] )). [2] (ln δ t=1 Clearly this term is bounded by the desired complexity. −2 −1 ∗ When log100 ∆−2 [2] ≤ t ≤ t , we choose rmax = log 2 ∆[2] = log4 ∆[2] . Note that in fact the algorithm may not always terminate before or at round rmax . However, since the probability that the algorithm lasts rmax + k rounds is bounded by O(100−k ), the contribution of those rounds to total complexity is also dominated. Thus we have t0 X O(4rmax (ln δ−1 + ln t)(ct − rmax )6 ) t=log100 ∆−2 [2] = t0 X 6 −1 + ln t)(ct − log4 ∆−2 O(∆−2 [2] ) ) [2] (ln δ t=log100 ∆−2 [2] −2 6 −2 −1 ∗ ∗ ) (ln δ + ln t )(ct − log ∆ · O ∆ =O t∗ − log100 ∆−2 4 [2] [2] [2] )7 (ln δ−1 + ln t∗ )(ct∗ − log4 ∆−2 =O ∆−2 [2] [2] −2 −1 7 −1 ) + ln Ent + ln δ ) (ln δ + ln ln H)(ln(H/∆ =O ∆−2 [2] [2] −1 −1 −1 7 (ln δ + ln ln n)(ln n + ln δ ) ln ln ∆ =O ∆−2 [2] [2] −1 −1 polylog(n, δ ) . ln ln ∆ =O ∆−2 [2] [2] The fourth step follows from O(ln t∗ ) = O(ln ln(H(Ent + ln δ−1 ))) = O(ln ln H + ln ln Ent + ln ln ln δ−1 ), while the last two terms are dominated by ln δ−1 + ln ln H. The fifth step follows from the simple observation that H/∆−2 [2] ≤ n and Ent = O(ln ln n). 49 C HEN L I Q IAO Appendix D. Lower Bound In this section, we prove Lemma 4.1. We restate it here for convenience. Lemma 4.1. (restated) Suppose δ ∈ (0, 0.04), m ∈ N and A is a δ-correct algorithm for SIGN-ξ. P is a probability distribution on {2−1 , 2−2 , . . . , 2−m } defined by P (2−k ) = pk . Ent(P ) denotes the Shannon entropy of distribution P . Let TA (µ) denote the expected number of samples taken by A when it runs on an arm with distribution N (µ, 1) and ξ = 0. Define αk = TA (2−k )/4k . Then, m X pk αk = Ω(Ent(P ) + ln δ−1 ). k=1 D.1. Change of Distribution We introduce a lemma that is essential for proving the lower bound for SIGN-ξ in Lemma 4.1, which is a special case of (Kaufmann et al., 2015, Lemma 1). In the following, KL stands for the Kullback-Leibler divergence, while D(x\|\|y) = x ln(x/y)+(1−x) ln[(1−x)/(1−y)] is the relative entropy function. Lemma D.1 (Change of Distribution) Let A be an algorithm for SIGN-ξ. Let A and A′ be two instances of SIGN-ξ (i.e., two arms). PrA and PrA′ (EA and EA′ ) denote the probability law (expectation) when A runs on instance A and A′ respectively. Random variable τ denotes the number of samples taken by the algorithm. For all event E in Fσ , where σ is a stopping time with respect to the filtration {Ft }, we have ′ EA [τ ]KL(A, A ) ≥ D Pr[E] Pr′ [E] . A A D.2. Proof of Lemma 4.1 We start with an overview of our proof of Lemma 4.1. For each k, we consider the number of samples taken by Algorithm A when it runs on an arm with mean 2−k . We first show that with high probability, this number is between Ω(4k ) and O(4k αk ). Then we apply Lemma D.1 to show that the same event happens with probability at least e−αk when the input is an arm with mean zero. Since the probability of an event is at most 1, we would like to bound the sum of e−αk by 1, yet the problem is that the events for different k may not be disjoint. avoid this difficulty, we carefully PTo m select P a collection of disjoint events denoted by S. We bound k=1 e−dαk (for appropriate constant d) by k∈S e−αk based on the way Pwe construct S. After that, we use the “change of distribution” argument (Lemma D.1) to bound k∈S e−αk by 1. As a result, we have the following inequality for appropriate constant M , which is reminiscent of Kraft’s inequality in coding theory. m X e−dαk ≤ M . (10) k=1 Once we obtain (10), the desired bound directly follows from a simple calculation. Proof [Proof of Lemma 4.1] Fix m ∈ N. Recall that all arms are normal distributions with a standard deviation of 1 and ξ is always equal to zero. 4k αk is the expected number of samples taken 50 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION by A on an arm A with distribution N (2−k , 1). It is well-known that to distinguish N (2−k , 1) from N (−2−k , 1) with confidence level δ, Ω(4k ln δ−1P ) samples are required in expectation. Therefore, −1 −1 we have αk = Ω(ln δ ) for P all k. It follows that m k=1 pk αk = Ω(ln δ ). m It remains to prove that k=1 pk αk = Ω(Ent(P )) for all 0.04-correct algorithms. For each µ ∈ R, let Prµ and Eµ denote the probability and expectation when A runs on an arm with mean µ (i.e., N (µ, 1)). Define random variable τA as the number of samples taken by A. Let c = 1/64. Let Ek denote the event that A outputs “µ > 0” and τA ∈ [4k c, 16 · 4k αk ]. The following lemma gives a lower bound of Pr0 [Ek ]. Lemma D.2 1 Pr[Ek ] ≥ e−αk . 0 4 Our second step is to choose a collection of disjoint events from {Ek : 1 ≤ k ≤ m}. We have the following lemma. Lemma D.3 There exists a set S ⊆ {1, 2, . . . , m} such that: • {Ek : k ∈ S} is a collection of disjoint events. P Pm −dα −αk for universal constants d and M independent of m and A. k ≤ M • k∈S e k=1 e It follows that m X e−dαk ≤ M k=1 X e−αk ≤ 4M k∈S X k∈S Pr[Ek ] = 4M . 0 Here the first two steps follow from Lemma D.3 and Lemma D.2, respectively. The last step follows from the fact that {Ek : k ∈ S} is a disjoint collection of events. Finally, for a distribution P on {2−1 , 2−2 , . . . , 2−m } defined by P (2−k ) = pk , we consider the following optimization problem with variables α1 , α2 , . . . , αm : minimize m X pk αk k=1 subject to m X e−dαk ≤ 4M k=1 P −dαk = The method of Lagrange multipliers yields that the minimum value is obtained when m k=1 e 1 4M and e−dαk is proportional to pk . It follows that αk = − ln(4M pk ) and consequently d m X k=1 m pk αk ≥ 1 1X pk (ln(4M )−1 + ln p−1 ) = (Ent(P ) − ln(4M )) . k d d k=1 Note that d and M are constants independent of m, distribution P and algorithm A. This completes the proof. 51 C HEN L I Q IAO D.3. Proofs of Lemma D.2 and Lemma D.3 Finally, we prove the two technical lemmas. Proof [Proof of Lemma D.2] Recall that our goal is to lower bound Pr0 [Ek ]. We first show that Pr2−k [Ek ] ≥ 1/2 and then prove the desired lower bound by applying change of distribution. Recall that Ek = (A outputs µ > 0) ∧ (τA ∈ [4k c, 16 · 4k αk ]). We have h i h i Pr [Ek ] ≥ Pr [A outputs µ > 0] − Pr τA > 16 · 4k αk − Pr τA < 4k c 2−k 2−k 2−k 2−k h i ≥ 1 − 0.04 − 1/16 − Pr τA < 4k c 2−k h i k ≥ 0.8 − Pr τA < 4 c . 2−k Here the second step follows fromMarkov’sinequality and the fact that E2−k [τA ] = 4k αk . It remains to show that Pr2−k τA < 4k c ≤ 0.3. Suppose towards a contradiction this does not hold. Then we consider the algorithm A′ that simulates A in the following way: if A terminates within 4k c samples, A′ outputs the same answer; otherwise A′ outputs nothing. Let PrA′ ,µ denote the probability when A′ runs on an arm of mean µ. Moreover, let Ekbad denote the event that the output is “µ > 0”. Then we have i h i h Pr [Ekbad ] = Pr Ekbad ∧ τA < 4k c ≥ Pr τA < 4k c − 0.04 > 0.26. 2−k 2−k A′ ,2−k On the other hand, when we run A′ on an arm with mean −2−k , we have i i h h Pr Ekbad ≤ Pr Ekbad ≤ 0.04. −2−k A′ ,−2−k Since A′ never takes more than 4k c samples, it follows from Lemma D.1 that 2c = 4k c · KL(N (2−k , 1), N (−2−k , 1)) ≥ EA′ ,2−k [τA′ ] · KL(N (2−k , 1), N (−2−k , 1)) bad bad Pr [Ek ] ≥D Pr [Ek ] A′ ,2−k A′ ,−2−k ≥ D(0.26\|\|0.04) ≥ 0.2, which leads to a contradiction as c = 1/64. In the following, we lower bound Pr0 [Ek ] using change of distribution. Note that D Pr [Ek ] Pr[Ek ] ≤ 4k αk · KL(N (2−k , 1), N (0, 1)) 2−k 0 ≤ 4k αk · Let θk = e−αk /4. We have D(1/2\|\|θk ) = 1 −k 2 2 = αk /2. 2 1 1 1 1 ln ≥ ln = αk /2. 2 4θk (1 − θk ) 2 4θk 52 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Since we have shown Pr2−k [Ek ] ≥ 1/2, the two inequalities above imply 1 Pr[Ek ] ≥ θk = e−αk . 0 4 Proof [Proof of Lemma D.3] We map each event Ek to an interval Ik = [log4 (4k c) + 3, log4 (16 · 4k αk ) + 3] = [k, k + log4 αk + 5]. By construction, two events Ei and Ej are disjoint if and only if their corresponding intervals, Ii and Ij , are disjoint. We construct a subset of {1, 2, . . . , m} using the following greedy algorithm: • Sort (1, 2, . . . , m) into a list (l1 , l2 , . . . , lm ) such that αl1 ≤ αl2 ≤ · · · ≤ αlm . • While the list is not empty, we add the first element x in the list into set S. Let Sx = {y : y is in the current list, and Ix ∩ Iy 6= ∅}. We remove all elements in Sx from the list. Note that the way we construct S ensures that {Ek : k ∈ S} is indeed a disjoint collection of events, which proves the first part of the lemma. Moreover, {Sk : k ∈ S} is a partition of {1, 2, . . . , m}. Thus we have m X XX e−dαj . (11) e−dαk = k=1 k∈S j∈Sk It suffices to bound j∈Sk e−dαj by M e−αk for appropriate constants d and M . Summing over all k yields the desired bound m X X e−dαk ≤ e−αk . P k=1 k∈S According to our construction of S, for all j ∈ Sk we have αj ≥ αk . For each integer l ≥ ⌊log4 αk ⌋, we consider the values of j such that log4 αj ∈ [l, l + 1). Recall that the interval corresponding to event Ek is Ik = [k, k + log4 αk + 5]. In order for Ij to intersect Ik , we must have j ∈ [k − log4 αj − 5, k + log4 αk + 5]. Since log4 αj < l + 1, j must be contained in [k − l − 6, k + log4 αk + 5], and thus there are at most (log4 αk + l + 12) such values of j. Recall that since Ik = [k, k + log4 αk + 5] is nonempty, we have αk ≥ 4−5 . In the following calculation, we assume for simplicity that αk ≥ 1 for all k, since it can be easily verified that the contribution of the terms with αk < 1 (i.e., l = −5, −4, . . . , −1) is a constant, and thus can be 53 C HEN L I Q IAO covered by a sufficiently large constant M in the end. Then we have X e−dαj ≤ j∈Sk ≤ ∞ X X l=⌊log4 αk ⌋ j∈Sk ∞ X exp(−dαj )I[log4 αj ∈ [l, l + 1)] exp(−d4l )(log4 αk + l + 12) l=⌊log4 αk ⌋ = (log4 αk + 12) ∞ X l=⌊log 4 αk ⌋ exp −d4l + ∞ X l=⌊log4 αk ⌋ l exp −d4l (12) = (log4 αk + 12) · O(exp(−dαk )) + O(exp(−dαk ) · log4 αk ) ≤ M (log4 αk + 12) exp(−dαk ) = M exp(−dαk + ln log4 αk + ln 12) ≤ M e−αk . The first step rearranges the summation based on the value of l. The second step follows from the observation that Sk contains at most log4 αk + l + 12 values of j corresponding to each l. The fourth step holds since both summations decrease double-exponentially, and thus can be bounded by their respective first terms. Then we find a sufficiently large constant M (which depends on d) to cover the hidden constant in the big-O notation. Finally, the last step holds for sufficiently large d. In fact, we first choose d according to the last step, and then find the appropriate constant M . Clearly the choice of M and d is independent of the value of m and the algorithm A. Remark D.4 Recall that all distributions are assumed to be Gaussian distributions with a fixed variance of 1. In fact, our proof of Lemma 4.1 only uses the following property: the KL-divergence between two distributions with mean µ1 and µ2 is Θ((µ1 − µ2 )2 ). Note that this property is indeed essential to the “change of distribution” argument in the proof of Lemma D.2. In general, suppose U is a set of real numbers and D = {Dµ : µ ∈ U } is a family of distributions with the following two properties: (1) the mean of distribution Dµ is µ; (2) KL(Dµ1 , Dµ2 ) ≤ C(µ1 − µ2 )2 for fixed constant C > 0. Then Lemma 4.1 also holds for distributions from D. For instance, suppose D = {B(1, µ) : µ ∈ [1/2 − ε, 1/2 + ε]}, where ε ∈ (0, 1/2) is a constant and B(1, µ) denotes the Bernoulli distribution with mean µ. Since KL(B(1, p), B(1, q)) ≤ (p − q)2 (p − q)2 ≤ , q(1 − q) 1/4 − ε2 4 . It follows that Lemma 4.1 1 − 4ε2 also holds for Bernoulli distributions with means sufficiently away from 0 and 1. distribution family D satisfies the condition above with C = Appendix E. Missing Proofs in Section 5 In this section, we present the technical details in the proofs of Lemma 5.5 and Lemma 5.6. These are essentially identical to the proofs of Lemmas B.4 and B.5, which either use a potential function or apply a charging argument. 54 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION E.1. Proof of Lemma 5.5 Proof [Proof of Lemma 5.5 (continued)] Recall that P (r, Sr ) is defined as the probability that, given the value of Sr at the beginning of round r, at least one call to Elimination returns incorrectly at round r or later rounds, while Unif-Sampl and Frac-Test always return correctly. We prove inequality (2) by induction: for any Sr that contains the optimal arm A1 , P (r, Sr ) ≤ where C(r, Sr ) := δ 128C(r, Sr ) + 16M (r, Sr )ε−2 , r Ĥ ∞ X \|Sr ∩ Gi \| ε−2 j + rmax X+1 ε−2 i , i=r j=r i=r−1 and i+1 X M (r, Sr ) := \|Sr ∩ G≤r−2 \|. Note that if \|Sr \| = 1, the algorithm directly terminates at round r, and the inequality clearly holds. Thus, we assume \|Sr \| ≥ 2 in the following. Base case. We prove the base case r = rmax + 2, where rmax = maxGr 6=∅ r. Note that C(r, S) = 0 and M (r, S) = \|S\| − 1 for r = rmax + 2 and any S ⊆ I with A1 ∈ S. Let random variable r ∗ be the smallest integer greater than or equal to r, such that Med-Elim is correct at round r ∗ . Note that for k ≥ r, Pr [r ∗ = k] ≤ 0.01k−r . We claim that conditioning on r ∗ = k, if Elimination is correct between round r and round k, the algorithm will terminate at round k + 1. Consequently, the probability that Elimination fails in some round is bounded by the probabilityP that it fails between round r and k. This allows us to upper bound the conditional probability by ki=r δi′ . Now we prove the claim. By Observation 5.8, the lower threshold used in Frac-Test at round k, denoted by clow k , is greater than or equal to µ[1] − εk . Since k ≥ r = rmax + 2, \|{A ∈ S : µA < clow k }\| ≥ \|{A ∈ S : µA < µ[1] −εk }\| = \|S∩G≤k−1 \| ≥ \|S∩G≤rmax +1 \| = \|S\|−1 ≥ 0.5\|S\|. Thus by Fact 5.3, Frac-Test is guaranteed to return True in round k, and the algorithm calls Elimination. Then, by Observation 5.9, it holds that dlow k ≥ µ[1] −0.5εk . Assuming that Elimination returns correctly at round k, the set returned by Elimination, denoted by Sk+1 , satisfies \|{A ∈ Sk+1 : µA < dlow k }\| < 0.1\|Sk+1 \|, which implies \|Sk+1 \| − 1 ≤ \|Sk+1 ∩ G≤k \| = \|{A ∈ Sk+1 : µA < µ[1] − 0.5εk }\| < 0.1\|Sk+1 \|. Thus we have \|Sk+1 \| = 1, which proves the claim. Summing over all possible k yields that the probability that Elimination returns incorrectly is upper bounded by ∞ X k=r Pr [r ∗ = k] k X δj′ ≤ j=r ∞ X 0.01k−r j=r ∞ X \|Sr \|ε−2 r δ k=r 4 ≤ · 3 ≤ k X \|Sj \|ε−2 j Ĥ 2\|Sr \|ε−2 r δ Ĥ Ĥ δ 0.01k−r · 4k−r k=r ≤ δ Ĥ 55 128C(r, Sr ) + 16M (r, Sr )ε−2 . r C HEN L I Q IAO Inductive step. Assuming that the inequality holds for r + 1 and all Sr+1 that contains A1 , we bound the probability P (r, Sr ). We first note that both C and M are monotone in the following sense: C(r, S) ≤ C(r, S ′ ) and M (r, S) ≤ M (r, S ′ ) for S ⊆ S ′ . Moreover, we have C(r, Sr ) − C(r + 1, Sr ) = ∞ X −2 −2 \|Sr ∩ Gi \|ε−2 r + εr = εr (\|Sr ∩ G≥r−1 \| + 1). (13) i=r−1 We consider the following three cases separately: • Case 1. Med-Elim is correct and Frac-Test returns True. • Case 2. Med-Elim is correct and Frac-Test returns False. • Case 3. Med-Elim is incorrect. Let P1 through P3 denote the conditional probability of the event that Elimination fails at some round while Unif-Sampl and Frac-Test are correct in Case 1 through Case 3. Upper bound P1 . Assuming that Frac-Test returns True, procedure Elimination will be called at round r. By a union bound, we have P1 ≤ P (r + 1, Sr+1 ) + δr′ , where Sr+1 is the set of arms returned by Elimination. According to the inductive hypothesis, the monotonicity of C, and identity (13), P (r + 1, Sr+1 ) ≤ ≤ = δ Ĥ δ Ĥ δ Ĥ 128C(r + 1, Sr+1 ) + 16M (r + 1, Sr+1 )ε−2 r+1 128C(r + 1, Sr ) + 64M (r + 1, Sr+1 )ε−2 r 128C(r, Sr ) + ε−2 r (64M (r + 1, Sr+1 ) − 128\|Sr ∩ G≥r−1 \| − 128) . (14) ≤ µ[1] − εr . If Elimination returns correctly at round r, we have By Observation 5.9, dlow r M (r+1, Sr+1 ) = \|{A ∈ Sr+1 : µA < µ[1] −εr }\| ≤ \|{A ∈ Sr+1 : µA < dlow r }\| < 0.1\|Sr+1 \| ≤ 0.1\|Sr \|. For brevity, let Nsma , Ncur and Nbig denote \|Sr ∩ G≤r−2 \|, \|Sr ∩ Gr−1 \| and \|Sr ∩ G≥r \|, respectively. Note that \|Sr \| = Nsma + Ncur + Nsma + 1. Then we have P1 ≤ P (r + 1, Sr+1 ) + δr′ δ ≤ 128C(r, Sr ) + ε−2 r (\|Sr \| + 64M (r + 1, Sr+1 ) − 128\|Sr ∩ G≥r−1 \| − 128) Ĥ δ 128C(r, Sr ) + ε−2 ≤ r (7.4(Nsma + Ncur + Nbig + 1) − 128(Ncur + Nbig + 1)) Ĥ δ 128C(r, Sr ) + 7.4ε−2 ≤ r Nsma . Ĥ 56 T OWARDS I NSTANCE O PTIMAL B OUNDS FOR B EST A RM I DENTIFICATION Upper bound P2 . Since Frac-Test returns True, procedure Elimination is not called. Then P2 ≤ P (r + 1, Sr+1 ) = P (r + 1, Sr ). By inequality (14), P2 ≤ ≤ ≤ δ Ĥ δ Ĥ δ 128C(r, Sr ) + ε−2 r (64M (r + 1, Sr ) − 128\|Sr ∩ G≥r−1 \| − 128) 128C(r, Sr ) + ε−2 r (64(Nsma + Ncur ) − 128(Ncur + Nbig + 1)) δ 128C(r, Sr ) + 64ε−2 · 128C(r, Sr ). r (Nsma − Ncur − Nbig − 1) ≤ Ĥ Ĥ Here the last step holds since by Observation 5.8, clow ≥ µ[1] − 2εr , and thus Frac-Test returns r False implies that Nsma = \|Sr ∩ G≤r−2 \| = \|{A ∈ Sr : µA < εr−1 }\| < 0.5\|Sr \| = (Nsma + Ncur + Nbig + 1)/2. Upper bound P3 . By (14) and M (r + 1, Sr+1 ) ≤ M (r + 1, Sr ), we have P3 ≤ P (r + 1, Sr+1 ) + δr′ δ 128C(r, Sr ) + ε−2 ≤ r (64M (r + 1, Sr ) − 128\|Sr ∩ G≥r−1 \| − 128 + \|Sr \|) Ĥ δ = 128C(r, Sr ) + ε−2 r (64(Nsma + Ncur ) − 128(Ncur + Nbig ) − 128 + (Nsma + Ncur + Nbig + 1)) Ĥ δ 128C(r, Sr ) + 65ε−2 ≤ r Nsma . Ĥ Recall that Case 3 happens with probability at most 0.01, and Nsma = \|Sr ∩G≤r−2 \| = M (r, Sr ). Therefore, we obtain the following bound on P (r, Sr ), which finishes the proof. δ δ 128C(r, Sr ) + 65ε−2 128C(r, Sr ) + 7.4ε−2 r Nsma + 0.99 · r Nsma Ĥ Ĥ δ 128C(r, Sr ) + 16ε−2 ≤ r Nsma Ĥ δ 128C(r, Sr ) + 16M (r, Sr )ε−2 . ≤ r Ĥ P (r, Sr ) ≤ 0.01 · E.2. Proof of Lemma 5.6 Proof [Proof of Lemma 5.6 (continued)] Recall that for each round i, ri is defined as the largest integer r such that \|G≥r \| ≥ 0.5\|Si \|, and   0, j < ri , Ti,j = H −2  , j ≥ ri ln δ−1 + ln εi \|Gj \|εi−2 is the number of samples that each arm in Gj is charged at round i. 57 C HEN L I Q IAO H We first show that j \|Gj \|Ti,j is an upper bound on + ln , the num\|Si \|ε−2 i ber of samples taken by Med-Elim and Elimination at round i. Recall that \|G≥ri \| ≥ 0.5\|Si \|. By definition of Ti,j , X X H −2 −1 \|Gj \|Ti,j = \|Gj \|εi ln δ + ln \|Gj \|εi−2 j j≥ri H −2 −1 ≥ \|G≥ri \|εi ln δ + ln \|Si \|ε−2 i H 1 −2 −1 . ln δ + ln ≥ \|Si \|εi 2 \|Si \|ε−2 i P Then we prove the upper bound on i E[Ti,j ], the expected number of samples that each arm in Gj is charged. For i ≤ j + 1, we have the straightforward bound H −2 −1 E[Ti,j ] ≤ εi ln δ + ln . (15) \|Gj \|ε−2 i \|Si \|ε−2 i P ln δ−1 For i ≥ j + 2, we note that Ti,j is non-zero only if j ≥ ri , which implies that \|G≥j+1 \| < 0.5\|Si \|. We claim that this happens only if Med-Elim fails between round j + 2 and round i − 1, which happens with probability at most 0.01i−j−1 . In fact, suppose Med-Elim is correct at some round k, ≥ µ[1] − 2εk and dlow ≥ µ[1] − εk , where j + 2 ≤ k ≤ i − 1. By Observations 5.8 and 5.9, clow k k low low where c and d are the two lower thresholds used in Frac-Test and Elimination. If Frac-Test returns False, by Fact 5.3, we have \|Sk ∩ G<k−1 \| = {A ∈ Sk : µA < µ[1] − 2εk } ≤ {A ∈ Sk : µA < clow k } < 0.5\|Sk \|. Since Sk+1 = Sk in this case, it follows that \|Sk+1 ∩ G<k−1 \| < 0.5\|Sk+1 \|. If Frac-Test returns True and the algorithm calls Elimination, by Fact 5.4, \|Sk+1 ∩ G<k \| = \|{A ∈ Sk+1 : µA < µ[1] − εk }\| ≤ \|{A ∈ Sk+1 : µA < dlow k }\| < 0.1\|Sk+1 \|. In either case, we have \|Sk+1 ∩ G≥k−1 \| > 0.5\|Sk+1 \|, and thus, \|G≥j+1 \| ≥ \|G≥k−1 \| ≥ \|Sk+1 ∩ G≥k−1 \| > 0.5\|Sk+1 \| ≥ 0.5\|Si \|, which contradicts \|G≥j+1 \| < 0.5\|Si \|. Therefore, for i ≥ j + 2, we have H −2 −1 E[Ti,j ] = Pr [Ti,j > 0] · εi ln δ + ln \|Gj \|εi−2 H −2 i−j−1 −1 . ≤ 0.01 · εi ln δ + ln \|Gj \|ε−2 i By (15) and (16), a direct summation gives X i E[Ti,j ] = O ε−2 j H ln δ−1 + ln \|Gj \|ε−2 j 58 !! . (16)	8
Combinatorial Multi-Armed Bandit with General Reward Functions arXiv:1610.06603v3 [cs.LG] 1 Feb 2017 Wei Chen∗ Wei Hu† Fu Li‡ Jian Li§ Yu Liu¶ Pinyan Luk Abstract In this paper, we study the stochastic combinatorial multi-armed bandit (CMAB) framework that allows a general nonlinear reward function, whose expected value may not depend only on the means of the input random variables but possibly on the entire distributions of these variables. Our framework enables a much larger class of reward functions such as the max() function and nonlinear utility functions. Existing techniques relying on accurate estimations of the means of random variables, such as the upper confidence bound (UCB) technique, do not work directly on these functions. We propose a new algorithm called stochastically dominant confidence bound (SDCB), which estimates the distributions of underlying random variables and their stochastically dominant confidence bounds.√We prove that SDCB can achieve O(log T ) distribution-dependent regret and Õ( T ) distribution-independent regret, where T is the time horizon. We apply our results to the K-MAX problem and expected utility maximization problems. In particular, for K-MAX, we provide the first polynomial-time √ approximation scheme (PTAS) for its offline problem, and give the first Õ( T ) bound on the (1 − ǫ)approximation regret of its online problem, for any ǫ > 0. 1 Introduction Stochastic multi-armed bandit (MAB) is a classical online learning problem typically specified as a player against m machines or arms. Each arm, when pulled, generates a random reward following an unknown distribution. The task of the player is to select one arm to pull in each round based on the historical rewards she collected, and the goal is to collect cumulative reward over multiple rounds as much as possible. In this paper, unless otherwise specified, we use MAB to refer to stochastic MAB. MAB problem demonstrates the key tradeoff between exploration and exploitation: whether the player should stick to the choice that performs the best so far, or should try some less explored alternatives that may provide better rewards. The performance measure of an MAB strategy is its cumulative regret, which is defined as the difference between the cumulative reward obtained by always playing the arm with the largest expected reward and the cumulative reward achieved by the learning strategy. MAB and its variants have been extensively studied in the literature, with classical √ results such as tight Θ(log T ) distribution-dependent and Θ( T ) distribution-independent upper and lower bounds on the regret in T rounds [19, 2, 1]. An important extension to the classical MAB problem is combinatorial multi-armed bandit (CMAB). In CMAB, the player selects not just one arm in each round, but a subset of arms or a combinatorial ∗ Microsoft Research, email: weic@microsoft.com. The authors are listed in alphabetical order. Princeton University, email: huwei@cs.princeton.edu. ‡ The University of Texas at Austin, email: fuli.theory.research@gmail.com. § Tsinghua University, email: lapordge@gmail.com. ¶ Tsinghua University, email: liuyujyyz@gmail.com. k Shanghai University of Finance and Economics, email: lu.pinyan@mail.shufe.edu.cn. † 30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain. object in general, referred to as a super arm, which collectively provides a random reward to the player. The reward depends on the outcomes from the selected arms. The player may observe partial feedbacks from the selected arms to help her in decision making. CMAB has wide applications in online advertising, online recommendation, wireless routing, dynamic channel allocations, etc., because in all these settings the action unit is a combinatorial object (e.g. a set of advertisements, a set of recommended items, a route in a wireless network, and an allocation between channels and users), and the reward depends on unknown stochastic behaviors (e.g. users’ click through behaviors, wireless transmission quality, etc.). Therefore CMAB has attracted a lot of attention in online learning research in recent years [12, 8, 22, 15, 7, 16, 18, 17, 23, 9]. Most of these studies focus on linear reward functions, for which the expected reward for playing a super arm is a linear combination of the expected outcomes from the constituent base arms. Even for studies that do generalize to non-linear reward functions, they typically still assume that the expected reward for choosing a super arm is a function of the expected outcomes from the constituent base arms in this super arm [8, 17]. However, many natural reward functions do not satisfy this property. For example, for the function max(), which takes a group of variables and outputs the maximum one among them, its expectation depends on the full distributions of the input random variables, not just their means. Function max() and its variants underly many applications. As an illustrative example, we consider the following scenario in auctions: the auctioneer is repeatedly selling an item to m bidders; in each round the auctioneer selects K bidders to bid; each of the K bidders independently draws her bid from her private valuation distribution and submits the bid; the auctioneer uses the first-price auction to determine the winner and collects the largest bid as the payment.1 The goal of the auctioneer is to gain as high cumulative payments as possible. We refer to this problem as the K-MAX bandit problem, which cannot be effectively solved in the existing CMAB framework. Beyond the K-MAX problem, many expected utility maximization (EUM) problems are studied in stochastic P optimization literature [27, 20, 21, 4]. The problem can be formulated as maximizing E[u( i∈S Xi )] among all feasible sets S, where Xi ’s are independent random variables and u(·) is a utility function. For example, Xi could be the random delay of edge ei in a routing graph, S is a routing path in the graph, and the objective is maximizing the utility obtained from any routing path, and typically the shorter the delay, the larger the utility. The utility function u(·) is typically nonlinear to model risk-averse or risk-prone behaviors of users (e.g. a concave utility function is often used to model risk-averse behaviors). The non-linear utility function makes the objective function much more complicated: in particular, it is no longer a function of the means of the underlying random variables Xi ’s. When the distributions of Xi ’s are unknown, we can turn EUM into an online learning problem where the distributions of Xi ’s need to be learned over time from online feedbacks, and we want to maximize the cumulative reward in the learning process. Again, this is not covered by the existing CMAB framework since only learning the means of Xi ’s is not enough. In this paper, we generalize the existing CMAB framework with semi-bandit feedbacks to handle general reward functions, where the expected reward for playing a super arm may depend more than just the means of the base arms, and the outcome distribution of a base arm can be arbitrary. This generalization is non-trivial, because almost all previous works on CMAB rely on estimating the expected outcomes from base arms, while in our case, we need an estimation method and an analytical tool to deal with the whole distribution, not just its mean. To this end, we turn the problem into estimating the cumulative distribution function (CDF) of each arm’s outcome distribution. We use stochastically dominant confidence bound (SDCB) to obtain a distribution that stochastically dominates the true distribution with high probability, and hence √ we also name our algorithm SDCB. We are able to show O(log T ) distribution-dependent and Õ( T ) distribution-independent regret bounds in T rounds. Furthermore, we propose a more efficient algorithm called Lazy-SDCB, which first executes a discretization step √ and then applies SDCB on the discretized problem. We show that Lazy-SDCB also achieves Õ( T ) distribution-independent regret bound. Our regret bounds are tight with respect to their dependencies on T (up to a logarithmic factor for distribution-independent bounds). To make our scheme work, we make a few reasonable assumptions, including boundedness, monotonicity and Lipschitz-continuity2 of the reward function, and independence among base arms. We apply our algorithms to the K-MAX and EUM problems, and provide efficient solutions with 1 We understand that the first-price auction is not truthful, but this example is only for illustrative purpose for the max() function. 2 The Lipschitz-continuity assumption is only made for Lazy-SDCB. See Section 4. 2 concrete regret bounds. Along the way, we also provide the first polynomial time approximation scheme (PTAS) for the offline K-MAX problem, which is formulated as maximizing E[maxi∈S Xi ] subject to a cardinality constraint \|S\| ≤ K, where Xi ’s are independent nonnegative random variables. To summarize, our contributions include: (a) generalizing the CMAB framework to allow a general reward function whose expectation may depend on the entire distributions of the input random variables; (b) proposing the SDCB algorithm to achieve efficient learning in this framework with near-optimal regret bounds, even for arbitrary outcome distributions; (c) giving the first PTAS for the offline K-MAX problem. Our general framework treats any offline stochastic optimization algorithm as an oracle, and effectively integrates it into the online learning framework. Related Work. As already mentioned, most relevant to our work are studies on CMAB frameworks, among which [12, 16, 18, 9] focus on linear reward functions while [8, 17] look into nonlinear reward functions. In particular, Chen et al. [8] look at general non-linear reward functions and Kveton et al. [17] consider specific non-linear reward functions in a conjunctive or disjunctive form, but both papers require that the expected reward of playing a super arm is determined by the expected outcomes from base arms. The only work in combinatorial bandits we are aware of that does not require the above assumption on the expected reward is [15], which is based on a general Thompson sampling framework. However, they assume that the joint distribution of base arm outcomes is from a known parametric family within known likelihood function and only the parameters are unknown. They also assume the parameter space to be finite. In contrast, our general case is non-parametric, where we allow arbitrary bounded distributions. Although in our known finite support case the distribution can be parametrized by probabilities on all supported points, our parameter space is continuous. Moreover, it is unclear how to efficiently compute posteriors in their algorithm, and their regret bounds depend on complicated problem-dependent coefficients which may be very large for many combinatorial problems. They also provide a result on the K-MAX problem, but they only consider Bernoulli outcomes from base arms, much simpler than our case where general distributions are allowed. There are extensive studies on the classical MAB problem, for which we refer to a survey by Bubeck and Cesa-Bianchi [5]. There are also some studies on adversarial combinatorial bandits, e.g. [26, 6]. Although it bears conceptual similarities with stochastic CMAB, the techniques used are different. Expected utility maximization (EUM) encompasses a large class of stochastic optimization problems and has been well studied (e.g. [27, 20, 21, 4]). To the best of our knowledge, we are the first to study the online learning version of these problems, and we provide a general solution to systematically address all these problems as long as there is an available offline (approximation) algorithm. The K-MAX problem may be traced back to [13], where Goel et al. provide a constant approximation algorithm to a generalized version in which the objective is to choose a subset S of cost at most K and maximize the expectation of a certain knapsack profit. 2 Setup and Notation Problem Formulation. We model a combinatorial multi-armed bandit (CMAB) problem as a tuple (E, F , D, R), where E = [m] = {1, 2, . . . , m} is a set of m (base) arms, F ⊆ 2E is a set of subsets of E, D is a probability distribution over [0, 1]m , and R is a reward function defined on [0, 1]m × F . The arms produce stochastic outcomes X = (X1 , X2 , . . . , Xm ) drawn from distribution D, where the i-th entry Xi is the outcome from the i-th arm. Each feasible subset of arms S ∈ F is called a super arm. Under a realization of outcomes x = (x1 , . . . , xm ), the player receives a reward R(x, S) when she chooses the super arm S to play. Without loss of generality, we assume the reward value to be nonnegative. Let K = maxS∈F \|S\| be the maximum size of any super arm. Let X (1) , X (2) , . . . be an i.i.d. sequence of random vectors drawn from D, where X (t) = (t) (t) (X1 , . . . , Xm ) is the outcome vector generated in the t-th round. In the t-th round, the player (t) chooses a super arm St ∈ F to play, and then the outcomes from all arms in St , i.e., {Xi \| i ∈ St }, are revealed to the player. According to the definition of the reward function, the reward value in the t-th round is R(X (t) , St ). The expected reward for choosing a super arm S in any round is denoted by rD (S) = EX∼D [R(X, S)]. 3 We also assume that for a fixed super arm S ∈ F , the reward R(x, S) only depends on the revealed outcomes xS = (xi )i∈S . Therefore, we can alternatively express R(x, S) as RS (xS ), where RS is a function defined on [0, 1]S .3 A learning algorithm A for the CMAB problem selects which super arm to play in each round based on the revealed outcomes in all previous rounds. Let StA be the super arm selected by A to maximize the expected cumulative reward in T rounds, which in the t-th round.4 The goal hP i is PT T (t) A is E , St ) = t=1 E rD (StA ) . Note that when the underlying distribution D is t=1 R(X known, the optimal algorithm A∗ chooses the optimal super arm S ∗ = argmaxS∈F {rD (S)} in every round. The quality of an algorithm A is measured by its regret in T rounds, which is the difference between the expected cumulative reward of the optimal algorithm A∗ and that of A: ∗ RegA D (T ) = T · rD (S ) − T X t=1 E rD (StA ) . For some CMAB problem instances, the optimal super arm S ∗ may be computationally hard to find even when the distribution D is known, but efficient approximation algorithms may exist, i.e., an α-approximate (0 < α ≤ 1) solution S ′ ∈ F which satisfies rD (S ′ ) ≥ α · maxS∈F {rD (S)} can be efficiently found given D as input. We will provide the exact formulation of our requirement on such an α-approximation computation oracle shortly. In such cases, it is not fair to compare a CMAB algorithm A with the optimal algorithm A∗ which always chooses the optimal super arm S ∗ . Instead, we define the α-approximation regret of an algorithm A as ∗ RegA D,α (T ) = T · α · rD (S ) − T X t=1 E rD (StA ) . As mentioned, almost all previous work on CMAB requires that the expected reward rD (S) of a super arm S depends only on the expectation vector µ = (µ1 , . . . , µm ) of outcomes, where µi = EX∼D [Xi ]. This is a strong restriction that cannot be satisfied by a general nonlinear function RS and a general distribution D. The main motivation of this work is to remove this restriction. Assumptions. Throughout this paper, we make several assumptions on the outcome distribution D and the reward function R. Assumption 1 (Independent outcomes from arms). The outcomes from all m arms are mutually independent, i.e., for X ∼ D, X1 , X2 , . . . , Xm are mutually independent. We write D as D = D1 × D2 × · · · × Dm , where Di is the distribution of Xi . We remark that the above independence assumption is also made for past studies on the offline EUM and K-MAX problems [27, 20, 21, 4, 13], so it is not an extra assumption for the online learning case. Assumption 2 (Bounded reward value). There exists M > 0 such that for any x ∈ [0, 1]m and any S ∈ F , we have 0 ≤ R(x, S) ≤ M . Assumption 3 (Monotone reward function). If two vectors x, x′ ∈ [0, 1]m satisfy xi ≤ x′i (∀i ∈ [m]), then for any S ∈ F , we have R(x, S) ≤ R(x′ , S). Computation Oracle for Discrete Distributions with Finite Supports. We require that there exists an α-approximation computation oracle (0 < α ≤ 1) for maximizing rD (S), when each Di (i ∈ [m]) has a finite support. In this case, Di can be fully described by a finite set of numbers (i.e., its support {vi,1 , vi,2 , . . . , vi,si } and the values of its cumulative distribution function (CDF) Fi on the supported points: Fi (vi,j ) = PrXi ∼Di [Xi ≤ vi,j ] (j ∈ [si ])). The oracle takes such a representation of D as input, and can output a super arm S ′ = Oracle(D) ∈ F such that rD (S ′ ) ≥ α · maxS∈F {rD (S)}. 3 SDCB Algorithm [0, 1]S is isomorphic to [0, 1]\|S\| ; the coordinates in [0, 1]S are indexed by elements in S. Note that StA may be random due to the random outcomes in previous rounds and the possible randomness used by A. 3 4 4 Algorithm 1 SDCB (Stochastically dominant confidence bound) 1: Throughout the algorithm, for each arm i ∈ [m], maintain: (i) a counter Ti which stores the number of times arm i has been played so far, and (ii) the empirical distribution D̂i of the observed outcomes from arm i so far, which is represented by its CDF F̂i 2: // Initialization 3: for i = 1 to m do 4: // Action in the i-th round 5: Play a super arm Si that contains arm i 6: Update Tj and F̂j for each j ∈ Si 7: end for 8: for t = m + 1, m + 2, . . . do 9: // Action in the t-th round 10: For each i ∈ [m], let Di be a distribution whose CDF Fi is Fi (x) = ( max{F̂i (x) − 1, q 3 ln t 2Ti , 0}, 0≤x<1 x=1 11: Play the super arm St ← Oracle(D), where D = D1 × D2 × · · · × Dm 12: Update Tj and F̂j for each j ∈ St 13: end for We present our algorithm stochastically dominant confidence bound (SDCB) in Algorithm 1. Throughout the algorithm, we store, in a variable Ti , the number of times the outcomes from arm i are observed so far. We also maintain the empirical distribution D̂i of the observed outcomes from arm i so far, which can be represented by its CDF F̂i : for x ∈ [0, 1], the value of F̂i (x) is just the fraction of the observed outcomes from arm i that are no larger than x. Note that F̂i is always a step function which has “jumps” at the points that are observed outcomes from arm i. Therefore it suffices to store these discrete points as well as the values of F̂i at these points in order to store the whole function F̂i . Similarly, the later computation of stochastically dominant CDF Fi (line 10) only requires computation at these points, and the input to the offline oracle only needs to provide these points and corresponding CDF values (line 11). The algorithm starts with m initialization rounds in which each arm is played at least once5 (lines 27). In the t-th round (t > m), the algorithm consists of three steps. First, it calculates for each i ∈ [m] a distribution Di whose CDF Fi is obtained by lowering the CDF F̂i (line 10). The second step is to call the α-approximation oracle with the newly constructed distribution D = D1 × · · · × Dm as input (line 11), and thus the super arm St output by the oracle satisfies rD (St ) ≥ α·maxS∈F {rD (S)}. Finally, the algorithm chooses the super arm St to play, observes the outcomes from all arms in St , and updates Tj ’s and F̂j ’s accordingly for each j ∈ St . The idea behind our algorithm is the optimism in the face of uncertainty principle, which is the key principle behind UCB-type algorithms. Our algorithm ensures that with high probability we have Fi (x) ≤ Fi (x) simultaneously for all i ∈ [m] and all x ∈ [0, 1], where Fi is the CDF of the outcome distribution Di . This means that each Di has first-order stochastic dominance over Di .6 Then from the monotonicity property of R(x, S) (Assumption 3) we know that rD (S) ≥ rD (S) holds for all S ∈ F with high probability. Therefore D provides an “optimistic” estimation on the expected reward from each super arm. √ Regret Bounds. We prove O(log T ) distribution-dependent and O( T log T ) distributionindependent upper bounds on the regret of SDCB (Algorithm 1). 5 Without loss of generality, we assume that each arm i ∈ [m] is contained in at least one super arm. We remark that while Fi (x) is a numerical lower confidence bound on Fi (x) for all x ∈ [0, 1], at the distribution level, Di serves as a “stochastically dominant (upper) confidence bound” on Di . 6 5 We call a super arm S bad if rD (S) < α · rD (S ∗ ). For each super arm S, we define ∆S = max{α · rD (S ∗ ) − rD (S), 0}. Let FB = {S ∈ F \| ∆S > 0}, which is the set of all bad super arms. Let EB ⊆ [m] be the set of arms that are contained in at least one bad super arm. For each i ∈ EB , we define ∆i,min = min{∆S \| S ∈ FB , i ∈ S}. Recall that M is an upper bound on the reward value (Assumption 2) and K = maxS∈F \|S\|. Theorem 1. A distribution-dependent upper bound on the α-approximation regret of SDCB (Algorithm 1) in T rounds is 2 X 2136 π 2 M K ln T + + 1 αM m, ∆i,min 3 i∈EB and a distribution-independent upper bound is √ 93M mKT ln T + π2 + 1 αM m. 3 The proof of Theorem 1 is given in Appendix A.1. The main idea is to reduce our analysis on general reward functions satisfying Assumptions 1-3 to the one in [18] that deals with the summation reward P function R(x, S) = i∈S xi . Our analysis relies on the Dvoretzky-Kiefer-Wolfowitz inequality [10, 24], which gives a uniform concentration bound on the empirical CDF of a distribution. Applying Our Algorithm to the Previous CMAB Framework. Although our focus is on general reward functions, we note that when SDCB is applied to the previous CMAB framework where the expected reward depends only on the means of the random variables, it can achieve the same regret bounds as the previous combinatorial upper confidence bound (CUCB) algorithm in [8, 18]. Let µi = EX∼D [Xi ] be arm i’s mean outcome. In each round CUCB calculates (for each arm i) an upper confidence bound µ̄i on µi , with the essential property that µi ≤ µ̄i ≤ µi +Λi holds with high probability, for some Λi > 0. In SDCB, we use Di as a stochastically dominant confidence bound of Di . We can show that µi ≤ EYi ∼Di [Yi ] ≤ µi + Λi holds with high probability, with the same interval length Λi as in CUCB. (The proof is given in Appendix A.2.) Hence, the analysis in [8, 18] can be applied to SDCB, resulting in the same regret bounds.We further remark that in this case we do not need the three assumptions stated in Section 2 (in particular the independence assumption on Xi ’s): the summation reward case just works as in [18] and the nonlinear reward case relies on the properties of monotonicity and bounded smoothness used in [8]. 4 Improved SDCB Algorithm by Discretization In Section 3, we have shown that our algorithm SDCB achieves near-optimal regret bounds. However, that algorithm might suffer from large running time and memory usage. Note that, in the t-th round, an arm i might have been observed t − 1 times already, and it is possible that all the observed values from arm i are different (e.g., when arm i’s outcome distribution Di is continuous). In such case, it takes Θ(t) space to store the empirical CDF F̂i of the observed outcomes from arm i, and both calculating the stochastically dominant CDF Fi and updating F̂i take Θ(t) time. Therefore, the worst-case space usage of SDCB in T rounds is Θ(T ), and the worst-case running time is Θ(T 2 ) (ignoring the dependence on m and K); here we do not count the time and space used by the offline computation oracle. In this section, we propose an improved algorithm Lazy-SDCB which reduces the worst-case √ √ mem3/2 ory usage and running time to O( T ) and O(T ), respectively, while preserving the O( T log T ) distribution-independent regret bound. To this end, we need an additional assumption on the reward function: Assumption 4 (Lipschitz-continuous reward function). There exists C > 0 such that for any S ∈ F ′ ′ m ′ ′ and P any x, x ′ ∈ [0, 1] , we have \|R(x, S) − R(x , S)\| ≤ CkxS − xS k1 , where kxS − xS k1 = \|x − x \|. i i i∈S 6 Algorithm 2 Lazy-SDCB with known time horizon Input: time √ horizon T 1: s ← ⌈ T ⌉ 1 j=1 [0, s ], 2: Ij ← j−1 j ( s , s ], j = 2, . . . , s 3: Invoke SDCB (Algorithm 1) for T rounds, with the following change: whenever observing an outcome x (from any arm), find j ∈ [s] such that x ∈ Ij , and regard this outcome as sj Algorithm 3 Lazy-SDCB without knowing the time horizon 1: q ← ⌈log2 m⌉ 2: In rounds 1, 2, . . . , 2q , invoke Algorithm 2 with input T = 2q 3: for k = q, q + 1, q + 2, . . . do 4: In rounds 2k + 1, 2k + 2, . . . , 2k+1 , invoke Algorithm 2 with input T = 2k 5: end for We first describe the algorithm when the time horizon T is known in advance. The algorithm is summarized in Algorithm 2. We perform a discretization on the distribution D = D1 × · · · × Dm to obtain a discrete distribution D̃ = D̃1 × · · · × D̃m such that (i) for X̃ ∼ D̃, X̃1 , . . . , X̃m are also mutually independent, and (ii) √ every D̃i is supported on a set of equally-spaced values { 1s , 2s , . . . , 1}, where s is set to be ⌈ T ⌉. Specifically, we partition [0, 1] into s intervals: I1 = s−1 s−1 [0, 1s ], I2 = ( 1s , 2s ], . . . , Is−1 = ( s−2 s , s ], Is = ( s , 1], and define D̃i as Pr [X̃i = j/s] = X̃i ∼D̃i Pr [Xi ∈ Ij ] , Xi ∼Di j = 1, . . . , s. For the CMAB problem ([m], F , D, R), our algorithm “pretends” that the outcomes are drawn from D̃ instead of D, by replacing any outcome x ∈ Ij by js (∀j ∈ [s]), and then applies SDCB to the problem ([m], F , D̃, R). Since each D̃i has a known support { 1s , 2s , . . . , 1}, the algorithm only needs to maintain the number of occurrences of each support value in order to obtain the empirical CDF of all the observed outcomes from arm i. Therefore, all the operations in a round can be done √ using O(s) = O( T ) time and space, and the total time and space used by Lazy-SDCB are O(T 3/2 ) √ and O( T ), respectively. The discretization parameter s in Algorithm 2 depends on the time horizon T , which is why Algorithm 2 has to know T in advance. We can use the doubling trick to avoid the dependency on T . We present such an algorithm (without knowing T ) in Algorithm 3. It is easy to see that Algorithm 3 has the same asymptotic time and space usages as Algorithm 2. √ Regret Bounds. We show that both Algorithm 2 and Algorithm 3 achieve O( T log T ) distribution-independent regret bounds. The full proofs are given in Appendix B. Recall that C is the coefficient in the Lipschitz condition in Assumption 4. Theorem 2. Suppose the time horizon T is known in advance. Then the α-approximation regret of Algorithm 2 in T rounds is at most 2 √ √ π + 1 αM m. 93M mKT ln T + 2CK T + 3 Proof Sketch. The regret consists of two parts: (i) the regret for the discretized CMAB problem ([m], F , D̃, R), and (ii) the error due to discretization. We directly apply Theorem 1 for the first part. For the second part, a key step is to show \|rD (S) − rD̃ (S)\| ≤ CK/s for all S ∈ F (see Appendix B.1). Theorem 3. For any time horizon T ≥ 2, the α-approximation regret of Algorithm 3 in T rounds is at most √ √ 318M mKT ln T + 7CK T + 10αM m ln T. 7 5 Applications We describe the K-MAX problem and the class of expected utility maximization problems as applications of our general CMAB framework. The K-MAX Problem. In this problem, the player is allowed to select at most K arms from the set of m arms in each round, and the reward is the maximum one among the outcomes from the selected arms. In other words, the set of feasible super arms is F = S ⊆ [m] \|S\| ≤ K , and the reward function is R(x, S) = maxi∈S xi . It is easy to verify that this reward function satisfies Assumptions 2, 3 and 4 with M = C = 1. Now we consider the corresponding offline K-MAX problem of selecting at most K arms from m independent arms, with the largest expected reward. It can be implied by a result in [14] that finding the exact optimal solution is NP-hard, so we resort to approximation algorithms. We can show, using submodularity, that a simple greedy algorithm can achieve a (1 − 1/e)-approximation. Furthermore, we give the first PTAS for this problem. Our PTAS can be generalized to constraints other than the cardinality constraint \|S\| ≤ K, including s-t simple paths, matchings, knapsacks, etc. The algorithms and corresponding proofs are given in Appendix C. Theorem 4. There exists a PTAS for the offline K-MAX problem. In other words, for any constant ǫ > 0, there is a polynomial-time (1 − ǫ)-approximation algorithm for the offline K-MAX problem. We thus can apply our SDCB√algorithm to the K-MAX bandit problem and obtain O(log T ) distribution-dependent and Õ( T ) distribution-independent regret bounds according to Theorem 1, √ or can apply Lazy-SDCB to get Õ( T ) distribution-independent bound according to Theorem 2 or 3. Streeter and Golovin [26] study an online submodular maximization problem in the oblivious adversary model. In particular, their result can cover the stochastic K-MAX bandit problem as a special √ case, and an O(K mT log m) upper bound on the (1 − 1/e)-regret can be shown. While the techniques in [26] can√only give a bound on the (1 − 1/e)-approximation regret for K-MAX, we can obtain the first Õ( T ) bound on the (1 − ǫ)-approximation regret for any constant ǫ > 0, using our PTAS as the offline oracle. Even when we use the simple greedy algorithm as the oracle, our experiments show that SDCB performs significantly better than the algorithm in [26] (see Appendix D). Expected Utility Maximization. Our framework can also be applied to reward functions of the P utility function. The corresponding offline form R(x, S) = u( i∈S xi ), where u(·) is an increasing P problem is to maximize the expected utility E[u( i∈S xi )] subject to a feasibility constraint S ∈ F . Note that if u is nonlinear, the expected utility may not be a function of the means of the arms in S. Following the celebrated von Neumann-Morgenstern expected utility theorem, nonlinear utility functions have been extensively used to capture risk-averse or risk-prone behaviors in economics (see e.g., [11]), while linear utility functions correspond to risk-neutrality. Li and Deshpande [20] obtain a PTAS for the expected utility maximization (EUM) problem for several classes of utility functions (including for example increasing concave functions which typically indicate risk-averseness), and a large class of feasibility constraints (including cardinality constraint, s-t simple paths, matchings, and knapsacks). Similar results for other utility functions and feasibility constraints can be found in [27, 21, 4]. In the online problem, we can apply our algorithms, using their PTASs as the offline oracle. Again, we can obtain the first tight regret bounds on the (1 − ǫ)-approximation regret for any ǫ > 0, for the class of online EUM problems. Acknowledgments Wei Chen was supported in part by the National Natural Science Foundation of China (Grant No. 61433014). Jian Li and Yu Liu were supported in part by the National Basic Research Program of China grants 2015CB358700, 2011CBA00300, 2011CBA00301, and the National NSFC grants 61033001, 61361136003. The authors would like to thank Tor Lattimore for referring to us the DKW inequality. 8 References [1] Jean-Yves Audibert and Sébastien Bubeck. Minimax policies for adversarial and stochastic bandits. In COLT, pages 217–226, 2009. [2] Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multiarmed bandit problem. 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In Section A.1.1, we review the L1 distance between two distributions and present a property of it. In Section A.1.2, we review the DvoretzkyKiefer-Wolfowitz (DKW) inequality, which is a strong concentration result for empirical CDFs. In Section A.1.3, we prove some key technical lemmas. Then we complete the proof of Theorem 1 in Section A.1.4. A.1.1 The L1 Distance between Two Probability Distributions For simplicity, we only consider discrete distributions with finite supports – this will be enough for our purpose. Let P be a probability distribution. For any x, let P (x) = PrX∼P [X = x]. We write P = P1 ×P2 × · · · × Pn if the (multivariate) random variable X ∼ P can be written as X = (X1 , X2 , . . . , Xn ), where X1 , . . . , Xn are mutually independent and Xi ∼ Pi (∀i ∈ [n]). For two distributions P and Q, their L1 distance is defined as X L1 (P, Q) = \|P (x) − Q(x)\|, x where the summation is taken over x ∈ supp(P ) ∪ supp(Q). The L1 distance has the following property. It is a folklore result and we provide a proof for completeness. Lemma 1. Let P = P1 ×P2 ×· · ·×Pn and Q = Q1 ×Q2 ×· · ·×Qn be two probability distributions. Then we have n X L1 (Pi , Qi ). (1) L1 (P, Q) ≤ i=1 Proof. We prove (1) by induction on n. When n = 2, we have XX L1 (P, Q) = \|P (x, y) − Q(x, y)\| x = x ≤ = y XX y XX x X y \|P1 (x)P2 (y) − Q1 (x)Q2 (y)\| (\|P1 (x)P2 (y) − P1 (x)Q2 (y)\| + \|P1 (x)Q2 (y) − Q1 (x)Q2 (y)\|) P1 (x) x X y \|P2 (y) − Q2 (y)\| + = 1 · L1 (P2 , Q2 ) + 1 · L1 (P1 , Q1 ) = 2 X X y Q2 (y) X x \|P1 (x) − Q1 (x)\| L1 (Pi , Qi ). i=1 Here the summation is taken over x ∈ supp(P1 ) ∪ supp(Q1 ) and y ∈ supp(P2 ) ∪ supp(Q2 ). Suppose (1) is proved for n = k − 1 (k ≥ 3). When n = k, using the results for n = k − 1 and n = 2, we get L1 (P, Q) ≤ k−2 X i=1 L1 (Pi , Qi ) + L1 (Pk−1 × Pk , Qk−1 × Qk ) 10 ≤ = k−2 X L1 (Pi , Qi ) + L1 (Pk−1 , Qk−1 ) + L1 (Pk , Qk ) i=1 k X L1 (Pi , Qi ). i=1 This completes the proof. A.1.2 The DKW Inequality Consider a distribution D with CDF F (x). Let F̂n (x) be the empirical CDF of n i.i.d. samples Pn X1 , . . . , Xn drawn from D, i.e., F̂n (x) = n1 i=1 1{Xi ≤ x} (x ∈ R).7 Then we have: Lemma 2 (Dvoretzky-Kiefer-Wolfowitz inequality [10, 24]). For any ǫ > 0 and any n ∈ Z+ , we have 2 Pr sup F̂n (x) − F (x) ≥ ǫ ≤ 2e−2nǫ . x∈R h i Note that for any fixed x ∈ R, from the Chernoff bound we have Pr F̂n (x) − F (x) ≥ ǫ ≤ 2 2e−2nǫ . The DKW inequality states a stronger guarantee that the Chernoff concentration holds simultaneously for all x ∈ R. A.1.3 Technical Lemmas The following lemma describes some properties of the expected reward rP (S) = EX∼P [R(X, S)]. ′ be two probability distributions over Lemma 3. Let P = P1 × · · · × Pm and P ′ = P1′ × · · · × Pm m ′ ′ [0, 1] . Let Fi and Fi be the CDFs of Pi and Pi , respectively (i = 1, . . . , m). Suppose each Pi (i ∈ [m]) is a discrete distribution with finite support. (i) If for any i ∈ [m], x ∈ [0, 1] we have Fi′ (x) ≤ Fi (x), then for any super arm S ∈ F , we have rP ′ (S) ≥ rP (S). (ii) If for any i ∈ [m], x ∈ [0, 1] we have Fi (x) − Fi′ (x) ≤ Λi (Λi > 0), then for any super arm S ∈ F , we have X rP ′ (S) − rP (S) ≤ 2M Λi . i∈S Proof. It is easy to see why (i) is true. If we have Fi′ (x) ≤ Fi (x) for all i ∈ [m] and x ∈ [0, 1], then for all i, Pi′ has first-order stochastic dominance over Pi . When we change the distribution from Pi into Pi′ , we are moving some probability mass from smaller values to larger values. Recall that the reward function R(x, S) has a monotonicity property (Assumption 3): if x and x′ are two vectors in [0, 1]m such that xi ≤ x′i for all i ∈ [m], then R(x, S) ≤ R(x′ , S) for all S ∈ F . Therefore we have rP (S) ≤ rP ′ (S) for all S ∈ F . Now we prove (ii). Without loss of generality, we assume S = {1, 2, . . . , n} (n ≤ m). Let P ′′ = ′′ P1′′ × · · · × Pm be a distribution over [0, 1]m such that the CDF of Pi′′ is the following: max{Fi (x) − Λi , 0}, 0 ≤ x < 1, ′′ Fi (x) = (2) 1, x = 1. It is easy to see that Fi′′ (x) ≤ Fi′ (x) for all i ∈ [m] and x ∈ [0, 1]. Thus from the result in (i) we have (3) rP ′ (S) ≤ rP ′′ (S). 7 We use 1{·} to denote the indicator function, i.e., 1{H} = 1 if an event H happens, and 1{H} = 0 if it does not happen. 11 Let supp(Pi ) = {vi,1 , vi,2 , . . . , vi,si } where 0 ≤ vi,1 < · · · < vi,si ≤ 1. Define PS = P1 × P2 × · · · × Pn , and define PS′ and PS′′ similarly. Recall that the reward function R(x, S) can be written as RS (xS ) = RS (x1 , . . . , xn ). Then we have rP ′′ (S) − rP (S) X X RS (x1 , . . . , xn )PS (x1 , . . . , xn ) RS (x1 , . . . , xn )PS′′ (x1 , . . . , xn ) − = x1 ,...,xn x1 ,...,xn = X x1 ,...,xn ≤ X x1 ,...,xn RS (x1 , . . . , xn ) · (PS′′ (x1 , . . . , xn ) − PS (x1 , . . . , xn )) M · \|PS′′ (x1 , . . . , xn ) − PS (x1 , . . . , xn )\| = M · L1 (PS′′ , PS ), where the summation is taken over xi ∈ {vi,1 , . . . , vi,si } (∀i ∈ S). Then using Lemma 1 we obtain X rP ′′ (S) − rP (S) ≤ M · L1 (Pi′′ , Pi ). (4) i∈S ′′ Now we give an upper bound on L1 (Pi′′ , Pi ) for each i. Let Fi,j = Fi (vi,j ), Fi,j = Fi′′ (vi,j ), and ′′ Fi,0 = Fi,0 = 0. We have L1 (Pi′′ , Pi ) = si X j=1 = = si X j=1 si X j=1 \|Pi′′ (vi,j ) − Pi (vi,j )\| ′′ ′′ (Fi,j − Fi,j−1 ) − (Fi,j − Fi,j−1 ) (5) ′′ ′′ (Fi,j − Fi,j ) − (Fi,j−1 − Fi,j−1 ) . ′′ ′′ In fact, for all 1 ≤ j < si , we have Fi,j − Fi,j ≥ Fi,j−1 − Fi,j−1 . To see this, consider two cases: ′′ ′′ = Fi,j−1 = • If Fi,j < Λi , then we have Fi,j−1 ≤ Fi,j < Λi . By definition (2) we have Fi,j ′′ ′′ 0. Thus Fi,j − Fi,j = Fi,j ≥ Fi,j−1 = Fi,j−1 − Fi,j−1 . ′′ ′′ • If Fi,j ≥ Λi , then by definition (2) we have Fi,j − Fi,j = Λi ≥ Fi,j−1 − Fi,j−1 . Therefore (5) becomes L1 (Pi′′ , Pi ) = sX i −1 j=1 ′′ ′′ ′′ (Fi,j − Fi,j ) − (Fi,j−1 − Fi,j−1 ) + (1 − 1) − (Fi,si −1 − Fi,s ) i −1 ′′ ′′ = Fi,si −1 − Fi,s + Fi,si −1 − Fi,s i −1 i −1 ′′ = 2 Fi,si −1 − Fi,s i −1 (6) ≤ 2Λi , where the last inequality is due to (2). We complete the proof of the lemma by combining (3), (4) and (6): X X rP ′ (S) − rP (S) ≤ rP ′′ (S) − rP (S) ≤ M · L1 (Pi′′ , Pi ) ≤ 2M Λi . i∈S i∈S The following lemma is similar to Lemma 1 in [18]. We will use some additional notation: • For t ≥ m + 1 and i ∈ [m], let Ti,t be the value of counter Ti right after the t-th round of SDCB. In other words, Ti,t is the number of observed outcomes from arm i in the first t rounds. 12 • Let St be the super arm selected by SDCB in the t-th round. Lemma 4. Define an event in each round t (m + 1 ≤ t ≤ T ): s ( ) X 3 ln t Ht = 0 < ∆St ≤ 4M · . 2Ti,t−1 (7) i∈St Then the α-approximation regret of SDCB in T rounds is at most " T # X π2 E + 1 αM m. 1{Ht }∆St + 3 t=m+1 Proof. Let Fi be the CDF of Di . Let F̂i,l be the empirical CDF of the first l observations from arm i. For m + 1 ≤ t ≤ T , define an event s ( ) 3 ln t Et = there exists i ∈ [m] such that sup F̂i,Ti,t−1 (x) − Fi (x) ≥ , 2Ti,t−1 x∈[0,1] which means that the empirical CDF F̂i is not close enough to the true CDF Fi at the beginning of the t-th round. Recall that we have S ∗ = argmaxS∈F {rD (S)} and ∆S = max{α · rD (S ∗ ) − rD (S), 0} (S ∈ F ). We bound the α-approximation regret of SDCB as RegSDCB D,α (T ) = T X t=1 =E " E [α · rD (S ∗ ) − rD (St )] ≤ m X # ∆St + E t=1 " where ¬Et is the complement of event Et . T X t=m+1 T X E [∆St ] t=1 # 1{Et }∆St + E " T X t=m+1 # (8) 1{¬Et }∆St , We separately bound each term in (8). (a) the first term The first term in (8) can be trivially bounded as "m # m X X E α · rD (S ∗ ) ≤ m · αM. ∆St ≤ (9) t=1 t=1 (b) the second term By the DKW inequality we know that for any i ∈ [m], l ≥ 1, t ≥ m + 1 we have # " r 3 ln t 3 ln t ≤ 2e−2l· 2l = 2e−3 ln t = 2t−3 . Pr sup F̂i,l (x) − Fi (x) ≥ 2l x∈[0,1] Therefore E " T X t=m+1 # 1{Et } ≤ ≤ T m X t−1 X X t=m+1 i=1 l=1 T m X t−1 X X t=m+1 i=1 l=1 ≤ 2m ≤ T X t−2 t=m+1 π2 m, 3 13 Pr " 2t−3 F̂i,j,l − Fi,j ≥ r 3 ln t 2l # and then the second term in (8) can be bounded as " T # X π2 π2 E 1{Et }∆St ≤ m · (α · rD (S ∗ )) ≤ αM m. 3 3 t=m+1 (10) (c) the third term q 3 ln t We fix t > m and first assume ¬Et happens. Let ci = 2Ti,t−1 for each i ∈ [m]. Since ¬Et happens, we have F̂i,Ti,t−1 (x) − Fi (x) < ci ∀i ∈ [m], x ∈ [0, 1]. (11) Recall that in round t of SDCB (Algorithm 1), the input to the oracle is D = D1 × · · · × Dm , where the CDF Fi of Di is 0 ≤ x < 1, max{F̂i,Ti,t−1 (x) − ci , 0}, Fi (x) = (12) 1, x = 1. From (11) and (12) we know that Fi (x) ≤ Fi (x) ≤ Fi (x) + 2ci for all i ∈ [m], x ∈ [0, 1]. Thus, from Lemma 3 (i) we have rD (S) ≤ rD (S) ∀S ∈ F , (13) and from Lemma 3 (ii) we have rD (S) ≤ rD (S) + 2M X 2ci i∈S ∀S ∈ F . (14) Also, from the fact that the algorithm chooses St in the t-th round, we have rD (St ) ≥ α · max{rD (S)} ≥ α · rD (S ∗ ). (15) S∈F From (13), (14) and (15) we have α · rD (S ∗ ) ≤ α · rD (S ∗ ) ≤ rD (St ) ≤ rD (St ) + 2M X 2ci , i∈St which implies ∆St ≤ 4M X ci . i∈St P Therefore, when ¬Et happens, we always have ∆St ≤ 4M i∈St ci . In other words, s ) ( X 3 ln t . ¬Et =⇒ ∆St ≤ 4M 2Ti,t−1 i∈St This implies {¬Et , ∆St > 0} =⇒ ( 0 < ∆St ≤ 4M X i∈St s 3 ln t 2Ti,t−1 ) = Ht . Hence, the third term in (8) can be bounded as " T # " T # " T # X X X E 1{¬Et }∆St = E 1{¬Et , ∆St > 0}∆St ≤ E 1{Ht }∆St . t=m+1 t=m+1 t=m+1 Finally, by combining (8), (9), (10) and (16) we have " T # X π2 SDCB RegD,α (T ) ≤ E 1{Ht }∆St + + 1 αM m, 3 t=m+1 completing the proof of the lemma. 14 (16) A.1.4 Finishing the Proof of Theorem 1 Lemma 4 is very similar to Lemma 1 in [18]. We now apply the counting argument in [18] to finish the proof of Theorem 1. hP i T From Lemma 4 we know that it remains to bound E t=m+1 1{Ht }∆St , where Ht is defined in (7). Define two decreasing sequences of positive constants 1 = β0 >β1 > β2 > . . . α1 > α2 > . . . such that limk→∞ αk = limk→∞ βk = 0. We choose {αk } and {βk } as in Theorem 4 of [18], which satisfy ∞ √ X βk−1 − βk 6 ≤1 (17) √ αk k=1 and ∞ X αk k=1 βk < 267. For t ∈ {m + 1, . . . , T } and k ∈ Z+ , let ( 2 ln T αk 2MK ∆St mk,t = +∞ (18) ∆St > 0, ∆St = 0, and Ak,t = {i ∈ St \| Ti,t−1 ≤ mk,t }. Then we define an event Gk,t = {\|Ak,t \| ≥ βk K}, which means “in the t-th round, at least βk K arms in St had been observed at most mk,t times.” Lemma 5. In the t-th round (m + 1 ≤ t ≤ T ), if event Ht happens, then there exists k ∈ Z+ such that event Gk,t happens. Proof. Assume that Ht happens and that none of G1,t , G2,t , . . . happens. Then \|Ak,t \| < βk K for all k ∈ Z+ . Let A0,t = St and Āk,t = St \ Ak,t for k ∈ Z+ ∪{0}. It is easy to see Āk−1,t ⊆ Āk,t for all k ∈ Z+ . Note that limk→∞ exists N ∈ Z+ such that Āk,t = St for all k ≥ N , and then S∞mk,t = 0. Thus there we have St = k=1 Āk,t \ Āk−1,t . Finally, note that for all i ∈ Āk,t , we have Ti,t−1 > mk,t . Therefore ∞ X i∈St X 1 p = Ti,t−1 k=1 ∞ X ∞ X i∈Āk,t \Āk−1,t X 1 p ≤ Ti,t−1 k=1 ∞ X X i∈Āk,t \Āk−1,t 1 √ mk,t ∞ Āk,t \ Āk−1,t \|Ak−1,t \ Ak,t \| X \|Ak−1,t \| − \|Ak,t \| = = √ √ √ mk,t mk,t mk,t k=1 k=1 k=1 ∞ X 1 \|St \| 1 + \|Ak,t \| √ =√ −√ m1,t mk+1,t mk,t k=1 ∞ X K 1 1 <√ + βk K √ −√ m1,t mk+1,t mk,t = k=1 ∞ X (βk−1 − βk )K = . √ mk,t k=1 15 Note that we assume Ht happens. Then we have s X X √ 1 3 ln t p ∆St ≤ 4M · ≤ 2M 6 ln T · 2Ti,t−1 Ti,t−1 i∈S i∈S t t ∞ ∞ X √ √ X (βk−1 − βk )K βk−1 − βk = · ∆St ≤ ∆St , 6 < 2M 6 ln T · √ √ mk,t αk k=1 k=1 where the last inequality is due to (17). We reach a contradiction here. The proof of the lemma is completed. By Lemma 5 we have T X t=m+1 ∞ T X X 1{Ht }∆St ≤ 1{Gk,t , ∆St > 0}∆St . k=1 t=m+1 For i ∈ [m], k ∈ Z+ , t ∈ {m + 1, . . . , T }, define an event Gi,k,t = Gk,t ∧ {i ∈ St , Ti,t−1 ≤ mk,t }. Then by the definitions of Gk,t and Gi,k,t we have 1{Gk,t , ∆St > 0} ≤ Therefore T X t=m+1 1{Ht }∆St ≤ 1 X 1{Gi,k,t , ∆St > 0}. βk K i∈EB ∞ T X X X 1{Gi,k,t , ∆St > 0} i∈EB k=1 t=m+1 ∆St . βk K B B B For each arm i ∈ EB , suppose i is contained in Ni bad super arms Si,1 , Si,2 , . . . , Si,N . Let ∆i,l = i ∆Si,l B (l ∈ [Ni ]). Without loss of generality, we assume ∆i,1 ≥ ∆i,2 ≥ . . . ≥ ∆i,Ni . Note that 2 = 0. Then we ∆i,Ni = ∆i,min . For convenience, we also define ∆i,0 = +∞, i.e., αk 2MK ∆i,0 have T X t=m+1 ≤ ≤ = 1{Ht }∆St Ni T ∞ X X X X i∈EB k=1 t=m+1 l=1 Ni ∞ T X X X X i∈EB k=1 t=m+1 l=1 Ni ∞ T X X X X i∈EB k=1 t=m+1 l=1 = B 1{Gi,k,t , St = Si,l } B 1{Ti,t−1 ≤ mk,t , St = Si,l } ( 1 Ti,t−1 ≤ αk Ni X ∞ T l X X X X i∈EB k=1 t=m+1 l=1 j=1 ≤ ≤ ≤ Ni X ∞ T l X X X X i∈EB k=1 t=m+1 l=1 j=1 Ni X Ni ∞ T X X X X i∈EB k=1 t=m+1 l=1 j=1 Ni ∞ T X X X X i∈EB k=1 t=m+1 j=1 ∆St βk K ( 2M K ∆i,l ( 1 αk 2M K ∆i,j−1 2 ( 1 αk 2M K ∆i,j−1 2 ( 2M K ∆i,j−1 2 1 αk 1 αk 2M K ∆i,j−1 2 2 ∆i,l βk K B ln T, St = Si,l ) ∆i,l βk K B ln T, St = Si,l ) ∆i,l βk K ln T, St = B Si,l ) ∆i,j βk K 2 ln T, St = B Si,l ) ∆i,j βk K ln T ) ln T < Ti,t−1 ≤ αk 2M K ∆i,j 2 ln T < Ti,t−1 ≤ αk 2M K ∆i,j 2 ln T < Ti,t−1 ≤ αk 2M K ∆i,j ln T < Ti,t−1 ≤ αk 16 2M K ∆i,j 2 ∆i,j βk K ≤ Ni ∞ X X X αk i∈EB k=1 j=1 ∞ X αk 2M K ∆i,j 2 ln T − αk 2M K ∆i,j−1 2 Ni X X ! 1 1 ln T · − 2 =4M K 2 βk ∆i,j ∆i,j−1 i∈EB j=1 k=1 ! Ni X X 1 1 ≤1068M 2K ln T · ∆i,j , 2 − ∆2 ∆ i,j i,j−1 j=1 2 ln T ! ! ∆i,j βk K ∆i,j i∈EB where the last inequality is due to (18). Finally, for each i ∈ EB we have Ni X 1 1 − 2 ∆2i,j ∆i,j−1 j=1 ! ∆i,j = ≤ 1 ∆i,Ni 1 ∆i,Ni + N i −1 X j=1 + Z 1 (∆i,j − ∆i,j+1 ) ∆2i,j ∆i,1 ∆i,Ni 2 1 dx x2 1 − ∆i,Ni ∆i,1 2 . < ∆i,min = It follows that T X t=m+1 1{Ht }∆St ≤ 1068M 2K ln T · X i∈EB 2 ∆i,min = M 2K X 2136 ln T. ∆i,min (19) i∈EB Combining (19) with Lemma 4, the distribution-dependent regret bound in Theorem 1 is proved. To prove the distribution-independent bound, we decompose T X t=m+1 1{Ht }∆St = T X t=m+1 ≤ ǫT + PT 1{Ht , ∆St ≤ ǫ}∆St + T X t=m+1 T X 1{Ht }∆St into two parts: 1{Ht , ∆St > ǫ}∆St t=m+1 (20) 1{Ht , ∆St > ǫ}∆St , t=m+1 where ǫ > 0 is a constant to be determined. The second term can be bounded in the same way as in the proof of the distribution-dependent regret bound, except that we only consider the case ∆St > ǫ. Thus we can replace (19) by T X t=m+1 1{Ht , ∆St > ǫ}∆St ≤ M 2 K It follows that T X t=m+1 Finally, letting ǫ = q T X i∈EB ,∆i,min >ǫ 2136 2136 ln T ≤ M 2 Km ln T. ∆i,min ǫ 1{Ht }∆St ≤ ǫT + M 2 Km 2136M 2 Km ln T T t=m+1 X (21) 2136 ln T. ǫ , we get √ √ 1{Ht }∆St ≤ 2 2136M 2KmT ln T < 93M mKT ln T . Combining this with Lemma 4, we conclude the proof of the distribution-independent regret bound in Theorem 1. 17 Algorithm 4 CUCB [8, 18] 1: For each arm i, maintain: (i) µ̂i , the average of all observed outcomes from arm i so far, and (ii) Ti , the number of observed outcomes from arm i so far. 2: // Initialization 3: for i = 1 to m do 4: // Action in the i-th round 5: Play a super arm Si that contains arm i, and update µ̂i and Ti . 6: end for 7: for t = m + 1, m + 2, . . . do 8: // Action in the t-th q round ln t 9: µ̄i ← min{µ̂i + 32T , 1} i ∀i ∈ [m] 10: Play the super arm St ← Oracle(µ̄), where µ̄ = (µ̄1 , . . . , µ̄m ). 11: Update µ̂i and Ti for all i ∈ St . 12: end for A.2 Analysis of Our Algorithm in the Previous CMAB Framework We now give an analysis of SDCB in the previous CMAB framework, following our discussion in Section 3. We consider the case in which the expected reward only depends on the means of the random variables. Namely, rD (S) only depends on µi ’s (i ∈ S), where µi is arm i’s mean outcome. In this case, we can rewrite rD (S) as rµ (S), where µ = (µ1 , . . . , µm ) is the vector of means. Note that the offline computation oracle only needs a mean vector as input. We no longer need the three assumptions (Assumptions 1-3) given in Section 2. In particular, we do not require independence among outcome distributions of all arms (Assumption 1). Although we cannot write D as D = D1 × · · · × Dm , we still let Di be the outcome distribution of arm i. In this case, Di is the marginal distribution of D in the i-th component. We summarize the CUCB algorithm [8, 18] in Algorithm 4. It maintains the empirical mean µ̂i of the outcomes from each arm i, and stores the number of observed outcomes from arm i in a variable Ti . In each round, it calculates an upper confidence bound (UCB) µ̄i of µi , Then it uses the UCB vector µ̄ as the input to the oracle, and plays the super arm output by the oracle. In the t-th round (t > m), each UCB µ̄i has the key property that s 3 ln t µi ≤ µ̄i ≤ µi + 2 (22) 2Ti,t−1 holds with high probability. (Recall q that Ti,t−1 is the value of Ti after t − 1 rounds.) To see this, ln t with high probability (by Chernoff bound), and then (22) note that we have \|µi − µ̂i \| ≤ 2T3i,t−1 follows from the definition of µ̄i in line 9 of Algorithm 4. We prove that the same property as (22) also holds for SDCB. Consider a fixed t > m, and let D = D1 × · · · × Dm be the input to the oracle in the t-th round of SDCB. Let νi = EYi ∼Di [Yi ]. We can think that SDCB uses the mean vector ν = (ν1 , . . . , νm ) as the input to the oracle used by CUCB. We now show that for each i, we have s 3 ln t µi ≤ ν i ≤ µi + 2 (23) 2Ti,t−1 with high probability. To show (23), we first prove the following lemma. Lemma 6. Let P and P ′ be two distributions over [0, 1] with CDFs F and F ′ , respectively. Consider two random variables Y ∼ P and Y ′ ∼ P ′ . (i) If for all x ∈ [0, 1] we have F ′ (x) ≤ F (x), then we have E[Y ] ≤ E[Y ′ ]. (ii) If for all x ∈ [0, 1] we have F (x) − F ′ (x) ≤ Λ (Λ > 0), then we have E[Y ′ ] ≤ E[Y ] + Λ. 18 Proof. We have E[Y ] = Z 1 x dF (x) = (xF (x)) 0 1 0 − Z 1 0 F (x) dx = 1 − Z 1 F (x) dx. 0 Similarly, we have ′ E[Y ] = 1 − Then the lemma holds trivially. Z 1 F ′ (x) dx. 0 Now we prove (23). According to the DKW inequality, with high probability we have s 3 ln t ≤ Fi (x) ≤ Fi (x) Fi (x) − 2 2Ti,t−1 (24) for all i ∈ [m] and x ∈ [0, 1], where Fi is the CDF of Di used in round t of SDCB, and Fi is the CDF of Di . Suppose (24) holds for all i, x, then qfor any i, the two distributions Di and Di ln t satisfy the two conditions in Lemma 6, with Λ = 2 2T3i,t−1 ; then from Lemma 6 we know that q ln t µi ≤ νi ≤ µi + 2 2T3i,t−1 . Hence we have shown that (23) holds with high probability. The fact that (23) holds with high probability means that the mean of Di is also a UCB of µi with the same confidence as in CUCB. With this property, the analysis in [8, 18] can also be applied to SDCB, resulting in exactly the same regret bounds. B Missing Proofs from Section 4 B.1 Analysis of the Discretization Error The following lemma gives an upper bound on the error due to discretization. Refer to Section 4 for the definition of the discretized distribution D̃. Lemma 7. For any S ∈ F , we have \|rD (S) − rD̃ (S)\| ≤ CK . s To prove Lemma 7, we show a slightly more general lemma which gives an upper bound on the discretization error of the expectation of a Lipschitz continuous function. Lemma 8. Let g(x) be a Lipschitz continuous function on P [0, 1]n such that for any x, x′ ∈ [0, 1]n , ′ ′ ′ we have \|g(x) − g(x )\| ≤ Ckx − x k1 , where kx − x k1 = ni=1 \|xi − x′i \|. Let P = P1 × · · · × Pn be a probability distribution over [0, 1]n . Define another distribution P̃ = P̃1 × · · ·× P̃n over [0, 1]n as follows: each P̃i (i ∈ [n]) takes values in { 1s , 2s , . . . , 1}, and Pr [X̃i = j/s] = Pr [Xi ∈ Ij ] , X̃i ∼P̃i Xi ∼Pi j ∈ [s], s−1 s−1 where I1 = [0, 1s ], I2 = ( 1s , 2s ], . . . , Is−1 = ( s−2 s , s ], Is = ( s , 1]. Then EX∼P [g(X)] − EX̃∼P̃ [g(X̃)] ≤ C ·n . s (25) Proof. Throughout the proof, we consider X = (X1 , . . . , Xn ) ∼ P and X̃ = (X̃1 , . . . , X̃n ) ∼ P̃ . Let vj = j s (j = 0, 1, . . . , s) and pi,j = Pr[X̃i = vj ] = Pr[Xi ∈ Ij ] We prove (25) by induction on n. 19 i ∈ [n], j ∈ [s]. (1) When n = 1, we have E[g(X1 )] = X j∈[s],p1,j >0 p1,j · E g(X1 ) X1 ∈ Ij . (26) Since g is continuous, for each j ∈ [s] such that p1,j > 0, there exists ξj ∈ [vj−1 , vj ] such that E [g(X1 )\|X1 ∈ Ij ] = g(ξj ) From the Lipschitz continuity of g we have \|g(vj ) − g(ξj )\| ≤ C\|vj − ξj \| ≤ C\|vj − vj−1 \| = C . s Hence E[g(X1 )] − E[g(X̃1 )] = X p1,j · E[g(X1 )\|X1 ∈ Ij ] − X p1,j · g(ξj ) − j∈[s],p1,j >0 = j∈[s],p1,j >0 ≤ ≤ = j∈[s],p1,j >0 X p1,j · \|g(ξj ) − g(vj )\| X p1,j · j∈[s],p1,j >0 j∈[s],p1,j >0 C . s X X j∈[s],p1,j >0 p1,j · g(vj ) p1,j · g(vj ) C s This proves (25) for n = 1. (ii) Suppose (25) is correct for n = 1, 2, . . . , k − 1. Now we prove it for n = k (k ≥ 2). We define two functions on [0, 1]k−1 : h(x1 , . . . , xk−1 ) = EXk [g(x1 , . . . , xk−1 , Xk )] and h̃(x1 , . . . , xk−1 ) = EX̃k [g(x1 , . . . , xk−1 , X̃k )]. For any fixed x1 , . . . , xk−1 ∈ [0, 1], the function g(x1 , . . . , xk−1 , x) on x ∈ [0, 1] is Lipschitz continuous. Therefore from the result for n = 1 we have h(x1 , . . . , xk−1 ) − h̃(x1 , . . . , xk−1 ) ≤ 20 C s ∀x1 , . . . , xk−1 ∈ [0, 1]. Then we have E[g(X)] − E[g(X̃)] = EX1 ,...,Xk−1 [E[g(X)\|X1 , . . . , Xk−1 ]] − E[g(X̃)] = EX1 ,...,Xk−1 [h(X1 , . . . , Xk−1 )] − E[g(X̃)] ≤ EX1 ,...,Xk−1 [h(X1 , . . . , Xk−1 )] − EX1 ,...,Xk−1 [h̃(X1 , . . . , Xk−1 )] + EX1 ,...,Xk−1 [h̃(X1 , . . . , Xk−1 )] − E[g(X̃)] h i ≤ EX1 ,...,Xk−1 h(X1 , . . . , Xk−1 ) − h̃(X1 , . . . , Xk−1 ) + EX1 ,...,Xk−1 ,X̃k [g(X1 , . . . , Xk−1 , X̃k )] − E[g(X̃)] h i C + EX̃k E[g(X1 , . . . , Xk−1 , X̃k )\|X̃k ] − E[g(X̃1 , . . . , X̃k−1 , X̃k )\|X̃k ] ≤ EX1 ,...,Xk−1 s h i C ≤ + EX̃k E[g(X1 , . . . , Xk−1 , X̃k )\|X̃k ] − E[g(X̃1 , . . . , X̃k−1 , X̃k )\|X̃k ] s X C = + pk,j · E[g(X1 , . . . , Xk−1 , vj )] − E[g(X̃1 , . . . , X̃k−1 , vj )] . s j∈[s],pk,j >0 (27) For any j ∈ [s], the function g(x1 , . . . , xk−1 , vj ) on (x1 , . . . , xk−1 ) ∈ [0, 1] uous. Then from the induction hypothesis at n = k − 1, we have E[g(X1 , . . . , Xk−1 , vj )] − E[g(X̃1 , . . . , X̃k−1 , vj )] ≤ k−1 C(k − 1) s is Lipschitz contin- ∀j ∈ [s]. (28) From (27) and (28) we have E[g(X)] − E[g(X̃)] ≤ C + s X j∈[s],pk,j >0 pk,j · C(k − 1) s C(k − 1) C + s s Ck . = s = This concludes the proof for n = k. Now we prove Lemma 7. Proof of Lemma 7. We have rD (S) = EX∼D [R(X, S)] = EX∼D [RS (XS )] = EXS ∼DS [RS (XS )], where XS = (Xi )i∈S and DS = (Di )i∈S . Similarly, we have rD̃ (S) = EX̃S ∼D̃S [RS (X̃S )]. According to Assumption 4, the function RS defined on [0, 1]S is Lipschitz continuous. Then from Lemma 8 we have \|rD (S) − rD̃ (S)\| = EXS ∼DS [RS (XS )] − EX̃S ∼D̃S [RS (X̃S )] ≤ This completes the proof. 21 C·K C · \|S\| ≤ . s s B.2 Proof of Theorem 2 Proof of Theorem 2. Let S ∗ = argmaxS∈F {rD (S)} and S̃ ∗ = argmaxS∈F {rD̃ (S)} be the optimal super arms in problems ([m], F , D, R) and ([m], F , D̃, R), respectively. Suppose Algorithm 2 selects super arm St in the t-th round (1 ≤ t ≤ T ). Then its α-approximation regret is bounded as Alg. 2 RegD,α (T ) = T · α · rD (S ∗ ) − T X E [rD (St )] t=1 T X = T · α rD (S ∗ ) − rD̃ (S̃ ∗ ) + E [rD̃ (St ) − rD (St )] + t=1 ≤ T · α (rD (S ∗ ) − rD̃ (S ∗ )) + T X t=1 T · α · rD̃ (S̃ ∗ ) − T X E [rD̃ (St )] t=1 ! Alg. 1 (T ). E [rD̃ (St ) − rD (St )] + RegD̃,α where the inequality is due to rD̃ (S̃ ∗ ) ≥ rD̃ (S ∗ ). Then from Lemma 7 and the distribution-independent bound in Theorem 1 we have 2 √ CK CK π Alg. 2 +T · + 93M mKT ln T + + 1 αM m RegD,α (T ) ≤ T · α · s s 3 2 √ CKT π (29) ≤2· + 93M mKT ln T + + 1 αM m s 3 2 √ √ π ≤ 93M mKT ln T + 2CK T + + 1 αM m. 3 √ √ Here in the last two inequalities we have used α ≤ 1 and s = ⌈ T ⌉ ≥ T . The proof is completed. B.3 Proof of Theorem 3 Proof of Theorem 3. Let n = ⌈log2 T ⌉. Then we have 2n−1 < T ≤ 2n . If n ≤ q = ⌈log2 m⌉, then T ≤ 2m and the regret in T rounds is at most 2m · αM . The regret bound holds trivially. Now we assume n ≥ q + 1. Using Theorem 2, we have Alg. 3 (T ) RegD,α Alg. 3 ≤ RegD,α (2n ) Alg. 2 q = RegD,α (2 ) + n−1 X 2 k RegAlg. D,α (2 ) k=q 2 ≤ RegAlg. D,α (2m) ≤ 2m · αM + + n−1 X Alg. 2 k RegD,α (2 ) k=q n−1 X k=q 2 √ √ π k k k + 1 αM m 93M mK · 2 ln 2 + 2CK 2 + 3 2 n−1 X√ √ π n−1 k ≤ 2αM m + 93M mK ln 2 + 2CK · 2 + (n − 1) · + 1 αM m 3 k=1 2 √2n √ π n−1 + 2CK · √ + + 3 (n − 1) · αM m ≤ 93M mK ln 2 3 2−1 √ 2 √ 2T π ≤ 93M mK ln T + 2CK · √ + 3 log2 T · αM m + 3 2−1 22 Algorithm 5 Greedy-K-MAX 1: S ← ∅ 2: for i = 1 to K do 3: k ← argmaxj∈[m]\S rD (S ∪ {j}) 4: S ← S ∪ {k} 5: end for Output: S √ √ ≤ 318M mKT ln T + 7CK T + 10αM m ln T. C The Offline K-MAX Problem In this section, we consider the offline K-MAX problem. Recall that we have m independent random variables {Xi }i∈[m] . Xi follows the discrete distribution Di with support {vi,1 , . . . , vi,si } ⊂ [0, 1], and D = D1 × · · · × Dm is the joint distribution of X = (X1 , . . . , Xm ). Let pi,j = Pr[Xi = vi,j ]. Define rD (S) = EX∼D [maxi∈S Xi ] and OPT = maxS:\|S\|=K rD (S). Our goal is to find (in polynomial time) a subset S ⊆ [m] of cardinality K such that rD (S) ≥ α · OPT (for certain constant α). First, we show that rD (S) can be calculated in polynomial time given any S ⊆ [m]. Let S = {iS 1 , i2 , . . . , in }. Note that for X ∼ D, maxi∈S Xi can only take values in the set V (S) = i∈S supp(Di ). For any v ∈ V (S), we have Pr max Xi = v X∼D i∈S = Pr [Xi1 = v, Xi2 ≤ v, . . . , Xin ≤ v] X∼D + Pr [Xi1 < v, Xi2 = v, Xi3 ≤ v, . . . , Xin ≤ v] (30) X∼D + ··· + Pr [Xi1 < v, . . . , Xin−1 < v, Xin = v]. X∼D Since Xi1 , . . . , Xin are mutually independent, each probability appearing in (30) can be calculated in polynomial time. Hence for any v ∈ V (S), PrX∼D [maxi∈S Xi = v] can be calculated in polynomial time using (30). Then rD (S) can be calculated by X v · Pr max Xi = v rD (S) = v∈V (S) X∼D i∈S in polynomial time. C.1 (1 − 1/e)-Approximation We now show that a simple greedy algorithm (Algorithm 5) can find a (1 − 1/e)-approximate solution, by proving the submodularity of rD (S). In fact, this is implied by a slightly more general result [13, Lemma 3.2]. We provide a simple and direct proof for completeness. Lemma 9. Algorithm 5 can output a subset S such that rD (S) ≥ (1 − 1/e) · OPT. Proof. For any x ∈ [0, 1]m , let fx (S) = maxi∈S xi be a set function defined on 2[m] . (Define fx (∅) = 0.) We can verify that fx (S) is monotone and submodular: • Monotonicity. For any A ⊆ B ⊆ [m], we have fx (A) = maxi∈A xi ≤ maxi∈B xi = fx (B). • Submodularity. For any A ⊆ B ⊆ [m] and any k ∈ [m] \ B, there are three cases (note that maxi∈A xi ≤ maxi∈B xi ): (i) If xk ≤ maxi∈A xi , then fx (A ∪ {k}) − fx (A) = 0 = fx (B ∪ {k}) − fx (B). 23 (ii) If maxi∈A xi < xk ≤ maxi∈B xi , then fx (A ∪ {k}) − fx (A) = xk − maxi∈A xi > 0 = fx (B ∪ {k}) − fx (B). (iii) If xk > maxi∈B xi , then fx (A ∪ {k}) − fx (A) = xk − maxi∈A xi ≥ xk − maxi∈B xi = fx (B ∪ {k}) − fx (B). Therefore, we always have fx (A ∪ {k}) − fx (A) ≥ fx (B ∪ {i}) − fx (B). The function fx (S) is submodular. For any S ⊆ [m] we have rD (S) = s2 s1 X X j1 =1 j2 =1 ··· sm X f(v1,j1 ,...,vm,jm ) (S) m Y pi,ji . i=1 jm =1 Since each set function f(v1,j1 ,...,vm,jm ) (S) is monotone and submodular, rD (S) is a convex combination of monotone submodular functions on 2[m] . Therefore, rD (S) is also a monotone submodular function. According to the classical result on submodular maximization [25], the greedy algorithm can find a (1 − 1/e)-approximate solution to maxS⊆[m],\|S\|≤K {rD (S)}. C.2 PTAS Now we provide a PTAS for the K-MAX problem. In other words, we give an algorithm which, given any fixed constant 0 < ε < 1/2, can find a solution S of cardinality \|K\| such that rD (S) ≥ (1 − ε) · OPT in polynomial time. We first provide an overview of our approach, and then spell out the details later. 1. (Discretization) We first transform each Xi to another discrete distribution X̃i , such that all X̃i ’s are supported on a set of size O(1/ε2 ). 2. (Computing signatures) For each Xi , we can compute from X̃i a signature Sig(X P i ) which is a vector of size O(1/ε2 ). For a set S, we define its signature Sig(S) to be i∈S Sig(Xi ). We show that if two sets S1 and S2 have the same signature, their objective values are close (Lemma 12). 3. (Enumerating signatures) We enumerate all possible signatures (there are polynomial number of them when treating ε as a constant) and try to find the one which is the signature of a set of size K, and the objective value is maximized. C.2.1 Discretization We first describe the discretization step. We say that a random variable X follows the Bernoulli distribution B(v, q) if X takes value v with probability q and value 0 with probability 1 − q. For any discrete distribution, we can rewrite it as the maximum of a set of Bernoulli distributions. Definition 1. Let X be a discrete random variable with support {v1 , v2 , . . . , vs }(v1 < v2 < · · · < vs ) and Pr[X = vj ] = pj . We define a set of independent Bernoulli random variables {Zj }j∈[s] as ! pj Zj ∼ B vj , P . j ′ ≤j pj ′ We call {Zj } the Bernoulli decomposition of Xi . Lemma 10. For a discrete distribution X and its Bernoulli decomposition {Zj }, maxj {Zj } has the same distribution with X. Proof. We can easily see the following: Pr[max{Zj } = vi ] = Pr[Zi = vi ] j = P pi i′ ≤i pi′ 24 Y Pr[Zi′ = 0] i′ >i Y h>i 1− P ph h′ ≤h p h′ ! Algorithm 6 Discretization 1: We first run Greedy-K-MAX to obtain a solution SG and let W = rD (SG ). 2: for i = 1 to m do 3: Compute the Bernoulli decomposition {Zi,j }j of Xi . 4: for all Zi,j do 5: Create another Bernoulli variable Z̃i,j as follows: 6: if vi,j > W/ε then ε 7: Let Z̃i,j ∼ B W ε , E[Zi,j ] W (Case 1) 8: else Zi,j 9: Let Z̃i,j = ⌊ εW ⌋εW (Case 2) 10: end if 11: end for 12: Let X̃i = maxj {Z̃ij } 13: end for = P pi i′ ≤i pi′ Y h>i Hence, Pr[maxj {Zj } = vi ] = Pr[X = vi ] for all i ∈ [s]. P h′ ≤h−1 ph′ P = pi . h′ ≤h ph′ Now, we describe how to construct the discretization X̃i of Xi for all i ∈ [m]. The pseudocode can be found in Algorithm 6. We first run Greedy-K-MAX to obtain a solution SG . Let W = rD (SG ). By Lemma 9, we know that W ≥ (1 − 1/e)OPT. Then we compute the Bernoulli decomposition {Zi,j }j of Xi . For each Zi,j , we create another Bernoulli variable Z̃i,j as follows: Recall that vi,j is the nonzero possible value of Zij . We distinguish two cases. Case 1: If vi,j > W/ε, then we let ε , E[Z ] . It is easy to see that E[Z̃ij ] = E[Zij ]. Case 2: If vi,j ≤ W/ε, then we Z̃i,j ∼ B W i,j ε W Zi,j ⌋εW. We note that more than one Z̃ij ’s may have the same support, and all Z̃ij ’s let Z̃i,j = ⌊ εW are supported on DS = {0, εW, 2εW, . . . , W/ε}. Finally, we let X̃i = maxj {Z̃ij }, which is the discretization of Xi . Since X̃i is the maximum of a set of Bernoulli distributions, it is also a discrete distribution supported on DS. We can easily compute Pr[X̃i = v] for any v ∈ DS. Now, we show that the discretization only incurs a small loss in the objective value. The key is to show that we do not lose much in the transformation from Zi,j ’s to Z̃i,j ’s. We prove a slightly more general lemma as follows. Lemma 11. Consider any set of Bernoulli variables {Zi ∼ B(ai , pi )}1≤i≤n . Assume that E[maxi∈[n] Zi ] < cW, where c is a constant such that cε < 1/2. For each Zi , we create a Bernoulli variable Z̃i in the same way as Algorithm 6. Then the following holds: E[max Zi ] ≥ E[max Z̃i ] ≥ E[max Zi ] − (2c + 1)εW. Proof. Assume a1 is the largest among all ai ’s. If a1 < W/ε, all Z̃i are created in Case 2. In this case, it is obvious to have that E[max Zi ] ≥ E[max Z̃i ] ≥ E[max Zi ] − εW. If a1 ≥ W/ε, the proof is slightly more complicated. Let L = {i \| ai ≥ W/ε}. We prove by induction on n (i.e., the number of the variables) the following more general claim: X E[max Zi ] ≥ E[max Z̃i ] ≥ E[max Zi ] − εW − c εai pi . (31) i∈L Consider the base case n = 1. The lemma holds immediately in Case 1 as E[Z1 ] = E[Z˜1 ]. Assuming the lemma is true for n = k, we show it also holds for n = k + 1. Recall we have Z̃1 ∼ B( W ε , εE[Z1 ]/W). Thus E[max Zi ] − E[max Z̃i ] =a1 p1 + (1 − p1 )E[max Zi ] − a1 p1 − (1 − εE[Z1 ]/W)E[max Z̃i ] i≥1 i≥1 i≥2 25 i≥2 ≥(1 − p1 )E[max Z̃i ] − (1 − εE[Z1 ]/W)E[max Z̃i ] i≥2 i≥2 =(εa1 p1 /W − p1 )E[max Z̃i ] ≥ 0, i≥2 where the first inequality follows from the induction hypothesis and the last from a1 ≥ W/ε. The other direction can be seen as follows: E[max Z̃i ] − E[max Zi ] =a1 p1 + (1 − εE[Z1 ]/W)E[max Z̃i ] − (a1 p1 + (1 − p1 )E[max Zi ]) i≥1 i≥1 i≥2 i≥2 X εai pi ≥(1 − εE[Z1 ]/W)E[max Zi ] − (1 − p1 )E[max Zi ] − εW − c i≥2 i≥2 ≥(−εE[Z1 ]/W)E[max Zi ] − εW − c i≥2 ≥ − εW − c X X i∈L\{1} εai pi i∈L\{1} εai pi , i∈L where the last inequality holds since E[maxi≥2 Zi ] ≤ cW. This finishes the proof of (31). P Now, we show that i∈L ai pi ≤ 2W. This can be seen as follows. First, we can see from Markov inequality that Pr[max Zi > W/ε] ≤ cε. Q Equivalently, we have i∈L (1 − pi ) ≥ 1 − cε. Then, we can see that W≥ X ai i∈L X Y 1X (1 − pj )pi ≥ (1 − cε) ai p i ≥ ai p i . 2 j<i i∈L i∈L Plugging this into (31), we prove the lemma. Corollary 1. For any set S ⊆ [m], suppose E[maxi∈S Xi ] < cW, where c is a constant such that cε < 1/2. Then the following holds: E[max Xi ] ≥ E[max X̃i ] ≥ E[max Xi ] − (2c + 1)εW. i∈S i∈S i∈S C.2.2 Signatures For each Xi , we have created its discretization X̃i = maxj {Z̃ij }. Since X̃i is a discrete distribution, we can define its Bernoulli decomposition {Yij }j∈[h] where h = \|DS\|. Suppose Yij ∼ B(jεW, qij ). Now, we define the signature of Xi to be the vector Sig(Xi ) = (Sig(Xi )1 , . . . , Sig(Xi )h ) where 4 − ln (1 − qij ) ε ln(1/ε4 ) Sig(Xi )j = min , · j ∈ [h]. ε4 /m ε4 /m m For any set S, define its signature to be Sig(S) = X Sig(Xi ). i∈S Define the set SG of signature vectors to be all nonnegative h-dimensional vectors, where each coordinate is an integer multiple of ε4 /m and at most m ln(1/ε4 ). Clearly, the size of SG is 2 h−1 O mε−4 log(h/ε2 ) = Õ(mO(1/ε ) ), which is polynomial for any fixed constant ε > 0 (recall h = \|DS\| = O(1/ε2 )). Now, we prove the following crucial lemma. Lemma 12. Consider two sets S1 and S2 . If Sig(S1 ) = Sig(S2 ), the following holds: E[max X̃i ] − E[max X̃i ] ≤ O(ε)W. i∈S1 i∈S2 26 Algorithm 7 PTAS-K-MAX 1: U ← ∅ 2: for all signature vector sg ∈ SG do 3: Find a set S such that \|S\| = K and Sig(S) = sg 4: if rD (S) > rD (U ) then 5: U ←S 6: end if 7: end for Output: U Proof. Suppose {Yij }j∈[h] is the Bernoulli decomposition of X̃i . For any set S, we define Yk (S) = maxi∈S Yik (it is the max of a set of Bernoulli distributions). Q It is not hard to see that Yk (S) has a Bernoulli distribution B(kεW, pk (S)) with pk (S) = 1 − i∈S (1 − qik ). As Sig(S1 ) = Sig(S2 ), we have that Y Y \|pk (S1 ) − pk (S2 )\| = \| (1 − qik ) − (1 − qik )\| i∈S1 = exp i∈S2 X i∈S1 ≤ 2ε4 ! ln(1 − qik ) − exp ∀k ∈ [h]. X i∈S2 ! ln(1 − qik ) Noticing maxi∈S X̃i = maxk Yk (S), we have that E[max X̃i ] − E[max X̃i ] = E[max Yk (S1 )] − E[max Yk (S2 )] k k i∈S2 i∈S1 ! X W \|pk (S1 ) − pk (S2 )\| ≤ ε k ≤4hε3 W = O(ε)W where the first inequality follows from Lemma 1. For any signature vector sg, we associate to it a set of random variables {Bk ∼ B(kεW, 1 − e−sgk )}hk=1 .8 Define the value of sg to be Val(sg) = E[maxk∈[h] Bk ]. Corollary 2. For any feasible set S with Sig(S) = sg, \|E[maxi∈S X̃i ] − Val(sg)\| ≤ O(ε)W. Moreover, combining with Corollary 1, we have that \|E[maxi∈S Xi ] − Val(sg)\| ≤ O(ε)W. C.2.3 Enumerating Signatures Our algorithm enumerates all signature vectors sg in SG. For each sg, we check if we can find a set S 2 of size K such that Sig(S) = sg. This can be done by a standard dynamic program in Õ(mO(1/ε ) ) ′ ′ time as follows: We use Boolean variable R[i][j][sg ] to represent whether signature vector sg ∈ SG can be dominated by i variables in set {X1 , . . . , Xj }. The dynamic programming recursion is R[i][j][sg′ ] = R[i][j − 1][sg′ ] ∧ R[i − 1][j − 1][sg′ − Sig(Xj )]. If the answer is yes (i.e., we can find such S), we say sg is a feasible signature vector and S is a candidate set. Finally, we pick the candidate set with maximum rD (S) and output the set. The pseudocode can be found in Algorithm 7. Now, we are ready to prove Theorem 4 by showing Algorithm 7 is a PTAS for the K-MAX problem. Proof of Theorem 4. Suppose S ∗ is the optimal solution and sg∗ is the signature of S ∗ . By Corollary 2, we have that \|OPT − Val(sg∗ )\| ≤ O(ε)W. 8 It is not hard to see the signature of maxk∈[h] Bk is exactly sg. 27 Algorithm 8 Online Submodular Maximization [26] 1: Let A1 , A2 , . . . , AK be K instances of Exp3 2: for t = 1, 2, . . . do 3: // Action in the t-th round 4: for i = 1 to K do 5: Use Ai to select an arm at,i ∈ [m] 6: end for SK 7: Play the super arm St ← i=1 {at,i } 8: for i = 1 to K do Si−1 Si 9: Feed back ft ( j=1 {at,j }) − ft ( j=1 {at,j }) as the payoff Ai receives for choosing at,i 10: end for 11: end for When Algorithm 7 is enumerating sg∗ , it can find a set S such that Sig(S) = sg∗ (there exists at least one such set since S ∗ is one). Therefore, we can see that \|E[max Xi ] − E[max∗ Xi ]\| ≤ \|Val(sg∗ ) − max Xi \| + \|Val(sg∗ ) − E[max∗ Xi ]\| ≤ O(ε)W. i∈S i∈S i∈S i∈S Let U be the output of Algorithm 7. Since W ≥ (1 − 1/e)OPT, we have rD (U ) ≥ rD (S) = E[maxi∈S Xi ] ≥ (1 − O(ε))OPT. The running time of the algorithm is polynomial for a fixed constant ε > 0, since the number of signature vectors is polynomial and the dynamic program in each iteration also runs in polynomial time. Hence, we have a PTAS for the K-MAX problem. Remark. In fact, Theorem 4 can be generalized in the following way: instead of the cardinality constraint \|S\| ≤ K, we can have more general combinatorial constraint on the feasible set S. As long as we can execute line 3 in Algorithm 7 in polynomial time, the analysis wound be the same. Using the same trick as in [20], we can extend the dynamic program to a more general class of combinatorial constraints where there is a pseudo-polynomial time for the exact version9 of the deterministic version of the corresponding problem. The class of constraints includes s-t simple paths, knapsacks, spanning trees, matchings, etc. D Empirical Comparison between the SDCB Algorithm and Online Submodular Maximization on the K-MAX Problem We perform experiments to compare the SDCB algorithm with the online submodular maximization algorithm in [26], on the K-MAX problem. Online Submodular Maximization. First we briefly describe the online submodular maximization problem considered in [26] and the algorithm therein. At the beginning, an oblivious adversary sets a sequence of submodular functions f1 , f2 , . . . , fT on 2[m] , where ft will be used to determine the reward in the t-th round. In the t-th round, if the player selects a feasible super arm St , the reward will be ft (St ). This model covers the K-MAX problem as an instance: suppose (t) (t) X (t) = (X1 , . . . , Xm ) ∼ D is the outcome vector sampled in the t-th round, then the func(t) tion ft (S) = maxi∈S Xi is submodular and will determine the reward in the t-th round. We summarize the algorithm in Algorithm 8. It uses K copies of the Exp3√algorithm (see [3] for an introduction). For the K-MAX problem, Algorithm 8 achieves an O(K mT log m) upper bound on the (1 − 1/e)-approximation regret. Setup. We set m = 9 and K = 3, i.e., there are 9 arms in total and it is allowed to select at most 3 arms in each round. We compare the performance of SDCB/Lazy-SDCB and the online submodular maximization algorithm on four different distributions. Here we use the greedy algorithm Greedy-K-MAX (Algorithm 5) as the offline oracle. 9 In the exact version of a problem, we ask for a feasible set S such that total weight of S is exactly a given target value B. For example, in the exact spanning tree problem where each edge has an integer weight, we would like to find a spanning tree of weight exactly B. 28 Let Xi ∼ Di (i = 1, . . . , 9). We consider the following distributions. For all of them, the optimal super arm is S ∗ = {1, 2, 3}. • Distribution 1: All Di ’s have the same support {0, 0.2, 0.4, 0.6, 0.8, 1}. For i ∈ {1, 2, 3}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.1 and Pr[Xi = 1] = 0.5. For i ∈ {4, 5, 6, . . . , 9}, Pr[Xi = 0] = 0.5 and Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = Pr[Xi = 1] = 0.1. • Distribution 2: All Di ’s have the same support {0, 0.2, 0.4, 0.6, 0.8, 1}. For i ∈ {1, 2, 3}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.1 and Pr[Xi = 1] = 0.5. For i ∈ {4, 5, 6, . . . , 9}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.12 and Pr[Xi = 1] = 0.4. • Distribution 3: All Di ’s have the same support {0, 0.2, 0.4, 0.6, 0.8, 1}. For i ∈ {1, 2, 3}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.1 and Pr[Xi = 1] = 0.5. For i ∈ {4, 5, 6}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.12 and Pr[Xi = 1] = 0.4. For i ∈ {7, 8, 9}, Pr[Xi = 0] = Pr[Xi = 0.2] = Pr[Xi = 0.4] = Pr[Xi = 0.6] = Pr[Xi = 0.8] = 0.16 and Pr[Xi = 1] = 0.2. • Distribution 4: All Di ’s are continuous distributions on [0, 1]. For i ∈ {1, 2, 3}, Di is the uniform distribution on [0, 1]. For i ∈ {4, 5, 6, . . . , 9}, the probability density function (PDF) of Xi is 1.2 x ∈ [0, 0.5], f (x) = 0.8 x ∈ (0.5, 1]. These distributions represent several different scenarios. Distribution 1 is relatively “easy” because the suboptimal arms 4-9’s distribution is far away from arms 1-3’s distribution, whereas distribution 2 is “hard” since the distribution of arms 4-9 is close to the distribution of arms 1-3. In distribution 3, the distribution of arms 4-6 is close to the distribution of arms 1-3’s, while arms 7-9’s distribution is further away. Distribution 4 is an example of a group of continuous distributions for which Lazy-SDCB is more efficient than SDCB. We use SDCB for distributions 1-3, and Lazy-SDCB (with known time horizon) for distribution 4. Figure 1 shows the regrets of both SDCB and the online submodular maximization algorithm. We plot the 1-approximation regrets instead of the (1 − 1/e)-approximation regrets, since the greedy oracle usually performs much better than its (1 − 1/e)-approximation guarantee. We can see from Figure 1 that our algorithms achieve much lower regrets in all examples. 29 1000 SDCB Online Submodular Maximization 900 450 SDCB Online Submodular Maximization 800 400 700 350 600 Regret Regret 300 500 400 250 200 300 150 200 100 100 0 50 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 10000 Time 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 8000 9000 10000 Time (a) Distribution 1 (b) Distribution 2 600 600 Lazy-SDCB Online Submodular Maximization 500 500 400 400 Regret Regret SDCB Online Submodular Maximization 300 300 200 200 100 100 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 10000 Time 0 1000 2000 3000 4000 5000 6000 7000 Time (c) Distribution 3 (d) Distribution 4 Figure 1: Regrets of SDCB/Lazy-SDCB and Algorithm 8 on the K-MAX problem, for distributions 1-4. The regrets are averaged over 20 independent runs. 30	8
Published as a conference paper at ICLR 2018 C ONTINUOUS A DAPTATION VIA M ETA -L EARNING IN N ONSTATIONARY AND C OMPETITIVE E NVIRONMENTS Maruan Al-Shedivat∗ CMU Trapit Bansal UMass Amherst arXiv:1710.03641v2 [cs.LG] 23 Feb 2018 Igor Mordatch OpenAI Yura Burda OpenAI Ilya Sutskever OpenAI Pieter Abbeel UC Berkeley A BSTRACT The ability to continuously learn and adapt from limited experience in nonstationary environments is an important milestone on the path towards general intelligence. In this paper, we cast the problem of continuous adaptation into the learning-to-learn framework. We develop a simple gradient-based meta-learning algorithm suitable for adaptation in dynamically changing and adversarial scenarios. Additionally, we design a new multi-agent competitive environment, RoboSumo, and define iterated adaptation games for testing various aspects of continuous adaptation. We demonstrate that meta-learning enables significantly more efficient adaptation than reactive baselines in the few-shot regime. Our experiments with a population of agents that learn and compete suggest that meta-learners are the fittest. 1 I NTRODUCTION Recent progress in reinforcement learning (RL) has achieved very impressive results ranging from playing games (Mnih et al., 2015; Silver et al., 2016), to applications in dialogue systems (Li et al., 2016), to robotics (Levine et al., 2016). Despite the progress, the learning algorithms for solving many of these tasks are designed to deal with stationary environments. On the other hand, real-world is often nonstationary either due to complexity (Sutton et al., 2007), changes in the dynamics or the objectives in the environment over the life-time of a system (Thrun, 1998), or presence of multiple learning actors (Lowe et al., 2017; Foerster et al., 2017a). Nonstationarity breaks the standard assumptions and requires agents to continuously adapt, both at training and execution time, in order to succeed. Learning under nonstationary conditions is challenging. The classical approaches to dealing with nonstationarity are usually based on context detection (Da Silva et al., 2006) and tracking (Sutton et al., 2007), i.e., reacting to the already happened changes in the environment by continuously fine-tuning the policy. Unfortunately, modern deep RL algorithms, while able to achieve super-human performance on certain tasks, are known to be sample inefficient. Nevertheless, nonstationarity allows only for limited interaction before the properties of the environment change. Thus, it immediately puts learning into the few-shot regime and often renders simple fine-tuning methods impractical. A nonstationary environment can be seen as a sequence of stationary tasks, and hence we propose to tackle it as a multi-task learning problem (Caruana, 1998). The learning-to-learn (or meta-learning) approaches (Schmidhuber, 1987; Thrun & Pratt, 1998) are particularly appealing in the few-shot regime, as they produce flexible learning rules that can generalize from only a handful of examples. Meta-learning has shown promising results in the supervised domain and have gained a lot of attention from the research community recently (e.g., Santoro et al., 2016; Ravi & Larochelle, 2016). In this paper, we develop a gradient-based meta-learning algorithm similar to (Finn et al., 2017b) and suitable for continuous adaptation of RL agents in nonstationary environments. More concretely, our agents meta-learn to anticipate the changes in the environment and update their policies accordingly. While virtually any changes in an environment could induce nonstationarity (e.g., changes in the physics or characteristics of the agent), environments with multiple agents are particularly challenging ∗ Correspondence: maruan.alshedivat.com. Work done while MA and TB interned at OpenAI. 1 Published as a conference paper at ICLR 2018 due to complexity of the emergent behavior and are of practical interest with applications ranging from multiplayer games (Peng et al., 2017) to coordinating self-driving fleets Cao et al. (2013). Multi-agent environments are nonstationary from the perspective of any individual agent since all actors are learning and changing concurrently (Lowe et al., 2017). In this paper, we consider the problem of continuous adaptation to a learning opponent in a competitive multi-agent setting. To this end, we design RoboSumo—a 3D environment with simulated physics that allows pairs of agents to compete against each other. To test continuous adaptation, we introduce iterated adaptation games—a new setting where a trained agent competes against the same opponent for multiple rounds of a repeated game, while both are allowed to update their policies and change their behaviors between the rounds. In such iterated games, from the agent’s perspective, the environment changes from round to round, and the agent ought to adapt in order to win the game. Additionally, the competitive component of the environment makes it not only nonstationary but also adversarial, which provides a natural training curriculum and encourages learning robust strategies (Bansal et al., 2018). We evaluate our meta-learning agents along with a number of baselines on a (single-agent) locomotion task with handcrafted nonstationarity and on iterated adaptation games in RoboSumo. Our results demonstrate that meta-learned strategies clearly dominate other adaptation methods in the few-shot regime in both single- and multi-agent settings. Finally, we carry out a large-scale experiment where we train a diverse population of agents with different morphologies, policy architectures, and adaptation methods, and make them interact by competing against each other in iterated games. We evaluate the agents based on their TrueSkills (Herbrich et al., 2007) in these games, as well as evolve the population as whole for a few generations—the agents that lose disappear, while the winners get duplicated. Our results suggest that the agents with meta-learned adaptation strategies end up being the fittest. Videos that demonstrate adaptation behaviors are available at https://goo.gl/tboqaN. 2 R ELATED W ORK The problem of continuous adaptation considered in this work is a variant of continual learning (Ring, 1994; 1997) and is related to lifelong (Thrun & Pratt, 1998; Silver et al., 2013) and neverending (Mitchell et al., 2015) learning. Life-long learning systems aim at solving multiple tasks sequentially by efficiently transferring and utilizing knowledge from already learned tasks to new tasks while minimizing the effect of catastrophic forgetting (McCloskey & Cohen, 1989). Neverending learning is concerned with mastering a fixed set of tasks in iterations, where the set keeps growing and the performance on all the tasks in the set keeps improving from iteration to iteration. The scope of continuous adaptation is narrower and more precise. While life-long and never-ending learning settings are defined as general multi-task problems (Silver et al., 2013; Mitchell et al., 2015), continuous adaptation targets to solve a single but nonstationary task or environment. The nonstationarity in the former two problems exists and is dictated by the selected sequence of tasks. In the latter case, we assume that nonstationarity is caused by some underlying dynamics in the properties of a given task in the first place (e.g., changes in the behavior of other agents in a multiagent setting). Finally, in the life-long and never-ending scenarios the boundary between training and execution is blurred as such systems constantly operate in the training regime. Continuous adaptation, on the other hand, expects a (potentially trained) agent to adapt to the changes in the environment at execution time under the pressure of limited data or interaction experience between the changes1 . Nonstationarity of multi-agent environments is a well known issue that has been extensively studied in the context of learning in simple multi-player iterated games (such as rock-paper-scissors) where each episode is one-shot interaction (Singh et al., 2000; Bowling, 2005; Conitzer & Sandholm, 2007). In such games, discovering and converging to a Nash equilibrium strategy is a success for the learning agents. Modeling and exploiting opponents (Zhang & Lesser, 2010; Mealing & Shapiro, 2013) or even their learning processes (Foerster et al., 2017b) is advantageous as it improves convergence or helps to discover equilibria of certain properties (e.g., leads to cooperative behavior). In contrast, each episode in RoboSumo consists of multiple steps, happens in continuous time, and requires learning a good intra-episodic controller. Finding Nash equilibria in such a setting is hard. Thus, fast adaptation becomes one of the few viable strategies against changing opponents. 1 The limited interaction aspect of continuous adaptation makes the problem somewhat similar to the recently proposed life-long few-shot learning (Finn et al., 2017a). 2 Published as a conference paper at ICLR 2018 Policy ... τφ τθ ... T φi−1 φi+1 φi τi−1 τi τi+1 Ti−1 Ti Ti+1 Trajectory + + ... deterministic (a) Loss ... Intermediate steps φ θ (b) stochastic gradient (c) Fig. 1: (a) A probabilistic model for MAML in a multi-task RL setting. The task, T , the policies, π, and the trajectories, τ , are all random variables with dependencies encoded in the edges of the given graph. (b) Our extended model suitable for continuous adaptation to a task changing dynamically due to non-stationarity of the environment. Policy and trajectories at a previous step are used to construct a new policy for the current step. (c) Computation graph for the meta-update from φi to φi+1 . Boxes represent replicas of the policy graphs with the specified parameters. The model is optimized via truncated backpropagation through time starting from LTi+1 . Our proposed method for continuous adaptation follows the general meta-learning paradigm (Schmidhuber, 1987; Thrun & Pratt, 1998), i.e., it learns a high-level procedure that can be used to generate a good policy each time the environment changes. There is a wealth of work on meta-learning, including methods for learning update rules for neural models that were explored in the past (Bengio et al., 1990; 1992; Schmidhuber, 1992), and more recent approaches that focused on learning optimizers for deep networks (Hochreiter et al., 2001; Andrychowicz et al., 2016; Li & Malik, 2016; Ravi & Larochelle, 2016), generating model parameters (Ha et al., 2016; Edwards & Storkey, 2016; Al-Shedivat et al., 2017), learning task embeddings (Vinyals et al., 2016; Snell et al., 2017) including memory-based approaches (Santoro et al., 2016), learning to learn implicitly via RL (Wang et al., 2016; Duan et al., 2016), or simply learning a good initialization (Finn et al., 2017b). 3 M ETHOD The problem of continuous adaptation in nonstationary environments immediately puts learning into the few-shot regime: the agent must learn from only limited amount of experience that it can collect before its environment changes. Therefore, we build our method upon the previous work on gradient-based model-agnostic meta-learning (MAML) that has been shown successful in the fewshot settings (Finn et al., 2017b). In this section, we re-derive MAML for multi-task reinforcement learning from a probabilistic perspective (cf. Grant et al., 2018), and then extend it to dynamically changing tasks. 3.1 A PROBABILISTIC VIEW OF MODEL - AGNOSTIC META - LEARNING (MAML) Assume that we are given a distribution over tasks, D(T ), where each task, T , is a tuple: T := (LT , PT (x) , PT (xt+1 \| xt , at ) , H) (1) LT is a task-specific loss function that maps a trajectory, τ := (x0 , a1 , x1 , R1 , . . . , aH , xH , RH ) ∈ T , to a loss value, i.e., LT : T 7→ R; PT (x) and PT (xt+1 \| xt , at ) define the Markovian dynamics of the environment in task T ; H denotes the horizon; observations, xt , and actions, at , are elements (typically, vectors) of the observation space, X , and action space, A, respectively. The loss of a PH trajectory, τ , is the negative cumulative reward, LT (τ ) := − t=1 Rt . The goal of meta-learning is to find a procedure which, given access to a limited experience on a task sampled from D(T ), can produce a good policy for solving it. More formally, after querying K trajectories from a task T ∼ D(T ) under policy πθ , denoted τθ1:K , we would like to construct a new, task-specific policy, πφ , that would minimize the expected subsequent loss on the task T . In particular, MAML constructs parameters of the task-specific policy, φ, using gradient of LT w.r.t. θ: K 1 X φ := θ − α∇θ LT τθ1:K , where LT τθ1:K := LT (τθk ), and τθk ∼ PT (τ \| θ) K k=1 3 (2) Published as a conference paper at ICLR 2018 Algorithm 1 Meta-learning at training time. Algorithm 2 Adaptation at execution time. input Distribution over pairs of tasks, P(Ti , Ti+1 ), learning rate, β. 1: Randomly initialize θ and α. 2: repeat 3: Sample a batch of task pairs, {(Ti , Ti+1 )}n i=1 . 4: for all task pairs (Ti , Ti+1 ) in the batch do 5: Sample traj. τθ1:K from Ti using πθ . Compute φ = φ(τθ1:K , θ, α) as given in (7). 6: 7: Sample traj. τφ from Ti+1 using πφ . 8: end for 9: Compute ∇θ LTi ,Ti+1 and ∇α LTi ,Ti+1 using τθ1:K and τφ as given in (8). 10: Update θ ← θ + β∇θ LT (θ, α). 11: Update α ← α + β∇α LT (θ, α). 12: until Convergence output Optimal θ∗ and α∗ . input A stream of tasks, T1 , T2 , T3 , . . . . 1: Initialize φ = θ. 2: while there are new incoming tasks do 3: Get a new task, Ti , from the stream. 4: Solve Ti using πφ policy. 1:K 5: While solving Ti , collect trajectories, τi,φ . 1:K ∗ ∗ 6: Update φ ← φ(τi,φ , θ , α ) using importance-corrected meta-update as in (9). 7: end while Policy parameter space We call (2) the adaptation update with a step α. The adaptation update is parametrized by θ, which we optimize by minimizing the expected loss over the distribution of tasks, D(T )—the meta-loss: min ET ∼D(T ) [LT (θ)] , where LT (θ) := Eτθ1:K ∼PT (τ \|θ) Eτφ ∼PT (τ \|φ) LT (τφ ) \| τθ1:K , θ (3) θ where τθ and τφ are trajectories obtained under πθ and πφ , respectively. In general, we can think of the task, trajectories, and policies, as random variables (Fig. 1a), where φ is generated from some conditional distribution PT (φ\| θ, τ1:k ). The meta-update(2) is equivalent to PK 1 2 assuming the delta distribution, PT (φ \| θ, τ1:k ) := δ θ − α∇θ K k=1 LT (τk ) . To optimize (3), we can use the policy gradient method (Williams, 1992), where the gradient of LT is as follows: " " ## K X k ∇θ LT (θ) = Eτθ1:K ∼PT (τ \|θ) LT (τφ ) ∇θ log πφ (τφ ) + ∇θ log πθ (τθ ) (4) τφ ∼PT (τ \|φ) k=1 The expected loss on a task, LT , can be optimized with trust-region policy (TRPO) (Schulman et al., 2015a) or proximal policy (PPO) (Schulman et al., 2017) optimization methods. For details and derivations please refer to Appendix A. 3.2 C ONTINUOUS ADAPTATION VIA META - LEARNING In the classical multi-task setting, we make no assumptions about the distribution of tasks, D(T ). When the environment is nonstationary, we can see it as a sequence of stationary tasks on a certain timescale where the tasks correspond to different dynamics of the environment. Then, D(T ) is defined by the environment changes, and the tasks become sequentially dependent. Hence, we would like to exploit this dependence between consecutive tasks and meta-learn a rule that keeps updating the policy in a way that minimizes the total expected loss encountered during the interaction with the changing environment. For instance, in the multi-agent setting, when playing against an opponent that changes its strategy incrementally (e.g., due to learning), our agent should ideally meta-learn to anticipate the changes and update its policy accordingly. In the probabilistic language, our nonstationary environment is equivalent to a distribution of tasks represented by a Markov chain (Fig. 1b). The goal is to minimize the expected loss over the chain of tasks of some length L: " L # X min EP(T0 ),P(Ti+1 \|Ti ) LTi ,Ti+1 (θ) (5) θ 2 i=1 Grant et al. (2018) similarly reinterpret adaptation updates (in non-RL settings) as Bayesian inference. 4 Published as a conference paper at ICLR 2018 Here, P(T0 ) and P(Ti+1 \| Ti ) denote the initial and the transition probabilities in the Markov chain of tasks. Note that (i) we deal with Markovian dynamics on two levels of hierarchy, where the upper level is the dynamics of the tasks and the lower level is the MDPs that represent particular tasks, and (ii) the objectives, LTi ,Ti+1 , will depend on the way the meta-learning process is defined. Since we are interested in adaptation updates that are optimal with respect to the Markovian transitions between the tasks, we define the meta-loss on a pair of consecutive tasks as follows: h i 1:K 1:K ∼P LTi ,Ti+1 (θ) := Eτi,θ Eτi+1,φ ∼PTi+1 (τ \|φ) LTi+1 (τi+1,φ ) \| τi,θ ,θ (6) Ti (τ \|θ) 1:K The principal difference between the loss in (3) and (6) is that trajectories τi,θ come from the current task, Ti , and are used to construct a policy, πφ , that is good for the upcoming task, Ti+1 . Note that even though the policy parameters, φi , are sequentially dependent (Fig. 1b), in (6) we always start from the initial parameters, θ 3 . Hence, optimizing LTi ,Ti+1 (θ) is equivalent to truncated backpropagation through time with a unit lag in the chain of tasks. To construct parameters of the policy for task Ti+1 , we start from θ and do multiple4 meta-gradient steps with adaptive step sizes as follows (assuming the number of steps is M ): φ0i := θ, τθ1:K ∼ PTi (τ \| θ), m−1 1:K m−1 LT φm := φ − α ∇ τ , m = 1, . . . , M − 1, m−1 m i i i (7) φi i,φ i −1 1:K φi+1 := φM − αM ∇φM −1 LTi τi,φ M −1 i i i where {αm }M m=1 is a set of meta-gradient step sizes that are optimized jointly with θ. The computation graph for the meta-update is given in Fig. 1c. The expression for the policy gradient is the same as in (4) but with the expectation is now taken w.r.t. to both Ti and Ti+1 : ∇θ,α LTi ,Ti+1 (θ, α) = " " ## K X (8) k 1:K E τi,θ LTi+1 (τi+1,φ ) ∇θ,α log πφ (τi+1,φ ) + ∇θ log πθ (τi,θ ) ∼PT (τ \|θ) i τi+1,φ ∼PTi+1 (τ \|φ) k=1 More details and the analog of the policy gradient theorem for our setting are given in Appendix A. Note that computing adaptation updates requires interacting with the environment under πθ while computing the meta-loss, LTi ,Ti+1 , requires using πφ , and hence, interacting with each task in the sequence twice. This is often impossible at execution time, and hence we use slightly different algorithms at training and execution times. Meta-learning at training time. Once we have access to a distribution over pairs of consecutive tasks5 , P(Ti−1 , Ti ), we can meta-learn the adaptation updates by optimizing θ and α jointly with a gradient method, as given in Algorithm 1. We use πθ to collect trajectories from Ti and πφ when interacting with Ti+1 . Intuitively, the algorithm is searching for θ and α such that the adaptation update (7) computed on the trajectories from Ti brings us to a policy, πφ , that is good for solving Ti+1 . The main assumption here is that the trajectories from Ti contain some information about Ti+1 . Note that we treat adaptation steps as part of the computation graph (Fig. 1c) and optimize θ and α via backpropagation through the entire graph, which requires computing second order derivatives. Adaptation at execution time. Note that to compute unbiased adaptation gradients at training time, we have to collect experience in Ti using πθ . At test time, due to environment nonstationarity, we usually do not have the luxury to access to the same task multiple times. Thus, we keep acting according to πφ and re-use past experience to compute updates of φ for each new incoming task (see Algorithm 2). To adjust for the fact that the past experience was collected under a policy different from πθ , we use importance weight correction. In case of single step meta-update, we have: K 1 X πθ (τ k ) φi := θ − α ∇θ LTi−1 (τ k ), τ 1:K ∼ PTi−1 (τ \| φi−1 ), (9) K πφi−1 (τ k ) k=1 3 This is due to stability considerations. We find empirically that optimization over sequential updates from φi to φi+1 is unstable, often tends to diverge, while starting from the same initialization leads to better behavior. 4 Empirically, it turns out that constructing φ via multiple meta-gradient steps (between 2 and 5) with adaptive step sizes tends yield better results in practice. 5 Given a sequences of tasks generated by a nonstationary environment, T1 , T2 , T3 , . . . , TL , we use the set of all pairs of consecutive tasks, {(Ti−1 , Ti )}L i=1 , as the training distribution. 5 Published as a conference paper at ICLR 2018 (a) (b) (c) Fig. 2: (a) The three types of agents used in experiments. The robots differ in the anatomy: the number of legs, their positions, and constraints on the thigh and knee joints. (b) The nonstationary locomotion environment. The torques applied to red-colored legs are scaled by a dynamically changing factor. (c) RoboSumo environment. where πφi−1 and πφi are used to rollout from Ti−1 and Ti , respectively. Extending importance weight correction to multi-step updates is straightforward and requires simply adding importance weights to each of the intermediate steps in (7). 4 E NVIRONMENTS We have designed a set of environments for testing different aspects of continuous adaptation methods in two scenarios: (i) simple environments that change from episode to episode according to some underlying dynamics, and (ii) a competitive multi-agent environment, RoboSumo, that allows different agents to play sequences of games against each other and keep adapting to incremental changes in each other’s policies. All our environments are based on MuJoCo physics simulator (Todorov et al., 2012), and all agents are simple multi-leg robots, as shown in Fig. 2a. 4.1 DYNAMIC First, we consider the problem of robotic locomotion in a changing environment. We use a six-leg agent (Fig. 2b) that observes the absolute position and velocity of its body, the angles and velocities of its legs, and it acts by applying torques to its joints. The agent is rewarded proportionally to its moving speed in a fixed direction. To induce nonstationarity, we select a pair of legs of the agent and scale down the torques applied to the corresponding joints by a factor that linearly changes from 1 to 0 over the course of 7 episodes. In other words, during the first episode all legs are fully functional, while during the last episode the agent has two legs fully paralyzed (even though the policy can generate torques, they are multiplied by 0 before being passed to the environment). The goal of the agent is to learn to adapt from episode to episode by changing its gait so that it is able to move with a maximal speed in a given direction despite the changes in the environment (cf. Cully et al., 2015). Also, there are 15 ways to select a pair of legs of a six-leg creature which gives us 15 different nonstationary environments. This allows us to use a subset of these environments for training and a separate held out set for testing. The training and testing procedures are described in the next section. 4.2 C OMPETITIVE Our multi-agent environment, RoboSumo, allows agents to compete in the 1-vs-1 regime following the standard sumo rules6 . We introduce three types of agents, Ant, Bug, and Spider, with different anatomies (Fig. 2a). During the game, each agent observes positions of itself and the opponent, its own joint angles, the corresponding velocities, and the forces exerted on its own body (i.e., equivalent of tactile senses). The action spaces are continuous. Iterated adaptation games. To test adaptation, we define the iterated adaptation game (Fig. 3)—a game between a pair of agents that consists of K rounds each of which consists of one or more fixed length episodes (500 time steps each). The outcome of each round is either win, loss, or draw. The agent that wins the majority of rounds (with at least 5% margin) is declared the winner of the game. There are two distinguishing aspects of our setup: First, the agents are trained either via pure self-play or versus opponents from a fixed training collection. At test time, they face a new opponent from a testing collection. Second, the agents are allowed to learn (or adapt) at test time. In particular, an 6 To win, the agent has to push the opponent out of the ring or make the opponent’s body touch the ground. 6 Published as a conference paper at ICLR 2018 Round 1 Round 2 Round 3 Round K version 1 version 2 version 3 version K Agent: Episodes: Opponent: Fig. 3: An agent competes with an opponent in an iterated adaptation games that consist of multi-episode rounds. The agent wins a round if it wins the majority of episodes (wins and losses illustrated with color). Both the agent and its opponent may update their policies from round to round (denoted by the version number). agent should exploit the fact that it plays against the same opponent multiple consecutive rounds and try to adjust its behavior accordingly. Since the opponent may also be adapting, the setup allows to test different continuous adaptation strategies, one versus the other. Reward shaping. In RoboSumo, rewards are naturally sparse: the winner gets +2000, the loser is penalized for -2000, and in case of a draw both opponents receive -1000 points. To encourage fast learning at the early stages of training, we shape the rewards given to agents in the following way: the agent (i) gets reward for staying closer to the center of the ring, for moving towards the opponent, and for exerting forces on the opponent’s body, and (ii) gets penalty inversely proportional to the opponent’s distance to the center of the ring. At test time, the agents continue having access to the shaped reward as well and may use it to update their policies. Throughout our experiments, we use discounted rewards with the discount factor, γ = 0.995. More details are in Appendix D.2. Calibration. To study adaptation, we need a well-calibrated environment in which none of the agents has an initial advantage. To ensure the balance, we increased the mass of the weaker agents (Ant and Spider) such that the win rates in games between one agent type versus the other type in the non-adaptation regime became almost equal (for details on calibration see Appendix D.3). 5 E XPERIMENTS Our goal is to test different adaptation strategies in the proposed nonstationary RL settings. However, it is known that the test-time behavior of an agent may highly depend on a variety of factors besides the chosen adaptation method, including training curriculum, training algorithm, policy class, etc. Hence, we first describe the precise setup that we use in our experiments to eliminate irrelevant factors and focus on the effects of adaptation. Most of the low-level details are deferred to appendices. Video highlights of our experiments are available at https://goo.gl/tboqaN. 5.1 T HE SETUP Policies. We consider 3 types of policy networks: (i) a 2-layer MLP, (ii) embedding (i.e., 1 fully-connected layer replicated across the time dimension) followed by a 1-layer LSTM, and (iii) RL2 (Duan et al., 2016) of the same architecture as (ii) which additionally takes previous reward and done signals as inputs at each step, keeps the recurrent state throughout the entire interaction with a given environment (or an opponent), and resets the state once the latter changes. For advantage functions, we use networks of the same structure as for the corresponding policies and have no parameter sharing between the two. Our meta-learning agents use the same policy and advantage function structures as the baselines and learn a 3-step meta-update with adaptive step sizes as given in (7). Illustrations and details on the architectures are given in Appendix B. Meta-learning. We compute meta-updates via gradients of the negative discounted rewards received during a number of previous interactions with the environment. At training time, meta-learners interact with the environment twice, first using the initial policy, πθ , and then the meta-updated policy, πφ . At test time, the agents are limited to interacting with the environment only once, and hence always act according to πφ and compute meta-updates using importance-weight correction (see Sec. 3.2 and Algorithm 2). Additionally, to reduce the variance of the meta-updates at test time, the agents store the experience collected during the interaction with the test environment (and the corresponding importance weights) into the experience buffer and keep re-using that experience 7 Published as a conference paper at ICLR 2018 Total episodic reward Back two legs Middle two legs Front two legs Policy + adaptation method 1000 1000 1000 500 500 0 500 0 0 1 3 5 7 MLP MLP + PPO-tracking MLP + meta-updates LSTM LSTM + PPO-tracking LSTM + meta-updates RL2 1 3 5 7 Consecutive episodes 1 3 5 7 Fig. 4: Episodic rewards for 7 consecutive episodes in 3 held out nonstationary locomotion environments. To evaluate adaptation strategies, we ran each of them in each environment for 7 episodes followed by a full reset of the environment, policy, and meta-updates (repeated 50 times). Shaded regions are 95% confidence intervals. Best viewed in color. to update πφ as in (7). The size of the experience buffer is fixed to 3 episodes for nonstationary locomotion and 75 episodes for RoboSumo. More details are given in Appendix C.1. Adaptation baselines. We consider the following three baseline strategies: (i) naive (or no adaptation), (ii) implicit adaptation via RL2 , and (iii) adaptation via tracking (Sutton et al., 2007) that keeps doing PPO updates at execution time. Training in nonstationary locomotion. We train all methods on the same collection of nonstationary locomotion environments constructed by choosing all possible pairs of legs whose joint torques are scaled except 3 pairs that are held out for testing (i.e., 12 training and 3 testing environments for the six-leg creature). The agents are trained on the environments concurrently, i.e., to compute a policy update, we rollout from all environments in parallel and then compute, aggregate, and average the gradients (for details, see Appendix C.2). LSTM policies retain their state over the course of 7 episodes in each environment. Meta-learning agents compute meta-updates for each nonstationary environment separately. Training in RoboSumo. To ensure consistency of the training curriculum for all agents, we first pre-train a number of policies of each type for every agent type via pure self-play with the PPO algorithm (Schulman et al., 2017; Bansal et al., 2018). We snapshot and save versions of the pretrained policies at each iteration. This lets us train other agents to play against versions of the pre-trained opponents at various stages of mastery. Next, we train the baselines and the meta-learning agents against the pool of pre-trained opponents7 concurrently. At each iteration k we (a) randomly select an opponent from the training pool, (b) sample a version of the opponent’s policy to be in [1, k] (this ensures that even when the opponent is strong, sometimes an undertrained version is selected which allows the agent learn to win at early stages), and (c) rollout against that opponent. All baseline policies are trained with PPO; meta-learners also used PPO as the outer loop for optimizing θ and α parameters. We retain the states of the LSTM policies over the course of interaction with the same version of the same opponent and reset it each time the opponent version is updated. Similarly to the locomotion setup, meta-learners compute meta-updates for each opponent in the training pool separately. A more detailed description of the distributed training is given in Appendix C.2. Experimental design. We design our experiments to answer the following questions: • When the interaction with the environment before it changes is strictly limited to one or very few episodes, what is the behavior of different adaptation methods in nonstationary locomotion and competitive multi-agent environments? • What is the sample complexity of different methods, i.e., how many episodes is required for a method to successfully adapt to the changes? We test this by controlling the amount of experience the agent is allowed to get form the same environment before it changes. 7 In competitive multi-agent environments, besides self-play, there are plenty of ways to train agents, e.g., train them in pairs against each other concurrently, or randomly match and switch opponents each few iterations. We found that concurrent training often leads to an unbalanced population of agents that have been trained under vastly different curricula and introduces spurious effects that interfere with our analysis of adaptation. Hence, we leave the study of adaptation in naturally emerging curricula in multi-agent settings to the future work. 8 Published as a conference paper at ICLR 2018 Agent: Opponent: 0.6 Win rate RL2 Ant Ant LSTM + PPO-tracking Opponent: 0.6 Bug 0.2 0.2 0 25 Agent: 0.8 Win rate Spider 0.4 0.2 50 75 100 0.0 0 25 50 75 Consecutive rounds RL2 Bug Opponent: 0.8 Ant 100 0 LSTM + PPO-tracking Opponent: 0.6 0.6 0.4 0.4 0.4 0.2 0 25 Agent: 50 75 25 50 75 Consecutive rounds RL2 Ant Bug 0.2 0.2 0.2 75 100 0 25 50 75 Consecutive rounds 25 100 Spider 100 50 75 100 LSTM + meta-updates Opponent: 0.6 0.4 50 Opponent: 0 0.4 25 75 LSTM + meta-updates LSTM + PPO-tracking Opponent: 0.6 100 0.4 0 50 0.2 0 Spider Opponent: 0.6 100 25 0.8 Bug 0.6 0.2 Win rate Opponent: 0.6 0.4 0.4 LSTM + meta-updates 0 25 50 Spider 75 100 Fig. 5: Win rates for different adaptation strategies in iterated games versus 3 different pre-trained opponents. At test time, both agents and opponents started from versions 700. Opponents’ versions were increasing with each consecutive round as if they were learning via self-play, while agents were allowed to adapt only from the limited experience with a given opponent. Each round consisted of 3 episodes. Each iterated game was repeated 100 times; shaded regions denote bootstrapped 95% confidence intervals; no smoothing. Best viewed in color. Additionally, we ask the following questions specific to the competitive multi-agent setting: • Given a diverse population of agents that have been trained under the same curriculum, how do different adaptation methods rank in a competition versus each other? • When the population of agents is evolved for several generations—such that the agents interact with each other via iterated adaptation games, and those that lose disappear while the winners get duplicated—what happens with the proportions of different agents in the population? 5.2 A DAPTATION IN THE FEW- SHOT REGIME AND SAMPLE COMPLEXITY Few-shot adaptation in nonstationary locomotion environments. Having trained baselines and meta-learning policies as described in Sec. 5.1, we selected 3 testing environments that corresponded to disabling 3 different pairs of legs of the six-leg agent: back, middle, and front legs. The results are presented on Fig. 4. Three observations: First, during the very first episode, the meta-learned initial policy, πθ? , turns out to be suboptimal for the task (it underperforms compared to other policies). However, after 1-2 episodes (and environment changes), it starts performing on par with other policies. Second, by the 6th and 7th episodes, meta-updated policies perform much better than the rest. Note that we use 3 gradient meta-updates for the adaptation of the meta-learners; the meta-updates are computed based on experience collected during the previous 2 episodes. Finally, tracking is not able to improve upon the baseline without adaptation and sometimes leads to even worse results. Adaptation in RoboSumo under the few-shot constraint. To evaluate different adaptation methods in the competitive multi-agent setting consistently, we consider a variation of the iterated adaptation game, where changes in the opponent’s policies at test time are pre-determined but unknown to the 9 Published as a conference paper at ICLR 2018 Ant vs Ant Win rate 0.6 Bug vs Bug 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0 25 50 75 0.2 0 Spider vs Spider 0.6 25 50 75 Episodes per round 0.2 0 25 50 RL2 LSTM + PPO-tracking LSTM + meta-updates No adaptation 75 Fig. 6: The effect of increased number of episodes per round in the iterated games versus a learning opponent. agents. In particular, we pre-train 3 opponents (1 of each type, Fig. 2a) with LSTM policies with PPO via self-play (the same way as we pre-train the training pool of opponents, see Sec. 5.1) and snapshot their policies at each iteration. Next, we run iterated games between our trained agents that use different adaptation algorithms versus policy snapshots of the pre-trained opponents. Crucially, the policy version of the opponent keeps increasing from round to round as if it was training via self-play8 . The agents have to keep adapting to increasingly more competent versions of the opponent (see Fig. 3). This setup allows us to test different adaptation strategies consistently against the same learning opponents. The results are given on Fig. 5. We note that meta-learned adaptation strategies, in most cases, are able to adapt and improve their win-rates within about 100 episodes of interaction with constantly improving opponents. On the other hand, performance of the baselines often deteriorates during the rounds of iterated games. Note that the pre-trained opponents were observing 90 episodes of self-play per iteration, while the agents have access to only 3 episodes per round. Sample complexity of adaptation in RoboSumo. Meta-learning helps to find an update suitable for fast or few-shot adaptation. However, how do different adaptation methods behave when more experience is available? To answer this question, we employ the same setup as previously and vary the number of episodes per round in the iterated game from 3 to 90. Each iterated game is repeated 20 times, and we measure the win-rates during the last 25 rounds of the game. The results are presented on Fig. 6. When the number of episodes per round goes above 50, adaptation via tracking technically turns into “learning at test time,” and it is able to learn to compete against the self-trained opponents that it has never seen at training time. The meta-learned adaptation strategy performed near constantly the same in both few-shot and standard regimes. This suggests that the meta-learned strategy acquires a particular bias at training time that allows it to perform better from limited experience but also limits its capacity of utilizing more data. Note that, by design, the meta-updates are fixed to only 3 gradient steps from θ? with step-sizes α? (learned at training), while tracking keeps updating the policy with PPO throughout the iterated game. Allowing for meta-updates that become more flexible with the availability of data can help to overcome this limitation. We leave this to future work. 5.3 E VALUATION ON THE POPULATION - LEVEL Combining different adaptation strategies with different policies and agents of different morphologies puts us in a situation where we have a diverse population of agents which we would like to rank according to the level of their mastery in adaptation (or find the “fittest”). To do so, we employ TrueSkill (Herbrich et al., 2007)—a metric similar to the ELO rating, but more popular in 1-vs-1 competitive video-games. In this experiment, we consider a population of 105 trained agents: 3 agent types, 7 different policy and adaptation combinations, and 5 different stages of training (from 500 to 2000 training iterations). First, we assume that the initial distribution of any agent’s skill is N (25, 25/3) and the default distance that guarantees about 76% of winning, β = 4.1667. Next, we randomly generate 1000 matches between pairs of opponents and let them adapt while competing with each other in 100-round iterated adaptation games (states of the agents are reset before each game). After each game, we 8 At the beginning of the iterated game, both agents and their opponent start from version 700, i.e., from the policy obtained after 700 iterations (PPO epochs) of learning to ensure that the initial policy is reasonable. 10 Published as a conference paper at ICLR 2018 record the outcome and updated our belief about the skill of the corresponding agents using the TrueSkill algorithm9 . The distributions of the skill for the agents of each type after 1000 iterated adaptation games between randomly selected players from the pool are visualized in Fig. 7. TrueSkill 40 Sub-population: Ants Sub-population: 40 Bugs 30 30 30 20 20 20 MLP LSTM MLP no adaptation Sub-population: 40 LSTM PPO-tracking MLP meta-updates RL2 Spiders LSTM Fig. 7: TrueSkill for the top-performing MLP- and LSTM-based agents. TrueSkill was computed based on outcomes (win, loss, or draw) in 1000 iterated adaptation games (100 consecutive rounds per game, 3 episodes per round) between randomly selected pairs of opponents from a population of 105 pre-trained agents. There are a few observations we can make: First, recurrent policies were dominant. Second, adaptation via RL2 tended to perform equally or a little worse than plain LSTM with or without tracking in this setup. Finally, agents that meta-learned adaptation rules at training time, consistently demonstrated higher skill scores in each of the categories corresponding to different policies and agent types. Proportion (%) Finally, we enlarge the population from 105 to 1050 agents by duplicating each of them 10 times and evolve it (in the “natural selection” sense) for several generations as follows. Initially, we start with a balanced population of different creatures. Next, we randomly match 1000 pairs of agents, make them play iterated adaptation games, remove the agents that lost from the population and duplicate the winners. The same process is repeated 10 times. The result is presented in Fig 8. We see that many agents quickly disappear form initially uniform population and the meta-learners end up dominating. Policy + adaptation 100 75 50 25 0 MLP MLP + PPO-tracking MLP + meta-updates LSTM LSTM + PPO-tracking LSTM + meta-updates RL2 Spiders Bugs Ants 0 1 2 3 4 5 6 Creature generation (#) 7 8 9 10 Fig. 8: Evolution of a population of 1050 agents for 10 generations. Best viewed in color. 6 C ONCLUSION AND F UTURE D IRECTIONS In this work, we proposed a simple gradient-based meta-learning approach suitable for continuous adaptation in nonstationary environments. The key idea of the method is to regard nonstationarity as a sequence of stationary tasks and train agents to exploit the dependencies between consecutive tasks such that they can handle similar nonstationarities at execution time. We applied our method to nonstationary locomotion and within a competitive multi-agent setting. For the latter, we designed the RoboSumo environment and defined iterated adaptation games that allowed us to test various aspects of adaptation strategies. In both cases, meta-learned adaptation rules were more efficient than the baselines in the few-shot regime. Additionally, agents that meta-learned to adapt demonstrated the highest level of skill when competing in iterated games against each other. The problem of continuous adaptation in nonstationary and competitive environments is far from being solved, and this work is the first attempt to use meta-learning in such setup. Indeed, our meta-learning algorithm has a few limiting assumptions and design choices that we have made mainly due to computational considerations. First, our meta-learning rule is to one-step-ahead update of the 9 We used an implementation from http://trueskill.org/. 11 Published as a conference paper at ICLR 2018 policy and is computationally similar to backpropagation through time with a unit time lag. This could potentially be extended to fully recurrent meta-updates that take into account the full history of interaction with the changing environment. Additionally, our meta-updates were based on the gradients of a surrogate loss function. While such updates explicitly optimized the loss, they required computing second order derivatives at training time, slowing down the training process by an order of magnitude compared to baselines. Utilizing information provided by the loss but avoiding explicit backpropagation through the gradients would be more appealing and scalable. 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Machine learning, 8(3-4):229–256, 1992. Chongjie Zhang and Victor R Lesser. Multi-agent learning with policy prediction. In AAAI, 2010. 14 Published as a conference paper at ICLR 2018 A D ERIVATIONS AND THE POLICY GRADIENT THEOREM In this section, we derive the policy gradient update for MAML as give in (4) as well as formulate and equivalent of the policy gradient theorem (Sutton et al., 2000) in the learning-to-learn setting. Our derivation is not bound to a particular form of the adaptation update. In general, we are interested in meta-learning a procedure, fθ , parametrized by θ, which, given access to a limited experience on a task, can produce a good policy for solving it. Note that fθ is responsible for both collecting the initial experience and constructing the final policy for the given task. For example, in case of MAML (Finn et al., 2017b), fθ is represented by the initial policy, πθ , and the adaptation update rule (4) that produces πφ with φ := θ − α∇θ LT (τθ1:K ). More formally, after querying K trajectories, τθ1:K , we want to produce πφ that minimizes the expected loss w.r.t. the distribution over tasks: h i L(θ) := ET ∼D(T ) Eτθ1:K ∼pT (τ \|θ) Eτφ ∼pT (τ \|φ) LT (τφ ) \| τθ1:K (10) Note that the inner-most expectation is conditional on the experience, τθ1:K , which our meta-learning procedure, fθ , collects to produce a task-specific policy, πφ . Assuming that the loss LT (τθ1:K ) is linear in trajectories, and using linearity of expectations, we can drop the superscript 1 : K and denote the trajectory sampled under φθ for task Ti simply as τθ,i . At training time, we are given a finite sample of tasks from the distribution D(T ) and can search for θ̂ close to optimal by optimizing over the empirical distribution: θ̂ := argmin L̂(θ), where L̂(θ) := θ N h i 1 X Eτθ,i ∼pTi (τ \|θ) Eτφ,i ∼pTi (τ \|φ) [LTi (τφ,i ) \| τθ,i ] (11) N i=1 We re-write the objective function for task Ti in (11) more explicitly by expanding the expectations: h i LTi (θ) := Eτθ,i ∼pTi (τ \|θ) Eτφ,i ∼pTi (τ \|φ) [LTi (τφ,i ) \| τθ,i ] = Z (12) LTi (τφ,i ) PTi (τφ,i \| φ) PTi (φ \| θ, τθ,i ) PTi (τθ,i \| θ) dτφ,i dφ dτθ,i Trajectories, τφ,i and τθ,i , and parameters φ of the policy πφ can be thought as random variables that we marginalize out to construct the objective that depends on θ only. The adaptation update rule (4) assumes the following PTi (φ \| θ, τθ,i ): ! K 1 X k LTi (τθ,i ) (13) PTi (φ \| θ, τθ,i ) := δ θ − α∇θ K k=1 Note that by specifying PTi (φ \| θ, τθ,i ) differently, we may arrive at different meta-learning algorithms. After plugging (13) into (12) and integrating out φ, we get the following expected loss for task Ti as a function of θ: h i LTi (θ) = Eτθ,i ∼pTi (τ \|θ) Eτφ,i ∼pTi (τ \|φ) [LTi (τφ,i ) \| τθ,i ] = ! Z K (14) 1 X k LTi (τφ,i ) PTi τφ,i \| θ − α∇θ LTi (τθ,i ) PTi (τθ,i \| θ) dτφ,i dτθ,i K k=1 The gradient of (14) will take the following form: Z ∇θ LTi (θ) = [LTi (τφ,i ) ∇θ log PTi (τφ,i \| φ)] PTi (τφ,i \| φ) PTi (τθ,i \| θ) dτ dτθ,i + Z [LTi (τ ) ∇θ log PTi (τθ,i \| θ)] PTi (τ \| φ) PTi (τθ,i \| θ) dτ dτθ,i (15) 1:K where φ = φ(θ, τθ,i ) as given in (14). Note that the expression consists of two terms: the first term is the standard policy gradient w.r.t. the updated policy, πφ , while the second one is the policy 1:K gradient w.r.t. the original policy, πφ , that is used to collect τθ,i . If we were to omit marginalization 15 Published as a conference paper at ICLR 2018 1:K of τθ,i (as it was done in the original paper (Finn et al., 2017b)), the terms would disappear. Finally, the gradient can be re-written in a more succinct form: " " ## K X 1:K ∇θ LTi (θ) = Eτθ,i log πθ (τk ) (16) ∼PT (τ \|θ) LTi (τ ) ∇θ log πφ (τ ) + ∇θ i τ ∼PTi (τ \|φ) k=1 The update given in (16) is an unbiased estimate of the gradient as long as the loss LTi is simply the sum of discounted rewards (i.e., it extends the classical REINFORCE algorithm (Williams, 1992) to meta-learning). Similarly, we can define LTi that uses a value or advantage function and extend the policy gradient theorem Sutton et al. (2000) to make it suitable for meta-learning. Theorem 1 (Meta policy gradient theorem). For any MDP, gradient of the value function w.r.t. θ takes the following form: ∇θ VTθ (x0 ) = " # X φ X ∂πφ (a \| x) φ Eτ1:K ∼pT (τ \|θ) dT (x) QT (a, x) + ∂θ (17) x a " ! # K X ∂ X log πθ (τk ) πφ (a \| x0 )QφT (a, x0 ) , Eτ1:K ∼pT (τ \|θ) ∂θ a k=1 where dφT (x) is the stationary distribution under policy πφ . Proof. We define task-specific value functions under the generated policy, πφ , as follows: "H # X φ t VT (x0 ) = Eτ ∼pT (τ \|φ) γ RT (xt ) \| x0 , t=k " QφT (x0 , a0 ) = Eτ ∼pT (τ \|φ) H X t=k # (18) t γ RT (xt ) \| x0 , a0 , where the expectations are taken w.r.t. the dynamics of the environment of the given task, T , and the policy, πφ . Next, we need to marginalize out τ1:K : ## " "H X t θ VT (x0 ) = Eτ1:K ∼pT (τ \|θ) Eτ ∼pT (τ \|φ) γ RT (xt ) \| x0 , (19) t=k and after the gradient w.r.t. θ, we arrive at: ∇θ VTθ (x0 ) = # " X ∂πφ (a \| x0 ) φ ∂QφT (a, x0 ) Eτ1:K ∼pT (τ \|θ) QT (a, x0 ) + πφ (a \| x0 ) + ∂θ ∂θ a " K ! # X X ∂ φ Eτ1:K ∼pT (τ \|θ) log πθ (τk ) πφ (a \| x0 )QT (a, x0 ) , ∂θ a (20) k=1 where the first term is similar to the expression used in the original policy gradient theorem (Sutton et al., 2000) while the second one comes from differentiating trajectories τ1:K that depend on θ. Following Sutton et al. (2000), we unroll the derivative of the Q-function in the first term and arrive at the following final expression for the policy gradient: ∇θ VTθ (x0 ) = " # X φ X ∂πφ (a \| x) φ Eτ1:K ∼pT (τ \|θ) dT (x) QT (a, x) + ∂θ (21) x a " ! # K X ∂ X Eτ1:K ∼pT (τ \|θ) log πθ (τk ) πφ (a \| x0 )QφT (a, x0 ) ∂θ a k=1 Remark 1. The same theorem is applicable to the continuous setting with the only changes in the distributions used to compute expectations in (17) and (18). In particular, the outer expectation in (17) should be taken w.r.t. pTi (τ \| θ) while the inner expectation w.r.t. pTi+1 (τ \| φ). 16 Published as a conference paper at ICLR 2018 A.1 M ULTIPLE ADAPTATION GRADIENT STEPS All our derivations so far assumed single step gradient-based adaptation update. Experimentally, we found that the multi-step version of the update often leads to a more stable training and better test time performance. In particular, we construct φ via intermediate M gradient steps: φ0 := θ, τθ1:K ∼ PT (τ \| θ), φm := φm−1 − αm ∇φm−1 LT τφ1:K , m−1 φ := φM −1 − αM ∇φM −1 LT τφ1:K M −1 m = 1, . . . , M − 1, (22) where φm are intermediate policy parameters. Note that each intermediate step, m, requires interacting with the environment and sampling intermediate trajectories, τφ1:K m . To compute the policy gradient, we need to marginalize out all the intermediate random variables, πφm and τφ1:K m , m = 1, . . . , M . The objective function (12) takes the following form: LT (θ) = Z LT (τ ) PT (τ \| φ) PT φ \| φM −1 , τφ1:K dτ dφ× M −1 M −2 Y PT m=1 τφ1:K m+1 \|φ m+1 (23) PT φ m+1 \|φ m , τφ1:K m m+1 dτφ1:K × m+1 dφ PT τ 1:K \| θ dτ 1:K Since PT φm+1 \| φm , τφ1:K at each intermediate steps are delta functions, the final expression m for the multi-step MAML objective has the same form as (14), with integration taken w.r.t. all intermediate trajectories. Similarly, an unbiased estimate of the gradient of the objective gets M additional terms: " " ## M −1 K X X k M −1 ∇θ LT = E{τ 1:K ∇θ log πφm (τφm ) , (24) ,τ LT (τ ) ∇θ log πφ (τ ) + m } φ m=0 m=0 k=1 where the expectation is taken w.r.t. trajectories (including all intermediate ones). Again, note that at training time we do not constrain the number of interactions with each particular environment and do rollout using each intermediate policy to compute updates. At testing time, we interact with the environment only once and rely on the importance weight correction as described in Sec. 3.2. 17 Published as a conference paper at ICLR 2018 surrogate loss surrogate loss surrogate loss at Vt at Vt at Vt MLP MLP LSTM LSTM LSTM LSTM 64 x 64 64 x 64 64 x 64 64 x 64 xt xt (a) MLP (b) LSTM 64 x 64 xt 64 x 64 dt-1 rt-1 (c) RL2 Fig. 9: Policy and value function architectures. B A DDITIONAL DETAILS ON THE ARCHITECTURES The neural architectures used for our policies and value functions are illustrated in Fig. 9. Our MLP architectures were memory-less and reactive. The LSTM architectures had used a fully connected embedding layer (with 64 hidden units) followed by a recurrent layer (also with 64 units). The state in LSTM-based architectures was kept throughout each episode and reset to zeros at the beginning of each new episode. The RL2 architecture additionally took reward and done signals from the previous time step and kept the state throughout the whole interactions with a given environment (or opponent). The recurrent architectures were unrolled for T = 10 time steps and optimized with PPO via backprop through time. C C.1 A DDITIONAL DETAILS ON META - LEARNING AND OPTIMIZATION M ETA - UPDATES FOR CONTINUOUS ADAPTATION Our meta-learned adaptation methods were used with MLP and LSTM policies (Fig. 9). The metaupdates were based on 3 gradient steps with adaptive step sizes α were initialized with 0.001. There are a few additional details to note: 1. θ and φ parameters were a concatenation of the policy and the value function parameters. 2. At the initial stages of optimization, meta-gradient steps often tended to “explode”, hence we clipped them by values norms to be between -0.1 and 0.1. 3. We used different surrogate loss functions for the meta-updates and for the outer optimization. For meta-updates, we used the vanilla policy gradients computed on the negative discounted rewards, while for the outer optimization loop we used the PPO objective. C.2 O N PPO AND ITS DISTRIBUTED IMPLEMENTATION As mentioned in the main text and similar to (Bansal et al., 2018), large batch sizes were used to ensure enough exploration throughout policy optimization and were critical for learning in the competitive setting of RoboSumo. In our experiments, the epoch size of the PPO was set 32,000 episodes and the batch size was set to 8,000. The PPO clipping hyperparameter was set to = 0.2 and the KL penalty was set to 0. In all our experiments, the learning rate (for meta-learning, the learning rate for θ and α) was set to 0.0003. The generalized advantage function estimator (GAE) (Schulman et al., 2015b) was optimized jointly with the policy (we used γ = 0.995 and λ = 0.95). To train our agents in reasonable time, we used a distributed implementation of the PPO algorithm. To do so, we versioned the agent’s parameters (i.e., kept parameters after each update and assigned it a version number) and used a versioned queue for rollouts. Multiple worker machines were generating rollouts in parallel for the most recent available version of the agent parameters and were pushing them into the versioned rollout queue. The optimizer machine collected rollouts from the queue and made a PPO optimization steps (see (Schulman et al., 2017) for details) as soon as enough rollouts were available. 18 Published as a conference paper at ICLR 2018 We trained agents on multiple environments simultaneously. In nonstationary locomotion, each environment corresponded to a different pair of legs of the creature becoming dysfunctional. In RoboSumo, each environment corresponded to a different opponent in the training pool. Simultaneous training was achieved via assigning these environments to rollout workers uniformly at random, so that the rollouts in each mini-batch were guaranteed to come from all training environments. D D.1 A DDITIONAL DETAILS ON THE ENVIRONMENTS O BSERVATION AND ACTION SPACES Both observation and action spaces in RoboSumo continuous. The observations of each agent consist of the position of its own body (7 dimensions that include 3 absolute coordinates in the global cartesian frame and 4 quaternions), position of the opponent’s body (7 dimensions), its own joint angles and velocities (2 angles and 2 velocities per leg), and forces exerted on each part of its own body (6 dimensions for torso and 18 for each leg) and forces exerted on the opponent’s torso (6 dimensions). All forces were squared and clipped at 100. Additionally, we normalized observations using a running mean and clipped their values between -5 and 5. The action spaces had 2 dimensions per joint. Table 1 summarizes the observation and action spaces for each agent type. Table 1: Dimensionality of the observation and action spaces of the agents in RoboSumo. Agent Coordinates Observation space Self Opponent Velocities Forces Coordinates Forces Action space Ant 15 14 78 7 6 8 Bug Spider 19 23 18 22 114 150 7 7 6 6 12 16 Note that the agents observe neither any of the opponents velocities, nor positions of the opponent’s limbs. This allows us to keep the observation spaces consistent regardless of the type of the opponent. However, even though the agents are blind to the opponent’s limbs, they can sense them via the forces applied to the agents’ bodies when in contact with the opponent. D.2 S HAPED REWARDS In RoboSumo, the winner gets 2000 reward, the loser is penalized for -2000, and in case of draw both agents get -1000. In addition to the sparse win/lose rewards, we used the following dense rewards to encourage fast learning at the early training stages: • Quickly push the opponent outside. The agent got penalty at each time step proportional to exp{−dopp } where dopp was the distance of the opponent from the center of the ring. • Moving towards the opponent. Reward at each time step proportional to magnitude of the velocity component towards the opponent. • Hit the opponent. Reward proportional to the square of the total forces exerted on the opponent’s torso. • Control penalty. The l2 penalty on the actions to prevent jittery/unnatural movements. D.3 R O B O S U M O CALIBRATION To calibrate the RoboSumo environment we used the following procedure. First, we trained each agent via pure self-play with LSTM policy using PPO for the same number of iterations, tested them one against the other (without adaptation), and recorded the win rates (Table 2). To ensure the balance, we kept increasing the mass of the weaker agents and repeated the calibration procedure until the win rates equilibrated. 19 Published as a conference paper at ICLR 2018 Table 2: Win rates for the first agent in the 1-vs-1 RoboSumo without adaptation before and after calibration. E Masses (Ant, Bug, Spider) Ant vs. Bug Ant vs. Spider Bug vs. Spider Initial (10, 10, 10) Calibrated (13, 10, 39) 25.2 ± 3.9% 50.6 ± 5.6% 83.6 ± 3.1% 51.6 ± 3.4% 90.2 ± 2.7% 51.7 ± 2.8% A DDITIONAL DETAILS ON EXPERIMENTS E.1 AVERAGE WIN RATES Table 3 gives average win rates for the last 25 rounds of iterated adaptation games played by different agents with different adaptation methods (win rates for each episode are visualized in Figure 5). Table 3: Average win-rates (95% CI) in the last 25 rounds of the 100-round iterated adaptation games between different agents and different opponents. The base policy and value function were LSTMs with 64 hidden units. Adaptation Strategy LSTM + PPO-tracking LSTM + meta-updates Agent Opponent Ant Ant Bug Spider 24.9 (5.4)% 21.0 (6.3)% 24.8 (10.5)% 30.0 (6.7)% 15.6 (7.1)% 27.6 (8.4)% 44.0 (7.7)% 34.6 (8.1)% 35.1 (7.7)% Bug Ant Bug Spider 33.5 (6.9)% 28.6 (7.4)% 45.8 (8.1)% 26.6 (7.4)% 21.2 (4.2)% 42.6 (12.9)% 39.5 (7.1)% 43.7 (8.0)% 52.0 (13.9)% Spider Ant Bug Spider 40.3 (9.7)% 38.4 (7.2)% 33.9 (7.2)% 48.0 (9.8)% 43.9 (7.1)% 42.2 (3.9)% 45.3 (10.9)% 48.4 (9.2)% 46.7 (3.8)% E.2 RL2 T RUE S KILL RANK OF THE TOP AGENTS Rank Agent TrueSkill rank* Bug+LSTM-meta - 1 0.50 0.59 0.69 0.70 0.70 0.89 0.93 0.94 0.97 0.97 1 2 3 4 5 Bug + LSTM-meta Ant + LSTM-meta Bug + LSTM-track Ant + RL2 Ant + LSTM 31.7 30.8 29.1 28.6 28.4 Ant+LSTM-meta - 2 0.41 0.50 0.60 0.62 0.62 0.85 0.90 0.92 0.96 0.95 Bug+LSTM-track - 3 0.31 0.40 0.50 0.51 0.51 0.78 0.84 0.88 0.93 0.92 Ant+LSTM-RL2 - 4 0.30 0.38 0.49 0.50 0.50 0.77 0.83 0.87 0.92 0.91 Ant+LSTM - 5 0.30 0.38 0.49 0.50 0.50 0.77 0.83 0.87 0.92 0.91 6 7 8 9 10 Bug + MLP-meta Ant + MLP-meta Spider + MLP-meta Spider + MLP Bug + MLP-track 23.4 21.6 20.5 19.0 18.9 Bug+MLP-meta - 6 0.11 0.15 0.22 0.23 0.23 0.50 0.60 0.65 0.75 0.74 Ant+MLP-meta - 7 0.07 0.10 0.16 0.17 0.17 0.40 0.50 0.56 0.66 0.65 Spd+MLP-meta - 8 0.06 0.08 0.12 0.13 0.13 0.35 0.44 0.50 0.61 0.60 Spd+MLP - 9 0.03 0.04 0.07 0.08 0.08 0.25 0.34 0.39 0.50 0.49 Bug+MLP-track - 10 0.03 0.05 0.08 0.09 0.09 0.26 0.35 0.40 0.51 0.50 2 3 4 5 6 7 8 9 10 * The rank is a conservative estimate of the skill, r = µ − 3σ, to ensure that the actual skill of the agent is higher with 99% confidence. 1 Table 4 & Fig. 10: Top-5 agents with MLP and LSTM policies from the population ranked by TrueSkill. The heatmap shows a priori win-rates in iterated games based on TrueSkill for the top agents against each other. Since TrueSkill represents the belief about the skill of an agent as a normal distribution (i.e., with two parameters, µ and σ), we can use it to infer a priori probability of an agent, a, winning against 20 Published as a conference paper at ICLR 2018 its opponent, o, as follows (Herbrich et al., 2007): ! µa − µo 1 x P (a wins o) = Φ p , where Φ(x) := 1 + erf √ 2 2 2β 2 + σa2 + σo2 (25) The ranking of the top-5 agents with MLP and LSTM policies according to their TrueSkill is given in Tab. 1 and the a priori win rates in Fig. 10. Note that within the LSTM and MLP categories, the best meta-learners are 10 to 25% more likely to win the best agents that use other adaptation strategies. E.3 I NTOLERANCE TO LARGE DISTRIBUTIONAL SHIFTS Continuous adaptation via meta-learning assumes consistency in the changes of the environment or the opponent. What happens if the changes are drastic? Unfortunately, the training process of our meta-learning procedure turns out to be sensitive to such shifts and can diverge when the distributional shifts from iteration to iteration are large. Fig. 11 shows the training curves for a meta-learning agent with MLP policy trained against versions of an MLP opponent pre-trained via self-play. At each iteration, we kept updating the opponent policy by 1 to 10 steps. The meta-learned policy was able to achieve non-negative rewards by the end of training only when the opponent was changing up to 4 steps per iteration. # of opponent updates per iteration 1 step 4 steps 7 steps 10 steps Avg. reward per episode 2000 1000 0 −1000 −2000 −3000 0 500 1000 Iteration 1500 Fig. 11: Reward curves for a meta-learning agent trained against a learning opponent. Both agents were Ants with MLP policies. At each iteration, the opponent was updating its policy for a given number of steps using self-play, while the meta-learning agent attempted to learn to adapt to the distributional shifts. For each setting, the training process was repeated 15 times; shaded regions denote 90% confidence intervals. 21	2
A TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFi arXiv:1702.02046v1 [cs.CE] 6 Feb 2017 Xuyu Wang, Auburn University, Auburn, AL Chao Yang, Auburn University, Auburn, AL Shiwen Mao, Auburn University, Auburn, AL Breathing signal monitoring can provide important clues for human’s physical health problems. Comparing to existing techniques that require wearable devices and special equipment, a more desirable approach is to provide contact-free and long-term breathing rate monitoring by exploiting wireless signals. In this paper, we propose TensorBeat, a system to employ channel state information (CSI) phase difference data to intelligently estimate breathing rates for multiple persons with commodity WiFi devices. The main idea is to leverage the tensor decomposition technique to handle the CSI phase difference data. The proposed TensorBeat scheme first obtains CSI phase difference data between pairs of antennas at the WiFi receiver to create CSI tensor data. Then Canonical Polyadic (CP) decomposition is applied to obtain the desired breathing signals. A stable signal matching algorithm is developed to find the decomposed signal pairs, and a peak detection method is applied to estimate the breathing rates for multiple persons. Our experimental study shows that TensorBeat can achieve high accuracy under different environments for multi-person breathing rate monitoring. General Terms: Health sensing, breathing rate estimation, tensor decomposition, stable roommate matching, commodity WiFi, channel state information, phase difference 1. INTRODUCTION It is estimated that 100 million Americans suffer chronic health conditions such as lung disorders, diabetes, and heart disease [Boric-Lubeke and Lubecke 2002]. About three-fourths of the total US healthcare cost is spent on dealing with these health conditions. To reduce such costs, there is an increasing demand for longterm health monitoring in indoor environments. By tracking vital signs such as breathing and heart beats, the patient’s physical health can be timely evaluated and meaningful clues for medical problems can be provided [Rashidi and Cook 2013]. For example, monitoring breathing signals can help identify sleep disorders or anomalies for patients, as well as decreasing sudden infant death syndrome (SIDS) for sleeping infants [Hunt and Hauck 2006]. Traditional approaches for monitoring vital signs require patients to wear special devices, such as a capnometer [Mogue and Rantala 1988] to estimate breathing rate, or a pulse oximeter [Shariati and Zahedi 2005] on the finger to track heart beats. Recently, smartphones are used to estimate breathing rate by employing the built-in gyroscope, accelerometer [Aly and Youssef 2016], and microphone [Ren et al. 2015], and for physical activity recognition using accelerometer [Tao et al. 2016]. This requires the patient to place a smartphone near-by and wear sensors in the monitoring environment. Moreover, the readily available smartphone sensors such as accelerometer and gyroscope This work is supported in part by the U.S. NSF under Grants CNS-1247955 and CNS-1320664, and by the Wireless Engineering Research and Education Center (WEREC) at Auburn University. Author’s addresses: X. Wang, C. Yang, and S. Mao, Department of Electrical and Computer Engineering, Auburn University, Auburn, AL 36849-5201 USA. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. c YYYY ACM. 2157-6904/YYYY/01-ARTA $15.00 DOI: http://dx.doi.org/10.1145/0000000.0000000 ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:2 X. Wang et al. can only monitor breathing rate for single person. Such existing approaches could be expensive, inconvenient to use, and annoying even for a short period of time. An alternative approach is in need to provide a contact-free and long-term breathing monitoring at low costs. Recently, several RF based systems for vital signs tracking are proposed, which employ wireless signal to monitor breathing-induced chest movements and are mainly based on radar and WiFi techniques. For radar based vital signals monitoring, several techniques, such as the Doppler radar [Droitcour et al. 2009] and ultra-wideband radar [Salmi and Molisch 2011], are used to track vital signs, which require special hardware operated at high frequency and at a high cost. Moreover, the Vital-Radio system employs a frequency modulated continuous wave (FMCW) radar to track breathing and heart rates [Adib et al. 2015]; it requires a customized hardware using a wide bandwidth from 5.46 GHz to 7.25 GHz. For WiFi based vital signs monitoring, mmVital [Yang et al. 2016] can exploit the received signal strength (RSS) of 60 GHz millimeter wave (mmWave) signals for breathing and heart rate estimation. mmVital also operates with a larger bandwidth of about 7 GHz, and uses a customized hardware with a mechanical rotator. Another technique UbiBreathe uses WiFi RSS for breathing rate estimation, which, however, requires the device be placed in the line of sight (LOS) path between the transmitter and the receiver [Abdelnasser et al. 2015]. Compared with RSS, channel state information (CSI) provides fine-grained channel information in the physical layer, which can now be read from modified device drivers for several off-the-shelf WiFi network interface cards (NIC), such as the Intel WiFi Link 5300 NIC [Halperin. et al. 2010] and the Atheros AR9580 chipset [Xie et al. 2015]. Moreover, CSI includes both amplitude and phase values at the subcarrier-level for orthogonal frequency-division multiplexing (OFDM) channels, which is a more stable and accuracy representation of channel characteristics than RSS, including the non-LOS (NLOS) components for small-scale fading. Recently, the authors in [Liu et al. 2015] use the CSI amplitude data to monitor breathing and heart signals, which requires the person to remain in the sleeping mode. However, the measured CSI phase data has not been fully exploited in prior works, largely due to random phase fluctuation resulting from asynchronous times and frequencies of the transmitter and receiver. For multiple person breathing monitoring, because the reflected components in the received signal are from the chests of multiple persons, each moves slightly due to breathing and the movements are independent. Thus, vital signs monitoring and estimation for multiple persons still remains a challenging and open problem. In this paper, we propose to utilize CSI phase difference data between antenna pairs to monitor the breathing rates of multiple persons. First, we show that when the person is in a stationary state, such as standing, sitting, or sleeping, the CSI phase difference data is highly stable in consecutively received packets, which can be leveraged for extracting the small, periodic breathing signal hidden in the received WiFi signal. In fact, phase difference is more robust than amplitude, which usually exhibits large fluctuations because of the attenuation over the link distance, obstacles, and the multipath effect. Moreover, the phase difference data captures and preserves the periodicity of breathing, when the wireless signal is reflected from the patients’ chests. To extract the weak breathing signal, and more important, to distinguish among multiple persons, we propose to employ a tensor decomposition method to handle the phase difference data [Papalexakis et al. 2016; Luo et al. 2015; Hu et al. 2014]. We create the CSI tensor data by increasing the dimension of CSI data from one to three, which can be used to effectively separate different breathing signals in different clusters. We present a system termed TensorBeat, Tensor decomposition for estimating multiple persons breathing Beats, by exploiting CSI phase difference data. TensorBeat ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:3 operates as follows. First, it obtains 60 CSI phase difference data from antenna pairs 1 and 2, and 2 and 3, at the receiver. Next, a data preprocessing procedure is applied to the measured phase difference data, including data calibration and Hankelization. In the data calibration phase, the direct current (DC) component and high frequency noises are removed. In the Hankelization phase, a two dimensional Hankel matrix is created based on the calibrated phase difference data from every subcarrier, and the rank of the Hankel matrix is analyzed. Then, we adopt Canonical Polyadic (CP) decomposition for estimating multiple persons breathing signs, and prove the uniqueness of the proposed CSI tensor. After CP decomposition, we obtain twice amount of breathing signals, which, however, are randomly indexed. We thus design a stable signal matching algorithm (for the stable roommate problem [Irving 1985]) to identify the decomposed signal pairs for each person. Finally, we combine the decomposed signals in each pair and employ a peak detection method to estimate the breathing rate for each person. We implement TensorBeat on commodity 5 GHz WiFi devices and verify its performance with five persons over six months in different indoor environments, such as a computer laboratory, a through-wall scenario, and a long corridor. The results show that the proposed TensorBeat system can achieve high accuracy and high success rates for multiple persons breathing rates estimation. Moreover, we demonstrate the robustness of the proposed TensorBeat system for monitoring multiple persons’ breathing beats under a wide range of environmental parameters. The main contributions of this paper are summarized as follows. (1) We theoretically and experimentally verify the feasibility of leveraging CSI phase difference for breathing monitoring. In particular, we analyze the measured phase errors in detail and demonstrate that phase difference data is stable and can be used to extract breathing signs. To the best of our knowledge, we are the first to leverage phase difference for multiple persons breathing rate estimation. (2) We are also the first to apply tensor decomposition for RF sensing based vital signs monitoring. We use the phase difference data to create a CSI tensor for all subcarrier at the three antennas of the WiFi receiver. We then incorporate CP decomposition to obtain the desired breathing signals. A stable signal matching algorithm is developed to match the decomposed signals for each person, while a peak detection method is used to estimate multiple persons’ breathing rates. (3) We prototype the TensorBeat system with commodity 5 GHz WiFi devices and demonstrate its superior performance in different indoor environments with extensive experiments. The results show that the proposed TensorBeat system can achieve very high accuracy and high success rates for multiple persons breathing rate estimation. The remainder of this paper is organized as follows. The preliminaries and phase difference analysis are provided in Section 2. We present the TensorBeat system design and performance analysis in Section 3 and verify its performance with extensive experiments in Section 4. We provide the related work in Section 5. Section 6 concludes this paper. 2. PRELIMINARIES AND PHASE DIFFERENCE INFORMATION 2.1. Tensor Decomposition Preliminaries A tensor is considered as a multidimensional array [Kolda and Bader 2009]. The dimensions of the tensor are called as modes, and the order of the tensor is the number of the modes. For example, the N -order tensor is a N -mode tensor. Moreover, It is noticed that a first-order tensor is a vector, a second-order tensor is a matrix, and ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:4 X. Wang et al. a third-order tensor is a cubic structure. Higher-order tensors with (N ≥ 3) have a wide range of applications such as data mining, brain data analysis, recommendation systems, wireless communications, computer vision, and healthcare and medical applications [Papalexakis et al. 2016]. For higher-order tensors, they face various computational challenging because of the exponential increase in time and space complexity with the orders increase of tensors. This leads to the curse of dimensionality. Fortunately, tensor decomposition as one powerful tool is leveraged for alleviating the curve by decomposing high-order tensors into a limited number of factors. Also, it can obtain hidden feature components, thus extracting physical insight of higher-order tensors. Two main tensor decompositions are tucker decomposition and CP decomposition [Kolda and Bader 2009]. We consider CP decomposition for multiple persons breathing rate estimation because it can easily obtain the unique solution [Kolda and Bader 2009]. On the other hand, we will provide some necessary definitions and equations of tensor decomposition, which can be used for our proposed algorithm. Definition 2.1. (Frobenius Norm of a Tensor). The Frobenius norm of a tensor χ ∈ KI1 ×I2 ×···×IN is the square root of the sum of the squares of all its elements, which is defined by v u I1 I2 IN X uX X t kχkF = (1) x2i1 ,i2 ···iN . ··· i1 =1 i2 =1 iN =1 where K stands for R or C. Definition 2.2. (Kronecker Product). The Kronecker product of matrics A ∈ KI×J and B ∈ KM×N is denoted as A ⊗ B. The result is an (IM ) × (JN ) matrix, which is defined by   a11 B a12 B . . . a1J B a21 B a22 B . . . a2J B  A⊗B =  . (2) .. .. ..   ... . . .  aI1 B aI2 B . . . aIJ B Definition 2.3. (Khatri-Rao Product). The Khatri-Rao product of A ∈ KI×J and B ∈ KM×J is denoted as A ⊙ B. It is the column-wise Kronecker product with the size (IM ) × J, which is defined by (3) A ⊙ B = [a1 ⊗ b1 , a2 ⊗ b2 , · · · , aJ ⊗ bJ ]. I×J Definition 2.4. (Hadamard product). The Hadamard product of A ∈ K and B ∈ KI×J is denoted as A ∗ B. It is the elementwise matrix product with the size I × J, which is defined by   a11 b11 a12 b12 . . . a1J b1J a21 b21 a22 b22 . . . a2J b2J  A∗B = . (4) .. .. ..   ... . . .  aI1 bI1 aI2 bI2 . . . aIJ bIJ 2.2. Channel State Information Preliminaries OFDM is an effective wireless transmission technique widely used in many wireless systems, including WiFi (such as IEEE 802.11 a/g/n) and LTE [Wang et al. 2016; Xu et al. 2014]. The OFDM system partitions the wireless channel into multiple ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:5 orthogonal subcarriers, where data is transmitted over all the subcarriers by using the same modulation and coding scheme (MCS) to combat frequency selective fading. With modified device driver for off-the-shelf NICs, such as the Intel 5300 NIC [Halperin. et al. 2010] and the Atheros AR9580 chipset [Xie et al. 2015], the CSI data can be extracted, which represents fine-grained physical (PHY) information. Moreover, CSI captures rich wireless channel characteristics such as shadowing fading, distortion, and the multipath effect. The WiFi OFDM channel can be regarded as a narrowband flat fading channel, which can be expressed in the frequency domain as ~ =H·X ~ +N ~, Y (5) ~ denote the received and transmitted wireless signal vectors, respecwhere Y~ and X ~ is the additive white Gaussian noise, and H represents the channel frequency tively, N ~ and X. ~ response, which can be estimated from Y Although the WiFi OFDM system can use 56 subcarriers for data transmission on a 20 MHz channel, the Intel 5300 NIC device driver can only provide CSI for 30 out of the 56 subcarriers using the channel bonding technique. The channel frequency response of subcarrier i, denoted by Hi , is a complex value, given as Hi = Ii + jQi = \|Hi \| exp (j 6 Hi ), (6) where Ii and Qi are the in-phase component and quadrature component, respectively; \|Hi \| and 6 Hi are the amplitude and phase response of subcarrier i, respectively. For indoor environments with multipath components, the channel frequency response of subcarrier i, Hi , can also be written as Hi = K X rk · e−j2πfi τk , (7) k=0 where K is the number of multipath components, and rk and τk are the attenuation and propagation delay on the kth path, respectively. Traditionally, the multipath components are regarded as harmful for wireless communications, since they cause the delay spread (requires guard times) and large fluctuation of received wireless signal (harder to demodulate). For indoor localization systems, multiple signals will be received from a single transmission, including one LOS signal and many reflected signals. It is a challenging problem to detect the LOS signal from the mixed multipath components, which is indicative of the direction of the transmitter [Yang et al. 2013; Wang et al. 2014]. In this paper, however, we take a different view and show that the multiple signals reflected from the chests of multiple persons can be useful for estimating their breathing rates simultaneously. 2.3. Phase Difference Information As discussed, we exploit phase difference information for breathing rate estimation. We verify that the phase difference values between two adjacent antennas are stable for consecutively received packets in this section. In fact, the extracted phase information from the Intel 5300 NIC is high random and cannot be used for breathing monitoring. This is because of the asynchronous times and frequencies of transmitter and receiver NICs. Recently, two effective techniques are proposed for CSI phase calibration, to remove the unknown random components in CSI phase data. The first technique is to take a linear transformation for the CSI phase data over all the subcarirers [Qian et al. 2014; Wang et al. 2016; Wang et al. 2015]. The other technique is to use the phase difference between two adjacent antennas in the 2.4 GHz band, and to ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:6 X. Wang et al. remove the measured average of phase difference for LOS recognition [Wu et al. 2015]. It can be seen that these techniques only obtain the stable phase information and phase difference data with a zero mean, respectively, but none of these are useful for breathing rate estimation. To prove the stability of measured CSI phase difference in the 5 GHz band, we write b i , as [Speth et al. 1999] the measured phase of subcarrier i, denoted as 6 H 6 b i = 6 Hi + (λp + λs )mi + λc + β + Z, H (8) where Hi is the true phase of CSI data, mi is the subcarrier index of subcarrier i, β is the initial phase offset at the phase-locked loop (PLL), Z is the measurement environment noise, and λp , λs and λc are the phase errors from the packet boundary detection (PBD), the sampling frequency offset (SFO), and central frequency offset (CFO), respectively [Speth et al. 1999], which are given by   λp = 2π ∆t N′ (9) λ = 2π( T T−T ) TTus n  s λc = 2π∆f Ts n, 6 where ∆t is the packet boundary detection delay, N is the FFT size, T ′ and T are the sampling periods from the receiver and the transmitter, respectively, Tu is the length of the data symbol, Ts is the total length of the data symbol and the guard interval, n is the sampling time offset for the current packet, and ∆f is the center frequency ′ difference between the transmitter and receiver. In fact, the values of ∆t, T T−T , n, ∆f , and β in (8) and (9) are unknown, and the values of λp , λs , and λc can be different for different packets. Thus, we cannot obtain the true phase 6 Hi of CSI data from measured phase values. However, the measured phase difference on subcarrier i is stable, which can be employed for breathing rate estimation. Since the three antennas (radios) of the Intel 5300 NIC are on the same NIC, they use the same system clock and the same downconverter frequency. The measured CSI phases on subcarrier i from two adjacent antennas have the same λp , λs , λc , and mi . The phase difference can be computed as b i = ∆6 Hi + ∆β + ∆Z, ∆6 H (10) where ∆6 Hi is the true phase difference of subcarrier i, ∆β is the unknown difference in phase offsets, which is a constant [Gjengset et al. 2014], and ∆Z is the noise difference. Since in (10), the random values ∆t, ∆f , and n are all removed, the phase difference becomes more stable for back-to-back received packets. As an example, we plot in Fig. 1 the phase differences (marked as red dots) and the single antenna phases (marked as gray crosses) read from the 3rd subcarrier for 500 consecutively received packets. It can be seen that the single antenna phase is nearly uniformly distributed between 0◦ and 360◦ . However, all the phase difference data concentrate in a small sector between 330◦ and 340◦ , which is significantly more stable than phase data. Breathing rate estimation for multiple persons is a challenging problem, because the reflected components in the received signal are from the chests of multiple persons, each moves slightly due to breathing and the movements are independent. Thus, the peak-to-peak detection method cannot be effective for detecting the multiple breathing signals from the received signal. The aggregated breathing signal from multiple persons is not a clearly periodic signal anymore. Fig. 2 shows the detected breathing signals for one person (the upper plot) and three persons (the lower plot). We can see that for one person, the breathing signal exhibits a noticeable periodicity. So the breathing rate can be estimated by peak detection after removing the noise. However, the aggregated breathing signal of three persons does not show noticeable periodicity for ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:7 90 20 120 15 60 10 150 30 5 180 0 330 210 300 240 270 Fig. 1. The phase differences (marked by red dots) and the single antenna phases (marked by gray crosses) of the 3rd subcarrier for 500 back-to-back packets, plotted in the polar coordinate system. packet 400 to 600. Traditional FFT based methods can transform the received signal from the time domain to the frequency domain to estimate the breathing frequencies from multiple persons. Fig. 3 shows the breathing rate estimation for one person (the upper plot) and three persons (the lower plot) with the FFT method. We can see that the estimated frequency for one person is 0.2 Hz, which is almost the same as the true breathing rate. However, for three-person breathing rate estimation, the FFT curve only has two peaks, and the estimated breathing rates are much less accurate. In particular, the third peak cannot be estimated. This is because FFT based methods require a larger window size to improve the frequency resolution. We show that the proposed tensor decomposition based method is highly effective for multi-person breathing rate estimation in the following section. 3. THE TENSORBEAT SYSTEM 3.1. TensorBeat System Architecture The main idea of the proposed TensorBeat system is to estimate multi-person breathing rates by employing a tensor decomposition method. To obtain CSI tensor data, we first create a two dimensional Hankel matrix with phase difference data from backto-back received packets extracted from each subcarrier at each antenna. Then, by leveraging the phase differences from the 60 subcarriers, i.e., that between antennas 1 and 2, and between antennas 2 and 3, we can construct the third dimension of the CSI tensor data. The TensorBeat system will then leverage the created CSI tensor to estimate multi-person breathing signs. Our approach is motivated by two observations. First, for stationary modes of a person, such as standing, sitting, or sleeping, CSI phase difference from consecutively received packets is highly stable. It can thus be useful for extracting the periodic breathing signals. Second, the tensor decomposition method can effectively estimate multi-person breathing beats. We create the CSI tensor data by increasing the dimension of CSI data, from one dimension to three dimensions. The higher dimension CSI data is helpful to effectively separate ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:8 X. Wang et al. Breathing Curve of One Person 0.8 0.7 0.6 0.5 0 100 200 300 400 500 600 500 600 Breathing Curve of Three Persons -1 -1.5 -2 0 100 200 300 400 Packet Index FFT Magnitude FFT Magnitude Fig. 2. Detected breathing signals for one person (the upper plot) and three persons (the lower plot). Breathing Rate Estimation for One Person 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.7 Breathing Rate Estimation for Three Persons 40 20 0 0 0.1 0.2 0.3 0.4 0.5 Frequency (Hz) Fig. 3. Breathing rate estimation for one person (the upper plot) and three persons (the lower plot) based on FFT. different breathing signals by forming different clusters. This strategy is similar to the kernel method in traditional machine learning, such as support vector machine (SVM) [Wu et al. 2007] or multiple hidden layers in deep learning [LeCun et al. 2015; Wang et al. 2015; Wang et al. 2017]. As shown in Fig. 4, the TensorBeat system consists of four main modules: Data Extraction, Data Preprocessing, CP Decomposition, Signal Matching, and Breathing Rate Estimation. For Data Extraction, TensorBeat obtains 60 CSI phase difference data, 30 between antennas 1 and 2, and 30 between antennas 2 and 3, at the receiver with an off-the-shelf WiFi device. The Data Preprocessing module includes data calibration and Hankelization. Data calibration is implemented to remove the DC component and high frequency noises. Hankelization is to create a two dimensional Hankel matrix with phase difference data from each subcarrier for back-to-back received packets. The rank of the constructed Hankel matrix is then analyzed. We next apply CP decomposition to estimate multiple persons’ breathing signals, and prove the uniqueness of the proposed CSI tensor. For Signal Matching, we first compute the autocorrelation function of the decomposed signals, and incorporate a stable roommate matching algorithm to identify the decomposed signal pairs for each person, where a preference ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:9 )&1&#$,(3,2.(''-4 )&1&#/01,&.12!"#$%&'(#)++(,(-.(# /01,&.12- '; 6. &, ,( ,' < < < )&1&# &56,&12- 3&.8(1' < $#)(.2:32'12< '; 6. &, ,( ,' 7&-8(59&12- 7&-8(5# :&1,0 =#͑ ͑ ͑ ͑ ͑ = !?@ !> ͑͑͑͑͑͑͑͑͑͑ !4-&5#B&1.%-4#A542,1%: D,(&1%-4#/'1:&12!4-&5#C;'2- !1&65(#@22::&1(# B&1.%-4 $(&8#)(1(.12- !?@ +> ͑͑͑͑͑͑ A;12.2,,(5&12- ͑͑͑͑͑͑ !> +@ Fig. 4. The TensorBeat system architecture. list is computed with the dynamic time warping (DTW) values of the autocorrelation signals. For Breathing Rate Estimation, we combine the decomposed signals in each pair and use the peak detection method to compute the breathing rate for each person. In the remainder of this section, we present the design and analysis of each module of the TensorBeat system in detail. 3.2. Data Preprocessing 3.2.1. Data Calibration. We use a 20 Hz sampling rate to obtain 60 CSI phase difference data, 30 between antennas 1 and 2, and 30 between antennas 2 and 3, at the receiver with an off-the-shelf WiFi device at 5 GHz for data extraction. Then, data calibration is applied to remove the DC component and high frequency noises. Because the DC component is also considered as a kind of signal, which may affect CSI tensor decomposition, TensorBeat adopts the Hampel filter to remove the DC component. Unlike traditional data calibration approaches that only remove the high frequency noise, we use the Hampel Filter for detrending the original CSI phase difference data to remove DC component. In fact, the Hampel Filter, which is set as a large sliding window with 150 samples wide and a small threshold of 0.001, is firstly used to extract the basic trend of the original data. Then, the detrended data is generated by subtracting the basic trend data from the original data. We also utilize the Hampel Filter to reduce ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:10 X. Wang et al. Original Phase Difference -0.5 -1 -1.5 0 100 200 300 400 500 600 500 600 Calibrated Phase Difference 0.2 0 -0.2 -0.4 0 100 200 300 400 Packet Index Fig. 5. Data calibration: an example. the high frequency noise by using a sliding window of 6 samples wide and a threshold of 0.01. Fig. 5 presents an example of data calibration. We can see that the original phase differences of all the subcarriers have both a DC component and high frequency noises. With the proposed data calibration approach, it can be seen that the DC components are readily removed and all the subcarriers demonstrate a similar calibrated signal over the 600 packet range with low noise. Such calibrated signal will then be used for estimating the breathing rates of multiple persons. 3.2.2. Hankelization. After data calibration, we obtain the CSI phase difference data matrix with a dimension of (number of packets × number of subcarriers). We then employ a Hankelization method to transform the large CSI matrix into a CSI tensor by expanding the packets into an additional dimension [Lathauwer 2011]. Specifically, we rearrange the signals of each subcarrier into a 2-D Hankel matrix, so that the signals from all the 60 subcarriers can be considered as a 3-Dimensional tensor. Define Hr as the constructed Hankel matrix with the size I × J for subcarrier r, which is created by mapping N packets onto the Hankel matrix with N = I +J −1. We consider the Hankel matrix with size I = J = N2+1 . We thus obtain the Hankel matrix Hr for subcarrier r, as   hr (0) hr (1) . . . hr ( N2+1 − 1)  hr (1) hr (2) . . . hr ( N2+1 )    Hr =  (11) , .. .. .. ..   . . . . hr ( N2+1 − 1) hr ( N2+1 ) . . . hr (N − 1) where hr (i) is the calibrated phase difference data from subcarrier r for packet i. In our experiments, we set N = 599 and I = J = 300. To determine the number of components needed for CSI tensor decomposition, we provide the following theorem for estimating R breathing signals. T HEOREM 3.1. If there are R breathing signals in an indoor monitoring environment, the constructed Hankel matrix H r for subcarrier r has a rank of 2R when noise is negligible. P ROOF. When analyzing signal data structure, we assume the noise is negligible. Moreover, let the ith breathing signal be represented as Si (t) = Ai cos(wi t + ϕi ). The ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:11 observed signal from a subcarrier can be represented by [Cichocki et al. 2015] Y (t) = i=R X Ki Si (t) = i=R X (12) K̂i cos(wi t + ϕi ), i=1 i=1 where Ki is the coefficient for breathing signal i and the new coefficient K̂i = Ki Ai . The ith component of Y (t), K̂i cos(wi t + ϕi ), can be decomposed using Euler’s formula. We have K̂i K̂i exp(j(wi t + ϕi )) + exp(j(−wi t − ϕi )) 2 2 K̂i K̂i exp(jϕi ) exp(jwi t) + exp(−jϕi ) exp(−jwi t). (13) = 2 2 Each breathing signal can be separated into two exponential signals with different coefficients. Combining all the R breathing signals, we have ! R X K̂i K̂i Y (t) = exp(jϕi ) exp(jwi t) + exp(−jϕi ) exp(−jwi t) 2 2 i=1 K̂i cos(wi t + ϕi ) = = 2R X K̃i Zit , (14) i=1 where the updated signal Zit is denoted as Zit = exp(±jwi t), and K̃i = K̂2i exp(±jϕi ) is its coefficient. For packets received at discrete times, we represent the received P2R n signal as Y (n) = i=1 K̃i Zi . Note that the combined signal can be considered as an exponential polynomial with 2R different exponential terms. Map signal Y (n) for n = 1, 2, · · · , N into a Hankel matrix with size I = J = N2+1 , we have  P  N +1 P2R P2R −1 2R K̃i Zi1 · · · K̃i Zi 2 K̃i Zi0 i=1 i=1 i=1  P N +1  P2R P2R 2R   2 1 2 ···   i=1 K̃i Zi i=1 K̃i Zi i=1 K̃i Zi Hr =  (15) . . . .   .. .. . · · · .   N +1 N +1 P2R P2R P2R N −1 2 −1 2 K̃ Z · · · K̃ Z K̃ Z i i i i i i i=1 i=1 i=1 We can see that the Hankel matrix can be decomposed with Vandermonde decomposition [Lathauwer 2011], as T H r = V r · diag(K̃1 , K̃1 , · · · , K̃2R ) · Ṽ r , where the Vandermode matrices V r ∈ K  1  Z1  .. V r = Ṽ r =  .  N +1 2 −1 Z1 N +1 2 ×2R and Ṽ r ∈ K 1 Z2 .. . N +1 2 −1 Z2 ··· ··· ··· N +1 2 ×2R 1 Z2R .. . N +1 2 −1 · · · Z2R (16) are given by    .  (17) Because a Vandermode matrix is full rank, which is obtained by different poles, the rank of the Hankel matrix generated by R breathing signals is 2R. According to Theorem 3.1, 2R signal components is required to separate the R breathing signals. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:12 X. Wang et al. Next we consider the influence of measurement noise on the Hankel matrix H r . Because of noise, the Hankel matrix Hr is actually a full-rank matrix. However, Theorem 3.1 shows that the rank of the combined breathing signal is 2R, meaning that the first 2R weighted decomposed components are much stronger than the remaining ones as long as the signal to noise ratio (SNR) is not very low. This shows that the Hankel matrix structure can be used to effectively separate breathing signals from white noise. Actually, the different signals will be well denoised and separated by using tensor decomposition, as to be discussed in Section 3.3. 3.3. Canonical Polyadic Decomposition Once the CSI tensor is ready, we apply CP decomposition to estimate multiple persons’ breathing signals. With CP decomposition, the CSI tensor data can be approximated as the sum of 2R rank-one tensors according to Theorem 3.1. Denote χ ∈ KI×J×K as a third-order CSI tensor, which can be obtained by the sum of three-way outer products as [Kolda and Bader 2009; Papalexakis et al. 2016] χ≈ 2R X (18) ar ◦ b r ◦ cr , r=1 where ar , br , cr are the vectors at the rth position for the first, second, and third dimension, respectively, and 2R is the number of decomposition components, which is the approximation rank of the tensor based on CP decomposition [Sun and Kumar 2014; Sun and Kumar 2015]. Their outer product is defined by for all i, j, k. (ar ◦ br ◦ cr )(i, j, k) = ar (i)br (j)cr (k), (19) I×2R We consider factor matrices A = [a1 , a2 , · · · , a2R ] ∈ K , B = [b1 , b2 , · · · , b2R ] ∈ KJ×2R , and C = [c1 , c2 , · · · , c2R ] ∈ KK×2R as the combination of vectors from rankone components. Moreover, define X(1) ∈ KI×JK , X(2) ∈ KJ×IK , and X(3) ∈ KK×IJ as 1-mode, 2-mode, and 3-mode matricization of CSI tensor χ ∈ KI×J×K , respectively, which are obtained by fixing one mode and arranging the slices of the rest of the modes into a long matrix [Kolda and Bader 2009]. Then, we can write the three matricized forms as X(1) ≈ A(C ⊙ B)T , (20) T (21) T (22) X(2) ≈ B(C ⊙ A) , X(3) ≈ C(B ⊙ A) , where ⊙ denotes the Khatri-Rao product. When the number of components 2R is given, we apply the Alternating Least Squares (ALS) algorithm, the most widely used algorithm for CP decomposition [Kolda and Bader 2009]. To decompose the CSI tensor, we minimize the square sum of the differences between the CSI tensor χ and the estimated tensor. min A,B,C χ− 2R X 2 ar ◦ b r ◦ cr r=1 . (23) F Note that (23) is not convex. However, the ALS algorithm can effectively solve the problem by fixing two of the factor matrices, to reduce the problem to a linear least squares problem with the third factor matrix as variable. If we fix B and C, we can rewrite problem (23) as min X(1) − A(C ⊙ B)T A 2 F . (24) ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:13 We can derive the optimal solution to problem (24) as A = X(1) [(C ⊙ B)T ]† . Applying the property of pseudoinverse of the Khatri-Rao product, it follows that A = X(1) (C ⊙ B)(C T C ∗ B T B)† , (25) where ∗ denotes the Hadamard product. This equation only requires computing the pseudoinverse of a 2R × 2R matrix rather than a JK × 2R matrix. Note that R is much smaller than J and K, thus the computing complexity can be greatly reduced. Similarly, we can obtain the optimal solutions for B and C as B = X(2) (C ⊙ A)(C T C ∗ AT A)† T T † C = X(3) (B ⊙ A)(B B ∗ A A) . (26) (27) Applying ALS to CP decomposition, we obtain matrices A, B, and C. To guarantee the effectiveness of the decomposed components, we next examine the uniqueness of CP decomposition. The basic theorem on the uniqueness of CP decomposition is given in [Kolda and Bader 2009], which is provided in the following. FACT 1. For tensor χ with rank L, if kA + kB + kC ≥ 2L + 2, then the CP decomposition of χ is unique, where kA , kB , and kC denote the k-rank of matrix A, B, C, respectively. Here k-rank means the maximum value k such that any k columns are linearly independent [Kolda and Bader 2009]. Based on Fact 1, we have the following theorem for the CSI tensor. T HEOREM 3.2. For the proposed CSI tensor χ with rank 2R, the CP decomposition of χ is unique. P ROOF. The proposed CSI tensor χ is created by K Hankel matrix, where the rth Hankel matrix H r is rank-2R according to Theorem 3.1. Thus, for the k-rank of the matrices A and B, we have kA = 2R and kB = 2R. On the other hand, because phase differences of subcarriers between antennas 1 and 2, and antennas 2 and 3 are independent, the k-rank of matrix C has kC ≥ 2. Thus, the expression is kA + kB + kC ≥ 2R + 2R + 2 = 2(2R) + 2, which satisfies the conditions in Theorem 1. This proofs the theorem. Theorem 3.2 indicates that the CP decomposition of the created CSI tensor is unique, which can be used to effectively estimate multiple breathing rates. In the proposed TensorBeat system, we leverages the matrix A = [a1 , a2 , · · · , a2R ] as decomposed signals S1 , S2 , · · · , S2R . For example, Fig. 6 shows the results of CP decomposition for CSI tensor data from three persons (R = 3). We can see that there are six signals. Moreover, signals 1 and 2 are similar, signals 3 and 5 are similar, and signals 4 and 6 are similar. This is because CP decomposition cannot guarantee that similar signals are located in adjacent locations (i.e., the output signals are randomly indexed). Thus, we need to identify the signal pairs among the decomposed signals for each person, which will be addressed in Section 3.4. 3.4. Signal Matching Algorithm The CP decomposition of CSI tensor data yields 2R decomposed signals, i.e., S1 , S2 , · · · , S2R , which, however, are randomly indexed. In this section, we propose a signal matching algorithm to pair the two similar decomposed signals that belong to the same person. The main idea is to leverage the autocorrelation to strengthen the periodicity of decomposed signals and use the Dynamic Time Warping (DTW) method to compute the similarity value for any pair of signals. Finally, we apply the stableroommate matching algorithm to pair the decomposed signals for each person, using ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:14 X. Wang et al. 0.2 6 0 5 4 -0.2 0 3 100 2 200 Packet Index 300 Signal Index 1 Fig. 6. CP decomposition results for a CSI tensor of three persons. 2 6 0 -2 0 5 4 100 3 200 300 400 Packet Index 2 500 600 Signal Index 1 Fig. 7. Autocorrelation of the decomposed breathing signals. the DTW values as the closeness metric. We introduce the proposed signal matching algorithm in the following. 3.4.1. Autocorrelation and Dynamic Time Warping. After CP decomposition of CSI tensor data, we first compute the autocorrelation function of the 2R decomposed signals to strengthen their periodicity. We evaluate the autocorrelation function of the decomposed signals for two reasons. The first is that the autocorrelation of a decomposed signal can increase the data length, which helps to improve the accuracy of the peak detection. Second, because the decomposed signals have phase shift and nonalignment, using the autocorrelation of decomposed signals can reduce such shifts and strengthen the periodicity of the decomposed signals. Fig. 7 shows the autocorrelation of the decomposed breathing signals produced by CP decomposition. We can see that each autocorrelation signal exhibits a more obvious periodicity than that of the original decomposition signals. Moreover, the data length is increased from 300 to 600. Furthermore, we employ the DTW approach to measure the distance between any pair of autocorrelation signals, which is different from the Euclidean distance method that computes the sum of distances from each value on one curve to the corresponding ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:15 value on the other curve. Moreover, the Euclidean distance method believes that two autocorrelation signals with the same length are different as long as one of them has a small shift. However, DTW can automatically identify these shifts and provide the similar distance measurement between two autocorrelation signals by aligning the corresponding time series, thus overcoming the limitation of the Euclidean distance method. With the autocorrelation signals, we design the DTW method for measuring their pairwise distance. Given two autocorrelation signals and a cost function, the DTW method seeks an alignment by matching each point in the first autocorrelation signal to one or more points in the second signal, thus minimizing the cost function for all points [Wang and Katabi 2013; Salvador and Chan 2004; Salvador and Chan 2007]. To reduce the computational complexity of DTW, we apply downsampling to the two autocorrelation signals, which leads to a reduced number of packets N ′ . Then, consider two downsampled autocorrelation signals Pi = [Pi (0), Pi (1), · · · , Pi (N ′ − 1)] and Pj = [Pj (0), Pj (1), · · · , Pj (N ′ − 1)], we need to find a warping path W = [w1 , w2 , · · · , wL ], where L is the length of the path, and the lth element of the warping path is wl = (ml , nl ), where m and n are the packet index for the two downsampled autocorrelation signals. The objective is to minimize the total cost function by implementing the non-linear mapping between two downsampled autocorrelation signals Pi and Pj . The formulated problem is given by min L X kPi (ml ) − Pj (nl )k (28) l=1 s.t. (m1 , n1 ) = (0, 0) (mL , nL ) = (N ′ − 1, N ′ − 1) ml ≤ ml+1 ≤ ml + 1 nl ≤ nl+1 ≤ nl + 1. (29) (30) (31) (32) The objection function is to minimize the distance between two downsampled autocorrelation signals. The first and second constraints are boundary constraints, which require that the warping path starts at Pi (0) and Pj (0) and ends at Pi (N ′ − 1) and Pj (N ′ − 1). This can guarantee all points of the two downsampled autocorrelation signals are used for measuring their distance, thus avoiding to use only local data. Furthermore, the third and fourth constraints are monotonic and marching constraints, which require that there be no cycles for wi and wj in the warping path and the path is increased with the maximum 1 at each step. We apply dynamic programming to solve problem (28), to obtain the minimum distance warping path between two downsampled autocorrelation signals. We consider a two-dimensional cost matrix C with size N ′ × N ′ , whose element C(ml , nl ) is the minimum distance warping path for two downsampled autocorrelation signals Pi = [Pi (0), Pi (1), · · · , Pi (ml )] and Pj = [Pj (0), Pj (1), · · · , Pj (nl )]. We design the recurrence equation in dynamic programming as follows. C(ml , nl ) = kPi (ml ) − Pj (nl )k + min [C(ml − 1, nl ), C(ml , nl − 1), C(ml − 1, nl − 1)]. (33) By filling all elements of the cost matrix C, the value C(N ′ − 1, N ′ − 1) can be computed as the DTW value between the two downsampled autocorrelation signals. The time complexity is O(N ′2 ). Fig. 8 shows the DTW results for downsampled autocorrelation signals 4 and 6 (the upper plot), and downsampled autocorrelation signals 4 and 3 (the N lower plot), where we set the downsampling number of packets as N ′ = 10 = 60. It can be seen that downsampled autocorrelation signals 4 and 6 have a smaller DTW value (i.e., 3.65) than downsampled autocorrelation signals 4 and 3 (i.e., 13.7). That is, ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:16 X. Wang et al. DTW Value with 3.6500 3 2 1 0 0 10 20 30 40 50 60 50 60 DTW Value with 13.7013 3 2 1 0 0 10 20 30 40 Packet Index Fig. 8. DTW results for downsampled autocorrelation signals 4 and 6 (the upper plot), and downsampled autocorrelation signal 4 and 3 (the lower plot), respectively. signals 4 and 6 are more similar, and more likely to belong to the same person. We also find that the downsampled autocorrelation signals have a high similarity in the center than that on the boundary, and it can reduce the phase shift values. Thus, the DTW value is a good measure of the distance between two downsampled autocorrelation signals. We need to compute the DTW values for all the downsampled autocorrelation signal pairs, which are then used in stable roommate matching. 3.4.2. Stable Roommate Matching. Since the CP decomposed signals are randomly indexed (see Fig. 7), we need to identify the pair for each person. With the DTW values for all downsampled autocorrelation signal pairs, we can model this problem as a stable roommate matching problem [Irving 1985; Feng et al. 2015b; Feng et al. 2015a]. There are a group of 2R signals, and each signal maintains a preference list of all other signals in the group, where the preference value for another signal is the inverse of the corresponding DTW value (i.e., distance). The problem is to pair the signals, such that there is no such a pair of signals that both of them have a more desired selection than their current selection, i.e., to find a stable matching [Irving 1985; Feng et al. 2015b; Feng et al. 2015a]. The proposed signal matching algorithm is presented in Algorithm 1. We first compute the autocorrelation of all decomposed signals. Then each autocorrelation signal populates its preference list with other autocorrelation signals according to the DTW values. The stable roommate matching algorithm is executed in two steps. In step 1, each signal proposes to other signals according to its preference list. If a signal m receives a proposal from another signal n, we implement the following strategy: (i) signal m rejects signal n if it has a better proposal from another signal; (ii) signal m accepts signal n’s proposal if it is better than all other proposals that signal m currently holds. Moreover, signal n stops to propose when its proposal is accepted, while it needs to continue to propose to other signals if being rejected. This strategy is implemented in step 1 of the signal matching algorithm, where we use f inish f lag to mark whether the current signal num is accepted or not. Moreover, variables accept num and propose num are used to record the current signal’s proposed number and proposing number, respectively. Also, variable scan num is used to record the current scanning signal number. After completing step 1, every signal holds a proposal or one signal has been rejected by other signals (this case hardly happens in TensorBeat, because the CP decomposition produces two very similar signals with high probability for each ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:17 ALGORITHM 1: Signal Matching Algorithm Input: Decomposed signals: S1 , S2 , · · · , S2R . Output: Matched signal pairs. Compute autocorrelation of all decomposed signals; Compute the DTW values for every pair of autocorrelation signals; Each autocorrelation signal sets its preference list using the DTW values; //step1; for signal num = 1 : 2R do Set f inish f lag = 0; Set scan num = signal num; while f inish f lag = 0 do if the proposal is the first one then Proposing signal’s propose num=the current choice; Set f inish f lag = 1; Proposed signal’s accept num = scan num; else if the signal prefers the former proposal then Reject the current proposal symmetrically; Propose to the next choice; else Accept the current proposal; Reject the former proposal symmetrically; scan num = proposed signal’s accept num; Propose to the next choice; end end end end for signal num = 1 : 2R do Reject signals that have less than accept num in every preference list symmetrically; end //step2; signal num = 1; while signal num < 2R + 1 do if propose num = accept num then signal num=signal num+1; else Let p1 be a signal whose preference list contains more than one element; while p sequence is not cyclic do qi = the second preference of pi ’s current list; pi+1 = the last preference of qi ’s current list; end Denote ps as the first element in the p sequence to be repeated and r as the length of the circle; for i = 1 : r do Reject matching (qs+i−2 , ps+i−1 ) symmetrically; end signal num = 1; end end Obtain signal matching pairs based on all processed preference lists; person). Then, we need to delete some elements in all the preference lists based on the following method, which is that if signal m is the first on signal n’s list , then signal n ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:18 X. Wang et al. is the last on signal m’s list. For the proposed algorithm, every signal can reject signals that have less than accept num in its preference list symmetrically (reject each other). An example is shown in Fig. 6. According to the DTW values, signals 1, 2, 3, 4, 5, and 6 have their preference lists as (2, 3, 5, 6, 4), (1, 3, 5, 6, 4), (5, 1, 2, 6, 4), (6, 5, 3, 2, 1), (3, 2, 1, 4, 6), and (4, 5, 3, 2, 1), respectively. When step 1 is executed, we have: Signal 1 proposes to 2, and signal 2 holds 1; Signal 2 proposes 1, and signal 1 holds; Signal 3 proposes to signal 5, and signal 5 holds; Signal 4 proposes to signal 6, and signal 6 holds; Signal 5 proposes to signal 3, and signal 3 holds; Signal 6 proposes to signal 4, and signal 4 holds. It is easy to find three pairs (1,2), (3,5), and (4,6). Although most of decomposed signals are paired in step 1, step 2 will still be necessary for the more challenging cases of much more breathing signals and NLOS environments. In step 2, we consider the reduced preference lists, where some of the lists have more than one signals. By implementing step 2, we can reduce the preference lists such that each signal only holds one proposal. The main idea is that we need to find some all-or-nothing cycles and symmetrically delete signals in the cycle sequence by rejecting the first and last choice pairs. The signal in the cycle accepts the secondary choice, thus obtaining a stable roommate matching. To find all-or-nothing cycles, let p1 be a signal with a preference list that contains more than one element, and generate the sequences such that qi = the second preference of pi ’s current list, and pi+1 = the last preference of qi ’s current list. After the cycle sequence generation, denote ps as the first element in the p sequence to be repeated. Then, we reject matching (qs + i − 2, ps + i − 1) for i = 1 to r symmetrically, where r is the length of the cycle. Finally, we can obtain signal matching pairs based on all processed preference lists. The computational complexity of Algorithm 1 is O(R2 ), because steps 1 and 2 each has a complexity of O(R2 ), respectively. 3.5. Breathing Rate Estimation 3.5.1. Signal Fusion and Autocorrelation. After obtaining the outcomes of the signal matching algorithm, TensorBeat next applies peak detection to estimate the breathing rates for multiple persons. Comparing to the FFT method, a higher resolution in the time domain can be achieved. To implement peak detection, we first need to combine the decomposed signal pairs for each person into a single signal, by taking the average of the signal pairs. Averaging can decrease the variance of the decomposed signals while preserving the same period. For example, Fig. 9 shows the fusion results based on the outcome of the signal matching algorithm, where three smoothly decomposed signals with different periods are obtained. To strengthen the accuracy of peak detection, we compute the autocorrelation function again for every fused signal. Fig. 10 shows the autocorrelation of the three fused signals. It can be seen that the length of data is increased from 300 to 600 and the number of peaks of every signal are also increased, which help to improve the estimation accuracy. 3.5.2. Peak Detection. Although breathing signal is generated by the small periodic chest movement of inhaling and exhaling, the phase difference data can effectively capture the breathing rate. Traditionally, estimation of breathing rates is achieved with FFT based methods. However, the FFT approach may have limited accuracy, because the frequency resolution of breathing signals is based on the window size of FFT. When the window size becomes larger, the accuracy will be higher, but the time domain resolution will be reduced. Also see Figs. 2 and 3 for the limitation of the FFT based approach for the multi-person scenario. Therefore, we leverage peak detection instead in TensorBeat system to achieve accurate breathing rate estimation for each of autocorrelations of fused signals. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:19 0.2 3 0 -0.2 0 2 50 100 Signal Index 150 200 Packet Index 250 300 1 Fig. 9. Fusion results based on the outcomes of the signal matching algorithm. 5 3 0 -5 0 2 100 200 300 Signal Index 400 Packet Index 500 600 1 Fig. 10. Autocorrelation of fused signals. For peak detection, the traditional method based on amplitude needs to detect the fake peak, which is not a real peak but has larger values than its two immediate neighboring points. To avoid the fake peak, a large moving window can be used to identify the real peak based on the maximum breathing periodicity. This method is not robust, which requires adjusting the window size. In TensorBeat, we only consider a smaller moving window of 7 samples wide. This is because we leverage the Hankel matrix and CP decomposition to smooth out the breathing curves, which hardly contains any fake peaks. Then, for the ith autocorrelation curve of fused signal, we seek all the peaks by determining whether or not the medium of the 7 samples in the moving window is the maximum value. Finally, we consider the median of all peak-to-peak intervals as the final period of the ith breathing signal, which is denoted as Ti . Finally, the estimated breathing rates can be computed as fi = 60/Ti , for i = 1 to R. 4. EXPERIMENTAL STUDY 4.1. Experiment Configuration In this section, we validate the TensorBeat performance with an implementation with 5 GHz Wi-Fi devices. To obtain 5 GHz CSI data, we use a desktop computer and a Dell ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:20 X. Wang et al. laptop as access point and mobile device, respectively, both of which are equipped with an Intel 5300 NIC. We use the desktop computer instead of the commodity routers, because none is equipped with the Intel 5300 NIC. The operating system is Ubuntu destop 14.04 LTS OS for both the access point and the mobile device. The PHY is the IEEE 802.11n OFDM system with QPSK modulation and 1/2 coding rate. Moreover, the access point is set in the monitor model and the distance between its two adjacent antennas is approximately 2.68 cm, which is half of the wavelength of 5GHz WiFi. Also, the mobile device is set in the injection model and uses one antenna to transmit data. Moreover, we use omnidirectional antennas for both the receiver and transmitter to estimate breathing signs beats. With the packet injection technique with LORCON version 1, we can obtain 5 GHz CSI data from the three antennas of the receiver. Our experimental study is with up to five persons over a period of six months. The experimental scenarios include a computer laboratory, a through-wall scenario, and a corridor, as shown in Fig. 11. The first scenario is within a 4.5 × 8.8 m2 laboratory, where both single person and multi-person breathing rate estimation experiments are conducted. There are lots of tables and desktop computers crowded in the laboratory, which block parts of the LOS paths and form a complexity radio propagation environment. The second setup is a through-wall environment, where single person breathing rate estimation is tested due to the relatively weaker signal reception. The person is on the transmitter side, and the receiver is behind a wall in this experiment. The third scenario is a long corridor of 20 m, where the maximum distance between the receiver and transmitter is 11 m in the experiment. This scenario is still considered for single person breathing rate monitoring. We use a NEULOG Respiration to record the ground truths for single person breathing rates. The single person breathing rates estimation can be easily implemented by removing the signal matching algorithm, because there are only two decomposed signals after CP decomposition in this case. For muti-person breathing rate estimation in the first scenario, all persons participating in the experiment record their breathing rates by using a metronome smartphone application with 1 bpm accuracy at the same time. We consider five persons are stationary for LOS and NLOS environments for breathing monitoring. Moreover, there are no other persons in the breathing measurement area. For multi-person breathing rate estimation, we need to define a proper metric for evaluating TensorBeat’s performance. For R estimated breathing rates [f1 , f2 , ...fR ], the ith breathing rate estimation error, Ei , is defined as Ei = fi − fˆi , for i = 1, 2, · · · , R, (34) where fˆi is the ground truth of the ith breathing rate. We also define a new metric termed success rate, denoted as SR, which is defined as SR = N {maxi {Ei } < 2bpm} × 100%, N {E} (35) where N {maxi {Ei } < 2bpm} means the number of repeated experiments of the maximum breathing rate error less than 2 bpm, and N {E} is the number of repeated experiments. We adopt the success rate metric because there are weak signals for multi-person experiments in indoor experiments at different locations, and sometimes a breathing signal may not be successfully detected [Wang et al. 2016]. 4.2. Performance of Breathing Estimation In Fig. 12, we present the cumulative distribution functions (CDF) of the estimation errors for single person breathing rate detection for three different experiment scenarios. We can see that for TensorBeat, high estimation accuracy of breathing rates ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:21 '()+#,--.(/01/+,-2)-,0/, &!&# !"# %!$# 3.4,.5.(/01/+,-2)-,0*/, $!$# Fig. 11. Experimental setup: computer laboratory, through-wall, and long corridor scenarios. 1 0.8 Laboratory Corridor Through-wall CDF 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Estimation Error (bpm) Fig. 12. Performance of single person breathing rate estimation in the computer laboratory, through-wall, and long corridor scenarios. can be achieved in all the three scenarios. The maximum estimation error is less than 0.9 bpm. Moreover, it is noticed that 50% of the tests for the computer laboratory experiment have errors less than aboout 0.19 bpm, while the tests for the corridor and through-wall scenarios have errors less than approximately 0.25 bpm and 0.35 bpm, respectively. Thus, the performances in the laboratory setting is better than that in the corridor and through-wall scenarios. This is because the laboratory has a smaller space and the breathing signal is stronger than that of other two cases with larger attenuation due to the long distance and the wall. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:22 X. Wang et al. 1 0.8 CDF 0.6 One Person Two Persons Three Persons Four Persons Five Persons 0.4 0.2 0 0 0.5 1 1.5 2 Estimation Error (bpm) Fig. 13. Performance of breathing rate estimation for different number of persons (computer laboratory). Fig. 13 presents the performance of breathing rate estimation for different number of persons. It is noticed that higher accuracy is achieved for the single person test, where approximately 96% of the test data have an estimation error less than 0.5 bpm. The five-person test has the worse performance, where approximately 62% of the test data have an estimation error less than 0.5 bpm. Moreover, we fine that the performances of the two-person and three-person tests are similar, both of which can have an error smaller than 0.5 bpm for 93% of the test data. Generally, when the number of persons is increased, the performance of breathing rate estimation gets worse. In fact, when the number of breathing signals is increased, the distortion of the mixed received signal will become larger, thus leading to high estimation errors. Fig. 14 plots the success rates for different number of persons. We find that although the success rate for one person is the highest, there are still few of test data that cannot obtain high accuracy breathing rates estimation. These test data should come from different locations in the indoor environments, where parts of the received signals are severely distorted. In fact, we find that low phase difference usually occurs when the SNR is low. On the other hand, we can see that breathing rate estimation for two persons also has a high success rate, because the probability for two persons to have exactly the same breathing rate is very low. When the number of persons is increased, the chance of getting two close breathing rates becomes higher. Even in this case, TensorBeat can still effectively separate them with a high success probability. With the increase of the number of persons, the success rate for TensorBeat system decreases. The reason is that each breathing rate is more likely to cover each other and the strength of the received signal becomes lower. From Fig. 14, we can see that the success rate is about 82.4% when the number of persons is five. Fig. 15 shows the success rate for different sampling rates. In this experiment, there are four persons and the window size is set to 30 s. From Fig. 15, we can see that with the increase of the number of sampling rates, the success rate is also increased. It is noticed that the success rates for 5 Hz and 30 Hz are approximately 70% and 90%, respectively. As the sampling rate is increased, the length of the data for CP decomposition is increased for the 30 s window size case, which helps to improve the estimation accuracy. Furthermore, we find that the performance becomes stable when the sampling rate exceeds 20 Hz, indicating that a sampling rate of 20 Hz is sufficient ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:23 100 Success Rate (%) 80 60 40 20 0 1 2 3 4 5 The Number of Persons Fig. 14. Success rates for different number of persons (computer laboratory). 100 Success Rate (%) 80 60 40 20 0 5 10 15 20 25 30 Sampling Rate (Hz) Fig. 15. Success rates for different sampling rates (computer laboratory). for CP decomposition. Thus, we set the sampling rate to 20 Hz for the TensorBeat experiments. Fig. 16 plots the success rates for different window sizes. This experiment is for the computer laboratory scenario with four persons and the sampling rate is set to 20 Hz. From Fig. 16, we can see that the success rate is greatly increased by increasing the window size of the Hankel matrix from 15 s to 30 s. This is because Hankelizaiton will take half of the data to smooth the phase difference signal, which reduces the resolution in the time domain. Thus, we need to increase the window size to improve the estimation accuracy. Furthermore, the change of success rate is small for window sizes from 30 s to 45 s. Thus, we select the window size of 30 s for the TensorBeat experiments. Finally we examine the impact of LOS and NLOS scenarios. The success rates are plotted in Fig. 17. In this experiment, we consider the challenge condition of the NLOS scenario, where all the persons stay on the LOS path between the transmitter and receiver, i.e., they form a straight line and block each other. From Fig. 15, we find ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:24 X. Wang et al. 100 Success Rate (%) 80 60 40 20 0 15 20 25 30 35 40 45 The Size of Window (s) Fig. 16. Success rates for different window sizes (computer laboratory). 100 LOS NLOS Success Rate (%) 80 60 40 20 0 2 3 4 5 The Number of Persons Fig. 17. Success rates for (i) when multiple persons form a line in the LOS path between the transmitter and receiver; (ii) when multiple persons are in scattered around (computer laboratory). that the performances for LOS and NLOS are nearly the same for the cases of two or three persons, where high estimation accuracy can be achieved. This is due to the WiFi multipath effect, which is regarded harmful in general but becomes helpful in breathing rate estimation when tensor decomposition is used. The breathing signal of every person can still be captured at the receiver from the phase difference data. However, when the number of the persons is further increased, the success rate will decrease quickly. In fact, the strength of the breathing signals for some persons will become too weak to be detected when there are too many people blocking each other. 5. RELATED WORK This work is closely related to sensor based and RF signal based vital signs monitoring, as well as CSI based indoor localization and human activity recognition, which are discussed in the following. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. TensorBeat: Tensor Decomposition for Monitoring Multi-Person Breathing Beats with Commodity WiFiA:25 Sensor-based vital signs monitoring with wearable and smart devices employs the special hardware attached the person body to monitor breathing and heart rate data. The capnometer can measure carbon dioxide (CO2) concentrations in respired gases, which are employed to monitor patients breathing rate in hospital. However, it is uncomforted for patients to wear them, which are thus leveraged in clinical environments [Mogue and Rantala 1988]. Photoplethysmography (PPG) is an optical technique to monitor the blood volume changes in the tissues by measuring light absorption changes, thus requiring the sensors attached to persons finger such as pulse oximeters [Shariati and Zahedi 2005]. In addition, smartphone can use the camera to detect light changes from the video frames which can extract the PPG signal to monitor the heart rate [Scully et al. 2010]. Moreover, the smartphones can also estimate breathing rate by using the built-in accelerometer, gyroscope [Aly and Youssef 2016] and microphone [Ren et al. 2015], which require persons to put smartphones near-by and wear sensors for breathing monitoring. However, these techniques based on sensors cannot be applied for remote monitoring vital signs. RF based systems for vital signs monitoring use wireless signals to track the breathing-induced chest change of a person, which are mainly based on radar and WiFi techniques. For radar based vital signals monitoring, Vital-Radio employs frequency modulated continuous wave (FMCW) radar to estimate breathing and heart rates, even for two person subjects in parallel [Adib et al. 2015]. But the system requires a custom hardware with a large bandwidth from 5.46 GHz to 7.25 GHz. For WiFi based vital signs monitoring, UbiBreathe system employ WiFi RSS for breathing rate monitoring, which, however, requires the device placed in the line of sight path between the transmitter and the receiver for estimating the breathing rate [Abdelnasser et al. 2015]. Moreover mmVital based on RSS can use 60 GHz millimeter wave (mmWave) signal for breathing and heart rates monitoring with the larger bandwidth about 7GHz, which cannot monitor the longer distance and require high gain directional antennas for the transmitter and the receiver [Yang et al. 2016][Gong et al. 2010]. Recently, the authors leverage the amplitudes of CSI data to monitoring vital signs [Liu et al. 2015]. This work is mainly to track the vital signs when a person is sleeping, which is limited for monitoring a maximum of two persons at the same time. In additional to vital signs monitoring, recently, CSI based sensing systems have also been used for indoor localization and human activity recognition [Cushman et al. 2016]. CSI-based fingerprinting systems have been proposed to obtain high localization accuracy. FIFS is the first work to uses the weighted average of CSI amplitude values over multiple antennas for indoor localization [Xiao et al. 2012]. To exploit the diversity among the multiple antennas and subcarriers, DeepFi leverage 90 CSI amplitude data from the three antennas with a deep autoencoder network for indoor localization [Wang et al. 2015]. Also, PhaseFi leverages calibrated CSI phase data for indoor localization based on deep learning [Wang et al. 2015]. Different from CSI-based fingerprinting techniques, SpotFi system leverages a super-resolution algorithm to estimate the angle of arrival (AoA) of multipath components for indoor localization based on CSI data from three antennas [Kotaru et al. 2015]. On the other hand, E-eyes system leverages CSI amplitude values for recognizing household activities such as washing dishes and taking a shower [Wang et al. 2014]. WiHear system employs specialized directional antennas to measure CSI changes from lip movement for determining spoken words [Wang et al. 2014]. CARM system considers a CSI based speed model and a CSI based activity model to build the correlation between CSI data dynamics and a given human activity [Wang et al. 2015]. Although CSI based sensing are effective for indoor localization and activity recognitions, there are few works for using CSI phase difference data to detect multiple persons behaviors at the same time. ACM Transactions on Intelligent Systems and Technology, Vol. V, No. N, Article A, Publication date: January YYYY. A:26 X. Wang et al. The TensorBeat system is motivated by these interesting prior works. To the best of our knowledge, we are the first to leverage CSI phase difference data for multiple persons breathing rate estimation. We are also the first to employ tensor decomposition for RF sensing based vital signs monitoring, which can be also employed for indoor localization and human activity recognition. 6. 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SAMPLE-LEVEL CNN ARCHITECTURES FOR MUSIC AUTO-TAGGING USING RAW WAVEFORMS Taejun Kim1 , Jongpil Lee2 , Juhan Nam2 arXiv:1710.10451v2 [cs.SD] 14 Feb 2018 1 School of Electrical and Computer Engineering, University of Seoul, 2 Graduate School of Culture Technology, KAIST, ktj7147@uos.ac.kr, {richter, juhannam}@kaist.ac.kr ABSTRACT Recent work has shown that the end-to-end approach using convolutional neural network (CNN) is effective in various types of machine learning tasks. For audio signals, the approach takes raw waveforms as input using an 1-D convolution layer. In this paper, we improve the 1-D CNN architecture for music auto-tagging by adopting building blocks from state-of-the-art image classification models, ResNets and SENets, and adding multi-level feature aggregation to it. We compare different combinations of the modules in building CNN architectures. The results show that they achieve significant improvements over previous state-of-the-art models on the MagnaTagATune dataset and comparable results on Million Song Dataset. Furthermore, we analyze and visualize our model to show how the 1-D CNN operates. Index Terms— convolutional neural networks, music auto-tagging, raw waveforms, multi-level learning 1. INTRODUCTION Time-frequency representations based on short-time Fourier transform, often scaled in a log-like frequency such as melspectrogram, are the most common choice of input in the majority of state-of-the-art music classification algorithms [1, 2, 3, 4, 5]. The 2-dimentional input represents acoustically meaningful patterns well but requires a set of parameters, such as window size/type and hop size, which may have different optimal settings depending on the type of input signals. In order to overcome the problem, there have been some efforts to directly use raw waveforms as input particularly for convolutional neural networks (CNN) based models [6, 7]. While they show promising results, the models used large filters, expecting them to replace the Fourier transform. Recently, Lee et. al. [8] addressed the problem using very small filters and successfully applied the 1D CNN to the music autotagging task. Inspired from the well-known VGG net that uses very small size of filters such as 3 × 3, [9], the sample-level CNN model was configured to take raw waveforms as input and have filters with such small granularity. A number of techniques to further improve performances of CNNs have appeared recently in image domain. He et. al. introduced ResNets which includes skip connections that enables a very deep CNN to be effectively trained and makes gradient propagation fluent [10]. Using the skip connections, they could successfully train a 1001-layer ResNet [11]. Hu et. al proposed SENets [12] which includes a building block called Squeeze-and-Excitation (SE). Unlike other recent approaches, the block concentrates on channel-wise information, not spatial. The SE block adaptively recalibrates feature maps using a channel-wise operation. Most of the techniques were developed in the field of computer vision but they are not fully adopted for music classification tasks. Although there were a few approaches to readily apply them to audio domain [7, 13]. They used 2D representations as input [13] or used large filters for the first 1D convolutional layer [7]. On the other hand, some methods are concerned with overall architecture of the model rather than designing a fine-grained building block [2, 14, 15, 16, 17]. Specifically, multi-level feature aggregation combines several hidden layer representations for final prediction [2, 14]. They significantly improved the performance in music auto-tagging by taking different levels of abstractions of tag labels into account. In this paper, we explore the building blocks of advanced CNN architectures, ResNets and SENets, based on the sample-level CNN for music auto-tagging. Also, we observe how the multi-level feature aggregation affects the performance. The results show that they achieve significant improvements over previous state-of-the-art models on the MagnaTagATune dataset and comparable results on Million Song Dataset. Furthermore, we analyze and visualize our model built with the SE blocks to show how the 1D CNN operates. The results show that the input signals are processed in a different manner depending on the level of layers. 2. ARCHITECTURES All of our models are based on the sample-level 1D CNN model [8], which is constructed with the basic block shown in Figure 1(b). Every filter size of the convolution layers is fixed raw waveform strided conv 19683 × 128 1D convolutional block ×9 2187× 128 729 × 256 243 × 256 81 × 256 FC ×2 9 × 256 3 × 256 1 × 512 512 global max pooling global max pooling 27 × 256 multi-level feature aggregation 6551 × 128 256 Conv1D Conv1D BatchNorm BatchNorm relu relu Conv1D Conv1D BatchNorm BatchNorm 59049 × 1 MaxPool relu relu MaxPool T×C GlobalAvgPool 1×C FC 1×αC relu Dropout Dropout Conv1D Conv1D BatchNorm BatchNorm T×C FC T×C 1×C GlobalAvgPool sigmoid relu Scale 1×C FC MaxPool T×C 1×αC relu FC T×C 1×C sigmoid Scale 256 T×C 50 relu MaxPool tag prediction (a) Overview of the architecture (b) Basic block [8] (c) SE block (d) Res-n block (e) ReSE-n block Fig. 1. The proposed architecture for music auto-tagging. (a) The models consist of a strided convolutional layer, 9 blocks, and two fully-connected (FC) layers. The outputs of the last three blocks are concatenated and then used as input of the last two FC layers. Output dimensions of each block (or layer) are denoted inside of them (temporal×channel). (b-e) The 1D convolutional building blocks that we evaluate. to three. The differences between the sample-level CNN and ours are the use of advanced building blocks and multi-level feature aggregation. In this section, we describe the details. 2.1. 1D convolutional building blocks 2.1.1. SE block We utilize the SE block from SENets to increase representational power of the basic block. As shown in Figure 1(c), we simply attached the SE block to the basic block. The SE block recalibrates feature maps from the basic block through two operations. One is squeeze operation that aggregates a global temporal information into channel-wise statistics using global average pooling. The operation reduces the temporal dimensionality (T ) to one, averaging outputs from each channel. The other is excitation operation that adaptively recalibrates feature maps of each channel using the channel-wise statistics from the squeeze operation and a simple gating mechanism. The gating mechanism consists of two fully-connected (FC) layers that compute nonlinear interactions among channels. Finally, the original outputs from the basic block are rescaled by channel-wise multiplication between the feature map and the sigmoid activation of the second FC layer. Unlike the original SE block in SENets, our excitation operation does not form a bottleneck. On the contrary, we expand the channel dimensionality (C) to αC at the first FC layer, and then reduce the dimensionality back to C at the second layer. We set the amplifying ratio α to be 16, after a grid search with α = [2−3 , 2−2 , ..., 26 ]. 2.1.2. Res-n block Inspired by skip connections from ResNets, we modified the basic block by adding a skip connection as shown in Figure 1(d). Res-n denotes that the block uses n convolutional layers where n is one or two. Specifically, Res-2 is a block that has the additional layers denoted by the dotted line in Figure 1(d), and Res-1 is a block that has a skip connection only. When the block uses two convolutional layers (Res-2), we add a dropout layer (with a drop ratio of 0.2) between two convolutions to avoid overfitting. This technique was firstly introduced at WideResNets [18]. 2.1.3. ReSE-n block The ReSE-n block is a combination of the SE and Res-n blocks as shown in Figure 1(e). n denotes the number of convolutional layers in the block, where n is also one or two. A dropout layer is inserted when n is two. 2.2. Multi-level feature aggregation Fig. 1(a) shows the multi-level feature aggregations that we configured. The outputs of the last three blocks are concate- Table 1. AUCs of CNN architectures on MTAT. “multi” and “no multi” indicates if the multi-level feature aggregation is used or not. † denotes using a weight decay of 10−4 . MTAT Block Basic [8] SE Res-1 Res-2 ReSE-1 ReSE-2 multi no multi 0.9077 0.9111 0.9037 0.9098 0.9053 0.9113† 0.9055 0.9083 0.9048 0.9061 0.9066 0.9102† nated and then delivered to the FC layers. Before the concatenation, temporal dimensions of the outputs are reduced to one by a global max pooling. Unlike [2], the concatenation occurs while training the CNN and the average pooling over the whole audio clip (i.e. 29 second long), which followed by the global max pooling, is not included. 3. EXPERIMENTS 3.1. Datasets We evaluated the proposed architectures on two datasets, MagnaTagATune (MTAT) dataset [19] and Million Song Dataset (MSD) annotated with the Last.FM tags [20]. We split and filtered both of the datasets, following the previous work [5, 6, 8]. We used the 50 most frequent tags. All songs are trimmed to 29 seconds long, and resampled to 22050Hz as needed. The song is divided into 10 segments of 59049 samples. To evaluate the performance of music autotagging which is a multi-class and multi-label classification task, we computed the Area Under the Receiver Operating Characteristic curve (AUC) for each tag and computed the average across all 50 tags. During the evaluation, we average predictions across all segments. 3.2. Implementation details All the networks were trained using SGD with Nesterov momentum of 0.9 and mini-batch size 23. The initial learning rate is set to 0.01, decayed by a factor of 5 when a validation loss is on a plateau. None of the regularizations are used on MSD. A dropout layer of 0.5 was inserted before the last FC layer on MTAT. For all building blocks, we evaluated either with or without the multi-level feature aggregation. Since the training for MSD takes much time longer than MTAT, we explored the architectures mainly on MTAT, and then trained the two best models on MSD. Code and models built with TensorFlow and Keras are available at the link1 . 1 https://github.com/tae-jun/resemul Table 2. AUCs of state-of-the-art models on MTAT and MSD. † denotes that the model used an ensemble of three. Model Bag of multi-scaled features [3] End-to-end [6] Transfer learning [4] Persistent CNN [21] Time-Frequency CNN [22] Timbre CNN [23] 2D CNN [5] CRNN [1] Multi-level & multi-scale [2] SampleCNN multi-features [14] SampleCNN [8] SE [This work] ReSE [This work] MTAT 0.8980 0.8815 0.8800 0.9013 0.9007 0.8930 0.8940 0.9017† 0.9064† 0.9055 0.9111 0.9113 MSD 0.8510 0.8620 0.8878† 0.8842 0.8812 0.8840 0.8847 4. RESULTS AND DISCUSSION 4.1. Comparison of the architectures Table 1 summarizes the evaluation results of compared CNN architectures on the MTAT dataset. They show that the SE block is more effective than the Res-n blocks, increasing the performance of the basic block for all cases. In the Res-n block, only adding the skip connection to the basic block (Res-1) actually decreases the performance. The combination of the SE and the Res-2 improves it slightly more. However, a training time of the ReSE-2 is 1.8 times longer than the basic block whereas the SE block only 1.08 times longer. Thus, if the training or prediction time of the models is important, the SE model will be preferred to the ReSE-2. The effect of the multi-level aggregation is valid for the majority of the models. We obtained two best results in Table 1 by using the multi-level aggregation. 4.2. Comparison with state-of-the-arts Table 2 compares previous state-of-the-art models in music auto-tagging with our best models, the SE block and ReSE2 block, each with multi-level aggregation. On the MTAT dataset, our best models outperform all the previous results. On MSD, they are not the best but are comparable to the second-tier. 5. ANALYSIS OF EXCITATION To lay the groundwork for understanding how 1D CNNs operate, we analyze the sigmoid activations of excitations in the SE blocks at different levels graphically and quantitatively. In this section, we observe how the SE blocks recalibrate channels, depending on which level they exist. The blocks used for the analysis are from the SE model using the multi-level Table 3. Co-occurrence matrix of the tags used in Figure 2 1st block classical metal dance classical 704 0 1 metal 0 166 0 dance 1 0 153 5th (mid) block std 0.05 0.04 0.03 0.02 1 2 3 4 5 6 7 8 9 block level 9th (last) block Fig. 2. Visualization of the sigmoid activations of excitations in the SE model. The channel index was sorted by the average of the activations. feature aggregation and they were trained on MTAT. The activations were extracted from its test set. The activations were averaged over all segments separately for each tag. 5.1. Graphical analysis For this analysis, we chose three tags, classical, metal, and dance that are not similar to each other as shown in Table 3. Figure 2 shows the average sigmoid activations in the SE blocks for the songs with the three tags. The different levels of activations indicate that the SE blocks process input audio differently depending on the tag (or genre) of the music. That is, every block in Figure 2 fires different patterns of activations for each tag at a specific channel. This trend is strongest at the first block (top), weakest at the mid block (middle), and becomes stronger again at the last block (bottom). This trend is somewhat different from what are observed in the image domain [12], where the exclusiveness of average excitation for input with different labels are monotonically increasing along the layers. Specifically, the first block fires high activations for classical, low ones for dance, and even lower ones for metal for the majority of the channels. On the other hand, the activations of the last block vary depending on the tags. For example, the activations of metal are high at some channels but low at the others, which makes the activations noisy even though they are sorted. We can interpret this result as follows. The first block normalizes the loudness of the audios because the block fires high activations for classical music, which tend to have small volume, and low activa- Fig. 3. Standard deviations (std) of the activations of excitations across all tags along each layer. tions for metal music, which tend to have large volume. Also, the middle block processes common features among them as they have similar levels of activations. Finally, the noisy exclusiveness in the last block indicates that they effectively discriminate the music with different tags. 5.2. Quantitative analysis We assure the exclusiveness trend by measuring standard deviations of the activations across all tags at every level. Figure 3 shows that the higher the standard deviation is, the more the block responses to the song differently according to its tag. The result shows that the standard deviation is highest at the first block, it drops and stays low up to the 5th block and then increases gradually until the last block. That is, the four lower blocks except the the bottom one (2 to 5) tend to handle general features whereas the four upper blocks (6 to 9) tend to progressively more discriminative features. 6. CONCLUSION We proposed 1D convolutional building blocks based on the previous work, the sample-level CNN, ResNets, and SENets. The ReSE block, which is a combination of the three models, showed the best performance. Also, the multi-level feature aggregation showed improvements on the majority of the building blocks. Through the experiments, we obtained stateof-the-art performance on the MTAT dataset and high-ranked results on MSD. In addition, we analyzed the activations of excitation in SE model to understand the effect. With this analysis, we could observe that the SE blocks process nonsimilar songs exclusively and how the different levels of the model process the songs in a different manner. 7. REFERENCES [1] Keunwoo Choi, György Fazekas, Mark Sandler, and Kyunghyun Cho, “Convolutional recurrent neural networks for music classification,” in ICASSP. IEEE, 2017, pp. 2392–2396. [2] Jongpil Lee and Juhan Nam, “Multi-level and multiscale feature aggregation using pretrained convolutional neural networks for music auto-tagging,” IEEE Signal Processing Letters, vol. 24, no. 8, pp. 1208–1212, 2017. [3] Sander Dieleman and Benjamin Schrauwen, “Multiscale approaches to music audio feature learning,” in International Society of Music Information Retrieval Conference (ISMIR), 2013, pp. 116–121. 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On the linear quadratic problem for systems with time reversed Markov jump parameters and the duality with filtering of Markov jump linear systems arXiv:1611.07558v1 [cs.SY] 22 Nov 2016 Daniel Gutierrez and Eduardo F. Costa Abstract— We study a class of systems whose parameters are driven by a Markov chain in reverse time. A recursive characterization for the second moment matrix, a spectral radius test for mean square stability and the formulas for optimal control are given. Our results are determining for the question: is it possible to extend the classical duality between filtering and control of linear systems (whose matrices are transposed in the dual problem) by simply adding the jump variable of a Markov jump linear system. The answer is positive provided the jump process is reversed in time. I. I NTRODUCTION In this note we study a class of systems whose parameters are driven by a time reversed Markov chain. Given a time horizon ℓ and a standard Markov chain {η(t), t = 0, 1, . . .} taking values in the set {1, 2, . . . , N }, we consider the process θ(t) = η(ℓ − t), 0 ≤ t ≤ ℓ, (1) and the system ( x(t + 1) = Aθ(t) x(t) + Bθ(t) u(t), Φ: x(0) = x0 , 0 ≤ t ≤ ℓ − 1, (2) where, as usual, x represents the state variable of the system and u is the control variable. These systems may be encountered in real world problems, specially when a Markov chain interacts with the system parameters via a first in last out queue. An example consists of drilling sedimentary rocks whose layers can be modelled by a Markov chain from bottom to top as a consequence of their formation process. The first drilled layer is the last formed one. Another example is a DC-motor whose brush is grind by a machine subject to failures, leaving a series of imprecisions on the brush width that can be described by a Markov chain, so that the last failure will be the first to affect the motor collector depending on how the brush is installed. One of the most remarkable features of system Φ is that it provides a dual for optimal filtering of standard Markov jump linear systems (MJLS). In fact, if we consider a quadratic cost functional for system Φ with linear state feedback, leading to an optimal control problem [4] that we call time reversed Markov jump linear quadratic problem (TRM-JLQP), then we show that the solution is identical to the gains of the linear minimum mean square estimator This work was supported by FAPESP, CNPQ and Capes. Authors are with the Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Brasil dgutip@icmc.usp.br, efcosta@icmc.usp.br (LMMSE) formulated in [8], [9], with time-reversed gains and transposed matrices. In perspective with existing duality relations, the one obtained here is a direct generalization of the well known relation between control and filtering of linear time varying systems as presented for instance in [10, Table 6.1], or also in [3], [16], [17] in different contexts. As for MJLS, the duality between control and filtering have been considered e.g. in [2], [5], [7], [8], [12], [13], [15], while purely in the context of standard MJLS, thus leading to more complex relations involving certain generalized coupled Riccati difference equations. Here, the duality follows naturally from the simple reversion of the Markov chain given in (1), with no extra assumptions nor complex constructions. Another interesting feature of Φ is that the variable E{x(t)x(t)′ · 1{θ(t)=i} }, which is commonly used in the literature of MJLS, [1], [8], [11], [14], evolves along time t according to a time-varying linear operator, as shown in Remark 1, in a marked dissimilarity with standard MJLS. This motivated us to employ X(k), the conditioned second moment of x(k), leading to time-homogeneous operators. The contents of this note are as follows. We present basic notation in Section II. In Section III we give the recursive equation describing X, which leads to a stability condition involving the spectral radius of a time-homogeneous linear operator. In Section IV, we formulate and solve the TRMJLQ problem, following a proof method where we decompose X into two components as to handle θ that are visited with zero probability. The duality with the LMMSE then follows in a straightforward manner, as presented in Section V. Concluding remarks are given in Section VI. II. N OTATION AND THE SYSTEM SETUP Let ℜn be the n-dimensional euclidean space and ℜm,n be the space formed by real matrices of dimension m by n. We write C m,n to represent the Hilbert space composed of N real matrices, that is Y = (Y1 , . . . , YN ) ∈ C m,n , where Yi ∈ ℜm,n , i = 1, . . . , N . The C n,m equipped Pspace N ′ with the inner product hY, Zi = Tr(Y i Zi ), where i=1 Tr(·) is the trace operator and the superscript ′ denotes the transpose, is a Hilbert space. The inner product induces the norm kY k = hY, Y i1/2 . If n = m, we write simply C n . The mathematical operations involving elements of C n,m , are used in element-wise fashion, e.g. for Y and Z in C n we have Y Z = (Y1 Z1 , . . . , YN ZN ), where Yi Zi is the usual matrix multiplication. Similarly, for a set of scalars α = (α1 , . . . , αN ) ∈ C 1 we write αY = (α1 Y1 , . . . , αN YN ). Regarding the system setup, it is assumed throughout the paper that x0 ∈ ℜn is a random variable with zero mean satisfying E{x0 x′0 } = ∆. We have x(t) ∈ ℜn and u(t) ∈ ℜm . The system matrices belong to given sets A, E ∈ C n , B ∈ C n,m , C ∈ C s,n and D ∈ C s,m with Ci′ Di = 0 and Di′ Di > 0 for each i = 1, . . . , N . We write πi (t) = Pr(θ(t) = i), where Pr(·) is the probability measure; π(t) is considered as an element of C 1 , that is, π(t) = (π1 (t), . . . , πN (t)). πi stands for the limiting distribution of the Markov chain η when it exists, in such a manner that πi = limℓ→∞ πi (0). Also, we denote by P = [pij ], i, j = 1, . . . , N the transition probability matrix of the Markov chain η, so that for any t = 1, . . . , ℓ, Pr(θ(t − 1) = j \| θ(t) = i) = = Pr(η(ℓ − t + 1) = j \| η(ℓ − t) = i) = pij . UZ,i (Y ) = VZ,i (Y ) = Di (Y ) = pij Zj Yj Zj′ j=1 Zi′ Di (Y N X )Zi , (3) pji Yj . OF SYSTEM Φ Let E{·} be the expected value of a random variable. We consider the conditioned second moment of x(t) defined by Xi (t) = E{x(t)x(t)′ \| θ(t) = i}, t = 0, 1, . . . . (4) Lemma 3.1: Consider the system Φ with u(t) = 0 for each t. The conditioned second moment X(t) ∈ C n is given by X(0) = (∆, . . . , ∆) and X(t + 1) = UA (X(t)), t = 0, 1, . . . ℓ − 1. (5) Proof: For a fixed, arbitrary i ∈ {0, . . . , N }, note that Xi (0) = E{x0 x′0 \| θ(0) = i} = E{x0 x′0 } = ∆. From (2), (4) and the total probability law we obtain: Xi (t + 1) = E{Aθ(t) x(t)x(t)′ A′θ(t) \| θ(t + 1) = i} = N X ′ E{Aθ(t) x(t)x(t) A′θ(t) = E{Γ(η(ℓ), . . . , η(ℓ − t))\|η(ℓ − t) = j, η(ℓ − t − 1) = i} · Prob(η(ℓ − t) = j \| η(ℓ − t − 1) = i} = E{Γ(η(ℓ), . . . , η(ℓ − t))\|η(ℓ − t) = j}pij = E{Γ(θ(0), . . . , θ(t)) \| θ(t) = j}pij , then by replacing Γ with Aθ (t)x(t)x(t)′ A′θ(t) and applying the above in (6) yields Xi (t + 1) = · 1θ(t)=j \| θ(t + 1) = i}. N X pij Aj Xj (t)A′j = UA,i (X(t)), j=1 which completes the proof. Remark 1: Let W (t) ∈ C n , t = 0, . . . , ℓ be given by Wi (t + 1) = N X pij j=1 πi (t + 1) Aj Wj (t)A′j . πj (t) j=1 In order to compute the right hand side of (6), we need the following standard Markov chain property: for any function (8) Note that the Markov chain measure appears explicitly, leading to a time-varying mapping from W (t) to W (t + 1). The only exception is when the Markov chain is reversible, in which case the facts that πj pji = πi pij and that the Markov chain starts with the invariant measure (by definition) yield πi (t + 1) πi = pji , = pij πj (t) πj in which case W evolves exactly as in a standard MJLS. The following notion is adapted from [8, Chapter 3]. Definition 3.1: We say that the system Φ with u(t) = 0 is mean square stable (MS-stable), whenever lim E kx(ℓ)k2 = 0. ℓ→∞ This is equivalent to say that the variable X(ℓ) converges to zero as ℓ goes to infinity, leading to the following result. Theorem 3.1: The system Φ with u(t) = 0 is MS-stable if and only if the spectral radius of UA is smaller than one. IV. T HE TRM-JLQ PROBLEM Let the output variable y given by y(ℓ) = Eθ(ℓ) x(ℓ) and y(t) = Cθ(t) x(t) + Dθ(t) u(t), t = 0, . . . , ℓ − 1. The TRM-JLQ consists of minimizing the mean square of y with ℓ stages, as usual in jump linear quadratic problems, ) ( ℓ X 2 (9) min E ky(t)k . u(0),...,u(ℓ−1) (6) (7) This variable is commonly encountered in the majority of papers dealing with (standard) MJLS. However, calculations similar to that in Lemma 3.1 lead to pij j=1 III. P ROPERTIES E{Γ(θ(0), . . . , θ(t)) · 1θ(t)=j \| θ(t + 1) = i} = E{Γ(η(ℓ), . . . , η(ℓ − t)) · 1η(ℓ−t)=j \| η(ℓ − t − 1) = i} Wi (t) = E{x(t)x(t)′ · 1θ(t)=i }. No additional assumption is made on the Markov chain, yielding a rather general setup that includes periodic chains, important for the duality relation given in Remark 2. We shall deal with linear operators UZ , VZ , D : C n → C n . We write the i-th element of UZ (Y ) by UZ,i (Y ) and similarly for the other operators. For each i = 1, . . . , N , we define: N X Γ : θ(0), . . . , θ(t) → ℜn,n we have t=0 Regarding the information structure of the problem, we assume that θ(t) is available to the controller, that the control is in linear state feedback form, u(t) = Kθ(t) (t)x(t), t = 0, 1, . . . ℓ − 1, (10) where K(t) ∈ C m,n is the decision variable, and that one should be able to compute the sequence K(0), . . . , K(ℓ − 1) prior to the system operation, that is, K(t) is not a function of the observations (x(s), θ(s)), 0 ≤ s ≤ ℓ. The conditioned second moment X for the closed loop system is of much help in obtaining the solution. The recursive formula for X follows by a direct adaptation of Lemma 3.1, by replacing A ∈ C n with its closed loop version Ai (t) = Ai + Bi Ki (t). (11) Then, V t (X) = hP (t), Xi (19) op and K (t) = M (t), t = 0, . . . , ℓ. Proof: We apply the dynamic programming approach for the costs defined in (16) and the system in (12), whose state is the variable X := X(t). It can be checked that U is the adjoint operator of V, and consequently V t (X) = min hπ(t)Q(t), Xi + hP (t + 1), UA(t) (X)i, K(t) = min hπ(t)Q(t) + A(t)′ D(P (t + 1))A(t), Xi. (20) K(t) Lemma 4.1: The conditioned second moment X(t) ∈ C is given by X(0) = (∆, . . . , ∆) and X(t + 1) = UA(t) (X(t)), t = 0, 1, . . . , ℓ − 1. In what follows, for brevity we denote Q(ℓ) = π(ℓ)E ′ E, Q(t) = C ′ C + K(t)′ D′ DK(t), n N t = 0, . . . , ℓ − 1. t=0 Proof: The mean square of the terminal cost, y(ℓ) is: E{ky(ℓ)k2 } = N X ′ E{x(ℓ)′ Eθ(ℓ) Eθ(ℓ) x(ℓ) · 1θ(ℓ)=i } i=1 = N X (14) i=1 Now, by a calculation similar as above leads to ′ ′ Substituting (14) and (15) into (9) we obtain (13). Let us denote the gains attaining (13) by K op (t). From a dynamic programming standpoint, we introduce value functions V t : C n → ℜ by: V ℓ = hπ(ℓ)Q(ℓ), X(ℓ)i and for t = ℓ − 1, ℓ − 2, . . . , 0, ( ℓ ) X t (16) hπ(τ )Q(τ ), X(τ )i , V (X) = min τ =t where X(t) = X and X(τ ), τ = t + 1, . . . , ℓ, satisfies (12). Theorem 4.1: Define P (t) ∈ C n and M (t) ∈ C m,n , t = 0, . . . , ℓ − 1, as follows. Let P (ℓ) = π(ℓ)E ′ E and for each t = ℓ − 1, . . . , 0 and i = 1, . . . , N , compute: if πi (t) = 0, Mi (t) = 0 and Pi (t) = 0, Ri (t) = (Bi′ Di (P (t + 1))Bi + πi (t)Di′ Di ), Pi (t) = πi (t)Ci′ Ci + A′i Di (P (t + 1))Ai − A′i Di (P (t + 1))Bi Ri (t)−1 Bi′ Di (P (t is zero irrespectively of K(t). First, note that for i ∈ Nt we have πi (t)Qi (t)XN i = 0. Second, for i ∈ Nt one can check that πj (t + 1) = 0 for all j such that pji > 0, so that Pj (t + 1) = 0. This yields Di (P (t + 1)) = 0. Bringing these facts together and recalling that by construction XN i = 0 for all i ∈ / Nt , we evaluate (21) By substituting (21) into (20) we write V t (X) = min hπ(t)Q(t) + A(t)′ D(P (t + 1))A(t), XP i. K(t) X Tr (πi (t)Qi (t) = min {i∈N / t} + Ai (t)′ Di (P (t + 1))Ai (t))XPi . (22) By expanding some terms and after some algebra to complete the squares, we have X Tr (πi (t)Ci′ Ci + A′i Di (P (t + 1))Ai V t (X) = min K(t) {i∈N / t} +(Ki (t) − Oi (t))′ (Bi′ Di (P (t + 1))Bi + πi (t)Di′ Di ) ·(Ki (t) − Oi (t)) − Oi (t)′ (Bi′ Di (P (t + 1))Bi + πi (t)Di′ Di )Oi (t))XP i , (23) where Oi (t) = Mi (t) as given in (17). This makes clear that the minimal cost is achieved by setting Kiop (t) = Mi (t), op i∈ / Nt . Now, by replacing Ki (t) with Ki (t) in (23), X Tr Pi (t)XP V t (X) = i {i∈N / t} else (if πi (t) > 0), Mi (t) = Ri (t)−1 Bi′ Di (P (t + 1))Ai , hπ(t)Q(t) + A(t)′ D(P (t + 1))A(t), XN i K(t) ′ E{ky(t)k } = hπ(t)(C C + K(t) D DK(t)), X(t)i. (15) K(t),...,K(ℓ−1) We write X = X + XP , where XN is such that XN i = 0 for any i ∈ / Nt , and in a similar fashion XPi = 0 for i ∈ Nt . We now show that the term hπ(t)Q(t) + A(t)′ D(P (t + 1))A(t), XN i = 0. Tr(πi (ℓ)Ei′ Ei Xi (ℓ) = hπ(ℓ)E ′ E, X(ℓ)i. 2 Nt = {i : πi (t) = 0}. (12) Lemma 4.2: The TRM-JLQ problem can be formulated as ( ℓ ) X (13) min hπ(t)Q(t), X(t)i . K(0),...,K(ℓ−1) Let us decompose X as follows; let the set of states θ having zero probability of being visited at time t be denoted by (17) with Pi (t) as given in (18), i ∈ / Nt . Finally, by choosing Pi (t) = 0, i ∈ Nt , we write X X Tr Pi (t)XN + Tr Pi (t)XP V t (X) = i , i {i∈N / t} + 1))Ai . (18) P {i∈Nt } N = hP (t), X i + hP (t), X i = hP (t), Xi, which completes the proof. V. T HE DUALITY BETWEEN THE TRM-JLQ LMMSE FOR STANDARD MJLS AND THE We consider the LMMSE for standard MJLS as presented in [6], [9]. The problem consists of finding the sequence of sets of gains K f (t), t = 0, . . . , ℓ, that minimizes the covariance of the estimation error z̃(t) = ẑ(t) − z(t) when the estimate is given by a Luenberger observer in the form f ẑ(t + 1) = Aη(t+1) ẑ(t) + Kη(t+1) (t)(y(t) − Lη(t) ẑ(t)), where y(t) is the output of the MJLS   z(t + 1) = Fη(t+1) z(t) + Gη(t+1) ω(t) y(t) = Lη(t+1) z(t) + Hη(t+1) ω(t)   z(0) = z0 , (24) and ω(t) and z0 are i.i.d. random variables satisfying E{ω(t)} = 0, E{ω(t)ω(t)′ } = I and E{z0 z0′ } = Σ. Moreover, it is assumed that Li Hi′ = 0 and Hi Hi′ > 0. We write υi (t) = Prob(η(t) = i), i = 1, . . . , N , so that it is the time-reverse of π, υ(t) = π(ℓ − t). Note that we are considering the same problem as in [6], [9], though our notation is slightly different: here we assume that (y(t), η(t + 1)) is available for the filter to obtain ẑ(t + 1) and the system matrices are indexed by η(t+1), while in the standard formulation (y(t), η(t)) are observed at time t and the system matrices are indexed by η(t). This “time shifting” in η avoids a cluttering in the duality relation. Along the same line, instead of writing the filter gains as a function of the variable Yi (t) = E{z̃(t)z̃(t)′ · 1{η(t+1)=i} }, given by the coupled Riccati difference equation [9, Equation 24] Yi (t + 1) = N X j=1 pji Fj Yj (t)Fj′ + υj (t)Gj G′j − Fj Yj (t)L′j · Lj Yj (t)L′j + υj (t)Hj Hj′ −1 Lj Yj (t)Fj′ whenever υi (t) > 0 and Yi (t + 1) = 0 otherwise, in this note we use the variable Si (t) = E{z̃(t)z̃(t)′ · 1{η(t)=i} } defined in [7], leading to Y (t) = D(S(t)). Replacing this in the above equation, after some algebraic manipulation one obtains [7, Equation 8]: Si (t + 1) = υi (t)Gi G′i + Fi Di (S(t))Fi′ − Fi Di (S(t))L′i · (Li Fi (S(t))L′i + υi (t)Hi Hi′ )−1 Li Di (S(t))Fi′ , (25) whenever υi (t) > 0 and Si (t + 1) = 0 otherwise, with initial condition Si (0) = E{z̃(0)z̃(0)′ · 1{η(0)=i} } = υi (0)Σ. The optimal gains are given for t = 0, . . . , ℓ by −1 Kif (t) = Fi Di (S(t))L′i (Li Di (S(t))L′i + υi (t)Hi Hi′ ) (26) whenever υi (t) > 0 and Kif (t) = 0 otherwise. The duality relations between the filtering and control problems are now evident by direct comparison between (18) and (25). Fi , Li , Gi and Hi are replaced with A′i , Bi′ , Ci′ and Di′ , respectively. Moreover, comparing the initial conditions of the coupled Riccati difference equations, we see Σ replaced with E ′ E. Also, we note that P (0), P (1), . . . , P (ℓ) are equivalent to S(ℓ), S(ℓ−1), . . . , S(0), with a similar relation op for the gains Kif and Ki . The Markov chains driving the filtering and control systems are time-reversed one to each other. Remark 2: Time-varying parameters can be included both in standard MJLS and in Φ by augmenting the Markov state as to describe the pair (θ, t), 1 ≤ θ ≤ N , 0 ≤ t ≤ ℓ, and considering a suitable matrix P of higher dimension N × (ℓ + 1). Although this reasoning leads to a matrix P of high dimension, periodic and sparse, it is useful to make clear that our results are readily adaptable to plants whose matrices are in the form Aθ(t) (t). Either by this reasoning or by re-doing all computations given in this note for timevarying plants, we obtain the following generalization of [10, Table 6.1]. FILTERING of MJLS CONTROL of Φ Fi (t) Li (t) Gi (t) Hi (t) A′i (t) Bi′ (t) Ci′ (t) Di′ (t) Kif (t) Si (0) = υi (0)Σ Si (t) ηi (t) Ki (ℓ − t) Pi (ℓ) = πi (ℓ)Ei′ Ei Pi (ℓ − t) θi (ℓ − t) op ′ TABLE I S UMMARY OF THE F ILTERING /C ONTROL D UALITY. t = 0, . . . , ℓ. VI. C ONCLUDING REMARKS We have presented an operator theory characterization of the conditional second moment X, an MS stability test and formulas for the optimal control of system Φ. The results have exposed some interesting relations with standard MJLS. For system Φ it is fruitful to use the true conditional second moment X whereas for standard MJLS one has to resort to the variable W given in (7) to obtain a recursive equation similar to the ones expressed in the Lemmas 3.1 and 4.1. Moreover, these classes of systems are equivalent if and only if the Markov chain is revertible, as indicated in Remark 1. The solution of the TRM-JLQ problem is given in Theorem 4.1 in the form of a coupled Riccati equation that can be computed backwards prior to the system operation, as usual in linear quadratic problems for linear systems. The result beautifully extends the classic duality between filtering and control into the relations expressed in Table 1. R EFERENCES [1] E. F. Costa A. N. 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Offline and Online Models of Budget Allocation for Maximizing Influence Spread arXiv:1508.01059v3 [cs.DS] 12 Mar 2018 Noa Avigdor-Elgrabli∗ Gideon Blocq† Iftah Gamzu‡ Ariel Orda§ Abstract The research of influence propagation in social networks via word-of-mouth processes has been given considerable attention in recent years. Arguably, the most fundamental problem in this domain is the influence maximization problem, where the goal is to identify a small seed set of individuals that can trigger a large cascade of influence in the network. While there has been significant progress regarding this problem and its variants, one basic shortcoming of the underlying models is that they lack the flexibility in the way the overall budget is allocated to different individuals. Indeed, budget allocation is a critical issue in advertising and viral marketing. Taking the other point of view, known models allowing flexible budget allocation do not take into account the influence spread in social networks. We introduce a generalized model that captures both budgets and influence propagation simultaneously. For the offline setting, we identify a large family of natural budget-based propagation functions that admits a tight approximation guarantee. This family extends most of the previously studied influence models, including the well-known Triggering model. We establish that any function in this family implies an instance of a monotone submodular function maximization over the integer lattice subject to a knapsack constraint. This problem is known to admit an optimal 1 − 1/e ≈ 0.632-approximation. We also study the price of anarchy of the multi-player game that extends the model and establish tight results. For the online setting, in which an unknown subset of agents arrive in a random order and the algorithm needs to make an irrevocable budget allocation in each step, we develop a 1/(15e) ≈ 0.025-competitive algorithm. This setting extends the celebrated secretary problem, and its variant, the submodular knapsack secretary problem. Notably, our algorithm improves over the best known approximation for the latter problem, even though it applies to a more general setting. ∗ Yahoo Labs, Haifa, Israel. Email: noaa@oath.com. Technion, Haifa, Israel. Email: gideon@alumni.technion.ac.il. This work was done when the author was an intern at Yahoo Labs, Haifa, Israel. ‡ Amazon, Israel. Email: iftah.gamzu@yahoo.com. This work was done when the author was affiliated with Yahoo Labs, Haifa, Israel. § Technion, Haifa, Israel. Email: ariel@ee.technion.ac.il. † 1 Introduction The study of information and influence propagation in societies has received increasing attention for several decades in various areas of research. Recently, the emergence of online social networks brought forward many new questions and challenges regarding the dynamics by which information, ideas, and influence spread among individuals. One central algorithmic problem in this domain is the influence maximization problem, where the goal is to identify a small seed set of individuals that can trigger a large word-of-mouth cascade of influence in the network. This problem has been posed by Domingos and Richardson [13, 36] in the context of viral marketing. The premise of viral marketing is that by targeting a few influential individuals as initial adopters of a new product, it is possible to trigger a cascade of influence in a social network. Specifically, those individuals are assumed to recommend the product to their friends, who in turn recommend it to their friends, and so on. The influence maximization problem was formally defined by Kempe, Kleinberg and Tardos [26, 27]. In this setting, we are given a social network graph, which represents the individuals and the relationships between them. We are also given an influence function that captures the expected number of individuals that become influenced for any given subset of initial adopters. Given some budget b, the objective is to find a seed set of b initial adopters that will maximize the expected number of influenced individuals. Kempe et al. studied several operational models representing the step-by-step dynamics of propagation in the network, and analyzed the influence functions that are derived from them. While there has been significant progress regarding those models and related algorithmic issues, one shortcoming that essentially has not been treated is the lack of flexibility in the way that the budget is allocated to the individuals. Indeed, budget allocation is a critical factor in advertising and viral marketing. This raises some concerns regarding the applicability of current techniques. Consider the following scenario as a motivating example. A new daily deals website is interested in increasing its exposure to new audience. Consequently, it decides to provide discounts to individuals who are willing to announce their purchases in a social network. The company has several different levels of discounts that it can provide to individuals to incentivize them to better communicate their purchases, e.g., making more enthusiastic announcements on their social network. The company hopes that those announcements will motivate the friends of the targeted individuals to visit the website, so a word-of-mouth process will be created. The key algorithmic question for this company is whom should they offer a discount, and what amount of discounts should be offered to each individual. Alon et al. [1] have recently identified the insufficiency of existing models to deal with budgets. They introduced several new models that capture issues related to budget distribution among potential influencers in a social network. One main caveat in their models is that they do not take into account the influence propagation that happens in the network. The main aspect of our work targets this issue. 1.1 Our results We introduce a generalized model that captures both budgets and influence propagation simultaneously. Our model combines design decisions taken from both the budget models [1] and propagation models [26]. The model interprets the budgeted influence propagation as a two-stage process consisting of: (1) influence related directly to the budget allocation, in which the seed set of targeted individuals influence their friends based on their budget, and (2) influence resulting from a secondary word-of-mouth process in the network, in which no budgets are involved. Note that the two stages give rise to two influence functions whose combination designates the overall influence function of the model. We study our model in both offline and online settings. An offline setting. We identify a large family of natural budget-based propagation functions that admits a tight approximation guarantee. Specifically, we establish sufficient properties for the two influence functions 1 mentioned above, which lead to a resulting influence function that is both monotone and submodular. It is important to emphasize that the submodularity of the combined function is not the classical set-submodularity, but rather, a generalized version of submodularity over the integer lattice. Crucially, when our model is associated with such an influence function, it can be interpreted as a special instance of a monotone submodular function maximization over the integer lattice subject to a knapsack constraint. This problem is known to have an efficient algorithm whose approximation ratio of 1 − 1/e ≈ 0.632, which is best possible under P 6= NP assumption [40]. We then focus on social networks scenario, and introduce a natural budget-based influence propagation model that we name Budgeted Triggering. This model extends many of the previously studied influence models in networks. Most notably, it extends the well-known Triggering model [26], which in itself generalizes several models such as the Independent Cascade and the Linear Threshold models. We analyze this model within the two-stage framework mentioned above, and demonstrate that its underlying influence function is monotone and submodular. Consequently, we can approximate this model to within a factor of 1 − 1/e. We also consider a multi-player game that extends our model. In this game, there are multiple players, each of which is interested to spend her budget in a way that maximizes her own network influence. We establish that the price of anarchy (PoA) of this game is equal to 2. This result is derived by extending the definition of a monotone utility game on the integer lattice [32]. Specifically, we show that one of the conditions of the utility game can be relaxed, while still maintaining a PoA of at most 2, and that the refined definition captures our budgeted influence model. An online setting. In the online setting, there is unknown subset of individuals that arrive in a random order. Whenever an individual arrives, the algorithm learns the marginal influence for each possible budget assignment, and needs to make an irrevocable decision regarding the allocation to that individual. This allocation cannot be altered later on. Intuitively, this setting captures the case in which there is an unknown influence function that is partially revealed with each arriving individual. Similarly to before, we focus on the case that the influence function is monotone and submodular. Note that this setting is highly-motivated in practice. As observed by Seeman and Singer [37], in many cases of interest, online merchants can only apply marketing techniques on individuals who have engaged with them in some way, e.g., visited their online store. This gives rise to a setting in which only a small unknown sample of individuals from a social network arrive in an online fashion. We identify that this setting generalizes the submodular knapsack secretary problem [5], which in turn, extends the well-known secretary problem [14]. We develop a 1/(15e) ≈ 0.025-competitive algorithm for the problem. Importantly, our results not only apply to a more general setting, but also improve the best known competitive bound for the former problem, which is 1/(20e) ≈ 0.018, due to Feldman, Naor and Schwartz [17]. 1.2 Related work Models of influence spread in networks are well-studied in social science [22] and marketing literature [18]. Domingos and Richardson [13, 36] were the first to pose the question of finding influential individuals who will maximize adoption through a word-of-mouth effect in a social network. Kempe, Kleinberg and Tardos [26, 27] formally modeled this question, and proved that several important models have submodular influence functions. Subsequent research have studied extended models and their characteristics [29, 34, 6, 24, 8, 39, 15, 37, 28, 12], and developed techniques for inferring influence models from observable data [20, 19, 31]. Influence maximization with multiple players has also been considered in the past [6, 21, 25]. Kempe et al. [26], and very recently, Yang et al. [43], studied propagation models that have a similar flavor to our budgeted setting. We like to emphasize that there are several important distinctions between their models and ours. Most importantly, their models assume a strong type of fractional diminishing returns property that our integral model does not need to satisfy. Therefore, their models cannot capture the scenarios we describe. The reader may refer to the cited papers and the references therein for a broader review of the literature. 2 Alon et al. [1] studied models of budget allocation in social networks. As already mentioned, our model follows some of the design decisions in their approach. For example, their models support constraints on the amount of budget that can be assigned to any individual. Such constraints are motivated by practical marketing conditions set by policy makers and regulations. Furthermore, their models focus on a discrete integral notion of a budget, which is consistent with common practices in many organizations (e.g., working in multiplications of some fixed value) and related simulations [38]. We include those considerations in our model as well. Alon et al. proved that one of their models, namely, the budget allocation over bipartite influence model, admits an efficient (1 − 1/e)-approximation algorithm. This result was extended by Soma et al. [40] to the problem of maximizing a monotone submodular function over the integer lattice subject to a knapsack constraint. The algorithm for the above problems is a reminiscent of the algorithm for maximizing a monotone submodular set function subject to a knapsack constraint [41]. Note that none of those papers have taken into consideration the secondary propagation process that occurs in social networks. The classical secretary problem was introduced more than 50 years ago (e.g., [14]). Since its introduction, many variants and extension of that problem have been proposed and analyzed [30, 2, 3, 4]. The problem that is closest to the problem implied from our online model is the submodular knapsack secretary problem [5, 23, 17]. An instance of this problem consists of a set of n secretaries that arrive in a random order, each of which has some intrinsic cost. An additional ingredient of the input is a monotone submodular set function that quantifies the value gained from any subset of secretaries. The objective is to select a set of secretaries of maximum value under the constraint that their overall cost is no more than a given budget parameter. Note that our model extends this setting by having a more general influence function that is submodular over the integer lattice. Essentially, this adds another layer of complexity to the problem as we are not only required to decide which secretaries to select, but we also need to assign them budgets. 2 Preliminaries We begin by introducing a very general budgeted influence propagation model. This model will be specialized later when we consider the offline and online settings. In our model, there is a set of n agents and an influence function f : Nn → R+ . Furthermore, there is a capacity vector c ∈ Nn+ and a budget B ∈ N+ . Our objective is to compute a budget assignment to the agents b ∈ Nn , which maximizes the influence f (b). The vector b must Pn (1) respect the capacities, that is, 0 ≤ bi ≤ ci , for every i ∈ [n], (2) respect the total budget, namely, i=1 bi ≤ B. In the following, we assume without loss of generality that each ci ≤ B. We primarily focus on influence functions that maintain the properties of monotonicity and submodularity. A function f : Nn → R+ is called monotone if f (x) ≤ f (y) whenever x ≤ y coordinate-wise, i.e., xi ≤ yi , for every i ∈ [n]. The definition of submodularity for functions over the integer lattice is a natural extension of the classical definition of submodularity over sets (or boolean vectors): Definition 2.1. A function f : Nn → R+ is said to be submodular over the integer lattice if f (x) + f (y) ≥ f (x ∨ y) + f (x ∧ y), for all integer vectors x and y, where x ∨ y and x ∧ y denote the coordinate-wise maxima and minima, respectively. Specifically, (x ∨ y)i = max{xi , yi } and (x ∧ y)i = min{xi , yi }. In the remainder of the paper, we abuse the term submodular to describe both set functions and functions over the integer lattice. We also make the standard assumption of a value oracle access for the function f . A value oracle for f allows us to query about f (x), for any vector x. The question of how to compute the function f in an efficient (and approximate) way has spawned a large body of work in the context of social networks (e.g., [26, 10, 9, 33, 11, 7]). Notice that for the classical case of sets, the submodularity condition implies that f (S) + f (T ) ≥ f (S ∪ T ) + f (S ∩ T ), for every S, T ⊆ [n], and the monotonicity property implies that f (S) ≤ f (T ) if S ⊆ T . 3 An important distinction between the classical set setting and the integer lattice setting can be seen when we consider the diminishing marginal returns property. This property is an equivalent definition of submodularity of set functions, stating that f (S ∪ {i}) − f (S) ≥ f (T ∪ {i}) − f (T ), for every S ⊆ T and every i ∈ / T. However, this property, or more accurately, its natural extension, does not characterize submodularity over the integer lattice, as observed by Soma et al. [40]. For example, there are simple examples of a submodular function f for which f (x + χi ) − f (x) ≥ f (x + 2χi ) − f (x + χi ) does not hold. Here, χi is the characteristic vector of the set {i}, so that x + kχi corresponds to an update of x by adding an integral budget k to agent i. Note that a weaker variant of the diminishing marginal returns does hold for submodular functions over the integer lattice. Lemma 2.2 (Lem 2.2 [40]). Let f be a monotone submodular function over the integer lattice. For any i ∈ [n], k ∈ N, and x ≤ y, it follows that f (x ∨ kχi ) − f (x) ≥ f (y ∨ kχi ) − f (y) . 3 An Offline Model In this section, we study the offline version of the budgeted influence propagation model. As already noted, we consider budgeted influence propagation to be a two-stage process consisting of (1) direct influence related to the budget assignment, followed by (2) influence related to a propagation process in the network. In particular, in the first stage, the amount of budget allocated to an individual determines her level of effort and success in influencing her direct friends. This natural assumption is consistent with previous work [1]. Then, in the second stage, a word-of-mouth propagation process takes place in which additional individuals in the network may become affected. Note that the allocated budgets do not play role at this stage. We identify a large family of budget-based propagation functions that admit an efficient solution. Specifically, we first identify sufficient properties of the influence functions of both stages, which give rise to a resulting (combined) influence function that is monotone and submodular. Consequently, our model can be interpreted as an instance of a monotone submodular function maximization over the integer lattice subject to a knapsack constraint. This problem is known to have an efficient (1 − 1/e)-approximation [40], which is best possible under the assumption that P 6= NP. This NP-hardness bound of 1 − 1/e already holds for the special case of maximum coverage [16, 1]. We subsequently develop a natural model of budgeted influence propagation in social networks that we name Budgeted Triggering. This model generalizes many settings, including the well-known Triggering model. Note that the Triggering model already extends several models used to capture the spread of influence in networks, like the Independent Cascade, Linear Threshold, and Listen Once models [26]. We demonstrate that the influence function defined by this model is monotone and submodular, and thus, admits an efficient (1 − 1/e)-approximation. Technically, we achieve this result by demonstrating that the two-stage influence functions that underlie this model satisfy the sufficient properties mentioned above. Finally, we study an extension of the Budgeted Triggering model to a multi-player game. In this game, there are multiple self-interested players (e.g., advertisers), each of which is interested to spend her budget in a way that maximizes her own network influence. We establish that the price of anarchy (PoA) of the game is exactly 2. In fact, we prove that this result holds for a much more general type of games. Maehara et al. [32] recently defined the notion of a monotone utility game on the integer lattice, and demonstrated that its PoA is at most 2. Their utility game definition does not capture our multi-player game. We show that one of the conditions in their game definition can be relaxed while still maintaining the same PoA. Crucially, this relaxed definition captures our model. 4 3.1 A two-stage influence composition The two-stage process can be formally interpreted as a composition of two influence functions, f = h◦g. The first function g : Nn → {0, 1}n captures the set of influenced agents for a given budget allocation, while the second function h : {0, 1}n → R+ captures the overall number (or value) of influenced agents, given some seed agent set for a propagation process. In particular, the influenced agents of the first stage are the seed set for the second stage. We next describe sufficient conditions for the functions g and h which guarantee that their composition is monotone and submodular over the integer lattice. Note that we henceforth use notation related to sets and their binary vector representation interchangeably. Definition 3.1. A function g : Nn → {0, 1}n is said to be coordinate independent if it satisfies g(x ∨ y) ≤ g(x) ∨ g(y), for any x, y ∈ Nn . Definition 3.2. A function g : Nn → {0, 1}n is said to be monotone if g(x) ≤ g(y) coordinate-wise whenever x ≤ y coordinate-wise. Many natural influence functions are coordinate independent. One such example is the family of functions in which the output vector is a coordinate-wise disjunction over a set of n vectors, each of which captures the independent influence implied by some agent. Specifically, the ith vector in the disjunction is the result of some function fi : N → {0, 1}n indicating the affected agents as a result of any budget allocation assigned only to agent i. We are now ready to prove our composition lemma. Lemma 3.3. Given a monotone coordinate independent function g : Nn → {0, 1}n and a monotone submodular function h : {0, 1}n → R+ , the composition f = h ◦ g : Nn → R+ is a monotone submodular function over the integer lattice. Proof. The coordinate independence properties of g and the monotonicity of h imply that h(g(x) ∨ g(y)) ≥ h(g(x ∨ y)). In addition, from the monotonicity of g we know that g(x ∧ y) ≤ g(x) and g(x ∧ y) ≤ g(y). Thus, together with the monotonicity of h, we get that h(g(x) ∧ g(y)) ≥ h(g(x ∧ y)). Utilizing the above results, we attain that f is submodular since f (x) + f (y) = h(g(x)) + h(g(y)) ≥ h(g(x) ∨ g(y)) + h(g(x) ∧ g(y)) ≥ h(g(x ∨ y)) + h(g(x ∧ y)) = f (x ∨ y) + f (x ∧ y) , where the first inequality is by the submodularity of h. We complete the proof by noting that f is monotone since both g and h are monotone. Formally, given x ≤ y then it follows that f (x) = h(g(x)) ≤ h(g(y)) = f (y) by h’s monotonicity and since g(x) ≤ g(y) by g’s monotonicity. As a corollary of the lemma, we get the following theorem. 5 Theorem 3.4. Given a monotone coordinate independent function g : Nn → {0, 1}n and a monotone submodular function h : {0, 1}n → R+ , there is a (1 − 1/e)-approximation algorithm for maximizing the influence function f = h ◦ g : Nn → R+ under capacity constraints c ∈ Nn+ and a budget constraint B ∈ N+ , whose running time is polynomial in n, B, and the query time of the value oracle for f . Proof. We know by Lemma 3.3 that f is monotone and submodular. Consequently, we attain an instance of maximizing a monotone submodular function over the integer lattice subject to a knapsack constraint. Soma et al. [40] recently studied this problem, and developed a (1 − 1/e)-approximation algorithm whose running time is polynomial in n, B, and the query time for the value oracle of the submodular function. 3.2 The budgeted triggering model We now focus on social networks, and introduce a natural budget-based influence model that we call the Budgeted Triggering model. This model consists of a social network, represented by a directed graph G = (V, E) with n nodes (agents) and a set E of directed edges (relationships between agents). In addition, there is a function f : Nn → R+ that quantifies the influence of any budget allocation b ∈ Nn to the agents. The concrete form of f strongly depends on the structure of the network, as described later. The objective is to find a budget allocation b that maximizes the numberPof influenced nodes, while respecting the feasibility constraints: (1) bi ≤ ci , for every node i ∈ V , and (2) i∈V bi ≤ B. For ease of presentation, we begin by describing the classic Triggering model [26]. Let N (v) be the set of neighbors of node v in the graph. The influence function implied by a Triggering model is defined by the following simple process. Every node v ∈ V independently chooses a random triggering set T v ⊆ N (v) among its neighbors according to some fixed distribution. Then, for any given seed set of nodes, its influence value is defined as the result of a deterministic cascade process in the network which works in steps. In the first step, only the selected seed set is affected. At step ℓ, each node v that is still not influenced becomes influenced if any of its neighbors in T v became influenced at time ℓ − 1. This process terminates after at most n rounds. Our generalized model introduces the notion of budgets into this process. Specifically, the influence function in our case adheres to the following process. Every node v independently chooses a random triggering vector tv ∈ N\|N (v)\| according to some fixed distribution. Given a budget allocation b ∈ Nn , the influence value of that allocation is the result of the following deterministic cascade process. In the first step, every node v that was allocated a budget bv > 0 becomes affected. At step ℓ, every node v that is still not influenced becomes influenced if any of its neighbors u ∈ N (v) became influenced at time ℓ − 1 and bu ≥ tvu . One can easily verify that the Triggering model is a special case of our Budgeted Triggering model, where the capacity vector c = 1n , and each tvu = 0 if u ∈ T v , and tvu = B + 1, otherwise. Intuitively, the triggering vectors in our model capture the amount of effort that is required from each neighbor of some agent to affect her. Of course, the underlying assumption is that the effort of individuals correlates with the budget they receive. As an example, consider the case that a node v selects a triggering value tvu = 1 for some neighbor u. In this case, u can only influence v if it receives a budget of at least 1. However, if v selects a value tvu = 0 then it is enough that u becomes affected in order to influence v. In particular, it is possible that u does not get any budget but still influences v after it becomes affected in the cascade process. Given a budget allocation b, the value of the influence function f (b) is the expected number of nodes influenced in the cascade process, where the expectation is taken over the random choices of the model. Formally, let σ be some fixed choice of the triggering vectors of all nodes (according to the model distribution), and let Pr(σ) be the probability of this outcome. Let fσ (b) be the (deterministic) number of Pnodes influenced when the triggering vectors are defined by σ and the budget allocation is b. Then, f (b) = σ Pr(σ) · fσ (b). 6 Theorem 3.5. There is a (1 − 1/e)-approximation algorithm for influence maximization under the Budgeted Triggering model whose running time is polynomial in n, B, and the query time of the value oracle for the influence function. Proof. Consider an influence function f : Nn → R+ resulting from the Budgeted Triggering model. We next show that the function f is monotone and submodular over the integer lattice. As a result, our model can be interpreted as an instance of maximizing a monotone submodular function over the integer lattice subject to a knapsack constraint, which admits an efficient (1 − 1/e)-approximation. Notice that it is sufficient to prove that each (deterministic) function fσ is monotone submodular function over the integer lattice. This follows as f is a non-negative linear combination of all fσ . One can easily validate that submodularity and monotonicity are closed under non-negative linear combinations. Consider some function fσ : Nn → R+ . For the purpose of establishing that fσ is monotone and submodular, we show that fσ can be interpreted as a combination of a monotone coordinate independent function gσ : Nn → {0, 1}n , and a monotone submodular function hσ : {0, 1}n → R+ . The theorem then follows by utilizing Lemma 3.3. We divide the diffusion process into two stages. In the first stage, we consider the function gσ , which given a budget allocation returns (the characteristic vector of) the set S of all the nodes that were allocated a positive budget along with their immediate neighbors that were influenced according to the Budgeted Triggering model. Formally, gσ (b) = S , v : bv > 0 ∪ u : ∃v ∈ N (u), bv > 0, bv ≥ tuv . In the second stage, we consider the function hσ that receives (the characteristic vector of) S as its seed set, and makes the (original) Triggering model interpretation of the vectors. Specifically, the triggering set of each node v is considered to be T v = {u : tvu = 0}. Intuitively, the function gσ captures the initial budget allocation step and the first step of the propagation process, while the function hσ captures all the remaining steps of the propagation. Observe that fσ = hσ ◦ gσ by our construction. Also notice that hσ is trivially monotone and submodular as it is the result of a Triggering model [26, Thm. 4.2]. Therefore, we are left to analyze the function gσ , and prove that it is monotone and coordinate independent. The next claim establishes these properties, and completes the proof of the theorem. Claim 3.6. The function gσ is monotone and coordinate independent. Proof. Let x, y ∈ Nn , and denote w = x ∨ y. We establish coordinate independence by considering each influenced node in gσ (w) separately. Recall that gσ (w) consist of the union of two sets {v : wv > 0} and {u : ∃v ∈ N (u), wv > 0, wv ≥ tuv }. Consider a node v for which wv > 0. Since wv = max{xv , yv }, we know that at least one of {xv , yv } is equal to wv , say wv = xv . Hence, v ∈ gσ (x). Now, consider a node u ∈ gσ (w) having wu = 0. It must be the case that u is influenced by one of its neighbors v. Clearly, wv > 0 and wv ≥ tuv . Again, we can assume without loss of generality that wv = xv , and get that u ∈ gσ (x). This implies that for each v ∈ gσ (x ∨ y), either v ∈ gσ (x) or v ∈ gσ (y), proving coordinate independence, i.e., gσ (x ∨ y) ≤ gσ (x) ∨ gσ (y). We prove monotonicity in a similar way. Let x ≤ y. Consider a node v ∈ gσ (x) for which xv > 0. Since yv ≥ xv > 0, we know that v ∈ gσ (y). Now, consider a node u ∈ gσ (x) having xu = 0. There must be a node v ∈ N (u) such that xv > 0, and xv ≥ tuv . Accordingly, we get that yv ≥ tuv , and hence, u ∈ gσ (y). This implies that gσ (x) ≤ gσ (y), which completes the proof. 3.3 A multi-player budgeted influence game We now focus on a multi-player budgeted influence game. In the general setting of the game, which is i M formally defined by the tuple (M, (Ai )M i=1 , (f )i=1 ), there are M self-interested players, each of which needs 7 to decide how to allocate its budget B i ∈ N+ among n agents. Each player has a capacity vector ci ∈ Nn+ that bounds the amount of budget she may allocate to every agent. The budget assignment of player i is denoted The strategy of player i is feasible if it respects the constraints: by bi ∈ Nn+ , and is referred to as its strategy. P (1) bij ≤ cij , for every j ∈ [n], and (2) nj=1 bij ≤ B i . Let Ai be the set of all feasible strategies for player i. Note that we allow mixed (randomized) strategies. Each player has an influence function f i : NM ×n → R+ that designates her own influence (payoff) in the game. Specifically, f i (b) is the payoff of player i for the budget allocation b of all players. This can also be written as f i (bi , b−i ), where the strategy of i is bi and the strategies of all the other players are marked as b−i . Note that the goal of each player is to maximize PM i her own influence, given her feasibility constraints and the strategies of other players. Let F (b) = i=1 f (b) be the social utility of all players in the game. One of the most commonly used notions in game theory is Nash equilibrium (NE) [35]. This notion translates to our game as follows: A budget allocation b is said to be in a NE if f i (bi , b−i ) ≥ f i (b̃i , b−i ), for every i and b̃i ∈ Ai . Monotone utility game on the integer lattice. We begin by studying a monotone utility game on the integer lattice, and establish that its PoA is no more than 2. Later on, we demonstrate that our Budgeted Triggering model can be captured by this game. Utility games were defined for submodular set functions by Vetta [42], and later extended to submodular functions on the integer lattice by Maehara et al. [32]. We build on the latter work, and demonstrate that one of the conditions in their utility game definition, namely, the requirement that the submodular function satisfies component-wise concavity, can be neglected. Note that component-wise concavity corresponds to the diminishing marginal returns property, which does not characterize submodularity over the integer lattice, as noted in Section 2. Therefore, removing this constraint is essential for proving results for our model. We refine the definition of a monotone utility game on the integer lattice [32], so it only satisfies the following conditions: (U1) F (b) is a monotone submodular function on the integer lattice. P i (U2) F (b) ≥ M i=1 f (b). (U3) f i (b) ≥ F (bi , b−i ) − F (0, b−i ), for every i ∈ [M ]. Theorem 3.7. The price of anarchy of the monotone utility game designated by U1-U3 is at most 2. 1 M Proof. Let b∗ = (b1∗ , . . . , bM ∗ ) be the social optimal budget allocation, and let b = (b , . . . , b ) be a budget i 1 i allocation in Nash equilibrium. Let b̃ = (b∗ , . . . , b∗ , 0, . . . , 0) be the optimal budget allocation restricted to the first i players. Notice that F (b∗ ) − F (b) ≤ F (b∗ ∨ b) − F (b) = M X F (b̃i ∨ b) − F (b̃i−1 ∨ b) i=1 ≤ M X F (bi∗ ∨ bi , b−i ) − F (bi , b−i ) , i=1 where the first inequality is due to the monotonicity of F , the equality holds by a telescoping sum, and the last inequality is due to submodularity of F . Specifically, submodularity implies that inequality since F (bi∗ ∨ bi , b−i ) + F (b̃i−1 ∨ b) ≥ F (b̃i ∨ b) + F (bi , b−i ) . 8 Now, observe that F (bi , b−i ) + F (bi∗ , b−i ) ≥ F (bi∗ ∨ bi , b−i ) + F (bi∗ ∧ bi , b−i ) ≥ F (bi∗ ∨ bi , b−i ) + F (0, b−i ) . Here, the first inequality holds by the submodularity of F , while the last inequality follows from the monotonicity of F . Consequently, we derive that F (b∗ ) − F (b) ≤ M X F (bi∗ , b−i ) − F (0, b−i ) ≤ M X i=1 i=1 f i (bi∗ , b−i ) ≤ M X f i (bi , b−i ) ≤ F (b) , i=1 where the second inequality is by condition (U3) of the utility game, the third inequality holds since b is a Nash equilibrium, and the last inequality is by condition (U2) of the utility game. This completes the proof as F (b∗ ) ≤ 2F (b). A multi-player Budgeted Triggering model. We extend the Budgeted Triggering model to a multi-player setting. As before, we have a social network, represented by a directed graph G = (V, E), such that every node v has an independent random triggering vector tv ∈ N\|N (v)\| . Each player i has a budget B i ∈ N+ , and a function f i : NM ×n → R+ that quantifies her influence, given the budget allocation of all players. The objective of each player i is to find a budget allocation bi , given the budget allocations of other players, that maximizes the number of nodes that she influences, while respecting the feasibility constraints: (1) bij ≤ cij , P for every node j ∈ V , and (2) j∈V bij ≤ B. The process in which nodes become affected is very similar to that in Budgeted Triggering, but needs some refinement for the multi-player setting. We follow most design decisions of Bharathi et al. [6]. Specifically, whenever a player influences a node, this node is assigned the color of that player. Once a node become influenced, its color cannot change anymore. If two or more players provide positive budgets to the same node, then the node is given the color of the player that provided the highest budget. In case there are several such players, the node is assigned a color uniformly at random among the set of players with the highest budget assignment. If a node u becomes influenced at step ℓ, it attempts to influence each of its neighbors. If the activation attempt from u to its neighbor v succeeds, which is based on the triggering vector of v, then v becomes influenced with the same color as u at step ℓ + Tuv , assuming that it has not been influenced yet. All Tuv ’s are independent positive continuous random variables. This essentially prevents simultaneous activation attempts by multiple neighbors. PM i Lemma 3.8. The social function F (b) = i=1 f (b) is a monotone submodular function on the integer lattice. Proof. We prove this lemma along similar lines to those in the proof of Theorem 3.5, which attains to the single-player scenario. Let σ be some fixed choice of triggering vectors of all nodes and all the activation times Tuv . We also assume that σ encodes other random decisions in the model, namely, all tie-breaking choices related to equal (highest) budget assignments for nodes. Let Fσ (b) be the deterministic number of P influenced nodes for the random choices σ and the budget allocation b, and note that F (b) = σ Pr(σ)·Fσ (b). Similar to Theorem 3.5, it is sufficient to prove that Fσ is monotone submodular function on the integer lattice. Again, we view the social influence as a two-stage process. In the first step, we consider a function Gσ : NM ×n → {0, 1}n that given the budget allocation of all players returns a set S of immediate influenced nodes. Formally, Gσ (b) = S , v : ∃i, biv > 0 ∪ u : ∃v ∈ N (u), ∃i, biv > 0, biv ≥ tuv . 9 In the second stage, we consider the function Hσ that receives S as its seed set, and makes the original Triggering model interpretation of the vectors, that is, it sets each T v = {u : tvu = 0}. Notice that the fact that there are multiple players at this stage does not change the social outcome, i.e., the number of influenced nodes, comparing to a single-player scenario. The only difference relates to the identity of the player that affects every node. This implies that Hσ is monotone and submodular as its result is identical to that of the original (single-player) Triggering model [26]. Observe that Fσ = Hσ ◦ Gσ by our construction. Therefore, by Lemma 3.3, we are left to establish that the function Gσ is monotone and coordinate independent. The next claim proves that. Claim 3.9. The function Gσ is monotone and coordinate independent. We prove this claim using almost identical line of argumentation to that in Claim 3.6. Let x, y ∈ NM ×n , and denote w = x ∨ y. We establish coordinate independence by considering every affected node in Gσ (w) separately. Recall that Gσ (w) consist of the union of two sets {v : ∃i, wvi > 0} and {u : ∃v ∈ N (u), ∃i, wvi > 0, wvi ≥ tuv }. Consider a node v that has some player i with wvi > 0. Since wvi = max{xiv , yvi }, we know that at least one of {xiv , yvi } is equal to wvi , say wvi = xiv . Hence, v ∈ Gσ (x), since in particular, player i competes on influencing v. Now, consider a node u ∈ Gσ (w) with wui = 0, for all i. It must be the case that u is influenced by one of its neighbors v. Clearly, there exists some player i such that wvi > 0 and wvi ≥ tuv . Again, we can assume without loss of generality that wvi = xiv , and get that u ∈ Gσ (x), since in particular, player i competes on influencing u via v. This implies that for each v ∈ Gσ (x ∨ y), either v ∈ Gσ (x) or v ∈ Gσ (y), proving coordinate independence, i.e., Gσ (x ∨ y) ≤ Gσ (x) ∨ Gσ (y). We prove monotonicity in a similar way. Let x ≤ y. Consider a node v ∈ Gσ (x) such that there is a player i for which xiv > 0. Since yvi ≥ xiv > 0, we know that player i competes on influencing v, and thus, v ∈ Gσ (y). Now, consider a node u ∈ Gσ (x) with xiu = 0, for all i. There must be a node v ∈ N (u) and a player i such that xiv > 0 and xiv ≥ tuv . Accordingly, we get that yvi ≥ tuv . Therefore, player i also competes on influencing u via v, and thus, u ∈ Gσ (y). This implies that Gσ (x) ≤ Gσ (y), which completes the proof. Theorem 3.10. The Budgeted Triggering model with multiple players has a PoA of exactly 2. Proof. We begin by demonstrating that the model satisfies conditions U1-U3 of the monotone utility game on the integer lattice. As a result, we can apply Theorem 3.7 to attain an upper bound of 2 on the PoA of the model. Notice that condition (U1) holds by Lemma 3.8. Also, condition (U2) trivially holds by the definition of the social function F . For the purpose of proving that the model satisfies condition (U3), let σ be some fixed choice of triggering vectors of all nodes and all the activation times Tuv . We also assume that σ encodes other random decisions in the model, namely, all tie-breaking choices related to equal (highest) budget assignments for nodes. Let Fσ (b) be the deterministic number of influenced nodes for the random choices σ and the budget allocation b. Finally, let fσi (b) be the deterministic number of nodes influenced by player i for the random choices σ and the budget allocation b. We next argue that fσi (b) ≥ Fσ (bi , b−i ) − Fσ (0, b−i ), for any σ. Notice that this implies condition (U3) since X X f i (b) = Pr(σ)fσi (b) ≥ Pr(σ) Fσ (bi , b−i ) − Fσ (0, b−i ) = F (bi , b−i ) − F (0, b−i ) . σ σ We turn to prove the above argument. Notice that it is sufficient to focus only on cases that Fσ (bi , b−i ) > Fσ (0, b−i ), since otherwise, the argument is trivially true as fσi (b) ≥ 0. We concentrate on all nodes u that are not influenced by any player when the mutual strategy is (0, b−i ), but became influenced for a strategy (bi , b−i ). We claim that all those nodes must be assigned the color of player i. It is easy to verify that 10 increasing the budget assignment of a player to any node can only negatively affect other players, that is, they may only influence a subset of the nodes. This follows as all the activation results are deterministically encoded in the choices σ, so adding a competition can only make the outcome worse, i.e., players may not affect a node that they previously did. This implies the claim. As a result, fσi (b) ≥ Fσ (bi , b−i ) − Fσ (0, b−i ). This completes the proof that the model is an instance of the monotone utility game on the integer lattice, and thus, has a PoA of at most 2. We proceed by proving the tightness of the PoA result. We show there is an instance of the multi-player Budget Triggering model whose PoA is 2N/(N + 1). Notice that as N → ∞, the lower bound on the PoA tends to 2. This instance has been presented in a slightly different context by He and Kempe [25, Proposition 1]. Concretely, the input graph is a union of a star with one center and N leaves, and N additional (isolated) nodes. The triggering vectors are selected from a degenerate distribution that essentially implies that each activated node also activates all of its neighbors. Every player has one unit of budget. One can easily verify that the solution in which all players assign their unit budget to the center of the star is a NE. This follows since the expected payoff for each player is (N + 1)/N , while unilaterally moving the budget to any other node leads to a payoff of 1. However, the strategy that optimizes the social utility is to place one unit of budget at the center of the star graph, and the remaining budget units at different isolated nodes. 4 An Online Model We study the online version of the budgeted influence propagation model. This setting can capture scenarios in which the social influences in a network are known in advance, but the (subset of) agents that will arrive and their order is unknown. The input for this setting is identical to that of the offline variant with the exception that the n agents arrive in an online fashion. This intuitively means that we do not know the monotone submodular influence function f : Nn → R+ in advance, but rather, it is revealed to us gradually with time. More specifically, upon the arrival of the ith agent, we can infer the (constrained) function fi , which quantifies the influence of f for the set of the first i agents, while fixing the budget of all other agents to 0. Note that we also learn the maximum budget ci that can be allocated to agent i whenever she arrives. For every arriving agent i, the algorithm needs to make an irrevocable decision regarding the amount of budget bi allocated to that agent without knowing the potential contribution of future arriving agents. As mentioned in the introduction, this problem is a generalization of the classical secretary problem. This immediately implies that any online algorithm preforms very poorly under an unrestricted adversarial arrival of the agents. We therefore follow the standard assumption that the agents and their influence are fixed in advanced, but their order of arrival is random. Note that the overall influence of some budget allocation to the agents is not affected by the arrival order of the agents. We analyze the performance of our algorithm, ON, using the competitive analysis paradigm. Note that competitive analysis focuses on quantifying the cost that online algorithms suffer due to their complete lack of knowledge regarding the future, and it does not take into account computational complexity. Let OPT be an optimal algorithm for the offline setting. Given an input instance I for the problem, we let OPT(I) and ON(I) be the influence values that OPT and ON attain for I, respectively. We say that ON is c-competitive if inf I E[ON(I)]/OPT(I) ≥ c, where E[ON(I)] is the expected value taken over the random choices of the algorithm and the random arrival order of the agents. We like to note that our algorithm and its analysis are inspired by the results of Feldman et al. [17] for the submodular knapsack secretary problem. However, we make several novel observations and identify some interesting structural properties that enable us to simultaneously generalize and improve their results. Also note that in the interests of expositional simplicity, we have not tried to optimize the constants in our analysis. 11 Theorem 4.1. There is an online randomized algorithm that achieves 1/(15e) ≈ 0.025-competitive ratio for the budgeted influence maximization problem. Proof. Recall that an instance of the online budgeted influence maximization problem consists of a set of n agents that arrive in a random order, a budget constraint B ∈ N+ , capacity constraints c ∈ Nn+ , and a monotone submodular influence function over the integer lattice f : Nn → R+ . We begin by describing the main component of our algorithm. This component is built to address the case that the contribution of each agent is relatively small with respect to the optimal solution. That is, even when one assigns the maximum feasible budget to any single agent, the contribution of that agent is still small compared to the optimum. We refer to this component as light influence algorithm (abbreviated, LI). This component will be later complemented with another component, derived from the classical secretary algorithm, to deal with highly influential agents. Let ha1 , a2 , . . . , an i be an arbitrary fixed ordering of the set of agents. This is not necessarily the arrival order of the agents. Algorithm light influence, formally described below, assumes that each agent ai is assigned a uniform continuous random variable ti ∈ [0, 1) that determines its arrival time. Note that this assumption does not add restrictions on the model since one can create a set of n samples of the uniform distribution from the range [0, 1) in advance, and assign them by a non-decreasing order to each arriving agent (see, e.g., the discussion in [17]). The algorithm begins by exploring the first part of the agent sequence, that is, the agents in L = {ai : ti ≤ 1/2}. Note that it does not allocate any budget to those agents. Let bL be an optimal (offline) budget allocation for the restricted instance that only consists of the agents in L, and let f (bL ) be its influence value. Furthermore, let f (bL )/B be a lower bound on the average contribution of each unit of budget in that solution. The algorithm continues by considering the remainder of the sequence. For each arriving agent, it allocates a budget of k if the increase in the overall influence value is at least αkf (bL )/B, for some fixed α to be determined later. That is, the average influence contribution of an each unit of budget is (up to the α-factor) at least as large as the average unit contribution in the optimal solution for the first part. If there are several budget allocations that satisfy the above condition then the algorithm allocates the maximal amount of budget that still satisfies the capacity and budget constraints. Prior to formally describing our algorithm, we like to remind that χi corresponds to the characteristic vector of ai , i.e., (χi )i = 1 and (χi )j = 0 for every j 6= i. Accordingly, if b ∈ Nn is a budget allocation vector in which the ith coordinate represents the allocation to agent ai , and bi = 0, then the allocation b ∨ kχi corresponds to an update of b by adding a budget k to agent ai . We say that the marginal value of assigning k units of budget to ai is f (b ∨ kχi ) − f (b), and the marginal value per unit is (f (b ∨ kχi ) − f (b))/k. Having described our main component, we are now ready to complete the description of our algorithm. As already , we randomly combine algorithm LI with the classical algorithm for the secretary problem. Specifically, algorithm LI is employed with probability 5/8 and the classical secretary algorithm with probability 3/8. This latter algorithm assigns a maximal amount of budget to a single agent ai to attain an influence value of f (ci χi ). The algorithm selects ai by disregarding the first n/e agents that arrive, and then picking the first agent whose influence value is better than any of the values of the first n/e agents. This optimal algorithm is known to succeed in finding the single agent with the best influence with probability of 1/e [14]. 4.1 Analysis We begin by analyzing the performance guarantee of algorithm LI, and later analyze the complete algorithm. Let OPT∗ = [OPT∗1 , . . . , OPT∗n ] be the optimal budget allocation for a given instance, and let OPTL be the L ∗ budget allocation for the agents in L, that is, OPTL i = OPTi whenever i ∈ L and OPTi = 0, otherwise. R R Similarly, OPT is the budget allocation for the agents in R = [n] \ L, i.e., OPTi = OPT∗i for i ∈ R, and 12 Algorithm 1: Light Influence (LI) Input : an online sequence of n agents, a budget constraint B ∈ N+ , capacity constraints c ∈ Nn+ , a monotone submodular function f : Nn → R+ , a parameter α ∈ R+ Output: A budget allocation b b ← (0, 0, . . . , 0) f (bL ) ← value of the optimal budget allocation for agents in L = {ai : ti ≤ 1/2} for every agent ai such that ti ∈ (1/2, 1] do P Ki ← k ≤ ci : f (b ∨ kχi ) − f (b) ≥ αkf (bL )/B ∪ k + j6=i bj ≤ B if Ki 6= ∅ then k ← maxk {Ki } b ← b ∨ kχi end end return b Algorithm 2: Online Influence Maximization Input : an online sequence of n agents, a budget constraint B ∈ N+ , capacity constraints c ∈ Nn+ , a monotone submodular function f : Nn → R+ , a parameter α ∈ R+ Output: A budget allocation b r ← random number in [0, 1] if r ∈ [0, 3/8] then b ← run the classical secretary algorithm with (n, B, c, f ) else if r ∈ (3/8, 1] then b ← run algorithm LI with (n, B, c, f, α) end return b OPTR / R. Recall that algorithm LI attends to the case in which no single agent has a significant i = 0 for i ∈ influence contribution compared to the optimal value. More formally, let β = maxi f (ci χi )/f (OPT∗ ) be the ratio between the maximal contribution of a single agent and the optimal value. Lemma 4.2. If α ≥ 2β then f (b) ≥ min{αf (OPTL )/2, f (OPTR ) − αf (OPT∗ )}. Proof. We prove this lemma by bounding the expected influence value of the algorithm in two cases and taking the minimum of them: Case I: Algorithm LI allocates a budget of more than B/2 units. We know that the algorithm attains a value of at least αf (bL )/B from each allocated budget unit by the selection rule f (b ∨ ki χi ) − f (b) ≥ αkf (bL )/B. Hence, the total influence of this allocation is at least f (b) > αf (bL ) αf (OPTL ) B αf (bL ) · = ≥ . 2 B 2 2 Case II: Algorithm LI allocates at most B/2 budget units. We utilize the following lemma proven in [40, Lem 2.3]. Lemma 4.3. Let f be a monotone submodular function over the integer lattice. For arbitrary x, y, X f (x ∨ y) ≤ f (x) + f (x ∨ yi χi ) − f (x) . i∈[n]: yi >xi 13 This lemma applied to our case implies that f (b ∨ OPTR ) ≤ f (b) + X i∈[n]: OPTR i >bi f (b ∨ OPTR i χi ) − f (b) . (1) We consider two sub-cases: Subcase A: There is ℓ ∈ [n] such that OPTR ℓ > B/2. Clearly, there can only be one agent ℓ having this ∗ property. One can easily validate that f (b ∨ OPTR ℓ χℓ ) − f (b) ≤ β · f (OPT ) by the definition of β and R Lemma 2.2. Now, consider any agent i 6= ℓ with OPTi > bi . The reasonP that the optimal solution allocated more budget to i than our algorithm cannot be the lack of budget since i bi < B/2 and OPTR i < B/2. Hence, it must be the case that f (bL ) f (b ∨ OPTR i χi ) − f (b) < α , (2) B OPTR i by the selection rule of the algorithm. Note that b in the above equation designates the budget allocation at the time that the agent ai was considered and not the final allocation. However, due to the weak version of marginal diminishing returns that was described in Lemma 2.2, the inequality also holds for the final allocation vector. As a result, f (OPTR ) ≤ f (b ∨ OPTR ) ≤ f (b) + f (b ∨ OPTR ℓ χℓ ) − f (b) + X i∈[n]\{ℓ}: OPTR i >bi f (b ∨ OPTR i χi ) − f (b) f (bL ) B · ≤ f (b) + βf (OPT∗ ) + α B 2 α , ≤ f (b) + f (OPT∗ ) · β + 2 where the first inequality follows due to the monotonicity of f , and the third inequality uses the sub-case assumption that there is one agent that receives at least half of the overall budget in OPTR , and thus, P R R ∗ i6=ℓ OPTi ≤ B/2. Recall that α ≥ 2β, and thus, f (b) ≥ f (OPT ) − αf (OPT ). Subcase B: OPTR i ≤ B/2, for every i ∈ [n]. The analysis of this sub-case follows the same argumentation of the previous sub-case. Notice that for every agent i ∈ [n] such that OPTR i > bi , we can apply inequality (2). Consequently, we can utilize inequality (1), and get that f (OPTR ) ≤ f (b ∨ OPTR ) ≤ f (b) + X i∈[n]: OPTR i >bi f (bL ) f (b ∨ OPTR ·B , i χi ) − f (b) ≤ f (b) + α B which implies that f (b) ≥ f (OPTR ) − αf (OPT∗ ). Recall that we considered some arbitrary fixed ordering of the agents ha1 , a2 , . . . , an i that is not necessary their arrival order. Let wi the marginal contribution of agent ai to the optimal value when calculated according to this order. Namely, let OPT∗<i = [OPT∗1 , . . . , OPT∗i−1 , 0, . . . , 0] be the allocation giving the same budget as OPT∗ for every agent aj with j < i, and 0 for the rest, and define wi = f (OPT∗<i ∨OPT∗i χi )−f (OPT<i ). This point of view allow us to associate fixed parts of the optimal value to the agents in a way that is not affected by their order of arrival. Let Xi be a random indicator for the event that ai ∈ L, and let W = Pn i=1 wi Xi . Let α ≥ 2β to be determined later. 14 By the weak version of marginal in Lemma 2.2, it holds that f (OPTL ) ≥ Pn diminishing returns specified R ∗ W , and similarly, f (OPT ) ≥ i=1 wi (1 − Xi ) = f (OPT ) − W . Using this observation, in conjunction with Lemma 4.2, we get that f (b) ≥ min{αW/2, f (OPT∗ )·(1−α−W/f (OPT∗ ))}. Let Y = W/f (OPT∗ ), and observe that f (b) ≥ f (OPT∗ ) · min{αY /2, 1 − α − Y } . (3) Note that Y ∈ [0, 1] captures the ratio between the expected optimum value associated with the agents in L and the (overall) optimum value. We continue to bound the expected value of f (b) by proving the following lemma. Lemma 4.4. Let α = 2/5 and assume that β ≤ 1/5, then, E[f (b)] ≥ p 2 f (OPT∗ ) · 1− β . 20 Proof. By assigning α = 2/5 to the bound in inequality 3, we obtain that Y 3 ∗ , −Y . f (b) ≥ f (OPT ) · min 5 5 Notice that the expected value of f (b) is ∗ E[f (b)] ≥ f (OPT ) Z 3 5 ′ [Pr(Y ≤ γ)] · min 0 γ 3 , − γ dγ , 5 5 since [Pr(Y ≤ γ)]′ is the probability density function of Y . Now, observe that we can split the integral range into two parts ∗ E[f (b)] ≥ f (OPT ) f (OPT∗ ) ≥ 5 Z 1 2 [Pr(Y ≤ γ)] 0 Z ′γ 1 2 5 ∗ dγ + f (OPT ) Z 3 5 1 2 ′ [Pr(Y ≤ γ)] [Pr(Y ≤ γ)]′ γdγ. 3 − γ dγ 5 (4) 0 To bound Pr(Y ≤ γ), we use Chebyshev’s inequality, while noting that E[Y ] = n X wi E[Xi ]/f (OPT∗ ) = W/(2f (OPT∗ )) = 1/2 , i=1 since E[Xi ] = 1/2 and W = f (OPT∗ ). Now, 1 β Var[Y ] Pr Y − ≤ 2 , ≥c ≤ 2 c2 4c where the last inequality follows from [17, Lem B.5]. For completeness, the proof of this lemma appears as Lemma A.1 in the Appendix. Now, observe that Y is symmetrically distributed around 1/2, and therefore, Pr(Y ≤ 12 − c) = Pr(Y ≥ 21 + c) ≤ β/(8c2 ). This implies that for every γ ≤ 1/2, Pr(Y ≤ γ) ≤ 15 β . 8( 21 − γ)2 Note that we cannot simply plug this upper bound on the cumulative distribution function into inequality (4). R 1/2 The fact that Y is symmetrically distributed around 1/2 implies that 0 [Pr(Y ≤ γ)]′ dγ = 1/2, and this does hold with this bound. To bypass this issue, and maintain the later constrain, we decrease the integration range. One can easily verify that #′ Z 1−√β " 2 β 1 dγ = , 1 2 2 8( 2 − γ) 0 and as a result, we can infer that Z 1 2 ′ [Pr[Y ≤ γ]] γdγ ≥ 0 Z √ 1− β 2 0 " β 1 8( 2 − γ)2 #′ γdγ. Specifically, this inequality holds since we essentially considered the worst distribution (from an algorithms analysis point of view) by shifting probability from higher values of Y to smaller values (note that multiplication by γ). The proof of the lemma now follows since #′ Z 1−√β " 2 β f (OPT∗ ) γdγ E[f (b)] ≥ 5 8( 12 − γ)2 0 √ ∗ Z 1−2 β β · f (OPT ) γ = dγ 1 20 ( 2 − γ)3 0 1−√β 2 4γ − 1 β · f (OPT∗ ) = 20 (1 − 2γ)2 0 ∗ p 2 f (OPT ) · 1− β = . 20 Recall that β = maxi f (ci χi )/f (OPT∗ ). We next consider two cases depending on the value of β. When β > 1/5, our algorithm executes the classical secretary algorithm with probability 3/8. This algorithm places a maximal amount of budget on the agent having maximum influence, maxi f (ci χi ), with probability 1/e. Consequently, 3 β · f (OPT∗ ) 3f (OPT∗ ) f (OPT∗ ) E[f (b)] ≥ · > > . 8 e 40e 15e When β ≤ 1/5, we know that our algorithm executes the classical secretary algorithm with probability 3/8, and algorithm LI with probability 5/8. 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P 2 P 2 P w V ar[ i wi Xi ] i wi V ar[Xi ] = = 2 i i ∗ V ar[Y ] = ∗ ∗ 2 2 f (OPT ) f (OPT ) 4f (OPT ) P maxi wi · i wi maxi wi · f (OPT∗ ) β = ≤ . ≤ ∗ ∗ 2 2 4f (OPT ) 4f (OPT ) 4 19	8
arXiv:1403.6920v3 [math.AC] 10 Apr 2014 GRÖBNER BASES OF BALANCED POLYOMINOES JÜRGEN HERZOG, AYESHA ASLOOB QURESHI AND AKIHIRO SHIKAMA Abstract. We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a squarefree Gröbner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced. Introduction Polyominoes are, roughly speaking, plane figures obtained by joining squares of equal size edge to edge. Their appearance origins in recreational mathematics but also has been subject of many combinatorial investigations including tiling problems. A connection of polyominoes to commutative algebra has first been established by the second author of this paper by assigning to each polyomino its ideal of inner minors, see [14]. This class of ideals widely generalizes the ideal of 2-minors of a matrix of indeterminates, and even that of the ideal of 2-minors of two-sided ladders. It also includes the meet-join ideal of plane distributive lattices. Those classical ideals have been extensively studied in the literature, see for example [2], [3] and [7]. Typically one determines for such ideals their Gröbner bases, determines their resolution and computes their regularity, checks whether the rings defined by them are normal, Cohen-Macaulay or Gorenstein. A first step in this direction for the inner minors of a polyomino (also called polyomino ideals) has been done by Qureshi in the afore mentioned paper. Very recently those convex polyominoes have been classified in [5] whose ideal of inner minors is linearly related or has a linear resolution. For some special polyominoes also the regularity of the ideal of inner minors is known, see [8]. In this paper, for balanced polyominoes, we provide some positive answers to the questions addressed before. To define a balanced polyomino, one labels the vertices of a polyomino by integer numbers in a way that row and column sums are zero along intervals that belong to the polyomino. Such a labeling is called admissible. There is a natural way to attach to each admissible labeling of a polyomino P a binomial. The given polyomino is called balanced if all the binomials arising from admissible labelings belong to the ideal of inner minors IP of P. It turns out that P is balanced if and only if IP coincides with the lattice ideal determined by P, 1991 Mathematics Subject Classification. 13C05, 05E40, 13P10. Key words and phrases. polyominoes, Gröbner bases, lattice ideals. This paper was partially written during the visit of the second and third author at Universität Duisburg-Essen, Campus Essen. The second author wants to thank the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy for supporting her. The third author wants to thank Professor Hibi who made his visit to Essen possible. 1 see Proposition 1.2. This is the main observation of Section 1 where we provide the basic definitions regarding polyominoes and their ideals of inner minors. An important consequence of Proposition 1.2 is the result stated in Corollary 1.3 which asserts that for any balanced polyomino P, the ideal IP is a prime ideal and that its height coincides with the number of cells of P. It is conjectured in [14] by the second author of this paper that IP is a prime ideal for any simple polyomino P. A polyomino is called simple if it has no ‘holes’, see Section 1 for the precise definition. We expect that simple polyominoes are balanced. This would then imply Qureshi’s conjecture. In Section 2 we identify the primitive binomials of a balanced polyomino (Theorem 2.1) and deduce from this that for a balanced polyomino P the ideal IP has a squarefree initial ideal for any monomial order. This then implies, as shown in Corollary 2.3, that the residue class ring of IP is a normal Cohen-Macaulay domain. Finally, in Section 3 we show that all row or column convex, and all tree-like polyminoes are simple and balanced. This supports our conjecture that simple polyominoes are balanced. 1. The ideal of inner minors of a polyomino In this section we introduce polyominoes and their ideals of inner minors. Given a = (i, j) and b = (k, l) in N2 we write a ≤ b if i ≤ k and j ≤ l. The set [a, b] = {c ∈ N2 : a ≤ c ≤ b} is called an interval, and an interval of the from C = [a, a + (1, 1)] is called a cell (with left lower corner a). The elements of C are called the vertices of C, and the sets {a, a+(1, 0)}, {a, a+(0, 1)}, {a+(1, 0), a+(1, 1)} and {a + (0, 1), a + (1, 1)} the edges of C. Let P be a finite collection of cells of N2 , and let C and D be two cells of P. Then C and D are said to be connected, if there is a sequence of cells C = C1 , . . . , Cm = D of P such that Ci ∩ Ci+1 is an edge of Ci for i = 1, . . . , m − 1. If in addition, Ci 6= Cj for all i 6= j, then C is called a path (connecting C and D). The collection of cells P is called a polyomino if any two cells of P are connected, see Figure 1. Figure 1. A polyomino Let P be a polyomino, and let K be a field. We denote by S the polynomial over K with variables xij with (i, j) ∈ V (P). Following [14] a 2-minor xij xkl − xil xkj ∈ S is called an inner minor of P if all the cells [(r, s), (r + 1, s + 1)] with i ≤ r ≤ k − 1 and j ≤ s ≤ l − 1 belong to P. In that case the interval [(i, j), (k, l)] is called an 2 inner interval of P. The ideal IP ⊂ S generated by all inner minors of P is called the polyomino ideal of P. We also set K[P] = S/IP . Let P be a polyomino. An interval [a, b] with a = (i, j) and b = (k, l) is called a horizontal edge interval of P if j = l and the sets {r, r + 1} for r = i, . . . , k − 1 are edges of cells of P. Similarly one defines vertical edge intervals of P. According to [14], an integer value function α : V (P) → Z is called admissible, if for all maximal horizontal or vertical edge intervals I of P one has X α(a) = 0. a∈I −3 −1 −2 5 −2 2 −2 0 1 −4 4 0 3 0 0 −2 −1 0 3 −1 2 −2 Figure 2. An admissible labeling In Figure 2 an admissible labeling of the polyomino displayed Figure 1 is shown. Given an admissible labeling α we define the binomial fα = Y xaα(a) − a∈V (P) α(a)>0 Y xa−α(a) , a∈V (P) α(a)<0 Let JP be the ideal generated by the binomials fα where α is an admissible labeling of P. It is obvious that IP ⊂ JP . We call a polyomino balanced if for any admissible labeling α, the binomial fα ∈ IP . This is the case if and only if IP = JP . L Consider the free abelian group G = (i,j)∈V (P) Zeij with basis elements eij . To any cell C = [(i, j), (i + 1, j + 1)] of P we attach the element bC = eij + ei+1,j+1 − ei+1,j − ei,j+1 in G and let Λ ⊂ G be the lattice spanned by these elements. Lemma 1.1. The elements bC form a K-basis of Λ and hence rankZ Λ = \|P\|. Moreover, Λ is saturated. In other words, G/Λ is torsionfree. Proof. We order the basis elements eij lexicographically. Then the lead term of bC is eij . This shows that the elements bC are linearly independent and hence form a Z-basis of Λ. We may complete this basis of Λ by the elements eij for which (i, j) is not a left lower corner of a cell of P to obtain a basis of G. This shows that G/Λ is free, and hence torsionfree. The lattice ideal IΛ attached to the lattice Λ is the ideal generated by all binomials fv = Y xvaa − Y a∈V (P) va <0 a∈V (P) va >0 with v ∈ Λ. 3 xa−va Proposition 1.2. Let P be a balanced polyomino. Then IP = IΛ . Proof. The assertion follows once we have shown that for any v ∈ Λ there exists an admissible labeling α of P such that va = α(a) for all a ∈ V (P). Indeed, since the elements bC ∈ Λ form a Z-basis of Λ, there exist integers zC ∈ Z such that P P v = C zC bC . We set α = C∈P zC αC where for C = [(i, j), (i + 1, j + 1)], αC ((k, l)) =    1, if (k, l) = (i, j) or (k, l) = (i + 1, j + 1), −1, if (k, l) = (i + 1, j) or (k, l) = (i, j + 1),   0, otherwise. Then α(a) = va for all a ∈ V (P). Since each αC is an admissible labeling of P and since any linear combination of admissible labelings is again an admissible labeling, the desired result follows. Corollary 1.3. If P is a balanced polyomino, then IP is a prime ideal of height \|P\|. Proof. By Proposition 1.2, IP = IΛ and by Lemma 1.1, Λ is saturated. It follows that IP is a prime ideal, see [9, Theorem 7.4]. Next if follows from [12, Corollary 2.2] (or [9, Proposition 7.5]) that height IP = rankZ Λ. Hence the desired conclusion follows from Lemma 1.1. Let P be a polyomino and let [a, b] an interval with the property that P ⊂ [a, b]. According to [14], a polyomino P is called simple, if for any cell C not belonging to P there exists a path C = C1 , C2 , . . . , Cm = D with Ci 6∈ P for i = 1, . . . , m and such that D is not a cell of [a, b]. It is conjectured in [14] that IP is a prime ideal if P is simple. There exist examples of polyominoes for which IP is a prime ideal but which are not simple. Such an example is shown in Figure 3. On the other hand, we conjecture that a polyomino is simple if and only it is balanced. This conjecture implies Qureshi’s conjecture on simple polyominoes. Figure 3. Not simple but prime 2. Primitive binomials of balanced polyominoes The purpose of this section is to identify for any balanced polyomino P the primitive binomials in IP . This will allow us to show that the initial ideal of IP is a squarefree monomial ideal for any monomial order. The primitive binomials in P are determined by cycles. A sequence of vertices C = a1 , a2 , . . . , am in V (P) with am = a1 and such that ai 6= aj for all 1 ≤ i < j ≤ m − 1 is a called a cycle in P if the following conditions hold: (i) [ai , ai+1 ] is a horizonal or vertical edge interval of P for all i = 1, . . . , m − 1; 4 (ii) for i = 1, . . . , m one has: if [ai , ai+1 ] is a horizonal interval of P, then [ai+1 , ai+2 ] is a vertical edge interval of P and vice versa. Here, am+1 = a2 . a1 a2 a5 a9 a7 a8 a3 a1 a2 a4 a3 a4 Figure 4. A cycle and a non-cycle in P It follows immediately from the definition of a cycle that m − 1 is even. Given a cycle C, we attach to C the binomial (m−1)/2 fC = Y (m−1)/2 xa2i−1 − i=1 Y xa2i i=1 Theorem 2.1. Let P be a balanced polyomino. (a) Let C be a cycle in P. Then fC ∈ IP . (b) Let f ∈ IP be a primitive binomial. Then there exists a cycle C in P such that each maximal interval of P contains at most two vertices of C and f = ±fC . Proof. (a) Let C = a1 , a2 , . . . , am be the cycle in P. We define the labeling α of P by setting α(a) = 0 if a 6∈ C and α(ai ) = (−1)i+1 for i = 1, . . . , m, and claim that α is an admissible labeling of P. To see this we consider a maximal horizontal edge interval I of P. If I ∩ C = ∅, then α(a) = 0 for all a ∈ I. On the other hand, if I ∩ C = 6 ∅, then there exist integers i such that ai , ai+1 ∈ I (where ai+1 = a1 if i = m − 1), and P no other vertex of I belongs C. It follows that a∈I α(a) = 0. Similarly, we see that P a∈I α(a) = 0 for any vertical edge interval. It follows form the definition of α that fC = fα , and hence since P is balanced it follows that fC ∈ IP . (b) Let f ∈ IP be a primitive binomial. Since P is balanced and f is irreducible, [14, Theorem 3.8(a)] implies that there exists an admissible labeling α of P such that Y Y f = fα = xaα(a) − xa−α(a) . a∈V (P) α(a)>0 a∈V (P) α(a)<0 Choose a1 ∈ V (P) such that α(a1 ) > 0. Let I1 be the maximal horizontal edge interval with a1 ∈ I1 . Since α is admissible, there exists some a2 ∈ I1 with α(a2 ) < 0. Let I2 be the maximal vertical edge interval containing a2 . Then similarly as before, there exists a3 ∈ I2 with α(a3 ) > 0. In the next step we consider the maximal horizontal edge interval containing and a3 and proceed as before. Continuing in this way we obtain a sequence a1 , a2 , a3 . . . . , of vertices of P such that α(a1 ), α(a2 ), α(a3 ), . . . is a sequence with alternating signs. Since V (P) is a finite set, there exist a number m such that ai 6= aj for all 1 ≤ i < j ≤ m and am = ai for some i < m. If follows that α(am ) = α(ai ) which implies that m − i is even. Then the sequence C = ai , ai+1 , . . . am is a cycle in P, and hence by (a), fC ∈ IP . For any binomial g = u − v we set g (+) = u and g (−) = v. Now if i is odd, then (+) (−) (+) fC divides f (+) and fC divides f (−) , while if i is even, then fC divides f (−) and (−) fC divides f (+) . Since f is primitive, this implies that f = ±fC , as desired. 5 Corollary 2.2. Let P be a balanced polyomino. Then IP admits a squarefree initial ideal for any monomial order. Proof. By Corollary 1.3, IP is a prime ideal, since P is a balanced polyomino. This implies that IP is a toric ideal, see for example [4, Theorem 5.5]. Now we use the fact (see [10, Lemma 4.6] or [6, Corollary 10.1.5]) that the primitive binomials of a toric ideal form a universal Gröbner basis. Since by Theorem 2.1, the primitive binomials of IP have squarefree initial terms for any monomial order, the desired conclusion follows. Corollary 2.3. Let P be a balanced polyomino. Then K[P] is a normal CohenMacaulay domain of dimension \|V (P)\| − \|P\|. Proof. A toric ring whose toric ideal admits a squarefree initial ideal is normal by theorem of Sturmfels [10, Chapter 8], and by a theorem of Hochster ([1, Theorem 6.3.5]) a normal toric ring is Cohen–Macaulay. Since P is balanced, we know from L Proposition 1.2 that IP = IΛ where Λ is the lattice in (i,j)∈V (P) Zeij spanned by the elements bC = eij + ei+1,j+1 − ei+1,j − ei,j+1 where C = [(i, j), (i + 1, j + 1)] is a cell of P. By [9, Proposition 7.5], the height of IΛ is equal to the rank of Λ. Thus we see that height IP = \|P\|. It follows that the Krull dimension of K[P] is equal to \|V (P)\| − \|P\|, as desired. 3. Classes of balanced polyominoes In this section we consider two classes of balanced polyominoes. As mentioned in Section 1 one expects that any simple polyomino is balanced. In this generality we do not yet have a proof of this statement. Here we want to consider only two special classes of polyominoes which are simple and balanced, namely the row and column convex polyominoes, and the tree-like polyominoes. Let P be a polyomino. Let C = [(i, j), (i + 1, j + 1)] be a cell of P. We call the vertex a = (i, j) the left lower corner of C. Let C1 and C2 be two cells with left lower corners (i1 , j1 ) and (i2 , j2 ), respectively. We say that C1 and C2 are in horizontal (vertical) position if j1 = j2 (i1 = i2 ). The polyomino P is called row convex if for any two horizontal cells C1 and C2 with lower left corners (i1 , j) and (i2 , j) and i1 < i2 , all cells with lower left corner (i, j) with i1 ≤ i ≤ i2 belong to P. Similarly one defines column convex polyominoes. For example, the polyomino displayed in Figure 5 is column convex but not row convex. Figure 5. Column convex but not row convex A neighbor of a cell C in P is a cell D which shares a common edge with C. Obviously, any cell can have at most four neighbors. We call a cell of P a leaf, if 6 it has an edge which does not has a common vertex with any other cell. Figure 6 illustrates this concept. Figure 6. A polyomino with three leaves The polyomino P is called tree-like each subpolyomino of P has a leaf. The polyomino displayed in Figure 6 is not tree-like, because it contains a subpolyomino which has no leaf. On the other hand, Figure 7 shows a tree-like polyomino. Figure 7. A tree-like polyomino A free vertex of P is a vertex which belongs to exactly one cell. Notice that any leaf has two free vertices. We call a path of cells a horizontal (vertical) cell interval, if the left lower corners of the path form a horizontal (vertical) edge interval. Let C be a leaf, and let I by the maximal cell interval to which C belongs, and assume that I is a horizontal (vertical) cell interval. Then we call C a good leaf, if for one of the free vertices of C the maximal horizontal (vertical) edge interval which contains it has the same length as I. We call a leaf bad if it is not good, see Figure 8. Theorem 3.1. Let P be a row or column convex, or a tree-like polyomino. Then P is balanced and simple. good bad good Figure 8. Bad and good leaves 7 Proof. Let P be a tree-like polyomino. We first show that P is balanced. Let α be an admissible labeling of P. We have to show that fα ∈ IP . To prove this we first show that P has a good leaf. If \|P\| = 1, then the assertion is trivial. We may assume that \|P\| ≥ 2. Let ni = \|{C ∈ P : deg C = i}\|. Observe that C∈P deg C = n1 + 2n2 + 3n3 + 4n4 , where deg C denotes the number of neighbor cells of P. Let P be any polyomino with cells C1 , . . . , Cn . Associated to P, is the so-called connection graph on vertex [n] with the edge set {{i, j} : E(Ci ) ∩ E(Cj ) 6= ∅}. It is easy to see that connection graph of a tree-like polyomino is a tree. Therefore, using some elementary facts from graph theory, we obtain that P n1 + 2n2 + 3n3 + 4n4 = 2(\|P\| − 1) = 2(n1 + n2 + n3 + n4 − 1). This implies that n1 = n3 + 2n4 + 2. Let g(P) be the number of good leaves in P and b(P) be the number of bad leaves in P. It is obvious that n1 = g(P) + b(P). Then we have (1) g(P) = n3 + 2n4 + 2 − b(P). Next, we show that b(P) ≤ n3 . Suppose that C is a bad leaf in P and I the unique maximal cell interval to which C belongs. Let DC be the end cell of the interval I. Observe that C 6= DC . We claim that deg DC = 3. Indeed, since C is bad, the length of the maximal intervals containing the two free vertices of C is bigger than the length of the interval I. See Figure 9 where the cells belonging to I are marked with dots and E is the cell next to DC which does not belong to P. Since E ∈ / P, the cells D1 and D2 belong to P. Suppose D3 6∈ P. Since P is a polyomino there exists a path C connecting D1 and DC . Since E, D3 6∈ P the path C (which is a subpolyomino of P) does not have a leaf, contradicting the assumption that P is tree-like. Therefore, D3 ∈ P. Similarly one shows that D4 ∈ P. This shows that deg DC = 3. D3 D1 E D4 D2 Figure 9. Moreover, DC cannot be the end cell of any other cell interval, because D3 and D4 are neighbors of DC . Thus we obtain a one-to-one correspondence between the bad leafs C and the cells DC as defined before. It follows that n3 ≤ b(P). Therefore, by (1) we obtain g(P) = n3 + 2n4 + 2 − b(P) ≥ 2n4 + 2 ≥ 2. 8 Thus, we have at least 2 good end cell for every tree-like polyomino. Now we show P is balanced. Indeed, let α be an admissible labeling of P. We want to show that fα ∈ IP . Let C be a good leaf of P with free vertices a1 and a2 . Then α(a1 ) = −α(a2 ). If α(ai ) = 0 for i = 1, 2, then α restricted to P ′ is an admissible labeling of P ′ , where P ′ is obtained from P by removing the cell C from P. Inducting on the number of cells we may then assume that fα ∈ IP ′ . Since IP ′ ⊂ IP , we are done is this case. Assume now that α(a1 ) 6= 0. We proceed by induction on \|α(a1)\|. Note that α(a2 ) 6= 0, too. Since C is a good leaf, we may assume that the maximal interval [a2 , b] to which a2 belongs has the same length as the cell interval [C, DC ] and α(a1 ) > 0. Then α(a2 ) < 0 and hence, since α is admissible, there exists a c ∈ [a2 , b] with α(c) > 0. The cells of the interval [c, a1 ] all belong to [C, DC ]. Therefore, g = xc xa1 − xd xa2 ∈ IP . Here d is the vertex as indicated in Figure 10. a2 c b DC C d a1 Figure 10. Notice that the labeling β of P defined by β1 (e) =    α(e) − 1, if e = a1 or e = c, α(e) + 1, if e = a2 or e = d,   α(e), elsewhere. is admissible and \|β(a1 )\| < \|α(a1 )\|. By induction hypothesis we have that fβ ∈ IP . Then the following relation (2) fα − (fα(+) /xc xa1 )g = (xa2 xd )fβ gives that fα ∈ IP , as well. Next, we show that P is simple by applying induction on number of cells. The assertion is trivial if P consists of only one cell. Suppose now \|P\| > 1, and let D be a cell not belonging to P and I an interval such V (P) ⊂ I. Since P is tree-like, P admits a leaf cell C. Let P ′ be the polyomino which is obtained from P by removing the cell C. Then P ′ is again tree-like and hence is simple by induction. Therefore, since D 6∈ P ′ , there exists a path D ′ : D = D1 , D2 , · · · , Dm of cells with Di 6∈ P ′ for all i and such that Dm is a border cell of I. We let D = D ′ , if Di 6= C for all i. Note that C 6= D1 , Dm . Suppose Di = C for some i with 1 < i < m. Since C a leaf, it follows that the cells C1 , C2 , C3 , C4 and C5 as shown in Figure 11 do not belong to P. Since Di = C and since D ′ is a path of cells it follows that Di−1 ∈ {C1 , C3 , C5 } and Di+1 ∈ {C1 , C3 , C5 }. If Di−1 = C1 and Di+1 = C3 , then we let D : D1 , . . . , Di−1 , C2 , Di+1 , . . . , Dm , 9 C1 C2 C C3 C5 C4 Figure 11. and if Di−1 = C1 and Di+1 = C5 , then we let D : D1 , . . . , Di−1 , C2 , C3 , C4 , Di+1 , . . . , Dm . Similarly one defines D in all the other cases that my occur for Di−1 and Di+1 , so that in any case D ′ can be replaced by the path of cells D which does not contain any cell of P c and connects D with the border of I. This shows that P is simple. Now we show that if P is row or column convex then P is balanced and simple. We may assume that P be column convex. We first show that P is balanced. For this part of the proof we follow the arguments given in [14, Proof of Theorem 3.10]. Let C1 = [C1 , Cn ] be the left most column interval of P and a be lower left corner of C1 . Let α be an admissible labeling for P. If α(a) = 0, then α restricted to P ′ is an admissible labeling of P ′ , where P ′ is obtained from P by removing the cell C1 from P. By applying induction on the number of cells we have that fα ∈ IP ′ ⊂ IP , and we are done is this case. Now we may assume that α(a) 6= 0. We may assume that α(a) > 0. By following the same arguments as in case of tree-like polyominoes, it suffice to show that there exist an inner interval [(i, j), (k, l)] with i < k and j < l such that α((i, j))α((k, l)) > 0 or α((k, j))α((i, l)) > 0. Let [a, b] and [a, c] be the maximal horizontal and vertical edge intervals of P containing a. Then there exist e ∈ [a, b] and f ∈ [a, c] such that α(e), α(f ) < 0 because α is admissible. Let [f, g] be the maximal horizontal edge interval of P which contains f . Then there exists a vertex h ∈ [f, g] such that α(h) > 0. If size([f, g]) ≤ size([a, b]), then column convexity of P gives that [a, h] is an inner interval of P and α(a)α(h) > 0. Otherwise, if size([f, g]) ≥ size([a, b]), then again by using column convexity of P, xa xq − xe xf ∈ IP where q is as shown in following figure. By following the same arguments as in case of tree-like polyominoes, we conclude that fα ∈ IP . Now we will show that P is simple. We may assume that V (P) ⊂ [(k, l), (r, s)] with l > 0. Let C 6∈ P be a cell with with lower left corner a = (i, j), and consider the infinite path C of cells where C = C0 , C1 , . . . and where the lower left corner of Ck is (i, k) for k = 0, 1, . . .. If C ∩ P = ∅, then C is connected to C0 . On the other hand, if C ∩ P = 6 ∅, then, since P is column convex, there exist integers k1 , k2 with l ≤ k1 ≤ k2 ≤ s such that Ck ∈ P if and only if k1 ≤ k ≤ k2 . Since C 6∈ P it follows that j < k1 or j > k2 . In the first case, C is connected to C0 and in the second case C is connected to Cs . 10 Corollary 3.2. Let P be a row or column convex, or a tree-like polyomino. Then K[P] is a normal Cohen–Macaulay domain. 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Jürgen Herzog, Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany E-mail address: juergen.herzog@uni-essen.de Ayesha Asloob Qureshi, The Abdus Salam International Center of Theoretical Physics, Trieste, Italy E-mail address: ayesqi@gmail.com Akihiro Shikama, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 5600043, Japan E-mail address: a-shikama@cr.math.sci.osaka-u.ac.jp 11	0
arXiv:1802.02568v1 [cs.CV] 7 Feb 2018 V I S E R: Visual Self-Regularization Hamid Izadinia∗ University of Washington Pierre Garrigues Flickr, Yahoo Research izadinia@cs.uw.edu garp@yahoo-inc.com Abstract problem. However, image recognition remains an area of active research. ImageNet is indeed biased towards single objects appearing in the middle of the image, which is in contrast with the photos we take with our mobile phones that typically contain a range of objects that appear in context. Also, the list of object categories in ImageNet is a subset of the lexical database WordNet [29]. This makes ImageNet biased towards certain categories such as breeds of dogs, and does not match the scope of more general image recognition tasks such as object detection and localization in context. In this work, we propose the use of large set of unlabeled images as a source of regularization data for learning robust visual representation. Given a visual model trained by a labeled dataset in a supervised fashion, we augment our training samples by incorporating large number of unlabeled data and train a semi-supervised model. We demonstrate that our proposed learning approach leverages an abundance of unlabeled images and boosts the visual recognition performance which alleviates the need to rely on large labeled datasets for learning robust representation. To increment the number of image instances needed to learn robust visual models in our approach, each labeled image propagates its label to its nearest unlabeled image instances. These retrieved unlabeled images serve as local perturbations of each labeled image to perform Visual Self-Regularization (V I S E R). To retrieve such visual self regularizers, we compute the cosine similarity in a semantic space defined by the penultimate layer in a fully convolutional neural network. We use the publicly available Yahoo Flickr Creative Commons 100M dataset as the source of our unlabeled image set and propose a distributed approximate nearest neighbor algorithm to make retrieval practical at that scale. Using the labeled instances and their regularizer samples we show that we significantly improve object categorization and localization performance on the MS COCO and Visual Genome datasets where objects appear in context. Datasets such as MS COCO [25] or Visual Genome [21] have been constructed such that photos are typically composed of multiple objects appearing at a variety of positions and scales. They provide a more realistic benchmark for image recognition systems that are intended for consumer photography products such as Flickr or Google Photos. MS COCO currently contains 300K images and 80 object categories, whereas Visual Genome contains 100K images and thousands of object categories. CNNs are also showing the best performance on these datasets [34, 21]. As training deep neural networks requires a large amount of data and the size of MS COCO and Visual Genome is an order of magnitude smaller than ImageNet, the CNN weights are initialized using the weights of a model that was originally trained on ImageNet. In this paper we focus on improving image recognition performance on MS COCO and Visual Genome. The labels in MS COCO and Visual Genome are obtained via crowdsourcing platforms such as Amazon Mechanical Turk. Hence it is time-consuming and expensive to obtain additional labels. However, we have access to huge quantities of unlabeled or weakly labeled images. For example, the Yahoo Flickr Creative Commons 100M dataset (YFCC) [40] is comprised of a hundred million Flickr photos with userprovided annotations such as photo tags, titles, or descriptions. 1. Introduction Image recognition has rapidly progressed in the last five years. It was shown in the ground-breaking work of [22] that deep convolutional neural networks (CNNs) are extremely effective at recognizing objects and images. The development of deeper neural networks with over a hundred layers has kept improving performance on the ImageNet dataset [7], and we have arguably achieved human performance on this task [35]. These developments have become mainstream and may lead to the perception of image recognition as a solved In this paper, we present a simple yet effective semisupervised learning algorithm that is able to leverage labeled and unlabeled data to improve classification accuracy on the MS COCO and Visual Genome datasets. We first train a fully convolutional network using the multi-labeled data ∗ Work was done while the author was an intern at Flickr, Yahoo Research. 1 Figure 1. The t-SNE [27] map of the whole set of images (including MS COCO and YFCC images) labeled as ‘Bus’ category after applying our proposed V I S E R approach. Can you guess whether green or blue background correspond to the human annotated images of MS COCO dataset? Answer key: blue: MS COCO, green: YFCC (e.g. MS COCO or Visual Genome). Then, we retrieve for each training sample the nearest samples in YFCC using the cosine similarity in a semantic space of the penultimate layer in the trained fully convolutional network. We call these Regularizer samples which can be considered as real perturbed samples compared to the Gaussian noise perturbation considered in virtual adversarial training [31]. Having access to a large set of unlabeled data is critical for finding representative regularizer samples for each training instance. For making this approach practical at scale, we propose an approximate distributed algorithm to find the images with semantically similar attention activation. We then fine-tune the network using the labeled instances and Regularizer samples. Our experimental results show that we significantly improve performance over previous methods where models are trained using only the labeled data. We also demonstrate how our approach is applicable to object-in-context retrieval. 2. Related work The recognition and detection of objects that appear “in context” is an active area of research. The most common benchmarks for this task are the PASCAL VOC [10] and MS COCO datasets. Deep convolutional neural networks have been shown to provide optimal performance in this setting with state-of-the-art performance results for object detection in [34]. It has recently been shown in [33, 38, 9, 3, 42] that it is possible to accurately classify and localize objects using training data that does not contain any object bounding box information. We refer to this training data that does not contain the location information of the object as weaklysupervised. The size of labeled "objects in context" datasets is typically small. For example, MS COCO has around 300,000 images and Visual Genome has over 100,000 images. However, we have access to large amounts of unlabeled web images. The Yahoo Flickr Creative Commons 100M dataset has one hundred million images that have user annotations such as tags, titles, and description. There has been some recent efforts to leverage this user annotation to build object classifiers. For instance, [18] proposes a noise model that is able to better capture the uncertainty in the user annotations and improve the classification performance. It is shown in [19] that it is possible to learn state-of-the-art image features when training a convolutional neural network from a random initialization using user annotations as target labels. In [12], the authors also train deep neural networks from scratch and use the output layers as classifiers directly. However, classifier performance is lower when training on noisy data. Contrary to these approaches, we propose a form of curriculum learning [4] where we first train a model on a small set of clean data, and then augment the training set by mining instances from a large set of unlabeled images. While it is shown that by making small perturbations to the input it is possible to make adversarial examples which can fool machine learning models [39, 23, 24], adversarial examples can be used as a means for data augmentation to improve the regularization capability of the deep models. Our method is related to adversarial training techniques [13, 31, 30] in the sense that additional training instances with small perturbations are created and added to the training data. However, in contrast to those methods, we retrieve real adversarial examples from a large set of person person, sports ball, tennis racket 3 sports ball tennis racket High 64 128 256 512 Conv(3,3,512) 512 Low 1024 Conv(3,3,512) 2048 N N Noisy-Or Conv(3,3,256) Conv(3,3,64) Conv(3,3,128) Figure 2. We use a Fully Convolutional Network to simultaneously categorize images and localize the objects of interest in a single forward pass. The last layer of the network produces an tensor of N heatmaps for localizing objects where each corresponds to one of the Nth object. The green areas correspond to regions with high probability for the object produced by our network. unlabeled images. Our examples are thus real image instances which possess high correlation to the labeled data in the semantic space determined by the penultimate layer of the neural network after the first phase of training. Such instances usually correspond to large perturbations in the input space but follow the natural distribution of the data which is analogous to the adversarial perturbations. We call our retrieved image instances as Regularizer and show that the Regularizer instances can be used to re-train the model and further improve performance. other modalities. For example, in [8] the performance of several nearest neighbor methods is examined on the image captioning task. By conducting extensive experiments, the results of [8] have shown that nearest neighbor approaches can perform as good as state-of-the-art methods for image captioning. Semi-supervised learning is the class of algorithms where classifiers are trained using labeled and unlabeled data. A number of approaches have been proposed in this setting such as Naive Bayes and EM algorithm [32], ensemble methods [5] and propagating labels based on similarity as in [43]. In our case the size of the unlabeled set is three orders of magnitude larger than the size of the labeled set. Existing methods are therefore impractical, and we propose a simple method to propagate labels using a nearest neighbor search. The metric is the cosine distance in the space defined by the penultimate layer of the fully convolutional neural network after it has been trained on the clean dataset. We argue that the size of our unlabeled set is critical in order for the label propagation to work effectively, and we propose approximations using MapReduce to make the search practical [6]. Most recent developments in image recognition have been driven by optimizing performance on the ImageNet dataset. However, images in this dataset have a bias for single objects appearing in the center of the image. In order to increase performance on photos where multiple objects may appear at different scales and position, we adopt a fully convolutional neural network architecture inspired by [26]. Each fully connected layer is replaced with a convolutional layer. Hence, the output of the network is a H × W × N tensor where the width and height depend on the input image size and N is the number of object classes. For each object class the corresponding heatmap provides information about the object’s location as illustrated in Figure 2. In our experiments we use the base architecture of VGG16 [36] shown in Figure 2. Large-scale nearest neighbor search is commonly used in the computer vision community for a variety of tasks such as scene completion [16], image editing with the PatchMatch algorithm [2], or image annotation with the TagProp algorithm [15]. Techniques such as TagProp [15] have been proposed to transfer tags from labeled to unlabeled images. In this work we take advantage of the powerful image representation from a deep neural network to transfer labels as well as regularize training. Similarly labels can be propagated using semantic segmentation [14]. This method is applied on ImageNet which has a bias towards a single object appearing in the center of the image. We focus here on images where objects appear in context. 3.2. Multiple instance learning for multilabel classification Nearest neighbor search has also been shown to be successful in other computer vision applications which involve 3. Proposed Method 3.1. Fully Convolutional Network Architecture We are given a set of annotated images A = {(xi , yi )}i=1...n , where xi is an image and yi = (yi1 , . . . , yiN ) ∈ {0, 1}N is a binary vector determining which object category labels are present in xi . Let f l be the object heatmap for the lth label in the final layer of the network. The probability at location j is given by applying a sigmoid unit to the logits f l , e.g. plj = σ(fjl ). We do not have access to the location information of the objects since we are in a weakly labeled setting. Therefore, to compute the probability score for the lth object category to appear at the jth location, we incorporate a multiple instance learning approach with Noisy-OR operation [28, 41, 11]. The probability for label l is given by Equation 1. Also, for learning the parameters of the FCN, we use stochastic gradient descent to minimize the cross-entropy loss L formalized in Equation 2. Y pl = 1 − (1 − plj ). (1) L= N X j −y l log pl − (1 − y l ) log (1 − pl ) (2) l=1 3.3. Visual Self-Regularization It has been observed that deep neural networks are vulnerable to adversarial examples [39]. Let x be an image and η a small perturbation such that kηk∞ ≤ . If the perturbation is aligned with the gradient of the loss function η = sign(∇x L) which is the most discriminative direction in the image space, then the output of the network may change dramatically, even though the perturbed image x̃ = x + η is virtually indistinguishable from the original. Goodfellow et al. suggest that this is due to the linear nature of deep neural networks. They also show that augmenting the training set with adversarial examples results in regularization similar to dropout[13]. In Virtual Adversarial Training [31] the perturbation is produced by maximizing the smoothness of the local model distribution around each data point. This method does not require the labels for data perturbations and can also be used in semi-supervised learning. The virtual adversarial example is the point in an ball around the datapoint that maximally perturbs the label distribution around that point as measured by the Kullback-Leibler divergence η = arg min KL[p(y\|x, θ) \|\| p(y\|x + r, θ)]. (3) r:krk2 ≤ We propose to draw perturbations from a large dataset of unlabeled images U whose cardinality is much higher than A. For each example x, we use the example x̃ that is nearby in the space defined by the penultimate layer in our fully convolutional network. This layer contains spatial and semantic information about the objects present in the image, and therefore x and x̃ have similar semantics and composition while they may be far away in pixel space. We consider the cosine similarity metric to find samples which are close to each other in the feature space and for efficiency we compute the dot product of the L2 normalized feature vectors. Let θ denote the optimal parameters found after minimizing the cross-entropy loss using the training data in A, and fθ (x) be the L2 normalized feature vector obtained from the penultimate layer of our network(Conv(1,1,2048)). The similarity between two images x and x0 is then computed by their dot product S(x, x0 ) = fθ (x)T fθ (x0 ). For each training sample (xi , yi ) in A, we find the most similar item in U x̃i = arg max S(xi , x), (4) x∈U Algorithm 1 Distributed Regularizer Sample Search function M AP(k, x) . k: sample index in U, x: image data Compute network output layer fθ (x) Compute similarities with samples in A: si = fθ (x)T fθ (xi ), ∀i = 1 . . . n Sort s by descending similarity values si1 ≥ si2 . . . ≥ sin for l = 1 to km do EMIT il , (k, sil ) end for end function function R EDUCE(k, v) . k: sample index in A, v: Iterator over (sample index in U, similarity score) tuples Let v = ((i1 , c1 ), (i2 , c2 ) . . . Sort v by descending similarity values cj1 ≥ cj2 . . . ... for l = 1 to kr do EMIT k, ijl , cjl end for end function and transfer the labels from xi , to generate a new, Real Adversarial (Regularizer), training sample (x̃i , yi ). Similar to adversarial and virtual adversarial training, our method improves the classification performance. We interpret our sample perturbation as a form of adversarial training where additional examples are sampled from a similar semantic distribution as opposed to noise. We also used the perturbation of each labeled sample in the gradient direction (similar to adversarial training) to find the nearest neighbor in unlabeled set and observed similar performance. Therefore, in this paper our focus is on using the labeled samples for finding Regularizer instances to improve performance. 3.4. Large scale approximate regularizer sample search In our experiments we use the YFCC dataset as our set of unlabeled images. Since it contains 100 million images, an exhaustive nearest neighbor search is impractical. Hence we use the MapReduce framework [6] to find approximate nearest neighbors in a distributed fashion. Our approach is outlined in Algorithm 1. We first pre-compute the feature representations fθ (xi ) for xi ∈ A. The size of A for datasets such as MS COCO or Visual Genome is small enough that it is possible for each mapper to load a copy into memory. A mapper then iterates over samples x in U, computes the feature representation fθ (x) and its inner product with the pre-computed features in A. It emits tuples for the top km matches that are keyed by the index in A, and also contain the index in U and similarity score. After the shuffling phase, the reducers can select for each sample in A the kr closest samples in U. We use km = 1000 and kr = 10. We are able to run the search in a few hours, with the majority of Figure 3. Top regularizer examples from unlabeled YFCC dataset (row 2-6) that are retrieved for multi-label image queries in several of the MS COCO categories (first row). the time being in the mapper phase where we compute the image feature representation. Note that our method does not guarantee that we can retrieve the nearest neighbor for each sample in A. Indeed, if for a sample xi there exists km samples xj such that fθ (xj )T fθ (x̃i ) ≥ fθ (xi )T fθ (x̃i ), then the algorithm will output either no nearest neighbor or another sample in U. However we found our approximate method to work well in practice. 4. Experiments 4.1. Semi-Supervised Multilabel Object Categorization and Localization We use the MS COCO [25] and Visual Genome [21] datasets as our source of clean training data as well as for evaluating our algorithms. MS COCO has 80 object categories and is a common benchmark for evaluating object detectors and classifiers in images where objects appear in context. The more recent Visual Genome dataset has annotations for a larger number of categories than MS COCO. Applying our proposed method on the Visual Genome dataset is important to understand whether the algorithm scales to a larger number of categories, as it is ultimately important to recognize thousands of object classes in real world applications. All images for both datasets come from Flickr. In all experiments we only use the image labels for training our models and discard image captions and bounding box annotations. For the MS COCO dataset we use the standard split used in [25] for training and evaluating the models. The training set contains 82,081 images and validation set has 40,504 images. For the Visual Genome dataset we only use object category annotations for images. The images are labeled as a positive instance for each object if the area ratio of the bounding box with regards to the image area is more than 0.025. We only consider the 1,432 object categories for which there are at least 80 image instances in the training set. The Visual Genome test set is the intersection of Visual Genome with the MS COCO validation set which is comprised of 17,471 images. We use the remaining 90,606 images for training our models. As for the source of unlabeled images, we use the tie bottle umbrella refrigerator truck remote Figure 4. Object localization comparison between “FCN,N-OR”(mid row) and “FCN,N-OR,V I S E R”(last row). YFCC dataset [40] and discard the images that are present in Visual Genome or MS COCO. The data is 14TB and is stored in the Hadoop Distributed File System. We use the TensorFlow software library [1] to implement our neural networks and conduct our experiments. To conduct the distributed nearest-neighbor search, we use a CPU cluster. We use VGG16 architecture pre-trained for the ImageNet classification task as our base network. We resize images to 500×500. Our initial learning rate is 0.01 and we apply the 0.1 decay factor for adapting the learning rate two times during the training after 20K and 40K mini-batches. We run stochastic gradient descent for 60K iterations with mini-batches of size 15 which corresponds to 11 epochs. We conduct our experiments on the object classification and point-based object localization tasks. As for the object evaluation metric, we use the mean Average Precision (AP) metric where we first compute the precision for each class and then take the average over classes. For evaluating our object localization, we use the point localization metric introduced in [33], where the location for a given class is given by the location with maximum response in the corresponding object heatmap. The location is considered correct if it is located inside the bounding box associated with that class. Similar to [33] we use a tolerance of 18 pixels. Tables 1 and 2 summarize our results with the mean AP for the classification and localization tasks on the MS COCO and Visual Genome datasets. We compare our performance with three state-of-the-art methods for object localization and classification [33, 38, 3]. In [33], to handle the uncertainty in object localization, the last fully connected layers of the network are considered as convolution layers and a maxpooling layer is used to hypothesize the possible location of the object in the images. In contrast, we use Noisy-OR as our pooling layer. In [38], a multi-scale fully convolutional neural network called ProNet is proposed that aims to zoom into promising object-specific boxes for localization and classification. We compare against the different variants of ProNet with chain and tree cascades. Our method uses a single fully convolutional network, is simpler and has a lighter architecture as compared to ProNet. In all tables ‘FullyConn’ refers to the standard VGG16 architecture while ’FullyConv’ refers to the fully convolutional version of our network (see Figure 2). The Noisy-OR loss is abbreviated as ’N-OR’, and we denote our algorithm with V I S E R. We can see in Table 1 that our proposed algorithm reaches 50.64% accuracy in the object localization task on the MS COCO dataset which is more than a 4% boost over [38] and a 9.5% boost over [33]. Also, without doing any regularization and by only using Noisy OR (N-OR) paired with a fully convolutional network, we obtain higher localization accuracy than Oquab et al. [33] and different variants of ProNet [38]. In the object classification task, our proposed V I S E R approach outperforms other state-of-the-art baselines of [38, 33] by a margin of more than 4.5% and gains an accuracy of 75.48% for the MS COCO dataset. In addition, other variants of [38] are less accurate than our fully convolutional network architecture with Noisy-OR pooling (‘FullyConv, NOR’). This result is consistent with the results we obtained in the object localization task. While the method of [3] obtains competitive performance on the MS COCO localization task, our method outperforms it in the classification task by a large margin of 21.4%. A recent method [9] obtains classification and localization accuracy of 80.7% and 53.4% respectively using the deeper ResNet [17] architecture. Hence it is not directly comparable with ProNet [38], [3] and our method which use the VGG [36] as base network architecture. In addition our proposed method has a label propagation step which produces a large set of labeled images with object level localization in “object in context” scenes and can be used in Table 1. Mean AP for classification and localization tasks on the MS COCO dataset (higher is better). Table 2. Mean AP for classification and localization tasks on the Visual Genome dataset (higher is better). Method Classification Localization Method Classification Localization Oquab et al. [33] ProNet (proposal) [38] ProNet (chain cascade) [38] ProNet (tree cascade) [38] Bency et al. [3] 62.8 67.8 69.2 70.9 54.1 41.2 43.5 45.4 46.4 49.2 FullyConn FullyConv,N-OR FullyConv,N-OR,AT [13] FullyConv,N-OR,VAT [31] FullyConv,N-OR,V I S E R 9.94 12.35 13.96 13.95 14.82 – 7.55 9.05 9.06 9.74 FullyConn FullyConv,N-OR FullyConv,N-OR,AT [13] FullyConv,N-OR,VAT [31] FullyConv,N-OR,V I S E R 66.68 72.52 74.38 74.30 75.48 – 47.47 49.75 49.42 50.64 other learning methods. Also, the method of [9] is based on a new pooling mechanism while our method proposes a better regularization for training ConvNets using a large scale set of unlabeled images in a semi-supervised setting and therefore is orthogonal to [9]. We also perform an ablation study and compare against other forms of regularization using our fully convolutional network architecture with Noisy-OR pooling (‘FullyConv, N-OR’). In Table 1 and 2, we compare three forms of regularization: adversarial training (‘AT’) [13], virtual adversarial training (‘VAT’) [31], and our proposed Visual Self-Regularization (V I S E R) using the YFCC dataset as source of unlabeled images. To conclude, our proposed approach outperforms state-ofthe-art methods as well as several baselines by a substantial margin in object classification and localization tasks according to the results shown in Table 1 and Table 2. Hence, the regularization mechanism of our proposed method results in a performance boost compared to the other forms of adversarial example data augmentation. We show that visual self-regularizers (V I S E R) make our learning robust to noise and provides better generalization capabilities. 4.2. Object-in-Context Retrieval To qualitatively evaluate V I S E R, we show several examples of the Regularizer instances retrieved using our approach in Figure 3. For each of the labeled images shown in the first row of Figure 3, we show the top 5 retrieved images. As we can see, the unlabeled images retrieved by our approach have high similarity with the queried labeled image. Furthermore, most of the objects in the labeled images also appear in the retrieved images. This observation qualitatively demonstrates the effectiveness of our label propagation approach. It is worth mentioning that Figure 3 shows that the relative location of the objects in the retrieved images is fairly consistent with that of the query images. This suggests that our simultaneous categorization and localization approach can also be used for propagating bounding box annotations. Figure 1 shows the results of our V I S E R approach on the ‘Bus’ category. We visualize the t-SNE [27] map of the whole set of images labeled as ‘Bus’ which includes instances from both the labeled images in the MS COCO and unlabeled instances from the YFCC dataset. To produce the t-SNE visualization we take the output of the penultimate layer of our network as explained in Section 3. We L2 normalize the feature vectors to compute the pairwise cosine similarity between images using a dot product. We visualize the t-SNE map using a grid [20]. A different background color (blue vs. green) is assigned to images depending on whether they are from the labeled or unlabeled set. Notice that it is challenging to determine the color corresponding to each dataset as photos are from a similar domain. The images with a blue background belong to the MS COCO dataset and the images with a green background belong to the YFCC dataset. This visualization reveals that there are many images in the large unlabeled web resources that can potentially be used to populate the fully annotated dataset with more examples. This is a step forward for improving object categorization as well as decreasing human effort for supervision. Figure 6 demonstrates the qualitative performance of “FCN,N-OR,V I S E R” for multi-label localization. We visualize the object localization score maps where the localization regions with high probability are shown in green. We also display the localized objects using red dots. The score maps show that our approach can accurately localize small and big objects even in extreme cases where a big portion of the object is occluded. In the first row of Figure 6 ‘dog’ and ‘laptop’ are localized quite accurately while they are largely occluded and truncated. Similarly, the third row shows the accurate localization of a ‘chair’ although it appears in a small region of the image and is largely occluded. When there are multiple instances of an object category, such as ‘person’ in the second row, ‘potted plant’ in the third row, and ‘car’ in the sixth row, all regions corresponding to these instances get a high score. The failure cases of our approach are distinguished via red boxes in Figure 6. For instance, the ‘skateboard’ in row 6 is localized around the region close to the person’s leg. In row 8, although the ‘backpack’ region gets a high score map, it fails to contain the highest peak and thus the localization metric considers it as a mistake. 0.95 0.80 0.65 0.50 0.35 0.20 Cross entropy Dropout Adversarial training VAT VISER(ours) 0.05 Figure 5. Generalization comparison on a synthetic dataset between proposed V I S E R , dropout, adversarial training, and virtual adversarial training (VAT). Training samples are shown with black borders and rest of instances are test set. Each plot demonstrates the contour of the p(y = 1\|x, θ), from p = 0 (blue) to p = 1 (red). Table 3. Classification error on test synthetic dataset (lower is better). Error(%) cross entropy dropout [37] AT [13] VAT [31] VISER 9.244±0.651 9.262±0.706 8.960±0.849 8.940±0.393 8.508±0.493 We show several examples of the localization score maps produced by “FCN,N-OR” and “FCN,N-OR,V I S E R” in Figure 4. By comparing the localized regions in green, we see that “FCN,N-OR,V I S E R” can locate both small and big objects more accurately. For example, in localizing small objects such as ‘tie’, ‘bottle’ and ‘remote’, the peak of the localization region produced by “FCN,N-OR” has a large distance with the correct location of the object while “FCN,N-OR,V I S E R” localizes these objects precisely. Also, “FCN,N-OR” fails to be as accurate as “FCN,N-OR,V I S E R” in localizing big objects such as ‘umbrella’ and ‘refrigerator’. 4.3. Classification on Synthetic Data In order to evaluate the ability of our algorithm to leverage unlabeled data to regularize learning, we generate a synthetic two-class dataset with a multimodal distribution. The dataset contains 16 training instances (each class has 8 modes with random mean and covariance for each mode and 1 random sample per mode is selected), 1000 unlabeled and 1000 test samples. We linearly embed the data in 100 dimensions. Since the data has different modes, we can mimic the object categorization task where each object category appears in a variety of shapes and poses, each of which can be considered as a mode in the distribution. We use a multi layer neural network with two fully connected layers of size 100, each followed by a ReLU activation and optimized via the cross-entropy loss. We compare the generalization behavior of V I S E R with the following regularization methods: dropout [37], adversarial training [13], and virtual adversarial training (VAT) [31]. The contour visualization of the estimated model distribution is shown in Figure 5. We can see that both adversarial training and virtual adversarial training are vulnerable to the location of the training sample of each mode. These regularization techniques can learn a good boundary of the class when the training instance is at the center of the mode, but they over-smooth the boundary whenever the training instance is off-center. However, our proposed V I S E R sampling from unlabeled data learns a better local class distribution as adversarial samples follow the true distribution of the data and are less biased to the training instances. The dropout technique is also learning a good regularization, but it is less smooth at the boundaries of the local modes. Table 3 summarizes the misclassification error on test data over 50 independent runs on the synthetic dataset. 5. Conclusion and Future Work In this paper we have presented a simple yet effective method to leverage a large unlabeled dataset in addition to a small labeled dataset to train more accurate image classifiers. Our semi-supervised learning approach is able to find Regularizer examples from a large unlabeled dataset. We have achieved significant improvements on the MS COCO and Visual Genome datasets for both the classification and localization tasks. The performance of our approach could be further improved in future work by incorporating user provided data such as ‘tags’. Also, having access to a large set of unlabeled data is fairly common in other domains and hence we believe our approach could be applicable beyond visual recognition. References [1] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous systems. 2015. 6 [2] C. Barnes, E. Shechtman, A. Finkelstein, and D. Goldman. Patchmatch: A randomized correspondence algorithm for structural image editing. ACM TOG, 2009. 3 [3] A. J. Bency, H. Kwon, H. Lee, S. Karthikeyan, and B. Manjunath. Weakly supervised localization using deep feature maps. In ECCV, 2016. 2, 6, 7 Figure 6. Localization results of the proposed model on MS COCO validation set. 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Article Antenna Arrays for Line-Of-Sight Massive MIMO: Half Wavelength Is Not Enough Daniele Pinchera , Marco Donald Migliore, Fulvio Schettino and Gaetano Panariello DIEI and ELEDIA@UNICAS, University of Cassino and Southern Lazio, via G. Di Biasio 43, 03043 Cassino, Italy; mdmiglio@unicas.it (M.D.M.); schettino@unicas.it (F.S.); panariello@unicas.it (G.P.) Correspondence: pinchera@unicas.it; Tel.: +39-0776-299-4348 arXiv:1705.06804v2 [cs.IT] 1 Sep 2017 Received: 22 July 2017; Accepted: 8 August 2017; Published: 10 August 2017 Abstract: The aim of this paper is to analyze the array synthesis for 5 G massive MIMO systems in the line-of-sight working condition. The main result of the numerical investigation performed is that non-uniform arrays are the natural choice in this kind of application. In particular, by using non-equispaced arrays, we show that it is possible to achieve a better average condition number of the channel matrix and a significantly higher spectral efficiency. Furthermore, we verify that increasing the array size is beneficial also for circular arrays, and we provide some useful rules-of-thumb for antenna array design for massive MIMO applications. These results are in contrast to the widely-accepted idea in the 5 G massive MIMO literature, in which the half-wavelength linear uniform array is universally adopted. Keywords: MIMO; antenna arrays; line-of-sight propagation; millimeter wave propagation; 5 G mobile communication 1. Introduction Massive MIMO is going to become one of the key technologies of the incoming next generation 5 G wireless systems [1–12]. In particular, the use of frequencies above 6 GHz seems to be particularly appealing, because of two main reasons: first of all, the availability of wide contiguous frequency bands to exploit for communications; second, the relative compactness of the resulting arrays [12–15]. If we briefly focus on this second aspect, we would observe that the use of a frequency of the order of 60 GHz would mean a wavelength λ of 5 mm; thus, a 100λ array would be just half a meter in size. The use of mm-length frequencies is not, unfortunately, without drawbacks. The well-known Friis formula for wireless connections shows a power attenuation that increases with the square of the wavelength, which significantly reduces the area that can be covered by a transmitting antenna. Furthermore, the propagation of the waves at mm-wave frequencies is much different from the propagation at lower frequencies, resembling more a quasi-optical connection between the terminals: the rich scattering condition is really difficult to realize, and the Line-of-Sight (LoS) component becomes dominant (if not the only present) [16,17]. This fact has posed some questions on the actual possibility to achieve the so-called “favorable propagation” in the Multi-User MIMO (MU-MIMO) channel matrix, but both measurements and simulations have confirmed the advantages of massive MIMO also in the extreme LoS condition [18–24]. One of the key points to understand why massive MIMO keeps on working with a very good spectral efficiency also in this case is that the massive antenna array, because of its huge size, does not work in the far-field condition, so that the wave fronts are not planar, but spherical: we could be able to distinguish between two terminals positioned at the same angle with respect to the Base Station (BS) by means of their different distances [25–27]. Electronics 2017, 6, 57; doi:10.3390/electronics6030057 www.mdpi.com/journal/electronics Electronics 2017, 6, 57 2 of 20 Among the many aspects of massive MIMO systems, the optimization of the radiation system has been the object of relatively little research, and equispaced half-wavelength arrays are commonly considered for massive MIMO antennas. However, as pointed out above, massive MIMO antennas work in the near-field condition. This characteristic does not seem to have been exploited in the literature. A further point to be better explored is the reduction of the cost and the complexity of the overall antenna, which is critically related to the number of radiating elements. Accordingly, a fundamental consideration is now in order: is it possible to exploit this near-field behavior with a fixed complexity of the antenna in terms of the number of elements without affecting, or hopefully increasing, the performance? This paper will try to answer this question, showing that optimization of the antenna array using non-uniform arrays in the case of linear antennas and properly equispaced arrays in circular antennas can allow a significant improvement of the communication system performance. 2. Antennas and Propagation Model In this paper, we will use a simplified model for antennas and propagation; in particular, we will consider isotropic radiating elements; we will neglect the effect of mutual coupling and polarization, and we will consider a simple 2D propagation model, i.e., BS antennas and user terminals will lie on the same plane. Furthermore, we will consider a fixed frequency f and the corresponding wavelength λ. These assumptions have a two-fold reason; first, the model we will discuss, and the achieved results, could be easily reproduced by any interested reader; second, the simplified model will allow us to remove the aspects that play a minor role in the definition of the channel matrix (polarization and mutual coupling), focusing only on the aspects, like the antenna positioning, that play the main role in the overall system performance. In particular, we will consider a linear antenna array of N identical elements, with the antennas aligned along the x axis, whose abscissas are stored in the vectors xs = [ xs,1 ; · · · ; xs,N ]. The position of the K terminal antennas will be instead described by the vectors xt = [ xt,1 ; · · · ; xt,K ] and yt = [yt,1 ; · · · ; yt,K ] (see Figure 1). The transfer function between the n-th antenna of the BS and the k-th terminal will be achieved as: e− jβRn,k hn,k = γ (1) Rn,k where γ is an inessential multiplicative constant, β = 2π/λ is the free space wavenumber and: Rn,k = q ( xs,n − xt,k )2 + y2t,k (2) The MU-MIMO channel matrix can be thus built as: H = [ h1 · · · h K ] (3) where the column vector hk represents the channel response of the k-th terminal. This channel matrix can be properly normalized, according to the guidelines in [20] before performing its singular value decomposition or calculating the spectral efficiency (thus making the constant γ inessential). It has also to be underlined that the use of Equation (1) allows us to take into account the different power received by the antennas at the base station because of their different distance from the terminals, as well as the spherical shape of the wavefronts. Electronics 2017, 6, 57 3 of 20 Figure 1. Massive MIMO system scenario. Circles: base station antennas; squares: terminals. 3. User Correlation Analysis From this section on, we will choose f = 60 GHz as the working frequency and a base station array of 200 elements. The choice of considering a fixed number of antennas at the BS has been made just to compare different antenna geometries with the same system complexity. In order to get better insight into the effect of the antenna shape on the system performance, we will start considering the simple case of only two terminals, and we will calculate the correlation coefficient between the two columns of H, as: χ1,2 = \|h1 · h2∗ \| . \|\|h1 \|\| \|\|h2 \|\| (4) The use of this performance metric is due to the fact that it provides a simple way to check the effectiveness of a maximal ratio combining approach at the BS; furthermore, as will be clear in the following, it allows us to provide a simple and effective graphical description of the correlation between user terminal positions. Let us first consider the case of an equispaced array, with the inter-element distance d = λ/2; in Figure 2, it is possible to see the case in which the two terminals are aligned along the y axis, and have only different distances (r1 and r2 ) from the BS. In the left subplot, we can see an image map representing the correlation coefficient of Equation (4) for variable r1 and r2 , while in the right subplot, we can see the plot of the correlation coefficient for some fixed values of r2 . This figure confirms that it is possible to distinguish between two users sharing the same angular position with respect to the center of the BS, but this feature can be exploited only when one terminal is very close to the BS with respect to the other. It has to be underlined that this behavior has been obtained when the two terminals lie on the y axis; for other directions, it is possible to obtain different plots, showing a slightly increased correlation, not reported here for the lack of space. Let us now consider an equispaced array with a larger inter-element distance, say d = 2λ. From an antenna point of view, increasing the inter element distance, without changing the number of elements, increases the far-field distance of the array, which is conventionally calculated as: r FAR = 2[( N − 1)d]2 . λ (5) Electronics 2017, 6, 57 4 of 20 This means that an increase of the inter-element distance of a four factor increases the far field distance of a 16 factor. In Figure 3, we can see the image map and the plot of the correlation for the larger array. It is clear that we are now allowed to easily distinguish two users that share the same angular position with respect to the BS, much more than in the previous case. Figure 2. Two terminal correlation analysis for the λ/2 equispaced array. (Left) Image map with the correlation level for variable r1 and r2 ; (right): plot of the correlation coefficient for some fixed values of r2 . Figure 3. Two terminal correlation analysis for the 2λ equispaced array. (Left) Image map with the correlation level for variable r1 and r2 ; (right): plot of the correlation coefficient for some fixed values of r2 . The previous result has shown that the increase of the inter-element distance seems to have a beneficial effect on the capability of decorrelating users. To confirm this feature, let us see what happens if we consider the first terminal in a fixed position with respect to the array, and we move the second terminal on the xy plane. In Figure 4, we can see the case of a λ/2 equispaced array: apart from an angular sector around the broadside direction of the array, the correlation is very low, thus confirming the very good results, in terms of favorable propagation, found by other researchers. If we instead employ the 2λ equispaced array, we obtain the result depicted in Figure 5: apart from the broadside direction, which shows a slightly lower correlation level with respect to the λ/2 case, there are other directions in space that present a non-negligible correlation. These regions are related to the unavoidable grating lobes, appearing when we consider equispaced elements with an inter-element distance greater than half-wavelength. Clearly, the presence of grating lobes causes ambiguities and must be avoided. As an example, let us imagine a communication system with just two user terminals: when user Terminal A is in the angular position of a grating lobe relative to user Terminal B, it could be very difficult to distinguish them. Electronics 2017, 6, 57 5 of 20 Figure 4. Image map of the correlation coefficient when the first terminal is fixed in the position marked with the circle and the second user is moved on the plane for the λ/2 equispaced array. Figure 5. Image map of the correlation coefficient when the first terminal is fixed in the position marked with the circle and the second user is moved on the plane for the 2λ equispaced array. We would like to specify that in an array not working in the far field region, we could not rigorously talk of grating lobes, but for the sake of simplicity, we will keep on using this definition for the rest of the manuscript. According to the correlation analysis, it seems that the possible advantage of distinguish aligned user terminals could be negated by the effect of grating lobes. These results suggest that a non-equispaced array, not affected by grating lobes, could get the advantages of a larger size of the array, without the drawbacks of the grating lobes, but to confirm this hypothesis, we are first required to analyze what happens with multiple terminals. 4. Multi User Channel Matrix Let us now consider a number K of terminals, uniformly randomly displaced in an angular sector ϕ ∈ [− ϕs , ϕs ], with a minimum distance rmin and a maximum distance rmax from the center of the BS array. For all of the simulations that will be discussed in this section, we will use ϕs = π/3, rmin = 5 m and rmax = 200 m; the choice of these parameters has been done in order to simulate a sectorized coverage of a large planar area, like a plaza or a park. It has to be underlined that differently from [28], we will consider a fixed number of antennas at the BS, in order to compare systems with the same complexity (and cost). Electronics 2017, 6, 57 6 of 20 In order to better emphasize the possibility to discriminate different users according to their distance, we will employ “Normalization 1” from [20]. In this way, all of the columns of the channel matrix H will have the same norm, thus removing the unbalance of channel attenuation between users due to their different relative distance from the BS. M = 1000 different random scenarios have been considered in the numerical simulations. As a first test, in Figure 6, we show as a function of the inter-element distance d and the number of users K the average condition number of the channel matrix, defined as: ρdB = 20 ∗ log10 σ1 σK (6) where σ1 and σK are the greatest and smallest singular values of the channel matrix. The choice of the condition number is due to the fact that it provides a direct indication of the orthogonality of the columns of the channel matrix, and it is related to the stability of its inversion [29], needed in many multi-user MIMO transmission schemes [30]. Figure 6. Mean condition number of the channel matrix as a function of the inter-element distance d and the number of users K. The main advantage in using the condition number is that the presented results are independent of the actual SNR at the receiver; it is indeed true that the condition number provides only a rough indication of the channel quality, since it gives only the spread of the singular values and not their actual distribution, but its value is strongly related with the performances of MIMO precoders and detectors [31–33]. We see that the average condition number is an increasing function of the number of users, but it is interesting to note that (besides some oscillations), it is also a decreasing function of the inter-element distance d at the base station, confirming the advantages of a larger array at the BS foreseen in the previous paragraph. A brief discussion of the oscillations is now needed. Taking as a reference the K = 100 curve of Figure 6, we can see that the condition number increases up to about d = 0.7λ, then it decreases up to d = 1.1λ, then it increase again up to d = 1.3λ, and so on. As we have seen in Figure 5, the presence of grating lobes implies that a number of angular regions of the scenario should be avoided for an effective communication. The number of grating lobes that affect the results is indeed an increasing discrete function of d, while the reduction of the beamwidth and the correlation among users sharing the same angle ϕ with respect to the BS are instead a continuous decreasing function of d. These two effects together contribute to the aforementioned oscillating/decreasing behavior seen in Figure 6. Electronics 2017, 6, 57 7 of 20 It is important to recall that a very low condition number is necessary to apply the Maximal Ratio Combining (MRC) algorithm at the BS, and a low condition number is also necessary to correctly calculate the matrix inversion needed to efficiently use Zero Forcing (ZF) at the BS [11,34]. To get better insight into this result, let us focus on the case of K = 50; in Figure 7, we show the distribution of the condition number as a function of the distance d (each column of the matrix in the image map is a histogram of the condition number). The increase of d from λ/2 to 2λ reduces the average value of the condition number of about 15 dB, and also, its spreading decreases. Figure 7. Distribution of the condition number as a function of the distance d for K = 50. Each column of the matrix in the image map is a histogram of the condition number. With reference to the two extreme cases of d = λ/2 and d = 2λ, let us now analyze the distribution of the singular values for K = 50. In Figures 8 and 9, we have the image map of the distribution of the singular values, as well as the plot of their average values. The larger spacing improves in particular the lower singular values, leading to a much more stable inversion. Electronics 2017, 6, 57 8 of 20 Figure 8. Singular values analysis for d = λ/2 and K = 50. (Left) Image map of the distribution of each one of the 50 singular values; (right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. It has to be underlined that the results achieved in this paragraph hold also when we do not consider the normalization. Since we would just see a slightly larger spreading of the singular values for all of the considered cases, we do not report these results because of the limited space of the paper. Figure 9. Singular values analysis for d = 2λ and K = 50. (Left) Image map of the distribution of each one of the 50 singular values. (Right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. 5. Exploiting Sparse Arrays In the previous section, we have shown that increasing the size of the array is beneficial to the overall MU-MIMO channel matrix condition number: even if we have to deal with the effect of grating lobes, the possibility to discriminate sources that share a similar angular direction, as well as the reduced width of the main beam lead to a better conditioned matrix. It is trivial to recognize that, if we could get rid of the grating lobes, we should expect a further improvement of the performances. To this aim, we would have two different possibilities: in the first Electronics 2017, 6, 57 9 of 20 case, we could employ a sub-array architecture [35], but we should also deal with the reduced scanning capability of our array (and hence, the dimension of the sector we could cover); the second possibility would be to employ a non-equispaced, sparse array. The synthesis of sparse arrays is one of the most studied antenna topics in the last 50 years, and a very large number of methods and algorithms has been developed for linear and planar arrays [36–47]. Unfortunately, none of these algorithms can be directly applied to an MU-MIMO system: our aim is to obtain the antenna positions, within a given range, that allow the best performances of a multi-user scenario, and this problem cannot be directly translated into a classical antenna pattern synthesis problem. To solve this issue, we propose a novel paradigm for antenna array synthesis, inspired by the results presented in the previous paragraph: we will be seeking for the antenna positions providing the best performances on a relatively large set of user terminal scenarios. We could, for example, optimize the antenna array to provide the lowest average condition number of the channel matrix, or we could look for the antenna configuration that guarantees the lowest 0.99-quantile of the condition number. Once we have defined our optimization objective, we could use an evolutionary search method to achieve the sought antenna positions. Actually, the direct application of this approach could be very ineffective, since the number of unknowns we have to deal with is of the order of the number of antennas, and evolutionary algorithms are known to require a computational time that increases exponentially with the number of unknowns [48]. A much more convenient, yet sub-optimal, choice is to define a proper parametrization of the antenna positions, thus reducing the search space of the evolutionary algorithm to a small number of parameters and leading to a reasonable computational time. A simple and effective parametrization of the antenna positions can be found using Tchebyshev polynomials. If we are looking for a symmetric array, the position of the n-th antenna xs,n could be found has: ! ! P xs,n = L 1− ∑ αp p =1 with: L= P un + ∑ α p T2p+1 (un ) (7) p =1 d0 2( N − 1) (8) where Tp (·) is the Tchebyshev polynomial of order p, [α1 , · · · , α P ] is the set of P parameters defining the position of the antenna elements, d0 is the desired average inter-element distance, N is the overall number of antennas and [u1 ; · · · ; u N ] is an equispaced ordered sampling of the interval [−1, 1]. Not all of the combinations of [α1 ; · · · ; α P ] would lead to a monotonically-increasing vector of positions xs in the range [− L, L]; for instance, it is easy to demonstrate that if P = 1, any value of α1 in the range: − 1 1 < α1 < 8 4 (9) would lead to a set of positions in the range [− L, L] satisfying the monotonicity constraint, but for higher values of P, it would be non-trivial to achieve a formula like Equation (9) in closed form. In the remaining part of the paper, we will focus on the relatively easy case of P = 1; this choice is justified by the results of some numerical tests, which have shown only a marginal improvement when using P > 1. In particular, values of α1 < 0 would lead to antenna arrays with a higher density of elements at the extrema, while values of α1 > 0 would lead to antenna arrays with a higher density of elements at the center. Following the guidelines discussed before, we will consider now a parametric analysis of the performances, in terms of average condition number, on a set of 1000 configurations of K = 50 user terminals, as a function of α1 . The result is depicted in Figure 10: apart from values very close to α1 = 0, corresponding to the equispaced array, we achieve a significant advantage in the mean condition number for both positive and negative values of α1 (with a slight advantage for the latter). Electronics 2017, 6, 57 10 of 20 Figure 10. Average condition number as a function of α1 for a number of users K = 50 and d0 = 2λ. To get better insight into this surprising result and to better understand the effect of the non-linear spacing on the array performances, we will consider the same analysis developed in the previous two paragraphs on a test array obtained with α = −0.03 with an average inter-element distance d0 = 2λ. The choice of this particular value of α1 is driven by the desire of reducing the spreading of the inter-element distance, in particular in this case, we have a minimum inter-element distance (at the edges) of 1.53λ and a maximum inter-element distance (at the center) of 2.24λ. In Figure 11, we can see that the capability of the array to distinguish terminals sharing the same angular position with respect to the BS has been marginally improved with respect to the case of Figure 3. The difference with respect to the corresponding equispaced array is instead much clearer from the analysis of Figure 12: now, the effect of the grating lobes is replaced by some sectors of the plane that show a lower (but non-null) correlation coefficient. Figure 11. Two terminal correlation analysis for the 2λ non-equispaced array with α1 = −0.03. (Left) Image map with the correlation level for variable r1 and r2 ; (right) plot of the correlation coefficient for some fixed values of r2 . Electronics 2017, 6, 57 11 of 20 Figure 12. Image map of the correlation coefficient when the first terminals are fixed in the position marked with the circle and the second user is moved on the plane for the 2λ non-equispaced array with α1 = −0.03. In Figure 13, we have instead depicted the effect of the non-linear spacing for a variable number of the average inter-element distance d and the number of users K: the oscillatory behavior of Figure 6 has now disappeared, and the condition number is now a monotonic function of the number of terminals and the average inter-element distance. The improvement with respect to the equispaced array is even more significant when the number of user terminals is higher. It is worth underlining that the use of a non-equispaced array is beneficial even when using an inter-element distance only slightly higher than half-wavelength, so to achieve a performance improvement, we are not required to employ large array sizes. It is also interesting to see that the spreading of the condition number is reduced when using a non-equispaced array: the result in Figure 14, where the case with K = 50 is analyzed, shows a conditioning significantly better than the one presented in the equispaced case of Figure 7, for any average inter-element spacing. Finally, also the analysis of the distribution of the singular values for the K = 50 case confirms the advantages of the non-equispaced array (see Figure 15). Figure 13. Mean condition number of the channel matrix as a function of the average inter-element distance d and the number of users K for a non-equispaced array with α1 = −0.03. Electronics 2017, 6, 57 12 of 20 Figure 14. Distribution of the condition number as a function of the average distance d for K = 50 and the sparse array with α1 = −0.03. Each column of the matrix in the image map is a histogram of the condition number. Figure 15. Singular values analysis for d0 = 2λ, α1 = −0.03 and K = 50. (Left) Image map of the distribution of each one of the 50 singular values; (right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. It is important to underline that, whatever the optimization approach used to define the position of the elements in the sparse array, those positions are fixed: their choice has to be done only once, when designing the array. It is also worth underlining that in the simulations shown, no smart terminal selection techniques [18] have been employed. This means that we can achieve an even better conditioned channel matrix by the application of these approaches. 6. Circular Arrays In the previous sections, we have considered linear BS arrays for a sectoral coverage; if the terminals are displaced in a circular area, it could be convenient to use a circular array. The increase of the antenna dimension, without changing the number of active elements, can be very beneficial also in this case. A circular array does not show grating lobes, so it will not be necessary to consider non-equispaced positioning of the array elements on the circle, but the increase in size would allow a better discrimination of the users, because of the increased far field distance of the array. Electronics 2017, 6, 57 13 of 20 To better compare the results of the present section to the results of the previous ones, we will always consider N = 200 radiating elements, displaced on a circle of radius R = Nd/(2π ) where d is the (approximate) inter-element distance. For the sake of compactness of the paper, we will skip the correlation analysis, and we will only focus on the condition number analysis of the MU-MIMO channel matrix. In particular, we will consider a set of 1000 different propagation scenarios, where a variable number K of randomly-positioned user terminals is displaced between a distance rmin = 5 m and rmax = 200 m from the center of the BS antenna. In Figure 16, we can see the average condition number that we can obtain for a variable inter-element distance d and a variable number of users K. This result confirms the importance of increasing the size of the antenna array as much as possible if we are interested in improving the conditioning of the channel matrix. It is also worth noting that this result is surprisingly very similar to the result obtained with the non-equispaced antenna array of Figure 13. In Figure 17, we instead depict the distribution of the condition number for a variable antenna distance d for the specific case K = 50; even in this case, it is surprising to see the behavior of the condition number that is very similar to the result depicted in Figure 14. Finally, in Figures 18 and 19, we depict the distribution of the singular values for the two specific cases of d = λ/2 and d = 2λ, confirming the advantage of an increased antenna array dimension. Figure 16. Mean condition number of the channel matrix as a function of the average inter-element distance d and the number of users K for a circular array. Figure 17. Distribution of the condition number as a function of the average distance d for the circular array with K = 50. Each column of the matrix in the image map is a histogram of the condition number. Electronics 2017, 6, 57 14 of 20 Figure 18. Singular values analysis for the circular array with d = λ/2 and K = 50. (Left) Image map of the distribution of each one of the 50 singular values. (Right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. Figure 19. Singular values analysis for the circular array with d = 2λ and K = 50. (Left) Image map of the distribution of each one of the 50 singular values; (right) solid line, average singular values; dashed lines, average singular values ± the standard deviation of the distribution. 7. Evaluating the Spectral Efficiency Up to this point, we have focused on the singular values of the channel matrix and its condition number. As stated before, this choice provides results independent of the SNR at the receiver. A further figure of merit that is often used in MIMO system analysis is spectral efficiency. In this section, we show how the better conditioning of the channel matrix influences the spectral efficiency of massive MIMO systems. In particular, we will evaluate the average maximum achievable rate using a Zero Forcing (ZF) receiver [49], considering a set of 1000 different scenarios for the user terminals: K CZF = ∑ k =1 log 1 + ρ N [(H H H)−1 ]kk (10) where ρ is the average SNR at the receivers and [A]ij takes the (i, j)-entry of the matrix A. In Figure 20, we compare the performances of the equispaced array, with inter-element distances (λ/2, λ, 2λ), with the performances of the non-equispaced array, calculated with a α1 = −0.03 and average inter-element distance of (λ, 2λ); the SNR at the receiver is 5 dB. It is worth noting that Electronics 2017, 6, 57 15 of 20 the maximum spectral efficiency is increased from C = 146.1 bit/s/Hz/cell for K = 45 of the λ/2 equispaced array to C = 162.6 bit/s/Hz/cell for K = 55 for the 2λ non equispaced array. We can obtain an average spectral efficiency gain of about 11% just positioning the radiating element of the BS in an optimal fashion. In Figure 21, we present the same results, but comparing only the λ/2 equispaced array and the 2λ non-equispaced array. In this plot, we also represent, by means of two colored areas, the spreading of the spectral efficiency: the areas represent a range of a standard deviation with respect to the mean. It is interesting to compare the “top” border of these two areas, which gives us an indication of the performances achievable by a smart user scheduling scheme: the advantage of the non-equispaced architecture is again clear. As a final remark, in our numerical tests, we have also considered the Dirty Paper Coding (DPC) capacity [30], which is calculated supposing the use of the optimum receiver. In this case, the results are stable with respect to the inter-element distance and marginally depend on the use of non-equispaced architectures. The reason for this fact is that only the greatest singular values of the channel matrix affect the DPC, and from the plots in Figures 8, 9 and 15, we can see that the distribution of the elements of the BS array influences mostly the distribution of the smallest singular values. Figure 20. Average maximum achievable Zero Forcing (ZF) rate as a function of the number of user terminals K. Electronics 2017, 6, 57 16 of 20 Figure 21. Average maximum achievable ZF rate as a function of the number of user terminals K; the areas represent the range of the standard deviation with respect to the mean. Actually, communication schemes that achieve the DPC capacity are very hard to implement in the massive MIMO framework, where linear processing (like ZF) is preferred for computing reasons, and for this reason, we decided to show only the ZF performances. 8. Some Considerations and Antenna Array Design Guidelines In this section, we would like to sum up some design consideration that can be inferred by the shown results. All of the simulations have been performed focusing on arrays with N = 200 radiating elements, but we have also performed other simulations, with a different number of radiators, confirming the advantages of sparse arrays for massive MIMO applications. We have indeed chosen a very simple antenna model for the simulations (an array of isotropic elements without mutual coupling), but in the examples reported in the paper, the inter-element spacing is always larger than 0.5 wavelengths, and in some cases, it is much larger. For such inter-element distances, it is well known that mutual coupling usually plays a minor role. Anyway, we have performed some numerical tests including the mutual coupling (not reported here for lack of space), and we found that there was no significant modification of the distribution of the singular values. Similarly, the element pattern and the polarization will provide a different amplification of the signal received by the terminals, but the overall impact on the singular values distribution would be small. For these reasons, we preferred the use of the architecture of isotropic elements without mutual coupling, providing more easily repeatable results by interested readers. We would also like to underline that the proposed increase in size of the array can have an effect on the overall cost of the antenna system, but this increment would not be significant when dealing with mm-wave frequencies. It is indeed true that the optimization of the BS antenna that we propose has to be realized only when setting up the system; no optimizations/antenna selections are required during operation. On the contrary, the use of non-equispaced architectures, instead of equispaced ones, should not have, in general, a direct impact on the overall cost or complexity of the communication system. The only relevant effect that should be taken into account in an architecture trade-off is the difficulty of employing modular BS arrays (i.e., antenna arrays divided in equal modules of a fixed number of elements): in this case, it could be advantageous to design non-equispaced modules for the BS array and placing them in a non-regular fashion, but the discussion of this possibility goes beyond the aims of this paper. Electronics 2017, 6, 57 17 of 20 A final consideration is now needed. In the numerical tests shown, we have considered the LoS component only; even if this is a reasonable assumption, we could argue if the shown results hold also when a non-LoS propagation component is present. Some preliminary simulations show that the advantage of using larger, non-equispaced arrays is still present, but smaller. Anyway, the analysis of this case deserves particular attention and will be the main topic of a forthcoming work. Summing up, we would like to list some rules-of-thumb for designing massive MIMO arrays that work in LoS propagation. 1. 2. 3. The antenna array geometry needs to be matched to the expected user terminal displacement. For sectoral coverage, a linear array is preferable, while for circular coverage, a circular antenna works better, since a linear array is not capable of discriminating between two terminals in specular positions with respect to the array. Given the number of radiating elements, and hence transceivers, that can be used in the MIMO system, the BS antenna array should be as large as possible. Once the maximum dimension is used, in a linear array, the radiating elements should be possibly arranged in a non-equispaced fashion, by using for instance the positions provided by Equation (7). 9. Conclusions In this paper, we have discussed the impact of the array on the performances of a massive MIMO system. In particular, we have analyzed the LoS case, showing that increasing the size of the array can be very beneficial to the condition number of the MU-MIMO channel matrix: since the BS antenna is working in the near-field region of the array, a larger element distance allows one to exploit the spherical shape of the wavefronts. We have also shown that using a non-equispaced array, we are able to further improve the resulting condition number, since by getting rid of the grating lobes, we are able to achieve a significant lowering of the average condition number also for small increases of the antenna array dimension. Such an improvement in the conditioning of the channel matrix results in an improvement in the spectral efficiency of the massive MIMO systems when using linear processing of the signals. We have also analyzed the results achievable by means of circular arrays and a full circle coverage, finding results perfectly aligned with the ones of the non-equispaced linear array. It is worth underlining that all of the performed analyses have been done considering a fixed BS complexity. It is trivial to understand that with a proper sparse array design, we could reduce the number of antennas at the BS, achieving the same average conditioning of the channel matrix for a desired number of user terminals. It is fundamental to underline that the improvement that can be achieved by optimizing the antenna array geometry has a minimum impact on the overall system cost and does not increase the computational effort of the communication schemes in any way. As a development of the present work, we are working on the design of more complex planar and conformal antenna array geometries; we are also working towards the use of a ray-tracing simulator, in order to check the achieved results when the line of sight is the dominant, but not the only, propagation term present. 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On Reachable Sets of Hidden CPS Sensor Attacks arXiv:1710.06967v1 [cs.SY] 19 Oct 2017 Carlos Murguia and Justin Ruths Abstract— For given system dynamics, observer structure, and observer-based fault/attack detection procedure, we provide mathematical tools – in terms of Linear Matrix Inequalities (LMIs) – for computing outer ellipsoidal bounds on the set of estimation errors that attacks can induce while maintaining the alarm rate of the detector equal to its attack-free false alarm rate. We refer to these sets to as hidden reachable sets. The obtained ellipsoidal bounds on hidden reachable sets quantify the attacker’s potential impact when it is constrained to stay hidden from the detector. We provide tools for minimizing the volume of these ellipsoidal bounds (minimizing thus the reachable sets) by redesigning the observer gains. Simulation results are presented to illustrate the performance of our tools. I. I NTRODUCTION There has recently been significant interest and work in the broad area of security of cyber-physical systems (CPS), see for example [1]-[8]. This topic investigates the properties of conventional control systems in the presence of adversarial disturbances. Control theory has shown great ability to robustly deal with disturbances and uncertainties [9]. However, adversarial attacks raise all-new issues due to the aggressive and strategic nature of the disturbances that attackers might inject into the system. This paper focuses on attack detection and attack capabilities in CPSs. A majority of the work on attack detection leverages the established literature of fault detection [1],[2],[10],[11]. A fault detection approach uses an estimator to forecast the evolution of the system dynamics. When the residual (the difference between what is measured and the estimation), or some function of the residual, is larger than a predetermined threshold, an alarm is raised. Arguably the most insidious attacks are those that occur without our knowledge. Fault detectors impose limits on the attacker, if the attacker aims to avoid being identified. Beyond retooling these existing methods for the new attack detection context, a fundamental question is: given a chosen fault detection approach, how does this method constrain the influence of an attacker? More specifically, what is an attacker able to accomplish when a system employs certain fault detection procedure? Different methodologies exist for evaluating the impact of attacks. Most of the existing work uses some measure This work was supported by the National Research Foundation (NRF), Prime Minister’s Office, Singapore, under its National Cybersecurity R&D Programme (Award No. NRF2014NCR-NCR001-40) and administered by the National Cybersecurity R&D Directorate. C. Murguia is with the Engineering Systems and Design Pillar, Singapore University of Technology and Design. J. Ruths is with the Departments of Mechanical and Systems Engineering, University of Texas at Dallas. emails: murguia_rendon@sutd.edu.sg & jruths@utdallas.edu. of state (or state estimate) deviation. In [2], the authors identify that if the attacker can take advantage of the zero dynamics of a (noise-free) input-output system, it can modify the system dynamics without reflecting its influence in the residual variables. This type of attacks are stealthy to any fault detector. A number of groups have studied the system response when the attacks are constrained by the detector. An important distinction between the collection of existing work – and the work discussed here – is the definition of how the attacker is constrained. We suggest the following terminology. While the term stealthy attack is used very broadly, we suggest that this refer to the zero-dynamics case, as discussed in [2], because these attacks do not propagate to the residual. Some work has investigated the case of system response due to what we here call zero-alarm attacks, i.e., attacks such that the detector threshold is never crossed [12]-[16]. Because real systems (with noise) always have a nonzero rate of false alarms raised by the detector, this attack model yields a relatively obvious attack signature because the alarms stop as soon as the attack starts. Other papers identify attacks that mimic the false alarm rate, thus making the alarm rate during the attack very close to the false alarm rate before the attack started [17],[18]. These attacks we call hidden attacks because although they do change the distribution of the residual, these changes are hidden from the way the detector evaluates the distribution. A majority of this work uses state bounds or steady-state limits to quantify the impact that an attacker can have. The exceptions to this are [17],[18], which quantify the reachable set of states and estimation errors when driven by the attack input. This paper fuses several of these successful lines of research with a more strict interpretation of hidden attacks. The papers [17],[18] consider hidden attacks, however, they permit the alarm rate to change by a small value; the attacker capabilities that are derived are associated with this small deviation rather than the full scope of allowable attacks. Here, we fix the alarm rate exactly to study true hidden attacks (i.e., alarm rate exactly equal to the false alarm rate), and characterize the reachable sets on the estimation error dynamics associated with this broader definition of possible attack vectors. In this work, we characterize the hidden reachable sets that the attacker can induce through manipulation of sensor data. Because in general, it is quite difficult to compute these sets exactly, for given system dynamics and attack detection scheme, we derive ellipsoidal bounds on the hidden reachable sets using Linear Matrix Inequalities (LMIs) [19]. Then, we provide synthesis tools for minimizing these bounds (minimizing thus the hidden reachable set) by properly redesigning the detectors. This builds off of our previous work in [18]. The strict interpretation of hidden attacks requires more direct handling of the effect of noise. To derive finite ellipsoidal bounds, we introduce the notion of p-probable reachable sets, which provides a nested set of ellipsoidal bounds based on the probability of the driving random sequences taking certain values. Because the derivation of the reachable set of states from the reachable set of estimation errors is captured in [18] (for a class of observer-based output feedback controllers), and similar techniques can be used in this paper, we report here only on estimation error reachable sets. Note that the problem formulation in this paper, while seemingly similar, requires an entirely different characterization from [18]. II. S YSTEM D ESCRIPTION & ATTACK D ETECTION We study LTI stochastic systems of the form: ( x(tk+1 ) = F x(tk ) + Gu(tk ) + v(tk ), y(tk ) = Cx(tk ) + η(tk ), (1) with sampling time-instants tk , k ∈ N, state x ∈ Rn , measured output y ∈ Rm , control input u ∈ Rl , matrices F , G, and C of appropriate dimensions, and i.i.d. multivariate zero-mean Gaussian noises v ∈ Rn and η ∈ Rm with covariance matrices R1 ∈ Rn×n , R1 ≥ 0 and R2 ∈ Rm×m , R2 ≥ 0, respectively. The initial state x(t1 ) is assumed to be a Gaussian random vector with covariance matrix R0 ∈ Rn×n , R0 ≥ 0. The processes v(tk ), k ∈ N and η(tk ), k ∈ N and the initial condition x(t1 ) are mutually independent. It is assumed that (F, G) is stabilizable and (F, C) is detectable. At the time-instants tk , k ∈ N, the output of the process y(tk ) is sampled and transmitted over a communication network. The received output ȳ(tk ) is used to compute control actions u(tk ) which are sent back to the process, see Fig. 1. The complete control-loop is assumed to be performed instantaneously, i.e., the sampling, transmission, and arrival time-instants are supposed to be equal. In this paper, we focus on attacks on sensor measurements. That is, in between transmission and reception of sensor data, an attacker may replace the signals coming from the sensors to the controller, see Fig. 1. After each transmission and reception, the attacked output ȳ takes the form: ȳ(tk ) := y(tk ) + δ(tk ) = Cx(tk ) + η(tk ) + δ(tk ), (2) where δ(tk ) ∈ Rm denotes additive sensor attacks. Denote xk := x(tk ), uk := u(tk ), vk := v(tk ), ȳk := ȳ(tk ), ηk := η(tk ), and δk := δ(tk ). Using this new notation, the attacked system is written as follows xk+1 = F xk + Guk + vk , (3) ȳk = Cxk + ηk + δk . A. Observer In order to estimate the state of the process, we use the following Luenberger observer [20] (4) x̂k+1 = F x̂k + Guk + L ȳk − C x̂k , with estimated state x̂k ∈ Rn , x̂1 = E[x(t1 )], where E[ · ] denotes expectation, and observer gain matrix L ∈ Rn×m . Define the estimation error ek := xk − x̂k . Given the system Fig. 1. Cyber-physical system under attacks on the sensor measurements. dynamics (3) and the observer (4), the estimation error is governed by the following difference equation ek+1 = F − LC ek − Lηk − Lδk + vk . (5) The pair (F, C) is detectable; hence, the observer gain L can be selected such that (F − LC) is Schur. Moreover, under detectability of (F, C), if there are no attacks (i.e., δk = 0), where 0 denotes the zero matrix of appropriate dimensions, the covariance matrix Pk := E[ek eTk ] converges to steady state in the sense that limk→∞ Pk = P exists, see [21]. For a given L and δk = 0, it can be verified that the asymptotic covariance matrix P = limk→∞ Pk is given by the solution P of the following Lyapunov equation: (F − LC)P (F − LC)T − P + R1 + LR2 LT = 0. (6) It is assumed that the system has reached steady state before an attack occurs. B. Residuals and Hypothesis Testing In this manuscript, we characterize the effect that output injection attacks can induce in the system with being detected by fault detection techniques. The main idea behind fault detection theory is the use of an estimator to forecast the evolution of the system. If the difference between what it is measured and the estimation is larger than expected, there may be a fault in or attack on the system. Although the notion of residuals and model-based detectors is now routine in the fault detection literature, the primary focus has been on detecting and isolating failures that have known signatures in the degradation of measurement quality, i.e., faults with specific structures. Now, in the context of an intelligent adversarial attacker for which there is no known attack signature, new challenges arise to understand the effect that an adaptive intruder can have on the system without being detected. In this paper, we use the linear observer introduced in the previous section as our estimator. Define the residual sequence rk , k ∈ N, as rk := ȳk − C x̂k = Cek + ηk + δk , (7) which evolves according to the difference equation: ek+1 = F − LC ek − Lηk − Lδk + vk , rk = Cek + ηk + δk . If there are no attacks, the steady state mean of rk is E[rk+1 ] = CE[ek+1 ] + E[ηk+1 ] = 0m×1 , and its asymptotic covariance matrix is given by T Σ := E[rk+1 rk+1 ] = CP C T + R2 . (8) (9) (10) It is assumed that Σ ∈ Rm×m is positive definite. For this residual, we identify two hypotheses to be tested: H0 the normal mode (no attacks) and H1 the faulty mode (with faults/attacks). Then, we have E[rk ] = 0m×1 , E[rk ] 6= 0m×1 , or H0 : H1 : T E[rk rk ] = Σ, E[rk rkT ] 6= Σ, where 0m×1 denotes an m-dimensional vector composed of zeros only. In this manuscript, we use the chi-squared procedure for examining the residual and subsequently detecting attacks. C. Distance Measure and Chi-squared Procedure The input to any detection procedure is a distance measure zk ∈ R, k ∈ N, i.e., a measure of how deviated the estimator is from the sensor measurements. We employ distance measures any time we test to distinguish between H0 and H1 . The chi-squared test uses a quadratic form on the residual as distance measure to test for substantial variations in mean and variance of the error between the measured output and the estimate. Consider the residual sequence rk , (8), and its covariance matrix Σ, (10). The chisquared procedure is defined as follows. Chi-squared procedure: If zk := rkT Σ−1 rk > α, k̃ = k. (11) Design parameter: threshold α ∈ R>0 . Output: alarm time(s) k̃. Thus, the procedure is designed so that alarms are triggered if zk exceeds the threshold α. The normalization by Σ−1 makes setting the value of the threshold α system independent. This quadratic expression leads to a sum of the squares of m normally distributed random variables which implies that the distance measure zk follows a chi-squared distribution with m degrees of freedom, see, e.g., [22] for details. D. False Alarms The occurrence of an alarm in the chi-squared procedure when there are no attacks to the CPS is referred to as a false alarm. The threshold α must be selected to fulfill a desired false alarm rate A∗ . Let A ∈ [0, 1] denote the false alarm rate of the chi-squared procedure defined as the expected proportion of observations which are false alarms, i.e., A := pr[zk ≥ α], where pr[·] denotes probability, see [23] and [24]. Proposition 1 [13]. Assume that there are no attacks on the system and consider the chi-squared procedure (11) with residual rk ∼ N (0, Σ) and threshold α ∈ R>0 . Let −1 ∗ (·, ·) denotes the α = α∗ := 2P−1 ( m 2 , 1 − A ), where P inverse regularized lower incomplete gamma function (see [22]), then A = A∗ . III. H IDDEN R EACHABLE S ETS In this section, we provide tools for quantifying (for given L) and minimizing (by selecting L) the impact of the attack sequence δk on the estimation error ek when the chisquared procedure is used for attack detection. To quantify the effect of attacks, we need to introduce some measure of impact. However, because malicious adversaries may launch any arbitrary attack, we need a measure which can capture all possible trajectories that the attacker can induce in the estimation error dynamics, given how it accesses the dynamics (i.e., through residual variables by tampering with sensor measurements). We propose to use the reachable set of the attack sequence δk as our measure of impact. We are interested in attacks that do not change the false alarm rate of the detector A, i.e., Ā = A, where Ā denotes the alarm rate under the attacker’s action. This class of attacks is what we refer to as hidden attacks and the trajectories that hidden attacks can induce in the system are referred to as hidden reachable sets. In this section, we provide tools based on Linear Matrix Inequalities (LMIs) for computing outer ellipsoidal bounds on the hidden reachable sets induced by the attack sequence δk given the system dynamics, the chi-squared procedure, the noise, and the false alarm rate A. A. Attack Model and Hidden Reachable Sets We assume that the attacker has perfect knowledge of the system dynamics, the observer, measurements, and detection procedure (chi-squared). It is further assumed that all the sensors can be compromised by the attacker at each time step (the case where not all the sensors are attacked is left as future work). By considering this strong, worst-case attacker, we are able to construct an upper bound on the abilities of the attacker. Consider the estimation error dynamics (8), the residual sequence rk = Cek + ηk + δk , and the distance measure 1 1 (12) zk = \|\|Σ− 2 rk \|\|2 = \|\|Σ− 2 (Cek + ηk + δk )\|\|2 , − 21 denotes the symmetric squared root matrix of where Σ Σ−1 . The set of feasible attack sequences that the opponent can launch while satisfying Ā = A can be written as the following constrained control problem on δk : ( δk ∈ R m ek satisfies (8), and 1 pr[\|\|Σ− 2 (Cek + ηk + δk )\|\|2 > α ] = A, ) , (13) for k ∈ N. We are interested in the error trajectories that the attacker can induce in the system restricted to satisfy (13). Note that, as long as Ā = A, the attacker may induce any arbitrary random sequence δk . This and the fact that vk and ηk are Gaussian (thus having infinite support) imply that deterministic reachable sets induced by δk and the noise sequences are generally unbounded. To overcome this obstacle, we introduce the notion of p-probable hidden 1 reachable sets Rpα . Define ζk := Σ− 2 (Cek + ηk + δk ) and note that the estimation error dynamics (8) can be written in terms of ζk as: 1 ek+1 = F ek − LΣ 2 ζk + vk , (14) 1 ζk = Σ− 2 (Cek + ηk + δk ). For given false alarm rate A and probability p ∈ (0, 1), the p-probable hidden reachable set of the attack sequence δk in (14), Rpα , is defined as the set of ek ∈ Rn , k ∈ N that can be reached from the origin e1 = 0 due to the the attacker’s action δk restricted to satisfy Ā = A and 2 pr[\|\|ζk \|\|2 ≤ ζ̄p ] = pr[kvk k ≤ v̄p ] = p, (15) for some constants ζ̄p , v̄p ∈ R>0 , i.e., p n e1 = 0, . (16) Rα := ek ∈ R ek , δk , vk satisfy (13)-(15), By restricting the probabilities in (15), we are delimiting the support of the attack and noise sequences to compact sets. Then, the p-probable hidden reachable sets correspond to the trajectories of the system when the driving random sequences are restricted to satisfy Ā = A and (15). For delimited vk and δk , we can characterize reachable sets using deterministic tools. In general, it is analytically intractable to compute Rpα exactly. Instead, using LMIs, for some positive definite matrix Pαp ∈ Rn×n , we derive outer ellipsoidal bounds of the form Eαp := {ek ∈ Rn \|eTk Pαp ek ≤ 1} containing Rpα . Remark 1 Note, from (14), that if for some k = k ∗ , ek∗ 6= 0 and ρ[F ] > 1, where ρ[·] denotes spectral radius, then \|\|ek \|\| diverges to infinity as k → ∞ for any non-stabilizing ζk . That is, Rpα is unbounded if the system is open-loop unstable. If ρ[F ] ≤ 1, then \|\|ek \|\| may or may not diverge to infinity depending on algebraic and geometric multiplicities of the eigenvalues with unit modulus of F (a known fact from stability of LTI systems), see [21] for details. Given Remark 1, in what follows, we consider open-loop stable systems (ρ[F ] < 1). The following result is used to compute the ellipsoidal bounds Eαp . Lemma 1 [25] Let ξk ∈ Rn , ξ1 = 0, Vk := ξkT Pξk , for some positive definite matrix P ∈ Rn×n , and ωkT ωk ≤ ω̄, ω̄ ∈ R>0 . If there exists a constant b ∈ (0, 1) such that 1−b T Vk+1 − bVk − ωk ωk ≤ 0, ∀ k ∈ N, (17) ω̄ T then, Vk = ξk Pξk ≤ 1. B. Case 1: p ∈ [0, 1 − A] Because the attack sequence is restricted to satisfy (13), we start computing the ellipsoidal bounds corresponding to p = 1 − A, i.e., Eα1−A . It is easy to verify using Lemma 1 that Eαp ⊆ Eα1−A for p ∈ [0, 1 − A] because ζ̄p ≤ ζ̄1−A = α and v̄p ≤ v̄1−A in (15); i.e., all p-probable ellipsoidal bounds for p ∈ [0, 1 − A] lie within the 1 − A-probable ellipsoidal bound. It follows that Rpe ⊆ Eα1−A for p ∈ [0, 1−A], i.e., for p in this interval, we only need to compute the ellipsoidal bound corresponding to p = 1−A. Characterizing p-probable sets for small p values is of little interest because they do not provide a informative bound on system trajectories (since the smaller p is, the more trajectories lie outside the p-probable ellipsoidal bound). We work with the data available in this setting, namely the number of alarms raised by the detector, to bound the most informative p = 1 − A probable reachable set; in Case 2, we extend these results for larger p values. Theorem 1 For given system matrix F , observer gain L, residual covariance matrix Σ, and false alarm rate A, consider the set R1−A in (16). If there exists a positive α definite matrix P ∈ Rn×n and b ∈ (0, 1) satisfying the following matrix inequality:   bP FTP 0 0 00 1 PF P P −PLΣ 2 0 0   1−b  0 0 0 0 P  ≥ 0; ω̄ I  (18) 1 1−b  0 −Σ 2 LT P 0 0 0   ω̄ I  0 0 0 0 I 0 0 0 0 0 0I 1−A 1−A for ω̄ = α + v̄1−A ; then, Rα ⊆ Eα with Pα1−A = P, i.e., the (1 − A)-probable hidden reachable set is contained in the ellipsoid Eα1−A = {ek ∈ Rn \|eTk Pα1−A ek ≤ 1}. Proof : For a positive definite matrix P ∈ Rn×n , consider the function Vk := eTk Pek , then, from (16), inequality (17) takes the form:   1 bP − F T PF F T PLΣ 2 −F T P 1 1 1 1 T 2 2 Σ 2 LT P  ϑk = −ϑTk  Σ 2 LT PF 1−b ω̄ I − Σ L1 PLΣ 1−b 2 −PF PLΣ ω̄ I − P T =: −ϑk Qe ϑk ≤ 0, where ϑ := (eTk , ζkT , vkT )T . The above inequality is satisfied if and only if Qe ≥ 0. Matrix Qe can be written as the Schur complement of a higher dimensional matrix Q′e ; hence, Qe ≥ 0 ↔ Q′e ≥ 0, i.e., Qe ≥ 0 ↔   bP 0 0 F T P 00 1  0 1−b I 0 −Σ 2 LT P 00 ω̄   1−b  0 (19) I P 00 0 ′  ≥ 0. ω̄  Qe :=  1  2 P 00 PF −PLΣ P  0 0 0 0 I 0 0 0 0 0 0I Finally, inequality (18) follows from (19) by a simple reordering of rows and columns.The result follows now from Lemma 1 by taking Pα1−A = P and ω̄ = α + v̄1−A . The result in Theorem 1 provides a tool for computing ellipsoidal bounds on R1−A . To make the bounds most α useful, we next construct ellipsoids with minimal volume, i.e., the tightest possible ellipsoid bounding R1−A . In this α case, we have to minimize det P −1 subject to (18) (because det P −1 is proportional to the volume of eTk Pek = 1). This is formally stated in the following corollary of Theorem 1, see [19] for further details. Corollary 1 For given matrices (F, L, Σ), false alarm rate A, and b ∈ (0, 1), the solution P of the following convex optimization: ( minP − log det P, (20) s.t. P > 0 and (18), for ω̄ = α + v̄1−A , minimizes the volume of the ellipsoid Eα1−A (with Pα1−A = P) bounding R1−A . α See [26] for an example of how to solve (26) using YALMIP. As we now move toward redesigning L to minimize the ellipsoids, we note that as \|\|L\|\| → 0, the volume of Eα1−A goes to zero because the attack-dependent term in 1 (14), LΣ 2 ζk , vanishes. In other words, without any other considered criteria, the observer gain leading to the minimum volume ellipsoid is trivially given by L = 0. While this is effective at eliminating the impact of the attacker, it implies that we discard the observer altogether and, therefore, forfeit any ability to build a reliable estimate of the system state. If we impose a performance criteria that the observer must satisfy in the attack-free case (e.g., convergence speed, noiseoutput gain, and minimum asymptotic variance), it has to be added into the minimization problem (26) so as to minimize the volume of Eα1−A while still achieving the observer performance in the attack-free case. For completeness, in the following proposition, we provide an LMI criteria for ensuring that the H∞ gain from the noise to the residual rk in (8) is less than or equal to some γ ∈ R>0 . Then, using this criteria and Theorem 1, we provide a synthesis tool for minimizing the volume of Eα1−A while ensuring a desired H∞ performance in the attack-free case. Proposition 2 For given matrices (F, C, L), if there exist a positive definite matrix P ∈ Rn×n and constant γ ∈ R>0 satisfying the following matrix inequality:   P 0 0 (F − LC)T P C T  0 γ2I 0 −LT P I    2  0 0 γ I P 0   ≥ 0, (21)  P(F − LC) −PL P P 0  C I 0 0 I then, the H∞ gain from the noise νk := (ηkT , vkT )T to the residual rk = Cek + ηk of the estimation error dynamics (8) is less than or equal to γ. P) bounding R1−A and guarantees that the H∞ gain from α the noise νk = (ηkT , vkT )T to the residual rk of (8) is less than or equal to γ in the attack-free case. Proof : This follows from Theorem 1, Proposition 2, and the linearizing change of variables M = PL. C. Case 2: p ∈ (1 − A, 1] Note that, for p ∈ (1 − A, 1], ζ̄p > ζ̄1−A = α according 2 to (15). Then, we can write ζ̄p = α + ǫp and pr[kζk k ≤ α + ǫp ] = 1 − A + ap, for some ǫp ∈ (0, ∞) and ap ∈ (0, A]. To be able to compute ellipsoidal bounds, the constant ǫp corresponding to a given probability 1−A+ap is required. If ǫp is available, we can restrict ζk to compact sets as in Case 1. Note, however, that the distribution of the attack sequence δk (and thus the one of ζk ) is generally unknown. Actually, the attacker may induce any arbitrary (and possibly) nonstationary random sequence ζk in (14) as long as Ā = A. Nevertheless, we can obtain bounds on ǫp using Markov’s inequality [22] to link the statistical properties of ζk with ǫp . This is stated in the following proposition. Proposition 3 Denote Mk := E[ζk ζkT ] and µk := E[ζk ]. For given false alarm rate A, probability p = 1 − A + ap , and ap ∈ (0, A), the following is satisfied:  2  (23a)  pr[kζk k ≤ α + ǫp ] ∈ [1 − A + ap , 1], tr[Mk ] + µTk µk   − α. for all ǫp ≥ ǫp := A − ap (23b) 2 The proof of Proposition 2 is omitted here due to the page limit. However, this is a standard result and details about the proof can be found in, e.g., [9] and references therein. In the following corollary of Theorem 1 and Proposition 2, we formulate the optimization problem for designing the observer gain L such that the volume of the ellipsoid Eα1−A is minimized and a desired H∞ performance is achieved in the attack-free case. Corollary 2 For given system matrices (F, C), residual covariance matrix Σ, false alarm rate A, b ∈ (0, 1), and γ ∈ R>0 , if there exist matrices P ∈ Rn×n and M ∈ Rn×m solution to the following convex optimization:  minP,M − log det P,         bP F T P 0 0 00   1  PF  P P −M Σ 2 0 0     1−b   0 P I 0 0 0  ω̄  s.t. P > 0,  ≥ 0, and  1  T 1−b   0 −Σ 2 M 0 I 0 0  ω̄   0 0 0 0 I0  0 0 0 0 0 I        P 0 0 F T P − CT M T CT   2 T   0 γ I 0 −M I     0 0 γ2I P 0    ≥ 0,     PF − M C −M P P 0  C I 0 0 I (22) for ω̄ = α + v̄1−A ; then, the observer gain L = P −1 M minimizes the volume of the ellipsoid Eα1−A (with Pα1−A = Proof : The probability pr[kζk k ≤ α + ǫp ] can be written as 2 2 pr[kζk k ≤ α + ǫp ] = 1 − pr[kζk k > α + ǫp ]. Then, using Markov’s inequality [22], we can write the following 2 E[kζk k ] pr[kζk k2 > α + ǫp ] = 1 − pr[kζk k2 ≤ α + ǫp ] ≤ . α + ǫp Therefore, if ǫp satisfies E[kζk k2 ]/(α + ǫp ) ≤ A − ap , then 2 2 pr[kζk k > α+ǫp ] ≤ A−ap and hence pr[kζk k ≤ α+ǫp ] ∈ [1 − A + ap , 1]. The expectation of the quadratic form ζkT ζk 2 is given by E[kζk k ] = E[ζkT ζk ] = tr[Mk ] + µTk µk [22]; 2 then, E[kζk k ]/(α + ǫp ) ≤ A − ap is satisfied for all ǫp ≥ ǫp with ǫp as defined in (23b), and the assertion follows. Using Proposition 3, for given false alarm rate A and probability p = 1 − A + ap ∈ (1 − A, 1], ap ∈ (0, A), we can characterize p-probable hidden reachable sets, R̃pα , 2 by using the lower bounds on pr[kζk k ≤ α + ǫp ] and ǫp in (23). Specifically, for p > 1 − A, the set R̃pα of the sequence δk is defined as the set of ek ∈ Rn that can be reached from e1 = 0 restricted to satisfy Ā = A and pr[kvk k2 ≤ v̄p ] = 1 − A + ap and (24) ǫp = ǫp → pr[kζk k2 ≤ α + ǫp ] ∈ [1 − A + ap , 1], for some constant v̄p ∈ R>0 and ǫp as defined in (24), i.e., R̃pα := e k ∈ Rn e1 = 0, ek , ζk , vk , satisfy (13)-(14),(24), . (25) Remark 2 For a p-probable reachable set, we select ap such that p = 1 − A + ap , then determine ǫp using (23b). Note that, because the attacker can induce an attack sequence with Fig. 2. 1−A Ellipsoid Eα for different values of false alarm rate A. Fig. 3. The improvement in the (1 − A)-probable hidden reachable set 1−A ellipsoidal bound Eα , for A = 0.01, through application of Corollary 2 to design the optimal observer gain. arbitrarily large covariance Mk and mean µk , the lower bound on ǫp , ǫp , in (23b) can be made arbitrarily large for any ap . Therefore, if δk (and thus ζk ) is only restricted to satisfy Ā = A, the opponent can induce arbitrarily large reachable sets R̃pα . Remark 2 implies that if we only monitor the alarms raised by the detector, the attacker can inject arbitrarily large signals in the residual sequence rk without changing the alarm rate. Consequently, the sets R̃pα can be made arbitrarily large for arbitrarily small ap . If we place additional assumptions on the attacker, namely that the mean and covariance of the attack sequence ζk are finite, the reachable sets will be bounded by Proposition 3. In particular, if we assume the attacker maintains the mean and covariance of the attack-free scenario, i.e., E[ζk ] = µk = 0 and E[ζk ζkT ] = Mk = Im , m − α. Hence, if in addition to imposing Ā = then ǫp = A−a p A, the attack is restricted to keep the statistical properties of ζk in the attack-free case, i.e., µk = 0 and Mk = Im , the reachable sets R̃pα are bounded for each ap ∈ (0, A) (because ǫp is bounded); and therefore, in this case, we can compute ellipsoidal bounds on R̃pα . This additional assumption could be enforced by adding detectors that identify anomalies in the sample mean and sample covariance of the residual. Such detectors would force the attacker to avoid arbitrarily large attack values in order to avoid detection by these additional mean and covariance detectors. As before, we characterize, for some positive definite matrix P̃αp ∈ Rn×n , outer ellipsoidal bounds of the form Ẽαp := {ek ∈ Rn \|eTk P̃αp ek ≤ 1} containing R̃pα . The results corresponding to Theorem 1, and Corollary 1 for Case 1 are stated in the following corollary. Corollary 3 For given false alarm rate A, probability p = m − α, 1 − A + ap , ap ∈ (0, A), threshold ǫp = ǫp = A−a p p and matrices (F, L, Σ), consider the set R̃α in (25). Then, for given b ∈ (0, 1), if there exists a matrix P ∈ Rn×n solution of the following convex optimization: ( minP − log det P, (26) s.t. P > 0 and (18), for ω̄ = α + ǫp + v̄p ; then, R̃pα ⊆ Ẽαp (with P̃αp = P) and Ẽαp has minimum volume, i.e., the p-probable hidden reachable set R̃pα is contained in the minimum volume ellipsoid Eαp = {ek ∈ Rn \|eTk Pαp ek ≤ 1}. p Fig. 4. Ellipsoidal bound Ẽα for different values of ap obtained using Corollary 3. A result for redesigning the observer gain for minimizing the volume of the above ellipsoids, as in Corollary 2 for Case 1, can be stated in a similar manner as the corollary above; however, this is omitted here due to the page limit. IV. S IMULATION E XPERIMENTS Consider the closed-loop system (3)-(4) with matrices:  0.84 0.23 0.07  F = , G = ,C= 10 ,   −0.47 0.12 0.23      1.16 0.45 −0.11 L= , R1 = , (27) −0.69 −0.11 0.45        R = 1 0 , R = 1, Σ = 3.26.  0 2 01 We start with Case 1. Using Proposition 2, the observer gain L is designed such that the H∞ gain from the noise to the residual rk of (8) is less than or equal to γ = 1.86 in the attack-free case. Consider the false alarm rates A = {0.01, 0.05, 0.10, 0.20} and the corresponding α = {6.63, 3.84, 2.70, 1.64}, obtained using Proposition 1. The 2 thresholds v̄1−A in (15) are computed such that pr[kvk k ≤ v̄1−A ] = 1−A. Because the entries on the diagonal of R1 are 2 equal and vk ∼ N (0, R1 ), the random sequence kvk k , k ∈ N follows a gamma distribution, Γ(κ, θ), with shape parameter κ = 1 and scale parameter θ = 0.90, see [22]. It follows that, for these A, v̄1−A = {4.14, 2.69, 2.07, 1.44}. For these values of v̄1−A and α, in Figure 2, we depict the ellipsoidal bounds Eα1−A on the (1 − A)-probable hidden reachable sets obtained using Theorem 1 and Corollary 1. Next, for R1−A α A = 0.01, using Corollary 2, we redesign the observer gain L to minimize the volume of Eα1−A while maintaining the H∞ performance below γ = 1.86. The obtained optimal ellipsoidal bound, Eα1−A , is depicted in Figure 3 for the optimal observer gain L = (0.1272, −0.0160)T . For Case 2, let A = 0.05, p = 1 − A + ap , ap = {0.01, 0.03}, and L as in (27); then, the corresponding v̄p are v̄p = {2.8970, 3.5208} and the ǫp , computed through (23), are given by ǫp = {21.16, 46.16}. In Figure 4, we show the ellipsoidal bounds Ẽαp on the reachable sets R̃pα obtained using Corollary 3. Remark 3 Many numerical results considering hidden attacks with different distributions are presented in the accompanying paper [27] (Section 4). Also, extensive Monte-Carlo simulations showing the tightness of the bounds presented here are given in [27]. V. C ONCLUSION In this paper, for a class of discrete-time LTI systems subject to sensor/actuator noise, we have provided tools for quantifying and minimizing the negative impact of sensor attacks on the estimation error dynamics performance given how the opponent accesses the dynamics (i.e., through the controller by tampering with sensor measurements). We have proposed to use the reachable set as a measure of the impact of an attack given a chosen detection method. For given system dynamics and attack detection scheme, we have derived ellipsoidal bounds on these reachable sets using LMIs. Then, we have provided synthesis tools for minimizing these bounds (minimizing thus the reachable sets) by properly redesigning the detectors. R EFERENCES [1] A. Cárdenas, S. Amin, Z. Lin, Y. Huang, C. Huang, and S. 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arXiv:1710.11354v1 [cs.CV] 31 Oct 2017 Spatio-temporal interaction model for crowd video analysis Neha Bhargava Indian Institute of Technology Bombay India Subhasis Chaudhuri Indian Institute of Technology Bombay India neha@ee.iitb.ac.in sc@ee.iitb.ac.in Abstract scopic modeling methods. They also present a critical survey on crowd event detection. Julio et al. cover various vision techniques applicable to crowd analysis such as tracking, density estimation, and computer simulation [3]. Zhan et al. discuss various vision based techniques used in crowd analysis. They also discuss crowd analysis from the perspective of different disciplines − psychology, sociology and computer graphics [4]. At the top level, the techniques used in crowd motion analysis can be divided into two major classes − holistic and particle based. The holistic methods consider crowd as a single entity and analyze the overall behavior. These methods fail to provide much insight at an individual or intermediate level. On the other hand, particle based methods consider crowd as a collection of individuals. But their performance degrades with the increase in crowd density due to occlusion and tracking problems. The analysis at intermediate level i.e. at group level might provide more insights at individual and overall levels. We present an unsupervised approach to analyze crowd at various levels of granularity − individual, group and collective. We also propose a motion model to represent the collective motion of the crowd. The model captures the spatio-temporal interaction pattern of the crowd from the trajectory data captured over a time period. Furthermore, we also propose an effective group detection algorithm that utilizes the eigenvectors of the interaction matrix of the model. We also show that the eigenvalues of the interaction matrix characterize various group activities such as being stationary, walking, splitting and approaching. The algorithm is also extended trivially to recognize individual activity. Finally, we discover the overall crowd behavior by classifying a crowd video in one of the eight categories. Since the crowd behavior is determined by its constituent groups, we demonstrate the usefulness of group level features during classification. Extensive experimentation on various datasets demonstrates a superlative performance of our algorithms over the state-of-the-art methods. We believe that a moderately dense crowd consists of various groups which form a primary entity of a crowd [6, 7] whereas a highly dense crowd can be considered to form a single group and a highly sparse crowd might have groups with cardinality of one. Together, they guide the overall behavior of the crowd and individually influence the actions of the members. Therefore, group level analysis and hence group detection becomes important in crowd analysis. We define a group as a set of individuals (agents) having some sort of interactions e.g. the group members are walking together. Spatial proximity is also necessary to form a group; if there are agents with a similar motion pattern but are far away from each other, they do not form a group as per our definition. Each group has its own set of goals that leads to various interaction patterns among the members of the group. The collective behavior of these constituent groups identifies the global crowd behavior which can vary from a highly structured to a completely unstructured pat- Understanding human behavior at an individual level, at a group level and at a crowd level in different scenarios has always attracted the researchers. The variability and complexity in the behavior make it a highly challenging task. However, this decade is witnessing a huge interest of researchers in the area of crowd motion analysis due to its various applications in surveillance, safety, public place management, hazard prevention, and virtual environments. This interest has resulted in many interesting papers in the area. We are aware of at least four survey papers on the subject of crowd analysis that indicate the amount of attention, it has drawn in this and the previous decade [1],[2],[3],[4]. The latest survey paper [1] by Chang et al. encapsulates the recent works published after 2009, covering topics of motion pattern segmentation, crowd behavior and anomaly detection. Thida et al. [2] provide a review on macroscopic and micro1 (a) Uniform crowd (d) Walking (b) Mixed crowd (c) Stationary group (e) Approaching (f) Splitting Figure 1: (a) and (b) give examples of structured and unstructured crowd. Output of the proposed algorithm: (c) - (f) show groups with different activities: Standing (St), Walking (W), Splitting (Sp) and Approaching (A). Tracklets for some of the agents over past few frames are also shown. Each color represents a detected group (Best viewed in color). The videos are from BEHAVE [5] and CUHK [6] datasets. tern. In case of a structured crowd, for example − marching of soldiers, all groups are in coordination and share the same goal (see Fig.1a); whereas in an unstructured crowd, for example − at railway station or at a shopping complex, there are multiple groups with different goals (see Fig.1b). We are interested in understanding these different types of crowd behaviors at various levels by exploiting motion information of individuals. The paper makes the following contributions: strates the effectiveness of the algorithm. 3. We also demonstrate how the activities can be classified at three different levels − at atomic (individual) level, at group level and at crowd level. The eigenvalues of the interaction matrix characterize various group and individual activities − Fig 1c-1f show examples of activities at group level. At crowd level, we employ group level features to identify the behavior of the crowd. We classify the crowd videos in one of the 8 categories as defined by [6] and demonstrate its performance in terms of classification accuracy. 1. A framework is proposed to model the collective motion of the crowd by a first order dynamical system. The model captures the interaction patterns among the individuals. Although, the proposed model does not capture any possible nonlinear relations, its usefulness for short-term analysis has been verified experimentally. We also provide an optimization formulation for the estimation of the interaction matrix under the constraints of spatial proximity, temporal continuity and sparsity of inter-agent relationship. The remaining part of the paper is organized as follows. Next section reviews the related literature. Section 2 explains the proposed mathematical formulation followed by group detection algorithm in Section 3. Detection of group activity and atomic activity is discussed in Section 4. We look at crowd video classification in Section 5. The experimental results are presented in Section 6 followed by conclusions in Section 7. 2. Since the interaction matrix is learned from the trajectory data, it captures the spatio-temporal patterns present among the agents. We observe that the eigenvectors of the interaction matrix reflect the spatio-temporal patterns. Thus, we propose a spectral clustering [8] based algorithm to identify the groups present in the scene. Extensive experimentation on various datasets demon- 1. Related Work There are numerous research papers in the challenging and interesting area of crowd behavior analysis. There are several holistic approaches (e.g. [9], [10], [11], [12], [13]) as well as particle based algorithms (e.g. [14], [15], [7], [16], [17]) in the literature. Holistic methods analyze crowd as a single en2 2. Mathematical Formulation tity and ignore individuals or groups. In many papers, a dense crowd is considered analogous to fluid and hence concepts from fluid mechanics are applied for analysis. Mehran et al. in [9] present streakline representation of crowd flow for behavior analysis. Solmaz et al. recognize crowd behaviors such as bottlenecks, fountainheads, lanes, arches and blocks through stability analysis of a dynamical system [10]. Benabbas et al. detect motion patterns and events in the crowded scenes by modeling motion and velocity at each spatial location [11]. In [12], Lin et al. find a coherent motion regions in the video by generating thermal energy field. We define a group as a set of agents having spatial proximity and some sort of interaction. In general, such interactions are complex and non-linear in nature. We approximate these interactions locally in time by a first order dynamical model. Note that we refer by agent an individual entity (represented by a point to be tracked) in the crowd. 2.1. Proposed Interaction Model We model the collective relationship among the agents by a first order affine system. Our hypothesis is based on the intuition that each agent takes into consideration (a) the movement of other agents present nearby and (b) her/his desired goal, while taking the next step. To capture these two intuitions, our model relates the next position of each agent to the current positions of all the agents including herself/himself. Let x(k ) = [ x1 (k), x2 (k), ..., x N (k )] T , then The agent based approaches analyze each individual or group to discover the global behavior. Solera et al. propose correlation clustering based group detection which uses socially constrained features. Shao et al. introduce a collective transition prior in [6] and represent each group by a Markov chain. They define interesting group descriptors which proved to be useful in group state analysis and crowd classification. In [15], Sethi and Chowdhury propose a phase space algorithm to identify pairwise correlation between the motion patterns. Ge et al. find groups by hierarchical clustering based on pairwise velocities and distance [18], [7]. Zhou et al. find groups by using coherent filtering [16]. They propose a coherent neighbor invariance property which characterizes coherently moving individuals. Sochman et al. [19] infer groups based on social force model [14]. They define a pairwise group activity confidence to identify groups. Srikrishnan and Chaudhuri in [20] define a linear cyclic pursuit based framework for collective motion modelling with the goal of short-term prediction. But they do not explore group detection and there is no analysis of crowd behavior. In the interesting work of [17], they consider group detection as a clustering problem and learn a socially meaningful pairwise affinity under Structural SVM framework. x ( k + 1) = [ A \| a ] x(k) = A0 x0 (k ) 1 (1) where N is the total number of agents, A ∈ R N × N , A0 ∈ R N ×( N +1) , a ∈ R N ×1 is the bias, x0 (k) ∈ R( N +1)×1 and xi (k) ∈ R is the location of the ith agent at time instant k along the x-axis. We call A as the interaction matrix which captures the evolution of an agent as a function of all agents present in the scene. Note that A has no assumption on its form and entries. It need not be symmetric i.e. agent i may not depend on agent j in the same way as agent j depends on agent. For example, consider a case where agent i is stationary and agent j approaches him/her. Since their behaviors are not symmetric with respect to each other, we assume that it implies aij 6= a ji . In this paper, it is assumed that the motions along x and y directions are independent and hence can be analyzed independently. The corresponding model along y direction is y(k + 1) = By(k) + b. In the rest of the paper, we discuss the solution for matrix A noting this fact that the same process is also carried out for B. In the end, the outputs from both the models are combined appropriately to get the final output. We expect matrices A and B to be dependent on crowd motion. Since crowd behavior might change with time, the interaction matrix is time varying in nature, which we represent as Ak where k is a time instant. Assuming A has N independent eigenvectors, the general solution Most of the particle based algorithms compute pairwise velocity and spatial cues to find the groups hierarchically. They do not model spatio-temporal patterns of the agents collectively which might capture more complex interactions. Additionally, most of the methods assume a constant velocity motion model which is not valid for many scenarios. To address these limitations in the paper, we propose to model motion trajectories collectively instead of individually or pairwise. Also instead of relying on spatio-temporal information directly (which is prone to noise) for group detection, we use spectral clustering to identify groups. 3 to Eq.(1) is given as N x(k) = ∑ {ci λik ei + di i =1 λ i 6 =1 (λik − 1) e }+ λi − 1 i A and a with each incoming frame as interaction patterns may change over the time. In addition, sudden changes in these interactions are unlikely. Therefore it is desired that the entries of A and a do not change drastically in consecutive time instants − we assume them to be varying smoothly over time. We incorporate this constraint by minimizing l2 norm of the difference between the current matrix A0k and the previous estimate at (k − 1)th instant. Furthermore for crowded scenes, it is unlikely that an agent’s motion depends on all the agents present in the neighborhood. We capture this sparse relationship in A0k by minimizing l1 norm of A0k . Adding these constraints to the cost function, the final formulation at kth time instant becomes: N ∑ (ci + kdi )ei , i =1 λ i =1 (2) where λi is the ith eigenvalue, ei is the corresponding normalized eigenvector, ci and di are the corresponding constant coefficients that depend on the initial condition and a respectively. Different values of λi and ei generate various motion patterns for an agent. These patterns can be associated to different motion tracks generated by an agent while walking, approaching, splitting or being stationary. For example, an agent is stationary if λ1 = 1 and d1 = 0 at location c1 e1 or an agent is moving with a constant speed if λ1 = 1 and d1 6= 0. Hence, this more generalized model is appropriate for modeling temporally localized complex motions. Â0k j k N (n + i ) − x j ∑ ∑ \|x actual j pred A0k ∈R N ×( N +1) n 1 k 2 \|\|A0k Xkk− − L − Xk− L+1 \|\|2 (4) where Xi ∈ R N +1× L contains the positions of all N agents from ith to jth frames concatenated together with an appended row of ones to account for the bias, A0k−1 is the estimate at the previous frame and r1 and r2 are appropriate regularization parameters. Note that we will use A0 instead of A0k for notational convenience. One requires at least L ≥ ( N + 1) past positions to solve the Eq. 4. Therefore the interaction pattern is assumed to remain constant over L frames. Hence we want L to be small enough to capture the shortterm linear relationship among the agents. A large N (in crowded scenes) leads to two major problems: (i) longer trajectories (i.e. higher L) are required to learn the interaction matrix A0 as L ≥ N + 1 which may not be available and (ii) the interaction may not remain constant over past L positions for high values of L as discussed before and we would like to keep L ≤ 30 as discussed in the previous section. To address these problems, we identify spatial neighbors of each agent separately and learn only the corresponding entries in the matrix (one row at a time). The neighborhood is defined as follows − the agent p is a neighbor to the agent q if dist(p, q) < Rp . The value of Rp is decided so as to satisfy the constraint L ≤ 30. The intuition for enforcing the neighborhood criteria is that it is unlikely that far away agents influence the motion of an agent. The advantage is that the shorter trajectories are now sufficient as the number of entries of A0 to be learned are lesser. Note that we estimate matrix A0 in a row-wise manner where the ith row has number of entries to be estimated as equal to one more than We use an average k-step prediction error as a measure to test the validity of the proposed model on real videos. Fig. 2a shows average errors for different step size prediction on videos from BEHAVE and CUHK datasets, each curve corresponding to a different video. The k-step prediction error at any time instant n is calculated as follows: 1 kN min o + r1 \|\|A0k − A0k−1 \|\|22 + r2 \|\|A0k \|\|1 , 2.2. Validation of the Model En (k ) = = arg (n + i )\|, (3) i =1 j =1 It may be noted that matrix A is estimated from the latest video frames up to n and then Eq. 1 is used to pred obtain x j . The k-step prediction error for the video is obtained by averaging En (k ) over all the frames of the video. As expected, error increases with k but with a marginal increment. We observe that, for both the databases, prediction is quite valid up to 1-1.5 seconds (about 30 frames). Since the model assumes that the interaction remains same over L frames, Fig 2a suggests that one can select L upto 30 frames without introducing much error. These error plots show that the proposed model is suitable for short-term analysis, which is the underlying theme of the proposed algorithm. 2.3. Estimation of Interaction Model Parameters The matrix A and vector a at any time instant are learned from the immediate past trajectory data of all the agents in a least squares framework. We update 4 (a) On videos from BEHAVE dataset Error (in pixels) 30 20 10 0 5 10 15 20 25 30 35 40 45 50 45 50 (b) On videos from CUHK dataset Error (in pixels) 30 20 10 0 5 10 15 20 25 30 35 40 Number of frames predicted (k) (a) (b) Figure 2: (a) Illustration of suitability of the proposed model: Average k-step prediction error for sample videos from BEHAVE and CUHK datasets, each curve corresponds to a different video. (b) Neighborhood criteria: Spatial neighborhoods around agents p and r are represented as circles around them. There are a total of 20 agents in the scene out of which only 8 are neighbors of p. Estimation of elements of a row of A corresponding to agent p, considering all agents present in the scene requires 2.5 × 20 = 50 previous video frames (assuming L = 2.5N). While the use of neighborhood constraint reduces this to 2.5 ∗ 9 ≈ 23 frames. the number of the neighbors of agent i. Further, there could be an agent within the spatial proximity of another agent but there may not be any interaction between them. Hence it is required that the corresponding entry in the matrix A0 should be zero. This is enforced by adding sparsity constraint in Eq. 4. We use L1General package developed by Schmidt [21] for solving L1-regularization problems. For an illustration, see Fig.2b. There are a total of N = 20 agents present in the scene. Estimation of the row of matrix A corresponding to agent p requires 50 previous frames (assuming L = 2.5N) whereas the neighborhood based estimation reduces this to 23. Also consider a case where agents p and r interact with each other but are not within the spatial proximity owing to neighborhood constraint. The interaction is captured when intersection of neighborhoods of p and r has at least one interacting agent, in this case its q who is in the spatial proximity of both. Let the eigenvector matrix contain all the eigenvectors column-wise. We define a mapping for ith agent as f ( xi ) : xi ∈ R → zi = (ei1 , ei2 , . . . , eir ) T ∈ Rr×1 where e ji is the ith entry of jth eigenvector of interaction matrix A and r is the number of significant eigenvalues. A clustering algorithm is applied on the points {zi }, ∀i = 1, 2, . . . , N to identify the groups. The clustering algorithm runs on the components of eigenvectors, therefore this algorithm falls in the category of spectral clustering [8]. Since the number of groups is unknown, we apply a threshold based clustering. The adaptive threshold used for the ith point is c\|zi \|, where \|zi \| is its magnitude and c is found empirically. For example, all the agents within the distance of c\|z1 \| from z1 will form a group with agent 1. In this way, all the groups are obtained. We consider only significant eigenvectors (with \|λ\| ≥ 0.90) of A for group detection since the response from the eigenvectors with \|λ\| < 0.9 dies down to an insignificant level within the period of L frames. It may be noted that this group detection algorithm remains the same in the case where A does not have N independent eigenvectors. In such a case, the clustering algorithm runs on generalized eigenvectors. 3. Group Detection In this section, we discuss an algorithm for identifying the groups present in the scene. As seen from Eq. 2, the general solution is a linear combination of eigenvectors at any time instant k. Notice that if the corresponding entries of any two rows of the eigenvector matrix are similar, the corresponding agents form a group. This group information is not available from the position vector alone at a particular time instant x(k) because temporal evolution is also an important factor in deciding the groups. Since the eigenvectors are learned from the trajectories collectively, it encapsulates spatio-temporal evolution of the agents and hence can be exploited for group detection. 4. Group Activity Identification While the eigenvectors identify the groups, the eigenvalues can be used to determine the activity of a group. We employ the same model mentioned in Eq. 1 for the group g to estimate its interaction matrix Ag and ag . We do not use the submatrix formed by the agents of the group g in the previously learned 0 matrix A0 = [A\|a] to get Ag = [Ag \|ag ]. This is to get a refined matrix for the group and avoid any possi5 this corresponds to j = 1. Initially the two agents were standing together and then the second agent starts moving away from the first one leading to split of the group. ble interference from the outside agents in the estimag g g tion. Let xg (k) = [ x1 (k), x2 (k ), . . . , x M (k)] T , where M g is the cardinality of the group g and xi (k) is the position of the ith agent of the group at time instant k. To learn matrix Ag’ at kth time instant, we define a similar optimization framework as follows, where the second term enforces temporal continuity in the activity but unlike Eq. 4, there is no need for sparsity constraint as, by definition, all agents in a group interact. Therefore, g’ Âk = arg g’ min n Ak ∈R M×( M+1) g’ g’ + λ\|\|Ak − Ak−1 \|\|22 This group activity detection method is dependent on eigenvalues and hence sensitive to perturbations in the measurements. To address this, we define threshold bands for crucial values of eigenvalues. For example, if 0.995 < µ < 1.005, we consider µ to be 1 and if µ < 0.5 then it is considered as 0. 4.1. Atomic Activity Detection g’ 1 k 2 \|\|Ak Xkk− − L − Xk− L+1 \|\|2 o This algorithm is now extended to identification of individual’s activity as follows. Let x (k ) denotes position of an agent at time k, then (5) x (k + 1) = µx (k) + b Assuming Ag to be again diagonalizable, the general solution is similar as given in Eq. 2. The velocity vector v(k) for the group g can be written as M v(k) = ∑ i =1 {ci (µi − 1)µik−1 + di µik−1 }ui , (7) The velocity v(k) is as follows: v ( k ) = (1 − µ ) µ k −1 x (0 ) + µ k −1 b (6) (8) Note that there is no longer a activity called splitting as one needs at least two agents to define it. We identify following activities based on the value of µ: where \|µ1 \| ≥ \|µ2 \| . . . ≥ \|µ M \| are the eigenvalues of Ag . Since some of the coefficients ci and di could be zero, let µ j be the largest eigenvalue for which at least one of the coefficients ci or di is non-zero. Now we state how different values of µ j characterize various activities: 1. Stationary: An agent is stationary if \|µ\| = 0 at the location given by b. It is also stationary when \|µ\| = 1 and b = 0. 2. Stopping: 0 < \|µ\| < 1 indicates that the agent is stopping soon. 1. Stationary: A group is stationary if \|µ j \| = 0 indicating all the eigenvalues (with at least one nonzero coefficient) to be zero. That corresponds to zero velocity vector and hence the agents are stationary. In the illustration shown in Fig. 3(a), the deciding eigenvalue is µ2 which is 0. The two agents are stationary at locations 140 and 120 respectively. 3. Walking: An agent is walking if \|µ\| > 1. Further, an agent is walking with a constant velocity if \|µ\| = 1 and b 6= 0. Note that the group detection and activity recognition algorithms run in x and y directions independently and results need to be combined together. For group detection, a group is formed only if it is formed in both the directions. For example, let Zx = [1, 1, 2, 1] and Zy = [2, 1, 2, 2] be the label vectors (indicating assigned group number for all the four agents) obtained along x and y directions respectively. It says that agents {1,2,4} form a group along x direction while {1,3,4} form a group along y axis. Combining both the labels will result in the final label vector as Z = [1, 2, 3, 1] i.e. out of 4 agents, 1 and 4 are grouped together while agents 2 and 3 are singleton groups. To identify the final group activity from the two separate group activity estimates along x and y directions, we merge the two decisions according to the following priority sequence − Splitting>Walking>Approaching>Stationary. For example, 2. Approaching: A group has an approaching members if \|µ j \| < 1 as limk→∞ v(k ) → 0. In the example shown in Fig. 3(b), j = 2. One agent is stationary at 120 while the other agent starts from the location 100 and approaches to the first one. 3. Walking: If \|µ j \| = 1 then the group is walking with a constant velocity of d j u j . In Fig. 3(c), both the agents walk together and deciding eigenvalue corresponds to j = 2. Note that we do not discriminate between walking and running in this work. 4. Splitting: A group has a tendency for divergence if \|µ j \| > 1 as limk→∞ v(k) → ∞. In Fig. 3(d), 6 150 145 125 µ1 = 1, d1 = 0 µ2* = 0 120 125 µ1 = 1, d1 = 0 µ2* = 0.9, d2 ≠ 0 x(k) 100 115 140 110 115 135 130 50 105 100 110 0 95 125 90 105 120 115 0 150 120 85 5 10 100 0 k 5 10 k 80 0 µ1 = 1, d1 = 0 µ2* = 1, d2 ≠ 0 5 10 k −50 15 −100 0 µ1* >1, c1 ≠ 0 5 10 k Figure 3: Illustration of group activity - Stationary, Approaching, Walking and Splitting respectively from the estimated model parameters for a group consisting of two members. Eigenvalue with ∗ is the activity deciding eigenvalue. See the text for details. if a group has splitting and approaching activities in x and y directions respectively, the final group activity is splitting. We employ group level features that cover lowlevel details such as motion information to high-level information such as group activities. The features are described as follows: 5. Crowd Video Classification 1. Group density (GD): It is the ratio of number of groups by the total number of agents in the scene. A low value of GD indicates highly structured crowd. For example, GD for a group of marching soldiers is small whereas a mixed crowd has a higher group density. Having group level information in hand, we can use them in identifying the overall crowd behavior. Ability to identify crowd behavior enables crowd management systems to design and manage public places effectively to ensure safety and smooth operation. The overall crowd behavior is determined by how each group behaves. Depending on the synchronization among the groups, the behavior of crowd varies from being structured to unstructured. In this section, we define group level features that are useful for crowd video classification. We classify crowd videos into 8 classes as defined by [6]. The dataset containing 474 video clips covers a variety of videos. The eight classes are as follows: 2. Histogram of λmax : The histogram has three bins which are λmax ≥ 1, λmax < 1 and λmax = 0, where λmax is the largest eigenvalue of the interaction matrix for a group (µ j from the last section). The value at a particular bin is the number of groups in a scene having λmax as defined by that bin. Left skewed histogram i.e. towards λmax ≥ 1 indicates moving crowd whereas right skewed histogram suggests more or less stationary crowd. C1 : Mixed crowd C2 : Well organized crowd following mainstream: 3. Histogram of motion direction: The motion direction of each member of a group is calculated from its trajectory data and the mean direction is assigned to the group. This histogram has eight bins covering 0◦ to 360◦ with a bin size of 45◦ . The bin value is the number of groups falling in that particular bin. The uniform histogram indicates a mixed crowd whereas a skewed histogram indicates directionality in the crowd movement. C3 : Not well organized crowd following any mainstream C4 : Crowd merge C5 : Crowd split C6 : Crowd crossing in opposite directions Since the analysis is conducted independently in x and y directions; we get two histograms for λmax , leading to final feature vector of length 1 + 2 × 3 + 8 = 15. C7 : Intervened escalator traffic C8 : Smooth escalator traffic 7 Table 1: Performance comparison of different group detection algorithms on CUHK dataset. We use random forest (RF) as a classifier [22]. It consists of a multitude of decision trees that are trained from randomly sampled subsets of training dataset (bootstrap aggregating). This bootstrapping increases the performance by reducing the variance of the classifier. Also the split at each node of a tree is decided by m features selected randomly out of n features where m << n. We use RF to classify a crowd video by training it with the above mentioned features. The classification results are discussed in next section. NMI Purity RI CF [16] 0.66 0.71 0.67 CT [6] 0.69 0.72 0.69 Proposed 0.86 0.90 0.85 for each video where we compare the proposed algorithm with other methods and the ground truth instead of manually deciding on the instants when the performance has to be evaluated. We use Normalized Mutual Information (NMI) [26], Purity [27] and Rand Index (RI) [28] which are widely used for evaluation of clustering algorithms. Table 1 shows the comparison on these measures. It is quite evident from the table that the performance of the proposed algorithm far surpasses those of [6] and [16]. Fig. 4 demonstrates a visual comparison for different scenarios. Since Zhou et al. in [16] find coherent motion patterns at one time and then update them over time, it is sensitive to tracking errors and has the possibility of accumulation of errors if any frame has tracking error. Shao et al. [6] assign every agent to a collective transition prior. They have spatial proximity constraint only at the initial time instant which might not be effective as time progresses. Their algorithm groups all the agents moving in the same direction giving less importance to their spatial relationships. This can be observed from the output figures in column (b) of Fig. 4. Further in 4th row, a person with red hat is moving faster than the group behind him but CT and CF fail to capture this difference in velocity while the proposed algorithm could capture it. The groups in last row have small changes in their directions of movement which is again not captured by these two methods while the proposed method detects such small changes. We also compare the proposed group detection algorithm with the method of [17] on the videos VEIIG, student003 and eth. To compare with [17], we also use G-MITRE precision P and recall R as proposed by them. Table 2 shows the quantitative results that indicate an improved performance by the proposed method. The proposed algorithm outperforms these stateof-the-art methods because it is more robust to tracking errors since we extract groups from the eigenvectors rather than directly using the tracklets. It is quite evident from Fig. 4 where the tracklets for various agents are marked with different colors to indicate the group they belong to, that the proposed algorithm is able to detect agents in a group much better than the 6. Experiments and Results In this section, we discuss the performance of the proposed algorithms for group detection, group activity recognition and crowd video classification. We have tested our algorithms on various publicly available datasets containing real videos. We first discuss these datasets followed by performance evaluation of the proposed algorithms. 6.1. Datasets We tested our algorithms on different videos from various datasets contributed by several researchers namely CUHK [6], BEHAVE [5], BIWI Walking Pedestrians [23], Crowds-By-Example (CBE) [24] and Vittorio Emanuele II Gallery (VEIIG) [25]. CUHK dataset is a comprehensive crowd video dataset containing 474 video clips covering various crowd behaviors with varying crowd density. BEHAVE dataset has video clips with low crowd density and covering various group activities. BIWI dataset contains two low density crowd videos (namely eth and hotel). CBE has a medium density crowd video (student003) recorded outside a university. These datasets collectively cover a large variety of crowd videos. 6.2. Group Detection We tested group detection algorithm on all the 474 videos from CUHK dataset and 3 video clips (having duration of more than 10 minutes in total) from BEHAVE dataset. In case of videos from CUHK dataset, we restricted our algorithm to run only on those data that have sufficiently long tracks, since some of the clips are too short to accommodate for an analysis of a large number of agents. We compared the proposed algorithm with other methods on these selected agents. The ground truth for CUHK dataset was obtained manually. We compare the proposed algorithm for group detection with state-of-the-art methods by Shao et al. [6] and Zhou et al. [16]. For quantitative analysis on CUHK videos, we randomly select two time instants 8 Table 2: Performance comparison of the proposed group detection with [17] . BIWI eth CBE student003 VEIIG Baseline [17] P R 72.4 ± 4.4 65.2 ± 3.4 59.9 ± 2.9 53.5 ± 6.8 49.2 ± 9.9 34.4 ± 6.7 [17] P R 91.8 ± 1.2 94.2 ± 0.9 81.7 ± 0.2 82.5 ± 0.2 84.12 ± 0.6 84.11 ± 0.5 other existing methods. Also the proposed algorithm yields NMI = 0.92, Purity = 0.94 and RI = 0.93 on video clips from BEHAVE dataset whereas the corresponding measures for [6] and [16] have very low values (e.g. Purity for CF is only 0.35). It shows that these methods do not perform well in videos of a sparse crowd whereas the proposed method can also handle a sparse crowd effectively. Proposed P R 95.78 96.15 77.58 85.90 82.97 84.70 for Class 4 (Class Merge) is difficult, while the rest of the classes could be categorized quite easily using the proposed method. The OOB error, which indicates the generalized error, converges to a value 30% as shown in Fig. 7b. The importance plots, which show the significance of each group level feature in the classification are shown in Fig. 7c. It shows that the group density and histogram of eigenvalues are important for classification. 6.3. Group Activity Recognition We use BEHAVE and CUHK datasets for testing the algorithm for group activity identification. Here, we have excluded the clips containing other activities such as fight. We compared the activity results with the ground truth at regular intervals. Table 8 shows the confusion matrix for the proposed algorithm on BEHAVE dataset. The algorithm gives an accuracy of 70% for Walking and Stationary activities whereas it is less for the other two activities. We observed that the algorithm gets confused between these two activities. We suspect that the confusion is due to the fact that Splitting and Approaching are more abrupt in the motion dynamics than Walking and Stationary, which results in a poorer estimate of eigenvalues over the window of L frames. In CUHK dataset, since groups in most of the videos are walking, we obtain an accuracy of 85%. Some of the qualitative results on the videos from BEHAVE and CUHK dataset are given in Fig. 5 and Fig. 6, respectively. 7. Conclusions In this work, we presented a framework for analysis of medium dense crowd videos at various levels. We proposed a first order dynamical system to model agent trajectories collectively and subsequently demonstrated the effectiveness of this interaction model for group detection. We also show how eigenvalues of the model characterize group activities. We then showed the effectiveness of group level features in crowd video classification. Our algorithm assumes the availability of tracks which itself is a challenge in many crowded videos due to occlusion and other tracking problems. As a next goal, we aspire to define a unified framework where the proposed model and a tracker work together to improve each other’s performance in crowded videos by incorporating group interaction cues. 6.4. Crowd Video Classification References Since we update the interaction model with each incoming frame as explained in Section 6, we collect group level features at regular intervals. From each class, we randomly pick 70% feature vectors to train the classifier and the remaining for testing. As discussed before, we use random forest as a classifier with n = 17 and m = 4. We run the classifier 100 times with random splits of dataset for training and testing. To avoid over-fitting, the training data and testing data do not contain features from the same video. The average accuracy obtained is around 74%, an improvement over [6] where the reported accuracy is 70%. The confusion matrix is shown in Fig. 7a. From this figure, it is seen that classification of the crowd [1] T. Li, H. Chang, M. Wang, B. Ni, R. Hong, S. Yan, Crowded scene analysis: A survey, IEEE Transactions on Circuits and Systems for Video Technology 25 (3) (2015) 367–386. [2] M. Thida, Y. L. Yong, P. Climent-Pérez, H.-l. Eng, P. Remagnino, A literature review on video analytics of crowded scenes, in: Intelligent Multimedia Surveillance, Springer, 2013, pp. 17–36. [3] J. S. J. Junior, S. Musse, C. 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Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers∗ arXiv:1608.01700v3 [cs.DS] 3 Nov 2016 Sara Ahmadian† Chaitanya Swamy† Abstract We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {Li }i∈F and a set D of clients located in a common metric space {c(i, j)}i,j∈F ∪D , and bounds k, m. A feasible solution is a pair S ⊆ F, σ : D 7→ S ∪ {out} , where σ specifies the client assignments, such that \|S\| ≤ k, \|σ −1 (i)\| ≥ Li for all i ∈ S, and \|σ −1 (out)\| ≤ m.P In the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, the objective is to minimize i∈S maxj∈σ−1 (i) c(i, j), and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize maxi∈S maxj∈σ−1 (i) c(i, j). We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the nonoutlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-ofradii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the \|S\| ≤ k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplierpreserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds. 1 Introduction Clustering is an ubiquitous problem that arises in many applications in different fields such as data mining, machine learning, image processing, and bioinformatics. Many of these problems involve finding a set S of at most k “cluster centers”, and an assignment σ mapping an underlying set D of data points located in some metric space {c(i, j)} to S, to minimize some objective P function; examples include the k-center (minimize maxj∈D P c(σ(j), j)) [21, 22], k-median (minimize j∈D c(σ(j), j)) [10, 23, 26, 7], and min-sumof-radii (minimize i∈S maxj:σ(j)=i c(i, j)) [16, 12] problems. Viewed from this perspective, clustering problems can often be viewed as facility-location problems, wherein an underlying set of clients that require service need to be assigned to facilities that provide service in a cost-effective fashion. Both clustering and facility-location problems have been extensively studied in the Computer Science and Operations Research literature; see, e.g., [28, 30] in addition to the above references. We consider clustering problems with (non-uniform) lower-bound requirements on the cluster sizes, and where a bounded number of points may be designated as outliers and left unclustered. One motivation for considering lower bounds comes from an anonymity consideration. In order to achieve data privacy, [29] proposed an anonymization problem where we seek to perturb (in a specific way) some of (the attributes of) ∗ A preliminary version [3] appeared in the Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP), 2016. † {sahmadian,cswamy}@uwaterloo.ca. Dept. of Combinatorics and Optimization, Univ. Waterloo, Waterloo, ON N2L 3G1. Supported in part by NSERC grant 327620-09 and an NSERC Discovery Accelerator Supplement award. 1 the data points and then cluster them so that every cluster has at least L identical perturbed data points, thus making it difficult to identify the original data from the clustering. As noted in [2, 1], this anonymization problem can be abstracted as a lower-bounded clustering problem where the clustering objective captures the cost of perturbing data. Another motivation comes from a facility-location perspective, where (as in the case of lower-bounded facility location), the lower bounds model that it is infeasible or unprofitable to use services unless they satisfy a certain minimum demand (see, e.g., [27]). Allowing outliers enables one to handle a common woe in clustering problems, namely that data points that are quite dissimilar from any other data point can often disproportionately (and undesirably) degrade the quality of any clustering of the entire data set; instead, the outlier-version allows one to designate such data points as outliers and focus on the data points of interest. Formally, adopting the facility-location terminology, our setup is as follows. We have a set F of facilities with lower bounds {Li }i∈F and a set D of clients located in a common metric space {c(i, j)}i,j∈F ∪D , and bounds k, m. A feasible solution chooses a set S ⊆ F of at most k facilities, and assigns each client j to a facility σ(j) ∈ S, or designates j as an outlier by setting σ(j) = out P so that \|σ −1 (i)\| ≥ Li for all i ∈ S, and −1 \|σ (out)\| ≤ m. We consider two clustering objectives: minimize i∈S maxj:σ(j)=i c(i, j), which yields the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, and minimize maxi∈S maxj:σ(j)=i c(i, j), which yields the lower-bounded k-supplier with outliers (LBkSupO) problem. (k-supplier denotes the facility-location version of k-center; the latter typically has F = D.) We refer to the non-outlier versions of the above problems (i.e., where m = 0) as LBkSR and LBkSup respectively. Our contributions. We obtain the first results for clustering problems with non-uniform lower bounds and outliers. We develop various techniques for tackling these problems using which we obtain constantfactor approximation guarantees for LBkSRO and LBkSupO. Note that we need to ensure that none of the three types of hard constraints involved here—at most k clusters, non-uniform lower bounds, and at most m outliers—are violated, which is somewhat challenging. We obtain an approximation factor of 12.365 for LBkSRO (Theorem 2.8, Section 2.2), which improves to 3.83 for the non-outlier version LBkSR (Theorem 2.7, Section 2.1). These also constitute the first approximation results for the min-sum-of-radii objective when we consider: (a) lower bounds (even uniform bounds) but no outliers (LBkSR); and (b) outliers but no lower bounds. Previously, an O(1)-approximation was known only in the setting where there are no lower bounds and no outliers (i.e., Li = 0 for all i, m = 0) [12]. For the k-supplier objective (Section 3), we obtain an approximation factor of 5 for LBkSupO (Theorem 3.2), and 3 for LBkSup (Theorem 3.1). These are the first approximation results for the k-supplier problem with non-uniform lower bounds. Previously, [1] obtained approximation factors of 4 and 2 respectively for LBkSupO and LBkSup for the special case of uniform lower bounds and when F = D. Complementing our approximation bounds, we prove a factor-3 hardness of approximation for LBkSup (Theorem 3.3), which shows that our approximation factor of 3 is optimal for LBkSup. We also show (Appendix C) that LBkSupO is equivalent to the k-center version of the problem (where F = D). Our techniques. Our main technical contribution is an O(1)-approximation algorithm for LBkSRO (Section 2.2). Whereas for the non-outlier version LBkSR (Section 2.1), one can follow an approach similar to that of Charikar and Panigrahi [12] for the min-sum-of-radii problem without lower bounds or outliers, the presence of outliers creates substantial difficulties whose resolution requires various novel ingredients. As in [12], we view LBkSRO as a k-ball-selection (k-BS) problem of picking k suitable balls (see Section 2) and consider its LP-relaxation (P2 ). Let OPT denote its optimal value. Following the Jain-Vazirani (JV) template for k-median [23], we move to the version where we may pick any number of balls but incur a fixed cost of z for each ball we pick. The dual LP (D2 ) has αj dual variables for the clients, which “pay” for (i, r) pairs (where (i, r) denotes the ball {j ∈ D : c(i, j) ≤ r}). For LBkSR (where m = 0), as observed 2 in [12], it is easy to adapt the JV primal-dual algorithm for facility location to handle this fixed-cost version of k-BS: we raise the αj s of uncovered clients until all clients are covered by some fully-paid (i, r) pair (see PDAlg). This yields a so-called Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm: if F P is the primal solution constructed, then 3 j αj can pay for cost(F ) + 3\|F \|z; hence, by varying z, one can find two solutions F1 , F2 for nearby values of z, and combine them to extract a low-cost k-BS-solution. The presence of outliers in LBkSRO significantly complicates things. The natural adaptation of the primal-dual algorithm is to now stop when at least \|D\| − m clients are covered by fully-paid (i, r) pairs. But now, the dual objective involves a −m · γ term, where γ = maxj αj , which potentially cancels the dual contribution of (some) clients that pay for the last fully-paid (i, r) pair, say f . Consequently, we do not obtain anPLMP-approximation: if F is the primal solution we construct, we can only say that (loosely speaking) 3( j αj − m · γ) pays for cost(F \ f ) + 3\|F \ f \|z (see Theorem 2.9 (ii)). In particular, this means that even if the primal-dual algorithm returns a solution with k pairs, its cost need not be bounded, an artifact that never arises in LBkSR (or k-median). This in turn means that by combining the two solutions F1 , F2 found for z1 , z2 ≈ z1 , we only obtain a solution of cost O(OPT + z1 ) (see Theorem 2.14). Dealing with the case where z1 = Ω(OPT ) is technically the most involved portion of our algorithm (Section 2.2.2). We argue that in this case the solutions F1 , F2 (may be assumed to) have a very specific structure: \|F1 \| = k + 1, and every F2 -ball intersects at most one F1 -ball, and vice versa. We utilize this structure to show that either we can find a good solution in a suitable neighborhood of F1 and F2 , or F2 itself must be a good solution. We remark that the above difficulties (i.e., the inability to pay for the last “facility” and the ensuing complications) also arise in the k-median problem with outliers. We believe that our ideas also have implications for this problem and should yield a much-improved approximation ratio for this problem. (The current approximation ratio is a large (unspecified) constant [13].) For the k-supplier problem, LBkSupO, we leverage the notion of skeletons and pre-skeletons defined by [15] in the context of capacitated k-supplier with outliers, wherein facilities have capacities instead of lower bounds limiting the number of clients that can be assigned to them. Roughly speaking, a skeleton F ⊆ F ensures there is a low-cost solution (F, σ). A pre-skeleton satisfies some of the properties of a skeleton. We show that if F is a pre-skeleton, then either F is a skeleton or F ∪ {i} is a pre-skeleton for some facility i. This allows one to find a sequence of facility-sets such that at least one of them is a skeleton. For a given set F , one can check if F admits a low-cost assignment σ, so this yields an O(1)-approximation algorithm. Related work. There is a wealth of literature on clustering and facility-location (FL) problems (see, e.g., [28, 30]); we limit ourselves to the work that is relevant to LBkSRO and LBkSupO. The only prior work on clustering problems to incorporate both lower bounds and outliers is by Aggarwal et al. [1]. They obtain approximation ratios of 4 and 2 respectively for LBkSupO and LBkSup with uniform lower bounds and when F = D, which they consider as a means of achieving anonymity. They also consider an alternate cellular clustering (CellC) objective and devise an O(1)-approximation algorithm for lower-bounded CellC, again with uniform lower bounds, and mention that this can be extended to an O(1)-approximation for lower-bounded CellC with outliers. More work has been directed towards clustering problems that involve either outliers or lower bounds (but not both), and here, clustering with outliers has received more attention than lower-bounded clustering problems. Charikar et al. [11] consider (among other problems) the outlier-versions of the uncapacitated FL, k-supplier and k-median problems. They devise constant-factor approximations for the first two problems, and a bicriteria approximation for the k-median problem with outliers. They also proved a factor-3 approximation hardness result for k-supplier with outliers. This nicely complements our factor-3 hardness result for k-supplier with lower bounds but no outliers. Chen [13] obtained the first true approximation for k-median with outliers via a sophisticated combination of the primal-dual algorithm for k-median and 3 local search that yields a large (unspecified) O(1)-approximation. As remarked earlier, the difficulties that we overcome in designing our 12.365-approximation for LBkSRO are similar in spirit to the difficulties that arise in k-median with outliers, and we believe that our techniques should also help and significantly improve the approximation ratio for this problem. Cygan and Kociumaka [15] consider the capacitated ksupplier with outliers problem, and devise a 25-approximation algorithm. We leverage some of their ideas in developing our algorithm for LBkSupO. Lower-bounded clustering and FL problems remain largely unexplored and are not well understood. Besides LBkSup (which has also been studied in Euclidean spaces [17]), another such FL problem that has been studied is lower-bounded facility location (LBFL) [24, 20], wherein we seek to open (any number of) facilities (which have lower bounds) and assign each client j to an open facility σ(j) so as to miniP mize j∈D c(σ(j), j). Svitkina [31] obtained the first true approximation for LBFL, achieving an O(1)approximation; the O(1)-factor was subsequently improved by [4]. Both results apply to LBFL with uniform lower bounds, and can be adapted to yield O(1)-approximations to the k-median variant (where we may open at most k facilities). We now discuss work related to our clustering objectives, albeit that does not consider lower bounds or outliers. Doddi et al. [16] introduced the k-clustering min-sum-of-diameters (kSD) problem, which is closely related to the k-clustering min-sum-of-radii (kSR) problem: the kSD-cost is at least the kSR-cost, and at most twice the kSR-cost. The former problem is somewhat better understood than the latter one. Whereas the kSD problem is APX-hard even for shortest-path metrics of unweighted graphs (it is NP-hard to obtain a better than 2 approximation [16]), the kSR problem is only known to be NP-hard for general metrics, and its complexity for shortest-path metrics of unweighted graphs is not yet settled with only a quasipolytime (exact) algorithm known [18]. On the positive side, Charikar and Panigrahi [12] devised the first (and current-best) O(1)-approximation algorithms for these problems, obtaining approximation ratios of 3.504 and 7.008 for the kSR and kSD problems respectively, and Gibson et al. [18] obtain a quasi-PTAS for the kSR problem when F = D. Various other results are known for specific metric spaces and when F = D, such as Euclidean spaces [19, 8] and metrics with bounded aspect ratios [18, 6]. The k-supplier and k-center (i.e., k-supplier with F = D) objectives have a rich history of study. Hochbaum and Shmoys [21, 22] obtained optimal approximation ratios of 3 and 2 for these problems respectively. Capacitated versions of k-center and k-supplier have also been studied: [25] devised a 6approximation for uniform capacities, [14] obtained the first O(1)-approximation for non-uniform capacities, and this O(1)-factor was improved to 9 in [5]. Finally, our algorithm for LBkSRO leverages the template based on Lagrangian relaxation and the primal-dual method to emerge from the work of [23, 9] for the k-median problem. 2 Minimizing sum of radii with lower bounds and outliers Recall that in the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, we have a facility-set F and client-set D located in a metric space {c(i, j)}i,j∈F ∪D, lower bounds {Li }i∈F , and bounds k and m. A feasible solution is a pair S ⊆ F, σ : D 7→ S ∪ {out} , where σ(j) ∈ S indicates that j is assigned −1 to facility σ(j), and σ(j) = out designates j as an outlier, such P that \|σ (i)\| ≥ Li for all i ∈ S, and −1 \|σ (out)\| ≤ m. The cost of such a solution is cost(S, σ) := i∈S ri , where ri := maxj∈σ−1 (i) c(i, j) denotes the radius of facility i; the goal is to find a solution of minimum cost. We use LBkSR to denote the non-outlier version where m = 0. It will be convenient to consider a relaxation of LBkSRO that we call the k-ball-selection (k-BS) problem, which focuses on selecting at most k balls centered at facilities of minimum total radius. More precisely, let B(i, r) := {j ∈ D : c(i, j) ≤ r} denote the ball of clients centered at i with radius r. Let S cmax = maxi∈F ,j∈D c(i, j). Let Li := {(i, r) : \|B(i, r)\| ≥ Li }, and L := i∈F Li . The goal in k-BS is to 4 P S find a set F ⊆ L with \|F \| ≤ k and D \ (i,r)∈F B(i, r) ≤ m so that cost(F ) := (i,r)∈F r is minimized. (When formulating the LP-relaxation of the k-BS-problem, we equivalently view L as containing only pairs of the form (i, c(i, j)) for some client j, which makes L finite.) It is easy to see that any LBkSRO-solution yields a k-BS-solution of no greater cost. The key advantage of working with k-BS is that we do not explicitly consider the lower bounds (they are folded into the Li s) and we do not require the balls B(i, r) for (i, r) ∈ F to be disjoint. While a k-BS-solution F need not directly translate to a feasible LBkSROsolution, one can show that it does yield a feasible LBkSRO-solution of cost at most 2 · cost(F ). We prove a stronger version of this statement in Lemma 2.1. In the following two sections, we utilize this relaxation to devise the first constant-factor approximation algorithms for for LBkSR and LBkSRO. To our knowledge, our algorithm is also the first O(1)-approximation algorithm for the outlier version of the min-sum-of-radii problem without lower bounds. We consider an LP-relaxation for the k-BS-problem, and to round a fractional k-BS-solution to a good integral solution, we need to preclude radii that are much larger than those used by an (integral) optimal solution. We therefore “guess” the t facilities in the optimal solution with the largest radii, and Otheir 2t radii, where t ≥ 1 is some constant. That is, we enumerate over all O (\|F\| + \|D\|) S choices F = {(i1 , r1 ), . . . , (it , rt )} of t (i, r) pairs from L. For each such selection, we set D0 = D \ (i,r)∈F O B(i, r), L0 = {(i, r) ∈ L : r ≤ min(i,r)∈F O r} and k 0 = k − \|F O \|, and run our k-BS-algorithm on the modified k-BS-instance (F, D0 .L0 , c, k 0 , m) to obtain a k-BS-solution F . We translate F ∪ F O to an LBkSROsolution, and return the best of these solutions. The following lemma, and the procedure described therein, is repeatedly used to bound the cost of translating F ∪ F O to a feasible LBkSRO-solution. We call pairs (i, r), (i0 , r0 ) ∈ F × R≥0 non-intersecting, if c(i, i0 ) > r + r0 , and intersecting otherwise. Note that B(i, r) ∩ B(i0 , r0 ) = ∅ if (i, r) and (i0 , r0 ) are non-intersecting. For a set P ⊆ F × R≥0 of pairs, define µ(P ) := {i ∈ F : ∃r s.t. (i, r) ∈ P }. Lemma 2.1. Let F O ⊆ L, and D0 , L0 , k 0 be as defined above. Let F ⊆ L be a k-BS-solution for the k-BS-instance (F, D0 , L0 , c, kS0 , m). Suppose for each i ∈ µ(F ), we have a radius ri0 ≤ maxr:(i,r)∈F r and U ⊆ L. Then there exists a feasible such that the pairs in U := i∈µ(F ) (i, ri0 ) are non-intersecting P LBkSRO-solution (S, σ) with cost(S, σ) ≤ cost(F ) + (i,r)∈F O 2r. Proof. Pick a maximal subset P ⊆ F O to add to U such that all pairs in U 0 = U ∪ P are non-intersecting. For each (i, r) ∈ F O \ P , define κ(i, r) to be some intersecting pair (i0 , r0 ) ∈ U 0 . Define S = µ(U 0 ). Assign each client j to σ(j) ∈ S as follows. If j ∈ B(i, r) for some (i, r) ∈ U 0 , set σ(j) = i. Note that U 0 ⊆ L, so this satisfies the lower bounds for all i ∈ S. Otherwise, if j ∈ B(i, r) for some (i, r) ∈ F , set σ(j) = i. Otherwise, if j ∈ B(i, r) for some (i, r) ∈ F O \ P and (i0 , r0 ) = κ(i, r), set σ(j) = i0 . Any remaining unassigned client is not covered by the balls corresponding to pairs in F ∪ F O . There are at most m such clients, and we set σ(j) = out for each such client j. Thus (S, σ) is a feasible LBkSRO-solution. For any i ∈ S and j ∈ σ −1 (i) either j ∈ B(i, r) for some (i, r) ∈ F ∪ U 0 , or j ∈ B(i0 , r0 ) where P κ(i0 , r0 ) = (i, r) ∈ U 0 , in which case c(i, j) ≤ r + 2r0 . So cost(S, σ) ≤ cost(F ) + (i,r)∈F O 2r. 2.1 Approximation algorithm for LBkSR We now present our algorithm for the non-outlier version, LBkSR, which will introduce many of the ideas underlying our algorithm for LBkSRO described in Section 2.2. Let O∗ denote the cost of an optimal solution to the given LBkSR instance. As discussed above, for each selection of (i1 , r1 ), . . . , (it , rt ) of t pairs, we do the following. We set S D0 = D \ tp=1 B(ip , rp ), L0 = {(i, r) ∈ L : r ≤ R∗ := minp=1,...,t rp }, k 0 = k − t, and consider the k-BS-problem of picking at most k 0 pairs from L0 whose corresponding balls cover D0 incurring minimum cost (but our algorithm k-BSAlg will return pairs from L). We consider the following natural LP-relaxation 5 (P1 ) of this problem, and its dual (D1 ). X r · yi,r min (P1 ) max X yi,r ≥ 1 ∀j ∈ D0 s.t. (D1 ) X αj − z ≤ r ∀(i, r) ∈ L0 (2) j∈B(i,r)∩D0 (i,r)∈L0 :j∈B(i,r) X αj − k 0 · z j∈D0 (i,r)∈L0 s.t. X yi,r ≤ k 0 α, z ≥ 0. (1) (i,r)∈L0 y ≥ 0. If (P1 ) is infeasible then we discard this choice of t pairs and move on to the next selection. So we assume (P1 ) is feasible in the remainder of this section. Let OPT denote the common optimal value of (P1 ) and (D1 ). As in the JV-algorithm for k-median, we Lagrangify constraint (1) and consider the unconstrained problem where we do not bound the number of pairs we may pick, but we incur a fixed cost z for each pair (i, r) that we pick (in addition to r). It is easy to adapt the JV primal-dual algorithm for facility location [23] to devise a simple Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for this problem (see PDAlg and Theorem 2.3). We use this LMP algorithm within a binary-search procedure for z to obtain two solutions F1 and F2 with \|F1 \| ≤ k < \|F2 \|, and show that these can be “combined” to extract a k-BSsolution F of cost at most 3.83 · OPT + O(R∗ ) (Lemma A.4). This combination step is more involved than in k-median. The main idea here is to use the F2 solution as a guide to merge some F1 -pairs. We cluster the F1 pairs around the F2 -pairs and setup a covering-knapsack problem whose solution determines for each F2 -pair (i, r), whether to “merge” the F1 -pairs clustered around (i, r) or select all these F1 -pairs (see step B2). Finally, we add back the pairs (i1 , r1 ), . . . (it , rt ) selected earlier and apply Lemma 2.1 to obtain an LBkSR-solution. As required by Lemma 2.1, to aid in this translation, our k-BS-algorithm returns, along with F , a suitable radius rad(i) for every facility i ∈ µ(F ). This yields a (3.83 + )-approximation algorithm (Theorem 2.7). While our approach is similar to the one in [12] for the min-sum-of-radii problem without lower bounds (although our combination step is notably simpler), an important distinction that arises is the following. In the absence of lower bounds, the ball-selection problem k-BS is equivalent to the min-sum-of-radii problem, but (as noted earlier) this is no longer the case when we have lower bounds since in k-BS we do not insist that the balls we pick be disjoint. Consequently, moving from overlapping balls in a k-BS-solution to an LBkSR-solution incurs, in general, a factor-2 blowup in the cost (see Lemma 2.1). It is interesting that we are able to avoid this blowup and obtain an approximation factor that is quite close to the approximation factor (of 3.504) achieved in [12] for the min-sum-of-radii problem without lower bounds. We now describe our algorithm in detail and proceed to analyze it. We describe a slightly simpler (6.183 + )-approximation algorithm below (Theorem 2.2). We sketch the ideas behind the improved approximation ratio at the end of this section and defer the details to Appendix A. Algorithm 1. Input: An LBkSR-instance I = F, D, {Li }, {c(i, j)}, k , parameter > 0. Output: A feasible solution (S, σ). A1. Let t = min k, 1 . For each set F O ⊆ L with \|F O \| = t, do the following. S A1.1. Set D0 = D \ (i,r)∈F O B(i, r), L0 = {(i, r) ∈ L : r ≤ R∗ = min(i,r)∈F O r}, and k 0 = k − t. O 0 0 0 0 A1.2. If (P1 ) is infeasible, then reject this guess and move to the next set F . If D 6= ∅, run k-BSAlg(D , L , k , ) to obtain F, {rad(i)}i∈F ; else set (F, rad) = (∅, ∅). A1.3. Apply the procedure in Lemma 2.1 taking ri0 = rad(i) for all i ∈ µ(F ) to obtain (S, σ). A2. Among all the solutions (S, σ) found in step A2, return the one with smallest cost. 6 Figure 1: An example of stars formed by F1 and F2 where F1 = {u1 , u2 , . . . , u11 } and F2 = {v1 , v2 , . . . , v6 } depicted by squares and circles, respectively. Algorithm k-BSAlg(D 0 , L0 , k0 , ). Output: F ⊆ L with \|F \| ≤ k 0 , a radius rad(i) for all i ∈ µ(F ). B1. Binary search for z. B1.1. Set z1 = 0 and z2 = 2k 0 cmax . For p = 1, 2, let(Fp , {radp (i)}, αp ) ← PDAlg(D0 , L0 , zp ), and let kp = \|Fp \|. If k1 ≤ k 0 , stop and return F1 , {rad1 (i)} . We prove in Theorem 2.3 that k2 ≤ k 0 ; if k2 = k 0 , stop and return F2 , {rad2 (i)} . z1 +z2 B1.2. Repeat the following until z2 − z1 ≤ δz = OPT 3n , where n = \|F\| + \|D\|. Set z = 2 . Let 0 0 0 (F, {rad(i)}, α) ← PDAlg(D , L , z). If \|F \| = k , stop and return F, {rad(i)} ; if \|F \| > k 0 , update z1 ← z and (F1 , rad1 , α1 ) ← (F, rad, α), else update z2 ← z and (F2 , rad2 , α2 ) ← (F, rad, α). B2. Combining F1 and F2 . Let π : F1 7→ F2 be any map such that (i0 , r0 ) and π(i0 , r0 ) intersect ∀(i0 , r0 ) ∈ F1 . (This exists since every j ∈ D0 is covered by B(i, r) for some (i, r) ∈ F2 .) Define star Si,r = π −1 (i, r) for all (i, r) ∈ F2 (see Fig. 1). Solve the following covering-knapsack LP. X P P min xi,r (2r + (i0 ,r0 )∈Si,r 2r0 ) + (1 − xi,r ) (i0 ,r0 )∈Si,r r0 (C-P) (i,r)∈F2 s.t. X xi,r + \|Si,r \|(1 − xi,r ) ≤ k, 0 ≤ xi,r ≤ 1 ∀(i, r) ∈ F2 . (i,r)∈F2 Let x∗ be an extreme-point optimal solution to (C-P). The variable x(i,r) has the following interpretation. If x∗i,r = 0, then we select all pairs in Si,r . Otherwise, if Si,r 6= ∅, we pick a pair in (i0 , r0 ) ∈ Si,r , and include (i0 , 2r + r0 + max(i00 ,r00 )∈Si,r \{(i0 ,r0 )} 2r00 ) in our solution. the radius of i0 to 2r + r0 + S Notice that by expanding 00 00 00 max(i00 ,r00 )∈Si,r \{(i0 ,r0 )} 2r , we cover all the clients in (i00 ,r00 )∈Si,r B(i , r ). Let F 0 be the resulting set of pairs. B3. If cost(F2 ) ≤ cost(F ), return (F2 , rad2 ), else return F 0 , {rad1 (i)}i∈µ(F 0 ) . Algorithm PDAlg(D 0 , L0 , z). Output: F ⊆ L, radius rad(i) for all i ∈ µ(F ), dual solution α. P1. Dual-ascent phase. Start with αj = 0 for all j ∈ D0 , D0 as the set of active clients, and the set T of tight pairs initialized to ∅. We repeat the following until all clients become inactive: we raise the αj s of all active clients uniformly until constraint (2) becomes tight for some (i, r); we add (i, r) to T and mark all active clients in B(i, r) as inactive. P2. Pruning phase. Let TI be a maximal subset of non-intersecting pairs in T picked by a greedy algorithm that scans pairs in T in non-increasing order of radius. Note that for each i ∈ µ(TI ), there is exactly one pair (i, r) ∈ TI . We set rad(i) = r, and ri = max {c(i, j) : j ∈ B(i0 , r0 ), (i0 , r0 ) ∈ T, r0 ≤ r, (i0 , r0 ) intersects (i, r) ((i0 , r0 ) could be (i, r))}. Let F = {(i, ri )}i∈µ(TI ) . Return F , {rad(i)}i∈µ(TI ) , and α. Analysis. We prove the following result. Theorem 2.2. For any > 0, Algorithm 1 returns a feasible LBkSR-solution of cost at most 6.1821 + O() O∗ in time nO(1/) . We firstP prove that PDAlg is an LMP 3-approximation algorithm, i.e., its output (F, α) satisfies cost(F )+ 3\|F \|z ≤ 3 j∈D0 αj (Theorem 2.3). Utilizing this, we analyze k-BSAlg, in particular, the output of the combination step B2, and argue that k-BSAlg returns a feasible solution of cost at most 6.183 + O() · OPT + O(R∗ ) (Theorem 2.5). For the right choice of F O , combining this with Lemma 2.1 yields Theorem 2.2. 7 Theorem 2.3. Suppose PDAlg(D0 , L0 , z) returns (F, {rad(i)}, α). Then (i) the balls corresponding to F cover D0 , P (ii) cost(F ) + 3\|F \|z ≤ 3 j∈D0 αj ≤ 3(OPT + k 0 z), (iii) (i, rad(i)) i∈µ(F ) ⊆ L0 , is a set of non-intersecting pairs, and rad(i) ≤ ri ≤ 3R∗ ∀i ∈ µ(F ), (iv) if \|F \| ≥ k 0 then cost(F ) ≤ 3 · OPT ; if \|F \| > k 0 , then z ≤ OPT . (Hence, k2 ≤ k 0 in step B1.1.) Proof. We prove parts (i)—(iii) first. Note that (i, rad(i)) i∈µ(F ) is TI (by definition). Consider a client j ∈ D0 and let (i0 , r0 ) denote the pair in T that causes j to become inactive. Then there must be a pair (i, r) ∈ TI that intersects (i0 , r0 ) such that r ≥ r0 (we could have (i, r) = (i0 , r0 )). Since by definition ri ≥ c(i, j), j ∈ B(i, ri ). Also, c(i, i0 ) ≤ r + r0 . So if j is the client that determines ri , then ri = c(i, j) ≤ c(i0 , i) + c(i, j) ≤ 2r0 + r ≤ 3r ≤ 3R∗ . All pairs in TI are tight and non-intersecting. So for every i ∈ µ(F ), there must be some j ∈ B(i, rad(i)) ∩ D0 with c(i, j) = rad(i), so rad(i) ≤ ri . Since \|F \| = \|TI \|, X X X X (ri + 3z) ≤ 3αj ≤ 3(OPT + k 0 z). cost(F ) + 3\|F \|z = (3r + 3z) = 3αj ≤ (i,r)∈TI (i,r)∈TI (i,r)∈TI j∈B(i,r)∩D0 j∈D0 The last inequality above follows since (α, z) is a feasible solution to (D1 ). Rearranging the bound yields 3(\|F \|−k 0 )z ≤ 3·OPT −cost(F ), so when \|F \| ≥ k 0 , we have cost(F ) ≤ 3 · OPT , and when \|F \| > k 0 , we have z ≤ OPT . Recall that in step B1.1, k2 is the number of pairs returned by PDAlg for z = 2k 0 cmax . So the last 0 0 statement follows Psince OPT ≤ k0 cmax , as all balls in L have radius at most cmax and any feasible solution to (P1 ) satisfies (i,r)∈L0 yi,r ≤ k . Let F, {rad(i)} = k-BSAlg(D0 , L0 , k 0 ). If k-BSAlg terminates in step B1, then cost(F ) ≤ 3 · OPT due to part (ii) of Theorem 2.3, so assume otherwise. Let a, b ≥ 0 be such that ak1 + bk2 = k 0 , a + b = 1. Let C1 = cost(F1 ) and C2 = cost(F2 ). Recall that (F1 , rad1 , α1 ) and (F2 , rad2 , α2 ) are the outputs of PDAlg for z1 and z2 respectively. Claim 2.4. We have aC1 + bC2 ≤ (3 + )OPT . Proof. By part (ii) of Theorem 2.3, we have C1 +3k1 z1 ≤ 3(OPT +k 0 z1 ) and C2 +3k2 z2 ≤ 3(OPT +k 0 z2 ). Combining these, we obtain aC1 +bC2 ≤ 3OPT +3k 0 (az1 +bz2 )−3(ak1 z1 +bk2 z2 ) ≤ 3(OPT +k 0 z2 )−3k 0 z2 +3ak1 δz ≤ (3+)OPT . The second inequality follows since 0 ≤ z2 − z1 ≤ δz . 0 , k 0 ) returns a feasible solution F, {rad(i)} with cost(F ) ≤ 6.183 + Theorem 2.5. k-BSAlg(D0 , L O() · OPT + O(R∗ ) where (i, rad(i))}i∈µ(F ) ⊆ L0 is a set of non-intersecting pairs. Proof. The radii {rad(i)}i∈µ(F ) are simply the radii obtained from some execution of PDAlg, so (i, rad(i)) i∈µ(F ) ⊆ L0 and comprises non-intersecting pairs. If k-BSAlg terminates in step B1, we have argued a better bound on cost(F ). If not, and we return F2 , the cost incurred is C2 . Otherwise, we return the solution F 0 found in step B2. Since (C-P) has only one constraint in addition to the bound constraints 0 ≤ xi,r ≤ 1, the extreme-point x∗ has at most one fractional P optimal solution ∗ component, and if it has a fractional component, then (i,r)∈F2 xi,r + \|Si,r \|(1 − x∗i,r ) = k 0 . For any (i, r) ∈ F2 with x∗i,r ∈ {0, 1}, the number of pairs we include is exactly x∗i,r + \|Si,r \|(1 − x∗i,r ), and the total cost of these pairs is at most the contribution to the objective function of (C-P) from the x∗i,r and (1 − x∗i,r ) 8 terms. If x∗ has a fractional component (i0 , r0 ) ∈ F2 , then x∗i0 ,r0 + \|Si0 ,r0 \|(1 − x∗i0 ,r0 ) is a positive integer. Since we include at most one pair for (i0 , r0 ), this implies that \|F 0 \| ≤ k 0 . The cost of the pair we include is at most 15R∗ , since all (i, r) ∈ F1 ∪ F2 satisfy r ≤ 3R∗ . Therefore, cost(F 0 ) ≤ OPT C-P + 15R∗ . Also, OPT C-P ≤ 2bC2 + (2b + a)C1 = 2bC2 + (1 + b)C1 , since setting xi,r = b for all (i, r) ∈ F2 yields a feasible solution to (C-P) of this cost. So when we terminate in step B3, we return a solution F with cost(F ) ≤ min{C2 , 2bC2 + (1 + b)C1 + 15R∗ }. The following claim (Claim 2.6) shows that min{C2 , 2bC2 + (1 + b)C1 } ≤ 2.0607(aC1 + bC2 ) for all a, b ≥ 0 with a + b = 1. Combining this with Claim 2.4 yields the bound in the theorem. Claim 2.6. min{C2 , 2bC2 + (1 + b)C1 } ≤ ( 3b2b+1 )(aC1 + bC2 ) ≤ 2.0607(aC1 + bC2 ) for all a, b ≥ 0 −2b+1 such that a + b = 1. Proof. Since the minimum is less than any convex combination, min(C2 , 2bC2 + bC1 + C1 ) ≤ = 3b2 − b 1−b C2 + 2 (2bC2 + bC1 + C1 ) 2 3b − 2b + 1 3b − 2b + 1 (1 − b)(1 + b) b2 + b b+1 C1 + 2 C2 = 2 ((1 − b)C1 + bC2 ) 2 3b − 2b + 1 3b − 2b + 1 3b − 2b + 1 Since a = 1 − b, the first inequality in the claim follows.√ 3 The expression 3b2b+1 ≈ 2.0607, which yields is maximized at b = −1 + 2, and has value 1 + 2√ −2b+1 2 the second inequality in the claim. Now we have all the ingredients needed for proving the main theorem of this section. Proof of Theorem 2.2. It suffices to show that when the selection F O = {(i1 , r1 ), . . . (it , rt )} in step A1 corresponds to the t facilities in an optimal solution with largest radii, we obtain the desired approximation ∗ bound. In this case, if t = k, then F O is an optimal solution. Otherwise, t ≥ 1 , so we have R∗ ≤ Ot ≤ O∗ P and OPT ≤ O∗ − tp=1 rp . Combining Theorem 2.5 and Lemma 2.1 then yields the theorem. Improved approximation ratio. The improved approximation ratio comes from a better way of combining F1 and F2 in step B2. The idea is that we can ensure that the dual solutions α1 and α2 are componentwise quite close to each other by setting δz in the binary-search procedure to be sufficently small. Thus, we may essentially assume that if T1,I , T2,I denote the tight pairs yielding F1 , F2 respectively, then every pair in T1,I intersects some pair in T2,I , because we can augment T2,I to include non-intersecting pairs of T1,I . This yields dividends when we combine solutions as in step B2, because we can now ensure that if π(i0 , r0 ) = (i, r), then the pairs of T2,I and T1,I yielding (i, r) and (i0 , r0 ) respectively intersect, which yields an improved bound on ci,i0 . This yields an improved approximation of 3.83 for the combination step (Lemma A.4), and hence for the entire algorithm (Theorem 2.7); we defer the details to Appendix A. Theorem 2.7. For any > 0, our algorithm returns a feasible LBkSR-solution of cost at most (3.83 + O())O∗ in time nO(1/) . 2.2 Approximation algorithm for LBkSRO We now build upon the ideas in Section 2.1 to devise an O(1)-approximation algorithm for the outlier version LBkSR. The high-level approach is similar to the one in Section 2.1. We again “guess” the t (i, r) pairs F O corresponding to the facilities with largest radii in an optimal solution, and consider the modified k-BS-instance (D0 , L0 , k 0 , m) (where D0 , L0 , k 0 are defined as before). We design a primal-dual algorithm for the Lagrangian relaxation of the k-BS-problem where we are allowed to pick any number of pairs from 9 L0 (leaving at most m uncovered clients) incurring a fixed cost of z for each pair picked, utilize this to obtain two solutions F1 and F2 , and combine these to extract a low-cost solution. However, the presence of outliers introduces various difficulties both in the primal-dual algorithm and in the combination step. We consider the following LP-relaxation of the k-BS-problem and its dual (analogous to (P1 ) and (D1 )). X X max αj − k 0 · z − m · γ (D2 ) min r · yi,r (P2 ) j∈D0 (i,r)∈L0 s.t. X yi,r + wj ≥ 1 ∀j ∈ D0 s.t. αj − z ≤ r ∀(i, r) ∈ L0 (3) j∈B(i,r)∩D0 (i,r)∈L0 :j∈B(i,r) X X yi,r ≤ k 0 , (i,r)∈L0 X αj ≤ γ wj ≤ m ∀j ∈ D0 α, z, γ ≥ 0. j∈D0 y, w ≥ 0. As before, if (P2 ) is infeasible, we reject this guess; so we assume (P2 ) is feasible in the remainder of this section. Let OPT denote the optimal value of (P2 ). The natural modification of the earlier primal-dual algorithm PDAlg is to now stop the dual-ascent process when the number of active clients is at most m and set γ = maxj∈D0 αj . This introduces the significant complication that one may not be able to pay for the (r + z)-cost of non-intersecting tight pairs selected in the pruning phase by the dual objective value P j∈D0 αj − m · γ, since clients with αj = γ may be needed to pay for both the r + z-cost of the last tight pair f = (if , rf ) but their contribution gets canceled by the −m · γ term. This issue affects us in various guises. First, we no longer obtain an LMP-approximation for the unconstrained problem since we have to account for the (r + z)-cost of f separately. Second, unlike Claim 2.4, given solutions F1 and F2 obtained via binary search for z1 , z2 ≈ z1 respectively with \|F2 \| ≤ k 0 < \|F1 \|, we now only obtain a fractional k-BS-solution of cost O(OPT + z1 ). While one can modify the covering-knapsack-LP based procedure in step B2 of k-BSAlg to combine F1 , F2 , this only yields a good solution when z1 = O(OPT ). The chief technical difficulty is that z1 may however be much larger than OPT . Overcoming this obstacle requires various novel ideas and is the key technical contribution of our algorithm. We design a second combination procedure that is guaranteed to return a good solution when z1 = Ω(OPT ). This requires establishing certain structural properties for F1 and F2 , using which we argue that one can find a good solution in the neighborhood of F1 and F2 . We now detail the changes to the primal-dual algorithm and k-BSAlg in Section 2.1 and analyze them to prove Theorem 2.18, which states the performance guarantee we obtain for the modified k-BSAlg. As before, for the right guess of F O , combining this with Lemma 2.1 immediately yields the following result. Theorem 2.8. There exists a 12.365 + O() -approximation algorithm for LBkSRO that runs in time nO(1/) for any > 0. Modified primal-dual algorithm PDAlgo (D 0 , L0 , z). This is quite similar to PDAlg (and we again return pairs from L). We stop the dual-ascent process when there are at most m active clients. We set γ = maxj∈D0 αj . Let f = (if , rf ) be the last tight pair added to the tight-pair set T , and Bf = B(if , rf ). We sometimes abuse notation and S use (i, r) to also denote the singleton set {(i, r)}. For a set P of (i, r) pairs, define uncov(P ) := D0 \ (i,r)∈P B(i, r). Note that \|uncov(T \ f )\| > m ≥ \|uncov(T )\|. Let Out be a set of m clients such that uncov(T ) ⊆ Out ⊆ uncov(T \ f ). Note that αj = γ for all j ∈ Out. The pruning phase is similar to before, but we only use f if necessary. Let TI be a maximal subset of non-intersecting pairs picked by greedily scanning pairs in T \ f in non-increasing order of radius. For i ∈ µ(TI ), set rad(i) to be the unique r such that (i, r) ∈ TI , and let ri be the smallest radius ρ such that B(i, ρ) ⊇ B(i0 , r0 ) for every (i0 , r0 ) ∈ T \ f such that r0 ≤ rad(i) and (i0 , r0 ) intersects (i, rad(i)). Let F 0 = {(i, ri )}i∈µ(TI ) . If uncov(F 0 ) ≤ m, set F = F 0 . If uncov(F 0 ) > m and ∃i ∈ µ(F 0 ) such 10 that c(i, if ) ≤ 2R∗ , then increase ri so that B(i, ri ) ⊇ Bf and let F be this updated F 0 . Otherwise, set F = F ∪ f and rif = rad(if ) = rf . We return (F, f, Out, {rad(i)}i∈µ(F ) , α, γ). The proof of Theorem 2.9 dovetails the proof of Theorem 2.3. Theorem 2.9. Let (F, f, Out, {rad(i)}, α, γ) = PDAlgo (D0 , L0 , z). Then (i) uncov(F ) ≤ m, P (ii) cost(F \ f ) + 3\|F \ f \|z − 3R∗ ≤ 3( j∈D0 αj − mγ) ≤ 3(OPT + k 0 z), (iii) (i, rad(i)) i∈µ(F ) ⊆ L0 , is a set of non-intersecting pairs, and rad(i) ≤ ri ≤ 3R∗ ∀i ∈ µ(F ), (iv) if \|F \ f \| ≥ k 0 then cost(F ) ≤ 3 · OPT + 4R∗ , and if \|F \ f \| > k 0 then z ≤ OPT . Proof. We first prove parts (i)–(iii). Let F 0 = {(i, ri0 )}i∈µ(TI ) be the set of pairs obtained from the set TI in the pruning phase. By the same argument as in the proof of Theorem 2.3, we have ri0 ≤ 3rad(i) ≤ 3R∗ for all i ∈ µ(TI ), and uncov(F 0 ) ⊆ uncov(T \ f ). If we return F = F 0 , then \|uncov(F )\| ≤ m by definition. If uncov(F 0 ) > m and we increase the radius of some i ∈ µ(F 0 ) with c(i, if ) ≤ 2R∗ , then we have ri ≤ max{ri0 , 3R∗ } ≤ 3R∗ and uncov(F ) ⊆ uncov(T ), so \|uncov(F )\| ≤ m. If f ∈ F , then we again have uncov(F ) ⊆ uncov(T ). This proves part (i). P The above argument shows that cost(F \ f ) ≤ i∈µ(TI ) 3 · rad(i) + 3R∗ . All pairs in TI are tight and non-intersecting and \|F \ f \| = \|TI \|. Also, Out ⊆ uncov(T \ f ) ⊆ uncov(TI ). (Recall that \|Out\| = m and αj = γ for all j ∈ Out.) So X X cost(F \ f ) + 3\|F \ f \|z − 3R∗ ≤ (3 · rad(i) + 3z) = 3αj i∈µ(TI ) i∈µ(TI ) j∈B(i,rad(i))∩D0 X X X ≤3 αj − αj = 3 αj − mγ ≤ 3(OPT + k 0 z). (4) j∈D0 j∈Out j∈D0 The last inequality follows since (α, γ, z) is a feasible solution to (D2 ). This proves part (ii). Notice that (i, rad(i)) i∈µ(F ) is TI if f ∈ / F , and TI + f otherwise. In the latter case, we know that ∗ c(i, if ) > 2R for all i ∈ µ(TI ), so f does not intersect (i, rad(i)) for any i ∈ µ(TI ). Thus, all pairs in (i, rad(i)) i∈µ(F ) are non-intersecting. The claim that rad(i) ≤ ri for all i ∈ µ(F ) follows from exactly the same argument as that in the proof of Theorem 2.3. Part (iv) follows from part (ii) and (4). The bound on cost(F P ) follows from part (ii) since that cost(F ) ≤ cost(F \ f ) + R∗ . Inequality (4) implies that \|F \ f \|z ≤ i∈µ(TI ) (rad(i) + z) ≤ OPT + k 0 z, and so z ≤ OPT if \|F \ f \| > k 0 . Modified algorithm k-BSAlgo (D 0 , L0 , k0 , ). As before, we use binary search to find solutions F1 , F2 and extract a low-cost solution from these. The only changes to step B1 are as follows. We start with z1 = 0 and z2 = 2nk 0 cmax ; for this z2 , we argue below that PDAlgo returns at most k 0 pairs. We stop o 0 when z2 − z1 ≤ δz := OPT 3n2n . We do not stop even if PDAlg returns a solution (F, . . .) with \|F \| = k for 2 some z = z1 +z 2 , since Theorem 2.9 is not strong enough to bound cost(F ) even when this happens!. If 0 \|F \| > k , we update z1 ← z and the F1 -solution; otherwise, we update z2 ← z and the F2 -solution. Thus, we maintain that k1 = \|F1 \| > k 0 , and k2 = \|F2 \| ≤ k 0 . Claim 2.10. When z = z2 = 2nk 0 cmax , PDAlgo returns at most k 0 pairs. Proof. Let (F, f, out, {rad(i)}i∈µ(F ) , α, γ) be the output of PDAlgo for this z. Let T be the sight of tight pairs after the dual-ascent process. Observe that γ ≥ 2k 0 cmax , since for any tight pair (i, r) ∈ T , we have 11 P P that nγ ≥ j∈B(i,r)∩D0 αj ≥ z. We have j∈D0 αj − mγ ≤ OPT + k 0 z ≤ k 0 cmax + k 0 z. On the other hand, since uncov(T \ f ) \ out 6= ∅ and αj = γ for all j ∈ uncov(T \ f ), we also have the lower bound X X αj − mγ ≥ αj + γ ≥ \|F \ f \|z + γ. j∈D0 i∈µ(F \f ) j∈B(i,rad(i))∩D0 So if \|F \| > k 0 , we arrive at the contradiction that γ ≤ k 0 cmax . The main change is in the way solutions F1 , F2 are combined. We adapt step B2 to handle outliers (procedure A in Section 2.2.1), but the key extra ingredient is that we devise an alternate combination procedure B (Section 2.2.2) that returns a low-cost solution when z1 = Ω(OPT ). We return the better of the solutions output by the two procedures. We summarize these changes at the end in Algorithm k-BSAlgo (D0 , L0 , k 0 , ) and state the approximation bound for k-BSAlgo (Theorem 2.18). Combining this with Lemma 2.1 (for the right selection of t (i, r) pairs) immediately yields Theorem 2.8. We require the following continuity lemma, which is essentially Lemma 6.6 in [12]; we include a proof in Appendix B for completeness. Lemma 2.11. Let (Fp , . . . , αp , γ p ) = PDAlgo (D0 , L0 , zp ) for p = 1, 2, where 0 ≤ z2 − z1 ≤ δz . Then, 1 2 n 1 2 n 0 kα P j − αj k∞ ≤p2 δz and \|γ −n γ \| ≤ 2 δz . Thus, if (3) is tight for some (i, r) ∈ L in one execution, then j∈B(i,r)∩D0 αj ≥ r + z1 − 2 δz for p = 1, 2. 2.2.1 Combination subroutine A (F1 , rad1 ), (F2 , rad2 ) As in step B2, we cluster the F1 -pairs around F2 -pairs in stars. However, unlike before, some (i0 , r0 ) ∈ F1 may remain unclustered and and we may not pick (i0 , r0 ) or some pair close to it. Since we do not cover all clients covered by F1 , we need to cover a suitable number of clients from uncov(F1 ). We again setup an LP to obtain a suitable collection of pairs. Let ucp denote uncov(Fp ) and Dp := D0 \ ucp for p = 1, 2. Let π : F1 → F2 ∪ {∅} be defined as follows: for each (i0 , r0 ) ∈ F1 , if (i0 , r0 ) ∈ F1 intersects some F2 -pair, pick such an intersecting (i, r) ∈ F2 and set π(i0 , r0 ) = (i, r); otherwise, set π(i0 , r0 ) = ∅. In the latter case, (i0 , r0 ) is unclustered, and B(i0 , r0 ) ⊆ uc2 . Define Si,r = π −1 (i, r) for all (i, r) ∈ F2 . Let Q = π −1 (∅). Let {uc1 (i, r)}(i,r)∈F2 be a partition of uc1 ∩ D2 such that uc1 (i, r) ⊆ uc1 ∩ B(i, r) for all (i, r) ∈ F2 . Similarly, let {uc2 (i0 , r0 )}(i0 ,r0 )∈F1 be a partition of uc2 ∩ D1 such that uc2 (i0 , r0 ) ⊆ uc2 ∩ B(i0 , r0 ) for all (i0 , r0 ) ∈ F1 . We consider the following 2-dimensional covering knapsack LP. X X P P min xi,r (2r + (i0 ,r0 )∈Si,r 2r0 ) + (1 − xi,r ) (i0 ,r0 )∈Si,r r0 + qi0 ,r0 · r0 (2C-P) (i0 ,r0 )∈Q (i,r)∈F2 X s.t. xi,r + \|Si,r \|(1 − xi,r ) + (i,r)∈F2 (1 − xi,r )\|uc1 (i, r)\| + qi0 ,r0 ≤ k (5) (i0 ,r0 )∈Q (i,r)∈F2 X X X (1 − qi0 ,r0 )\|uc2 (i0 , r0 )\| ≤ m − \|uc1 ∩ uc2 \| (6) (i0 ,r0 )∈Q 0 ≤ xi,r ≤ 1 ∀(i, r) ∈ F2 , 0 ≤ qi0 ,r0 ≤ 1 ∀(i0 , r0 ) ∈ Q. The interpretation of the variable xi,r is similar to before. If xi,r = 0, or xi,r = 1, Si,r 6= ∅, we proceed as in step B2 (i.e., select all pairs in Si,r , or pick some (i0 , r0 ) ∈ Si,r and expand its radius suitably). But if xi,r = 1, Si,r = ∅, then we may also pick (i, r) (see Theorem 2.14). Variable qi0 ,r0 indicates if we pick (i0 , r0 ) ∈ F1 . The number of uncovered clients in such a solution is at most \|uc1 ∩ uc2 \| + (LHS of (6)), and (6) enforces that this is at most m. 12 Let (x∗ , q ∗ ) be an extreme-point optimal solution to (2C-P). The number of fractional components in is at most the number of tight constraints from (5), (6). We exploit this to round (x∗ , q ∗ ) to an integer solution (x̃, q̃) of good objective value (Lemma 2.13), and then use (x̃, q̃) to extract a good set of pairs as sketched above (Theorem 2.14). Recall that k1 = \|F1 \|, k2 = \|F2 \|. Let a, b ≥ 0 be such that ak1 + bk2 = k 0 , a + b = 1. Let C1 = cost(F1 ) and C2 = cost(F2 ). (x∗ , q ∗ ) Lemma 2.12. The following hold. (i) aC1 + bC2 ≤ (3 + )OPT + 4R∗ + 3z1 , (ii) OPT2C-P ≤ 2bC2 + (1 + b)C1 . Proof. Part (i) follows easily from part (ii) of Theorem 2.9 and since cost(Fp ) ≤ cost(Fp \ fp ) + R∗ for p = 1, 2. So we have C1 + 3(k1 − 1)z1 ≤ 3(OPT + k 0 z1 ) + 4R∗ and C2 + 3(k2 − 1)z2 ≤ 3(OPT + k 0 z2 ) + 4R∗ . Combining these, we obtain aC1 + bC2 ≤ 3OPT + 3k 0 (az1 + bz2 ) − 3(ak1 z1 + bk2 z2 ) + 3(az1 + bz2 ) + 4R∗ ≤ 3(OPT + k 0 z2 ) − 3k 0 z2 + 3ak1 δz + 3z1 + 3bδz + 4R∗ ≤ (3 + )OPT + 4R∗ + 3z1 . The second inequality follows since 0 ≤ z2 − z1 ≤ δz . For part (ii), we claim that setting xi,r = b for all (i, r) ∈ F2 , and qi0 ,r0 = a for all (i0 , r0 ) ∈ Q yields 0 a feasible solution to (2C-P). The P LHS of (5) evaluates to ak1 + bk2 , which is exactly k . The first term on the LHS of (6) evaluates to a (i,r)∈F2 \|uc1 (i, r)\| = a\|uc1 ∩ D2 \| = a\|uc1 \ uc2 \| since {uc1 (i, r)}(i,r)∈F2 is a partition of uc1 ∩ D2 . Similarly, the second term on the LHS of (6) evaluates to at most b\|uc2 ∩ D1 \| = b\|uc2 \ uc1 \|. So we have (LHS of (6)) + \|uc1 ∩ uc2 \| = a\|uc1 \| + b\|uc2 \| ≤ m since \|uc1 \|, \|uc2 \| ≤ m. The objective value of this solution is 2bC2 + 2bC1 + (1 − b)C1 = 2bC2 + (1 + b)C1 . Let P = {(i, r) ∈ F2 : Si,r = ∅}. Lemma 2.13. (x∗ , q ∗ ) can be rounded to a feasible integer solution (x̃, q̃) to (2C-P) of objective value at most OPT2C-P + O(R∗ ). Proof. Let S be the set of fractional components of (x∗ , q ∗ ). As noted earlier, \|S\| is at most the number of tight constraints from (5), (6). Let X X l∗ := x∗i,r + \|Si,r \|(1 − x∗i,r ) + qi∗0 ,r0 (i0 ,r0 )∈S∩Q (i,r)∈S∩F2 denote the contribution of the fractional components of (x∗ , q ∗ ) to the LHS of (5). Note that if (5) is tight, then l∗ must be an integer. For a vector v = (vj )j∈I where I is some index-set, let dve denote dvj e j∈I . We round (x∗ , q ∗ ) as follows. • If l∗ ≥ 2 or \|S\| ≤ 1 or \|S ∩ (F2 \ P)\| ≥ 1, set (x̃, q̃) = d(x∗ , q ∗ )e. • Otherwise, we set x̃i,r = x∗i,r , q̃i0 ,r0 = qi∗0 ,r0 for all the integer-valued coordinates. We set the fractional component with larger absolute coefficient value on the LHS of (6) equal to 1 and the other fractional component to 0. 13 We prove that (x̃, q̃) is a feasible solution to (2C-P). Note that (6) holds for (x̃, q̃) since we always have X X (1 − q̃i0 ,r0 )\|uc2 (i0 , r0 )\| (1 − x̃i,r )\|uc1 (i, r)\| + (i0 ,r0 )∈Q (i,r)∈F2 ≤ X (1 − x∗i,r )\|uc1 (i, r)\| + X (1 − qi∗0 ,r0 )\|uc2 (i0 , r0 )\|. (i0 ,r0 )∈Q (i,r)∈F2 Clearly, the contribution to the LHS of (5) from the components not in S is the same in both (x̃, q̃) and (x∗ , q ∗ ). Let l denote the contribution from (x̃, q̃) to the LHS of (5) from the components in S. Clearly, l is an integer. If l∗ ≥ 2, then l = 2. If \|S\| ≤ 1, then l = 1. If l∗ ≥ 1, then in these cases the LHS of (5) evaluated at (x̃, q̃) is at most the LHS of (5) evaluated at (x∗ , q ∗ ). If l∗ < 1 and \|S\| ≤ 1 (so l = 1), then since l∗ is fractional, we know that (5) is not tight for (x∗ , q ∗ ). So despite the increase in LHS of (5), we have that (5) holds for (x̃, q̃). If \|S\| = 2 and \|S ∩ (F2 \ P)\| ≥ 1, then we actually have l∗ > 1 and l = 2. Again, since l∗ is fractional, we can conclude that (x̃, q̃) satisfies (5) despite the increase in LHS of (5). Finally, suppose l∗ < 2, \|S\| P P = 2, and S ∩ (F2 \ P) = ∅. Then the contribution from S to the LHS of (5) is x + (i,r)∈S∩F2 i,r (i0 ,r0 )∈S∩Q qi0 ,r0 , and at most one of the components in S is set to 1 in (x̃, q̃). So l = 1, ∗ ∗ and either l ≤ l or l < 1, and in both cases (5) holds for (x̃, q̃). To bound the objective value of (x̃, q̃), notice that compared to (x∗ , q ∗ ), the solution (x̃, q̃) pays extra only for the components that are rounded up. There are at most two such components, and their objectivefunction coefficients are bounded by 15R∗ , so the objective value of (x̃, q̃) is at most OPT2C-P + 30R∗ . Theorem 2.14. The integer solution (x̃, q̃) returnedby Lemma 2.13 yields a solution F, {rad(i)} to i∈µ(F ) ∗ the k-BS-problem with cost(F ) ≤ 6.1821 + O() (OPT + z1 ) + O(R ) where (i, rad(i)) i∈µ(F ) ⊆ L0 is a set of non-intersecting pairs. Proof. Unlike in step B2 of k-BSAlg, we will not simply pick a subset of pairs of F1 and expand their radii. We will sometimes need to pick pairs from F2 in order to ensure that we have at most m outliers, but we need to be careful in doing so because we also need to find suitable radii for the facilities we pick so that we obtain non-intersecting pairs. We first construct F 00 as follows. If q̃i0 ,r0 = 1, we include (i0 , r0 ) ∈ F 00 and set rad(i0 ) = rad1 (i0 ). If x̃i,r = 0, we include all pairs in Si,r in F 00 and set rad(i0 ) = rad1 (i0 ) for all (i0 , r0 ) ∈ Si,r . If x̃i,r = 1 and Si,r 6= ∅, we pick a pair in (i0 , r0 ) ∈ Si,r , and include (i0 , 2r + r0 + max(i00 ,r00 )∈Si,r \{(i0 ,r0 )} 2r00 ) in F 00 . We set rad(i0 ) = rad1 (i0 ). Now we initialize F 0 = F 00 and consider all (i, r) ∈ P with x̃i,r = 1. If (i, r) does not intersect any (i0 , r0 ) ∈ F 00 then we add (i, r) to F 0 , and set rad(i) = rad2 (i). Otherwise, if (i, r) intersects some (i0 , r0 ) ∈ F 00 , then we replace (i0 , r0 ) ∈ F 0 with (i0 , r0 + 2r). We have thus ensured that (i, rad(i)) i∈µ(F 0 ) ⊆ L0 and consists of non-intersecting pairs. Note that in all the cases above, the total cost of the pairs we include when we process some q̃i0 ,r0 or x̃i,r term is at most the total contribution to the objective function from the q̃i0 ,r0 term, or the x̃i,r and 1 − x̃i,r terms. Therefore, cost(F 0 ) is at most the objective value of (x̃, q̃). Finally, we argue that \|uncov(F 0 )\| ≤ m. We have \|uncov(F 0 )\| ≤ \|uc1 ∩ uc2 \| + \|uncov(F 0 ) ∩ D1 \| + \|uncov(F 0 ) ∩ D2 ∩ uc1 \|. Observe that for every client j ∈ uncov(F 0 ) ∩ D1 and every 0 , r0 ) (i0 , r0 ) ∈ F1 such that j ∈ B(i0 , r0 ), it must be that (i0 , r0 ) ∈ Q and q̃i0 ,r0 = P0. It follows that j ∈ uc2 (i 0 0 0 0 for some (i , r ) ∈ Q with q̃i0 ,r0 = 0. Therefore, \|uncov(F ) ∩ D1 \| ≤ (i0 ,r0 )∈Q (1 − q̃i0 ,r0 )\|uc2 (i , r0 )\|. Similarly, for every j ∈ uncov(F 0 ) ∩ D2 ∩ uc1 and every (i, r) ∈ F2 such that j ∈ B(i, r), we must have (i, r) ∈ P and x̃i,r P = 0; hence, j ∈ uc1 (i, r) for some (i, r) ∈ P with x̃i,r = 0. Therefore, 0 \|uncov(F ) ∩ D2 ∩ uc1 \| ≤ (i,r)∈P (1 − x̃i,r )\|uc1 (i, r)\|. Thus, since (x̃, q̃) is feasible, constraint (6) implies that \|uncov(F 0 )\| ≤ m. We return (F2 , rad2 ) if cost(F2 ) ≤ cost(F 0 ), and F 0 , {rad(i)}i∈µ(F 0 ) otherwise. Combining the above bound on cost(F 0 ) with part (ii) of Lemma 2.12 and Lemma 2.13, we obtain that the cost of the 14 solution returned is at most min C2 , 2bC2 + (1 + b)C1 + 30R∗ ≤ 2.0607 aC1 + bC2 + 30R∗ ≤ 2.0607 (3 + )OPT + 4R∗ + 3z1 + 30R∗ ≤ (6.1821 + 3)(OPT + z1 ) + 39R∗ . The first inequality follows from Claim 2.6, and the second follows from part (i) of Lemma 2.12. 2.2.2 Subroutine B (F1 , f1 , Out 1 , rad1 , α1 , γ 1 ), (F2 , f2 , Out 2 , rad2 , α2 , γ 2 ) Subroutine A in the previous section yields a low-cost solution only if z1 = O(OPT ). We complement subroutine A by now describing a procedure that returns a good solution when z1 is large. We assume in this section that z1 > (1 + )OPT . Then \|F1 \ f1 \| ≤ k 0 (otherwise z1 ≤ OPT by part (iv) of Theorem 2.9), so \|F1 \ f1 \| ≤ k 0 < \|F1 \|, which means that k1 = k 0 + 1 and f1 ∈ F1 . Hence, αj1 = γ 1 for all j ∈ Bf1 ∩ D0 . First, we take care of some simple cases. If there exists (i, r) ∈ F1 \f1 such that \|uncov F1 \{f1 , (i, r)}∪ (i, r + 12R∗ ) \| ≤ m, then set F = F1 \ {f1 , (i, r)} ∪ (i, r + 12R∗ ). We have cost(F ) = cost(F1 \ f1 ) + 12R∗ ≤ 3 · OPT + 15R∗ (by part (ii) of Theorem 2.9). If there exist pairs (i, r), (i0 , r0 ) ∈ F1 such that c(i, i0 ) ≤ 12R∗ , take r00 to be the minimum ρ ≥ r such that B(i0 , r0 ) ⊆ B(i, ρ) and set F = ∗ ∗ F1 \ {(i, r), (i0 , r0 )} ∪ (i, r00 ). We have cost(F ) ≤ cost(F1 \ f1 ) + 13R ≤ 3 · OPT + 16R . In both cases, we return F, {rad1 (i)}i∈µ(F ) . So we assume in the sequel that neither of the above apply. In particular, all pairs in F1 are wellP separated. Let AT = {(i, r) ∈ L0 : j∈B(i,r)∩D0 αj1 ≥ r + z1 − 2n δz } and AD = {j ∈ D0 : αj1 ≥ γ 1 − 2n δz }. By Lemma 2.11, AT includes the tight pairs of PDAlgo (D0 , L0 , zp ) for both p = 1, 2, and Out 1 ∪ Out 2 ⊆ AD. Since the tight pairs T2 used for building solution F2 are almost tight in (α1 , γ 1 , z1 ), we swap them in and swap out pairs from F1 one by one while maintaining a feasible solution. Either at some point, we will be able to remove f , which will give us a solution of size k 0 , or we will obtain a bound on cost(F2 ). The following lemma is our main tool for bounding the cost of the solution returned. Lemma 2.15. Let F ⊆ L0 , and let TF = (i, ri0 ) i∈µ(F ) where ri0 ≤ r for each (i, r) ∈ F . Suppose S TF ⊆ AT and pairs in TF are non-intersecting. If \|F \| ≥ k 0 and \|AD \ (i,r)∈F B(i, r))\| ≥ m then cost(TF ) ≤ (1 + )OPT . Moreover, if \|F \| > k 0 then z1 ≤ (1 + )OPT . S Proof. Let Out F be a subset of exactly m of clients from AD \ (i,r)∈F B(i, r). Since the pairs in TF are P P non-intersecting and almost tight, i∈µ(F ) (ri0 + z1 ) ≤ j∈D0 \Out F (αj1 + 2n δz ), so X i∈µ(F ) (ri0 +z1 ) ≤ X j∈D0 (αj1 +2n δz )−m(γ 1 −2n δz ) ≤ X αj1 −mγ 1 +(m+\|D0 \|)2n δz ≤ (1+)OPT +k 0 z1 j∈D0 where the last inequality follows since (α1 , γ 1 , z1 ) is a feasible solution to (D2 ). So cost(TF ) ≤ (1+)OPT if \|TF \| = \|F \| ≥ k 0 , and z1 ≤ (1 + )OPT if \|F \| > k 0 . Define a mapping ψ : F2 → F1 \ f1 as follows. Note that any (i, r) ∈ F2 may intersect with at most one F1 -pair: if it intersects (i0 , r0 ), (i00 , r00 ) ∈ F1 , then we have c(i0 , i00 ) ≤ 12R∗ . First, for each (i, r) ∈ F2 that intersects with some (i0 , r0 ) ∈ F1 , we set ψ(i, r) = (i0 , r0 ). Let M ⊆ F2 be the F2 -pairs mapped by ψ this way. For every (i, r) ∈ F2 \ M , we arbitrarily match (i, r) with a distinct (i0 , r0 ) ∈ F1 \ ψ(M ). We claim that ψ is in fact a one-one function. Lemma 2.16. Every (i, r) ∈ F1 \ f1 intersects with at most one F2 -pair. 15 Proof. Suppose two pairs (i1 , r1 ), (i2 , r2 ) ∈ F2 intersect with a common pair (i, r) ∈ F1 \ f1 . Let T1,I be the tight pairs corresponding to F1 \ f1 obtained from (the pruning phase of) PDAlgo (D0 , L0 , z1 ). Let (i, rad1 (i)) ∈ T1,I be the tight pair corresponding to (i, r). Let (i1 , rad2 (i1 )), (i2 , rad2 (i2 )) be the tight pairs corresponding to (i1 , r1 ), (i2 , r2 ) obtained from PDAlgo (D0 , L0 , z2 ). Let F 00 = F1 \ {f1 , (i, r)} ∪ (i, r + 12R∗ ). We show that either z1 ≤ OPT or \|uncov(F 00 )\| ≤ m, both of which lead to a contradiction. Define F 0 = F1 \ {f1 , (i, r)} ∪ {(i1 , r1 ), (i2 , r2 )}, so \|F 0 \| = k + 1. Consider the set TF 0 = T1,I \ {(i, rad1 (i))} ∪ {(i1 , rad2 (i1 )), (i2 , rad2 (i2 ))}. Since (i1 , rad2 (i1 ) and (i2 , rad2 (i2 )) are non-intersecting and they do not intersect with any pair in TS1,I \ (i, rad1 (i)), the pairs in TF 0 are non-intersecting. Also, TF 0 ⊆ AT . If \|AD ∩ uncov(F 0 )\| = \|AD \ (i0 ,r0 )∈F 0 B(i0 , r0 )\| ≥ m, then z1 ≤ OPT by Lemma 2.15. Otherwise, note that every client in B(i1 , r1 ) ∪ B(i2 , r2 ) is at distance at most r + 2 max{r1 , r2 } ≤ r + 6R∗ from i. So we have uncov(F 00 ) ⊆ uncov(F ) ∪ Bf1 ⊆ AD and uncov(F 00 ) ⊆ uncov(F 0 ). So \|uncov(F 00 )\| ≤ \|AD ∩ uncov(F 0 )\| ≤ m. Let F20 be the pairs (i, r) ∈ F2 such that if (i0 , r0 ) = ψ(i, r), then r0 < r. Let P = F20 ∩ M and Q = F20 \ M . For every (i0 , r0 ) ∈ ψ(Q) and j ∈ B(i0 , r0 ), we have j ∈ uncov(F2 ) ⊆ AD (else (i0 , r0 ) would lie in ψ(M )). Starting with F = F1 \ f1 , we iterate over (i, r) ∈ F20 and do the following. Let (i0 , r0 ) = ψ(i, r). If (i, r) ∈ P , we update F ← F \ (i0 , r0 ) ∪ (i, r + 2r0 ) (so B(i, r + 2r0 ) ⊇ B(i0 , r0 )), else we update F ← F \ (i0 , r0 ) ∪ (i, r). Let TF = {(i, rad1 (i))}(i,r)∈F ∩F1 ∪ {(i, rad2 (i))}(i,r)∈F \F1 . Note that \|F \| = k 0 and uncov(F ) ⊆ AD at all times. Also, since (i, r) intersects only (i0 , r0 ), which we remove when (i, r) is added, we maintain that TF is a collection of non-intersecting pairs and a subset of AT ⊆ L0 . This process continues until \|uncov(F )\| ≤ m, or when all pairs of F20 are swapped in. In the former case, we argue that cost(F ) is small and return F, {rad1 (i)}(i,r)∈F ∩F1 ∪ {rad2 (i)}(i,r)∈F \F1 . In the latter case, we show that cost(F20 ), and hence cost(F2 ) is small, and return (F2 , rad2 ). Lemma 2.17. (i) If the algorithm stops with \|uncov(F )\| ≤ m, then cost(F ) ≤ (9 + 3)OPT + 18R∗ . (ii) If case (i) does not apply, then cost(F2 ) ≤ (3 + 3)OPT + 9R∗ . (iii) The pairs corresponding to the radii returned are non-intersecting and form a subset of L0 . Proof. Part (iii) follows readily from the algorithm description and the discussion above. Consider part (i). Let (i, r) ∈ F20 be the last pair scanned by the algorithm before it terminates, and (i0 , r0 ) = ψ(i, r). Let F 0 be the set F before the last iteration. So F 0 = F \ (i, r + 2r0 ) ∪ (i0 , r0 ) if (i, r) ∈ P , and F 0 = F \ (i, r) ∪ (i0 , r0 ) if (i, r) ∈ Q. Note that r + 2r0 ≤ 9R∗ . Since uncov(F 0 ) ⊆ AD and \|uncov(F 0 )\| > m, by Lemma 2.15, we have cost(TF 0 ) ≤ (1 + )OPT . For all (i, r) ∈ F1 , we have r ≤ 3rad1 (i) (since f1 ∈ F1 ). For all but at most one (i, r) ∈ F2 , we have r ≤ 3rad2 (i) and for the one possible exception, we have r ≤ 3R∗ . Therefore, cost(F ) ≤ cost(F 0 ∩ F1 ) + cost(F 0 \ F1 ) + 9R∗ ≤ 3 · cost(TF 0 ) + 3R∗ + 2 · cost(F1 \ F 0 ) + 9R∗ ≤ 3(1 + )OPT + 3R∗ + 2(3 · OPT + 3R∗ ) + 9R∗ = 9 + 3)OPT + 18R∗ . P The second inequality above follows since cost(F 0 ∩ F1 ) ≤ (i,r)∈F 0 ∩F1 3rad1 (i) and cost(F 0 \ F1 ) ≤ P ∗ 0 (i,r)∈F 0 \F1 3rad2 (i) + 3R + 2cost(F1 \ F ). Forpart (ii), Lemma 2.15 shows that cost(TF ) ≤ (1 + )OPT , and so cost(F20 ) + cost F1 \ (f1 ∪ ψ(F20 )) ≤ 3 · cost(TF ) + 3R∗ . Now cost(F2 ) = cost(F20 ) + cost(F2 \ F20 ) ≤ cost(F20 ) + cost ψ(F2 \ F20 ) = cost(F20 ) + cost F1 \ (f1 ∪ ψ(F20 ) ≤ 3(1 + ) · OPT + 3R∗ where the first inequality follows by the definition of F20 . 16 Algorithm k-BSAlgo (D 0 , L0 , k0 , ). Output: F ⊆ L with \|F \| ≤ k 0 , a radius rad(i) for all i ∈ µ(F ). C1. Binary search. Let (F1 , rad1 , . . .) = PDAlgo (D0 , L0 , 0). If \|F1 \| ≤ k 0 pairs, return (F1 , rad1 ). Else perform binary-search in the range [0, ncmax ] to find z1 , z2 with 0 ≤ z2 − z1 ≤ δz = OPT 3n2n such that letting (Fp , fp , Out p , radp , αp , γ p ) = PDAlg(D0 , L0 , zp ) for p = 1, 2, we have \|F2 \| ≤ k 0 < \|F1 \|. C2. Let FA , {radA (i)}i∈µ(FA ) = A (F1 , rad1 ), (F2 , rad2 ) (Section 2.2.1). If \|F1 \ f1 \| > k 0 , return (FA , radA ). C3. If ∃(i, r) ∈ F1 \f1 such that \|uncov F1 \{f1 , (i, r)}∪(i, r +12R∗ ) \| ≤ m, then set F = F1 \{f1 , (i, r)}∪(i, r + 0 0 12R∗ ). If ∃(i, r), (i0 , r0 ) ∈ F1 such that c(i, i0 ) ≤ 12R∗ , let r00 be the minimum ρ ≥ r such that B(i ,r ) ⊆ 0 0 00 B(i, ρ); set F = F1 \ {(i, r), (i , r )} ∪ (i, r ). If either of the above apply, return F, {rad1 (i)}i∈µ(F ) . C4. Let FB , {radB (i)}i∈µ(FB ) be the output of subroutine B (Section 2.2.2). C5. If cost(FA ) ≤ cost(FB ), return (FA , radA ), else return (FB , radB ). Theorem 2.18. k-BSAlgo (D0 , L0 , k 0 ) returns a solution (F, rad) with cost(F ) ≤ 12.365 + O() · OPT + O(R∗ ) where (i, rad(i)) i∈µ(F ) ⊆ L0 comprises non-intersecting pairs. Proof. This follows essentially from Theorem 2.14 and Lemma 2.17. When z1 ≤ (1 + ) · OPT , Theorem 2.14 yields the above bound on cost(FA ). Otherwise, if none of the cases in step C3 apply, then Lemma 2.17 bounds cost(FB ). In the boundary cases, when we terminate in step C1 or C3, we have cost(F ) ≤ cost(F1 \ f1 ) + cost(f1 ) + 12R∗ , which is at most the expression in the theorem due to part (ii) of Theorem 2.9. 3 Minimizing the maximum radius with lower bounds and outliers The lower-bounded k-supplier with outliers (LBkSupO) problem is the min max-radius version of LBkSRO. The input and the set of feasible solutions are the same as in LBkSRO: the input is an instance I = F, D, {Li }, {c(i, j)}, k 0 , m , and a feasible solution is S ⊆ F, σ : D 7→ S ∪ {out} with \|S\| ≤ k, \|σ −1 (i)\| ≥ Li for all i ∈ S, and \|σ −1 (out)\| ≤ m. The cost of (S, σ) is now maxi∈S maxj∈σ−1 (i) c(i, j). The special case where m = 0 is called the lower-bounded k-supplier (LBkSup) problem, and the setting where D = F is often called the k-center version. Let τ ∗ denote the optimal value; note that there are only polynomially many choices for τ ∗ . As is common in the study of min-max problems, we reduce the problem to a “graphical” instance, where given some value τ , we try to find a solution of cost O(τ ) or deduce that τ ∗ > τ . We construct a bipartite unweighted graph Gτ = Vτ = D ∪ Fτ , Eτ ), where Fτ = {i ∈ F : \|B(i, τ )\| ≥ Li }, and Eτ = {ij : c(i, j) ≤ τ, i ∈ Fτ , j ∈ D}. Let dist τ (i, j) denote the shortest-path distance in Gτ between i and j, so c(i, j) ≤ distτ (i, j) · τ . We say that an assignment σ : D 7→ Fτ ∪ {out} is a distance-α assignment if dist τ (j, σ(j)) ≤ α for every client j with σ(j) 6= out. We call such an assignment feasible, if it yields a feasible LBkSupO-solution, and we say that Gτ is feasible if it admits a feasible distance-1 assignment. It is not hard to see that given F ⊆ Fτ , the problem of finding a feasible distance-α-assignment σ : D 7→ F ∪ {out} in Gτ (if one exists) can be solved by creating a network-flow instance with lower bounds and capacities. Observe that an optimal solution yields a feasible distance-1 assignment in Gτ ∗ . We devise an algorithm that for every τ , either finds a feasible distance-α assignment in Gτ for some constant α, or detects that Gτ is not feasible. This immediately yields an α-approximation algorithm since the smallest τ for which the algorithm returns a feasible LBkSupO-solution must be at most τ ∗ . We obtain Theorems 3.1 and 3.2 via this template. Theorem 3.1. There is a 3-approximation algorithm for LBkSup. Theorem 3.2. There is a 5-approximation algorithm for LBkSupO. 17 We complement our approximation results via a simple hardness result (Theorem 3.3) showing that our approximation factor for LBkSup is tight. We also show that LBkSupO is equivalent to the k-center version (i.e., where F = D) of the problem (Appendix C); a similar equivalence is known to hold for the capacitated versions of k-supplier and k-center with outliers [15]. Theorem 3.3. It is NP-hard to approximate LBkSup within a factor better than 3, unless P = N P . Proof. The result is shown via a reduction from set cover problem. Suppose we have a set cover instance 0 with set U = [n] of elements and collection S = ∪np=1 {Sp } of subsets of U, and we want to know if there exists k subsets of U in S that cover all elements of U. Let j1 , j2 , · · · , jn represent the elements and i1 , i2 , · · · , in0 represent subsets of U in S. Construct an LBkSup instance I with client set D = ∪np=1 {jp }, 0 facility set F = ∪nq=1 {iq }, define c(jp , iq ) for jp ∈ D, iq ∈ F to be 1 if p ∈ Sq , 3 otherwise, and let Li = 1 for each i ∈ F. Suppose there exists a collection F of k subsets in S that cover all elements. First, remove any set i in F , if i does not cover an element that is not covered by F \ i. Let σ : D → F be defined for element j to be some set in F that covers j. Since each set i in F covers at least one element that is not covered by F \ i, \|σ −1 (i)\| ≥ 1, so (F, σ) is a feasible solution to I with radius 1. If no collection of k subsets of U in S covers all elements, then there does not exist k facilities in F that all elements are at distance at most 1 from them, so optimal solution of I has cost at least 3. Therefore, it is NP-hard to approximate LBkSup with a factor better than 3 as otherwise the algorithm can be used to answer the decision problem. Finding a distance-3 assignment for LBkSup. Consider the graph Gτ ∗ . Note that there exists an optimal center among the neighbors of each client in G. Moreover, two clients at distance at least 3 are served by two distinct centers. These insights motivate the following algorithm. Let N (v) denote the neighbors of vertex v in the given graph Gτ . Find a maximal subset Γ of clients with distance at least 3 from each other. If \|Γ\| > k or there exists a client j with N (j) = ∅, then return Gτ is not feasible. For each j ∈ Γ, let ij denote the center S in N (j) with minimum lower bound. If there exists a feasible distance-3 assignment σ of clients to F = j∈Γ {ij }, return σ, otherwise return Gτ is not feasible. The following lemma yields Theorem 3.1. Lemma 3.4. The above algorithm finds a feasible distance-3 assignment in Gτ if Gτ is feasible. Proof. Let σ ∗ : D 7→ F ∗ be a feasible distance-1 assignment in Gτ . So F ∗ ⊆ Fτ and every client has a non-empty neighbor set. Since each client in Γ has to be served by a distinct center in F ∗ , \|Γ\| ≤ \|F ∗ \| ≤ k. For each client j ∈ Γ, let i∗j = σ ∗ (j). Note that i∗j ∈ N (j), so Lij ≤ Li∗j by the choice of ij , and every client in σ ∗−1 (i∗j ) is at distance at most 3 from ij . We show that there is a feasible distance-3 assignment σ : D 7→ F . For each j ∈ Γ, we assign all clients in σ ∗−1 (i∗j ) to ij . As argued above this satisfies the lower bound of ij . For any unassigned client j, let j 0 ∈ Γ be a client at distance at most 2 from j (which must exist by maximality of Γ). We assign j to ij 0 . Finding a distance-5 assignment for LBkSupO. The main idea here is to find a set F ⊆ Fτ of at most k centers that are close to the centers in F ∗ ⊆ Fτ for some feasible distance-1 assignment σ ∗ : D 7→ F ∗ ∪ {out} in Gτ . The non-outlier clients of (F ∗ , σ ∗ ) are close to F , so there are at least \|D\| − m clients close to F . If centers in F do not share a neighbor in Gτ , then clients in N (i) can be assigned to i for each i ∈ F to satisfy the lower bounds. We cannot check if F satisfies the above properties, but using an idea similar to that in [15], we will find a sequence of facility sets such that at least one of these sets will have the desired properties when Gτ is feasible. 18 Definition 3.5. Given the bipartite graph Gτ , a set F ⊆ F is called a skeleton if it satisfies the following properties. (a) (Separation property) For i, i0 ∈ F , i 6= i0 , we have dist τ (i, i0 ) ≥ 6; (b) There exists a feasible distance-1 assignment σ ∗ : D 7→ F ∗ ∪ {out} in Gτ such that • (Covering property) For all i∗ ∈ F ∗ , dist τ (i∗ , F ) ≤ 4, where dist τ (i∗ , F ) = mini∈F dist τ (i∗ , i). • (Injection property) There exists f : F 7→ F ∗ such that dist τ (i, f (i)) ≤ 2 for all i ∈ F . If F satisfies the separation and injection properties, it is called a pre-skeleton. Note that if F ⊆ Fτ is a skeleton or pre-skeleton, then Gτ is feasible. Suppose F ⊆ Fτ is a skeleton and satisfies the properties with respect to a feasible distance-1 assignment (F ∗ , σ ∗ ). The separation property ensures that the neighbor sets of any two locations i, i0 ∈ F are disjoint. The covering property ensures that F ∗ is at distance at most 4 from F , so there are at least \|D\| − m clients at distance at most 5 from F . Finally, the injection and separation properties together ensure that \|F \| ≤ k since no two locations in F can be mapped to the same location in F ∗ . Thus, if F is a skeleton, then we can obtain a feasible distance-5 assignment σ : D 7→ F ∪ {out}. Lemma 3.6. Let F be a pre-skeleton in Gτ . Define U = {i ∈ Fτ : dist τ (i, F ) ≥ 6} and let i = arg maxi0 ∈U \|N (i0 )\|. Then, either F is a skeleton, or F ∪ {i} is a pre-skeleton. Proof. Suppose F is not a skeleton and F ∪ {i} is not a pre-skeleton. Let σ ∗ : D 7→ F ∗ ∪ {out} be a feasible distance-1 assignment in Gτ such F satisfies the injection property with respect to (F ∗ , σ ∗ ). Let f : F 7→ F ∗ be the mapping given by the injection property. Since F ∪ {i} is not a pre-skeleton and dist τ (i, F ) ≥ 6, this implies that dist τ (i, F ∗ ) > 2, and hence, dist τ (i, F ∗ ) ≥ 4 as Gτ is bipartite. This means that all clients in N (i) are outliers in (F ∗ , σ ∗ ). Moreover, since F is not a skeleton, there exists a center i∗ ∈ F ∗ with dist τ (i∗ , F ) > 4, and so dist(i∗ , F ) ≥ 6. Therefore, i∗ ∈ U . By the choice of i, we know that \|N (i)\| ≥ \|N (i∗ )\|. Now consider F 0 = F ∗ \{i∗ }∪{i}, and define σ 0 : D 7→ F 0 ∪{out} as follows: σ 0 (j) = σ ∗ (j) for all j ∈ / N (i) ∪ N (i∗ ), σ 0 (j) = i for all j ∈ N (i), and σ 0 (j) = out for all j ∈ N (i∗ ). Note that the F covers as many clients as F ∗ , and so σ 0 : D 7→ F 0 ∪ {out} is another feasible distance-1 assignment. But this yields a contradiction since F ∪ {i} now satisfies the injection property with respect to (F 0 , σ 0 ) as certified by the function f 0 : F → F 0 defined by f 0 (s) = f (s) for s ∈ F , f 0 (i) = i. If Gτ is feasible, then ∅ is a pre-skeleton. A skeleton can have size at most k. 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A Improved Approximation Ratio for LBkSR We now describe in detail the changes to algorithm k-BSAlg and its analysis leading to Theorem 2.7. First, we set δz = OPT 3n2n in the binary-search procedure (step B1); note that the binary search still takes polynomial time. By Lemma 2.11 (specialized to the non-outlier setting), we have kα1 − α2 k∞ ≤ 2n δz , which implies 21 that every (i, r) ∈ T1 ∪ T2 is almost tight with respect to (αp , zp ) for p = 1, 2. To obtain the improved guarantee, we construct the mapping π : F1 7→ F2 , and hence, our stars, based on whether pairs (i0 , rad1 (i0 )) and (i, rad2 (i)) intersect for i0 ∈ µ(F1 ), i ∈ µ(F2 ). To ensure that every (i0 , r0 ) ∈ F1 belongs to some star, we first modify F2 and T2,I by including non-intersecting pairs from T1,I (which are almost tight in (α2 , z2 )). We consider pairs in F1 in arbitrary order. For each (i, r) ∈ F1 , if (i, rad1 (i)) does not intersect any pair in T2,I , we add (i, rad1 (i)) to T2,I , add (i, r) to F2 , and set rad2 (i) = rad1 (i). We continue this process until all pairs in F1 are scanned or \|F2 \| = k 0 . Lemma A.1. If \|F2 \| = k 0 after the above process, then F2 is a feasible k-BS solution with cost(F2 ) ≤ (3 + )OPT , and T2,I ⊆ L0 is a set of non-intersecting pairs. Proof. All clients in D0 are covered by balls corresponding to the F2 -pairs since this holds even before any 0 pairs P are added to F2 . It is clear Using Lemma 2.11, we P that T12,I ⊆nL and consists P of non-intersecting pairs. have (î,r̂)∈T2,I (r̂ + z1 ) ≤ j∈D0 αj + 2 δz \|T2,I \|, so (î,r̂)∈T2,I r̂ ≤ 1 + 3 OPT . For every (i, r) ∈ F2 P we have r ≤ 3rad2 (i), so cost(F2 ) ≤ (î,r̂)∈T2,I 3r̂ ≤ (3 + )OPT . So if \|F2 \| = k 0 after the above preprocessing, we simply return (F2 , rad2 ). Otherwise, we combine solutions F1 and F2 using an LP similar to (C-P). We construct a map π : F1 → F2 similar to before, but with the small modification that we set π(i0 , r0 ) = (i, r) only if (i0 , rad1 (i0 )) intersects with (i, rad2 (i)). Due to our preprocessing, π is well-defined. As before, let star Si,r = π −1 (i, r) for each (i, r) ∈ F2 . Figure 2: Old combination method. Si,r = {(i1 , r1 ), (i2 , r2 ), (i3 , r3 )} Figure 3: New combination method. Si,r = {(i1 , r1 ), (i2 , r2 ), (i3 , r3 )} The LP again has an indicator variable xi,r . If xi,r = 0,Pwe select all pairs in Si,r . Otherwise, if Si,r 6= ∅, we select a pair (i0 , r0 ) ∈ Si,r and include i0 , 2rad2 (i) + (i00 ,r00 )∈Si,r 4rad1 (i00 ) in our solution; note that S the corresponding ball covers all clients in (i00 ,r00 )∈Si,r B(i00 , r00 ). So we consider the following LP. min X xi,r 2rad2 (i) + P (i0 ,r0 )∈S P 0 0 4rad (i ) + (1 − x ) 3rad (i ) 0 0 1 i,r 1 (i ,r )∈Si,r i,r (i,r)∈F2 s.t. X xi,r + \|Si,r \|(1 − xi,r ) ≤ k, (i,r)∈F2 22 0 ≤ xi,r ≤ 1 ∀(i, r) ∈ F2 . (C-P’) Let x∗ be an extreme point of (C-P’). Let F 0 be the pairs obtained by picking the pairs corresponding to dx∗ e as described above. Since x∗ has at most one fractional component, it follows as before that \|F 0 \| ≤ k 0 . 0 As before, we return F , {rad(i)}µ(i)∈F 0 or P (F2 , {rad2 (i)}), whichever has lower cost. P 0 = 0 Let C10 = rad (i) and C 1 2 (i,r)∈F1 (i0 ,r0 )∈F2 rad2 (i ). The following claims are analogous to Claims 2.4 and 2.6. Claim A.2. We have aC10 + bC20 ≤ 1 + 3 OPT . Proof. Using Lemma 2.11, we have aC10 + bC20 = a X (i,r)∈F1 rad1 (i) + b X rad2 (i) ≤ a · X j∈D0 (i,r)∈F2 X αj1 − k1 z1 + b · k2 2n δz + αj1 − k2 z2 j∈D0 · OPT 3 j∈D0 X = αj1 − k 0 · z1 + · OPT ≤ 1 + OPT . 3 3 0 ≤ X (aαj1 + bαj1 ) − (ak1 + bk2 ) · z1 + j∈D Claim A.3. min{3C20 , 2bC20 + (3 + b)C10 } ≤ such that a + b = 1. 3(b+3) (aC10 3b2 −2b+3 + bC20 ) ≤ 3.83(aC10 + bC20 ) for all a, b ≥ 0 Proof. Since the minimum is less than any convex combination, min(3C20 , 2bC20 + bC10 + 3C10 ) ≤ = = 3b2 + b −3b + 3 (3C20 ) + 2 (2bC20 + bC10 + 3C10 ) 3b2 − 2b + 3 3b − 2b + 3 3b(b + 3) 3(1 − b)(b + 3) 0 (C1 ) + 2 C0 3b2 − 2b + 3 3b − 2b + 3 2 3(b + 3) ((1 − b)C10 + bC20 ). 3b2 − 2b + 3 Since a = 1 − b, the first inequality in the claim follows. The expression 3b3(b+3) 2 −2b+3 is maximized at √ √ 3 b = −3 + 2 3, and has value 8 (5 + 3 3) ≈ 3.8235, which yields the second inequality in the claim. Lemma A.4. The cost of the solution (F, {rad(i)}) returned by the above combination subroutine is at most (3.83 + O())OP T + O(R∗ ) where {(i, rad(i))}i∈µ(F ) ⊆ L0 is a set of non-intersecting pairs. Proof. First note that {rad(i)} correspond to {rad2 (i)} if F = F2 and {rad(i)} ⊆ {rad1 (i)} if F = F 0 , so in both cases it consists of non-intersecting pairs from L0 . The cost of the pair included in F 0 corresponding to a fractional component of x∗ is at most 7R∗ as each radp (i) is bounded by R∗ for p ∈ {1, 2}. Since x∗ has at most one fractional component, cost(F 0 ) ≤ OPT C-P’ + 7R∗ . Also, OPT C-P’ ≤ 2bC20 + (4b + 3a)C10 = 2bC20 + (3 + b)C10 , since setting xi,r = b for all (i, r) ∈ F2 yields a feasible solution to (C-P’) of this cost. Therefore, cost(F ) ≤ min{3C20 , 2bC20 + (b + 3)C10 + 7R∗ }, which is at most 3.83(aC10 + bC20 ) + 7R∗ by Claim A.3. Combining this with Claim A.2 yields the bound in the lemma. Proof of Theorem 2.7. It suffices to show that when the selection F O = {(i1 , r1 ), . . . (it , rt )} in step A1 corresponds to the t facilities in an optimal solution with largest radii, we obtain the desired approximation ∗ bound. In this case, if t = k, then F O is an optimal solution; otherwise, we have R∗ ≤ Ot ≤ O∗ and P OPT ≤ O∗ − tp=1 rp . Combining Lemma A.4 and Lemma 2.1 then yields the theorem. 23 B Proof of Lemma 2.11 We abbreviate PDAlgo (D0 , L0 , z) to PDAlgo (z). We use x− to denote a quantity infinitesimally smaller than x. Consider the dual-ascent phase of PDAlgo for z1 and z2 . First, suppose that m = 0. Sort clients with respect to their αj0 = min(αj1 , αj2 ) value. Let this ordering be α10 ≤ α20 ≤ · · · ≤ αn0 . We prove by induction that \|αj1 − αj2 \| ≤ 2j−1 δz . For the base case, assume without loss of generality that αj0 = αj1 , and let (i, r) be the tight pair that 0 caused j to become inactive in PDAlgo (z1 ). Consider time point t = Pα1 in the two executions. By definition − o all clients are active at time t in PDAlg (z2 ). So the contribution j∈B(i,r)∩D0 αj of clients to the LHS of (3) at time t− is at least as much as their contribution in PDAlgo (z1 ) at time t− . Therefore, we can increase α1 by at most δz beyond time t in PDAlgo (z2 ) as z2 − z1 = δz . Suppose we have shown that for all clients j = 1, 2, · · · , ` − 1 (where ` ≥ 2), Now consider client ` and let (i, r) be the tight pair that makes ` inactive at time α`0 in PDAlgo (zp ), where p ∈ {1, 2}. Consider time point t = α`0 in both executions. By definition, all clients j > ` are still active at time t− in both o executions PDAlgo (z1 ) and PPDAlg (z2 ). (They might become inactive at time t but can not become inactive earlier.) The contribution j∈B(i,r)∩D0 αj of clients to the LHS of (3) in the execution other than p at time P j−1 δ . The values of z in the t− is at least their contribution in PDAlgo (zp ) at time t− minus `−1 z j=1 2 two executions differs by at most δ , so in the execution other than p, α can grow beyond t by at most z ` P`−1 j−1 2 )δz ≤ 2` δz . (1 + j=1 P Now if we consider a tight pair (i, r) in one of the execution, the value of RHS and LHS of j∈B(i,r) αj ≤ P r + z for the other execution can differ by at most (1 + nj=1 2j−1 )δz ≤ 2n δz . Now consider the case where m > 0. Note that in this case, we can assume that we have the execution for m = 0, pick the first time at which there are at most m active clients, i.e., time γ in PDAlgo , and set αj = γ for every active client at this time point. Let γ 0 = min(γ 1 , γ 2 ), suppose γ 0 = γp , where p ∈ {1, 2}. Note that by time γ 0 + 2n δz , all pairs that are tight in the p-th execution by time γ 0 are also tight in the other execution. So the number of active clients after this time point is at most m. Therefore \|γ 1 −γ 2 \| ≤ 2n δz . C Equivalence of lower-bounded k-supplier with outliers and lower-bounded k-center with outliers Let LBkCentO denote the special case of LBkSupO where F = D. In this section, we show that if there exists an α-approximation for LBkCentO, then there exists an α-approximation for LBkSupO. Let I = (k, F, D, c, L, m) be an instance of LBkSupO with N = \|F\| + 1 and \|D\| = n. Define an instance I 0 = (k 0 , D0 , c0 , L0 , m0 ) as follows: let k 0 = k and D0 = (D × {1, 2, · · · , N }) ∪ F. Let c0 ((j, p), i) = c(j, i) for each j ∈ D, p ∈ [N ], i ∈ F, and let c0 be the metric completion of these distances (i.e., c0 (q, q 0 ) is the shortest-path distance between q and q 0 with respect to these distances for q, q 0 ∈ D0 ). Define L0i = N Li for i ∈ F and L0(j,p) = N (n + 1), and let m0 = N · m + (N − 1). Clearly I 0 can be constructed from I in polynomial time. The lower-bounds for (j, p), j ∈ D, p ∈ [N ] are set so that L0(j,p) < \|D0 \|, so (j, p) cannot be opened as a center in any feasible solution to I 0 . Let OP T (I 0 ) denote the value of optimal solution of I 0 and OP T (I) denote the value of optimal solution of I. We claim that OP T (I 0 ) ≤ OP T (I). Let (F ∗ , σ ∗ ) denote an optimal solution of I. Let solution (F̂ , σ̂) for I be constructed as follows: let F̂ = F ∗ , for each p ∈ [N ], define σ(q) = i for q = (j, p) if σ ∗ (j) = i, and σ(q) = out otherwise. Note that since there are at most m outliers in solution (F ∗ , σ ∗ ) then there are at most N m + \|F\| = N m + (N − 1) outliers in (F̂ , σ̂). Clearly the radius of the opened centers is the same as before, so OP T (I 0 ) ≤ OP T (I). Now suppose there exists an α-approximation algorithm A for LBkCentO problem. Use A to generate 24 a solution (F̂ , σ̂) for I 0 with maximum radius R. As noted above, we have F̂ ⊆ F. We construct a solution (F̂ , σ 0 ) for I of maximum radius at most R using Algorithm 1. Algorithm 1 Constructing a feasible assignment σ 0 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: Construct network N = (V, E) where V = {s, t} ∪ D ∪ F̂ and E = {si : i ∈ F̂ } ∪ {ij : i ∈ F̂ , j ∈ D, c(i, j) ≤ r} ∪ {jt : j ∈ D}. Set lij = 0, uij = ∞ for each ij ∈ E, i ∈ F̂ , j ∈ D. Set lsi = Li , usi = ∞ for each si ∈ E, i ∈ F̂ . Set ljt = 0, ujt = 1 for each jt ∈ E, j ∈ D. Let f ← max-flow(N ) respecting lower-bounds (l) and upper-bounds (u) on edges. if value of f is ≥ n − m then, set σ 0 (j) = i if fjt = 1 and fij = 1 for i ∈ F̂ . set σ 0 (j) = out if fjt = 0. return f . return σ 0 = ∅. Lemma C.1. Solution (F̂ , σ 0 ) is a feasible solution to I with maximum radius at most R, where σ 0 is the output of Algorithm 1. P Proof. Consider any set S ⊆ F̂ . There are at least Pi∈S N Li clients in D0 assigned to S. Since there are P i∈S N Li −(N −1) at most N − 1 facilities among D0 , there are at least > i∈S Li − 1 clients at distance at N P most R from S. So there are P at least i∈S Li clients in neighbor set of S in N . It follows that every s-t cut in N has capacity at least i∈F̂ Li , so there exists a flow f that satisfies the lower-bounds and upper-bounds on the edges. It remains to show that value of f is at least \|D\| − m. If there is an incoming edge to a client in N , then a flow of 1 can be sent through j. So we want to bound the number of clients with no incoming edge in N . If any copy of client j is served by some facility in the solution (F̂ , σ̂) then j is at distance at most R from some facility in F̂ . Since there are at most N m + (N − 1) outliers in (F̂ , σ̂), there are at most N m+(N −1) < m + 1 clients with no incoming edge in N . N Since algorithm A is an α-approximation algorithm, wehave R ≤ α · OP T (I 0 ) ≤ αOP T (I). 25	8
Quintic algebras over Dedekind domains and their resolvents arXiv:1511.03162v1 [math.NT] 10 Nov 2015 Evan O’Dorney November 11, 2015 This is an addendum to [4], which classified quadratic, cubic, and quartic rings over a Dedekind domain. 1 A coordinate-free description of resolvents Let Q be a quintic ring over a Dedekind domain R, and let L = Q/R. Our first task is to generalize the notion of a sextic resolvent, developed by Bhargava in [2] in the case R = Z. Following the approach of [6] and the present author’s senior thesis, we expect the resolvent to consist of a rank-5 lattice M (to be thought of as S/R, where S is a sextic ring) with two linear maps relating certain multilinear expressions in L and M . The orientation map θ—which relates the top exterior powers of L and M —is easy to guess. The discriminant of an R-algebra T naturally lies in (Λtop (T ))⊗−2 . Just as the equality Disc Q = Disc C between the discriminants of a quartic ring and its cubic resolvent(s) suggests an identification of the top exterior powers of the two rings, so the relation Disc S = (16 Disc Q)3 (Bhargava’s (33) of [2]) linking the discriminants of a quintic ring and its sextic resolvent(s) suggests an isomorphism θ : Λ5 M →(Λ4 L)⊗3 . The second piece of data—that which contains the 40 integers that actually parametrize resolvents over Z—is slightly trickier to work out. Bhargava presents it as a map φ from L to Λ2 M (equivalently, from Λ2 M ∗ to L∗ ), but this does not have the correct properties in our situation. The correct construction, foreshadowed somewhat by the mysterious constant factor in Bhargava’s fundamental resolvent ((28) in [2]), is to take a map φ : Λ4 L ⊗ L → Λ2 M. Finally, we must find the fundamental relations that link φ and θ to the ring structure. Just as Lemma 9 of [1] provided the inspiration for Bhargava’s coordinate-free description of resolvents of a quartic ring ([1], section 3.9), so we begin at Lemma 4(a), which, after eliminating the references to S5 -closure, states that 1 φ(y) φ(x) φ(y) φ(x) Pfaff − Pfaff = 1 ∧ y ∧ x ∧ z ∧ yz. φ(x) φ(z) φ(x) −φ(z) 2 The Pfaffians are to be interpreted by writing φ(x), etc., as a 5 × 5 skew-symmetric matrix with regard to any convenient basis (i.e. viewing it as a skew bilinear form on Λ2 M , once a generator of Λ4 L is fixed. Then we paste together four of these to make a 10 × 10 skew-symmetric matrix and take the Pfaffian. This is a clever way to manufacture certain degree-5 integer polynomials in the 40 coefficients of φ. To re-express them in a way that is coordinate-free (and applicable in characteristic 2), we consider two preliminary multilinear constructions. 1 1.1 The quadratic map µ 7→ µ Let V be a 5-dimensional vector space over a field K (which we will soon take to be Frac R). We examine the constructions that can be made starting with elements of Λ2 V . We have a bilinear map ∧ : Λ2 V × Λ2 V → Λ4 V . However, the most fundamental map from Λ2 V to Λ4 V is not the bilinear map ∧ but the quadratic map from which it arises. It is defined by ! n X X vi ∧ wi = vi ∧ wi ∧ vj ∧ wj . (1) i=1 1≤i<j≤n It is not hard to prove that this is well defined. Note that if char K 6= 2, then µ can be described more simply by 1 µ = µ ∧ µ. 2 Moreover, the bilinear map ∧ can always be recovered from • via µ ∧ ν = (µ + ν) − µ − ν . 1.2 (2) The contraction µ(α, β) The second construction takes one element µ ∈ Λ2 V and two elements α, β ∈ Λ4 V and outputs an element of a suitable one-dimensional vector space as follows. First, the perfect pairing ∧ : Λ4 V × V → Λ5 V allows us to identify α and β as elements of Λ5 V ⊗ V ∗ . These have a wedge product α ∧ β ∈ Λ2 (Λ5 V ⊗ V ∗ ) ∼ = (Λ5 V )⊗2 ⊗ Λ2 V ∗ . We now use the duality between Λ2 V ∗ and Λ2 V , described explicitly by (f ∧ g)(v ∧ w) = f v · gw − f w · gv, to obtain an element µ(α, β) ∈ (Λ5 V )⊗2 . 1.3 The definition We are now ready to state the definition of a sextic resolvent. Definition 1.1. Let Q be a quintic ring over a Dedekind domain R, and let L = Q/R. A resolvent for Q consists of a rank-5 lattice M and a pair of linear maps φ : Λ4 L ⊗ L → Λ2 M and θ : Λ5 M →(Λ4 L)⊗3 satisfying the identity θ⊗2 [φ(λ1 x)(φ(λ2 y) , φ(λ3 z) ] = λ1 λ22 λ23 (x ∧ y ∧ z ∧ yz) (3) where x, y, z ∈ L and λi ∈ Λ4 L are formal variables. The resolvent is called numerical if θ is an isomorphism. Note that the expression within square brackets lies in (Λ5 M )⊗2 ; applying θ⊗2 , one ends up in (Λ4 L)⊗6 which is where the right-hand side also resides. It should also be remarked that the product yz is carried out in Q; translating the lifts ỹ, z̃ by constants in R simply changes the product ỹ z̃ by multiples of ỹ, z̃, and 1, thereby not changing the product y ∧ z ∧ yz. 2 2 Resolvent to ring Our first task is to show that the resolvent maps φ and θ uniquely encode the multiplication data of the ring Q. Theorem 2.1. Let L and M be lattices over R of ranks 4 and 5 respectively, and let φ : Λ4 L ⊗ L → Λ2 M and θ : Λ5 M →(Λ4 L)⊗3 be maps. There is a quintic ring Q with a quotient map Q/R ∼ = L, unique up to isomorphism, such that (M, φ, θ) is a resolvent of Q. Proof. Let (e1 , e2 , e3 , e4 ) be a basis for L, by which we mean that there is a decomposition L = a1 e1 ⊕ · · · ⊕ a4 e4 for some fractional ideals ai of R. To place a ring structure on the module −1 Q = L ⊕ R, it is then necessary to choose the coefficient ckij ∈ ak a−1 i aj of ek in the product ei ej . We allow k = 0, with the conventions e0 = 1 and a0 = R. On the other hand, allowing i = 0 or j = 0 gives no useful information. Hence the ring structure is given by the 40 coefficients ckij , 1 ≤ i ≤ j ≤ 4, 0 ≤ k ≤ 5. Some of these coefficients are immediately determined by the resolvent. For instance, if {i, j, k, ℓ} is a permutation of {1, 2, 3, 4}, and ǫ = ±1 its sign, then −6 ⊗2 ckij = −ǫe−1 [φ(eℓ etop )(φ (ei etop ), φ (ej etop )], top · eℓ ∧ ei ∧ ej ∧ ei ej = −ǫetop · θ (4) where etop = e1 ∧ e2 ∧ e3 ∧ e4 = ǫ · ei ∧ ej ∧ ek ∧ eℓ is the natural generator of Λ4 L. This determines the values of all ckij where i, j, and k are nonzero and distinct. Likewise, the following expressions are determined, for i, j, k distinct: j cjii = ǫe−1 top · eℓ ∧ ei ∧ (ei + ek ) ∧ ei (ei + ek ) − cik j k ckik − cjij = ǫe−1 top · eℓ ∧ ei ∧ (ej + ek ) ∧ ei (ej + ek ) − cik + cij ciii − cjij − ckik = ǫe−1 top · eℓ ∧ (ei + ek ) ∧ (ei + ej ) ∧ (ei + ej )(ei + ek ) (5) − cijk + cjik + ckij + cjii + ckii + (cjkj − ciki ) + (ckjk − ciji ). The reader familiar with ring parametrizations will recognize the left-hand sides of (4) and (5) as the linear expressions in the ckij that are invariant under translations ei 7→ ei + ti (ti ∈ a−1 i ) of the ring basis elements. If we normalize our basis so that, say, c112 = c212 = c334 = c434 = 0, then all the ckij are now uniquely determined, except for the c0ij . The c0ij can be computed by comparing the coefficients of k in (ei ej )ek and ei (ej ek ) for any k 6= i, yielding formula (22) of [2]: 4 X c0ij = (crjk ckri − crij ckrk ). r=1 The theorem is now reduced to three verifications. −1 1. That all ckij belong to the correct ideals ak a−1 i aj . This is routine. 2. ThatP the c0ij are well defined, and more generally that the associative law holds on the ring Q = ai ei that we have just constructed. This is a family of integer polynomial identities in the 40 free coefficients of φ in the chosen basis; as such, it was proved in the course of Bhargava’s parametrization of quintic rings over Z. 3. That the original maps φ and θ indeed form a resolvent of Q, i.e. that the identity (3) holds. This can probably also be proved by appeal to results over Z, but here a direct proof is not difficult. We can assume that λ1 = λ2 = λ3 = etop and x is a basis element eℓ , since the equation (3) is linear in those variables. We can also assume that each of y and z is a basis element or a sum of two different basis elements, since (3) is quadratic in those variables. Now we have a finite set of cases, some of which are the relations (4) and (5), and the rest of which will be reduced to them using the following properties of the underlying multilinear operations: 3 Lemma 2.2. Let V be a 5-dimensional vector space, and let µ, ν, ξ ∈ Λ2 V and α ∈ Λ4 V . Then (a) µ(µ ∧ ν, α) = ν(µ , α) (b) µ(µ , α) = 0 (c) ν(µ , µ ∧ ξ) = −ξ(µ , µ ∧ ν). Proof. Calculation, although only (a) need be checked directly, as (b) follows by setting µ = ν and (c) by the derivation ν(µ , µ ∧ ξ) = µ(µ ∧ ν, µ ∧ ξ) = −µ(µ ∧ ξ, µ ∧ ν) = −ξ(µ , µ ∧ ν). Now we return to proving θ⊗2 [φ(etop x)(φ(etop y) , φ(etop z) ] = e5top (eℓ ∧ y ∧ z ∧ yz) (6) for x = eℓ and y, z ∈ {ei }i ∪ {ei + ej }i<j . The cases where eℓ does not appear in y or z are all subsumed by the definitions (4) and (5), with one exception: the expression for cjii is not visibly symmetric under switching k and ℓ. This can be seen by writing cjii = ǫe−1 top (eℓ ∧ ei ∧ (ei + ek ) ∧ ei (ei + ek ) − eℓ ∧ ei ∧ ek ∧ ei ek ) = ǫe−5 top φ(eℓ )(φ (ei ), φ (ei + ek )) − φ(eℓ )(φ (ei ), φ (ek )) = ǫe−5 top φ(eℓ )(φ (ei ), φ (ei ) + φ(ei ) ∧ φ(ek ) + φ (ek )) − φ(eℓ )(φ (ei ), φ (ek )) = ǫe−5 top φ(eℓ )(φ (ei ), φ(ei ) ∧ φ(ek )) and using Lemma 2.2(c). It remains to dispose of the cases where eℓ does appear in y or z. The key is to use Lemma 2.2(a) to reduce the case (x, x + y, z) of (6) to the cases (x, y, z) and (y, x, z). The details are left to the reader. Remark. Over Z, assuming that θ is an isomorphism, the resolvent devolves into the basis representation of φ. This has 40 independent entries which can be arranged into a quadruple of 5 × 5 skew-symmetric matrices, representing the values φ(x) (as x runs through a basis) as skew bilinear forms on M ∗ . The coefficients ckij of the ring we have constructed are certain degree-5 polynomials in these 40 entries which are easily identified with the formulas given in (21) of [2]. Thus our definition of resolvent is compatible with Bhargava’s (Definition 10), which justifies our invocation of his computations in our situation, despite the dissimilarities of the definitions. 2.1 The sextic ring It ought to be remarked that, given any resolvent (L, M, φ, θ), the rank-6 lattice M ⊕ R also picks up a canonical ring structure, whose structure coefficients dkij are integer polynomials in the coefficients of φ of degree 12 (for k 6= 0) and 24 (for k = 0). As the construction given by Bhargava in [2], Section 6 works without change over a Dedekind domain, we will not discuss it further. 3 3.1 Constructing resolvents Resolvents over a field Let K be a field. We will first investigate what sort of family of resolvents a quintic K-algebra has. In the quartic case, it was the trivial ring T = K[x, y, z]/(x, y, z)2 that had a large family, all 4 other rings having a unique resolvent. Here, if a ring has multiple resolvents, it is not necessarily trivial, but as we will see, it is in a sense minimally far from being trivial. The appropriate definition is as follows: Definition 3.1. A quintic algebra Q over K is very degenerate if it has subspaces Q4 ⊆ Q3 , of dimension 4 and 3 respectively, such that Q4 Q3 = 0 (that is, the product of any element of Q4 and any element of Q3 is zero). This implies that Q has a multiplication table × 1 α β γ δ 1 1 α β γ δ α α u 0 0 0 β β 0 0 0 0 γ γ 0 0 0 0 δ δ 0 0 0 0 (7) in which 15 of the 16 non-forced entries are zero. We prove: Theorem 3.2. Every not very degenerate quintic K-algebra has a unique resolvent up to isomorphism. Proof. The first few steps are easy: let M be a K-vector space of dimension 5, and let θ : Λ5 M →(Λ4 L)⊗3 be any isomorphism. So far we have not made any choices. (The choice θ = 0 works only for the trivial ring.) We will first try to construct the map φ = φ(•) , a quadratic map from Λ4 L ⊗ L to Λ4 M . For this purpose we concoct a corollary of (3) that involves only φ . Lemma 3.3. Let V be a 5-dimensional vector space. Let µ ∈ Λ2 M and α, β, γ, δ ∈ Λ4 M . Then µ ∧ α ∧ β ∧ γ ∧ δ = µ(α, β)µ(γ, δ) + µ(α, γ)µ(δ, β) + µ(α, δ)µ(β, γ) in Λ5 (Λ4 V ) ∼ = (Λ5 V )⊗4 . Proof. Write the general µ as u ∧ v + w ∧ x (u, v, w, x ∈ V ) and expand. As a corollary, we get that if (M, φ, θ) is a resolvent of a quintic ring Q, then for all a, b, c, d, e ∈ Λ4 L ⊗ L, Motivated by this, we define for any quintic ring Q the pentaquadratic form F (a, b, c, d, e) = (a ∧ b ∧ c ∧ bc)(a ∧ d ∧ e ∧ de) + (a ∧ b ∧ d ∧ bd)(a ∧ e ∧ c ∧ ec) + (a ∧ b ∧ e ∧ be)(a ∧ c ∧ d ∧ cd) (8) from L5 to (Λ4 L)⊗2 , or equivalently from (Λ4 L ⊗ L)5 to (Λ4 L)⊗12 . We get that for any resolvent (M, φ, θ) of Q, θ⊗4 (φ (a) ∧ φ (b) ∧ φ (c) ∧ φ (d) ∧ φ (e)) = F (a, b, c, d, e). (9) We claim the following: Lemma 3.4. F is identically zero if and only if Q is very degenerate. Proof. We prove that the property of being very degenerate is invariant under base-changing to the algebraic closure K̄ of K; then the lemma can be proved by checking the finitely many quintic algebras over an algebraically closed field (see [3, 5]). Let Q̄ = Q ⊗K K̄ be the corresponding 5 K̄-algebra. Clearly if Q is very degenerate, so is Q̄, so assume that Q̄ is very degenerate. Then the subsets M = {x ∈ Q′ \| dim ker x ≥ 3} and N = {x ∈ Q′ \|M x = 0} are, by reference to the multiplication table (7), vector spaces with dim M = 4 and dim N ∈ {3, 4}. Moreover, because they are canonically defined, they are invariant under the Galois group Gal(K̄/K). This shows that M ∩ Q and N ∩ Q are K-vector spaces of the same dimensions with (M ∩ Q)(N ∩ Q) = 0, so Q is very degenerate. Picking a1 , . . . , a5 ∈ Λ4 L ⊗ L such that F (a1 , a2 , a3 , a4 , a5 ) = f0 6= 0, we get that the five vectors vi = φ (ai ) must form a basis such that θ⊗4 (v1 ∧ v2 ∧ v3 ∧ v4 ∧ v5 ) = f0 . Any such basis is as good as any other—they are all related by elements of SL(∧4 M ), which is canonically isomorphic to SL(M ) (although GL(∧4 M ) ∼ 6 GL(M ) in general). Once the vi are = fixed, there is at most one candidate for the map φ up to SL(M )-equivalence, namely φ (a) = 5 X F (a1 , . . . , âi , a, . . . , a5 ) F (a1 , a2 , a3 , a4 , a5 ) i=1 vi (10) Then the relations φ(x)(φ (ai ), φ (aj )) = x ∧ ai ∧ aj ∧ ai aj , for 1 ≤ i < j ≤ 5, determine the map φ uniquely. So the resolvent map φ, if it exists, must be given by a predetermined formula, or rather by any one of a finite number of such formulas, inasmuch as the ai in (10) can be chosen from the finite set {e1 , e2 , e3 , e4 , e1 + e2 , e1 + e3 , . . . , e3 + e4 } for any basis {e1 , e2 , e3 , e4 } of Λ4 L ⊗ L. It remains to prove that the (M, φ, θ) we have hereby constructed is actually a resolvent; this is a collection of integer polynomial identities, not in a family of free variables as in the previous lemma, but in the coefficients ckij of the given ring Q, which are restricted by the associative law. If Q has nonzero discriminant—the most common case—the theorem can be proved by base-changing to the algebraic closure K̄ and noting that K̄ ⊕5 , the unique nondegenerate quintic K̄-algebra, does have a resolvent (Example 4.1). The general case can be handled by a limiting argument, appealing to the known fact that all quintic K̄-algebras can be deformed to K̄ ⊕5 , at least in characteristic zero (see [3]). 3.2 From field to Dedekind domain Let Q be a quintic ring over a Dedekind domain R. We will assume that Q is not very degenerate and hence that the corresponding K-algebra QK = Q ⊗R K has a unique resolvent (MK , φ, θ). Resolvents of Q are now in bijection with lattices M in the vector space MK such that φ(Λ4 L ⊗ L) ⊆ Λ2 M (11) ⊗3 (12) 5 4 θ(Λ M ) ⊆ (Λ L) . For any resolvent M , note that we must have M∗ ∼ = Λ4 M ⊗ (M 5 )⊗−1 ⊇ φ (Λ4 L ⊗ L) ⊗ (θ((Λ4 L)⊗3 ))⊗−1 . Since Q is not very degenerate, the right-hand side is a lattice of full rank and we may take its dual, which we denote by M0 . Then any resolvent is contained in M0 . Condition (11) is vacuous for M = M0 , since φ(λx)(φ (λ′ y), φ (λ′′ z)) = θ⊗2 (λλ′ λ′′ (x ∧ y ∧ z ∧ yz)) ∈ (θ(Λ3 L))⊗2 6 for all λ′ , λ′′ ∈ Λ3 L and y, z ∈ L. On the other hand, condition (12) is generally not satisfied by M = M0 ; indeed, one readily finds that θ−1 ((Λ4 L)⊗3 ) ⊆ Λ5 M0 using (9), so if M0 is a resolvent, it is numerical. The classification of resolvents is now reduced to a local problem. Any M determines a family of resolvents (Mp , φ, θ) of the quintic algebras Qp over the DVR’s Rp ⊆ K, and conversely an arbitrary choice of resolvents Mp of the Rp can be glued together to form the resolvent T M = p Mp . The choice Mp = M0,p = M0 ⊗ Rp is forced for all but finitely many primes p, namely those dividing the ideal c = [Λ5 M0 : θ−1 ((Λ4 L)⊗3 )] = [(Λ4 L)⊗2 : hF (a, b, c, d, e) : a, b, c, d, e ∈ Li]. (13) In the lucky case that c is the unit ideal, M0 is the only resolvent. This occurs in one important instance: Theorem 3.5. If Q is a maximal quintic ring, that is, is not contained in any strictly larger quintic ring, then Q has a unique resolvent, which is numerical. Proof. Suppose that c were not the unit ideal, so there is a prime p such that p\|F (a, b, c, d, e) for all a, b, c, d, e ∈ L. We will prove that Q is not maximal at p. It is convenient to localize and to assume that R = Rp is a DVR with uniformizer π. Note that Q/pQ, a quintic algebra over R/p, has its associated pentaquadratic form F identically zero, so by Lemma 3.4, it is very degenerate. So Q has an R-basis (1, x, ǫ1 , ǫ2 , ǫ3 ) such that (x, ǫ1 , ǫ2 , ǫ3 )(ǫ1 , ǫ2 , ǫ3 ) ⊆ pR. We claim that the lattice Q′ with basis (1, x, π −1 ǫ1 , π −1 ǫ2 , π −1 ǫ3 either is a quintic ring or is contained in a quintic ring, showing that Q is not maximal. Set M = hπ, x, ǫ1 , ǫ2 , ǫ3 i and N = hπ, πx, ǫ1 , ǫ2 , ǫ3 i. Then Q ⊇ M ⊇ N ⊇ πQ and M N ⊆ πQ. Consider, for any i, j ∈ {1, 2, 3}, the multiplication maps Q/N ǫi h1, xi / N/πQ ǫj hǫ1 , ǫ2 , ǫ3 i ǫi / πQ/πN hπ, πxi / πN /π 2 Q hπǫ1 , πǫ2 , πǫ3 i . These are all linear maps of R/p-vector spaces. Denote by f the composition of the left two maps and by g the composition of the right two. Write f (1) = π(a + bx), where a, b ∈ R/p. Then g(ǫi ) = aπǫi , since xǫi ∈ πQ. Thus g is given in the bases above by the scalar matrix a. But g has rank at most 2, since it factors through the two-dimensional space πQ/πN ; hence a = 0. So N 2 ⊆ πM . Now consider the following multiplication maps: Q/M h1i ǫi / N/πQ hǫ1 , ǫ2 , ǫ3 i ǫj / πM /πN ǫk / π 2 Q/π 2 M π2 hπxi ǫi / π 2 N /π 3 Q π 2 ǫ1 , π 2 ǫ2 , π 2 ǫ3 . Similarly to the previous argument, the composition of the first three maps must be zero, or else the composition of the last three would be a nonzero scalar. Since the images of the first map (as i varies) span N/πQ, the composition of the middle two maps is always zero. There are two cases: (a) The second map is always zero, that is, N 2 ⊆ πN . This implies that π −1 N is a quintic ring, as desired. 7 (b) The third map is always zero, that is, M N ⊆ πM . We get that π −1 ǫi is integral over R (look at the characteristic polynomial of its action on M ), so R[π −1 ǫ1 , π −1 ǫ2 , π −1 ǫ3 ] is finitely generated and thus a quintic ring, as desired. Note that, in this proof, if the resolvent is not unique, then the extension Q′ ) Q has (R/p)3 ⊆ Q′ /Q. So the following stronger theorem holds: Theorem 3.6. If Q is a quintic ring such that the R/p-vector space of congruence classes in π −1 Q/Q whose elements are integral over R has dimension at most 2, for each prime p, then Q has a unique resolvent, which is numerical. 3.3 Bounds on the number of numerical resolvents Finally, we examine bounds on the number of numerical resolvents a not very degenerate quintic ring can have. A lower bound of 1 was proved over Z in [2], Theorem 12; the method is adaptable to our situation, and we do not attempt to sharpen the bound. Instead, let us prove a complementary upper bound in terms of the invariant c of (13). Theorem 3.7. A not very degenerate ring Q has at most Y N (p)5 − 1 n N (p) − 1 p prime, pn kc numerical resolvents, provided that the absolute norms N (p) = \|R/p\| are finite. In particular, a not very degenerate quintic ring over the ring of integers of a number field has finitely many numerical resolvents. Proof. Since all numerical resolvents have index c in M0 , it suffices to bound the number of sublattices of index c in a fixed lattice M0 . By localization we may reduce to the case c = pn , where p is prime. Now a fixed lattice M has (N (p)5 − 1)/(N (p) − 1) sublattices of index p, the kernels of the nonzero linear functionals ℓ : M/pM → R/p mod scaling. A sublattice Mn of index pn has a filtration M0 ( M1 ( · · · ( Mn where the quotients are R/p; given Mi , there are at most (N (p)5 − 1)/(N (p) − 1) possibilities for Mi+1 , giving the claimed bound. 4 Examples Example 4.1. The most fundamental example of a sextic resolvent is as follows. Let Q = R⊕5 , with basis e1 , e2 , . . . , e5 , and let M = R5 with basis f1 , . . . , f5 . Then the map φ(ei ) = fi ∧ (fi−1 + fi+1 ) (indices mod 5), supplemented by the natural orientation θ(ftop ) = e3top , is verified to be a resolvent for Q (indeed the unique one, as Q is maximal). The automorphism group S5 of Q acts on M by the 5-dimensional irreducible representation obtained (in characteristic not 2) by restricting to the image of the exceptional embedding S5 ֒→ S6 the standard representation of S6 , permuting the six vectors fi−2 − fi−1 + fi − fi+1 + fi+2 (1 ≤ i ≤ 5) and f1 + f2 + f3 + f4 + f5 . 8 Example 4.2. For the subring Q = {x1 e1 + · · · + x5 e5 ∈ Z⊕5 : x1 ≡ x2 ≡ x3 ≡ x4 mod p}, the bounding module M0 of Section 3.2 is no longer a resolvent, as can be seen by observing that Q/pQ ∼ = Fp [t, ǫ1 , ǫ2 , ǫ3 ]/h{t2 − t, tǫi , ǫi ǫj }i is very degenerate. We have L = hpe1 , pe2 , pe3 , e5 i and thus Λ4 L = hp3 etop i. One computes that M0 = p(f1 + f4 ), p2 f2 , p2 f3 , p2 f4 , pf5 , and thus c = [Λ5 M0 : θ−1 ((Λ4 L)⊗3 )] = [hp8 ftop i : hp9 ftop i] = p. Consequently a numerical resolvent of Q is a submodule M of index p in M0 having the property that φ(Λ4 L ⊗ L) ⊆ Λ2 M . Writing M as the kernel of some linear functional ℓ : M0 /pM0 → Fp , the condition is that ℓ lies in the kernel of each of the skew-symmetric bilinear forms obtained by reducing φ(x) ∈ Λ2 M0 mod p for all x ∈ Λ4 L ⊗ L). Let f1′ = p(f1 + f4 ), f2′ = p2 f2 , f3′ = p2 f3 , f4′ = p2 f4 , f5′ = pf5 be the basis elements of M0 listed above. We compute φ(p4 etop e1 ) = (pf1′ − f4′ ) ∧ (pf5′ + f2′ ) φ(p4 etop e2 ) = f2′ ∧ (pf1′ − f4′ + f3′ ) φ(p4 etop e3 ) = f3′ ∧ (pf2′ + f4′ ) φ(p3 etop e5 ) = f5′ ∧ (pf1′ ). So, letting f¯i′ denote the basis vector of M0 /pM0 corresponding to fi′ and f¯i′∗ the corresponding vector of the dual basis, we have ℓ ∈ ker(f¯2′ ∧ f¯4′ ) ∩ ker(f¯2′ ∧ f¯3′ ) ∩ ker(f¯3′ ∧ f¯4′ ) = f¯1′∗ , f¯5′∗ . Since ℓ can take any value in the last-named vector space, up to scaling, we get p + 1 numerical resolvents (and, as it turns out, no nonnumerical ones). Example 4.3. The ring Q = Z ⊕ Z ⊕ Z[x, y]/(x, y)2 is a curious example of Theorem 3.6. Although Q is infinitely far from being maximal (Z ⊕ Z ⊕ Z[n−1 x, n−1 y]/(n−2 (x, y)2 ) is a quintic extension ring for any n > 0), the extensions are only in two directions, as it were, and the resolvent is accordingly unique. Example 4.4. The simplest example of a non-numerical resolvent is given by the ring Q = Z + p2 Z⊕5 = {x1 e1 + · · · + x5 e5 ∈ Z⊕5 : x1 ≡ · · · ≡ x5 mod p2 }. We recognize L = p2 L1 , and we set M = p5 M1 , φ = φ1 \|Λ4 L⊗L , and θ = θ1 \|M , where the subscript 1 denotes the corresponding entity in Example 4.1. Here φ(Λ4 L ⊗ L) = φ(p8 Λ4 L1 ⊗ p2 L1 ) = p10 Λ2 M1 = Λ2 M, while θ(Λ5 M ) = θ(p25 Λ5 M1 ) = p25 (Λ4 L1 )⊗3 ( p24 (Λ4 L1 )⊗3 = (p8 Λ4 L1 )⊗3 = (Λ4 L)⊗3 . The ring Q also has a large number of numerical resolvents, including for instance any supermodule of index p over M . 9 Example 4.5. Very degenerate rings. Over a field K, a very degenerate ring has a multiplication table (7) with a single indeterminate entry u = α2 . By changing basis, we can reduce to the case that u is either α, β, or 0, giving three very degenerate rings up to isomorphism; in Mazzola’s nomenclature [3], they are A18 = K ⊕ K[x, y, z]/(x, y, z)2 , A19 = K[x, y, z]/(x3 , xy, y 2 , xz, yz, z 2) A20 = K[x, y, z, w]/(x, y, z, w)4 . Each of these has a large family of resolvents. Utilizing Bhargava’s representation quadruple of 5 × 5 skew-symmetric matrices, the maps        0 0 0 0 ∗ 0 0 0 0 ∗ 0 0 −1 0 ∗ 0 1 0 0 0 0 0 0 ∗ 0 0 1 0 ∗ 0 0 0 0 ∗ −1 0 0 0        0 0 0 0 ∗ , 0 −1 0 0 ∗ , 1 0 0 0 ∗ ,  0 0 0 0        0 0 0 0 1 0 0 0 0 0 0 0 0 0 0  0 0 0 0 ∗ ∗ ∗ −1 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ 0 (where ∗ represents any  0 0 0 0 0 0 1 0  0 −1 0 0  0 0 0 0 ∗ ∗ ∗ −1 element)   ∗ 0 0 ∗    ∗  , 0 1 0 0 ∗ are resolvents for A18 ,   0 0 0 ∗ 0 0 0 0 0 0 0 ∗    0 0 0 ∗  , 1 0 0 0 0 0 0 0 ∗ ∗ 0 0 ∗ ∗ while −1 0 0 0 ∗ 0 0 0 0 0   ∗ 0 −1 ∗    ∗ , 0 0  0 0 ∗ 1 0 0 0 ∗ 0 0 0 0 ∗ 0 0 0 0 0 of φ as a  ∗  ∗   ∗  0 0  ∗  ∗   ∗  0 0 works for A19 . The trivial ring A20 has an even larger family of resolvents, namely those where φ lands in Λ2 N , for any hyperplane N ⊆ M , or in V ∧ M for any 2-plane V , or where θ = 0. Are these all the resolvents? The classification of resolvents of very degenerate rings, even over fields, is an inviting problem which does not readily yield to the φ -based method of Theorem 3.2. References [1] Manjul Bhargava. Higher composition laws. III. The parametrization of quartic rings. Ann. of Math. (2), 159(3):1329–1360, 2004. [2] Manjul Bhargava. Higher composition laws. IV. The parametrization of quintic rings. Ann. of Math. (2), 167(1):53–94, 2008. [3] Guerino Mazzola. Generic finite schemes and Hochschild cocycles. Comment. Math. Helv., 55(2):267–293, 1980. [4] Evan O’Dorney. Rings of small rank over a Dedekind domain. http://arxiv.org/abs/1508.02777 (accessed 11/10/2015). Awaiting publication. [5] Bjorn Poonen. The moduli space of commutative algebras of finite rank. J. Eur. Math. Soc. (JEMS), 10(3):817–836, 2008. [6] Melanie Matchett Wood. Parametrizing quartic algebras over an arbitrary base. Algebra Number Theory, 5(8):1069–1094, 2011. 10	0
arXiv:1710.07177v1 [cs.CL] 19 Oct 2017 Findings of the Second Shared Task on Multimodal Machine Translation and Multilingual Image Description Desmond Elliott School of Informatics University of Edinburgh d.elliott@ed.ac.uk Stella Frank Centre for Language Evolution University of Edinburgh stella.frank@ed.ac.uk Loı̈c Barrault and Fethi Bougares LIUM University of Le Mans first.last@univ-lemans.fr Lucia Specia Department of Computer Science University of Sheffield l.specia@sheffield.ac.uk Abstract We present the results from the second shared task on multimodal machine translation and multilingual image description. Nine teams submitted 19 systems to two tasks. The multimodal translation task, in which the source sentence is supplemented by an image, was extended with a new language (French) and two new test sets. The multilingual image description task was changed such that at test time, only the image is given. Compared to last year, multimodal systems improved, but text-only systems remain competitive. 1 Introduction The Shared Task on Multimodal Translation and Multilingual Image Description tackles the problem of generating descriptions of images for languages other than English. The vast majority of image description research has focused on Englishlanguage description due to the abundance of crowdsourced resources (Bernardi et al., 2016). However, there has been a significant amount of recent work on creating multilingual image description datasets in German (Elliott et al., 2016; Hitschler et al., 2016; Rajendran et al., 2016), Turkish (Unal et al., 2016), Chinese (Li et al., 2016), Japanese (Miyazaki and Shimizu, 2016; Yoshikawa et al., 2017), and Dutch (van Miltenburg et al., 2017). Progress on this problem will be useful for native-language image search, multilingual ecommerce, and audio-described video for visually impaired viewers. The first empirical results for multimodal translation showed the potential for visual context to improve translation quality (Elliott et al., 2015; Hitschler et al., 2016). This was quickly followed by a wider range of work in the first shared task at WMT 2016 (Specia et al., 2016). The current shared task consists of two subtasks: • Task 1: Multimodal translation takes an image with a source language description that is then translated into a target language. The training data consists of parallel sentences with images. • Task 2: Multilingual image description takes an image and generates a description in the target language without additional source language information at test time. The training data, however, consists of images with independent descriptions in both source and target languages. The translation task has been extended to include a new language, French. This extension means the Multi30K dataset (Elliott et al., 2016) is now triple aligned, with English descriptions translated into both German and French. The description generation task has substantially changed since last year. The main difference is that source language descriptions are no longer observed for test images. This mirrors the realworld scenario in which a target-language speaker wants a description of image that does not already have source language descriptions associated with it. The two subtasks are now more distinct because multilingual image description requires the use of the image (no text-only system is possible because the input contains no text). Another change for this year is the introduction of two new evaluation datasets: an extension of the existing Multi30K dataset, and a “teaser” evaluation dataset with images carefully chosen to contain ambiguities in the source language. This year we encouraged participants to submit systems using unconstrained data for both tasks. Training on additional out-of-domain data is underexplored for these tasks. We believe this setting will be critical for future real-world improvements, given that the current training datasets are small and expensive to construct.1 2 2.1 Tasks & Datasets Tasks The Multimodal Translation task (Task 1) follows the format of the 2016 Shared Task (Specia et al., 2016). The Multilingual Image Description Task (Task 2) is new this year but it is related to the Crosslingual Image Description task from 2016. The main difference between the Crosslingual Image Description task and the Multilingual Image Description task is the presence of source language descriptions. In last year’s Crosslingual Image Description task, the aim was to produce a single target language description, given five source language descriptions and the image. In this year’s Multilingual Image Description task, participants received only an unseen image at test time, without source language descriptions. 2.2 En: A group of people are eating noddles. De: Eine Gruppe von Leuten isst Nudeln. Fr: Un groupe de gens mangent des nouilles. Datasets The Multi30K dataset (Elliott et al., 2016) is the primary dataset for the shared task. It contains 31K images originally described in English (Young et al., 2014) with two types of multilingual data: a collection of professionally translated German sentences, and a collection of independently crowdsourced German descriptions. This year the Multi30K dataset has been extended with new evaluation data for the Translation and Image Description tasks, and an additional language for the Translation task. In addition, we released a new evaluation dataset featuring ambiguities that we expected would benefit from visual context. Table 1 presents an overview of the new evaluation datasets. Figure 1 shows an example of an image with an aligned English-German-French description. In addition to releasing the parallel text, we also distributed two types of ResNet-50 visual features 1 All of the data and results are available at http://www. statmt.org/wmt17/multimodal-task.html Figure 1: Example of an image with a source description in English, together with German and French translations. (He et al., 2016) for all of the images, namely the ‘res4 relu’ convolutional features (which preserve the spatial location of a feature in the original image) and averaged pooled features. Multi30K French Translations We extended the translation data in Multi30K dataset with crowdsourced French translations. The crowdsourced translations were collected from 12 workers using an internal platform. We estimate the translation work had a monetary value of e9,700. The translators had access to the source segment, the image and an automatic translation created with a standard phrase-based system (Koehn et al., 2007) trained on WMT’15 parallel text. The automatic translations were presented to the crowdworkers to further simplify the crowdsourcing task. We note that this did not end up being a post-editing task, that is, the translators did not simply copy and paste the suggested translations. To demonstrate this, we calculated text-similarity metric scores between the phrase-based system outputs and the human translations on the training corpus, resulting in 0.41 edit distance (measured using the TER metric), meaning that more than 40% of the words between these two versions do not match. Multi30K 2017 test data We collected new evaluation data for the Multi30K dataset. We sampled new images from five of the six Flickr groups used to create the original Flickr30K dataset using MMFeat (Kiela, 2016)2 . 2 Strangers!, Wild Child, Dogs in Action, Action Photography, and Outdoor Activities. Training set Development set Images Sentences Images Sentences Translation 29,000 29,000 1,014 1,014 Description 29,000 145,000 1,014 5,070 2017 test COCO Images Sentences Images Sentences Translation 1,000 1,000 461 461 Description 1,071 5,355 — Table 1: Overview of the Multi30K training, development, 2017 test, and Ambiguous COCO datasets. Task 1 Task 2 Strangers! 150 154 Wild Child 83 83 remaining 1,071 images were used for the Multilingual Image Description task. We collected five additional independent German descriptions of those images from Crowdflower. Dogs in Action 78 92 Ambiguous COCO Action Photography 238 259 Flickr Social Club 241 263 Everything Outdoor 206 214 Outdoor Activities 4 6 As a secondary evaluation dataset for the Multimodal Translation task, we collected and translated a set of image descriptions that potentially contain ambiguous verbs. We based our selection on the VerSe dataset (Gella et al., 2016), which annotates a subset of the COCO (Lin et al., 2014) and TUHOI (Le et al., 2014) images with OntoNotes senses for 90 verbs which are ambiguous, e.g. play. Their goals were to test the feasibility of annotating images with the word sense of a given verb (rather than verbs themselves) and to provide a gold-labelled dataset for evaluating automatic visual sense disambiguation methods. Altogether, the VerSe dataset contains 3,518 images, but we limited ourselves to its COCO section, since for our purposes we also need the image descriptions, which are not available in TUHOI. The COCO portion covers 82 verbs; we further discarded verbs that are unambiguous in the dataset, i.e. although some verbs have multiple senses in OntoNotes, they all occur with one sense in VerSe (e.g. gather is used in all instances to describe the ‘people gathering’ sense), resulting in 57 ambiguous verbs (2,699 images). The actual descriptions of the images were not distributed with the VerSe dataset. However, given that the ambiguous verbs were selected based on the image descriptions, we assumed that in all cases at least one of the original COCO description (out of the five per image) should contain the ambiguous verb. In cases where more than one description contained the verb, we Group Table 2: Distribution of images in the Multi30K 2017 test data by Flickr group. We sampled additional images from two thematically related groups (Everything Outdoor and Flickr Social Club) because Outdoor Activities only returned 10 new CC-licensed images and Flickr-Social no longer exists. Table 2 shows the distribution of images across the groups and tasks. We initially downloaded 2,000 images per Flickr group, which were then manually filtered by three of the authors. The filtering was done to remove (near) duplicate images, clearly watermarked images, and images with dubious content. This process resulted in a total of 2,071 images. We crowdsourced five English descriptions of each image from Crowdflower3 using the same process as Elliott et al. (2016). One of the authors selected 1,000 images from the collection to form the dataset for the Multimodal Translation task based on a manual inspection of the English descriptions. Professional German translations were collected for those 1,000 English-described images. The 3 http://www.crowdflower.com En: A man on a motorcycle is passing another vehicle. De: Ein Mann auf einem Motorrad fährt an einem anderen Fahrzeug vorbei. Fr: Un homme sur une moto dépasse un autre véhicule. En: A red train is passing over the water on a bridge De: Ein roter Zug fährt auf einer Brücke über das Wasser Fr: Un train rouge traverse l’eau sur un pont. Figure 2: Two senses of the English verb ”to pass” in their visual contexts, with the original English and the translations into German and French. The verb and its translations are underlined. randomly selected one such description to be part of the dataset of descriptions containing ambiguous verbs. This resulted in 2,699 descriptions. As a consequence of the original goals of the VerSe dataset, each sense of each ambiguous verb was used multiple times in the dataset, which resulted in many descriptions with the same sense, for example, 85 images (and descriptions) were available for the verb show, but they referred to a small set of senses of the verb. The number of images (and therefore descriptions) per ambiguous verb varied from 6 (stir) to 100 (pull, serve). Since our intention was to have a small but varied dataset, we selected a subset of a subset of descriptions per ambiguous verb, aiming at keeping 1-3 instances per sense per verb. This resulted in 461 descriptions for 56 verbs in total, ranging from 3 (e.g. shake, carry) to 26 (reach) (the verb lay/lie was excluded as it had only one sense). We note that the descriptions include the use of the verbs in phrasal verbs. Two examples of the English verb “to pass” are shown in Figure 2. In the German translations, the source language verb did not require disambiguation (both German translations use the verb “fährt”), whereas in the French translations, the verb was disambiguated into “dépasse” and “traverse”, respectively. 3 Participants This year we attracted submissions from nine different groups. Table 3 presents an overview of the groups and their submission identifiers. AFRL-OHIOSTATE (Task 1) The AFRLOHIOSTATE system submission is an atypical Machine Translation (MT) system in that the image is the catalyst for the MT results, and not the textual content. This system architecture assumes an image caption engine can be trained in a target language to give meaningful output in the form of a set of the most probable n target language candidate captions. A learned mapping function of the encoded source language caption to the corresponding encoded target language captions is then employed. Finally, a distance function is applied to retrieve the “nearest” candidate caption to be the translation of the source caption. CMU (Task 2) The CMU submission uses a multi-task learning technique, extending the baseline so that it generates both a German caption and an English caption. First, a German caption is generated using the baseline method. After the LSTM for the baseline model finishes producing a German caption, it has some final hidden state. Decoding is simply resumed starting from that final state with an independent decoder, separate vocabulary, and this time without any direct access to the image. The goal is to encourage the model to keep information about the image in the hidden state throughout the decoding process, hopefully improving the model output. Although the model is trained to produce both German and English cap- ID AFRL-OHIOSTATE Participating team Air Force Research Laboratory & Ohio State University (Duselis et al., 2017) CMU Carnegie Melon University (Jaffe, 2017) CUNI Univerzita Karlova v Praze (Helcl and Libovický, 2017) DCU-ADAPT Dublin City University (Calixto et al., 2017a) LIUMCVC Laboratoire d’Informatique de l’Université du Maine & Universitat Autonoma de Barcelona Computer Vision Center (Caglayan et al., 2017) NICT National Institute of Information and Communications Technology & Nara Institute of Science and Technology (Zhang et al., 2017) OREGONSTATE SHEF UvA-TiCC Oregon State University (Ma et al., 2017) University of Sheffield (Madhyastha et al., 2017) Universiteit van Amsterdam & Tilburg University (Elliott and Kádár, 2017) Table 3: Participants in the WMT17 multimodal machine translation shared task. tions, at evaluation time the English component of the model is ignored and only German captions are generated. den state. Each image has one corresponding feature vector, obtained from the activations of the FC7 layer of the VGG19 network, and consist of a 4096D real-valued vector that encode information about the entire image. CUNI (Tasks 1 and 2) For Task 1, the submissions employ the standard neural MT (NMT) scheme enriched with another attentive encoder for the input image. It uses a hierarchical attention combination in the decoder (Libovický and Helcl, 2017). The best system was trained with additional data obtained from selecting similar sentences from parallel corpora and by back-translation of similar sentences found in the SDEWAC corpus (Faaß and Eckart, 2013). The submission to Task 2 is a combination of two neural models. The first model generates an English caption from the image. The second model is a text-only NMT model that translates the English caption to German. LIUMCVC (Task 1) LIUMCVC experiment with two approaches: a multimodal attentive NMT with separate attention (Caglayan et al., 2016) over source text and convolutional image features, and an NMT where global visual features (2048dimensional pool5 features from ResNet-50) are multiplicatively interacted with word embeddings. More specifically, each target word embedding is multiplied with global visual features in an elementwise fashion in order to visually contextualize word representations. With 128-dimensional embeddings and 256-dimensional recurrent layers, the resulting models have around 5M parameters. DCU-ADAPT (Task 1) This submission evaluates ensembles of up to four different multimodal NMT models. All models use global image features obtained with the pre-trained CNN VGG19, and are either incorporated in the encoder or the decoder. These models are described in detail in (Calixto et al., 2017b). They are model IMGW , in which image features are used as words in the source-language encoder; model IMGE , where image features are used to initialise the hidden states of the forward and backward encoder RNNs; and model IMGD , where the image features are used as additional signals to initialise the decoder hid- NICT (Task 1) These are constrained submissions for both language pairs. First, a hierarchical phrase-based (HPB) translation system s built using Moses (Koehn et al., 2007) with standard features. Then, an attentional encoder-decoder network (Bahdanau et al., 2015) is trained and used as an additional feature to rerank the n-best output of the HPB system. A unimodal NMT model is also trained to integrate visual information. Instead of integrating visual features into the NMT model directly, image retrieval methods are employed to obtain target language descriptions of images that are similar to the image described by the source sentence, and this target description information is integrated into the NMT model. A multimodal NMT model is also used to rerank the HPB output. All feature weights (including the standard features, the NMT feature and the multimodal NMT feature) were tuned by MERT (Och, 2003). On the development set, the NMT feature improved the HPB system significantly. However, the multimodal NMT feature did not further improve the HPB system that had integrated the NMT feature. OREGONSTATE (Task 1) The OREGONSTATE system uses a very simple but effective model which feeds the image information to both encoder and decoder. On the encoder side, the image representation was used as an initialization information to generate the source words’ representations. This step strengthens the relatedness between image’s and source words’ representations. Additionally, the decoder uses alignment to source words by a global attention mechanism. In this way, the decoder benefits from both image and source language information and generates more accurate target side sentence. UvA-TiCC (Task 1) The submitted systems are Imagination models (Elliott and Kádár, 2017), which are trained to perform two tasks in a multitask learning framework: a) produce the target sentence, and b) predict the visual feature vector of the corresponding image. The constrained models are trained over only the 29,000 training examples in the Multi30K dataset with a source-side vocabulary of 10,214 types and a target-side vocabulary of 16,022 types. The unconstrained models are trained over a concatenation of the Multi30K, News Commentary (Tiedemann, 2012) parallel texts, and MS COCO (Chen et al., 2015) dataset with a joint source-target vocabulary of 17,597 word pieces (Schuster and Nakajima, 2012). In both constrained and unconstrained submissions, the models were trained to predict the 2048D GoogleLeNetV3 feature vector (Szegedy et al., 2015) of an image associated with a source language sentence. The output of an ensemble of the three best randomly initialized models - as measured by BLEU on the Multi30K development set - was used for both the constrained and unconstrained submissions. SHEF (Task 1) The SHEF systems utilize the predicted posterior probability distribution over the image object classes as image features. To do so, they make use of the pre-trained ResNet-152 (He et al., 2016), a deep CNN based image network that is trained over the 1,000 object categories on the Imagenet dataset (Deng et al., 2009) to obtain the posterior distribution. The model follows a standard encoder-decoder NMT approach using softdot attention as described in (Luong et al., 2015). It explores image information in three ways: a) to initialize the encoder; b) to initialize the decoder; c) to condition each source word with the image class posteriors. In all these three ways, non-linear affine transformations over the posteriors are used as image features. Baseline — Task 1 The baseline system for the multimodal translation task is a text-only neural machine translation system built with the Nematus toolkit (Sennrich et al., 2017). Most settings and hyperparameters were kept as default, with a few exceptions: batch size of 40 (instead of 80 due to memory constraints) and ADAM as optimizer. In order to handle rare and OOV words, we used the Byte Pair Encoding Compression Algorithm to segment words (Sennrich et al., 2016b). The merge operations for word segmentation were learned using training data in both source and target languages. These were then applied to all training, validation and test sets in both source and target languages. In post-processing, the original words were restored by concatenating the subwords. Baseline — Task 2 The baseline for the multilingual image description task is an attention-based image description system trained over only the German image descriptions (Caglayan et al., 2017). The visual representation are extracted from the so-called res4f relu layer from a ResNet-50 (He et al., 2016) convolutional neural network trained on the ImageNet dataset (Russakovsky et al., 2015). Those feature maps provide spatial information on which the model focuses through the attention mechanism. 4 Text-similarity Metric Results The submissions were evaluated against either professional or crowd-sourced references. All submissions and references were pre-processed to lowercase, normalise punctuation, and tokenise the sentences using the Moses scripts.4 The evaluation was performed using MultEval (Clark et al., 2011) with the primary metric of Meteor 4 https://github.com/moses-smt/ mosesdecoder/blob/master/scripts/ 1.5 (Denkowski and Lavie, 2014). We also report the results using BLEU (Papineni et al., 2002) and TER (Snover et al., 2006) metrics. The winning submissions are indicated by •. These are the topscoring submissions and those that are not significantly different (based on Meteor scores) according the approximate randomisation test (with p-value ≤ 0.05) provided by MultEval. Submissions marked with * are not significantly different from the Baseline according to the same test. 4.1 Task 1: English → German 4.1.1 Multi30K 2017 test data Table 4 shows the results on the Multi30K 2017 test data with a German target language. It interesting to note that the metrics do not fully agree on the ranking of systems, although the four best (statistically indistinguishable) systems win by all metrics. All-but-one submission outperformed the text-only NMT baseline. This year, the best performing systems include both multimodal (LIUMCVC MNMT C and UvA-TiCC IMAGINATION U) and text-only (NICT NMTrerank C and LIUMCVC MNMT C) submissions. (Strictly speaking, the UvATiCC IMAGINATION U system is incomparable because it is an unconstrained system, but all unconstrained systems perform in the same range as the constrained systems.) 4.1.2 Ambiguous COCO Table 5 shows the results for the out-of-domain ambiguous COCO dataset with a German target language. Once again the evaluation metrics do not fully agree on the ranking of the submissions. It is interesting to note that the metric scores are lower for the out-of-domain Ambiguous COCO data compared to the in-domain Multi30K 2017 test data. However, we cannot make definitive claims about the difficulty of the dataset because the Ambiguous COCO dataset contains fewer sentences than the Multi30K 2017 test data (461 compared to 1,000). The systems are mostly in the same order as on the Multi30K 2017 test data, with the same four systems performing best. However, two systems (DCU-ADAPT MultiMT C and OREGONSTATE 1NeuralTranslation C) are ranked higher on this test set than on the in-domain Flickr dataset, indicating that they are relatively more robust and possibly better at resolving the ambiguities found in the Ambiguous COCO dataset. 4.2 4.2.1 Task 1: English → French Multi30K Test 2017 Table 6 shows the results for the Multi30K 2017 test data with French as target language. A reduced number of submissions were received for this new language pair, with no unconstrained systems. In contrast to the English→German results, the evaluation metrics are in better agreement about the ranking of the submissions. Translating from English→French is an easier task than English→German systems, as reflected in the higher metric scores. This also includes the baseline systems where English→French results in 63.1 Meteor compared to 41.9 for English→German. Eight out of the ten submissions outperformed the English→French baseline system. Two of the best submissions for English→German remain the best for English→French (LIUMCVC MNMT C and NICT NMTrerank C), the text-only system (LIUMCVC NMT C) decreased in performance, and no UvA-TiCC IMAGINATION U system was submitted for French. An interesting observation is the difference of the Meteor scores between text-only NMT system (LIUMCVC NMT C) and Moses hierarchical phrase-based system with reranking (NICT NMTrerank C). While the two systems are very close for the English→German direction, the hierarchical system is better than the text-only NMT systems in the English→French direction. This pattern holds for both the Multi30K 2017 test data and Ambiguous COCO test data. 4.2.2 Ambiguous COCO Table 7 shows the results for the out-of-domain Ambiguous COCO dataset with the French target language. Once again, in contrast to the English→German results, the evaluation metrics are in better agreement about the ranking of the submissions. The performance of all the models is once again in mostly agreement with the Multi30K 2017 test data, albeit lower. Both DCU-ADAPT MultiMT C and OREGONSTATE 2NeuralTranslation C again perform relatively better on this dataset. 4.3 Task 2: English → German The description generation task, in which systems must generate target-language (German) captions for a test image, has substantially changed since •LIUMCVC MNMT C •NICT NMTrerank C •LIUMCVC NMT C •UvA-TiCC IMAGINATION U UvA-TiCC IMAGINATION C CUNI NeuralMonkeyTextualMT U OREGONSTATE 2NeuralTranslation C DCU-ADAPT MultiMT C CUNI NeuralMonkeyMultimodalMT U CUNI NeuralMonkeyTextualMT C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyMultimodalMT C SHEF ShefClassInitDec C SHEF ShefClassProj C Baseline (text-only NMT) AFRL-OHIOSTATE-MULTIMODAL U BLEU ↑ Meteor ↑ TER ↓ 33.4 31.9 33.2 33.3 30.2 31.1 31.0 29.8 29.5 28.5 29.7 25.8 25.0 24.2 19.3 6.5 54.0 53.9 53.8 53.5 51.2 51.0 50.6 50.5 50.2 49.2 48.9 47.1 44.5 43.4 41.9 20.2 48.5 48.1 48.2 47.5 50.8 50.7 50.7 52.3 52.5 54.3 51.6 56.3 53.8 55.9 72.2 87.4 Table 4: Official results for the WMT17 Multimodal Machine Translation task on the English-German Multi30K 2017 test data. Systems with grey background indicate use of resources that fall outside the constraints provided for the shared task. •LIUMCVC NMT C •LIUMCVC MNMT C •NICT 1 NMTrerank C •UvA-TiCC IMAGINATION U DCU-ADAPT MultiMT C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyTextualMT U UvA-TiCC IMAGINATION C OREGONSTATE 2NeuralTranslation C CUNI NeuralMonkeyMultimodalMT U CUNI NeuralMonkeyTextualMT C CUNI NeuralMonkeyMultimodalMT C SHEF ShefClassInitDec C SHEF ShefClassProj C Baseline (text-only NMT) BLEU ↑ Meteor ↑ TER ↓ 28.7 28.5 28.1 28.0 26.4 27.4 26.6 26.4 26.1 25.7 23.2 22.4 21.4 21.0 18.7 48.9 48.8 48.5 48.1 46.8 46.5 46.0 45.8 45.7 45.6 43.8 42.7 40.7 40.0 37.6 52.5 53.4 52.9 52.4 54.5 52.3 54.8 55.4 55.9 55.7 59.8 60.1 56.5 57.8 66.1 Table 5: Results for the Multimodal Translation task on the English-German Ambiguous COCO dataset. Systems with grey background indicate use of resources that fall outside the constraints provided for the shared task. •LIUMCVC MNMT C •NICT NMTrerank C DCU-ADAPT MultiMT C LIUMCVC NMT C OREGONSTATE 2NeuralTranslation C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyMultimodalMT C CUNI NeuralMonkeyTextualMT C Baseline (text-only NMT) SHEF ShefClassInitDec C SHEF ShefClassProj C BLEU ↑ Meteor ↑ TER ↓ 55.9 55.3 54.1 53.3 51.9 51.0 49.9 50.3 44.3 45.0 43.6 72.1 72.0 70.1 70.1 68.3 67.2 67.2 67.0 63.1 62.8 61.5 28.4 28.4 30.0 31.7 32.7 33.6 34.3 33.6 39.6 38.4 40.5 Table 6: Results for the Multimodal Translation task on the English-French Multi30K Test 2017 data. •LIUMCVC MNMT C •NICT NMTrerank C •DCU-ADAPT MultiMT C OREGONSTATE 2NeuralTranslation C LIUMCVC NMT C CUNI NeuralMonkeyTexutalMT C CUNI NeuralMonkeyMultimodalMT C OREGONSTATE 1NeuralTranslation C SHEF ShefClassInitDec C SHEF ShefClassProj C Baseline (text-only NMT) BLEU ↑ Meteor ↑ TER ↓ 45.9 45.1 44.5 44.1 43.6 43.0 42.9 41.2 37.2 36.8 35.1 65.9 65.6 64.1 63.8 63.4 62.5 62.5 61.6 57.3 57.0 55.8 34.2 34.7 35.2 36.7 37.4 38.2 38.2 37.8 42.4 44.5 45.8 Table 7: Results for the Multimodal Translation task on the English-French Ambiguous COCO dataset. Baseline (target monolingual) CUNI NeuralMonkeyCaptionAndMT C CUNI NeuralMonkeyCaptionAndMT U CMU NeuralEncoderDecoder C CUNI NeuralMonkeyBilingual C BLEU ↑ Meteor ↑ TER ↓ 9.1 4.2 6.5 9.1 2.3 23.4 22.1 20.6 19.8 17.6 91.4 133.6 91.7 63.3 112.6 Table 8: Results for the Multilingual Image Description task on the English-German Multi30K 2017 test data. last year. The main difference is that source language descriptions are no longer observed for images at test time. The training data remains the same and contains images with both source and target language descriptions. The aim is thus to leverage multilingual training data to improve a monolingual task. Table 8 shows the results for the Multilingual image description task. This task attracted fewer submissions than last year, which may be because it was no longer possible to re-use a model designed for Multimodal Translation. The evaluation metrics do not agree on the ranking of the submissions, with major differences in the ranking using either BLEU or TER instead of Meteor. The main result is that none of the submissions outperform the monolingual German baseline according to Meteor. All of the submissions are statistically significantly different compared to the baseline. However, the CMU NeuralEncoderDecoder C submission marginally outperformed the baseline according to TER and equalled its BLEU score. asked to evaluate the semantic relatedness between the source sentence in English and the target sentence in German or French. The image was shown along with the source sentence and the candidate translation and evaluators were told to rely on the image when necessary to obtain a better understanding of the source sentence (e.g. in cases where the text was ambiguous). Note that the reference sentence is not displayed during the evaluation, in order to avoid influencing the assessor. Figure 3 shows an example of the direct assessment interface used in the evaluation. The score of each translation candidate ranges from 0 (meaning that the meaning of the source is not preserved in the target language sentence) to 100 (meaning the meaning of the source is “perfectly” preserved). The human assessment scores are standardized according to each individual assessor’s overall mean and standard deviation score. The overall score of a given system (z) corresponds to the mean standardized score of its translations. 5 The French outputs were evaluated by seven assessors, who conducted a total of 2,521 DAs, resulting in a minimum of 319 and a maximum of 368 direct assessments per system submission, respectively. The German outputs were evaluated by 25 assessors, who conducted a total of 3,485 DAs, resulting in a minimum of 291 and a maximum of 357 direct assessments per system submission, respectively. This is somewhat less than the recommended number of 500, so the results should be considered preliminary. Tables 9 and 10 show the results of the human evaluation for the English to German and the English to French Multimodal Translation task (Multi30K 2017 test data). The systems are ordered by standardized mean DA scores and clustered ac- Human Judgement Results This year, we conducted a human evaluation in addition to the text-similarity metrics to assess the translation quality of the submissions. This evaluation was undertaken for the Task 1 German and French outputs for the Multi30K 2017 test data. This section describes how we collected the human assessments and computed the results. We would like to gratefully thank all assessors. 5.1 Methodology The system outputs were manually evaluated by bilingual Direct Assessment (DA) (Graham et al., 2015) using the Appraise platform (Federmann, 2012). The annotators (mostly researchers) were 5.2 Results Figure 3: Example of the human direct assessment evaluation interface. cording to the Wilcoxon signed-rank test at p-level p ≤ 0.05. Systems within a cluster are considered tied. The Wilcoxon signed-rank scores can be found in Tables 11 and 12 in Appendix A. When comparing automatic and human evaluations, we can observe that they globally agree with each other, as shown in Figures 4 and 5, with German showing better agreement than French. We point out two interesting disagreements: First, in the English→French language pair, CUNI NeuralMonkeyMultimodalMT C and DCUADAPT MultiMT C are significantly better than LIUMCVC MNMT C, despite the fact that the latter system achieves much higher metric scores. Secondly, across both languages, the text-only LIUMCVC NMT C system performs well on metrics but does relatively poorly on human judgements, especially as compared to the multimodal version of the same system. 6 Discussion Visual Features: do they help? Three teams provided text-only counterparts to their multimodal systems for Task 1 (CUNI, LIUMCVC, and OREGONSTATE), which enables us to evaluate the contribution of visual features. For many systems, visual features did not seem to help reliably, at least as measured by metric evaluations: in German, the CUNI and OREGONSTATE text-only systems outperformed the counterparts, while in French, there were small improvements for the CUNI multimodal system. However, the LIUMCVC multimodal system outperformed their text-only system across both languages. The human evaluation results are perhaps more promising: nearly all the highest ranked systems (with the exception of NICT) are multimodal. An intruiging result was the text-only LIUMCVC NMT C, which ranked highly on metrics but poorly in the human evaluation. The LIUMCVC systems were indistinguishable from each other in terms of Meteor scores but the standardized mean direct assessment score showed a significant difference in performance (see Tables 11 and 12): further analysis of the reasons for humans disliking the text-only translations will be necessary. The multimodal Task 1 submissions can be broadly categorised into three groups based on how they use the images: approaches useing double-attention mechanisms, initialising the hidden state of the encoder and/or decoder networks with the global image feature vector, and alternative uses of image features. The doubleattention models calculate context vectors over the source language hidden states and locationpreserving feature vectors over the image; these vectors are used as inputs to the translation decoder (CUNI NeuralMonkeyMultimodalMT). Encoder and/or decoder initialisation involves initialising the recurrent neural network with an affine transformation of a global image feature vector (DCU-ADAPT MultiMT, OREGONSTATE 1NeuralTranslation) or initialising the encoder and decoder with the 1000 dimension softmax probability vector over the object classes in ImageNet object recognition challenge NICT NMTrerank C 54 LIUMCVC NMT C LIUMCVC MNMT C UvA-TiCC IMAGINATION U 52 Meteor 50 48 UvA-TiCC IMAGINATION C CUNI NeuralMonkeyTextualMT U OREGONSTATE 2NeuralTranslation C DCU-ADAPT MultiMT C CUNI NeuralMonkeyMultimodalMT U CUNI NeuralMonkeyTextualMT C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyMultimodalMT C 46 SHEF ShefClassInitDec C 44 42 40 −1 SHEF ShefClassProj C Baseline −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 Human judgments (z) 0.6 0.8 1 Figure 4: System performance on the English→German Multi30K 2017 test data as measured by human evaluation against Meteor scores. The AFRL-OHIOSTATE-MULTIMODAL U system has been ommitted for readability. 74 LIUMCVC MNMT C 72 NICT NMTrerank C LIUMCVC NMT C Meteor 70 DCU-ADAPT MultiMT C OREGONSTATE 2NeuralTranslation C 68 66 CUNI NeuralMonkeyMultimodalMT C OREGONSTATE 1NeuralTranslation C CUNI NeuralMonkeyTextualMT C 64 Baseline 62 60 −1 SHEF ShefClassInitDec C SHEF ShefClassProj C −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 Human judgments (z) 0.6 0.8 1 Figure 5: System performance on the English→French Multi30K 2017 test data as measured by human evaluation against Meteor scores. # Raw z English→German System 1 77.8 0.665 LIUMCVC MNMT C 2 74.1 0.552 UvA-TiCC IMAGINATION U 3 70.3 68.1 68.1 65.1 60.6 59.7 55.9 54.4 54.2 53.3 49.4 46.6 0.437 0.325 0.311 0.196 0.136 0.08 -0.049 -0.091 -0.108 -0.144 -0.266 -0.37 NICT NMTrerank C CUNI NeuralMonkeyTextualMT U DCU-ADAPT MultiMT C LIUMCVC NMT C CUNI NeuralMonkeyMultimodalMT U UvA-TiCC IMAGINATION C CUNI NeuralMonkeyMultimodalMT C OREGONSTATE 2NeuralTranslation C CUNI NeuralMonkeyTextualMT C OREGONSTATE 1NeuralTranslation C SHEF ShefClassProj C SHEF ShefClassInitDec C 15 39.0 36.6 -0.615 -0.674 Baseline (text-only NMT) AFRL-OHIOSTATE MULTIMODAL U Table 9: Results of the human evaluation of the WMT17 English-German Multimodal Translation task (Multi30K 2017 test data). Systems are ordered by standardized mean DA scores (z) and clustered according to Wilcoxon signed-rank test at p-level p ≤ 0.05. Systems within a cluster are considered tied, although systems within a cluster may be statistically significantly different from each other (see Table 11). Systems using unconstrained data are identified with a gray background. (SHEF ShefClassInitDec). The alternative uses of the image features include element-wise multiplication of the target language embeddings with an affine transformation of a global image feature vector (LIUMCVC MNMT), summing the source language word embeddings with affine-transformed 1000 dimension softmax probability vector (SHEF ShefClassProj), using the visual features in a retrieval framework (AFRL-OHIOSTATE MULTIMODAL), and learning visually-grounded encoder representations by learning to predict the global image feature vector from the source language hidden states (UvATiCC IMAGINATION). Overall, the metric and human judgement results in Sections 4 and 5 indicate that there is still a wide scope for exploration of the best way to integrate visual and textual information. In particular, the alternative approaches proposed in the LIUMCVC MNMT and UvA-TiCC IMAGINATION submissions led to strong performance in both the metric and human judgement results, surpassing the more common approaches using initialisation and double attention. Finally, the text-only NICT system ranks highly across both languages. This system uses hierarchical phrase-based MT with a reranking step based on a neural text-only system, since their multimodal system never outperformed the text-only variant in development (Zhang et al., 2017). This is in line with last year’s results and the strong Moses baseline (Specia et al., 2016), and suggests a continuing role for phrase-based MT for small homogeneous datasets. Unconstrained systems The Multi30k dataset is relatively small, so unconstrained systems use more data to complement the image description translations. Three groups submitted systems using external resources: UvA-TiCC, CUNI, and AFRLOHIOSTATE. The unconstrained UvA-TiCC and CUNI submissions always outperformed their respective constrained variants by 2–3 Meteor points and achieved higher standardized mean DA scores. These results suggest that external parallel text corpora (UvA-TiCC and CUNI) and external monolingual image description datasets (UvA-TiCC) can usefully improve the quality of multimodal translation models. However, tuning to the target domain remains important, even for relatively simple image captions. # Raw z English→French System 1 79.4 74.2 74.1 0.446 0.307 0.3 NICT NMTrerank C CUNI NeuralMonkeyMultimodalMT C DCU-ADAPT MultiMT C 4 71.2 65.4 61.9 60.8 60.5 0.22 0.056 -0.041 -0.078 -0.079 LIUMCVC MNMT C OREGONSTATE 2NeuralTranslation C CUNI NeuralMonkeyTextualMT C OREGONSTATE 1NeuralTranslation C LIUMCVC NMT C 9 54.7 54.0 -0.254 -0.282 SHEF ShefClassInitDec C SHEF ShefClassProj C 11 44.1 -0.539 Baseline (text-only NMT) Table 10: Results of the human evaluation of the WMT17 English-French Multimodal Translation task (Multi30K 2017 test data). Systems are ordered by standardized mean DA score (z) and clustered according to Wilcoxon signed-rank test at p-level p ≤ 0.05. Systems within a cluster are considered tied, although systems within a cluster may be statistically significantly different from each other (see Table 12). We ran the best-performing English→German WMT’16 news translation system (Sennrich et al., 2016a) on the English→German Multi30K 2017 test data to gauge the performance of a state-ofthe-art text-only translation system trained on only out-of-domain resources5 . It ranked 10th in terms of Meteor (49.9) and 11th in terms of BLEU (29.0), placing it firmly in the middle of the pack, and below nearly all the text-only submissions trained on the in-domain Multi30K dataset. The effect of OOV words The Multi30k translation training and test data are very similar, with a low OOV rate in the Flickr test set (1.7%). In the 2017 test set, 16% of English test sentences include a OOV word. Human evaluation gave the impression that these often led to errors propagated throughout the whole sentence. Unconstrained systems may perform better by having larger vocabularies, as well as more robust statistics. When we evaluate the English→German systems over only the 161 OOV-containing test sentences, the highest ranked submission by all metrics is the unconstrained UvA-TiCC IMAGINATION submission, with +2.5 Metor and +2.2 BLEU over the second best system (LIUMCVC NMT; 45.6 vs 43.1 Meteor and 24.0 vs 21.8 BLEU). The difference over non-OOV-containing sen5 http://data.statmt.org/rsennrich/ wmt16_systems/en-de/ tences is not nearly as stark, with constrained systems all performing best (both LIUMCVC systems, MNMT and NMT, with 56.6 and 56.3 Meteor, respectively) but unconstrained systems following close behind (UvA-TiCC with 55.4 Meteor, CUNI with 53.4 Meteor). Ambiguous COCO dataset We introduced a new evaluation dataset this year with the aim of testing systems’ ability to use visual features to identify word senses. However, it is unclear whether visual features improve performance on this test set. The text-only NICT NMTrerank system performs competitively, ranking in the top three submissions for both languages. We find mixed results for submissions with text-only and multimodal counterparts (CUNI, LIUMCVC, OREGONSTATE): LIUMCVC’s multimodal system improves over the text-only system for French but not German, while the visual features help for German but not French in the CUNI and OREGONSTATE systems. We plan to perform a further analysis on the extent of translation ambiguity in this dataset. We will also continue to work on other methods for constructing datasets in which textual ambiguity can be disambiguated by visual information. Multilingual Image Description It proved difficult for Task 2 systems to use the English data to improve over the monolingual German baseline. In future iterations of the task, we will consider a lopsided data setting, in which there is much more English data than target language data. This setting is more realistic and will push the use of multilingual data. We also hope to conduct human evaluation to better assess performance because automatic metrics are problematic for this task (Elliott and Keller, 2014; Kilickaya et al., 2017). No. ANR-15-CHR2-0006-01 – Loı̈c Barrault and Fethi Bougares), and by the MultiMT project (EU H2020 ERC Starting Grant No. 678017 – Lucia Specia). Desmond Elliott acknowledges the support of NWO Vici Grant No. 277-89-002 awarded to K. Sima’an, and an Amazon Academic Research Award. We thank Josiah Wang for his help in selecting the Ambiguous COCO dataset. 7 A Significance tests Conclusions We presented the results of the second shared task on multimodal translation and multilingual image description. The shared task attracted submissions from nine groups, who submitted a total of 19 systems across the tasks. The Multimodal Translation task attracted the majority of the submissions. Human judgements for the translation task were collected for the first time this year and ranked systems broadly in line with the automatic metrics. The main findings of the shared task are: (i) There is still scope for novel approaches to integrating visual and linguistic features in multilingual multimodal models, as demonstrated by the winning systems. (ii) External resources have an important role to play in improving the performance of multimodal translation models beyond what can be learned from limited training data. (iii) The differences between text-only and multimodal systems are being obfuscated by the well-known shortcomings of text-similarity metrics. Multimodal systems often seem to be prefered by humans but not rewarded by metrics. Future research on this topic, encompassing both multimodal translation and multilingual image description, should be evaluated using human judgements. In future editions of the task, we will encourage participants to submit the output of single decoder systems to better understand the empirical differences between approaches. We are also considering a Multilingual Multimodal Translation challenge, where the systems can observe two language inputs alongside the image to encourage the development of multi-source multimodal models. Acknowledgements This work was supported by the CHIST-ERA M2CR project (French National Research Agency Tables 11 and 12 show the Wilcoxon signed-rank test used to create the clustering of the systems. UvA-TiCC IMAGINATION U 2.8e-2 3.1e-4 NICT NMTrerank C 4.9e-2 4.8e-5 CUNI NeuralMonkeyTextualMT U - 1.8e-2 1.1e-6 DCU-ADAPT MultiMT C - 1.2e-3 4.5e-8 LIUMCVC NMT C 1.3e-2 4.3e-2 - 6.8e-5 6.6e-10 CUNI NeuralMonkeyMultimodalMT U 1.6e-3 1.3e-2 4.0e-2 - 3.5e-6 6.2e-13 8.1e-5 6.8e-4 4.9e-3 - 2.0e-8 7.9e-18 CUNI NeuralMonkeyMultimodalMT C 9.6e-8 2.4e-6 3.0e-5 5.7e-3 1.7e-2 - 3.2e-12 1.0e-18 OREGONSTATE 2NMT C 1.0e-8 4.9e-7 3.3e-6 1.4e-3 5.7e-3 2.8e-2 - 2.1e-13 2.9e-17 CUNI NeuralMonkeyTextualMT C 4.6e-8 1.3e-6 1.3e-5 2.2e-3 5.6e-3 3.3e-2 - 4.7e-12 3.2e-18 OREGONSTATE 1NMT C 9.8e-9 3.1e-7 2.2e-6 7.5e-4 2.8e-3 1.3e-2 - 5.6e-13 5.2e-26 SHEF ShefClassProj C 1.3e-13 1.6e-11 1.4e-10 2.2e-06 4.5e-06 6.2e-05 6.5e-03 1.9e-02 4.7e-02 - 3.2e-19 1.2e-30 SHEF ShefClassInitDec C 2.2e-17 6.8e-15 5.3e-14 8.5e-09 2.7e-08 3.3e-07 1.5e-04 1.1e-03 3.3e-03 1.2e-02 - 9.1e-24 2.2e-26 3.8e-23 9.8e-23 2.0e-15 1.2e-15 3.5e-14 1.5e-10 2.0e-09 8.3e-08 9.1e-07 4.7e-05 2.0e-03 3.9e-33 5.5e-41 BASELINE LIUMCVC MNMT C Table 11: English → German Wilcoxon signed-rank test at p-level p ≤ 0.05. ‘-’ means that the value is higher than 0.05. BASELINE AFRL-OHIOSTATE MULTIMODAL C NICT NMTrerank C CUNI NMT U DCU-ADAPT MultiMT C LIUMCVC NMT C CUNI MNMT U UvA-TiCC IMAGINATION C CUNI NMMT C OREGONSTATE 2NMT C CUNI NMT C OREGONSTATE 1NMT C SHEF ShefClassProj C SHEF ShefClassInitDec C UvA-TiCC IMAGINATION U LIUMCVC MNMT C UvA-TiCC IMAGINATION C English→German - 4.0e-28 4.7e-25 2.4e-24 4.0e-17 9.7e-18 5.0e-16 1.7e-12 3.6e-11 2.1e-09 2.6e-08 1.3e-06 1.2e-04 1.2e-34 3.1e-43 AFRL-OHIOSTATE MULTIMODAL C CUNI NeuralMonkeyMultimodalMT C - DCU-ADAPT MultiMT C - LIUMCVC MNMT C 1.0e-03 8.8e-03 2.2e-02 - 2.5e-05 2.9e-04 1.3e-03 CUNI NeuralMonkeyTextualMT C 1.7e-02 - 5.5e-08 5.2e-07 6.7e-06 OREGONSTATE 1NeuralTranslation C 2.7e-03 - 1.5e-09 5.0e-08 7.9e-07 LIUMCVC NMT C 1.2e-02 - 1.8e-07 2.9e-06 2.3e-05 SHEF ShefClassInitDec C 5.3e-07 1.7e-04 7.7e-03 2.4e-02 1.8e-02 2.2e-16 2.4e-14 2.2e-12 - 2.0e-06 1.8e-04 7.7e-03 1.8e-02 1.6e-02 6.8e-15 1.1e-13 1.3e-11 SHEF ShefClassProj C NICT NMTrerank C 3.2e-04 1.8e-03 5.9e-14 7.3e-11 5.0e-08 2.3e-07 5.6e-07 5.8e-26 2.3e-24 3.9e-21 Table 12: English → French Wilcoxon signed-rank test at p-level p ≤ 0.05. ‘-’ means that the value is higher than 0.05. 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Noname manuscript No. (will be inserted by the editor) Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ arXiv:1106.1194v2 [cs.NE] 19 Sep 2013 Angelos A. Anastassi Abstract A methodology that can generate the optimal coefficients of a numeri- cal method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a specific system of ordinary differential equations numerically. The case we are examining here concerns an explicit two-stage Runge-Kutta method for the numerical solution of the two-body problem. Following the implementation of the network, the latter is trained to obtain the optimal values for the coefficients of the Runge-Kutta method. The comparison of the new method to others that are well known in the literature proves its efficiency and demonstrates the capability of the network to provide efficient algorithms for specific problems. Keywords Feedforward artificial neural networks, Gradient descent, Backprop- agation, Initial value problems, Ordinary differential equations, Runge-Kutta methods 1 Introduction The literature that involves solving ordinary or partial differential equations with the use of artificial neural networks is quite limited, but has grown significantly in the past decade. To name a few, Dissanayake et al. used a ”universal approximator” (see [1][2]) to transform the numerical problem of solving partial differential equations to an unconstrained minimization problem of an objective function [3]. Meade et al. demonstrated feedforward neural networks that could approximate linear and nonlinear ordinary differential equations [4][5]. Puffer et al. constructed cellular neural networks that were able to approximate the solution of various ⋆ The final publication is available at http://link.springer.com/article/10.1007%2Fs00521013-1476-x A.A. Anastassi Department of Informatics, University of Piraeus, Karaoli & Dimitriou 80, 18534 Piraeus, GREECE E-mail: ang.anastassi@gmail.com 2 Angelos A. Anastassi partial differential equations [6]. Lagaris et al. introduced a method for the solution of initial and boundary value problems, that consists of an invariable part, which satisfies by construction the initial/boundary conditions and an artificial neural network, which is trained to satisfy the differential equation [7]. S. He et al. used feedforward neural networks to solve a special class of linear first-order partial differential equations [8]. The radial basis function (RBF) network architecture [10] has also been used for the solution of differential equations in [9], where Jianyu et al. demonstrated a method for solving linear ordinary differential equations based on multiquadric RBF networks. Ramuhalli et al. proposed an artificial neural network with an embedded finite-element model, for the solution of partial differential equations [11]. Tsoulos et al. demonstrated a hybrid method for the solution of ordinary and partial differential equations, which employed neural networks, based on the use of grammatical evolution, periodically enhanced using a local optimization procedure [12]. In all the aforementioned cases, the neural networks functioned as direct solvers of differential equations. In this work, however, we use the constructed neural network, not as a direct solver, but as a means to generate proper Runge-Kutta coefficients. In this aspect, there has been little relevant published material. For instance in [13], Tsitouras constructs a multilayer feedforward neural network that uses input data associated with an initial value problem and trains the network to produce the solution of this problem as output data, thus generating (by construction of the network) coefficients for a predefined number of Runge-Kutta stages. In this work we construct an artificial neural network that can generate the coefficients of two-stage Runge-Kutta methods. We consider the two-body problem, which is a typical case of an initial value problem where Runge-Kutta methods apply, and therefore the resulting method is specialized in solving it. The comparison shows that the new method is more efficient than the classical methods and thus proves the capability of the constructed neural network to create new Runge-Kutta methods. The structure is as follows: in Sections 2 and 3 we present the basic theory for Runge-Kutta methods and Artificial Neural Networks respectively. In Section 4 we present the implementation of the neural network that applies to a two-stage Runge-Kutta method which solves the two-body problem numerically, while in Section 5 the derivation of the method is provided. In Section 6 we demonstrate the final results along with the comparison of the new method to other well-known methods and finally in Section 7 we reach some conclusions about this work. 2 Runge-Kutta methods We consider a two-stage Runge-Kutta method to solve the first order initial value problem v′ (t) = f (t, v) and v(t0 ) = v0 (1) At each step of the integration interval, we approximate the solution of the initial value problem at tn+1 , where tn+1 = tn + h. For the two-stage Runge-Kutta method, the approximate solution vn+1 is given by v n + 1 = v n + h ( b1 k 1 + b2 k 2 ) (2) Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 3 where k 1 = f ( tn , v n ) (3) k2 = f (tn + c2 h, vn + h a21 k1 ) (4) The coefficients for this set of methods can be presented by the Butcher tableau given below c2 a21 b1 b2 An explicit two-stage Runge-Kutta method can be of second algebraic order at most (see [14]). In order for that to hold, the following conditions must be satisfied b1 + b2 = 1 (5) 1 (6) 2 The extra condition that needs to be satisfied in order for the method to be consistent is a21 = c2 (7) b2 c2 = 3 Artificial Neural Networks An artificial neural network (ANN) is a network of interconnected artificial processing elements (called neurons) that co-operate with each other in order to solve specific problems. ANNs are inspired by the structure and functional aspects of biological nervous systems and therefore present a resemblance. Haykin in [15] defines an artificial neural network as ”a massively parallel distributed processor that has a natural propensity for storing experiential knowledge and making it available for use”. ANNs, similarly to brains, acquire knowledge through a learning process, which is called training. That knowledge is stored in the form of synaptic weights, whose values express the strength of the connection between two neurons. There are many types of artificial neural networks, depending on the structure and the means of training. An ANN, in its simplest form, consists of a single neuron, in what we call the Perceptron model. The Perceptron is connected with a number of inputs: x1 , x2 , ..., xn . A weight corresponds to each of the neuron’s connections with the inputs and expresses the specific input’s significance to the calculation of the Perceptron’s output. Therefore, there is a number of weights equal to the number of inputs, with wi being the weight that corresponds to input xi , and wi xi being the combined weighted input. The neuron is also connected with a weight, which is not connected with any input and is called bias (symbolized by b). The weighted input data are essentially being mapped to an output value, through the transfer function of the neuron. The transfer function consists of the net input function and the activation function. The first function undertakes the task of combining the various weighted inputs into a scalar value. Typically, the summation function is used for this purpose, thus the net input in that case is described by the following expression 4 Angelos A. Anastassi Fig. 1 An artificial neuron (Perceptron) u= n X wi x i + b (8) i=1 For ease of use, the bias can be treated as any other weight and be represented as w0 , with x0 = 1, so the previous expression becomes u= n X wi x i (9) i=0 The activation function receives the output u of the net input function and produces its own output y (also known as the activation value of the neuron), which is a scalar value as well. The general form of the activation function is y = f (u) (10) What is required of a neural network, such as the Perceptron, is the ability to learn. That means the ability to adjust its weights in order to be able to produce a specific set of output data for a specific set of input data. In the supervised type of learning (which is used to train the Perceptron and other types of neural networks), besides the input data, we provide the network with target data, which are the data that the output should ideally converge to, with the given set of input. The Perceptron uses an estimation of the error as a measure that expresses the divergence between the actual output and the target. In that sense, training is essentially the learning process of adjusting the synaptic weights, in order to minimize the aforementioned error. To achieve this, a gradient descent Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 5 algorithm called delta rule is employed, also known as the Least Mean Square (LMS) method or the Widrow-Hoff rule, developed by Widrow and Hoff in [16]. The training process that uses the delta rule has the following steps: 1. Assign random values to the weights 2. Generate the output data for the set of the training input data 3. Calculate the error E , which is given by a norm of the differences between the target and the output data 4. Adjust the weights according to the following rule ∆wi = −η ∂E ∂u ∂E = −η = ηδxi ∂wi ∂u ∂wi (11) Where η is a small positive constant, called learning step or learning rate, and δ (delta error) is equal to the following expression δ=− ∂E ∂u (12) 5. Repeat from step 2, unless the error E is less than a small predefined constant ǫ or the maximum number of iterations has been reached With each iteration of the learning method, the weights are adjusted in such manner as to reduce the error. Notice that the correction ∆wi of the weights is ∂E , since the desired goal is the minimization of proportional to the negative of ∂w i the error. Strictly speaking, the Perceptron is not technically a network, since it consists of a single neuron. An actual network and the most common form of ANNs is the multilayer Perceptron (MLP), which falls in the general category of feedforward neural networks (FFNNs). Feedforward are the neural networks that contain no feedback connections, i.e. the connections do not form a loop, but are instead all directed towards the output of the network. A multilayer Perceptron consists of various layers, namely the input layer, the output layer, and one or more hidden layers. The hidden layers are not visible externally of the network, hence the ”hidden” property. Each of the layers consists of one or more neurons, except for the input layer, whose input nodes simply function as an entry point for the input data. Each layer is connected to the previous and the next layer, thus providing a pathway for the data to travel throughout the network. When a layer is connected to another, each neuron of the first layer is connected to every neuron of the second layer. In this way, the output of one layer becomes the input for the next one. Therefore, to calculate the output of the MLP, the data (either input data or activation values) are being propagated forward through the neural network. To train the MLP, the backpropagation method is used. Backpropagation is a generalization of the delta rule that is used to train the Perceptron. The method, essentially, functions in the same way for each neuron of the MLP, as simple delta rule does for the Perceptron. The difference lies in the fact that, since the learning procedure requires the calculation of the δ error, the neurons that reside inside the hidden layers must be provided with the δ errors of the neurons of the next layer. For this reason, the δ errors are first calculated for the neurons of the output layer and are propagated backwards (hence the backpropagation term), all across the network. The procedure is the following: 6 Angelos A. Anastassi Fig. 2 A typical multilayer Perceptron 1. Assign random values to the weights 2. Generate the activation values of each neuron, starting from the first hidden layer and continue until the output layer 3. Calculate the error E , which is given by a norm of the differences between the target and the output data 4. Calculate the δ errors of each neuron of the output layer, according to (12) 5. Calculate the δ errors of each neuron of the previous layer, according to δ = f ′ (u) M X w i δi (13) i=1 Where f (u) is the activation function of the neuron, M is the number of the neurons in the next layer, and δi is the δ error of the ith neuron of the next layer 6. Repeat step 5 for the previous layer, until all δ errors for all the hidden layers have been calculated 7. Adjust the weights according to (11) 8. Repeat from step 2, unless the error E is less than a small predefined constant ǫ or the maximum number of iterations has been reached 4 Implementation We introduce a feedforward neural network that can generate the coefficients of two-stage Runge-Kutta methods, specifically designed to apply to the needs of the two-body problem. The two-body problem is described by the following system of equations p x y x′′ = − 3 , y ′′ = − 3 , r = x2 + y 2 (14) r r whose analytical solution is given by x = cos t, y = sin t (15) Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 7 Since a Runge-Kutta method can only solve a system of first order ordinary differential equations, v according to the notation of equation (1), can be given by v = [v1 , v2 , v3 , v4 ] = x, y, x′ , y ′ (16) and h f (t, v) = v′ (t) = v1′ , v2′ , v3′ , v4′ = v3 , v4 , − p v2 v1 , − 3 , r = v12 + v22 r3 r i (17) From now on we will be using x, y, x′ , y ′ , for being more straightforward than v1 , v2 , v3 , v4 . The neural network we have constructed has six inputs: The coordinates of the second body: x and y , the derivatives of the coordinates with respect to the independent variable t: x′ , y ′ , the steplength h and a ”dummy” input with the constant value of 1. The input data consist of a matrix, where each column is a vector, intended to be used by the corresponding input of the network. The matrix satisfies the following equation ′ xn , yn , x′n , yn , h, 1 = [cos tn , sin tn , − sin tn , cos tn , h, 1] (18) With tn+1 = tn + h, where n = 1, 2, ..., N and N is the number of total steps. Similarly, the target matrix satisfies the following equation [xn+1, yn+1 , x′n+1 , yn′ +1 ] = [cos tn+1 , sin tn+1 , − sin tn+1 , cos tn+1 ] (19) The input and the target data are generated through a certain procedure. In this procedure, we set a number of parameters, namely the integration interval [ts , te ], the x0 , y0 , x′0 , y0′ values that correspond to v0 as notated in equation (1), and the constant steplength h. With the use of the integration interval and the steplength, the number of steps N is calculated. For each step, the process generates the input and target data that correspond to the starting and ending point of the integration step respectively. Thus, two matrices are generated by using the theoretical solution of the problem and consist of x, y, x′ , y ′ at the beginning and at the end of each integration step. Apart from the input layer, the network comprises of thirteen additional layers, each consisting of a single neuron. Each layer is connected to one or more other layers, always in a feedforward fashion. Most of the connections have a fixed, unadjustable weight, whose value is equal to one. Practically, that means that the significance of the specific connections does not vary in the progress of the training phase. The connections that do have an adjustable weight are those related with the three coefficients of the Runge-Kutta method, namely a21 , b1 , and b2 . The coefficient c2 is equal to a21 , therefore it does not need to be explicitly calculated through the use of the neural network. The neural network is constructed in such manner as to replicate the actual Runge-Kutta methodology and produce the numerical solution for the two-body problem for each step of the integration. In contrast to the actual methodology, the approximate solution at the end of each step is not used as a starting point for the next step, but instead of this, theoretical values are used as starting points for each step’s calculations. 8 Angelos A. Anastassi Fig. 3 The constructed neural network Most of the layers do not use the standard summation function as the net input function, but a custom one, different in most of the cases. The activation function in every neuron is the identity function, described by the following equation y = f (u) = u (20) Therefore, the transfer function in each case is, in essence, the net input function. Below, there is the list of the inputs and the layers of the network, their input and output connections (with any of the 6 inputs or 13 layers) and in parentheses the expression that they represent: Input 1: – Output connections: layers 1 (k1x′ ), 2 (k1y ′ ), 6 (k2x′ ), 7 (k2y ′ ), 10 (xn+1 ) – Value: xn Input 2: – Output connections: layers 1 (k1x′ ), 2 (k1y ′ ), 6 (k2x′ ), 7 (k2y ′ ), 11 (yn+1 ) – Value: yn Input 3: – Output connections: layers 8 (k2x ), 10 (xn+1 ), 12 (x′n+1 ) Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 9 – Value: x′n = k1x Input 4: ′ – Output connections: layers 9 (k2y ), 11 (yn+1 ), 13 (yn +1 ) ′ – Value: yn = k1y Input 5: ′ – Output connections: layers 3 (h a21 ), 10 (xn+1 ), 11 (yn+1 ), 12 (x′n+1 ), 13 (yn +1 ) – Value: h Input 6: – Output connections: layers 4 (b1 ), 5 (b2 ) – Value: 1 Layer 1: – Input connections: inputs 1 (xn ), 2 (yn ) – Output connections: layers 6 (k2x′ ), 7 (k2y ′ ), 12 (x′n+1 ) – Net input function: fx′ (xn , yn ) = − √ 2xn 2 3 ( xn +yn ) – Output value: k1x′ Layer 2: – Input connections: inputs 1 (xn ), 2 (yn ) ′ – Output connections: layers 6 (k2x′ ), 7 (k2y ′ ), 13 (yn +1 ) – Net input function: fy ′ (xn , yn ) = − √ 2yn 2 3 ( xn +yn ) – Output value: k1y ′ Layer 3: – – – – Input connections: input 5 (h) Output connections:P layers 6 (k2x′ ), 7 (k2y′ ), 8 (k2x ), 9 (k2y ) Net input function: Output value: h a21 Layer 4: – – – – Input connections: input 6 (1) Output connections:P layers 10 (xn+1), 11 (yn+1), 12 (x′n+1), 13 (yn′ +1 ) Net input function: Output value: b1 Layer 5: – – – – Input connections: input 6 (1) Output connections:P layers 10 (xn+1), 11 (yn+1), 12 (x′n+1), 13 (yn′ +1 ) Net input function: Output value: b2 Layer 6: – – – – Input connections: inputs 1 (xn ), 2 (yn ), layers 1 (k1x′ ), 2 (k1y′ ), 3 (h a21 ) Output connections: layer 12 (x′n+1) Net input function: f2x′ = fx′ (xn + h a21 k1x′ , yn + h a21 k1y′ ) Output value: k2x′ 10 Angelos A. Anastassi Layer 7: – – – – Input connections: inputs 1 (xn ), 2 (yn ), layers 1 (k1x′ ), 2 (k1y′ ), 3 (h a21 ) Output connections: layer 13 (yn′ +1 ) Net input function: f2y′ = fy′ (xn + h a21 k1x′ , yn + h a21 k1y′ ) Output value: k2y′ Layer 8: – – – – Input connections: input 3 (x′n = k1x ), layer 3 (h a21 ) Output connections: layer 10 (xn+1) Net input function: f2 = x′n + h a21 k1x = x′n · (1 + h a21 ) Output value: k2x Layer 9: – – – – Input connections: input 4 (yn′ = k1y ), layer 3 (h a21 ) Output connections: layer 11 (yn+1 ) Net input function: f2 = yn′ + h a21 k1y = yn′ · (1 + h a21 ) Output value: k2y Layer 10: – Input connections: inputs 1 (xn ), 3 (x′n = k1x ), 5 (h), layers 4 (b1 ), 5 (b2 ), 8 ( k2 x ) – Output connections: network output – Net input function: f3 = xn + h · (b1 k1x + b2 k2x ) – Output value: xn+1 Layer 11: ′ – Input connections: inputs 2 (yn ), 4 (yn = k1y ), 5 (h), layers 4 (b1 ), 5 (b2 ), 9 ( k2 y ) – Output connections: network output – Net input function: f3 = yn + h · (b1 k1y + b2 k2y ) – Output value: yn+1 Layer 12: – Input connections: inputs 3 (x′n = k1x ), 5 (h), layers 1 (k1x′ ), 4 (b1 ), 5 (b2 ), 6 ( k2 x ′ ) – Output connections: network output – Net input function: f3 = x′n + h · (b1 k1x′ + b2 k2x′ ) – Output value: x′n+1 Layer 13: ′ – Input connections: inputs 4 (yn = k1y ), 5 (h), layers 2 (k1y′ ), 4 (b1 ), 5 (b2 ), 7 ( k2 y ′ ) – Output connections: network output ′ – Net input function: f3 = yn + h · ( b 1 k1 y ′ + b 2 k2 y ′ ) ′ – Output value: yn +1 Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks ⋆ 11 5 Method derivation The derivation of the coefficients for the RK method described in (2), includes certain subroutines: – Data generation – Neural network training – Conversion to fraction During the data generation phase, we generate the input and target data that are going to be used to train the neural network. We have used various integration intervals [ts , te ], from [0, 2π ] up to [0, 200π ] and various steplengths, from h = π4 π down to 512 . As we mentioned before, the input and the target data are described by the equations (18) and (19) respectively. Therefore, after the data generation, we obtain an input matrix of size 6 × N and a target matrix of size 4 × N . Where N is the number of steps and the following equation holds N= te − ts h (21) During the training phase, we use the generated data to train the neural network. The method used to conduct the training is a variation of the typical backpropagation method, called Gradient Descent with Momentum and Adaptive Learning Rate Backpropagation or GDX. The back-propagation algorithm and its numerous variants constitute the most popular learning technique for feedforward neural networks [15]. However, a shortcoming of the original backpropagation algorithm is that it can be easily trapped in local minima. In order to deal with this disadvantage, the addition of a momentum term has been proposed. GDX includes the momentum term in order to be able to escape from local minima, but also has been found to present faster convergence and lower training times compared to other competing methods [17]. Instead of a standard error estimation function, we use a custom one for this purpose, which is shown below \|\|Output − Target\|\|∞ + \|b1 + b2 − 1\| + \|a21 b2 − 1 \| 2 (22) The error to be minimized is the maximum absolute difference between the output and the target data, plus some added expressions to satisfy the algebraic conditions. The terms \|b1 + b2 − 1\| and \|a21 b2 − 12 \| are used to satisfy the conditions (5) and (6) respectively. The coefficient a21 is used interchangeably with c2 , due to (7). As a result of the training phase, the obtained coefficients a21 , b1 and b2 satisfy the conditions to a certain degree of accuracy. Additionally, the Runge-Kutta method constructed with the generated coefficients can produce an output, sufficiently close to the target data. During the last phase, the coefficient a21 is converted into a fraction to simplify the method, with an insignificant loss of accuracy. The rest of the coefficients are calculated with the use of the fractional a21 , in order for the algebraic and consistency conditions, (5), (6) and (7) respectively, to be satisfied. The coefficients of the new method are presented in Table 1. For the current implementation, the number of maximum iterations was set at 10000. That value was selected empirically, as the neural network provided a 12 Angelos A. Anastassi solution after approximately 5000 iterations and effectively completed the training, as it could not improve on the performance any further. 6 Results The best method was provided by using the integration interval [0, 2π ] and the π . Training with other integration intervals and steplengths have steplength h = 128 returned similar results. 2π represents a full oscillation for the two-body problem, which in part explains why training over wider intervals does not yield better results. Apart from the new method, the training has also generated some well known classical methods, some of which are given later in this section. We compare the new method with three classical two-stage explicit RungeKutta methods. The corresponding Butcher tableaus of all methods are shown in Tables 1-4. 11 26 11 26 2 − 11 13 11 Table 1 New method 1 1 1 2 1 2 Table 2 Heun’s method 1 2 1 2 0 1 Table 3 Midpoint method 2 3 2 3 1 4 3 4 Table 4 Third classical method A comparison between the classical methods and the new method is presented in figure 4, which regards the numerical integration of the two-body problem over the interval [0, 1000]. The vertical axis represents the accuracy as expressed by −log10 (maximum absolute error), while the horizontal axis represents the total number of function evaluations that are required for the computation. The latter is provided by the formula F.E. = s · N , where s = 2 and stands for the number of stages of the RK method and N is the number of total steps. We can see that the new method is more efficient among all methods compared. Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks 13 ⋆ Two-body Problem 8 7 New Method Midpoint Classical 3 Heun -log10(error) 6 5 4 3 2 1 2.5E+05 2.5E+06 2.5E+07 2.5E+08 Function Evaluations Fig. 4 Efficiency of the compared methods for the integration of the two-body problem 7 Conclusions We have constructed an artificial neural network that can generate the optimal coefficients of an explicit two-stage Runge-Kutta method. The network is specifically designed to apply to the needs of the two-body problem, and therefore the resulting method is specialized in solving that problem. Following the implementation of the network, the latter is trained to obtain the optimal values for the coefficients of the method. The comparison has shown that the new method developed in this work is more efficient than other well known methods and thus proved the capability of the constructed neural network to create new efficient numerical algorithms. References 1. Cybenko G (1989) Approximations by superpositions of sigmoidal functions. Mathematics of Control, Signals, and Systems 2:303–314 2. Hornik K (1991) Approximation Capabilities of Multilayer Feedforward Networks. Neural Networks 4:251–257 3. Dissanayake MWMG, Phan-Thien N (1994) Neural-network-based approximations for solving partial differential equations. Communications in Numerical Methods in Engineering 10:195–201 4. 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SPATIAL DIFFUSENESS FEATURES FOR DNN-BASED SPEECH RECOGNITION IN NOISY AND REVERBERANT ENVIRONMENTS Andreas Schwarz, Christian Huemmer, Roland Maas, Walter Kellermann Multimedia Communications and Signal Processing Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) Cauerstr. 7, 91058 Erlangen, Germany arXiv:1410.2479v2 [cs.CL] 16 Feb 2015 {schwarz, huemmer, maas, wk}@lnt.de ABSTRACT We propose a spatial diffuseness feature for deep neural network (DNN)-based automatic speech recognition to improve recognition accuracy in reverberant and noisy environments. The feature is computed in real-time from multiple microphone signals without requiring knowledge or estimation of the direction of arrival, and represents the relative amount of diffuse noise in each time and frequency bin. It is shown that using the diffuseness feature as an additional input to a DNN-based acoustic model leads to a reduced word error rate for the REVERB challenge corpus, both compared to logmelspec features extracted from noisy signals, and features enhanced by spectral subtraction. Index Terms— Speech Recognition, Reverberation, Diffuse Noise, Deep Neural Networks 1. INTRODUCTION In automatic speech recognizers (ASR) based on Gaussian Mixture Models and Hidden Markov Models (GMM-HMM), a wide variety of transformations and feature extraction steps is currently being employed with the aim of extracting and normalizing the information contained in the time-domain input signal as efficiently as possible. Recently, with the development of effective training methods for acoustic models based on multiple-layer neural networks, which are often summarized under the term “deep neural networks” (DNN) [1], it has become possible for the acoustic model to learn relationships between features and phonemes to a higher degree than it is possible with manually implemented feature transformation steps. For example, it has been found that simple filterbank features outperform mel-frequency cepstral coefficients (MFCCs) [2, 3], and it is conceivable that, given large amounts of training data and sufficiently complex network structures, time-domain signals may at some point even be directly used as inputs to a neural network. Although the trend in ASR goes towards replacing explicit processing stages by implicit learning, for noise- and reverberation-robust ASR using microphone arrays, spatial information is still predominantly being exploited in a separate speech The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for supporting this work (contract number KE 890/4-2). enhancement preprocessor, e.g., in the form of beamforming [4], multichannel linear prediction [5], blocking matrix-based postfilters [6] or coherence-based postfilters [7]. The single-channel output of the preprocessor is then used to compute features for ASR. In some GMM-HMM-based systems, spatial information is exploited indirectly in uncertainty decoding-based approaches, e.g., in [8], where the feature uncertainty is derived from a noise estimate obtained in a multichannel signal enhancement stage. For DNN-based acoustic models, “noise-aware training” has been proposed [3], where a noise estimate is appended to the noisy feature vector. This has been evaluated for stationary noise estimates [3] and noise-estimates derived from time-frequency masking [9], but may in principle also be used for noise estimates obtained from spatial processing. In [10] and [11], feature vectors from multiple microphones are concatenated to form the input of a DNN-based acoustic model, however, no spatial phase information is exploited. Inspired by the trend towards moving more explicit feature processing steps into the DNN, we propose to exploit spatial information about the diffuseness of the sound field directly by incorporating it into the acoustic model of a DNN-based speech recognizer. The diffuseness estimate is derived from the complex coherence between two omnidirectional microphones and has been used for signal enhancement based on the assumption that late reverberation and noise components can be modeled as diffuse noise [7]. Using the diffuseness as a feature is motivated by the fact that humans exploit similar spatial information for speech recognition in reverberant and noisy environments [12, 13], as it was found that the human auditory system treats spectro-temporal variations in the interaural coherence as “a perceptual surrogate for spectro-temporal variations in the energy of speech signals” [13]. The aim is to learn similar behavior in a DNN-based acoustic model. We first describe the signal model for the estimation of the diffuseness from the instantaneous spatial coherence of a reverberated and noisy speech signal. Then, we show how this estimate is integrated into a feature extraction scheme for ASR, and describe the structure of the DNN-based speech recognizer. Finally, we evaluate the proposed feature on the two-channel task of the REVERB challenge [14], showing that the proposed approach outperforms both noisy multi-condition training and multichannel spectral subtraction-based signal enhancement. 2. BLIND DIFFUSENESS ESTIMATION We consider a reverberated and noisy speech signal recorded by two omnidirectional microphones. The signal xi (t) recorded at the i-th microphone is composed of the desired signal component si (t) and the undesired noise component ni (t) comprising additive noise and late reverberation, i.e., xi (t) = si (t) + ni (t), i = 1, 2. The microphone, desired, and noise signals are represented in the short-time Fourier transform (STFT) domain by the corresponding uppercase letters, i.e., Xi (k, f ), Si (k, f ) and Ni (k, f ), respectively, with the discrete frame index k and continuous frequency f , and the auto- and cross-power spectra Φxi xj (k, f ), Φsi sj (k, f ), Φni nj (k, f ). Note that the continuous frequency f is used here for generality; in practice, f denotes discrete values along the frequency axis. It is assumed that the auto-power spectra of all signal components are identical at both microphones, i.e., Φsi si (k, f ) = Φs (k, f ), Φni ni (k, f ) = Φn (k, f ). The timeand frequency-dependent signal-to-noise ratio (SNR) of the microphone signals can then be defined as SNR(k, f ) = Φs (k, f ) . Φn (k, f ) (1) The aim in the following is to estimate the SNR from the coherence of the mixed sound field Γx (k, f ). This coherence is first estimated as Φ̂x1 x2 (k, f ) , Γ̂x (k, f ) = q Φ̂x1 x1 (k, f )Φ̂x2 x2 (k, f ) where the spectral estimates Φ̂xi xj (k, f ) are obtained by recursive averaging: Φ̂xi xj (k, f ) = λΦ̂xi xj (k−1, f ) + (1−λ)Xi (k, f )Xj∗ (k, f ), (7) with a constant forgetting factor λ between 0 and 1. In [15, 7], it was shown that (3) can be solved for the SNR without requiring knowledge of Γs , using only the assumption that the desired signal is fully coherent, i.e., \|Γs \| = 1. This yields a “blind” estimator for the SNR (or coherent-to-diffuse ratio, CDR) from the mixture coherence Γ̂x (k, f ) which does not require knowledge or estimation of the signal DOA. The estimator is given in (8) at the bottom of this page (the indices k and f are omitted for brevity). The CDR can be transformed into the diffuseness [16] d D̂(k, f ) = [CDR(k, f ) + 1]−1 , The complex spatial coherence functions of the desired signal and noise components are given by Φs1 s2 (k, f ) Φn1 n2 (k, f ) Γs (f ) = , Γn (f ) = , Φs (k, f ) Φn (k, f ) (2) and are assumed to be time-invariant, i.e., dependent only on the spatial characteristics of the signal components. It is furthermore assumed that signal and noise components are orthogonal, such that Φx (k, f ) = Φs (k, f ) + Φn (k, f ). The complex spatial coherence of the mixed sound field can then be written as a function of the SNR and the signal and noise coherence functions: Γx (k, f ) = SNR(k, f )Γs (f ) + Γn (f ) . SNR(k, f ) + 1 (3) The direct sound is now modeled as a plane wave with an unknown direction of arrival (DOA) and therefore unknown time difference of arrival ∆t, while the undesired noise and late reverberation component is modeled as a diffuse (spherically isotropic) sound field. The corresponding spatial coherence functions for the direct and diffuse sound components are then given by Γs (f ) = ej2πf ∆t , (4) d Γn (f ) = Γdiffuse (f ) = sinc(2πf ), c (5) respectively. The direct signal coherence has a magnitude of one with an unknown phase determined by the DOA, while the diffuse noise coherence only depends on the known microphone spacing d. 2 d CDR(k, f) = Γn Re{Γ̂x } − \|Γ̂x \| − (6) (9) which can be thought of as the relative amount of diffuse signal power in the respective time and frequency bin. Since the diffuseness is bounded between 0 and 1, it is more convenient to use as basis for feature computation than the CDR itself. 3. FEATURE EXTRACTION FOR ASR Fig. 1 shows the block diagram of the proposed feature extraction scheme. The microphone signals are first windowed and transformed into the STFT domain. The upper path then corresponds to a classical feature extraction of NMel -dimensional logmelspec (often termed “log-filterbank” or “Log FBank”) features, where the two microphone signals are combined by averaging the spectral powers computed from each microphone, and NMel triangular Mel-scaled weighting filters are applied. The second path shows the extraction of enhanced logmelspec features, where signal enhancement based on the diffuseness estimate is performed by multiplication in the STFT domain with a gain factor G(k, f ), which is computed as described in [7] according to the spectral magnitude subtraction rule. The third path illustrates the computation of the proposed “meldiffuseness” features: the diffuseness D̂(k, f ) is estimated as described in the previous section, and the same NMel triangular Mel weighting filters that are used in the logmelspec feature extraction are applied to create an output vector of the dimensionality NMel . Finally, for comparison, the Mel-weighted magnitude-squared coherence (“melmsc”) is computed as a feature. While the magnitude-squared coherence of a q 2 2 2 Γ2n Re{Γ̂x } − Γ2n \|Γ̂x \| + Γ2n − 2 Γn Re{Γ̂x } + \|Γ̂x \| 2 \|Γ̂x \| − 1 (8) Window X1(k, f ) STFT \|⋅\| 2 logmelspec NSTFT Window avg X2(k, f ) STFT log Mel weighting NMel \|⋅\|2 G(k, f ) NSTFT enhanced logmelspec log Mel weighting NMel Coherence estimation Γˆ x(k, f ) Diffuseness estimation ˆ f) D(k, meldiffuseness Mel weighting NMel melmsc \|⋅\|2 Mel weighting NMel Fig. 1. Feature extraction of logmelspec, enhanced logmelspec, meldiffuseness and melmsc features from 2-channel signals. 1 http://www.lms.lnt.de/files/publications/icassp2015-diffuseness.zip Mel band a) logmelspec 20 10 100 200 300 0 −5 −10 −15 −20 Mel band b) enhanced logmelspec 20 10 100 200 300 0 −5 −10 −15 −20 c) meldiffuseness Mel band mixed sound field is also related to the amount of diffuse noise, this relationship is strongly dependent on the signal DOA and the microphone spacing, therefore the melmsc feature is expected to perform worse than the proposed diffuseness estimate. The interesting question is now how using concatenated logmelspec and meldiffuseness features as input to the neural network compares to using logmelspec features which have been enhanced in the STFT domain. Since the trend in DNN-based acoustic modeling goes towards replacing explicit feature preprocessing and normalization steps by implicit learning, one might consider using the complex spatial coherence directly as feature. Note, however, that the proposed diffuseness feature has two significant advantages over the complex coherence. The complex coherence depends on two additional variables, namely the DOA and the microphone spacing, both of which would need to be sufficiently represented in the training data. Moreover, the diffuseness is a characteristic of the sound field which is independent of the microphone array geometry, and may therefore also be estimated from microphone arrays with other geometries, e.g., spherical arrays [17] or arrays consisting of directional microphones [18], without requiring adaptation of the acoustic model. It is interesting to note that the additional temporal smoothing which is required for the estimation of the coherence (and therefore the diffuseness) has parallels in the human auditory system, where reaction to changes in interaural coherence was found to be more sluggish than reaction to changes in energy [19]. For the results presented in this paper, the time-domain signals (sampled at 16 kHz) are windowed using a 25 ms Hann window with a frame shift of 10 ms and transformed using a 512point DFT, resulting in NSTFT = 257 subbands in the STFT domain. The spatial coherence is estimated using the forgetting factor λ = 0.68. NMel = 24 triangular Mel-scale weighting filters are used, covering a frequency range from 64 to 8000 Hz. MATLAB code for the feature computation is provided online1 . Fig. 2 illustrates the features computed from a noisy and reverberated speech signal taken from the multi-condition training set of the REVERB challenge corpus (LargeRoom2). The coherence-based spectral enhancement visibly reduces the noise floor and the smearing of the speech features over time. The meld- 20 10 100 200 300 1 0.8 0.6 0.4 0.2 0 frame Fig. 2. Features for the reverberated utterance “The statute allows for a great deal of latitude”. iffuseness clearly highlights portions of the signal where noise or reverberation components are dominant. 4. DNN-BASED SPEECH RECOGNITION We employ the Kaldi toolkit [20] as ASR back-end system using the WSJ0 trigram 5k language model of the REVERB challenge and 3551 context-dependent triphone-states in the acoustic model. In a first step, we set up a GMM-HMM baseline system based on Weninger et al. [4] by extracting 13 mean and variance normalized MFCCs (including the zeroth cepstral coefficient), followed by ±4 frame splicing, linear discriminant analysis (LDA), maximum likelihood linear transform (MLLT), and feature-space maximum likelihood linear regression (fMLLR) (see [4, 21] for a detailed description). After conventional maximum likelihood training, discriminative training is performed with the boosted maximum mutual information (bMMI) criterion [4]. The GMM- Table 1. ASR Word Error Rate for the REVERB challenge evaluation and development test sets. Recognizer Feature GMM-HMM MFCC-LDA-MLLT-fMLLR logmelspec+∆+∆∆ enhanced logmelspec+∆+∆∆ DNN-HMM logmelspec+∆+meldiffuseness logmelspec+∆+melmsc Room 1 near far 6.61 7.50 5.74 6.67 6.61 7.12 5.91 6.06 6.17 6.32 SimData Room 2 near far 9.42 16.59 7.65 13.92 7.65 12.18 6.94 10.96 7.04 12.25 HMM system is trained on the clean WSJCAM0 Cambridge Read News REVERB corpus [22]. The alignment of the training data to the HMM states is then extracted from the clean training data and used for the later multi-condition training of the DNN-HMM system. This technique is known to yield better results than a multi-condition state-frame alignment [9, 23]. The hybrid DNN-HMM Kaldi system is based on “Dan’s implementation” [20] using a maxout network with 2-norm nonlinearities/activation functions and 4 hidden layers, each one with an input dimension of 2000 and an output dimension of 400. In accordance with [2, 3], and as described in the previous section, we extract NMel = 24 static logmelspec coefficients, with or without applying coherence-based spectral subtraction enhancement in the STFT domain. Depending on the particular setup in Table 1, also Delta (∆), acceleration (∆∆), melmsc, and/or the proposed meldiffuseness features are derived. Mean and variance normalization and ±5 frame splicing is applied to the entire resulting feature vector. The training is performed on the REVERB multi-condition training set [14], consisting of 7861 noisy and reverberated utterances from the WSJCAM0 corpus, using greedy layer-wise supervised training, preconditioned stochastic gradient descent, “mixing up” [24] as well as final model combination [24]. 5. EVALUATION RESULTS We evaluate the proposed system using the two-channel task of the REVERB challenge [14]. The REVERB evaluation test set consists of ∼5000 reverberated and noisy utterances, partially created by convolution of clean WSJCAM0 utterances with impulse responses and mixing with recorded noise sequences (“SimData”), and partially consisting of multichannel recordings of speakers in a reverberant and noisy room from the MC-WSJAV corpus (“RealData”). For SimData, the reverberation times of the three rooms are approx. 0.25 s, 0.5 s and 0.7 s and the sourcemicrophone spacing is 0.5 m (near) or 2 m (far). For RealData, the reverberation time is approx 0.7 s and the source-microphone distance is 1 m (near) or 2.5 m (far). In both cases, an 8-channel circular microphone array with a diameter of 20 cm was used, of which two microphones with a spacing of d = 8 cm are selected for the two-channel recognition task which is evaluated here. First, we evaluate the word error rate (WER) obtained from the GMM-based recognizer with MFCC features, which is used to obtain the alignment. For the DNN-based recognizer, we compare logmelspec features extracted from the noisy signals, and Evaluation Set Room 3 near far 11.05 20.69 8.65 14.62 8.32 14.57 8.24 12.88 8.17 13.87 Avg 11.98 9.54 9.41 8.50 8.97 RealData Room 1 near far 31.17 30.15 28.45 29.14 28.46 29.07 27.82 26.27 27.34 27.99 Development Set SimData RealData Avg 30.66 28.80 28.77 27.05 27.67 Avg 12.14 9.68 9.13 7.92 8.68 Avg 31.61 24.93 25.29 24.19 24.73 enhanced logmelspec features. In both cases, the feature vector is extended by first- (∆) and second-order (∆∆) derivatives. Then, we evaluate the combination of noisy logmelspec features with spatial meldiffuseness or melmsc features; in this case, only firstorder derivatives (∆) are computed for the logmelspec features, in order to keep the overall dimension of the feature vectors the same (3NMel ). Table 1 shows the WER results for the REVERB challenge evaluation test set, and the average WER for the development test set. As expected, the DNN-based acoustic model achieves a lower WER than the GMM-based model. The diffuseness-based signal enhancement has a negligible effect on WER. This seems to contradict [15], where the same signal enhancement method led to a significantly lower WER, however, there, acoustic models were trained on clean speech. Apparently the effect of the multichannel spectral subtraction for signal enhancement is compensated by noisy multi-condition training. Using the combined noisy logmelspec and diffuseness features as input to the neural network however yields a significantly reduced WER. This confirms that the spatial information extracted from the coherence can be exploited more successfully by the DNN than by speech enhancement using spectral subtraction, even though, in this case, the frequency resolution of the meldiffuseness features is reduced compared to the diffuseness estimate used for spectral subtraction. The melmsc feature also leads to a reduced WER compared to noisy logmelspec features, although the improvement is smaller than with meldiffuseness features. 6. CONCLUSION It has been shown that spatial information extracted from multiple microphones does not necessarily have to be exploited in a signal enhancement front-end, but may be used more effectively as an additional feature input for a DNN-based speech recognizer. 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Multiparameter spectral analysis for aeroelastic instability problems Arion Pons1,* and Stefanie Gutschmidt2 1 Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK. Department of Mechanical Engineering, University of Canterbury, Christchurch 8140, New Zealand. 2 Key words AMS classification multiparameter eigenvalue problems; instability; aeroelasticity. 97M50; 65F15; 15A18; 15A69 This paper presents a novel application of multiparameter spectral theory to the study of structural stability, with particular emphasis on aeroelastic flutter. Methods of multiparameter analysis allow the development of new solution algorithms for aeroelastic flutter problems; most significantly, a direct solver for polynomial problems of arbitrary order and size, something which has not before been achieved. Two major variants of this direct solver are presented, and their computational characteristics are compared. Both are effective for smaller problems arising in reduced-order modelling and preliminary design optimization. Extensions and improvements to this new conceptual framework and solution method are then discussed. 1. Introduction Predicting and controlling aeroelastic instability forms a major part of the discipline of aeroelasticity. Many different physical systems show flutter instability, and many models exist to describe them. In a linear system, or the linearization of a nonlinear system, the onset of flutter or divergence can be formulated in the well-known stability criterion: Im(𝜒) > 0 for stability (1) where 𝜒 are the time-eigenvalues of the system according to the Fourier transform 𝑞(𝑡) = 𝑞̂𝑒 𝜄𝜒𝑡 for the system coordinate 𝑞 [1]. These eigenvalues may be nondimensionalised. Flutter occurs when the system parameters are such that the system is on the stability boundary, Im(𝜒f ) = 0. The flutter point may then be described as a tuple which includes the set of system parameters (particularly, an airspeed parameter) and the modal frequency of instability, 𝜒f . Typically only one or two flutter or divergence points are of industrial relevance. Eq. 1 is not however the only criterion that can be used to characterize instability. This leads us into the study of aeroelastic methods. Apart from the four established methods – the pmethod, classical flutter analysis, the k-method (or U-g method, or V-g method) and the p-k method; all of which are detailed and discussed in a number of reference works [1–3] – recent years have seen a proliferation of new aeroelastic methods. Several authors have refined the existing established methods for particular scenarios or applications [4–6]. The application of concepts from robust control theory have yielded a series of methods, including the 𝜇-method by Lind and Brenner [7,8], the 𝜇-k method by Borglund [9–12], and others [13–15]. The prime advantage of these 𝜇-type methods is that they facilitate the propagation of uncertainty * Corresponding author, adp53@cam.ac.uk, tel. +44 755 366 3296 Multiparameter analysis for instability problems distributions through the system, allowing a worst-case flutter speed estimate to be made in a system with high uncertainty. Other developments have come from other fields: Afolabi [16,17] characterized coupled-mode flutter as a loss of eigenvector orthogonality, using methods from catastrophe theory. Irani and Sazesh [18] used stochastic methods, while Gu et al. [19] devised a genetic algorithm, and a number of authors [20–23] have applied neural networks to the detection of flutter points. It is in the context of these developments that we propose our method of analyzing flutter problems. The central methodological contribution of this paper is the concept that the solution of an aeroelastic system for its flutter points is nothing other than a multiparameter eigenvalue problem. We will show the simple link between the aeroelastic stability problem and multiparameter spectral theory, and how this allows for direct solution of a variety of flutter problems. The purpose of this paper is to detail this link and to explore the mechanisms by which this direct solution may be accomplished. For this purpose we will apply our analysis to simple example problems – however, the method does extend to problems that are significantly more complex; these are considered in Pons and Gutschmidt [24, 25]. 2. Aeroelastic flutter as a multiparameter eigenvalue problem (MEP) Consider a linear finite-dimensional system with eigenvector 𝐱 ∈ ℂ𝑛 and arbitrary continuous dependence on both an eigenvalue parameter 𝜒 ∈ ℂ, and another structural or environmental parameter 𝑝 ∈ ℝ: (2) A(𝜒, 𝑝)𝐱 = 𝟎 𝑛×𝑛 where A ∈ ℂ . Any complex structural parameter can of course be split into two real parameters. The stability problem for this system (with respect to parameter 𝑝) is to find 𝑝 such that an eigenvalue of the problem (𝜒) has zero imaginary part. This point is the ‘stability boundary’: for a system with multiple structural parameters, the stability boundary may be a line or other higher-dimensional surface. We then note that the condition Im(𝜒) = 0 is equivalent to modifying the original definition of the problem such that 𝜒 ∈ ℝ and not 𝜒 ∈ ℂ. Such a manoeuvre does not seem to be immediately useful: under 𝜒 ∈ ℝ, a solution to Eq. 2 only exists on the stability boundary, and nowhere else. In order to develop, for example, iterative methods for flutter point calculation, we need to be able to define some form of solution in the subcritical and supercritical areas (above and below the stability boundary, respectively). There is an easy way of doing this. Following [26], we take the complex conjugate of Eq. 2 as another equation: A(𝜒, 𝑝)𝐱 = 𝟎, ̅ (𝜒, 𝑝)𝐱̅ = 𝟎. A (3) As 𝑝 ∈ ℝ and 𝜒 ∈ ℝ are unaffected by the conjugation, this operation enforces these conditions. This procedure has been utilized before in the analysis of delay differential equations [26], and (in a limited form) in the context of Hopf bifurcation prediction [27], but has never been applied to aeroelastic or other structural stability problems. Equation 3 is nothing other than a multiparameter eigenvalue problem (MEP): an eigenvalue problem in which the eigenvalue point is not simply defined by a scalar and an eigenvector, but by an 𝑛tuple and an eigenvector. A number of methods of analysis have been developed for such problems, and in this paper we will explore some of these. However, as the methods that are available depend strongly on the structure of matrix A, we will first define a system to work with. Page 2 of 20 Multiparameter analysis for instability problems 3. An Example Section model 3.1. Formulation Consider first the simple section model shown in Figure 1. This model has two degrees of freedom: plunge ℎ and twist 𝜃. The governing equations for this model are easy to derive; they are: 𝑚ℎ̈ + 𝑑ℎ ℎ̇ + 𝑘ℎ ℎ − 𝑚𝑥𝜃 𝜃̈ = −𝐿(𝑡), 𝐼𝑃 𝜃̈ + 𝑑𝜃 𝜃̇ + 𝑘𝜃 𝜃 − 𝑚𝑥𝜃 ℎ̈ = 𝑀(𝑡), (4) where 𝑚 and 𝐼𝑃 are the section mass and polar moment of inertia, 𝑘ℎ and 𝑘𝜃 are the section plunge and twist stiffnesses, 𝑑ℎ and 𝑑𝜃 are the section plunge and twist damping coefficients, and 𝑥𝜃 is the section’s static imbalance – defined as the distance along the 𝑥-axis from the pivot point to the centre of mass. Taking the Fourier transform, [ℎ(𝑡), 𝜃(𝑡)] = [ℎ̂, 𝜃̂]𝑒 𝜄𝜒𝑡 , of this model, we obtain: (−𝑚𝜒 2 + 𝜄𝑑ℎ 𝜒 + 𝑘ℎ )ℎ̂ + 𝑚𝑥𝜃 𝜒 2 𝜃̂ = 𝐿(𝜒, ℎ̂, 𝜃̂), 𝑚𝑥𝜃 𝜒 2 ℎ̂ + (−𝐼𝑃 𝜒 2 + 𝜄𝑑𝜃 𝜒 + 𝑘𝜃 )𝜃̂ = 𝑀(𝜒, ℎ̂, 𝜃̂). (5) To model the aerodynamic loads in the frequency domain we use Theodorsen’s unsteady aerodynamic theory [3]: 𝐿 = −𝜒 2 (𝐿ℎ ℎ̂ + 𝐿𝜃 𝜃̂), 𝑀 = 𝜒 2 (𝑀ℎ ℎ̂ + 𝑀𝜃 𝜃̂ ). (6) The aerodynamic coefficients {𝐿ℎ , 𝐿𝜃 , 𝑀ℎ , 𝑀𝜃 } are complex functions of 𝜅 – the reduced frequency; an aerodynamic parameter related to the airspeed (𝑈) by 𝜅 = 𝑏𝜒⁄𝑈. Other structural and environmental parameters involved are the air density (𝜌), the semichord (𝑏) and the distance along the 𝑥-axis from the midchord to the pivot point, as a fraction of the semichord (𝑎). See Pons [24] or Hodges and Pierce [3] for details. We assume that the flow over the airfoil is quasisteady: that is, that Theodorsen’s function takes a value of 1 universally [1,3]. We will deal with general Theodorsen aerodynamics in a later paper, as this requires iterative or approximate multiparameter solution methods which we will not cover here. We also assume without loss of generality a lift-angle of attack coefficient of 𝐶𝐿𝛼 = 2𝜋, as per thin-airfoil theory [28]. It is then customary to nondimensionalise Eq. 5. Further details of this are given in Pons [24]. The final result is a flutter problem of the form 1 1 ((M0 + G0 + G1 + G2 2 ) 𝜒 2 − D0 𝜒 − K 0 ) 𝐱 = 𝟎. 𝜅 𝜅 (7) with dimensionless parameters defined as in Table 1 and the nomenclature, and the matrix coefficients 𝑎 1 1 G0 = [𝑎 (1 + 𝑎2 )], 𝜇 8 (8) 1 −2𝜄 2𝜄(1 − 𝑎) G1 = [ ], 𝜇 −𝜄(1 + 2𝑎) 𝜄𝑎(1 − 2𝑎) Page 3 of 20 Multiparameter analysis for instability problems Table 1: Dimensionless parameter values for the section model Parameter Value mass ratio – 𝜇 20 radius of gyration – 𝑟 0.4899 0.5642 rad/s bending nat. freq. – 𝜔ℎ 1.4105 rad/s torsional nat. freq. – 𝜔𝜃 bending damping – 𝜁ℎ 1.4105 % torsional damping – 𝜁𝜃 2.3508 % −0.1 static imbalance – 𝑟𝜃 pivot point location – 𝑎 −0.2 Figure 1: Diagram of section model G2 = 1 0 [ 𝜇 0 2𝜄𝜁ℎ 𝜔ℎ D0 = [ 0 2 ], 1 + 2𝑎 M0 = [ 0 ], 2 2𝜄𝑟 𝜁𝜃 𝜔𝜃 1 −𝑟𝜃 𝜔2 K0 = [ ℎ 0 −𝑟𝜃 ], 𝑟2 0 2 2 ], 𝑟 𝜔𝜃 (8) Note that this nondimensionalisation is a convenience and not necessary prerequisite for using multiparameter solution methods. However something that will be of great use is to rearrange Eq. 7 into two polynomial forms by defining new eigenvalue parameters: Υ = 𝑈⁄𝑏, 𝜏 = 1⁄𝜅 , and 𝜆 = 1⁄𝜒. These two forms are then ((M0 + G0 )𝜒 2 + G1 Υ𝜒 + G2 Υ 2 − D0 𝜒 − K 0 )𝐱 = 𝟎, (9) which we term the Υ-𝜒 form, and ((M0 + G0 ) + G1 𝜏 + G2 𝜏 2 − D0 𝜆 − K 0 𝜆2 )𝐱 = 𝟎, (10) which we term the 𝜏-𝜆 form. Both are preferable to the 𝑘-𝜒 form, Eq. 7. We should note at this point that, although we have here derived these two forms for a small 2-DOF section model, these forms arise in many other models; often with significantly larger matrices. The solution methods we develop for this form will thus be applicable to a wide variety of problems. 3.2. Undamped system In many aeroelastic systems structural damping is negligible, in which case D0 = 0. While this may seem to be a trivial case of the preceeding systems, the omission of the structural damping term does allow us to change the structure of the system, leading to a faster computation time (as we will show later). In the case of Eq. 10, we have ((M0 + G0 ) + G1 𝜏 + G2 𝜏 2 − K 0 Λ)𝐱 = 𝟎, (11) where Λ = 𝜆2. This system is now linear in Λ, whereas it was previously quadratic in 𝜆. Note that the 𝜏-𝜒 form is the only polynomial form presented which allows us to make one parameter linear in this situation. Page 4 of 20 Multiparameter analysis for instability problems 3.3. Transformation into MEPs The three polynomial systems that we have introduced in this section (Eq. 9, 10 and 11) are not yet fully constrained MEPs, as they only consist of one equation. To constrain the system we employ the method introduced in Section 2 – the addition of an equation representing the conjugate of the initial equation. We obtain: ((M0 + G0 ) + G1 𝜏 + G2 𝜏 2 − K 0 Λ)𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝜏 2 − K ̅0 + G ̅ 0 Λ)𝐱̅ = 𝟎, ((M (12) ((M0 + G0 ) + G1 𝜏 + G2 𝜏 2 − D0 𝜆 − K 0 𝜆2 )𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝜏 2 − D ̅0 + G ̅0𝜆 − K ̅ 0 𝜆2 )𝐱̅ = 𝟎, ((M (13) ((M0 + G0 )𝜒 2 + G1 Υ𝜒 + G2 Υ 2 − D0 𝜒 − K 0 )𝐱 = 𝟎, ̅0 )𝜒 2 + G ̅1 Υ𝜒 + G ̅2 Υ 2 − D ̅0 + G ̅0𝜒 − K ̅ 0 )𝐱̅ = 𝟎. ((M (14) and These systems are all polynomial multiparameter eigenvalue problems [29]. A number of solution methods are known for such systems. 4. Direct solution via linearization 4.1. Linearization Any polynomial MEP can be made into a linear one (consisting of a zeroth-order term and first-order terms in each eigenvalue) via the process of linearization [30]. This process bears resemblance to the linearization of single-parameter polynomial eigenvalue problems; a process which is well-known [31]. Multiparameter linearization is relevant because a number of direct solution methods exist for linear MEPs. Consider first Eq. 12. This equation contains only one quadratic variable (𝜏), as Λ is already linear. To linearize this system, we define a new eigenvector which contains a factor of 𝜏: 𝐪 = [𝐱; 𝜏𝐱]. By expanding the system’s coefficient matrices and using this factor of 𝜏 in the eigenvector to reduce the order of the quadratic term, we obtain a linear problem of double the size: M0 + G0 0 ̅0 ̅ M +G ([ 0 0 ([ 0 −K ]+[ 0 −𝐼𝑛 0 ̅ 0 ] + [−K 0 −𝐼𝑛 0 G 0 ]Λ + [ 1 𝐼𝑛 0 ̅ 0] Λ + [G1 𝐼𝑛 0 G2 ] 𝜏) 𝐪 = 𝟎 0 ̅2 G ̅=𝟎 ] 𝜏) 𝐪 0 (15) The upper rows of Eq. 15 represents Eq. 12 directly, and the lower row represents the identity 𝐼𝑛 (𝜏𝐱) = 𝜏𝐼𝑛 𝐱. Note that other linearizations of similar form are possible. However, irrespective of the exact linearization used, the resulting system will be of the form: (A + BΛ + C𝜏)𝐪 = 𝟎, ̅+B ̅Λ + C̅𝜏)𝐪 (A ̅ = 𝟎. (16) Note that it is not actually necessary to linearize the second (i.e. conjugate) equation of the aeroelastic system, because the linearized conjugate equation will be the conjugate of the linearized first equation. This is a property of this method of linearization. This same linearization process can be applied to any quadratic MEP. Any problem of the form (A + B𝜆 + C𝜏 + D𝜆𝜏 + E𝜆2 + F𝜏 2 )𝐱 = 𝟎, (17) Page 5 of 20 Multiparameter analysis for instability problems can be linearized as A B ([ 0 −𝐼𝑛 0 0 C 0 0 ] + [𝐼𝑛 −𝐼𝑛 0 0 0 +[0 0 𝐼𝑛 0 D E 0 0] 𝜆 0 0 F 𝐱 0] 𝜏) [𝜆𝐱] = 𝟎. 0 𝜏𝐱 (18) We have A = M0 + G0 D = −K 0 (19) B = G1 E=0 C = −D0 F = G2 for Eq. 13, and changing 𝜆 → 𝜒 and 𝜏 → Υ, A = −K 0 D = M0 + G0 (20) B = −D0 E = G1 C=0 F = G2 for Eq. 14. These linearised systems have matrix coefficients triple the size of the original problem. 4.2. Direct solution Consider Eq. 16. Post-multiplying the first equation in this system by C̅𝐲 and premultiplying the second by C𝐱, we have (A + BΛ + C𝜏)𝐱 ⊗ (C̅𝐲) = 0, (21) ̅+B ̅Λ + C̅𝜏)𝐲 = 0. (C𝐱) ⊗ (A These two equations are both equal to zero so we may equate them. After cancelling the terms in 𝜏, the expression can be manipulated into: (22) Δ1 𝐳 = ΛΔ0 𝐳 and an enlarged eigenvector 𝐳 = 𝐱 ⊗ 𝐲 and the operator determinants ̅ Δ0 = B ⊗ C̅ − C ⊗ B ̅ Δ1 = C ⊗ A − A ⊗ C̅ ̅. ̅−B⊗A Δ2 = A ⊗ B (23) Eq. 22 is a generalized eigenvalue problem (GEP), in the single parameter 𝜆. Solvers for the generalized eigenvalue problem are very widely available. If the linear system has square coefficients of size 𝑚 then the operator determinants are of size 𝑚2 . The operator determinants can also be used to define a second GEP in 𝜏. By multiplying the ̅𝐲 and B𝐱 respectively, we can also show that: first and second equations of Eq. 16 by B (24) Δ2 𝐳 = 𝜏Δ0 𝐳. However, it is only necessary to solve one of Eq. 22 or Eq. 24: once one has been solved, then its solutions can be substituted back into the original polynomial system, which yields another smaller GEP. Alternatively, if the eigenvalue is simple then it may be computed more cheaper via Rayleigh quotients in 𝐳 or (decomposing 𝐳 = 𝐱 ⊗ 𝐲), 𝐱 and 𝐲. The system is thus completely solved for its flutter points. This direct solution method is known in mathematical literature as the operator determinant method. Its computational complexity is 𝒪(𝑛6 ), irrespective of whether 𝑛 is the size of the linear coefficients (A, B, etc.) or the polynomial coefficients (M0 , G0 , etc.) [30,32,33]. This large complexity arises from solving the generalized eigenvalue problem, an 𝒪(𝑚3 ) process by the QZ algorithm [34], with operator Page 6 of 20 Multiparameter analysis for instability problems determinants of size 𝑚 = 𝒪(𝑛2 ). The operator determinant method has not previously been used in aeroelasticity, and has only rarely seen engineering application in the study of dynamic model updating [35,36]. 4.3. Singularity One important caveat of the operator determinant approach is that the matrix Δ0 must not be singular. If it is, then the eigenvalues of the original polynomial system (e.g. Eq. 12-14) will not generally coincide with those of the GEPs of the linearized problem (Eq. 22 and 24) [30,37,38]. A linear MEP with singular Δ0 is said to be singular MEP. A large proportion of the linear flutter problems that arise in the study of aircraft aeroelasticity are singular. This is largely because the linearization of polynomial problems tends to generate singular linear problems, even if the original polynomial problem has all its matrix coefficients at full rank – compare Eq. 18. Some smaller linearizations are not necessarily singular: for example Eq. 15. However, here we find that (for our aerodynamic model) the coefficient matrix G2 is not full rank and so the linearized problem is singular anyway. In many circumstances we will be dealing with a singular problem, and for a long time the lack of a working solver for such singular problems has been a major obstacle to the application of multiparameter methods to real-world problems. However, recently a solution has been proposed. Muhič and Plestenjak [29] proved that the eigenvalues of a polynomial system are equivalent to the finite regular eigenvalues of the pair of singular operator determinant GEPs constructed via linearization. The finite regular eigenvalues of a linear two-parameter system are the pairs (𝜆, 𝜇) such that 𝑖 ∈ {1,2} [37]: rank(A𝑖 + B𝑖 𝜆 + C𝑖 𝜇) < max 2 rank(A𝑖 + B𝑖 𝑠 + C𝑖 𝑡), (𝑠,𝑡)∈ℂ (25) that is, the finite regular eigenvalues are the set of points that cause the singular problem to have its maximum rank, even though this is not full rank. On the basis of this proof, Muhič and Plestenjak [29] devised a set of algorithms which would extract the common regular part of the singular matrix pencils (i.e. matrix-valued functions polynomial in a variable 𝜆) Δ1s − 𝜆Δ0s and Δ2s − 𝜆Δ0s . This common regular part is represented by two smaller nonsingular matrix pencils (Δ1 − 𝜆Δ0 and Δ2 − 𝜆Δ0 ), the eigenvalues of which are the finite regular eigenvalues of the singular problem and thus the eigenvalues of the polynomial problem. In practical terms, this can be seen as a compression of the singular operator determinant matrices into smaller full-rank matrices. The operator determinant method, as presented in Section 4.2, can be applied to these compressed pencils. The algorithms involved in the extraction of the common regular part are presented in [29] and published also in MATLAB code [39]. We will not detail them here as they are complex. 5. Direct solution via Quasi-linearization 5.1. Quasi-linearization Hochstenbach et al. [30] recently presented another method of linearization for polynomial MEPs. Instead of increasing the size of the coefficient matrices, this method of linearization increases the number of parameters and equations in the system. To differentiate it from strict linearization, Hochstenbach et al. [30] term this new method quasi-linearization. The process is as follows. Considering again Eq. 12, we define a new eigenvalue parameter 𝛼 = 𝜏 2 . The equations then become Page 7 of 20 Multiparameter analysis for instability problems ((M0 + G0 ) + G1 𝜏 + G2 𝛼 − K 0 Λ)𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝛼 − K ̅0 + G ̅ 0 Λ)𝐱̅ = 𝟎. ((M (26) This is now a linear three-parameter eigenvalue problem, but with only two equations. We need a third equation to constrain the system. We have the relation 𝛼 − 𝜏 2 = 0, but this is nonlinear. However, noticing that we can write this relation as 𝛼 𝜏 (27) det ([ ]) = 0 𝜏 1 we can recast it as a new MEP to add to Eq. 26 (using an arbitrary eigenvector 𝐲): ((M0 + G0 ) + G1 𝜏 + G2 𝛼 − K 0 Λ)𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝛼 − K ̅0 + G ̅ 0 Λ)𝐱̅ = 𝟎, ((M 0 0 0 1 1 0 ([ ]+[ ]𝜏 + [ ] 𝛼) 𝐲 = 𝟎. 0 1 1 0 0 0 (28) In a similar way, we can linearize Eq. 13 with the definitions 𝛼 = 𝜏 2 and 𝛽 = 𝜆2 : ((M0 + G0 ) + G1 𝜏 + G2 𝛼 − D0 𝜆 − K 0 𝛽)𝐱 = 𝟎, ̅0 ) + G ̅1 𝜏 + G ̅2 𝛼 − D ̅0 + G ̅ 0𝜆 − K ̅ 0 𝛽)𝐱̅ = 𝟎, ((M 0 0 0 1 1 0 ([ ]+[ ]𝜏 +[ ] 𝛼) 𝐲𝟏 = 𝟎, 0 1 1 0 0 0 0 0 0 1 1 0 ([ ]+[ ]𝜆 + [ ] 𝛽) 𝐲𝟐 = 𝟎, 0 1 1 0 0 0 (29) and Eq. 14 with the definitions 𝛼 = 𝜒 2 , 𝛽 = Υ 2 , 𝛾 = Υ𝜒: ((M0 + G0 )𝛼 + G2 𝛽 + G1 𝛾 − D0 𝜒 − K 0 )𝐱 = 𝟎, ̅0 )𝛼 + G ̅2𝛽 + G ̅1 𝛾 − D ̅0 + G ̅0𝜒 − K ̅ 0 )𝐱 = 𝟎, ((M 1 0 0 1 0 0 ([ ]𝛼 + [ ]𝜒 +[ ]) 𝐲𝟏 = 𝟎, 0 0 1 0 0 1 0 1 0 0 1 0 ([ ]𝛼 + [ ]𝛽 + [ ] 𝛾) 𝐲𝟐 = 𝟎, 0 0 1 0 0 1 (30) These systems can be solved via the operator determinant method, in a more general form than that presented in Section 4.2. 5.2. The general operator determinant method The general form of a nonhomogeneous MEP is 𝑁 𝑊𝑖 (𝜼) = A𝑖0 𝐱 𝑖 + ∑ 𝜂𝑗 A𝑖𝑗 𝐱 𝑖 , 𝑖 = 1, … , 𝑛 (31) 𝑗=1 where 𝜼 is the vector of eigenvalues (𝜂𝑗 being the individual eigenvalues), A𝑖𝑗 are the coefficient matrices, which can be complex and of different sizes for each equation, and 𝐱 𝑖 are the eigenvectors. Eq. 31 can be visualized as a non-square array of matrices: A10 A20 [ ⋮ A𝑁0 A11 A21 ⋮ A𝑁1 A12 A22 ⋮ A𝑁2 ⋯ A1𝑁 ⋯ A2𝑁 ]. ⋱ ⋮ ⋯ A𝑁𝑁 Page 8 of 20 (32) Multiparameter analysis for instability problems In a process analogous to that presented in Section 4.4, we can construct a series of operator determinants for this system. There are 𝑁 + 1 such operator determinants (Δ0 through to Δ𝑁 ), of size 𝑛𝑁 for constant coefficient size 𝑛. They correspond to taking determinants of Eq. 32, with certain columns removed or inserted, and with the normal scalar multiplication operation replaced by a Kronecker product between matrices. Definitions and derivations of these determinants may be found in [40–44], and one software implementation in [39]. In the two-parameter case this analysis collapses into that of Section 4.2. These general operator determinants allow us to compute the eigenvalues of the Eq. 31 via generalized eigenvalue problems. Providing Δ0 is nonsingular, it holds that Δ𝑖 𝐱 = 𝜂𝑖 Δ0 𝐱. (33) In this way we are able to solve the quasi-linearized systems given in Section 5.1 (we will discuss singularity in Section 5.4). However, note that already by using this quasilinearization process we have made the operator determinants smaller than they were with standard linearization. Using Eq. 12, the operator determinants are square and of size 2𝑛2 , and using Eq. 13 or 14 they are of size 4𝑛2 . With standard linearization they are of size 4𝑛2 and 9𝑛2 , respectively. 5.3. Computing operator determinants In the two-parameter case, the computation of the operator determinants is trivial. However in the case of larger systems we must find some other approach. In the general case, the operator determinants may be computed by modifying the Leibniz formula for the determinant [40]: 𝑁 Δ(M) = ∑ sgn(𝐬) ⊗ M𝑖,𝐬𝑖 , 𝐬 ∈ 𝑆𝑁 𝑖=1 (34) where 𝑆𝑁 is the set of permutations of the set {1, … , 𝑁}, sgn(𝐬) is the sign of the permutation vector 𝐬 ∈ 𝑆𝑁 . M is the square matrix array (a modification of Eq. 32) corresponding to the operator determinant desired. The notation ⊗𝑁 𝑖 denotes the repeated application of the Kronecker product. Note that the tensor determinant definition in [40] is slightly erroneous as the factor (−1)sgn(𝜎) in their tensor determinant expressions should be either sgn(𝜎) or (−1)𝑁(𝜎) , where 𝑁(𝜎) is the number of inversions in 𝜎. Alternatively, Muhič and Plestenjak [39] devised an operator determinant Laplace expansion [45] that is able to compute the operator determinants of a system of arbitrary size, via a process of recursion: 𝑁 Δ(M) = ∑(−1)𝑖+1 M𝑖1 ⊗ 𝑚𝑖1 , (35) 𝑖=1 where 𝑚𝑖1 denotes the minor entry corresponding to M𝑖1 . The summation in Eq. 35 follows the first column of M, though, of course, many other summation paths could be used. It should be noted that the use of the Laplace or Leibniz methods for computing the determinant of an ordinary matrix have very high non-polynomial computational complexity costs – 𝒪(𝑛!) and 𝒪(𝑛! 𝑛), respectively [46,47]. 5.4. Singularity One major advantage of the direct solver based on quasi-linearization is that it generates linearized problems that are not always singular – when the coefficient matrices are of full rank, Eq. 28-30 are in general nonsingular. However in our case the coefficient G2 is singular, and so it happens that all these three equations are singular anyway. The theory of the Page 9 of 20 Multiparameter analysis for instability problems compression process noted in Section 4.3 has never been extended to the general multiparameter case, and so we have no rigorous method of solving singular MEPs with 𝑁 > 2. However, we do have a practical method of doing so. Numerical experiments (some of which are detailed in Section 6) indicate that, although the compression algorithm takes only Δ0 , Δ1 and Δ2 as an input, it will successfully compress these three operator determinants for the general multiparameter problem (ignoring all the others). In the case of Eq. 28-30, with these three operator determinants we can solve for all the eigenvalue parameters of the system, irrespective of the order in which we arrange the eigenvalues. We should note that there is no justification for this procedure other than the experimental evidence that the algorithm works – evidence confirmed also by the nonrigourous 𝑁 > 2 compression process utilized in [39]. The common regular part relationship proved by [29] applies only to two-parameter eigenvalue problems arising from linearization, and indeed it is not clear how the concept of the common regular part would extend to an 𝑁 > 2 problem with more than two pencils. That the two-parameter algorithm does work for the first three operator determinants of an 𝑁 > 2 problem implies that the algorithm will generalize – it is a question of working out what this generalization is. This is an interesting area for further research. For the purposes of this paper, however, we will use the compression algorithm for the quasi-linearized Δ0 , Δ1 and Δ2 without proof. 6. Numerical experiments 6.1. Undamped model We are now in a position to compute the flutter points of the models we introduced in Section 3. We have devised two direct solvers to do so: a direct solver with linearization and a direct solver with quasi-linearization. Consider first the undamped section model (Eq. 12), with parameter values as per Table 1. To validate our direct solution methods, we first produce a modal damping plot (Figure 2). Three points of neutral stability may be observed 𝜏 = ±1.00 and 𝜏 = 0. The point 𝜏𝐹 = 1.00 is the only physical flutter point. We can link the modal damping curve for this flutter point with the lower modal frequency path in the Re(𝜒) plot (this requires data not observed directly on Figure 2). We thus can estimate that 𝜒𝐹 = 1.32 rad/s (Λ 𝐹 = 0.57s2 /rad2 ). The point at 𝜏 = −1.00 along the same mode corresponds to a nonphysical flutter event occurring at negative airspeed. Finally, at 𝜏 = 0 both modes have neutral stability (at 𝜒 = 0.548 rad/s and 𝜒 = 1.426 rad/s or Λ = 3.33 s2 /rad2 and Λ = 0.492s2 /rad2 ). This, however, merely represents the fact that the structure is undamped at zero airspeed due to the lack of any structural damping in the model. The divergence point of this system does not appear on this plot, as it occurs at infinite 𝜏 and Λ. However, the computation of divergence points does not require multiparameter methods anyway, as if the frequency 𝜒 is assumed to be zero in the governing equation (e.g. Eq. 9) then the problem becomes a single parameter eigenvalue problem in the airspeed parameter (e.g. Υ). This problem can then be solved with single-parameter solvers. We then compute the flutter points of the system via our two direct solution methods, yielding flutter points identical to those detailed above for the presented accuracy. With the solver based on linearization, the uncompressed operator determinants are of size 16 and the compressed ones of size 4. With the solver based on quasi-linearization the size of the uncompressed determinants can be reduced significantly – down to 8 – while producing compressed determinants of the same size. The quasi-linearization method thus reduces the time required for the compression process, as the finite-regular part extraction algorithm Page 10 of 20 Multiparameter analysis for instability problems works by successively increasing the rank of the system (not all at once). However, for this problem the computation time is too small for any meaningful assessment: we will investigate computation time more fully in Section 6.3. Figure 3 shows the system flutter points superimposed on a contour plot [24]. As can be seen, there is an exact agreement between the direct solutions and the contour plot solutions (the intersections of Re(𝑑) = 0 and Im(𝑑) = 0). Figure 2: Modal damping plot for the undamped section model. Figure 3: Contour plot for the undamped section model, showing identical solutions from the two direct solvers. Page 11 of 20 Multiparameter analysis for instability problems 6.2. Damped model We now simulate the damped system, in both forms (Eq. 13 and 14). Figure 4 shows a contour plot for the 𝜏-𝜆 form of this system (Eq. 16), with direct solutions from both solution methods. There is a single physical flutter point at 𝜏𝐹 = 1.650, 𝜆𝐹 = 0.8336 s/rad, and a nonphysical flutter point at a small negative 𝜏. The divergence point occurs at infinite 𝜏 and 𝜆. As expected, the solutions from both direct solution methods are identical to each other and the solutions from the contour plot. Figure 5 shows a contour plot of the Υ-𝜒 form (Eq. 14) with direct solutions from both solution methods. This physical flutter point can be located at Υ𝐹 = 1.98 Hz, 𝜒𝐹 = 1.20 rad/s and the divergence point at Υ𝐷 = 3.99 Hz. Again, the flutter points computed with direct solvers agree exactly with those seen on the contour plot, and the results from the system as a whole agree with those of the 𝜏-𝜆 form (𝜆𝐹 = 1⁄𝜒𝐹 = 0.83 s/rad and 𝜏𝐹 = Υ𝐹 ⁄𝜆𝐹 = 1.65). 6.3. Computation time Given the high computational complexity of these direct solution methods – 𝒪(𝑛6 ) – we are interested in the maximum system size for which a direct solution is practical. We have already found that for 𝑛 = 2 the computational effort required is tiny, and so at the very least these direct solvers are useful for small reduced-order models as might be used in a preliminary design analysis. This alone is of use, as the directness of these solvers makes them ideal for use in optimization routines or other applications in which a large number of computations must be performed with limited user guidance. To gain a better understanding of the computational complexity characteristics of our methods, we simulate a series of systems of increasing matrix coefficient size. We generate random complex-valued matrices for the polynomial coefficients (M0 , G0 , etc.), of size 𝑛 = 2𝑘 with 𝑘 ∈ {1, 2, … , 5}. For robustness, we average the results for 𝑘 = 1 over 50 random matrices, for 𝑘 = 2 over 10 random matrices, for 𝑘 = 3 over 5 random matrices; and for systems larger than this we generate only one matrix. Figure 6 shows the wall-clock solution time against system size for the direct solver with linearization, for a 64-bit Intel i7-4770 with 3.4 GHz processor and 16 GB RAM, running MATLAB R2014b. Note that the 𝜆 solution time denotes the time required to compute the 𝜆 components of the solution via a series of one-parameter eigenvalue problems. As can be seen, the compression process is the most expensive component of the algorithm, making up a constant fraction of about 65% of the total computation time over the entire range of 𝑛. The GEP solution time is initially completely negligible but becomes more significant as system size increases: by 𝑛 = 32 it makes up 34% of the total computation time. The 𝜆-solution and setup process are never of much significance. We expect from computational complexity theory that the gradient of the GEP-solution time curve in log-log units will be approximately 6.0 (Section 4.2). Fitting a linear curve through this data yields a gradient of 5.46, with an 𝑅 2 of 0.994. However, the GEP solution-time curve is slightly concave, with a maximum gradient of 6.47. The algorithm can then be said to have complexity 𝒪(𝑛6 ) overall. Figure 7 shows the wall-clock solution time against system size for the direct solver with quasi-linearization. This simulation was run on the same Intel i7-4770 platform with MATLAB R2014b. The random coefficient matrices used are identical to those in Figure 6.And although now the quasi-linearized system in no longer singular, we still apply the compression algorithm to capture some of its overhead costs.. As can be seen, these costs are not large but are still more significant than the GEP and 𝜆 solution times for some system sizes. Page 12 of 20 Multiparameter analysis for instability problems Figure 4: Contour plot for the damped section model (𝜏-𝜆 form), showing identical solutions from the two direct solvers. Figure 5: Contour plot for the damped section model (Υ-𝜒 form), showing identical solutions from the two direct solvers. Page 13 of 20 Multiparameter analysis for instability problems Figure 6: Wall-clock solution time against system size for the direct solver with linearization. Figure 7: Wall-clock solution time against system size for the direct solver with quasi-linearization. Page 14 of 20 Multiparameter analysis for instability problems These GEP and 𝜆 solution times are effectively identical to those of the direct solver with linearization, as the algorithm is solving a GEP of exactly the same size, yielding exactly the same eigenvalues. However, the most striking aspect of Figure 7 is the fact that the setup time occupies the majority of the required computation time right up to 𝑛 = 16 (after which it is surpassed by the GEP solution time). The setup process involved using defining the system array (Eq. 32) in a cell array, and computing the necessary operator determinants of this array (Δ0 and Δ2 ) using the modified Leibniz formula (Eq. 34). On investigating these two procedures we find that it is the computation of the Kronecker products within the operator determinant which occupies the greatest time. However, despite this, the actual computational complexity of the algorithm as a whole is lower than that of the direct solver with linearization. GEP solution time is still approximately 𝒪(𝑛6 ), though the concavity that was noted in Figure 6 is still present. Finally, Figure 8 shows a comparison of the total computation times for the two algorithms. As can be seen, below a system coefficient size of approximately 10, linearisation is faster than quasi-linearisation (at 𝑛 = 2, over an order of magnitude faster). Above this coefficient size, quasi-linearisation is faster (at 𝑛 = 32, twice as fast). To keep computation times below 10s, the system size must be below 11; to keep it below 1s, 8, and below 0.1s, 5. As it stands, the operator determinant method is not suitable for use with finite-element models or any other systems with a large number of degrees of freedom. It is, however, useful for the solution of simple reduced order models, e.g. in a preliminary design optimization. Figure 8: Total wall-clock solution time against system size for the two direct solver variants. Page 15 of 20 Multiparameter analysis for instability problems 7. Discussion 7.1. Alternative solution methods We have so far discussed the operator determinant method – a solution method which is direct but computationally expensive. However, two other classes of method are also available. The Sylvester-Arnoldi type methods are a set of closely-related algorithms that are only valid for two-parameter problems, but are fast and can handle large systems [48]. They still use operator determinants to reformulate the system into a set of generalized eigenvalue problems, but instead of solving this set of GEPs directly, the solution procedure is optimized based on the given knowledge that the operator determinants consist of a difference of two Kronecker products. For example, each step of a Krylov subspace procedure solution procedure for the GEP reduces to the solution of a Sylvester equation. Several related algorithms may be devised this way, and in some cases the computational complexity of the solution process can be reduced all the way down to 𝒪(𝑛3 ) [48]. However, this technique shows little potential for generalization to 𝑁-parameter systems, as it relies on being able to reformulate the operator determinant GEP into a simple and well-known matrix equation. As the operator determinant definition becomes more complex, the resulting matrix equation changes and efficient solvers may not be available. There is also no easy extension for singular problems. Subspace methods for one-parameter eigenvalue problems are based around generating a series of linear spaces that eventually approximate one of the system’s eigenspaces (the linear space of eigenvectors corresponding to a given eigenvalue). The Jacobi-Davidson and Rayleigh-Ritz methods are well-known one-parameter subspace methods, which can be generalized to apply to two-parameter systems [26,32,49–51], though this is not without difficulties [49]. These methods do not invoke the operator determinants, and show potential for generalization both to 𝑁-parameter and to polynomial systems. The Jacobi-Davidson is applicable to singular systems and has previously been tested on singular linearized aeroelastic stability problems [25]; but its performance was observed to be poorer than the operator determinant method. However the solution of singular MEPs has only been achieved very recently (2010 [29]), and so it is likely that the next few years will bring significant developments in this area. 7.2. Extensions to the concept The core methodology presented in this paper can be extended to a wide variety of problems. Not only more complex two-parameter flutter models – considered in Pons [24] and Pons and Gutschmidt [25] – but also stability problems from a wide variety of fields, including systems with entirely different eigenvalue definitions. Indeed, any combination of model parameters in a stability model can indeed be treated as multiparameter eigenvalues; and scalar constraints on these parameters can cast as eigenvalue problems by introducing a scalar eigenvector [24]. For example, in aeroelastic model, given the location of a flutter point, we could solve for the sets of parameter values that could generate such a flutter point – allowing us to perform model identification based on flutter point information. This could pave the way for a least-squares approach for overconstrained multiparameter systems. Alternatively, we could introduce a flight altitude / air density / Mach number parameter and compute points on the aeroelastic flutter envelope of the aircraft. The multiparameter formulation provides a versatile way of analysing stability problems in any combination of parameters. Page 16 of 20 Multiparameter analysis for instability problems 8. Conclusion In this paper we have demonstrated and discussed the use of multiparameter solution techniques for the solution of aeroelastic and related stability problems. We have introduced the link between multiparameter spectral theory and stability analysis, and we showed how this link can be used to reformulate stability problems with a complex-valued stability metric and a pertinent environmental parameter into a two-parameter eigenvalue problem. We demonstrated that this allows the direct solution of stability problems that are linear or polynomial in these parameters, and we discussed aspects of the solution process, including the linearization and quasi-linearization of polynomial problems, general N-parameter problems, computational costs and approaches to problem singularity. 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Cluster Synchronization of Coupled Systems with Nonidentical Linear Dynamics Zhongchang Liu∗ and Wing Shing Wong arXiv:1502.07481v2 [cs.SY] 30 Dec 2015 Abstract This paper considers the cluster synchronization problem of generic linear dynamical systems whose system models are distinct in different clusters. These nonidentical linear models render control design and coupling conditions highly correlated if static couplings are used for all individual systems. In this paper, a dynamic coupling structure, which incorporates a global weighting factor and a vanishing auxiliary control variable, is proposed for each agent and is shown to be a feasible solution. Lower bounds on the global and local weighting factors are derived under the condition that every interaction subgraph associated with each cluster admits a directed spanning tree. The spanning tree requirement is further shown to be a necessary condition when the clusters connect acyclicly with each other. Simulations for two applications, cluster heading alignment of nonidentical ships and cluster phase synchronization of nonidentical harmonic oscillators, illustrate essential parts of the derived theoretical results. Index Terms Cluster synchronization; Coupled linear systems; Nonidentical systems; Graph topology I. I NTRODUCTION Understanding the interaction of coupled individual systems continues to receive interest in the engineering research community [1]. Recently, more attention has been paid to cluster synchronization problems which have much wide applications, such as segregation into small subgroups for a robotic team [2] or physical particles [3], predicting opinion dynamics in social networks [4], and cluster phase synchronization of coupled oscillators [5], [6]. In the models reported in most of the literature, the clustering pattern is predefined and fixed; research focuses are on deriving conditions that can enforce cluster synchronization for This work is supported by the Hong Kong RGC Earmarked Grant CUHK 14208314. The authors are with Department of Information Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. ∗ Corresponding author. E-mail: zcliu@ie.cuhk.edu.hk. various system models [7]–[17]. Preliminary studies in [7]–[10] reported algebraic conditions on the interaction graph for coupled agents with simple integrator dynamics. Subsequently, a cluster-spanning tree condition is used to achieve intra-cluster synchronization for firstorder integrators (discrete time [11] or continuous time [12]), while inter-cluster separations are realized by using nonidentical feed-forward input terms. For more complicated system models, e.g., nonlinear systems ([13]–[15]) and generic linear systems ([16], [17]), both control designs and inter-agent coupling conditions are responsible for the occurrence of cluster synchronization. For coupled nonlinear systems, e.g., chaotic oscillators, algebraic and graph topological clustering conditions are derived for either identical models ([13]) or nonidentical models ([14], [15]) under the key assumption that the input matrix of all systems is identical and it can stabilize the system dynamics of all individual agents via linear state feedback (i.e., the so-called QUAD condition). For identical generic linear systems which are partial-state coupled [16], [17], a stabilizing control gain matrix solved from a Ricatti inequality is utilized by all agents, and agents are pinned with some additional agents so that the interaction subgraph of each cluster contains a directed spanning tree. The system models introduced above can describe a rich class of applications for multiagent systems. A common characteristic is that the uncoupled system dynamics of all the agents can be stabilized by linear state feedback attenuated by a unique matrix (i.e., static state feedback). This simplification allows the derivation of coupling conditions to be independent of the control design of any agent, and thus offers scalability to a static coupling strategy. This kind of benefit still exists for nonidentical nonlinear systems which are full-state coupled, as all the system dynamics can be constrained by a common Lipchitz constant (Lipchitz can imply the QUAD condition [18]). However, for the class of partial-state coupled nonidentical linear systems, the stabilizing matrices for distinct linear system models are usually different. Then the coupling conditions under static couplings will be correlated with the control designs of all individual systems. This correlation not only harms the scalability of a coupling strategy but also increases the difficulty in specifying a graph topological condition on the interaction graph. The goal of this paper is to achieve state cluster synchronization for partial-state coupled nonidentical linear systems, where agents with the same uncoupled dynamics are supposed to synchronize together. This is a problem of practical interest, for instance, maintaining different formation clusters for different types of interconnected vehicles, providing different synchronization frequencies for different groups of clocks using coupled nonidentical harmonic oscil- lators, reaching different consensus values for people with different opinion dynamics, and so on. In order to relieve the difficulties in using the conventional static couplings, couplings with a dynamic structure is proposed by introducing a vanishing auxiliary variable which facilitates interactions among agents. With the proposed dynamic couplings, an algebraic necessary and sufficient condition, which is independent of the control design, is derived. This newly derived algebraic condition subsumes those published for integrator systems in [7]–[10] as special cases. Due to the entanglement between nonidentical system matrices and the parameters from the interaction graph, the algebraic condition is not straightforward to check. Thus, a graph topological interpretation of the algebraic condition is provided under the assumption that the interaction subgraph associated with each cluster contain a directed spanning tree. We also derive lower bounds for the local coupling strengths in different clusters, which are independent of the control design due to the dynamic coupling structure. This spanning tree condition is further shown to be a necessary condition when the clusters and the intercluster links form an acyclic structure. This conclusion reveals the indispensability of direct links among agents belonging to the same cluster, and further strengthens the sufficiency statement presented initially in [16]. Another contribution of the proposed dynamic couplings in comparison to those static couplings in [17] is that the lower bound of a global factor which weights the whole interaction graph is also independent of the control design. For this reason, the least exponential convergence rate of cluster synchronization is characterized more explicitly than that in [17]. The derived results in this paper are illustrated by simulation examples for two applications: cluster heading alignment of nonidentical ships and cluster phase synchronization of nonidentical harmonic oscillators. The organization of this paper is as follows: Following this section, the problem formulation is presented in Section II. In Section III, both algebraic and graph topological conditions for cluster synchronization are developed. Simulation examples are provided in Section IV. Concluding remarks and discussions for potential future investigations follow in Section V. II. P ROBLEM S TATEMENT Consider a multi-agent system consisting of L agents, indexed by I = {1, . . . , L}, and S N ≤ L clusters. Let C = {C1 , . . . , CN } be a nontrivial partition of I, that is, N i=1 Ci = I, Ci 6= ∅, and Ci ∩ Cj = ∅, ∀i 6= j. We call each Ci a cluster. Two agents, l and k in I, belong to the same cluster Ci if l ∈ Ci and k ∈ Ci . Agents in the same cluster are described by the same linear dynamic equation: ẋl (t) = Ai xl (t) + Bi ul (t), l ∈ Ci , i = 1, . . . , N (1) where xl (t) ∈ Rn with initial value, xl (0), is the state of agent l and ul (t) ∈ Rmi is the control input; Ai ∈ Rn×n and Bi ∈ Rn×mi are constant system matrices which are distinct for different clusters. A. Interaction graph topology and graph partitions A directed interaction graph G = (V, E, A) is associated with system (1) such that each agent l is regarded as a node, vl ∈ V, and a link from agent k to agent l corresponds to a directed edge (vk , vl ) ∈ E. An agent k is said to be a neighbor of l if and only if (vk , vl ) ∈ E. The adjacency matrix A = [alk ] ∈ RL×L has entries defined by: alk 6= 0 if (vk , vl ) ∈ E, and alk = 0 otherwise. In addition, all = 0 to avoid self-links. Note that alk < 0 means that the influence from agent k to agent l is repulsive, while links with alk > 0 are cooperative. P Define L = [blk ] ∈ RL×L as the Laplacian of G, where bll = Lk=1 alk and blk = −alk for any k 6= l. Corresponding to the partition C = {C1 , . . . , CN }, a subgraph Gi , i = 1, . . . , N , of G contains all the nodes with indexes in Ci , and the edges connecting these nodes. See Fig. 1 for illustration. Without loss of generality, we assume that each cluster Ci , i = 1, . . . , N , consists P of li ≥ 1 agents ( N i=1 li = L), such that C1 = {1, . . . , l1 }, . . ., Ci = {σi + 1, . . . , σi + li }, Pi−1 . . ., CN = {σN + 1, . . . , σN + lN } where σ1 = 0 and σi = j=1 lj , 2 ≤ i ≤ N . Then, the Laplacian L of the graph G can be partitioned  L11 L12  L  21 L22 L= . ..  .. .  LN 1 LN 2 into the following form:  · · · L1N  · · · L2N   , .. ...  .  · · · LN N (2) where each Lii ∈ Rli ×li specifies intra-cluster couplings and each Lij ∈ Rli ×lj with i 6= j, specifies inter-cluster influences from cluster Cj to Ci , i, j = 1, · · · , N . Note that Lii is not the Laplacian of Gi in general. Construct a new graph by collapsing any subgraph of G, Gi , into a single node and define a directed edge from node i to node j if and only if there exists a directed edge in G from a node in Gi to a node in Gj . We say G admits an acyclic partition with respect to C, if G¹1 G1 1 2 01 1 3 4 02 3 4 G¹2 G2 Fig. 1. 2 A graph topology partitioned into two subgraphs. the newly constructed graph does not contain any cyclic components. If the latter holds, by 4 1 2 relabeling the clusters and the nodes in G, we can represent the Laplacian L in a lower -a triangular form 1 a  -1  1 L 0  11   ..  . . L= . . , 3   -1 LN 1 · · · LN N -1 (3) 4 so that each cluster Ci receives no input from clusters Cj if j > i. In Fig. 1, the two subgraphs -1 1 G1 and G2 illustrate an acyclic partition of the whole graph. -1 1 1 c1 B. The cluster synchronization problem 1 5 3 -1 c1 c2 The main task in this paper is to achieve cluster synchronization for the states of systems 2 6 -1 1 4 in (1) via distributed couplings through the control inputs ul (t) which is defined as follows: for l ∈ Ci , i = 1, . . . , N ul (t) = Ki ηl (t) " 1 X η̇l (t) = (Ai + Bi Ki )ηl + c ci c1 c1 2 3 alk (ηk − ηl + xl − xk ) -1 (4a) 4 1 k∈Ci  + X alk (ηk − ηl +1xl −-1xk ) , 1 -1 (4b) k∈C / i 1 n c c 3 where Ki is the control gain matrix to be specified; the7 vector3 ηl (t)8 ∈ R , l ∈ I is an5 6 -1 auxiliary control variable with initial value, ηl (0); c > 0 is the global weighting factor for the whole interaction graph G; each ci > 0 is a local weighting factor used to adjust the intra-cluster coupling strength of cluster Ci . Note that the couplings in (4) takes a dynamic structure. The reasons why conventional static couplings (e.g., those in [13]–[17]) are not c2 preferred will be explained in details in the main part. The cluster synchronization problem is defined below. Definition 1: A linear multi-agent system in (1) with couplings in (4) is said to achieve N -cluster synchronization with respect to the partition C if the following holds: for any xl (0) and ηl (0), l ∈ I, limt→∞ kxl (t) − xk (t)k = 0 ∀k, l ∈ Ci , i = 1, . . . , N , limt→∞ ηl (t) = 0 ∀l ∈ I, and for any set of xl (0), l ∈ I there exists a set of ηl (0), l ∈ I such that lim supt→∞ kxl (t) − xk (t)k > 0 ∀l ∈ Ci , ∀k ∈ Cj , ∀i 6= j. In the definition, all auxiliary variables, ηl (t), l ∈ I are required to decay to zero so as to guarantee that the control effort of every agent is essentially of finite duration. For state separations among distinct clusters, one should not expect them to happen for any set of xl (0)’s and ηl (0)’s; an obvious counterexample is that all system states will stay at zero when xl (0) = ηl (0) = 0 for all l ∈ I. Some assumptions throughout the paper are in order. Assumption 1: Each of the pairs (Ai , Bi ), i = 1, . . . , N is stabilizable. Assumption 2: Each Ai has at least one eigenvalue on the closed right half plane. This assumption excludes trivial scenarios where all system states synchronize to zero. To deal with stable Ai ’s, one may introduce distinct feed-forward terms in ul (t) as studied in [11], [12]. In order to segregate the system states according to the uncoupled system dynamics in (1), an additional mild assumption is made on the system matrices Ai ’s, namely, they can produce distinct trajectories. Rigorously, for any i 6= j, the solutions xi (t) and xj (t) to the linear differential equations ẋi (t) = Ai xi (t) and ẋj (t) = Aj xj (t), respectively satisfy lim supt→∞ kxi (t) − xj (t)k > 0 for almost all initial states xi (0) and xj (0) in the Euclidean space Rn . Assumption 3: Every block Lij of L defined in (2) has zero row sums, i.e., Lij 1lj = 0. This assumption guarantees the invariance of the clustering manifold {x(t) = [xT1 (t), . . . , xTL (t)]T : x1 (t) = · · · = xl1 (t), . . . , xσN +1 (t) = · · · = xL (t)}. It is imposed frequently in the literature to result in cluster synchronization for various multi-agent systems (see [7]–[10], [13], [14], [16], [17]). To fulfill it, one can let positive and negative weights be balanced for all of the links directing from one cluster to any agent in another cluster. The negative weights for inter-cluster links is supposed to provide desynchronizing influences. Note also that with Assumption 3 each Lii is the Laplacian of a subgraph Gi , i = 1, . . . , N . Notation: 1n = [1, 1, . . . , 1]T ∈ Rn . The identity matrix of dimension n is In ∈ Rn×n . The symbol blockdiag{M1 , . . . , MN } represents the block diagonal matrix constructed from the N matrices M1 , . . . , MN . “⊗” stands for the Kronecker product. A symmetric positive (semi-) definite matrix S is represented by S > 0(S ≥ 0). Reλ(A) is the real part of the eigenvalue of a square matrix A, and σ(A) is the spectrum of A. III. C ONDITIONS FOR ACHIEVING C LUSTER S YNCHRONIZATION In this section, we first present a necessary and sufficient algebraic clustering condition that entangles parameters from the Laplacian L and the system matrices Ai ’s. Then, we present some graph topological conditions which offer more intuitive interpretations. The following discussion makes use  cL  1 11  .. Lc =  .  LN 1 of the weighted graph Laplacian  ··· L1N   .. ...  ∈ RL×L , .  · · · cN L N N (5) and the following matrix:  c L̂ ···  1 11  .. ... L̂c =  .  L̂N 1 · · · L̂1N .. . cN L̂N N     ∈ R(L−N )×(L−N ) ,  (6) where each L̂ij , i, j = 1, . . . , N is a block matrix defined as L̂ij = L̃ij − 1li γijT , (7) with γij = [bσi +1,σj +2 , · · · , bσi +1,σj +lj ]T ∈ Rlj −1 ,   b · · · bσi +2,σj +lj  σi +2,σj +2    .. .. . . L̃ij =  .  ∈ R(li −1)×(lj −1) . . .   bσi +li ,σj +2 · · · bσi +li ,σj +lj The two matrices Lc and L̂c have the following relation, whose proof is shown in Appendix I. Lemma 1: Under Assumption 3, each diagonal block Lii in Lc has exactly one zero eigenvalue if and only if the corresponding matrix L̂ii defined in (7) is nonsingular. Moreover, Lc defined in (5) has exactly N zero eigenvalues if and only if the matrix L̂c defined in (6) is nonsingular. A. Algebraic clustering conditions Under Assumption 1, for each i = 1, . . . , N there exists a matrix Pi > 0 satisfying the Riccati equation Pi Ai + ATi Pi − Pi Bi BiT Pi = −I. (8) Choose the control gain matrices as Ki = −BiT Pi , and denote Â = blockdiag{Il1 −1 ⊗ A1 , . . . , IlN −1 ⊗ AN }. Then, we have the following algebraic condition to check the cluster synchronizability. Theorem 1: Under Assumptions 1 to 3, the multi-agent system in (1) with couplings in (4) achieves N -cluster synchronization if and only if the matrix Â − cL̂c ⊗ In is Hurwitz, where L̂c is defined in (6). The proof is given in Appendix II. The matrix Â − cL̂c ⊗ In contains parameters from the interaction graph that entangle intimately with those from the system dynamics. In general, it is not possible to verify the above synchronization condition by simply comparing the eigenvalues of L̂ with those of Ai ’s. However, one can do so for a homogeneous multi-agent system as stated in the following corollary. Corollary 1: Under Assumptions 1 to 3, and with identical system parameters: Ai = A, Bi = B, Ki = K, for all i = 1, . . . , N , a multi-agent system in (1) with couplings in (4) achieves N -cluster synchronization if and only if the following holds: min Reλ(cL̂c ) > max Reλ(A). σ(L̂c ) σ(A) (9) A sketch of the proof for this corollary is given in Appendix III. Remark 1: In words, the algebraic condition (9) states that the weighted graph Laplacian Lc has exactly N zero eigenvalues, and all the nonzero eigenvalues have large enough positive real parts to dominate the unstable system dynamics described by A. This condition implies that related results in [7]–[9] are special cases with A = 0, B = 1 and K = 1. It also includes part of the results in [10], which are obtained for identical double integrators. Note that with identical system parameters, one can use static controllers without involving the auxiliary variables ηl ’s. However, in that case the synchronized state in each cluster depends linearly on the initials states xl (0)’s only. For certain initial state sets, state separations in the limit cannot be guaranteed. B. Graph topological conditions The matrix Â − cL̂c ⊗ In in Theorem 1 can be proven to be Hurwitz for certain graph topologies in conjunction with some lower bounds on the weighting factors. To do so, the following well-known result for subgraphs will be useful. Lemma 2 ([19]): Let Gi be a non-negatively weighted subgraph. Then, the Laplacian Lii of Gi has a simple zero eigenvalue and all the nonzero eigenvalues have positive real parts if and only if Gi contains a directed spanning tree. By Lemma 1, there exists a positive definite matrix Ŵi ∈ R(li −1)×(li −1) such that Ŵi L̂ii + L̂Tii Ŵi > 0, i = 1, . . . , N, (10) if the corresponding subgraph Gi satisfies the conditions in Lemma 2. Denote Ŵ = blockdiag{Ŵ1 , . . . , ŴN }, and let L̂o = L̂c − L̂d , (11) with L̂d = blockdiag{c1 L̂11 , . . . , cN L̂N N }. The following theorem states the main result of this subsection. Theorem 2: Under Assumptions 1 to 3, a multi-agent system in (1) with couplings in (4) achieves N -cluster synchronization exponentially fast with the least rate of 21 [c − λmax (Â + ÂT )], if each subgraph, Gi , contains only cooperative edges and has a directed spanning tree, and the weighting factors satisfy c> max λmax (Ai + ATi ), (12) i∈{1,...,N } and for each i = 1, . . . , N ci ≥ λmax (Ŵ) − λmin (Ŵ L̂o + L̂To Ŵ) λmin (Ŵi L̂ii + L̂Tii Ŵi ) , (13) where each Ŵi satisfies (10). Proof: Following the proof of the sufficiency part of Theorem 1, we need to show that the system ζ̇(t) = (Â − cL̂c ⊗ In )ζ(t) (14) is exponentially stable under the conditions in Theorem 2. First, these conditions guarantee the existence of positive definite matrices, Ŵi ’s, satisfying (10). Hence, (13) can be written as ci λmin (Ŵi L̂ii + L̂Tii Ŵi ) + λmin (Ŵ L̂o + L̂To Ŵ) ≥ λmax (Ŵ) for i = 1, . . . , N . These inequalities together with Weyl’s eigenvalue theorem ([20]) yield the following: λmin (Ŵ L̂c + L̂Tc Ŵ) = λmin (Ŵ L̂d + L̂Td Ŵ + Ŵ L̂o + L̂To Ŵ) ≥ λmin (Ŵ L̂d + L̂Td Ŵ) + λmin (Ŵ L̂o + L̂To Ŵ) ≥ λmax (Ŵ), which further implies that Ŵ L̂c + L̂Tc Ŵ ≥ Ŵ. (15) Now, consider the Lyapunov function candidate V (t) = ζ(t)T (Ŵ ⊗ In )ζ(t) for the system (14). Taking time derivative on both sides of V (t), one gets V̇ (t) = ζ T (t)(Ŵ ⊗ In )(Â − cL̂c ⊗ In ) + (Â − cL̂c ⊗ In )T (Ŵ ⊗ In )ζ(t) = ζ T (t)[(Ŵ ⊗ In )(Â + ÂT ) − c(Ŵ L̂c + L̂Tc Ŵ) ⊗ In ]ζ(t) ≤ ζ T (t)[(Ŵ ⊗ In )(Â + ÂT ) − cŴ ⊗ In ]ζ(t) ≤ −[c − λmax (Â + ÂT )]V (t), where the last inequality follows from (12). This confirms the exponential stability of system (14), and therefore cluster synchronization can be achieved exponentially fast with the least rate of 21 [c − λmax (Â + ÂT )]. We have the following comments on the condition in (12): 1) From the above proof, one can find another lower bound for c as follows: c> λmax ((Ŵ ⊗ In )(Â + ÂT )) λmin (Ŵ L̂c + L̂Tc Ŵ) . (16) This bound is tighter than that in (12) since the inequality in (15) and λmax (Ŵi ) > 0, λmax (Ai + ATi ) ≥ 0 for any i imply that the right-hand side (RHS) of (16) ≤ λmax (Ŵ)λmax (Â + ÂT ) = RHS of (12). However, this tighter bound only guarantees λmax (Ŵ) that V̇ (t) < 0, and does not provide a lower bound on the convergence rate, which could be quite slow. Moreover, the RHS of (16) involves all the ci ’s in L̂c , and no known distributed algorithm is available for the computation. 2) Note that the role of c is more essential in stabilizing the unstable modes of the system matrices, Ai ’s, than in strengthening the connective ability of the interaction graph. A global weighting factor similar to c is utilized in a related paper [17] where the clustering problem for identical linear systems are solved. In that paper, the global factor serves as a parameter in a Ricatti inequality so as to result in a control gain matrix. However, using a larger value for that factor does not necessarily increase the convergence rate. In contrast, the selection of c in this paper is independent of the control design in (8). And the rate of convergence can be improved definitely by using a larger value for c. The following two remarks explain why we prefer the dynamic couplings in (4) than static couplings when dealing with nonidentical linear systems. Remark 2: To achieve state cluster synchronization for a group of generic linear systems, a natural choice of static couplings is the following (slightly modified from couplings of homogeneous linear systems in [16], [17]): for each l ∈ Ci , i = 1, . . . , N   X X ul (t) = Ki ci blk xk (t) + blk xk (t) k∈Ci (17) k∈C / i However, following a similar procedure as in [17], one will need the following condition ci λmin ((Ŵi L̂ii + L̂Tii Ŵi ) ⊗ Pi Bi BiT Pi ) ≥ ρ, (18) for every i = 1, . . . , N , where ρ = λmax ((Ŵ ⊗ In )PBBT P) − λmin (PBBT P(ŴLo ⊗ In ) + (LTo Ŵ ⊗ In )PBBT P). To compute ρ, one needs information on the control design of all agents, i.e., BT P = blockdiag{Il1 −1 ⊗ B1 P1 , . . . , IlN −1 ⊗ BN PN }. This fact renders the selection of local weighting factors, ci ’s, a centralized decision. Moreover, (18) cannot be satisfied by any ci in the nontrivial case that ρ > 0, and Pi Bi BiT Pi is singular for some i. In contrast to (18), the condition (13) specifies explicitly the requirements for ci ’s, and it is independent of the design of control gain matrices. In this sense, the dynamic couplings in (4) are preferable to the static ones in (17). Remark 3: For nonidentical nonlinear systems of the form, ẋl (t) = fi (xl , t), l ∈ Ci , static couplings are used to result in closed-loop systems as follows ([14], [15]):   X X ẋl (t) = fi (xl , t) − Γ ci blk xk (t) + blk xk (t) , k∈Ci k∈C / i where Γ is a constant (usually nonnegative-definite) matrix. It was shown that clustering conditions involve the graph Laplacian only (see [15]) if all individual self-dynamics are constrained by the so-called QUAD condition: for any x, y ∈ Rn , (x − y)T [fl (x) − fl (y) − Γ(x−y)] ≤ −ω(x−y)T (x−y), where ω > 0 is a prescribed positive scalar. For generic linear systems with static couplings in (17), this QUAD condition requires that for any x ∈ Rn , xT (Ai − Γ)x ≤ −ωxT x with Γ = Bi Ki for all i = 1, . . . , N . Given a Γ, for the existence of control gains Ki ’s, one needs all Bi ’s to satisfy Rank(Bi ) = Rank([Bi Γ]). However, this rank condition is too restrictive. For example, for the models in (21), an applicable choice of Γ is I2 , but then Rank(Bi ) < Rank([Bi Γ]) and thus no Ki can be solved from Γ = Bi Ki . In contrast, the dynamic couplings in (4) do not impose such constraints on the system models. Generally, it is not always necessary to let every subgraph contain a directed spanning tree. In fact, agents belonging to a common cluster may not need to have direct connections at all as long as the algebraic condition in Theorem 1 is satisfied. This point is illustrated by a simulation example in the next section. Nevertheless, the spanning tree condition turns out to be necessary under some particular graph topologies as stated by the corollary below. Corollary 2: Let G be an interaction graph with an acyclic partition as in (3), and let the edge weights of every subgraph Gi be nonnegative. Under Assumptions 1 to 3, a multi-agent system (1) with couplings in (4) achieves N -cluster synchronization if and only if every Gi contains a directed spanning tree, and the weighting factors satisfy c · ci > maxσ(Ai ) Reλ(Ai ) minσ(L̂ii ) Reλ(L̂ii ) , ∀i = 1, . . . , N, (19) where each L̂ii is defined in (7). Proof: By Theorem 1, we can examine the stability of Â − cL̂c ⊗ In . Let Ti ∈ R(li −1)×(li −1) , i = 1, . . . , N , be a set of nonsingular matrices such that Ti−1 L̂ii Ti = Ji , where Ji is the Jordan form of L̂ii . Denote T = blockdiag{T1 ⊗ In , . . . , TN ⊗ In }. Then, the 3 4 3 02 c1 4 G¹2 G2 1 G1 2 G1 1 1 2 1 2 -1 1 -5 -1 1 5 -1 -4 1 4 -1 1 1 1 3 4 1 4 G2 (b) (a) Fig. 2. Interaction graph partitioned into two clusters C1 = {1, 2} and C2 = {3, 4} (a) cyclic partition (b) acyclic partition. 1 3 5 -1 1 1 1 3 G2 c2 -1 4 -1 G1 c1 c2 −1 s1 1 1 2 block triangular matrix T (Â − cL̂c ⊗1 In )T has diagonal blocks Ai − c̃i λk (L̂ii )In , where 6 -1 4 -5 5 c̃i = c ·1 ci , k = 1, . . . , li − 1, i = 1, . . . , N . Hence, the matrix Â − cL̂c ⊗ In is Hurwitz if 1 and only if c̃i mink Reλk (L̂ii ) > maxm Reλ1m (Ai )-1for any i. This claim is equivalent to the c1 2 3 -1 1 4 3 the first 1claim of Lemma 2 conclusion of4 this corollary due to sLemma 2, 1, and Assumption 1 2 that requires maxm Reλm (Ai ) ≥ 0. -1 1 G2 c2 reveals the indispensability of direct links among agents in the same cluster This corollary under 1 an acyclicly partitioned interaction graph. Note that such direct communication require7 c3 8 5 ments -1 for intra-cluster agents is not necessary under a nonnegatively weighted interaction graph (see [11], [12], [15] for references). Remark 4: It is worth mentioning for the condition in (19) that one can set ci = 1 for all i, and adjust the global factor c only to result in cluster synchronization. In contrast, without the acyclic partitioning structure, the local weighting factors ci ’s need to satisfy the lower bound conditions in (13). Note that (19) specifies the tightest lower bound for c, while a lower bound reported in [16] for identical linear systems via Lyapunov stability analysis can be quite loose. IV. S IMULATION E XAMPLES In this section, we provide application examples for cluster synchronization of nonidentical linear systems. We also conduct numerical simulations using these models to illustrate the derived theoretical results. A. Example 1: Heading alignment of nonidentical ships Consider a group of four ships with the interaction graph described by Fig. 2(a), where ship 1 and 2 (respectively, ship 3 and 4) are of the same type. The purpose is to synchronize the heading angles for ships of the same type. The steering dynamics of a ship is described by the well-known Nomoto model [21]: ψ̇l (t) = vl (t) 1 κi v̇l (t) = − vl (t) + ul (t) τi τi (20) where ψl is the heading angle (in degree) of a ship l ∈ I, vl (deg/s) is the yaw rate, and ul is the output of the actuator (e.g., the rudder angle). The parameter τi is a time constant, and κi is the actuator gain, both of which are related to the type of a ship. Define for i = 1, 2 the system matrices     0 1 0  , Bi =   , Ai =  κi 0 − τ1i τi (21) and assume that τ1 = 42.21, τ2 = 107.3, κ1 = 0.181, κ2 = 0.185. The solutions to the   22.3 233.2 34 580  and P2 =  , Riccati equations in (8) are given by P1 =  233.2 3915.4 580 16875 which lead to the control gain matrices K1 = −[1 16.79] and K2 = −[1 29.09]. Since maxi=1,2 λmax (Ai + ATi ) = 0.99, we set c = 1 according to (12). The weighted graph Laplacian is given by   0 0 5 −5   −c c   1 1 1 −1 Lc =  ,  −1 1  0 0   0 0 −c2 c2  c1 4   using the definition in (6). So, L̂11 = 1, L̂22 = 1, and for any which yields L̂c =  −1 c2 Ŵ1 > 0 and Ŵ2 > 0, the inequalities in (10) hold. We choose Ŵ1 = Ŵ2 = 1. It follows that λmax (Ŵ) = 1, λmin (Ŵ L̂o + L̂To Ŵ) = −3, and λmin (Ŵi L̂ii + L̂Tii Ŵi ) = 2 for i = 1, 2. Then, we can choose c1 = c2 = 2 so that the inequalities in (13) are satisfied. Simulation result in Fig. 3(a) shows that cluster synchronization is achieved for the heading angles (the velocity vl (t) of every agent will converge to zero as shown in Fig. 3(b)). Now, let c1 = 0 so that agents 1 and 2 in cluster C1 have no direct connection. Cluster synchronization is still achieved as shown in Fig. 3(c). This example illustrates that intracluster connections are not necessary for cluster synchronization under a cyclicly partitioned ψ(t) (deg) 40 20 ψ1 ψ2 ψ3 ψ4 0 −20 −40 0 v1 v2 v3 v4 2 v(t) (deg/s) 60 1 0 −1 −2 100 200 t (sec) 300 0 100 200 t (sec) 300 (a) Under the graph in Fig. 2(a), cluster synchro- (b) Under the graph in Fig. 2(a), the velocity nization is achieved with c1 = c2 = 2. vl (t) of every ship converges to zero. 60 ψ(t) (deg) ψ1 ψ2 ψ3 ψ4 20 0 −20 −40 0 ψ(t) (deg) 50 40 ψ1 ψ2 ψ3 ψ4 0 −50 100 200 t (sec) 300 0 100 200 t (sec) 300 (c) Under the graph in Fig. 2(a), cluster synchro- (d) Under the acyclic partitioning graph in Fig. nization is achieved with c1 = 0, c2 = 2. 2(b), ψl ’s in the first cluster cannot synchronize together. Fig. 3. Evolutions of ψl (t) and vl (t) for the ship heading steering dynamics in (20) connected with graphs in Fig. 2. interaction graph. However, under an acyclic partition as in Fig. 2(b), the first cluster of agents, having no direct connections, cannot achieve state synchronization as shown in Fig. 3(d). B. Example 2: Cluster synchronization of oscillators The studied cluster synchronization problem for nonidentical linear systems may find applications in the coexistence of oscillators with different frequencies. To see this, let us consider two clusters of coupled harmonic oscillators with graph topology in Fig. 4(a), where the first cluster contains a sender s1 and two receivers r1 and r2 , the second cluster contains a sender s2 and two receivers r3 and r4 , and the four receivers are coupled by some directed links. Assume the angular frequencies of the two clusters of oscillators are w1 = 2 rad/s and 1 3 4 1 3 G2 G2 (b) (a) G1 3 s1 1 1 r1 -1 1 4 1 s2 1 r4 1 r3 x1l (t) -5 5 10 0 −10 0 G2 (a) c2 s1 r1 r2 s2 r3 r4 20 r2 c1 4 4 π 2π t (sec) 3π (b) Fig. 4. (a) Interaction graph partitioned into two clusters C1 = {s1 , r1 , r2 } and C2 = {s2 , r3 , r4 }. (b) Evolutions of the first components of the nonidentical harmonic oscillators under the graph in Fig. 4(a). 5 w2 = 2.5 rad/s, respectively. So, the dynamic equation of each oscillator is ẋ1l (t) = x2l (t), ẋ2l (t) = −wi2 x1l (t) + ul (t), l ∈ Ci , i = 1, 2 (22) which corresponds to the following system matrices:     0 1 0  , Bi =   , i = 1, 2. Ai =  −wi2 0 1 The objective is to let the receivers of each cluster follow the state of the sender. By a similar design procedure as in the previous example, we can set K1 = −[0.1231 1.1163], K2 = −[0.0554 1.0539], c = 6, and c1 = c2 = 13. Simulation result in Fig. 4(b) shows the synchronous oscillations of the harmonic oscillators with two distinct angular frequencies. V. C ONCLUSIONS This paper investigates the state cluster synchronization problem for multi-agent systems with nonidentical generic linear dynamics. By using a dynamic structure for coupling strategies, this paper derives both algebraic and graph topological clustering conditions which are independent of the control designs. For future studies, cluster synchronization which can only be achieved for the system outputs is a promising topic, especially for linear systems with parameter uncertainties or for heterogeneous nonlinear systems. For completely heterogenous linear systems, research works following this line are conducted by the authors in [22] and others in [23]. For nonlinear heterogeneous systems, the new theory being established for complete output synchronization problems [24], [25] may be further extended. Another interesting challenge existing in cluster synchronization problems is to discover other graph topologies that meet the algebraic conditions. A PPENDIX I P ROOF OF L EMMA 1  1 0   ∈ Rli ×li for i = Proof: Let S = blockdiag{S1 , . . . , SN }, where Si =  1li −1 Ili −1   1 0 . By direct computation 1, . . . , N . Clearly, Si has the inverse matrix Si−1 =  −1li −1 Ili −1 one can show that   0 γij . Si−1 Lij Sj =  0 L̂ij This implies the first claim when i = j. For the second claim, consider that  0 γ11  0 c L̂ 1 11  . .. −1 . S Lc S =  . .  0 γN 1  0 L̂N 1 ··· 0 γ1N  ··· ... 0 .. . L̂1N .. . ··· 0 γN N     .     ··· 0 cN L̂N N Rearrange the columns and rows of S−1 Lc S by permutation and similarity transformations to get the following block upper-triangular  01×N  ..  .    01×N  0(L−N )×N matrix γ11 .. . ··· .. . γN 1 · · ·  γ1N ..  .   , γN N   L̂c where L̂c is defined in (6). Then, the second claim of this lemma follows immediately. A PPENDIX II P ROOF OF T HEOREM 1 Proof: The closed-loop system equations for (1) using couplings (4) are given as   X X żl = Aci zl − c ci blk Ezk + blk Ezk  , (23) k∈C / i k∈Ci for all l ∈ Ci , i = 1 . . . , N , where zl = [xTl , ηlT ]T and     Ai Bi Ki 0 0 , E =  . Aci =  0 Ai + Bi Ki −In In (24) Let el (t) := zl (t) − zσi +1 (t) for l ∈ Ci and l 6= σi + 1, i = 1, . . . , N . It follows from (23) and Assumption 3 that " ėl (t) = Aci el (t) − c ci X (blk − bσi +1,k )Eek (t) k∈Ci  + X (blk − bσi +1,k )Eek (t) . (25) k∈C / i Define a nonsingular transformation matrix Q as follows:     In 0 I 0  , Q−1 =  n , Q= In In −In In (26) and let εl (t) := [ξlT (t), ζlT (t)]T = Q−1 el (t). Clearly, ξl = xl − xσi +1 and ζl = ηl − ησi +1 − xl + xσi +1 . By (25), one can obtain the following dynamic equations: ξ˙l (t) = (Ai + Bi Ki )ξl (t) + Bi Ki ζl (t), " X (blk − bσi +1,k )ζk (t) ζ̇l (t) = Ai ζl (t) − c ci k∈Ci  + X (blk − bσi +1,k )ζk (t) , k∈C / i for l ∈ Ci and l 6= σi + 1, i = 1, . . . , N . Since Ki stabilizes (Ai , Bi ), the variable εl (t) tends to zero as t → ∞ if and only if ζl (t) tends to zero. Denote ζ(t) = [ζσT1 +2 (t), . . . , ζσT1 +l1 (t), · · · , ζσTN +2 (t), . . . , ζσTN +lN (t)]T , which evolves with the following differential equation ζ̇(t) = Â − cL̂c ⊗ In ζ(t). (27) Clearly, ζ(t) and every εl (t) (hence every el (t)) all converge to zero if and only if Â−cL̂c ⊗In is Hurwitz. That is, we have shown that limt→∞ kxl (t) − xk (t)k = 0 and limt→∞ kηl (t) − ηk (t)k = 0, ∀l, k ∈ Ci , i = 1, . . . , N . Next, we prove that ηl (t) for any l ∈ I vanishes as t → ∞. To this end, for each i = 1, . . . , N , let ηi (t) be the solution of η̇i (t) = (Ai + Bi Ki )ηi (t) with an arbitrary initial P value ηi (0). Since k∈Cj blk = 0 ∀l ∈ I by Assumption 3, we have that η̇i (t) = (Ai + Bi Ki )ηi (t) " = (Ai + Bi Ki )ηi (t) − c ci ( X blk )(ησi +1 − xσi +1 ) k∈Ci  N X X + ( blk )(ησj +1 − xσj +1 ) , j=1,j6=i k∈Cj for any l ∈ Ci . Subtracting the above from (4b) yields η̇l (t) − η̇i (t) = (Ai + Bi Ki )(ηl (t) − ηi (t))   N X X X − c ci blk ζk + blk ζk  . k∈Ci j=1,j6=i k∈Cj The above system is exponentially stable and driven by inputs which all converge to zero exponentially fast. Therefore, for any ηl (0), l ∈ I, we have ηl (t) → ηi (t) → 0 ∀l ∈ Ci , as t → ∞. Lastly, we show that inter-cluster state separations can be achieved for any initial states xl (0)’s by selecting ηl (0)’s properly. Given any set of xl (0), l ∈ I, choose ηl (0), l ∈ I such that xl (0)−ηl (0) = xσi +1 (0)−ησi +1 (0) for all l ∈ Ci , i = 1 . . . , N , and lim supt→∞ keAi t [xl (0)− ηl (0)] − eAj t [xl (0) − ηl (0)]k = 6 0 for any i 6= j. Considering the definition of ζl and the linear differential equation (27), one has xl (t) − ηl (t) = xσi +1 (t) − ησi +1 (t) for all t > 0. This together with (4) lead to the following dynamics ẋl (t) − η̇l (t) = Ai (xl (t) − ηl (t)), ∀l ∈ Ci . It follows that xl (t) = eAi t [xl (0) − ηl (0)] + ηl (t) → eAi t [xl (0) − ηl (0)], ∀l ∈ Ci as t → ∞. Therefore, lim supt→∞ kxl (t) − xk (t)k = 6 0 ∀l ∈ Ci , ∀k ∈ Cj , ∀i 6= j. This completes the proof. A PPENDIX III P ROOF OF C OROLLARY 1 Proof: The proof for the necessity and sufficiency of (9) is straightforward using the results in Lemma 1 and Theorem 1, and thus is omitted for simplicity. We only show that state separations are possible for any initial states xl (0), l ∈ I by using the dynamic couplings even for systems with identical parameters. Constellate the states zl (t) = [xTl (t), ηlT (t)]T of all L agents to form z(t) := [z1T (t), z2T (t), . . . , zLT (t)]T . It follows that ż(t) = (IL ⊗ Ac − cLc ⊗ E)z(t), (28)     A BK 0 0  and E =   . 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ArbiText: Arbitrary-Oriented Text Detection in Unconstrained Scene Daitao Xing1,3 , Zichen Li1,3 , Xin Chen4 , and Yi Fang1,2,3∗ arXiv:1711.11249v1 [cs.CV] 30 Nov 2017 1 NYU Multimedia and Visual Computing Lab 2 NYU Abu Dhabi, UAE 3 New York University, Brooklyn, NY, USA 4 HERE Technologies Abstract Arbitrary-oriented text detection in the wild is a very challenging task, due to the aspect ratio, scale, orientation, and illumination variations. In this paper, we propose a novel method, namely Arbitrary-oriented Text (or ArbText for short) detector, for efficient text detection in unconstrained natural scene images. Specifically, we first adopt the circle anchors rather than the rectangular ones to represent bounding boxes, which is more robust to orientation variations. Subsequently, we incorporate a pyramid pooling module into the Single Shot MultiBox Detector framework, in order to simultaneously explore the local and global visual information, which can therefore generate more confidential detection results. Experiments on established scene-text datasets, such as the ICDAR 2015 and MSRA-TD500 datasets, have demonstrated the superior performance of the proposed method, compared to the state-of-the-art approaches. Figure 1: Text detection results of the proposed ArbiText method. Images in the first row shows examples from ICDAR2015 dataset, the second row shows examples MSRATD500 dataset 1. Introduction Understanding texts in the wild plays an important role in many real-world applications such as PhotoOCR [2], road sign detection in intelligent vehicles [3], license plate detection [17], and assistive technology for the visually impaired [6] [1]. To achieve this goal, the task of accurate arbitraryoriented text detection becomes extremely important. Conventionally, when dealing with horizontal texts under controlled environments, this task can be accomplished through character-based methods such as [9], [10], [18], and [24] considering that individual letters can be easily segmented and distinguished. However, in an unconstrained naturalscene image, text detection becomes rather challenging due ∗ 5 Metrotech Center LC024, Brooklyn, NY, 11201; yfang@nyu.edu; Tel: +1-646-854-8866 to uncontrolled text variations and uncertainties, such as multi-orientation, text distortion, background noise, occlusion, and illumination changes. To address these problems, a lot of recent efforts have been devoted to employing state-of-the-art generic object detectors, such as the Fully Convolutional Network(FCN) [20], Region-based Convolutional Neural Network(R-CNN) [19], and Single Shot Detector(SSD) [14], for the purpose of text detection in the wild. Despite their promising performance in generic object detection, these methods suffered from bridging gaps between the data distributions of texts and generic objects. To enhance the generative abilities of existing deep mod- Email: 1 Figure 2: The Framework of the Proposed Method. Given an input image with a size of 384×384,VGG-16 base network outputs the first feature map from conv4 3 layer. More feature maps with cascading sizes are extracted from extra layers following the first feature map. The first feature map is also used to produce different sub-region representations through the Pyramid Pooling Module. These representation layers are then concatenate feature maps with same size to output the final feature maps. Finally, those maps are fed into a convolution layer to get the final. els, [7] proposed to naturally blend rendered words onto wild images for training data augmentation. The trained model based on such training data is robust to noises and uncontrolled variations. [31] and [13] attempted to integrate region-proposal layers into the deep neural networks, which can generate text-specific proposals (e.g. bounding boxes with larger aspect ratios). In [16], the bounding-box rotation proposals were introduced to make the proposed model more adaptive for unknown orientations of texts in naturalscene images. Nevertheless, the aforementioned scene-text detectors either needed to consider a lot of proposal hypotheses, thus dramatically decreasing the computational efficiency, or utilize insufficient preset bounding-box characteristics to handle severe visual variations of scene-texts in unconstrained natural images. the conventional rectangular ones. The Single Shot Detector (SSD), one of the state-of-art object detectors, is employed, considering its fast detection speed and promising accuracy in generic object detection. Besides the feature maps generated by the original SSD, we additionally incorporate a pyramid pooling module, which can build multiple feature representations on different spatial scales. By merging those different kinds of feature maps, both the local and global information can be preserved, such that texts in unconstrained natural scenes can be more reliably detected. Subsequently, the merged feature maps are fed into a text detection module, consisting of several fully connected convolutional layers, to predict confidential circle anchors. Furthermore, in order to overcome the difficulty of deciding positive points caused by unfixed sizes of circle anchors, we introduce a novel mask loss function by assigning those ambiguous points to a new class. To obtained the final detection results, the Locality-Aware Non-MaximumSuppression (LANMS) scheme [33] is employed. It should be noted that we do not utilize any proposal, which makes the proposed method more computationally efficient. To address the aforementioned drawbacks of existing methods, we propose a novel proposal-free model for arbitrary-oriented text detection in natural images based on the circle anchors and the Single Shot Detector (SDD) framework. More specifically, we adopt circle anchors to represent the bounding boxes, which are more robust to orientation, aspect ratio, and scale variations, compared to In summary, the contributions of our work mainly lie in 2 three-fold: text-line hypothesis. Although these methods can achieve state-of-art results even with scene-text detection in the wild, the requirement for a sophisticated post-processing step of word partitioning and false positive removal can be too time consuming and computationally intensive for realworld applications. A more recent method proposed in [33], however, seeks to make dramatic improvement in efficiency over [29] and [26] by eliminating intermediate steps such as candidate aggregation and word partitioning in the neural network. Nevertheless, the inherent nature of the segmentation approach dense per-pixel processing and prediction - is still a bottleneck that prevents segmentation-based methods from outperforming its competitors. Region-Proposal-based Methods: Although regionproposal based models like R-CNN have already been a state-of-art object detector, it can not be implemented for the purpose of text detection without modifications since its anchor box design is not ideal for the large aspect ratio of words/text-lines. [31] addresses this problem by proposing a novel Region Proposal Network(RPN) called InceptionRPN, which contains a preset of text characteristic prior bounding boxes to generate text-specific proposals and thus filtering out low-quality word regions. However, [31] only performs well on horizontal texts since bounding box characteristics are extremely unpredictable for scene-texts in the wild; multi-oriented and distorted texts can create countless possibilities and variations of bounding-box size, shape, and orientation. To address this challenge, some researchers designed novel region-proposal methods: the rotation proposal method in [16] has the ability to predict the orientation of a text line and thus generate inclined bounding-boxes for oriented texts, while the quadrilateral sliding windows in [15] create a much tighter bounding-box fit around text regions, thus dramatically reduce background noise and interference. On the other hand, some researchers propose methods to modify model architecture like the one proposed in [5], which adds 2D offsets in the standard convolution to enable free form deformation of the sampling grid, and the one proposed in [8], which utilizes direct bounding-box regression originating from a center anchor point in a proposal region. SSD-based Methods. SSD-based method is highly stable and efficient in generating word proposals because SSD is one of the fastest object detector that is also as accurate as slower region-proposal based models like R-CNN. However, SSD possesses similar shortcomings in terms of anchor box design when it comes to scene-text detection. Thus, [13] supplements SSD with ”textbox layers” that can generate bounding-boxes with larger aspect ratios and simultaneously predict text presence and bounding boxes. Unfortunately, this method only works on horizontal texts, • We propose a novel proposal-free method for detecting arbitrary-oriented texts in unconstrained natural scene images, based on the circular anchor representation and the Single Shot Detector framework. The circular anchors are more robust to different aspect ratio, scale, and orientation variations, compared to conventional rectangular ones. • We incorporate a pyramid pooling module into SSD, which can explore both the local and global visual information for robust text detection. • We develop a new mask loss function to overcome the difficulty of deciding positive points caused by unfixed sizes of circle anchors, which can therefore improve the final detection accuracy. 2. Related Works Character-based detection methods have already achieved state-of-art results on horizontal texts in relatively controlled and stable environments. Methods like those proposed in [9], [10], [18], and [24] either detect individual characters by classification of sliding windows or utilize some form of connected-component and regionbased framework such as the Maximally Stable Extremal Regions(MSER) detector. However, some of these methods might not be ideal for detecting scene-texts or multi-oriented texts as more and more environmental variations and uncertainties in terms of text distortion, orientation, occlusion, and noise are introduced. Detecting individual characters in close clusters or ones that blend into the background can also be challenging. Hence, many researchers decide to tackle the problem by approaching the task of text detection as object detection: treating words and/or text-lines as the target object. Region-based Convolutional Neural Network(RCNN)[19], Single Shot Detector(SSD)[14], and segmentation-based Fully Convolutional Network(FCN)[20] are frequently re-purposed for text detection because their superior speed and accuracy are better suited for the time and resource-constraining nature of the task. Expectedly, a majority of the cutting-edge research in scene-text detection, including ours, are based on one of the aforementioned object detection models, which we will analyze below. Segmentation-based Methods: Both [29] and [26] accomplish semantic segmentation of text lines by utilizing Fully Convolutional Network(FCN), which has achieved great performance in pixel-level classification tasks. In [29], for example, pixel-wise text/non-text salient map is first produced via the FCN and subsequently, geometric and character processing is implemented to generate and filter 3 conv10 2 and global layers. However, local information is lost as layer goes deeper and deeper, which results in poor detection precisions especially on texts with complex contextual information. Inspired by [30], we introduced the Pyramid Pooling Module to leverage low-level visual information even in deeper layers. This module fuses feature maps with different pyramid scales. As shown in Fig. 2, the first feature map from the based network is separated on pyramid levels into different sub-regions and output pooled representations after a 1 ∗ 1 convolution layer. Thus, the low-level information of the original image could be preserved in multi-scale level feature maps, which will be further concatenated with ones of the same size to form the final feature map for text detection. By merging these two types of features, both the local and global visual information can be explored. Finally, the text detection layer applies a 3∗3 convolution kernel on the fused feature map to output prediction on the text bounding box. Figure 3: Matching of Circle Anchors The red circle is generated from the ground-truth coordinates, and the blue circle anchor is the one with same size with red circle on the feature point. The blue circle which associates the red one by a vector(cx, cy, area, diameter, angle) where cx and cy are offset between centers of circles. 3.2. Circle Anchors and not scene-texts. Thus in this paper, we attempt to solve the aforementioned limitations of previous detection models by utilizing a proposal-free method based on circular anchors and the SSD framework. Our method is computationally more efficient than both segmentation-based and Region-Proposalbased models because the removal of the region-proposal layer in our network. Our method also improves upon the existing SSD-based method by having the ability to detect both arbitrary-oriented texts and generic objects. As illustrated in Fig. 3, instead of the traditional rectangular anchors, we use circle anchors to represent the bounding box. Specifically, a bounding box can be represented by a 5-dimensional vector (x, y, a, r, θ), where a, r, and θ denotes the area, radius, and rotated angle of a circle anchor. a 1 arcsin( 2 ) 2 (2r ) x2 = r · cos(α + θ), y2 = r · sin(α + θ) α= 3. Method x3 = r · cos(α − θ), y3 = r · sin(α − θ) (1) x1 , y1 = −x3 , −y3 , x4 , y4 = −x2 , −y2 In this section, we will describe the details of our proposed model - ArbiText. We will first introduce the framework and network architecture of our method. Subsequently, we will elaborate on the key components such as the circle anchor representation and the proposed loss function. x1 , x2 , x3 , x4 = x1 + x, x2 + x, x3 + x, x4 + x y1 , y2 , y3 , y4 = y1 + y, y2 + y, y3 + y, y4 + y On a feature map of size (w ∗ w), location, denoted (i, j), associates a circle anchor C0 with (c, ∆x, ∆y, ∆a, ∆r, ∆θ), indicating that a unique circle anchor, represented by (x, y, a, r, θ), is detected with confidence c, where 3.1. Model Framework Our proposed method, in essence, is a multi-scale, proposal-free framework based on the Single Shot Detector. As shown in Fig.2, our model mainly consists of the following four components: 1) the backbone-based network for converting original images into dense feature representations; 2) the feature maps component with cascading map size for detecting multi-scale texts; 3) the Pyramid Pooling Module [30] for extracting sub-region feature representations; and 4) the final text detection layer for circle anchor prediction. We adopt VGG-16 [22] as our base network, and utilize the 6 feature maps at the conv4 3, conv7, con8 2, conv9 2, ra ra + j, y = ∆y · +i w w ra r = exp(∆r) · w ra a = exp(∆a) · w θ = ∆θ x = ∆x · (2) Here, we use the area a and radius r for computational stability. Also, we multiply each value by a factor rwa where ra =1.5. 4 (a) (b) Figure 4: (a) shows the the coordinates of a rectangle. Given the area and diagonal of rectangle, p2 and p3 can be calculated by rotation angles. p1 and p4 are diagonal points of p3 and p1, respectively, which also have negative values. (b) shows the predictable vertical flag, which is 0 if the angles of the bounding box are between −45◦ and 45◦ ; otherwise, it is set to 1. Figure 6: The Selection of Ground Truth. The blue rectangle is the bounding box and the 10 ∗ 10 grid represents a feature map with the same size. Only the feature points in the red eclipse are labeled as positives; the points outside of the yellow eclipse are labeled as negatives. The feature points in the yellow region will be labeled as their own separate class. Figure 5: Score Distribution. (a) shows a bounding box the text. The red and yellow rectangles are possible bounding boxes that have maximum IOU overlap scores with ground truth . (b) shows the eclipse-shape like score function we use in ArbiText. For a SSD-based method, all points on a feature map will potentially be used for minimizing a specific loss function. Each feature point needs to be labeled as either positive or negative. Specifically, in SSD, the points that are labeled as positive are chosen from the regions where the overlap between the default anchor and ground-truth bounding box is larger than 0.5. However, there is no default anchor in our method, but we can still calculate a confidence score for each point. As illustrated in Fig. 5 (a), the feature point on the edge of the bounding box can have a maximum overlap of 0.5(bounding boxes are colored in red and yellow). So the score follows an eclipse distribution (as illustrated in Fig. 5 (b)). The scores at the center of the eclipse have a maximum value of 1.0 and it decreases to 0.5 when the points reach the edge. We use a semi-ellipse as the function to compute the score for each point. As a result, the points outside of the eclipse have a score value 0. Imagine an eclipse score function which has rotation angle θ, the semi-major axes and semi-minor axes have length of w2 and h2 , respectively, where w and h are the width and height of the bounding box. Thus, the score function can be The angle θ is the intersection angle between the long edge of the bounding box and the horizontal axis. Thus, the value of θ ranges from −90◦ to 90◦ . In a deep neural network, each layer has a receptive field that indicates how much contextual information we can utilize. Although the circle anchor representation is invariant to scale variations, [32] has shown that feature maps have limited receptive fields that are much smaller than theoretical ones, especially on high-level layers. As a result, if we do not utilize multi-scale feature maps, the detection scope of the proposed circle anchor representation will be restricted. And considering the size of the extra feature layers, this operation only adds a small amount of computational cost. 3.3. Training Labels Rebuilding and the Loss Function Formulation . 5 4. Experiments represented as: A · (x)2 + B · (x) · (y) + C · (y)2 + F · s2 = F w h a = ;b = 2 2 2 2 2 A = a · sin (θ) + b · cos2 (θ) 4.1. Datasets In order to evaluate the performance of the proposed method, we ran experiments on two benchmark datasets: the ICDAR 2015 dataset and the MSRA-TD500(TD500) dataset. SynthText in the Wild[7] dataset contains more than 800,000 synthetic images created by blending rendered words on wild images. Only samples with width larger than 10 pixels are chosen for training. ICDAR 2015[34] incidental text dataset is from Challenge 4 of ICDAR 2015 Robust Reading Competition that includes 1000 training images and 500 testing images. Since those images are collected by Google Glasses, they suffer from motion blur. The blurry texts have a label of ”###” and are excluded from our experiment. We also included training and testing images from the ICDAR 2013 dataset[12], which helps us in building a more robust text detector. MSRA-TD500(TD500)[4] is a multilingual dataset that includes oriented texts in both Chinese and English. Unlike ICDAR 2015, texts in MSRA-TD500 are annotated at the text-line level and the images were captured more formally, thus texts are much clearer and standardized. There are a total of 500 images, 300 of them were used as training data and 200 were used as testing data. (3) B = −2(a2 − b2 ) · sin(θ) · cos(θ) C = a2 · cos2 (θ) + b2 · sin2 (θ) F = a2 · b2 where x,y are the distances between a feature point and the center of the bounding box respectively,s is score. According to the score function, all points inside the eclipse have a score greater than 0.5. However, the closer the point is to the edge of the bounding box, the large the noise outside of the bounding box will be, which could make the training of networks harder to converge. Thus, only the points with score large than a threshold α will be treated as positives (as shown in Fig. 6, only points inside the red zone are labeled as ”positive”). Points outside of the bounding box will be labeled as ”negative”. For those points with a score between 0.5 and α, we assign them an additional label. Thus, there will be a total of (N + 1) classes(without background). This additional class will only be involved in calculating the classification loss. The feature maps with different sizes can detect texts with different scales. A default box will be labeled as positive if hg 1< < 4. (4) ch 4.2. Implementation Details Base network In our experiment, we uses a pre-trained VGG-16 as our based network. This network is widely used in object detection tasks. All images are resized to 384 ∗ 384 after data augmentation. We extracted five layers with cascading resolution as our feature maps, which are conv4 3, conv7, conv8, conv9, and conv10. We first trained our model on the SynthText dataset for 50,000 iterations with a learning rate of 0.001. Then, we fine-tuned our model using the other datasets with a 0.0005 learning rate. The details of training different datasets are described in later sections. We tested our model on different α values and we discovered that our model achieves optimal performance when α is set to 0.7 for text detection. Locality-Aware NMS In the post-processing stage, bounding boxes with a confidence score greater than 0.5 will be used to produce the final output by NMS merging. The naive NMS has O(n2 ) computational complexity, which is not ideal for real-world applications. We adopt LocalityAware NMS [33] to improve speed of merging bounding boxes. Hard Negative Mining Hard Negative Mining is essential for SSD-based methods because of the imbalance between positive and negative training samples. We adopt the same configuration in SSD[14] by selecting the top 3K where hg is the height of bounding box and ch is the height of cell on feature map. For training, we use the following objective loss function: L(mask, c, l, g) = λ1 λ2 N +1 1 X maski · Lcls (ci )+ Ncls i=0 N 1 X maski · Lloc (l, g)+ Nreg i=1 (5) N 1 X maski · Lcls (vertical) Nreg i=1 where N is the number of object categories, l is the prediction location, and g is the ground-truth location. For each class, maski = 1 if the corresponding point is labeled as ”positive” and belongs to the i-th class. is the loss for vertical bounding box classification that only includes the positive points. We adopt L1 loss for smoothing and Softmax loss as the classification losses. 6 Figure 7: Results of ArtTex on the ICDAR2015 dataset. Blue rectangles are detected text regions by using ArtTex. Table 1: Comparison results of various methods on the ICDAR 2015 Incidental Text dataset negative training samples, where k is number of positive training samples. Thus, we adopt Locality-Aware NMS[33] in our experiment. This algorithm can produce bounding boxes with greater precision in shorter time. Data Augmentation We utilize a data augmentation pipeline that is similar to the one in SSD to make our model more robust against different text variations. The original image is randomly cropped into patches. The crop size is chosen from [0.1, 1] of original image size. Each sample patch will be horizontally flipped with a probability of 0.5. In order to balance samples of different orientations, we also augmented datasets by randomly rotating images by θ degrees. θ is randomly chosen from the following angle set: (-90, -75, -60, -45, -30, -15, 0, 15, 30, 45, 60, 75, 90). Method HUST MCLAB NJU Text StradVision-2 MCLAB FCN[29] CTPN[23] Megcii-Image++ Yao et al.[26] Seglink [21] ArbiText Precision 47.5 72.7 77.5 70.8 51.6 72.4 72.3 73.1 79.2 Recall 34.8 35.8 36.7 43.0 74.2 57.0 58.7 76.8 73.5 F-score 40.2 48.0 49.8 53.6 60.9 63.8 64.8 75.0 75.9 4.3. Detection Results 4.3.1 Detecting Oriented English Text official off-line evaluation scripts. We list the results of our model along with other state-of-art object and text detection methods. The results were obtained from the original papers. The best result of this dataset was obtained by Seglink[21], which achieved a F-measure score of 75.0%. However, our model obtained a score of 75.9%. The improvement comes from the high precision rate we obtained, which outperforms the second First, our model is tested on ICDAR 2015 dataset. The pre-trained model is fine-tuned using both the ICDAR 2013 and the ICDAR 2015 training datasets after 20k iterations. Considering all images in ICDAR 2015 have high resolution, testing images are first resized to 768 ∗ 768. The threshold α is set to 0.7, similar to the one used in the pre-training stage. Performance is evaluated using the 7 Figure 8: Results of ArtTex on the MSRA-TD500 dataset. Blue rectangles are detected text regions by using ArtTex. Table 2: Comparison results of various methods on the MSRA-TD500 dataset highest model by 6.1% Figure.7 shows several detection results taken from the testing dataset of ICDAR 2015. Our proposed method ArbiText can distinguish and localize all kinds of scene text in noisy backgrounds. 4.3.2 Method Kang et al.[11] Yao et al.[25] Yin et al.[27] Yin et al.[28] Zhang et al.[29] Yao et al.[26] Seglink [21] ArbiText Detecting Multi-Lingual Text in Long Lines We further tested our method on the TD500 dataset consists of long text in English and non-Latin scripts. We augmented this dataset by doing the following: 1) Randomly place an image on a canvas of n times of the original image size filled by mean values where n ranges from 1 to 3. 2)We applies random crop according to the overlap strategy described in section 4.3. Thus, we obtained enough images for training. The pre-trained model is fine-tuned for 20K iterations. All images are resized to 384 ∗ 384, which is consistent with the training stage. The experiment has demonstrated that this technique can dramatically increase detection speed without losing much precision. As illustrated in Table 2, ArbiText achieved comparable F-measure scores with other state-of-the-art methods. However, benefiting from lighter network architecture and simplified anchors mechanism, ArbiText has the highest FPS of 12.1. Figure.8 shows ArbiText can detect long lines of text in mixed languages(English and Chinese) without changing any parameters or structures. Precision 71 63 81 71 83 77 86 78 Recall 62 63 63 61 67 75 70 72 F-score 66 60 74 65 74 76 77 75 FPS 0.14 0.71 1.25 0.48 ∼1.61 8.9 12.1 4.4. Limitations As shown in Figure.9.a,b, curved texts can’t be represented by circle anchors. Moreover, Figure.9.c shows our model’s weakness in detecting hand-written texts. 5. Conclusion We have presented ArbiText, a novel, proposal-free object detection method that can be utilized to detect both arbitrary-oriented texts and generic objects simultaneously. Its outstanding performance on different benchmarks demonstrates that ArbiText is accurate, robust, and flexible for real-world applications. In the future, we will extend 8 [13] (a) (b) (c) [14] Figure 9: Failure Cases On MSRA-TD500 The blue rectangles are true positives. The red ones are false negatives and yellow ones are false positives. In (a) and (b), ArbiText fails to detect curved texts. 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arXiv:1609.06782v2 [cs.CV] 22 Feb 2018 1 Deep Learning for Video Classification and Captioning Zuxuan Wu (University of Maryland, College Park), Ting Yao (Microsoft Research Asia), Yanwei Fu (Fudan University), Yu-Gang Jiang (Fudan University) 1.1 Introduction Today’s digital contents are inherently multimedia: text, audio, image, video, and so on. Video, in particular, has become a new way of communication between Internet users with the proliferation of sensor-rich mobile devices. Accelerated by the tremendous increase in Internet bandwidth and storage space, video data has been generated, published, and spread explosively, becoming an indispensable part of today’s big data. This has encouraged the development of advanced techniques for a broad range of video understanding applications including online advertising, video retrieval, video surveillance, etc. A fundamental issue that underlies the success of these technological advances is the understanding of video contents. Recent advances in deep learning in image [Krizhevsky et al. 2012, Russakovsky et al. 2015, Girshick 2015, Long et al. 2015] and speech [Graves et al. 2013, Hinton et al. 2012] domains have motivated techniques to learn robust video feature representations to effectively exploit abundant multimodal clues in video data. In this chapter, we review two lines of research aiming to stimulate the comprehension of videos with deep learning: video classiﬁcation and video captioning. While video classiﬁcation concentrates on automatically labeling video clips based on their semantic contents like human actions or complex events, video captioning 4 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning Benchmarks and challenges Video classiﬁcation Video captioning … Basketball Graduation … A woman is playing Frisbee with a black dog on the grass. Basic deep learning modules Figure 1.1 An overview of the organization of this chapter. attempts to generate a complete and natural sentence, enriching video classiﬁcation’s single label to capture the most informative dynamics in videos. There have been several efforts surveying the literature on video content understanding. Most of the approaches surveyed in these works adopted handcrafted features coupled with typical machine learning pipelines for action recognition and event detection [Aggarwal and Ryoo 2011, Turaga et al. 2008, Poppe 2010, Jiang et al. 2013]. In contrast, this chapter focuses on discussing state-of-the-art deep learning techniques not only for video classiﬁcation but also video captioning. As deep learning for video analysis is an emerging and vibrant ﬁeld, we hope this chapter could help stimulate future research along the line. Figure 1.1 shows the organization of this chapter. To make it self-contained, we ﬁrst introduce the basic modules that are widely adopted in state-of-the-art deep learning pipelines in Section 1.2. After that, we discuss representative works on video classiﬁcation and video captioning in Section 1.3 and Section 1.4, respectively. Finally, in Section 1.5 we provide a review of popular benchmarks and challenges in that are critical for evaluating the technical progress of this vibrant ﬁeld. 1.2 Basic Deep Learning Modules In this section, we brieﬂy review basic deep learning modules that have been widely adopted in the literature for video analysis. 1.2 Basic Deep Learning Modules 5 1.2.1 Convolutional Neural Networks (CNNs) Inspired by the visual perception mechanisms of animals [Hubel and Wiesel 1968] and the McCulloch-Pitts model [McCulloch and Pitts 1943], Fukushima proposed the “neocognitron” in 1980, which is the ﬁrst computational model of using local connectivities between neurons of a hierarchically transformed image [Fukushima 1980]. To obtain the translational invariance, Fukushima applied neurons with the same parameters on patches of the previous layer at different locations; thus this can be considered the predecessor of convolutional neural networks (CNN. Further inspired by this idea, LeCun et al. [1990] designed and trained the modern framework of CNNs LeNet-5, and obtained the state-of-the-art performance on several pattern recognition datasets (e.g., handwritten character recognition). LeNet-5 has multiple layers and is trained with the back-propagation algorithm in an end-toend formulation, that is, classifying visual patterns directly by using raw images. However, limited by the scale of labeled training data and computational power, LeNet-5 and its variants [LeCun et al. 2001] did not perform well on more complex vision tasks until recently. To better train deep networks, Hinton et al. in 2006 made a breakthrough and introduced deep belief networks (DBNs) to greedily train each layer of the network in an unsupervised manner. And since then, researchers have developed more methods to overcome the difﬁculties in training CNN architectures. Particularly, AlexNet, as one of the milestones, was proposed by Krizhevsky et al. in 2012 and was successfully applied to large-scale image classiﬁcation in the well-known ImageNet Challenge. AlexNet contains ﬁve convolutional layers followed by three fully connected (fc) layers [Krizhevsky et al. 2012]. Compared with LeNet-5, two novel components were introduced in AlexNet: 1. ReLUs (Rectiﬁed Linear Units) are utilized to replace the tanh units, which makes the training process several times faster. 2. Dropout is introduced and has proven to be very effective in alleviating overﬁtting. Inspired by AlexNet, several variants, including VGGNet [Simonyan and Zisserman 2015], GoogLeNet [Szegedy et al. 2015a], and ResNet [He et al. 2016b], have been proposed to further improve the performance of CNNs on visual recognition tasks: VGGNet has two versions, VGG16 and VGG19, which contain 16 and 19 layers, respectively [Simonyan and Zisserman 2015]. VGGNet pushed the depth of CNN architecture from 8 layers as in AlexNet to 16–19 layers, which greatly 6 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning improves the discriminative power. In addition, by using very small (3 × 3) convolutional ﬁlters, VGGNet is capable of capturing details in the input images. GoogLeNet is inspired by the Hebbian principle with multi-scale processing and it contains 22 layers [Szegedy et al. 2015a]. A novel CNN architecture commonly referred to as Inception is proposed to increase both the depth and the width of CNN while maintaining an affordable computational cost. There are several extensions upon this work, including BN-Inception-V2 [Szegedy et al. 2015b], Inception-V3 [Szegedy et al. 2015b], and Inception-V4 [Szegedy et al. 2017]. ResNet, as one of the latest deep architectures, has remarkably increased the depth of CNN to 152 layers using deep residual layers with skip connections [He et al. 2016b]. ResNet won the ﬁrst place in the 2015 ImageNet Challenge and has recently been extended to more than 1000 layers on the CIFAR-10 dataset [He et al. 2016a]. From AlexNet, VGGNet, and GoogLeNet to the more recent ResNet, one trend in the evolution of these architectures is to deepen the network. The increased depth allows the network to better approximate the target function, generating better feature representations with higher discriminative power. In addition, various methods and strategies have been proposed from different aspects, including but not limited to Maxout [Goodfellow et al. 2013], DropConnect [Wan et al. 2013], and Batch Normalization [Ioffe and Szegedy 2015], to facilitate the training of deep networks. Please refer to Bengio et al. [2013] and Gu et al. [2016] for a more detailed review. 1.2.2 Recurrent Neural Networks (RNNs) The CNN architectures discussed above are all feed-forward neural networks (FFNNs) whose connections do not form cycles, which makes them insufﬁcient for sequence labeling. To better explore the temporal information of sequential data, recurrent connection structures have been introduced, leading to the emergence of recurrent neural networks (RNNs). Unlike FFNNs, RNNs allow cyclical connections to form cycles, which thus enables a “memory” of previous inputs to persist in the network’s internal state [Graves 2012]. It has been pointed out that a ﬁnite-sized RNN with sigmoid activation functions can simulate a universal Turing machine [Siegelmann and Sontag 1991]. The basic RNN block, at a time step t, accepts an external input vector x (t) ∈ Rn and generates an output vector z(t) ∈ Rm via a sequence of hidden states 1.2 Basic Deep Learning Modules 7 h(t) ∈ Rr : h(t) = σ Wx x (t) + Whh(t−1) + bh (1.1) z(t) = softmax Wz h(t) + bz where Wx ∈ Rr×n, Wh ∈ Rr×r , and Wz ∈ Rm×r are weight matrices and bh and bz are biases. The σ is deﬁned as sigmoid function σ (x) = 1+e1−x and softmax (.) is the softmax function. A problem with RNN is that it is not capable of modeling long-range dependencies and is unable to store information about past inputs for a very long period [Bengio et al. 1994], though one large enough RNN should, in principle, be able to approximate the sequences of arbitrary complexity. Speciﬁcally, two well-known issues—vanishing and exploding gradients, exist in training RNNs: the vanishing gradient problem refers to the exponential shrinking of gradients’ magnitude as they are propagated back through time; and the exploding gradient problem refers to the explosion of long-term components due to the large increase in the norm of the gradient during training sequences with long-term dependencies. To solve these issues, researchers introduced Long short-term memory models. Long short-term memory (LSTM) is an RNN variant that was designed to store and access information in a long time sequence. Unlike standard RNNs, nonlinear multiplicative gates and a memory cell are introduced. These gates, including input, output, and forget gates, govern the information ﬂow into and out of the memory cell. The structure of an LSTM unit is illustrated in Figure 1.2. h(t–1) f (t) Memory cell o(t) i (t) Figure 1.2 × c (t) × Output x(t) Input × h(t) The structure of an LSTM unit. (Modiﬁed from Wu et al. [2016]) 8 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning More speciﬁcally, given a sequence of an external input vector x (t) ∈ Rn, an LSTM maps the input to an output vector z(t) ∈ Rm by computing activations of the units in the network with the following equations recursively from t = 1 to t = T : i(t) = σ (Wxi x (t) + Whi h(t−1) + Wci c(t) + bi ), (1.2) f (t) = σ (Wxf x (t) + Whf h(t) + Wcf c(t) + bf ), c(t) = f (t)c(t−1) + it tanh(Wxc x (t) + Whc h(t−1) + bc ), o(t) = σ (Wxo x (t) + Who h(t−1) + Wco c(t) + bo ), h(t) = o(t) tanh(c(t)), where x (t) , h(t) are the input and hidden vectors with the subscription t denoting the t-th time step, while i(t) , f (t) , c(t) , o(t) are, respectively, the activation vectors of the input gate, forget gate, memory cell, and output gate. Wαβ denotes the weight matrix between α and β. For example, the weight matrix from the input x (t) to the input gate i(t) is Wxi . In Equation 1.2 and Figure 1.2 and at time step t, the input x (t) and the previous states h(t−1) are used as the input of LSTM. The information of the memory cell is updated/controlled from two sources: (1) the previous cell memory unit c(t−1) and (2) the input gate’s activation it . Speciﬁcally, c(t−1) is multiplied by the activation from the forget gate f (t), which learns to forget the information of the previous states. In contrast, the it is combined with the new input signal to consider new information. LSTM also utilizes the output gate o(t) to control the information received by hidden state variable h(t). To sum up, with these explicitly designed memory units and gates, LSTM is able to exploit the long-range temporal memory and avoids the issues of vanishing/exploding gradients. LSTM has recently been popularly used for video analysis, as will be discussed in the following sections. 1.3 Video Classification The sheer volume of video data has motivated approaches to automatically categorizing video contents according to classes such as human activities and complex events. There is a large body of literature focusing on computing effective local feature descriptors (e.g., HoG, HoF, MBH, etc.) from spatio-temporal volumes to account for temporal clues in videos. These features are then quantized into bagof-words or Fisher Vector representations, which are further fed into classiﬁers like 1.3 Video Classiﬁcation 9 support vector machines (SVMs). In contrast to hand crafting features, which is usually time-consuming and requires domain knowledge, there is a recent trend to learn robust feature representations with deep learning from raw video data. In the following, we review two categories of deep learning algorithms for video classiﬁcation, i.e., supervised deep learning and unsupervised feature learning. 1.3.1 Supervised Deep Learning for Classification 1.3.1.1 Image-Based Video Classification The great success of CNN features on image analysis tasks [Girshick et al. 2014, Razavian et al. 2014] has stimulated the utilization of deep features for video classiﬁcation. The general idea is to treat a video clip as a collection of frames, and then for each frame, feature representation could be derived by running a feed-forward pass till a certain fully-connected layer with state-of-the-art deep models pre-trained on ImageNet [Deng et al. 2009], including AlexNet [Krizhevsky et al. 2012], VGGNet [Simonyan and Zisserman 2015], GoogLeNet [Szegedy et al. 2015a], and ResNet [He et al. 2016b], as discussed earlier. Finally, frame-level features are averaged into video-level representations as inputs of standard classiﬁers for recognition, such as the well-known SVMs. Among the works on image-based video classiﬁcation, Zha et al. [2015] systematically studied the performance of image-based video recognition using features from different layers of deep models together with multiple kernels for classiﬁcation. They demonstrated that off-the-shelf CNN features coupled with kernel SVMs can obtain decent recognition performance. Motivated by the advanced feature encoding strategies in images [Sánchez et al. 2013], Xu et al. [2015c] proposed to obtain video-level representation through vector of locally aggregated descriptors (VLAD) encoding [Jégou et al. 2010b], which can attain performance gain over the trivial averaging pooling approach. Most recently, Qiu et al. [2016] devised a novel Fisher Vector encoding with Variational AutoEncoder (FV-VAE) to quantize the local activations of the convolutional layer, which learns powerful visual representations of better generalization. 1.3.1.2 End-to-End CNN Architectures The effectiveness of CNNs on a variety of tasks lies in their capability to learn features from raw data as an end-to-end pipeline targeting a particular task [Szegedy et al. 2015a, Long et al. 2015, Girshick 2015]. Therefore, in contrast to the imagebased classiﬁcation methods, there are many works focusing on applying CNN models to the video domain with an aim to learn hidden spatio-temporal patterns. Ji et al. [2010] introduced the 3D CNN model that operates on stacked video frames, 10 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning extending the traditional 2D CNN designed for images to the spatio-temporal space. The 3D CNN utilizes 3D kernels for convolution to learn motion information between adjacent frames in volumes segmented by human detectors. Karpathy et al. [2014] compared several similar architectures on a large scale video dataset in order to explore how to better extend the original CNN architectures to learn spatiotemporal clues in videos. They found that the performance of the CNN model with a single frame as input achieves similar results to models operating on a stack of frames, and they also suggested that a mixed-resolution architecture consisting of a low-resolution context and a high-resolution stream could speed up training effectively. Recently, Tran et al. [2015] also utilized 3D convolutions with modern deep architectures. However, they adopted full frames as the inputs of 3D CNNs instead of the segmented volumes in Ji et al. [2010]. Though the extension of conventional CNN models by stacking frames makes sense, the performance of such models is worse than that of state-of-the-art handcrafted features [Wang and Schmid 2013]. This may be because the spatio-temporal patterns in videos are too complex to be captured by deep models with insufﬁcient training data. In addition, the training of CNNs with inputs of 3D volumes is usually time-consuming. To effectively handle 3D signals, Sun et al. [2015] introduced factorized spatio-temporal convolutional networks that factorize the original 3D convolution kernel learning as a sequential process of learning 2D spatial kernels in the lower layer. In addition, motivated by the fact that videos can naturally be decomposed into spatial and temporal components, Simonyan and Zisserman [2014] proposed a two-stream approach (see Figure 1.3), which breaks down the learning of video representation into separate feature learning of spatial and temporal clues. More speciﬁcally, the authors ﬁrst adopted a typical spatial CNN to model appearance information with raw RGB frames as inputs. To account for temporal clues among adjacent frames, they explicitly generated multiple-frame dense optical ﬂow, upon which a temporal CNN is trained. The dense optical ﬂow is derived from computing displacement vector ﬁelds between adjacent frames (see Figure 1.4), which represent motions in an explicit way, making the training of the network easier. Finally, at test time, each individual CNN generates a prediction by averaging scores from 25 uniformly sampled frames (optical ﬂow frames) for a video clip, and then the ﬁnal output is produced by the weighted sum of scores from the two streams. The authors reported promising results on two action recognition benchmarks. As the two-stream approach contains many implementation choices that may affect the performance, Ye et al. [2015b] evaluated different options, including dropout ratio and network architecture, and discussed their ﬁndings. Very recently, there have been several extensions of the two-stream approach. Wang et al. utilized the point trajectories from the improved dense trajectories 1.3 Video Classiﬁcation ·· ·· ·· ·· 11 ·· ··· Individual frame Spatial stream CNN Score fusion ·· ·· ·· ·· ·· ··· Stacked optical ﬂow Motion stream CNN Figure 1.3 Two-stream CNN framework. (From Wu et al. [2015c]) Figure 1.4 Examples of optical ﬂow images. (From Simonyan and Zisserman [2014]) [Wang and Schmid 2013] to pool two-stream convolutional feature maps to generate trajectory-pooled deep-convolutional descriptors (TDD) [Wang et al. 2015]. Feichtenhofer et al. [2016] improved the two-stream approach by exploring a better fusion approach to combine spatial and temporal streams. They found that two streams could be fused using convolutional layers rather than averaging classiﬁcation scores to better model the correlations of spatial and temporal streams. Wang et al. [2016b] introduced temporal segment networks, where each segment is used as the input of a two-stream network and the ﬁnal prediction of a video clip is produced by a consensus function combining segment scores. Zhang et al. [2016] proposed to replace the optical ﬂow images with motion vectors with an aim to achieve real-time action recognition. More recently, Wang et al. [2016c] proposed to learn feature representation by modeling an action as a transformation from an initial state (condition) to a new state (effect) with two Siamese CNN networks, operating on RGB frames and optical ﬂow images. Similar to the original two-stream approach, they then fused the classiﬁcation scores from two streams linearly to obtain ﬁnal predictions. They reported better results on two challenging benchmarks 12 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning than Simonyan and Zisserman [2014], possibly because the transformation from precondition to effect could implicitly model the temporal coherence in videos. Zhu et al. [2016] proposed a key volume mining approach that attempts to identify key volumes and perform classiﬁcation at the same time. Bilen et al. [2016] introduced the dynamic image to represent motions with rank pooling in videos, upon which a CNN model is trained for recognition. 1.3.1.3 Modeling Long-Term Temporal Dynamics As discussed earlier, the temporal CNN in the two-stream approach [Simonyan and Zisserman 2014] explicitly captures the motion information among adjacent frames, which, however, only depicts movements within a short time window. In addition, during the training of CNN models, each sweep takes a single frame (or a stacked optical frame image) as the input of the network, failing to take the order of frames into account. This is not sufﬁcient for video analysis, since complicated events/actions in videos usually consist of multiple actions happening over a long time. For instance, a “making pizza” event can be decomposed into several sequential actions, including “making the dough,” “topping,” and “baking.” Therefore, researchers have recently attempted to leverage RNN models to account for the temporal dynamics in videos, among which LSTM is a good ﬁt without suffering from the “vanishing gradient” effect, and has demonstrated its effectiveness in several tasks like image/video captioning [Donahue et al. 2017, Yao et al. 2015a] (to be discussed in detail later) and speech analysis [Graves et al. 2013]. Donahue et al. [2017] trained two two-layer LSTM networks (Figure 1.5) for action recognition with features from the two-stream approach. They also tried to ﬁne-tune the CNN models together with LSTM but did not obtain signiﬁcant performance gain compared with only training the LSTM model. Wu et al. [2015c] fused the outputs of LSTM models with CNN models to jointly model spatio-temporal clues for video classiﬁcation and observed that CNNs and LSTMs are highly complementary. Ng et al. [2015] further trained a 5-layer LSTM model and compared several pooling strategies. Interestingly, the deep LSTM model performs on par with single frame CNN on a large YouTube video dataset called Sports-1M, which may be because the videos in this dataset are uploaded by ordinary users without professional editing and contain cluttered backgrounds and severe camera motion. Veeriah et al. [2015] introduced a differential gating scheme for LSTM to emphasize the change in information gain to remove redundancy in videos. Recently, in a multi-granular spatio-temporal architecture [Li et al. 2016a], LSTMs have been utilized to further model the temporal information of frame, motion, and clip streams. Wu et al. [2016] further employed a CNN operating on spectrograms derived from 1.3 Video Classiﬁcation Figure 1.5 13 LSTM LSTM yt LSTM LSTM yt+1 LSTM LSTM yt+2 LSTM LSTM yt+n Utilizing LSTMs to explore temporal dynamics in videos with CNN features as inputs. soundtracks of videos to complement visual clues captured by CNN and LSTMs, and demonstrated strong results. 1.3.1.4 Incorporating Visual Attention Videos contain many frames. Using all of them is computationally expensive and may degrade the performance of recognizing a class of interest as not all the frames are relevant. This issue has motivated researchers to leverage the attention mechanism to identify the most discriminative spatio-temporal volumes that are directly related to the targeted semantic class. Sharma et al. [2015] proposed the ﬁrst attention LSTM for action recognition with a soft-attention mechanism to attach higher importance to the learned relevant parts in video frames. More recently, Li et al. [2016c] introduced the VideoLSTM, which applied attention in convolutional LSTM models to discover relevant spatio-temporal volumes. In addition to soft-attention, VideoLSTM also employed motion-based attention derived from optical ﬂow images for better action localization. 1.3.2 Unsupervised Video Feature Learning Current remarkable improvements with deep learning heavily rely on a large amount of labeled data. However, scaling up to thousands of video categories presents signiﬁcant challenges due to insurmountable annotation efforts even at video level, not to mention frame-level ﬁne-grained labels. Therefore, the utilization of unsupervised learning, integrating spatial and temporal context information, is a promising way to ﬁnd and represent structures in videos. Taylor et al. [2010] 14 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning proposed a convolutional gated boltzmann Machine to learn to represent optical ﬂow and describe motion. Le et al. [2011] utilized two-layer independent subspace analysis (ISA) models to learn spatio-temporal models for action recognition. More recently, Srivastava et al. [2015] adopted an encoder-decoder LSTM to learn feature representations in an unsupervised way. They ﬁrst mapped an input sequence into a ﬁxed-length representation by an encoder LSTM, which would be further decoded with single or multiple decoder LSTMs to perform different tasks, such as reconstructing the input sequence, or predicting the future sequence. The model was ﬁrst pre-trained on YouTube data without manual labels, and then ﬁne-tuned on standard benchmarks to recognize actions. Pan et al. [2016a] explored both local temporal coherence and holistic graph structure preservation to learn a deep intrinsic video representation in an end-to-end fashion. Ballas et al. [2016] leveraged convolutional maps from different layers of a pre-trained CNN as the input of a gated recurrent unit (GRU)-RNN to learn video representations. Summary The latest developments discussed above have demonstrated the effectiveness of deep learning for video classiﬁcation. However, current deep learning approaches for video classiﬁcation usually resort to popular deep models in image and speech domain. The complicated nature of video data, containing abundant spatial, temporal, and acoustic clues, makes off-the-shelf deep models insufﬁcient for videorelated tasks. This highlights the need for a tailored network to effectively capture spatial and acoustic information, and most importantly to model temporal dynamics. In addition, training CNN/LSTM models requires manual labels that are usually expensive and time-consuming to acquire, and hence one promising direction is to make full utilization of the substantial amounts of unlabeled video data and rich contextual clues to derive better video representations. 1.4 Video Captioning Video captioning is a new problem that has received increasing attention from both computer vision and natural language processing communities. Given an input video, the goal is to automatically generate a complete and natural sentence, which could have a great potential impact, for instance, on robotic vision or on helping visually impaired people. Nevertheless, this task is very challenging, as a description generation model should capture not only the objects, scenes, and activities presented in the video, but also be capable of expressing how these objects/scenes/activities relate to each other in a natural sentence. In this section, we elaborate the problem by surveying the state-of-the-art methods. We classify exist- 1.4 Video Captioning 15 Input video: … Video tagging: baby, boy, chair Figure 1.6 … Video (frame) captioning: Two baby boys are in the chair. Video captioning: A baby boy is biting ﬁnger of another baby boy. Examples of video tagging, image (frame) captioning, and video captioning. The input is a short video, while the output is a text response to this video, in the form of individual words (tags), a natural sentence describing one single image (frame), and dynamic video contents, respectively. ing methods in terms of different strategies for sentence modeling. In particular, we distill a common architecture of combining convolutional and recurrent neural networks for video captioning. As video captioning is an emerging area, we start by introducing the problem in detail. 1.4.1 Problem Introduction Although there has already been extensive research on video tagging [Siersdorfer et al. 2009, Yao et al. 2013] and image captioning [Vinyals et al. 2015, Donahue et al. 2017], video-level captioning has its own characteristics and thus is different from tagging and image/frame-level captioning. A video tag is usually the name of a speciﬁc object, action, or event, which is recognized in the video (e.g., “baby,” “boy,” and “chair” in Figure 1.6). Image (frame) captioning goes beyond tagging by describing an image (frame) with a natural sentence, where the spatial relationships between objects or object and action are further described (e.g., “Two baby boys are in the chair” generated on one single frame of Figure 1.6). Video captioning has been taken as an even more challenging problem, as a description should not only capture the above-mentioned semantic knowledge in the video but also express the spatio-temporal relationships in between and the dynamics in a natural sentence (e.g., “A baby boy is biting ﬁnger of another baby boy” for the video in Figure 1.6). Despite the difﬁculty of the problem, there have been several attempts to address video caption generation [Pan et al. 2016b, Yu et al. 2016, Xu et al. 2016], which are mainly inspired by recent advances in machine translation [Sutskever 16 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning et al. 2014]. The elegant recipes behind this are the promising developments of the CNNs and the RNNs. In general, 2D [Simonyan and Zisserman 2015] and/or 3D CNNs [Tran et al. 2015] are exploited to extract deep visual representations and LSTM [Hochreiter and Schmidhuber 1997] is utilized to generate the sentence word by word. More sophisticated frameworks, additionally integrating internal or external knowledge in the form of high-level semantic attributes or further exploring the relationship between the semantics of sentence and video content, have also been studied for this problem. In the following subsections we present a comprehensive review of video captioning methods through two main categories based on the strategies for sentence generation (Section 1.4.2) and generalizing a common architecture by leveraging sequence learning for video captioning (Section 1.4.3). 1.4.2 Approaches for Video Captioning There are mainly two directions for video captioning: a template-based language model [Kojima et al. 2002, Rohrbach et al. 2013, Rohrbach et al. 2014, Guadarrama et al. 2013, Xu et al. 2015b] and sequence learning models (e.g., RNNs) [Donahue et al. 2017, Pan et al. 2016b, Xu et al. 2016, Yu et al. 2016, Venugopalan et al. 2015a, Yao et al. 2015a, Venugopalan et al. 2015b, Venugopalan et al. 2015b]. The former predeﬁnes the special rule for language grammar and splits the sentence into several parts (e.g., subject, verb, object). With such sentence fragments, many works align each part with detected words from visual content by object recognition and then generate a sentence with language constraints. The latter leverages sequence learning models to directly learn a translatable mapping between video content and sentence. We will review the state-of-the-art research along these two dimensions. 1.4.2.1 Template-based Language Model Most of the approaches in this direction depend greatly on the sentence templates and always generate sentences with syntactical structure. Kojima et al. [2002] is one of the early works that built a concept hierarchy of actions for natural language description of human activities. Tan et al. [2011] proposed using predeﬁned concepts and sentence templates for video event recounting. Rohrbach et al.’s conditional random ﬁeld (CRF) learned to model the relationships between different components of the input video and generate descriptions for videos [Rohrbach et al. 2013]. Furthermore, by incorporating semantic unaries and hand-centric features, Rohrbach et al. [2014] utilized a CRF-based approach to generate coherent video descriptions. In 2013, Guadarrama et al. used semantic hierarchies to choose an appropriate level of the speciﬁcity and accuracy of sentence fragments. Recently, a 1.4 Video Captioning 17 deep joint video-language embedding model in Xu et al. [2015b] was designed for video sentence generation. 1.4.2.2 Sequence Learning Unlike the template-based language model, sequence learning-based methods can learn the probability distribution in the common space of visual content and textual sentence and generate novel sentences with more ﬂexible syntactical structure. Donahue et al. [2017] employed a CRF to predict activity, object, and location present in the video input. These representations were concatenated into an input sequence and then translated to a natural sentence with an LSTM model. Later, Venugopalan et al. [2015b] proposed an end-to-end neural network to generate video descriptions by reading only the sequence of video frames. By mean pooling, the features over all the frames can be represented by one single vector, which is the input of the following LSTM model for sentence generation. Venugopalan et al. [2015a] then extended the framework by inputting both frames and optical ﬂow images into an encoder-decoder LSTM. Inspired by the idea of learning visual-semantic embedding space in search [Pan et al. 2014, Yao et al. 2015b], [Pan et al. 2016b] additionally considered the relevance between sentence semantics and video content as a regularizer in LSTM based architecture. In contrast to mean pooling, Yao et al. [2015a] proposed to utilize the temporal attention mechanism to exploit temporal structure as well as a spatio-temporal convolutional neural network to obtain local action features. Then, the resulting video representations were fed into the text-generating RNN. In addition, similar to the knowledge transfer from image domain to video domain [Yao et al. 2012, 2015c], Liu and Shi [2016] leveraged the learned models on image captioning to generate a caption for each video frame and incorporate the obtained captions, regarded as the attributes of each frame, into a sequence-to-sequence architecture to generate video descriptions. Most recently, with the encouraging performance boost reported on the image captioning task by additionally utilizing high-level image attributes in Yao et al. [2016], Pan et al. [2016c] further leveraged semantic attributes learned from both images and videos with a transfer unit for enhancing video sentence generation. 1.4.3 A Common Architecture for Video Captioning To better summarize the frameworks of video captioning by sequence learning, we illustrate a common architecture as shown in Figure 1.7. Given a video, 2D and/or 3D CNNs are utilized to extract visual features on raw video frames, optical ﬂow images, and video clips. The video-level representations are produced by mean 18 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning … … a small red #end … Mean pooling/ soft-attention LSTM LSTM LSTM … … #start Figure 1.7 LSTM 3D CNN … Optical ﬂow images … 2D CNN … Raw video frames a small street A common architecture for video captioning by sequence learning. The video representations are produced by mean pooling or soft-attention over the visual features of raw video frames/optical ﬂow images/video clips, extracted by 2D/3D CNNs. The sentence is generated word by word in the following LSTM, based on the video representations. pooling or soft attention over these visual features. Then, an LSTM is trained for generating a sentence based on the video-level representations. Technically, suppose we have a video V with Nv sample frames/optical images/clips (uniform sampling) to be described by a textual sentence S, where S = {w1 , w2 , . . . , wNs } consisting of Ns words. Let v ∈ RDv and wt ∈ RDw denote the Dv dimensional visual features of a video V and the Dw -dimensional textual features of the t-th word in sentence S, respectively. As a sentence consists of a sequence of words, a sentence can be represented by a Dw × Ns matrix W ≡ [w1 , w2 , . . . , wNs ], with each word in the sentence as its column vector. Hence, given the video representations v, we aim to estimate the conditional probability of the output word sequence {w1 , w2 , . . . , wNs }, i.e., Pr (w1 , w2 , . . . , wNs \|v). (1.3) Since the model produces one word in the sentence at each time step, it is natural to apply the chain rule to model the joint probability over the sequential words. Thus, the log probability of the sentence is given by the sum of the log probabilities over the words and can be expressed as: log Pr (W\|v) = Ns t=1 log Pr wt v, w1 , . . . , wt−1 . (1.4) 1.5 Benchmarks and Challenges 19 In the model training, we feed the start sign word #start into LSTM, which indicates the start of the sentence generation process. We aim to maximize the log probability of the output video description S given the video representations, the previous words it has seen, and the model parameters θ, which can be formulated as ∗ θ = arg max θ Ns log Pr wt v, w1 , . . . , wt−1; θ . (1.5) t=1 This log probability is calculated and optimized over the whole training dataset using stochastic gradient descent. Note that the end sign word #end is required to terminate the description generation. During inference, we choose the word with maximum probability at each time step and set it as the LSTM input for the next time step until the end sign word is emitted. Summary The introduction of the video captioning problem is relatively new. Recently, this task has sparked signiﬁcant interest and may be regarded as the ultimate goal of video understanding. Video captioning is a complex problem and has been initially forwarded by the fundamental technological advances in recognition that can effectively recognize key objects or scenes from video contents. The developments of RNNs in machine translation have further accelerated the growth of this research direction. The recent results, although encouraging, are still indisputably far from practical use, as the forms of the generated sentences are simple and the vocabulary is still limited. How to generate free-form sentences and support open vocabulary are vital issues for the future of this task. 1.5 Benchmarks and Challenges We now discuss popular benchmarks and challenges for video classiﬁcation (Section 1.5.1) and video captioning (Section 1.5.2). 1.5.1 Classification Research on video classiﬁcation has been stimulated largely by the release of the large and challenging video datasets such as UCF101 [Soomro et al. 2012], HMDB51 [Kuehne et al. 2011], and FCVID [Jiang et al. 2015], and by the open challenges organized by fellow researchers, including the THUMOS challenge [Jiang et al. 2014b], the ActivityNet Large Scale Activity Recognition Challenge [Heilbron et al. 2015], and the TRECVID multimedia event detection (MED) task [Over et al. 2014]. 20 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning Table 1.1 Popular benchmark datasets for video classification, sorted by the year of construction Dataset KTH #Video #Class Released Year Background 600 6 2004 Clean Static 81 9 2005 Clean Static 1,358 25 2007 Dynamic 430 8 2008 Dynamic Hollywood2 1,787 12 2009 Dynamic MCG-WEBV 234,414 15 2009 Dynamic 800 16 2010 Dynamic HMDB51 6,766 51 2011 Dynamic CCV 9,317 20 2011 Dynamic UCF-101 13,320 101 2012 Dynamic THUMOS-2014 18,394 101 2014 Dynamic ≈31,000 20 2014 Dynamic Sports-1M 1,133,158 487 2014 Dynamic ActivityNet 27,901 203 2015 Dynamic EventNet 95,321 500 2015 Dynamic MPII Human Pose 20,943 410 2014 Dynamic FCVID 91,223 239 2015 Dynamic Weizmann Kodak Hollywood Olympic Sports MED-2014 (Dev. set) In the following, we ﬁrst discuss related datasets according to the list shown in Table 1.1, and then summarize the results of existing works. 1.5.1.1 Datasets KTH dataset is one of the earliest benchmarks for human action recognition [Schuldt et al. 2004]. It contains 600 short videos of 6 human actions performed by 25 people in four different scenarios. Weizmann dataset is another very early and simple dataset, consisting of 81 short videos associated with 9 actions performed by 9 actors [Blank et al. 2005]. Kodak Consumer Videos dataset was recorded by around 100 customers of the Eastman Kodak Company [Loui et al. 2007]. The dataset collected 1,358 1.5 Benchmarks and Challenges 21 video clips labeled with 25 concepts (including activities, scenes, and single objects) as a part of the Kodak concept ontology. Hollywood Human Action dataset contains 8 action classes collected from 32 Hollywood movies, totaling 430 video clips [Laptev et al. 2008]. It was further extended to the Hollywood2 [Marszalek et al. 2009] dataset, which is composed of 12 actions from 69 Hollywood movies with 1,707 video clips in total. This Hollywood series is challenging due to cluttered background and severe camera motion throughout the datasets. MCG-WEBV dataset is another large set of YouTube videos that has 234,414 web videos with annotations on several topic-level events like “a conﬂict at Gaza” [Cao et al. 2009]. Olympic Sports includes 800 video clips and 16 action classes [Niebles et al. 2010]. It was ﬁrst introduced in 2010 and, unlike in previous datasets, all the videos were downloaded from the Internet. HMDB51 dataset comprises 6,766 videos annotated into 51 classes [Kuehne et al. 2011]. The videos are from a variety of sources, including movies and YouTube consumer videos. Columbia Consumer Videos (CCV) dataset was constructed in 2011, aiming to stimulate research on Internet consumer video analysis [Jiang et al. 2011]. It contains 9,317 user-generated videos from YouTube, which were annotated into 20 classes, including objects (e.g., “cat” and “dog”), scenes (e.g., “beach” and “playground”), sports events (e.g., “basketball” and “soccer”), and social activities (e.g., “birthday” and “graduation”). UCF-101 & THUMOS-2014 dataset is another popular benchmark for human action recognition in videos, consisting of 13,320 video clips (27 hours in total) with 101 annotated classes such as “diving” and “weight lifting” [Soomro et al. 2012]. More recently, the THUMOS-2014 Action Recognition Challenge [Jiang et al. 2014b] created a benchmark by extending the UCF-101 dataset (used as the training set). Additional videos were collected from the Internet, including 2,500 background videos, 1,000 validation videos, and 1,574 test videos. TRECVID MED dataset was released and annually updated by the task of MED, created by NIST since 2010 [Over et al. 2014]. Each year an extended dataset based on datasets from challenges of previous years is constructed and released for worldwide system comparison. For example, in 2014 the MED 22 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning dataset contained 20 events, such as “birthday party,” “bike trick,” etc. According to NIST, in the development set, there are around 8,000 videos for training and 23,000 videos used as dry-run validation samples (1,200 hours in total). The MED dataset is only available to the participants of the task, and the labels of the ofﬁcial test set (200,000 videos) are not available even to the participants. Sports-1M dataset consists of 1 million YouTube videos in 487 classes, such as “bowling,” “cycling,” “rafting,” etc., and has been available since 2014 [Karpathy et al. 2014]. The video annotations were automatically derived by analyzing online textual contexts of the videos. Therefore the labels of this dataset are not clean, but the authors claim that the quality of annotation is fairly good. ActivityNet dataset is another large-scale video dataset for human activity recognition and understanding and was released in 2015 [Heilbron et al. 2015]. It consists of 27,801 video clips annotated into 203 activity classes, totaling 849 hours of video. Compared with existing datasets, ActivityNet contains more ﬁne-grained action categories (e.g., “drinking beer” and “drinking coffee”). EventNet dataset consists of 500 events and 4,490 event-speciﬁc concepts and was released in 2015 [Ye et al. 2015a]. It includes automatic detection models for its video events and some constituent concepts, with around 95,000 training videos from YouTube. Similarly to Sports-1M, EventNet was labeled by online textual information rather than manually labeled. MPII Human Pose dataset includes around 25,000 images containing over 40,000 people with annotated body joints [Andriluka et al. 2014]. According to an established taxonomy of human activities (410 in total), the collected images (from YouTube videos) were provided with activity labels. Fudan-Columbia Video Dataset (FCVID) dataset contains 91,223 web videos annotated manually into 239 categories [Jiang et al. 2015]. The categories cover a wide range of topics, such as social events (e.g., “tailgate party”), procedural events (e.g., “making cake”), object appearances (e.g., “panda”), and scenes (e.g., “beach”). 1.5.1.2 Challenges To advance the state of the art in video classiﬁcation, several challenges have been introduced with the aim of exploring and evaluating new approaches in realistic settings. We brieﬂy introduce three representative challenges here. 1.5 Benchmarks and Challenges 23 THUMOS Challenge was ﬁrst introduced in 2013 in the computer vision community, aiming to explore and evaluate new approaches for large-scale action recognition of Internet videos [Idrees et al. 2016]. The three editions of the challenge organized in 2013–2015 made THUMOS a common benchmark for action classiﬁcation and detection. TRECVID Multimedia Event Detection (MED) Task aims to detect whether a video clip contains an instance of a speciﬁc event [Awad et al. 2016, Over et al. 2015, Over et al. 2014]. Speciﬁcally, based on the released TRECVID MED dataset each year, each participant is required to provide for each testing video the conﬁdence score of how likely one particular event is to happen in the video. Twenty pre-speciﬁed events are used each year, and this task adopts the metrics of average precision (AP) and inferred AP for event detection. Each event was also complemented with an event kit, i.e., the textual description of the event as well as the potentially useful information about related concepts that are likely contained in the event. ActivityNet Large Scale Activity Recognition Challenge was ﬁrst organized as a workshop in 2016 [Heilbron et al. 2015]. This challenge is based on the ActivityNet dataset [Heilbron et al. 2015], with the aim of recognizing highlevel and goal-oriented activities. By using 203 activity categories, there are two tasks in this challenge: (1) Untrimmed Classiﬁcation Challenge, and (2) Detection Challenge, which is to predict the labels and temporal extents of the activities present in videos. 1.5.1.3 Results of Existing Methods Some of the datasets introduced above have been popularly adopted in the literature. We summarize the results of several recent approaches on UCF-101 and HMDB51 in Table 1.2, where we can see the fast pace of development in this area. Results on video classiﬁcation are mostly measured by the AP (for a single class) and mean AP (for multiple classes), which are not introduced in detail as they are well known. 1.5.2 Captioning A number of datasets have been proposed for video captioning; these commonly contain videos that have each been paired with its corresponding sentences annotated by humans. This section summarizes the existing datasets and the adopted evaluation metrics, followed by quantitative results of representative methods. 24 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning Table 1.2 Comparison of recent video classification methods on UCF-101 and HMDB51 datasets Methods Table 1.3 LRCN [Donahue et al. 2017] 82.9 — LSTM-composite [Srivastava et al. 2015] 84.3 — FST CN [Sun et al. 2015] 88.1 59.1 C3D [Tran et al. 2015] 86.7 — Two-Stream [Simonyan and Zisserman 2014] 88.0 59.4 LSTM [Ng et al. 2015] 88.6 — Image-Based [Zha et al. 2015] 89.6 — Transformation CNN [Wang et al. 2016c] 92.4 63.4 Multi-Stream [Wu et al. 2016] 92.6 — Key Volume Mining [Zhu et al. 2016] 92.7 67.2 Convolutional Two-Stream [Feichtenhofer et al. 2016] 93.5 69.2 Temporal Segment Networks [Wang et al. 2016b] 94.2 69.4 Comparison of video captioning benchmarks Dataset Context MSVD Multi-category AMT workers — TV16-VTT Multi-category Humans 2,000 YouCook Cooking AMT workers 88 — 2,668 TACoS-ML Cooking AMT workers 273 14,105 52,593 Movie DVS 92 48,986 55,905 94 68,337 68,375 653,467 7,180 10,000 200,000 1,856,523 M-VAD MPII-MD MSR-VTT-10K 1.5.2.1 UCF-101 HMDB51 Sentence Source #Videos Movie Script+DVS 20 categories AMT workers #Clips 1,970 — #Sentences 70,028 4,000 #Words 607,339 — 42,457 — 519,933 Datasets Table 1.3 summarizes key statistics and comparisons of popular datasets for video captioning. Figure 1.8 shows a few examples from some of the datasets. Microsoft Research Video Description Corpus (MSVD) contains 1,970 YouTube snippets collected on Amazon Mechanical Turk (AMT) by requesting workers to pick short clips depicting a single activity [Chen and Dolan 2011]. Annotators then label the video clips with single-sentence descrip- 1.5 Benchmarks and Challenges (a) MSVD dataset Sentences: • A dog walks around on its front legs. • The dog is doing a handstand. • A pug is trying for balance walk on two legs. Sentences: • A man lights a match book on ﬁre. • A man playing with ﬁre sticks. • A man lights matches and yells. (b) M-VAD dataset Sentence: • Later he drags someone through a jog. Sentence: • A waiter brings a pastry with a candle. (c) MPII-MD dataset Sentence: • He places his hands around her waist as she • opens her eyes. Sentence: • Someone’s car is stopped by a couple of • uniformed police. (d) MSR-VTT-10K dataset Sentences: • People practising volleyball in the play ground. • A man is hitting a ball and he falls. • A man is playing a football game on green land. Figure 1.8 Sentences: • A cat is hanging out in a bassinet with a baby. • The cat is in the baby bed with the baby. • A cat plays with a child in a crib. Examples from (a) MSVD, (b) M-VAD, (c) MPII-MD, and (d) MSR-VTT-10K datasets. 25 26 Chapter 1 Deep Learning for Video Classiﬁcation and Captioning tions. The original corpus has multi-lingual descriptions, but only the English descriptions are commonly exploited on video captioning tasks. Specifically, there are roughly 40 available English descriptions per video and the standard split of MSVD is 1,200 videos for training, 100 for validation, and 670 for testing, as suggested in Guadarrama et al. [2013]. YouCook dataset consists of 88 in-house cooking videos crawled from YouTube and is roughly uniformly split into six different cooking styles, such as baking and grilling [Das et al. 2013]. All the videos are in a third-person viewpoint and in different kitchen environments. Each video is annotated with multiple human descriptions by AMT. Each annotator in AMT is instructed to describe the video in at least three sentences totaling a minimum of 15 words, resulting in 2,668 sentences for all the videos. TACoS Multi-Level Corpus (TACoS-ML) is mainly built [Rohrbach et al. 2014] based on MPII Cooking Activities dataset 2.0 [Rohrbach et al. 2015c], which records different activities used when cooking. TACoS-ML consists of 185 long videos with text descriptions collected via AMT workers. Each AMT worker annotates a sequence of temporal intervals across the long video, pairing every interval with a single short sentence. There are 14,105 distinct intervals and 52,593 sentences in total. Montreal Video Annotation Dataset (M-VAD) is composed of about 49,000 DVD movie snippets, which are extracted from 92 DVD movies [Torabi et al. 2015]. Each movie clip is accompanied by one single sentence from semiautomatically transcribed descriptive video service (DVS) narrations. The fact that movies always contain a high diversity of visual and textual content, and that there is only one single reference sentence for each movie clip, has made the video captioning task on the M-VAD dataset very challenging. MPII Movie Description Corpus (MPII-MD) is another collection of movie descriptions dataset that is similar to M-VAD [Rohrbach et al. 2015b]. It contains around 68,000 movie snippets from 94 Hollywood movies and each snippet is labeled with one single sentence from movie scripts and DVS. MSR Video to Text (MSR-VTT-10K) is a recent large-scale benchmark for video captioning that contains 10K Web video clips totalling 41.2 hours, covering the most comprehensive 20 categories obtained from a commercial video 1.5 Benchmarks and Challenges 27 search engine, e.g., music, people, gaming, sports, and TV shows [Xu et al. 2016]. Each clip is annotated with about 20 natural sentences by AMT workers. The training/validation/test split is provided by the authors with 6,513 clips for training, 2,990 for validation, and 497 for testing. The TRECVID 2016 Video to Text Description (TV16-VTT) is another recent video captioning dataset that consists of 2,000 videos randomly selected from Twitter Vine videos [Awad et al. 2016]. Each video has a total duration of about 6 seconds and is annotated with 2 sentences by humans. The human annotators are asked to address four facets in the generated sentences: who the video is describing (kinds of persons, animals, things) and what the objects and beings are doing, plus where it is taking place and when. 1.5.2.2 Evaluation Metrics For quantitative evaluation of the video captioning task, three metrics are commonly adopted: BLEU@N [Papineni et al. 2002], METEOR [Banerjee and Lavie 2005], and CIDEr [Vedantam et al. 2015]. Speciﬁcally, BLEU@N is a popular machine translation metric which measures the fraction of N-gram (up to 4-gram) in common between a hypothesis and a reference or set of references. However, as pointed out in Chen et al. [2015], the N-gram matches for a high N (e.g., 4) rarely occur at a sentence level, resulting in poor performance of BLEU@N especially when comparing individual sentences. Hence, a more effective evaluation metric, METEOR, utilized along with BLEU@N , is also widely used in natural language processing (NLP) community. Unlike BLEU@N, METEOR computes unigram precision and recall, extending exact word matches to include similar words based on WordNet synonyms and stemmed tokens. Another important metric for image/video captioning is CIDEr, which measures consensus in image/video captioning by performing a Term Frequency Inverse Document Frequency (TF-IDF) weighting for each N-gram. 1.5.2.3 Results of Existing Methods Most popular methods of video captioning have been evaluated on MSVD [Chen and Dolan 2011], M-VAD [Torabi et al. 2015], MPII-MD [Rohrbach et al. 2015b], and TACoS-ML [Rohrbach et al. 2014] datasets. We summarize the results on these four datasets in Tables 1.4, 1.5, and 1.6. As can be seen, most of the works are very recent, indicating that video captioning is an emerging and fast-developing research topic. Table 1.4 Reported results on the MSVD dataset, where B@N , M, and C are short for BLEU@N , METEOR, and CIDEr-D scores, respectively Methods B@1 B@2 B@3 B@4 M C FGM [Thomason et al. 2014] — — — 13.7 23.9 — LSTM-YT [Venugopalan et al. 2015b] — — — 33.3 29.1 — MM-VDN [Xu et al. 2015a] — — — 37.6 29.0 — S2VT [Venugopalan et al. 2015a] — — — — 29.8 — S2FT [Liu and Shi 2016] — — — — 29.9 — 80.0 64.7 52.6 41.9 29.6 51.7 — — — 42.1 31.4 — 78.8 66.0 55.4 45.3 31.0 — SA [Yao et al. 2015a] Glove+Deep Fusion [Venugopalan et al. 2016] LSTM-E [Pan et al. 2016b] GRU-RCN [Ballas et al. 2016] h-RNN [Yu et al. 2016] — — — 43.3 31.6 68.0 81.5 70.4 60.4 49.9 32.6 65.8 All values are reported as percentages (%). Table 1.5 Reported results on (a) M-VAD and (b) MPII-MD datasets, where M is short for METEOR M-VAD dataset Methods M SA [Yao et al. 2015a] 4.3 Mean Pool [Venugopalan et al. 2015a] 6.1 Visual-Labels [Rohrbach et al. 2015a] 6.4 S2VT [Venugopalan et al. 2015a] 6.7 Glove+Deep Fusion [Venugopalan et al. 2016] 6.8 LSTM-E [Pan et al. 2016b] 6.7 MPII-MD dataset Methods M SMT [Rohrbach et al. 2015b] 5.6 Mean Pool [Venugopalan et al. 2015a] 6.7 Visual-Labels [Rohrbach et al. 2015a] 7.0 S2VT [Venugopalan et al. 2015a] 7.1 Glove+Deep Fusion [Venugopalan et al. 2016] 6.8 LSTM-E [Pan et al. 2016b] 7.3 All values are reported as percentages (%). 1.6 Conclusion Table 1.6 29 Reported results on the TACoS-ML dataset, where B@N , M, and C are short for BLEU@N , METEOR, and CIDEr-D scores, respectively Methods B@1 B@2 B@3 B@4 M C CRF-T [Rohrbach et al. 2013] 56.4 44.7 33.2 25.3 26.0 124.8 CRF-M [Rohrbach et al. 2014] 58.4 46.7 35.2 27.3 27.2 134.7 LRCN [Donahue et al. 2017] 59.3 48.2 37.0 29.2 28.2 153.4 h-RNN [Yu et al. 2016] 60.8 49.6 38.5 30.5 28.7 160.2 All values are reported as percentages (%). 1.6 Conclusion In this chapter, we have reviewed state-of-the-art deep learning techniques on two key topics related to video analysis, video classiﬁcation and video captioning, both of which rely on the modeling of the abundant spatial and temporal information in videos. In contrast to hand crafted features that are costly to design and have limited generalization capability, the essence of deep learning for video classiﬁcation is to derive robust and discriminative feature representations from raw data through exploiting massive videos with an aim to achieve effective and efﬁcient recognition, which could hence serve as a fundamental component in video captioning. Video captioning, on the other hand, focuses on bridging visual understanding and language description by joint modeling. We also provided a review of popular benchmarks and challenges for both video classiﬁcation and captioning tasks. Though extensive efforts have been made in video classiﬁcation and captioning with deep learning, we believe we are just beginning to unleash the power of deep learning in the big video data era. 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DOI: 10.1017 /S0956792515000492. 209 Index 1-bit compressive sensing for sketch similarity, 121 1 Million Song Corpus for audition, 37–38 3D CNN model, 9–10 3DMark Ice Storm Benchmark, 305–306 Abstraction in situations, 166, 168 Accelerometers in multimodal analysis of social interactions, 55–56 Acoustic backgrounds, 41 Acoustic events, 41 Acoustic intelligence, 32–33 Action Control in EventShop platform, 179 Actionable situations, 168 Actions of audition objects, 47 ActivityNet dataset for video classiﬁcation, 22 ActivityNet Large Scale Activity Recognition Challenge, 23 Actuators in eco-systems, 162 AdaPtive HFR vIdeo Streaming (APHIS) for cloud gaming, 309–310 Additive and multiplicative homomorphism, 88–90 Additive quantization in similarity searches, 131 ADMM (alternating-direction method of multipliers), 63–66 Advanced Micro Devices (AMD) for cloud gaming, 289 AES (Advanced Encryption Standard), 98 Affective ratings in emotion and personality type recognition, 240, 243–247 Aggregation in situation recognition, 175– 176 Agreeableness personality dimension, 237 Alerts in EventShop platform, 182 AlexNet, 5, 9 Algebraic homomorphic property in encrypted content, 101 All-in-one software pack for GamingAnywhere, 294–295 Alternating-direction method of multipliers (ADMM), 63–66 AMD (Advanced Micro Devices) for cloud gaming, 289 amtCNN (asymmetric multi-task CNN) model, 149 Animation rendering crowdsourced, 273–275 resource requirements, 256 ANN (approximate nearest neighbors) strategies, 105, 107–111 Annotations eye ﬁxations as, 226–236 user cues, 220 Anti-sparse coding for sketch similarity, 126–127 APHIS (AdaPtive HFR vIdeo Streaming) for cloud gaming, 309–310 Applications for audition, 49 Approximate nearest neighbors (ANN) strategies, 105, 107–111 Approximate search algorithms, 107–108 Architecture for video captioning, 17–18 Arousal dimensions in emotion and 380 Index personality type recognition, 236– 237, 243–247 Arrival times in Poisson processes, 195 Asthma/allergy risk recommendations, 163, 183–184 Asymmetric multi-task CNN (amtCNN) model, 149 Asymmetric sketch similarity schemes, 123 Attention mechanism for LSTM, 13 Audio emotion recognition, 221 multimodal analysis of social interactions, 55–56 multimodal pose estimation, 60 segment grouping, 43–44 Audio processing in encrypted domain audio editing quality enhancement, 99–100 speech/speaker recognition, 95–99 Audition for multimedia computing applications, 49 background, 35–39 Computer Audition ﬁeld, 32–35 conclusion, 49–50 data for, 39–40 generative models of sound, 45 generative structure of audio, 44–45 grouping audio segments, 43–44 nature of audio data, 40–41 NELS, 47–48 overview, 31–32 peculiarities of sound, 41–49 representation and parsing of mixtures, 42–43 structure discovery in audio, 46–47 weak and opportunistic supervision, 48–49 Augmented Lagrangian in ADMM, 64 Augmented-reality cloud gaming, 313 Average query time for similarity searches, 111 B-trees in similarity searches, 107 Bag-of-words representation object recognition, 232–233 similarity searches, 116 Basic linear algebra subprograms (BLAS) for similarity searches, 111 Behavior analysis in user-multimedia interaction, 146 Benchmarks in deep learning, 19–29 Best-bin-ﬁrst strategy in similarity searches, 107 BGN (Boneh-Goh-Nissim) cryptosystem, 99 BGV (Brakerski-Gentry-Vaikuntanathan) scheme, 101 Big-ﬁve marker scale (BFMS) questionnaire for emotion and personality type recognition, 238 Big-ﬁve model for emotion recognition, 237 Binary regularization term in multimodal pose estimation, 62 Biometric data, SPED for, 82–83 Biometric recognition for image processing, 88 BLAS (basic linear algebra subprograms) for similarity searches, 111 BLEU@N metric for video classiﬁcation, 27–29 Blind source separation (BSS) in audition, 36 BOINC platform, 258 Boneh-Goh-Nissim (BGN) cryptosystem, 99 Bottlenecks in Hawkes processes, 208–209 Bounding box annotations in scene recognition, 228–229 Brakerski-Gentry-Vaikuntanathan (BGV) scheme, 101 Branching structure in Hawkes processes, 200–202, 213 Broadcasts in situation-aware applications, 164 BSS (blind source separation) in audition, 36 Building blocks in situation recognition, 171 Business models in fog computing, 261 Index C-ADMM (coupled alternating direction method of multipliers), 64–66 Caller/callee pairs of streams in speech/ speaker recognition, 97 Captioning video, 14–19, 23–29 Capture servers for GamingAnywhere, 296 CASA (computational auditory scene analysis), 36–37 Cascade size in Hawkes processes, 213–215 CBIR (content-based information retrieval), 85 CCV (Columbia Consumer Videos) dataset, 21 CDC (cloud data center) architecture, 80 Cell-probe model and algorithms for similarity searches description, 108 hash functions, 113–118 introduction, 113–114 query mechanisms, 118–120 CF (collaborative ﬁltering) recommender systems, 151–152 SPED for, 82 CG (computationally grounded) factor deﬁnitions for situations, 167 Characterization in situation recognition, 175–176 Chi-Square in similarity searches, 133 CHIL Acoustic Event Detection campaign, 38 Children events in Hawkes processes, 212–213 Chroma features in audition, 37–38 CIDEr metric for video classiﬁcation, 27–29 Classiﬁcation audition, 38 situation recognition, 175–176 Client-server privacy-preserving image retrieval framework, 86 Cloud computing. See Fog computing Cloud data center (CDC) architecture, 80 Cloud gaming, 256 adaptive transmission, 308–310 cloud deployment, 298–302 381 communication, 307–310 conclusion, 314 future paradigm, 310–314 GamingAnywhere, 291–298 hardware decoders, 306 interaction delay, 302–304 introduction, 287–289 multiplayer games, 310–311 research, 289–291 thin client design, 302–306 Cloudlet servers, 259 Clusters of offspring in Hawkes processes, 201–202 CNNs (convolutional neural networks) end-to-end architectures, 9–12 similarity searches, 107 video, 5–6 Collaborative ﬁltering (CF) recommender systems, 151–152 SPED for, 82 Color SIFT (CSIFT) features in object recognition, 234–236 Columbia Consumer Videos (CCV) dataset, 21 Column-wise regularization in C-ADMM, 64 Common architecture for video captioning, 17–18 Common principles of interactions between users and multimedia data, 155 Communications cloud gaming, 307–310 fog computing, 257 Complementary hash functions, 117–118 Complex mathematical operations in encrypted multimedia analysis, 102–103 Complexity encrypted multimedia content, 101 similarity searches, 109–111 Compositional models of sound, 42–43 Compressed-domain distance estimation in similarity searches, 127–128 Compressed speaker recognition (CSR) systems, 97 382 Index Compression in cloud gaming, 307–308 Computational auditory scene analysis (CASA), 36–37 Computational bottlenecks in Hawkes processes, 208–209 Computational complexity in encrypted multimedia content, 101 Computational/conditional security in encrypted multimedia content, 103 Computational cost in similarity searches, 109 Computationally grounded (CG) factor deﬁnitions for situations, 167 Computations in fog computing, 257 Computer Audition. See Audition for multimedia computing Conditional random ﬁeld (CRF) in video captioning, 17–18 Conﬁdentiality of data, SPED for, 83–84 Connectivity for cloud gaming, 311 Conscientiousness personality dimension, 237, 246 Containers in fog computing, 261, 282–284 Content-based information retrieval (CBIR), 85 Content-centric computing, 138, 143 Content delay in CrowdMAC framework, 266–267 “Content Is Dead; Long Live Content!” panel, 142 Content quality in Hawkes processes, 210 Context-aware cloud gaming, 313–314 Controlled minions in fog computing, 257 Conversion operator in EventShop platform, 182 Convolutional neural networks (CNNs) end-to-end architectures, 9–12 similarity searches, 107 video, 5–6 Cooperative component sharing in cloud gaming, 312 Correctness in fog computing, 262 Cosine similarity and indexing similarity searches, 133 sketch similarity, 121–124 Cost CrowdMAC framework, 266–267 crowdsourced animation, 273 “Coulda, Woulda, Shoulda: 20 Years of Multimedia Opportunities” panel, 142 Counter-Strike game, 302 Coupled alternating direction method of multipliers (C-ADMM), 64–66 CRF (conditional random ﬁeld) in video captioning, 17–18 Crowdedness monitors in fog computing, 284 CrowdMAC framework, 263–267 Crowdsensing in SAIS, 268 Crowdsourced animation rendering services, 273–275 Cryptology. See Encrypted domain multimedia content analysis CSIFT features in object recognition, 234–236 CSR (compressed speaker recognition) systems, 97 CubeLendar system, 188 D-CASE (Detection and Classiﬁcation of Acoustic Scenes and Events) challenge, 38 DAGs (direct acyclic graphs) in fog computing, 276–278 Data Box project, 188 Data-centric computing, 137–138, 143 Data collection of social media, 144–145 Data compression in cloud gaming, 307– 308 Data ingestion EventShop platform, 179, 181 situation recognition, 173 Data overhead in encrypted domain multimedia content analysis, 101 Data representation and analysis multimedia analysis, 154 situation recognition, 174, 185–186 Index Data-source Panel in EventShop platform, 178 Data uniﬁcation in situation recognition, 173 Datasets in video classiﬁcation, 19–23 DBNs (deep belief networks) in convolutional neural networks, 5 DCT sign correlation for images, 88 dE-mages, 185 Decoders in cloud gaming, 306 Decomposition in Hawkes processes, 204–205 Deep belief networks (DBNs) in convolutional neural networks, 5 Deep learning benchmarks and challenges, 19–29 conclusion, 29 introduction, 3–4 modules, 4–8 video captioning, 14–19 video classiﬁcation, 8–14, 19–23 Delay bounded admission control algorithm, 266 Delays cloud gaming, 302–304 CrowdMAC framework, 266–267 Demographic information inference from user-generated content, 147 Descriptors in situation deﬁnitions, 167 Detection and Classiﬁcation of Acoustic Scenes and Events (D-CASE) challenge, 38 Device-aware scalable applications for cloud gaming, 291 Difference-of-Gaussian (DoG) transforms in SIFT, 86–87 Digital watermarking, SPED for, 81 Direct acyclic graphs (DAGs) in fog computing, 276–278 Direct memory access (DMA) channels for cloud gaming, 300 Discrete Fourier transformation for probabilistic cryptosystems, 80 383 Distance embeddings in similarity searches, 133 Distance metric learning, intentionoriented, 149–150 Distributed principal component analysis, 275–280 DMA (direct memory access) channels for cloud gaming, 300 Docker containers for fog computing, 283–284 DoG (Difference-of-Gaussian) transforms in SIFT, 86–87 Dropout in AlexNet, 5 E-mages EventShop platform, 178–181 situation recognition, 173 E2LSH techniques in similarity searches, 131 ECGs (electrocardiograms) in emotion and personality type recognition, 238, 241–242 Eco-systems for situation recognition, 162–164 Edge effects in Hawkes processes, 208 EEG (electroencephalogram) devices emotion and personality type recognition, 238, 241–242 user cues, 220–221 Efﬁciency encrypted domain multimedia content analysis, 103 situation recognition, 174 SPED for, 84 Efﬁcient Task Assignment (ETA) algorithm in SAIS, 269–272 EigenBehaviors in situation recognition, 187 Electrocardiograms (ECGs) in emotion and personality type recognition, 238, 241–242 Electroencephalogram (EEG) devices emotion and personality type recognition, 238, 241–242 384 Index Electroencephalogram (EEG) devices (continued) user cues, 220–221 Electronic voting, SPED for, 81 ElGamal cryptosystem, 79 Emerging applications in encrypted domain multimedia content analysis, 102– 103 Emotion and personality type recognition introduction, 236–238 materials and methods, 238–240 personality scores vs. affective ratings, 243–247 physiological feature extraction, 240–243 physiological signals, 247–250 user cues, 221 Encoder-decoder LSTM, 14 Encoding strategies in sketch similarity, 125–126 Encrypted domain multimedia content analysis audio processing, 95–100 conclusion, 104 future research and challenges, 101–104 image processing, 84–90 introduction, 75–78 SPED, 78–84 video processing in encrypted domain, 91–96 End-to-end CNN architectures in imagebased video classiﬁcation, 9–12 Energy usage in CrowdMAC framework, 266, 272 Enhancement layer in cloud gaming, 308 Environmental sound classiﬁcation in audition, 38 Epidemic type aftershock-sequences (ETAS) model, 209 Equivalent counting point processes, 194 ESP game, 220 ETA (Efﬁcient Task Assignment) algorithm in SAIS, 269–272 ETAS (epidemic type aftershock-sequences) model, 209 Ethical issues in situation recognition, 188 Euclidean case in similarity searches, 106–107, 114–115 Evaluation criteria for similarity searches, 109–113 Event-driven servers for GamingAnywhere, 296 EventNet dataset for video classiﬁcation, 22 Events, point processes for. See Point processes for events EventShop platform asthma/allergy risk recommendation, 183–184 heterogeneous data, 180–182 operators, 181–182 overview, 177–178 situation-aware applications, 182–185 system design, 178–180 Exact search algorithms, 107–108 Exhaustive search algorithms, 107–108 Exogenous events in Hawkes processes, 198 Expectation in similarity searches, 128 Expected number of future events in Hawkes processes, 212–213 Expected value in Poisson processes, 194 Exploding gradients in recurrent neural networks, 7 Expressive power concept in situation recognition, 170 Extraversion personality dimension, 237, 244 Eye ﬁxations and movements in object recognition discussion, 232 emotion recognition, 237 ﬁxation-based annotations, 232–236 free-viewing and visual search, 227–232 introduction, 226–227 materials and methods, 227 scene semantics inferences from, 222– 226 user cues, 220 Eysenck’s personality model, 237 F-formation detection in head and body pose estimation, 69–73 Index Face detectors in fog computing, 283–284 Face swapping in video surveillance systems, 92 Facial expressions, differentiating via eye movements in, 224–226 Facial landmark trajectories in emotion and personality type recognition, 242–243 Fairness in cloud gaming, 311 FCGs. See Free-standing conversational groups (FCGs) FCVID (Fudan-Columbia Video Dataset) dataset for video classiﬁcation, 22 Feature extraction in image processing, 86–88 Feed-forward neural networks (FFNNs), 6 Filtering probabilistic cryptosystems, 80 recommender systems, 151–152 situation recognition, 175 SPED for, 82 Fisher Vector encoding with Variational AutoEncoder (FV-VAE), 9 Five-factor model for emotion recognition, 237 Fixed-base annotations for object recognition, 232–236 Flickr videos in audition, 39 user favorite behavior patterns, 148 Fog computing challenges, 260–262 conclusion, 285–286 CrowdMAC framework, 263–267 crowdsourced animation rendering service, 273–275 introduction, 255–258 open-source platforms, 280–285 related work, 258–260 scalable and distributed principal component analysis, 275–280 Smartphone-Augmented Infrastructure Sensing, 268–272 Fraud in fog computing, 262 385 Frechet distances in similarity searches, 133 Free-standing conversational groups (FCGs), 51 conclusion, 73–74 F-formation detection, 69–73 head and body pose estimation, 56–57, 66–69 introduction, 52–55 matrix completion for multimodal pose estimation, 59–66 matrix completion overview, 57–58 multimodal analysis of social interactions, 55–56 SALSA dataset, 58–59 Free-viewing (FV) tasks eye movements, 227–232 scene recognition, 226 Freelance minions in fog computing, 257 Fudan-Columbia Video Dataset (FCVID) dataset for video classiﬁcation, 22 Fully homomorphic encryption (FHE) techniques, 88–90 Function hiding, SPED for, 81 Future actions (FA) in situation deﬁnitions, 166 Future events in Hawkes processes, 212–213 FV-VAE (Fisher Vector encoding with Variational AutoEncoder), 9 G-cluster cloud gaming company, 290 Galvanic skin response (GSR) in emotion and personality type recognition, 238, 241–242 Games as a Service (GaaS), 291 GamingAnywhere community participation, 297–298 environment setup, 294–295 execution, 296–297 introduction, 291–292 research, 297 system architecture, 293–294 target users, 292–293 Gaussian distribution in situation recognition, 185 386 Index Gaussian mixture model (GMM) CSR systems, 97–99 speech recognition, 36 Generative models Hawkes processes, 213–215 sound, 45 Generative structure of audio, 44–45 Geolocation in audition, 39 GIST descriptors in similarity searches, 132 GMM (Gaussian mixture model) CSR systems, 97–99 speech recognition, 36 Goal based (GB) factor in situation deﬁnitions, 166 Google search engine, 151 GoogLeNet, 6, 9 GPUs (graphical processing units) for cloud gaming, 298–302 Gradients in recurrent neural networks, 7 Graph-based approaches for similarity searches, 132–134 Graph-cut approach in F-formation detection, 69–70 Graphical processing units (GPUs) for cloud gaming, 298–302 Graphics compression in cloud gaming, 307–308 Graphs in social graph modeling, 145 Grouping audio segments, 43–44 GSR (galvanic skin response) in emotion and personality type recognition, 238, 241–242 Hamming space and embedding similarity searches, 114–115, 132 sketch similarity, 121–122 Hardware decoders in cloud gaming, 306 Hash functions similarity searches, 113–118 sketch similarity, 123–124 Hawkes processes, 197 branching structure, 200–202 conclusion, 217–218 estimating, 211–212 expected number of future events, 212–213 generative model, 213–215 hands-on tutorial, 215–217 Hawkes model for social media, 209–217 information diffusion, 210–211 intensity function, 198–200 introduction, 192–193 likelihood function, 205–206 maximum likelihood estimation, 207– 209, 211–212 parameter estimates, 205–209 sampling by decomposition, 204– 205 self-exciting processes, 197–198 simulating events, 202–205 thinning algorithm, 202–204 Head and body pose estimates (HBPE) experiments, 66–69 F-formation detection, 69–73 free-standing conversational groups, 53–54 overview, 56–57 SALSA dataset, 59 Heart rate in emotion and personality type recognition, 238, 241–243 Heterogeneous data EventShop platform, 180–182 user-multimedia interaction, 146 Hidden Markov models (HMM) in speech/ speaker recognition, 95–98 Hierarchical structure in audition, 46–47 High-intensity facial expressions in eye movements, 224–226 High-rate quantization theory in similarity searches, 117 Histogram of oriented gradient (HOG) descriptors, 59 HMDB51 dataset for video classiﬁcation, 21, 24 HMM (Hidden Markov models) in speech/ speaker recognition, 95–98 HOG (Histogram of oriented gradient) descriptors, 59 Index Hollywood Human Action dataset for video classiﬁcation, 21 Homomorphic encryption image search and retrieval, 86 SPED, 79 speech/speaker recognition, 99 video processing in encrypted domain, 94 Homophily hypothesis, 154 Honest-but-curious model, 80–81 Hotspots in mobile Internet, 264–268 Hough voting method (HVFF-lin), 69–70 Hough voting method multi-scale extension (HVFF-ms), 69–70 Householder decomposition in sketch similarity, 124 Human actuators in eco-systems, 162 Human behavior in situation recognition, 187 Human-centric computing, 138 Human pose in free-standing conversational groups, 53 Human sensors in eco-systems, 162 HVFF-lin (Hough voting method), 69–70 HVFF-ms (Hough voting method multi-scale extension), 69–70 Hybrid approaches to similarity searches, 131–132 Hybrid compression in cloud gaming, 307–308 Hyper-diamond E8 lattices in similarity searches, 115 Hypervisors for cloud gaming, 299 IARPA Aladdin Please program, 38 Image-based video classiﬁcation, 9 Image collectors in fog computing, 283–284 Image processing in encrypted domain biometric recognition, 88 feature extraction, 86–88 image search and retrieval, 84–86 quality enhancement, 88–90 Image representation learning, intentionoriented, 148–149 387 ImageNet, 9 Imbalance factor (IF) in similarity searches, 116–117 Immigrant events in Hawkes processes, 198, 200, 204, 210 Impact sounds, 45 In-situ sensing in SAIS, 270–271 Independent sample-wise processing in SPED, 79–80 Indexing schemes image search and retrieval, 85 similarity searches, 107–109, 112, 133 Inferring scene semantics from eye movements, 222–226 Inﬂuence-based recommendation, 152 Information diffusion in Hawkes processes, 210–211 Information theoretic/unconditional security in encrypted multimedia content, 103 Infrared detection in multimodal pose estimation, 60 Infrastructure sensing in fog computing, 268–272 Input/output memory management units (IOMMUs) for cloud gaming, 300 Instance-level inferences in audition, 33 Integrity of data encrypted domain multimedia content analysis, 103–104 SPED for, 83 Intensity Hawkes processes, 198–200, 210 Poisson processes, 195 Intention-oriented learning distance metric, 149–150 image representation, 148–149 Inter-arrival times in Poisson processes, 194–196 Interaction delay in cloud gaming, 302–304 Interactions of objects in audition, 47 Interactive scenes, saccades in, 223 Intermediate Query Panel in EventShop platform, 178 388 Index Internal Storage in EventShop platform, 179 Interoperable common principles in social media, 155 Interpolation operator in EventShop platform, 182 Intuitive query and mental model in situation recognition, 174 Inverse transform sampling technique for Hawkes processes, 202–203 IOMMUs (input/output memory management units) for cloud gaming, 300 Job assignment algorithms in SAIS, 269 Johnson-Lindenstrauss Lemma, 107 Joint estimation in multimodal pose estimation, 62 JPEG 2000 images in encrypted domains, 88 K-means in similarity searches, 117, 127– 129 k-NN graphs for similarity searches, 109 KD-trees in similarity searches, 107, 116– 117 Kernel function in Hawkes processes, 198–200, 204–205 Kernel PAC (KPCA) in similarity searches, 133 Kernelized LSH (KLSH) in similarity searches, 133 Keyword recognizers (KWR) in speech/ speaker recognition, 95–97 Keyword searches for images, 85 Kgraph method for similarity searches, 133 KLSH (kernelized LSH) in similarity searches, 133 Kodak Consumer Videos dataset for video classiﬁcation, 20–21 KPCA (kernel PAC) in similarity searches, 133 Kronecker product in multimodal pose estimation, 62 KTH dataset for video classiﬁcation, 20 Kubernetes platform cloud applications, 256 fog computing, 281–283 Kusanagi project, 290 KVM technology in fog computing, 282 KWR (keyword recognizers) in speech/ speaker recognition, 95–97 LabelMe database, 220 Lagrangian in ADMM, 64 Laplacian matrix in multimodal pose estimation, 62 Large margin nearest neighbor (LMNN), 149–150 Last-mile technology, 142 Latent variable analysis (LVA) approach in audition, 37 Lattices in similarity searches, 115 Learned quantizers in similarity searches, 115–116 Leech lattices in similarity searches, 115 Legal issues in cloud gaming, 291 LeNet-5 framework, 5 Likelihood function in Hawkes processes, 205–206 Limb tracking with visual-inertial sensors, 57 Linear ﬁltering in probabilistic cryptosystems, 80 Local maxima in Hawkes processes, 207– 208 Local minima in Hawkes processes, 212 Locality-Sensitive Hashing (LSH) hash functions, 114–115 query-adaptive, 119–120 similarity searches, 105–106 sketch similarity, 122, 126–127 Long short-term memory (LSTM) modeling, 12–13 recurrent neural networks, 7–8 video classiﬁcation, 17–19 Long-term temporal dynamics modeling, 12–13 Look-up operations in similarity searches, 110 Low-intensity facial expressions in eye movements, 224–226 Index Lower the ﬂoor concept in situation recognition, 171 LSH. See Locality-Sensitive Hashing (LSH) LSTM (long short-term memory) modeling, 12–13 recurrent neural networks, 7–8 video classiﬁcation, 17–19 LXC containers for fog computing, 282 M-VAD (Montreal Video Annotation Dataset) for video classiﬁcation, 26 Macroscopic behavior analysis, 146 Magnetoencephalogram (MEG) signals in emotion recognition, 221 Magnitude of inﬂuence in Hawkes processes, 210 Mahout-PCA library for fog computing, 279–280 Malicious model in SPED, 81 MapReduce platform, 279–280 Marked Hawkes processes, 210–211 Marked memory kernel in Hawkes processes, 211 Markov process in Hawkes processes, 204–205 Massively multiplayer online role-playing games (MMORPGs), 310 Matching biometric data, SPED for, 82–83 Matrix completion (MC) free-standing conversational groups, 54 head and body pose estimation, 57–58 multimodal pose estimation, 59–66 Matrix completion for head and body pose estimation (MC-HBPE), 60–61, 66–69 Maximum likelihood estimation in Hawkes processes, 207–209, 211–212 McCulloch-Pitts model for convolutional neural networks, 5 MCG-WEBV dataset for video classiﬁcation, 21 MDA (Module Deployment Algorithm), 282, 284 Mean average precision in similarity searches, 112 389 MediaEval dataset for audition, 39 Medical data, SPED for, 83 MEG (magnetoencephalogram) signals in emotion recognition, 221 Mel-Frequency Cepstral Coefﬁcients (MFCCs) in audition, 37 Memory over time in Hawkes processes, 210 Memory reads in similarity searches, 110–111 Memorylessness property in Poisson processes, 195–196 Mesoscopic behavior analysis, 146 Meta inferences in audition, 33 METEOR metric for video classiﬁcation, 27–29 MFCCs (Mel-Frequency Cepstral Coefﬁcients) in audition, 37 Microscopic behavior analysis, 146 Microsoft Research Video Description Corpus (MSVD) dataset for video classiﬁcation, 24–28 Minions in fog computing, 257–258 MIREX evaluations in audition, 37 Missing and uncertain data in situation recognition, 185–186 Mixtures in audition, 42–43 MMORPGs (massively multiplayer online role-playing games), 310 Mobile Internet, CrowdMAC framework for, 263–267 Mobile ofﬂoading in fog computing, 258– 259 Modeling approach in situation recognition, 171–172 Module Deployment Algorithm (MDA), 282, 284 Montreal Video Annotation Dataset (M-VAD) for video classiﬁcation, 26 MPEG-X standard, 156–157 MPII Human Pose dataset for video classiﬁcation, 22 MPII Movie Description Corpus (MPII-MD) for video classiﬁcation, 26 MSR Video to Text (MSR-VTT-10K) dataset for video classiﬁcation, 26–27 390 Index MSVD (Microsoft Research Video Description Corpus) dataset for video classiﬁcation, 24–28 Multi-probe LSH similarity searches, 118–119 sketch similarity, 122 Multi-task learning approach in head and body pose estimation, 57 Multimedia Commons initiative for audition, 39 Multimedia data social attributes for, 154–155 user interest modeling from, 147–148 Multimedia fog platforms, 257–258 Multimedia-sensed social computing, 156 Multimodal analysis encrypted domain multimedia content analysis, 102 free-standing conversational groups. See Free-standing conversational groups (FCGs) sentiment, 150 social interactions, 55–56 Multimodal pose estimation matrix completion, 59–66 model, 60–63 optimization method, 63–66 Multiplayer games, 310–311 Music signals in audition, 37–38 Natural audio, 33–34 Near neighbors in similarity searches, 114 Need gap vs. semantic gap, 139–141 Neighbors in similarity searches, 106–111, 114 NELL (Never Ending Language Learner) system, 48 NELS (Never-Ending Sound Learner) proposal, 47–48 Neuroticism personality dimension, 237, 244–245 NN-descent algorithm for similarity searches, 133 Non-Euclidean metrics for similarity searches, 132–134 Non-homogeneous Poisson process, 195– 197 Non-interactive scenes, saccades in, 223 Normalized deviation in crowdsourced animation rendering service, 274– 275 NP-hard problem in multimodal pose estimation, 61 Object interactions with saccades, 222–224 Object recognition with eye ﬁxations as implicit annotations, 226–236 Observed situations, 168 Observers in GamingAnywhere, 293 Offspring events in Hawkes processes, 200–201 Olympic Sports dataset for video classiﬁcation, 21 OnLive gaming, 290 Open-source platforms cloud gaming systems, 292 fog computing, 280–285 OpenCV package in EventShop platform, 179 Openness personality dimension, 237, 246 OpenPDS project, 188 OpenStack platform cloud applications, 256 fog computing, 281 Operators Panel in EventShop platform, 178 Opportunistic supervision in audition, 48–49 Optimization techniques cloud gaming, 288 hash functions, 117–118 multimodal pose estimation, 63–66 OTT service for cloud gaming, 310 P2P (peer-to-peer) paradigm CrowdMAC framework, 265 fog computing, 259 Paillier cryptosystem homomorphic encryption, 79 SIFT, 86–87 speech/speaker recognition, 99 Index ParaDrop for fog computing, 259 Parsing of mixtures in audition, 42–43 Pascal animal classes Eye Tracking (PET) database, 226–228 PASCAL Visual Object Classes for user cues, 220 Pass-through GPUs for cloud gaming, 300 Pattern matching in situation recognition, 175–176 PCA (principal component analysis), 275– 280 peer-to-peer (P2P) paradigm CrowdMAC framework, 265 fog computing, 259 Percepts in audition, 43 Personality scores in emotion and personality type recognition vs. affective ratings, 243–247 description, 240 Personality trait recognition, 247–248 Personalization in EventShop platform, 182 Personalized alerts in situation-aware applications, 164 Persuading user action, 187–188 PET (Pascal animal classes Eye Tracking) database, 226–228 Physiological signals in emotion and personality type recognition experiments, 249–250 feature extraction, 240–243 materials and methods, 238–240 overview, 236–238 personality scores vs. affective ratings, 243–247 responses, 238–239 trait recognition, 247–249 Play-as-you-go services in cloud gaming, 298 Point processes for events conclusion, 217–218 deﬁning, 193–194 Hawkes processes. See Hawkes processes introduction, 191–193 Poisson processes, 193–197 Poisson processes deﬁnition, 194–195 391 memorylessness property, 195– 196 non-homogeneous, 196–197 points, 193–194 POSIX platforms for GamingAnywhere, 295–296 Power consumption in cloud gaming, 305–306 Power-law kernel function for Hawkes processes, 215–216 Pre-binarized version vectors in sketch similarity, 126 Principal component analysis (PCA), 275– 280 Privacy encryption for. See Encrypted domain multimedia content analysis fog computing, 262 situation recognition, 174, 188 Probabilistic cryptosystems, 80 Probability density function in Poisson processes, 195 Product quantization in similarity searches, 128–131 Project+take sign approach for sketch similarity, 124–127 Protocols in emotion and personality type recognition, 240 Proximity sensors in multimodal analysis of social interactions, 55–56 Public video surveillance systems, 91 Quality enhancement in image processing in encrypted domain, 88–90 Quality of Experience (QoE) metrics in cloud gaming, 288 Quality of Service (QoS) cloud gaming, 288 fog computing, 262 Quantization in similarity searches, 106, 127–131 Quantization-optimized LSH in sketch similarity, 126–127 Quantizers in similarity searches, 115–116, 128–129 392 Index Queries in similarity searches mechanisms, 118–120 preparation cost, 110 query-adaptive LSH, 119–120 times, 109, 111 Query plan trees in EventShop platform, 180 Query Processing Engine in EventShop platform, 179–180 R-trees in similarity searches, 107 Raise the ceiling concept in situation recognition, 171 Rank-ordered image searches, 85 Rapid prototyping toolkit in situation recognition, 172 Raspberry Pis for fog computing, 283–284 Raw vectors in similarity searches, 112 Real time strategy (RTS) games, 303 Real time streaming protocol (RTSP), 293–294 Real time transport protocol (RTP), 293–294 Recognition problem in situation deﬁnitions, 168 Recommendation in social-sensed multimedia, 151–152 Rectiﬁed Linear Units (ReLUs) in AlexNet, 5 Recurrence Quantiﬁcation Analysis (RQA) in scene recognition, 229–230 Recurrent neural networks (RNNs), 6–8 Redundant computation in fog computing, 278 Region-of-interest rectangles, 225 Registered Queries in EventShop platform, 178 Reliable data collection of social media, 144–145 ReLUs (Rectiﬁed Linear Units) in AlexNet, 5 RenderStorm cloud rendering platform, 273 Representation and parsing of mixtures in audition, 42–43 Request delays in CrowdMAC framework, 266–267 Residual vectors in similarity searches, 129 ResNet, 6 Resource management in fog computing, 261–262 Results Panel in EventShop platform, 178 Revenue in CrowdMAC framework, 266–267 RNNs (recurrent neural networks), 6–8 Round trip time (RTT) jitter in cloud gaming, 308 Rowe, Larry, 142 Rowley eye detectors, 225 RQA (Recurrence Quantiﬁcation Analysis) in scene recognition, 229–230 RSA cryptosystem, 79 RSA-OPRF protocol, 95 RTP (real time transport protocol), 293–294, 303 RTSP (real time streaming protocol), 293– 294 SaaS (Software as a Service) for cloud gaming, 290 Saccades object interactions, 222–224 scene recognition, 229 SAIS (Smartphone-Augmented Infrastructure Sensing), 268–272 SALSA (Synergistic sociAL Scene Analysis) dataset free-standing conversational groups, 58–59 head and body pose estimation, 66–69 SaltStack platform cloud applications, 256 fog computing, 281–282 Sampling by decomposition in Hawkes processes, 204–205 Scalable and distributed principal component analysis, 275–280 Scalable solutions, SPED for, 84 Scalable video coded (SVC) videos, 94–95 Scale invariant feature transform (SIFT) fog computing, 275–276 image processing in encrypted domain, 86–87 object recognition with ﬁxation-based annotations, 234–236 Index similarity searches, 107, 132 Scene understanding from eye movements, 222–226 user cues, 220 Searches images, 84–86 similarity. See Similarity searches SPED, 82 speech/speaker recognition, 97 visual, 226–232 Secure domains for video encoding, 92–93 Secure processing of medical data, SPED for, 83 Secure real time transport protocol (SRTP), 98 Security vs. accuracy, 102–103 encryption. See Encrypted domain multimedia content analysis fog computing, 262 Selectivity in similarity searches, 110, 112 Self-exciting processes Hawkes processes, 197–198 point processes, 192 Semantic gap bridging, 145 vs. need gap, 139–141 Semantic inferences in audition, 33 Semi-controlled minions in fog computing, 257 Semi-honest model for SPED, 80–81 Semi-supervised hierarchical structure in audition, 46 Sensors eco-systems, 162 fog computing, 257 Sequence learning for video captioning, 17 SETI@Home application, 258 SEVIR (Socially Embedded VIsual Representation Learning) approach, 149 Shaderlight cloud rendering platform, 273 Shallow models and approaches audition, 46 393 intention-oriented image representation learning, 148–149 Shannon Information Theory, 138 Shape gain in similarity searches, 115, 128 SIDL (Social-embedding Image Distance Learning) approach, 149–150 SIFT. See Scale invariant feature transform (SIFT) Signal processing in encrypted domain (SPED), 80 applications, 81–83 background, 78–81 beneﬁts, 83–84 introduction, 76–78 Similarity estimation for sketches, 121–123 Similarity searches background, 106–107 cell-probe algorithms, 113–120 conclusion, 134 evaluation criteria, 109–113 hash functions, 113–118 hybrid approaches, 131–132 introduction, 105–106 non-Euclidean metrics and graph-based approaches, 132–134 quantization, 127–131 query mechanisms, 118–120 sketches and binary embeddings, 120– 127 types, 107–108 Simulating events in Hawkes processes, 202–205 Situation awareness in SAIS, 268 Situation deﬁnitions data, 168–169 existing, 165–167 features, 169 overview, 164 proposed, 167–168 recognition problem, 168 Situation recognition using multimodal data asthma risk-based recommendations, 163 challenges and opportunities, 185–188 394 Index Situation recognition using multimodal data (continued) conclusion, 188–189 eco-system, 162–164 EventShop platform. See EventShop platform framework, 170–177 individual behavior, 186–187 introduction, 159–161 missing and uncertain data, 185–186 persuading user action, 187–188 privacy and ethical issues, 188 situation-aware applications, 163–164 situation deﬁned, 164–170 situation estimation, 186 situation evaluation, 174–176 situation responses, 177 workﬂow, 172–176 Sketches and binary embeddings hash function design, 123–124 introduction, 120–121 project+take sign approach, 124–127 similarity estimation, 121–123 similarity search, 120–127 SLA in fog computing, 262 Smartphone-Augmented Infrastructure Sensing (SAIS), 268–272 Social attributes for users and multimedia, 154–155 Social-embedding Image Distance Learning (SIDL) approach, 149–150 Social graph modeling, 145 Social interactions in multimodal analysis, 55–56 Social knowledge on user-multimedia interactions, 143 Social representation of multimedia data, 145 Social-sensed multimedia computing conclusion, 157 demographic information inference from user-generated content, 147 exemplary applications, 150–153 future directions, 153–157 intention-oriented distance metric learning, 149–150 intention-oriented image representation learning, 148–149 introduction, 137–139 MPEG-X, 156–157 multimedia sentiment analysis, 150 overview, 142–144 recent advances, 146–150 reliable data collection, 144–145 semantic gap vs. need gap, 139–141 social attributes for users and multimedia, 154–155 social representation of multimedia data, 145 social-sensed multimedia recommendation, 151–152 social-sensed multimedia search, 151 social-sensed multimedia summarization, 152–153 social-sensed video communication, 153 user interest modeling from multimedia data, 147–148 user-multimedia interaction behavior analysis, 146 user proﬁling and social graph modeling, 145 Socially Embedded VIsual Representation Learning (SEVIR) approach, 149 Sociometric badges in SALSA dataset, 58–59 Software as a Service (SaaS) for cloud gaming, 290 Sound. See Audition for multimedia computing Source separation in audition, 36 Space and time (ST) in situation deﬁnitions, 166 Space complexity in similarity searches, 111–112 Space-Time ARIMA (STARIMA) models, 186 Sparse coding in object recognition, 234 Spatial inferences in audition, 33 Spatial pyramid histogram representation in object recognition, 232–234 Index Spatio-temporal aggregation in situation recognition, 173–174 Spatio-temporal convolutional networks, 10 Spatio-temporal situations, 168 sPCA algorithm, 277–279 SPED. See Signal processing in encrypted domain (SPED) Speech/speaker recognition, 36, 95–99 Sports-1M dataset LSTM, 12 video classiﬁcation, 22 ST (space and time) in situation deﬁnitions, 166 STARIMA (Space-Time ARIMA) models, 186 Stimuli in emotion and personality type recognition, 240 Stop-words in similarity searches, 116 Storage in fog computing, 257 Stream selection in situation recognition, 172–173 Structural questions in audition, 33 Structure discovery in audio, 46–47 Structured quantizers in similarity searches, 115–116 STTPoints EventShop platform, 180–181 situation recognition, 173 Subcritical regime in Hawkes processes, 201 Sum-product of two signals in probabilistic cryptosystems, 80 Super-bits in sketch similarity, 124 Supercritical regime in Hawkes processes, 201 Supervised deep learning for video classiﬁcation, 9–13 Surface information in audition, 47 Surveillance systems, 91 SVC (scalable video coded) videos, 94–95 Synergistic sociAL Scene Analysis (SALSA) dataset free-standing conversational groups, 58–59 head and body pose estimation, 66–69 395 System architecture in GamingAnywhere, 293–294 System design in EventShop platform, 178–180 TACoS Multi-Level Corpus (TACoS-ML) dataset for video classiﬁcation, 26 Tagging video, 14–19 TDD (trajectory-pooled deep-convolutional descriptors), 11 Template-based language model for video captioning, 16–17 Temporal inferences in audition, 33 Temporal segment networks, 11 Temporary engagement in cloud gaming, 311 Thin client design for cloud gaming, 302–306 Thinning algorithm for Hawkes processes, 202–204 Third-party service providers security and privacy concerns, 75 Throughput-driven GPUs for cloud gaming, 299 THUMOS Challenge, 23 Time complexity in similarity searches, 109–110 Trajectory-pooled deep-convolutional descriptors (TDD), 11 Transductive support vector machines (TSVMs), 68 Transferable common principles in social media, 155 Transmission efﬁciency in encrypted multimedia content analysis, 104 TRECVID 2016 Video to Text Description (TV16-VTT) dataset for video classiﬁcation, 27 TRECVID MED dataset for video classiﬁcation, 21–23 Trust-based approaches in recommender systems, 152 TSVM (transductive support vector machines), 68 396 Index TV16-VTT (TRECVID 2016 Video to Text Description) dataset for video classiﬁcation, 27 Tweets dataset for fog computing, 278–280 Two-layer LSTM networks, 12 Two-stream approach for end-to-end CNN architectures, 10–11 UCF-101 & THUMOS-2014 dataset for video classiﬁcation, 21, 24 Unicast alerts in situation-aware applications, 164 Unimodal approaches for wearable devices, 52 Unitary regularization term in multimodal pose estimation, 62 Unsupervised hierarchical structure in audition, 46–47 Unsupervised video feature learning, 13–14 User-centric multimedia computing, 143– 144 User cues conclusion, 250–251 emotion and personality type recognition, 236–250 eye ﬁxations as implicit annotations for object recognition, 226–236 introduction, 219–222 scene semantics inferences from eye movements, 222–226 User-generated content, demographic information inference from, 147 User interest modeling from multimedia data, 147–148 User-multimedia interaction behavior analysis, 146 User proﬁling, 145 Users, social attributes for, 154–155 VA-ﬁles (vector approximation ﬁles) for similarity searches, 109–110 Valence dimensions in emotion and personality type recognition, 236– 237, 243–247 Vanishing gradients in recurrent neural networks, 7 Variable bit rate (VBR) encoding in speech/speaker recognition, 98 Vector approximation ﬁles (VA-ﬁles) for similarity searches, 109–110 Vector ARMA (VARMA) models, 186 Vector identiﬁers in similarity searches, 112 Vector of locally aggregated descriptors (VLAD) encoding, 9 in similarity searches, 132 Vegas over Access Point (VoAP) algorithm in cloud gaming, 309 VGGNet, 5–6, 9 Video convolutional neural networks, 5–6 introduction, 3–4 recurrent neural networks, 6–8 social-sensed communication, 153 Video captioning approaches, 16–17 common architecture, 17–18 overview, 14–15 problem, 15–16 research, 23–29 Video classiﬁcation overview, 8–9 research, 19–23 supervised deep learning, 9–13 unsupervised video feature learning, 13–14 Video in cloud gaming compression, 307 decoders, 305–306 sharing, 311–312 Video processing in encrypted domain introduction, 91 making video data unrecognizable, 91–92 secure domains for video encoding, 92–93 security implementation, 93–96 VideoLSTM, 13 Viola-Jones face detectors, 225 Index Virality in Hawkes processes, 201, 210 Virtualization cloud gaming, 298–302, 312–313 fog computing, 261, 282 Visual data and inertial sensors in head and body pose estimation, 57 Visual Objects Classes (VOC) challenge in scene recognition, 226 Visual search (VS) tasks eye movements, 227–232 scene recognition, 226 Visualization EventShop platform, 182 situation recognition, 173 VLAD (vector of locally aggregated descriptors) encoding, 9 in similarity searches, 132 VoAP (Vegas over Access Point) algorithm in cloud gaming, 309 VOC (Visual Objects Classes) challenge in scene recognition, 226 VOC2012 dataset, 227 Vocal sounds, 45 Voice over Internet protocol (VoIP) trafﬁc, 95–99 Volunteer computing in fog computing, 258 Voronoi diagrams for similarity searches, 109 Voting, SPED for, 81 397 Watermarking, SPED for, 81 Weak and opportunistic supervision in audition, 48–49 Wearable sensing devices, 52 head and body pose estimation, 58 multimodal analysis of social interactions, 55–56 SALSA dataset, 58 Weizmann dataset for video classiﬁcation, 20 Windows platforms for GamingAnywhere, 295–296 Wisdom sources in eco-systems, 162 Wonder Shaper for fog computing, 283 Word-of-mouth diffusion Hawkes processes, 210 point processes, 192 Wrappers in EventShop platform, 181 Xen technology in fog computing, 282 XFINITY Games, 310 Yahoo Flickr Creative Commons 100 Million dataset (YFCC100M) for audition, 39 YouCook dataset for video classiﬁcation, 26 Zero-knowledge watermark detection protocol, 81 ZYNC cloud rendering platform, 273 Editor Biography Shih-Fu Chang Shih-Fu Chang is the Richard Dicker Professor at Columbia University, with appointments in both Electrical Engineering Department and Computer Science Department. His research is focused on multimedia information retrieval, computer vision, machine learning, and signal processing. A primary goal of his work is to develop intelligent systems that can extract rich information from the vast amount of visual data such as those emerging on the Web, collected through pervasive sensing, or available in gigantic archives. His work on content-based visual search in the early 90s—VisualSEEk and VideoQ—set the foundation of this vibrant area. Over the years, he continued to develop innovative solutions for image/video recognition, multimodal analysis, visual content ontology, image authentication, and compact hashing for large-scale indexing. His work has had major impacts in various applications like image/video search engines, online crime prevention, mobile product search, AR/VR, and brain machine interfaces. His scholarly work can be seen in more than 350 peer-reviewed publications, many best-paper awards, more than 30 issued patents, and technologies licensed to seven companies. He was listed as the Most Inﬂuential Scholar in the ﬁeld of Multimedia by Aminer in 2016. For his long-term pioneering contributions, he has been awarded the IEEE Signal Processing Society Technical Achievement Award, ACM Multimedia Special Interest Group Technical Achievement Award, Honorary Doctorate from the University of Amsterdam, the IEEE Kiyo Tomiyasu Award, and IBM Faculty Award. For his contributions to education, he received the Great Teacher Award from the Society of Columbia Graduates. He served as Chair of ACM SIGMM 400 Editor Biography (2013–2017), Chair of Columbia Electrical Engineering Department (2007–2010), the Editor-in-Chief of the IEEE Signal Processing Magazine (2006–2008), and advisor for several international research institutions and companies. In his current capacity as Senior Executive Vice Dean at Columbia Engineering, he plays a key role in the School’s strategic planning, special research initiatives, international collaboration, and faculty development. He is a Fellow of the American Association for the Advancement of Science (AAAS), a Fellow of the IEEE, and a Fellow of the ACM.	1
1 FLiER: Practical Topology Update Detection Using Sparse PMUs arXiv:1409.6644v3 [cs.SY] 22 Jul 2016 C. Ponce D. S. Bindel Abstract—In this paper, we present a Fingerprint Linear Estimation Routine (FLiER) to identify topology changes in power networks using readings from sparsely-deployed phasor measurement units (PMUs). When a power line, load, or generator trips in a network, or when a substation is reconfigured, the event leaves a unique “voltage fingerprint” of bus voltage changes that we can identify using only the portion of the network directly observed by the PMUs. The naive brute-force approach to identify a failed line from such voltage fingerprints, though simple and accurate, is slow. We derive an approximate algorithm based on a local linearization and a novel filtering approach that is faster and only slightly less accurate. We present experimental results using the IEEE 57-bus, IEEE 118-bus, and Polish 19992000 winter peak networks. Index Terms—topology changes, phasor measurement units, voltage fingerprint, approximation, linearization, filtering I. I NTRODUCTION A. Motivation Detection of topology changes is an important network monitoring function, and is a key part of the power grid state estimation pipeline, either as a pre-processing step or as an integrated part of a generalized state estimator. If a topology error processing module fails to detect an error in the model topology, poor and even dangerous control actions may result [1], [2], as unexpected topology changes, such as those due to failed lines, may put stress on the remaining lines and destabilize the network. Thus, it is important to identify topology changes quickly in order to take appropriate control actions. Substations and transmission lines in transmission networks have sensors that directly report failures (or switch open/closed status). However, if a sensor malfunctions, then finding the topology change is again difficult. This can happen due to normal equipment malfunctions, or because a cyber-attacker wishes to mislead network operators. Although failure to The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award Number de-ar0000230. The information, data, or work presented herein was funded in part by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. This research was conducted with Government support under and awarded by DoD, Air Force Office of Scientific Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a correctly identify a topology error is less common in a transmission network, the stakes are higher: state estimation based on incorrect topology assumptions can lead to incorrect estimates, causing operators to overlook system instability, and in the worst case, leading to avoidable blackouts. Thus, it is important to have more than one way to monitor network topology. B. Prior work State estimation and topology detection have co-evolved since at least the late 1970s [3]; see [4] for an overview of the state of the art as of 2000. These industry-standard methods use low time-resolution SCADA data about power flows and digital status, together with a reference model, to infer system voltages and currents as well as the current topology. Since the turn of the century, researchers have increasingly proposed PMU-based methods for state estimation, model calibration, fault detection, and wide area monitoring and control [5], [6], [7]. Some PMU-based methods yield state estimates [8], [9], information about oscillations [10], [11], [12], [13], and indicators of faults or contigencies [14], [15] without SCADA data and with little or no reference to a system model during regular operation. These “model-free” methods are attractive because power system models are often not wholly correct. Current PMU deployments do not provide complete observability of most of the power grids, which has lead to a broad literature on optimally placing what PMUs are available [16], [17], [18]. And where PMU data is available, it is much faster than other data sources: within an operating area, SCADA sensors typically report at most every few seconds; and the system data exchange (SDX) model of the North American Electric Reliability Corporation (NERC) provides inter-area topology information only on an hourly basis [19]. According to the IEEE specification, data from a correctly functioning PMU also satisfy tight phase and magnitude error tolerances [20]. Hence, many methods combine PMU data, SCADA data, and model information for state estimation [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33] and for detection and localization of faults [34], [35], [36], [37], [38], [14], [39], [15], [40], [41], [42], [43]. Conventional state estimators use loss functions such as least absolute variation (LAV) or Huber loss [44], [21] for robustness to inconsistent outlier equations caused by bad data or by errors in the model topology. The shape of residuals associated with the outlier equations reflects topology errors, and a quarter century of work on topology estimators exploits this [45], [46], [47], [48], [49], [50]. Robust regression is also used in many hybrid state estimators [21], [22], [25], [31], 2 as deployed PMUs sometimes have errors greater than those nominally allowed by the IEEE standard [?], [?], [?], [?], whether due to poor GPS synchronization (which may cause persistent phase errors), deficiencies in the signal processing algorithms (which may cause oscillations in the PMU output), or other issues. The methods in this paper use differences between PMU predictions, so are insensitive to absolute phase errors; and we filter the signal to reduce sensitivity to deficiencies in signal processing algorithms and to ambient oscillations. Nonetheless, we also use robust regression to defend against other types of sensor errors. Because PMUs measure current and voltage phasors directly, PMU-assisted methods can observe topology changes even in the sparse case when there is not a measurement directly incident on the affected element. This is the basis for several methods for line outage detection; the closest to our work is [34], though other closely related work includes [35], [36], [37]. Our work extends these prior approaches by using PMU signals to identify not only line trips, but also load trips, generator trips, and substation reconfigurations. As with many SCADA topology estimation techniques, we model possible substation reconfigurations using a breaker-level model [51], [52], [53], [44], [54]. In the SCADA literature, one sometimes uses expanded bus-section models only in regions where a substation reconfiguration is suspected [55], but we use a fixed breaker-level model and an extended system with a minimal set of multipliers corresponding to the edges in trees associated with each connected set of bus sections [56], [57]. C. Our work We present here an efficient method to identify topology changes in networks with a (possibly small) number of PMUs. We assume that a complete state estimate is obtained shortly before a topology change, e.g. through conventional SCADA measurements, and we use discrepancies between this state estimate and PMU measurements to identify failures. Our method does not require complete observability from PMU data; it performs well even when there are few PMUs in the network, though having more PMUs does improve the accuracy. Though our approach is similar in spirit to [34], this paper makes three key novel contributions: • We treat not only line outages, but also load trips, generator trips, and substation reconfigurations. This is not new for standard topology estimators, but to our knowledge it is new for PMU-based methods. • We describe a novel subspace-based filtering method to rule out candidate topology changes at low cost. • We take advantage of existing state estimation procedures by linearizing the full AC power flow about a previously estimated state. This increases accuracy compared to the DC approximation used in prior work. II. P ROBLEM F ORMULATION AND F INGERPRINTS Let yik = gik + bik denote the elements of the admittance matrix Y ∈ Cn×n in a bus-branch network model; let Pℓ and Qℓ denote the real and reactive power injections at bus ℓ; and let vℓ = \|vℓ \| exp(θℓ ) denote the voltage phasor at bus ℓ. These quantities are related by the power flow equations H(v; Y ) − s = 0 (1) where X n gℓh Hℓ = \|vℓ \|\|vh \| −bℓh Hn+ℓ h=1 with θℓh = θℓ − θh and s = P1 · · · Pn Q1 bℓh gℓh cos(θℓh ) , sin(θℓh ) (2) T (3) ··· Qn . We note that H is quadratic in v, but linear Y . In a breaker-level model, we use a similar system in which variables are associated with bus sections, and H represents the power flows when all breakers are open. We then write the power flow equations as H(v; Y ) + Cλ − s = 0 C T v = b, (4) where the constraint equations C T v = b have the form cTk v = (ei − ej )T v = vi − vj = bk = 0, i.e. voltage variable j for a “slave” bus section is constrained to be the same as voltage variable i for a “master” bus section. In addition, we include constraints of the form cTk v = eTi v = bk to assign a voltage magnitude at a PV bus or the phase angle at a slack bus. We could trivially eliminate these constraints, but keep them explicit for notational convenience. Our goal is to use the power flow equations to diagnose topology changes such as single line failures, substation reconfigurations, or load or generator trips. We assume the network remains stable and the state shifts from one quasisteady state to another. In practice, of course, there may be oscillations, whether due to ringdown after a topology change or ambient forcing; hence we recommend applying a low-pass filter to extract the mean behavior at each quasi-steady state. Under a topology update, the voltage vector shifts from v to v̂ = v + ∆v. We assume m voltage phasor components, indicated by the rows of E ∈ {0, 1}m×n, are directly observed by PMUs, with appropriate filtering. Assuming the loads and generation vary slowly, modulo high-frequency fluctuations removed by the low-pass filter, we can predict what E∆v should be for each possible contingency. That is, we can match the observed voltage changes E∆v to a list of voltage fingerprints to identify simple topology changes. We note that the same approach used to produce fingerprints for load and generator trips can also be used to identify significant changes in load or generation at a single source. Multiple contingencies can have the same or practically indistinguishable fingerprints. For example, one of two parallel lines with equal admittance may fail, or two lines that are distant from all PMUs but near each other may yield similar fingerprints. But even when a contingency is not identifiable, our method still produces valuable information. When multiple lines have the same effect on the network, our technique can be used to identify a small set of potential lines or breakers to inspect more closely. 3 III. A PPROXIMATE F INGERPRINTS To compute the exact fingerprint for a contingency, we require a nonlinear power flow solve. In a large network with many possible contingencies, this computation becomes expensive. We approximate the changing voltage in each contingency by linearizing the AC power flow equations about the pre-contingency state. As in methods based on the DC approximation, we use the structure of changes to the linearized system to compute voltage change fingerprints for each contingency with a few linear solves. By using information about the current state, we observe better diagnostic accuracy with our AC linearization than with the DC approximation. We consider three different types of contingencies: bus merging or bus splitting due to substation reconfiguration, and line failure. In each case, we assume the pre-contingency state is x = (v, λ) satisfying (4). We denote the post-contingency state by primed variables x′ = (v ′ , λ′ ); we assume in general that the power injections s are the same before and after the contingency. The exact shift in state is ∆x′ = x′ − x, and our approximate fingerprints are based from the approximation δx′ ≈ ∆x to the shift in state. The computation of δx′ for each contingency involves the pre-contingency Jacobian matrix ∂H (v; Y ) C ∂v . A= CT 0 We assume a factorization of A is available, perhaps from a prior state estimate. A. Bus Merging Fingerprints In the case of two bus sections becoming electrically tied due to a breaker closing, we augment C by two additional constraints C ′ to tie together the voltage magnitudes and phase angles of the previously-separate bus sections. That is, the post-contingency state satisfies the augmented system H(v ′ ; Y ) + Cλ′ + C ′ γ − s = 0 C T v′ = b (5) ′T ′ C v = 0. We linearize (5) about the original state x (with γ = 0); because the first two equations are satisfied at this state, we have the approximate system ′ A U δx′ 0 C = − , U = (6) UT 0 γ C ′T v 0 The slack variables γ let the voltage phasor for a “breakaway” group of previously-slaved sections differ from a phasor at the former master section. The two columns of F ∈ {0, 1}n×2 indicate rows of C T that constrain the breakaway voltage magnitudes and the phase angles, respectively. The third equation says no power flows across the open breaker. We linearize (9) about the original state x (with γ = 0); because the first two equations are satisfied at this state, we have the approximate system A U δx′ 0 0 = − , U = . (10) UT 0 γ FTλ F The bordered systems (10) has the same form as (6); and, as before, block Gaussian elimination requires only two solves with A, some dot products, and a 2 × 2 system solve. C. Load/Generator Trip Fingerprints When a load or generator trips offline, that bus becomes a zero-injection PQ nodes. In the case of a PQ load tripping offline, the network itself does not change, so we do not need to augment the matrix A as is done in Equations (6) and (10), but we compute the approximate fingerprint with just A. Rather, it is the power injection vector s that changes. In the case of a generator at a PV bus tripping offline, we need to convert that bus into a PQ bus with zero power injection. We write the augmented system H(v ′ ; Y ) + Cλ′ − s′ = 0 C T v′ + f γ = b T (11) ′ f λ = 0. The slack variable γ lets the voltage magnitude of the bus of interest shift. The vector f is an indicator vector such that the third equation constrains the reactive power injection slack variable to zero. Note that s has also been changed to s′ to represent the real power injection shifting to zero. The resulting system is A u δx′ 0 0 = − , u = . (12) uT 0 γ fT λ f The bordered system (12) has the same form as (10) and (6). In this case, only one solve with A is required. We then solve the system by block elimination to obtain γ = (U T A−1 U )−1 (C ′T v) ′ −1 δx = −A Uγ (7) (8) The formulas (7)–(8) only require two significant linear solves (to evaluate A−1 U ), some dot products, and a 2 × 2 solve. B. Bus Splitting Fingerprints When a bus splits after a breaker opens, the postcontingency state satisfies the augmented system H(v ′ ; Y ) + Cλ′ − s = 0 C T v′ + F γ = b T ′ F λ = 0. (9) D. Line Failure Fingerprints In principle, line failures can be handled in the same way as substation reconfigurations that lead to bus splitting: explicitly represent two nodes on a line that are normally connected (physically corresponding to two sides of a breaker) with a multiplier that forces them to be equal, and compute the fingerprint by an extended system that negates the effect of that multiplier. In practice, we may prefer to avoid the extra variables in this model. The following formulation requires no explicit extra variables in the base model, and can be used with either a breaker-level model or a bus-branch model with no breakers (i.e. C an empty matrix). 4 For line failures, the admittance changes to Y ′ = Y + ∆Y ′ where ∆Y ′ is a rank-one update. The post-contingency state satisfies the system More concretely, we have Uik =   ′ ′ \|vi \|2 P̌i − gii \|vi \|2 P̌i + gii −Q̌i − b′ii \|vi \|2  Q̌k + b′ \|vk \|2 P̌k − g ′ \|vk \|2 P̌k + g ′ \|vk \|2  kk kk kk    P̌i − g ′ \|vi \|2 Q̌i + b′ii \|vi \|2  Q̌i − b′ii \|vi \|2 ii ′ −P̌k + gkk \|vk \|2 Q̌k + b′kk \|vk \|2 Q̌k − b′kk \|vk \|2   1 −1 0 0 0 . VikT = 0 0 \|vi \|−1 0 0 0 \|vk \|−1 H(v ′ ; Y ′ ) + Cλ′ − s = 0 C T v ′ = b, and linearization about x gives ∂H ′ ′ δv H(v; ∆Y ′ ) ∂v (v; Y ) C =− 0 CT 0 δλ′ (13) where H(v; ∆Y ′ ) = H(v; Y ′ ) − H(v; Y ). As we show momentarily, Because H(v, ∆Y ′ ) does not depend on any voltage phasors other than those at nodes i and j, we may write ∂H ∂H ∂H (v; Y ′ ) − (v; Y ) = (v; ∆Y ′ ) = U 0 (V 0 )T . ∂v ∂v ∂v ∂H(v; ∆Y ′ ) T = Eik Dik Eik = U 0 (V 0 )T ∂v e ek ∈ R2n×4 . Eik = i ei ek where U 0 and V 0 each have three columns. That is, the matrix in the system (13) is a rank-three update to A. We can solve such a system by the Sherman-Morrison-Woodbury update formula, also widely known as the Inverse Matrix Modification Lemma [58], [59]. We use the equivalent extended system ′ A U δx r = − , (14) V T −I γ 0 where U= U0 , 0 V = V0 , 0 r= H(v; ∆Y ′ ) . 0 γ = (I + V T A−1 U )−1 (V T r) δx′ = −A−1 (r + U γ). and U 0 = Eik Uik , (16) ′ = P̌ik + gii \|vi \|2 Q̌i ≡ Hi+n = Q̌ik − b′ii \|vi \|2  P̌i  P̌k  0  H(v, ∆Y ′ ) = Eik   Q̌i  = U z, Q̌k  ′ = P̌ki + gkk \|vk \|2 where ′ P̌ik gik ≡ \|vi \|\|vk \| −b′ik Q̌ik b′ik ′ gik cos(θik ) , sin(θik ) and P̌ki , Q̌ki are defined similarly. Let Dik ≡ ∂(P̌i , P̌k , Q̌i , Q̌k ) ∈ R4×4 ; ∂(θi , θk , \|v\|i , \|v\|k ) by the chain rule, we can write Dik = Uik VikT where ∂(P̌i , P̌k , Q̌i , Q̌k ) ∈ R4×3 ∂(θik , log \|v\|i , log \|v\|k ) ∂(θik , log \|v\|i , log \|v\|k ) ∈ R3×4 . ≡ ∂(θi , θk , \|v\|i , \|v\|k ) Uik ≡ VikT (19)   0 z = 1/2 , 1/2 δx′ = −A−1 U (z + γ). (20) IV. F ILTERING In Section III we discussed how to approximate voltage shifts δv ′ associated with several types of contingencies. This approach to predicting voltage changes costs less than a nonlinear power flow solve, but may still be costly for a large network with many contingencies to check. In the current section we show how to rule out contingencies without any solves by computing a cheap lower bound on the discrepancy between the observed voltage changes and the predicted voltage changes under the contingencies. For each contingency, we define the fingerprint score t = kE∆v − Eδv ′ k Q̌k ≡ Hk+n = Q̌ki − b′kk \|vk \|2 , V 0 = Eik Vik . Moreover, we note that (15) The work to evaluate (15)–(16) is three linear solves (for A−1 U ), some dot products, and a small 3 × 3 solve. We will show momentarily how to avoid the solve involving r. We now show that the Jacobian matrix changes by a rank-3 update. For a failed line between nodes i and k, the vector H(v, ∆Y ′ ) has only four nonzero entries: P̌k ≡ Hk (18) so that we may rewrite (16) as We again solve by block elimination: P̌i ≡ Hi where (17) (21) where ∆v is the observed voltage shift and δv ′ is the voltage shift predicted for the contingency. For the contingencies we have described, Eδv ′ has the form Eδv ′ = ĒA−1 U γ (22) where Ē = E 0 simply ignores the multiplier variables λ, and γ is some short vector of slack variables. The expression ĒA−1 does not depend on the contingency, and can be precomputed at the cost of m linear solves (one per observed phasor component). After this computation, the main cost in evaluating (22) is the computation of γ, which involves a contingency-dependent linear system with A as an intermediate step. However, we do not need γ for the filter score τ = min kE∆v − ĒA−1 U µk ≤ t. µ (23) 5 PMUs FLiER FLiER+noise DC Approx DC Approx+noise Algorithm 1 FLiER Compute and store ĒA−1 . For each contingency i, compute τi via (23). Order the contingencies in ascending order by τ . for ℓ = 2, 3, . . . do Compute fingerprint score tℓ Break if tℓ < τℓ+1 end for Return contingencies with computed tℓ tk < τi ≤ ti , without ever computing ti . Exploiting this fact leads to the FLiER method (Algorithm 1)1 . We note that PQ load trips require no filtering, as they involve no change to the system matrix. Filter score computations are embarrassingly parallel and can be spread across processors. Nonetheless, for huge networks with many PMUs and many contingencies, the filter computations might be deemed too expensive for very rapid diagnosis (e.g. in less than a second). However, the concept of a filter subspace can be adapted to these cases. First, one can define a coarse filtering subspaces that is the sum of the filtering subspaces for a set of contingencies. For example, one might define a coarse filtering subspace associated with all possible breaker reconfigurations inside a substation. The coarse subspace filter score provides a lower bound on the filter scores (and hence the fingerprint scores) for all contingencies in the set. Hence, it may not even be necessary to compute individual filter scores for all contingencies considered. Second, one can work with a projected filtering subspace W T (EA−1 U ) where W is a matrix with orthonormal columns. The distance from a projected measurement vector to the projected filtering subspace again gives a lower bound on the full filter score. In addition to reducing the cost of filter score computations, projections can also be used to eliminate faulty or missing PMU measurements from consideration. V. E XPERIMENTS Our standard experimental setup is as follows. For each possible topology change, we compute and pass to FLiER both the full pre-contingency state and the subset of the postcontingency state that would be observed by the PMUs. We test both with no noise and with independent random Gaussian noise with standard deviation 1.7 · 10−3 (≈ 0.1 degrees for phase angles) added to both the initial state estimate and the can be Sparse 68(77) 66(78) 52(74) 49(64) All 78(78) 78(78) 72(77) 66(68) TABLE I IEEE 57- BUS NETWORK ACCURACY COMPARISON FOR 78 LINE FAILURE CONTINGENCIES . W E REPORT COUNTS OF LINE FAILURES CORRECTLY IDENTIFIED AND THOSE SCORED IN THE TOP THREE ( IN PARENTHESES ). In the Euclidean norm, τ is simply the size of the residual in a least squares fit of E∆v to the columns of ĒA−1 U , which can be computed quickly due to the sparsity of U . If U is the augmentation matrix associated with contingency i, we refer to ĒA−1 U as its filtering subspace. Filter score computations are cheap; and if the filter score τi for contingency i exceeds the fingerprint score tk for contingency k, then we know 1 Example Python code of this algorithm https://github.com/cponce512/FLiER Test Suite Single 55(73) 40(65) 17(40) 5(22) found at PMU readings. In [34], 0.1 degrees of Gaussian random noise was applied to phase angles, then smoothed by passing a simulated time-domain signal through a low pass filter; we apply the noise without filtering, so the effect is more drastic. One of the possibilities FLiER checks is that there has been no change; in this case, we use the norm of the fingerprint as both the fingerprint score and the filter score. By including this possibility among those checked, FLiER acts simultaneously as a method for topology change detection and identification. We run tests on the IEEE 57 bus and 118 bus networks, with three different PMU arrangements on each: • Single: Only one PMU is placed in the network, at a lowdegree node (bus 35 in the 57-bus network and 65 in the 118-bus network, providing a total of 3 and 5 bus voltage readings, respectively). This represents a near-worst-case deployment for our method. • Sparse: A few PMUs are placed about the network (on buses 4, 13, and 34 in the 57-bus network and on buses 5, 17, 37, 66, 80, and 100 in the 118-bus network, providing a total of 15 and 40 bus voltage readings, respectively). We consider this a realistic scenario in which sparselydeployed PMUs do not offer full network observability. • All: PMUs are placed on all buses. Any error is due purely to the linear approximation. We did not test changes that cause convergence failure in our power flow solver. We assume such contingencies result in collapse without some control action. A. Accuracy 1) Line Failures: Figure 1 shows the accuracy of FLiER in identifying line failures in the IEEE 57-bus test network. For each PMU deployment, we show the cumulative distribution function of ranks, i.e. the ranks of each simulated contingency in the ordered list produced by FLiER. We show further results in Table I. With PMUs everywhere, the correct answer was chosen in all 78 of 78 cases, even in the presence of noise. The case with three PMUs is also quite robust to noise. In the test with a single unfavorably-placed PMU, FLiER typically ranks the correct line among the top three in the absence of noise; with noise, the accuracy degrades, though not completely. In Figure 2, we repeat the experiment of Figure 1, but with the DC approximation used in [34] rather than the AC linearization used in FLiER. We also present comparisons in Table I. With PMUs everywhere, there is little difference in accuracy. With fewer PMUs, FLiER is more accurate. In the sparse case, the DC approximation without noise behaves similarly to FLiER with noise, while in the single PMU 1.0 0.8 0.8 0.6 0.4 0.2 0.0 Fraction of Tests All Sparse Single 1 2 3 4 5 6 Rank 7 8 Fraction of Tests 1.0 0.6 0.4 1.0 1.0 0.8 0.8 0.6 All Sparse Single Single BF 0.4 0.2 0.0 1 2 3 4 5 6 Rank 7 8 Fig. 1. Cumulative distribution function showing the fraction of line failures where FLiER assigned the correct line at most a given rank (up to 10). Top: Noise-free case. Bottom: Entries with Gaussian noise with σ = 0.0017. “Single BF” is the result from using a brute-force approach with a single PMU and represents the signal-to-noise ratio in that PMU’s information. 1 2 3 4 5 6 Rank 7 8 9 10 0.6 All Sparse Single Single BF 0.4 0.2 0.0 9 10 All Sparse Single 0.2 0.0 9 10 Fraction of Tests Fraction of Tests 6 1 2 3 4 5 6 Rank 7 8 9 10 Fig. 2. CDF of line failures where the DC approximation of [34] assigned the correct line at most a given rank (up to 10). Top: Noise-free case. Bottom: Entries with Gaussian noise with σ = 0.0017. “Single BF” is the result from using a brute-force approach with a single PMU and represents the signal-tonoise ratio in that PMU’s information. deployment the DC results without noise are much worse than those from FLiER even with noise. Figure 3 shows the raw scores computed by FLiER with three PMUs. In this plot, each column represents the fingerprint scores computed for one line failure scenario. The black crosses represent the scores of lines that get past the filter, while the green circles and yellow triangles represent the scores for the correct answer. If there is a green circle, then our algorithm correctly identified the actual line that failed. If there is a yellow triangle, the correct line was not chosen but was among the top three lines selected by the algorithm. In Figure 4, we show one case that FLiER misidentifies. PMUs are deployed on buses marked with blue squares, and lines are colored and thickened according to the FLiER score. The best-scoring line is adjacent to the line that failed. 2) Substation Reconfigurations: Next, we show the accuracy of FLiER as it applies to substation reconfigurations. For these tests, we suppose that every bus in the IEEE 57-bus test network is a ring substation with each bus section on the ring possessing either load, generation, or a branch. We then suppose a substation splits when two of its circuit breakers open. We do not consider cases that isolate a node with a nonzero power injection. Line failures are a subset of this scenario: if the breakers on either side of a section with a branch open, that section becomes a zero-injection leaf bus, which disappears in the quasi-static setting. Figure 5 shows the accuracy of FLiER on substation re- Approximation error 100 10−1 10−2 10−3 10−4 10−5 0 10 20 30 40 50 60 70 80 Line number (sorted) Fig. 3. Test of our algorithm on the IEEE 57-bus network with the sparse PMU deployment. Each column represents one test. Black crosses are fingerprint scores for incorrect lines. Green dots and yellow triangles indicate the scores of the correct line in the case of correct diagnosis or diagnosis in the top three, respectively. configurations with and without noise. With three PMUs and no noise, FLiER is right in 164 of 193 possibilities, and ranks the correct answer among the top three scores in 160 cases. With PMUs everywhere, FLiER is right 181 times, but gets the answer in the top three every single time. With few PMUs, FLiER is more susceptible to noise when diagnosing substation reconfigurations. This is expected, as there are significantly more possibilities to choose from in this case. Also, FLiER sometimes filters out the correct answer in the presence of noise. One possible remedy for this would be to be 7 Contingency Correct % Top 3 % 26 75.2 95.4 Substation of contingency 85.4 96.5 TABLE II A CCURACY OF FL I ER WITH 100 RANDOMLY- PLACED PMU S ON THE P OLISH NETWORK . R ESULTS ARE OUT OF 6283 TESTS . 24 27 1.0 Fig. 4. Line (24, 26) isqthe line removed in this test. Lines are colored and thickened according to t−1 ik . Line (26, 27) was chosen by the algorithm. Fraction of Tests 28 0.8 0.6 0.4 All Sparse Single 0.2 1.0 1 2 3 4 5 6 Rank 7 9 10 1.0 0.4 All Sparse Single 0.2 0.0 1 2 3 4 5 6 Rank 7 8 9 10 0.8 0.6 0.4 All Sparse Single 0.2 1.0 0.0 Fraction of Tests 8 0.6 Fraction of Tests Fraction of Tests 0.0 0.8 0.8 1 2 3 4 5 6 Rank 7 8 9 10 0.6 All Sparse Single Single BF 0.4 0.2 0.0 1 2 3 4 5 6 Rank 7 8 9 10 Fig. 5. Rank CDF for substation reconfigurations without noise (top) and with noise (bottom). “Single BF” is the result from using a brute-force approach with a single PMU and represents the signal-to-noise ratio in that PMU’s information. more lenient with filtering, only throwing a possibility away if τℓ is greater than the kth smallest tℓ , for example. Finally, we demonstrate the effectiveness of using FLiER for substation reconfigurations on a large-scale network by running FLiER on the 400, 220, and 100 kV subset of the Polish network during peak conditions of the 1999-2000 winter, taken from [60]. This is a larger network with 2,383 buses. We placed 100 PMUs randomly around the network, and tested every substation reconfiguration contingency. We summarize the results in Table II. We could likely further improve the accuracy with a thoughtful deployment of PMUs. 3) Including Load and Generator Trips: Here we include load and generator trips in our results. In Figure 6, we show Fig. 6. Rank CDF for generator trips without noise (top) and with noise (bottom). the results of testing for generator trips, in Figure 7, we show the results of testing for load trips, and in Figure 8, we show the results of including load and generator trips in the set of contingencies to test and check for along with substation splits. All tests use the IEEE 57-bus network. In the noisy case for Figure ??, we allow some extra slack in the filtering procedure. In particular, rather than checking if tk < τi , we check if tk < τi − σ, where σ is the standard deviation of the included noise. We did this because the filtering method sometimes filtered out the correct answer for load trips. In a real-world setting one would need to choose that slack value intelligently, but this test shows that slightly less-stringent filtering can improve the method in some cases. 4) PMU Placement: Here we demonstrate the robustness of FLiER to PMU placement. We tested this by sampling three busses from the IEEE 57-bus network uniformly at random, running FLiER for the substation reconfiguration case (as in Figure 5) and recording the fraction of contingencies FLiER identified correctly, and the fraction of contingencies with the solution ranked in the top 3. We performed this test 200 times. We show cumulative distribution functions for the fraction correct and fraction in the top 3 in Figure 9. 1.0 0.8 0.8 0.6 0.4 All Sparse Single 0.2 0.0 1 2 3 4 5 6 Rank 7 8 0.6 0.4 1.0 1.0 0.8 0.8 0.6 All Sparse Single Single BF 0.4 0.2 0.0 1 2 3 4 5 6 Rank 7 8 Fig. 7. Rank CDF for load trips without noise (top) and with noise (bottom). In the noisy case, we allow a filtering slack equal to one noise standard deviation, as otherwise the filter is too stringent. “Single BF” is the result from using a brute-force approach with a single PMU and represents the signalto-noise ratio in that PMU’s information. We do not include zero-injection busses in this test. This figure shows that FLiER tends to be quite robust to PMU placement. In fact, the median fraction correct and median fraction in top 3 are only slightly below those for the noiseless test in Figure 5, where we attempted to place the three PMUs favorably. Furthermore, the standard deviation for fraction correct is only 3.65%, while the standard deviation for fraction in top 3 is 2.67%. 5) Robustness to Bad Data: Here we demonstrate how one can easily modify FLiER for enhanced robustness to bad data and heavy-tailed noise. In Equations (21) and (23), we use the L2 loss function to measure the distance between an estimated fingerprint or subspace and the actual fingerprint. The L2 loss function, while simple and useful, is sensitive to outliers. A single large error component drastically increases the t or τ value, even if all other components have very small error. The result is using the L2 loss can make FLiER sensitive to bad data and heavy-tailed noise. An alternative is to use a loss function that is robust to large outliers. One popular such loss function is the Huber loss [44]: kek2H = Lδ (ei ) m X Lδ (ei ) i=1 1 2 2 ei = δ \|ei \| − 12 δ (24) \|ei \| ≤ δ otherwise (25) This function behaves like the L2 loss for error components near zero, but for error components larger than δ, the loss 1 2 3 4 5 6 Rank 7 8 9 10 0.6 All Sparse Single Single BF 0.4 0.2 0.0 9 10 All Sparse Single 0.2 0.0 9 10 Fraction of Tests Fraction of Tests Fraction of Tests 1.0 1 2 3 4 5 6 Rank 7 8 9 10 Fig. 8. Rank CDF for substation reconfigurations and load/generator trips without noise (top) and with noise (bottom). “Single BF” is the result from using a brute-force approach with a single PMU and represents the signal-tonoise ratio in that PMU’s information. 1.0 Fraction of Samples Fraction of Tests 8 0.8 Correct Top 3 0.6 0.4 0.2 0.0 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Fraction Correct/Fraction Top3 Fig. 9. CDF for fraction of substation reconfiguration contingencies that FLiER gets correct (blue) and in the top 3 (green) with random uniform placement of three PMUs. Noise is not included in this test. grows linearly rather than quadratically. The result is a loss function that is robust to outliers in data. In Figures 10 and 11 we show the usefulness of the Huber loss in certain cases. In Figure 10, we run FLiER on the line failures test (as in the noisy case of Figure 1), but with the PMU reading from bus 4 given a large post-event bias, always returning an after-event angle reading that is 5 degrees too large. As shown in the figure, this single bad reading can cause major problems with FLiER accuracy using L2 loss. With Huber loss, however, that bad data is almost completely ignored, resulting in accuracy essentially identical to that of the noisy case in Figure 1. 9 0.8 0.6 0.4 L2 Loss Huber Loss 0.2 0.0 1 2 3 4 5 6 Rank 7 8 1.0 0.8 0.8 0.6 0.4 All Sparse Single 0.2 0.0 0.0 9 10 Fig. 10. CDF for line failures with a single PMU reading that systematically gives angle readings that are too large by 5 degrees, along with Gaussian random noise with standard deviation 0.0017. PMUs are at locations 4, 13, and 34. Huber scale parameter δ = 1.365 · 0.0017 Fraction of Tests Fraction of lines filtered 1.0 0.2 0.4 0.6 0.8 Fraction of failure scenarios 0.6 0.8 0.6 0.4 All Sparse Single 0.2 0.0 0.0 0.2 0.4 0.6 0.8 Fraction of failure scenarios 0.4 L2 Loss Huber Loss 0.2 0.0 1 2 3 4 5 6 Rank 7 8 9 10 1.0 1.0 Fraction of lines filtered Fraction of Tests 1.0 1.0 Fig. 12. Cumulative distribution function of fraction of lines for which tik need not be computed when a line in the IEEE 57-bus (top) or 118-bus (bottom) network fails uniformly at random. Fig. 11. CDF for substation reconfigurations with Cauchy noise with a scale parameter of 0.001. In Figure 11, we show the utility of the Huber loss in the presence of heavy-tailed noise. Here we test substation reconfigurations with noise given by a Cauchy distribution with scale parameter 0.001. As shown in the figure, the use of the Huber loss significantly improves FLiER’s performance under this type of noise. It is unclear if either of these situations are likely to arise in practice. In the former case (with a bad PMU), it is likely that this bad data stream would be identified in an earlier state estimation stage and already removed from the fingerprint. In the latter case, the IEEE specification [20] requires that the PMU noise profile not have heavy tails, so systematic heavytailed noise is unlikely. B. Filter Effectiveness and Speed The cost of FLiER depends strongly on the effectiveness of the filtering procedure. In Figure 12, we show how often the filter saves us from computing fingerprint scores in experiments on the IEEE 57-bus and 118-bus networks when checking for line failures. For each PMU deployment, we show the cumulative distribution function of the fraction of lines for which fingerprint scores need not be computed for each line failure. The filter performs well even for the sparse PMU deployments; we show a typical case in Figure 13. Approximation error 100 10−1 10−2 10−3 10−4 0 10 20 30 40 50 60 70 80 Line number (sorted) Fig. 13. Example of effective filtering in Algorithm 1. Each column represents a line checked. Blue dots are the lower bounds τik , while red squares are true scores tik . Columns are sorted by τik . In this case, tik only needs to be computed for eight lines. Finally, we demonstrate the importance of the filter by running FLiER on the large Polish network [60] with 100 randomly placed PMUs. Table III shows FLiER run times with and without the filter on ten randomly selected branches. The code is unoptimized Python, so these timings do not indicate of how fast FLiER would run in a performance setting. However, they give a sense of the speedup one expects from filtering. Note also that FLiER correctly identified the failed branch in 9 of 10 cases. In the one case in which it failed, on branch (2346, 2341), tℓ for the correct answer was 6.25 · 10−5 ; this suggests the failure had a negligible impact on the network. We also performed this timing test for a randomly selected 10 Line (1502, 917) (1502, 1482) (557, 556) (2346, 2341) (909, 1155) (644, 629) (591, 737) (559, 542) (378, 336) (101, 94) FLiER (s) 0.27 0.27 0.27 2.95 0.26 0.29 0.29 0.32 0.28 0.26 Solution rank 1 1 1 14 1 1 1 1 1 1 # t’s computed 2 2 4 502 2 7 6 13 6 2 FLiER n.f. (s) 14.14 15.30 14.75 14.40 14.32 14.59 17.27 16.59 16.16 15.29 TABLE III FL I ER RUN TIMES FOR TEN LINE FAILURES WITH AND WITHOUT FILTERING . A BOUT 3000 CONTINGENCIES ARE CONSIDERED . Bus / Split nodes 86/1,2,3 176/1,2 539/7 702/4,5 754/2,3,4,5 994/2,3,4,5 1131/2 1513/4,5,6 1663/1,2,3,4 2164/5 FLiER (s) 4.54 4.70 8.11 4.66 4.97 4.86 5.22 6.65 7.83 4.56 Sol. rank / Sol. bus rank 1/1 3/3 1/1 1/1 1/1 1/1 1/1 4/1 1/1 1/1 # t’s computed 2 4 38 4 7 6 9 23 35 3 FLiER n.f. (s) 823.0 836.9 829.8 820.1 829.8 835.8 850.3 862.3 928.1 875.3 TABLE IV FL I ER RUN TIMES FOR TEN SUBSTATION RECONFIGURATIONS WITH AND WITHOUT FILTERING . N EARLY 7000 CONTINGENCIES ARE CONSIDERED . set of substation reconfigurations, shown in Table IV. Again, the results give a sense of the speedup one expects from filtering in the substation reconfiguration case. VI. C ONCLUSION AND F UTURE W ORK We have presented FLiER, a new algorithm to identify topology changes involving load and generator trips, line outages, and substation reconfigurations using a sparse deployment of PMUs. Our method uses a linearization of the power flow equations together with a novel subspace-based filtering approach to provide fast diagnosis. Unlike prior approaches based on DC approximation, our approach takes advantage of a state estimate obtained shortly before the topology changes, assuming that the network specifications remain unchanged or change in a known way as a result of the failure. 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arXiv:1503.02436v2 [math.GR] 15 Mar 2017 RATIONAL DISCRETE COHOMOLOGY FOR TOTALLY DISCONNECTED LOCALLY COMPACT GROUPS I. CASTELLANO AND TH. WEIGEL Abstract. Rational discrete cohomology and homology for a totally disconnected locally compact group G is introduced and studied. The Hom-⊗ identities associated to the rational discrete bimodule Bi(G) allow to introduce the notion of rational duality groups in analogy to the discrete case. It is shown that semi-simple groups defined over a non-discrete, non-archimedean local field are rational t.d.l.c. duality groups, and the same is true for certain topological Kac-Moody groups. However, Y. Neretin’s group of spheromorphisms of a locally finite regular tree is not even of finite rational discrete cohomological dimension. For a unimodular t.d.l.c. group G of type FP it is possible to define an Euler-Poincaré characteristic χ(G) which is a rational multiple of a Haar measure. This value is calculated explicitly for Chevalley groups defined over a non-discrete, non-archimedean local field K and some other examples. 1. Introduction For a topological group G several cohomology theories have been introduced and studied in the past. In many cases the main motivation was to obtain an interpretation of the low-dimensional cohomology groups in analogy to discrete groups (cf. [3], [33], [34], [35]). The main subject of this paper is rational discrete cohomology for totally disconnected, locally compact (= t.d.l.c.) groups. Discrete cohomology can be defined in a very general frame work. We will call a topological group G to be of type OS, if (1.1) Osgrp(G) = { O ⊆ G \| O an open subgroup of G } is a basis of neighbourhoods of 1 in G, e.g., any profinite group is of type OS, and, by van Dantzig’s theorem (cf. [48]), any t.d.l.c. group is of type OS. For any topological group G of type OS and any commutative ring R the category R[G] dis of discrete left R[G]-modules is an abelian category with enough injectives. This allows to define dExt•R[G] ( , ) as the right derived functors of HomR[G] ( , ), and dH• (R[G], ) as the right derived functors of the fixed point functor G . Even in this general context one has a Hochschild-Lyndon-Serre spectral sequence for short exact sequences of topological groups of type OS (cf. §2.8), and a generalized Eckmann-Shapiro type lemma in the first argument (cf. §2.9). In case that G is a t.d.l.c. group it turns out that Q[G] dis is also an abelian category with enough projectives (cf. Prop. 3.2). Moreover, any projective rational discrete right Q[G]-module is flat with respect to short exact sequences of rational Date: March 16, 2017. 2010 Mathematics Subject Classification. 20J05, 22D05, 57T99, 22E20, 51E42, 17B67. Key words and phrases. Discrete cohomology, totally disconnected locally compact groups, duality groups, discrete actions on simplicial complexes, E-spaces. 1 2 I. CASTELLANO AND TH. WEIGEL discrete left Q[G]-modules (cf. (4.4)). Thus for a discrete left Q[G]-module M one may define rational discrete cohomology with coefficients in M by (1.2) dHk (G, M ) = dExtkG (Q, M ), k ≥ 0, and discrete rational homology with coefficients in M by (1.3) dHk (G, M ) = dTorG k (Q, M ), k≥0 in analogy to discrete groups (cf. §4). The rational discrete cohomological dimension cdQ (G) (cf. (3.11)) reflects structural information on a t.d.l.c. group G, e.g., (1) G is compact if, and only if, cdQ (G) = 0 (cf. Prop. 3.7(a)); (2) the flat rank of G is bounded by cdQ (G) (cf. Prop. 3.10(b)); (3) if G is acting discretely on a tree with compact stabilizers then cdQ (G) ≤ 1 (cf. Prop. 5.4(b)); (4) more generally, if G is acting discretely on a contractable simplical complex Σ with compact stabilizers, then cdQ (G) ≤ dim(Σ) (cf. Prop. 6.6(a)); (5) a nested t.d.l.c. group G satisfies cdQ (G) ≤ 1 (cf. Prop. 5.8). The existence of finitely generated projective rational discrete Q[G]-modules allows to define the FPk -property, k ∈ N0 ∪ {∞}, (cf. §3.6) in analogy to discrete groups (cf. [14, §VIII.5]). A t.d.l.c. group G is compactly generated if, and only if, it is of type FP1 (cf. Thm. 5.3). Using the theory of rough Cayley graphs due to H. Abels (cf. [1]) and Bass-Serre theory, one may also introduce the notion of a compactly presented t.d.l.c. groups (cf. §5.8). Such a group must be of type FP2 . For a t.d.l.c. group G there is not necessarily a group algebra but two natural rational discrete Q[G]-bimodules Bi(G) and Cc (G, Q) and both turn out to be useful in this context. If G is unimodular, they are isomorphic, but not necessarily canonically isomorphic; if G is not unimodular, they are non-isomorphic (cf. Prop. 4.11(b)). The Q[G]-bimodule Cc (G, Q) just consists of the continuous functions with compact support from G to the discrete field Q. The rational discrete left Q[G]-module Bi(G) is defined by (1.4) Bi(G) = lim −→ Q[G/O], O∈CO(G) where CO(G) = { O ∈ Osgrp(G) \| O compact } and the direct limit is taken along a rational multiple of the transfers (cf. (4.15)). It carries naturally also the structure of a discrete right Q[G]-module. The rational discrete Q[G]-bimodule Bi(G) can be used to establish Hom-⊗ identities which are well known for discrete groups (cf. §4.3). This allows to define the notion of a rational t.d.l.c. duality group. In more detail, a t.d.l.c. group G will be said to be a rational duality group of dimension d ≥ 0, if (i) G is of type FP∞ ; (ii) cdQ (G) < ∞; (iii) dHk (G, Bi(G)) = 0 for k 6= d. Note that any discrete virtual duality group in the sense of R. Bieri and B. Eckmann (cf. [9]) of virtual cohomological dimension d ≥ 0 is indeed a rational t.d.l.c. duality group of dimension d ≥ 0, and the same is true for discrete duality groups of dimension d over Q (cf. [8, §9.2]). For these t.d.l.c. groups one has a nontrivial relation between discrete cohomology and discrete homology (cf. Prop. 4.10). RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 3 However, it differs slightly from the usual form. A t.d.l.c. group G satisfying (i) and (ii) will be said to be of type FP. Note that a t.d.l.c. group G satisfying (i), (ii) and (iii) must also satisfy cdQ (G) = d (cf. Prop. 4.7). The rational discrete right Q[G]-module DG = dHd (G, Bi(G)) (1.5) will be called the rational dualizing module. A compact t.d.l.c. group G is a rational t.d.l.c. duality group of dimension 0 with dualizing module isomorphic to the trivial right Q[G]-module Q. If G is a compactly generated t.d.l.c. group satisfying cdQ (G) = 1, then G is a rational t.d.l.c. duality group of dimension 1. Provided that G acts discretely on a locally finite tree T with compact stabilizers one has an isomorphism of rational discrete left Q[G]-modules (1.6) × DG ≃ Hc1 (\|T \|, Q) ⊗ Q(∆) (cf. §6.10), where \|T \| denotes the topological realization1 of T (cf. §A.2), and Hc1 ( , Q) denotes cohomology with compact support with coefficients in the discrete field Q. If Σ is a locally finite C-discrete simplicial G-complex (cf. §6.4), then Hck (\|Σ\|, Q) carries naturally the structure of a rational discrete left Q[G]-module (cf. §A.4). Here Q(∆) denotes the 1-dimensional rational discrete left Q[G]-module which action is given by the modular function2 ∆ : G → Q+ of G (cf. Remark 4.13). For a right Q[G]-module M we denote by × M the associated left Q[G]-module, i.e., for m ∈ M and g ∈ G one has g · m = m · g −1 . Let G be a t.d.l.c. group. The family of compact open subgroups C of G is closed under conjugation and finite intersections. Hence there exists a G-classifying space E C (G) for the family C (cf. [32, §2], [47, Chap. 1, Thm. 6.6]) which is unique up to G-homotopy equivalence. E.g., if Π = π1 (A, Λ, x0 ) is the fundamental group of a graph of profinite groups, then the tree T = T (A, Λ, Ξ, E + ) arising in Bass-Serre theory (cf. [41, §I.5.1, Thm. 12]) is an E C (Π)-space (cf. §6.10). For algebraic groups defined over a non-discrete non-archimedean t.d.l.c. field K a celebrated theorem of A. Borel and J-P. Serre yields the following (cf. §6.11). Theorem A. Let G be a simply-connected semi-simple algebraic group defined over a non-discrete non-archimedian local field K, let (C, S) be the affine building associated to G, and let \|Σ(C, S)\| denote the topological realization of the Bruhat-Tits realization of (C, S). Then G(K) is a rational t.d.l.c. duality group of dimension d = rk(G) = dim(Σ(C, S)), and (1.7) × DG(K) ≃ Hcd (\|Σ(C, S)\|, Q). Moreover, \|Σ(C, S)\| is an E C (G(K))-space. Here rk(G) denotes the rank of G as algebraic group defined over K, i.e., rk(G) coincides with the rank of a maximal split torus of G. As already mentioned in [12] one may think of × DG(K) as a generalized Steinberg module of G(K). Another source of rational t.d.l.c. duality groups arises in the context of topological Kac-Moody groups as introduced by B. Rémy and M. Ronan in [40, §1B]. In principal, these t.d.l.c. groups behave very similarly as the examples described 1Although this topological space is also known as the realization of the tree T (cf. [41, §I.2, p. 14]), we have chosen this name in order to avoid the linguistic ambiguity with the geometric realization of buildings. 2Throughout this paper Haar measures are considered to be left invariant. 4 I. CASTELLANO AND TH. WEIGEL in Theorem B, but there are also quite notable differences. Any split Kac-Moody group G(F) defined over a finite field F has an action on the positive part Ξ+ of the associated twin building. Assuming that the associated Weyl group (W, S) is infinite one defines Ḡ(F) as the permutation closure of G(F) on its action on the chambers of Ξ+ . Then Ḡ(F) acts also on Σ(Ξ+ ), the Davis-Moussang realization of Ξ+ , which is a finite dimensional locally finite simplicial complex. The subgroup Ḡ(F)◦ of Ḡ(F), which is the closure of the image of the type preserving automorphisms of Ξ, is of finite index in Ḡ(F). This group has the following properties. Theorem B. Let F be a finite field, let Ḡ(F) be a topological Kac-Moody group obtained by completing the split Kac-Moody group G(F), and let (W, S) denote its associated Weyl group. (a) Then Σ(Ξ) is a tame C-discrete simplicial Ḡ(F)◦ -complex, and \|Σ(Ξ)\| is an E C (Ḡ(F)◦ )-space. (b) cdQ (Ḡ◦ (F)) = cdQ (Ḡ(F)) = cdQ (W ). (c) For d = cdQ (Ḡ◦ (F)) one has × DḠ(F)◦ ≃ Hcd (\|Σ(Ξ)\|, Q). (d) Ḡ(F)◦ (and hence Ḡ(F)) is a rational t.d.l.c. duality group if, and only if, W is a rational duality group. One may verify easily that whenever W is a hyperbolic Coxeter group, the topological Kac-Moody group Ḡ(F)◦ is indeed a rational t.d.l.c. duality group (cf. Remark 6.12). However, one may also construct examples of topological Kac-Moody groups which are not rational t.d.l.c. duality groups (cf. Remark 6.13). We close our investigations with a short discussion of the Euler-Poincaré characteristic χ(G) one may define for a unimodular t.d.l.c. group G of type FP. The value χ(G) is a rational multiple of a left invariant Haar measure µO , µO (O) = 1, for some compact open subgroup O of G. Indeed, for a discrete group Π of type FP it is straightforward to verify that χ(Π) = χΠ · µ{1} , where χΠ denotes the classical Euler-Poincaré characteristic (cf. [14, §XI.7]). Although we present some explicit calculations in section 7.2, we are still very far away from understanding the generic behaviour of the Euler-Poincaré characteristic. We conjecture that a unimodular compactly generated t.d.l.c. group G satisfying cdQ (G) = 1 should have non-positive Euler-Poincaré characteristic (cf. Question 6). An affirmative answer to this question would resolve the accessibility problem, and thus would yield a complete description of unimodular compactly generated t.d.l.c. groups of rational discrete cohomological dimension 1. Acknowledgements: Sections 2, 3, 4 and 5 are part of the first author’s PhD thesis (cf. [20]). The authors would like to thank M. Bridson for some helpful discussions concerning CAT(0)-spaces. 2. Discrete cohomology 2.1. Topological groups of type OS. Let G be a topological group of type OS. If H is a closed subgroup of G, then { O ∩ H \| O ∈ Osgrp(G) } is a basis of neighbourhood of 1 in H. Hence H is also of type OS. Moreover, if π : G → Ḡ is a continuous, open, surjective homomorphism of topological groups, then { π(O) \| O ∈ Osgrp(G) } is a basis of neighbourhoods of 1 in Ḡ. Thus Ḡ is also of type OS. Finally, if G is homeomorphic to an open subgroup of a topological group Y , then Y is of type OS as well. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 5 Any compact totally-disconnected topological group is profinite and therefore, by definition, of type OS (cf. [42, Prop. 1.1.0]). By van Dantzig’s theorem (cf. [48]), every t.d.l.c. group contains a compact open subgroup. Hence from the previously mentioned properties one conlcudes the following. Fact 2.1. Any totally-disconnected locally-compact group is of type OS. 2.2. Discrete modules of topological groups of type OS. Let R be a commutative ring with 1 ∈ R. For a group G we denote by R[G] its R-group algebra, and by R[G] mod the abelian category of left R[G]-modules. Let G be a topological group of type OS. For M ∈ ob(R[G] mod) the subset (2.1) d(M ) = { m ∈ M \| stabG (m) open in G }, where stabG (m) = { g ∈ G \| g · m = m }, is an R[G]-submodule of M , the largest discrete R[G]-submodule of M . Here we call a left R[G]-module B discrete, if the map · : G × B → B is continuous, where B carries the discrete topology. If φ : B → M is a morphism of left R[G]-modules and B is discrete, one concludes that im(φ) ⊆ d(M ). This has the following consequences (cf. [49, Prop. 2.3.10]). Fact 2.2. Let G be a topological group of type OS, and let R[G] dis denote the full subcategory of R[G] mod the objects of which are discrete left R[G]-modules. Then (a) R[G] dis is an abelian category; (b) d : R[G] mod → R[G] dis is a covariant additive left exact functor, which is the right-adjoint of the forgetful functor f : R[G] dis → R[G] mod; (c) d is mapping injectives to injectives; (d) R[G] dis has enough injectives. The abelian category R[G] mod admits minimal injective envelopes, i.e., for M ∈ ob(R[G] mod) the maximal essential extension iM : M → I(M ) is a minimal injective envelope (cf. [31, §III.11]). Thus, if M ∈ ob(R[G] dis), then d(iM ) : M → d(I(M )) is a minimal injective envelope in R[G] dis. This has the following consequence. Fact 2.3. The abelian category R[G] dis admits minimal injective resolutions. If M and N are left R[G]-modules, then M ⊗ N = M ⊗R N is again a left R[G]-module. Moreover, if G is a topological group of type OS and M and N are discrete left R[G]-modules, then M ⊗ N is discrete as well. 2.3. Closed subgroups. Let H be a closed subgroup of the topological group G of type OS. Then one has a restriction functor (2.2) resG H( ): R[G] dis −→ R[H] dis which is exact. The discrete coinduction functor (2.3) G dcoindG H ( ) = d ◦ coindH ( ) : R[H] dis −→ R[G] dis where coindG H (M ) = HomR[H] (R[G], M ), M ∈ ob(R[H] mod), is the coinduction functor, is left exact. It is the right-adjoint of the restriction functor. In particular, dcoindG H ( ) is mapping injectives to injectives (cf. [49, Prop. 2.3.10]). Let N be a closed normal subgroup of the topological group G of type OS. The canonical projection π : G → Ḡ, where Ḡ = G/N , is surjective, continuous and open. One has an inflation functor (2.4) inf G Ḡ ( ) : R[Ḡ] dis −→ R[G] dis 6 I. CASTELLANO AND TH. WEIGEL which is exact. The inflation functor is left-adjoint to the N -fixed point functor (2.5) N : R[G] dis −→ R[Ḡ] dis which is left exact. In particular, one has the following (cf. [49, Prop. 2.3.10]). Fact 2.4. Let G be a topological group of type OS, and let N be a closed normal subgroup of G. Then the N -fixed point functor N : R[G] dis −→ R[Ḡ] dis, Ḡ = G/N , is mapping injectives to injectives. 2.4. Open subgroups. Let H ⊆ G be an open subgroup P of the topological group G of type OS, and let M ∈ ob(R[H] dis). Then for y = 1≤i≤n ri gi ⊗mi ∈ indG H (M ) T one has 1≤i≤n g stabH (mi ) ⊆ stabG (y). In particular, M is a discrete left R[G]module, and one has an exact induction functor (2.6) indG H( ): R[H] dis −→ R[G] dis. which is the left adjoint of the restriction functor resG H ( ). Hence one has the following (cf. [49, Prop. 2.3.10]). Fact 2.5. Let H ⊆ G be an open subgroup of the topological group G of type OS. Then resG H ( ) is mapping injectives to injectives. 2.5. Discrete cohomology. Let G be a topological group of type OS. For M ∈ ob(R[G] dis) we denote by (2.7) dExtkR[G] (M, ) = Rk HomR[G] dis (M, ) the right derived functors of HomR[G] (M, ) in bifunctors (2.8) dExtkR[G] ( , ) : op R[G] dis R[G] dis. The contra-/covariant × R[G] dis −→ R mod, k ≥ 0, admit long exact sequences in the first and in the second argument. Moreover, dExt0R[G] (M, ) ≃ HomR[G] (M, ), and dExtkR[G] (M, I) = 0 for every injective, discrete left R[G]-module I. We define the k th discrete cohomology group of G with coefficients in R[G] dis by (2.9) dHk (R[G], ) = dExtkR[G] (R, ), k ≥ 0, where R denotes the trivial left R[G]-module. Example 2.6. Let R = Z. (a) If G is a discrete group, G is of type OS and Z[G] dis = Z[G] mod. In particular, dH• (Z[G], ) = H • (G, ) coincides with ordinary group cohomology (cf. [14]). (b) If G is a profinite group, G is of type OS and Z[G] dis coincides with the abelian category of discrete G-modules. Thus dH• (Z[G], ) = H • (G, ) coincides with the Galois cohomology groups discussed in [42]. 2.6. The restriction functor. Let H ⊆ G be a closed subgroup of the topological group G of type OS, and let M ∈ ob(R[G] dis). Let (I • , ∂ • , µG ) be an injective resolution of M in ob(R[G] dis), and let (J • , δ • , µH ) be an injective resolution of resG H (M ) in R[H] dis. By the comparison theorem in homological algebra, one has a RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 7 • • morphism of cochain complexes of left R[H]-modules φ• : resG H (I ) → J such that the diagram (2.10) resG H (M ) µG / resG (I 0 ) H ∂0 / resG (I 1 ) H φ0 resG H (M ) / J0 µH ∂1 / resG (I 2 ) H φ1 δ 0 / J1 ∂2 / ... φ2 δ / J2 1 δ2 / ... commutes. Moreover, φ• is unique up to chain homotopy equivalence with respect to this property. Let φ̃k : (I k )G (2.11) / (I k )H (φk )H / (J k )H . Then φ̃• : ((I • )G , ∂˜• ) → (J • )H , δ̃ • ) is a mapping of cochain complexes of R-modules. Applying the cohomology functor yields a mapping of cohomological functors (2.12) res•G,H (M ) : dH• (R[G], M ) −→ dH• (R[H], resG H (M )). The map res•G,H ( ) will be called the restriction functor in discrete cohomology. 2.7. The inflation functor. Let N be a closed normal subgroup of the topological group of type OS, and let π : G → Ḡ, Ḡ = G/N , denote the canonical projection. Let M ∈ ob(R[Ḡ] dis), and let (I • , ∂ • , µḠ ) be an injective resolution of M in R[Ḡ] dis. Let (J • , δ • , µG ) be an injective resolution of inf G Ḡ (M ) in R[G] dis. By the comparison theorem in homological algebra, one has a mapping of cochain complexes of left • • R[G]-modules φ• : inf G Ḡ (I ) → J which is unique up to chain homotopy equivalence such that the diagram (2.13) inf G Ḡ (M ) inf G (µḠ ) Ḡ / inf G (I 0 ) Ḡ ∂0 φ0 inf G Ḡ (M ) µG / J0 / inf G (I 1 ) Ḡ φ1 δ0 / J1 commutes. Applying first the fixed point functor functor one obtains a functor (2.14) ∂1 / inf G (I 2 ) Ḡ ∂2 / ... φ2 δ1 G / J2 δ2 / ... and then the cohomology inf •Ḡ,G (M ) : dH• (R[Ḡ], M ) −→ dH• (R[G], inf G Ḡ (M )), which will be called the inflation functor in discrete cohomology. 2.8. The Hochschild - Lyndon - Serre spectral sequence. Let N be a closed normal subgroup of the topological group G of type OS, and let π : G → Ḡ, Ḡ = G/N , denote the canonical projection. The functor G : R[G] dis −→ R mod can be decomposed by G = ( Ḡ ) ◦ ( N ). Moreover, N : R[G] dis −→ R[Ḡ] dis is mapping injectives to injectives (cf. Fact 2.4). Thus for M ∈ ob(R[G] dis) the Grothendieck spectral sequence (cf. [49, Thm. 5.8.3]) yields a Hochschild-Lyndon-Serre spectral sequence (2.15) s+t E2s,t (M ) = dHs (R[Ḡ], dHt (R[N ], resG (R[G], M ) N (M ))) =⇒ dH 8 I. CASTELLANO AND TH. WEIGEL of cohomological type concentrated in the first quadrant which is functorial in M . The edge homomorphisms are given by esh : dHs (R[Ḡ], M N ) (2.16) inf sḠ,G (M N ) 0,t etv : E∞ (M ) / E s,0 (M ) 2 / E 0,t (M ) 2 restG,N (M)N s,0 / E∞ (M ), / dHt (R[N ], M )G/N , and one has a 5-term exact sequence (2.17) 0 / dH1 (R[Ḡ], M N ) e1h / dH1 (R[G], M ) e1v / dH1 (R[N ], M )G/N d0,1 2 dH2 (R[G], M ) o e2h dH2 (R[Ḡ], M N ) 2.9. An Eckmann-Shapiro type lemma. Let H be an open subgroup of the topological group G of type OS, and let M ∈ ob(R[G] dis). Let (I • , ∂ • , µG ) be • G • G an injective resolution of M in R[G] dis. Then (resG H (I ), resH (∂ ), resH (µG )) is an G injective resolution of resH (M ) in R[H] dis (cf. Fact 2.5). For B ∈ ob(R[H] dis) one has natural isomorphisms k G k HomR[G] (indG H (B), I ) ≃ HomR[H] (B, resH (I )) (2.18) k ≥ 0, which yield isomorphisms k G dExtkR[G] (indG H (B), M ) ≃ dExtR[H] (B, resH (M )), (2.19) k ≥ 0. In particular, for B = R one obtains natural isomorphisms k εk : dExtkR[G] (indG H (R), ) −→ dH (R[H], ) (2.20) such that (2.21) dExt•R[G] (indG H (R), ) ❚❚❚❚ ❧5 ❧ ❧ ❚❚❚ε❚• ❧❧❧ ❧ ❚❚❚❚ ❧ ❧ ❧ ❚❚❚) ❧ • ❧❧❧ resG,H ( ) • / dExtR[G] (R, ) dH• (R[H], resG H ( )) dExt• R[G] (ε) is a commutative diagram of cohomological functors with values in ε : indG H (R) → R is the canonical map. R[G] dis, where 3. Rational discrete cohomology for t.d.l.c. groups Throughout this section we will assume that G is a t.d.l.c. group. Furthermore, we choose R = Q to be the field of rational numbers. From now on we will omit the appearance of the field Q in the notation, i.e., we put dH• (G, ) = dH• (Q[G], ), dExt•Q[G] ( , ) = dExt•G ( , ), etc. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 9 3.1. Profinite groups. It is well known that a t.d.l.c. group is compact if, and only if, it is profinite (cf. [42, Prop. 1.1.0]). The main goal of this subsection is to establish the following property which will turn out to be essential for our purpose. Proposition 3.1. Let G be a profinite group. Then every discrete left Q[G]-module M is injective and projective in Q[G] dis. Proof. The proposition is a consequence of several facts. Claim 3.1.1. One has dHk (G, M ) = 0 for every M ∈ ob(Q[G] dis) and k ≥ 1. In particular, every finite dimensional discrete left Q[G]-module is projective. Proof of Claim 3.1.1. By [42, §2.2, Prop. 8], one has (3.1) dHk (G, M ) = limU⊳ G dHk (G/U, M U ), −→ ◦ where the projective limit is running over all open normal subgroups U of G. By Maschke’s theorem, one has dHk (G/U, M U ) = 0 for k ≥ 1, and thus, by (3.1), dHk (G, M ) = 0 for all k ≥ 1. Let U be an open subgroup of G. Since dExt1U (Q, ) = 0, HomU (Q, ) : Q[U] dis → Q mod is an exact functor. In particuG lar, Q ∈ ob(Q[U] dis) is projective. Thus, as resG U ( ) is exact, Q[G/U ] ≃ indU (Q) is projective for every open subgroup U of G (cf. [49, Prop. 2.3.10]). By Maschke’s theorem, every finite dimensional discrete left Q[G]-module is a direct summand of Q[G/U ] for some open subgroup U of G. This yields the claim. As a consequence one obtains the following. Claim 3.1.2. Let S ∈ ob(Q[G] dis) be irreducible, and let M ∈ ob(Q[G] dis). Then dExtkG (S, M ) = 0 for all k ≥ 1. Proof of Claim 3.1.2. Note that every irreducible discrete left Q[G]-module S is of finite dimension, and therefore projective (cf. Claim 3.1.1). For a discrete left Q[G]-module M we denote by soc(M ) ⊆ M the socle of M , i.e., soc(M ) is the sum of all irreducible left Q[G]-submodules of M . Moreover, as m ∈ M is contained in the finite dimensional discrete left Q[G]-module Q[G] · m, one has that soc(M ) = 0 if, and only if, M = 0. Claim 3.1.3. Every discrete left Q[G]-module M is injective in ob(Q[G] dis). Proof of Claim 3.1.3. Let (I • , δ • ) be a minimal injective resolution of M in Q[G] dis (cf. Fact 2.3). By Claim 3.1.2, for any irreducible S ∈ ob(Q[G] dis) one has for k ≥ 1 that 0 = dExtkG (S, M ) = HomG (S, I k ) (cf. [6, Cor. 2.5.4(ii)]). In particular, soc(I k ) = 0, and therefore I k = 0. Hence M is injective. Claim 3.1.4. The category Q[G] dis has enough projectives. Proof of Claim 3.1.4. Let M ∈ ob(Q[G] dis). By definition, for m ∈ M the left Q[G]submodule Q[G] · m ⊆ M is finite dimensional and discrete. Thus, by Claim 3.1.1, ` Q[G] · m is projective in Q[G] dis, and therefore, P = m∈M Q[G] · m is projective in Q[G] dis. The canonical map π : P → M is surjective, and this yields the claim. Claim 3.1.5. Every discrete left Q[G]-module is projective in Q[G] dis. 10 I. CASTELLANO AND TH. WEIGEL Proof of Claim 3.1.5. Let M ∈ ob(Q[G] dis). By Claim 3.1.4, there exist a projective discrete left Q[G]-module P and a map π : P → M such that (3.2) 0 / ker(π) /P π /M /0 is a short exact sequence in Q[G] dis. By Claim 3.1.3, ker(π) is injective. Hence (3.2) splits, and M is isomorphic to a direct summand of P , i.e., M is projective. 3.2. Discrete permutation modules. Let G be a t.d.l.c. group. A left G-set Ω is said to be discrete, if stabG (ω) is an open subgroup of G for any ω ∈ Ω. In this case · : G × Ω → Ω is continuous, where Ω is considered to be a discrete topological space and G × Ω carries the product topology. We say that Ω is a discrete left G-set with compact stabilizers, if stabG (ω) is a compact and open subgroup for any ω ∈ Ω. Let Ω be a left G-set. Then Q[Ω] - the free Q-vector space over the set Ω carries canonically the structure of left Q[G]-module. Moreover, Ω is a discrete left G-set if, and only if, Q[Ω] is a discrete left Q[G]-module. For a discrete left G-set Ω, Q[Ω] is also called a discrete left Q[G]-permutation module. Additionally, one has the following property. Proposition 3.2. Let G be a t.d.l.c. group, and let Ω be a discrete left G-set with compact stabilizers. Then Q[Ω] is projective in Q[G] dis. In particular, the abelian category Q[G] dis has enough projectives. F Proof. Let Ω = i∈I Ωi be the decomposition of Ω into G-orbits. Since Q[Ω] = ` i∈I Q[Ωi ] and as the coproduct of projective objects remains projective, it suffices to prove the claim for transitive discrete left G-sets Ω with compact stabilizers, i.e., we may assume that for ω ∈ Ω one has an isomorphism of left G-sets Ω ≃ G/O, and an isomorphism of left Q[G]-modules Q[Ω] ≃ indG O (Q), where O = stabG (ω). The induction functor is left adjoint to the restriction functor which is exact. Since the trivial left Q[O]-module is projective in Q[O] dis (cf. Prop. 3.1), and as indG O ( ) is mapping projectives to projectives (cf. §2.4 and [49, Prop. 2.3.10]), one concludes that Q[Ω] is projective. Let M ∈ ob(Q[G] dis). For each element m in M let Om be a compact open subgroup of stabG (m). Then πm : indG Om (Q) → M given by πm (g ⊗ 1) = g · m is a mapping of left Q[G]-modules. It is straightforward to verify that ` ` (3.3) π = m∈M πm : m∈M indG Om (Q) −→ M ` is F a surjective mapping of discrete left Q[G]-modules, and m∈M indG Om (Q) ≃ Q[ m∈M G/Om ] is projective. It is somehow astonishing that for a t.d.l.c. group G, the category Q[G] dis is a full subcategory of Q[G] mod with enough projectives and enough injectives (cf. Fact 2.2), but the reader may have noticed that this is a direct consequence of Proposition 3.1. Corollary 3.3. Let G be a t.d.l.c. group. A discrete left Q[G]-module M is projective if, and only if, it is a direct summand of a discrete left Q[G]-permutation module Q[Ω] for some discrete left G-set Ω with compact stabilizers. From Proposition 3.2 and Corollary 3.3 one deduces the following properties. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 11 Proposition 3.4. Let G be a t.d.l.c. group and, let H ⊆ G be a closed subgroup of G. Then resG H : Q[G] dis → Q[H] dis is mapping projectives to projectives. Moreover, if H is open in G, then indG H : Q[H] dis → Q[G] dis is mapping projectives to projectives. Proof. By Corollary 3.3, it suffices to prove that resG H (Q[Ω]) is projective for any discrete left Q[G]-permutation with compact stabilizers. Since resG H ( ) commutes with coproducts, we may also assume that Ω is a transitive discrete left G-set with compact stabilizers, i.e., for ω ∈ Ω and O = stabG (ω) one has Ω ≃ G/O.FLet R ⊆ G be a set of representatives of the (H, O)-double cosets in G, i.e., G = r∈R H r O. Then ` r (3.4) resG H (Q[G/O]) ≃ r∈R Q[H/(H ∩ O)]. Since H is closed and O compact and open, H ∩ r O is compact and open in H. Hence, by Proposition 3.2, resG H (Q[G/O]) is a projective discrete left Q[H]-module. Assume that H is open in G. Since indG H ( ) : Q[H] dis → Q[G] dis is the left adjoint G of the exact functor resG ( ) : dis → Q[G] Q[H] dis, indH ( ) is mapping projectives H to projectives (cf. [49, Prop. 2.3.10]). We close this subsection with the following characterization of compact t.d.l.c. groups. Proposition 3.5. Let G be a t.d.l.c. group, and let O be a compact open subgroup of G. Then HomG (Q, Q[G/O]) 6= 0 if, and only if, G is compact. Moreover, if G is compact, then HomG (Q, Q[G/O]) ≃ Q. Proof. If G is not compact, then \|G : O\| = ∞. Hence HomG (Q, Q[G/O]) = 0 (cf. [14, §III.5, Ex. 4]). If G is compact, then \|G : O\| < ∞, and Q[G/O] coincides with the coinduced module coindG O (Q) (cf. [14, §III, Prop. 5.9]). Hence HomG (Q, Q[G/O]) ≃ HomO (Q, Q) ≃ Q. This yields the claim. 3.3. Signed rational discrete permutation modules. For some application it will be useful to consider signed discrete permutation modules. Let (Ω, ¯ , ·) be a signed discrete left G-set, i.e., (Ω, ¯) is a signed set (cf. §A.5), and (Ω, ·) is a left G-set satisfying g · ω = g · ω̄, (3.5) for all g ∈ G and ω ∈ Ω. Then (cf. (A.16)) (3.6) Q[Ω] = Q[Ω]/spanQ { ω + ω̄ \| ω ∈ Ω } is a discreteF left Q[G]-module. For ω ∈ Ω let ω denote its canonical image in Q[Ω]. Let Ω = j∈J Ωr be a decomposition of Ω in G × C2 -orbits, where C2 = Z/2Z and σ = 1 + 2Z ∈ C2 acts by application of ¯ . We say that G acts without inversion on Ωj , if g · ω 6= ω for all g ∈ G and ω ∈ Ωj , otherwise we say that G acts with inversion. In this case, there exists g ∈ G and ω ∈ Ωj such that g ·ω = ω̄. Moreover, the action of G±ω = stabG ({ω, ω̄}) on Q · ω ⊆ Q[Ω] is given by a sign character sgnω : G±ω −→ {±1}. (3.7) Moreover, by choosing representatives of the G × C2 -orbits, one obtains an isomorphism of discrete left Q[G]-modules a a (3.8) Q[Ω] ≃ indG indG G±ω (Q(sgnω )) ⊕ G̟ (Q)). ω∈Rw ̟∈Rwo 12 I. CASTELLANO AND TH. WEIGEL Note that by the ± -construction (cf. §A.5) any rational discrete permutation module Q[Ω] can be considered as the signed rational discrete permutation module Q[Ω± ]. 3.4. The rational discrete cohomological dimension. Let M be a discrete left Q[G]-module for a t.d.l.c. group G. Then M is said to have projective dimension less or equal to n ≤ ∞, if M admits a projective resolution (P• , ∂• , ε) in Q[G] dis satisfying Pk = 0 for k > n, i.e., the sequence (3.9) / Pn 0 ∂n ∂n−1 / Pn−1 / ... ∂2 / P1 ∂1 / P0 ε /M /0 is exact. The projective dimension projQ dim(M ) ∈ N0 ∪ {∞} of M ∈ ob(Q[G] dis) is defined to be the minimum of such n ≤ ∞. As in the discrete case one has the following property. Lemma 3.6. Let G be a t.d.l.c. group, and let M be a discrete left Q[G]-module. Then the following are equivalent. (i) projQ dim(M ) ≤ n; (ii) dExti (M, ) = 0 for all i > n; (iii) dExtn+1 (M, ) = 0; (iv) In any exact sequence (3.10) 0 in /K Q[G] dis / Pn−1 ∂n−1 / ... ∂2 / P1 ∂1 / P0 ε /M /0 with Pi , 0 ≤ i ≤ n − 1, projective, one has that K is projective. Proof. The proof of [14, Chap. VIII, Lemma 2.1] can be transferred verbatim. For a t.d.l.c. group G (3.11) cdQ (G) = projQ dim(Q) will be called the rational discrete cohomological dimension of G, i.e., one has cdQ (G) =projQ dim(Q) (3.12) = min({ n ∈ N0 \| dHi (G, ) = 0 for all i > n } ∪ {∞}) = sup{ n ∈ N0 \| dHn (G, M ) 6= 0 for some M ∈ ob(Q[G] dis) }. Proposition 3.7. Let G be a t.d.l.c. group. (a) One has cdQ (G) = 0 if, and only if, G is compact. (b) If N is a closed normal subgroup of G, then (3.13) cdQ (G) ≤ cdQ (N ) + cdQ (G/N ). (c) If H is a closed subgroup of G, then (3.14) cdQ (H) ≤ cdQ (G). In particular, if H is normal and co-compact, then equality holds in (3.14). Proof. (a) If G is compact, then G is profinite, and thus, by Proposition 3.1, cdQ (G) = projQ dim(Q) = 0. Suppose that G satisfies cdQ (G) = 0, i.e., Q ∈ ob(Q[G] dis) is projective. By van Dantzig’s theorem, there exists a compact open subgroup O of G. By hypothesis, the canonical projection ε : Q[G/O] → Q admits a section. In particular, HomG (Q, Q[G/O]) 6= 0. Hence one concludes that RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 13 \|G : O\| < ∞, and G is compact. (b) is a direct consequence of the HochschildLyndon-Serre spectral sequence (cf. (2.15)) and (3.12). (c) By Proposition 3.4, the restriction of a projection resultion of Q in Q[G] dis is a projective resolution of Q in Q[H] dis showing the inequality (3.14). The final remark is a consequence of (3.13), (3.14) and part (a). Remark 3.8. Let G be a t.d.l.c. group, and let H ⊆ G be such that \|G : H\| < ∞. One concludes from Proposition 3.7(c) that cdQ (H) = cdQ (G). However, the following question will remain unanswered in this paper. Question 1. Suppose that H is a closed co-compact subgroup of the t.d.l.c. group G. Is it true that cdQ (H) = cdQ (G)? Remark 3.9. Proposition 3.7(c) implies that for any discrete group D of a t.d.l.c. group G one has (3.15) cdQ (D) ≤ cdQ (G). 3.5. The flat rank of a t.d.l.c. group. Let G be a t.d.l.c. group, and let H be a closed subgroup of G. Then H is called a flat subgroup of G, if there exists a compact open subgroup O of G which is tidy for all h ∈ H (cf. [5, §2, Def. 1]). For such a subgroup H(1) = { h ∈ H \| s(h) = 1 } is a normal subgroup, and H/H(1) is a free abelian group over some set IH (cf. [51, Cor. 6.15]). Moreover, rk(H) = card(IH ) ∈ N ∪ {∞} is called the rank of the flat subgroup H. The flat rank frk(G) of G is defined by (3.16) frk(G) = sup({ rk(H) \| H a flat subgroup of G } ∪ {0}) ∈ N0 ∪ {∞} (cf. [4, §1.3]). We call the t.d.l.c. group G to be N -compact, if for any compact open subgroup O of G, the open subgroup NG (O) is also compact. One has the following property. Proposition 3.10. Let G be an N -compact t.d.l.c. group. (a) Let H be a flat subgroup of G. Then rk(H) = cdQ (H). (b) frk(G) ≤ cdQ (G). Proof. (a) For any abelian group A which is a free over a set IA one has (3.17) cdQ (A) = cdZ (A) = card(IA ) ∈ N0 ∪ {∞}. Let O be a compact open subgroup such that O is tidy for all h ∈ H. By hypothesis, NG (O) is compact, and therefore C = H ∩ NG (O) is compact. Then, by Proposition 3.7(a) and the Hochschild-Lyndon-Serre spectral sequence (cf. (2.15)), one has natural isomorphisms dHk (H, ) ≃ dHk (H/C, C ) for all k > 0. Hence, by (3.17), cdQ (H) = cdQ (H/C) = rk(H). (b) is a direct consequence of (a). Remark 3.11. (Neretin’s group of spheromorphisms) Let Td+1 be a d+1-regular tree, and let G = Hier◦ (Td+1 ) = Nd+1,d be the group of all almost automorphisms (or spheromorphisms) of Td+1 introduced by Y. Neretin in [37]. It is well known that G is a compactly generated simple (in particular topologically simple) t.d.l.c. group. The group G contains a discrete subgroup D which is isomorphic to the HigmanThompson group Fd+1,d . From this fact one concludes that G contains discrete subgroups isomorphic to Zn for all n ≥ 1. Hence, by Remark 3.9, cdQ (G) = ∞. 14 I. CASTELLANO AND TH. WEIGEL 3.6. Totally disconnected locally compact groups of type FP∞ . Let G be a t.d.l.c. group, and let M be a discrete left Q[G]-module. Then M is said to be finitely generated, if there exist ` compact open subgroups O1 , . . . , On of G and a surjective homomorphism π : 1≤j≤n Q[G/Oj ] → M . A partial projective resolution (3.18) Pn ∂n / Pn−1 ∂n−1 / ... ∂2 / P1 ∂1 ε / P0 /M /0 of M will be called to be of finite type, if Pj is finitely generated for all 0 ≤ j ≤ n. One has the following property. Proposition 3.12. Let G be a t.d.l.c. group, and let M be a discrete left Q[G]module. Then the following are equivalent: (i) There is a partial projective resolution (Pj , ∂j , ε)0≤j≤n of finite type of M in Q[G] dis; (ii) the discrete left Q[G]-module M is finitely generated and for every partial projective resolution of finite type (3.19) Qk in Q[G] dis ðk / Qk−1 ðk−1 / ... ð2 / Q1 ð1 / Q0 ε′ /M /0 with k < n, ker(ðk ) is finitely generated. Proof. The proposition is a direct consequence of generalized form of Schanuel’s lemma (cf. [14, Chap. VIII, Lemma 4.2 and Prop. 4.3]). The discrete left Q[G]-module M will be said to be of type FPn , n ≥ 0, if M satisfies either of the hypothesis (i) or (ii) in Proposition 3.12, i.e., M is of type FP0 if, and only if, M is finitely generated. If M is of type FPn for all n ≥ 0, then M will be said to be of type FP∞ . The t.d.l.c. group G will be called to be of type FPn , n ∈ N ∪ {∞}, if the trivial left Q[G]-module Q is of type FPn . 4. Rational discrete homology for t.d.l.c. groups β α For a profinite group O any short exact sequence 0 → L → M → N → 0 of discrete left Q[O]-modules splits (cf. Prop. 3.1). Hence, if Q ∈ ob(disQ[O] ) denotes the trivial right Q[O]-module, the sequence (4.1) 0 / Q ⊗O L idQ ⊗α / Q ⊗O M idQ ⊗β / Q ⊗O N /0 is exact. Let G be a t.d.l.c. group, let O ⊆ G be a compact open subgroup, and β α let 0 → L → M → N → 0 be a short exact sequence of discrete left Q[G]-modules. Since indG O ( ) : disQ[O] → disQ[G] is an exact functor, and as indG O (Q) ⊗G (4.2) ≃ Q ⊗O resG O( ) : Q[G] dis −→ Q mod are naturally isomorphic additive functors, the sequence (4.3) 0 / indG O (Q) ⊗G L id⊗α / indG O (Q) ⊗G M id⊗β / indG O (Q) ⊗G N /0 : Q[G] dis → Q mod is an exact functor. Moreover, as is exact, i.e., indG O (Q) ⊗G ⊗G commutes with direct limits in the first and in the second argument (cf. [14, §VIII.4, p. 195]), the sequence (4.4) 0 / Q ⊗G L idQ ⊗α / Q ⊗G M idQ ⊗β / Q ⊗G N /0 RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 15 is exact for any projective discrete right Q[G]-module Q (cf. Cor. 3.3), i.e., Q is flat. For a discrete right Q[G]-module C, we denote by dTorG k (C, ) : (4.5) Q[G] dis −→ Q mod the left derived functors of the right exact functor C ⊗G define the rational discrete homology of G by : Q[G] dis → Q mod, and dHk (G, ) = dTorG k (Q, ). (4.6) By definition, one has the following properties. Proposition 4.1. Let G be a t.d.l.c. group, let C be a discrete right Q[G]-module, and let M be a discrete left Q[G]-module. Then (a) dHk (G, M ) = 0 for all k > cdQ (G); (b) if P ∈ ob(Q[G] dis) is projective, one has dTorG k (C, P ) = 0 for all k ≥ 1; (c) if G is compact, one has dTorG (C, M ) = 0 for all k ≥ 1. k Proof. (a) is a direct consequence of the definition of cdQ (G) (cf. (3.11)). (b) Using the previously mentioned arguments one concludes that ⊗G P : disQ[G] → Q mod is an exact functor. This property implies that for the left derived functors LM k of G M P ⊗G M one has that dTork (C, M ) ≃ Lk (C). Since P is projective, Lk = 0 for all k > 1. This yields the claim. (c) is a direct consequence of Proposition 3.1. 4.1. Invariants and coinvariants. Let O be a profinite group, and let M be a discrete left Q[O]-module. The Q-vector space M O = { m ∈ M \| g · m = m for all g ∈ O } ⊆ M (4.7) is called the space of O-invariants of M , and MO = Q ⊗O M, (4.8) is called the space of O-coinvariants of M . By definition, one has a canonical map φM,O : M O −→ MO (4.9) given by φM,O (m) = 1 ⊗O m for m ∈ M . One has the following. Proposition 4.2. Let O be a profinite group, and let M be a discrete left Q[O]module. Then φM,O is an isomorphism. Moreover, if U ⊆ O is an open subgroup, one has commutative diagrams MO (4.10) φM,O ρO,U MU where φM,U MO O / MO φM,O / MO O λU,O / MU MU φM,U / MU X 1 r · m, \|O : U \| r∈R X 1 ρO,U (1 ⊗O n) = 1 ⊗U r−1 · n, \|O : U \| λU,O (m) = (4.11) (4.12) r∈R U m ∈ M , n ∈ N , and R ⊆ O is any set of coset representatives of O/U . 16 I. CASTELLANO AND TH. WEIGEL Proof. The space of O-invariants M O ⊆ M is a discrete left Q[O]-submodule of M . Hence, by Claim 3.1.3, M = M O ⊕ C for some discrete left Q[O]-submodule C of M . Since C is a discrete left Q[O]-module, one has C = limi∈I Ci , where Ci are −→ finite-dimensional discrete left Q[G]-submodules of C. By construction, CiO = 0, and thus, by Maschke’s theorem, (Ci )O = 0. Since Q ⊗O commutes with direct limits, this implies that CO = 0. Hence C is contained in the kernel of the canonical map τM : M → MO . Let φM,O = (τM )\|M O denote the restriction of τM to the Q[O]-submodule M O . Since dH1 (O, ) = 0 (cf. Prop 3.7(a), Prop. 4.1(a)), the long exact sequence for dH• (O, ) implies that the map (M O )O → MO is injective. As τM O : M O → (M O )O is a bijection, the commutativity of the diagram /M MO ◆ ◆◆◆ ◆◆φ◆M,O τM O τM ◆◆◆ ◆◆& / MO (M O )O (4.13) yields that φM,O is injective. As τM is surjective and C ⊆ ker(τM ), one concludes that φM,O is bijective, and C = ker(τM ). The commutativity of the diagrams (4.10) can be verified by a straightforward calculation. 4.2. The rational discrete standard bimodule Bi(G). For a t.d.l.c. group G the set of compact open subgroups CO(G) of G with the inclusion relation ”⊆” is a directed set, i.e., (CO(G), ⊆) is a partially ordered set, and for U, V ∈ CO(G) one has also U ∩ V ∈ CO(G). For U, V ∈ CO(G), V ⊆ U , one has an injective mapping of discrete left Q[U ]-modules X 1 r V, (4.14) η̃U,V : Q −→ Q[U/V ], η̃U,V (1) = \|U : V \| r∈R G U where R ⊂ U is any set of coset representatives of U/V . As indG V = indU ◦ indV , η̃U,V induces an injective mapping X 1 x r V, x ∈ G. (4.15) ηU,V : Q[G/U ] −→ Q[G/V ], ηU,V (x U ) = \|U : V \| r∈R By construction, one has for W ∈ CO(G), W ⊆ V ⊆ U , that ηU,W = ηV,W ◦ ηU,V . For g ∈ G one has an isomorphism of discrete left Q[G]-modules cU,g : Q[G/U ] −→ Q[G/U g ], (4.16) g where U = g −1 cU,g (x U ) = x g U g , x ∈ G, U g. Moreover, for g, h ∈ G and U, V ∈ CO(G), V ⊆ U , one has (4.17) cV,g ◦ ηU,V = ηU g ,V g ◦ cU,g , (4.18) cU g ,h ◦ cU,g = cU,gh . Let Bi[G] = limU∈CO(G) (Q[G/U ], ηU,V ). Then, by definition, Bi[G] is a discrete left −→ Q[G]-module. By (4.17), the assignment (4.19) (xU ) · g = cU,g (x U ) = xg U g , g, x ∈ G, U ∈ CO(G), defines a Q-linear map · : Bi(G) × Q[G] → Bi(G), and, by (4.18), this map defines a right Q[G]-module structure on Bi(G). By (4.19), Bi(G) is a discrete right Q[G]-module making Bi(G) a rational discrete Q[G]-bimodule. Therefore, we RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 17 will call Bi(G) the rational discrete standard bimodule of G. It has the following straightforward properties. Proposition 4.3. Let G be a t.d.l.c. group. (a) Bi(G) is a flat rational discrete right Q[G]-module, and a flat rational discrete left Q[G]-module, i.e., (4.20) G dTorG k (A, Bi(G)) = dTork (Bi(G), B) = 0 for all A ∈ ob(disQ[G] ), B ∈ ob(Q[G] dis) and k ≥ 1. (b) One has ( Q if G is compact, (4.21) HomG (Q, Bi(G)) ≃ 0 if G is not compact. Proof. (a) The direct limit of flat objects is flat (cf. [14, §VIII.4, p. 195]). Thus, by Proposition 4.1(b), Bi(G) is a flat rational discrete left Q[G]-module. A similar argument shows that Bi(G) is also a flat rational discrete right Q[G]-module. (b) is a direct consequence of Proposition 3.5. Let M ∈ ob(Q[G] dis) and U ∈ CO(G). The composition of maps (cf. Prop. 4.2) (4.22) θM,U : Q[U \G] ⊗G M ξM,U / Q ⊗U M φ−1 M,U / MU , where ξM,U (U g ⊗G m) = 1 ⊗U gm, g ∈ G, m ∈ M , is an isomorphism. Moreover, if V ∈ CO(G), V ⊆ U , one has by (4.10) a commutative diagram (4.23) Q[U \G] ⊗G M θM,U / MU νU,V ⊗G idM Q[V \G] ⊗G M / MV θM,V which is natural in M , where νU,V : Q[U \G] → Q[V \G] is given by X 1 V r−1 x, (4.24) νU,V (U x) = \|U : V \| r∈R x ∈ G, and R ⊆ U is a set of coset representatives of U/V , i.e., U = Thus the family of maps (θM,U )U∈CO(G) induces an isomorphism (4.25) θM : Bi(G) ⊗G M −→ M F r∈R rV . which is natural in M . Note that the map θM can be described quite explicitly. For g ∈ G and m ∈ M the subgroup W = stabG (g · m) is open. In F particular, for U ∈ CO(G) the intersection U ∩W is of finite index in U . Let U = r∈R r·(U ∩W ). Then X 1 (4.26) θM (U g ⊗G m) = · r · m. \|U : (U ∩ W )\| r∈R From the commutative diagram (4.23) one concludes the following property. Proposition 4.4. Let G be a t.d.l.c. group. Then one has a natural isomorphism θ : Bi(G) ⊗G −→ idQ[G] dis . 18 I. CASTELLANO AND TH. WEIGEL Remark 4.5. Let G be a t.d.l.c. group. One verifies easily using (4.26) that · (4.27) = θBi(G) : Bi(G) ⊗G Bi(G) → Bi(G) defines the structure of an associative algebra on (Bi(G), ·). However, (Bi(G), ·) contains a unit 1Bi(G) ∈ Bi(G) if, and only if, G is compact. Nevertheless, the algebra (Bi(G), ·) has an augmentation or co-unit, i.e., the canonical map ε : Bi(G) → Q, (4.28) x ∈ G, U ∈ CO(G) ε(xU ) = 1, is a mapping of Q[G]-bimodules satisfying ε(a · b) = ε(a) · ε(b) for all a, b ∈ Bi(G). The algebra Bi(G) comes also equipped with an antipode (4.29) S= × : Bi(G) → Bi(G), S(U g) = g −1 U, g ∈ G, U ∈ CO(G). Again by (4.26) one verifies also that θM : Bi(G) ⊗G M → M defines the structure of a Bi(G)-module for any rational discrete Q[G]-module M . 4.3. The Hom-⊗-identity. Let G be a t.d.l.c. group, and let Q, M ∈ ob(Q[G] dis). Then one has a canonical map (cf. (4.25)) (4.30) ζQ,M : HomG (Q, Bi(G)) ⊗G M −→ HomG (Q, M ), ζQ,M (h ⊗G m)(q) = θM (h(q) ⊗G m), for h ∈ HomG (Q, Bi(G)), q ∈ Q, m ∈ M . Let U ⊆ G be a compact open subgroup of G. From now on we will assume that the canonical embedding ηU : Q[G/U ] −→ Bi(G) (4.31) is given by inclusion. By the Eckmann-Shapiro-type lemma (cf. §2.9), the map (4.32) ˆ : HomG (Q[G/U ], Bi(G)) −→ Bi(G), ĥ = h(U ), h ∈ HomG (Q[G/U ], Bi(G)), is injective satisfying im(ˆ ) = spanQ { x U x \| x ∈ G } ⊆ Bi(G). (4.33) In particular, one has an isomorphism jU : HomG (Q[G/U ], Bi(G)) → Q[U \G] (4.34) induced by ˆ . Hence, by the Eckmann-Shapiro-type lemma (cf. §2.9), (4.22), and the commutativity of the diagram (4.35) HomG (Q[G/U ], Bi(G)) ⊗G M ζQ[G/U ],M / HomG (Q[G/U ], M ) O jU ⊗G idM Q[U \G] ⊗G M / MU φ−1 M,U / MU one concludes that ζQ[G/U],M is an isomorphism for all M ∈ ob(Q[G] dis). The additivity HomG ( , M ) yields the following. Proposition 4.6. Let G be a t.d.l.c. group, let M, P ∈ ob(Q[G] dis), and assume further that P is finitely generated and projective. Then (4.36) ζP,M : HomG (P, Bi(G)) ⊗G M −→ HomG (P, M ) is an isomorphism. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 19 4.4. Dualizing finitely generated projective rational discrete Q[G]-modules. From (4.34) one concludes that for any finitely generated projective rational discrete left Q[G]-module P , HomG (P, Bi(G)) is a finitely generated projective rational discrete right Q[G]-module. In order to obtain a functor ⊛ : Q[G] disop → Q[G] dis we put for M ∈ ob(Q[G] dis) (4.37) M ⊛ = Hom× G (M, Bi(G)) = { ϕ ∈ HomQ (M, Bi(G)) \| ∀g ∈ G ∀m ∈ M : ϕ(g · m) = m · g −1 }. Obviously, × ◦ : HomG (M, Bi(G)) → M ⊛ (cf. (4.29)) yields an isomorphism of Q-vector spaces. Note that for g ∈ G, ϕ ∈ M ⊛ and m ∈ M one has (g · ϕ)(m) = g · ϕ(m). (4.38) By construction, if P is a finitely generated projective rational discrete left Q[G]module, then P ⊛ is a finitely generated projective rational discrete left Q[G]-module as well. Let M ∈ ob(Q[G] dis), let m ∈ M and let ϕ ∈ M ⊛ . Put ϑM (m)(ϕ) = ϕ(m)× . (4.39) Then for g ∈ G one has ϑM (m)(g · ϕ) = ((g · ϕ)(m))× = (g · ϕ(m))× = ϕ(m)× · g −1 = ϑM (m)(ϕ) · g −1 , (4.40) i.e., ϑM (m) ∈ M ⊛⊛ , and thus one has a Q-linear map ϑM : M → M ⊛⊛ . As ϑM (g · m)(ϕ) = ϕ(g · m)× = (ϕ(m) · g −1 )× = g · ϕ(m)× = (g · ϑM (m))(ϕ), (4.41) ϑM : M → M ⊛⊛ is a mapping of left Q[G]-modules. It is straightforward to verify that one obtains a natural transformation ϑ : idQ[G] dis −→ (4.42) ⊛⊛ . By (4.32) and (4.34) one has for U ∈ CO(G) that ϑQ[G/U] (U ) = jU× , where jU× is the map making the diagram (4.43) HomG (Q[G/U ), Bi(G)) × / Q[U \G] × ◦... Q[G/U ]⊛ jU × jU / Bi(G) commute. This yields that ϑQ[G/U] is an isomorphism. Hence the additivity of ϑ and the functors idQ[G] dis and ⊛⊛ imply that ϑP : P → P ⊛⊛ is an isomorphism for every finitely generated projective rational discrete Q[G]-module P . 4.5. T.d.l.c. groups of type FP. A t.d.l.c. group G is said to be of type FP, if (i) cdQ (G) = d < ∞; (ii) G is of type FP∞ (cf. [14, §VIII.6]). By Lemma 3.6 and Proposition 3.12, for such a group G the trivial left Q[G]-module Q possesses a projective resolution (P• , ∂• , ε) which is finitely generated and concentrated in degrees 0 to d. One has the following property. 20 I. CASTELLANO AND TH. WEIGEL Proposition 4.7. Let G be a t.d.l.c. group of type FP. Then (4.44) cdQ (G) = max{ k ≥ 0 \| dHk (G, Bi(G)) 6= 0 }. Proof. Let m = max{ k ≥ 0 \| dHk (G, Bi(G)) 6= 0 }. Then m ≤ d = cdQ (G). By definition, there exists a discrete left Q[G]-module M such that dHd (G, M ) 6= 0. As G is of type FP∞ , the functor dHd (G, ) commutes with direct limits (cf. [14, §VIII.4, Prop. 4.6]). Since M = limi∈I Mi , where Mi are finitely generated Q[G]−→ submodules of M , we may assume also that M is finitely generated, i.e., there P exist finitely many elements m1 , . . . , mn ∈ M such that M = 1≤j≤n Q[G] · mj . Let Oj be a compact open subgroup contained in stabG (mj ). Then one has a ` canonical surjective map 1≤j≤n Q[G/Oj ] → M . Thus, as dHd (G, ) is right ` exact, dHd (G, 1≤j≤n Q[G/Oj ]) 6= 0, and therefore dHd (G, Q[G/Oj ]) 6= 0 for some element j ∈ {1, . . . , d}. Let O = Oj . For any pair of compact open subgroups U, V of G, V ⊆ U , which are contained in O the map ηU,V : Q[G/U ] → Q[G/V ] (cf. (4.15)) is split injective (cf. Prop. 3.1). From the additivity of dHd (G, ) one concludes that dHd (ηU,V ) : dHd (G, Q[G/U ]) → dHd (G, Q[G/V ]) is injective. Hence (4.45) dHd (G, Bi(G)) = limU∈CO (G) dHd (G, Q[G/U ]) 6= 0, −→ O i.e., m ≥ d, and this yields the claim. For a t.d.l.c. group G of type FP with d = cdQ (G) we will call the rational discrete right Q[G]-module DG = dHd (G, Bi(G)) (4.46) the rational dualizing module of G. It has the following fundamental property. Proposition 4.8. Let G be a t.d.l.c. group of type FP with d = cdQ (G). Then there exists a canonical isomorphism υ : dHd (G, ) −→ DG ⊗G (4.47) of covariant additive right exact functors. Proof. Let (P• , ∂•P , εQ ) be a projective resolution of the trivial left Q[G]-module Q, which is finitely generated and concentrated in degrees 0 to d, and let (Q• , ð• , εDG ) be a projective resolution of the discrete right Q[G]-module DG . Let (C• , ∂•C ) be the chain complex of discrete right Q[G]-modules given by (4.48) Ck = HomG (Pd−k , Bi(G)), ∂kC = HomG (∂d−k+1 , Bi(G)) : Ck → Ck−1 , 0 ≤ k ≤ d, (cf. (4.37)). In particular, by construction, H0 (C• , ∂•C ) is canonically isomorphic to DG . Since (C• , ∂• ) is a chain complex of projectives, and as (Q• , ð• , εDG ) is exact, the comparison theorem in homological algebra implies that there exists a mapping of chain complexes χ• : (C• , ∂•C ) → (Q• , ð• ) which is unique up to chain homotopy equivalence. For M ∈ ob(Q[G] dis), χ• induces a chain map (4.49) χ•,M = χ• ⊗G idM : (C• ⊗G M, ∂•C ⊗G idM ) −→ (Q• ⊗G M, ð• ⊗G idM ). The right exactness of (4.50) ⊗G M implies that H0 (χ•,M ) : H0 (C• ⊗G M ) −→ DG ⊗G M RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 21 is an isomorphism. By the Hom-⊗ identity (4.36), one has a natural isomorphism of chain complexes (4.51) C• ⊗G M ≃ HomG (Pd−• , M ). Hence H0 (C• ⊗G M ) ≃ dHd (G, M ), and this yields the claim. Remark 4.9. A straightforward modification of the proof of Proposition 4.8 shows that for a t.d.l.c. group G of type FP one has natural isomorphisms (4.52) χ̄•, : dHd−• (G, ) −→ dTorG • (C• , [[0]]), where dTorG • ( , ) denotes the hyper-homology (cf. [6, §2.7], [14, §VIII.4, Ex. 7]). 4.6. Rational duality groups. A t.d.l.c. group G is said to be a rational duality group of dimension d ≥ 0, if (i) G is of type FP; (ii) dHk (G, Bi(G)) = 0 for all k 6= d. By Proposition 4.7, for such a group one has d = cdQ (G). These groups have the following fundamental property. Proposition 4.10. Let G be t.d.l.c. group which is a rational duality group of dimension d ≥ 0, and let DG denote its rational dualizing module. Then one has natural isomorphisms of (co)homological functors (4.53) dH• (G, ) ≃ dTorG d−• (DG , ), × dH• (G, ) ≃ dExtd−• G ( DG , ). Proof. Let (P• , ∂• , εQ ) be a projective resolution of the trivial left Q[G]-module Q, which is finitely generated and concentrated in degrees 0 to d. Then (4.54) Qk = HomG (Pd−k , Bi(G)), ðk = HomG (∂d−k+1 , Bi(G)), k ∈ {0, . . . , d}, is a chain complex of projective rational discrete right Q[G]-modules concentrated in degrees 0, . . . , d which satisfies ( DG for k = 0, (4.55) Hk (Q• , ð• ) ≃ 0 for k 6= 0, By Proposition 4.6, one has for M ∈ ob(Q[G] dis) isomorphisms (4.56) d−k Hk (Q• ⊗G M, ð• ⊗G idM ) ≃ dTorG k (DG , M ) ≃ dExtG (Q, M ) which are natural in M . This yields the first natural isomorphism in (4.53). Let (P•⊛ [d], ∂•⊛ [d]) be the chain complex (P•⊛ , ∂•⊛ ) concentrated in homological degrees −d, . . . , −0 moved d-places to the left such that it is concentrated in homological degrees 0, . . . , d, i.e., ( × DG for k = 0, ⊛ ⊛ (4.57) Hk (P• [d], ∂• [d]) ≃ 0 for k 6= 0. In particular, (P•⊛ [d], ∂•⊛ [d]) is a projective resolution of the rational discrete left Q[G]-module × DG . Then, by (4.42) and the remarks following it, one has an isomorphism of chain complexes of projective rational discrete left Q[G]-modules (4.58) (P•⊛ [d]⊛ [d], ∂•⊛ [d]⊛ [d]) ≃ (P• , ∂• ). 22 I. CASTELLANO AND TH. WEIGEL By the same arguments as used before, one obtains for M ∈ ob(Q[G] dis) and (4.59) Rk = HomG (P ⊛ [d]d−k , Bi(G)), δk = HomG (∂ ⊛ [d]d−k+1 , Bi(G)), k ∈ {0, . . . , d}, isomorphisms (4.60) d−k × Hk (R• ⊗G M, δ• ⊗G idM ) ≃ dTorG k (Q, M ) ≃ dExtG ( DG , M ) which are natural in M . This yields the claim. 4.7. Locally constant functions with compact support. Let G be a t.d.l.c. group, and let C(G, Q) denote the Q-vector space of continuous functions from G to Q, where Q is considered as a discrete topological space. Then C(G, Q) is a Q[G]-bimodule, where the G-actions are given by (4.61) (g · f )(x) = f (g −1 x), (f · g)(x) = f (x g −1 ), g, x ∈ G, f ∈ C(G, Q). For any compact open set Ω ⊆ G let IΩ : G → Q denote the continuous function given by IΩ (x) = 1 for x ∈ Ω and IΩ (x) = 0 for x ∈ G \ Ω. Then g · IΩ = Ig Ω and IΩ · g = IΩ g . Moreover, (4.62) Cc (G, Q) = spanQ { IgU \| g ∈ G, U ∈ CO(G) } ⊆ C(G, Q), coincides with the set of locally constant functions from G to Q with compact support. It is a rational discrete Q[G]-bisubmodule of C(G, Q). The rational discrete left Q[G]-module Bi(G) is isomorphic to the left Q[G]module Cc (G, Q), but the isomorphism is not canonical. Let O ⊆ G be a fixed compact open subgroup of G, and put COO (G) = { U ∈ CO(G) \| U ⊆ O }. The (O) Q-linear map ψU : Q[G/U ] → Cc (G, Q) given by (4.63) (O) ψU (xU ) = \|O : U \| · IxU is an injective homomorphism of rational discrete left Q[G]-modules. For U, V ∈ (O) (O) (O) COO (G), V ⊆ U , one has ψV ◦ ηU,V = ψU (cf. (4.15)), and thus (ψU )U∈COO (G) induces an isomorphism of left Q[G]-modules (4.64) (O) ψ (O) : Bi(G) → Cc (G, Q), ψ (O) \|Q[G/U] = ψU . This isomorphism has the following properties. Proposition 4.11. Let G be a t.d.l.c. group with modular function ∆ : G → Q+ , i.e., if µ : Bor(G) → R+ 0 ∪ {∞} is a left invarinat Haar measure on G, one has µ(S · g) = ∆(g) · µ(S) for all S ∈ Bor(G). Let O ∈ CO(G). (a) If U is a compact open subgroup of G containing O, then ψ (U ) = \|U : O\| · ψ (O) . (4.65) (b) For all g, h ∈ G and u ∈ Bi(G) one has (4.66) ψ (O) (g · W · h) = ∆(h−1 ) · g · ψ (O) (W ) · h. Proof. (a) Let g ∈ G and W ∈ CO(G). Then, by subsection 4.2, X 1 (4.67) gW = · g · r · (W ∩ O). \|W : W ∩ O\| r∈R Here we omitted the mappings ηW,W ∩O in the notation. Hence (4.68) ψ (O) (gW ) = µ(O) \|O : W ∩ O\| · IgW = · IgW , \|W : W ∩ O\| µ(W ) RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 23 for some left invariant Haar measure µ on G, and (4.69) ψ (U ) (gW ) = µ(U) · IgW . µ(W ) As µ(U) = \|U : O\| · µ(O), this yields the claim. (b) As ψ (O) is a homomorphism of left Q[G]-modules, it suffices to prove the claim F for g = 1 and u = W ∈ CO(G). If W h = h−1 · W · h = s∈S s · (W h ∩ O), one obtains by (a) that ψ (O) (W h) = h · ψ (O) (W h ) X 1 =h· · ψ (O) (s · (W h ∩ O)). h h \|W : W ∩ O\| s∈S h (4.70) = µ(O) \|O : W ∩ O\| · IW h = · IW h . \|W h : W h ∩ O\| µ(W h ) In particular, ψ (O) (W h) = µ(W )/µ(W h ) · ψ (O) (W ) · h. Since (4.71) µ(W h ) = µ(h−1 W h) = µ(W h) = ∆(h) · µ(W ), this completes the proof. L Remark 4.12. With a particular choice of Haar measure µ : Bor(G) → R+ 0 ∪ {∞} on G one can make (Cc (G, Q), ∗µ ) an associative algebra where ∗µ is convolution with respect to µ. However, we have seen that Bi(G) carries an algebra structure which is independent of the choosen Haar measure µ (cf. Remark 4.5). Both rational discrete Q[G]-bimodules Bi(G) and Cc (G, Q) will turn out to be useful. The standard rational Q[G]-bimodule Bi(G) seem to be the canonical choice for studying Hom-⊗ identities or dualizing functors (cf. §4.3, §4.4); while Cc (G, Q) seem to be the right choice for relating the cohomology groups dH• (G, Cc (G, Q)) with the cohomology with compact support of certain topological spaces associated to G (cf. §6.8). Remark 4.13. Let G be a t.d.l.c. group, and let ∆ : G → Q+ denote its modular function. Let Q(∆) denote the rational discrete left Q[G]-module which is isomorphic - as abelian group - to Q and which G-action is given by (4.72) g · q = ∆(g) · q, g ∈ G, q ∈ Q(∆). Let Q(∆)× denote the rational discrete right Q[G]-module associated to Q(∆), i.e., Q(∆)× = Q(∆) and for g ∈ G and q ∈ Q(∆)× one has q · g = g −1 · q. Then, by Proposition 4.11(b), one has a (non-canonical) isomorphism (4.73) Bi(G) ≃G Cc (G) ⊗ Q(∆)× of rational discrete right Q[G]-modules. Considering Q(∆)× as a trivial left Q[G]module, the isomorphism (4.73) can be interpreted as an isomorphism of rational discrete Q[G]-bimodules. 24 I. CASTELLANO AND TH. WEIGEL 4.8. The trace map. For a compact open subgroup O of G let µO denote the left-invariant Haar measure on G satisfying µO (O) = 1, i.e., if U is a compact open subgroup of G containing O, then µO = \|U : O\| · µU . We also denote by h(G) = Q · µO (4.74) the 1-dimensional Q-vector space generated by all Haar measures µO , O ∈ CO(G). One has a Q-linear map tr = ψ (O) ( )(1) · µO : Bi(G) −→ h(G), (4.75) which is independent of the choice of the compact open subgroup O of G, i.e., for all u ∈ Bi(G) and all O, U ∈ CO(G) one has (4.76) tr(u) = ψ (O) (u)(1) · µO = ψ (U ) (u)(1) · µU (cf. (4.65)). In particular, by definition, ( µO (4.77) tr(Og) = 0 if Og = O, if Og = 6 O. In case that G is unimodular one has the following. Proposition 4.14. Let G be a unimodular t.d.l.c. group. Then one has (4.78) tr(g · u) = tr(u · g) for all u ∈ Bi(G) and g ∈ G. In particular, putting Bi(G) = Bi(G)/h g · u − u · g \| u ∈ Bi(G), g ∈ G iQ , the Q-linear map tr induces a Q-linear map tr : Bi(G) −→ h(G). Proof. Since (g · f )(1) = (f · g)(1) = f (g −1 ) for all g ∈ G and f ∈ C(G, Q), one concludes from (4.66) that (4.79) tr(g · u) − tr(u · g) = (ψ (O) (u)(g −1 ) − ψ (O) (u)(g −1 )) · µO = 0. This yields the claim. 4.9. Homomorphism into the bimodule Cc (G, Q). Let G be a t.d.l.c. group. In order to simplify notations we put Cc (G) = Cc (G, Q) (cf. (4.62)). For a rational discrete left Q[G]-module M we put also M ⊙ = HomG (M, Cc (G)), (4.80) and consider M ⊙ as left Q[G]-module, i.e., for g ∈ G and h ∈ M ⊙ one has (g · h)(m) = h(m) · g −1 . (4.81) One has a homomorphism of Q-vector spaces (4.82) ∨ M : HomG (M, Cc (G)) −→ HomQ (M, Q), which is given by evaluation in 1 ∈ G, i.e., for m ∈ M and h ∈ M ⊙ one has (4.83) h∨ M (m) = h(m)(1). The Q-vector space M ∗ = HomQ (M, Q) carries canonically the structure of a left Q[G]-module, i.e., for g ∈ G and f ∈ M ∗ one has (4.84) (g · f )(m) = f (g −1 · m). RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 25 However, M ∗ is not necessarily a discrete Q[G]-module. For m ∈ M , g ∈ G and h ∈ HomG (M, Cc (G)) one concludes from (4.81) and (4.84) that −1 (g · h)∨ )(1) = h(m)(g) M (m) = (g · h)(m) (1) = (h(m) · g −1 −1 −1 (4.85) = g · h(m) (1) = h(g · m)(1) = h∨ · m), M (g i.e., ∨ M is a homomorphism of left Q[G]-modules. This homomorphism has the following property. Proposition 4.15. Let G be a t.d.l.c. group, and let M ∈ ob(Q[G] dis). Then ∨ ⊙ → M ∗ is injective. M: M Proof. Let h ∈ M ⊙ , h 6= 0, i.e., there exists m ∈ M with h(m) 6= 0. In particular, there exists g ∈ G such that h(m)(g) 6= 0. Since (4.86) −1 h(m)(g) = (g −1 · h(m))(1) = h(g −1 · m)(1) = h∨ · m), M (g this implies h∨ M 6= 0. Let α : B → M be a homomorphism of rational discrete left Q[G]-modules. Then for h ∈ M ⊙ and b ∈ B one has (cf. (4.83)) ⊙ α⊙ (h)∨ B (b) = α (h)(b) (1) = h(α(b))(1) ∗ ∨ = h∨ M (α(b)) = α (hM )(b), (4.87) i.e., the diagram M⊙ (4.88) α⊙ / B⊙ ∨ M M∗ ∨ B ∗ α / B∗ is commutative. 4.10. Proper signed discrete left G-sets. Let G be a t.d.l.c. group. A signed discrete left G-set Ω (cf. §3.3) will be said to be proper, if G has finitely many orbits on Ω and stabG (ω) is compact for all ω ∈ Ω. In particular, Q[Ω] is a finitely generated projective rational discrete left Q[G]-module (cf. (3.8)). For ω ∈ Ω define ω ∗ ∈ Q[Ω]∗ by   if ξ = ω, 1 ∗ (4.89) ω (ξ) = −1 if ξ = ω̄, and   0 if ξ 6∈ {ω, ω̄}. In particular, if g ∈ G then (4.90) g · ω ∗ = (g · ω)∗ . For a proper signed discrete left G-set Ω we put also (4.91) Q[Ω∗ ] = spanQ { ω ∗ \| ω ∈ Ω} ⊆ Q[Ω]∗ . In particular, as Ω∗ = { ω ∗ \| ω ∈ Ω } is a proper signed discrete left G-set, Q[Ω∗ ] is a finitely generated projective discrete left Q[G]-module. One has the following property. 26 I. CASTELLANO AND TH. WEIGEL Proposition 4.16. Let G be a t.d.l.c. group, and let Ω be a proper signed discrete ∗ ∨ left G-set. Then im( ∨ Q[Ω] ) = Q[Ω ]. In particular, Q[Ω] induces an isomorphism ∗ ⊙ ∨ ]. −→ Q[Ω : Q[Ω] Ω Proof. In suffices to prove the claim for a proper signed discrete left G-set Ω with one G × C2 -orbit (cf. (3.8)). We distinguish the two cases. Case 1: G acts without inversion on Ω. Let ω ∈ Ω and put O = stabG (ω). By hypothesis, Ω = G · ω ⊔ G · ω̄. Let R ⊆ G be a set of representatives for G/O. For x ∈ R there exists hx ∈ Q[Ω]⊙ = HomG (Q[Ω], Cc (G)) given by hx (ω) = IOx−1 . Moreover, as O Cc (G) = spanQ { IOx−1 \| x ∈ R } (cf. (4.34) and (4.64)), one has Q[Ω]⊙ = spanQ { hx \| x ∈ R }. For ̟ ∈ G · ω there exists a unique element y ∈ R such that ̟ = y · ω. Hence ∨ (4.92) (hx )∨ Q[Ω] (̟) = (hx )Q[Ω] (y · ω) = hx (y · ω)(1) = IyOx−1 (1) = IyO (x) = δx,y . ∨ ∗ Thus (hx )∨ Q[Ω] = (x·ω) . From this fact one concludes that Q[Ω] induces a mapping ∗ ⊙ −→ Q[Ω ], and that Ω is surjective. Hence the claim follows from Ω : Q[Ω] Proposition 4.15. Case 2: G acts with inversion on Ω. Let ω ∈ Ω and σ ∈ G be such that σ·ω = ω̄. Put U = stabG ({ω, ω̄}) and O = stabG (ω). In particular, O is normal in U and σ ∈ U. Let S ⊆ G be a set of representatives for G/U, and put R = S ⊔ Sσ. Then R is a set of representatives for G/O. For x ∈ S there exists kx ∈ Q[Ω]⊙ = HomG (Q[Ω], Cc (G)) given by kx (ω) = IOx−1 − IOσ−1 x−1 . Moreover, as before one has O Cc (G) = spanQ { IOz−1 \| z ∈ R }, and spanQ { kx \| x ∈ S } coincides with the eigenspace of the endomorphism σ◦ ∈ End(O Cc (G)) with respect to the eigenvalue −1. Thus Q[Ω]⊙ = spanQ { kx \| x ∈ S }. By hypothesis, Ω = G · ω. Let Ω+ = S · ω. Then Ω = Ω+ ⊔ Ω− , where − Ω = {̟ ¯ \| ̟ ∈ Ω+ }. Hence for ̟ ∈ Ω+ there exists a unique element y ∈ S such that ̟ = y · ω. This yields ∨ (kx )∨ Q[Ω] (̟) = (kx )Q[Ω] (y · ω) = kx (y · ω)(1) = IyOx−1 (1) − IyOσ−1 x−1 (1). (4.93) = IyOx−1 (1) = δx,y Here we used the fact that IyOσ−1 x−1 (1) = IyU x−1 (1) − IyOx−1 (1) = 0. Hence ∗ (kx )∨ Q[Ω] = (x · ω) and the claim follows by the same argument as in the previous case. One also concludes the following. Proposition 4.17. Let G be a t.d.l.c. group, let Ξ and Ω be proper signed discrete left G-sets, and let α : Q[Ξ] → Q[Ω] be a proper homomorphism of left Q[G]-modules (cf. (A.20)). Then one has a commutative diagram (4.94) Q[Ω]⊙ α⊙ ∨ Ω Q[Ω∗ ] / Q[Ξ]⊙ ∨ Ξ α ∗ / Q[Ξ∗ ], where the vertical maps are isomorphisms. Proof. This is an immediate consequence of Proposition 4.16 and the commutativity of the diagram (4.88). RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 27 5. Discrete actions of t.d.l.c. groups on graphs The notion of graph which will be used throughout the paper coincides with the notion used by J-P. Serre in [41], i.e., a graph Γ = (V(Γ), E(Γ)) will consist of a set of vertices V(Γ), a set of edges E(Γ), an origin mapping o : V(Γ) → E(Γ), a terminus mapping t : V(Γ) → E(Γ) and an edge inversion mapping ¯ : E(Γ) → E(Γ) satisfying ¯ = e, ē ē 6= e (5.1) t(ē) = o(e), o(ē) = t(e), for all e ∈ E(Γ) (cf. [41, §I.2.1]). In particular, E(Γ) is a signed set (cf. §A.5). Such a graph is said to be combinatorial, if the map (t, o) : E(Γ) −→ V(Γ) × V(Γ) (5.2) is injective. Moreover, Γ is said to be locally finite, if stΓ (v) = { e ∈ E(Γ) \| o(e) = v } is finite for every vertex v ∈ V(Γ). 5.1. The exact sequence associated to a graph. For a graph Γ let (5.3) V(Γ) = Q[V(Γ)] denote the free Q-vector space over the set of vertices of Γ, and let E(Γ) = Q[E(Γ)] = Q[E(Γ)]/h e + ē \| e ∈ E(Γ) i, (5.4) (cf. (A.16)), i.e., if e ∈ E[Γ] denotes the image of e ∈ E(Γ) in E(Γ), one has ē = −e. One has a canonical Q-linear mapping ∂ : E(Γ) → V(Γ) given by ∂(e) = t(e) − o(e), (5.5) e ∈ E(Γ), which has the following well known properties (cf. [41, §2.3, Cor. 1]). Fact 5.1. Let Γ = (V(Γ), E(Γ)) be a graph, and let ∂ : E(Γ) → V(Γ) be the map given by (5.5). Then (a) ker(∂) ≃ H1 (\|Γ\|, Q), where \|Γ\| denotes the topological realization of Γ. (b) coker(∂) ≃ Q[V(Γ)/ ∼], where ∼ is the connectedness relation, i.e., Γ is connected, if and only if, coker(∂) ≃ Q. In particular, Γ is a tree if, and only if, ker(∂) = 0 and coker(∂) ≃ Q. Let L(Γ) = ker(∂), i.e., if Γ is a connected graph the sequence (5.6) 0 / L(Γ) / E(Γ) ∂ / V(Γ) ε /Q /0 is exact. 5.2. Rough Cayley graphs. Let G be a t.d.l.c. group, let O be a compact open subgroup of G, and let S ⊆ G \ O be a symmetric subset of G intersecting O trivially, i.e., s ∈ S implies s−1 ∈ S. The rough Cayley graph Γ = Γ(G, S, O) associated with (G, S, O) is the graph given by V(Γ) = G/O and (5.7) E(Γ) = { (gO, gsO) \| g ∈ G, s ∈ S }. The origin mapping is given by the projection on the first coordinate, the terminus mapping is given by the projection on the second coordinate, while the edge inversion mapping permutes the first and second coordinate. The definition we have chosen here follows the approach used in [29, §2]. However, in our setup edges of a graph are directed, and we do not claim Γ to be connected. By construction, G has a discrete left action on Γ = Γ(G, S, O). It is straightforward to verify that Γ is combinatorial. One has the following properties. 28 I. CASTELLANO AND TH. WEIGEL Proposition 5.2. Let G be a t.d.l.c. group, let O be a compact open subgroup of G, let S be a symmetric subset of G \ O, and let Γ = Γ(G, S, O) be the rough Cayley graph associated with (G, S, O). (a) G acts transitively on the set of vertices of Γ; (b) if S is a finite set, then Γ is locally finite; (c) Γ is connected if, and only if, G is generated by O and S, i.e., G = h S, O i. Proof. (a) is obvious. (b) Suppose that S is finite. By (a), it suffices to show that stΓ (O) is a finite set. By definition, stΓ (O) = { (ωO, ωsO \| ω ∈ O, s ∈ S }. Since O is compact open in G, any double coset OsO is the union of finitely many right cosets g1 (s)O,. . . , gα(s) (s)O, i.e., t(stΓ (O)) = { gj (s)O \| s ∈ S, 1 ≤ j ≤ α(s) }. As Γ is combinatorial, the restriction of t to stΓ (O) is injective. This yields the claim. (c) Suppose that Γ is connected, and let g ∈ G. Then there exists a path p = e0 · · · en from O to gO, i.e., o(e0 ) = O, t(en ) = gO and o(ej ) = t(ej−1 ) for 1 ≤ j ≤ n. Let gj ∈ G, sj ∈ S such that ej = (gj O, gj sj O) for 0 ≤ j ≤ n, i.e., g0 = ω0 ∈ O. By induction, one concludes that gj ∈ h S, O i for all 1 ≤ j ≤ n. By construction, there exists ωn ∈ O such that gn sn ωn = g. In particular, g ∈ h S, O i, and thus G = h S, O i. Suppose now that G = h S, O i. By (a), it suffices to show that for any vertex gO ∈ V(Γ), there is a path from O to gO. By hypothesis, there exists elements s1 , . . . , sn ∈ S and ω0 , . . . , ωn ∈ O such that g = ω0 s1 · · · sn ωn . Hence for e0 = (ω0 O, ω0 s1 O) and ek = (ω0 s1 · · · sk−1 ωk−1 O, ω0 s1 · · · sk O) for 1 ≤ k ≤ n one verifies easily that p = e0 · · · en is a path from O to gO. Thus Γ is connected. 5.3. T.d.l.c. groups of type FP1 . It is well known that a discrete group is finitely generated if, and only if, it is of type FP1 (cf. [14, Ex. 1d of §I.2 and Ex. 1 of §VIII]). For t.d.l.c. groups one has the following. Theorem 5.3. Let G be a t.d.l.c. group. Then the following are equivalent. (i) The group G is compactly generated. ε (ii) The discrete Q[G]-module ker(Q[G/O] → Q) is finitely generated for any compact open subgroup O of G, where ε is the augmentation map. ε (iii) There exists a compact open subgroup O of G such that ker(Q[G/O] → Q) is finitely generated. (iv) The group G is of type FP1 . Proof. The implication (ii)⇒(iii) is trivial, and, by Proposition 3.12, (iii) and (iv) are equivalent. Suppose that G is compactly generated, and let O be any compact open subgroup of G. Then there exists a finite symmetric subset S ⊆ G \ O such that G = h S, O i. Let Γ = Γ(G, S, O) be the rough Cayley graph associated with (G, S, O). Then, as Γ is connected (cf. Prop. 5.2), one has a short exact sequence (cf. (5.6)) (5.8) E(Γ) ∂ / Q[G/O] ε /Q / 0. Moreover, as E(Γ) is a rational discrete left Q[G]-module being generated by the finite subset { (O, sO) \| s ∈ S }, ker(ε) is finitely generated. This shows (i)⇒(ii). Let O be a compact open subgroup of G such that K = ker(εO : Q[G/O] → Q) is a finitely generated discrete left Q[G]-module, i.e., there exist finitely many elements u1 , . . . , un ∈ K generating K as Q[G]-module. The set B = { gO −O \| g ∈ G\O } is RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 29 a generating set of K as Q-vector space, i.e., any element uj is a linear combination of finitely many elements of B. Hence there exists a finite subset S ⊂ G \ O such that any element uj is a linear combination of Σ = { sO − O \| s ∈ S }. Adjoining the inverses of elements in S if necessary, we may also assume that S is symmetric, i.e., s ∈ S implies s−1 ∈ S. Note that, by construction, one has P P (5.9) K = 1≤j≤n Q[G] · uj = s∈S Q[G] · (sO − O). Let Γ = Γ(G, S, O) be the rough Cayley graph associated with (G, S, O), and let ∂ : E(Γ) → Q[G/O] denote the canonical map, where we have identified V(Γ) with Q[G/O]. Then im(∂) ⊆ ker(εO ), and sO − O ∈ im(∂) for all s ∈ S (cf. (5.5), (5.7)). In particular, by (5.9), im(∂) = K, and coker(∂) ≃ Q, i.e., Γ is connected (cf. Fact 5.1(b)). Therefore, G = h S, O i (cf. Prop. 5.2(c)), and G is compactly generated. This yields the implication (iii)⇒(i) and completes the proof. 5.4. T.d.l.c. groups acting discretely on a tree. Let G be a group acting discretely on a graph Γ, i.e., all vertex stabilizers Gv = stabG (v), v ∈ V(Γ), and all edge stabilizers Ge = stabG (e), e ∈ E(Γ) are open. For e ∈ E(Γ) the set {e, ē} will be called a geometric edge of Γ. By E g (Γ) we denote the set of all geometric edges of Γ, i.e., E g (Γ) = E(Γ)/C2 , where C2 is the cyclic group of order 2 where the involution is acting by edge inversion. Let G{e} denote the stabilizer of the geometric edge {e, ē}, i.e., one has a sign character sgne : G{e} −→ {±1}. (5.10) By Qe we denote the 1-dimensional rational discrete left Q[G{e} ]-module with action given by (5.10), i.e., for g ∈ G{e} and q ∈ Qe one has g · q = sgne (g) · q. For a tree T one has L(T ) = 0. Hence the exact sequence (5.6) specializes to a short exact sequence. This fact implies the following version of a result of I. Chiswell (cf. [14, p. 179], [23]). Proposition 5.4. Let G be a t.d.l.c. group acting discretely on a tree T . Let RV ⊆ V(T ) be a set of representatives of the G-orbits on V(T ), and let RE ⊆ E(T ) be a set of representatives of the G × C2 -orbits on E(T ). (a) For M ∈ ob(Q[G] dis) one has a long exact sequence (5.11) Q Q k k / / / dHk (G, M ) ... v∈RV dH (Gv , M ) e∈RE dH (G{e} , M ⊗ Qe ) / ... (b) Suppose that Gv is compact for all v ∈ V(T ). Then cdQ (G) ≤ 1. In particular, (5.11) specializes to an exact sequence Q 0 / dH0 (G, M ) / (5.12) 0 v∈RV dH (Gv , M ) 0o dH1 (G, M ) o Q e∈RE dH0 (G{e} , M ⊗ Qe ) Proof. (a) By hypothesis, one has isomorphisms ` V(T )≃ v∈RV indG Gv (Q), (5.13) ` E(T )≃ e∈RE indG G{e} (Qe ), 30 I. CASTELLANO AND TH. WEIGEL and a short exact sequence (5.14) 0 / E(T ) ∂ / V(T ) ε /Q / 0. of rational discrete left Q[G]-modules. Hence (5.11) is a direct consequence of the long exact sequence associated to dExt•G ( , M ) and (5.14), the Eckmann-Shapiro lemma (cf. §2.9), and the isomorphisms dExtkG{e} (Qe , M ) ≃ dHk (G{e} , M ⊗ Qe ) which are natural in M . (b) By hypothesis, Gv is compact open, and Ge is open for all v ∈ V(T ) and e ∈ E(T ). As Ge ⊆ Gt(e) , this implies that Ge and G{e} are also compact and open. Hence (5.14) is a projective resolution of Q in Q[G] dis, and thus cdQ (G) ≤ 1. The exact sequence (5.12) follows from (5.11) and the fact that in this case one has cdQ (Gv ) = cdQ (G{e} ) = 0 (cf. Prop. 3.7(a)). Remark 5.5. If G is a compactly generated t.d.l.c. group, then dH1 (G, Bi(G)) must have dimension 0, 1 or ∞. It is shown in [21] that in `the latter case G is either isomorphic to a free product with amalgamation H U O, where H is an open subgroup of G, O is a compact open subgroup of G and \|O : O ∩ H\| 6= 1, or G is isomorphic to an HNN-extension HNN(H, ϕ) for some open subgroup H, O and U are compact open subgroups of H with at least one of them different from H and ϕ : O → U is an isomorphism, i.e., Stallings’ decomposition theorem holds for compactly generated t.d.l.c. groups. As a consequence a compactly generated t.d.l.c. group G satisfying dim(dH1 (G, Bi(G))) ≥ 2 must have a non-trivial discrete action on some tree T . 5.5. Fundamental groups of graphs of profinite groups. Let Λ be a connected graph. A graph of profinite groups (A, Λ) based on the graph Λ consists of (i) a profinite group Av for every vertex v ∈ V(Λ); (ii) a profinite group Ae for every edge e ∈ E(Λ) satisfying Ae = Aē ; (iii) an open embedding ιe : Ae → At(e) for every edge e ∈ E(Λ). For a graph of groups (A, Λ) one defines the group F (A, Λ) (cf. [41, §I.5.1]) as the group generated by Av , v ∈ V(Λ), and e ∈ E(Λ) subject to the relations (5.15) e−1 = ē and e ιe (a)e−1 = ιē (a) for all e ∈ E(Λ), a ∈ Ae . Let Ξ ⊆ Λ be a maximal subtree of Λ. The fundamental group π1 (A, Λ, Ξ) of the graph of groups (A, Λ) with respect to Ξ is given by (5.16) Π = π1 (A, Λ, Ξ) = F (A, Λ)/h e \| e ∈ E(Ξ) i (cf. [41, §I.5.1]). Choosing an orientation E + ⊂ E(Λ) of Λ one may construct a tree T = T (A, Λ, Ξ, E + ) with a canonical left Π-action (cf. [41, §I.5.1, Thm. 12]). The fundamental group of a graph of profinite groups carries naturally the structure of t.d.l.c. group. Indeed, the set of all open subgroups of vertex stabilizers are a neighborhood basis of the identity element of the topology. Thus from the Main Theorem of Bass-Serre theory one concludes the following. Proposition 5.6. Let (A, Λ) be a graph of profinite groups, let Ξ ⊆ Λ be a maximal subtree of Λ, and let E + ⊂ E(Λ) be an orientation of Λ. Then Π = π1 (A, Λ, Ξ) carries naturally the structure of a t.d.l.c. group, and its action on T = T (A, Λ, Ξ, E + ) is discrete and proper, i.e., vertex stabilizers and edge stabilizers are compact and open. In particular, cdQ (Π) ≤ 1. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 31 Proof. This follows from the Main Theorem of Bass-Serre theory and Proposition 5.4(b). Remark 5.7. The Main Theorem of Bass-Serre theory would yield more information than stated in Proposition 5.6. Indeed, a discrete and proper action of a t.d.l.c. group G on a tree T can be used to define a graph of profinite groups (A, Λ) such that G ≃ π1 (A, Λ, Ξ). 5.6. Nested t.d.l.c. groups. A t.d.l.c. group G is said to be nested, if there exists a countable ascending sequence of compact open subgroups (Ok )k≥0 of G such that S (5.17) G = k≥0 Ok . Such a group can be represented as the fundamental group of the graph of profinite groups (5.18) (A, Λ) : O1 • O1 O2 • O2 O3 • O3 O4 • O4 O5 • O5 ... based on the straight infinite half line Λ, where all injections are the canonical ones, i.e., one has (5.19) G ≃ π1 (A, Λ, Λ). Thus from Proposition 5.6 one concludes the following. Proposition 5.8. Let G be a nested t.d.l.c. group. Then cdQ (G) ≤ 1. S Remark 5.9. Let p be a prime number. Then Qp = k≥0 p−k Zp , the additive group of the p-adic numbers is a nested t.d.l.c. group, but Qp is not compact. Hence cdQ (Qp ) = 1. 5.7. Generalized presentations of a t.d.l.c. groups. Let G be a t.d.l.c. group. A generalized presentation of G is a graph of profinite groups (A, Λ) together with a continuous open surjective homomorphism (5.20) φ : π1 (A, Λ, Ξ) −→ G, such that φ\|Av is injective for all v ∈ V(Λ). Proposition 5.10. Let G be a t.d.l.c. group. (a) G has a generalized presentation ((A, Λ0 ), φ), where Λ0 is a graph with a single vertex. (b) For any generalized presentation ((A, Λ), φ), K = ker(φ) is a discrete free group contained in the quasi-center (5.21) QZ(Π) = { g ∈ Π \| CΠ (g) is open in Π }. of Π = π1 (A, Λ, Ξ). In particular, (5.22) R(φ) = K/[K, K] ⊗Z Q is a rational discrete left Q[G]-module. (c) If ((Ai , Λi ), φi ), i = 1, 2, are two presentations of G, then R(φ1 ) and R(φ2 ) are stably equivalent, i.e., there exist projective rational discrete Q[G]-modules P1 and P2 such that R(φ1 ) ⊕ P1 ≃ R(φ2 ) ⊕ P2 . 32 I. CASTELLANO AND TH. WEIGEL Proof. (a) By van Dantzig’s theorem there exists a compact open subgroup O of G. Choose a subset S of G such that G =< O, S >. Without loss of generality we may assume that S = S −1 , and choose a subset S + ⊆ S such that S = S + ∪ (S + )−1 . Let Λ0 be the graph with a single vertex v with a loop attached for any s ∈ S + , i.e., V(Λ0 ) = {v} and E(Λ0 ) = { es , ēs \| s ∈ S + }. Let (A, Λ0 ) be the graph of profinite groups based on Λ0 defined as follows: - Av = O; - Aes = Aēs = O ∩ s−1 Os for all s ∈ S + ; is Av for all s ∈ S + , where is is left conjugation by - αes : Aes −→ s−1 Os −→ s; - αēs : Aes → Av is the canonical inclusion for all s ∈ S + . Clearly, (A, Λ0 ) is a graph of profinite groups, and for the canonical map (5.23) φ0 : E(Λ0 ) ∪ O → G, φ(es ) = φ(ēs )−1 = s, φ0 (w) = w, w ∈ O, the relations (5.15) are satisfied. The edge set of the maximal subtree Ξ is empty. Hence F (A, Λ0 ) → π1 (A, Λ0 , Ξ) is an isomorphism. This show that there exists a unique group homomorphism φ : π1 (A, Λ0 , Ξ) → G such that φ\|E(Λ0 )∪O = φ0 . By construction, φ is surjective and φ\|Av is injective. Hence ((A, Λ0 ), φ) is a generalized presentation. (b) Let T = T (A, Λ, Ξ, E + ). Then, by construction, any non-trivial element of K acts without inversion of edges and without fixed points on T . Thus, by Stallings’ theorem (cf. [41, §I.3.3, Thm. 4]), K is a free discrete group. As K is normal in Π and discrete, K must be contained in QZ(Π). This implies that R(φ) is a rational discrete left Q[G]-module. (c) Let Πi = π1 (Ai , Λi , Ξi ), and let Ti = T (Ai , Λi , Ξi , Ei+ ) be trees associated to (Ai , Λi ). In particular, Πi is acting without inversion on Ti . Put Ki = ker(φi ). Then for the quotient graphs Γi = Ti //Ki , (5.24) E(Γi ) ∂i / V(Γi ) εi /Q are partial projective resolutions of Q in Q[G] dis. Hence, by Schanuel’s lemma (cf. [14, §VIII.4, Lemma 4.2]), ker(∂1 ) and ker(∂2 ) are stably equivalent. By Fact 5.1(a), one has ker(∂1 ) ≃ R(φ1 ) and ker(∂2 ) ≃ R(φ2 ). This yields the claim. Remark 5.11. Let ((A, Λ), φ) be a presentation of the t.d.l.c. group G. One should think of R(φ) as the relation module of the presentation ((A, Λ), φ). By Proposition 5.10, its stable isomorphism type is an invariant of the t.d.l.c. group G. 5.8. Compactly presented t.d.l.c. groups. A t.d.l.c. group G is said to be compactly presented, if there exists a presentation ((A, Λ), φ), such that (i) Λ is a finite connected graph, and (ii) K = ker(φ) is finitely generated as a normal subgroup of Π = π1 (A, Λ, Ξ). Condition (i) implies that such a group G must be compactly generated. Hence G is of type FP1 (cf. Thm. 5.3). From condition (ii) follows that R(φ) is a finitely generated rational discrete Q[G]-module. Thus, as R(φ) ≃ L(Γ), where T = T (A, Λ, Ξ, E + ) and Γ = T //K (cf. (5.6)), G must be of type FP2 (cf. Prop. 3.12). It has been an open problem for a long time whether discrete groups of type FP2 are indeed finitely presented. Finally, it has been answered negatively by M. Betsvina and N. Brady in [7]. In our context the following question arises. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 33 Question 2. Does there exist a non-discrete t.d.l.c. group G which is of type FP2 , but which is not compactly presented? 6. G-spaces for topological groups of type OS Throughout this section we assume that G is a topological group of type OS, and that F ⊆ Osgrp(G) is a non-empty subset of open subgroups of G satisfying (F1 ) for A ∈ F and g ∈ G one has g A = gAg −1 ∈ F; (F2 ) for A, B ∈ F one has A ∩ B ∈ F; In some cases it will be helpful to assume that the class F satisfies additionally the following condition. (F3 ) for A, B ∈ Osgrp(G), B ⊆ A, \|A : B\| < ∞ and B ∈ F one has A ∈ F. 6.1. F-discrete G-spaces. Let G be a topological group of type OS. We fix a nonempty set of open subgroups F satisfying (F1 ) and (F2 ). A non-trivial topological space X together with a continuous left G-action · : G × X → X will be called a left G-space. Such a space will be said to be F-discrete, if stabG (x) ∈ F for all x ∈ X. A continuous map f : X → Y of left G-spaces which commutes with the G-action is called a mapping of G-spaces. The non-empty G-space X together with an increasing filtration (X n )n≥0 of closed subspaces X n ⊆ X is called a (G, F)CW-complex, if S (C1 ) X = n≥0 X n ; (C2 ) X 0 is an F-discrete subspace of X; (C3 ) for n ≥ 1 there is an F-discrete G-space Λn , G-maps f : S n−1 × Λn → X n−1 and fb: B n × Λn → X n such that (6.1) S n−1 × Λn f / X n−1 B n × Λn fb / Xn is a pushout diagram, where S n−1 denotes the unit sphere and B n the unit ball in euclidean n-space (with trivial G-action); (C4 ) a subspace Y ⊂ X is closed if, and only if, Y ∩ X n is closed for all n ≥ 0. The dimension of X is defined by (6.2) dim(X) = min({ k ∈ N0 \| X k = X } ∪ {∞}). Moreover, X is said to be of type Fn , n ∈ N0 ∪ {∞}, if G has finitely many orbits on Λk for 0 ≤ k ≤ n. 6.2. The F-universal G-CW-complex. Let G be a topological group of type OS, and let F ⊆ Osgrp(G) be a non-empty set of open subgroups satisfying (F1 ) and (F2 ). A left G-CW-complex X said to be F-universal or an E F (G)-space, if (U1 ) X is contractable; (U2 ) for all x ∈ X one has stabG (x) ∈ F; (U3 ) for all A ∈ F the set of A-fixed points X A is non-empty and contractable. It is well known that an F-universal G-CW-complex E F (G) exists and is unique up to G-homotopy. The G-CW-complex E F (G) is universal with respect to all G-CW-complexes with point stabilizers in F, i.e., if X is a G-CW-complex with 34 I. CASTELLANO AND TH. WEIGEL stabG (x) ∈ F for all x ∈ X, then there exists a G-map X → E F (G) which is unique up to G-homotopy (cf. [32, Remark 2.5], [47, Chap. 1, Ex. 6.18.7]). This space can be explicitly constructed as the union of the n-fold joins of the discrete G-space F ΩF = A∈F G/A (cf. [32, §2], [47, Chap. 1, §6]). This notion leads to an obvious notion of dimension, i.e., (6.3) dimF (G) = min{ dim(X) \| X an E F (G)-space } ∈ N0 ∪ {∞} may be considered as the topological F-dimension of G. Although this might be the most natural dimension to be considered, we prefered to introduce and study a combinatorial version of this dimension, the simplicial F-dimension of G. This choice is motivated by the fact, that for the examples we are interested in the E F (G)spaces arise naturally as the topological realization of certain simplicial complexes. 6.3. Simplicial G-complexes. A simplicial complex (cf. §A.1) Σ with a left Gaction will be called a simplicial G-complex. The simplicial G-complex Σ is said to be of type Fn , n ∈ N0 , if G has finitely many orbits on Σq for all q ∈ {0, . . . , n }, and of type F∞ , if it is of type Fn for all n ∈ N0 . For a simplicial G-complex Σ and a q-simplex ω = {x0 , . . . , xq } ∈ Σq we denote by T stabG (ω) = 0≤j≤q stabG (xj ), (6.4) stabG {ω} = { g ∈ G \| g · ω = g }, the pointwise and setwise stabilizer of ω, respectively. In particular, one has stabG (ω) ⊆ stabG {ω}, and \|stabG {ω} : stabG (ω)\| < ∞. The simplicial G-complex Σ will be said to be G-tame3 if stabG (ω) = stabG {ω} for all ω ∈ Σ. For the convenience of the reader some basic properties of simplicial complexes are recollected in §A. 6.4. F-discrete simplical G-complexes. Let G be a topological group of type OS, and let F ⊆ Osgrp(G) be a non-empty set of open subgroups satisfying (F1 ), (F2 ) and (F3 ). A simplicial G-complex Σ will be said to be F-discrete, if stabG (x) ∈ F for all x ∈ Σ0 . Thus, by (F2 ), for an F-discrete simplicial G-complex one has stabG (ω) ∈ F and, by (F3 ), stabG {ω} ∈ F for all ω ∈ Σ. This property has the following consequence. Fact 6.1. Let G be topological group of type OS, and let F ⊆ Osgrp(G) be a nonempty set of open subgroups of G satisfying (F1 ), (F2 ) and (F3 ). Then for every F-discrete simplicial G-complex Σ, its topological realization \|Σ\| (cf. §A.2) is an F-discrete G-CW-complex. Proof. The property of being a G-CW-complex is straightforward. For any point z ∈ \|Σ\|, there exists a unique q-simplex ω ∈ Σ such that z is an interior point of \|Σ(ω)\| (cf. Fact A.3). Hence stabG (ω) ⊆ stabG (z) ⊆ stabG {ω}, and thus stabG (z) ∈ F by (F3 ). 6.5. Contractable F-discrete simplical G-complexes. The F-discrete simplical G-complex Σ will be said to be contractable, if \|Σ\| is contractable. But note that we do not claim that \|Σ\| is G-homotopic to a point (cf. [32, Thm. 2.2]). 3In case that Σ = Σ(Γ) for a graph Γ (cf. Ex. A.2), then Σ is a G-tame simplicial G-complex if, and only if, G acts without inversion of edges on the graph Γ. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 35 Example 6.2. Let G be a topological group of type OS, and let F ⊆ Osgrp(G) be a non-empty set of open subgroups satisfying (F1 ), (F2 ) and (F3 ). Let A ∈ F, and put Ω(A) = Σ[G/A] (cf. Ex. A.1). Then Ω(A) is an F-discrete simplical G-complex. By Fact A.4, \|Ω(A)\| is contractable, i.e., Ω(A) is a contractable F-discrete simplicial G-complex. We define the simplicial F-dimension of G by (6.5) sdimF (G) = min({ n ∈ N0 \| ∃ a contractable F-discrete simplicial G-complex Σ: dim(Σ) = n } ∪ {∞}) (cf. (A.4)). For n ∈ N0 ∪ {∞} we call G to be of type sFn (F), if there exists a contractable F-discrete simplicial G-complex Σ which is of type Fn (cf. §6.3). Moreover, G will be said to be of type sF(F), if there exists a contractable F-discrete simplicial G-complex Σ of type F∞ satisfying dim(Σ) < ∞. 6.6. The Bruhat-Tits property. The F-discrete simplicial G-complex Σ is said to have the Bruhat-Tits property, if for all A ∈ F one has \|Σ\|A 6= ∅. For every point z ∈ \|Σ\|A there exists a unique q-simplex ω(z) ∈ Σ such that z is contained in the interior of \|Σ(ω(z))\| (cf. Fact A.3). Hence A ⊆ stabG (z) ⊆ stabG {ω(z)}. 6.7. Classes of open subgroups for t.d.l.c. groups. Let G be a t.d.l.c. group. Obviously, the class of all open subgroups O = Osgrp(G) and the class C = CO(G) of all compact open subgroups satisfy the conditions (F1 ), (F2 ) and (F3 ). The same is true for (6.6) vN = { O ∈ Osgrp(G) \| ∃ U ⊆ O, U open and nested, \|O : U\| < ∞ }, the class of all virtually nested open subgroups of G (cf. §5.6). Proposition 6.3. Let G be a t.d.l.c. group. Then vN satisfies the conditions (F1 ), (F2 ) and (F3 ). Proof. Obviously, vN satisfies (F1 ) and (F3 ). Let S O ∈ vN, and let U ⊆ O be an open, nested subgroup of finite index, i.e., U = k≥0 Uk , Uk ⊆ Uk+1 , Uk ∈ CO(G). Since \|O : U\| < ∞, we may also assume that U is normal in O. Let V ∈ vN. Then V ∩ U is a nested open subgroup of G. Moreover, as V ∩ U = ker(V → O/U), one has that \|V : V ∩ U\| < ∞. Hence vN satisfies (F2 ) as well. Remark 6.4. The proof of Proposition 5.8 show also that N - the class of all nested open subgroups of a t.d.l.c. group G - satisfies the conditions (F1 ) and (F2 ). Remark 6.5. It is a remarkable fact, that although the Higman-Thompson groups Fn,r , Tn,r , Gn,r are not of finite cohomological dimension, they still satisfy the FP∞ -condition (cf. [15]). The Neretin group Nn,r of spheromorphism of a regular rooted tree Tn,r can be seen as a t.d.l.c. analogue of the Higman-Thompson groups (cf. Remark 3.11). In particular, this group has a factorization (6.7) Nn,r = O · Fn,r , O ∩ Fn,r = {1}, where Fn,r is a discrete subgroup isomorphic to the Richard Thompson group Fn,r and O is a nested open subgroup of Nn,r . Hence the following questions arise. Question 3. Is Nn,r of type sF(N)∞ or even of type sF(C)∞ ? Question 4. Does there exist an E N (Nn,r )-space which is of type F∞ ? 36 I. CASTELLANO AND TH. WEIGEL The finiteness conditions we discussed in subsection §6.5 for C-discrete simplicial G-complexes for a t.d.l.c. group G have the following consequence for its cohomological finiteness conditions. Proposition 6.6. Let G be a t.d.l.c. group. Then one has the following. (a) sdimC (G) ≥ cdQ (G). (b) Let n ∈ N0 ∪ {∞}. If G is of type sFn (C), then G is of type FPn . (c) If G is of type sF(C), then G is of type FP. Proof. Let Σ be a C-discrete simplicial G-complex, let ω = {x0 , . . . , xq } ∈ Σq , and e q (cf. (A.8)). Then stabG {ω} is acting on {±ω}, i.e., one let ω = x0 ∧ · · · ∧ xq ∈ Σ has a continuous group homomorphism (6.8) sgnω : stabG {ω} −→ {±1}. Let Qω denote the 1-dimensional rational discrete left stabG {ω}-module associated to sgnω , i.e., for g ∈ stabG {ω} and z ∈ Qω one has g · z = sgnω (g) · z. Thus if Rq ⊆ Σq is a set of representatives for the G-orbits on Σq , one has an isomorphism ` (6.9) Cq (Σ) ≃ ω∈Rq indG stabG {ω} (Qω ) of rational discrete left Q[G]-modules (cf. §A.3). In particular, Cq (Σ) is a projective rational discrete left Q[G]-module. If Σ is contractable, one has H0 (\|Σ\|, Q) ≃ Q and Hk (\|Σ\|, Q) = 0 for k 6= 0. Hence (C• (Σ), ∂• ) is a projective resolution of the rational discrete left Q[G]-module Q (cf. Fact A.6). In particular, cdQ (G) ≤ dim(Σ). This yields (a). If Σ is contractable and of type Fn , n ∈ N0 ∪ {∞}, then, by (6.9), Ck (Σ) is a finitely generated projective rational discrete left Q[G]-module for 0 ≤ k ≤ n. This yields (b). Part (c) follows by similar arguments. 6.8. Simplicial t.d.l.c. rational duality groups. As in the discrete case it is possible to detect from a group action that a t.d.l.c. group G is indeed a rational duality group. Theorem 6.7. Let G be a t.d.l.c. group, and let Σ be a C-discrete simplicial G-complex such that (D1 ) Σ is contractable; (D2 ) dim(Σ) < ∞ and Σ is of type F∞ ; (D3 ) Σ is locally finite. Then d = cdQ (G) < dim(Σ) and one has (6.10) × DG = Hcd (\|Σ\|, Q) ⊗ Q(∆). In particular, if (D4 ) Hck (\|Σ\|, Q) = 0 for k 6= d, then G is a t.d.l.c. rational duality group of dimension d. Proof. By (D1 ), (C• (Σ), ∂• ) is a projective resolution of Q in Q[G] dis, and by (D2 ), Cq (Σ) is a finitely generated projective rational discrete Q[G]-modules for all q ≥ 0. As Σ is locally finite by (D3 ), ∂q : Cq (Σ) → Cq−1 (Σ), q ≥ 1, has an adjoint map e ∗ ] → Q[Σ e ∗ ], where one has to replace Σ e 0 by Σ e ± (cf. §A.5). Moreover, ∂q∗ : Q[Σ q−1 q 0 (6.11) ∨ eq Σ ∗ eq] : Cq (Σ)⊙ −→ Q[Σ RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 37 is an isomorphism (cf. Prop. 4.16), and one has the commutative diagram (6.12) ∨ eq Σ Cq (Σ)⊙ / Q[Σ e ∗] q Ccq (Σ) ∂∗ q+1 ⊙ ∂q+1 ∨ e Σ Cq+1 (Σ)⊙ q+1 e ∗q+1 ] / Q[Σ ðq Ccq+1 (Σ) (cf. Prop. 4.17). In particular, one obtains canonical isomorphisms (cf. Fact A.7) dHk (G, Cc (G)) ≃ Hck (\|Σ\|, Q)× (6.13) of rational discrete right Q[G]-modules, and non-canonical isomorphisms × (6.14) dHk (G, Bi(G)) ≃ Hck (\|Σ\|, Q) ⊗ Q(∆) of rational discrete left Q[G]-modules (cf. Remark 4.13). In particular, as G is of type FP (cf. Prop. 6.6(c)), one has (6.15) cdQ (G) = max{ k ∈ N0 \| Hck (\|Σ\|, Q) 6= 0 } (cf. Prop. 4.7). Thus for d = cdQ (G) one has an isomorphism (6.16) × DG = Hcd (\|Σ\|, Q) ⊗ Q(∆). This yields the claim. A t.d.l.c. group G admitting a C-discrete simplicial G-complex Σ satisfying (Dj ), 1 ≤ j ≤ 4, will be called a simplicial t.d.l.c. rational duality group. It should be mentioned that the dimension of Σ as simplicial complex does not have to coincide with d = cdQ (G). 6.9. Universal C-discrete simplicial G-complexes. Let Σ be a C-discrete simplical G-complex, and let δ : \|Σ\| × \|Σ\| → R+ 0 be a G-invariant metric on its topological realization. Then (Σ, δ) is called a C-discrete CAT(0) simplicial G-complex, if (E1 ) (\|Σ\|, δ) is a complete CAT(0)-space4, (E2 ) Σ is G-tame. Note that property (E1 ) implies that \|Σ\| is contractable (cf. [2, Prop. 11.7]). Let O be a compact open subgroup of G. By hypothesis, any orbit O · x, x ∈ \|Σ\|, is finite and thus in particular bounded. In particular, O has a fixed point z in \|Σ\| (cf. [2, Thm. 11.23]). Let ω(z) ∈ Σ denote the unique simplex such that z is contained in the interior of ω(z) (cf. Fact A.3). Then, by property (E2 ), O ⊆ stabG (z) = stabG {ω(z)} = stabG (ω(z)), i.e., Σ has the Bruhat-Tits property (cf. §6.6). By property (E2 ), ΣO is a simplicial subcomplex of Σ, and, by (E1 ), (\|ΣO \|, δ) is a complete CAT(0)-space. In particular, \|ΣO \| is contractable. This shows the following. Proposition 6.8. Let G be a t.d.l.c. group, and let Σ be a C-discrete CAT(0) simplical G-complex. Then \|Σ\| is an E C (G)-space. 4For details on CAT(0)-spaces see [13]. 38 I. CASTELLANO AND TH. WEIGEL 6.10. Compactly generated t.d.l.c. groups of rational discrete cohomological dimension 1. Let G be a compactly generated t.d.l.c. group satisfying cdQ (G) = 1. Then G is of type FP (cf. Thm. 5.3). As G is not compact (cf. Prop. 3.7(a)), one has (6.17) dH0 (G, Bi(G)) = HomG (Q, Bi(G)) = 0 (cf. (4.21)). Hence G is a rational t.d.l.c. duality group. Suppose that Π = π1 (A, Λ, Ξ) is the fundamental group of a finite graph of profinite groups (A, Λ), i.e., Λ is a finite connected graph. Moreover, Π acts on the tree T = T (A, Λ, Ξ, E + ) without inversion of edges. L We also assume that Π is not compact. Then cdQ (G) = 1 (cf. Prop. 5.6). Hence the simplicial version of the tree T - which we will denote by the same symbol - has the following properties: (1) T is a locally finite simplicial complex of dimension 1; (2) T is a C-discrete simplicial Π-complex; (3) Π has finitely many orbits on the simplicial complex T , i.e., T is of type F∞ ; (4) T is Π-tame; (5) the standard metric δ : \|T \| × \|T \| → R+ 0 gives (\|T \|, δ) the structure of a complete CAT(0)-space. (6) Hc0 (\|T \|, Q) = 0. Hence (1), (2), (3) and (5) imply that Π is a simplicial t.d.l.c. duality group. Moreover, by Theorem 6.7, one has (6.18) × DΠ ≃ Hc1 (\|T \|, Q) ⊗ Q(∆). From (4) and (5) one concludes that T is a C-discrete CAT(0)-simplicial Π-complex, and thus an E C (Π)-space (cf. Prop. 6.8). 6.11. Algebraic groups defined over a non-discrete, non-archimedean local field. Let K be a non-discrete non-archimedean local field5 with residue field F; in particular, F is finite. Let G be a semi-simple, simply-connected algebraic group defined over K, and let G(K) denote the group of K-rational points. In particular, G(K) carries naturally the structure of a t.d.l.c. group. Suppose that G(K) is not compact. Then G(K) has a type preserving action on an affine building (C, S), the Bruhat-Tits building, where (W, S) is an affine Coxeter group. Moreover, d = \|S\| − 1 coincides with the algebraic K-rank of G. For further details see [17], [18], [46] and [50]. The Tits realization Σ(C, S) of the Bruhat-Tits building (C, S) is a discrete simplicial G(K)-complex of dimension d with the following properties: (1) For ω ∈ Σ(C, S) its stabilizer stabG(K) (ω) is a parahoric subgroup and hence compact, i.e., Σ(C, S) is a C-discrete simplicial G(K)-complex. (2) G(K) has finitely many orbits on Σ(C, S), i.e., Σ(C, S) is of type F∞ . (3) As G(K) acts type preserving, Σ(C, S) is G(K)-tame. (4) Σ(C, S) is locally finite. 5By a local field we understand a non-discrete locally compact field. Such a field comes equipped with a valuation v, and, in case that v is non-archimedean, v is also discrete. Hence such a field is either isomorphic to a finite extension of the p-adic numbers Qp or isomorphic to a Laurent power series ring Fq [t−1 , t]] for some finite field Fq . RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 39 (5) There exists a G(K)-invariant metric δ : \|Σ(C, S)\| × \|Σ(C, S)\| → R+ 0 making (\|Σ(C, S)\|, δ) a complete CAT(0)-space (cf. [2, Thm. 11.16]). In particular, \|Σ(C, S)\| is contractable. (6) By a theorem of A. Borel and J-P. Serre, one has Hck (\|Σ(C, S)\|, Q) = 0 for k 6= d and Hcd (\|Σ(C, S)\|, Q) 6= 0 (cf. [10], [12]). From these properties one concludes the following. Theorem 6.9. Let G be a semi-simple, simply-connected algebraic group defined over the non-discrete non-archimedean local field K. (a) The topological realization \|Σ(C, S)\| of the Tits realization of the affine building (C, S) is an E C (G(K))-space. (b) G(K) is a rational t.d.l.c. duality group of dimension rk(G) = dim(Σ(C, S)). (c) For d = cdQ (G(K)) one has × DG(K) ≃ Hcd (\|Σ(C, S)\|, Q). Proof. (a) is a direct consequence of the properties (1), (3), (5) and Proposition 6.8. (b) and (c) follow from the properties (1), (2), (4), (5), (6) and Theorem 6.7. Note that G(K) is in particular a simplicial t.d.l.c. rational duality group and (6.19) dimC (G(K)) = sdimC (G(K)) = cdQ (G(K)). 6.12. Topological Kac-Moody groups. Let F be a finite field, and let G(F) denote the rational points of an almost split Kac-Moody group defined over F (cf. [38, 12.6.3]) with infinite Weyl group (W, S). Such a group acts naturally on a twin building Ξ (cf. [40, 1.A]). Let Σ(Ξ+ ) denote the Davis-Moussang realization of the positive part of Ξ (cf. [24], [36]). In particular, (1) Σ(Ξ+ ) is a locally finite simplicial complex of finite dimension. Moreover, G(F) is acting on Ξ+ and hence on Σ(Ξ+ ), and thus one has a homomorphism of groups π : G(F) → Aut(Σ(Ξ+ )). As Σ(Ξ+ ) is locally finite, Aut(Σ(Ξ+ )) carries naturally the structure of a t.d.l.c. group. The closure Ḡ(F) of im(π) is called the topological Kac-Moody group associate to G(F) (cf. [40, 1.B and 1.C]). By construction, (2) Σ(Ξ+ ) is a C-discrete Ḡ(F)-simplicial complex. Indeed, if F is of positive characteristic p, then stabḠ(F) (ω) is a virtual pro-p group for all ω ∈ Σ(Ξ+ ) (cf. [40, p. 198, Theorem]). Let G(F)◦ denote the subgroup of G(F) which elements are type preserving on Ξ, and let Ḡ(F)◦ denote the closure of π(G(F)◦ ). Then Ḡ(F)◦ is open and of finite index in Ḡ(F). As G(F)◦ is type preserving, one concludes that (3) Σ(Ξ+ ) is Ḡ(F)◦ -tame. Since G(F) (and thus Ḡ(F)) acts δ-2-transitive on chambers of Ξ, one has that (4) Σ(Ξ+ ) is a C-discrete simplicial Ḡ(F)◦ -complex of type F∞ (cf. §6.3). By a remarkable result of M.W. Davis (cf. [24]) the topological realization \|Σ(Ξ+ )\| admits a Ḡ(F)◦ -invariant CAT(0)-metric d : \|Σ(Ξ+ )\| × \|Σ(Ξ+ )\| → R+ 0 (cf. [2, Thm. 12.66]). Hence Proposition 6.8 implies the following. Theorem 6.10. Let F be a finite field, let G(F) denote the rational points of an almost split Kac-Moody group defined over F with infinite Weyl group, and let Ḡ(F) be the associated topological Kac-Moody group. Then the topological realization \|Σ(Ξ+ )\| of the David-Moussang realization Σ(Ξ+ ) is an E C (Ḡ(F)◦ )-space. 40 I. CASTELLANO AND TH. WEIGEL Note that by Davis’ theorem, (5) Σ(Ξ+ ) is a contractable C-discrete simplicial Ḡ(F)-complex. Hence Theorem 6.7 implies that (6.20) d = cdQ (Ḡ(F)) = cdQ (Ḡ(F)◦ ) ≤ dim(Σ(Ξ+ )) < ∞, Ḡ(F) is of type FP, and since Ḡ(F) is unimodular (cf. [39, p. 811, Thm.(i)]), × (6.21) DḠ(F) ≃ Hcd (\|Σ(Ξ+ )\|, Q). Moreover, by (6.15), (6.22) cdQ (Ḡ(F)) = max{ k ≥ 0 \| Hck (\|Σ(Ξ+ )\|, Q) 6= 0 }. However, Ḡ(F) may or may not be a rational t.d.l.c. duality group (cf. Remark 6.12 and 6.13). The cohomology with compact support of \|Σ(Ξ+ )\| with coefficients in Z was computed in [26]. In more detail, a bT (Ξ+ ), Hck (\|Σ(Ξ+ )\|, Z) = (6.23) H k (K, K S−T ) ⊗ A T ∈S and a similar formula holds for the Davis realization of the Coxeter complex Σ(C) associated to (W, S) (cf. [25]), i.e., a bT (C), Hck (\|Σ(C)\|, Z) = (6.24) H k (K, K S−T ) ⊗ A T ∈S Here S denotes the set of non-trivial spherical subsets of S, and K is a simplicial complex associated to (W, S) build up from the spherical subsets of S. A more bT (Ξ+ ) and A bT (C) yields the following. detailed analysis of the coeffcient modules A Theorem 6.11. Let F be a finite field, let G(F) denote the rational points of an almost split Kac-Moody group defined over F with infinite Weyl group (W, S), and let Ḡ(F) be the associated topological Kac-Moody group. Then (6.25) cdQ (Ḡ(F)◦ ) = cdQ (W ). Moreover, Ḡ(F)◦ (and thus Ḡ(F)) is a rational t.d.l.c. duality group if, and only if, W is a rational discrete duality group. bT (Ξ+ ) and A bT (C) are free Z-modules. Moreover, for T ∈ S one Proof. Note that A bT (C) 6= 0 and A bT (Ξ+ ) 6= 0 (cf. [26, §6, p. 570, Remark and Definition 7.4]). has A As the abelian groups H k (K, K S−T ) are finitely generated, the universal coefficient theorem implies that cdQ (W ) = max{ k ≥ 0 \| ∃ T ∈ S : H k (K, K S−T ) is not torsion } = max{ k ≥ 0 \| Hck (\|Σ(Ξ+ )\|, Q) 6= 0 } (6.26) = cdQ (Ḡ(F)) (cf. Thm. 6.7). Let d = cdQ (W ) = cdQ (Ḡ(F)). Then W is a rational duality group if, and only if, H k (K, K ST ) is a torsion group for all T ∈ S and all k 6= d. By (6.24), this is equivalent to Hck (\|Σ(Ξ+ )\|, Q) = 0 for all k 6= d. Thus Theorem 6.7 yields the claim. Remark 6.12. Let F be a finite field, let G(F) denote the rational points of a split Kac-Moody group defined over F, and let Ḡ(F) be the associated topological KacMoody group. Suppose further that RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 41 (1) the associated generalized Cartan matrix is symmetrizable, and that (2) the associated Weyl group (W, S) is hyperbolic (cf. [28, §6.8]). Condition (2) implies that W is a lattice in the real Lie group O(n, 1), \|S\| = n + 1 (cf. [28, p. 140, and the references therein]). Moreover, as W is the Weyl group of a (split) Kac-Moody Lie algebra with property (1), the Tits representation of the Weyl group W is integral (cf. [19, §16.2]). Hence W is an arithmetic lattice in O(n, 1). Thus by A. Borel’s and J-P. Serre’s theorem (cf. [11]), W is a virtual duality group, and thus, in particular, a rational duality group. Hence Theorem 6.11 implies that Ḡ(F) is a rational discrete t.d.l.c. duality group. Remark 6.13. Consider the Coxeter group (W, S) associated to the Coxeter diagram •✤✷ ☞☞✤ ✷✷✷ ☞ ✷ ☞ ☞☞ ✤ ∞ ✷✷✷ ☞ ✤ ☞ ✷✷ ✤ ☞☞ ✷✷ ☞ ☞ ✷✷ ✤▼ ☞ • (6.27) q ✷ ☞ ▼ ▼∞ ✷✷ ☞☞ ∞q q ▼ ☞ ✷ q ☞q ▼ ✷✷ ▼• ☞☞ •q ` e2 ) W (A1 ). In particular, cdQ (W ) = 2. By Stallings’ decomposiThen W ≃ W (A tion theorem (cf. [44]), H 1 (W, Q[W ]) 6= 0. Thus W is not a rational duality group. Let G(F) denote the rational points of a split Kac-Moody group defined over F with associated Coxeter group is (W, S), and let Ḡ(F) be the associated topological Kac-Moody group. Then Theorem 6.11 implies, that Ḡ(F) is of rational cohomological dimension 2, but that Ḡ(F) is not a rational t.d.l.c. duality group, i.e., dH1 (Ḡ(F), Bi(Ḡ(F)) 6= 0. The t.d.l.c. version of Stalling’s decomposition theorem established in [21] implies that Ḡ(F) can be decomposed non-trivially as a free product with amalgamation in a compact open subgroup (where one factor is a compact open subgroup as well), or as an HNN-extension with amalgamation in a compact open subgroup. It would be interesting to understand how this decomposition is related to the root datum of G(F). We close this section with a question which is motivated by (6.19). Question 5. For which topological Kac-Moody groups Ḡ(F), F a finite field, does dimC (Ḡ(F)) = cdQ (Ḡ(F)) hold (cf. §6.3)? 7. The Euler-Poincaré characteristic of a unimodular t.d.l.c. group of type FP In this subsection we indicate how one can associate to any unimodular t.d.l.c. group G of type FP an Euler-Poincaré characteristic (7.1) χ(G) ∈ h(G) = Q · µO , where O is a compact open subgroup, and µO denotes the left invariant Haar measure on G satisfying µO (O) = 1. If h ∈ Q+ · µO we simply write h > 0; similarly h < 0 shall indicate that h ∈ Q− · µO . In some particular cases discussed in §6.10 and §6.11 it will be possible to calculate the Euler-Poincaré characteristic χ(G) explicitly. It is quite likely, that similar calculation can be done also for the examples decribed in §6.12. However, one of the 42 I. CASTELLANO AND TH. WEIGEL most interesting question from our perspective which was our original motivation to introduce and study this notion we were not able to answer. Question 6. Let G be a compactly generated t.d.l.c. group satisfying cdQ (G) = 1. Does this imply that χ(G) ≤ 0? An affirmative answer of Question 6 would resolve the problem of accessibility for compactly generated t.d.l.c. groups of rational discrete cohomological dimension 1 in analogy to the discrete case (cf. [30]). 7.1. The Hattori-Stallings rank of a finitely generated projective rational discrete Q[G]-module. Let G be a t.d.l.c. group, and let P be a finitely generated projective rational discrete Q[G]-module. The evaluation map (7.2) evP : HomG (P, Bi(G)) ⊗Q P → Bi(G), φ ∈ HomG (P, Bi(G)), evP (φ ⊗ p) = φ(p), p ∈ P, induces a mapping evP : HomG (P, Bi(G)) ⊗G P → Bi(G) such that the diagram (7.3) HomG (P, Bi(G)) ⊗Q P / HomG (P, Bi(G)) ⊗G P evP evP / Bi(G) Bi(G) commutes, where the horizontal maps are the canonical ones and Bi(G) is as defined in §4.8. Then - provided that G is unimodular - one has a map (7.4) ρP : HomG (P, P ) −1 ζP,P / HomG (P, Bi(G)) ⊗G P evP / Bi(G) tr / h(G). (cf. Prop. 4.6, Prop. 4.14). The value rP = ρP (idP ) ∈ h(G) will be called the Hattori-Stallings rank of P . Proposition 7.1. Let G be a unimodular t.d.l.c. group, and let O ⊆ G be a compact open subgroup. Then rQ[G/O] = µO . Proof. Let ηO : Q[G/O] → Bi(G) be the canonical embedding (cf. (4.31)). Then jO (ηO ) = O ∈ O\G (cf. (4.34)). Hence, (7.5) (jO ⊗G idQ[G/O] )(ηO ⊗ O) = O ⊗G O and O φ−1 Q[G/O],O (O ⊗G O) = O ∈ Q[G/O] . (7.6) Under the Eckmann-Shapiro isomorphism α: Q[G/O]O → HomG (Q[G/O], Q[G/O]) the element O is mapped to idQ[G/O] . Hence, by (4.35), (7.7) ζQ[G/O],Q[G/O](ηO ⊗G O) = idQ[G/O] , −1 (idQ[G/O] )) = O, where O denotes the canonical image and evQ[G/O] (ζQ[G/O],Q[G/O] of O in Bi(G). Thus, by the definition of tr : Bi(G) → h(G) (cf. (4.77)), (7.8) and this yields the claim. rQ[G/O] = tr(O) = µO RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 43 Remark 7.2. Let G be a unimodular t.d.l.c. group, and let P ∈ ob(Q[G] dis) be finitely generated and projective. In [22] it is shown, that rP ≥ 0, and rP = 0 if, and only if, P = 0. 7.2. The Euler-Poincaré characteristic. Let G be a unimodular t.d.l.c. group, and let P1 , P2 ∈ ob(Q[G] dis) be finitely generated and projective. Then, by (7.4), one has ρP1 ⊕P2 (idP1 ⊕P2 ) = ρP1 (idP1 ) + ρP2 (idP2 ), i.e., rP1 ⊕P2 = rP1 + rP2 . (7.9) Let (P• , δ• ) be a finite projective resolution of Q in Q[G] dis, i.e., there exists N ≥ 0 such that Pk = 0 for k > N , and Pk is finitely generated for all k ≥ 0. From the identity (7.10) one concludes that the value P (7.10) χ(G) = k≥0 (−1)k · rPk ∈ h(G) is independent from the choice of the projective resolution (P• , δ• ) (cf. [45, Proposition 4.1]) for a similar argument). We will call χ(G) the Euler-Poincaré characteristic of G. 7.3. Examples. 7.3.1. Compact t.d.l.c. groups. Let O be a compact t.d.l.c. group. Then Proposition 7.1 implies that χ(G) = rQ = µO . 7.3.2. Fundamental groups of finite graphs of profinite groups. Let (A, Λ) be a finite graph of profinite groups. Assume further that Π = π1 (A, Λ, x0 ) is unimodular6. Then, by Proposition 7.1, X X µAv − µAe . (7.11) χ(Π) = v∈V(Λ) e∈E + (Λ) In particular, if (A, Λ) is a finite graph of finite groups one obtains X X 1 1 (7.12) χ(Π) = · µ = χΠ · µ{1} , − \|Av \| \|Ae \| + v∈V(Λ) e∈E (Λ) where χΠ denotes the Euler characteristic of the discrete group Π (cf. [41, §II.2.6, Ex. 3]). One has the following: Proposition 7.3. Let (A, Λ) be a finite graph of profinite groups, such that Π = π1 (A, Λ, x0 ) is unimodular and non-compact. Then χ(Π) ≤ 0. Proof. Suppose that one of the open embeddings αe : Ae → At(e) is surjective. Then removing the edge e form Λ, idetifying t(e) with o(e) and taking the induced graph of profinite groups (A′ , Λ′ ) does not change the fundamental group, i.e., one has a topological isomorphism π1 (A, Λ, x0 ) ≃ π1 (A′ , Λ′ , x′0 ). Thus we may assume that the finite graph of profinite groups has the property that none of the injections αe : Ae → At(e) is surjective. Since Π is not compact, Λ cannot be a single vertex. Thus, as \|At(e) : αe (Ae )\| ≥ 2 one has µAe ≥ µAt(e) + µAo(e) for any edge e ∈ E(Λ). Choosing a maximal subtree of Λ then yields the claim. 6For a given finite graph of profinite groups it is possible to decide when Π is unimodular. However, the complexity of this problem will depend on the rank of H 1 (\|Λ\|, Z). E.g., if Λ is a finite tree, then Π will be unimodular. 44 I. CASTELLANO AND TH. WEIGEL 7.3.3. The automorphism group of a locally finite regular tree. Let Td+1 be a (d+1)regular tree, 1 ≤ d < ∞, and let G = Aut(Td+1 )◦ denote the group of automorphism not inverting the orientation of edges. Then, as G is the fundamental group of a finite graph of profinite groups based on a tree with 2 vertices, G is unimodular. One concludes from Bass-Serre theory and §7.3.2 that χ(Aut(Td+1 )◦ ) = (7.13) 1−d · µG e , 1+d where Ge is the stabilizer of an edge. Moreover, χ(Aut(Td+1 )◦ ) < 0. 7.3.4. Chevalley groups over non-discrete non-archimedean local fields. Let X be a simply-connected simple Chevalley group scheme, and let K be a non-archimedean local field with finite residue field F. Put q = \|F\|. Let W be the finite (or spherical) f denote the associated affine Weyl group Weyl group associated to X, and let W f . Let n = associated to W . We also fix a Coxeter generating system ∆ of W f rk(W ) = rk(W ) − 1 = \|∆\| − 1 denote the rank of W as Coxeter group. It is wellknown that X(K) modulo its center is simple. In particular, X(K) is a unimodular t.d.l.c. group. Let Σ be the affine building associated to X(K) (cf. §6.11). The stablizer Iw = stabX(K) (ω) of a simplex ω ∈ Σn of maximal dimension is also called a Iwahori subgroup of X(K). Let Σ(ω) = { ̟ ∈ Σ \| ̟ ⊆ ω } = P(ω) \ {∅}. For ̟ ∈ Σ(ω) let P̟ = stabX(K) (̟) denote the parahoric subgroup associated to ̟, i.e., Pω = Iw. Then X (−1)\|̟\| · µP̟ , (7.14) χ(X(K)) = ̟∈Σ(ω) where \|̟\| is the degree of ̟, i.e., ̟ ∈ Σ\|̟\| (cf. §6.11). Any parahoric subgroup P̟ , ̟ ∈ Σ(ω), corresponds to a unique proper subset ∆(̟) of ∆ of cardinality \|ω\| − \|̟\|, i.e., ∆(ω) = ∅. Moreover, \|P̟ : Iw\| = pW (∆(̟)) (q), (7.15) where pW (∆(̟)) (t) ∈ Z[t] is the Poincaré polynomial associated to the finite Weyl group W (∆(̟)). Thus, by [28, §5.12, Proposition], one obtains that χ(X(K)) = X ̟∈Σ(ω) (7.16) (−1)\|̟\| · µIw \|P̟ : Iw\| = (−1)n+1 · X (−1)\|I\|−1 (−1)n+1 · µIw = · µIw . pW (I) (q) pW f (q) I⊆∆ I6=∆ Q mi ), where mi = di − 1 By R. Bott’s theorem, one has pW f (t) = pW (t)/ 1≤i≤n (1 − t are the exponents of W (cf. [28, §8.9, Theorem]). Thus χ(X(K)) can be rewritten as (7.17) χ(X(K)) = − Y 1 · (q mi − 1) · µIw . pW (q) 1≤i≤n In particular, χ(X(K)) < 0. RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 45 Remark 7.4. One may speculate whether in a situation which is controlled by geometry the Euler-Poincaré characteristic should be related to a meromorphic ζfunction evaluated in −1. There are some phenomenon supporting this idea which arise even in a completely different context (cf. [16]). Indeed, using Hecke algebras one may define a formal Dirichlet series ζG,O (s) for certain t.d.l.c. groups G and any compact open subgroup O of such a group G. It turns out that for the example discussed in §7.3.4 one obtains χ(X(K)) = (−1)n+1 · ζX(K),Iw (−1)−1 · µIw (cf. [22]). Nevertheless, further investigations seem necessary in order to shed light on such a possible connection. We close the discussion of the Euler-Poincaré characteristic with a question which is motivated by the examples discussed in §7.3.2 and §7.3.4. Question 7. Suppose that G is a t.d.l.c. group admitting a C-discrete simplicial G-complex Σ such that (1) (2) (3) (4) (5) dim(Σ) = cdQ (G) < ∞; Σ is of type F∞ ; Σ is locally finite; Σ is G-tame; \|Σ\| admits a G-invariant metric δ : \|Σ\| × \|Σ\| → R+ 0 making (\|Σ\|, δ) a complete CAT(0)-space (cf. §6.9). What further condition ensures that χ(G) ≤ 0? The most trivial examples (cf. §7.3.1) show that the conditions (1)-(5) are not sufficient (for G compact Σ can be chosen to consist of 1 point). Appendix A. Simplicial complexes A.1. Simplicial complexes. A simplicial complex Σ is a non-empty set of nontrivial finite subsets of a set X with the property that if B ∈ Σ and F A ⊆ B then A ∈ Σ. Any simplicial complex Σ carries a canonical grading Σ = q≥0 Σq , where (A.1) Σq = { A ∈ Σ \| card(A) = q + 1 }. Example A.1. Let X be a set, and let Σ[X] ⊆ P(X) \ {∅} denote the set of all non-trivial finite subsets of X. Then Σ[X] is a simplicial complex - the simplicial complex generated by X. Example A.2. Let Γ = (V(Γ), E(Γ)) be a combinatorial graph (cf. §5.4). Then Σ(Γ) given by Σ0 (Γ) = V(Γ) and Σ1 (Γ) = { {o(e), t(e)} \| e ∈ E(Γ) } is a simplicial complex - the simplicial complex associated with the graph Γ. In a simplicial complex Σ and n ≥ 0 the subset (A.2) Σ(n) = { A ∈ Σ \| card(A) ≤ n + 1 } is again simplicial subcomplex of Σ - the n-skeleton of Σ. A subset Y ⊆ Σ0 defines a simplicial subcomplex (A.3) Σ(Y ) = { C ∈ Σ \| C ⊆ Y }. For a simplicial complex Σ one defines the dimension by (A.4) dim(Σ) = min({ q ∈ N0 \| Σq+1 = ∅ } ∪ {∞}). 46 I. CASTELLANO AND TH. WEIGEL In particular, dim(Σ(X)) = card(X) − 1. A simplicial complex of dimension 0 is just a set, while simplicial complexes of dimension 1 are unoriented graphs (cf. Ex. A.2). For A ∈ Σq one defines the set of neighbours NΣ (A) of A by NΣ (A) = { B ∈ Σq+1 \| A ⊂ B }. (A.5) Moreover, Σ is called locally finite if for all A ∈ Σq one has card(NΣ (A)) < ∞. A.2. The topological realization of a simplicial complex. With any simplicial complex Σ ⊆ P(X) \ {∅} one can associate a topological space \|Σ\| - the topological realization of Σ - which is a CW-complex. The space \|Σ\| can be constructed as follows: Let V = R[X] denote the free R-vector space over the set X, and for A ∈ Σq , A = { a0 , . . . , aq }, put Pq Pq (A.6) \|A\| = { j=0 tj · aj \| 0 ≤ tj ≤ 1, j=0 tj = 1 } ⊂ R[X]. S Then \|Σ\| = A∈Σ \|A\| ⊂ R[X], and \|Σ\| carries the weak topology with respect to the embeddings iA : \|A\| → \|Σ\|, where \|A\| ⊂ R[A] carries the induced topology (cf. [43, §3.1]). By construction, one has the following property. Fact A.3. Let Σ be a simplical complex, and let z ∈ \|Σ\|. Then there exists a unique simplex A ∈ Σ such that z is an interior point of \|Σ(A)\| ⊆ \|Σ\|. The topological space \|Σ\| has the following well-known property. Fact A.4. Let Σ be a simplicial complex, and let C be a compact subspace of \|Σ\|. Then there exists a finite subset Y ⊆ Σ1 such that C ⊆ \|Σ(Y )\|. Proof. (cf. [27, p. 520, Prop. A.1]). The just mentioned property has the following consequence. Fact A.5. Let X be a non empty set. Then \|Σ[X]\| is contractable. Proof. Put Σ = Σ[X]. As \|Σ\| is a connected CW-complex, one concludes from Whitehead’s theorem (cf. [27,SThm. 4.5]) that it suffices to show that πi (\|Σ\|, x0 ) = 1 for all i ≥ 1. Note that Σ = A∈Px Σ(A), where 0 (A.7) Px0 = { A ⊆ X \| x0 ∈ A, card(A) < ∞ }. By Fact A.4, for any compact subset C of \|Σ\|, there exists a finite set A ∈ Px0 such that C ⊆ \|Σ(A)\|. Hence πi (\|Σ\|, x0 ) = limA∈P πi (\|Σ(A)\|, x0 ). As \|Σ(A)\| coincides −→ x0 with the standard (card(A) − 1)-simplex, one has πi (\|Σ(A)\|, x0 ) = 1. A.3. The chain complex of a simplicial complex. Let Σ ⊆ P(X) \ {∅} be a simplicial complex. For q ≥ 0 put e q = { x0 ∧ · · · ∧ xq \| {x0 , . . . , xq } ∈ Σq } ⊂ Λq+1 (Q[X]), (A.8) Σ where Λ• (Q[X]) denotes the exterior algebra of the free Q-vector space over the set X, and define e q ) ⊆ Λq+1 (Q[X]). (A.9) Cq (Σ) = span (Σ Q Then (C• (Σ), ∂• ), where (A.10) ∂q (x0 ∧ · · · ∧ xq ) = P j 0≤j≤q (−1) x0 ∧ · · · ∧ xj−1 ∧ xj+1 ∧ · · · ∧ xq is a chain complex of Q-vector spaces - the rational chain complex of the simplicial complex Σ (cf. [43, §4.1]). It has the following well-known property (cf. [43, §4.3, Thm. 8]). RATIONAL DISCRETE COHOMOLOGY FOR T.D.L.C. GROUPS 47 Fact A.6. Let Σ be a simplicial complex. Then one has a canonical isomorphism (A.11) H• (C• (Σ), ∂• ) ≃ H• (\|Σ\|, Q). A.4. Cohomology with compact support. Let Σ ⊆ P(X) \ {∅} be a locally finite simplicial complex. Put X ∗ = { x∗ \| x ∈ X }, and think of X ∗ as a subset of the Q-vector space Cc (X, Q), the functions from X to Q with finite support, i.e., for x, y ∈ X one has x∗ (y) = δx,y . Here δ.,. denotes Kronecker’s function. For q ≥ 0 put (A.12) and define (A.13) b q = { x∗ ∧ · · · ∧ x∗ \| {x0 , . . . , xq } ∈ Σq } ⊂ Λq (Q[X ∗ ]), Σ 0 q b q ) ⊆ Λq (Q[X ∗ ]). Ccq (Σ) = spanQ (Σ By definition, for A ∈ Σq the set (A.14) IΣ (A) = { x ∈ Σ1 \ A \| A ∪ {x} ∈ Σq+1 } is finite. Then (Cc• (Σ), ð• ), where (A.15) ðq (x∗0 ∧ · · · ∧ x∗q ) = P z∈IΣ ({x0 ,...,xq }) z ∧ x∗0 ∧ · · · ∧ x∗q b q , is a cochain complex. It has the following well-known property x∗0 ∧ · · · ∧ x∗q ∈ Σ (cf. [27, p. 242ff]). Fact A.7. Let Σ be a locally finite simplicial complex. Then one has a canonical isomorphism H • (Cc• (Σ), ð• ) ≃ Hc• (\|Σ\|, Q), where Hc• ( , Q) denotes cohomology with compact support with coefficients in Q. A.5. Signed sets. A set X together with a map ¯ : X → X satisfying x̄¯ = x and x̄ 6= x for all x ∈ X will be called a signed set. A map of signed sets φ : X → Y is a map satisfying φ(x̄) = φ(x) for all x ∈ X. Every signed set X defines a Q-vector space (A.16) Q[X] = Q[X]/spanQ { x + x̄ \| x ∈ X }. For a signed set X let X ∗ = { x∗ \| x ∈ X } denote the signed set satisfying x̄∗ = x∗ . Then we may consider Q[X ∗ ] as a Q-subspace of Q[X]∗ = HomQ (Q[X], Q), i.e.,   if y = x, 1 ∗ (A.17) x (y) = −1 if y = x̄,   0 if y 6∈ {x, x̄}. Any map of signed sets φ : X → Y defines a Q-linear map φQ : Q[X] → Q[Y ], but it does not necessarily possess an adjoint map φ∗Q : Q[Y ∗ ] → Q[X ∗ ]. However, if φ is proper, i.e., φ has finite fibres, then there exists a unique map φ∗Q satisfying (A.18) hv ∗ , φQ (u)iY = hφ∗Q (v), uiX , u ∈ Q[X], v ∈ Q[Y ∗ ], where h., .iX and h., .iY denote the evaluation mappings, respectively. Let X + ⊂ X and Y + ⊂ Y be a set of representative for the Z/2Z-orbits on X and Y , respectively. For a map ψ : Q[X] → P Q[Y ] and x ∈ Q[X] - the canonical image of x ∈ X + in Q[X] - one has ψ(x) = y∈Y + λy (x) · y for λy (x) ∈ Q. Put (A.19) supp(ψ, x) = { y ∈ Y + \| λy (x) 6= 0 }. 48 I. CASTELLANO AND TH. WEIGEL We call the map ψ : Q[X] → Q[Y ] proper, if for all y ∈ Y + the set (A.20) csupp(ψ, y) = { x ∈ X + \| y ∈ supp(φ, x) } is finite. 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Generalized Preconditioning and Network Flow Problems Jonah Sherman∗ University of California, Berkeley June 28, 2016(preliminary draft) arXiv:1606.07425v2 [cs.DS] 27 Jun 2016 Abstract We consider approximation algorithms for the problem of finding x of minimal norm kxk satisfying a linear system Ax = b, where the norm k · k is arbitrary and generally non-Euclidean. We show a simple general technique for composing solvers, converting iterative solvers with residual error kAx−bk ≤ t−Ω(1) into solvers with residual error exp(−Ω(t)), at the cost of an increase in kxk, by recursively invoking the solver on the residual problem b̃ = b − Ax. Convergence of the composed solvers depends strongly on a generalization of the classical condition number to general norms, reducing the task of designing algorithms for many such problems to that of designing a generalized preconditioner for A. The new ideas significantly generalize those introduced by the author’s earlier work on maximum flow, making them more widely applicable. As an application of the new technique, we present a nearly-linear time approximation algorithm for uncapacitated minimum-cost flow on undirected graphs. Given an undirected graph with m edges labelled with costs, and n vertices labelled with demands, the algorithm takes −2 m1+o(1) -time and outputs a flow routing the demands with total cost at most (1 + ) times larger than minimal, along with a dual solution proving near-optimality. The generalized preconditioner is obtained by embedding the cost metric into `1 , and then considering a simple hierarchical routing scheme in `1 where demands initially supported on a dense lattice are pulled from a sparser lattice by randomly rounding unaligned coordinates to their aligned neighbors. Analysis of the generalized condition number for the corresponding preconditioner follows that of the classical multigrid algorithm for lattice Laplacian systems. 1 Introduction A fundamental problem in optimization theory is that of finding solutions x of minimum-norm kxk to rectangular linear systems Ax = b. Such solvers have extensive applications, due to the fact that many practical optimization problems reduce to minimum norm problems. When the norm is Euclidean, classical iterative solvers such as steepest descent and conjugate gradient methods produce approximately optimal solutions with residual error kAx − bk exponentially small in iteration count, The rate of exponential convergence depends strongly on the condition number of the square matrix AA∗ . Therefore, the algorithm design process for such problems typically consists entirely of efficiently constructing preconditioners: easily computable left-cancellable operators P, for which the transformed problem PA = Pb is well-conditioned so iterative methods converge rapidly. In a seminal work, Spielman and Teng[17] present a nearly-linear-time algorithm to construct preconditioners with condition number polylogarithmic in problem dimension for a large class of operators A, including Laplacian systems and others arising from discretization of elliptic PDEs. There are many fundamental optimization problems that can be expressed as minimum-norm problems with respect to more general, non-Euclidean norms, including some statistical inference problems and network flow problems on graphs. In particular, the fundamental maximum-flow and uncapacitated minimum-cost flow problems reduce respectively to `∞ and `1 minimum-norm problems in the undirected case. For such problems, there is presently no known black-box iterative solver analagous to those for `2 that converge exponentially with rate independent of problem dimension. Therefore, one is forced to choose between interior point methods to obtain zero or exponentially small error[4], at the cost of iteration count depending strongly on problem dimension, or alternative methods that have error only polynomially small with respect to iterations[15, 9, 14]. ∗ Research supported by NSF Grant CCF-1410022 An important idea implicitly used by classical iterative `2 solvers is residual recursion: reducing the error of an approximate solution x by recursively applying the solver to b̃ = b − Ax. Such recursion is implicit in those solvers due to the fact that in `2 , the map b 7→ xopt is linear, with xopt = (AA∗ )+ b. Therefore, the classical methods are effectively recursing on every iteration, with no distinction between iterations and recursive solves. In general norms, an xopt may not be unique, and there is generally no linear operator mapping b to some xopt . In this paper, we make two primary contibutions. Our first contribution is to extend analysis of residual recursion to arbitrary norms. We present a general and modular framework for efficiently solving minimumnorm problems by composing simple well-known base solvers. The main tool is the composition lemma, describing the approximation parameters of a solver composed via residual recursion of two black-box solvers. The resulting parameters depend on a generalization of the condition number to arbitrary norms; therefore, much like `2 , the algorithm design process is reduced to constructing good generalized preconditioners. Our second contribution is to present, as a practical application of those tools, a nearly-linear time approximation algorithm for the uncapacitated minimum-cost flow problem on undirected graphs. Having established the former framework, the latter algorithm is entirely specified by the construction of a generalized preconditioner, and analyzed by bounding its generalized condition number. Our framework builds upon earlier work[16], where we leveraged ideas introduced by Spielman and Teng to obtain nearly-linear time approximation algorithms for undirected maximum flow. We have since realized that some of the techniques applied to extend `2 -flows to `∞ -flows are in no way specific to flows, and indeed the composition framework presented here is a generalization of those ideas to non-Euclidean minimum-norm problems. Moreover, a significantly simpler max-flow algorithm may be recovered using black-box solvers and general composition as presented here, requiring only the truly flow-specific part of the earlier work: the congestion approximator, now understood to be a specific instance of a more widely-applicable generalized preconditioner. We proceed to discuss those contributions separately in more detail, and outline the corresponding sections of the paper. We also discuss some related work. 1.1 Minimum Norm Problems In section 2, we define the minimum norm problem more precisely, and introduce a useful bicriteria notion of approximation we call (α, β)-solutions. In that notion, α quantifies how much kxk relatively exceeds kxopt k, and β quantifies the residual error kAx − bk, with an appropriate relative scale factor. We may succinctly describe some frequently used algorithms as (α, β)-solvers; we discuss some examples in section 4, including steepest descent and conjugate gradient for `2 , and multiplicative weights for `1 and `∞ . In section 3, we present the composition lemma, which shows that by composing a (α1 , β1 )-solver with a (α2 , β2 )-solver in a black-box manner, we obtain a (α3 , β3 )-solver with different parameter tradeoffs. The composed algorithm’s parameters (α3 , β3 ) depends crucially on a generalization of the classical condition number to rectangular matrices in general norms; Demko[7] provides essentially the exact such generalization needed, so we briefly recall that definition before stating and proving the composition lemma. The proof of the composition lemma is not difficult; it follows quite easily once the right definitions of (α, β)-solutions and a generalized condition number have been established. Nevertheless, the modularity of the composition lemma proves to quite useful. By recursively composing a solver with itself t times, we may decrease β exponentially with t, at the cost of a single fixed increase in α. By composing solvers with different parameters, better parameters may be obtained than any single solver alone. A particularly useful example uses a (M, 0)-solver, where M is uselessly-large on its own, to terminate a chain of t solvers and obtain zero error. As the final step in the chain, we get all of the benefits (zero error), and almost none of the costs, as it contributes M 2−t instead of M to the final solver. By composing the existing solvers discussed in section 4, we obtain solvers for `1 and `∞ problems with residual error exponentially small in t. We state the final composed algorithms parameters. As a result, the algorithm design problem for such problems is reduced to that of designing a generalized preconditioner for A. We briefly discuss such preconditioning in section 5. 1.2 Minimum Cost Flow The second part of this paper applies generalized preconditioning to solve a fundamental problem in network optimization. In the uncapacitated minimum-cost flow problem on undirected graphs, we are given a connected graph G with m edges, each annotated with a cost, and a specified demand at each vertex. The 2 problem is to find an edge flow with vertex divergences equal to the specified demands, of minimal total cost. Our main result is a randomized algorithm that, given such a problem, takes −2 m1+o(1) time and outputs a flow meeting the demands with cost at most (1 + )-times that of optimal. We define various flow-related terms and give a more precise statement of the problem and our result in section 6. We describe and analyze the algorithm in section 7. Having built up our general tools in earlier sections, our task is reduced to the design and analysis of a generalized preconditioner for min-cost flow. We begin by interpreting the edge costs as lengths inducing a metric on the graph, and then apply Bourgain’s smalldistortion embedding[3] into `1 . That is, by paying a small distortion factor in our final condition number, we may focus entirely on designing a preconditioner for min-cost flow in `1 space. The construction of our precondtitioner is based on an extremely simple hierarchical routing scheme, inspired by a combination of the multigrid algorithm for grid Laplacians, and the Barnes-Hut algorithm for n-body simulation[2]. The routing scheme proceeds on a sequence of increasingly dense lattices V0 , V1 , . . . in `1 , where Vt are the lattice points with spacing 2−t , and the input demands are specified on the densest level VT . Starting at level t = T , the routing scheme sequentially eliminates the demand supported on Vt \ Vt−1 by pulling the demand from level t − 1 using a random-rounding based routing: each vertex x ∈ Vt picks a random nearest neighbor in Vt−1 , corresponding to randomly rounding its unaligned coordinates, and then pulls a fractional part of its demand from that neighbor via any shortest path. Afterwards, all demands on Vt \ Vt−1 are met, leaving a reduced problem with demands supported on Vt−1 . The scheme then recurses on the reduced problem. At level 0, the demands are supported on the corners of a hypercube, for which the simple routing scheme of pulling all demand from a uniformly random corner performs well-enough. Having described the routing scheme, we need not actually carry it out. Instead, our preconditioner crudely estimates the cost of that routing. Furthermore, while the routing has been described on an infinite lattice, we’ll observe that if the demands are only initially supported on a small set of n vertices, the support remains small throughout. 1.3 Related Work Spielman and Teng present a nearly-linear time algorithm for preconditioning and solving symmetric diagonally dominant matrices, including those of graph Laplacians[17]. The ideas introduced in that work have led to breakthroughs in various network cut and flow algorithms for the case of undirected graphs. Christiano et. al. apply the solver as a black-box to approximately solve the maximum flow problem in Θ(m4/3 ) time. Madry[13] opens the box and uses the ideas directly to give a family of algorithms for approximating cut-problems, including a mo(1) -approximation in m1+o(1) time. Both the present author[16] and Kelner et. al.[11] combine madry’s ideas with `∞ optimization methods to obtain (1 − )-approxmations to maximum flow in m1+o(1) −O(1) time. While those two algorithms use similar ideas, the actual path followed to achieve the result is rather different. Kelner et. al. construct an oblivious routing scheme that is no(1) -competetive, and then show how to use it to obtain a flow. The algorithm maintains a demand-respecting flow at all times, and minimizes a potential function measuring edge congestion. In contrast, our earlier algorithm short-cuts the need to explicitly construct the routing scheme by using Madry’s construction directly, maintaining a flow that is neither demand nor capacity respecting, aiming to minimize a certain potential function that measures both the congestion and the demand error. For min-cost flow, earlier work considers the more general directed, capacitated case, with integer capacities in {1, . . . , U }. The Θ(nm log log U )-time double-scaling algorithm of Ahuja, Goldberg, Orlin, and Tarjan[1] remained the fastest algorithm for solving min-cost flow for 25 years between its publication and the Laplacian breakthrough. Shortly after that breakthrough, Daitch and Spielman[5] showed how to use the Laplacian solver with interior-point methods to obtain an Θ(m3/2 log2 U )-time algorithm. Lee and Sidford √ further reduce this to Θ(m n logO(1) U ) by a general improvement in interior point methods[12]. We are not aware of prior work 2 Approximate Solutions Let X , Y be finite dimensional vector spaces, where X is also a Banach space, and let A ∈ Lin(X , Y) be fixed throughout this section. We consider the problem of finding a minimal norm pre-image of a specified b in the image of A; that is, finding x ∈ X with Ax = b and kxkX minimal. Let xopt be an exact solution, with Axopt = b, and kxopt kX minimal. There are multiple notions of 3 approximate solutions for this problem. The most immediate is x ∈ X with Ax = b and kxk ≤α kxopt k (1) . We call such x an (α, 0)-solution; we shall be interested in finding (1 + , 0)-solutions for small . A weaker notion of approximation is obtained by further relaxing Ax = b to Ax u b. Quantifying that requires more structure on Y, so let us further assume Y to also be a Banach space. In that case, we say x is an (α, β)-solution if equation 1 holds and kAx − bk ≤β (2) kAkkxopt k The practical utility of such solutions depends on the application. However, the weaker notion has the distinct advantage of being approachable by a larger family of algorithms, such as penalty and dual methods, by avoiding equality constraints. We discuss such existing algorithms in section 4, but mention that they typically yield (1, )-solutions after some number of iterations. For `2 , the iteration dependency on is O(log(1/)), while for some more general norms it is −O(1) . 3 Recursive Composition and Generalized Condition Numbers We now consider how to trade an increase in α for a decrease in β. Residual recursion suggests a natural strategy: after finding an (α, β)-solution, recurse on b̃ = b − Ax. More precisely, we define the composition of two algorithms as follows. Definition 3.1. Let Fi be an (αi , βi )-algorithm for A, for i ∈ {1, 2}. The composition F2 ◦ F1 takes input b, and first runs F1 on b to obtain x. Next, setting b̃ = b − Ax, it runs F2 on input b̃ to obtain x̃. Finally, it outputs x + x̃. Success of composition depends on kb̃kY being small implying b̃ has a small-norm pre-image. The extent to which that is true is quantified by the condition number of A. The condition number in `2 may be defined several ways, resulting in the same quantity. When generalized to arbitrary norms, those definitions differ. We recall two natural definitions, following Demko[7]. Definition 3.2. The non-linear condition number of A : X → Y is kAkX →Y kxkX κ̃X →Y (A) = min : Ax 6= 0 kAxkY The linear condition number is defined by κX →Y (A) = min {kAkX →Y kGkY→X : G ∈ Lin(Y, X ) : AGA = A} Of course, κ̃ ≤ κ. Having defined the condition number, we may now state how composition affects the approximation parameters. Theorem 3.3 (Composition). Let Fi be an (αi , βi /κ̃)-algorithm for A : X → Y, where A has non-linear condition number κ̃ Then, the composition F2 ◦ F1 is an (α1 + α2 β1 , β1 β2 /κ̃-algorithm for the same problem. Before proving lemma 3.3, we state two useful corollaries. The first concerns the result of recursively composing a (α, β/κ̃)-algorithm with itself. Corollary 3.4. Let F be a (α, β/κ̃)-algorithm for β < 1. Let F t be the sequence formed by iterated composition, with F 1 = F , F t+1 = F t ◦ F . Then, F t is a (α/(1 − β), β t /κ̃)-algorithm. Proof. By induction on t. To start, an (α, β/κ̃)-algorithm is trivially a (α/(1−β), β/κ̃)-algorithm. Assuming the claim holds for F t , lemma 3.3 implies F t+1 is a (α + αβ/(1 − β), β t+1 /κ̃)-algorithm. We observe that to obtain (1 + O(), δ)-solution, only the first solver in the chain need be very accurate. Corollary 3.5. Let F be a (1+, /2κ̃-solver and G be a (2, 1/2κ̃)-solver. Then, Gt ◦F is a (1+5, 2−t−1 /κ̃solver. For some problems, there is a very simple (M, 0)-solver known. A final composition with that solver serves to elimate the error. Corollary 3.6. Let F be a (1 + , δ/κ̃)-solver and G be a (M, 0)-solver. Then G ◦ F is a (1 + (1 + δM ), 0)solver. 4 3.1 Proof of lemma 3.3 Since F1 is an (α1 , β1 /κ̃)-algorithm, we have kxk kb̃k kAk ≤ α1 kxopt k β1 kxopt k κ̃ ≤ By the definition of κ̃, kx̃opt k ≤ κ̃ kb̃k ≤ β1 kxopt k kAk Since F2 is an (α2 , β2 /κ̃)-algorithm, kx̃k kAx̃ − b̃k kAk ≤ α2 kx̃opt k ≤ α2 β1 kxkopt ≤ β2 β 1 β2 kx̃opt k ≤ κ̃ κ̃ The conclusion follows from kx + x̃k ≤ kxk + kx̃k. 4 Solvers for `p The recursive composition technique takes existing (α, β)-approximation algorithms and yields new algorithms with different approximation parameters. In this section, we discuss the parameters of existing well-known base solvers. Let A : Rm → Rn be linear and fixed throughout this section. To concisely describe results and ease comparison, for a norm k · kX on Rm and a norm k · kY in Rn , we write (α, β)X →Y to denote an (α, β) solution with respect to A : X → Y. The classical `2 solvers provide a useful goalpost for comparison. Theorem 4.1 ([4]). The steepest-descent algorithm produces a (1, δ)2→2 -solution after O(κ2→2 (A)2 log(1/δ)) simple iterations The conjugate gradient algorithm produces a (1, δ)2→2 -solution after O(κ2→2 (A) log(1/δ)) simple iterations. There are many existing algorithms for p-norm minimization, and composing them yields algorithms with exponentially small error. For this initial manuscript, we simply state the `1 version required for min-cost flow. The general cases consist of describing the many existing algorithms[14] in terms of (α, β)-solvers. Theorem 4.2. There is a (1 + , δ)1→1 -solver with simple iterations totalling O κ̃1→1 (A)2 log(m) −2 ) + log(δ −1 ) Theorem 4.3. There is a (1 + , δ)∞→∞ -solver with simple iterations totalling O κ̃∞→∞ (A)2 log(n) −2 + log(δ −1 ) The preceding theorems follow by combining the multiplicative weights algorithm[15, 9] with the composition lemma. The former algorithm applies in a more general online setting; for our applications, the algorithm yields the “weak” approximation algorithms that shall be composed. A support oracle for a compact-convex set C takes input y and returns some x ∈ C maximizing x · C. Let 4n be the unit simplex in Rn (i.e., the convex hull of the standard basis). The multiplicative weights method finds approximate solutions to the saddle point problem, max min w · z w∈4n z∈C In particular, it obtains an additive -approximation in O(ρ2 −2 log n) iterations with each iteration taking O(n) time plus a call to a support oracle for C. 5 Let Bp be the unit ball in `p . A point w ∈ B1 is represented as w = w+ − w− for (w+ , w− ) ∈ 42n . The base solvers follow by considering the saddle-point problems, max min y ∗ · (Ax − µb) max min y ∗ · (Ax − µb) y ∗ ∈b1 x∈b∞ x∈b1 y ∗ ∈b∞ where µ is scaled via binary-search as needed. The search adds a factor O(log κ̃) to the complexity. This factor may be avoided by using `p -norm for p = log n and p = (log m)/ regularization, respectively, instead. 5 Generalized Preconditioning Let A : X → Y be linear. The minimum-norm problem for A does not intrinsically require Y to be normed; nor does the definition of a (α, 0)-solution. Thus, an algorithm designer seeking to solve a class of minimum norm problems may freely choose how to norm Y. The best choice requires balancing two factors. First, the norm should be sufficiently “simple” that (1, ) solutions for polynomially-small are easy to find. Second, the norm should be chosen so that κ̃(A)X →Y is small. If only the latter constraint existed, the ideal choice would be the optimal A pre-image norm,1 kbkopt(A) = min{kxkX : Ax = b} By construction, κ̃(A)X →opt(A) = 1. Of course, simply evaluating that norm is equivalent to the original problem we aim to solve. It follows that Y should be equipped with the relatively closest norm to opt(A) that can be efficiently minimized. Following the common approach in `2 , when X = `p , we propose to choose a generalized pre-conditioner P : Y → `p injective and consider the norm kPbkp . Again, P should be chosen such that κ̃(PA)p→p is small, and P, P∗ are easy to compute. 6 Graph Problems In this section we discuss some applications of generalized preconditioning to network flow problems. Let G = (V, E) be an undirected graph with n vertices and m edges. While undirected, we assume the edges are oriented arbitrarily. We denote by V ∗ , E ∗ the spaces of real-valued functions f : V → R and f : E → R on vertices and edges, respectively. We denote by V, E the corresponding dual spaces of demands and flows. For S ⊆ V , we write 1S ∈ V ∗ for the indicator function on S. The discrete derivative operator D∗ : V ∗ → E ∗ is defined by (D∗ f )(xy) = f (y) − f (x) for each oriented edge xy. The constant function has zero derivative, D∗ 1V = 0. The (negative) adjoint −D : E → V is the discrete divergence operator ; for a flow  ∈ E and vertex x ∈ V , −(D)(x) is the net quantity being transported away from vertex x by the flow. A single-commodity flow problem is specified by a demand vector b ∈ V, and requires finding a flow  ∈ E satisfying D = b, minimizing some cost function. When the cost-function is a norm on E, the problem is a minimum-norm pre-image problem. Assuming G is connected, b has a pre-image iff the total demand 1V · b is zero. We hereafter assume G to be connected and total demands equal to zero. Having established the generalized preconditioning framework, we may now design fast algorithms for fundamental network flow problems by designing generalized preconditioners for the corresponding minimumnorm problems. As a result, we obtain nearly-linear time algorithms for max-flow and uncapacitated min-cost flow in undirected graphs. The max-flow algorithm was presented previously[16], and involved a combination of problem-specific definitions (e.g. congestion approximators) and subroutines (e.g. gradient-based `∞ minimization). The brief summary in subsection 6.1 shows how the same result may be immediately obtained by combining theorem ?? with only small part of the earlier flow-specific work[16]: the actual construction of the preconditioner. In section 7, we present a nearly-linear time algorithm for uncapacitated min-cost flows, another fundamental network optimization problem. Once again, all that is needed is description and analysis of the preconditioner. 1 We remark this is only defined on the image space of A, rather than the entire codomain. However, we only consider b in that image space 6 6.1 Max-Flow A capacitated graph associates with each edge e, a capacity c(e) > 0. Given capacities, we define the capacity P norm on E ∗ by kf kc∗ = e c(e)\|f (e)\|; the dual congestion norm on E is defined by kkc = maxe \|(e)\| c(e) . ∗ The boundary capacity of a set S ⊆ V is kD 1S kc , which we abbreviate as c(∂S). The single-commodity flow problem with capacity norm is undirected maximum flow problem, and is a fundamental problem in network design and optimization. Following the work of Spielman and Teng for `2 -flows[17], and Madry[13] for cut problems, the author has obtained a nearly-linear time algorithm for maximum flow[16]. We have since realized that only certain parts of the ideas introduced in that paper are actually specific to maximum-flow, and the present work has been obtained through generalizing the other parts. The (previously presented as) flow-specific composition and optimization parts of that paper are entirely subsumed by the earlier sections of this paper. The truly flow-specific part of that work required to complete the algorithm is the generalized preconditioner construction, which we have previously called a congestion-approximator. We briefly describe the main ideas of the construction, and some simple illustrative examples. For the full details, we refer the reader to the third section of [16]. Let F ⊆ 2V be a family of subsets. Such a family, together with a capacity norm on E, induces a cut\|1S ·b\| . That is, for each set in the family, consider the congestion semi-norm on V via kbkc(F) = maxS∈F c(∂S) ratio of the total aggregate demand in that set to the total capacity of all edges entering that set. If the indicator functions of sets in F span V, then it is a norm. Note also that we may write kbkc(F) = kPbk∞ , 1S ·b . where P : Rn → RF is defined by (Pb)S = c(∂S) V If F is taken to be the full power set F = 2 , the max-flow, min-cut theorem is equivalent to the statement that the non-linear condition number κ̃c→c(F) (D∗ ) is exactly one. If F is taken to be the family of singleton sets, the non-linear condition number is the inverse of the combinatorial conductance of G. That is, for highconductance graphs, composition yields a fast algorithm requiring only a simple diagonal preconditioner. This is closely analagous to the situation for `2 , where diagonal preconditioning yields fast algorithms for graphs of large algebraic conductance. A√particularly illustrative example consists of a unit-capacity 2t × 2t 2-D grid. With low (Θ(2−t ) = Θ(1/ n) conductance, the singleton family F performs poorly. A simple but dramatically better family consists of taking F to be all power-of-two size, power-of-two aligned subgrids. The latter family yields linear condition number O(t) = O(log n). Furthermore, computing the aggregate demand in all such sets requires only O(n) time, with each sub-grid aggregate equal to the sum of its four child aggregates. This algorithm is analagous to an `∞ version of multi-grid [?]. Note that the indicator “step” functions 1S are different from the bilinear “pyramid” functions used by multigrid. Theorem 6.1 ([16]). There is an algorithm that given a capacitated graph (G, c) with n vertices and m edges, takes m1+o(1) time and outputs a data structure of size n1+o(1) that efficiently represents a preconditioner P with κ(PD)c→∞ ≤ no(1) . Given the data structure, P and P∗ can be applied in n1+o(1) time. Let C : Rm → E be the diagonal capacity matrix; (Cx)(e) = c(e)x(e). The nearly-linear time algorithm follows by using the preconditioner P in the preceding theorem, applying theorem ?? to the problem of minimizing kxk∞ subject to PDCx = b. A final minor flow-specific part is needed to achieve zero error: the simple (m, 0)-approximation algorithm that consists of routing all flow through a maximum-capacity spanning tree. 7 Uncapacitated Min-Cost Flow In this section, we present a nearly-linear time algorithm for the uncapacitated minimum-cost flow problem on undirected graphs. A cost graph associates P a cost l(e) > 0 with each edge e in G. The cost norm on the space of flows E is defined by kkl = e k(e)kl(e). The undirected uncapacitated minimum-cost flow problem is specified by the graph G, lengths `, and demands b ∈ V, and consists of finding  ∈ E with D = b minimizing kkl . Note that, unlike max-flow, the single-source uncapacitated min-cost flow problem is significantly easier than the distributed source case: the optimal solution is to route along a single-source shortest path away from the source. The problem only becomes interesting with distributed sources and sinks. As a new application of generalized preconditioning, we obtain a nearly-linear time algorithm for approximately solving this problem. 7 Theorem 7.1. There is a randomized algorithm that, given a length-graph G = (V, E, l), takes and outputs a (1 + , 0)-solution to the undirected uncapacitated min-cost flow problem. m1+o(1) 2 time As expected, the bulk of the algorithm lies in constructing a good preconditioner. After introducing some useful definitions and lemmas, we spend the majority of this section constructing and analyzing a preconditioner for this problem. Theorem 7.2. There is a randomized algorithm that, given a length-graph G, takes O(m log2 n + n1+o(1) ) time and outputs a n1+o(1) × n matrix P with κlto∞ (PD) ≤ no(1) . Every column of P has no(1) non-zero entries. Dual to cost is the stretch norm on E ∗ is kf kl∗ = maxe \|f (e)\|/l(e). We say a function φ ∈ V is L-Lipschitz iff kD∗ φkl∗ ≤ L; that is, \|φ(x) − φ(y)\| ≤ Ll(xy) for all edges xy. The dual problem is to maximize φ · b over 1-Lipschitz φ ∈ V. A length function l induces an intrinsic metric d : V × V → R, with distances determined by shortest paths. The preceding definition of Lipschitz is equivalent to \|φ(x) − φ(y)\| ≤ Ld(x, y) for all x, y ∈ V . That is, the Lipschitz constant of φ depends only on the metric, and is otherwise independent of the particular graph or edge lengths inducing that metric. As the dual problem depends only on the induced metric, so must the value of the min-cost flow. Therefore, we may unambiguously write kbkopt(d) = max{φ · b : φ is 1-Lipschitz w.r.t. d} The monotonicity of cost with distance follows immediately. Lemma 7.3. If d(x, y) ≤ d̃(x, y) for all x, y ∈ V, then for all b ∈ V⊥1 , kbkopt(d) ≤ kbkopt(d̃) We recall that in some cases, the identity operator is a good preconditioner; for max-flow, such is the case in constant-degree expanders. For min-cost flow, the analagous case is a metric where distances between any pair of points differ relatively by small factors. Definition 7.4. Let U ⊆ V . We say U is r-separated if d(x, y) ≥ r for all x, y ∈ U with x 6= y. We say U is R-bounded if d(x, y) ≤ R for all x, y ∈ U . Lemma 7.5. Let U ⊂ V be R-bounded and r-separated with respect to d. Let b ∈ V⊥1 be demands supported on U . Then, R r kbk1 ≤ kbkopt(d) ≤ kbk1 2 2 Proof. Note 2r kbk1 is exactly the min-cost of routing b in the metric where all distances are exactly r (consider the star graph with edge lengths r/2, with a leaf for each x ∈ U ). The inequalities follow by monotonicity. To complete the proof of theorem 7.1, assuming theorem 7.2, we also need a simple approximation algorithm that yields a poor approximation, but with zero error. This is easily achieved by the algorithm that routes all flow through a minimum cost (i.e., total length) spanning tree T . Lemma 7.6. Let T be a minimum cost spanning tree with respect to l on G. Consider the algorithm that, given b ∈ V⊥1 , outputs the flow  ∈ E routing b using only the edges in T . Then, kk ≤ nkbkopt(d) Proof. The flow on T is unique, and therefore, optimal for the metric d̃ induced by paths restricted to lie in T . As T is a minimum cost spanning tree, d̃(x, y) ≤ nd(x, y). We now prove theorem 7.1. Let L : Rm → E be the diagonal length matrix (Lx)(e) = `(e)x(e), and let P be the preconditioner from theorem 7.2. Computing P takes O(m log2 n) + n1+o(1) time. The minimum-cost flow problem is equivalent to the problem of minimizing kxk1 subject to PDL−1 x = Pb. As P has n1+o(1) 1+o(1) non-zero entries and κ1→1 (PDL−1 ) ≤ no(1) , the total running time is m 2 . 8 8 Preconditioning Min-Cost Flow A basic concept in metric spaces is that of an embedding: ψ : V → Ṽ , with respective metrics d, d̃. An embedding has distortion L if for some µ > 0, 1≤µ d̃(ψ(x), ψ(y)) ≤L d(x, y) Metric embeddings are quite useful for preconditioning; if d embeds into d̃ with distortion L, then the monotonicity lemma implies the relative costs of flow problems differ by a factor L. It follows that if we can construct a preconditioner P for the min-cost flow problem on d̃, the same P may be used to precondition min-cost flow on d, with condition number at most L-factor worse. Our preconditioner uses embeddings in two ways. The preconditioner itself is based on a certain hierarchical routing scheme. Hierarchical routing schemes based on embeddings into tree-metrics have been studied extensively(see e.g. [8]), and immediately yield preconditioners with polylogarithmic condition number. However, we are unable to efficiently implement those schemes in nearly-linear time. Instead, we apply Bourgain’s O(log n)-distortion embedding[3] into `1 . That is, by paying an O(log n) factor in condition number, we may completely restrict our attention to approximating min-cost flow in `1 space. Theorem 8.1 (Bourgain’s Embedding[3]). Any n-point metric space embeds into `p space with distortion O(log n). If d is the intrinsic metric of a length-graph G = (V, E) with m edges, there is a randomized algorithm that takes O(m log2 n) time and w.h.p. outputs such an embedding, with dimension O(log2 n). The complexity of constructing and evaluating√our preconditioner is exponential in the embedding’s √ dimension. Therefore, we reduce the dimension to Θ( log n), incuring another distortion factor of exp(O( log n)). Theorem 8.2 (Johnson-Lindenstrauss Lemma[10, 6]). Any n points in Euclidean space embed into kdimensional Euclidean space√with distortion nO(1/k) √ , for k ≤ Ω(log n). In particular, k = O(log n) yields O(1) distortion, and k = O( log n) yields exp(O( log n)) distortion. If the original points are in d > k dimensions, there is a randomized algorithm that w.h.p outputs such an embedding in O(nd) time. Our√reduction consists of first embedding the vertices into `2 via theorem 8.1, reducing the dimension to k = O( log √ n) using lemma 8.2, and then using the identity embedding of `2 into `1 , for a total distortion of exp(O( log n)). To estimate routing costs in `1 , we propose an extremely simple hierarchical routing algorithm in `1 , and show it achieves a certain competetive ratio. The routing scheme applies to the exponentially large graph corresponding to a lattice discretization of the k-dimensional cube in `1 ; after describing such a scheme, we show that if the demands in that graph are supported on n vertices, the distributed routing algorithm finds no demand to route in most of the graph, and the same routing algorithm may be carried out in a k − d-tree instead of the entire lattice. On the lattice, the routing algorithm consists of sequentially reducing the support set of the demands from a lattice with spacing 2−t−1 to a lattice with spacing 2−t , until all demand is supported on the corners of a cube containing the original demand points. On each level, the demand on a vertex not appearing in the coarser level is eliminated by pushing its demand to a distribution over the aligned points nearby. 8.1 Lattice Algorithm For t ∈ Z, let Vt = (2−t Z)k be the lattice with spacing 2−t . We design a routing scheme and preconditioner for min-cost flow in `1 , with initial input demands bT supported on a bounded subset of VT . For t = T, T −1, . . ., the routing scheme recursively reduces a min-cost flow problem with demands supported on Vt to a min-cost flow problem with demands supported on the sparser lattice Vt−1 . Given demands bt supported on Vt , the scheme consists of each vertex x ∈ Vt pulling its demand bt (x) uniformly from the closest points to x in Vt−1 along any shortest path, the lengths of which are at most k2−t . For example, a point x = (x1 , . . . , xk ) ∈ V1 with j ≤ k non-integer coordinates and demand b1 (x) pulls b1 (x)2−j units of flow from each of the 2j points in V0 corresponding to rounding those j coordinates in any way. By construction, all points in Vt \ Vt−1 have their demands met exactly, and we are left with a reduced problem residual demands bt−1 supported on Vt−1 . As we show, eventually the demands are supported entirely on the 2k corners of a hypercube forming a 9 bounding box of the original support set, at which point the the simple scheme of distributing the remaining demand uniformly to all corners is a k-factor approximation. We do not actually carry out the routing; rather, we state a crude upper-bound on its cost, and then show that bound itself exceeds the true min-cost by a factor of some κ. For the preconditioner, we shall only require the sequence of reduced demands bt , bt−1 , . . ., and not the flow actually routing them. Theorem 8.3. kbt kopt ≤ t X k2−s kbs k1 ≤ 2k(t + 1)kbt kopt s=0 Assuming theorem 8.3 holds, we take our preconditioner to be a matrix P such that, kPbT k1 = T X k2−t kbt k1 t=0 The proof of theorem 8.3 uses two essential properties of the routing scheme described. The first is that the cost incurred in the reduction from bt to bt−1 is not much larger than the min-cost of routing bt . The second is the reduction does not increase costs. That is, the true min-cost for routing the reduced problem bt−1 is at most that of routing the original demands bt . Lemma 8.4. Let bt−1 be the reduced demands produced when the routing scheme is given bt . Then, 1. The cost incurred by that level of the scheme is at most k2−t kbt k1 ≤ 2kkbt kopt 2. The reduction is non-increasing with respect to min-cost: kbt−1 kopt ≤ kbt kopt Assuming lemma 8.4, we now prove 8.3. We argue the sum is an upper bound on the total cost incurred by the routing scheme, when applied to bt . The left inequality follows immediately, as the true min-cost is no larger. The total cost of the scheme is at most the cost incurred on each level, plus the cost of the final routing of b0 . Lemma 8.4(a) accounts for all terms above s = 0. For the final routing, the demands b0 are supported in {0, 1}k , so the cost of routing all demand to a random corner is at most 12 kkb0 k1 . For the right inequality, we observe t X k2−t kbt k1 ≤ s=0 t X 2kkbs kopt s=0 The inequality follows termwise, using lemma 8.4(a) for s ≥ 1; the s = 0 case follows from lemma 7.5 because V0 is 1-separated. 8.2 Proof of Lemma 8.4 For the first part, each vertex x ∈ Vt routes its demand bt (x) to the closest points in Vt−1 any shortest path. As such paths have length at most k2−t , each point x contributes at most k2−t \|bt (x)\|. For the second part, by scale-invariance it suffices to consider t = 1. Moreover, it suffices to consider the case where b1 consists of a unit demand from between two points x, y ∈ V1 with kx − yk1 = 21 . To see this, we observe any flow routing arbitrary demands b1 consists of a sum of such single-edge flows. As the reduction is linear, if the cost of each single-edge demand-pair is not increased, then the cost their sum is not increased. By symmetry, it suffices to consider x, y with x = (0, z) and y = ( 21 , z). Then, the reduction distributes x’s demand to a distribution over (0, z 0 ) where z 0 ∼ Z 0 ; y’s demand is split over (0, z 0 ) and (1, z 0 ) where z 0 ∼ Z 0 . Therefore, half of the demand at (0, z 0 ) is cancelled, leaving the residual problem of routing 1/2 unit from (0, z 0 ) to (1, z 0 ). That problem has cost 1/2, by routing each fraction directly from (0, z 0 ) to (1, z 0 ). References [1] Ravindra K Ahuja, Andrew V Goldberg, James B Orlin, and Robert E Tarjan. Finding minimum-cost flows by double scaling. Mathematical programming, 53(1-3):243–266, 1992. 10 [2] Josh Barnes and Piet Hut. A hierarchical o (n log n) force-calculation algorithm. nature, 324(6096):446– 449, 1986. [3] J. Bourgain. On lipschitz embedding of finite metric spaces in hilbert space. Israel Journal of Mathematics, 52(1):46–52, 1985. [4] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, New York, NY, USA, 2004. [5] Samuel I Daitch and Daniel A Spielman. Faster approximate lossy generalized flow via interior point algorithms. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 451–460. ACM, 2008. [6] Sanjoy Dasgupta and Anupam Gupta. An elementary proof of a theorem of johnson and lindenstrauss. Random Structures & Algorithms, 22(1):60–65, 2003. [7] Stephen Demko. Condition numbers of rectangular systems and bounds for generalized inverses. Linear Algebra and its Applications, 78:199–206, 1986. [8] Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 448–455. ACM, 2003. [9] Yoav Freund and Robert E Schapire. Adaptive game playing using multiplicative weights. Games and Economic Behavior, 29(1):79–103, 1999. [10] William B Johnson and Joram Lindenstrauss. Extensions of lipschitz mappings into a hilbert space. Contemporary mathematics, 26(189-206):1, 1984. [11] Jonathan A Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 217–226. Society for Industrial and Applied Mathematics, 2014. [12] Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming: Solving linear programs in o (vrank) iterations and faster algorithms for maximum flow. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 424–433. IEEE, 2014. [13] Aleksander Madry. Fast approximation algorithms for cut-based problems in undirected graphs. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 245–254. IEEE, 2010. [14] Yu Nesterov. Smooth minimization of non-smooth functions. Math. Program., 103(1):127–152, May 2005. [15] Serge A. Plotkin, David B. Shmoys, and Eva Tardos. Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research, 20:257–301, 1995. [16] Jonah Sherman. Nearly maximum flows in nearly linear time. In Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on, pages 263–269. IEEE, 2013. [17] Daniel A. Spielman and Shang-Hua Teng. Nearly-linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. CoRR, abs/cs/0607105, 2006. 11	8
Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time ⋆ Seok-Hee Hong1 and Hiroshi Nagamochi2 arXiv:1608.07662v2 [cs.CG] 2 Sep 2016 1 University of Sydney, Australia seokhee.hong@sydney.edu.au 2 Kyoto University, Japan nag@amp.i.kyoto-u.ac.jp Abstract. Thomassen characterized some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph is drawable in straight-lines if and only if it does not contain the configuration [C. Thomassen, Rectilinear drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988]. In this paper, we characterize some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph can be reembedded into a straight-line drawable 1-plane embedding of the same graph if and only if it does not contain the configuration. Re-embedding of a 1-plane embedding preserves the same set of pairs of crossing edges. We give a linear-time algorithm for finding a straight-line drawable 1plane re-embedding or the forbidden configuration. 1 Introduction Since the 1930s, a number of researchers have investigated planar graphs. In particular, a beautiful and classical result, known as Fáry’s Theorem, asserts that every plane graph admits a straight-line drawing [5]. Indeed, a straight-line drawing is the most popular drawing convention in Graph Drawing. More recently, researchers have investigated 1-planar graphs (i.e., graphs that can be embedded in the plane with at most one crossing per edge), introduced by Ringel [13]. Subsequently, the structure of 1-planar graphs has been investigated [4, 12]. In particular, Pach and Toth [12] proved that a 1-planar graph with n vertices has at most 4n − 8 edges, which is a tight upper bound. Unfortunately, testing the 1-planarity of a graph is NP-complete [6, 11], however linear-time algorithms are available for special subclasses of 1-planar graphs [1, 3, 7]. Thomassen [14] proved that every 1-plane graph (i.e., a 1-planar graph embedded with a given 1-plane embedding) admits a straight-line drawing if and only if it does not contain any of two special 1-plane graphs, called the Bconfiguration or W-configuration, see Fig. 1. ⋆ Research supported by ARC Future Fellowship and ARC Discovery Project DP160104148. This is an extended abstract. For a full version with omitted proofs, see [9]. u2 outer face u2 v1 u4 c u1 outer face c s u3 u3 u1 w4c w3c u1 u3 v2 (a) u4 v3 c w2c w1c u2 (c) (b) Fig. 1. (a) B-configuration with three edges u1 u2 , u2 u3 and u3 u4 and one crossing c made by an edge pair {u1 u2 , u3 u4 }, where edge u2 u3 may have a crossing when the configuration is part of a 1-plane embedding; (b) W-configuration with four edges u1 u2 , u2 u3 , v1 v2 and v2 v3 and two crossings c and s made by edge pairs {u1 u2 , v2 v3 } and {u2 u3 , v1 v2 }, where possibly u1 = v1 and u3 = v3 ; (c) Augmenting a crossing c ∈ χ made by edges u1 u3 and u2 u4 with a new cycle Qc = (u1 , w1c , u2 , w2c , u3 , w3c , u4 , w4c ) depicted by gray lines. Recently, Hong et al. [8] gave an alternative constructive proof, with a lineartime testing algorithm and a drawing algorithm. They also showed that some 1-planar graphs need an exponential area with straight-line drawing. We call a 1-plane embedding straight-line drawable (SLD for short) if it admits a straight-line drawing, i.e., it does not contain a B- or W-configuration by Thomassen [14]. In this paper, we investigate a problem of “re-embedding” a given non-SLD 1-plane embedding γ into an SLD 1-plane embedding γ ′ . For a given 1-plane embedding γ of a graph G, we call another 1-plane embedding γ ′ of G a cross-preserving embedding of γ if exactly the same set of edge pairs make the same crossings in γ ′ . More specifically, we first characterize the forbidden configuration of 1-plane embeddings that cannot admit an SLD cross-preserving 1-plane embedding. Based on the characterization, we present a linear-time algorithm that either detects the forbidden configuration in γ or computes an SLD cross-preserving 1-plane embedding γ ′ . Formally, the main problem considered in this paper is defined as follows. Re-embedding a 1-Plane Graph into a Straight-line Drawing Input: A 1-planar graph G and a 1-plane embedding γ of G. Output: Test whether γ admits an SLD cross-preserving 1-plane embedding γ ′ , and construct such an embedding γ ′ if one exists, or report the forbidden configuration. To design a linear-time implementation of our algorithm in this paper, we introduce a rooted-forest representation of non-intersecting cycles and an efficient procedure of flipping subgraphs in a plane graph. Since these data structure and procedure can be easily implemented, it has advantage over the complicated decomposition of biconnected graphs into triconnected components [10] or the SPQR tree [2]. 2 2 Plane Embeddings and Inclusion Forests Let U be a set of n elements, and let S be a family of subsets S ⊆ U . We say that two subsets S, S ′ ⊆ U are intersecting if none of S ∩ S ′ , S − S ′ and S ′ − S is empty. We call S a laminar if no two subsets in S are intersecting. For a laminar S, the inclusion-forest of S is defined to be a forest I = (S, E) of a disjoint union of rooted trees such that (i) the sets in S are regarded as the vertices of I, and (ii) a set S is an ancestor of a set S ′ in I if and only if S ′ ⊆ S. Lemma 1. For a cyclic sequence (u1 , u2 , . . . , uδ ) of δ ≥ 2 elements, define an interval (i, j) to be the set of elements uk with i ≤ k ≤ j if i ≤ j and (i, j) = (i, δ) ∪ (1, j) if i > j. Let S be a set of intervals. A pair of two intersecting intervals in S (when S is not a laminar) or the inclusion-forest of S (when S is a laminar) can be obtained in O(δ + \|S\|) time. Throughout the paper, a graph G = (V, E) stands for a simple undirected graph. The set of vertices and the set of edges of a graph G are denoted by V (G) and E(G), respectively. For a vertex v, let E(v) be the set of edges incident to v, N (v) be the set of neighbors of v, and deg(v) denote the degree \|N (v)\| of v. A simple path with end vertices u and v is called a u, v-path. For a subset X ⊆ V , let G − X denote the graph obtained from G by removing the vertices in X together with the edges in ∪v∈X E(v). A drawing D of a graph G is a geometric representation of the graph in the plane, such that each vertex of G is mapped to a point in the plane, and each edge of G is drawn as a curve. A drawing D of a graph G = (V, E) is called planar if there is no edge crossing. A planar drawing D of a graph G divides the plane into several connected regions, called faces, where a face enclosed by a closed walk of the graph is called an inner face and the face not enclosed by any closed walk is called the outer face. A planar drawing D induces a plane embedding γ of G, which is defined to be a pair (ρ, ϕ) of the rotation system (i.e., the circular ordering of edges for each vertex) ρ, and the outer face ϕ whose facial cycle Cϕ gives the outer boundary of D. Let γ = (ρ, ϕ) be a plane embedding of a graph G = (V, E). We denote by F (γ) the set of faces in γ, and by Cf the facial cycle determined by a face f ∈ F , where we call a subpath of Cf a boundary path of f . For a simple cycle C of G, the plane is divided by C in two regions, one containing only inner faces and the other containing the outer area, where we say that the former is enclosed by C or the interior of C, while the latter is called the exterior of C. We denote by Fin (C) the set of inner faces in the interior of C, by Ein (C) the set of edges in E(Cf ) with f ∈ Fin (C), and by Vin (C) the set of end-vertices of edges in Ein (C). Analogously define Fex (C), Eex (C) and Vex (C) in the exterior of C. Note that E(C) = Ein (C) ∩ Eex (C) and V (C) = Vin (C) ∩ Vex (C). For a subgraph H of G, we define the embedding γ\|H of γ induced by H to be a sub-embedding of γ obtained by removing the vertices/edges not in H, keeping the same rotation system around each of the remaining vertices/crossings and the same outer face. 3 2.1 Inclusion Forests of Inclusive Set of Cycles In this and next subsections, let (G, γ) stand for a plane embedding of γ = (ρ, ϕ) of a biconnected simple graph G = (V, E) with n = \|V \| ≥ 3. Let C be a simple cycle in G. We define the direction of C to be an ordered pair (u, v) with uv ∈ E(C) such that the inner faces in Fin (C) appear on the right hand side when we traverse C in the order that we start u and next visit v. For simplicity, we say that two simple cycles C and C ′ are intersecting if Fin (C) and Fin (C ′ ) are intersecting. Let C be a set of simple cycles in G. We call C inclusive if no two cycles in C are intersecting, i.e., {Fin (C) \| C ∈ C} is a laminar. When C is inclusive, the inclusion-forest of C is defined to be a forest I = (C, E) of a disjoint union of rooted trees such that: (i) the cycles in C are regarded as the vertices of I, and (ii) a cycle C is an ancestor of a cycle C ′ in I if and only if Fin (C ′ ) ⊆ Fin (C). Let I(C) denote the inclusion-forest of C. For a vertex subset X ⊆ V , let C(X) denote the set of cycles C ∈ C such that x ∈ V (C) for some vertex x ∈ X, where we denote C({v}) by C(v) for short. Lemma 2. For (G, γ), let C be a set ofP simple cycles of G. Then any of the following tasks can be executed in O(n + C∈C \|E(C)\|) time. (i) Decision of the directions of all cycles in C; (ii) Detection of a pair of two intersecting cycles in C when C is not inclusive, and construction of the inclusion-forests I(C(v)) for all vertices v ∈ V when C is inclusive; and (iii) Construction of the inclusion-forest I(C) when C is inclusive. 2.2 Flipping Spindles A simple cycle C of G is called a spindle (or a u, v-spindle) of γ if there are two vertices u, v ∈ V (C) such that no vertex in V (C) − {u, v} is adjacent to any vertex in the exterior of C, where we call vertices u and v the junctions of C. Note that each of the two subpaths of C between u and v is a boundary path of some face in F (γ). Given (G, γ), we denote the rotation system around a vertex v ∈ V by ργ (v). For a spindle C in γ, let J(C) denote the set of the two junctions of C. Flipping a u, v-spindle C means to modify the rotation system of vertices in Vin (C) as follows: (i) For each vertex w ∈ Vin (C) − J(C), reverse the cyclic order of ργ (w); and (ii) For each vertex u ∈ J(C), reverse the order of subsequence of ργ (u) that consists of vertices N (u) ∩ Vin (C). Every two distinct spindles C and C ′ in γ are non-intersecting, and they always satisfy one of Ein (C) ∩ Ein (C ′ ) = ∅, Ein (C) ⊆ Ein (C ′ ), and Ein (C ′ ) ⊆ Ein (C). Let C be a set of spindles in γ, which is always inclusive, and let I(C) denote the inclusion-forest of C. 4 When we modify the current embedding γ by flipping each spindle in C, the resulting embedding γC is the same, independent from the ordering of the flipping operation to the spindles, since for two spindles C and C ′ which share a common junction vertex u ∈ J(C) ∩ J(C ′ ), the sets N (u) ∩ Vin (C) and N (u) ∩ Vin (C ′ ) do not intersect, i.e., they are disjoint or one is contained in the other. Define the depth of a vertex v ∈ V in I to be the number of spindles C ∈ C such that v ∈ Vin (C) − J(C), and denote by p(v) the parity of depth of vertex v, i.e., p(v) = 1 if the depth is odd and p(v) = −1 otherwise. For a vertex v ∈ V , let C[v] denote the set of spindles C ∈ C such that v ∈ J(C), and let γC[v] be the embedding obtained from γ by flipping all spindles in C[v]. Let revhσi mean the reverse of a sequence σ. Then we see that ργC (v) = ργC[v] (v) if p(v) = 1; and ργC (v) = revhργC[v] (v)i otherwise. To obtain the embedding γC from the current embedding γ by flipping each spindle in C, it suffices to show how to compute each of p(v) and ργC[v] (v) for all vertices v ∈ V . Lemma 3. Given (G, γ), let C be a set P of spindles of γ. Then any of the following tasks can be executed in O(n + C∈C \|E(C)\|) time. (i) Decision of parity p(v) of all vertices v ∈ V ; and (ii) Computation of ργC[v] (v) for all vertices v ∈ V . 3 Re-embedding 1-plane Graph and Forbidden Configuration A drawing D of a graph G = (V, E) is called a 1-planar drawing if each edge has at most one crossing. A 1-planar drawing D of graph G induces a 1-plane embedding γ of G, which is defined to be a tuple (χ, ρ, ϕ) of the crossing system χ of E, the rotation system ρ of V , and the outer face ϕ of D. The planarization G(G, γ) of a 1-plane embedding γ of graph G is the plane embedding obtained from γ by regarding crossings also as graph vertices, called crossing-vertices. The set of vertices in G(G, γ) is given by V ∪χ. For a notational convenience, we refer to a subgraph/face of G(G, γ) as a subgraph/face in γ. Let γ = (χ, ρ, ϕ) be a 1-plane embedding of graph G. We call another 1-plane embedding γ ′ = (χ′ , ρ′ , ϕ′ ) of graph G a cross-preserving 1-plane embedding of γ when the same set of edge pairs makes crossings, i.e., χ = χ′ . In other words, the planarization G(G, γ ′ ) is another plane embedding of G(G, γ) such that the alternating order of edges incident to each crossing-vertex c ∈ χ is preserved. To eliminate the additional constraint on the rotation system on each crossingvertex c ∈ χ, we introduce “circular instances.” We call an instance (G, γ) of 1-plane embedding circular when for each crossing c ∈ χ, the four end-vertices of the two crossing edges u1 u3 and u2 u4 that create c (where u1 , u2 , u3 and u4 appear in the clockwise order around c) are contained in a cycle Qc = (u1 , w1c , u2 , w2c , u3 , w3c , u4 , w4c ) of eight crossing-free edges for some vertices wic , i = 1, 2, 3, 4 of degree 2, as shown in Fig. 1(c). By definition, c and each wic not necessarily appear along the same facial cycle in the planarization G(G, γ). For example, path (v, w, u) is part of such a cycle Qs for the crossing s in the 5 circular instance in Fig. 2(a), but c and w are not on the same facial cycle in the planarization. A given instance can be easily converted into a circular instance by augmenting the end-vertices of each pair of crossing edges as follows. In the plane graph, G(G, γ), for each crossing-vertex c ∈ χ and its neighbors u1 , u2 , u3 and u4 that appear in the clockwise order around c, we add a new vertex wic , i = 1, 2, 3, 4 and eight new edges ui wic and wic ui+1 , i = 1, 2, 3, 4 (where u5 means u1 ) to form a cycle Qc of length 8 whose interior contains no other vertex than c. Let H be the resulting graph augmented from G, and let Γ be the resulting 1-plane embedding of H augmented from γ. Note that \|V (H)\| ≤ \|V (G)\| + 4\|χ\| holds. We easily see that if γ admits an SLD cross-preserving embedding γ ′ then Γ admits an SLD cross-preserving embedding Γ ′ . This is because a straight-line drawing Dγ ′ of γ ′ can be changed into a straight-line drawing DΓ ′ of some crosspreserving embedding Γ ′ of Γ by placing the newly introduced vertices wic within the region sufficiently close to the position of c. We here see that cycle Qc can be drawn by straight-line segments without intersecting with other straight-line segments in Dγ ′ . Note that the instance (G, γ ′ ) remains circular for any cross-preserving embedding γ ′ of γ. In the rest of paper, let (G, γ) stand for a circular instance (G = (V, E), γ = (χ, ρ, ϕ)) with n ≥ 3 vertices and let G denote its planarization G(G, γ). Fig. 2 shows examples of circular instances (G, γ), where the vertexconnectivity of G is 1. As an important property of a circular instance, the subgraph G(0) with crossing-free edges is a spanning subgraph of G and the four end-vertices of any two crossing edges are contained in the same block of the graph G(0) . The biconnectivity is necessary to detect certain types of cycles by applying Lemma 2. outer face ϕ outer face ϕ u’ u’ w s c’ v s u s’ s’ c u v c’ c z v’ v’ (a) (b) Fig. 2. Circular instances (G, γ) with a cut-vertex u of G, where the crossing edges are depicted by slightly thicker lines: (a) hard B-cycles C = (u, c, v, s) and C ′ = (u′ , c′ , v ′ , s′ ), (b) hard B-cycle C = (u, c, v, s) and a nega-cycle C ′ = (u′ , c′ , v ′ , s′ ) whose reversal is a hard B-cycle, where vertices u, v, u′ , v ′ ∈ V and crossings c, s, c′ , s′ ∈ χ. 6 3.1 Candidate Cycles, B/W Cycle, Posi/Nega Cycle, Hard/Soft Cycle For a circular instance (G, γ), finding a cross-preserving embedding of γ is effectively equivalent to finding another plane embedding of G so that all the current B- and W-configurations are eliminated and no new B- or W-configurations are introduced. To detect the cycles that can be the boundary of a B- or Wconfiguration in changing the plane embedding of G, we categorize cycles containing crossing vertices in G. A candidate posi-cycle (resp., candidate nega-cycle) in G is defined to be a cycle C = (u, c, v) or C = (u, c, v, s) in G with u, v ∈ V and c, s ∈ χ such that the interior (resp., exterior) of C does not contain a crossing-free edge uv ∈ E and any other crossing vertex c′ adjacent to both u and v. outer face ϕ ϕ u e c’ v (a) C C p c s c’ ϕ ϕ u c’ c e c v v (c) C C n (b) C C p u u c’ c s v (d) C C n Fig. 3. Candidate posi- and nega-cycles C = (u, c, v) and C = (u, c, v, s) in G, where white circles represent vertices in V while black ones represent crossings in χ: (a) candidate posi-cycle of length 3, (b) candidate posi-cycle of length 4, (c) candidate nega-cycle of length 3, and (d) candidate nega-cycle of length 4. Fig. 3(a)-(b) and (c)-(d) illustrate candidate posi-cycles and candidate negacycles, respectively. Let C p and C n be the sets of candidate posi-cycles and candidate nega-cycles, respectively. By definition we see that the set C p ∪ C n ∪ {Cf \| f ∈ F (γ)} is inclusive, and hence \|C p ∪ C n ∪ {Cf \| f ∈ F (γ)}\| = O(n). A candidate posi-cycle C with C = (u, c, v) (resp., C = (u, c, v, s)) is called a B-cycle if (a)-(B): the exterior of C contains no vertices in V − {u, v} adjacent to c (resp., contains exactly one vertex in V − {u, v} adjacent to c or s). Note that uv ∈ E when C = (u, c, v) is a B-cycle, as shown in Fig. 4(a). Fig. 4(b) and (d) illustrate the other types of B-cycles. A candidate posi-cycle C = (u, c, v, s) is called a W-cycle if (a)-(W): the exterior of C contains no vertices in V − {u, v} adjacent to c or s. Fig. 4(c) and (e) illustrate W-cycles. Let CW (resp., CB ) be the set of W-cycles (resp., B-cycles) in γ. Clearly a W-cycle (resp., B-cycle) gives rise to a W-configuration (resp., B-configuration). 7 outer face ϕ ϕ u e c (a) C C B-C+ c v (f) C C- v (b) C C B-C+ (c) C C W-C+ c c v (e) C C W C+ ϕ u u c c s v (h) C C n-C- (g) C C- c v s v u s (d) C C B C+ ϕ u ϕ u s c s s ϕ u v ϕ u s c s v ϕ ϕ u v (i) C C n-C- Fig. 4. Illustration of types of cycles C = (u, c, v) and C = (u, c, v, s) in G, where white circles represent vertices in V while black ones represent crossings in χ: (a) B-cycle of length 3, which is always soft, (b) soft B-cycle of length 4, (c) soft W-cycle, (d) hard B-cycle of length 4, (e) hard W-cycle, (f) nega-cycle whose reversal is a hard B-cycle, (g) nega-cycle whose reversal is a hard W-cycle, (h) candidate nega-cycle of length 4 that is not a nega-cycle whose reversal is a hard B-cycle, and (i) candidate nega-cycle of length 4 that is not a nega-cycle whose reversal is a hard W-cycle. Conversely, by choosing a W-configuration (resp., B-configuration) so that the interior is minimal, we obtain a W-cycle (resp., B-cycle). Hence we observe that the current embedding γ admits a straight-line drawing if and only if CW = CB = ∅. A W- or B-cycle C is called hard if (b): length of C is 4, and the interior of C = (u, c, v, s) contains no inner face f whose facial cycle Cf contains both vertices u and v, i.e., some path connects c and s without passing through u or v. On the other hand, a W- or B-cycle C = (u, c, v, s) of length 4 that does not satisfy condition (b) or a B-cycle of length 3 is called soft. We also call a hard B- or W-cycle a posi-cycle. Fig. 4(d) and (e) illustrate a hard B-cycle and a hard W-cycles, respectively, whereas Fig. 4(a) and (b) (resp., (c)) illustrate soft B-cycles (resp., a soft Wcycle). A cycle C = (u, c, v, s) is called a nega-cycle if it becomes a posi-cycle when an inner face in the interior of C is chosen as the outer face. In other words, a nega-cycle is a candidate nega-cycle C = (u, c, v, s) of length 4 that satisfies the following conditions (a’) and (b’), where (a’) (resp., (b’)) is obtained from the above conditions (a)-(B) and (a)-(W) (resp., (b)) by exchanging the roles of “interior” and “exterior”: (a’): the interior of C contains at most one vertex in V − {u, v} adjacent to c or s; and 8 (b’): the exterior of C contains no face f whose facial cycle Cf contains both vertices u and v. Fig. 4(f) and (g) illustrate nega-cycles, whereas Fig. 4(h) and (i) illustrate candidate nega-cycles that are not nega-cycles. Let C + (resp., C − ) denote the set of posi-cycles (resp., nega-cycles) in γ. By definition, it holds that C + ⊆ CW ∪ CB ⊆ C p and C − ⊆ C n . 3.2 Forbidden Cycle Pairs We define a forbidden configuration that characterizes 1-plane embeddings, which cannot be re-embedded into SLD ones. A forbidden cycle pair is defined to be a pair {C, C ′ } of a posi-cycle C = (u, c, v, s) and a posi- or nega-cycle C ′ = (u′ , c′ , v ′ , s′ ) in G with u, v, u′ , v ′ ∈ V and c, s, c′ , s′ ∈ χ to which G has a u, u′ -path P1 and a v, v ′ -path P2 such that: (i) when C ′ ∈ C + , paths P1 and P2 are in the exterior of C and C ′ , i.e., V (P1 ) − {u, u′ }, V (P2 ) − {v, v ′ } ⊆ Vex (C) ∩ Vex (C ′ ), where C and C ′ cannot have any common inner face; and (ii) when C ′ ∈ C − , paths P1 and P2 are in the exterior of C and the interior of C ′ , i.e., V (P1 ) − {u, u′ }, V (P2 ) − {v, v ′ } ⊆ Vex (C) ∩ Vin (C ′ ), where C is enclosed by C ′ . In (i) and (ii), P1 and P2 are not necessary disjoint, and possibly one of them consists of a single vertex, i.e., u = u′ or v = v ′ . The pair of cycles C and C ′ in Fig. 5(a) (resp., Fig. 5(b)) is a forbidden cycle pair, because there is a pair of a u, u′ -path P1 = (u, x, z, y, u′ ) and a v, v ′ -path P2 = (v, x′ , z, y ′ , v ′ ) that satisfy the above conditions (i) (resp., (ii)). Note that the pair of cycles C and C ′ in Fig. 2(a)-(b) is not forbidden cycle pair, because there are no such paths. Our main result of this paper is as follows. Theorem 1. A circular instance (G, γ) admits an SLD cross-preserving embedding if and only if it has no forbidden cycle pair. Finding an SLD cross-preserving embedding of γ or a forbidden cycle pair in G can be computed in linear time. Proof of necessity: The necessity of the theorem follows from the next lemma. For a cycle C = (u, c, v, s) ∈ C + (resp., C − ) with u, v ∈ V and c, s ∈ χ in G, we call a vertex z ∈ V an in-factor of C if the exterior of C ∈ C + (resp., the interior of C ∈ C − ) has a z, u-path Pz,u and a z, v-path Pz,v , i.e., V (Pz,u − {u}) ∪ V (Pz,v − {v}) is in Vex (C) (resp., Vin (C)). Paths Pz,u and Pz,v are not necessarily disjoint. Lemma 4. Given G = G(G, γ), let γ ′ be a cross-preserving embedding of γ. Then: (i) Let z ∈ V be an in-factor of a cycle C ∈ C + ∪ C − in G. Then cycle C is a posi-cycle (resp., a nega-cycle) in G(G, γ ′ ) if and only if z is in the exterior (resp., interior) of C in γ ′ ; 9 outer face ϕ outer face ϕ u’ y u’ u x u x y c’ s’ s z c z s’ s x’ x’ v v’ c’ c y’ v y’ v’ (a) (b) Fig. 5. Illustration of circular instances (G, γ) with a cut-vertex z of G, where the crossing edges are depicted by slightly thicker lines: (a) forbidden cycle pair with hard B-cycles C = (u, c, v, s) and C ′ = (u′ , c′ , v ′ , s′ ) (b) forbidden cycle pair with a hard B-cycle C = (u, c, v, s) and a nega-cycle C ′ = (u′ , c′ , v ′ , s′ ) whose reversal is a hard B-cycle, where vertices u, v, u′ , v ′ ∈ V and crossings c, s, c′ , s′ ∈ χ. (ii) For a forbidden cycle pair {C, C ′ }, one of C and C ′ is a posi-cycle in G(G, γ ′ ) (hence any cross-preserving embedding of γ contains a B- or W-configuration and (G, γ) admits no SLD cross-preserving embedding). Proof of sufficiency: In the rest of paper, we prove the sufficiency of Theorem 1 by designing a linear-time algorithm that constructs an SLD crosspreserving embedding of an instance without a forbidden cycle pair. 4 Biconnected Case In this section, (G, γ) stands for a circular instance such that the vertex-connectivity of the plane graph G is at least 2. In a biconnected graph G, any two posi-cycles C = (u, c, v, s), C ′ = (u′ , c′ , v ′ , s′ ) ∈ C + with u, v, u′ , v ′ ∈ V give a forbidden cycle pair if they do not share an inner face, because there is a pair of u, u′ -path and v, v ′ -path in the exterior of C and C ′ . Analogously any pair of a posi-cycle C and a nega-cycle C ′ such that C ′ encloses C is also a forbidden cycle pair in a biconnected graph G. To detect such a forbidden pair in G in linear time, we first compute the sets Cp , Cn , CW , CB , C + and C − in γ in linear time by using the inclusion-forest from Lemma 2. Lemma 5. Given (G, γ), the following in (i)-(iv) can be computed in O(n) time. (i) (ii) (iii) (iv) The sets Cp , Cn and the inclusion-forest I of Cp ∪ Cn ∪ {Cf \| f ∈ F (γ)}; The sets CW and CB ; The sets C + , C − and the inclusion-forest I ∗ of C + ∪ C − ; and A set {fC \| C ∈ (CW ∪CB )− C + } such that fC is an inner face in the interior of a soft B- or W-cycle C with V (Cf ) ⊇ V (C). 10 Given (G, γ), a face f ∈ F (γ) is called admissible if all posi-cycles enclose f but no nega-cycle encloses f . Let A(γ) denote the set of all admissible faces in F (γ). Lemma 6. Given (G, γ), it holds A(γ) 6= ∅ if and only if no forbidden cycle pair exists in γ. A forbidden cycle pair, if one exists, and A(γ) can be obtained in O(n) time. By the lemma, if (G, γ) has no forbidden cycle pair, i.e., A(γ) 6= ∅, then any new embedding obtained from γ by changing the outer face with a face in A(γ) is a cross-preserving embedding of γ which has no hard B- or W-cycle. 4.1 Eliminating Soft B- and W-cycles Suppose that we are given a circular instance (G, γ) such that G is biconnected and C + = ∅. We now show how to eliminate all soft B- and W-cycles in G in linear time using the inclusion-forest from Lemma 2 and the spindles from Lemma 3. Lemma 7. Given (G, γ) with C + = ∅, there exists an SLD cross-preserving embedding γ ′ = (χ, ρ′ , ϕ′ ) of γ such that V (Cϕ′ ) ⊇ V (Cϕ ) for the facial cycle Cϕ (resp., Cϕ′ ) of the outer face ϕ (resp., ϕ′ ), which can be constructed in O(n) time. Given an instance (G, γ) with a biconnected graph G, we can test whether it has either a forbidden cycle pair or an admissible face by Lemmas 5 and 6. In the former, it cannot have an SLD cross-preserving embedding by Lemma 4. In the latter, we can eliminate all hard B- and W-cycles by choosing an admissible face as a new outer face, and then eliminate all soft B- and W-cycles by a flipping procedure based on Lemma 7. All the above can be done in linear time. To treat the case where the vertex-connectivity of G is 1 in the next section, we now characterize 1-plane embeddings that can have an SLD cross-preserving embedding such that a specified vertex appears along the outer boundary. For a vertex z ∈ V in a graph G, we call a 1-plane embedding γ of G z-exposed if vertex z appears along the outer boundary of γ. We call (G, γ) z-feasible if it admits a z-exposed SLD cross-preserving embedding γ ′ of γ. Lemma 8. Given (G, γ) such that A(γ) 6= ∅, let z be a vertex in V . Then: (i) The following conditions are equivalent: (a) γ admits no z-exposed SLD cross-preserving embedding; (b) A(γ) contains no face f with z ∈ V (Cf ); and (c) G has a posi- or nega-cycle C to which z is an in-factor; (ii) A z-exposed SLD cross-preserving embedding or a posi- or nega-cycle C to which z is an in-factor can be computed in O(n) time. 11 5 One-connected Case In this section, we prove the sufficiency of Theorem 1 by designing a linear-time algorithm claimed in the theorem. Given a circular instance (G, γ), where G may be disconnected, obviously we only need to test each connected component of G separately to find a forbidden cycle pair. Thus we first consider a circular instance (G, γ) such that the vertex-connectivity of G is 1; i.e., G is connected and has some cut-vertices. A block B of G is a maximal biconnected subgraph of G. For a biconnected graph G, we already know how to find a forbidden cycle pair or an SLD crosspreserving embedding from the previous section. For a trivial block B with \|V (B)\| = 2, there is nothing to do. If some block B of G with \|V (B)\| ≥ 3 contains a forbidden cycle pair, then (G, γ) cannot admit any SLD cross-preserving embedding by Lemma 4. We now observe that G may contain a forbidden cycle pair even if no single block of G has a forbidden cycle pair. Lemma 9. For a circular instance (G, γ) such that the vertex-connectivity of G is 1, let B1 and B2 be blocks of G and let P1,2 be a z1 , z2 -path of G with the minimum number of edges, where V (Bi ) ∩ V (P1,2 ) = {zi } for each i = 1, 2. If γ\|Bi has a posi- or nega-cycle Ci to which zi is an in-factor for each i = 1, 2, then {C1 , C2 } is a forbidden cycle pair in G. For a linear-time implementation, we do not apply the lemma for all pairs of blocks in B. A block of G is called a leaf block if it contains only one cut-vertex of G, where we denote the cut-vertex in a leaf block B by vB . Without directly searching for a forbidden cycle pair in G, we use the next lemma to reduce a given embedding by repeatedly removing leaf blocks. Lemma 10. For a circular instance (G, γ) such that the vertex-connectivity of G = G(G, γ) is 1 and a leaf block B of G such that γ\|B is vB -feasible, let H = G−(V (B)−{vB }) be the graph obtained by removing the vertices in V (B)−{vB }. Then (i) The instance (H, γ\|H ) is circular; and ∗ (ii) If (H, γ\|H ) admits an SLD cross-preserving embedding γH , then an SLD ∗ cross-preserving embedding γ of γ can be obtained by placing a vB -exposed ∗ SLD cross-preserving embedding γB of γ\|B within a space next to the cut∗ vertex vB in γH . Given a circular instance (G, γ) such that G = G(G, γ) is connected, an algorithm Algorithm Re-Embed-1-Plane for Theorem 1 is designed by the following three steps. The first step tests whether G has a block B such that γ\|B has a forbidden cycle pair, based on Lemma 8. If one exists, the algorithm outputs a forbidden cycle pair and halts. After the first step, no block has a forbidden cycle pair. In the current circular instance (G, γ), one of the following holds: 12 (i) the number of blocks in G is at least two and there is at most one leaf block B such that γ\|B is not vB -feasible; (ii) G has two leaf blocks B and B ′ such that γ\|B is not vB -feasible and γ\|B ′ is not vB ′ -feasible; and (iii) the number of blocks in G is at most one. In (ii), vB is an in-factor of a cycle C in γ\|B and vB ′ is an in-factor of a cycle C ′ in γ\|B ′ by Lemma 8, and we obtain a forbidden cycle pair {C, C ′ } by Lemma 9. Otherwise if (i) holds, then we can remove all leaf blocks B such that γ\|B is not vB -feasible by Lemma 10. The second step keeps removing all leaf blocks B such that γ\|B is not vB -feasible until (ii) or (iii) holds to the resulting embedding. If (i) occurs, then the algorithm outputs a forbidden cycle pair and halts. When all the blocks of G can be removed successfully, say in an order of B 1 , B 2 , . . . , B m , the third step constructs an embedding with no B- or Wcycles by starting with such an SLD embedding of B m and by adding an SLD embedding of B i to the current embedding in the order of i = m−1, m−2, . . . , 1. By Lemma 10, this results in an SLD cross-preserving embedding of the input instance (G, γ). ∗ Note that we can obtain an SLD cross-preserving embedding γH 1 of γ in the third step when the first and second step did not find any forbidden cycle pair. Thus the algorithm finds either an SLD cross-preserving embedding of γ or a forbidden cycle pair. This proves the sufficiency of Theorem 1. By the time complexity result from Lemma 8, we see that the algorithm can be implemented in linear time. References 1. Auer, C., Bachmaier, C., Brandenburg, F. J. , Gleißner, A., Hanauer, K., Neuwirth D., Reislhuber J.: Outer 1-Planar Graphs, Algorithmica, 74(4), 1293–1320 (2016) 2. 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Action selection in growing state spaces: Control of Network Structure Growth arXiv:1606.07777v2 [cs.SI] 27 Dec 2016 Dominik Thalmeier1 , Vicenç Gómez 2 , and Hilbert J. Kappen 1 1 Donders Institute for Brain, Cognition and Behaviour Radboud University Nijmegen, the Netherlands 2 Department of Information and Communication Technologies Universitat Pompeu Fabra. Barcelona, Spain Abstract The dynamical processes taking place on a network depend on its topology. Influencing the growth process of a network therefore has important implications on such dynamical processes. We formulate the problem of influencing the growth of a network as a stochastic optimal control problem in which a structural cost function penalizes undesired topologies. We approximate this control problem with a restricted class of control problems that can be solved using probabilistic inference methods. To deal with the increasing problem dimensionality, we introduce an adaptive importance sampling method for approximating the optimal control. We illustrate this methodology in the context of formation of information cascades, considering the task of influencing the structure of a growing conversation thread, as in Internet forums. Using a realistic model of growing trees, we show that our approach can yield conversation threads with better structural properties than the ones observed without control. Keywords: control, complex Networks, sampling, conversation threads 1 Introduction Many complex systems can be described as dynamic processes which are characterized by the topology of an underlying network. Examples of such systems are human interaction networks, where the links may represent transmitting opinions Olfati-Saber et al. [2007], Dai and Mesbahi [2011], Centola and Baronchelli [2015], habits Centola [2010], Farajtabar et al. [2014], money Gai and Kapadia [2010], Amini et al. [2016], Giudici and Spelta [2016] or viruses Pastor-Satorras and Vespignani [2001], Eguı́luz and Klemm [2002]. Being able to control, or just influence in some way, the dynamics of such complex networks may lead to important progress, for example, avoiding financial crises, preventing epidemic outbreaks or maximizing information spread in marketing campaigns. The control of the dynamics on networks is a very challenging problem that has attracted significant interest recently Liu et al. [2011], Cornelius et al. [2013], Gao et al. [2014], Yan et al. [2015]. Existing approaches typically consider network controllability as the controllability of 1 the dynamical system induced by the underlying network structure. While it is agreed that network controllability critically depends on the network structure, the problem of how to control the network structure itself while it is evolving remains open. The network structure is determined by the dynamics of addition and deletion of nodes and links over time. In this paper, we address the problem of influencing this dynamics in the framework of stochastic optimal control. The standard way to address these problems is through the Bellman equation and dynamic programming. Dynamic programming is only feasible in small problems and requires approximations when the state and action spaces are large. In the setting of network growth, this problem is more severe, since the state space increases (super-)exponentially with the number of nodes. In order to deal with this curse of dimensionality, we propose to approximate the network growth control problem by a special class of stochastic optimal control problems, known as Kullback-Leibler (KL) control or Linearly-Solvable Markov Decision Problems (LMDPs) Todorov [2009], Kappen et al. [2012]. For this class of problems, one can use efficient adaptive importance sampling methods that scale well in high dimensions. The optimal solution for the KL-control problem tends to be sparse, so that only a few next states become relevant, effectively reducing the branching factor of the original problem. The obtained solution of the KL-control problem is then used to compute the optimal action in the original problem that does not belong to the KL-control class. In the next section we present our proposed general methodology. We then apply it to a realistic problem: influencing the growth process of cascades in online forums, in order to maximize structural network measures that are connected to the quality of an online conversation thread. We conclude the paper with a discussion. 2 Optimal Network Growth as a Control Problem We now formulate the network growth control problem as a stochastic optimal control problem. Let xt ∈ X , with X being the set of all possible network structures, denote the growing structure (state) of the network at time t and let P (x0 \|x, u) describe the network dynamics, where the control variable u ∈ U denotes possible actions we can perform in order to manipulate the network. Let us label the default action, which means not interacting with the system, with u = 0. We denote the corresponding dynamics without control as the uncontrolled process p(x0 \|x) := P (x0 \|x, u = 0). At each time-step t, we incur an arbitrary cost function on the network state r(x, t) which is assigned when the state is reached. The state cost r(x, t) penalizes network structures that are not convenient in the particular context under consideration. For example, if one wants to favour networks with large average clustering coefficient C(x), then r(x, t) = −C(x). Alternatively, one can consider more complex functions, such as the structural virality or Wiener index Mohar and Pisanski [1988], as proposed recently Goel et al. [2015], to maximize the influence in a social network. In general, any measure that can be (efficiently) computed from x fits the presented framework. Our objective is to find the control function u(x, t) : X × N 7→ U which minimizes the total cost over a time horizon T, starting at state x at initial time t = 0 * T + X 0 C (x, t = 0, u(·)) = r (x, 0) + r xt0 , t , (1) t0 =t+1 2 P (x1:T \|x,u(·),t=0) where the expectation is taken with respect to the probability P (x1:T \|x, u(·), t = 0) over paths x1:T in the state space, given state x at time Q t = 0 using the control-function u(·). The probability of a path is given by P (xt+1:T \|x, u(·), t = 0) = T−1 s=t P (xs+1 \|xs , u(xs , s), s). Computing the optimal control can be done by dynamic programming Bertsekas [1995]. We introduce the optimal cost-to-go J(x, t) = min C (x, t, u(·)) , (2) u(·) which is an expectation of the cumulative cost starting at state x and time t and acting optimally thereafter. This can be computed using the Bellman equation J(x, t) = min r(x, t) + J(x0 , t + 1) P (x0 \|x,u,t) . (3) u From J(x, t), the optimal control is obtained by a greedy local optimization: u∗ (x, t) = argminu r(x, t) + J(x0 , t + 1) P (x0 \|x,u,t) . (4) In general, the solution to equation (3) can be computed recursively using dynamic programming Bertsekas [1995] for all possible states. This is however infeasible for controlling network growth, as the computation is of polynomial order in the number of states and the state space of networks increases super-exponentially on the number of nodes. E.g. for directed unweighed networks, there 2 are 2N possible networks with N labelled nodes. 3 Approximating the network growth problem by a Kullback-Leibler control problem In this section we present our main approach, which first computes the optimal cost-to-go on a relaxed problem and then uses it as a proxy for the original optimal cost-to-go. In the next subsection, we introduce the class of KL-control problems that we use as a relaxation. We then illustrate KL-control using a tractable example of tree growth. In subsection 3.3, we explain how can we approximate the KL-control solution using the cross-entropy method. Finally, in subsection 3.4 we show how can we use that result to compute the action selection in the original problem. 3.1 Kullback-Leibler control In order to efficiently compute the optimal cost-to-go, we make the assumption that our controls directly specify the transition probabilities between two subsequent network structures, e.g. P (x0 \|x, u(t)) ≈ u(x0 \|x, t). Further, we define the natural growth process of the network (the uncontrolled dynamics) as a Markov chain with transition probabilities p(x0 \|x). Because our influence on the network dynamics is limited, we add a regularization term to the total cost defined in equation (1) that penalizes deviations from p(x0 \|x). The approximated control cost becomes * T + X λ CKL (x, t, u(·)) =λKL [u (xt+1:T \|x, t) k p (xt+1:T \|x, t)] + r (x, t) + r xt0 , t0 , (5) t0 =t+1 3 u(xt+1:T \|x,t) with the KL-divergence KL [u (xt+1:T \|x, t) k p (xt+1:T \|x, t)] = u (xt+1:T \|x, t) log p (xt+1:T \|x, t) , u(xt+1:T \|x,t) which measures the closeness of the two path distributions, p (xt+1:T \|x, t) and u (xt+1:T \|x, t). The parameter λ thereby regulates the strength of this penalization. λ w.r.t. the control u(x0 \|x, t) With this assumption, the control problem consisting in minimizing CKL belongs to the KL-control class and has a closed form solution Todorov [2009], Kappen et al. [2012]. The probability distribution of an optimal path u∗KL (xt+1:T \|x, t) that minimizes equation (5) is u∗KL (xt+1:T \|x, t) = p (xt+1:T \|x, t) φ(xt+1:T ), hφ(xt+1:T )ip(xt+1:T \|x,t) with φ(xt+1:T ) := exp −λ −1 T X (6) ! 0 r(xt0 , t ) . (7) t0 =t+1 Plugging this into equation (5) and minimizing gives the optimal cost-to-go λ JKL (x, t) = r(x, t) − λ log hφ(xt+1:T )ip(xt+1:T \|x,t) , (8) which can be numerically approximated using paths sampled from the uncontrolled dynamics p(xt+1:T \|x, t). The optimal control corresponding to equation (4) corresponds to a state transition probability distribution that is obtained by marginalization in equation (6). It is expressed in terms of the uncontrolled transition probability p(x0 \|x) and the (exponentiated) optimal cost-to-go: λ (x0 , t + 1) X JKL ∗ 0 ∗ 0 0 uKL (x \|x, t) = uKL xt+1 = x , xt+2:T \|x, t ∝ p(x \|x) exp − . (9) λ x t+2:T This resembles a Boltzmann distribution with temperature λ where the optimal cost-to-go takes the role of an energy. The effect of the temperature becomes clear: for high values of λ, u∗KL (x0 \|x, t) deviates only a little from the uncontrolled dynamics p(x0 \|x), thus the optimal control has a weak influence on the system. In contrast, for low values of λ, the exponential in equation (9) becomes very λ (x0 , t + 1), suppressing the transition pronounced for the state(s) x0 with the smallest cost-to-go JKL 0 probabilities to suboptimal states x . Thus the control has a very strong effect on the process. In λ (x0 , t + 1) is not the limit of λ going to zero, the controlled process becomes deterministic, if JKL 0 degenerate (meaning there is a unique state x which minimizes the optimal cost-to-go). In this case the control is so strong that it overpowers the noise completely. We thus approximate our original (possibly difficult) control problem as a KL-control problem, parametrized by the temperature λ. The approximated optimal cost-to-go J(x0 , t + 1) of equation λ (x0 , t + 1) of (4) is replaced by the corresponding optimal cost-to-go of the KL-control problem JKL equation (9) and used to compute the action selection in the original problem. 3.2 A Tractable Example We now present a tractable example amenable for exact optimal control computation. This example already belongs to the KL-control class, so no approximation is made. The purpose of this analysis it 4 to show how different values of the temperature λ may lead to qualitatively different optimal solutions and other interesting phenomena. Let’s consider a tree that grows at discrete time-steps, starting with the root node at time t = 0. We represent the tree at time t as a vector xt = (x0 , x1 , ..., xt ), where xt indicates the label of the parent of the node attached at time t. At every time-step, either the tree remains the same or a new node is attached to it. The root node has label 1 and the label 0 is specially used to indicate that no node was added at a given time-step (it is also the label of the parent of the root node). The nodes are labelled in increasing order as they arrive to the tree, so that at time-step t, for a tree with k nodes, k ≤ t, xt = 0, 1, . . . , k corresponds to the parent of node k + 1 if a node is added or zero otherwise. Thus, the parent vector at time t = 1 is always x1 = (0, 1). Our example is a finite horizon task of T = 10 time-steps and end-cost only. The end-cost implements two control objectives: it prefers trees of large Wiener index while penalising trees with many nodes (more than five, in this case). The Wiener index is the sum of the lengths of the shortest paths between all nodes in a graph. It is maximal for a chain and minimal for a star. The uncontrolled process is biased to the root: new nodes choose to link the root with probability 3/5 and uniformly otherwise. More precisely ( 3 for j = 1 p(xt+1 = j\|xt ) = 5 2 (10) for (j = 0) or j ∈ {2, . . . , kxt k0 } 5kxt k0 ( −Wiener(xt )δt,T if kxt k0 < 5 r(xt , t) = (11) δt,T otherwise where kxk0 denotes the number of non-zero elements in x and Wiener the (normalized) Wiener index. In this setting, the uncontrolled process p tends to grow trees with more than five nodes with many of them attached to the root node, i.e. with low Wiener index. We want to influence this dynamics so that the target configuration, a chain of five nodes (maximal Wiener index) is more likely to be obtained. Figure 1 (top) shows the state cost r of the final tree that results from choosing the most probable control (MAP solution) as a function of the temperature λ. The exact solution is calculated using dynamic programming Kappen et al. [2012]. We can differentiate three types of solutions, denoted as A, B and C in the figure. For low temperatures (region A) the control aims to fulfil both control objectives: to find a small network with maximal Wiener index. The optimal strategy does not add nodes initially and then builds a tree of maximal Wiener index (see inset of initial controls in left column of the figure). This type of control (to wait while the target is far in the future) is reminiscent of the delayed choice mechanism described previously Kappen [2005]. This initial waiting period makes sense because if the chain of length 5 would be grown immediately, then at time 6 the size of 5 is already reached. If now an additional node attaches, then the final cost would be zero. However if one first waits and then grows the chain, an accidental node insertion before time 6 would not be so disastrous (actually it may help), as one can then just wait until time 7 to start growing the rest of the tree. So delaying the decision when to start growing the tree helps compensating accidental events. For intermediate temperatures (region B), the initial control becomes less extreme, as we observe if we compare the left plots between regions A and B. For λ ≈ 0.07, the solution that builds the tree with maximal Wiener index is no longer optimal, since it deviates too much from the uncontrolled 5 Most Probable solution 0 state cost C -0.5 B A -1 ∗ 1 A u 0.5 0 1 B 10 -2 (λ = 0.01) 0 u∗ 1 10 -1 10 0 λ 1 1 1 1 1 1 2 3 4 5 1 1 1 1 1 1 2 3 4 5 1 0 2 1 6 1 2 2 3 7 1 1 2 4 2 4 3 3 5 8 9 u∗ (λ = 100) 3 1 0 1 1 2 6 1 2 7 3 3 1 8 2 10 2 1 4 9 3 4 2 1 5 10 3 1 C 0.5 0 1 (λ = 0.29) 0.5 0 10 1 1 2 1 2 1 3 2 4 3 2 1 4 4 5 3 3 4 2 4 2 3 3 2 4 2 4 2 5 1 9 5 1 10 2 4 1 1 5 1 9 8 67 8 6 5 5 7 6 1 6 6 7 8 7 3 5 6 7 8 9 10 Figure 1: Example of optimal control of tree growth. (Top): the state cost of the most probable solutions as a function of the temperature λ. In region A, the optimal strategy waits until the last time-steps and then grows a tree with maximal Wiener index. In region B, it builds a star of five nodes. Finally, in region C, it follows the uncontrolled dynamics and builds a star of ten nodes. (Left): for each region, the optimal probabilities u∗ (xt+1 \|xt ) at t = 1 for the two actions which are initially available: no node addition (0) and adding a node to the root (1). In regions A, B the optimal control favours not adding a new node initially. The sequences on the right show how the tree grows. When a new node is added to the tree, it is coloured in red. dynamics. In region B, the control aims to build a network of five nodes or less, but no longer aims to maximize the Wiener index. The control is characterized by an initial waiting period and the subsequent growth of a tree of five nodes, which are in this case all attached to the root node. Finally, for high temperatures (region C, λ > 0.4), the control essentially ignores the cost r and the optimal strategy is to add one node to the root at every time-step, following the uncontrolled process. From these results we conclude that KL-control as a mechanism for controlling network growth can capture complex phenomena such as transitions between qualitatively different optimal solutions and delayed choice effects. 3.3 Sampling from the KL-optimally controlled dynamics In this subsection, we explain how we can sample from the optimally controlled dynamics and thereby λ (x, t) of equation (8). obtain an estimate of the optimal cost-to-go JKL The probability of an optimally controlled path, equation (6), corresponds to the product of the uncontrolled dynamics by the exponentiated state costs. Hence a naive way to obtain samples from the optimal dynamics, would consist in sampling paths from the uncontrolled dynamics p(x0 \|x) and 6 weight them by their exponentiated state costs. Using these samples we can then compute expectations from the optimally controlled dynamics. We use that for any function f (xt+1:T ) we have: + * φ(xt+1:T ) . hf (xt+1:T )iu∗ (xt+1:T \|x,t) = f (xt+1:T ) KL hφ(xt+1:T )ip(xt+1:T \|x,t) p(xt+1:T \|x,t) More precisely, provided a learned model or a simulator of the uncontrolled dynamics p(x0 \|x), we (i) (i) generate M sample paths xt+1:T , i = 1, . . . , M from p(x0 \|x) and compute the weights denominator thereby gives with equation (8) an estimate of the optimal cost-to-go as φ(xt+1:T ) φ̂ . The M hφ(xt+1:T )ip(xt+1:T \|x,t) 1 X (i) ≈ φ̂ := φ xt+1:T . M i=1 This method can be combined with resampling techniques Douc and Cappé [2005], Hol et al. [2006] opt,(i) to obtain unweighted samples xt+1:T from the optimal dynamics (for the numerical methods in this article, we have used structural resampling Douc and Cappé [2005], Hol et al. [2006]). Using such a naive sampling method, however, can be inefficient, specially for low temperatures. (i) φ(x ) t+1:T are more or less equal, for low temperWhile for high temperatures λ basically all weights φ̂ atures only a few samples with very large weights contribute to the approximation, resulting in very poor estimates. This is a standard problem in Monte Carlo sampling and can be addressed using the Cross-Entropy (CE) method De Boer et al. [2005], Kappen and Ruiz [2016], which is an adaptive importance sampling algorithm that incrementally updates a baseline sampling policy or sequence of controls. Here we propose to use the CE method in the discrete formulation and use a parametrized Markov process e uω (x0 \|x, t), with parameters ω, to approximate u∗KL . The CE method in our setting alternates the following steps: 1. In the first step, the optimal control is estimated using M sample paths drawn from a parametrized proposal distribution e uω (x0 \|x, t). 2. In the second step, the parameters ω are updated so that the proposal distribution becomes closer to the optimal probability distribution. As a proposal distribution e uω (x0 \|x, t), we use JeKL (x0 , ω(t)) e uω (x0 \|x, t) ∝ p(x0 \|x) exp − λ ! , (12) which has the same functional form as the optimally controlled transition probabilities in equation (9). The KL-optimal cost-to-go is thereby approximated by a linear sum of time-dependent feature vectors ψkt (x) X JeKL (x, ω(t)) = ωk (t)ψkt (x). (13) k The probability distribution of an optimally controlled path, equation (6), can be written as ! (i) T p xt+1:T \|x, t X (i) (i) (i) 0 ∗ −1 uKL xt+1:T \|x, t ∝ e uω (xt+1:T \|x, t) exp −λ r(xt0 , t ) . (14) (i) e uω (xt+1:T \|x, t) 0 t =t+1 7 This shows that we can draw samples from the proposal distribution and reweight them with the combined weights (i) p(xt+1:T \|x, t) (i) w(i) = φ xt+1:T . (i) e uω (xt+1:T \|x, t) The parameters ωk (t) of the importance sampler are initialized with zeros, which makes the initial proposal distribution equivalent to the uncontrolled dynamics. The procedure requires the gradients of e uω (x0 \|x, t) at each iteration. We describe the details of the CE method in A. We measure the efficiency of an obtained proposal control using the effective sample size (EffSS), which estimates how many effective samples can be drawn from the optimal distribution. Given M samples with weights w(i) , the EffSS is given by 1 PM (i) 2 i=1 w M EffSS = P (15) 2 . M 1 (i) i=1 w M If the weights w(i) are all about the same value, the EffSS is high, indicating that many samples contribute to statistical estimates using the weighted samples. If all weights are equal, the EffSS is equal to the number of samples M. Conversely if the weights w(i) have a large spread, the EffSS is low, indicating that only few independent samples contribute to statistical estimates. In the extreme case, when one weight is much larger then all others, the EffSS approaches 1. 3.4 Action selection using the KL-approximation λ , we need to select an action u ∈ U in the original Once we have an estimate of the cost-to-go JKL control problem, which is not of the KL-control type. We select the optimal action according to D E λ u∗ (x, t) ≈ argminu r(x, t) + JKL (x0 , t + 1) , (16) 0 P (x \|x,u,t) λ (x which requires the computation of JKL t+1 , t + 1) for every reachable state xt+1 . In growing networks, the number of possible next states (the branching factor) increases quickly, and visiting all of them soon becomes infeasible. In this subsection we highlight an important benefit of using the KL-approximation as a relaxation of the original problem: the optimally controlled process tends to discard many irrelevant states, specially for small values of λ. This means that u∗KL (x0 \|x, t) is sparse on x0 (only a few next λ (x0 , t) is very large for the corresponding x0 where states are relevant for the task), since the cost JKL u∗KL (x0 \|x, t) ≈ 0. opt Let xt+1:T denote a trajectory sampled from the optimally controlled process, as described in the previous section. We compute u∗KL (x0 \|x, t) using: û∗KL (x0 \|x, t) = δxopt (t+1),x0 u∗KL (xt+1:T \|x,t) , (17) where xopt (t + 1) is the first element of the trajectory and δxopt (t+1),x0 is the Kronecker delta which is equal one if xopt (t + 1) is equal to x0 , and zero otherwise. We then compute the optimal cost using equation (9): ∗ ûKL (x0 \|x, t) λ 0 JKL (x , t + 1) ∼ − log , (18) p(x0 \|x, t) 8 Figure 2: Task illustration: example of an Internet news forum. News are posted periodically and users can write comments either to the original post or to other user’s comments, forming a cascade of messages. The figure shows an example of conversation thread taken from Slashdot about Google’s AlphaGo. The control task is to influence the structure of the conversation thread (shown as a growing tree in the top-right). where we dropped a term which does not depend on x0 and therefore plays no role in the minimization of equation (16). The KL-approximation can help reducing the branching factor because it needs only λ (x0 , t + 1) only for the x0 where u∗ (x0 \|x, t) > 0 and thus J λ (x0 , t) a few samples to calculate JKL KL KL has a finite value. As mentioned earlier, u∗KL (x0 \|x, t) tends to be more sparse for small values of λ, when the KLcontrol problem is less noisy. In B we provide analytical details of the two extreme conditions, when λ is zero or infinite, respectively. 4 Application to Conversation Threads We have described a framework for controlling growing graphs. We now illustrate this framework in the context of growing information cascades. In particular, we focus on the task of controlling the growth of online conversation threads. These are information cascades that occur, for example, in online forums such as weblogs Leskovec et al. [2007], news aggregators Gómez et al. [2008] or the synthesis of articles of Wikipedia Laniado et al. [2011]. In conversation threads, after an initial post appears, different users react writing comments either to the original post or to comments from other users. 9 Figure 3: Our proposed control mechanism: in addition to the the threaded conversation, we highlight a comment (red node in the growing tree), suggested to be replied by the user. The choice of suggested comment, shown at the bottom of the page, is calculated using the method described in section 3.3. Figure 2 shows an example of a conversation thread, taken from Slashdot (www.slashdot.org). Users see a conversation thread using a similar hierarchical interface. The task we consider is to optimize the structure of the generated conversation thread while it grows. The state is thus defined as a growing tree. We assume an underlying (not observed) population of users that keep adding nodes to this tree. Since we can not control directly what is the node that will receive the next comment, we propose the user interface as a control mechanism to influence indirectly the growth process. This can be done in different ways, for example, manipulating the layout of the comments. In our case, the control signal will be to recommend a comment (by highlighting it) to which the next user can reply. Figure 3 illustrates such a mechanism. The action selection strategy introduced in section 3.4 is used to select the comment to highlight. Our goal is thus to modify the structure of a cascade in certain way while it evolves, by influencing its growth indirectly. It is known that the structure of online threads is strongly related with the complexity of the underlying conversation Gómez et al. [2008], Gonzalez-Bailon et al. [2010]. To fully define our control problem, we need to specify the structural cost function, the uncontrolled dynamics, i.e. the equivalent of equations (10) and (11) for this task, and a model of how an action (highlighting a node) changes the dynamics. Globally, this application differs from the toy example of subsection 3.2 in some important ways: 1. The state-space is larger (threads typically receive more than 10 comments). 2. We choose as state-cost function the Hirsch index (h-index), which makes the control task 10 highly non-trivial. 3. The original problem is not a KL-control problem. We use the action selection method described in section 3.4 to control the growth of the conversation thread. 4.1 Structural Cost Function We propose to optimize the Hirsch index (h-index) as structural measure. In our context, a cascade with h-index h has h comments each of which have received at least h replies. It is a sensible quantity to optimize, since it measures how distributed the comments of users on previous comments are. A high h-index prevents two extreme cases that occur in a rather poor conversation: the case where a small number of posts attract most of the replies, thus there is no interaction, and the case with deep chains, characteristic of a flame war of little interest for the community. Both cases have a low h-index, while a high h-index spreads the conversation over multiple levels of the cascade. The h-index is a function of the degree sequence of all nodes in the tree, where the degree of a node is this case is the number of replies plus one, as there is also a link to the parent (replied comment or post). Therefore we use the degree histogram as features ψkt (x) for the parametrized form of the optimal cost-to-go, equation (13). That is, feature ψkt (x) is the number of nodes with degree k in the tree x at time-step t. We model the problem as a finite horizon task with end-cost. Thus, the state cost is defined as r(x, t) = −δt,T · h(x), where h(x) is the h-index of the tree x. 4.2 Uncontrolled Dynamics for Online Conversation Threads As uncontrolled dynamics, we use a realistic model that determines the probability of a comment to attract the replies of other users at any time, by means of an interplay between the following features: • Popularity α: number of replies that a comment has already received. • Novelty τ : the elapsed time since the comment appeared in the thread. • Root node bias β: characterizes the level of trendiness of the main post. Such a model has proven to be successful in capturing the structural properties and the temporal evolution of discussion threads present in very diverse platforms Gómez et al. [2013]. Notice that these features θ = (α, τ, β) should not be confused with the features ψkt (x) used to encode the costto-go. We represent the conversation thread as a vector of parents xt = (x0 , x1 , ..., xt ). Given the current state of the thread xt , the uncontrolled dynamics attaches a new node t + 1 to an existing node j with probability 1 pθ (xt+1 = j\|xt ) = degj,t α + δj,1 β + τ t+1−j (19) Zt+1 with Zt+1 a normalization constant, degj,t the degree of node j at time t and δj,1 the Kronecker delta function, so parameter β is only nonzero for the root. Given a dataset composed of S threads D := {x(1) , . . . , x(S) } with respective sizes \|x(k) \|, k ∈ {1, . . . S}, the parameter vector θ can be learned by minimizing (k) − log L(D; θ) = − S \|x X X\| k=1 t=2 11 (k) (k) log pθ (xt+1 \|xt ). We learn the parameters using the Slashdot dataset, which consists of S = 9, 820 threads, containing more than 2 · 106 comments among 93, 638 users. In Slashdot, the most relevant feature is the preferential attachment, as detailed in Gómez et al. [2013]. This will have implications in the optimal control solution, as we show later. 4.3 Control interaction The control interaction is done by highlighting a single comment of the conversation. We assume a behavioural model for the user inspired by Craswell et al. [2008], where the user looks at the highlighted comment and decides to reply or not. For simplicity, we assume that the user chooses the highlighted comment with a fixed probability p0 = α/(1 + α) and with probability 1 − p0 she chooses to ignore it. If the highlighting of the comment is ignored, the thread grows according to the uncontrolled process. Therefore, α parametrizes the strength of the influence the controller has on the user. For α → ∞, we can fully control the behaviour and for α = 0, the thread evolves according to the uncontrolled process. A typical control would have a small α as usually the influence of an controlling agent on a social systems is weak. 4.4 Experimental Setup To evaluate the proposed framework we use a simulated environment, without real users. We consider a finite horizon task with T = 50 with the goal to maximize the h-index at end-time, starting from a thread with a single node as initial condition. The state-space consists of 50! ≈ 364 states. The thread grows in discrete time-steps. At each time-step, a new node is added to the thread by a (simulated) user. For that, we first choose which node to highlight (optimal action) as described in section 3.4 using equation (16). We then simulate the user as described in section 4.3, so the highlighted node is selected with probability p0 = α/(1 + α) as the parent of the new node. Otherwise, with probability 1 − p0 , the user ignores the highlighted node and the parent of the new node is chosen according to the Slashdot model, equation (19). This is repeated until the end time. 4.5 Experimental Results We first analyse the performance of the adaptive importance sampling algorithm described in section 3.3 for different fixed values of λ. Figure 4 shows the effective sample size (EffSS), equation (15) as a function of the number of iterations of the CE method. We observe that the EffSS increases to reach a stable value. As expected, large temperature (easier) problems result in higher values of EffSS. We can also see that, even for hard problems with low temperature, the obtained EffSS is significantly larger than zero, which allows us to compute the KL-optimal control. In general, the curves are less smooth for smaller values of λ, because a few qualitatively better samples dominate the EffSS, resulting in higher variance. On the other hand we also observe that the EffSS never reaches 100%. This is expected, as this would mean that our parametrized importance sampler perfectly resembles the optimal control, and this is not possible due to the approximation error introduced by the use of features. We can better understand the learned control by analysing the linear coefficients of the parametrized optimal cost-to-go, equation (13), for this problem. Figure 5 shows the feature weights ωk (t), at different times t = 1, . . . , T, after convergence of the CE method. Feature k corresponds to the number of nodes with degree k in the tree, after a new node arrives. The parent node to which the new node 12 100 λ=1 λ = 1/3 λ = 0.2 λ = 0.1 90 80 EffSS in % 70 60 50 40 30 20 10 0 20 40 60 80 Iteration 100 120 140 Figure 4: Evaluation of the inference step: The Effective sampling size (EffSS) increases after several iterations of the cross-entropy method. As expected, large values of the temperature λ result in higher values of EffSS. We use M = 105 samples to compute the EffSS. The EffSS is measured here in percent of the maximum number of samples M . feature weight forλ=0.2 0.1 5 10 0.05 15 0 Time 20 -0.05 25 30 -0.1 35 -0.15 40 -0.2 45 5 10 15 20 feature degree of parent 25 30 Figure 5: The learned importance sampler: The figure shows the time-dependent parameters of the learned expected cost-to-go for λ = 0.2. Each pixel is the parameter of a feature at a certain time. The features are the degrees of the parent node after the new child attaches. The colour represents the weight of the parameter. Large negative weights (pixels in blue colour) stand for a low cost and thus a desirable state, while large positive weights (red pixels) stand for high cost and thus undesirable states. At all times there is a desirable degree which the parent should have and higher as well as lower degrees are inhibited. This desirable degree is small at early times and becomes larger at later times. 13 has attached is thereby the only node whose degree changes (the degree increases by 1). Thus a high weight for a feature which measures the number of nodes with a certain degree k results in a low probability of attaching to a node with degree k − 1. Conversely low, or large negative weights thus correspond to nodes which have a high probability of becoming the parent of the next node which is added. We observe that there is an intermediate preferred degree (large negative weight, in blue). This is the preferred degree of the parent of the new node, and this preferred degree increases with time, reaching a value of 5 at t = 50. Does this strategy make sense? The maximum h-index of a tree of 50 nodes is 7, and it is achieved if 6 nodes have exactly 7 children and one node has 8. However, achieving such a configuration requires a very precise control. For example, increasing too much the degree of a node, say up to 9, prevents the maximum h-index to be reached, as there are not enough links left, due to the finite horizon. Thus, in this setting, steering for the maximal possible h-index is not optimal. The controller prefers all parents to have a degree of 5 and not less, but also not much more. As having more than five parents with degree at least five will result in an h-index of 5 we conclude that the control seems to aim for a target h-index of 5, while preventing wasting links to higher or lower degree nodes, which would not contribute to achieve that target. The interpretation of why the preferred degree increases with time involves the uncontrolled dynamics. Remember that the most relevant term in equation (19) for the considered dataset corresponds to the preferential attachment, parametrized by α. This term boosts high-degree nodes to get more links. If this happens, most of the links end up attached to a few parents, and this effect can only be suppressed by a strong control. The controller prevents that self-amplifying effect by aiming initially for an overall low degree, preventing a high impact of the preferential attachment. This keeps the process controllable and allows for a more equal distribution of the links. After having evaluated the sampling algorithm, we evaluate the proposed mechanism for actual control of the conversation thread. As described in section 3.4, in our simulated scenario, we highlight the node as the parent which minimizes the computed expected cost-to-go. Figure 6 shows the evolution of the h-index using different control mechanisms. The blue curve shows how the h-index changes under the uncontrolled dynamics. On average, it reaches a maximum of about 3.7 after 50 time steps. In green, we show the evolution of the h-index under a KL-optimal controlled case, for temperature λ = 0.2. As expected, we observe a faster increase, on average, than using the uncontrolled dynamics. The maximum is about 4.7. The red and black curves show the evolution of the h-index using the control mechanism described λ of the KLin subsections 3.4 and 4.3, where we select actions using the expected cost-to-go JKL optimal control with λ = 0.2, for α = 1 and α = 0.5, respectively. In both cases the obtained h-index is even higher than the one obtained with the KL-control relaxation. Therefore, the objective for this task, to increase the h-index, can be achieved through our action selection strategy. As expected, a stronger interaction strength α = 1 leads to higher h-indices than a lower strength α = 0.5. Finally, in Figure 7 we show examples of a real discussion thread from the dataset (Slashdot), a thread generated from the learned model (uncontrolled process) and one resulting from applying our action selection strategy. The latter has higher h-index. 5 Discussion We have addressed the problem of controlling the growth process of a network using stochastic optimal control with the objective to optimize a structural cost that depends on the topology of the 14 6 p 5.5 u max-costtogo α=1 5 max-costtogo α=0.5 4.5 h-index 4 3.5 3 2.5 2 1.5 1 0 5 10 15 20 25 time 30 35 40 45 50 Figure 6: Evaluation of the actual control: uncontrolled dynamics (blue), KL-optimally controlled dynamics (green) action selection based control for α = 1 (red) and α = 0.5 (black). The KLoptimally controlled dynamics, which optimize the sum of the λ-weighted KL-term and the end cost, shifts the final mean value from about 3.7 to about 4.7. The action selection based control, which is aiming to optimize the end cost only, is able to shift the h-index to even higher values then the KL-optimal control. For the controlled dynamics, λ = 0.2 for all three cases. To compute the control in each time-step we sample 1000 trajectories. The statistics where computed using 1000 samples for each of the three cases. growing network. The main difficulty of such a problem is the exploding size of the state space, which grows (super-)exponentially with the number of nodes in the network and renders exact dynamic programming infeasible. We have shown that a convenient way to address this problem is using KL-control, where a regularizer is introduced which penalizes deviations from the natural network growth process. One advantage of this approach is that the optimal control can be solved by sampling. The difficulty of the sampling is controlled by the strength of the regularization, which is parametrized by a temperature parameter λ: for high temperatures the sampling is easy, while for low temperatures, it becomes hard. This is in contrast to standard dynamic programming, whose complexity is directly determined by the number of states and independent of λ. In order to tackle the more challenging low temperature case, we have introduced a featurebased parametrized importance sampler and used adaptive importance sampling for optimizing its parameters. This allows us to sample efficiently in the low temperature regime. For control problems which cannot directly be formulated as KL-control problems, we have proposed to use the solution of a related KL-control problem as a proxy to estimate the effective values of possible next network states. These expected effective values are subsequently used in a greedy strategy for action selection in the original control problem. This action selection mechanism benefits from the sparsity induced by the optimal KL-control solution. We have illustrated the effectiveness of our method on the task of influencing the growth of 15 Slashdot thread Uncontrolled thread Controlled thread Figure 7: Examples of threads. A thread from the data (Slashdot), an uncontrolled thread generated from the model and a controlled thread. The nodes that contribute to the h-index are coloured in yellow. The h-index for the data and the uncontrolled thread is 4 and 6 for the controlled one. conversation cascades. Our control seeks to optimize the structure of the cascade, as it evolves in time, to maximize the h-index at a final time. This task is non-trivial and characterized by a sparse, delayed reward, since the h-index remains constant during most of the time, and therefore a greedy strategy is not possible. Our approach for controlling network growth is inspired in recent approaches to optimal decisionmaking with information-processing constraints Todorov [2009], Tishby and Polani [2011], Kappen et al. [2012], Theodorou and Todorov [2012], Rawlik et al. [2012]. The Cross-Entropy method has been explored previously in the continuous case Kappen and Ruiz [2016]. The continuous formulation of this class of problems has been used in robotics, using parametrized policies Theodorou et al. [2010], Levine and Koltun [2013], Gómez et al. [2014]. In economics, the question of altering social network structure in order to optimize utility has been addressed mainly from a game theoretical point of view, under the name of strategic network formation Jackson and Watts [2002], Bloch and Jackson [2007]. To the best of our knowledge, the problem of network formation has not yet been addressed from a stochastic optimal control perspective. The standard approach to address the problem of controlling a complex, networked system is to directly try to control the dynamics on the network Liu et al. [2011], Cornelius et al. [2013]. This approach considers the classical notion of structural controllability as the capability of being driven from any initial state to any desired final state within finite time. Optimal control in thus referred to the situation where a network can be fully controlled using only one driving signal. This idea is also prevalent in the influence maximization problem in social networks Kempe et al. [2003], Farajtabar et al. [2014, 2015], which consists in finding the subset of driver (most influential) nodes in a network. Since the controllability of the dynamics on the network depends crucially on the topology, several works have considered the idea of changing the network structure is some way that favours structural controllability. For example, the perturbation approach introduced in Wang et al. [2012] looks for the minimum 16 number of links that needs to be added so that the perturbed network can be fully controlled using a single input signal. In Hou et al. [2015], a method to enhance structural controllability of a directed network by changing the direction of a small fraction of links is proposed. More recently, Wang et al. [2016] analyzed node augmentation of directed networks while insisting that the minimum number of drivers remains unchanged. The main difference between our approach and these approaches is that, rather than considering the controllability of the dynamical system on the underlying network, our optimal control task is defined on the structure of the network itself, regardless of the dynamical system defined on it. In some sense, our results complement these approaches. For example, one could use our optimal control approach to shape the growth of the network in a way that the structural controllability, understood as the state cost function, is optimized. Acknowledgements This project is co-financed by the Marie Curie FP7-PEOPLE-2012-COFUND Action, Grant agreement no: 600387, the Marie Curie Initial Training Network NETT, project N. 289146 and the Spanish Ministry of Economy and Competitiveness under the Marı́a de Maeztu Units of Excellence Programme (MDM-2015-0502). A Adaptive Importance Sampling for KL-Optimal Control Computation using the Cross-Entropy method Here we show how the time-dependent weights ωk (t) of the importance sampler are updated such that e uω (x0 \|x, t) becomes closer to the optimal sampling distribution. This corresponds to the second step of the Cross-Entropy method described in subsection 3.3. For clarity in the derivations, we will replace p(x1:T \|x, 0) and u∗KL (x1:T \|x, 0) by p and u∗KL , respectively, in the expectations. The closeness of the two distributions e uω (x0 \|x, t) and u∗KL (x0 \|x, t) can be measured as the cross entropy between the path x1:T probabilities under these two Markov processes: u∗KL (x1:T \|x, 0) ∗ KL [uKL (x1:T \|x, 0) k e uω (x1:T \|x, 0)] = log e uω (x1:T \|x, 0) u∗ KL = − hlog e uω (x1:T \|x, 0)iu∗ + const. =: −D(ω), KL (20) where the constant term hlog u∗KL (x1:T \|x, 0)iu∗ is dropped. KL We minimize equation (20) by gradient descent. At iteration l, the gradient D(ω (l) ) with respect to ωk (t) is given by ∂D(ω (l) ) =− ∂ωk (t) ∂ log e uω(l) (x1:T \|x, 0) ∂ωk (t) u∗ KL 17 where T−1 Y 1 JeKL (xt+1 , ω(t)) e uω(l) (x1:T \|x, 0) = p(x1:T \|x, 0) exp − Z λ t=0 T−1 !+ Y JeKL (xt0 +1 , ω(t0 )) Z= exp − λ 0 t =0 ! p with the normalization constant Z. This leads to !+ T−1 e X ∂D(ω (l) ) ∂ JKL (xt0 +1 , ω(t0 )) log p(x1:T \|x, 0) − =− − log Z ∂ωk (t) ∂ωk (t) λ 0 (21) u∗KL t =0 where we can drop the first term as it is independent of ω(t). The second term can be evaluated using the definition of JeKL , equation (13). Further, plugging in Z we get * T−1 !+ + Y ∂ JeKL (xt0 +1 , ω(t0 )) ∂D(ω (l) ) −1 t =λ ψk (xt+1 ) u∗ + log exp − KL ∂ωk (t) ∂ωk (t) λ 0 t =0 = λ−1 ψkt (xt+1 ) = λ−1 ψkt (xt+1 ) u∗KL u∗KL p T−1 Y ∂ JeKL (xt0 +1 , ω(t0 )) log exp − ∂ωk (t) λ t0 =0 T−1 !+ Y 1 ∂ JeKL (xt0 +1 , ω(t0 )) + exp − Z ∂ωk (t) 0 λ + t =0  u∗KL !+ p p !+  0 e 0 1 JKL (xt +1 , ω(t ))  = λ−1  ψkt (xt+1 ) u∗ − ψkt (xt+1 ) exp − KL Z λ 0 t =0 p = λ−1 ψkt (xt+1 ) u∗ − ψkt (xt+1 ) eu (x \|x,0) KL ω (l) 1:T  D E p(x1:T \|x,0) t (x φ (x ) ψ ) 1:T k t+1   euω (x1:T \|x,0) e uω(l) (x1:T \|x,0) D E = λ−1  − ψkt (xt+1 ) eu (x \|x,0)  ,  p(x1:T \|x,0) ω (l) 1:T φ (x ) 1:T e uω (x1:T \|x,0) T−1 Y e uω(l) (x1:T \|x,0) (22) where we have used the estimates from the importance sampling step and equation (9). The update rule for the parameters becomes (l+1) ωk (l) (t) = ωk (t) + η ∂D(ω (l) ) , ∂ωk (t) for some learning rate η. Algorithm 1 summarizes the CE method applied to this context. 18 (23) Algorithm 1 Cross-Entropy Method for KL-control Require: importance sampler e uω , feature space ψ(·), number of samples M, learning rate η l←0 (l) ωk (t) ← 0, Initialize weights for all k, t, l (i) xt+1:T ← draw M sample trajectories ∼ e uω(l) , i = 1, . . . , M repeat (l) ) compute gradient ∂D(ω ∂ωk (t) using equation (21) (l+1) ωk (l) (l) ) (t) ← ωk (t) + η ∂D(ω ∂ωk (t) for all k, t, l (i) xt+1:T ← draw M samples ∼ e uω(l+1) l ←l+1 until convergence B Analyzing the KL-optimal cost-to-go based action selection We have introduced an action selection framework which is based on an approximation of the optimal λ (x0 , t + 1) of a parametrized family of KL-control cost-to-go J(x0 , t) by the optimal cost-to-go JKL problems which share the same state cost r(x, t). Why is this a good idea? Consider the two extreme cases where the temperature λ, which parametrizes the family of equivalent KL-control problems, is zero or infinite, respectively. Extreme case λ → 0 (zero temperature): The total cost in the KL-control problem becomes equal to the total cost in the original control problem, equation (1), as the KL term vanishes. The KL-optimal control becomes deterministic: λ 0 ( J (x ,t+1) λ (x0 , t + 1) p(x0 \|x) exp − KL λ 1 for x0 = argmin JKL lim u∗KL (x0 \|x, t) = lim = , (24) λ→0 λ→0 Z 0 otherwise where Z is a normalization constant. Thus, for λ → 0, the KL-control problem becomes identical to the original problem if the system is fully controllable, i.e. for every t, x and x̃ there is a ux̃ ∈ U such that p(x0 \|x, t, ux̃ ) = δx̃,x0 . Extreme case λ → ∞ (infinite temperature): For this case, using equation (8) we get ∞ λ JKL (x, t) = lim JKL (x, t) λ→∞ * T X = r(x, t) − lim λ log  exp −λ−1 λ→∞ * = r(x, t) + T X t0 =t+1 t0 =t+1 + r(xt0 , t0 ) . p(xt+1:T \|x,t) 19 !+ r(xt0 , t0 )   p(xt+1:T \|x,t) Using equation (1) and the definition of the uncontrolled dynamics, we can write * T + X ∞ JKL (x, t) = r (x, t) + r xt0 , t0 = C (x, t, 0) . t0 =t+1 (25) P (xt+1:T \|x,0,t) Thus, for λ → ∞, the KL-optimal cost-to-go becomes equal to the total cost in the original control problem under the uncontrolled dynamics (using u = 0). Having this equation (16) can be written as u∗ (x, t) ≈ argminu r(x, t) + C x0 , t + 1, 0 P (x0 \|x,u,t) . (26) In this case, the action selection is equivalent to optimize an expected total cost assuming the system will evolve according to the free dynamics in the future. 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Submitted Spectral Precoding for Out-of-band Power Reduction under Condition Number Constraint in OFDM-Based System Lebing Pan 1 No.50 Research Institute of China Electronic Technology Group Corporation, Shanghai 200311, China. forza@aliyun.com Abstract: Due to the flexibility in spectrum shaping, orthogonal frequency division multiplexing (OFDM) is a promising technique for dynamic spectrum access. However, the out-of-band (OOB) power radiation of OFDM introduces significant interference to the adjacent users. This problem is serious in cognitive radio (CR) networks, which enables the secondary system to access the instantaneous spectrum hole. Existing methods either do not effectively reduce the OOB power leakage or introduce significant biterror-rate (BER) performance deterioration in the receiver. In this paper, a joint spectral precoding (JSP) scheme is developed for OOB power reduction by the matrix operations of orthogonal projection and singular value decomposition (SVD). We also propose an algorithm to design the precoding matrix under receive performance constraint, which is converted to matrix condition number constraint in practice. This method well achieves the desirable spectrum envelope and receive performance by selecting zero-forcing frequencies. Simulation results show that the OOB power decreases significantly by the proposed scheme under condition number constraint. Keyword: Spectral precoding, Out-of-band, Orthogonal frequency division multiplexing (OFDM), Sidelobe suppression, Condition number constraint. 1. Introduction Dynamic spectrum access [1, 2] technology is extensively studied as an effective scheme to achieve high spectral efficiency, which is a crucial step for cognitive radio (CR) networks. Due to the flexible operability over non-continuous bands, orthogonal frequency division multiplexing (OFDM) is considered as a candidate transmission technology for CR system [3]. However, due to the use of rectangular pulse shaping, the power attenuation of its sidelobe is slow by the square of the distance to the main lobe in the 1 Submitted frequency domain. Therefore, the out-of-band (OOB) power radiation or sidelobe leakage of OFDM causes severe interference to the adjacent users. Furthermore, this problem is serious in CR networks, which enables the secondary users to access the instantaneous spectrum hole. Therefore, these secondary users need to ensure that the interference level of the power emission is acceptable for primary users. In practice, typically of the order of 10% guard-band is needed for an OFDM signal in the long term evolution (LTE) system [4]. Therefore, the spectrum efficiency is significantly reduced. The traditional method of sidelobe suppression is based on windowing techniques [5], such as the raised cosine windowing [6], which is applied to the time-domain signal wave. However, this scheme requires an extended guard interval to avoid signal distortion, and the spectrum efficiency is also reduced for large guard intervals. The cancellation carriers (CCs) [7, 8] technique inserts a few carriers at the edge of the spectrum in order to cancel the sidelobe of the data carriers. However, this technique degraded the signalto-noise ratio (SNR) at the receiver. The subcarrier weighting method [9] is based on weighting the individual subcarriers in a way that the sidelobe of the transmission signal are minimized according to an optimization algorithm. However, the bit-error-rate (BER) increases in the receiver, and when the number of subcarriers is large, it is difficult to implement in real-time scenario. The multiple choice sequence method [10] maps the transmitted symbol into multiple equivalent transmit sequences. Therefore, the system throughput is reduced when the size of sequences set grows. Constellation adjustment [11] and constellation expansion [12] are difficult to implement when the order of quadrature amplitude modulation (QAM) is high. Strikingly, the methods [10, 11] require the transmission of side information. The adaptive symbol transition [13] scheme usually provides weak sidelobe suppression in frequency ranges closely neighboring the secondary user occupied band. However, the schemes [7-14] depend heavily on the transmitted data symbols. The precoding technology is widely used in OFDM system to enhance the performance of transmission reliability over the wireless environment, in which there are many precoding methods 2 Submitted proposed for OOB power reduction [15]. There are two main optimization schemes with obtaining large suppression performance. One is to force the frequency response of some frequency points to be zero by orthogonal projection [16-19]. These frequency points are regarded as zero-forcing frequencies. The other method [20, 21] is to minimize the power leakage in an optimization frequency region by adopting the quadratic optimization method, using matrix singular value decomposition (SVD). The precoding scheme in [16] is designed to satisfy the condition that the first N derivatives of the signals are continuous at the edges of symbols. However, this method introduces an error floor in error performance. The sidelobe suppression with orthogonal projection (SSOP) method in [19] adopts one reserved subcarrier for recovering the distorted signal in the receiver. To maintain the BER performance, data cost is introduced in [17] by exploiting the redundant information in the subsequent OFDM symbol. The sidelobe suppression in [21] is based on minimizing the OOB power by selecting some frequencies in an optimized region. The suppression problem is first treated as a matrix Frobenius norm minimization problem, and the optimal orthogonal precoding matrix is designed based on matrix SVD. In [20], an approach is proposed for multiuser cognitive radio system. This method ensures user independence by constructing individual precoder to render selected spectrum nulls. Unlike the methods that focus on minimizing or forcing the sidelobe to zero, the mask compliant precoder in [22] forces the spectrum below the mask by solving an optimization problem. However, the algorithm leads to high complexity. In this paper, a spectral precoding scheme is proposed with matrix orthogonal projection and SVD for OOB reduction in OFDM-based system. The main idea of this scheme is to reduce the OOB power under the receive quality. The condition number of the precoding matrix indicates the BER loss in the receiver. Therefore, we develop an iteration algorithm to design the precoding matrix under matrix condition number constraint. The proposed method has an appropriate balance among suppression performance, spectral efficiency and receive quality. Consequently, it is flexible for practical implementation. 3 Submitted The rest of the paper is organized as follows. In Section 2, we introduce the system model of OOB power reduction by the spectral precoding method. In Section 3, we present the proposed spectral precoding approach. Next, in Section 4, we provide an iteration algorithm to design the precoding matrix according to the desirable spectrum envelope, spectral efficiency and BER performance. Simulations are presented in Section 5 to demonstrate the performance of the proposed method, followed by a summary in Section 6. s d Precoding Add Guard IFFT DAC&UP Conversion Channel d s Decoding Remove Guard FFT Down Conversion &ADC Fig. 1 System diagram of spectrally precoded OFDM. 2. System Model The block diagram of a typical OFDM system using a precoding technique is illustrated in Fig. 1. The number of total carriers used in the transmitter is M . The digital spectral precoding process before inverse fast Fourier transform (IFFT) operation is expressed as s  Pd, (1) where d is the original OFDM symbol of size N 1 , and s is the precoded vector. The size of the precoding matrix P is M  N ( M  N ) and the coding redundancy R  M  N is usually small. The spectral precoding method achieves better suppression performance in zero-padding (ZP) OFDM system than cyclic-prefix (CP) OFDM [19, 21]. What’s more, the ZP scheme has already been proposed as an alternative to the CP in OFDM transmissions [23] and particularly for cognitive radio [24]. The proposed method also can be directly applied to CP systems with degraded performance on sidelobe suppression. 4 Submitted Conventionally, power spectral density (PSD) analysis for multicarrier systems is based on an analog model with a sinc kernel function [25]. The PSD converges to the sinc function with the sampling rate increasing. In addition, the oversampling constraint presented in [26] ensures the desirable power spectral sidelobe envelope property after precoding for DFT-based OFDM. In a general OFDM system, the time-domain signal can be defined by a rectangular function (baseband-equivalent) as 0t T elsewhere 1, g (t )   0, (2) where T is the symbol duration. The frequency domain representation of m-th subcarrier is written as Gm ( )  e jT /2 sin((  m )T / 2) , (  m ) / 2 (3) where m is the center frequency of m-th subcarrier. Therefore, the magnitude envelope in the OOB region (   m ) is expressed by Gm ( )  sin((  m )T / 2) (  m ) / 2 2  .   m (4) Then we define the function Cm ( ) to indicate the magnitude envelope by Cm ( )  1 .   m (5) The expression (5) is obtained from (4) by ignoring a constant factor. This operation does not affect the mathematical analysis for the design of the precoding matrix [19]. The complex computing by (3) is converted to real operation, which reduce the complexity of spectral precoding in the following. Based on (5), using the superposition of M carriers, the target function indicating the PSD at OOB frequency  is defined by 5 Submitted A( )  M 1 s C m0 m m 2 ( ) , (6) where sm is the m-th element in the precoded OFDM vector s . In OFDM system, the power spectrum of its sidelobe decays slowly as ( )2 , where     m is the frequency distance to the mainlobe. From (1) and (6), the PSD target function of the OFDM signal is expressed in the matrix form as P( )    2 1 1   A( )   cT Pd , Ts Ts (7) where c  C0 ( ), C1 ( ),...CM 1 ( ) , (.)T and  denote transpose operation and expectation respectively. T P( ) and C ( ) are not the PSD presentation of the OFDM signal and the frequency response respectively, but indicate their envelope character. The goal of the precoding method is to design P to reduce the emission in OOB region. Simultaneously, the process is irrespective of the value of vector d . 3. The Proposed Joint Spectral Precoding (JSP) Method In this paper, a joint spectral precoding (JSP) method is proposed with two times orthogonal projection and one time SVD. The process is given by three steps: Inner orthogonal projection → SVD → Outside orthogonal projection. As presented in the previous papers [19, 21], in each step, the main idea of orthogonal projection is to force the power of zero-forcing frequency points to be zero, and the SVD operation is to minimize the power in optimized region. However, the precoding matrix derived from the final step does not achieve the goal in each step again. Unlike the methods that focus on minimizing or forcing the sidelobe to zero, we reduce the OOB power under the receive quality constraint by selecting the frequency points in each step. Therefore, the proposed method has an appropriate balance between the suppression performance and the receive quality. In addition, when the number of the reserved carriers is small, such as only one, then the number of zero-forcing frequencies in orthogonal projection method is one. But in the JSP, we set the zero-forcing frequency in inner and outside orthogonal projection to be different and better suppression performance is obtained. 6 Submitted data carriers reserved carriers zero-forcing frequency … optimized region … …  Fig. 2 The spectrum diagram for OOB power reduction. Generally, the subcarriers of OFDM-based system are considered to be continuous. The proposed method also can be used for non-continuous multi-carriers system. The number of employed carriers is M and the index is from 0 to M  1 . Fig. 2 illustrates the frequency region of   m . The R reserved carriers at the upper and lower spectral edges, which are used to achieve sidelobe suppression and maintain the receive quality. Two groups of zero-forcing frequency points ω a and ωb are selected in inner and outside orthogonal projection respectively. The corresponding number is N a and N b ( N a  R , Nb  R ). K frequency points in the optimized region are chosen to reduce the OOB power by SVD operation. We first give the process of JSP in this section, and the detail of how to select the parameters is presented in the next. In the first step, the group of zero-forcing frequency points ω a ( ωa  [a1 ,...ai ,...aN ] , ai  0 or a a  M 1 ) are chosen to achieve i Ca Padˆ  0, (8) Where d̂ is a vector of size M , derived from the original data symbol d N1 by adding zero in the location of the reserved carriers, i.e., dˆ  [0, 0,...dT ,...0, 0]T . Ca is the magnitude response matrix of size N a  M , whose element Ca (i, j )  C j (ai ) computed by (5). j is the index of the carriers. The solution of (8) is equivalent to map d̂ to the nullspace of Ca , In [27], the orthogonal projector Pa mapping vector onto Ca  ( Ca  is the nullspace of Ca ) along Ca is given by 7 Submitted Pa  I M  Ca (CTa Ca )1 CTa , (9) where I M is a unit matrix of size M  M . This projector has a property that the precoded data vector Pa dˆ , in the nullspace of Ca , is closest to d̂ as min Padˆ  dˆ . Padˆ  Ca  (10) Obviously, Pa is non-orthogonal. Therefore, it will cause significant BER performance degradation in the receiver. After the step of the inner orthogonal projection, the precoded data is given by s a  Padˆ . (11) Then in the second step, the magnitude response matrix of K frequency points ω o in the optimized region is written as Cop  Co Pa , where Co is a magnitude response matrix of the K optimized frequency points. The elements in Co is computed by (5). We then use the SVD operation to minimize the power leakage in the optimized region. The problem is determined by Po  arg min Cop P . (12) P By decomposing the matrix Cop into Cop  Uc Σc VcT using SVD, the optimal precoding matrix Po to achieve (12) is derived from Po  [VcR , VcR1 ,...VcM 1 ], (13) where Vci is the i-th column of Vc . The matrix Po of size M  N is composed of the last N columns in Vc . Po is an orthogonal matrix that PoT Po  I N , where I N is a unit matrix of size N  N . The original data d is mapped to Pod , while the length is extended from N to M . The average power of each symbol of M 1 N 1 i 0 i 0 before and after precoding are Ps  i2 and Ps  i2 , where i is the i-th diagonal element in Σ c and Ps 8 Submitted is the average power of each symbol d . The second step is to abandon the largest R singular values in Σ c , then the OOB power is reduced after precoding by Pod . In the third step, the group of zero-forcing frequency points ωb ( ωb  [b1 ,...bi ,...bN ] , bi  0 or b b  M 1 ) are chosen. The orthogonal projection matrix Pb is obtained similar to (9) by i Pb  I M  Cb (CTb Cb )1 CTb , (14) where the matrix Cb of size Nb  M , is the magnitude response matrix of the frequency points b . Finally, the precoding matrix is obtained by P  Pb Po . (15) After precoding, the minimum error problem of decoding in the receiver, is given by ˆ -d , min PPd (16) where P̂ is the decoding matrix. The orthogonal projector Pb map the vector onto Cb  , so rank (Pb )  M  Nb  N , where rank (.) denotes the rank of a matrix. Due to rank (Po )  N and the value of N b is small, the case of that P is full column rank, is easy to be achieved in practice. Then the pseudo- ˆ  I , which is given by Pˆ  (PT P)1 PT . inverse matrix of P is the optimal solution for (16) to achieve PP N This JSP scheme is unlike the methods that minimize the OOB power or forcing the sidelobe to zero. In the third step, we map Pod to the nullspace of Cb rather than the nullspace of Cb Po . If the nullspace of Cb Po is selected, the JSP method turns to the traditional orthogonal projection method, which introduces large deterioration in the receiver. Furthermore, after the operations in the last two steps, the precoding matrix P does not achieve the goal in the first step of mapping d to the nullspace of Ca . In addition, after the operation in the third step, P also does not achieve the goal of minimizing the power leakage in the optimized region in the second step. 9 Submitted As presented in the SVD operation [21] or orthogonal projection method [19], the receive quality and suppression performance have not been well balanced. We also had some tests to examine other combinations that only employing the first two steps or the last two steps in the JSP method. The results indicate that the suppression performance is similar to only using orthogonal projection or SVD operation. The reason is that the better OOB power reduction performance by the JSP method is obtained by two times orthogonal projection. In addition, the BER performance is improved by using reserved carriers in the second step. What’s more, the desirable spectrum envelope also can be achieved by selecting the frequency points or optimized region in the three steps independently. 4. Design of The Precoding Matrix Under Condition Number Constraint In this section, we develop an algorithm to design the precoding matrix, obtaining the desirable spectrum envelope under receive performance constraint. As illustrated in Fig.1, the data after FFT process in the receiver is expressed as s  Qd + n, (17) where Q = HP , H is a complex diagonal matrix with the channel frequency response of M subcarriers and n is complex additive white Gaussian noise (AWGN) vector with zero mean. Q = P for AWGN channel. s is the received signal from noise measurement. The matrix P is full column but PT P  I N . Thus, this non-orthogonal precoding matrix will introduce BER loss in the receiver. If some singular values of P are too small, compared to the other values, low additive noise will result of large errors. Therefore, the condition number  of P is introduced, which is to measure the sensitivity of the solution of linear equations to errors in the data [28].  is given by   Con(P)= max , min (18) 10 Submitted where max and min is the largest and the smallest singular value of P . Con(.) denotes the condition number of a matrix. The value of  indicates the BER loss in the precoding.  [1, ) and larger value of  leads to worse BER performance. Fig. 3 The condition number of transition matrix Q in AWGN channel and Rayleigh fading channel. The transition matrix Q significantly influence the receive quality. We examine the average condition number of Q through different channels in Fig. 3. The results of through Rayleigh channel are averaged over 2000 realizations. Two Rayleigh channel is selected: 3GPP extended pedestrian A (EPA) model [29], whose excess tap delay = (0 30 70 90 110 190 410)ns with the relative power = (0 -1 -2 -3 -8 17.2 -20.8)dB, and a ten-tap Rayleigh block fading channel with exponentially decaying powers set as E (\| hl (i) \|2 )  el /3  9 j 0 e j /3 , j  0,1,...,9 . As illustrated in Fig. 3, the receive performance is better through the AWGN channel than the Rayleigh fading channel. Compared to the transmission without precoding, the condition number of Q linearly increases with condition number constraint  0 in AWGN channel, but decrease in the Rayleigh fading channel when the value of  0 is small. The zero-forcing equalizer is used for AWGN channel and fading channel respectively by 11 Submitted (PT P)1 PT s, d =   H 1 H  (Q Q) Q s, AWGN channel (19) Fading channel where (.) H and d denotes conjugate transpose and the decoded data respectively. In this part, we develop an algorithm to obtain P under a condition number constraint by selecting the frequency points. In order to keep the receive performance decreases slightly, the number of zeroforcing frequency points in the first orthogonal projection is chosen as small as possible. In practice, we select N a  1 for single-side suppression or N a  2 for double-side suppression to keep the spectrum symmetric. If the special case of R  1 is selected, then we fix N a  1 . The envelope of the precoded PSD curve is mainly determined by the outside orthogonal projection. Thus, Nb  R is selected for high suppression performance. The two group zero-forcing frequencies ω a and ωb are arranged in the different OOB region, far from or close to the mainlobe. The one far from the mainlobe is fixed, for the amplitude response is weak by (5). The one closed to the mainlobe is used to maintain the receive quality by adjusting its location. In the SVD operation, in order to effectively abandon the largest R singular values in Σ c , K should larger or equal to R . We select ωo  ωb for simple implementation and K  R . Then the variables for obtaining P are only the group closed to the mainlobe. Thus, we change these frequency points to achieve P  maxCon(P) P s.t. Con(P)   0 , (20) where  0 is the given condition number according to the BER constraint. The signal estimation performance under condition number constraint in a communication system is illustrated in [28, 30]. In addition, as presented in [16, 19], the zero-forcing frequency point close to mainlobe leads to quicker power reduction on the edge of the mainlobe, but the power far from the mainlobe is larger. Inversely, if these points are far from the mainlobe, the power decreases slowly on the edge, while the emission from the mainlobe decreases largely. 12 Submitted Table 1 The correlation between OFDM performance and the parameters in the proposed method. BER Case 0 R Na ― ↓ ― ↓ Case A: Quicker power reduction on the edge of the mainlobe Na=1: Single-side suppression ↑ ↑ Case B: Lower power leakage far Na=2: Double-side suppression from the mainlobe ↓: Negative correlation; ↑: Positive correlation; ―: Weak or no effect. OOB power reduction With summarizing the properties analyzed above, we first give the correlation between the parameters and the performance of OFDM transceiver in Table 1, which is also presented in the simulation section. In the following, we develop an algorithm to obtain the precoding matrix P by selecting the frequencies ωb and ω a , according to the summary in Table 1. The main spectrum envelope is decided in the initialization process step by step as (a.) The value of N a and Case is selected according to the power spectrum envelope property. (b.) R is chosen according to sidelobe suppression performance and spectral efficiency. (c.)  0 is the given condition number according to the BER quality constraint. If Case A is required, we fix ω a far from the mainlobe and adjust ωb . The process to solve (20) is given in the algorithm. 1. Algorithm. 1: Initialization:  0 , R , N a , iteration increment  , ωb  ωb0 , ωa  ωa0 ,   0 , i  0 . 1. i : i  i +1 . i-th iteration. 2. Design Pi by Section 3 and compute the condition number   Con(Pi ) . 3. if (  0 ) ωb :  ωb   , go back to Step 1. else Stop. Output: P  Pi1 . where ωb0 is the initializing frequency points close to the mainlobe, and ω a0 is the fixed frequencies far from the mainlobe. If the number of reserved carriers of R is large, a little adjusting of ωb will lead to 13 Submitted large change to  . Therefore, the iteration increment  also should be small, and the distance between the frequency points in ωb should not be too small. In practice, we select R  4 . If Case B is required, we fix ωb far from the mainlobe and adjust ωa :  ωa   in the Step 3. The distance between the frequency points in ωb also should not be too small or too far from the mainlobe. Otherwise, the matrix Cop may be close to singular or badly scaled. That may lead to that the results may be inaccurate in (13) by SVD in practice. 5. Simulation Results and Discussions In this section, some numerical results are presented to demonstrate the performance of the proposed method. An instance in LTE is selected that the subcarrier spacing is 15 kHz. The number of the subcarriers is N  300 in 5 MHz bandwidth [4]. The frequency axis is normalized to the spacing 2 / T . The index of data subcarriers is from -150 to 150, while the direct current carrier   0 is not employed. To illustrate the OOB power reduction effect, the PSD is obtained by computing the power of DFT coefficients of time-domain OFDM signal over a time span and averaging over thousands of symbols. The frequency-domain oversampling rate is eight and the QPSK modulation is employed. The simulated are mainly ZP-OFDM system, unless noted otherwise. A. Sidelobe Suppression Performance In the first experiment, the OOB reduction performance comparison is presented using different precoding technologies. The number of reserved subcarriers is selected as R  2 . The SSOP [19], the orthogonal projection (OP) [16], the optimal orthogonal precoding (OOP) [21] and CC [8] methods are selected as a comparison. The zero-forcing frequencies are fixed at ω  180 for OP and SSOP method. The optimized region for OOP and CC scheme is also selected nearby the frequencies ω  180 . N a  2 for double-side suppression to keep the spectrum symmetric in the proposed JSP method. The spectral 14 Submitted coding rate is 300/302, ωa  4000 and ωb  180 . The condition number of the precoding matrix is   3.2279 . Fig. 4 PSD of the OFDM signal with different precoding techniques. As the PSD curves illustrated in Fig. 4, all the transmitted signals are ZP-OFDM, besides CP-OFDM signal is presented by the proposed JSP method. It is obviously that the OOB power using the JSP method is lower than other approaches. The OOP/OP/SSOP methods have similar suppression performance. The CC, which is not a spectral precoding method, achieves less OOB power reduction. In addition, the results show the performance degradation of sidelobe suppression in CP systems. Fig. 5 Quicker power reduction on the edge of the mainlobe (Case A) with different condition number constraint. 15 Submitted Fig. 6 Lower power leakage far from the mainlobe (Case B) with different condition number constraint. In Fig. 5, the precoding matrix is designed under a condition number constraint of 0  {1.5,3,5,10, 20} respectively. The PSD curves of Case A is presented with N a  2 . ω a is fixed far from the mainlobe and ωa  4000 , R  2 and   0.25 . Obviously, the suppression performance is improved by relaxing the condition number constraint while the distance between ωb and mainlobe increasing. In Fig. 6, the Case B is presented with N a  2 . We fix ωb far from the mainlobe and ωb  4000 , and the other parameters are same with in Fig. 5. The suppression performance is also improved by relaxing  0 . The OOB power in the region far from the mainlobe is lower than Case A. However, the power on the edge of mainlobe decreases slowly. Fig. 7 Single-side suppression with different number of revered carriers, Na=1. Fig. 8 PSD of Multi-band OFDM signal. In Fig. 7, the Case B is selected, N a  1 and  0  1.5 . The right-side suppression is presented by choosing ω a and ωb in the right side of the mainlobe. The distance between two adjacent frequencies in ω a is five when R  1 . As the symmetry presented in (5), (9) and (14), the difference between adopting the OOB frequency ω and ω to design the precoding matrix is only in the second step of SVD. Therefore, the power on the left side OOB region is also reduced, while the power emission is higher than 16 Submitted on the right side. The suppression performance is improved by increasing the number of revered carriers. Although the simulation is not illustrated, this property is also presented in the case of N a  2 by adjusting R. In cognitive radio system, the secondary users access the spectrum hole dynamically. In addition, carrier aggregation (CA) [31] is a critical technology in LTE system, which enable a user to employ noncontiguous subbands. Therefore, in Fig. 8, the PSD of two non-contiguous subbands occupied by one user or two is presented. The condition number is  0  10 for all the cases. The number of the carriers of each subband is 150. The design of the precoding matrix for two users is independent, while the data transmitted by one user through two subbands is dependent. The results indicate that the proposed scheme is also suitable for multi-users transmitting through non-contiguous subbands. B. BER performance Fig. 9 BER performance with different spectral precoding techniques through AWGN channel. Fig. 10 BER performance with same condition number constraint through AWGN channel, Na=1. A comparison of the BER performance using spectral precoding techniques is shown in Fig.9. The condition number constraint is  0  2 for the JSP. The optimized region or zero-forcing frequencies for OOP, SSOP and OP method is same with the JSP method. As illustrated in Fig. 4, the sidelobe suppression 17 Submitted performance by OOP, SSOP and OP method is similar. The difference is that OOP scheme introduces no quality loss in the receiver and the OP method leads to large error as presented in Fig. 9. The SSOP method improves the receive quality by adopting a reserved subcarrier to decode the distorted signal. The BER quality loss by the JSP method is not notable. In Fig. 10, the condition number constraint is fixed as  0  10 , as well as single-side suppression is selected as N a  1 . The results illustrated that the receive quality is approximate identical when the condition number and N a are fixed, although both of the number of reserved carriers R and the Case are different. This property indicates that if N a has been selected according to single-side suppression or double-side suppression, the BER performance constraint can be converted to the condition number constraint in practice. Fig. 11 BER performance with difference condition number constraint and Na through AWGN channel, R=2. In Fig. 11, the results demonstrate that the BER performance of N a  1 is better than N a  2 under same constraint. This is why we select the value of N a as small as possible to maintain the receive quality. 18 Submitted Fig. 12 BER performance with difference condition number constraint through AWGN channel. Fig. 13 BER performance with difference condition number constraint through 3GPP EPA fading channel. In Fig. 12, the receive quality is presented with N a  2 . The number of revered carriers is R  2 . The results show that the BER performance decreases by relaxing  0 , but the suppression performance is improved shown in Fig. 5-6. For example, the power in the OOB region can be reduced by nearly 40dB when  0  10 , while the SNR loss is less than 1dB at BER=10−7. When the condition number of precoding matrix is small, such as  0  1.5 in Fig. 12, the BER performance is slightly better than that of without precoding. Because the number of the modulated carriers of each symbol is extended from N to M , the receive quality is improved by frequency diversity. However, this property is not notable when the value of  0 is large. In Fig. 13, the parameters are set same with in Fig.12. Comparing to without precoding, the BER performance through the fading channel is similar to AWGN channel, when SNR is small. The receive quality decreases by relaxing  0 . But with SNR increasing, the BER line converges to without precoding. 6. Conclusions In this paper, a joint spectral precoding (JSP) method is proposed to reduce the OOB power emission in OFDM-based system, as well as an iteration algorithm is given to obtain the precoding matrix. We also 19 Submitted convert the BER performance constraint to the condition number constraint of the precoding matrix. As summarized in Table I and presented in the simulations, the proposed method has an appreciate balance between suppression performance and spectral efficiency, under a receive quality constrain. We also can obtain the desirable spectrum envelope by parameters configuration. Therefore, this method is flexible for spectrum shaping in both conventional OFDM systems and non-continuous multi-carriers cognitive radio networks. 7. References 1. Xing, Y.P., Chandramouli, R., Mangold, S., and Shankar, N.S., 'Dynamic Spectrum Access in Open Spectrum Wireless Networks', IEEE Journal on Selected Areas in Communications, 2006, 24, (3), pp. 626-637. 2. Zhao, Q. and Sadler, B.M., 'A Survey of Dynamic Spectrum Access', IEEE Signal Processing Magazine, 2007, 24, (3), pp. 79-89. 3. Mahmoud, H.A., Yucek, T., and Arslan, H., 'Ofdm for Cognitive Radio: Merits and Challenges', IEEE Wireless Communications, 2009, 16, (2), pp. 6-14. 4. Dahlman, E., Parkvall, S., and Skold, J., 4g: Lte/Lte-Advanced for Mobile Broadband, (Academic Press, 2013) 5. Faulkner, M., 'The Effect of Filtering on the Performance of Ofdm Systems', IEEE Transactions on Vehicular Technology, 2000, 49, (5), pp. 1877-1884. 6. Lin, Y.P. and Phoong, S.M., 'Window Designs for Dft-Based Multicarrier Systems', IEEE Transactions on Signal Processing, 2005, 53, (3), pp. 1015-1024. 7. Schmidt, J.F., Costas-Sanz, S., and Lopez-Valcarce, R., 'Choose Your Subcarriers Wisely: Active Interference Cancellation for Cognitive Ofdm', IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2013, 3, (4), pp. 615-625. 8. Brandes, S., Cosovic, I., and Schnell, M., 'Reduction of out-of-Band Radiation in Ofdm Systems by Insertion of Cancellation Carriers', IEEE Communications Letters, 2006, 10, (6), pp. 420-422. 9. Cosovic, I., Brandes, S., and Schnell, M., 'Subcarrier Weighting: A Method for Sidelobe Suppression in Ofdm Systems', IEEE Communications Letters, 2006, 10, (6), pp. 444-446. 10. Cosovic, I. and Mazzoni, T., 'Suppression of Sidelobes in Ofdm Systems by Multiple-Choice Sequences', European Transactions on Telecommunications, 2006, 17, (6), pp. 623-630. 11. Li, D., Dai, X.H., and Zhang, H., 'Sidelobe Suppression in Nc-Ofdm Systems Using Constellation Adjustment', IEEE Communications Letters, 2009, 13, (5), pp. 327-329. 12. Liu, S., Li, Y., Zhang, H., and Liu, Y., 'Constellation Expansion-Based Sidelobe Suppression for Cognitive Radio Systems', Communications, IET, 2013, 7, (18), pp. 2133-2140. 20 Submitted 13. Mahmoud, H.A. and Arslan, H., 'Sidelobe Suppression in Ofdm-Based Spectrum Sharing Systems Using Adaptive Symbol Transition', IEEE Communications Letters, 2008, 12, (2), pp. 133-135. 14. Xu, R., Chen, M., Zhang, J., Wu, B., and Wang, H., 'Spectrum Sidelobe Suppression for Discrete Fourier Transformation-Based Orthogonal Frequency Division Multiplexing Using Adjacent Subcarriers Correlative Coding', IET Communications, 2012, 6, (11), pp. 1374-1381. 15. Huang., X., Zhang., J.A., and Guo, Y.J., 'Out-of-Band Emission Reduction and a Unified Framework for Precoded Ofdm', IEEE Communications Magazine, 2015, 53, (6), pp. 151-159. 16. van de Beek, J. and Berggren, F., 'N-Continuous Ofdm', IEEE Communications Letters, 2009, 13, (1), pp. 1-3. 17. Zheng, Y.M., Zhong, J., Zhao, M.J., and Cai, Y.L., 'A Precoding Scheme for N-Continuous Ofdm', IEEE Communications Letters, 2012, 16, (12), pp. 1937-1940. 18. van de Beek, J., 'Sculpting the Multicarrier Spectrum: A Novel Projection Precoder', IEEE Communications Letters, 2009, 13, (12), pp. 881-883. 19. Zhang, J.A., Huang, X.J., Cantoni, A., and Guo, Y.J., 'Sidelobe Suppression with Orthogonal Projection for Multicarrier Systems', IEEE Transactions on Communications, 2012, 60, (2), pp. 589599. 20. Zhou, X.W., Li, G.Y., and Sun, G.L., 'Multiuser Spectral Precoding for Ofdm-Based Cognitive Radio Systems', IEEE Journal on Selected Areas in Communications, 2013, 31, (3), pp. 345-352. 21. Ma, M., Huang, X.J., Jiao, B.L., and Guo, Y.J., 'Optimal Orthogonal Precoding for Power Leakage Suppression in Dft-Based Systems', IEEE Transactions on Communications, 2011, 59, (3), pp. 844853. 22. Tom, A., Sahin, A., and Arslan, H., 'Mask Compliant Precoder for Ofdm Spectrum Shaping', IEEE Communications Letters, 2013, 17, (3), pp. 447-450. 23. Muquet, B., Wang, Z.D., Giannakis, G.B., de Courville, M., and Duhamel, P., 'Cyclic Prefixing or Zero Padding for Wireless Multicarrier Transmissions?', IEEE Transactions on Communications, 2002, 50, (12), pp. 2136-2148. 24. Lu, H., Nikookar, H., and Chen, H., 'On the Potential of Zp-Ofdm for Cognitive Radio', Proc. WPMC’09, 2009, pp. 7-10. 25. Van Waterschoot, T., Le Nir, V., Duplicy, J., and Moonen, M., 'Analytical Expressions for the Power Spectral Density of Cp-Ofdm and Zp-Ofdm Signals', Signal Processing Letters, IEEE, 2010, 17, (4), pp. 371-374. 26. Wu, T.W. and Chung, C.D., 'Spectrally Precoded Dft-Based Ofdm and Ofdma with Oversampling', IEEE Transactions on Vehicular Technology, 2014, 63, (6), pp. 2769-2783. 27. Meyer, C.D., Matrix Analysis and Applied Linear Algebra, (SIAM, 2000) 28. Tong, J., Guo, Q.H., Tong, S., Xi, j.T., and Yu, Y.G., 'Condition Number-Constrained Matrix Approximation with Applications to Signal Estimation in Communication Systems', IEEE Communications Letters, 2014, 21, (8), pp. 990-993. 29. 3GPP TS 36.141: 'E-Utra Base Station (Bs) Conformance Testing', 2015 21 Submitted 30. Aubry, A., De Maio, A., Pallotta, L., and Farina, A., 'Maximum Likelihood Estimation of a Structured Covariance Matrix with a Condition Number Constraint', Signal Processing, IEEE Transactions on, 2012, 60, (6), pp. 3004-3021. 31. Yuan, G.X., Zhang, X., Wang, W.B., and Yang, Y., 'Carrier Aggregation for Lte-Advanced Mobile Communication Systems', IEEE Communications Magazine, 2010, 48, (2), pp. 88-93. 22	5
arXiv:1604.00656v1 [math.AC] 3 Apr 2016 DEPTH, STANLEY DEPTH AND REGULARITY OF IDEALS ASSOCIATED TO GRAPHS S. A. SEYED FAKHARI Abstract. Let K be a field and S = K[x1 , . . . , xn ] be the polynomial ring in n variables over K. Let G be a graph with n vertices. Assume that I = I(G) is the edge ideal of G and J = J(G) is its cover ideal. We prove that sdepth(J) ≥ n−νo (G) and sdepth(S/J) ≥ n − νo (G) − 1, where νo (G) is the ordered matching number of G. We also prove the inequalities sdepth(J k ) ≥ depth(J k ) and sdepth(S/J k ) ≥ depth(S/J k ), for every integer k ≫ 0, when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg(S/I) ≤ νo (G). 1. Introduction and Preliminaries Let K be a field and let S = K[x1 , . . . , xn ] be the polynomial ring in n variables over K. Let M be a finitely generated Zn -graded S-module. Let u ∈ M be a homogeneous element and Z ⊆ {x1 , . . . , xn }. The K-subspace uK[Z] generated by all elements uv with v ∈ K[Z] is called a Stanley space of dimension \|Z\|, if it is a free K[Z]-module. Here, as usual, \|Z\| denotes the number of elements of Z. A decomposition D of M as a finite direct sum of Stanley spaces is called a Stanley decomposition of M. The minimum dimension of a Stanley space in D is called the Stanley depth of D and is denoted by sdepth(D). The quantity sdepth(M) := max sdepth(D) \| D is a Stanley decomposition of M is called the Stanley depth of M. We say that a Zn -graded S-module M satisfies Stanley’s inequality if depth(M) ≤ sdepth(M). In fact, Stanley [22] conjectured that every Zn -graded S-module satisfies Stanley’s inequality. This conjecture has been recently disproved in [1]. However, it is still interesting to find the classes of Zn -graded S-modules which satisfy Stanley’s inequality. For a reader friendly introduction to Stanley depth, we refer to [18] and for a nice survey on this topic, we refer to [11]. Let G be a graph with vertex set V (G) = x1 , . . . , xn and edge set E(G) (by abusing the notation, we identify the vertices of G with the variables of S). For a vertex xi , the neighbor set of xi is NG (xi ) = {xj \| xi xj ∈ E(G)} and We set NG [xi ] = NG (xi ) ∪ {xi } and call it the closed neighborhood of xi . For every subset 2000 Mathematics Subject Classification. Primary: 13C15, 05E99; Secondary: 13C13. Key words and phrases. Cover ideal, Edge ideal, Ordered matching, Regularity, Stanley depth, Stanley’s inequality. 1 2 S. A. SEYED FAKHARI A ⊂ V (G), the graph G \ A is the graph with vertex set V (G \ A) = V (G) \ A and edge set E(G \ A) = {e ∈ E(G) \| e ∩ A = ∅}. A bipartite graph is one whose vertex set is partitioned into two (not necessarily nonempty) disjoint subsets in such a way that the two end vertices for each edge lie in distinct partitions. A matching in a graph is a set of edges such that no two different edges share a common vertex. A subset W of V (G) is called an independent subset of G if there are no edges among the vertices of W . A subset C of V (G) is called a vertex cover of the graph G if every edge of G is incident to at least one vertex of C. A vertex cover C is called a minimal vertex cover of G if no proper subset of C is a vertex cover of G. Next, we define the notion of ordered matching for a graph. It was introduced in [5] and plays a central role in this paper. Definition 1.1. Let G be a graph, and let M = {{ai , bi } \| 1 ≤ i ≤ r} be a nonempty matching of G. We say that M is an ordered matching of G if the following hold: (1) A := {a1 , . . . , ar } ⊆ V (G) is a set of independent vertices of G; and (2) {ai , bj } ∈ E(G) implies that i ≤ j. The ordered matching number of G, denoted by νo (G), is defined to be νo (G) = max{\|M\| \| M ⊆ E(G) is an ordered matching of G}. The edge ideal I(G) of G is the ideal of S generated by the squarefree monomials xi xj , where {xi , xj } is an edge of G. The Alexander dual of the edge ideal of G in S, i.e., the ideal \ J(G) = I(G)∨ = (xi , xj ), {xi ,xj }∈E(G) is called the cover ideal of G in S. The reason for this name is due to the well-known fact that the generators of J(G) correspond to minimal vertex covers of G. The main goal of This paper is to study the Stanley depth of cover ideals and their power. In Theorem 2.4, we prove that for every graph G, the inequalities sdepth(J(G)) ≥ n−νo (G) and sdepth(S/J(G)) ≥ n−νo (G)−1 hold. In that theorem, we also prove that the same inequalities hold, if one replaces sdepth by depth. Then, in Corollary 2.5, we conclude that for every graph G we have reg(S/I) ≤ νo (G). This inequality was previously proved by Constantinescu and Varbaro [5, Remark 4.8]. However, our proof is more elementary. In Section 3, we consider the Stanley depth of powers of cover ideal of bipartite graphs. Let G be a bipartite graph. In [20, Corollary 3.6], the author proved that the k ∞ sequences {sdepth(J(G)k )}∞ k=1 and {sdepth(S/J(G) )}k=1 are non-increasing. Thus the both sequences are convergent. In Theorem 3.3, we provide lower bounds for the limit value of theses sequences. Indeed, we prove that for every bipartite graph G, we have lim sdepth(J(G)k ) ≥ n − νo (G) and k→∞ lim sdepth(S/J(G)k ) ≥ n − νo (G) − 1. k→∞ Then we conclude in Corollary 3.4 that J(G)k and S/J(G)k satisfy the Stanley’s inequality, for every integer k ≫ 0. Theorem 3.3 also shows that a conjecture of the DEPTH, SDEPTH AND REGULARITY 3 author is true for the powers of cover ideal of bipartite graphs (see Conjecture 3.5 and the paragraph after it). 2. First Power The first main result of this paper is Theorem 2.4, which provides a lower bound for the depth and the Stanley depth of cover ideal of graphs. We first need the following three simple lemmas. The first one shows that the ordered matching number of a graph strictly decreases when we delete the closed neighborhood of a non-isolated vertex. Lemma 2.1. Let G be a graph and x be a non-isolated vertex of G. Then we have νo (G \ NG [x]) ≤ νo (G) − 1. Proof. Assume that νo (G\NG [x]) = t and let M = {{ai , bi } \| 1 ≤ i ≤ t} be an ordered matching of G \ NG [x]. Since x is not isolated, we may choose a vertex y ∈ NG (x). Set at+1 = x and bt+1 = y. Then {a1 , . . . , at+1 } is a set of independent vertices of G, because a1 , . . . , at are vertices of G\NG [x]. By the same reason, at+1 is not adjacent to b1 , . . . , bt . This shows that M ∪{at+1 , bt+1 } is an ordered matching of G and therefore, νo (G) ≥ t + 1. The next Lemma shows that how the cover ideal of a graph G and that of G \ NG [x] are related, when x is an arbitrary vertex of G. Lemma 2.2. Let G ba a graph with vertex set V (G) = {x1 , . . . , xn }. Assume that Q x ∈ V (G) is a vertex of G. Set u = xi ∈NG (x) xi and J ′ = J(G \ NG [x])S. Then J(G) + (x) = uJ ′ + (x). Proof. Let C be a vertex cover of G with x ∈ / C. Then NG (x) ⊆ C and C \ NG (x) is a vertex cover of G \ NG [x]. This shows that J(G) + (x) ⊆ uJ ′ + (x). For the converse inclusion, assume that D is a vertex cover of G \ NG [x]. Then D ∪ NG (x) is a vertex cover of G. This shows that uJ ′ + (x) ⊆ J(G) + (x) and completes the proof. The following lemma provides a combinatorial description for the colon of cover ideals. Lemma 2.3. Let G ba a graph with vertex set V (G) = {x1 , . . . , xn }. Assume that x ∈ V (G) is a vertex of G. Set J ′ = J(G \ x)S. Then (J(G) : x) = J ′ . Proof. If C is a vertex cover of G, then C \ {x} is a vertex cover of G \ x. This shows that (J(G) : x) ⊆ J ′ . On the other hand, if D is a vertex cover of G \ x, then D ∪ {x} is a vertex cover of G. This shows that J ′ ⊆ (J(G) : x). We are now ready to prove the first main result of this paper. As we mentioned in introduction, the second part of this theorem is known by [5, Remark 4.8]. But our argument is completely different and provides a simple proof for it. Theorem 2.4. Let G be a graph and J(G) be its cover ideal. Then 4 S. A. SEYED FAKHARI (i) sdepth(J(G)) ≥ n − νo (G) and sdepth(S/J(G)) ≥ n − νo (G) − 1, (ii) depth(S/J(G)) ≥ n − νo (G) − 1. Proof. We prove (i) and (ii) simultaneously by induction on the number of edges of G. If G has only one edge, then νo (G) = 1 and J(G) is generated by two variables. Then depth(S/J(G)) = n − 2. Also, sdepth(S/J(G)) = n − 2 by [19, Theorem 1.1] and sdepth(J(G)) ≥ n − 1 by [11, Corollary 24] and [13, Lemma 3.6]. Therefore, in these cases, the inequalities in (i) and (ii) are trivial. We now assume that G has at least two edges. Note that, G has at least one nonisolated vertex. Without loss of generality, we may assume that x1 is a non-isolated vertex of G. Let S ′ = K[x2 , . . . , xn ] be the polynomial ring obtained from S by deleting the variable x1 and consider the ideals J ′ = J(G) ∩ S ′ and J ′′ = (J(G) : x1 ). Now J(G) = J ′ S ′ ⊕ x1 J ′′ S and S/J(G) = (S ′ /J ′ S ′ ) ⊕ x1 (S/J ′′ S) (as vector spaces) and therefore by definition of the Stanley depth we have sdepth(J(G)) ≥ min{sdepthS ′ (J ′ S ′ ), sdepthS (J ′′ )}, (1) and (2) sdepth(S/J(G)) ≥ min{sdepthS ′ (S ′ /J ′ S ′ ), sdepthS (S/J ′′ )}. On the other hand, by applying the depth lemma on the exact sequence 0 −→ S/(J(G) : x1 ) −→ S/J(G) −→ S/(J(G), x1 ) −→ 0 we conclude that (3) depth(S/J(G)) ≥ min{depthS ′ (S ′ /J ′ S ′ ), depthS (S/J ′′ )}. Using Lemma 2.3, it follows that J ′′ = J(G \ x1 )S. Hence our induction hypothesis implies that depthS (S/J ′′ ) = depthS ′ (S ′ /J ′′ ) + 1 ≥ n − 1 − νo (G \ x1 ) − 1 + 1 ≥ n − νo (G) − 1. Also, it follows from [13, Lemma 3.6] that sdepthS (S/J ′′ ) = sdepthS ′ (S ′ /J ′′ ) + 1 ≥ n − 1 − νo (G \ x1 ) − 1 + 1 ≥ n − νo (G) − 1, and sdepthS (J ′′ ) = sdepthS ′ (J ′′ ) + 1 ≥ n − 1 − νo (G \ x1 ) + 1 ≥ n − νo (G). On the other hand, it follows from Lemma 2.2 that there exists a monomial u ∈ S ′ such that J ′ S ′ = uJ(G \ NG [x1 ])S ′ . Since uJ(G \ NG [x1 ])S ′ and J(G \ NG [x1 ])S ′ , (up to a shift) are isomorphic as graded S ′ -Modules, we conclude that depthS ′ (J ′ S ′ ) = depthS ′ (J(G \ NG [x1 ])S ′ ). On the other hand, it follows from [7, Theorem 1.1] that sdepthS ′ (J ′ S ′ ) = sdepthS ′ (J(G\NG [x1 ])S ′ ) and sdepthS ′ (S ′ /J ′ S ′ ) = sdepthS ′ (S ′ /J(G\ NG [x1 ])S ′ ). Therefore by [13, Lemma 3.6], Lemma 2.1 and the induction hypothesis we conclude that sdepthS ′ (J ′ S ′ ) = sdepthS ′ (J(G \ NG [x1 ])S ′ ) ≥ n − 1 − νo (G \ NG [x1 ]) ≥ n − νo (G), DEPTH, SDEPTH AND REGULARITY 5 and similarly sdepthS ′ (S ′ /J ′ S ′ ) ≥ n−νo (G) −1 and depthS ′ (S ′ /J ′ S ′ ) ≥ n−νo (G) −1. Now the assertions follow by inequalities (1), (2) and (3). Let M be a finitely generated graded S-Module. The Castelnuovo-Mumford regularity (or simply, regularity) of M, denoted by reg(M), is defined as follows: reg(M) = max{j − i\| TorSi (K, M)j 6= 0}. The regularity of a module is one of the most important homological invariants of it. Computing the regularity of edge ideals or finding bounds for it has been studied by a number of researchers (see for example [8], [10], [14], [15], [23]). An immediate consequence of the second part of theorem 2.4 is the following corollary. Corollary 2.5. For every graph G, we have reg(S/I(G)) ≤ νo (G). Proof. It follows from Theorem 2.4 and the Auslander-Buchsbaum Formula that the projective dimension of J(G) is at most νo (G). Then it follows from Terai’s theorem [12, Theorem 8.1.10] that reg(S/I(G)) ≤ νo (G). Remark 2.6. One can give a direct proof for the above corollary, by applying [16, Corollary 18.7] on the following exact sequence. 0 −→ S/(I(G) : x1 ) −→ S/I(G) −→ S/(I(G), x1 ) −→ 0 However, this proof is essentially the same as given above. In [10], Hà and Van Tuyl proved that the for every graph G, the regularity of S/I(G) is less than or equal to the maximum cardinality of matchings of G. In fact, it follows from their proof (and was explicitly stated in [24]) that the reg(S/I(G)) is at most the minimum cardinality of maximal matchings of G. The following examples show that this bound is not comparable with the bond given in Corollary 2.5. Examples 2.7. (1) Let G = C4 be the 4-cycle-graph. Then one can easily check that νo (G) = 1 and the cardinality of every maximal matching of G is equal to 2. Thus, in this example, νo (G) is strictly less than the minimum cardinality of maximal matchings of G. We also have reg(S/I(G)) = 1 = νo (G). (2) Let G = P4 be the path with 4 vertices. Then one can easily check that νo (G) = 2, while the minimum cardinality of maximal matchings of G is equal to 1. Thus, in this example, the minimum cardinality of maximal matchings of G is strictly less than νo (G). We also have reg(S/I(G)) = 1 is equal to the minimum cardinality of maximal matchings of G. 3. High Powers The aim of this section is to prove that the high powers of cover ideal of bipartite graphs satisfy the Stanley’s inequality. To do this, in Theorem 3.3, we provide a lower bound for the Stanley depth of cover ideal of bipartite graphs. Before that, in Lemma 3.2, we prove that the different powers of cover ideal of a bipartite graphs, can be 6 S. A. SEYED FAKHARI obtained from each other by taking colon with respect to a suitable monomial. To prove Lemma 3.2, we need to remind the definition of symbolic powers. Definition 3.1. Let I be a squarefree monomial ideal in S and suppose that I has the irredundant primary decomposition I = p1 ∩ . . . ∩ pr , where every pi is an ideal of S generated by a subset of the variables of S. Let k be a positive integer. The kth symbolic power of I, denoted by I (k) , is defined to be I (k) = pk1 ∩ . . . ∩ pkr . The proof of the following lemma is based on the fact that the symbolic and the ordinary powers of cover ideal of bipartite graphs coincide. Lemma 3.2. Let G be a bipartite graph Q and assume that V (G) = U ∪ W is a bipartition for the vertex set of G. Set u = xi ∈U xi . Then for every integer k ≥ 1, we have (J(G)k : u) = J(G)k−1 . Proof. It follows from [9, Corollary 2.6] that for every integer k ≥ 1 we have J(G)k = J(G)(k) . On the other hand, for every edge e = {xi , xj } of G, we have \| e ∩ U \|= 1. Thus, ((xi , xj )k : u) = (xi , xj )k−1 , for every integer k ≥ 1. Hence \ (J(G)k : u) = (J(G)(k) : u) = ((xi , xj )k : u) {xi ,xj }∈E(G) = \ (xi , xj )k−1 = J(G)(k−1) = J(G)k−1 . {xi ,xj }∈E(G) As we mentioned in the the first section, the sequences {sdepth(J(G)k )}∞ k=1 and {sdepth(S/J(G)k )}∞ are convergent. In the following theorem, we provide lower k=1 bounds for the limit of theses sequences. Theorem 3.3. Let G be a bipartite graph. Then for every integer k ≥ 1, the inequalities sdepth(J(G)k ) ≥ n − νo (G) and sdepth(S/J(G)k ) ≥ n − νo (G) − 1 hold. Proof. Assume that V (G) = U ∪ W is a bipartition for the vertex set of G. Without loss of generality, we may assume that U = {x1 , . . . , xt } and W = {xt+1 , . . . , xn }, for some integer t with 1 ≤ t ≤ n. Let m be the number of edges of G. We prove the assertions by induction on m + k. First, we can assume that G has no isolated vertex. Because deleting the isolated vertices does not change the cover ideal and the ordered matching number of G. For k = 1, the assertions follow from Theorem 2.4. If m = 1, then G has two vertices and νo (G) = 1. In this case, the first inequality follows from [11, Corollary 24] and the DEPTH, SDEPTH AND REGULARITY 7 second inequality is trivial. Therefore, assume that k, m ≥ 2. Let S1 = K[x2 , . . . , xn ] be the polynomial ring obtained from S by deleting the variable x1 and consider the ideals J1 = J(G)k ∩ S1 and J1′ = (J(G)k : x1 ). Now J(G)k = J1 ⊕ x1 J1′ and S/J(G)k = (S1 /J1 ) ⊕ x1 (S/J1′ ) (as vector spaces) and therefore by definition of the Stanley depth we have (†) sdepth(J(G)k ) ≥ min{sdepthS1 (J1 ), sdepthS (J1′ )}, and (‡) sdepth(S/J(G)k ) ≥ min{sdepthS1 (S1 /J1 ), sdepthS (S/J1′ )}. Notice that J1 = (J(G) ∩ S1 )k . Hence, by Lemma 2.2 we conclude that there exists a monomial u1 ∈ S1 such that J1 = uk1 J(G \ NG [x1 ])k S1 . It follows from [7, Theorem 1.1] that sdepthS1 (J1 ) = sdepthS1 (J(G \ NG [x1 ])k S1 ) and sdepthS1 (S1 /J1 S1 ) = sdepthS1 (S1 /J(G \ NG [x1 ])k S1 ). Therefore, by [13, Lemma 3.6], Lemma 2.1 and the induction hypothesis, we conclude that sdepthS1 (J1 ) = sdepthS1 (J(G \ NG [x1 ])k S1 ) ≥ n − 1 − νo (G \ NG [x1 ]) ≥ n − νo (G), and similarly sdepthS1 (S1 /J1 ) ≥ n−νo (G)−1. Thus, using the inequalities (†) and (‡), it is enough to prove that sdepthS (J1′ ) ≥ n−νo (G) and sdepthS (S/J1′ ) ≥ n−νo (G)−1. For every integer i with 2 ≤ i ≤ t, let Si = K[x1 , . . . , xi−1 , xi+1 , . . . , xn ] be the polynomial ring obtained from S by deleting the variable xi and consider the ideals ′ ′ ∩ Si . : xi ) and Ji = Ji−1 Ji′ = (Ji−1 Claim. For every integer i with 1 ≤ i ≤ t − 1 we have ′ sdepth(Ji′ ) ≥ min{n − νo (G), sdepth(Ji+1 )} and ′ sdepth(S/Ji′ ) ≥ min{n − νo (G) − 1, sdepth(S/Ji+1 )}. Proof of the Claim. For every integer i with 1 ≤ i ≤ t − 1, we have Ji′ = Ji+1 ⊕ ′ ′ ) (as vector spaces) and therefore by and S/Ji′ = (Si+1 /Ji+1 ) ⊕ xi+1 (S/Ji+1 xi+1 Ji+1 definition of the Stanley depth we have (∗) ′ sdepth(Ji′ ) ≥ min{sdepthSi+1 (Ji+1 ), sdepthS (Ji+1 )}, and (∗∗) ′ )}. sdepth(S/Ji′ ) ≥ min{sdepthSi+1 (Si+1 /Ji+1 ), sdepthS (S/Ji+1 Notice that for every integer i with 1 ≤ i ≤ t−1, we have Ji′ = (J(G)k : x1 x2 . . . xi ). Thus Ji+1 = Ji′ ∩ Si+1 = ((J(G)k ∩ Si+1 ) :Si+1 x1 x2 . . . xi ). Hence, it follows from [17, Proposition 2] and [6, Proposition 2.7] (see also [20, Proposition 2.5]) that (∗ ∗ ∗) sdepthSi+1 (Ji+1 ) ≥ sdepthSi+1 (J(G)k ∩ Si+1 ). and (∗ ∗ ∗∗) sdepthSi+1 (Si+1 /Ji+1 ) ≥ sdepthSi+1 (Si+1 /(J(G)k ∩ Si+1 )). 8 S. A. SEYED FAKHARI By Lemma 2.2 we conclude that there exists a monomial ui+1 ∈ Si+1 such that J(G) ∩ Si+1 = ui+1 J(G \ NG [xi+1 ])Si+1 . Therefore J(G)k ∩ Si+1 = uki+1 J(G \ NG [xi+1 ])k Si+1 and it follows from [7, Theorem 1.1] that sdepthSi+1 (J(G)k ∩ Si+1 ) = sdepthSi+1 (J(G \ NG [xi+1 ])k Si+1 ) and sdepthSi+1 (Si+1 /(J(G)k ∩ Si+1 )) = sdepthSi+1 (Si+1 /J(G \ NG [xi+1 ])k Si+1 ). Therefore by [13, Lemma 3.6], Lemma 2.1 and the induction hypothesis we conclude that sdepthSi+1 (J(G)k ∩ Si+1 ) ≥ n − 1 − νo (G \ NG [xi+1 ]) ≥ n − νo (G), and similarly sdepthSi+1 (Si+1 /(J(G)k ∩ Si+1 )) ≥ n − νo (G) − 1. Now the claim follows by inequalities (∗), (∗∗), (∗ ∗ ∗) and (∗ ∗ ∗∗). Now, Jt′ = (J(G)k : x1 x2 . . . xt ) and hence, Lemma 3.2 implies that Jt′ = J(G)k−1 and thus, by induction hypothesis we conclude that sdepth(Jt′ ) ≥ n − νo (G) and sdepth(S/Jt′ ) ≥ n − νo (G) − 1. Therefore, using the claim repeatedly implies that sdepth(J1′ ) ≥ n − νo (G) and sdepth(S/J1′ ) ≥ n − νo (G) − 1. This completes the proof of the theorem. Let I ⊂ S be a monomial ideal. A classical result by Burch [3] states that min depth(S/I k ) ≤ n − ℓ(I), k where ℓ(I) analytic spread of I, that is, the dimension of R(I)/mR(I), where L is the n I = S[It] ⊆ S[t] is the Rees ring of I and m = (x1 , . . . , xn ) is the maxR(I) = ∞ n=0 imal ideal of S. By a theorem of Brodmann [2], depth(S/I k ) is constant for large k. We call this constant value the limit depth of I, and denote it by limk→∞ depth(S/I k ). Brodmann improved the Burch’s inequality by showing that (♯) limk→∞ depth(S/I k ) ≤ n − ℓ(I). Let I ⊂ S be an arbitrary ideal. An element f ∈ S is integral over I, if there exists an equation f k + c1 f k−1 + . . . + ck−1 f + ck = 0 with ci ∈ I i . The set of elements I in S which are integral over I is the integral closure of I. The ideal I is integrally closed, if I = I. It is known that the equality holds, in inequality (♯), if I is a normal ideal. By [9, Corollary 2.6] and [12, Theorem 1.4.6], we know that J(G) is a normal ideal, for every bipartite graph G. Also, it follows from [5, Theorem 2.8] that for every bipartite graph G, we have ℓ(J(G)) = νo (G) + 1. Thus, we conclude that lim depth(S/J(G)k ) = n − 1 − νo (G). k→∞ DEPTH, SDEPTH AND REGULARITY 9 (This equality is explicitly stated in [4, Theorem 4.5].) Therefore, Theorem 3.3 implies the following result Corollary 3.4. Let G be a bipartite graph and J(G) be its edge ideal. Then there exists an integer n0 ≥ 1 such that J(G)k and S/J(G)k satisfy the Stanley’s inequality, for every integer k ≥ n0 . In [21], the author proposed the following conjecture regarding the Stanley depth of integrally closed monomial ideals. Conjecture 3.5. ([21, Conjecture 2.6]) Let I ⊂ S be an integrally closed monomial ideal. Then sdepth(S/I) ≥ n − ℓ(I) and sdepth(I) ≥ n − ℓ(I) + 1. Let G be a bipartite graph. As we mentioned above J(G) is a normal ideal. Thus, every power of J(G) is integrally closed. Therefore, Theorem 3.3 shows that Conjecture 3.5 is true for the powers of cover ideal of bipartite graphs. References [1] A. M. Duval, B. Goeckner, C. J. Klivans, J. L. Martin, A non-partitionable Cohen-Macaulay simplicial complex, preprint. [2] M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 1, 35–39. [3] L. Burch, Codimension and analytic spread, Proc. Cambridge Philos. Soc. 72 (1972), 369–373. [4] A. Constantinescu, M. R. Pournaki, S. A. Seyed Fakhari, N.Terai, S. Yassemi, CohenMacaulayness and limit behavior of depth for powers of cover ideals, Comm. Algebra, 43 (2015), no. 1, 143–157. [5] A. Constantinescu, M. Varbaro, Koszulness, Krull dimension, and other properties of graphrelated algebras, J. Algebraic Combin. 34 (2011), no. 3, 375–400. [6] M. Cimpoeaş, Several inequalities regarding Stanley depth, Romanian Journal of Math. and Computer Science 2, (2012), 28–40. [7] M. Cimpoeaş, Stanley depth of monomial ideals with small number of generators, Central European Journal of Mathematics, 7 (2009), 629–634. [8] H. Dao, C. Huneke, J. Schweig, Bounds on the regularity and projective dimension of ideals associated to graphs, J. Algebraic Combin. 38 (2013), 37–55. [9] I. Gitler, E. Reyes, R. H. Villarreal, Blowup algebras of ideals of vertex covers of bipartite graphs, Contemp. Math. 376 (2005), 273–279. [10] H. T. Hà, A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27 (2008), 215–245. [11] J. Herzog, A survey on Stanley depth. In ”Monomial Ideals, Computations and Applications”, A. Bigatti, P. Giménez, E. Sáenz-de-Cabezón (Eds.), Proceedings of MONICA 2011. Lecture Notes in Math. 2083, Springer (2013). [12] J. Herzog, T. Hibi, Monomial Ideals, Springer-Verlag, 2011. [13] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra 322 (2009), no. 9, 3151–3169. [14] M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Algebraic Combin. 30 (2009), 429–445. [15] E. Nevo, Regularity of edge ideals of C4 -free graphs via the topology of the lcm-lattice, J. Combin. Theory Ser. A 118 (2011), 491–501. [16] I. Peeva, Graded syzygies, Algebra and Applications, vol. 14, Springer-Verlag London Ltd., London, 2011. 10 S. A. SEYED FAKHARI [17] D. Popescu, Bounds of Stanley depth, An. St. Univ. Ovidius. Constanta, 19(2),(2011), 187–194. [18] M. R. Pournaki, S. A. Seyed Fakhari, M. Tousi, S. Yassemi, What is . . . Stanley depth? Notices Amer. Math. Soc. 56 (2009), no. 9, 1106–1108. [19] A. Rauf, Stanley decompositions, pretty clean filtrations and reductions modulo regular elements, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 50(98) (2007), no. 4, 347–354. [20] S. A. Seyed Fakhari, Stanley depth and symbolic powers of monomial ideals, Math. Scand., to appear. [21] S. A. Seyed Fakhari, Stanley depth of the integral closure of monomial ideals, Collect. Math. 64 (2013), 351–362. [22] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175–193. [23] A. Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs: vertex decom posability and regularity, Arch. Math. (Basel) 93 (2009), 451–459. [24] R. Woodroofe, Matchings, coverings, and Castelnuovo-Mumford regularity, J. Commut. Algebra 6 (2014), 287–304. S. A. Seyed Fakhari, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran. E-mail address: fakhari@khayam.ut.ac.ir URL: http://math.ipm.ac.ir/∼fakhari/	0
arXiv:1206.5648v3 [cs.PL] 19 Sep 2013 C to O-O Translation: Beyond the Easy Stuff∗ Marco Trudel · Carlo A. Furia · Martin Nordio Bertrand Meyer · Manuel Oriol 19 April 2013 Abstract Can we reuse some of the huge code-base developed in C to take advantage of modern programming language features such as type safety, object-orientation, and contracts? This paper presents a source-to-source translation of C code into Eiffel, a modern object-oriented programming language, and the supporting tool C2Eif. The translation is completely automatic and handles the entire C language (ANSI, as well as many GNU C Compiler extensions, through CIL) as used in practice, including its usage of native system libraries and inlined assembly code. Our experiments show that C2Eif can handle C applications and libraries of significant size (such as vim and libgsl), as well as challenging benchmarks such as the GCC torture tests. The produced Eiffel code is functionally equivalent to the original C code, and takes advantage of some of Eiffel’s features to produce safe and easy-to-debug translations. 1 Introduction Programming languages have significantly evolved since the original design of C in the 1970’s as a “system implementation language” [29] for the Unix operating system. C was a high-level language by the standards of the time, but it is pronouncedly low-level compared with modern programming paradigms, as it lacks advanced features—static type safety, encapsulation, inheritance, and contracts [20], to mention just a few—that can have a major impact on programmer’s productivity and on software quality and maintainability. C still fares as the most popular general-purpose programming language [38], and countless C applications are being actively written and maintained, that take advantage of the language’s conciseness, speed, ubiquitous support, and huge code-base. An automated solution to translate and integrate C code into a modern language would combine the large availability of C programs in disparate domains with the integration in a modern language that facilitates writing safe, robust, and easy-to-maintain applications. The present paper describes the fully automatic translation of C programs into Eiffel, an object-oriented programming language, and the tool C2Eif, which implements ∗ This work was partially supported by ETH grant “Object-oriented reengineering environment”. 1 the translation. While the most common approaches to re-use C code in other host languages are based on “foreign function APIs” (see Section 5 for examples), sourceto-source translation solves a different problem, and has some distinctive benefits: the translated code can take advantage of the high-level nature of the target language and of its safer runtime. Main features of C2Eif. Translating C to a high-level object-oriented language is challenging because it requires adapting to a more abstract memory representation, a tighter type system, and a sophisticated runtime that is not directly accessible. There have been previous attempts to translate C into an object-oriented language (see the review in Section 5). A limitation of the resulting tools is that they hardly handle the trickier or specialized parts of the C language [7], which it is tempting to dismiss as unimportant “corner cases”, but figure prominently in real-world programs; examples include calls to pre-compiled C libraries (e.g., for I/O), inlined assembly, and unrestricted branch instructions including setjmp and longjmp. One of the distinctive features of the present work is that it does not stop at the core features but extends over the often difficult “last mile”: it covers the entire C language as used in practice. The completeness of the translation scheme is attested by the set of example programs to which the translation was successfully applied, as described in Section 4, including major applications such as the vim editor (276 KLOC), major libraries such as libgsl (238 KLOC), and the challenging “torture” tests for the GCC C compiler. C2Eif is available at http://se.inf.ethz.ch/research/c2eif. The webpage includes C2Eif’s sources, pre-compiled binaries, source and binaries of all translated programs of Table 1, and a user guide. Sections 2–3 describe the distinctive features of the translation: it supports the complete C language (including pointer arithmetic, unrestricted branch instructions, and function pointers) with its native system libraries; it complies with ANSI C as well as many GNU C Compiler extensions through the CIL framework [22]; it is fully automatic, and it handles complete applications and libraries of significant size; the generated Eiffel code is functionally equivalent to the original C code (as demonstrated by running thorough test suites), and takes advantage of some advanced features, such as strong typing and contracts, to facilitate debugging some programming mistakes. In our experiments, C2Eif translated completely automatically over 900,000 lines of C code from real-world applications, libraries, and testsuites, producing functionally equivalent1 Eiffel code. Safer code. Translating C code to Eiffel with C2Eif is quite useful to reuse C programs in a modern environment, and is the main motivation behind this work. However, it also implies other valuable side-benefits, which we demonstrate in Section 4. First, the translated code blends reasonably well with hand-written Eiffel code because it is not a mere transliteration from C; it is thus modifiable with Eiffel’s native tools and environments (EiffelStudio and related analysis and verification tools). Second, the 1 As per standard regression testsuites and general usage. 2 translation automatically introduces simple contracts, which help detect recurring mistakes such as out-of-bound array access or null-pointer dereferencing. To demonstrate this, Section 4.4 discusses how we easily discovered a few unknown bugs in widely used C programs (such as libgmp) just by translating them into Eiffel and running standard tests. While the purpose of C2Eif is not to debug C programs, the source of errors is usually more evident when executing programs translated into Eiffel—either because a contract violation occurs, or because the Eiffel program fails sooner, before the effects of the error propagate to unrelated portions of the code. The translated C code also benefits from the tighter Eiffel runtime, so that certain buffer overflow errors are harder to exploit than in native C environments. Why Eiffel? We chose Eiffel as the target language not only out of our familiarity with it, but also because it offers features that complement C’s, such as an emphasis on correctness [44] through the native support of contracts. Another reason for the choice has to do with the size of Eiffel’s community of developers, which is quite small compared to those of other object-oriented languages such as Java or C#. While Java users can avail of countless libraries and frameworks offering high-quality implementations of a wide array of functionalities, Eiffel users often have to implement such functionalities themselves from scratch. An automatic translation tool could therefore bring immediate benefits to reuse the rich C code-base in Eiffel; in fact, translations produced by C2Eif are already being used by the Eiffel community. Finally, Eiffel uncompromisingly epitomizes the object-oriented paradigm; hence translating C into it cannot take the shortcut of merely transliterating similar constructs (as it would have been possible, for example, with C++). The results of the paper are thus applicable to other object-oriented languages.2 An abridged version of this paper was presented at WCRE 2012 [42]; a companion paper presents the tool C2Eif from the user’s point of view [40]. 2 Overview and Architecture C2Eif is a compiler with graphical user interface that translates C programs to Eiffel programs. The translation is a complete Eiffel program which replicates the functionality of the C source program. C2Eif is implemented in Eiffel. High-level view. Figure 1 shows the overall picture of how C2Eif works. C2Eif inputs C projects (applications or libraries) processed with the C Intermediate Language (CIL) framework. CIL [22] is a C front-end that simplifies programs written in ANSI C or using the GNU C Compiler extensions into a restricted subset of C amenable to program transformations; for example, there is only one form of loop in CIL. Using CIL input to C2Eif ensures complete support of the whole set of C statements, without having to deal with each of them explicitly (although it also introduces some limitations, discussed in Section 4.5). C2Eif then translates CIL programs to Eiffel projects 2 As proof of this, the first author has made some progress towards applying the technique described in this paper to automatic translations from C to Java; and has recently been granted startup funding to support this development and turn it into a commercial product. 3 Helper Classes C application or library CIL CIL C file C2Eif Eiffel application or library Eiffel Compiler Binary Figure 1: Usage workflow of C2Eif. consisting of collections of classes that rely on a small set of Eiffel helper classes (described below). Such projects can be compiled with any standard Eiffel compiler. Incremental translation. C2Eif implements a translation T from CIL C to Eiffel as a series T1 , . . . , Tn of successive incremental transformations on the Abstract Syntax Tree. Every transformation Ti targets exactly one language aspect (for example, loops or inlined assembly code) and produces a program in an intermediate language Li which is a mixture of C and Eiffel constructs: the code progressively morphs from C to Eiffel code. The current implementation uses around 45 such transformations (i.e., n = 45). Combining several simple transformations improves the decoupling among different language constructs and facilitates reuse (e.g., to implement a translator of C to Java) and debugging: the intermediate programs are easily understandable by programmers familiar with both C and Eiffel. Helper classes. The core of the translation from C to Eiffel must ensure that Eiffel applications have access to objects with the same capabilities as their counterparts in C; for example, an Eiffel class that translates a C struct has to support field access and every other operation defined on structs. Conversely, C external pre-compiled code may also have to access the Eiffel representations of C constructs; for example, the Eiffel translation of a C program calling printf to print a local string variable str of type char∗ must grant printf access to the Eiffel object that translates str, in conformance with C’s conventions on strings. To meet these requirements, C2Eif includes a limited number of hand-written helper Eiffel classes that bridge the Eiffel and C environments; their names are prefixed by CE for C and Eiffel. Rather than directly replicating or wrapping commonly used external libraries (such as stdio and stdlib), the helper classes target C fundamental language features and in particular types and type constructors. This approach works with any external library, even non-standard ones, and is easier to maintain because involves only a limited number of classes. We now give a concise description of the most important helper classes; Section 3 shows detailed examples of their usage. • CE POINTER [G] represents C pointers of any type through the generic parameter G. It includes features to perform full-fledged pointer arithmetic and to get pointer representations which C can access but the Eiffel’s runtime will not modify (in particular, the garbage collector will not modify pointed addresses nor relocate memory areas). 4 • CE CLASS defines the basic interface of Eiffel classes that correspond to unions and structs. It includes features (members) that return instances of class CE POINTER pointing to a memory representation of the structure that C can access. • CE ARRAY [G] extends CE POINTER and provides consistent array access to both C and Eiffel (according to their respective conventions). It includes contracts that check for out-of-bound access. • CE ROUTINE represents function pointers. It supports calls to Eiffel routines through agents—Eiffel’s construct for function objects (closures or delegates in other languages)—and calls to (and callbacks from) external C functions through raw function pointers. • CE VA LIST supports variadic functions, using the Eiffel class TUPLE (sequences of elements of heterogeneous type) to store a variable number of arguments. It offers an Eiffel interface that extends the standard C’s (declared in stdarg.h), as well as output in a format accessible by external C code. 3 Translating C to Eiffel This section presents the major details of the translation T from C to Eiffel implemented in C2Eif, and illustrates the general rules with a number of small examples. The presentation breaks down T into several components that target different language aspects (for example, TTD maps C type declarations to Eiffel classes); these components mirror the incremental transformations Ti of C2Eif (mentioned in Section 2) but occasionally overlook inessential details for presentation clarity. External functions in Eiffel. Eiffel code translated from C normally includes calls to external C pre-compiled functions, whose actual arguments correspond to objects in the Eiffel runtime. This feature relies on the external Eiffel language construct: Eiffel routines can be declared as external and directly execute C code embedded as Eiffel strings3 or call functions declared in header files. For example, the following Eiffel routine (method) sin twice returns twice the sine of its argument by calling the C library function sin (declared in math.h): sin twice (arg: REAL 32): REAL 32 external C inline use <math.h> alias return 2sin($arg); end Calls using external can exchange arguments between the Eiffel and the C runtimes only for a limited set of primitive type: numeric types (that have the same underlying machine representation in Eiffel and C) and instances of the Eiffel system class POINTER that corresponds to raw untyped C pointers (not germane to Eiffel’s pointer representation, unlike CE POINTER). In the sin twice example, argument arg of numeric type REAL 32 is passed to the C runtime as $arg. Every helper class (described in Section 2) includes an attribute c pointer of type POINTER that offers access to a C-conforming representation usable in external calls. 3 For readability, we will omit quotes in external strings. 5 The mechanism of external calls is used not only in the translations produced by C2Eif but also in the Eiffel standard libraries, wherever interaction with the operating system is required, such as for input/output, file operations, and memory management. Supporting calls to native code is also the only strict requirement to be able to implement, in languages other than Eiffel, this paper’s approach to automatic translation from C. Java, for example, offers similar functionalities through the Java Native Interface (JNI). 3.1 Types and Type Constructors C declarations T v of a variable v of type T become Eiffel declarations v :TTY (T), where TTY is the mapping from C types to Eiffel classes described in this section. Numeric types. C numeric types correspond to Eiffel classes INTEGER (signed integers), NATURAL (unsigned integers), REAL (floating point numbers) with the appropriate bit-size as follows4 : C TYPE T char short int int, long int long long int float double long double E IFFEL CLASS TTY (T ) INTEGER 8 INTEGER 16 INTEGER 32 INTEGER 64 REAL 32 REAL 64 REAL 96 Unsigned variants follow the same size conventions as signed integers but for class NATURAL; for example TTY (unsigned short int) is NATURAL 16. Pointers. Pointer types are translated using class CE POINTER [G] with the generic parameter G instantiated with the pointed type: TTY (T ∗) = CE POINTER [TTY (T)] with the convention that TTY (void) maps to Eiffel class ANY, ancestor to every other class (Object in Java). The definition works recursively for multiple indirections; for example, CE POINTER[CE POINTER[REAL 32]] stands for TTY (float ∗∗). Function pointers. Function pointers are translated to Eiffel using class CE ROUTINE: TTY (T0 (∗) (T1 , ...,Tn )) = CE ROUTINE CE ROUTINE inherits from CE POINTER [ANY], and hence it behaves as a generic pointer, but customizes its behavior using references to agents that wrap the functions pointed to; Section 3.2 describes this mechanism. Arrays. Array types are translated to Eiffel using class CE ARRAY[G] with the generic parameter G instantiated with the array base type: TTY (T [n]) = CE ARRAY[TTY (T)]. 4 We implemented class REAL 96 specifically to support long double on Linux machines. 6 The size parameter n, if present, does not affect the declaration, but initializations of array variables use it (see Section 3.2). Multi-dimensional arrays are defined recursively as arrays of arrays: TTY (T [n1 ][n2 ]...[nm ]) is then CE ARRAY[TTY (T[n2 ]...[nm ])]. Enumerations. For every type enum E defined or used, the translation introduces an Eiffel class E defined by the translation TTD (for type definition): TTD (enum E {v1 = k1 , . . . , vm = km }) = class E feature v1 : INTEGER 32 =k1 ; . . . ; vm : INTEGER 32 =km end Class E has as many attributes as the enum type has values, and each attribute is an integer that receives the corresponding value in the enumeration. Every C variable of type E also becomes an integer variable in Eiffel (that is, TTY (enum E ) = INTEGER 32), and the class E is only used to assign constant values according to the enum naming scheme. Structs and unions. For every compound type struct S defined or used, the translation introduces an Eiffel class S: class S inherit CE CLASS feature TF (T1 v1 ) . . . TF (Tm vm ) end for TTD (struct S {T1 v1 ; . . . ; Tm vm }). Correspondingly, TTY (S) = S; that is, variables of type S become references of class S in Eiffel. The translation TF (T v) of each field v of the struct S introduces an attribute of the appropriate type in class S, and a setter routine set v which also updates the underlying C representation of v: v: TTY (T) assign set v −− declares ‘set v’ as the setter of v set v (a v: TTY (T)) do v := a v ; update memory field (”v”) end Class CE CLASS, from which S inherits, implements update memory field using reflection, so that the underlying C representation is created and updated dynamically only when needed during execution (for example, to pass a struct instance to a native C library), thus avoiding any data duplication overhead whenever possible. Example 1. Consider a C struct car that contains an integer field plate num and a string field brand: typedef struct { unsigned int plate num; char∗ brand; } car; The translation TTD introduces a class CAR as follows: class CAR inherit CE CLASS feature plate num: NATURAL 32 assign set plate num brand: CE POINTER [INTEGER 8] assign set brand set plate num (a plate num: NATURAL 32) do plate num := a plate num; update memory field (”plate num”) end set brand (a brand: CE POINTER [INTEGER 8]) do brand := a brand; update memory field (”brand”) end end 7 The translation of union types follows the same lines as that of structs, with the only difference that classes translating unions generate the underlying C representation in any case upon initialization, even if the union is not passed to the C runtime; calls to update memory field update all attributes of the class to reflect the correct memory value. We found this to be a reasonable compromise between performance and complexity of memory management of union types where, unlike structs, fields share the same memory space. 3.2 Variable Initialization and Usage Initialization. Eiffel variable declarations v : C only allocate memory for a reference to objects of class C, and initialize it to Void (null in Java). The only exceptions are, once again, numeric types: a declaration such as n: INTEGER 64 reserves memory for a 64-bit integer and initializes it to zero. Therefore, every C local variable declaration T v of a variable v of type T may also produce an initialization, consisting of calls to creation procedures of the corresponding helper classes, as specified by the declaration mapping TDE :   (NT) v : TTY (T ) TDE (T v;) = v : TTY (T ); create v.make( n1 , . . . , nm ) (AT)   v : TTY (T ); create v.make (OT) where definition (NT) applies if T is a numeric type; (AT) applies if T is an array type S[n1 ],. . ., [nm ]; and (OT) applies otherwise. The creation procedure make of CE ARRAY takes a sequence of integer values to allocate the right amount of memory for each array dimension; for example int a[2][3] is initialized by create a.make(2, 3). Memory management. Helper classes are regular Eiffel classes; therefore, the Eiffel garbage collector disposes instances when they are no longer referenced (for example, when a local variable gets out of scope). Upon collection, the dispose finalizer routines of the helper classes ensure that the C memory representations are also appropriately deallocated; for example, the finalizer of CE ARRAY frees the array memory area by calling free on the attribute c pointer. To replicate the usage of malloc and free, we offer wrapper routines that emulate the syntax and functionalities of their C homonym functions, but operate on CE POINTER: they get raw C pointers by external calls to C library functions, convert them to CE POINTER, and record the dynamic information about allocated memory size. The latter is used to check that successive usages conform to the declared size (see Section 4.4). Finally, the creation procedure make cast of the helper classes can convert a generic pointer returned by malloc to the proper pointed type, according to the following translation scheme: C CODE T∗ p; p = (T ∗)malloc(sizeof(T)); free(p); T RANSLATED E IFFEL CODE p: CE POINTER[TTY (T)] create p.make cast (malloc (σ(T))) free(p) 8 where σ is an encoding of the size information. Variable usage. The translation of variable usage is straightforward: variable reads in expressions are replicated verbatim, and C assignments (=) become Eiffel assignments (:=); the latter is, for CE ARRAY, CE POINTER, and classes translating C structs and unions, syntactic sugar for calls to setter routines that achieve the desired effect. The only exceptions occur when implicit type conversions in C must become explicit in Eiffel, which may spoil the readability of the translated code but is necessary with strong typing. For example, the C assignment cr = ’s’—assigning character constant ’s’ to variable cr of type char—becomes the Eiffel assignment cr := (’s’).code.to integer 8 that encodes ’s’ using the proper representation. Variable address. Whenever the address &v of a C variable v of type T is taken, v is translated as an array of unit size and type T: TDE (T v) = TDE (T v[1]), and every usage of v is adapted accordingly: &v becomes just v, and occurrences of v in expressions become ∗v. This little hack makes it possible to have Eiffel assignments translate C assignment uniformly; otherwise, usages of v should have different translations according to whether the information about v’s memory location is copied around (with &) or not. Dereferencing and pointer arithmetic. The helper class CE POINTER features a query item which translates dereferencing (∗) of C pointers. Pointer arithmetic is translated verbatim, because class CE POINTER overloads the arithmetic operators to be aliases of proper underlying pointer manipulations, so that an expression such as p + 3 in Eiffel, for references p of type CE POINTER, hides the explicit expression c pointer + 3 ∗ element size. Example 2. Consider an integer variable num, a pointer variable p, and a double pointer variable carsp with target type struct car (defined in Example 1). The following C code fragment declares num, p, and carsp, allocates space for an array of num consecutive cars pointed to by carsp, and makes p point to the array’s third element. int num; car ∗ p; car ∗∗ carsp; ∗carsp = malloc(num ∗ sizeof(car)); p = ∗carsp + 2; C2Eif translates the C fragment as follows, where the implicit C conversion from signed to unsigned int become explicit in Eiffel (calls to routine to natural 32). num: INTEGER 32 p: CE POINTER [CAR] carsp: CE POINTER [CE POINTER [CAR]] carsp.item := create {CE POINTER [CAR]}.make cast (malloc (num.to natural 32 ∗ (create {CAR}.make).structure size. to natural 32)) p := carsp.item + 2 9 2:libffi 1:agent Eiffel C 3:libffi Figure 2: Calls through function pointers. Using function pointers. Class CE ROUTINE, which translates C function pointers, is usable both in the Eiffel and in the C environment (see Figure 2). On the Eiffel side, its instances wrap Eiffel routines using agents—Eiffel’s mechanism for function objects. A private attribute routine references objects of type ROUTINE [ANY, TUPLE], an Eiffel system class that corresponds to agents wrapping routines, with any number of arguments and argument types stored in a tuple. Thus, Eiffel code can use the agent mechanism to create instances of class ROUTINE. For example, if foo denotes a routine of the current class and fp has type CE ROUTINE, create fp.make agent (agent foo) makes fp’s attribute routine point to foo. On the C side, when function pointers are directly created from C pointers (e.g., references to external C functions), CE ROUTINE behaves as a wrapper of raw C function pointers, and dynamically creates invocations to the pointed functions using the library libffi [9]. The Eiffel interface to CE ROUTINE will then translate calls to wrapped functions into either agent invocations or external calls with libffi according to how the class has been instantiated. Assume, for example, that fp is an object of class CE ROUTINE that wraps a procedure with one integer argument. If fp has been created with an Eiffel agent foo as above, calling fp.call ([42]) wraps the call foo (42) (edge 1 in Figure 2); if, instead, fp only maintains a raw C function pointer, the same instruction fp.call ([42]) creates a native C call using libffi (edge 2 in Figure 2). The services of class CE ROUTINE behave as an adapter between the procedural and object-oriented representation of routines: the signatures of C functions must change when they are translated to Eiffel routines, because routines in object-oriented languages include references to a target object as implicit first argument. Calls from external C code to Eiffel routines are therefore intercepted at runtime with libffi callbacks (edge 3 in Figure 2) and dynamically converted to suitable agent invocations. 3.3 Conditionals, Loops, and Functions The translation TCF takes care of constructs to structure the control flow of computations: conditionals, loops, and function definitions and calls. Conditionals and loops. Sequential composition, conditional instructions, and loop instructions are similar in C and Eiffel; therefore, their translation is straightforward.5 5 Eiffel loops have the general form: from init until exit loop B end. init is executed once before the ac- 10 TCF (I1 ; I2 ) TCF (if (c) {TB} else {EB}) = = TCF (while (c) {LB}) = T(I1 ) ; T(I2 ) if T(c) then T(TB) else T(EB) end from until not T(c) loop T(LB) end Note that Eiffel has only one type of loop, which translates the unique type of C loops produced by CIL. Functions. Function declarations and function calls in C translate to routine declarations and routine calls in Eiffel. Given a function foo with arguments a1 , . . . , an of type T1 , . . . , Tn and return type T0 , the translation: TCF T0 foo (T1 a1 , . . . , Tn an ){ B } is defined as follows: ( foo(a1: TTY (T1 ); . . . ; an: TTY (Tn )) do T(B) end if T0 is void foo(a1: TTY (T1 ); . . . ; an: TTY (Tn )) : TTY (T0 ) do T(B) end otherwise Correspondingly, function calls in C become routine calls in Eiffel with the actual arguments also recursively translated: TCF (foo (e1 , . . . , en )) = foo (T(e1 ), . . . , T(en )) Variadic functions. The C language supports variadic functions, also known as varargs functions, which work on a variable number of arguments. The Eiffel translation of variadic function declarations uses the class TUPLE to wrap lists of arguments: the translation of a variadic function var foo with n ≥ 0 required arguments a1 , . . . , an of type T1 , . . . , Tn , optional additional arguments, and return type T0 : TCF T0 var foo (T1 a1 , . . . , Tn an , . . .){ B } is defined as: var foo (args: TUPLE[a1 : TTY (T1 ); . . . ; an : TTY (Tn )]): TTY (T0 ) assuming T0 is not void; otherwise, the return type is omitted, as in the translation of standard functions. Calls to variadic functions with n required arguments may provide m ≥ 0 additional actual arguments en+1 , . . . , en+m after the n required ones e1 , . . . , en . The translation combines all arguments, required and additional, into an (n + m)-TUPLE denoted by square brackets: TCF (var foo (e1 , . . . , en , en+1 , . . . , en+m )) = var foo ([T(e1 ), . . . , T(en ), T(en+1 ), . . . , T(en+m )]) tual loop; the loop body B is executed until the Boolean condition exit becomes true; since the exit condition is tested before each iteration, the body may not be executed. Body B and init may be empty. 11 The correctness of this translation relies on Eiffel’s type conformance rule for instances of the TUPLE class: every (n + m)-TUPLE with types [T1 , . . . , Tn , Tn+1 , . . . , Tn+m ], for n, m ≥ 0, conforms to any shorter n-TUPLE with types [T1 , . . . , Tn ]. Thus, translated calls are type-safe because the assignment of actual to formal arguments is covariant [12]. The bodies of variadic functions can refer to the required arguments by name using the usual syntax. The C type va list and functions va start and va arg in library stdarg provide a means to access the optional arguments: va start(argp, last) initializes a variable argp of type va list to point to the first optional argument after last (the name of the last declared argument). Every successive invocation of va arg(argp, T) returns the argument of type T pointed to by argp, and then moves argp right after it. The Eiffel translation relies on the helper class CE VA LIST which replicates stdarg’s functionality: TDE ( va list argp ) T( va start(argp, last) ) T( va arg(argp, T) ) = = = argp: CE VA LIST create argp.make (args, index) argp.TTY (T ) item where args is the name of the argument TUPLE in the enclosing variadic function and index is the index of the first optional argument. While CE VA LIST provides a uniform interface to both Eiffel and C, the actual arguments of a variadic function may also be accessed in the Eiffel translation using the standard syntax for TUPLEs: in a variadic function with n required arguments, the expression args.ai refers to the ith named argument ai , for 1 ≤ i ≤ n; and args.Tj item(j) refers to the jth argument aj , for any 1 ≤ j ≤ n (required argument) or j > n (optional argument), where Tj is aj ’s type. Example 3. Continuing Examples 1 and 2, consider a variadic function init cars that takes a pointer carsp to an array of cars and num pairs of unsigned integers and strings (passed as optional arguments), and initializes the array with num cars whose plate numbers and brands respectively correspond to the integer and string in each pair. void init cars(car ∗∗carsp, int num, . . .) { va list argp; car ∗ccar; int n; ∗carsp = malloc(num ∗ sizeof(car)); ccar = ∗carsp; va start(argp, num); for(n = num ; n >0; n−−) { ccar →plate num = va arg(argp, unsigned int); ccar →brand = va arg(argp, char∗); ccar++; } } The translation into Eiffel uses the class CE VA LIST and accesses the optional unsigned integer arguments with argp.natural 32 item, and the optional strings with 12 argp.pointer item. The latter returns a generic pointer, which is cast to char ∗ with an explicit cast (line 15). 13 init cars (args: TUPLE [carsp: CE POINTER [CE POINTER [CAR]]; num: INTEGER 32]) local argp: CE VA LIST ccar: CE POINTER [CAR] n: INTEGER 32 do args.carsp.item := create {CE POINTER [CAR]}.make cast ( malloc ( num.to natural 32 ∗ (create {CAR}.make).structure size. to natural 32 ) ) ccar := args.carsp.item create argp.make (args, 3) n := num from until not n >0 loop ccar.item.plate num := argp.natural 32 item 15 ccar.item.brand := (create {CE POINTER [INTEGER 8]}.make cast (argp. pointer item)) ccar := ccar + 1 n := n − 1 end end 3.4 Unstructured Control Flow In addition to constructs for structured programming, C offers control-flow breaking instructions such as jumps. This section discusses their translation to Eiffel. Jumps. Eiffel enforces structured programming, and hence it lacks control-flow breaking instructions such as goto, break, continue, and return. The translation TCF eliminates such instructions along the lines of the global version—using Harel’s terminology [13]—of the structured programming theorem. The body of a function using goto determines a list of instructions s0 , s1 , . . . , sn , where each si is a maximal sequential block of instructions, with no labels after the first instruction or jumps before the last one in the block. TCF translates the body into a single loop over an auxiliary integer variable pc that emulates a program counter:6 6 Eiffel’s inspect/when instructions corresponds to a restricted form of switch/case without fall through. 14 from pc := 0 until pc = −1 loop inspect pc when 0 then T(s0 ) ; upd(pc) when 1 then T(s1 ) ; upd(pc) TCF (hs0 , s1 , . . . , sn i) = ..  .     when n then T(sn ) ; upd(pc)     end    end              Variable pc is initially zero; every iteration of the loop body executes spc for the current value of pc, and then updates pc (with upd(pc)) to determine the next instruction to be executed: blocks ending with jumps modify pc directly, other blocks increment it by one, and exit blocks set it to −1, which makes the overall loop terminate (whenever the original function terminates). This translation supports all control-flow breaking instructions, and in particular continue, break, and return, which are special cases of goto. TCF , however, improves the readability in these special simpler cases by directly using auxiliary Boolean flag variables with the same names as the instruction they replace. The flags are tested as part of the translated exit conditions for the loops where the control-flow breaking instructions appear. Using this alternative translation scheme where the generality of gotos is not required makes for translations with little changes to the code structure, which are usually more readable. The loop while (n > 0){ if (n == 3) break; n−−; }, for example, becomes: from until break or not n >0 loop if n = 3 then break := True end if not break then n := n − 1 end end Long jumps. The C library setjmp provides functions setjmp and longjmp to save an arbitrary return point and jump back to it across function call boundaries. The wrapping mechanism used for external functions (see Section 3.5) does not work to replicate long jumps, because the return values saved by setjmp wrapped as an external function are no longer valid after execution leaves the wrapper. Therefore, C2Eif translates setjmp and longjmp by means of the helper class CE EXCEPTION. As the name suggests, CE EXCEPTION uses Eiffel’s exception propagation mechanism to go back in the call stack to the allocation frame of the function that called setjmp. There, translated goto instructions jump to the specific point saved with setjmp within the function body. Example 4. Consider a C function target that executes setjmp(buf) to save a location using a global variable buf of type jmp buf. In the translated Eiffel code, buf becomes an attribute of type CE EXCEPTION, and the call to setjmp in target becomes a creation of an instance attached to buf: buf := create {CE EXCEPTION}.make (. . .); make’s actual arguments (omitted for simplicity) contain references to the stack frame of the current instance of target, and to the specific instruction where to jump. 15 Later during the execution, another function source called from target performs a longjmp(buf, 3), which diverts execution to the location previously marked by setjmp and returns value 3. In Eiffel, the longjmp becomes a raising of the exception object buf enclosing the return value: buf.raise (3). The flow of what happens next depends on the semantics of Eiffel exceptions, which is significantly different than that of other object-oriented languages such as Java and C#. The raised exception propagates to the recipient target, where it is handled by a rescue clause. Such exception handling blocks are routine-specific in Eiffel, rather than being associated to arbitrary scopes such as in Java’s try /catch blocks. The translation of setjmp also took care of setting up target’s rescue clause: if the raised exception stores a reference to the current instance of routine target, the code in the rescue clause modifies a variable pc local to target to point to the location of the setjmp and, using Eiffel’s retry instruction, executes target’s body again with this new value. Based on the value, the new execution of target’s body jumps to the correct location following a mechanism similar to the previously discussed translation of goto instructions. Finally, if the handled exception references a routine instance other than the current execution of target, the rescue clause propagates the exception through the call stack, so that it can reach the correct stack frame. 3.5 Externals and Encapsulation The translation T uses classes to wrap header and source files, which give a simple modular structure to C programs. Externals. For every included system header header.h, T defines a class HEADER with wrappers for all external functions and variables declared in header.h. The wrappers are routines using the Eiffel external mechanism and performing the necessary conversions between the Eiffel and the C runtimes. In particular, external functions using only numeric types, which are interoperable between C and Eiffel, directly map to wrapper routines; for example, exit in stdlib.h becomes: exit (status: INTEGER 32) external C inline use <stdlib.h> alias exit($status); end When external functions involve types using helper classes in Eiffel, a routine passes the underlying C representation to the external calls; for example, fclose in stdio.h generates: fclose (stream: CE POINTER [ANY]): INTEGER 32 do Result := c fclose (stream.c pointer) end c fclose (stream: POINTER): INTEGER 32 external C inline use <stdio.h> alias return fclose($stream); end In some complex cases—typically, with variadic external functions—the wrapper can only assemble the actual call on the fly at runtime; this is done using CE ROUTINE. For example, printf is wrapped as: 16 printf (args: TUPLE[format: CE POINTER[INTEGER 8]]): INTEGER 32 do Result := (create {CE ROUTINE}.make shared (c printf)).integer 32 item (args) end c printf: POINTER external C inline use <stdio.h> alias return &printf; end The translation can also inline assembly code, using the same mechanisms as external function calls. Globals. For every source file source.c, T defines a class SOURCE that includes translations of all function definitions (as routines) and global variables (as attributes) in source.c. Class SOURCE also inherits from the classes translating the other system header files that source.c includes, to have access to their declarations. For example, if foo.c includes stdio.h, FOO is declared as class FOO inherit STDIO. 3.6 Formatted Output Optimization The library function printf is the standard C output function, which displays values according to format strings passed as argument. In contrast, Eiffel provides the command Io.put string to put plain text strings on standard output, as well as type-specific formatter classes, such as FORMAT INTEGER, to produce string representations of arbitrary types. With the goal of making the translated code as close as possible to standard Eiffel, C2Eif tries to replace calls to printf with equivalent calls to Io.put string and routines of the formatter classes. Whenever printf’s returned value is not used and the format string is a constant literal, C2Eif parses the literal format string and encodes it as equivalent calls to Io.put string and formatters. This process may find mismatches between some format specifiers and the types of the corresponding value arguments, which C2Eif reports as warnings. Another situation where C2Eif can replace calls to printf with calls to put string is when only one argument is passed to the former, which is then interpreted verbatim as a string. Whenever a warning is issued or the usage of printf cannot be rendered using put string, C2Eif falls back to calling a wrapper of the native printf function (described in Section 3.5). The translation also implements similar optimizations for variants of printf such as fprintf. 4 Evaluation and Discussion This section evaluates the translation T and its implementation in C2Eif. The bulk of the evaluation assesses correctness (Section 4.1) and performance (Section 4.2) based on experiments with 14 open-source programs. We also qualitatively discuss other aspects that influence maintainability (Section 4.3), the advantages in terms of safety deriving from switching to the Eiffel runtime (Section 4.4), as well as the current limitations of the C2Eif approach (Section 4.5). 17 4.1 Correct Behavior To experimentally assess the translations produced by C2Eif, we applied it to 14 opensource C programs, including 7 applications, 6 libraries, and one testsuite; most of them are widely-used in Linux and other “nix” distributions. hello world is the only toy application, which is however useful to have a baseline of translating from C to Eiffel with C2Eif. The other applications are: micro httpd 12dec2005, a minimal HTTP server; xeyes 1.0.1, a widget for the X Windows System which shows two googly eyes following the cursor movements; less 382-1, a text terminal pager; wget 1.12, a command-line utility to retrieve content from the web; links 1.00, a simple web browser; vim 7.3, a powerful text editor. The libraries are: libSDL mixer 1.2, an audio playback library; libmongoDB 0.6, a library to access MongoDB databases; libpcre 8.31, a library for regular expressions; libcurl 7.21.2, a URL-based transfer library supporting protocols such as FTP and HTTP; libgmp 5.0.1, for arbitrary-precision arithmetic; libgsl 1.14, a powerful numerical library. The gcc “torture tests” are short but semantically complex pieces of C code, used as regression tests for the GCC compiler. Table 1 shows the results of translating the 14 programs into Eiffel using C2Eif, running on a GNU/Linux box (kernel 2.6.37) with a 2.66 GHz Intel dual-core CPU and 8 GB of RAM, GCC 4.5.1, CIL 1.3.7, EiffelStudio 7.1.8. For each application, library, and testsuite Table 1 reports: (1) the size (in lines of code) of the CIL version of the C code and of the translated Eiffel code; (2) the number of Eiffel classes created; (3) the time (in seconds) spent by C2Eif to perform the source-to-source translation (not including compilation from Eiffel source to binary); (4) the size of the binaries (in MBytes) generated by EiffelStudio.7 We ran extensive trials on the translated programs to verify that they behave as in their original C version, thus validating the correctness of the translation T and its implementation in C2Eif. The trials comprised informal usage, systematic performance tests for some of the applications, and running the standard testsuites available for the three biggest libraries (libcurl, libgmp, and libgsl). The rest of this section describes the translated testsuites and their behavior with respect to correctness; Section 4.2 discusses the quantitative performance results. Library libcurl comes with a client application and a testsuite of 583 tests defined in XML and executed by a Perl script calling the client; libgmp and libgsl respectively include testsuites of 145 and 46 tests, consisting of client C code using the libraries. All tests execute and pass on both the C and the translated Eiffel versions of the libraries, with the same logged output. For libcurl, C2Eif translated the library and the client application. For libgmp and libgsl, it translated the test cases as well as the libraries. The gcc torture testsuite includes 1116 tests; the GCC version we used fails 4 of them; CIL (which depends on GCC) fails another 110 tests among the 1112 that GCC passes; finally, C2Eif (which depends on CIL) passes 989 (nearly 99%) and fails 13 of the 1002 tests passed by CIL. Given the challenging nature of the torture testsuite, this result is strong evidence that C2Eif handles the complete C language used in practice, and produces correct translations. 7 We do not give a binary size for libraries, because EiffelStudio cannot compile them without a client. 18 Table 1: Translation of 14 open-source programs. S IZE (LOCS) CIL E IFFEL hello world micro httpd xeyes less wget links vim libSDL mixer libmongoDB libpcre libcurl libgmp libgsl gcc (torture) T OTAL 8 565 1,463 16,955 46,528 70,980 276,635 7,812 7,966 18,220 37,836 61,442 238,080 147,545 932,035 15 1,934 10,661 22,545 57,702 100,815 395,094 11,553 10,341 24,885 65,070 79,971 344,115 256,246 1,380,947 #E IFFEL CLASSES T IME (S) B INARY SIZE (MB) 1 16 78 75 183 211 663 47 43 38 289 370 978 2,569 5,561 1 1 1 5 25 33 144 3 3 14 18 21 85 79 433 1.3 1.5 1.8 2.6 4.5 13.9 24.2 – – – – – – 1,576 1,626 The 13 torture tests failing after translation to Eiffel target the following unsupported features. One test reads an int from a va list (variadic function list of arguments) which actually stores a struct whose first field is a double; the Eiffel typesystem does not allow this, and inspection suggests that it is probably a copy-paste error rather than a feature. Two tests exercise GCC-specific optimizations, which are immaterial after translation to Eiffel. Six tests target exotic GCC built-in functions, such as builtin frame address; one test performs explicit function alignment; and three rely on special bitfield operations. 4.2 Performance We analyzed the performance of 5 applications, the GCC torture testsuite, and the standard testsuites of 3 libraries (described in Section 4.1) translated to Eiffel using C2Eif. Table 2 shows the result of the performance trials, running on the same system used for the experiments of Section 4.1.8 For each program or testsuite, Table 2 reports the execution time (in seconds), the maximum percentage of CPU, and the maximum amount of RAM (in MBytes) used while running. The table compares the performance of the original C versions (columns “C”) against the Eiffel translations with C2Eif (columns “T”), and, for the simpler examples, against manually written Eiffel implementations (columns “E”) that transliterate the original C implementations using the closest Eiffel constructs (for example, putchar becomes Io.put character) with as little changes as possible to the code structure (we manually wrote these transliterations 8 We compiled all CIL-processed C programs with the GCC options -O2 -s; and all Eiffel programs with disabled void checking, inlining size 100, and stripped binaries. 19 Table 2: Performance comparison for 5 applications and 4 testsuites. hello world micro httpd less wget vim libcurl libgmp libgsl gcc (torture) E XECUTION TIME ( S ) C T E M AX % CPU C T E M AX MB RAM C T E 0 5 36 16 85 199 44 25 0 0 99 – 22 – – – – – 1.3 2.3 – 4.4 – – – – – 0 37 36 16 85 212 728 1501 5 0 46 – – – – – – – 30 99 – 22 – – – – – 30 99 – – – – – – – 5.5 7.8 – 69 – – – – – 5.3 5.6 – – – – – – – ourselves). Maximum CPU and RAM usages are immaterial for the testsuites (torture and libraries), because their execution consists of a large number of separate calls. The rest of this section discusses the performance results together with other qualitative performance assessments. The performance of hello world demonstrates the base overhead, in terms of CPU and memory usage, of the default Eiffel runtime (objects, automatic memory management, and contract checking—which can however be disabled for applications where sheer performance is more important than having additional checks). The test with micro httpd consisted in serving the local download of a 174 MB file (the Eclipse IDE); this test boils down to a very long sequence (approximately 200 million iterations) of inputting a character from file and outputting it to standard output. The translated Eiffel version incurs a significant overhead with respect to the original C version, but is faster than the manually written Eiffel transliteration. This might be due to feature lookups in Eiffel or to the less optimized implementation of Eiffel’s character services. As a side note, we did the same exercise of manually transliterating micro httpd using Java’s standard libraries; this Java translation ran the download example in 170 seconds, using up to 99% of CPU and 150 MB of RAM. The test with wget downloaded the same 174 MB Eclipse package over the SWITCH Swiss network backbone. The bottleneck is the network bandwidth, and hence differences in performance are negligible, except for memory consumption, which is higher in Eiffel due to garbage collection (memory is deallocated only when necessary, thus the maximum memory usage is higher in operational conditions). The test with libcurl consisted in running all 583 tests from the standard testsuite mentioned before. The total runtime is comparable in translated Eiffel and C. The tests with libgmp and libgsl ran their respective standard testsuites. The overall slow-down seems significant, but a closer look shows that the large majority of tests run in comparable time in C and Eiffel: 30% of the libgmp tests take up over 95% of the running time; and 26% of the libgsl tests take up almost 99% of the time. The GCC torture tests incur only a moderate slow-down, concentrated in 3 tests that take 97% of the time. In all these experiments, the tests responsible for the conspicuous 20 slow-down target operations that execute slightly slowlier in the translated Eiffel than in the native C (e.g., accessing a struct field) and repeat it a huge number of times, so that the basic slow-down increases many-fold. These bottlenecks are an issue only in a small fraction of the tests and could be removed manually in the translation. The interactive applications (xeyes, less, links, and vim) run smoothly with good responsiveness, comparable to their original implementations. The test for less and vim consisted in scrolling through large text files one line at a time (by holding “arrow down” in less) or one page at a time (by holding “page down” in vim). Table 2 shows the running times, which are the same in C and Eiffel; the screen refresh rate is also indistinguishable. In all, the performance overhead in switching from C to Eiffel significantly varies with the program type but, even when it is significant, does not preclude the usability of the translated application or library in normal conditions—as opposed to the behavior in a few specific test cases. 4.3 Usability and Maintainability A tool such as C2Eif, which provides automatic translation between languages, is applicable in different contexts within general software maintenance and evolution processes. This section discusses some of these applications and how suitable C2Eif can be for each of them. Reuse in clients. The first, most natural application is using C2Eif to automatically reuse large C code-bases in Eiffel. This is not merely a possibility, but something extremely valuable for Eiffel, whose user community is quite small compared to those of other mainstream languages such as C, Java, or C++. Since we released C2Eif to the public as open-source, we have been receiving several requests from the community to produce Eiffel versions of C libraries whose functionalities are sorely missed in Eiffel, and whose native implementation would require a substantial effort to get to software of quality comparable to the widely tested and used C implementations. This was the case, in particular, of libSDL mixer, libmongoDB, and libpcre, whose automatic translations created using C2Eif were requested from the community and are now being used in Eiffel applications. We are also aware of some Eiffel programmers directly trying to use C2Eif to translate useful C libraries and deploy them in their own software. This form of reuse mainly entails writing Eiffel client code that accesses translated C components. Supporting it requires tools that handle the full C language as used in practice, and that produce translated APIs understandable and usable with a programming style sufficiently close to what is the norm in Eiffel, without requiring in-depth understanding of the C conventions. To give an idea of how C2Eif fares in this respect, consider writing a simple class PRINT SOURCE SEHOME, which uses the API of cURL to retrieve and print the HTML code of the home page at http://se.inf.ethz.ch. Table 3 shows two versions of this client class. The one on the left uses the translation of libcurl automatically generated by C2Eif; the one on the right uses the wrapper of libcurl part of the Eiffel standard libraries included with the EiffelStudio IDE, written by EiffelSoftware programmers. The two 21 solutions are quite similar in structure and style. The version based on C2Eif still requires adhering to a couple of conventions that are a legacy of the translation from C: the arguments passed to routine curl easy setopt must be listed in a tuple, and the routine call itself includes a do nothing which is needed whenever a function that returns a value is used as an instruction (Eiffel enforces separation between functions and procedures). On the other hand, the “native” solution has a slightly more complex control structure, because it has to check that the dynamically linked cURL library is actually accessible at runtime; this is unnecessary with the implementation based on C2Eif, which does not depend on dynamically linked libraries since libcurl is available translated to Eiffel. When a C library undergoes maintenance, the introduced changes have the same impact on the C and on the Eiffel clients of the library. In particular, if the changes to the library do not break client compatibility (that is, the API does not change), one should simply run C2Eif again on the new library version and replace it in the Eiffel projects that depends on it. If the API changes, clients may have to change too, independent of the language they are written in. Evolution of translated libraries. Once a program is translated from C to Eiffel, one may decide it has become part of the Eiffel ecosystem, and hence it will undergo maintenance and evolution as any other piece of Eiffel code. In this scenario, C2Eif class PRINT SOURCE SEHOME class PRINT SOURCE SEHOME inherit LIBCURL CONSTANTS inherit CURL OPT CONSTANTS create make create make feature feature make local easy: P EASY STATIC handle: CE POINTER [ANY] ret: NATURAL 32 do create easy handle := easy.curl easy init easy.curl easy setopt ([handle, Curlopt url, ce string (”se.inf.ethz.ch”)]).do nothing ret := easy.curl easy perform (handle) easy.curl easy cleanup (handle) end end make local easy: CURL EASY EXTERNALS handle: POINTER ret: INTEGER 32 do create easy if easy.is dynamic library exists then handle := easy.init easy.setopt string (handle, Curlopt url, ”se.inf.ethz.ch”) ret := easy.perform (handle) easy.cleanup (handle) else Io.error.put string (”cURL not found.%N”) end end end Table 3: Two Eiffel clients of cURL: using libcurl translated with C2Eif (left) and using the library wrapper provided by EiffelStudio (right). 22 provides an immediately applicable solution to port C code to Eiffel, whereas the ensuing maintenance effort is distributed over an entire lifecycle and devoted to improve the automatic translation (for example, removing the application-dependent performance bottlenecks highlighted in Section 4.2) and completely conforming it to the Eiffel style. Such maintenance of translations is the easier the closer the generated code follows Eiffel conventions and, more generally, the object-oriented paradigm. Providing a convincing empirical evaluation of the readability and maintainability of the code generated by C2Eif (not just from the perspective of writing client applications) is beyond the scope of the present work. Notice, however, that C2Eif already follows numerous Eiffel conventions such as for the names of classes, types, and local variables, which might look verbose to hard-core C programmers but are de rigueur in Eiffel. Follow-up work, which we briefly discuss in Section 6, has targeted the object-oriented reengineering of C2Eif translations. In all, while the translations produced by C2Eif still retain some “C flavor”, we believe they are overall understandable and modifiable in Eiffel with reasonable effort. Two-way propagation of changes. One more maintenance scenario occurs if one wants to be able to independently modify a C program and its translation to Eiffel, while still being able to propagate the changes produced in each to the counterpart. For example, this scenario applies if a C library is being extended with new functionality, while its Eiffel translation produced by C2Eif undergoes refactoring to optimize it to the Eiffel environment. This scenario is the most challenging of those discussed in this section; it poses problems similar to those of merging different development branches of the same project. While merge conflicts are still a bane of collaborative development, modern version control systems (such as Git or Mercurial) have evolved to provide powerful support to ease the process of conflict reconciliation. Thus, they could be very useful also in combination with automatic translators such as C2Eif to be able to integrate changes in C with other changes in Eiffel. 4.4 Safety and Debuggability Besides the obvious advantage of reusing the huge C code-base, translating C code to Eiffel using C2Eif leverages some high-level feature which may improve safety and make debugging easier in some conditions. Uncontrolled format string is a well-known vulnerability [4] of C’s printf library function, which permits malicious clients to access data in the stack by supplying special format strings. Consider for example the C program: int main (int argc, char ∗ argv[]) { char ∗secret = ”This is secret!”; if (argc >1) printf(argv[1]); return 0; } If we call it with: ./example "{Stack: %x%x%x%x%x%x} --> %s", we get the output {stack: 0b7[. . .]469} --> This is secret!, which reveals the local string secret. The safe way to achieve the intended behavior would be the instruction printf (”%s”, argv[1]) instead of printf (argv[1]), so that the input string is interpreted literally. 23 What is the behavior of code vulnerable to uncontrolled format strings, when translated to Eiffel with C2Eif? In simple usages of printf with just one argument as in the example, the translation replaces calls to printf with calls to Eiffel’s Io.put string, which prints strings verbatim without interpreting them; therefore, the translated code is not vulnerable in these cases. The replacement was possible in 65% of all the printf calls in the programs of Table 1. C2Eif translates more complex usages of printf (for example, with more than one argument and no literal format string such as printf (argv[1], argv[2])) into wrapped calls to the external printf function, and hence the vulnerability still exists. However, it is less extensive or more difficult to exploit in Eiffel: primitive types (such as numeric types) are stored on the stack in Eiffel as they are in C, but Eiffel’s bulkier runtime typically stores them farther up the stack, and hence longer and more complex format strings must be supplied to reach the stack data (for instance, a variation of the example with secret requires 386 %x’s in the format string to reach local variables). On the other hand, non-primitive types (such as strings and structs) are wrapped by Eiffel classes in C2Eif, which are stored in the heap, and hence unreachable directly by reaching stack data. In these cases, the vulnerability vanishes in the Eiffel translation. Debugging format strings. C2Eif also parses literal format strings passed to printf and detects type mismatches between format specifiers and actual arguments. This analysis, necessary when moving from C to a language with a stronger type system, helps debug incorrect and potentially unsafe usages of format strings. Indeed, a mismatch detected while running the 145 libgmp tests revealed a real error in the library’s implementation of macro TESTS REPS: char ∗envval, ∗end; /∗ ... ∗/ long repfactor = strtol(envval, &end, 0); if(∗end \|\| repfactor ≤ 0) fprintf (stderr, ”Invalid repfactor: %s.\n”, repfactor); String envval should have been passed to fprintf instead of long repfactor. GCC with standard compilation options does not detect this error, which may produce garbage or even crash the program at runtime. Interestingly, version 5.0.2 of libgmp patches the code in the wrong way, changing the format specifier %s into %ld. This is still incorrect because when envval does not encode a valid “repfactor”, the outcome of the conversion into long is unpredictable. Finally, notice that C2Eif may also report false positives, such as long v = ”Hello!”; printf(”%s”, v) which is acceptable (though probably not commendable) usage. Out-of-bound error detection. C arrays translate to instances of class CE ARRAY (see Section 3.1), which includes contracts that signal out-of-bound accesses to the array content. Therefore, out-of-bound errors are much easier to detect in Eiffel applications using components translated with C2Eif. Simply by translating and running the libgmp testsuite, we found an off-by-one error causing out-of-bound access (our patch has been included in more recent versions of the library); the error does not manifest itself when running the original C version. More generally, contracts help detect the precise location of array access errors. Consider, for example: 24 int ∗ buf = (int ∗) malloc(sizeof (long long int) ∗ 10); buf = buf − 10; buf = buf + 29; ∗buf = ’a’; buf++; ∗buf = ’b’; 1 2 3 4 5 buf is an array that stores 20 elements of type int (which has half the size of long long int). The error is on line 5, when buf points to position 20, out of the array bounds; line 2 is instead OK: buf points to an invalid location, but it is not written to. This program executes without errors in C; the Eiffel translation, instead, stops exactly at line 5 and signals an out-of-bound access to buf. Array bound checking may be disabled, which is necessary in borderline situations where out-of-bound accesses do not crash because they assume a precise memory layout. For example, links and vim use statements of the following form block ∗p = (block ∗)malloc(sizeof(struct block)+ len), with len >0, to allocate struct datatypes of the form struct block { /∗... ∗/char b[1]; }. In this case, p points to a struct with room for 1 + len characters in p→b; the instruction p→b[len]=‘i’ is then executed correctly in C, but the Eiffel translation assumes p→b has the statically declared size 1, hence it stops with an error. Another borderline situation is with multidimensional arrays, such as double a[2][3]. An iteration over a’s six elements with double ∗p = &a[0][0] translated to Eiffel fails to go past the third element, because it sees a[0][0] as the first element of an array of length 3 (followed by another array of the same length). A simple cast double ∗ p = (double∗)a achieves the desired result without fooling the compiler, hence it works without errors also in translated code. These situations are examples of unsafe programming more often than not. More safety in Eiffel. Our experiments found another bug in libgmp, where function gmp sprintf final had three formal input arguments, but was only called with one actual through a function pointer. Inspection suggests it is a copy-paste error of the other function gmp sprintf reps. The Eiffel version found the mismatch when calling the routine and reported a contract violation. Easily finding such bugs demonstrates the positive side-effects of translating widely-used C programs into a tighter, higher-level language. 4.5 Limitations The only significant limitations of the translation T implemented in C2Eif in supporting C programs originate in the introduction of strong typing: programming practices that implicitly rely on a certain memory layout may not work in C applications translated to Eiffel. Section 4.4 mentioned some examples in the context of array manipulation (where, however, the checks on the Eiffel side can be disabled). Another example is a function int trick (int a, int b) that returns its second argument through a pointer to the stack, with the instructions int ∗p = &a; return ∗(p+1). C2Eif’s translation assumes p points to a single integer cell and cannot guarantee that b is stored in the next cell. Another limitation is the fact that C2Eif takes input from CIL, and hence it does not support legacy C such as K&R C. The support can, however, be implemented by di25 rectly extending the pre-processing CIL front-end. Similarly, the GCC torture testsuite highlighted a few exotic GCC features currently unsupported by C2Eif (Section 4.1), which may be handled in future work. Code using unrestricted gotos poses the biggest hurdles to producing readable Eiffel code. This is arguably unavoidable when translating to any programming language that does not have jumps. The translation T does, however, avoid the most complicated general translation scheme with the simpler control-flow breaking instructions continue, break, and return instructions, whose translation is normally much more readable than when using unrestricted gotos (see the example in Section 4). 5 Related Work There are two main approaches to reuse source code written in a “foreign” language (e.g., C) in a different “host” language (e.g., Eiffel): define wrappers for the components written in the foreign language; and translate the foreign source code into functionally equivalent host code. We discuss related work pursuing these approaches in Sections 5.1 and 5.2. In Section 5.3, we review the major solutions to automate object-oriented reengineering, which is a natural follow-up of automatic translation into object-oriented languages. 5.1 Wrapping Foreign Code Wrappers enable the reuse of foreign implementations through the API of bridge libraries. This approach (e.g., [6, 5, 30]) does not modify the foreign code, whose functionality is therefore not altered; moreover, the complete foreign language is supported. On the other hand, the type of data that can be retrieved through the bridging API is often restricted to a simple subset common to the host and foreign language (e.g., primitive types). C2Eif uses wrappers only to translate external functions and assembly code. 5.2 Translating Foreign Code Industrial practices have long encompassed manual migrations of legacy code. Some semi-automated tools exist that help translate code written in legacy programming languages such as old versions of COBOL [37, 21], Fortran-77 [1, 34], and K&R C [46]. Terekhov et al. [37] review how automated language conversion is applied in industry; based on their experience, they conclude that “creating 100% automated conversion tools is neither possible, nor desirable”. Our experience with C2Eif, however, suggests that such a conclusion has only relative validity: in the rich design space of automatic translators, there are scenarios where trading off some performance for complete automation is possible and desirable. Rather than imposing an upfront heavyweight burden on developers in charge of migration, we suggest to start with an automatically translated version which is suboptimal but works out of the box, and then devote the manual programming effort to improving and adapting what is necessary— 26 incrementally with an agile approach which also depends on the specific application domain and requirements. Some translators focus on the adaptation of code into an extension (superset) of the original language. Examples include the migration of legacy code to object-oriented code, such as Cobol to OO-Cobol [25, 32, 45], Ada to Ada95 [35], and C to C++ [15, 47]. Some of such efforts try to go beyond the mere hosting of the original code, and introduce refactorings that take advantage of the object-oriented paradigm. Most of these refactorings are, however, limited to restructuring modules into classes (see the focused discussion in Section 5.3). C2Eif follows a similar approach, but also takes advantage of some advanced features (such as contracts) to improve the reliability of translated code. Ephedra [18] is a tool that translates legacy C to Java. It first translates K&R C to ANSI C; then, it maps data types and type casts; finally, it translate the C source code to Java. Ephedra handles a significant subset of C, but cannot translate frequently used features such as unrestricted gotos, external pre-compiled libraries, and inlined assembly code. A case study evaluating Ephedra [19] involved three small programs: the implementation of fprintf, a monopoly game (about 3 KLOC), and two graph layout algorithms. The study reports that the source programs had to be manually adapted to be processable by Ephedra. In contrast, C2Eif is completely automatic, and works with significantly larger programs. Other tools (proprietary or open-source) to translate C (and C++) to Java or C# include: C2J++ [39], C2J [27], and C++2C# and C++2Java [36]. Table 4 shows a feature comparison among the currently available tools that translate C to an objectoriented language, showing: • The target language. • Whether the tool is completely automatic, that is whether it generates translations that are ready for compilation. • Whether the tool is available for download and usable. In a couple of cases we could only find papers describing the tool but not a version of the implementation working on standard machines. • An (subjective to a certain extent) assessment of the readability of the code produced. In each case, we tried to evaluate if the translated code is sufficiently similar to the C source to be readily understandable by a programmer familiar with the latter. We judged C2J’s readability negatively because the tool puts data into a single global array to support pointer arithmetic. This is quite detrimental to readability and also circumvents type checking in the Java translation. • Whether the tool supports unrestricted calls to external libraries, unrestricted pointer arithmetic, unrestricted gotos, and inlined assembly code. The table demonstrates that C2Eif is arguably the first completely automatic tool that handles the complete C language as used in practice. In previous work, we developed J2Eif, an automatic source-to-source translator from Java to Eiffel [43]; translating between two object-oriented languages does not 27 completely automatic available readability external libraries pointer arithmetic gotos inlined assembly Ephedra Convert2Java C2J++ C2J C++2Java C++2C# C2Eif target language Table 4: Tools translating C to O-O languages. Java Java Java Java Java C# Eiffel no no no no no no yes no no no yes yes yes yes + + + − + + + no no no no no no yes no no no yes no no yes no no no no no no yes no no no no no no yes pose some of the formidable problems of bridging wildly different abstraction levels, which C2Eif had to deal with. TXL [2] is an expressive programming language designed to support source code analysis and transformation. It has been used to implement several language translation frameworks such as from Java to TCL and to Python. We could have used TXL to implement C2Eif; using regular Eiffel, however, allowed us to be independent of third-party closed-source tools, and to retain complete control over the implemented functionalities. Safer C. Many techniques exist aimed at ameliorating the safety of existing C code; for example, detection of format string vulnerability [3], out-of-bound array accesses and other memory errors [28, 24], or type errors [23]. C2Eif has a different scope, as it offers improved safety and debuggability as side-benefits of automatically porting C programs to Eiffel. This shares a little similarity with Ellison and Rosu’s formal executable semantics of C [7], which also finds errors in C program as a “side effect” of a rigorous translation. 5.3 Object-Oriented Reengineering After code written in a procedural language has been migrated to an object-oriented environment, it is natural to reengineer it to conform to the object-oriented design style, taking full advantage of features such as inheritance; this is the goal of object-oriented reengineering. Object-oriented reengineering is beyond the scope of this paper; and we tackled it in follow-up work based on C2Eif which we mention in Section 6. Nonetheless, it is still useful to compare existing approaches to reengineering with C2Eif, solely based on features such as degree of automation and tool support; see our follow-up work [41] for a discussion focused on the object-oriented reengineering techniques. Table 5 summarizes some features of the main approaches to object-oriented reengineering of procedural code: • The source and the target languages (or if it is a generic methodology). 28 full language evaluated methodology methodology C–C++ C–C++ C–C++ C++–C++ Cobol–OOSM Cobol–Java Cobol–Java PL/I–Java C–Eiffel completely automatic Gall [11] Jacobson [14] Livadas [17] Kontogiannis [16] Frakes [10] Fanta [8] Newcomb [26] Mossienko [21] Sneed [33] Sneed [31] C2Eif tool support source–target Table 5: Comparison of approaches to O-O reengineering. no no yes yes yes yes yes yes yes yes no no no no no no yes no yes yes yes – – no ? no no no no no no yes yes yes no 10KL 2KL 120KL 168KL 25KL 200KL 10KL 932KL YES • Whether tool support was developed, that is whether there exists a tool or the paper explicitly mentions the implementation of a tool. A YES in small caps denotes the only currently publicly available tool, namely C2Eif. • Whether the approach is completely automatic, that is does not require any user input other than providing a source procedural program. • Whether the approach supports the full source language (as used in practice) or only a subset thereof. • Whether the approach has been evaluated, that is whether the paper mentions evidence, such as a case study, that the approach was tried on real programs. If available, the table indicates the size of the programs used in the evaluation. Newcomb’s [26] and Sneed’s [33] are the only automatic tools which have been evaluated on programs of significant size. Newcomb’s tool [26], however, produces hierarchical object-oriented state machine models (OOSM); the mapping from OOSM to an object-oriented language is out of the scope of the work. Sneed’s tool [33] translates Cobol to Java; the paper reports that manual corrections of the automatically generated code were needed to get to a correct translation. While these corrections have successively been incorporated as extensions of the tool, the full Cobol language remains unsupported, according to the paper. 6 Conclusions and Future Work This paper presented the complete translation of C applications into Eiffel, and its implementation into the freely available automatic tool C2Eif. C2Eif supports the com29 plete C language, including unrestricted pointer arithmetic and pre-compiled libraries. Experiments in the paper showed that C2Eif correctly translates complete applications and libraries of significant size, and takes advantage of some of Eiffel’s advanced features to produce code of good quality. Future work. Future work will improve the readability and maintainability of the generated code. CIL, in particular, optimizes the code for program analysis, which is sometimes detrimental to readability of the Eiffel code generated by C2Eif. For example, CIL does not preserve comments, which are therefore lost in translation. We will also optimize the helper classes to improve on the few performance bottlenecks mentioned in Section 4.2. A major follow-up to the work described in this paper is the object-oriented reengineering of C code translated to Eiffel. In recent work [41], we have developed an automatic reengineering technique, and implemented it atop the translation produced by C2Eif and described in the present paper. The technique encapsulates functions and type definitions into classes that achieve low coupling and high cohesion, and introduces inheritance and contracts when possible. Other future work still remains in the direction of reengineering, such as in automatically replacing C data structure implementations (e.g., hash tables) with their Eiffel equivalents. 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arXiv:1704.01302v1 [math.ST] 5 Apr 2017 Extremes in Random Graphs Models of Complex Networks Natalia Markovich1 V.A.Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya 65, 117997 Moscow, Russia, Moscow Institute of Physics and Technology State University, Kerchenskaya 1, 117303 Moscow, Russia, (E-mail: nat.markovich@gmail.com) Abstract. Regarding the analysis of Web communication, social and complex networks the fast finding of most influential nodes in a network graph constitutes an important research problem. We use two indices of the influence of those nodes, namely, PageRank and a Max-linear model. We consider the PageRank as an autoregressive process with a random number of random coefficients that depend on ranks of incoming nodes and their out-degrees and assume that the coefficients are independent and distributed with regularly varying tail and with the same tail index. Then it is proved that the tail index and the extremal index are the same for both PageRank and the Max-linear model and the values of these indices are found. The achievements are based on the study of random sequences of a random length and the comparison of the distribution of their maxima and linear combinations. Keywords: Extremal Index, PageRank, Max-Linear Model, Branching Process, Autoregressive Process, Complex Networks. 1 Introduction Regarding the analysis of Web communication, social and complex networks the fast finding of most influential nodes in a network graph constitutes an important research problem. PageRank remains the most popular characteristic of such influence. We aim to find an extremal index of PageRank whose reciprocal value determines the first hitting time, i.e. a minimal time to reach the first influential node by means of a PageRank random walk. The extremal index θ ∈ [0, 1] has many other interpretations and plays a significant role in the theory of extreme values. Particularly, the limit distribution of maxima of stationary random variables (r.v.s) depends on θ. For independent r.v.s θ = 1 holds. θ has a connection to the tail index that shows the heaviness of the tail of a stationary distribution of an underlying process. Google’s PageRank defines the rank R(Xi ) of the Web page Xi as R(Xi ) = c X Xj ∈N (Xi ) R(Xj ) + (1 − c)qi , Dj i = 1, ..., n, (1) where N (Xi ) is the set of pages that link to Xi (in-degree), Dj is the number of outgoing links of page Xj (out-degree), c ∈ (0, 1) is a damping factor, q = (q1 , q2 , ..., qn ) is P a personalization probability vector or user preference such n that qi ≥ 0 and i=1 qi = 1, and n is the total number of pages, [11]. We omit in (1) the term with dangling nodes for simplicity. PageRank of a randomly selected page (a node in the graph) with random inand out-degrees may be considered as a branching process (Cf. [4], [5], [14]) Ri = Ni X (j) Aj Ri + Qi , i = 1, ..., n, (2) j=1 (j) denoting Ri = R(Xi ), Aj =d c/Dj , Qi = (1 − c)qi , [14]. Ri are ranks of descendants of node i, i.e. nodes with incoming links to node i. The r.v. Ni determines an in-degree, i.e. a number of directed edges to the ith node, and a number of nodes in the first generation of descendants belonging to the ith node as a parent, {Qi } is a sequence of i.i.d. r.v.s. Starting from the initial page (node) X0 , a PageRank random walk determines a regenerative process or Harris recurrent process {Xt }, letting it visits pages-followers of the underlying node with probability c and it restarts with probability 1 − c by jumping to a random independent node. A Max-linear model can be considered as an alternative characteristic of the node influence. This model is obtained by a substitution of sums in Google’s definition of PageRank by maxima, i.e. Ri = Ni _ (j) Aj Ri ∨ Qi , i = 1, ..., n, (3) j=1 is proposed in [6]. Formally, (2) can be considered as an autoregressive process with the random number Ni of random coefficients and the independent random term Qi . The extremal index of AR(1) processes with regularly varying stationary distribution and its relation to the tail index were considered in [9]. The extremal index of AR(q), q ≥ 1 processes with q random coefficients was obtained in [10] in a form which is not convenient for calculations. In [7] the results by [9] were extended to multivariate regularly varying distributed random sequences and the extremal and tail indices of sum and maxima of such sequences with l ≥ 1 r.v.s were derived. Our achievements extend and adapt the results by [7] to PageRank and Maxlinear processes. The problem concerns the finding of the extremal index of a random graph that models a real network where incoming nodes of the root node may be linked and, hence, be dependent. Such a random graph is called a Thorny Branching Tree (TBT) since any node may have outbound stubs (teleportations) to arbitrary nodes of the network, [4]. In this respect, such a graph cannot be considered as a pure Galton-Watson branching process where descendants of any node are mutually independent and teleportations are impossible. The paper is organized as follows. In Section 2 we recall necessary results regarding the relation between the tail and extremal indices obtained in [7] for multivariate random sequences which are regularly varying distributed (Theorems 1 and 2). Linear combinations and maxima of the random sequences of a fixed length are considered and it is derived that they have the same tail and extremal indices. In Section 3 we extend Theorem 2 to the case of unequal tail indices assuming r.v.s of a random sequence (Theorem 3). In Section 4 we consider sequences of random lengths and obtain the tail and extremal indices of their linear combinations and maxima (Theorem 4). We further discuss how these results can be applied to PageRank and the Max-linear processes in Section 5. 2 Related Work Let {Rj } be a stationary sequence with distribution function F (x) and maxima Mn = max1≤j≤n Rj . We shall interpret {Rj } as PageRanks of Web pages. Definition 1. A stationary sequence {Rn }n≥1 is said to have extremal index θ ∈ [0, 1] if for each 0 < τ < ∞ there is a sequence of real numbers un = un (τ ) such that lim n(1 − F (un )) = τ and (4) n→∞ lim P {Mn ≤ un } = e−τ θ (5) n→∞ hold ([12], p.53). In [7] the following theorems are proved which we will use to find the extremal (l) (2) (1) and tail indices of PageRank and a Max-linear model. Let Yn , Yn , ..., Yn , n ≥ 1, l ≥ 1 be sequences of r.v.s having stationary distributions with tail indices k1 , ..., kl and extremal indices θ1 , ..., θl , respectively, i.e. P {Yn(i) > x} ∼ c(i) x−ki as x → ∞, (i) where c are some real positive constants. Let us consider the weighted sum Yn (z) = z1 Yn(1) + z2 Yn(2) + ... + zl Yn(l) , z1 , ..., zl > 0 (6) and denote its tail index by k(z) and extremal index by θ(z). Supposing that there is a minimal tail index among k1 , ..., kl , the following theorem states the corresponding k(z) and θ(z). Theorem 1. ([7]) Let k1 < ki , i = 2, ..., l hold. Then Yn (z) has the tail index k(z) = k1 and the extremal index θ(z) = θ1 . (1) (2) (l) In the next theorem it is assumed that sequences Yn , Yn , ..., Yn are mutually independent with equal tail indices k1 = ... = kl = k. We denote (7) Yn∗ (z) = max z1 Yn(1) , z2 Yn(2) , ..., zl Yn(l) . Theorem 2. ([7]) The sequences Yn∗ (z) and Yn (z) have the same tail index k and the same extremal index equal to θ(z) = c(1) z1k c(l) zlk θ + ... + θl . 1 c(1) z1k + ... + c(l) zlk c(1) z1k + ... + c(l) zlk 3 Generalization of Theorem 2 Theorem 3 is a generalization of Theorem 2 to the case of unequal tail indices. (j) Theorem 3. Let {Yn }, n ≥ 1, j = 1, ..., l be mutually independent regularly varying r.v.s with tail indices k1 , ..., kl , respectively. Let km < ki , i = 1, ..., l, i 6= m hold. Then r.v.s Yn∗ (z) and Yn (z) have the same tail index k(z) = km and the same extremal index θ(z) = θm . Proof. First we show that P {Yn∗ (z) > x} ∼ c(z)x−km , Pl where c(z) = i=1 x → ∞, (8) c(i) ziki 1{ki = km }. Similar to [7] and as P {zi Yn(i) > x} ∼ c(i) ziki x−ki (9) holds, we have P {Yn∗ (z) > x} = P {max(z1 Yn(1) , .., zl Yn(l) ) > x} = 1 − P {max(z1 Yn(1) ≤ x} · ... · P {zl Yn(l) ) ≤ x} = l X + l X (−1) ∼ l X c(i) ziki x−ki + l X (−1)k−1 P {zi Yn(i) > x} i=1 k−1 l X P {zi1 Yn(i1 ) > x} · ... · P {zik Yn(ik ) > x} l X c(i1 ) zi1 1 x−ki1 · ... · c(ik ) zik k x−kik i1 <i2 <...<ik ;i1 ,i2 ,...,ik =1 k=2 i=1 ki ki i1 <i2 <...<ik ;i1 ,i2 ,...,ik =1 k=2 −km ∼ c(z)x + o(x−km ), x → ∞. (10) Thus, P {Yn (z) > x} ∼ c(z)x−km follows from Theorem 1. Now we show that Yn∗ (z) and Yn (z) have the same extremal index θ(z) = θm . We use the same notations as in [7] (i) (i) Mn(i) = max{Y1 , Y2 , ..., Yn(i) }, i = 1, .., l; Mn (z) = max{Y1 (z), Y2 (z), ..., Yn (z)}, Mn∗ (z) = max{Y1∗ (z), Y2∗ (z), ..., Yn∗ (z)}, n ≥ 1. By (7) it holds (1) (l) Mn∗ (z) = max{z1 Y1 , ..., z1 Yn(1) , ..., zl Y1 , ..., zl Yn(l) } = max{z1 Mn(1) , ..., zl Mn(l) }. Then we get P {Mn∗ (z)n−1/k ≤ x} = P {z1 Mn(1) n−1/k ≤ x, ..., zl Mn(l) n−1/k ≤ x} (11) Since km is the minimal tail index we have P {zi Mn(i) n−1/km ≤ x} = P {zi Mn(i) n−1/ki ≤ xn1/km −1/ki }. It implies zi Mn(i) n−1/km →P 0, i = 1, ..., l, i 6= m as n → ∞ (12) (i) since limn→∞ P {zi Mn n−1/ki ≤ x} = exp(−c(i) θi ziki x−ki ). By (11) it holds km θm x−km ), P {Mn∗ (z)n−1/km ≤ x} → exp(−c(m) zm n → ∞. Now we have to show that P {Mn∗ (z)n−1/km ≤ x} ∼ P {Mn (z)n−1/km ≤ x}. Let us denote un = xn1/km . Note that the event {Mn∗ (z) ≤ un } follows from {Mn (z) ≤ un }. Then, as in [7], we obtain 0 ≤ P {Mn∗ (z) ≤ un } − P {Mn (z) ≤ un } (13) ∗ ∗ = P {Mn (z) ≤ un } − P {Mn (z) ≤ un , Mn (z) ≤ un } n X = P {Mn∗ (z) ≤ un , Mn (z) > un } ≤ P {Mn∗ (z) ≤ un , Yk (z) > un } k=1 ≤ n X P {Yk∗ (z) ≤ un , Yk (z) > un } = nP {Yn∗ (z) ≤ un , Yn (z) > un } k=1 due to the stationarity of the sequences Yn∗ (z) and Yn (z). Lemma 1 in [7] states that P {Yn∗ (z) ≤ un \|Yn (z) > un } → 0, n → ∞, (14) (j) for i.i.d. regularly varying {Yn } with equal tail index. This can be extended to the case of unequal k1 , ..., kl . Since nP {Yn (z) > un } → c(z)x−km , n → ∞, (15) and (14) hold, it follows lim (P {Mn∗ (z) ≤ un } − P {Mn (z) ≤ un }) = 0. n→∞ 4 Extremal Index of PageRank and the Max-Linear Processes (j) (j) (j) We denote in (2) Ri as Yi (z) and Aj Ri = cRi /Dj , j = 1, ..., Ni as zj Yi . Then we can represent (2) in the form (6) as Yi (z) = Ni X j=1 (j) zj Yi + Qi , i = 1, ..., n, (16) where Ni is a nonnegative integer-valued r.v.. In the context of PageRank zj = c, j = 1, 2, ..., Ni , Qi = z ∗ qi with z ∗ = 1 − c and Ni represents the node in-degree. It is realistic to assume that Ni is a power law distributed r.v. with parameter α > 0, i.e. P {Ni = ℓ} ∼ ℓ−α (17) and Ni is bounded by a total number of nodes in the network. The distribution of Ni is in the domain of attraction of the Fréchet distribution with shape parameter α > 0 and P {Ni > x} = x−α ℓ(x), ∀x > 0, where ℓ(x) is a slowly varying function, since it satisfies a sufficient condition for this property, i.e. the von Mises type condition limn→∞ nP {Ni = n}/P {Ni > n} = α, [1]. Theorem 4 is an extension of Theorems 2 and 3 to maxima and sums of multivariate random sequences of random lengths, that can be applied to PageRank and the Max-linear processes. Let us turn to (16) and denote YN∗n (z) = max(z1 Yn(1) , .., zNn Yn(Nn ) , Qn ), YNn (z) = z1 Yn(1) + .. + zNn Yn(Nn ) + Qn . (j) Theorem 4. Let {Yn }, n ≥ 1, j = 1, ..., Nn and qn = Qn /z ∗ be mutually independent regularly varying i.i.d. r.v.s with tail indices k > 0 and β > 0, respectively, and Nn be regularly varying r.v. with tail index α > 0. Let (1) (N ) Yn ,.., Yn n have extremal indices θ1 , ..., θNn , respectively. Then r.v.s YN∗n (z) and YNn (z) are regularly varying distributed with the same tail index k(z) = min(k, α, β) and the same extremal index θ(z) such that θ(z) = (z ∗ )β , if k ≥ β, ∞ X c(i) θi zik /c(z), if k < β, θ(z) = (18) i=1 where c(z) = P∞ i=1 c(i) zik holds. Proof. We shall show first that P {YN∗n (z) > x} ∼ P {YNn (z) > x} ∼ x− min(k,α,β) . (19) (j) Since r.v.s {Yn }j≥1 are subexponential and i.i.d. it holds P {z1 Yn(1) + .. + z⌊x⌋ Yn(⌊x⌋) > x} ∼ P {max(z1 Yn(1) , .., z⌊x⌋ Yn(⌊x⌋) ) > x} ∼ xP {z1 Yn(1) > x}, x → ∞, (20) (j) [8]. Due to mutual independence of Qn and {Yn } and similar to (10) we get P {YN∗n (z) > x} = P {YN∗n (z) > x, Nn ≤ x} + P {YN∗n (z) > x, Nn > x} ∗ (z) > x} + P {Nn > x} ≤ P {Y⌊x⌋ = 1 − P {max(z1 Yn(1) , .., z⌊x⌋ Yn(⌊x⌋) ) ≤ x}P {Qn ≤ x} + P {Nn > x} ∼ cN x−α + cq (z ∗ )β x−β + c(z)x−k ∼ x− min{k,α,β} , (21) as x → ∞, where cN , cq > 0, c(z) = P∞ i=1 c(i) zik . On the other hand, P {YN∗n (z) > x} ≥ 0 + P {YN∗n (z) > x, Nn > x} ∗ ∗ (z) ≤ x, Nn ≤ x} − 1 (z) > x} + P {Nn > x} + P {Y⌈x⌉ ≥ P {Y⌈x⌉ ∼ x− min{k,α,β} (22) ∗ holds, since P {Y⌈x⌉ (z) ≤ x, Nn ≤ x} → 1 as x → ∞. Due to (21) and (22) we obtain P {YN∗n (z) > x} ∼ x− min{k,α,β} . The same is valid for YNn (z) by substitution of the maximum by the sum due to (20). Hence, (19) follows. Let us prove that YN∗n (z) and YNn (z) have the same extremal index θ(z). Let us denote ∗ (z) = max{YN∗1 (z), YN∗2 (z), ..., YN∗n (z)} MN n (N1 ) (1) = max{z1 Y1 , ..., zN1 Y1 (23) , Q1 , ..., z1 Yn(1) , ..., zNn Yn(Nn ) , Qn } and MNn (z) = max{YN1 (z), YN2 (z), ..., YNn (z)} (1) = max{z1 Y1 (N1 ) + ... + zN1 Y1 + Q1 , ..., z1 Yn(1) + ... + zNn Yn(Nn ) + Qn }. Without loss of generality we may assume that Nn = max{N1 , ..., Nn }. Then (N ) (1) we can complete vectors (z1 Yi , ..., zNi Yi i ), i = 1, 2, ..., n by zeros up to the dimension Nn and separate the vector (Q1 , ..., Qn ). We rewrite (23) as (1) ∗ (z) = max{z1 Y1 , ..., z1 Yn(1) , ..., zNn · 0, ..., zNn · 0, ..., zNn Yn(Nn ) , MN n Q1 , ..., Qn } = max(z1 Mn(1) , z2 Mn(2) , ..., zNn Mn(Nn ) , Mn(Q) ). (Q) Here, Mn = max{Q1 , ..., Qn } relates to the second term in the rhs of (16) corresponding to the user preference term Qi in (2). Following the same arguments as after (11) in Section 3 the statement follows. Really, denoting ∗ k ∗ = min{k, β} and un = xn1/k , x > 0, we get ∗ (z) > un } P {MN n ∗ ∗ (z) > un , Nn ≤ un } = P {MNn (z) > un , Nn > un } + P {MN n ∗ (z) > un } + P {Nn > un }. ≤ P {M⌈u n⌉ On the other hand, ∗ ∗ (z) > un } ≥ P {M⌈u (z) > un , Nn > un } P {MN n n⌉ ∗ ∗ (z) ≤ un , Nn ≤ un } − 1. = P {Nn > un } + P {M⌈un ⌉ (z) > un } + P {M⌈u n⌉ ∗ Note that P {M⌈u (z) ≤ un , Nn ≤ un } − 1 tends to zero as n → ∞. Hence, it n⌉ holds ∗ ∗ (z) > un } + P {Nn > un }, n → ∞. (24) (z) > un } ∼ P {M⌈u P {MN n n⌉ (Q) (i) ∗ If k < β holds, then Mn · n−1/k →P 0 as n → ∞ since P {zi Mn n−1/k ≤ x} → exp(−c(i) θi zik x−k ), i = 1, 2, .... Since P {Nn > un } ∼ u−α n → 0 as n → ∞ holds, then by (24) it follows ∗ ∗ (z)n−1/k ≤ x} = exp{−c(z)θ∗ (z)x−k }, lim P {MN n n→∞ (25) P∞ P∞ where θ∗ (z) = i=1 c(i) θi zik /c(z) and c(z) = i=1 c(i) zik . ∗ (i) (Q) If k ≥ β holds, then zi Mn ·n−1/k →P 0, i = 1, 2, ... follows since P {Mn n−1/β ≤ ∗ β −β x} → exp(−cq (z ) x ) as n → ∞. Thus, we obtain ∗ ∗ (z)n−1/k ≤ x} = exp(−cq (z ∗ )β x−β ). lim P {MN n n→∞ (26) Since {qi } are i.i.d., its extremal index is equal to one. Then by (25) and (26) the extremal index of YN∗n (z) satisfies (18) irrespectively of α. It remains to show that YN∗n (z) and YNn (z) have the same extremal index. Similarly to [7], we have to derive that ∗ ∗ ∗ (z)n−1/k ≤ x}. lim P {MNn (z)n−1/k ≤ x} = lim P {MN n n→∞ n→∞ (27) ∗ Since from the event {MNn (z) ≤ un } it follows {MN (z) ≤ un }, and P {MNn (z) ≤ n ∗ un } ≤ P {MNn (z) ≤ un } holds, we obtain similarly to (13) ∗ (z) ≤ un } − P {MNn (z) ≤ un } 0 ≤ P {MN n ∗ ∗ (z) ≤ un , MNn (z) ≤ un } (z) ≤ un } − P {MN = P {MN n n ∗ = P {MNn (z) ≤ un , MNn (z) > un } ∗ (z) ≤ un , MNn (z) > un , Nn > un } = P {MN n ∗ (z) ≤ un , MNn (z) > un , Nn ≤ un } + P {MN n ∗ (z) ≤ un , M⌊un ⌋ (z) > un , Nn ≤ un } ≤ P {Nn > un } + P {MN n ⌊un ⌋ ≤ P {Nn > un } + X P {Yk∗ (z) ≤ un , Yk (z) > un } k=1 = P {Nn > un } + ⌊un ⌋P {Yk∗ (z) ≤ un , Yk (z) > un } due to the stationarity of {Yk∗ (z)} and {Yk (z)}. (1) (N ) Completing vectors (z1 Yk , ..., zNk Yk k ) by zeroes (28) up to the maximal dimen- sion ⌊un ⌋, we get (1) (⌊un ⌋) P {Yk∗ (z) ≤ un , Yk (z) > un } = P {max(z1 Yk , ..., z⌊un ⌋ Yk (1) z1 Yk (⌊un ⌋) + ... + z⌊un ⌋ Yk , Qk ) ≤ un , + Qk > un } Then (27) follows from (14) and (15) since in (28) P {Yk∗ (z) ≤ un , Yk (z) > un } = P {Yk (z) > un }P {Yk∗ (z) ≤ un \|Yk (z) > un } holds. 5 Application to Indices of Complex Networks Theorem 4 can be applied to PageRank and the Max-linear processes. These processes then have the same tail index and the same extremal index. Theorem 4 is in the agreement with statements in [5] and [14], namely, that the stationary PNi (j) + Qi is regularly varying and distribution of PageRank R =d j=1 Aj Ri its tail index is determined by a most heavy-tailed distributed term in the (j) triple (Ni , Qi , Ai Ri ). This is derived if all terms in the triple are mutually independent. In contrast, Theorem 4 is valid for an arbitrary dependence (j) structure between Nn and {Yn } as well as Nn and Qn , and {Ni } are not necessarily independent. The novelty of Theorem 4 is that the extremal index of both PageRank and the Max-linear processes is the same and it depends on (j) the tail indices in the couple (Qi , Ai Ri ), irrespective of the tail index of Ni . The assumptions of both Theorem 4 and the statements in [5] and [14] do not reflect properly the complicated dependence between node ranks due to the entanglement of links in a real network. For better understanding let us consider the matrix   (N ) (1) (2) 0 0 Q1 z1 Y1 z2 Y1 ...zN1 Y1 1    z1 Y2(1) z2 Y2(2) ...zN1 Y2(N1 ) ...zN2 Y2(N2 ) 0 Q2    ... ... ... ... ...   ... (N ) (N ) (N ) (1) (2) z1 Yn z2 Yn ...zN1 Yn 1 ...zN2 Yn 2 ...zNn Yn n Qn (k, θ1 ) (k, θ2 ) ...(k, θN1 ) ...(k, θN2 ) ......(k, θNn ) ...(β, (z ∗ )β ) corresponding to (16) and completed by zeros up to the maximal dimension, let’s say Nn . Strings of the matrix correspond to generations of descendants of nodes with numbers 1, 2, ..., n. Each column may contain descendants of different nodes having the same extremal index θi , i = 1, 2, ..., Nn . All columns apart of the last one are identically regularly varying distributed with the same tail index k. The columns are mutually independent. In terms of some network, the conditions of Theorem 4 imply that ranks of all nodes with incoming links to a root node (i.e. its followers) are mutually independent, but followers of different nodes may be dependent and, thus, they are combined into clusters. The reciprocal of the extremal index approximates the mean cluster size, [12]. The statement (18) implies that the extremal index of PageRank is equal to θ(z) = (1 − c)β if the user preference dominates (i.e. its distribution tail is heavier than the tail of ranks of followers). If the damping factor c is close to one, then θ(z) is close to zero. The latter means the huge-sized cluster of nodes around a root-node in the presence of rare teleportations. If c is close to zero, then θ(z) is close to one due to the independence of frequent teleportations. If k < β holds, then roughly, the mean size of the cluster is determined by the consolidation of all clusters related to the followers of the underlying root. In practice, the followers of a node may be linked and their ranks can therefore be dependent. The future work will focus on the extremal index of PageRank (j) process when the terms {Yi } in (16) are mutually dependent. References 1. C.W. Anderson. Local limit theorems for the maxima of discrete random variables. 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Issue 3, 42–51, 2013, Moscow state university ISBN 9785211056527 (In russian) 8. C.M. Goldie and C. Klüppelberg. Subexponential Distributions, A Practical Guide to Heavy Tails, eds R. Adler, R. Feldman and M.S. Taqqu, Birkhäuser, Boston, MA, 435–459, 1998 9. L. de Haan, S. Resnick, H. Rootzén and G. de Vries. Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stochastic Processes and Applications, 32, 213–224, 1989. 10. C. Klüppelberg and S. Pergamenchtchikov. Extremal behaviour of models with multivariate random recurrence representation. Stochastic Processes and their Applications, 117, 432-456, 2007. 11. A. N. Langville and C. D. Meyer. Google’s PageRank and Beyond: The Science of Search Engine Rankings, Princeton University Press, 2006. 12. M.R. Leadbetter, G. Lingren and H. Rootzén. Extremes and Related Properties of Random Sequence and Processes, ch.3, Springer, New York, 1983. 13. H. Rootzén. Maxima and exceedances of stationary markov chains. Advances in Applied Probability, 20, 371–390, 1988. 14. Y.V. Volkovich and N. Litvak. Asymptotic analysis for personalized web search. Advances in Applied Probability, 42(2), 577–604, 2010.	10
Item Recommendation with Evolving User Preferences and Experience ∗ Subhabrata Mukherjee† , Hemank Lamba‡ and Gerhard Weikum† Planck Institute for Informatics, ‡ Carnegie Mellon University Email: {smukherjee, weikum}@mpi-inf.mpg.de, hlamba@cs.cmu.edu arXiv:1705.02519v1 [cs.AI] 6 May 2017 † Max Abstract—Current recommender systems exploit user and item similarities by collaborative filtering. Some advanced methods also consider the temporal evolution of item ratings as a global background process. However, all prior methods disregard the individual evolution of a user’s experience level and how this is expressed in the user’s writing in a review community. In this paper, we model the joint evolution of user experience, interest in specific item facets, writing style, and rating behavior. This way we can generate individual recommendations that take into account the user’s maturity level (e.g., recommending art movies rather than blockbusters for a cinematography expert). As only item ratings and review texts are observables, we capture the user’s experience and interests in a latent model learned from her reviews, vocabulary and writing style. We develop a generative HMM-LDA model to trace user evolution, where the Hidden Markov Model (HMM) traces her latent experience progressing over time — with solely user reviews and ratings as observables over time. The facets of a user’s interest are drawn from a Latent Dirichlet Allocation (LDA) model derived from her reviews, as a function of her (again latent) experience level. In experiments with five real-world datasets, we show that our model improves the rating prediction over state-of-the-art baselines, by a substantial margin. We also show, in a use-case study, that our model performs well in the assessment of user experience levels. more than recommendations for new blockbusters. Also, the facets of an item that a user focuses on change with experience. For example, a mature user pays more attention to narrative, light effects, and style rather than actors or special effects. Similar observations hold for ratings of wine, beer, food, etc. Our approach advances state-of-the-art by tapping review texts, modeling their properties as latent factors, using them to explain and predict item ratings as a function of a user’s experience evolving over time. Prior works considering review texts (e.g., [12], [15], [18], [23], [24]) did this only to learn topic similarities in a static, snapshot-oriented manner, without considering time at all. The only prior work [16], considering time, ignores the text of user-contributed reviews in harnessing their experience. However, user experience and their interest in specific item facets at different timepoints can often be observed only indirectly through their ratings, and more vividly through her vocabulary and writing style in reviews. Use-cases: Consider the reviews and ratings by two users on a “Canon DSLR” camera about the facet camera lens. • User 1: My first DSLR. Excellent camera, takes great pictures in I. I NTRODUCTION • User 2: The EF 75-300 mm lens is only good to be used outside. HD, without a doubt it brings honor to its name. [Rating: 5] The 2.2X HD lens can only be used for specific items; filters are Motivation and State-of-the-Art: Collaborative filtering algouseless if ISO, AP,... are correct. The short 18-55mm lens is cheap rithms are at the heart of recommender systems for items like and should have a hood to keep light off lens. [Rating: 3] movies, cameras, restaurants and beer. Most of these methods The second user is clearly more experienced than the first exploit user-user and item-item similarities in addition to the one, and more reserved about the lens quality of that camera history of user-item ratings — similarities being based on model. Future recommendations for the second user should take latent factor models over user and item features [11], and more into consideration the user’s maturity. As a second use-case, recently on explicit links and interactions among users [6][25]. consider the following reviews of Christopher Nolan movies All these data evolve over time leading to bursts in item where the facet of interest is the non-linear narrative style. popularity and other phenomena like anomalies[7]. State-ofthe-art recommender systems capture these temporal aspects by • User 1 on Memento (2001): “Backwards told is thriller noir-art empty ultimately but compelling and intriguing this.” introducing global bias components that reflect the evolution • User 2 on The Dark Knight (2008): “Memento was very compliof the user and community as a whole[10]. A few models also cated. The Dark Knight was flawless. Heath Ledger rocks!” consider changes in the social neighborhood of users[14]. What • User 3 on Inception (2010): “Inception is a triumph of style over is missing in all these approaches, though, is the awareness of substance. It is complex only in a structural way, not in terms of how experience and maturity levels evolve in individual users. plot. It doesn’t unravel in the way Memento does.” Individual experience is crucial in how users appreciate items, and thus react to recommendations. For example, a mature The first user does not appreciate complex narratives, making cinematographer would appreciate tips on art movies much fun of it by writing her review backwards. The second user prefers simpler blockbusters. The third user seems to appreciate ∗ This is an extended version of the paper published in ICDM 2015 [20]. the complex narration style of Inception and, more of, Memento. Refer to [19] for a general (continuous) version of this model with fine-grained temporal evolution of user experience, and resulting language model using We would consider this maturity level of the more experienced Geometric Brownian Motion and Brownian Motion, respectively. User 3 to generate future recommendations to her. Approach: We model the joint evolution of user experience, interests in specific item facets, writing style, and rating behavior in a community. As only item ratings and review texts are directly observed, we capture a user’s experience and interests by a latent model learned from her reviews, and vocabulary. All this is conditioned on time, considering the maturing rate of a user. Intuitively, a user gains experience not only by writing many reviews, but she also needs to continuously improve the quality of her reviews. This varies for different users, as some enter the community being experienced. This allows us to generate individual recommendations that take into account the user’s maturity level and interest in specific facets of items, at different timepoints. We develop a generative HMM-LDA model for a user’s evolution, where the Hidden Markov Model (HMM) traces her latent experience progressing over time, and the Latent Dirichlet Allocation (LDA) model captures her interests in specific item facets as a function of her (again, latent) experience level. The only explicit input to our model is the ratings and review texts upto a certain timepoint; everything else – especially the user’s experience level – is a latent variable. The output is the predicted ratings for the user’s reviews following the given timepoint. In addition, we can derive interpretations of a user’s experience and interests by salient words in the distributional vectors for latent dimensions. Although it is unsurprising to see users writing sophisticated words with more experience, we observe something more interesting. For instance in specialized communities like beeradvocate.com and ratebeer.com, experienced users write more descriptive and fruity words to depict the beer taste (cf. Table V). Table I shows a snapshot of the words used at different experience levels to depict the facets beer taste, movie plot and bad journalism, respectively. We apply our model to 12.7 million ratings from 0.9 million users on 0.5 million items in five different communities on movies, food, beer, and news media, achieving an improvement of 5% to 35% for the mean squared error for rating predictions over several competitive baselines. We also show that users at the same (latent) experience level do indeed exhibit similar vocabulary, and facet interests. Finally, a use-case study in a news community to identify experienced citizen journalists demonstrates that our model captures user maturity fairly well. Contributions: To summarize, this paper introduces the following novel elements: a) This is the first work that considers the progression of user experience as expressed through the text of item reviews, thereby elegantly combining text and time. b) An approach to capture the natural smooth temporal progression in user experience factoring in the maturing rate of the user, as expressed through her writing. c) Offers interpretability by learning the vocabulary usage of users at different levels of experience. d) A large-scale experimental study in five real world datasets from different communities like movies, beer and food; and an interesting use-case study in a news community. Experience Beer Movies News Level 1 Level 2 Level 3 stupid, bizarre storyline, epic realism, visceral, nostalgic bad, stupid biased, unfair opinionated, fallacy, rhetoric bad, shit sweet, bitter caramel finish, coffee roasted TABLE I: Vocabulary at different experience levels. II. OVERVIEW A. Model Dimensions Our approach is based on the intuition that there is a strong coupling between the facet preferences of a user, her experience, writing style in reviews, and rating behavior. All of these factors jointly evolve with time for a given user. We model the user experience progression through discrete stages, so a state-transition model is natural. Once this decision is made, a Markovian model is the simplest, and thus natural choice. This is because the experience level of a user at the current instant t depends on her experience level at the previous instant t-1. As experience levels are latent (not directly observable), a Hidden Markov Model is appropriate. Experience progression of a user depends on the following factors: • Maturing rate of the user which is modeled by her activity in the community. The more engaged a user is in the community, the higher are the chances that she gains experience and advances in writing sophisticated reviews, and develops taste to appreciate specific facets. • Facet preferences of the user in terms of focusing on particular facets of an item (e.g., narrative structure rather than special effects). With increasing maturity, the taste for particular facets becomes more refined. • Writing style of the user, as expressed by the language model at her current level of experience. More sophisticated vocabulary and writing style indicates higher probability of progressing to a more mature level. • Time difference between writing successive reviews. It is unlikely for the user’s experience level to change from that of her last review in a short time span (within a few hours or days). • Experience level difference: Since it is unlikely for a user to directly progress to say level 3 from level 1 without passing through level 2, the model at each instant decides whether the user should stay at current level l, or progress to l+1. In order to learn the facet preferences and language model of a user at different levels of experience, we use Latent Dirichlet Allocation (LDA). In this work, we assume each review to refer to exactly one item. Therefore, the facet distribution of items is expressed in the facet distribution of the review documents. We make the following assumptions for the generative process of writing a review by a user at time t at experience level et : • A user has a distribution over facets, where the facet preferences of the user depend on her experience level et . • A facet has a distribution over words where the words used to describe a facet depend on the user’s vocabulary at Level 1: film will happy people back supposed good wouldnt cant Level 2: storyline believable acting time stay laugh entire start funny Level 3 & 4: narrative cinema resemblance masterpiece crude undeniable admirable renowned seventies unpleasant myth nostalgic Level 5: incisive delirious personages erudite affective dramatis nucleus cinematographic transcendence unerring peerless fevered E1 E1 E2 E3 E4 E5 0 348 504 597 768 720 E1 E2 E3 E4 E5 E1 0 141 126 204 371 E2 146 0 36 129 321 E3 129 39 0 39 202 E4 225 153 42 0 106 E5 430 393 240 125 0 640 E2 384 0 449 551 742 560 480 E3 437 318 0 111 190 400 320 E4 509 352 109 0 138 240 E5 452 164 129 Experience Levels 0 80 0 300 250 200 150 100 160 577 400 350 Experience Levels Experience Levels Level 1: stupid people supposed wouldnt pass bizarre totally cant Level 2:storyline acting time problems evil great times didnt money ended simply falls pretty Level 3: movie plot good young epic rock tale believable acting Level 4: script direction years amount fast primary attractive sense talent multiple demonstrates establish Level 5: realism moments filmmaker visual perfect memorable recommended genius finish details defined talented visceral nostalgia Experience Levels 50 0 (a) Divergence of language model (b) Divergence of facet preference as a function of experience. as a function of experience. Fig. 1: KL Divergence as a function of experience. Hypothesis 2: Facet Preferences Depend on Experience Level. The second hypothesis underlying our work is that users experience level et . Table II shows salient words for two at similar levels of experience have similar facet preferences. facets of Amazon movie reviews at different levels of user In contrast to the LM’s where words are observed, facets are experience, automatically extracted by our latent model. The latent so that validating or falsifying the second hypothesis is facets are latent, but we can interpret them as plot/script not straightforward. We performed a three-step study: and narrative style, respectively. • We use Latent Dirichlet Allocation (LDA) [2] to compute a As a sanity check for our assumption of the coupling between latent facet distribution hfk i of each review. user experience, rating behavior, language and facet prefer- • We run Support Vector Regression (SVR) [4] for each user. ences, we perform experimental studies reported next. The user’s item rating in a review is the response variable, with the facet proportions in the review given by LDA as B. Hypotheses and Initial Studies features. The regression weight wkue is then interpreted as Hypothesis 1: Writing Style Depends on Experience Level. the preference of user ue for facet fk . We expect users at different experience levels to have • Finally, we aggregate these facet preferences for each divergent Language Models (LM’s) — with experienced users experience level e to get the facet preference P corresponding ue ue exp(wk ) having a more sophisticated writing style and vocabulary distribution given by < >. #ue than amateurs. To test this hypothesis, we performed initial Figure 1b shows the KL divergence between the facet prefstudies over two popular communities1 : 1) BeerAdvocate erences of users at different experience levels in BeerAdvocate. (beeradvocate.com) with 1.5 million reviews from 33, 000 We see that the divergence clearly increases with the difference users and 2) Amazon movie reviews (amazon.com) with 8 in user experience levels; this confirms the hypothesis. The million reviews from 760, 000 users. Both of these span a heatmap for Amazon is similar and omitted. period of about 10 years. Note that Figure 1 shows how a change in the experience In BeerAdvocate, a user gets points on the basis of likes level can be detected. This is not meant to predict the experience received for her reviews, ratings from other users, number of level, which is done by the model in Section IV. posts written, diversity and number of beers rated, time in the community, etc. We use this points measure as a proxy for the III. B UILDING B LOCKS OF OUR M ODEL user’s experience. In Amazon, reviews get helpfulness votes Our model, presented in the next section, builds on and from other users. For each user, we aggregate these votes over compares itself against various baseline models as follows. all her reviews and take this as a proxy for her experience. We partition the users into 5 bins, based on the points A. Latent-Factor Recommendation / helpfulness votes received, each representing one of the According to the standard latent factor model (LFM) [9], experience levels. For each bin, we aggregate the review texts the rating assigned by a user u to an item i is given by: of all users in that bin and construct a unigram language model. rec(u, i) = βg + βu + βi + hαu , φi i (1) The heatmap of Figure 1a shows the Kullback-Leibler (KL) where h., .i denotes a scalar product. βg is the average rating divergence between the LM’s of different experience levels, for the BeerAdvocate case. The Amazon reviews lead to a very of all items by all users. βu is the offset of the average rating similar heatmap, which is omitted here. The main observation given by user u from the global rating. Likewise βi is the rating is that the KL divergence is higher — the larger the difference bias for item i. αu and φi are the latent factors associated with is between the experience levels of two users. This confirms our user u and item i, respectively. These latent factors are learned hypothesis about the coupling of experience and user language. using gradient descent by minimizing the mean squared error (M SE) between observed P ratings r(u, i) and predicted ratings 1 Data available at http://snap.stanford.edu/data/ rec(u, i): M SE = \|U1 \| u,i∈U (r(u, i) − rec(u, i))2 TABLE II: Salient words for two facets at five experience levels in movie reviews. B. Experience-based Latent-Factor Recommendation The most relevant baseline for our work is the “user at learned rate” model of [16], which exploits that users at the same experience level have similar rating behavior even if their ratings are temporarily far apart. Experience of each user u for item i is modeled as a latent variable eu,i ∈ {1...E}. Different recommenders are learned for different experience levels. Therefore Equation 1 is parameterized as: receu,i (u, i) = βg (eu,i )+βu (eu,i )+βi (eu,i )+hαu (eu,i ), φi (eu,i )i (2) The parameters are learned using Limited Memory BFGS with the additional constraint that experience levels should be non-decreasing over the reviews written by a user over time. However, this is significantly different from our approach. All of these models work on the basis of only user rating behavior, and ignore the review texts completely. Additionally, the smoothness in the evolution of parameters between experience levels is enforced via L2 regularization, and does not model the natural user maturing rate (via HMM) as in our model. Also note that in the above parametrization, an experience level is estimated for each user-item pair. However, it is rare that a user reviews the same item multiple times. In our approach, we instead trace the evolution of users, and not user-item pairs. C. User-Facet Model In order to find the facets of interest to a user, [22] extends Latent Dirichlet Allocation (LDA) to include authorship information. Each document d is considered to have a distribution over authors. We consider the special case where each document has exactly one author u associated with a Multinomial distribution θu over facets Z with a symmetric Dirichlet prior α. The facets have a Multinomial distribution φz over words W drawn from a vocabulary V with a symmetric Dirichlet prior β. Exact inference is not possible due to the intractable coupling between Θ and Φ. Two ways for approximate inference are MCMC techniques like Collapsed Gibbs Sampling and Variational Inference. The latter is typically much more complex and computationally expensive. In our work, we thus use sampling. Fig. 2: Supervised model for user facets and ratings. Fig. 3: Supervised model for user experience, facets, and ratings. hyper-parameter α for the document-facet Multinomial distribution Θ is parametrized as αd,z = exp(xTd λz ). The model is trained using stochastic EM which alternates between 1) sampling facet assignments from the posterior distribution conditioned on words and features, and 2) optimizing λ given the facet assignments using L-BFGS. Our approach, explained in the next section, follows a similar approach to couple the User-Facet Model and the Latent-Factor Recommendation Model (depicted in Figure 2). IV. J OINT M ODEL : U SER E XPERIENCE , FACET P REFERENCE , W RITING S TYLE We start with a User-Facet Model (UFM) (aka. AuthorTopic Model [22]) based on Latent Dirichlet Allocation (LDA), where users have a distribution over facets and facets D. Supervised User-Facet Model have a distribution over words. This is to determine the The generative process described above is unsupervised and facets of interest to a user. These facet preferences can be does not take the ratings in reviews into account. Supervision interpreted as latent item factors in the traditional Latent-Factor is difficult to build into MCMC sampling where ratings are Recommendation Model (LFM) [9]. However, the LFM is continuous values, as in communities like newstrust.net. supervised as opposed to the UFM. It is not obvious how to For discrete ratings, a review-specific Multinomial rating incorporate supervision into the UFM to predict ratings. The distribution πd,r can be learned as in [13], [21]. Discretizing the user-provided ratings of items can take continuous values (in continuous ratings into buckets bypasses the problem to some some review communities), so we cannot incorporate them into extent, but results in loss of information. Other approaches [12], a UFM with a Multinomial distribution of ratings. We propose [15], [18] overcome this problem by learning the feature an Expectation-Maximization (EM) approach to incorporate supervision, where the latent facets are estimated in an Eweights separately from the user-facet model. An elegant approach using Multinomial-Dirichlet Regression Step using Gibbs Sampling, and Support Vector Regression is proposed in [17] to incorporate arbitrary types of observed (SVR) [4] is used in the M-Step to learn the feature weights continuous or categorical features. Each facet z is associated and predict ratings. Subsequently, we incorporate a layer for with a vector λz whose dimension equals the number of features. experience in the UFM-LFM model, where the experience Assuming xd is the feature vector for document d, the Dirichlet levels are drawn from a Hidden Markov Model (HMM) in the E-Step. The experience level transitions depend on the evolution of the user’s maturing rate, facet preferences, and writing style over time. The entire process is a supervised generative process of generating a review based on the experience level of a user hinged on our HMM-LDA model. A. Generative Process for a Review example, a user at a high level of experience may choose to write on the beer “hoppiness” or “story perplexity” in a movie. The word that she writes depends on the facet chosen and the language model for her current experience level. Thus, she draws a word from the multinomial distribution φetd ,zdi with a symmetric Dirichlet prior δ. For example, if the facet chosen is beer taste or movie plot, an experienced user may choose to use the words “coffee roasted vanilla” and “visceral”, whereas an inexperienced user may use “bitter” and “emotional” resp. Algorithm 1 describes this generative process for the review; Figure 3 depicts it visually in plate notation for graphical models. We use MCMC sampling for inference on this model. Consider a corpus with a set D of review documents denoted by {d1 . . . dD }. For each user, all her documents are ordered by timestamps t when she wrote them, such that tdi < tdj for i < j. Each document d has a sequence of Nd words denoted by d = w1 . . . .wNd . Each word is drawn from a vocabulary V having unique words indexed by {1 . . . V }. Consider a set of U users involved in writing the documents in the corpus, Algorithm 1: Supervised Generative Model for a User’s where ud is the author of document d. Consider an ordered set Experience, Facets, and Ratings of experience levels {e1 , e2 , ...eE } where each ei is from a set for each facet z = 1, ...Z and experience level e = 1, ...E do E, and a set of facets {z1 , z2 , ...zZ } where each zi is from a choose φe,z ∼ Dirichlet(β) set Z of possible facets. Each document d is associated with end a rating r and an item i. for each review d = 1, ...D do At the time td of writing the review d, the user ud has Given user ud and timestamp td experience level etd ∈ E. We assume that her experience level /Current experience level depends on previous level/ transitions follow a distribution Π with a Markovian assumption 1. Conditioned on ud and previous experience etd−1 , choose and certain constraints. This means the experience level of ud etd ∼ πetd−1 at time td depends on her experience level when writing the /User’s facet preferences at current experience level are previous document at time td−1 . influenced by supervision via α – scaled by hyper-parameter ρ controlling influence of supervision/ πei (ej ) denotes the probability of progressing to experience 2. Conditioned on supervised facet preference αud ,etd of ud level ej from experience level ei , with the constraint ej ∈ at experience level etd scaled by ρ, choose {ei , ei + 1}. This means at each instant the user can either stay θud ,etd ∼ Dirichlet(ρ × αud ,etd ) at her current experience level, or move to the next one. for each word i = 1, ...Nd do The experience-level transition probabilities depend on the /Facet is drawn from user’s experience-based facet interests/ rating behavior, facet preferences, and writing style of the user. 3. Conditioned on ud and etd choose a facet The progression also takes into account the 1) maturing rate of zdi ∼ M ultinomial(θud ,etd ) ud modeled by the intensity of her activity in the community, /Word is drawn from chosen facet and user’s and 2) the time gaps between writing consecutive reviews. We vocabulary at her current experience level/ incorporate these aspects in a prior for the user’s transition 4. Conditioned on zdi and etd choose a word wdi ∼ M ultinomial(φetd ,zdi ) rates, γ ud , defined as: end D u d γ ud = + λ(td − td−1 ) /Rating computed via Support Vector Regression with Dud + Davg chosen facet proportions as input features to learn α/ Dud and Davg denote the number of reviews written by ud 5. Choose rd ∼ F (hαud ,etd , φetd ,zd i) and the average number of reviews per user in the community, end respectively. Therefore the first term models the user activity with respect to the community average. The second term reflects the time difference between successive reviews. The B. Supervision for Rating Prediction user experience is unlikely to change from the level when writing the previous review just a few hours or days ago. λ The latent item factors φi in Equation 2 correspond to controls the effect of this time difference, and is set to a very the latent facets Z in Algorithm 1. Assume that we have small value. Note that if the user writes very infrequently, the some estimation of the latent facet distribution φe,z of each second term may go up. But the first term which plays the document after one iteration of MCMC sampling, where e dominating role in this prior will be very small with respect denotes the experience level at which a document is written, to the community average in an active community, bringing and let z denote a latent facet of the document. We also have down the influence of the entire prior. Note that the constructed an estimation of the preference of a user u for facet z at HMM encapsulates all the factors for experience progression experience level e given by θu,e (z). outlined in Section II. For each user u, we compute a supervised regression function At experience level etd , user ud has a Multinomial facet- Fu for the user’s numeric ratings with the – currently estimated preference distribution θud ,etd . From this distribution she draws – experience-based facet distribution φe,z of her reviews as input a facet of interest zdi for the ith word in her document. For features and the ratings as output. e The learned feature weights hαu,e (z)i indicate the user’s Let mei−1 denote the number of transitions from experience i preference for facet z at experience level e. These feature level ei−1 to ei over all users in the community, with the weights are used to modify θu,e to attribute more mass to the constraint ei ∈ {ei−1 , ei−1 + 1}. Note that we allow selffacet for which u has a higher preference at level e. This is transitions for staying at the same experience level. The counts reflected in the next sampling iteration, when we draw a facet z capture the relative difficulty in progressing between different from the user’s facet preference distribution θu,e smoothed by experience levels. For example, it may be easier to progress αu,e , and then draw a word from φe,z . This sampling process to level 2 from level 1 than to level 4 from level 3. is repeated until convergence. The state transition probability depending on the previous In any latent facet model, it is difficult to set the hyper- state, factoring in the user-specific activity rate, is given by: e m i−1 +I(ei−1 =ei )+γ u parameters. Therefore, most prior work assume symmetric i P (ei \|ei−1 , u, e−i ) = meei−1 u +I(e i−1 =ei )+Eγ . Dirichlet priors with heuristically chosen concentration paramwhere I(.) is an indicator function taking the value 1 when eters. Our approach is to learn the concentration parameter α the argument is true, and 0 otherwise. The subscript −i denotes of a general (i.e., asymmetric) Dirichlet prior for Multinomial th the value of a variable excluding the data at the i position. distribution Θ – where we optimize these hyper-parameters to All the counts of transitions exclude transitions to and from learn user ratings for documents at a given experience level. ei , when sampling a value for the current experience level ei during Gibbs sampling. The conditional distribution for the C. Inference experience level transition is given by: We describe the inference algorithm to estimate the distriP (E\|U, Z, W ) ∝ P (E\|U ) × P (Z\|E, U ) × P (W \|Z, E) (4) butions Θ, Φ and Π from observed data. For each user, we Here the first factor models the rate of experience progression factoring in user activity; the second and third factor models the facet-preferences of user, and language model at a specific level of experience respectively. All three factors combined decide whether the user should stay at the current level of experience, or has matured enough to progress to next level. In Gibbs sampling, the conditional distribution for each hidden variable is computed based on the current assignment of other hidden variables. The values for the latent variables are sampled repeatedly from this conditional distribution until convergence. In our problem setting we have two sets of latent Du Y du U Y E Y Z N variables corresponding to E and Z respectively. Y Y P (U, E, Z, W, θ, φ, π; α, δ, γ) = { We perform Collapsed Gibbs Sampling [5] in which we first u=1 e=1 i=1 z=1 j=1 sample a value for the experience level ei of the user for the × P (θu,e ; αu,e ) × P (zi,j \|θu,ei ) P (πe ; γ u ) × P (ei \|πe ) current document i, keeping all facet assignments Z fixed. In {z } \| {z } \| experience transition distribution user experience facet distribution order to do this, we consider two experience levels ei−1 and × P (φe,z ; δ) × P (wi,j \|φei ,zi,j ) } ei−1 + 1. For each of these levels, we go through the current {z } \| document and all the token positions to compute Equation 4 — experience facet language distribution (3) and choose the level having the highest conditional probability. Let n(u, e, d, z, v) denote the count of the word w occurring Thereafter, we sample a new facet for each word wi,j of the in document d written by user u at experience level e belonging document, keeping the currently sampled experience level of to facet z. In the following equation, (.) at any position in a the user for the document fixed. The conditional distributions for Gibbs sampling for the joint distribution indicates summation of the above counts for the update of the latent variables E and Z are given by: respective argument. compute the conditional distribution over the set of hidden variables E and Z for all the words W in a review. The exact computation of this distribution is intractable. We use Collapsed Gibbs Sampling [5] to estimate the conditional distribution for each hidden variable, which is computed over the current assignment for all other hidden variables, and integrating out other parameters of the model. Let U, E, Z and W be the set of all users, experience levels, facets and words in the corpus. In the following, i indexes a document and j indexes a word in it. The joint probability distribution is given by: Exploiting conjugacy of the Multinomial and Dirichlet ui = u, {zi,j = zj }, {wi,j = wj }, e−i ) ∝ distributions, we can integrate out Φ from the above distribution E-Step 1: P (ei = e\|ei−1 ,Y P (ei \|u, ei−1 , e−i ) × P (zj \|ei , u, e−i ) × P (wj \|zj , ei , e−i ) ∝ to obtain the posterior distribution P (Z\|U, E; α) of the latent j variable Z given by: e U Y E Y u=1 e=1 P Q Γ( z αu,e,z ) z Γ(n(u, e, ., z, .) + αu,e,z ) P P Q z αu,e,z ) z Γ(αu,e,z )Γ( z n(u, e, ., z, .) + where Γ denotes the Gamma function. Similarly, by integrating out Θ, P (W \|E, Z; δ) is given by E Y Z Y e=1 z=1 P Q Γ( v δv ) v Γ(n(., e, ., z, v) + δv ) Q P P v Γ(δv )Γ( v n(., e, ., z, v) + v δv ) Y P j zj mei−1 + I(ei−1 = ei ) + γ u i × ei−1 m. + I(ei−1 = ei ) + Eγ u n(u, e, ., zj , .) + αu,e,zj n(., e, ., zj , wj ) + δ P ×P n(u, e, ., zj , .) + zj αu,e,zj wj n(., e, ., zj , wj ) + V δ E-Step 2: P (zj = z\|ud = u, ed = e, wj = w, z−j ) ∝ n(u, e, ., z, .) + αu,e,z n(., e, ., z, w) + δ P P ×P n(u, e, ., z, .) + α z z u,e,z w n(., e, ., z, w) + V δ (5) The proportion of the z th facet in document d with words {wj } written at experience level e is given by: PNd j=1 φe,z (wj ) φe,z (d) = Nd For each user u, we learn a regression model Fu using these facet proportions in each document as features, along with the user and item biases (refer to Equation 2), with the user’s item rating rd as the response variable. Besides the facet distribution of each document, the biases < βg (e), βu (e), βi (e) > also depend on the experience level e. We formulate the function Fu as Support Vector Regression [4], which forms the M -Step in our problem: 1 M-Step: min αu,e T αu,e + C× αu,e 2 Du X (max(0, \|rd − αu,e T < βg (e),βu (e), βi (e), φe,z (d) > \| − ))2 d=1 The total number of parameters learned is [E×Z+E×3]×U . Our solution may generate a mix of positive and negative real numbered weights. In order to ensure that the concentration parameters of the Dirichlet distribution are positive reals, we take exp(αu,e ). The learned α’s are typically very small, whereas the value of n(u, e, ., z, .) in Equation 5 is very large. Therefore we scale the α’s by a hyper-parameter ρ to control the influence of supervision. ρ is tuned using a validation set by varying it from {100 , 101 ...105 }. In the E-Step of the next iteration, we choose θu,e ∼ Dirichlet(ρ × αu,e ). We use the LibLinear2 package for Support Vector Regression. V. E XPERIMENTS Setup: We perform experiments with data from five communities in different domains: BeerAdvocate (beeradvocate.com) and RateBeer (ratebeer.com) for beer reviews, Amazon (amazon.com) for movie reviews, Yelp (yelp.com) for food and restaurant reviews, and NewsTrust (newstrust.net) for reviews of news media. Table III gives the dataset statistics3 . We have a total of 12.7 million reviews from 0.9 million users from all of the five communities combined. The first four communities are used for product reviews, from where we extract the following quintuple for our model < userId, itemId, timestamp, rating, review >. NewsTrust is a special community, which we discuss in Section VI. For all models, we used the three most recent reviews of each user as withheld test data. All experience-based models consider the last experience level reached by each user, and corresponding learned parameters for rating prediction. In all the models, we group light users with less than 50 reviews in training data into a background model, treated as a single user, to avoid modeling from sparse observations. We do not ignore any user. During the test phase for a light user, we take her parameters from the background model. We set Z = 20 for BeerAdvocate, RateBeer and Yelp facets; and Z = 100 for Amazon movies and NewsTrust which have much richer 2 http://www.csie.ntu.edu.tw/ cjlin/liblinear http://www.yelp.com/dataset challenge/ 3 http://snap.stanford.edu/data/, Dataset #Users #Items #Ratings Beer (BeerAdvocate) Beer (RateBeer) Movies (Amazon) Food (Yelp) Media (NewsTrust) 33,387 40,213 759,899 45,981 6,180 66,051 110,419 267,320 11,537 62,108 1,586,259 2,924,127 7,911,684 229,907 134,407 TOTAL 885,660 517,435 12,786,384 TABLE III: Dataset statistics. Models Beer Rate Advocate Beer News Trust AmazonYelp Our model (most recent experience level) f) Our model (past experience level) e) User at learned rate c) Community at learned rate b) Community at uniform rate d) User at uniform rate a) Latent factor model 0.363 0.309 0.373 1.174 1.469 0.375 0.362 0.470 1.200 1.642 0.379 0.383 0.391 0.394 0.409 0.336 0.334 0.347 0.349 0.377 0.575 0.656 0.767 0.744 0.847 1.293 1.203 1.203 1.206 1.248 1.732 1.534 1.526 1.613 1.560 TABLE IV: MSE comparison of our model versus baselines. latent dimensions. For experience levels, we set E = 5 for all. However, for NewsTrust and Yelp datasets our model categorizes users to belong to one of three experience levels. A. Quantitative Comparison Baselines: We consider the following baselines for our work, and use the available code4 for experimentation. a) LFM: A standard latent factor recommendation model [9]. b) Community at uniform rate: Users and products in a community evolve using a single “global clock” [10][27][26], where the different stages of the community evolution appear at uniform time intervals. So the community prefers different products at different times. c) Community at learned rate: This extends (b) by learning the rate at which the community evolves with time, eliminating the uniform rate assumption. d) User at uniform rate: This extends (b) to consider individual users, by modeling the different stages of a user’s progression based on preferences and experience levels evolving over time. The model assumes a uniform rate for experience progression. 4 http://cseweb.ucsd.edu/ jmcauley/code/ 60 50 40 30 20 10 0 BeerAdvocate RateBeer NewsTrust User at learned rate Community at learned rate User at uniform rate Latent factor model Amazon Yelp Community at uniform rate Fig. 4: MSE improvement (%) of our model over baselines. Experience Level 1: drank, bad, maybe, terrible, dull, shit e) User at learned rate: This extends (d) by allowing each user to evolve on a “personal clock”, so that the time to Experience Level 2: bottle, sweet, nice hops, bitter, strong light, head, smooth, good, brew, better, good reach certain experience levels depends on the user [16]. f) Our model with past experience level: In order to determine Expertise Level 3: sweet alcohol, palate down, thin glass, malts, poured thick, pleasant hint, bitterness, copper hard how well our model captures evolution of user experience over time, we consider another baseline where we randomly Experience Level 4: smells sweet, thin bitter, fresh hint, honey end, sticky yellow, slight bit good, faint bitter beer, red brown, sample the experience level reached by users at some timepoint good malty, deep smooth bubbly, damn weak previously in their lifecycle, who may have evolved thereafter. We learn our model parameters from the data up to this time, Experience Level 5: golden head lacing, floral dark fruits, citrus sweet, light spice, hops, caramel finish, acquired taste, and again predict the user’s most recent three item ratings. Note hazy body, lacing chocolate, coffee roasted vanilla, creamy that this baseline considers textual content of user contributed bitterness, copper malts, spicy honey reviews, unlike other baselines that ignore them. Therefore it is better than vanilla content-based methods, with the notion TABLE V: Experience-based facet words for the illustrative of past evolution, and is the strongest baseline for our model. beer facet taste. Discussions: Table IV compares the mean squared error (MSE) 0.6 for rating predictions, generated by our model versus the six baselines. Our model consistently outperforms all baselines, 0.5 reducing the MSE by ca. 5 to 35%. Improvements of our model 0.4 over baselines are statistically significant at p-value < 0.0001. 0.3 Our performance improvement is most prominent for the 0.2 NewsTrust community, which exhibits strong language features, and topic polarities in reviews. The lowest improvement (over 0.1 the best performing baseline in any dataset) is achieved 0 RateBeer BeeeAdvocate NewsTrust Amazon Yelp for Amazon movie reviews. A possible reason is that the Level 1 Level 2 Level 3 Level 4 Level 5 community is very diverse with a very wide range of movies and that review texts heavily mix statements about movie plots Fig. 5: Proportion of reviews at each experience level of users. with the actual review aspects like praising or criticizing certain facets of a movie. The situation is similar for the food and movies (e.g., by genre or production studios) that may better restaurants case. Nevertheless, our model always wins over the distinguish experienced users from amateurs or novices in best baseline from other works, which is typically the “user at terms of their refined taste and writing style. MSE for different experience levels: We observe a weak trend learned rate” model. Evolution effects: We observe in Table IV that our model’s that the MSE decreases with increasing experience level. Users predictions degrade when applied to the users’ past experience at the highest level of experience almost always exhibit the level, compared to their most recent level. This signals that lowest MSE. So we tend to better predict the rating behavior for the model captures user evolution past the previous timepoint. the most mature users than for the remaining user population. Therefore the last (i.e., most recent) experience level attained by This in turn enables generating better recommendations for the a user is most informative for generating new recommendations. “connoisseurs” in the community. Experience progression: Figure 5 shows the proportion of B. Qualitative Analysis reviews written by community members at different experience Salient words for facets and experience levels: We point levels right before advancing to the next level. Here we plot out typical word clusters, with illustrative labels, to show the users with a minimum of 50 reviews, so they are certainly not variation of language for users of different experience levels “amateurs”. A large part of the community progresses from and different facets. Tables II and V show salient words to level 1 to level 2. However, from here only few users move describe the beer facet taste and movie facets plot and narrative to higher levels, leading to a skewed distribution. We observe style, respectively – at different experience levels. Note that the that the majority of the population stays at level 2. facets being latent, their labels are merely our interpretation. User experience distribution: Table VI shows the number of users per experience level in each domain, for users with Other similar examples can be found in Tables I and VII. BeerAdvocate and RateBeer are very focused communities; > 50 reviews. The distribution also follows our intuition of so it is easier for our model to characterize the user experience a highly skewed distribution. Note that almost all users with evolution by vocabulary and writing style in user reviews. We < 50 reviews belong to levels 1 or 2. observe in Table V that users write more descriptive and fruity Language model and facet preference divergence: Figure 6b words to depict the beer taste as they become more experienced. and 6c show the KL divergence for facet-preference and For movies, the wording in reviews is much more diverse language models of users at different experience levels, as and harder to track. Especially for blockbuster movies, which computed by our model. The facet-preference divergence tend to dominate this data, the reviews mix all kinds of aspects. increases with the gap between experience levels, but not as A better approach here could be to focus on specific kinds of smooth and prominent as for the language models. On one hand, Datasets e=1 e=2 e=3 e=4 e=5 Models F1 N DCG BeerAdvocate RateBeer NewsTrust Amazon Yelp 0.05 0.03 - 0.59 0.42 0.72 - 0.19 0.35 0.15 0.13 0.30 0.10 0.18 0.60 0.10 0.68 0.07 0.02 0.25 0.05 0.02 User at learned rate [16] Our model 0.68 0.75 0.90 0.97 TABLE VI: Distribution of users at different experience levels. Level 1: bad god religion iraq responsibility Level 2: national reform live krugman questions clear jon led meaningful lives california powerful safety impacts Level 3: health actions cuts medicare nov news points oil climate major jobs house high vote congressional spending unemployment strong taxes citizens events failure TABLE VII: Salient words for the illustrative NewsTrust topic US Election at different experience levels. this is due to the complexity of latent facets vs. explicit words. On the other hand, this also affirms our notion of grounding the model on language. Baseline model divergence: Figure 6a shows the facetpreference divergence of users at different experience levels computed by the baseline model “user at learned rate” [16]. The contrast between the heatmaps of our model and the baseline is revealing. The increase in divergence with increasing gap between experience levels is very rough in the baseline model, although the trend is obvious. TABLE VIII: Performance on identifying experienced users. based on community engagement, time in the community, other users’ feedback on reviews, profile transparency, and manual validation. We use these member levels to categorize users as experienced or inexperienced. This is treated as the ground truth for assessing the prediction and ranking quality of our model and the baseline “user at learned rate” model [16]. Table VIII shows the F1 scores of these two competitors. We also computed the Normalized Discounted Cumulative Gain (NDCG) [8] for the ranked lists of users generated by the two models. NDCG gives geometrically decreasing weights to predictions at various positions of ranked list: N DCGp = DCGp IDCGp where DCGp = rel1 + Pp reli i=2 log2 i Here, reli is the relevance (0 or 1) of a result at position i. As Table VIII shows, our model clearly outperforms the baseline model on both F1 and N DCG. VII. R ELATED W ORK State-of-the-art recommenders based on collaborative filtering [9][11] exploit user-user and item-item similarities by latent factors. Explicit user-user interactions have been exploited in trust-aware recommendation systems [6][25]. The temporal VI. U SE -C ASE S TUDY aspects leading to bursts in item popularity, bias in ratings, or So far we have focused on traditional item recommendation the evolution of the entire community as a whole is studied for items like beers or movies. Now we switch to a different in [10][27][26]. Other papers have studied temporal issues kind of items - newspapers and news articles - tapping into the for anomaly detection [7], detecting changes in the social NewsTrust online community (newstrust.net). NewsTrust neighborhood [14] and linguistic norms [3]. However, none of features news stories posted and reviewed by members, many these prior work has considered the evolving experience and of whom are professional journalists and content experts. behavior of individual users. The recent work[16], which is one of our baselines, modeled Stories are reviewed based on their objectivity, rationality, and the influence of rating behavior on evolving user experience. general quality of language to present an unbiased and balanced However, it ignores the vocabulary and writing style of users in narrative of an event. The focus is on quality journalism. reviews, and their natural smooth temporal progression. In conIn our framework, each story is an item, which is rated trast, our work considers the review texts for additional insight and reviewed by a user. The facets are the underlying topic into facet preferences and smooth experience progression. distribution of reviews, with topics being Healthcare, Obama Prior work that tapped user review texts focused on other Administration, NSA, etc. The facet preferences can be mapped issues. Sentiment analysis over reviews aimed to learn latent to the (political) polarity of users in the news community. Recommending News Articles: Our first objective is to topics [13], latent aspects and their ratings [12][24], and userrecommend news to readers catering to their facet preferences, user interactions [25]. [15][23] unified various approaches viewpoints, and experience. We apply our joint model to this to generate user-specific ratings of reviews. [18] further task, and compare the predicted ratings with the ones observed leveraged the author writing style. However, all of these for withheld reviews in the NewsTrust community. The mean prior approaches operate in a static, snapshot-oriented manner, squared error (MSE) results are reported in Table IV in without considering time at all. Section V. Table VII shows salient examples of the vocabulary From the modeling perspective, some approaches learn a by users at different experience levels on the topic US Election. document-specific discrete rating [13][21], whereas others Identifying Experienced Users: Our second task is to find learn the facet weights outside the topic model (e.g., [12], experienced members of this community, who have potential [15], [18]). In order to incorporate continuous ratings, [1] for being citizen journalists. In order to find how good our proposed a complex and computationally expensive Variational model predicts the experience level of users, we consider the Inference algorithm, and [17] developed a simpler approach following as ground-truth for user experience. In NewsTrust, using Multinomial-Dirichlet Regression. The latter inspired our users have Member Levels calculated by the NewsTrust staff technique for incorporating supervision. E3 E4 E5 138 123 93 18 135 105 E2 E3 134 0 33 8 65 90 75 112 34 0 21 43 60 E4 89 8 18 0 30 45 30 E5 19 63 48 31 0 E1 120 15 E1 E2 E3 E4 E5 0 180 71 140 125 E2 38 0 175 47 150 175 125 E3 83 149 5 0 88 100 75 E4 E5 86 155 1 E2 E3 0 180 48 203 100 140 180 150 259 E2 0 73 120 90 60 65 83 0 30 E3 160 2217 E2 181 0 1910 E3 601 309 0 1750 1500 1250 1000 750 500 0 Experience Levels 25 0 E2 0 2000 210 50 116 E1 E1 240 E3 86 0 E1 175 E1 0 Experience Levels 200 Experience Levels E2 0 Experience Levels E1 Experience Levels Experience Levels E1 250 0 Experience Levels 0 Experience Levels (a) User at learned rate [16]: Facet preference divergence with experience. E3 E4 E5 0 104 76 141 114 E1 E2 E3 E4 E5 0 192 199 324 161 135 E2 120 105 98 0 78 77 91 90 E3 75 78 0 89 106 75 60 E4 320 E1 132 74 90 0 52 45 109 86 106 52 0 E2 200 0 20 267 167 E3 211 20 0 271 244 160 120 E4 332 284 296 0 129 80 E5 15 240 200 163 0 Experience Levels E2 E3 0 216 234 E1 E2 E3 0 19 34 225 E1 30 E5 E1 54 280 Experience Levels Experience Levels E1 66 45 23 0 200 E1 175 Experience Levels E2 Experience Levels E1 150 E2 204 0 162 125 100 75 E3 156 222 50 0 42 36 E2 20 0 57 18 E3 55 34 0 12 6 0 Experience Levels 30 24 25 40 48 0 Experience Levels 0 Experience Levels (b) Our model: Facet preference divergence with experience. E3 E4 E5 0 193 209 154 201 200 E1 E1 E2 E3 E4 E5 0 571 563 1242 1440 197 0 71 100 162 150 125 E3 E4 213 71 0 77 140 100 75 156 98 76 0 102 50 E5 203 161 138 102 Experience Levels 0 25 0 RateBeer Experience Levels Experience Levels E2 1350 E2 575 0 78 303 390 1050 900 E3 593 105 0 333 362 750 600 E4 1264 326 329 0 213 E1 E2 E3 0 72 96 450 E1 1441 393 339 194 Experience Levels Amazon Movies 0 E2 E3 0 32 210 200 70 60 E2 72 0 50 76 40 30 E3 96 76 150 E1 80 20 0 0 Experience Levels 175 150 125 E2 33 0 132 E3 209 132 0 100 75 50 10 300 E5 E1 90 1200 175 Experience Levels E2 Experience Levels E1 E1 Experience Levels 25 0 0 Yelp NewsTrust (c) Our model: Language model divergence with experience. Fig. 6: Facet preference and language model KL divergence with experience. A general (continuous) version of this work is presented in [19] with fine-grained temporal evolution of user experience, and resulting language model using Geometric Brownian Motion and Brownian Motion, respectively. VIII. C ONCLUSION Current recommender systems do not consider user experience when generating recommendations. In this paper, we have proposed an experience-aware recommendation model that can adapt to the changing preferences and maturity of users in a community. We model the personal evolution of a user in rating items that she will appreciate at her current maturity level. We exploit the coupling between the facet preferences of a user, her experience, writing style in reviews, and rating behavior to capture the user’s temporal evolution. Our model is the first work that considers the progression of user experience as expressed in the text of item reviews. Our experiments – with data from domains like beer, movies, food, and news – demonstrate that our model substantially reduces the mean squared error for predicted ratings, compared to the state-of-the-art baselines. This shows our method can generate better recommendations than those models. 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Learning Invariants using Decision Trees Siddharth Krishna, Christian Puhrsch, and Thomas Wies arXiv:1501.04725v1 [cs.PL] 20 Jan 2015 New York University, USA Abstract. The problem of inferring an inductive invariant for verifying program safety can be formulated in terms of binary classification. This is a standard problem in machine learning: given a sample of good and bad points, one is asked to find a classifier that generalizes from the sample and separates the two sets. Here, the good points are the reachable states of the program, and the bad points are those that reach a safety property violation. Thus, a learned classifier is a candidate invariant. In this paper, we propose a new algorithm that uses decision trees to learn candidate invariants in the form of arbitrary Boolean combinations of numerical inequalities. We have used our algorithm to verify C programs taken from the literature. The algorithm is able to infer safe invariants for a range of challenging benchmarks and compares favorably to other ML-based invariant inference techniques. In particular, it scales well to large sample sets. 1 Introduction Finding inductive invariants is a fundamental problem in program verification. Many static analysis techniques have been proposed to infer invariants automatically. However, it is often difficult to scale those techniques to large programs without compromising on precision at the risk of introducing false alarms. Some techniques, such as abstract interpretation [9], are effective at striking a good balance between scalability and precision by allowing the analysis to be fine-tuned for a specific class of programs and properties. This fine-tuning requires careful engineering of the analysis [11]. Instead of manually adapting the analysis to work well across many similar programs, refinement-based techniques adapt the analysis automatically to the given program and property at hand [8]. A promising approach to achieve this automatic adaptation is to exploit synergies between static analysis and testing [15,17,34]. Particularly interesting is the use of Machine Learning (ML) to infer likely invariants from test data [14,31,32]. In this paper, we present a new algorithm of this type that learns arbitrary Boolean combinations of numerical inequalities. In most ML problems, one is given a small number of sample points labeled by an unknown function. The task is then to learn a classifier that performs well on unseen points, and is thus a good approximation to the underlying function. Binary classification is a specific instance of this problem. Here, the sample data is partitioned into good and bad points and the goal is to learn a predicate that separates the two sets. Invariant inference can be viewed as a binary classification problem [31]. If the purpose of the invariant is to prove a safety property, then the good points are the forward-reachable safe states of the program and the bad points are the backward-reachable unsafe states. These two sets are sampled using program testing. The learned classifier then represents 2 Siddharth Krishna, Christian Puhrsch, and Thomas Wies a candidate invariant, which is proved safe using a static analysis or theorem prover. If the classifier is not a safe invariant, the failed proof yields a spurious counterexample trace and, in turn, new test data to improve the classifier in a refinement loop. Our new algorithm is an instance of this ML-based refinement scheme, where candidate invariants are inferred using a decision tree learner. In this context, a decision tree (DT) is a binary tree in which each inner node is labeled by a function f from points to reals, called a feature, and a real-valued threshold t. Each leaf of the tree is labeled with either “good” or “bad”. Such a tree encodes a predicate on points that takes the form of a Boolean combination of inequalities, f pxq ď t, between features and thresholds. Given sets of features and sample points, a DT learner computes a DT that is consistent with the samples. In our algorithm, we project the program states onto the numerical program variables yielding points in a d-dimensional space. The features describe distances from hyperplanes in this space. The DT learner thus infers candidate invariants in the form of arbitrary finite unions of polyhedra. However, the approach also easily generalizes to features that describe nonlinear functions. Our theoretical contribution is a probabilistic completeness guarantee. More precisely, using the Probably Approximately Correct model for learning [33], we provide a bound on the sample size that ensures that our algorithm successfully learns a safe inductive invariant with a given probability. We have implemented our algorithm for specific classes of features that we automatically derive from the input program. In particular, inspired by the octagon abstract domain [22], we use as features the set of all hyperplane slopes of the form ˘xi ˘ xj , where 1 ď i ă j ď d. We compared our implementation to other invariant generation tools on benchmarks taken from the literature. Our evaluation indicates that our approach works well for a range of benchmarks that are challenging for other tools. Moreover, we observed that DT learners often produce simpler invariants and scale better to large sample sets compared to other ML-based invariant inference techniques such as [14, 30–32]. 2 Overview In this section, we discuss an illustrative example and walk through the steps taken in our algorithm to compute invariants. To this end, consider the program in Fig. 1. Our goal is to find an inductive invariant for the loop on line 4 that is sufficiently strong to prove that the assertion x ‰ 0 on line 12 is always satisfied. Good and Bad States. We restrict ourselves to programs over integer variables x “ px1 , . . . , xd q without procedures. Then a state is a point in Zd that corresponds to some assignment to each of the variables. For simplicity, we assume that our example program has a single control location corresponding to the head of the loop. That is, its states are pairs pv1 , v2 q where v1 is the value of x and v2 the value of y. When our program begins execution, the initial state could be p0, 1q or p0, ´3q, but it cannot be p2, 3q or p0, 0q because of the precondition specified by the assume statement in the program. A good state is defined as any state that the program could conceivably reach when it is started from a state consistent with the precondition. If we start execution at Learning Invariants using Decision Trees 1 2 var x, y: Int ; assume x “ 0 ^ y ‰ 0; 3 4 3 5 6 7 8 9 10 while (y ‰ 0) { if (y ă 0) { x := x ´ 1; y := y + 1; } else { x := x + 1; y := y ´ 1; } } 2 y 4 0 2 4 11 12 assert x ‰ 0; 4 2 0 x 2 4 Fig. 1: Example program. The right side shows some of the good and bad states of the program, in blue and red respectively p0, ´3q, then the states we reach are tp´1, ´2q, p´2, ´1q, p´3, 0qu and thus these are all good states. Similarly, bad states are defined to be the states such that if the program execution was to be started at that point (if we ran the program from the loop head, with those values) then the loop will exit after a finite point and the assertion will fail. For example, p0, 0q is a bad state, as the loop will not run, and we directly go to the assertion and fail it. Similarly, p´2, ´2q is a bad state, as after one iteration of the loop the state becomes p´1, ´1q, and after another iteration we reach p0, 0q, which fails the assertion. The right-hand side of Fig. 1 shows some of the good and bad states of the program. A safe inductive invariant can be expressed in terms of a disjunction of the indicated hyperplanes, which separate the good from the bad states. Our algorithm automatically finds such an invariant. Overview. The high-level overview of our approach is as follows: good and bad states are sampled by running the program on different initial states. Next, a numeric abstract domain that is likely to contain the invariant is chosen manually. We use disjunctions of octagons by default. For each hyperplane (up to translation) in the domain we add a new “feature” to each of the sample points corresponding to the distance to that hyperplane. A Decision Tree (DT) learning algorithm is then used to learn a DT that can separate the good and bad states in the sample, and this tree is converted into a formula that is a candidate invariant. Finally, this candidate invariant is passed to a theorem prover that verifies the correctness of the invariant. We now discuss these steps in detail. Sampling. The first step in our algorithm is to sample good and bad states of this program. We sample the good states by picking states satisfying the precondition, running the program from these states and collecting all states reached. To sample bad states, we look at all points close to good states, run the program from these. If the loop exits within a bounded number of iterations and fails the assert, we mark all states reached as bad states. The sampled good and bad states are shown in Fig. 1. 4 Siddharth Krishna, Christian Puhrsch, and Thomas Wies Features. The next step is to choose a candidate hyperplane set for the inequalities in the invariant. For most of our benchmarks, we used the octagon abstract domain, which consists of all linear inequalities of the form: bi ¨ xi ` bj ¨ xj ď c where 1 ď i ă j ď d, bi , bj P t´1, 0, 1u, and c P Z. We then let H “ tw1 , w2 , . . . u be the set of hyperplane slopes for this domain. Then we transform our sample points (both good and bad) according to these slopes. For each sample point x, we get a new point z given by zi “ x ¨ wi . In our example, the octagon slopes H and some of our transformed good and bad points are fi » fi » 0 » fi 1 0 —0 1ffi , H“– 1 ´1fl 1 1 and 0 1 1 ´1 1 — 1 0 1 1ffi — 1 0ffi ffi — —´1 0ffi ffi „ —  —´1 0 ´1 ´1ffi ffi — ffi — ... . . . ffi 1 0 1 1 ffi — X ¨ HT “ — — 0 0ffi ¨ 0 1 ´1 1 “ — 0 0 0 0ffi . ffi — ffi — — 2 ´2 4 0ffi — 2 ´2ffi fl – fl – ´1 1 ... ´1 1 ´2 ... 0 Learning the DT. After this transformation, we run a Decision Tree learning algorithm on the processed data. A DT (see Fig. 2 for an example) is a concise way to represent a set of rules as a binary tree. Each inner node is labeled by a feature and a threshold. Given a sample point and its features, we evaluate the DT by starting at the root and taking the path given by the rules: if the features is less than or equal to the threshold, we go to the left child, otherwise the right. Leaves specify the output label on that point. Most DT learning algorithms start at an empty tree, and greedily pick the best feature to split on at each node. From the good and bad states listed above, we can easily see that a good feature to split on must be the last one, as all bad states have the last column 0. Indeed, the first split made by the DT is to split on z4 at ´0.5. Since w4 “ p1, 1q, this split corresponds to the linear inequality x ` y ď ´0.5. Now, half the good states are represented in the left child of the root (corresponding to z4 ď ´0.5). The right child contains all the bad states and the other half of the good states. So the algorithm leaves the left child as is and tries to find the best split for the right child. Again, we see the same pattern with z4 , and so the algorithm picks the split z4 ă 1. Now, all bad states fall into the left child, and all good states fall into the right child, and we are done. The computed DT is shown in Fig. 2. Finally, we need to convert the DT back into a formula. To do this, we can follow all paths from the root that lead to good leaves, and take the conjunction of all inequalities on the path, and finally take the disjunction of all such paths. In our example, we get the candidate invariant: px ` y ď ´0.5q _ px ` y ą ´0.5 ^ x ` y ą 0.5q . This can be simplified to px ` y ‰ 0q. Verifying the Candidate Invariant. Our program is then annotated with this invariant and passed to a theorem prover to verify that the invariant is indeed sufficient to prove the program correct. Learning Invariants using Decision Trees 5 FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE Good TRUE TRUE Good Good Good Good FALSE TRUE TRUE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE Bad TRUE Bad Bad Bad Bad Bad Good Good FALSE FALSE Good Good Good Good Good Good Good Fig. 2: Decision tree learned from the program and sample data in Fig. 1 3 Preliminaries Problem Statement. We think of programs as transition systems P “ pS, R, init, safeq, where S is a set of states, R Ď S ˆ S is a transition relation on the states, init Ď S is a set of initial states, and safe Ď S is the set of safe states that we want our program to remain within. For any set of states X, the set postpXq represents all the successor states with respect to R. More formally, postpXq “ t x1 \| Dx P X. Rpx, x1 q u . Then, we can define the set of reachable states of the program (the good states) to be the least fixed point, good “ lfppλX. init Y postpXqq . Similarly, we can define prepXq “ t x \| Dx1 P X. Rpx, x1 q u and bad “ lfppλX. error Y prepXqq, where error is the complement of safe. Then we see that the program is correct, with respect to the safety property given by safe, if good X bad “ H. Thus, our task is to separate the good states from the bad states. One method to show that these sets are disjoint is to show the existence of a safe inductive invariant. A safe inductive invariant is a set inv that satisfies the following three properties: – init Ď inv, – postpinvq Ď inv, – inv Ď safe. It is easy to see that these conditions imply, in particular, that inv separates the good and bad states: good Ď inv and inv X bad “ H. 6 Siddharth Krishna, Christian Puhrsch, and Thomas Wies Invariant Generation as Binary Classification. The set up for most machine learning problems is as follows. We have an input space X and an output space Y, and one is given a set of samples X Ď X that are labeled by some unknown function f : X Ñ Y. We fix a hypothesis set H Ď Y X , and the aim is to find the hypothesis h P H that most closely approximates f . The samples are often given in terms of feature vectors, and thus a sample x can be thought of as a point in some d-dimensional space. Binary classification is a common instance of this problem, where the labels are restricted to a binary set Y “ t0, 1u. Following [31], we view the problem of computing a safe inductive invariant for a program P “ pS, R, init, safeq as a binary classification problem by defining the input space as the set of all program states X “ S. Sample states x P bad are labeled by 0 and x P good are labeled by 1. Thus, the unknown function f is the characteristic function of the set of good states. The hypothesis to be learned is a safe inductive invariant. Note that if sampling the program shows that some state x is both good and bad, there exists no safe inductive invariant, and the program is shown to be unsafe. Hence, we assume that the set of sample states can be partitioned into good and bad states. Decision Trees. Instead of considering a hypothesis space that can represent arbitrary invariants, we restrict it to a specific abstract domain, namely those invariants that can be represented using decision trees. A Decision Tree (DT) [23] is a binary tree that represents a Boolean function. Each inner node v of T is labeled by a decision of the form xi ď t, where xi is one of the input features and t is a real valued threshold. We denote this inequality by v.cond. We denote the left and right children of an inner node by v.left and v.right respectively. Each leaf v is labeled by an output v.label. To evaluate an input, we trace a path down from the root node T.root of the tree, going left at each inner node if the decision is true, and right otherwise. The output of the tree on this input is the label of the leaf reached by this process. The hypothesis set corresponding to all DTs is thus arbitrary Boolean combinations of linear inequalities of the form xi ď t (axis-aligned hyperplanes). As one can easily see, many DTs can represent the same underlying function. However, the task of finding the smallest (in terms of number of nodes) DT for a particular function can be shown to be NP-complete [19]. Standard algorithms to learn DTs work by greedily selecting at each node the co-ordinate and threshold that separates the remaining training data best [7,27]. This procedure is followed recursively until all leaves have samples labeled by a single class. The criterion for separation is normally a measure such as conditional entropy. Entropy is a commonly used measure of uncertainty in a system. It is a function that is low when the system is homogeneous (in this case, think of when all samples reaching a node have the same label), and high otherwise. Conditional entropy, analogously, measures how homogeneous the samples are after choosing a particular co-ordinate and threshold. More formally, at each node, we look at the samples that reach that node, and define the conditional entropy of splitting feature xi at threshold t as Hpy\|xi : tq “ ppxi ă tqHpy\|xi ă tq ` ppxi ě tqHpy\|xi ě tq, ÿ where Hpy\|xi ă tq “ ´ ppy “ a\|xi ă tq log ppy “ a\|Xi ă tq, aPY Learning Invariants using Decision Trees 7 Algorithm 1: DTInv: Invariant generation algorithm using DT learning def DTInv (P : program): safe inductive invariant for P or fail “ val X, y “ Sampler(P) val H “ Slopes(P) val Z “ X ¨ H T val T “ LearnDT(Z, y) val ϕ “ DTtoForm(T.root) if IsInvariant(P , ϕ) then ϕ else fail def DTtoForm (v: node of a decision tree): formula represented by subtree rooted at v “ if v is a leaf then v.label else pv.cond ^ DTtoForm(v.left) _ v.cond ^ DTtoForm(v.right)q and ppAq is the empirical probability, i.e., the fraction of sample points reaching this node that satisfy the condition A. Hpy\|xi ě tq is defined similarly to Hpy\|xi ă tq. Note that if a particular split perfectly separates good and bad samples, then the conditional entropy is 0. The greedy heuristic is to pick the feature and split that minimize the conditional entropy. There are other measures as well, such as the Gini index, which is used by the DT learner we used in our experiments [25]. 4 Algorithm We now present our DT-learning algorithm. We assume that we have black box procedures for sampling points from the given program, and for getting a set of slopes from a chosen abstract domain. To this end, let Sampler be a procedure that takes a program and returns an n ˆ d matrix X of n sample points, and an n-dimensional vector y corresponding to the label of each sample point (1 for good states, 0 otherwise). Similarly, let Slopes be a procedure that takes a program and returns an m ˆ d matrix H of m hyperplane slopes. We describe the actual procedures used in our experiments in Section 5.1. Our final algorithm is surprisingly simple, and is given in Algorithm 1. We get the sample points and the slopes from the helper functions mentioned above, and then transform the sample points according to the slopes given. We run a standard DT learning algorithm on the transformed sample to obtain a tree that classifies all samples correctly. The tree is then transformed into a formula that is a candidate invariant, by a simple procedure DTtoForm. Finally, the program is annotated with the candidate invariant and verified. This final step is realized by another black box procedure IsInvariant, which checks that the invariant satisfies the three conditions necessary to be a safe inductive invariant. For example, this can be done by encoding init, inv, post and safe as SMT formulas and feeding the three conditions into an SMT solver. To convert the DT into a formula, we note that the set of states that reach a particular leaf is given by the conjunction of all predicates on the path from the root to that leaf. Thus, the set of all states classified as good by the DT is the disjunction of the sets of states that reach all the good leaves. A simple conversion is then to take the disjunction 8 Siddharth Krishna, Christian Puhrsch, and Thomas Wies over all paths to good leaves of the conjunction of all predicates on such paths. The procedure DTtoForm computes this formula recursively by traversing the learned DT. Since the tree was learnt on the transformed sample data, the predicate at each node of the tree will be of the form zi ď c where zi is one of the columns of Z and c is a constant. Since Z “ X ¨ H T , we know that zi pxq “ x ¨ wi for a sample x. Thus, the predicate is equivalent to x ¨ wi ď c, which is a linear inequality in the program variables. Combining this with the conversion procedure above, we see that our algorithm outputs an invariant which is a Boolean combination of linear inequalities over the numerical program variables. Soundness. From the above discussion, we see that the formula ϕ returned by the procedure DTtoForm is of the required format. Moreover, we assume that the procedure IsInvariant is correct. Thus, we can say that our invariant generation procedure DTInv is sound: if it terminates successfully, it returns a safe inductive invariant. Probabilistic Completeness. It is harder to prove that such invariant generation algorithms are complete. We see that the performance of our algorithm depends heavily on the sample set, if the sample is inadequate, it is impossible for the DT learner to learn the underlying invariant. One could augment our algorithm with a refinement loop, which would make the role of the sampler less pronounced, for if the invariant is incorrect, the theorem prover will return a counterexample that could potentially be added to the sample set and we can re-run learning. However, we find in practice that we do not need a refinement loop if our sample set is large enough. We can justify this observation using Valiant’s PAC (probably approximately correct) model [33]. In this model, one can prove that an algorithm that classifies a large enough sample of data correctly has small error on all data, with high probability. It must be noted that one key assumption of this model is that the sample data is drawn from the same distribution as the underlying data, an assumption that is hard to justify in most of its applications, including this one. In practice however, PAC learning algorithms are empirically successful on a variety of applications where the assumption on distribution is not clearly true. Formally, we can give a generalization guarantee for our algorithm using this result of Blumer et al. [6]: Theorem 1. A learning algorithm for a hypothesis class H that outputs a hypothesis h P H will have true error at most with probability at least 1 ´ δ if h is consistent log 13 with a sample of size maxp 4 log 2δ , 8V CpHq q. In the above theorem, a hypothesis is said to be consistent with a sample if it classifies all points in the sample correctly. The quantity V CpHq is a property of the hypothesis class called the Vapnik-Chervonenkis (VC) dimension, and is a measure of the expressiveness of the hypothesis class. As one might imagine, more complex classes lead to a looser bound on the error, as they are more likely to over-fit the sample and less likely to generalize well. Thus, it suffices for us to bound the VC dimension of our hypothesis class, which is all finite Boolean combinations of hyperplanes in d dimensions. The VC dimension of a class H is defined as the cardinality of the largest set of points that H can shatter. A set of points is said to be shattered by a hypothesis class H if for every possible labeling of the points, there exists a hypothesis in H that is consistent with it. Unfortunately, Learning Invariants using Decision Trees 9 DTs in all their generality can shatter points of arbitrarily high cardinality. Given any set of m points, we can construct a DT with m leaves such that each point ends up at a different leaf, and now we can label the leaf to match the labeling given. Since in practice we will not be learning arbitrarily large trees, we can restrict our algorithm a-priori to stop growing the tree when it reaches K nodes, for some fixed K independent of the sample. Now one can use a basic, well-known lemma from [24] combined with Sauer’s Lemma [29] to get that the VC dimension of bounded decision trees is OpKd log Kq. Combining this with Theorem 1, we get the following polynomial bound for probabilistic completeness: Theorem 2. Under the assumptions of the PAC model, the algorithm DTInv returns an invariant that has true error at most with probability at least 1 ´ δ, given that its sample size is Op 1 Kd log K log 1δ q . Complexity. The running time of our algorithm depends on many factors such as the running time of the sampling (which in turn depends on the benchmark being considered), and so is hard to measure precisely. However, the running time of the learning routine for DTs is Opmn logpnqq, where m is the number of hyperplane slopes in H and n is the number of sample points [7, 25]. The learning algorithm therefore scales well to large sets of sample data. Nonlinear invariants. An important property of our algorithm is that it generalizes elegantly to nonlinear invariants as well. For example, if a particular program requires invariants that reason about px mod 2q for some variable x, then we can learn such invariants as follows: given the sampled states X, we add to it a new column that corresponds to a variable x1 , such that x1 “ x mod 2. We then run the rest of our algorithm as before, but in the final invariant, replace all occurrences of x1 by px mod 2q. As is easy to see, this procedure correctly learns the required nonlinear invariant. We have added this feature to our implementation, and show that it works on benchmarks requiring these nonlinear features (see Section 5.2). 5 5.1 Implementation and Evaluation Implementation We implemented our algorithm in Python, using the scikit-learn library’s decision tree classifier [25] as the DT learner LearnDT. This implementation uses the CART algorithm from [7] which learns in a greedy manner as described in Section 3, and uses the Gini index. We implemented a simple and naive Sampler: we considered all states that satisfied the precondition where the value of every variable was in the interval r´L, Ls. For these states, we ran the program with a bound I on the number of iterations of loops, and collected all states reached as good states. To find bad states, we looked at all states that were a margin M away from every good state, ran the program from this state (again with at most I iterations), and if the program failed an assertion, we collected all the states on this path as bad states. The bounds L, I, M were initialized to low values and increased until we had sampled enough states to prove our program correct. 10 Siddharth Krishna, Christian Puhrsch, and Thomas Wies For the Slopes function, we found that for most of the programs in our benchmarks it sufficed to consider slopes in the octagonal abstract domain. This consists of all vectors in t´1, 0, 1ud with at most two non zero elements. In a few cases, we needed additional slopes (see Table 1). For the programs hola15 and hola34, we used a class of slopes of vectors in C d where C is a small set of constants that appear in the program, and their negations. We also developed methods to learn the class of slopes needed by a program by looking at the states sampled. In [31], the authors suggest working in the null space of the good states, viewed as a matrix. This is because if the good states lie in some lower dimensional space, this would automatically suggest equality relationships among them that can be used in the invariant. It also reduces the running time of the learning algorithm. Inspired by this, we propose using Principal Component Analysis [20] on the good states to generate slopes. PCA is a method to find the basis of a set of points so as to maximize the variance of the points with respect to the basis vectors. For example, if all the good points lie along the line 2x ` 3y “ 4, then the first PCA vector will be p2, 3q, and intuition suggests that the invariant will use inequalities of the form 2x ` 3y ă c. Finally, for the IsInvariant routine, we used the program verifier Boogie [4], which allowed us to annotate our programs with the invariants we verified. Boogie uses the SMT solver Z3 [12] as a back-end. 5.2 Evaluation We compared our algorithm with a variety of other invariant inference tools and static analyzers. We mainly focused on ML-based algorithms, but also considered tools based on interpolation and abstract interpretation. Specifically, we considered: – ICE [14]: an ML algorithm based on ICE-learning that uses an SMT solver to learn numerical invariants. – MCMC [30]: an ML algorithm based on Markov Chain Monte Carlo methods. There are two versions of this algorithm, one that uses templates such as octagons for the invariant, and one with all constants in the slopes picked from a fixed bag of constants. The two algorithms have very similar characteristics. We ran both versions 5 times each (as they are randomized) and report the better average result of the two algorithms for each benchmark. – SC [31]: an ML algorithm based on set cover. We only had access to the learning algorithm proposed in [31] and not the sampling procedure. To obtain a meaningful comparison, we combined it with the same sampler that we used in the implementation of our algorithm DTInv. – CPAchecker [5]: a configurable software model checker. We chose the default analysis based on predicate abstraction and interpolation. – UFO [2]: a software model checker that combines abstract interpretation and interpolation (denoted CPA). – InvGen [18]: an inference tool for linear invariants that combines abstract interpretation, constraint solving, and testing. Learning Invariants using Decision Trees 11 For our comparison, we chose a combination of 22 challenging benchmarks from various sources. In particular, we considered a subset of the benchmarks from [13, 14, 18, 31]. We chose the benchmarks at random among those that were hard for at least one other tool to solve. Due to this bias in the selection, our experimental results do not reflect the average performance of the tools that we compare against. Instead, the comparison should be considered as an indication that our approach provides a valuable complementary technique to existing algorithms. We ran our experiments on a machine with a quad-core 3.40GHz CPU and 16GB RAM, running Ubuntu GNU/Linux. For the analysis of each benchmark, we used a memory limit of 8GB and a timeout of 5 minutes. The results of the experiment are summarized in Table 1. Here are the observations from our experiments (we provide more in-depth explanations for these observations in the next section, where we discuss related work in more detail): – Our algorithm DTInv seems to learn complex Boolean invariants as easily as simple conjunctions. – ICE seems to struggle on programs that needed large invariants. For some of these (gopan, popl07), this is because the constraint solver runs out of time/memory, as the constants used in the invariants are also large. For hola19 and prog4, ICE stops because the tool has an inbuilt limit for the complexity of Boolean templates. – However, ICE solves sum1 and trex3 quickly, even though they need many predicates, because the constant terms are small, so this space is searched first by ICE. – Similarly, we notice that MCMC has difficulty finding large invariants, again because the search space is huge. – SC’s learning algorithm is consistently slower than DTInv, due to its higher running time complexity. It also runs out of memory for large sample sizes. – DTInv is able to easily handle benchmarks that CPA, UFO and InvGen struggle on. This is mainly because they are specialized for reasoning about linear invariants, and have issues dealing with invariants that have complicated Boolean structure. We also learned some of the weak points in our current approach: – DTInv is slow in processing fig1 and prog4, both of which are handled by at least one other tool without much effort. However, we note that most of this time is spent in the sampling routine, which is currently a naive implementation. We therefore believe that DT learning could benefit from a combination with a static analysis that provides approximations of the good and bad states to guide the sampler. – We also note that the method of “constant slopes” which we used to handle the non octagonal benchmarks (hola15, hola34) is ad-hoc, and might not work well for larger benchmarks. Beyond octagons. As mentioned in Section 4, we implemented a feature to learn certain nonlinear invariants. We were able to verify some benchmarks that needed reasoning about the modulus of certain variables, as shown in Table 1. Finally, we show one example where we were able to infer a nonlinear invariant (specifically s “ i2 ^ i ď n for square). 12 Siddharth Krishna, Christian Puhrsch, and Thomas Wies Name Vars Type Octagonal: ex23 [14] 4 conj. fig6 [14] 2 conj. fig9 [14] 2 conj. hola10 [13] 4 conj. nested2 [31] 4 conj. nested5 [18] 4 conj. fig1 [14] 2 disj. test1 [31] 4 disj. cegar2 [14] 3 ABC gopan [31] 2 ABC hola18 [13] 3 ABC hola19 [13] 4 ABC popl07 [31] 2 ABC prog4 [31] 3 ABC sum1 [14] 3 ABC trex3 [14] 8 ABC Non octagonal: hola15 [13] 3 conj. Modulus: hola02 [13] 4 conj. hola06 [13] 4 conj. hola22 [13] 4 conj. Non octagonal modulus: hola34 [13] 4 ABC Quadratic: square 3 conj. \|ϕ\| Samp DT SC DTInv ICE MCMC 3 2 2 8 3 4 3 2 5 8 6 7 7 8 6 9 0.10 0.00 0.00 0.03 0.03 2.48 14.61 0.90 0.03 0.03 1.60 0.19 0.03 2.32 0.01 8.44 0.01 0.00 0.01 0.00 0.00 0.02 0.01 0.01 0.01 0.00 0.04 0.01 0.00 0.02 0.01 0.06 4.73 2.61 2.31 2.45 2.39 MO F 7.86 2.64 2.54 MO 3.47 2.72 MO 2.61 MO 0.11 0.00 0.01 0.03 0.03 2.50 14.62 0.91 0.04 0.03 1.64 0.20 0.03 2.34 0.02 8.50 CPA UFO InvGen 8.82 0.01 19.77 1.50 0.30 0.00 1.68 0.13 0.33 0.00 1.73 0.13 49.21 TO 2.03 F 62.02 0.09 1.86 0.12 60.95 31.28* 2.08 0.35 0.38 5.13 1.75 1.64 0.39 TO 1.71 F 4.86 17.30 1.97 0.18 F TO 63.85 58.29 TO 21.93* TO 8.38 F TO F F F TO 110.81 15.20 F 0.13 F F 1.32 29.04* F 0.17 4.51 NA F 0.18 0.02 0.01 0.01 F 0.03 0.03 F 0.04 F F F F F F F F 2 0.52 0.02 0.01 0.54 0.53 4 3 5 0.03 0.03 F 1.79 0.03 MO 0.04 0.00 F 0.06 1.82 0.04 F F F NA NA NA F F F F F F F F F 6 1.14 0.03 3.10 1.17 F NA F F F 3 0.27 0.24 MO 0.51 F NA F F F 0.04 MO 0.13 0.02 Table 1: Results of comparison. The table is divided according to what kinds of invariants the benchmark needed. The “Type” column denotes the Boolean structure of the invariant - conjunctive, disjunctive and arbitrary Boolean combination are denoted as conj., disj. and ABC respectively. The column \|ϕ\| contains the number of predicates in the invariant found by DTInv. The columns ’Samp’ and ’DT’ show the running time in seconds of our sampling and DT learning procedures respectively. We show SC next, as we only compare learning times with SC. Then follows the total time of our tool (DTInv), followed by those for other tools. Each entry of the tool columns shows the running time in seconds if a safe invariant was found, or otherwise one of the following entries. ’NA’: program contains arithmetic operations that are not supported by the tool; ’F’: analysis terminated without finding a safe invariant; ’TO’: timeout; ’MO’: out of memory. The times for MCMC have an asterisk if at least one of the repetitions timed out. In this case the number shown is the average of the other runs. Learning Invariants using Decision Trees 13 We believe our experiments show that Decision Trees are a natural representation for invariants, and that the greedy learning heuristics guide the algorithm to discover simple invariants of complex structures without additional overhead. 6 Related Work and Conclusions Our experimental evaluation compared against other algorithms for invariant generation. We discuss these algorithms in more detail. Sharma et. al. [31] used the greedy set cover algorithm SC to learn invariants in the form of arbitrary Boolean combinations of linear inequalities. Our algorithm based on decision trees is simpler than the set cover algorithm, works better on our benchmarks (which includes most of the benchmarks from [31]), and scales much better to large sample sets of test data. The improved scalability is due to the better complexity of DT learners. The running time of our learning algorithm is Opmn logpnqq where m is the number of features/hyperplane slopes that we consider, and n the number of sample points. On the other hand, the set cover algorithm has a running time of Opmn3 q. This is because the greedy algorithm for set cover takes time Ophn2 q where h is the number of hyperplanes, and [31] considers one hyperplane for every candidate slope and sample point, yielding h “ mn. When invariant generation is viewed as binary classification, then there is a problem in the refinement loop: if the learned invariant is not inductive, it is unclear whether the counterexample model produced by the theorem prover should be considered a “bad” or a “good” state. The ICE-learning framework [14] solves this problem by formulating invariant generation as a more general classification problem that also accounts for implication constraints between points. We note that our algorithm does not fit within this framework, as we do not have a refinement loop that can handle counterexamples in the form of implications. However, we found in our experiments that we did not need any refinement loop as our algorithm was able to infer correct invariants directly after sampling enough data. Nevertheless, considering an ICE version of DT learning is interesting as sampling without a refinement loop becomes difficult for more complex programs. The paper [14] also proposes a concrete algorithm for inferring linear invariants that fits into the ICE-learning framework (referred to as ICE in our evaluation). If we compare the complexity of learning given a fixed sample, our algorithm performs better than [14] both in terms of running time and expressiveness of the invariant. The ICE algorithm of [14] iterates through templates for the invariant. This iteration is done by dovetailing between more complex Boolean structures and increasing the range of the thresholds used. For a fixed template, it formulates the problem of this template being consistent with all given samples as a constraint in quantifier-free linear integer arithmetic. Satisfiability of this constraint is then checked using an SMT solver. We note that the size of the generated constraint is linear in the sample size, and that solving such constraints is NP-complete. In comparison, our learning is sub-quadratic time in the sample size. Also, we do not need to fix templates for the Boolean structure of the invariant or bound the thresholds a priori. Instead, the DT learner automatically infers those parameters from the sample data. 14 Siddharth Krishna, Christian Puhrsch, and Thomas Wies Another ICE-learning algorithm based on randomized search was proposed in [30] (the algorithm MCMC in our evaluation). This algorithm searches over a fixed space of invariants S that is chosen in advance either by bounding the Boolean structure and coefficients of inequalities, or by picking some finite sub-lattice of an abstract domain. Given a sample, it randomly searches using a combination of random walks and hill climbing until it finds a candidate invariant that satisfies all the samples. There is no obvious bound on the time of this search other than the trivial bound of \|S\|. Again, we have the advantage that we do not have to provide templates of the Boolean structure and the thresholds of the hyperplanes. These parameters have to be fixed for the algorithm in [30]. Furthermore, the greedy nature of DT learning is a heuristic to try simpler invariants before more complex ones, and hence the invariants we find for these benchmarks are often much simpler than those found by MCMC. Decision trees have been previously used for inferring likely preconditions of procedures [28]. Although this problem is related to invariant generation, there are considerable technical differences to our algorithm. In particular, the algorithm proposed in [28] only learns formulas that fall into a finite abstract domain (Boolean combinations of a given finite set of predicates), whereas we use decision trees to learn more general formulas in an infinite abstract domain (e.g., unions of octagons). We believe that the main value of our algorithm is its ability to infer invariants with a complex Boolean structure efficiently from test data. Other techniques for inferring such invariants include predicate abstraction [16] as well as abstract interpretation techniques such as disjunctive completion [10]. However, for efficiency reasons, many static analyses are restricted to inferring conjunctive invariants in practice [3, 11]. There exist techniques for recovering loss of precision due to imprecise joins using counterexample-guided refinement [1, 21, 26]. 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Belief Propagation Min-Sum Algorithm for Generalized Min-Cost Network Flow arXiv:1710.07600v1 [stat.ML] 20 Oct 2017 Andrii Riazanov1, Yury Maximov2 and Michael Chertkov3 Abstract— Belief Propagation algorithms are instruments used broadly to solve graphical model optimization and statistical inference problems. In the general case of a loopy Graphical Model, Belief Propagation is a heuristic which is quite successful in practice, even though its empirical success, typically, lacks theoretical guarantees. This paper extends the short list of special cases where correctness and/or convergence of a Belief Propagation algorithm is proven. We generalize formulation of Min-Sum Network Flow problem by relaxing the flow conservation (balance) constraints and then proving that the Belief Propagation algorithm converges to the exact result. I. INTRODUCTION Belief Propagation algorithms were designed to solve optimization and inference problems in graphical models. Since a variety of problems from different fields of science (communication, statistical physics, machine learning, computer vision, signal processing, etc.) can be formulated in the context of graphical model, Belief Propagation algorithms are of great interest for research during the last decade [1], [2]. These algorithms belong to message-passing heuristic, which contains distributive, iterative algorithms with little computation performed per iteration. There are two types of problems in graphical models of the greatest interest: computation of the marginal distribution of a random variable, and finding the assignment which maximizes the likelihood. Sum-product and Min-sum algorithms were designed for solving these two problems under the heuristic of Belief Propagation. Originally, the sum-product algorithm was formulated on trees ([3], [4], [5]), for which this algorithm represents the idea of dynamic programming in the message-passing concept, where variable nodes transmit messages between each other along the edges of the graphical model. However, these algorithms showed surprisingly good performance even when applied to the graphical models of non-tree structure ([6], [7], [8], [9], [10]). Since Belief Propagation algorithms can be naturally *The work was supported by funding from the U.S. Department of Energy’s Office of Electricity as part of the DOE Grid Modernization Initiative. 1 Andrii Riazanov is with the Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA. riazanov@cs.cmu.edu 2 Yury Maximov is with Skolkovo Institute of Science and Technology, Center for Energy Systems, and Los Alamos National Laboratory, Theoretical Division T-4 & CNLS, Los Alamos, NM 87544, USA yury@lanl.gov 3 Michael Chertkov is with Skolkovo Institute of Science and Technology, Center for Energy Systems, and Los Alamos National Laboratory, Theoretical Division T-4, Los Alamos, NM 87544, USA chertkov@lanl.gov implemented and paralleled using the simple idea of message passing, these instruments are widely used for finding approximate solutions to optimization of inference problems. However, despite the good practical performance of this heuristic in many cases, the theoretical grounding of these algorithm remain unexplored (to some extend), so there are no actual proofs that the algorithms give correct (or even approximately correct) answers for the variety of problem statements. That’s why one of important tasks is to explore the scope of the problems, for which these algorithms indeed can be applied, and to justify their practical usage. In [11] the authors proved that the Belief Propagation algorithm give correct answers for Min-Cost Network Flow problem, regardless of the underlying graph being a tree or not. Moreover, the pseudo-polynomial time convergence was proven for these problems, if some additional conditions hold (the uniqueness of the solution and integral input). This work significantly extended the set of problems, for which Belief Propagation algorithms are justified. In this paper we formulate the extension of Min-Cost Network Flow problem, which we address as Generalized Min-Cost Network Flow problem (GMNF). This problem statement is much broader then the original formulation, but we amplify the ideas of [11] to prove that Belief Propagation algorithms also give the correct answers for this generalization of the problem. This extension might find a lot of applications in various fields of study, since GMNF problem is the general problem of linear programming with additional constraints on the cycles of the underlying graphs (more precise, on the coefficient of corresponding vertices), which might be natural for some practical formulations. II. GENERALIZED MIN-COST NETWORK FLOW A. Problem statement Let G = (V, E) be a directed graph, where V is the set of vertices and E is the set of edges, \|V \| = n, \|E\| = m. For any vertex v we denote Ev as the set of edges incident to v, and aev is the coefficient related to this pair (v, e), such that aev > 0 if e is an out-arc with respect to v (e.g. e = (v, w) for some vertex w), and aev < 0 if e is an in-arc with respect to v (e.g. e = (w, v) for some vertex w). For any vertex v and edges e1 , e2 incident to v we ae1 define δ(v, e1 , e2 ) , ev2 . Then we consider the following av property of the graph: Definition 2.1: The graph G is called ratio-balanced if for every non-directed cycle C which consists of vertices {v1 , v2 , v3 , . . . , vk } and edges {e1 , e2 , e3 , . . . , ek } it holds: k Y δ(vi , ei , vi−1 ) = δ(v1 , ek , e1 ) · δ(v2 , e1 , e2 ) · . . . · × i=1 ×δ(vk−1 , ek−2 , ek−1 ) · δ(vk , ek−1 , ek ) = 1. (1) follows: c(C) , c1 + δ(v2 , e1 , e2 )× ! × c2 + δ(v3 , e2 , e3 ) c3 + · · · + δ(vk , ek−1 , ek )ck · · · = = c1 + Here by non-directed cycle we mean that for every pair (vi , vi−1 ) and the pair (v1 , vk ) it holds that either (vi , vi−1 ) ∈ E or (vi−1 , vi ) ∈ E. It is not hard to verify that it suffices for equation (1) to hold only for every simple non-directed cycle of G, since then the equation (1) can be easily deduced to hold for arbitrary non-directed cycle. To check that the given graph is ratio-balances, one then need to check whether (1) holds for any simple cycle. If m is the number of edges, and C is the number of simple cycles of G, one obviously needs at least O(m + C) time to iterate trough all simple cycles. In fact, the optimal algorithm for this task was introduced in [12], which runs for O(m + C) time. Then, to check whether a graph is ratio-balances, one may use this algorithm to iterate through all simple cycles and to check (1) for every one of them. We formulate the Generalized Min-Cost Network Flow problem for ratio-balanced graph G as follows: minimize X ce xe e∈E subject to X aev xe = fv , ∀v ∈ V, (GMNF) e∈Ev 0 ≤ xe ≤ ue , ∀e ∈ E. Here the first set of constrains are balance constraints which must hold for each vertex. The second set of constrains consists of capacity constraints on each edge of G. Coefficients ce and ue , defined for each edge e ∈ E, are called the cost and the capacity of the edge, respectively. Any assignment of x in this problem which satisfies the balance and capacity constraints P is referred as flow. Finally, the objective function g(x) = e∈E ce xe is called the total flow. B. Definitions and properties For the given (GMNF) problem on the graph G and flow x on this graph, the residual network G(x) is defined as follows: G(x) has the same vertex set as G, and for each edge e = (v, w) ∈ E if xe < ue then e is an arc in G(x) with the cost cxe = ce and coefficients (aev )x = aev , (aew )x = aew . Finally, if xe > 0 then there is an arc e′ = (w, v) in G(x) with the cost cxe′ = −ce and coefficients (aev )x = −aev , (aew )x = −aew . It is not hard to see that G(x) is ratiobalanced whenever G is, since only the absolute values of the coefficients aev occur in the definition of this property. Then for each directed cycle C = ({v1 , v2 , v3 , . . . , vk }, {e1 , e2 , e3 , . . . , ek }) we define the cost of this cycle as k X i=2 ci i Y δ(vj , ej−1 , ej ) j=2 It is easy to see that c(C) is properly defined whenever the graph G is ratio-constrained. Then we define σ(x) , min{cx (C)}, where the minimum C is taken over all cycles C in the residual network G(x). Lemma 2.1: If (GMNF) has a unique solution x∗ , then σ(x∗ ) > 0. Proof: We will show that for every directed cycle C from G(x∗ ) we can push the additional flow through the edges of this cycle such that the linear constraints in (GMNF) will still be satisfied, but the total flow in the cycle will change by ε · cx (C) for some ε > 0. Let C = ({v1 , v2 , v3 , . . . , vk }, {e1 , e2 , e3 , . . . , ek }). From the definition of the residual network it follows that we can increase the flow in every edge of the cycle for some positive quantity such that the capacity constraints will still be satisfied. Let’s push additional flow ε > 0 trough e1 . In order to satisfy the balance constraint in v1 , we need to adjust the flow in e2 . We have 4 cases: either one of e1 , e2 can be in G, or their opposites can be in G. If e ∈ C is in G, we will say that it is ’direct’ arc, otherwise it will be ’opposite’. Then there are four cases: 1) e1 , e2 are direct arcs. Then we have the new flow on edge e − 1: y1 = x∗1 + ε. In order to satisfy the balance constraint for the vertex v2 , is must hold aev12 x∗1 + aev22 x∗2 = aev12 y1 + aev22 y2 = aev12 x∗1 + aev12 ε + aev22 y2 ⇒ aev22 (x∗2 − y2 ) = aev12 ε ⇒ y2 = e1 a v2 ∗ x2 − ε e2 . Since both e1 and e2 are direct, it means av2 that e1 = (v1 , v2 ) ∈ E, and e2 = (v2 , v3 ) ∈ E, and thus, by definition, aev12 < 0, and aev22 > 0. Therefore, we have y2 = x∗2 + εδ(v2 , e1 , e2 ). 2) e1 is direct, e2 is opposite. Then again, y1 = x∗1 + ε, but now ’pushing’ the flow through e2 (as the edge in the residual network) means decreasing x2 . The same equalities holds, so ae1 y2 = x∗2 − ε ev22 = x∗2 − εδ(v2 , e1 , e2 ). Since e2 av2 is opposite arc, that means that we should push additional εδ(v2 , e1 , e2 ) through e2 . 3) e1 is opposite, e2 is direct – similar to the case 2). 4) e1 , e2 are opposite arcs – similar to the case 1). So, if we push ε through e1 , we need to push ε2 = εδ(v2 , e1 , e2 ) through e2 to keep the balance in v2 . Then, analogically, to maintain the balance in v3 , we need to push additional ε3 = ε2 δ(v3 , e2 , e3 ) = εδ(v2 , e1 , e2 )δ(v3 , e2 , e3 ) through e3 . Then, consequently adjusting the balance in all the vertexes of C, we will retrieve that to keep the balance Qk in vk , we need to push εk = εk−1 δ(vk , ek−1 , ek ) = ε i=2 δ(vi , ei−1 , ei ) through ek . Now it suffices to show that the balance in v1 is also satisfied. Similarly, we know that if we push εk Q in ek , then we need to push ε1 = εk δ(v1 , ek , e1 ) = ε ki=1 δ(vi , ei−1 , e1 ) = ε (since G is ratio-balanced) through e1 , and that is exactly the amount which we assumed to push at the beginning of this proof. So we indeed push consistent flow through all the edges of C in such a way that the balance constraints in all the vertices is satisfied. We now only need to mention that we can take ε > 0 as small as it is needed to satisfy also all the capacity constraints in the cycle. Now the additional total cost of such adjusting will be k X Proof: l(R) = c1 + δ(v2 , e1 , e2 ) c2 + δ(v3 , e2 , e3 )× ! = × · · · ck−2 + δ(vk−1 , ek−2 , ek−1 )ck−1 · · ·  =  c1 +  + \| cxi εi = cx1 ε + cx2 εδ(v2 , e1 , e2 )+ +cx2 εδ(v2 , e1 , e2 )δ(v3 , e2 , e3 ) + · · · + cxk ε δ(vi , ei−1 , ei ) = × ci i=2 δ(vj , ej−1 , ej ) j=2 We also define the ’reducer’ of the path as follows: t(S) , min j=2,...,k−1 j Y \| m X i=2 p−1 Y j=2 " c′i i Y } # + δ(vj′ , e′j−1 , e′j ) j=2 {z × cp + }   δ(vj′ , e′j−1 , e′j ) ·δ(vp , e′m , ep )× {z k−1 X i=p+1 " } i Y ci " p−1 X i=2 j=p+1 ci i Y j=2  k−1 X i=p+1 " i Y ci # δ(vj , ej−1 , ej )  ≥ # δ(vj , ej−1 , ej )  + δ(vj , ej−1 , ej ) ·  δ(vi , ei−1 , ei ) {z c(C) ≥  c1 + +  (δ(v1′ ,e′m ,e′1 ))−1  i=2 To prove the main result of this paper we will use the following crucial lemma: Lemma 2.2: Let G be any ratio-balanced graph, or a residual network of some ratio-balanced graph (as we already mentioned, the residual network will also be ratio-balanced in this case). Let S = ({v1 , · · · , vk }, {e1, · · · , ek−1 }) ′ be a directed path in G, and C = ({v1′ , v2′ , · · · , vm }, {e′1 , e′2 , · · · , e′m }) be a cycle with v1′ = vp . Let R be ′ , v1′ = the path R = {v1 , v2 , · · · , vp = v1′ , v2′ , . . . , vm vp , vp+1 , . . . , vk }. Then l(R) ≥ l(S) + T c(C), where T = minS t(S) is the minimum of all the reducers among all directed paths S in G. m Y × cp +  δ(vj , ej−1 , ej )  + δ(vj , ej−1 , ej ) · δ(vp , ep−1 , e′1 )× j=2 ! × · · · ck−2 + δ(vk−1 , ek−2 , ek−1 )ck−1 · · · = = c1 + j=2  l(S) , c1 + δ(v2 , e1 , e2 ) c2 + δ(v3 , e2 , e3 )× i Y  +  j=2 # ≥T p−1 Y ∗ k−1 X j=2  Now it is obvious that if cx (C) ≤ 0 for some C, we can change the flow in G such that the total cost will not increase. It means that either x∗ is not an optimal flow, or it is not the unique solution of (GMNF). Next we define the cost of a directed path in G or G(x): Definition 2.2: Let S = ({v1 , · · · , vk }, {e1, · · · , ek−1 }) be a directed path. Then the cost of this path is defined as ci i Y δ(vj , ej−1 , ej ) · δ(vp , ep−1 , e′1 ) × \| i=2 = ε · cx (C) i=2 × c′1 + i=1 k Y p−1 Y " p−1 X e′ e av1′ avp−1 p e′ av1p \| j=p+1 1 e′m v1′ a {z e′ avm p × e avpp } δ(vp ,ep−1 ,ep ) # δ(vj , ej−1 , ej )  + +T c(C) = # " p−1 i Y X δ(vj , ej−1 , ej )  + ci =  c1 +  j=2 i=2 + p Y δ(vj , ej−1 , ej )× j=2  × cp + k−1 X i=p+1 " ci i Y j=p+1 # δ(vj , ej−1 , ej )  + T c(C) = # " i p−1 Y X δ(vj , ej−1 , ej ) + ci = c1 + B. Computation trees One of the important notions used for proving correctness and/or convergence for BP algorithm is the computation tree # " i k−1 ([11], [13], [14], [15]) (unwrapped tree in some sources). The Y X δ(vj , ej−1 , ej ) + T c(C) = ci + idea under this construction is the following: for the fixed j=2 i=p edge e of the graph G, one might want to build a tree of depth N , such that performing N iterations of BP on graph = l(S) + T c(C) G gives the same estimation of flow on e, as the optimal solution of the appropriately defined (GMNF) problem on the computation tree TeN . Since the proof of our result is based on the compuIII. B ELIEF P ROPAGATION ALGORITHM FOR GMNF tation trees approach, in this subsection we describe the construction in details. We will use the same notations for A. Min-Sum algorithm computation tree, as in [11] (section 5). Algorithm 1 represents the Belief Propagation Min-Sum In this paper we consider the computation trees, correalgorithm for (GMNF) from [11]. In the algorithm the func- sponding to edges of G. We say that e ∈ E is the ”root” tions φe (z) and ψv (z) are the variable and factor functions, for N -level computation tree TeN . Each vertex or edge of respectively, defined for v ∈ V, e ∈ E as follows: TeN is a duplicate of some vertex or edge of G. Define the ′ N N ( mapping ΓN e : V (Te ) → V (G) such that if v ∈ V (Te ) is N ′ ce z e if 0 ≤ ze ≤ ue , a duplicate of v ∈ V (G), then Γe (v ) = v. In other words, φe (z) = this function maps each duplicate from V (TeN ) to its inverse +∞ otherwise. ( P in V (G). e 0 if e∈Ev av ze = fv , The easiest way to describe the construction is inductively. ψv (z) = +∞ otherwise. Let e = (v, w) ∈ E(G). Then the tree Te0 consists of two vertices v ′ , w′ , such that Γ0e (v ′ ) = v, Γ0e (w′ ) = w, and an edge e′ = (v ′ , w′ ). We say that v ′ , w′ belong to 0-level Algorithm 1 BP for (GMNF) of Te0 . Note that for any two vertices v ′ , w′ ∈ V (Te0 ) it 0 0 holds that (v ′ , w′ ) ∈ E(Te0 ) ⇔ (Γ0e (v ′ ), Γ0e (w′ )) ∈ E(G), 1: Initialize t = 0, messages me→v (z) = 0, me→w (z) = so the vertices in a tree are connected with an edge if and 0, ∀z ∈ R for each e = (v, w) ∈ E. only if their inverse in the initial graph are connected. This 2: for t = 1, 2, . . . N do property will hold for all trees TeN . Now assume that we 3: For each e = (v, w) ∈ E update messages as defined a tree TeN , such that for any v ′ , w′ ∈ V (TeN ) it follows: ′ N ′ holds that (v ′ , w′ ) ∈ E(TeN ) ⇔ (ΓN e (v ), Γe (w )) ∈ E(G). N N mte→v (z) = φe (z)+ Denote by L(Te ) the set of leafs of Te (vertices which X are connected by edge with exactly one another vertex). For mt−1 zẽ ) , ∀z ∈ R ψw (~z) + + min e→w (~ \|E \| w any u′ ∈ L(TeN ), denote by P (u′ ) the vertex, with which ~ z ∈R ,~ ze =z ẽ∈Ew \e ′ u is connected by edge (so either (u′ , P (u′ )) ∈ E(TeN ) or (P (u′ ), u′ ) ∈ E(TeN )). We now build TeN +1 by extending mte→w (z) = φe (z)+ the three TeN as follows: for every u′ ∈ L(TeN ) let u = X N ′ N ′ mt−1 (~zẽ ) , ∀z ∈ R Γe (u ), and consider the set Bu′ = Su \ {Γe (P (u ))}, ψv (~z) + + min e→v ~ z ∈R\|Ev \| ,~ ze =z where Su is the set of neighbors of u in G. Then for every ẽ∈Ev \e vertex w ∈ Bu′ add vertex w′ to expand V (TeN ) and an edge 4: t := t + 1 (u′ , w′ ) if (u, w) ∈ E or an edge (w′ , u′ ) if (w, u) ∈ E to +1 5: end for expand E(TeN ), and set ΓN (w′ ) = w. Also set the level e ′ 6: For each e = (v, w) ∈ E, set the belief function as of w to be equal (N + 1). N N N So, the tree TeN +1 contains TeN as an induced subtree, and be = φe (z) + me→v (z) + me→w (z) also contains vertices on level N + 1, which are connected 7: Calculate the belief estimate by finding x̂N ∈ to leafs of T N (in fact, it is easy to see that new vertices e e arg min bN e (z) for each e ∈ E. are connected only with leafs from N th level). From the 8: Return x̂N as an estimation of the optimal solution of construction, one may see that for any v ′ , w′ ∈ V (TeN +1 ) it +1 ′ +1 (GMNF). holds that (v ′ , w′ ) ∈ E(TeN +1 ) ⇔ (ΓN (v ), ΓN (w′ )) ∈ e e E(G). In fact, any vertex of TeN +1 with level less then We address the reader to the article [11] for more details, (N + 1) is a local copy of the corresponding vertex from G. intuition and justifications on the Belief Propagation algo- More precisely: let v ′ ∈ V (TeN +1 ) and the level of v ′ is less +1 ′ rithm for general optimization problems, linear programs, or or equal then N . Denote v = ΓN (v ). Then for any vertex e Min-Cost Network Flow in particular. w ∈ V (G) such that either (v, w) ∈ E(G) or (w, v) ∈ E(G), i=2 j=2 there exist exactly one vertex w′ ∈ V (TeN +1 ) such that +1 ΓN (w′ ) = w and v ′ is connected with w′ in the same e way (direction) as v and w are connected in G. Then it is clear that we can extend the mapping Γ on edges by saying +1 ′ +1 ′ +1 ΓN (e = (v ′ , w′ )) = (ΓN (v ), ΓN (w′ )). Now for e e e ′ N +1 every vertex v ∈ V (Te ) and an incident edge ẽ, we +1 ′ can define the coefficient aẽv′ = aev , where v = ΓN (v ), e N +1 and e = Γe (ẽ). We also set the cost and capacity on the computation tree correspondingly to the initial graph, so +1 +1 cẽ = cΓN (ẽ) and uẽ = uΓN (ẽ) e e Now assume there is a (GMNF) problem stated for a graph G. We define the induced (GMNF)N e problem on a computation tree TeN in the following way. Let V 0 (TeN ) ⊂ V (TeN ) be a set of vertices with levels less than N . Then consider the problem: X minimize cẽ xẽ ẽ∈E(TeN ) subject to X aẽv′ xẽ = fv′ , ∀v ′ ∈ V 0 (TeN ), ẽ∈Ev′ ∀ẽ ∈ E(TeN ). (GMNFN e ) N Roughly speaking, (GMNFe ) is just a simple (GMNF) on a computation tree, except that there are no balance constraints for the vertices of N th level. Keeping in mind that the computation tree is locally equivalent to the initial graph, and that Min-Sum algorithm belongs to messagepassing heuristic, which means that the algorithm works locally at each step, one can intuitively guess that BP for (GMNFN e ) works quite similar as BP for the initial (GMNF). This reasoning can be formalized in the following lemma from [11]. Lemma 3.1: Let x̂N e be the value produced by BP for (GMNF) at the end of iteration N for the flow value on edge e ∈ E. Then there exists an optimal solution y ∗ of ∗ ′ N N (GMNFN e ) such that ye′ = x̂e , where e is the root of Te . Though this lemma was proven only for ordinary MinCost Network Flow problem, where \|aev \| = 1 for all v, e, its proof doesn’t rely on these coefficients at any point, which allows us to extend it for any values of these coefficients. 0 ≤ xẽ ≤ uẽ , C. Main results We will now use lemma 3.1 to prove our main result of correctness of BP Min-Sum for (GMNF). The following theorem is the generalization of Theorem 4.1 from [11], and our proof shares the ideas from the original proof. Let n = \|V (G)\|, and denote by x̂N the estimation of flow after N iterations of Algorithm 1. Theorem 3.2: Suppose (GMNF) has a unique solution x∗ . Define L to be the maximum absolute value of the cost of a simple directed path in G(x∗ ), and T as the minimum of the reducersamong all such paths. Then for any N ≥ L + 1 n, x̂N = x∗ . 2σ(x∗ )T Proof: Suppose to the contrary that there exists e0 = L (vα , vβ ) ∈ E and N ≥ + 1 n such that 2σ(x∗ )T ∗ x̂N e0 6= xe0 . By Lemma 3.1, there exist an optimal solution ∗ ∗ N ∗ ∗ y of GMNFN e0 such that ye0 = x̂e0 , and thus ye0 = xe0 . ∗ ∗ Then, without loss of generality, assume ye0 > xe0 . We will show that it it possible to adjust y ∗ in such way, that the flow in GMNFN e0 will decrease, which will contradict to the optimality of y ∗ . Let e′0 = (vα′ , vβ′ ) be the root edge of the computation tree ∗ N Te0 . Since y ∗ is a feasible solution of GMNFN e0 and x is a feasible solution of GMNF: X X e′ fΓ(vα′ ) = aẽvα′ yẽ∗ = av0α′ ye∗′0 + aẽvα′ yẽ∗ ẽ∈Ev′ ẽ∈Ev′ α \e0 α fΓ(vα′ ) = X e′ ∗ 0 aẽΓ(vα′ ) x∗ẽ = aΓ(v ′ ) xe0 + α ẽ∈EΓ(v′ α) X aẽΓ(vα′ ) x∗ẽ ẽ∈Ev′ α \e0 Since the nodes and the edges in the computation tree TeN0 are copies of nodes and vertexes in G, aẽΓ(v′ ) = aẽvα′ . Then α from above equalities it follows that there exists e′1 6= e′0 ′ e incident to vα′ in TeN0 such that av1α′ (x∗Γ(e′ ) − ye∗1 ′ ) > 0. If e′ 1 av1α′ > 0, then e′1 is an in-arc for vα′ , and we say that e′1 has the same orientation, as e′0 . In such case, x∗Γ(e′ ) > ye∗1 ′ . 1 Otherwise, we say that e′1 has the opposite orientation, and x∗Γ(e′ ) < ye∗1 ′ . Using the similar arguments, we will find 1 e′−1 6= e′0 incident to vβ′ satisfying similar condition. Then we can apply the similar reasoning for the other ends of e′1 , e′−1 , using the balance constraints and inequalities on components of x∗ and y ∗ for corresponding vertexes. In the end, we will have a non-directed path starting and ending in leaves of TeN0 : X = {e′−N , e′−N +1 , . . . , e′−1 , e′0 , e′1 , . . . , e′N } such that for −N ≤ i ≤ N one of two cases holds: ∗ ∗ ′ ′ ′ • ye′ > xΓ(e′ ) . Then ei = (v , w ) has the same orientation as e0 . In this case, define Aug(e′ ) = (v ′ , w′ ) and e′ ∈ D. ∗ ∗ ′ ′ ′ • ye′ < xΓ(e′ ) . Then ei = (v , w ) and e0 have opposite i i ′ orientations. Define Aug(e ) = (w′ , v ′ ) and e′ ∈ O. Note that the capacity constraints are similar for corresponding vertices from GMNFN e0 and GMNF, and since , hence, every e′ ∈ X it holds y ∗ is feasible for GMNFN e0 0 ≤ ye∗′ ≤ ue′ = uΓe′ . Then, for any e′ ∈ D, we have x∗Γ(e′ ) < ye∗′ ≤ ue′ , which means that Γ(e′ ) ∈ G(x∗ ) (from the definition of the residual network). Note that in this case e′ = Aug(e′ ), so Γ(Aug(e′ )) ∈ E(G(x∗ )). Next, let’s now e′ = (v ′ , w′ ) ∈ O, and thus x∗Γ(e′ ) > ye∗′ ≥ 0. Again, out of the definition of the residual network, (Γ(w′ ), Γ(v ′ )) ∈ E(G(x∗ )). But since e′ ∈ O, we have Aug(e′ ) = (w′ , v ′ ), thus Γ(Aug(e′ )) ∈ E(G(x∗ )). Therefore, for every edge e′ ∈ X it holds Γ(Aug(x∗ )) ∈ E(G(x∗ )). From the definition of Aug(e′ ) for e′ ∈ X one can see that all Aug(e′ ) have the same direction as e0 . Therefore, W = {Aug(e′−N ), Aug(e′1−N ), . . . , Aug(e′−1 ), Aug(e′0 ), Aug(e′1 ), . . . , Aug(e′N −1 ), Aug(e′N )} is the directed path in TeN0 , and we will call it augmenting path of y ∗ with respect to x∗ . Γ(W ) is also a directed walk on G(x∗ ), which can be decomposed into a simple directed path P and a collection of k simple directed cycles C1 , C2 , . . . , Ck . Since each simple cycle or path has at most n edges and W has 2N + 1 edges, it holds l(W ) ≤ n + kn. On the other Ln + n + 1. Then we obtain hand, l(W ) = 2N + 1 = σ(x∗ )T L . Further we would denote by c∗ (·) and l∗ (·) k > σ(x∗ )T costs of cycles and paths in the residual network G(x∗ ). For each cycle Ci we have c∗ (Ci ) ≥ σ(x∗ ) > 0 by Lemma (2.1), while the cost of P is at least −L. Then using Lemma (2.2) we have: L l∗ (W ) ≥ l∗ (P ) + kT σ(x∗ ) > −L + T σ(x∗ ) = 0 σ(x∗ )T We will now ”extract” flow from W in TeN0 . Let’s redefine the numeration of arcs in W for convenience: W = {w1 , w2 , . . . , w2N +1 }. This edges correspond to X = ′ ′ {w1′ , w2′ , . . . , w2N +1 }, where wi and wi have the same ′ orientation if wi ∈ D, and have the opposite orientation if wi′ ∈ O. Hence extraction of flow from wi means decreasing ∗ ∗ if wi′ ∈ O. if wi′ ∈ D, and increasing yw yw i i Then similarly to the proof of Lemma 2.1, to keep the balance in all the vertexes of W (except the start and the end, since there are no balance constraints for them in GMNFN e0 ), we have to adjust the flow in the following way: ( ∗ yw if wi′ ∈ D ′ − λ, 1 ỹw1′ = (2) ∗ yw if wi′ ∈ O ′ + λ, 1 ( Qi ∗ ′ ′ yw if wi′ ∈ D ′ − λ j=2 δ(vj , wi−1 , wi ), i ỹwi′ = (3) Qi ∗ ′ ′ yw′ + λ j=2 δ(vj , wi−1 , wi ), if wi′ ∈ O i for i = 2, 3, . . . , 2N + 1. Obviously, there exist small enough λ > 0 such that ỹe > x∗Γ(e) ≥ 0 ∀e ∈ D, and ỹe < x∗Γ(e) ≤ uΓ(e) ∀e ∈ O, so the capacity constraints are satisfied for ỹ. (The balance constraints are satisfied by the construction). So, ỹ is a feasible solution of GMNFN e0 . Now we explore how the total cost changes after such transformation of the flow: X cΓ(e′ ) ye∗′ − e′ ∈E(TeN0 ) = X cΓ(e′ ) ỹe′ = e′ ∈E(TeN0 ) X cΓ(e′ ) (ye∗′ − ỹe′ ) = e′ ∈E(TeN0 ) = X cΓ(e′ ) λe′ − e′ ∈D = X e′ ∈D = 2N +1 X i=1 c∗Γ(w ) λ i cΓ(e′ ) λe′ = e′ ∈O c∗Γ(e′ ) λe′ +  X X e′ ∈O i Y j=2 c∗Γ(e′ ) λe′ = X c∗Γ(e′ ) λe′ = e′ ∈W  δ(vj , wj−1 , wj ) = l(W )λ > 0 Here we used that c∗Γ(e′ ) = cΓ(e′ ) for e′ ∈ D and c∗Γ(e′ ) = −cΓ(e′ ) for e′ ∈ O, that is obvious from the construction of W, D, and O. So we found the feasible solution of GMNFN e0 with the total cost less then of y ∗ . It means that y ∗ is not an optimal solution of GMNFN e0 , which leads us to the contradiction. IV. CONCLUSIONS The proved correctness of the Belief Propagation algorithms for General Min-Cost Network Flow problems may serve as justification of applying these algorithms in practice for real problems. The statement of GMNF problem is broad enough, and thus many practical problems may fall under this formulation, which means that these problems may be solved correctly using BP algorithms. The future research is required to determine the speed of convergence of BP for this Generalized Min-Cost Network Flow, since in our paper we have only proved that after the finite iterations of the algorithms the answer will not change and be the correct one. However, since σ(x∗ ) in 3.2 may be arbitrary small for some problems, the number of iterations until the algorithm will give the correct answer may be arbitrary big, and further analysis is needed to reasonably bound the number of steps that should be done. R EFERENCES [1] M. J. Wainwright, M. I. Jordan et al., “Graphical models, exponential families, and variational inference,” Foundations and Trends R in Machine Learning, vol. 1, no. 1–2, pp. 1–305, 2008. [2] K. P. Murphy, Y. Weiss, and M. I. 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Neural Distributed Autoassociative Memories: A Survey V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov Abstract Introduction. Neural network models of autoassociative, distributed memory allow storage and retrieval of many items (vectors) where the number of stored items can exceed the vector dimension (the number of neurons in the network). This opens the possibility of a sublinear time search (in the number of stored items) for approximate nearest neighbors among vectors of high dimension. The purpose of this paper is to review models of autoassociative, distributed memory that can be naturally implemented by neural networks (mainly with local learning rules and iterative dynamics based on information locally available to neurons). Scope. The survey is focused mainly on the networks of Hopfield, Willshaw and Potts, that have connections between pairs of neurons and operate on sparse binary vectors. We discuss not only autoassociative memory, but also the generalization properties of these networks. We also consider neural networks with higher-order connections and networks with a bipartite graph structure for non-binary data with linear constraints. Conclusions. In conclusion we discuss the relations to similarity search, advantages and drawbacks of these techniques, and topics for further research. An interesting and still not completely resolved question is whether neural autoassociative memories can search for approximate nearest neighbors faster than other index structures for similarity search, in particular for the case of very high dimensional vectors. Keywords: distributed associative memory, sparse binary vector, Hopfield network, Willshaw memory, Potts model, nearest neighbor, similarity search. DOI: https://doi.org/10.15407/kvt188.02.005 This front page has been added to assist automated indexing. The paper as published begins on the next page. Please cite as: V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov. Neural distributed autoassociative memories: A survey. Cybernetics and Computer Engineering. 2017. N 2 (188). P. 5–35. Информатика и информационные технологии DOI: https://doi.org/10.15407/kvt188.02.005 УДК 004.22 + 004.93'11 V.I. GRITSENKO1, Corresponding Member of NAS of Ukraine, Director, e-mail: vig@irtc.org.ua D.A. RACHKOVSKIJ 1, Dr (Engineering), Leading Researcher, Dept. of Neural Information Processing Technologies, e-mail: dar@infrm.kiev.ua A.A. FROLOV2, Dr (Biology), Professor, Faculty of Electrical Engineering and Computer Science FEI, e-mail: docfact@gmail.com R. GAYLER3, PhD (Psychology), Independent Researcher, e-mail: r.gayler@gmail.com D. KLEYKO4 graduate student, Department of Computer Science, Electrical and Space Engineering, e-mail: denis.kleyko@ltu.se E. OSIPOV4 PhD (Informatics), Professor, Department of Computer Science, Electrical and Space Engineering, e-mail: evgeny.osipov@ltu.se 1 International Research and Training Center for Information Technologies and Systems of the NAS of Ukraine and of Ministry of Education and Science of Ukraine, ave. Acad. Glushkova, 40, Kiev, 03680, Ukraine 2 Technical University of Ostrava, 17 listopadu 15, 708 33 Ostrava-Poruba, Czech Republic 3 Melbourne, VIC, Australia 4 Lulea University of Technology, 971 87 Lulea, Sweden NEURAL DISTRIBUTED AUTOASSOCIATIVE MEMORIES: A SURVEY Introduction. Neural network models of autoassociative, distributed memory allow storage and retrieval of many items (vectors) where the number of stored items can exceed the vector dimension (the number of neurons in the network). This opens the possibility of a sublinear time search (in the number of stored items) for approximate nearest neighbors among vectors of high dimension. The purpose of this paper is to review models of autoassociative, distributed memory that can be naturally implemented by neural networks (mainly with local learning rules and iterative dynamics based on information locally available to neurons). Scope. The survey is focused mainly on the networks of Hopfield, Willshaw and Potts, that have connections between pairs of neurons and operate on sparse binary vectors. We ã V.I. GRITSENKO, D.A. RACHKOVSKIJ, A.A. FROLOV, R. GAYLER, D. KLEYKO, E. OSIPOV, 2017 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 5 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov discuss not only autoassociative memory, but also the generalization properties of these networks. We also consider neural networks with higher-order connections and networks with a bipartite graph structure for non-binary data with linear constraints. Conclusions. In conclusion we discuss the relations to similarity search, advantages and drawbacks of these techniques, and topics for further research. An interesting and still not completely resolved question is whether neural autoassociative memories can search for approximate nearest neighbors faster than other index structures for similarity search, in particular for the case of very high dimensional vectors. Keywords: distributed associative memory, sparse binary vector, Hopfield network, Willshaw memory, Potts model, nearest neighbor, similarity search. INTRODUCTION In this paper, we review some artificial neural network variants of distributed autoassociative memories (denoted by Neural Associative Memory, NAM) [1– 159]. Varieties of associative memory [93] (or content addressable memory) can be considered as index structures performing some types of similarity search. In autoassociative memory, the output is the word of memory, most similar to the key at the input. We restrict our initial attention to systems where the key and memory words are binary vectors. Therefore, autoassociative memory answers nearest neighbor queries for binary vectors. In distributed memory, different vectors (items to be stored) are stored in shared memory cells. That is, each item to be stored consists of a pattern of activation across (potentially) all the memory cells of the system and each memory cell of the system contributes to the storage and recall of many (potentially all) stored items. Some of types of distributed memory have attractive properties of parallelism, resistance to noise and malfunctions, etc. However, exactly correct answers to the nearest neighbor queries from such memories are not guaranteed, especially when too many vectors are stored in the memory. Neurobiologically plausible variants of distributed memory can be represented as artificial neural networks. These typically perform one-shot memorization of vectors by a local learning rule modifying connection weights and retrieve a memory vector in response to a query vector by an iterative procedure of activity propagation between neurons via their connections. In the first Section, we briefly introduce Hebb's theory of brain functioning based on cell assemblies because it has influenced many models of NAM. Then we introduce a generic scheme of NAMs and their characteristics (discussed in more details in the other sections). The following three Sections discuss the widespread matrix-type NAMs (where each pair of neurons is connected by two symmetric connections) of Hopfield, Willshaw, and Potts that work best with sparse binary vectors. The next Section is devoted to the function of generalization, which differs from the function of autoassociative memory and emerges in some NAMs. The following Section discusses NAMs with higherorder connections (more than two neurons have a connection) and NAMs without connections. Then some recent NAMs with a bipartite graph structure are considered. The last Section provides discussion and concludes the paper. 6 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey CELL ASSEMBLIES AND GENERIC NAM Hebb's paradigm of cell assemblies. According to Hebb [65], nerve cells of the brain are densely interconnected by excitatory connections, forming a neural network. Each neuron determines its membrane potential as the sum of other active neurons' outputs weighted by connection weights. A neuron becomes active if this potential (the input sum) exceeds the threshold value. During network functioning, connection weights between simultaneously active neurons (encoding various items) are increased (the Hebbian learning rule). This results in organization of neurons into cell assemblies — groups of nerve cells most often active together and consequently mutually excited by connection weights between neurons in the assembly. At the same time, the process of increased connection within assemblies leads to mutual segregation of assemblies. When a sufficient part of a cell assembly is activated, the assembly becomes active as a whole because of the strong excitatory connection weights between the cells within the assembly. Cell assemblies may be regarded as memorized representations of items encoded by the distributed patterns of active neurons. The process of assembly activation by a fragment of the memorized item may be interpreted as the process of pattern completion or the associative retrieval of similar stored information when provided with a partial or distorted version of the memorized item. Hebb's theory of brain functioning — interpretation of various mental phenomena in terms of cell assemblies — has turned out to be one of the most profound and generative approaches to brain modeling and has influenced the work of many researchers in the fields of artificial intelligence, cognitive psychology, modeling of neural structures, and neurophysiology (see also reviews in [39, 40, 54, 75, 98, 104, 120, 121, 134]). A generic scheme and characteristics of NAMs. Let us introduce a generic model of the NAM type, inspired by Hebb's paradigm, that will be elaborated in the sections below devoted to specific NAMs. We mainly consider NAMs of the distributed and matrix-type, which are fully connected networks of binary neurons (but see Sections "NAMs with Higher-Order Connections and without Connections", "NAMs with a Bipartite Graph Structure for Nonbinary Data with Constraints" for other NAM types). Each of the neurons (their number is D ) represents a component of the binary vector z . That is, each of the D neurons can be in the state 0 or 1. Each pair of neurons has two mutual connections (one in each direction). The elements of the connection matrix W( D ´ D) represent the weights of all these connections. In the learning mode, the vectors y from the training or memory set (which we call the "base") are "stored" (encoded or memorized) in the matrix W by using some learning rule that changes the values of wij (initially each wij is usually zero). In the retrieval mode, an input binary vector x (probe or key or query vector) is fed to the network by activation of its neurons: z = x . The input sum of the i -th neuron si = å j =1, D wij z j ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 7 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov is calculated. The neuron state is determined as zi (t + 1) = 1 (active) for si (t ) ³ Ti (t ) and zi (t + 1) = 0 (inactive) for si (t ) < Ti (t ) ; Ti is the value of the neuron threshold. For parallel (synchronous) network dynamics, the input sums and the states of all D neurons are calculated (updated) at each step t of iterative retrieval. For sequential (asynchronous) dynamics, zi is calculated for one neuron i , selected randomly. For simplicity, let us consider random selection without replacement, and one step of the asynchronous dynamics to consist of update of the states of all D neurons. The parameters W and T are set so that after a single, or several, steps of dynamics the state of the network (neurons) reaches a stable state (typically, the state vector does not change with t , but cyclic state changes are also considered as "stable"). At the stable state, z is the output of the network. The query vector x is usually a modified version of one of the stored vectors y . In the literature, this might be referred to as a noisy, corrupted, or distorted version of a vector. While the number of stored vectors is not too high, the output z is the stored y closest to x (in terms of dot product simdot (x, y ) º á x, y ñ ). That is, z is the base vector y with the maximum value of á x, y ñ . In this case, NAM returns the (exact) nearest neighbor in terms of simdot . For binary vectors with the same number of unit (i.e. with value equal to 1) components, this is equivalent to the nearest neighbor by the Hamming distance ( dist Ham ). The time complexity (runtime) of one step of the network dynamics is O( D 2 ) . Thus, if a NAM can be constructed that stores a base of N > D vectors so that they can be successfully retrieved from their distorted versions, then the retrieval time via the NAM could be less than the O( DN ) time required for linear search (i.e. the sequential comparison of all base vectors y to x ). Since the memory complexity of this NAM type is O( D 2 ) , as D increases, one can expect an increasing in the size N of the bases that could be stored and retrieved by NAM. Unfortunately, the vector at the NAM output may not be the nearest neighbor of the query vector, and possibly not even a vector of the base. (Note that if one was not concerned with biological plausibility, one can quickly check whether the output vector is in the base set by using a hash table to store all base vectors.) In some NAMs, it is only possible to store many fewer vectors N than D , with high probability of accurate retrieval, especially if the query vectors are quite dissimilar to the base vectors. For NAM analysis, base vectors are typically selected randomly independently from some distribution of binary vectors (e.g., vectors with the probability p of 1-components equal to 1/2, or vector with pD 1-components, 8 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey for some p from interval (0,1)). The assumption of independence simplifies analytical approaches, but is likely unrealistic for real applications of NAMs. The query vectors are typically generated by as modifications of the base vectors. Distortion by deletion randomly changes some of the 1s to 0s (the remaining components are guaranteed to agree). A more complex distortion by noise randomly changes some 1s to 0s, and some 0s to 1s while (exactly or approximately) preserving the total number of 1s. For a random binary vector of dimension D with the probability p of a component to be 1, the Shannon entropy H = Dh( p) , Where h( p) = - p log p - (1 - p)log(1 - p) . For D >> 1 , a random vector with pD of 1s has approximately the same entropy. The entropy of N vectors is NDh( p) . When N vectors are stored in NAM, the entropy per connection is [40, 41]. a = NDh( p) / D 2 = Nh( p) / D . Knowing h( p) , it is easy to determine N for a given a . When too many vectors are stored, NAM becomes overloaded and the probability of accurate retrieval drops (even to 0). The value of a for which a NAM still works reliably depends on the mode of its use (in addition to the NAM design and distributions of base and query vectors). The mode where the undistorted stored base vectors are still stable NAM states (or stable states differ by few components from the intended base vectors), has the largest value of a . (We denote the largest value of a for this mode as "critical", a crit , and the corresponding N as N crit .) For a > a crit the stored vectors become unstable. Note that checking if an input vector is stable does not allow one to extract information from the NAM, since vectors not stored can also be stable. The information (in the Shannon information-theoretic sense) that can be extracted from a NAM is determined by the information efficiency (per connection) E . This quantity is bounded above by some specific a (the entropy per connection that still permits information extraction), which in its turn is bounded above by a crit . A NAM may work in recognition or correction mode. In recognition mode, the NAM distinguishes whether the input (query) vector is from the base or not, yielding extracted information quantified by Erecog . When NAM answers the nearest neighbor queries (correction mode), information quantified by Ecorr is extracted from the NAM by correction (completion) of the distorted query vectors. The more distorted the base vector used as the query at the NAM input, the more information Ecorr is extracted from the NAM (provided that the intended base vector is sufficiently accurately retrieved). However, more distorted input vectors lower the value of a corr at which the NAM is still able to retrieve the correct base vectors, and so lowers Ecorr (which is constrained to be less than a corr ). We refer to these informationISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 9 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov theoretic properties of NAMs as their information characteristics. Let us now consider specific variants of the generic NAM. Hereafter we use the terms "NAM" and "(neural) network" interchangeably. HOPFIELD NAMs Hopfield networks with dense vectors. In the Hopfield NAM, "dense" random binary vectors (with the components from {0,1} with the probability p = 1 / 2 of 1) are used [68]. The learning procedure forms a symmetric matrix W of connection weights with positive and negative elements. The connection matrix is constructed by successively storing each of the base vectors y according to the Hopfield learning rule: wij = wij + ( yi - q)( y j - q) with parameter q = p = 1 / 2 ; wii = 0 . (For brevity, we use the same name for generalization of this rule with q < 1 / 2 , though Hopfield did not propose it, Subsection "Hopfield networks with sparse vectors"). The dynamics in [68] is sequential (in many subsequent studies and implementations it is parallel) with the threshold T = 0 . It was shown [68] that each neuron state update decreases the energy function -(1 / 2) å i , j =1, D zi wij z j , so that a (local) minimum of energy is eventually reached and such a network comes into a stable activation state. As D ® ¥ , various methods of analysis and approximation of experimental (modeling) data obtain a crit » 0.14 [68, 8, 5, 71, 35] which gives N crit » 0.14 D since h(1 / 2) = 1 . Note that similar values of a crit are achieved at rather small finite D . For rigorous proofs of (smaller) a crit see refs in [107]. max for distortion by noise, 0.092 was obtained by the method of As for Ecorr approximate dynamical equations of the mean field [71], and 0.095 by approximating the experiments to D ® ¥ [35]. By the coding theory methods in [112] it was shown that asymptotically (as D ® ¥ ) it is possible to retrieve (with probability approaching 1) exact base vectors with query vectors distorted by noise (so that their dist Ham < D / 2 from the base vectors), for N = D / (2ln D) stored base vectors if non-retrieval of some is permitted. If one requires the exact retrieval of all stored base vectors, the maximum number of vectors which can be stored decreases to N = D / (4ln D) . These values of N were shown to be the lower and upper bounds in [25, 20]. Note that in [47] a crit = 2 was obtained for "optimal" W (obtained by a non-Hebbian learning rule); a pseudoinverse rule (e.g. [125, 140]) gives a crit = 1 . For correlated base vectors, the storage capacity N crit depends on the structure of the correlation. When the base vectors are generated by a onedimensional Markov chain [107], N crit is somewhat higher than it is for 10 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey independent vectors. This and other correlation models were considered in [108]. Hopfield networks with sparse vectors. Hopfield NAMs operating with sparse vectors p < 1 / 2 appeared to have better information characteristics [154] (see also Sections "Willshaw NAMs", "Potts NAMs") than those operating with dense vectors ( p = 1 / 2 ). For example, they attain values N > D . In the usual Hopfield NAM and learning rule (with q = p = 1 / 2 and threshold T = 0 ) the number of active neurons is kept near D / 2 by the balance of negative and positive connections in W . Using the Hopfield rule with q = p < 1 / 2 one can not set T = 0 . This is especially evident for the Hebb learning rule (which we obtain from the Hopfield rule by setting q = 0 ). All connections become non-negative, and T = 0 eventually activates all neurons. Similar behavior is demonstrated by the Hopfield NAM and learning rule with q = p < 1 / 2 . The problem of network activity control (i.e. maintaining some average activity level chosen by the designer) can be solved by applying an appropriate uniform activation bias to all neurons [9, 21]. This is achieved by setting an appropriate positive value of the time-varying threshold T (t ) [21] to ensure, for example, pD < D / 2 active neurons (to match pD in the stored vectors) for parallel dynamics. Note that the Hopfield rule with q = p < 1 / 2 provides better information characteristics than the pure Hebb rule with q = 0 [41, 35]. However, the Hebb rule requires modification of only ( pD)2 connections per vector, whereas the Hopfield rule modifies all connections per vector. As D ® ¥ and p ® 0 ( pD >> 1 , and often p ~ ln D / D ) the theoretical analysis (e.g., [154, 41, 34] and others) gives a crit = (log e) / 2 = 1 / (2ln 2) » 0.72 for the Hopfield rule with q = p , the Hebb rule, and the "optimal" W [47]. In [34] they use a scaled sparseness parameter e = (ln p) -1/2 to investigate max max . For e << 1 they obtained a crit » 0.72 . However convergence of a crit to a crit for e = 0.1 (corresponding to p = 10-8 and to D > 109 ), a crit = 0.43 only. max In [122] it was shown that Erecog = 1 / (4ln 2) » 0.36 (by the impractical exhaustive enumeration procedure of checking that all vectors of the base are stable and all other vectors with the same number of 1-components are not stable). This empirical estimate coincides with the estimate [41]. For retrieval by a single step of dynamics, max Ecorr = 1 / (8ln 2) » 0.18 for distortion by deletion of half the 1-components of a base vector [40, 41, 119, 146]. Let us note again, all these results are obtained for D ® ¥ and p ® 0 . For these conditions, multi-step retrieval (the usual mode of NAM retrieval as explained in Subsection "A generic scheme and characteristics of NAMs") is not ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 11 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov required since NAM reaches a stable state after a single step. In terms of N , since h( p) ® 0 for p ® 0 , it follows that N >> D , that is N crit = a crit D / h ( p ) ® ¥ much faster than D ® ¥ . The same is valid for N corresponding to E . In the experiments [146], for the Hebb rule and multi-step retrieval Ecorr values up to 0.09 were obtained. Detailed studies of the information characteristics of the finite- and infinite-dimensional networks with the Hopfield rule, can be found in [34, 35]. Different degrees of distortion by noise for vectors with pD unit components were used. The dynamic threshold ensured pD active neurons at each step of the parallel dynamics. It was shown [35] that with this choice of threshold the stable states are static (some vector) or cycles of length 2 (two alternating vectors on adjacent steps of the dynamics). (This is the same behavior as for the fixed static threshold and is valid for all networks with symmetric connections.) It has been demonstrated experimentally [35] that even if after the first step of dynamics simdot (y , z ) < simdot (y , x ) (where y is the correct base vector, z is the network state, and x is the distorted input), the correct base vector can sometimes be retrieved by the subsequent steps of the dynamics. Conversely, increasing simdot (y , z ) at the first step of the dynamics does not guarantee correct retrieval [5]. These results apply to both the dense and sparse vector cases. The study [35] used analytical methods developed for the dense Hopfield network and adapted for sparse vectors, including the statistical neurodynamics (SN) [5, 34], the replica method (RM) [8], and the single step method (SS) [80]. All these analytical methods rather poorly predicted the behavior of finite networks for highly sparse vectors, at least for the parallel dynamics studied. (Note that all these methods (SN, RM, SS) provide accurate results for D ® ¥ and p ® 0 , where retrieval by a single step of dynamics is enough.) Empirical experimentation avoids the shortcomings of these analytical methods by directly simulating the behavior of the networks. These simulations allow a corr and information efficiency, Ecorr , to be estimated as a functions of p, D and the level of noise in the input vectors. The value of Ecorr monotonically increases as D increases for a constant p . For p = 0.001 – 0.01, which corresponds to the activity level of neurons in the brain, the maximum value of Ecorr » 0.205 was obtained by approximating experimental results to the case D ® ¥ [35] (higher max than Ecorr » 0.18 for p ® 0 ). In [38] the time of retrieval in the Hopfield network was investigated (using the number of retrieval steps observed in simulation experiments; this number somewhat increases with D ). They conclude that for random vectors with small p , large D , and large N , Hopfield networks may be faster than some other algorithms (e.g., the inverted index) for approximate and exact nearest neighbor 12 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey querying. An increase in the number N crit of stable states corresponding to the stored vectors proportional to ( D / ln D) 2 for p ~ ln D / D is shown asymptotically in [4] (although non-rigorously, see also [34]). Note, this result also follows from N = a D / h( p) by approximating h( p) » - p log p for small enough p ( - ln p >> 1 ). In [67] they give a rigorous analysis of a Hopfield network variant (neurons are divided into parts, see Section "Potts NAMs"), with the Hebb learning rule and p slightly less than ln D / D , for retrieval by a single step of parallel dynamics with fixed T . The lower and upper bounds of N were obtained for which the memory vectors are stable states (with probability approaching 1 as D ® ¥ ), and can also be exactly retrieved from query vectors distorted by noise. The lower and upper bounds of N found in [67] are of the same order as those found in [4]. For this mode of network operation, if we approximate the number of retrieval steps as ln D , we may estimate speed-up as D / (ln D)3 relative to linear search (see Subsection "A generic scheme and characteristics of NAMs"). For both dense and sparse vectors, the NAM capacity N max grows with increasing D . Also, in order to maintain an adequate information content for a sparse vector ( Dh( p) for h( p) << 1 ), it is necessary to have a sufficiently high D . The number of connections grows as D squared (because Hopfield networks are fully connected), which is unattainable even on modern computers at D of millions. Besides, the neurobiologically plausible number of connections per neuron is on the order of 10,000. Therefore, the development of "diluted" networks that perform NAM functions without being fully connected is attractive, e.g. [105, 150, 151, 41, 142]. This partial connectivity can be used to reduce the memory complexity of NAM from quadratic to linear in D [98, 99]. WILLSHAW NAMs Willshaw networks with sparse vectors. NAMs with binary connections from {0,1} are promising since they require only one bit per connection. Such networks were proposed both in heteroassociative [157] and autoassociative versions (e.g. [118, 156, 16, 152, 49, 115, 119, 122, 24, 48, 41, 42, 43, 44, 81]). The learning rule (let's call it the Willshaw rule) becomes: wij = wij Ú ( yi Ù y j ) , where Ú is disjunction, Ù is conjunction. Various strategies for threshold setting can be used, e.g., setting threshold T to ensure pD active neurons, as in Subsection "Hopfield networks with sparse vectors". Note that this NAM can not work with dense vectors, since storing only a small number of dense vectors will set almost all the connection weights to 1. Moreover, for the same reason, the Willshaw networks (unlike the Hopfield networks) cease to work at any constant p and a as D ® ¥ . The number N of random binary vectors able to be stored and retrieved in the Willshaw NAM ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 13 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov grows with decreasing vector density. Even for not very large networks and not very sparse vectors ( p ~ D / D ), N can exceed D (e.g. D = 4096 allowed storage and retrieval of up to N = 7000 vectors distorted by noise in the experiments of [16]). The particular N values reached (for D and p fixed) vary depending on the degree of the query vector distortion and on the desired probability of retrieval. The maximum theoretical a crit = ln 2 » 0.69 is reached as D ® ¥ for p ~ ln D / D [40, 119, 41]. In [122] they obtained max Erecog = ln 2 / 2 » 0.346 (using a computationally expensive exhaustive enumeration procedure). In [42, max 43] the same a crit and Erecog were obtained analytically for the sparseness parameter b = log(1 / p) / ( pD) equal to 1, i.е. for pD somewhat less than log D . (The probability of a connection to be modified after storing N vectors is 1 - (1 - p 2 ) N » 1 - exp( - Np 2 ) = 1 - exp( -a Dp 2 / h( p)) = 1 - exp( -a p log(1 / p) / ( b h( p))) » 1 - exp( -a / b ) <1 (we used p ® 0 ); thus the network can be analyzed for fixed a and b at D ® ¥ .) The same upper bound of E is given as the maximum entropy of W learnt by the Willshaw rule. In [119] the efficiency max Ecorr = ln 2 / 4 » 0.173 was theoretically shown for single-step (as well as multi-step) retrieval and distortion by deletion. For multi-step retrieval in finite Willshaw NAMs (with distortion by deletion) Ecorr up to 0.19 (at D = 20000) was obtained experimentally in [146]. Experiments in [146, 44] show that in the Willshaw NAM (unlike the Hopfield NAM), the values a corr and Ecorr for not too large D are higher than for D ® ¥ (see also [42]). Note that the quality of retrieval in the Willshaw NAM is higher than in the Hopfield NAM; the retrieved vectors more often coincide exactly with the stored vectors of the base. From the detailed analytical and experimental study of the values of a corr in [44] (at various levels of sparsely, parameterized as b , degrees of distortion by noise, and D up to 100000), it was found that Ecorr » 0.13 per connection can be reached in the experiments (for small networks, N = 640, pD = 20, b = 0.25 ). It was also shown that the results of the analytical methods SS [80] and GR [48] are far from the experimental results (in most cases, worse than them). Due to the connections being binary, the efficiency per bit of connection implemented in computer memory is higher than that for the Hopfield network (where Ecorr » 0.205 per connection [41]). A review of NAM studies in [81] concludes that for Willshaw networks having connection matrix W with probability of a nonzero element close to 0 or 1, compression of W improves information characteristics compared with the 14 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey usual uncompressed Willshaw NAM. Such compressed W are obtained when the base vectors have the number of 1s sublogarithmic or superlogarithmic in D . Their comparison of the retrieval time in compressed Willshaw networks and the inverted index has shown the advantage of the inverted index for most parameters. An analytical and experimental comparison of the Willshaw, the GB (Subsection "Willshaw-Potts network"), and the Hopfield networks (with the Hebb rule [4]) for vectors with p of the order of ln D / D and distortion by deletion was carried out in [59]. They investigate single-step retrieval theoretically (asymptotically, for D ® ¥ with probability approaching 1). For all models, the lower bound of N of the order ( D / ln D) 2 is obtained, and for the Willshaw network the matching upper bound is shown. In experiments, the results are worse for a fixed threshold than for a variable threshold. The Hopfield network performed worse, in terms of empirical probability of retrieval versus N , compared with the other NAMs, probably because of the non-optimal Hebb learning rule and non-optimal threshold selection. For the diluted Willshaw networks [24, 6, 41, 99] the optimal pD (providing approximately the same capacity N ) is higher than for the fully connected networks. Willshaw networks in the index structures for nearest neighbor search. In [159] the base of binary sparse vectors is divided into disjoint sets (of the same cardinality) and each is stored in a Willshaw NAM with its own W . When the query vector x is input, simdot (x, Wx) is calculated for the matrices W of all sets, and the vectors of sets with the maximum similarity are used as the nearest neighbor candidates (verified by linear search). Analysis and experiments for bases of random vectors with small random distortions of query vectors showed that up to a certain number of vectors in each network the nearest neighbor is found with a high probability (in experiments, without error) and faster than by linear search only. If this number of vectors per network is exceeded, both the probability of finding an incorrect nearest neighbor and its distance to the correct vector increase. A somewhat lower speedup relative to linear search is shown for real, nonrandom data, versus synthetic, random data. In [60] similar results were obtained analytically (asymptotically for D ® ¥ and error probability approaching zero) and experimentally for bases of sparse and dense random vectors (for the Hopfield rule). POTTS NAMs Potts networks. The NAM from [77] can be considered as a network of neurons that are divided into non-overlapping parts ("columns"), with d neurons in each column and only one active neuron in the state z = d - 1 , for the remaining column neurons z = -1 . That is, the sum of activations over all the neurons is zero in each column. The Hebb rule is used for learning. For the more convenient version of this model with the neurons having the states from {0,1} and single active neuron per column, the Hopfield rule is used. The connection matrix W for the entire network is constructed so that wij = 0 for neurons i and j in the same column (this implies that wii = 0 ). That is, the ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 15 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov network is structured as a multipartite graph. Network dynamics (parallel or sequential) activates the one neuron in each column with the maximum input sum s (one of these neurons is randomly activated for neurons with equal s , but see the GB network below). For the number of columns D , the value of N crit / D of the Potts network was estimated by [77] to be d (d - 1) / 2 times more than 0.14 (i.e., a crit for the Hopfield network with p = 1 / 2 ). However, to approximate the number of connections in the Hopfield network, the Potts network must have D / d columns. Note also that one "Potts vector" contains only ( D / d )log d bits of information [79]. The Potts network with parallel and sequential dynamics, and with singlestep and multi-step retrieval was analytically explored in [109]. For exact retrieval (asymptotically, as D ® ¥ , with probability approaching 1), the upper and lower bounds of N were estimated both for the mode of querying with stable stored vectors and for the correction mode querying with distorted query vectors. In both cases, N = cD / ln D , where the constant c increases quadratically in d , but with different c depending on the degree of distortion and the desired probability of state to be stable or vector to be retrieved. Willshaw-Potts network. For binary connections with the Willshaw learning rule, the Potts network becomes the Willshaw-Potts network [79]. When a vector is stored, a clique (a complete subgraph) is created in the connection graph. As for the Hopfield network, only static stable states or cycles of length 2 were experimentally observed for parallel dynamics. According to [79], the information characteristics of this network are close to the Willshaw network at the same vector sparsity. Since the information content of Willshaw-Potts vectors is low, the N crit is higher than for the Willshaw network. This network was rediscovered as the GB network in [58] with various modifications [3, 59] and hardware implementations (for example, [117] with non-binary connections). The GB network is oriented for exact retrieval of vectors with distortion by deletion (columns without values activate all neurons). The peculiarities of the GB network include: connections of neurons with themselves; the possibility for several neurons in a column (with the maximum input sum) to be in the active state; the contribution from each column to the input sum of a neuron is not more than 1; various options for threshold management; the possibility of working with vectors having all zero components in some columns, etc. A theoretical GB analysis for single-step retrieval, as well as an experimental comparison with the Hopfield and the Willshaw networks for multi-step retrieval is given in [59], see also Subsection "Willshaw networks with sparse vectors". Processing of realistic data. To represent arbitrary binary vectors in the Potts network, they are divided into segments of dimension log d and each segment is encoded by the activation of one neuron of its d -dimensional column [97]. Components of integer vectors can be represented in a similar way. Simulated data are typically generated as independent random samples. This 16 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey ensures that the vectors to be stored are very nearly equally and maximally dissimilar. However, data generated from situations in the world is very unlikely to be so neatly distributed. When working with (unevenly distributed) real data, NAM is used non-optimally (many connections are not modified, others are "oversaturated"). To overcome this in the GB network, a free column neuron is allocated when the number of connections of a neuron (encoding some value) exceeds the threshold [19]. During retrieval, all column neurons that encode a certain value are activated. For better balancing the number of connections, in [64] the number of neurons in the column allocated to represent a vector component is proportional to the frequency of its 1-value in the vectors of the base. During storage, the various neurons representing the component are activated in turn. The authors of [64] also propose an algorithm for finding (with a high probability) all vectors of the base closest to the query vector distorted by deletion; the algorithm often significantly reduces the number of required queries. GENERALIZATION IN NAMs The Hebbian learning in matrix-type distributed memories naturally builds a kind of correlation matrix where the frequencies of joint occurrence of active neurons are accumulated in the updated connection weights. The neural assemblies thus formed in the network may have a complex internal structure reflecting the similarity structure of stored data. This structure can be revealed as stable states of the network — in the general case, different from the stored data vectors. That is, it is possible for vectors retrieved from a network to not be identical to any of the vectors stored in the network (which is generally undesirable). Similarity preserving binary vectors. Similarity of patterns of active neurons (represented as binary vectors) are assumed to reflect the similarity of items (of various complexity and generality) they encode. The similarity value is measured in terms of the number or fraction of common active neurons (or overlap, i.e. normalized dot product of the representing binary vectors). Moreover, the similarity "content" is available as the identities of common active neurons (the IDs of the common 1-components of the representing binary vectors). Note that such data representation schemes by similarity preserving binary vectors have been developed for objects represented by various data types, mainly for (feature) vectors (see survey in [131]), but also for structured data types such as sequences [102, 72, 85,86] and graphs [127, 128, 148, 136, 62, 134]. A significant part of this research is developed in the framework of distributed representations [45, 76, 106, 126, 89], including binary sparse distributed representations [102, 98, 103, 127, 128, 113, 114, 137, 138, 139, 148, 135, 136, 61, 134, 129, 130, 131, 132, 31, 33] and dense distributed representations [75, 76] (see [82, 84, 87, 88, 83] for examples of their applications). Complex internal structure of cell assemblies for graded connections. When binary vectors reflecting similarity of real objects are stored in a NAM by variants of the Hebb (or Hopfield) rule, the weights of connections between the neurons frequently activated together will be greater than the mean value of all ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 17 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov the weights. On the other hand, rare combinations of active neurons will have smaller weights. Thus, neuron assemblies (cell assemblies in terms of Hebb [65]) formed in the network may have a very complicated structure. Hebb and Milner introduced the notions of "cores" and "fringes" ([65] pp. 126–134; for more recent research see [101]) to characterize qualitatively such complex internal structure of assemblies. The notions of core (kernel, nucleus) and fringe (halo) of assemblies have attracted attention to the function of assemblies distinct from the function of associative memory. This different function is not just memorization of individual activity patterns (vectors), but emergence of some generalized internal representations, that were not explicitly presented to the network as vectors for memorization. Some assembly cores may correspond to "prototypes" representing subsets of attributes (encoded by the active neurons represented by the 1-components of the corresponding vectors) often present together in some input vectors. Note that some of these subsets of components/attributes may never be present in any single vector. Stronger cores, corresponding to stable combinations of a small number of typical attributes present in many vectors employed for learning, may correspond to some more abstract or general object category (class). Cores formed by more attributes may represent more specific categories, or object prototypes. However, object tokens (category instances) may also have strong cores if they were often presented to the network for learning. Note that some mechanisms may exist to prevent repeated learning of vectors that are already "familiar" to the network. Also, the rate of weight modification may vary based on the "importance" of the input vector. The representations of real objects have different degrees of similarity with each other. Similarities in various combinations of features often form different hierarchies of similarities that reflect hierarchies of categories of different degrees of generalization (object — class of objects — a more general class, etc.). So, assemblies formed in the network (cores and fringes of different "strength") may have a complex and rich hierarchical structure, with multiple overlapping hierarchies reflecting the structure of different contents and values of similarity implicitly present in the base of vectors used for the unsupervised network learning by the employed variant of the Hebb rule. Thus, many types of the category-based hierarchies (also known as generalization or classification or type-token hierarchies) may naturally emerge in the internal structure of assemblies formed in a single assembly neural network (NAM). Complex internal structure of a neural assembly allows a virtually continuous variety of hierarchical transitions. To reveal various types of categories and prototypes and instances formed in the network, the corresponding assemblies should be activated. To activate only stronger cores, higher values of threshold should be used. Lower threshold values may additionally activate fringes. Research of generalization function in NAMs. Additional stable states that emerge in NAM after memorizing random base vectors and do not coincide with the base vectors are known as false or spurious states or memories, e.g., [69, 155]. In [155] they regarded the emergence of spurious attractors in the Hopfield networks as a side effect of the main function of distributed NAMs, consisting 18 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey not in memorizing individual patterns, but in formation of prototypes. Such an interpretation is close to the earlier work [13, 11, 12] that considered formation of concepts, prototypes, and taxonomic hierarchies as a natural generalization of correlated patterns memorized in a distributed memory. Research of hierarchically correlated patterns and states in Hopfield networks has been initiated by physicists who studied the "ultrametric" organization of ground states in spin-glasses (e.g., [123, 32, 63]; see also [6] and its references). While these earlier works required explicit representation of patterns at various hierarchical levels to be used in the learning rule, more neurobiologically plausible and practical Hebb and Hopfield rules applied to (hierarchically) correlated sparse binary patterns themselves (obtained with some simple correlation model) were considered, e.g., in [153, 66]. [153] theoretically showed "natural" formation of stable cores and fringes as well as traveling through different levels of hierarchies by uniform changing of the threshold. More complex probabilistic neuron dynamics and threshold control expressing neuronal fatigue was modeled in [66]. Dynamics of transitions between stable memory states that models human free recall data and can also be used with hierarchically organized data was considered in [143]. The "neurowindow" approach of [74] may be considered as using multiple thresholds to activate cores or fringes. Revealing the stable states corresponding to emergent assemblies is used for data mining (binary factor analysis) in [36, 37]. Generalization in NAMS with binary connections. The Willshaw learning rule does not form assemblies with the complex internal structure needed for generalization functions, such as emergence of generalization (type-token) hierarchies. The Willshaw learning rule causes the connectivity of an assembly (corresponding to a vector) to become full after a single learning act (vector storage) and not change thereafter. To preserve the capability of forming assemblies with a non-uniform connectivity in NAMs with binary connections, a stochastic analogue of the Hebbian learning rule for binary connections was proposed in [100, 98]: wij = wij Ú ( yi Ù y j Ù xij ) , where xij is a binary random variable equal to 1 with the probability that determines the learning rate. The connectivity value for some set of neurons is determined here by the number of their 1-weight connections. Neurons that have more than some fraction of 1-weight connections with the other neurons of the same assembly may be attributed to the core part of the assembly. In [15] they experimentally studied formation of assemblies with cores and fringes using the above mentioned "stochastic Willshaw" rule ( D = 4096, pD = 120 – 200, about 60 neurons in the core and 60–140 neurons in the fringe). Tests have been performed on retrieving a core by its part; a core and its full fringe by the core and a part of the fringe (the most difficult test); and, a core by a part of its fringe. As expected, experiments with correlated base vectors have shown a substantial decrease of storage capacity compared to random independent vectors. A special learning rule was proposed to increase the stability of fringes. ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 19 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov Formation of prototypes with the stochastic Willshaw rule was also investigated in [7, 23]; a model of paired-associate learning in humans is considered in [141]. Generalization in modular NAMs. A modular structure of neural networks where the Hebb assemblies are formed inside the modules was proposed and developed in [50–56]. The modular assembly neural network is intended for recognition of a limited number of classes. The network is artificially partitioned into several modules (sub-networks) according to the number of classes that the network is required to recognize. Each module network is full-connected, connections are graded. The features extracted from all objects of a certain class are encoded into activation of the patterns of neurons within the corresponding sub-network. After learning, the Hebb assemblies are formed in each module network. In this modular structure, the network acquires the capability to generalize the description of each class within the corresponding module (subnetwork), i.e. separately and completely independently from all other classes. In [56] it was shown that the number of connections in each module can be reduced without loss of the recognition capability. NAMs WITH HIGHER-ORDER CONNECTIONS AND WITHOUT CONNECTIONS Neural networks in the previous sections have connections of order n = 2 (а connection is between two neurons). In this section we consider values of n other than 2, for the NAMs with the structure of the Hopfield network (unlike Section "Hopfield NAMs" where we only considered the case n = 2 ). Neural networks with higher-order connections. In the higher-order (order n > 2 ) generalization of the Hopfield network, n neurons are connected by single connection instead of just two (for example, [124, 17, 46, 1, 70, 94, 26, 95, 96, 30]). For the neuron with states from {–1, + 1} ( p = 1/2), the network dynamics can be defined as zi = sign(å j ... j ¹i wij1 ... jn z j1 ... jn ) . 1 n The analogue of the Hebb learning rule becomes wi1 ...in = 1 D n -1 å m =1, N y im y im2 ... y im . 1 n Other learning rules can also be used. The number of stable states corresponding to the stored random binary vectors (possibly slightly different from them) is estimated in the mentioned papers to be N crit » a crit (n) D n -1 . As in NAMs with connections between pairs of neurons, a crit depends on the specific type of learning rule and network dynamics. a crit does not exceed 2 and decreases with increasing n [94]. For the absence of errors (with a probability approaching 1), the number of stored vectors N » 1 D n -1 / ln D cn 20 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey (for example, [46, 26, 95, 30]). In [95] they obtain cn > 2(2 n - 3)!! So, the exponential in n growth of N is due to the exponential growth of the connection number, and the characteristics per connection deteriorate with increasing n . The generalization of Krotov-Hopfield. For networks with higher-order connections, the network energy in [95] is written as -å m =1, N F (á z, y m ñ) with a smooth function F (u) . For polynomial F (u) and n = 2 , this gives the energy of the usual Hopfield network ([68] and Subsection "Hopfield networks with dense vectors"). For small n , many memory vectors y m have approximately the same values of F (u) and make a comparable contribution to the energy. For n ® ¥ , the main contribution to the energy is given by the memory vector y with the largest á z , yñ . For intermediate n , a large contribution is made by several nearest memory vectors. In [30] it is proved that for F (u) = exp(u) this memory allows one to retrieve N = exp(a D) randomly distorted vectors (within dist Ham < D / 2 from the stored vectors) by a single step of the sequential dynamics, for some 0 < a < ln 2 / 2 , depending on the distortion, with probability converging to 1 for D®¥. In [95] they consider the operation of such a network in the classification mode, where each stored base vector corresponds to one of the categories to be recognized. In particular, to classify the handwritten digits of the MNIST base into 10 classes, in addition to the "visible" neurons to which 28x28 images (with pixel values in [–1, + 1]) are input, there are 10 "classification" neurons. The value of the output is obtained by a non-linearity g ( s ) applied to the input sum s , for example, tanh( s) (instead of the sign( s) function used in the memory mode). The outputs of visible neurons are fixed, and the outputs of classification neurons are determined by a single step of the dynamics. Vectors of N memory states are formed by learning on the training set. The N = 2000 memory vectors minimizing the classification error for the 60,000 images of the MNIST training set were obtained with the stochastic gradient descent algorithm. For a single step of the dynamics this structure is equivalent to a perceptron with one layer of N hidden neurons [95]. The nonlinearity at the output of the visible neurons is f (u ) = F (u) , and that at the output of hidden neurons is g (u ) . The learned memory vectors (with components normalized to [–1, + 1]) are encoded in the weight vectors of the connections between the visible neurons and the hidden neurons. It is shown that when n changes the visualized memory vectors change. For small n , the memorized vectors correspond to the features of the digit images, and for large n they become prototypes of individual digits [95, 96]. Neural networks with first-order connections. In a neural network with connection order n = 1 , each neuron is connected only with itself. They can be ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 21 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov considered as networks without connections, where learning changes the state of the neurons themselves (with "neuron plasticity" [39]). Thus, memory is a single vector of the dimension of the vectors of the base. For binary connections, we get the Bloom filter (see the reviews [22, 149]), which exactly recognizes the absence of an undistorted query vector in the stored database by absence of at least one of its 1-components in the memory vector. If the 1-components of the query vector are a subset of 1-components of the memory vector, the vector is recognized as the base vector, but it is necessary to check this, since there is a false positive probability due to "ghosts" (vectors not from the base, the 1-components of which belong to the memory vector). Ghosts can be considered as analogous to spurious memories (Subsection "Research of generalization function in NAMs"). An analysis of the probability of their appearance under certain restrictions on stored random vectors is given in [144]. In [158], they reduce the probability of false positives. In [57], a Bloom filter version is analyzed which recognizes the absence of distorted query vectors. The autoscaling Bloom filter approach proposed in [92] suggests a generalization of the counting Bloom filter approach based on the mathematics of sparse hyperdimensional computing and allows elastic adjustment of its capacity with probabilistic bounds on false positives and true positives. In [90], the formation of sparse memory vectors (with an additional operation of context-dependent thinning [134]) is considered, and in [91] the probability of correct recognition is estimated. The use of graded connections (the formation of the memory vector is done by addition), including subsequent binarization, and the classification problem for vectors not from the base, are considered in [89, 91]. For real-valued vectors and connections, the recognition of random undistorted vectors is analyzed in [10, 126]. In [73] they allow distortion of vectors. In [126, 73], the analysis of non-random base vectors is given. NAMs WITH A BIPARTITE GRAPH STRUCTURE FOR NONBINARY DATA WITH CONSTRAINTS In some recent papers (e.g., [78, 145, 110, 111]), in order to create NAMs which can store and retrieve (from rather noisy input vectors) the number N of (not always binary) vectors with N near exponential in D , the vectors considered are not arbitrary random but satisfy (linear) constraints. The neural network has the structure of a bipartite graph. One set of neurons (not connected with each other) is used to represent the vectors of the base, neurons of the other set represent constraints. A rectangular matrix of connections between these two sets is learned on the vectors of the base. The connection vector of each constraint neuron represents the vector of that particular constraint. Iterative algorithms with local neuron computations are used for retrieval. Iterative algorithms for learning constraint matrix and vector recovery . In [78, 145], they consider the problem of the exact retrieval (with high probability) of vectors that belong to a subspace of dimension less than D . The graded weights of the bipartite graph connections representing linear constraints are learned from the vectors of the base (which have only non-negative integer components). Iterative algorithms are used for learning. The weights are constrained to be sparse, which is required for analyzing the retrieval algorithm. 22 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey The input (query) vectors x are obtained from the vectors y of the base by additive noise: x = y + e , where e are random sparse vectors with (bipolar) integer components. During retrieval, activity propagates first from the data neurons to the constraint neurons and then in the opposite direction, and so on for multistep retrieval. Non-linear transformations are used in neurons. In a stable state, the data neurons represent a base vector, and all constraint neurons obtain a total weighted zero input from the associated data neurons. In [78] the vectors are divided into intersecting parts. Any part of the vector belongs to a subspace of smaller dimension than the vector dimension of that part. А subset of the constraint neurons corresponds to each part. They are not looking for an orthogonal basis of constraints, but for vectors orthogonal to the corresponding parts of the data vectors from the base: W ( k ) y ( k ) = 0 , where k is the part number. To do this, the objective function is formulated and optimized with a stochastic gradient descent (several times for each part). During retrieval, they first independently correct errors in each part by performing several steps of the network dynamics. The correction is based on the fact that W ( k ) y ( k ) = W ( k ) e( k ) . Then, exploiting intersection of the parts, the parts without errors are used to correct the parts with errors. In [145], y from a subspace of dimension d < D are considered. Training forms a matrix W of D - d non-zero linearly independent vectors orthogonal to the vectors y of the base: Wy = 0 for all y of the base. An iterative algorithm of activity propagation in the network retrieves y . Algorithms [78, 145], described above, are claimed to store the number N of vectors (generated from their respective data models) exponential in D ( O( a D ) , a > 1 ) with the possibility of correcting a number of random errors that is linear in the vector dimension, D . However, to ensure a high probability of retrieval, a graph with a certain structure must be obtained, which is not guaranteed by the learning algorithms used. NAMs based on sparse recovery algorithms. To create autoassociative memory on the basis of a bipartite graph, in [110, 111] they use connection matrices W which allow them to reconstruct a sparse noise vector e which additively distorts the vector y of the base to form the query vector x . Then the required base vector is obtained as y = x - e . The noise vector is calculated using sparse recovery methods (that is, methods that find the solution vector with the least number of non-zero components). These methods require knowledge of the linear constraints matrix W such that Wy = 0 for all vectors y of the base. For some models of vectors (i.e. constraints or generative processes for the base vectors), such W can be obtained in polynomial time from the base of vectors generated by the model. In contrast to [78, 145], finding W is guaranteed with high probability, and adversarial rather than random errors are used as noise. In [110] real-valued vectors are used as the base, satisfying a set of nonsparse linear constraints. The data model, where the vectors of the base are given ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 23 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov by linear combinations of vectors with sub-Gaussian components, allows storing the number of vectors N up to exp( D3/4 ) . The data model with a basis of orthonormal vectors provides N up to exp(d ) , where 1 £ d £ D . Both models allow for accurate recovery from vectors with significant noise. In [111], as in [145], the vectors of the base are from a subspace defined by sparse linear constraints. They consider both real-valued vectors and binary vectors from {-1, +1}D satisfying W models of a certain type (sparse-subGaussian model). Learning is based on solving the dictionary learning problem with a square dictionary [111]. An iterative retrieval algorithm uses the fact that W is an expander graph with good properties [111]. The memory capacity and resistance to distortion is increased relative to [110]. Note the drawback of the methods considered in this section is that bases of real data may not correspond to the data models used. DISCUSSION In addition to being an interesting model of biological memory, neural network autoassociative distributed memories (NAMs) have also been considered as index structures that give promise to speed up nearest neighbor search relative to linear search (and, hopefully, to some other index structures). This mainly concerns sparse binary vectors of high dimension, because the number of such vectors that it is possible to memorize and retrieve from a significantly distorted version may far exceed the dimensionality of the vectors in some matrix-type NAMs, and the ratio N / D may be similar to the speed-up relative to linear search (see the first four Sections). Distributed NAMs have some drawbacks relative to traditional computer science methods for nearest neighbor search. The vector retrieved by a NAM may not be the nearest neighbor of the query vector. This could be tolerable if the output vector is an approximate nearest neighbor from the set of stored vectors. However, in NAMs the output vector may not even be a vector of the base set ("spurious memories"). (Up to a certain number of stored vectors and query vector distortion these problems remain insignificant.) For dense binary vectors, the number of vectors able to be reliably stored and retrieved is (much) smaller than the vector dimension. Also, NAMs are usually analyzed for the average case of random vectors and distortions, whereas real data are not like that, which results in poorer performance. However, available comparisons with the inverted index for sparse binary vectors in the average case do not clearly show the advantage of one or other algorithm in query time (Subsections "Hopfield networks with sparse vectors", "Willshaw networks with sparse vectors") An obvious approach to improve the memory and time complexity of the matrix-type NAMs from quadratic to linear in vector dimension is the use of incompletely connected networks with constant (but rather large) number of connections per neuron. An interesting direction is index structures for similarity search in which NAM modules are used at some stages. The index structure of Subsection "Willshaw networks in the index structures for nearest neighbor search" uses several NAMs to memorize parts of the base, and the similarity of the result of 24 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey single-step retrieval with the query vector is used to select the "best" NAM on which to perform an exact linear search against its stored vectors. Some studies are aimed at more efficient use of NAMs when working with real data. For example, the GB network with binary connections uses different neurons to represent the same component of the source vector, which allows for more balanced use of connections. In Section "Generalization in NAMs" we discussed the use of NAMs for generalization rather than exact retrieval from associative memory. In the NAMs that use versions of the Hebb learning rule, storage of vectors (even random ones) is accompanied by emergence of additional stable states. For correlated vectors, their common 1-components become "tightly" connected and stable states corresponding to them arise. Revealing these stable states can be used for data mining, e.g. for binary factor analysis [36, 37]. Research of complex (possibly hierarchical) structure of stable states (discussed in terms of cores and fringes of neural assemblies) may appear useful both for modeling brain function and for applications. Real data in many cases are not binary sparse vectors of high dimension with which the NAMs considered in the first four Sections work best. So, similarity preserving transformations to that format are required (Subsection "Similarity preserving binary vectors"). However, the obtained vectors (as well as the initial real data) are not random and independent, so the analytical and experimental results available for random, independent vectors usually can not predict NAM characteristics for real data. Using data vectors (often non-binary) that satisfy some linear constraints (instead of random independent vectors) allows bipartite graph based NAM construction with capacity near exponential in the vector dimension (Section "NAMs with a Bipartite Graph Structure for Nonbinary Data with Constraints"). However, again, this requires data from specific vector models (to which real data often do not fit). In NAMs with higher-order connections, connections are not between a pair, but between a larger number of neurons (this number being the order). So, the NAM becomes of tensor-type instead of matrix-type. These NAMs (Section "NAMs with Higher-Order Connections and without Connections") allow storing the number of dense vectors exponential in the order. However, this is achieved by the corresponding increase in the number of connections, and therefore in memory and in query time. The higher-order NAMs are generalized in [95, 96], where, roughly, the sum of polynomial functions of the dot products between all memory vectors and the network state is used as the input sum of a neuron. Such a treatment makes it possible to draw interesting analogies with perceptrons and kernel methods in the classification problem. However, for nearest neighbor search, this seems impose a query time exceeding that of linear search. Overcoming these and other drawbacks and knowledge gaps, and improving NAMs are promising topics for further research. Let's note other directions of research in fast similarity search of binary (non- sparse) vectors. Examples of index structures for exact search are [28] (with a fixed query radius and analysis for worst-case data; however impractical due to the small query radius required for sub-linear query time and moderate ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 25 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov memory costs) and [116] (practical, with variable radius of the query and analysis for random data). Theoretical algorithms for approximate search (providing: sublinear search time, a specified maximum difference of the result from the result of the exact search, and no false negatives) in [2] are modifications of more practical algorithm classes related to Locality Sensitive Hashing and Locality Sensitive Filtering (see [18, 14, 147]). However, the latter allow false negatives (with low probability). Unlike NAMs, these algorithms provide guarantees for the worstcase data, but require a separate index structure for each degree of distortion of the query vector. 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Фролов2, д-р биол.наук, проф., факультет электротехники и информатики, e-mail: docfact@gmail.com Р. Гейлер3 PhD (психология), исследователь, e-mail: r.gayler@gmail.com Д. Клейко4, аспирант, факультет информатики, электрической и космической техники. e-mail: denis.kleyko@ltu.se Е. Осипов4, PhD (информатика), проф., факультет информатики, электрической и космической техники e-mail: evgeny.osipov@ltu.se 1 Международный научно-учебный центр информационных технологий и систем НАН Украины и МОН Украины, пр. Академика Глушкова, 40, г. Киев, 03187, Украина 2 Технический университет Остравы, 17 listopadu 15, 708 33 Острава-Поруба, Чешская Республика 3 Мельбурн, штат Виктория, Австралия 4 Технологический университет Лулео, 971 87 Лулео, Швеция НЕЙРОСЕТЕВАЯ РАСПРЕДЕЛЕННАЯ АВТОАССОЦИАТИВНАЯ ПАМЯТЬ: ОБЗОР В настоящем обзоре рассмотрены модели автоассоциативной распределенной памяти, которые могут быть естественным образом реализованы нейронными сетями. Модели используют для запоминания векторов в основном локальном правиле обучения путем модификации значений весов межнейронных связей, которые существуют между всеми нейронами (полносвязные сети). В распределенной памяти различные векторы запоминают в одних и тех же ячейках памяти, которым в рассматриваемом случае нейронной сети соответствуют одни и те же связи. Обычно исследуют запоминание векторов, случайно выбранных из некоторого распределения. При подаче на вход автоассоциативной памяти искаженных вариантов запомненных в ней векторов осуществляется извлечение (восстановление) ближайшего запомненного вектора. Это реализуется за счет итеративной динамики нейронной сети на основе локально доступной в нейронах информации, полученной по связям от других нейронов сети. Вплоть до определенного количества запомненных в сети векторов и степени их искажения на входе, в результате динамики сеть с симметричными связями приходит в устойчивое состояние, соответствующее запомненному в сети вектору, имеющему наибольшее сходство с входным вектором (сходство обычно измеряют в терминах скалярного произведения). Такие нейросетевые варианты автоассоциативной памяти позволяют запомнить с возможностью восстановления такого количества векторов, которое может превышать размерность векторов (совпадающую с количества нейронов в сети). Для векторов большой размерности это открывает возможность поиска приближенного ближайшего соседа с временной сложностью, сублинейной от количества запомненных в нейронной сети векторов. К недостаткам такой памяти относится то, что восстановленный динамикой сети вектор может не быть ближайшим ко входному или даже может вообще не принадлежать к множеству запомненных векторов и значительно отличаться от любого из них. Исследования различных типов нейросетевой автоассоциативной памяти направлены на выявление диапазонов параметров, при которых указанные недостатки проявляются с малой вероятностью, а достоинства выражены в максимальной степени. ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) 33 V.I. Gritsenko, D.A. Rachkovskij, A.A. Frolov, R. Gayler, D. Kleyko, E. Osipov Основное внимание уделено сетям с парными связями типа Hopfield, Willshaw, Potts и работе с бинарными разреженными векторами (векторами с количеством единичных компонентов, малым по сравнению с количеством их нулевых компонентов), т.к. только для таких векторов удается запомнить с возможностью восстановления большое количество векторов. Помимо функции автоассоциативной памяти, для этих сетей также обсуждается функция обобщения. Обсуждаются также неполносвязные сети. Кроме того, рассмотрена автоассоциативная память в нейронных сетях со связями высшего порядка — то есть со связями не между парами, а между большим количеством нейронов. Рассмотрена также автоассоциативная память в нейронных сетях со структурой двудольного графа, где одно множество нейронов представляет запоминаемые векторы, а другое — линейные ограничения, которым они подчиняются. Эти сети выполняют функцию автоассоциативной памяти и для небинарных данных, удовлетворяющих заданной модели ограничений. Обсуждаются отношение рассмотренных в обзоре моделей нейросетевой автоассоциативной распределенной памяти к проблематике поиска по сходству, достоинства и недостатки рассмотренных методов, направления дальнейших исследований. Один из интересных и все еще не полностью разрешенных вопросов заключается в том, может ли нейронная автоассоциативная память искать приближенных ближайших соседей быстрее других индексных структур для поиска по сходству, в частности, для случая векторов очень больших размерностей. Ключевые слова: распределенная ассоциативная память, разреженный бинарный вектор, сеть Хопфилда, память Уиллшоу, модель Поттса, ближайший сосед, поиск по сходству. В.І. Гриценко1, член-кореспондент НАН України, директор, e-mail: vig@irtc.org.ua Д.А. Рачковський1, д-р техн. наук, пров. наук. співроб. відд. нейромережевих технологій оброблення інформації, e-mail: dar@infrm.kiev.ua А.А. Фролов2, д-р біол. наук, проф., факультет електротехніки та інформатики, e-mail: docfact@gmail.com Р. Гейлер3, PhD (психологія), дослідник, e-mail: r.gayler@gmail.com Д. Клейко4, аспірант, факультет інформатики, електричної та космичної техніки. e-mail: denis.kleyko@ltu.se Е. Осипов4, PhD (інформатика), проф., факультет інформатики, електричної та космичної техніки e-mail: evgeny.osipov@ltu.se 1 Міжнародний науково-учбовий центр інформаціоних технологій та систем НАН України та МОН України, пр. Академіка Глушкова, 40, м. Київ, 03187, Україна 2 Технічний університет Острави, 17 listopadu 15, 708 33 Острава-Поруба, Чеська Республіка 3 Мельбурн, штат Вікторія, Австралія 4 Технологічний університет Лулео, 971 87 Лулео, Швеція НЕЙРОМЕРЕЖНА РОЗПОДІЛЕНА АВТОАССОЦІАТИВНА ПАМ’ЯТЬ: ОГЛЯД У цьому огляді розглянуто моделі автоасоціативної розподіленої пам’яті, які можуть бути природним чином реалізовані нейронними мережами. Моделі використовують для запам’ятовування векторів в основному локальному правилі навчання шляхом 34 ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) Neural Distributed Autoassociative Memories: A Survey модифікації значень ваг міжнейронних зв’язків, які існують між всіма нейронами (повнозв’язні мережі). У розподіленій пам’яті різні вектори запам’ятовуються в одних і тих самих елементах пам’яті, яким в цьому випадку нейронної мережі відповідають одні і ті ж зв’язки. Зазвичай досліджують запам’ятовування векторів, випадково вибраних з деякого розподілу. Якщо на вхід автоасоціативної пам’яті подаються спотворені варіанти запам’ятованих в ній векторів, здійснюється витяг (відновлення) найближчого раніше запам’ятованого вектора. Це реалізується за рахунок ітераційної динаміки нейронної мережі на основі локально доступної в нейронах інформації, отриманої від інших нейронів мережі. До певної кількості запам’ятованих в мережі векторів і ступеня їх спотворення на вході, в результаті динаміки мережа із симетричними зв’язками приходить в стійкий стан, відповідний запам’ятованому в мережі вектору, який має найбільшу схожість з вхідним вектором (схожість зазвичай вимірюють як скалярний добуток). Такі нейромережні варіанти автоасоціативної пам’яті дозволяють запам’ятати з можливістю відновлення таку кількість векторів, яка може перевищувати розмірність векторів (що збігається з кількістю нейронів в мережі). Для векторів великої розмірності це відкриває можливість пошуку наближеного найближчого сусіда з складністю, сублінейною від кількості запам’ятованих в нейронній мережі векторів. До недоліків такої пам’яті відноситься те, що відновлений динамікою мережі вектор може не бути найближчим до вхідного або навіть може взагалі не належати до множини запам’ятованих векторів і значно відрізнятися від будь-якого з них. Дослідження різних типів нейромережної автоасоціативної пам’яті спрямовано на виявлення діапазонів параметрів, при яких зазначені недоліки проявляються з малою імовірністю, а достоїнства виражені в максимальному ступені. Основну увагу приділено мережам з парними зв’язками типу Hopfield, Willshaw, Potts і роботі з бінарними розрідженими векторами (векторами з кількістю одиничних компонентів, яке є малим у порівнянні з кількістю їх нульових компонентів), так як тільки для таких векторів вдається запам’ятати з можливістю відновлення велику кількість векторів. Крім функції автоасоціативної пам’яті, для цих мереж також обговорюється функція узагальнення. Обговорюються також неповнозв’язкові мережі. Крім того, розглянуто автоасоціативну пам’ять в нейронних мережах зі зв’язками вищого порядку — тобто зі зв’язками не між парами, а між великою кількістю нейронів. Розглянуто також автоасоціативна пам’ять в нейронних мережах зі структурою двудольного графа, де одна множина нейронів надає вектори, які запам’ятовуються, а інша — лінійні обмеження, яким вони підкорюються. Ці мережі виконують функцію автоасоціативної пам’яті також для небінарних даних, які відповідають заданій моделі обмежень. Обговорюються можливості використання розглянутих в огляді моделей нейромережної автоасоціативної розподіленої пам’яті у проблематиці пошуку за схожістю, достоїнства і недоліки розглянутих методів, напрямки подальших досліджень. Один із цікавих і все ще не повністю вирішених питань полягає в тому, чи може нейронна автоасоціативна пам’ять шукати наближених найближчих сусідів швидше інших індексних структур для пошуку за схожістю, зокрема, у випадку векторів дуже великих розмірностей. Ключові слова: розподілена асоціативна пам’ять, розріджений бінарний вектор, мережа Хопфілда, пам’ять Уілшоу, модель Потса, найближчий сусід, пошук за схожістю. ISSN 2519-2205 (Online), ISSN 0454-9910 (Print). Киб. и выч. техн. 2017. № 2 (188) View publication stats 35	9
Multi-Rate Control over AWGN Channels via Analog Joint Source–Channel Coding arXiv:1609.07715v3 [cs.SY] 28 Oct 2016 Anatoly Khina, Gustav M. Pettersson, Victoria Kostina and Babak Hassibi Abstract— We consider the problem of controlling an unstable plant over an additive white Gaussian noise (AWGN) channel with a transmit power constraint, where the signaling rate of communication is larger than the sampling rate (for generating observations and applying control inputs) of the underlying plant. Such a situation is quite common since sampling is done at a rate that captures the dynamics of the plant and which is often much lower than the rate that can be communicated. This setting offers the opportunity of improving the system performance by employing multiple channel uses to convey a single message (output plant observation or control input). Common ways of doing so are through either repeating the message, or by quantizing it to a number of bits and then transmitting a channel coded version of the bits whose length is commensurate with the number of channel uses per sampled message. We argue that such “separated source and channel coding” can be suboptimal and propose to perform joint source– channel coding. Since the block length is short we obviate the need to go to the digital domain altogether and instead consider analog joint source–channel coding. For the case where the communication signaling rate is twice the sampling rate, we employ the Archimedean bi-spiral-based Shannon– Kotel’nikov analog maps to show significant improvement in stability margins and linear-quadratic Gaussian (LQG) costs over simple schemes that employ repetition. Index Terms— Networked control, Gaussian channel, joint source–channel coding. I. I NTRODUCTION Networked control systems, especially those for which the links connecting the different components of the system (plant, observer, and controller, say) are noisy, are increasingly finding applications and, as a result have been the subject of intense recent investigations [1]–[3]. In many of these applications the rate at which the output of the plant is sampled and observed, as well as the rate at which control inputs are applied to the plant, is different from the signaling rate with which communication occurs. We shall henceforth A. Khina, V. Kostina and B. Hassibi are with the Dept. of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA (E-mails: {khina, vkostina, hassibi}@caltech.edu). G. M. Pettersson is with the Dept. of Aeronautical and Vehicle Engineering, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden (E-mail: gupet@kth.se). The work of A. Khina was supported in part by a Fulbright fellowship, Rothschild fellowship and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłlodowska-Curie grant agreement No 708932. The work of G. M. Pettersson at Caltech was supported by The Boeing Company under the SURF program. The work of B. Hassibi was supported in part by the National Science Foundation under grants CNS-0932428, CCF-1018927, CCF-1423663 and CCF-1409204, by a grant from Qualcomm Inc., by NASA’s Jet Propulsion Laboratory through the President and Directors Fund, and by King Abdullah University of Science and Technology. call such systems multi-rate networked control systems. The rate at which the plant is sampled and controlled is often governed by how fast the dynamics of the plant is, whereas the signaling rate of the communication depends on the bandwidth available, the noise levels, etc. As a result, there is no inherent reason why these two rates should be related and, in fact, the communication rate is almost always higher than the sampling rate. This latest fact clearly gives us the opportunity to improve the performance of the system by having the possibility to convey information about each sampled output of the plant, and/or each control signal, through multiple uses of the communication channel. An obvious strategy is to simply repeat the transmitted signal (so-called repetition coding). In analog communication this simply adds a linear factor to the SNR (3 dB for a single repetition); in digital communication over a memoryless packet erasure link, say, it simply reduces the probability of packet loss exponentially in the number of retransmissions. A more sophisticated solution would be to first quantize the analog message (the sampled output or the control signal) and then protect the quantized bits with an error-correcting channel code whose block length is commensurate with the number of channel uses available per sample. A yet more sophisticated solution would be to use a tree code which collectively encodes the quantized bits in a causal fashion over all channel uses [4]–[6]. The latter two solutions implicitly assume what is called the “separation between source and channel coding”, i.e., that quantization of the messages and channel coding of the quantized bits (using either a block code or a tree code) can be done independently of one another. While this is asymptotically true in communication systems (where it is a celebrated result), it is not true for control systems where the overall objective is to minimize a linear-quadratic Gaussian (LQG) cost [7]. To minimize an LQG cost what is needed is joint source–channel coding (JSCC). Unfortunately, in its full generality, this is known to be a notoriously difficult problem and so it has rarely been attempted (especially, in a control context). Nonetheless, this is what we shall attempt in this paper. We assume the communication links are AWGN (additive white Gaussian noise) channels with a certain signal-to-noise ratio (SNR). As we show below, this SNR puts an upper limit on the size of the maximum unstable eigenvalue of the plant that can be stabilized. We further assume that the signaling rate of the communication channel is not much larger than the sampling rate of the plant, say only a factor of 2 to 10 larger. Thus, if one sets aside the (daunting) task of performing coding over multiple messages (a la tree codes) then one is left with constructing a joint source–channel code of relatively short length — something that could very well be feasible. In particular, since both the message and transmitted signals are analog, in this short block regime it is not even clear whether it is necessary to go through a digitization process. Thus, we shall focus on analog JSCC, originally proposed by Shannon [8] and Kotel’nikov [9], which can simply be viewed as an appropriately chosen nonlinear mapping from the analog message to the analog transmitted signal(s). Finally, we should mention that we view this work as a first step and the results as preliminary. Nonetheless, these already indicate that one can obtain substantial gains (in the LQG cost) over simple schemes, such as repetition, by using the ideas mentioned above. The design of more sophisticated JSCC schemes, as well as a comprehensive comparison of different schemes will be deferred to future work. II. P ROBLEM S ETUP We now formulate the control–communication setting that will be treated in this work, depicted also in Fig. 1. We concentrate on the simple case of a scalar full observable state and a scalar AWGN channel. The model and solutions can be extended to more complex cases of vector states and multi-antenna channels. Consider the scalar system with the plant evolution: xt+1 = αxt + wt + ut , (1) where xt is the (scalar) state at time t, wt is an AWGN of power W , α > 1 is a known scalar, and ut is the control signal. Assume further that x0 is Gaussian with power P0 . The measured output is equal to the state corrupted by noise: y t = x t + vt , (2) where vt is an AWGN of power V . In contrast to classical control settings, the observer and the controller are not co-located, and are connected instead via an AWGN channel bi = ai + ni , (3) where bi is the channel output, ai is the channel input subject to a unit power constraint, and ni is an AWGN of power 1/SNR.1 We assume an integer ratio KC between the sample rate of (2) and the signaling rate over the channel (3). That is, KC channel uses (3) are available per one control sample (1), (2). In this work we further assume that the observer knows all past control signals {ui \|i = 1, . . . , t − 1}; for a discussion of the case when such information is not available at the observer, see Section V. 1 This representation is w.l.o.g., as the case of an average power P and C noise power N , can always be transformed to an equivalent channel with average power 1 and √ noise power N/PC , 1/SNR by multiplying both sides of (3) by 1/ PC . vt wt Plant xt xt+1 = αxt + wt + ut yt ut Channel Controller/ Receiver ai Observer/ Transmitter bi ni Fig. 1. Scalar control system with white driving and observation AWGNs and an AWGN communication channel. The dashed line represents the assumption that the past control signals are available at the transmitter. Similarly to the classical LQG control (in which the controller and the observer are co-locatedco-located controller and observer), we wish to minimize the average stage LQG cost after the total number of observed samples T : " # T X 1 2 2 2 Qxt + Rut , J¯T , E F xT +1 + T t=1 for some non-negative constants F , Q and R, by designing appropriate operations at the observer [which also plays the role of the transmitter over (3)] and the controller [which also serves as the receiver of (3)]. The infinite horizon cost is defined as J¯∞ , lim J¯T . T →∞ To that end, we recall next known results from information theory for joint source–channel coding design with low delay. III. L OW-D ELAY J OINT S OURCE –C HANNEL C ODING In this section, we review known results from information theory and communications for transmitting an i.i.d. zeromean Gaussian source st of power PS over an AWGN channel (3). The number of source samples generated per a time instant is not necessarily equal to the channel uses of (3) per the same time. In general, consider the case where KC channel uses of bi are available for every KS source samples of st . The goal of the transmitter is to convey the source st to the receiver with a minimal possible average distortion, where the appropriate distortion measure for our case of interest is the mean square error distortion. To that end, the transmitter applies a mapping E that transforms every KS source samples to KC channel inputs: (a1 , . . . , aKC ) = E (s1 , . . . , sKS ) , such that the input power constraint is satisfied: i 1 h 2 E {E (s1 , . . . , sKS )} ≤ 1. KS The receiver, upon receiving the KC channel outputs of (3) — corresponding to the KC transmitted channel inputs — applies a mapping to these measured outputs to recover estimates ŝt of the source samples: (ŝ1 , . . . , ŝKS ) = D (b1 , . . . , bKC ) . The resulting average distortion of this scheme is D= KS h i 1 X 2 E (st − ŝt ) , KS t=1 and the corresponding (source) signal-to-distortion ratio (SDR) is defined as PS . D Our results here are more easily presented in terms of unbiased errors, as these can be regarded as uncorrelated additive noise in the sequel (when used as part of the developed control scheme). Therefore, we consider the use of (samplewise) correlation-sense unbiased estimators (CUBE), namely, estimators that satisfy SDR , SDRlin = 2SNR, E [st (st − ŝt )] = 0. We note that any estimator ŝB t can be transformed into a CUBE ŝt by multiplying by a suitable constant: E s2 ; ŝt = tB ŝB E st ŝt t for a further discussion of such estimators and their use in communications the reader is referred to [10]. Shannon’s celebrated result [11] states that the minimal achievable distortion, using any transmiter–receiver scheme, is dictated, in the case of a Gaussian source, by2 KS 2 log (1 + SDR) = KS R(D) ≤ KC C = KC 2 log (1 + SNR) where R(D) is the rate–distortion function of the source and C is the channel capacity [11]. Thus, the optimal SDR, commonly referred to as optimum performance theoretically achievable (OPTA) SDR, is given by SDROPTA = (1 + SNR) For finite blocklengths, (4) cannot be exactly attained, except for specific cases in which the source and the distortion measure are probabilistically matched to the channel [12], and strictly tighter outer bounds on the distortion can be derived [13]–[15]. One eminent case where such a matching occurs is that of a Gaussian source and a Gaussian channel with matching number of samples/uses KC = KS . In this case, sending each source sample as is, up to a possible power adjustment, proves optimal and achieves (4) with KC = KS = 1 (and hence also any other positive integer) [16]. Unfortunately, this breaks down when KC 6= KS , and consequently led to the proposal and study of various techniques for low-delay JSCC schemes.3 We next concentrate on the simple case of KS = 1 and KC = 2. That is, the case in which one source sample is conveyed over two channel uses. A naı̈ve approach is to send the source as is over both channel uses, up to a power adjustment. The corresponding unbiased SDR in this case is KC /KS − 1. (4) Shannon’s proof (for a more general case of not necessarily Gaussian source or channel) is based upon the separation principle, according to which the source samples are partitioned into blocks and quantized together, resulting in (approximately) uniform independent bits. These bits are then partitioned again into blocks and encoded together to form the channel inputs. At the receiver first the coded bits are recovered, followed by the reconstruction of the source samples from these bits. However, this compression–coding separation-based technique is optimal only in the limit when the blocklengths KS and KC grow to infinity for a fixed ratio between the two, which implies, in turn, very large delays. 2 The rate–distortion function here is written in terms of the unbiased SDR, in contrast to the more common biased SDR expression log(SDR). a linear improvement rather than an exponential one as in (4). This scheme approaches (4) for very low SNRs, but suffers great losses at high SNRs. We note that the linear factor 2 comes from the fact that the total power available over both channel uses has doubled, and the same performance can be attained by allocating all of the available power to the first channel use and remaining silent during the second channel use. This suggests that better mappings that truly exploit the extra channel use can be constructed. The first to propose an improvement for the 1:2 case were Shannon [8] and Kotel’nikov [9], in the late 1940s. In their works, the source sample is viewed as a point on a singledimensional line, whereas the two channel uses correspond to a two-dimensional space. In these terms, the linear scheme corresponds to mapping the one-dimensional source line to a straight line in the two-dimensional channel space (see Fig. 2), and hence clearly cannot provide any improvement (since AWGN is invariant to rotations). However, by mapping the one-dimensional source line into a twodimensional curve that fills better the space, a great boost in performance can be attained. Specifically, consider the Archimedean bi-spiral, which was considered in several works [17]–[20] (depicted in Fig. 2): ( reg areg s cos(ωs) = creg \|s\| cos(ω\|s\|) sign(s) 1 (s) = c reg a2 (s) = creg s sin(ωs) sign(s) = creg \|s\| sin(ω\|s\|) sign(s) (5) where ω determines the rotation frequency, the factor creg is chosen to satisfy the power constraint, and the sign(s) term is needed to avoid overlap of the curve for positive and 3 The term JSCC is somewhat misleading, as in many of these schemes there is no use of digital components, let alone coding, including the Shannon–Kotel’nikov (SK) maps which are described in detail and used in the sequel. mance (4) for a specific design SNR (4), and improve linearly for higher SNRs. Similar behavior is observed also in Fig. 3 where the optimal ω value varies with the (design) SNR, and mimics closely the quadratic growth in the SDR. Above the design SNR, linear growth is achieved for a particular choice of ω. We further note that the distortion component incurred when a threshold event happens, grows with \|s\|. To avoid stretch this behavior, instead of increasing the magnitude a stretch proportionally to the phase ∠ a as in (6), we increase it slightly faster at a pace that guarantees that the incurred distortion does not grow with \|s\|: ( abounded (s) = cbounded \|s\|λβ cos ω\|s\|λ sign(s) 1 (7) abounded (s) = cbounded \|s\|λβ sin ω\|s\|λ sign(s) 2 Fig. 2. Linear repetition and Archimedean spiral curves. negative values of s (for each of which now corresponds a distinct spiral, and the two meet only at the origin). This spiral allows to effectively improve the resolution w.r.t. small noise values, since the one-dimensional source line is effectively stretched compared to the noise, and hence the noise magnitude shrinks when the source curve is mapped (contracted) back. However, for large noise values, a jump to a different branch — referred to as a threshold effect — may occur, incurring a large distortion. Thus, the value ω needs to be chosen to be as large as possible to allow maximal stretching of the curve for the same given power, while maintaining a low threshold event probability. The SDRs for different values of ω are depicted in Fig. 3a. Another ingredient that is used in conjunction with (5) is stretching s prior to mapping it to a bi-spiral using φλ (s) , sign(s)\|s\|λ : ( stretch astretch (s) = areg \|s\|λ cos ω\|s\|λ sign(s) 1 1 (φλ (s)) = c stretch astretch (s) = areg \|s\|λ sin ω\|s\|λ sign(s) 2 2 (φλ (s)) = c (6) The choice λ = 0.5 promises great boost in performance in the region of high SNRs, as is seen in Fig. 3b. We further note that although the optimal decoder is a minimum mean square error (MMSE) estimator E [s\|b1 , b2 ], in this case, the maximum-likelihood (ML) decoder, p(b1 , b2 \|s), achieves similar performance for moderate and high SNRs. A joint optimization of λ and ω for each SNR, for both ML and MMSE decoding, was carried out in [19] and is depicted in Fig. 3. A desired property of the linear JSCC schemes is their SDR proportional improvement with the channel SNR (“SNR universality”). Such an improvement is not allowed by the separation-based technique, as it fails when the actual SNR is lower than the design SNR, and does not promise any improvement for SNRs above it. This motivated much work in designing JSCC schemes whose performance improves with the SNR, even for the case of large blocklengths [21]– [23]. The schemes in these works achieve optimal perfor- for some β > 1. This has only a slight effect on the resulting SDRs, as is illustrated in Fig. 3. Finally note that in no way do we claim that the spiralbased Shannon–Kotel’nikov (SK) scheme is optimal. Various other techniques exist, most using a hybrid of digital and analog components [24]–[26], which outperform the spiral-based scheme for various parameters. Nevertheless, this scheme is the earliest technique to be considered and gives good performance boosts which suffice for our demonstration. IV. C ONTROL VIA L OW-D ELAY JSCC In this section we construct a Kalman-filter-like solution [27] by employing JSCC schemes. We note that the additional complication here is due to the communication channel (3) and its inherent input power constraint. Denote by x̂rt1 \|t2 the estimate of xt1 at the receiver given {bi \|i = 1, . . . , t2 }, where ‘r’ stands for ‘receiver’, and by x̂tt1 \|t2 the estimate of xt1 given {yi \|i = 1, . . . , t2 }, where ‘t’ stands for ‘transmitter’. Denote their mean square erh further i h i rors (MSEs) by Ptr1 \|t2 , E x̃rt1 \|t2 and Ptt1 \|t2 , E x̃tt1 \|t2 , where x̃tt1 \|t2 , xt1 − x̂tt1 \|t2 and x̃rt1 \|t2 , xt1 − x̂rt1 \|t2 . Then, the scheme works as follows. At time instant t, the controller constructs an estimate x̂rt\|t of xt . It then applies the control signal ut = −Lt x̂rt\|t to the plant, for a pre-determined gain Lt 6= 0. Note that, since both the controller and the observer know the previously applied control signals {uj \|j = 1, . . . , t}, they also know x̂rt\|t and x̂rt+1\|t . Hence, in order to describe xt , the controller aims to convey its best estimate of the state x̂tt\|t . To that end, it can save transmit power by transmitting the error signal (x̂tt\|t − x̂rt\|t−1 ), instead of x̂tt\|t . The controller can then add back x̂rt\|t−1 to the received signal to construct x̂rt\|t . Remark 1: Note that even in the case of a fully observable state, i.e., when V = 0, the state is corrupted by nt when conveyed over the AWGN channel (3) to the controller. The performance of the transmission and the estimation processes applied by the observer and the controller, respectively, determine in turn, the total effective observation noise. The general scheme used throughout this work is detailed below. • Sends the KC channel inputs ai over the channel (3). Controller/Receiver: At time t • Receives the KC channel outputs bi corresponding to time sample t. ˆt = s̄t +neff • Recovers a CUBE of the source signal s̄t : s̄ t , eff where nt ⊥ s̄t is an additive noise of power of (at most) 1/SDR0 . ˆt to construct an estimate of st : • Unnormalizes s̄ q t s̄ r ˆt (10a) − Pt\|t ŝt = Pt\|t−1 q r t = Pt\|t−1 − Pt\|t s̄t + neff (10b) t q t neff . (10c) r − Pt\|t = x̂tt\|t − x̂rt\|t−1 + Pt\|t−1 t • (a) λ = 1. Constructs an estimate x̂rt\|t of xt from all received channel outputs until and including at time t. Since ŝt ⊥ x̂rt\|t−1 , the linear MMSE estimate amounts to4 x̂rt\|t = x̂rt\|t−1 + SDR0 ŝt , 1 + SDR0 (11) with an MSE of 1 r t Pt\|t−1 + SDR0 Pt\|t . 1 + SDR0 Generates the control signal (Lt is given next): r Pt\|t = • (12) ut = −Lt x̂rt\|t , and the receiver prediction of the next system state x̂rt\|t−1 = αx̂rt−1\|t−1 + ut−1 . (b) λ = 0.5. Fig. 3. Performances of the JSCC linear repetition scheme, OPTA bound, and the JSCC SK spiral scheme for optimized λ and ω, for the standard case (β = 1) and distortion-bounded case. The solid lines depict the performance of the standard spiral for various values of ω for two stretch parameters λ = 0.5 and 1, which perform better at high and low SNRs, respectively. Observer/Transmitter: At time t • Generates the desired error signal • • St = α2 R St+1 + Q, St+1 + R (13b) ST = F . (13c) Using (12) and (1), the prediction error at the decoder is given by the following recursion: α2 r t r Pt+1\|t = Pt\|t−1 + SDR0 Pt\|t + W. (14) 1 + SDR0 Scheme 1: st = x̂tt\|t − x̂rt\|t−1 (8a) = x̃rt\|t−1 − x̃tt\|t (8b) r Pt\|t−1 The control (LQG) signal gain Lt is given by (see, e.g., [27]): αSt+1 , (13a) Lt = St+1 + R t Pt\|t of average power − (determined in the sequel). Since the channel input is subject to a unit power constraint, st is normalized: 1 s̄t = q st . (9) r t Pt\|t−1 − Pt\|t Constructs KC channel inputs ai corresponding to s̄t , using a bounded-distortion JSCC scheme of choice of rate ratio 1 : KC with (maximum given any input) average distortion 1/SDR0 for the given channel SNR. The estimates x̂tt\|t can be generated via Kalman filtering (see, e.g., [27]): ỹt = yt − αx̂tt−1\|t−1 − ut−1 , x̂tt\|t = αx̂tt−1\|t−1 + ut−1 + Ktt ỹt (15a) , (15b) where the Kalman filter coefficients are generated via the recursion [27]: Ktt = t Pt\|t−1 t Pt\|t−1 +V , (16a) 4 If the resulting effective noise neff is not an AWGN with power that t does not depend on the channel input, then a better estimator than that in (11) may be constructed. t t = α2 Pt\|t−1 1 − Ktt + W, Pt+1\|t t Pt\|t = Ktt V. (16b) (16c) The recursive relations (14) and (16) lead to the following condition for the stabilizability of the control system. Theorem 1 (Achievable): The scalar control system of Section II is stabilizable using Scheme 1 if α2 < 1 + SDR0 , and its infinite-horizon average stage LQG cost J¯r is upper bounded by Q + α2 − 1 S P t − P̄ t , (17a) J¯r ≤ J¯t + 2 1 + SDR0 − α J¯t = QP̄ t + S P t − P̄ t , (17b) where J¯t is the average stage cost achievable at the observer,5 and P t and P̄ t are the infinite-horizon values of t t Pt\|t−1 and Pt\|t , respectively; S and P t are given as the positive solutions of S 2 − Q + α2 − 1 R S − QR = 0 and Pt 2 − α2 − 1 V + W P t − V W = 0 respectively, and P tV . Pt + V The following theorem is an adaptation of the lower bound in [28] to our setting of interest. Theorem 2 (Lower bound): The scalar control system of Section II is stabilizable only if α2 < 1 + SDROPTA , and the optimal achievable infinite-horizon average stage LQG cost is lower bounded by Q + α2 − 1 S r t ¯ ¯ J ≥J + P t − P̄ t , (18) 1 + SDROPTA − α2 P̄ t = where P t , P̄ t and S are as in Theorem 1, and SDROPTA is given in (4). By comparing (17a) and (18) we see that the possible gap between the two bounds stems from the gap between the bounds on the achievable SDR over the AWGN channel (3). It is interesting to note that in this case, in stark contrast to the classical LQG setting in which the system is stabilizable for any values of α, V and W , low values of SDR render the system unstable. Hence, it provides, among others, the minimal required transmit power for the system to remain stable. The difference from the classical LQG case stems from the additional input power constraint, which effectively couples the power of the observation noise with that of the estimation error, and was previously observed in, e.g., [7], [28]–[30] for the fully-observed setting. Remark 2: In the limit SNR → ∞, we attain also SDR → ∞. In this case, the estimate (11) and the prediction error (14) at the decoder coincide with those at the encoder, (15b) and (16b), respectively, recovering the renowned results of classical (without a communication channel) LQG control. 5 Alternatively, Fig. 4. Optimal average stage LQG cost J¯r of a single representative run for KC = KS = 1, α = 2, and SNRs 2 and 4 which correspond to a stabilizable and an unstabilizable systems. The driving noise and observation noise powers and the LQG penalty coefficients are Q = R = F = W = V = 1. this is the cost J¯r in the limit SNR → ∞. We next discuss the special cases of KC = 1 and KC = 2 channel uses per sample in Sections IV-A and IV-B, respectively. A. Source–Channel Rate Match In this subsection we treat the case of KC = 1, namely, where the sample rate of the control system and the signaling rate of the communication channel match. As we saw in Section III, analog linear transmission of a Gaussian source over an AWGN channel achieves optimal performance (even when infinite delay is allowed), namely, the OPTA SDR (4), and given any input value. Thus, the JSCC scheme that we use in this case is linear transmission — the source is transmitted as is, up to a power adjustment [recall (8) and (9)]: at = s̄t 1 =q st . r t Pt\|t−1 − Pt\|t Since in this case SDR0 = SDROPTA , the upper and lower bounds of Theorems 1 and 2 coincide, establishing the optimum performance in this case. Corollary 1: The scalar control system of Section II with KC = KS = 1 is stabilizable if only if α2 < 1 + SNR, and the optimal achievable infinite-horizon average stage LQG cost satisfies (17a) with equality where SDR0 = SNR. Remark 3: The stabilizability condition and optimum MMSE performance were previously established in [29], [30] for the case of no observation noise V = 0. The optimal averate stage LQG cost is illustrated in Fig. 4, where the normalized in time LQG cost J¯r is evaluated for a system with α = 2 and two SNRs — 2 and 4. SNR = 4 satisfies the stabilizability condition α2 < 1 + SNR, whereas SNR = 2 fails to do so. Unit LQG penalty coefficients Fig. 5. Average stage LQG costs when using the (distortion-bounded) SK Archimedean bi-spiral, repetition and the lower bound of Theorem 2 for α = 3, W = 1, V = 0, Q = 1, R = 0. The vertical dotted lines represent the minimum SNR below which the cost diverges to infinity. Q = R = F = 1 and unit driving noise and observation noise powers W = V = 1 are used. B. Source–Channel Rate Mismatch We now consider the case of KC = 2 channel uses per sample. As we saw in Section III, linear schemes are suboptimal outside the low-SNR region. Instead, by using non-linear maps, e.g., the (modified) Arcimedean spiralbased SK maps (7), better performance can be achieved. We note that the improvement in the SDR of the JSCC scheme is substantial when α2 is of the order of SDR. That is, when the SDR of the linear scheme is close to α2 −1, using an improved scheme with better SDR improves substantially the LQG cost. Unfortunately, the spiral-based SK schemes do not promise any improvement for SNRs below 5dB under maximum-likelihood (ML) decoding. Remark 4: By replacing the ML decoder with an MMSE one, strictly better performance can be achieved over the linear scheme for all SNR values. The effect of the SDR improvement is illustrated in Fig. 5 for a fully-observable (V = 0) system with α = 3 and W = 1, for Q = 1 and R = 0, by comparing the achievable costs and lower bound of Theorems 1 and 2. Remark 5: The resulting effective noise at the output of the JSCC receiver is not necessarily Gaussian, and hence the resulting system states xt , are not necessarily Gaussian either. Nevertheless, for the bounded-distortion scheme (7), this has no effect on the resulting performance. V. D ISCUSSION AND F UTURE R ESEARCH In this paper we considered the simplest case of scalar systems, and KS = 1 and KC = 2. Clearly, an (exponentially) large gain in performance can be achieved for KC > 2. We further note that the results of Theorems 1 and 2 readily extend to systems with vector states xt and vector control signals ut but scalar observed outputs yt . Interestingly, for the case of vector observed, state and control signals, even if the signaling rate of the channel and the sample rate of the observer are equal (rate matched case), conveying several analog observations over a single channel input may be of the essence. This is achieved by a compression JSCC scheme, e.g., by reversing the roles of the source and the channel inputs in the SK spiral-based scheme and similarly promises exponentially growing gains with the SNR and dimension; see [8], [9], [17]–[19], [26], [31]. In this work, we assumed that the observer knows all past control signals. This case can be viewed as a two-sided side-information scenario. Nevertheless, although this is a common situation in practice, there are scenarios in which the observer is oblivious of the control signal applied or has only a noisy measurement of control signal generated by the controller. 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arXiv:1711.09860v1 [math.GR] 27 Nov 2017 TWISTED LAWRENCE-KRAMMER REPRESENTATIONS ANATOLE CASTELLA Abstract. Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected. Here, we investigate an analogue of these representations — introduced by Digne in the spherical cases — for every Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under a group of graph automorphisms, and for the corresponding Artin-Tits group. Under the same assumptions on the parameters as in the small type cases, we first show that these so-called “twisted Lawrence-Krammer representations” are faithful, and we then prove, by computing their formulas when the group of graph automorphisms is of order two or three, their irreducibility in all the spherical and connected cases but one. Introduction Lawrence-Krammer representations (shortened to LK-representations in what follows) are a family of linear representations of Artin-Tits monoids and groups of small type, introduced by Lawrence [17] and Krammer [15] for the braid groups, and intensively studied since then (see the references listed below). Under some assumptions on the defining ring and on the parameters (see condition (⋆) of theorem 6 below), they are known to be faithful for the monoids [16, 1, 8, 12, 22, 14, 6] which ensure their faithfulness for the groups when the type is spherical, irreducible when the type is connected [23, 18, 7, 6], and have proved to be useful in the study of several other properties of Artin-Tits groups [18] and related objects [7, 19]. It is therefore an interesting question to ask if there exists some analogous linear representations for Artin-Tits monoids and groups of non-small type. In [12, end of section 3], Digne defines such objects for Artin-Tits monoids and groups of spherical type Bn , F4 and G2 , using the fact that they appear as the submonoids and subgroups of fixed points of Artin-Tits monoids and groups of small and spherical type (namely of type A2n−1 , E6 and D4 respectively) under the action of a graph automorphism. These new linear representations — that I call “twisted” LK-representations — share the same good combinatorial properties with respect to the associated root systems as in the small type cases, and Digne proves their faithfulness for his choice of parameters in [12, Cor. 3.11]. Date: November 28, 2017. 1 2 ANATOLE CASTELLA The aim of this article is to generalize this construction to every Artin-Tits monoid that appears as the submonoid of fixed elements of an Artin-Tits monoid of small type under a group of graph automorphisms, and to the associated ArtinTits group. Under the same condition (⋆) on the defining ring and on the parameters as in the small type cases, we prove their faithfulness for the monoids, which again ensure their faithfulness for the groups when the type is spherical, and we prove their irreducibility for all the connected and spherical types but one. The paper is organized as follows. We recall the basic needed notions on Coxeter groups, Artin-tits monoids an groups, standard root systems and graph automorphisms in section 1. We recall the definition and the main properties of LK-representations in the small type cases, as stated in [6], in subsection 2.1, and turn to the definition of the twisted LK-representations in subsection 2.2. We prove our faithfulness result in subsection 2.3 (theorem 13). We explicit the formulas of these twisted LK-representations when the group of graph automorphisms is of order two or three in section 3. We use these formulas in section 4 to study the spherical cases. We prove our irreducibility result in subsection 4.1 (theorem 35), and conclude in subsection 4.2 by giving a sufficient condition on the parameters for non-equivalence in the family of twisted LK-representations of a given type. 1. Coxeter matrices and related objects A Coxeter matrix is a matrix Γ = (mi,j )i,j∈I over an arbitrary set I with mi,j = mj,i ∈ N>1 ∪ {∞}, and mi,j = 1 ⇔ i = j for all i, j ∈ I. As usual, we encode the data of Γ by its Coxeter graph, i.e. the graph with vertex set I, an edge between the vertices i and j if mi,j > 3, and a label mi,j on that edge when mi,j > 4. We say that a Coxeter matrix Γ is connected when so is its Coxeter graph. In this paper, we will always assume that I is finite. This condition could be removed at the cost of some refinements in certain statements below (see [4, Ch. 11] for some of them), which are left to the reader. 1.1. Coxeter groups and Artin-Tits monoids and groups. To a Coxeter matrix Γ = (mi,j )i,j∈I , we associate the Coxeter group W = WΓ , the Artin-Tits group B = BΓ and the Artin-Tits monoid B + = BΓ+ , given by the following presentations : W = h si , i ∈ I \| si sj si · · · = sj si sj · · · if mi,j 6= ∞, and s2i = 1 i, \| {z } \| {z } mi,j terms B = h si , i ∈ I \| si sj si · · · = sj si sj · · · if mi,j 6= ∞ i, \| {z } \| {z } mi,j terms B + = mi,j terms mi,j terms h si , i ∈ I \| si sj si · · · = sj si sj · · · if mi,j 6= ∞ i+ . \| {z } \| {z } mi,j terms mi,j terms We denote by ℓ the length function on W and on B + relatively to their generating sets {si , i ∈ I} and {si , i ∈ I} respectively. TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 3 We denote by 4 the left divisibility in the monoid B + , i.e. for g, h ∈ B + , we write h 4 g if there exists h′ ∈ B + such that g = hh′ . This leads to the natural notions of left gcd ’s and right lcm’s in B + . Let J be a subset of I. We denote by ΓJ the submatrix (mi,j )i,j∈J of Γ and by WJ the subgroup hsj , j ∈ Ji of W . We say that J and ΓJ are spherical if WJ is finite, or equivalently if the elements sj , j ∈ J, have a common right multiple in B + (see [3, Thm. 5.6]). In that case, there exists a unique element rJ of maximal length in WJ , and the elements sj (j ∈ J) have a unique right lcm in B + that we denote by ∆J (see [3, Props. 4.1 and 5.7]). 1.2. Standard root system. We refer the reader to [11] for the basic notions on standard root systems. We denote by Φ = ΦΓ = {w(αi ) \| w ∈ W, i ∈ I} the standard root system associated with Γ in the real vector space E = ⊕i∈I Rαi , where the action of W on E is defined, for a generator si and a basis element αj , by si (αj ) = αj +2 cos mπi,j αi . T It is known that Φ = Φ+ ⊔ Φ− , where Φ+ = Φ (⊕i∈I R+ αi ) and Φ− = −Φ+ . As in [6], we will represent Φ+ — and any of its subsets — as a graded graph, where two elements α and β of Φ+ are linked by an edge labeled i if α = si (β), and where the grading is by the depth function on Φ+ defined, for α ∈ Φ+ , by dp(α) = min{ℓ(w) \| w ∈ W such that w(α) ∈ Φ− }. By convention in those graded graphs, we choose to place roots of great depth above roots of small depth, so edges like the following ones will all mean that α = si (β) and dp(β) > dp(α) : sβ sβ sβ i ❏i i✡ ✡ sα , s α , ❏s α . . . In particular in section 3, we will focus on some particular subsets of Φ+ , closure of subsets of Φ+ under the action of some subgroup WJ of W , that we call, after Hée, meshes : Notation For J ⊆ I, we call J-mesh of a subset Ψ of Φ+ the set MJ (Ψ) = T 1. + WJ (Ψ) Φ = {w(α) \| w ∈ WJ , α ∈ Ψ such that w(α) ∈ Φ+ }. 1.3. Graph automorphisms. We call graph automorphism of a Coxeter matrix Γ = (mi,j )i,j∈I every permutation σ of I such that mσ(i),σ(j) = mi,j for all i, j ∈ I, and we denote by Aut(Γ) the group they constitute. Any graph automorphism σ of Γ acts by an automorphism on W (resp. on B and B + ) by permuting the generating set {si \| i ∈ I} (resp. {si \| i ∈ I}). If Σ is a subgroup of Aut(Γ), we denote by W Σ (resp. B Σ , resp. (B + )Σ ) the subgroup of W (resp. subgroup of B, resp. submonoid of B + ) of fixed points under the action of the elements of Σ. It is known that W Σ (resp. (B + )Σ ) is a Coxeter group (resp. Artin-Tits monoid), and that the analogue holds for B Σ when Γ is spherical, or more generally of FC-type (see [13, 21] for the Coxeter case, [20, 9, 10, 5] for the Artin-Tits case). More precisely, if we denote by I Σ the set of spherical orbits of I under Σ, then Σ W (resp. (B + )Σ , and B Σ when Γ is of FC-type) has a Coxeter (resp. ArtinTits) presentation associated with a Coxeter matrix ΓΣ = (mΣ J,K )J,K∈I Σ easily 4 ANATOLE CASTELLA computable from Γ, where the generators are the elements rJ (resp. ∆J ) for J running through I Σ (see for example [9, 10, 5]). Similarly, any graph automorphism σ of Γ acts by a linear automorphism on the vector space E = ⊕i∈I Rαi by permuting the basis (αi )i∈I . This action stabilizes Φ and Φ+ , and the induced action on those sets is w(αi ) 7→ (σ(w))(ασ(i) ). It is easily seen on the definition that the action of Aut(Γ) on Φ+ thus defined respects the depth function on Φ+ . In particular, we can define the depth of an orbit Θ of Φ+ under Σ as the depth of any element α ∈ Θ. 2. Twisted Lawrence-Krammer representations From now on, we fix a Coxeter matrix Γ = (mi,j )i,j∈I of small type, i.e. with mi,j ∈ {2, 3} for all i, j ∈ I with i 6= j. 2.1. Lawrence-Krammer representations - the small type case. Let R be a (unitary) commutative ring and V be a free R-module with basis (eα )α∈Φ+ . We denote by V ⋆ the dual of V and by R× the group of units of R. For f ∈ V ⋆ and e ∈ V , we denote by f ⊗ e the endomorphism of V defined by v 7→ f (v)e. 4 Definition 2 ([6, Def. 7]). For (a, b, c, d) ∈ R and i ∈ I, we denote by ϕi = ϕi,(a,b,c,d) the endomorphism of V given on the basis (eα )α∈Φ+ by  ϕi (eα ) = 0 if α = αi ,     ϕ (e ) = de if α s ✐i , i α α (  β s  ϕi (eβ ) = beα   i in Φ+ . if  ϕi (eα ) = aeα + ceβ αs For a linear form fi ∈ V ⋆ , we define the Lawrence-Krammer map — or the LK-map for short — associated with (a, b, c, d) and fi to be the endomorphism ψi = ψi,(a,b,c,d),fi of V given by ψi = ϕi + fi ⊗ eαi . Proposition 3 ([6, Lem. 9 and Prop. 12]). Assume that d2 − ad − bc = 0 and that the family (fi )i∈I ∈ (V ⋆ )I satisfies the following properties : (i) for i, j ∈ I with i 6= j, fi (eαj ) = 0, (ii) for i, j ∈ I with mi,j = 2, fi ϕj = dfi , (iii) for i, j ∈ I with mi,j = 3, fi ϕj = fj ϕi . Then si 7→ ψi defines a linear representation ψ : B + → L (V ). Moreover if b, c, d and fi (eαi ), i ∈ I, belong to R× , then the image of ψ is included in GL(V ) and hence ψ induces a linear representation ψgp : B → GL(V ). Definition 4 ([6, Def. 11]). We call LK-family (relatively to (a, b, c, d)) any family (fi )i∈I ∈ (V ⋆ )I satisfying conditions (i), (ii) and (iii) of proposition 3 above, and we call LK-representation (relatively to (a, b, c, d) and (fi )i∈I ) the induced representation ψ and, when appropriate, ψgp . Remark 5. When b is invertible in R, an LK-family (fi )i∈I necessarily satisfies fi (eαi ) = fj (eαj ) for every i, j ∈ I with mi,j = 3 (see [6, Prop. 34]). In particular when Γ is connected, we thus get that the elements fi (eαi ), i ∈ I, are all equal. For sake of brevity when this is the case, we will simply denote by f the common value of the fi (eαi ), i ∈ I. TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 5 Moreover when Γ is connected and spherical, one can show that the LK-family (fi )i∈I is entirely determined by this common value f (see [6, Section 3.2]). The first main properties of these representations can be stated as follows : Theorem 6 ([6, Thm. A]). Let ψ : B + → L (V ) be an LK-representation over R associated with parameters (a, b, c, d) ∈ R4 and (fi )i∈I ∈ (V ⋆ )I . Assume that the following condition holds : (⋆) R is an integral domain, fi (eαi ) 6= 0 for all i ∈ I, and there exists a totally ordered integral domain R0 and a ring homomorphism ρ : R → R0 , such that ρ(t) > 0 for t ∈ {a, b, c, d}, and Im(fi ) ⊆ ker(ρ) for all i ∈ I. By extension of the scalars from R to its field of fractions K, consider ψ as acting on the K-vector space VK = K ⊗R V . Then ψ has its image included in GL(VK ), hence induces an LK-representation ψgp : B → GL(VK ), and the following holds. (i) The LK-representation ψ is faithful, and so is ψgr if Γ is spherical. (ii) If Γ is connected, the LK-representations ψ and ψgp are irreducible on VK . (iii) Assume that Γ has at least one edge and that ψ ′ is an LK-representation of B + associated with parameters (a′ , b′ , c′ , d′ ) ∈ R4 and (fi′ )i∈I ∈ (V ⋆ )I that satisfy the conditions of the preamble for the fixed ρ. Then if a′ 6= a, or if d′ 6= d or if fi′ (eαi ) 6= fi (eαi ) for some i ∈ I, the LK-representations ψ and ψ ′ are not equivalent on VK . 2.2. Definition of the twisted Lawrence-Krammer representations. Let us fix a subgroup Σ of Aut(Γ) and an LK-representation ψ : B + → L (V ), g 7→ ψg , of B + associated with an LK-family (fi )i∈I ∈ (V ⋆ )I . Recall that Σ naturally acts on B + , and that the submonoid (B + )Σ of fixed points of B + under Σ is the Artin-Tits monoid BΓ+Σ associated with a certain Coxeter matrix ΓΣ (see subsection 1.3). Moreover, Σ acts on Φ+ and this action induces an action of Σ on V by permutation of the basis (eα )α∈Φ+ , which we denote by Σ → GL(V ), σ 7→ σV . We denote by V Σ the submodule of fixed points of V under the action of Σ. Let us denote by ϕ : B + → L (V ), g 7→ ϕg , the LK-representation of B + associated with the trivial LK-family (i.e. where fi is the zero form for all i ∈ I). Lemma 7. For all (g, σ) ∈ B + × Aut(Γ), we get σV ϕg = ϕσ(g) σV . In particular, for every g ∈ (B + )Σ , ϕg stabilizes V Σ and hence ϕ induces a linear representation ϕΣ : (B + )Σ → L (V Σ ), g 7→ ϕΣ g = ϕg \|V Σ . Proof. The action of Aut(Γ) on Φ+ respects the depth, and this clearly implies that σV (ϕi (eα )) = ϕσ(i) (eσ(α) ) in view of the formulas of definition 2. The result follows by linearity and induction on ℓ(g). Proposition 8. Assume that fi = fσ(i) σV in V ⋆ , for every (i, σ) ∈ I ×Σ. Then for every (g, σ) ∈ B + × Σ, we get σV ψg = ψσ(g) σV . In particular, for every g ∈ (B + )Σ , ψg stabilizes V Σ and hence ψ induces a linear representation ψ Σ : (B + )Σ → L (V Σ ), g 7→ ψgΣ = ψg \|V Σ . Moreover if the images of ψ are invertible, then so are the images of ψ Σ . 6 ANATOLE CASTELLA Proof. We have σV ψi = σV ϕi + σV (fi ⊗ eαi ) = σV ϕi + fi ⊗ eασ(i) and ψσ(i) σV = ϕσ(i) σV + (fσ(i) ⊗ eασ(i) )σV = ϕσ(i) σV + (fσ(i) σV ) ⊗ eασ(i) (thanks to the formulas of [6, Rem. 1]), whence σV ψi = ψσ(i) σV by the previous lemma and assumption on fi and fσ(i) . The first point follows by induction on ℓ(g). Moreover if the images −1 of ψ are invertible, then the equality σV ψg σV−1 = ψσ(g) implies σV ψg−1 σV−1 = ψσ(g) , −1 Σ + Σ and hence ψg stabilizes V for every g ∈ (B ) . This gives the result. Definition 9 (twisted LK-representations). Under the assumption of the previous proposition, we call twisted LK-representation the linear representation ψ Σ : (B + )Σ → L (V Σ ) of the Artin-Tits monoid (B + )Σ = BΓ+Σ , and, when appropriate, Σ the induced representation ψgp : BΓΣ → GL(V Σ ) of the Artin-Tits group BΓΣ . The assumption fi = fσ(i) σV for every (i, σ) ∈ I × Σ is equivalent to fi (eα ) = fσ(i) (eσ(α) ) for every (i, α, σ) ∈ I × Φ+ × Σ. It is not always satisfied : for example if i and σ(i) are not in the same connected component of Γ, then fi (eαi ) and fσ(i) (eασ(i) ) can be chosen to be distinct (see [6, Section 3.1]). I do not know if this assumption is always satisfied when Γ is connected, but we have the following partial result : Proposition 10. Let (fi )i∈I be an LK-family. Assume that b, c, d are invertible in R and that we are in one of the following cases : (i) Γ is spherical and connected (i.e. of type ADE), or (ii) Γ is affine (i.e. of type ÃD̃Ẽ), or (iii) Γ has no triangle and (fi )i∈I is the LK-family of Paris, as in [6, Def. 42]. Then fi = fσ(i) σV for every (i, σ) ∈ I × Aut(Γ). Proof. The condition fi (eα ) = fσ(i) (eσ(α) ) for every (i, α, σ) ∈ I × Φ+ × Aut(Γ), for the three situations (note that the first one is a consequence of the third one), is easy to see by induction on dp(α), using the inductive construction of the fi (eα ), (i, α) ∈ I × Φ+ , and the independence results at the inductive steps, of [6, Sections 3.2, 3.3 and 3.4] respectively, and using the fact that the action of Aut(Γ) on Φ+ respects the depth. 2.3. Twisted faithfulness criterion. The aim of this subsection is to prove that the classical faithfulness criterion of [14], as stated in theorem 6 above, also works in the twisted cases. For g ∈ B + , we set I(g) = {i ∈ I \| si 4 g}. Lemma 11. Let ψ : (B + )Σ → M be a monoid homomorphism where M is left cancellative. If ψ satisfies ψ(g) = ψ(g ′ ) ⇒ I(g) = I(g ′ ) for all g, g ′ ∈ (B + )Σ , then ψ is injective. Proof. Let g, g ′ ∈ (B + )Σ be such that ψ(g) = ψ(g ′ ). We prove by induction on ℓ(g) that g = g ′ . If ℓ(g) = 0, i.e. if g = 1, then I(g) = I(g ′ ) = ∅, hence g ′ = 1 and we are done. If ℓ(g) > 0, fix i ∈ I(g) = I(g ′ ). Since the action of Σ on B + respects the divisibility and since g is fixed by Σ, the orbit J of i under Σ is included in I(g) = I(g ′ ), but then J is spherical and there exist g1 , g1′ ∈ B + such that g = ∆J g1 and g ′ = ∆J g1′ . Since the elements g, g ′ and ∆J are fixed by Σ (recall that ∆J is a generator of (B + )Σ ), so are g1 and g1′ by cancellation in B + . We thus get ψ(∆J )ψ(g1 ) = ψ(∆J )ψ(g1′ ) in M , whence ψ(g1 ) = ψ(g1′ ) by cancellation in M . We therefore get g1 = g1′ by induction and finally g = g ′ . TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 7 + + Notation 12. We denote every Pby Φ /Σ the set of orbits of Φ under Σ and, for + Θ ∈ Φ /Σ, we set eΘ = α∈Θ eα . The family (eΘ )Θ∈Φ+ /Σ is a basis of V Σ . Theorem 13. Assume that fi = fσ(i) σV for every (i, σ) ∈ I × Σ, so that the twisted LK-representation ψ Σ : (B + )Σ → L (V Σ ) is defined (see proposition 8), and assume that condition (⋆) holds : (⋆) R is an integral domain, fi (eαi ) 6= 0 for all i ∈ I, and there exists a totally ordered integral domain R0 and a ring homomorphism ρ : R → R0 , such that ρ(t) > 0 for t ∈ {a, b, c, d}, and Im(fi ) ⊆ ker(ρ) for all i ∈ I. By extension of the scalars from R to its field of fractions K, consider ψ Σ as acting on the K-vector space VKΣ = K⊗R V Σ , so that ψ Σ has its image included in GL(VKΣ ), Σ and hence induces a twisted LK-representation ψgp : BΓΣ → GL(VKΣ ). Then the Σ Σ twisted LK-representation ψ is faithful, and so is ψgp if ΓΣ is spherical. Proof. Under condition (⋆), the elements b, c, d and fi (eαi ), i ∈ I, are non-zero in R, so become units of the field of fractions K of R. By propositions 3 and 8, Im(ψ Σ ) Σ is then included in GL(VKΣ ) — hence is cancellative — and ψgp : BΓΣ → GL(VKΣ ) Σ Σ is defined. Moreover when Γ is spherical, the faithfulness of ψ implies the one of Σ ψgp by [6, Lem. 6]. In order to prove the theorem, it then suffices to see that ψ Σ satisfies the assumption of lemma 11. So let g, g ′ ∈ (B + )Σ be such that ψgΣ = ψgΣ′ and let us show that I(g) = I(g ′ ). We need for that some notations of [6, Section 2.2]. If we denote by V0 the free R0 -module with basis (eα )α∈Φ+ , then the morphism ρ : R → R0 induces a natural monoid homomorphism ρ̃ : L (V ) → L (V0 ), ϕ 7→ ϕ. By (⋆), the homomorphism ρ̃ sends Im(ψ) into L + (V0 ), the submonoid of L (V0 ) composed of the endomorphisms of V0 whose matrix in the basis (eα )α∈Φ+ has non-negative coefficients. Now for h ∈ (B + )Σ and β ∈ Φ+ , let us denote by Rh (β) the support of ψh (eβ ) in the basis (eα )α∈Φ+ , and for Ψ ⊆ Φ+ , let us set S Rh (Ψ) = β∈Ψ Rh (β). Since the coefficients of the matrix of ψh in the basis (eα )α∈Φ+ of V0Pare non-negative, the set Rh (Ψ), when Ψ is finite, is precisely the support of ψh ( β∈Ψ eβ ) in the basis (eα )α∈Φ+ . In order to show that I(g) = I(g ′ ), it suffices to prove, in view of [14, Prop. 2] (see [6, Lem. 20]), that Rg (Φ+ ) = Rg′ (Φ+ ). But since ψg and ψg′ coincide on V Σ , we get in particular that ψg (eΘ ) = ψg′ (eΘ ), S and hence Rg (Θ) = Rg′ (Θ) for every Θ ∈ Φ+ /Σ. And finally since Φ+ = Θ∈Φ+ /Σ Θ, we get Rg (Φ+ ) = S S + Θ∈Φ+ /Σ Rg′ (Θ) = Rg′ (Φ ), whence the result. Θ∈Φ+ /Σ Rg (Θ) = 3. Case of an automorphism of order two or three Let Γ = (mi,j )i,j∈I be a Coxeter matrix of small type and fix Σ = hσi 6 Aut(Γ) of order two or three. Recall that the submonoid (B + )Σ of fixed points of B + under Σ is generated by the elements ∆J , for J running through the spherical orbits of I under Σ. We fix four parameters (a, b, c, d) ∈ R such that d2 − ad − bc = 0 and consider an LK-family (fi )i∈I associated with (a, b, c, d) that satisfy fi = fσ(i) σV for every (i, σ) ∈ I × Σ, so that the associated twisted LK-representation ψ Σ : (B + )Σ → L (V Σ ) is defined. 8 ANATOLE CASTELLA Σ The aim of this section is to compute and study the map ψ∆ for a spherical J Σ Σ Σ orbit J of I under Σ. For sake of brevity, we set ψJ := ψ∆J and ϕΣ J := ϕ∆J for any spherical orbit J. Notation 14. Any spherical orbit J of I under Σ is of one of the following types : (i) type A : J = {i}, (ii) type B : J = {i, j} with mi,j = 2, (iii) type C : J = {i, j, k} with mi,j = mj,k = mk,i = 2, (iv) type D : J = {i, j} with mi,j = 3. Notice that types B and D only occur when Σ is of order two, and type C only occurs when Σ is of order three. We will make a constant use of these types A to D in the following. Notation 15. Let J be a spherical orbit of I under Σ. • We denote by ΘJ the set {αi \| i ∈ J} ; it is an orbit of Φ+ under Σ. • In case D, {αi + αj } is also an orbit of Φ+ under Σ, that we denote by Θ′J . Notation 16. Let J be a spherical orbit of I under Σ. Since we are assuming that fi = fσ(i) σV for every i ∈ I, the linear forms fi , for i ∈ J, coincide on V Σ . Depending on J, we define the following linear forms on V Σ : (i) for type A : fJ = fi \|V Σ , where J = {i}, (ii) for type B : fJ = dfi \|V Σ , for any i ∈ J, (iii) for type C : fJ = d2 fi \|V Σ , for any i ∈ J, (iv) for type D : fJ = fj (bc Id +aϕi )\|V Σ and fJ′ = cfj ϕi \|V Σ , where J = {i, j}. Notice that the two linear forms for case D really only depend on J (not on the choice of i and j in J) since fi \|V Σ = fj \|V Σ by assumption and since fj ϕi = fi ϕj by definition of an LK-family. Lemma 17. If i, j ∈ I with i 6= j, then fj (bc Id +aϕi ) = fj ϕ2i in V ⋆ . Moreover if mi,j = 3, then fj ϕ2i = fi ϕj ϕi in V ⋆ . Proof. By [6, Lem. 11], the endomorphism ϕ2i − aϕi − bc Id has its image included in Reαi , which is included in ker(fj ) if i 6= j since then fj (eαi ) = 0 by definition of an LK-family. So fj (ϕ2i − aϕi − bc Id) is the zero form and this gives the first point. The second point is clear since then fj ϕi = fi ϕj by definition of an LK-family. Proposition 18. Let J be a spherical orbit of I under Σ. Then (i) ψJΣ = ϕΣ J + fJ ⊗ eΘJ if J is of type A, B or C, ′ (ii) ψJΣ = ϕΣ J + fJ ⊗ eΘJ + fJ ⊗ eΘ′J if J is of type D. Proof. Recall that for every i ∈ I, ψi is given by ψi = ϕi + fi ⊗ eαi . The result for type A is trivial, since then ψJΣ = ψi \|V Σ . For type B, we have ψJΣ = ψi ψj \|V Σ . But in that case, we also have ϕi (eαj ) = deαj (by definition of ϕi ) and fi (eαj ) = 0 (by definition of an LK-family), thus we get, thanks to the formulas of [6, Rem. 1], ψi ψj = ϕi ϕj + fi ϕj ⊗ eαi + dfj ⊗ eαj , whence the result since fi ϕj = dfi by definition of an LK-family. Similarly for type C, we have ψJΣ = ψi ψj ψk \|V Σ and since ϕi (eαj ) = deαj , ϕi (eαk ) = ϕj (eαk ) = deαk , and fi (eαj ) = fi (eαk ) = fj (eαk ) = 0, we get, thanks to [6, Rem. 1], ψi ψj ψk = ϕi ϕj ϕk + fi ϕj ϕk ⊗ eαi + dfj ϕk ⊗ eαj + d2 fk ⊗ eαk , TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 9 whence the result since fi ϕj ϕk = dfi ϕk = d2 fi and fj ϕk = dfj by definition of an LK-family. Finally for type D, we have ψJΣ = ψi ψj ψi \|V Σ and ϕi (eαj ) = aeαj + ceαi +αj , ϕi ϕj (eαi ) = bceαj , fi (eαj ) = fj (eαi ) = 0, and fi ϕj (eαi ) = 0 (since fi ϕj = fj ϕi and ϕi (eαi ) = 0), hence we get, thanks to [6, Rem. 1], ψi ψj ψi = ϕi ϕj ϕi + fi ϕj ϕi ⊗ eαi + [bcfi + afj ϕi ] ⊗ eαj + cfj ϕi ⊗ eαi +αj , whence the result by lemma 17 and by the fact that fi \|V Σ = fj \|V Σ . Remark 19. Let J be a spherical orbit of I under Σ and let Θ be an orbit of Φ+ under T Σ. Recall that we denote by MJ (Θ) the J-mesh of Θ, i.e. the set WJ (Θ) Φ+ . Notice that this subset of Φ+ is a union of orbits of Φ+ under Σ. It is clear that all the maps ϕi , for i ∈ J, stabilize the submodule of V generated by the elements eβ for β running through MJ (Θ). So the map ϕΣ J stabilizes the submodule of V Σ generated by the elements eΘ for Θ running through the orbits Σ included in MJ (Θ), and hence the matrix of ϕΣ J in the basis (eΘ )Θ∈Φ+ /Σ of V + is block diagonal, relatively to the partition of Φ /Σ induced by those J-meshes MJ (Θ) for Θ ∈ Φ+ /Σ. In subsections 3.1 to 3.4 below, we explicit the different possible blocks of ϕΣ J, for the four possible types of J and for the different possible J-meshes MJ (Θ) when Θ runs through Φ+ /Σ. For each such J-mesh X = MJ (Θ), we denote by MX the corresponding block in ϕΣ J and by PX its characteristic polynomial. Notation 20. For sake of brevity in what follows, we set dˇ := a − d in R. In ˇ and ddˇ = −bc. particular, the roots of the polynomial X 2 − aX − bc are d and d, 3.1. Type A. We assume here that J = {i}, and hence that ϕΣ J = ϕi \|V Σ . • Configuration A1 : Θ = ΘJ . The corresponding block is MA1 = 0 . i • Configuration A2 : Θ s✐ or Θ The corresponding block is Θ2 • Configuration A3 : MA2 = d . s Θ2 i Θ1 s✐ s✐ i i or Θ or s Θ1 s s i i s s or Θ2 Θ1 The corresponding block is MA3 eΘ1 eΘ2 a b , = c 0 whose characteristic polynomial is ˇ PA3 = (X − d)(X − d). si✐ si✐ si✐. s s i i s i s s s . 10 ANATOLE CASTELLA 3.2. Type B. We assume here that J = {i, j} with mi,j = 2, and hence that ϕΣ J = (ϕi ϕj )\|V Σ . • Configuration B1 : Θ = ΘJ . The corresponding block is MB1 = 0 . si✐ j✐ • Configuration B2 : Θ s✐ s✐ j✐ i j✐ i . or Θ MB2 • Configuration B3 : Θ2 s j✐ i s i✐ j Θ1 s j✐ s i✐ The corresponding block is = d2 . . The corresponding block is MB3 whose characteristic polynomial is eΘ1 eΘ2 ad bd , = cd 0 ˇ PB3 = (X − d2 )(X − dd). Θ3 i • Configuration B4 : Θ2 Θ1 s ❅j ❅s s ❅ j ❅s MB4 whose characteristic polynomial is . The corresponding block is i 3  eΘ2 1 eΘ2 e2Θ a 2ab b = ac bc 0 , c2 0 0 ˇ PB4 = (X − d2 )(X − dd)(X − dˇ2 ). Θ4 s s ❅ j ❅i s ❅s ❅s Θ3 s . The corresponding block is ❅i ❅ j j ❅s ❅s i • Configuration B5 : Θ2 Θ1 MB5  eΘ21 a ac = ac c2 whose characteristic polynomial is eΘ2 eΘ3 eΘ4  2 ab ab 0 bc bc 0 0 0 b 0 , 0 0 ˇ 2 (X − dˇ2 ). PB5 = (X − d2 )(X − dd) Moreover, it is easily checked that the polynomial PB4 already annihilates MB5 . TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 11 3.3. Type C. We assume here that J = {i, j, k} with mi,j = mj,k = mk,i = 2, and hence that ϕΣ J = (ϕi ϕj ϕk )\|V Σ . • Configuration C1 : Θ = ΘJ . The corresponding block is MC1 = 0 . k✐ s j✐i✐ or • Configuration C2 : Θ k✐ k✐ k✐ s s s j✐i✐ j✐i✐ j✐i✐. Θ The corresponding block is MC2 = d3 . • Configuration C3 : Θ2 sk✐ i✐ sk✐ j✐si✐ j✐ Θ1 sk✐ i✐ sk✐ j✐si✐ j✐ i j MC3 k . The corresponding block is eΘ21 eΘ22 ad bd = , cd2 0 whose characteristic polynomial is ˇ PC3 = (X − d3 )(X − d2 d). Θ4 • Configuration C4 : Θ2 Θ1 k✐ i✐ j✐ s s s ✑◗✑j ◗ k ✑ j✑ ◗ k i ✑◗✑◗ ◗ ✑ s i✐✑ s ✐s✑◗s ✐◗s ✐◗s ✐ k✐ ◗ ◗ j j◗✑k k ✑ i ✑ j. Θ3 i◗ ◗✑◗✑ ✑ i ◗✑ s ◗s✑◗s✑ k✐ i✐ j✐ The corresponding block is MC4 eΘ4 eΘ3 eΘ1 eΘ2  a2 d abd abd b2 d acd 0 bcd 0  , = acd bcd 0 0  c2 d 0 0 0  whose characteristic polynomial is ˇ 2 (X − ddˇ2 ). PC4 = (X − d3 )(X − d2 d) ˇ −ddˇ2 ) already Moreover, one can check that the polynomial (X −d3 )(X −d2 d)(X annihilates MC4 . s Θ4 Θ3 • Configuration C5 : Θ2 Θ1 k j❅i s s ❅s k i j❅ ❅ j i s ❅s k❅s ❅ i j ❅s k 12 ANATOLE CASTELLA The corresponding block is MC5  eΘ1 a3 a2 c = ac2 c3 eΘ2 3a2 b 2abc bc2 0 eΘ3 3ab2 b2 c 0 0 whose characteristic polynomial is eΘ4  3 b 0 , 0 0 ˇ − ddˇ2 )(X − dˇ3 ). PC5 = (X − d3 )(X − d2 d)(X s ✏P s ✏sP ✏P PP ✏✏ ✏P ✏P ✏✏ ✏P PP PPPP ✏ ✏ P ✏ ✏ ✏ s s s ✏P s ✏P s ✏sP✏ Ps P ✏ ✏ ✏ P P ✏s✏P ✏s PP P P P P P✏ ✏ ✏ ✏ ✏ P✏ P✏ P✏ P✏ P✏ P✏ P P P P P P ✏ ✏ ✏ ✏ ✏ ✏ P P P P✏ sP✏✏ s ✏✏ s ✏ s ✏ s ✏ P✏ P✏ Ps✏ PsP✏ PsPPs ✏ P ✏ PP PP ✏✏ ✏✏✏✏ PP P ✏ PPPP✏ P✏ ✏ ✏ P s P✏ s ✏ P✏ Ps✏✏ Θ8 Θ5 • Configuration C6 : Θ2 Θ1 Θ6 , Θ7 Θ3 , Θ4 (the labels i, j, k on the edges are omitted). The corresponding block is MC6 eΘ3 2 eΘ2 2 eΘ31 a a2 c  2 a c  2 a c = ac2  2 ac  2 ac c3 a b 0 abc abc 0 0 bc2 0 a b abc 0 abc 0 bc2 0 0 eΘ4 2 a b abc abc 0 bc2 0 0 0 eΘ5 2 ab 0 0 b2 c 0 0 0 0 eΘ6 2 ab 0 b2 c 0 0 0 0 0 eΘ7 2 ab b2 c 0 0 0 0 0 0 whose characteristic polynomial is eΘ8  3 b 0  0  0 , 0  0  0 0 ˇ 3 (X − ddˇ2 )3 (X − dˇ3 ). PC6 = (X − d3 )(X − d2 d) And one can check that the polynomial PC5 already annihilates MC6 . 3.4. Type D. We assume here that J = {i, j} with mi,j = 3, and hence that ϕΣ J = (ϕi ϕj ϕi )\|V Σ . • Configuration D1 : ΘJ and Θ′J . The corresponding block is 0 0 MD1 = . 0 0 • Configuration D2 : Θ si✐ j✐ or Θ s✐ si✐. j✐ i j✐ MD2 Θ3 • Configuration D3 : Θ2 Θ1 s j✐ s i✐ j i s The corresponding block is = d3 . s . The corresponding block is i j s j✐ s i✐ TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 13  eΘ21 eΘ2 eΘ23  ad abd b d = acd bcd 0 , c2 d 0 0 MD3 whose characteristic polynomial is ˇ 3 ). PD3 = (X − d3 )(X 2 + (dd) Θ4 j s Θ3 • Configuration D4 : Θ 2 s ❅i ❅s i j s s . The corresponding block is ❅ j ❅s i Θ1 eΘ4 eΘ3 eΘ2  2eΘ1  a(a + bc) 2a2 b 2ab2 b3  a2 c 2abc b2 c 0  , MD4 =  2  ac bc2 0 0 c3 0 0 0 whose characteristic polynomial is ˇ 3 )(X − dˇ3 ). PD4 = (X − d3 )(X 2 + (dd) Θ6 i s Θ4 • Configuration D5 : Θ2 Θ1  MD5 s s ❅j❅i s ❅s ❅s Θ5 j i j i s s s ❅j ❅ i ❅s ❅s s Θ3 . The corresponding block is i eΘ2 eΘ1 eΘ3 a(a2 + bc) a2 b a2 b  a2 c abc abc   a2 c abc abc =  ac2 0 bc2   ac2 bc2 0 3 c 0 0 eΘ4 ab2 0 b2 c 0 0 0 eΘ5 ab2 b2 c 0 0 0 0 eΘ6  3 whose characteristic polynomial is b 0  0 , 0  0 0 ˇ 3 )2 (X − dˇ3 ). PD5 = (X − d3 )(X 2 + (dd) Moreover, it is easily checked that the polynomial PD4 already annihilates MD5 . 3.5. Consequences on annihilating polynomials. Lemma 21. Let ϕ ∈ L (V ), f, f ′ ∈ V ⋆ and e, e′ ∈ V be such that ϕ(e) = ϕ(e′ ) = 0 and f (e′ ) = f ′ (e) = 0. We set fe = f (e) and fe′ ′ = f ′ (e′ ). Then we get the following identities in L (V ), for all n ∈ N : "n−1 # X (i) (ϕ + f ⊗ e)n = ϕn + f fek ϕn−1−k ⊗ e, k=0 14 ANATOLE CASTELLA ′ ′ n n (ii) (ϕ+f ⊗e+f ⊗e ) = ϕ +f "n−1 X k=0 fek ϕn−1−k # ⊗e+f ′ "n−1 X Proof. By induction on n, using the identities of [6, Rem. 1]. k=0 fe′k′ ϕn−1−k # ⊗e′ . Lemma 22. Let J = {i, j} be an orbit of I under Σ, with mi,j = 3. Then fJ (eΘ′J ) = fJ′ (eΘJ ) = 0 and fJ (eΘJ ) = fJ′ (eΘ′J ). Proof. With the formulas of notation 16 and lemma 17, we get for the first point : fJ (eΘ′J ) = fi ϕj ϕi (eαi +αj ) = fi ϕj (beαj ) = 0, and fJ′ (eΘJ ) = cfi ϕj (eαi + eαj ) = cfj ϕi (eαi ) + cfi ϕj (eαj ) = 0, and for the second point : fJ (eΘJ ) = fi ϕj ϕi (eαi + eαj ) = fi ϕj ϕi (eαj ) = fi (bceαi ) = bcfi (eαi ), and fJ′ (eΘ′J ) = cfi ϕj (eαi +αj ) = cfi (beαi ) = bcfi (eαi ). Proposition 23. Let J be a spherical orbit of I under Σ, and consider P ∈ R[X]. Then there exists a polynomial Q ∈ R[X] (that depends on J and P ) such that : Σ (i) P (ψJΣ ) = P (ϕΣ J ) + fJ Q(ϕJ ) ⊗ eΘJ if J is of type A, B or C, ′ Σ Σ Σ (ii) P (ψJ ) = P (ϕJ ) + fJ Q(ϕΣ J ) ⊗ eΘJ + fJ Q(ϕJ ) ⊗ eΘ′J if J is of type D. Proof. For types A, B and C, the result follows from proposition 18 and lemma 21 (i), which applies since ϕΣ J (eΘJ ) = 0 (cf. configurations A1, B1 and C1). For type D, lemma 21 (ii) applies to the decomposition of ψJΣ given in proposition 18 Σ (ii) since ϕΣ J (eΘJ ) = ϕJ (eΘ′J ) = 0 (cf. configuration D1), and since fJ (eΘ′J ) = ′ fJ (eΘJ ) = 0 by the previous lemma. Hence we get two polynomials Q and Q′ such ′ ′ Σ Σ that P (ψJΣ ) = P (ϕΣ J ) + fJ Q(ϕJ ) ⊗ eΘJ + fJ Q (ϕJ ) ⊗ eΘ′J . But the two polynomials Q and Q′ thus obtained are in fact equal since, again by the previous lemma, fJ (eΘJ ) = fJ′ (eΘ′J ). Pd n Remark 24. More precisely, if P = in proposition 23, then Q = n=0 pn X Pd−1 n q X with q = p and q = p + q f (e d−1 d n−1 n n J ΘJ ) for every 1 6 n 6 d − 1. n=0 n Notation 25. Let J be a spherical orbit of I under Σ. Depending on J, we define a polynomial PJ ∈ R[X] by : ˇ if J is of type A, (i) PJ = (X − d)(X − d) 2 ˇ (ii) PJ = (X − d )(X − dd)(X − dˇ2 ) if J is of type B, 3 2ˇ (iii) PJ = (X − d )(X − d d)(X − ddˇ2 )(X − dˇ3 ) if J is of type C, ˇ 3 )(X − dˇ3 ) if J is of type D. (iv) PJ = (X − d3 )(X 2 + (dd) Lemma 26. Let J be a spherical orbit of I under Σ. Then ′ (i) PJ (ϕΣ J )(eΘ ) = 0 when Θ 6= ΘJ (resp. when Θ 6= ΘJ and Θ 6= ΘJ ) if J is of type A, B or C (resp. D), ′ (ii) PJ (ϕΣ J )(eΘ ) = PJ (0)eΘ when Θ = ΘJ (resp. when Θ = ΘJ or Θ = ΘJ ) if J is of type A, B or C (resp. D). Proof. In view of the results of subsections 3.1 to 3.4 above, the polynomial PJ Σ annihilates the non-zero blocks of ϕΣ J , hence we get that PJ (ϕJ )(eΘ ) = 0 if Θ 6= ′ ΘJ (resp. if Θ 6= ΘJ and Θ 6= ΘJ ) for types A, B and C (resp. D). Moreover Σ since ϕΣ J (eΘJ ) = 0 for all cases, and since ϕJ (eΘ′J ) = 0 for type D, we get that Σ PJ (ϕJ )(eΘJ ) = PJ (0)eΘJ for all cases, and that PJ (ϕΣ J )(eΘ′J ) = PJ (0)eΘ′J for type D. TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 15 Proposition 27. Let J be a spherical orbit of I under Σ. The image of PJ (ψJΣ ) is included in ReΘJ (resp. in ReΘJ ⊕ ReΘ′J ) if J is of type A, B or C (resp. D). As a consequence, the polynomial (X − fJ (eΘJ ))PJ annihilates ψJΣ . Proof. The previous lemma shows that the image of PJ (ϕΣ J ) is included in ReΘJ (resp. in ReΘJ ⊕ ReΘ′J ) for types A, B and C (resp. D). But by proposition 23, Σ PJ (ψJΣ ) is the sum of PJ (ϕΣ whose image is J ) and of some endomorphism of V included in ReΘJ (resp. in ReΘJ ⊕ ReΘ′J ) for types A, B and C (resp. D), whence the first point of the proposition. The second point for types A, B and C (resp. D) follows from proposition 18 (i) (resp. proposition 18 (ii) and lemma 22), and the studies of configurations A1, B1, and C1 (resp. D1), which show that eΘJ (resp. each of eΘJ and eΘ′J ) is an eigenvector of ψJΣ for the eigenvalue fJ (eΘJ ). Remark 28. Let us denote by f the common value of the fi (eαi ) for i ∈ J (recall that we are assuming that fi = fσ(i) σV for all i ∈ I). Then it is easily checked on the definitions that the value of fJ (eΘJ ) is f (resp. df , resp. d2 f , resp. bcf ) if J is of type A (resp. B, resp. C, resp. D). 4. The spherical case We assume here that Γ is of spherical type An (n > 2), Dn (n > 4) or E6 . In all the cases but D4 , we fix Σ = Aut(Γ) (of order two), and when Γ is of type D4 , we take for Σ a subgroup of Aut(Γ) (which is of order six) of order two or three. The Coxeter matrix ΓΣ = (mΣ J,K )J,K∈I Σ — which encodes the presentations of the Coxeter group W Σ , the Artin-Tits monoid (B + )Σ and the Artin-Tits group B Σ — is then of type Bn if Γ = A2n−1 , A2n or Dn+1 with \|Σ\| = 2, of type F4 if Γ = E6 , or of type G2 if Γ = D4 with \|Σ\| = 3 (see for example [9, 5]). We fix an integral domain R and (a, b, c, d) ∈ R4 such that d2 − ad − bc = 0. Let (fi )i∈I ∈ (V ⋆ )I be an LK-family relatively to (a, b, c, d) and consider the associated LK-representation ψ : B + → L (V ) of B + . We will always assume in this section that the elements b, c and d are non-zero in R, so that they become invertible in the field of fractions K of R. By extension of the scalars from R to K, we will consider ψ as acting on the K-vector space VK = V ⊗R K. It then follows from proposition 10 that ψ induces a twisted LKrepresentation ψ Σ : (B + )Σ → L (VKΣ ) of (B + )Σ over VKΣ = K ⊗R V Σ . By [6, Section 3.2], the LK-family (fi )i∈I , seen as an element of (VK⋆ )I , is entirely determined by the common value f of the elements fi (eαi ), i ∈ I. Moreover when f 6= 0, then ψ has its image included in GL(VK ) and hence induces an LK-representation ψgp : B → GL(VK ) of the Artin-Tits group B, and a twisted Σ LK-representation ψgp : B Σ → GL(VKΣ ) of the Artin-Tits group B Σ . In view of the results of the previous section, in order to really understand the Σ maps ψJΣ = ψ∆ , J ∈ I Σ , we need to list the possible configurations of the J-meshes J + MJ (Θ) in Φ , when Θ runs through Φ+ /Σ. This is done in the following lemma. Lemma 29. Fix an orbit J of I under Σ. We list, in the following tables, the number NX of occurrences of configuration X (with the notations of subsections 3.1 to 3.4) among the J-meshes MJ (Θ) in Φ+ , when Θ runs through Φ+ /Σ. (The configurations that do not occur are omitted). 16 ANATOLE CASTELLA • If Γ = A2n−1 , or Dn with \|Σ\| = 2, or E6 , then J can be of type A or B and we get : A2n−1 Dn (\|Σ\| = 2) E6 NA1 1 1 1 if J is of type A NA2 NA3 (n − 1)2 n−1 n2 −6n+10 2n−5 9 7 NB1 1 1 1 if J is of type B NB2 NB3 NB4 (n − 2)2 2n − 4 1 (n−2)(n−3) 0 n−2 6 4 3 • If Γ = A2n , then J can be of type B (if n > 2) or D and we get : NB1 1 if J is of type B NB2 NB3 (n − 1)(n − 2) 2n − 3 NB4 1 if J is of type D ND1 ND2 ND3 1 (n − 1)2 n − 1 • If Γ = D4 with \|Σ\| = 3, then J can be of type A or C and we get : if J is of type A NA1 NA2 NA3 1 1 2 Proof. Left to the reader. if J is of type C NC1 NC2 NC5 1 1 1 4.1. Irreducibility. In this subsection, we exclude the case Γ = A2n , so that there is no orbit of type D in I, and hence no configurations D1 to D5 in Φ+ . Lemma 30. For all Θ ∈ Φ+ /Σ with dp(Θ) > 2, there exists J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ). Proof. Fix α ∈ Θ and i ∈ I such that dp(si (α)) < dp(α), and let us denote by J the orbit of i under Σ. Necessarily, the J-mesh MJ (Θ) is in configuration A3, B3, B4 or C5, and Θ is not of minimal depth in MJ (Θ). In case of a configuration A3, B3, B4 with Θ of maximal depth in MJ (Θ), or C5 with Θ of maximal or second-maximal depth in MJ (Θ), then dp(rJ (Θ)) < dp(Θ) and we are done. In the other cases, then (dp(rJ (Θ)) > dp(Θ) and) a look at the graph of Φ+ in the different possible situations reveals that there exists i′ ∈ I with dp(si′ (α)) < dp(α) for which, if we denote by J ′ its orbit under Σ, then MJ ′ (Θ) is in configuration A3 or B3, so here again we are done, up to changing J for J ′ . In the following proposition, we prove the first part of our irreducibility criterion. Notation 31. If J if an orbit of I under Σ, we consider the polynomial PJ as in notation 25. In view of proposition 23, there exist a polynomial QJ ∈ R[X] such Σ that PJ (ψJΣ ) = PJ (ϕΣ J ) + fJ QJ (ϕJ ) ⊗ eΘJ . + By lemma 26, we get PJ (ψJΣ )(eΘ ) = fJ QJ (ϕΣ J )(eΘ )eΘJ for every Θ ∈ Φ /Σ with ) = 0 for J, K ∈ I Σ with )(e Θ 6= ΘJ . Moreover it can be checked that fJ QJ (ϕΣ ΘK J = 2. mΣ J,K Σ with Proposition 32. Assume that fJ QJ (ϕΣ J )(eΘK ) 6= 0 for every J, K ∈ I Σ Σ Σ Σ mJ,K > 3. If U is an invariant subspace of VK under ψ such that PK (ψK )(U ) 6= {0} for some K ∈ I Σ , then U = VKΣ . TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 17 T Proof. By proposition 27 and since U is invariant, we get PJ (ψJΣ )(U ) ⊆ U ReΘJ Σ )(U ) 6= {0}. Now if J ∈ I Σ is distinct for all J ∈ I Σ , whence eΘK ∈ U since PK (ψK Σ from K, we get that PJ (ψJ )(eΘK ) = fJ QJ (ψJΣ )(eΘK )eΘJ belongs to U , whence Σ eΘJ ∈ U if mΣ J,K > 3 since then, by assumption, fJ QJ (ϕJ )(eΘK ) 6= 0. We thus get that eΘJ ∈ U for all J ∈ I Σ by connectivity of ΓΣ . Now consider Θ ∈ Φ+ /Σ, of depth p > 2, such that every eΘ′ belongs to U whenever dp(Θ′ ) < p. By lemma 30, there exists J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ), so in particular erJ (Θ) ∈ U . Necessarily, the J-mesh MJ (Θ) is in configuration A3, B3, B4 with Θ of maximal depth in MJ (Θ), or C5 with Θ of maximal or second-maximal depth in MJ (Θ), and hence ψJΣ (erJ (Θ) ) is a linear combination of eΘ , erJ (Θ) , eΘJ and possibly some other eΘ′ for Θ′ ⊆ MJ (Θ) with dp(Θ′ ) < p, in which the coefficient of eΘ is c, cd, c2 , c3 or bc2 respectively. Since U is invariant by ψJΣ and since bcd 6= 0 by assumption, we thus get, in all the cases, the element eΘ as a linear combination of elements of U , hence eΘ ∈ U and we conclude by induction that U = VKΣ . The following lemma explicits the condition on fJ QJ (ϕΣ J )(eΘK ) of the previous proposition (recall that f is the common value of the fi (eαi ) for i ∈ I, and that we set dˇ = a − d) : Lemma 33. Consider two orbits J, K ∈ I Σ with mΣ J,K > 3. Then the coefficient Σ fJ QJ (ϕJ )(eΘK ) is equal to : (i) −\|K\|af if J is of type A, (ii) ad2 f (dˇ2 − df ) if J is of type B, (iii) ad5 f (−d3 f 2 + addˇ2 f − dˇ5 ) if J is of type C. Proof. First notice that the assumption mΣ J,K > 3 implies that mj,k = 3 for some (j, k) ∈ J × K (see [9, Section 3] or [5, Section 4.2]). Now it is easy to see that if J is of type A (resp. B, resp. C), then ΘK necessarily appears at the bottom of a configuration A3 (resp. B3 or B4, resp. C5). The result then follows from direct computations, using the expression of QJ deduced from remark 24 and notation 25, and the expressions of the encountered fJ (eΘ ) as multiples of f deduced from [6, Table 1]. Now we turn to the second part of the irreducibility criterion. Proposition 34. Assume that there exists a prime ideal Q of R with bcd 6∈ Q and Im(fi ) ⊆ Q for all i ∈ I. If U is an invariant subspace of VKΣ under ψ Σ such that PJ (ψJΣ )(U ) = {0} for all J ∈ I Σ , then U = {0}. Proof. Recall that the image of PJ (ψJΣ ) is included in KeΘJ . Since U is invariant, we get PJ (ψJΣ )ψgΣ (U ) = {0} for every (J, g) ∈ I Σ × (B + )Σ , and hence, if we denote by L(J,g),Θ the element of R such that PJ (ψJΣ )ψgΣ (eΘ ) = L(J,g),Θ eΘJ , and by L the matrix (L(J,g),Θ )(J,g)∈I Σ ×(B + )Σ ,Θ∈Φ+ /Σ , we get that U is included in ker(L). Now we claim that under the assumptions of the proposition, the matrix L is non-singular, which thus implies that U = {0}. To prove our claim, it suffices to construct some pairs (JΘ , gΘ ) ∈ I Σ ×(B + )Σ for Θ running through Φ+ /Σ, such that the square submatrix L′ = (L(JΘ ,gΘ ),Ω )Θ,Ω∈Φ+ /Σ of L is invertible. We construct the pairs (JΘ , gΘ ) ∈ I Σ × (B + )Σ by induction on dp(Θ) as follows. 18 ANATOLE CASTELLA If dp(Θ) = 1, i.e. if Θ = ΘJ for some J ∈ I Σ , we set JΘ = J and gΘ = 1. Now by induction if dp(Θ) > 2, then we fix some J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ) (which exists by lemma 30) and we set JΘ = JrJ (Θ) and gΘ = grJ (Θ) ∆J . We are going to see that k l m Σ (i) PJΘ (ϕΣ JΘ )ϕgΘ (eΘ ) = ±b c d eΘJΘ for some k, l, m ∈ N, Σ Σ (ii) PJΘ (ϕJΘ )ϕgΘ (eΩ ) = 0 for Ω ∈ Φ+ /Σ with dp(Ω) > dp(Θ) and Ω 6= Θ. Σ Since PJ (ψJΣ )ψgΣ (eΘ ) ≡ PJ (ϕΣ J )ϕg (eΘ ) mod Q, points (i) and (ii) will show that ′ the matrix L is lower triangular modulo Q — if we choose an order on Φ+ /Σ that is consistent with depth — with non-zero diagonal coefficients modulo Q, since bcd 6∈ Q. Hence this will show that the determinant of L′ is non-zero modulo Q (recall that Q is prime), and therefore is non-zero in R, whence the result. So it remains to prove (i) and (ii). Let us prove (i). If Θ = ΘJ for some J ∈ I Σ , then by lemma 26 we get 3 6 PJ (ϕΣ J )(eΘJ ) = PJ (0)eΘJ , with PJ (0) = −bc (resp. (bc) , resp. (bc) ) if J is of type A (resp. B, resp. C), whence the result when dp(Θ) = 1. Now assume that dp(Θ) > 2 and consider the fixed J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ). If MJ (Θ) is in configuration A3, B3, B4 or C5, with Θ of maximal depth in 2 3 MJ (Θ), then ϕΣ J (eΘ ) = λerJ (Θ) with λ = b, bd, b or b respectively, whence the result by induction. The only other possible configuration is C5 with Θ of second maximal depth in MJ (Θ), which occurs when Γ = D4 with \|Σ\| = 3. If we label the vertices of D4 as in [2, Planche IV], we thus get I Σ = {J, K} with J = {1, 3, 4} and K = {2}, Θ = {α1 + α2 + α3 , α1 + α2 + α4 , α2 + α3 + α4 }, gΘ = ∆K ∆J , and Σ 9 7 Σ 3 JΘ = J, whence ϕΣ gΘ (eΘ ) = b ceΘJ and PJΘ (ϕJΘ )ϕgΘ (eΘ ) = b c eΘJΘ . Let us prove (ii). By lemma 26, it suffices to see that ϕΣ gΘ (eΩ ) is a linear combi′ ′ nation of some eΘ with Θ 6= ΘJΘ . This is clear if Θ = ΘJ since then gΘ = 1. So assume that dp(Θ) > 2 and consider the fixed J ∈ I Σ such that dp(rJ (Θ)) < dp(Θ). In view of the results of section 3, ϕΣ J (eΩ ) is a linear combination of some elements eΩ′ where Ω′ is an orbit included in MJ (Ω), which hence satisfies dp(Ω′ ) > dp(Ω)−p, where p = 1, 2 or 3 when J is of type A, B or C respectively. If dp(Ω′ ) > dp(rJ (Θ)) and Ω′ 6= rJ (Θ) for every orbit Ω′ ⊆ MJ (Ω), we get by ′ induction that ϕΣ grJ (Θ) (eΩ′ ) is a linear combination of elements eΘ′ with Θ distinct from ΘJrJ (Θ) = ΘJΘ , whence the result. The only obstructions to this situation are the two following configurations : • MJ (Θ) in configuration B3, MJ (Ω) in configuration B4, and dp(Θ) = dp(Ω) maximal in MJ (Θ) and in MJ (Ω). • MJ (Θ) = MJ (Ω) in configuration C5, Ω of maximal depth, and Θ of second maximal depth, in MJ (Θ). The second point occurs if Γ = D4 with Ω = {α1 + α2 + α3 + α4 } and Θ, J and K 3 Σ as in the proof of (i) above (hence gΘ = ∆K ∆J ), whence ϕΣ gΘ (eΩ ) = b ϕK (eΘK ) = 0 and the result in that case. The first point occurs once if Γ = E6 , for J = {3, 5}, Ω = {α2 + α3 + α4 + α5 } and Θ = {α1 + α3 + α4 + α5 , α3 + α4 + α5 + α6 } with the notations of [2, Planche V]. But the possibilities for (JΘ , gΘ ) are then ({1, 6}, ∆J ∆{2} ∆J ), (J, ∆{1,6} ∆{2} ∆J ) 3 2 or (J, ∆{2} ∆{1,6} ∆J ), whence ϕΣ gΘ (eΩ ) = b d e{α2 } and the result in that case. The first point also occurs if Γ = A2n−1 (n > 4), for J = {n − j, n + j}, Pn+3j Pn−j Pn+j Ω = { i=n−j αi } and Θ = { i=n−3j αi , i=n+j αi } where 1 6 j 6 n−1 3 , with the notations of [2, Planche I]. But in that case we necessarily get gΘ = grK rJ (Θ) ∆K ∆J TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 19 where K = {n − k, n + k} for some j + 1 6 k 6 3j. Hence the K-mesh MK (rJ (Ω)) Σ 2 is in configuration B2 and we get ϕΣ K ϕJ (eΩ ) = (bd) erJ (Ω) , with dp(rJ (Ω)) = dp(Θ) − 2 = dp(rK rJ (Θ)) and rJ (Ω) 6= rK rJ (Θ), whence the result in those cases by induction on rK rJ (Θ). Since there is no configuration B3 in Φ+ for Γ = Dn , the proof is completed. Theorem 35. Assume that condition (⋆) holds : (⋆) R is an integral domain, fi (eαi ) 6= 0 for all i ∈ I, and there exists a totally ordered integral domain R0 and a ring homomorphism ρ : R → R0 , such that ρ(t) > 0 for t ∈ {a, b, c, d}, and Im(fi ) ⊆ ker(ρ) for all i ∈ I. Σ Then the LK-representations ψ Σ and ψgp are irreducible over VKΣ . Proof. The result follows from propositions 32 and 34 : the second one applies with Q = ker(ρ), and the first one applies since under condition (⋆), we get a, d and ˇ < 0 (since ρ(dd) ˇ = ρ(−bc) < 0), so f = fi (eαi ) non-zero in R, f ∈ ker(ρ) and ρ(d) Σ Σ lemma 33 shows that the coefficients fJ QJ (ϕJ )(eΘK ), for mJ,K > 3, are non-zero in R (look at the image of the parenthesized expressions by ρ). Remark 36. This result extends to all the spherical and connected cases but A2n what was done for Γ = E6 in [18, Section 6.1], by a totally different method. The question for Γ = A2n (n > 2) is still open, as far as I know. The strategy used here does not directly apply because of the orbit J of type D in Γ : first, lemma 33 above is not true in this case (look at the orbit Θ′J ), and second, the image of PJ (ψJΣ ) is not of dimension one, but two (at least when f 6= 0). The case Γ = A2 is trivial however, since then I Σ = {I}, VKΣ = KeΘI ⊕ KeΘ′I , and ψIΣ is the homothety of ratio bcf , hence in this case, ψ Σ is not irreducible. 4.2. Non-equivalence. In corollary 39 and theorem 41 below, we study the case of two twisted LKrepresentations ψ Σ and ψ ′Σ of BΓ+Σ ≈ (BΓ+ )Σ induced by two LK-representations ψ and ψ ′ of BΓ+ for a fixed Coxeter matrix Γ. On the contrary in proposition 42 below, we focus on the particular case of the Coxeter type Bn , n > 3, and study the case of three twisted LK-representations ′ ′′ + ψ Σ , ψ ′Σ and ψ ′′Σ of BB induced by three LK-representations ψ, ψ ′ and ψ ′′ of n + + + BΓ , BΓ′ and BΓ′′ respectively, where Γ = A2n−1 , Γ′ = Dn+1 and Γ′′ = A2n . But let us begin with a result on the diagonalizability of the maps ψJΣ . Proposition 37. Let J be an orbit of I under Σ. We set dˇ = a − d. (i) If J is of type A and if d, dˇ and f are pairwise distinct, then ψJΣ is diagonalizable over K, with eigenvalues d, dˇ and f of multiplicity NA2 + NA3 , NA3 and NA1 respectively. ˇ dˇ2 and df are pairwise distinct, then ψ Σ (ii) If J is of type B and if d2 , dd, J ˇ dˇ2 and df of multiplicity is diagonalizable over K, with eigenvalues d2 , dd, NB2 + NB3 + NB4 , NB3 + NB4 , NB4 and NB1 respectively. Proof. Let us prove (i). In view of proposition 27, the map ψJΣ is diagonalizable ˇ But we more precisely claim over K and its eigenvalues can be fJ (eΘJ ) = f , d or d. that for any J-mesh M in configuration A1 (resp. A2, resp. A3), some suitable linear combinations of eΘJ and eΘ , Θ ⊆ M , are eigenvectors of ψJΣ for the value f ˇ whence the result. (resp. d, resp. d and d), 20 ANATOLE CASTELLA The claim is obvious for A1 since eΘJ is clearly an eigenvector of ψJΣ for the value f . For configuration A2, the linear combination eΘ + λeΘJ is suitable if λ satisfies fJ (eΘ ) + λf = dλ, and such a λ exists in K since f 6= d by assumption. Now since ˇ the block of ϕΣ for a J-mesh M in configuration A3 is certainly similar d 6= d, J ˇ i.e. there certainly exists a linear combination to the diagonal matrix Diag(d, d), ẽ of eΘ and erJ (Θ) , where M = Θ ∪ rJ (Θ), that is an eigenvector of ϕΣ J for the ˇ eigenvalue d (resp. d). The linear combination ẽ + λeΘJ is then an eigenvector of ˇ whenever fJ (ẽ) + λf = dλ (resp. dλ), ˇ and such a ψJΣ for the eigenvalue d (resp. d) ˇ λ exists in K since f 6= d (resp. f 6= d) by assumption. The proof of (ii) is similar (recall that there is no configuration B5 in Φ+ ). Remark 38. One can prove an analogue of the previous results for the orbit J of type C in Γ = D4 (when \|Σ\| = 3), but we will not need it in what follows. The ˇ ddˇ2 , eigenvalues in this case would be d3 (of multiplicity NC2 + NC5 = 2), and d2 d, 3 2 ˇ d and d f (of multiplicity NC5 = NC1 = 1). There is also an analogue for the orbit J of type D in Γ = A2n , if we assume ˇ 3 of PJ splits in K[X]. that −ddˇ = bc is a square in K, so that the factor X 2 + (dd) Now let us fix another LK-representation ψ ′ : B + → L (V ) associated with a quadruple (a′ , b′ , c′ , d′ ) ∈ R4 (such that d′2 − a′ d′ − b′ c′ = 0) and an LK-family (fi′ )i∈I relatively to (a′ , b′ , c′ , d′ ). We assume that b′ , c′ and d′ are non-zero in R. As in the preamble of the current section, the LK-family (fi′ )i∈I , seen as an element of (VK⋆ )I , is entirely determined by the common value f ′ of the fi′ (eαi ) for i ∈ I, and the LK-representation ψ ′ induces a twisted LK-representation ψ ′Σ : (B + )Σ → L (VKΣ ) of (B + )Σ . Corollary 39. We set dˇ = a − d and ď′ = a′ − d′ . (i) Assume that Γ 6= A2n , A3 and that d, dˇ and f (resp. d′ , ď′ and f ′ ) are pairwise distinct. Then for any orbit J ∈ I Σ of type A, the maps ψJΣ and ˇ f ) = (d′ , ď′ , f ′ ). ψJ′Σ are similar over K if and only if (d, d, ˇ dˇ2 and df (resp. d′2 , d′ ď′ , (ii) Assume that Γ = A2n , n > 3, and that d2 , dd, ď′ 2 and d′ f ′ ) are pairwise distinct. Then for any orbit J of type B, the ˇ f ) = ±(d′ , ď′ , f ′ ). maps ψJΣ and ψJ′Σ are similar over K if and only if (d, d, ˇ f ) 6= (d′ , ď′ , f ′ ) (resp. if Γ = A2n , n > 3, In particular, if Γ 6= A2n , A3 and (d, d, ′ ′ ′ ˇ and (d, d, f ) 6= ±(d , ď , f )), then the twisted LK-representations ψ Σ and ψ ′Σ are not equivalent over K. Proof. Of course if ψ Σ and ψ ′Σ are equivalent, then the maps ψJΣ and ψJ′Σ are similar for every J ∈ I Σ . But in view of proposition 37 above, the maps ψJΣ and ψJ′Σ are similar if and only if they have the same eigenvalues, with same multiplicity. In case (i), the three multiplicities are NA2 + NA3 > NA3 > NA1 (the second inequality is strict since we avoid the case Γ = A3 ), whence the result. In case (ii), the four multiplicities are NB2 + NB3 + NB4 > NB3 + NB4 > NB4 = NB1 (the first inequality is strict since we avoid the case Γ = A4 ), so ψJΣ and ψJ′Σ are similar if and only if d2 = d′2 , ddˇ = d′ ď′ and {dˇ2 , df } = {ď′ 2 , d′ f ′ }. And these ˇ f ) = ±(d′ , ď′ , f ′ ). three equalities are clearly equivalent to (d, d, Remark 40. The case Γ = A3 (resp. A4 ) can be dealt with as in case (i) (resp. (ii)) of the previous proposition and its proof, but the criterion of similarity of ψJΣ and TWISTED LAWRENCE-KRAMMER REPRESENTATIONS 21 ˇ f } = {ď′ , f ′ } ψJ′Σ for J of type A (resp. B) as then to be relaxed to d = d′ and {d, 2 ′ ′ 2 2 ′2 ′ ′ ′ ˇ ˇ (resp. {d , dd} = {d , d ď } and {d , df } = {ď , d f }). The case Γ = A2 is trivial since then I Σ = {I}, VKΣ = KeΘI ⊕ KeΘ′I , and ψIΣ (resp. ψI′Σ ) is the homothety of ratio bcf (resp. b′ c′ f ′ ). Hence in this case, the twisted LK-representations ψ Σ and ψ ′Σ are equivalent if and only if bcf = b′ c′ f ′ . Theorem 41. Assume that Γ 6= A2 and that condition (⋆) holds for both ψ and ˇ f ) 6= ψ ′ , for the same morphism ρ : R → R0 if Γ = A3 or Γ = A2n . Then if (d, d, ′ ′ ′ Σ ′Σ (d , ď , f ), the twisted LK-representations ψ and ψ are not equivalent over K. Proof. Under condition (⋆), the elements d, dˇ and f are pairwise distinct since ˇ the result their image by ρ are positive, negative and zero respectively (for d, 2 ˇ follows from the identity dd = −bc). Similarly, the elements df , d , ddˇ and dˇ2 are ˇ < 0, ρ(d2 ) > 0 and ρ(dˇ2 ) > 0, pairwise distinct : indeed, we get ρ(df ) = 0, ρ(dd) 2 2 2 2 ˇ ˇ ˇ and d 6= d since d − d = (d − d)a and hence ρ(d2 − dˇ2 ) > 0. The analogue occurs for ψ ′ and hence the results of corollary 39 and remark 40 applies. This gives the result if Γ 6= A2n , A3 . If Γ = A2n , n > 3, then ˇ f ) 6= ±(d′ , ď′ , f ′ ), but we clearly have the condition stated in corollary 39 is (d, d, ′ ′ ′ ′ ˇ (d, d, f ) 6= −(d , ď , f ) since ρ(d) and ρ(d ) are positive, whence the result in that case. Similarly for Γ = A3 (resp. A4 ), the condition of non-equivalence derived ˇ f } 6= {ď′ , f ′ } (resp. {d2 , dd} ˇ = from remark 40 should be d 6= d′ or {d, 6 {d′2 , d′ ď′ } or 2 ′ ′ ′ 2 ˇ f } = {ď′ , f } is equivalent to (d, ˇ f ) = (ď′ , f ′ ) {dˇ , df } 6= {ď′ , d f }). But here {d, ′ ′ since ρ(d) and ρ(d ) are positive and ρ(f ) = ρ(f ) = 0, whence the result for Γ = A3 . Finally for Γ = A4 , since the images by ρ of d2 , dˇ2 , d′2 and ď′ 2 (resp. of ddˇ and d′ ď′ , resp. of df and d′ f ′ ) are positive (resp. negative, resp. zero), the ˇ = {d′2 , d′ ď′ } and {dˇ2 , df } = {ď′ 2 , d′ f ′ } of remark 40 is equivalent condition {d2 , dd} 2 2 ˇ f ) = (d′ , ď′ , f ′ ) since, again, ρ(d) ˇ ˇ to (d , dd, d , df ) = (d′2 , d′ ď′ , ď′ 2 , d′ f ′ ), i.e. (d, d, ′ and ρ(d ) are positive. Let us finally turn to the case of the Coxeter type Bn by considering three twisted ′ ′′ + LK-representations ψ Σ , ψ ′Σ and ψ ′′Σ of BB induced by three LK-representations n + + + ′ ′′ ψ, ψ and ψ of BΓ , BΓ′ and BΓ′′ respectively, where Γ = A2n−1 , Γ′ = Dn+1 and Γ′′ = A2n . ′′ The twisted LK-representation ψ ′′Σ is clearly non-equivalent to the two others ′′ since it is of dimension \|Φ+ Γ′′ /Σ \| = n(n+1) whereas the two others are of dimension ′ + + ′ 2 \|ΦΓ /Σ\| = \|ΦΓ′ /Σ \| = n . The following proposition proves that ψ Σ ans ψ ′Σ are not equivalent, under the usual assumptions on the parameters (a, b, c, d) and (fi )i∈I of ψ, and (a′ , b′ , c′ , d′ ) and (fi′ )i∈I ′ of ψ ′ . We set as above dˇ = a − d and f = fi (eαi ) for any i ∈ I. Proposition 42. Assume that d, dˇ and f (resp. df , d2 , ddˇ and dˇ2 ) are pairwise distinct, and that the analogue holds for ψ ′ , which is the case for example if condition (⋆) holds for both ψ and ψ ′ . Then the twisted LK-representations ψ Σ and ′ ψ ′Σ are not equivalent over K. Proof. The fact that condition (⋆) implies the given conditions on the parameters has been shown in the proof of theorem 42. But then proposition 37 applies and + shows that the images by ψ and ψ ′ of a given standard generator of BB do not n have the same number of eigenvalues, whence the result. 22 ANATOLE CASTELLA References [1] S. Bigelow. Braid groups are linear. J. Amer. Math. Soc. 14 (2001), 471–486. [2] N. Bourbaki. 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THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC BEN HAYES AND ANDREW SALE arXiv:1601.03286v1 [math.GR] 13 Jan 2016 Abstract. Given sofic approximations for countable, discrete groups G, H, we construct a sofic approximation for their wreath product G ≀ H. Sofic groups, introduced by Gromov [10] and developed by Weiss [19], are a large class of groups which can be approximated, in some sense, by finite groups. There are many examples, including all amenable groups, all residually finite groups, and all linear groups (by Malcev’s Theorem). However, because of the weakness of the approximation by finite groups, few permanence properties of soficity are properly understood. Relatively straight-forward examples include closure under direct product and increasing unions, and the soficity of residually sofic groups. More substantial results generally require some amenability assumption. For example, an amalgamated product of two sofic groups is know to be sofic if the amalgamated subgroup is amenable (see [9],[15],[4],[17]). This was extended to encompass the fundamental groups of all graphs of groups with sofic vertex groups and amenable edge groups [3]. In the same paper, it is shown that the graph product of sofic groups is sofic. Also, if H is sofic and is a coamenable subgroup of G, then G is sofic too [8]. We prove a new permanence result, namely that soficity is closed under taking wreath products: Theorem 1. Let G, H be countable, discrete, sofic groups. Then G ≀ H is sofic. We remark that our result is general and requires no amenability or residual finiteness assumptions. The special case of Theorem 1 when G is abelian was proved by Paunescu [15], who used methods of analysis and the notion of sofic equivalence relations developed by Elek and Lippner [5]. While finishing this paper, we learnt that Holt and Rees have dealt with the case when H is residually finite and G sofic [12]. We prove the general result directly, by constructing a sofic approximation for the wreath product, giving a proof that is constructive, quantitative (see Proposition 2.1), and entirely self-contained. The notion of hyperlinearlity gives a class of groups defined in a similar vein to sofic groups, but where they are approximated instead by unitary groups. Our construction extends to the situation where G is hyperlinear and H sofic, showing that G ≀ H is hyperlinear, see [11]. Via their approximations by finite groups, sofic groups have applications to problems of current mathematical interest in a wide area of fields. Sofic groups are relevant to ergodic theory because they are the largest class of groups for which Bernoulli shifts are classified by their base entropy (see [1],[13]) and for which Gottschalk’s surjuncitivity conjecture holds (see e.g. [10],[13]). In the study of group rings they are useful because they are the largest class of groups for which Kaplansky’s direct finiteness conjecture (see [6]) is known. In the field of L2 –invariants, they are the largest class of groups for which the determinant conjecture is known (see [7]), which is necessary to define L2 –torsion (see [14] Conjecture 3.94). They are also the largest class of groups for which an analogue of Lück approximation is known (see [18]). We refer the reader to [14] for applications of L2 –invariants to geometry and group theory. See also [16],[2] for a survey of sofic groups. Date: April 11, 2018. 1 THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 2 Acknowledgments. The first named author would like to thank Jesse Peterson for asking him if wreath products of sofic groups are sofic at the NCGOA Spring Institute in 2012 at Vanderbilt University. 1. Preliminaries We begin with the necessary definitions, as well as a useful lemma to help us identify sofic approximations in wreath products. Definition 1. Let A be a finite set. The normalized Hamming distance, denoted dHamm , on Sym(A) is defined by dHamm (σ, τ ) = 1 \|{a ∈ A : σ(a) 6= τ (a)}\|. \|A\| Definition 2. Let G be a countable discrete group, F a finite subset of G, and ε > 0. Fix a finite set A and a function σ : G → Sym(A). We say that σ is (F, ε)–multiplicative if max dHamm (σ(g)σ(h), σ(gh)) < ε. g,h∈F We say that σ is (F, ε)–free if min dHamm (σ(g), Id) > 1 − ε. g∈F \{1} We say that σ is an (F, ε)–sofic approximation if it is (F, ε)–multiplicative, (F, ε)–free, and furthermore σ(1) = Id . Lastly, we say that G is sofic if for every finite F ⊆ G and ε > 0, there is a finite set A and an (F, ε)–sofic approximation σ : G → Sym(A). Our aim is to use sofic approximations for G and H and build a sofic approximation for G ≀ H. First recall that the wreath product is defined as M G≀H = G⋊H H L where the action of h ∈ H is given via αh ∈ Aut ( H G), defined by αh (gx )x∈H = (gh−1 x )x∈H . A homomorphism π : L G ≀ H → K, for some group K, can be decomposed into a pair of homomorphisms π1 : H G → K, π2 : H → K which satisfy the following equivariance condition: L π2 (h)π1 (g) = π1 (αh (g))π2 (h), for all h ∈ H, g ∈ H G. The following lemma gives an “approximate analogue” to this situation. Lemma 1.1. Let G, H beLcountable, discrete groups. For every finite set F0 ⊆ G ≀ H there are finite sets E1 ⊆ H G, E2 ⊆ H such that the following holds: Let ε > 0 and Ω be a finite set. Suppose σ : G ≀ H → Sym(Ω) is a map such that M • the restriction of σ to G is (E1 , ε/6)–multiplicative, H • the restriction of σ to H is (E2 , ε/6)–multiplicative, • • max g∈E1 ,h∈E2 max g∈E1 ,h∈E2 dHamm σ(g, h), σ(g, 1)σ(1, h) < ε/6, dHamm (σ(1, h)σ(g, 1), σ(αh (g), 1)σ(1, h)) < ε/6. Then σ is (F0 , ε)–multiplicative. THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 3 Proof. After making the right definitions for E1 , E2 , we apply the triangle inequality several times to obtain the result. We require that if (g, h), (ĝ, ĥ) are in F0 , then g, ĝ, αh (ĝ) ∈ E1 , and h, ĥ ∈ E2 . This is true if we define E1 = αh (g) : h ∈ πH (F0 ) ∪ {1}, g ∈ πG (F0 ) , E2 = πH (F0 ). We leave verification that this is sufficient to the reader. To see how this gives an “approximate analogue” of the situation for homomorphisms, notice that the above lemma says that an approximate homomorphismL σ : G≀H → Sym(A) can be thought of as a pair of approximate homomorphisms σ1 : H G → Sym(A), σ2 : H → Sym(A) so that σ2 (h)σ1 (g) ≈ σ1 (αh (g))σ2 (g) L for all g in a large enough finite subset of H G and all h in a large enough finite subset of H. (Here we use ≈ to indicate that both maps agree on a sufficiently large subset of A). It is not hard to see that the above lemma is valid with G≀H replaced with any semidirect product and Sym(A) replaced with any group equipped with a bi-invariant metric. 2. The sofic approximation for wreath products To facilitate our proof, we need to introduce some notation. Let G, H be countable discrete groups and σA : G → Sym(A), σB : H → Sym(B) be two functions (not assumed to be homomorphisms). For h ∈ H, b0 ∈ B, define ! M (h) σA,b0 : G → Sym A B by (h) ab )b∈B , σA,b0 (g) (ab )b∈B = (b where b ab = ( σA (g)(ab ), ab , if b = σB (h)b0 , otherwise. Now suppose that E ⊆ H is finite, and take a subset B0 ⊆ {b ∈ B : σB (h1 )b 6= σB (h2 )b for all h1 , h2 ∈ E with h1 6= h2 .}. (h ) (h ) Note that σA,b1 0 (g1 ) and σA,b2 0 (g2 ) commute if b ∈ B0 , g1 , g2 ∈ G, h1 , h2 ∈ H and h1 6= h2 . Thus it makes sense to define, for b ∈ B0 , ! M M σA,b : G → Sym A E by B Y (x) σA,b (gx ). σA,b (gx )x∈E = x∈E In our applications σB will be a sofic approximation, so we can take B0 to make up the majority of B. Thus σA,b will be defined for “most” b ∈ B. We package all these maps together as a single map ! ! M M σc G → Sym A ⊕B A: E by σc A (g)(a, b) = B ( σA,b (g)(a), b , if b ∈ B0 (a, b), if b ∈ B \ B0 . THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC We extend σc A to define L H 4 L L / E G. Now G by declaring that σc A (g) = Id if g ∈ H G, but g ∈ ! ! M σc A ⊕B B : H → Sym B by σc B (h)(a, b) = a, σB (h)b . Finally, define σ b : G ≀ H → Sym by M ! A B ⊕B ! σ b (g, h) = σc c A (g)σ B (h). Note that σ b is determined by the choice of E, B0 and the two maps σA and σB . Proposition 2.1. Let F ⊆ G ≀ H be finite and ε > 0. Then there are finite sets EA ⊆ G, EB ⊆ H and an ε′ > 0 so that if σA : G → Sym(A) and σB : H → Sym(B) are (EA , ε′ ), (EB , ε′ )–sofic approximations for G, H respectively, then σ b is an (F, ε)–sofic approximation. The remainder of this section is dedicating to proving Proposition 2.1. We will see below that it is possible to compute an explicit upper bound on ε′ . It will depend only on ε and the set F . We remark that σ b(1, 1) = Id by construction. We need to show it is (F, ε)–multiplicative and (F, ε)–free. First we explain how to define the sets E, EA and EB . L Let F ⊆ G ≀ H be finite and ε > 0. Define projections πG : G ≀ H → H G and πH : G ≀ H → H by πG (g, h) = g, πH (g, h) = h. Let E1 , E2 be as in Lemma 1.1 for the finite set F0 = F ∪ {1} ∪ F −1 . As in the proof of Lemma 1.1, we have E1 = αh (g) : h ∈ πH (F0 ), g ∈ πG (F0 ) , E2 = πH (F0 ). Recall that for g = (gx )x∈H ∈ ⊕H G the support of g, denoted Supp(g), is the set of x ∈ H with gx 6= 1. We set [ E = E2 ∪ h Supp(g), g∈E1 h∈E2 EA = gx ∈ G : (gx )x∈H ∈ E1 , EB = E −1 E. Then our collections of finite sets satisfy the following properties, each of which we need later on: EA ⊇ {gx : (gx ) ∈ E1 , x ∈ E}, E E ⊇ ⊇ h Supp(g) for all h ∈ E2 , g ∈ E1 , E2 , EB ⊇ E ∪ E −1 ∪ E −1 E. ε Let σA : G → Sym(A), σB : H → Sym(B) be Choose ε′ so that 0 < ε′ < 48\|E\| 2. (EA , ε′ ), (EB , ε′ )–sofic approximations respectively. Set B01 = {b ∈ B : σB (h1 )b 6= σB (h2 )b for all h1 , h2 ∈ E, h1 6= h2 }, B02 = {b ∈ B : σB (h1 h2 )b = σB (h1 )σB (h2 )b for all h1 , h2 ∈ E}, B0 = B01 ∩ B02 . THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 5 Since σB is a sofic approximation, we can intuitively think of B0 as making up most of the set B. Indeed, Lemma 2.2 below confirms this. Use the sets E ⊂ H, B0 ⊆ B and the maps σA , σB to define the maps σc c b, as constructed at the start of this section. A, σ B, σ We claim that the map σ b is an (F, ε)–sofic approximation. We first make the following preliminary observation. Lemma 2.2. Let κ > 0. If ε′ < Proof. Note that κ 4\|E\|2 \ B01 = then \|B0 \| ≥ (1 − κ)\|B\|. {b ∈ B : σB (h1 )b 6= σB (h2 )b}, h1 ,h2 ∈E h1 6=h2 B02 = \ {b ∈ B : σB (h1 )σB (h2 )b = σB (h1 h2 )b}. h1 ,h2 ∈E h1 6=h2 So \|B \ B01 \| ≤ \|B\| X h1 ,h2 ∈E h1 6=h2 \|{b ∈ B : σB (h1 )b 6= σB (h2 )b}\| . 1− \|B\| We have for all h1 , h2 ∈ E with h1 6= h2 : \|{b ∈ B : σB (h1 )b 6= σB (h2 )b}\| \|B\| = dHamm (σB (h1 ), σB (h2 )) = dHamm (σB (h2 )−1 σB (h1 ), Id). Since dHamm is a invariant under left multiplication, and EB ⊇ E ∪ E −1 we have that −1 ′ dHamm (σB (h2 )−1 , σB (h−1 2 )) = dHamm (Id, σB (h2 )σB (h2 )) < ε . Inserting this into the above two inequalities and using that dHamm is invariant under right multiplication we see that: \|{b ∈ B : σB (h1 )b 6= σB (h2 )b}\| \|B\| = dHamm (σB (h1 ), σB (h2 )) = dHamm (σB (h2 )−1 σB (h1 ), Id) > ′ dHamm (σB (h−1 2 )σB (h1 ), Id) − ε ≥ dHamm (σB (h2−1 h1 ), Id) − 2ε′ > 1 − 3ε′ , where in the last two lines we again use that EB ⊇ E ∪ E −1 ∪ E −1 E. Thus \|B01 \| ≥ (1 − 3 \|E\|2 ε′ ). \|B\| Similarly, (EB , ε′ )–multiplicativity of σB gives X \|B02 \| ≥1− 1 − dHamm σB (h1 h2 ), σB (h1 )σB (h2 ) ≥ 1 − \|E\|2 ε′ . \|B\| h1 ,h2 ∈E This proves the Lemma. ε ε and choose ε′ > 0 as in Lemma 2.2, so ε′ < 48\|E\| Fix κ > 0 with κ < 12 2 . To complete the proof of Proposition 2.1, we break it up into two steps: we first show that this choice of ε′ gives us that σ b is (F, ε)–multiplicative, and then show it is (F, ε)–free. Step 1. We show that σ b is (F, ε)–multiplicative. THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 6 To prove Step 1, we apply Lemma 1.1, verifying below the four necessary conditions. We first check that the restriction to ⊕H G is (E1 , ε/6)–multiplicative. Let g, g ′ ∈ E1 , then ′ ′ dHamm σc c c A (gg ), σ A (g)σ A (g ) M (a, b) : b ∈ B0 , σA,b (g)σA,b (g ′ ) 6= σA,b (gg ′ ) 1 A ⊕ (B \ B ) + ≤ 0 \|A\|\|B\| \|B\| B \|A\|\|B\| \|B\| 1 X ≤κ+ dHamm (σA,b (g)σA,b (g ′ ), σA,b (gg ′ )). \|B\| b∈B0 Q (x) Write g = (gx ), g ′ = (gx′ ). Recall, σA,b (g) = x∈E σA,b (gx ). In the following, we use that the different terms in the product commute, to be precise: for x1 6= x2 ∈ E we have (x ) (x ) [σA,b1 (gx1 ), σA,b2 (gx′ 2 )] = 1. This gives ! Y (x) Y (x) (x) ′ ′ ′ ′ dHamm (σA,b (g)σA,b (g ), σA,b (gg )) = dHamm σA,b (gx )σA,b (gx ), σA,b (gx gx ) . x∈E x∈E Next, using the bi-invariance of the Hamming distance, the triangle inequality, and the (EA , ε′ )–multiplicativity of σA to see that X (x) (x) (x) dHamm (σA,b (g)σA,b (g ′ ), σA,b (gg ′ )) ≤ dHamm σA,b (gx )σA,b (gx′ ), σA,b (gx gx′ ) x∈E = X dHamm (σA (gx )σA (gx′ ), σA (gx gx′ )) x∈E < \|E\| ε′ . Thus we get the required multiplicativity: ′ ′ ′ dHamm (σc c c A (gg ), σ A (g)σ A (g )) < κ + \|E\| ε < ε . 6 The fact that the restriction to H is (E2 , ε/6)–multiplicative is more straight-forward. Indeed, for h, h′ ∈ E2 we have ′ ′ ′ ′ ′ dHamm (σc c c B (hh ), σ B (h)σ B (h )) = dHamm (σB (hh ), σB (h)σB (h )) < ε , where we note that we can use the multiplicative property of σB since E2 ⊆ EB . The third condition of Lemma 1.1 is automatically satisfied by σ b, by construction. We finish this step by verifying the bound on the Hamming distance between σ b (1, h)b σ (g, 1) and σ b (αh (g), 1)b σ (1, h) for h ∈ E2 , g ∈ E1 . Indeed, for such g, h we have dHamm (σc c c c B (h)σ A (g), σ A (αh (g))σ B (h)) ! M 1 ≤ A ⊕ B \ (B0 ∩ σB (h)−1 (B0 )) \|B\| \|A\| \|B\| B 1 (a, b) : b ∈ B0 ∩ σB (h)−1 B0 , σA,b (g)a 6= σA,σB (h)b (αh (g))a \|A\|\|B\| \|B\| 1 (a, b) : b ∈ B0 ∩ σB (h)−1 B0 , σA,b (g)a 6= σA,σB (h)b (αh (g))a . ≤ 2κ + \|B\| \|A\| \|B\| + Since Supp(αh (g)) = h Supp(g), and E contains both Supp(g) and h Supp(g), it follows that for every b ∈ B0 ∩ σB (h)−1 (B0 ) we have Y Y (hx) (x) σA,σB (h)b (gx ). σA,σB (h)b (gh−1 x ) = σA,σB (h)b (αh (g)) = x∈h Supp(g) x∈Supp(g) THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 7 Note that we have used that σA (1) = Id to restrict the number of terms in the product. (x) (hx) We use that for h ∈ E (and hence for h ∈ E2 ) we have that σA,σB (h)b (g) = σA,b (g). Inserting this into the above equation we see that Y (x) σA,b (g) = σA,b (g). σA,σB (h)b (αh (g)) = x∈Supp(g) Returning to the above inequality, we have shown that the remaining sets involved are empty, implying that ε dHamm (σc c c c B (h)σ A (g), σ A (αh (g))σ B (h)) < 2κ < . 6 The hypotheses of Lemma 1.1 have been checked, completing Step 1. Step 2. We show that σ b is (F, ε)–free. Suppose (g, h) ∈ F. If h 6= 1, then Thus (a, b) : σ b(g, h)(a, b) 6= (a, b) ⊇ M ! A B ⊕ b ∈ B : σB (h)b 6= b . dHamm σ b(g, h), Id ≥ dHamm σB (h), Id ≥ 1 − ε′ > 1 − ε. We may therefore assume that h = 1. When b ∈ B0 , the permutation σ b(g, 1) will act by (a, b) 7→ (σA,b (g)a, b), for all a ∈ ⊕B A. We then see that ! ! M (a, b) : σ b (g, 1)(a, b) = (a, b) ⊆ A ⊕ B \ B0 ∪ (a, b) : b ∈ B0 , σA,b (g)a = a . B Assume g = (gx ) 6= 1 and consider the proportion of elements fixed by σ b(g, 1). By the above we have L 1 X a ∈ B A : σA,b (g)a = a 1 (a, b) : σ b(g, 1)(a, b) = (a, b) ≤ κ + . \|B\| \|A\|\|B\| \|B\| \|A\|\|B\| b∈B0 Since g 6= 1, we can find x0 ∈ Supp(g). For every b ∈ B0 , we have that σB (h1 )b 6= σB (h2 )b for every h1 , h2 ∈ Supp(g) with h1 6= h2 . It follows that the (σB (x0 )b)–coordinate of σA,b (g)a is σA (gx0 )aβ , where aβ is the (σB (x0 )b)–coordinate of a. Thus  ( )  M M A ⊕ a ∈ A : σA (gx0 )a = a . a∈ A : σA,b (g)a = a ⊆  B So B\{σB (x0 )b} a∈ L B A : σA,b (g)a = a \|A\|\|B\| ≤ 1 − dHamm (σA (gx0 ), 1). Since gx0 ∈ EA we can use the freeness of σA , yielding 1 \|A\|\|B\| \|B\| \|{(a, b) : σ b (g, 1)(a, b) = (a, b)}\| ≤ κ + \|B0 \| (1 − dHamm (σA (g), 1)) ≤ κ + ε′ . \|B\| Equivalently, dHamm (b σ (g, 1), Id) ≥ 1 − κ − ε′ > 1 − ε. This verifies that σ b is (F, ε)–free, and thus completes the proof of Proposition 2.1, and hence of Theorem 1. THE WREATH PRODUCT OF TWO SOFIC GROUPS IS SOFIC 8 References [1] L. Bowen. 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Hall-Littlewood-PushTASEP and its KPZ limit Promit Ghosal arXiv:1701.07308v1 [math.PR] 25 Jan 2017 Columbia University Department of Statistics, 1255 Amsterdam Avenue, New York, NY 10027 e-mail: pg2475@columbia.edu Abstract: We study a new model of interactive particle systems which we call the randomly activated cascading exclusion process (RACEP). Particles wake up according to exponential clocks and then take a geometric number of steps. If another particle is encountered during these steps, the first particle goes to sleep at that location and the second is activated and proceeds accordingly. We consider a totally asymmetric version of this model which we refer as Hall-Littlewood-PushTASEP (HL-PushTASEP) on Z≥0 lattice where particles only move right and where initially particles are distributed according to Bernoulli product measure on Z≥0 . We prove KPZ-class limit theorems for the height function fluctuations. Under a particular weak scaling, we also prove convergence to the solution of the KPZ equation. Keywords and phrases: Stochastic six vertex model, KPZ universality, Interacting particle system. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model and main results . . . . . . . . . . . . . . . . . . . 3 Transition Matrix & Laplace Transform . . . . . . . . . . 4 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Weak Scaling Limit . . . . . . . . . . . . . . . . . . . . . . A Fredholm Determinant . . . . . . . . . . . . . . . . . . . . B eigenfunction of the Transition Matrix in HL-PushTASEP C Moment Formula . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 14 21 43 61 63 66 69 1. Introduction We introduce and study a special class of interacting particle systems which we call randomly activated cascading exclusion processes (RACEP). Fix a directed graph G = (V, E) with conductances ce (i.e., non-negative weights) along directed edges e ∈ E and define a random walk measure as the Markov chain on vertices with transition probability from v to v ′ given by cv→v′ normalized by the sum of all conductances out of v. The state space for RACEP is {0, 1}V where 1 denotes a particle and 0 denotes a hole. Each particle is activated according to independent Poisson clocks. Once active, a particle moves randomly to one its adjacent sites. Therein, it chooses an independent random number of steps according to a geometric distribution and then performs an independent random walk (according to the random walk measure just described) of that many steps. However, if along that random walk trajectory, the first particle arrives at a site occupied by a second particle, then, the first particle stays 1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 2 at that site (and goes back to sleep) and the second particle becomes active and proceeds as if its Poisson clock had rung (according to the above rules). The Poisson clock activation processes are supposed to model a much slower process than the random walk cascades, so for simplicity we assume that these random walk cascades occur instantaneously. There is some work necessary to prove well-definedness of this process in general, and we do not pursue that here since we will mainly focus on one concrete case. RACEP is a special type of exclusion process (see, e.g. [Lig05, Lig99]) in which the jump distribution depends on the state around the particle in a rather non-trivial way. Exclusion processes are important models of lattice gases, transport (such as in various ecological or biological contexts), traffic, and queues in series (see for example [SCN10, Cor12] and references therein). The prototypical example of an exclusion process is the asymmetric simple exclusion process (ASEP) which was introduced in the biology literature in 1968 [MGP68] (see also more recent applications such as in [CSN05, GS10]) and two years later in the probability literature [Spi70]. RACEP can also be thought of as a variant of a frog model (see, e.g. [AMP02, AMPR01]) in which particles do not fall asleep once active but continue to move and wake up other particles. A natural interpolation between RACEP and the frog model is to have the probability that a particle goes back to sleep be in (0, 1). These relations to exclusion processes and frog models suggest a number of natural probabilistic questions for RACEP, such as understanding its hydrodynamic and fluctuation limit theorems for various families of infinite graphs with simple conductances. We attack these questions exactly for the special case of RACEP in which the underlying graph is Z≥0 and the random walk is totally asymmetric (in the positive direction). From here on out, we will only focus on this one-dimensional case of RACEP. ASEP and RACEP on Z are siblings in that they both arise as (different) special limits of the stochastic six vertex model [GS92, BCG16] – see Section 2.2. They are, in fact, part of a broader hierarchy of integrable probability particle systems which are solvable due to connections to quantum integrable systems – see [CP16, BP16a, BP16b]. A totally asymmetric version of RACEP also comes up as a marginal of certain continuous time RSK-type dynamics which preserve the class of Hall-Littlewood processes – see [BBW16] wherein they refer to RACEP as the t-pushTASEP due to its similarity with the model of q-PushTASEP [BP16a, CP15]. Here, we are mostly interested in this one dimensional totally asymmetric specialization of RACEP. To avoid the confusion with the time parameter t, we refer RACEP as HallLittlewood-PushTASEP (HL-PushTASEP) throughout the rest of the paper. These systems enjoy many concise and exact formulas from which one can readily perform asymptotics. Such results for ASEP go back to the now seminal papers [TW08, TW11] (and earlier to [Joh00] for TASEP), and results for the stochastic six vertex model were worked out in [BCG16, AB16]. In our present paper, we work out the analogous asymptotics for the HLPushTASEP on Z≥0 using the approach of [BCG16]. (We also provide some additional details in the asymptotics compared to the previous works.) Owing to the recent work [Bor16, BO16] such asymptotics could alternatively be performed via reduction to Schur process asymptotics. We do not pursue that route here and instead opt for the more direct (albeit technically demanding) route. The asymptotic results we show for HL-PushTASEP on Z≥0 (as well as those previously shown for ASEP and the stochastic six vertex model) demonstrate its membership in the Kardar-Parisi-Zhang universality class (see, e.g. the review [Cor12] and references therein). In particular, in Theorem 2.4 we show that for step Bernoulli initial data, HL-PushTASEP imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 3 has fluctuations in its rarefaction fan of order t1/3 and with statistics determined by the GUE Tracy-Widom distribution. At the edge of the rarefaction fan, we also demonstrate the occurrence the Baik-Ben Arous-Péché crossover distributions in [BBAP05] which arises also in ASEP [TW09b], TASEP [BBAP05, BAC11], and the stochastic six vertex model [AB16]. See [Cor14] for further references to asymptotics of other integrable probabilistic systems. The centering and scaling that our result demonstrates agrees with the predictions of KPZ scaling theory from the physics literature – see Section 2.4. The KPZ scaling theory also suggests that under certain weak scalings a broad class of interacting particle systems should converge to the Cole-Hopf solution to the KPZ stochastic PDE (i.e., the KPZ equation). There are, in fact, many different choices of weak scalings under which the KPZ equation remains statistically invariant (see, e.g.[QS15, Section 2.4]). The main motivation for our present investigation into the KPZ equation limit for HL-PushTASEP was our desire to solve the analogous question for the stochastic six vertex model. Let us briefly explain why that question is difficult and what our HL-PushTASEP results suggest regarding it. For ASEP under weak asymmetry scaling, the KPZ equation limit was proved in [BG97] (see also [ACQ11, DT16, CST16]). That work relies upon two main identities. The first is the Gärtner (or microscopic Cole-Hopf) transform which turns ASEP into a discrete stochastic heat equation (SHE). The second is a non-trivial key identity (Proposition 4.8 and Lemma A.1 in [BG97]) which allows one to identify white-noise as the limit of the martingale part of the discrete SHE. For the stochastic six vertex model, one still has an analogous Gärtner transform (in fact, this holds for all higher-spin vertex models in the hierarchy of [CP16] due to the duality shown therein). The discrete time nature of the stochastic six vertex model, however, renders the identification of the white-noise much more complicated and presently it is unclear how to proceed. For higher-spin vertex models with unbounded particle occupation capacity, [CT15] proved convergence to the KPZ equation under certain weak scalings. Even though these models are still discrete time, the scaling was such that the overall particle density goes to zero with the scaling parameter ǫ. Owing to that fact, there was no need for an analog of the key estimate used in the case of ASEP. Our present analysis of HL-PushTASEP also involves a weak scaling under which the local density of particles goes to zero. Consequently, we are able to prove the KPZ limit without the key estimate. This suggests that it may be simpler to derive the KPZ equation directly from the stochastic six vertex model (or other finite-spin integrable stochastic vertex models) when the weak scaling facilitates local density decay to zero. Indeed, there are many different weak scalings which should all lead to the KPZ equation. This can be anticipated, for instance, from analyzing moment or one-point distribution formulas, see e.g. [BO16, Section 12]. We intend on investigating such zero-density KPZ equation limits of the stochastic six vertex model in subsequent work. There is another approach for proving KPZ limit of particle system (when started from their invariant measure) via energy solutions. This approach was introduced by Assing [Ass02] and then significantly developed in [GJ14, GP15]. The invariant measure is Bernoulli product measure for HL-PushTASEP (as well as for the stochastic six vertex model), so it would be interesting to see if these methods apply. Presently, the energy solution approach is only developed for continuous time systems, so an analysis of the stochastic six vertex model results would require developing a discrete time variant. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 4 Outline We introduced above RACEP in an arbitrary infinite directed graph. In Section 2, we describe the dynamics of HL-PushTASEP in one dimension and its connection with stochastic six vertex model. Further, we state our results on asymptotic fluctuation and KPZ limit of HLPushTASEP in Section 2. At the end of this section, we explain the KPZ scaling theory for HL-PushTASEP in brief. In Section 3, we first derive the eigenfunctions of the transition matrix of HL-PushTASEP and then use it to get Fredholm determinant formulas. Section 4 contains the proof of the limiting fluctuation results under the assumption of step Bernoulli initial data. Interestingly, we see a phase transition in the limiting behavior of the fluctuation. In Section 5, we prove the KPZ limit theory for HL-PushTASEP. First, we start with the construction of a discrete SHE. Next, we prove few estimates on the discrete heat kernel and subsequently, show the tightness of the rescaled SHE. At the end, we elaborate on the equivalence of all the limit points by solving the martingale problem for HL-PushTASEP. Acknowledgements Special thanks are due to Ivan Corwin for suggesting the problem and guiding throughout this work. We thank Guillaume Barraquand and Peter Nejjar for several constructive comments on the first draft of this paper. We would also like to thank Li-Cheng Tsai and Rajat Subhra Hazra for helpful discussions. 2. Model and main results We now define the HL-PushTASEP on Z≥0 , describe its connection to the stochastic six vertex model, and then state our main results. Theorems 2.4 and 2.5 give the fluctuations limits for HL-PushTASEP started with step Bernoulli initial data. Theorem 2.9 and 2.10 contain the KPZ equation limits for HL-PushTASEP under weak scaling given in Definition 2.7. 2.1. HL-PushTASEP on Z≥0 We will consider HL-PushTASEP with a finite number of particles supported on Z≥0 . The dynamics of HL-PushTASEP are such that the behavior of particles restricted to the interval [0, L] is Markov. Thus, even if we are interested in HL-PushTASEP with an infinite number of particles, if we only care about events involving the restriction to finite intervals, then it suffices to consider the finite particle number version we define here. The N -particle HL-PushTASEP on Z≥0 has state at time t given by (x1 (t), . . . , xN (t)) where xi (t) ∈ Z≥0 for all t ∈ R+ and x1 (t) < . . . < xN (t). For notational simplicity we add two virtual particles fixed for all time so that x0 = −1 and xN +1 = +∞. For later reference, we denote the state space of N -particle HL-PushTASEP on Z≥0 by X N := ~x = (−1 = x0 < x1 < x2 < . . . < xN < xN +1 = ∞) : ∀i, xi ∈ Z≥0 . We illustrate typical moves in HL-PushTASEP on Z≥0 in Fig 1 and Fig 2. Each of the particles in HL-PushTASEP carries an exponential clock of rate 1. When the clock in a particle rings, it jumps to the right. Unlike the simple exclusion processes, if the particle at xm (t) gets imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit xm−2 (t) xm−1 (t) xm (t) xm+1 (t) 5 xm+2 (t) Fig 1: Assume that particle at xm (t) (colored green) becomes active at time t. It immediately takes a step to the right and then continues to jump according to a geometric distribution with parameter b. Thus, the probability that it takes a jump of exactly size 2 is b(1 − b). With probability b2 it reaches the position of xm+1 (t) (colored red). In that scenario, the green particle occupies the position xm+1 (t) and activates the red particle as shown in Fig. 2. xm−2 (t) xm−1 (t) xm (t) xm+1 (t) xm+2 (t) Fig 2: Once red particle has been activated, it moves by one to the right, and then proceeds according to the same rules which had governed the green particle. For instance, it can jump by j steps with probability (1 − b)bj−1 for 1 ≤ j < xm+2 (t) − xm+1 (t) and with probability bj−1 if j = xm+2 (t) − xm+1 (t). We illustrated here the cases when j equals xm+2 (t) − xm+1 (t) and xm+2 (t) − xm+1 (t) − 1. excited at time t, then it can jump j steps to the right with a geometric probability (1− b)bj−1 for 1 ≤ j < xm+1 (t) − xm (t). It can also knock its neighbor on the right side at xm+1 (t) out from its position with probability bxm+1 (t)−xm (t)−1 . In that case, particle at position xm+1 (t) will also get excited and jumps in the same way independently to others. To be precise, the HL-PushTASEP shares a large extent of resemblance with the particle dynamics in the case of stochastic six vertex model (SSVM) (see, [BCG16, Section 2.2]). Although the dynamics in latter case is governed by a discrete time process, but pushing effect of the particles comes into play in the same way. We embark more upon the connection between these two model later in the following section. One can see similar pushing mechanism in the case of Push-TASEP [BF08], q-PushASEP[CP15] which are also continuous time countable state space Markov processes. But in contrast with the HL-PushTASEP, each of the particles in those cases can only jump by one step when their clocks ring. For HL-PushTASEP on Z≥0 , we are interested in studying the evolution of the empirical particles counts, i.e, for any ν ∈ R, what is the limit shape and fluctuation of the number of particles in an interval [0, ⌊νt⌋] as the time evolves to infinity. One can note that empirical particle counts fits into the stereotype of the height function for the exclusion processes. In what follows, we provide the explicit definitions of the height function and the step Bernoulli initial condition. Definition 2.1. Fix a positive integer N . Define a function Nx (t; .) and ηt (x; .) of ~x = (x1 < x2 < . . . < xN ) ∈ X N via 1 xi = x for some i. Nx (t; ~x) = #{i; xi ≤ x}, ηx (t; ~x) = 0 otherwise. For the rest of the paper, we prefer to use Nx (t) and ηt (x) instead of Nx (t; ~x) and ηt (x; ~x) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 6 with the understanding that those are evaluated at ~x. We call Nx (t) as the height function of HL-PushTASEP and ηt (x) as the occupation variable. Definition 2.2 (Step Bernoulli Initial Data). Fix any positive integers L. In step Bernoulli initial condition with parameter ρ over [0, L], each integer lattice site is occupied by a particle with probability ρ independently of others. Whenever ρ = 1, we refer it as step initial condition. 2.2. Connection with Stochastic Six Vertex Model Six vertex model was first studied in [Lie67] to compute the residual entropy of the ice structure. One can think of six vertex model as a lattice model where the configurations are the assignments of six different types of H2 O molecular structure to the vertices of a square grid so that O atoms are at the vertices of the grid. To each O atoms, there are two H atoms attached so that they are at the angles 90◦ or 180◦ to each other. Later, a stochastic version of the six vertex model was introduced by Gwa and Spohn in [GS92]. Recently, several new discoveries related to the stochastic six vertex model came into light in works including [BCG16, BP16a, BP16b, AB16, Agg16]. Furthermore, there are three equivalent ways of describing stochastic six vertex model: (i) as a ferro-electric asymmetric six vertex model on a long rectangle with specific vertex weights (see, [GS92] or [BCG16, Section 2.2]); (ii) as an interacting particle system subject to the pushing effect and exclusion of mass (see, [BCG16, Section 2.2], [GS92]); or (iii) as a probability measure on directed path ensemble (see, [BP16a, Section 1], [AB16, Section 1.1]). In what follows, we elaborate in details on the second description of stochastic six vertex model. Particle dynamics of stochastic six vertex model is governed by a discrete time Markov chain with local interactions. Fix b1 , b2 ∈ [0, 1]. Consider a particle configuration Yb1 ,b2 (t) = (y1 (t) < y2 (t) < . . .) at time t where 0 < b1 , b2 < 1. Then, at time t + 1, each particle decides with probability b1 whether it will stay in its position, or not. If not, then it moves to the right for j steps with probability (1 − b2 )b2j−1 where j can be any integer in between 1 and the distance of the right neighbor from the particle. Further, the particle at yi (t) can jump to yi+1 (t) with probability byi+1 (t)−yi (t)−1 . In that case, the particle which was initially at yi+1 (t) is pushed towards right by one step and starts to move in the same way. It has been noted in [BCG16, Section 2.2] that the dynamics of the limit Y b (t) := lim Y1−ǫ,b (tǫ−1 ) ǫ→0 (2.1) are those of HL-PushTASEP on Z≥0 . Similarly, they show that the dynamics of Z p,q (t) := lim {Yǫp,ǫq (tǫ−1 ) − tǫ−1 } ǫ→0 are those of ASEP with jump parameters p and q (see also [Agg16]). For any finite particle configuration, it is easy to show that the limiting dynamics in (2.1) indeed matches with HLPushTASEP on Z≥0 . Heuristically, this is attributed to the facts that in an time interval [t, t+ ∆t], (i) the expected number of jumps for each of the particles of the limiting process Y b (t) varies linearly with ∆t and (ii) the probability of more than one jumps decays quadratically with ∆t. In particular, (i) shows the number of jumps of each of the particles follows a Poisson point process with intensity 1 and (ii) shows that the Poisson clocks in all particles are imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 7 independent of each other. In the case of countable particle configurations, one can complete the proof of the convergence of stochastic six vertex model to HL-PushTASEP on Z≥0 using the similar arguments as in [Agg16]. 2.3. Main Results In the following theorems, we state the asymptotic phase transition result for the height function Nνt (t) of HL-PushTASEP on Z≥0 under the step Bernoulli initial data. Let us explain the scalings used in the paper. Assume that initial condition for the HL-PushTASEP dynamics is step Bernoulli with parameter ρ. Thus, it is expected that the macroscopic density of the particles varies between 0 to ρ. In fact, it will be shown that whenever ν ≤ (1 − b)−1 , then macroscopic density around νt will be 0. Furthermore, in the case when (1 − b)−1 < ν < (1 − ρ)−2 (1 − b)−1 , macroscopic density varies as 1 − (ν(1 − b))−1/2 around the position νt. Consequently, one can expect KPZ scaling theory to hold in such scenario. On the contrary, when ν > (1 − ρ)−2 (1 − b)−1 , density profile around the position νt turns out to be flat, thus yielding the gaussian fluctuation. Before stating the theorems, we must define three distributions, namely the GUE Tracy-Widom distribution, square of GOE Tracy-Widom distribution and Gaussian distribution which will arise as limits in three different scenario. Definition 2.3. (1) Consider a piecewise linear curve Γ(1) consists of two linear segments: from ∞e−iπ/3 to 0 and from 0 to ∞eiπ/3 . The distribution function FGU E (x) of the GUE Tracy-widom distributions is defined by FGU E (x) = det(I + KAi )L2 (Γ(1) ) where KAi is the Airy kernel. For any w, w′ ∈ Γ(1) , Airy kernel is expressed as Z e2πi/3 ∞ w3 /3−sw 1 e 1 ′ dv KAi (w, w ) = 3 /3−sv v 2πi e−2πi/3 ∞ e (v − w)(v − w′ ) where the contour of v is piecewise linear from −1 + e−2πi/3 ∞ to −1 to −1 + e2πi/3 ∞. It is important to note that the real part of v − w is always negative. (2) Let δ > 0. Consider a piecewise linear curve Γ(2) which is composed of two linear segments: 2 from −δ + ∞e−iπ/3 to −δ and from −δ to −δ + ∞eiπ/3 . Then, the FGOE is defined by 2 FGOE (xs) = det(I + KGOE,2 )L2 (Γ(2) ) where KGOE,2 is given by 1 KGOE,2 (w, w ) = 2πi ′ Z −2δ+∞e2πi/3 −2δ+∞e−2πi/3 3 v ew /3−sw 1 dv 3 /3−sv ′ v (v − w)(v − w ) w e for aany w, w′ ∈ Γ(2) . The contour of v is piecewise linear from −2δ + ∞e−2πi/3 to −2δ to −2δ + ∞e2πi/3 . (3) For completeness, we also add here a very brief description of the Gaussian distribution. Density of the standard Gaussian distribution function Φ(x) is given by 2 x 1 √ exp − . 2 2π Theorem 2.4. Consider HL-PushTASEP on Z≥0 . Assume initial data is step Bernoulli data. −1 Set α = 1−ρ ρ . Let us also fix a real number ν > (1 − b) . Define the parameters, p p p b1/3 ( ν(1 − b) − 1)2/3 ( ν(1 − b) − 1)2 , σν := , ̺ := b( ν(1 − b) − 1), mν := 1−b (1 − b)1/2 ν 1/6 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 1 ν − , m̃ν := 1 + α α(1 − b) σ̃ν := s 8 ν(1 − b)(1 − ρ)2 − 1 . α2 (1 − b) (a) If ρ > 1 − (ν(1 − b))−1/2 , then lim P t→∞ = FGU E (s). (2.2) 2 = FGOE (s). (2.3) (2.4) mν t − Nνt (t) ≤s σν t1/3 mν t − Nνt (t) ≤s σν t1/3 (b) If ρ = 1 − (ν(1 − b))−1/2 , then lim P t→∞ (c) If ρ < 1 − (ν(1 − b))−1/2 , then lim P t→∞ m̃ν t − Nνt (t) ≤s σ̃ν t1/2 = Φ(s). Let us denote the position of m-th particle at time t by xm (t). To this end, one can note that for any y, t ∈ R+ and m ∈ Z≥0 , the event {Ny (t) ≥ m} is same as the event {xm (t) ≤ y}. Thus, Theorem 2.4 translates to the following results (noted down below) in terms of the particle position xm (t). Theorem 2.5. Continuing with all the notations introduced in Theorem 2.4 above, we have (a) if ρ > 1 − (ν(1 − b))−1/2 , then lim P xtmν (t) ≤ νt + t→∞ ∂mν ∂ν −1 ∂mν ∂ν −1 1/3 ! = FGU E (s), (2.5) σν st1/3 ! 2 = FGOE (s), (2.6) ! (2.7) σν st (b) if ρ = 1 − (ν(1 − b))−1/2 , then lim P xtmν (t) ≤ νt + t→∞ (c) if ρ < 1 − (ν(1 − b))−1/2 , then lim P xtm̃ν (t) ≤ tν + t→∞ ∂ m̃ν ∂ν −1 σ̃ν st 1/2 = Φ(s). Using the theorems above, we derive the following law of large numbers for HL-PushTASEP. Corollary 2.6 (Law of large numbers). Consider HL-PushTASEP on Z≥0 with step initial condition. Denote ν0 := (1 − b)−1 . Then, ( √ ( ν(1−b)−1)2 Nνt (t) when ν > ν0 , (1−b) = lim t→∞ t 0 otherwise. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 9 Interactive particle systems which obey KPZ class conjecture (see, Section 2.4) form the KPZ universality class. Models is the KPZ universality class are in a sense characterized by the KPZ equation which is written as ∂t H(t, x) = √ 1 2 1 δ∂x H(t, x) + κ(∂x H(t, x))2 + Dξ(t, x), 2 2 for δ, κ ∈ R and D > 0 where ξ(., .) denotes the space time white noise. In a seminal work [KPZ86], Karder, Parisi and Zhang introduced the KPZ equation as a continuum model for the randomly growing interface which can be subjected to the local dynamics like smoothing, lateral growth and space-time noise. Most prominent feature of the models in KPZ universality class is that the scaling exponents of the fluctuation, the space and the time follow a ratio 1 : 2 : 3. Further, it is expected that the long time asymptotics of any model in the KPZ universality class coincides with the corresponding statistics of the KPZ equation when initial condition of the model corresponds with the initial condition in KPZ equation. To illustrate, consider ASEP on Z. Initially, if all the sites in Z≤0 are occupied and the rest are left empty, then [TW09b] shows that the height fluctuation in t1/3 scale converges to the Tracy-Widom GUE distribution. Later, it is verified in the work of [ACQ11], [Dot10], [CDR10] and [SS10] that under narrow wedge initial condition the height fluctuation of the KPZ equation in t1/3 scale has the same asymptotic distribution. It is now natural to ask what are the other distributions which arise as the long time fluctuation limits of stochastic models in KPZ universality class. There are only very few instances where this question has been addressed. In the case of ASEP under the step Bernoulli initial condition (sites in Z≤0 are occupied w.p ρ and other places are empty), Tracy and Widom [TW09b] demonstrates that the limiting behavior of the height fluctuation undergoes a phase transition. In particular, whenever ρ is close to 1, scaling exponent of the fluctuations is still t1/3 and limiting distribution is Tracy-Widom GUE. On the contrary, when ρ nears 0, the fluctuation converges to the Gaussian distribution under t1/2 scaling. In between those two regime, there lies a critical point where the limiting distribution of the fluctuation is F12 . Nevertheless, the scaling exponent of the fluctuation at the critical point remains 1/3. It is shown in [CQ13] that the asymptotic limit of the fluctuation (in ASEP) at the critical point corresponds to the Half-Brownian initial data of the KPZ equation. Similar phase transition was in fact first unearthed in the work of Baik, Ben Arous and Péché [BBAP05], where they observed a transition phenomenon (now known as BBP transition) in the limiting law of the largest eigenvalue λ1 of large spiked covariance matrices. In a follow up paper, [Péc06] proved the same for the finite rank perturbation of the GUE matrices. When the perturbation has rank one, then the asymptotic distribution of λ1 at criticality coincides with F12 . In the context of interacting particle systems, Barraquand [Bar15] established the BBP transition in the case of q − T ASEP with few slower particles. More recently, Aggarwal and Borodin [AB16] showed that the same phenomenon also holds in the case of stochastic six vertex model when subjected to generalized Bernoulli initial data. Our next result is on the convergence of an exponential transform of the height function Nx (t) to a mild solution of the stochastic heat equation (SHE). The stochastic heat equation is written as κ√ 1 DZ(t, x)ξ(t, x) ∂t Z(t, x) = δ∂x2 Z(t, x) + 2 δ imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 10 ξ(·, ·) denotes the space time white noise. Let us call Z(t, r) ∈ C([0, ∞), C(R)) a mild solution of the SHE starting from the initial condition Zin if Z tZ Z Pt−s (r − r ′ )Zin (s, r − r ′ )ξ(ds, dr ′ ) (2.8) Ps (r − r ′ )Zin (s, r ′ )dr ′ + Z(t, r) = 0 R √1 2πt R exp(−r 2 /2t) where Pt (r) := is the standard heat kernel. For existence, uniqueness, continuity and positivity of the solutions to (2.8), see [Cor12, Proposition 2.5] and [BG97], [BC95], [Mue91]. Interestingly, the Cole - Hopf solution H(t, x) of the KPZ equation is defined by H(t, x) = κδ log Z(t, x) whenever Z is a mild solution (see, (2.8)) of the SHE. Bertini and Giacomin [BG97] proved that ASEP under weak asymmetric scaling and near equilibrium initial condition converges to the Cole - Hopf solution of the KPZ equation. Later, the same result has been shown for the narrow wedge initial data in [ACQ11]. In both of these works, Gärtner transformation which is a discrete analogue of Cole - Hopf transform plays crucial roles. For weakly asymmetric exclusion process with hopping range m more than 1, similar result has first appeared in the work of Dembo and Tsai [DT16]. But, their proof breaks down when m ≥ 4. In the following discussion, we show that under a particular weak scaling, HL-PushTASEP on Z≥0 started from the step initial condition converges to the KPZ equation. Definition 2.7 (Gärtner Tranform & Weak Noise Scaling). Here, we turn to define explicitly the exponential transformation (or, Gärtner transformation) N νǫ when x + ⌊t/(1 − bνǫ )⌋ ∈ Z≥0 b x+⌊t/(1−bνǫ )⌋ (t)−(1−νǫ )(x+⌊t/(1−b )⌋) exp(−µǫ t) Z(t, x) = 0 o.w (2.9) √ where t ∈ R+ , νǫ = ( 5 − 1)/2 and µǫ = (1 − νǫ ) log b b−1 − 1 . + b−νǫ − 1 1 − bν ǫ (2.10) In the rest of our discussion, we will keep on making an abuse of notation by writing t/(1−bνǫ ) 1/2 instead of ⌊t/(1 − bνǫ )⌋. Let us mention that we use a weak noise scaling b = bǫ := e−λǫ ǫ , −3/2 where ǫ → 0 and λǫ = νǫ . We extend the process Z(t, x), defined for t ∈ R+ and x ∈ Z, to a continuous process in R+ × R by linearly interpolating in x. Let us introduce the scaled process Zǫ (t, x) := Z(ǫ−1 t, ǫ−1 x). (2.11) Furthermore, we endow the space C(R+ ), and C(R+ × R) with the topology of uniform convergence and use ⇒ to denote the weak convergence. Definition 2.8 (Near-Equilibrium Condition). Discrete SHE Zǫ (., .) is defined to have near equilibrium initial condition when the initial data Zǫ (0, x) satisfies the following two conditions: kZǫ (0, x)k2k ≤ Ceǫτ \|x\|, kZǫ (0, x) − Zǫ (0, x′ )k2k v ′ ≤ C ǫ\|x − x′ \| eǫτ (\|x\|+\|x \|) , (2.12) (2.13) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 11 for some v ∈ (0, 1) and τ > 0. Theorem 2.9. Let Z be the unique C(R+ × R)- valued solution of the SHE starting from some initial condition ∆ and let Zǫ (t, x) ∈ C(R+ × R) as defined in (2.11) evolved from some near equilibrium initial condition. If Zǫ (0, .) converges to Z(0, .) whenever ǫ tends to 0, then Zǫ (., .) ⇒ Z(., .) on C(R+ × R) as ǫ → 0. It can be noted that the step initial condition, i.e, xn (0) = n for n ≥ Z+ and xn (0) = −∞ for n ∈ Z− , doesn’t belong to the family of near equilibrium initial conditions in Definition 2.8. Nevertheless, the rescaled height function of HL-PushTASEP started with step initial condition also has KPZ limit. We illustrate this further in the following theorem. Theorem 2.10. Let Z be the unique C(R+ × R)- valued solution of the SHE starting from delta initial measure. Consider Z̃ǫ (t, x) := ǫ−1 (1 − exp(−λǫ νǫ ))Zǫ (t, x) in C(R+ × R) evolving from the step initial condition. Then, Z̃ǫ (., .) ⇒ Z(., .) on C(R+ × R), as ǫ tends to 0. Corollary 2.11. Let Z be as in the Theorem 2.9 so that H(t, x) := log Z(t, x) be the unique Cole-Hopf solution of the KPZ equation starting from the delta initial measure. Then, log Z̃ǫ (., .) ⇒ H(., .) on C(R+ × R) as ǫ → 0. P Proof of Theorem 2.10. To begin with, note that ǫ ζ∈Ξ(0) Z̃ǫ (0, ζ) = 1. Thus, Z̃ǫ (0, .) converges to the delta initial measure as ǫ goes to 0. Below, in Proposition 2.12 which we prove in Section 5, we show the moment estimates for Z̃ǫ at any future time point t. Thus, using Theorem 2.9, one can complete the proof in the same way as in [ACQ11, Section 2]. Proposition 2.12. Fix any T > 0. Consider Z̃(t, x) = ǫ−1 (1 − exp(−λǫ νǫ ))Z(t, x) evolving from the step initial condition. Then, for any t ∈ (0, ǫ−1 T ], ζ1 , ζ2 ∈ Ξ(t) and v ∈ (0, 1/2), we have kZ̃(t, ζ)k ≤ C min{ǫ−1/2 , (ǫt)−1/2 }, ′ ′ v −(1+v)/2 kZ̃(t, ζ) − Z̃(t, ζ )k ≤ (ǫ\|ζ − ζ \|) (ǫt) . (2.14) (2.15) where the constant C depend only on T and v. 2.4. KPZ Scaling Theory Here, we explain how the asymptotic fluctuation results in Theorem 2.4 confirms the KPZ scaling theory (see, [KMHH92], [Spo14]) of the physics literature. To start with, we present the predictions of KPZ scaling theory in the context of exclusion process following [Spo14]. For imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 12 that, one need to assume that spatially ergodic and time stationary measures of the exclusion process on Z are precisely labelled by the density of the particles ρ, where n X 1 η(n). n→∞ 2n + 1 ρ := lim k=−n We denote the stationary measure arising out of the density ρ by νρ . Consider the average steady state current j(ρ) which counts the average number of particles transported from 0 to 1 in unit time where the system is distributed as νρ . Further, denote the integrated covariance P j∈Z Cov(η0 , ηj ) by A(ρ) where the covariance between the occupation variables η0 and ηj is computed under the stationary measure νρ . One expects that on the microscopic scale the rescaled particle density ̺(x, t) given as ̺(x, τ ) = lim P (there is a particle at site ⌊xt⌋ at time ⌊τ t⌋) t→∞ satisfies the conservation equation ∂ ∂ ̺(x, τ ) + j(̺(x, τ )) = 0 ∂τ ∂x when the system is started with the step initial condition, i.e, 1 x≥0 ̺(x, 0) = 0 x < 0. (2.16) (2.17) This result can also be phrased as a hydrodynamic limit theory. For any exclusion process, consider the height function h(., .) : Z × R+ → Z given by  Pj if j > 0  Nt + i=1 ηt (j) h(j, t) := N if j = 0 P t  Nt − −j η (−i) if j < 0. i=1 t Then, using the arguments in [Spo91, Part II, Section 3.3], one can prove that (2.16) implies the following law of large numbers h(νt, t) = φ(ν) t→∞ t lim where the limit shape φ(.) is given by φ(y) = sup {yρ − j(ρ)}. ρ∈[0,1] Furthermore, Let us denote λ(ρ) = −j ′′ (ρ). Under such parametrization, we have the following KPZ class conjecture. KPZ Class Conjecture:([Spo14]) Let y be such that φ is twice differentiable at y with φ′′ 6= 0. Set ρ0 = φ′ (y). If 0 ≤ ρ0 < 1, A(ρ0 ) < ∞ and λ(ρ0 ) 6= 0, then under the step initial condition in (2.17), one has ! tφ(y) − h(⌊yt⌋, t) ≤ s = FGU E (s). lim P t→∞ (− 12 λA2 )1/3 t1/3 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 13 In the case when φ(y) is linear or cusp like, then KPZ conjecture doesn’t hold. It is likely that the scaling of the fluctuations would be different in those cases. Historically, the conjecture on the scaling exponent t1/3 of the fluctuation above dates back to the the work [KPZ86]. To that end, the magnitude of the fluctuation was obtained in [KMHH92] and the limit distribution was first surfaced in [Joh00]. Lemma 2.13. Fix 0 ≤ ρ ≤ 1. Consider the HL-PushTASEP model on Z≥0 . Further, assume that a jet of particles is entering into the region Z≥0 through 0 at rate τ where τ := ρ(1 − ρ)−1 (1 − b)−1 . Then, the product Bernoulli measure νρ will be the invariant measure for the HL-PushTASEP model. In particular, each of the occupation variables {ηt (x)\|x ∈ Z≥0 } follows Ber(ρ) independent of others. Proof. Assume, at time t = 0, each site on Z+ is occupied by at most one particle with probability ρ independently of others. First, we show νρ (ηt (0) = 1) = ρ for any t ≥ 0. Note that influx rate of particles at site x = 0 is equal to τ . Then, rate at which particle leaves site 0 is given by χ1 + χ2 + χ3 χ1 := τ ρ, χ2 := τ b(1 − ρ), and , χ3 := ρ. To begin with, consider the case when initially the site 0 was occupied. Probability of this event is ρ. In such scenario, χ1 captures the rate of out flux of any particle to the right from the site 0 after a particle arrives there. On the contrary, probability that the site 0 was initially unoccupied is 1 − ρ. Thus, if a particle arrives when the site 0 is empty, it moves further one step towards the right with probability b. Henceforth, the value of χ2 synchronizes perfectly with the rate of outflux from the site 0 whenever initially it was unoccupied. Finally, the particle which was initially at the site 0 jumps to the right at rate 1 and this contribution has added up in the total rate out flux through χ3 . Due to the specific choice of τ , we get τ = χ1 + χ2 + χ3 . Consequently, the rate of in flux matches with the rate of out flux of the particles from the site 0. Therefore, probability that the site 0 is occupied doesn’t change over time. More importantly, the rate at which particles will move into the site 1 is also equal to τ . This further implies P(ηt (1) = 1) = ρ for any t ≥ 0. Using induction, one can now prove that the occupation variable ηt (x) for any x ∈ Z+ and t ≥ 0 follows Ber(ρ). In fact, using similar argument, one can show that at any time t, the rate of influx of the particles into any interval [x1 , x2 ] of finite length over Z+ is equal to the rate at which the particles will come out of the interval [x1 , x2 ]. Now, we turn to prove the independence of any two occupation variables ηt (x1 ) and ηt (x2 ) for x1 < x2 ∈ Z+ . Let us consider (ǫx1 , ǫx1 +1 , . . . , ǫx2 ) ∈ {0, 1}x2 −x1 +1 . Further, denote ηt (a) if ǫa = 1, ǫa ηt (a) := 1 − ηt (a) if ǫa = 0. To this end, one can note that E x2 Y a=x1 ! ηtǫa (a) =E x2 Y a=x1 ! η0ǫa (a) for any choice of the variables (ǫx1 , ǫx1 +1 , . . . , ǫx2 ) in the space {0, 1}x2 −x1 +1 . This is again due to the principle of conservation of mass, i.e, the influx rate into the interval [x1 , x2 ] is imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 14 same as the rate of out flux of the particles. Thus, we get   xY 2 −1 X E(ηt (x1 )ηt (x2 )) = E  ηtǫa (a) ηt (x1 )ηt (x2 )  = E ǫx1 +1 ,...,ǫx2 −1 a=x1 +1 X xY 2 −1 η0 (x1 )η0 (x2 ) ǫx1 +1 ,...,ǫx2 −1 a=x1 +1  η0ǫa (a) = E (η0 (x1 )η0 (x2 )) = ρ2 . To that effect, covariance of ηt (x1 ) and ηt (x2 ) becomes 0. As these are Bernoulli random variables, thus the independence is proved. Now, we turn to showing how Theorem 2.4 verifies the KPZ class conjecture in the context of HL-PushTASEP. In Lemma 2.13, it is proved that all the stationary translation invariant measure of HL-PushTASEP are given by product measure νρ where each site follows Ber(ρ) distribution. Further, it has been also shown that under stationary measure νρ steady state current will be ρ(1 − ρ)−1 (1 − b)−1 . To that effect, we have ρ φ(y) = sup ρy − . (2.18) (1 − ρ)(1 − b) ρ∈[0,1] If y(1 − b) > 1, then the function g(ρ) = yρ − ρ(1 − ρ)−1 (1 − b)−1 is maximized when ρ = 1 − (y(1 − b))−1/2 . In this scenario, we get φ(y) = p 2 y(1 − b) − 1 (1 − b) . One can further derive that the equilibrium density ρ corresponding to y is 1 − (y(1 − b))−1/2 . To this end, we have λ(ρ) = −j ′′ (ρ) = −2y 3/2 b(1 − b)1/2 . Further, stationarity of the product Bernoulli measure implies A(ρ) = ρ(1 − ρ) = (y(1 − b))−1/2 (1 − (y(1 − b))−1/2 ). Finally, it can be noted that 2/3 p 1/3 1/3 y(1 − b) − 1 b 1 = σy . = − λ(ρ)A2 (ρ) 2 y 1/6 (1 − b)1/2 Thus, KPZ class conjecture implies that the height function N⌊yt⌋ (t) of HL-PushTASEP started from the step initial condition has a limit shape given by tφ(y) in (2.18) and more importantly, the fluctuation around the limit shape under the scaling of (−2−1 λA2 t)1/3 converges to the Tracy-Widom GUE distribution. As one can see, Theorem 2.4 verifies the conjecture and at the same time, identifies the form of the limit shape and the correct scaling of the fluctuation. 3. Transition Matrix & Laplace Transform For proving the asymptotic results, we need to have concrete knowledge on the distribution of the positions of the particles at any fixed time. In most of the cases of interactive particle systems (see, [TW11] for ASEP, [BCG16] for the stochastic six vertex model), transition imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 15 matrices for the underlying Markov processes play important roles towards that goal. In particular, transition matrices assists in getting suitable Fredholm determinant formulas for the Laplace transforms of some of the important observables like in the case of several quantum integrable interacting particle systems (see, [BC14] for a survey). In the following subsections, we elaborate on the derivation of the transition matrix of the HL-PushTASEP and its use to proclaim the formulas for the Laplace transforms. 3.1. Transition Matrix & Transition Probabilities Primary goal of this section is to compute the transition matrix in the case of finite particle configurations of the HL-PushTASEP. Later in this section, we use the transition matrix to specialize on the transition probabilities of a single particle in a system of finite particle configuration. At first, fix some positive integer N . For any A ⊆ X N define X (N ) P~x (A; t) := Tt (~x → ~y ) ~ x∈A (N ) where Tt (~x → ~y ) denotes the probability of transition to ~y ∈ X N from ~x ∈ X N . In the (N ) following proposition, we derive the eigenfunctions of the transfer matrix Tt . Proposition 3.1. For N > 0, fix N small complex numbers z1 , z2 , . . . , zN such that \|bzi \| < 1 for 1 ≤ i ≤ N and 1 − (2 + b)zj + bzi zj 6= 0 for all 1 ≤ i 6= j ≤ N . Fix a permutation σ ∈ S(N ). Define, Y 1 − (1 + b)zσ(i) + bzσ(i) zσ(j) Aσ = (−1)σ . 1 − (1 + b)zi + bzi zj i<j Then the function Φ(x1 , x2 , . . . , xN ; z1 , z2 , . . . , zN ) = X σ∈S(n) Aσ N Y xi zσ(i) i=1 is an eigenfunction of the transfer matrix T (N ) , i.e X y ∈X N ~ (N ) Tt (~x → ~y )Φ(~y ; z1 , z2 , . . . , zn ) = exp N X i=1 1 − zi −t 1 − bzi ! Φ(~y ; z1 , z2 , . . . , zn ). (3.1) As pointed out in Section 2.2, HL-PushTASEP can be considered as a continuum version of the stochastic six vertex model for which the transfer matrix has been worked in full details in the past. See [Lie67], [Nol92] for the derivation in the case of the stochastic six vertex model under periodic boundary condition and [BCG16, Theorem 3.4] for the derivation on Z. Let us note that one can derive the relation (3.1) by taking appropriate limit (see, (2.1)) of the transfer matrix of stochastic six vertex model in [BCG16, Eq.16]. Although, one has to be careful because the limit might not exist. For completeness, we present a self contained proof of the result in Appendix B. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 16 Proposition 3.2. For N > 0, fix N small complex numbers z1 , z2 , . . . , zN such that \|bzi \| < 1 for 1 ≤ i ≤ N and 1 − (2 + b)zj + bzi zj 6= 0 for all 1 ≤ i 6= j ≤ N . For any permutation σ ∈ S(N ), define Y 1 − (1 + b)zσ(j) + bzσ(i) zσ(j) A′σ = (−1)σ 1 − (1 + b)zj + bzi zj i<j and A′′σ = (−1)σ Y 1 − (1 + b−1 )zσ(i) + b−1 zσ(i) zσ(j) 1 − (1 + b−1 )zi + b−1 zi zj i<j . Let us assume ~x = (x1 , . . . , xN ) and ~y = (y1 , . . . , yN ). Denote the transition matrix for N (N ) particles in time t by Tt . Then, we have (N ) Tt (~x → ~y) = Z X (Cr )N σ∈S(N ) A′σ n Y −xi yi −1 zσ(i) zi exp xi zσ(i) zi−yi −1 exp i=1 1 − zi −t 1 − bzi dz1 . . . dzN (3.2) (3.3) and (N ) Tt (~x → ~y ) = Z X (CR )N σ∈S(N ) A′′σ n Y i=1 1 − zi−1 −t 1 − bzi−1 dz1 . . . dzN where Cr is small positively oriented circle leaving all the singularities outside and CR is large positively oriented circle containing all the singularities inside. Proof. Using the relation (3.1), both the formulas in (3.2) and (3.3) can be proved in the same way as in [BCG16, Theorem 3.6] (see also [TW11, Section II.4.b]). We omit here further details of the proof. Theorem 3.3. Fix an integer N > 0 and a positive real number t. Let us assume that we have N particles which are initially at the position ~y = (y1 , y2 , . . . , yN ) ∈ X N evolve according the dynamics of HL-PushTASEP. Then, for 1 ≤ m ≤ N , we have X X k−1 bκ(S,Z>0 )−mk−k(k−1)/2 P~y (xm = x; t) = (−1)m−1 bm(m−1)/2 m−1 b m≤k≤N \|S\|=k I I Y zj − zi 1 . . . × (2πi)k 1 − (1 + b−1 )zi + b−1 zi zj i,j∈S,i<j Q Y 1 − i∈S zi 1 − zi−1 x−yi −1 dzi zi exp −t ×Q 1 − bzi−1 i∈S (1 − zi ) i∈S where the summation goes over S ⊂ {1, 2, . . . , N } of size k, κ(S, Z>0 ) is the sum of the elements in S, and contours are positively oriented large circles of equal radius which contains all the singularities. Proof. This result is in the same spirit of [BCG16, Theorem 4.9] where they found out explicitly the probability of the similar event in the case of the stochastic six vertex model. They closely followed the techniques used in [TW08, Section 6] in the context of N-particle ASEP. It begins with the realization that P(xm (t) = x; 1, b) is the sum of the probabilities imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 17 (N ) Tt (~y → ~x) over all ~x = (x1 , x2 , . . . , xN ) ∈ X N such that −∞ < x1 < x2 < . . . < xm−1 < xm = x and x < xm+1 < . . . < xN < ∞. Now, one have to make use of the formulas for (N ) Tt (~y → ~x) mentioned in (3.2) and (3.3). Notice that for the sum over all ~x ∈ X N such that −∞ < x1 < x2 < . . . < xm−1 < xm = x contours of integration is Cr and for other part of the sum contours will be CR . For brevity, we are skipping here the details of the proof. 3.2. Moments Formulas In this subsection, we aim to present a series of results on moments of the height function of HL-PushTASEP. One can see [BCG16, Section 4.4] for similar results in the case of the stochastic six vertex model. In Section 4, these formulas turns out to be instrumental for obtaining the asymptotics. For a function f : X N → R, let E~y (f ; t) represent the expectation of f at time t, i.e X (N ) E~y (f ; t) = Tt (~y → ~x). ~ x∈X N Also, let us introduce (a; q)k and (a; q)∞ by defining (a; q)k := k Y (1 − aq j−1 ) and (a; q)k := j=1 ∞ Y (1 − aq j−1 ). j=1 Furthermore, there is a q - analogue of binomials defined as follows (q N −k+1 ; q)k n (1 − q N )(1 − q N −1 ) . . . (1 − q N −k+1 ) = . := (1 − q)(1 − q 2 ) . . . (1 − q k ) (q; q)k k q In both of the above definitions, we assume k is a non-negative integers. Proposition 3.4. Consider HL-PushTASEP on Z≥0 . Fix a positive real number t. Assume that the process has been started from the step Bernoulli initial condition with parameter ρ. Denote the initial data by stepb which belongs to X N . Then for L = 0, 1, 2, . . . and any positive integer x, we have ! L k−2 Y X b−k(k−1)/2+L−k ρk (1 − b1−k+L bi ) Estepb (bLNx ; t) = 1 + (b−L − 1) (b−1 ; b−1 )k I Ik=1 Yi=1 zj − zi 1 × ... k (2πi) 1 − (1 + b−1 )zi + b−1 zi zj 1≤i<j≤k × k Y i=1 (ρ + (1 − zix (b−1 − 1) ρ)b−1 − zi b−1 )(1 1 − zi−1 exp −t − zi ) 1 − bzi−1 dzi (3.4) where the contours of integrations are largely oriented circles with equal radius and contain all the singularities of the integrand. Proof. One can find a similar result in [BCG16, Proposition 4.11] in the case of stochastic six vertex model with step initial data. For the sake of clarity, we present an independent proof in C. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 18 Proposition 3.5. Consider HL-PushTASEP on Z≥0 . Fix a positive number t. As usual stepb ∈ X N denotes the step Bernoulli initial condition with parameter ρ. Then for L = 0, 1, 2, . . . and for any integer x, we have Estepb (b LNx bL(L−1)/2 ; t) = (2πi)L I x I Y L tui dui uA − uB Y 1 + ui exp , ... −1 uA − buB b 1 + ui b ui (1 − αui b−1 ) A<B i=1 where the positively oriented contour for uA includes 0, −b, but does not include bρ(1 − ρ)−1 or {buB }B>A . Note that α := (1 − ρ)ρ−1 . Proof. To prove this result, one needs to substitute zi = (1 + ui )/(1 + ui b−1 ) into the formula (3.4). Consequently, one can notice ui − uj zj − zi = −1 −1 1 − (1 + b )zi + b zi zj ui − buj , dui ρ(b−1 − 1)dzi = −1 −1 (ρ + (1 − ρ)b − zi b )(1 − zi ) ui (1 − αui b−1 ) where α = (1 − ρ)/ρ. Let us mention that a similar result is proved in the case of stochastic six vertex model in [BCG16, Theorem 4.12]. See [BCS14, Theorem 4.20] for a different proof in the case of ASEP. Rest of the calculation after the substitution can be completed in the same way as in [BCG16, Theorem 4.12]. We omit any further details from here. 3.3. Fredholm Determinant Formula In this section, we discuss the representation of the moment formulas of HL-PushTASEP in terms of Fredholm determinant. In the last ten years, a large number of works had been put forward in the literature of KPZ universality class and Fredholm determinant formulas play a crucial role in reaching out such results. For similar exposition in stochastic six vertex model, see [BCG16, Section 4.5]. See [BC14, Section 3.2] and [BCS14, Section 3 and 5] for corresponding formulas in the case of q-Whittaker process and q-TASEP. Moreover, various other instances can be found in the works like [CP16, Theorem 4.2], [BC16, Section 3.3], [BCF14]. Definition and some of the useful properties of the Fredholm determinants are discussed in Appendix A. Definition 3.6. Here, we define the contours DR,d,δ and DR,d,δ;κ . Let us fix positive numbers R, δ, d such that d, δ < 1. Then the contour DR,d,δ is composed of five linear sections: DR,d,δ := (R − i∞, R − id] ∪ [R − id, δ − id] ∪ [δ − id, δ + id] ∪ [δ + id, R + id] ∪ [R + id, R + i∞). Let κ be an integer which is greater than R. Denote two points at which circle of radius κ + δ and centre at 0 crosses (R − i∞, R − id] and (R − i∞, R − id] by ζ and ζ ′ . Let us call that minor arc joining ξκ and ξκ′ to be Iκ . Then DR,δ,d;κ is defined to be a positively oriented closed contour formed by the union DR,d,δ := (ξκ , R − id] ∪ [R − id, δ − id] ∪ [δ − id, δ + id] ∪ [δ + id, R + id] ∪ [R + id, ξκ′ ) ∪ Iκ . See Figure 3 for details. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 19 R + i∞ ζ δ + id R + id κ 1 R − id δ − id ζ′ R − i∞ Fig 3: Plot of the contour DR,d,δ and DR,d,δ;κ . Theorem 3.7. Consider HL-PushTASEP on Z≥0 . Fix a positive number t. Let stepb ∈ WN stands for the step Bernoulli initial data with parameter ρ. Let Cr be a positively oriented circle centered at some negative real number r such that Cr include 0 and −b, but does not contain bα−1 . Furthermore, Cr does not intersect with bCr , i.e, bCr is contained in the interior of Cr . Then, there exist a sufficiently large real number R > 1, and sufficiently small real numbers δ, d < 1 such that for all ζ ∈ C\R≥0 , we have (A) inf ′ w,w ∈Cr κ∈(2R,∞)∩Z s∈DR,d,δ;κ \|bs w − w′ \| > 0, (B) sup w∈Cr κ∈(2R,∞)∩Z s∈DR,d,δ;κ g(w) < ∞. g(bs w) Moreover, for any ζ ∈ C\R≥0 , we have 1 b , Estepb ; t = det I + K ζ (ζbNx ; b)∞ L2 (Cr ) (3.5) (3.6) where the kernel Kζb is given by Kζb (w, w′ ) 1 = 2πi Z DR,d,δ g(w; 1, b, x, t) ds (−ζ)s · · s , s sin(πs) g(b w; 1, b, x, t) b w − w′ (3.7) integration contour DR,d,δ is oriented from bottom to top and x 1 tz 1 g(z; 1, b, x, t) = exp . 1 + zb−1 b(1 − b) (αb−1 z; b)∞ where α = ρ−1 (1 − ρ). imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 20 Proof. We first prove the existence of R, d and δ such that (A) and (B) in (3.5) are satisfied. Note that Cr doesn’t contain 0 on it and it is a bounded closed set in C, thus, compact. Hence, \|w′ w−1 \| has a lower bound whenever w, w′ ∈ Cr . Let us choose R in such a way that bR becomes less than the lower bound of \|w′ w−1 \|. This ensures that the infimum of \|bs w − w′ \| is bounded away from 0 along (ξk , R − id] ∪ [R + id, ξk′ ) ∪ Ik . Further, note that bCr never touches Cr . Thereafter, using boundedness of Cr , we can found δ sufficiently close to 1 such that bs Cr remains disjoint from Cr for all s ∈ [δ − id, δ + id]. However, it implies somewhat much stronger statement. On can say further that bs Cr escapes from Cr for all s ∈ [δ − id, R − id] ∪ [δ + id, R + id]. Hence, condition (A) in (3.5) is satisfied. Furthermore, one can see g(w) 1 + bs−1 w tw(1 − bs ) (αbs−1 w; b)∞ = . exp g(bs w) 1 + b−1 w b(1 − b) (αb−1 w; b)∞ Choice of Cr and R, δ, d implies immediately that condition (B) in (3.5) is also met. Now, we turn into proving (3.6). Techniques for similar determinantal formula is quite well known by now, thanks to the vast literature mentioned above. We will sketch here the basic outline of such technique. For brevity, we skip major portion of details in the proof. We first look into the expansion of 1/(ζbNx ; b)∞ . Using [AAR99, Corollary 10.2.2 a], one can say ∞ X ζ L bLNx 1 = (ζbNx ; b)∞ (b; b)L L=0 where (b; b)L = QL i=1 (1 − b · bi−1 ). Recall that in Lemma 3.5, we have bL(L−1)/2 µL := Estepb (bLNx ; t) = (2πi)L where f (u) := exp tu b I ... I Y L uA − uB Y f (ui ) dui , uA − buB ui A<B 1+u 1 + ub−1 x i=1 1 1 − αub−1 and the integration contours do include 0, −b inside, but, not α−1 b. Moreover, one can notice that f doesn’t have a pole in an open neighborhood of the line joining 0 P and −b. Let us denote any partition λ = (λ1 , λ2 , . . .) of any positive integer k by λ ⊢ k where ∞ i=1 λi = k. Further, denote number of positive λi ’s in λ by l(λ). Thus, using [BCF14, Proposition 5.2], one can conclude that X 1 1 µL = (b; b)L m1 !m2 ! . . . (2πi)l(λ) λ⊢L 1m1 2m2 ... × Z ... C Z C det −1 bλi wi − wj l(λ) l(λ) Y f (wj )f (bwj ) . . . f (bλj −1 wj )dwj i,j=1 j=1 where the contour C also avoids 0, −b and doesn’t intersect with bn C for any n ∈ N. To this end, one can conform the contour C to the the circle Cr without passing any singularity of the integrand. Further, using the form of the function g, it is easy to see g(wj ; 1, b, x, t) = f (wj )f (bwj ) . . . f (bλj −1 wj ). g(bλj wj ; 1, b, x, t) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 21 Consequently, we can write ∞ ∞ X X ζ L µL = (b; b)L L=0 L=0 X λ⊢L 1m1 2m2 ... 1 ζL m1 !m2 ! . . . (2πi)l(λ) × Z Cr ... Z det Cr (3.8) g(wj ; 1, b, x, t) −1 λ i b wi − wj g(bλj wj ; 1, b, x, t) l(λ) l(λ) Y dwj . i,j=1 j=1 At this point, we will make an interchange of two summations above. Let us fix a positive integer k and first sum over all partitions λ which has size k, i.e l(λ) = k. Interchange of summations is justified by the linearity of the integrals. Furthermore, one can write ζ L = ζ λ1 +...+λl(λ) . To that effect, the right side of (3.8) simplifies to !k Z Z ∞ k ∞ X Y X g(wj ; 1, b, x, t) (−ζ)s 1 det . . . dwj . k!(2πi)k C bL wi − wj g(bL wj ; 1, b, x, t) C k=0 i,j=1 j=1 L=1 Now, it remains to show that ∞ X L=1 g(wj ; 1, b, x, t) (−ζ)s 1 = L L b wi − wj g(b wj ; 1, b, x, t) 2πi Z DR,d,δ (−ζ)s g(w; 1, b, x, t) ds · · s . (3.9) s sin(πs) g(b w; 1, b, x, t) b w − w′ Note that the function 1/ sin(πs) decays exponentially as \|s\| → ∞ along the segments (R − i∞, R − id] and [R + id, R + i∞). On the flip side, (bs w − w′ )−1 g(w)/g(bs w) is uniformly bounded on DR,d,δ,κ for all k ≤ R. Thus, integrand in (3.9) goes to zero at at both ends of the line Re(z) = R. Instead of integrating over DR,d,δ , if we choose to integrate over DR,d,δ;κ , then using residue theorem, we obtain Z 1 (−ζ)s g(w; 1, b, x, t) ds · · s (3.10) s 2πi DR,d,δ;κ sin(πs) g(b w; 1, b, x, t) b w − w′ κ X g(w; 1, b, x, t) 1 (−ζ)s · · . Ress=L = sin(πs) g(bs w; 1, b, x, t) bs w − w′ L=1 Moreover, if we let κ grows to infinity, then the integral along the semi-circular arc Iκ tends 0 thanks again to the exponential growth of sin(πs). Thus, integral over DR,d,δ;κ converges to the right side in (3.9) whereas when κ → ∞, the right side in (3.10) recovers the other side of the equality in (3.9). This completes the proof. 4. Asymptotics Main aim of this section is to prove the Theorem 2.4 and Theorem 2.5. Asymptotic analysis that we present here adapts closely to the similar expositions in [BCG16, Section 5]. The crux of the proof of Theorem 2.4 essentially is governed by the the steepest descent analysis of the kernel of the Fredholm determinant in Theorem 3.7. In the following propositions, large time limit of the Fredholm determinant will be provided. We conclude the proof of Theorem 2.4 after Lemma 4.2 modulo the proof of these propositions. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 22 Proposition 4.1. Recall the definition of the contour Cr from Theorem 3.7. Adopt the notations used in Theorem 2.4. 1/3 (a) Assume ρ > 1 − (ν(1 − b))−1/2 . Set x(t) = ⌊νt⌋ and ζt = −b−mν t+sσν t lim det 1 + Kζbt 2 = FGU E (s). t→∞ . Then, we have (4.1) L (Cr ) (b) Let ρ = 1 − (ν(1 − b))−1/2 . For the same choices of x(t) and ζt as above, we have b 2 lim det 1 + Kζt 2 = FGOE (s). t→∞ L (Cr ) ′ 1/3 (c) In the case when ρ < 1 − (ν(1 − b))−1/2 , set x(t) = ⌊ν ′ t⌋ and ζt = −b−mν t+sσ νt one can get lim det 1 + Kζbt 2 = Φ(s). t→∞ L (Cr ) . Then, (4.2) Lemma 4.2. ([BC14, Lemma 4.1.39]) Consider a set of functions {ft }t≥0 mapping R → [0, 1] such that for each t, ft (x) is strictly decreasing in x with a limit of 1 at x = −∞ and 0 at x = ∞ and for each δ > 0, on R\[−δ, δ], ft converges uniformly to 1(x ≤ 0). Define the r-shift of ft as fnr (x) = fn (x − r). Consider a set of random variables Xt indexed by t ∈ R+ such that for each r ∈ R, t→∞ E[ftr (Xt )] −→ p(r) and assume that p(r) is a continuous probability distribution. Then Xt weakly converges in distribution to a random variable X which is distributed according to P(X ≤ r) = p(r). Proof of Theorem 2.4. For part (a) and (b) of Theorem 2.4, we use Lemma 4.2 with the functions 1 , t ∈ R+ . ft (z) := 1/3 z −t (−b ; b)∞ Note that ft (z) is a monotonously decreasing function of the real argument of z, yielding limt→∞ ft (z) = 0 if z > 0 and limt→−∞ ft (Z) = 1 if z ≤ 0. Set Xt = σν−1 t−1/3 (mν t − Nνt (t)). Thus, for any s ∈ R, we can write fts (Xt ) = 1 (ζt bNνt (t) ; b)∞ 1/3 where ζt = b−mν t+sσν t . To this end, part (a) and (b) of Proposition 4.1 shows that the limit of E (fts (Xt )) are indeed probability distribution function. Hence, this proves part (a) and (b) of Theorem 2.4. For part (c), we replace ft by gt which is defined as gt (z) := 1 (−b −t1/2 z ; b)∞ , t ∈ R+ . 1/3 Setting Xt = (σ̃ν )−1 t−1/2 (m̃ν t − Nνt (t)) and ζt = b−mν t+sσ̃ν t , one can note gts (Xt ) = (ζt bNνt (t) ; b)−1 ∞ . Therefore, rest of the proof follows from the part (c) of Proposition 4.1. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 23 Proof of Proposition 4.1 We start with some necessary pre-processing of the kernel Kζbt . Notice that in the statements of all three parts of Proposition 4.1, ζt is substituted with −bβ and x is substituted with ⌊νt⌋ where β is some function of the time variable t and ν > (1 − b)−1 . Under these substitutions, the form of the kernel Kζbt effectively reduces to 1 2i Kζbt (w, w′ ) = Z DR,d,δ g(w; 1, b, νt, t) dh bhβ · · sin(πh) g(bh w; 1, b, νt, t) bh w − w′ where the contour ofthe integration is oriented from the bottom to top and g(z; 1, b, x, t) = x 1 t exp b(1−b) (αb−1 z; b)−1 ∞ . Like in [BCG16], the particular form of the integrand in 1+z/b (3.7) calls for the substitution bh w = v. Unfortunately, the latter substitution is not injective. Thus, to study how this effect the integral, we have to divide the contour DR,d,δ into countably many parts which have one-to-one images under the above substitution. For instance, each of the linear sections Il := [R + i(d + 2π log(b)−1 l), R + i(d + 2π(log(b))−1 (l + 1))] of the contour of h for l ∈ Z\g{−1} maps to the same image. Furthermore, other parts of DR,d,δ , i.e., the union [R − i(d + 2π log(b)−1 ), R − id] ∪ [R − id, δ − id] ∪ [δ − id, δ + id] ∪ [δ + id, R + id] maps to a closed contour DR,d,δ,w composed of four different portions. For completeness, we present here a formal definition of the contour DR,d,δ,w . Definition 4.3. For each w, define a contour DR,d,δ,w as the union T1 ∪ T2 ∪ T3 ∪ T4 . We describe Ti ’s successively. To begin with, T1 is a major arc of a small circle around origin with radius bR \|w\|. To be exact, T1 is the image of the segment [R − i(d + 2π log(b)−1 ), R − id] under the map h 7→ bh w. Similarly, T2 , image of [δ − id, δ + id], is a minor arc of a larger circle of radius bδ \|w\|. Apparently, these two arcs of two different circle are connected by two linear segments T2 and T3 . Precisely, T2 connects bR−id w to bδ−id w and T3 connects bδ+id w to bR+id w. It is evident from the definition of T2 (T3 ) that it is the image of [R − id, δ − id] ([δ + id, R + id]) under the above map. Due to the bottom-to-top orientation of the contour DR,d,δ in Theorem 3.7, DR,d,δ,w will be anti clockwise oriented. See Figure 4 for further details. For notational convenience, let us denote the circle of radius s with center at origin in the complex plane C by Cs (note the difference from Cr ). In fact, CbR \|w\| is the image of each linear segments Il for l ∈ Z\{−1}. Thus, one can write Kζbt (w, w′ ) Z ∞ X 1 = 2i C R k=−∞ k6=−1 1 + 2i Z DR,d,δ,w b (v/w)β \|w\| sin π log(b) (log(v/w)) (v/w)β + 2πik · dv g(w; 1, b, νt, t) · g(v; 1, b, νt, t) v log(b)(v − w′ ) dv g(w; 1, b, νt, t) · · . (4.3) π g(v; 1, b, νt, t) v log(b)(v − w′ ) (log(v/w)) − 2πi sin log(b) At this point, we would make a surgery over the contour DR,d,w,k . Let us connect a minor arc −1 from bR−id w to bR−i(d+2π log(b) ) w in the circle CbR \|w\| (major arc being T1 ) and call it T1′ . We imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit T4 T3 24 wbh R + id T2 R − id T1 R − i(d + 2π log(b)−1 ) Fig 4: Figure on the right side indicates the section of the contour DR,d,δ which under the map h 7→ wbh pertains to the contour T1 ∪ T2 ∪ T3 ∪ T4 shown in the left. use −T1′ to indicate clockwise orientation. Therefore, D̃R,d,w,k := T2 ∪ T3 ∪ T4 ∪ −T1′ denotes a closed contour oriented anti clockwise. Using the properties of the contour integration, it can be said that integral over DR,d,w,k in (4.3) is a sum of the integrals over D̃R,d,w,k and CbR \|w\| . Further, notice that there is no pole of the integrand sitting inside the contour D̃R,d,w,k for any w ∈ Cr (see, Theorem 3.7 for w) thanks to the conditions (A) and (B) in (3.5). Thus, integral over D̃R,d,w,k vanishes. To this effect, one can write Kζbt (w, w′ ) Z ∞ X 1 = 2i C R k=−∞ b (v/w)β \|w\| sin π log(b) (log(v/w)) + 2πik · dv g(w; 1, b, νt, t) · . g(v; 1, b, νt, t) v log(b)(v − w′ ) (4.4) 4.1. Case ρ > 1 − (ν(1 − b))−1/2 , Tracy-Widom Fluctuations (GUE). Fix ν > (1 − b)−1 . Set β = −mν t + sσν t1/3 . Define a function Λ as Λ(z) = z − ν log(1 + z/b) + mν log(z). b(1 − b) (4.5) Thus, one can further simplify (4.4) into Kζbt (w, w′ ) ∞ Z X 1 = 2i log(b) C R k=−∞ b exp(t(Λ(w) − Λ(v)) + t1/3 sσν (log(v) − log(w)))dv π ′ ) sin (log(v/w)) + 2πik (v − w \|w\| log(b) × dv (αb−1 v; b)∞ × . −1 (αb w; b)∞ v (4.6) In a nutshell, the asymptotic behavior of the kernel is governed by the variations of the real part of Λ. In what follows, we exhibit the steepest descent contours, which helps in localizing the main contribution in a neighborhood of a critical point of Re(Λ). Thereafter, using Taylor imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 25 expansion of the arguments of the exponential inside the kernel in (4.6) and appropriate estimates for the kernel in the region away from the critical points, we prove the limit. To this end, notice that νb−1 mν 1 − + . Λ′ (z) = b(1 − b) 1 + z/b z √ ( ν(1−b)−1)2 Plugging mν = , we get 1−b Λ′ (z) = (z − ̺)2 , b2 (1 − b)(1 + ̺/b)̺ p where ̺ = b( ν(1 − b) − 1). This shows that Λ′ has an unique double critical point at ̺. In particular, in a neighborhood of ̺, we have Λ(z) = Λ(̺) + σν ̺ 3 (z − ̺)3 + R((z − ̺)) where (z − ̺)−3 R((z − ̺)) → 0 when z → ̺. thanks to fact that σν ̺ 3 p −1 ν(1 − b) − 1 (1 − b)3/2 ν 1/2 = b2 = Λ′′′ (̺). Now, we would like to deform the contours of v and w suitably so that Re(Λ(w) − λ(v)) < −c for some c > 0 whenever v and w are away from the critical point ̺. In the following lemma, we describe the level curves of Re(Λ(z)) = Re(Λ(̺)) in details. We call a closed contour (let’s say C) star shaped if for any φ ∈ [0, 2π), there exists exactly one point z ∈ C such that Arg(z) = φ. Lemma 4.4. There are exactly three level curves L1 , L2 and L3 of Re(Λ(z)) = Re(Λ(̺)) in the complex plane C satisfying the following properties: (a) both L1 and L2 are simple closed contours meeting with L3 at the point ̺. Further, L1 (L2 ) cuts the negative half of the real line at some point d1 (d2 ) in the interval (−b, 0) ((−∞, −b)), (b) L3 resides in the right half of the complex plane, escapes to infinity as \|z\| → ∞, (c) both L1 and L2 are star shaped, (d) L1 \{̺} is completely contained inside the interior of the region enclosed by L2 . Furthermore, L3 meet the contours L1 and L2 at no other point point except ̺, (e) L1 (L2 ) leaves the x-axis at an angle 5π/6 (π/2) from the point ̺ and meets again at −5π/6 (−π/2). Moreover, upper half (lower half ) of the contour L3 leaves the real line from ̺ at an angle π/6 (−5π/6). Proof. We closely adapt here the proofs of the related results in the context of stochastic six vertex model (see, [BCG16, Section 5.1]). (a) It can be observed in (4.5) that Λ(z) has singularities at 0 and −b on R. Thus, there are at least two level curves L1 and L2 of Re(Λ(z)) = Re(Λ(̺)) which originate from the point ̺ and in the way back to their origin, they cut R− at some points d1 ∈ (−b, 0) and d2 ∈ (−∞, −b) respectively. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 26 (b) Tracing the changes in sign of Re(Λ(z)) − Re(Λ(̺)) in Figure 5, one can infer that outside L2 , there should be a region where Re(Λ(z)) − Re(Λ(̺)) < 0. But, if we take a large circle CM of radius M in C, then for z ∈ CM and Re(z) < 0, we have Re(z) − ν log \|zb−1 (1 + z −1 b)\| + mν log \|z/b\| b(1 − b) Re(z) − (ν − mν ) log \|z\| + ν log(b) − mν log(b) + o(1) < 0 = b(1 − b) Re(Λ(z)) = (4.7) because mν < ν. Consequently, there doesn’t exists any other level curve which contains 0 or −b inside. But, there are another level curve L3 escaping to the infinity in the right half of the complex plane after taking off from ̺. This behavior can also be predicted from (4.7). Let us sketch the necessary arguments behind such predictions. Note that if we increase the value Re(z) to the infinity with a fixed value of Im(z), then the right side of (4.7) goes to infinity as well. This shows that at least in the right half of C, some more solutions (other than those on L1 and L2 ) of the equation Re(Λ(z)) = Re(Λ(̺)) exist. This verifies the second claim. (c) To show that both L1 and L2 are star shaped, we need to prove that any line from origin can cut L1 or L2 exactly at two points. Let us fix a line rz0 in C where z0 is fixed with Re(z0 ) = 1 and r varies over the real line. To this end, we have (z0 ) νz0 b−1 1 ∂ Re(Λ(z)) = Re − + mν . (4.8) ∂r b(1 − b) 1 + rzo b−1 r One can note that solutions of the equation that the right side of (4.8) equals to some fixed value is given by a cubic polynomial. But any cubic polynomial over R can have either one ∂ or three real roots. Next, we show ∂r Re(Λ(rz0 ))) = 0 has exactly three real roots unless Im(z0 ) = 0. It can be readily verified that (i) when r ↓ 0, then right side of (4.8) increases ∂ Re(Λ(rz0 ))) → b−1 (1 − b)−1 and when r = ̺, then to infinity, (ii) when r → ∞, then ∂r ∂ ∂ ∂r Re(Λ(rz0 ))) < 0 unless Im(z0 ) = 0. This shows the equation ∂r Re(Λ(rz0 ))) = 0 has at + least two distinct solutions on R unless Im(z0 ) = 0. Further, co-efficient of r 3 and the constant term in the denominator of (4.8) have same sign. This proves the existence of three real roots out of which two must be positive and last one should be negative. Even those two positive roots are distinct unless Im(z0 ) = 0. Thus, except for the case when Im(z0 ) = 0, none of the roots is an inflection point. We have seen Re(Λ(z)) = Re(Λ(̺)) has at least five roots along the line rz0 (four on L1 & L2 and fifth one on L3 ). As ∂ Re(Λ(rz0 )) vanishes in between any two roots of the same sign, therefore, we know ∂r number of roots of Re(Λ(rz0 )) = Re(Λ(̺)) cannot be greater than five. Otherwise, pairs of consecutive roots with the same sign must be greater than three contradicting the fact that the numerator of the right hand side in (4.8) is a polynomial of degree 3 in r. Thus, the third claim is proved. ∂ (d) Fourth claim again follows from the fact that the equation ∂r Re(Λ(rz0 ))) = 0 has exactly three solutions in r except when Im(z0 ) = 0. If L1 meets L2 at some point ̺′ 6= ̺, ∂ Re(Λ(r̺′ ))) = 0 is supposed to have at most two real solutions. Thus, ̺′ cannot then ∂r be different from ̺. But in the latter case, Re(Λ(r̺)) = (Λ(̺)) is satisfied for three distinct values of r, namely, 1, d1 ̺−1 and d2 ̺−1 . As a matter of fact, we know d1 6= d2 . Consequently, L1 has no other point of intersection with L2 except at ̺. Similarly, one imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 27 can prove L3 doesn’t meet L1 or L2 at any other point besides ̺. Moreover, positions of d1 and d2 over the real line implies L1 \{̺} is completely contained inside the interior of the region bounded by L2 . (e) To begin with, note that Λ(z) in a neighbouhood of the point ̺ behaves as Λ(̺) + σν3 ̺−3 (z − ̺)3 . To that effect, possible choices of the tangents for any of the level curves of the equation Re(Λ(z)) = Re(Λ(̺)) are given by {π/6 ± 2π/3} ∪ {π/6 ± 4π/3} = {±π/6, ±π/2, ±5π/6}. Moreover, we know L1 \{0} is completely contained inside the interior of the region bounded by the contour L2 . Also, L3 stays outside of both the contours L1 and L2 . These together justify the claims in part (e). Definition 4.5. We define two new contours Lw and Lv . Both Lv and Lw consist of a (1) (1) (2) (2) piecewise linear segment (Lv and Lw ) and a curved segment (Lv and Lw ). To begin with, (1) (1) Lw extends linearly from ̺ + ǫe−iπ/3 to ̺ and from there to ̺ + ǫeiπ/3 . Similarly, Lv goes from ̺−̺σν−1 t−1/3 +ǫe−2iπ/3 to ̺−̺σν−1 t−1/3 to ̺−̺σν−1 t−1/3 +ǫe2iπ/3 . Here, ǫ will be chosen (1) (1) in such a way that (i) Lv is completely contained inside L2 and Lw \{̺} does not intersect L2 anywhere else, (ii) \|R((z − ̺))\| is bounded above by c\|z − ̺\|3 for some small constant c. (2) Next, Lw starts from the point ̺ + ǫeiπ/3 and encircles around L2 to join ̺ + ǫe−iπ/3 . On the (2) contrary, Lv starts from the point ̺ − ̺σν−1 t−1/3 + ǫe2iπ/3 and curves around 0 and −b to meet ̺ − ̺σν−1 t−1/3 + ǫe−2iπ/3 . But, it never goes out of the contour L2 . It can be ascertained that both Lv and Lw are bounded away from L2 in a ǫ-neighborhood of ̺ by choosing ǫ appropriately. L3 Lw L2 + + Lv − −b L1 ̺ bα−1 − L3 Fig 5: In the above plot, dashed lines denote the level curves of the equation Re(Λ(z)) = Re(Λ(̺)) whereas solid lines indicate the contour Lv and Lw respectively. Signature of the function Re(Λ(z) − Λ(̺)) has been shown in the respective regions with ± signs. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 28 We deform the contours of v and w to Lv and Lw respectively. Such deformations is possible because Lv sits completely inside Lw whereas previously, contour for v was CbR \|w\| . It can be also noted that we didn’t allow any poles to cross in between in doing the deformations of the contours. Consequently, we have Re(Λ(w)) < Re(Λ(v)) for all w ∈ Lw and for all v ∈ Lv . In what follows, we show how the contribution of the integrals of Kb (ζt ) over the contour Lw dominates the rest in the Fredholm determinant in Proposition 4.1 (4.1). Lemma 4.6. There exists some c, C > 0 and t0 > 0, such that for all t ≥ t0 \|Kζbt (w, w′ )\| ≤ C max{ǫ−1 , ̺−1 σν t1/2 } exp −c(t + \|w\|) + sσ̃ν t1/3 \|w\| ∨ \|w\|−1 log(1 + ǫ) (4.9) (2) where w ∈ Lw , w′ ∈ Lw and ǫ is defined in Definition 4.5. Moreover, the following convergence holds lim det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) = 0 t→∞ w (4.10) Proof. To show (4.9), we need to bound the integrand in (5.13) with suitable integrable functions. In the following discussion, we enlist the upper bounds on the absolute value of each of the components of the integrand. (2) (i) For any two points v and w on the contours Lv and Lw respectively, we know Re(Λ(w)) < Re(Λ(v)). In fact, using the compactness of the respective contours, we can further say that there exists c > 0 such that Re(Λ(w)) − Re(Λ(v)) < −c whenever v ∈ Lv and (1) w ∈ Lw . Furthermore, for any complex number z, we have log \|z\| ≤ \|z\| ∨ \|z\|−1 . To sum up, we get exp(t(Λ(w) − Λ(v)) + t1/3 sσν (log(v) − log(w))) ≤ exp(−tc + sσν t1/3 \|w\| ∨ \|w\|−1 ). (4.11) (ii) For any k ∈ Z\{0}, a crude upper bound on the sine function in the integrand is given by π (log(v/w)) + 2πik \| ≥ C ′ exp(2π\|k\|) (4.12) \| sin log(b) for some constant C ′ > 0. For the case when k = 0, \| sin((π/ log(b)) log(v/w))\| is bounded (2) below by C ′′ minv∈L ,w∈L(2) \| log \|(v/w)\|\|. Recall that Lw stays at least ǫ away from the v w contour L2 . Thus, we have min (2) v∈Lv ,w∈Lw \| log \|(v/w)\|\| ≥ log(1 + ǫ). (4.13) Henceforth, (4.12) and (4.13) implies that X k∈Z 1 sin π log(b) (log(v/w)) + 2πik ≤ ∞ X 2 exp(−2πk). log(1 + ǫ) (4.14) k=0 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 29 (iii) Among the other components, the following bounds can be readily proved. −1 αb (αb−1 v; b)∞ ≤ exp (\|v\| + \|w\|) , (αb−1 w; b)∞ 1−b 1 1 ≤ max{ǫ−1 , ̺−1 σν t1/2 }. ≤ max v∈Lv ,w ′ ∈Lw v − w ′ v−w (4.15) Thus, these completes the proof of (4.9). Next, we show the limit in (4.10). For that, decompose Kζb as Kζbt (w, w′ ) := Kζb,1 (w, w′ ) + Kζbt (w, w′ ), t (4.16) b (2) ′ = Kζbt (w, w′ )1(w ∈ L(1) w ) + Kζt (w, w )1(w ∈ Lw ). Using the definition of the Fredholm determinant in (A.1), one can write det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) (4.17) w Z Z X 1 \| det(Kζbt (wi , wj )) − det(Kζb,1 (wi , wj ))\|dw1 . . . dwn . ≤ ... t n!(2πi)n Lw Lw n≥0 Furthermore, (4.16) implies \| det(Kζbt (wi , wj )) − det(Kζb,1 (wi , wj ))\| ≤ t X τ1 ,...,τn ∈{1,2} (τ1 ,...,τn ) 6=(1,...,1) b,τ \| det(Kζt j (wi , wj ))\| (4.18) b,τ where for any fixed (τ1 , . . . , τn ) ∈ {1, 2}n , (Kζt j (wi , wj ))ni,j=1 denotes a matrix whose j-th b,τ column is given by the j-th column of Kζt j . Using the bound on the kernel in (4.9), we have ! n n X X ǫ−n tn/2 b,τj −1 1/2 . \| det(Kζt (wi , wj ))\| ≤ C \|wi \| ∨ \|wi \| \|wi \| + sσν t exp −tc + c (log(1 + ǫ))n i=1 i=1 Moreover, there are 2n − 1 many terms in the sum in the right side of (4.18). These implies one can further bound the right side of (4.17) by det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) ≤ C exp(−ct + c′ t1/2 ) w (4.19) for some constant C, c, c′ > 0. Notice that the right side of (4.19) converges to 0 as t tends to ∞. This implies that the difference of two Fredholm determinant in (4.10) is exponentially decaying to 0. We have decomposed the kernel Kζbt in (4.16). Apart from that, there is another kind of canonical decomposition of Kζbt based on the division of the integral in (4.6) into two parts. For instance, let us consider (1) (2) Kζbt (w, w′ ) = Kζ1 (w, w′ ) + Kζt (w, w′ ) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 30 where for l = 1, 2 (1) Kζt ∞ Z exp t(Λ(v) − Λ(w)) + sσν t1/3 (log(v) − log(w)) 1 X := (l) π 2i (log(v) − log(w)) + 2πik sin log(b) k=−∞ Lv αb−1 v; b ∞ exp (s (log(v) − log(w))) dv × × × . ′ −1 log(b)(v − w ) (αb w; b)∞ v (2) In what follows, we show that the contribution of the kernel Kζ2 in det(1 + Kζbt )L2 (Lw ) fades way as t grows up. Lemma 4.7. There exist c, C > 0 such that ǫ−1 (2) \|Kζt (w, w′ )\| ≤ exp −c(t + \|w\|) + sσν t1/3 \|w\| ∨ \|w\|−1 log(1 + ǫ) (4.20) (1) holds for all w, w′ ∈ Lw . Thus, one would get (1) lim det(1 + Kζbt )L2 (L(1) ) − det(1 + Kζt )L2 (L(1) ) = 0. t→∞ w w (4.21) Proof. Like in the proof of (4.9), (4.20) can be proved using the bound on each of the com(2) ponents of the integrand of Kζt . Bound for the exponential term in the integrand would be exactly same as what had been derived in (4.11). For the series over inverse sin function, we use the bounds in (4.12) and (4.13) together to get (4.14). We must point out here that mini(2) (1) mum value of \| log \|v/w\|\| for any v ∈ Lv and w′ ∈ Lw is again bounded below by log(1 + ǫ), thus, verifying (4.13) in the present scenario. Bound on the infinite product term in (4.15) will (2) (1) again be the same. Minimum distance between any two points on the contour Lv and Lw is ǫ. Thus, bound on \|1/(v − w′ )\| for the present scenario is only ǫ−1 . These all together establish the inequality in (4.20). Using the expansion of the Fredholm determinant as in (4.17) and bound on difference between two determinants like in (4.18), one can complete the proof of (2) (4.21) given the bound on the kernel Kζt . As the bounds in (4.9) and (4.20) are almost same, thus, it implies that difference of two Fredholm determiant in (4.21) also decays exponentially fast as t escapes to ∞. Now, we make a substitutions of the variables v, w and w′ in the ǫ-neighborhood of ̺. In particular, we transform v 7→ ̺ + t−1/3 ̺σν−1 ṽ, w 7→ ̺ + t−1/3 ̺σν−1 w̃, w′ 7→ ̺ + t−1/3 ̺σν−1 w̃′ . (4.22) (1) To the effect of these substitutions, transformed kernel Kζt looks like (1) Kζt ∞ Z t−1/3 ̺σν−1 X = (1) 2i Lṽ sin k=−∞ exp t(Λ(1 + t−1/3 σν−1 ṽ) − Λ(1 + t−1/3 σν−1 w̃)) + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃)) + 2πik exp sσν t1/3 log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) × log(b)(ṽ − w̃′ ) αb−1 (̺ + t−1/3 ̺σν−1 ṽ); b ∞ dṽ × × . (4.23) −1 −1 −1/3 −1/3 αb (̺ + t ̺σν w̃); b ∞ ̺ + t ̺σν−1 ṽ π log(b) (log(1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit (1) 31 (2) We must point out here that Lṽ , Lṽ have been used to denote the contours for the transformed variable ṽ. Similar change in notations for the contours of w̃ and w̃′ will also be made in the subsequent discussion. Next, we state two lemmas which essentially prove part (a) of Proposition 4.1. (1) Lemma 4.8. For all w̃, w̃′ ∈ Lw̃ , we have (1) \|Kζt (w̃, w̃′ )\| ≤ C exp −c1 \|Re(w̃3 )\| − sσν \|w\| (4.24) for some constants C, c1 , c2 > 0. Moreover, the following convergence Z ∞eiπ/3 exp w̃3 − ṽ3 + sṽ − sw̃ dṽ 3 3 1 (1) lim Kζt (w̃, w̃′ ) = t→∞ 2πi ∞e−iπ/3 (ṽ − w̃′ )(ṽ − w̃) (4.25) holds. (1) (1) Proof. For all w̃ ∈ Lw̃ and ṽ ∈ Lṽ , we have w̃3 ṽ 3 t Λ(̺ + t−13 ̺σν−1 w̃) − Λ(̺ + t−13 ̺σν−1 w̃) = − (4.26) 3 3 + t R(t−1/3 ̺σν−1 w̃) − R(t−1/3 ̺σν−1 ṽ) where t(\|R(t−1/3 ̺σν−1 w̃)\| + \|R(t−1/3 ̺σν−1 ṽ)\|) converges to 0 as t → ∞. One can note that ǫ (1) (1) in Definition 4.5 has been chosen in such a way that for ṽ ∈ Lṽ and w̃ ∈ Lw̃ , we have t(\|R(t−1/3 ̺σν−1 w̃)\| + \|R(t−1/3 ̺σν−1 ṽ)\|) ≤ c \|ṽ\|3 + \|w̃\|3 . (1) (1) for some small constant c > 0. Further, we know Re(w3 − v 3 ) < 0 for w ∈ Lw and v ∈ Lv . That implies real part of the right side in (4.26) is bounded above by −c1 \|Re(ṽ)3 \| + \|Re(w̃)3 \| for some constant c1 > 0. Among the other components of the integrand in (4.23), we have (i) ≤ exp (sσν (\|v\| − \|w\|)) exp sσν t1/3 log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) (4.27) (ii) and, as t → ∞, exp sσν t1/3 log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) → exp(s(ṽ − w̃)),(4.28) t1/3 sin and, π log(b) log(b) sin πt−1/3 ̺σν−1 σ \| log(b)\| ≤ ν 2\|ṽ − w̃\| log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) (4.29) π t→∞ −1/3 −1 −1/3 −1 log(1 + t σν ṽ) − log(1 + t σν w̃) −→ ṽ − w̃, log(b) (4.30) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 32 (iii) t −1/3 ∞ X k=−∞ k6=0 1 sin π log(b) log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃) + 2πik t→∞ −→ 0, (4.31) (iv) (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)∞ → 1, (αb−1 (̺ + ̺σν−1 t−1/3 w̃); b)∞ (4.32) whenever t → ∞. Convergence in (4.28) and (4.30) are easy to see. To prove the inequality in (4.27), note that we have Re t1/3 (log(1 + t−1/3 σν−1 ṽ) − log(1 + t−1/3 σν−1 w̃)) ≤ σν−1 (\|v\| − \|w\|) for all large t. For showing (4.29), we used the fact that \| sin(x)\| ≥ 2x/π for all x ∈ −1 −1 [−π/2, π/2]. For large values of t, log(1 + t−/3σν ṽ ) and log(1 + t−/3σν ṽ ) will be close enough to 0. This validates the bound in (4.29). In the third case, when k 6= 0, sin(. + 2πik) grows exponentially fast as \|k\| increases. This makes the series in (4.31) convergent and hence, when multiplied with t−1/3 , it goes to 0. To see (4.32), notice that αb−1 ̺ < 1 thanks to the condition ρ > 1 − (ν(1 − b))−1/2 . Therefore, (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)m t→∞ −→ 1. (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)m uniformly over m. Moreover, using some simple inequalities like \|(1 + z)\| ≤ exp(\|z\|) for any z ∈ C and (1 + z)−1 ≤ exp(\|z\|) for z ∈ C with \|Re(z)\| < 1, it is also easy to see following crude bound (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)∞ ≤ exp(σν−1 t−1/3 (\|v\| + \|w\|)). (αb−1 (̺ + ̺σν−1 t−1/3 ṽ); b)∞ Thus, the integrand in (4.23) is uniformly bounded by an integrable function C 1 exp(−c1 (\|Re(ṽ 3 )\| + \|Re(w̃)3 \|) + c2 (\|v\| − \|w\|)). 1 + \|ṽ\| This integrable envelop proves the bound on the kernel in (4.24). Further, using dominated convergence theorem, we get the convergence in (4.25) whereas (4.28)-(4.32) shows the limiting value. Proof of Proposition 4.1(a). Using Lemma 4.6 and 4.7, we have lim det(1 + Kζbt )L2 (Cr ) = lim det(1 + Kζbt )L2 (L(1) ) . t→∞ Furthermore, the kernel (1) (1) Kζt t→∞ w̃ has an integrable upper bound as shown in (4.24) for all w̃, w̃′ ∈ Lw̃ . Thus, using Corollary A.3, one can establish that the limit of the Fredholm determinant det(1 + Kζbt )L2 (L(1) ) is same as the Fredholm determinant of the limiting kernel in (4.25). This w̃ completes the proof. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 33 2 Fluctuations. 4.2. Case ρ = 1 − (ν(1 − b))−1/2 , FGOE In this scenario, the double critical point ̺ of the function Λ becomes equal to bα−1 . Thus, the deformation of the contour of w to Lw is no more viable because in Definition 4.5, Lw is described to pass through ̺ whereas the contour of w must avoid the pole at bα−1 as stated in Theorem 3.7. Furthermore, the result in (4.32) is also not going hold. In what follows, we make a mild change in the definition of contours Lw and Lv to remedy those issues. Definition 4.9. Recall all the notations from Definition 4.5. To avoid the pole at bα−1 , we (1) redefine Lw to go from ̺ − t−1/3 δ̺σν−1 + ǫe−iπ/3 to ̺ − t−1/3 δ̺σν−1 to ̺ − t−1/3 δ̺σν−1 + ǫeiπ/3 . (1) Similarly, we shift the contour Lv to the left in the complex plane by t−1/3 δ̺σν−1 . For instance, (1) in this new settings, Lv extends from ̺ − t−1/3 (1 + δ)̺σν−1 + ǫe−i2π/3 to ̺ − t−1/3 δ̺σν−1 to ̺ − t−1/3 (1 + δ)̺σν−1 + ǫei2π/3 . Here, δ can be any positive real number. Furthermore, ǫ must be chosen is such a way that following conditions, given as (1) (1) both end points ̺ − t−1/3 δ̺σν−1 + ǫe−iπ/3 and ̺ − t−1/3 δ̺σν−1 + ǫeiπ/3 of Lw lie outside the level curve L2 (see, Theorem 4.4 for definition), (1) (1) (2) Re(Λ(w)) < Re(Λ(w)) for all v ∈ Lv and Lw , 3 (3) \|R((z − ̺))\| < c\|z − ̺\| for any z in an ǫ-neighbourhood of ̺, (2) (2) are satisfied. The residual parts, namely Lw and Lv are defined exactly the same way as in Definition 4.5 except the necessary adjustments of their origins and end points. For instance, (2) Lw now starts from ̺ − t−1/3 δ̺σν−1 + ǫe−iπ/3 and ends at ̺ − t−1/3 δ̺σν−1 + ǫeiπ/3 . Thanks to the new definitions above, we can deform of the contour of v and w to Lv and Lw without crossing any singularities. Moreover, one can note that bα−1 lies outside contour Lw . To this end, one can prove the results like Lemma 4.6 and Lemma 4.7 almost in the same way in the present −1context except possibly with a different treatment for the quantity αb−1 v; b ∞ αb−1 w; b ∞ in (4.15). In what follows, we carry out such changes. Lemma 4.10. Assume ρ = 1 − (ν(1 − b))−1/2 . There exist c′ , C ′ > 0 such that whenever max{\|w − ̺\|, \|v − ̺\|} < c′ , then αb−1 v; b ∞ v−̺ < C′ . −1 (αb w; b)∞ w−̺ (4.33) (1) (1) Furthermore, consider the substitutions in (4.22). Denote the contours Lv and Lw after the (1) (1) (1) (1) substitution by Lṽ and Lw̃ respectively. Then, for any ṽ ∈ Lṽ and w̃ ∈ Lw̃ , we have αb−1 (̺ + t−1/3 ̺σν−1 ṽ); b ∞ ṽ = . lim −1 t→∞ αb−1 (̺ + t−1/3 ̺σν w̃); b w̃ ∞ (4.34) Proof. To prove (4.33), let us write αb−1 v; b ∞ =A·B·C (αb−1 w; b)∞ imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit where v−̺ , A := w−̺ Qm j−1 v) j=1 (1 − αb B := Qm , j−1 w) j=1 (1 − αb C := 34 (αbm v; b)∞ . (αbm w; b)∞ Further, m is chosen in such a way that the inequality bm α (̺ + c′ ) < 1/2 is satisfied. To that effect, we have      ∞ ∞ m+j−1 j−1 X X 1 + αb v  b (v − w)  \|C\| = exp  log = exp αbm Re  m+j−1 1 + αb v 1 + ψ(v, w, j) j=1 j=1    ∞ j−1 X b \|v − w\|  2αbm \|v − w\| m  ≤ exp αb . (4.35) ≤ exp 1−b 1 − 21 j=1 In the second equality above, we have used the mean value theorem for the complex variables. Thus, ψ(v, w, j) denotes a complex number in the line joining αbm+j v and αbm+j w. Consequently, one have v−w \|v − w\| \|v − w\| Re ≤ . ≤ 1 + ψ(v, w, j) 1 − \|ψ(v, w, j)\| 1 − bm α (̺ + c′ ) This shows the inequalities in (4.35). Lastly, B involves product of finitely many terms which are bounded above. Hence, this completes the proof of (4.33). Further, one can write α(̺ + t−1/3 b̺σν−1 ṽ); b ∞ αb−1 (̺ + t−1/3 ̺σν−1 ṽ); b ∞ (−t−1/3 ̺σν−1 ṽ) = . (4.36) × (−t−1/3 ̺σν−1 w̃) αb−1 (̺ + t−1/3 ̺σν−1 w̃); b ∞ α(̺ + t−1/3 b̺σν−1 w̃); b ∞ Note that bn α̺ is bounded away from 1 for all non-negative powers of b. Thus, the second term in the product of the right side in (4.36) converges to 1 as t → ∞. This shows the limit in (4.34). Now, we note down two results in the same spirit of Lemma 4.6 and Lemma 4.7. Proofs are exactly same except one has to use the bound in (4.33) instead of (4.15). For brevity, we omit the proofs. Lemma 4.11. There exists some c, C > 0 and t0 > 0, such that for all t ≥ t0 \|Kζbt (w, w′ )\| ≤ C 1 max{ǫ−1 , ̺−1 σν t1/3 } exp −c(t + \|w\|) + sσν t1/3 \|w\| ∨ \|w\|−1 \|w − ̺\| log(1 + ǫ) (2) where w ∈ Lw , w′ ∈ Lw and ǫ is defined in Definition 4.5. Moreover, the following convergence lim det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L(1) = 0 t→∞ w also holds in the present scenario. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 35 Lemma 4.12. There exist c, C > 0 such that (2) \|Kζt (w, w′ )\| ≤ ǫ−1 1 exp −c(t + \|w\|) + sσν t1/3 \|w\| ∨ \|w\|−1 \|w − ̺\| log(1 + ǫ) (1) holds for all w, w′ ∈ Lw . Thus, one would get (1) lim det(1 + Kζbt )L2 (L(1) ) − det(1 + Kζt )L2 (L(1) ) = 0. t→∞ w w Lemma 4.13. Recall all the notations from the previous section and Definition 4.9. For all (1) w̃, w̃′ ∈ Lw̃ , we have 1 (1) \|K̃ζt (w̃, w̃′ )\| ≤ C exp −c1 \|Re(w̃3 )\| − sσν \|w\| δ (4.37) for some positive constants c1 , c2 , C and (1) lim K̃ζt (w̃, w̃′ ) = t→∞ 1 2πi Z −(1+δ)+ei2π/3 exp w̃ 3 3 − ṽ3 3 + s(ṽ − w̃) (ṽ − w̃)(ṽ − w̃′ ) −(1+δ)+e−i2π/3 × ṽdṽ . w̃ (4.38) Proof. To get the bound in (4.37), note that integrand in (4.23) is bounded above by 1 exp(−c1 (\|Re(ṽ 3 )\| + \|Re(w̃)3 \|) + c2 (\|v\| − \|w\|)) C (1 + \|ṽ\|) αb−1 (̺ + t−1/3 ̺σν−1 ṽ); b ∞ . αb−1 (̺ + t−1/3 ̺σν−1 w̃); b ∞ We have illustrated this fact in Lemma 4.8. Thus, we omit further details on it. Let us recall that Lw is shifted to the left by t−1/3 δ̺σν−1 from ̺. To this effect, \|w̃\| & δ and henceforth, the ratio of the infinite products in (4.34) gets bounded above by C ′ \|ṽ\|/δ thanks to (4.33). This accounts for the term 1/δ in the bound on the right side in (4.37). Proof of (4.38) follows from the similar arguments like in Lemma 4.8 except the limit in (4.32) should be replaced by the result (4.34) of Lemma 4.10. To summarize, all these results above prove lim det(1 + Kζbt )L2 (Cr ) = det(1 + Kcrit )L2 (L̄w̃ ) t→∞ where Kcrit (w̃, w̃′ ) := 1 2πi Z −(1+δ)+ei2π/3 −(1+δ)+e−i2π/3 exp w̃ 3 3 − ṽ3 3 + s(ṽ − w̃) (ṽ − w̃)(ṽ − w̃′ ) × ṽdṽ w̃ and L̄w̃ denotes a piecewise linear contour which extends linearly from −δ + ∞e−iπ/3 to −δ and from there again linearly to −δ + ∞eiπ/3 . Hence, the proof of Proposition 4.1 (b) follows. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 36 4.3. Case ρ < 1 − (ν(1 − b))−1/2 , Gaussian Fluctuation In this section, we prove part (c) of Proposition 4.1. Main reason behind a separate analysis is that whenever ρ < 1 − (ν(1 − b))−1/2 , contours like Lv and Lw in Definition 4.5 or even their modifications in Definition 4.9 contain bα−1 inside. To circumvent this illegal inclusion, we adjust the choice of mν in the function Λ (see (4.5)) in such a way that bα−1 turns out to be a new critical point of Λ. To begin with, the form of the kernel Kζbt is given by Kζbt (w, w′ ) ∞ Z X 1 = 2i log(b) C R k=−∞ b exp(t(Λ(w) − Λ(v)) + t1/2 sσ̃ν (log(v) − log(w)))dv π (v − w′ ) sin log(b) (log(v/w)) + 2πik \|w\| × dv (αb−1 v; b)∞ × . −1 (αb w; b)∞ v (4.39) 1/2 and v respectively. Under the these where ζt and bh w are substituted with b−m̃ν t+sσ̃ν t recalibration, the explicit form of the function Λ is given by Λ(z) = z − ν log(1 + zb−1 ) + m̃ν log(z). b(1 − b) (4.40) In what follows, we perform some calculations de rigueur the asymptotic analysis. Plugging 1 1 ν − α(1−b) in (4.40), one can write m̃ν = 1+α Λ′ (z) = (z − bα−1 )(z − ̺′ ) . zb(z + b)(1 − b) where ̺′ := b (ν(1 − ρ)(1 − b) − 1). Under the condition ρ < 1− (ν(1− b))−1/2 , it can be noted that ̺′ > ̺. To this end, we have Λ′′ (bα−1 ) = − (1 − ρ)2 ν(1 − b) − 1 . b2 ρ(1 − b) Further, using Taylor’s expansion, one would get −1 Λ(z) = Λ(bα 1 )− 2 σ̃ν bα−1 2 (z − bα−1 )2 + R̃(z − bα−1 ) (4.41) where \|z − bα−1 \|2 R̃(z − bα−1 ) converges to 0 whenever z → bα−1 . In the following lemma, we illustrate on the properties of the level curves Re(Λ(z)) = Re(Λ(bα−1 )). Lemma 4.14. There exists two simple closed contours L1 and L2 such that for all z ∈ L1 ∪L2 , we have Re(Λ(z)) = Re(Λ(bα−1 )). They further satisfy the following properties. (a) Both the contours originate at the point bα−1 and encircle around 0 to meet again at their origin. For instance, L1 (L2 ) in its way back to bα−1 crosses the real line for the second time at a point d1 ∈ (−b, 0) (d2 ∈ (−∞, −b)). (b) Contour L1 \{bα−1 } is completely contained in the interior of the region bounded by L2 . (c) Contour L1 (L2 ) leaves x axis from the point bα−1 at an angle 3π/4 (π/4) and meet again at angle −3π/4 (−π/4). imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 37 Apart from L1 and L2 , there exists another level curve L3 of the equation Re(Λ(z)) = Re(Λ(bα−1 )) lying completely in the right half of the complex plane. Furthermore, L3 doesn’t intersect L1 or L2 and escapes to infinity whenever Re(z) → ∞. Proof. Proof of the existence of L1 , L2 and L3 are similar to that in Lemma 4.4. Moreover, property (a) and (b) can also be proved in the same way. To see part (c), note that −1 −1 2 −1 )2 in a neighborhood around bα−1 . Λ(z) − Λ(bα−1 ) behaves as −2 (σ̃ν αb ) ν(z − bα −1 −1 Thus, Re Λ(z) − Λ(bα ) = 0 implies Arg(z − bα ) = ± π4 ± 2π 2 . This indicates part (c). We are left to show that L3 doesn’t intersect L1 or L2 . To start with, note that L3 doesn’t pass through bα−1 . This is because there are only four possible choices of the angles (± π4 ± π2 ) −1 that any level curve of the equation Re Λ(z) − Λ(bα ) = 0 can make at bα−1 . Thus, there can be at most two level curves which pass through the point bα−1 . In fact, L3 crosses the real line at a point further right of bα−1 . To see this, notice that Re Λ(z) − Λ(bα−1 ) < 0 if Im(z) = 0 and bα−1 < Re(z) < bα−1 + δ for some δ > 0. Additionally, Re(Λ(z)) increases to infinity with increasing Re(z). Thus, there must exists another point on the real line to the right of bα−1 where once again Re Λ(z) − Λ(bα−1 ) = 0 holds. This shows the claim. Moreover, one can mimick almost the same proof as in Lemma 4.4 to show that L3 doesn’t intersect L1 or L2 at any other point. This completes the proof. We must point out here that L2 is no more star shaped (see Figure 6). This urges for the necessary changes in the definitions of the contours Lv and Lw in the present scenario. In what follows, we enunciate those changes formally. (1) Definition 4.15. We define the linear segment Lv of Lv to extend linearly from −ǫi + Intbα−1 ,2δ to ǫi + Intbα−1 ,2δ crossing the real line at Intbα−1 ,2δ where Intx,τ = x − t−1/3 τ xδ(σ̃ν )−1 for any x ∈ C, τ ∈ R+ . (1) On the flip side, we define the piecewise linear segment Lw to extend linearly from ǫe−iπ/6 + Intbα−1 ,δ to Intbα−1 ,δ and from there to ǫeiπ/6 + Intbα−1 ,δ . At this junction, it is worthwhile to point out that any particular choice of ǫ must satisfies the following properties: (a) ǫe−iπ/6 + Intbα−1 ,δ and ǫeiπ/6 + Intbα−1 ,δ should lie outside the contour L2 , (b) in the ǫ - neighborhood of the point bα−1 , \|R(z − bα−1 )\| in (4.41) must be less than c\|z − bα−1 \|2 for small constant (2) (2) c. In particular, c = 1/4 suitably fits into our analysis. Curved segments Lv and Lw are (2) (1) defined similarly as in Definition 4.5. For instance, Lv starts from the top end point of Lv (1) and encircles around 0 to meet at the bottom end point of Lv . To be precise, it remains (1) (1) (2) always inside of the contour L2 . Unlike Lv , starting from the top end point Lw , Lw circles (1) outside the contour L2 to meet the bottom end point of Lw . In the next lemma, we show that the contribution of the kernel Kζbt in the Fredholm (2) determinant det(1 + Kζbt )L2 (Lw ) over the segment Lw decays down exponentially. (2) Lemma 4.16. For any w ∈ Lw and w′ ∈ Lw , we have \|Kζbt (w, w′ )\| ≤ C max{ǫ−1 , δ−1 t1/2 } exp −tc + sσ̃ν t1/2 \|w\| ∨ \|w\|−1 + α \|w\| (4.42) 1−b imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit − L2 − + α−1 b L1 L3 Lw Lv + −b 38 ̺ − L3 Fig 6: Solid lines are used to denote the contour Lw and Lv whereas dotted lines indicate the level curves of the equation Re(Λ(z) − Λ(bα−1 )) = 0. for some constants c, C > 0. Here, ǫ is same as in Definition 4.15. To that effect, we have lim det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) = 0. t→∞ w (4.43) (2) Proof. To begin with, note that for all w ∈ Lw and all v ∈ Lv there exists a constant c > 0 (2) such that Re (Λ(w) − Λ(v)) < −c. This holds due to the particular way the contours Lw , Lv are chosen and the compactness of each of them. Other components of the integrand in (4.39) are bounded in the following ways (i) exp sσ̃ν t1/2 (log(v) − log(w)) ≤ exp sσ̃ν t1/2 (\|v\| ∨ \|v\|−1 + \|w\| ∨ \|w\|−1 ) ,(4.44) (ii) ∞ X k=−∞ 1 sin π log(b) log(v/w) + 2πik ≤ ∞ X 2 exp(−2πk) exp k=0 π −1 −1 (\|v\| ∨ \|v\| + \|w\| ∨ \|w\| ) , log(b) (4.45) (iii) 1 ≤C v(v − w) minv∈L 1 (2) ′ v ,w ∈Lw \|v − w′ \| ≤ C max{ǫ−1 ∨ δ−1 t1/2 } (4.46) for some large constant C, imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 39 (iv) ∞ m ∞ Y Y Y (αb−1 v; b)∞ k−1 m−1 −1 ≤ (1 + b α\|v\|) \|(1 − b α\|w\|)\| (1 + 2bk−1 α\|w\|) αb−1 w; b)∞ k=1 k=0 k=m+1 α ≤ C exp (\|v\| + 2\|w\|) (4.47) 1−b where for all k ≥ m, one has (1 + 2bk−1 α\|w\|)(1 − bk−1 α\|w\|) ≤ 1. Inequality in (4.44) is a simple consequence of the fact that \| exp(log(v) − log(w))\| ≤ exp Re(v) ∨ Re(v −1 ) + Re(w) ∨ Re(w−1 ) ≤ exp \|v\| ∨ \|v\|−1 + \|w\| ∨ \|w\|−1 for any two complex numbers v, w. To see the bound in the right side of (4.45), note that sin(u + 2πik) is bounded above by 2 exp(\|u\| − 2πk) for all u ∈ C and k ∈ Z. It is easy to comprehend the first inequality in (4.46). For the second inequality, recall that both the end (1) points of Lw are at a distance ǫ away from the point bα−1 for large enough t. Thus, it is (2) possible to construct the contour Lw at a distance of ǫ far away from L2 . As we know Lv is contained in the region enclosed by L2 , thus the minimum value of \|v − w′ \| must be greater (2) (1) than ǫ for all v ∈ Lv and w′ ∈ Lw . On the flip side, if w′ ∈ Lw , then minimum value of \|v − w\| should be greater than δt−1/2 , thus shows the claim in (4.46). First inequality in straightforward. To prove the second inequality, note that the finite product (4.47) Qm is quite m−1 α\|w\|)\|−1 is bounded by a large constant. To bound the other components \|(1 − b k=0 apart from the finite product term, we use a simple inequality (1 + x) ≤ ex which holds for any x ∈ R. These all together imply the inequality in (4.42). Next, we prove the limit in (4.43). Let us write down b (2) ′ Kζbt (w, w′ ) = Kζbt (w, w′ )1(w ∈ L(1) w ) + Kζt (w, w )1(w ∈ Lw ). . Then, using the inequality in (4.18) and the bound Denote Kζbt (w, w′ )1(w ∈ Lw ) by Kζb,res t (1) on Kζbt (w, w′ )1(w ∈ Lw ) from (4.42), we have (2) n det Kζbt (wi , wj ) i,j=1 n (w , w ) − det Kζb,res i j t i,j=1 ≤ C2n max{ǫ−n , (δ−2 t)n/2 } exp(−ct). Further, using series expansion of Fredholm determinant, we get the following bound det(1 + Kζbt )L2 (Lw ) − det(1 + Kζbt )L2 (L(1) ) = det(1 + Kζbt )L2 (Lw ) − det(1 + Kζb,res )L2 (Lw ) t w ≤ C(1 + exp(−2ǫ−1 )) exp(−ct + 2δ−1 t1/2 ) (4.48) where the right side of the inequality above goes to 0 as t → ∞. This completes the proof. Likewise in (4.23), we further represent Kζbt as the sum of two different kernels to identify the leading contribution in the computation of the Fredholm determinant det(1 + Kζbt )L2 (L(1) ) . w In particular, we set (1) (2) Kζbt = Kζt + Kζt imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 40 where for l = 1, 2, (l) Kζt ∞ Z 1 X := (l) 2i Lv sin k=−∞ exp (t(Λ(v) − Λ(w))) π log(b) (log(v) − log(w)) + 2πik (4.49) αb−1 v; b ∞ exp (s (log(v) − log(w))) dv × × . × ′ −1 log(b)(v − w ) (αb w; b)∞ v Following lemma which is in the same spirit of the last Lemma 4.48 proves that one can even (2) ignore the effect of Kζt as t increases to infinity. (1) Lemma 4.17. For any w, w′ ∈ Lw , we have b ′ −1 1/2 \|Kζt (w, w )\| ≤ C(ǫδ) t exp −tc + sσ̃ν t1/2 \|w\| ∨ \|w\|−1 + α \|w\| 1−b (4.50) for some constants c, C > 0. Thus, we claim (1) lim det(1 + Kζbt )L2 (L(1) ) − det(1 + Kζt )L2 (L(1) ) = 0. t→∞ (4.51) w w (2) Proof. Proof of (4.50) hinges upon the fact that Re(Λ(w) − Λ(v)) < −c for all v ∈ Lv and all (1) w ∈ Lw . This again follows from the compactness of both of the contours, continuity of the map Re(Λ(.)) and more importantly, strategic position of the contours around L2 . Bounds on the other component of the integrand in (4.49) can be derived as in (4.44) - (4.47). For instance, bounds in (4.44) and (4.45) work perfectly for the current result. In what follows, we recollect bounds on other two terms and sharpen it for the present scenario. (i) 1 1 ≤C ≤ Cǫ−1 v(v − w) minv∈L(2) ,w′ ∈L(1) \|v − w′ \| v (4.52) w for some large constant C (ii) ∞ m ∞ Y Y Y (αb−1 v; b)∞ k−1 m−1 −1 ≤ (1 + b α\|v\|) \|(1 − b α\|w\|)\| (1 + 2bk−1 α\|w\|) αb−1 w; b)∞ k=1 k=0 k=m+1 α ≤ Cδ−1 t1/2 exp (\|v\| + 2\|w\|) (4.53) 1−b where for all k ≥ m, one has (1 + 2bk−1 α\|w\|)(1 − bk−1 α\|w\|) ≤ 1. One can prove the inequality in (4.52) by recalling that the minimum value of \|v − w′ \| over (2) (1) all v ∈ Lv , w′ ∈ Lw is bounded below by ǫ (see, Definition 4.15). In (4.53), we cannot now bound the finite product term by a large constant uniformly over all choices of t and w. This is because min \|1 − αb−1 w\| = δt−1/2 . (1) w∈Lw imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 41 This causes the factor δ−1 t1/2 to come in the right side of the second inequality. To show (4.51), we use the inequality in (4.18) to conclude n n (1) ≤ C2n (δǫ)−n tn/2 exp(−ct). − det Kζt (wi , wj ) det Kζbt (wi , wj ) i,j=1 i,j=1 (1) whenever w1 , . . . , wn ∈ Lw . To that effect, Fredholm determinant expansion provides the sim(1) ilar bound as in (4.48) on the difference between det(1 + Kζbt )L2 (L(1) ) and det(1 + Kζt )L2 (L(1) ) . w w This ends the proof. Above two lemmas show that in order to find out the limit of the Fredholm determinant (1) (1) in (4.2), it suffices to study Kζt over the segment Lw . Thus, like in (4.22), we make the following substitutions v 7→ bα−1 + t−1/2 b(ασ̃ν )−1 ṽ, w 7→ bα−1 + t−1/2 b(ασ̃ν )−1 w̃, w′ 7→ bα−1 + t−1/2 b(ασ̃ν )−1 w̃′ . (4.54) (1) To this end, one can further write Kζt in the following form - (1) Kζt = t−1/2 b(ασ̃ ν )−1 2i Z 2 2 exp ṽ2 − w̃2 + t R̃(t−1/2 b(ασ̃ν )−1 w̃) − R̃(t−1/2 b(ασ̃ν )−1 w̃) (1) π −1/2 (σ̃ )−1 ṽ) − log(1 + t−1/2 (σ̃ )′ w̃)) + 2πik Lṽ k=−∞ sin (log(1 + t ν ν log(b) ∞ X × × (1) exp sσ̃ν t1/2 (4.55) log(1 + t−1/2 (σ̃ν )−1 ṽ) − log(1 + t−1/2 (σ̃ν )−1 w̃) log(b)(ṽ − w̃′ ) αb−1 (bα−1 + t−1/2 b(ασ̃ν )−1 ṽ); b αb−1 (bα−1 (1) + t−1/2 b(ασ̃ (1) ν )−1 w̃); b ∞ × ∞ dṽ bα−1 + t−1/2 b(ασ̃ −1 ν ) ṽ (1) where Lṽ and Lw̃ denote the segments Lv and Lw respectively after the substitutions made in (4.54). In the next result, we show that in fact the Fredholm determinant of the (1) kernel Kζ1 converges to the normal distribution. Theorem 4.18. For all s ∈ R, we have (1) lim det 1 + Kζt t→∞ (1) L2 (Lw ) = G(s). Proof. To begin with, we enlist below how each of the components of the integrand in (4.55) is bounded above by some integrable function and what are their limits as t → ∞. (i) We first look at the inequalities shown below 2 w̃2 ṽ −1/2 −1 −1/2 −1 − + t R̃(t b(ασ̃ν ) w̃) − R̃(t b(ασ̃ν ) w̃) (4.56) exp 2 2 2 w̃2 ṽ 2 2 − + c\|ṽ\| + c\|w̃\| ≤ C ′ exp −c′1 \|ṽ\|2 + c′2 Re(w̃2 ) ≤ exp Re 2 2 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 42 for some constants C ′ , c′1 > 0 and c′2 . To see the first inequality, recall that \|R̃(z − bα−1 )\| is less than c\|z −bα−1 \|2 in the ǫ-neighborhood of bα−1 (see, Definition 4.15). For showing the second inequality in (4.56), one can argue that Re(ṽ 2 ) + c\|ṽ\|2 ≤ −c′1 \|ṽ\|2 for some c′1 > 0. This is because the complex variable ṽ varies over −2δ + iR. Thus, Re(ṽ 2 ) converges to −∞ as \|ṽ 2 \| → ∞. In fact, we get Re(ṽ 2 )/\|ṽ\|2 → −1 whenever \|ṽ\| goes to ∞. This completes the proof of the inequalities above. Furthermore, as t → ∞, both tR̃(t−1/2 b(ασ̃ν )−1 ṽ) and R̃(t−1/2 b(ασ̃ν )−1 w̃) goes to 0. Thus, we get 2 2 ṽ w̃2 w̃2 ṽ exp − + t R̃(t−1/2 b(ασ̃ν )−1 w̃) − R̃(t−1/2 b(ασ̃ν )−1 w̃) − . → exp 2 2 2 2 (ii) Next, we illustrate on the limit of the infinite sum over k ∈ Z. Note that for any k ∈ Z, we have π −1/2 −1 −1/2 ′ sin (log(1 + t (σ̃ν ) ṽ) − log(1 + t (σ̃ν ) w̃)) + 2πik log(b) ≥ C(1 + t−1/2 \|ṽ\|) exp(2π\|k\|). Thus, it shows the series in (4.55) converges absolutely and uniformly over ṽ. Hence, as t goes to infinity, lim t→∞ ∞ X k=−∞ t−1/2 π(log(b))−1 sin π log(b) (log(1 + t−1/2 (σ̃ν )−1 ṽ) − log(1 + t−1/2 (σ̃ν )′ w̃)) + 2πik = 1 . ṽ − w̃ (iii) Likewise in Lemma 4.10, one can prove that for all large t, there exists constants C ′ such that αb−1 (bα−1 + t−1/2 b(ασ̃ν )−1 ṽ); b ∞ ṽ . ≤ C′ −1 −1 −1/2 −1 w̃ αb (bα + t b(ασ̃ν ) w̃); b ∞ Moreover, we have αb−1 (bα−1 + t−1/2 b(ασ̃ν )−1 ṽ); b αb−1 (bα−1 + t−1/2 b(ασ̃ν )−1 w̃); b ∞ t→∞ −→ ∞ ṽ . w̃ . (iv) Lastly, we can put a very crude bound like exp sσ̃ν t1/2 log(1 + t−1/2 (σ̃ν )−1 ṽ) − log(1 + t−1/2 (σ̃ν )−1 w̃) ≤ C ′′ exp(c(\|ṽ\| + \|w̃\|)) bα−1 bα−1 + t−1/2 b(ασ̃ν )−1 ṽ 1 . 1 + \|ṽ\| for some positive constant C ′′ , c′′ . Notice that even this crude bound doesn’t affect integrability of the product of all bounds together because of the dominating term exp(−c′ \|ṽ\|2 ) in (4.56). Further, we have the following limit bα−1 exp sσ̃ν t1/2 log(1 + t−1/2 (σ̃ν )−1 ṽ) − log(1 + t−1/2 (σ̃ν )−1 w̃) bα−1 + t−1/2 b(ασ̃ν )−1 ṽ t→∞ −→ exp (s(ṽ − w̃)) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 43 for the remaining component of the integrand in (4.55). To this end, using dominated convergence theorem, one can write 2 ṽ2 w̃ Z − + s( w̃ − ṽ) exp 2 2 ṽ 1 (1) · · dṽ lim K (w̃, w̃′ ) = t→∞ ζt 2πi −2δ+iR (ṽ − w̃′ )(ṽ − w̃) w̃ (4.57) (1) for all w̃, w̃′ in the limiting form of the contour Lw̃ . In Appendix A, we prove that Fredholm determinant of the limiting kernel is indeed Gaussian distribution function. Proof of Theorem 2.5. To show (2.2), note that {xmν t ≤ νt + ∂mν sLt1/3 } = {Nνs (t) ≥ mν t}. ∂ν ν −2/3 ) and L, s ∈ R. It is easy to see that νs equals to ν when where νs := ν(1 + sLσν ∂m ∂ν t s = 0. Moreover, using Taylor’s theorem on mνs , one can obtain mνs = mν + sLσν Let us choose L = ∂mν −2/3 Lt + o(t−2/3 ). ∂ν ∂mν −1 . ∂ν To that effect, we have P (Nνs (t) ≥ mν t) = P mνs t − Nνs t ≤ sσν t1/3 + o(t1/3 ) (4.58) Theorem 2.2 implies that the right side in (4.58) converges to FGU E (s). Thus, it completes the proof of (2.5). Similarly, the proofs of (2.6) and (2.7) follow from (2.3) and (2.4) respectively. Proof of Corollary 2.6 Theorem 2.4 implies (t−1 Nνt (t) − mν ) = Op (t−2/3 ) when ν > ν0 . Thus, it proves the first case. Further, note that for any ǫ > 0, there exists δǫ > 0 such that whenever ν ≤ ν0 + δǫ , limit shape mν < ǫ. To that effect, for any ν ≤ ν0 , we have lim sup t→∞ N(ν0 +δǫ )t (t) Nνt (t) ≤ lim ≤ǫ t→∞ t t with high probability. Henceforth, this shows the second claim. 5. Weak Scaling Limit Our goal here is to prove the Theorem 2.9. Following two propositions provide the necessary arguments needed to complete the proof. Proposition 5.1. The collection of processes {Zǫ }ǫ is tight in C(R+ × R). Proposition 5.2. Any limiting point Z of {Zǫ }ǫ solves the SHE. To prove Proposition 5.1-5.2, we first show that the Gärtner transform (recall Definition 2.7) of HL-PushTASEP satisfies a discrete SHE. This step is in harmony with the other works like [BC95, Section 3.2], [DT16, Section 2], [CT15, Section 3]. Let us note that here onwards, constants will change from line to line. Unless specified, they do not depend on anything else. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 44 5.1. Discrete SHE First, we derive a dynamical equation for Z(t, x). Recall that any particle which is at left side of any site x when gets excited can move across x with certain probability. For instance, once the clock of the particle at site y(< x) rings at time t, it jumps across x with probability bx−y−Nx (t)+Ny (t) . For notational convenience, let us introduce the following notation κ(x, t) = x + t . 1 − bν ǫ (5.1) From the definition of Z(t, x) in (2.9), it is clear that contributions of the all the right hops across x in dZ(t, x) during the interval [t, t + dt] is given by (b−1 − 1)Z(t, x)dPtx where {Ptx }−∞<x<∞ denotes a sequence of Poisson point processes with rates κ(x,t) X ηt (l)bκ(x,t)−l−Nκ(x,t) (t)+Nl (t) l=1 and ηt (l) is the occupation variable for the position l (see, Definition 2.1). There is also a contribution coming from the exponential growth term exp(−µǫ t). Collecting all the terms together, we can write down the following   κ(x,t) X (1 − νǫ ) log b − µǫ  Z(t, x)dt dZ(t, x) = (b−1 − 1) ηt (l)bκ(x,t)−l−Nκ(x,t) (t)+Nl (t) − (1 − bνǫ ) l=1   κ(x,t) X + (b−1 − 1)Z(t, x) dPtx − ηt (l)bκ(x,t)−l−Nκ(x,t) (t)+Nl (t) dt . l=1 For the sake of simplicity, denote dM (t, x) = (b −1 − 1) dPtx − X ηt (l)b κ(x,t)−l−Nκ(x,t) (t)+Nl (t) l=1 ! dt . (5.2) Further, let us define Ωx := (b−1 − 1) X ηt (l)bκ(x,t)−l−Nκ(x,t) (t)+Nl (t) . (5.3) l=1 Lemma 5.3. We have Ωx Z(t, x) = (b−1 − 1) X l<x bνǫ (x−l) Z(t, l) − Z(t, x). (5.4) Proof. Using the form of Ωx given in (5.3), we get X Ωx Z(t, x) = (b−1 − 1) ηt (l)bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l exp(−µǫ t). l≤κ(x,t) We will show X X bνǫ (κ(x,t)−l) bNl (t)−l/2 .(5.5) bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l + exp(µǫ t)Ωx Z(x, t) = b−1 l≤κ(x,t) l<κ(x,t) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 45 One can easily P see that above result implies (5.4). For proving (5.5), we have Ωx Z(t, x) = √ √ λǫ ǫ(1 + C ǫ) l≤κ(x,t) ηt (l)bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l by expanding b−1 − 1 in terms of ǫ. Note P λn+1 ǫn ǫ that the value of C is ∞ n=1 (n+1)! . To this effect, we have X bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l + Ωx Z(x, t) exp(µb t) l≤κ(x,t) = X (1 + l≤κ(x,t) = X l≤κ(x,t) = b−1 √ ǫλǫ ηt (l))bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l + Cλǫ ǫ √ Ωx Z(t, x) 1+C ǫ √ [exp(λǫ ǫηt (l)) − Cλǫ ǫηt (l)]bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l Cλǫ ǫ √ Ωx Z(t, x) + 1+C ǫ X bνǫ (κ(x,t)−l) bNl (t)−(1−νǫ )l . l<κ(x,t) √ √ We wrote down the equality in the third line by identifying (1+ηt (l)λǫ ǫ) as exp(ηt (l)λǫ ǫ)− Cλǫ ǫηt (l). In the last line, we used the fact Nx (t) − ηt (x) = Nx−1 (t) for any x ∈ Z≥0 . This completes the proof. Using the definition of µǫ in (2.10), one can further write down ! ∞ X 1−b dZ(t, x) = (1 − bνǫ )bνǫ k Z(t, x − k) − Z(t, x) dt + Z(t, x)dM (t, x). b(1 − bνǫ ) k=0 Let {Ri }i≥1 be iid random variables such that 1 , supp(Ri ) = Z≥0 − 1 − bν ǫ and P Ri = k − 1 1 − bν ǫ = (1 − bνǫ )bνǫ k for k ∈ Z≥0(5.6) . Note that Ri satisfies E(Ri ) = 0. Under weak noise scaling, one can note that for small ǫ, √ √ −1 Ri + (νǫ λǫ ǫ)−1 follows Geo(λǫ νǫ ǫ). Let us denote a Poisson point process of rate b1−b−1 νǫ associated with Ri ’s by Γt . Its increments over the interval [t1 , t2 ] will be denoted as Γt1 ,t2 . Further, denote Ξ(t1 , t2 ) := Z≥0 − φǫ for t1 ≤ t2 where φǫ := (b−1 − 1)(t2 − t1 ) . (1 − bνǫ )2 (5.7) In this context, define the transition probability   Γt1 ,t2 X Γt1 ,t2 Ri + pǫ (t1 , t2 , ξ) := P  − φǫ = ξ  1 − bν ǫ i=1 for ξ ∈ Ξ(t1 , t2 ). Note that support space of the transition probability pǫ (t1 , t2 , .) is Ξ(t1 , t2 ). Moreover, one can see that {pǫ (t1 , t2 , ξ)}t1 ≤tP 2 ,ξ∈t1 ,t2 form a semi-group. We use the notation [pǫ (t1 , t2 ) ∗ f (t1 )] to denote the convolution ζ∈Ξ(t1 ,t2 ) pǫ (t1 , t2 , ξ − ζ)f (t1 , ζ) of any function imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 46 f with the semi-group. One can further note that pǫ (t, .) satisfy the following differential equation, ! ∞ X (1 − b) d νǫ νǫ k pǫ (t, x) = (1 − b )b pǫ (t, x − k) − pǫ (t, x) (5.8) dt b(1 − bνǫ ) k=0 pǫ (0, x) = δ0 (x). To put this into words, pǫ behaves as a discrete heat kernel. To that effect, we get the following simple form of a discrete SHE Z t Z(t, x) = [pǫ (t1 , t) ∗ Z(t1 )](x) + [pǫ (s, t) ∗ Z(s)dM (s)](x) where 0 ≤ t1 < t. (5.9) t1 Further, utilizing the particle dynamics of HL-PushTASEP, we derive the time derivative of the quadratic variation of the martingale M (t, x) defined in (5.2). Lemma 5.4. We have (b−1 −1)−2 Z(t, x1 )Z(t, x2 )dhM (t, x1 ), M (t, x2 )i (5.10) bνǫ \|x1 −x2 \| (1 − bνǫ )b−1 = Z(t, x1 ∧ x2 )dt Z(t, x1 ∧ x2 ) [pǫ (t, t + dt) ∗ Z(t)](x1 ∧ x2 ) − 1 − bν ǫ b−1 − 1 Proof. We know that dhM (t, x1 ), M (t, x2 )i equals to the rate at which both the Poisson process associated with M (t, x1 ) and M (t, x2 ) acknowledges a jump in the interval [t, t + dt]. Thus, it must be equals to the rate at which any particle to the left of κ(x1 ∧ x2 , t) hops across κ(x1 , t) and κ(x2 , t) at time t. Here, we denote min{x1 , x2 } and max{x1 , x2 } by x1 ∧ x2 and x1 ∨ x2 respectively. To this effect, one can write X ηt (l)bκ(x1 ∨x2 ,t)−l−Nκ(x1 ∨x2 ,t) (t)+Nl (t) dt. dhM (t, x1 ), M (t, x2 )i = (b−1 − 1)2 l≤κ(x1 ∧x2 ,t) Notice that for l ≤ κ(x1 ∧ x2 , t), we have bκ(x1 ∨x2 ,t)−l−Nκ(x1 ∨x2 ,t) (t)+Nl (t) = Z(t, x1 ∧x2 )Z −1 (x1 ∨x2 )bνǫ \|x1 −x2 \| bκ(x1 ∧x2 ,t)−l−Nκ(x1 ∧x2 ,t) (t)+Nl (t) . Therefore, we can say (b−1 − 1)−2 Z(t, x1 )Z(t, x2 )dhM (t, x1 ), M (t, x2 )i X ηt (l)bκ(x1 ,x2 ,l)l−Nκ(x1 ∧x2 ,t) (t)+Nl (t) dt. = Z 2 (t, x1 ∧ x2 )bνǫ \|x1 −x2 \| l≤κ(x1 ∧x2 ,t) One can recognize the following X ηt (l)bκ(x1 ,x2 ,l)l−Nκ(x1 ∧x2 ,t) (t)+Nl (t) = (b−1 − 1)−1 Ωx1 ∧x2 Z(t, x). l≤κ(x1 ∧x2 ,t) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 47 Thus, using Lemma 5.3, we can further say that (b−1 −1)−2 Z(t, x1 )Z(t, x2 )dhM (t, x1 ), M (t, x2 )i equals to ! ∞ νǫ X 1 − b Z(t, x1 ∧ x2 ) dt. (1 − bνǫ )bνǫ k Z(t, x1 ∧ x2 − k) − −1 (1−bνǫ )−1 bνǫ \|x1 −x2 \| Z(t, x1 ∧x2 ) b −1 k=1 Note that one can simplifies the term inside the bracket in the above expression to (1 − bνǫ )b−1 [pǫ (t, t + dt) ∗ Z(t)](x1 ∧ x2 ) − Z(t, x1 ∧ x2 )dt . b−1 − 1 This shows the claim. 5.2. Heat Kernel Estimates & Tightness Now, we aim at proving the tightness of the sequence {Zǫ }ǫ . For that, we need to specialize on certain properties the discrete heat kernel pǫ (see, (5.8) for definition). In the following proposition, we provides the necessary estimates for the heat kernel. Proposition 5.5. Given any T > 0, u ∈ R and v ∈ (0, 1], there exists C depending on T and u such that (i) X ζ∈Ξ(t1 ,t2 ) pǫ (t1 , t2 , ζ) exp(uǫ\|ζ\|) ≤ C, (5.11) (ii) X ζ∈Ξ(t1 ,t2 ) p(t1 , t2 , ζ)\|ζ\|v exp(uǫ\|ζ\|) ≤ Cǫ−v/2 (\|t2 − t1 \|)v/2 , (iii) √ pǫ (t1 , t2 , v) ≤ C ǫ min{1, (\|t2 − t1 \|)−1/2 }, (5.12) (iv) \|pǫ (t1 , t2 , ζ) − pǫ (t1 , t2 , ζ ′ )\| ≤ Cǫ(1+v)/2 \|ζ − ζ ′ \|v min{1, (t2 − t1 )−(1+v)/2 }, (5.13) for all t1 ≤ t2 ∈ (0, ǫ−1 T ] and ζ, ζ ′ ∈ Ξ(t1 , t2 ). Proof. (i) Recall that φǫ (see, (5.7)) is the mean of a random walk at time t of iid Geometric random variable with parameter (1 − bνǫ ) driven by a Poisson point process of rate (1 − bνǫ )−1 (b−1 − 1) over the interval [t1 , t2 ]. We have noted that the support of the transition probability pǫ (t1 , t2 , .) is given by Z≥0 − φǫ . Also define u′ := ǫ1/2 (νǫ λǫ )−1 u. For any ζ ∈ Ξ(t1 , t2 ) and n ∈ N, let us denote n + ζ + φǫ − 1 B(n, ζ) := . ζ + φǫ − 1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 48 For small ǫ, (1 − bνǫ )−1 (b−1 − 1) ≤ νǫ−1 . Hence, Poisson process associated with the semigroup {pǫ (t1 , t2 , .)} will be dominated by a Poisson process of rate νǫ−1 . For notational simplicity, we introduce the following ν −1 (t2 − t1 )n . m(νǫ , t2 − t1 , n) = exp −νǫ−1 (t2 − t1 ) n n P To that effect, we can bound the sum ζ∈Ξ(t1 ,t2 ) pǫ (t1 , t2 , ζ) exp(uǫ\|ζ\|) by ∞ X X n=0 ζ∈Ξ(t1 ,t2 ) m(νǫ , t2 − t1 , n)B(n, ζ)(1 − bνǫ )n bνǫ (ζ+φǫ ) exp(uǫ\|ζ\|) Using a simple fact euǫ\|ζ\| ≤ euǫζ + e−uǫζ and taking the sum over all ζ ∈ Ξ(t1 , t2 ), one can further bound the sum above by (A) + (B) where !n ∞ X (1 − bνǫ ) m(νǫ , t2 − t1 , n) (A) : = exp(−uǫφǫ ) −1 (1 − bνǫ (1−u′ ) ) n=0 1 − bν ǫ −1 −1 = exp νǫ (t2 − t1 ) − νǫ (t2 − t1 ) − uǫφǫ , (5.14) 1 − bνǫ (1−u′ ) n ∞ X (1 − bνǫ ) exp(uǫφǫ ) m(νǫ , t2 − t1 , n) (B) : = (1 − bνǫ (1+u′ ) ) n=0 1 − bν ǫ −1 −1 = exp νǫ (t2 − t1 ) − νǫ (t2 − t1 ) + uǫφǫ . (5.15) 1 − bνǫ (1+u′ ) Now recall b = e−ǫ 1/2 . Henceforth, one can say 1 − bν ǫ −1 1/2 + O(ǫ) ′ = 1 + uνǫ ǫ 1 − bνǫ (1−u ) and 1 − bν ǫ −1 1/2 + O(ǫ). ′ = 1 − uνǫ ǫ 1 − bνǫ (1+u ) Also, we have uǫφǫ = uǫ(1 − bνǫ )−2 (b−1 − 1)(t2 − t1 ) = u(νǫ−2 ǫ1/2 + O(ǫ))(t2 − t1 ). 1/2 1/2 This enforces νǫ−1 1−b1−b (t2 − t1 ) − νǫ−1 (t2 − t1 ) − uǫφǫ and νǫ−1 1−b1−b (t2 − t1 ) − (1−u′ )/2 (1+u′ )/2 −1 −1 νǫ (t2 − t1 ) + uǫφǫ to be bounded by some constant whenever t1 , t2 ∈ (0, ǫ T ]. Thus, one can note right side in the last line of (5.14) and (5.15) are bounded by some constant C which depends P only on u and T . (ii) We can bound ζ∈E(t1 ,t2 ) pǫ (t1 , t2 , ζ)\|ζ\|v exp(uǫ\|ζ\|) by ∞ X n=0 m(νǫ , t2 − t1 , n) X B(n, ζ)(1 − bνǫ )n bνǫ (ζ+φǫ ) \|ζ\|v exp(uǫ\|ζ\|) ≤ (A′ ) + (B ′ ) ζ∈Ξ(t1 ,t2 ) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 49 where (A′ ) := ∞ X n=0 ≤ (B ′ ) := ∞ X n=0 ≤ n=0 X B(n, ζ)(1 − bνǫ )n bνǫ (ζ+φǫ ) \|ζ\|v exp(uǫζ) ζ∈Ξ(t1 ,t2 ) m(νǫ , t2 − t1 , n) n=0 ∞ X ∞ X m(νǫ , t2 − t1 , n) m(νǫ , t2 − t1 , n) v n 1 − bνǫ (1−u′ ) X − φǫ 1 − bν ǫ 1 − bνǫ (1−u′ ) n exp(−uǫφǫ ), (5.16) B(n, ζ)(1 − b1/2 )n bνǫ (ζ+φǫ ) \|ζ\|v exp(−uǫζ) ζ∈Ξ(t1 ,t2 ) m(νǫ , t2 − t1 , n) v n 1 − bνǫ (1+u′ ) − φǫ 1 − bν ǫ 1 − bνǫ (1+u′ ) n exp(uǫφǫ ). (5.17) Notice g(x) = \|x\|v is a concave function for v ∈ (0, 1). We used Jensen’s inequality for concave function in (5.16) - (5.17) of the above computation. To this end, it can be noted that n (1 − bνǫ )n = + (±C + O(ǫ1/2 ))n ′ ′ 1 − bνǫ (1±u ) (1 − bνǫ (1±u ) )2 for some constant C. Also, φǫ = νǫ−1 (t2 −t1 )(1−bνǫ ) ′ (1−bνǫ (1±u ) )2 ± C ′ (t1 − t1 ) + O(ǫ) for some constant C ′ . For any v < 1, using Hölder’s inequality, we can write E (\|aX + b\|v ) ≤ (E((ax + b)2 )v/2 for any random variable X. Thus, we have ∞ X n=0 m(νǫ , t2 − t1 , n) n v 1 − bν ǫ 1 − bνǫ (1±u′ ) n exp(±uǫφǫ ) − φǫ 1 − bνǫ (1−u′ ) 1 − bν ǫ −1 −1 (t2 − t1 ) − νǫ (t2 − t1 ) ± uǫφǫ ≤ exp νǫ 1 − bνǫ (1±u′ ) v/2 −1 νǫ (t2 − t1 )(1 − bνǫ )2 2 + C1 (t2 − t1 ) . (5.18) × (1 − bνǫ (1±u′ ) )4 To this end, we have νǫ−1 (t2 − t1 )(1 − bνǫ )2 + C1 (t2 − t1 )2 . ǫ−1 (t2 − t2 ). ′ (1 − bνǫ (1±u ) )4 Also, one can note that that other part of the product on the right side of (5.18) appeared in (5.14) and (5.15). Thus, it is bounded by some constant which only depends on T . This completes the proof. (iii) Using the inversion formulae of the characteristic function, we can write down the following pǫ (t1 , t2 , ζ) ≤ ∞ X n=0 exp(−νǫ−1 (t2 ν −n (t2 − t1 )n 1 − t1 )) ǫ n! 2πi Z π −π \|φ(s, ǫ)\|n ds imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 50 where φ(s, ǫ) = ∞ X ξ=0 exp is ξ − (1 − bνǫ )−1 Now, using the estimate \|φ(s, ǫ)\| ≤ Z π −π n \|φ(s, ǫ)\| ds ≤ √ √ ǫ ǫ+s2 √ Z [−π,π] (1 − bνǫ )bνǫ ξ . (5.19) (obtained by Taylor series expansion), we get 2 √ 1 + \|s\|/ ǫ !n √ ds ≤ 2 ǫ Z ∞ 0 √ !n 2 ds. 1+s (5.20) Notice that when n = 1, transition probability is just (1 − b)bζ+φǫ which is anyway less that ǫ. But for n > 1, using (5.20) one can further say Z π √ 1 \|φ(s, ǫ)\|n ds ≤ 21+n/2 ǫ n−1 −π which implies " # ∞ √ −1 n X √ ( 2νǫ ) (t2 − t1 )n −1 −1 pǫ (t1 , t2 , ζ) ≤ C ǫ exp(−νǫ (t2 − t1 )) νǫ (t2 − t1 ) + 2 . n!(n − 1) n=2 (5.21) If X ∼ Poisson(τ ), then using Bernstein’s inequality, we have √ P \|X − τ \| ≥ t τ ≤ 2 exp(−Ct2 ) for some constant. This being said, one can deduce that E X1 ≤ C min{1, τ −1/2 }. One can now see second term in (5.21) has an upper bound min{1, (t2 − t1 )−1/2 }. Also, (t2 − t1 ) exp(−2(t2 − t1 )) can be bounded by (t2 − t1 )−1/2 upto some multiplicative constant when ǫ is small enough. Thus, we get √ pǫ (t1 , t2 , ζ) ≤ C ǫ min{1, (t2 − t1 )−1/2 }. (iv) Using uniform v-Holder continuity of the map x 7→ eix for x ∈ R and inversion formulae of the characteristic functions, one can write down Z π ∞ X (t2 − t1 )n 1 \|s(ζ − ζ ′ )\|v \|φ(s, ǫ)\|n ds. exp(−(t2 − t1 )) \|pǫ (t1 , t2 , ζ) − pǫ (t1 , t2 , ζ ′ )\| ≤ n! 2πi −π n=0 (5.22) where φ(., .) is defined in (5.19). Using the estimate of the characteristic function in the last part, we can have the following bound √ !n Z π Z ∞ 2 v n (1+v)/2 v ds (5.23) s \|φ(s, ǫ)\| ds ≤ 2ǫ s 1 + s −π 0 when n > 2. Note that (n−1)(1+s)−n is a density on R+ . It is easy to see that expectation under that density is 1/(n − 2) whenever n > 2. Thus, using Jensen’s inequality, one can imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 51 conclude that right side of (5.23) is bounded above by 21+n/2 ǫ(1+v)/2 ((n − 1)(n − 2)v )−1 when n exceeds 2. For n = 1, 2, ′ νǫ−1 exp(−νǫ−1 (t2 −t1 ))(t2 −t1 )n (1−b)\|bnζ −bnζ \| ≤ C min{1, (t2 −t1 )−(1+v)/2 }ǫ(1+v)/2 \|ζ−ζ ′ \|v for small enough ǫ. Once again, using the property of the confidence band of Poisson distribution, ∞ X n=2 exp(−νǫ−1 (t2 − t1 )) 2n/2 νǫ−n (t2 − t1 )n ≤ C ′ min{1, (t2 − t1 )−(1+v)/2 }, n!(n − 1)(n − 2)v one can bound right side in (5.22) by C\|ζ − ζ ′ \|v ǫ(1+v)/2 min{1, (t2 − t1 )−(1+v)/2 }. This completes the proof. For the notational convenience, we use Z(t, x) instead of Zǫ (t, x) for the next two lemmas. Let us note that one can modify the scaling appropriately to get necessary results for Zǫ (t, x). In a nutshell, the following results portray the key role in showing the tightness. One can also find similar result in the context of ASEP [BC95, Lemma 4.2, 4.3], higher spin exclusion process [CT15, Lemma 4.3]. Recall the decomposition of Z(t, x) in (5.9). We define Zmg (t1 , t2 , x) and Z∇,mg (t1 , t2 , x) as follows Zmg (t1 , t2 , x) : = ′ Z∇,mg (t1 , t2 , x, x ) : = Z t2 t1 Z t2 t1 [pǫ (s, t2 ) ∗ Z(s)dM (s)](x) [pǫ (s, t2 ) ∗ Z(s)dM (s)](x) − Z t2 t1 [pǫ (s, t2 ) ∗ Z(s)dM (s)](x′ ) In the next lemma, we probe into the higher order norms of the processes Zmg (t1 , t2 , x) and Z∇,mg (t1 , t2 , x). Lemma 5.6. For any k ≥ 1 and v ∈ (0, 1], there exists a large constant C which depends only on k such that for all t1 ≤ t2 ∈ N and ζ, ζ ′ ∈ Ξ(t1 , t2 ), kZmg (t1 ,t2 , ζ)k22k ≤ Cǫ 1/2 Z t2 min{1, (t2 − s)−1/2 } (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZ(s)k22k ] (ζ)ds t1 ′ 2 kZ∇,mg (t1 , t2 , ζ, ζ )k2k (1+v)/2 ′ v Z t2 ≤ Cǫ \|ζ − ζ \| min{1, (t2 − s)−(1+v)/2 } t1 + (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZ(s)k22k ] (ζ ′ ) ds (5.24) (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZ(s)k22k ] (ζ) (5.25) where δ0 is the probability mass function of the distribution which is degenerate at 0 and Pǫ denotes probability mass function of the random variables Ri (see, (5.6) for definition). Proof. Fix any t ∈ (t1 , t2 ). Using Burkholder-Davis-Gundy’s inequality, we have \|Zmg (t1 , t, ζ)\|22k ≤ k[Zmg (t1 , ., ζ), Zmg (t1 , ., ζ)]t kk , (5.26) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 52 and \|Z∇,mg (t1 , t, ζ)\|22k ≤ k[Z∇,mg (t1 , ., ζ), Z∇,mg (t1 , ., ζ)]t kk , (5.27) where [., .] denotes the quadratic variation. Let us denote set all of all time points at which particles hop across ζ, ζ ′ in the interval (s1 , s2 ] by S(s1 ,s2 ] (ζ, ζ ′ ). Recall once again the form of the Martingale in (5.2) and the function κ(., .) in (5.1). Define X ηs (y)Z −1 (ξ1 ∨ ξ2 , s)bνǫ κ(ξ1 ∨ξ2 ,s)+Ny (s)−y Q(ξ1 , ξ2 , y, s) = (b−1 − 1)2 y≤κ(ξ1 ∧ξ2 ,s) which controls the contribution coming out of the Martingale M (t, x) in the total variation terms on right side of (5.26) and (5.27). Using interdependence of the Poisson processes {Pl }−∞<l<∞ , one can say Q(ξ1 , ξ2 , y, s) captures the hopping effect of particles at any position y across κ(ξ1 , s) and κ(ξ2 , s) at time s. Thus, we have X X pξǫ1 ,ξ2 (t1 , t, ζ)Z ξ1 ,ξ2 (s)Q(ξ1 , ξ2 , y, s) (5.28) [Zmg (t1 , ., ζ), Zmg (t1 , ., ζ)]t = ξ1 ,ξ2 s∈S(t1 ,t] (ξ1 ,ξ2 ) and [Z∇,mg (t1 , ., ζ, ζ ′ ), Z∇,mg (t1 , ., ζ, ζ ′ )]t = X X ξ1 ,ξ2 s∈S(t1 ,t] (ξ1 ,ξ2 ) 1 ,ξ2 pξ∇,ǫ (t1 , t, ζ, ζ ′ )Z ξ1 ,ξ2 (s)Q(ξ1 , ξ2 , y, s) (5.29) where Z ξ1 ,ξ2 (s) = Z(ξ1 , s)Z(ξ2 , s), pξǫ1 ,ξ2 (s, t2 , ζ) = pǫ (s, t2 , ζ − ξ1 )pǫ (s, t2 , ζ − ξ2 ) and 1 ,ξ2 pξ∇,ǫ (s, t, ζ) = (pǫ (s, t2 , ζ − ξ1 ) − pǫ (s, t2 , ζ ′ − ξ1 ))(pǫ (s, t2 , ζ − ξ2 ) − pǫ (s, t2 , ζ ′ − ξ2 )). For the notational simplicity, for any two point m > n on the lattice Z≥0 and s ∈ R+ h(m, n, s) := m − n − Nn (s) + Nm (s). On simplifying the summands in (5.28) and (5.29), one can get X ηs (y)bh(y,κ(ξ1 ∧ξ2 ,s),s) bνǫ \|ξ1 −ξ2 \| Z(ξ1 , s)Z(ξ, s)Q(ξ1 , ξ2 , y, s) = ǫZ(ξ1 ∧ ξ2 , s)2 y≤κ(ξ1 ∧ξ2 ,s) where h(y, κ(ξ1 ∧ ξ2 , s), s) counts the number of holes in between y and κ(ξ1 ∧ ξ2 , s) at time √ s. Notice that pξǫ1 ,ξ2 (s, t2 , ζ) ≤ C ǫ max{1, (t2 − s)−1/2 }p (s, t , ζ − ξ ∧ ξ2 ) thanks to the √ P ǫ νǫ 2\|ξ1 −ξ2 \| 1 is constant. Thus, we Proposition 5.5. For any fixed ξ1 , one can note ǫ ξ2 ≥ξ1 b bound right side of (5.28) by X X X ηs (y)bh(y,κ(ξ,s),s) . Cǫ min{1, (t2 − s)−1/2 } pǫ (s, t2 , ζ − ξ)Z(ξ, s)2 ξ y≤κ(ξ,s) s∈S(t1 ,t] (ζ) Divide the interval (t1 , t] into disjoint intervals ∩m i=1 (si , si+1 ] where each of them is of length less than or equals to 1. Next, recall that X dhM (s, ξ), M (s, ξ)i = ηs (y)bh(y,κ(ξ,s),s) dt. (5.30) y≤κ(ξ,s) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 53 Using Lemma 5.10, one can bound Z 2 (s, ξ) times the right side of (5.30) by 1 (1 − bνǫ )b−1 Z(s, ξ) [Pǫ ∗ Z(s)](ξ) − Z(s, ξ)dt . 1 − bν ǫ b−1 − 1 √ Thus, recalling that b = e−λǫ ǫ , we get the following bound √ [Pǫ ∗ kZ(s)k22k ](ξ) + kZ(s, ξ)k22k (νǫ λǫ ǫ)−1 + νǫ kZ(s, ξ)k22k 2 P on the k-th norm of Z(ξ, s)2 y≤κ(ξ,s) ηs (y)bh(y,κ(ξ,s),s) . Also note that, √ max Z(s, x) ≤ exp 2 ǫλǫ N(si ,si+1 ] (x) + 4(νǫ )−1 Z(si , x) s∈(si ,si+1 ] and min s∈(si ,si+1 ] √ Z(s, x) ≥ exp −2 ǫλǫ N(si ,si+1 ] (x) − 4(νǫ )−1 max s∈(si ,si+1 ] Z(si , x) where N(si ,si+1] (x) is independent of Z(si , x) and dominated by a Poisson point process of P rate y≤κ(x,si+1 ) ηsi+1 (y)bh(y,κ(x,si+1 ),si+1 ) . Once again, using the same analysis as above one can show that the rate is bounded above by Cǫ−1/2 . Thus, we have √ E exp 2 ǫkλǫ N(si ,si+1 ] (x) + 4k(νǫ )−1 ≤ C ′ where C ′ is a large constant which only depends on k. To this end, we can bound right side of (5.26) by m X √ X min{1, (t2 − s)−1/2 } pǫ (s, t1 , ζ − ξ) C ǫ ξ k=1 inf s∈(si ,si+1 ] [Pǫ ∗ kZ(s)k22k ](ξ) + kZ(s, ξ)k22k . Recall that the length of each of the intervals (siR, si+1 ] is less than Ror equals to 1. Applying the bound inf s∈(si ,si+1 ] kZ(ξ, s)k2k ≤ (si −sk+1 )−1 kZ(ξ, s)k2k ds = kZ(ξ, s)k2k ds and using the semi-group property of pǫ , we get Z t2 kZmg (t1 , t2 , ζ)k22k ≤ Cǫ1/2 min{1, (t2 − s)−1/2 } (δ0 + Pǫ ) ∗ [pǫ (s, t2 ) ∗ kZ(s)k22k ] (ζ) ds t1 where δ0 is the probability mass function which puts all its mass in 0. Note that we also had made of the fact that under convolution operation, pǫ (t1 , t, .) and Pǫ commute with each other. This holds because pǫ is essentially the semi-group of the random walk of the iid random variables coming from the distribution Pǫ . 1 ,ξ2 (t1 , s, ζ, ζ ′ ) in (5.29) by In the case of Z∇,mg , using Proposition 5.13, we can bound pξ∇,ǫ Cǫ(1+v)/2 min{1, (t2 − s)−(1+v)/2 }\|ζ − ζ ′ \|v pǫ (t1 , s, ζ − ξ1 ∧ ξ2 ) + pǫ (t1 , s, ζ ′ − ξ1 ∧ ξ2 ) . Rest of the terms in (5.28) can be controlled exactly in the same fashion as we have done for bounding kZmg (t1 , t2 , ζ)2 k2k . Thus, one can write down kZ∇,mg (t1 , t2 , ζ)k22k Z t2 ≤ Cǫ(1+v)/2 \|ζ − ζ ′ \|v min{1, (t2 − s)−(1+v)/2 } (δ0 + Pǫ ) ∗ [pǫ (s, t2 ) ∗ kZ(s)k22k ] (ζ) t1 + (δ0 + Pǫ ) ∗ [pǫ (s, t2 ) ∗ kZ(s)k22k ] (ζ ′ ) ds. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 54 In the next result, we present a chaos-series type bound for Z(t, ζ). Lemma 5.7. For any k ≥ 1, t > 0 and ζ ∈ Ξ(t), (5.31) kZ(t, ζ)k2 ≤ 2([pǫ (0, t) ∗ kZ(0)k22k ](ζ)) ∞ Z X hǫ (s1 , s2 ) . . . hǫ (sn , sn+1 )Ψ(sn+1 , ζ)ds1 ds2 . . . dsn+1 +2 n=1 ~s∈∆n+1 (t) where hǫ (si , si+1 ) = Cǫ1/2 min{1, (si − s)−1/2 }1si ≥si+1 , Ψn (sn+1 , ζ) = (δ0 + Pǫ )n ∗ [pǫ (0, sn+1 ) ∗ kZ 2 (0)k2k ] (ζ) and ∆n (t) = {~s ∈ (R≥0 )n \|t ≥ s1 ≥ . . . ≥ sn+1 ≥ 0}. Here, Pǫ denotes probability mass function of Ri (see, (5.6) for definition). Note that here C is a finite constant depended only on k. Proof. Recall the decomposition of Z(t, x) in (5.9). Using one simple inequality, \|x + y\|2 ≤ 2(x2 + y 2 ), we have kZǫ (t, ζ)k22k 2 ≤ 2(k[pǫ (0, t) ∗ Zǫ (0)]k2k ) + 2 Z 2 t 0 [pǫ (s, t) ∗ Zǫ (s)dM (s)](ζ)ds . (5.32) 2k We use Jensen’s inequality to bound the first term on the right side of (5.32) by 2([pǫ (0, t) ∗ kZǫ (0)k22k ]). Moreover, one can also bound second term in (5.32) using (5.24). Thus, we get kZǫ (t, ζ)k22k ≤ 2([pǫ (0, t) ∗ kZǫ (0)k22k ]) (5.33) Z t + Cǫ1/2 min{1, (t − s)−1/2 } (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZǫ (s)k22k ] (ζ)ds. 0 To this end, using successive recursion on kZ(s, .)k22k in RHS of (5.33), one can get an asymptotic expansion. Furthermore, semi-group property [pǫ (t1 , s) ∗ pǫ (s, t2 )](.) = pǫ (t1 , t2 , .) and the fact that Pǫ commutes with the semi-group {pǫ } helps in concluding the final step of the inequality. Our next goal is to prove Proposition 5.1 with some refined estimates of the norms of Zǫ (t, x). Below, we illustrate the idea in details. Proof of Proposition 5.1: First, we prove the following moment estimates: kZǫ (t, x)k2k ≤ Ceǫτ \|x\| , kZǫ (t, x) − Zǫ (t, x′ )k2k ′ kZǫ (t, x) − Zǫ (t , x)k2k v ′ ≤ C ǫ\|x − x′ \| eτ ǫ(\|x\|+\|x \|) , α ≤ C ǫ\|t − t′ \| eτ ǫ\|x\| . (5.34) (5.35) (5.36) for some α := v/2 ∧ 1/4, t, t′ ∈ (t1 , ǫ−1 T ], x, x′ ∈ Ξ(t) where T ∈ (0, ∞) is fixed apriori and v is exactly same as in (2.13). Once those are proved, we can make use of Kolmogorov-Chentsov criteria to conclude the tightness of the sequence {Zǫ }ǫ . imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 55 Proof of (5.34): We have kZ(0, x)k2k ≤ eτ ǫ\|x\| from (2.12). One can further bound \|ξ\| by \|x − ξ\| + \|x\|. Thus, we have X pǫ (0, t, x − ξ) exp(2τ ǫ\|x − ξ\|) exp(2τ ǫ\|x\|) ≤ C exp(2τ ǫ\|x\|). [pǫ (0, t) ∗ kZ(0)k22k ](x) ≤ ξ∈Ξ(t) thanks to (5.11). Consequently, first term in RHS of (5.31) is bounded above by C exp(2τ ǫ\|x\|). Using the same arguments, one can also bound [pǫ (0, sn+1 ) ∗ kZ(0)k22k ](x) of the second term in (5.31) by C1 exp(2τ ǫ\|x\|). Furthermore, we know x X ξ=−∞ ∞ √ X √ √ Pǫ (x − ξ) exp(2τ ǫ\|ξ\|) ≤ exp(2τ ǫ\|x\|) ǫ exp(−k ǫ(νǫ − 2 ǫτ )) ≤ C2 exp(2τ ǫ\|x\|) k=0 for some constant C2 when ǫ is small enough. Thus, we can conclude that [(δ0 + Pǫ )n ∗ [pǫ (0, sn+1 ) ∗ Z(s)]](x) is bounded by C1 C2n exp(2τ ǫ\|x\|) for all small ǫ. Interestingly, we can now easily compute left over integral of the second term in the RHS of (5.31). It turns out that n/2 Z n Y (CΓ(1/2))2 ǫt hǫ (si , si+1 )ds1 ds2 . . . dsn+1 ≤ . (5.37) Γ(n/2) ~ s∈∆n+1 (t) i=1 This makes the sum in (5.31) finite and thus helps us to conclude (5.34). Proof of (5.35): We can decompose Z(t, x) − Z(t, x′ ) as X Z(t, x) − Z(t, x′ ) = pǫ (0, t, ξ) Z(t1 , x − ξ) − Z(t1 , x′ − ξ) ξ∈t t + Z X 0 ξ∈Ξ(s,t) pǫ (s, t, x − ξ) − pǫ (s, t, x′ − ξ) Z(s, ξ)dM (s, ξ) (5.38) Notice that we have the bound on pǫ (0, t, ζ) − pǫ (0, t, ζ ′ ) from Lemma 5.5. One can use (2.13) to bound the first term on the RHS of (5.38) as noted down below. k[p(0, t) ∗ Z(0)](x) − [p(0, t) ∗ Z(0)](x′ )k2k X ≤ pǫ (0, t, ξ)kZ(0, x − ξ) − Z(0, x′ − ξ)k2k ξ∈Ξ(0,t) ≤ X ξ∈Ξ(0,t) v pǫ (0, t, ξ) ǫ\|x − x′ \| exp τ ǫ(\|x − ξ\| + \|x′ − ξ\|) v ≤ ǫ\|x − x′ \| exp τ ǫ(\|x\| + \|x′ \|) X pǫ (0, t, ξ) exp(2τ ǫ\|ξ\|) 2ξ∈Ξ(0,t) v ≤ C ǫ\|x − x \| exp τ ǫ(\|x\| + \|x′ \|) . ′ One can identify the term in the last line of (5.38) with Z∇,mg (t1 , t, x, x′ ). It follows from (5.34) that kZ(s, ξ)k2k ≤ C exp(τ ǫ\|ξ\|) for all ξ ∈ Ξ(t1 , s) and 0 ≤ s ≤ ǫ−1 T where C only depends on T . Using the bound \|x − ξ\| + \|x\| for \|ξ\| in exp(τ ǫ\|ξ\|), one can bound [pǫ (s, t) ∗ kZ(s)k22k ](x) above by C exp(2τ ǫ\|x\|). Similarly, C exp(2τ ǫ\|x′ \|) is also valid as an upper bound imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 56 to [pǫ (s, t) ∗ kZ(s)k22k ](x′ ). So, the integrand in (5.24) can be bounded above by min{1, (t2 − s)−(1+v)/2 }1s≤t2 C exp (2τ ǫ(\|x\| + \|x′ \|)) where t2 is equal to t and s ∈ (0, t). Consequently, we can bound the kZ∇,mg (0, t, x, x′ )k2k by C(ǫ\|x − x′ \|)v (ǫt)(1−v)/2 exp (τ ǫ(\|x\| + \|x′ \|)). This proves (5.35) thanks to the fact t ∈ (0, ǫ−1 T ]. Proof of (5.36): Assume t′ < t. We can write down the following: X X Z(t, x) − Z(t′ , x) = p(0, t, ξ)Z(t1 , x − ξ) − p(0, t′ , ξ ′ )Z(0, x − ξ ′ ) ξ ′ ∈Ξ(t′ ) ξ∈Ξ(0,t) + Z t t′ [pǫ (s, t) ∗ Z(s)dM (s)](x). (5.39) Notice that the first term in the last line of (5.39) is exactly Zmg (t′ , t, x). Furthermore, using semigroup property of pǫ , one can say X X X pǫ (t1 , t, ξ)Z(0, x − ξ) = pǫ (0, t′ , ξ ′ )pǫ (t′ , t, ξ − ξ ′ )Z(0, x − ξ) ξ∈Ξ(t) ξ ′ ∈Ξ(t′ ) ξ∈Ξ(0,t) X = pǫ (0, t′ , ξ ′ ) ξ ′ ∈Ξ(0,t′ ) X ζ∈Ξ(t′ ,t) pǫ (t′ , t, ζ)Z(0, x − ξ ′ − ζ). This helps us to analyse the first term in RHS of (5.39) in the following way. X ξ∈Ξ(t) p(0, t, ξ) Z(t1 , x − ξ) − = X ξ ′ ∈Ξ(t′ ) ≤C X ξ ′ ∈Ξ(t′ )  pǫ (0, t′ , ξ ′ )  X p(0, t′ , ξ ′ )Z(0, x − ξ ′ ) 2k ζ∈Ξ(t′ ,t) pǫ (0, t′ , ξ ′ ) ξ ′ ∈Ξ(t′ ) X X pǫ (t′ , t, ζ) Z(0, x − ξ ′ − ζ) − Z(0, x − ξ ′ ) ζ∈Ξ(t′ ,t) ≤ C(ǫ\|t − t′ \|)v/2 exp(2τ ǫ\|x\|). ǫ\|ζ\|v pǫ (t′ , t, ζ) exp τ ǫ(\|ζ\| + 2\|x\| + 2\|ξ ′ \|) 2k   Note that we have used (5.13) in Proposition 5.5 in the last inequality. One can bound kZmg (t′ , t, x)k2k using (5.24). Moreover, integrand in the RHS of (5.24) is bounded above by √ C ǫ1t′ ≤s≤t min{1, (t′ − s)−1/2 } exp(2τ ǫ\|x\|) thanks to (5.34). This enforces kZmg (t′ , t, x)k22k to be bounded by C(ǫ\|t − t′ \|)1/2 exp(2τ ǫ\|x\|). Hence, the claim follows. Proof of Proposition 2.12: First, we show (2.14). In Lemma 5.7, we multiply both side with ǫ−2 (1 − exp(−λǫ νǫ ))2 . Consequently, we get 2 kZ̃(t, ζ)k22k ≤2 [pǫ (0, t) ∗ Z̃(0)](ζ) ∞ Z X +2 h(s1 , s2 ) . . . h(sn , sn+1 )Ψ̃n (sn+1 , ζ)ds1 . . . dsn+1 n=1 ~s∈∆n+1 (t) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 57 where i h Ψ̃n (sn+1 , ζ) := (δ0 + Pǫ )n ∗ ([pǫ (0, sn+1 ) ∗ Z̃(0)])2 (ζ). Note that Z̃(0, .) is now deterministic. Using the relation ǫ bound on pǫ (0, t, .) from (5.12), we further have P ζ∈Ξ(0) Z̃(0) (5.40) = 1 and the upper [pǫ (0, t) ∗ Z̃(0)](x) ≤ C min{ǫ−1/2 , (ǫt)−1/2 }. Similarly, the term [pǫ (0, sn+1 ) ∗ Z̃(0)] inside Ψ̃n (sn+1 , ζ) in (5.40) is also bounded above by min{ǫ−1/2 , (ǫsn+1 )−1/2 }. To that effect, one can bound the rest of the integral as Z −1/2 h(s1 , s2 ) . . . h(sn , sn+1 )(ǫsn+1 ) ~s∈∆n+1 (t) ds1 . . . dsn+1 (CΓ(1/2))2 ǫt ≤ Γ(n/2) n/2 (5.41) in the same way as in (5.37). Thus, for all t ∈ (0, ǫ−1 T ], summing (5.41) over n ∈ Z+ , we get kZ̃(t, x)k22k ≤ C([pǫ (0, t) ∗ Z̃ǫ (0)](ζ))2 + exp((ǫt)1/2 ) ≤ C ′ min{ǫ−1 , (ǫt)−1 } (5.42) where the constants C, C ′ > 0 depend only on T . Thus the claim follows. Now, we turn to show (2.15). To this end, multiplying Z(t, ζ)−Z(t, ζ ′ ) by ǫ−1 (1−exp(−λǫ νǫ )), we get 2  X \|pǫ (0, t, ζ − ξ) − pǫ (0, t, ζ − ξ)\|Z̃(0, ξ) kZ̃(t, ζ) − Z̃(t, ζ ′ )k22k ≤ 2  ξ∈Ξ(0) + ckZ̃∇,mg (0, t, ζ1 , ζ1′ )k22k for some constant c. Using the bound on \|pǫ (0, t, ζ − ξ) − pǫ (0, t, ζ − ξ)\| from (5.13) and the P relation ǫ ζ∈Ξ(0) Z̃(0, ζ) = 1, one can get the following X ξ∈Ξ(0) \|pǫ (0, t, ζ − ξ) − pǫ (0, t, ζ − ξ)\|Z̃(0, ξ) ≤ C(ǫ\|ζ − ζ ′ \|)v min{ǫ− 1+v 2 , (ǫt)− 1+v 2 }. Moreover, substituting 2v in place of v and taking (t1 , t2 ) = (0, t) in (5.25), we get Z h i 1+2v ′ 2 2v kZ̃∇,mg (0, t, ζ, ζ )k2k ≤ǫ 2 \|ζ − ζ\| min{1, (t − s)−(1+2v)/2 } Ψ̃(s, t, ζ) + Ψ̃(s, t, ζ ′ ) ds (5.43) i h where Ψ̃(s, t, ζ) := (δ0 + Pǫ ) ∗ [pǫ (s, t) ∗ kZ̃(s)k22k ] (ζ). One can further bound the integrand on the right side in (5.43) using bound on kZ̃(s, ζ)k22k from (5.42). Henceforth, this implies kZ̃∇,mg (0, t, ζ, ζ ′ )k22k ≤ C(ǫ\|ζ − ζ ′ \|)2v (ǫt)−v+(1/2) . (5.44) But, we know ǫt ≤ T for all t ∈ (0, ǫ−1 T ]. Thus, we can improve the bound on the right side of (5.44) to C ′ (ǫ\|ζ − ζ\|)2v (ǫt)−(1+v) where C ′ depends only on T . imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 58 5.3. Proof of Proposition 5.2: The Martingale Problem Before going into the details of the proof, we present a brief exposure on the martingale problem from [BG97]. Definition 5.8. Let Z be a C([0, ∞) × (−∞, ∞)) valued process such that for any given T > 0, there exists u < ∞ such that sup sup e−u\|x\| E(Z(t, x)) < ∞. t∈[0,T ] x∈R R For such Z and ψ ∈ Cc∞ (R), let hZ(t), ψi := R Z(t, x)ψ(x)dx. We say Z solves the martingale problem with initial condition Z ic ∈ C(R), if Z(0, .) = Z ic (.) in distribution and Z 1 t ∂2ψ t 7→ Nψ (t) := hZ(t), ψi − hZ(0), ψi − Z(τ ), 2 dτ 2 0 ∂x Z t Z 2 (τ ), ψ 2 dτ t 7→ Nψ′ (t) := (Nψ (t))2 − 0 are local martingales for any ψ ∈ Cc∞ (R). It has been stated in [BG97], that for any initial condition Z ic satisfying kZ ic (x)k2k ≤ Cea\|x\| for some a > 0, the solution of the martingale problem stated above coincides with the solution of the SHE with initial condition Z ic . In our case, if we can show any limit point of {Zǫ }ǫ started from Z ic solves the martingale problem, then it follows from [BG97] that {Zǫ }ǫ converges to a unique process which solves SHE with initial condition being Z ic . For solving the martingale problem, we have to have a discrete analogue of hZ(t), ψi for any Zǫ . Note that we can take X hZ(s), ψiǫ = ǫ Z(t, ξ)ψ(ǫξ) ξ∈Z as the discrete analogue for hZǫ (t), ψi. Let us divide (0, ∞) into a number of disjoint subintervals ∪∞ i=1 (si , si+1 ) where 0 = s1 < s2 < . . . and each has length ǫ. Let us assume that sm ≤ t < sm+1 . On the basis of decomposition of Z(t, x) in (5.9), one can say the following:   m−1 X X X  hZ(t), ψiǫ − hZ(0), ψiǫ = ǫ pǫ (si , si+1 , ξ − ζ)ψ(ǫξ) − ψ(ǫζ) Z(si , ζ) i=1 ζ∈Ξ(si ) X +ǫ ζ∈Ξ(si ) +ǫ  Z t ξ∈Ξ(si ,si+1 )+ζ X  ξ∈Ξ(sm ,t)+ζ m−1 X Z si+1 i=1 +ǫ  si X sm ζ∈Ξ(s) X ζ∈Ξ(s)    pǫ (sm , t, ξ − ζ)ψ(ǫξ) − ψ(ǫζ) Z(sm , ζ)  X X ξ∈Ξ(s,si+1 )+ζ ξ∈Ξ(s,t)+ζ  pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) Z(s, ζ)dM (s, ζ)  pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) Z(s, ζ)dM (s, ζ). imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 59 Let us define Rǫ (t) as Rǫ (t) :=ǫ m−1 X X i=1 ζ∈Ξ(si ) X +ǫ ζ∈Ξ(si ) Also, define Nǫ (t) :=ǫ m Z X i=1 +ǫ Z t    si+1 si  X ξ∈Ξ(si ,si+1 )+ζ X ξ∈Ξ(sm ,t)+ζ X ζ∈Ξ(s) X sm ζ∈Ξ(s)     pǫ (si , si+1 , ξ − ζ)ψ(ǫξ) − ψ(ǫζ) Z(s, ζ)ds  pǫ (sm , t, ξ − ζ)ψ(ǫξ) − ψ(ǫζ) Z(sm , ζ). X ξ∈Ξ(s,si+1 ) X  ξ∈Ξ(s,t)+ζ  pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) Z(s, ζ)dM (s, ζ)  pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) Z(s, ζ)dM (s, ζ). Clearly, Nǫ (t) is a local martingale. Let us denote the quadratic variation of Nǫ (t) by N̂ǫ (t). So, for showing that indeed any limit point of {Zǫ }ǫ solves the martingale problem, it is enough to show the following: E hZ(ǫ−1 t), ψiǫ − hZǫ (t), ψi → 0, Z t ∂2ψ −1 E Rǫ (ǫ t) − Zǫ (s), 2 ds → 0, ∂x 0 Z t E N̂ǫ (ǫ−1 t) − hZǫ2 (s), ψ 2 ids → 0. (5.45) (5.46) (5.47) 0 To prove (5.45) - (5.47), we need the following lemma. Proof of (5.45): Here, we have to show that Z Zǫ (t, x)ψ(x)dx hZǫ (t), ψi = R and hZ(ǫ−1 t), ψiǫ are asymptotically same in L1 . Notice that for x such that \|x − ζ\| ≤ 1 where ζ ∈ Ξ(ǫ−1 t), using smoothness of ψ and moment estimates from (5.34)-(5.36), one can say that the sequences ǫ−v Z(ǫ−1 t, ζ)ψ(ǫζ) − Z(ǫ−1 t, x)ψ(ǫx) are L2k bounded for all k ∈ N. This forces hZ(ǫ−1 t), ψiǫ − hZǫ (t), ψi to go to 0 in L1 as ǫ → 0. Proof of (5.46): Recall that subintervals {[si , si+1 )}∞ i=1 form a partition of (0, ∞) such that length of each of the interval is ǫ. Assume here that ǫsm ≤ t < ǫsm+1 . Using the smoothness of ψ, one can get X X pǫ (si ,si+1 , ξ − ζ)ψ(ǫξ) − ψ(ǫζ) = ǫψ ′ (ǫζ) pǫ (si , si+1 , ξ − ζ)(ξ − ζ) ξ∈Ξ(si ,si+1 )+ζ ξ∈Ξ(si ,si+1 ) + X ǫ2 1 2 ξ∈Ξ(si ,si+1 ) ∂2ψ ∂x (ǫζ) + o(ǫ)B(ζ) pǫ (si , si+1 , ξ − ζ)(ξ − ζ)2 2 (5.48) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 60 P where B belongs to L2 . Note that ξ∈Ξ(s,t) pǫ (si , si+1 , ξ − ζ)(ξ − ζ) = 0. This goes back to P the fact that mean under pǫ (si , si+1 , .) is 0. Moreover, ξ∈Ξ(si ,si+1 ) pǫ (si , si+1 , ξ − ζ)(ξ − ζ)2 = −1 1 −1 + O(ǫ))(ν λ )−2 ǫ−1 . Recall from (si+1 − si )σǫ2 = (si+1 − si ) b1−b−1 νǫ (1−bνǫ )2 = (si+1 − si )(νǫ ǫ ǫ −3/2 Definition 2.7 that λǫ = νǫ Rǫ1 (ǫ−1 t) = ǫ . To this effect, we have 2 m X ∂ ψ 1X (si+1 − si )ǫ Z(si , ζ) (ǫζ) + o(ǫ) 2 ∂x2 i=1 (5.49) ζ∈Ξ(si ) tǫ−1 Z tǫ−1 1 ∂2ψ = ǫ hZ(s), Biǫ ds + D. Z(s), 2 ds + o(ǫ) 2 0 ∂x ǫ 0 E RtD 2 Simple change of variable reduces first term in (5.49) to 2−1 0 Z(s), ∂∂xψ2 ds at a cost of Z ǫ an error D. It follows from the moment estimates in (5.34)-(5.36) that D in (5.49) indeed converges to 0 in L1 as ǫ → 0. Further, we know the fact that B is in L2 . Thus, second term also converegs to L1 . Thereafter, one can R t 0 in −1 R tuse DCT along with (5.45) to conclude 1 L distance between 0 hZ(ǫ s), ∂ 2 ψ/∂x2 iǫ ds and 0 hZǫ (s), ∂ 2 ψ/∂x2 ids converges to 0, thus shows the claim. Proof of (5.47): One can write N̂ (ǫ −1 t) = ǫ 2 + ǫ2 m−1 X Z si+1 i=1 t Z si X X ψ ζ1 ,ζ2 (s, si+1 )Z(s, ζ1 )Z(s, ζ2 )dhM (s, ζ1 ), M (s, ζ2 )i ζ1 ,ζ2 ψ ζ1 ,ζ2 (s, t)Z(s, ζ1 )Z(s, ζ2 )dhM (s, ζ1 ), M (s, ζ2 )i sm ζ ,ζ 1 2 where  ψ ζ1 ,ζ2 (s1 , s2 ) =  X ξ∈Ξ(s1 ,s2 ) Recall from (5.10) that  pǫ (s1 , s2 , ξ − ζ1 )ψ(ǫξ)  X ξ∈Ξ(s1 ,s2 )  pǫ (s1 , s2 , ξ − ζ2 )ψ(ǫξ) . (b−1 − 1)−2 Z(t, ζ1 )Z(t, ζ2 )dhM (t, ζ1 ), M (t, ζ2 )i bνǫ \|ζ1 −ζ2 \| (1 − bνǫ )b−1 = Z(t, ζ1 ∧ ζ2 ) [pǫ (t, t + dt) ∗ Z(t)](ζ1 ∧ ζ2 ) − Z(t, ζ1 ∧ ζ2 )dt . 1 − bν ǫ b−1 − 1 Furthermore, one can write ∞ X [pǫ (t, t + dt) ∗ Z(t)](x) − Z(t, x)dt = (1 − bνǫ )bkνǫ (Z(t, x − k) − Z(t, x)) dt. k=0 Using (5.35), we have ∞ ∞ X X √ √ νǫ kνǫ ǫ\|ǫk\|v exp −λǫ νǫ k ǫ + kǫ = O(ǫv/2 ) (1 − b )b (Z(t, x − k) − Z(t, x)) ≤ k=0 k=0 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 61 for any t > 0. Moreover, one can deduce from (5.48) that X ξ∈Ξ(s,si+1 ) pǫ (s, si+1 , ξ − ζ)ψ(ǫξ) = ψ(ǫζ) + ǫ 1 ∂2ψ (ǫζ)(si+1 − s) + o(ǫ2 )B, 2 ∂x2 where B is in L2 (R). Thus, we get N̂ (ǫ−1 t) equal to m−1 X Z si+1 i=1 si ǫ 2 X ψ ζ1 ,ζ2 (s, si+1 )b νǫ \|ζ1 −ζ2 \| (b −1 ζ1 ,ζ2 − 1) bνǫ −1 − 1 2 v/2 Z (s, ζ1 ∧ ζ2 ) + O(ǫ ) ds. 1 − bν ǫ where ψ ζ1 ,ζ2 (s, si+1 ) = ψ(ǫζ1 )ψ(ǫζ2 ) + O(ǫ2 )B1 . Further, the random variable B1 belongs to L2 . Note that νǫ > 0 (see Definition 2.7) is chosen in way so that it satisfies 1 − νǫ = νǫ2 . For this specific choice of νǫ , we have X ζ2 ≥ζ1 bνǫ \|ζ1 −ζ2 \| (b−1 − 1) √ bνǫ −1 − 1 = 1 + O( ǫ). 1 − bν ǫ Rt Thereafter, using continuity of ψ, one can say further N̂ǫ (ǫ−1 t) − 0 hZ 2 (ǫ−1 ), ψ 2 iǫ ds converges Rt Rt to 0 in L1 as ǫ → 0. To this end, (5.45) shows 0 hZ 2 (ǫ−1 s), ψ 2 iǫ ds and 0 hZǫ2 (s), ψ 2 ids both has same L1 limit and hence, completes the proof. Appendix A: Fredholm Determinant We start with the definition of Fredholm determinant. Definition A.1. Let K(x, y) be the meromorphic function of two complex variable, invariably referred as kernel in the literature. Let Γ ⊂ C be a curve. Assume that K has no singularities on Γ × Γ. Then the Fredholm determinant det(I + K) of the kernel K is defined as the sum of the series of the complex integrals Z Z X 1 det(I + K) = (A.1) . . . det (K(zi , zj ))ni,j=1 dz1 , . . . dzn . n!(2πi)n Γ Γ n≥0 Remark 1. In the usual definition of the kernel of Fredholm determinant (see, [Sim05, Chapter 5]), there is no such pre-factor of 1/(2πi)n as in (A.1). Furthermore, to show the absolute convergence of the series in (A.1), one need to have suitable control on supx∈Γ \|K(x, z)\|. To that end, following inequality (see, [Bha97, Exercise 1.1.3]) proves the absolute convergence. Lemma A.2 (Hadamard Inequality). If D is a N × N complex matrix, then v N uX Y u n 2. t Dij \|det(D)\| ≤ i=1 j=1 Corollary A.3. Let Γ be a curve. Kernel K is defined over Γ×Γ. Furthermore, supx∈Γ \|K(x, z)\| ≤ R K(z), where K(z) satisfies Γ K(z)dz < ∞. Then series in (A.1) converges absolutely. imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 62 In the next lemma, we show that Fredholm determinant of the limiting kernel in (4.57) is the Gaussian distribution function. Although, we couldn’t find a direct proof that result, we believe that it is quite well known. For the sake of clarity, we present here a short proof. Lemma A.4. Consider a piecewise linear infinite curve Γ which extends linearly from −δ + ∞e−π/6 to −δ and from −δ to −δ + ∞eπ/6 . Let us define the kernel KG for any two points w, w′ ∈ Γ as 2 w2 v Z − + s(v − w) exp 2 2 v 1 dv. KG := 2πi −2δ+R (v − w)(v − w′ ) w Then, det(1 + KG )L2 (Γ) = Z s −∞ 1 2 √ e−z /2 dz. 2π Proof. To begin with, notice that for all v on the line R ∞ −2δ + iR, Re(v − w) < 0 whenever w ∈ Γ. Thus, we can write exp(s(v − w)) = (v − w) s exp(z(v − w))dz. Fix any σ ∈ S(n). To this end, we have Z Z (A.2) . . . K(w1 , wσ(1) )K(w2 , wσ(2) ) . . . K(wn , wσ(n) )dw1 , . . . dwn Γ Γ P 2 vi wi2 n Z Z n exp − + s(v − w ) Y i i i=1 2 2 vi Y 1 Q dwi dv = i n (2πi)n Γn (−2δ+iR)n wi i=1 (vi − wi )(vi − wσ(i) ) i=1 i=1 P 2 vi wi2 n Z Z Z n n n exp Y Y i=1 2 − 2 + zi (vi − wi ) 1 vi Y Q dwi = dv dz i i n (2πi)n Γn (−2δ+iR)n [−s,∞)n wi i=1 (vi − wσ(i) ) i=1 = 1 (2πi)n Z [s,∞)n Z (−2δ+iR)n Z exp Γn P 2 vi n i=1 i=1 wi2 2 n n n + z (v − w ) Y Y i i i i=1 2 vi Y Qn dwi dvi dzi . wi i=1 (vi − wσ(i) ) − i=1 i=1 i=1 Define a piecewise linear curve Γr which extends linearly from −δ + re−iπ/6 to −δ and from −δ to −δ + reiπ/6 . Notice that Γr tends to Γ as r → ∞. Denote the closed contour formed by Γ and the line joining −δ + re−iπ/6 to −δ + reiπ/6 as Γ̃r . As r increases, integrand inside the paranthesis of the last line in (A.2) exponentially decays to zero uniformly for all w ∈ Γ̃r \Γr . Consequently, the integral over Γ̃r will converges to the integral over Γ. Thus, computing the integral over Γn boils down to finding out the residue of the integrand at z1 = . . . = zn = 0 because these are the only poles sitting inside the contour Γ̃nr for all large value of r. Thereafter, 2 vi wi2 ! Z exp Pn n 2 n − + z (v − w ) X i i i i=1 2 2 vi Y vi Qn dwi = exp + zi vi . wi 2 Γn i=1 (vi − wσ(i) ) i=1 i=1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 63 Using the analytic behavior of the function exp(2−1 v 2 + vz), one can argue further ! Y ! Y Z Z n 2 n n n 2 X X vi vi exp + zi vi + zi vi dvi = dvi exp 2 2 (−i∞,i∞)n (−2δ+iR)n i=1 i=1 i=1 i=1 ! n X √ 2 n = ( 2πi) exp − zi . i=1 To this end, multiple integral in (A.2) becomes equal to (1 − Φ(s))n where Φ(.) is the distribution function of standard normal distribution. This implies for all n ≥ 2 Z Z . . . det(K(wi , wj ))ni,j=1 dw1 . . . dwn = 0. Γ Γ Thus, we have det(1 + KG )L2 (Γ) = 1 − (1 − Φ(s)) = Φ(s). Hence, this completes the proof. Appendix B: eigenfunction of the Transition Matrix in HL-PushTASEP Proof of Proposition 3.1: Here, we sketch a proof of the present proposition extending the arguments in [BCG16, Theorem 3.4] in the continuous case. Let us define N functions gi (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN ) as follows X (t,t+dt) yN gi (xi , xi+1 , . . . , xN ; z1 , z2 , . . . , zN ) = Ti ((xi , . . . , xN ) → (yi , . . . , yN )) ziyi . . . zN (yi ,...,yN ) where Tit,t+dt (.) denotes the probability of the transition from the configuration (xi , . . . , xN ) to (yi , . . . , yN ) once the particle at position xi gets excited in the time interval (t, t + dt). For convenience, let us consider gi (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN ) := g (1) (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN ) (2) + gi (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN ). (B.1) Subsequently, we describe two new functions introduced in (B.1) as follows. In the expansion of gi , one can note the contribution of the event that the particle at position xi moves by j-steps x −1 xi+1 xN zi where 1 ≤ j < xi+1 − xi in the time gap (t, t + dt) is 1−bz (zixi − bxi+1−xi −1 zi i+1 )zi+1 . . . . zN i Thus, define (1) gi (xi , . . . , xN ; zi , . . . , zN ) = zi (1 − b) xi x −1 xi+1 xN . . . zN . (zi − bxi+1 −xi −1 zi i+1 )zi+1 1 − bzi Lastly, g(2) takes account of the event that particle at xi jumps to the position xi+1 and consequently, particle which was previously there in xi+1 starts to hop to the right in the same fashion. To this end, one can write (2) x gi (xi , . . . , xN ; zi , . . . , zN ) = bxi+1 −xi −1 zi i+1 gi+1 (xi+1 , . . . , xN ; zi , . . . , zN ). imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 64 Let us define a function g as X g(x1 , . . . , xN ; z1 , . . . , zN ) := (y1 ,...,yN ) yN T (t,t+dt) ((x1 , . . . , xN ) → (y1 , . . . , yN )) ziyi . . . zN where T (t,t+dt) denotes the transfer matrix in the interval (t, t + dt). Naively, g acts as a probability generating function in the infinitesimal time interval (t, t + dt). In what follows, it is easy to see how g and gi ’s are connected g(x1 , . . . , xN ; z1 , . . . , zN ) = N X x i−1 z1x1 . . . zi−1 gi (xi , xi+1 , . . . , xN ; zi , zi+1 , . . . , zN )dt. i=1 If we take N = 2, then we get g(x1 , x2 ; z1 , z2 ) = z x2 +1 (1 − b) x2 −x1 −1 x2 z1 (1 − b) x1 (z1 − bx2 −x1 −1 z1x2 −1 )z2x2 dt + 2 (b z1 + z1x1 )dt. 1 − bz1 1 − bz2 Let us consider S12 g(x1 , x2 ; z1 , z2 )+S21 g(x1 , x2 ; z2 , z1 ) for some functions S12 and S21 of z1 , z2 . One can write z1 (1 − b) z2 (1 − b) + (S12 z1x1 z2x2 + S21 z2x1 z1x2 )dt S12 g(x1 , x2 ; z1 , z2 ) + S21 g(x1 , x2 ; z2 , z1 ) = 1 − bz1 1 − bz2 x2 x2 −x1 −1 + (1 − b)(z1 z2 ) b S12 1 1 z1 z2 − − + S21 dt. 1 − bz2 1 − bz1 1 − bz1 1 − bz2 Thus, on taking S12 = 1 − (1 + b)z1 + bz1 z2 and S12 = −1 + (1 + b)z2 − bz1 z2 , one can note that S12 z1x1 z2x2 + S21 z1x2 z2x1 is an eigenfunction of the transfer matrix T with the eigenvalue z2 (1−b) z1 (1−b) 1−bz1 + 1−bz2 in the interval (t, t+dt). One can also scale S12 and S21 by 1−(1+b)z1 +bz1 z2 without any major consequences. To obtain the eigenfunction and eigenvalues for any N , we use induction. Assume N = n − 1 and X Aσ g(x1 , x2 , . . . , xn−1 ; zσ(1) , zσ(2) , . . . , zσ(n−1) ) (B.2) σ∈S(n−1) = (1 − b) n−1 X i=1 zi 1 − bzi where Aσ := Y (−1)σ 1≤i<j≤n−1 !  X σ∈S(n−1) x  x1 n−1  Aσ zσ(1) . . . zσ(n−1) dt 1 − (1 + b)zσ(i) + bzσ(i) zσ(j) . 1 − (1 + b)zi + bzi zj We need to show that (B.2) holds even when N = n. Consider the following X Aσ g(x1 , x2 , . . . , xn ; zσ(1) , zσ(2) , . . . , zσ(n) ) = (I) + (II) + (III) σ∈S(n) imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 65 where  X (I) : = (y1 ,y2 ,...,yn ) y1 =x1 =  n X j=1  T (t,t+dt) ((x1 , . . . , xn ) → (y1 , . . . , yn ))    (1 − b)  X (II) : = (y1 ,y2 ,...,yn ) x1 <y1 <x2 = (1 − b) n X i=1 i6=j  n X j=1 Aσ σ∈S(n) σ(1)=j Y  zi   X x1 x2 xn  Aσ zσ(1) zσ(2) . . . zσ(n) dt,  1 − bzi  X Aσ σ∈S(n) Aσ σ∈S(n) yi   , (B.3) zσ(i)  σ∈S(n) T (t,t+dt) ((x1 , . . . , xn ) → (y1 , . . . , yn ))  X X  x2 −x1 bx2 −x1 −1 zσ(1) zσ(1) − 1 − bzσ(1) 1 − bzσ(1) ! Y  yi  zσ(i) x1 x2 xn dt, zσ(1) zσ(2) . . . zσ(n) and (III) : = X (y1 ,y2 ,...,yn ) y1 =x2 = (1 − b)b + X  T (t,t+dt) ((x1 , . . . , xn ) → (y1 , . . . , yn ))  (y1 ,y2 ,...,yn ) Aσ σ∈S(n) Y  yi  zσ(i) x3 −x2 ! bx3 −x2 −1 zσ(2) zσ(2) x1 x2 xn − dt zσ(1) zσ(2) . . . zσ(n) Aσ 1 − bzσ(2) 1 − bzσ(2) σ∈S(n)   Y y X i  zσ(i) Aσ . T (t,t+dt) ((x1 , . . . , xn ) → (y1 , . . . , yn ))  x2 −x1 −1 y1 =x2 ,y2 =x3 X X σ∈S(n) For any σ ∈ S(n), one can note that " x2 −x1 # # " x2 −x1 zσ(1) zσ′ (1) zσ(2) zσ′ (2) x2 Aσ − (zσ(1) zσ(2) ) = −Aσ′ − (zσ′ (1) zσ′ (2) )x2 1 − bzσ(1) 1 − bzσ(2) 1 − bzσ′ (1) 1 − bzσ′ (2) where σ ′ = (1, 2) ∗ σ. Thus, there will be telescopic cancellations of large number of terms in the sum (II) + (III). To that effect, we would have (II) + (III) = (1 − b) X σ∈S(n) Aσ zσ(1) xn dt z x1 z x2 . . . zσ(n) 1 − bzσ(1) σ(1) σ(2) (B.4) and consequently, adding (B.3) and (B.4) shows (B.2) when N = n. To this end, denoting (0,t) the transfer matrix for exactly m-many clock counts in the interval (0, t) by Tm (.), one can imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 66 deduce X (y1 ,...,yN ) = Z  Tm(0,t) ((x1 , . . . , xN ) → (y1 , . . . , yN ))  (t0 ,...,tm+1 )∈∆(t) 1 = exp(−N t) m! t N X (1 − b)zi i=1 (1 − bzi ) N X (1 − b)zi i=1 (1 − bzi ) !m   X X σ∈S(n)  y1 yN  Aσ zσ(1) . . . zσ(N ) x1 Aσ zσ(1) σ∈S(n)  xN  . . . zσ(N ) m Y k=1 (B.5) exp(−N (tk − tk−1 ))dtk  !m  X xN  x1  Aσ zσ(1) . . . zσ(N ) σ∈S(n) where ∆(t1 , . . . , tm+1 ) = {0 = t0 < t1 < t2 < . . . < tm+1 = t}. Note that exp(−N (tk − tk−1 )) term inside the integral above accounts for the probability that N independent clocks associated with N particles do not ring in the interval (tk−1 , tk ). Once any of the N clock rings, it P (1−b)zi results in a factor of N i=1 (1−bzi ) at the front. Thus, m many clock counts add upto m such factors. Using (B.5), one can further deduce   X X (N ) y1 yN  Aσ zσ(1) . . . zσ(N Tt ((x1 , . . . , xN ) → (y1 , . . . , yN ))  ) σ∈S(N ) (y1 ,...,yn ) = ∞ X X m=0 (y1 ,...,yn ) = ∞ X m=0  Tm(0,t) ((x1 , . . . , xN ) → (y1 , . . . , yN ))  1 exp(−N t) m! X σ∈S(N ) t N X (1 − b)zi i=1 (1 − bzi ) !m   X σ∈S(n)  y1 yN  Aσ zσ(1) . . . zσ(N )  xN  x1 Aσ zσ(1) . . . zσ(N )  ! N X 1 − zi  X xN  x1 = exp − Aσ zσ(1) . . . zσ(N . ) 1 − bzi i=1 σ∈S(n) This completes the proof. Appendix C: Moment Formula Proof of Proposition 3.4: In what follows, we first find out the moment formula with the initially N many particles. First, we claim that under step Bernoulli initial condition the probability of transition of m-th particle to x by time t in HL-PushTASEP is given as X X κ(S,Z>0 )−mk−k(k−1)/2 Sk k − 1 m−1 m(m−1)/2 b ρ Pstepb (xm (t) = x; t) = (−1) b m−1 b m≤k≤N \|S\|=k !Si −Si−1 I I k Y Y zj − zi 1 1 × ... (2πi)k 1 − (1 + b−1 )zi + b−1 zi zj ) 1 − (1 − ρ)zi−1 · · · zk−1 i=1 1≤i<j≤k Q k 1 − zi−1 1 − ki=1 zi Y x−Si −1 zi exp −t dzi , (C.1) × Qk 1 − bzi−1 i=1 (1 − zi ) i=1 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 67 where S = {S1 , . . . , Sk } with 1 ≤ S1 < S2 < . . . < Sk ≤ N and κ(S, Z>0 ) counts the sum of the elements in S. One can note that when HL-PushTASEP is started with particles at N ordered places (y1 , . . . , yn ) (call it Y) in Z≥0 , we can have an explicit expression of P~y (xm (t) = x; t) given in Theorem 3.3. To put all the ingredients together, let us just recall the formula of P~y (xm (t) = x; t) given there. For a fixed N ≥ 0, when the initial data is ~y = (y1 , . . . , yN ), then X X κ(S,Z>0 )−mk−k(k−1)/2 k − 1 m−1 m(m−1)/2 b P~y (xm (t) = x; t) = (−1) b m−1 b m≤k≤N \|S\|=k I I Y zj − zi 1 . . . × (2πi)k 1 − (1 + b−1 )zi + b−1 zi zj ) i,j∈S,i<j Q Y 1 − i∈S zi 1 − zi−1 x−yi −1 ×Q exp −t zi dzi , (C.2) 1 − bzi−1 i∈S (1 − zi ) i∈S where κ(S, Z>0 ) is the sum of the element in S. The term which relates initial data with P~y (xm (t) = x; t) in (C.2), is zi−yi for i ∈ S. To obtain the result in case of step initial condition, we have to sum over all ~y by multiplying P~y (xm (t) = x; t) with appropriate probability. Let us introduce k-many new variables t1 , . . . , tk which are defined as t1 := S1 − 1, t2 := S1 − S2 − 1, . . . , tk := Sk − Sk−1 − 1. P To this end, we can write down Si in terms of ti by Si = ij=1 ti + i. So, initially, we would P have the configuration yS1 = S1 + ii , . . . , ySk = Sk + kl=1 il with probability Sk ρ k Y tl + il tl l=1 (1 − ρ)il . One can note that k k Pj Y X Y tj + ij −S ij −Sj + l=1 il zSj j = (1 − ρ) zSj tj j=1 i1 ,...,ik ≥0 j=1 1 · · · zS−1 1 − (1 − ρ)zS−1 j k !tj +1 .(C.3) Q Q Q −S Now, kj=1 zSj j in (C.3) counts for the replacement of i∈S zix−yi −1 in (C.2) with ki=1 zix−Si −1 in (C.1), whereas extra factor in second line of (C.1) comes from the second term at the end of the right side in (C.3). Thus, the claim is proved. Now, we turn to showing the moment formula. Given any L ∈ N, we multiply (C.1) by bmL and sum over all 1 ≤ m ≤ N . Let us note down the following result (see, [AAR99, Theorem 10.2.1]) known as q- binomial Theorem. For any q, t ∈ C such that \|q\| = 6 1 and any positive integer n, we have n n−1 X Y i k(k−1)/2 n (1 + q t) = q tk . (C.4) k q i=0 k=0 Thus, using (C.4), one can write k X m=1 m−1 m(m−1)/−mk+mL (−1) b k−2 Y k−1 (1 − b1−k+L .b). = bL−k m−1 b i=0 imsart-generic ver. 2011/11/15 file: HL-PushTASEPandItsKPZLimit.tex date: January 26, 2017 Ghosal/HL-PushTASEP & its KPZ limit 68 To that effect, we get min(L,N X ) Estepb bLNx (t) ηx (t) = k=1 1 × (2πi)k 1− × Qk I ... Qk i=1 zi i=1 (1 I Y 1≤i<j≤k k Y − zi ) i=1 k−2 Y i=0 (1 − b 1−k+L ! .b) X bκ(S,Z>0 )−k(k−1)/2+L−k ρSk \|S\|=k k Y zj − zi 1 − (1 + b−1 )zi + b−1 zi zj 1 1 − (1 − ρ)zi−1 · · · zk−1 i=1 zix−Si −1 exp −t 1 − zi−1 1 − bzi−1 !Si −Si−1 dzi , (C.5) where S = {S1 , . . . , Sk } with 1 ≤ S1 < S2 < . . . < Sk ≤ N and κ(S, Z>0 ) counts the sum of the elements in S. Letting N → ∞, we sum (C.5) over all possible k ordered subsets 1 ≤ S1 < S2 < . . . < Sk of various sizes. This infinite sum can be simplified in the following way X b k Y S1 +...+Sk Sk ρ zi−Si i=1 1≤S1 <S2 <...<Sk 1 1 − (1 − ρ)zi−1 · · · zk−1 = bk(k+1)/2 ρk !Si −Si−1 = k Y i=1 1 ρ(b/zi )···(b/zk ) 1−(1−ρ)zi−1 ···zk−1 ρ(b/zi )···(b/zk ) − 1−(1−ρ)z −1 −1 i ···zk 1 . zi · · · zk − (1 − ρ) − bk−i+1 ρ In what follows, we use a symmetrization identity from [TW09a, Section III] to conclude that ! L k−2 X Y b−k(k−1)/2+L−k ρk LNx (t) 1−k+L i Estepb b ηx (t) = (1 − b b) (b−1 ; b−1 )k k=1 i=1 ! I I k Y Y zj − zi 1 × zi ... 1− (2πi)k 1 − (1 + b−1 )zi + b−1 zi zj i=1 1≤i<j≤k × k Y 1 − zi−1 zix (b−1 − 1) exp −t (ρ + (1 − ρ)b−1 − zi b−1 )(1 − zi ) 1 − bzi−1 i=1 dzi . 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Image Captioning using Deep Neural Architectures Parth Shah (orcid.org/0000-0002-7880-2228), Vishvajit Bakarola (orcid.org/0000-0002-6602-9194), and Supriya Pati (orcid.org/0000-0001-5528-6104) arXiv:1801.05568v1 [cs.CV] 17 Jan 2018 Chhotubhai Gopalbhai Patel Institute of Technology, Bardoli, India parthpunita@yahoo.in vishvajit.bakrola@utu.ac.in supriya.pati@utu.ac.in Abstract. Automatically creating the description of an image using any natural languages sentence like English is a very challenging task. It requires expertise of both image processing as well as natural language processing. This paper discuss about different available models for image captioning task. We have also discussed about how the advancement in the task of object recognition and machine translation has greatly improved the performance of image captioning model in recent years. In addition to that we have discussed how this model can be implemented. In the end, we have also evaluated the performance of model using standard evaluation matrices. Keywords: Deep Learning, Deep Neural Network, Image Captioning, Object Recognition, Machine Translation, Natural Language Processing, Natural Language Generation 1 Introduction A single image can contain large amount of information in it. Humans have ability to parse this large amount of information by single glance of it. Humans normally communicate though written or spoken language. They can use languages for describing any image. Every individual will generate different caption for same image. If we can achieve same task with machine it will be greatly helpful for variety of tasks. However, generating captions for an image is very challenging task for machine. In order to perform caption generation task by machine, it requires brief understanding of natural language processing and ability to identify and relate objects in an image. Some of the early approaches that tried to solve these challenge are often based on hard-coded features and well defined syntax. This limits the type of sentence that can be generated by any given model. In order to overcome this limitation the main challenge is to make model free of any hard-coded feature or sentence templates. Rule for forming models should be learned from the training data. Another challenge is that there are large number of images available with their associated text available in the ever expanding internet. However, most of them are noisy hence it can not be directly used in image captioning model. Training an image captioning model requires huge dataset with properly available annotated image by multiple persons. In this paper, we have studied collections of different existing natural image captioning models and how they compose new caption for unseen images. We have also presented results of our implementation of these model and compared them. Section 2 of this paper describes Related Work in detail. Show & Tell model in detailed is described in Section 3. Section 4 contains details about implementation environment and dataset. Results and Discussion is provided in detail in Section 5. At the end we provided our concluding remarks in section 6. 2 Related work Creating captioning system that accurately generate captions like human depends on the connection between importance of object in image and how they will be related to other objects in image. Image can be described using more than one sentence but to efficiently train the image captioning model we requires only single sentence that can be provided as a caption. This leads to problem of text summarization in natural language processing. There are mainly two different way to perform the task of image captioning. These two types are basically retrieval based method and generative method. From that most of work is done based on retrieval based method. One of the best model of retrieval based method is Im2Txt model [1]. It was proposed by Vicente Ordonez, Girish Kulkarni and Tamara L Berg. Their system is divided into mainly two part 1) Image matching and 2) Caption generation. First we will provide our input image to model. Matching image will be retrieved from database containing images and its appropriate caption. Once we find matching images we will compare extracted high level objects from original image and matching images. Images will then reranked based on the content matched. Once it is reranked caption of top-n ranked images will be returned. The main limitation of these retrieval based method is that it can only produce captions which are already present in database. It can not generate novel captions. This limitation of retrieval based method is solved in generative models. Using generative models we can create novel sentences. Generative models can be of two types either pipeline based model or end to end model. Pipeline type models uses two separate learning process, one for language modeling and and one for image recognition. They first identify objects in image and provides the result of it to language modeling task. While in end-to-end models we combine both language modeling and image recognition models in single end to end model [2]. Both part of model learn at the same time in end-to-end system. They are typically created by combination of convolutional and recurrent neural networks. Show & Tell model proposed by Vinyals et al. is of generative type end-toend model. Show & Tell model uses recent advancement in image recognition and neural machine translation for image captioning task. It uses combination of Inception-v3 model and LSTM cells [3]. Here Inception-v3 model will provides object recognition capability while LSTM cell provides it language modeling capability [4][5]. 3 Show & Tell Model Recurrent neural networks generally used in neural machine translation [6]. They encodes the variable length inputs into a fixed dimensional vectors. Then it uses these vector representation to decode to the desired output sequence [7][8]. Instead of using text as input to encoder Show & Tell model uses image as input. This image is then converted to word vector and then this word vector is translated to caption using Recurrent neural networks as decoder. To achieve this goal, Show & Tell model is created by hybridizing two different models. It takes input as the image and provides it to Inception-v3 model. At the end of Inception-v3 model single fully connected layer is added. This layer will transform output of Inception-v3 model into word embedding vector. We input this word embedding vector into series of LSTM cell. LSTM cell provides ability to store and retrieve sequential information through time. This helps to generate the sentences with keeping previous words in context. Training of Show & Tell Fig. 1: Architecture of Show & Tell Model model can be divided into two part. First part is of training process where model learns its parameters. While second part is of testing process. In testing process we infer the captions and we compare and evaluate these machine generated caption with human generated captions. 3.1 Training During training phase we provides pair of input image and its appropriate caption to Show & Tell model. Inception-v3 part of model is trained to identify all possible objects in an image. While LSTM part of model is trained to predict every word in the sentence after it has seen image as well as all previous words. For any given caption we add two additional symbols as start word and stop word. Whenever stop word is encountered it stop generating sentence and it marks end of string. Loss function for model is calculated as L(I, S) = − N X log pt (St ) . (1) t=1 where I represent input image and S represent generated caption. N is length of generated sentence. pt and St represent probability and predicted word at the time t respectively. During the process of training we will try to minimize this loss function. 3.2 Inference From various approaches to generate caption a sentence from given image Show & Tell model uses Beam Search to find suitable words to generate caption. If we keep beam size as K, it recursively consider K best word at each output of the word. At each step it will calculate joint probability of word with all previously generated word in sequence. It will keep producing the output until end of sentence marker is predicted. It will select sentence with best probability and outputs it as caption. 4 Implementation For evaluation of image captioning model we have implemented Show & Tell model. Details about dataset,implementation tool and implementation environment is given as follows: 4.1 Datasets For task of image captioning there are several annotated images dataset are available. Most common of them are Pascal VOC dataset and MSCOCO Dataset. In this work MSCOCO image captioning dataset is used. MSCOCO is a dataset developed by Microsoft with the goal of achieving the state-of-the-art in object recognition and captioning task. This dataset contains collection of day-to-day activity with theri related captions. First each object in image is labeled and after that description is added based on objects in an image. MSCOCO dataset contains image of around 91 objects types that can be easily recognizable by even a 4 year old kid. It contains around 2.5 million objects in 328K images. Dataset is created by using crowdsourcing by thousonds of humans [9]. 4.2 Implementation tool and environment For the implementation of this experiment we have used machine with Intel Xeon E3 processor with 12 cores and 32GB RAM running CentOS 7. Tensorflow liberary is used for creating and training deep neural networks. Tensorflow is a deep learning library developed by Google[10]. It provides heterogeneous platform for execution of algorithms i.e. it can be run on low power devices like mobile as well as large scale distributed system containing thousands of GPUs. To define structure of our network tensorflow uses graph definition. Once graph is defined it can be executed on any supported devices. 5 5.1 Results and Discussion Results By the implementation of the Show & Tell model we can able to generate moderately comparable captions with compared to human generated captions. First of all it model will identify all possible objects in image. Fig. 2: Generated word vector from Sample Image As shown in Fig. 2 Inception-v3 model will assign probability of all possible object in image and convert image into word vector. This word vector is provided as input to LSTM cells which will then form sentence from this word vector as shown in Fig. 3 using beam search as described in previous section. Fig. 3: Generated caption from word vector for Sample Image 5.2 Evaluation Matrices To evaluating of any model that generate natural language sentence BLEU (Bilingual Evaluation Understudy) Score is used. It describes how natural sentence is compared to human generated sentence [11]. It is widely used to evaluate performance of Machine translation. Sentences are compared based on modified n-gram precision method for generating BLEU score [12]. Where precision is calculated using following equation: P C∈{Candidates} pn = P P C 0 ∈{Candidates} ngram∈C P Countclip (ngram) ngram0 ∈C 0 Count(ngram0 ) (2) To evaluate our model we have used image from validation dataset of MSCOCO Dataset. Some of captions generated by Show & Tell model is shown as follows: (a) Input image (b) Generated caption Fig. 4: Experiment Result As you can see in Fig. 4, generated sentence is “a woman sitting at a table with a plate of food.”, while actual human generated sentence are “The young woman is seated at the table for lunch, holding a hotdog.”, “a woman is eatting a hotdog at a wooden table.”, “there is a woman holding food at a table.”, “a young woman holding a sandwich at a table.” and “a woman that is sitting down holding a hotdog.”. This result in BLEU score of 63 for this image. (a) Input image (b) Generated caption Fig. 5: Experiment Result Similarly in Fig. 5, generated sentence is “a woman holding a cell phone in her hand.” while actual human generated sentence are “a woman holding a Hello Kitty phone on her hands”, “a woman holds up her phone in front of her face”, “a woman in white shirt holding up a cellphone”, “a woman checking her cell phone with a hello kitty case” and “the asian girl is holding her miss kitty phone”. This result in BLEU score of 77 for this image. While calculating BLEU score of all image in validation dataset we get average score of 65.5. Which shows that our generated sentence are very similar compared to human generated sentence. 6 Conclusion We can conclude from our findings that we can combine recent advancement in Image Labeling and Automatic Machine Translation into an end-to-end hybrid neural network system. This system is capable to autonomously view an image and generate a reasonable description in natural language with better accuracy and naturalness. References 1. V. Ordonez, G. Kulkarni, and T. L. Berg, “Im2text: Describing images using 1 million captioned photographs,” in Advances in Neural Information Processing Systems, pp. 1143–1151, 2011. 2. A. Karpathy and L. Fei-Fei, “Deep visual-semantic alignments for generating image descriptions,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3128–3137, 2015. 3. O. Vinyals, A. Toshev, S. Bengio, and D. Erhan, “Show and tell: Lessons learned from the 2015 mscoco image captioning challenge,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. PP, no. 99, pp. 1–1, 2016. 4. C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–9, 2015. 28. 5. C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” arXiv preprint arXiv:1512.00567, 2015. 37. 6. D. Britz, “Introduction to rnns.” WILDML, http://www.wildml.com/, 2016. [Accessed 4-September-2016]. 7. Y. Wu, M. Schuster, Z. Chen, Q. V. Le, M. Norouzi, W. Macherey, M. Krikun, Y. Cao, Q. Gao, K. Macherey, J. Klingner, A. Shah, M. Johnson, X. Liu, L. Kaiser, S. Gouws, Y. Kato, T. Kudo, H. Kazawa, K. Stevens, G. Kurian, N. Patil, W. Wang, C. Young, J. Smith, J. Riesa, A. Rudnick, O. Vinyals, G. Corrado, M. Hughes, and J. Dean, “Google’s neural machine translation system: Bridging the gap between human and machine translation,” CoRR, vol. abs/1609.08144, 2016. 8. I. Sutskever, O. Vinyals, and Q. V. Le, “Sequence to sequence learning with neural networks,” in Advances in neural information processing systems, pp. 3104–3112, 2014. 9. T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick, “Microsoft coco: Common objects in context,” in European Conference on Computer Vision, pp. 740–755, Springer, 2014. 10. M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, et al., “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv preprint arXiv:1603.04467, 2016. 11. K. Papineni, S. Roukos, T. Ward, and W.-J. Zhu, “Bleu: a method for automatic evaluation of machine translation,” in Proceedings of the 40th annual meeting on association for computational linguistics, pp. 311–318, Association for Computational Linguistics, 2002. 12. D. Jurafsky, Speech & language processing. Pearson Education India, 2000.	1
Learning-based Model Predictive Control for Safe Exploration and Reinforcement Learning arXiv:1803.08287v1 [cs.SY] 22 Mar 2018 Torsten Koller, Felix Berkenkamp, Matteo Turchetta and Andreas Krause Abstract— Learning-based methods have been successful in solving complex control tasks without significant prior knowledge about the system. However, these methods typically do not provide any safety guarantees, which prevents their use in safety-critical, real-world applications. In this paper, we present a learning-based model predictive control scheme that provides provable high-probability safety guarantees. To this end, we exploit regularity assumptions on the dynamics in terms of a Gaussian process prior to construct provably accurate confidence intervals on predicted trajectories. Unlike previous approaches, we do not assume that model uncertainties are independent. Based on these predictions, we guarantee that trajectories satisfy safety constraints. Moreover, we use a terminal set constraint to recursively guarantee the existence of safe control actions at every iteration. In our experiments, we show that the resulting algorithm can be used to safely and efficiently explore and learn about dynamic systems. I. I NTRODUCTION In model-based reinforcement learning (RL, [1]), we aim to learn the dynamics of an unknown system from data, and based on the model, derive a policy that optimizes the long-term behavior of the system. Crucial to the success of such methods is the ability to efficiently explore the state space in order to quickly improve our knowledge about the system. While empirically successful, current approaches often use exploratory actions during learning, which lead to unpredictable and possibly unsafe behavior of the system, e.g. in exploration approaches based on the optimism in the face of uncertainty principle [2]. Applying such approaches to the exploration of a real-world safety-critical system, such as an autonomous car, is undesirable. In this paper we introduce S AFE MPC, a safe model predictive control (MPC) scheme that guarantees the existence of feasible return trajectories to a safe region of the state space at every time step with high-probability. These return trajectories are identified through a novel uncertainty propagation method that, in combination with constrained MPC, allows for formal safety guarantees in learning control. Related Work: One area that has considered safety guarantees is robust MPC. There, we iteratively optimize the performance along finite-length trajectories at each time step, based on a known model that incorporates uncertainties and disturbances acting on the system [3]. In a constrained robust This work was supported by SNSF grant 200020 159557, a fellowship within the FITweltweit program of the German Academic Exchange Service (DAAD), and the Max Planck ETH Center for Learning Systems Torsten Koller is with the Department of Computer Science, University of Freiburg, Germany. Email: torsten.koller@merkur.uni-freiburg.de Felix Berkenkamp, Matteo Turchetta and Andreas Krause are with the Learning & Adaptive Systems Group, Department of Computer Science, ETH Zurich, Switzerland. Email: {befelix, matteotu, krausea}@inf.ethz.ch Fig. 1. Propagation of uncertainty over multiple time steps based on a wellcalibrated statistical model of the unknown system. We iteratively compute ellipsoidal over-approximations (purple) of the intractable image (green) of the learned model for uncertain ellipsoidal inputs. MPC setting, we optimize these local trajectories under additional state and control constraints. Safety is typically defined in terms of recursive feasibility and robust constraint satisfaction. In [4], this definition is used to safely control urban traffic flow, while [5] guarantees safety by switching between a standard and a safety mode. However, these methods are conservative since they do not update the model. In contrast, learning-based MPC approaches (LBMPC, [6]) adapt their models online based on observations of the system. This allows the controller to improve over time, given limited prior knowledge of the system. Theoretical safety guarantees in LBMPC are established in [6], which enforces robust feasibility and constraint satisfaction with learned statistical models. However, the model errors are assumed to be deterministically bounded in a polytope and it is not clear how to obtain this outer bound in the first place. MPC based on Gaussian process (GP, [7]) models is proposed in a number of works, e.g. [8], [9]. The difficulty here is that trajectories have complex dependencies on states and unbounded stochastic uncertainties. Safety through probabilistic chance constraints is considered in [10] and [11] based on approximate uncertainty propagation. While often being empirically successful, these approaches do not theoretically guarantee safety of the underlying system. Another area that has considered learning for control is model-based RL. There, we aim to learn global policies based on data-driven modeling techniques, e.g., by explicitly trading-off between finding locally optimal closed-loop policies (exploitation) and learning the behavior of the system globally through exploration [1]. This results in data-efficient learning of policies in unknown systems [12]. In contrast to MPC, where we optimize finite-length trajectories, in RL we typically aim to find an infinite horizon optimal policy. Hence, enforcing hard constraints in RL is challenging. Control-theoretic safety properties such as Lyapunov stability or robust constraint satisfaction are only considered in a few works [13]. In [14], safety is guaranteed by optimizing parametric policies under stability constraints, while [15] guarantees safety in terms of constraint satisfaction through reachability analysis. Our Contribution: We combine ideas from robust control and GP-based RL to design a MPC scheme that recursively guarantees the existence of a safety trajectory that satisfies the constraints of the system. In contrast to previous approaches, we use a novel uncertainty propagation technique that can reliably propagate the confidence intervals of a GP-model forward in time. We use results from statistical learning theory to guarantee that these trajectories contain the system with high probability jointly for all time steps. In combination with a constrained MPC approach with a terminal set constraint, we then prove the safety of the system. We apply the algorithm to safely explore the dynamics of an inverted pendulum simulation. II. P ROBLEM S TATEMENT We consider a nonlinear, discrete-time dynamical system xt+1 = f (xt , ut ) = h(xt , ut ) + g(xt , ut ) , \| {z } \| {z } prior model (1) unknown error where xt ∈ Rnx is the state and ut ∈ Rnu is the control input to the system at time step t ∈ N. We assume that we have access to a twice continuously differentiable prior model h(xt , ut ), which could be based on a first principles physics model. The model error g(xt , ut ) is a priori unknown and we use a statistical model to learn it by collecting observations from the system during operation. In order to provide guarantees, we need reliable estimates of the modelerror. In general, this is impossible for arbitrary functions g. We make the following additional regularity assumptions. assume that the model-error g is of the form g(z) = PWe ∞ nx α × Rn u , a i=0 i k(z, zi ), αi ∈ R, z = (x, u) ∈ R weighted sum of distances between inputs z and representer points zi = (xi , ui ) ∈ Rnx × Rnu as defined through a symmetric, positive definite kernel k. This class of functions is well-behaved in the sense that they form a reproducing kernel Hilbert space (RKHS, [16]) Hk equipped with an inner-product h·, ·ik . The induced norm \|\|g\|\|2k = hg, gik is a measure of the complexity of a function g ∈ Hk . Consequently, the following assumption can be interpreted as a requirement on the smoothness of the model-error g w.r.t the kernel k. Assumption 1 The unknown function g has bounded norm in the RKHS Hk , induced by the continuously differentiable kernel k, i.e. \|\|g\|\|k ≤ Bg . In the case of a multi-dimensional output nx > 1, we follow [17] and redefine g as a single-output function g̃ such that g̃(·, i) = gi (·) and assume that \|\|g̃\|\|k ≤ Bg . We further assume that the system is subject to polytopic state and control constraints X = {x ∈ Rnx \|Hx x ≤ hx , hx ∈ Rmx }, U = {u ∈ R nu u \|Hu u ≤ h , hu ∈ R mu }, (2) (3) which are bounded. For example, in an autonomous driving scenario, the state region could correspond to a highway lane and the control constraints could represent the physical limits on acceleration and steering angle of the car. Lastly, we assume access to a backup controller that guarantees that we remain inside a given safe subset of the state space once we enter it. In the autonomous driving example, this could be a simple linear controller that stabilizes the car in a small region in the center of the lane at slow speeds. Assumption 2 We are given a controller πsafe (·) and a polytopic safe region Xsafe := {x ∈ Rnx \|Hs x ≤ hs } ⊂ X , (4) which is (robust) control positive invariant (RCPI) under πsafe (·). Moreover, the controller satisfies the control constraints inside Xsafe , i.e. πsafe (x) ∈ U ∀x ∈ Xsafe . This assumption allows us to gather initial data from the system inside the safe region even in the presence of significant model errors, since the system remains safe under the controller πsafe . Moreover, we can still guarantee constraint satisfaction asymptotically outside of Xsafe , if we can show that a finite sequence of control inputs eventually steers the system back to the safe set Xsafe . In principle, it is sufficient that we have an arbitrary convex and bounded RCPI set XRCPI ⊂ X . This set can be inner-approximated by the safety polytope Xsafe to arbitrary precision [18]. Given a controller π, ideally we want to enforce the stateand control constraints at every time step, ∀t ∈ N : fπ (xt ) ∈ X , π(xt ) ∈ U, (5) where xt+1 = fπ (xt ) = f (xt , π(xt )) denotes the closedloop system under π. However, due to limited knowledge of the system, it is in general impossible to design a controller that enforces (5) without additional assumptions. Instead, we slightly relax this requirement to safety with high probability. Definition 1 Let π : Rnx → Rnu be a controller for (1) with the corresponding closed-loop system fπ . Let x0 ∈ X and δ ∈ (0, 1]. A system is δ−safe under the controller π iff: Pr [ ∀t ∈ N : fπ (xt ) ∈ X , π(xt ) ∈ U] ≥ 1 − δ. (6) Based on Definition 1, the goal is to design a control scheme that guarantees δ-safety of the system (1). At the same time, we want to improve our model by learning from observations during operation, which increase the performance of the controller over time. III. BACKGROUND In this section, we introduce the necessary background on GPs and set-theoretic properties of ellipsoids that we need to model our system and perform multi-step ahead predictions. A. Gaussian Processes (GPs) We want to learn the unknown model-error g from data using a GP model. A GP(m, k) is a distribution over functions, which is fully specified through a mean function m : Rd → R and a covariance function k : Rd × Rd → R, where d = nx + nu . Given a set of noisy observations yi = f (zi ) + wi , wi ∼ N (0, λ2 ), i = 1, . . . , n, we choose a zero-mean prior on g as m ≡ 0 and regard the differences ỹn = [y1 − h(z1 ), . . . , yn − h(zn )]T between prior model h and observed system response at input locations Z = [z1 , .., zn ]T . The posterior distribution at z is then given as a Gaussian N (µn (z), σn2 (z)) with mean and variance µn (z) = kn (z)T [Kn + λ2 In ]−1 ỹn σn2 (z) T (7) 2 −1 = k(z, z) − kn (z) [Kn + λ In ] kn (z), (8) where [Kn ]ij = k(zi , zj ), [kn (z)]j = k(z, zj ), and In is the n−dimensional identity matrix. In the case of multiple outputs nx > 1, we model each output dimension with an independent GP, GP(mj , kj ), j = 1, .., nx . We then redefine (7) and (8) as µn (·) = (µn,1 (·), .., µn,nx (·)) and σn (·) = (σn,1 (·), .., σn,nx (·)) corresponding to the predictive mean and variance functions of the individual models. Based on Assumption 1, we can use GPs to model the unknown part of the system (1), which provides us with reliable confidence intervals on the model-error g. Lemma 1 [14, Lemma 2]: Assume \|\|g\|\|k ≤ Bg and that measurements are pcorrupted by λ-sub-Gaussian noise. Let βn = Bg + 4λ γn + 1 + ln(1/δ), where γn is the information capacity associated with the kernel k. Then with probability at least 1−δ we have for all i = 1 ≤ i ≤ nx , z ∈ X × U that \|µn−1,i (z) − gi (z)\| ≤ βn · σn−1,i (z). In combination with the prior model h(z), this allows us to construct reliable confidence intervals around the true dynamics of the system (1). The scaling βn depends on the number of data points n that we gather from the system through the information capacity, γn = maxA⊂Z̃,\|A\|=ñ I(g̃A ; g), Z̃ = X × U × I, ñ = n · nx , i.e. the maximum mutual information between a finite set of samples A and the function g. Exact evaluation of γn is in general NPhard, but γn can be greedily approximated and has sublinear dependence on n for many commonly used kernels [19]. The regularity assumption 1 on our model-error and the smoothness assumption on the covariance function k additionally imply that the function g is Lipschitz. Lemma 2 [20, Lemma 2]: Let g have bounded RKHS norm \|\|g\|\|k ≤ Bg induced by a continuously differentiable kernel k. Then g is Lg -Lipschitz continuous. B. Ellipsoids We use ellipsoids to give an outer bound on the uncertainty of our system when making multi-step ahead predictions. Due to appealing geometric properties, ellipsoids are widely used in the robust control community to compute reachable sets [21], [22]. These sets intuitively provide an outer approximation on the next state of a system considering all possible realizations of uncertainties when applying a controller to the system at a given set-valued input. We briefly review some of these properties and refer to [23] for an exhaustive introduction to ellipsoids and to the derivations for the following properties. We use the basic definition of an ellipsoid, E(p, Q) := {x ∈ Rn \|(x − p)T Q−1 (x − p) ≤ 1}, (9) with center p ∈ R and a symmetric positive definite (s.p.d) shape matrix Q ∈ Rn×n . Ellipsoids are invariant under affine subspace transformations such that for A ∈ Rn×r , r ≤ n with full column rank and b ∈ Rr , we have that n A · E(p, Q) + b = E(p + b, AQAT ). (10) The Minkowski sum E(p1 , Q1 ) ⊕ E(p2 , Q2 ), i.e. the pointwise sum between two arbitrary ellipsoids, is in general not an ellipsoids anymore, but we have that E(p1 , Q1 )⊕E(p2 , Q2 ) ⊂ E p1 +p2 , (1+c−1 )Q1 +(1+c)Q2 (11) for all c > 0. Moreover, the minimizer of the trace of the resulting shape matrix is analytically given as c = p T r(Q1 )/T r(Q2 ). A particular problem that we encounter is finding the maximum distance r to the center of an ellipsoid E := E(0, Q) under a special transformation, i.e. r(Q, S) = max \|\|S(x − p)\|\|2 , x∈E(p,Q) (12) where S ∈ Rm×n with full column rank. We show in the appendix that this is a generalized eigenvalue problem of the pair (Q, S T S) and the optimizer is given as the square-root of the largest generalized eigenvalue. IV. S AFE M ODEL P REDICTIVE C ONTROL In this section, we use the assumptions in Sec. II to design a control scheme that fulfills our safety requirements in Definition 1. We construct reliable, multi-step ahead predictions based on our GP model and use MPC to actively optimize over these predicted trajectories under safety constraints. Using Assumption 2, we use a terminal set constraint to theoretically prove the safety of our method. A. Multi-step Ahead Predictions From Lemma 1 and our prior model h(xt , ut ), we directly obtain high-probability confidence intervals on f (xt , ut ) uniformly for all t ∈ N. We extend this to over-approximate the system after a sequence of inputs (ut , ut+1 , ..). The result is a sequence of set-valued confidence regions that contain the true dynamics of the system with high probability. a) One-step ahead predictions: We compute an ellipsoidal confidence region that contains the next state of the system with high probability when applying a control input, given that the current state is contained in an ellipsoid. In order to approximate the system, we linearize our prior model h(xt , ut ) and use the affine transformation property (10) to compute the ellipsoidal next state of the linearized model. Next, we approximate the unknown model-error g(xt , ut ) using the confidence intervals of our GP model. We finally input u ∈ Rnu , by over-approximating the output of the system f (R, u) = {f (x, u)\|x ∈ R} for ellipsoidal inputs R. Here, we choose p as the linearization center of the state and choose ū = u, i.e. z̄ = (p, u). Since the function f˜µ is affine, we can make use of (10) to compute Fig. 2. Decomposition of the over-approximated image of the system (1) under an ellipsoidal input R0 . The exact, unknown image of f (right, green area) is approximated by the linearized model f˜µ (center, top) ˜ which accounts for the confidence interval and the remainder term d, and the linearization errors of the approximation (center, bottom). The resulting ellipsoid R1 is given by the Minkowski sum of the two individual approximations. f˜µ (R, u) = E(h(z̄) + µ(z̄), AQAT ), resulting again in an ellipsoid. This is visualized in Fig. 2 by the upper ellipsoid in the center. To upper-bound the confidence hyper-rectangle on the right hand side of (21), we upper-bound the term kz − z̄k2 by l(R, u) := apply Lipschitz arguments to outer-bound the approximation errors. We sum up these individual approximations, which result in an ellipsoidal approximation of the next state of the system. This is illustrated in Fig. 2. We formally derive the necessary equations in the following paragraphs. The reader may choose to skip the technical details of these approximations, which result in Lemma 3. We first regard the system f in (1) for a single input vector z = (x, u), f (z) = h(z) + g(z). We linearly approximate f around z̄ = (x̄, ū) via f (z) ≈ h(z̄) + Jh (z̄)(z − z̄) + g(z̄) = f˜(z), (13) where Jh (z̄) = [A, B] is the Jacobian of h at z̄. Next, we use the Lagrangian remainder theorem [24] on the linearization of h and apply a continuity argument on our locally constant approximation of g. This results in an upper-bound on the approximation error, L∇h,i \|\|z − z̄\|\|22 + Lg \|\|z − z̄\|\|2 , (14) \|fi (z) − f˜i (z)\| ≤ 2 where fi (z) is the ith component of f , 1 ≤ i ≤ nx , L∇h,i is the Lipschitz constant of the gradient ∇hi , and Lg is the Lipschitz constant of g, which is defined through Lemma 2. The function f˜ depends on the unknown model error g. We approximate g with the statistical GP model, µ(z̄) ≈ g(z̄). From Lemma 1 we have \|gi (z̄) − µn,i (z̄)\| ≤ βn σn,i (z̄), 1 ≤ i ≤ nx , (15) with high probability. We combine (14) and (15) to obtain L∇h,i \|fi (z) − f˜µ(z),i \| ≤ βn σi (z̄) + \|\|z − z̄\|\|22 + Lg \|\|z − z̄\|\|2 , 2 (16) where 1 ≤ i ≤ nx and f˜µ (z) = h(z̄t ) + Jh (z̄t )(z − z¯t ) + µn (z̄). (17) We can interpret (16) as the edges of the confidence hyperrectangle L∇h m̃(z) = f˜µ (z) ± [βn σn−1 (z̄) + \|\|z − z̄\|\|22 + Lg \|\|z − z̄\|\|2 ], 2 (18) where L∇h = [L∇h,1 , .., L∇h,nx ] and we use the shorthand notation a ± b := [a1 ± b1 ] × [anx ± bnx ], a, b ∈ Rnx . We are now ready to compute a confidence region based on an ellipsoidal state R = E(p, Q) ⊂ Rnx and a fixed (19) max z(x)=(x,u), x∈R \|\|z(x) − z̄\|\|2 , (20) which leads to ˜ u) = βn σn−1 (z̄) + L∇h l2 (R, u)/2 + Lg l(R, u). (21) d(R, Due to our choice of z, z̄, we have that \|\|z(x) − z̄\|\|2 = \|\|x − p\|\|2 and we can use (12) to get l(R, u) = r(Q, Inx ), which corresponds to the largest eigenvalue of Q−1 . Using (20), we can now over-approximate the right side of (21) for inputs R by an ellipsoid ˜ u) ⊂ E(0, Q ˜(R, u)), 0 ± d(R, d (22) where we obtain Qd˜(R, u) by over-approximating the hyper˜ u) with the ellipsoid E(0, Q ˜(R, u)) through rectangle d(R, d √ a ± b ⊂ E(a, nx · diag([b1 , .., bnx ])), ∀a, b ∈ Rnx . This is illustrated in Fig. 2 by the lower ellipsoid in the center. Combining the previous results, we can compute the final over-approximation using (11), R+ = m̃(R, u) = f˜µ (R, u) ⊕ E(0, Qd˜(R, u)). (23) Since we carefully incorporated all approximation errors and extended the confidence intervals around our model predictions to set-valued inputs, we get the following generalization of Lemma 1. Lemma 3 Let δ ∈ (0, 1] and choose βn as in Lemma 1. Then, with probability greater than 1 − δ, we have that: ∀x ∈ R : f (x, u) ∈ m̃(R, u), (24) uniformly for all R = E(p, Q) ⊂ X , u ∈ U. Proof: Define m(x, u) = h(x, u)+µn (x, u)±βn σn−1 + n(x, u). From Lemma 1 we have ∀SR ⊂ X , u ∈ U that, with S high probability, x∈R f (x, u)S⊂ x∈R m(x, u). Due to the over-approximations, we have x∈R m(x, u) ⊂ m̃(R, u). Lemma 3 allows us to compute confidence ellipsoid around the next state of the system, given that the current state of the system is given through an ellipsoidal belief. b) Multi-step ahead predictions: We now use the previous results to compute a sequence of ellipsoids that contain a trajectory of the system with high-probability, by iteratively applying the one-step ahead predictions (23). Given an initial ellipsoid R0 ⊂ Rnx and control input ut ∈ U, we iteratively compute confidence ellipsoids as Rt+1 = m̃(Rt , ut ). (25) We can directly apply Lemma 3 to get the following result. Corollary 1 Let δ ∈ (0, 1] and choose βn as in Lemma 1. Choose x0 ∈ R0 ⊂ X . Then the following holds jointly for all t ≥ 0 with probability at least 1 − δ: xt ∈ Rt , where zt = (xt , ut ) ∈ X × U, R0 , R1 , .. is computed as in (25) and xt is the state of the system (1) at time step t. Proof: Since Lemma 3 holds uniformly for all ellipsoids R ⊂ X and u ∈ U, this is a special case that holds uniformly for all control inputs ut , t ∈ N and for all ellipsoids Rt , t ∈ N obtained through (25). Corollary 1 guarantees that, with high probability, the system is always contained in the propagated belief (25). Thus, if we provide safety guarantees for the propagated belief, we obtain high-probability safety guarantees for the trajectories of (1). c) Predictions under state-feedback control laws: When applying multi-step ahead predictions under a sequence of feed-forward inputs ut ∈ X , the individual sets of the corresponding reachability sequence can quickly grow unreasonably large. This is because these open loop input sequences do not account for future control inputs that could correct deviations from the model predictions. Hence, we extend (23) to affine state-feedback control laws of the form πt (xt ) := Kt (xt − pt ) + ut , (26) where Kt ∈ Rnu ×nx is a feedback matrix and ut ∈ Rnu is the open-loop input. The parameter pt is determined through the center of the current ellipsoid Rt = E(pt , Qt ). Given an appropriate choice of Kt , the control law actively contracts the ellipsoids towards their center. Similar to the derivations (13)-(23), we can compute the function m̃ for affine feedback controllers (26) πt and ellipsoids Rt = E(pt , Qt ). The resulting ellipsoid is m̃(Rt , πt ) = E(h(z̄t )+µ(z̄t ), Ht Qt HtT )⊕E(0, Qd˜(Rt , πt )), (27) where z¯t = (pt , ut ) and Ht = At + Bt Kt . The set E(0, Qd˜(Rt , πt )) is obtained similarly to (20) as the ellipsoidal over-approximation of 0 ± [βn σ(z̄) + L∇h l2 (Rt , St ) + Lg l(Rt , St )], 2 (28) with St = [Inx , KtT ] and l(Rt , St ) = maxx∈Rt \|\|St (z(x) − z¯t )\|\|2 . The theoretical results of Lemma 3 and Corollary 1 directly apply to the case of the uncertainty propagation technique (27). B. Safety constraints The derived multi-step ahead prediction technique provides a sequence of ellipsoidal confidence regions around trajectories of the true system f through Corollary 1. We can guarantee that the system is safe by verifying that the computed confidence ellipsoids are contained inside the polytopic constraints (2) and (3). That is, given a sequence of feedback controllers πt , t = 0, .., T − 1 we need to verify Rt+1 ⊂ X , πt (Rt ) ⊂ U, t = 0, .., T − 1, (29) where (R0 , .., RT ) is given through (25). Since our constraints are polytopes, we have that X = Tm x nx x i=1 Xi , Xi = {x ∈ R \|[Hx ]i,· x − hi ≤ 0}, where x [Hx ]i,· is the ith row of H . We can now formulate the state constraints through the condition Rt = E(pt , Qt ) ⊂ X as mx individual constraints Rt ⊂ Xi , i = 1, .., mx , for which an analytical formulation exists [25], q [Hx ]i,· pt + [Hx ]i,· Qt [Hx ]Ti,· ≤ hxi , ∀i ∈ {1, .., mx }. (30) Moreover, we can use the fact that πt is affine in x to obtain πt (Rt ) = E(kt , Kt Qt , KtT ), using (10). The corresponding control constraint πt (Rt ) ⊂ U is then equivalently given by q u [Hu ]i,· ut + [Hu ]i,· Kt Qt KtT [Hu ]T i,· ≤ hi , ∀i ∈ {1, .., mu }. (31) C. The SafeMPC algorithm Based on the previous results, we formulate a MPC scheme that optimizes the long-term performance of our system, while satisfying the safety condition in Definition 1: minimize Jt (R0 , .., RT ) (32a) subject to Rt+1 = m̃(Rt , πt ), t = 0, .., T − 1 (32b) πt (Rt ) ⊂ U, t = 0, .., T − 1 (32d) π0 ,..,πT −1 Rt ⊂ X , t = 1, .., T − 1 RT ⊂ Xsafe , (32c) (32e) where R0 := {xt } is the current state of the system and the intermediate state and control constraints are defined in (30), (31). The terminal set constraint RT ⊂ Xsafe has the same form as (30) and can be formulated accordingly. The objective Jt can be chosen to suit the given control task. Due to the terminal constraint RT ⊂ Xsafe , a solution to (32) provides a sequence of feedback controllers π0 , .., πT that steer the system back to the safe set Xsafe . We cannot directly show that a solution to MPC problem (32) exists at every time step (this property is known as recursive feasibility) without imposing additional assumption, e.g. on the safety controller πsafe . However, we guarantee that such a sequence of feedback controllers exists at every time step through Algorithm 1 as follows: Given a feasible solution Πt = (πt0 , .., πtT −1 ) to (32) at time t, we apply the first feedback control πt0 . In case we do not find a feasible solution to (32) at the next time step, we shift the previous solution in a receding horizon fashion and append πsafe to the sequence to obtain Πt+1 = (πt1 , .., πtT −1 , πsafe ). We repeat this process until a new feasible solution exists that replaces the previous input sequence. We now state the main result of the paper that guarantees the safety of our system under Algorithm 1. Theorem 2 Let π be the controller defined through algorithm 1 and x0 ∈ Xsafe . Then the system (1) is δ−safe under the controller π. Proof: By induction. Base case: If (32) is infeasible, we are δ-safe using the backup controller πsafe of Assumption 2, Algorithm 1 Safe Model Predictive Control (S AFE MPC) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: Input: Safe policy πsafe , dynamics model h, statistical model GP(0, k). Π0 ← {πsafe , .., πsafe } with \|Π0 \| = T for t = 0, 1, .. do Jt ← objective from high-level planner feasible, Π ← solve MPC problem (32) if feasible then Πt ← Π else Πt ← (Πt−1,1:T −1 , πsafe ) xt+1 ← apply ut = Πt,0 (xt ) to the system (1) since x0 ∈ Xsafe . Otherwise the controller returned from (32) is δ-safe as a consequence of Corollary 1 and the terminal set constraint that leads to xt+T ∈ Xsafe . Induction step: let the previous controller πt be δ-safe. At time step t + 1, if (34) is infeasible then Πt leads to a state xt+T ∈ Xsafe , from which the backup-controller is δ-safe by Assumption 2. If (34) is feasible, then the return path is δ-safe by Corollary 1. D. Optimizing long-term behavior In MPC, we aim to optimize the long-term behavior of our system based on finite-length trajectories. In contrast, the S AFE MPC algorithm guarantees the existence of return strategies. This is undesirable in exploration scenarios, where one aims to explore uncertain areas of the state space. In order to optimize the long-term behavior of the system while maintaining the safety properties of our algorithm, we propose to simultaneously plan a performance trajectory perf s0 , .., sH under a sequence of inputs π0perf , .., πH−1 using a performance-model mperf along with the return strategy that we obtain when solving (32). We do not make any assumptions on the performance model and could be given by one of the approximate uncertainty propagation methods proposed in the literature (see, e.g. [10] for an overview). In order to maintain the safety of our system, we enforce that the first r ∈ {1, ., min{T, H}} controls are the same for both trajectories, i.e. we have that πk = πkperf , k = 0, .., r−1. This extended MPC problem is minimize πt ,..,πt+T −1 perf πtperf ,..,πt+H−1 subject to Jt (st , .., st+H ) (32b) − (32e), t = 0, .., T − 1 st+1 = mperf (st , πtperf ), t = 0, .., H − 1 πt = πtperf , t = 0, .., r − 1, (33) where we replace (32) with this problem in Algorithm 1. The safety guarantees of Theorem 2 directly translate to this setting, since we can always fall back to the return strategy. E. Discussion Algorithm 1 theoretically guarantees that the system remains safe, while actively optimizing for performance via the MPC problem (33). This problem can be solved by commonly used, nonlinear programming (NLP) solvers, such as the Interior Point OPTimizer (Ipopt, [26]). Due to the solution of the eigenvalue problem (12) that is required to compute (23), our uncertainty propagation scheme is not analytic. However, we can still obtain exact function values and derivative information by means of algorithmic differentiation, which is at the core of many state-of-the-art optimization software libraries [27]. One way to further reduce the conservatism of the multistep ahead predictions is to linearize the GP mean prediction µn (xt , ut ), which we omitted for clarity. V. E XPERIMENTS In this section, we evaluate the proposed S AFE MPC algorithm to safely explore the dynamics of an inverted pendulum system. The continuous-time dynamics of the pendulum are given by ml2 θ̈ = gml sin(θ) − η θ̇ + u, where m = 0.15kg and l = 0.5m are the mass and length of the pendulum, respectively, η = 0.1Nms/rad is a friction parameter, and g = 9.81m/s2 is the gravitational constant. The state of the system x = (θ, θ̇) consists of the angle θ and angular velocity θ̇ of the pendulum. The system is controlled by a torque u that is applied to the pendulum. The origin of the system corresponds to the pendulum standing upright. The system is underactuated with control constraints U = {u ∈ R\| − 1 ≤ u ≤ 1}. Due to these limits, the pendulum becomes unstable and falls down beyond a certain angle. We do not impose state constraints, X = R2 . However the terminal set constraint (32e) of the MPC problem (32) acts as a stability constraint and prevents the pendulum from falling. As in [14], our prior model h consists of a linearized and discretized approximation to the true system with a lower mass and neglected friction. The safety controller πsafe is a discrete-time, infinite horizon linear quadratic regulator (LQR, [28]) of the approximated system h with cost matrices Q = diag([1, 2]), R = 20. The corresponding safety region XSafe is given by a conservative polytopic innerapproximation of the true region of attraction of πsafe . We use the same mixture of linear and Matérn kernel functions for both output dimensions, albeit with different hyperparameters. We initially train our model with a dataset (Z0 , ỹ0 ) sampled inside the safe set using the backup controller πSafe . That is, we gather n0 = 25 initial samples Z0 = {z10 , .., zn0 0 } with zi0 = (xi , πsafe (xi )), xi ∈ Xsafe , i = 1, .., n and observed next states ỹ0 = {y00 , .., yn0 0 } ⊂ XSafe . The theoretical choice of the scaling parameter βn for the confidence intervals in Lemma 1 can be conservative and we choose a fixed value of βn = 2 instead. We aim to iteratively collect the most informative samples of the system, while preserving its safety. To evaluate the exploration performance, we use the mutual information I(gZn , g) between the collected samples Zn = {z0 , .., zn } ∪ Z0 and the GP prior on the unknown model-error g, which can be computed in closed-form [19]. 1 1 0 0 0 −1 −1 −1 −1.0 −0.5 0.0 0.5 1.0 Angular velocity θ̇ −1.0 −0.5 0.0 0.5 1.0 −1.0 Angular velocity θ̇ 300 200 100 −0.5 0.0 0.5 Iteration Angle θ 1 1.0 Angular velocity θ̇ Fig. 3. Visualization of the samples acquired in the static exploration setting in Sec. V-A for T ∈ {1, 4, 5}. The algorithm plans informative paths to the safe set Xsafe (red polytope in the center). The baseline sample set for T = 1 (left) is dense around origin of the system. For T = 4 (center) we get the optimal trade-off between cautiousness due to a long horizon and limited length of the return trajectory due to a short horizon. The exploration for T = 5 (right) is too cautious, since the propagated uncertainty at the final state is too large. For a first experiment, we assume that the system is static, so that we can reset the system to an arbitrary state xn ∈ R2 in every iteration. In the static case and without terminal set constraints, a provably close-to-optimal exploration strategy is to, at each iteration n, select state-action pair zn+1 with the largest predictive standard deviation [19] X zn+1 = arg max σn,j (z), (34) z∈X ×U 1≤j≤nx where is the predictive variance (8) of the ith GP(0, ki ) at the nth iteration. Inspired by this, at each iteration we collect samples by solving Pnxthe MPC problem (32) with cost function Jn = − i=1 σn,i , where we additionally optimize over the initial state xn ∈ X . Hence, we visit high-uncertainty states, but only allow for stateaction pairs zn that are part of a feasible return trajectory to the safe set XSafe . Since optimizing the initial state is highly non-convex, we solve the problem iteratively with 25 random initializations to obtain a good approximation of the global minimizer. After every iteration, we update the sample set Zn+1 = Zn ∪{zn }, collect an observation (zn , yn ) and update the GP models. We apply this procedure for varying horizon lengths. The resulting sample sets are visualized for varying horizon lengths T ∈ {1, .., 5} with 300 iterations in Fig. 3, while Fig. 4 shows how the mutual information of the sample sets Zi , i = 0, .., n for the different values of T . For short time horizons (T = 1), the algorithm can only slowly explore, since it can only move one step outside of the safe set. This is also reflected in the mutual information gained, which levels off quickly. For a horizon length of T = 4, the algorithm is able to explore a larger part of the state-space, which means that more information is gained. For larger horizons, the predictive uncertainty of the final state is too large to explore effectively, which slows down exploration initially, when we do not have much information about our system. The results suggest that our approach could further benefit from adaptively choosing the horizon during operation, e.g. by employing a variable horizon MPC approach [29], or by carefully monitoring when the mutual information starts to saturate for the current horizon. 2 σn,i (·) I(gZn , g) A. Static Exploration 100 50 T=1 T=2 0 50 100 Iteration T=3 T=4 150 T=5 200 Fig. 4. Mutual information I(gZn , g), n = 1, .., 200 for horizon lengths T ∈ {1, .., 5}. Exploration settings with shorter horizon gather more informative samples at the beginning, but less informative samples in the long run. Longer horizon lengths result in less informative samples at the beginning, due to uncertainties being propagated over long horizons. However, after having gathered some knowledge they quickly outperform the smaller horizon settings. The best trade off is found for T = 4. B. Dynamic Exploration As a second experiment, we collect informative samples during operation; without resetting the system at every iteration. Starting at x0 ∈ Xsafe , we apply the S AFE MPC,Algorithm 1, over 200 iterations. We consider two settings. In the first, we solve the MPC problem (32) with −Jn given by (34), similar to the previous experiments. In the second setting, we additionally plan a performance trajectory as proposed in Sec. IV-D. We define the states of the performance trajectory as Gaussians st = N (mt , St ) ∈ Rnx × Rnx ×nx and the next state is given by the predictive mean and variance of the current state mt and applied action ut . That is, st+1 = N(mt+1 , St+1 ) with mt+1 = µn (mt , ut ), St+1 = Σn (mt , ut ), t = 0, .., H − 1, (35) where Σn (·) = diag(σn2 (·)) and m0 = xn . This simple approximation technique is known as mean-equivalent uncertainty propagation. We define the cost-function −Jt = PH PT 1/2 T t=1 (mt − pt ) Qperf (mt − pt ), which t=0 trace(St ) + maximizes the confidence intervals along the trajectory s1 , .., sH , while penalizing deviation from the safety trajectory. We choose r = 1 in the problem (33), i.e. the first action of the safety trajectory and performance trajectory are the same. As in the static setting, we update our GP models after every iteration. I(gZ200 , g) 150 100 50 0 T=1 T=2 T=3 T=4 T=5 Fig. 5. Comparison of the information gathered from the system after 200 iterations for the standard setting (blue) and the setting where we plan an additional performance trajectory (green). We evaluate both settings for varying T ∈ {1, .., 5} and fixed H = 5 in terms of their mutual information in Fig. 5. 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Generalized eigenvalue problem Starting at (12), we compute the similar problem r2 (Q, S) = max \|\|S(x − p)\|\|22 = x∈E(p,Q) max sT Q−1 s≤1 sT S T Ss, (36) where we used the definition of ellipsoids and the 2-norm under the variable shift s = x−p. Since S has full column rank and S T S is symmetric positive definite, the maximum of (36) is at the boundary, r2 (Q, S) = maxsT Q−1 s=1 sT S T Ss, which is a generalized eigenvalue problem.	3
PCA from noisy, linearly reduced data: the diagonal case arXiv:1611.10333v1 [math.ST] 30 Nov 2016 Edgar Dobriban∗, William Leeb†, and Amit Singer‡ Abstract Suppose we observe data of the form Yi = Di (Si + εi ) ∈ Rp or Yi = Di Si + εi ∈ Rp , i = 1, . . . , n, where Di ∈ Rp×p are known diagonal matrices, εi are noise, and we wish to perform principal component analysis (PCA) on the unobserved signals Si ∈ Rp . The first model arises in missing data problems, where the Di are binary. The second model captures noisy deconvolution problems, where the Di are the Fourier transforms of the convolution kernels. It is often reasonable to assume the Si lie on an unknown low-dimensional linear space; however, because many coordinates can be suppressed by the Di , this low-dimensional structure can be obscured. We introduce diagonally reduced spiked covariance models to capture this setting. We characterize the behavior of the singular vectors and singular values of the data matrix under high-dimensional asymptotics where n, p → ∞ such that p/n → γ > 0. Our results have the most general assumptions to date even without diagonal reduction. Using them, we develop optimal eigenvalue shrinkage methods for covariance matrix estimation and optimal singular value shrinkage methods for data denoising. Finally, we characterize the error rates of the empirical Best Linear Predictor (EBLP) denoisers. We show that, perhaps surprisingly, their optimal tuning depends on whether we denoise in-sample or out-of-sample, but the optimally tuned mean squared error is the same in the two cases. 1 Introduction Principal component analysis (PCA) is a classical statistical method that decomposes a collection of datapoints s1 , . . . , sn ∈ Rp as a linear combination of vectors that account for the most variability (e.g., Jolliffe, 2002; Anderson, 2003). More formally, if s1 , . . . , sn are drawn from a probability distribution with mean zero and covariance matrix ΣS , then the principal components (PCs) of the distribution are the eigenvectors u1 , . . . , up of ΣS . Typically, we approximate the distribution by projecting it onto the PCs with the largest eigenvalues. ∗ Department of Statistics, Stanford University. E-mail: dobriban@stanford.edu. Supported in part by NSF grant DMS-1407813, and by an HHMI International Student Research Fellowship. † Program in Applied and Computational Mathematics, Princeton University. E-mail: wleeb@math.princeton.edu. Supported by the Simons Collaborations on Algorithms and Geometry. ‡ Department of Mathematics, and Program in Applied and Computational Mathematics, Princeton University. Email: amits@math.princeton.edu. Partially supported by Award Number R01GM090200 from the NIGMS, FA955012-1-0317 from AFOSR, Simons Foundation Investigator Award and Simons Collaborations on Algorithms and Geometry, and the Moore Foundation Data-Driven Discovery Investigator Award. 1 A more specific model arising in many applications is the spiked covariance model (Johnstone, 2001). First, the signal Si is a linear combination of r fixed but unobserved orthonormal PCs uk : Si = r X 1/2 `k zik uk , (1) k=1 where r is a fixed parameter (independent of n and p) and zik are iid standardized random variables. Here `k are the eigenvalues, or equivalently the variances along the PCs uk . Second, the observations are Xi = Si + εi , where εi is noise with iid standardized entries. The spiked covariance model has been widely studied in probability and statistics (e.g., Baik et al., 2005; Baik and Silverstein, 2006; Paul, 2007; Benaych-Georges and Nadakuditi, 2012, etc); see also Paul and Aue (2014); Yao et al. (2015). This paper considers the setting when the vectors Si are not only corrupted by noise, but also linearly reduced. This means that for given matrices Di ∈ Rqi ×p , we observe either Yi = Di Xi = Di Si + Di εi . (2) Yi = Di Si + εi . (3) or We think of the reduction matrix Di as either a projection matrix or a linear filter reducing the information that we observe. In general, it will not be possible to reconstruct a vector v from Di v. We refer to model (2) as the reduced-noise model, and to model (3) as the unreduced-noise model. In the reduced-noise model, both the signal and noise are reduced, while in the unreduced-noise case, only the signal is. These models generalize the spiked covariance model, and arise naturally in several settings. For instance: 1. Missing data: For diagonal matrices Di with zeros or ones the reduced-noise model from (2) corresponds to missing data problems widely encountered in statistics (e.g., Schafer, 1997; Little and Rubin, 2014). For random Di independent of other variables, we are under the assumption of missing completely at random (MCAR). 2. Deconvolution and image restoration: In image processing, an image might be corrupted by “blurring”—convolution with a linear filter—followed by noise. After taking the Fourier transform, this can be modeled as a coordinate-wise multiplication by a diagonal matrix Di , followed by adding noise. This corresponds to the unreduced-noise model from (3). For example, the image formation model in cryo-electron microscopy (cryo-EM) under the linear, weak phase approximation leads to such a model (Frank, 1996). A closely related model was recently used by Bhamre et al. (2016), where Si are Fourier transforms of projection images of molecules, and Di are contrast transfer functions. 3. Structural variability in cryo-EM: In cryo-EM, Si is the three-dimensional structure of a molecule, and Di is a tomographic projection of this volume onto a randomly selected plane. Since the molecule typically has only a few degrees of freedom, such as different conformations or states, it is reasonable to model Si to lie on some unknown low-dimensional space (e.g., Katsevich et al., 2015; Andén et al., 2015). This corresponds to the unreduced-noise model from (3): the data Yi = Di Si + εi are tomographic projections Di Si with added noise εi . 2 4. Signal acquisition and compressed sensing: In some signal acquisition tasks such as hyperspectral imaging, due to resource constraints it is convenient to acquire reduced or compressed measurements of signals. The reductions are often taken to be random projections. It is of interest to reconstruct the PCs of the original measurements (e.g., Chang, 2003; Fowler, 2009). This falls under the reduced-noise model from (2). There may certainly be many other applications fitting this framework. In the above examples it is natural to posit that the distribution of the signal vectors Si is of low effective dimensionality. In this paper we will assume that the distribution lies on some unknown linear space of small dimension r, as in equation (1). Given observations of the form (2) or (3), we address several natural statistical questions: 1. Covariance estimation: How to estimate the covariance matrix of the signals Si ? This is both a fundamental statistical problem, and has numerous applications, including classification and denoising. 2. PCA: How should we estimate the principal components of Si ? This question is of special interest due to the importance of PCA for exploratory data analysis and visualization. 3. Denoising: How can we denoise—or predict—the individual signal vectors Si ? This is a central question both in the missing data problems, where it corresponds to imputation, as well as in the image processing problems, where it amounts to noise reduction. In this paper, we develop new methods for a special class of models, where the matrices Di ∈ Rp×p are diagonal. We will call the observations Yi from (2) or (3) diagonally reduced. The missing data and deconvolution problems belong to this class. In the high-dimensional asymptotic regime where p, n both grow to infinity and p/n → γ > 0, we develop methods that provide clear, quantitative answers to all questions posed above, under quite weak assumptions. Related work by Katsevich et al. (2015) and Andén et al. (2015) develops methods for covariance estimation when the Di ’s are projection matrices mapping a 3-D electron density to its integral on a randomly chosen plane. In this data acquisition model for cryo-EM, the authors propose consistent estimators of the covariance of the electron density. However, their observation models are different from our diagonally reduced models. In Bhamre et al. (2016), the questions of covariance estimation and denoising are studied empirically when the Di ’s come from the contrast transfer function of a microscope and the Si ’s are Fourier transforms of clean tomographic projections. Nadakuditi (2014) develops methods for low-rank matrix estimation with missing data. Our results are more general, and also include methods for covariance estimation and denoising individual datapoints, see Sec. 2.3 for more details. Lounici (2014) develops eigenvalue soft thresholding methods for covariance estimation with missing data. In our somewhat more specialized models, we instead find the optimal eigenvalue shrinkers, and they are different from soft thresholding. Cai and Zhang (2016) develop minimax rate-optimal covariance matrix estimators for missing data, focusing on bandable and sparse models. We instead focus on the spiked covariance model. We next give a brief overview of our results. 1.1 Probabilistic Results A lot of work in random matrix theory studies the asymptotic spectral theory of the spiked covariance model and its variants, see e.g., Paul and Aue (2014); Yao et al. (2015). In Sec. 2, we 3 introduce two general diagonally reduced spiked covariance models corresponding to (2) and (3). We characterize the limiting eigenvalues of the data matrix Y (with rows Yi> ), and the limiting angles of its singular vectors with the population singular vectors (of the matrix S with rows Si> ). More specifically, we show in Thm. 2.1 that the eigenvalue distribution of n−1 Y > Y converges to a general Marchenko-Pastur distribution (Marchenko and Pastur, 1967), while the top few eigenvalues have well-defined almost sure limits. This mirrors the behavior known in unreduced spiked models, (e.g., Baik et al., 2005; Baik and Silverstein, 2006; Benaych-Georges and Nadakuditi, 2012); however, our assumptions in Sec. 2 are very general, and in fact lead to the most general results to date even in the unreduced case when Di = Ip (see Cor. 2.3 and Sec. 2.3 for discussion). In the special case where the entries of Di are iid, the limiting spectrum and the angles between the population and empirical singular vectors are given by explicit formulas related to the standard Marchenko-Pastur law, as described in Cor. 2.2. For general Di , we can compute numerically the quantities specified by Thm. 2.1 with the Spectrode method (Dobriban, 2015); see Sec. 2.4. All computational results of this paper are reproducible with software publicly available at github.com/dobriban/diagonally_reduced/. 1.2 Covariance estimation In Sec. 3, we apply the probabilistic results from Sec. 2 to develop methods for covariance estimation from diagonally reduced data, under the additional assumption that the diagonal entries of the reduction matrices Di are iid. The prototypical example is the missing data problem with uniform missingness, where in the reduced-noise model (2) the entries are Bernoulli(δ). Already for unreduced data from the spiked covariance model, the sample covariance matrix is a poor estimator of the population covariance, since neither the empirical PCs nor the empirical spectrum converge to their population counterparts. Though little can be done about correcting the PCs, one can develop optimal shrinkage estimators of the eigenvalues. In the unreduced case, Donoho et al. (2013) consider estimators of the form U η(Λ)U > , where U is the orthogonal matrix of PCs, Λ is the matrix of eigenvalues of the sample covariance, η : R → R is a shrinkage function, and η(Λ) replaces every diagonal element λ of Λ by η(λ). In Sec. 3, we take a similar approach to the problem of covariance estimation in reduced models. We first define an unbiased estimator of the population covariance of the unreduced signals (Eq. 10). Building on the results of Sec. 2, we describe the asymptotic spectral theory of this estimator, and finally derive shrinkers of the spectrum that are asymptotically optimal for certain loss functions. We explicitly derive the optimal shrinkers in the case of operator norm loss (Eq. (15) for reducednoise and Eq. (23) for unreduced-noise) and Frobenius norm loss (Eq. (18) for reduced-noise and Eq. (24) for unreduced-noise). We also derive the asymptotic errors of the optimal shrinkers (Eq. (17) and (19)), and give a recipe for deriving the optimal shrinkers for a broad class of loss functions. 1.3 Denoising In Sec. 4, we consider denoising, the task of predicting the signal vectors Si based on the observations Yi . For this we study the Best Linear Predictor (BLP) well-known from random effects models (e.g., Searle et al., 2009). The general form of a BLP is Ŝi = ΣS (ΣS + Σε )−1 Yi , where Σε is the covariance matrix of the noise. This is also known as a “linear Bayesian” method (Hartigan, 1969). In other areas such as electrical engineering and signal processing, it is known as the “Wiener 4 filter”, “(linear) Minimum Mean Squared Estimator (MMSE)” (Kay, 1993, Ch. 12), “linear Wiener estimator” (Mallat, 2008, p. 538), or “optimal linear filter” (MacKay, 2003, p. 550-551). The BLP is an “oracle” method, because it depends on unknown population parameters. In practice we can use the empirical BLP (EBLP), Σ̂S (Σ̂S + Σ̂ε )−1 Yi , where the unknown parameters are estimated using the data. Due to the inconsistency of PCA in high dimensions, this is suboptimal to the BLP. However, we can find the asymptotically optimal method for estimating the covariance matrix ΣS using the empirical PCs. This estimator holds several surprises. In particular, the optimal EBLP coefficients are different for in-sample and out-of-sample denoising—but the optimal mean squared error ends up identical! See Thms. 4.1, 4.3 for reduced-noise and Sec. 4.4 for unreduced-noise. It also turns out that the formula for in-sample EBLP, applied to all Yi , is identical to optimal singular value shrinkage estimators (Sec. 4.2.2). This analysis involves characterizing random quantities with an intricate dependence structure, such Yi> Di ûk , where ûk is the k-th PC of the sample covariance matrix of Yi . For this we extend significantly the technique introduced by Benaych-Georges and Nadakuditi (2012) to study the angles between uk and ûk . We call this approach the outlier equation method (see Sec. 4). 2 2.1 Probabilistic results Main probabilistic results This section presents a new result in random matrix theory, which will be the key tool for our work on covariance estimation. Recall that we have diagonally reduced observations Yi = Di (Si + εi ) or Yi = Di Si + εi , i = 1, . . . , n, where Si ∈ Rp are unobserved signals and Di ∈ Rp×p are diagonal Pr 1/2 matrices. The signals have the form Si = k=1 `k zik uk . Here uk are deterministic signal directions with kuk k = 1. We will assume that uk are delocalized, so that \|uk \|∞ ≤ C log(p)B /p1/2 for some constants B, C > 0. The scalars zik are standardized independent random variables, specifying the variation in signal strength from sample to sample. For simplicity we assume that the deterministic spike strengths are different and sorted: `1 > `2 > . . . > `r > 0. Finally εi = Γ1/2 αi is sampling noise, where Γ = diag(g12 , . . . , gp2 ) is diagonal and deterministic, and αi = (αi1 , . . . , αip )> has independent standardized entries. The diagonal matrices Di = diag(Di1 , . . . , Dip ) have the form Di = µ + Σ1/2 Ei , (4) where Ei have independent standardized entries. Here µ = diag(µ1 , . . . , µp ) ∈ Rp×p is the deterministic diagonal mean and Σ = diag(σ12 , . . . , σp2 ) ∈ Rp×p is the deterministic diagonal covariance matrix of the entries of the reduction matrices. Let Hp be the uniform distribution on the p scalars 2 gj2 · EDij = gj2 · (µ2j + σj2 ), j = 1, . . . , p, and let Gp be the analogous object for gj2 , j = 1, . . . , p. It turns out that these are the distributions of noise variances relevant for our models. For dealing with model (2), we assume that as p → ∞, Hp converges to a compactly supported limit distribution H: Hp ⇒ H. For dealing with model (3), we assume that Gp converges to a compactly supported limit distribution G: Gp ⇒ G. We will consider the high-dimensional regime where n, p → ∞ such that p/n → γ > 0. In this setup, our answers will depend on the general Marchenko-Pastur distribution (Marchenko and Pastur, 1967). This law describes the behavior of empirical eigenvalue distributions of sample covariance matrices: If N is an n × p matrix with iid standardized entries, and T is a p × p positive 5 Table 1: Definitions Name Definition Defined in Reduced Observation (2), (3) Signal Yi = Di (Si + εi ) or Yi = Di Si + εi Pr 1/2 Si = k=1 `k zik uk Reduction Matrix Di = µ + Σ1/2 Ei (4) (1) µ = diag(µ1 , . . . , µp ), Σ = diag(σ12 , . . . , σp2 ) Noise Noise Variances εi = Γ1/2 αi , where Γ = diag(g12 , . . . , gp2 ) Pp Hp = p−1 j=1 δgj2 (µ2j +σj2 ) , Hp ⇒ H Pp Gp = p−1 j=1 δgj2 , Gp ⇒ G Stieltjes Transform Fγ,H , F γ,H (x) = γFγ,H (x) + (1 − γ)δ0 R dF γ,H (x) R dFγ,H (x) mγ,H (z) = x−z , mγ,H (z) = x−z R mH (z) = dH(x) x−z D-Transform Dγ,H (x) = x · mγ,H (x) · mγ,H (x) Upper Edge b2H = sup supp(Fγ,H ) General MP Law semidefinite matrix with eigenvalue distribution converging to H, then the eigenvalue distribution of the p × p matrix n−1 T 1/2 N > N T 1/2 converges almost surely (a.s.) to the Marchenko-Pastur distribution Fγ,H (see e.g., Bai and Silverstein, 2009, for a reference). When T = Ip is the identity, Fγ,H is known as the standard Marchenko-Pastur distribution, and has density (if γ ∈ (0, 1)): p (g+ − x)(x − g− ) fγ (x) = I(x ∈ [g− , g+ ]) 2πx √ where g± = (1 ± γ)2 . For general H, Fγ,H does not have a closed form, but it can be studied numerically (see e.g., Dobriban, 2015). Closely related to Fγ,H is the distribution F γ,H (x) = γFγ,H (x) + (1 − γ)δ0 . This is the limit of the eigenvalue distribution Rof the n×n matrix n−1 N > T N . We will also need the Stieltjes transform mγ,H of Fγ,H , mγ,H (z) = (x − z)−1 dFγ,H (x), and the Stieltjes transform mγ,H of F γ,H . Based on these, one can define the D-transform of Fγ,H by Dγ,H (x) = x · mγ,H (x) · mγ,H (x). Up to the change of variables x = y 2 , this agrees with the D-transform defined in BenaychGeorges and Nadakuditi (2012). Let b2H be the supremum of the support of Fγ,H , and Dγ,H (b2H ) = limt↓b Dγ,H (t2 ). It is easy to see that this limit is well defined, and is either finite or +∞. Let us denote the support of a distribution H on R by supp(H). Denote the normalized data matrix Ỹ = n−1/2 Y , with the n × p matrix Y having rows Yi> . Our main probability result, proved later in Sec. 5.1, is the following. Theorem 2.1 (Diagonally reduced spiked models). Consider the observation models (2) and (3), under the above assumptions. Suppose that 6 4 4 1. Eαij < C, EEij < C, and E\|zi \|4+φ < C for some φ > 0 and C < ∞. 2. Under model (2), sup supp(Hp ) → sup supp(H). Under model (3), sup supp(Gp ) → sup supp(G). 3. The squared norms kµuk k2 converge to τk > 0. 4. Under model (2), µuk are generic with respect to M = Γ(Σ + µ2 ) in the sense that −1 u> µuk → I(j = k) · τk · mH (z) j µ(M − zIp ) for all z ∈ C+ , where mH is the Stieltjes transform of H. −1 Under model (3), µuk are generic with respect to M = Γ, i.e., u> µuk → I(j = j µ(M − zIp ) k) · τk · mG (z). Then 1. Under model (2), the eigenvalue distribution of Ỹ > Ỹ converges to the general MarchenkoPastur law Fγ,H a.s. In addition, the k-th largest singular value of Ỹ converges, σk (Ỹ ) → tk > 0 a.s., where −1 if `k > 1/[τk Dγ,H (b2H )], Dγ,H ( τk1`k ) (5) t2k = 2 otherwise. bH Moreover, let νj = µuj /kµuj k be the normalized reduced signals and let ûk be the right singular vector of Ỹ corresponding to σk (Ỹ ). Then (νj> ûk )2 → c2jk a.s., where ( mγ,H (t2k ) if j = k and `k > 1/[τk Dγ,H (b2H )], 0 2 2 D γ,H (tk )τk `k cjk = 0 otherwise. 2. Under model (3), the analogous results hold with G replacing H everywhere. Assumption 4 needs explanation. This assumption generalizes the existing conditions for spiked models. In particular, it is easy to see that it holds when the vectors uk are random with independent coordinates. Specifically, let x, y are two independent random vectors with iid zero-mean entries with variance 1/p. Then Ex> µ(M − zIp )−1 µx = p−1 tr µ(M − zIp )−1 µ. Assumption 4 requires that this converges to τ · mH (z), which happens for instance when the vector µ itself has random independent coordinates with variance τ /p, or when it equals a multiple of the identity. However, Assumption 4 is more general, as it does not require any kind of randomness in uk . Thm 2.1 gives the limiting angles of the empirical eigenvectors ûk with the reduced population eigenvectors νj = µuj /kµuj k. These are in general different from the true eigenvectors uj . However, in our main application (Cor. 2.2 and the following sections) they are the same, because µ is a multiple of identity. One can gain some insight into the result in the simpler case where the noise is uncolored, so that Γ = Ip . In that case, before reduction we have a spike strength ` and an average noise level of unity. P After reduction under model (2), we have a spike strength `kµuk2 , and an average noise 2 level p−1 j EDij = p−1 (kµk2 + kσk2 ), where σ = (σ1 , . . . , σp ). For a delocalized u, we expect 2 −1 2 kµuk ≈ p kµk . Therefore, reduction in model (2) typically decreases the signal strength by a factor of kµk2 . kµk2 + kσk2 7 However, the spike strength after reduction—`kµuk2 —depends on the correlation between µ and u. In particular, the reduction can vary among the different PCs. In contrast it is easy to see in a similar way that reduction in model (3) may increase or decrease the signal strength. The key strength of Thm. 2.1 is its generality. Specifically, there is essentially only one previous result on reduced spiked models, appearing in Nadakuditi (2014). However, that only studies iid Bernoulli projections under restrictive conditions on the noise, whereas we allow for (1) a general diagonal covariance structure Σ in the reduction matrices, as well as (2) a general diagonal noise structure Γ, and (3) more general moment conditions. Moreover, even in unreduced spiked models, our results are already the most general results to date (see Sec. 2.3). We think that the generality is important for several reasons: first, for practical reasons it is good to have results that require as few assumptions as possible, especially unverifiable conditions like “randomness” in uk or “orthogonal invariance” of the noise. As a consequence of these general results, existing tools like singular value shrinkage are shown to apply more generally, so this is a direct improvement. Second, from a theoretical perspective it is good to understand the reason for the “spiking” behavior; our results clarify for instance that “orthogonal invariance” of the noise is not needed. 2.1.1 Comments on the proof The broad outline of the proof is inspired by the argument presented in Benaych-Georges and Nadakuditi (2012) for unreduced spiked models. However, there are several new steps. First, the proof in Benaych-Georges and Nadakuditi (2012) concerns only the unreduced case, and the dependence introduced by the random reduction matrices Di is a new challenge. The observations in model (2) are Di Xi = Di Si + Di εi , so the “signal” Di Si and “noise” Di εi are dependent. However, we show that the dependence is asymptotically negligible. For non-diagonal reduction matrices Di , the depencence may be asymptotically non-negligible; this explains why we currently need the diagonal assumption. As a second novelty, the proof involves finding the limits of certain quadratic forms u> k R(z)uj , where R(z) is a specific resolvent matrix with complex argument z. Since the uk are deterministic, the concentration arguments of Benaych-Georges and Nadakuditi (2012) are not available. Instead, we adapt the “deterministic equivalents” approach of Bai et al. (2007). For this we need to take the imaginary part of the complex argument z to zero, which appears to be a new argument in this context. 2.2 Reduced standard spiked models We will later use the following corollary for the reduced standard spiked model where the reduction coefficients and the noise entries are iid random variables. Suppose that the noise variances are equal to unity, so Γ = Ip and thus εi have independent standardized entries. Moreover assume that the reduction matrices Di have iid random diagonal entries Dij with mean µ = EDij and variance σ 2 = Var[Dij ]. Note that previously µ was a matrix, but from now on it will be a scalar, and there will be no possibility for confusion. Let m = µ2 + σ 2 and δ = µ2 /m. In this case it is easy to see that the reduced eigenvectors are the same as the unreduced ones, i.e., νk = uk . Moreover, Assumption 4 from Thm. 2.1 reduces to u> k ul → 0 as n → ∞, if k 6= l. Our answers can be expressed in terms of the well-known characteristics of the standard spiked model. The asymptotic location of the top singular values will depend on the spike forward map 8 (e.g., Baik et al., 2005): λ(`) = λ(`; γ) = (1 + `) 1 + (1 + γ 1/2 )2 γ ` if ` > γ 1/2 , otherwise. The asymptotic angle between singular vectors will depend on the cosine forward map c(`; γ) ≥ 0 given by (e.g., Paul, 2007; Benaych-Georges and Nadakuditi, 2011, etc): ( 1−γ/`2 if ` > γ 1/2 , 2 2 1+γ/` (6) c(`) = c(`; γ) = 0 otherwise. Corollary 2.2 (Reduced standard spiked models). Under observation model (2), in the above setting, the eigenvalue distribution of m−1 Ỹ > Ỹ converges to the standard Marchenko-Pastur law with aspect ratio γ, a.s. Moreover, m−1/2 σk (Ỹ ) → tk > 0 a.s., where t2k = λ(δ`k ) (7) 2 2 Finally, let ûk be the right singular vector of Ỹ corresponding to σk (Ỹ ). Then (u> j ûk ) → cjk a.s., where 2 c (δ`k ) if j = k c2jk = (8) 0 otherwise. 2 Under observation model (3) in the above setting, we have σk (Ỹ )2 → t2k = λ(µ2 `), while (u> j ûk ) → 2 2 2 2 cjk , where cjk = c (µ `k ) if j = k and 0 otherwise. For the proof, see Sec. 5.3. While in the current paper we only use this corollary of Thm. 2.1, the proof in the special case is essentially as involved as in the general case. For this reason, and for potential future applications, we prefer to state Thm. 2.1 as well. In this special case, reduction in model (2) lowers the spike strength from ` to δ`. This result is related to Thm 2.4 of Nadakuditi (2014) on missing data, but we have the following advantages: (1) our result works under a 4-th order moment condition instead of requiring all bounded moments; (2) our result admits arbitrary diagonal reductions, not just missing data (see Sec. 2.3 for details). Finally, when there is no reduction, i.e., when Di = Ip for all i, then it turns out we do not need the delocalization of uk . Indeed that is only needed to show that the diagonal reductions introduce a negligible amount of dependence, but we do not need this when Di = Ip . Hence, we can state the following corollary for unreduced spiked models: Corollary 2.3 (Standard spiked models). Suppose we observe unreduced signals Yi = Si + εi , and we do not assume of the PCs uk . Suppose that the other assumptions of Prthe delocalization 1/2 Cor. 2.2 hold: Si = k=1 `k zik uk , where u> k uj → δkj , while zik , εij are iid standardized with E\|zi \|4+φ < C, Eε4ij < C for some φ > 0 and C < ∞. Then the conclusions of Cor. 2.2 hold. Specifically, the spectrum of n−1 Y > Y converges to a standard MP law, its spikes converge a.s. to λ(`k ) and the squared cosines between population and sample eigenvectors converge a.s. to c2 (`k ). Again, the key strength of this corollary is its generality, specifically that it only has 4-th moment assumptions, not orthogonal invariance. 9 Figure 1: The effect of reduction on sample spikes and correlations between PCs. Formulas from Thm. 2.1 computed with Spectrode (Dobriban, 2015) overlaid with simulations. See Sec. 2.4. 2.3 Related work There is substantial earlier work on unreduced spiked models, and even in this case our result leads to an improvement. We refer to Paul and Aue (2014); Yao et al. (2015) for general overviews of the area. The paper of Benaych-Georges and Nadakuditi (2012) is closely related to our approach, and we essentially follow their novel technique, relying on controlling certain bilinear forms. When the signal direction u is fixed, their results require the distribution of the noise matrix to be bi-unitarily invariant, which essentially reduces to Gaussian distributions. Our model is more general since it only requires a fourth-moment condition on the noise. The technique introduced by Benaych-Georges and Nadakuditi (2011, 2012) was adapted to non-white Gaussian signal-plus noise matrices Xi = `1/2 zi u + Γ1/2 εi in Chapon et al. (2012). They rely on an integration by parts formula for functionals of Gaussian vectors and the Poincaré-Nash inequality. A Poincaré inequality was also assumed in Capitaine (2013). Our result is stronger, since we only require fourth moment conditions. Previous extensions to the setting of missing data are found in Nadakuditi (2014), which describes the OptShrink method for singular value shrinkage matrix denoising. The method is extended to data missing at random, and in particular the limit spectrum of the data matrix with zeroed-out missing data is found (his Thm. 2.4). This is related to Thm. 2.1, but we have the following advantages: (1) our result works under the optimal 4-th order moment condition instead of requiring all moments to be bounded; (2) our result admits arbitrary reductions, not just binary projection matrices, (3) we extend to reduction matrices Di that have non-iid entries, (4) we allow heteroskedastic diagonal noise εi = Γ1/2 αi ; and (5) we also consider the unreduced-noise model from Eq. (3) (whereas Nadakuditi (2014) considers the reduced-noise model from Eq. (2)). 2.4 A numerical study We report the results of a numerical study to gain insight into our theoretical results. We consider model (2), where Xi = `1/2 zi u + εi , where the noise is heteroskedastic, εi ∼ N (0, Γ), and Γ is 10 the diagonal matrix of eigenvalues of a p × p autoregressive (AR) covariance matrix of order 1, with entries Σij = ρ\|i−j\| . We set Yi = Di Xi , where Di have iid Bernoulli(δ) entries. We set the missingness parameter δ to the values 1/3, 2/3 and 1. We choose the AR autocorrelation coefficient ρ = 0.5, and vary the spikes ` from 0 to 3.5. We compute numerically the formulas in Thm. 2.1, using the recent Spectrode method (Dobriban, 2015), see Sec. 5.1.6 for the details. We compare this with a Monte Carlo simulation with n = 200, γ = 1/2, zi , u generated as Gaussian random variables, and the results averaged over 10 Monte Carlo trials. The results—displayed on Fig. 1—allow us to study the effect of reduction on spiked models. In particular, we observe the following phenomena: • The theory and simulations show good agreement. For eigenvalues, the results are very accurate. For the cosine, the results are more variable, and especially so for small δ. • In the left plot of Fig. 1, we see that the empirical spike is an increasing function of the population spike `. Moreover, the location of the phase transition (PT) decreases with δ, i.e., reduction degrades the critical signal strength. • Similarly, in the right plot of Fig. 1, we see that the cosine between population and empirical PCs increases with the population spike `. For a given `, the cosine decreases as δ → 0. It is not hard, but beyond our scope, to formalize the last two observations into theorems. 3 Covariance matrix estimation In this section, we develop methods for covariance estimation in the reduced-noise model Yi = Di Xi + Di εi (Secs. 3.2, 3.3) and in the unreduced-noise model Yi = Di Xi + εi (Sec. 3.4). We also discuss some related work in Sec. 3.5. Finally, we present numerical experiments illustrating the results in Sec. 3.6. We restrict our attention to a special case of the diagonally reduced model we considered in Sec. 2. We assume as in Sec. 2.2 that the entries of the reduction matrices Di are independently and identically distributed. We also suppose that the noise εi is white, with variance 1 on each coordinate; that is, Cov(εi ) = Ip . We will also require that the coordinates of εi and the diagonal entries of Di both have finite eighth moments. Recall that in the setting of Cor. 2.2, we have 2 µ = EDij and also that σ 2 = Var(Dij ). The second moment is m = EDij = µ2 + σ 2 , and δ = µ2 /m. For data missing uniformly at random, m = µ = δ is the probability that each entry is observed. 3.1 The reduced-noise model In the reduced-noise model, we observe n samples of the random vector Y = D(S + ε). It is then easy to see that we have the following formulas relating the covariance matrix of the signal ΣS and the covariance matrix of the observation ΣY : ΣY = µ2 ΣS + σ 2 diag(ΣS ) + mIp 2 ΣS = σ 1 ΣY − diag(ΣY ) − Ip µ2 mµ2 11 (9) Pn These equations make it clear that the sample covariance matrix Σ̂Y = n−1 i=1 Yi Yi> is a biased estimator of the signal covariance matrix ΣS . Based on the second equation, we consider the following debiased estimator of ΣS : Σ̂S = 1 σ2 Σ̂ − diag(Σ̂Y ) − Ip . Y µ2 mµ2 (10) Here we assume for simplicity that µ, σ 2 are known; but these scalar parameters are straightforward to estimate from the observed Di . In the special case of data missing completely at random, i.e., of iid sampling of entries with probability δ, we have µ2 = δ 2 and m = δ, so this formula becomes 1 1 1 (11) Σ̂S = 2 Σ̂Y + − 2 diag(Σ̂Y ) − Ip . δ δ δ If our goal is to estimate ΣX = ΣS + Ip instead of ΣS , the corresponding unbiased estimator is Σ̂X = Σ̂Y /δ 2 + (δ − 1) diag(Σ̂Y )/δ 2 . This recovers the unbiased estimator of ΣX proposed by Lounici (2014). That paper proposes to estimate ΣX by applying the soft-thresholding function ητ (λ) = (λ − τ )+ to the empirical eigenvalues λ of the covariance Σ̂X . Lounici (2014) proves error bounds for this estimator in both operator and Frobenius norm losses, for covariance matrices ΣX of small effective rank ref f (Σ) = tr(Σ)/kΣkop . In contrast, we want to estimate the covariance matrix ΣS of the signal. For this different task, in the spiked covariance model, the function ητ is not optimal, as we will show in Section 3.3. In the next section, we employ the probabilistic results from Sec. 2 to determine the asymptotic spectral theory of Σ̂Y . In particular, we find asymptotic formulas for the eigenvalues, and the angles between its PCs and those of the population covariance ΣS . Next, we show how to use these results in conjunction with the theory of Donoho et al. (2013) to derive optimal non-linearities η of the spectrum of Σ̂Y to estimate ΣS for a variety of loss functions. 3.2 The asymptotic spectral theory of Σ̂S in reduced-noise In this section, we will analyze the asymptotic spectral theory of the debiased estimator Σ̂S ; that is, the limiting eigenvalue distribution, spikes, and limiting angles of its top eigenvectors with those of ΣS . We will rely on Corollary 2.2 from Section 2 and an argument controlling the diagonal terms in the proof in Sec. 5.4.1. √ Corollary 3.1. Let 1 ≤ k ≤ r. Suppose that `k satisfies `k > γ/δ. Then in the limit p, n → ∞ and p/n → γ, the k th largest eigenvalue of Σ̂S , k = 1, . . . , r, converges almost surely to 1 γ 1 (δ`k + 1) 1 + − . (12) δ δ`k δ The distribution of the bottom p − r eigenvalues of Σ̂S converges to a shifted and scaled Marchenko√ √ Pastur distribution (µM P − 1)/δ supported on the interval [(1 − γ)2 − 1, (1 + γ)2 − 1]/δ. If û0k is the k th eigenvector of Σ̂Y and ûk is the k th eigenvector of Σ̂S , then almost surely we have limn→∞ hû0k , ûk i2 = 1 and lim hûk , uk i2 = n→∞ 12 1 − γ/(δ`k )2 . 1 + γ/(δ`k ) Table 2: Optimal covariance shrinkagepin unreduced-noise model. `˜ is the function defined by equation (16), δ = µ2 /m, and sk = 1 − c2k is the asymptotic sine of the angle between the empirical and population PCs. Loss function Eigenvalue Operator ˜ `(δλ+1) δ Squared Frobenius 2 ˜ ˜ `(δλ+1)·c (`(δλ+1)) δ Asymptotic loss References `1 s1 Pr (15), (17) k=1 (1 − c4k )`2k (18), (19) √ If `k ≤ γ/δ, then the top eigenvalue converges to the upper edge of the shifted MP distribution, and the cosine converges to 0. The shifted Marchenko-Pastur distribution arises as the limiting empirical spectral distribution of the eigenvalues corresponding to noise. This is also the case for the available case sample covariance of pure noise (Jurczak and Rohde, 2015). We will discuss the available-case estimator in Secs. 3.5 and 3.6. 3.3 Optimal shrinkage of the spectrum of Σ̂S in reduced-noise Having characterized the asymptotic spectrum of the debiased estimator Σ̂S , we can apply the technique of Donoho et al. (2013) to derive optimal shrinkers of the eigenvalues of Σ̂S to minimize various loss functions. Any of the 26 loss functions found in Donoho et al. (2013) can be adapted to the setting of diagonally reduced data. In Sec. 5.4.2, we carefully check the details of this program. Write the eigendecomposition of ΣS as ΣS = U ΛU > and the eigendecomposition of the debiased estimator Σ̂S as Σ̂S = Û Λ̂Û . For a given function η : R → [0, ∞), define the matrix Σ̂ηS by Σ̂ηS = Û η(Λ̂)Û > where η(Λ̂) is the diagonal matrix that replaces the k th diagonal element λ̂k of Λ̂ with η(λ̂k ). For any value of p, let Lp (A, B) denote a loss function between two p-by-p symmetric matrices A and B. We consider loss functions with two key properties: first, they must be orthogonally invariant; that is, Lp (A, B) = Lp (U AV, U BV ) for any orthogonal matrices U and V . Second, they must decompose over blocks. This means that if A1 , B1 ∈ Rp1 ×p1 and A2 , B2 ∈ Rp2 ×p2 where p = p1 + p2 , then either Lp (A1 ⊕ A2 , B1 ⊕ B2 ) = max{Lp1 (A1 , B1 ), Lp2 (A2 , B2 )}, in which case we say Lp is max-decomposable; or Lp (A1 ⊕ A2 , B1 ⊕ B2 ) = Lp1 (A1 , B1 ) + Lp2 (A2 , B2 ), in which case we say Lp is sum-decomposable. Operator norm loss Lp (A, B) = kA − Bkop is max-decomposable, whereas squared Frobenius norm loss Lp (A, B) = kA − Bk2F is sum-decomposable. Our goal is to find the function η that minimizes the asymptotic loss over certain classes; that is, we seek: η ∗ = arg min L∞ (ΣS , Σ̂ηS ) η where L∞ (ΣS , Σ̂ηS ) is the almost sure limit of Lp (ΣS , Σ̂ηS ) as n, p → ∞ and p/n → γ. We will show from first principles that this limit is well defined. As in Donoho et al. (2013), we will consider only those functions η that collapse the vicinity of the bulk to 0; that is, for which there is an ε > 0 such 13 that η(λ) = 0 whenever λ ≤ (1 + given in Cor. 3.1). In Sec. 5.4.2, we will show: √ γ)2 /δ − 1/δ + ε (this is the value of the upper bulk edge, as L∞ (ΣS , Σ̂ηS ) = L2r M r A2 (`k ), k=1 r M B2 (η(λk ), ck , sk ) (13) k=1 p ` 0 1 − c2 (`) ≥ 0 the , and ck = c(δ`k ), sk = s(δ`k ), where s(`) = 0 0 asymptotic sine of the angle between the empirical and population PCs, η(λ)c2 (δ`) η(λ)c(δ`)s(δ`) B2 (η(λ), c, s) = . η(λ)c(δ`)s(δ`) η(λ)s2 (δ`) where A2 (`) = Since the loss function is either max-decomposable or sum-decomposable, for η to minimize the right side, it is sufficient that it minimize every individual term L2 (A2 (`k ), B2 (η(λk ), ck , sk )). That is, the asymptotically optimal η minimizes the two-dimensional loss: η ∗ = argmin L2 (A2 (`), B2 (η(λ), c, s)). (14) η This dramatically simplifies the problem, as this minimization can often be done explicitly. Deriving the optimal η now depends on the particular choice of loss function. We consider two representative cases where a simple closed formula is easily found: operator norm loss, and squared Frobenius norm loss. The same recipe of explicitly solving the problem (14) can be used for any orthogonally-invariant and max- or sum-decomposable loss function, including those found in Donoho et al. (2013). 3.3.1 Operator norm loss/max-decomposable losses Since operator norm loss Lp (A, B) = kA − Bkop is max-decomposable, equation (13) implies that L∞ (ΣS , Σ̂ηS ) = max L2 (A(`k ), B(η(λk ), ck , sk )). 1≤k≤r Consequently, the asymptotically optimal η is the one that minimizes the two-dimensional loss function L2 (A(`k ), B(η(λk ), ck , sk )) = kA(`k ) − B(η(λk ), ck , sk )kop . Repeating the derivation in Donoho et al. (2013), the optimal η(λk ) sends λk back to its population value, `k . From formula (12) in Cor. 3.1, η ∗ (λ) = ˜ `(δλ + 1) δ where `˜ inverts the spike forward map ` 7→ λ(`; γ) defined in Sec. 2.2, p 2 ˜ = y − 1 − γ + (y − 1 − γ) − 4γ . `(y) 2 (15) (16) Direct computation shows that L2 (A(`k ), B(η ∗ (λk ), ck , sk )) = `k sk ; consequently, the asymptotic loss is given by the formula: ∗ L∞ (ΣS , Σ̂ηs ) = `1 s1 . 14 (17) ˜ Table 3: Optimal covariance p shrinkage in unreduced-noise model. ` is the function defined by 2 equation (16), and sk = 1 − ck is the asymptotic sine of the angle between the empirical and population PCs. 3.3.2 Loss function Eigenvalue Operator ˜ `(µ λ+1) µ2 Squared Frobenius ˜ 2 λ+1)·c2 (`(µ ˜ 2 λ+1)) `(µ µ2 2 Asymptotic loss References `1 s1 Pr (23), (17) k=1 (1 − c4k )`2k (24), (19) Frobenius norm loss/sum-decomposable losses Since the squared Frobenius loss Lp (A, B) = kA − Bk2F is sum-decomposable, equation (13) implies that L∞ (ΣS , Σ̂ηS ) = r X L2 (A(`k ), B(η(λk ), ck , sk )). k=1 Consequently, the asymptotically optimal η is the one that minimizes the two-dimensional loss function L2 (A(`k ), B(η(λk ), ck , sk )) = kA(`k ) − B(η(λk ), ck , sk )k2F . As derived in Donoho et al. ˜ k + 1)/δ, (2013), the value of η(λk ) that minimizes this is `k c2k . We have already seen that `k = `(δλ 2 2 ˜ where ` is the function defined by (16). Consequently, with c (`) = c (`; γ) being the cosine forward map, the formula for η ∗ (λ) is η ∗ (λ) = ˜ ˜ `(δλ + 1) · c2 (`(δλ + 1)) . δ (18) A straightforward computation shows that L2 (A(`k ), B(`k c2k , ck , sk )) = (1 − c4k )`2k , and consequently, the asymptotic loss is given by the formula: ∗ L∞ (ΣS , Σ̂ηs ) = r X (1 − c4k )`2k . (19) k=1 3.4 The unreduced-noise model In the unreduced-noise model Yi = Di si + εi we can develop similar methods for covariance estimation. The formulas relating ΣS to ΣY are ΣY = µ2 ΣS + σ 2 diag(ΣS ) + Ip (20) 2 ΣS = σ 1 1 ΣY − diag(ΣY ) − Ip . µ2 mµ2 m Consequently, the analogous debiased covariance estimator is: Σ̂S = 1 σ2 1 Σ̂Y − diag(Σ̂Y ) − Ip . 2 2 µ mµ m 15 (21) In this section, we will derive optimal shrinkers of the spectrum of Σ̂S in the unreduced-noise model using the same technique as for the reduced-noise model in Sec. 3.3. Our analysis rests on the following result, proved in Sec. 5.4.1: √ Corollary 3.2. Let 1 ≤ k ≤ r. Suppose that `k satisfies `k > γ/µ2 . Then in the limit p, n → ∞ and p/n → γ, the k th largest eigenvalue of Σ̂S converges almost surely to 1 2 γ 1 (µ ` + 1) 1 + (22) − 2. k 2 2 µ µ `k µ The distribution of the bottom p − r eigenvalues of Σ̂S converges to a shifted Marchenko-Pastur √ √ distribution supported on the interval [(1 − γ)2 − 1, (1 + γ)2 − 1]/µ2 . If û0k is the k th eigenvector of Σ̂Y and ûk is the k th eigenvector of Σ̂S , then almost surely we have limn→∞ hû0k , ûk i2 = 1 and lim hûk , uk i2 = n→∞ 1 − γ/(µ2 `k )2 . 1 + γ/(µ2 `k ) √ If `k ≤ γ/µ2 , then the top eigenvalue converges to the upper edge of the shifted MP distribution, and the cosine converges to 0. ˜ 2 λ̂k + Given an empirical eigenvalue λ̂k of Σ̂S , we estimate the population eigenvalue `k by `(µ 2 1)/µ where `˜ is the function given by formula (16). This estimator converges almost surely to the √ true value `k if `k exceeds the threshold γ/µ2 . This also gives us an estimator of the squared ˜ 2 λ̂k + 1)). We can now derive the optimal non-linear functions on the cosine, by the formula c(`(µ spectrum. For operator norm loss, we have η ∗ (λ) = ˜ 2 λ + 1) `(µ , µ2 (23) ∗ which incurs an asymptotic loss of L∞ (ΣS , Σ̂ηs ) = `1 s1 . For squared Frobenius norm loss, the optimal non-linearity is ˜ 2 λ + 1) · c(`(µ ˜ 2 λ + 1)) `(µ 2 µ Pr ∗ and the asymptotic loss is L∞ (ΣS , Σ̂ηs ) = k=1 (1 − c4k )`2k . η ∗ (λ) = 3.5 (24) Alternative linear systems for estimating ΣS The optimal shrinkers derived in Secs. 3.3 and 3.4 for estimating the covariance ΣS from the reduced-noise observations start from the debiased estimators (10) for reduced-noise and (21) for unreduced noise. Another way of viewing these estimators is as the solution to a linear system: for reduced-noise, this system is given by equation (9), and for unreduced-noise by equation (20). Of course, there are other linear systems yielding unbiased estimators whose spectrum we could shrink. The papers Katsevich et al. (2015); Andén et al. (2015); Bhamre et al. (2016) consider such an estimator, which we will briefly discuss here. 16 By definition, for any k = 1, . . . , n, ΣS = ES,ε [Sk Sk> ], where ES,ε denotes the expectation with respect to the random Sk and εk , but not the Dk . We can therefore write in the unreduced-noise model Yk = Dk Sk + εk : Dk ΣDk = ES,ε [(Dk Sk )(Dk Sk )> ] = ES,ε [(Yk − εk )(Yk − εk )> ] = ES,ε [Yk Yk> ] + Ip . (25) If we knew the values of ES,ε [Yk Yk> ] for every k, we could derive an unbiased estimator of ΣS by solving the n equations given by (25). The papers Katsevich et al. (2015); Andén et al. (2015); Bhamre et al. (2016) instead substitute the observed value Yk Yk> for its expected value, and derive an unbiased estimator of ΣS by the minimization problem n Σ̂S = arg min Σ 1X kDk ΣDk> − Yk Yk> − Ip k2F . n k=1 Differentiating in Σ, we see that Σ̂S must satisfy the linear system: n n n k=1 k=1 k=1 1X > 1X > 1X > Dk Dk ΣDk> Dk = (Dk Yk )(Dk> Yk )> − Dk Dk . n n n We now consider another linear system, defined by averaging the n equations (25): n n k=1 k=1 1X 1X Dk ΣS Dk = ES,ε [Yk Yk> ] − Ip . n n Note that the matrices Dk are fixed in this equation. By replacing ES,ε [Yk Yk> ] with the estimate Yk Yk> , we define a new estimator of ΣS as the solution to the equation n n k=1 k=1 1X 1X Dk ΣDk = Yk Yk> − Ip . n n (26) In the case of reduced-noise, we can repeat the same derivation and arrive at an unbiased estimator that satisfies the system n n n k=1 k=1 k=1 1X 1X 1X 2 Dk ΣDk = Yk Yk> − Dk . n n n (27) We will denote the estimator solving (26) (in the unreduced-noise case) and (27) (in the reduced noise case) by Σ̂0s . Taking the expectation of each side of (27), we arrive at the linear system (20), which defines our estimator Σ̂S in the unreduced-noise model. Similarly, taking the expectation of each side of (26), we arrive at the linear system (9), which defines our estimator Σ̂S in the reducednoise model. Consequently, we expect that, in the limit n, p → ∞, Σ̂S and Σ̂0S will be close. In fact, we can show that the relative error of the estimators converges to 0, as stated in the folowing proposition (proved in Sec. 5.4.3): Proposition 3.3. In both the reduced-noise and unreduced-noise models, the relative difference kΣ̂S − Σ̂0S kF /kΣ̂S kF → 0 almost surely as n, p → ∞ and p/n → γ. 17 3.6 Numerical experiments We perform experiments with missing data, where the diagonal entries of the reduction matrices Di are independent Bernoulli(δ) random variables. In this case, µ = m = δ, and σ 2 = δ(1 − δ). The unbiased estimator to which we apply shrinkage is given by the formula (11). In all experiments, both the signal and the noise are drawn from Gaussian distributions. 3.6.1 The errors in estimating the covariance In the first experiment, we illustrate the dependence of the asymptotic errors on the parameters δ and γ. The clean signal vectors Si are drawn from a rank 1 Gaussian. The noise is also Gaussian noise, of unit variance; the ambient dimension p is fixed at p = 1200 in this experiment, while the number of samples n varies with γ. The top two rows of Fig. 2 shows the errors in estimation for Frobenius loss and operator norm loss, as functions of the parameters γ and δ. The error bars cover the empirical mean error, plus/minus two standard deviations over 200 runs of the experiment. Several phenomena are apparent in these plots. First, the empirical mean of the errors is wellapproximated by the asymptotic error formulas (17) and (19), especially as the number of samples grows (corresponding to smaller γ, as n = p/γ = 1200/γ). Second, the errors decay as δ approaches 1, which is expected as larger δ increases the effective signal strength. Third, the errors grow as γ approaches 1; this is also expected, since large γ leads to higher dimensional problems. 4 4.1 Denoising Setup It is often of interest to denoise the observations Yi and predict the signal components Si . We envision a scenario where the data (Yi , Di ), i = 1, . . . , n is already collected, and we construct the denoisers using this dataset. With in-sample denoising we denoise Yi to predict the signal components Si . This makes sense in many applications where we want to use the entire dataset to construct the denoiser. A closely related scenario is out-of-sample denoising, where we want to denoise a new datapoint (Y0 , D0 ). This arises in applications where new samples are made available after an initial preprocessing of Y1 , . . . , Yn is performed, and it is not desired or not feasible to repeat this processing on the augmented dataset (Y0 , D0 ), (Y1 , D1 ), . . . , (Yn , Dn ) for every new data point. While these two settings are very closely related, it turns out, perhaps surprisingly, that the optimal way to construct the denoisers differs substantially between the two. The reason turns out to be closely related to the observation that in high dimensions, the addition of a single datapoint changes the direction of the PCs. We will explain this phenomenon in detail below. We first study the reduced-noise model from (2) in the setting of Cor. 2.2, where the diagonal entries of Di are drawn iid from a distribution with mean µ and variance σ 2 , and the observations are Yi = Di (Si + εi ). is a special type of random effects model, as the “effects” zik of the PrThis1/2 “factors” uk in Si = k=1 `k zik uk are random from sample to sample. As usual in random effects models, the optimal way to predict Si from a mean squared error (MSE) perspective is to use the Best Linear Predictor—or BLP—(e.g., Searle et al., 2009, Sec. 7.4). The BLP of Si is the predictor Ŝi = M Yi that minimizes EkŜi − Si k2 . −1 It is well known that the BLP is ŜiBLP = Cov [Si , Yi ] Cov [Yi , Yi ] Yi . Under the assumptions of Cor. 2.2, we can show (see Sec. 5.5) that the BLP has the same asymptotic MSE properties as 18 500 γ=.9 γ=.6 γ=.3 450 γ=.9 γ=.6 γ=.3 18 16 errors (operator norm loss) errors (Frobenius loss) 400 350 300 250 200 150 100 14 12 10 8 6 4 2 50 0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 δ 0.6 0.8 1 δ 500 δ=.2 δ=.5 δ=.8 450 δ=.2 δ=.5 δ=.8 18 16 errors (operator norm loss) errors (Frobenius loss) 400 350 300 250 200 150 100 14 12 10 8 6 4 2 50 0 0 0.2 0.4 0.6 0.8 1 γ 0.2 0.4 0.6 0.8 1 γ Figure 2: Top row : Estimation error of the covariance as a function of δ. Left: Frobenius shrinker. Right: operator norm shrinker. For each δ, 200 Monte Carlo tests were averaged. The error bars cover two standard deviations; the lines go through the predicted errors. Bottom row : Same plot as a function of γ. 19 Table 4: Denoising in the single-spiked case. BLP: population Best Linear Predictor using u. EBLP: in-sample empirical Best Linear Predictor using û. EBLP-OOS: out-of-sample empirical Best Linear Predictor using û. Here we abbreviate λ = λ(δ`; γ), c2 = c2 (δ`; γ), s2 = 1 − c2 , β = 1 + γ/(δ`). Name Definition Asy MSE Asy Opt η Asy Opt MSE Ref BLP η · uu> Yi (1 − ηµ)2 ` + η 2 m µ` µ2 `+m m` µ2 `+m Thm. 4.1 EBLP η · ûû> Yi ` + η 2 · mλ − 2η · µ`c2 · β µ`c2 µ2 `+m `· EBLP-OOS η · ûû> Yi ` + η 2 · (µ2 `c2 + m) − 2η · µ`c2 2 µ`c µ2 `c2 +m `· µ2 `c2 s2 +m µ2 `c2 +m 2 2 2 µ `c s +m µ2 `c2 +m Thm. 4.1 Prop. 4.3 a denoiser of the following simpler form: Ŝiτ,B = r X τk uk u> k Yi . (28) k=1 The denoisers Ŝiτ,B are indexed by τ = (τ1 , . . . , τr ), and our argument shows that with the choice τk = µ`k /(µ2 `k + m) they are in fact asymptotically equivalent to BLP denoisers. In practice, the true PCs uk are not known, so we use the Empirical BLP (EBLP), where we estimate the unknown parameters uk using the entire dataset. Here we will use the k-th top right singular vector ûk of the n × p matrix Y with rows Yi> as an estimator of uk . In analogy with the simplified form of the denoisers in (28), we will consider EBLPs scaled by η = (η1 , . . . , ηr ) having the form: r X Ŝiη = ηk ûk û> (29) k Yi . k=1 Our goal is to find the optimal scalars ηk , and characterize their MSE. 4.2 In-sample denoising First, we will study in-sample denoising, where the data to be denoised are also used to construct the denoisers. Our main findings for the asymptotic MSE (AMSE) are summarized in Table 4 (in the single-spiked case) and in the following theorem, proved in Sec. 5.6. Theorem 4.1 (In-sample denoising). In the setting of Cor. 2.2 consider in-sample best linear predictors (BLP) of the signals Si based on the observations Yi . Pr 1. The BLP denoisers Ŝiτ,B = k=1 τk · uk u> k Yi based on the population singular vectors have an AMSE limn,p→∞ EkSi − Ŝiτ,B k2 of AM SE B (τ1 , . . . , τr ; `1 , . . . , `r , γ) = r X k=1 20 AM SE B (τr ; `r , γ), where AM SE B (τ ; `, γ) = (1 − τ µ)2 ` + τ 2 m is the AMSE of the BLP Ŝiτ,B = τ · uu> Yi in a single-spiked model with spike strength ` under the assumptions of Cor. 2.2. The asymptotically optimal coefficients are µ`k τk∗ = 2 . µ `k + m Pr 2. The EBLP denoisers Ŝiη = k=1 ηk · ûk û> k Yi , based on the empirical singular vectors have an AMSE limn,p→∞ EkSi − Ŝiη k2 of AM SE E (η1 , . . . , ηr ; `1 , . . . , `r , γ) = r X AM SE E (ηr ; `r , γ), (30) k=1 where AM SE E (η; `, γ) = ` + η 2 · m · λ(δ`; γ) − 2η · µ` · c2 (δ`; γ) · β is the AMSE of EBLP Ŝiη = η · ûû> Yi in a single-spiked model with spike strength ` under the assumptions of Cor. 2.2. Here λ(δ`; γ) is the limit empirical spike, while c2 = c2 (δ`; γ) is the squared cosine, both corresponding to spike strength δ`, defined in Cor. 2.2. The asymptotically optimal coefficients are ηk∗ = µ`k c2k , µ2 `k + m (31) where c2k = c2 (δ`; γ). The basic discovery is that the optimal coefficients for empirical PCs are different from those for population PCs. The coefficients using empirical PCs are reduced by a squared cosine compared to the coefficients using population PCs: ηk∗ = c2k τk∗ . Note that we chose the optimal coefficients to minimize the limiting MSE. However, the limiting MSE, and thus the optimal coefficients, depend on the unknown parameters δ, `k , c2k . To make this a practical method, we can estimate the unknown parameters. The missingness parameter δ can be estimated by plug-in. Based on Cor. 2.2, the estimation of `k and c2k can be done by inverting the spike forward map `k → λ(`k ), see e.g., Bai and Ding (2012); Donoho et al. (2013). It is worth pointing out that the squared error kSi − Ŝi k2 for each individual column Si of BLP and EBLP does not converge in probability or a.s. In fact, its variability is of unit order, and does not decrease as n, p → ∞. However, as shown in Thm 4.1, its expectation—the MSE—does converge. Furthermore, we will show in Sec. 4.2.2 that the average error of EBLP over the entire data matrix converges almost surely (to the AMSE for a single column). As we will show, this is because EBLP, when applied to all columns of the data matrix, is a singular value shrinkage estimator, defined by modifying the singular values of the data matrix while leaving the empirical singular vectors fixed. As a consequence, in the case of missing data the optimal AMSE agrees with that achieved by optimal singular value shrinkage described in Nadakuditi (2014) and Gavish and Donoho (2014). We emphasize, however, that Thm. 4.1 applies to individual data points, not just to the entire matrix. 4.2.1 Comments on the proof Part 2 of Thm. 4.1 is a nontrivial result, because the AMSE is determined by stochastically dependent random quantities such as u> k Di ûk . These are challenging to study, because Di and ûk are dependent random variables. We will analyze these quantities from first principles. 21 While we will explain our method in detail later, briefly, we use the outlier equation approach (see Lemma 5.14), which reduces studying the inner products w> ûk to certain inner products w> r, where r = r(Y ) are vectors that depend on the entire dataset but not directly on the singular vector ûk . As we will see, these inner products are more convenient to study. The basic method was introduced by Benaych-Georges and Nadakuditi (2012), who used it to study the angles between uk and ûk . We extend their approach to other angles, which are more challenging to study. 4.2.2 Singular value shrinkage and the almost sure convergence of the error in the reduced-noise model A well-studied approach to matrix denoising is known as singular value shrinkage. Here, the singular values of the data matrix Y are replaced with shrunken versions, analogous to the eigenvalue shrinkage of covariance matrices studied in Sec. 3. When applied to every Prrow of the data matrix Y , EBLP is a singular value shrinkage algorithm: indeed, since Ŝiη = k=1 ηk · ûk û> k Yi , we can write the entire denoised matrix in the form Ŝ η = r X ηk · ûk û> kY = k=1 r X ηk σk (Y ) · ûk v̂k> . (32) k=1 In other words, the denoised matrix Ŝ η has the same singular vectors as the data matrix Y , where the singular values have been moved from σk (Y ) to ηk σk (Y ) for k = 1, . . . , r (and the remaining ones set to 0). It turns out that for the value of ηk∗ given by equation (31), ηk σk (Y ) is the optimal singular value of the denoised matrix if we seek to minimize the asymptotic Frobenius loss kS − Ŝ η k2F . The proof of this fact follows easily from the analysis of the optimal singular value shrinkers given by Gavish and Donoho (2014). This paper derives optimal shrinkers only when there is no missing data, and the task is to recover a low rank matrix from noisy observations of its entries. The k th singular value of the asymptotically optimal shrunken matrix (with respect to Frobenius loss) 1/2 is `k ck c̃k , where ck is the asymptotic cosine of the angle between the right population singular vector and the right empirical singular vector, and c̃k is the asymptotic cosine of the angle between the left population singular vector and the left empirical singular vector. In the models considered in Gavish and Donoho (2014), these values are ck = c(`k ; γ), where c is the cosine forward map from (6), and c̃k = c̃(`k ; γ) ≥ 0, where c̃2 (`; γ) = (1 − γ/`2 )/(1 + 1/`). As we can see from Cor. 2.2, in our data model the reduced model behaves like the original model, but with spike strengths reduced from `k to δ`k . In particular, the value of c2k is c2k = c2 (δ`; γ). In fact, it is easy to see from the proof of that this is true for the left singular vectors as well; that is, c̃2 = c̃2 (δ`k ; γ). We now quote the result of Gavish and Donoho (2014) that the optimal singular value is equal 1/2 to `k ck c̃k ; since formula (32) shows that this is equal to ηk∗ σk (Y ), the optimal coefficient ηk∗ is: 1/2 ηk∗ = `k c(δ`k ; γ)c̃(δ`k ; γ) c̃(δ`k ; γ) 1/2 = `k c2 (δ`k ; γ) 1/2 σk (Y ) ` σk (Y )c(δ`k ; γ) k Substituting the asymptotic value for σk (Y ) given from (7), a straightforward algebraic manipulation shows that 1/2 1/2 ηk∗ = `k c2 (δ`k ; γ) `k `k c2k = . δ`k + 1 δ`k + 1 22 This agrees with the value for the optimal coefficient ηk for EBLP derived in Thm. 4.1. In particular, the EBLP estimator we derive for the entire data matrix and the optimal singular value shrinkage estimator are identical. Furthermore, from the analysis of Gavish and Donoho (2014), the error of the entire denoised matrix converges almost surely to the expression given in equation (30). We have proved the following theorem: Theorem 4.2. If Y = [Y1 , . . . , Yn ]> is the n-by-p matrix of observations in the reduced-noise model, then the asymptotically optimal singular value shrinkage estimator of the n-by-p signal matrix S = [S1 , . . . , Sn ]> is equal to the EBLP estimator defined in Thm. 4.1. Furthermore, the squared Frobenius error of this estimator converges a.s. to the formula (30). 4.2.3 Comparison with matrix completion algorithms In the case when the reductions matrices are binary, the task of denoising the data matrix Y to approximate S is a matrix completion problem – see, for instance, Candès and Recht (2009), Recht (2011), Keshevan et al. (2009) and Keshevan et al. (2010). A typical model for matrix completion is a low-rank matrix with eigenvectors satisfying an incoherence condition, with order O(n·poly(log n)) entries revealed uniformly at random. In the setting we study in this paper, the number of observed entries of the matrix Y will be O(n2 ), almost an order of magnitude more. When the noise level is very small compared to the magnitude of the entries in the clean matrix (or put differently, when `r 1), then the smaller number of samples is sufficient to recover the low-rank matrix to high accuracy. The references listed above provide several methods with these guarantees. We compare the in-sample EBLP denoiser to the OptSpace method for matrix completion found in Keshevan et al. (2009) and Keshevan et al. (2010). Briefly, their method removes rows/columns with too many observations, truncates the singular values of the data matrix Y , and then cleans up the resulting matrix by an iterative algorithm. In Figure 3 we plot the squared Frobenius errors in reconstructing a rank 1 matrix, for different choices of spike size ` and missingness parameter δ. In this experiment, both the signal and noise are Gaussian, γ = .8 and the dimension p = 400. Each data point plotted is the average error over 100 Monte Carlo runs of the experiment. We observe that the in-sample EBLP outperforms OptSpace when ` is small relative to the noise level. As the size of ` grows, OptSpace’s performance improves, and for small δ and large ` it outperforms EBLP. This is consistent with the guarantees provided for OptSpace; it does well in the low-noise regime, with a small number of samples. 4.3 Out-of-sample denoising We now study out-of-sample denoising in the reduced-noise model, where we denoise new datapoints from the same distribution using a denoiser constructed on an existing dataset. This is typically faster than recomputing the denoiser on the entire dataset. For the oracle BLP denoiser, which assumes knowledge of u, this is the same as in-sample denoising. For the EBLP, however, it turns out that the optimal shrinkage coefficients in this case are different. To analyze this case, let Y0 = D0 X0 be the new sample from the same distribution. We evaluate the limit of the out-of-sample mean squared prediction error EkS0 − Ŝ0η k2 of the EBLP Pr Ŝ0η = k=1 ηk ûk û> k Y0 , where ûk were formed based on Yi , i ≥ 1. 23 δ = .1 1.8 δ = .7 0.8 EBLP error OptSpace error 1.6 EBLP error OptSpace error 0.7 1.4 0.6 average error average error 1.2 1 0.8 0.6 0.4 0.3 0.2 0.4 0.1 0.2 0 20 0.5 40 60 80 100 120 0 20 140 ℓ 40 60 80 100 120 ℓ Figure 3: Comparison of EBLP and OptSpace for completing/denoising the matrix, for different signal strengths. Left; δ = 1/10. Right: δ = 7/10. Errors are measured as the squared Frobenius norm between the estimator and the full, clean, rank 1 matrix. Every experiment was averaged over 100 runs, with γ = .8 and p = 400. Theorem 4.3. In the setting of Cor. 2.2 consider out-of-sample denoising of a new sample Y0 using the empirical BLP Ŝ0η based on the observations Yi , i = 1, . . . , n. Then the limit of the out-of-sample prediction error is r X η,o E (η1 , . . . , ηr ; `1 , . . . , `r , γ) = E ηk ,o (ηr ; `r , γ), k=1 where E η,o (η; `, γ) = ` + η 2 · (m + µ2 `c2 ) − 2η · µ`c2 is the out-of-sample prediction error of EBLP in a single-spiked model with spike ` under the assumptions of Cor. 2.2. The asymptotically optimal shrinkage coefficients are µ`k c2k ηk∗ = . m + µ2 `k c2k See Sec. 5.10 for the proof. The key point is that the optimal shrinkage for out-of-sample prediction is different from both of the shrinkers from in-sample denoising. We also mention that in the special case when Di = Ip for all i, the optimal shrinkage formula matches the one obtained by Singer and Wu (2013), under slightly more restrictive assumptions. It may be counterintuitive that the optimal coefficient changes when a single data point Y0 is added. However, this can be understood because in high-dimensions, a single data point can drastically change the empirical eigenvectors. For an illustration Pn in a simpler setting, consider the sample covariance matrix based on n samples, Σ̂Y,n = n−1 k=1 Yk Yk> and the corresponding 24 140 sample covariance Σ̂Y,n+1 based on the n + 1 samples Y0 , Y1 , . . . , Yn . Their difference is Σ̂Y,n − Σ̂Y,n+1 = n+1 X 1 1 1 1 Y0 Y0> + Yk Yk> = Y0 Y0> + Σ̂Y,n+1 . n n(n + 1) n n k=1 Since the operator norm of Σ̂Y,n+1 converges a.s. to a finite quantity, the term Σ̂Y,n+1 /n is asymptotically negligible. However, the operator norm of Y0 Y0> /n of size kY0 k2 /n, which converges a.s. to δ. Since the spectral distributions of Σ̂Y,n and Σ̂Y,n+1 converge to the same value, the fact that kΣ̂Y,n − Σ̂Y,n+1 kop is of order 1 is due to the fact that the addition of a single data point Y0 completely changes the direction of the empirical PCs of the data. To summarize and better understand our findings, we plot the optimal shrinkage coefficients and MSE for the three scenarios (BLP, in-sample optimal empirical BLP, and out-of-sample optimal empirical BLP) in a single-spiked model with γ = 1/2 and Di = Ip for all i, on Fig. 4. The optimal shrinkage for out-of-sample EBLP is intermediate between the stronger in-sample EBLP and the weaker out-of-sample BLP shrinkage coefficients. However, perhaps unexpectedly the MSE for the two EBLP scenarios agrees exactly! This prompts us to state the following result, proved in Sec. 5.11. Proposition 4.4 (In-sample vs out-of-sample EBLP). The asymptotically optimal shrinkage coefficient for in-sample EBLP is smaller than the asymptotically optimal shrinkage coefficient for the out-of-sample EBLP. However, the asymptotically optimal MSEs are equal in the two cases. This result is interesting, because it shows that same MSE can be achieved out-of-sample as in-sample. Out-of-sample denoising should be harder, because it involves a new datapoint never seen before. The ”hardness” of out-of-sample denoising should be observed in the leading order finite sample (n) correction to the asymptotic MSE. However, the above result shows that this correction vanishes as n, p → ∞. Using the right amount of shrinkage, the same MSE can be achieved asymptotically even out of sample. 4.4 Unreduced-noise We now study the denoising problem under the unreduced-noise model (3), where Yi = Di Si + εi under the assumptions of Cor. 2.2. The analysis is similar to the reduced-noise model (2). The key conclusions are summarized in Table 5. −1 The BLP of Si based on Yi is ŜiBLP = Cov [Si , Yi ] Cov [Yi , Yi ] Yi . Under the conditions of Cor. Pr 2.2, we can show (see Sec. 5.12.1) that this is asymptotically equivalent to Ŝiτ = k=1 τk uk u> k Yi , where τk = µ`k /(µ2 `k + 1). As before, the AMSE of Siτ with arbitrary τ decouples into the AMSEs over the different spikes `k , and those are equal to the AMSEs for the single-spiked model with spikes equal to `k . For a single-spiked model with spike ` we obtain in Sec. 5.12.1 that EkSi − Ŝiτ k2 → ` + τ 2 (`µ2 + 1) − 2τ `µ. The optimal coefficient is τ ∗ = `µ/(`µ2 + 1)—as it should be, based on the above discussion—and it has an AMSE of `/(`µ2 + 1). The advantage of this calculation is that it provides the MSE for any coefficient τ . Next, for the EBLP in the multispiked case, we use ûk as estimators of uk . Since the form of the BLP is the same as before, the EBLP scaled by η = (η1 , . . . , ηr ) have the form in (29). To compute 25 Figure 4: Optimal shrinkage coefficients (left) and MSE (right) for the three scenarios (BLP, insample optimal empirical BLP, and out-of-sample optimal empirical BLP) for γ = 1/2 and δ = 1. Table 5: Unreduced-noise: Denoising in the single-spiked case. BLP: population Best Linear Predictor using u. EBLP: in-sample empirical Best Linear Predictor using û. EBLP-OOS: out-of-sample empirical Best Linear Predictor using û. Here we abbreviate λ = λ(µ2 `; γ), c2 = c2 (µ2 `; γ). Name Definition Asy MSE Asy Opt η Asy Opt MSE BLP η · uu> Yi ` + η 2 · (µ2 ` + 1) − 2η · µ` µ` µ2 `+1 ` µ2 `+1 EBLP η · ûû> Yi ` + η 2 · λ − 2η · µ`c2 · [1 + γ/(µ2 `)] µ`c2 µ2 `+1 `−λ EBLP-OOS η · ûû> Yi ` + η 2 · (µ2 `c2 + 1) − 2η · µ`c2 µ`c2 µ2 `c2 +1 `· µ`c2 µ2 `+1 2 µ2 `c2 s2 +1 µ2 `c2 +1 the AMSE, it is again not hard to see that it decouples into the corresponding single-spiked AMSEs. In the single-spiked case, we find in Sec. 5.12.2 that with λ = λ(µ2 `; γ), c2 = c2 (µ2 `; γ), EkSi − Ŝiη k2 → ` + η 2 · λ − 2η · µ`c2 · [1 + γ/(µ2 `)]. This shows that the optimal coefficient is η ∗ = µ`c2 /(µ2 ` + 1). Finally, for out-of-sample EBLP denoising, it is again not hard to see that the AMSE decouples over the different spikes, and each term equals the AMSE in the single-spiked case. For the AMSE in the single-spiked case, we let (Y0 , D0 ) be a new sample, and find (Sec. 5.12.3) EkS0 − ηûû> Y0 k2 → ` + η 2 (1 + µ2 `c2 ) − 2ηµ`c2 . The optimal coefficient is η ∗ = µ`c2 /(1+µ2 `c2 ), while the optimal MSE is `(1+µ2 `c2 s2 )/(1+µ2 `c2 ). These findings are summarized in Table 5. 26 4.4.1 Singular value shrinkage and the almost sure convergence of the error in the unreduced-noise model As we discussed in Sec. 4.2.2 for the reduced-noise model, in-sample EBLP applied to every column of the data matrix Y = [Y1 , . . . , Yn ]> is equal to the asymptotically optimal singular value shrinkage estimator of the clean matrix S = [S1 , . . . , Sn ]> . The same reasoning applies verbatim to the unreduced-noise model, replacing the almost sure limits of the empirical eigenvalues and angles with their counterparts for unreduced-noise model. From Cor. 2.2, the value of the asymptotic value of the cosine of the angle between the right empirical singular vector and the right population singular vector is c2k is c2k = c2 (µ2 `; γ); and the same proof of this easily shows that the asymptotic cosine of the angle between the left singular vectors is c̃2 = c̃2 (δ`k ; γ). We now quote the formula from Gavish and Donoho (2014), which says that the optimal singular 1/2 value is equal to `k ck c̃k ; since formula (32) shows that this is equal to ηk∗ σk (Y ), the optimal coefficient ηk∗ is: 1/2 ηk∗ = `k c(δ`k ; γ)c̃(δ`k ; γ) c̃(δ`k ; γ) 1/2 = `k c2 (δ`k ; γ) 1/2 σk (Y ) ` σk (Y )c(δ`k ; γ) k Substituting the asymptotic value for σk (Y ) given from Cor. 2.2, it is easy to see that 1/2 1/2 ηk∗ = `k c2 (δ`k ; γ) `k `k c2k = . δ`k + 1 δ`k + 1 This agrees with the value for the optimal coefficient ηk for EBLP derived in Thm. 4.1. In particular, the EBLP estimator we derive for the entire data matrix and the optimal singular value shrinkage estimator are identical. Furthermore, from the analysis of Gavish and Donoho (2014), the error of the entire denoised matrix converges almost surely to the expression given in equation (30). We have proved the following theorem: Theorem 4.5. If Y = [Y1 , . . . , Yn ]> is the n-by-p matrix of observations in the unreduced-noise model, then the asymptotically optimal singular value shrinkage estimator of the n-by-p signal matrix S = [S1 , . . . , Sn ]> is equal to the EBLP estimator for the unreduced-noise model defined in Sec. 4.4. Furthermore, the squared Frobenius error of this estimator converges a.s. to the AMSE in the unreduced-noise model. 4.5 Simulations Next we perform a simulation to check the finite-sample accuracy of our formulas for denoising. In the reduced-noise model with Di = Ip , we consider a single-spiked model with n = 1000, p = 500, and generate iid Gaussian random variables zi , εi , as well as a standardized iid Gaussian random vector u. We vary the spike strength on a grid, and compare our formulas for in-sample theoretical MSE to those obtained by averaging the denoising error in the first sample kŜ1 − S1 k22 over 50 Monte Carlo simulations. The results in Fig. 5 show that the formulas are accurate up to the sampling error. This validates our results from Thm. 4.1. 27 Figure 5: In-sample MSE of denoising schemes: Theoretical results overlaid with Monte Carlo (MC) results. BLP: Best Linear Predictor assuming known population eigenvector u. Emp BLP: Empirical Best Linear Predictor, using the empirical eigenvector û, with the sub-optimal shrinkage coefficient from the BLP. Opt Emp BLP: Empirical Best Linear Predictor using optimal shrinkage coefficient. No missing data (δ = 1). γ = 1/2. Acknowledgements The authors wish to thank Joakim Anden, Tejal Bhamre, Xiuyuan Cheng, David Donoho, and Iain Johnstone for helpful discussions. 5 5.1 Proofs Proof of Thm. 2.1 The proof of Thm. 2.1 spans multiple sections, until Sec. 5.2.1. The proof of the claims under observation models (2) and (3) are very similar. Therefore, we present the proof of the result under model (2), and outline the argument for model (3) in Sec. 5.1.5. Moreover, to illustrate the idea of the proof, we first prove the single-spiked case, i.e., when r = 1. The proof of the multispiked extension is provided in Sec. 5.2. The form Di = µ + Σ1/2 Ei of the reduction matrices implies the following decomposition for the observations Yi : Yi = (µ + Σ1/2 Ei )Xi = µ(`1/2 zi u + εi ) + Σ1/2 Ei Xi = `1/2 zi µu + [µεi + Σ1/2 Ei Xi ]. This suggests a “signal+noise” decomposition for the reduced vectors Yi . Let us denote by ν = µu/ξ 1/2 the normalized reduced signal, where ξ = kµuk2 → τ , and by ε∗i = µεi +Σ1/2 Ei Xi the noise component. The noise has two parts: µεi is due to sampling, while Σ1/2 Ei Xi is due to projection. 28 In matrix form, with the n × p matrix Y having rows Yi> : Y = (ξ`)1/2 Z̃ν > + E ∗ . (33) This suggests that after reduction, the signal strength ` changes to ξ`, while the noise structure changes from εi to ε∗i . This is not obvious, however, because the noise ε∗i is functionally dependent on the signal Xi . Therefore we cannot rely on existing results. Instead, we will analyze the model from first principles, and show that the dependence is asymptotically negligible. For non-diagonal reduction matrices Di , the depencence may be asymptotically non-negligible; this explains why we currently need the diagonal assumption. 5.1.1 Proof outline We will extend the technique of Benaych-Georges and Nadakuditi (2012) to characterize the spiked eigenvalues in the model (33). We denote the normalized vector Z = n−1/2 Z̃, the normalized noise N = n−1/2 E ∗ and the normalized observable matrix Ỹ = n−1/2 Y . Then, our model is Ỹ = (ξ`)1/2 · Zν > + N. (34) We will assume that n, p → ∞ such that p/n → γ > 0. For simplicity of notation, we will first assume that n ≤ p, implying that γ ≥ 1. It is easy to see that everything works when n ≥ p. By Lemma 4.1 of Benaych-Georges and Nadakuditi (2012), the singular values of Ỹ that are not singular values of N are the positive reals t such that the 2-by-2 matrix t · Z > (t2 In − N N > )−1 Z Z > (t2 In − N N > )−1 N ν 0 (ξ`)−1/2 Mn (t) = > > 2 − ν N (t In − N N > )−1 Z t · ν > (t2 Ip − N > N )−1 ν (ξ`)−1/2 0 is not invertible, i.e., det[Mn (t)] = 0. We will find almost sure limits of the entries of Mn (t), to show that it converges to a deterministic matrix M (t). Solving the equation det[M (t)] = 0 will provide an equation for the almost sure limit of the spiked singular values of Ỹ . For this we will prove the following results: Lemma 5.1 (The noise matrix). The noise matrix N has the following properties: 1. The eigenvalue distribution of N > N converges almost surely (a.s.) to the Marchenko-Pastur distribution Fγ,H with aspect ratio γ ≥ 1. 2. The top eigenvalue of N > N converges a.s. to the upper edge b2H of the support of Fγ,H . This is proved in Sec. 5.1.2. For brevity we write b = bH . Compared to Benaych-Georges and Nadakuditi (2012), the key technical innovation here is to show that the contribution of the reduced signal component Σ1/2 Ei Si to E ∗ is negligible. This is accomplished by an ad-hoc bound on the operator norm of the contribution. Since Ỹ is a rank-one perturbation of N , it follows that the eigenvalue distribution of Ỹ > Ỹ also converges to the MP law Fγ,H . This proves the first claim of Thm 2.1. Moreover, since N N > has the same n eigenvalues as the nonzero of N > N , the two R eigenvalues −1 2 > −1 2 −1 facts in Lemma 5.1 imply that when t > b, n tr(t In −N N ) → (t −x) dF γ,H (x) = −m(t2 ). Here F γ,H (x) = γFγ,H (x) + (1 − γ)δ0 and m = mγ,H is the Stieltjes transform of F γ,H . Clearly this convergence is uniform in t. As a special note, when t is a singular value of the random matrix N , we formally define (t2 Ip − N > N )−1 = 0 and (t2 In − N N > )−1 = 0. When t > b, the complement of this event happens a.s. In fact, from Lemma 5.1 it follows that (t2 Ip − N > N )−1 has a.s. bounded operator norm. Next we control the quadratic forms in the matrix Mn . 29 Lemma 5.2 (The quadratic forms). When t > b, the quadratic forms in the matrix Mn (t) have the following properties: 1. Z > (t2 In − N N > )−1 Z − n−1 tr(t2 In − N N > )−1 → 0 a.s. 2. Z > (t2 In − N N > )−1 N ν → 0 a.s. 3. ν > (t2 Ip − N > N )−1 ν → −m(t2 ) a.s., where m = mγ,H is the Stieltjes transform of the Marchenko-Pastur distribution Fγ,H . Moreover the convergence of all three terms is uniform in t > b + c, for any c > 0. This is proved in Sec. 5.1.3. The key technical innovation is the proof of the third part. Most results for controlling quadratic forms x> Ax are concentration bounds for random x. Here x = ν is fixed, and matrix A = (t2 Ip − N > N )−1 is random instead. For this reason we adopt the “deterministic equivalents” technique of Bai et al. (2007) for quantities x> (zIp − N > N )−1 x, with the key novelty that we can take the imaginary part of the complex argument to zero. The latter observation is nontrivial, and mirrors similar techniques used recently in universality proofs in random matrix theory (see e.g., the review by Erdős and Yau, 2012). Lemmas 5.1 and 5.2 will imply that for t > b, the limit of Mn (t) is −t · m(t2 ) −(τ `)−1/2 . M (t) = −(τ `)−1/2 −t · m(t2 ) By the Weyl inequality, σ2 (Ỹ ) ≤ σ2 ((ξ`)1/2 · Zν > ) + σ1 (N ) = σ1 (N ). Since σ1 (N ) → b a.s. by Lemma 5.1, we obtain that σ2 (Ỹ ) → b a.s. Therefore for any ε > 0, a.s. only σ1 (Ỹ ) can be a singular value of Ỹ in (b + ε, ∞) that is not a singular value of N . It is easy to check that D(x) = x · m(x)m(x) is strictly decreasing on (b2 , ∞). Hence, denoting h = limt↓b D(t2 ), for τ ` > h, the equation D(t2 ) = 1/(τ `) has a unique solution t ∈ (b, ∞). By Lemma A.1 of Benaych-Georges and Nadakuditi (2012), we conclude that for τ ` > h, σ1 (Ỹ ) → t a.s., where t solves the equation det[M (t)] = 0, or equivalently, t2 · m(t2 )m(t2 ) = 1 . τ` If τ ` ≤ h, then we note that det[Mn (t)] → det[M (t)] uniformly on t > b + ε. Therefore, if det[Mn (t)] had a root σ1 (Ỹ ) in (b + ε, ∞), det[M (t)] would also need to have a root there, which is a contradiction. Therefore, we conclude σ1 (Ỹ ) ≤ b+ε a.s., for any ε > 0. Since σ1 (Ỹ ) ≥ σ2 (Ỹ ) → b, we conclude that σ1 (Ỹ ) → b a.s., as desired. This finishes the spike convergence claim in Thm. 2.1. Next, we turn to proving the convergence of the angles between the population and sample eigenvectors. Let Ẑ and û be the singular vectors associated with the top singular value σ1 (Ỹ ) of Ỹ . Then, by Lemma 5.1 of Benaych-Georges and Nadakuditi (2012), if σ1 (Ỹ ) is not a singular value of X, then the vector η = (η1 , η2 ) = (u> û, Z > Ẑ) belongs to the kernel of the matrix Mn (σ1 (Ỹ )). By the above discussion, this 2-by-2 matrix is of course singular, so this provides one linear equation for the vector r (with R = (t2 In − N N > )−1 ) tη1 · Z > RZ + η2 [Z > RN ν − (ξ`)−1/2 ] = 0. By the same lemma cited above, it follows that we have the norm identity (with t = σ1 (Ỹ )) t2 η12 · Z > R2 Z + η22 · ν > N > R2 N ν + 2tη1 η2 · Z > R2 N ν = (ξ`)−1 . 30 (35) This follows from taking the norm of the equation tη1 · RZ + η2 · RN ν = (ξ`)−1/2 Ẑ (see Lemma 5.1 in Benaych-Georges and Nadakuditi (2012)). We will find the limits of the quadratic forms below. Lemma 5.3 (More quadratic forms). The quadratic forms in the norm identity have the following properties: 1. Z > (t2 In − N N > )−2 Z − n−1 tr(t2 In − N N > )−2 → 0 a.s. 2. Z > (t2 In − N N > )−2 N ν → 0 a.s. 3. ν > N > (t2 In − N N > )−2 N ν → m(t2 ) + t2 m0 (t2 ) a.s., where m is the Stieltjes transform of the Marchenko-Pastur distribution Fγ,H . The proof is in Sec. 5.1.4. Again, the key novelty is the proof of the third claim. The standard concentration bounds do not apply, because u is non-random. Instead, we use an argument from complex analysis constructing a sequence of functions fn (t) such that their derivatives are fn0 (t) = ν > N > (t2 In − N N > )−2 N ν, and deducing the convergence fn0 (t) from that of fn (t). R of −1 2 > −2 2 Lemma 5.3 implies that n tr(t In − N N ) → (t − x)−2 dF γ,H (x) = m0 (t2 ) for t > b. Solving for η1 in terms of η2 from the first equation, plugging in to the second, and taking the limit as n → ∞, we obtain that η22 → c2 , where m0 (t2 ) 1 2 2 0 2 c2 + m(t ) + t m (t ) = . 2 2 τ `m(t ) τ` Using D(x) = x · m(x)m(x), we find c2 = m(t2 )/[D0 (t2 )τ `], where t solves (5). From the first equation, we then obtain η12 → c1 , where c1 = m(t2 )/[D0 (t2 )τ `], where t is as above. This finishes the proof of Thm. 2.1 in the single-spiked case. The proof of the multispiked case is a relatively simple extension of the previous argument, so we present it in Sec. 5.2. 5.1.2 Proof of Lemma 5.1 Recall that N = n−1/2 E ∗ , where E ∗ has rows ε∗i = µεi + Σ1/2 Ei Xi . Note ε∗ij = µj εij + σj Eij Xij = [µj + σj Eij ]εij + σj `1/2 Eij zi uj . Since εi = Γ1/2 αi , the terms aij = [µj + σj Eij ]gj εij are independent random variables with 2 2 variance gj2 EDij = gj2 (µ2j + σj2 ). Recall that we assumed that the distribution Hp of gj2 EDij converges weakly to the distribution H. Hence the eigenvalue distribution of the matrix A> A, where A = (n−1/2 aij )ij , converges to Marchenko-Pastur distribution Fγ,H (Bai and Silverstein, 2009, Thm. 4.3). Moreover, since 4 4 2 Eαij < ∞ and EDij < ∞, we have Ea4ij < ∞. In addition, by assumption sup gj2 EDij → sup supp(H). > 2 Thus the largest eigenvalue of A A converges a.s. to the upper edge b of the support of Fγ,H , see Bai and Silverstein (1998) and (Bai and Silverstein, 2009, Cor. 6.6). Therefore, since σj are bounded, it is enough to show that the operator norm of the error matrix E ∗∗ with entries n−1/2 Eij zi uj converges to zero a.s. This will ensure that N = A + E ∗∗ has the same two properties as A above, namely its ESD and operator norm converge. Now, denoting by elementwise products kE ∗∗ k = sup a> E ∗∗ c = n−1/2 kak=kck=1 sup kak=kck=1 31 (a z)> E(c u). We have ka zk ≤ kak max \|zi \| = max \|zi \| and kc uk ≤ kck max \|ui \| = max \|ui \|, hence kE ∗∗ k ≤ kEk max \|zi \| max \|ui \|. Since z has iid standardized entries, and C := Ezi4+φ < ∞, we can derive that P r(max \|zi \| ≥ a) ≤ E max \|zi \|4+φ /a4+φ ≤ nC/a4+φ . 0 0 Taking a = n1/2−φ for φ0 small enough, we obtain, max \|zi \| ≤ n1/2−φ a.s. However, since Eij are iid standardized random variables with bounded 4-th moment, kEk → √ 1 + γ a.s. (Bai and Silverstein, 2009). Since kuk∞ ≤ C log(p)B /p1/2 , we obtain kE ∗∗ k ≤ C(1 + 0 √ γ) · log(p)B n1/2−φ p−1/2 → 0 a.s., as required. 5.1.3 Proof of Lemma 5.2 Since N = A + E ∗∗ , and kE ∗∗ k → 0 a.s., it is enough to show the same concentration statements for A instead of N . Indeed, it is easy to see that the error terms are all negligible. Part 1: For Z > (t2 In − AA> )−1 Z, note that Z has iid entries —with mean 0 and variance 1/n—that are independent of A. We will use the following result: Lemma 5.4 (Concentration of quadratic forms, consequence of Lemma B.26 in Bai and i h √ Silverstein k (2009)). Leth x ∈ R be ia random vector with i.i.d. entries and E [x] = 0, for which E ( kxi )2 = 1 √ and supi E ( kxi )4+φ < C for some φ > 0 and C < ∞. Moreover, let Ak be a sequence of random k × k symmetric matrices independent of x, with a.s. uniformly bounded eigenvalues. Then the quadratic forms x> Ak x concentrate around their means: x> Ak x − k −1 tr Ak →a.s. 0. We apply this lemma with x = Z, k = p and Ap = (t2 In − AA> )−1 . To get almost sure convergence, here it is required that zi have finite 4 + φ-th moment. This shows the concentration of Z > (t2 In − AA> )−1 Z. Part 2: To show Z > (t2 In − AA> )−1 Aν concentrates around 0, we note that w = (t2 In − AA> )−1 Aν is a random vector independent of Z, with a.s. bounded norm. Hence, conditional on w: X X 4 2 2 P r(\|Z > w\| ≥ a\|w) ≤ a−4 E\|Z > w\|4 = a−4 [ EZni wi4 + EZni EZnj wi2 wj2 ] i −4 ≤a X 4 EZn1 ( wi2 )2 i6=j −4 −2 =a n EZ14 · kwk42 i For any C we can write P r(\|Z > w\| ≥ a) ≤ P r(\|Z > w\| ≥ a\|kwk ≤ C) + P r(kwk > C). For sufficiently large C, the second term, P r(kwk > C) is summable in n. By the above bound, the first term is summable for any C. Hence, by the Borel-Cantelli lemma, we obtain \|Z > w\| → 0 a.s. This shows the required concentration. Part 2: Finally we need to show that ν > (t2 Ip − A> A)−1 ν concentrates around a definite value. This is probably the most interesting part, because the vector u is not random. Most results for controlling expressions of the above type are designed for random u; however here the matrix A is random instead. For this reason we will adopt a different approach. 32 Under our assumption we have ν > (Γ(Σ + µ2 ) − zIp )−1 ν → mH (z), for z = t2 + iv with v > 0 fixed. Therefore, Thm 1 of Bai et al. (2007) shows that ν > (zIp − A> A)−1 ν → −m(z) a.s., where m(z) is the Stieltjes transform of the Marchenko-Pastur distribution Fγ,H . A close examination of their proofs reveals that their result holds when v → 0 sufficiently slowly, for instance v = n−α for α = 1/10. The reason is that all bounds in the proof have the rate N −k v −l for some small k, l > 0, and hence they converge to 0 for v of the above form. For instance, the very first bounds in the proof of Thm 1 of Bai et al. (2007) are in Eq. (2.2) on page 1543. The first one states a bound of order O(1/N r ). The inequalities leading up to it show that the bound is in fact O(1/(N r v 2r )). Similarly, the second inequality, stated with a bound of order O(1/N r/2 ) is in fact O(1/(N r/2 v r )). These bounds go to zero when v = n−α with small α > 0. In a similar way, the remaining bounds in the theorem have the same property. To get the convergence for real t2 from the convergence for complex z = t2 + iv, we note that \|ν > (zIp − A> A)−1 ν − ν > (t2 Ip − A> A)−1 ν\| = v\|ν > (zIp − A> A)−1 (t2 Ip − A> A)−1 ν\| ≤ ≤ vk(t2 Ip − A> A)−1 k2 · u> u. As discussed above, when t > b, the matrices (t2 Ip − A> A)−1 have a.s. bounded operator norm. Hence, we conclude that if v → 0, then a.s. ν > (zIp − A> A)−1 ν − ν > (t2 Ip − A> A)−1 ν → 0. Finally, m(z) → m(t2 ) by the continuity of the Stieltjes transform for all t2 > 0 (Bai and Silverstein, 2009). We conclude that ν > (t2 Ip − A> A)−1 ν → −m(t2 ) a.s. This finishes the analysis of the last quadratic form. 5.1.4 Proof of Lemma 5.3 As in Lemma 5.2, it is enough to show the same concentration statements for A instead of N . Parts 1 and 2: The proof of Part 1 and 2 are exactly analogous to those in Lemma 5.2. Indeed, the same arguments work despite the change from (t2 Ip − N > N )−1 to (t2 Ip − N > N )−2 , because the only properties we used are its independence from Z, and its a.s. bounded operator norm. These also hold for (t2 Ip − N > N )−2 , so the same proof works. Part 3: We start with the identity ν > N > (t2 In − N N > )−2 N ν = −ν > (t2 Ip − N > N )−1 ν + 2 > 2 t ν (t Ip − N > N )−2 u. Since in Lemma 5.2 we have already established ν > (t2 Ip − N > N )−1 ν → −m(t2 ), we only need to show the convergence of ν > (t2 Ip − N > N )−2 u. For this we will employ the following derivative trick (see e.g., Dobriban and Wager, 2015). We will construct a function with two properties: (1) its derivative is the quantity ν > (t2 Ip − N > N )−2 u that we want, and (2) its limit is convenient to obtain. The following lemma will allow us to get our answer by interchanging the order of limits: Lemma 5.5 (see Lemma 2.14 in Bai and Silverstein (2009)). Let f1 , f2 , . . . be analytic on a domain D in the complex plane, satisfying \|fn (z)\| ≤ M for every n and z in D. Suppose that there is an analytic function f on D such that fn (z) → f (z) for all z ∈ D. Then it also holds that fn0 (z) → f 0 (z) for all z ∈ D. Accordingly, consider the function fp (r) = −ν > (rIp − N > N )−1 ν. Its derivative is fp0 (r) = ν (rIp − N > N )−2 u. Let S := {x + iv : x > b + ε} for a sufficiently small ε > 0, and let us work on > 33 the set of full measure where kN > N k < b+ε/2 eventually, and where fp (r) → m(r). By inspection, fp are analytic functions on S bounded as \|fp \| ≤ 2/ε. Hence, by Lemma 5.5, fp0 (r) → m0 (r). In conclusion, ν > N > (t2 Ip − N > N )−2 N ν → m(t2 ) + t2 m0 (t2 ), finishing the proof. 5.1.5 Proof of Thm. 2.1: Model (3) The proof for model (3) is very similar to that for model (2). Therefore, we only present the outline. Working again in the single-spiked case for simplicity, we have the following decomposition for the observations Yi : Yi = (µ + Σ1/2 Ei )Si + εi = `1/2 zi µu + [µEi Si + εi ]. Denoting ν = µu/ξ 1/2 , where ξ = kµuk2 → τ , and ε∗i = µEi Si + εi , we have in matrix form Y = (ξ`)1/2 Z̃ν > + E ∗ . As in the proof of Lemma 5.1, it is not hard to see that the operator norm kE ∗ − Ekop → 0. Therefore, the spectral properties of Y are equivalent to those of Ỹ = (ξ`)1/2 Z̃ν > + E. However, this is now a spiked model where the signal component is independent of the noise component. It follows immediately from Thm 4.3 of Bai and Silverstein (2009) that the singular value distribution of E = [α1 , . . . , αn ]> Γ1/2 converges to the general Marchenko-Pastur distribution Fγ,G , where G is the limit of the distributions Gp of gj2 , j = 1, . . . , p. Similarly the top singular value of E converges to the upper edge bG of Fγ,G . Moreover, it also follows that the analogues of Lemmas 5.2 and 5.3 hold in our case. Indeed, the same arguments carry through, because the same assumptions hold. This allows the entire argument from Sec. 5.1.1 to carry through, finishing the single-spiked case of Thm. 2.1. The extension to the multispiked case is analogous to that in model (2). 5.1.6 Numerical computation of the quantities from Thm. 2.1 Spectrode computes the Marchenko-Pastur forward map: given an input limit population spectrum H and an aspect ratio γ, it outputs an accurate numerical approximation to the limit empirical spectral distribution (ESD) Fγ,H . Dobriban (2015) established the numerical convergence of the method, and showed in experiments that it is much faster than previous proposals. The method is publicly available at http://github.com/dobriban/eigenedge. The output of Spectrode includes a numerical approximation m̂ to the Stieltjes transform m of the limit ESD, computed over a dense grid xi on the real line. It also includes an approximation b̂2 of the upper edge b2 of the ESD. From this, we compute an approximation of the D-transform as D̂(xi ) = xi · m̂(xi ) · m̂(xi ), where m̂(xi ) = γ · m̂(xi ) + (γ − 1)/xi . Since D is monotone decreasing on (b2 , ∞), we find the smallest grid point xi such that D̂(xi ) ≤ 1/` to approximately compute D−1 (1/`). Finally, the derivative m0 (x) can be expressed as a function m0 (x) = F(m(x)) by differentiating the Marchenko-Pastur fixed-point equation (see e.g., Dobriban, 2015). Therefore, we compute a numerical approximation to D0 (x) = m(x) · m(x) + x[m(x) · m0 (x) + m0 (x) · m(x)] by approximating m̂0 (x) via the same function m̂0 (x) = F(m̂(x)). Similarly we approximate m̂0 . With these steps, we obtain a full numerical implementation of Thm. 2.1. 34 5.2 Proof of Thm. 2.1 - Multispiked extension 1/2 Let us denote by ui = µui /ξi the normalized reduced signals, where ξiP = kµui k2 → τi , and by r ∗ 1/2 εi = µεi +Σ Ei Xi . For the proof we start as in Sec. 5.1.1, obtaining Yi = k=1 (ξk `k )1/2 zik νk +ε∗i . Defining the r × r diagonal matrices L, ∆ with diagonal entries `k , ξk (respectively), and the n × r, p × r matrices Z, V, with columns Zk = n−1/2 (z1k , . . . , znk )> and uk respectively, we have Ỹ = Z(∆L)1/2 U > + E ∗ . The matrix Mn (t) is now 2r × 2r, and has the form t · Z > (t2 In − N N > )−1 Z Z > (t2 In − N N > )−1 N V 0r Mn (t) = − V > N > (t2 In − N N > )−1 Z t · V > (t2 Ip − N > N )−1 V (∆L)−1/2 (∆L)−1/2 . 0r It is easy to see that Lemma 5.1 still holds in this case. To find the limits of the entries of Mn , we need the following additional statement. Lemma 5.6 (Multispiked quadratic forms). The quadratic forms in the multispiked case have the following properties for t > b: 1. Zk> Rα Zj → 0 a.s. for α = 1, 2, if k 6= j. 2. νk> (t2 Ip − N > N )−α νj → 0 a.s. for α = 1, 2, if k 6= j. This lemma is proved in Sec. 5.2.1, using similar techniques as those in Lemma 5.1. Defining the r × r diagonal matrices T with diagonal entries τk , we conclude that for t > b, Mn (t) → M (t) a.s., where now −t · m(t2 )Ir −(T L)−1/2 . M (t) = −(T L)−1/2 −t · m(t2 )Ir As before, by Lemma A.1 of Benaych-Georges and Nadakuditi (2012), we get that for τk `k > 1/D(b2 ), σk (Ỹ ) → tk a.s., where t2k · m(t2k )m(t2k ) = 1/(τk `k ). This finishes the spike convergence proof. To obtain the limit of the angles for ûk for a k such that `k > τk D(b2 ), consider the left singular vectors Ẑk associated to σk (Ỹ ). Define the 2r-vector (∆L)1/2 V > ûk β α= 1 = . β2 (∆L)1/2 Z > Ẑk The vector α belongs to the kernel of Mn (σk (Ỹ )). As argued by Benaych-Georges and Nadakuditi (2012), the fact that the projection of α into the orthogonal complement of M (tk ) tends to zero, implies that αj → 0 for all j ∈ / {k, k + r}. This proves that νj> ûk → 0 for j 6= k, and the analogous claim for the left singular vectors. The linear equation Mn (σk (Ỹ ))α = 0 in the k-th coordinate, where k ≤ r, reads (with t = σk (Ỹ )): X tαk Zk> RZk − αr+k (ξk `k )−1/2 + Mn (σk (Ỹ ))ik αk = 0. i6=k 35 Only the first two terms are non-negligible due to the behavior of Mn , so we obtain tαk Zk> RZk = αr+k (ξk `k )−1/2 + op (1). Moreover taking the norm of the equation Ẑk = R(tZβ1 + N Vβ2 ) (see Lemma 5.1 in Benaych-Georges and Nadakuditi (2012)), we get X X X t2 αi αj Zi> R2 Zj + αk+i αk+j νi> N > R2 N νj + αi αk+j Zi R2 N νj = 1. i,j≤r i,j≤r i,j≤r 2 From Lemma 5.6 and the discussion above, only the terms αk2 Zk> R2 Zk and αr+k νk> N > R2 N νk are non-negligible, so we obtain 2 t2 αk2 Zk> R2 Zk + αr+k νk> N > R2 N νk = 1 + op (1). Combining the two equations above, Zk> R2 Zk > > 2 2 + ν N R N ν αr+k k = 1 + op (1). k ξk `k (Zk> RZk )2 Since this is the same equation as in the single-spiked case, we can take the limit in a completely analogous way. This finishes the proof. 5.2.1 Proof of Lemma 5.6 As in Lemma 5.2, it is enough to show the same concentration statements for A instead of N . Part 1: The convergence Zk> Rα Zj → 0 a.s. for α = 1, 2, if k 6= j, follows directly from the following well-known lemma, cited from Couillet and Debbah (2011): Lemma 5.7 (Proposition 4.1 in Couillet and Debbah (2011)). Let xn ∈ Rn and yn ∈ Rn be independent sequences of random vectors, such that for each n the coordinates of xn and yn are independent random variables. Moreover, suppose that the coordinates of xn are identically distributed with mean 0, variance C/n for some C > 0 and fourth moment of order 1/n2 . Suppose the same conditions hold for yn , where the distribution of the coordinates of yn can be different from those of xn . Let An be a sequence of n × n random matrices such that kAn k is uniformly bounded. Then x> n An yn →a.s. 0. Part 2: To show νk> (t2 Ip − N > N )−α νj → 0 a.s. for α = 1, 2, if k 6= j, the same technique cannot be used, because the vectors uk are deterministic. However, it is straightforward to check that the method of Bai et al. (2007) that we adapted in proving Part 3 of Lemma 5.2 extends to proving νk> (t2 Ip − N > N )−1 νj → 0. Indeed, it is easy to see that all their bounds hold unchanged. In the final step, as a deterministic equivalent for νk> (t2 Ip − N > N )−1 νj → 0, one obtains νk> (t2 Ip − t2 m(t2 )Σ)−1 νj , which tends to 0 by our assumption, showing νk> (t2 Ip − N > N )−1 νj → 0. Then νk> (t2 Ip − N > N )−2 νj → 0 follows from the derivative trick employed in Part 3 of Lemma 5.3. This finishes the proof. 5.3 Proof of Corollary 2.2 This corollary is a special case of our previous results, so the convergence results hold in this case. We only need to check that the limits are given by the formulas provided. Since very similar analysis has been performed by Benaych-Georges and Nadakuditi (2012) and Nadakuditi (2014), we only give part of the proof. 36 Under model (2), since we consider the singular values of the normalized matrix m−1/2 Ỹ instead of Ỹ as in Thm 2.1, it is easy to see that the relevant equation for the limiting singular values in this case is t2 · m0 (t2 )m0R(t2 ) = 1/δ` instead of t2 · m(t2 )m(t2 ) = 1/τ `. Note that m0 (t2 ) = (x − t2 )−1 dF γ,H (x) = γ · m0 (t2 ) + (γ − 1)t−2 . Thus the equation for t2 reads (γt2 m0 (t2 )+γ −1)m0 (t2 ) = (δ`)−1 . However, it is well-known that m0 (t2 ) obeys the equation γt2 m20 + (t2 + γ − 1)m0 + 1 = 0. From these two relations we obtain t2 · m0 (t2 ) = −1 − 1/(δ`). Plugging this back into the second equation and simplifying, we obtain t2 = (δ` + 1)(1 + γ/(δ`)), as required. Finally, under model (3), the proof is analogous. 5.4 5.4.1 Proofs for covariance estimation Proof of Cor. 3.2 From the spectral analysis of Y contained in Corollary 2.2 from Section 2 and the formula (10), the proof of this corollary is immediate from the first part of the following lemma, proved below: Lemma 5.8. We have the following limits (in operator norm) of the diagonals: 1. limn→∞ k diag(Σ̂Y ) − m · Ip k = 0 a.s. 2. limn→∞ k diag(Σ̂S ) − Ip k = 0 a.s. First, note that the second statement is an immediate corollary of the first. In the proof of the first statement, for notational convenience only, we will assume that r = 1; however, an identical proof goes through for any fixed r. We will therefore drop the subscript k for the proof. Decompose Xi into signal and noise, writing Xi = Si + εi , where Si = `1/2 zi u and Cov(ε) = Ip , and Si and εi are independent. So if Si = (si1 , . . . , sip )> , Xi = (Xi1 , . . . , Xip )> , Yi = (Yi1 , . . . , Yip )> , and εi = (εi1 , . . . , εip )> , we can write Yij = Dij Xij = Dij sij + Dij εij and con2 2 2 2 2 sequently Yij2 = Dij sij + 2Dij sij εij + Dij εij . th The (i, i) element of diag(Σ̂Y ) is then n n n n 1X 2 2 1X 1X 2 2 1X 2 2 Yij = Dij sij + 2Dij sij εij + D ε n j=1 n j=1 n j=1 n j=1 ij ij and we will controleach of the three sums on the right side separately. P n 2 2 sij = m`u2i and so Observe that E n1 j=1 Dij X n 1 1 log4B (n) 2 2 Dij sij = (E[d4 ]E[z 4 ]`2 u4i − m2 `u4i ) ≤ c Var n j=1 n n3 where c > 0 is a constant. Chebyshev’s inequality then gives X n 1 p log4B (n) 2 2 2 2 P Dij sij − m · ` · ui ≥ ε for some i ≤ c . n j=1 ε2 n3 Since u2i ≤ C logB (p)/p, this shows that n−1 p/n → γ. Pn j=1 37 2 2 Dij sij converges a.s. to 0 as p, n → ∞ and 2 2 For the sum of terms Dij εij , observe that En−1 Pn j=1 2 2Dij sij εij = 0, and 2 E\|2Dij sij εij \|4 = 16E(d8 )`2 u4i ≤ c log4B (n) n2 √ where c > 0 is a constant and we have used the estimate \|ui \| ≤ C logB (p)/ p. By using Markov’s inequality for the fourth moment (see Petrov (2012)), we then obtain: X n 1 log4B (n) 2 2Dij P sij εij ≥ ε for some i ≤ c 4 2 . n j=1 ε n Pn 2 s ε converges a.s. to 0 as p, n → ∞ and p/n → γ. This proves that n−1 j=1 2Dij Pn ij ij2 2 −1 2 2 2 2 4 8 8 Finally, to deal with n j=1 Dij εij , observe that EDij εij = m, and that E\|Dij εij \| = Ed Eε which is assumed finite. Therefore using again Markov’s inequality for the fourth moment, X n 1 Ed8 Eε8 2 2 P Dij εij − m ≥ ε for some i ≤ cp 4 4 . n j=1 ε n This proves that n−1 proof. 5.4.2 Pn j=1 2 2 Dij εij converges a.s. to 0 as p, n → ∞ and p/n → γ. This finishes the Proof of (13) The following analysis is adapted from that in Donoho et al. (2013). We start with a fact from linear algebra. Lemma 5.9. Suppose A and B are two p-by-p symmetric matrices, with eigenvectors u1 , . . . , up and v1 , . . . , vp . Suppose that each one has eigenvalue 0 with multiplicity p−r for some 1 ≤ r < p/2, with corresponding eigenvectors ur+1 , . . . , up and vr+1 , . . . , vp . Suppose that no eigenvector v1 , . . . , vr lies in the span of u1 , . . . , ur . Then there is an orthogonal matrix W such that W > AW = diag(`1 , . . . , `r ) ⊕ 0(p−r)×(p−r) (36) W > BW = B̃ ⊕ 0(p−2r)×(p−2r) (37) and where B̃ is a 2r-by-2r matrix whose entries are continuous functions of the inner products u> i vj , 1 ≤ i, j ≤ r, on R2r \ {±1}r . Furthermore, if we assume in addition that u> i vj = ci δi=j for some numbers ci ∈ (−1, 1), and p we let si = 1 − c2i , then the matrix B̃ is of the form: diag(η1 c21 , . . . , ηr c2r ) diag(η1 c1 s1 , . . . , ηr cr sr ) diag(η1 c1 s1 , . . . , ηr cr sr ) diag(η1 s21 , . . . , ηr s2r ) Proof. Form vectors ṽ1 , . . . , ṽp−2r , by performing Gram-Schmidt orthogonalization of v1 , . . . , ṽp−2r against the vectors u1 , . . . , ur . If the columns of W are the vectors in this basis, then clearly (36) holds, since Auk = `k uk for k = 1, . . . , r, and Aṽk = 0 for k = 1, . . . , p − 2r. Furthermore, since 38 Bṽk = 0 for k = r + 1, . . . , p − 2r, (37) holds for some 2r-by-2r block B̃. We need only check that 2r the entries of B̃ are continuous functions of u> \ {±1}r . i vj , 1 ≤ i, j ≤ r, on R Let W2r denote the p-by-2r matrix containing the first 2r columns of W , Ur denote the matrix with columns u1 , . . . , ur , Vr denote the matrix with columns v1 , . . . , vr , and Ṽr the matrix with columns ṽ1 , . . . , ṽr . We can therefore write W2r = [Ur Ṽr ]; By the Gram-Schmidt construction, every vector vk , k = 1, . . . , r, can be written as a linear combination of the vectors u1 , . . . , ur , ṽ1 , . . . , ṽk ; in matrix form, these linear combinations can be expressed as: Vr = Ur (Ur> Vr ) + Ṽr (Ṽ > Vr ) Now, we observe that the 2r-by-2r block B̃ is > U V > W2r BW2r = r> diag(η1 , . . . , ηr )[Ur> V Ṽr> V ] Ṽr V We need only show that the entries of Ṽr> Vr are continuous functions of the inner products 1 ≤ i, j ≤ r. Let R = Ṽr> V . Then Rk,l = 0 whenever k > l. When k = l, we have: v u r k−1 u X X 2− (u> v ) (ṽi> vk )2 Rk,k = ṽk> vk = t1 − i k u> i vj , i=1 i=1 and when k < l we have Rk,l = − Pr Pk−1 > > > > i=1 (ui vl )(ui vk ) + i=1 (ṽi vl )(ṽi vk ) Pr . kvk − i=1 (u> i vk )uk k Since for any l = 2, . . . , r, we have R1,l = ṽ1> vl Pr − i=1 (u> v1 )(u> vl ) Pr i > i = kv1 − i=1 (ui v1 )ui k which is obviously a continuous function of the inner products v1> ui on Rr \ {±1}r , an induction argument easily shows all entries of R are continuous in the inner products u> k vl . Finally, if u> v = c δ , the final assertion follows immediately, finishing the proof. j i i=j i If we apply the permutation π2r = (1, r + 1, 2, r + 2, . . . , r, 2r) to L the rows and columns of the r block matrix B̃ from Lemma 5.9, the corresponding matrix becomes i=1 B2 (ηi , ci , si ) where 2 ηc ηcs B2 (η, c, s) = . (38) ηcs ηs2 Applying the same permutation to the top 2r-by-2r block of the matrix A from Lemma 5.9 turns Lr ` 0 this block into: . i=1 A2 (`i ) where A2 (`) = 0 0 Now, let B̃ be given by the formula (38), but where c = ci = c(δ`i ) is the asymptotic inner product between the population eigenvector ui and the empirical eigenvector ûi , and η = ηi = η(λi ) is the almost sure limit of the ith eigenvalue η(λ̂i ) of Σ̂ηS . Since the shrinkers η collapse the vicinity 39 of the bulk to 0, it follows from Cor. 3.1 that the shrunken estimator has rank at most r a.s. Hence, by Lemma 5.9 and Cor. 3.1, if Lp (A, B) is any orthogonally-invariant loss function that decomposes over blocks, we have the almost sure convergence of Lp (ΣS , Σ̂ηS ) to the quantity L∞ (ΣS , Σ̂ηS ) = L2r M r A2 (`k ), k=1 r M B2 (η(λk ), ck , sk ) . k=1 This is the desired result. 5.4.3 Proof of Props. 3.3 We will only provide a proof for the reduced-noise model; the proof for the unreduced-model is 2 2 even simpler. Define the operator Lp acting Pnon p-by-p matrices by Lp (Σ) = µ Σ + σ diag(Σ) 1 and define the operator Lp by Lp (Σ) = n k=1 Dk ΣDk . Also, define define the p-by-p matrices Pn Bp = Σ̂Y − n1 k=1 Dk2 and Bp = Σ̂Y − mIp . Let ∆Bp = Bp − Bp , and ∆Lp = Lp − Lp . The operator ∆Lp taking p-by-p matrices to p-by-p matrices is diagonal. In this proof, when we refer to the (i, j)th diagonal entry of a diagonal operator on Rp×p , we mean the entry that multipliesPthe (i, j)th coordinate of a matrix. When i 6= j, the n 2 −1 th (i, j)th diagonal entry of ∆L diagonal k=1 Di,k Dj,k Σij ; and when i = j, the (i, i) Pnp (Σ) is2 µ −n −1 . entry of ∆Lp is m − n D k=1 i,k A proof nearly identical to the proof of Cor. 3.1 shows that the maximum of all the p2 diagonal entries of ∆Lp converges almost surely to 0; in other words, if Rp×p is equipped with Frobenius norm, then the operator norm of ∆Lp : Rp×p → Rp×p converges to 0 almost surely. Pn The p-by-p matrix ∆Bp is diagonal, with ith diagonal entry equal to m − n−1 k=1 Dk,i . Again, a proof like the proof of Cor. 3.1 shows that the supremum of these elements converges to zero almost surely as n, p → ∞. Also, note that √ the Frobenius norm of the matrix Σ̂S , which is the sum of squares of its eigenvalues, is of size ≈ n. To see this, observe that p p−r 1X 1X 1 1 kΣ̂S k2F = σk (Σ̂S )2 = σk (Σ̂S )2 + n n n n k=1 k=1 p X σk (Σ̂S )2 . k=p−r+1 Pp−r The first term n1 k=1 σk (Σ̂S )2 converges almost surely to the second moment of the MarchenkoPp Pastur law, which is finite since the distribution has finite support. The second term n1 k=p−r+1 σk (Σ̂S )2 converges to 0. Observe that Lp (Σ̂S ) − ∆Lp (Σ̂S ) = Lp (Σ̂S ) = Bp = Bp + ∆Bp = Lp (Σ̂0S ) + ∆Bp or in other words, Σ̂S − Σ̂0S kΣ̂S k = L−1 p (∆Lp Σ̂S ) kΣ̂S k The result follows immediately. 40 − L−1 p (∆Bp ) kΣ̂S k . 5.5 Proof that BLP is asymptotically diagonalized in PC basis (Sec. 4.1) Pr Pr −1 First, it is easy to check that Cov [Si , Yi ] = µ k=1 `k uk u> = [µ2 j=1 `j uj u> j + k , and M = Cov [Yi , Yi ] Pr Pr mIp +E]−1 , where E = σ 2 j=1 `j diag(uj uj ). Therefore the BLP equals ŜiBLP = µ k=1 `k uk u> M Yi . k Moreover, since uj are delocalized, the operator norm kEk → 0, so that it is easy to check that kŜiBLP − Ŝi0 k2 → 0, where r X Ŝi0 = µ `k uk u> k M0 Yi , k=1 Pr > −1 and M0 = [µ . Therefore, it is enough to show that kŜi0 − Ŝi k2 → 0, where j=1 `j uj uj + mIp ] P r > 2 recall that Ŝi = k=1 µ`k /(µ2 `k +m)uk u> k Yi . We can write, with mk = uk (M0 −Ip /(µ `k +m))Yi . 2 kŜi0 − Ŝi k2 = kµ = kµ r X k=1 r X `k uk u> k M0 Yi − `k uk mk k2 = µ2 k=1 r X 2 µ`k /(µ2 `k + m)uk u> k Yi k k=1 r X `k `j mk mj u> k uj . k,j=1 Therefore, to show kŜi0 − Ŝi k2 → 0, it is enough to show mk → 0. For this, using the formula u> (uu> + T )−1 = u> T −1 /(1 + u> T −1 u), we can write  −1  −1   −1  r r r X X X 2 > 2 µ2  = u> µ2  /  uk  u> `j uj u> `j uj u> ` j uj u> 1 + µ uk µ . k j + mIp k j + mIp j + mIp j=1 j6=k j6=k Under the assumptions of Cor. 2.2, we have u> k uj → δkj , hence it is easy to see that the denominator converges a.s. to 1 + µ2 /m. Indeed by using the formula (V V > + mI)−1 = [I − 1/2 1/2 V (V > V + mI)−1 V > ]/m, for V = µ[`1 u1 , . . . , `r ur ] (excluding uk )  −1 r X > > −1 > µ2  uk = u> u> `j uj u> V uk /m. k j + mI k uk /m − uk V (V V + mI) j6=k Now the entries of the r − 1-dimensional vector vk = V > uk are µ`1/2 u> j uk for j 6= k. Therefore, > −1 they converge to zero a.s. Since the operator norm of (V V + mI) is bounded above by 1/m, this shows that the second term converges to zero a.s. This shows that the denominator converges to 1 + µ2 /m a.s. Finally, by using the formula (V V > + mI)−1 = [I − V (V > V + mI)−1 V > ]/m once again, we conclude similarly that  −1 r X > > −1 > µ2  Yi = u> `j uj u> V Yi /m. u> k j + mI k Yi /m − uk V (V V + mI) j6=k h Denoting Mk = µ2 Pr > j6=k `j uj uj + I i−1 u> u> u> k Yi /m k Yi k Mk Yi − = mk = > m 1 + uk Mk uk 1 + µ2 /m · `k , we have > > 1 1 vk (V V + mI)−1 vk /m − − . > 1 + µ2 /m · `k 1 + uk Mk uk 1 + u> k Mk uk 41 Based on our previous calculations, both terms tend to 0, implying mk → 0. Therefore, we have kŜiBLP − Ŝi k2 → 0. By the triangle inequality, this immediately implies that all MSE properties for ŜiBLP also hold for Ŝi . This proves the desired claim. 5.6 5.6.1 Proof of Thm. 4.1 BLP For the BLP, consider first the single-spiked case where the squared prediction error of Ŝiτ = Ŝiτ,B = τ uu> Yi , with Yi = Di (Si + εi ) = Di (`1/2 zi u + εi ) is kSi − Ŝiτ k2 = k`1/2 zi u − τ u> Yi uk2 = (`1/2 zi − τ u> Yi )2 = [`1/2 zi − τ (`1/2 zi u> Di u + u> Di εi )]2 = (1 − τ u> Di u)2 `zi2 + τ 2 (u> Di εi )2 − 2τ (1 − τ u> Di εi )`1/2 zi u> Di εi . Now, zi and εi are independent of Di , hence, conditional on Di we have E[kSi − Ŝiτ k2 \|Di ] = (1 − τ u> Di u)2 ` + τ 2 u> Di2 u. Moreover, Eu> Di u = µ, E(u> Di u)2 = µ2 + σ 2 kuk44 → µ2 —since u is delocalized—and Eu> Di2 u = m, so the overall expectation converges to EkSi − Ŝiτ k2 → (1 − 2τ µ + τ 2 µ2 )` + τ 2 m. This proves that the asymptotic MSE of BLP is AM SE B (τ ; `, γ) = (1 − τ µ)2 ` + τ 2 m. The minimum is achieved for τ ∗ = µ`/(µ2 ` + m), and equals m`/(µ2 ` + m). In the multispiked case, the prediction error is kSi − Ŝiτ k2 = k r X 1/2 2 `k zik uk − τk u> k Yi uk k k=1 = r X 1/2 2 (`k zik − τk u> k Yi ) + k=1 X 1/2 1/2 > > (`k zik − τk u> k Yi )(`j zij − τj uj Yi ) · uk uj k6=j Now, zik , zij are independent if k 6= j, hence we have if k 6= j 1/2 1/2 > > > E(`k zik − τk u> k Yi )(`j zij − τj uj Yi ) = τk τj Euk Yi uj Yi . Using Yi = Di Xi , Xi = have 1/2 k=1 `k zik uk Pr + εi , and the independence properties described above, we > > > > Eu> k Yi uj Yi = Euk Di Xi Xi Di uj = Euk Di ( r X `m um u> m + Ip )Di uj m=1 = = r X m=1 r X > > 2 `m · E(u> k Di um · uj Di um ) + Euk Di uj > 2 `m · [µ2 (u> k um ) · (uj um ) + σ (uk um )> · (uj um )] + mu> k uj . m=1 Above we denoted by c = a b the vector with entries cj = aj bj . Since the vectors uk are delocalized and we assumed u> k ul → 0 for k 6= l, it is easy to see that the entire expression converges to zero. 42 This shows that EkSi − Ŝiτ k2 − E r X 1/2 2 (`k zik − τk u> k Yi ) → 0. k=1 Therefore, the asymptotic MSE decouples over the different PCs. Therefore, we can use our previous results about the asymptotic MSE in the single-spiked model, as soon as we can show that the same formulas for the MSE of specific coordinates given in Sec. 5.6.1 hold in this setting. However, this is easy to see using a calculation similar to the one given above. Indeed, we have X 1/2 1/2 2 > > > 2 E(`k zik − τk u> τk `1/2 k Yi ) = E(`k (1 − τk uk Di uk )zik − m zim uk Di um − τk uk Di εi ) m6=k = `k E(1 − 2 τk u> k Di uk ) + τk2 X 2 2 > 2 `m E(u> k Di um ) + τk Euk Di uk . m6=k 2 2 > 2 2 For k 6= m, the term E(u> um k2 → 0, while the other terms can k Di um ) = µ (uk um ) + σ kuk be evaluated as before, leading to the same formulas. Therefore, the asymptotic limit of the MSE is the same as that in the single-spiked case. This finishes the claims about BLP. 5.6.2 Empirical BLP The squared error of a general EBLP denoiser is kSi − Ŝiη k2 = k r X 1/2 `k zik uk − = 2 ηk ûk û> k Yi k k=1 k=1 r X r X 1/2 2 k`k zik uk − ηk ûk û> k Yi k + k=1 X 1/2 > 1/2 > (`k zik uk − ηk ûk û> k Yi ) (`j zij uj − ηj ûj ûj Yi ) k6=j The following lemma, proved in Sec. 5.6.3, shows that the cross terms vanish: Lemma 5.10 (Vanishing cross term in MSE). The cross terms in the MSE of EBLP denoisers 1/2 > 1/2 > vanish, i.e., for all k 6= j, E(`k zik uk − ηk ûk û> k Yi ) (`j zij uj − ηj ûj ûj Yi ) → 0 Therefore, the limit of the MSE in the multispiked case decouples into the MSEs for the individual spikes: r X 1/2 2 Ek`k zik uk − ηk ûk û> EkSi − Ŝiη k2 − k Yi k → 0. k=1 To evaluate the limiting MSE, we rely on the following lemma (proved in Sec. 5.7), which finds limiting expectations of inner products of the empirical singular vector ûB with the samples Yi and the population singular vector u. Lemma 5.11 (Denoising risk limit). We have the following convergences: 2 2 2 2 1. E(û> k Yi ) → mλk , where λk = t (δ`k ; γ) is defined in Eq. (7). 1/2 > 2 2 2 2. Ezi (û> k Yi )(uk ûk ) → µ`k ck · βk , where ck = c (δ`k ; γ) is defined in Eq. (8), and β =1+ 43 γ . δ`k (39) The key technical innovation is the proof of the second part. The main argument is an extension of the technique introduced by Benaych-Georges and Nadakuditi (2012) for characterizing the limits of the inner products u> j ûk between population and sample eigenvectors. For our proof, we need to extend this technique to characterize limits w> ûk for arbitrary random vectors w. Since this technique relies on an equation for the “outliers” among the eigenvalues of a finite-rank perturbation of a matrix in the terminology commonly used in random matrix theory (see e.g., Tao, 2013), we call this the outlier equation method. Applying the outlier equation method is nontrivial in our case, because the random vectors w to which we need to apply it—e.g., Xi —are dependent with ûk . For this reason, we need to use rank-one perturbation formulas once again to show that the dependence is negligible. We envision that the proof method could have several other applications. Going back to our main argument, Pr based on Lemma 5.11, the limit of the MSE is the following deterministic quantity: AM SE = k=1 `k + ηk2 · mt2k − 2ηk · µ`k c2k · βk . The optimal η minimizing the AMSE has ηk∗ = µ`k c2k · βk /[mt2k ] = µ`k c2k /[m(1 + δ`k )]. This finishes the EBLP analysis. 5.6.3 Proof of Lemma 5.10 1/2 1/2 > > We need to show that E(`k zik uk − ηk ûk û> k Yi ) (`j zij uj − ηj ûj ûj Yi ) → 0. We expand the paran1/2 1/2 theses and note that the first term is E(`k zik uk )> `j zij uj = 0, while the last term is a multiple > > > of Eû> k ûj · ûk Yi · ûj Yi = 0 (because ûk ûj = 0 for k 6= j). Thus it is enough to show the following claim for k 6= j: > Ezij û> k uj · ûk Yi → 0 > For this, we have u> k ûj → 0 a.s., by Cor. 2.2, thus also zij uk ûj → 0, so by convergence reduction 2 (Lemma 5.12) it is enough to show that E(û> Y ) is uniformly bounded. However, as in the proof k i 2 Y ) is uniformly bounded. This finishes of Part 1 of Lemma 5.11 in Sec. 5.7.1, we obtain that E(û> k i the proof of Lemma 5.10. 5.7 Proof of Lemma 5.11 For simplicity of exposition, we first prove the single-spiked case, when r = 1. The extension to the multispiked case is presented in Sec. 5.9. 5.7.1 Part 1 Similar to Lee et al. (2010) in the unreduced case, we use the following exchangeability argument: E(û> Yi )2 = Eû> Yi Yi> û = n−1 n X Eû> Yi Yi> û = n−1 Eû> Y > Y û = Eσ1 (Ỹ )2 . i=1 Now σ1 (Ỹ )2 → m · λ(δ`; γ) a.s. by Cor. 2.2. Moreover, by a bound on the expectation of top eigenvalues of sample covariance matrices such as that in Srivastava and Vershynin (2013), it is easy to see that Eσ1 (Ỹ )2 is uniformly bounded. Hence, Eσ1 (Ỹ )2 → m · λ(δ`; γ), finishing the proof. 44 5.7.2 Part 2 As a preliminary remark for Part 2, denoting t2 = λ(δ`; γ) we note that β = 1 + γ/(δ`) = 1 − m · γm(t2 )/[1 + m · γm(t2 )]. By a simple calculation, this is equivalent to m · m(t2 ) = −1/(γ + δ`). This can be checked as in the proof of Cor. 2.2 in Sec. 5.3. Therefore it is enough to show that −m · γm(t2 ) −m · γm(t2 ) > > 1/2 2 1/2 2 1/2 2 Ezi (û Yi )(u û) → ` c µ + = µ` c + ` c · . 1 + m · γm(t2 ) 1 + m · γm(t2 ) Expanding by using Yi = Di Xi , Xi = `1/2 zi u + εi , we have Ezi (û> Yi )(u> û) = `1/2 Ezi2 (û> Di u)(u> û) + Ezi (û> Di εi )(u> û). Ezi2 (û> Di u)(u> û) > (40) > → µc and Ezi (û Di εi )(u û) → `1/2 · Therefore, it is enough to show that 2 2 c · [−m · γm(t )]/[1 + m · γm(t )]. Since we already know that (u> û)2 → c2 a.s., our first step is to reduce these two claims by “getting rid” of the u> û term. For this and similar arguments, we will rely repeatedly on the following simple lemma: 2 2 Lemma 5.12 (Convergence Reduction). Suppose Wn ,Yn are random variables such that EWn → D for a constant D, EWn2 is uniformly bounded, Yn → C a.s. for some constant C, and Yn is a.s. uniformly bounded. Then EWn Yn → DC. In the special case when C = 0, the statement EWn → D is not needed. Proof. We have EWn Yn = EWn (Yn − C) + CEWn , so it is enough to show EWn (Yn − C) → 0. Therefore we can assume without loss of generality that C = 0. In that case, \|EWn Yn \| ≤ \|EWn2 \|1/2 \|EYn2 \|1/2 ≤ MW \|EYn2 \|1/2 → 0, because EWn2 are uniformly bounded; and because Yn → 0 a.s. and Yn are bounded, so E(Yn )2 → 0 by the dominated convergence theorem. Clearly, once we assumed C = 0 we did not use EWn → D. This finishes the proof. To use this lemma, let us choose the orientation of û such that u> û ≥ 0. Therefore, we know from our main results (e.g., Thm. 2.1), that u> û → c a.s. Moreover, since kûk = 1, it is easy to see that we can apply Lemma 5.12 to both terms in (40). Specifically, for the first term, we apply it with Wn = zi2 (û> Di u) and Yn = u> û; while for the second term we apply it with Wn = zi (û> Di εi ) and Yn = u> û. As mentioned above, this effectively removes the u> û terms, and thus simplifies the analysis considerably. Indeed, by Lemma 5.12 we conclude that Part 2 follows from the following auxiliary convergence results, proved in Sec. 5.8: Lemma 5.13 (Denoising risk auxiliary limits: Part 2). We have the following convergence results: 1. Ezi2 (û> Di u) → µc, where c2 is defined in Eq. (8). 2 −m·γm(t ) 2 2. Ezi (û> Di εi ) → `1/2 · c · 1+m·γm(t is defined in Eq. (7), and m is the Stieltjes 2 ) , where t transform of the standard Marchenko-Pastur law. Based on the above lemma, we completed the proof of Part 2 of Lemma 5.11. We will prove Part 1 of Lemma 5.11 later in Sec. 5.7.1, because it re-uses many of the techniques and results established in the proof of Part 1. This finishes the proof of Lemma 5.11. 45 5.8 Proof of Lemma 5.13 In this lemma, we need to understand the asymptotics of inner products such as u> Di û. Previous results by Benaych-Georges and Nadakuditi (2012) characterized the asymptotics of the cosines u> û. However, these results do not allow us to understand the asymptotics of the inner products u> Di û, due to the dependence of û and Di . Instead, we must go back to first principles, and extend the outlier equation technique introduced by Benaych-Georges and Nadakuditi (2012) to our setting. We will see that the current case is more challenging than the one handled in Benaych-Georges and Nadakuditi (2012). Let us recall the notation from Sec. 5.1.1. According to (34), the normalized data matrix m−1/2 Ỹ = (nm)−1/2 Y can be written as m−1/2 Ỹ = `1/2 · Zu> + N . Note here that u = ν. Let here t = tp be the singular value of m−1/2 Ỹ with left and right singular vectors Ẑ, û, and suppose that t is not a singular value of N . As in Lemmas 4.1 and 5.1 of Benaych-Georges and Nadakuditi (2012), we then have the following outlier equation that provides an equation for the singular vectors of the perturbed matrix m−1/2 Ỹ : 1 Ẑ t · (t2 In − N N > )−1 (t2 In − N N > )−1 N (u> û) · Z (41) = N > (t2 In − N N > )−1 t · (t2 Ip − N > N )−1 (Z > Ẑ) · u `1/2 û The idea is to take inner products of the right hand side with the quantity we want to characterize (e.g., inner product with Di u to understand u> Di û), and then evaluate the limit of the quantities on the left hand side. This extends the technique of Benaych-Georges and Nadakuditi (2012), who took inner products only with the population and sample singular vectors (u and û). In contrast, by using other vectors, such as Di u and Di εi , we are able to extend vastly the reach of their technique. To formalize this, let w be a p-dimensional vector. By taking the inner product of the outlier equation’s last p coordinates with w, we obtain the following scalar equation: (u> û) · w> N > (t2 In − N N > )−1 Z + t(Z > Ẑ) · w> (t2 Ip − N > N )−1 u = 1 `1/2 w> û. (42) Therefore, we can formalize the outlier equation technique as follows: Lemma 5.14 (Outlier equation method). Under the assumptions of Cor. 2.2, suppose that û, Ẑ are chosen so that û> u ≥ 0, Ẑ > Z ≥ 0. Suppose moreover that w = wp is a sequence of random vectors such that the assumptions of Lemma 5.12 for Wn apply to Wn1 = w> N > (t2 In − N N > )−1 Z and Wn2 = w> (t2 Ip − N > N )−1 u. Specifically, suppose that EWn1 → w1∗ and EWn2 → w2∗ . Then the random variables w> û also converge in expectation, namely Ew> û → `1/2 (c(δ`; γ)w1∗ + t · c̃(δ`; γ)w2∗ ). (43) Here c ≥ 0 where c2 is defined in Eq. (8), while c̃(δ`; γ) ≥ 0 is the limit cosine between Z, Ẑ, c̃(`; γ)2 = (1 − γ/`2 )/(1 + 1`) if ` > γ 1/2 and c̃2 = 0 otherwise. As an important special case, suppose that w2∗ = −κm(t2 ), while w1∗ = 0. Then Ew> û → κc. (44) Proof. The first part follows from the discussion before the lemma. For the second part, from Lemma 5.2, it folows that for w = u, we have Ew> (t2 Ip − N > N )−1 u → −m(t2 ), so that in this case the claim holds with κ = 1. For other vectors w, the result follows by linearity. An analogous version of Lemma 5.14 holds with convergence in expectation replaced by convergence a.s. Next, we will use this lemma to prove the two parts of Lemma 5.13. 46 5.8.1 Part 1 Recall the notation R = (t2 In −N N > )−1 and define R̃ = (t2 Ip −N > N )−1 . To show Ezi2 (û> Di u) → µc, we use Lemma 5.14 with the sequence of vectors w = zi2 Di u. To conclude our result by the second part of Lemma 5.14, it is enough to establish the following claims 1. Ezi2 u> Di N > RZ → 0 2. Ezi2 u> Di R̃u → −µ · m(t2 ). Claim 1 : As in the Proof of Lemma 5.1 in Sec. 5.1.2, we can write N = A + E ∗∗ , where E ∗∗ is independent of z and kE ∗∗ k → 0. Therefore, it is easy to see that it is enough to prove the claim with N replaced by A. Let us define R(A) = (t2 In − AA> )−1 . Then, as in the proof of Part 2 of Lemma 5.2 in Sec. 5.1.3, we obtain u> Di A> R(A)Z → 0 a.s. Clearly, these random variables are bounded a.s., since by the proof of Lemma 5.1, R(A) has a.s. uniformly bounded operator norm. Therefore, by the convergence reduction Lemma 5.12 applied to Wn = zi and Yn = u> Di A> R(A)Z—valid since Ezi2 = 1 and Ezi4 is uniformly bounded—we conclude that Ezi2 u> Di A> R(A)Z → 0. As discussed, this implies the desired Claim 1. Claim 2 : As in Claim 1, it is enough to prove the result with N replaced by A. From now on in this claim, we will only work with A, not N ; therefore, we denote for brevity R = R(A), R̃ = R̃(A) = (t2 Ip − A> A)−1 (and no confusion will arise). Also similarly to Claim 1, it will be enough to prove that u> Di R̃u → −µm(t2 ) a.s. For this, note that Di of course depends on the i-th data vector Yi = Di Xi . Therefore, we cannot use the concentration of quadratic forms directly. However, since the only dependence occurs in the i-th sample, we can use a rank-one perturbation formula to separate the i-th sample, and then control the two resulting terms separately. P −1 For this, we define aj = n−1/2 Dj εj to be the rows of A, so that R̃ = (t2 Ip − j aj a> . We j ) P 2 > −1 also define the perturbed matrix R̃i = (t Ip − j6=i aj aj ) . By the matrix inversion formula (M + aa> )−1 = M −1 − M −1 aa> M −1 /(1 + a> M −1 a), we have R̃ = R̃i + R̃i ai a> i R̃i > 1 − ai R̃i ai (45) Therefore, u> Di R̃u = u> Di R̃i u + (u> Di R̃i ai ) · (u> R̃i ai ) . 1 − a> i R̃i ai (46) > > 2 As in the proof of Part P 3 of Lemma 5.2 in Sec. 5.1.3, we obtain u Di R̃i u + u Di u · m(t ) → 0 a.s. However, u> Di u = j u2j Fj , where Fj are iid random variables with mean µ = EDij . Hence, it is easy to see that u> Di u → µ a.s. This shows that the first term in (46) converges to −µ · m(t2 ) a.s. and in expectation. It remains to show that the second term in (46) converges to 0 in expectation. Since 1/(1 − > > a> i R̃i ai ) is uniformly bounded a.s., it is enough to show that E(u Di R̃i ai )·(u R̃i ai ) → 0. However, > > by independence of εi from Di , R̃i , we have E(u Di R̃i ai ) · (u R̃i ai ) = n−1 Eu> Di R̃i Di2 R̃i u = O(n−1 ) a.s., showing the desired claim. This finishes the proof of Claim 2, and hence that of Part 1 of Lemma 5.13. 47 5.8.2 Part 2 For Part 2 of Lemma 5.13, we need to show that Ezi (û> Di εi ) → −`1/2 · cm · γm(t2 )/[1 + m · γm(t2 )]. Using the Outlier Equation method, Lemma 5.14, as in Part 1, with the sequence of vectors w = zi Di εi , it is enough to establish the following claims > 1. Ezi ε> i Di N RZ → −m·γm(t2 ) 1+m·γm(t2 ) 2. Ezi ε> i Di R̃u → 0. As before, we can work with A instead of N , and with R, R̃ being the resolvents of A. The second claim follows immediately, because zi is independent of ε> i Di R̃u, since R̃ does not depend > on zi . Therefore, Ezi ε> D R̃u = Ez Eε D R̃u = 0. i i i i i P 1/2 For the first claim, noting that A> R = R̃A> , and expanding A> Z = , where j zj aj /n −1/2 aj = n Dj εj , we can write X > > > 2 > Ezi ε> Ezi zj ε> i Di A RZ = Ezi εi Di R̃A Z = Ezi εi Di R̃ai + i Di R̃aj j6=i =n −1 Eε> i Di R̃Di εi = Ea> i R̃ai since zi are independent of ε, R̃, and have mean 0. To control this last term, we use a technique similar to that in Sec. 5.8.1. Using the rank one perturbation formula (45), and denoting βi = −1 > a> εi Di R̃i Di εi , we find i R̃i ai = n > a> i R̃ai = ai R̃i ai + 2 βi (a> i R̃i ai ) = . 1 − βi 1 − βi By independence and the concentration of quadratic forms, βi − n−1 m · tr(R̃i ) → 0, while by the Marchenko-Pastur law, n−1 tr(R̃i ) → −γ · m(t2 ). By the convergence reduction lemma, Ea> i R̃ai → −m · γm(t2 )/[1 + m · γm(t2 )] This finishes the proof of Part 2 of Lemma 5.13. Therefore, the proof of Lemma 5.13 is complete. 5.9 Multispiked EBLP MSE We now show that the formulas from Lemma 5.11 are also valid in the multispiked case. Indeed, 1/2 1/2 2 2 > 2 > > Ek`k zik uk − ηk ûk û> k Yi k = `k + ηk E(ûk Yi ) − 2ηk `k Ezik (ûk Yi )(uk ûk ). We have already showed in the proof of Lemma 5.10 that the limit of the second term is the same as in the single-spiked case. It remains to characterize the third term. For this, we follow a similar pattern to that used in the proof of Lemma 5.10 in Sec. 5.7, using the multispiked outlier equation. We sketch the argument below. As in the proof of Thm. 2.1 in Sec. 5.2, the normalized data matrix m−1/2 Ỹ = (nm)−1/2 Y can be written as m−1/2 Ỹ = ZL1/2 U > + N , where Z is the n × r matrix with entries zik , U is the n × r matrix with columns uk , while and L is the r × r diagonal matrix with entries `k . Let tk be the singular value of m−1/2 Ỹ with left and right singular vectors Ẑk , ûk , and suppose that tk is not a singular value of N . The multispiked outlier equation is now: Z · L1/2 · (U > ûk ) tk · (t2k In − N N > )−1 (t2k In − N N > )−1 N Ẑ = k (47) N > (t2k In − N N > )−1 tk · (t2k Ip − N > N )−1 U · L1/2 · (Z > Ẑk ) ûk 48 For a sequence of random vectors w = wp , denoting the r-vectors Wn1 = w> N > (t2k In − N N > )−1 Z and Wn2 = w> (t2k Ip − N > N )−1 u, the analogue of Eq. (42) is obtained by taking the inner product of the last p coordinates of the multispiked equation with w: Wn1 · L1/2 · (U > ûk ) + tk · Wn2 · L1/2 · (Z > Ẑk ) = w> ûk The outlier equation method formalized in Lemma 5.14 also extends in the natural way, by taking the limits of the above equation. Going back to our main argument about EBLP risks, it remains only to characterize the > limit of Ezik (û> k Yi )(uk ûk ). By the convergence reduction lemma, it is enough to show that 1/2 > Ezik (ûk Yi ) → µ`k ck βk , where ck ≥ 0 is the cosine, while βk = 1 + γ/(δ`k ). Examining closely the proof of Lemma 5.13, we see that all claims hold unchanged, and the proofs go through either unchanged or with minimal modifications (similar to the extension of the spike behavior to the multispiked case). We omit the details. 5.10 Proof of Thm. 4.3 Similarly to our previous calculations, the MSE equals Enη,o = EkS0 − Ŝ0η k2 = r X 1/2 2 Ek`k z0k uk − ηk ûk û> k Y0 k k=1 + X 1/2 1/2 > > E(`k z0k uk − ηk ûk û> k Y0 ) (`j z0j uj − ηj ûj ûj Y0 ). k6=j The cross terms vanish because z0k are independent zero-mean random variables, and û> k ûj = 0. To find the limit, we expand the main term as follows: 1/2 1/2 2 2 > 2 > > Ek`k z0k uk − ηk ûk û> k Y0 k = `k + ηk E(ûk Y0 ) − 2ηk `k Ez0k (ûk Y0 )(uk ûk ) Now, because Y0 are independent of ûj , we can take expectation over the randomess in Y0 to see that   r r X X > > 2 2 Eû> µ ` j uj u> `j diag(uj uj ) ûk k Y0 · ûk Y0 = Eûk j + mIp + σ j=1 = m + µ2 r X 2 2 `j E(û> k uj ) + σ j=1 j=1 r X `j Eû> k diag(uj uj )ûk → m + µ2 `k c2k . j=1 On the last line we used the results of Lemma 2.2, as well as the delocalization of uk , and denoted c2k = c2kk the asymptotic cosine of the k-th singular vectors. Moreover ! r X > > 1/2 > > Ez0k (ûk Y0 )(uk ûk ) = Ez0k `m z0m ûk D0 um + ûk D0 ε0 · (u> k ûk ) = m=1 1/2 > `k Eûk D0 uk 1/2 2 · (u> k ûk ) → µ`k ck . Hence, we have the following convergence: 1/2 2 ηk ,o Ek`k z0k uk − ηk ûk û> (`k ) = `k + ηk2 · (m + µ2 `k c2k ) − 2ηk · (µ`k c2k ) k Y0 k → E Pr Therefore, the limit prediction error is E η,o = k=1 E ηk ,o (`k ), finishing the proof of Thm. 4.3. 49 5.11 Proof of Prop. 4.4 First, clearly the optimal shrinkage for in-sample denoising is smaller, because µ`c2 /(µ2 ` + m) ≤ µ`c2 )/(µ2 `c2 + m). To check that the MSE of in-sample and out-of sample EBLP agree, we verify that ` − µ(µ`c2 · β)2 /(mt2 ) = ` − µ(µ`c2 )2 /(µ2 `c2 + m). Clearly this holds whenever c2 = 0. Considering the case c2 > 0, it is enough to show that β 2 /(mt2 ) = 1/(µ2 `c2 + m), or also that (δ` + γ)2 /(δ`)2 = t2 /(δ`c2 + 1). Since t2 = (δ` + 1)[1 + γ/(δ`)], using the formula for c2 , this follows immediately, finishing the proof. 5.12 Proofs for denoising in the unreduced-noise model (Sec. 4.4) The proofs for denoising in the unreduced-noise model from (3) are very similar to those in the reduced-noise model from (2). Therefore, while we give the full outline of the proofs, we skip some of the more technical parts that are essentially identical to those in the proof of Thms. 4.1 and 4.3. 5.12.1 BLP Pr Pr 1/2 Since Si = k=1 `k zik uk and Di has iid entries with mean µ, we have Cov [Si , Yi ] = µ k=1 `k uk u> k. Moreover, Cov [Yi , Yi ] = EYi Yi> = EDi Si Si> Di + Eεi ε> i = EDi r X ` k uk u> k Di + Ip . k=1 Pr Since the entries σ 2 , the a, b-th entry of the first component equals k=1 `k uka ukb · Pr of Di have variance EDia Dib = k=1 `k uka ukb [µ2 + σ 2 I(a = b)]. Therefore, denoting diag(uk uk ) the diagonal matrix with entries {u2ki }, i = 1, . . . , p, Cov [Yi , Yi ] = µ2 r X 2 `k uk u> k + Ip + σ k=1 r X `k diag(uk uk ). k=1 Since uk are delocalized, the operator norm of the last term converges to zero, k diag(uk uk )kop ≤ −1 −1 C 2 log2 B(p)/p → 0. Hence by using the matrix = Pr inversion difference formula A − B A−1 (B − A)B −1 , for A = Cov [Yi , Yi ] and B = µ2 k=1 `k uk u> + I , so that kA − Bk → 0, we see p k that k Cov [Si , Yi ] A−1 Yi − Cov [Si , Yi ] B −1 Yi k → 0. Thus the BLP is asymptotically equivalent to 2 Pr −1 Pr > µ k=1 `k uk u> Yi . As in the reduced-noise model in Sec. 5.5, it is not hard k µ k=1 `k uk uk + Ip Pr 2 to show that this is asymptotically equivalent to Ŝiτ = k=1 τk uk u> k Yi , where τk = µ`k /(µ `k + 1). τ To calculate the MSE of Si for general τ in the single-spiked case, we write as in the proof of Thm 4.1 in Sec. 5.6 EkSi − Ŝiτ k2 = E(`1/2 zi − τ u> Yi )2 = ` + τ 2 E(u> Yi )2 − 2τ `1/2 Ezi u> Yi . But Yi = Di Si + εi and in the single-spiked case Si = `1/2 zi u, so Ezi u> Yi = E`1/2 zi2 u> Di u + Ezi u> εi = `1/2 µ. Moreover, E(u> Yi )2 = E`zi2 (u> Di u)2 + 2`1/2 Ezi u> Di uu> εi + E(u> εi )2 = `E(u> Di u)2 + 1. Now, denoting by u(k) the entries of u X X X X E(u> Di u)2 = E( u(k)2 Dik )( u(l)2 Dil ) = µ2 u(k)2 u(l)2 + (µ2 + σ 2 ) u(k)4 . k l k6=l 50 k P Since u is delocalized, k u(k)4 → 0, so that E(u> Di u)2 → µ2 . In conclusion, we obtain as required EkSi − Ŝiτ k2 → ` + τ 2 (`µ2 + 1) − 2τ `µ. 5.12.2 EBLP To find the MSE of EBLP in the single-spiked case, we have EkSi − Ŝiη k2 = Ek`1/2 zi u − ηûû> Yi k2 = ` + η 2 E(û> Yi )2 − 2η`1/2 Ezi û> u · û> Yi . As in Lemma 5.11, we find that E(û> Yi )2 → t2 (µ2 `; γ), where t2 (µ2 `; γ) is defined in Eq. (7). Also, by the convergence reduction Lemma 5.12, choosing the orientation of û such that û> u ≥ 0, for the last term it is enough to characterize the limit of Ezi û> Yi . By the same argument as in Lemma 5.11, we find that Ezi · û> Yi → µ`1/2 ·c(µ2 `; γ)·[1+γ/(µ2 `)]. Hence, EkSi − Ŝiη k2 → ` + η 2 · t2 (µ2 `; γ) − 2η · µ` · c2 (µ2 `; γ) · [1 + γ/(µ2 `)], as claimed. This also shows that the optimal coefficient is η ∗ = µ` · c2 (µ2 `; γ)/[µ2 ` + 1]. 5.12.3 EBLP Out-of-sample Finally, for out-of-sample EBLP AMSE in the single-spiked case, we let (Y0 , D0 ) be a new sample, and expand Ek`1/2 z0 u − ηûû> Y0 k2 = ` + η 2 E(û> Y0 )2 − 2η`1/2 Ez0 (û> Y0 )(u> û) Now, because Y0 is independent of û, we can take expectation over the randomess in Y0 to see that Eû> Y0 · û> Y0 = Eû> [Ip + µ2 `uu> + σ 2 diag(u 2 > 2 2 > = 1 + µ `E(u û) + σ Eû diag(u u)]û u)û → 1 + µ2 `c2 . On the last line we used that u is delocalized, and the results of Lemma 2.2. Next, Ez0 (û> Y0 )(u> û) = Ez0 `1/2 z0 û> D0 u + û> ε0 · (u> û) = `1/2 Eû> D0 u · (u> û) → `1/2 µc2 . 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1 Multiplicative Updates for Elastic Net Regularized Convolutional NMF Under β -Divergence arXiv:1803.05159v1 [cs.LG] 14 Mar 2018 Pedro J. Villasana T., Stanislaw Gorlow, Member, IEEE and Arvind T. Hariraman Abstract—We generalize the convolutional NMF by taking the β-divergence as the loss function, add a regularizer for sparsity in the form of an elastic net, and provide multiplicative update rules for its factors in closed form. The new update rules embed the β-NMF, the standard convolutional NMF, and sparse coding alias basis pursuit. We demonstrate that the originally published update rules for the convolutional NMF are suboptimal and that their convergence rate depends on the size of the kernel. Index Terms—Convolution, nonnegative matrix factorization, multiplicative update rules, sparsity, elastic net, β-divergence. I. I NTRODUCTION N ONNEGATIVE matrix factorization finds its application in the fields of machine learning and in connection with inverse problems, mostly. It became immensely popular after Lee and Seung derived multiplicative update rules that made the up until then additive steps in the direction of the negative gradient obsolete [1]. In [2], they gave empirical evidence of their convergence to a stationary point, using (a) the squared Euclidean distance, and, (b) the generalized Kullback–Leibler divergence as the contrast function. The factorization’s origins can be traced back to [3], [4]. To better deal with noisy data, the notion of a basis is (most commonly) abandoned in favor of an overcomplete frame or dictionary, and sparsity becomes a desired property of the coefficient matrix. Early works that encourage such a factorization by means of L1 -penalty are [5], [6]. A convolutional variant of the factorization is introduced in [7] based on the Kullback–Leibler divergence. The main idea is to exploit temporal dependencies in the neighborhood of a point in the time-frequency plane. In their original form, the update rules lead to a biased factorization, because the frame of each time translation is updated separately but only the last frame is considered when updating the coefficient matrix. To provide a remedy, multiple coefficient matrices are computed in [8], one for each translation frame, and the final update is by taking the average over the time-aligned matrices. A nonnegative matrix factor deconvolution in 2D based on the Kullback–Leibler divergence is found in [9]. Not only do the authors give a derivation of the update rules, they show a simple way of making the update rules multiplicative. It may be pointed out that the update rule for the coefficient matrix is different from the one in [8]. The same guiding principles are applied to derive the convolutional factorization based on P. Villasana is a graduate student at the KTH Royal Institute of Technology, 114 28 Stockholm, Stockholm County, Sweden and a holder of a scholarship from the Mexican National Council for Science and Technology CONACYT e-mail: pjvt@kth.se. S. Gorlow is with Dolby Sweden. A. Hariraman is enrolled at the KTH Royal Institute of Technology. the squared Euclidean distance in [10]. But in the attempt to give a formal proof for their update rules, the authors largely rederive a biased factorization comparable to [7]. In [11], nonnegative matrix factorization is generalized to a family of α-divergences under the constraints of sparsity and smoothness, while the unconstrained β-divergence is brought into focus in [12]. For both cases multiplicative update rules are given. In this letter, we derive multiplicative update rules for the factorial deconvolution under the β-divergence and an elastic net regularizer. Furthermore, we show that the updates in [7], [8] are suboptimal and that their convergence rate and stability depend on the size of the convolution kernel. II. N ONNEGATIVE M ATRIX FACTORIZATION The nonnegative matrix factorization (NMF) is an umbrella term for a low-rank matrix approximation of the form V ' WH (1) K×I with V ∈ RK×N , W ∈ R>0 , and H ∈ RI×N >0 >0 , where I is the predetermined rank of the factorization. The letters above help distinguish between visible (v) and hidden variables (h) that are put in relation through weights (w). The factorization is usually formulated as a convex minimization problem with a dedicated cost function Cλ according to minimize Cλ (W, H) W, H subject to wki , hin > 0 (2) with Cλ (W, H) = L(V, W H) + λ R(H), (3) where L is a loss function that assesses the error between V and the factorization W H, R is a regularization term, and λ (λ > 0) is a tuning parameter that controls the importance of R. For λ equal to 0, we obtain a non-regularized NMF. A. β-Divergence The loss in (3) can be expressed by means of a contrast or distance function between the elements of V and W H. Due to its robustness with respect to outliers for certain values of the input parameter β, we resort to the β-divergence [13] as a subclass of the Bregman divergence [14], which for the points p and q in a closed convex set is given by Dβ (p, q) = β 1 p + (β − 1) q β − β p q β−1 , β (β − 1) (4) including the prominent special cases DLS (p, q) = Dβ (p, q)\|β=2 = 1 2 (p − q) , 2 (5) 2 for the squared Euclidean distance or the least-squares loss, p DKL (p, q) = lim Dβ (p, q) = p log − p + q, (6) β→1 q for the generalized Kullback–Leibler divergence, and p p (7) DIS (p, q) = lim Dβ (p, q) = − log − 1, β→0 q q for the Itakura–Saito distance. Accordingly, the β-divergence for matrices V and W H can be defined entrywise, as X X def Dβ (V, W H) = Dβ vkn , wki hin . (8) k,n i B. Elastic Net Regularization The factorization in (1) is an inverse problem, which is ill posed. As such, it can benefit from a regularization term. We set in an L1 -regularizer to favor the solution with the fewest hidden variables, i.e. the lowest model order. More precisely, the regularizer that imposes sparsity on H is defined as X def R(H) = kHk1 = hin with hin > 0, (9) i,n which is simply the sum of the entries in H. To overcome the limitations of the L1 -penalty, we add a quadratic term, which is known as elastic net regularization [15]: III. S PARSE β-CNMF Ensuing from the preliminary considerations in Section II, we adopt the formulation of the CNMF from (13) and derive multiplicative update rules for gradient descent, while taking the entrywise β-divergence from (8) as the loss function and imposing sparsity on H as in (10). The result is referred to as the “sparse β-CNMF”, or β-SCNMF for short. A. Problem Statement Under the premise that V is factorizable into {Wm } and H, m = 0, 1, . . . , M − 1, and given the cost function X m Cλ ({Wm }, H) = Dβ V, Wm H−→ +λ R(H), (14) m we seek to find the multiplicative equivalents of the iterative update rules for gradient descent: t ∂ t t+1 t C W , H , (15a) Wm = Wm −κ λ m t ∂Wm t ∂ C λ Wm , Ht , (15b) Ht+1 = Ht − µ t ∂H where (15a) and (15b) alternate at each iteration (t > 0). The step sizes κ and µ are allowed to change at every iteration. 2 def λ R(H) = λ2 kHkF + λ1 kHk1 , (10) P 2 2 where k·kF = i,n hin is the squared Frobenius norm. B. Multiplicative Updates Rules C. Discrete Convolution As can be seen explicitly in (8), the weight wki for the ith hidden variable hin at point (k, n) is applied using the scalar product. Given that hi evolves with n, we can assume that hi is correlated with its past and future states. We can take this into account by extending the dot product to a convolution in our model. Postulating causality and letting the weights have finite support of cardinality M , the convolution writes Computing the partial derivatives of Cλ w.r.t. Wm and H, and by choosing appropriate values for κ and µ, the iterative update rules from (15) become multiplicative: h ◦(β−1) T i◦−1 t+1 t Wm = Wm ◦ Ut Ht−→ m h i T ◦(β−2) ◦ V ◦ Ut Ht−→ (16a) m , hX tT t◦(β−1) Ht+1 = Ht ◦ Wm U←− + 2λ2 Ht m m h i X ◦−1 tT t◦(β−2) m ◦ U m + λ1 1I×N ] ◦ Wm V←− , (16b) ←− def (wki ∗ hi )(n) = M −1 X m wki (m) hi,n−m . (11) m=0 The operation can be converted to a matrix multiplication by lining up the states hin in a truncated Toeplitz matrix:   hi,1 hi,2 hi,3 · · · hi,N −1 hi,N  0 hi,1 hi,2 · · · hi,N −2 hi,N −1    (12)  .. . .. .. . .. .. ..  .  . . . . 0 0 0 ··· hi,N −M hi,N −M +1 In accordance with (1), the convolutional NMF (CNMF) can be formulated as follows to accommodate the structure given in (12), see also [7], [8]: V' M −1 X m , Wm H−→ (13) m=0 m where ·−→ is a columnwise right-shift operation similar to a logical shift in programming languages that shifts all the columns of H by m positions to the right, and fills the vacant positions with zeros. The operation is size-preserving. It can be seen that the CNMF has M times as many weights as the regular NMF, whereas the number of hidden states is equal. for m = 0, 1, . . . , M − 1, with X t Ut = Wm Ht−→ m , m where ◦ denotes the Hadamard, i.e. entrywise, product, ·◦p is equivalent to entrywise exponentiation and ·◦−1 , respectively, m operator is the leftstands for the entrywise inverse. The ·←− shift counterpart of the right-shift operator. The details of the derivation of (16) can be found in the Appendix. C. Interpretation The two update rules given in (16) are of significant value because: • They are multiplicative, and thus, they converge fast and are easy to implement. • Eq. (16a) extends the update rule of the β-NMF for W to a set of M weight matrices that are linked through a convolution operation. It also extends the corresponding update rule of the existing convolutional NMFs with the squared Euclidean distance or the generalized Kullback– Leibler divergence to the family of β-divergences. 3 Eq. (16b) is even more important: Apart from providing a means to control the sparsity of H through λ1 and λ2 , it yields a complete update of the hidden states at every iteration taking all M weight matrices into account. The update rule in (16b) can be viewed as an equivalent to a descent in the direction of the Reynolds gradient, which is to take the average over the partial derivatives under the group of time translations (in m). Note that this is not the same as what was originally proposed by Smaragdis [8]. The average operator reduces the gradient spread as a function of Wm at each step t and makes the descent converge to a single point that is most likely. • D. Uniqueness and Normalization It is understood that the factorization is not unique. This is easily shown by the equivalence V' M −1 X m Wm H−→ m=0 ≡ M −1 X m = Wm B B−1 H−→ M −1 X (17) fm H e m W −→ m=0 m=0 f m = Wm B and H e = B−1 H, for any B that has an with W I×I inverse, B ∈ R in general. Non-negativity still holds for f m and H e if B is a nonnegative diagonal matrix, which in W this case results in a scaling of the columns of Wm and the corresponding rows of H. We can use this to enforce insight the same p-norm on the matrices Wi = wkm i ∈ RK×M : −1 −1 B = diag kW1 k−1 . (18) p , kW2 kp , . . . , kWI kp In Fig. 1 it can be seen that the mean loss is bounded from below by our updates. Furthermore, one can observe that our updates converge to a local minimum after the same number of iterations, whereas the convergence rate of the updates by Smaragdis depends on the number of time translations M in the convolution: the greater M , the slower the rate. It can be added that the average updates produce a steadily decreasing loss, while the biased updates, where H is updated based on WM only, can diverge as the iteration continues. For M = 2, the loss distributions of our and Smaragdis’ average updates seem to have the same mean but a slightly different standard deviation. To test the hypothesis that the two populations are statistically equivalent w.r.t. the mean, we use Welch’s t-test. The shown p-values indicate that already after the first cycle of updates the null hypothesis can be rejected almost surely, i.e. the two update rules converge to different minima. Quite similar results were obtained for other values of K and N . V. C ONCLUSION To the best of our knowledge, with the exception of [9] in the 2D unconstrained case, our letter is the only to provide a complete and correct derivation of the multiplicative updates for the convolutional NMF. In addition, the contrast function is generalized to the family of β-divergences and a penalty is added in the form of an elastic net for sparse modeling. It is shown by simulation that the originally proposed updates for the convolutional NMF are suboptimal and that convergence rate of the latter, and stability alike, are subject to the size of the convolution kernel. A PPENDIX Let ukn = i,m wki (m) hi,n−m and U = ukn ∈ RK×N . Then, for any p ∈ {1, 2, . . . , K}, q ∈ {1, 2, . . . , I}, and r ∈ {0, 1, . . . , M − 1}: P e If all Wm are multiplied by B before (16b), (16b) yields H immediately. In this way, we can put the significances of the hidden variables in relation to each other. IV. S IMULATION In this section, we compare our update rules with the ones by Smaragdis in terms of convergence performance. For this, we generate 100 different V-matrices from M χ2 -distributed Wm -matrices, wki (m) = 2 X ∂ Cλ ({Wm }, H) ∂wpq (r) ! X uβpn vpn uβ−1 ∂ pn = − n ∂wpq (r) β β−1 X β−1 β−2 = upn − vpn upn hq,n−r . n (21) Choosing κ from (15a) as 2 wki (m) p ∼ χ22 wkip (m) ∼ N (0, 1), (19) κ= P p=1 n and a uniformly distributed H-matrix, hin ∼ U(0, 1). wpq (r) uβ−1 pn hq,n−r , (22) leads to the first update rule (20) The Wm -matrices, which represent random spectrograms, are normalized by the Frobenius norm as per (18). The value for β is set to 1 (Kullback–Leibler) and the regularization term is omitted (λ2 = λ1 = 0). The factorizations are simulated with 10 random initializations of {Wm,0 } and H0 (with non-zero entries). The results shown in Fig. 1 are thus computed over ensembles of 1000 losses at each iteration. In our simulation, the number of visible and hidden variables is K = 1000 and I = 10, and the number of time samples (realizations) is N = 100. = t wki (m) β−2 vkn utkn hti,n−m . P tβ−1 t n ukn hi,n−m P t+1 wki (m) n (23) Further, for any p ∈ {1, 2, . . . , I} and q ∈ {1, 2, . . . , N }: ∂ Cλ ({Wm }, H) ∂hpq ! β−1 uβkn vkn ukn ∂ X = − k,n ∂hpq β β−1 + λ2 ∂ ∂ h2 + λ1 hpq . ∂hpq pq ∂hpq (24) 4 M=2 10 3 10 0 10 -4 10 0 10 1 Mean Villasana et al. Smaragdis (biased) Smaragdis (average) 10 1 10 0 Std 10 1 10 1 10 1 10 0 Villasana et al. Smaragdis (biased) Smaragdis (average) 10 -6 10 0 Villasana et al. Smaragdis (biased) Smaragdis (average) 10 0 10 -4 10 0 10 2 M = 16 10 -6 10 0 10 2 1 10 2 Villasana et al. Smaragdis (biased) Smaragdis (average) Std Mean 10 3 10 1 10 2 1 p-value Villasana et al. vs. Smaragdis (average) Significance level p-value Villasana et al. vs. Smaragdis (average) Significance level 0.05 0.05 10 0 10 1 Iteration 10 2 10 0 10 1 Iteration 10 2 Fig. 1. Simulation results including the mean and the standard deviation of the loss distribution and the p-value from Welch’s t-test for the hypothesis that our and Smaragdis’ update rules on average converge to the same minimum as a function of iteration (M is the number of time translations). It is straightforward to show that ∂ ukn = wkp (n − q) ∂hpq (25) by setting n − m = q m = n − q. As a result, plugging in q + m for n in (24) and using (25), we finally obtain ∂ Cλ ({Wm }, H) ∂hpq X β−2 = wkp (m) uβ−1 − v u k,q+m k,q+m k,q+m k,m + 2λ2 hpq + λ1 . (26) Choosing µ from (15b) as hpq β−1 k,m wkp (m) uk,q+m µ= P + 2λ2 hpq + λ1 , leads to the second update rule P t tβ−2 k,m wki (m) vk,n+m uk,n+m t+1 t hin = hin P . t tβ−1 t k,m wki (m) uk,n+m + 2λ2 hin + λ1 (27) (28) R EFERENCES [1] D. D. Lee and H. S. Seung, “Learning the parts of objects by nonnegative matrix factorization,” Nature, vol. 401, pp. 788–791, 1999. [2] ——, “Algorithms for non-negative matrix factorization,” in Advances in Neural Information Processing Systems, 2001, pp. 556–562. [3] P. Paatero and U. Tapper, “Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values,” Environmetrics, vol. 5, no. 2, pp. 111–126, 1994. [4] P. Paatero, “Least squares formulation of robust non-negative factor analysis,” Chemometrics and Intelligent Laboratory Systems, vol. 37, no. 1, pp. 23–35, 1997. [5] P. O. Hoyer, “Non-negative sparse coding,” in Neural Networks for Signal Processing, 2002, pp. 557–565. [6] J. Eggert and E. Korner, “Sparse coding and NMF,” in Neural Networks, vol. 4, 2004, pp. 2529–2533. [7] P. Smaragdis, “Non-negative matrix factor deconvolution; extraction of multiple sound sources from monophonic inputs,” in Independent Component Analysis and Blind Signal Separation, 2004, pp. 494–499. [8] ——, “Convolutive speech bases and their application to supervised speech separation,” IEEE Transactions on Audio, Speech, and Language Processing, vol. 15, no. 1, pp. 1–12, 2007. [9] M. N. Schmidt and M. Mørup, “Nonnegative matrix factor 2-D deconvolution for blind single channel source separation,” in Independent Component Analysis and Blind Signal Separation, 2006, pp. 700–707. [10] W. Wang, A. Cichocki, and J. A. Chambers, “A multiplicative algorithm for convolutive non-negative matrix factorization based on squared Euclidean distance,” IEEE Transactions on Signal Processing, vol. 57, no. 7, pp. 2858–2864, 2009. [11] A. Cichocki, R. Zdunek, and S. Amari, “Csiszár’s divergences for nonnegative matrix factorization: Family of new algorithms,” in Independent Component Analysis and Blind Signal Separation, 2006, pp. 32–39. [12] C. Févotte and J. Idier, “Algorithms for nonnegative matrix factorization with the β-divergence,” Neural Computation, vol. 23, no. 9, pp. 2421– 2456, 2011. [13] A. Basu, I. R. Harris, N. L. Hjort, and M. C. Jones, “Robust and efficient estimation by minimising a density power divergence,” Biometrika, vol. 85, no. 3, pp. 549–559, 1998. [14] L. M. Bregman, “The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming,” USSR Computational Mathematics and Mathematical Physics, vol. 7, no. 3, pp. 200–217, 1967. [15] H. Zou and T. Hastie, “Regularization and variable selection via the elastic net,” Journal of the Royal Statistical Society, vol. 67, no. 2, pp. 301–320, 2005.	8
Accepted as a workshop contribution at ICLR 2015 L EARNING C OMPACT C ONVOLUTIONAL N ETWORKS WITH N ESTED D ROPOUT N EURAL arXiv:1412.7155v4 [cs.CV] 10 Apr 2015 Chelsea Finn, Lisa Anne Hendricks & Trevor Darrell Department of Computer Science UC Berkeley Berkeley, CA 94704, USA {cbfinn, lisa anne, trevor}@eecs.berkeley.edu A BSTRACT Recently, nested dropout was proposed as a method for ordering representation units in autoencoders by their information content, without diminishing reconstruction cost (Rippel et al., 2014). However, it has only been applied to training fully-connected autoencoders in an unsupervised setting. We explore the impact of nested dropout on the convolutional layers in a CNN trained by backpropagation, investigating whether nested dropout can provide a simple and systematic way to determine the optimal representation size with respect to the desired accuracy and desired task and data complexity. 1 M OTIVATION Supervised convolutional neural networks (CNNs) learn representations that are effective on a wide range of tasks. A drawback of current such approaches, however, is that the selection of such architectures is largely optimized by hand, with researchers explicitly searching architecture hyperparameters via cross-validation. Model selection for network architectures has been explored in the context of learning network connectivity dating back to Optimal Brain Damage from LeCun et al. (1989) and has been continued to be explored in the context of learning optimal sparse models. To our knowledge there is no deep visual network capable of increasing its representation capacity based on the complexity of available data or tasks. The recently proposed nested dropout method implicitly accomplishes this, by learning deep representation units in an incremental fashion. We investigate whether such an approach is applicable to visual CNN models, and propose a visual CNN model which can learn to scale its capacity according to the complexity of the data presented to the network during training. In standard dropout, units in the layer are independently dropped out with probability p, namely the output of that unit is set to zero. This is traditionally applied to convolutional and fully-connected layers during training time, and has been shown to act as a regularizer, discouraging over-fitting to the training data (Hinton et al., 2012), though it has also been applied to entire channels of a convolutional layer output by Tompson et al. (2014). Empirically, when training large networks, such as those trained on ImageNet, drop out is necessary to avoid over fitting (Krizhevsky et al., 2012). Nested dropout, on the other hand, randomly draws unit indices from a geometric distribution and drops out all of the units that follow the number drawn, e.g. if a number k is drawn, then the units 0 through k are kept and the remaining units are dropped. When applied to a single layer semi-linear autoencoder, this technique has been proven to enforce an ordering of the units by their information capacity, while not decreasing the flexibility of the representation nor the quality of the resulting solution (Rippel et al., 2014). The primary contributions of this paper are to (1) demonstrate that nested dropout can successfully be applied to convolutional layers trained by back-propagation, (2) propose nested dropout as an advantageous method to learn CNNs that adapt to task and data complexity in a deep learning setting, and (3) provide our implementation in Caffe (Jia et al., 2014), a widely used deep learning framework, upon publication. 1 Accepted as a workshop contribution at ICLR 2015 2 N ESTED D ROPOUT ON CNN S The nested dropout algorithm for a convolutional layer with n channels is as follows: for each sample in a mini-batch, we draw a number k from a geometric distribution and drop out the latter n − k channels of the output of the layer. Nested dropout can also be applied to multiple layers in a network by applying nested dropout to each layer iteratively. First, the number of filters ni for layer i is determined through nested dropout. After fixing the number of filters ni in layer i, nested dropout can then be used to determine the number of filters, ni+1 , for layer i + 1. We trained our CNNs using Stochastic Gradient Descent (SGD) and mini-batches of 100 samples. Because dropped out units are determined by drawing from a geometric distribution, units with a low index are rarely dropped out and thus converge quickly, whereas latter units are frequently dropout out and thus learned very slowly. Filters are incrementally fixed once they have converged, and only the remaining filters are considered when drawing numbers from the geometric distribution. Though incrementing the sweeping index could be done upon filter convergence, we achieved satisfactory results by simply incrementing the unit sweeping index after a set number of iterations. To implement this in the Caffe framework, we added a nested dropout layer that can follow any layer (e.g. convolutional, fully-connected) in the same way as the standard dropout layer. We also added customizations to the solver to support unit sweeping during training. 3 E XPERIMENTS Figure 1: Accuracy as a function of the number of filters in conv1, with and without nested dropout. The oracle consists of 32 separate networks, whereas the nested dropout curve only requires training a single network. The “brain damaged” baseline is trained without nested dropout and only the k best filters are used during test. The nested dropout network achieves maximum accuracy with a compact representation of 23 filters while requiring considerably fewer iterations than a brute force approach. We apply nested dropout to the first convolutional layer of a CNN trained to classify images in the CIFAR-10 dataset, using the default Caffe architecture and training with a fixed learning rate. In Figure 1, we show the test accuracy of a single network trained with nested dropout as a function 2 Accepted as a workshop contribution at ICLR 2015 of the number of conv1 filters. We report the accuracy with k filters by using the partially-trained network after the kth filter has converged, and testing it with the last n − k filters dropped out. We compare to two naive approaches to select the number of filters in conv1. The first approach simply trains 32 separate networks which differ in the number of conv1 filters. The second approach trains a single network with 32 filters, but at test time, a varying number of conv1 filters are used with the remaining dropped out. Note that because there is no inherent ordering to the learned representation without nested dropout, removing a filter severely damages the network resulting in low accuracies. Our experiments in Figure 1 demonstrate that nested dropout efficiently determines the relationship between model capacity and test accuracy. The nested dropout implementation requires 90,000 iterations of training, whereas the brute force approach requires significantly more, up to millions iterations. For example, training 32 separate networks to completion requires 2,880,000 iterations. (a) (b) Figure 2: conv1 filters of networks trained with nested dropout (a) and without (b). Note that the latter filters of (a) carry very little information, yet both networks achieve similar performance, giving an indication for how many units are necessary for this model. In Figure 2, we visualize the filters learned with and without nested dropout. Though the latter filters carry very little information, the test accuracy of the two sets of filters are comparable with 0.787 test accuracy for the network trained with nested dropout and 0.786 test accuracy for the baseline network. Applying nested dropout to the second convolutional layer yielded similar results to our experiments with conv1. After training a network with nested dropout applied to conv1 filters, we determined only 23 conv1 filters are necessary to achieve the maximum classification accuracy for this network. We next train a network in which nested dropout follows conv2 and with a unit sweeping index of 5,000. After learning 25 conv2 filters, the training accuracy converges to 78%. Thus, by using nested dropout we can reduce the total number of parameters in the first two layers by 25% from 64 filters to 48, while maintaining similar accuracy. 4 D ISCUSSION In summary, we have provided a simple method for determining a more compact representation of the convolutional layers. In our experiments, we learned a representation that achieved the same classification accuracy using 23 conv1 filters and 25 conv2 filters rather than the baseline 32 each, within the same optimization framework. A main advantage of our method is that it enables the network to gradually increase network capacity during training. Additionally, we hope that, in the future, ordering parameters may provide insights into optimization of deep convolutional neural networks and how the network architecture impacts performance. 3 Accepted as a workshop contribution at ICLR 2015 ACKNOWLEDGMENTS This work was supported in part by DARPA’s MSEE and SMISC programs, NSF awards IIS1427425, IIS-1212798, and IIS-1116411, Toyota, and the Berkeley Vision and Learning Center. Chelsea Finn was supported by a Berkeley EECS Fellowship and Lisa Anne Hendricks by an NDSEG Fellowship. R EFERENCES Hinton, Geoffrey E., Srivastava, Nitish, Krizhevsky, Alex, Sutskever, Ilya, and Salakhutdinov, Ruslan. Improving neural networks by preventing co-adaptation of feature detectors. CoRR, abs/1207.0580, 2012. URL http://arxiv.org/abs/1207.0580. Jia, Yangqing, Shelhamer, Evan, Donahue, Jeff, Karayev, Sergey, Long, Jonathan, Girshick, Ross, Guadarrama, Sergio, and Darrell, Trevor. Caffe: Convolutional architecture for fast feature embedding. In Proceedings of the ACM International Conference on Multimedia, pp. 675–678. ACM, 2014. Krizhevsky, Alex, Sutskever, Ilya, and Hinton, Geoffrey E. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012. LeCun, Yann, Denker, John S, Solla, Sara A, Howard, Richard E, and Jackel, Lawrence D. Optimal brain damage. In NIPs, volume 2, pp. 598–605, 1989. Rippel, Oren, Gelbart, Michael A, and Adams, Ryan P. Learning ordered representations with nested dropout. arXiv preprint arXiv:1402.0915, 2014. Tompson, Jonathan, Goroshin, Ross, Jain, Arjun, LeCun, Yann, and Bregler, Christoph. Efficient object localization using convolutional networks. CoRR, abs/1411.4280, 2014. URL http: //arxiv.org/abs/1411.4280. 4	9
IDENTITIES IN PLACTIC, HYPOPLACTIC, SYLVESTER, BAXTER, AND RELATED MONOIDS arXiv:1611.04151v2 [math.CO] 5 Mar 2017 ALAN J. CAIN AND ANTÓNIO MALHEIRO Abstract. This paper considers whether non-trivial identities are satisﬁed by certain ‘plactic-like’ monoids that, like the plactic monoid, are closely connected to combinatorics. New results show that the hypoplactic, sylvester, Baxter, stalactic, and taiga monoids satisfy identities, and indeed give shortest identities satisﬁed by these monoids. The existing state of knowledge is discussed for the plactic monoid and left and right patience sorting monoids. 1. Introduction The ubiquitous plactic monoid, whose elements can be viewed as semistandard Young tableaux, and which appears in such diverse contexts as symmetric functions [Mac08], representation theory [Ful97], algebraic combinatorics [Lot02], Kostka–Foulkes polynomials [LS81, LS78], Schubert polynomials [LS85, LS90], and musical theory [Jed11], is one of a family of ‘placticlike’ monoids that are closely connected with combinatorics. These monoids include the hypoplactic monoid [KT97, Nov00], the sylvester monoid [HNT05], the taiga monoid [Pri13], the stalactic monoid [HNT07, Pri13], the Baxter monoid [Gir12], and the left and right patience sorting monoids [Rey07, CMS]. Each of these monoids is obtained by factoring the free monoid A∗ over the infinite ordered alphabet A = {1 < 2 < 3 < . . .} by a congruence that arises from a so-called insertion algorithm that computes a combinatorial object from a word. For instance, for the plactic monoid, the corresponding combinatorial objects are (semistandard) Young tableaux; for the sylvester monoid, they are binary search trees. An identity is a formal equality between two words in the free monoid, and is non-trivial if the two words are distinct. A monoid M satisfies such an identity if the equality in M holds under every substitution of letters in the words by elements of M . For example, any commutative monoid satisfies the non-trivial identity xy = yx. A finitely generated group has polynomial growth if and only if it is virtually nilpotent [Gro81], and a virtually nilpotent group satisfies a non-trivial identity [Mal53, NT63]. Thus it is natural to ask whether every finitely generated semigroup or monoid with polynomial growth satisfies a non-trivial identity. Schneerson [Shn93] provided the first counterexample, but it remains to be seen whether there is a The ﬁrst author was supported by an Investigador FCT fellowship (IF/01622/2013/CP1161/CT0001). For both authors, this work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações), and the project PTDC/MHC-FIL/2583/2014. 1 Symbol Plactic Hypoplactic Sylvester #-sylvester Baxter Stalactic Taiga Left patience sorting Right patience sorting plac hypo sylv sylv# baxt stal taig lPS rPS Identity satisfied ? xyxy = yxyx xyxy = yxxy yxyx = yxxy yxxyxy = yxyxxy xyx = yxx xyx = yxx None None Discussed in Original result Subsec. Subsec. Subsec. Subsec. Subsec. Subsec. Subsec. Subsec. Subsec. 3.7 3.2 3.4 3.4 3.5 3.3 3.3 3.6 3.6 — Present Present Present Present Present Present [CMS] [CMS] paper paper paper paper paper paper ALAN J. CAIN AND ANTÓNIO MALHEIRO Monoid 2 Table 1. Examples of non-trivial identities satisfied by ‘plactic-like’ monoids. The stated identities are always shortest non-trivial identities satisfied by the corresponding monoid, but there may be other identities of the same length. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 3 ‘natural’ finitely generated semigroup with polynomial growth that does not satisfy a non-trivial identity. It is easy to see that the finite-rank analogues of the plactic monoid and the other related monoids discussed above have polynomial growth. This naturally leads to the question of whether these monoids satisfy non-trivial identities, for if any of them failed to do so, it would certainly be a very natural example of a polynomial-growth monoid that does not satisfy a non-trivial identity. Further motivation for this question comes from a result of Jaszuńska & Okniński, who proved that the Chinese monoid [CEK+ 01], which has the same growth type as the plactic monoid [DK94] but which does not arise from such a natural combinatorial object, satisfies Adian’s identity xyyxxyxyyx = xyyxyxxyyx [JO11, Corollary 3.3.4]. (This is the shortest non-trivial identity satisfied by the bicyclic monoid [Adi66, Chapter IV, Theorem 2]). The goal of this paper is to present new results showing that some of these monoids satisfy non-trivial identities, and to survey the state of knowledge for other monoids in this family. New results show that the hypoplactic, sylvester, baxter, stalactic, and taiga monoids satisfy non-trivial identities. A discussion of the situation for the left and right patience sorting monoids and plactic monoids completes the paper. Table 1 summarizes the results. 2. ‘Plactic-like’ monoids In this section, we recall only the definition and essential facts about the various monoids; for further background, see [Lot02, Ch. 5] on the plactic monoid, [Nov00] on the hypoplactic monoid, [HNT05] on the sylvester monoid; [Pri13, § 5] on the taiga monoid; [Pri13] on the stalactic monoid; [Gir12] on the Baxter monoid; and [CMS] on the patience sorting monoids. 2.1. Alphabets and words. For any alphabet X, the free monoid (that is, the set of all words, including the empty word) on the alphabet X is denoted X ∗ . The empty word is denoted ε. For any u ∈ X ∗ , the length of u is denoted \|u\|, and, for any x ∈ X, the number of times the symbol x appears in u is denoted \|u\|x . Throughout the paper, A = {1 < 2 < 3 < . . .} is the set of natural numbers viewed as an infinite ordered alphabet, and An = {1 < 2 < . . . < n} is set of the first n natural numbers viewed as a finite ordered alphabet. 2.2. Combinatorial objects and insertion algorithms. A Young tableau is a finite array of symbols from A, with rows non-decreasing from left to right and columns strictly increasing from top to bottom, with shorter rows below longer ones, and with rows left-justified. An example of a Young tableau is (2.1) 1 1 1 2 5 . 3 3 5 6 6 The following algorithm takes a Young tableau and a symbol from A and yields a new Young tableau: 4 ALAN J. CAIN AND ANTÓNIO MALHEIRO Algorithm 2.1 (Schensted’s algorithm). Input: A Young tableau T and a symbol a ∈ A. (1) If a is greater than or equal to every entry in the topmost row of T , add a as an entry at the rightmost end of T and output the resulting tableau. (2) Otherwise, let z be the leftmost entry in the top row of T that is strictly greater than a. Replace z by a in the topmost row and recursively insert z into the tableau formed by the rows of T below the topmost. (Note that the recursion may end with an insertion into an ‘empty row’ below the existing rows of T .) A quasi-ribbon tableau is a finite array of symbols from A, with rows non-decreasing from left to right and columns strictly increasing from top to bottom, that does not contain any 2 × 2 subarray (that is, of the form ). An example of a quasi-ribbon tableau is: (2.2) 1 1 1 2 3 3 5 5 . 6 6 Notice that the same symbol cannot appear in two different rows of a quasiribbon tableau. There is also an insertion algorithm for quasi-ribon tableau: Algorithm 2.2 ([Nov00, Algorithm 4.4]). Input: A quasi-ribbon tableau T and a symbol a ∈ A. If there is no entry in T that is less than or equal to a, output the tableau obtained by putting a and gluing T by its top-leftmost entry to the bottom of a. Otherwise, let x be the right-most and bottom-most entry of T that is less than or equal to a. Put a new entry a to the right of x and glue the remaining part of T (below and to the right of x) onto the bottom of the new entry a. Output the new tableau. A right strict binary search tree is a labelled rooted binary tree where the label of each node is greater than or equal to the label of every node in its left subtree, and strictly less than every node in its right subtree. An example of a binary search tree is 5 2 (2.3) 1 1 1 6 5 . 6 3 3 The insertion algorithm for right strict binary search trees adds the new symbol as a leaf node in the unique place that maintains the property of being a right strict binary search tree: Algorithm 2.3 ([HNT05, § 3.3]). Input: A right strict binary search tree T and a symbol a ∈ A. If T is empty, create a node and label it a. If T is non-empty, examine the label x of the root node; if a ≥ x, recursively insert a into the right subtree IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 5 of the root node; otherwise recursively insert a into the left subtree of the root note. Output the resulting tree. A left strict binary search tree is a labelled rooted binary tree where the label of each node is strictly greater than the label of every node in its left subtree, and less than or equal to every node in its right subtree; see the left tree shown in (2.4) below for an example. The insertion algorithm for left strict binary search trees is dual to Algorithm 2.3 above: Algorithm 2.4. Input: A left strict binary search tree T and a symbol a ∈ A. If T is empty, create a node and label it a. If T is non-empty, examine the label x of the root node; if a ≤ x, recursively insert a into the left subtree of the root node; otherwise recursively insert a into the right subtree of the root note. Output the resulting tree. The canopy of a (labelled or unlabelled) binary tree T is the word over {0, 1} obtained by traversing the empty subtrees of the nodes of T from left to right, except the first and the last, labelling an empty left subtree by 1 and an empty right subtree by 0. (See (2.4) below for examples of canopies.) A pair of twin binary search trees consist of a left strict binary search tree TL and a right strict binary search tree TR , such that TL and TR contain the same symbols, and the canopies of TL and TR are complementary, in the sense that the i-th symbol of the canopy of TL is 0 (respectively 1) if and only if the i-th symbol of the canopy of TR is 1 (respectively 0). The following is an example of a pair of twin binary search trees, with the complementary canopies 0110101 and 1001010 shown in grey:   3 (2.4) 5  1 6     3 6  11 1 1   1 5   1 1   2 5 1 0 1 0  2 1 , 5 0 1 3 0 1 0 3 0 1 0 6 0 1 0 6      .       A stalactic tableau is a finite array of symbols of A in which columns are top-aligned, and two symbols appear in the same column if and only if they are equal. For example, (2.5) 3 1 2 6 5 3 1 6 5 1 is a stalactic tableau. The insertion algorithm is very straightforward: Algorithm 2.5 ([HNT07, § 3.7]). Input: A stalactic tableau T and a symbol a ∈ A. 6 ALAN J. CAIN AND ANTÓNIO MALHEIRO If a does not appear in T , add a to the left of the top row of T . If a does appear in T , add a to the bottom of the (by definition, unique) column in which a appears. Output the new tableau. A binary search tree with multiplicities is a labelled binary search tree in which each label appears at most once, and where a non-negative integer called the multiplicity is assigned to each node label. An example of a binary search tree is: 52 (2.6) 1 62 2 13 . 32 (The superscripts on the labels in each node denote the multiplicities.) Algorithm 2.6. ([Pri13, Algorithm 3]) Input: A binary search tree with multiplicities T and a symbol a ∈ A. If T is empty, create a node, label it by a, and assign it multiplicity 1. If T is non-empty, examine the label x of the root node; if a < x, recursively insert a into the left subtree of the root node; if a > x, recursively insert a into the right subtree of the root note; if a = x, increment by 1 the multiplicity of the node label x. An lPS tableau is a finite array of symbols of A in which columns are top-aligned, the entries in the top row are non-decreasing from left to right, and the entries in each column are strictly increasing from top to bottom. For example, (2.7) 1 1 1 2 5 3 3 5 6 6 is an lPS tableau. The insertion algorithm is very straightforward: Algorithm 2.7 ([TY11, § 3.2]). Input: An lPS tableau T and a symbol a ∈ A. If a is greater than or equal to every symbol that appears in the top row of T , add a to the right of the top row of T . Otherwise, let C be the leftmost column whose topmost symbol is strictly greater than a. Slide column C down by one space and add a as a new entry on top of C. Output the new tableau. An rPS tableau is a finite array of symbols of A in which columns are topaligned, the entries in the top row are increasing from left to right, and the entries in each column are non-decreasing from top to bottom. For example, (2.8) 1 2 5 6 1 3 5 1 6 3 is an rPS tableau. Again, the insertion algorithm is very straightforward: Algorithm 2.8 ([TY11, § 3.2]). Input: An rPS tableau T and a symbol a ∈ A. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 7 If a is greater than every symbol that appears in the top row of T , add a to the right of the top row of T . Otherwise, let C be the leftmost column whose topmost symbol is greater than or equal to a. Slide column C down by one space and add a as a new entry on top of C. Output the new tableau. 2.3. Monoids from insertion. Let M ∈ {plac, hypo, sylv, sylv# , baxt, stal, taig, lPS, rPS} and u ∈ A∗ . Using the insertion algorithms described above, one can compute from u a combinatorial object PM (u) of the type associated to M in Table 2. If M 6= baxt, one computes PM (u) by starting with the empty combinatorial object and inserting the symbols of u one-by-one using the appropriate insertion algorithm and proceeding through the word u either left-to-right or right-to-left as shown in the table. For M = baxt, one uses a slightly different procedure: Pbaxt (u) is the pair of twin binary search trees Psylv# (u), Psylv (u) . For each M ∈ {plac, hypo, sylv, sylv# , baxt, stal, taig, lPS, rPS}, define the relation ≡M by u ≡M v ⇐⇒ PM (u) = PM (v). In each case, the relation ≡M is a congruence on A∗ , and so the factor monoid M = A∗/≡M can be formed, and is named as in Table 2. The rankn analogue is the factor monoid Mn = A∗n /≡M , where the relation ≡M is naturally restricted to A∗n × A∗n . It follows from the definition of ≡M (for any M ∈ {plac, hypo, sylv, sylv# , baxt, stal, taig, lPS, rPS}) that each element [u]≡M of the factor monoid M can be identified with the combinatorial object PM (u). The evaluation (also called the content) of a word u ∈ A∗ , denoted ev(u), is the infinite tuple of non-negative integers, indexed by A, whose a-th element is \|u\|a ; thus this tuple describes the number of each symbol in A that appears in u. It is immediate from the definition of the monoids above that if u ≡M v, then ev(u) = ev(v), and hence it makes sense to define the evaluation of an element p of one of these monoids to be the evaluation of any word representing it. Note that Mn is a submonoid of M for all n, and that Mm is a submonoid of Mn for all m ≤ n. In each case, M1 is a free monogenic monoid and thus commutative, but that Mn is non-commutative for n ≥ 2 (and thus M is also non-commutative). 3. Identities Subsections 3.2 to 3.5 find shortest non-trivial identities satisfied by hypo, stal, taig, sylv, sylv# , and baxt, in that order. Subsection 3.6 discusses the situation for lPS and rPS, and Subsection 3.7 discusses the current state of knowledge for plac. 3.1. Preliminaries. An identity over an alphabet X is a formal equality u = v, where u and v are words in the free monoid X ∗ , and is non-trivial if u and v are not equal. The elements of X are called variables. A monoid M satisfies the identity u = v if, for every homomorphism φ : X ∗ → M , the equality φ(u) = φ(v) holds in M . It is often useful to think of this in the informal way mentioned in the introduction: M satisfies the identity u = v if equality in M holds under every substitution of variables in the words u Symbol Combinatorial object Algorithm Direction Example Plactic Hypoplactic Sylvester #-sylvester Baxter Stalactic Taiga Left patience sorting Right patience sorting plac hypo sylv sylv# baxt stal taig lPS rPS Young tableau Quasi-ribbon tableau Right strict binary search tree Left strict binary search tree Pair of twin BSTs Stalactic tableau BST with multiplicities lPS tableau rPS tableau 2.1 2.2 2.3 2.4 — 2.5 2.6 2.7 2.8 Pplac (3613151265) is (2.1) Phypo (3613151265) is (2.2) Psylv (3613151265) is (2.3) Psylv (3613151265) is (2.3) Pbaxt (3613151265) is (2.4) Pstal (3613151265) is (2.5) Ptaig (3613151265) is (2.6) PlPS (3613151265) is (2.7) PrPS (3613151265) is (2.8) L-to-R L-to-R R-to-L L-to-R — R-to-L R-to-L L-to-R L-to-R ALAN J. CAIN AND ANTÓNIO MALHEIRO Monoid 8 Table 2. Insertion algorithms used to compute combinatorial objects, and the corresponding monoids. ‘L-to-R’ and ‘R-to-L’ abbreviate ‘left-to-right’ and ‘right-to-left’, respectively. The algorithm used to compute Pbaxt (·) is a special case and is discussed in the main text. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 9 and v by elements of M . Note that if a monoid satisfies an identity u = v, all of its submonoids and all of its homomorphic images satisfy u = v. Lemma 3.1. Let M be a monoid satisfying a non-trivial identity. Then the shortest (in terms of the sums of the lengths of the two sides) non-trivial identity satisfied by M is an identity over an alphabet with at most two variables. Proof. It is sufficient to show that, for any non-trivial identity over an alphabet with at most three variables, there is a strictly shorter identity with at most two variables. So suppose that u = v is a non-trivial identity over X satisfied by M , where \|X\| ≥ 3. Interchanging u and v if necessary, assume \|u\| ≤ \|v\|. Let eM be the identity (that is, the multiplicative neutral element) of M . First consider the case where u is a prefix of v. By non-triviality, u is strictly shorter than v, so that v = uw for some non-empty word w. Let x ∈ X be a variable that appears in w. So x\|u\|x = x\|v\|x is a nontrivial identity. To see that it is satisfied by M , proceed as follows: Let φ′ : {x}∗ → M be a homomorphism. Extend φ′ to a homomorphism φ : X → M by defining φ(z) = eM for all z ∈ X \ {x}. Note that φ′ (x\|u\|x ) = φ(u) and φ′ (x\|v\|x ) = φ(v) by the definition of φ′ , and that φ′ (u) = φ′ (v) since M satisfies u = v. Combining these equaltities shows that φ(x\|u\|x ) = φ(x\|v\|x ). Hence M satisfies the non-trivial identity x\|u\|x = x\|v\|x over the alphabet {x}. Now suppose u is not a prefix of v. Suppose u = u1 · · · u\|u\| and v = v1 · · · v\|v\| . Let i be minimal such that the variables ui and vi are different; suppose these variables are x and y. Let u′ and v ′ be obtained from u and v respectively by deleting all variables other than x and y. By the choice of i, u′ = v ′ is a non-trivial identity. To see that it is satisfied by M , proceed as follows: Let φ′ : {x, y}∗ → M be a homomorphism. Extend φ′ to a homomorphism φ : X ∗ → M by defining φ(z) = eM for all z ∈ X \ {x, y}. Note that φ(u) = φ′ (u′ ) and φ(v) = φ′ (v ′ ) by the definitions of the words u′ and v ′ and the homomorphism φ; note also that φ(u) = φ(v) since M satisfies the identity u = v. Combining these equalities shows that φ′ (u′ ) = φ′ (v ′ ). So M satisfies the non-trivial identity u′ = v ′ over the alphabet {x, y}. Note that in both cases the resulting identity is strictly shorter than the original one. Lemma 3.2. Let M be a monoid that contains a free monogenic submonoid and suppose that M satisfies an identity u = v over an alphabet X. Then \|u\| = \|v\| and \|u\|x = \|v\|x for all x ∈ X. Consequently, if u = v is non-trivial, then \|X\| ≥ 2. Proof. Let a ∈ M generate a free monogenic submonoid of M and let eM be the identity of M . Let x ∈ X. Define φx : X ∗ → M, x 7→ a, z 7→ eM for all z ∈ X \ {x}. Then a\|u\|x = φx (u) = φx (v) = a\|v\|x since M satisfies u = v. Since a generates a free monogenic submonoid, \|u\|x = \|v\|x . Since this holds for all x ∈ X, it follows that \|u\| = \|v\|. 10 ALAN J. CAIN AND ANTÓNIO MALHEIRO The length of an identity u = v with \|u\| = \|v\| is defined to be \|u\| = \|v\|; note that this applies to any identity satisfied by a monoid that contains a free monogenic submonoid by Lemma 3.2. Two identities u = v and u′ = v ′ are equivalent if one can be obtained from the other by possibly renaming variables and swapping the two sides of the equality. For example, xyxy = yxxy and xyyx = yxyx are equivalent, since the second can be obtained from the first by interchanging the variables x and y and swapping the two sides. Lemma 3.3. (1) Up to equivalence, the only length-2 non-trivial identity that can be satisfied by a monoid that contains a free monogenic submonoid is xy = yx; thus any homogeneous monoid satisfying a length-2 identity is commutative. (2) Up to equivalence, the only length-3 non-trivial identities over a twoletter alphabet that can be satisfied by a monoid that contains a free monogenic submonoid are xxy = xyx, xxy = yxx, xyx = yxx. (1) Let u1 u2 = v1 v2 be a non-trivial identity satisfied by some monoid that contains a free monogenic submonoid. Suppose u1 and v1 are the same variable. Then Lemma 3.2, implies that u2 and v2 are also the same variable, contradicting non-triviality. Thus u1 and v1 are different variables. Then Lemma 3.2, implies that u2 is the same variable as v1 and v2 is the same variable as u1 . Thus, up to equivalence, the identity is xy = yx. (2) Let u1 u2 u3 = v1 v2 v3 (where ui , vi ∈ {x, y}) be a non-trivial identity satisfied by some homogeneous monoid. Let I be the set of indices i ∈ {1, 2, 3} where the variables ui and vi are the same. Clearly, non-triviality shows that \|I\| < 3. If \|I\| = 2, then, interchanging x and y if necessary, there is a unique i ∈ {1, 2, 3} where ui is x and vi is y. Then \|u\|x = 1 + \|v\|x , since the words u and v differ only at the i-th symbol; this contradicts Lemma 3.2. If \|I\| = 0, then every symbol of u differs from every symbol of v. Since \|u\| = 3, at least one of \|u\|x and \|u\|y is at least 2. Interchanging x and y if necessary, assume \|u\|x ≥ 2. Then \|v\|x = \|u\|y ≤ 1, which is a contadiction. The only remaining possibility is \|I\| = 1. Interchanging x and y if necessary, assume that ui and vi are both x for exactly one i ∈ 1, 2, 3. For each of the three possibilities, the conditions \|u\|x = \|v\|x and \|u\|y = \|v\|y require the other symbols in u must be x and y, with the corresponding symbols in v being the opposite. Each possibility gives one of the given identities. Proof. Let M ∈ {plac, hypo, sylv, taig, stal, baxt, lPS, rPS}. Then M contains the free monogenic submonoid M1 . (Note that M1 thus satisfies the non-trivial identity xy = yx.) By Lemma 3.2, if M satisfies a non-trivial identity, it must be over an alphabet with at least two variables. The aim is to find shortest non-trivial identities satisfies by each monoid M: Lemmata 3.1 and 3.2 show that it will suffice to consider identities u = v over {x, y} with IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 11 \|u\| = \|v\|. Note that since none of the monoids M is commutative, any identity must have length at least 3. 3.2. Hypoplactic monoid. The authors proved the following result using a quasi-crystal structure for the hypoplactic monoid [CM17, Theorem 9.3]; this subsection presents a direct proof. Proposition 3.4. The hypoplactic monoid satisfies the following non-trivial identities xyxy = xyyx = yxxy = yxyx; xxyx = xyxx. Furthermore, up to equivalence, these are the shortest non-trivial identities satisfied by the hypoplactic monoid. Proof. The first goal is to show that these identities are satisfied by hypo. Let s, t ∈ hypo, and let p, q ∈ A∗ be words representing s, t, respectively. Let B = {a1 < . . . < ak } be the set of symbols in A that appear in at least one of p and q. By Algorithm 2.2, symbols ai+1 and ai are on the same row of Phypo (w) (for any w ∈ B ∗ ) if and only if the word w does not contain a symbol ai somewhere to the right of a symbol ai+1 (since inserting this symbol ai results in the part of the quasi-ribbon tableau that contains ai+1 being glued below the entry ai ). Suppose pqpq contains a symbol ai somewhere to the right of a symbol ai+1 . Then, regardless of whether these symbols both lie in u, both lie in v, or one lies in u and the other lies in v, the word pqqp also contains a symbol ai to the right of a symbol ai+1 . Similar reasoning establishes that if any of the words pqpq, qppq, qpqp, pqqp, ppqp, or pqpp, contains a symbol ai somewhere to the right of a symbol ai+1 , so do all the others. Hence either the symbols ai and ai+1 are on the same row in all of Phypo (pqpq), Phypo (pqqp), Phypo (qppq), Phypo (qpqp), Phypo (ppqp), and Phypo (pqpp), or on different rows in all of them. A quasi-ribbon tableau is clearly determined by its evaluation and by knowledge of whether adjacent different entries are on the same row. Thus, noting that ev(pqpq) = ev(qppq) = ev(qpqp) = ev(pqqp), it follows from the previous paragraph that stst = Phypo (pqpq) = tsst = Phypo (qppq) = tsts = Phypo (qpqp) = stts = Phypo (pqqp). Similarly, noting that ev(ppqp) = ev(pqpp), it follows that ssts = Phypo (ppqp) = Phypo (pqpp) = stss. The next goal is to show that, up to equivalence, these are the only length-4 non-trivial identities satisfied by hypo. Let u = v be a length-4 non-trivial identity over {x, y} satisfied by hypo. By Lemma 3.2, \|u\|x = \|v\|x and \|u\|y = \|v\|y . By Algorithm 2.2, symbols 1 and 2 are on different rows of Phypo (w) (for any w ∈ {1, 2}∗ ) if and only if the word w contain a symbol 1 somewhere to the right of a symbol 2. Suppose u contains a factor xy. Let φ : {x, y}∗ → 12 ALAN J. CAIN AND ANTÓNIO MALHEIRO hypo map x to 2 and y to 1; then Phypo (φ(u)) has 1 and 2 on different rows. The same must hold for Phypo (φ(v)), and thus v must contain a factor xy. The converse is similar, and the reasoning is parallel for factors yx. Thus u contains a factor xy (respectively, yx) if and only if v contains a factor xy (respectively, yx). If both u and v contain only factors xy, then u = x\|u\|x y \|u\|y = x\|v\|x y \|v\|y = v, which contradicts non-triviality. Similarly, it is impossible for u and v to contain only factors yx. So both u and v contain factors xy and yx. Interchanging x and y if necessary, assume \|u\|x = \|v\|x ≥ \|u\|y = \|v\|y . First consider the case where \|u\|x = \|v\|x = 3 and \|u\|y = \|v\|y = 1. Since u and v both contain xy and yx, this implies that u, v ∈ {xxyx, xyxx}. Now consider the case \|u\|x = \|v\|x = 2 and \|u\|y = \|v\|y = 2. Again, since u and v both contain xy and yx, this implies that u, v ∈ {xyxy, xyyx, yxxy, yxyx}. Thus u = v is equivalent to one of the identities in the statement. The final goal is to prove that no length-3 non-trivial identity is satisfied by hypo. Consider the length-3 identities from Lemma 3.3(2): xxy = xyx, xxy = yxx, xyx = yxx. If one puts x = 1 and y = 2 into the first two identities and x = 2 and y = 1 in the second, one sees that hypo satisfies none of them: Phypo (112) = 1 1 2 6= 1 1 = Phypo (121); 2 Phypo (112) = 1 1 2 6= 1 1 = Phypo (211); 2 Phypo (212) = 1 6= 1 2 2 = Phypo (122). 2 2 Thus hypo satisfies no length-3 identity. 3.3. Stalactic and taiga monoids. Lemma 3.5. The stalactic monoid satisfies the non-trivial identity xyx = yxx. Proof. Let x, y ∈ stal and let u, v ∈ A∗ be words representing x, y, respectively. Since the tree Pstal (w) is computed by applying Algorithm 2.5 to each symbol in the word w ∈ A∗ , proceeding right to left, it is clear that the order of the rightmost appearances of each symbol in w determines order of symbols in the top row of the stalactic tableau Pstal (w). Since the order of rightmost appearances of each symbol in the words uvu and vuu are the same, the order of symbols in the top row of Pstal (uvu) and Pstal (vuu) is also the same. Since ev(uvu) = ev(vuu), the length of corresponding columns in Pstal (uvu) and Pstal (vuu) are equal. Hence xyx = Pstal (uvu) = Pstal (vuu) = yxx. Lemma 3.6. The taiga monoid does not satisfy either of the identities xxy = xyx or xxy = yxx, and does not satisfy a non-trivial identity of length 2. Proof. To see that taig does not satisfy either of the identities xxy = xyx or xxy = yxx, note that the rightmost symbol of w ∈ A∗ labels the root node IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 13 of Ptaig (w), but the rightmost symbols of both these identities are different. Thus substituting 1 for x and 2 for y shows that neither identity can be satisfied by taig. Finally, taig does not satisfy a non-trivial identity of length 2 since it is clearly non-commutative. Since the taiga monoid is a homomorphic image of the stalactic monoid [Pri13, § 5], any identity that is satisfied by stal is satisfied by taig. Combining this with Lemmata 3.3, 3.5, and 3.6 gives the following results: Proposition 3.7. The stalactic monoid satisfies the non-trivial identity xyx = yxx. Furthermore, this is the unique shortest non-trivial identity satisfied by the stalactic monoid. Proposition 3.8. The taiga monoid satisfies the non-trivial identity xyx = yxx. Furthermore, this is the unique shortest non-trivial identity satisfied by the taiga monoid. 3.4. Sylvester monoid. By the definition of a binary search tree, if a tree has a node with label a, then any other node with label a is either above it or in its left subtree. This shows that there is a unique path from the root to a leaf node that contains all nodes labelled a. In particular, there is at most one leaf node with label a. Furthermore, there is a unique node a that is both the leftmost node with label a and the node labelled a that is furthest from the root. Similarly, there is a unique node a that is both the rightmost node with label a and the node labelled a that is closest to the root. Lemma 3.9. The sylvester monoid is left-cancellative. Proof. Let u, v ∈ A∗ and a ∈ A. Suppose that Psylv (au) = Psylv (av). That is, the binary search trees Psylv (au) and Psylv (av) are equal. Note further that the last symbol a inserted into both binary search trees is a leaf node. Deleting this (unique) leaf node labelled a from the equal trees Psylv (au) and Psylv (av) leaves equal binary search trees. That is, Psylv (u) and Psylv (v) are equal. Since u and v are arbitrary words and a represents an arbitrary generator, it follows that sylv is left-cancellative. Lemma 3.10. The sylvester monoid does not satisfy an identity equivalent to one of the form uxx = vyx, for any u, v ∈ {x, y}∗ . Proof. Using the Algorithm 2.3, one sees that 2 Psylv (· · · 22) = 2 2 and Psylv (· · · 12) = 1 ; note that these binary search trees differ in the left child of the root node. Thus to see that the identity uxx = vyx cannot be satisfied by sylv, one can substitute 2 for x and 1 for y. Lemma 3.11. Let p, q, r ∈ sylv be such that ev(p) = ev(q) = ev(r). Then pr = qr. 14 ALAN J. CAIN AND ANTÓNIO MALHEIRO Proof. Let u, v, w ∈ A∗ be words representing p, q, r, respectively. The tree Psylv (x) is computed by applying Algorithm 2.3 to each symbol in x, proceeding right to left. That is, Psylv (uw) and Psylv (vw) are obtained by inserting u and v (respectively) into Psylv (w). Since ev(u) = ev(v) = ev(w), every symbol that appears in u or v also appears in w and thus in Psylv (w). Consider how further symbols from u or v are inserted into Psylv (w): • Let a be the smallest symbol that appears in u, v, and w. Then the leftmost node of Psylv (w) must be labelled by a. Thus, during the computation of Psylv (uw) and Psylv (vw), all symbols a in u and v will certainly be inserted into the left subtree of this leftmost node in Psylv (w), and no other symbols are inserted into this left subtree by the minimality of a. • Let c be some symbol other than a that appears in u, v, and w, and let b be the maximum symbol less than c appearing in u, v, and w. Let Nc be the leftmost node in Psylv (w) labelled by c and let Nb be the rightmost node in Psylv (w) labelled by b. Suppose v is the label of the lowest common ancestor Nv of the nodes Nb and Nc . If Nv is neither Nb not Nc , then b ≤ v < c, which implies v = b by the choice of b, which contradicts the fact that Nb is the rightmost node labelled b. Thus the lowest common ancestor of Nb and Nc must be one of Nb or Nc . That is, one of the following must hold: – Nb is above Nc . Then Nc is in the right subtree of Nb , since b < c. Since there is no node that is to the right of Nb and to the left of Nc , it follows that Nc has an empty left subtree. Thus any symbol c in u or v will be inserted into this currently empty left subtree of Nc , and no other symbols will be inserted into this subtree by the maximality of b among the symbols less than c. – Nc is above Nb . Then Nb is in the left subtree of Nc , since b < c. Since there is no node that is to the right of Nb and to the left of Nc , it follows that Nb is the left child of Nc , and that Nb has an empty right subtree. Thus any symbol c in u or v will be inserted into this currently empty right subtree of Nb , and no other symbols will be inserted into this subtree by the maximality of b among the symbols less than c. Combining these cases, one sees that every symbol d from u or v is inserted into a particular previously empty subtree of Psylv (w), dependent only on the value of the symbol d (and not on its position in u or v), and that unequal symbols are inserted into different subtrees. Since ev(u) = ev(v), the same number of symbols d are inserted, for each such symbol d. Hence Psylv (uw) = Psylv (vw) and thus pr = Pstal (uw) = Pstal (vw) = qr. Proposition 3.12. The sylvester monoid satisfies the non-trivial identity xyxy = yxxy. Furthermore, up to equivalence, this is the unique shortest identity satisfied by the sylvester monoid. Proof. Let x, y ∈ sylv. Let p = r = xy and q = yx. Then ev(p) = ev(q) = ev(r), so pr = qr by Lemma 3.11. Thus sylv satisfies xyxy = yxxy. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 15 Since hypo is a homomorphic image of sylv [Pri13, Example 6], and hypo satisfies no length-3 identity by Proposition 3.4, it follows that sylv cannot satisfy a length-3 identity. So let u = v be some length-4 identity satisfied by sylv. Suppose u = u1 u2 u3 u4 and v = v1 v2 v3 v4 , where ui , vi ∈ {x, y}. Since the rightmost symbol in a word w determines the root node of Psylv (w), it follows that u4 and v4 are the same symbol; interchanging x and y if necessary, assume this is y. If u1 and v1 were the same symbol, then the left-cancellativity of sylv (Lemma 3.9) would imply that sylv satisfied the length-3 identity u2 u3 u4 = v2 v3 v4 , which contradicts the previous paragraph. Thus u1 and v1 are different symbols. Interchanging u and v if necessary, assume that u1 is x and v1 is y. The condition \|u\|y = \|v\|y implies that at least one of u2 and u3 is y, which yields the following three possibilities for u: xyyy, xxyy, xyxy. The conditions \|u\|x = \|v\|x and \|u\|y = \|v\|y now imply the following four possibilities for the identity u = v (with the first possibility for u above giving two different identities): xyyy = yxyy, xyyy = yyxy, xxyy = yxxy, xyxy = yxxy. The last of these is the identity already proven to hold in sylv. The second and third cannot hold in sylv by Lemma 3.10. In the first identity, putting x = 1 and y = 2 proves that it is not satisfied by sylv: 2 Psylv (1222) = 2 2 1 2 6= 2 = Psylv (2122) 1 2 Thus xyxy = yxxy is the unique shortest identity satisfied by sylv. Since the sylvester and #-sylvester monoids are anti-isomorphic, the following results follow by dual reasoning: Lemma 3.13. The #-sylvester monoid is right-cancellative. Lemma 3.14. The #-sylvester monoid does not satisfy an identity equivalent to one of the form xxu = xyv, for any u, v ∈ {x, y}∗ . Lemma 3.15. Let p, q, s ∈ sylv# be such that ev(p) = ev(q) = ev(s). Then sp = sq. Proposition 3.16. The #-sylvester monoid satisfies the non-trivial identity yxyx = yxxy. Furthermore, up to equivalence, this is the unique shortest identity satisfied by the #-sylvester monoid. 3.5. Baxter monoid. Lemma 3.17. Let p, q, r, s ∈ baxt be such that ev(p) = ev(q) = ev(r) = ev(s). Then spr = sqr. Proof. Let t, u, v, w ∈ A∗ represent p, q, r, s, respectively. Let t′ = wt and u′ = wu; note that ev(t′ ) = ev(u′ ). By Lemma 3.11 applied to the elements Psylv (t′ ), Psylv (u′ ), Psylv (v), it follows that Psylv (t′ v) = Psylv (u′ v); thus Psylv (wtv) = Psylv (wuv) 16 ALAN J. CAIN AND ANTÓNIO MALHEIRO Let t′′ = tv and u′′ = uv; note that ev(t′′ ) = ev(u′′ ). By Lemma 3.15 applied to the elements Psylv# (t′ ), Psylv# (u′ ), Psylv# (w), it follows that Psylv# (wt′′ ) = Psylv# (wu′′ ); thus Psylv# (wtv) = Psylv# (wuv). Hence spr = Pbaxt (wuv) = Psylv# (wtv), Psylv (wtv) = Psylv# (wuv), Psylv (wuv) = Pbaxt (wuv) = sqp. Proposition 3.18. The Baxter monoid satisfies the identities yxxyxy = yxyxxy and xyxyxy = xyyxxy. Furthermore, up to equivalence, these are the unique shortest non-trivial identities satisfied by the Baxter monoid. Proof. Let x, y ∈ baxt. Let p = r = xy and q = s = yx. Then ev(p) = ev(q) = ev(r) = ev(s). Thus, by Lemma 3.17, yxxyxy = spr = sqr = yxyxxy. The same reasoning with s = xy shows that xyxyxy = xyyxxy. The next step is to show that, up to equivalence, these are the only identities of length 6 satisfied by baxt. So suppose that u = v is an identity of length 6 satisfied by baxt, with u = u1 u2 · · · u6 and v = v1 v2 · · · v6 , where ui , vi ∈ {x, y}. Since the first symbol in a word w determines the root symbol of the left-hand tree in the pair Pbaxt (u), and since the last symbol determines the root symbol of the right-hand tree, it follows that u1 and v1 must be the same variable, and that u6 and v6 must the same variable. By Lemmata 3.10 and 3.14, u2 and v2 must be the same variable, and u5 and v5 must be the same variable. Since sylv and sylv# are both homomorphic images of baxt [Gir12, Proposition 3.7], the identity u = v is also satified by sylv and sylv# . Since sylv is left-cancellative by Lemma 3.9, u3 u4 u5 u6 = v3 v4 v5 v6 is satisfied by sylv and is thus equivalent to xyxy = yxxy. The aim is to characterize u = v up to equivalence, so assume that u3 u4 u5 u6 = v3 v4 v5 v6 actually is the identity xyxy = yxxy. Since sylv# is right-cancellative by Lemma 3.13, u1 u2 u3 u4 = v1 v2 v3 v4 is satisfied by sylv# and is thus equivalent to yxyx = yxxy, which is equivalent (by swapping the two sides) to yxxy = yxyx and (by interchaning x and y) to xyxy = xyyx. Combining these with u3 u4 u5 u6 = v3 v4 v5 v6 from the previous paragraph shows that u = v is (up to equivalence) either yxxyxy = yxyxxy or xyxyxy = xyyxxy. Finally, it is necessary to show that no length-5 identity is satisfied by baxt. Suppose u = v is an identity of length 5 satisfied by the Baxter monoid. Suppose that u = u1 u2 · · · u5 and v = v1 v2 · · · v5 , where ui , vi ∈ {x, y}. As before, considering the root symbols of the left-hand and right-hand trees in Pbaxt (w) shows that u1 and v1 must be the same symbol, and that u5 and v5 must the same symbol. Since sylv and sylv# are both homomorphic images of baxt [Gir12, Proposition 3.7], both of them satisfy the identity u = v. Furthermore, sylv# is right-cancellative and so satisfies the identity u1 u2 u3 u4 = v1 v2 v3 v4 ; while sylv is left-cancellative and so satisfies the identity u2 u3 u4 u5 = v2 v3 v4 v5 . By Proposition 3.12, the unique length-4 identity satisfied by sylv is xyxy = yxxy, so the identity u = v is either xxyxy = xyxxy or yxyxy = yyxxy. Deleting the rightmost symbol y from each of these identities yields xxyx = IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS 17 xyxx and yxyx = yyxx, one of which must the identity u1 u2 u3 u4 = v1 v2 v3 v4 satisfied by sylv# . This is a contradiction, since by Proposition 3.16 the only length-4 identity satisfied by sylv# is yxyx = yxxy. 3.6. Left and right patience sorting monoids. The present authors and Silva [CMS, § 3.8] have shown that the elements PlPS (21) and PlPS (1) generate a free submonoid of lPS , and that the elements PrPS (2) and PrPS (31) generate a free submonoid of rPS. Since free monoids of rank at least 2 satisfy no non-trivial identities, it follows that neither lPS nor rPS satisfy a non-trivial identity. Let lPSn = A∗n /≡lPS (where ≡lPS is naturally restricted to A∗n ×A∗n ) be the left patience sorting monoid of rank n. Let rPSn = A∗n /≡rPS (where ≡rPS is naturally restricted to A∗n ×A∗n ) be the right patience sorting monoid of rank n. Clearly lPSn is a submonoid of lPS and rPSn is a submonoid of rPS. Then by the previous paragraph, lPSn for n ≥ 2 satisfies no non-trivial identities, while lPS1 , as a monogenic monoid, is of course commutative and satisfies the identity xy = yx. Similarly, rPSn for n ≥ 3 satisfies no non-trivial identities, while rPS1 , as a monogenic monoid, is of course commutative and satisfies the identity xy = yx. Finally, rPS2 satisfies the identity xyxy = xyyx, and, up to equivalence, this is the shortest identity satisfied by rPS2 [CMS, Proposition 3.29]. 3.7. Plactic monoid. Whether the plactic monoid satisfies a non-trivial identity remains an open question, but some partial results are known. Let placn = A∗n /≡plac (where ≡plac is naturally restricted to A∗n × A∗n ) be the plactic monoid of rank n; clearly placn is a submonoid of plac. First, plac1 is monogenic and thus commutative and satisfies xy = yx. Kubat & Okniński [KO14] have shown that plac2 satisfies Adian’s identity xyyxxyxyyx = xyyxyxxyyx, and that plac3 satisfies the identity pqqpqp = pqpqqp, where p(x, y) and q(x, y) are respectively the left and right side of Adian’s identity (and so the identity pqqpqp = pqpqqp has sixty variables x or y on each side). Furthermore, plac3 does not satisfy Adian’s identity [KO14, p. 111–2]. Conjecture 3.19. For each n ≥ 4, there is a non-trivial identity satisfied by placn . Conjecture 3.20. There is no non-trivial identity satisfied by plac. References [Adi66] S. Adian. ‘Deﬁning relations and algorithmic problems for groups and semigroups’. Trudy Mat. 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Ser. A, 274 (1963), pp. 1–4. J.-B. Priez. ‘A lattice of combinatorial Hopf algebras: Binary trees with multiplicities’. In Formal Power Series and Algebraic Combinatorics, Nancy, 2013. The Association. Discrete Mathematics & Theoretical Computer Science. url: http://www.dmtcs.org/pdfpapers/dmAS0196.pdf. M. Rey. ‘Algebraic constructions on set partitions’. In Formal Power Series and Algebraic Combinatorics, 2007. url: http://www-igm.univ-mlv.fr/~ rey/articles/rey-fpsac07.pdf. IDENTITIES IN PLACTIC, HYPOPLACTIC, AND RELATED MONOIDS [Shn93] [TY11] 19 L. Shneerson. ‘Identities in ﬁnitely generated semigroups of polynomial growth’. J. Algebra, 154, no. 1 (1993), pp. 67–85. doi: 10.1006/jabr.1993.1004. H. Thomas & A. Yong. ‘Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm’. Advances in Applied Mathematics, 46, no. 1 (2011), pp. 610–642. doi: 10.1016/j.aam.2009.07.005. Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829–516 Caparica, Portugal E-mail address: a.cain@fct.unl.pt Departamento de Matemática & Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829–516 Caparica, Portugal E-mail address: ajm@fct.unl.pt	4
A NOTE ON NILPOTENT SUBGROUPS OF AUTOMORPHISM GROUPS OF RAAGS arXiv:1803.08284v1 [math.GR] 22 Mar 2018 JAVIER ARAMAYONA & ANTHONY GENEVOIS Abstract. We observe that automorphism groups of right-angled Artin groups contain nilpotent non-abelian subgroups, namely H3 (Z) the threedimensional integer Heisenberg group, provided they admit a certain type of element, called an adjacent transvection. This represents a (minor) extension of a result of Charney-Vogtmann. 1. Introduction Automorphism groups of right-angled Artin groups (RAAGs) are normally studied through their comparison with linear groups and automorphism groups of free groups (and thus, by extension, with mapping class groups). One fact that distinguishes the linear group GL(n, Z) from the automorphism group Aut(Fn ) of the free group Fn , and from the mapping class group Mod(S), is that every solvable subgroup of the latter two is virtually abelian; see [2, 1] and [3], respectively. In sharp contrast, arbitrary automorphism groups of RAAGs may contain a copy of GL(n, Z) with n ≥ 3. Even when this is not the case, Charney-Vogtmann [6] proved that (outer) automorphism groups of RAAGs contain torsion-free nilpotent and non-abelian subgroups, whenever they contain at least two adjacent transvections; see below for a definition. Further examples may be found in [7]. The purpose of this short note is to observe that the presence of a single adjacent transvection suffices for the (full) automorphism group of a centerless RAAG to contain a nilpotent non-abelian subgroup. We remark that Charney-Vogtmann showed [6], if there are no adjacent transvections then every solvable subgroup is virtually abelian. Before we state our result, denote by ≤ the usual partial order in the vertex set V (Γ) of Γ; see Section 2. Also, let H3 (Z) be the three-dimensional integer Heisenberg group, which has presentation H3 (Z) = hA, B, C \| [A, C] = [B, C] = 1, [A, B] = Ci. With this notation, we will observe: Theorem 1.1. Let Γ be a simplicial graph. If there are adjacent vertices a, b ∈ V (Γ) with a ≤ b, such that b is not adjacent to all the vertices of Γ, then H3 (Z) is a subgroup of Aut(AΓ ). As a consequence, for such Γ the group Aut(AΓ ) does not embed in Mod(S) or Aut(Fn ). A further immediate application, surely well-known to experts, is that no finite-index subgroup of such Aut(AΓ ) may act nicely on non-positively curved space: 1 2 JAVIER ARAMAYONA & ANTHONY GENEVOIS Corollary 1.2. If Γ is as in Theorem 1.1 , then no finite-index subgroup of Aut(AΓ ) can act properly by semi-simple isometries on a CAT(0) space. In sharp contrast, Aut(AΓ ) and Out(AΓ ) are sometimes commensurable to a right-angled Artin group [5, 7], and therefore they act nicely on CAT(0) spaces. 2. Definitions In order to make this note as concise and self-contained as possible, we only introduce the objects that we will need in the proof of our results. We refer the reader to the various papers in the bibliography below for a thorough introduction to automorphism groups of right-angled Artin groups. Let Γ be a simplicial graph, and denote by V (Γ) (resp. E(Γ)) its set of vertices (resp. edges). The right-angled Artin group AΓ defined by Γ is the group given by the presentation AΓ = hv ∈ V (Γ) \| [v, w] = 1 ⇐⇒ vw ∈ E(Γ)i . Given a right-angled Artin group AΓ , we consider its automorphism group Aut(AΓ ). This is a finitely presented group with an explicit generating set [11, 12] and an explicit (although in slightly different terms) presentation [9]. Here, we will need a specific type of element of Aut(AΓ ) and Out(AΓ ), called a tranvection. Given vertices v, w ∈ V (Γ), the transvection tvw is the self-map of AΓ defined by tvw : v 7→ vw and tvw (u) = u for every u 6= v. An easy observation is that tvw ∈ Aut(AΓ ) if and only if lk(v) ⊂ st(w); here, lk(·) denotes the link of a vertex in Γ, while st(·) denotes its star, namely the link union the vertex. As usual in the literature, we will write v ≤ w to mean lk(v) ⊂ st(w), noting that the relation ≤ is in fact a partial order. Finally, we say that tvw is an adjacent transvection if v and w are adjacent in Γ. 3. A homomorphic image of the Heisenberg group We now prove Theorem 1.1: Proof of Theorem 1.1. Let a, b be adjacent vertices of Γ, with a ≤ b and which are not adjacent to all the vertices of Γ. We write ca and cb for the automorphisms of AΓ given by conjugation by a and b, respectively. Finally, let t = tab the transvection that sends a to ab and fixes the rest of generators. First, observe that ca and cb commute since a and b are adjacent. Next, we claim that [cb , t] = 1 also. Indeed, if v 6= a, tcb t−1 (v) = tcb (v) = t(bvb−1 ) = bvb−1 = cb (v), and tcb t−1 (a) = tcb (ab−1 ) = t(bab−2 ) = babb−2 = cb (a). Finally, we claim that tca t−1 = cb ca . Indeed, if v 6= a then tca t−1 (v) = tca (v) = t(ava−1 ) = abvb−1 a−1 = cb ca (v), A NOTE ON NILPOTENT SUBGROUPS OF AUTOMORPHISM GROUPS OF RAAGS 3 while tca t−1 (a) = tca (ab−1 ) = t(a2 b−1 a−1 ) = ababb−1 b−1 a−1 = cb ca (a), as desired. In light of the above, and using notation for the presentation of H3 (Z) given in the introduction, the map H3 (Z) → Aut(AΓ ) given by A 7→ ca , B 7→ t, and C 7→ cb is a homomorphism. Now, we want to prove that this is a monomorphism. From the presentation of H3 (Z), it clearly follows that any of its elements can be written as Am B n C p for some m, n, p ∈ Z. Furthermore, Am B n C p = 1 if and only if m = n = p = 0. Indeed, we first deduce from the equality Am B n C p = 1 that m = n = 0 by looking at the image of Am B n C p into the abelianization of H3 (Z); therefore, our equality reduces to C p = 1, and finally p = 0 follows from the torsion-freeness of H3 (Z). Thus, in order to prove that our homomorphism H3 (Z) → Aut(ΓG) is injective, we only have n p to verify that, for every m, n, p ∈ Z, cm a t cb = 1 implies m = n = p = 0. n p So let m, n, p ∈ Z and suppose that cm a t cb = 1. Noticing that n p m n p −p m n m n m n −m cm = abn , a t cb (a) = ca t (b ab ) = ca t (a) = ca (ab ) = a ab a p we first deduce that n = 0. Therefore, cm a cb = 1. This precisely means that m p a b belongs to the center of AΓ . On the other, the center of AΓ corresponds exactly to the subgroup generated by the vertices which are adjacent to all the vertices of Γ. Because we supposed that neither a nor b is adjacent to all the vertices of Γ, it follows that am bp = 1. Therefore, we deduce that am = bp = 1, and finally that m = p = 0 by torsion-freeness of AΓ . 4. Actions on CAT(0) spaces We prove Corollary 1.2: Proof. Let Γ as in Theorem 1.1, so there exist adjacent vertices a, b ∈ V (Γ) with a ≤ b and which are not adjacent to all the vertices of Γ. As above, write ca and cb for the conjugations on a and b, respectively, and t for the transvection tab . Notice that ca and cb are infinite-order automorphisms because a and b do not belong to the center of AΓ . Consider a finite-index subgroup G < Aut(AΓ ); as such, there exists some m m m ≥ 1 such that cm a , cb , t ∈ G. It is immediate to check that, for all n ∈ N, (1) −n mn tn cm = cm a t a cb . Suppose now that G acts properly by semi-simple isometries on some CAT(0) m space X. As cm a and cb commute, the Flat Torus Theorem [4] implies that X contains an isometrically embedded copy of a Euclidean plane, on which m cm a and cb act by translations with quotient a 2-torus. It follows that, as mn must tend a transformation of this plane, the translation length of cm a cb to infinity as n grows; on the other hand, equation (1) implies that this translation length must be equal to that of cm a . This is a contradiction, and the result follows. A possible interpretation of the previous proof is the following. As Theorem 1.1 proves, Aut(ΓG) contains a copy of the three-dimensional integer 4 JAVIER ARAMAYONA & ANTHONY GENEVOIS Heisenberg group H3 (Z) = hA, B, C \| [A, C] = [B, C] = 1, [A, B] = Ci. It is not difficult to notice that, for any k ≥ 1, the subgroup hAk , B k , C k i ≤ H3 (Z) is naturally isomorphic to H3 (Z) itself, so that any finite-index subgroup of Aut(ΓG) has to contain a copy of H3 (Z). Therefore, Corollary 1.2 follows from the fact that H3 (Z) does not act properly by simple-isometries on a CAT(0) space. This is essentially what we have shown in the previous proof. An alternative argument can be found in [10, Corollary 5.1], where it is furthermore proved that, for any proper action of H3 (Z) on a CAT(0) space, C is necessarily parabolic. It is worth noticing that H3 (Z) has a proper parabolic action on the complex hyperbolic plance H2C , which is a proper finite-dimensional CAT(−1) space [10, Corollary 5.1]. References [1] E. Alibegović, Translation lengths in Out(Fn ), Geom. Dedicata 92 (1), pp. 87–93 (2002). [2] M. Bestvina, M. Feighn, M. Handel, Solvable subgroups of Out(Fn ) are virtually Abelian, Geom. Dedicata 104 (1), pp. 71–96 (2004). [3] J. Birman, A. Lubotzky, J. McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (4), pp. 1107–1120 (1983). [4] M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer-Verlag, Berlin (1999). [5] R. Charney, M. Farber, Random groups arising as graph products, Alg. Geom. Topol. 12, pp. 979–995 (2012). [6] R. Charney, K. Vogtmann. Subgroups and quotients of automorphism groups of RAAGs. In Low-dimensional and symplectic topology, pp. 9–27, Proc. Sympos. Pure Math., 82, Amer. Math. Soc., Providence, RI (2011). [7] M. Day, Finiteness of outer automorphism groups of random right-angled Artin groups, Algebr. Geom. Topol. 12 (3), pp. 1553–1583 (2012). [8] M. Day, On solvable subgroups of automorphism groups of right-angled Artin groups, Internat. J. Algebra Comput. 21, pp. 61–70 (2011). [9] M. Day. Peak reduction and finite presentations for automorphism groups of right angled Artin groups, Geom. Topol. 13, pp. 817–855 (2009). [10] K. Fujiwara, T. Shioya, S. Yamagata, Parabolic isometries of CAT(0) spaces and CAT(0) dimensions, Algebr. Geom. Topol. 4, pp. 861–892 (2004). [11] M. R. Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. 52 (2), pp. 318–334 (1995). [12] H. Servatius, Automorphisms of graph groups, J. Algebra 126 (1), pp. 34–60 (1989).	4
arXiv:1711.04052v1 [math.AC] 11 Nov 2017 BIG COHEN-MACAULAY MODULES, MORPHISMS OF PERFECT COMPLEXES, AND INTERSECTION THEOREMS IN LOCAL ALGEBRA LUCHEZAR L. AVRAMOV, SRIKANTH B. IYENGAR, AND AMNON NEEMAN Abstract. There is a well known link from the first topic in the title to the third one. In this paper we thread that link through the second topic. The central result is a criterion for the tensor nilpotence of morphisms of perfect complexes over commutative noetherian rings, in terms of a numerical invariant of the complexes known as their level. Applications to local rings include a strengthening of the Improved New Intersection Theorem, short direct proofs of several results equivalent to it, and lower bounds on the ranks of the modules in every finite free complex that admits a structure of differential graded module over the Koszul complex on some system of parameters. 1. Introduction A big Cohen-Macaulay module over a commutative noetherian local ring R is a (not necessarily finitely generated) R-module C such that some system of parameters of R forms a C-regular sequence. In [16] Hochster showed that the existence of such modules implies several fundamental homological properties of finitely generated R-modules. In [17], published in [18], he proved that big Cohen-Macaulay modules exist for algebras over fields, and conjectured their existence in the case of mixed characteristic. This was recently proved by Y. André in [2]; as a major consequence many “Homological Conjectures” in local algebra are now theorems. A perfect R-complex is a bounded complex of finite projective R-modules. Its level with respect to R, introduced in [6] and defined in 2.3, measures the minimal number of mapping cones needed to assemble a quasi-isomorphic complex from bounded complexes of finite projective modules with differentials equal to zero. The main result of this paper, which appears as Theorem 3.3, is the following Tensor Nilpotence Theorem. Let f : G → F be a morphism of perfect complexes over a commutative noetherian ring R. If f factors through a complex whose homology is I-torsion for some ideal I of R with height I ≥ levelR HomR (G, F ), then the induced morphism ⊗nR f : ⊗nR G → ⊗nR F is homotopic to zero for some non-negative integer n. Date: November 15, 2017. 2010 Mathematics Subject Classification. 13D22 (primary); 13D02, 13D09 (secondary). Key words and phrases. big Cohen-Macaulay module, homological conjectures, level, perfect complex, rank, tensor nilpotent morphism. Partly supported by NSF grants DMS-1103176 (LLA) and DMS-1700985 (SBI). 1 2 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN Big Cohen-Macaulay modules play an essential, if discreet role in the proof, as a tool for constructing special morphisms in the derived category of R; see Proposition 3.7. In applications to commutative algebra it is convenient to use another property of morphisms of perfect complexes: f is fiberwise zero if H(k(p) ⊗R f ) = 0 holds for every p in Spec R. Hopkins [21] and Neeman [25] have shown that this is equivalent to tensor nilpotence; this is a key tool for the classification of the thick subcategories of perfect R-complexes. It is easy to see that the level of a complex does not exceed its span, defined in 2.1. Due to these remarks, the Tensor Nilpotence Theorem is equivalent to the Morphic Intersection Theorem. If f is not fiberwise zero and factors through a complex with I-torsion homology for an ideal I of R, then there are inequalities: span F + span G − 1 ≥ levelR HomR (G, F ) ≥ height I + 1 . In Section 4 we use this result to prove directly, and sometimes to generalize and sharpen, several basic theorems in commutative algebra. These include the Improved New Intersection Theorem, the Monomial Theorem and several versions of the Canonical Element Theorem. All of them are equivalent, but we do not know if they imply the Morphic Intersection Theorem; a potentially significant obstruction to that is that the difference span F − levelR F can be arbitrarily big. Another application, in Section 5, yields lower bounds for ranks of certain finite free complexes, related to a conjecture of Buchsbaum and Eisenbud, and Horrocks. In [7] a version of the Morphic Intersection Theorem for certain tensor triangulated categories is proved. This has implications for morphisms of perfect complexes of sheaves and, more generally, of perfect differential sheaves, over schemes. 2. Perfect complexes Throughout this paper R will be a commutative noetherian ring. This section is a recap on the various notions and construction, mainly concerning perfect complexes, needed in this work. Pertinent references include [6, 26]. 2.1. Complexes. In this work, an R-complex (a shorthand for ‘a complex of Rmodules’) is a sequence of homomorphisms of R-modules X := ∂X X ∂n−1 n → Xn−1 −−−→ Xn−2 −→ · · · · · · −→ Xn −−− such that ∂ X ∂ X = 0. We write X ♮ for the graded R-module underlying X. The ith i X suspension of X is the R-complex Σi X with (Σi X)n = Xn−i and ∂nΣ X = (−1)i ∂n−i for each n. The span of X is the number span X := sup{i \| Xi 6= 0} − inf{i \| Xi 6= 0} + 1 Thus span X = −∞ if and only if X = 0, and span X = ∞ if and only if Xi 6= 0 for infinitely many i ≥ 0. The span of X is finite if and only if span X is an natural number. When span X is finite we say that X is bounded. Complexes of R-modules are objects of two distinct categories. In the category of complexes C(R) a morphism f : Y → X of R-complexes is a family (fi : Yi → Xi )i∈Z or R-linear maps satisfying ∂iX fi = fi−1 ∂iY . It is a quasiisomorphism if H(f ), the induced map in homology, is bijective. Complexes that can be linked by a string of quasi-isomorphisms are said to be quasi-isomorphic. TENSOR NILPOTENT MORPHISMS 3 The derived category D(R) is obtained from C(R) by inverting all quasi-isomorphisms. For constructions of the localization functor C(R) → D(R) and of the derived functors ?⊗LR ? and RHomR (¿, ?), see e.g. [13, 31, 24]. When P is a complex of projectives with Pi = 0 for i ≪ 0, the functors P ⊗LR ? and RHomR (P, ?) are represented by P ⊗R ? and HomR (P, ?), respectively. In particular, the localization functor induces for each n a natural isomorphism of abelian groups ∼ HomD(R) (P, Σn X) . (2.1.1) H−n (RHomR (P, X)) = 2.2. Perfect complexes. In C(R), a perfect R-complex is a bounded complex of finitely generated projective R-modules. When P is perfect, the R-complex P ∗ := HomR (P, R) is perfect and the natural biduality map P −→ P ∗∗ = HomR (HomR (P, R), R) is an isomorphism. Moreover for any R-complex X the natural map P ∗ ⊗R X −→ HomR (P, X) is an isomorphism. In the sequel these properties are used without comment. 2.3. Levels. A length l semiprojective filtration of an R-complex P is a sequence of R-subcomplexes of finitely generated projective modules 0 = P (0) ⊆ P (1) ⊆ · · · ⊆ P (l) = P ♮ ♮ such that P (i − 1) is a direct summand of P (i) and the differential of P (i)/P (i−1) is equal to zero, for i = 1, . . . , l. For every R-complex F , we set F is a retract of some R-complex P that R . level F = inf l ∈ N has a semiprojective filtration of length l By [6, 2.4], this number is equal to the level F with respect to R, defined in [6, 2.1]. In particular, levelR F is finite if and only if F is quasi-isomorphic to some perfect complex. When F is quasi-isomorphic to a perfect complex P , one has levelR F ≤ span P . (2.3.1) Indeed, if P := 0 → Pb → · · · → Pa → 0, then consider the filtration by subcomplexes P (n) := P<n+a . The inequality can be strict; see 2.7 below. When R is regular, any R-complex F with H(F ) finitely generated satisfies levelR F ≤ dim R + 1 . (2.3.2) For R-complexes X and Y one has (2.3.3) levelR (Σi X) = levelR X for every integer i, and R level (X ⊕ Y ) = max{levelR X, levelR Y }. These equalities follow easily from the definitions. Lemma 2.4. The following statements hold for every perfect R-complex P . (1) levelR (P ∗ ) = levelR P . (2) For each perfect R-complex Q there are inequalities levelR (P ⊗R Q) ≤ levelR P + levelR Q − 1 . levelR HomR (P, Q) ≤ levelR P + levelR Q − 1 . 4 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN Proof. (1) If P is a retract of P ′ , then levelR P ≤ levelR P ′ and P ∗ is a retract of (P ′ )∗ . Thus, we can assume P itself has a finite semiprojective filtration {P (n)}ln=0 . The inclusions P (l − i) ⊆ P (l − i + 1) ⊆ P define subcomplexes ∗ P ∗ (i) := Ker(P ∗ −→ P (l − i) ) ⊆ Ker(P ∗ −→ P (l − i + 1)) =: P ∗ (i + 1) of finitely generated projective modules. They form a length l semiprojective filtration of P ∗ , as P ∗ (i − 1)♮ is a direct summand of P ∗ (i)♮ and there are isomorphisms ∗ P ∗ (n) ∼ P (l − n + 1) . = P ∗ (n − 1) P (l − n) This gives levelR P ∗ ≤ levelR P , and the reverse inequality follows from P ∼ = P ∗∗ . (2) Assume first that P has a semiprojective filtration {P (n)}ln=0 and Q has a semiprojective filtration {Q(n)}m n=0 . For all h, i, we identity P (h) ⊗R Q(i) with a subcomplex of P ⊗R Q. For each non-negative integer n ≥ 0 form the subcomplex X C(n) := P (j + 1) ⊗R Q(n − j) j>0 of P ⊗R Q. A direct computation yields an isomorphism of R-complexes Q(n − j) C(n) ∼ X P (j + 1) ⊗R . = C(n − 1) P (j) Q(n − j − 1) j>0 l+m−1 {C(n)}n=0 Thus is a semiprojective filtration of P ⊗R Q. The second inequality in (2) follows from the first one, given (1) and the isomorphism HomR (P, Q) ∼ = P ∗ ⊗R Q. Next we verify the first inequality. There is nothing to prove unless the levels of P and Q are finite. Thus we may assume that P is a retract of a complex P ′ with a semiprojective filtration of length l = levelR P and Q is a retract of a complex Q′ with a semiprojective filtration of length m = levelR Q. Then P ⊗R Q is a retract of P ′ ⊗R Q′ , and—by what we have just seen—this complex has a semiprojective filtration of length l + m − 1, as desired. 2.5. Ghost maps. A ghost is a morphism g : X → Y in D(R) such that H(g) = 0; see [11, §8]. Evidently a composition of morphisms one of which is ghost is a ghost. The next result is a version of the “Ghost Lemma”; cf. [11, Theorem 8.3], [29, Lemma 4.11], and [5, Proposition 2.9]. Lemma 2.6. Let F be an R-complex and c an integer with c ≥ levelR F . When g : X → Y is a composition of c ghosts the following morphisms are ghosts F ⊗LR g : F ⊗LR X −→ F ⊗LR Y and RHomR (F, g) : RHomR (F, X) −→ RHomR (F, Y ) Proof. For every R-complex W there is a canonical isomorphism ≃ RHomR (F, R) ⊗LR W −−→ RHomR (F, W ) , so it suffices to prove the first assertion. For that, we may assume that F has a semiprojective filtration {F (n)}ln=0 , where l = levelR F . By hypothesis, g = h ◦ f where f : X → W is a (c − 1)fold composition of ghosts and h : W → Y is a ghost. Tensoring these maps with the exact sequence of R-complexes ι π 0 −→ F (1) −−→ F −−→ G −→ 0 TENSOR NILPOTENT MORPHISMS 5 where G := F/F (1), yields a commutative diagram of graded R-modules H(F (1) ⊗R X) // H(F ⊗R X) H(F (1)⊗f ) // H(G ⊗R X) H(F ⊗f ) H(F (1) ⊗R W ) H(ι⊗W ) // H(F ⊗R W ) H(F (1)⊗h) H(G⊗f ) H(π⊗W ) // H(G ⊗R W ) H(F ⊗h) H(F (1) ⊗R Y ) H(ι⊗Y ) // H(F ⊗R Y ) // H(G ⊗R Y ) where the rows are exact. Since levelR G ≤ l − 1 ≤ c − 1, the induction hypothesis implies G ⊗ f is a ghost; that is to say, H(G ⊗ f ) = 0. The commutativity of the diagram above and the exactness of the middle row implies that Im H(F ⊗ f ) ⊆ Im H(ι ⊗ W ) This entails the inclusion below. Im H(F ⊗ g) = H(F ⊗ h)(Im H(F ⊗ f )) ⊆ H(F ⊗ h)(Im H(ι ⊗ W )) ⊆ Im H(ι ⊗ Y ) H(F (1) ⊗ h)) =0 The second equality comes from the commutativity of the diagram. The last one holds because F (1) is graded-projective and H(h) = 0 imply H(F (1) ⊗ h) = 0. 2.7. Koszul complexes. Let x := x1 , . . . , xn be elements in R. ♮ We write K(x) for the Koszul complex on x. Thus K(x) is the exterior algebra K on a free R-module K(x)1 with basis {e x1 , . . . , x en }, and ∂ is the unique R-linear map that satisfies the Leibniz rule and has ∂(e xi ) = xi for i = 1, . . . , n. In particular, K(x) is a DG (differential graded) algebra, and so its homology H(K(x)) is a graded algebra with H0 (K(x)) = R/(x). This implies (x) H(K(x)) = 0. Evidently K(x) is a perfect R-complex; it is indecomposable when R is local; see [1, 4.7]. As K(x)i is non-zero precisely for 0, . . . , n, from (2.3.1) one gets levelR K(x) ≤ span K(x) = n + 1 . Equality holds if R is local and x is a system of parameters; see Theorem 4.2 below. However, span K(x) − levelR K(x) can be arbitrarily large; see [1, Section 3]. For any Koszul complex K on n elements, there are isomorphisms of R-complexes n M n Σi K ( i ) . K∗ ∼ = = Σ−n K and K ⊗R K ∼ i=0 See [8, Propositions 1.6.10 and 1.6.21]. It thus follows from (2.3.3) that (2.7.1) levelR HomR (K, K) = levelR (K ⊗R K) = levelR K . In particular, the inequalities in Lemma 2.4(2) can be strict. 3. Tensor nilpotent morphisms In this section we prove the Tensor Nilpotence Theorem announced in the introduction. We start by reviewing the properties of interest. 6 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN 3.1. Tensor nilpotence. Let f : Y → X be a morphism in D(R). The morphism f is said to be tensor nilpotent if for some n ∈ N the morphism f ⊗L · · · ⊗L f : Y ⊗LR · · · ⊗LR Y −→ X ⊗LR · · · ⊗LR X \| R {z R } n is equal to zero in D(R); when the R-complexes X, Y are perfect this means that the morphism ⊗n f : ⊗nR Y → ⊗nR X is homotopic to zero. When X is perfect and f : X → Σl X is a morphism with ⊗n f homotopic to zero the n-fold composition Σl f Σ2l f Σnl X −→ Σl X −−−→ Σ2n X −−−−→ · · · −−−→ Σnl X is also homotopic to zero. The converse does not hold, even when R is a field for in that case tensor nilpotent morphisms are zero. 3.2. Fiberwise zero morphisms. A morphism f : Y → X that satisfies k(p) ⊗LR f = 0 in D(k(p)) for every p ∈ Spec R is said to be fiberwise zero. This is equivalent to requiring k ⊗LR f = 0 in D(k) for every homomorphism R → k with k a field. In D(k) a morphism is zero if (and only if) it is a ghost, so the latter condition is equivalent to H(k ⊗LR f ) = 0. In D(k), a morphism is tensor nilpotent exactly when it is zero. Thus if f is tensor nilpotent, it is fiberwise zero. There is a partial converse: If a morphism f : G → F of perfect R-complexes is fiberwise zero, then it is tensor nilpotent. This was proved by Hopkins [21, Theorem 10] and Neeman [25, Theorem 1.1]. The next result is the Tensor Nilpotence Theorem from the Introduction. Recall that an R-module is said to be I-torsion if each one of its elements is annihilated by some power of I. Theorem 3.3. Let R be a commutative noetherian ring and f : G → F a morphism of perfect R-complexes. If for some ideal I of R the following conditions hold (1) f factors through some complex with I-torsion homology, and (2) levelR HomR (G, F ) ≤ height I , then f is fiberwise zero. In particular, f is tensor nilpotent. The proof of the theorem is given after Proposition 3.7. Remark 3.4. Lemma 2.4 shows that the inequality (2) is implied by levelR F + levelR G ≤ height I + 1 ; the converse does not hold; see (2.7.1). On the other hand, condition (2) cannot be weakened: Let (R, m, k) be a local ring and G the Koszul complex on some system of parameters of R and let f : G −→ (G/G6d−1 ) ∼ = Σd R be the canonical surjection with d = dim R. Then G is an m-torsion complex and levelR G = d + 1; see 2.7. Evidently H(k ⊗R f ) 6= 0, so f is not fiberwise zero. In the proof of Theorem 3.3 we exploit the functorial nature of I-torsion. TENSOR NILPOTENT MORPHISMS 7 3.5. Torsion complexes. The derived I-torsion functor assigns to every X in D(R) an R-complex RΓI X; when X is a module it computes its local cohomology: HnI (X) = H−n (RΓI X) holds for each integer n. There is a natural morphism t : RΓI X −→ X in D(R) that has the following universal property: Every morphism Y → X such that H(Y ) is I-torsion factors uniquely through t; see Lipman [24, Section 1]. It is easy to verify that the following conditions are equivalent. (1) H(X) is I-torsion. (2) H(X)p = 0 for each prime ideal p 6⊇ I. (3) The natural morphism t : RΓI X → X is a quasi-isomorphism. When they hold, we say that X is I-torsion. Note a couple of properties: (3.5.1) (3.5.2) If X is I-torsion, then X ⊗LR Y is I-torsion for any R-complex Y . There is a natural isomorphism RΓI (X ⊗L Y ) ∼ = (RΓI X) ⊗L Y . R R Indeed, H(Xp ) ∼ = H(X)p = 0 holds for each p 6⊇ I, giving Xp = 0 in D(R). Thus (X ⊗LR Y )p ∼ = Xp ⊗LR Y ∼ =0 holds in D(R). It yields H(X ⊗LR Y )p ∼ = H((X ⊗LR Y )p ) = 0, as desired. A proof of the isomorphism in (3.5.2) can be found in [24, 3.3.1]. 3.6. Big Cohen-Macaulay modules. Let (R, m, k) be a local ring. A (not necessarily finitely generated) R-module C is big Cohen-Macaulay if every system of parameters for R is a C-regular sequence, in the sense of [8, Definition 1.1.1]. In the literature the name is sometimes used for R-modules C that satisfy the property for some system of parameters for R; however, the m-adic completion of C is then big Cohen-Macaulay in the sense above; see [8, Corollary 8.5.3]. The existence of big Cohen-Macaulay was proved by Hochster [16, 17] in case when R contains a field as a subring, and by André [2] when it does not; for the latter case, see also Heitmann and Ma [15]. In this paper, big Cohen-Macaulay modules are visible only in the next result. Proposition 3.7. Let I be an ideal in R and set c := height I. When C is a big Cohen-Macaulay R-module the following assertions hold. (1) The canonical morphism t : RΓI C → C from the I-torsion complex RΓI C (see 3.5) is a composition of c ghosts. (2) If a morphism g : G → C of R-complexes with levelR G ≤ c factors through some I-torsion complex, then g = 0. Proof. (1) We may assume I = (x), where x = {x1 , . . . , xc } is part of a system of parameters for R; see [8, Theorem A.2]. The morphism t factors as RΓ(x1 ,...,xc ) (C) −→ RΓ(x1 ,...,xc−1 ) (C) −→ · · · −→ RΓ(x1 ) (C) −→ C . Since the sequence x1 , . . . , xc is C-regular, we have Hi (RΓ(x1 ,...,xj ) (C)) = 0 for i 6= −j; see [8, (3.5.6) and (1.6.16)]. Thus every one of the arrows above is a ghost, so that t is a composition of c ghosts, as desired. (2) Suppose g factors as G → X → C with X an I-torsion R-complex. As noted in 3.5, the morphism X → C factors through t, so g factors as g′ g′′ t G −→ X −→ RΓI C − →C. 8 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN In view of the hypothesis levelR G ≤ c and part (1), Lemma 2.6 shows that RHomR (G, t) : RHomR (G, RΓI C) −→ RHomR (G, C) is a ghost. Using brackets to denote cohomology classes, we get [g] = [tg ′′ g ′ ] = H0 (RHomR (G, t))([g ′′ g ′ ]) = 0 . Due to the isomorphism (2.1.1), this means that g is zero in D(R). Lemma 3.8. Let f : G → F be a morphism of perfect R-complexes, where G is finite free with Gi = 0 for i ≪ 0 and F is perfect. Let f ′ : F ∗ ⊗R G → R denote the composed morphism in the next display, where e is the evaluation map: F ∗ ⊗R f e F ∗ ⊗R G −−−−−−→ F ∗ ⊗R F −−→ R . If f factors through some I-torsion complex, then so does f ′ . The morphism f ′ is fiberwise zero if and only if so is f . Proof. For the first assertion, note that if f factors through an I-torsion complex X, then F ∗ ⊗R f factors through F ∗ ⊗R X, and the latter is I-torsion. For the second assertion, let k be field and R → k be a homomorphism of rings. Let (−) and (−)∨ stand for k ⊗R (−) and Homk (−, k), respectively. The goal is to prove that f = 0 is equivalent to f ′ = 0. Since F is perfect, there are canonical isomorphisms ∼ ∨ = F ∗ ⊗ k −−→ HomR (F, k) ∼ = Homk (k ⊗R F, k) = (F ) . Given this, it follows that f ′ can be realized as the composition of morphisms ∨ (F ) ⊗k f ∨ ∨ e (F ) ⊗k G −−−−−−→ (F ) ⊗k F −−→ k . If F is zero, then f = 0 and f ′ = 0 hold. When F is nonzero, it is easy to verify 6 0 is equivalent to f ′ 6= 0, as desired. that f = Proof of Theorem 3.3. Given morphisms of R-complexes G → X → F such that F and G are perfect and X is I-torsion for an ideal I with levelR HomR (G, F ) ≤ height I , we need to prove that f is fiberwise zero. This implies the tensor nilpotence of f , as recalled in 3.1. By Lemma 3.8, the morphism f ′ : F ∗ ⊗R G → R factors through an I-torsion complex, and if f ′ is fiberwise zero, so if f . The isomorphisms of R-complexes (F ∗ ⊗R G)∗ ∼ = G∗ ⊗R F ∼ = HomR (G, F ) and Lemma 2.4 yield levelR (F ∗ ⊗R G) = levelR HomR (G, F ). Thus, replacing f by f ′ , it suffices to prove that if f : G → R is a morphism that factors through an I-torsion complex and satisfies levelR G ≤ height I, then f is fiberwise zero. Fix p in Spec R. When p 6⊇ I we have Xp = 0, by 3.5(2). For p ⊇ I we have levelRp Gp ≤ levelR G ≤ height I ≤ height Ip , where the first inequality follows directly from the definitions; see [6, Proposition 3.7]. It is easy to verify that Xp is Ip -torsion. Thus, localizing at p, we may further assume (R, m, k) is a local ring, and we have to prove that H(k ⊗R f ) = 0 holds. Let C be a big Cohen-Macaulay R-module. It satisfies mC 6= C, so the canonical γ ε f γ map π : R → k factors as R − →C − → k. The composition G − →R− → C is zero in TENSOR NILPOTENT MORPHISMS 9 D(R), by Proposition 3.7. We get πf = εγf = 0, whence H(k ⊗R π) H(k ⊗R f ) = 0. Since H(k ⊗R π) is bijective, this implies H(k ⊗R f ) = 0, as desired. The following consequence of Theorem 3.3 is often helpful. Corollary 3.9. Let (R, m, k) be a local ring, F a perfect R-complex, and G an R-complex of finitely generated free modules. If a morphism of R-complexes f : G → F satisfies the conditions (1) f factors through some m-torsion complex, (2) sup F ♮ − inf G♮ ≤ dim R − 1, and then H(k ⊗R f ) = 0. Proof. An m-torsion complex X satisfies k(p) ⊗LR X = 0 for any p in Spec R \ {m}. Thus a morphism, g, of R-complexes that factors through X is fiberwise zero if and only if k ⊗LR g = 0. This remark will be used in what follows. Condition (2) implies Gn = 0 for n ≪ 0. Let f ′ denote the composition F ∗ ⊗R f e F ∗ ⊗R G −−−−−−→ F ∗ ⊗R F −−→ R , where e is the evaluation map. Since inf (F ∗ ⊗R G)♮ = − sup F ♮ +inf G♮ , Lemma 3.8 shows that it suffices to prove the corollary for morphisms f : G → R. As f factors through some m-torsion complex, so does the composite morphism f G60 ⊆ G −−→ R It is easy to check that if the induced map H(k ⊗R G60 ) → H(k ⊗R R) = k is zero, then so is H(k ⊗R f ). Thus we may assume Gn = 0 for n 6∈ [−d + 1, 0], where d = dim R. This implies levelR G ≤ d, so Theorem 3.3 yields the desideratum. For some applications the next statement, with weaker hypothesis but also weaker conclusion, suffices. The example in Remark 3.4 shows that the result cannot be strengthened to conclude that f is fiberwise zero. Theorem 3.10. Let R be a local ring and f : G → F a morphism of R-complexes. If there exists an ideal I of R such that (1) f factors through an I-torsion complex, and (2) levelR F ≤ height I, then H(C ⊗LR f ) = 0 for every big Cohen-Macaulay module C. Proof. Set c := height I and let t : RΓI C → C be the canonical morphism. It follows from (3.5.1) that C ⊗LR f also factors through an I-torsion R-complex. Then the quasi-isomorphism (3.5.2) and the universal property of the derived I-torsion functor, see 3.5, implies that C ⊗LR f factors as a composition of the morphisms: t⊗L F R C ⊗LR G −→ (RΓI C) ⊗LR F −−−− −→ C ⊗LR F By Proposition 3.7(1) the morphism t is a composition of c ghosts. Thus condition (2) and Lemma 2.6 imply t ⊗LR F is a ghost, and hence so is C ⊗LR f . 10 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN 4. Applications to local algebra In this section we record applications the Tensor Nilpotence Theorem to local algebra. To that end it is expedient to reformulate it as the Morphic Intersection Theorem from the Introduction, restated below. Theorem 4.1. Let R be a commutative noetherian ring and f : G → F a morphism of perfect R-complexes. If f is not fiberwise zero and factors through a complex with I-torsion homology for an ideal I of R, then there are inequalities: span F + span G − 1 ≥ levelR HomR (G, F ) ≥ height I + 1 . Proof. The inequality on the left comes from Lemma 2.4 and (2.3.1). The one on the right is the contrapositive of Theorem 3.3. Here is one consequence. Theorem 4.2. Let R be a local ring and F a complex of finite free R-modules: F := 0 → Fd → Fd−1 → · · · → F0 → 0 For each ideal I such that I · Hi (F ) = 0 for i ≥ 1 and I · z = 0 for some element z in H0 (F ) \ m H0 (F ), where m is the maximal ideal of R, one has d + 1 ≥ span F ≥ levelR F ≥ height I + 1 . Proof. Indeed, the first two inequalities are clear from definitions. As to third one, pick ze ∈ F0 representing z in H0 (F ) and consider the morphism of complexes f : R → F given by r 7→ re z . Since z is not in m H0 (F ), one has H0 (k ⊗R f ) = k ⊗R H0 (f ) 6= 0 for k = R/m. In particular, k⊗R f is nonzero. On the other hand, f factors through the inclusion X ⊆ F , where X is the subcomplex defined by ( Fi i≥1 Xi = Re z + d(F1 ) i = 0 By construction, we have Hi (X) = Hi (F ) for i ≥ 1 and I H0 (X) = 0, so H(X) is I-torsion. The desired inequality now follows from Theorem 4.1 applied to f . The preceding result is a stronger form of the Improved New Intersection Theorem1 of Evans and Griffith [12]; see also [19, §2]. First, the latter is in terms of spans of perfect complexes whereas the one above is in terms of levels with respect to R; second, the hypothesis on the homology of F is weaker. Theorem 4.2 also subsumes prior extensions of the New Intersection Theorem to statements involving levels, namely [6, Theorem 5.1], where it was assumed that I · H0 (F ) = 0 holds, and [1, Theorem 3.1] which requires Hi (F ) to have finite length for i ≥ 1. In the influential paper [18], Hochster identified certain canonical elements in the local cohomology of local rings, conjectured that they are never zero, and proved that statement in the equal characteristic case. He also gave several reformulations that do not involve local cohomology. Detailed discussions of the relations between these statements and the histories of their proofs are presented in [28] and [20]. 1This and the other statements in this section were conjectures prior to the appearance of [2]. TENSOR NILPOTENT MORPHISMS 11 Some of those statements concern properties of morphisms from the Koszul complex on some system of parameters to resolutions of various R-modules. This makes them particularly amenable to approaches from the Morphic Intersection Theorem. In the rest of this section we uncover direct paths to various forms of the Canonical Element Theorem and related results. We first prove a version of [18, 2.3]. The conclusion there is that fd is not zero, but the remarks in [18, 2.2(6)] show that it is equivalent to the following statement. Theorem 4.3. Let (R, m, k) be a local ring, x a system of parameters for R, and K the Koszul complex on x, and F a complex of free R-modules. If f : K → F is a morphism of R-complexes with H0 (k ⊗R f ) 6= 0, then one has Hd (S ⊗R f ) 6= 0 for S = R/(x) and d = dim R . Proof. Recall from 2.7 that ∂(K) lies in (x)K, that K1 has a basis x e1 , . . . , x ed and K ♮ is the exterior algebra on K1 . Thus Kd is a free R-module with basis x = x e1 · · · x ed and Hd (S ⊗R K) = S(1 ⊗ x), so we need to prove f (Kd ) 6⊆ (x)Fd + ∂(Fd+1 ). Arguing by contradiction, we suppose the contrary. This means f (x) = x1 y1 + · · · + xd yd + ∂ F (y) with y1 , . . . , yd ∈ Fd and y ∈ Fd+1 . For i = 1, . . . , d set x∗i := x e1 · · · x ei−1 x ei+1 · · · x ed ; thus {x∗1 , . . . , x∗d } is a basis of the R-module Kd−1 . Define R-linear maps hd−1 : Kd−1 → Fd by hd : Kd → Fd+1 by hd−1 (x∗i ) = (−1)i−1 yi for i = 1, . . . , d . hd (x) = y . Extend them to a degree one map h : K → F with hi = 0 for i 6= d − 1, d. The map g := f − ∂ F h − h∂ K : K → F is easily seen to be a morphism of complexes that is homotopic to f and satisfies gd = 0. This last condition implies that g factors as a composition of morphisms g′ K −−→ F<d ⊆ F . The complex K is m-torsion; see 2.7. Thus Corollary 3.9, applied to g ′ , yields H(k ⊗R g ′ ) = 0. This gives the second equality below: H0 (k ⊗R f ) = H0 (k ⊗R g) = H0 (k ⊗R g ′ ) = 0 . The first one holds because f and g are homotopic, and the last one because g0 = g0′ . The result of the last computation contradicts the hypotheses on H0 (k ⊗R f ). A first specialization is the Canonical Element Theorem. Corollary 4.4. Let I be an ideal in R containing a system of parameters x1 , . . . , xd . With K the Koszul complex on x and F a free resolution of R/I, any morphism f : K → F of R-complexes lifting the surjection R/(x) → R/I has fd (K) 6= 0. As usual, when A is a matrix, Id (A) denotes the ideal of its minors of size d. Corollary 4.5. Let R be a local ring, x a system of parameters for R, and y a finite subset of R with (y) ⊇ (x). If A is a matrix such that Ay = x, then Id (A) 6⊆ (x) for d = dim R. 12 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN Proof. Let K and F be the Koszul complexes on x and y, respectively. The matrix A defines a unique morphism of DG R-algebras f : K → F . Evidently, H0 (k ⊗R f ) is the identity map on k, and hence is not zero. Since fd can be represented by a column matrix whose entries are the various d×d minors of A, the desired statement is a direct consequence of Theorem 4.3. A special case of the preceding result yields the Monomial Theorem. Corollary 4.6. When y1 , . . . , yd is a system of parameters for local ring, one has (y1 · · · yd )n 6∈ (y1n+1 , . . . , ydn+1 ) Proof. Apply Corollary 4.5 to the inclusion  n y1 0  0 y2n  A:= . ..  .. . 0 0 for every integer n ≥ 1. (y1n+1 , . . . , ydn+1 ) ⊆ (y1 , . . . , yd ) and  ··· 0 ··· 0   ..  . .. . . · · · ydn We also deduce from Theorem 4.3 another form of the Canonical Element Theorem. Roberts [27] proposed the statement and proved that it is equivalent to the Canonical Element Theorem; a different proof appears in Huneke and Koh [22]. Recall that for any pair (S, T ) of R-algebras the graded module TorR (S, T ) carries a natural structure of graded-commutative R-algebra, given by the ⋔-product of Cartan and Eilenberg [10, Chapter XI.4]. Lemma 4.7. Let R be a commutative ring, I an ideal of R, and set S := R/I. Let G → S be some R-free resolution, K be the Koszul complex on some generating set of I, and g : K → G a morphisms of R-complexes lifting the identity of S. For every surjective homomorphism ψ : S → T of of commutative rings there is a commutative diagram of strictly graded-commutative S-algebras S ⊗R K V S H1 (S ⊗R K) α V ψ // // V TorR 1 (S, S) S µS TorR 1 (S,ψ) V T TorR 1 (S, T ) // TorR (S, S) TorR (S,ψ) µT // TorR (S, T ) where α1 = H1 (S⊗R g), the map α is defined by the functoriality of exterior algebras, and the maps µ? are defined by the universal property of exterior algebras. V Proof. The equality follows from ∂ K (K) ⊆ IK and K ♮ = R K1 . The resolution G can be chosen to have G61 = K61 ; this makes α1 surjective, and the surjectivity of α follows. The map TorR 1 (S, ψ) is surjective because it can be identified with the natural map I/I 2 → I/IJ, where J = Ker(R → T ); the surjectivity of V R ψ Tor1 (S, ψ) follows. The square commutes by the naturality of ⋔-products. Theorem 4.8. Let (R, m, k) be a local ring, I a parameter ideal, and S := R/I. For each surjective homomorphism S → T the morphism of graded T -algebras V R µT : T TorR 1 (S, T ) −→ Tor (S, T ) has the property that µT ⊗T k is injective. In particular, µk is injective. TENSOR NILPOTENT MORPHISMS 13 Proof. The functoriality of the construction of µ implies that the canonical surjection π : T → k induces a commutative diagram of graded-commutative algebras V T V π TorR 1 (S, T ) µT // TorR (S, T ) TorR 1 (S,π) V k TorR (S,π) TorR 1 (S, k) µk // TorR (S, k) It is easy to verify that π induces a bijective map R R TorR 1 (S, π) ⊗T k : Tor1 (S, T ) ⊗T k → Tor1 (S, k) , so (∧π TorR (S, π)) ⊗T k is an isomorphism. Thus it suffices to show µk is injective. ≃ Let K be the Koszul complex on a minimal generating set of I. Let G − → S and ≃ F − → k be R-free resolutions of S and k, respectively. Lift the identity map of S and the canonical surjection ψ : S → k to morphisms g : K → G and h : G → F , respectively. We have µS α = H(S ⊗R g) and TorR (S, ψ) = H(S ⊗R h). This implies the second equality in the string R S µkd TorR d (S, π)αd = Tord (S, ψ)µd αd = Hd (S ⊗R hg) 6= 0 . The first equality comes from Lemma 4.7, with T = k, and the non-equality from Theorem 4.3, with f = hg. In particular, we get µkd 6= 0. We have an isomorphism ∼ V k d of graded k-algebras, so µk is injective by the next remark. TorR 1 (S, k) = k V Remark 4.9. If Q is a field, d is a non-negative integer, and λ : Q Qd → B is a homomorphism of graded Q-algebras with λd 6= 0, then λ is injective. V Vd Indeed, the graded subspace Q Qd of exterior algebra Q Qd is contained in every every non-zero ideal and has rank one, so λd 6= 0 implies Ker(λ) = 0. 5. Ranks in finite free complexes This section is concerned with DG modules over Koszul complexes on sequences of parameters. Under the additional assumptions that R is a domain and F is a resolution of some R-module, the theorem below was proved in [3, 6.4.1], and earlier for cyclic modules in [9, 1.4]; background is reviewed after the proof. The Canonical Element Theorem, in the form of Theorem 4.3 above, is used in the proof. Theorem 5.1. Let (R, m, k) be a local ring, set d = dim R, and let F be a complex of finite free R-modules with H0 (F ) 6= 0 and Fi = 0 for i < 0. If F admits a structure of DG module over the Koszul complex on some system of parameters of R, then there is an inequality d (5.1.1) rankR (Fn ) ≥ for each n ∈ Z . n Proof. The desired inequality is vacuous when d = 0, so suppose d ≥ 1. Let x be the said system of parameters of R and K the Koszul complex on x. Since F is a DG K-module, each Hi (F ) is an R/(x)-module, and hence of finite length. First we reduce to the case when R is a domain. To that end, let p be a prime ideal of R such that dim(R/p) = d. Evidently, the image of x in R is a system of 14 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN parameters for R/p. By base change, (R/p) ⊗R F is a DG module over (R/p) ⊗R K, the Koszul complex on x with coefficients in R/p, with ∼ R/p ⊗ H0 (F ) 6= 0 . H0 ((R/p) ⊗R F ) = ♮ Moreover, the rank of F ♮ as an R-module equals the rank of (R/p) ⊗R F as an R/p-module. Thus, after base change to R/p we can assume R is a domain. Choose a cycle z ∈ F0 that maps to a minimal generator of the R-module H0 (F ). Since F is a DG K-module, this yields a morphism of DG K-modules f: K →F with f (a) = az . This is, in particular, a morphism of complexes. Since k ⊗R H0 (F ) 6= 0, by the choice of z, Theorem 4.3 applies, and yields that f (Kd ) 6= 0. Since R is a domain, this implies f (Q ⊗R Kd ) is non-zero, where Q is the field of fractions of R. ♮ Set Λ := (Q ⊗R K) and consider the homomorphism of graded Λ-modules V λ := Q ⊗R f ♮ : Λ → Q ⊗R F ♮ . Qd , Remark 4.9 gives the inequality in the display d rankR (Fn ) = rankQ (Q ⊗R Fn ) ≥ rankQ (Λn ) = . n As Λ is isomorphic to Q Both equalities are clear. The inequalities (5.1.1) are related to a major topic of research in commutative algebra. We discuss it for a local ring (R, m, k) and a bounded R-complex F of finite free modules with F<0 = 0, homology of finite length, and H0 (F ) 6= 0. 5.2. Ranks of syzygies. The celebrated and still open Rank Conjecture of Buchsbaum and Eisenbud [9, Proposition 1.4], and Horrocks [14, Problem 24] predicts that (5.1.1) holds whenever F is a resolution of some module ofPfinite length. That conjecture is known for d ≤ 4. Its validity would imply n rankR Fn ≥ 2d . For d = 5 and equicharacteristic R, this was proved in [4, Proposition 1] by using Evans and Griffth’s Syzygy Theorem [12]; in view of [2], it holds for all R. M. Walker [30] used methods from K-theory to prove that P In a breakthrough, d rank F ≥ 2 holds when R contains 12 and is complete intersection (in particR n n ular, regular), and when R is an algebra over some field of positive characteristic. 5.3. Obstructions for DG module structures. Theorem 5.1 provides a series of obstruction for the existence of any DG module structure on F . In particular, it implies that if rankR F < 2d holds with d = dim R, then F supports no DG module structure over K(x) for any system of parameters x. Complexes satisfying the restriction on ranks were recently constructed in [23, 4.1]. These complexes have nonzero homology in degrees 0 and 1, so they are not resolutions of R-modules. 5.4. DG module structures on resolutions. Let F be a minimal resolution of an R-module M of nonzero finite length and x a parameter set for R with xM = 0. When F admits a DG module structure over K(x) the Rank Conjecture holds, by Theorem 5.1. It was conjectured in [9, 1.2′ ] that such a structure exists for all F and x. An obstruction to its existence was found in [3, 1.2], and examples when that obstruction is not zero were produced in [3, 2.4.2]. On the other hand, by [3, 1.8] the obstruction vanishes when x lies in m annR (M ). TENSOR NILPOTENT MORPHISMS 15 It is not known if F supports some DG K(x)-module structure for special choices of x; in particular, for high powers of systems of parameters contained in annR (M ). References [1] H. Altmann, E Grifo, J. Montaño, W. Sanders, T. Vu, Lower bounds on levels of perfect complexes, J. Algebra 491 (2017), 343–356. [2] Y. André, La conjecture du facteur direct, preprint, arXiv:1609.00345. [3] L. L. Avramov, Obstructions to the existence of multiplicative structures on minimal free resolutions, Amer. J. Math. 103 (1987), 1–31. [4] L. L. Avramov, R.-O. Buchweitz, Lower bounds on Betti numbers, Compositio Math. 86 (1993), 147–158. [5] L. L. Avramov, R.-O. Buchweitz, S. 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Manin, Methods of Homological Algebra, Annals of Math. (2) 114 (1981), 323–333. [14] R. Hartshorne, Algebraic vector bundles on projective spaces: a problem list, Topology 18 (1979), 117–128. [15] R. Heitmann, L. Ma, Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic, preprint, arXiv:1703.08281. [16] M. Hochster, Cohen-Macaulay modules, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., 311, Springer, Berlin, 1973; pp. 120–152. [17] M. Hochster, Deep local rings, Aarhus University preprint series, December 1973. [18] M. Hochster, Topics in the Homological Theory of Modules over Commutative Rings, Conf. Board Math. Sci. 24, Amer. Math. Soc., Providence, RI, 1975. [19] M. Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), 503–553. [20] M. Hochster, Homological conjectures and lim Cohen-Macaulay sequences, Homological and Computational Methods in Commutative Algebra (Cortona, 2016), Springer INdAM Series, Springer, Berlin, to appear. [21] M. Hopkins, Global methods in homotopy theory. Homotopy Theory (Durham, 1985), 73–96, London Math. Soc. Lecture Note Ser. 117, Cambridge Univ. Press, Cambridge, 1987. [22] C. Huneke, J. Koh, Some dimension 3 cases of the Canonical Element Conjecture, Proc. Amer. Math. Soc. 98 (1986), 394–398. [23] S. B. Iyengar, M. E. Walker, Examples of finite free complexes of small rank and homology, preprint, arXiv:1706.02156. [24] J. Lipman, Lectures on local cohomology and duality, Local Cohomology and its Applications (Guanajuato, 1999), Lecture Notes Pure Appl. Math. 226, Marcel Dekker, New York, 2002; pp. 39–89. [25] A. Neeman, The chromatic tower for D(R), Topology 31 (1992), 519–532. [26] P. C. Roberts, Homological Invariants of Modules over Commutative Rings, Sém. Math. Sup. 72, Presses Univ. Montréal, Montréal, 1980. [27] P. C. Roberts, The equivalence of two forms of the Canonical Element Conjecture, undated manuscript. [28] P. C. Roberts, The homological conjectures, Progress in Commutative Algebra 1, de Gruyter, Berlin, 2012; pp. 199–230. 16 L. L. AVRAMOV, S. B. IYENGAR, AND A. NEEMAN [29] R. Rouquier, Dimensions of triangulated categories, J. K-Theory 1 (2008), 193–256, 257–258. [30] M. E. Walker, Total Betti numbers of modules of finite projective dimension, Annals of Math., to appear; preprint, arXiv:1702.02560. [31] C. Weibel, An Introduction to Homological Algebra, Cambridge Stud. Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1994. Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A. E-mail address: avramov@math.unl.edu Department of Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A. E-mail address: iyengar@math.utah.edu Centre for Mathematics and its Applications, Mathematical Sciences Institute Australian National University, Canberra, ACT 0200, Australia. E-mail address: Amnon.Neeman@anu.edu.au	0
arXiv:1802.03734v1 [math.OC] 11 Feb 2018 The Use of Presence Data in Modelling Demand for Transportation Jonathan Epperlein1 , Jaroslaw Legierski2,3 , Marcin Luckner3 , Jakub Mareček1∗ and Rahul Nair1 February 13, 2018 Abstract We consider the applicability of the data from operators of cellular systems to modelling demand for transportation. While individual-level data may contain precise paths of movement, stringent privacy rules prohibit their use without consent. Presence data aggregate the individual-level data to information on the numbers of transactions at each base transceiver station (BTS) per each time period. Our work is aimed at demonstrating value of such aggregate data for mobility management while maintaining privacy of users. In particular, given mobile subscriber activity aggregated to short time intervals for a zone, a convex optimisation problem estimates most likely transitions between zones. We demonstrate the method on presence data from Warsaw, Poland, and compare with official demand estimates obtained with classical econometric methods. 1 Introduction Estimating travel demand is a key step of the transportation planning process. Cities typically build a modeling framework and use surveys along with observations to characterize how, where and by what means citizens move. These estimates are then used for management of existing infrastructure or for design of new services. Such an approach is resource-intensive, time consuming, and therefore can only be performed infrequently providing estimates that may be out-of-date. Alternatively, current movement patterns can be estimated based on data from mobile devices. Several works have studied digital trajectories, or “breadcrumbs”, which provide a very rich demand profile. Nevertheless, legislation and privacy regulation ∗ Jakub can be reached at jakub.marecek@ie.ibm.com. 1: IBM Research – Ireland, B3 IBM Campus Damastown, Dublin 15, Ireland. 2: Orange Labs Poland, 02-691 Warsaw, ul. Obrzezna 7, Poland. 3: Warsaw University of Technology, Faculty of Mathematics and Information Science, 00-662 Warsaw, ul. Koszykowa 75, Poland. 1 often restrict the use of such methods. In Europe, in particular, requirements on processing telecommunications transmission data are set by European directives: I Personal data should be protected, as per directive on e-Privacy(2002/58/EC) and directive on personal data protection (95/46/EC). I The consent of the data subject is required for processing any subscriber personal data. I The anonymized data can be processed without consent only if they are aggregated in the first step of processing and de-anonymization of the data is not possible. The anonymized record cannot not be connected with any subscriber in any case. The use and processing of CDR records without the permission of the subscriber is hence not possible in the Europe, and one may need to rely on aggregate data. In this paper, we present methods to estimate trip generation and trip distribution rates from aggregate presence data. In particular, we consider mobile subscriber’s location statistics aggregated in accordance with the above-mentioned regulations. We the present a linear program to be solved for each period of the discretisation of time to obtain the most likely estimate of movements since the previous period. We also present algorithms for obtaining the estimates of movements of likely movements over a number of periods, or consider a distribution of trip durations. Crucially, we show that the linear program can be solved in linear time, which makes the approach applicable in practice. 2 2.1 The Problem Data The data comes in the form of time-stamped averages of event counts for each of a number of geographical zones: (Zone, Timestamp, Event Count), where the zones are defined by the local public transport operator. More specifically: Say there are NZ zones Z1 , . . . , ZNZ , and we have a total of Nt time intervals [t0 ,t1 ], [t1t2 ], . . . , [tNt −1 ,tNt ] for which the event counts are reported; here, the intervals are all of equal length T , i.e. t` − t`−1 = T , but this is not necessary for the applicability of what follows. Thus, the data consists of tuples of the form (Zk ,t` , Ek,` ) for k = 1, . . . , NZ , ` = 1, . . . , Nt , which encode that “In zone Zk during the time interval [t`−1 ,t` ], Ek,` events were observed.” Note that the timestamp of the end of the interval is reported in the dataset, and that there are NZ · Nt such tuples. For the purposes of this paper, we consider the number of events as a measure of how many unique terminals (phones, pagers, etc.) were present in the give zone over the given time interval. 2 2.2 Modelling user movement The given data suggests how many terminals were present at any zone, but, due to privacy restrictions, not which terminals were present where at what time. Hence, the information as to how users move from one zone to another is lost, and the problem we address in this paper is how to estimate this information from the aggregate data that we are given. The problems treated are as follows: Problem 1 (One-Step Transitions). Given aggregate presence data for disjoint geographical zones at two points in time, and a cost function mapping each pair of geographical zones to a real number, estimate for each zone the proportion of users travelling to each other zone, within the time interval between the two points in time. Problem 2 (k-Step Transitions). Given aggregate presence data for disjoint geographical zones at two points t1 ,t2 in time and a cost function mapping each pair of geographical zones to a real number, estimate for each zone the proportion of users travelling to each other zone, within time k(t2 − t1 ). Problem 3 (Duration-L Transitions). Given aggregate presence data for disjoint geographical zones, for two or more points t1 ,t2 , . . . in time, and a cost function mapping each pair of geographical zones to a real number, and a given trip duration L, estimate for each zone the proportion of users travelling to each other zone, with trip duration being L. Problem 4 (Realistic Transitions). Given aggregate presence data for disjoint geographical zones, for two or more points t1 ,t2 , . . . in time, and a cost function mapping each pair of geographical zones to a real number, and a distribution of trip durations, as a histogram approximation of a probability density function fL , estimate for each zone the proportion of users travelling to each other zone. Absent further information, we have to make assumptions about user behaviour, and the main assumption is that the users redistribute themselves in an optimal way – optimal with respect to a cost function, the design of which requires insights into the underlying transportation structure. This is best explained via a trivial-seeming example: Assume first, that there are only two zones, Z1 and Z2 , and that we have the following data: (Z1 ,t1 , 3), (Z2 ,t1 , 1), (Z1 ,t2 , 2), (Z2 ,t2 , 2). Intuitively, it seems obvious to expect that one user went from Zone 1 to Zone 2, whereas all other stayed put. Of course, it is also possible that two users went from Zone 1 to Zone 2 and the sole user in Zone 2 went to Zone 1. However, if we associate a cost to users moving, then the first, intuitive solution, is clearly the optimal one. Assume now, that there is a third zone, Z3 , and that we additionally have the following data: (Z3 ,t1 , 1), (Z3 ,t2 , 1). A natural assumption is now that the user in Zone 3 simply stayed there. But what if Zone 3 is connected to Zones 1 and 2 through rapid transit, whereas the only convenient way of getting from Zone 1 to Zone 2 is through Zone 3, and this journey takes longer than t2 − t1 ? In this case, the cost of travel from Z1 to Z2 would be quite high, and the 3 most reasonable and optimal solution would be to expect one user to move from Z1 to Z3 and one user to move from Z3 to Z2 . This example illustrates the idea of optimal user movement and highlights, how knowledge of the transportation system needs to be incorporated in the design of the cost function. One can, for instance, benefit from the estimates of the travel-times provided by Google Maps [36], use the Euclidean distance, or a discrete metric, in which adjacent pairs of zones have cost 0 and all other pairs have cost 1. Either way, we will solve that problem by casting it as a linear program, which will be formally defined in the next section. Finally, in Problem 4, note that we treat the travel time of all users of the road network as a random variable L. For this random variable L, one assumes there exists probability density function fX , i.e., a non-negative function: Pr[a ≤ L ≤ b] = Z b a fL (x) dx. (1) On the input, we assume a histogram approximation thereof, with the bins of the histogram centered at multiples of the interval at which we sample the input data, i.e., t2 − t1 = tk+1 − tk . For example for the sampling at 1 minute, we would be given a list: Pr[0 ≤ L < 1.5], Pr[1.5 ≤ L < 2.5], . . . (2) We note that such distributional data are regularly obtained by most operators of public transport, either by surveys or by studying traces of individuals using personal seasonal tickets, although the alignment of the bins may not readily match the sampling interval of the presence data. 3 The Linear Program First, we show that Problem 1 can be solved by a linear program. 3.1 Notation and preliminaries Throughout, Z+ denotes the nonnegative integers, n and k are positive integers, 1n denotes an n-dimensional column vector of all 1s, and In is the n-dimensional identity matrix; the dimension n is omitted if it is clear from the context. We denote matrices by capital letters, and their elements by the corresponding lower-case letters, e.g. for a matrix M, mi j denotes the element in the ith row, jth column; M T denotes the transpose of M; tr(M) = ∑i mii denotes the trace of the square matrix M. Note that with this notation, we obtain the vector η of row-sums of M, i.e. η with ηi = ∑ j mi j , as η = M 1 and analogously we have γ T = 1T M for the vector of column-sums. To keep the notation compact, we consider user movement between two consecutive time intervals only; else, we’d have to include an index corresponding to the time stamp 4 in everything, which is unnecessary for the developments. In other words, we consider a data set of the form (Z1 ,t1 , E11 ), (Z2 ,t1 , E21 ), . . . , (3) (ZNz ,t1 , ENz 1 ), (Z1 ,t2 , E12 ), . . . , (ZNz ,t2 , ENz 2 ). Let E` denote the (column) vector of users present in all zones at time t` , i.e. E1 = T E11 E21 · · · ENZ 1 and so on. The number of users moving from Zi to Z j is denoted by xi j and the matrix of flows is X ∈ RNz ×Nz ; this is the quantity we are interested in finding. We denote the costs of moving from Zi to Z j by ci j ∈ R and collect them in the matrix C ∈ RNz ×Nz . 3.2 Cost functions The costs of moving from Zi to Z j is, in some sense, a parameter to the model. We have conducted our experiments with three natural choices of the cost matrix C ∈ RNz ×Nz , but we do not claim these are original. Let us have each zone Zi associated with a polygon, which is defined by a sequence of corner-points in the Euclidean plane. Let us denote ki corner-points associated with Zi x , zy ), (Px , Py ), . . . , (Px , Py ). In the adjacency metric, we consider: by (Pi,1 i,2 i,2 i,1 i,ki i,ki  0     0.1 Ca (Zi , Z j ) :=     1 ifi = j if ∃1 ≤ k1 ≤ ki , 1 ≤ k2 ≤ k j x , Py ) = (Px , Py ) s.t. ∃(Pi,k j,k2 j,k2 i,k1 1 otherwise (4) Considering the centroid of Zi : k centroid(Zi ) := k y i i Pi,x j ∑ j=1 Pi, j ∑ j=1 , ki ki ! (5) The centroidal distance Cc (Zi , Z j ) is the Euclidean distance between centroids centroid(Zi ) and centroid(Z j ) of the two zones. Finally, considering the closest corner-point associated with the other zone: NN(Zi , Z j ) := arg D(A, B) := q min min y y x ,P x (Pi,k i,k )1≤k1 ≤ki (Pj,k ,Pj,k )1≤k2 ≤ki 1 1 2 D(Pi,k1 , Pj,k2 ) 2 (Ax − Bx )2 + (Ay − By )2 . (6) Distance Cd (Zi , Z j ) is then the Euclidean distance between NN(Zi , Z j ) and NN(Z j , Zi ). One can clearly define many other cost matrices, e.g., considering free-flow travel times, and introducing randomisation. 5 3.3 Constraints assuming a constant number of users over time N N z z Ei1 = ∑i=1 Ei2 = N, i.e. that the number N of users in As a first step, assume that ∑i=1 the entire network is the same in both time intervals. In order for a flow X to explain the observations, it must “remove” Ei1 users from, and “place” E j2 users back in each zone Zi (note that the number of users staying in Zi is xii ). The number of users moving from Zi is then given by the row-sum ∑ j xi j , whereas the number of users moving to Zi is given by the column-sum ∑k xki . Of course, flows have to be nonnegative, too. Using the above notations, this translates to X 1 = E1 (7) T X 1 = E2 (8) X ≥ 0. (9) We note here that this set of constraints defines what is known as the (Nz , Nz )-transportation polytope with marginals E1 and E2 . A further constraint can be added: in reality, only integer numbers of users can move, hence we could also require X ∈ ZNz ×Nz . Since we are only approximating anyway, this might or might not be a good idea. Note also Lemma 5, which suggests that for integer marginals, the minimizer is also integral. N N z z The total cost associated with a flow X is ∑i=1 ci j xi j = tr(CT X), and we find the ∑ j=1 flow X minimizing the overall cost – and hence an estimate of the true movement of users between zones – as the solution to the following optimization problem: z = tr(CT X) minimize X ≥0 subject to 3.4 (LP-C) X 1 = E1 T X 1 = E2 . Constraints allowing for time-varying numbers of users We cannot assume that the number of users stays constant: Users are arriving in the covered area and leaving it, and of course devices are being turned off and on. To address that, we introduce a sink/source zone and index it by Nz + 1. For this zone we have ci,Nz +1 = cost of user disappearing from Zi cNz +1,i = cost of user spawning in Zi The number of users in the sink/source zone can only be computed after the data at t2 is available. Then, we let Nz Nz δ12 = ∑ Ei1 − ∑ Ei2 i=1 i=1 and ENz +1,1 = u, ENz +1,2 = u + δ12 , 6 where selecting a parameter u > 0 allows for consideration of users disappearing from Zi and spawning in Z j instead of travelling from Zi to Z j . One might also consider adding another sink/source for this specific purpose to gain more control over the cost: for instance, if ci,Nz +1 and cNz +1, j are too small compared to ci j , the optimization problem will always prefer disappearing/spawning over travel. 4 The Matrix Manipulations Second, one should like go beyond a snapshot of transitions within a single interval between two points in time, at which the presence data are available. We show that one can manipulate the solution of the linear program (LP) of the previous section to derive first the k-step transitions of Problem 2 and subsequently the solutions to Problems 3 and 4 by straightforward matrix manipulation. 4.1 Extrapolating the most recent one-step transitions One possible solution to Problem 2 involves considering the output of the LP as a matrix and raising it to the appropriate power. Let us return to our example, where we had (Z1 ,t1 , 3), (Z2 ,t1 , 1), (Z1 ,t2 , 2), (Z2 ,t2 , 2). We expect the flow of users to be X = 20 11 , in other words two users remain in Z1 , one goes from Z1 to Z2 , and the user already in Z2 remains there. Another interpretation is in terms of proportions: One third of the users in Z1 move to Z2 , whereas two thirds stay put, and so on. Mathematically, that corresponds to normalizing each of X to h 2column i 1 sum up to one (i.e. making X into a row-stochastic matrix), say Xe = 3 3 . 0 1 If this trend were to continue, then we’d have for the number of users in Z1 and Z2 at time t3 : E13 = xe11 E12 + xe21 E22 and E23 = xe12 E12 + xe22 E22 . This can be written compactly, and more generally, as the matrix-vector product E`+1 = XeT E` , and with E1 given, we have E2 = XeT E1 and E3 = XeT E2 = XeT XeT E1 and eventually we get T E`+1 = Xe` E1 . In the context of probability distributions (where the elements Ek` denote probability mass instead of users), this is a well-known and thoroughly researched Markov chain, and all results apply here. The most important one is that, should the trend continue forever, the system will (under mild conditions) settle into a steady state. More specifically, there is a distribution of users f such that XeT f = f and as ` → ∞, we have E` → f . 7 4.2 Extrapolating k recent one-step transitions If instead of just event counts at two time steps, we have event counts at several consecutive time steps, another solution to Problem 2 would be to multiply by the estimated one-step flow matrices in turn. To return to our example, let us extend it to: (Z1 ,t1 , 3), (Z2 ,t1 , 1), (Z1 ,t2 , 2), (Z2 ,t2 , 2), (Z1 ,t3 , 3), (Z2 ,t3 , 1). The flows between times t1 and t2 would be as before, and from t2 to t3 we’d have X2 = 21 01 , or after normalization: Xe1 = 2 3 1 3 0 1 and 1 e X2 = 1 2 0 1 2 . Now the estimate for E4 would be E4 = Xe1T E3 = Xe1T Xe2T E2 = Xe1T Xe2T Xe1T E1 and in general  `/2   XeT XeT E1 if ` even 1 2 E`+1 = (`−1)/2   XeT XeT Xe1T E1 if ` odd. 1 2 Because the product of row-stochastic matrices is again row-stochastic, this new Markov chain may have a stationary distribution appearing every k time steps. The apparent difficulty with a naive implementation of these approaches is runtime, in particular when k is large. As we will show in the next section, there are surprisingly efficient algorithms, though. 4.3 Approximating duration-L transitions In order to solve Problem 3, it suffices to consider a convex combination of 2 solutions to Problem 2 for suitable k. Consider the computation of duration-L transitions, where there exist presence data at times tk ,tk+1 with tk ≤ L ≤ tk+1 . Consider the k-step transition matrix Sk and (k + 1)-step transition matrix Sk+1 computed as a solution to Problem 2. Clearly, S= L − tk tk+1 − L Sk + Sk+1 tk+1 − tk tk+1 − tk (10) is the solution to Problem 3. 4.4 Approximating realistic transitions Finally, in order to solve Problem 4, one has to notice that the solution is a convex combination of H solutions to Problem 2 for H bins of the histogram approximation. 8 Method Run-time Ref. General-purpose interior-point method Hungarian method Augmenting-path algorithm O(n7 ln(1/ε)) O(n4 ) O(n3 ) [10] [17] [15] The proposed algorithm O(n) Theorem 6 Table 1: The complexity of solving the linear program (LP-C) in dimension n × n. We note that it makes it possible to obtain the objective function value at cost O(n), while retrieving the n × n matrix, at which this value is attained requires time O(n2 ). Consider the H bins of the histogram with values: h1 = Pr[0 ≤ L < 1.5] h2 = Pr[1.5 ≤ L < 2.5] ... hH = Pr[max supp L − 1 ≤ L < max supp L], where max supp L is the largest possible realisation of L. It is clear that hi , 1 ≤ i ≤ H can be seen as a probability mass function on a discretisation of L, with ∑H i=1 hi = 1. One can use hi as weights in a convex combination of the transition matrices, which is a sum. For the first bin, we consider a solution S1 of Problem 1 with weight h1 as the first summand. For the second bin, we consider a 2-step transition matrix S2 computed as a solution to Problem 2 with weight h2 as the second summand. Subsequently, we consider k-step transition matrix Sk , 2 < k ≤ H computed as a solution to Problem 2 with weight hk to obtain: H S = ∑ (hi Si ), (11) i=1 which is the solution to Problem 4. 5 Run-Time Analysis 5.1 Linear Programming Next, let us consider whether a faster algorithm for solving the class of linear programming problems (13) are possible. We make use of: Lemma 5 (E.g. [7, Lemma 2.2 and Corollary 2.11]). For the (p, q) transportation polytope defined by γ and η we have: 1. it is nonempty if and only if 1Tq η = 1Tp γ, in other words if the sums of the marginals are equal; 9 Method Run-time Ref. Standard dense matrix multiplication Present-best dense matrix multiplication Fourier-transform-based methods O(n3 ) Standard [18] [26] O(n2.3728639 ) Õ(k + nb) Table 2: The complexity of multiplying two sparse matrices, each in dimension n × n, each with k non-zero coefficients, where the product has b non-zero coefficients. 2. all its vertices are integral, if η ∈ Zq and γ ∈ Z p , i.e. if the marginals are integral. but we stress that the polytope has been studied very extensively over the past 70 years and refer to [29, 30, 33] as standard references. Now we state the main result: Theorem 6. Given η, γ ∈ Zn+ , with 1T η = 1T γ = k, consider z := max tr(X) s.t. X 1 = γ, X T 1 = η. X∈Zn×n + (IP2) Then, z = ∑ni=1 min ηi , γi . This is computable in time linear in n. For the proof, please see the Appendix. 5.2 Sparse Matrix Multiplication In addition to the complexity of the solving of the instance of the linear programming problem, we may need to solve a number of matrix-matrix multiplication problems, where the two matrices are sparse, each in dimension n × n, each with k non-zero coefficients, and where the product has b non-zero coefficients. Using traditional dense linear algebra, one can obtain the result in time O(n3 ), using more sophisticated methods for dense matrices, one can improve this to O(nω ), ω ≈ 2.37. Considering the sparsity of matrices A, B, however, Pagh [26] has shown methods that there are based on conversion to polynomial: n p(x) = n n ∑ ( ∑ Aik s1 (i)xh1 (i) )( ∑ Bk j s2 ( j)xh2 ( j) ) k=1 i=1 j=1 and utilisation of fast Fourier transform for polynomial multiplication, which can compute c0 , ..., cb−1 such that ∑i ci xi = (p(x) mod xb ) + (p(x) ÷ xb ) in time (n2 + nb), with (AB)i j = s1 (i)s2 ( j)c(h1 (i)+h2 ( j)) mod b. Eventually, Pagh [26] shows that the algorithm computes AB exactly in time (N + nb) with high probability. Although these algorithms may not be easy to implement, related algorithms with simpler hashing functions are readily implementable. 10 Table 3: Types of registered events: First two types are generated by a user (“active”), while the following eight are generated by the network (“passive”). I A cell identifier change reported in the access request procedure of any mobile originated (MO) or mobile terminating (MT) transaction I A cell identifier change reported when a cell identifier different from the one stored in the visitor location register (VLR) is used in response to any mobile terminating (MT) transaction I Location update performed by a mobile station I International mobile subscriber identity (IMSI) attached to a mobile station changes I A subscriber is deleted from the visitor location register I The subscriber switches off the mobile station I Mobile station is considered as detached by the network after a long period of inactivity I Cell identifier change is detected during GPRS activity I Cell identifier change is detected during the provision of subscriber information I During long calls, the visitor location register (VLR) may receive request messages to refresh subscriber data. In this way, the VLR notices that the cell identifier arriving in the message differs from the one stored in a database. 6 Computational Experiments For our preliminary experiments, we have used data collected from the Public Land Mobile Network (PLMN) of Orange Polska within the municipal area of Warsaw. For each base transceiver station (BTS) of each separate cellular system (2G/3G), we have received the total numbers of connections from unique terminals per a unit of time. No data other than the aggregate statistics were collected. In particular, the data were collected by an internal transmission system, in which some types of network events are associated with a location. Such events can be subdivided into events generated by the user (“active”) or generated by the network (“passive”). In Table 3, the first two bullet points capture events generated by the user. These events relate to mobile terminating (MT) transactions, i.e., events registered during an active use of a mobile terminal by the user. The following bullet points list event types generated by the network. These events are generated without an active participation of the user and include operations of the so-called visitor location register (VLR), which tracks subscriber roaming within a mobile switching centre (MSC) location area, cell identifier changes, International Mobile Subscriber Identity (IMSI) changes, location updates performed by a mobile station, and the switching of mobile stations. Within the one week of data collected, the most frequent events were location updates, while the cell identifier change was very rare. Figure 1 presents the evolution of the total number of events during the period. The volumes of both active and passive events are correlated with the time of day; the cyclical nature of the evolution of the volumes of events over time is clear. The evolution of the numbers of passive events is more uneven 11 12 104 11 active passive 104 10 10 Number of events 9 8 8 6 7 6 4 5 2 4 0 Jan 09, 08:00 Jan 10, 08:00 Jan 11, 08:00 Jan 12, 08:00 3 08:00:00 09:00:00 10:00:00 Jan 13, 08:00 2017 Figure 1: Total number of events vs time stamps, split by passive (magenta) and active (black) events. The spike in passive events at 3:15am is an artefact of the data acquisition system’s operation. than that of the active events. The local maxima of the numbers of passive events at 3:15 a. m. are related with the operations of the data acquisition and processing system. Further, note that the observed events are not equally distributed in the city. Figure 2 presents statistics of the number of events observed across all BTSs at a given time. The large difference between mean and median, the wide interquartile range, and the very large difference between maximum and mean number of events at all times suggest a large variability in the distribution of events over the BTSs. This disproportion is the result of differing population density, differing ranges of base transceiver stations, and differing telecommunication technologies. For comparison, we have used the estimates presently employed by the public transport operator in Warsaw (ZTM). One should like to point out that the estimates of ZTM imply a rather different distribution of the population across the zones than the density of events within the Orange network. See Figure 3 for the density of events within the Orange network for three representative time periods. In Figure 4, we compare it to the implied presence data within the ZTM estimate: with each data point represents one zone and the two axes correspond to the presence implied by ZTM for the morning peak and the numbers of events within Orange network during the same time period, averaged over the 5 work days. As can be seen, our model starts with rather different data, and any direct comparison of our estimates of trip distributions with the estimates of the public transport operator (ZTM) will hence be imperfect. For further comparison, we have implemented the doubly-constrained gravity model [9]. There, the presence vectors E` across two consecutive time steps are seen as production and attraction vectors within the framework of a travel-demand model, which aims to generate trip distribution rates. An appropriate normalization can give yield matrix X similar: Ai B j Ei1 E j2 xi j = , (12) cαi j 12 Figure 2: The evolution of some statistics of the active (left) and passive (right) events observed. The statistics are computed over all BTSs and all 5 available days (Monday – Friday), e.g. the mean at 9:00 is the mean of the number of events at all BTSs on all 5 days at 9am. The inner (dark blue) band is the interquartile range, whereas the outer (light blue) band extends between the 5th percentile and the maximum event count at that time. (a) 6 a. m. (b) 9:30 a. m. (c) 4:30 p. m. Figure 3: Density of events across zones for different time periods. Color scale uses a power law normalization. where Ai and B j are zone-specific parameters for the origin and destination, respectively, which are learned from the data, and α is a parameter, which we assume to be α = 1.0. An iterative algorithm [9] can then be used to determine the flow, subject to flow conservation constraints. To test the approaches, data were aggregated spatially to 820 zones defined by the public 13 Figure 4: Presence by zone implied by the estimates of ZTM and the density of events in the Orange network. transport operator (ZTM), and temporally to 15-minute units, prior to the application of any methods described in the present paper. Independently, for each cost function, an 820 × 820 matrix has been calculated, based on a description of each zone as a polygon with points defined by their latitude and longitude. Additionally, we have obtained randomised variants of the objective function by adding uniformly-distributed noise U[0, 0.0001) to each element, e.g., Ca (Zi , Z j ) for all i, j. (Note that the introduction of a small amount of noise makes it possible to obtain multiple optimizers with very similar objective function value.) The data have been normalised to the number N = 1, 000, 000 of users, i.e., such that the marginals, and hence the 672, 400 scalar variables sum up to 1, 000, 000. Subsequently, the linear program (LP-C) of Section 3.3 with 672, 400 scalar variables and 1, 640 dense constraints has been formulated and solved for each pair of consecutive time steps between 8 a. m. and 9.30 a. m. on each of the five work days and each of 4 randomisation of the objective function. The sparse, integral solutions of all linear programs have been averaged to reduce the sparsity in our estimate of the 1-step transition matrix S1 . Subsequently, we have obtained i-step transition matrices for i = 3, 4, . . . 7, i.e., corresponding to trips of 30, 45, 60, 75, and 90 minutes of duration, by the matrix-multiplication procedure of Section 4.1. Finally, we have computed a convex combination of the i-step transition matrices as suggested in Section 4.4, using weights detailed in the caption of Figure 5. Notice that we did not have an accurate estimate of the trip-duration distribution, which Figure 5 suggests has a considerable impact. Still, the match seems encouraging, considering the simplicity of the methods applied: we have considered the simplest cost function Ca , we have assumed the constant number N of users throughout in (LP-C), and the crudest method of arriving at the multi-step transition matrices. 14 (a) 1/6, 1/6, 1/6, 1/6, 1/6, 1/6 (b) 0.5, 0.1, 0.1, 0.1, 0.1, 0.1 (c) 1 − 5ε, ε, ε, ε, ε, ε (d) Gravity model Figure 5: Solutions to Problem 4 obtained using (LP-C) and three variants of the vector h1 , h2 , h3 , h4 , h5 , h6 , compared against the estimates of the public transport operator (ZTM). 7 7.1 Related Work The Demand Modelling There is a long history of the use of individual mobile phone subscriber data in transportation modelling. Outside of Europe, the most common study is based on CDR (Call Data Record). For example in [4], the CDR are used to analyse flows between the city and the suburban area of New York. In [19], researchers analyse the CDR from the main mobile operator in Mexico, and one of the papers focuses on users movements analysis. In [37], CDR data including phone calls, SMS messages, and web browsing 15 of customers of one US mobile operator in the San Francisco Bay area are used to determine patterns of urban road use. The Orange company organized two editions of the D4D Challenge (http://www.d4d.orange.com/en/Accueil), during which the anonymous CDR of Abidjan in Cote d’Ivoire data was made available to developers and researchers. Some of the present authors [23, 28] used the data set to optimize public transport. In a related paper [11], the data from the competition is used to estimate the travel demand and network allocation. The access to the CDR data is particularly limited in Europe, though, where privacy-protection regulation make access to the CDR data impossible without their anonymization and user’s consent, in most cases. Within Europe, [2] discusses the use of GPS data, CDR, and cellular data (details of base stations) to determine the trajectory of subscriber mobility. The data for this study was collected by a group of volunteer users in cooperation with one of the French operators, and the study covers the metropolitan area of Paris (Ile-de-France). The use of anonymous mobile data and their use for locating subscribers anonymously is described in [13]. [35] discusses the use of data from Italy’s Vodafone users to compute mobility patterns in Italian cities. In this case, the real-time events (due to voice, data, and SMS communication) observed between A Interface and IU-CS interface in 2G and 3G mobile networks have been used. In [31], the authors describe the usage of aggregated mobile phone data from Rome, including the information about the base station telephone traffic, expressed in Erlang units. These records are combined with the location and trajectory data of callers and data from the two city transportation companies (taxi and public bus company). [24] describe the aggregation procedure used to derive the dataset we use. 7.2 The Algorithms Notice that the linear program (LP-C) has a well-known structure, in that the feasible set is the transportation polytope: Definition 7 (Transportation Polytope). Given γ ∈ R p , η ∈ Rq , the set of matrices X ∈ R p×q , Xi j ≥ 0 satisfying p ∑i=1 Xi j = η j q ∑ j=1 Xi j = γi ∀ j = 1, . . . , q ∀i = 1, . . . , p (13) is called the (p, q)-transportation polytope defined by the marginals γ and η. Related algorithms for solving special-cases of linear programming are employed throughout pattern recognition and content-based content retrieval. The commonly used statistical distances include the Earth mover’s distance (EMD) [32], also known as the transportation cost metric [16], and Mallows distance [25], which are equivalent [20], in the sense of being dual of each other. These are discretisations of the metrics of Wasserstein and Monge-Kantorovich [8], which are also dual of each other. Computing the distances amounts to solving a certain problem over the so-called transportation polytope. Perhaps the best overview is provided by the two-volume work of Rachev and Rüschendorf [29, 30]. 16 In Computer Vision [27, 34, 12], a variety of exact and approximate methods have been considered. Perhaps the best known exact method [27] uses the min-cost flow algorithm, yielding run-time of O(n2 log n) for n bins. Approximations based on gradient flows of certain partial differential equations [12] can be computed faster than earth mover’s distance, empirically, but do not come with any guarantees, i.e., the approximation ratio. Approximations based on the sum of absolute values of weighted wavelet coefficients of the difference of the histograms [34] have been proposed. Such an approximation can be computed in time linear in the number of bins, but does not come with a bound on its quality, i.e., the approximation ratio. Papers on approximations based on space-filling curves [14] have been withdrawn, recently. In Machine Learning [6], Sinkhorn distances, which are a regularised variant of the Earth mover’s distance, have been explored. Although the algorithms based on the Sinkhorn distance are substantially faster than the network simplex for Earth mover’s distance, It has also been realised [22] that one can employ the following representation with O(n) variables for n bins: leading to run-time cubic in the number of bins. In Theoretical Computer Science, [16] have studied the limits of approximability of earth mover’s distance, [1] have proposed logarithmic approximation in sub-linear time, while [3, 5] proposed algorithms approximation algorithms, whose run-time depends on the dimension of the domain space and the quality of the result required, possibly sub-linear in the number of bins. A similar in spirit is a recent method [21], which considers dithering of the domain space, followed by max-flow computations [27]. We stress that our algorithms are much easier to implement and provide the exact solution. Overall, the present-best algorithms [27] had run-time of O(n2 log n) for n bins, compared to O(n) we present in Theorem 6. Across all the references listed, only the value of the objective function, rather than the value of the argument at which it is achieved, is sought. 8 Conclusions Across many transport-engineering applications, from planning of public transport to balancing of a fleet of shared vehicles, one needs an up-to-date model of demand for transportation. Further, one may be interested in higher moments of the demand, rather than just the commonly-used expectation. The use of mobile phone data makes it possible to address both issues. For the first time, we have formalised the problem of extraction of a model of demand for transport from the aggregate data on the use of the base stations. We have shown that the associated optimisation problems have the form of a trace maximisation over the transportation polytope, which allows for much faster algorithms than the usual simplex or interior point methods. 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Step 1: Since Xi j ≥ 0, we certainly have Xi j ≤ ∑n`=1 X` j = η j and Xi j ≤ ∑n`=1 Xi` = γi . Hence in particular Xii ≤ ηi and Xii ≤ γi , and so by choosing the tightest bound at each i, y = ∑ni=1 Xii ≤ ∑ni=1 min ηi , γi follows. Step 2: Before we proceed with Step 2 proper, we make two assumptions and explain that these are without any loss of generality. Assumption 1: η 6= γ. Notice that if η = γ, one lets Xii = ηi = γi for i = 1, . . . , n and Xi j = 0 for all off-diagonalelements Such an X trivially satisfies all constraints and tr(X) = ∑ni=1 ηi = ∑ni=1 min ηi , γi = k, and hence the result holds for this case. We can hence assume η 6= γ without the loss of generality. Assumption 2:The marginals are ordered such that: ηi ≤ γi for i = 1, . . . , m ηi > γi for i = m + 1, . . . , n. If that is not the case, then let P denote the permutation matrix so that Pη and Pγ are ordered.1 Then, if X̄ is feasible for (IP2) with the modified marginals Pη and Pγ, then X = PX̄P is feasible for (IP2) and achieves the identical objective, since tr(PX̄P) = tr(PPX̄) = tr(X̄). Note that we certainly have 1 ≤ m ≤ n − 1, because m = 0 implies ηi > γi for all i, which in turn implies ∑ni=1 ηi > ∑ni=1 γi , which contradicts ∑ni=1 ηi = k = ∑ni=1 γi , hence m ≥ 1; m = n together with Assumption 1 (η 6= γ) leads to an analogous contradiction. The construction of Step 2 fixes all values on the diagonal of X, the first m columns and the last n − m rows: Xii = ηi , Xi j = 0 for j = 1, . . . , m, Xii = γi , Xi j = 0 for i = m + 1, . . . , n, i 6= j j 6= i. e := η − min{η, γ}, γe := γ − min{η, γ}, Denote the remainders of the marginals by η where the min{·} is taken element-wise. 1 P can be constructed easily: let C := {i \| η ≤ γ }. Then m = card(C ) and we obtain P by rearranging i i the identity matrix In so that the columns with indices in C are the first m columns, in any order. 22 γf = η1 γ1 − η1 η2 γ2 − η2 Y =: γe 0 ηm γm − ηm γm+1 0 γn ηfT = 0 0 0 ηm+1 −γm+1 0 ηn − γn =: ηe 0 Figure 6: Illustration of the proof of Theorem 6: After possibly reordering the marginals, the diagonals are filled in, and the hatched regions are filled with zeros. The elements in the rectangular block Y are then used to satisfy the remaining constraints, which are e 0. collected in γe0 and η Let us now consider several properties of this construction; see also Figure 6 for an illustration: P1 tr X = ∑ni=1 min ηi , γi , which matches the bound of Step 1. P2 ∑ni=1 Xi j = η j for j = 1, . . . , m and ∑nj=1 Xi j = γi for i = m + 1, . . . , n, i.e., all constraints but the m first row sums and last n − m column sums are already satisfied. P3 Only Xi j for i = 1, . . . , m and j = m + 1, . . . , n are unassigned. Denote this m × (n − m) matrix by Y . e = 1T γe =: e e , γe ≥ 0, and η ei = γej = 0 for i = 1, . . . , m and j = m + 1, . . . , n. P4 1T η k, η e and γe by η e 0 and γe0 respectively, i.e. η e 0 ∈ Zn−m Denote the remaining elements of η + 23 and γe0 ∈ Zm + and 0 e= 0 , η e η 0 γe γe = . 0 e 0 = 1Tm γe0 = e k, since we only removed zero elements. We still have 1Tn−m η m×(n−m) By property P1 and P3, if we are able to find a matrix Y ∈ Z+ satisfying Y 1n−m = e 0 , we can recover a feasible X that achieves the bound of Step 1. By γe0 , and Y T 1m = η P3 and P4, this amounts to finding a point in the (m, n − m) transportation polytope e 0 . Since by P4 we have that 1Tn−m η e 0 = 1Tm γe0 , it follows defined by the marginals γe0 and η from Lemma 5 that such a Y does exist, which concludes Step 2. 24	8
Interface Reconciliation in Kahn Process Networks using CSP and SAT arXiv:1503.00622v4 [cs.PL] 12 Jul 2015 Pavel Zaichenkov, Olga Tveretina, Alex Shafarenko Compiler Technology and Computer Architecture Group, University of Hertfordshire, United Kingdom {p.zaichenkov,o.tveretina,a.shafarenko}@ctca.eu Abstract. We present a new CSP- and SAT-based approach for coordinating interfaces of distributed stream-connected components provided as closed-source services. The Kahn Process Network (KPN) is taken as a formal model of computation and a Message Definition Language (MDL) is introduced to describe the format of messages communicated between the processes. MDL links input and output interfaces of a node to support flow inheritance and contextualisation. Since interfaces can also be linked by the existence of a data channel between them, the match is generally not only partial but also substantially nonlocal. The KPN communication graph thus becomes a graph of interlocked constraints to be satisfied by specific instances of the variables. We present an algorithm that solves the CSP by iterative approximation while generating an adjunct Boolean SAT problem on the way. We developed a solver in OCaml as well as tools that analyse the source code of KPN vertices to derive MDL terms and automatically modify the code by propagating type definitions back to the vertices after the CSP has been solved. Techniques and approaches are illustrated on a KPN implementing an image processing algorithm as a running example. Keywords: coordination programming, component programming, Kahn Process Networks, interface coordination, constraint satisfaction, satisfiability 1 Introduction The software intensive systems have reached unprecedented scale by every measure: number of lines of code; number of people involved in the development; number of dependencies between software components, and amount of data stored and manipulated [1]. Many of them include heterogeneous elements, which come from variety of different sources: parts of them are written in different languages and tuned for different hardware/software platforms. Furthermore, when the software is developed and modified by dispersed teams, inconsistencies in the design, implementation and usage are unavoidable. This leads to clashes of assumptions about operation cost, resource availability and algorithm processing rate. Last but not least, parts of the system are constantly changing. Many elements need to be replaced without negative effects on performance or behaviour of the rest of the system. One way to attack the software challenge is to suggest a component-based design: a program is designed as a set of components, each represented by an interface that specifies how they can be used in an application, and one or more implementations which define their actual behaviour. When a designer of the application uses a component, they agree to rely only on the interface specification. Similarly, a developer who creates an implementation for a component is unaware of the context where the component will be used. An algorithm that specifies the behaviour depends solely on self-contained input and its result is produced in the form of a message without a specific destination address. The process network, introduced by G. Kahn (KPNs) [2], is a collection of stream-connected algorithmic building blocks, which are fully independent single-threaded processes that lack a global state. The execution of the network generally requires a supervisory coordination program that manages the progress of the blocks and which provides a message-communication infrastructure for the streams. Since all domain-specific computations are performed by the sequential processes inside the blocks, programming is naturally separated into algorithm and concurrency engineering [3]. The coordination language is responsible for component orchestration, namely 1) dynamic load control and adaptivity for a changing environment; 2) access control to shared resources; and 3) communication safety between components. This paper focuses on the last aspect. Component-based design requires an implementation of a single component to be independent from the rest of the network. It raises a number of software engineering issues: components’ interfaces are required to be specific enough so that components are aware of data structures communicated between them and, at the same time, generic enough to facilitate decontextualisation and software reuse. In this paper we present a solution to the interface reconciliation problem for an interface definition language specifically designed for KPNs. We demonstrate a static mechanism (based on solving Constraint Satisfaction Problem (CSP) and SAT) that checks compatibility of component interfaces connected in a network with support of overloading and structural subtyping. This allows one to design completely decontextualised components, so that they may be reused in different contexts without changing the code. This is especially important when the components are provided as a compiled library and its source is either private or unavailable. The components are compatible with a potentially unlimited number of input/output data formats coming from the environment. We also introduce a flow inheritance [4] mechanism: put simply, a message sent from one component to another may also be required to contain additional data which, although not needed by the recipient itself, is nevertheless required by a component that the recipient sends its own messages to (Fig. 1). We propose a Message Definition Language (MDL) that enables components’ generic interfaces as well as subtyping and flow inheritance; we then recast the interface reconciliation problem as a CSP for the interface variables and propose an original solution algorithm that solves it by iterative approximation while generating an adjunct Boolean SAT problem on the way. We designed and implemented a communication protocol1 for components coded in C++ to demonstrate the capabilities of MDL. We developed tools that 1) automatically derive MDL interfaces from the source code; 2) generate a set of constraints given a netlist2 that describes the topology of the network; 3) solve the CSP; and 4) based on the solution of the CSP generate compilable code for every component with some API provided for run-time support. The process is similar to template specialisation in C++, however, in our case, constraints that are produced by a pair of vertices may affect the whole network, and, consequently, a global constraint satisfaction procedure is required. In this paper we provide a formal description of MDL, the CSP definition and the algorithm designed to solve the CSP. Throughout the paper we demonstrate the utility of the proposed approach on a practical example: an image segmentation algorithm based on k-means clustering (Fig. 2). Related work. Linda [5] is the first language designed to separate computation and coordination models. It is based on a simple tuplespace model. One of the disadvantages of the model is that the knowledge about the communication protocol is required while implementing the processes. The problem of separation of concerns has not been solved in Concurrent Collections from Intel [6] (Linda’s successor). Therefore, generic components, which may be reused in multiple contexts without being modified, are not supported in the language. In the programming language Reo [7] components are communicating through hierarchical connectors that coordinate their activities and manipulate message dataflow. Similar to our approach, a constraint satisfaction engine, which finds a solution that specifies a valid interaction between components, is implemented. S. Kemper describes a SAT-based verification of Timed Constraint Automata that is used for coordination of communicating components [8] as well as in Reo. However, the research mostly focuses on the design of reusable interaction protocols and lacks the description of reusable component interfaces. In previous years there were attempts to design efficient programming languages and run-time systems for parallel programming based on KPN [4,9], however, the interface reconciliation problem stemming from nonlocal inheritance in KPNs has not been given enough attention. 2 Motivation Kahn Process Networks is a concurrency model that introduces data streams in the form of sequential channels that connect independent processes into a network. Decontextualisation of processes is an advantage of the model. Since processes do not share any data, a process’s conformity with the context is defined by its interface, which describes the kinds of message that the process 1 2 for the avoidance of doubt we state that the term “protocol” is used here in the sense of ‘convention governing the structure and interpretation of messages’ and not in any state-transition sense a textual representation of a graph A B Fig. 1: Illustration of flow inheritance. A component A can process a value of type X and return a value of type Y as a result. However, an input message contains not only an element of type X, but also an element of type Z. The latter can be processed by a component B. Flow inheritance provides a mechanism for partial message processing in a pipelined fashion. can send and receive. Our goal is to provide a means of interface coordination that supports genericity, i.e. the ability of an interface to function correctly in a variety of contexts. The distributed components are commonly provided as closed-source services. Each service contains multiple processing functions compatible with a variety of contexts. The input interface of a component is specialised based on the message format that the message producer is capable to produce, and, symmetrically, the output interface is specialised based on the consumer’s requirements. For example, one can define a component that contains two functions with type signatures Int -> Int and Int -> String. The functions implement algorithms that compute different values given the same input. The task is to statically choose the algorithm based the consumer’s requirements. This also demonstrates that input and output types in the interfaces of the KPN components are treated in the same manner. The latter makes services fundamentally different from functions. In functional languages, a type signature that corresponds to the interface of the component in the example can be defined by the intersection type Int -> Int ∧ Int -> String, which is unsound due to its ambiguity. In functional languages, the return type of a function depends solely on input argument types. In contrast, the interfaces of the KPN components form context-dependent relations that offer a selection of output types to a consumer. This makes the typing decisions essentially nonlocal and genuinely multiple. The problem being solved can be seen as a type inference problem; however, it cannot be solved using conventional type inference mechanisms based on firstorder unification due to the presence of polymorphic output types and potential cyclic dependencies in the network (the example in Fig. 2 contains a back edge). A common communication pattern in streaming networks is a pipeline, where a message travels along a chain of components that work on its content. The component can accept a subtype of the input type, but part of the message may be bypassed to another component down the pipeline if the message contains the data the further component will need to use (Fig. 1). Two modes of flow inheritance are envisaged, considered next. Flow inheritance for records. The fundamental type of a message in a variety of systems is record, which is a collection of label-value pairs. Each component processes only a specific set of pairs, however the pairs that the component does not require may be bypassed to the output, so they can be processed in the read 1 init 1 2 1 kMeans 1 2 1 1 2 3 segmented image logs/errors Fig. 2: An image segmentation algorithm based on k-means clustering that is implemented as a Kahn Process Network next stages of the pipeline if they are required. For example, a message that represents a geometric shape and has a type {x:float, y:float, radius:float} may be processed in two steps: the first component processes the position of the shape as defined by the pairs x and y, and the second one needs the pair labelled radius. Flow inheritance for variants. OOP extensively uses overloading [10] to improve modularity and reusability of code. Similarly, support of polymorphic components in KPNs facilitates their reuse in different contexts. At the top level we see a component’s interface as a collection of alternative label- record pairs, called variants, where the label corresponds to the particular implementation that can process the message defined by the given record, e.g. (: cart: {x:float, y:float}, polar: {r:float, phi:float} :). Here the colonised parentheses delimit the collection of variants and each variant is associated with a record written as a set of label-value pairs. Any message that does not belong to one of the accepted variants must cause an error unless there exists another component further down in the pipeline that can process it. In this case, the message should be bypassed to the recipient. In streaming networks flow inheritance alleviates the problem and makes configuration of individual interfaces independent from each other. In our work we developed a solution for the interface reconciliation based on the CSP with support of flow inheritance for records and variants. Our mechanism statically detects implementation variants in components that are not required in the context, which is important for applications running in the Cloud where a user is charged proportionally to the amount of resources their application uses. Example. As a running example we use our implementation [11] of an image segmentation algorithm based on k-means clustering [12]. The applications’s KPN graph is shown in Fig. 2. The network represents a pipeline composed of three components: – The component read opens an image file using an input message Mr1 with the file name, and sends it to the first output channel. The component contains 3 functions that overload component’s behaviour (i.e. the input interface of the component is defined by 3 variants): 1) Vr1 loads the colour image in RGB format; 2) Vr2 loads the greyscale image as an intensity one; and 3) Vr3 loads the image as it is stored in the file. – The component init sets initial parameters for the k-means algorithm. The component contains one processing function Vi1 . The input message can either come from the component read or from the environment with an input message Mi1 if it has been opened and preprocessed before. The input message must contain the number of clusters K and the image itself. – The kMeans component represents an iterative implementation (defined as a function Vk1 ) of the k-means algorithm. The result of each iteration is sent to the first output channel, which is circuited back to the input channel of the component itself. This kind of design gives an opportunity to manage system load in the run-time and execute the next iteration only when sufficient resources are available. Once the cluster centres have converged, the algorithm yields the result to the second output channel. Using flow inheritance for variants Mi1 is routed directly to the init component bypassing a component read. Using flow inheritance for records a parameter K that is contained in Mr1 is implicitly bypassed through read to init. The interface reconciliation algorithm is capable of finding out that Vr2 and Vr3 are not used with the provided input, and functions containing the implementations will be excluded from the generated code. 3 Message Definition Language Now we define the Message Definition Language (MDL) that describes component interfaces. Each component has its associated input and output interface terms. A message is a collection of data entities, defined by a corresponding collection of terms that can contain term variables, Boolean variables and Boolean expressions. Each term is either atomic or a collection in its own right. Atomic terms are symbols, which are identifiers used to represent standard C++ types, such as int or string. To account for subtyping (including the kinds that are not present in C++) we include three categories of collections (see Fig. 3): tuples that demand exact match and thus admit no structural subtyping, records that are subtyped covariantly (a larger record is a subtype) and choices that are contravariantly subtyped using set inclusion (a smaller choice is a subtype). The intention of these terms is to represent 1. extensible data records [13,14], where additional named fields can be introduced without breaking the match between the producer and the consumer and where fields can also be inherited from input to output records by lowering the output type, which is always safe; 2. data-record variants, where generally more variants can be accepted by the consumer than the producer is aware of, and where such additional variants can be inherited from the output back to the input of the producer — hence contravariance — again by raising the input type, which is always safe also. Term variables correspond to four categories of terms. However, for the correctness of the algorithm it is important to distinguish variables that represent choices from variables that represent other term categories (due to two kinds of subtyping defined by the seniority relation in Definition 1). We use an up-coerced term variable, e.g. ↑a, to represent a choice term and a down-coerced term variable, e.g. ↓a, to represent any other term, i.e. a symbol, a tuple or a record. Formally, hterm variablei ::= ↑identifier \| ↓identifier We use symbol 8 instead of ↑ or ↓ symbols in the context where the sort is unimportant, e.g. 8a is a term variable that can be either up-coerced or downcoerced. For brevity, term variables are called variables, Boolean variables are called flags and Boolean expressions are called guards. The following grammar specifies the guards: hbool i ::= (hbool i ∧ hbool i) \| (hbool i ∨ hbool i) \| ¬hbool i \| true \| false \| flag MDL terms are built recursively using the constructors: tuple, record, choice and switch, according to the following grammar: htermi ::= hsymbol i \| hterm variablei \| htuplei \| hrecord i \| hchoicei \| hswitchi htuplei ::= (htermi [htermi]∗ ) hrecord i ::= {[hlabel i(hbool i):htermi[,hlabel i(hbool i):htermi]∗ [\|↓identifier ]]} hchoicei ::= (:[hlabel i(hbool i):htermi[,hlabel i(hbool i):htermi]∗ [\|↑identifier ]]:) hlabel i ::= hsymbol i Informally, a tuple is an ordered collection of terms and a record is an extensible, unordered collection of guarded labeled terms, where labels are arbitrary symbols, which are unique within a single record. A choice is a collection of alternative terms. The syntax of choice is the same as that of record except for the delimiters. The difference between records and choices is in subtyping and will become clear below when we define seniority on terms. We use choices to represent polymorphic messages and component interfaces. Records and choices are defined in tail form. The tail is denoted by a variable that represents a term of the same kind as the construct in which it occurs. For example, in the term {l1 (true): t1 , . . . , ln (true): tn \|↓v} the variable ↓v represents the tail of the record, i.e. its members with labels li : li 6= l1 , . . . li 6= ln . A switch is a set of unlabeled (by contrast to a choice) guarded alternatives. hswitchi ::= <hbool i:htermi[, hbool i:htermi]∗ > Exactly one guard must be true for any valid switch. The switch is substitutionally equivalent to the term marked by the true guard: hfalse: t1 , . . . , true: ti , . . . , false: tn i = htrue: ti i = ti . The switch is an auxiliary construct intended for building conditional terms. For example, ha: int, ¬a: stringi represents the symbol int if a = true, and the symbol string otherwise. For each term t, we use V ↑ (t) to denote the set of up-coerced term variables that occur in t, V ↓ (t) to denote the set of the down-coerced ones, and F (t) to denote the set of flags. A term t is called semi-ground if it does not contain variables, i.e. V ↑ (t) ∪ ↓ V (t) = ∅. A term t is called ground if it is semi-ground and does not contain flags, i.e. V ↑ (t) ∪ V ↓ (t) ∪ F (t) = ∅. A term t is well-formed if it is ground and one of the following holds: 1. t is a symbol. 2. n > 0 and t is a tuple (t1 . . . tn ) where all ti , 0 < i ≤ n, are well-formed. 3. n ≥ 0 and t is a record {l1 (b1 ): t1 , . . . , ln (bn ): tn } where for all 0 ≤ i 6= j ≤ n, bi ∧ bj → li 6= lj and all ti for which bi are true are well-formed. 4. n ≥ 0 and t is a choice (:l1 (b1 ): t1 , . . . , ln (bn ): tn :) where for all 0 ≤ i 6= j ≤ n, bi ∧ bj → li 6= lj and all ti for which bi are true are well-formed. 5. n > 0 and t is a switch hb1 : t1 , . . . , bn : tn i where for some 1 ≤ i ≤ n, bi = true and ti is well-formed and where bj = false for all j 6= i. If an element of a record, choice or switch has a guard that is equal to false, then the element can be omitted, e.g. {l1 (b1 ): t1 , l2 (false): t2 , l3 (bn ): t3 } = {l1 (b1 ): t1 , l3 (b3 ): t3 } . If an element of a record or a choice has a guard that is true, the guard can be syntactically omitted, e.g. {l1 (b1 ): t1 , l2 (true): t2 , l3 (bn ): t3 } = {l1 (b1 ): t1 , l2 : t2 , l3 (bn ): t3 } . We define the canonical form of a well-formed collection as a representation that does not include false guards, and we omit true guards anyway. The canonical form of a switch is its (only) term with a true guard, hence any term in canonical form is switch-free. Next we introduce a seniority relation on terms for the purpose of structural subtyping. In the sequel we use nil to denote the empty record { }, which has the meaning of void type in C++ and represents a message without any data. Similarly, we use none to denote the empty choice (: :). Definition 1 (Seniority relation). The seniority relation ⊑ on well-formed terms is defined in canonical form as follows: 1. 2. 3. 4. none ⊑ t if t is a choice. t ⊑ nil if t is any term but a choice. t ⊑ t. t1 ⊑ t2 , if for some k, m > 0 one of the following holds: i i each 1 ≤ i ≤ k; (a) t1 = t11 . . . tk1 , t2 = t12 . . . tk2 and 1 t11⊑ t2 for k 1 1 k (b) t1 = l1 : t1 , . . . , l1 : t1 and t2 = l2 : t2 , . . . , l2m : tm 2 , where k ≥ m and j i for each j ≤ m there is i ≤ k such that l1 = l2 and ti1 ⊑ tj2 ; (c) t1 = (:l11 : t11 , . . . , l1k : tk1 :) and t2 = (:l21 : t12 , . . . , l2m : tm 2 :), where k ≤ m and for each i ≤ k there is j ≤ m such that l1i = l2j and ti1 ⊑ tj2 ; ... nil symbol tuple subtype record choice ... none Fig. 3: Two semilattices representing the seniority relation for terms of different categories. The lower terms are the subtypes of the upper ones. Given the relation t ⊑ t′ , we say that t′ is senior to t and t is junior to t′ . Proposition 1. The seniority relation ⊑ is trivially a partial order, and (T, ⊑) is a pair of upper and lower semilattices (Fig. 3). The seniority relation represents the subtyping relation on terms. If a term t′ describes the input interface of a component, then the component can process any message described by a term t, such that t ⊑ t′ . Although the seniority relation is straightforwardly defined for ground terms, terms that are present in the interfaces of components can contain variables and flags. Finding such ground term values for the variables and such Boolean values for the flags that the seniority relation holds represents a CSP problem, which is formally introduced next. 4 Constraint Satisfaction Problem for KPN In this section we define a Constraint Satisfaction Problem for Kahn Process Networks (CSP-KPN). We regard a KPN network as a directed weakly connected labeled graph G = (V, E, L), where 1. V is a set of vertices. The vertices correspond to individual Kahn processes. 2. E is a set of edges, where each edge e ∈ E is an ordered pair of vertices (v, v ′ ), v, v ′ ∈ V. The edges correspond to channels connecting Kahn processes. 3. A function L : E → Term × Term assigns a label3 to each edge e ∈ E which represents a pair of MDL terms L(e) = t ⊑ t′ called a constraint 4 . It defines the input requirements and the output properties associated with the channel. Given a graph G = (V, E, L) we define the set of constraints as [ C(G) = L(e), e∈E 3 4 we use the same word “label” to refer to the mark on a graph edge and the symbol that labels a term in a record or a choice; however our intention is always clear from the context. in the rest of the paper symbol ⊑ denotes the seniority relation for a pair ground terms; alternatively, if the terms are not ground, ⊑ specifies a constraint. the sets of up-coerced term variables V ↑ (G) and down-coerced term variables as V ↓ (G) [ [ V ↓ (t) ∪ V ↓ (t′ ), V ↑ (t) ∪ V ↑ (t′ ) and V ↓ (G) = V ↑ (G) = t⊑t′ ∈C(G) t⊑t′ ∈C(G) and the set of flags F (G) as F (G) = [ F (t) ∪ F (t′ ). t⊑t′ ∈C(G) Assume a vector of flags f~ = (f1 , . . . , fl ), a vector of term variables 8~v = (8v1 , . . . , 8vm ), a vector of Boolean values ~b = (b1 , . . . , bl ) and a vector of terms ~s = (s1 , . . . , sm ). Then for each term t 1. t[f~/~b] denotes the vector obtained as a result of the simultaneous replacement of fi with bi for each 1 ≤ j ≤ l; 2. t[8~v /~s] denotes the vector obtained as a result of the simultaneous replacement of 8vi with si for each 1 ≤ i ≤ m; 3. t[f~/~b, 8~v /~s] is a shortcut for t[f~/~b][8~v /~s]. Assume a KPN graph G = (V, E, L) such that \|F (G)\| = l, \|V ↑ (G)\| = m, \|V (G)\| = n and for some l, m, n ≥ 0. ↓ Definition 2 (CSP-KPN). We define a CSP for a KPN graph G (CSP-KPN) as follows: for each t ⊑ t′ ∈ C(G) find a vector of Boolean values ~b = (b1 , . . . , bl ), a vectors of ground terms ~t = (t1 , . . . , tm ), ~t′ = (t′1 , . . . , t′n ) such that t[f~/~b, ↑~v /~t, ↓~v /~t′ ] ⊑ t′ [f~/~b, ↑~v /~t, ↓~v /~t′ ], where f~ = (f1 , . . . , fl ), ↑~v = (↑v1 , . . . , ↑vm ), ↓~v = (↓v1 , . . . , ↓vn ). The tuple (~b, ~t, ~t′ ) is called a solution. A CSP-KPN is decidable since the message definition language we introduced can be seen as a term algebra, and decision problems for term algebras are decidable [15]. 5 Adjunct SAT The CSP-KPN solution algorithm presented in the next section is iterative and takes advantage of the order-theoretical structure of the MDL (Proposition 1). Let B0 ⊆ B1 ⊆ · · · ⊆ Bs be sets of Boolean constraints, and ~a↑ and ~a↓ be vectors of semiground terms such that \|~a↑ \| = \|V ↑ (G)\| and \|~a↓ \| = \|V ↓ (G)\|. The vectors ~a↑ and ~a↓ are conditional approximations of the solution. We seek the solution as a fixed point of a series of approximations in the following form: (B0 , ~a↑0 , ~a↓0 ), . . . , (Bs−1 , ~a↑s−1 , ~a↓s−1 ), (Bs , ~a↑s , ~a↓s ), Algorithm 1 CSP-KPN(G) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: c ← \|C(G)\| i←0 B0 ← ∅ ~a↑0 ← (none, . . . , none) ~a↓0 ← (nil, . . . , nil) repeat for 1 ≤ j ≤ c : tj ⊑ t′j ∈ C(G) do (Bi·c+j , ~a↑i·c+j , ~a↓i·c+j ) ← Solve(Bi·c+j−1 , ~a↑i·c+j−1 , ~a↓i·c+j−1 , true, tj , t′j ) end for i←i+1 until (SAT(Bi·c ), ~a↑i·c , ~a↓i·c ) = (SAT(B(i−1)·c ), ~a↑(i−1)·c , ~a↓(i−1)·c ) if Bi·c is unsatisfiable then return Unsat else return (SATSol(Bi·c ), ~a↑i·c [f~/~b], ~a↓i·c [f~/~b]) end if where for every 1 ≤ k ≤ s and a vector of Boolean values ~b that is a solution to SAT(Bk ) (by SAT(Bk ) we mean a set of Boolean vector satisfying Bk ): ~a↑k−1 [f~/~b] ⊑ ~a↑k [f~/~b] and ~a↓k [f~/~b] ⊑ ~a↓k−1 [f~/~b], (1) where the elements of the vectors are compared pairwise. The starting point is B0 = ∅, ~a↑0 = (none, . . . , none), ~a↓0 = (nil, . . . , nil) and the series terminates as soon as SAT(Bs ) = SAT(Bs−1 ), ~a↑s = ~a↑s−1 , ~a↓s = ~a↓s−1 . The adjunct set of Boolean constraints potentially expands at every iteration of the algorithm by inclusion of further logic formulas called assertions into its conjunction as the algorithm processes constraints C(G). Whether the set of Boolean constraints actually expands or not can be determined by checking the satisfiability of SAT(Bk ) 6= SAT(Bk−1 ) for the current iteration k. We argue below that if the original CSP-KPN is satisfiable then so is SAT(Bs ) and that the tuple of vectors (b~s , ~a↑s [f~/b~s ], ~a↓s [f~/b~s ]) is a solution to the former, where b~s is a solution of SAT(Bs ). In other words, the iterations terminate when the conditional approximation limits the term variables, and when the adjunct SAT constrains the flags enough to ensure the satisfaction of all CSP-KPN constraints. In general, the set SAT(Bs ) can have more than one solution. We select one of them, denoted by SATSol(Bs ) in the algorithm. 6 Algorithm In this section we present Algorithm 1 which solves CSP-KPN for a given KPN graph G = (V, E, L). It performs the following steps. The algorithm iterates over the set of constraints C(G) and at each step it builds a closer approximation of the solution. The relation between two consequent approximations satisfies formulas (1). The function Solve() solves the constraint tj ⊑ t′j (see equation (2) in Lemma 1) and updates the vectors ~a↑i·c+j and ~a↓i·c+j with new values. Furthermore, it adds Boolean assertions presented below that ensure 1) satisfaction of the constraint for any ~b ∈ SAT(Bi·c+j ) as provided by Definition 1; and 2) well-formedness of the terms occurring in it. The algorithm terminates if Bi·c ≡ B(i−1)·c , ~a↑i·c = ~a↑(i−1)·c and ~a↓i·c = ~a↓(i−1)·c . Well-formedness assertions for records and choices. Any pair of elements in a well-formed record/choice cannot have equal labels. Therefore, for each record {l1 (b1 ): t1 , . . . , l1 (bn ): tn } and each choice (:l1 (b1 ): t1 , . . . , l1 (bn ): tn :) occurring anywhere in C(G) the following assertion is added to the SAT: ^ ¬(bi ∧ bj ). ∀1≤i,j≤n : li =lj Well-formedness assertions for switches. A well-formed switch term must have exactly one positive guard. Hence, for each switch hb1 : t1 , . . . , bn : tn i occurring anywhere in C(G) the following assertion is added to the SAT: ^ (b1 ∨ · · · ∨ bn ) ∧ ¬(bi ∧ bj ). ∀1≤i,j≤n : i6=j Order assertions. We generate two kinds of order assertions. 1. If a variable 8x is junior to two incommensurable, identically guarded terms 8x ⊑ h. . . : b : t1 . . .i and 8x ⊑ h. . . : b : t2 . . .i, where neither t1 ⊑ t2 nor t2 ⊑ t1 , the assertion ¬b is added to the adjunct SAT. 2. For each c ∈ C(G) of the form hb1 : t1 , . . . , bn : tn i ⊑ hb′1 : t′1 , . . . , b′m : t′m i , the assertion ¬(bi ∧ b′j ) is added to the adjunct SAT if ti 6⊑ t′j . Further details are found in Appendix A. Lemma 1 (Loop invariant). Algorithm 1 finds a series of approximations in the form of (Bk0 , ~a↑k0 , ~a↓k0 ), . . . , (Bks−1 , ~a↑ks−1 , ~a↓ks−1 ), (Bks , ~a↑ks , ~a↓ks ), where ki = i · \|C(G)\|, and such that the following holds for any ~bki ∈ SAT(Bki ). 1. Bki ⊇ Bki−1 ; 2. ~a↑ki−1 [f~/~bki ] ⊑ ~a↑ki [f~/~bki ] and ~a↓ki [f~/~bki ] ⊑ ~a↓ki−1 [f~/~bki ]; 3. ∄ (~a′ , ~a′′ ): (a) ~a↑ki−1 [f~/~bki ] ⊑ ~a′ [f~/~bki ], ~a′ [f~/~bki ] ⊑ ~a↑ki [f~/~bki ]; (b) ~a↓ [f~/~bk ] ⊑ ~a′′ [f~/~bk ], ~a′′ [f~/~bk ] ⊑ ~a↓ [f~/~bk ]. ki i i i ki−1 i given (Bki−1 , ~a↑ki−1 , ~a↓ki−1 ), the Proof. Let c = \|C(G)\|. To construct algorithm iteratively calls Solve() function (see Appendix A). (Bki , ~a↑ki , ~a↓ki ) (Br , ~a↑r , ~a↓r ) ← Solve(Br−1 , ~a↑r−1 , ~a↓r−1 , true, tj , t′j ) where r = ki−1 + j, 1 ≤ j ≤ c and ki = ki−1 + c. For any tj ⊑ t′j , Solve() constructs the Boolean constraints Br , and finds ~a↑r and ~a↓r by solving the equation t[f~/~br−1, ↑~v /~a↑r−1 , ↓~v /~a↓r ] = t′ [f~/~br−1 , ↑~v /~a↑r , ↓~v /~a↓r−1 ], (2) 1. Bki ⊇ Bki−1 by construction: Solve() only adds new Boolean constraints to the existing set. 2. Solve() iteratively constructs the local approximation for each constraint. The series of local approximations converges to the global approximation. 3. Proof by contradiction. Assume that (~a↑r , ~a↓r ) is a solution of (2) and there exists another solution (~a′ , ~a′′ ), such that ~a′ 6= ~a↑r and ~a′′ 6= ~a↓r . Then t[f~/~br−1 , ↑~v /~a↑r−1 , ↓~v /~a′′ ] = t′ [f~/~br−1 , ↑~v /~a′ , ↓~v /~a↓r−1 ]. Two ground terms are equal only if they represent the same term, and, therefore, ~a′ = ~a↑r and ~a′′ = ~a↓r , which contradicts the initial assumption. ⊓ ⊔ Theorem 1 (Termination). CSP-KPN(G) terminates after a finite number of steps for any KPN graph G. Proof. For a given graph G the number of flags, variables and labels for records and choices is bounded. There are two ways to produce new terms: either to add entries with new labels to records and choices, or to substitute subterms for terms. 1. The number of new terms constructed by adding new entries is bounded because the number of labels in a given G is finite. 2. The number of terms constructed by substituting subterms for other terms is bounded because a) the number of variables is finite (the algorithm does not generate new variables); b) after the variables have been instantiated, the category of the term cannot be changed, otherwise, the seniority relation would be violated. It implies that for each ↑v ∈ V ↑ (G) there exists a ground term t̄, such that ↑v ⊑ t̄, and for each ↓v ∈ V ↓ (G) there exists a ground term t, such that t ⊑ ↓v. Providing that \|SAT(Bki )\| ≤ \|SAT(Bki−1 )\|, the algorithm terminates after a finite number of steps. ⊓ ⊔ Theorem 2. Assume a KPN graph G = (V, E, L). The set of constraints C(G) is inconsistent if and only if CSP-KPN(G) returns Unsat. Proof. As the initial approximation the algorithm selects the weakest approximation (∅, (none, . . . , none), (nil, . . . , nil)), it follows from Lemma 1 that the algorithm iterates over all possible approximations in consecutive order starting from (Bk0 , ~a↑k0 , ~a↓k0 ). Therefore, the algorithm cannot skip a solution if one exists. By Theorem 1 the algorithm terminates after a finite number of steps. Hence, it returns Unsat only if and only if the set of constraints C(G) cannot be satisfied. ⊓ ⊔ message message variant variant variant _1_init(vector<vector<double> img); _2_error(string msg); _1_read_color(string fname) { _1_init(...); ... _2_error(...); } _1_read_grayscale(string fname) {...} _1_read_unchanged(string fname) {...} (a) The source code IN 1: (: read_color(c): {fname: string \| $_rc}, read_grayscale(g): {fname: string \| $_rg}, read_unchanged(u): {fname: string \| $_ru} \| $^r :) OUT 1: (: init(or c g u): {img: vector<vector<double>>, \| $_ro1 } \| $^r :) 2: (: error(or c g u): {msg: string \| $_ro2 } :) $_rc <= $_ro1; $_rg <= $_ro1; $_ru <= $_ro1; $_rc <= $_ro2; $_rg <= $_ro2; $_ru <= $_ro2; (b) The interface Fig. 4: The source code and the interface of the component read of the image processing algorithm 7 Communication Protocol In this section we demonstrate interfaces with flow inheritance and code customisation using the example from Section 2. The interfaces are defined as choiceof-records terms. Labels in the choice term of the input interface correspond to function names that can process messages tagged with corresponding labels. Output messages are produced by calling special functions called salvos. The name of a salvo corresponds to one of the labels in the output choice term. The compatibility of two components connected by a channel is defined by the seniority relation. Consider the source code and the interface of the component read in Fig. 4. Integers that have been added as prefixes to functions in the code specify the channels that messages are received from and sent to. In the interfaces we use prefixes $^ and $_ before identifiers to denote up- and down-coerced variables, respectively. A tail variable $^r in the interface enables flow inheritance for choices: variants from the input channel that cannot be processed by the component (i.e. all variants but read_color, read_grayscale or read_unchanged), are absorbed by $^r. Thus, the messages of type Mr1 that contain the name of the image file are processed by the component and the messages of type Mi1 are inherited to the output and forwarded directly to the component init. Flow inheritance for records is realised by down-coerced variables $_rc, $_rg, $_ru, $_ro1, $_ro2, and a set of auxiliary constraints. A record in the input message contains an entry with the label K, which a processing function does not expect. After solving the CSP, the entry is added to the tail variable $_ro1, because the solver deduces that the element with the label K is required by the component init. Furthermore, we use flags c, g and u to exclude the code that is not used in the context. The guards in the output interface employ the joint set of flags from the input variants that can fire salvos specified in the output interface. In the example all three functions can fire init and error salvos; accordingly, the salvos’ guards are c ∨ g ∨ u. The solver deduces that that the variants read_grayscale and read_unchanged cannot receive any messages, and, therefore, their respective processing functions can be excluded from the code. To facilitate decontextualisation we introduce a wrapper for every component called a shell : an auxiliary configuration file that provides facilities for renaming labels in output records and choices and changing the routing of output messages. The source code and the interfaces for the other two components are available in the repository [11]. 8 Implementation We implemented the solver [16] for the CSP-KPN in OCaml. It works on top of the PicoSAT [17] library, the latter used as a subsolver dealing with Boolean assertions. The input for the solver is a set of constraints and the output is in the form of assignments to flags and term variables. We also developed a toolchain in C++ and OCaml that performs the interface reconciliation in five steps: 1. Given a set of C++ sources (the components), augment them with macros acting as placeholders for the code that enables flow-inheritance. 2. Derive the interfaces from the code. 3. Given the interfaces and a netlist that specifies a KPN graph, construct the constraints to be passed on to the CSP-KPN algorithm. 4. Run the solver. 5. Based on the solution, generate header files for every component with macro definitions. In addition, the tool generates the API functions to be called when a component sends or receives a message. Advantages of the presented design are the following: – Interfaces and the code behind them can be generic as long as they are sufficiently configurable. No communication between component designers is necessary to ensure consistency in the design. – Configuration and compilation of every component is separated from the rest of the application. This prevents source code leaks in proprietary software running in the Cloud.5 5 which is otherwise a serious problem. For example, proprietary C++ libraries that use templates cannot be distributed in binary form due to restrictions of the language’s static specialisation mechanism. 9 Conclusion and Future Work We have presented a new static mechanism for coordinating component interfaces based on CSP and SAT that checks compatibility of component interfaces connected in a network with support of overloading and structural subtyping. We developed a fully decoupled Message Definition Language that can be used in the context of KPN for coordinating components written in any programming language. We defined the interface of C++ components to demonstrate the binding between the MDL and message processing functions. Our techniques support genericity, inheritance and structural subtyping, thanks to the order relation defined on MDL terms. On the theory side, we presented the CSP solution algorithm, showed its correctness and identified the termination condition. Although we assume that the algorithm is NP-complete because of the SAT problem, which needs to be solved as a subproblem, the complexity of the algorithm will be evaluated in further research. The next step will be to support multiple flow inheritance in the MDL, to enable combined structures with inheritance (for example, (union ↓a ↓b) represents a record that contains a union of entries associated with records ↓a and ↓b). This would allow one to design components that perform synchronisation and merge multiple messages into one while preserving the inheritance mechanism of a vertex. In the context of Cloud, our results may prove useful to the software-asservice community since we can support much more generic interfaces than are currently available without exposing the source code of proprietary software behind them. Building KPNs the way we do could enable service providers to configure a solution for a network customer based on components that they have at their disposal as well as those provided by other providers and the customer themselves, all solely on the basis of interface definitions and automatic tuning to nonlocal requirements. References 1. Northrop, L., Feiler, P., Gabriel, R.P., Goodenough, J., Linger, R., Longstaff, T., Kazman, R., Klein, M., Schmidtd, D., Sullivan, K., Wallnau, K.: Ultra-LargeScale Systems: The Software Challenge of the Future. Technical report, Software Engineering Institute, Carnegie-Mellon (2006) 2. Kahn, G.: The Soemantics of Simple Language for Parallel Programming. In: IFIP Congress. (1974) 471–475 3. 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Vrba, v., Halvorsen, P., Griwodz, C., Beskow, P., Espeland, H., Johansen, D.: The Nornir run-time system for parallel programs using Kahn process networks on multi-core machines — a flexible alternative to MapReduce. The Journal of Supercomputing 63(1) (2013) 191–217 10. Strachey, C.: Fundamental concepts in programming languages. Higher-order and symbolic computation 13(1-2) (2000) 11–49 11. Zaichenkov, P.: Image Segmentation using K-means Clustering. https://github.com/zayac/joule/tree/kmeans (June 2015) (accessed June 16, 2015). 12. MacQueen, J.: Some methods for classification and analysis of multivariate observations (1967) 13. Gaster, B.R., Jones, M.P.: A Polymorphic Type System for Extensible Records and Variants. Technical report (1996) 14. Leijen, D.: Extensible records with scoped labels (September 2005) 15. Ferrante, J., Rackoff, C.W.: The computational complexity of logical theories. Lecture notes in mathematics. Springer-Verlag (1979) 16. 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Journal on Satisfiability, Boolean Modeling and Computation (JSAT) (2008) A Appendix: Solve function Algorithm 2 Solve(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: B̂i ← AssertWellFormed(AssertWellFormed(Bi , b, t), b, t′ ) if ((t = none or t′ = nil) and (t 6= none or t′ 6= nil)) or (t and t′ are ground, and t = t′ ) then return (B̂i , ~a↑i , ~a↓i ) else if t = ↑v and t′ = ↑v ′ then Bi+1 , ~a↑i+1 ← set a new approximation for ↑v ′ equal to the one of ↑v else if t = ↓v and t′ = ↓v ′ then Bi+1 , ~a↓i+1 ← set a new approximation for ↓v equal to the one of ↓v ′ return (Bi+1 , ~a↑i , ~a↓i+1 ) else if t is nil, symbol, tuple or record, and t′ = ↓v ′ then return Solve(B̂i , ~a↑i , ~a↓i , b, t, t′ [↑~v /~a↑i , ↓~v /~a↓i ]) else if t is a choice and t′ = ↑v ′ then Bi+1 , ~a↑i+1 ← set a new approximation for ↑v ′ as t[↑~v /~a↑i , ↓~v /~a↓i ] when b return (Bi+1 , ~a↑i+1 , ~a↓i ) else if t = ↓v, and t′ is nil, symbol, tuple or record then Bi+1 , ~a↓i+1 ← set a new approximation for ↓v as t′ [↑~v /~a↑i , ↓~v /~a↓i ] when b return (Bi+1 , ~a↓i , ~a↓i+1 ) else if t = ↑v and t′ is a choice then return Solve(B̂i , ~a↑i , ~a↓i , b, t[↑~v /~a↑i , ↓~v /~a↓i ], t′ ) else if t and t′ are tuples then return SolveTupleTuple(B̂i , ~a↑i , ~a↓i , b, t, t′ ) else if t = nil and t′ is a record then return SolveNilRecord(B̂i , ~a↑i , ~a↓i , b, t′ ) else if t and t′ are records then return SolveRecordRecord(B̂i , ~a↑i , ~a↓i , b, t, t′ ) else if t and t′ are choices then return SolveChoiceChoice(B̂i , ~a↑i , ~a↓i , b, t, t′ ) else if t or t′ is a switch then return SolveSwitch(B̂i , ~a↑i , ~a↓i , b, t, t′ ) else return (B̂i ∪ {¬b}, ~a↑i , ~a↓i ) end if Algorithm 3 AssertWellFormed(Bi , b, t) 1: if t is a record {l1 (b1S): t1 , . . . , ln (bn ): tn } or (:l1 (b1 ): t1 , . . . , ln (bn ): tn :) then {b → ¬(bi ∧ bj )} 2: Bi+1 ← Bi ∀1≤i,j≤n : li =lj 3: else if t is a switch hb1 : t1 , . . . , bn : tS n i then 4: Bi+1 ← Bi ∪ {b1 ∨ · · · ∨ bn } {b → ¬(bi ∧ bj )} ∀1≤i,j≤n : i6=j 5: end if 6: return Bi+1 Algorithm 4 SolveTupleTuple(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: Let t be of the form (t1 . . . tn ) g ← t′ [↑~v /~a↑i , ↓~v /~a↓i ] if g = (t′1 . . . t′m ) and n = m then for i : 1 ≤ j ≤ n do Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b, tj , t′j ) end for return (Bi+n , ~a↑i+n , ~a↓i+n ) else return (Bi ∪ {¬b}, ~a↑i+n , ~a↓i+n ) end if Algorithm 5 SolveNilRecord(Bi , ~a↑i , ~a↓i , b, t′ ) 1: g ← t′ [↑~v /~a↑i , ↓~v /~a↓i ] ′ 2: Let g be of the form {l1′ (b′1 ): t′1 , . . . , lm (b′m ): t′m } n V ¬b′j }, ~a↑i , ~a↓i ) 3: return (Bi ∪ {b → j=1 Algorithm 6 SolveRecordRecord(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: g ← t′ [↑~v /~a↑i , ↓~v /~a↓i ] if t = {l1 (b1 ): t1 , . . . , ln (bn ): tn } then for j : 1 ≤ j ≤ m do if ∃k : lk ∈ t, lk = lj′ then Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b → b′j → bk , tk , t′j ) else Bi+j ← Bi+j−1 ∪ {b → ¬b′j } end if end for else if t = {l1 (b1 ): t1 , . . . , ln (bn ): tn \|↓v} then for j : 1 ≤ j ≤ m do if ∃k : lk ∈ t, lk = lj′ then Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b → b′j → bk , tk , t′j ) else Bi+j , ~a↑i+j , ~a↓i+j ← set a new approximation for ↓v as lj′ (b′j ): t′j when b end if end for end if return (Bi+m , ~a↑i+m , ~a↓i+m ) Algorithm 7 SolveChoiceChoice(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: g ← t[↑~v /~a↑i , ↓~v /~a↓i ] if t′ = (:l1 (b1 ): t1 , . . . , ln (bn ): tn :) then for j : 1 ≤ j ≤ m do if ∃k : lk ∈ t, lk′ = lj then Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b → bj → b′k , tj , t′k ) else Bi+j ← Bi+j−1 ∪ {b → ¬b′j } end if end for else if t′ = (:l1 (b1 ): t1 , . . . , ln (bn ): tn \|↑v:) then for j : 1 ≤ j ≤ m do if ∃k : lk ∈ t, lk = lj′ then Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b → bj → b′k , tj , t′k ) else Bi+j , ~a↑i+j , ~a↓i+j ← set a new approximation for ↑v as (:lj (bj ): tj :) when b end if end for end if return (Bi+m , ~a↑i+m , ~a↓i+m ) Algorithm 8 SolveSwitch(Bi , ~a↑i , ~a↓i , b, t, t′ ) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: if t = hb1 : t1 , . . . , bn : tn i then for j : 1 ≤ i ≤ n do Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b, tj , t′ ) end for else if t′ = hb′1 : t′1 , . . . , b′n : t′n i then for i : 1 ≤ j ≤ n do Bi+j , ~a↑i+j , ~a↓i+j ← Solve(Bi+j−1 , ~a↑i+j−1 , ~a↓i+j−1 , b, t, t′j ) end for end if return (Bi+n , ~a↑i+n , ~a↓i+n )	6
1 Isogeometric Analysis Simulation of TESLA Cavities Under Uncertainty arXiv:1711.01828v1 [physics.comp-ph] 6 Nov 2017 Jacopo Corno∗† , Carlo de Falco†‡ , Herbert De Gersem∗ , Sebastian Schöps∗ ∗ Institut für Theorie Elektromagnetischer Felder, TU Darmstadt, Germany † MOX Modeling and Scientific Computing, Politecnico di Milano, Italy ‡ CEN - Centro Europeo di Nanomedicina, Milano, Italy Abstract—In the design of electromagnetic devices the accurate representation of the geometry plays a crucial role in determining the device performance. For accelerator cavities, in particular, controlling the frequencies of the eigenmodes is important in order to guarantee the synchronization between the electromagnetic field and the accelerated particles. The main interest of this work is in the evaluation of eigenmode sensitivities with respect to geometrical changes using Monte Carlo simulations and stochastic collocation. The choice of an IGA approach for the spatial discretization allows for an exact handling of the domains and their deformations, guaranteeing, at the same time, accurate and highly regular solutions. I. I NTRODUCTION The performance of electromagnetic devices such as, for example, energy transducers, magnetrons, waveguides, antennas and linear accelerators is strongly related to the shape of the devices themselves. In particle accelerator cavities, in particular, the acceleration of the particle beam is achieved by exciting specific eigenmodes in the cavity and the resonant frequencies need to be synchronized with the flying particles to guarantee the acceleration. Even small mechanical deformations, either due to manufacturing imperfections or to the electromagnetic pressure (Lorentz detuning) may cause a nonnegligible frequency shift [2], [7]. A correct representation and handling of the domain is then of great importance when implementing a simulation scheme. It has been shown in [7] that the Isogeometric Analysis (IGA) discretization methods introduced in [3], [4] can produce a highly accurate solution, for example for the coupled electromagnetic-mechanic problem modeling Lorenz detuning. The focus of this paper is on quantifying the uncertainty of linear accelerator superconducting cavities in frequency domain, or more precisely the sensitivity of the solution of Maxwell’s eigenvalue problem ∇ × µ−1 ∇ × E = ω 2 εE (1) with respect to (randomly) perturbed domains Ω(y), as in e.g. [1], where E is the electric field, y are geometry parameters, µ, ε and ω are the permeability, permittivity and angular frequency, respectively. The paper is structured as follows, Section 2 and 3 discuss the main ideas behind IGA and Uncertainty Quantification (UQ). In Section 4 and 5 the methods are applied to a benchmark example (pillbox) and the TESLA cavity [6]. The sensitivity of the eigenfrequencies are investigated with respect to uncertain design parameters. Furthermore, non axis-symmetric deformations are taken into consideration. II. I SOGEOMETRIC A NALYSIS In classical discretization methods such as the Finite Element Method (FEM), two steps are usually required: first, the meshing step, i.e., the discretization of the geometry from a Computer Aided Design (CAD) representation, in, typically, hexahedral or tetrahedral elements. Secondly the discretization of the set of equations describing the problem to solve. In [3], Hughes et. al proposed a new approach, called Isogeometric Analysis, were the same basis functions that CAD uses for the description of geometries (e.g. B-Splines and NURBS) are used also as a basis for the solution of the partial differential equations. The IGA paradigm guarantees the Fig. 1: Cylindrical pillbox cavity. c 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. 2 exact description of the computational domain throughout all the analysis, even at the coarsest level of refinement. d-dimensional B-Splines basis functions are defined in the reference space [0, 1]d following a tensor product approach. Along each dimension let p be the degree and Ξ = [ξ0 , ..., ξn+p+1 ] be a knot vector subdividing the unit segment [0, 1]. The Cox-de Boor formula then defines the n + 1 basis functions {Bip }n0 . A B-spline in the physical space is obtained by a mapping 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Fig. 2: B-Spline basis functions of 2nd degree in 1D obtained with knots Ξ = [0, 0, 0, 0.2, 0.4, 0.6, 0.8, 1, 1, 1]. x= n X Bip Pi (2) i=0 where Pi are called control points and acts as degrees of freedom for the curve. Surfaces and volumes are the result of tensorization. As shown in Fig. 2, B-Splines basis functions have higher regularity properties with respect to classical FEM (a B-Spline of degree p can have in general up to p − 1 global derivatives whereas FEM basis are only C 0 across the element boundaries). As a consequence of this, IGA has proven to have a higher accuracy w.r.t. the number of degrees of freedom than classical FEM. The features of IGA are particulary beneficial for quantification of uncertainty related to shape. The CAD representation is not lost in the meshing process, it is possible to easily modify the geometry and for each modified domain there is no need to perform remeshing when deforming the geometry. are a sufficient measure but often one is interested in statistics of outputs. They can be quantified by stochastic moments as expected value and variance Z ∞ E(f ) = f (E(y))ρ(y) dy (5) −∞ Z ∞ 2 f (E(y)) − E(f ) ρ(y) dy var(f ) = (6) −∞ p or the standard deviation std(f0 ) := var(f0 ). However, those integrals can rarely be solved exactly and thus one relies on numerical methods. One way out is the well-known Monte Carlo (MC) sampling method [5]. In this case the set of equations describing the problem is solved M times, for M realizations yi . The results obtained for E are then used to estimate expectations values of the desired quantity of interest by sample averages: E(f ) ≈ III. U NCERTAINTY Q UANTIFICATION (3) The question is then how the input uncertainty E affects the quantity of interest f (E), e.g., a cavity’s eigenfrequency. In many applications classical (local) sensitivities, i.e., Df (E) = ∂ f (E) ∂E (7) i=1 When dealing with mathematical models, e.g. partial differential equations, one has to consider that the input parameters might be affected by uncertainty. Let θ be a random event and y(θ) a vector containing N random input parameters yi (θ) with density ρ(y). Supposing the problem under consideration, i.e., (1), is well defined for any y, then the solution is itself a random variable E = E(y(θ)). M 1 X f (E(yi )). M (4) More sophisticated approaches exploit the regularity of the solution to increase the speed of convergence, e.g., in the generalized Polynomial Chaos (gPC) the mapping (3) from parameter to solution space is approximated by polynomials of degree w [9]. This approximation can be constructed by a basis of distribution dependent orthonormal polynomials {ψp (y)}w 0 and a grid of points yp in the parameter space (called collocation points), i.e., E(y) ≈ w X E(yp )ψp (y). (8) p=0 If the exact solution has a sufficiently smooth dependency w.r.t. its parameters this method converges exponentially. Unfortunately the rate depends heavily on the number of random parameters such that the advantage is lost for large N . 3 −2 10 −3 RelativeSError 10 −4 10 −5 10 GeoPDEsS−SFD CSTSStudioS−SFD CSTSStudioS−SSens.SAnalysis −6 10 0.495 0.5 RadiusS[m] Relative Error in Mean Frequency 100 10-2 10-4 10-6 10-8 10-10 0.505 Closed-form Solution CST Studio (accuracy 1e-6) GeoPDEs (accuracy 1e-6) 0 2 4 6 8 10 Number of Quadrature Nodes (a) Sensitivity of the fundamental frequency f0 in the cylindrical cavity w.r.t. a change in the radius. (b) Convergence of E(f0 ) using gPC, i.e., stochastic Gauss quadrature, for various models. (c) Design parameters of the TESLA cavity half-cell [6]. Fig. 3: Pillbox sensitivities, convergence of stochastic quadrature and TESLA cavity design 3 IV. P ILLBOX C AVITY As a benchmark example we consider the case of a cylindrical pillbox cavity with uncertain radius r (see Fig. 1). We are interested in the resonant frequency of the fundamental (accelerating) mode f0 and its sensitivity w.r.t. the change in radius close to the design value rd = 0.5 m. In this settings one may exactly characterize the frequency and its derivative as Mean Frequency [GHz] 2.5 2 1.5 1 0 f0 (r) = Gc , 2πr df0 (r) Gc =− dr 2πr2 where G ≈ 2.405 is the first zero of the Bessel function of order 0 and c is the speed of light. In Figure 3a we compare implementations of IGA (GeoPDEs [8]) to FEM (CST EM Studio [10]). First, we compute the fundamental frequency for small perturbation of the radius across the design value, using second order IGA and second order FEM with the same level of accuracy. Secondly we use Finite Differences (FD) to estimate the sensitivity. It is clear from the figure that the FEM approximation suffer heavily from the need of remeshing the geometry each time the radius changes, while, even for this naı̈ve approach, IGA obtains a smooth solution, since the parametrization of the cylinder doesn’t change. Finally, the magenta curve in Fig. 3a shows the results obtained by the automated sensitivity analysis tool available in CST: by using more sophisticated methods CST can achieve higher precision but oscillations are still present. To study the global sensitivity of f0 (r) an (artificial) random radius r ∼ U(0.2 m, 0.8 m) is considered. We compare the numerical approach (gPC with FEM and Monopole Modes Dipole Modes Quadrupole Modes 0.5 0 2 4 6 8 10 12 14 16 # Mode Fig. 4: First 16 modes of one-cell TESLA cavity. IGA) to the analytical solution E(f0 ) = 0.2651115... Hz, std(f0 ) = 0.1095555... Hz. Fig. 3b depicts the rapid convergence until the accuracy of the spatial discretization is reached. V. TESLA C AVITY Let us now consider the single cell TESLA cavity geometry whose seven design parameters are described in Figure 3c. Following [11], we apply UQ for design parameters albeit this restricts possible deviations significantly and more realistic cases, such as nonaxissymmetric deformations, bumps/kinks due to welding, mechanical deformation due to Lorentz forces or misalignment of the irisis cannot be represented. We consider the parameters to be uncertain with mean values equal to the TESLA mid-cell design [6] and deviations yi ∼ ( − 0.125 mm, 0.125 mm). MC sampling 4 6 1.5 Standard Deviation [MHz] 5 Standard Deviation [MHz] Monopole Modes Dipole Modes Quadrupole Modes 4 3 2 Monopole Modes Dipole Modes Quadrupole Modes 1 0.5 1 0 0 0 2 4 6 8 10 12 14 16 # Mode (a) Eigenmodes due to uncertain design parameters 0 2 4 6 8 10 12 14 16 # Mode (b) Eigenmodes due to elliptic deformation Fig. 5: Standard deviations for the first 16 modes of the one-cell TESLA Cavity. is applied to estimate mean values and standard deviation of the resonant frequencies for the first 16 modes. With respect to [11], where a 2D solver was used to compute the accelerating frequency by exploiting the cylindrical symmetry of the geometry, GeoPDEs is able to solve the full 3D cavity, thus allowing for the computation of the full spectrum of frequencies, Fig. 4. In Fig. 5a the standard deviation of these modes are depicted. The values are in the MHz range with a maximum deviation in correspondence of the third monopole mode. A second case investigates elliptic deformations of the cavity. For this the domain is deformed in such a way that the cross section is no more a circle but an ellipse, i.e., breaking symmetry. The deviation with respect to the design geometry is chosen to be a uniformly distributed random variable with zero mean and σ = 6.6667 · 10−5 , i.e., Req ∼ U(103.1mm, 103.5mm). The solution of the eigenproblem is computed on a 10×10 grid of collocation points for the first 16 modes as in the previous case. Results (in Fig. 5b) suggest that possible elliptical deformations of the cavity play a slightly weaker role than the design parameters. Nevertheless the higher order modes are, once again, more effected than the fundamental one. Future studies on the full 9-cell cavity should take into account the necessity of including these modes in the analysis. VI. C ONCLUSIONS In this paper the usage of IGA and gPC for the uncertainty quantification of cavities was proposed since those methods exploit the smoothness of the solution in spatial and parameter domain. Academic and realistic geometries underline that this methodology allows to accurately quantify the impact of uncertainties. ACKNOWLEDGMENTS This work is supported by the “Excellence Initiative” of the German Federal and State Governments. the Graduate School CE at TU Darmstadt and the DFG network “UQ for Cavities” (SCHM3127/21). C. de Falco’s work is partially funded by the “Start-up Packages and PhD Program project”, co-funded by Regione Lombardia through “Fondo per lo sviluppo e la coesione 2007-2013” (FAS program). R EFERENCES [1] L. Xiao et al., Modeling imperfection effects on dipole modes in TESLA cavity, IEEE PAC 2007, 2454–2456, 2007. [2] J. Deryckere, T. Roggen, B. Masschaele, and H. De Gersem, Stochastic response surface method for studying microphoning and lorentz detuning of accelerators cavities, ICAP2012, 158160, 2012. [3] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comp. Meth. in App. Mech. and Eng., 194(39), 41354195, 2005. [4] A. Buffa, G. Sangalli, and R. Vzquez, Isogeometric analysis in electromagnetics: B-splines approximation, Comp. Meth. in App. Mech. and Eng., 199, 1143-1152, 2010. [5] Liu, J.S., Monte Carlo Strategies In Scientific Computing, Harvard University, 2002. [6] B. Aune et al., Superconducting tesla cavities. Physical Review Special Topics - Accelerators and Beams, 3(9):092001, 2000. [7] J. Corno, C. de Falco, H. De Gersem and S. Schöps, Isogeometric Simulation of Lorentz Detuning in Superconducting Accelerator Cavities, MOX Report 31/2014. [8] C. de Falco, A. Reali, R. Vázquez, GeoPDEs: A research tool for Isogeometric Analysis of PDEs, Advances in Eng. Soft., 12(42), 1020-1034, 2011. [9] D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, 2010. [10] CST EM STUDIO R , Computer Simulation Technology AG, www.cst.com. [11] C. Schmidt, T. Flisgen, J. Heller, U. van Rienen, Comparison of techniques for uncertainty quantification of superconducting radio frequency cavities, ICEAA 2014, 117–120, 2014. [12] J. Gravesen, A. Evgrafov und D.-M. Nguyen, On the sensitivities of multiple eigenvalues, Structural and Multidisciplinary Optimization 44.4, 583-587, 2011.	5
0 Truthful Mechanisms with Implicit Payment Computation MOSHE BABAIOFF, Microsoft Research, Herzeliya, Israel. ROBERT D. KLEINBERG, Computer Science Department, Cornell University, Ithaca, NY, USA. ALEKSANDRS SLIVKINS, Microsoft Research, New York, NY, USA. arXiv:1004.3630v5 [cs.GT] 15 Nov 2015 First version: April 2010 This version: November 2015 It is widely believed that computing payments needed to induce truthful bidding is somehow harder than simply computing the allocation. We show that the opposite is true: creating a randomized truthful mechanism is essentially as easy as a single call to a monotone allocation rule. Our main result is a general procedure to take a monotone allocation rule for a single-parameter domain and transform it (via a black-box reduction) into a randomized mechanism that is truthful in expectation and individually rational for every realization. The mechanism implements the same outcome as the original allocation rule with probability arbitrarily close to 1, and requires evaluating that allocation rule only once. We also provide an extension of this result to multi-parameter domains and cycle-monotone allocation rules, under mild star-convexity and non-negativity hypotheses on the type space and allocation rule, respectively. Because our reduction is simple, versatile, and general, it has many applications to mechanism design problems in which re-evaluating the allocation rule is either burdensome or informationally impossible. Applying our result to the multi-armed bandit problem, we obtain truthful randomized mechanisms whose regret matches the information-theoretic lower bound up to logarithmic factors, even though prior work showed this is impossible for truthful deterministic mechanisms. We also present applications to offline mechanism design, showing that randomization can circumvent a communication complexity lower bound for deterministic payments computation, and that it can also be used to create truthful shortest path auctions that approximate the welfare of the VCG allocation arbitrarily well, while having the same running time complexity as Dijkstra’s algorithm. Categories and Subject Descriptors: J.4 [Social and Behavioral Sciences]: Economics; K.4.4 [Computers and Society]: Electronic Commerce; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems General Terms: theory, algorithms, economics Additional Key Words and Phrases: algorithmic mechanism design, single-parameter mechanisms, multiarmed bandits, regret, multi-parameter mechanisms This is a merged and revised version of the conference papers [Babaioff et al. 2010, 2013] that have appeared in the ACM Conf. on Electronic Commerce (ACM EC) in 2010 and 2013, respectively. This paper contains all results from [Babaioff et al. 2010] and the main result from [Babaioff et al. 2013] (in Section 8). This version is updated to reflect the current status of the follow-up work and open questions. Parts of this research have been done while R. Kleinberg was a Consulting Researcher at Microsoft Research Silicon Valley. He was also supported by NSF Awards CCF-0643934 and AF-0910940, an Alfred P. Sloan Foundation Fellowship, and a Microsoft Research New Faculty Fellowship. 0:2 M. Babaioff et al. 1. INTRODUCTION Algorithmic Mechanism Design studies the problem of implementing the designer’s goal under computational constraints. Multiple hurdles stand in the way for such implementation. Computing the desired outcome might be hard (as in the case of combinatorial auctions) or truthful payments implementing the goal might not exist (as when exactly minimizing the make-span in machine scheduling [Archer and Tardos 2001]). Even when payments that will generate the right incentives do exist, finding such payments might be computationally costly or impossible due to online constraints. It is widely believed that computing payments needed to induce truthful bidding is somehow harder than simply computing the allocation. For example, the formula for payments in a VCG mechanism involves recomputing the allocation with one agent removed in order to determine that agent’s payment; this seemingly increases the required amount of computation by a factor of n + 1, where n is the number of agents. Likewise, for truthful single-parameter mechanisms the formula for payments of a given agent includes integrating the allocation rule over this agent’s bid [Myerson 1981; Archer and Tardos 2001]. In some contexts with incomplete observable information, such as online pay-per-click auctions, computing these “counterfactual allocations” may actually be information-theoretically impossible. This calls into question the mechanism designer’s ability to compute payments that make an allocation rule truthful, even when such payment functions are known to exist. Rigorous lower bounds based on these observations have been established for the communication complexity [Babaioff et al. 2013] and regret [Babaioff et al. 2014; Devanur and Kakade 2009] of truthful deterministic mechanisms. In contrast to these negative results, we show that the opposite is true for randomized single-parameter mechanisms that are truthful-in-expectation: computing the allocation and payments is essentially as easy as a single call to the allocation rule. This allows for positive results that circumvent the lower bounds for deterministic mechanisms cited earlier. 1.1. Single-parameter mechanisms We consider an arbitrary single-parameter domain. The paradigmatic example is an auction that allocates items between agents whose utility is linear in the number of items they receive. The private information of each agent is expressed by a single parameter: her value per item.1 Each agent submits a bid, then the mechanism performs the allocation and charges payments. A mechanism is called “truthful” if each agent maximizes her utility by submitting her true value per item. The allocation rule in a truthful mechanism is called “truthfully implementable”. It is known that an allocation rule is truthfully implementable if and only if it is “monotone”: increasing one agent’s bid while keeping all other bids the same does not decrease this agent’s allocation [Myerson 1981; Archer and Tardos 2001]. A similar property holds for randomized mechanisms and truthfulness-in-expectation. Our contributions. Our main result is a general procedure to take any monotone-inexpectation allocation rule A and transform it into a randomized mechanism that is truthful-in-expectation, implements the same outcome as A with probability arbitrarily close to 1, and requires evaluating that allocation rule only once. (We refer to this 1 In a general single-parameter domain, the allocation rule selects an outcome from some arbitrary collection of feasible outcomes. Each agent has her own type of “good”, and for each agent there is an arbitrary, publicly known mapping from feasible outcomes to a real-valued amount of the corresponding good. The agent’s utility is linear in this amount; the value per unit amount of good is her private information. Truthful Mechanisms with Implicit Payment Computation 0:3 procedure as the generic transformation.) The allocation rule A is accessed only as a function call, so our result applies even if A is an online algorithm. Moreover, for each realization of randomness an agent never loses by participating in the mechanism and bidding truthfully; thus the agents are protected from undesirable random deviations. We make a distinction between randomness in the mechanism and randomness in “nature”: the environment that the mechanism interacts with. Randomness in nature is subject to modeling assumptions and hence is less “reliable”; moreover, agents’ beliefs about nature may be different from the mechanism’s. On the other hand, randomness in the mechanism is fully controlled by the mechanism. Therefore it is desirable to design mechanisms that are truthful in a stronger sense: in expectation over the mechanism’s random seed, for every realization of randomness in nature; we will call such mechanisms ex-post truthful. It is easy to see from [Myerson 1981; Archer and Tardos 2001] that in any ex-post truthful mechanism the allocation rule must satisfy ex-post monotonicity (which is defined similarly to ex-post truthfulness). In the generic transformation described above, if the original allocation rule A is ex-post monotone then the resulting randomized mechanism is ex-post truthful. Similarly, our result extends to Bayesian incentive-compatibility: if A is monotone in expectation with respect to a Bayesian prior over other agents’ bids, then the mechanism is truthful in expectation over this prior. Our generic transformation is particularly useful for mechanism design problems in which re-evaluating the allocation rule is either burdensome or informationtheoretically impossible. 1.2. Bandit mechanisms A leading problem for which only a single call to the allocation rule can be evaluated is the multi-armed bandit (MAB) mechanism design problem [Babaioff et al. 2014; Devanur and Kakade 2009]. In this problem information about the state of the world is dynamically revealed during the allocation; the particular information that is revealed depends on the prior choices of the allocation, and in turn may impact the future choices. Simulating the allocation rule on different inputs may therefore require information that was not observed on the actual run. This “informational obstacle” (insufficient observable information) is a crucial obstacle for deterministic ex-post truthful MAB mechanisms; it is used in [Babaioff et al. 2014] to derive that the appropriate payments cannot be computed unless the allocation rule is very “naı̈ve” (and therefore suboptimal). To put more context, MAB mechanisms are motivated by online pay-per-click ad auctions, and were suggested in [Babaioff et al. 2014; Devanur and Kakade 2009] as a simple model which combines strategic bidding by agents and online learning by the mechanism. Each agent has a single ad that she wants to display to users, and derives utility only if her ad is clicked. The value per click is her private information. The allocation rule proceeds in rounds: in each round the mechanism allocates one ad to be shown to a user and observes whether this ad was clicked. The click probabilities (also known as “click-through rates”, or CTRs) are unknown to the mechanism, and need to be estimated during the run of the allocation rule. All bids are submitted before the allocation starts, and all payments are assigned after it ends. MAB mechanisms are related to MAB algorithms: the allocation rule is essentially an MAB algorithm whose “rewards” are clicks weighted by the corresponding bids. Moreover, welfare of an MAB mechanism is precisely the same as the total reward of its allocation rule.2 Therefore one could directly compare the performance of truthful 2 This is because payments cancel out: the total amount paid by the agents is equal to the total amount received by the mechanism. 0:4 M. Babaioff et al. MAB mechanisms with that of MAB algorithms; both can be quantified using regret: the loss in welfare compared to the benchmark which always picks the best ad. Following [Babaioff et al. 2014; Devanur and Kakade 2009], we focus on the stochastic version of the problem, i.e. we assume that the CTRs do not change over time. Then the “randomness in nature” corresponds to the random clicks, and ex-post truthfulness means truthfulness for every realization of the clicks (but in expectation over the randomness in the mechanism). Note that ex-post truthfulness is a very strong property which holds even if the clicks are chosen by an oblivious adversary. As discussed in [Babaioff et al. 2014; Devanur and Kakade 2009], this property is highly desirable, compared to the weaker notion of “truthfulness in expectation over clicks”, even if the corresponding mechanism has regret guarantees that only apply to the stochastic setting. Our contributions. Applying our generic transformation to the MAB problem we derive that the problem of designing truthful MAB mechanisms reduces to the problem of designing monotone MAB allocation rules. Such a problem has not been previously studied in the rich literature on MAB. Our main result in this direction is a randomized MAB mechanism that is ex-post truthful and has regret O(T 1/2 ) for the stochastic version. This upper bound on regret matches the information-theoretic lower bound for algorithms in the same setting (i.e., the lower bound holds even in the absence of incentive constraints). This stands in contrast to the lower bound of [Babaioff et al. 2014], where it was shown that deterministic ex-post truthful MAB mechanisms must suffer a larger regret of Ω(T 2/3 ). On a technical level, we design a new MAB allocation rule that is ex-post monotone and has regret O(T 1/2 ) for the stochastic setting. (We use it to obtain a randomized ex-post truthful MAB mechanism with the same regret.) Moreover, we show that UCB1 [Auer et al. 2002a] (and a number of similar MAB algorithms) give rise to MAB allocations that are monotone in expectation over clicks, and therefore can be transformed to randomized MAB mechanisms that are truthful in the same sense and have optimal regret. The new ex-post monotone MAB allocation rule is deterministic, which rigorously confirms the intuition from [Babaioff et al. 2014; Devanur and Kakade 2009] that the impossibility results for deterministic MAB mechanisms are caused by the “informational obstacle” (insufficient observable information about clicks) rather than ex-post monotonicity. 1.3. Other contributions Power of randomization. As a by-product of our analysis of MAB mechanisms, we obtain an unconditional separation between the power of randomized vs. deterministic ex-post truthful mechanisms for welfare maximization, in the online setting. (The separation result is unconditional in the sense that it considers exactly the same setting for both classes of mechanisms.) This complements the result of Dobzinski and Dughmi [2009], which gives a separation between these two classes of mechanisms in the offline setting, under a polynomial communication complexity constraint. It is worth noting that the separation in Dobzinski and Dughmi [2009] applies to a rather unnatural problem (two-player multi-unit auctions in which if at least one item is allocated, then all items are allocated and each player receives at least one item) whereas our separation result is for a natural problem: online pay-per-click ad auctions for a single slot, with unknown click-through rates. For the objective of revenue maximization, separations between randomized and deterministic mechanisms have been known for much longer [Thanassoulis 2004; Manelli and Vincent 2006; Dobzinski et al. 2012; Briest et al. 2014] and are in some Truthful Mechanisms with Implicit Payment Computation 0:5 sense less surprising. Randomization allows the mechanism to access a larger set of possible allocations, i.e. the set of all probability distributions over pure allocations, and in some cases this leads to greater revenue, for example by permitting more finegrained price discrimination between agent types. This is not the case for the objective of maximizing welfare (because VCG mechanisms are deterministic and they maximize welfare pointwise while obeying incentive constraints). For welfare maximization, randomized mechanisms are sometimes more powerful than deterministic ones due to other reasons, such as computational power or informational limitations (as in the problem we study). Offline mechanisms. Our main result also has implications for offline mechanism design. Nisan and Ronen, in their seminal paper [Nisan and Ronen 2001] which started the field of algorithmic mechanism design, cite the apparent n-fold computational overhead of computing VCG payments and pose the open question of whether payments can be computed faster than solving n versions of the original problem, e.g. for VCG path auctions. Our result shows that the answer is affirmative, if one adopts the truthful-in-expectation solution concept and tolerates a mechanism that outputs an outcome whose welfare is a (1 + ǫ)-approximation to that of the VCG allocation, for arbitrarily small ǫ > 0. Babaioff et al. [2013] present a social choice function f in an n-player single-parameter domain such that the deterministic communication complexity required for truthfully implementing f exceeds that required for evaluating f by a factor of n. Our result shows that no such lower bound holds when one considers randomized mechanisms, again allowing for a small amount of random error in the allocation. Extension to multi-parameter mechanisms. We extend our generic transformation from single-parameter to multi-parameter mechanisms. It is known that a multiparameter allocation rule is truthfully implementable if and only if it satisfies a property called “cycle-monotonicity”. (This is a rather strong property which specializes to monotonicity in the single-parameter case.) Similar to the single-parameter case, we present a general procedure to take any cycle-monotone allocation rule A and transform it into a randomized mechanism that is truthful-in-expectation, implements the same outcome as A with probability arbitrarily close to 1, and requires evaluating that allocation rule only once. The technical contribution here is that we find a reduction from the multi-parameter setting to the single-parameter case. While much more general that our single-parameter transformation, this result may be more difficult to apply. This is because cycle-monotonicity is known to be a very restrictive property. However, the follow-up work already provides two applications, see Section 2.1 for details. 1.4. Map of the paper This paper makes four high-level contributions: the generic transformation for singleparameter mechanisms (Sections 4 and 5), the two applications to off-line mechanism design (Section 6), the results on MAB mechanisms (Section 7), and an extension to multi-parameter mechanisms (Section 8). Presenting these results requires a significant amount of preliminaries on mechanisms design (Section 3), multi-armed bandits (Section 7.1), and multi-parameter mechanism (Section 8.1). We conclude with open questions (Section 9). A considerable amount of work followed up on the initial conference publication [Babaioff et al. 2010] of this paper. This work is discussed in Section 2.1. 0:6 M. Babaioff et al. 2. RELATED WORK AND FOLLOW-UP WORK The characterization of truthful mechanisms for single-parameter domains, given by Myerson [1981] for single-item auctions and by Archer and Tardos [2001] for a more general class of single-parameter problems, states that a mechanism is truthful if and only if its allocation rule is monotone and its payment rule charges each agent its value for the realized outcome, minus a correction term expressed as an integral over all types lower than the agent’s declared type. Exact computation of this correction term may be intractable, but Archer et al. [2004] developed a clever workaround: one can use random sampling to compute an unbiased estimator of the correction term, at the cost of evaluating the allocation rule once more. Thus, for n agents, the allocation rule must be evaluated n + 1 times: once to determine the actual allocation, and once more per agent to determine that agent’s payment. Our generic transformation relies on a generalization of this random sampling technique, but we show how to avoid recomputing the allocation rule when determining each agent’s payment, by coupling payment generation with the allocation itself. The question of whether computing payments is computationally harder than computing the allocation was raised by Nisan and Ronen [2001] in the context of VCG path auctions. The most significant progress to date was the communication complexity lower bound of Babaioff et al. [2013] mentioned above. Payment computation in online mechanism design is a central issue in the analysis of truthful MAB mechanisms in Babaioff et al. [2014] and Devanur and Kakade [2009]. The main result of [Babaioff et al. 2014] is a characterization of deterministic ex-post truthful mechanisms. It is more restrictive than the Myerson and ArcherTardos characterization. The reason is that computing an agent’s payment requires knowing how many clicks she would have received if she had submitted a lower bid value, which may require the mechanism to hypothetically go back into the past and allocate impressions to a different agent for the purpose of seeing whether a user would have clicked on that agent’s advertisement. Such counterfactual information is typically impossible to obtain in an online setting. Babaioff et al. [2014] focus on welfare maximization. Using the above characterization, they prove that any deterministic ex-post truthful MAB mechanism must incur regret Ω(T 2/3 ), whereas MAB algorithms for the same setting can achieve regret O(T 1/2 ). Devanur and Kakade [2009] consider revenue maximization, and derive a similar Ω(T 2/3 ) lower bound on loss of revenue compared to the VCG payments.3 Dynamic auctions [Athey and Segal 2013; Bergemann and Välimäki 2010; Bergemann and Said 2011] constitute another setting in which information is revealed “dynamically” (over time). However, while in MAB auctions all information from the agents (the bids) is submitted only once and then information is revealed to the mechanism by the environment over time, in dynamic auctions the agents continuously observe private “signals” from the environment and submit “actions” to the mechanism. Accordingly, providing the right incentives becomes much more challenging. On the other hand, existing work has focused on a fully Bayesian setting with known priors on the signals, whereas all of our results do not rely on priors. Finally, several recent papers have explored the theme of reductions in algorithmic mechanism design. Unlike our work which requires mechanisms to be truthful for every realization of the agents’ types, these papers focus on Bayesian settings and adopt Bayesian incentive-compatibility as their solution concept. A reduction converting any allocation rule into a Bayesian incentive-compatible mechanism with approx3 For revenue-maximizing MAB mechanisms, there is no clear comparison with the performance of MAB algorithms. Truthful Mechanisms with Implicit Payment Computation 0:7 imately the same expected social welfare was developed in [Hartline and Lucier 2010; Bei and Huang 2011; Hartline et al. 2011]. Chawla et al. [2012] considered black-box reductions of mechanism design problems to algorithmic problems with the same objective, and demonstrated significant limitations of this approach. The breakthrough results of Cai et al. [2012; 2013a; 2013b] and Daskalakis and Weinberg [2014] circumvented these limitations by instead reducing to algorithmic problems with a modified objective. In particular, reductions from revenue-maximizing mechanisms to welfaremaximizing algorithms are presented in [Cai et al. 2012, 2013a], whereas Cai et al. [2013b] and Daskalakis and Weinberg [2014] present reductions for non-linear objective functions, such as makespan in scheduling.4 2.1. Follow-up work (subsequent to [Babaioff et al. 2010]) Our generic transformation exhibits high variability in payments, and includes an explicit tradeoff between the variability in payments and the loss in performance. Formally, variability can be expressed as variance, maximal absolute value, or (for positive types) maximal rebate. Performance can be expressed as welfare or revenue. Wilkens and Sivan [2012] have proved this tradeoff to be optimal in a certain worstcase sense: our transformation achieves the optimal worst-case variance in payments for any given worst-case loss in performance, where the worst case is over all monotone allocation rules. Their result applies to any single-parameter domain and any of the above notions of variability and performance. Our generic transformation is likely to be very useful in single-parameter settings which exhibit the “informational obstacle” (insufficient observable information) such as the one found for deterministic MAB mechanisms. The follow-up work describes three additional settings. First, Wilkens and Sivan [2012] observe that the same obstacle arises in offline pay-per-click ad auctions with multiple ad slots, where the CTRs have slot-specific multipliers. In conjunction with our generic transformation, an obvious welfare-maximizing allocation rule for that setting results in a truthfulin-expectation mechanism. Second, Shnayder et al. [2012] describe a packet scheduling problem in a network router, where the “informational obstacle” arises due to the potentially missing information about packet arrival times. (As they observe, this information may be missing not only because it is not observed by the router but also because the router simply does not have space to store it.) They design a monotone allocation rule for their setting, and use our generic transformation to convert it to a truthful-in-expectation mechanism. Third, Gatti et al. [2012] consider an extension of MAB mechanisms to multiple ad slots. While they provide truthful mechanisms based on the simple MAB mechanism from [Babaioff et al. 2014; Devanur and Kakade 2009], our generic transformation could give rise to more efficient truthful mechanisms. Wilkens and Sivan [2012] obtain a similar “single-call reduction” (i.e., a reduction from allocation rules to truthful-in-expectation mechanisms which calls the allocation rule only once) for multi-parameter allocation rules that are maximal-indistributional-range (MIDR). MIDR allocation rules [Dobzinski and Dughmi 2009] pick a welfare-maximizing distribution over outcomes from some fixed collection of distributions; they are precisely the allocation rules for which VCG payments produce a truthful mechanism. This result is an independent work with respect to, and a special case of, the multi-parameter reduction in Section 8. The multi-parameter generic transformation in Section 8 has been used in two recent papers. First, Jain et al. [2011] used it to speed up the payment computation for a mechanism that allocates batch jobs in a cloud system. Second, Huang and Kannan 4 All papers discussed in this paragraph, except [Hartline and Lucier 2010], have appeared after the conference publication of this paper [Babaioff et al. 2010]. 0:8 M. Babaioff et al. [2012] used it to compute payments for their privacy-preserving procurement auction for spanning trees, which is based on the well-known “exponential privacy mechanism” from prior work [McSherry and Talwar 2007]. Simplified payment computation. Our generic transformation is most useful if the allocation rule cannot be invoked more than once, as in “bandit mechanisms” or other examples provided in follow-up work. Segal [2010] has observed that any truthful single-parameter mechanism can be implemented in a much simpler way, as long as two calls to the allocation rule are allowed: one computes the allocation, and the other one generates random payments with the correct expectation. In the first call one uses the original bids. For the second call, one selects an agent uniformly at random, and uses the random sampling trick from Archer and Tardos [2001] described above to compute the payment for this agent, and then scales the payment appropriately.5 Also, a simpler generic transformation is possible if one settles for a weaker notion of Bayesian incentive-compatibility [Hartline 2012]. 3. PRELIMINARIES Single-parameter domains. We present the single parameter model for which we apply our procedure. The model is very similar to the model of Archer and Tardos [Archer and Tardos 2001], yet it is slightly more general. We state the model is terms of values and not costs and allow the values to be both positive and negative. We also allow randomization by nature. All these changes are minor and do not change the fundamental characterization, yet are helpful to later derive our results. Let n be the number of agents and let N = [n] be the set of agents. Each agent i ∈ N has some private type consisting of a single parameter xi ∈ Ti that describes the agent, and is known only to i, everything else is public knowledge. We assume that the domain Ti is an open subset of R which is an interval with positive length (possibly starting from −∞ or going up to ∞). Let T = T1 × T2 × ... × Tn denote the domain of types and let t ∈ T denote the vector of true types. There is some set of outcomes O. For single-parameter domains, agents evaluate outcomes in a particular way that we describe next. For each agent i ∈ N there is a function ai : O → R+ specifying the allocation to agent i. The value of an outcome o ∈ O for an agent i ∈ N with type xi is xi · ai (o). The utility that agent i ∈ N derives from outcome o ∈ O when he is charged pi is quasi-linear: ui = xi · ai (o) − pi . For instance, consider the allocation of k identical units of good to agents with additive valuations: agent i has a value of xi per unit. An outcome o specifies how many items each agent receives: ai (o) is the number of items i receives. His valuation for that outcome is his value per-unit times the number of units he receives. A (direct revelation) deterministic mechanism M consists of the pair (A, P), where A : T → O is the allocation rule and P : T → Rn is the payment rule, i.e. the vector of payment functions Pi : T → R for each agent i. Each agent is required to report a type bi ∈ Ti to the mechanism, and bi is called the bid of agent i. We denote the vector of bids by b ∈ T . The mechanism picks an outcome A(b) and charges agent i payment of Pi (b). The allocation for agent i when the bids are b is Ai (b) = ai (A(b)) and he is charged Pi (b). Agent i’s utility when the agents bid b ∈ T and his type is xi ∈ Ti is ui (xi , b) = xi · Ai (b) − Pi (b) 5 However, (1) more work is needed for domains with negative agents’ types, such as VCG shortest path auctions (see Section 6 for more details). In particular, one needs to carefully define the random sampling of the bid for payment computation, using a version of the argument in Section 5.2 to bound the loss in welfare. Truthful Mechanisms with Implicit Payment Computation 0:9 We also consider randomized mechanisms, which are distributions over deterministic mechanisms. For a randomized allocation rule Ai (b) and Pi (b) will denote the expected allocation and payment charged from agent i, when the bids are b. The expectation is taken over the randomness of the mechanism. Sometimes it will be helpful to explicitly consider the deterministic allocation and payment that is generated for specific random seed. in this case we use w to denote the random seed and use Ai (b; w) and Pi (b; w) to denote allocation and payment when the seed is w. There may be some outside randomization that influences the outcome and is not controlled by the mechanism, e.g. randomness in the realization of clicks in sponsored search auction. We call this randomization by nature. With such randomization Ai (b) and Pi (b) also encapsulate expectations over nature’s randomization. Finally, we use the notation Ai (b; w, r) and Pi (b; w, r) to denote the allocation and payment charged from agent i, when the bids are b, the mechanism random seed is w and nature’s random seed is r. Allocation and Mechanism Properties. Let b−i denote the vector of bids of all agents but agent i. We can now write the vector of bids as b = (b−i , bi ). Similar notation will be used for other vectors. We next list two central properties, truthfulness and individual rationality. — Mechanism M is truthful if for every agent i truthful bidding is a dominant strategy: for every agent i, bidding xi always maximizes her utility, regardless of what the other agents bid. Formally, xi · Ai (b−i , xi ) − Pi (b−i , xi ) ≥ xi · Ai (b) − Pi (b) (2) xi · Ai (b−i , xi ) − Pi (b−i , xi ) ≥ 0 (3) holds for every agent i ∈ N , type xi ∈ Ti , bids of others b−i ∈ T−i and bid bi ∈ Ti of agent i. — Mechanism M is individually rational (IR) if an agent never receives negative utility by participating in the mechanism and bidding truthfully. Formally, holds for every agent i ∈ N , type xi ∈ Ti and bids of others b−i ∈ T−i . It will be helpful to establish terminology for the case that the above hold not only in expectation but also for specific realizations. For example, we will say that a mechanism is universally truthful if Equation (2) holds not only in expectation over the mechanism’s randomness, but rather for every realization of that randomness. In general, every property that we define is defined by some inequality, and if the inequality holds for every realization of the mechanism randomness we say that it holds universally, and if it holds for every realization of nature randomness we say that it holds ex-post. When we want to emphasize that the property holds only in expectation over the nature’s randomness we say that it holds stochastically. Note that in an individually rational mechanism an agent is ensured not to incur any loss in expectation. That is rather unsatisfying as for some realizations the agent might suffer a huge loss. It is more desirable to design mechanisms that are universally ex-post individually rational, that is a truthful agent should incur no loss for every bids of the others and every realization of the random events (not only in expectation). If all types are positive, then in addition to individual rationality it is desirable that all agents are charged a non-negative amount; this is known as the no-positivetransfers property. Finally, the welfare of a truthful mechanism is defined to be the P total utility i xi · Ai (t). Characterization. The following characterization of truthful mechanisms, due to Archer and Tardos [Archer and Tardos 2001], is almost identical to the characteriza- 0:10 M. Babaioff et al. tion presented by Myerson [Myerson 1981] for truthful mechanisms in the special case of single item auctions. The crucial property of an allocation that yields truthfulness is monotonicity, defined as follows: Definition 3.1. Allocation rule A is monotone if for every agent i ∈ N , bids b−i ∈ T−i − and two possible bids of i, bi ≥ b− i , we have Ai (b−i , bi ) ≥ Ai (b−i , bi ). Recall that monotonicity of an allocation rule is also defined universally and/or ex-post. We next present the characterization of truthful mechanisms. In the theorem statement, the expression Ai (b−i , u) is interpreted to equal zero when u 6∈ Ti . T HEOREM 3.2. [Myerson 1981; Archer and Tardos 2001] Consider an arbitrary single-parameter domain. An allocation rule A admits a payment rule P such that the mechanism (A, P) is truthful if and only if A is monotone and moreover for each R bi agent i and bid vector b it holds that −∞ Ai (b−i , u) du < ∞. In this case the payment Pi (b) for each agent i must satisfy R bi Pi (b) = Pi0 (b−i ) + bi Ai (b−i , bi ) − −∞ Ai (b−i , u) du, (4) where Pi0 (b−i ) does not depend on bi . A mechanism is called normalized if for each agent i and every bid vector b, zero allocation implies a zero payment: Ai (b) = 0 ⇒ Pi (b) = 0. C OROLLARY 3.3. The truthful mechanism in Theorem 3.2 is normalized if and only if Pi0 (b−i ) ≡ 0, in which case the mechanism is also individually rational and for positive-only types (T ⊂ Rn+ ) it moreover satisfies the no-positive-transfers property. Both Theorem 3.2 and Corollary 3.3 hold in the “ex-post” sense (resp., “universal” sense), if Ai , Pi and Pi0 (b−i ) are interpreted to mean their respective values for a specific random seed of nature (resp., mechanism). In Corollary 3.3, the mechanism is normalized in the same sense as it is truthful. 4. THE GENERIC TRANSFORMATION FOR SINGLE-PARAMETER DOMAINS This section presents a generic procedure which takes any monotone allocation rule for a single-parameter domain and creates a randomized truthful-in-expectation mechanism which attains the same outcome as the original allocation rule with high probability. The resulting mechanism uses the allocation rule as a “black box,” calls it only once, and allocates according to the this call. Henceforth, we will refer to this procedure as the generic transformation. Our main result — the existence of the generic transformation with the desired properties — can be stated informally as follows. T HEOREM 4.1 (I NFORMAL ). Consider an arbitrary single-parameter domain with n agents. Let A be a monotone allocation rule for this domain. Then for each µ ∈ [0, 1] e P) e with the following properties: there exists a truthful mechanism M = (A, — M executes a single call to A(b̃) to compute the allocation, with a pre-processing step to compute the modified bid vector b̃, and a post-processing step to compute the payments. Both pre- and post-processing steps take O(n) time and do not depend on A. e and A(b) — For any bid vector b and any fixed random seed of nature allocations A(b) are identical with probability at least 1 − nµ. — M is universally ex-post individually rational. If all types are positive, then M is ex-post no-positive-transfers, and never pays any agent i more than bi · Ai (x) · ( µ1 − 1). Truthful Mechanisms with Implicit Payment Computation 0:11 Presenting the formal version of this result (Theorem 4.5) requires defining the generic transformation. We begin with an informal description thereof. As evidenced by Equation (4), the payment for agent i is a difference of two terms: the agent’s reported utility (i.e., the product of her bid and her allocation), minus the integral of the allocation assigned to every smaller bid value. We charge the agent for her reported utility, and we give her a random rebate whose expectation equals the required integral. When integrating a function over a finite interval, an unbiased estimator of the integral can be obtained by sampling a uniformly random point of that interval and evaluating the function at the sampled point. This idea was applied, in the context of mechanism design, by Archer et al. [2004]. Below, we show how to generalize the transformation to allow for integrals over unbounded intervals, as required by Equation (4). Using this transformation it is easy to transform any monotone allocation rule into a randomized mechanism that is truthful in expectation and only evaluates the allocation rule n + 1 times: once to determine the actual allocation, and once more per agent to obtain an unbiased estimate of that agent’s payment. Our main innovation is a transformation that uses the same random sampling trick, but only needs to evaluate the allocation rule once during the entire mechanism. (In other words, it does not require additional calls to the allocation rule to compute the payments.) Assume that a parameter µ ∈ (0, 1) is given. For every player, with probability 1 − µ, we leave their bid unchanged; with probability µ, we sample a smaller bid value. The allocation rule is invoked on these bids. An agent is always charged her reported value of the outcome, but if her bid was replaced with a smaller bid value then we refund her an amount equal to an unbiased estimator of the integral in Equation (4), scaled by 1/µ to counterbalance the fact that the refund is only being applied with probability µ. A naı̈ve application of this plan suffers from the following defect: the random resampling of bids modifies the expected allocation vector, so we need to obtain an unbiased estimator of the integral of the modified allocation rule. However, if we change our sampling procedure to obtain such an estimate, then this modifies the allocation rule once again, so we will still be estimating the wrong integral! What we need is a “fixed point” of this process of redefining the sampling procedure. Below, we give a definition of self-resampling procedures that satisfy the requisite fixed point property, and we give two simple constructions of self-resampling procedures. A self-resampling procedure transforms the bid bi of a given agent i bid into two correlated random values (xi , yi ), where xi is the modified bid presented to the original allocation rule, and yi is used (together with the allocation itself) in computing the payment for this agent. More specifically, yi is needed to correctly normalize the unbiased estimator of the integral in Equation (4) for the modified allocation rule, according to Theorem 4.2 below. For agents with positive types we define a simpler self-resampling procedure for which the unbiased estimator does not depend on yi , and therefore, strictly speaking, the procedure only needs to output xi (more details can be found in Section 4.5). However, we explicitly return the yi even for the positive types so as to be consistent with the general definitions and (perhaps more importantly) because we use it to define self-resampling procedures with general support (see Section 4.4). Thus, the formal description of our generic transformation consists of three parts: (1) a method for estimating integrals by evaluating the integrand at a randomly sampled point, (2) the definition and construction of self-resampling procedures, (3) the generic transformation that uses the previous two ingredients to convert any monotone allocation rule into a truthful-in-expectation randomized mechanism. We now specify the details of each of these three parts. 0:12 M. Babaioff et al. 4.1. Estimating integrals via random sampling Let I be a nonempty open interval in R (possibly with infinite endpoints) and let g be R a function defined on I. Let us describe a procedure for estimating the integral g(z) dz by evaluating g at a single randomly sampled point of I. The procedure is I well known; we describe it here for the purpose of giving a self-contained exposition of our algorithm. T HEOREM 4.2. Let F : I → [0, 1] be any strictly increasing function that is differentiable and satisfies inf z∈I F (z) = 0 and supz∈I F (z) = 1. If Y is a random variable with cumulative distribution function F , then Z g(Y ) . g(z) dz = E F ′ (Y ) I P ROOF. Since inf z∈I F (z) = 0 and supz∈I F (z) = 1, it follows that the random variable Y is supported on the entire interval I. Our assumption that F is differentiable implies that Y has a probability density function, namely F ′ (z). Thus, for any funcR ′ tion h, the expectation of h(Y ) is given by I h(z)F (z) dz. Applying this formula to the function h(z) = g(z)/F ′ (z) one obtains the theorem. 4.2. Self-resampling procedures The basic ingredient of our generic transformation is a procedure for taking a bid bi and a random seed wi , and producing two random numbers xi (bi ; wi ), yi (bi ; wi ). The mechanism will use {xi (bi ; wi )}i∈N for determining the allocation and additionally yi (bi ; wi ) for determining the payment it charges agent i. To prove that the mechanism is truthful in expectation we will require the following properties.6 Definition 4.3. Let I be a nonempty interval in R. A self-resampling procedure with support I and resampling probability µ ∈ (0, 1) is a randomized algorithm with input bi ∈ I, random seed wi , and output xi (bi ; wi ), yi (bi ; wi ) ∈ I, that satisfies the following properties: (1) For every fixed wi , xi (bi ; wi ) and yi (bi ; wi ) are non-decreasing functions of bi . (2) With probability 1 − µ, xi (bi ; wi ) = yi (bi ; wi ) = bi . Otherwise xi (bi ; wi ) ≤ yi (bi ; wi ) < bi . (3) The conditional distribution of xi (bi ; wi ), given that yi (bi ; wi ) = b′i < bi , is the same as the unconditional distribution of xi (b′i ; wi ). In other words, Pr[ xi (bi ; wi ) < ai \| yi (bi ; wi ) = b′i ] = Pr[ x(b′i ; wi ) < ai ], ∀ai ≤ b′i < bi . (4) Consider the two-variable function F (ai , bi ) = Pr[yi (bi ; wi ) < ai \| yi (bi ; wi ) < bi ], which we will call the distribution function of the self-resampling procedure. For each bi , the function F (·, bi ) must be differentiable and strictly increasing on the interval I ∩ (−∞, bi ). As it happens, it is easier to construct self-resampling procedures with support R+ , and one such construction that we call the canonical self-resampling procedure (Algorithm 1) forms the basis for our general construction. We defer the discussion of self-resampling procedures with general support until after we have described and analyzed the generic transformation. 6 To keep the notation consistent, we state Definition 4.3 for a given agent i. Strictly speaking, the subscript i is not necessary. Truthful Mechanisms with Implicit Payment Computation 0:13 Algorithm 1: The canonical self-resampling procedure. 1: Input: bid bi ∈ [0, ∞), parameter µ ∈ (0, 1). 2: Output: (xi , yi ) such that 0 ≤ xi ≤ yi ≤ bi . 3: with probability 1 − µ 4: xi ← bi , yi ← bi . 5: else 6: Pick b′i ∈ [0, bi ] uniformly at random. 7: xi ← Recursive(b′i ), yi ← b′i . 8: 9: 10: 11: 12: 13: function Recursive(bi ) with probability 1 − µ return bi . else Pick b′i ∈ [0, bi ] uniformly at random. return Recursive(b′i ). P ROPOSITION 4.4. Algorithm 1 is a self-resampling procedure with support R+ and resampling probability µ. The distribution function for this procedure is F (ai , bi ) = ai /bi . P ROOF. Properties 1 and 2 in Definition 4.3 are immediate from the description of the algorithm. The random seed wi for the algorithm can be defined as a countably infinite sequence of real numbers drawn independently and uniformly at random from [0, 1] interval. Then in order pick a random number in some range [0, r], the algorithm takes the next number in this sequence and multiplies it by r. Property 3 follows from the recursive nature of the sampling procedure: the event yi (bi ; wi ) = b′i < bi implies that the algorithm has followed the “else” branch on Line 5, and has chosen b′i in Line 6. Finally, the distribution function is F (ai , bi ) = ai /bi since conditional on the event yi (bi ; wi ) < bi , the distribution of yi (bi ; wi ) is uniform in the interval [0, bi ]. Property 4 follows trivially. 4.3. The generic transformation Suppose we are given a monotone allocation rule A and for each agent i ∈ N a self-resampling procedure that has resampling probability µ ∈ (0, 1), support Ti , and output values fi = (xi , yi ). Let Fi (ai , bi ) denote the distribution function of the self-resampling procedure for agent i, and let Fi′ (ai , bi ) denote the partial derivative ∂Fi (ai ,bi ) . Our generic transformation combines these ingredients into a randomized ∂ai mechanism M = AllocToMech(A, µ, f ) that works as follows: If A itself is randomized or if there is randomness arising from nature, then we allocate according to A(x; w, r) and we assume that the algorithm’s random seed w and the nature’s random seed r are independent of the random seeds wi used in the resampling step. We are now ready to present our main result: T HEOREM 4.5. Consider an arbitrary single-parameter domain. Let A be a monotone allocation rule. Suppose we are given an ensemble f of self-resampling procedures fi = (xi , yi ) for each agent i, each with resampling probability µ ∈ (0, 1). Then the e P) e = AllocToMech(A, µ, f ) has the following properties. mechanism M = (A, (a) M is truthful, universally ex-post individually rational, 0:14 M. Babaioff et al. Mechanism 2: Generic transformation M = AllocToMech(A, µ, f ) 1: Solicit bid vector b ∈ T . 2: Execute each agent’s self-resampling procedure using an independent random seed wi , to obtain two vectors of modified bids x = (x1 (b1 ; w1 ) , . . . , xn (bn ; wn )), y = (y1 (b1 ; w1 ) , . . . , yn (bn ; wn )). 3: 4: Allocate according to A(x). Each agent i is charged the amount bi · Ai (x) − Ri , where Ri is the rebate ( 1 i (x) · A if yi < bi , ′ Ri = µ Fi (yi ,bi ) 0 otherwise. (5) (b) For n agents and any bid vector b (and any fixed random seed of nature) allocations e and A(b) are identical with probability at least 1 − nµ. A(b) (c) If T = Rn+ (all types are positive), and each fi is the canonical self-resampling procedure, then mechanism M is ex-post no-positive-transfers, and never pays any agent i more than bi · Ai (x) · ( µ1 − 1). Several remarks are in order. — The mechanism never explicitly computes the payment for each agent i (Equation (4)) but rather implicitly creates the correct expected payments through its randomization of the bids. — The mechanism only invokes the original allocation rule A once. This property is very useful when it is impossible to invoke the allocation rule more than once, e.g. for multi-armed bandit allocations. — The mechanism M is randomized even if A is deterministic. It is truthful in expectation over the randomness used by the self-resampling procedures. — If A is ex-post monotone, then M will be ex-post truthful. To see this, fix nature’s random seed r and apply Theorem 4.5 to the allocation rule Ar induced by this r. — If agents’ types are positive then by part (b), the welfare of M is at least 1 − nµ times that of A. Further results on bounding the welfare loss are presented in Section 5. — By definition of the payment rule, the mechanism is universally ex-post normalized. We will not explicitly mention this property in the subsequent applications. Parameter µ controls the trade-off between the loss in welfare and the variance in payments, as quantified by the rebate size Ri . If µ is very small and the mechanism issues rebate(s), then its revenue may be very low and possibly negative. However, this risk may be mitigated if the auction maker runs many independent auctions, as may be the case in practice. Further, the follow-up paper [Wilkens and Sivan 2012] proves that our welfare vs. variance trade-off is optimal. P ROOF OF T HEOREM 4.5. We start with some notation. Aei (b−i , bi ; q) denotes the allocation for agent i given the bid vector b = (b−i , bi ) and the combined random seed q = (w1 , . . . , wn , w, r). When we write Aei (b−i , u) without indicating the dependence on the q, we are referring to the unconditional expectation of Aei (b−i , u; q) over q. Truthful Mechanisms with Implicit Payment Computation 0:15 To prove that M is truthful, we need to prove two things: that the randomized alloe satisfies cation rule Ae is monotone, and that the expected payment rule P R ei (b) = bi Aei (b−i , bi ) − bi Aei (b−i , u) du. (6) P −∞ The monotonicity of randomized allocation rule Ae follows from the monotonicity of A and the monotonicity property 1 in the definition of a self-resampling procedure. To ei satisfies Equation (6), we begin by recalling that the payment charged to prove that P player i is bi Ai (x) − Ri , where the rebate Ri is defined by Equation (5). The expectation of bi Ai (x) is simply bi Aei (b−i , bi ), so to conclude the proof of truthfulness we must show that Rb E[Ri ] = i Aei (b−i , u) du. (7) −∞ Our proof of Equation (7) begins by observing that the conditional distribution of xi , given that yi = u < bi , is the same as the unconditional distribution of xi (u; wi ), by Property 3 of a self-resampling procedure. Combining this with the fact that the random seed wi is independent of {wj : j 6= i}, we find that the conditional distribution of the tuple x = (x−i , xi ), given that yi = u, is the same as the unconditional distribution of the vector x̂ of modified bids that M would input into the allocation rule A if the bid vector were (b−i , u) instead of (b−i , bi ). Taking expectations, this implies that for all e −i , u). u < bi , we have E[Ai (x) \| yi = u] = E[Ai (x̂)] = A(b Now apply Theorem 4.2 with the function g(u) = Aei (b−i , u). Recalling that Fi (·, bi ) is the cumulative distribution function of yi given that yi < bi , we apply the theorem to obtain " # Z bi ei (b−i , yi ) A Ai (x) e Ai (b−i , u) du = E yi < bi = E yi < bi Fi′ (yi , bi ) Fi′ (yi , bi ) −∞ = µ · E[Ri \| yi < bi ], (8) where the second equation follows from the equation derived at the end of the preceding paragraph, averaging over all u < bi . Observing that Ri = 0 unless yi < bi , an event that has probability µ, we see that E[Ri ] = µ · E[Ri \| yi < bi ]. Combined with Equation (8), this establishes Equation (7) and completes the proof that M is truthful. Mechanism M is universally ex-post individually rational because agent i is never charged an amount greater than bi Aei (b; q). Part (b) follows from the union bound: the probability that xi = bi for all i is at least 1 − nµ. For part (c), note that by Proposition 4.4, the canonical self-resampling procedure has distribution function F (ai , bi ) = ai /bi , hence Fi′ (yi , bi ) = 1/bi , for all i, yi , bi . The rebate Ri is equal either i (x) to 0 or to µ1 · F A = bi · Ai (x) · µ1 . We also charge bi · Ai (x) to agent i. The claimed ′ i (yi ,bi ) upper bound on the amount paid to agent i follows by combining these two terms. 4.4. Self-resampling procedures with general support To construct a self-resampling procedure with support in an arbitrary interval I, we can use the following technique. Suppose h : (0, 1] × I → I is a two-variable function such that the partial derivatives ∂h(zi , bi )/∂zi and ∂h(zi , bi )/∂bi are well-defined and strictly positive at every point (zi , bi ) ∈ (0, 1] × I. Suppose furthermore that h(1, bi ) = bi and inf zi ∈(0,1] {h(zi , bi )} = inf(I) for all bi ∈ I. Then we define the h-canonical selfresampling procedure (xhi , yih ) with support I, by specifying that h xi (bi ; wi ) = h(xi (1; wi ), bi ) (9) yih (bi ; wi ) = h(yi (1; wi ), bi ), 0:16 M. Babaioff et al. where (xi , yi ) is the canonical self-resampling procedure as defined in Algorithm 1. P ROPOSITION 4.6. (xhi , yih ) as defined in Equation (9) is a self-resampling procedure with support I and resampling probability µ. The distribution function for (xhi , yih ) is the unique two-variable function F (ai , bi ) such that h(F (ai , bi ), bi ) = ai for all ai , bi ∈ I, ai < bi . (10) P ROOF. Property 1 in Definition 4.3 holds because of the monotonicity of h, Property 2 holds because h(1, bi ) = bi for all bi , and Property 3 holds because the function h is deterministic and monotone. Let Fh (ai , bi ) and F0 (ai , bi ) be the distribution functions for the h-canonical and canonical self-resampling procedures, respectively. Recall that F0 (ai , bi ) = ai /bi by Proposition 4.4. Note that F (ai , bi ) in Equation (10) is unique (and hence well-defined) by the strict monotonicity of h. The claim that Fh (ai , bi ) = F (ai , bi ) easily follows from in Equation (9). By definition of h we have h(yi (1, wi ), bi ) < bi ⇐⇒ yi (1, wi ) < 1. Therefore, letting yi = yi (1, wi ) we have Fh (ai , bi ) , Pr[h(yi , bi ) < ai \| h(yi , bi ) < bi ] = Pr[yi < F (ai , bi ) \| yi < 1] = F0 ( F (ai , bi ) , 1) = F (ai , bi ). Our assumption that h is differentiable and strictly increasing in its first argument now implies that the same property holds for F , which verifies Property 4. 4.5. A simplified generic transformation for positive types We focus on the important special case of positive types, and present Mechanism 3, a simplified version of the generic transformation (Mechanism 2), for this case. Mechanism 3: A simplified generic transformation for positive types. 1: Parameter: resampling probability µ ∈ (0, 1). 2: 3: 4: 5: 6: 7: 8: Collect bid vector b ∈ (0, ∞)n . Independently for each agent i ∈ [n]: Sample: γi uniformly at random from [0, 1] 1/(1−µ) . Set χi = 1 with probability 1 − µ and otherwise χi = γi Construct the vector of modified bids x = (x1 , . . . , xn ), where xi = χi bi . Allocate according to A(x). ( 1 if χi = 1, For each agent i, assign payment bi · Ai (x) · . 1 1 − µ if χi < 1 We prove that Mechanism 3 is equivalent to the generic transformation (Mechanism 2) with a canonical self-resampling procedure (Algorithm 1). P ROPOSITION 4.7. The allocation and payments in Mechanism 3 coincide with those in Mechanism 2 with a canonical self-resampling procedure. Truthful Mechanisms with Implicit Payment Computation 0:17 To prove Proposition 4.7, we provide a non-recursive version of the canonical selfresampling procedure (Algorithm 1), which we call O NE S HOT . We argue that the output of Mechanism 3 is identical to the output of the mechanism obtained by plugging O NE S HOT into Mechanism 2.7 O NE S HOT is also essential for the analysis in Section 5. Algorithm 4: O NE S HOT : a non-recursive version of Algorithm 1. 1: Input: bid bi ∈ [0, ∞), parameter µ ∈ (0, 1). 2: Output: (xi , yi ) such that 0 ≤ xi ≤ yi ≤ bi . 3: with probability 1 − µ 4: xi ← bi , yi ← bi . 5: else 6: Pick γ1 , γ2 ∈ [0, 1] indep., uniformly at random. 1/(1−µ) 1/(1−µ) 1/µ 7: xi ← bi · γ1 , yi ← bi · max{γ1 , γ2 }. P ROPOSITION 4.8. Algorithm 1 and O NE S HOT generate the same output distribution: for any bid bi ∈ [0, ∞), the joint distribution of the pair (xi , yi ) = (xi (bi ; wi ), yi (bi ; wi )) is the same for both procedures. (Here wi denotes the random seed for each agent i.) The proof of Proposition 4.8 can be found in the Appendix. P ROOF OF P ROPOSITION 4.7. By Proposition 4.8, it suffices to compare Mechanism 3 to Mechanism 2 with self-resampling procedure O NE S HOT. To show that the two mechanisms are equivalent, we must show that they yield the same distribution over allocations and the same payments. First we argue about the allocations. In both mechanisms, each bidder’s bid bi is independently transformed into a random xi , and then the allocation rule A is applied to the vector x = (x1 , . . . , xn ). Furthermore, the conditional distribution of xi given bi is the same in both cases: xi = bi with probability 1 − µ, and otherwise xi = bi · γ 1/(1−µ) where γ is uniformly distributed in [0, 1]. Hence, the two mechanisms yield the same distribution over allocations. To see that the payment rules are the same, consider the distribution function of O NE S HOT, as defined in Definition 4.3: F (ai , bi ) = Pr[yi (bi ; wi ) < ai \| yi (bi ; wi ) < bi ]. By Proposition 4.8, Fi (ai , bi ) is also the distribution function for Algorithm 1. By Proposition 4.4 we have F (ai , bi ) = ai /bi , and consequently Fi′ (ai , bi ) , ∂Fi (ai , bi ) 1 = . ∂ai bi In particular, neither allocation nor payments in this mechanism depend on the yi ’s. Suppressing the yi ’s from mechanism M and plugging in Fi′ (yi , bi ) = b1i , we obtain Mechanism 3. This completes the proof of Proposition 4.7. 7 This observation is due to [Shnayder et al. 2012]. 0:18 M. Babaioff et al. 5. IMPROVED BOUNDS ON WELFARE We present improved bounds on the welfare obtained by our generic transformation. We consider two interesting special cases when the agents’ private types are, respectively, always positive and always negative. In the second case, agents are contractors who incur costs and get paid by the mechanism; one such example is a shortest paths mechanism considered in Section 6. We consider the approximation that is achieved by the mechanism as a function of the approximation of the original allocation rule. Recall that our generic transformation creates a mechanism with an allocation that is identical to the original allocation with probability at least 1 − nµ. For positive types this immediately implies a bound on the approximation which degrades with n, the number of agents. (For negative types such bound does not immediately follow since the cost in the low probability event might be prohibitively high.) For both settings, we present a similar bound that does not degrade with n. 5.1. Positive private types Assume that the agents’ types are always positive, more specifically that the type space is T = (0, ∞)n . Recall that for P agents’ types t ∈ T the social welfare of an outcome o is defined to SW(o, t) = i∈N ti ai (o). The optimal social welfare is OPT(t) = maxo∈O SW(o, t), where O is the set of all feasible outcomes. (A mechanism with) an allocation rule A is α-approximate if it holds that α · E[SW(A(t), t)] ≥ OPT(t) for every t. (11) n T HEOREM 5.1. Consider the setting in Theorem 4.5(c), so that T = (0, ∞) and each fi is the canonical self-resampling procedure. If allocation rule A is α-approximate, then µ mechanism AllocToMech(A, µ, f ) is α/(1 − 2−µ )-approximate. P ROOF. Fix a bid vector b, and let o∗ be the corresponding optimal allocation. Recall that our mechanism outputs allocation A(x), where x is the vector of randomly modified bids. As the original allocation rule A is α-approximate, by Equation (11) it holds that α · SW(A(x), x) ≥ OPT(x). We will show that µ bi for each agent i. (12) E[xi ] = 1 − 2−µ Thus when we evaluate o∗ with respect to bids x we get: P α · SW(A(x), x) ≥ OPT(x) ≥ SW(o∗ , x) = i∈N xi ai (o∗ ) P ∗ α · E[SW(A(x), x)] = E i∈N xi ai (o ) X µ µ ∗ = bi ai (o ) = 1 − OPT(b). 1− 2−µ 2−µ i∈N It remains to prove Equation (12). Let us use O NE S HOT to describe the canonical self-resampling procedure. Recall that O NE S HOT generates xi = xi (bi ; wi ) by setting xi = bi with probability 1 − µ, and otherwise sampling γ1 uniformly at random in [0, 1] 1/(1−µ) and outputting xi = bi · γ1 . Hence R1 1/(1−µ) 1 E[xi \| xi < bi ] = 0 bi · γ1 dγ1 = bi · 1+ 1 1 = bi · 1 − 2−µ 1−µ µ . E[xi ] = (1 − µ) · bi + µ · E[xi \| xi < bi ] = bi · 1 − 2−µ Truthful Mechanisms with Implicit Payment Computation 0:19 For arbitrary self-resampling procedures fi with support R+ , Equation (12) can be α replaced by E[xi ] ≥ (1 − µ) bi , which gives a slightly weaker result, namely an 1−µ approximation to the social welfare. 5.2. Negative private types Now assume that the agents’ types are always negative, more specifically that T = (−∞, 0)n . For negative types approximation is defined with respect to the social cost, which is the negation of the social welfare. An algorithm is α-approximate if for every input it outputs an outcome with cost at most α times the optimal cost. We present an approximation bound for an h-canonical self-resampling procedure, for a suitably chosen h. T HEOREM 5.2. Consider the setting in Theorem 4.5. Assume that T = (−∞, 0)n and √ that each fi is the h-canonical self-resampling procedure, where h(zi , bi ) = bi / zi . Suppose µ ∈ (0,12 ). If allocation rule A is α-approximate, then mechanism µ AllocToMech(A, µ, f ) is α 1 + 1−2µ -approximate. The proof of this theorem is almost identical to that of Theorem 5.1, and thus is omitted. The main modification is that Equation (12) is replaced by the following lemma: L EMMA 5.3. In the setting of Theorem 5.2, letting xh be the vector of modified types, it holds that µ E[xhi ] = bi 1 + 1−2µ for all i. P ROOF. Recall that xh is defined by Equation (9). As in the proof of Equation (12), we will use O NE S HOT to describe the canonical self-resampling procedure. It follows that R1 R1 − 1 1 , E[xhi \| xi < bi ] = 0 r bi 1 dγ1 = 0 bi · γ1 2(1−µ) dγ1 = bi · 1− 1 1 = bi · 1 + 1−2µ (1−µ) γ1 E[xhi ] = (1 − µ) · bi + µ · E[xhi \| xhi < bi ] = bi · 1 + 2(1−µ) µ 1−2µ . 6. APPLICATIONS TO OFFLINE MECHANISM DESIGN The VCG mechanism for shortest paths. The seminal paper Nisan and Ronen [2001] has presented the following question: is there a computational overhead in computing payments that will induce agents to be truthful, compared to the computation burden of computing the allocation. One of their examples is the VCG mechanism for the shortest path mechanism design problem, where a naive computation of VCG payments requires additional computation of n shortest path instances. Yet, an explicit payment computation is not the real goal, it is just a means to an end. The real goal is inducing the right incentives. Our procedure shows that without any overhead in computation, if we move to a randomized allocation rule and settle for truthfulness in expectation (and a small loss in performance) one can induce the right incentives. The shortest path mechanism design problem is the following. We are given a graph G = (V, E) and a pair of source-target nodes (vs , vt ). Each agent e controls an edge e ∈ E and has a cost ce > 0 if picked (thus ve = −ce < 0 and Te = (−∞, 0) for every e). That cost is private information, known only to agent e. The mechanism designer’s goal is to pick a path P from node vs to node vt in the graph with minimal total cost, 0:20 M. Babaioff et al. P that is e∈P ce is minimal. Assume that there is no edge that forms a cut between vs and vt . The VCG mechanism is an cost-optimal and truthful mechanism for this problem. It computes a shortest path P with respect to the reported costs and pays to an agent e the difference between the cost of the shortest path that does not contains e and the total cost shortest path excluding the cost of e. A naive implementation of the VCG mechanism requires computing \|P \| + 1 shortest path instances (where \|P \| denotes the number of edges in path P ). VCG is deterministic, truthful and cost-optimal. Let EFF an the cost-optimal allocation rule for the shortest path problem. We can use our general procedure to derive the following result (its proof follows directly from Theorem 4.5 and Theorem 5.2). T HEOREM 6.1. Fix any µ ∈ (0, 21 ). For each agent i, let fi be the h-canonical self√ resampling procedure, where h(zi , bi ) = bi / zi . Let M = AllocToMech(EFF, µ, {fi }) be the mechanism created by applying AllocToMech() to EFF. Then M has the following properties: • It is truthful and universally individually rational. • It only computes one shortest paths instance. µ • It outputs a path with expected length at most 1 + 1−2µ times the length of the shortest path. Recall that parameter µ controls the trade-off between approximation ratio and the rebate size Ri , which for a given random seed is proportional to µ1 . Communication overhead of payment computation. Babaioff et al. [2013] show that there exists a monotone deterministic allocation rule for which the communication required for computing the allocation is factor Ω(n) less than the communication required to computing prices. This implies that inducing the correct incentives deterministically has a large overhead in communication. Assume that instead of requiring explicit computation of payments we are satisfied with inducing the correct incentives using a randomized mechanism. In such case our reduction shows that the deterministic lower bound cannot be extended to randomized mechanisms, if we allow a small error in the allocation. More concretely, consider a single parameter domain with types that are positive, Ti = (0, ∞) (as in [Babaioff et al. 2013]). For all i, use the canonical self-resampling procedure. Consider any monotone allocation rule A. We can apply Theorem 4.5 to obtain a randomized mechanism that is truthful and only executes that allocation rule A once (thus has no communication overhead at all) and has exactly the same allocation with probability at least (1 − µ)n . For any ǫ > 0 we can find µ > 0 such that the error probability is less than ǫ. 7. MULTI-ARMED BANDIT MECHANISMS In this section we apply the main result to multi-armed bandit (MAB) mechanisms: single-parameter mechanisms in which the allocation rule is (essentially) an MAB algorithm parameterized by the bids. As in any single-parameter mechanism, agents submit their bids, then the allocation rule is run, and then the payments are assigned. This application showcases the full power of the main result, since in the MAB setting the allocation rule is only run once, and (in general) cannot be simulated as a computational routine without actually implementing the allocation. Focusing on the stochastic setting, we design truthful MAB mechanisms with the same regret guarantees as the best MAB algorithms such as UCB1 [Auer et al. 2002a]. First, we prove that allocation rules derived from UCB1 and similar MAB algorithms Truthful Mechanisms with Implicit Payment Computation 0:21 are in fact monotone, and hence give rise to truthful MAB mechanisms. Second, we provide a new allocation rule with the same regret guarantees that is ex-post monotone, and hence gives rise to an ex-post truthful MAB mechanism. Third, we use this new allocation rule to obtain an unconditional separation between the power of randomized and deterministic ex-post truthful MAB mechanisms. 7.1. Preliminaries: MAB mechanisms An MAB mechanism [Babaioff et al. 2014; Devanur and Kakade 2009] operates as follows. There are n agents. Each agent i has a private value vi and submits a bid bi . We assume that bi , vi ∈ [0, bmax ], where bmax is known a priori. The allocation consists of T rounds, where T is the time horizon. In each round t the allocation rule chooses one of the agents, call it i = i(t), and observes a click reward π(t) ∈ [0, 1] for this choice; the chosen agent i receives vi π(t) units of utility. Payments are assigned after the last round of the allocation. Note that the social welfare of the mechanism is equal to the PT total value-adjusted click reward: t=1 vi(t) π(t). The special case of 0-1 click rewards corresponds to the scenario in which agents are advertisers in a pay-per-click auction, and choosing agent i in a given round t means showing this agent’s ad. Then the click reward π(t) is the click bit: 1 if the ad has been clicked, and 0 otherwise. Following the web advertising terminology, we will say that in each round, an impression is allocated to one of the agents. Formally, an MAB allocation rule A is an online algorithm parameterized by n, T, bmax and the bids b. In each round it allocates the impression and observes the click reward. Absent truthfulness constraints, the objective is to maximize the reported P welfare: Tt=1 bi(t) π(t). This formulation generalizes MAB algorithms: the latter are precisely MAB allocation rules with all bids set to 1. Given an MAB algorithm Â, there is a natural way to transform it into an MAB allocation rule A. Namely, A runs algorithm Â with modified click rewards: if agent i is chosen in round t then the click reward reported to Â is π̂(t) = (bi /bmax ) π(t). We will say that algorithm Â induces allocation rule A. From now on we will identify an MAB algorithm with the induced allocation rule, e.g. allocation rule UCB1 is induced by algorithm UCB1 [Auer et al. 2002a]. We will focus on the stochastic MAB setting: in all rounds t in which an agent i is chosen, the click reward π(t) is an independent random sample from some fixed distribution on [0, 1] with expectation µi .8 Following the web advertisement terminology, we will call µi the click-through rate (CTR) of agent i. The CTRs are fixed, but no further information about them (such as priors) is revealed to the mechanism. Regret. The performance of an MAB allocation rule is quantified in terms of regret: PT R(T ; b; µ) , T maxi [ bi µi ] − E[ t=1 bi(t) µi(t) ], the difference in expected click rewards between the algorithm and the benchmark: the best agent in hindsight, knowing the µi ’s. We focus on R(T ) , max R(T ; b; µ), where the maximum is taken over all CTR vectors µ and all bid vectors b such that bi ≤ 1 for all i. 9 Regret guarantees from the vast literature on MAB algorithms easily translate to MAB allocation rules. In particular, allocation rule UCB1 has regret √ R(T ) = O( nT log T ) [Auer et al. 2002a], which is nearly matching the information8 The exact shape of this distribution is not essential. E.g. in the advertising example π(t) ∈ {0, 1}. define R(T ) with bmax = 1 merely to simplify the notation. All regret bounds (scaled up by a factor of bmax ) hold for an arbitrary bmax . 9 We 0:22 M. Babaioff et al. √ theoretically optimal regret bound Θ( nT ) [Auer et al. 2002b; Audibert and Bubeck 2010]. The stochastic MAB setting tends to be easier if the best agent is much better than the second-best one. Let us sort the agents so that b1 µ1 ≥ b2 µ2 ≥ . . . ≥ bn µn . The gap δ of the problem instance is defined as (b1 µ1 − b2 µ2 )/bmax . The δ-gap regret Rδ (T ) is defined as the worst-case regret over all problem instances with gap δ. Allocation rule UCB1 achieves Rδ (T ) = O( nδ log T ) [Auer et al. 2002a]; there is a √ lower bound Rδ (T ) = Ω(min( nδ log T, nT )) [Lai and Robbins 1985; Auer et al. 2002b; Kleinberg et al. 2008a]. Click realizations. A click realization is a n × T table ρ in which the (i, t) entry ρi (t) is the click reward (e.g., the click bit) that agent i receives if it is played in round t. Note that in order to fully define the behavior of any algorithm on all bid vectors one may need to specify all entries in the table, whereas only a subset thereof is revealed in any given run. We view ρ as a realization of nature’s random seed. Thus, we can now define ex-post truthfulness and other ex-post properties: informally, ex-post property is a property that holds for every given click realization. For each agent i, round t, bid vector b and click realization ρ, let Ati (b; ρ) denote the probability that MAB allocation rule A allocates the impression at round t to agent i. (If Â is deterministic, the probability Âti (ρ) is trivial: either 0 or 1.) For MAB algorithm Â, define Âti (ρ) similarly. 7.2. Truthfulness and monotonicity Theorem 4.5(c) reduces the problem of designing truthful MAB mechanisms to that of designing monotone MAB allocations. Let us state this reduction explicitly: T HEOREM 7.1. Consider the stochastic MAB mechanism design problem. Let A be a stochastically monotone (resp., ex-post monotone) MAB allocation rule. Applying the transformation in Theorem 4.5(c)10 to A with parameter µ, we obtain a mechanism M such that: (a) M is stochastically truthful (resp., ex-post truthful), ex-post no-positivetransfers, and universally ex-post individually rational. (b) for each click realization, the difference in expected welfare between A and M is at most µnT bmax . Note that the theorem provides two distinct types of guarantees: game-theoretic guarantees in part (a), and performance guarantees in part (b). We show that a very general class of deterministic MAB algorithms induces monotone MAB allocation rules (to which Theorem 7.1 can be applied). Definition 7.2. In a given run of an MAB algorithm, the round-t statistics is a pair of vectors (π, ν), where the i-th component of π (resp., ν) is equal to the total payoff (resp., the number of impressions) of agent i in rounds 1 to t − 1, for each agent i. Vectors π and ν are called p-stats vector and i-stats vector, respectively. Definition 7.3. A deterministic MAB algorithm Â is called well-formed if for each round t and agent i, letting (π, ν) be the round-t statistics, the following properties hold: — [Âti (ρ) is determined by (π, ν)] there is a function χi (π; ν) that depends only on the round-t statistics such that Âti (ρ) = χi (π; ν) for any click realization ρ and all t. — [χ-monotonicity] χi (π; ν) is non-decreasing in πi for any fixed (π−i , ν). 10 Theorem 4.5(c) is stated for the type space T = (0, ∞)n , but it trivially extends to the case T = (0, bmax )n . Truthful Mechanisms with Implicit Payment Computation 0:23 — [χ-IIA] for each round t, any three distinct agents {i, j, l} and any fixed (π−i , ν−i ), changing (πi , νi ) cannot transfer an impression from j to l. The χ-IIA property above is reminiscent of Independence of Irrelevant Alternatives (IIA) property in the Social Choice literature (hence the name). A similar but technically different property is essential in the analysis of deterministic MAB allocation rules in [Babaioff et al. 2014]. Remark 7.4. For a concrete example of a well-formed MAB algorithm, consider (a version of) UCB1.11 The algorithm is very simple: in each round t, it chooses agent p min arg max πi (t)/νi (t) + 8 log(T )/νi (t) . i L EMMA 7.5. In the stochastic MAB mechanism design problem, let A be a MAB allocation rule induced by a well-formed MAB algorithm. Then A is stochastically monotone. P ROOF. We will use an alternative way to define a realization of random click rewards: a stack-realization is a n × T table in which the (i, t) entry is the click bit that agent i receives the t-th time she is played. Clearly a stack-realization and a bid vector uniquely determine the behavior of A. We will show that: A is monotone for each stack-realization. (13) Then A is monotone in expectation over any distribution over stack-realizations, and in particular it is monotone in expectation over the random clicks in the stochastic MAB setting, so the Lemma follows. Let us prove Claim (13). Throughout the proof, fix stack-realization σ, agent i, and bid vector b−i . Consider two bids bi < b+ i . The claim asserts that agent i receives at least as many clicks with bid b+ than with bid bi . i Let us introduce some notation (letting bi be the bid of agent i). Let A(bi , t) be the agent selected by the allocation rule in round t. For each agent j, let νj (bi , t) and πj (bi , t) be, respectively, the total number of impressions and the total click reward of agent j in the first t rounds. Let π̂j (bi , t) = (bj /bmax ) πj (bi , t) be the corresponding total modified click reward. Let ν(bi , t) (resp., π(bi , t) and π̂(bi , t)) be the n-dimensional vector whose j-th component is νj (bi , t) (resp., πj (bi , t) and π̂j (bi , t)) for each agent j. Note that (π̂(bi , t), ν(bi , t)) is the round-t statistics for the MAB algorithm that A is induced by. For each agent j, νj (bi , t) uniquely determines πj (bi , t): Pνj σ(j, s) where νj = νj (bi , t). (14) πj (bi , t) = s=1 Let us overview the forthcoming technical argument. We will show by induction on t that νi (bi , t) ≤ νi (b+ i , t) for all t. For the induction step we only need to worry about the case when the claim holds for a given t with equality. In this case we show that ν−i (bi , t) = ν−i (b+ i , t). This is trivial for n = 2 agents; the general case requires a rather delicate argument that uses the χ-IIA property in Definition 7.3.12 Now let us carry out the proofs in detail. First, denote ν∗ (bi , t) , t − νi (bi , t), and let us show that for any two rounds t, s it holds that + ν∗ (bi , t) = ν∗ (b+ i , s) ⇒ ν−i (bi , t) = ν−i (bi , s). (15) 11 To ensure the χ-IIA property, we use a slightly modified version of UCB1: log T is used instead of log t, and min is used to break ties (instead of an arbitrary rule). This change does not affect regret guarantees. We will denote this version as UCB1 without further notice. 12 Also, we will use the fact that the probabilities χ (π̂, ν) in Definition 7.3 do not depend on the round (given j j and (π̂, ν)). This is the only place in any of the proofs where we invoke this fact. 0:24 M. Babaioff et al. Let us use induction on ν∗ (bi , t). For ν∗ (bi , t) = 0 the statement is trivial. For the induction step, suppose Equation (15) holds whenever ν∗ (bi , t) = ν∗ , and let us sup′ ′ pose ν∗ (bi , t) = ν∗ (b+ i , s) = ν∗ + 1. Let t and s be the latest rounds such that + ′ ′ ′ ν∗ (bi , t ) = ν∗ (bi , s ) = ν∗ . By the induction hypothesis, ν−i (bi , t′ ) = ν−i (b+ i , s ). It re+ ′ ′ mains to prove that A(bi , t + 1) = A(bi , s + 1), i.e. that the allocation rule’s selections in round t′ + 1 given bids (b−i , bi ), and in round s′ + 1 given bids (b−i , b+ i ), are the same.13 By Definition 7.3 these selections are uniquely determined (given the stackrealization) by the bids and the impression counts ν. By the choice of t′ and s′ , neither of the two selections is i, so by the χ-IIA condition in Definition 7.3 the selections are uniquely determined by b−i and ν−i , and hence are the same. This proves Equation (15). Now, to prove Claim (13) it suffices to show that for all t νi (bi , t) ≤ νi (b+ i , t). (16) Let us use induction on t. The claim is trivial for t = 1, since the impression of agent i in round 1 does not depend on (b; σ). For the induction step, assume that the assertion Equation (16) holds for some t, and let us prove it for t + 1. Note that (using the notation from Definition 7.3) νi (bi , t + 1) = νi (bi , t) + χi (π̂(bi , t); ν(bi , t)). Now, νi (bi , t) ≤ νi (b+ i , t) by induction hypothesis. If the inequality is strict then Equation (16) trivially holds for t+1. Now suppose νi (bi , t) = νi (b+ i , t). Then by Equation (15) we have ν(bi , t) = ν(b+ , t). Moreover, by Equation (14) we have π(bi , t) = π(b+ i i , t) and + + therefore π̂−i (bi , t) = π̂−i (bi , t) and π̂i (bi , t) < π̂i (bi , t). Thus, by the χ-monotonicity property in Definition 7.3 we have + χi (π̂(bi , t); ν(bi , t)) ≤ χi (π̂(b+ i , t); ν(bi , t)). This concludes the proof of Equation (16), and that Claim (13). 7.3. Truthfulness and regret In this subsection we focus on the stochastic MAB setting, and consider the trade-off between regret and various notions of truthfulness. Ideally, one would like an MAB mechanism to be truthful in the strongest possible sense (universally ex-post), and have the same regret bounds as optimal MAB algorithms. Let us start with some background. In Babaioff, Sharma and Slivkins [Babaioff et al. 2014] it was proved that any deterministic mechanism that is ex-post truthful and ex-post normalized (under very mild restrictions), and any distribution over such deterministic mechanisms, incurs much higher regret than an optimal MAB algorithm such as UCB1. Namely, the lower bound in [Babaioff √ et al. 2014] states that R(T ) = Ω(n1/3 T 2/3 ), whereas UCB1 has regret R(T ) = O( nT log T ). 14 For δ-gap instances the difference is even more pronounced: the analysis in [Babaioff et al. 2014] provides a polynomial lower bound of Rδ (T ) = Ω(δ T λ ) for some λ > 0, whereas UCB1 achieves logarithmic regret Rδ (T ) = O( nδ log T ). Our first result is that we can use the machinery from Section 7.2 to match the regret of UCB1 for truthful mechanisms. We apply Theorem 7.1 (with µ = T1 ) and Lemma 7.5 to UCB1 to obtain the following corollary: 13 Then ν (b , t) = ν (b+ , s) because in all rounds from t′ + 2 to t (resp., from s′ + 2 to s) agent i is played. −i i −i i 14 Following the literature on regret minimization, we are mainly interested in the asymptotic behavior of R(T ) as a function of T when n is fixed. Truthful Mechanisms with Implicit Payment Computation 0:25 C OROLLARY 7.6. In the stochastic MAB mechanism design problem, there exists a mechanism M such that (a) M is stochastically truthful, ex-post no-positive-transfers, universally ex-post individually rational. √ (b) M has regret R(T ) = O( nT log T ) and δ-gap regret Rδ (T ) = O( nδ log T ). Remark 7.7. The √ regret and δ-gap regret in the above theorem are within small factors (resp., O( log T ) and O(1)) of the best possible for any MAB allocation rule. Remark 7.8. [Babaioff et al. 2014] provides a weaker result which transforms any monotone MAB algorithm such as UCB1 into a truthful and normalized MAB mechanism with matching regret bounds. The guarantees in [Babaioff et al. 2014] are weaker for the following reasons. First, it only applies to 0-1 click rewards, whereas our setting allows for arbitrary click rewards in [0, 1]. Second, the individual rationality guarantee in [Babaioff et al. 2014] is much weaker: an agent may be charged more than her bid (which never happens in our mechanism), and the charge may be huge, as high as bi × (4n)T ; thus, a risk-averse agent may be reluctant to participate. Third, the nopositive-transfers guarantee is weaker: for some realizations of the click rewards the expected payment may be negative. Finally, the payment rule in [Babaioff et al. 2014] requires (as stated) a prohibitively expensive computation. The truthfulness in Corollary 7.6 is only in expectation over the random click rewards. Thus, after seeing a specific realization of the rewards an agent might regret having been truthful. Accordingly, we would like a stronger property: ex-post truthfulness, i.e. truthfulness for every given realization of the rewards. The main result of this section is an ex-post truthful MAB mechanism with optimal regret bounds. Unlike Corollary 7.6, this result requires designing a new MAB allocation rule.15 This allocation rule and its analysis are the main technical contributions. T HEOREM 7.9. In the stochastic MAB mechanism design problem, there is a mechanism M such that (a) M is ex-post truthful, ex-post no-positive-transfers, and universally ex-post individually rational. √ (b) M has regret R(T ) = O( nT log T ) and δ-gap regret Rδ (T ) = O( nδ log T ). The theorem follows from Theorem 7.1 (with µ = T1 ) if there exists an MAB allocation rule that is ex-post monotone and has the claimed regret bounds. Below we provide such allocation rule, called NewCB. L EMMA 7.10. NewCB is ex-post monotone and satisfies the regret bounds in Theorem 7.9(b). Remark 7.11. NewCB is deterministic. While not essential for Theorem 7.9, this fact confirms the intuition from [Babaioff et al. 2014] that the main obstacle for deterministic ex-post truthful MAB mechanisms is insufficient observable information to compute payments rather than ex-post monotonicity of an allocation rule. NewCB maintains a set of active agents; initially all agents are active. For each round t, there is a designated agent i = 1 + (t mod n). If this agent is active, then it is allo15 In particular, the allocation rule induced by UCB1 is not ex-post monotone and thus cannot be used to achieve ex-post truthfulness using the results of Section 7.2. To see that, consider a simple setting with two agents and two rounds, and a click realization in which both agents are not clicked at the first round, but are clicked at the second. With this click realization, an agent might be better off decreasing his bid in order to lose (i.e., not be selected in) the first round, and then win (i.e., be selected in) the second round. 0:26 M. Babaioff et al. cated. Else, an active agent is chosen (according to some fixed ordering on the agents) and allocated. For each agent i, lower and upper confidence bounds (Li , Ui ) on the product bi µi are maintained (recall that µi is the CTR of agent i). After each round, each agent is de-activated if its upper confidence bound is smaller than someone else’s lower confidence bound. The pseudocode is in Algorithm 5. Algorithm 5: NewCB: ex-post monotone MAB allocation rule. 1: Given: n = #agents, T = #rounds, upper bound bmax . 2: Solicit a bid vector b from the agents; b ← b/bmax . 3: Initialize: set of active agents Sact = {all agents}. 4: for all agent i do 5: ci ← 0; ni ← 0 {total click reward and #impressions} 6: {the totals are only over “designated” rounds} 7: Ui ← bi ; Li ← 0 {Upper and Lower Confidence Bounds} 8: {Main Loop} 9: for rounds t = 1, 2, . . . , T do 10: i ← 1 + (t mod n). {The “designated” agent} 11: if i ∈ Sact then 12: Allocate agent i. 13: ni ← ni + 1; ci ← ci + reward. {Update statistics.} 14: {Update confidence bounds.} 15: if Li < Ui then p 16: (L′i , Ui′ ) ← bi (ci /ni ∓ 8 log(T )/ni )). 17: if max(Li , L′i ) < min(Ui , Ui′ ) then 18: (Li , Ui ) ← (max(Li , L′i ), min(Ui , Ui′ )). 19: else i i 20: (Li , Ui ) ← ( Li +U , Li +U ). 2 2 21: else 22: Allocate agent i = min Sact . 23: for all agent i ∈ Sact do 24: if Ui < maxj∈Sact Lj then 25: Remove i from Sact . Fix realization ρ and bid vector b. Let Sact (t, b) be the set of active agents in the beginning of round t. For each agent i, let Li (t, b) and Ui (t, b) be the values of Li and Ui in the end of round t. The goal of the specific update rules for the confidence bounds (lines 15-20) and the statistics (lines 13) is to guarantee the following two properties: — the statistics are kept only for rounds when a designated agent is played. Moreover, for each agent i and round t, and any two bid vectors b and b′ we have Li (t, b)/bi = Li (t, b′ )/b′i ′ if i ∈ Sact (t, b) ∩ Sact (t, b ) then . (17) Ui (t, b)/bi = Ui (t, b′ )/b′i — for any fixed realization ρ and bid vector b, and each agent i: Li ≤ Ui , and from round to round Li is non-decreasing and Ui is non-increasing. In other words, for each round t it holds that Li (t − 1, b) ≤ Li (t, b) ≤ Ui (t, b) ≤ Ui (t − 1, b). (18) Truthful Mechanisms with Implicit Payment Computation 0:27 The ex-post monotonicity follows from these two properties and the de-activation rule (lines 24-25). Ex-post monotonicity. Let L∗ (t, b) , maxi∈Sact (t,b) Li (t, b). Fix agent i and b+ i > bi , + and let b+ = (b−i , b+ ) be the “alternative” bid vector. Let λ = b /b . i i i C LAIM 7.12. We establish the following sequence of claims: (C1) L∗ (t, b) is non-decreasing in t, for any fixed b. (C2) For each round t, L∗ (t, b) ≤ λ L∗ (t, b+ ). (C3) For each round t, Sact (t, b+ ) \ {i} ⊂ Sact (t, b) \ {i}. (C4) In each round t: if i ∈ Sact (t, b) then i ∈ Sact (t, b+ ). P ROOF. Let us prove the parts (C1-C4) one by one. (C1). We use property (18) and the de-activation rule. Throughout the proof, we omit the bid vector b from the notation. Fix round t ≥ 2. Let i ∈ Sact (t − 1) be an agent such that L∗ (t − 1) = Li (t − 1). If i ∈ Sact (t) then L∗ (t − 1) = Li (t − 1) ≤ Li (t) ≤ L∗ (t) Else i is de-activated in round t, so L∗ (t − 1) = Li (t − 1) ≤ Li (t) ≤ Ui (t) < L∗ (t). (C2). Suppose, for the sake of contradiction, that L∗ (t, b) > λ L∗ (t, b+ ). Let j ∈ Sact (t, b) be an agent such that Lj (t, b) = L∗ (t, b). If j ∈ Sact (t, b+ ) then by property (17) we have L∗ (t, b) = Lj (t, b) = λ Lj (t, b∗ ) ≤ λ L∗ (t, b+ ), contradiction. We conclude that j 6∈ Sact (t, b+ ). Thus with bid vector b+ agent j gets disqualified during some round s < t. Thus, Uj (s, b+ ) < L∗ (s, b+ ) ≤ L∗ (t, b+ ), (19) where the second inequality is by Part (C1). Now using property (18) and property (17) (for the right-most inequality), we get that L∗ (t, b) = Lj (t, b) ≤ Uj (t, b) ≤ Uj (s, b) = λ Uj (s, b+ ). Thus, L∗ (t, b) ≤ λ L∗ (t, b+ ) by Equation (19), the desired contradiction. (C3). Use induction on t. The claim trivially holds for t = 1. Assuming the claim holds for some t we prove it holds for t + 1. Fix agent j ∈ Sact (t + 1, b+ ) \ {i}. We need to prove that j ∈ Sact (t + 1, b). Note that j ∈ Sact (t, b+ ), and so j ∈ Sact (t, b) by the induction hypothesis. Therefore L∗ (t, b) ≤ λ L∗ (t, b+ ) ≤ λ Uj (t, b+ ) = Uj (t, b) (by Part (C2)) (by the de-activation rule) (by property (17)) So agent j is not deactivated in round t under bid vector b, i.e. j ∈ Sact (t + 1, b), completing the proof. (C4). Use induction on t. The base case t = 0 holds because initially all agents are active. For the induction step, assume that the statement holds for some round t ≥ 0. Suppose i ∈ Sact (t + 1, b). We need to prove that i ∈ Sact (t + 1, b+ ). Note that i ∈ Sact (t, b), and so i ∈ Sact (t, b+ ) by the induction hypothesis. Therefore L∗ (t, b) ≤ Ui (t, b) (by the de-activation rule) + = λ Ui (t, b ) (by property (17)). 0:28 M. Babaioff et al. If L∗ (t, b) = λ L∗ (t, b+ ), then L∗ (t, b+ ) = L∗ (t, b)/λ ≤ Ui (t, b+ ), and we are done. From here on, assume L∗ (t, b) 6= λ L∗ (t, b+ ). Then L∗ (t, b) < λ L∗ (t, b+ ) by Part (C2). Pick agent j which maximizes Lj (t, b+ ). If j 6= i then j ∈ Sact (t, b) by Part (C3), so L∗ (t, b) ≥ Lj (t, b) = λ Lj (t, b+ ) ∗ + = λ L (t, b ) by property (17)) (by the choice of j), contradicting our assumption. Then j = i, and so L∗ (t, b+ ) = Li (t, b+ ) ≤ Ui (t, b+ ). Ex-post monotonicity follows easily from (C3-C4). C LAIM 7.13. Consider a fixed round t. Suppose agent i is allocated with bid vector b. Then it is also allocated with bid vector b+ . P ROOF. Since i ∈ Sact (t, b), by (C4) we have i ∈ Sact (t, b+ ). If agent i is the designated agent in round t, then as such it is allocated under both bid vectors. If agent i is not the designated agent in round t, then i = min Sact (t, b). By (C3) it holds that Sact (t, b+ ) ⊂ Sact (t, b), which implies that i = min Sact (t, b+ ). So i is allocated under bid vector b+ , too. Regret analysis. The regret analysis is relatively standard, following the ideas in [Auer et al. 2002a]. For simplicity assume that bmax = 1. Fix a bid vector b. For each agent i, let ci (t) and ni (t) be, respectively, the number of clicks and impressions p in all rounds s ≤ t when it is allocated as the designated agent. Let ri (t) = 8 log(T )/ni (t). Then the event \|µi − ci (t)/ni (t)\| ≤ ri (t) for all rounds t (20) holds with probability at least 1 − T −2. 16 In what follows, let us assume that this event holds for all agents i. (The regret accumulated if this event fails is negligible.) Then it easily follows from the specs of NewCB that for each agent i, Li (t, b) ≤ bi µi ≤ Ui (t, b) Ui (t, b) − Li (t, b) ≤ 2 ri (t) Let i∗ ∈ argmaxi bi µi be a best agent. Note that Ui∗ (t, b) ≥ bi∗ µi∗ ≥ bi µi ≥ Li (b, t) for all agents i and rounds t. It follows that i∗ is never de-activated by the algorithm. Consider some agent i with ∆i , bi∗ µi∗ − bi µi > 0. Then ri (t) < ∆i after O(∆−2 i log T ) rounds in which this agent is allocated as the designated agent. After such round t, Ui (b, t) ≤ bi µi + ri (t) < bi µi + ∆i = bi∗ µi∗ ≤ Li∗ (b, t), and therefore agent i is deactivated. It follows that agent i is allocated as the designated agent at most O(∆−2 i log T ) times. Therefore it is de-activated after at most O(k ∆−2 log T ) rounds. This, in turn, implies the claimed regret bound. i 16 This follows from Azuma-Hoeffding inequality via a standard argument, one version of which we provide below. Fix agent i. For each s ∈ N , let Xs be the click bit for the s-th time this agent is allocated as the designated agent, if s ≤ ni (T ), and otherwise define Xs to be an independent 0-1 random variable with expectation µi . Then the random variables Ys = Xs − µs , s ∈ N form a martingale. Applying p AzumaP 8N log(T ) Hoeffding inequality to Y1 , . . . , YN , for any given N , we obtain that the event \| N s=1 Ys \| ≤ holds with probability at least 1 − T −3 . Taking the Union Bounds over all N ≤ T , and noting that ci (t) = Pni (t) −2 . s=1 Xs , it follows that the event (20) holds with probability at least 1 − T Truthful Mechanisms with Implicit Payment Computation 0:29 7.4. The power of randomization A by-product of Theorem 7.9 is a separation between the power of deterministic and randomized mechanisms, in terms of regret for MAB mechanisms that are ex-post truthful and ex-post normalized. The lower bound for deterministic mechanisms is from [Babaioff et al. 2014]. One challenge here is to ensure that the upper and lower bounds talk about exactly the same problem; as stated, Theorem 7.9 and the main lower bound result from [Babaioff et al. 2014] do not. To bypass this problem, we focus on the case of two agents, and use a more general version of the lower bound: Theorem C.1 in the full version of [Babaioff et al. 2014]. Further, to match [Babaioff et al. 2014] we extend the mechanism from Theorem 7.9 to a setting in which bmax is not known a priori. We formulate the separation theorem as follows. Denote R(T, bmax ) , max R(T ; b; µ), where the maximum is taken over all CTR vectors µ and all bid vectors b such that bi ≤ bmax for all i. T HEOREM 7.14. Consider the stochastic MAB mechanism design problem with two agents. Assume bmax is not known a priori to the mechanism. Suppose M is an MAB mechanism that is (i) ex-post truthful and ex-post normalized, and (ii) has regret R(T, bmax ) = Õ(bmax T γ ) for some γ and any bmax . Then: (a) [Babaioff et al. 2014] If M is deterministic then γ ≥ 23 . (b) There exists such randomized M with γ = 12 . P ROOF OF PART ( B ). Let A′ be the ex-post monotone MAB allocation rule in Theorem 7.9, for bmax = 1. Define an MAB allocation rule A as a rule that inputs the bid vector b and passes the modified bid vector b′ = b/(maxi bi ) to A′ . We claim that A is expost monotone, too. Indeed, w.l.o.g. assume b1 > b2 . If b2 increases (to a value ≤ b1 ), then b′2 increases while b′1 stays the same. Thus, the total click reward of agent 2 increases. If b1 increases then b′2 decreases while b′1 stays the same, so the total click reward of agent 2 does not increase, which implies that the total click reward of agent 1 does not decrease. Claim proved. Now part (b) follows from Theorem 7.1 (with µ = T1 ). 8. EXTENSION TO MULTI-PARAMETER DOMAINS Our general transformation from Section 4 can be extended to multi-parameter mechanisms.It is known that a multi-parameter allocation rule is truthfully implementable if and only if it satisfies a property called “cycle-monotonicity”. Similar to the singleparameter case, we present a general procedure to take any cycle-monotone allocation rule A and transform it into a randomized mechanism that is truthful-in-expectation, implements the same outcome as A with probability arbitrarily close to 1, and requires evaluating that allocation rule only once. The technical contribution here is that we find a reduction from the multi-parameter setting to the single-parameter case. This section is self-contained. For more background on multi-parameter mechanisms for a CS-oriented audience, please refer to [Archer and Kleinberg 2008a,b]. An Economics-oriented background for this area can be found in [Ashlagi et al. 2010]. 8.1. Preliminaries: multi-parameter domains Generalized types. In the full generality, multi-parameter mechanisms are defined as follows. There are n agents and a set O of outcomes. Each agent i is characterized by his type xi : O → R, where xi (o) is interpreted as the agent’s valuation for the outcome o ∈ O. For each agent i there is a set of feasible types, denoted Ti . Denote 0:30 M. Babaioff et al. T = T1 × . . . × Tn and call it the type space; call Ti the type space of agent i. The mechanism knows (n, O, T ), but not the actual types xi ; each type xi is known only to the corresponding agent i. Formally, a problem instance, also called a multi-parameter domain, is a tuple (n, O, T ). Using this general notion of types, we define truthful mechanisms in essentially the same way as in Section 3, with minimal syntactic changes. A (direct revelation) mechanism M consists of the pair (A, P), where A : T → O is the allocation rule and P : T → Rn is the payment rule. Both A and P can be randomized. Each agent i reports a type bi ∈ Ti to the mechanism, which is called the bid of this agent. We denote the vector of bids by b = (b1 , . . . , bn ) ∈ T . The mechanism receives the bid vector b ∈ T , selects an outcome A(b), and charges each agent i a payment of Pi (b). The utilities are quasi-linear and agents are risk-neutral: if agent i has type xi ∈ Ti and the bid vector is b ∈ T , then this agent’s utility is ui (xi ; b) = E [xi (A(b)) − Pi (b)] . M (21) For each type xi ∈ Ti of agent i we use a standard notation (b−i , xi ) to denote the bid vector b̂ such that b̂i = xi and b̂j = bj for every agent j 6= i. Special case: dot-product valuations. For intuition, consider dot-product valuations, an important special case where the type x ∈ Ti of each agent i can be decomposed as a dot product x(o) = βx · ai (o), for each outcome o ∈ O, where βx , ai (o) ∈ Rd are some finite-dimensional vectors. Here the term ai (o) is the same for all types x ∈ Ti (and known to the mechanism), whereas βx is the same for all outcomes o ∈ O and is known only to agent i. The term ai (o) is usually called an “allocation” of agent i for outcome o, and βx is called the “private value”. The single-parameter domains defined in Section 3 correspond to the case d = 1. Note that the type x of each agent i is determined by the corresponding private value βx , and his type space Ti is determined by Di = {βx : x ∈ Ti } ⊂ Rd . Because of this, in the literature on dot-product valuations the term “type” often refers to βx . To avoid ambiguity, in this section we will refer to βx as “private value” rather than “type”, and call D1 × . . . × Dn the private value space. Game-theoretic properties. Truthfulness and individual rationality are defined exactly as in Section 3 if expressed in terms of the agents’ utility: — A mechanism is truthful if for every agent i truthful bidding is a dominant strategy: ui (xi ; (b−i , xi )) ≥ ui (xi ; b) ∀xi ∈ Ti , b ∈ T . (22) An allocation rule is called truthfully implementable if it is the allocation rule in some truthful mechanism. — A mechanism is individually rational (IR) if each agent i never receives negative utility by participating in the mechanism and bidding truthfully: ui (xi ; (b−i , xi )) ≥ 0 ∀xi ∈ Ti , b−i ∈ T−i . (23) The right-hand side in (23) represents the maximal guaranteed utility of an “outside option” (i.e., from not participating in the mechanism). For example, our definition of IR is meaningful whenever this utility is 0, which is a typical assumption for most multi-parameter domains studied in the literature. Our assumptions. We make two assumptions on the type space T : — non-negative types: xi (o) ≥ 0 for each agent i, each type xi ∈ Ti , and each outcome o ∈ O. Truthful Mechanisms with Implicit Payment Computation 0:31 — rescalable types: λxi ∈ Ti for each agent i, each type xi ∈ Ti , and any parameter λ ∈ [0, 1]. For dot-product valuations, types are rescalable if and only if it holds that βx ∈ Di ⇒ λβx ∈ Di for each λ ∈ [0, 1]. Thus, assuming rescalable types is equivalent to assuming that the set Di is star-convex at 0. To ensure non-negative types, it suffices to assume that Di ⊂ Rd+ for each agent i, and all allocations are non-negative: ai (o) ∈ Rd+ for all o ∈ O. In particular, for each agent i there exists a zero type: a type xi ∈ Ti such that xi (·) ≡ 0. Let us say that a mechanism is normalized if for each agent i, the expected payment of this agent is 0 whenever she submits the zero type. Truthfulness characterization. We will use the following characterization of truthful mechanisms. A (randomized) allocation rule A is cycle-monotone if the following property holds: for each bid vector b ∈ T , each agent i, each k ≥ 2, and each k-tuple xi,0 , xi,1 , . . . , xi,k ∈ Ti of this agent’s types, we have   k X xi,j (oi,j ) − xi, (j−1) mod k (oi,j ) ≥ 0, where oi,j = A (b−i , xi,j ) ∈ O. E (24) A j=0 T HEOREM 8.1 (R OCHET [1987]). Consider an arbitrary multi-parameter domain (n, O, T ). A (randomized) allocation rule A is truthfully implementable if and only if it is cycle-monotone. Assuming rescalable types, for any cycle-monotone allocation rule A, a mechanism (A, P) is truthful and normalized if and only if Z 1 E [Pi (b)] = E bi (A(b)) − bi (A(b−i , t bi )) dt . (25) A A t=0 Note that this theorem generalizes Theorem 3.2 for single-parameter mechanisms, as applied to single-parameter domains with private value space [0, 1]n . In particular, Equation (25) generalizes the Myerson payment rule for single-parameter mechanisms. 8.2. The multi-parameter transformation Consider allocation rule A, bid vector b ∈ T , and the rescaling vector λ ∈ [0, 1]n . Denote λ ⊗ b = (λ1 b1 , . . . , λn bn ) ∈ T . In other words, λ ⊗ b is the “rescaled” bid vector where the bid of each agent i is λi bi ; this bid vector is well-defined because we assumed the rescalable types property. Note that for each b the subset Tb = {λ ⊗ b : λ ∈ [0, 1]n } ⊂ T forms a single-parameter type space where each agent i has private value λi ∈ [0, 1] and allocation bi (o) for every outcome o. By abuse of notation, let us treat the allocation / payment rules for Tb as functions from the private value space [0, 1]n rather than the type space Tb . Consider an allocation rule Ab (λ) = A(λ ⊗ b) for the single-parameter type space Tb . If the original allocation rule A is truthfully implementable for type space T using payment rule P, then Ab is truthfully implementable for type space Tb using payment rule Pb (λ) = P(λ ⊗ b), because restricting the allocation and payment rules to Tb only limits the set of possible misreports of an agent. Essentially, the idea will be to apply our single-parameter transformation to Ab . 0:32 M. Babaioff et al. Let fµ = (f1 , . . . , fn ) be the n-tuple of canonical self-resampling procedures with resampling probability µ, for some fixed µ ∈ (0, 1). (See Algorithm 1 on page 13.) For each bid vector b ∈ T , let eb ) = AllocToMech(Ab , µ, fµ ) (Aeb , P (26) be the single-parameter mechanism for type space Tb obtained by applying our singleparameter transformation from Section 4 to allocation Ab .17 The transformed multi-parameter mechanism is defined as e e eb (~1 ) for every b ∈ T . A(b), P(b) = Aeb (~1 ), P (27) This completes the description of our multi-parameter transformation. The useful properties of this transformation are captured in the theorem below. T HEOREM 8.2. Consider an arbitrary multi-parameter domain (n, O, T ) with rescalable, non-negative types. Let A be a cycle-monotone allocation rule. Let Mµ = e P) e be the transformed mechanism defined by Equations(26-27), for some parameter (A, µ ∈ (0, 1). Then Mµ has the following properties: (a) Mµ is truthful and normalized. (b) Mµ is universally ex-post individually rational and ex-post no-positive-transfers. Moreover, given a bid vector b, it never pays any agent i more than bi (o)( µ1 − 1), where o = A(b) ∈ O. e (c) For any bid vector b ∈ T (and any fixed random seed of nature) allocations A(b) and A(b) are identical with probability at least 1 − nµ. 2 -approximate. (d) If A is α-approximate (for social welfare) then Ae is α/ 1 − 1−µ P ROOF. Parts (b) and (c) follow immediately from Theorem 4.5, and part (d) follows immediately from Theorem 5.1. Thus, it remains to prove part (a). Note that the single-parameter allocation rule Aeb has the following property: for each agent i the single-parameter bid λi is rescaled by the (randomly chosen) factor χi ∈ [0, 1] which does not depend on the bid, and then Ab is called. Therefore, letting χ = (χ1 , . . . , χn ), it holds that Aeb (λ) = A(χ ⊗ (λ ⊗ b)) for all b ∈ T and λ ∈ [0, 1]n . (28) We claim that Ae is cycle-monotone. Indeed, fix bid vector b ∈ T , agent i, some k ≥ 2, and a k-tuple xi,0 , xi,1 , . . . , xi,k ∈ Ti of this agent’s types. Let us consider a fixed realization of the random vector χ ∈ [0, 1]n . For each type xi,j , note that (by Equations (28) and (27)) we have e i,j , b−i ) = Ae(x ,b ) ( ~1 ) = A (χ ⊗ (b−i , xi,j )) ∈ O. A(x i,j −i Denote this outcome by oi,j (χ). Apply the cycle-monotonicity of A for bid vector χ ⊗ (xi,j , b−i ):   k X xi,j (oi,j (χ)) − xi, (j−1) mod k (oi,j (χ)) ≥ 0. E (29) A j=0 17 Note that the transformed mechanism depends on µ. We do not make this dependence explicit, to simplify the notation. Truthful Mechanisms with Implicit Payment Computation 0:33 e i,j , b−i ), we observe that for this fixed realization of Recalling that oi,j (χ) = A(x χ, Equation (29) is exactly the inequality in the definition of cycle-monotonicity for e Therefore taking expectation over χ, we obtain the desired inequality (24) for A. e A. Claim proved.18 e P), e the payment rule satIt remains to prove that in the transformed mechanism (A, isfies Equation (25). Fix bid vector b and consider the transformed single-parameter eb ) for the single-parameter type space Tb . In the terminology of mechanism (Aeb , P single-parameter domains, each agent i receives an allocation Aeb, i (λ) = bi (Aeb (λ)) whenever the bid vector is λ ∈ [0, 1]n . Since this is a truthful and normalized singleparameter mechanism, it follows that # " Z λi h i e e e E Pb (λ) = E λi Ab, i (λ) − Ab, i (λ−i , u) du , ∀λ ∈ [0, 1]n . 0 Plugging in λ = ~1 and Equation (27), we obtain the desired Equation (25). 9. OPEN QUESTIONS This paper gives rise to a number of open questions. As discussed in Section 2.1, some of these questions have been partially addressed in the follow-up work. Here we present the current status. Variance vs. expectation tradeoff. Randomized mechanisms constructed via our general transformation exhibit an explicit tradeoff between the variance in payments and the loss in expected welfare compared to the optimal allocation rule. Since the variance in payments can be very high, it is desirable to optimize this tradeoff (to complement the expectation-only guarantees). The worst-case optimality result in Wilkens and Sivan [2012], discussed in Section 2.1, does not resolve this question, since it does not rule out a reduction which achieves a better tradeoff for some (but not all) monotone allocation rules. Further, the optimal tradeoff for a given domain could be achieved by a mechanism that cannot be presented as a reduction from some welfare-optimal allocation rule. Our informal conjecture is that the tradeoff in this paper is optimal for any given single-parameter domain with “informational obstacle”, i.e. whenever payment computation for welfare-optimal allocation rule is impossible due to the insufficient observable information. A specific formal conjecture is that our tradeoff is optimal for MAB mechanisms. To take an extreme version, what welfare loss can be achieved if no rebates (i.e., no positive transfers) are allowed? The power of randomization. We have a separation result for randomized vs. deterministic ex-post truthful MAB mechanisms. Can one obtain similar separation results for other single-parameter domains? The positive side for any such hypothetical separation result is provided by our general reduction, so it remains to produce the corresponding negative result for deterministic mechanisms. However, such negative results are not likely to be easy, considering the difficulties faced by [Babaioff et al. 2014; Devanur and Kakade 2009] for MAB mechanisms. One specific target would be the router scheduling problem proposed in [Shnayder et al. 2012]. 18 Note that the proof of cycle-monotonicity of A e did not use any other property of the canonical selfresampling procedures fµ other than Equation (28). The truthfulness properties of fµ are used in the forthcoming argument about payments. 0:34 M. Babaioff et al. MAB allocation rules. This paper opens up the problem of designing monotone MAB allocation rules, which is a new angle in the rich literature on MAB (also see [Slivkins 2011b]). While we have focused on stochastic MAB, many other MAB settings have been studied in the literature, making various assumptions on payoff evolution over time (e.g., [Auer et al. 2002b; Slivkins and Upfal 2008; Hazan and Kale 2009]), dependencies between arms (e.g., [Flaxman et al. 2005; Pandey et al. 2007; Kleinberg et al. 2008b; Srinivas et al. 2010]), and side information available to the algorithm (e.g., [Kleinberg et al. 2008b; Langford and Zhang 2007; Slivkins 2011a]). For most such settings one could meaningfully define the corresponding mechanism design problem; we have reduced this problem to that of designing monotone MAB allocation rules. In particular, for any given MAB setting one could ask whether monotone MAB allocation rules can achieve optimal regret. One appealing target here is the adversarial MAB setting (with oblivious adversary). The ex-post truthful mechanism in [Babaioff et al. 2014] achieves regret 1/3 2/3 Õ(k √ T ) for this setting, whereas the best known MAB algorithms achieve regret O( kT ) [Auer et al. 2002b; Audibert and Bubeck 2010]; it is not clear what is the tight regret bound. More applications. In addition to the applications presented in this paper and the follow-up work, what other domains can our general reduction (and the multiparameter extension thereof) be fruitfully applied to? In particular, one could consider two generalizations of MAB mechanisms: to multiple ads per agent and to multiple ad slots with slot-dependent values-per-click. APPENDIX: O NE S HOT is equivalent to Algorithm 1 (proof of Proposition 4.8) Let us compare the sampling procedures defined by Algorithm 1 and O NE S HOT . To simplify the notation, we will omit the subscript i from the description of the procedures. That is, a self-resampling procedure inputs a scalar bid b and a random seed w, and outputs two numbers (x, y). To prove that O NE S HOT is equivalent to Algorithm 1, we analyze a family of sampling rules that uses bounded-depth recursion to “interpolate” between O NE S HOT and Algorithm 1. Specifically, define BDRk to be the following family of sampling algorithms parameterized by k ∈ N∪{∞}, where k−1 is interpreted as ∞ when k = ∞. Algorithm 6: The sampling algorithm BDRk : Bounded Depth Recursion. 1: Input: bid b ∈ [0, ∞], parameter µ ∈ (0, 1). 2: Output: (x, y) such that 0 ≤ x ≤ y ≤ b. 3: with probability 1 − µ 4: x ← b, y ← b. 5: else 6: if k = 0 7: Pick γ1 , γ2 ∈ [0, 1] indep., uniformly at random. 1/(1−µ) 1/(1−µ) 1/µ 8: x ← b · γ1 , y ← b · max{γ1 , γ2 }. 9: else k>0 10: Pick b′ ∈ [0, b] uniformly at random. 11: (x′ , y ′ ) = BDRk−1 (b′ , µ). 12: x ← x′ , y ← b′ . Truthful Mechanisms with Implicit Payment Computation 0:35 The reader may easily verify that BDRk is equal to O NE S HOT when k = 0 and that it is equal to Algorithm 1 when k = ∞. Furthermore, for any k < k ′ (where k ′ ≤ ∞) there is an obvious coupling of BDRk with BDRk′ such that the two algorithms have probability at most µk+1 of outputting different results: simply let the two executions share the same randomness until the recursion depth equals k. Thus, as k → ∞, the output distribution of BDRk converges, in total variation distance, to that of BDR∞ . We will prove that for every finite k the algorithms BDRk and BDRk+1 have identical output distributions, from which it follows that their output distribution is identical to that of BDR0 and, therefore, that BDR∞ also has the same output distribution as BDR0 , confirming Proposition 4.8. Couple BDRk and BDRk+1 so that they use shared randomness until the two algorithms reach differing points in their control flow. This occurs when the first algorithm is executing a call to BDR0 and the second algorithm is executing a call to BDR1 on the same input (β, µ). (We are denoting the input in this step of the recursive algorithms by (β, µ) rather than (b, µ), to distinguish β from the value of b on which the two algorithms BDRk , BDRk+1 were originally called.) At this point, with probability 1 − µ both algorithms output (x, y) = (β, β). Conditional on this event not taking place, 1/q 1/p 1/p BDR0 outputs (x, y) = (γ1 β, max{γ1 , γ2 }β) where p = 1 − µ, q = µ. Instead BDR 1 ′ computes β = γ3 β where γ3 ∈ [0, 1] is uniformly random, and it outputs (x, y) = (β ′ , β ′ ) 1/p with probability p and otherwise (x, y) = (γ1 β ′ , β ′ ). Lemma A.1 tells us that these two output distributions are the same. L EMMA A.1. Let γ1 , γ2 , γ3 be mutually independent random variables, each uniformly distributed in [0, 1]. Let p, q > 0 be numbers such that p + q = 1. Define random variables x, y, z by: 1/p x = γ1 1/p 1/q y = max{γ1 , γ2 } ( γ3 if γ2 < p z= 1/p γ1 γ3 if γ2 ≥ p Then the pairs (x, y) and (z, γ3 ) are identically distributed. P ROOF. We will show, equivalently, that the pairs (y, x/y) and (γ3 , z/γ3 ) are identically distributed. The distribution of (γ3 , z/γ3) is completely characterized by the following facts which are immediate from the definition of z. (1) γ3 and z/γ3 are independent; (2) γ3 is uniformly distributed in [0, 1]; (3) z/γ3 is equal to 1 with probability p, and conditional on z/γ3 6= 1, the distribution of (z/γ3 )p is uniform on [0, 1). To finish the proof of the lemma, we shall prove the corresponding facts about y and x/y. Let I1 , I2 ⊆ [0, 1] be any pair of intervals (open, closed, or half-open). Let a, b be the endpoints of I1 and c, d the endpoints of I2 . To compute Pr(y ∈ I1 , x/y ∈ I2 ) it suffices 0:36 M. Babaioff et al. to make the following two observations: 1/p 1/q 1/p Pr(y ∈ I1 , x/y = 1) = Pr a ≤ γ1 ≤ b, 0 ≤ γ2 ≤ γ1 q/p = Pr ap ≤ γ1 ≤ bp , 0 ≤ γ2 ≤ γ1 Z bp Z bp = tq/p dt = t1/p−1 dt = p(b − a) ap ap 1/q Pr(y ∈ I1 , x/y ∈ I2 \ {1}) = Pr(a ≤ γ2 1/q ≤ b, cγ2 1/p ≤ γ1 1/q < dγ2 ) p/q p/q = Pr(aq ≤ γ2 ≤ bq , cp γ2 ≤ γ1 ≤ dp γ2 ) Z bq Z bq p p p/q p p = (d − c )t dt = (d − c ) t1/q−1 dt = q(dp − cp )(b − a). aq aq Therefore, Pr(y ∈ I1 , x/y ∈ I2 ) = (b − a) · q(dp − cp ) if 1 6∈ I2 . p p p + q(d − c ) if 1 ∈ I2 From this formula it follows that y and x/y are independent, y is uniformly distributed, Pr(x/y = 1) = p, and the distribution of (x/y)p conditional on x/y 6= 1 is uniform on [0, 1). Acknowledgments We are indebted to Tim Roughgarden for suggesting that for positive types, an improved bound on the social welfare is possible (see Section 5). 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arXiv:1710.05209v2 [cs.LG] 16 Feb 2018 Settling the Sample Complexity for Learning Mixtures of Gaussians Hassan Ashtiani University of Waterloo Shai Ben-David University of Waterloo Nick Harvey University of British Columbia Abbas Mehrabian McGill University Christopher Liaw University of British Columbia Yaniv Plan University of British Columbia Abstract We prove that Θ̃(kd2 /ε2 ) samples are necessary and sufficient for learning a mixture of k Gaussians in Rd , up to error ε in total variation distance. This improves both the known upper bound and lower bound for this problem. For mixtures of axis-aligned Gaussians, we show that Õ(kd/ε2 ) samples suffice, matching a known lower bound. Moreover, these results hold in an agnostic learning setting as well. The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in Rd has an efficient sample compression. 1 Introduction Estimating distributions from observed data is a fundamental task in statistics that has been studied for over a century. This task frequently arises in applied machine learning and it is very common to assume that the distribution can be modeled using a mixture of Gaussians. Popular software packages have implemented heuristics, such as the EM algorithm, for learning a mixture of Gaussians. The theoretical machine learning community also has a rich literature on distribution learning. For example, the recent survey 1 of Diakonikolas (2016) considers learning of structured distributions, and the survey of Kalai, Moitra, and Valiant (2012) focuses on mixtures of Gaussians. This paper develops a general technique for distribution learning, then employs these techniques in the important setting of mixtures of Gaussians. The theoretical model that we adopt is density estimation: given i.i.d. samples from an unknown target distribution, find a distribution that is close to the target distribution in total variation (TV) distance. Our focus is on sample complexity bounds: using as few samples as possible to obtain a good estimate of the target distribution. For background on this model see, e.g., Devroye and Lugosi (2001, Chapter 5) and Diakonikolas (2016). Our new technique for upper bounds on the sample complexity involves a form of sample compression. If it is possible to “encode” members of a class of distributions using a carefully chosen subset of the samples, then this yields an upper bound on the sample complexity of distribution learning for that class. In particular, by constructing compression schemes for mixtures of axis-aligned Gaussians and general Gaussians, we obtain new upper bounds on the sample complexity of learning with respect to these classes, which are optimal up to logarithmic factors. The compression framework can incorporate a notion of robustness, which leads to sample complexity bounds for agnostic learning. Namely, if the target distribution is close to a mixture of Gaussians (in TV distance), our method uses few samples to find a mixture of Gaussians that is close to the target distribution (in TV distance). 1.1 Main results In this section, all learning results refer to the problem of producing a distribution within total variation distance ε from the target distribution. Our first main result is an upper bound for learning mixtures of multivariate Gaussians. This bound is tight up to logarithmic factors. Theorem 1.1. The class of k-mixtures of d-dimensional Gaussians can be learned using e 2 /ε2 ) samples. This result generalizes to the agnostic setting. O(kd Previously, the best upper bounds on the sample complexity of this problem were e O(kd2 /ε4 ), due to Ashtiani, Ben-David, and Mehrabian (2017), and O(k 4 d4 /ε2 ), based on a VC-dimension bound discussed later. For the case of a single Gaussian (i.e., k=1), a sample complexity bound of O(d2 /ε2 ) is well known (see, e.g., Ashtiani et al. (2017, Theorem 13)). Our second main result is a lower bound matching Theorem 1.1 up to logarithmic factors. 2 Theorem 1.2. Any method for learning the class of k-mixtures of d-dimensional Gaus2 2 e sians has sample complexity Ω(kd /ε ). 2 e Previously, the best lower bound on the sample complexity was Ω(kd/ε ) (Suresh, Orlitsky, Acharya, and 2 2 e 2014). Even for a single Gaussian (i.e., k=1), an Ω(d /ε ) lower bound was not known prior to this work. Our third main result is an upper bound for learning mixtures of axis-aligned Gaussians, i.e., Gaussians with diagonal covariance matrix. This bound is tight up to logarithmic factors. Theorem 1.3. The class of k-mixtures of axis-aligned d-dimensional Gaussians can be 2 e learned using O(kd/ε ) samples. This result generalizes to the agnostic setting. 2 e A matching lower bound of Ω(kd/ε ) samples was shown by Suresh et al. (2014). 4 e Previously, the best known upper bounds were O(kd/ε ), due to Ashtiani et al. (2017), 4 2 3 3 2 and O((k d + k d )/ε ), based on a VC-dimension discussed later. Computational efficiency. Although our approach for proving sample complexity upper bounds is algorithmic, our focus is not on computational efficiency. The resulting algorithms are efficient in terms of sample complexity, but their runtime is exponential in the dimension d and the number of mixture components k. The existence of a polynomial time algorithm for density estimation is unknown even for the class of mixtures of axis-aligned Gaussians (Diakonikolas, Kane, and Stewart, 2017a, Question 1.1). Even for the case of a single Gaussian, the published proofs of the O(d2/ε2 ) bound are not algorithmically efficient. Using ideas from our proof of Theorem 1.1, we show in Appendix B that an algorithmically efficient proof for the single Gaussian case can be obtained simply by computing the empirical mean and covariance matrix of O(d2 /ε2 ) samples. 1.2 Related work Distribution learning is a vast topic and many approaches have been considered in the literature. We briefly review approaches that are most relevant to our problem. For parametric families of distributions, a common approach is to use the samples to estimate the parameters of the distribution, possibly in a maximum likelihood sense, or possibly aiming to approximate the true parameters. For the specific case of mixtures of Gaussians, there is a substantial theoretical literature on algorithms that approximate the mixing weights, means and covariances. Kalai et al. (2012) gave a recent survey of this literature. The strictness of this objective cuts both ways. On the one hand, a 3 successful learner uncovers substantial structure of the target distribution. On the other hand, this objective is clearly impossible when the means and covariances are extremely close. Thus, algorithms for parameter estimation of mixtures necessarily require some assumptions on the target parameters. Also, the basic definition of parameter estimation does not immediately extend to an agnostic setting, although there is literature on agnostic parameter estimation, e.g., Lai, Rao, and Vempala (2016). Density estimation has a long history in the statistics literature, where the focus is on the sample complexity question; see, e.g., Devroye (1987); Devroye and Lugosi (2001) for general background. It was first studied in the computational learning theory community under the name PAC learning of distributions by Kearns, Mansour, Ron, Rubinfeld, Schapire, and Sellie (1994), whose focus is on the computational complexity of the learning algorithm. For density estimation there are various possible measures of distance between distributions, the most popular ones being the TV distance and the Kullback-Leibler (KL) divergence. Here we focus on the TV distance since it has several appealing properties, such as being a metric and having a natural probabilistic interpretation. In contrast, KL divergence is not even symmetric and can be unbounded even for intuitively close distributions. For a detailed discussion on why TV is a natural choice, see Devroye and Lugosi (2001, Chapter 5). A popular method for distribution learning in practice is kernel density estimation (see, e.g., Devroye and Lugosi (2001, Chapter 9)). The few rigorously proven sample complexity bounds for this method require certain smoothness assumptions on the class of densities (e.g., Devroye and Lugosi (2001, Theorem 9.5)). The class of Gaussians is not universally Lipschitz and does not satisfy these assumptions, so those results do not apply to the problems we consider. Another elementary method for density estimation is using histogram estimators. Straightforward calculations show that histogram estimators for mixtures of Gaussians would result in a sample complexity that is exponential in the dimension. The same is true for estimators based on piecewise polynomials. The minimum distance estimate (Devroye and Lugosi, 2001, Section 6.8) is another approach for deriving sample complexity upper bounds for distribution learning. This approach is based on uniform convergence theory. In particular, an upper bound for any class of distributions can be achieved by bounding the VC-dimension of an associated set system, called the Yatracos class (see Devroye and Lugosi (2001, page 58) for the definition). For example, Diakonikolas, Kane, and Stewart (2017b) used this approach to bound the sample complexity of learning high-dimensional log-concave distributions. However, for mixtures of Gaussians and axis-aligned Gaussians in Rd , the best known VC-dimension bound (Anthony and Bartlett, 1999, Theorem 8.14) results in loose upper bounds of O(k 4 d4 /ε2 ) and O((k 4 d2 + k 3 d3 )/ε2 ) respectively. 4 Another approach is to first approximate the mixture class using a more manageable class such as piecewise polynomials, and then study the associated Yatracos class, see, e.g., Chan, Diakonikolas, Servedio, and Sun (2014). However, piecewise polynomials do a poor job in approximating d-dimensional Gaussians, resulting in an exponential dependence on d. For density estimation of mixtures of Gaussians, the current best sample complexity 4 e 2 /ε4 ) for general Gaussians and O(kd/ε e upper bounds (in terms of k and d) are O(kd ) for axis-aligned Gaussians, both due to Ashtiani et al. (2017). For the general Gaussian e 2 /ε2 ) and partitions this sample case, their method takes an i.i.d. sample of size O(kd 2 2 e in every possible way into k subsets. Based on those partitions, k O(kd /ε ) “candidate distributions” are generated. The problem is then reduced to learning with respect to that finite class of candidates. Their sample complexity has a suboptimal factor of 1/ε4, of which 1/ε2 arises in their approach for choosing the best candidate, and another factor 1/ε2 is due to the exponent in the number of candidates. Our approach via compression schemes also ultimately reduces the problem to learning with respect to finite classes. However, our compression technique leads to a more refined bound. In the case of mixtures of Gaussians, one factor of 1/ε2 is again incurred due to learning with respect to finite classes. The key is that the number of compressed samples has no additional factor of 1/ε2, so the overall sample complexity bound has only 2 e a O(1/ε ) dependence on ε. As for lower bounds on the sample complexity, much fewer results are known for learning mixtures of Gaussians. The only lower bound of which we are aware is due to 2 e Suresh et al. (2014), who show a bound of Ω(kd/ε ) for learning mixtures of axis-aligned Gaussians (and hence for general Gaussians as well). This bound is tight for the axisaligned case, as we show in Theorem 1.3, but loose in the general case, as we show in Theorem 1.2. 1.3 Our techniques We introduce a novel method for learning distributions via a form of sample compression. Given a class of distributions, suppose there is a method for “compressing” the samples generated by any distribution in the class. Further, suppose there exists a fixed decoder for the class, such that given the compressed set of instances and a sequence of bits, it approximately recovers the original distribution. In this case, if the size of the compressed set and the number of bits is guaranteed to be small, we show that the sample complexity of learning that class is small as well. More precisely, say a class of distributions admits (τ, t, m) compression if there exists a decoder function such that upon generating m i.i.d. samples from any distribution in 5 the class, we are guaranteed, with reasonable probability, to have a subset of size at most τ of that sample, and a sequence of at most t bits, on which the decoder outputs an approximation to the original distribution. Note that τ, t, and m can be functions of ε, the accuracy parameter. This definition is generalized to a stronger notion of robust compression, where the target distribution is to be encoded using samples that are not necessarily generated from the target itself, but are generated from a distribution that is close to the target. We prove that robust compression implies agnostic learning. In particular, if a class admits (τ, t, m) robust compression, then the sample complexity of agnostic learning with respect e + (τ + t)/ε2 ) (Theorem 3.5). to this class is bounded by O(m An attractive property of robust compression is that it enjoys two closure properties. Specifically, if a base class admits robust compression, then the class of k-mixtures of that base class, as well as the class of products of the base class, are robustly compressible (Lemmas 3.6 and 3.7). Consequently, it suffices to provide a robust compression scheme for the class of single Gaussian distributions in order to obtain a compression scheme for classes of mixtures of Gaussians (and therefore, to be able to bound their sample complexity). We prove e e 2 ), O(d)) e that the class of d-dimensional Gaussian distributions admits (O(d), O(d robust e compression (Lemma 4.2). The high level idea is that by generating O(d) samples from a Gaussian, one can get some rough sketch of the geometry of the Gaussian. In particular, the convex hull of the points drawn from a Gaussian enclose an ellipsoid centered at the mean and whose principal axes are the eigenvectors of the covariance matrix. Using ideas from convex geometry and random matrix theory, we show one can in fact encode the center of the ellipsoid and the principal axes using a convex combination of these samples. Then we discretize the coefficients and obtain an approximate encoding. The above results together imply tight (up to logarithmic factors) upper bounds of 2 e e O(kd2 /ε2 ) for mixtures of k Gaussians, and O(kd/ε ) for mixtures of k axis-aligned Gausd sians over R . The robust compression framework we introduce is quite flexible, and can be used to prove sample complexity upper bounds for other distribution classes as well. Lower bound. For proving our lower bound for mixtures of Gaussians, we first prove e 2 /ε2 ) for learning a single Gaussian. Although the approach is quite a lower bound of Ω(d intuitive, the details are intricate and much care is required to make a formal proof. The 2 main step is to construct a large family (of size 2Ω(d ) ) of covariance matrices such that the associated Gaussian distributions are well-separated in terms of their total variation distance while simultaneously ensuring that their Kullback-Leibler divergences are small. Once this is established, we can then apply a generalized version of Fano’s inequality to complete the proof. 6 2 To construct this family of covariance matrices, we sample 2Ω(d ) matrices from the following probabilistic process: start with an identity covariance matrix. Then choose a random subspace of dimension d/9 and slightly increase the eigenvalues corresponding √ to this eigenspace from 1 to roughly 1 + ε/ d. It is easy to bound the KL divergence between the constructed Gaussians. To lower bound the total variation, we show that for every pair of these distributions, there is some subspace for which a vector drawn from one Gaussian will have slightly larger projection than a vector drawn from the other Gaussian. Quantifying this gap will then give us the desired lower bound on the total variation distance. 1.4 Paper outline We set up our formal framework and notations in Section 2. In Section 3, we define compression schemes for distributions, prove their closure properties, and show their connection with density estimation. Theorem 1.1 and Theorem 1.3 are proved in Section 4. Theorem 1.2 is proved in Section 5. All omitted proofs can be found in the appendix. 2 Preliminaries A distribution learning method or density estimation method is an algorithm that takes as input a sequence of i.i.d. samples generated from a distribution g, and outputs (a description of) a distribution ĝ as an estimation for g. We work with continuous distributions in this paper, and so we identify a probability distribution by its probability density function. Let f1 and f2 be two probability distributions defined over the Borel σ-algebra B. The total variation (TV) distance between f1 and f2 is defined by Z 1 TV(f1 , f2 ) := sup (f1 (x) − f2 (x))dx = kf1 − f2 k1 , 2 B∈B B R where kf k1 := Rd \|f (x)\|dx is the L1 norm of f . The Kullback-Leibler (KL) divergence between f1 and f2 is defined by Z f1 (x) dx. KL(f1 k f2 ) := f1 (x) log f2 (x) Rd In the following definitions, F is a class of probability distributions, and g is a distribution (not necessarily in F ). Definition 2.1 (ε-approximation, (ε, C)-approximation). A distribution ĝ is an ε-approximation for g if kĝ − gk1 ≤ ε. A distribution ĝ is an (ε, C)-approximation for g with respect to F if kĝ − gk1 ≤ C · inf kf − gk1 + ε f ∈F 7 Definition 2.2 (PAC-learning distributions, realizable setting). A distribution learning method is called a (realizable) PAC-learner for F with sample complexity mF (ε, δ), if for all distribution g ∈ F and all ε, δ ∈ (0, 1), given ε, δ, and a sample of size mF (ε, δ) generated i.i.d. by that g, with probability at least 1 − δ (over the samples) the method outputs an ε-approximation of g. Definition 2.3 (PAC-learning distributions, agnostic setting). For C > 0, a distribution learning method is called a C-agnostic PAC-learner for F with sample complexity mC F (ε, δ), if for all distributions g and all ε, δ ∈ (0, 1), given ε, δ, and a sample of size mC F (ε, δ) generated i.i.d. from g, with probability at least 1 − δ the method outputs an (ε, C)approximation of g w.r.t. F . We sometimes say a class can be “C-learned in the agnostic setting” to indicate the existence of a C-agnostic PAC-learner for the class. The case C > 1 is sometimes called P wi = 1 } denote the semi -agnostic learning. Let ∆n := { (w1 , . . . , wn ) : wi ≥ 0, n-dimensional simplex. Definition 2.4 (k-mix(F )). Let F be a class of probability distributions. Then the class of k-mixtures of F , written k-mix(F ), is defined as k-mix(F ) := { Pk i=1 wi fi : (w1 , . . . , wk ) ∈ ∆k , f1 , . . . , fk ∈ F } Let d denote the dimension. A Gaussian distribution with mean µ ∈ Rd and covariance matrix Σ ∈ Rd×d is denoted by N (µ, Σ). If Σ is a diagonal matrix, then N (µ, Σ) is called an axis-aligned Gaussian. For a distribution g, we write X ∼ g to mean X is a random variable with distribution g, and we write S ∼ g m to mean that S is an i.i.d. sample of size m generated from g. Definition 2.5. A random variable X is said to be σ-subgaussian if Pr[\|X\| ≥ t] ≤ 2 exp(−t2 /σ 2 ) for any t > 0. Note that if X ∼ N (0, 1) then X is (1984, formula (7.1.13)). √ 2-subgaussian, see, e.g., Abramowitz and Stegun Definition 2.6. Let A, B be symmetric, positive definite matrices of the same size. The log-det divergence of A and B is defined as LD(A, B) := tr(B −1 A − I) − log det(B −1 A). We will use kvk or kvk2 to denote the Euclidean norm p of a vector v, kAk or kAk2 to denote the operator norm of a matrix A, and kAkF := tr(AT A) to denote the Frobenius norm of a matrix A. For x ∈ R, we will write (x)+ := max{0, x}. 8 3 Compression schemes and their connection with learning Let F be a class of distributions over a domain Z. Definition 3.1 (distribution decoder). A distribution decoder for F is a deterministic S∞ S n n function J : ∞ n=0 {0, 1} → F , which takes a finite sequence of elements of Z n=0 Z × and a finite sequence of bits, and outputs a member of F . Definition 3.2 (robust distribution compression schemes). Let τ, t, m : (0, 1) → Z≥0 be functions, and let r ≥ 0. We say F admits (τ, t, m) r-robust compression if there exists a decoder J for F such that for any distribution g ∈ F , and for any distribution q on Z with kg − qk1 ≤ r, the following holds: For any ε ∈ (0, 1), if the sample S is drawn from q m(ε) , then with probability at least 2/3, there exists a sequence L of at most τ (ε) elements of S, and a sequence B of at most t(ε) bits, such that kJ (L, B) − gk1 ≤ ε. Essentially, the definition asserts that with high probability, there should be a (small) subset of S and some (small number of) additional bits, from which g can be approximately reconstructed. We say that the distribution g is “encoded” with L and B, and in general we would like to have a compression scheme of a small size. This compression scheme is called “robust” since it requires g to be approximately reconstructed from a sample generated from q rather than g itself. Remark 3.3. In the definition above we required the probability of existence of L and B to be at least 2/3, but one can boost this probability to 1 − δ by generating a sample of size m(ε) log(1/δ). Next we show that if a class of distributions can be compressed, then it can be learned; thus we build the connection between robust compression and agnostic learning. We will need the following useful result about PAC-learning of finite classes of distributions, which immediately follows from Devroye and Lugosi (2001, Theorem 6.3) and a standard Chernoff bound. It states that a finite class of size M can be 3-learned in the agnostic setting using O(log(M/δ)/ε2 ) samples. Denote by [M] the set {1, 2, ..., M}. Throughout the paper, a/bc always means a/(bc). Theorem 3.4 (Devroye and Lugosi (2001)). There exists a deterministic algorithm that, given candidate distributions f1 , . . . , fM , a parameter ε > 0, and log(3M 2 /δ)/2ε2 i.i.d. samples from an unknown distribution g, outputs an index j ∈ [M] such that kfj − gk1 ≤ 3 min kfi − gk1 + 4ε, i∈[M ] with probability at least 1 − δ/3. 9 The proof of the following theorem appears in Appendix C.1. Theorem 3.5 (compressibility implies learnability). Suppose F admits (τ, t, m) r-robust compression. Let τ ′ (ε) := τ (ε/6) + t(ε/6). Then F can be max{3, 2/r}-learned in the agnostic setting using 1 τ ′ (ε) log(m( ε ) log(1/δ)) + log(1/δ) ′ τ (ε) ε ε 6 e m log + + 2 =O samples. O m 6 δ ε2 6 ε If F admits (τ, t, m) 0-robust compression, then F can be learned in the realizable setting using the same number of samples. We next prove two closure properties of compression schemes. First, Lemma 3.6 below implies that if a class F of distributions can be compressed, then the class of distributions that are formed by taking products of members of F can also be comQ pressed. If p1 , . . . , pd are distributions over domains Z1 , . . . , Zd , then di=1 pi denotes Qd the standard product distribution nQ o over i=1 Zi . For a class F of distributions, define d d := F i=1 pi : p1 , . . . , pd ∈ F . The following lemma is proved in Appendix C.2. Lemma 3.6 (compressing product distributions). If F admits (τ (ε), t(ε), m(ε)) r-robust compression, then F d admits (dτ (ε/d), dt(ε/d), m(ε/d) log(3d)) r-robust compression. Our next lemma implies that if a class F of distributions can be compressed, then the class of distributions that are formed by taking mixtures of members of F can also be compressed. The proof appears in Appendix C.3. Lemma 3.7 (compressing mixtures). If F admits (τ (ε), t(ε), m(ε)) r-robust compression, then k-mix(F ) admits (kτ (ε/3), kt(ε/3) + k log2 (4k/ε)), 48m(ε/3)k log(6k)/ε) r-robust compression. 4 4.1 Upper bound: learning mixtures of Gaussians by compression schemes Warm-up: learning mixtures of axis-aligned Gaussians by compression schemes In this section, we give a simple application of our compression framework to prove an 2 e upper bound of O(kd/ε ) for the sample complexity of learning mixtures of k axis-aligned Gaussians in the realizable setting. In the following section, we generalize these arguments to work for general Gaussians in the agnostic setting. 10 Lemma 4.1. The class of single-dimensional Gaussians admits a (3, O(log(1/ε)), 3) 0robust compression scheme. Proof. Let c < 1 < C be such that PrX∼N (0,1) [c < \|X\| < C] ≥ 0.99. Let N (µ, σ 2 ) be the target distribution. We first show how to encode σ. Let g1 , g2 ∼ N (µ, σ 2). Then g = √12 (g1 − g2 ) ∼ N (0, σ 2 ). So with probability at least 0.99, we have σc < \|g\| < σC. Conditioned on this event, this implies that there is a λ ∈ [−1/c, 1/c] such that λg = σ. We now choose λ̂ ∈ {0, ±ε/2C 2, ±2ε/2C 2 , ±3ε/2C 2 . . . , ±1/c} satisfying \|λ̂ − λ\| ≤ ε/(4C 2), and encode the standard deviation by (g1 , g2 , λ̂). The decoder then √ estimates σ̂ := λ̂(g1 − g2 )/ 2. Note that \|σ̂ − σ\| ≤ \|λ̂ − λ\|\|g\| ≤ σε/(4C) and that the encoding requires two sample points and O(log(C 2 /ε)) = O(log(1/ε)) bits (for encoding λ̂). Now we turn to encoding µ. Let g3 ∼ N (µ, σ 2 ). Then \|g3 − µ\| ≤ Cσ with probability at least 0.99. We will condition on this event, which implies existence of some η ∈ [−C, C] such that g3 +ση = µ. We choose η̂ ∈ {0, ±ε/2, ±2ε/2, ±3ε/2 . . . , ±C} such that \|η̂−η\| ≤ ε/4, and encode the mean by (g3 , η̂). The decoder estimates µ̂ := g3 + σ̂ η̂. Again, note that \|µ̂ − µ\| = \|ση − σ̂η̂\| ≤ \|ση − ση̂\| + \|ση̂ − σ̂ η̂\| ≤ σε/2. Moreover, encoding the mean requires one sample point and O(log(1/ε)) bits. To summarize, the decoder has \|µ̂ − µ\| ≤ σε/2 and \|σ̂ − σ\| ≤ σε/2. Plugging these bounds into Lemma A.4 gives kN (µ, σ 2) − N (µ̂, σ̂ 2 )k1 ≤ ε, as required. To complete the proof of Theorem 1.3 in the realizable setting, we note that Lemma 4.1 combined with Lemma 3.6 implies that the class of axis-aligned Gaussians in Rd admits a (O(d), O(d log(d/ε)), O(log(3d))) 0-robust compression scheme. Then, by Lemma 3.7, the class of mixtures of k axis-aligned Gaussians admit a (O(kd), O(kd log(d/ε) + k log(k/ε)), O(k log(k) log(3d)/ε)) 0-robust compression scheme. Applying Theorem 3.5 implies that 2 e the class of k-mixtures of axis-aligned Gaussians in Rd can be learned using O(kd/ε ) many samples in the realizable setting. 4.2 Agnostic learning mixtures of Gaussians by compression schemes e 2 /ε2) for the sample complexity of learning In this section we prove an upper bound of O(kd 2 e mixtures of k Gaussians in d dimensions, and an upper bound of O(kd/ε ) for the sample complexity of learning mixtures of k axis-aligned Gaussians, both in the agnostic sense. The heart of the proof is to show that Gaussians have robust compression schemes in any dimension. Lemma 4.2. For any positive integer d, the class of d-dimensional Gaussians admits an O(d log(2d)), O(d2 log(2d) log(d/ε)), O(d log(2d)) 2/3-robust compression scheme. 11 Remark 4.3. This lemma can be boosted to give an r-robust compression schemes for any r < 1 at the expense of worse constants hidden in the big Oh, but this will not yield any improvement in the final results. Remark 4.4. In the special case d = 1, there also exists a (4, 1, O(1/ε)) (i.e., constant size) 0.773-robust compression scheme using completely different ideas. The proof appears in Appendix D.4. Remarkably, this compression scheme has constant size, as the value of τ + t is independent of ε (unlike Lemma 4.2). This scheme could be used instead of Lemma 4.2 in the proof of Theorem 1.3, although it would not improve the sample complexity bound asymptotically. Proof of Theorem 1.1. Combining Lemma 4.2 and Lemma 3.7 implies that the class of k-mixtures of d-dimensional Gaussians admits a O(kd log(2d)), O(kd2 log(2d) log(d/ε) + k log(k/ε)), O(dk log k log(2d)/ε) e 2/3-robust compression scheme. Applying Theorem 3.5 with m(ε) = O(dk/ε) and τ ′ (ε) = e 2 k) shows that the sample complexity of learning this class is O(kd e 2 /ε2). This proves O(d Theorem 1.1. Proof of Theorem 1.3. Let G denote the class of 1-dimensional Gaussian distributions. By Lemma 4.2, G admits an (O(1), O(log(1/ε)), O(1)) 2/3-robust compression scheme. By Lemma 3.6, the class G d admits a (O(d), O(d log(d/ε)), O(log(3d))) 2/3-robust compression scheme. Then, by Lemma 3.7, the class k-mix(G d ) admits (O(kd), O(kd log(d/ε) + k log(k/ε)), O(k log(k) log(3d)/ε)) 2/3-robust compression. Applying Theorem 3.5 implies that the class of k-mixtures of axis-aligned Gaussians in Rd can be 3-agnostically learned 2 e using O(kd/ε ) many samples. 4.3 Proof of Lemma 4.2 Let Q denote the target distribution, which satisfies kQ − N (µ, Σ)k1 ≤ 2/3 for some Gaussian N (µ, Σ) which we are to encode. Note that this implies TV(Q, N (µ, Σ)) ≤ 1/3. Remark 4.5. The case of rank-deficient Σ can easily be reduced to the case of full-rank Σ. If the rank of Σ is k < d, any X ∼ N (µ, Σ) lies in some affine subspace S of dimension k. Thus, any X ∼ Q lies in S with probability at least 2/3. With high probability, after seeing 10d samples from Q, at least k + 1 points from S will appear in the sample. We encode S using these samples, and for the rest of the process we work in this affine space, and discard outside points. Hence, we may assume Σ has full rank d. We first prove a lemma that is similar to known results in random matrix theory (see Litvak, Pajor, Rudelson, and Tomczak-Jaegermann, 2005, Corollary 4.1), but is tailored 12 for our purposes. Its proof appears in Appendix D.1. Let S d−1 := and B2d := y ∈ Rd : kyk ≤ 1 . y ∈ Rd : kyk = 1 Lemma 4.6. Let q1 , . . . , qm be i.i.d. samples from a distribution Q where TV(Q, N (0, Id)) ≤ 2/3. Let √ T := { ±qi : kqi k ≤ 4 d }. Then for a large enough constant C > 0, if m ≥ Cd(1 + log d) then 1 d Pr B 6⊆ conv(T ) ≤ 1/6. 20 2 P P Suppose Σ = di=1 vi viT , where the vi vectors are orthogonal. Let Ψ := di=1 vi viT /kvi k. Note that both Σ and Ψ are positive definite, and that Σ = Ψ2 . Moreover, it is easy to P P see that Σ−1 = di=1 vi viT /kvi k4 and Ψ−1 = di=1 vi viT /kvi k3 . The following lemma is proved in Appendix D.2. Lemma 4.7. Let C > 0 be a sufficiently large constant. Given m = 2Cd(1 + log d) samples S from Q, where TV(Q, N (µ, Σ)) ≤ 1/3, with probability at least 2/3, one can encode vectors vb1 , . . . , b vd , µ b ∈ Rd satisfying kΨ−1 (b vj − vj )k ≤ ε/6d2 ∀j ∈ [d], and kΨ−1 (b µ − µ)k ≤ ε, using O(d2 log(2d) log(d/ε)) bits and the points in S. Lemma 4.2 now follows immediately from the following lemma, which is proved in Appendix D.3. P Lemma 4.8. Suppose Σ = Ψ2 = i∈[d] vi viT , where vi are orthogonal and Σ is full rank, and that kΨ−1 (b vj − vj )k ≤ ρ ≤ 1 ∀j ∈ [d], and kΨ−1 (b µ − µ)k ≤ ζ. Then TV(N (µ, X i∈[d] vi viT ), N (b µ, X i∈[d] 13 vbi b viT )) ≤ p 9d3 ρ2 + ζ 2/2. 5 The lower bound for Gaussians and their mixtures e 2 /ε2 ) for learning a single Gaussian, and In this section, we establish a lower bound of Ω(d 2 2 e then lift it to obtain a lower bound of Ω(kd /ε ) for learning mixtures of k Gaussians in d dimensions. Both our lower bounds consider the realizable setting (so they also hold in the agnostic setting). Our lower bound is based on the following lemma, which follows from Fano’s inequality in information theory (see Lemma E.1). Its proof appears in Appendix E.1. Lemma 5.1. Let F be a class of distributions such that for all small enough ε > 0 there exist N densities f1 , . . . , fN ∈ F with KL(fi k fj ) ≤ κ(ε) and TV(fi , fj ) = Ω(ε) ∀i 6= j ∈ [N]. Then any algorithm that learns F to within total variation distance ε with success probability at least 2/3 has sample complexity Ω (log N/(κ(ε) log(1/ε))). Theorem 5.2. Any algorithm that learns a general Gaussian in Rd in the realizable setting within total variation distance ε and with success probability 2/3 has sample complexity d2 Ω ε2 log3 (1/ε) . 2 Proof. Let r = 9 and λ = Θ(εd−1/2 log(1/ε)). Guided by Lemma 5.1, we will build 2Ω(d ) Gaussian distributions of the form fa := N (0, Σa ) where Σa = Id + λUa UaT , where each Ua is a d × d/r matrix with orthonormal columns. To apply Lemma 5.1, we need to give an upper bound on the KL-divergence between any two fa and fb , and a lower bound on their total variation distance. Upper bounding the KL divergence is easy: by Lemma A.1 and since kUaT Ub k2F ≥ 0, λ Ua UaT )(I + λUb UbT ) − I) 1+λ λ λ2 = Tr(λUb UbT − Ua UaT − Ua UaT Ub UbT ) 1+λ 1+λ λ2 λ (d/r) − kU T Ub k2F = λ(d/r) − 1+λ 1+λ a λ2 d ≤ ≤ λ2 d/(2r) = O(ε2 log2 (1/ε)), r + rλ 2 KL(fa k fb ) = Tr(Σ−1 a Σb − I) = Tr((I − as required. Our next goal is to give a lower bound on the total variation distance between fa and fb . For this, we would like the matrices {Ua } to be “spread out,” in the sense that their columns should be nearly orthogonal. This is formalized in Lemma 5.3 below, where we d show if we choose the Ua randomly, we can achieve kUaT Ub k2F ≤ 2r for any a 6= b. Then, 14 if Sa is the subspace spanned by the columns of Ua , then we expect that a Gaussian drawn from N (0, Σa ) should have a slightly larger projection onto Sa then a Gaussian drawn from N (0, Σb). This will then allow us to give a lower bound on the total variation distance between N (0, Σa ) and N (0, Σ b ). √Moreprecisely, in Lemma 5.4 we show that λ d/r d T 2 √ = Ω(ε), completing the proof. kUa Ub kF ≤ 2r implies TV(fa , fb ) = Ω log(r/λ d) We defer the proofs of the following lemmas to Appendix E.2 and Appendix E.3, respectively. 2 Lemma 5.3. Suppose d ≥ r ≥ 9. Then there exists 2Ω(d /r) orthonormal d × d/r matrices d {Ua } such that for any a 6= b we have kUaT Ub k2F ≤ 2r . √ d/r ∈ (0, 1/3). If kUaT Ub k2F ≤ d/(2r), then Lemma 5.4. Suppose that λ ≤ 1 ≤ r, and λ √ λ d/r √ TV(fa , fb ) = Ω log(r/λ . d) Finally, in Appendix E.4 we prove our lower bound for mixtures. Theorem 5.5. Any algorithm that learns a mixture of k general Gaussians in Rd in the realizable setting within total distance ε and with success probability at least 2/3 variation kd2 has sample complexity Ω ε2 log3 (1/ε) . 6 Further discussion A central open problem in distribution learning and density estimation is characterizing the sample complexity of learning a distribution class. An insight from supervised learning theory is that the sample complexity of learning a class (of concepts, functions, or distributions) may be proportional to some kind of intrinsic dimension of the class divided by ε2 , where ε is the error tolerance. For the case of agnostic binary classification, the intrinsic dimension is captured by the VC-dimension of the concept class (see Vapnik and Chervonenkis (1971); Blumer, Ehrenfeucht, Haussler, and Warmuth (1989)). For the case of distribution learning with respect to ‘natural’ parametric classes, we expect this dimension to be equal to the number of parameters. In this paper, we showed that this is indeed the case for the class of Gaussians, axis-aligned Gaussians, and their mixtures in any dimension. In binary classification, the combinatorial notion of Littlestone-Warmuth compression has been shown to be sufficient (Littlestone and Warmuth, 1986) and necessary (Moran and Yehudayoff, 2016) for learning. In this work, we showed that the new but related notion of robust distribution compression is sufficient for distribution learning. Whether the existence of compression schemes is necessary for learning an arbitrary class of distributions remains an intriguing open problem. 15 We would like to mention that while it may first seem that the VC-dimension of the Yatracos set associated with a class of distributions can characterize its sample complexity, it is not hard to come up with examples where this VC-dimension is infinite while the class can be learned with finite samples. Covering numbers do not work, either; for instance the class of Gaussians do not have a bounded covering number in the TV metric, nevertheless it is learnable with finite samples. A concept related to compression is that of core-sets. In a sense, core-sets can be viewed as a special case of compression, where the decoder is required to be the empirical error minimizer. See the work of (Lucic, Faulkner, Krause, and Feldman, 2017) for using core-sets in maximum likelihood estimation. A Standard results Lemma A.1 (Rasmussen and Williams (2006, Equation A.23)). For two full-rank Gaussians N (µ, Σ) and N (µ′ , Σ′ ), their KL divergence is KL(N (µ, Σ) k N (µ′, Σ′ )) = 1 Tr(Σ−1 Σ′ − I) + (µ − µ′ )T Σ−1 (µ − µ′ ) − log det(Σ′ Σ−1 ) . 2 Lemma A.2 (Pinsker’s Inequality (Tsybakov, 2009, Lemma 2.5)). For any two distributions A and B, we have 2 TV(A, B)2 ≤ KL(A k B). Lemma A.3. For two full-rank Gaussians N (µ, Σ) and N (µ′, Σ′ ), their total variation distance is bounded by 2 TV(N (µ, Σ), N (µ′, Σ′ ))2 ≤ KL(N (µ, Σ k N (µ′, Σ′ )) 1 = LD(Σ, Σ′ ) + (µ − µ′ )T Σ−1 (µ − µ′ ) . 2 Proof. Follows from Lemma A.1 and Lemma A.2. Lemma A.4. For any µ, σ, µ b, σ b ∈ R with \|b µ − µ\| ≤ εσ and \|b σ − σ\| ≤ εσ and ε ∈ [0, 2/3] we have kN (µ, σ 2) − N (b µ, σ b2 )k1 ≤ 2ε. Proof. By Lemma A.3, σ b2 4 TV(N (µ, σ 2), N (b µ, σ b2 ))2 ≤ 2 −1−log σ 2 2 ! σ b2 \|µ − µ b\|2 σ b σ b +ε2 . + ≤ −1−log 2 2 σ σ σ σ Since z := σ b/σ ∈ [1−ε, 1+ε] and ε ≤ 2/3, using the inequality x2 −1−log(x2 ) ≤ 3(x−1)2 valid for all \|x − 1\| ≤ 2/3, we find 1 1 TV(N (µ, σ 2), N (b µ, σ b2 ))2 ≤ (3(z − 1)2 + ε2 ) ≤ (4ε2) = ε2 . 4 4 16 And the lemma follows since the L1 distance is twice the TV distance. Fact A.5. Let X and Y be arbitrary random variables on the same space. For any function f , we have TV(f (X), f (Y )) ≤ TV(X, Y ). Proof. This follows from the observation that Pr [f (X) ∈ A] − Pr [f (Y ) ∈ A] = Pr X ∈ f −1 (A) − Pr Y ∈ f −1 (A) ≤ TV(X, Y ), so taking supremum of the left-hand side gives the result. Lemma A.6 (Laurent and Massart (2000, Lemma 1)). Let X have the chi-squared disP tribution with parameter d; that is, X = di=1 Xi2 where the Xi are i.i.d. standard normal. Then, √ Pr[X − d ≥ 2 dt + 2t] ≤ exp(−t) and √ Pr[d − X ≥ 2 dt] ≤ exp(−t). d. The first inequality above implies, in particular, that Pr[X ≥ 16d] ≤ exp(−3) for any Lemma A.7. Let g1 , . . . , gm ∈ Rd be independent samples from N (0, I). For ε ∈ [0, 1], Pr[ Proof. Note that X = 1 m Pm √1 m i=1 gi Pm i=1 gi 2 ≥ (1 + ε)d/m] ≤ exp(−ε2 d/9). 2 has the chi-squared distribution with parameter d. 2 Applying Lemma A.6 with t = ε d/9 shows that Pr[X ≥ (1 + ε)d] ≤ exp(−ε2 d/9). Lemma A.8 (Theorem 3.1.1 in Vershynin (2018)). Let g ∼ N (0, Id). Then (kgk2 − √ is O(1)-subgaussian. Consequently, (kgk2 − d)+ is also O(1)-subgaussian. √ d) Lemma A.9 (Proposition 2.5.2 in Vershynin (2018)). A random variable X is σ-subgaussian if and only if supp≥1 p−1/2 (E\|X\|p)1/p ≤ Cσ for some global constant C > 0. Lemma A.10 (Hoeffding’s Inequality, Proposition 2.6.1 in Vershynin (2018)). Let X1 , . . . , Xn be independent, mean-zero random variables and suppose Xi is σi -subgaussian. Then, for some global constant c > 0 and any t ≥ 0, # " X −ct2 Pr Xi > t ≤ 2 exp P 2 . i σi i 17 Lemma A.11 (Bernstein’s Inequality, Theorem 2.8.1 in Vershynin (2018)). Let g1 , . . . , gn ∼ N (0, 1) and a1 , . . . , an > 0. Then, there is a global constant c > 0 such that for every t ≥ 0, # " n n X X t2 t 2 Pr . ai gi − ai ≥ t ≤ 2 exp −c min Pn 2 , i=1 ai maxi ai i=1 i=1 Theorem A.12 (Gordon’s Theorem, Theorem 5.32 in Vershynin (2012)). Let G be a m×n √ √ matrix with entries independently drawn from N (0, 1). Then Eσmin (G) ≥ m − n. Lemma A.13 (Corollary 5.50 and Remark 5.51 in Vershynin (2012)). Let X1 , . . . , Xm ∼ N (0, Σ), where Σ is d × d, and let 0 < ε < 1 < t. If m ≥ C(t/ε)2 d, then with probability 1 − 2 exp(−t2 d) we have m 1 X Xi XiT − Σ ≤ εkΣk. m i=1 Lemma A.14 (Corollary 4.2.13 in Vershynin (2018)). For any ε ∈ (0, 1), there exists an ε-net for B2d of size (3/ε)d . B Efficient algorithm for learning a single Gaussian by empirical mean/covariance estimation In this section we give a simple algorithm for learning a single high dimensional Gaussian, with sample complexity O(d2/ε2 ) and computational complexity O(d4/ε2 ). Lemma B.1. Let v1 , . . . , vm ∈ Rd be independent samples from N (µ, Σ). Let v̄ = Pm 1 i=1 vi . Then m Pr[(v̄ − µ)T Σ−1 (v̄ − µ) ≥ 2d/m] ≤ exp(−d/9). Proof. Let gi = Σ−1/2 (vi − µ), so that g1 , . . . , gm are independent samples from N (0, I). Then Pr (v̄ − µ)T Σ−1 (v̄ − µ) ≥ (1 + ε)d/m = Pr 1 m Pm i=1 gi ≤ exp(−ε2 d/9), 2 ≥ (1 + ε)d/m by Lemma A.7. We write B A if A−B is a positive semidefinite matrix. Observe that, x−1−log x ≤ (x − 1)2 for any x ∈ [1/2, ∞). 18 Lemma B.2. Let A, B be symmetric, positive definite matrices, satisfying (1 − α)B A (1 + α)B for some α ∈ [0, 1/2]. Then LD(A, B) ≤ dα2. Proof. Let λ1 , . . . , λd be the eigenvalues of B −1 A. By the hypothesis, each λi ∈ [1 − α, 1 + α]. So, LD(A, B) = tr(B −1 A − I) − log det(B −1 A) = d X i=1 (λi − 1) − log d d X X (λi − 1)2 ≤ dα2 . (λi − 1 − log(λi )) ≤ = d Y λi i=1 i=1 i=1 Our result immediately follows from the following theorem. Theorem B.3. Let m = Cd2 /ε2 for a large enough constant C. Let v1 , . . . , vm be i.i.d. P Pm 1 T samples from N (µ, Σ). Let µ̃ = m1 m i=1 vi and Σ̃ = m i=1 vi vi respectively be the empirical mean and the empirical covariance matrix. Then TV(N (µ̃, Σ̃), N (µ, Σ)) ≤ ε with probability at least 1 − 3 exp(−d/9). Proof. We will show that, with probability at least 1 − 3 exp(−d/9), KL(N (µ̃, Σ̃) k N (µ, Σ)) ≤ ε2 , and the theorem follows from Pinsker’s inequality (Lemma A.2). By e −Σ ≤ standard concentration for Gaussian matrices (see Lemma A.13) we have Σ √ ε/ d =: α with probability at least 1 − 2 exp(−d). That is, (1 − α)Σ Σ̃ (1 + α)Σ. Applying Lemma B.2 shows that LD(Σ̃, Σ) ≤ dα2 = ε2 . Next, by Lemma B.1 we have (µ̃ − µ)T Σ−1 (µ̃ − µ) ≤ 2d/m = ε2 /18d with probability at least 1 − exp(−d/9). So KL(N (µ̃, Σ̃) k N (µ, Σ)) = 1 LD(Σ̃, Σ) + (µ̃ − µ)T Σ−1 (µ̃ − µ) ≤ ε2 , 2 with probability at least 1 − 3 exp(−d/9). It is easy to see that, by multiplying the sample size by log(1/δ), one can boost the success probability to 1 − δ, for any δ ∈ (0, 1). C C.1 Omitted proofs from Section 3 Proof of Theorem 3.5 We give the proof for the agnostic case. The proof for the realizable case is similar. Let q be the target distribution that the samples are being generated from. Let α = inf f ∈F kf −qk1 19 be the approximation error of q with respect to F . The goal of the learner is to find a distribution ĥ such that kĥ − qk1 ≤ max{3, 2/r} · α + ε. First, consider the case α ≤ r. In this case, we develop a learner that finds a distribuε tion ĥ such that kĥ−qk1 ≤ 3α+ε. Let g ∈ F be a distribution such that kg −qk1 ≤ α+ 12 (such a g exists by the definition of α). By assumption, F admits (τ, t, m) compression. Let J denote the corresponding decoder. Given ε, the learner first asks for an i.i.d. sample S ∼ q m(ε/6)·log(2/δ) . By the definition of robust compression, we know that with probability at least 1 − δ/2, there exist L ∈ S τ (ε/6) and B ∈ {0, 1}t(ε/6) such that kJ (L, B) − gk ≤ ε/6 (see Remark 3.3). Let h∗ := J (L, B). The learner is of course unaware of L and B. However, given the sample S, it can try all of the possibilities for L and B and create a candidate set of distributions. More concretely, let H = {J (L, B) : L ∈ S τ (ε/6) , B ∈ {0, 1}t(ε/6) }. Note that ′ \|H\| ≤ (m(ε/6) log(2/δ))τ (ε/6) 2t(ε/6) ≤ (m(ε/6) log(2/δ))τ (ε) . Since H is finite, we can use the algorithm of Theorem 3.4 to find a good candidate ĥ from H. In particular, we set the accuracy parameter in Theorem 3.4 to be ε/16 and the confidence parameter to be δ/2. In this case, Theorem 3.4 requires ′ τ (ε) log(m( 6ε ) log( 1δ )) + log( 1δ ) log(6\|H\|2/δ) e ′ (ε)/ε2 ) = O(τ =O 2 2 2(ε/16) ε additional samples, and its output ĥ will be an (ε,3)-approximation of q: kĥ − qk1 ≤ 3kh∗ − qk1 + 4 ε ε ε ≤ 3(kh∗ − gk1 + kg − qk1 ) + ≤ 3(ε/6 + (α + ε/12)) + 16 4 4 ≤ 3α + ε. ′ e Note that the above procedure uses O(m(ε/6) + τ ε(ε) 2 ) samples, and the probability of failure is at most δ (i.e., the probability of either H not containing a good h∗ , or the failure of Theorem 3.4 in choosing a good candidate among H, is bounded by δ/2 + δ/2 = δ). The other case, α > r, is trivial: the learner outputs some distribution b h. Since b h and 2 b q are density functions, we have kh − qk1 ≤ 2 < · α < max{3, 2/r} · α + ε. r C.2 Proof of Lemma 3.6 The following proposition is standard. Proposition C.1 (Lemma 3.3.7 in Reiss (1989)). For i ∈ [d], let pi and qi be probability P distributions over the same domain Z. Then kΠdi=1 pi − Πdi=1 qi k1 ≤ di=1 kpi − qi k1 . 20 Proof of Lemma 3.6. Let G = Πdi=1 gi be an arbitrary element of F d . Let Q be an arbitrary distribution over Z d , subject to kG − Qk1 ≤ r. Let q1 , . . . , qd be the marginal distributions of Q on the d components. First, observe that, since projection onto a coordinate cannot increase the total variation distance (see Fact A.5), we have kqj −gj k1 ≤ r for each j ∈ [d]. We know that F admits (τ, t, m) r-robust compression. Call the corresponding decoder J , and let m0 := m(ε/d) log(3d), and S ∼ Qm0 . The goal is then to encode an εapproximation of G using dτ (ε/d) elements of S and dt(ε/d) bits. Note that each element of S is a d-dimensional vector. For each i ∈ [d], let Si ∈ Z m0 be the set of the ith components of elements of S. By definition of qi , we have Si ∼ qim0 for each i. Thus, for each i ∈ [d], since kqi −gi k ≤ r, with probability at least 1−1/3d there exists a sequence Li of at most τ (ε/d) elements of Si , and a sequence Bi of at most t(ε/d) bits, such that kJ (Li , Bi ) − gi k1 ≤ ε/d. By the union bound, this assertion holds for all i ∈ [d], with probability at least 2/3. We may encode these L1 , . . . , Ld , B1 , . . . , Bd using dτ (ε/d) elements of S and dt(ε/d) bits. Our decoder for F d then extracts L1 , . . . , Ld , B1 , . . . , Bd Q from these elements and bits, and then outputs di=1 J (Li , Bi ) ∈ F d . Finally, ProposiP tion C.1 gives kΠdi=1 J (Li , Bi ) − Gk1 ≤ di=1 kJ (Li , Bi ) − gi k1 ≤ d × ε/d ≤ ε, completing the proof. C.3 Proof of Lemma 3.7 We will need the following standard proposition. Proposition C.2. Let g and g ∗ be probability densities with kg − g ∗k1 = ρ and g ∗ = P w f , with (w1 , . . . , wk ) ∈ ∆k and where each fi is a density. Then we may write i∈[k] P i i g = i∈[k] wi Gi , such that each Gi is a density, and for each i we have kfi − Gi k1 ≤ ρ. Proof. Write ∗ g =g +h= k X wi fi + h = i=1 k X wi (fi + h) (1) i=1 with khk1 = ρ. Note that fi + h is not necessarily a probability density function. Let D denote the set of probability density functions, that is, the set of nonnegative functions with unit L1 norm. Note that this is a convex set. Since projection is a linear operator, by P projecting both sides of (1) onto D we find g = ki=1 wi Gi , where Gi is the L1 projection of fi + h onto D (since g ∈ D, the projection of g onto D is itself). Also, since fi ∈ D and projection onto a convex set does not increases distances, we have kfi − Gi k1 ≤ kfi − (fi + h)k1 = khk1 = ρ, as required. 21 Proof of Lemma 3.7. Let g be the distribution from which we have 48m(ε/3)k log(6k)/ε samples, and suppose g ∗ ∈ k-mix(F ) is the distribution to be compressed, so kg−g ∗ k1 ≤ r. P Thus we have g ∗ = i∈[k] wi fi , with each fi ∈ F , and (w1 , . . . , wk ) ∈ ∆k . By ProposiP tion C.2, we also have g = i∈[k] wi Gi for some distributions G1 , . . . , Gk , such that for each i we have kfi − Gi k1 ≤ r. We view g as a mixture of these k distributions, so the samples from g can be partitioned into k parts, so that samples from the ith part have distribution Gi . We compress each of the parts individually. Moreover, we compress the mixing weights w1 , . . . , wk using bits, as follows. Consider an (ε/3k)-net in ℓ∞ for ∆k , of size (1 + 3k/ε)k . Such a net can be obtained from a mesh of grid-size ε/3k for [0, 1]k , and projecting each of its points onto ∆k . Let (w b1 , . . . , w bk ) be an element in the net that has k(w b1 , . . . , w bk ) − (w1 , . . . , wk )k∞ ≤ ε/3k; then, wi − wbi ≤ ε/3k for all i. Moreover, the particular element (w b1 , . . . , w bk ) of the net k can be encoded using log2 ((1 + 3k/ε) ) ≤ k log2 (4k/ε) bits. For any i ∈ [k], we say component i is negligible if wi ≤ ε/(6k). Since the total number of samples is 48m(ε/3)k log(6k)/ε, by a standard Chernoff bound combined with a union bound over the k components, with probability at least 5/6, for each non-negligible component i, we have at least m(ε/3) log(6k) samples from i. Let i be a non-negligible component. Since F admits (τ, t, m) robust compression and fi ∈ F and kfi − Gi k1 ≤ r, with probability at least 1 − 1/6k there exists τ (ε/3) samples from part i and t(ε/3) bits, from which the decoder can construct a distribution fbi with kfi −fbi k1 ≤ ε/3. Using a union bound over the k components, this is true uniformly over all non-negligible components, with probability at least 5/6. (Note that, for negligible components i, there is no guarantee P b about fbi .) Hence, given the mixing weights w b1 , . . . , w bk , the decoder outputs w bi fi . The total number of instances used to encode is kτ (ε/3). Similarly, the total number of used bits is not more than kt(ε/3) + k log2 (4k/ε). Thus to complete the proof of the P P b lemma, we need only show that k wi fi − w bi fi k1 ≤ ε. Let L ⊆ [k] denote the set of negligible components. We have X (w bi fbi − wi fi ) i∈[k] ≤ 1 ≤ ≤2 X i∈[k] X i∈L X wi (fbi − fi ) wi (fbi − fi ) wi + i∈L X + X (w bi − wi )fbi i∈[k] 1 + 1 X i∈L / wi (ε/3) + i∈L / wi (fbi − fi ) X i∈[k] completing the proof of the lemma. 1 + 1 X i∈[k] \|w bi − wi \| fbi 1 ε/3k × 1 ≤ ε/3 + ε/3 + ε/3 = ε, 22 D D.1 Omitted proofs from Section 4 Proof of Lemma 4.6 First we show the following proposition implies max\|h y, q i\| ≥ q∈S 1 20 1 Bd 20 2 ⊆ conv(T ): ∀y ∈ S d−1 . (2) For, let P := conv(T ). Its polar is P ◦ = y ∈ Rd : \|h y, q i\| ≤ 1 ∀q ∈ T . So (2) implies 1 )B2d . P ◦ ⊆ 20B2d . As polarity reverses containment, we obtain P ⊇ (20B2d )◦ = ( 20 We now bound the probability that (2) fails. For this, let g ∼ N (0, Id) and let we Xy := h y, g i. Notice that Xy ∼ N (0, 1). Since the pdf of Xy is bounded above by 1, √ 1 have Pr \|Xy \| ≤ 10 ≤ 1/5. Moreover, by Lemma A.6, the probability that kgk2 ≥ 4 d is ≤ exp(−3). Hence √ 1 ∨ kgk2 ≥ 4 d ≤ 1/5 + exp(−3) < 0.25. Pr \|Xy \| ≤ 10 √ 1 Now let Yy,i := h y, qi i and let Ey,i be the event {\|Yy,i \| ≤ 10 ∨ kqi k > 4 d}. As TV(Q, N (0, Id)) ≤ 2/3, we have Pr[Ey,i ] ≤ 0.25 + 2/3 < 0.92. Thus   ^ Pr  Ey,i  < (0.92)m i∈[m] √ √ Let N be an (1/80 d)-net of S d−1 with \|N\| ≤ (240 d)d . By a union since √ d bound, m m ≥ Cd(1+log d) for C large enough, with probability at least 1−(240 √ d) (0.92) ≥ 5/6, 1 for all y ∈ N there exists i ∈ [m] such that \|Yy,i \| ≥ 10 and kqi k ≤ 4 d. √ Suppose this event holds.√ Let y ∈ S d−1 , and let y ′ ∈ N satisfy ky − y ′k2 ≤ 1/80 d. 1 . These imply ±qi ∈ T and Let qi be such that kqi k ≤ 4 d and \|Yy′ ,i \| ≥ 10 √ 1 \|Yy,i \| ≥ \|Yy′ ,i \| − kqi k/80 d ≥ 1/10 − 1/20 = , 20 as required. D.2 Proof of Lemma 4.7 Let X1 , . . . , X2m be the samples, and let Yi := √12 Ψ−1 (X2i − X2i−1 ) for i ∈ [m]. Observe that if X2i and X2i−1 were N (µ, Σ), then √12 Ψ−1 (X2i − X2i−1 ) would have been N (0, I). Since X2i and X2i−1 have TV distance at most 1/3 from N (µ, Σ), Yi has TV distance at most 2/3 from N (0, I) (this can be seen, e.g., by the coupling characterization of the TV 23 √ distance). Let I := {i ∈ [m] : kYi k ≤ 4 d}. By Lemma 4.6, with probability ≥ 5/6 we have 1 d B ⊆ conv{±Yi : i ∈ I} C 2 with C = 20. We give the encoding for b vj conditioned on this event. Fix j ∈ [d]. Observe that Ψ−1 vj = vj /kvj k has unit norm, so we can write X Ψ−1 vj /C = θj,i Yi i∈[m] for some vector θj ∈ [−1, 1]m supported on I. Applying Ψ to both sides, we obtain C X θj,i (X2i − X2i−1 ). vj = √ 2 i∈[m] For encoding vbj , consider an (ε/24Cmd3)-net for [−1, 1]m in ℓ∞ distance. The size of the net is (48Cmd3 /ε)m , so any element of the net can be described using O(m log(d/ε)) bits. Let θbj be an element in the net that is closest to θj subject to support(θbj ) ⊆ I, and let P vj := √C2 i∈[m] θbj,i (X2i − X2i−1 ). We have, b m C X (θj,i − θbj,i )Ψ−1 (X2i − X2i−1 )k kΨ−1 (b vj − vj )k = √ k 2 i=1 √ C ≤ √ m(max \|θj,i − θbj,i \|)(max 2kYi k) i i∈I 2 √ √ C ≤ √ m(ε/24Cmd3 )(4 2 d) ≤ ε/6d2, 2 as required. The total number of bits used to encode each vbj is O(m log(d/ε)), giving a total number of O(d2 log(2d) log(d/ε)) bits for encoding them all. We next describe the encoding of µ b. Let Zi := Ψ−1 (Xi − µ) and observe that Zi has a distribution with TV distance at most 2/3 to N (0, I). So, using Lemma A.6, p √ Pr[kZi k ≥ 4 d] ≤ exp(−3) + 1/3 < 1/6, √ which means, with probability at least 5/6, min{kZ1 k, kZ2k} ≤ 4 d. We give an encoding for µ b provided this event occurs. √ P Without loss of generality, assume kZ1 k ≤ 4 d, and suppose Z1 = j∈[d] λj ej , {ei } P P 2 being the standard basis. This implies µ = X1 − j∈[d] λj vj , with λj ≤ 16d2 . Consider √ d an (ε/3d)-net for 4 dB2 of size (36d2/ε)d , and let b λ be the closest element to λ in this P b net. We encode µ b = X1 − j∈[d] λj vbj . Note that this requires O(d log(d/ε)) more bits, which is dominated by the number of bits for encoding the b vj . 24 Finally, observe that, kΨ−1 (b µ − µ)k = k ≤ X (λbj − λj )Ψ−1 (vj − vbj )k j X j bj )Ψ−1 vj k kλbj (Ψ−1 vj − Ψ−1 vbj ) + (λj − λ n o bj \|kΨ−1 vj k ≤ d max \|λbj \|kΨ−1vj − Ψ−1 b vj k + \|λj − λ j √ ε ≤ d · 4 d · 2 + d(ε/3d) ≤ ε, 6d as required. D.3 Proof of Lemma 4.8 b := Let Σ P i vi vbiT . We will show that b b ≤ 9d3 ρ2 LD(Σ, Σ) (3) If this is true, Lemma A.3 gives 1 b + (µ − µ b 2 ≤ 1 LD(Σ, Σ) b)T Σ−1 (µ − µ b) ≤ (9d3 ρ2 + ζ 2), TV(N (µ, Σ), N (b µ, Σ)) 4 4 completing the proof. For showing (3), note that we have b = LD(Ψ−1 ΣΨ−1 , Ψ−1ΣΨ b −1 ) LD(Σ, Σ) X X X = LD( Ψ−1 vi viT Ψ−1 , Ψ−1 vbi b viT Ψ−1 ) = LD(I, Ψ−1 b vi vbiT Ψ−1 ) i i i For the first equality we have used the fact that, if A and B are positive definite and C is invertible, then LD(A, B) = LD(CAC, CBC). Indeed, as we proved in Lemma B.2, LD(A, B) only depends on the spectrum of B −1 A. Observe that (v, λ) is an eigenvector/eigenvalue pair for B −1 A if and only if (C −1 v, λ) is an eigenvector/eigenvalue pair for (CBC)−1 CAC; hence these two matrices have the same spectrum. P Let B := i Ψ−1 b vi vbiT Ψ−1 . We shall show kB − Ik ≤ 3dρ, which implies −3dρI 4 b = LD(I, B) ≤ 9d3 ρ2 . We B − I 4 3dρI and together with Lemma B.2 implies LD(Σ, Σ) have X X kB − Ik = k (Ψ−1 b vi b viT Ψ−1 − Ψ−1 vi viT Ψ−1 )k ≤ kΨ−1 b vi vbiT Ψ−1 − Ψ−1 vi viT Ψ−1 k i = X i i kxi xTi − yi yiT k, vi and yi := Ψ−1 vi (here we have used the fact that Ψ−1 is symmetric). Note with xi := Ψ−1 b that kyi k = kΨ−1 vi k = 1, and that, by the lemma hypothesis, kxi − yi k ≤ ρ. Applying Lemma D.1 below concludes the proof. 25 Lemma D.1. Suppose x, y satisfy kyk = 1 and kx − yk ≤ ε ≤ 1. Then we have kxxT − yy Tk ≤ 3ε. Proof. Suppose x = y + z with kzk ≤ ε. Then, kxxT − yy Tk = kyz T + zy T + zz T k ≤ kyz T k + kzy T k + kzz T k ≤ ε + ε + ε2 ≤ 3ε. D.4 Proof of Remark 4.4 Recall that N (µ, σ 2) denotes a 1-dimensional Gaussian distribution with mean µ and standard deviation σ. We will need a lemma bounding the L1 distance of two Gaussians in terms of their parameters. Any vector (p1 , . . . , pn ) ∈ ∆n induces a discrete probability distribution over [n] defined by Pr(i) := pi . Let x ∨ y := max{x, y}. Lemma D.2. Let (p1 , . . . , p2n+1 ) ∈ ∆2n+1 and (q1 , . . . , q2n+1 ) ∈ ∆2n+1 be discrete probability distributions with ℓ1 distance between them ≤ t. Suppose we have 2n + 1 bins, numbered 1 to 2n + 1. We throw m balls into these bins, where each ball chooses a bin independently according to qi . We pair bin 1 with bin 2, bin 3 with bin 4, . . . , and bin 2n − 1 with bin 2n; so bin 2n + 1 is unpaired. The probability that, for all pairs of bins, at most one them gets a ball, is not more than !m n X max{p2i−1 , p2i } 2n t/2 + p2n+1 + i=1 Proof. Let P1 = {1, 2}, P2 = {3, 4}, ..., Pn = {2n − 1, 2n}, and let A := {A ⊂ [2n] : \|A ∩ Pi \| = 1 ∀i ∈ [n]}. Clearly \|A\| = 2n . For any A ∈ A, let EA be the event that, the first ball does not choose a bin in A, and let FA be the event that, none of the balls choose a bin in A. Then, X X X Pr[EA ] = qi ≤ TV(p, q) + pi ≤ t/2 + pi i∈[2n+1]\A ≤ t/2 + p2n+1 + i∈[2n+1]\A n X i=1 i∈A / max{p2i−1 , p2i }, P and so Pr[FA ] = Pr[EA ]m ≤ (t/2 + p2n+1 + ni=1 (p2i−1 ∨ p2i ))m . Finally, observe that, if for each pair of bins, at most one them gets a ball, then there exists at least one A ∈ A, such that none of the balls chooses a bin in A. The lemma is thus proved by applying the union bound over all events {FA }A∈A . 26 Theorem D.3. The class of all Gaussian distributions over the real line admits (4, 1, O(1/ε)) 0.773-robust compression. Proof. Let q be any distribution (not necessarily a Gaussian) such that there exists a Gaussian g = N (µ, σ 2) with kq − gk1 ≤ r ≤ 0.773. Our goal is to encode g using samples generated from q. Let m = C/ε for a large enough constant C to be determined, and let S ∼ q m be an i.i.d. sample. The goal is to approximately encode µ and σ using only four elements of S and a single bit. We start by defining the decoder J . Our proposed decoder takes as input four points x1 , x2 , y1 , y2 ∈ R, and one bit b ∈ {0, 1}. The decoder then outputs a Gaussian distribution based on the following rule:  x +x \|y −y \|2  N ( 1 2 2 , 1 9 2 ) : if b = 1 J (x1 , x2 , y1 , y2 , b) =  N ( x1 +x2 , \|y − y \|2 ) : if b = 0 1 2 2 Our goal is thus to show that, with probability at least 2/3, there exists x1 , x2 , y1, y2 ∈ S and b ∈ {0, 1} such that kJ (x1 , x2 , y1 , y2, b) − gk1 ≤ ε. Let M = 1/ε and partition the interval [µ − 2σ, µ + 2σ) into 4M subintervals of length εσ. Enumerate these intervals as I1 to I4M , i.e., Ii = [µ − 2σ + (i − 1)(εσ), µ − 2σ + i(εσ)). S Also let I4M +1 = R \ 4M i=1 Ii . We state two claims which will imply the theorem, and will be proved later. Claim 1. With probability at least 5/6, there exist y1 , y2 ∈ S such that at least one of the following two conditions holds: (a) y1 ∈ Ii and y2 ∈ Ii+M for some i ∈ {M + 1, 2M + 2, ..., 2M}. In this case, we let b = 0, and so J (x1 , x2 , y1 , y2 , b) will have standard deviation \|y1 − y2 \|. (b) y1 ∈ Ii and y2 ∈ Ii+3M for some i ∈ [M]. In this case, we let b = 1, and so 2\| J (x1 , x2 , y1 , y2 , b) will have standard deviation \|y1 −y . 3 Also, if both of (a) and (b) happen, we will go with the first rule. Note that if Claim 1 holds, and σ̂ is the standard deviation of J (x1 , x2 , y1 , y2, b), then we will have \|σ̂−σ\| ≤ εσ. Claim 2. With probability at least 5/6, there exist x1 , x2 ∈ S such that x1 ∈ Ii and 2 =: µ̂. x2 ∈ I4M −i+1 for some i ∈ [2M]. If so, J (x1 , x2 , y1 , y2, b) will have mean x1 +x 2 Also note that if Claim 2 holds, then \|µ̂ − µ\| ≤ εσ. Therefore, if both claims hold, Lemma A.4 gives that J (x1 , x2 , y1 , y2, b) = N (µ̂, σ̂ 2 ) is a 2ε-approximation for N (µ, σ 2) = g. In other words, g can be approximately reconstructed, up to error 2ε, using only four data points (i.e., {x1 , x2 , y1 , y2}) from a sample S of size O(1/ε) and a single bit b (the definition of robust compression requires error ≤ ε. For getting this, one just needs to refine the partition by a constant factor, which multiplies M by a constant factor, and as we will see below, this will only multiply m by a constant factor). Note also that the 27 probability of existence of such four points is at least 1 − (1 − 5/6) − (1 − 5/6) ≥ 2/3. Therefore, it remains to prove Claim 1 and Claim 2. We start with Claim 1. View the sets I1 , . . . , I4M , I4M +1 as bins, and consider the R i.i.d. samples as balls landing in these bins according to q. Let pi := Ii g(x)dx and R qi := Ii q(x)dx for i ∈ [4M + 1]. Note that, by triangle’s inequality, the ℓ1 distance between (p1 , . . . , p4M +1 ) and (q1 , . . . , q4M +1 ) is not more than the L1 distance between g and q, which is at most r. We pair the bins as follows: Ii is paired with Ii+M for i ∈ {M + 1, . . . , 2M}, and Ii is paired with Ii+3M for i ∈ [M]. Therefore, by Lemma D.2, the probability that Claim 1 does not hold can be bounded by !m M 2M X X r 22M (pi ∨pi+M )+ (pi ∨pi+3M )+p4M +1 + 2 i=1 i=M +1 m  5 7 M M M 2 2 X X X r pi+ = 22M pi+p4M +1 +  , pi+ 2 3 M 3M +1 i= 2 M +1 i= 2 +1 where we have used the fact that pi are coming from a Gaussian, and thus p1 ≤ · · · ≤ p2M = p2M +1 ≥ · · · ≥ p4M (we have also assumed, for simplicity, that M is even). Let X ∼ N (µ, σ 2) and Φ(A) := Pr[N(0, 1) ∈ A]. Then using known numerical bounds for Φ, we obtain 2.5M X pi + i=1.5M +1 M X i=M/2+1 pi + 3.5M X pi + p4M +1 + r/2 3M +1 = Pr[X ∈ [µ − σ/2, µ + σ/2]] + 2Pr[X ∈ [µ − 3σ/2, µ − σ]] + Pr[X ∈ / [µ − 2σ, µ + 2σ]] + r/2 r r = Φ([−0.5, 0.5])+2Φ([−1.5, −1])+2Φ((−∞, −2])+ < 0.383 + 0.184 + 0.046 + 2 2 = 0.613 + r/2 ≤ 0.9995. Therefore since M = Θ(1/ε), by making m = C/ε for a large enough C, we can make this probability arbitrarily small, completing the proof of Claim 1. Via a similar argument, the probability that Claim 2 does not hold can be bounded by 22M =2 2M X i=1 2M max{pi , p4M −i+1 } + p4M +1 + r/2 !m = 22M 2M X i=1 pi + p4M +1 + r/2 !m (Φ([−1, 1]) + Φ([2, ∞) + r/2)m < 22M (0.5 + 0.023 + r/2)m < 22M (0.91)m < 1/6, for m = C/ε with a large enough C. 28 Remark D.4. By using more bits and adding more scales, one can show that 1-dimensional Gaussians admit (4, b(r), O(1/ε)) r-robust compression for any fixed r < 1 (the number of required bits and the implicit constant in the O will depend on the value of r). E E.1 Omitted proofs from Section 5 Proof of Lemma 5.1 The proof of the following lemma, which is called the ‘generalized Fano’s inequality,’ uses Fano’s inequality in information theory (Cover and Thomas, 2006, Theorem 2.10.1). It was first proved in (Devroye, 1987, page 77). We write here a slightly stronger version, which appears in (Yu, 1997, Lemma 3). Lemma E.1 (generalized Fano’s inequality). Suppose we have M > 1 distributions f1 , . . . , fM with KL(fi k fj ) ≤ β and kfi − fj k1 > α ∀i 6= j ∈ [M]. Consider any density estimation method that gets n i.i.d. samples from some fi , and outputs an estimate fb (the method does not know i). For each i, define ei as follows: assume the method receives samples from fi , and outputs fb. Then ei := Ekfi − fbk1 . Then, we have max ei ≥ α(log M − nβ + log 2)/(2 log M) . i To prove Lemma 5.1, consider a distribution learning method for learning F with sample complexity m(ε), and consider M distributions f1 , . . . , fM satisfying the hypotheses. Suppose we are in the setup of generalized Fano’s inequality: we get samples from some unknown j ∈ [M], and we are to find which fj are the samples coming from. When m(ε) samples are given to the method, with probability ≥ 2/3 it outputs some g within distance ε to fj . Suppose we repeat this procedure for k times, and the method outputs k distributions. If more than half of the times the method’s output was ε-close to some fi , then we output that fi as the answer; otherwise we output f1 . Our error would be 0 with probability Pr [Bin(k, 2/3) > k/2], and at most 2 with the remaining probability. Thus, the expected error can be upper bounded by exp(−Ω(k)) by the Chernoff bound. Thus, generalized Fano’s inequality gives α(log M − (km(ε))κ(ε) + log 2)/(2 log M) ≤ exp(−Ω(k)). Choosing k = Θ(log(1/ε)) and rearranging gives m(ε) = Ω(log M/κ(ε) log(1/ε)), as required. 29 E.2 Proof of Lemma 5.3 We use the probabilistic method. We let the d/r columns of each Ua to be the first d/r columns of a uniformly random orthonormal basis of Rd . To complete the proof, we need 2 only show that for two such random matrices Ua and Ub , with probability 1 − 2−Ω(d /r) d . we have kUaT Ub k2F ≤ 2r d In the following, U = V means U and V have the same distribution. By rotation d invariance, we may assume Ua = [e1 , . . . , ed/r ], so that kUaT Ub k2F = kUd/r k2F , where Ud/r is the d/r × d/r principal submatrix of a uniformly random orthogonal matrix U (alternatively, the columns of Ud/r are the first d/r coordinates of d/r orthonormal vectors in Rd chosen uniformly at random). Hence, it suffices to show that kUd/r k2F ≤ d/(2r) with 2 probability at least 1 − 2−Ω(d /r) . We will do this indirectly by relating Ud/r to a matrix with independent Gaussian entries. To that end, let G be a d × d/r matrix with i.i.d. N (0, 1/d) entries. Let G = UG ΣG VGT be its SVD, where UG ∈ Rd×d/r and ΣG , VG ∈ Rd/r×d/r . Observe that, by rotation invariance of the Gaussian matrix G, the columns of UG are d/r uniformly random orthonormal vectors and hence, the top d/r rows of UG have the same distribution as Ud/r . Moreover, by rotation invariance again, ΣG is independent of UG . Now let Gd/r be the first d/r rows of G. Then, d kGd/r k = (UG )d/r ΣG VG = Ud/r ΣG VG , and, taking Frobenius norms of both sides, d kGd/r kF = kUd/r ΣG VG kF = kUd/r ΣG kF ≥ σmin (ΣG )kUd/r kF , (4) with σmin (ΣG ) being the smallest singular value of ΣG . Since kGd/r k2F is a sum of i.i.d. √ random variables and concentrates sharply around its mean, and Eσmin (ΣG ) ≥ 1 − 1/ r by Gordon’s Theorem (Theorem A.12), this allows us to control kUd/r kF . In particular, for any p ≥ 1, we can bound a suitably translated moment of kUd/r kF . Let (x)+ := max{0, x}. Then, from (4) we get √ √ EG (kGd/r kF − d/r)p+ ≥ EUd/r ,ΣG (σmin (ΣG )kUd/r kF − d/r)p+ √ = EUd/r EΣG (σmin (ΣG )kUd/r kF − d/r)p+ √ ≥ EUd/r (EΣG σmin (ΣG )kUd/r kF − d/r)p+ √ √ ≥ EUd/r ((1 − 1/ r)kUd/r kF − d/r)p+ , where the second inequality is Jensen’s inequality √ inequality is Gordon’s √ and the third Theorem. Lemma A.6 gives that (kG √d/r kF − d/r) is O(1/ d)-subgaussian, and since √ the moments of (1 − 1/ r)kUd/r kF − d/r)+ are bounded by the moments of this random 30 variable, by equivalence of having moment bounds and being subgaussian (Lemma A.9), p √ √ we find that (1 − 1/r)kUd/r kF − d/r is also O(1/ d)-subgaussian. Hence, for any + t > 0, we have h i p √ 2 Pr (1 − 1/r)kUd/r kF − d/r ≤ t ≥ 1 − 2−Ω(t d) . √ √ Choosing t = d/(12 r) and the assumption that r ≥ 9 gives kUd/r k2F ≤ d/(2r) with 2 probability at least 1 − 2−Ω(d /r) , completing the proof. E.3 Proof of Lemma 5.4 Assume that kUaT Ub k2F ≤ d/(2r). Our goal is to show that TV(fa , fb ) = Ω √ λ d/r √ . log(r/λ d) √ Recall that fa = N (0, Σa ) with Σa = Id + λUa UaT . Let g ∼ N (0, Id). Then Σa g ∼ fa . √ √ Thus our goal is to lower bound TV( Σa g, Σb g). Since total variation distance never increases under a mapping (Fact A.5), we need only show that ! √ p p d/r λ √ . TV(Ua UaT Σa g, Ua UaT Σb g) = Ω log(r/λ d) √ P Now let C := Ua UaT Σb and since C is symmetric, its SVD has form C = i∈[d/r] σi wi wiT for orthonormal {wi } (some σi may be zero). Let S denote the column space of Ua ; so w1 , . . . , wd/r ∈ S, and we may assume w1 , . . . , wd/r is an orthonormal basis for S. This √ d P means Ua UaT Σb g = i∈[d/r] σi gi wi , where the gi are the components of g. Note that X σi2 = kCk2F = Tr(CC T ) = Tr(Ua UaT (1 + λUb UbT )Ua UaT ) i∈[d/r] = Tr(Ua UaT ) + λ Tr(Ua UaT Ub UbT Ua UaT ) = = d + λ Tr(UaT Ua UaT Ub UbT Ua ) r d d + λkUaT Ub k2F ≤ (1 + λ/2). r r √ On the other hand, suppose u1 , . . . , ud/r are the columns of Ua . Then, (Ua UaT Σa )2 = √ √ P d √ (1+λ)Ua UaT and hence, Ua UaT Σa g = 1 + λ i∈[d/r] gi ui . That is, Ua UaT Σa g is a spher√ P d ical Gaussian in the subspace S. So, by its rotation invariance, we have 1 + λ i∈[d/r] gi ui = √ P 1 + λ i∈[d/r] gi wi . Hence our goal is to show √ TV( 1 + λ d/r X gi wi , i=1 provided P i∈[d/r] d/r X σi gi wi ) = Ω i=1 σi2 ≤ d(1 + λ/2)/r. 31 √ λ d/r √ log(r/λ d) ! , 2 By reordering the wi , we may assume σ12 ≤ · · · ≤ σd/r . At most half of the σi2 can be 2 twice their average, which means σ12 ≤ · · · ≤ σd/2r ≤ 2(1 + λ/2) ≤ 3. Now we may project the two random vectors onto the subspace generated by the d/(2r) smallest eigenvectors, and this can only decrease √ the total variation distance. Taking the norms of the projected vectors and dividing by d can only decrease the total variation distance; hence our new goal is to show ! √ d/2r d/2r X λ d/r gi 2 X 2 gi2 √ √ , , (5) σi √ ) = Ω TV((1 + λ) d d log(r/λ d) i=1 i=1 provided P i∈[d/2r] σi2 ≤ d(1 + λ/2)/2r and maxi σi ≤ 3. Observe that     d/2r d/2r 2 2 X X √ √ √ gi g √ −E E (1 + λ) σi2 √i  ≥ (1 + λ)( d/2r) − d(1 + λ/2)/2r = λ d/(4r). d d i=1 i=1 Moreover, by Bernstein’s inequality (Lemma A.11), there exists a global constant c > 0 (independent of λ) such that for any t > 0,   d/(2r) d/(2r) X g2 X g2 √ i i 2   √ − E(1 + λ) √ > t ≤ 2 exp −c min{t , t d} , Pr (1 + λ) d d i=1 i=1 and, since σi2 ≤ 3 for all i,   d/2r d/2r 2 2 X X √ g g Pr  σi2 √i − E σi2 √i > t ≤ 2 exp −c min{t2 , t d} . d d i=1 i=1 √ Applying Lemma E.2 gives (5) (note that ζ ≤ O(log(r/λ d)) in the lemma), as required. Lemma E.2. Let X, Y be continuous random variables such that \|EX − EY \| ≥ ∆. Suppose also there exist c, C, β such that for any t > 0 we have max{Pr [\|X − EX\| ≥ t] , Pr [\|Y − EY \| ≥ t]} ≤ C exp(−c min{t2 , βt}). Let ζ := max{1, ∆, log(4C)/cβ, Then, TV(X, Y ) ≥ ∆/8ζ. p p log(4C)/c, log(8C/cβ∆)/cβ, log(4C/c∆)/c}. 32 Proof. We shall show that, for any h > 0 we have TV(X, Y ) ≥ ∆ − C exp(−c min(h2 , βh))∆ − C exp(−ch2 )/ch − 2C exp(−cβh)/cβ , (6) h+∆ and the lemma will follow by choosing h = ζ. Without loss of generality, we may assume EY ≥ EX and by translating, we may assume EX = −∆/2 and EY = ∆/2. Let I = [EX − h, EY + h], I ′ = R \ I and let f (x), g(x) denote the densities of X and Y . Define random variable Z to be \|X − EX\| if \|X − EX\| > h, and 0 otherwise. Then, since I ′ ⊆ (−∞, EX − h) ∪ (EX + h, ∞), we have Z EZ = \|(x − EX)f (x)\|dx (−∞,EX−h)∪(EX+h,∞) Z ≥ (\|x\|f (x) − \|EX\|f (x))dx ′ ZI = \|xf (x)\|dx − (∆/2)Pr [\|X − EX\| > h] ′ ZI \|xf (x)\|dx − C exp(−c min(h2 , βh))∆/2 ≥ I′ On the other hand, Z ∞ Z ∞ EZ = Pr [Z > t] dt = Pr [\|X − EX\| > t] dt 0 h Z ∞ Z ∞ Z 2 2 ≤ C exp(−c min{t , βt})dt ≤ C exp(−ct )dt + h h ∞ C exp(−cβt)dt h ≤ C exp(−ch2 )/2ch + C exp(−cβh)/cβ, where for the last inequality we have used tail bounds for the standard normal distribution (see Abramowitz and Stegun, 1984, formula (7.1.13)). Thus we find Z \|xf (x)\|dx ≤ C exp(−ch2 )/2ch + C exp(−cβh)/cβ + C exp(−c min(h2 , βh))∆/2. I′ A similar calculation gives the same upper bound for Z Z ∆ = x(g(x) − f (x))dx + x(g(x) − f (x))dx ′ I I Z Z ≤ \|x\|\|g(x) − f (x)\|dx + \|x\|(\|g(x)\| + \|f (x)\|)dx I R I′ \|xg(x)\|dx. Finally, observe that I′ ≤ kf (x) − g(x)k1 (h + ∆)/2 + C exp(−ch2 )/ch + 2C exp(−cβh)/cβ + C exp(−c min(h2 , βh))∆, since \|x\| ≤ (h + ∆)/2 for all x ∈ I. Re-arranging and noting total variation distance is half the L1 distance gives (6). 33 E.4 Proof of Theorem 5.5 We begin with a combinatorial lemma. Let dH (·, ·) denote the Hamming distance between two tuples. Lemma E.3. Let T ≥ 2 and k ∈ N. There exists a set of tuples X ⊆ [T ]k such that \|X \| ≥ 2Ω(k log(T )) and dH (x, y) ≥ k/4 for any pair of distinct x, y ∈ X . Proof. Let x and y be strings of length k where each coordinate is drawn from [T ] independently and uniformly at random. Let Xi be the indicator random variable which is P 1 if xi = yi . Then k − dH (x, y) = ki=1 Xi and E[k − dH (x, y)] =p k/T ≤ k/2. Observe that Xi = 1 with probability 1/T and 0 otherwise; hence it is 1/ log(T )-subgaussian. By Hoeffding’s Inequality (Lemma A.10), " k # X Pr Xi ≥ 3k/4 ≤ 2 · exp(−ck log(T )) i=1 for some absolute constant c > 0. Thus, we conclude that there is some set X with \|X \| ≥ 2Ω(k log(T )) and dH (x, y) ≥ k/4 for any pair of distinct x, y ∈ X . We now prove Theorem 5.5, using Lemma 5.1 again. The proof of Theorem 5.2 2 promises a collection of T = 2Ω(d ) matrices Σ1 , . . . , ΣT ≺ 2Id with KL(N (0, Σi k N (0, Σi′ )) ≤ O(ε2 log2 (1/ε)) and TV(N (0, Σi ), N (0, Σi′ )) ≥ Ω(ε) for i 6= i′ . Choose µ1 , . . . , µk ∈ Rd such that kµi − µi′ k2 ≥ C(kd/ε)10 for all i 6= i′ , where C is a large enough constant. 2 By Lemma E.3, there exists a set X ⊆ [T ]k of size 2Ω(k log(T )) = 2Ω(kd ) such that dH (x, y) ≥ k/4 for any distinct x, y ∈ X . We now define a set of mixture distributions as 1 F = fx := (N (µ1, Σx1 ) + . . . + N (µk , Σxk )) : x ∈ X . k We shall show that for any x 6= y, we have KL(fx k fy ) ≤ O(ε2 log2 (1/ε)) and TV(fx , fy ) ≥ Ω(ε). Lemma 5.1 will then conclude the proof. The proof of upper bound for KL divergence simply follows from convexity of KLdivergence (see Cover and Thomas, 2006, Theorem 2.7.2). and the fact that, for each i, KL(N (µi , Σxi k N (µi , Σyi )) = KL(N (0, Σxi k N (0, Σyi )) ≤ O(ε2 log2 (1/ε)). Next we show that TV(fx , fy ) ≥ Ω(ε). Let o n ′ Aj ∈ argmax Prg∼N (µj ,Σxj ) [g ∈ A] − Prg∼N (µj ,Σyj ) [g ∈ A] . A⊆Rd Since Σi ≺ 2Id for all i, we have Prg∼N (µj ,Σxj ) [kg − µj k22 ≥ 2d + O(d log(k/ε))] ≤ ε2 /k 2 , 34 p and a similar bound holds for N (µj , Σyj ). Now define Aj = A′j ∩ B2d (µj , O( d log(k/ε))), where B2d (µ, r) denotes the ℓ2 ball centered at µ with radius r. Then, Prg∼N (µj ,Σxj ) [g ∈ Aj ] − Prg∼N (µj ,Σyj ) [g ∈ Aj ] ≥ TV(N (µj , Σxj ), N (µj , Σyj )) − ε2 /k 2 . Note that the separation of µ1 , . . . , µk implies that A1 , . . . , Ak are disjoint sets and Prg∼N (µi ,Σyi ) [g ∈ Aj ] ≤ ε2 /k 2 for i 6= j. Let A = ∪j Aj . Finally, to lower bound the total variation distance, we have TV(fx , fy ) ≥ Prg∼fx [g ∈ A] − Prg∼fy [g ∈ A] = k X j=1 Prg∼fx [g ∈ Aj ] − Prg∼fy [g ∈ Aj ] k k 1 XX Prg∼N (µi ,Σxi ) [g ∈ Aj ] − Prg∼N (µi ,Σyi ) [g ∈ Aj ] = k j=1 i=1 k i 1 Xh = Prg∼N (µj ,Σxj ) [g ∈ Aj ] − Prg∼N (µj ,Σyj ) [g ∈ Aj ] k j=1 k 1 XX Prg∼N (µi ,Σxi ) [g ∈ Aj ] − Prg∼N (µi ,Σyi ) [g ∈ Aj ] + k j=1 i6=j i 1 Xh ≥ Prg∼N (µj ,Σxj ) [g ∈ Aj ] − Prg∼N (µj ,Σyj ) [g ∈ Aj ] − ε2 k j=1 k k ≥ 1 X TV(N (µj , Σxj ), N (µj , Σyj )) − ε2 /k 2 − ε2 k j=1 1 (k/4)Ω(ε) − 2ε2 ≥ Ω(ε), k where the last inequality is because TV(N (µj , Σxj ), N (µj , Σyj )) ≥ Ω(ε) whenever xj 6= yj which is the case for at least k/4 of the indices j. ≥ References M. Abramowitz and I. A. Stegun, editors. Handbook of mathematical functions with formulas, graphs, and mathematical tables. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York; National Bureau of Standards, Washington, DC, 1984. ISBN 0-471-80007-4. 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Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model∗ Keren Censor-Hillel Seri Khoury Ami Paz May 17, 2017 arXiv:1705.05646v1 [cs.DC] 16 May 2017 Abstract We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires Ω(n2 / log2 n) rounds in the worst case in the CONGEST model, as well as any algorithm for χ-coloring a graph, where χ is the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and nearquadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairsshortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple Ω(n) lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a complexity of Θ(n/ log n). We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question. ∗ Department of Computer Science, Technion, {ckeren,serikhoury,amipaz}@cs.technion.ac.il. Supported in part by ISF grant 1696/14. 1 1 Introduction It is well-known and easily proven that many graph problems are global for distributed computing, in the sense that solving them necessitates communication throughout the network. This implies tight Θ(D) complexities, where D is the diameter of the network, for global problems in the LOCAL model. In this model, a message of unbounded size can be sent over each edge in each round, which allows to learn the entire topology in D rounds. Global problems are widely studied in the CONGEST model, in which the size of each message is restricted to O(log n) bits, where n is the size of the network. The trivial complexity of learning the entire topology in the CONGEST model is O(m), where m is the number of edges of the communication graph, and since m can be as large as Θ(n2 ), one of the most basic questions for a global problem is how fast in terms of n it can be solved in the CONGEST model. Some global problems admit fast O(D)-round solutions in the CONGEST model, such as √ constructing a breadth-first search tree [59]. Some others have complexities of Θ̃(D + n), such as constructing a minimum spanning tree, and various approximation and verification problems [32, 39, 45, 60, 61, 64]. Some problems are yet harder, with complexities that are nearlinear in n [1, 32, 41, 51, 60]. For some problems, no O(n) solutions are known and they are candidates to being even harder that the ones with linear-in-n complexities. A major open question about global graph problems in the CONGEST model is whether natural graph problems for which a super-linear number of rounds is required indeed exist. In this paper, we answer this question in the affirmative. That is, our conceptual contribution is that there exist super-linearly hard problems in the CONGEST model. In fact, the lower bounds that we prove in this paper are as high as quadratic in n, or quadratic up to logarithmic factors, and hold even for networks of a constant diameter. Our lower bounds also imply linear and near-linear lower bounds for the CLIQUE-BROADCAST model. We note that high lower bounds for the CONGEST model may be obtained rather artificially, by forcing large inputs and outputs that must be exchanged. However, we emphasize that all the problems for which we show our lower bounds can be reduced to simple decision problems, where each node needs to output a single bit. All inputs to the nodes, if any, consist of edge weights that can be represented by polylogn bits. Technically, we prove a lower bound of Ω(n2 / log2 n) on the number of rounds required for computing an exact minimum vertex cover, which also extends to computing an exact maximum independent set (Section 3.1). This is in stark contrast to the recent O(log ∆/ log log ∆)-round algorithm of [8] for obtaining a (2 + )-approximation to the minimum vertex cover. Similarly, we give an Ω(n2 / log2 n) lower bound for 3-coloring a 3-colorable graph, which extends also for deciding whether a graph is 3-colorable, and also implies the same hardness for computing the chromatic number χ or computing a χ-coloring (Section 3.2). These lower bounds hold even for randomized algorithms which succeed with high probability.1 An immediate question that arises is whether only NP-hard problems are super-linearly hard in the CONGEST model. In Section 4, we provide a negative answer to such a postulate, by showing two simple problems that admit polynomial-time sequential algorithms, but in the CONGEST model require Ω(n2 ) rounds (identical subgraph detection) or Ω(n2 / log n) rounds (weighted cycle detection). The latter also holds for randomized algorithms, while for the former we show a randomized algorithm that completes in O(D) rounds, providing the strongest possible separation between deterministic and randomized complexities for global problems in the CONGEST model. Finally, we address the intriguing open question of the complexity of computing exact weighted all-pairs-shortest-paths (APSP) in the CONGEST model. While the complexity of the unweighted version of APSP is Θ(n/ log n), as follows from [32, 42], the complexity of weighted 1 We say that an event occurs with high probability (w.h.p) if it occurs with probability c > 0. 1 1 , nc for some constant APSP remains largely open, and only recently the first sub-quadratic algorithm was given in [28]. With the current state-of-the-art, this problem could be considered as a suspect for having a super-linear complexity in the CONGEST model. While we do not pin-down the complexity of weighted APSP in the CONGEST model, we provide a truly linear lower bound of Ω(n) rounds for it, which separates its complexity from that of the unweighted case. Moreover, we argue that it is not a coincidence that we are currently unable to show super-linear lower bound for weighted APSP, by formally proving that the commonly used framework of reducing a 2-party communication problem to a problem in the CONGEST model cannot provide a super-linear lower bound for weighted APSP, regardless of the function and the graph construction used (Section 5). This implies that obtaining any super-linear lower bound for weighted APSP provably requires a new technique. 1.1 The Challenge Many lower bounds for the CONGEST model rely on reductions from 2-party communication problems (see, e.g., [1,17,25,27,32,41,56,57,61,64]). In this setting, two players, Alice and Bob, are given inputs of K bits and need to a single output a bit according to some given function of their inputs. One of the most common problem for reduction is Set Disjointness, in which the players need to decide whether there is an index for which both inputs are 1. That is, if the inputs represent subsets of {0, . . . , K − 1}, the output bit of the players needs to indicate whether their input sets are disjoint. The communication complexity of 2-party Set Disjointness is known to be Θ(K) [49]. In a nutshell, there are roughly two standard frameworks for reducing the 2-party communication problem of computing a function f to a problem P in the CONGEST model. One of these frameworks works as follows. A graph construction is given, which consists of some fixed edges and some edges whose existence depends on the inputs of Alice and Bob. This graph should have the property that a solution to P over it determines the solution to f . Then, given an algorithm ALG for solving P in the CONGEST model, the vertices of the graph are split into two disjoint sets, VA and VB , and Alice simulates ALG over VA while Bob simulates ALG over VB . The only communication required between Alice and Bob in order to carry out this simulation is the content of messages sent in each direction over the edges of the cut C = E(VA , VB ). Therefore, given a graph construction with a cut of size \|C\| and inputs of size K for a function f whose communication complexity on K bits is at least CC(f ), the round complexity of ALG is at least Ω(CC(f )/\|C\| log n). The challenge in obtaining super-linear lower bounds was previously that the cuts in the graph constructions were large compared with the input size K. For example, the graph construction for the lower bound for computing the diameter in [32] has K = Θ(n2 ) and \|C\| = Θ(n), which gives an almost linear lower bound. The graph construction in [32] for the lower bound √ for computing a (3/2 − )-approximation to the diameter has a smaller cut of \|C\| = Θ( n), but this comes at the price of supporting a smaller input size K = Θ(n), which gives a lower bound that is roughly a square-root of n. To overcome this difficulty, we leverage the recent framework of [1], which provides a bitgadget whose power is in allowing a logarithmic-size cut. We manage to provide a graph construction that supports inputs of size K = Θ(n2 ) in order to obtain our lower bounds for minimum vertex cover, maximum independent set and 3-coloring2 . The latter is also inspired by, and is a simplification of, a lower bound construction for the size of proof labelling schemes [33]. Further, for the problems in P that we address, the cut is as small as \|C\| = O(1). For one of the problems, the size of the input is such that it allows us to obtain the highest possible lower bound of Ω(n2 ) rounds. 2 It can also be shown, by simple modifications to our constructions, that these problems require Ω(m) rounds, for graphs with m edges. 2 With respect to the complexity of the weighted APSP problem, we show an embarrassingly simple graph construction that extends a construction of [56], which leads to an Ω(n) lower bound. However, we argue that a new technique must be developed in order to obtain any superlinear lower bound for weighted APSP. Roughly speaking, this is because given a construction with a set S of nodes that touch the cut, Alice and Bob can exchange O(\|S\|n log n) bits which encode the weights of all lightest paths from any node in their set to a node in S. Since the cut has Ω(\|S\|) edges, and the bandwidth is Θ(log n), this cannot give a lower bound of more than Ω(n) rounds. With some additional work, our proof can be carried over to a larger number of players at the price of a small logarithmic factor, as well as to the second Alice-Bob framework used in previous work (e.g. [64]), in which Alice and Bob do not simulate nodes in a fixed partition, but rather in decreasing sets that partially overlap. Thus, determining the complexity of weighted APSP requires new tools, which we leave as a major open problem. 1.2 Additional Related Work Vertex Coloring, Minimum Vertex Cover, and Maximum Independent Set: One of the most central problems in graph theory is vertex coloring, which has been extensively studied in the context of distributed computing (see, e.g., [9–14, 18, 20, 21, 29–31, 37, 53, 55, 62, 65] and references therein). The special case of finding a (∆ + 1)-coloring, where ∆ is the maximum degree of a node in the network, has been the focus of many of these studies, but is a local problem, which can be solved in much less than a sublinear number of rounds. Another classical problem in graph theory is finding a minimum vertex cover (MVC). In distributed computing, the time complexity of approximating MVC has been addressed in several cornerstone studies [5, 6, 8, 14, 34–36, 44, 46–48, 58, 63]. Observe that finding a minimum size vertex cover is equivalent to finding a maximum size independent set. However, these problems are not equivalent in an approximation-preserving way. Distributed approximations for maximum independent set has been studied in [7,15,22,52]. Distance Computations: Distance computation problems have been widely studied in the CONGEST model for both weighted and unweighted networks [1, 32, 38–42, 50, 51, 56, 60]. One of the most fundamental problems of distance computations is computing all pairs shortest paths. For unweighted networks, an upper bound of O(n/ log n) was recently shown by [42], matching the lower bound of [32]. Moreover, the possibility of bypassing this near-linear barrier for any constant approximation factor was ruled out by [56]. For the weighted case, however, we are still very far from understanding the complexity of APSP, as there is still a huge gap 2 5 between the upper and lower bounds. Recently, Elkin [28] showed an O(n 3 · log 3 (n)) upper bound for weighted APSP, while the previously highest lower bound was the near-linear lower bound of [56] (which holds also for any (poly n)-approximation factor in the weighted case). Distance computation problems have also been considered in the CONGESTED-CLIQUE model [16, 38, 40], in which the underlying communication network forms a clique. In this model [16] showed that unweighted APSP, and a (1 + o(1))-approximation for weighted APSP, can be computed in O(n0.158 ) rounds. Subgraph Detection: The problem of finding subgraphs of a certain topology has received a lot of attention in both the sequential and the distributed settings (see, e.g., [2–4, 16, 23–25, 43, 54, 66] and references therein). The problems of finding paths of length 4 or 5 with zero weight are also related to other fundamental problems, notable in our context is APSP [2]. 3 2 2.1 Preliminaries Communication Complexity In a two-party communication complexity problem [49], there is a function f : {0, 1}K × {0, 1}K → {TRUE, FALSE}, and two players, Alice and Bob, who are given two input strings, x, y ∈ {0, 1}K , respectively, that need to compute f (x, y). The communication complexity of a protocol π for computing f , denoted CC(π), is the maximal number of bits Alice and Bob exchange in π, taken over all values of the pair (x, y). The deterministic communication complexity of f , denoted CC(f ), is the minimum over CC(π), taken over all deterministic protocols π that compute f . In a randomized protocol π, Alice and Bob may each use a random bit string. A randomized protocol π computes f if the probability, over all possible bit strings, that π outputs f (x, y) is at least 2/3. The randomized communication complexity of f , CC R (f ), is the minimum over CC(π), taken over all randomized protocols π that compute f . In the Set Disjointness problem (DISJK ), the function f is DISJK (x, y), whose output is FALSE if there is an index i ∈ {0, . . . , K − 1} such that xi = yi = 1, and TRUE otherwise. In the Equality problem (EQK ), the function f is EQK (x, y), whose output is TRUE if x = y, and FALSE otherwise. Both the deterministic and randomized communication complexities of the DISJK problem are known to be Ω(K) [49, Example 3.22]. The deterministic communication complexity of EQK is in Ω(K) [49, Example 1.21], while its randomized communication complexity is in Θ(log K) [49, Example 3.9]. 2.2 Lower Bound Graphs To prove lower bounds on the number of rounds necessary in order to solve a distributed problem in the CONGEST model, we use reductions from two-party communication complexity problems. To formalize them we use the following definition. Definition 1. (Family of Lower Bound Graphs) Fix an integer K, a function f : {0, 1}K × {0, 1}K → {TRUE, FALSE} and a predicate P for graphs. The family of graphs {Gx,y = (V, Ex,y ) \| x, y ∈ {0, 1}K }, is said to be a family of lower bound graphs w.r.t. f and P if the following properties hold: ˙ B a fixed (1) The set of nodes V is the same for all graphs, and we denote by V = VA ∪V partition of it; (2) Only the existence or the weight of edges in VA × VA may depend on x; (3) Only the existence or the weight of edges in VB × VB may depend on y; (4) Gx,y satisfies the predicate P iff f (x, y) = TRUE. We use the following theorem, which is standard in the context of communication complexitybased lower bounds for the CONGEST model (see, e.g. [1, 25, 32, 40]) Its proof is by a standard simulation argument. Theorem 1. Fix a function f : {0, 1}K ×{0, 1}K → {TRUE, FALSE} and a predicate P . If there is a family {Gx,y } of lower bound graphs with C = E(VA , VB ) then any deterministic algorithm for deciding P in the CONGEST model requires Ω(CC(f )/ \|C\| log n) rounds, and any randomized algorithm for deciding P in the CONGEST model requires Ω(CC R (f )/ \|C\| log n) rounds. Proof. Let ALG be a distributed algorithm in the CONGEST model that decides P in T rounds. Given inputs x, y ∈ {0, 1}K to Alice and Bob, respectively, Alice constructs the part of Gx,y 4 for the nodes in VA and Bob does so for the nodes in VB . This can be done by items (1),(2) and (3) in Definition 1, and since {Gx,y } satisfies this definition. Alice and Bob simulate ALG by exchanging the messages that are sent during the algorithm between nodes of VA and nodes of VB in either direction. (The messages within each set of nodes are simulated locally by the corresponding player without any communication). Since item (4) in Definition 1 also holds, we have that Alice and Bob correctly output f (x, y) based on the output of ALG. For each edge in the cut, Alice and Bob exchange O(log n) bits per round. Since there are T rounds and \|C\| edges in the cut, the number of bits exchanged in this protocol for computing f is O(T \|C\| log n). The lower bounds for T now follows directly from the lower bounds for CC(f ) and CC R (f ). In what follows, for each decision problem addressed, we describe a fixed graph construction G = (V, E), which we then generalize to a family of graphs {Gx,y = (V, Ex,y ) \| x, y ∈ {0, 1}K }, which we show to be a family lower bound graphs w.r.t. to some function f and the required predicate P . By Theorem 1 and the known lower bounds for the 2-party communication problem, we deduce a lower bound for any algorithm for deciding P in the CONGEST model. Remark: For our constructions which use the Set Disjointness function as f , we need to exclude the possibilities of all-1 input vectors. This is for the sake of guaranteeing that the graphs are connected, in order to avoid trivial impossibilities. However, this restriction does not change the asymptotic bounds for Set Disjointness, since computing this function while excluding all-1 input vectors can be reduced to computing this function for inputs that are shorter by one bit (by having the last bit be fixed to 0). 3 3.1 Near-Quadratic Lower Bounds for NP-Hard Problems Minimum Vertex Cover The first near-quadratic lower bound we present is for computing a minimum vertex cover, as stated in the following theorem. Theorem 2. Any distributed algorithm in the CONGEST model for computing a minimum vertex cover or for deciding whether there is a vertex cover of a given size M requires Ω(n2 / log2 n) rounds. Finding the minimum size of a vertex cover is equivalent to finding the maximum size of a maximum independent set, because a set of nodes is a vertex cover if and only if its complement is an independent set. Thus, Theorem 3 is a direct corollary of Theorem 2. Theorem 3. Any distributed algorithm in the CONGEST model for computing a maximum independent set or for deciding whether there is an independent set of a given size requires Ω(n2 / log2 n) rounds. Observe that a lower bound for deciding whether there is a vertex cover of some given size M or not implies a lower bound for computing a minimum vertex cover. This is because computing the size of a given subset of nodes can be easily done in O(D) rounds using standard tools. Therefore, to prove Theorem 2 it is sufficient to prove its second part. We do so by describing a family of lower bound graphs with respect to the Set Disjointness function and the predicate P that says that the graph has a vertex cover of size M . We begin with describing the fixed graph construction G = (V, E) and then define the family of lower bound graphs and analyze its relevant properties. The fixed graph construction: Let k be a power of 2. The fixed graph (Figure 1) consists 5 Figure 1: The family of lower bound graphs for deciding the size of a vertex cover, with many edges omitted for clarity. The node ak−1 is connected to all the nodes in TA1 , and a12 is connected to t0A2 and 1 0 to all the nodes in FA2 \ {fA2 }. Examples of edges from b01 and b02 to the bit-gadgets are also given. An additional edge, which is among the edges corresponding to the strings x and y, is {b01 , b12 }, while the edge {a01 , a02 } does not exist. Here, x0,0 = 1 and y0,1 = 0. of four cliques of size k: A1 = {ai1 \| 0 ≤ i ≤ k − 1}, A2 = {ai2 \| 0 ≤ i ≤ k − 1}, B1 = {bi1 \| 0 ≤ i ≤ k − 1} and B2 = {bi2 \| 0 ≤ i ≤ k − 1}. In addition, for each set S ∈ {A1 , A2 , B1 , B2 }, there are two corresponding sets of nodes of size log k, denoted FS = {fSh \| 0 ≤ h ≤ log k − 1} and TS = {thS \| 0 ≤ h ≤ log k − 1}. The latter are called bit-gadgets and their nodes are bit-nodes. The bit-nodes are partitioned into 2 log k 4-cycles: for each h ∈ {0, . . . , log k − 1} and ` ∈ {1, 2}, we connect the 4-cycle (fAh` , thA` , fBh ` , thB` ). Note that there are no edges between pairs of nodes denoted fSh , or between pairs of nodes denoted thS . The nodes of each set S ∈ {A1 , A2 , B1 , B2 } are connected to nodes in the corresponding set of bit-nodes, according to their binary representation, as follows. Let si` be a node in a set S ∈ {A1 , A2 , B1 , B2 }, i.e. s ∈ {a, b}, ` ∈ {1, 2} and i ∈ {0, . . . , k − 1}, and let ihdenote the bit number h in the binary representation of i. For such a node si` define bin(si` ) = fSh \| ih = 0 ∪ h tS \| ih = 1 , and connect si` by an edge to each of the nodes in bin(si` ). The next two claims address the basic properties of vertex covers of G. Claim 1. Any vertex cover of G must contain at least k − 1 nodes from each of the clique A1 , A2 , B1 and B2 , and at least 4 log k bit-nodes. Proof. In order to cover all the edges of each if the cliques on A1 , A2 , B1 and B2 , any vertex cover must contain at least k − 1 nodes of the clique. For each h ∈ {0, . . . , log k − 1} and ` ∈ {1, 2}, in order to cover the edges of the 4-cycle (fAh` , thA` , fBh ` , thB` ), any vertex cover must contain at least two of the cycle nodes. Claim 2. If U ⊆ V is a vertex cover of G of size 4(k − 1) + 4 log k, then there are two indices i, j ∈ {0, . . . , k − 1} such that ai1 , aj2 , bi1 , bj2 are not in U . Proof. By Claim 1, U must contain k − 1 nodes from each clique A1 , A2 , B1 and B2 , and 4 log k 0 0 bit-nodes, so it must not contain one node from each clique. Let ai1 , aj2 , bi1 , bj2 be the nodes in A1 , A2 , B1 , B2 which are not in U , respectively. To cover the edges connecting ai1 to bin(ai1 ), U 0 must contain all the nodes of bin(ai1 ), and similarly, U must contain all the nodes of bin(bi1 ). If i 6= i0 then there is an index h ∈ {0, . . . , log k − 1} such that ih 6= i0h , so one of the edges 6 (fAh1 , thB1 ) or (thA1 , fBh 1 ) is not covered by U . Thus, it must hold that i = i0 . A similar argument shows j = j 0 . Adding edges corresponding to the strings x and y: Given two binary strings x, y ∈ 2 {0, 1}k , we augment the graph G defined above with additional edges, which defines Gx,y . Assume that x and y are indexed by pairs of the form (i, j) ∈ {0, . . . , k − 1}2 . For each such pair (i, j) we add to Gx,y the following edges. If xi,j = 0, then we add an edge between the nodes ai1 and aj2 , and if yi,j = 0 then we add an edge between the nodes bi1 and bj2 . To prove that {Gxy } is a family of lower bound graphs, it remains to prove the next lemma. Lemma 1. The graph Gx,y has a vertex cover of cardinality M = 4(k − 1) + 4 log k iff DISJ(x, y) = FALSE. Proof. For the first implication, assume that DISJ(x, y) = FALSE and let i, j ∈ {0, . . . , k − 1} be such that xi,j = yi,j = 1. Note that in this case ai1 is not connected to aj2 , and bi1 is not connected to bj2 . We define a set U ⊆ V as the union of the two sets of nodes (A1 \ {ai1 }) ∪ (A2 \ {aj2 }) ∪ (B1 \ {bi1 }) ∪ (B2 \ {bj2 }) and bin(ai1 ) ∪ bin(aj2 ) ∪ bin(bi1 ) ∪ bin(bj2 ), and show that U is a vertex cover of Gx,y . First, U covers all the edges inside the cliques A1 , A2 , B1 and B2 , as it contains k − 1 nodes from each clique. These nodes also cover all the edges connecting nodes in A1 to nodes in A2 and all the edges connecting nodes in B1 to nodes in B2 . Furthermore, U covers any edge connecting some node u ∈ (A1 \{ai1 })∪(A2 \{aj2 })∪(B1 \{bi1 })∪(B2 \{bj2 }) with the bit-gadgets. For each node s ∈ ai1 , aj2 , bi1 , bj2 , the nodes bin(s) are in U , so U also cover the edges connecting s to the bit gadget. Finally, U covers all the edges inside the bit gadgets, as from each 4-cycle (fAh` , thA` , fBh ` , thB` ) it contains two non-adjacent nodes: if ih = 0 then fAh1 , fBh 1 ∈ U and otherwise thA1 , thB1 ∈ U , and if jh = 0 then fAh2 , fBh 2 ∈ U and otherwise thA2 , thB2 ∈ U . We thus have that U is a vertex cover of size 4(k − 1) + 4 log k, as needed. For the other implication, let U ⊆ V be a vertex cover of Gx,y of size 4(k − 1) + 4 log k. As the set of edges of G is contained in the set of edges of Gx,y , U is also a cover of G, and by Claim 2 there are indices i, j ∈ {0, . . . , k − 1} such that ai1 , aj2 , bi1 , bj2 are not in U . Since U is a cover, the graph does not contain the edges (ai1 , aj2 ) and (bi1 , bj2 ), so we conclude that xi,j = yi,j = 1, which implies that DISJ(x, y) = FALSE. Having constructed the family of lower bound graphs, we are now ready to prove Theorem 2. Proof of Theorem 2: To complete the proof of Theorem 2, we divide the nodes of G (which are also the nodes of Gx,y ) into two sets. Let VA = A1 ∪ A2 ∪ FA1 ∪ TA1 ∪ FA2 ∪ TA2 and VB = V \ VA . Note that n ∈ Θ(k), and thus K = \|x\| = \|y\| = Θ(n2 ). Furthermore, note that the only edges in the cut E(VA , VB ) are the edges between nodes in {FA1 ∪ TA1 ∪ FA2 ∪ TA2 } and nodes in {FB1 ∪ TB1 ∪ FB2 ∪ TB2 }, which are in total Θ(log n) edges. Since Lemma 1 shows that {Gx,y } is a family of lower bound graphs, we can apply Theorem 1 on the above partition to deduce that because of the lower bound for Set Disjointness, any algorithm in the CONGEST model for deciding whether a given graph has a cover of cardinality M = 4(k − 1) + 4 log k requires at least Ω(K/ log2 (n)) = Ω(n2 / log2 (n)) rounds. 3.2 Graph Coloring Given a graph G, we denote by χ(G) the minimal number of colors in a proper vertex-coloring of G. In this section we consider the problems of coloring a graph in χ colors, computing χ and approximating it. We prove the next theorem. Theorem 4. Any distributed algorithm in the CONGEST model that colors a χ-colorable graph G in χ colors or compute χ(G) requires Ω(n2 / log2 n) rounds. 7 Figure 2: The family of lower bound graphs for coloring, with many edges omitted for clarity. The node Ca1 is connected to all the nodes in FA1 ∪ TA1 and Cb1 is connected to all the nodes in FB1 ∪ TB1 . The node Ca2 is connected to all the nodes in FA2 ∪ TA2 and Cb2 is connected to all the nodes in FB2 ∪ TB2 . Any distributed algorithm in the CONGEST model that decides if χ(G) ≤ c for a given integer c, requires Ω((n − c)2 /(c log n + log2 n)) rounds. We give a detailed lower bound construction for the first part of the theorem, by showing that distinguishing χ ≤ 3 from χ ≥ 4 is hard. Then, we extend our construction to deal with deciding whether χ ≤ c. The fixed graph construction: We describe a family of lower bound graphs, which builds upon the family of graphs defined in Section 3.1. We define G = (V, E) as follows (see Figure 2). There are four sets of size k: A1 = {ai1 \| 0 ≤ i ≤ k − 1}, A2 = {ai2 \| 0 ≤ i ≤ k − 1}, B1 = {bi1 \| 0 ≤ i ≤ k − 1} and B2 = {bi2 \| 0 ≤ i ≤ k − 1}. As opposed to the construction in Section 3.1, the nodes of these sets are not connected to one another. In addition, as in Section 3.1, for each set S ∈ {A1 , A2 , B1 , B2 }, there are two corresponding sets of nodes of size log k, denoted FS = {fSh \| 0 ≤ h ≤ log k − 1} and TS = {thS \| 0 ≤ h ≤ log k − 1}. For each h ∈ {0, . . . , log k − 1} and ` ∈ {1, 2}, the nodes (fAh` , thA` , fBh ` , thB` ) constitute a 4-cycle. Each node si` in a set S ∈ {A1 , A2 , B1 , B2 } is connected by to all nodes in bin(si` ). Up to here, the construction differs from the construction in Section 3.1 only by not having edges inside the sets A1 , A2 , B1 , B2 . We now add the following two gadgets to the graph. 1. We add three nodes Ca0 , Ca1 , Ca2 connected as a triangle, another set of three nodes Cb0 , Cb1 , Cb2 connected as a triangle, and edges connecting Cai to Cbj for each i 6= j ∈ {0, 1, 2}. We connect all the nodes of the form fAh1 , thA1 , h ∈ {0, . . . , log k − 1}, to Ca1 . Similarly, we 8 connect all the nodes fBh 1 , thB1 to Cb1 , the nodes fAh2 , thA2 to Ca2 and the nodes fBh 2 , thB2 to Cb2 . i i 2. For each set S ∈ A , A , B , B , we add two sets of nodes, S̄ = s̄` \| s` ∈ S and S̄¯ = 1 2 1 2 i i s̄¯` \| s` ∈ S . For each ` ∈ {1, 2} and i ∈ {0, . . . , k − 1} we connect a path (si` , s̄i` , s̄¯i` ), and for each ` ∈ {1, 2} and i ∈ {0, . . . , n − 2}, we connect s̄¯i` to s̄i+1 ` . In addition, we connect the gadgets by the edges: ¯i1 ), for each i ∈ {0, . . . , k − 1}; (Ca2 , ā01 ) and (Ca2 , ā ¯k−1 (a) (Ca2 , ai1 ) and (Ca1 , ā 1 ). (b) (Cb2 , bi1 ) and (Cb1 , ¯b̄i1 ), for each i ∈ {0, . . . , k − 1}; (Cb2 , b̄01 ) and (Cb2 , ¯b̄k−1 1 ). ¯i2 ), for each i ∈ {0, . . . , k − 1}; (Ca1 , ā02 ) and (Ca1 , ā ¯k−1 (c) (Ca1 , ai2 ) and (Ca2 , ā 2 ). (d) (Cb1 , bi2 ) and (Cb2 , ¯b̄i2 ), for each i ∈ {0, . . . , k − 1}; (Cb1 , b̄02 ) and (Cb1 , ¯b̄2k−1 ). Assume there is a proper 3-coloring of G. Denote by c0 , c1 and c2 the colors of Ca0 , Ca1 and respectively. By construction, these are also the colors of Cb0 , Cb1 and Cb2 , respectively. For the nodes appearing in Section 3.1, coloring a node by c0 is analogous to not including it in the vertex cover. Ca2 Claim 3. In each set S ∈ {A1 , A2 , B1 , B2 }, at least one node is colored by c0 . Proof. We start by proving the claim for S = A1 . Assume, towards a contradiction, that all nodes of A1 are colored by c1 and c2 . All these nodes are connected to Ca2 , so they must all be colored by c1 . Hence, all the nodes āi1 , i ∈ {0, . . . , k − 1}, are colored by c0 and c2 . The nodes ¯i1 , i ∈ {0, . . . , k − 1}, are connected to Ca1 , so they are colored by c0 and c2 as well. ā ¯k−1 ¯01 , ā11 , ā ¯11 , . . . āk−1 Hence, we have a path (ā01 , ā 1 , ā1 ) with an even number of nodes, starting k−1 0 ¯1 . This path must be colored by alternating c0 and c2 , but both ā01 and in ā1 and ending in ā k−1 ¯1 are connected to Ca2 , so they cannot be colored by c2 , a contradiction. ā A similar proof shows the claim for S = B1 . For S ∈ {A2 , B2 }, we use a similar argument but change the roles of c1 and c2 . Claim 4. For each i ∈ {0, . . . , k − 1}, the node ai1 is colored by c0 iff bi1 is colored by c0 and the node ai2 is colored by c0 iff bi2 is colored by c0 . Proof. Assume ai1 is colored by c0 , so all of its adjacent nodes bin(ai1 ) can only be colored by c1 or c2 . As all of these nodes are connected to C1a , they must be colored by c2 . Similarly, if a node bj1 in B1 is colored by c0 , then the nodes bin(bj1 ), which are also adjacent to C1b , must be colored by c2 . If i 6= j then there must be a bit i such that ih 6= jh , and there must be a pair of neighboring nodes (fAh1 , thB1 ) or (thA1 , fBh 1 ) which are colored by c2 . Thus, the only option is i = j. By Claim 3, there is a node in B1 that is colored by c0 , and so it must be bi1 . An analogous argument shows that if bi1 is colored by c0 , then so does ai1 . For ai2 and bi2 , similar arguments apply, where c1 plays the role of c2 . 2 Adding edges corresponding to the strings x and y: Given two bit strings x, y ∈ {0, 1}k , we augment the graph G described above with additional edges, which defines Gx,y . Assume x and y are indexed by pairs of the form (i, j) ∈ {0, . . . , k − 1}2 . To construct Gx,y , add edges to G by the following rules: if xi,j = 0 then add the edge (ai1 , aj2 ), and if yi,j = 0 then add the edge (bi1 , bj2 ). To prove that {Gx,y } is a family of lower bound graphs, it remains to prove the next lemma. Lemma 2. The graph Gx,y is 3-colorable iff DISJ(x, y) = FALSE. 9 Proof. For the first direction, assume Gx,y is 3-colorable, and denote the colors by c0 , c1 and c2 , as before. By Claim 3, there are nodes ai1 ∈ A1 and aj2 ∈ A2 that are both colored by c0 . Hence, the edge (ai1 , aj2 ) does not exist in Gx,y , implying xi,j = 1. By Claim 4, the nodes bi1 and bj2 are also colored c0 , so yi,j = 1 as well, giving that DISJ(x, y) = FALSE, as needed. For the other direction, assume DISJ(x, y) = FALSE, i.e, there is an index (i, j) ∈ {0, . . . , k − 1}2 such that xi,j = yi,j = 1. Consider the following coloring. 1. Color Cai and Cbi by ci , for i ∈ {0, 1, 2}. 0 0 2. Color the nodes ai1 , bi1 , aj2 and bj2 by c0 . Color the nodes ai1 and bi1 , for i0 6= i, by c1 , and 0 0 the nodes aj2 and bj1 , for j 0 6= j, by c2 . 3. Color the nodes of bin(ai1 ) by c2 , and similarly color the nodes of bin(bi1 ) by c2 . Color k−i the rest of the nodes in this gadget, i.e. bin(ak−i 1 ) and bin(b1 ), by c0 . Similarly, color k−j bin(aj2 ) and bin(bj2 ) by c0 and bin(ak−j 2 ) and bin(b2 ) by c1 . 4. Finally, color the nodes of the forms s̄i` and s̄¯i` as follows. 0 0 0 0 (a) Color āi1 and b̄i1 by c1 , all nodes āi1 and b̄i1 with i0 < i by c0 , and all nodes āi1 and b̄i1 with i0 > i by c2 . 0 0 (b) Similarly, color āi2 and b̄i2 by c2 , all nodes āi2 and b̄i2 with i0 < i by c0 , and all nodes 0 0 āi2 and b̄i2 with i0 > i by c1 . ¯i10 and ¯b̄i10 with i0 < i by c2 , and all nodes ā ¯i10 and ¯b̄i10 with i0 ≥ i by (c) Color all nodes ā c0 . ¯i20 and ¯b̄i20 with i0 < i by c1 , and all nodes ā ¯i20 and ¯b̄i20 with (d) Similarly, color all nodes ā i0 ≥ i by c0 . It is not hard to verify that the suggested coloring is indeed a proper 3-coloring of Gx,y , which completes the proof. Having constructed the family of lower bound graphs, we are now ready to prove Theorem 4. Proof of Theorem 4: To complete the proof of Theorem 4, we divide the nodes of G (which are also the nodes of Gx,y ) into two sets. Let VA = A1 ∪A2 ∪FA1 ∪TA1 ∪FA2 ∪TA2 ∪ Ca0 , Ca1 , Ca2 ∪ Ā1 ∪ Ā¯1 ∪ Ā2 ∪ Ā¯2 , and VB = V \ VA . Note that n ∈ Θ(k). The edges in the cut E(VA , VB ) are the 6 edges connecting Ca0 , Ca1 , Ca2 and Cb0 , Cb1 , Cb2 , and 2 edges for every 4-cycle of the nodes of FA1 ∪ TA1 ∪ FB1 ∪ TB1 and FA2 ∪ TA2 ∪ FB2 ∪ TB2 , for a total of Θ(log n) edges. Since Lemma 2 shows that {Gx,y } is a family of lower bound graphs with respect to DISJK , K = k 2 ∈ Θ(n2 ) and the predicate χ ≤ 3, we can apply Theorem 1 on the above partition to deduce that any algorithm in the CONGEST model for deciding whether a given graph is 3-colorable requires at least Ω(n2 / log2 n) rounds. Any algorithm that computes χ of the input graph, or produces a χ-coloring of it, may be used to deciding whether χ ≤ 3, in at most O(D) additional rounds. Thus, the lower bound applies to these problems as well. Our construction and proof naturally extend to handle c-coloring, for any c ≥ 3. To this end, we add to G (and to Gx,y ) new nodes denoted Cai , i ∈ {3, . . . , c − 1}, and connect them to all of VA , and new nodes denoted Cbi , i ∈ {3, . . . , c − 1}, and connect them to all of VB and also to Ca0 , Ca1 and Ca2 . The nodes Cai are added to Va , and the rest are added to Vb , which increases the cut size by Θ(c) edges. Assume the extended graph is colorable by c colors, and denote by ci the color of the node Cai (these nodes are connected by a clique, so their colors are distinct). The nodes Cbi , i ∈ {2, . . . , c − 1} form a clique, and they are all connected to the nodes Ca0 , Ca1 and Ca2 , so they are colored by the colors {c3 , . . . , cc−1 }, in some order. All the original nodes of VA are 10 connected to Cai , i ∈ {3, . . . , c − 1}, and all the original nodes of VB are connected to Cbi , i ∈ {3, . . . , c − 1}, so the original graph must be colored by 3 colors, which we know is possible iff DISJ(x, y) = FALSE. We added 2c − 6 nodes to the graph, so the inputs strings are of length K = n − 2c + 6. Thus, the new graphs constitute a family of lower bound graphs with respect to EQK and the predicate χ ≤ c, the communication complexity of EQK is in Ω(K 2 ) = Ω((n − c)2 ), the cut size is Θ(c + log n), and Theorem 1 completes the proof. A lower bound for (4/3 − )-approximation: Finally, we extend our construction to give a lower bound for approximate coloring. That is, we show a similar lower bound for computing a (4/3 − ε)-approximation to χ and for finding a coloring in (4/3 − ε)χ colors. Observe that since χ is integral, any (4/3 − )-approximation algorithm must return the exact solution in case χ = 3. Thus, in order to rule out the possibility for an algorithm which is allowed to return a (4/3 − ε)-approximation which is not the exact solution, we need a more general construction. For any integer c, we show a lower bound for distinguishing between the case χ ≤ 3c and χ ≥ 4c. Claim 5. Given an integer c, any distributed algorithm in the CONGEST model that distinguishes a graph G with χ(G) ≤ 3c from a graph with χ(G) ≥ 4c requires Ω(n2 /(c3 log2 n)) rounds. To prove Claim 5 we show a family of lower bound graphs with respect to the DISJK function, where K ∈ Θ(n2 /c2 ), and the predicate χ ≤ 3c (TRUE) or χ ≥ 4c (FALSE). The predicate is not defined for other values of χ. We create a graph Gcx,y , composed of c copies of Gx,y . The i-th copy is denoted Gx,y (i), and its nodes are partitioned into VA (i) and VB (i). We connect all the nodes of VA (i) to all nodes of VA (j), for each i 6= j. Similarly, we connect all the nodes of VB (i) to all the nodes of VB (j). This construction guarantees that each copy is colored by different colors, and hence if DISJ(x, y) = FALSE then χ(Gcx,y ) = 3c and otherwise χ(Gcx,y ) ≥ 3c. Therefore, Gcx,y is a family of lower bound graphs. Proof of Claim 5: Note that n ∈ Θ(kc). Thus, K = \|x\| = \|y\| = n2 /c2 . Furthermore, observe that for each Gx,y (i), there are O(log n) edges in the cut, so in total Gcx,y contains O(c log n) edges in the cut. Since we showed that Gcx,y is a family of lower bound graphs, we can apply Theorem 1 to deduce that because of the lower bound for Set Disjointness, any algorithm in the CONGEST model for distinguishing between χ ≤ 3c and χ ≥ 4c requires at least Ω(n2 /c3 log2 (n)) rounds. For any > 0 and any c it holds that (4/3 − )3c < 4c. Thus, we can choose c to be an arbitrary constant to achieve the following theorem. Theorem 5. For any constant ε > 0, any distributed algorithm in the CONGEST model that computes a (4/3 − ε)-approximation to χ requires Ω(n2 / log2 n) rounds. 4 Quadratic and Near-Quadratic Lower Bounds for Problems in P In this section we support our claim that what makes problems hard for the CONGEST model is not necessarily them being NP-hard problems. First, we address a class of subgraph detection problems, which requires detecting cycles of length 8 and a given weight, and show a nearquadratic lower bound on the number of rounds required for solving it, although its sequential complexity is polynomial. Then, we define a problem which we call the Identical Subgraphs 11 Figure 3: The family of lower bound graphs for detecting weighted cycles, with many edges omitted for clarity. Here, x0,1 = 1 and yk−1,k−1 = 1. Thus, a01 is connected to a12 by an edge of weight k 3 +k·0+1 = k 3 +1, and bk−1 is connected to bk−1 by an edge of weight k 3 −(k(k−1)+k−1) = k 3 −k 2 +1. 1 2 All the dashed edges are of weight 0. Detection problem, in which the goal is to decide whether two given subgraphs are identical. While this last problem is rather artificial, it allows us to obtain a strictly quadratic lower bound for the CONGEST model, with a problem that requires only a single-bit output. 4.1 Weighted Cycle Detection In this section we show a lower bound on the number of rounds needed in order to decide the graph contains a cycle of length 8 and weight W , such that W is a polylog(n)-bit value given as an input. Note that this problem can be solved easily in polynomial time in the sequential n setting by simply checking all of the at most 8 cycles of length 8. Theorem 6. Any distributed algorithm in the CONGEST model that decides if a graph with edge weights w : E → [0, poly(n)] contains a cycle of length 8 and weight W requires Ω(n2 / log2 n) rounds. Similarly to the previous sections, to prove Theorem 6 we describe a family of lower bound graphs with respect to the Set Disjointness function and the predicate P that says that the graph contains a cycle of length 8 and weight W . The fixed graph construction: The fixed graph construction G = (V, E) is defined as follows. The set of nodes contains four sets A1 , A2 , B1 and B2 , each of size k. To simplify our proofs in this section, we assume that k ≥ 3. For each set S ∈ {A1 , A2 , B1 , B2 } there is a node cS , which is connected to each of the nodes in S by an edge of weight 0. In addition there is an edge between cA1 and cB1 of weight 0 and an edge between cA2 and cB2 of weight 0 (see Figure 3). Adding edges corresponding to the strings x and y: Given two binary strings x, y ∈ 2 {0, 1}k , we augment the fixed graph G defined in the previous section with additional edges, which defines Gx,y . Recall that we assume that k ≥ 3. Let x and y be indexed by pairs of the form (i, j) ∈ {0, . . . , k − 1}2 . For each (i, j) ∈ {0, . . . , k − 1}2 , we add to Gx,y the following edges. If xi,j = 1, then we add an edge of weight k 3 + ki + j between the nodes ai1 and aj2 . If yi,j = 1, then we add an edge of weight k 3 − (ki + j) between the nodes bi1 and bj2 . We denote by InputEdges the set of edges {(u, v) \| u ∈ A1 ∧ v ∈ A2 } ∪ {(u, v) \| u ∈ B1 ∧ v ∈ B2 }, and we denote by w(u, v) the weight of the edge (u, v). 12 Observe that the graph does not contain edges of negative weight. Furthermore, the weight of any edge in InputEdges does not exceed k 3 + k 2 − 1, which is the weight of the edge k−1 (ak−1 1 , a2 ), in case xk−1,k−1 = 1. Similarly, the weight of an edge in InputEdges is not less k−1 3 than k − k 2 + 1, which is the weight of the edge (bk−1 1 , b2 ), in case yk−1,k−1 = 1. Using these two simple observations, we deduce the following claim. Claim 6. For any cycle of weight 2k 3 , the number of edges it contains that are in InputEdges is exactly two. Proof. Let C be a cycle of weight 2k 3 , and assume for the sake of contradiction that C does not contain exactly two edges from InputEdges. In case C contains exactly one edge from InputEdges, then the weight of C is at most k 3 + k 2 − 1 < 2k 3 , because all the other edges of C are of weight 0. Otherwise, in case C contains three or more edges from InputEdges, it holds that the weight of C is at least 3k 3 − 3k 2 + 3 > 2k 3 , because all the other edges on C are of non-negative weight. To prove that {Gx,y } satisfies the definition of a family of lower bound graphs, we prove the following lemma. Lemma 3. The graph Gx,y contains a cycle of length 8 and weight W = 2k 3 if and only if DISJ(x, y) = FALSE. Proof. For the first direction, assume that DISJ(x, y) = FALSE and let 0 ≤ i, j ≤ k − 1 be such that xi,j = 1 and yi,j = 1. Consider the cycle (ai1 , cA1 , cB1 , bi1 , bj2 , cB2 , cA2 , aj2 ). It is easy to verify that this is a cycle of length 8 and weight w(aj1 , ai2 ) + w(bi1 , bj2 ) = k 3 + ki + j + k 3 − ki − j = 2k 3 , as needed. For the other direction, assume that the graph contains a cycle C of length 8 and weight 2k 3 . By Claim 6, C contains exactly two edges from InputEdges. Denote these two edges by (u1 , v1 ) and (u2 , v2 ). Since all the other edges in C are of weight 0, the weight of C is w(u1 , v1 )+w(u2 , v2 ). The rest of the proof is by case analysis, as follows. First, it is not possible that (u1 , v1 ), (u2 , v2 ) ∈ {(u, v) \| u ∈ A1 ∧v ∈ A2 }, since in this case w(u1 , v1 )+w(u2 , v2 ) ≥ w(a01 , a02 )+w(a01 , a12 ) = 2k 3 +1. Similarly, it is not possible that (u1 , v1 ), (u2 , v2 ) ∈ {(u, v) \| u ∈ B1 ∧ v ∈ B2 }, since in this case w(u1 , v1 )+w(u2 , v2 ) ≤ w(b01 , b02 )+w(b01 , b12 ) = 2k 3 −1. Finally, suppose without loss of generality that (u1 , v1 ) ∈ {(u, v) \| u ∈ A1 ∧ v ∈ A2 } and (u2 , v2 ) ∈ {(u, v) \| u ∈ B1 ∧ v ∈ B2 }. Denote 0 0 0 0 u1 = ai1 , u2 = aj1 , v1 = bi1 and v2 = bj2 . It holds that w(ai1 , aj2 ) + w(bi1 , bj2 ) = 2k 3 if and only if i = i0 and j = j 0 , which implies that xi,j = 1 and yi,j = 1 and DISJ(x, y) = FALSE. Having constructed the family of lower bound graphs, we are now ready to prove Theorem 6. Proof of Theorem 6: To complete the proof of Theorem 6, we divide the nodes of G (which are also the nodes of Gx,y ) into two sets. Let VA = A1 ∪ A2 ∪ {cA1 , cA2 } and VB = V \ VA . Note that n ∈ Θ(k). Thus, K = \|x\| = \|y\| = Θ(n2 ). Furthermore, note that the only edges in the cut E(VA , VB ) are the edges (cA1 , cB1 ) and (cA2 , cB2 ). Since Lemma 3 shows that {Gx,y } is a family of lower bound graphs, we apply Theorem 1 on the above partition to deduce that because of the lower bound for Set Disjointness, any algorithm in the CONGEST model for deciding whether a given graph contains a cycle of length 8 and weight W = 2k 3 requires at least Ω(K/ log n) = Ω(n2 / log n) rounds. 4.2 Identical Subgraphs Detection In this section we show the strongest possible, quadratic lower bound, for a problem which can be solved in linear time in the sequential setting. Consider the following sequential specification of a graph problem. 13 Definition 2. (The Identical Subgraphs Detection Problem) Given a weighted input graph G = (V, E, w), with an edge-weight function w : E → {0, . . . , W − 1}, W ∈ poly n, such that the set of nodes V is partitioned into two enumerated sets of the same size, VA = {a0 , ..., ak−1 } and VB = {b0 , ..., bk−1 }, the Identical Subgraphs Detection problem is to determine whether the subgraph induced by the set VA is identical to the subgraph induced by the set VB , in the sense that for each 0 ≤ i, j ≤ k − 1 it holds that (ai , aj ) ∈ E if and only if (bi , bj ) ∈ E and w(ai , aj ) = w(bi , bj ) if these edges exist. The identical subgraphs detection problem can be solved easily in linear time in the sequential setting by a single pass over the set of edges. However, as we prove next, it requires a quadratic number of rounds in the CONGEST model, in any deterministic solution (note that this restriction did not apply in the previous sections). For clarity, we emphasize that in the distributed setting, the input to each node in the identical subgraphs detection problem is its enumeration ai or bi , as well as the enumerations of its neighbors and the weights of the respective edges. The outputs of all nodes should be TRUE if the subgraphs are identical, and FALSE otherwise. Theorem 7. Any distributed deterministic algorithm in the CONGEST model for solving the identical subgraphs detection problem requires Ω(n2 ) rounds. To prove Theorem 7 we describe a family of lower bound graphs. The fixed graph construction: The fixed graph G is composed of two k-node cliques on the node sets VA = {a0 , ..., ak−1 } and VB = {b0 , ..., bk−1 }, and one extra edge (a0 , b0 ). Adding edge weights corresponding to the strings x and y: Given two binary strings x k and y, each of size K = 2 log n, we augment the graph G with additional edge weights, which define Gx,y . For simplicity, assume that x and y are vectors of log n-bit numbers each having k 2 entries enumerated as xi,j and yi,j , with i < j, i, j ∈ {0, . . . , k − 1}. For each such i and j we set the weights of w(ai , aj ) = xi,j and w(bi , bj ) = yi,j , and we set w(a0 , b0 ) = 0. Note that {Gx,y } is a family of lower bound graphs with respect to EQK and the predicate P that says that the subgraphs are identical in the aforemention sense. Proof of Theorem 7: Note that n ∈ Θ(k), and thus K = \|x\| = \|y\| = Θ(n2 log n). Furthermore, the only edge in the cut E(VA , VB ) is the edge (a0 , b0 ). Since we showed that {Gx,y } is a family of lower bound graphs, we can apply Theorem 1 on the above partition to deduce that because of the lower bound for EQK , any deterministic algorithm in the CONGEST model for solving the identical subgraphs detection problem requires at least Ω(K/ log n) = Ω(n2 ) rounds. We remark that in a distributed algorithm for the identical subgraphs detection problem running on our family of lower bound graphs, information about essentially all the edges and weights in the subgraphs induced on VA and VB needs to be sent across the edge (a0 , b0 ). This might raise the suspicion that this problem is reducible to learning the entire graph, making the lower bound trivial. To argue that this is far from being the case, we present a randomized algorithm that solves the identical subgraphs detection problem in O(D) rounds and succeeds w.h.p. This has the additional benefit of providing the strongest possible separation between deterministic and randomized complexities for global problems in the CONGEST model, as the former is Ω(n2 ) and the latter is at most O(D). Our starting point is the following randomized algorithm for the EQK problem, presented in [49, Exersise 3.6]. Alice chooses a prime number p among the first K 2 primes uniformly PK−1 ` at random. She treats her input string x as a binary representation of an integer x̄ = `=0 2 x` , and sends p and x̄ (mod p) to Bob. Bob similarly computes ȳ, compares x̄ mod p with ȳ mod p, 14 and returns TRUE if they are equal and false otherwise. The error probability of this protocol is at most 1/K. We present a simple adaptation of this algorithm for the identical subgraph detection problem. Consider the following encoding of a weighted induced subgraph on VA : for each pair i, j of indices, we have dlog W e + 1 bits, indicating the existence of the edge and its weight (recall that W ∈ poly n is an upper bound on the edge weights). This weighted induced subgraph is thus represented by a K ∈ O(n2 log n) bit-string, denoted x = x0 , . . . , xK−1 , and each pair (i, j) has a set Si,j of indices representing the edge (ai , aj ). Note that the bits {x` \| ` ∈ si,j } are known to both ai and aj , and in the algorithm we use the node with smaller index in order to encode these bits. Similarly, a K ∈ O(n2 log n) bit-string, denoted y = y0 , . . . , yK−1 encodes a weighted induced subgraph on VB . The Algorithm. As standard, assume the input graph is connected. The nodes are enumerated as in Definition 2. The algorithm starts with some node, say, a0 , constructing a BFS tree, which completes in O(D) rounds. Then, a0 chooses a prime number p among the first K 2 primes uniformly at random and sends p to all the nodes over the tree, which takes O(D) rounds. P P Each node ai computes the sum j>i `∈Si,j x` 2` mod p, and the nodes then aggregate P P these local sums modulo p up the tree, until a0 computes the sum x̄ mod p = j6=i `∈Si,j x` 2` mod p. A similar procedure is then invoked w.r.t ȳ. Finally, a0 compares x̄ mod p and ȳ mod p, and downcasts over the BFS tree its output, which is TRUE if these values are equal and is FALSE otherwise. If the subgraphs are identical, a0 always returns TRUE, while otherwise their encoding differs in at least one bit, and as in the case of EQK , a0 returns TRUE falsely with probability at most 1/K ∈ O(1/n2 ). Theorem 8. There is a randomized algorithm in the CONGEST model that solves the identical subgraphs detection problem on any connected graph in O(D) rounds. 5 Weighted APSP In this section we use the following, natural extension of Definition 1, in order to address more general 2-party functions, as well as distributed problems that are not decision problems. For a function f : {0, 1}K1 × {0, 1}K2 → {0, 1}L1 × {0, 1}L2 , we define a family of lower bound graphs in a similar way as Definition 1, except that we replace item (4) in the definition with a generalized requirement that says that for Gx,y , the values of the of nodes in VA uniquely determine the left-hand side of f (x, y), and the values of the of nodes in VB determine the right-hand side of f (x, y). Next, we argue that theorem similar to Theorem 1 holds for this case. Theorem 9. Fix a function f : {0, 1}K1 × {0, 1}K2 → {0, 1}L1 × {0, 1}L2 and a graph problem P . If there is a family {Gx,y } of lower bound graphs with C = E(VA , VB ) then any deterministic algorithm for solving P in the CONGEST model requires Ω(CC(f )/ \|C\| log n) rounds, and any randomized algorithm for deciding P in the CONGEST model requires Ω(CC R (f )/ \|C\| log n) rounds. The proof is similar to that of Theorem 1. Notice that the only difference between the theorems, apart from the sizes of the inputs and outputs of f , are with respect to item (4) in the definition of a family of lower bound graphs. However, the essence of this condition remains the same and is all that is required by the proof: the values that a solution to P assigns to nodes in VA determines the output of Alice for f (x, y), and the values that a solution to P assigns to nodes in VB determines the output of Bob for f (x, y). 15 5.1 A Linear Lower Bound for Weighted APSP Nanongkai [56] showed that any algorithm in the CONGEST model for computing a poly(n)approximation for weighted all pairs shortest paths (APSP) requires at least Ω(n/ log n) rounds. In this section we show that a slight modification to this construction yields an Ω(n) lower bound for computing exact weighted APSP. As explained in the introduction, this gives a separation between the complexities of the weighted and unweighted versions of APSP. At a high level, while we use the same simple topology for our lower bound as in [56], the reason that we are able to shave off the extra logarithmic factor is because our construction uses O(log n) bits for encoding the weight of each edge out of many optional weights, while in [56] only a single bit is used per edge for encoding one of only two options for its weight. Theorem 10. Any distributed algorithm in the CONGEST model for computing exact weighted all pairs shortest paths requires at least Ω(n) rounds. The reduction is from the following, perhaps simplest, 2-party communication problem. Alice has an input string x of size K and Bob needs to learn the string of Alice. Any algorithm (possibly randomized) for solving this problem requires at least Ω(K) bits of communication, by a trivial information theoretic argument. Notice that the problem of having Bob learn Alice’s input is not a binary function as addressed in Section 2. Similarly, computing weighted APSP is not a decision problem, but rather a problem whose solution assigns a value to each node (which is its vector of distances from all other nodes). We therefore use the extended Theorem 9 above. The fixed graph construction: The fixed graph construction G = (V, E) is defined as follows. It contains a set of n − 2 nodes, denoted A = {a0 , ..., an−3 }, which are all connected to an additional node a. The node a is connected to the last node b, by an edge of weight 0. Adding edge weights corresponding to the string x: Given the binary string x of size K = (n − 2) log n we augment the graph G with edge weights, which defines Gx , by having each non-overlapping batch of log n bits encode a weight of an edge from A to a. It is straightforward to see that Gx is a family of lower bound graphs for a function f where K2 = L1 = 0, since the weights of the edges determine the right-hand side of the output (while the left-hand side is empty). Proof of Theorem 10: To prove Theorem 10, we let VA = A ∪ {a} and VB = {b}. Note that K = \|x\| = Θ(n log n). Furthermore, note that the only edge in the cut E(VA , VB ) is the edge (a, b). Since we showed that {Gx } is a family of lower bound graphs, we apply Theorem 9 on the above partition to deduce that because K bits are required to be communicated in order for Bob to know Alice’s K-bit input, any algorithm in the CONGEST model for computing weighted APSP requires at least Ω(K/ log n) = Ω(n) rounds. 5.2 The Alice-Bob Framework Cannot Give a Super-Linear Lower Bound for Weighted APSP In this section we argue that a reduction from any 2-party function with a constant partition of the graph into Alice and Bob’s sides is provable incapable of providing a super-linear lower bound for computing weighted all pairs shortest paths in the CONGEST model. A more detailed inspection of our analysis shows a stronger claim: our claim also holds for algorithms for the CONGEST-BROADCAST model, where in each round each node must send the same (log n)-bit message to all of its neighbors. The following theorem states our claim. Theorem 11. Let f : {0, 1}K1 × {0, 1}K2 → {0, 1}L1 × {0, 1}L2 be a function and let Gx,y be a family of lower bound graphs w.r.t. f and the weighted APSP problem. When applying 16 Theorem 9 to f and Gx,y , the lower bound obtained for the number of rounds for computing weighted APSP is at most linear in n. Roughly speaking, we show that given an input graph G = (V, E) and a partition of the set of vertices into two sets V = VA ∪ VB , such that the graph induced by the nodes in VA is simulated by Alice and the graph induced by nodes in VB is simulated by Bob, Alice and Bob can compute weighted all pairs shortest paths by communicating O(n log n) bits of information for each node touching the cut C = (VA , VB ) induced by the partition. This means that for any 2-party function f and any family of lower bound graphs w.r.t. f and weighted APSP according to the extended definition of Section 5.1, since Alice and Bob can compute weighted APSP which determines their output for f by exchanging only O(\|V (C)\|n log n) bits, where V (C) is the set of nodes touching C, the value CC(f ) is at most O(\|V (C)\|n log n). But then the lower bound obtained by Theorem 9 cannot be better than Ω(n), and hence no super-linear lower can be deduced by this framework as is. ˙ B , E) we denote C = E(VA , VB ). Let V (C) denote Formally, given a graph G = (V = VA ∪V the nodes touching the cut C, with CA = V (C) ∩ VA and CB = V (C) ∩ VB . Let GA = (VA , EA ) be the subgraph induced by the nodes in VA and let GB = (VB , EB ) be the subgraph induced by the nodes in VB . For a graph H, we denote the weighted distance between two nodes u, v by wdistH (u, v). ˙ B , E, w) be a graph with an edge-weight function w : E → Lemma 4. Let G = (V = VA ∪V {1, . . . , W }, such that W ∈ poly n. Suppose that GA , CB , C and the values of w on EA and C are given as input to Alice, and that GB , CA , C and the values of w on EB and C are given as input to Bob. Then, Alice can compute the distances in G from all nodes in VA to all nodes in V and Bob can compute the distances from all nodes in VB to all the nodes in V , using O(\|V (C)\| n log n) bits of communication. Proof. We describe a protocol for the required computation, as follows. For each node u ∈ CB , Bob sends to Alice the weighted distances in GB from u to all nodes in VB , that is, Bob sends {wdistGB (u, v) \| u ∈ CB , v ∈ VB } (or ∞ for pairs of nodes not connected in GB ). 0 , w 0 ) with the nodes V 0 = V ∪ C and edges Alice constructs a virtual graph G0A = (VA0 , EA A B A A 0 0 is defined by w 0 (e) = w(e) for each EA = EA ∪ C ∪ (CB × CB ). The edge-weight function wA A 0 (u, v) for u, v ∈ C is defined to be the weighted distance between u and e ∈ EA ∪ C, and wA B v in GB , as received from Bob. Alice then computes the set of all weighted distances in G0A , {wdistG0A (u, v) \| u, v ∈ VA0 }. Alice assigns her output for the weighted distances in G as follows. For two nodes u, v ∈ VA ∪ CB , Alice outputs their weighted distance in G0A , wdistG0A (u, v). For a node u ∈ VA0 and a node v ∈ VB \ CB , Alice outputs min{wdistG0A (u, x) + wdistGB (x, v) \| x ∈ CB }, where wdistG0A is the distance in G0A as computed by Alice, and wdistGB is the distance in GB that was sent by Bob. For Bob to compute his required weighted distances, for each node u ∈ CA , similar information is sent by Alice to Bob, that is, Alice sends to Bob the weighted distances in GA from u to all nodes in VA . Bob constructs the analogous graph G0B and outputs his required distance. The next paragraph formalizes this for completeness, but may be skipped by a convinced reader. 0 , w0 ) Formally, Alice sends {wdistGA (u, v) \| u ∈ CA , v ∈ VA }. Bob constructs G0B = (VB0 , EB B 0 = E ∪C ∪(C ×C ). The edge-weight function w 0 is defined with VB0 = VB ∪CA and edges EB B A A B 0 (e) = w(e) for each e ∈ E ∪ C, and w 0 (u, v) for u, v ∈ C is defined to be the weighted by wB B A B distance between u and v in GA , as received from Alice (or ∞ if they are not connected in GA ). Bob then computes the set of all weighted distances in G0B , {wdistG0B (u, v) \| u, v ∈ VB0 }. Bob assigns his output for the weighted distances in G as follows. For two nodes u, v ∈ VB ∪ CA , Bob outputs their weighted distance in G0B , wdistG0B (u, v). For a node u ∈ VB0 and a node 17 v ∈ VA \ CA , Bob outputs min{wdistG0B (u, x) + wdistGA (x, v) \| x ∈ CA }, where wdistG0B is the distance in G0B as computed by Bob, and wdistGA is the distance in GA that was sent by Alice. Complexity. Bob sends to Alice the distances from all nodes in CB to all node in VB , which takes O(\|CB \| \|VB \| log n) bits, and similarly Alice sends O(\|CA \| \|VA \| log n) bits to Bob, for a total of O(\|V (C)\| n log n) bits. Correctness. By construction, for every edge (u, v) ∈ CB ×CB in G0A with weight wdistG0A (u, v), there is a corresponding shortest path Pu,v of the same weight in GB . Hence, for any path P 0 = (v0 , v1 , . . . , vk ) in G0A between v0 , vk ∈ VA0 , there is a corresponding path Pv0 ,vk of the same weight in G, where P is obtained from P 0 by replacing every two consecutive nodes vi , vi+1 in P ∩ CB by the path Pvi ,vi+1 . Thus, wdistG0A (v0 , vk ) ≥ wdistG (v0 , vk ). On the other hand, for any shortest path P = (v0 , v1 , . . . , vk ) in G connecting v0 , vk ∈ VA0 , there is a corresponding path P 0 of the same weight in G0A , where P 0 is obtained from P by replacing any sub-path (vi , . . . , vj ) of P contained in GB and connecting vi , vj ∈ CB by the edge (vi , vj ) in G0A . Thus, wdistG (v0 , vk ) ≥ wdistG0A (v0 , vk ). Alice thus correctly computes the weighted distances between pairs of nodes in VA0 . It remains to argue about the weighted distances that Alice computes to nodes in VB \ CB . Any lightest path P in G connecting a node u ∈ VA0 and a nodev ∈ VB \CB must cross at least one edge of C and thus must contain a node in CB . Therefore, wdistG (u, v) = min{wdistG (u, x) + wdistG (x, v) \| x ∈ CB }. Recall that we have shown that wdistG0A (u, x) = wdistG (u, x) for any u, x ∈ VA0 . The sub-path of P connecting x and v is a shortest path between these nodes, and is contained in GB , so wdistGB (x, v) = wdistG (x, v). Hence, the distance min{wdistG0A (u, x) + wdistGB (x, v) \| x ∈ CB } returned by Alice is indeed equal to wdistG (u, v). The outputs of Bob are correct by the analogous arguments, completing the proof. Proof of Theorem 11: Let f : {0, 1}K1 ×{0, 1}K2 → {0, 1}L1 ×{0, 1}L2 be a function and let Gx,y be a family of lower bound graphs w.r.t. f and the weighted APSP problem. By Lemma 4, Alice and Bob can compute the weighted distances for any graph in Gx,y by exchanging at most O(\|V (C)\|n log n) bits, which is at most O(\|C\|n log n) bits. Since Gx,y is a family of lower bound graphs w.r.t. f and weighted APSP, condition (4) gives that this number of bits is an upper bound for CC(f ). Therefore, when applying Theorem 9 to f and Gx,y , the lower bound obtained for the number of rounds for computing weighted APSP is Ω(CC(f )/\|C\| log n), which is no higher than a bound of Ω(n). Extending to t players: We argue that generalizing the Alice-Bob framework to a sharedblackboard multi-party setting is still insufficient for providing a super-linear lower bound for weighted APSP. Suppose that we increase the number of players in the above framework to t players, P0 , . . . , Pt−1 , each simulating the nodes in a set Vi in a partition of V in a family of lower bound graphs w.r.t. a t-party function f and weighted APSP. That is, the outputs of nodes in Vi for an algorithm ALG for solving a problem P in the CONGEST model, uniquely determines the output of player Pi in the function f . The function f is now a function from {0, 1}K0 × · · · × {0, 1}Kt−1 to {0, 1}L0 × · · · × {0, 1}Lt−1 . The communication complexity CC(f ) is the total number of bits written on the shared blackboard by all players. Denote by C the set of cut edges, that is, the edge whose endpoints do not belong to the same Vi . Then, if ALG is a R-rounds algorithm, we have that writing O(R\|C\| log n) bits on the shared blackboard suffice for computing f , and so R = Ω(CC(f )/\|C\| log n). We now consider the problem P to be weighted APSP. Let f be a t-party function and let Gx0 ,...,xt−1 be a family of lower bound graphs w.r.t. f and weighted APSP. We first have the players write all the edges in C on the shared blackboard, for a total of O(\|C\| log n) bits. 18 Then, in turn, each player Pi writes the weighted distances from all nodes in Vi to all nodes in V (C) ∩ Vi . This requires no more than O(\|V (C)\|n log n) bits. It is easy to verify that every player Pi can now compute the weighted distances from all nodes in Vi to all nodes in V , in a manner that is similar to that of Lemma 4. This gives an upper bound on CC(f ), which implies that any lower bound obtained by a reduction from f is Ω(CC(f )/\|C\| log n), which is no larger than Ω((\|V (C)\|n+\|C\|) log n/\|C\| log n), which is no larger than Ω(n), since \|V (C)\| ≤ 2\|C\|. Remark 1: Notice that the t-party simulation of the algorithm for the CONGEST model does not require a shared blackboard and can be done in the peer-to-peer multiparty setting as well, since simulating the delivery of a message does not require the message to be known globally. This raises the question of why would one consider a reduction to the CONGEST model from the stronger shared-blackboard model to begin with. Notice that our argument above for t players does not translate to the peer-to-peer multiparty setting, because it assumes that the edges of the cut C can be made global knowledge within writing \|C\| log n bits on the blackboard. However, what our extension above shows is that if there is a lower bound that is to be obtained using a reduction from peer-to-peer t-party computation, it must use a function f that is strictly harder to compute in the peer-to-peer setting compared with the shared-blackboard setting. Remark 2: We suspect that a similar argument can be applied for the framework of non-fixed Alice-Bob partitions (e.g., [64]), but this requires precisely defining this framework which is not addressed in this version. 6 Discussion This work provides the first super-linear lower bounds for the CONGEST model, raising a plethora of open questions. First, we showed for some specific problems, namely, computing a minimum vertex cover, a maximum independent set and a χ-coloring, that they are nearly as hard as possible for the CONGEST model. However, we know that approximate solutions for some of these problems can be obtained much faster, in a polylogarithmic number of rounds or even less. A family of specific open questions is then to characterize the exact trade-off between approximation factors and round complexities for these problems. Another specific open question is the complexity of weighted APSP, which has also been asked in previous work [26,56]. Our proof that the Alice-Bob framework is incapable of providing super-linear lower bounds for this problem may be viewed as providing evidence that weighted APSP can be solved much faster than is currently known. Together with the recent subquadratic algorithm of [28], this brings another angle to the question: can weighted APSP be solved in linear time? Finally, we propose a more general open question which addresses a possible classification of complexities of global problems in the CONGEST model. Some such problems have complexities √ of Θ(D), such as constructing a BFS tree. Others have complexities of Θ̃(D + n), such as finding an MST. Some problems have near-linear complexities, such as unweighted APSP. 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Stein kernels and moment maps Max Fathi∗ arXiv:1804.04699v1 [math.PR] 12 Apr 2018 April 16, 2018 Abstract We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge-Ampère equation. As a consequence, we show how regularity bounds on these maps control the rate of convergence in the classical central limit theorem, and derive new rates in KantorovitchWasserstein distance in the log-concave situation, with explicit polynomial dependence on the dimension. 1 Introduction Stein’s method is a set of techniques introduced by Stein [33, 34] to estimate distances between probability measures. We refer to the survey [30] for an overview. We shall be interested in one particular way of implementing Stein’s method in the Gaussian setting, based on the notion of Stein kernels. Let µ be a probability measure on Rd . A matrix-valued function τµ : Rd −→ Md (R) is said to be a Stein kernel for µ (with respect to the standard Gaussian measure γ on Rd ) if for any smooth test function f taking values in Rd , we have Z Z x · f dν = hτµ , ∇f iHS dν. (1) For applications, it is Rgenerally enough to consider the restricted class of test functions f satisfying (\|f \|2 + k∇f k2HS )dµ < ∞, in which case both integrals in (1) are well-defined as soon as τµ ∈ L2 (µ), provided µ has finite second moments. The motivation behind the definition is that, since the Gaussian measure is the only probability distribution satisfying the integration by parts formula Z Z x · f dγ = div(f )dγ, (2) the Stein kernel τµ coincides with the identity matrix, denoted by Id, if and only if the measure µ is equal to γ. Hence, the Stein kernel can be used to control how far µ is from being a standard Gaussian measure in terms of how much it ∗ CNRS and Institut de Mathématiques max.fathi@math.univ-toulouse.fr 1 de Toulouse, Université de Toulouse, 2 violates the integration by parts formula (2). It appears implicitly in many works on Stein’s method, and has recently been the topic of more direct investigations [1, 27, 28, 21, 8]. The one-dimensional case, where Stein kernels can be explicitly constructed from the density, has been extensively studied [24]. It has applications to central limit theorems [26], concentration inequalities [29, 21, 32] and random matrix theory [7]. A related quantity is the Stein discrepancy Z 2 S(µ) := inf \|τ − Id \|2 dµ τ where the infimum is taken over all possible Stein kernels for µ, since they may not be unique. This quantity has two main interesting properties: it controls the L2 Kantorovitch-Wasserstein distance to the Gaussian [21], and is monotone along the central limit theorem [8]. The aim of this work is to describe how we can construct Stein kernels using a correspondence between centered measures and convex functions, known as the moment measure problem, or moment map problem, which we shall describe in Section 2.1. The main motivation was to give a construction of Stein kernels using optimal transport maps, of which these moment maps can be viewed as a variant. The Stein kernels we shall build have several nice properties that do not seem to be necessarily satisfied by previous constructions. Most notably they shall always takes values that are symmetric, nonnegative matrices. As an application, we shall derive in Section 3 new bounds on the rate of convergence in the multidimensional central limit theorem when the random variables are log-concave, with explicit dependence on the dimension. 2 2.1 Stein kernels and moment maps Moment maps In [9] (revisited in [31], and following earlier works [36, 11, 3, 23]), the following theorem was established: Theorem 2.1 (Cordero-Erausquin and Klartag 2015). Let µ be a centered measure, with finite first moment and that is not supported on a hyperplane. Then there exists a convex function ϕ such that µ is the pushforward of the probability measure with density e−ϕ by the map ∇ϕ. This result can be seen as a variant of the optimal transport problem, where instead of specifying two measures, we fix a target measure, and look for both an original measure and a transport map while imposing the constraint that the map should be the gradient of the potential of the measure. Indeed, here ∇ϕ is also the Brenier map from optimal transport theory [35] sending e−ϕ onto µ. The convex function given by this theorem may well not be smooth, most notably when µ is a combination of Dirac masses. For example, if µ is the uniform 3 measure on {−1, +1}, viewed as a subset of R, the convex function is ϕ(x) = \|x\| on R, which is not smooth at the origin. This will cause some issues later on. We can however assume it satisfies some weak continuity property on the boundary of its support (the notion of essential continuity, which is described in [9]). A smooth version of this theorem, under extra assumptions, was previously obtained by Berman and Berndtson [3], with earlier results due to Wang and Zhu [36] and Donaldson [11]: Theorem 2.2 (Berman and Berndtson 2013). Assume that µ is supported on a compact, open convex set, and that it has a smooth density ρ on its support. Assume moreover that C ≥ ρ ≥ C −1 on the whole support, for some positive constant C. Then the convex function ϕ of Theorem 2.1 is smooth and supported on the whole space Rd . In this result (which is based on Caffarelli’s regularity theory for MongeAmpère PDEs), the convexity of the support plays an essential role. We can reformulate those statements as pertaining to solutions of the PDE e−ϕ = ρ(∇ϕ) det(∇2 ϕ). (3) This PDE is a variant of the Monge-Ampère equation, sometimes called the toric Kähler-Einstein equation. It has been studied in complex geometry, where it is related to the construction of differential structures with specific properties on toric varieties (i.e. quotients of the complex space (C∗ )n ). More recently, it has been studied in [15, 16, 19, 20], where it was used to establish functional inequalities for log-concave measures. A relevant remark to the connection with Stein’s method that we shall describe in the next section is that the standard Gaussian measure is the only fixed point of the map µ → e−ϕ , where ϕ is the moment map of µ. So in some sense the moment map already contains some information on how far the measure is from being Gaussian. In general, unless the dimension is 1, solutions to (3) are not explicit. One particular case where it can be determined is for the uniform measure on the unit d P cube [−1, 1]d , where the moment map is of the form ϕ(x) = 2 log cosh(xi /2) + i=1 C. This can be generalized to uniform measures on centered parallelipipeds by composing this function with the appropriate linear map. 2.2 The connection with Stein kernels For now, assume that µ has a density with respect to the Lebesgue measure which is strictly positive on its support, and is such that the convex function ϕ given by Theorem 2.1 is C 2 . There exists an optimal transport map sending µ onto e−ϕ , which is necessarily ∇ϕ∗ , where ϕ∗ is the Legendre transform of ϕ. ϕ∗ is then also C 2 : since ∇ϕ∗ is the inverse of ∇ϕ (this is a property of the Legendre transform) and Hess ϕ is strictly positive on the whole space, ∇ϕ∗ inherits C 1 regularity from ∇ϕ. 4 Theorem 2.3. If µ has a density ρ with respect to the Lebesgue measure, and the solution ϕ to the PDE (3) is C 2 and supported on the whole space Rd , then Hess ϕ(∇ϕ∗ ) is a Stein kernel for µ. Moreover, the Stein discrepancy satisfies Z 2 S(µ) ≤ \| Hess ϕ − Id \|2HS e−ϕ dx. In particular, if µ is supported on a compact, convex set and has density bounded from above and below by positive constants, this result applies. R The regularity assumptions can be weakened, indeed if \| Hess ϕ − Id \|2HS e−ϕ dx is finite and µ has a continuous density, then the result will still hold. For general 2,1 measures with density and full support, the moment map is only in Wloc in the interior of its support [10], which is not enough to make the proof work. But this is not surprising, since for heavy-tailed random variables the CLT may fail, and this would rule out existence of a Stein kernel belonging to L2 (µ). For background on regularity theory for Monge-Ampère PDEs, we refer to the lecture notes [12]. Remark 2.1. An interesting byproduct of this result is that the Stein kernel obtained in this way takes values that are symmetric and positive matrices. In particular, this explains why the explicit formula for Stein kernels in dimension one defines a nonnegative function. Remark 2.2. The Stein kernel constructed this way seems to be in general different from the one constructed in [8]. Since when the density is supported on a compact, convex set and has density bounded from above and below by positive constants a Poincaré inequality holds, existence of a Stein kernel in that situation was already proven in [8]. It is the particular structure of the kernel we obtain here that makes it interesting, as we will see when obtaining new rates of convergence in the CLT. Proof. Since ϕ is smooth, we have the Stein equation Z Z −ϕ ∇ϕ · f e dx = Tr(∇f )e−ϕ dx. There is no boundary term remaining when integrating by parts because ϕ grows R at least linearly at infinity, since it is convex and e−ϕ dx < ∞ (see for example Lemma 2.1 in [14]). Fix g a smooth function, and take f (x) = g(∇ϕ(x)) in the above equation. We get Z Z ∇ϕ(x) · g(∇ϕ(x))e−ϕ dx = hHess ϕ, ∇g(∇ϕ)ie−ϕ dx. Applying the change of variable y = ∇ϕ∗ (x), which sends µ onto e−ϕ , we obtain Z Z x · g(x)dµ = hHess ϕ(∇ϕ∗ ), ∇gidµ which ensures that Hess ϕ(∇ϕ∗ ) = (Hess ϕ∗ )−1 is indeed a Stein kernel for µ. 5 The bound on the Stein discrepancy is an immediate consequence of the change of variable: since Hess ϕ(∇ϕ∗ ) is a Stein kernel, by definition of the Stein discrepancy we have Z Z 2 ∗ 2 S(µ) ≤ \| Hess ϕ(∇ϕ ) − Id \|HS dµ = \| Hess ϕ − Id \|2HS e−ϕ dx. The well-known Caffarelli contraction theorem [6] states that the Brenier map sending the standard gaussian map onto a uniformly log-concave measure is lipschitz. Klartag [15] proved an analogous estimate for moment maps, which leads to the following bound on Stein kernels in that setting: Corollary 2.4. Assume that µ is uniformly convex, that is it is of the form e−V dx with Hess V ≥ ǫ Id for some ǫ > 0. Then there exists a Stein kernel with values that are positive symmetric matrices, and which is uniformly bounded, that is \|\|τ \|\|op ≤ ǫ−1 . In dimension one, this result was pointed out in [32]. Such pointwise estimates can be used to derive properties of the density and concentration inequalities [29] and isoperimetric inequalities [32]. Proof. The Stein kernel described in this statement is the one built in Theorem 2.3, all that we need to do is to prove the uniform bound on its operator norm. In [15], it was shown that under the uniform convexity assumption, the moment map indeed satisfies the uniform bound \|\| Hess ϕ\|\|op ≤ ǫ−1 , and the conclusion follows. It would also be possible to build a Stein kernel using the construction of [7, 27] and the optimal transport map sending the standard Gaussian measure onto µ. Existence could be proved in the same setting, but there would be two main downsides: we do not have an analogue to Proposition 3.2 below for those maps, so we would only get a useful quantitative estimate in the uniformly convex setting, and due to the particular form of the construction of [7], even in the latter setting the quantitative estimates would get worse. But we would still get existence of a Stein kernel that is bounded for uniformly log-concave measures. 3 Application to rates of convergence in the central limit theorem We now show how the construction of Stein kernels discussed in the previous section leads to new estimates on the rate of convergence in the central limit theorem. The family of distances we shall consider to estimate the distance in the CLT are the Kantorovitch-Wasserstein distances from optimal transport theory, defined as Z Wp (µ, ν) := inf \|\|x − y\|\|p2 π(dx, dy) π 6 where the infimum runs over all couplings of the measures µ and ν. We refer to the textbook [35] for background on optimal transport. The following statement is a result of [21] on how Lp bounds on a Stein kernel control Wasserstein distances to the standard gaussian measure: Proposition 3.1. Let τ be a Stein kernel for the probability measure µ on Rd . Then for any p ≥ 2 we have 1/p  Z X Wp (µ, γ) ≤ Cp d1−2/p  \|τij − δij \|p dµ i,j 1/p R the value of the p-th moment for a one-dimensional with Cp = R \|x\|p dγ standard gaussian. These results mean that if we get estimates on Hess ϕ, averaged out against we can deduce estimates on transport distances. It turns out that when µ is log-concave and compactly supported, such an estimate was already obtained by Klartag [15]: e−ϕ , Proposition 3.2. Let µ be a log-concave probability measure, supported on an open bounded convex set, and with a density bounded from above and below. Then the essentially-continuous convex function ϕ for which µ is the moment measure is C 2 and satisfies for any p ≥ 1 and any θ ∈ S d−1 Z p Z ∗ p p 2p 2 \|hHess ϕ(∇ϕ )θ, θi\| dµ ≤ 8 p (x · θ) dµ . We shall give a proof of Klartag’s estimate in Section 3.1. It will be the same proof as in [15], reformulated in a different language, which may be of interest to some readers. Remark 3.1. The results of [15] assume C ∞ smoothness, relying on a result of [3] to deduce C ∞ smoothness of ϕ. Since we actually only need C 2 smoothness of ϕ, it turns out we only need continuity of the density. Combining our construction of Stein kernels, Klartag’s estimate and basic arguments from Stein’s method, we get the following application to rates of convergence in the CLT: Theorem 3.3. Let µ be an isotropic log-concave probability measure with strictly n P positive and continuous density on its support. Let µn be the law of n−1/2 Xi , i=1 where the Xi are i.i.d. random variables with law µ. Then for any p ≥ 2 we have Wp (µn , γ) ≤ C̃(p) d2−2/p n1/2 with C̃(p) is a constant that depends only on p (and which can be made explicit). In particular, this estimate does not depend on µ. 7 In the case p = 2, the main result of [8] combined with the best currentlyknown estimate on the Poincaré constant of log-concave measures [22] leads to a rate of convergence of the form Cd3/4 n−1/2 , which is better than the one we obtain here. The Kannan-Lovasz-Simonovits conjecture predicts that the Poincaré constant of isotropic log-concave measures is bounded by some universal constant, p independently of the dimension, so we expect a rate of order d/n. Bonis [5] proved a general rate of convergence for measures with finite fourth moment, again when p = 2. In the case p > 2, this result seems to be the first estimate with the sharp dependence on the number of variables in any dimension. In dimension one, Bonis [5] and Bobkov [4] obtained the sharp rate in a far more general setting. In the situation where µ is uniformly log-concave, the uniform estimate on the operator norm of our Stein kernel leads to an improved dependence on the dimension: Theorem 3.4. Let µ = e−V be an isotropic probability measure with strictly positive and continuous density on its support, and assume that Hess V ≥ ǫ Id for some ǫ > 0. Then for any p ≥ 2 we have d3/2−2/p . ǫn1/2 Proof of Theorem 3.3. We first work in the situation where µ has a compact sup2 ∗ −1 port and a density bounded away from zero. Let τ = know is a (∇n ϕ ) , which we n P P Stein kernel for µ. Then as is standard, τn (x) := E n1 τ (Xi ) \| √1n Xi = x Wp (µn , γ) ≤ C(p) k=1 k=1 is a Stein kernel for µn . Applying Jensen’s inequality, we have Z Z p 1X p τij (xi ) − δij dµ⊗n (x1 , .., xn ), \|(τn )ij − δij \| dµn ≤ n and Rosenthal’s inequality for sums of independent centered random variables [13] yields Z Z \|(τn )ij − δij \|p dµn ≤ Kp n−p/2 \|τij − δij \|p dµ. We then have, given an orthonormal basis (θ1 , .., θd ) of Rd , Z X p/2 X Z \|τij − δij \|2 dµ \|τij − δij \|p dµ ≤ i,j Z X p/2 ≤ 2p dp/2 + \|τij \|2 dµ !p ! Z X hτ θi , θi i dµ ≤ 2p dp/2 + i p/2 ≤ C(p) d (p−1) +d ≤ C(p)dp (1 + 8p p2p ). XZ p hτ θi , θi i dµ 8 Hence and therefore Z \|(τn )ij − δij \|p dµn ≤ Kp n−p/2 C(p)dp (1 + 8p p2p ), (4) d2−2/p . n1/2 For the general case, when the support of µ is not necessarily compact, we can take a sequence of compact sets Fℓ that converge to the support of µ, and apply our results to the restriction of µ to Fℓ (renormalized to remain a centered, isotropic probability measure, so that Fℓ has to be modified to take this into account, but this modification will remain convex and compact). The estimate on the Wasserstein distance does not depend on Fℓ , so that we can let ℓ go to infinity and the result remains valid. Wp (µn , γ) ≤ C̃(p) The proof of Theorem 3.4 follows the exact same argument except that we use the improved bound of Corollary 2.4, so we omit it. 3.1 Proof of Proposition 3.2 We shall now give a proof of Proposition 3.2, omiting many computations taken from [18, 16]. While it is not written in the same way as in [15], it is the same proof, and we stress it is not due to us. We describe it in this form so that it is more easily readable for people with a knowledge of Bakry-Emery calculus Proof of Proposition 3.2 . We introduce the Hessian metric on Rd given by the Riemannian metric tensor g = (∇ϕ)−1 . A result of Kolesnikov [18] asserts that when µ is log-concave and satisfies the regularity conditions of Theorem 2.2, then the metric-measure space M = (Rd , g, e−ϕ ) has Ricci curvature bounded from below by 1/2. Moreover, if we consider the Laplacian on M , which is given by the formula Lϕ f = Tr(∇2 f (∇ϕ)−1 ) + ∇ log ρ(∇ϕ) · ∇f then one can check that 2 Lϕ ∂e ϕ = −∂e ϕ; Γ(∂e ϕ) = ∂ee ϕ; where Γ is the squared norm of the gradient with respect to the metric g. These computations can be found in [18, 16]. We can then use tools from Bakry-Émery theory to obtain estimates on eigenfunctions of the Laplacian for spaces with positive curvature to deduce the desired bound. Indeed, if we denote by Pt the semigroup acting on functions induced by Lϕ , we have for any locally-lipschitz function f 1 Pt (f 2 ), Γ(Pt f ) ≤ t 2(e − 1) 9 see Theorem 4.7.2 in [2]. Taking f = ∂e ϕ, since it is an eigenfunction of Lϕ associated to the eigenvalue 1, we have Pt ∂e ϕ = e−t ∂e ϕ. Therefore Γ(Pt f ) = 2 ϕ and for any t > 0 and p ≥ 1 we have e−2t ∂ee p p 1 1 2 p −2pt 2 p (Pt ((∂e ϕ) )) ≤ Pt ((∂e ϕ)2p ). e (∂ee ϕ) ≤ 2(et − 1) 2(et − 1) Hence after integrating, for any t > 0 we have p Z Z e2t 2 p −ϕ (∂ee ϕ) e dx ≤ (∂e ϕ)2p e−ϕ dx. 2(et − 1) Taking t = ln 2, the result then follows from the bound \|\|f \|\|2p,e−ϕ ≤ 2p\|\|f \|\|2,e−ϕ for any eigenfunction of Lϕ associated with the eigenvalue −1, when a logarithmic Sobolev inequality with constant 1/2 holds, see Section 5.3 of [2]. 4 Transporting Stein kernels for other reference measures The abstract setup of Stein’s method can be generalized to cover non-gaussian reference measures. If we wish to compare some measure µ to a reference measure µ0 = e−V dx, say for a smooth function V that is finite everywhere, then µ0 is characterized by the integration by parts formula Z Z ∇V · f dµ0 = Tr(∇f )dµ0 which leads to a definition of Stein kernel as a matrix-valued function such that Z Z ∇V · f dµ = hτ, ∇f idµ. (5) Assume that V is convex, C 2 and that Hess V > 0, and let µV be the pushforward of µ by ∇V . Then for any vector-valued smooth function f , defining g(x) = f (∇V ∗ (x)), we have Z Z ∇V (x) · f (x)dµ = ∇V (x) · g(∇V (x))dµ Z = x · g(x)dµV , 10 so if we take τ̃V,γ to be a Stein kernel for µV with respect to the gaussian measure (assuming for now it exists), we get Z Z ∇V (x) · f (x)dµ = hτ̃V,γ , ∇gidµV Z = hτ̃V,γ (∇V (x)), ∇g(∇V (x))idµ Z = hτ̃V,γ (∇V (x)), (Hess V (x))−1 ∇f (x)idµ Z = hτ̃V,γ (∇V (x))(Hess V (x))−1 , ∇f (x)idµ and therefore τ̃V,γ (∇V (x))(Hess V (x))−1 is a Stein kernel for µ relative to µ0 . To be valid, in addition to the regularity and convexity assumptions on V , this arguments requires that τ̃V,γ exists. It is okay if it is only defined in the sense of distributions (since ∇V is smooth and bijective, composing a distribution with it is possible). Acknowledgments : This work was partly made while the author was in residence at the Institut Henri Poincaré in July 2017, and at the Mathematical Sciences Research Institute in Berkeley for part of the fall 2017 semester. The author benefited from support from the France Berkeley Fund, the ANR Project ANR-17-CE40-0030 - EFI, and ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. I thank Dimitri Shlyakhtenko for the discussion that initiated this work, as well as Dario Cordero-Erausquin, Thomas Courtade, Bo’az Klartag, Michel Ledoux and Emanuel Milman for their advice and comments, and Adrien Saumard for sharing a preliminary version of [32]. References [1] H. Airault, P. Malliavin, and F. Viens. Stokes formula on the Wiener space and n-dimensional Nourdin-Peccati analysis. Journal of Functional Analysis, 258(5):1763–1783, 2010. [2] D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators. 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arXiv:1702.00141v1 [math.ST] 1 Feb 2017 Reliability study of proportional odds family of discrete distributions Pradip Kundu and Asok K. Nanda∗ Department of Mathematics and Statistics, IISER Kolkata Mohanpur 741246, India Abstract The proportional odds model gives a method of generating new family of distributions by adding a parameter, called tilt parameter, to expand an existing family of distributions. The new family of distributions so obtained is known as Marshall-Olkin family of distributions or Marshall-Olkin extended distributions. In this paper, we consider Marshall-Olkin family of distributions in discrete case with fixed tilt parameter. We study different ageing properties, as well as different stochastic orderings of this family of distributions. All the results of this paper are supported by several examples. Keywords: Marshall-Olkin extended distribution, Stochastic ageing, Stochastic orders. 1 Introduction Since the work of Bennett [1] and Pettitt [25] on the proportional odds (PO) model in the survival analysis context, it has been used by many researchers. The assumption of constant hazard ratio is unreasonable in many practical cases as discussed by Bennett [1], Kirmani and Gupta [17] and Rossini and Tsiatis [26]. The PO model has been used by Bennett [1] to demonstrate the effectiveness of a cure, when the mortality rate of a group having some disease approaches that of a (disease-free) control group as time progresses. After Bennet’s [1] work, the PO model has found many practical applications, see, for instance, Collett [4], Dinse and Lagakos [8], Pettitt [25] and Rossini and Tsiatis [26]. Let X and Y be two random variables with distribution functions F (·), G(·), survival functions F̄ (·), Ḡ(·), probability density functions f (·), g(·) and hazard rate functions rX (·) = f (·)/F̄ (·), rY (·) = g(·)/Ḡ(·). Let the odds functions of X and Y be denoted respectively by θX (t) = F̄ (t)/F (t) and θY (t) = Ḡ(t)/G(t). The random variables X and Y are said to satisfy PO model with proportionality constant α if ∗ Corresponding author, e-mail: asok.k.nanda@gmail.com; asok@iiserkol.ac.in 1 θY (t) = αθX (t). For more discussion on PO models one may refer to Kirmani and Gupta [17]. It is observed that, in terms of survival functions, the PO model can be represented as Ḡ(t) = αF̄ (t) , 1 − ᾱF̄ (t) (1.1) where ᾱ = 1 − α. From the above representation we have rY (t) 1 G(t) = , = rX (t) F (t) 1 − ᾱF̄ (t) so that the hazard ratio is increasing (resp. decreasing) for α > 1 (resp. α < 1) and it convergence to 1 as t tends to ∞. This property of hazard functions makes the PO model reasonable in many practical applications. This is in contrast to the proportional hazards model where the ratio of the hazard rates remains constant with time. The model (1.1), with 0 < α < ∞; gives a method of introducing new parameter α to a family of distributions for obtaining more flexible new family of distributions as discussed by Marshall and Olkin [19]. The family of distributions so obtained is known as Marshall-Olkin family of distributions or Marshall-Olkin extended distributions (for details see [19, 20]). For more discussion and applications of Marshall-Olkin family of distributions one can see [3, 5, 6, 13]. The parameter α is called ‘tilt parameter’. This is because the hazard rate of the new family is shifted below or above the hazard rate of the underlying (baseline) distribution for α ≥ 1 and 0 < α ≤ 1 respectively. Thus, Marshall-Olkin family of distributions has implications both in terms of PO model as well as in generating a new family of flexible distributions, and hence it is worth to investigate this family of distributions. Kirmani and Gupta [17] studied some ageing properties of the PO model with fixed tilt parameter α. Nanda and Das [22] studied different ageing classes of Marshall-Olkin family of distributions taking the tilt parameter as random variable. Ghitany and Kotz [14] studied the reliability properties by taking F̄ as the reliability function of the linear failure-rate distribution. Gupta et al. [16] compared the Marshall-Olkin extended distribution and the original distribution with respect to some stochastic orderings. Nanda and Das [23] compared this family of distributions with respect to different stochastic orderings by taking the tilt parameter random. All the studies mentioned above consider the original (baseline) distribution to be continuous. However, not much work is available in the literature for discrete case. In this paper, we study different ageing properties, as well as different stochastic orderings of this family of distributions with discrete baseline distribution and with fixed tilt parameter. 2 Preliminaries Here we discuss the survival function, the hazard (failure) rate function, and the mean residual life of a discrete random variable X with support N = {1, 2, ...}. Let the probability mass 2 function (pmf) of X be given by f (k) = P {X = k}, and the distribution function be F (·) so that the reliability (survival) function F̄ (·) of X becomes F̄ (k) = P {X > k} = ∞ X f (j), k = 1, 2, ..., j=k+1 with F̄ (0) = 1. The failure rate function r(·) (Shaked et al. [28]) is given by r(k) = P {X = k\|X ≥ k} = P {X = k} f (k) = , P {X ≥ k} F̄ (k − 1) and the reversed hazard rate function is given by r̃(k) = f (k)/F (k). Below we give the definitions of different discrete ageing classes. Definition 2.1 A discrete random variable X is said to be (i) ILR (DLR) i.e. increasing (decreasing) in likelihood ratio if f (k) is log-concave (logconvex), i.e. if f (k + 2)f (k) ≤ (≥) f 2 (k + 1), k ∈ N (Dewan and Sudheesh [9]); (ii) IFR (DFR) i.e. increasing (decreasing) failure rate if r(k) is increasing (decreasing) in k ∈ N. This is equivalent to the fact that F̄ (k + 1)/F̄ (k) is decreasing (increasing) in k ∈ N (Salvia and Bollinger [27], Shaked et al. [28], Gupta et al. [15]); (iii) IFRA (DFRA) i.e. increasing (decreasing) in failure rate average if [F̄ (k)]1/k is decreasing (increasing) in k, i.e., [F̄ (k)]1/k ≥ (≤) [F̄ (k + 1)]1/(k+1) , k ∈ N (Esary et al. [10], Shaked et al. [28]); (iv) NBU (NWU) i.e. new better (worse) than used if F̄ (j + k) ≤ (≥) F̄ (j)F̄ (k), j, k ∈ N (Esary et al. [10], Shaked et al. [28]); (v) DRHR (decreasing reversed hazard rate) if r̃(k) is decreasing in k, i.e. if F (k) is logconcave, i.e. if [F (k + 1)]2 ≥ F (k)F (k + 2), k ∈ N (Nanda and Sengupta [24], Li and Xu [18]). (vi) NBAFR (new better than used in failure rate average) if [F̄ (k)]1/k ≤ F̄ (1), k ∈ N (Fagiuoli and Pellerey [11]). 2.1 Proportional odds family of discrete distributions Let X be a discrete random variable with support N = {1, 2, ...} having pmf f (·), distribution function F (·), survival function F̄ (·), hazard rate function rX (·), and reversed hazard rate function r̃X (·). Starting with the survival function F̄ , the survival function of the proportional odds family (also known as Marshall-Olkin family) of discrete distribution is given by Ḡ(k; α) = αF̄ (k) , k = 1, 2, ..., 0 < α < ∞, ᾱ = 1 − α, 1 − ᾱF̄ (k) 3 (2.1) with Ḡ(0; α) = 1. Let the corresponding random variable be denoted by Y . Now the distribution function of Y is given by G(k; α) = 1 − Ḡ(k; α) = F (k) , 1 − ᾱF̄ (k) (2.2) whereas the pmf is given by g(k; α) = Ḡ(k − 1; α) − Ḡ(k; α) = αf (k) . [1 − ᾱF̄ (k − 1)][1 − ᾱF̄ (k)] (2.3) The corresponding hazard rate and the reversed hazard rate functions are given by rX (k) , 1 − ᾱF̄ (k) (2.4) αr̃X (k) . 1 − ᾱF̄ (k − 1) (2.5) rY (k; α) = r̃Y (k; α) = It is to be mentioned here that different properties of (2.1) have been studied by Déniz and Sarabia [7] by taking F as the cdf of Poisson random variable. 3 Stochastic Ageing properties In this section we study how different ageing properties of X are transmitted to the random variable Y . With the following two counterexamples, one with α > 1 and the other with α < 1, we show that if X is ILR, then Y is neither ILR nor DLR. Counterexample 3.1 Consider the random variable X with the mass function given by   0, if k = 1;        0.1, if k = 2; f (k) = 0.25, if k = 3;     0.35, if k = 4;     0.3, if k = 5. Clearly X is ILR. For α = 5, we have the mass function of Y as   0, if k = 1;     1  if k = 2;   46 , 125 g(k; 5) = 1656 , if k = 3;   175   792 , if k = 4;     15 , if k = 5. 22 It is observed that Y is neither ILR nor DLR. 4 Counterexample 3.2 Consider the random variable X with mass function given by   0, if k = 1;        0.3, if k = 2; f (k) = 0.34, if k = 3;     0.26, if k = 4;     0.1, if k = 5. Here X is ILR. For α = 0.2, we have the mass function of Y as   0, if k = 1;     15  if k = 2;   22 , 425 g(k; 0.2) = 1958 , if k = 3;   325    4094 , if k = 4;    1, if k = 5. 46 It is observed that Y is neither ILR nor DLR. ✷ With the following two counterexamples, one for α > 1 and the other for α < 1, we show that if X is DLR, then Y is neither DLR nor ILR. Counterexample 3.3 Consider the random variable X with mass function given by   0.36, if k = 1;     0.26, if k = 2; f (k) =  0.21, if k = 3;     0.17, if k = 4. Here X is DLR. For α = 2, we have the mass function of Y as  9  if k = 1;  41 ,    650 , if k = 2; 2829 g(k; 2) = 700    2691 , if k = 3;   34 if k = 4. 117 , It is observed that Y is neither DLR nor ILR. Counterexample 3.4 Consider the random variable X with mass function given by   0.26, if k = 1;     0.18, if k = 2; f (k) =  0.24, if k = 3;     0.32, if k = 4. 5 Here X is DLR. For α = 0.4, we have the mass function of Y as  65  if k = 1;  139 ,    2250 , if k = 2; 11537 g(k; 0.4) = 1500  if k = 3;  8383 ,    16 if k = 4. 101 , It is observed that Y is neither DLR nor ILR. ✷ The following theorem gives the condition under which IFR/DFR property of X is transmit ted to the random variable Y . The proof follows from the fact that 1/ 1 − ᾱF̄ (k) is increasing (resp. decreasing) in k for α ≥ (resp. ≤)1. Theorem 3.1 If X is IFR (resp. DFR) and α ≥ (resp. ≤) 1, then Y is IFR (resp. DFR). ✷ The following counterexample shows that if α < 1, then the IFR property of X may not be transmitted to the random variable Y . Counterexample 3.5 Consider the random variable X following discrete IFR distribution (cf. Salvia and Bollinger [27]) with f (k) = (k − c)ck−1 /k!, k ∈ N, 0 < c ≤ 1. Here F̄ (k) = ck /k! and rX (k) = 1 − c/k so that X is IFR. Now rY (k; α) = 1 − c/k . 1 − ᾱck /k! It is observed that, for c = 0.8 and α = 0.2, we have rY (2; 0.2) = 0.8064516, rY (3; 0.2) = 0.7870635, and rY (4; 0.2) = 0.8110739. This shows that Y is neither IFR nor DFR. ✷ The following counterexample shows that the DFR property of X may not be transmitted to the random variable Y when α > 1. Counterexample 3.6 Let X follow the Type I discrete Weibull distribution (cf. Nakagawa and Osaki [21]) with pmf given by β β f (k) = q (k−1) − q k , k ∈ N, q ∈ (0, 1), β > 0. β Then the corresponding survival function is given by F̄ (k) = q k , and the hazard rate function is given by rX (k) = 1 − q k β −(k−1)β . Here X is DFR for 0 < β < 1. Note that β β 1 − q k −(k−1) rY (k; α) = . 1 − ᾱq kβ It is observed that, for β = 0.8, α = 5 and q = 0.5, we have rY (7; 5) = 0.2759209, rY (10; 5) = 0.2834942, and rY (13; 5) = 0.2793229. This shows that Y is neither DFR nor IFR. 6 ✷ Below we see that, for α ≥ (resp. ≤) 1, the NBU (resp. NWU) property of X is transmitted to the random variable Y . Theorem 3.2 If X is NBU (resp. NWU) and α ≥ (resp. ≤) 1, then Y is NBU (resp. NWU). Proof: Let X be NBU (resp. NWU). Then Y will be NBU (resp. NWU) if and only if 1 − ᾱF̄ (j + k) (1 − ᾱF̄ (j))(1 − ᾱF̄ (k)) ≥ (resp. ≤) . F̄ (j + k) αF̄ (j)F̄ (k) This is equivalent to the fact that 1 − ᾱ F̄ (j) + F̄ (k) + ᾱF̄ (k)F̄ (j) 1 , ≥ (resp. ≤) αF̄ (j)F̄ (k) F̄ (j + k) which holds if 1 1 ≥ (resp. ≤) . F̄ (j + k) F̄ (j)F̄ (k) The last inequality follows from the fact that, for α ≥ (resp. ≤) 1, 1 − ᾱ F̄ (j) + F̄ (k) + ᾱF̄ (k)F̄ (j) ≤ (resp. ≥) α. Hence the theorem follows. ✷ The following counterexamples show that, for α < (resp. >) 1, the NBU (resp. NWU) property of X may not be transmitted to the random variable Y . Counterexample 3.7 Consider X following the discrete S-distribution (cf. Bracuemond and Gaudoin [2]) with pmf given by f (k) = p(1 − ak ) k−1 Y (1 − p + pai ), k ∈ N, 0 < p ≤ 1, 0 < a < 1. i=1 This gives the survival function as F̄ (k) = rX (k) = p(1 − ak ). Here X is NBU. Now Ḡ(k; α) = α Qk i=1 (1 Qk 1 − ᾱ i=1 Q k − p + pai ), and hazard rate function as (1 − p + pai ) i=1 (1 − p + pai ) . For j = 2, k = 3, p = 0.3, a = 0.6, α = 0.2, we have Ḡ(j + k; α) = 0.075737 and Ḡ(j; α)Ḡ(k; α) = 0.063494. This shows that Y is not NBU. Counterexample 3.8 Let X follow the distribution as given in Counterexample 3.6. Then clearly X is NWU for β ∈ (0, 1). Now, for j = 2, k = 3, α = 5, q = 0.5, we have Ḡ(j + k; α) = 0.3062174 and Ḡ(j; α)Ḡ(k; α) = 0.3657684. This shows that Y is not NWU. 7 ✷ Kirmani and Gupta [17] have observed that if X is IFRA (DFRA), then Y is IFRA (DFRA) for α > (<) 1. Below we show that if X is IFRA, then Y may not be IFRA or DFRA when α < 1. Counterexample 3.9 Let X follow the distribution as given in Counterexample 3.7. Here X is IFRA. Now, for α = 0.2, p = 0.5, and a = 0.6, we have    0.44444, for k = 1;  1/k [Ḡ(k; 0.2)] = 0.438901, for k = 2;    0.457806, for k = 4. This shows that Y is neither IFRA nor DFRA. ✷ Below we show that if X is DFRA, then, for α > 1, Y may not be DFRA or IFRA. Counterexample 3.10 Let X follow the discrete Pareto distribution with survival function c d , k ∈ N, c, d > 0, F̄ (k) = k+d which is DFR and hence DFRA. Now α Ḡ(k; α) = 1 − ᾱ d k+d c d k+d c . For α = 6, d = 2, c = 3, we have [Ḡ(k; 6)]1/k =     0.7164179, for k = 1; 0.658037,    0.68081, for k = 4; for k = 8, which is neither increasing nor decreasing in k, i.e. Y is neither DFRA nor IFRA. ✷ Following theorem shows that, for α ≤ 1, the DRHR property of X is transmitted to the random variable Y . The proof follows from the fact that 1/ 1 − ᾱF̄ (k − 1) is decreasing in k, for α ≤ 1. Theorem 3.3 If X is DRHR, then Y is DRHR for α ≤ 1. ✷ The following counterexample shows that, for α > 1, DRHR property of X may not be transmitted to the random variable Y . Counterexample 3.11 Consider the random   0,     4    25 , 2 F (k) = 5,   2    3,    1, variable X having distribution function given by if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5. 8 Clearly X is DRHR. For α = 4, the distribution   0,     1    22 , 1 G(k; 4) = 7,     31 ,     1, function of Y is given by if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5, which is not DRHR. ✷ The following counterexample shows that, for α < 1, NBAFR property of X may not be transmitted to the random variable Y . Counterexample 3.12 Consider the random  4   5,    8, 13 F̄ (k) = 1   2,    0, variable X with reliability function given by if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4. Here X is NBAFR. For α = 0.4, we have the reliability function of Y as  8   13 , if 1 ≤ k < 2;    16 , if 2 ≤ k < 3; 41 Ḡ(k; 0.4) = 2  if 3 ≤ k < 4;  7,    0, if k ≥ 4. It is clear that Y is not NBAFR. ✷ We summarize the above findings in Table 1. 4 Stochastic Orderings Let X1 and X2 be two discrete random variables with support N = {1, 2, ...} having respective pmf f1 (·), f2 (·), distribution function F1 (·), F2 (·), and survival function F̄1 (·), F̄2 (·). Let the survival function of the Marshall-Olkin family of discrete distributions be given by Ḡi (k; α) = αF̄i (k) , 0 < α < ∞, ᾱ = 1 − α, 1 − ᾱF̄i (k) and let the corresponding random variable be Yi , i = 1, 2. The following theorem shows that the usual stochastic order between X1 and X2 and that of Y1 and Y2 are equivalent. Theorem 4.1 Y1 ≤st Y2 if and only if X1 ≤st X2 . 9 Table 1: Preservation of ageing classes Ageing properties α<1 α>1 ILR Not Preserved Not Preserved DLR Not Preserved Not Preserved IFR Not Preserved Preserved DFR Preserved Not Preserved NBU Not Preserved Preserved NWU Preserved Not Preserved IFRA Not Preserved Preserved DFRA Preserved Not Preserved DRHR Preserved Not Preserved NBAFR Not Preserved Proof: Note that Y1 ≤st Y2 if, and only if αF̄1 (k) 1 − ᾱF̄1 (k) αF̄2 (k) , 1 − ᾱF̄2 (k) ≤ which is equivalent to the fact that F̄1 (k) ≤ F̄2 (k). Hence the theorem follows. ✷ The following theorem gives condition on α, under which hazard rate order between X1 and X2 is transmitted to that between Y1 and Y1 . Theorem 4.2 If X1 ≤hr X2 , then Y1 ≤hr Y2 , provided α ≥ 1. Proof: Since hazard rate order is stronger than usual stochastic order, we have, for α ≥ 1, 1 1 ≥ . 1 − ᾱF̄1 (k) 1 − ᾱF̄2 (k) Now, using the hypothesis we have, from (2.4), rY1 (k; α) = rX2 (k) rX1 (k) ≥ = rY2 (k; α). 1 − ᾱF̄1 (k) 1 − ᾱF̄2 (k) Hence the theorem follows. ✷ The following counterexample shows that the above theorem does not hold if α < 1. Counterexample 4.1 Consider the random variables X1 and X2 with respective reliability function   1,     1, 2 F̄1 (k) = 2   5,    0, if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4, 10 and   1,     5, 8 F̄2 (k) =  11 ,  20    0, if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4. This shows that X1 ≤hr X2 . For α = 0.2, we have the respective reliability function of Y1 and Y2 as and This shows that Y1 hr Y2 .   1,     1, 6 Ḡ1 (k; α) = 2   17 ,    0, if 1 ≤ k < 2;   1,     1, 4 Ḡ2 (k; α) =  11 ,  56    0, if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4, if 2 ≤ k < 3; if 3 ≤ k < 4; if k ≥ 4. ✷ The following theorem gives the condition on α such that reversed hazard rate order between X1 and X2 is transmitted to that between Y1 and Y2 . Theorem 4.3 If X1 ≤rhr X2 , then Y1 ≤rhr Y2 , for 0 < α ≤ 1. Proof: Since reversed hazard rate order is stronger than usual stochastic order, we have, for α ≤ 1, 1 1 ≤ . 1 − ᾱF̄1 (k − 1) 1 − ᾱF̄2 (k − 1) Now, using the hypothesis we have, from (2.5), r̃Y1 (k; α) = αr̃X1 (k) αr̃X2 (k) ≤ = r̃Y2 (k; α). ¯ 1 − ᾱF1 (k − 1) 1 − ᾱF¯2 (k − 1) Hence the theorem follows. ✷ That the above theorem does not hold in case of α > 1 is shown in the following counterexample. Counterexample 4.2 Consider the random variables X1 and X2 with respective distribution function   0,     5    24 , 1 F1 (k) = 2,   3    4,    1, 11 if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5, and F2 (k) =   0,     1    6, if 1 ≤ k < 2; if 2 ≤ k < 3; 5 12 , 2 3,         1, if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5. Clearly X1 ≤rhr X2 . For α = 4, we have the distribution functions of Y1 and Y2 respectively as   0, if 1 ≤ k < 2;     5    81 , if 2 ≤ k < 3; 1 G1 (k; 4) = if 3 ≤ k < 4; 5,   3   if 4 ≤ k < 5;  7,    1, if k ≥ 5, and This shows that Y1 rhr Y2 .   0,     1    21 , 5 G2 (k; 4) = 33 ,   1    3,    1, if 1 ≤ k < 2; if 2 ≤ k < 3; if 3 ≤ k < 4; if 4 ≤ k < 5; if k ≥ 5. ✷ Following two counterexamples show that the likelihood ratio order between X1 and X2 is not necessarily transmitted to that between Y1 and Y2 . Counterexample 4.3 Let X1 and X2 have          f1 (k) =         and f2 (k) = the respective probability mass function 0, if k = 1; 0.3, if k = 2; 0.4, if k = 3; 0.2, if k = 4; 0.1, if k = 5,   0, if k = 1;        0.2, if k = 2; 0.3, if k = 3;     0.2, if k = 4;     0.3, if k = 5. 12 Clearly X1 ≤lr X2 . For α = 5, we have the mass functions of Y1 and Y2 respectively as   0, if k = 1;     3    38 , if k = 2; 50 g1 (k; 5) = 209 , if k = 3;     25 if k = 4;  77 ,    5 , if k = 5, 14 and   0,     1    21 , 5 g2 (k; 5) = 42 ,   5    33 ,    15 , 22 This shows that Y1 lr Y2 . if k = 1; if k = 2; if k = 3; if k = 4; if k = 5. ✷ Counterexample 4.4 Take the random variables X1 and X2 having respective mass functions   0, if k = 1;        0.3, if k = 2; f1 (k) = 0.3, if k = 3;     0.2, if k = 4;     0.2, if k = 5, and   0,        0.2, f2 (k) = 0.3,     0.24,     0.26, if k = 1; if k = 2; if k = 3; if k = 4; if k = 5. Clearly X1 ≤lr X2 . For α = 0.2, we have the mass functions of Y1 and Y2 as   0, if k = 1;     15    22 , if k = 2; 75 g1 (k; 0.2) = 374 , if k = 3;   25    357 , if k = 4;    1 , if k = 5, 21 13 and g2 (k; 0.2) = This shows that Y1 lr Y2 .   0,     5    9,         5 18 , 10 99 , 13 198 , if k = 1; if k = 2; if k = 3; if k = 4; if k = 5. ✷ We summarize the above findings in Table 2. Table 2: Preservation of stochastic orderings Stochastic orders between α<1 α>1 baseline distributions 5 Usual stochastic order Preserved Preserved Hazard rate order Not Preserved Preserved Reversed hazard rate order Preserved Not Preserved Likelihood ratio order Not Preserved Not Preserved Conclusion Marshall and Olkin [19] introduced a method of adding a new parameter, called tilt parameter, to a family of distributions for obtaining more flexible new families of distributions. 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